\nC-19 and Hot, Wide, Star Streams\nRaymond G. Carlberg, 1 Rodrigo Ibata, 2 Nicolas F. Martin, 2 Else Starkenburg, 3 David S. Aguado, 4 Khyati Malhan, 5 Kim Venn, 6 and Zhen Yuan ( 袁 珍 ) 7\n1 Department of Astronomy & Astrophysics, University of Toronto, Toronto, ON M5S 3H4, Canada\n2 Universit'e de Strasbourg, CNRS, Observatoire astronomique de Strasbourg, UMR 7550, F-67000 Strasbourg, France 3 Kapteyn Astronomical Institute, University of Groningen, Landleven 12, 9747 AD Groningen, The Netherlands 4\nInstituto de Astrof'ısica de Canarias, V'ıa L'actea, 38205 La Laguna, Tenerife, Spain\n5 DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 Copenhagen, Denmark\n6 Department of Physics and Astronomy, University of Victoria, Victoria, BC, V8W 3P2, Canada\n7 School of Astronomy and Space Science, Nanjing University, Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210093, China\nABSTRACT\nThe C-19 star stream has the abundance characteristics of an unusually metal poor globular cluster but kinematically is uncharacteristically hot and wide for a cluster stream, having a line of sight velocity dispersion of 6 km s -1 and a 1-sigma width of 240 pc. We show that the tidal dissolution of an old, lower mass, globular cluster in a CDM galactic halo naturally creates a hot, wide stream currently near orbital apocenter. More generally, simulations show that hot streams, which are all near their orbital apocenter, become thin, cool streams near pericenter. Furthermore, the wide streams from a population of dissolved clusters in the simulations have a mean galactocentric radial velocity dispersion of 7.8 ± 1.0 kms -1 in a CDM cosmology but only 4.1 ± 1.6 kms -1 in a WDM (5.5 keV) simulation. A detailed C-19 model in a simplified Milky Way halo potential with a CDM subhalo population provides a lower bound to stream heating, finding that the stream develops a line of sight velocity dispersion of 4.1 ± 1.1 kms -1 , whereas WDM (5.5 keV) subhalos give 3.1 ± 0.1 km s -1 . Known dwarf galaxies alone provide negligible heating. There are five other currently known streams wider than 200 pc that contain a globular cluster, all near their orbital apocenter.\n1. INTRODUCTION\nThe C-19 stream of extremely low metallicity stars (Martin et al. 2022) was discovered (Ibata et al. 2021) in the Gaia data (Gaia Collaboration et al. 2016) using the STREAMFINDER algorithm (Malhan & Ibata 2018). High resolution spectroscopy (Martin et al. 2022; Yuan et al. 2022) of photometrically identified low metallicity stream candidates (Starkenburg et al. 2017) established that C-19 has a metal abundance of [Fe/H] = -3.4 with a 95% confidence [Fe/H] spread of less than 0.2 (Martin et al. 2022). Further investigation has found additional stream candidates giving a length of more than 50 degrees (Yuan et al. 2022; Viswanathan et al. 2024). The spread in stellar abundances is close to the detection limit and the abundance patterns are characteristic of globular cluster stars (Yuan et al. 2022). The abundances and extrapolated luminosity of approximately\n10 4 L ⊙ led Martin et al. (2022) to argue that the progenitor was more likely to be a disrupted globular cluster than a disrupted dwarf galaxy, while considering that the 1-sigma width of 240 pc, and line of sight velocity dispersion of 6 km s -1 were more in keeping with a disrupted dwarf. A dynamical model of a disrupting dark matter dominated dwarf galaxy can account for the kinematic properties of the stream (Errani et al. 2022).\nThe question addressed here is under what conditions can a globular cluster progenitor be compatible with the C-19 kinematic data. Most of the known streams in the compilation Galstreams (Mateu 2023) are thinner than C-19 with typical FWHM of 100 pc or less. On the other hand, simulations of globular cluster streams in realistic cosmological conditions find that the dark matter subhalos present in the galactic halo cause the velocity dispersion of stream stars to increase with time (Carlberg & Agler 2023). The average velocity distribution of long, thin streams has a core with extended wings (Carlberg 2018; Malhan et al. 2019). The width of the\nwings increases with the numbers of subhalos present in the galactic halo (Carlberg et al. 2024). The velocity at which stars join the stream determines the width of the low velocity core of a stream. Subsequent encounters with subhalos perturb stars into the extended wings of the velocity and width distribution. The growing dataset for the GD-1 stream provides tentative evidence for the existence of a core-wing velocity structure (Ibata et al. 2024; Valluri et al. 2024).\nThe C-19 stream is on a nearly polar, inner halo orbit, 8-24 kpc, where the density of subhalos is relatively high, creating the circumstances for significant subhalo heating. The stream is composed of extremely metal poor, [Fe/H] ≃ -3.4, hence likely very old, stars in an inner halo orbit such that the moderate mass progenitor cluster likely dissolved relatively early and the stream has been subject to subhalo heating for much of the lifetime of the Milky Way.\nIn § 2 we measure the range of widths and velocity dispersions of the streams that form in both CDM and WDM cosmological simulations from low mass globular clusters. We then use the cosmological simulations to motivate an evolving potential model for the Milky Way in which model star clusters on the orbit of C-19 are integrated and their velocity dispersions are measured to compare to C-19 in § 3. The results are discussed § 4 to assess the accuracy of the model and its ability to constrain the subhalo content of the Milky Way. Other currently known wide streams from globular clusters are briefly discussed along with the likelihood that there are other wide streams yet to be discovered.\n2. LOW MASS STAR CLUSTERS IN COSMOLOGICAL SIMULATIONS\nThe total stellar mass of C-19 is estimated to be in the range 3 × 10 3 to 10 4 M ⊙ (Martin et al. 2022; Yuan et al. 2022; Ibata et al. 2024). There is no visible progenitor, as expected for an old star cluster with a half mass relaxation time of 1-2 Gyr orbiting in the Milky Way (Gnedin & Ostriker 1997; Binney & Tremaine 2008; Errani et al. 2022). The C-19 stream has an angular momentum of about 3100 kpc- km s -1 , and is on a moderate eccentricity, e ≃ 0 . 45, nearly polar, orbit within 25 kpc of the galactic center (Martin et al. 2022).\nTwo cosmological Milky Way-like simulations are seeded with low mass progenitor star clusters in the mass range 5-20 × 10 3 M ⊙ in a CDM cosmology and a WDM(5.5 keV) cosmology. The cosmological simulations of streams in Carlberg et al. (2024) contained progenitor clusters more massive than 4 × 10 4 M ⊙ , most of which survive to the present epoch. Other than the initial star cluster masses, the simulations here are iden-\n\n\n
Figure 1. The stream radial velocity dispersion (galactocentric) vs stream width as measured perpendicular to the stream track for low mass cluster progenitors in cosmological simulations at the final time. CDM streams are blue points, WDM red. The error bar gives the C-19 values.
\n\nto those in Carlberg et al. (2024) and the same stream analysis procedures are used. We examine the simulation streams above a minimum angular momentum of 2000 kpc- km s -1 , which excludes those that interact strongly with the disk. The streams are selected to be within 60 kpc of the galactic center. The velocity dispersion and width are measured for the central region of the stream above a surface density threshold of 20 M ⊙ deg -2 which approximately defines the readily detected part of streams. Figure 1 shows the stream radial velocity dispersion, after removing the mean, against stream width for the 32 CDM streams and 24 WDM streams longer than 10 degrees, the minimum angle for our search and analysis algorithm. The same plot for the more massive progenitor star clusters in Carlberg et al. (2024) shows a similar correlation of velocity dispersion and width for the streams which are about twice the length of the streams here but show less difference between CDM and WDM.\nThe distribution of stream stars projected onto the orbital plane varies from a narrow distribution at pericenter to extended 'feathers' at apocenter, as illustrated in Figure 2 of Carlberg (2015). The orbital variation is a consequence of the galactic tidal field accelerating stars away from the cluster in a narrow radial band near pericenter, the subsequent orbital spreading of the unbound stars at apocenter, then regrouping at pericenter again, somewhat similar to the orbital dependence of velocities of an accretion remnant (Helmi & White 1999).\n\n\n
Figure 5 shows that streams on high angular momentum orbits, those above 8000 kpc- km s -1 , have the velocity dispersion at which stars joined the streams, approximately 2 km s -1 . The medium size squares in Figure 5 identify the streams that are within 30 kpc of the galactic center. The mean of the radial velocity dispersion for streams inside 30 kpc are shown as the large squares, finding 7.8 ± 1 . 0 kms -1 for CDM and 4.1 ± 1 . 6 kms -1 for 5.5 keV WDM, where the error ranges are the spread of the population, not the error in the mean. Lower angular momentum streams orbit closer to the center of the galaxy where the subhalo density is higher than in the outskirts, and there are more subhalos in a CDM galactic halo than in a WDM version. Clearly, a stellar stream from a globular cluster progenitor can easily have a velocity dispersion of 6 km s -1 or more when the stream is near apocenter in a CDM galactic halo.
\n\n\n\n\n\n\n
Figure 2. The orbital evolution of the stream radial velocity dispersion (left panel) with time for a few of the CDM streams selected to have a high velocity dispersion and L ≃ 3000 kpc- km s -1 at the final time, 14.1 Gyr. The final time is marked with a dot in the two rightmost panels. The middle panel shows tracks the evolution of velocity dispersion with width. The left panel shows the evolution of the radial velocity dispersion with galactocentric distance.
\n\nThe variation of the radial velocity dispersion (galactocentric) and the average physical width with time for a few individual inner halo streams are shown in the left hand panel of Figure 2. The streams were selected to have angular momenta around 3000 kpc- km s -1 and velocity dispersion around 8 km s -1 , values comparable to those for C-19. The right hand panel shows the relation between the galactocentric σ r and the galactic radius of the center of the streams. The streams from clusters 154 and 179 are on moderately eccentric orbits with σ r varying from about 2-3 to 7-9 km s -1 as the stream orbits. Stream 86 is on a less eccentric orbit and shows smaller orbital variations in velocity dispersion and width. When near apocenter a stream's structure is often visibly less well organized than at pericenter, with offshoots and bifurcations.\nFigure 3 shows the radial velocity dispersion (left) and width (right) of all the CDM (top) and WDM (bottom) streams at the end of the simulation, over the 13-14.1 Gyr time interval, as a function of the fractional distance to apocenter. The 1.1 Gyr time range is just sufficient to capture the basic radial range of the most distant clusters in these samples, but not the full orbital complexity of the potentials. The point sizes are proportional to the orbital eccentricity. The figure shows that strong correlation of both σ r and σ width with the fractional distance to apocenter depends on orbital eccentricity, with a large scatter. The high eccentricity streams (larger point sizes) have the most extreme variation with apocenter distance and tend to have larger absolute velocity dispersions and widths.\nThe velocity distribution within a stream usually has an approximately Gaussian core with extended wings (Carlberg et al. 2024). The kurtosis of these velocity distributions is not very stable, so we characterize the velocity distribution using clipped variances of the distributions. The radial velocity dispersion is calculated along the length of the stream with 3-sigma clipping. The 6-sigma clipped velocity dispersion is typically a factor of two larger for streams with σ r ≤ 3 kms -1 as shown in Figure 4. As the streams pass near pericenter the velocity distribution narrows but leaves behind extended wings which are not present when the velocity distribution widens around apocenter.\n\n\n
Figure 3. The stream radial velocity dispersions and widths with fractional distance to apocenter for the CDM (top) and WDM(5.5 keV) (bottom) streams. The point sizes are proportional to the orbital eccentricity. The colors change with time from 13 to 14.1 Gyr (the final time), in steps of 0.1 Gyr with a cycle of 7 colors.
\n\nFigure 6 shows two streams in the CDM simulation with a final time velocity dispersion above 6 km s -1 that orbit within 30 kpc. The particle plots usefully reveal the entire distribution of the dispersed cluster stars, but only the 20-30 degree segments at a galactic Y coordinate of approximately 20 kpc have sufficiently high sky density to be readily visible. The orbital history of these two clusters is shown in Figure 7. Cluster 154, shown in red, accretes onto the main halo in the first few Gyr of the simulation. Cluster 179, shown in green, is accreted around 7 Gyr, about halfway through the simulation.\nDynamical friction in the aspherical halo (Binney 1977) draws the subhalo containing the cluster into the plane of the galaxy and reduces the orbital angular momentum of the cluster.\n3. MODELING THE C-19 STREAM\nThe globular cluster streams in the cosmological simulations have radial velocity dispersions that range from about 2 to 9 km s -1 , with the highest velocity dispersion at orbital apocenter. The highest density part of C-19 is just past orbital apocenter (Martin et al. 2022;\n\n\n
Figure 4. The ratio of the 6-sigma clipped radial velocity dispersion to the 3-sigma value vs the stream velocity dispersion. The 6-to-3 ratio is a measure of the size of the wings of the velocity distribution function. CDM streams are blue, WDM, red.
\n\n\n\n
Figure 5. The mean galactocentric σ r along the part of the stream above the density threshold with the (dissolved) progenitor angular momentum. All orbital phases are plotted. The CDM simulation streams are marked with blue and the WDM(5.5 keV) with red. The streams inside 30 kpc, with widths greater than 100 pc and angular momentum between 2000 and 4000 kpc- km s -1 are shown as squares with the averages shown with the large squares.
\n\nViswanathan et al. 2024) and is similar to the high velocity streams in the simulations. Detailed modeling of a star cluster on the C-19 orbit is useful to understand the conditions under which a wide, hot stream develops. In particular the subhalo heating of a stream depends on the number of subhalos present in the primary halo and the length of time that the stream has been orbiting after tidal removal from of its natal subhalo.\nOur C-1 stream model is a simplification of the cosmological simulation to follow a single star cluster in a precomputed version of the galactic potential. The model star cluster has a mass of 1 × 10 4 M ⊙ , composed of 1 M ⊙ star particles using the same star cluster simulation as in the full cosmological model. The initial half-mass radius is set to approximately 4 pc for the models reported here. The star particles have a softening of 1 pc and are integrated with the Gadget4 code (Springel et al. 2021). Two body interactions are included with small random velocities calculated from the relaxation time (Spitzer 1987; Binney & Tremaine 2008) added to star particles within the virial radius, about 5 pc, every 5 Myr (Carlberg & Agler 2023). The tidal acceleration increases with distance from the cluster, pumping the stars into more distant orbits until they are swept away from the cluster (Meiron et al. 2021). The cluster-centric orbits become increasingly complex with distance(Fukushige & Heggie 2000). The cluster-centric radial coordinates of a sample of star particles are shown in Figure 8.\nThe potential in which the star cluster orbits is drawn from the cosmological simulation. The primary mass component is a static, triaxial NFW potential, with mass, 9 . 2 × 10 11 M ⊙ , virial radius 200 kpc, scale of 14.4 kpc, and density triaxiality from the AHF halo finder (Gill et al. 2004; Knollmann & Knebe 2009) measurements at the final moment the simulation. The potential triaxiality is set to b/a=0.97, c/a=0.83 using Equation 2.72a of Binney & Tremaine (2008). A Miyamoto-Nagai galactic bulge-disk (Miyamoto & Nagai 1975) grows within the halo using the model of Carlberg et al. (2024). Although not important for the inner halo streams considered here, the LMC is included as a 1 . 8 × 10 11 M ⊙ (Garavito-Camargo et al. 2019) Hernquist sphere (Hernquist 1990), which means that the barycenter of the system is offset from the center of the primary halo. The framework could include a more detailed buildup of the primary halo and include a rotating galactic bar both of which would provide some additional stream heating. The current sparse C-19 data does not require more than the basic model. The positions and velocities of the population of subhalos above 10 6 M ⊙ as found in one of the full cosmological simulations at the chosen starting time is integrated in the\n\n\n
Figure 9 shows the model subhalo population in the inner halo along with the stream at a cosmological time of 13.1 Gyr, 1Gyr before the present time. For a CDM cosmology about 8200 subhalos are followed and 2800 for a WDM (5.5 keV) cosmology, both within the virial radius of 200 kpc. Most of the subhalos orbit well outside the region where C-19 orbits, but they are included for completeness. The dwarf galaxies are shown in green in Figure 9 although interactions are rare and usually do not influence inner halo streams (Bonaca et al. 2019). The XY projection of the stream is shown in grayscale at time 1 Gyr before present when stream particles are still passing through orbital apocenter. Subhalos that are within 2 scale radii of the stream are in blue and the stream particles within that radius are red. The interacting subhalo near the center of the image has a current mass of 2 . 2 × 10 7 M ⊙ and induces a velocity change of about 0.5 km s -1 in the nearest stream particles. The interactions that dominate the stream velocity variations are largely from more massive halos in the first 5-6 Gyr of the simulation, once the tidal tails have acquired appreciable length but before the subhalo interactions diminish as their masses decline.
\n\n\n\n
Figure 6. XY and XZ projections of two low angular momentum streams with radial velocity dispersion of 6.9 (red) and 9.2 kms -1 (green) that develop from two dissolved stars clusters with initial masses of 1.6 and 1.3 × 10 4 M ⊙ in the CDM simulation. The red stream is nearly polar whereas the green stream orbits near the galactic plane. Movies of the entire simulation and individually of these two streams are at CDM Low Mass Clusters.
\n\ncombined Milky Way dark halo, disk and LMC potential. The subhalos do not interact with each other. The subhalos' masses decrease in time with an exponential decay in time, with a timescale of 5.5 Gyr, measured from the subhalo numbers with time in the simulations. The decrease in subhalo mass with time means that the subhalo N ( > M ) relation within the main halo declines approximately as measured in the fully dynamical cosmological simulations. Subhalos over the mass range 10 6 M ⊙ to 3 × 10 8 M ⊙ are included. Lower mass halos have little dynamical effect on the stream and higher mass subhalos are expected to contain visible dwarf galaxies. All subhalos are modeled as Hernquist spheres with a scale radius equal to r max from AHF. As the subhalos lose mass the radii are adjusted as M 1 / 3 to keep the characteristic density of each subhalo constant. In addition to the subhalos the 55 dwarf galaxies with kinematics (McConnachie 2012) are included. The dwarf galaxy dark halos are assigned maximal masses, M = 5 × 10 8 ( L/L ⊙ ) 0 . 65 M ⊙ , which roughly match the high mass end of the subhalo mass distribution function. The dwarf galaxy halo radii are set to a = 1 . 0( M/ 10 8 M ⊙ ) 0 . 23 kpc, on the basis of the fit to the r max mass relation of the subhalos in the simulations. All the subhalo orbits are precomputed and define spline\ncoefficients for the positions of the masses which the star cluster calculation uses as an external potential.\n\n\n
Figure 7. The orbits of the two clusters in Figure 6 relative to the center of the primary halo. Note that the scale is larger in the lower panels. The coordinates are with respect to the center of the dominant halo. The colors indicate the time within the simulation, from a 1 Gyr start (violet) to the 14.1 Gyr end time (red).
\n\n\n\n\n100\nThe evolving potential model presented here does a reasonable job capturing the interactions of subhalos with the stream but misses the larger scale interactions that scatter the older, more distant ends of the stream over the halo as shown in Figure 6 for the cosmologi-\n\n\n
Figure 8. The distance from the star cluster center of star particles with time. The heating algorithm adds a random velocity, initially about 0.6 km s -1 , every 0.005 Gyr to the star particles between 3 to 4 pc. Tidal pumping raises particles over several orbits to the zero binding energy surface near 100 kpc after which they drift out to join the stream.
\n\n\n\n
Figure 9. The XY projection one Gyr before the final time of the stream (in greyscale), the dwarf galaxies (filled green circles) and the subhalos (grey circles). Subhalo-stream encounters are shown for subhalos within 2 scale radii of stream particles are blue circles and the particles within 2 scale radii are in red. The distances are in kpc.
\n\ncal simulation. Consequently the detailed stream model presented here is a lower bound on the effects of subhalos on the C-19 stream.\nThe C-19 stream track is defined as passing through the approximate mean of the stars in the ϕ 1 = [ -10 , 10] region in Yuan et al. (2022). Specifically the progenitor orbit passes through RA=355.0, Dec=25.0 degrees, with a radial velocity of -195 km s -1 , proper motion of 1.25 and -2.8 milli-arcsec per year in RAcos(Dec) and Dec at a heliocentric distance of 20 kpc. The progenitor is then integrated in the potential of the MW plus LMC model. No optimization of the stream track coordinates or potential model was undertaken. The resulting orbit is adequately close to the current data for the purposes of this paper. The stream simulations placed the current epoch progenitor position at -10 degrees along the stream track, then integrated the progenitor backward including the full subhalo distribution to the desired start time. The model star cluster was then put at that position and integrated forward to create the model stream. A consequence is that the model leaves the dissolved progenitor in the ϕ 1 = [ -40 , -10] region of the stream. The precise final time progenitor location depends on the details of encounters with subhalos, so varies from run to run.\nThe C-19 model starts the star cluster at a time of 3 Gyr after the Big Bang. Starting the model star cluster at later times means that there are less subhalos to heat the stream and less time for subhalos to encounter the stream, leading to insufficient heating. The model does a reasonable job of accumulating the subhalo interactions of the central region of the stream, but stream ends are more coherent here than they would be in a full cosmological simulation, as seen in Figure 6.\nThe subhalos in the mass decade around 10 7 M ⊙ dominate the perturbations to the stream (Carlberg & Agler 2023; Carlberg et al. 2024). Subhalo stream encounters are sufficiently infrequent that there are significant differences of detail in the stream morphology depending on the detailed stream collision history, hence the phases of the orbits of the subhalos. Therefore a set of simulations is done with the subhalo distribution rotated 30 degrees from 0 to 330 degrees between simulations. The 4 streams that are significantly bifurcated around ϕ 1 = 0 in the CDM subhalo distribution are not included in Figure 10 because they lead to artificially wide stream widths which the currently measured C-19 sky distribution does not support. The widths of the model streams, calculated as a standard deviation, are shown in the lower panel of Figure 10. At 20 kpc distance 0.5 degrees corresponds to 170 pc or a FWHM of 410 pc. The narrowing of the streams along their length is an effect of orbital motion.\nThe C-19 model streams have considerable structure along their length, but the structure in these model\n40\n30\n20\n10\n0\n10\n]\ne\ne\nr\ng\ne\nd\n[\n2\n40\n20\n0\n20\n40\n60\n\n\n
Figure 10. The model streams in C-19 stream coordinates (top) and the standard deviation of their spread about the mean position along the stream (bottom0. The subhalo distribution is rotated in steps of 30 degrees from 0 to 330 degrees for a dozen variants of the C-19 model stream. The plot does not show the streams which are bifurcated in the region around ϕ 1 = 0.
\n\nstreams is significantly less than the two streams drawn from the full cosmological simulation shown in Figure 6. The movies of the two example streams CDM Low Mass Clusters show that large scale merging disperses the 'ends' of the streams until the main halo settles down around 7 Gyr. The primary halo is not subject to major mergers (by design) for the last 7 Gyr of the simulation so at late times is fairly well represented with the halo potential model used here. At the C-19 start time of 3 Gyr the dominant halo of the full cosmological simulation is about 50% less massive, which mainly effects the orbit, not the subhalo density. The C-19 model potential does not capture the larger scale potential flucutations due to accretion and merger buildup between 3 and 7 Gyr, but does include the subhalo effects. Thefore the C-19 model is a lower bound on the heating effects.\nThe line of sight velocities and their dispersion along the streams in the CDM subhalo distribution are shown in Figure 11. The velocity dispersion of C-19, 6.2 kms -1 , and its 1-sigma errors from Yuan et al. (2022) is shown in black. The quadrature summed mean of the model streams over ϕ 1 = [ -10 , 10] is 4.1 ± 1.1 kms -1 . The mean velocity dispersion (excluding the bifurcated\n\n\n
Figure 11. The line of sight velocity dispersion (top) and its distribution along the streams (bottom) for the CDM models of the C-19 stream. The four bifurcated streams are not plotted. The dots show the velocities of the observed C-19 stars. The observational estimate of the velocity dispersion and its one-sigma errors is shown in black.
\n\nstreams) is within 1.2 sigma (combined) of the observed value. The same modeling procedure for a WDM (5.5 keV) model halo gives velocities shown in Figure 12. Its mean line of sight velocity dispersion of 3.1 ± 0.09 kms -1 , a 2.2 sigma difference from the observed value. The small increase in velocity dispersion in the WDM (5.5 keV) model is consistent with the cosmological simulation results (Carlberg et al. 2024). These simulations disfavor the WDM(5.5 keV) model. However given the significant scatter seen in the full cosmological simulation for clusters in the same orbital range, Figure 5, the result is not very strong and emphasizes that stronger conclusions require more than a single stream.\nA model stream with no subhalos and no dwarf galaxies (designated c84) is shown in Figure 12. It has a velocity dispersion of 3.6 km s -1 in the ϕ 1 = [ -10 , 10] region. Including the known dwarf galaxies (c85) also gives a line of sight velocity dispersion of 3.6 km s -1 . The streams velocity dispersion calculated with 3 σ clipping are 1.41 and 1.75 km s -1 , respectively, indicating unusually strong wings in the velocity profile. The numbers of particles in this section of the two streams are about 20% of those in the models with subhalos which\n1\n\n\n
Figure 12. The line of sight velocity distribution for the WDM (5.5 keV) models of the C-19 stream. The top two lines of points are for models with no subhalos and with no dwarf galaxies (purple, second line) and with known dwarf galaxies (yellow, top line). Streams that bifurcate over the plotted stream latitude range are not included.
\n\nis a consequence of the subalo perturbations gradually changing stream orbital phases.\n4. DISCUSSION AND CONCLUSIONS\nErrani et al. (2022) established, and we confirm, that the dissolution of a globular cluster in a smooth galactic halo cannot explain the the width and line of sight velocity dispersion of the C-19 stream. There are two generic solutions: either the C-19 stream was created from a stellar system hotter and larger than a globular cluster (Errani et al. 2022), or, the velocity dispersion of the stream was increased over the course of its orbital history. We have shown that an old, dissolved, globular cluster stream near apocenter in a galactic halo containing subhalos can explain the kinematics of the C-19 stream, under certain conditions. Detailed modeling of C-19 using a simplified model potential and the subhalos drawn from the cosmological simulations finds that heating C-19 requires the numbers of subhalos found in a CDM cosmology acting on the stream for ≃ 11 Gyr. The model finds that the numbers of WDM (5.5 kev) subhalos are insufficient. The subhalo heating of the stream leads to significant density and velocity variations along the stream, Figures 10 and 11, whereas a disrupted dwarf is smooth (Errani et al. 2022). The\ncurrently known numbers of stars do not allow a reliable density profile measurement.\nMore generally, cosmological Milky Way-like simulations find that dissolved globular cluster streams that are hot and wide near apocenter, like C-19, become thin and cool streams as they orbit through pericenter. The streams discussed here are from progenitor globular clusters with masses below 2 × 10 4 M ⊙ and half mass radii in the 3-5 pc range, which have dissolution times in the galactic tidal field of 1-2 Gyr so few stars are added to the streams in the last 5 Gyr.\nThe width of C-19 as a globular cluster stream is not unique. Streams wider than 0.2 kpc containing globular clusters are listed in Table 1 drawing from the Mateu (2023) compendium (see also Bonaca & Price-Whelan (2024)). There are a total of 27 streams wider than 0.2 kpc, with 5 containing globular clusters and the C-19 stream. The widths are all from Ibata et al. (2021) and therefore use a uniform measurement approach. C-19 may be part of an ancient, extremely metal poor group of streams (Malhan et al. 2022). The wide stream of the distant cluster NGC5466 is excluded because it is likely associated with a merger remnant (Malhan et al. 2022) with a complex orbital history. NGC288 is a complex wide stream (Grillmair 2024). All the listed globular clusters are near the apocenters of their orbits (Baumgardt et al. 2019). These clusters orbit within a few kpc of the bulge, whereas C-19 is less eccentric and has a ≃ 8 kpc pericenter. Most of the 22 other wide streams have the abundance spread of a dissolved dwarf galaxy. There should of course be other wide streams from dissolved globular clusters at orbital apocenter but their distance and spread in position and velocity make them harder to find. The distribution of stream widths with current distance in the simulations is shown in Figure 13. About half of the simulated streams are within 30 kpc, whereas about 90% of the Galstreams compendium (Mateu 2023) is within 30 kpc. The radial distribution of streams will have some dependence on the radii at which globular clusters form within pre-galactic subhalos, although that dependence is fairly weak for globular cluster formation within a dwarf galaxy (Carlberg & Keating 2022). On the average the streams are physically wider with galactocentric distance, but approximately the same angular width, around 0.2 degrees.\nThe hot, wide streams at apocenter become thin, cool streams at pericenter as a result of stream orbital dynamics. Stars are unbound from their progenitor cluster at pericenter in a narrow range of radii, joining the stream with a spread in velocities. The subhalo perturbations to the stream velocities are more likely to occur near orbital pericenter where the subhalo density\n
\n\n
Table 1. Wide Globular Cluster Streams
\n
\nis highest. The velocity differences lead to differences in pericenter angle, orbital tilt, and angular momentum which causes the stream particles to spread apart near apocenter. The streams wider than a σ w of 100 pc in a CDM simulation are on the average about a factor of two hotter than in a WDM (5.5 keV), 7.8 ± 1.0 kms -1 as compared to 4.1 ± 1.6 kms -1 .\nThe hot, wide C-19 and thin, cool, GD-1 (Grillmair & Dionatos 2006) streams cover similar radial ranges in their orbits, with the difference in their widths being at least partially explained with C-19 being near apocenter and GD-1 near pericenter. The C-19 model required that it be orbiting in the subhalos of the galaxy for 11 Gyr, which is consistent with the age that being extremely metal poor implies. That is, for C-19 there is very little room for stream age-subhalo abundance degeneracy. GD-1 is less metal poor which opens up the possibility of more age-abundance degeneracy in the stream properties. Finding a common model that explains the internal kinematics of a set of streams is a goal as the data improves.\nThere is nearly a factor of two velocity dispersion difference between CDM and WDM(5.5 keV) wide streams in the inner halo. The cosmology dependence of long, thin streams, which are generally near pericenter, is present in the wings of the velocity distribution (Carlberg et al. 2024) with a relatively insensitive core Gaussian velocity spread. On the other hand, the cosmology dependence of wide streams, which are generally near apocenter, is in the velocity dispersion itself, which is easier to measure than the core-wing structure of pericenter streams.\n\n\n
Figure 13. Stream widths with distance. Dissolved clusters are circles, remnant clusters squares, blue CDM, red WDM. Most of the known globular cluster streams are within 30 kpc.
\n\nAdrian Jenkins, Carlos Frenk and Andrew Cooper provided invaluable advice and support for computing. This work used the DiRAC@Durham facility managed by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded by BEIS capital funding via STFC capital grants ST/K00042X/1, ST/P002293/1, ST/R002371/1 and ST/S002502/1, Durham University and STFC operations grant ST/R000832/1. DiRAC is part of the National e-Infrastructure. This work used high-performance computing facilities operated by the Center for Informatics and Computation in Astronomy (CICA) at National Tsing Hua University. This equipment was funded by the Ministry of Education of Taiwan, the National Science and Technology Council of Taiwan, and National Tsing Hua University. Computations were performed on the niagara supercomputer at the SciNet HPC Consortium. SciNet is funded by: the Canada Foundation for Innovation; the Government of Ontario; Ontario Research Fund - Research Excellence; and the University of Toronto. CSF acknowledges support by the European Research Council (ERC) through Advanced Investigator grant, DMIDAS (GA 786910). ARJ and CSF acknowledge support from STFC Consolidated Grant ST/X001075/1. APC acknowledges the support of the Taiwan Ministry of Education Yushan Fellowship and Taiwan National Science and Technology Council grant 112-2112-M-007-017-MY3.\nSoftware: Gadget4: Springel et al. (2021), Amiga Halo Finder: (Gill et al. 2004; Knollmann & Knebe 2009), ROCKSTAR: (Behroozi et al. 2013), NumPy: (Harris et al. 2020).\nData Availability: Final snapshots, movies, images, and example scripts are at CDM Low Mass Clusters\nREFERENCES\nBaumgardt, H., Hilker, M., Sollima, A., & Bellini, A. 2019, MNRAS, 482, 5138, doi: 10.1093/mnras/sty2997 Behroozi, P. S., Wechsler, R. H., & Wu, H.-Y. 2013, ApJ, 762, 109, doi: 10.1088/0004-637X/762/2/109 Binney, J. 1977, MNRAS, 181, 735, doi: 10.1093/mnras/181.4.735 Binney, J., & Tremaine, S. 2008, Galactic Dynamics: Second Edition (Princeton University Press) Bonaca, A., Hogg, D. W., Price-Whelan, A. M., & Conroy, C. 2019, ApJ, 880, 38, doi: 10.3847/1538-4357/ab2873 Bonaca, A., & Price-Whelan, A. M. 2024, arXiv e-prints, arXiv:2405.19410, doi: 10.48550/arXiv.2405.19410 Carlberg, R. G. 2015, ApJ, 800, 133, doi: 10.1088/0004-637X/800/2/133\n-. 2018, ApJ, 861, 69, doi: 10.3847/1538-4357/aac88a\nCarlberg, R. G., & Agler, H. 2023, ApJ, 953, 99, doi: 10.3847/1538-4357/ace4be Carlberg, R. G., Jenkins, A., Frenk, C. S., & Cooper, A. P. 2024, arXiv e-prints, arXiv:2405.18522, doi: 10.48550/arXiv.2405.18522 Carlberg, R. G., & Keating, L. C. 2022, ApJ, 924, 77, doi: 10.3847/1538-4357/ac347e Errani, R., Navarro, J. F., Ibata, R., & Pe˜narrubia, J. 2022, MNRAS, 511, 6001, doi: 10.1093/mnras/stac476 Fukushige, T., & Heggie, D. C. 2000, MNRAS, 318, 753, doi: 10.1046/j.1365-8711.2000.03811.x Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016, A&A, 595, A1, doi: 10.1051/0004-6361/201629272\n
\n\n
\n
\n\n
\n"},"sections":{"kind":"list like","value":[{"title":"ABSTRACT","content":"The C-19 star stream has the abundance characteristics of an unusually metal poor globular cluster but kinematically is uncharacteristically hot and wide for a cluster stream, having a line of sight velocity dispersion of 6 km s -1 and a 1-sigma width of 240 pc. We show that the tidal dissolution of an old, lower mass, globular cluster in a CDM galactic halo naturally creates a hot, wide stream currently near orbital apocenter. More generally, simulations show that hot streams, which are all near their orbital apocenter, become thin, cool streams near pericenter. Furthermore, the wide streams from a population of dissolved clusters in the simulations have a mean galactocentric radial velocity dispersion of 7.8 ± 1.0 kms -1 in a CDM cosmology but only 4.1 ± 1.6 kms -1 in a WDM (5.5 keV) simulation. A detailed C-19 model in a simplified Milky Way halo potential with a CDM subhalo population provides a lower bound to stream heating, finding that the stream develops a line of sight velocity dispersion of 4.1 ± 1.1 kms -1 , whereas WDM (5.5 keV) subhalos give 3.1 ± 0.1 km s -1 . Known dwarf galaxies alone provide negligible heating. There are five other currently known streams wider than 200 pc that contain a globular cluster, all near their orbital apocenter.","pages":[1]},{"title":"C-19 and Hot, Wide, Star Streams","content":"Raymond G. Carlberg, 1 Rodrigo Ibata, 2 Nicolas F. Martin, 2 Else Starkenburg, 3 David S. Aguado, 4 Khyati Malhan, 5 Kim Venn, 6 and Zhen Yuan ( 袁 珍 ) 7 1 Department of Astronomy & Astrophysics, University of Toronto, Toronto, ON M5S 3H4, Canada 2 Universit'e de Strasbourg, CNRS, Observatoire astronomique de Strasbourg, UMR 7550, F-67000 Strasbourg, France 3 Kapteyn Astronomical Institute, University of Groningen, Landleven 12, 9747 AD Groningen, The Netherlands 4 Instituto de Astrof'ısica de Canarias, V'ıa L'actea, 38205 La Laguna, Tenerife, Spain 5 DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 Copenhagen, Denmark 6 Department of Physics and Astronomy, University of Victoria, Victoria, BC, V8W 3P2, Canada 7 School of Astronomy and Space Science, Nanjing University, Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210093, China","pages":[1]},{"title":"1. INTRODUCTION","content":"The C-19 stream of extremely low metallicity stars (Martin et al. 2022) was discovered (Ibata et al. 2021) in the Gaia data (Gaia Collaboration et al. 2016) using the STREAMFINDER algorithm (Malhan & Ibata 2018). High resolution spectroscopy (Martin et al. 2022; Yuan et al. 2022) of photometrically identified low metallicity stream candidates (Starkenburg et al. 2017) established that C-19 has a metal abundance of [Fe/H] = -3.4 with a 95% confidence [Fe/H] spread of less than 0.2 (Martin et al. 2022). Further investigation has found additional stream candidates giving a length of more than 50 degrees (Yuan et al. 2022; Viswanathan et al. 2024). The spread in stellar abundances is close to the detection limit and the abundance patterns are characteristic of globular cluster stars (Yuan et al. 2022). The abundances and extrapolated luminosity of approximately 10 4 L ⊙ led Martin et al. (2022) to argue that the progenitor was more likely to be a disrupted globular cluster than a disrupted dwarf galaxy, while considering that the 1-sigma width of 240 pc, and line of sight velocity dispersion of 6 km s -1 were more in keeping with a disrupted dwarf. A dynamical model of a disrupting dark matter dominated dwarf galaxy can account for the kinematic properties of the stream (Errani et al. 2022). The question addressed here is under what conditions can a globular cluster progenitor be compatible with the C-19 kinematic data. Most of the known streams in the compilation Galstreams (Mateu 2023) are thinner than C-19 with typical FWHM of 100 pc or less. On the other hand, simulations of globular cluster streams in realistic cosmological conditions find that the dark matter subhalos present in the galactic halo cause the velocity dispersion of stream stars to increase with time (Carlberg & Agler 2023). The average velocity distribution of long, thin streams has a core with extended wings (Carlberg 2018; Malhan et al. 2019). The width of the wings increases with the numbers of subhalos present in the galactic halo (Carlberg et al. 2024). The velocity at which stars join the stream determines the width of the low velocity core of a stream. Subsequent encounters with subhalos perturb stars into the extended wings of the velocity and width distribution. The growing dataset for the GD-1 stream provides tentative evidence for the existence of a core-wing velocity structure (Ibata et al. 2024; Valluri et al. 2024). The C-19 stream is on a nearly polar, inner halo orbit, 8-24 kpc, where the density of subhalos is relatively high, creating the circumstances for significant subhalo heating. The stream is composed of extremely metal poor, [Fe/H] ≃ -3.4, hence likely very old, stars in an inner halo orbit such that the moderate mass progenitor cluster likely dissolved relatively early and the stream has been subject to subhalo heating for much of the lifetime of the Milky Way. In § 2 we measure the range of widths and velocity dispersions of the streams that form in both CDM and WDM cosmological simulations from low mass globular clusters. We then use the cosmological simulations to motivate an evolving potential model for the Milky Way in which model star clusters on the orbit of C-19 are integrated and their velocity dispersions are measured to compare to C-19 in § 3. The results are discussed § 4 to assess the accuracy of the model and its ability to constrain the subhalo content of the Milky Way. Other currently known wide streams from globular clusters are briefly discussed along with the likelihood that there are other wide streams yet to be discovered.","pages":[1,2]},{"title":"2. LOW MASS STAR CLUSTERS IN COSMOLOGICAL SIMULATIONS","content":"The total stellar mass of C-19 is estimated to be in the range 3 × 10 3 to 10 4 M ⊙ (Martin et al. 2022; Yuan et al. 2022; Ibata et al. 2024). There is no visible progenitor, as expected for an old star cluster with a half mass relaxation time of 1-2 Gyr orbiting in the Milky Way (Gnedin & Ostriker 1997; Binney & Tremaine 2008; Errani et al. 2022). The C-19 stream has an angular momentum of about 3100 kpc- km s -1 , and is on a moderate eccentricity, e ≃ 0 . 45, nearly polar, orbit within 25 kpc of the galactic center (Martin et al. 2022). Two cosmological Milky Way-like simulations are seeded with low mass progenitor star clusters in the mass range 5-20 × 10 3 M ⊙ in a CDM cosmology and a WDM(5.5 keV) cosmology. The cosmological simulations of streams in Carlberg et al. (2024) contained progenitor clusters more massive than 4 × 10 4 M ⊙ , most of which survive to the present epoch. Other than the initial star cluster masses, the simulations here are iden- to those in Carlberg et al. (2024) and the same stream analysis procedures are used. We examine the simulation streams above a minimum angular momentum of 2000 kpc- km s -1 , which excludes those that interact strongly with the disk. The streams are selected to be within 60 kpc of the galactic center. The velocity dispersion and width are measured for the central region of the stream above a surface density threshold of 20 M ⊙ deg -2 which approximately defines the readily detected part of streams. Figure 1 shows the stream radial velocity dispersion, after removing the mean, against stream width for the 32 CDM streams and 24 WDM streams longer than 10 degrees, the minimum angle for our search and analysis algorithm. The same plot for the more massive progenitor star clusters in Carlberg et al. (2024) shows a similar correlation of velocity dispersion and width for the streams which are about twice the length of the streams here but show less difference between CDM and WDM. The distribution of stream stars projected onto the orbital plane varies from a narrow distribution at pericenter to extended 'feathers' at apocenter, as illustrated in Figure 2 of Carlberg (2015). The orbital variation is a consequence of the galactic tidal field accelerating stars away from the cluster in a narrow radial band near pericenter, the subsequent orbital spreading of the unbound stars at apocenter, then regrouping at pericenter again, somewhat similar to the orbital dependence of velocities of an accretion remnant (Helmi & White 1999). The variation of the radial velocity dispersion (galactocentric) and the average physical width with time for a few individual inner halo streams are shown in the left hand panel of Figure 2. The streams were selected to have angular momenta around 3000 kpc- km s -1 and velocity dispersion around 8 km s -1 , values comparable to those for C-19. The right hand panel shows the relation between the galactocentric σ r and the galactic radius of the center of the streams. The streams from clusters 154 and 179 are on moderately eccentric orbits with σ r varying from about 2-3 to 7-9 km s -1 as the stream orbits. Stream 86 is on a less eccentric orbit and shows smaller orbital variations in velocity dispersion and width. When near apocenter a stream's structure is often visibly less well organized than at pericenter, with offshoots and bifurcations. Figure 3 shows the radial velocity dispersion (left) and width (right) of all the CDM (top) and WDM (bottom) streams at the end of the simulation, over the 13-14.1 Gyr time interval, as a function of the fractional distance to apocenter. The 1.1 Gyr time range is just sufficient to capture the basic radial range of the most distant clusters in these samples, but not the full orbital complexity of the potentials. The point sizes are proportional to the orbital eccentricity. The figure shows that strong correlation of both σ r and σ width with the fractional distance to apocenter depends on orbital eccentricity, with a large scatter. The high eccentricity streams (larger point sizes) have the most extreme variation with apocenter distance and tend to have larger absolute velocity dispersions and widths. The velocity distribution within a stream usually has an approximately Gaussian core with extended wings (Carlberg et al. 2024). The kurtosis of these velocity distributions is not very stable, so we characterize the velocity distribution using clipped variances of the distributions. The radial velocity dispersion is calculated along the length of the stream with 3-sigma clipping. The 6-sigma clipped velocity dispersion is typically a factor of two larger for streams with σ r ≤ 3 kms -1 as shown in Figure 4. As the streams pass near pericenter the velocity distribution narrows but leaves behind extended wings which are not present when the velocity distribution widens around apocenter. Figure 6 shows two streams in the CDM simulation with a final time velocity dispersion above 6 km s -1 that orbit within 30 kpc. The particle plots usefully reveal the entire distribution of the dispersed cluster stars, but only the 20-30 degree segments at a galactic Y coordinate of approximately 20 kpc have sufficiently high sky density to be readily visible. The orbital history of these two clusters is shown in Figure 7. Cluster 154, shown in red, accretes onto the main halo in the first few Gyr of the simulation. Cluster 179, shown in green, is accreted around 7 Gyr, about halfway through the simulation. Dynamical friction in the aspherical halo (Binney 1977) draws the subhalo containing the cluster into the plane of the galaxy and reduces the orbital angular momentum of the cluster.","pages":[2,3,4]},{"title":"3. MODELING THE C-19 STREAM","content":"The globular cluster streams in the cosmological simulations have radial velocity dispersions that range from about 2 to 9 km s -1 , with the highest velocity dispersion at orbital apocenter. The highest density part of C-19 is just past orbital apocenter (Martin et al. 2022; Viswanathan et al. 2024) and is similar to the high velocity streams in the simulations. Detailed modeling of a star cluster on the C-19 orbit is useful to understand the conditions under which a wide, hot stream develops. In particular the subhalo heating of a stream depends on the number of subhalos present in the primary halo and the length of time that the stream has been orbiting after tidal removal from of its natal subhalo. Our C-1 stream model is a simplification of the cosmological simulation to follow a single star cluster in a precomputed version of the galactic potential. The model star cluster has a mass of 1 × 10 4 M ⊙ , composed of 1 M ⊙ star particles using the same star cluster simulation as in the full cosmological model. The initial half-mass radius is set to approximately 4 pc for the models reported here. The star particles have a softening of 1 pc and are integrated with the Gadget4 code (Springel et al. 2021). Two body interactions are included with small random velocities calculated from the relaxation time (Spitzer 1987; Binney & Tremaine 2008) added to star particles within the virial radius, about 5 pc, every 5 Myr (Carlberg & Agler 2023). The tidal acceleration increases with distance from the cluster, pumping the stars into more distant orbits until they are swept away from the cluster (Meiron et al. 2021). The cluster-centric orbits become increasingly complex with distance(Fukushige & Heggie 2000). The cluster-centric radial coordinates of a sample of star particles are shown in Figure 8. The potential in which the star cluster orbits is drawn from the cosmological simulation. The primary mass component is a static, triaxial NFW potential, with mass, 9 . 2 × 10 11 M ⊙ , virial radius 200 kpc, scale of 14.4 kpc, and density triaxiality from the AHF halo finder (Gill et al. 2004; Knollmann & Knebe 2009) measurements at the final moment the simulation. The potential triaxiality is set to b/a=0.97, c/a=0.83 using Equation 2.72a of Binney & Tremaine (2008). A Miyamoto-Nagai galactic bulge-disk (Miyamoto & Nagai 1975) grows within the halo using the model of Carlberg et al. (2024). Although not important for the inner halo streams considered here, the LMC is included as a 1 . 8 × 10 11 M ⊙ (Garavito-Camargo et al. 2019) Hernquist sphere (Hernquist 1990), which means that the barycenter of the system is offset from the center of the primary halo. The framework could include a more detailed buildup of the primary halo and include a rotating galactic bar both of which would provide some additional stream heating. The current sparse C-19 data does not require more than the basic model. The positions and velocities of the population of subhalos above 10 6 M ⊙ as found in one of the full cosmological simulations at the chosen starting time is integrated in the combined Milky Way dark halo, disk and LMC potential. The subhalos do not interact with each other. The subhalos' masses decrease in time with an exponential decay in time, with a timescale of 5.5 Gyr, measured from the subhalo numbers with time in the simulations. The decrease in subhalo mass with time means that the subhalo N ( > M ) relation within the main halo declines approximately as measured in the fully dynamical cosmological simulations. Subhalos over the mass range 10 6 M ⊙ to 3 × 10 8 M ⊙ are included. Lower mass halos have little dynamical effect on the stream and higher mass subhalos are expected to contain visible dwarf galaxies. All subhalos are modeled as Hernquist spheres with a scale radius equal to r max from AHF. As the subhalos lose mass the radii are adjusted as M 1 / 3 to keep the characteristic density of each subhalo constant. In addition to the subhalos the 55 dwarf galaxies with kinematics (McConnachie 2012) are included. The dwarf galaxy dark halos are assigned maximal masses, M = 5 × 10 8 ( L/L ⊙ ) 0 . 65 M ⊙ , which roughly match the high mass end of the subhalo mass distribution function. The dwarf galaxy halo radii are set to a = 1 . 0( M/ 10 8 M ⊙ ) 0 . 23 kpc, on the basis of the fit to the r max mass relation of the subhalos in the simulations. All the subhalo orbits are precomputed and define spline coefficients for the positions of the masses which the star cluster calculation uses as an external potential. 100 The evolving potential model presented here does a reasonable job capturing the interactions of subhalos with the stream but misses the larger scale interactions that scatter the older, more distant ends of the stream over the halo as shown in Figure 6 for the cosmologi- cal simulation. Consequently the detailed stream model presented here is a lower bound on the effects of subhalos on the C-19 stream. The C-19 stream track is defined as passing through the approximate mean of the stars in the ϕ 1 = [ -10 , 10] region in Yuan et al. (2022). Specifically the progenitor orbit passes through RA=355.0, Dec=25.0 degrees, with a radial velocity of -195 km s -1 , proper motion of 1.25 and -2.8 milli-arcsec per year in RAcos(Dec) and Dec at a heliocentric distance of 20 kpc. The progenitor is then integrated in the potential of the MW plus LMC model. No optimization of the stream track coordinates or potential model was undertaken. The resulting orbit is adequately close to the current data for the purposes of this paper. The stream simulations placed the current epoch progenitor position at -10 degrees along the stream track, then integrated the progenitor backward including the full subhalo distribution to the desired start time. The model star cluster was then put at that position and integrated forward to create the model stream. A consequence is that the model leaves the dissolved progenitor in the ϕ 1 = [ -40 , -10] region of the stream. The precise final time progenitor location depends on the details of encounters with subhalos, so varies from run to run. The C-19 model starts the star cluster at a time of 3 Gyr after the Big Bang. Starting the model star cluster at later times means that there are less subhalos to heat the stream and less time for subhalos to encounter the stream, leading to insufficient heating. The model does a reasonable job of accumulating the subhalo interactions of the central region of the stream, but stream ends are more coherent here than they would be in a full cosmological simulation, as seen in Figure 6. The subhalos in the mass decade around 10 7 M ⊙ dominate the perturbations to the stream (Carlberg & Agler 2023; Carlberg et al. 2024). Subhalo stream encounters are sufficiently infrequent that there are significant differences of detail in the stream morphology depending on the detailed stream collision history, hence the phases of the orbits of the subhalos. Therefore a set of simulations is done with the subhalo distribution rotated 30 degrees from 0 to 330 degrees between simulations. The 4 streams that are significantly bifurcated around ϕ 1 = 0 in the CDM subhalo distribution are not included in Figure 10 because they lead to artificially wide stream widths which the currently measured C-19 sky distribution does not support. The widths of the model streams, calculated as a standard deviation, are shown in the lower panel of Figure 10. At 20 kpc distance 0.5 degrees corresponds to 170 pc or a FWHM of 410 pc. The narrowing of the streams along their length is an effect of orbital motion. The C-19 model streams have considerable structure along their length, but the structure in these model 40 30 20 10 0 10 ] e e r g e d [ 2 40 20 0 20 40 60 streams is significantly less than the two streams drawn from the full cosmological simulation shown in Figure 6. The movies of the two example streams CDM Low Mass Clusters show that large scale merging disperses the 'ends' of the streams until the main halo settles down around 7 Gyr. The primary halo is not subject to major mergers (by design) for the last 7 Gyr of the simulation so at late times is fairly well represented with the halo potential model used here. At the C-19 start time of 3 Gyr the dominant halo of the full cosmological simulation is about 50% less massive, which mainly effects the orbit, not the subhalo density. The C-19 model potential does not capture the larger scale potential flucutations due to accretion and merger buildup between 3 and 7 Gyr, but does include the subhalo effects. Thefore the C-19 model is a lower bound on the heating effects. The line of sight velocities and their dispersion along the streams in the CDM subhalo distribution are shown in Figure 11. The velocity dispersion of C-19, 6.2 kms -1 , and its 1-sigma errors from Yuan et al. (2022) is shown in black. The quadrature summed mean of the model streams over ϕ 1 = [ -10 , 10] is 4.1 ± 1.1 kms -1 . The mean velocity dispersion (excluding the bifurcated streams) is within 1.2 sigma (combined) of the observed value. The same modeling procedure for a WDM (5.5 keV) model halo gives velocities shown in Figure 12. Its mean line of sight velocity dispersion of 3.1 ± 0.09 kms -1 , a 2.2 sigma difference from the observed value. The small increase in velocity dispersion in the WDM (5.5 keV) model is consistent with the cosmological simulation results (Carlberg et al. 2024). These simulations disfavor the WDM(5.5 keV) model. However given the significant scatter seen in the full cosmological simulation for clusters in the same orbital range, Figure 5, the result is not very strong and emphasizes that stronger conclusions require more than a single stream. A model stream with no subhalos and no dwarf galaxies (designated c84) is shown in Figure 12. It has a velocity dispersion of 3.6 km s -1 in the ϕ 1 = [ -10 , 10] region. Including the known dwarf galaxies (c85) also gives a line of sight velocity dispersion of 3.6 km s -1 . The streams velocity dispersion calculated with 3 σ clipping are 1.41 and 1.75 km s -1 , respectively, indicating unusually strong wings in the velocity profile. The numbers of particles in this section of the two streams are about 20% of those in the models with subhalos which 1 is a consequence of the subalo perturbations gradually changing stream orbital phases.","pages":[4,5,6,7,8,9]},{"title":"4. DISCUSSION AND CONCLUSIONS","content":"Errani et al. (2022) established, and we confirm, that the dissolution of a globular cluster in a smooth galactic halo cannot explain the the width and line of sight velocity dispersion of the C-19 stream. There are two generic solutions: either the C-19 stream was created from a stellar system hotter and larger than a globular cluster (Errani et al. 2022), or, the velocity dispersion of the stream was increased over the course of its orbital history. We have shown that an old, dissolved, globular cluster stream near apocenter in a galactic halo containing subhalos can explain the kinematics of the C-19 stream, under certain conditions. Detailed modeling of C-19 using a simplified model potential and the subhalos drawn from the cosmological simulations finds that heating C-19 requires the numbers of subhalos found in a CDM cosmology acting on the stream for ≃ 11 Gyr. The model finds that the numbers of WDM (5.5 kev) subhalos are insufficient. The subhalo heating of the stream leads to significant density and velocity variations along the stream, Figures 10 and 11, whereas a disrupted dwarf is smooth (Errani et al. 2022). The currently known numbers of stars do not allow a reliable density profile measurement. More generally, cosmological Milky Way-like simulations find that dissolved globular cluster streams that are hot and wide near apocenter, like C-19, become thin and cool streams as they orbit through pericenter. The streams discussed here are from progenitor globular clusters with masses below 2 × 10 4 M ⊙ and half mass radii in the 3-5 pc range, which have dissolution times in the galactic tidal field of 1-2 Gyr so few stars are added to the streams in the last 5 Gyr. The width of C-19 as a globular cluster stream is not unique. Streams wider than 0.2 kpc containing globular clusters are listed in Table 1 drawing from the Mateu (2023) compendium (see also Bonaca & Price-Whelan (2024)). There are a total of 27 streams wider than 0.2 kpc, with 5 containing globular clusters and the C-19 stream. The widths are all from Ibata et al. (2021) and therefore use a uniform measurement approach. C-19 may be part of an ancient, extremely metal poor group of streams (Malhan et al. 2022). The wide stream of the distant cluster NGC5466 is excluded because it is likely associated with a merger remnant (Malhan et al. 2022) with a complex orbital history. NGC288 is a complex wide stream (Grillmair 2024). All the listed globular clusters are near the apocenters of their orbits (Baumgardt et al. 2019). These clusters orbit within a few kpc of the bulge, whereas C-19 is less eccentric and has a ≃ 8 kpc pericenter. Most of the 22 other wide streams have the abundance spread of a dissolved dwarf galaxy. There should of course be other wide streams from dissolved globular clusters at orbital apocenter but their distance and spread in position and velocity make them harder to find. The distribution of stream widths with current distance in the simulations is shown in Figure 13. About half of the simulated streams are within 30 kpc, whereas about 90% of the Galstreams compendium (Mateu 2023) is within 30 kpc. The radial distribution of streams will have some dependence on the radii at which globular clusters form within pre-galactic subhalos, although that dependence is fairly weak for globular cluster formation within a dwarf galaxy (Carlberg & Keating 2022). On the average the streams are physically wider with galactocentric distance, but approximately the same angular width, around 0.2 degrees. The hot, wide streams at apocenter become thin, cool streams at pericenter as a result of stream orbital dynamics. Stars are unbound from their progenitor cluster at pericenter in a narrow range of radii, joining the stream with a spread in velocities. The subhalo perturbations to the stream velocities are more likely to occur near orbital pericenter where the subhalo density is highest. The velocity differences lead to differences in pericenter angle, orbital tilt, and angular momentum which causes the stream particles to spread apart near apocenter. The streams wider than a σ w of 100 pc in a CDM simulation are on the average about a factor of two hotter than in a WDM (5.5 keV), 7.8 ± 1.0 kms -1 as compared to 4.1 ± 1.6 kms -1 . The hot, wide C-19 and thin, cool, GD-1 (Grillmair & Dionatos 2006) streams cover similar radial ranges in their orbits, with the difference in their widths being at least partially explained with C-19 being near apocenter and GD-1 near pericenter. The C-19 model required that it be orbiting in the subhalos of the galaxy for 11 Gyr, which is consistent with the age that being extremely metal poor implies. That is, for C-19 there is very little room for stream age-subhalo abundance degeneracy. GD-1 is less metal poor which opens up the possibility of more age-abundance degeneracy in the stream properties. Finding a common model that explains the internal kinematics of a set of streams is a goal as the data improves. There is nearly a factor of two velocity dispersion difference between CDM and WDM(5.5 keV) wide streams in the inner halo. The cosmology dependence of long, thin streams, which are generally near pericenter, is present in the wings of the velocity distribution (Carlberg et al. 2024) with a relatively insensitive core Gaussian velocity spread. On the other hand, the cosmology dependence of wide streams, which are generally near apocenter, is in the velocity dispersion itself, which is easier to measure than the core-wing structure of pericenter streams. Adrian Jenkins, Carlos Frenk and Andrew Cooper provided invaluable advice and support for computing. This work used the DiRAC@Durham facility managed by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded by BEIS capital funding via STFC capital grants ST/K00042X/1, ST/P002293/1, ST/R002371/1 and ST/S002502/1, Durham University and STFC operations grant ST/R000832/1. DiRAC is part of the National e-Infrastructure. This work used high-performance computing facilities operated by the Center for Informatics and Computation in Astronomy (CICA) at National Tsing Hua University. This equipment was funded by the Ministry of Education of Taiwan, the National Science and Technology Council of Taiwan, and National Tsing Hua University. Computations were performed on the niagara supercomputer at the SciNet HPC Consortium. SciNet is funded by: the Canada Foundation for Innovation; the Government of Ontario; Ontario Research Fund - Research Excellence; and the University of Toronto. CSF acknowledges support by the European Research Council (ERC) through Advanced Investigator grant, DMIDAS (GA 786910). ARJ and CSF acknowledge support from STFC Consolidated Grant ST/X001075/1. APC acknowledges the support of the Taiwan Ministry of Education Yushan Fellowship and Taiwan National Science and Technology Council grant 112-2112-M-007-017-MY3. Software: Gadget4: Springel et al. (2021), Amiga Halo Finder: (Gill et al. 2004; Knollmann & Knebe 2009), ROCKSTAR: (Behroozi et al. 2013), NumPy: (Harris et al. 2020). Data Availability: Final snapshots, movies, images, and example scripts are at CDM Low Mass Clusters","pages":[9,10,11]},{"title":"REFERENCES","content":"Baumgardt, H., Hilker, M., Sollima, A., & Bellini, A. 2019, MNRAS, 482, 5138, doi: 10.1093/mnras/sty2997 Behroozi, P. S., Wechsler, R. H., & Wu, H.-Y. 2013, ApJ, 762, 109, doi: 10.1088/0004-637X/762/2/109 Binney, J. 1977, MNRAS, 181, 735, doi: 10.1093/mnras/181.4.735 Binney, J., & Tremaine, S. 2008, Galactic Dynamics: Second Edition (Princeton University Press) Bonaca, A., Hogg, D. W., Price-Whelan, A. M., & Conroy, C. 2019, ApJ, 880, 38, doi: 10.3847/1538-4357/ab2873 Bonaca, A., & Price-Whelan, A. M. 2024, arXiv e-prints, arXiv:2405.19410, doi: 10.48550/arXiv.2405.19410 Carlberg, R. G. 2015, ApJ, 800, 133, doi: 10.1088/0004-637X/800/2/133 -. 2018, ApJ, 861, 69, doi: 10.3847/1538-4357/aac88a Carlberg, R. G., & Agler, H. 2023, ApJ, 953, 99, doi: 10.3847/1538-4357/ace4be Carlberg, R. G., Jenkins, A., Frenk, C. S., & Cooper, A. P. 2024, arXiv e-prints, arXiv:2405.18522, doi: 10.48550/arXiv.2405.18522 Carlberg, R. G., & Keating, L. C. 2022, ApJ, 924, 77, doi: 10.3847/1538-4357/ac347e Errani, R., Navarro, J. F., Ibata, R., & Pe˜narrubia, J. 2022, MNRAS, 511, 6001, doi: 10.1093/mnras/stac476 Fukushige, T., & Heggie, D. C. 2000, MNRAS, 318, 753, doi: 10.1046/j.1365-8711.2000.03811.x Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016, A&A, 595, A1, doi: 10.1051/0004-6361/201629272","pages":[11]}],"string":"[\n {\n \"title\": \"ABSTRACT\",\n \"content\": \"The C-19 star stream has the abundance characteristics of an unusually metal poor globular cluster but kinematically is uncharacteristically hot and wide for a cluster stream, having a line of sight velocity dispersion of 6 km s -1 and a 1-sigma width of 240 pc. We show that the tidal dissolution of an old, lower mass, globular cluster in a CDM galactic halo naturally creates a hot, wide stream currently near orbital apocenter. More generally, simulations show that hot streams, which are all near their orbital apocenter, become thin, cool streams near pericenter. Furthermore, the wide streams from a population of dissolved clusters in the simulations have a mean galactocentric radial velocity dispersion of 7.8 ± 1.0 kms -1 in a CDM cosmology but only 4.1 ± 1.6 kms -1 in a WDM (5.5 keV) simulation. A detailed C-19 model in a simplified Milky Way halo potential with a CDM subhalo population provides a lower bound to stream heating, finding that the stream develops a line of sight velocity dispersion of 4.1 ± 1.1 kms -1 , whereas WDM (5.5 keV) subhalos give 3.1 ± 0.1 km s -1 . Known dwarf galaxies alone provide negligible heating. There are five other currently known streams wider than 200 pc that contain a globular cluster, all near their orbital apocenter.\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"C-19 and Hot, Wide, Star Streams\",\n \"content\": \"Raymond G. Carlberg, 1 Rodrigo Ibata, 2 Nicolas F. Martin, 2 Else Starkenburg, 3 David S. Aguado, 4 Khyati Malhan, 5 Kim Venn, 6 and Zhen Yuan ( 袁 珍 ) 7 1 Department of Astronomy & Astrophysics, University of Toronto, Toronto, ON M5S 3H4, Canada 2 Universit'e de Strasbourg, CNRS, Observatoire astronomique de Strasbourg, UMR 7550, F-67000 Strasbourg, France 3 Kapteyn Astronomical Institute, University of Groningen, Landleven 12, 9747 AD Groningen, The Netherlands 4 Instituto de Astrof'ısica de Canarias, V'ıa L'actea, 38205 La Laguna, Tenerife, Spain 5 DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 Copenhagen, Denmark 6 Department of Physics and Astronomy, University of Victoria, Victoria, BC, V8W 3P2, Canada 7 School of Astronomy and Space Science, Nanjing University, Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210093, China\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1. INTRODUCTION\",\n \"content\": \"The C-19 stream of extremely low metallicity stars (Martin et al. 2022) was discovered (Ibata et al. 2021) in the Gaia data (Gaia Collaboration et al. 2016) using the STREAMFINDER algorithm (Malhan & Ibata 2018). High resolution spectroscopy (Martin et al. 2022; Yuan et al. 2022) of photometrically identified low metallicity stream candidates (Starkenburg et al. 2017) established that C-19 has a metal abundance of [Fe/H] = -3.4 with a 95% confidence [Fe/H] spread of less than 0.2 (Martin et al. 2022). Further investigation has found additional stream candidates giving a length of more than 50 degrees (Yuan et al. 2022; Viswanathan et al. 2024). The spread in stellar abundances is close to the detection limit and the abundance patterns are characteristic of globular cluster stars (Yuan et al. 2022). The abundances and extrapolated luminosity of approximately 10 4 L ⊙ led Martin et al. (2022) to argue that the progenitor was more likely to be a disrupted globular cluster than a disrupted dwarf galaxy, while considering that the 1-sigma width of 240 pc, and line of sight velocity dispersion of 6 km s -1 were more in keeping with a disrupted dwarf. A dynamical model of a disrupting dark matter dominated dwarf galaxy can account for the kinematic properties of the stream (Errani et al. 2022). The question addressed here is under what conditions can a globular cluster progenitor be compatible with the C-19 kinematic data. Most of the known streams in the compilation Galstreams (Mateu 2023) are thinner than C-19 with typical FWHM of 100 pc or less. On the other hand, simulations of globular cluster streams in realistic cosmological conditions find that the dark matter subhalos present in the galactic halo cause the velocity dispersion of stream stars to increase with time (Carlberg & Agler 2023). The average velocity distribution of long, thin streams has a core with extended wings (Carlberg 2018; Malhan et al. 2019). The width of the wings increases with the numbers of subhalos present in the galactic halo (Carlberg et al. 2024). The velocity at which stars join the stream determines the width of the low velocity core of a stream. Subsequent encounters with subhalos perturb stars into the extended wings of the velocity and width distribution. The growing dataset for the GD-1 stream provides tentative evidence for the existence of a core-wing velocity structure (Ibata et al. 2024; Valluri et al. 2024). The C-19 stream is on a nearly polar, inner halo orbit, 8-24 kpc, where the density of subhalos is relatively high, creating the circumstances for significant subhalo heating. The stream is composed of extremely metal poor, [Fe/H] ≃ -3.4, hence likely very old, stars in an inner halo orbit such that the moderate mass progenitor cluster likely dissolved relatively early and the stream has been subject to subhalo heating for much of the lifetime of the Milky Way. In § 2 we measure the range of widths and velocity dispersions of the streams that form in both CDM and WDM cosmological simulations from low mass globular clusters. We then use the cosmological simulations to motivate an evolving potential model for the Milky Way in which model star clusters on the orbit of C-19 are integrated and their velocity dispersions are measured to compare to C-19 in § 3. The results are discussed § 4 to assess the accuracy of the model and its ability to constrain the subhalo content of the Milky Way. Other currently known wide streams from globular clusters are briefly discussed along with the likelihood that there are other wide streams yet to be discovered.\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"2. LOW MASS STAR CLUSTERS IN COSMOLOGICAL SIMULATIONS\",\n \"content\": \"The total stellar mass of C-19 is estimated to be in the range 3 × 10 3 to 10 4 M ⊙ (Martin et al. 2022; Yuan et al. 2022; Ibata et al. 2024). There is no visible progenitor, as expected for an old star cluster with a half mass relaxation time of 1-2 Gyr orbiting in the Milky Way (Gnedin & Ostriker 1997; Binney & Tremaine 2008; Errani et al. 2022). The C-19 stream has an angular momentum of about 3100 kpc- km s -1 , and is on a moderate eccentricity, e ≃ 0 . 45, nearly polar, orbit within 25 kpc of the galactic center (Martin et al. 2022). Two cosmological Milky Way-like simulations are seeded with low mass progenitor star clusters in the mass range 5-20 × 10 3 M ⊙ in a CDM cosmology and a WDM(5.5 keV) cosmology. The cosmological simulations of streams in Carlberg et al. (2024) contained progenitor clusters more massive than 4 × 10 4 M ⊙ , most of which survive to the present epoch. Other than the initial star cluster masses, the simulations here are iden- to those in Carlberg et al. (2024) and the same stream analysis procedures are used. We examine the simulation streams above a minimum angular momentum of 2000 kpc- km s -1 , which excludes those that interact strongly with the disk. The streams are selected to be within 60 kpc of the galactic center. The velocity dispersion and width are measured for the central region of the stream above a surface density threshold of 20 M ⊙ deg -2 which approximately defines the readily detected part of streams. Figure 1 shows the stream radial velocity dispersion, after removing the mean, against stream width for the 32 CDM streams and 24 WDM streams longer than 10 degrees, the minimum angle for our search and analysis algorithm. The same plot for the more massive progenitor star clusters in Carlberg et al. (2024) shows a similar correlation of velocity dispersion and width for the streams which are about twice the length of the streams here but show less difference between CDM and WDM. The distribution of stream stars projected onto the orbital plane varies from a narrow distribution at pericenter to extended 'feathers' at apocenter, as illustrated in Figure 2 of Carlberg (2015). The orbital variation is a consequence of the galactic tidal field accelerating stars away from the cluster in a narrow radial band near pericenter, the subsequent orbital spreading of the unbound stars at apocenter, then regrouping at pericenter again, somewhat similar to the orbital dependence of velocities of an accretion remnant (Helmi & White 1999). The variation of the radial velocity dispersion (galactocentric) and the average physical width with time for a few individual inner halo streams are shown in the left hand panel of Figure 2. The streams were selected to have angular momenta around 3000 kpc- km s -1 and velocity dispersion around 8 km s -1 , values comparable to those for C-19. The right hand panel shows the relation between the galactocentric σ r and the galactic radius of the center of the streams. The streams from clusters 154 and 179 are on moderately eccentric orbits with σ r varying from about 2-3 to 7-9 km s -1 as the stream orbits. Stream 86 is on a less eccentric orbit and shows smaller orbital variations in velocity dispersion and width. When near apocenter a stream's structure is often visibly less well organized than at pericenter, with offshoots and bifurcations. Figure 3 shows the radial velocity dispersion (left) and width (right) of all the CDM (top) and WDM (bottom) streams at the end of the simulation, over the 13-14.1 Gyr time interval, as a function of the fractional distance to apocenter. The 1.1 Gyr time range is just sufficient to capture the basic radial range of the most distant clusters in these samples, but not the full orbital complexity of the potentials. The point sizes are proportional to the orbital eccentricity. The figure shows that strong correlation of both σ r and σ width with the fractional distance to apocenter depends on orbital eccentricity, with a large scatter. The high eccentricity streams (larger point sizes) have the most extreme variation with apocenter distance and tend to have larger absolute velocity dispersions and widths. The velocity distribution within a stream usually has an approximately Gaussian core with extended wings (Carlberg et al. 2024). The kurtosis of these velocity distributions is not very stable, so we characterize the velocity distribution using clipped variances of the distributions. The radial velocity dispersion is calculated along the length of the stream with 3-sigma clipping. The 6-sigma clipped velocity dispersion is typically a factor of two larger for streams with σ r ≤ 3 kms -1 as shown in Figure 4. As the streams pass near pericenter the velocity distribution narrows but leaves behind extended wings which are not present when the velocity distribution widens around apocenter. Figure 6 shows two streams in the CDM simulation with a final time velocity dispersion above 6 km s -1 that orbit within 30 kpc. The particle plots usefully reveal the entire distribution of the dispersed cluster stars, but only the 20-30 degree segments at a galactic Y coordinate of approximately 20 kpc have sufficiently high sky density to be readily visible. The orbital history of these two clusters is shown in Figure 7. Cluster 154, shown in red, accretes onto the main halo in the first few Gyr of the simulation. Cluster 179, shown in green, is accreted around 7 Gyr, about halfway through the simulation. Dynamical friction in the aspherical halo (Binney 1977) draws the subhalo containing the cluster into the plane of the galaxy and reduces the orbital angular momentum of the cluster.\",\n \"pages\": [\n 2,\n 3,\n 4\n ]\n },\n {\n \"title\": \"3. MODELING THE C-19 STREAM\",\n \"content\": \"The globular cluster streams in the cosmological simulations have radial velocity dispersions that range from about 2 to 9 km s -1 , with the highest velocity dispersion at orbital apocenter. The highest density part of C-19 is just past orbital apocenter (Martin et al. 2022; Viswanathan et al. 2024) and is similar to the high velocity streams in the simulations. Detailed modeling of a star cluster on the C-19 orbit is useful to understand the conditions under which a wide, hot stream develops. In particular the subhalo heating of a stream depends on the number of subhalos present in the primary halo and the length of time that the stream has been orbiting after tidal removal from of its natal subhalo. Our C-1 stream model is a simplification of the cosmological simulation to follow a single star cluster in a precomputed version of the galactic potential. The model star cluster has a mass of 1 × 10 4 M ⊙ , composed of 1 M ⊙ star particles using the same star cluster simulation as in the full cosmological model. The initial half-mass radius is set to approximately 4 pc for the models reported here. The star particles have a softening of 1 pc and are integrated with the Gadget4 code (Springel et al. 2021). Two body interactions are included with small random velocities calculated from the relaxation time (Spitzer 1987; Binney & Tremaine 2008) added to star particles within the virial radius, about 5 pc, every 5 Myr (Carlberg & Agler 2023). The tidal acceleration increases with distance from the cluster, pumping the stars into more distant orbits until they are swept away from the cluster (Meiron et al. 2021). The cluster-centric orbits become increasingly complex with distance(Fukushige & Heggie 2000). The cluster-centric radial coordinates of a sample of star particles are shown in Figure 8. The potential in which the star cluster orbits is drawn from the cosmological simulation. The primary mass component is a static, triaxial NFW potential, with mass, 9 . 2 × 10 11 M ⊙ , virial radius 200 kpc, scale of 14.4 kpc, and density triaxiality from the AHF halo finder (Gill et al. 2004; Knollmann & Knebe 2009) measurements at the final moment the simulation. The potential triaxiality is set to b/a=0.97, c/a=0.83 using Equation 2.72a of Binney & Tremaine (2008). A Miyamoto-Nagai galactic bulge-disk (Miyamoto & Nagai 1975) grows within the halo using the model of Carlberg et al. (2024). Although not important for the inner halo streams considered here, the LMC is included as a 1 . 8 × 10 11 M ⊙ (Garavito-Camargo et al. 2019) Hernquist sphere (Hernquist 1990), which means that the barycenter of the system is offset from the center of the primary halo. The framework could include a more detailed buildup of the primary halo and include a rotating galactic bar both of which would provide some additional stream heating. The current sparse C-19 data does not require more than the basic model. The positions and velocities of the population of subhalos above 10 6 M ⊙ as found in one of the full cosmological simulations at the chosen starting time is integrated in the combined Milky Way dark halo, disk and LMC potential. The subhalos do not interact with each other. The subhalos' masses decrease in time with an exponential decay in time, with a timescale of 5.5 Gyr, measured from the subhalo numbers with time in the simulations. The decrease in subhalo mass with time means that the subhalo N ( > M ) relation within the main halo declines approximately as measured in the fully dynamical cosmological simulations. Subhalos over the mass range 10 6 M ⊙ to 3 × 10 8 M ⊙ are included. Lower mass halos have little dynamical effect on the stream and higher mass subhalos are expected to contain visible dwarf galaxies. All subhalos are modeled as Hernquist spheres with a scale radius equal to r max from AHF. As the subhalos lose mass the radii are adjusted as M 1 / 3 to keep the characteristic density of each subhalo constant. In addition to the subhalos the 55 dwarf galaxies with kinematics (McConnachie 2012) are included. The dwarf galaxy dark halos are assigned maximal masses, M = 5 × 10 8 ( L/L ⊙ ) 0 . 65 M ⊙ , which roughly match the high mass end of the subhalo mass distribution function. The dwarf galaxy halo radii are set to a = 1 . 0( M/ 10 8 M ⊙ ) 0 . 23 kpc, on the basis of the fit to the r max mass relation of the subhalos in the simulations. All the subhalo orbits are precomputed and define spline coefficients for the positions of the masses which the star cluster calculation uses as an external potential. 100 The evolving potential model presented here does a reasonable job capturing the interactions of subhalos with the stream but misses the larger scale interactions that scatter the older, more distant ends of the stream over the halo as shown in Figure 6 for the cosmologi- cal simulation. Consequently the detailed stream model presented here is a lower bound on the effects of subhalos on the C-19 stream. The C-19 stream track is defined as passing through the approximate mean of the stars in the ϕ 1 = [ -10 , 10] region in Yuan et al. (2022). Specifically the progenitor orbit passes through RA=355.0, Dec=25.0 degrees, with a radial velocity of -195 km s -1 , proper motion of 1.25 and -2.8 milli-arcsec per year in RAcos(Dec) and Dec at a heliocentric distance of 20 kpc. The progenitor is then integrated in the potential of the MW plus LMC model. No optimization of the stream track coordinates or potential model was undertaken. The resulting orbit is adequately close to the current data for the purposes of this paper. The stream simulations placed the current epoch progenitor position at -10 degrees along the stream track, then integrated the progenitor backward including the full subhalo distribution to the desired start time. The model star cluster was then put at that position and integrated forward to create the model stream. A consequence is that the model leaves the dissolved progenitor in the ϕ 1 = [ -40 , -10] region of the stream. The precise final time progenitor location depends on the details of encounters with subhalos, so varies from run to run. The C-19 model starts the star cluster at a time of 3 Gyr after the Big Bang. Starting the model star cluster at later times means that there are less subhalos to heat the stream and less time for subhalos to encounter the stream, leading to insufficient heating. The model does a reasonable job of accumulating the subhalo interactions of the central region of the stream, but stream ends are more coherent here than they would be in a full cosmological simulation, as seen in Figure 6. The subhalos in the mass decade around 10 7 M ⊙ dominate the perturbations to the stream (Carlberg & Agler 2023; Carlberg et al. 2024). Subhalo stream encounters are sufficiently infrequent that there are significant differences of detail in the stream morphology depending on the detailed stream collision history, hence the phases of the orbits of the subhalos. Therefore a set of simulations is done with the subhalo distribution rotated 30 degrees from 0 to 330 degrees between simulations. The 4 streams that are significantly bifurcated around ϕ 1 = 0 in the CDM subhalo distribution are not included in Figure 10 because they lead to artificially wide stream widths which the currently measured C-19 sky distribution does not support. The widths of the model streams, calculated as a standard deviation, are shown in the lower panel of Figure 10. At 20 kpc distance 0.5 degrees corresponds to 170 pc or a FWHM of 410 pc. The narrowing of the streams along their length is an effect of orbital motion. The C-19 model streams have considerable structure along their length, but the structure in these model 40 30 20 10 0 10 ] e e r g e d [ 2 40 20 0 20 40 60 streams is significantly less than the two streams drawn from the full cosmological simulation shown in Figure 6. The movies of the two example streams CDM Low Mass Clusters show that large scale merging disperses the 'ends' of the streams until the main halo settles down around 7 Gyr. The primary halo is not subject to major mergers (by design) for the last 7 Gyr of the simulation so at late times is fairly well represented with the halo potential model used here. At the C-19 start time of 3 Gyr the dominant halo of the full cosmological simulation is about 50% less massive, which mainly effects the orbit, not the subhalo density. The C-19 model potential does not capture the larger scale potential flucutations due to accretion and merger buildup between 3 and 7 Gyr, but does include the subhalo effects. Thefore the C-19 model is a lower bound on the heating effects. The line of sight velocities and their dispersion along the streams in the CDM subhalo distribution are shown in Figure 11. The velocity dispersion of C-19, 6.2 kms -1 , and its 1-sigma errors from Yuan et al. (2022) is shown in black. The quadrature summed mean of the model streams over ϕ 1 = [ -10 , 10] is 4.1 ± 1.1 kms -1 . The mean velocity dispersion (excluding the bifurcated streams) is within 1.2 sigma (combined) of the observed value. The same modeling procedure for a WDM (5.5 keV) model halo gives velocities shown in Figure 12. Its mean line of sight velocity dispersion of 3.1 ± 0.09 kms -1 , a 2.2 sigma difference from the observed value. The small increase in velocity dispersion in the WDM (5.5 keV) model is consistent with the cosmological simulation results (Carlberg et al. 2024). These simulations disfavor the WDM(5.5 keV) model. However given the significant scatter seen in the full cosmological simulation for clusters in the same orbital range, Figure 5, the result is not very strong and emphasizes that stronger conclusions require more than a single stream. A model stream with no subhalos and no dwarf galaxies (designated c84) is shown in Figure 12. It has a velocity dispersion of 3.6 km s -1 in the ϕ 1 = [ -10 , 10] region. Including the known dwarf galaxies (c85) also gives a line of sight velocity dispersion of 3.6 km s -1 . The streams velocity dispersion calculated with 3 σ clipping are 1.41 and 1.75 km s -1 , respectively, indicating unusually strong wings in the velocity profile. The numbers of particles in this section of the two streams are about 20% of those in the models with subhalos which 1 is a consequence of the subalo perturbations gradually changing stream orbital phases.\",\n \"pages\": [\n 4,\n 5,\n 6,\n 7,\n 8,\n 9\n ]\n },\n {\n \"title\": \"4. DISCUSSION AND CONCLUSIONS\",\n \"content\": \"Errani et al. (2022) established, and we confirm, that the dissolution of a globular cluster in a smooth galactic halo cannot explain the the width and line of sight velocity dispersion of the C-19 stream. There are two generic solutions: either the C-19 stream was created from a stellar system hotter and larger than a globular cluster (Errani et al. 2022), or, the velocity dispersion of the stream was increased over the course of its orbital history. We have shown that an old, dissolved, globular cluster stream near apocenter in a galactic halo containing subhalos can explain the kinematics of the C-19 stream, under certain conditions. Detailed modeling of C-19 using a simplified model potential and the subhalos drawn from the cosmological simulations finds that heating C-19 requires the numbers of subhalos found in a CDM cosmology acting on the stream for ≃ 11 Gyr. The model finds that the numbers of WDM (5.5 kev) subhalos are insufficient. The subhalo heating of the stream leads to significant density and velocity variations along the stream, Figures 10 and 11, whereas a disrupted dwarf is smooth (Errani et al. 2022). The currently known numbers of stars do not allow a reliable density profile measurement. More generally, cosmological Milky Way-like simulations find that dissolved globular cluster streams that are hot and wide near apocenter, like C-19, become thin and cool streams as they orbit through pericenter. The streams discussed here are from progenitor globular clusters with masses below 2 × 10 4 M ⊙ and half mass radii in the 3-5 pc range, which have dissolution times in the galactic tidal field of 1-2 Gyr so few stars are added to the streams in the last 5 Gyr. The width of C-19 as a globular cluster stream is not unique. Streams wider than 0.2 kpc containing globular clusters are listed in Table 1 drawing from the Mateu (2023) compendium (see also Bonaca & Price-Whelan (2024)). There are a total of 27 streams wider than 0.2 kpc, with 5 containing globular clusters and the C-19 stream. The widths are all from Ibata et al. (2021) and therefore use a uniform measurement approach. C-19 may be part of an ancient, extremely metal poor group of streams (Malhan et al. 2022). The wide stream of the distant cluster NGC5466 is excluded because it is likely associated with a merger remnant (Malhan et al. 2022) with a complex orbital history. NGC288 is a complex wide stream (Grillmair 2024). All the listed globular clusters are near the apocenters of their orbits (Baumgardt et al. 2019). These clusters orbit within a few kpc of the bulge, whereas C-19 is less eccentric and has a ≃ 8 kpc pericenter. Most of the 22 other wide streams have the abundance spread of a dissolved dwarf galaxy. There should of course be other wide streams from dissolved globular clusters at orbital apocenter but their distance and spread in position and velocity make them harder to find. The distribution of stream widths with current distance in the simulations is shown in Figure 13. About half of the simulated streams are within 30 kpc, whereas about 90% of the Galstreams compendium (Mateu 2023) is within 30 kpc. The radial distribution of streams will have some dependence on the radii at which globular clusters form within pre-galactic subhalos, although that dependence is fairly weak for globular cluster formation within a dwarf galaxy (Carlberg & Keating 2022). On the average the streams are physically wider with galactocentric distance, but approximately the same angular width, around 0.2 degrees. The hot, wide streams at apocenter become thin, cool streams at pericenter as a result of stream orbital dynamics. Stars are unbound from their progenitor cluster at pericenter in a narrow range of radii, joining the stream with a spread in velocities. The subhalo perturbations to the stream velocities are more likely to occur near orbital pericenter where the subhalo density is highest. The velocity differences lead to differences in pericenter angle, orbital tilt, and angular momentum which causes the stream particles to spread apart near apocenter. The streams wider than a σ w of 100 pc in a CDM simulation are on the average about a factor of two hotter than in a WDM (5.5 keV), 7.8 ± 1.0 kms -1 as compared to 4.1 ± 1.6 kms -1 . The hot, wide C-19 and thin, cool, GD-1 (Grillmair & Dionatos 2006) streams cover similar radial ranges in their orbits, with the difference in their widths being at least partially explained with C-19 being near apocenter and GD-1 near pericenter. The C-19 model required that it be orbiting in the subhalos of the galaxy for 11 Gyr, which is consistent with the age that being extremely metal poor implies. That is, for C-19 there is very little room for stream age-subhalo abundance degeneracy. GD-1 is less metal poor which opens up the possibility of more age-abundance degeneracy in the stream properties. Finding a common model that explains the internal kinematics of a set of streams is a goal as the data improves. There is nearly a factor of two velocity dispersion difference between CDM and WDM(5.5 keV) wide streams in the inner halo. The cosmology dependence of long, thin streams, which are generally near pericenter, is present in the wings of the velocity distribution (Carlberg et al. 2024) with a relatively insensitive core Gaussian velocity spread. On the other hand, the cosmology dependence of wide streams, which are generally near apocenter, is in the velocity dispersion itself, which is easier to measure than the core-wing structure of pericenter streams. Adrian Jenkins, Carlos Frenk and Andrew Cooper provided invaluable advice and support for computing. This work used the DiRAC@Durham facility managed by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded by BEIS capital funding via STFC capital grants ST/K00042X/1, ST/P002293/1, ST/R002371/1 and ST/S002502/1, Durham University and STFC operations grant ST/R000832/1. DiRAC is part of the National e-Infrastructure. This work used high-performance computing facilities operated by the Center for Informatics and Computation in Astronomy (CICA) at National Tsing Hua University. This equipment was funded by the Ministry of Education of Taiwan, the National Science and Technology Council of Taiwan, and National Tsing Hua University. Computations were performed on the niagara supercomputer at the SciNet HPC Consortium. SciNet is funded by: the Canada Foundation for Innovation; the Government of Ontario; Ontario Research Fund - Research Excellence; and the University of Toronto. CSF acknowledges support by the European Research Council (ERC) through Advanced Investigator grant, DMIDAS (GA 786910). ARJ and CSF acknowledge support from STFC Consolidated Grant ST/X001075/1. APC acknowledges the support of the Taiwan Ministry of Education Yushan Fellowship and Taiwan National Science and Technology Council grant 112-2112-M-007-017-MY3. Software: Gadget4: Springel et al. (2021), Amiga Halo Finder: (Gill et al. 2004; Knollmann & Knebe 2009), ROCKSTAR: (Behroozi et al. 2013), NumPy: (Harris et al. 2020). Data Availability: Final snapshots, movies, images, and example scripts are at CDM Low Mass Clusters\",\n \"pages\": [\n 9,\n 10,\n 11\n ]\n },\n {\n \"title\": \"REFERENCES\",\n \"content\": \"Baumgardt, H., Hilker, M., Sollima, A., & Bellini, A. 2019, MNRAS, 482, 5138, doi: 10.1093/mnras/sty2997 Behroozi, P. S., Wechsler, R. H., & Wu, H.-Y. 2013, ApJ, 762, 109, doi: 10.1088/0004-637X/762/2/109 Binney, J. 1977, MNRAS, 181, 735, doi: 10.1093/mnras/181.4.735 Binney, J., & Tremaine, S. 2008, Galactic Dynamics: Second Edition (Princeton University Press) Bonaca, A., Hogg, D. W., Price-Whelan, A. M., & Conroy, C. 2019, ApJ, 880, 38, doi: 10.3847/1538-4357/ab2873 Bonaca, A., & Price-Whelan, A. M. 2024, arXiv e-prints, arXiv:2405.19410, doi: 10.48550/arXiv.2405.19410 Carlberg, R. G. 2015, ApJ, 800, 133, doi: 10.1088/0004-637X/800/2/133 -. 2018, ApJ, 861, 69, doi: 10.3847/1538-4357/aac88a Carlberg, R. G., & Agler, H. 2023, ApJ, 953, 99, doi: 10.3847/1538-4357/ace4be Carlberg, R. G., Jenkins, A., Frenk, C. S., & Cooper, A. P. 2024, arXiv e-prints, arXiv:2405.18522, doi: 10.48550/arXiv.2405.18522 Carlberg, R. G., & Keating, L. C. 2022, ApJ, 924, 77, doi: 10.3847/1538-4357/ac347e Errani, R., Navarro, J. F., Ibata, R., & Pe˜narrubia, J. 2022, MNRAS, 511, 6001, doi: 10.1093/mnras/stac476 Fukushige, T., & Heggie, D. C. 2000, MNRAS, 318, 753, doi: 10.1046/j.1365-8711.2000.03811.x Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016, A&A, 595, A1, doi: 10.1051/0004-6361/201629272\",\n \"pages\": [\n 11\n ]\n }\n]"}}},{"rowIdx":4969,"cells":{"bibcode":{"kind":"string","value":"2024arXiv241100100C"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/2411.00100.pdf"},"content":{"kind":"string","value":"\nThe formation and stability of a cold disc made out of stellar winds in the Galactic Centre\n1,2, 3,4 5 6,7,8\nDiego Calderón ⋆ , Jorge Cuadra , Christopher M. P. Russell , Andreas Burkert , Stephan Rosswog 1,9 , and Mayura Balakrishnan 10\n\n1 Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, 21029 Hamburg, Germany\n2 Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Straße 1, 85748 Garching, Germany e-mail: calderon@mpa-garching.mpg.de\n3 Departamento de Ciencias, Facultad de Artes Liberales, Universidad Adolfo Ibáñez, Av. Padre Hurtado 750, Viña del Mar, Chile\n4 Millennium Nucleus on Transversal Research and Technology to Explore Supermassive Black Holes (TITANS)\n5 Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA\n6 Universitäts-Sternwarte, Ludwig-Maximilians-Universität München, Scheinerstr. 1, 81679 Munich, Germany\n7 Max-Planck Institute for Extraterrestrial Physics, Giessenbacherstr. 1, 85748 Garching, Germany\n8 Excellence Cluster ORIGINS, Boltzmannstrasse 2, 85748 Garching, Germany\n9 The Oskar Klein Centre, Department of Astronomy, AlbaNova, Stockholm University, SE-106 91 Stockholm, Sweden\n10 Department of Astronomy, University of Michigan, 1085 S. University, Ann Arbor, MI 48109, USA\n\nReceived November 4, 2024; accepted November 4, 2024\nABSTRACT\nContext. The reported discovery of a cold ( ∼ 10 4 K) disc-like structure around the super-massive black hole at the centre of the Milk Way, Sagittarius A* (Sgr A*), has challenged our understanding of the gas dynamics and thermodynamic state of the plasma in its immediate vicinity. State-of-the-art simulations do not agree on whether or not such a disc can indeed be a product of the multiple stellar wind interactions of the mass-losing stars in the region.\nAims. This study aims to constrain the conditions for the formation of a cold disc as a natural outcome of the system of the mass-losing stars orbiting around Sgr A*, to investigate if the disc is a transient or long-lasting structure, and to assess the validity of the model through direct comparisons with observations.\nMethods. We conduct a set of hydrodynamic simulations of the observed Wolf-Rayet (WR) stars feeding Sgr A* using the finitevolume adaptive mesh-refinement code Ramses. We focus, for the first time, on the impact of the chemical composition of the plasma emanating from the WR stars.\nResults. The simulations show that the chemical composition of the plasma a ff ects the radiative cooling enough to impact the properties of the medium such as density and temperature and, as a consequence, the rate at which the material inflows onto Sgr A*. We demonstrated that the formation of a cold disc from the stellar winds is possible for certain chemical compositions that are consistent with the current observational constraints. However, even in such a case, it is not possible to reproduce the reported properties of the observed disc-like structure, namely its inclination and hydrogen recombination line fluxes.\nConclusions. Weconclude that the stellar winds on their own cannot form the cold disc around Sgr A* inferred from the observations. Either relevant ingredients are still missing in the model, or the interpretation of the observed data needs to be revised.\nKey words. accretion, accretion discs - Galaxy: centre - hydrodynamics - Stars: winds, outflows - Stars: Wolf-Rayet\n1. Introduction\nThe Galactic Centre (GC) hosts the closest super-massive black hole to us, Sagittarius A* (Sgr A*; see Genzel et al. 2010, for a review). Unlike the black holes present in active galactic nuclei, Sgr A* is very underluminous. It does not seem to be currently accreting much material, or to have a standard accretion disc around it (Yuan & Narayan 2014). Murchikova et al. (2019) detected however a disc-like structure around Sgr A*. Using the 1.3-millimetre recombination line H30 α they observed a doublepeaked emission line with full velocity width of ∼ 2200 km s -1 . The centre of the emission coincides with Sgr A* and extends up to 0.11 '' ( ∼ 4.4 × 10 -3 pc) to both the redshifted and blueshifted sides. They interpreted this feature as a rotating disc of mass 10 -5 -10 -4 M ⊙ . Yusef-Zadeh et al. (2020) also reported the presence of broad hydrogen recombination lines, including\nthe double-peaked H30 α line, at the position of Sgr A* using the Very Large Array. However, they interpreted the signatures as a jet emanating from the central region. So far, there has not been any detection in the near infrared, despite many observations. Ciurlo et al. (2021) reports an upper limit for Br γ , which is two orders of magnitude below the extrapolation from the reported H30 α flux. Currently, there is no consensus on how such observations can be reconciled or from where the observed cold material comes from.\nThe gaseous environment around Sgr A* is dominated by the outflows from around 30 Wolf-Rayet (WR) stars, all located within a fraction of a parsec of the black hole (Paumard et al. 2006; Martins et al. 2007). At such close distances, the winds interact strongly in shocks that thermalise them and create a hot and di ff use X-ray emitting plasma (Quataert 2004). However, given the relatively low velocity of some of the outflows (450600 km s -1 ; Martins et al. 2007), the resulting plasma can be\nprone to radiative cooling and form dense, cold clumps (Cuadra et al. 2005; Calderón et al. 2016), which end up being embedded in the di ff use, hot plasma ( ∼ 10 7 K; Bagano ff et al. 2003). The complex interplay between the stellar winds around Sgr A* has been the subject of hydrodynamic simulations using a variety of numerical techniques. With smoothed particle hydrodynamics (SPH) simulations, Cuadra et al. (2006) showed that, if most of the slow-wind stars are located in a relatively compact stellar disc, many cold clumps form and quickly coalesce in a cold gaseous disc of radius ∼ 1 '' . However, using a stellar distribution closer to the one observed, Cuadra et al. (2008, 2015) showed that no conspicuous disc appears over the 1800 yr time-span of their models. Calderón et al. (2020b) performed finite-volume hydrodynamic simulations on a Cartesian grid of the same system, which are better suited to model shocks, the multi-phase medium, and subsequent cooling, and found that the formation of a cold disc is indeed possible. According to their model, the stellar wind of the star IRS 33E, which is fairly dense ( ∼ 10 -5 M ⊙ yr -1 ) and slow (450 km s -1 ; Martins et al. 2007) interacts with the medium creating a shell that is dense enough to radiate quickly its thermal energy and break into denser and smaller structures. These pieces manage to fall inwards forming a cold ( ∼ 10 4 K) disc with a total mass of ∼ 5 × 10 -3 M ⊙ within a simulation time of 3500 yr. Based on this result, Ciurlo et al. (2021) found that the properties of the modelled disc are consistent with the non-detection in Br γ . Nevertheless, this scenario has not been confirmed by analogous grid-based simulations. Ressler et al. (2020) revisited the same system using the same numerical approach, although a di ff erent code: athena++ (Stone et al. 2020), and found no disc formation even extending their simulation time up to 9000 yr. Building on this model, Solanki et al. (2023) also studied this system through hydrodynamic modelling but encompassing a larger region and evolving it for much longer timescales. They included, for the first time, the presence of the circumnuclear disc (CND), an observed gaseous structure located between 1 . 5-3 . 0 pc (Becklin et al. 1982; Genzel 1989) and with a total mass of 3-4 × 10 4 M ⊙ (Dinh et al. 2021; James et al. 2021). This work showed that the formation of a cold disc is possible on long timescales ( ≳ 300,000 yr) due to the interaction of the innermost boundary of the CND and the WR stellar winds, which results in the transport of CND material to smaller scales. However, the authors warned that the lack of certain physical mechanisms in the model on such timescales (e.g. magnetic fields, supernovae, thermal conduction) might a ff ect the robustness of this result. Thus, the exact formation process of the observed cold disc still remains unknown.\nAkey factor driving the formation of clumps and a disc in the model of Calderón et al. (2020b) is that radiative cooling allows the gas to get rid of its energy e ffi ciently. The strength of this process depends on the chemical composition of the gas, which is highly unusual in the GC given its origin as winds from evolved massive stars. In this work, we explore for the first time the impact of the radiative cooling through studying specific chemical compositions: Solar with 1 Z ⊙ , 3 Z ⊙ , and 5 Z ⊙ . More importantly, we develop a more realistic chemical setup that considers the atmospheric abundances for the di ff erent WR sub-types present in the region, in order to follow the thermodynamic evolution of the gas in a more appropriate manner. Our results show that the formation of the disc is indeed determined by the composition of the gas. However, the properties of such a disc do not agree with the observations when considering realistic wind compositions compatible with current observational constraints. We also improve the numerical setup in order to better assess the disc orientation and more directly compare our results to the work of\nRessler et al. (2018, 2020) as well as with SPH models (Cuadra et al. 2008, 2015; Russell et al. 2017).\nThis article is organised as follows: in Section 2, we present the numerical approach, the setup and the models investigated, Section 3 presents and describes the results of the numerical simulations. In Section 4, we present the synthetic observables obtained through post-processing the models and contrast them with observations. We compare our grid-based models with SPH models in Section 5. In Section 6, we present an analytic analysis of the stability of the observed cold disc. Section 7 discusses our findings and uncertainties in the parameters of our models. Finally, Section 8 presents conclusions and final remarks. Throughout this paper we use the mass and distance to Sgr A* of 4 . 3 × 10 6 M ⊙ and 8 . 33 kpc, respectively (Gillessen et al. 2017; GRAVITY Collaboration et al. 2019, 2021), so that 1 arcsec corresponds to a length of ∼ 0.04 pc ≈ 10 5 RSch, being RSch the Schwarzschild radius.\n2. Numerical simulations\n2.1. Equations\nThe simulations were performed using the adaptive-mesh refinement (AMR) hydrodynamic code Ramses (Teyssier 2002). This code uses a second-order Godunov method with a shockcapturing scheme to solve the Euler equations in their conservative form, i.e.\n∂ρ ∂ t + ∇ · ( ρ u ) = 0 , (1)\n∂ ∂ t ( ρ u ) + ∇ · ( ρ uu ) = -∇ p -ρ ∇ ϕ, (2)\n∂ ∂ t ( ρ e ) + ∇ · \" ρ u e + p ρ !# = -ρ u · ∇ ϕ -Q -tot , (3)\n∂ ∂ t ( ρ si ) + ∇ · ( ρ si u ) = 0 (4)\nwhere ( ρ, u , P ) are the primitive hydrodynamic variables: density, velocity, and pressure, respectively. The set of quantities si correspond to tracer scalar fields that are advected with the fluid whose usage we introduce in Section 2.3. The sink terms on the right-hand side correspond to the e ff ects of the gravitational potential ϕ = ϕ ( x ), assumed to be time independent, and the total radiative loses due to optically-thin radiative cooling Q -tot = Q -tot ( ρ, T , X i), with x the position vector, T the fluid temperature, and X i the chemical composition of the fluid. The total specific energy density e is given by\ne = 1 2 u · u + p ( γ -1) ρ , (5)\nwhere γ is the adiabatic index that is set to 5 / 3. Additionally, we consider the fluid can be described as an ideal gas so that the temperature can be calculated through P = ( ρ/µ ) k B T , being µ the mean molecular weight and k B the Boltzmann constant.\n2.2. Numerical setup\nThe model considers the system of WR stars blowing stellar winds while they move on their observed Keplerian orbits around Sgr A*. The setup is analogous to our previous work (Calderón et al. 2020b). The central black hole gravitational field is modelled as a point mass of 4 . 3 × 10 6 M ⊙ (GRAVITY Collaboration et al. 2019, 2021). The stars are simulated as test particles that only feel the gravitational pull of Sgr A*. Their initial\n
\n\n
Table 1. Atmospheric mass fractions (in percentage) and mean molecular weights.
\n
\nNotes. The Solar abundances were taken from Lodders (2003). The abundances for the three WR sub-types correspond to the compilation by Russell et al. (2017) based on previous studies (Herald et al. 2001; Crowther 2007; Onifer et al. 2008). Column 1: chemical mixture. Columns 2-7: hydrogen, helium, carbon, nitrogen, oxygen, and rest of the metals mass fractions, respectively. Column 8: mean molecular weigh assuming full ionisation.\n
\n\n
Table 2. Simulation runs and parameters.
\n
\nNotes. Column 1: simulation ID. Column 2: chemical composition of the winds. Column 3: Riemann solver. Column 4: Slope limiter. Column 5: whether or not the final state of the system shows the presence of a cold disc around Sgr A*.\nposition and velocity vectors are set so that they move on the orbits constrained by observations (Paumard et al. 2006; Cuadra et al. 2008; Gillessen et al. 2019; von Fellenberg et al. 2022). The stellar winds in the simulation are modelled following the procedure by Lemaster et al. (2007), which has been validated and used in our previous work (Calderón et al. 2020a,b). This consists in defining a 'masked region\" around each star that is a spherical volume where the hydrodynamic variables are reset in each timestep, in order to reproduce the free wind expansion solution of a spherical wind with certain mass-loss rate ˙ M w, terminal velocity V w, and temperature T w. For these quantities we used the values constrained through modelling of the infrared spectra (Martins et al. 2007; Cuadra et al. 2008). The wind temperature was set to the lowest temperature allowed in the simulation, T w = 10 4 K which is determined by the strong ultraviolet radiation produced by the hundreds of massive stars in the region.\nThe simulations were run in a Cartesian grid in a cubic domain of side 40 '' ( ∼ 1.6 pc) with outflow boundary conditions (zero gradients). We used the exact Riemann solver (e.g. Toro 2009) combined with the MinMod slope limiter, as these choices allow the modelling of hydrodynamic instabilities, which otherwise may be quenched due to numerical di ff usion. We investigated other choices such as the Harten-Lax-van Leer-Contact (HLLC; Toro et al. 1994) and / or the MonCen slope limiter but they produced unwanted numerical artifacts such as spurious oscillations. For a careful analysis of these choices we refer the reader to the Appendix A. Regarding the numerical resolution of our models, the coarse resolution was 64 3 cells, allowing adaptive refinement of four levels, i.e. e ff ectively ∆ x ≈ 0 . 0382 '' . However, the regions around the stars allowed an extra level of refinement ( ∆ x ≈ 0 . 0195 '' ). Additionally, the vicinity around the\ncentral black hole has a fixed nested grid with eight refinement levels above the coarse resolution ( ∆ x ≈ 2 . 44 mas). At the location of Sgr A*, we defined a spherical region where the hydrodynamic variables are reset to low values of density and pressure at rest, in order to avoid artificial accumulation of material. The refinement strategy is based on density gradients on top of the geometric criteria previously defined. In order to explore the role of the AMR potentially a ff ecting the results we tested di ff erent values of the smoothing parameter that controls the refinement in transitions between refinement levels. We tested quadrupling the smoothing parameter and the results remained unchanged 1 .\n2.3. Models\nThe main parameter explored in this work is the optically-thin radiative cooling function. This choice is determined by specifying the chemical composition of the fluid. Unfortunately, the metallicity of the young stars in the GC is still poorly known, yet it has been argued that it should be higher than Solar ( Z = 2 -3 Z ⊙ ; Genzel et al. 2010). Past numerical works have considered that the composition of the gas correspond to Solar abundances but with metallicity three times the Solar value, Z = 3 Z ⊙ (Cuadra et al. 2008; Calderón et al. 2016; Ressler et al. 2018, 2020; Solanki et al. 2023). However, more than the 'bulk' metallicity of the stars, the most appropriate composition choice would be the abundances of the WR stellar atmospheres, which normally di ff er significantly from Solar (Herald et al. 2001; Onifer et al. 2008). Although Ressler et al. (2018, 2020) used chemical compositions lacking hydrogen, the rest of the element abundances remained unchanged relative to Solar with 3 Z ⊙ . As a result, the cooling function varied at low temperature ( < 10 5 K) but at higher temperatures it remains unchanged. Moreover, the composition choice not only determines the cooling function but also the ion mean molecular weight as well as the electron to proton number densities n e / n p. These quantities are taken into account in the sink term in the energy equation (see equation 3), and following Schure et al. (2009) can be expressed as follows\nQ -i ( ρ, Ti , Xi ) = n 2 p n e n p ! i Λ i ( Ti ) , (6)\nwhere the subscript i stands for a given chemical abundance. Bear in mind that di ff erent chemical compositions also correspond to di ff erent temperature Ti , as this is obtained through the ideal gas expression that makes uses of the mean molecular weight of the fluid. In the general case, if we consider a fluid element composed of N mixtures of abundances, the total radiative loses will be given as a summation, i.e.\nQ -tot ( ρ, Ti , Xi ) = n 2 p P N i = 1 si GLYPH<18> n e n p GLYPH<19> i Λ i ( Ti ) P N i = 1 si , (7)\nwhere we have expressed the total radative cooling as a linear combination of the mixtures. Notice that we have introduced the passive scalar fields si , as we used them to quantify the fraction of a given chemical composition. In this work, we introduced three types of compositions that represent the WR subtypes based on their atmosphere abundances. In the region where the winds are generated, the corresponding passive scalar is set to 1 while the rest are set to 0. By doing so, it is possible to identify how much material of a given cell is supplied by which sub-group of WR stars. The mass fractions of each mixture are shown in Table 1.\nIn this work, we investigated five models with di ff erent compositions: Solar composition with metallicities Z ⊙ , 3 Z ⊙ , and 5 Z ⊙ , a mixture of three compositions based on the spectroscopic constraints of the sub-types of WR stars, and a variation of the latter one motivated by the uncertainty in the WR stellar atmospheres and, specifically on the H fraction in them that is set to X H = 40% (see Section 7.2, for a discussion). Figure 1 shows the cooling function Λ = Λ ( T ) as a function of temperature for different chemical mixtures. Each curve was calculated as a linear combination of the abundance of a given element and its contribution to the total radiative cooling. The values of the composition per element are shown in Table 1. The contribution of each element to the total cooling was taken from the plasma models developed by Štofanová et al. (2021). The cooling functions corresponding to WR subtypes compositions: WC8-9, WN5-7, and WN8-9 / Ofpe are shown with dashed blue, dotted orange, and dotted-dashed green lines, respectively. The Solar and Solar with 3 Z ⊙ are represented with thin and thick solid black lines, respectively. Furthermore, we added the cooling function used in previous works by Ressler et al. (2018, 2020) as a dotted-dashed red line. Here it can be observed that a Solar composition with 3 Z ⊙ (the typical used value) is between the range of the di ff erent WR compositions but the subtypes WC89 and WN89 / Ofpe are about a factor two higher.\nThe list of the runs investigated in this work are shown in Table 2. The initial setup was identical to our previous work (Calderón et al. 2020b), i.e. the medium across the whole domain was set to constant and low-enough values of density ρ = 10 -24 g cm -3 and pressure P = γ -1 ρ c 2 s,f with c s,f = 10 km s -1 , so that the stellar winds do not encounter impediments and fill quickly the domain. All models were run for a total of 3,500 yr but setting the starting state of the system in the past, so that the final state of the simulations corresponds to the current state of the system. This is achieved by integrating the current position and velocity vectors of the stars back in time, assuming that they have followed purely Keplerian orbits within this timescale due to the gravitational field of Sgr A* (Cuadra et al. 2008; Ressler et al. 2018; Calderón et al. 2020b).\n\n\n
Fig. 1. Comparison of cooling functions Λ ( T ) for di ff erent chemical abundances. The thin and thick lines show the radiative cooling for Solar and three times Solar abundances, respectively. The dashed blue, dotted orange, and dot-dashed green lines represent the cooling functions for atmosphere abundances corresponding to WR sub-types: WC8-9, WN5-7, WN8-9 and Ofpe / WN9 based on the compositions compiled shown in Table 1. The contributions of each element to the radiative cooling were taken from the plasma models by Štofanová et al. (2021). Additionally, the cooling function used in Ressler et al. (2018) is shown as a thick black line.
\n\n3. Results\nWe proceed to describe the evolution and final state of the simulations at t = 0 that corresponds to the present time. The runs with Solar composition, but varying metallicities, were used for understanding the impact of increasing and decreasing the e ff ect of radiative cooling in general. However, since they do not represent realistic compositions we opted not to discuss them in detail here. Instead, we chose to focus on the more physically motivated models, i.e. WR_f07 and WR_f1 which we also refer as fiducial and enhanced , respectively. For completeness, we give a brief description of the models A1, A2, and A3 in Appendix B.\n3.1. Hydrodynamics\nThe simulation evolution of all models is analogous, especially in the initial phases. At t = -3500 yr, the stars begin to orbit around Sgr A* that is located at the centre of the domain, while their winds quickly fill the whole computational domain. After ∼ 500 yr ( t = -3000 yr), the system reaches a quasi-steady state. At this point, the domain is full of di ff use ( ∼ 10 -22 -10 -21 g cm -3 ) and hot plasma ( ∼ 10 7 -10 8 K) due to the shocked stellar winds and their interactions. This is illustrated in Figure 2 that shows density maps across di ff erent simulation times. The maps show projected density fields along the line-of-sight direction ( z axis) weighted by density, i.e. R ρ 2 dz / R ρ dz , in order to highlight the densest regions with the highest resolution. The top panels show two models at t = -700 yr, respectively. In them, it is possible to observe the complex structure developed due to the stellar wind interactions such as bow shocks, instabilities, and dense clumps as a result of condensation through radiative cooling. Left- and right-hand side panels display models with di ff erent chemical compositions WR_f07 and WR_f1, respec-\n10 -24 10 -22 10 -20 10 -18 ∫ ρ 2 dz/ ∫ ρdz (g cm -3 )\n\n\n\n\n\n
Fig. 2. Comparison of the simulations with di ff erent chemical abundances at two di ff erent simulation times. The panels show projected density maps weighed by density along the z axis, i.e. R ρ 2 dz / R ρ dz , which is parallel to the line of sight. Top and bottom panels show the systems at t = -0 . 7 kyr and t = 0 (present), respectively. Left- and right-hand side panels show the runs WR_f07 and WR_f1, respectively. All maps display the full computational domain.
\n\ntively. On the left-hand side, it can be seen how mildy e ffi cient cooling a ff ects mostly one of the stellar winds. This star corresponds to IRS 33E and its wind is the slowest ( ∼ 450 km s -1 ). This fact causes its shocked temperature to be in an e ffi cient region of the radiative cooling functions that allows its material to cool down and become denser. Also, some of its material manages to reach the vicinity of Sgr A*, as the map shows an\nelongated clump being accreted and stretched. The right-hand side also portrays this picture but since the radiative cooling is enhanced more dense clumps can be observed, especially close to IRS 33E and Sgr A*. Finally, the bottom panels of Figure 2 show the models at time t = 0, i.e. the present time. One can clearly see that the amount of dense material that has accumulated around Sgr A* is di ff erent in both maps. Although in the\n\n\n
Fig. 3. Net mass flow rate across a sphere of radius 5 r in = 5 × 10 -4 pc (1 . 25 × 10 -2 '' ). At this radius, the direction of the mass flow is inwards throughout the entire simulation. The models WR_f07 and WR_f1 are shown in solid blue and orange lines, respectively.
\n\nmodel WR_f07 (see bottom left-hand side panel of Figure 2) some dense material is spiraling towards the black hole overall there are is no clear structure around it. In the case of WR_f1 (see bottom right-hand side panel of Figure 2), the dense material has settled at the centre in a a disc-like structure. This di ff erence is due to the e ffi ciency of the radiative cooling. Since model WR_f1 has enhanced cooling, more dense clumps and filaments form, and some of them manage to fall onto Sgr A*. Overall, the hydrodynamic evolution of the two models is analogous: the winds fill the domain during the first ∼ 500 yr, then the systems reach a quasi-steady state, and in the last 500 yr they diverge as the model WR_f1 creates much more dense, cool material that settles around Sgr A*. A quantitative analysis of the evolution of the properties of system is presented as follows.\nTo quantify the accretion rate at di ff erent spatial scales we calculated the mass flux as a function of both radial distance and time ˙ M ( r , t ) = 4 π r 2 ρ ( r , t ) vr ( r , t ) averaged over a spherical shell. Here, positive and negatives signs in the radial velocity refer to outflow and inflow mass fluxes, respectively. First, we analyse the net mass flux at the innermost radius we can resolve properly, which is equivalent to five times the inner boundary radius, i.e. 5 r in = 5 × 10 -4 pc ≈ 1 . 25 × 10 -2 '' . Figure 3 shows | ˙ M ( r = 5 r in , t ) | as a function of time for models WR_f07 and WR_f1 that are displayed as solid blue and orange lines, respectively. In both cases, at t > -3300 yr the net mass flux reaches levels of ∼ 10 -6 M ⊙ yr -1 with short variability episodes that increase the rate by factors of two to four. During this quasi-steady phase, both simulations display similar behaviour qualitatively, likely determined by the identical stellar wind configuration. This stage lasts until t ≈ -500 yr where the mass flow rates start to deviate from each other. The fiducial model continues to display a variability amplitude of the same order of magnitude. However, the WR_f1 model shows a transition to a strongly gas inflow dominated phase, with inflow rates that are enhanced by a factor four to eight. This stage of the evolution is what we refer to as the disc formation phase, analogously to our previous models reported in Calderón et al. (2020b).\nNext, we proceed to analyse the time-averaged behaviour of the simulations over the last 500 yr, as this can give us an idea\n\n\n\n\n\n
Fig. 4. Radial profiles of time-averaged volume-weighted density (top) and mass-weighted temperature (bottom) over the last 500 yr simulation time. The models WR_f07 and WR_f1 are shown in solid orange and dashed blue lines, respectively.
\n\nof the general state of the system minimising the e ff ects of the stochastic variability. First, we analyse the (volume-weighted) density and (mass-weighted) temperature radial profiles of the simulations, which are shown on the top and bottom panels in Figure 4, respectively. The dashed blue and solid orange lines represent the simulations WR_f07 and WR_f1, respectively. The orange lines in both panels clearly highlight the presence of the cold disc. The density profile decays with r 2 in both cases at large scales ( ≳ 1 '' ). This is the result of most of the material flowing outwards at these scales following a roughly isotropic spherical wind as seen in previous models (Ressler et al. 2018; Calderón et al. 2020b). However, the profiles di ff er at smaller scales ( < 1 '' ) where the fiducial case transitions to ρ ∝ r -1 , while the model WR_f1 shows a density enhancement due to the presence of the disc. This increase in density is more than one order of magnitude. Regarding the temperature profiles, both models match at larger scales ( ≳ 1 '' ) with a constant temperature of the order of 10 7 K, set by the stellar wind collisions. Again, the profiles differ at smaller scales where the fiducial case follows T ∝ r -1 , and the enhanced cooling case displays a temperature profile about\n\n\n
Fig. 5. Radial profiles of the mass inflow ˙ M in (solid lines) and outflow ˙ M out (dashed lines) rates averaged over the last 500 yr of the simulations. The models WR_f07 and WR_f1 are shown in solid orange and blue lines, respectively.
\n\ntwo orders of magnitudes lower on average. The minimum in temperature corresponds to the region where the disc contributes with most of the mass for a given spherical shell.\nTo quantify and characterise the mass inflow and outflow regimes in the simulation domain we calculated them as radial profiles. Figure 5 displays the absolute value of the mass flow rates as a function of distance from Sgr A*. The solid and dashed lines represent the mass inflow and outflow rates, respectively; while the colours follow the same convention as before. Thus, if the solid line is above the dashed line the net mass flow is inwards and vice-versa. Analogous to the model by Ressler et al. (2018), the fiducial model encompasses di ff erent regimes due to the dominant direction of the mass flow at given spatial scales. At r > 3 '' , the outflow component dominates and the inflow is negligible. We find a net mass outflow rate of ∼ 5 × 10 -4 M ⊙ yr -1 . Notice that the enhanced cooling model exhibits exactly the same behaviour at these scales. At smaller scales (1 '' -3 '' ), there is a transition region where | ˙ M in | ∼ | ˙ M out | that corresponds to the location of most mass-losing stars and wind interactions. For model WR_f1, the change is more abrupt due to the presence of the cold disc. In both models, at r < 1 '' the inflow component is larger than the outflow, so the material inflows across these scales and down to the innermost boundary. The net e ff ect makes the mass inflow rate about five times larger which generates the cold disc.\nThe origin of the infalling material can be traced in two ways: analysing the angular momentum of the material close to the inner boundary, and using the scalar tracer fields that we introduced to label the chemical abundances of the di ff erent WR sub-types. Figure 6 shows Hammer projections of the angular momentum direction of the gas enclosed in a sphere of radius 0 . 25 '' ( ∼ 0 . 01 pc) for the fiducial and enhanced cooling runs in the top and bottom panels, respectively. Each point represents a di ff erent simulation time over the last 1000 yr, and they are connected with dashed lines. For reference, we added the angular momentum direction of the orbits of the mass-losing stars as black star symbols, and the location of the clockwise disc based on the orbits we used as a grey shaded circle. This analysis shows that the angular momentum direction of the infalling\n\n\n\n-\n-\nSun\n\n\n
Fig. 6. Orientation of the angular momentum of the gas enclosed in a sphere of radius 0.01 pc ( ∼ 0.25 '' ), and the WR stars'. The vertical dimension represents inclination i , while the horizontal dimensions stand for the longitude of the ascending node Ω . Thus, a face-on star orbiting clockwise on the sky would be at the north pole of the graph. Top and bottom panels show the runs WR_f07 and WR_f1, respectively. Each dot corresponds to the analysis of a single snapshot. As reference, the grey shaded region corresponds to the average direction of the clockwise disk at (104 · , 126 · ) with a dispersion of 16 · (Yelda et al. 2014).
\n\n\n\n
Fig. 7. Radial profiles of the scalar tracer fields that represent the WR subtypes: WC89 (dashed lines), WN57 (solid lines), and WN89 / Ofpe (dot-dashed) averaged over the last 500 yr of the simulation. The models WR_f07 and WR_f1 are shown in solid orange and blue lines, respectively.
\n\n\n\n
Figure 7 shows radial profiles of the scalar tracers. The dashed, solid, and solid-dashed represent the scalars that correspond to the WR sub-types WC89, WN57, and WN89 / Ofpe, respectively. The blue and orange lines stand for the results for the models WR_f07 and WR_f1. In this analysis, the scalar tracer values represent the fraction of the mass density that comes from a given WR sub-type wind. Thus, in general most of the material supplied into the domain comes from the WN89 / Ofpe stars. At smaller scales ( r < 1 '' ), the dominance is even clearer as the material from these stars makes about 80-90% of the mass budget. This is also the result of these stars having slow winds and colder shocked material. The rest of the stars contribute mostly to the outflow of material, as their contribution is more relevant at larger scales. Notice that both models, fiducial and enhanced, display roughly the same behaviour across the entire domain.
\n\n10\n2.0\nFig. 8. Density and line-of-sight velocity maps of the central 2.5 '' × 2.5 '' of the model WR_f1 at t = 0. The left- and right-hand side panels show projected density weighed by density and line-of-sight velocity weighed by mass, respectively both integrated along the z-axis, which is parallel to the line of sight.\nmaterial varies stochastically but overall tends to align with the orientation of the orbits in the clockwise disc. However, the degree of variability depends on whether or not the model results in the formation of a disc. The fiducial model shows more variability while the enhanced cooling run displays that the angular momentum aligns more consistently with the orientation of the clockwise disc. The fact that the infalling material at this spatial scales comes from these stars agrees with previous numerical models (Cuadra et al. 2008; Ressler et al. 2018; Calderón et al. 2020b). Furthermore, this is not surprising since these stars orbit closer to Sgr A*, and have winds that are relatively slow ( ∼ 600 kms), which results into shocks more prone to radiative cooling and with smaller angular momentum.\nOverall, the fiducial model displays properties of the medium and gas dynamics consistent with the previous models that do not form a disc (Cuadra et al. 2008; Ressler et al. 2018, 2020; Calderón et al. 2020b, run for 1100 yr). The model with enhanced cooling is consistent with the simulations that show the formation of a cold disc as an outcome of this system (Calderón et al. 2020b). Before discussing if any of the models can be favoured given the current observational constraints we proceed to characterise the cold disc arising in the WR_f1, as its properties will aid us to do so.\n3.2. Properties of the cold disc\nAt the present time ( t = 0), the model WR_f1 shows the presence of a cold disc. Figure 8 shows zoomed maps of the central region of 2 . 5 '' × 2 . 5 '' . The left- and right-hand side panels show projected maps of density and line-of-sight velocity (both weighted by density), respectively. Here it is possible to observe that the projected diameter of the disc is roughly 1 '' . We estimated the mass of the disc by isolating the cold material ( T < 10 5 K) and integrating the cell mass content within a sphere of radius 1.0 '' . We found that the total mass of the disc is 0.005 M ⊙ which is consistent with the value in our previous work (Calderón et al. 2020b). However, the mass accumulated in the disc does not converge and keeps increasing at least for the next hundreds of years in the simulation.\nOn the right-hand side panel of Figure 8, we can see that the line-of-sight velocity peaks at ∼ 2000 km s -1 along both directions (towards and outwards the observed). This coincides with the maximum extension of the observed H30 α line (Murchikova et al. 2019). Additionally, the disc is observed to be ∼ 45 · tilted in projection. This is not in agreement with the observations since, at least in projection the di ff erence is about ∼ 90 · (see Figure 1 of Murchikova et al. 2019).\n4. Post-processing of observables\nIn order to assess the validity of our models for the Galactic centre we proceed to synthesise observational quantities that we could compare easily with observations. We focus on the recombination lines H30 α and Br γ . Afterwards, we calculate the X-ray spectrum from our models.\n4.1. Recombination lines: H30 α and Br γ\nThe cold gas at the location of Sgr A* has been observed through the H30 α recombination line (Murchikova et al. 2019; YusefZadeh et al. 2020), and only with an upper limit in the Br γ recombination line (Ciurlo et al. 2021). In order to explore if the models are consistent with these observations we have calculated the expected flux from these emission lines. To compute\n1e3\n\n\n
Figure 9 shows the maps over a region of 2 '' × 2 '' centred in Sgr A* that resulted from this procedure. Although the maps were also calculated for the model WR_f07 we chose not to show them here since the emission is negligible (four orders of magnitude smaller) due to the lack of cold gas that contributes to the recombination line flux. From these maps we estimated the flux as seen from Earth, i.e. at a distance of 8.33 kpc so that they could be compared directly with the observed values. These were calculated by integrating within a circular area of a projected radius p = 0 . 23 '' and p = 1 '' centred at location of Sgr A*. These radii were motivated by the observations that used an aperture of 0.23 '' for extracting the flux observed . However, our models show that the disc actually extends beyond this size, so to get an
\n\n10\n\n\n\n10\nFig. 9. Line-of-sight integrated maps of the emission of the H30 α and Br γ recombination lines shown on the left- and right-hand side panels. The maps correspond to the inner 2 '' × 2 '' of the model WR_f1.\nthe emission of the H30 α line we used the coe ffi cients given below fitting a piecewise linear function to them as a function of density and considering a decay ∝ T -1 (see Murchikova et al. 2019, supplementary information),\nϵ H30 α (10 4 K , 10 4 cm -3 ) = 1 . 05 × 10 -31 erg s -1 cm 3 , (8)\nϵ H30 α (10 4 K , 10 5 cm -3 ) = 1 . 08 × 10 -31 erg s -1 cm 3 , (9)\nϵ H30 α (10 4 K , 10 6 cm -3 ) = 1 . 25 × 10 -31 erg s -1 cm 3 , (10)\nϵ H30 α (10 4 K , 10 7 cm -3 ) = 1 . 36 × 10 -31 erg s -1 cm 3 . (11)\nIn the case of the Br γ line, its emissivity can be calculated following Schartmann et al. (2015) as\nj Br γ = 3 . 44 × 10 -27 GLYPH<18> T 10 4 K GLYPH<19> -1 . 09 erg s -1 cm 3 . (12)\nThen, in each cell of the domain we computed the respective emissivity values, multiplied them by their n p n e (assuming full ionisation), and then integrated the data cube along the z coordinate which is perpendicular to the line of sight. It is important to remark that in this calculation it is necessary to take into account the hydrogen nucleus n p and electron n e densities appropriately. Since our models consider di ff erent amounts of hydrogen in the plasma we include only the material that contains some, i.e. the material coming from the WN89 / Ofpe stars. Additionally, we only used the hydrogen fraction of that material which is X H = 11 . 5% and X H = 40% for the models WR_f07 and WR_f1.\nidea of the whole flux of the model we chose a size of p = 1 '' that encloses entirely the disc. The flux values obtained are shown in Table 3 where we also added the upper limit for Br γ from Ciurlo et al. (2021), and the flux of H30 α measured from integrating the spectrum across the velocity channels given by Murchikova et al. (2019). In the case of the Br γ line, we can see that only the model WR_f07 has a flux consistent with the upper limit reported for an aperture of p = 0 . 23 '' . The model WR_f1 has a flux that is ∼ 100 times higher than the upper limit. Also, we see that the disc in our models extends beyond this projected radius, yet its emission is dominated by the inner p = 0 . 23 '' (see Table 3). The situation is more complex for the H30 α line since the observed flux is in both cases higher than the synthetic emission. Although the model that forms a disc is closer it still remains 30% times fainter than the reported value. The model WR_f07 has negligible emission as it is ten order of magnitudes fainter compared to the observation. Thus, it is still not clear how to reconcile the models with both line emission simultaneously as none of them is capable to reproduce the observations. We discuss further the interpretation of these results in Section 7.1.\n4.2. Spectral X-ray emission\nThe X-ray emission from the central parsec is also an observable that has allowed validation of previous numerical models (see Russell et al. 2017; Ressler et al. 2018; Wang et al. 2020). Here we also post-processed our models aiming to obtain the Iron Kalpha spectrum of this region at ∼ 6 . 7 keV as this is the strongest X-ray feature from Sgr A* (e.g. Russell et al. 2017; Corrales et al. 2020). Our approach is based on the procedure outlined in Balakrishnan et al. (2024b) but adapted to our finite-volume grid-based hydrodynamic model, which is briefly described as follows. First, we assigned energy dependent X-ray emissivities j k E = n k e n k i Λ ( E , T k ) that were taken from the vvapec model (Smith et al. 2001), which was obtained with xspec (Arnaud 1996). The densities and temperatures were taken from the last output of our simulation, i.e. at t = 0. Additionally, the tracer fields were used to identify the composition fraction of the parcels of gas according to the WR sub-types. Since the optical depth through the simulation domain is optically thin in the X-ray, the radia-\n
\n\n
Table 3. Radiative flux of the recombination lines from central region.
\n
\nNotes. All fluxes correspond to quantities seen from Earth, i.e. at a distance of 8.33 kpc, and are given in units erg cm -2 s -1 . The observed H30 α flux was calculated integrating the reported flux density across the velocity channels, and taking into consideration the error propagation of the 0.3 mJy reported (see Figure 1 in Murchikova et al. 2019).\nv th. This function is given as follows\n\n\n
Fig. 10. Synthetic X-ray spectra computed from the numerical models in an annulus of 0.5 '' < p < 1.5 '' . The models WR_f07 and WR_f1 are shown in solid orange and dashed blue lines, respectively. Additionally, the thin solid black line corresponds to the results from the SPH model (Balakrishnan et al. 2024b).
\n\ntive transfer equation to solve is simply given by the integration along the line of sight of the emissivities of the respective cells. Given that we can trace the fraction of the material that comes from a given WR sub-type this actually corresponds to a linear combination of the tracer field si and the abundance dependent emissivity for each cell, i.e.\nIE ( x , y ) = 3 X i = 1 si Z j k E ( x , y , Xi ) dz (13) = 3 X i = 1 si Z n k ion n k e Λ ( E k , Ti , Xi ) ϕ k ( v los) dz , (14)\nwhere ϕ k represents a Gaussian function that it is used to obtain the emission across di ff erent velocity channels, and to take into account the thermal broadening represented with a velocity of\nϕ k ( v los) = 1 √ 2 π v th exp -( v k los -v los) 2 2 v 2 th . (15)\nWe performed calculations through velocity channels spanning line-of-sight velocities v los from -3000 km s -1 to 3000 km s -1 . These were later converted into energy space via Doppler broadening. Thus, for each energy bin from the emissivity tables we obtained 150 line profiles using a fine resolution of 6400 per dex. In order to sum them appropriately we interpolate them into a unique energy range so that we can obtain a single spectrum that includes both the continuum and line contributions. The result of this procedure is shown in Figure 10. This contains the X-ray spectrum emitted from a sky-projected annulus of 0 . 5 '' < p < 1 . 5 '' for models WR_f07 (solid blue line) and WR_f1 (dashed orange line). As a reference, we included the estimation from Balakrishnan et al. (2024b) computed from an analogous model that used the SPH approach, which we discuss in the following section. All spectra show qualitative agreement highlighting the di ff erent features of the Fe complex at 6.7 keV and 6.9 keV. This is also reflected in the broadening of the lines due to the velocity across the line of sight, which is expected since the models are based on the same stellar orbits and wind properties.\nIt is relevant to highlight the level of agreement between the finite-volume and SPH models. The qualitative agreement is reasonable and the level of the emission is of the same order of magnitude. The exact quantitative di ff erence among the models could be mainly attributed to the exact cooling table (composition) used in each simulation, which is the result of the di ff erent density and temperature distributions that can be observed in Figure 4. This is why the WR_f07 shows the highest flux level as it has the least e ffi cient cooling and, as a result it has hotter material. Analogously, the run WR_f1 with an enhanced cooling specified in the composition of its plasma results in a reduction of ∼ 40% of its flux. In case of the SPH model, the quantitative di ff erences could be a ff ected due to the intrinsic ability of the generic SPH to capture shocks which impacts directly the density, velocity structure, and the temperature of the medium. Notice that the flux level of the spectrum of the SPH model by Balakrishnan et al. (2024b) is within the cases explored in this work. Their cooling model is not the same as the ones explored here but this analysis show that it is equivalent to a cooling e ffi -ciency between both of our models.\nAlthough not shown here we also compared the X-ray spectra from simulation WR_f1 before and after the disc is formed, and found negligible di ff erences. Finally, we also analysed the slope of the continuum emission within this energy range and\n∫\n∫\n\n\n
Figure 11 shows density and line-of-sight velocity maps projected along the line of sight of the SPH model at t = 0 on the left- and right-hand side panels, respectively. Notice that these panels are analogous to the ones shown in Figure 8 for the Eulerian models presented in this work. First, let us compare the disc density maps between the two approaches. Simply through visual inspection it is possible to see that the structure of the discs is not exactly the same. The Eulerian disc has a more continu-
\n\n\n\n\n-\n2000\nFig. 11. Density and line-of-sight velocity maps of the central 2.5 '' × 2.5 '' of the SPH model at t = 0. The left- and right-hand side panels show projected density and line-of-sight velocity, respectively. These panels are analogous to the ones shown in Figure 8.\nfound no significant di ff erences among the models. Thus, in this work the presence of the disc does not imprint a feature in the X-ray spectrum that is resolvable by any current or forthcoming X-ray telescope, in agreement with Balakrishnan et al. (2024b).\n5. Comparison with Lagrangian models\nHydrodynamic simulations of the feeding of Sgr A* by the WR stellar winds have been conducted with di ff erent numerical tools. In this work, we have presented the results of using a finitevolume approach that solves the hydrodynamic equations in the Eulerian form. However, there has been extensive work on this problem using codes that solve the equations in the Lagrangian form. Specifically, the use of SPH has been the main choice for such a task (Cuadra et al. 2005, 2006, 2008, 2015; Russell et al. 2017; Wang et al. 2020). Despite the limitations of the generic SPH technique (e.g. capturing shocks) the models managed to reproduce the observed accretion rate at the Bondi radius (Cuadra et al. 2008) as well as the X-ray emission (Russell et al. 2017). In this context, we have conducted a comparison of the Eulerian models calculated with Ramses, and the Lagrangian models computed with the SPH code Gadget (Springel 2005). The SPH simulation setup is as similar as it can be to our setup. This corresponds to an updated version from the models in Russell et al. (2017). For more details on the setup of the SPH models, we refer the reader to Balakrishnan et al. (2024a,b). We focus the comparison on the analysis on the final state of the system, specifically in the cold discs formed after 3000 yr.\nous structure towards smaller scales while the Lagrangian disc displays a ring-like structure. This feature can be attributed to the spatial resolution and the inner boundary radius. The Eulerian models indeed manage to resolve smaller scales towards the centre, and the radius of the inner boundary is 5 mas while the Lagrangian models resolve ∼ 1 mas and have an inner boundary of 0 . 1 '' . As a result, the Lagrangian model obtained a disc 50 times lighter ( ∼ 10 -4 M ⊙ ) than in the Eulerian simulation. This di ff erence could be attributed to the radiative cooling employed in each simulation: WR sub-type abundance and Solar with 3 Z ⊙ in the Eulerian and the Lagrangian model, respectively. Regarding the projected orientation of the disc both models seem to be in agreement. This can be analysed more carefully on the lineof-sight velocity maps (see right-hand side panels of Figures 8 and 11). Both maps display that the discs indeed match their projected orientation as well as their most blue- and redshifted velocities are roughly -2000 km s -1 and 2000 km s -1 .\nOverall, the agreement on the outcome a cold disc around Sgr A* whose properties align between two di ff erent approaches indicates that this is a result independent of the numerical technique. Thus, this shows that the determinant factors to obtain this results lies in both the long-enough simulation time ( ≳ 3000 yr ) as well as e ffi cient radiative cooling.\n6. Disc stability analysis\nIn this section, we present an analysis of the stability of the observed cold disc in the Galactic Centre. Based on its reported properties, the disc should be depleted by accretion after its viscous timescale t ν ≈ 43 kyr, assuming an α = 0 . 1 thin disc (Shakura & Sunyaev 1973), a process much slower than its generation as modelled in our simulations, although faster than its formation out of CND material (Solanki et al. 2023). Nonetheless, there are several external mechanisms that could be capable of destroying the disc faster than through accretion. Among them are the gravitational and hydrodynamic interactions of the disc\nwith the stars, the stellar-wind-disc collisions due to the presence of the S-star cluster in its location, and / or the evaporation due to heat flowing between the hot medium and the cold disc. Although some of these physical scenarios have been studied analytically and numerically, there has not been any direct application to the cold disc discovered in our Galactic Centre.\n6.1. Stellar cluster perturbing a thin disc\nThe stability of a cold disc perturbed by stellar passages has been investigated by Ostriker (1983). Their work presented an analytical approach to calculate how the disc loses angular momentum due to the passages, considering the integrated e ff ect of a stellar cluster in stationary state. Besides this work, most e ff orts have been put on following how the stellar dynamics are a ff ected by the disc presence (e.g., Fabj et al. 2020, and references therein) or on the e ff ect of embedded accreting stars and / or black holes on the disc (e.g., Tagawa et al. 2021) rather than on the consequences to the disc properties.\nIn this Section, we will summarise the approach of Ostriker (1983) and apply it to the Galactic Centre case. Let us start considering a cold, thin accretion disc around a compact object. Within the thin disc we assume that its height H ( r ) is small compared to the radial coordinate r , i.e. H ( r ) / r ≪ 1. Also, we consider that the sound speed of the material in the disc is small compared to the speed of the stars at the same radius. In this scenario, a star passing through the disc will produce a torque that removes angular momentum from the disc. We consider that the star interacts with the disc gravitationally and hydrodynamically The gravitational interaction of a star crossing the disc can be described using the Bondi-Hoyle approach (Bondi & Hoyle 1944), while the hydrodynamic interaction is just a physical collision of a solid sphere with a gaseous disc.\nLet us consider a system of reference where the x axis is in the direction of rotation and z is in the direction of the rotation pole of the disc. Then, according to Ostriker (1983) the total change of linear momentum along the x direction ∆ px for a single stellar passage is\n∆ px = -π R 2 ∗ Σ d q + ln Λ D η 4 v 0 v rel ! 4 v cos ϕ -v d sin θ cos ϕ ! GLYPH<18> v rel v GLYPH<19> , (16)\nwhere R ∗ is the radius of the star, Σ d = R disc ρ d H ( r ) is the surface density of the disc, ρ d is the volume density of the disc, q is the hydrodynamic drag parameter that is set to q = 2 assuming a large Mach number collision, v 0 is the local stellar velocity, Λ D ≈ H ( r ) v 2 0 / ( R ∗ v 2 ∗ ) is the Coulomb logarithm, v ∗ is the escape velocity from the surface of the star, η = v ∗ / v 0 is the hardness parameter, v d is the disc velocity, v rel is the relative velocity vector, and ( θ, ϕ ) are the spherical coordinate angles along the polar and azimuthal directions.\nIn order to consider the e ff ect of the complete stellar cluster interacting with the disc we need to integrate over the cluster phase-space volume. First, we integrate over the local stellar velocity distribution. Let us define dN as the number of stars in the velocity range v → v + dv , i.e\ndN = 4 π n ∗ v -3 0 f ( v ) v 2 dv , (17)\nwhere f ( v ) is the velocity distribution function, n ∗ is the stellar number density. Then, the number of crossings per unit time and area in the ( θ, ϕ ) direction is\ndN cr = n ∗ v -3 0 f ( v ) v 2 dv sin θ d θ d ϕ sin θ cos ϕ. (18)\nArticle number, page 12 of 19\nNow, integrating over the phase space, the total transfer of momentum to the disc per unit area at r is\n˙ px , tot( r ) = -2 π 2 R ∗ v -3 0 Σ d( r ) n ∗ ( r ) Z ∞ 0 dvv 2 f ( v ) × Z + 1 -1 d µ ( v µ -v d) v rel h q + η 4 ln Λ D( v / v rel) 4 i , (19)\nwhere µ = cos ϕ .\nAs the local momentum per unit area in the disc is px , d( r ) = Σ ( r ) /τ d( r ) the inverse of the drag timescale to remove the angular momentum of the disc is τ d = -px , d / ˙ px , tot can be written as\nτ d = h n ∗ R 2 ∗ v 0 GLYPH<16> qI 0 + η 4 ln Λ D I 1 GLYPH<17>i -1 , (20)\nwhere the quantities ( I 0 , I 1) depend solely on the stellar distribution function. For instance, in the case of a Maxwellian distribution the values of these parameters can be obtained numerically and are I 0 = 12 . 553 and I 1 = 0 . 474.\nIn order to apply this model to the Galactic Centre we need to calculate the stellar density within the size of the disc. Following Schödel et al. (2007) the stellar mass density at r < 6 arcsec is described by\nρ ∗ ( r ) = 2 . 8 ± 1 . 3 × 10 7 M ⊙ pc -3 GLYPH<18> r 6 arcsec GLYPH<19> -1 . 2 . (21)\nAssuming that the stellar mass density is composed mainly by stars whose radii are R ∗ = 1 R ⊙ and move on their orbits typically at v 0 = 1000 km s -1 we can use Equation 20 to estimate the inverse of the drag timescale,\nτ d = 185 n ∗ 3 . 8 × 10 8 pc -3 ! -1 R ∗ 1R ⊙ ! -2 GLYPH<18> v 0 10 3 km s -1 GLYPH<19> -1 × \" 1 + 0 . 08 GLYPH<18> v ∗ 440 km s -1 GLYPH<19> 4 GLYPH<18> v 0 10 3 km s -1 GLYPH<19> -4 ln Λ D 10 !# -1 Myr . (22)\nBased on this calculation the angular momentum of the cold disc should be removed after ∼ 185 Myr of interactions of the stars. This is much longer than the formation time-scale found in our simulations. Furthermore, it is even much longer than the age of the Wolf-Rayet stars, which is about 6 Myr (Martins et al. 2007). Then, the gravitational and hydrodynamic interactions of the stars onto the disc should not a ff ect the stability of the disc.\n6.2. Stellar winds perturbing a disc\nIn the case of stellar irradiation a ff ecting the state of the accretion flow there have been many studies mainly motivated by the recent pericentre passages of the star S2 around Sgr A*. For instance, Cuadra et al. (2003) could rule out the existence of an optically-thick disc based on the lack of its thermallyreprocessed emission as it gets illuminated by S2. Nayakshin et al. (2004) estimated that a star as luminous as S2 could heat up and ionise an inner disc, which could enhance the accretion rate. Giannios & Sironi (2013); Christie et al. (2016) calculated the bremsstrahlung emission as a result of the stellar wind shocking a Radiatively Ine ffi cient Accretion Flow (RIAF) around Sgr A*, finding an X-ray luminosity comparable to quiescent emission of Sgr A*. In fact, no noticeable increase was detected during 2018 (see Table 1 of Andrés et al. 2022). Hosseini et al. (2020)\nconstrained the observed L ' -band variability of S2 to be about 2-3%, based on a model of bow shock of its stellar wind, implying an ambient density < 10 5 cm -3 , which rules out the presence of a standard accretion disc at S2's pericentre, in agreement with the previous studies. It is important to remark however that such a limit marginally allows the existence of both the disc reported by Murchikova et al. (2019), and also the one in our simulation.\nThe stars in the S cluster also have relatively powerful stellar winds. As the stars inhabit the black hole in the vicinity of the cold disc, it is expected that the winds interact with it. By either opening a bubble due to the high kinetic energy carried in the winds or depositing thermal energy they a ff ect the disc properties. We proceed to perform analytical estimates to quantify the impact of these processes.\n6.2.1. Bubbles in the disc\nLet us consider a star blowing an isotropic stellar wind with a mass-loss rate ˙ M w at a terminal speed v w. Then, its density ρ w at a distance r from the star is given by\nρ w( r ) = ˙ M w 4 π r 2 v w (23)\nIf such a star is immersed in a medium whose number density is n m at rest at a temperature T m, the radius of the bubble r b opened by the wind can be calculated equating the ram pressure of the wind and the thermal energy of the medium, i.e.\nr b = 1 . 48 × 10 15 ˙ M w 10 -8 M ⊙ yr -1 ! 1 / 2 GLYPH<18> v w 1000 km s -1 GLYPH<19> 1 / 2 × GLYPH<18> n m 10 6 cm -3 GLYPH<19> -1 / 2 GLYPH<18> T m 10 4 K GLYPH<19> -1 / 2 cm , (24)\nwhere we have used as fiducial values the disc parameters measured by Murchikova et al. (2019) and typical stellar wind properties for B stars (Oskinova et al. 2011; Krtiˇcka 2014). Then, the radius of the bubble is about 0.012 arcsec, which is approximately one tenth of the disc radius.\nIn order to open such a bubble a star would need to be within the disc fort at least the wind crossing time, i.e. r b / v w, which is about ∼ 0 . 5 yr. Let us estimate the duration of a stellar passage through the disc. Along the perpendicular direction a star crossing the disc would take at most 0 . 1 R d / v 0, being H ( r ) / r = 0 . 1. One S star moves typically at v 0 ≈ 1000 km s -1 , then 0 . 1 R d / v 0 ≈ 1 yr. As both timescales are comparable, we conclude that the bubble can be opened.\nFor such a perturbation to last we need to check if either shear or sound waves could eliminate it, i.e. close the bubble. In the case of shear, this is given by the orbital speed of the disc and the size of the bubble:\nt close = r b ∆ v kep = r r GM h r + r b + p r ( r + r b) i . (25)\nAs this expression increases with r , the largest value will be at the outer edge of the disc, then t close < 5 yr. This value should be interpreted as the longest duration of a perturbation of the wind onto the disc. As there are 12 S-stars on orbits whose pericentre distances are shorter than the radius of the disc and their orbital periods are of the order of ∼ 10 yr (Gillessen et al. 2017) we expect a bubble to be opened every year on average. However, as shear would close such bubbles in less than five years there would be at most four bubbles opened on average in a stationary\nstate. As the size of the bubbles is very small compared to the size of the disc (two orders of magnitude in area), four of them would not have an impact on its structure. The sound crossing timescale r b / c s ≈ 30 yr is longer than the shear timescale so we do not expect it to have an impact on erasing perturbations.\n6.2.2. Winds injecting thermal energy\nThe S stars are of spectral type B with masses of 10 M ⊙ (e.g. Habibi et al. 2017), thus their mass-loss rate must be ∼ 10 -8 M ⊙ yr -1 with terminal velocity of 1000 km s -1 . The supersonic nature of the winds should be enough to compress the shocked material at high temperature. Potentially, this thermal energy could be deposited onto the disc during each stellar passage. If we consider that most of the kinetic energy of the wind is transformed into thermal energy about ∼ 6 . 4 × 10 33 erg would be deposited in the disc during the ∼ 1 yr long passage (see above). In order to check if the wind energy can significantly impact the thermodynamic state of the disc let us estimate its total internal energy U int.\nU int = n d k B T d γ -1 V d , (26)\nwhere k B is the Boltzmann constant and V d is the disc volume. Replacing the observed disc properties we obtain that U int = 1 . 63 × 10 42 erg. From this calculation it is clear that even if all the energy from the wind in a stellar passage is deposited the contribution is still negligible to modify the state of the disc. Even if we consider the integrated e ff ect of all the S stars passages on a timescale comparable to the viscous timescale of the disc ( ∼ 43 kyr) the contribution is still < 0 . 1% of the total thermal energy of the disc. Thus, the energy supplied by the stellar winds into the disc is not enough to a ff ect its state.\n6.3. Thermal conduction\nThe vicinity of Sgr A* at the Bondi radius is made out of a diffuse (10 cm -3 ), hot medium (10 7 K; Bagano ff et al. 2003) due to the shocked stellar winds from the Wolf-Rayet stars in the region. At the disc vicinity ( ∼ 10 3 R Sch), the conditions are more extreme as the plasma has a density of about ∼ 5 × 10 3 cm -3 (Gillessen et al. 2019) and temperature of ∼ 10 8 K. As the disc is significantly colder it is expected that heat flows from the medium to the disc. The large temperature gradient should enable thermal conduction to take place and likely evaporate the disc.\nThe problem of evaporating a cold structure sitting in a hot environment due to thermal conduction was first studied by Cowie & McKee (1977). In that work, the authors derived an analytical expression for the mass-loss rate and evaporation timescale for a spherically symmetric gas cloud. Unfortunately, these expressions are not valid for more complex geometries. Meyer & Meyer-Hofmeister (1994) studied specifically the evaporation of a cold thin accretion disc in a hot corona. Although the model considered only one radial zone it was able to calculate the full perpendicular structure of the accretion disc and the corona by means of numerical simulations.\nLiu et al. (2004) used the model by Meyer & MeyerHofmeister (1994) to analyse which solutions were consistent with the observational data available at the time in case there were a cold disc in the center of the Galaxy. They found that any transient disc would evaporate quickly. With their obtained evaporation rate of ∼ 10 -4 M ⊙ yr -1 , the life-time of a disc with\nproperties such as found by Murchikova et al. (2019) or formed in our models would be ∼ 1 yr. Similar analysis have argued that small clumps ( ∼ M ⊕ ) in the Galactic Centre should also evaporate quickly ( ∼ 1 yr) due to thermal conduction (Burkert et al. 2012; Calderón et al. 2018). According to this, cold gas should not be able to last long if formed, yet we observe many cold gas structures in the central parsec: the many dusty cold clumps detected in the vicinity of IRS 13E (Fritz et al. 2010; Peißker et al. 2023), and a larger gaseous cloud X7 (Ciurlo et al. 2023). Thus, this regime of classical thermal conduction might not be applicable to the Galactic centre environment. Another possibility has been studied by Nayakshin (2004) who focused on the same problem and found solutions in a di ff erent regime, where the electron mean free path is long enough that an accretion disc does not evaporate but rather increases its mass by condensation. A more thorough analysis of the role of thermal conduction is beyond the scope of this work and is deferred to future study.\nOur findings shows that gravitational and hydrodynamic effects require timescales that are too long to have an impact on the disc stability. On the other hand, we find that thermal conduction in the classical case should evaporate the observed disc in about one year. Given the time-scale for disc formation found in our model, the Nayakshin (2004) condensation regime remains to our knowledge as the only viable physical scenario that would allow its continued existence and potential identification with Murchikova et al. (2019)'s reported disc.\n7. Discussion\nIn this section, we interpret our findings exploring the uncertainties in the abundances of WR stars in general and of the ones in the central parsec. Additionally, we contrast the properties of the simulated and observed cold discs as well as discuss further implications.\n7.1. Is the simulated disc the observed disc?\nIn order to compare the simulated and the observed discs, first we analyse their physical properties. The simulated disc has a total mass of ∼ 5 × 10 -3 M ⊙ , while the structure observed in H30 α was reported to have a mass of 10 -5 -10 -4 M ⊙ . However, comparing these quantities might not be appropriate since the mass of the observed structure is based on three assumptions: a thin disc with uniform density and temperature, and a masing factor of ∼ 100. All of them minimise the value of the mass inferred from the observations. For instance, dropping the maser assumption the inferred mass increases in a factor ten (see Equations 30-31 of Supplementary Material in Murchikova et al. 2019). Bearing this in mind, we could argue that the mass of the simulated and observed disc might be of the same order provided there was no maser. Although removing that assumption would create tension reconciling both the Br γ and H30 α observed fluxes.\nRegarding its extension, the simulated disc has a projected radius of ∼ 0 . 5 '' , while the observed disc has a radius of 0 . 23 '' . However, since most of the recombination line emission of the disc is originated from its central region ( p < 0 . 23 '' , see Table 3) the sizes of the simulated and observed discs are in agreement.\nA more direct comparison can be done through the analysis of the mock observational quantities computed from the simulations. In Section 4.1, we computed the H30 α and Br γ recombination line emission and estimated their fluxes. Here, we have found that both synthetic fluxes are in tension with the observations. First, the simulated Br γ flux is 100 times higher than\nthe upper limit. It is relevant to remark that this limit is already extinction corrected. Second, the simulated H30 α flux is 30% lower than the observed one. These results di ff er from the estimates by Ciurlo et al. (2021) from our previous models due to, again, the uniform disc assumptions. The simulated disc is not thin or possess uniform density or temperature. Although the mean density and temperature of the simulated disc is of the order of ∼ 10 -5 cm -3 and ∼ 10 4 K its geometry is more complex with a larger variance in its properties. As a result, both the synthetic Br γ and H30 α fluxes are much higher than the ones inferred by Ciurlo et al. (2021).\nBased on the line-of-sight velocity maps shown in Murchikova et al. (2019), the orientation of the observed disc displays the red- and blueshifted sides to be on the East (lefthand) and West (right-hand) sides, respectively. Although our model WR_f1 also shows this configuration the exact inclination of the discs does not match perfectly. Specifically, the discs are inclined in ∼ 90 · among each other, being the simulated disc tilted in the clockwise direction on the sky. In principle, this argues against the WR stars as the main responsible for the formation of the observed disc. But we cannot rule out completely their role in the process, since a combination of the WR stars with other gaseous structures, such as the minispiral or the CND, acting simultaneously might result in a net e ff ect that manages to reproduce the observed angular momentum. Solanki et al. (2023) showed that material from the circumnuclear disc could fall close to Sgr A* though on much longer timescales. Certainly, this scenario would be much more di ffi cult and challenging to simulate with the appropriate chemical abundances as well as with high-enough spatial resolution.\nIt is also relevant to notice that the observed accretion flow orientation at ≲ 10 RSch is consistent with the orientation of the clockwise stellar disc (GRAVITY Collaboration et al. 2020; Event Horizon Telescope Collaboration et al. 2022). Thus, our models reproduce that connection between the dynamics from large ( > 10 5 RSch) to small ( ≲ 10 RSch) scales. However, the observed cold disc does not seem to follow such a connection.\nFinally, an intriguing aspect of our findings is that the WR atmospheres with more H in them would favour the formation of the disc. At the same time, the disc indeed was detected in the H30 α recombination line, which obviously indicates the presence of H. Thus, it is sensible to ask how much H can be in the winds of WR stars but, specifically in the ones of the sub-type WN89 / Ofpe. This point is discussed in the following section.\n7.2. Uncertainties in abundances\nThe results of the hydrodynamic models demonstrate that the chemical abundances of the material in the WR winds are the key factor to determining whether or not a cold disc can be formed due to their action. Unfortunately, there are two large uncertainties in this context. First, the metallicity of the young population of the nuclear star cluster is not well constrained (e.g. Genzel et al. 2010). Although there is agreement that it is above Solar its exact value has not been established so far, with most studies quoting the range Z = 2 -3 Z ⊙ . However, even if the metallicity is determined more precisely, the problem would not be entirely solved, since the abundances of the plasma in the region are dominated by the material supplied by the WR star winds whose compositions are much more uncertain. Theoretically, the evolution of these stars is a topic of active research, and improved prescriptions for mass loss are changing the modelled properties of observed WR stars (see e.g., Gormaz-Matamala et al. 2023, and references therein). Simply based on observations, up to now, it\nhas been reported that some of these stars might possess from X H = 0 to X H = 50% for Galactic WR stars of the WN subtype (Hamann et al. 2006, 2019), and even as high as X H = 60% for extragalactic ones (Todt et al. 2015). In the vicinity of Sgr A*, Martins et al. (2007) characterised the wind properties of the sample of WR stars through the analysis of their infrared spectra. They reported abundance H / He ratios in the range 2-5 for these stars which does not allow us to rule out the scenario regarding this aspect. A variation on the hydrogen abundance in the chemical mixture in a factor five might change the net radiative cooling rate by a factor 1 . 1 -1 . 4, especially at lower temperatures. In our models, WR_f07 considered a fiducial value of hydrogen mass fraction of 11.5% (Russell et al. 2017) but the model WR_f1 assumed 40%. Such a change was enough to modify the cooling function by about 30%, which ended up affecting significantly the final result and the state of the plasma. Thus, within our current understanding of the abundances in the WR stellar atmospheres both scenario are plausible. Only better quality spectra and more sophisticated models of them could allow us to constrain the wind properties and their composition more precisely.\n8. Conclusions\nWe have presented a numerical study of the system of masslosing stars feeding Sgr A* focusing on the impact of the chemical composition of the plasma. Through studying simulations with di ff erent abundances we have found that the system can either form or not form a cold disc around the black hole, provided that the model is run for long-enough timescales ( ≳ 3000 yr) to create a dense enough medium in the inner parsec of the Galaxy. As in our past work, we have confirmed that if formed the cold disc can impact significantly the hydrodynamic and thermodynamic state of the plasma within the central parsec. As a result, the inferred mass in flow rate at 5 × 10 -4 pc can reach up to 8 × 10 -6 M ⊙ yr -1 , which is 4 -8 times higher than the value without the presence of such a disc. Our models still point to the mass-losing stars in the clockwise disc being the main source of the accreted material due to their dense and slow winds as previous models have shown (e.g. Cuadra et al. 2008; Ressler et al. 2018; Calderón et al. 2020b). In this work, we have followed more carefully the origin of the inflowing material through the use of passive scalar fields for di ff erent WR subtypes. This allowed us to confirm the origin of the accreted material but, more importantly led us to conclude that this is mostly provided by the winds of the stars of the WR sub-type WN89 / Ofpe. This provides evidence that the infalling material must have a nonnegligible fraction of hydrogen as these WR stars have not lost entirely their atmospheric hydrogen via winds.\nOur models also showed that the formation of a cold disc depends strictly on the radiative cooling employed, which is determined by the chemical abundances of the plasma. Previous models have only considered Solar abundances with 3 Z ⊙ but had not agreed whether or not a cold disc can be formed. From our results in Appendix B, we speculate that at Z ≈ 3 Z ⊙ there is a transition between the regimes of persistent quasi-steady accretion and disc formation, which would imply that small di ff erences in the cooling function or numerical implementation can result in models reproducing either regime. This may explain the di ff ering results obtained by Calderón et al. (2020b) and Ressler et al. (2020) with similar numerical and physical setups.\nIn the case a cold disc is formed around Sgr A* the structure may resemble the extension and the line-of-sight velocity structure of the observed disc. However, direct observational\nquantities could not be reproduced. Specifically, the exact skyprojected inclination of the simulated and observed discs di ff ers in ∼ 90 · . The Br γ emission line upper limit is 100 lower than the synthetic flux. The observed H30 α emission line flux is 30% higher than in our model.\nFurthermore, we have contrasted this work Eulerian (finitevolume grid-based) models with analogous (SPH) Lagrangian models in order to address the potential influence of the chosen approach in the outcome of the simulations. As it has been shown in Balakrishnan et al. (2024b), a cold disc is also formed around Sgr A* if the simulation is run for long timescales. Both cold structures agree on their sizes, their velocity structure, and their sky-projected inclination. Although there are di ff erences in the total mass and specific inner structure of the discs this could be attributed to the resolution employed and the exact assumption to consider the innermost boundary. Despite these di ff erences, the agreement is reasonable among the numerical approaches, especially when analysing the synthetic X-ray emission (see Figure 10). The spectral features agree qualitatively across the models presented in this work and the Lagrangian model. The level of the X-ray emission are all within the same order of magnitude but the exact base level depends on the radiative cooling chosen, which is determined by the chemical abundances in the WR atmospheres.\nWe also have explored the stability and long-term evolution of the putative disc under the e ff ect of perturbing agents: the nuclear star cluster, the wind of all the mass-losing stars (including the S stars) impacting the disc through opening bubbles and / or depositing thermal energy via shocks, and the potential e ff ect of thermal conduction. Our analysis showed that gravitational and hydrodynamic e ff ects would impact the disc on longer timescales then the viscous timescale. However, within the classical thermal conduction framework a cold disc in such a region should not exist. Based on this and the disc formation timescale from our simulations, we speculate that the condensation model of Nayakshin et al. (2004) remains a plausible mechanism consistent with the disc existence.\nIn conclusion, through this work we have been able to reconcile the discrepancy among the numerical simulations of the system of mass-losing stars feeding Sgr A*. The formation of a cold disc on ∼ 3000 yr timescale is indeed possible for certain chemical abundances which are consistent with the current observational constraints. Nonetheless, it is not possible to favour or disfavour this scenario due to the uncertainties in them. Thus, high-resolution spectra of the WR stars and more sophisticated modelling of them could allow us to discern the validity of such scenarios. But we must warn that even in the case that the cold disc is a result of action of the WR stars it is not possible to reproduce all its observed properties: its inclination and recombination line fluxes. More sophisticated models that include infalling material from other sources might be necessary to produce the exact inclination of the structure which could also impact the emission of the structure.\nAcknowledgements. We thank Prof. Sergei Nayakshin and Prof. Stanley Owocki for very helpful discussions about this project. We also thank Dr. Sean M. Ressler for sharing the radiative cooling function shown in Figure 1. Additionally, we are grateful to Dr. Álex Gormaz-Matamala for helping with references and discussions on the WR wind abundances. DC and JC acknowledge the support of the Kavli Foundation through its summer program at the Max Planck Institute for Astrophysics where fruitful discussions took place. In addition, DC thanks the warm hospitality of the Universidad Adolfo Ibañez in Chile and University of Delaware in the USA where part of this project was developed. We acknowledge the support from ANID in Chile through FONDECYT Regular grant 1211429 and Millennium Science Initiative Program NCN2023_002. The research of DC and SR was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - EXC 2121 -\n'Quantum Universe' - 390833306. Since 01.10.2024, DC has been funded by the Alexander von Humboldt Foundation. SR is also supported by the European Research Council (ERC) Advanced Grant INSPIRATION under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 101053985), the Swedish Research Council (VR) under grant number 202005044, and the research environment grant 'Gravitational Radiation and Electromagnetic Astrophysical Transients' (GREAT) funded by the Swedish Research Council (VR) under Dnr 2016-06012, by the Knut and Alice Wallenberg Foundation under grant Dnr. KAW 2019.0112. The numerical simulations of this work were run on the high-performance computing system cobra of the Max Planck Computing and Data Facility. Most of the analysis of the numerical data was carried out using the python package yt (Turk et al. 2011). Furthermore, this work made use of python libraries numpy (Harris et al. 2020) and matplotlib (Hunter 2007), as well as of the NASA's Astrophysics Data System.\nReferences\nAndrés, A., van den Eijnden, J., Degenaar, N., et al. 2022, MNRAS, 510, 2851 Arnaud, K. A. 1996, in Astronomical Society of the Pacific Conference Series, Vol. 101, Astronomical Data Analysis Software and Systems V, ed. G. H. Jacoby & J. Barnes, 17\n\nBagano ff , F. K., Maeda, Y., Morris, M., et al. 2003, ApJ, 591, 891\nBalakrishnan, M., Corrales, L., Marko ff , S., et al. 2024a, ApJ, 974, 98\nBalakrishnan, M., Russell, C. M. P., Corrales, L., et al. 2024b, ApJ, 974, 99\n\nBecklin, E. E., Gatley, I., & Werner, M. W. 1982, ApJ, 258, 135\n\nBondi, H. & Hoyle, F. 1944, MNRAS, 104, 273\nBurkert, A., Schartmann, M., Alig, C., et al. 2012, ApJ, 750, 58\nCalderón, D., Ballone, A., Cuadra, J., et al. 2016, MNRAS, 455, 4388\nCalderón, D., Cuadra, J., Schartmann, M., et al. 2018, MNRAS, 478, 3494\nCalderón, D., Cuadra, J., Schartmann, M., et al. 2020a, MNRAS, 493, 447\nCalderón, D., Cuadra, J., Schartmann, M., Burkert, A., & Russell, C. M. P. 2020b, ApJ, 888, L2\nChristie, I. M., Petropoulou, M., Mimica, P., & Giannios, D. 2016, MNRAS, 459, 2420\nCiurlo, A., Campbell, R. D., Morris, M. R., et al. 2023, ApJ, 944, 136\nCiurlo, A., Morris, M. R., Campbell, R. D., et al. 2021, ApJ, 910, 143\nCorrales, L., Bagano ff , F. K., Wang, Q. D., et al. 2020, ApJ, 891, 71\nCowie, L. L. & McKee, C. F. 1977, ApJ, 211, 135\nCrowther, P. A. 2007, ARA&A, 45, 177\nCuadra, J., Nayakshin, S., & Martins, F. 2008, MNRAS, 383, 458\nCuadra, J., Nayakshin, S., Springel, V., & Di Matteo, T. 2005, MNRAS, 360, L55\nCuadra, J., Nayakshin, S., Springel, V., & Di Matteo, T. 2006, MNRAS, 366, 358\nCuadra, J., Nayakshin, S., & Sunyaev, R. 2003, A&A, 411, 405\nCuadra, J., Nayakshin, S., & Wang, Q. D. 2015, MNRAS, 450, 277\nDinh, C. K., Salas, J. M., Morris, M. R., & Naoz, S. 2021, ApJ, 920, 79\nEvent Horizon Telescope Collaboration, Akiyama, K., Alberdi, A., et al. 2022, ApJ, 930, L12\nFabj, G., Nasim, S. S., Caban, F., et al. 2020, MNRAS, 499, 2608\nFritz, T. K., Gillessen, S., Dodds-Eden, K., et al. 2010, ApJ, 721, 395\nGenzel, R. 1989, in The Center of the Galaxy, ed. M. Morris, Vol. 136, 393\nGenzel, R., Eisenhauer, F., & Gillessen, S. 2010, Reviews of Modern Physics, 82, 3121\nGiannios, D. & Sironi, L. 2013, MNRAS, 433, L25\nGillessen, S., Plewa, P. M., Eisenhauer, F., et al. 2017, ApJ, 837, 30\nGillessen, S., Plewa, P. M., Widmann, F., et al. 2019, ApJ, 871, 126\nGormaz-Matamala, A. C., Cuadra, J., Meynet, G., & Curé, M. 2023, A&A, 673, A109\nGRAVITY Collaboration, Abuter, R., Amorim, A., et al. 2019, A&A, 625, L10\n\nGRAVITY Collaboration, Abuter, R., Amorim, A., et al. 2021, A&A, 647, A59\nGRAVITY Collaboration, Bauböck, M., Dexter, J., et al. 2020, A&A, 635, A143\n\nHabibi, M., Gillessen, S., Martins, F., et al. 2017, ApJ, 847, 120\nHamann, W. R., Gräfener, G., & Liermann, A. 2006, A&A, 457, 1015\nHamann, W. R., Gräfener, G., Liermann, A., et al. 2019, A&A, 625, A57\nHarris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357-362\nHerald, J. E., Hillier, D. J., & Schulte-Ladbeck, R. E. 2001, ApJ, 548, 932\n\nHosseini, S. E., Zajaˇcek, M., Eckart, A., Sabha, N. B., & Labadie, L. 2020, A&A,\n644, A105\n\nHunter, J. D. 2007, Computing in Science & Engineering, 9, 90\nJames, T. A., Viti, S., Yusef-Zadeh, F., Royster, M., & Wardle, M. 2021, ApJ, 916, 69\nKrtiˇcka, J. 2014, A&A, 564, A70 Lemaster, M. N., Stone, J. M., & Gardiner, T. A. 2007, ApJ, 662, 582 Liu, B. F., Meyer, F., & Meyer-Hofmeister, E. 2004, A&A, 421, 659 Lodders, K. 2003, ApJ, 591, 1220 Martins, F., Genzel, R., Hillier, D. J., et al. 2007, A&A, 468, 233 Meyer, F. & Meyer-Hofmeister, E. 1994, A&A, 288, 175\n\nArticle number, page 16 of 19\n\nMurchikova, E. M., Phinney, E. S., Pancoast, A., & Blandford, R. D. 2019, Nature, 570, 83\nNayakshin, S. 2004, arXiv e-prints, astro\nNayakshin, S., Cuadra, J., & Sunyaev, R. 2004, A&A, 413, 173\nOnifer, A., Heger, A., & Abdallah, J. 2008, in Astronomical Society of the Pacific Conference Series, Vol. 391, Hydrogen-Deficient Stars, ed. A. Werner & T. Rauch, 305\nOskinova, L. M., Todt, H., Ignace, R., et al. 2011, MNRAS, 416, 1456 Ostriker, J. P. 1983, ApJ, 273, 99\nPaumard, T., Genzel, R., Martins, F., et al. 2006, ApJ, 643, 1011\nPeißker, F., Zajaˇcek, M., Thomkins, L., et al. 2023, ApJ, 956, 70\nQuataert, E. 2004, ApJ, 613, 322\nRessler, S. M., Quataert, E., & Stone, J. M. 2018, MNRAS, 478, 3544\nRessler, S. M., Quataert, E., & Stone, J. M. 2020, MNRAS, 492, 3272\n\nRussell, C. M. P., Wang, Q. D., & Cuadra, J. 2017, MNRAS, 464, 4958\nSchartmann, M., Ballone, A., Burkert, A., et al. 2015, ApJ, 811, 155\n\nSchödel, R., Eckart, A., Alexander, T., et al. 2007, A&A, 469, 125\nSchure, K. M., Kosenko, D., Kaastra, J. S., Keppens, R., & Vink, J. 2009, A&A, 508, 751\nShakura, N. I. & Sunyaev, R. A. 1973, A&A, 500, 33\nSmith, R. K., Brickhouse, N. S., Liedahl, D. A., & Raymond, J. C. 2001, ApJ, 556, L91\nSolanki, S., Ressler, S. M., Murchikova, L., Stone, J. M., & Morris, M. R. 2023, ApJ, 953, 22\nSpringel, V. 2005, MNRAS, 364, 1105\nStone, J. M., Tomida, K., White, C. J., & Felker, K. G. 2020, ApJS, 249, 4\nTagawa, H., Kimura, S. S., Haiman, Z., et al. 2021, arXiv e-prints, arXiv:2112.01544\nTeyssier, R. 2002, A&A, 385, 337\nTodt, H., Sander, A., Hainich, R., et al. 2015, A&A, 579, A75\nToro, E. 2009, Riemann Solvers and Numerical Methods for Fluid Dynamics: A\n\nPractical Introduction (Berlin: Springer)\nToro, E. F., Spruce, M., & Speares, W. 1994, Shock Waves, 4, 25 Turk, M. J., Smith, B. D., Oishi, J. S., et al. 2011, ApJS, 192, 9 von Fellenberg, S. D., Gillessen, S., Stadler, J., et al. 2022, ApJ, 932, L6 Štofanová, L., Kaastra, J., Mehdipour, M., & de Plaa, J. 2021, A&A, 655, A2 Wang, Q. D., Li, J., Russell, C. M. P., & Cuadra, J. 2020, MNRAS, 492, 2481 Yelda, S., Ghez, A. M., Lu, J. R., et al. 2014, ApJ, 783, 131 Yuan, F. & Narayan, R. 2014, ARA&A, 52, 529\n\nYusef-Zadeh, F., Royster, M., Wardle, M., et al. 2020, MNRAS, 499, 3909\n\nAppendix A: Numerical setup choices\nCalderón et al. (2020b) simulated the system of WR stars feeding Sgr A* while orbiting around it following an analogous numerical setup compared to the models presented in this work. There, we had found that the resulting cold disc tended to align with the grid after its formation. In order to remedy this numerical artifact we investigated the use of a MinMod slope limiter, instead of MonCen. However, we warned that it was not straightforward to maintain the spherical symmetry of the winds using the MinMod slope limiter. In this work, we investigated more carefully these numerical choices in order to be confident that our results are robust.\nFirst, we investigated the choice of the MonCen slope limiter. To do this, we made use of the same setup presented in this work but paying attention to the values of the passive scalar fields. In principle, the values of these fields should be between zero and one. However, this experiment showed that the MonCen slope limiter produced non-physical oscillations that in some cases resulted in negative values of the passive scalar fields. This behaviour was not observed when using the MinMod slope limiter. Thus, we selected this choice throughout this work. To ensure the spherical symmetry of the stellar winds we had to impose a less restrictive refinement criterion based on density gradients as well as a smoother transition between refinement levels through the parameter nexpand .\nSecond, we tested the choice of the Riemann solver employed but using the MinMod slope limiter. Currently, the HLLC Riemann solver (Toro et al. 1994) is widely used in grid-based hydrodynamic simulations. In this work, we have found that this choice results in a cold disc that tends to align with the Cartesian grid as seen in Calderón et al. (2020b). This could indicate problems in conserving the angular momentum which is a known issue with simulations in Cartesian grids when the relevant regions are not well resolved. With the usage of the exact Riemann solver (Toro 2009), though more computationally expensive, it was possible to avoid this artificial alignment. Thus, the only sensible combination was the use of the exact Riemann solver with a MinMod slope limiter but we had to ensure to count with enough resolution and smoothness in the transition of the refinement levels. As a reference of these experiments, Figure A.1 shows density projection (weighted by density) maps of the final state of the system for two runs with Solar composition with 3 Z ⊙ but with di ff erent Riemann solvers. The left- and righthand side panels display the models using the HLLC and the exact Riemann solvers, respectively. Although overall the density structure looks similar there are subtle di ff erences. The most relevant is related to the disc orientation and structure. On the left-hand side, the disc tries to align with one of the Cartesian axes, while on the right-hand side the disc remains consistently with the shown orientation.\nAppendix B: Solar composition varying metallicity\nBefore investigating the models with the new implementation of the di ff erential cooling we ran some experiments simply modifying the contribution of the metals to the total radiative cooling. We simulated three models varying only the composition used: A1 used Solar composition, A3 used Solar composition with 3 Z ⊙ , and A5 used Solar composition with 5 Z ⊙ . The density projected (weighted by density) maps of the final state of the simulations are shown in Figure B.1. The left-hand side, central, and right-hand side panels display the runs A1, A3, and A5, respectively. A higher metallicity in the plasma increases the radiative cooling directly. As a result, more and denser filaments, clumps, and a denser central disc are obtained with increasing the metallicity. On the contrary, if the metallicity is decreased to a Solar value we observe almost no overdensities, and definitively no disc formation. We speculate that at Z ≈ 3 Z ⊙ there is a transition between the regimes of persistent quasi-steady accretion and disc formation, which would imply that small di ff erences in the cooling function or numerical implementation can result in models reproducing either regime. This may explain the di ff ering results obtained by Calderón et al. (2020b) and Ressler et al. (2020) with similar numerical and physical setups.\nIn order to perform a more quantitative comparison of these results Figure B.2 presents radial profiles of density (weighted by volume) and temperature (weighted by mass) on the top and bottom panels, respectively. The dashed blue, solid orange, and blue dotted-dashed lines represent the models A1, A3, and A5, respectively. The density profiles clearly shows the impact of cooling in the structure of the medium. More e ffi cient cooling, like in cases A3 and A5, results in an order of magnitude denser medium within 1 '' compared to the case with ine ffi cient cooling, i.e. A1. Notice that even higher metallicity in A5 extends the impact the denser region to slightly larger spatial scales. The e ff ect is similar in the temperature profile. In A1, the temperature profile decays with r within the central 1 '' . But when the metallicity is high enough to make cooling e ffi cient the temperature profile can decrease up to two orders of magnitude depending on the exact value.\nIn summary, these experiments allowed us to quantify the impact of radiative cooling by simply decreasing or increasing the radiative cooling influence. However, since the chemical composition in reality is much more complicated and di ff ers from Solar composition significantly these models should be interpreted only as general guides to assess the role of radiative cooling. In the main text of this work, we devoted to explore more physically motivated mixtures that, although depend on more uncertain parameters are more robust when attempting to build a physical model that can be constrasted with observational data.\n10 -24 10 -22 10 -20 10 -18 ∫ ρ 2 dz/ ∫ ρdz (g cm -3 )\n\n\n\n\n\n
Fig. A.1. Comparison of the final state (present time) of the simulations with di ff erent Riemann solvers. Left- and right-hand side panels correspond to runs with Harten-Lax-van Leer-Contact (HLLC) and exact Riemann solvers, respectively. Both models were run using the MinMod slope limiter.
\n\n\n\n\n\n\n\n\n\n\n\n\n
Fig. B.1. Comparison of the simulations with Solar composition varying the metallicity at two di ff erent simulation times. The panels show projected density maps weighed by density along the z axis, i.e. R ρ 2 dz / R ρ dz , which is parallel to the line of sight. Left-hand side, central, and right-hand side panels show the runs A1, A3, and A5, respectively. All maps display the full computational domain.
\n\n\n\n\n\n\n
Fig. B.2. Radial profiles of time-averaged volume-weighted density (top) and mass-weighted temperature (bottom) over the last 500 yr of simulation time. The models A1, A3, and A5 are shown in solid orange, dashed blue, and dashed-dotted green lines, respectively.
\n\n"},"sections":{"kind":"list like","value":[{"title":"ABSTRACT","content":"Context. The reported discovery of a cold ( ∼ 10 4 K) disc-like structure around the super-massive black hole at the centre of the Milk Way, Sagittarius A* (Sgr A*), has challenged our understanding of the gas dynamics and thermodynamic state of the plasma in its immediate vicinity. State-of-the-art simulations do not agree on whether or not such a disc can indeed be a product of the multiple stellar wind interactions of the mass-losing stars in the region. Aims. This study aims to constrain the conditions for the formation of a cold disc as a natural outcome of the system of the mass-losing stars orbiting around Sgr A*, to investigate if the disc is a transient or long-lasting structure, and to assess the validity of the model through direct comparisons with observations. Methods. We conduct a set of hydrodynamic simulations of the observed Wolf-Rayet (WR) stars feeding Sgr A* using the finitevolume adaptive mesh-refinement code Ramses. We focus, for the first time, on the impact of the chemical composition of the plasma emanating from the WR stars. Results. The simulations show that the chemical composition of the plasma a ff ects the radiative cooling enough to impact the properties of the medium such as density and temperature and, as a consequence, the rate at which the material inflows onto Sgr A*. We demonstrated that the formation of a cold disc from the stellar winds is possible for certain chemical compositions that are consistent with the current observational constraints. However, even in such a case, it is not possible to reproduce the reported properties of the observed disc-like structure, namely its inclination and hydrogen recombination line fluxes. Conclusions. Weconclude that the stellar winds on their own cannot form the cold disc around Sgr A* inferred from the observations. Either relevant ingredients are still missing in the model, or the interpretation of the observed data needs to be revised. Key words. accretion, accretion discs - Galaxy: centre - hydrodynamics - Stars: winds, outflows - Stars: Wolf-Rayet","pages":[1]},{"title":"The formation and stability of a cold disc made out of stellar winds in the Galactic Centre","content":"1,2, 3,4 5 6,7,8 Diego Calderón ⋆ , Jorge Cuadra , Christopher M. P. Russell , Andreas Burkert , Stephan Rosswog 1,9 , and Mayura Balakrishnan 10 Received November 4, 2024; accepted November 4, 2024","pages":[1]},{"title":"1. Introduction","content":"The Galactic Centre (GC) hosts the closest super-massive black hole to us, Sagittarius A* (Sgr A*; see Genzel et al. 2010, for a review). Unlike the black holes present in active galactic nuclei, Sgr A* is very underluminous. It does not seem to be currently accreting much material, or to have a standard accretion disc around it (Yuan & Narayan 2014). Murchikova et al. (2019) detected however a disc-like structure around Sgr A*. Using the 1.3-millimetre recombination line H30 α they observed a doublepeaked emission line with full velocity width of ∼ 2200 km s -1 . The centre of the emission coincides with Sgr A* and extends up to 0.11 '' ( ∼ 4.4 × 10 -3 pc) to both the redshifted and blueshifted sides. They interpreted this feature as a rotating disc of mass 10 -5 -10 -4 M ⊙ . Yusef-Zadeh et al. (2020) also reported the presence of broad hydrogen recombination lines, including the double-peaked H30 α line, at the position of Sgr A* using the Very Large Array. However, they interpreted the signatures as a jet emanating from the central region. So far, there has not been any detection in the near infrared, despite many observations. Ciurlo et al. (2021) reports an upper limit for Br γ , which is two orders of magnitude below the extrapolation from the reported H30 α flux. Currently, there is no consensus on how such observations can be reconciled or from where the observed cold material comes from. The gaseous environment around Sgr A* is dominated by the outflows from around 30 Wolf-Rayet (WR) stars, all located within a fraction of a parsec of the black hole (Paumard et al. 2006; Martins et al. 2007). At such close distances, the winds interact strongly in shocks that thermalise them and create a hot and di ff use X-ray emitting plasma (Quataert 2004). However, given the relatively low velocity of some of the outflows (450600 km s -1 ; Martins et al. 2007), the resulting plasma can be prone to radiative cooling and form dense, cold clumps (Cuadra et al. 2005; Calderón et al. 2016), which end up being embedded in the di ff use, hot plasma ( ∼ 10 7 K; Bagano ff et al. 2003). The complex interplay between the stellar winds around Sgr A* has been the subject of hydrodynamic simulations using a variety of numerical techniques. With smoothed particle hydrodynamics (SPH) simulations, Cuadra et al. (2006) showed that, if most of the slow-wind stars are located in a relatively compact stellar disc, many cold clumps form and quickly coalesce in a cold gaseous disc of radius ∼ 1 '' . However, using a stellar distribution closer to the one observed, Cuadra et al. (2008, 2015) showed that no conspicuous disc appears over the 1800 yr time-span of their models. Calderón et al. (2020b) performed finite-volume hydrodynamic simulations on a Cartesian grid of the same system, which are better suited to model shocks, the multi-phase medium, and subsequent cooling, and found that the formation of a cold disc is indeed possible. According to their model, the stellar wind of the star IRS 33E, which is fairly dense ( ∼ 10 -5 M ⊙ yr -1 ) and slow (450 km s -1 ; Martins et al. 2007) interacts with the medium creating a shell that is dense enough to radiate quickly its thermal energy and break into denser and smaller structures. These pieces manage to fall inwards forming a cold ( ∼ 10 4 K) disc with a total mass of ∼ 5 × 10 -3 M ⊙ within a simulation time of 3500 yr. Based on this result, Ciurlo et al. (2021) found that the properties of the modelled disc are consistent with the non-detection in Br γ . Nevertheless, this scenario has not been confirmed by analogous grid-based simulations. Ressler et al. (2020) revisited the same system using the same numerical approach, although a di ff erent code: athena++ (Stone et al. 2020), and found no disc formation even extending their simulation time up to 9000 yr. Building on this model, Solanki et al. (2023) also studied this system through hydrodynamic modelling but encompassing a larger region and evolving it for much longer timescales. They included, for the first time, the presence of the circumnuclear disc (CND), an observed gaseous structure located between 1 . 5-3 . 0 pc (Becklin et al. 1982; Genzel 1989) and with a total mass of 3-4 × 10 4 M ⊙ (Dinh et al. 2021; James et al. 2021). This work showed that the formation of a cold disc is possible on long timescales ( ≳ 300,000 yr) due to the interaction of the innermost boundary of the CND and the WR stellar winds, which results in the transport of CND material to smaller scales. However, the authors warned that the lack of certain physical mechanisms in the model on such timescales (e.g. magnetic fields, supernovae, thermal conduction) might a ff ect the robustness of this result. Thus, the exact formation process of the observed cold disc still remains unknown. Akey factor driving the formation of clumps and a disc in the model of Calderón et al. (2020b) is that radiative cooling allows the gas to get rid of its energy e ffi ciently. The strength of this process depends on the chemical composition of the gas, which is highly unusual in the GC given its origin as winds from evolved massive stars. In this work, we explore for the first time the impact of the radiative cooling through studying specific chemical compositions: Solar with 1 Z ⊙ , 3 Z ⊙ , and 5 Z ⊙ . More importantly, we develop a more realistic chemical setup that considers the atmospheric abundances for the di ff erent WR sub-types present in the region, in order to follow the thermodynamic evolution of the gas in a more appropriate manner. Our results show that the formation of the disc is indeed determined by the composition of the gas. However, the properties of such a disc do not agree with the observations when considering realistic wind compositions compatible with current observational constraints. We also improve the numerical setup in order to better assess the disc orientation and more directly compare our results to the work of Ressler et al. (2018, 2020) as well as with SPH models (Cuadra et al. 2008, 2015; Russell et al. 2017). This article is organised as follows: in Section 2, we present the numerical approach, the setup and the models investigated, Section 3 presents and describes the results of the numerical simulations. In Section 4, we present the synthetic observables obtained through post-processing the models and contrast them with observations. We compare our grid-based models with SPH models in Section 5. In Section 6, we present an analytic analysis of the stability of the observed cold disc. Section 7 discusses our findings and uncertainties in the parameters of our models. Finally, Section 8 presents conclusions and final remarks. Throughout this paper we use the mass and distance to Sgr A* of 4 . 3 × 10 6 M ⊙ and 8 . 33 kpc, respectively (Gillessen et al. 2017; GRAVITY Collaboration et al. 2019, 2021), so that 1 arcsec corresponds to a length of ∼ 0.04 pc ≈ 10 5 RSch, being RSch the Schwarzschild radius.","pages":[1,2]},{"title":"2.1. Equations","content":"The simulations were performed using the adaptive-mesh refinement (AMR) hydrodynamic code Ramses (Teyssier 2002). This code uses a second-order Godunov method with a shockcapturing scheme to solve the Euler equations in their conservative form, i.e. where ( ρ, u , P ) are the primitive hydrodynamic variables: density, velocity, and pressure, respectively. The set of quantities si correspond to tracer scalar fields that are advected with the fluid whose usage we introduce in Section 2.3. The sink terms on the right-hand side correspond to the e ff ects of the gravitational potential ϕ = ϕ ( x ), assumed to be time independent, and the total radiative loses due to optically-thin radiative cooling Q -tot = Q -tot ( ρ, T , X i), with x the position vector, T the fluid temperature, and X i the chemical composition of the fluid. The total specific energy density e is given by where γ is the adiabatic index that is set to 5 / 3. Additionally, we consider the fluid can be described as an ideal gas so that the temperature can be calculated through P = ( ρ/µ ) k B T , being µ the mean molecular weight and k B the Boltzmann constant.","pages":[2]},{"title":"2.2. Numerical setup","content":"The model considers the system of WR stars blowing stellar winds while they move on their observed Keplerian orbits around Sgr A*. The setup is analogous to our previous work (Calderón et al. 2020b). The central black hole gravitational field is modelled as a point mass of 4 . 3 × 10 6 M ⊙ (GRAVITY Collaboration et al. 2019, 2021). The stars are simulated as test particles that only feel the gravitational pull of Sgr A*. Their initial Notes. The Solar abundances were taken from Lodders (2003). The abundances for the three WR sub-types correspond to the compilation by Russell et al. (2017) based on previous studies (Herald et al. 2001; Crowther 2007; Onifer et al. 2008). Column 1: chemical mixture. Columns 2-7: hydrogen, helium, carbon, nitrogen, oxygen, and rest of the metals mass fractions, respectively. Column 8: mean molecular weigh assuming full ionisation. Notes. Column 1: simulation ID. Column 2: chemical composition of the winds. Column 3: Riemann solver. Column 4: Slope limiter. Column 5: whether or not the final state of the system shows the presence of a cold disc around Sgr A*. position and velocity vectors are set so that they move on the orbits constrained by observations (Paumard et al. 2006; Cuadra et al. 2008; Gillessen et al. 2019; von Fellenberg et al. 2022). The stellar winds in the simulation are modelled following the procedure by Lemaster et al. (2007), which has been validated and used in our previous work (Calderón et al. 2020a,b). This consists in defining a 'masked region\" around each star that is a spherical volume where the hydrodynamic variables are reset in each timestep, in order to reproduce the free wind expansion solution of a spherical wind with certain mass-loss rate ˙ M w, terminal velocity V w, and temperature T w. For these quantities we used the values constrained through modelling of the infrared spectra (Martins et al. 2007; Cuadra et al. 2008). The wind temperature was set to the lowest temperature allowed in the simulation, T w = 10 4 K which is determined by the strong ultraviolet radiation produced by the hundreds of massive stars in the region. The simulations were run in a Cartesian grid in a cubic domain of side 40 '' ( ∼ 1.6 pc) with outflow boundary conditions (zero gradients). We used the exact Riemann solver (e.g. Toro 2009) combined with the MinMod slope limiter, as these choices allow the modelling of hydrodynamic instabilities, which otherwise may be quenched due to numerical di ff usion. We investigated other choices such as the Harten-Lax-van Leer-Contact (HLLC; Toro et al. 1994) and / or the MonCen slope limiter but they produced unwanted numerical artifacts such as spurious oscillations. For a careful analysis of these choices we refer the reader to the Appendix A. Regarding the numerical resolution of our models, the coarse resolution was 64 3 cells, allowing adaptive refinement of four levels, i.e. e ff ectively ∆ x ≈ 0 . 0382 '' . However, the regions around the stars allowed an extra level of refinement ( ∆ x ≈ 0 . 0195 '' ). Additionally, the vicinity around the central black hole has a fixed nested grid with eight refinement levels above the coarse resolution ( ∆ x ≈ 2 . 44 mas). At the location of Sgr A*, we defined a spherical region where the hydrodynamic variables are reset to low values of density and pressure at rest, in order to avoid artificial accumulation of material. The refinement strategy is based on density gradients on top of the geometric criteria previously defined. In order to explore the role of the AMR potentially a ff ecting the results we tested di ff erent values of the smoothing parameter that controls the refinement in transitions between refinement levels. We tested quadrupling the smoothing parameter and the results remained unchanged 1 .","pages":[2,3]},{"title":"2.3. Models","content":"The main parameter explored in this work is the optically-thin radiative cooling function. This choice is determined by specifying the chemical composition of the fluid. Unfortunately, the metallicity of the young stars in the GC is still poorly known, yet it has been argued that it should be higher than Solar ( Z = 2 -3 Z ⊙ ; Genzel et al. 2010). Past numerical works have considered that the composition of the gas correspond to Solar abundances but with metallicity three times the Solar value, Z = 3 Z ⊙ (Cuadra et al. 2008; Calderón et al. 2016; Ressler et al. 2018, 2020; Solanki et al. 2023). However, more than the 'bulk' metallicity of the stars, the most appropriate composition choice would be the abundances of the WR stellar atmospheres, which normally di ff er significantly from Solar (Herald et al. 2001; Onifer et al. 2008). Although Ressler et al. (2018, 2020) used chemical compositions lacking hydrogen, the rest of the element abundances remained unchanged relative to Solar with 3 Z ⊙ . As a result, the cooling function varied at low temperature ( < 10 5 K) but at higher temperatures it remains unchanged. Moreover, the composition choice not only determines the cooling function but also the ion mean molecular weight as well as the electron to proton number densities n e / n p. These quantities are taken into account in the sink term in the energy equation (see equation 3), and following Schure et al. (2009) can be expressed as follows where the subscript i stands for a given chemical abundance. Bear in mind that di ff erent chemical compositions also correspond to di ff erent temperature Ti , as this is obtained through the ideal gas expression that makes uses of the mean molecular weight of the fluid. In the general case, if we consider a fluid element composed of N mixtures of abundances, the total radiative loses will be given as a summation, i.e. where we have expressed the total radative cooling as a linear combination of the mixtures. Notice that we have introduced the passive scalar fields si , as we used them to quantify the fraction of a given chemical composition. In this work, we introduced three types of compositions that represent the WR subtypes based on their atmosphere abundances. In the region where the winds are generated, the corresponding passive scalar is set to 1 while the rest are set to 0. By doing so, it is possible to identify how much material of a given cell is supplied by which sub-group of WR stars. The mass fractions of each mixture are shown in Table 1. In this work, we investigated five models with di ff erent compositions: Solar composition with metallicities Z ⊙ , 3 Z ⊙ , and 5 Z ⊙ , a mixture of three compositions based on the spectroscopic constraints of the sub-types of WR stars, and a variation of the latter one motivated by the uncertainty in the WR stellar atmospheres and, specifically on the H fraction in them that is set to X H = 40% (see Section 7.2, for a discussion). Figure 1 shows the cooling function Λ = Λ ( T ) as a function of temperature for different chemical mixtures. Each curve was calculated as a linear combination of the abundance of a given element and its contribution to the total radiative cooling. The values of the composition per element are shown in Table 1. The contribution of each element to the total cooling was taken from the plasma models developed by Štofanová et al. (2021). The cooling functions corresponding to WR subtypes compositions: WC8-9, WN5-7, and WN8-9 / Ofpe are shown with dashed blue, dotted orange, and dotted-dashed green lines, respectively. The Solar and Solar with 3 Z ⊙ are represented with thin and thick solid black lines, respectively. Furthermore, we added the cooling function used in previous works by Ressler et al. (2018, 2020) as a dotted-dashed red line. Here it can be observed that a Solar composition with 3 Z ⊙ (the typical used value) is between the range of the di ff erent WR compositions but the subtypes WC89 and WN89 / Ofpe are about a factor two higher. The list of the runs investigated in this work are shown in Table 2. The initial setup was identical to our previous work (Calderón et al. 2020b), i.e. the medium across the whole domain was set to constant and low-enough values of density ρ = 10 -24 g cm -3 and pressure P = γ -1 ρ c 2 s,f with c s,f = 10 km s -1 , so that the stellar winds do not encounter impediments and fill quickly the domain. All models were run for a total of 3,500 yr but setting the starting state of the system in the past, so that the final state of the simulations corresponds to the current state of the system. This is achieved by integrating the current position and velocity vectors of the stars back in time, assuming that they have followed purely Keplerian orbits within this timescale due to the gravitational field of Sgr A* (Cuadra et al. 2008; Ressler et al. 2018; Calderón et al. 2020b).","pages":[3,4]},{"title":"3. Results","content":"We proceed to describe the evolution and final state of the simulations at t = 0 that corresponds to the present time. The runs with Solar composition, but varying metallicities, were used for understanding the impact of increasing and decreasing the e ff ect of radiative cooling in general. However, since they do not represent realistic compositions we opted not to discuss them in detail here. Instead, we chose to focus on the more physically motivated models, i.e. WR_f07 and WR_f1 which we also refer as fiducial and enhanced , respectively. For completeness, we give a brief description of the models A1, A2, and A3 in Appendix B.","pages":[4]},{"title":"3.1. Hydrodynamics","content":"The simulation evolution of all models is analogous, especially in the initial phases. At t = -3500 yr, the stars begin to orbit around Sgr A* that is located at the centre of the domain, while their winds quickly fill the whole computational domain. After ∼ 500 yr ( t = -3000 yr), the system reaches a quasi-steady state. At this point, the domain is full of di ff use ( ∼ 10 -22 -10 -21 g cm -3 ) and hot plasma ( ∼ 10 7 -10 8 K) due to the shocked stellar winds and their interactions. This is illustrated in Figure 2 that shows density maps across di ff erent simulation times. The maps show projected density fields along the line-of-sight direction ( z axis) weighted by density, i.e. R ρ 2 dz / R ρ dz , in order to highlight the densest regions with the highest resolution. The top panels show two models at t = -700 yr, respectively. In them, it is possible to observe the complex structure developed due to the stellar wind interactions such as bow shocks, instabilities, and dense clumps as a result of condensation through radiative cooling. Left- and right-hand side panels display models with di ff erent chemical compositions WR_f07 and WR_f1, respec- tively. On the left-hand side, it can be seen how mildy e ffi cient cooling a ff ects mostly one of the stellar winds. This star corresponds to IRS 33E and its wind is the slowest ( ∼ 450 km s -1 ). This fact causes its shocked temperature to be in an e ffi cient region of the radiative cooling functions that allows its material to cool down and become denser. Also, some of its material manages to reach the vicinity of Sgr A*, as the map shows an elongated clump being accreted and stretched. The right-hand side also portrays this picture but since the radiative cooling is enhanced more dense clumps can be observed, especially close to IRS 33E and Sgr A*. Finally, the bottom panels of Figure 2 show the models at time t = 0, i.e. the present time. One can clearly see that the amount of dense material that has accumulated around Sgr A* is di ff erent in both maps. Although in the model WR_f07 (see bottom left-hand side panel of Figure 2) some dense material is spiraling towards the black hole overall there are is no clear structure around it. In the case of WR_f1 (see bottom right-hand side panel of Figure 2), the dense material has settled at the centre in a a disc-like structure. This di ff erence is due to the e ffi ciency of the radiative cooling. Since model WR_f1 has enhanced cooling, more dense clumps and filaments form, and some of them manage to fall onto Sgr A*. Overall, the hydrodynamic evolution of the two models is analogous: the winds fill the domain during the first ∼ 500 yr, then the systems reach a quasi-steady state, and in the last 500 yr they diverge as the model WR_f1 creates much more dense, cool material that settles around Sgr A*. A quantitative analysis of the evolution of the properties of system is presented as follows. To quantify the accretion rate at di ff erent spatial scales we calculated the mass flux as a function of both radial distance and time ˙ M ( r , t ) = 4 π r 2 ρ ( r , t ) vr ( r , t ) averaged over a spherical shell. Here, positive and negatives signs in the radial velocity refer to outflow and inflow mass fluxes, respectively. First, we analyse the net mass flux at the innermost radius we can resolve properly, which is equivalent to five times the inner boundary radius, i.e. 5 r in = 5 × 10 -4 pc ≈ 1 . 25 × 10 -2 '' . Figure 3 shows | ˙ M ( r = 5 r in , t ) | as a function of time for models WR_f07 and WR_f1 that are displayed as solid blue and orange lines, respectively. In both cases, at t > -3300 yr the net mass flux reaches levels of ∼ 10 -6 M ⊙ yr -1 with short variability episodes that increase the rate by factors of two to four. During this quasi-steady phase, both simulations display similar behaviour qualitatively, likely determined by the identical stellar wind configuration. This stage lasts until t ≈ -500 yr where the mass flow rates start to deviate from each other. The fiducial model continues to display a variability amplitude of the same order of magnitude. However, the WR_f1 model shows a transition to a strongly gas inflow dominated phase, with inflow rates that are enhanced by a factor four to eight. This stage of the evolution is what we refer to as the disc formation phase, analogously to our previous models reported in Calderón et al. (2020b). Next, we proceed to analyse the time-averaged behaviour of the simulations over the last 500 yr, as this can give us an idea of the general state of the system minimising the e ff ects of the stochastic variability. First, we analyse the (volume-weighted) density and (mass-weighted) temperature radial profiles of the simulations, which are shown on the top and bottom panels in Figure 4, respectively. The dashed blue and solid orange lines represent the simulations WR_f07 and WR_f1, respectively. The orange lines in both panels clearly highlight the presence of the cold disc. The density profile decays with r 2 in both cases at large scales ( ≳ 1 '' ). This is the result of most of the material flowing outwards at these scales following a roughly isotropic spherical wind as seen in previous models (Ressler et al. 2018; Calderón et al. 2020b). However, the profiles di ff er at smaller scales ( < 1 '' ) where the fiducial case transitions to ρ ∝ r -1 , while the model WR_f1 shows a density enhancement due to the presence of the disc. This increase in density is more than one order of magnitude. Regarding the temperature profiles, both models match at larger scales ( ≳ 1 '' ) with a constant temperature of the order of 10 7 K, set by the stellar wind collisions. Again, the profiles differ at smaller scales where the fiducial case follows T ∝ r -1 , and the enhanced cooling case displays a temperature profile about two orders of magnitudes lower on average. The minimum in temperature corresponds to the region where the disc contributes with most of the mass for a given spherical shell. To quantify and characterise the mass inflow and outflow regimes in the simulation domain we calculated them as radial profiles. Figure 5 displays the absolute value of the mass flow rates as a function of distance from Sgr A*. The solid and dashed lines represent the mass inflow and outflow rates, respectively; while the colours follow the same convention as before. Thus, if the solid line is above the dashed line the net mass flow is inwards and vice-versa. Analogous to the model by Ressler et al. (2018), the fiducial model encompasses di ff erent regimes due to the dominant direction of the mass flow at given spatial scales. At r > 3 '' , the outflow component dominates and the inflow is negligible. We find a net mass outflow rate of ∼ 5 × 10 -4 M ⊙ yr -1 . Notice that the enhanced cooling model exhibits exactly the same behaviour at these scales. At smaller scales (1 '' -3 '' ), there is a transition region where | ˙ M in | ∼ | ˙ M out | that corresponds to the location of most mass-losing stars and wind interactions. For model WR_f1, the change is more abrupt due to the presence of the cold disc. In both models, at r < 1 '' the inflow component is larger than the outflow, so the material inflows across these scales and down to the innermost boundary. The net e ff ect makes the mass inflow rate about five times larger which generates the cold disc. The origin of the infalling material can be traced in two ways: analysing the angular momentum of the material close to the inner boundary, and using the scalar tracer fields that we introduced to label the chemical abundances of the di ff erent WR sub-types. Figure 6 shows Hammer projections of the angular momentum direction of the gas enclosed in a sphere of radius 0 . 25 '' ( ∼ 0 . 01 pc) for the fiducial and enhanced cooling runs in the top and bottom panels, respectively. Each point represents a di ff erent simulation time over the last 1000 yr, and they are connected with dashed lines. For reference, we added the angular momentum direction of the orbits of the mass-losing stars as black star symbols, and the location of the clockwise disc based on the orbits we used as a grey shaded circle. This analysis shows that the angular momentum direction of the infalling - - Sun 10 2.0 material varies stochastically but overall tends to align with the orientation of the orbits in the clockwise disc. However, the degree of variability depends on whether or not the model results in the formation of a disc. The fiducial model shows more variability while the enhanced cooling run displays that the angular momentum aligns more consistently with the orientation of the clockwise disc. The fact that the infalling material at this spatial scales comes from these stars agrees with previous numerical models (Cuadra et al. 2008; Ressler et al. 2018; Calderón et al. 2020b). Furthermore, this is not surprising since these stars orbit closer to Sgr A*, and have winds that are relatively slow ( ∼ 600 kms), which results into shocks more prone to radiative cooling and with smaller angular momentum. Overall, the fiducial model displays properties of the medium and gas dynamics consistent with the previous models that do not form a disc (Cuadra et al. 2008; Ressler et al. 2018, 2020; Calderón et al. 2020b, run for 1100 yr). The model with enhanced cooling is consistent with the simulations that show the formation of a cold disc as an outcome of this system (Calderón et al. 2020b). Before discussing if any of the models can be favoured given the current observational constraints we proceed to characterise the cold disc arising in the WR_f1, as its properties will aid us to do so.","pages":[4,5,6,7,8]},{"title":"3.2. Properties of the cold disc","content":"At the present time ( t = 0), the model WR_f1 shows the presence of a cold disc. Figure 8 shows zoomed maps of the central region of 2 . 5 '' × 2 . 5 '' . The left- and right-hand side panels show projected maps of density and line-of-sight velocity (both weighted by density), respectively. Here it is possible to observe that the projected diameter of the disc is roughly 1 '' . We estimated the mass of the disc by isolating the cold material ( T < 10 5 K) and integrating the cell mass content within a sphere of radius 1.0 '' . We found that the total mass of the disc is 0.005 M ⊙ which is consistent with the value in our previous work (Calderón et al. 2020b). However, the mass accumulated in the disc does not converge and keeps increasing at least for the next hundreds of years in the simulation. On the right-hand side panel of Figure 8, we can see that the line-of-sight velocity peaks at ∼ 2000 km s -1 along both directions (towards and outwards the observed). This coincides with the maximum extension of the observed H30 α line (Murchikova et al. 2019). Additionally, the disc is observed to be ∼ 45 · tilted in projection. This is not in agreement with the observations since, at least in projection the di ff erence is about ∼ 90 · (see Figure 1 of Murchikova et al. 2019).","pages":[8]},{"title":"4. Post-processing of observables","content":"In order to assess the validity of our models for the Galactic centre we proceed to synthesise observational quantities that we could compare easily with observations. We focus on the recombination lines H30 α and Br γ . Afterwards, we calculate the X-ray spectrum from our models.","pages":[8]},{"title":"4.1. Recombination lines: H30 α and Br γ","content":"The cold gas at the location of Sgr A* has been observed through the H30 α recombination line (Murchikova et al. 2019; YusefZadeh et al. 2020), and only with an upper limit in the Br γ recombination line (Ciurlo et al. 2021). In order to explore if the models are consistent with these observations we have calculated the expected flux from these emission lines. To compute 1e3 10 10 the emission of the H30 α line we used the coe ffi cients given below fitting a piecewise linear function to them as a function of density and considering a decay ∝ T -1 (see Murchikova et al. 2019, supplementary information), In the case of the Br γ line, its emissivity can be calculated following Schartmann et al. (2015) as Then, in each cell of the domain we computed the respective emissivity values, multiplied them by their n p n e (assuming full ionisation), and then integrated the data cube along the z coordinate which is perpendicular to the line of sight. It is important to remark that in this calculation it is necessary to take into account the hydrogen nucleus n p and electron n e densities appropriately. Since our models consider di ff erent amounts of hydrogen in the plasma we include only the material that contains some, i.e. the material coming from the WN89 / Ofpe stars. Additionally, we only used the hydrogen fraction of that material which is X H = 11 . 5% and X H = 40% for the models WR_f07 and WR_f1. idea of the whole flux of the model we chose a size of p = 1 '' that encloses entirely the disc. The flux values obtained are shown in Table 3 where we also added the upper limit for Br γ from Ciurlo et al. (2021), and the flux of H30 α measured from integrating the spectrum across the velocity channels given by Murchikova et al. (2019). In the case of the Br γ line, we can see that only the model WR_f07 has a flux consistent with the upper limit reported for an aperture of p = 0 . 23 '' . The model WR_f1 has a flux that is ∼ 100 times higher than the upper limit. Also, we see that the disc in our models extends beyond this projected radius, yet its emission is dominated by the inner p = 0 . 23 '' (see Table 3). The situation is more complex for the H30 α line since the observed flux is in both cases higher than the synthetic emission. Although the model that forms a disc is closer it still remains 30% times fainter than the reported value. The model WR_f07 has negligible emission as it is ten order of magnitudes fainter compared to the observation. Thus, it is still not clear how to reconcile the models with both line emission simultaneously as none of them is capable to reproduce the observations. We discuss further the interpretation of these results in Section 7.1.","pages":[8,9]},{"title":"4.2. Spectral X-ray emission","content":"The X-ray emission from the central parsec is also an observable that has allowed validation of previous numerical models (see Russell et al. 2017; Ressler et al. 2018; Wang et al. 2020). Here we also post-processed our models aiming to obtain the Iron Kalpha spectrum of this region at ∼ 6 . 7 keV as this is the strongest X-ray feature from Sgr A* (e.g. Russell et al. 2017; Corrales et al. 2020). Our approach is based on the procedure outlined in Balakrishnan et al. (2024b) but adapted to our finite-volume grid-based hydrodynamic model, which is briefly described as follows. First, we assigned energy dependent X-ray emissivities j k E = n k e n k i Λ ( E , T k ) that were taken from the vvapec model (Smith et al. 2001), which was obtained with xspec (Arnaud 1996). The densities and temperatures were taken from the last output of our simulation, i.e. at t = 0. Additionally, the tracer fields were used to identify the composition fraction of the parcels of gas according to the WR sub-types. Since the optical depth through the simulation domain is optically thin in the X-ray, the radia- Notes. All fluxes correspond to quantities seen from Earth, i.e. at a distance of 8.33 kpc, and are given in units erg cm -2 s -1 . The observed H30 α flux was calculated integrating the reported flux density across the velocity channels, and taking into consideration the error propagation of the 0.3 mJy reported (see Figure 1 in Murchikova et al. 2019). v th. This function is given as follows tive transfer equation to solve is simply given by the integration along the line of sight of the emissivities of the respective cells. Given that we can trace the fraction of the material that comes from a given WR sub-type this actually corresponds to a linear combination of the tracer field si and the abundance dependent emissivity for each cell, i.e. where ϕ k represents a Gaussian function that it is used to obtain the emission across di ff erent velocity channels, and to take into account the thermal broadening represented with a velocity of We performed calculations through velocity channels spanning line-of-sight velocities v los from -3000 km s -1 to 3000 km s -1 . These were later converted into energy space via Doppler broadening. Thus, for each energy bin from the emissivity tables we obtained 150 line profiles using a fine resolution of 6400 per dex. In order to sum them appropriately we interpolate them into a unique energy range so that we can obtain a single spectrum that includes both the continuum and line contributions. The result of this procedure is shown in Figure 10. This contains the X-ray spectrum emitted from a sky-projected annulus of 0 . 5 '' < p < 1 . 5 '' for models WR_f07 (solid blue line) and WR_f1 (dashed orange line). As a reference, we included the estimation from Balakrishnan et al. (2024b) computed from an analogous model that used the SPH approach, which we discuss in the following section. All spectra show qualitative agreement highlighting the di ff erent features of the Fe complex at 6.7 keV and 6.9 keV. This is also reflected in the broadening of the lines due to the velocity across the line of sight, which is expected since the models are based on the same stellar orbits and wind properties. It is relevant to highlight the level of agreement between the finite-volume and SPH models. The qualitative agreement is reasonable and the level of the emission is of the same order of magnitude. The exact quantitative di ff erence among the models could be mainly attributed to the exact cooling table (composition) used in each simulation, which is the result of the di ff erent density and temperature distributions that can be observed in Figure 4. This is why the WR_f07 shows the highest flux level as it has the least e ffi cient cooling and, as a result it has hotter material. Analogously, the run WR_f1 with an enhanced cooling specified in the composition of its plasma results in a reduction of ∼ 40% of its flux. In case of the SPH model, the quantitative di ff erences could be a ff ected due to the intrinsic ability of the generic SPH to capture shocks which impacts directly the density, velocity structure, and the temperature of the medium. Notice that the flux level of the spectrum of the SPH model by Balakrishnan et al. (2024b) is within the cases explored in this work. Their cooling model is not the same as the ones explored here but this analysis show that it is equivalent to a cooling e ffi -ciency between both of our models. Although not shown here we also compared the X-ray spectra from simulation WR_f1 before and after the disc is formed, and found negligible di ff erences. Finally, we also analysed the slope of the continuum emission within this energy range and ∫ ∫ - 2000 found no significant di ff erences among the models. Thus, in this work the presence of the disc does not imprint a feature in the X-ray spectrum that is resolvable by any current or forthcoming X-ray telescope, in agreement with Balakrishnan et al. (2024b).","pages":[9,10,11]},{"title":"5. Comparison with Lagrangian models","content":"Hydrodynamic simulations of the feeding of Sgr A* by the WR stellar winds have been conducted with di ff erent numerical tools. In this work, we have presented the results of using a finitevolume approach that solves the hydrodynamic equations in the Eulerian form. However, there has been extensive work on this problem using codes that solve the equations in the Lagrangian form. Specifically, the use of SPH has been the main choice for such a task (Cuadra et al. 2005, 2006, 2008, 2015; Russell et al. 2017; Wang et al. 2020). Despite the limitations of the generic SPH technique (e.g. capturing shocks) the models managed to reproduce the observed accretion rate at the Bondi radius (Cuadra et al. 2008) as well as the X-ray emission (Russell et al. 2017). In this context, we have conducted a comparison of the Eulerian models calculated with Ramses, and the Lagrangian models computed with the SPH code Gadget (Springel 2005). The SPH simulation setup is as similar as it can be to our setup. This corresponds to an updated version from the models in Russell et al. (2017). For more details on the setup of the SPH models, we refer the reader to Balakrishnan et al. (2024a,b). We focus the comparison on the analysis on the final state of the system, specifically in the cold discs formed after 3000 yr. ous structure towards smaller scales while the Lagrangian disc displays a ring-like structure. This feature can be attributed to the spatial resolution and the inner boundary radius. The Eulerian models indeed manage to resolve smaller scales towards the centre, and the radius of the inner boundary is 5 mas while the Lagrangian models resolve ∼ 1 mas and have an inner boundary of 0 . 1 '' . As a result, the Lagrangian model obtained a disc 50 times lighter ( ∼ 10 -4 M ⊙ ) than in the Eulerian simulation. This di ff erence could be attributed to the radiative cooling employed in each simulation: WR sub-type abundance and Solar with 3 Z ⊙ in the Eulerian and the Lagrangian model, respectively. Regarding the projected orientation of the disc both models seem to be in agreement. This can be analysed more carefully on the lineof-sight velocity maps (see right-hand side panels of Figures 8 and 11). Both maps display that the discs indeed match their projected orientation as well as their most blue- and redshifted velocities are roughly -2000 km s -1 and 2000 km s -1 . Overall, the agreement on the outcome a cold disc around Sgr A* whose properties align between two di ff erent approaches indicates that this is a result independent of the numerical technique. Thus, this shows that the determinant factors to obtain this results lies in both the long-enough simulation time ( ≳ 3000 yr ) as well as e ffi cient radiative cooling.","pages":[11]},{"title":"6. Disc stability analysis","content":"In this section, we present an analysis of the stability of the observed cold disc in the Galactic Centre. Based on its reported properties, the disc should be depleted by accretion after its viscous timescale t ν ≈ 43 kyr, assuming an α = 0 . 1 thin disc (Shakura & Sunyaev 1973), a process much slower than its generation as modelled in our simulations, although faster than its formation out of CND material (Solanki et al. 2023). Nonetheless, there are several external mechanisms that could be capable of destroying the disc faster than through accretion. Among them are the gravitational and hydrodynamic interactions of the disc with the stars, the stellar-wind-disc collisions due to the presence of the S-star cluster in its location, and / or the evaporation due to heat flowing between the hot medium and the cold disc. Although some of these physical scenarios have been studied analytically and numerically, there has not been any direct application to the cold disc discovered in our Galactic Centre.","pages":[11,12]},{"title":"6.1. Stellar cluster perturbing a thin disc","content":"The stability of a cold disc perturbed by stellar passages has been investigated by Ostriker (1983). Their work presented an analytical approach to calculate how the disc loses angular momentum due to the passages, considering the integrated e ff ect of a stellar cluster in stationary state. Besides this work, most e ff orts have been put on following how the stellar dynamics are a ff ected by the disc presence (e.g., Fabj et al. 2020, and references therein) or on the e ff ect of embedded accreting stars and / or black holes on the disc (e.g., Tagawa et al. 2021) rather than on the consequences to the disc properties. In this Section, we will summarise the approach of Ostriker (1983) and apply it to the Galactic Centre case. Let us start considering a cold, thin accretion disc around a compact object. Within the thin disc we assume that its height H ( r ) is small compared to the radial coordinate r , i.e. H ( r ) / r ≪ 1. Also, we consider that the sound speed of the material in the disc is small compared to the speed of the stars at the same radius. In this scenario, a star passing through the disc will produce a torque that removes angular momentum from the disc. We consider that the star interacts with the disc gravitationally and hydrodynamically The gravitational interaction of a star crossing the disc can be described using the Bondi-Hoyle approach (Bondi & Hoyle 1944), while the hydrodynamic interaction is just a physical collision of a solid sphere with a gaseous disc. Let us consider a system of reference where the x axis is in the direction of rotation and z is in the direction of the rotation pole of the disc. Then, according to Ostriker (1983) the total change of linear momentum along the x direction ∆ px for a single stellar passage is where R ∗ is the radius of the star, Σ d = R disc ρ d H ( r ) is the surface density of the disc, ρ d is the volume density of the disc, q is the hydrodynamic drag parameter that is set to q = 2 assuming a large Mach number collision, v 0 is the local stellar velocity, Λ D ≈ H ( r ) v 2 0 / ( R ∗ v 2 ∗ ) is the Coulomb logarithm, v ∗ is the escape velocity from the surface of the star, η = v ∗ / v 0 is the hardness parameter, v d is the disc velocity, v rel is the relative velocity vector, and ( θ, ϕ ) are the spherical coordinate angles along the polar and azimuthal directions. In order to consider the e ff ect of the complete stellar cluster interacting with the disc we need to integrate over the cluster phase-space volume. First, we integrate over the local stellar velocity distribution. Let us define dN as the number of stars in the velocity range v → v + dv , i.e where f ( v ) is the velocity distribution function, n ∗ is the stellar number density. Then, the number of crossings per unit time and area in the ( θ, ϕ ) direction is Article number, page 12 of 19 Now, integrating over the phase space, the total transfer of momentum to the disc per unit area at r is where µ = cos ϕ . As the local momentum per unit area in the disc is px , d( r ) = Σ ( r ) /τ d( r ) the inverse of the drag timescale to remove the angular momentum of the disc is τ d = -px , d / ˙ px , tot can be written as where the quantities ( I 0 , I 1) depend solely on the stellar distribution function. For instance, in the case of a Maxwellian distribution the values of these parameters can be obtained numerically and are I 0 = 12 . 553 and I 1 = 0 . 474. In order to apply this model to the Galactic Centre we need to calculate the stellar density within the size of the disc. Following Schödel et al. (2007) the stellar mass density at r < 6 arcsec is described by Assuming that the stellar mass density is composed mainly by stars whose radii are R ∗ = 1 R ⊙ and move on their orbits typically at v 0 = 1000 km s -1 we can use Equation 20 to estimate the inverse of the drag timescale, Based on this calculation the angular momentum of the cold disc should be removed after ∼ 185 Myr of interactions of the stars. This is much longer than the formation time-scale found in our simulations. Furthermore, it is even much longer than the age of the Wolf-Rayet stars, which is about 6 Myr (Martins et al. 2007). Then, the gravitational and hydrodynamic interactions of the stars onto the disc should not a ff ect the stability of the disc.","pages":[12]},{"title":"6.2. Stellar winds perturbing a disc","content":"In the case of stellar irradiation a ff ecting the state of the accretion flow there have been many studies mainly motivated by the recent pericentre passages of the star S2 around Sgr A*. For instance, Cuadra et al. (2003) could rule out the existence of an optically-thick disc based on the lack of its thermallyreprocessed emission as it gets illuminated by S2. Nayakshin et al. (2004) estimated that a star as luminous as S2 could heat up and ionise an inner disc, which could enhance the accretion rate. Giannios & Sironi (2013); Christie et al. (2016) calculated the bremsstrahlung emission as a result of the stellar wind shocking a Radiatively Ine ffi cient Accretion Flow (RIAF) around Sgr A*, finding an X-ray luminosity comparable to quiescent emission of Sgr A*. In fact, no noticeable increase was detected during 2018 (see Table 1 of Andrés et al. 2022). Hosseini et al. (2020) constrained the observed L ' -band variability of S2 to be about 2-3%, based on a model of bow shock of its stellar wind, implying an ambient density < 10 5 cm -3 , which rules out the presence of a standard accretion disc at S2's pericentre, in agreement with the previous studies. It is important to remark however that such a limit marginally allows the existence of both the disc reported by Murchikova et al. (2019), and also the one in our simulation. The stars in the S cluster also have relatively powerful stellar winds. As the stars inhabit the black hole in the vicinity of the cold disc, it is expected that the winds interact with it. By either opening a bubble due to the high kinetic energy carried in the winds or depositing thermal energy they a ff ect the disc properties. We proceed to perform analytical estimates to quantify the impact of these processes.","pages":[12,13]},{"title":"6.2.1. Bubbles in the disc","content":"Let us consider a star blowing an isotropic stellar wind with a mass-loss rate ˙ M w at a terminal speed v w. Then, its density ρ w at a distance r from the star is given by If such a star is immersed in a medium whose number density is n m at rest at a temperature T m, the radius of the bubble r b opened by the wind can be calculated equating the ram pressure of the wind and the thermal energy of the medium, i.e. where we have used as fiducial values the disc parameters measured by Murchikova et al. (2019) and typical stellar wind properties for B stars (Oskinova et al. 2011; Krtiˇcka 2014). Then, the radius of the bubble is about 0.012 arcsec, which is approximately one tenth of the disc radius. In order to open such a bubble a star would need to be within the disc fort at least the wind crossing time, i.e. r b / v w, which is about ∼ 0 . 5 yr. Let us estimate the duration of a stellar passage through the disc. Along the perpendicular direction a star crossing the disc would take at most 0 . 1 R d / v 0, being H ( r ) / r = 0 . 1. One S star moves typically at v 0 ≈ 1000 km s -1 , then 0 . 1 R d / v 0 ≈ 1 yr. As both timescales are comparable, we conclude that the bubble can be opened. For such a perturbation to last we need to check if either shear or sound waves could eliminate it, i.e. close the bubble. In the case of shear, this is given by the orbital speed of the disc and the size of the bubble: As this expression increases with r , the largest value will be at the outer edge of the disc, then t close < 5 yr. This value should be interpreted as the longest duration of a perturbation of the wind onto the disc. As there are 12 S-stars on orbits whose pericentre distances are shorter than the radius of the disc and their orbital periods are of the order of ∼ 10 yr (Gillessen et al. 2017) we expect a bubble to be opened every year on average. However, as shear would close such bubbles in less than five years there would be at most four bubbles opened on average in a stationary state. As the size of the bubbles is very small compared to the size of the disc (two orders of magnitude in area), four of them would not have an impact on its structure. The sound crossing timescale r b / c s ≈ 30 yr is longer than the shear timescale so we do not expect it to have an impact on erasing perturbations.","pages":[13]},{"title":"6.2.2. Winds injecting thermal energy","content":"The S stars are of spectral type B with masses of 10 M ⊙ (e.g. Habibi et al. 2017), thus their mass-loss rate must be ∼ 10 -8 M ⊙ yr -1 with terminal velocity of 1000 km s -1 . The supersonic nature of the winds should be enough to compress the shocked material at high temperature. Potentially, this thermal energy could be deposited onto the disc during each stellar passage. If we consider that most of the kinetic energy of the wind is transformed into thermal energy about ∼ 6 . 4 × 10 33 erg would be deposited in the disc during the ∼ 1 yr long passage (see above). In order to check if the wind energy can significantly impact the thermodynamic state of the disc let us estimate its total internal energy U int. where k B is the Boltzmann constant and V d is the disc volume. Replacing the observed disc properties we obtain that U int = 1 . 63 × 10 42 erg. From this calculation it is clear that even if all the energy from the wind in a stellar passage is deposited the contribution is still negligible to modify the state of the disc. Even if we consider the integrated e ff ect of all the S stars passages on a timescale comparable to the viscous timescale of the disc ( ∼ 43 kyr) the contribution is still < 0 . 1% of the total thermal energy of the disc. Thus, the energy supplied by the stellar winds into the disc is not enough to a ff ect its state.","pages":[13]},{"title":"6.3. Thermal conduction","content":"The vicinity of Sgr A* at the Bondi radius is made out of a diffuse (10 cm -3 ), hot medium (10 7 K; Bagano ff et al. 2003) due to the shocked stellar winds from the Wolf-Rayet stars in the region. At the disc vicinity ( ∼ 10 3 R Sch), the conditions are more extreme as the plasma has a density of about ∼ 5 × 10 3 cm -3 (Gillessen et al. 2019) and temperature of ∼ 10 8 K. As the disc is significantly colder it is expected that heat flows from the medium to the disc. The large temperature gradient should enable thermal conduction to take place and likely evaporate the disc. The problem of evaporating a cold structure sitting in a hot environment due to thermal conduction was first studied by Cowie & McKee (1977). In that work, the authors derived an analytical expression for the mass-loss rate and evaporation timescale for a spherically symmetric gas cloud. Unfortunately, these expressions are not valid for more complex geometries. Meyer & Meyer-Hofmeister (1994) studied specifically the evaporation of a cold thin accretion disc in a hot corona. Although the model considered only one radial zone it was able to calculate the full perpendicular structure of the accretion disc and the corona by means of numerical simulations. Liu et al. (2004) used the model by Meyer & MeyerHofmeister (1994) to analyse which solutions were consistent with the observational data available at the time in case there were a cold disc in the center of the Galaxy. They found that any transient disc would evaporate quickly. With their obtained evaporation rate of ∼ 10 -4 M ⊙ yr -1 , the life-time of a disc with properties such as found by Murchikova et al. (2019) or formed in our models would be ∼ 1 yr. Similar analysis have argued that small clumps ( ∼ M ⊕ ) in the Galactic Centre should also evaporate quickly ( ∼ 1 yr) due to thermal conduction (Burkert et al. 2012; Calderón et al. 2018). According to this, cold gas should not be able to last long if formed, yet we observe many cold gas structures in the central parsec: the many dusty cold clumps detected in the vicinity of IRS 13E (Fritz et al. 2010; Peißker et al. 2023), and a larger gaseous cloud X7 (Ciurlo et al. 2023). Thus, this regime of classical thermal conduction might not be applicable to the Galactic centre environment. Another possibility has been studied by Nayakshin (2004) who focused on the same problem and found solutions in a di ff erent regime, where the electron mean free path is long enough that an accretion disc does not evaporate but rather increases its mass by condensation. A more thorough analysis of the role of thermal conduction is beyond the scope of this work and is deferred to future study. Our findings shows that gravitational and hydrodynamic effects require timescales that are too long to have an impact on the disc stability. On the other hand, we find that thermal conduction in the classical case should evaporate the observed disc in about one year. Given the time-scale for disc formation found in our model, the Nayakshin (2004) condensation regime remains to our knowledge as the only viable physical scenario that would allow its continued existence and potential identification with Murchikova et al. (2019)'s reported disc.","pages":[13,14]},{"title":"7. Discussion","content":"In this section, we interpret our findings exploring the uncertainties in the abundances of WR stars in general and of the ones in the central parsec. Additionally, we contrast the properties of the simulated and observed cold discs as well as discuss further implications.","pages":[14]},{"title":"7.1. Is the simulated disc the observed disc?","content":"In order to compare the simulated and the observed discs, first we analyse their physical properties. The simulated disc has a total mass of ∼ 5 × 10 -3 M ⊙ , while the structure observed in H30 α was reported to have a mass of 10 -5 -10 -4 M ⊙ . However, comparing these quantities might not be appropriate since the mass of the observed structure is based on three assumptions: a thin disc with uniform density and temperature, and a masing factor of ∼ 100. All of them minimise the value of the mass inferred from the observations. For instance, dropping the maser assumption the inferred mass increases in a factor ten (see Equations 30-31 of Supplementary Material in Murchikova et al. 2019). Bearing this in mind, we could argue that the mass of the simulated and observed disc might be of the same order provided there was no maser. Although removing that assumption would create tension reconciling both the Br γ and H30 α observed fluxes. Regarding its extension, the simulated disc has a projected radius of ∼ 0 . 5 '' , while the observed disc has a radius of 0 . 23 '' . However, since most of the recombination line emission of the disc is originated from its central region ( p < 0 . 23 '' , see Table 3) the sizes of the simulated and observed discs are in agreement. A more direct comparison can be done through the analysis of the mock observational quantities computed from the simulations. In Section 4.1, we computed the H30 α and Br γ recombination line emission and estimated their fluxes. Here, we have found that both synthetic fluxes are in tension with the observations. First, the simulated Br γ flux is 100 times higher than the upper limit. It is relevant to remark that this limit is already extinction corrected. Second, the simulated H30 α flux is 30% lower than the observed one. These results di ff er from the estimates by Ciurlo et al. (2021) from our previous models due to, again, the uniform disc assumptions. The simulated disc is not thin or possess uniform density or temperature. Although the mean density and temperature of the simulated disc is of the order of ∼ 10 -5 cm -3 and ∼ 10 4 K its geometry is more complex with a larger variance in its properties. As a result, both the synthetic Br γ and H30 α fluxes are much higher than the ones inferred by Ciurlo et al. (2021). Based on the line-of-sight velocity maps shown in Murchikova et al. (2019), the orientation of the observed disc displays the red- and blueshifted sides to be on the East (lefthand) and West (right-hand) sides, respectively. Although our model WR_f1 also shows this configuration the exact inclination of the discs does not match perfectly. Specifically, the discs are inclined in ∼ 90 · among each other, being the simulated disc tilted in the clockwise direction on the sky. In principle, this argues against the WR stars as the main responsible for the formation of the observed disc. But we cannot rule out completely their role in the process, since a combination of the WR stars with other gaseous structures, such as the minispiral or the CND, acting simultaneously might result in a net e ff ect that manages to reproduce the observed angular momentum. Solanki et al. (2023) showed that material from the circumnuclear disc could fall close to Sgr A* though on much longer timescales. Certainly, this scenario would be much more di ffi cult and challenging to simulate with the appropriate chemical abundances as well as with high-enough spatial resolution. It is also relevant to notice that the observed accretion flow orientation at ≲ 10 RSch is consistent with the orientation of the clockwise stellar disc (GRAVITY Collaboration et al. 2020; Event Horizon Telescope Collaboration et al. 2022). Thus, our models reproduce that connection between the dynamics from large ( > 10 5 RSch) to small ( ≲ 10 RSch) scales. However, the observed cold disc does not seem to follow such a connection. Finally, an intriguing aspect of our findings is that the WR atmospheres with more H in them would favour the formation of the disc. At the same time, the disc indeed was detected in the H30 α recombination line, which obviously indicates the presence of H. Thus, it is sensible to ask how much H can be in the winds of WR stars but, specifically in the ones of the sub-type WN89 / Ofpe. This point is discussed in the following section.","pages":[14]},{"title":"7.2. Uncertainties in abundances","content":"The results of the hydrodynamic models demonstrate that the chemical abundances of the material in the WR winds are the key factor to determining whether or not a cold disc can be formed due to their action. Unfortunately, there are two large uncertainties in this context. First, the metallicity of the young population of the nuclear star cluster is not well constrained (e.g. Genzel et al. 2010). Although there is agreement that it is above Solar its exact value has not been established so far, with most studies quoting the range Z = 2 -3 Z ⊙ . However, even if the metallicity is determined more precisely, the problem would not be entirely solved, since the abundances of the plasma in the region are dominated by the material supplied by the WR star winds whose compositions are much more uncertain. Theoretically, the evolution of these stars is a topic of active research, and improved prescriptions for mass loss are changing the modelled properties of observed WR stars (see e.g., Gormaz-Matamala et al. 2023, and references therein). Simply based on observations, up to now, it has been reported that some of these stars might possess from X H = 0 to X H = 50% for Galactic WR stars of the WN subtype (Hamann et al. 2006, 2019), and even as high as X H = 60% for extragalactic ones (Todt et al. 2015). In the vicinity of Sgr A*, Martins et al. (2007) characterised the wind properties of the sample of WR stars through the analysis of their infrared spectra. They reported abundance H / He ratios in the range 2-5 for these stars which does not allow us to rule out the scenario regarding this aspect. A variation on the hydrogen abundance in the chemical mixture in a factor five might change the net radiative cooling rate by a factor 1 . 1 -1 . 4, especially at lower temperatures. In our models, WR_f07 considered a fiducial value of hydrogen mass fraction of 11.5% (Russell et al. 2017) but the model WR_f1 assumed 40%. Such a change was enough to modify the cooling function by about 30%, which ended up affecting significantly the final result and the state of the plasma. Thus, within our current understanding of the abundances in the WR stellar atmospheres both scenario are plausible. Only better quality spectra and more sophisticated models of them could allow us to constrain the wind properties and their composition more precisely.","pages":[14,15]},{"title":"8. Conclusions","content":"We have presented a numerical study of the system of masslosing stars feeding Sgr A* focusing on the impact of the chemical composition of the plasma. Through studying simulations with di ff erent abundances we have found that the system can either form or not form a cold disc around the black hole, provided that the model is run for long-enough timescales ( ≳ 3000 yr) to create a dense enough medium in the inner parsec of the Galaxy. As in our past work, we have confirmed that if formed the cold disc can impact significantly the hydrodynamic and thermodynamic state of the plasma within the central parsec. As a result, the inferred mass in flow rate at 5 × 10 -4 pc can reach up to 8 × 10 -6 M ⊙ yr -1 , which is 4 -8 times higher than the value without the presence of such a disc. Our models still point to the mass-losing stars in the clockwise disc being the main source of the accreted material due to their dense and slow winds as previous models have shown (e.g. Cuadra et al. 2008; Ressler et al. 2018; Calderón et al. 2020b). In this work, we have followed more carefully the origin of the inflowing material through the use of passive scalar fields for di ff erent WR subtypes. This allowed us to confirm the origin of the accreted material but, more importantly led us to conclude that this is mostly provided by the winds of the stars of the WR sub-type WN89 / Ofpe. This provides evidence that the infalling material must have a nonnegligible fraction of hydrogen as these WR stars have not lost entirely their atmospheric hydrogen via winds. Our models also showed that the formation of a cold disc depends strictly on the radiative cooling employed, which is determined by the chemical abundances of the plasma. Previous models have only considered Solar abundances with 3 Z ⊙ but had not agreed whether or not a cold disc can be formed. From our results in Appendix B, we speculate that at Z ≈ 3 Z ⊙ there is a transition between the regimes of persistent quasi-steady accretion and disc formation, which would imply that small di ff erences in the cooling function or numerical implementation can result in models reproducing either regime. This may explain the di ff ering results obtained by Calderón et al. (2020b) and Ressler et al. (2020) with similar numerical and physical setups. In the case a cold disc is formed around Sgr A* the structure may resemble the extension and the line-of-sight velocity structure of the observed disc. However, direct observational quantities could not be reproduced. Specifically, the exact skyprojected inclination of the simulated and observed discs di ff ers in ∼ 90 · . The Br γ emission line upper limit is 100 lower than the synthetic flux. The observed H30 α emission line flux is 30% higher than in our model. Furthermore, we have contrasted this work Eulerian (finitevolume grid-based) models with analogous (SPH) Lagrangian models in order to address the potential influence of the chosen approach in the outcome of the simulations. As it has been shown in Balakrishnan et al. (2024b), a cold disc is also formed around Sgr A* if the simulation is run for long timescales. Both cold structures agree on their sizes, their velocity structure, and their sky-projected inclination. Although there are di ff erences in the total mass and specific inner structure of the discs this could be attributed to the resolution employed and the exact assumption to consider the innermost boundary. Despite these di ff erences, the agreement is reasonable among the numerical approaches, especially when analysing the synthetic X-ray emission (see Figure 10). The spectral features agree qualitatively across the models presented in this work and the Lagrangian model. The level of the X-ray emission are all within the same order of magnitude but the exact base level depends on the radiative cooling chosen, which is determined by the chemical abundances in the WR atmospheres. We also have explored the stability and long-term evolution of the putative disc under the e ff ect of perturbing agents: the nuclear star cluster, the wind of all the mass-losing stars (including the S stars) impacting the disc through opening bubbles and / or depositing thermal energy via shocks, and the potential e ff ect of thermal conduction. Our analysis showed that gravitational and hydrodynamic e ff ects would impact the disc on longer timescales then the viscous timescale. However, within the classical thermal conduction framework a cold disc in such a region should not exist. Based on this and the disc formation timescale from our simulations, we speculate that the condensation model of Nayakshin et al. (2004) remains a plausible mechanism consistent with the disc existence. In conclusion, through this work we have been able to reconcile the discrepancy among the numerical simulations of the system of mass-losing stars feeding Sgr A*. The formation of a cold disc on ∼ 3000 yr timescale is indeed possible for certain chemical abundances which are consistent with the current observational constraints. Nonetheless, it is not possible to favour or disfavour this scenario due to the uncertainties in them. Thus, high-resolution spectra of the WR stars and more sophisticated modelling of them could allow us to discern the validity of such scenarios. But we must warn that even in the case that the cold disc is a result of action of the WR stars it is not possible to reproduce all its observed properties: its inclination and recombination line fluxes. More sophisticated models that include infalling material from other sources might be necessary to produce the exact inclination of the structure which could also impact the emission of the structure. Acknowledgements. We thank Prof. Sergei Nayakshin and Prof. Stanley Owocki for very helpful discussions about this project. We also thank Dr. Sean M. Ressler for sharing the radiative cooling function shown in Figure 1. Additionally, we are grateful to Dr. Álex Gormaz-Matamala for helping with references and discussions on the WR wind abundances. DC and JC acknowledge the support of the Kavli Foundation through its summer program at the Max Planck Institute for Astrophysics where fruitful discussions took place. In addition, DC thanks the warm hospitality of the Universidad Adolfo Ibañez in Chile and University of Delaware in the USA where part of this project was developed. We acknowledge the support from ANID in Chile through FONDECYT Regular grant 1211429 and Millennium Science Initiative Program NCN2023_002. The research of DC and SR was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - EXC 2121 - 'Quantum Universe' - 390833306. Since 01.10.2024, DC has been funded by the Alexander von Humboldt Foundation. SR is also supported by the European Research Council (ERC) Advanced Grant INSPIRATION under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 101053985), the Swedish Research Council (VR) under grant number 202005044, and the research environment grant 'Gravitational Radiation and Electromagnetic Astrophysical Transients' (GREAT) funded by the Swedish Research Council (VR) under Dnr 2016-06012, by the Knut and Alice Wallenberg Foundation under grant Dnr. KAW 2019.0112. The numerical simulations of this work were run on the high-performance computing system cobra of the Max Planck Computing and Data Facility. Most of the analysis of the numerical data was carried out using the python package yt (Turk et al. 2011). Furthermore, this work made use of python libraries numpy (Harris et al. 2020) and matplotlib (Hunter 2007), as well as of the NASA's Astrophysics Data System.","pages":[15,16]},{"title":"References","content":"Andrés, A., van den Eijnden, J., Degenaar, N., et al. 2022, MNRAS, 510, 2851 Arnaud, K. A. 1996, in Astronomical Society of the Pacific Conference Series, Vol. 101, Astronomical Data Analysis Software and Systems V, ed. G. H. Jacoby & J. Barnes, 17 Becklin, E. E., Gatley, I., & Werner, M. W. 1982, ApJ, 258, 135 GRAVITY Collaboration, Abuter, R., Amorim, A., et al. 2021, A&A, 647, A59 GRAVITY Collaboration, Bauböck, M., Dexter, J., et al. 2020, A&A, 635, A143 Hosseini, S. E., Zajaˇcek, M., Eckart, A., Sabha, N. B., & Labadie, L. 2020, A&A, 644, A105 Article number, page 16 of 19 Russell, C. M. P., Wang, Q. D., & Cuadra, J. 2017, MNRAS, 464, 4958 Schartmann, M., Ballone, A., Burkert, A., et al. 2015, ApJ, 811, 155 Practical Introduction (Berlin: Springer) Toro, E. F., Spruce, M., & Speares, W. 1994, Shock Waves, 4, 25 Turk, M. J., Smith, B. D., Oishi, J. S., et al. 2011, ApJS, 192, 9 von Fellenberg, S. D., Gillessen, S., Stadler, J., et al. 2022, ApJ, 932, L6 Štofanová, L., Kaastra, J., Mehdipour, M., & de Plaa, J. 2021, A&A, 655, A2 Wang, Q. D., Li, J., Russell, C. M. P., & Cuadra, J. 2020, MNRAS, 492, 2481 Yelda, S., Ghez, A. M., Lu, J. R., et al. 2014, ApJ, 783, 131 Yuan, F. & Narayan, R. 2014, ARA&A, 52, 529","pages":[16]},{"title":"Appendix A: Numerical setup choices","content":"Calderón et al. (2020b) simulated the system of WR stars feeding Sgr A* while orbiting around it following an analogous numerical setup compared to the models presented in this work. There, we had found that the resulting cold disc tended to align with the grid after its formation. In order to remedy this numerical artifact we investigated the use of a MinMod slope limiter, instead of MonCen. However, we warned that it was not straightforward to maintain the spherical symmetry of the winds using the MinMod slope limiter. In this work, we investigated more carefully these numerical choices in order to be confident that our results are robust. First, we investigated the choice of the MonCen slope limiter. To do this, we made use of the same setup presented in this work but paying attention to the values of the passive scalar fields. In principle, the values of these fields should be between zero and one. However, this experiment showed that the MonCen slope limiter produced non-physical oscillations that in some cases resulted in negative values of the passive scalar fields. This behaviour was not observed when using the MinMod slope limiter. Thus, we selected this choice throughout this work. To ensure the spherical symmetry of the stellar winds we had to impose a less restrictive refinement criterion based on density gradients as well as a smoother transition between refinement levels through the parameter nexpand . Second, we tested the choice of the Riemann solver employed but using the MinMod slope limiter. Currently, the HLLC Riemann solver (Toro et al. 1994) is widely used in grid-based hydrodynamic simulations. In this work, we have found that this choice results in a cold disc that tends to align with the Cartesian grid as seen in Calderón et al. (2020b). This could indicate problems in conserving the angular momentum which is a known issue with simulations in Cartesian grids when the relevant regions are not well resolved. With the usage of the exact Riemann solver (Toro 2009), though more computationally expensive, it was possible to avoid this artificial alignment. Thus, the only sensible combination was the use of the exact Riemann solver with a MinMod slope limiter but we had to ensure to count with enough resolution and smoothness in the transition of the refinement levels. As a reference of these experiments, Figure A.1 shows density projection (weighted by density) maps of the final state of the system for two runs with Solar composition with 3 Z ⊙ but with di ff erent Riemann solvers. The left- and righthand side panels display the models using the HLLC and the exact Riemann solvers, respectively. Although overall the density structure looks similar there are subtle di ff erences. The most relevant is related to the disc orientation and structure. On the left-hand side, the disc tries to align with one of the Cartesian axes, while on the right-hand side the disc remains consistently with the shown orientation.","pages":[17]},{"title":"Appendix B: Solar composition varying metallicity","content":"Before investigating the models with the new implementation of the di ff erential cooling we ran some experiments simply modifying the contribution of the metals to the total radiative cooling. We simulated three models varying only the composition used: A1 used Solar composition, A3 used Solar composition with 3 Z ⊙ , and A5 used Solar composition with 5 Z ⊙ . The density projected (weighted by density) maps of the final state of the simulations are shown in Figure B.1. The left-hand side, central, and right-hand side panels display the runs A1, A3, and A5, respectively. A higher metallicity in the plasma increases the radiative cooling directly. As a result, more and denser filaments, clumps, and a denser central disc are obtained with increasing the metallicity. On the contrary, if the metallicity is decreased to a Solar value we observe almost no overdensities, and definitively no disc formation. We speculate that at Z ≈ 3 Z ⊙ there is a transition between the regimes of persistent quasi-steady accretion and disc formation, which would imply that small di ff erences in the cooling function or numerical implementation can result in models reproducing either regime. This may explain the di ff ering results obtained by Calderón et al. (2020b) and Ressler et al. (2020) with similar numerical and physical setups. In order to perform a more quantitative comparison of these results Figure B.2 presents radial profiles of density (weighted by volume) and temperature (weighted by mass) on the top and bottom panels, respectively. The dashed blue, solid orange, and blue dotted-dashed lines represent the models A1, A3, and A5, respectively. The density profiles clearly shows the impact of cooling in the structure of the medium. More e ffi cient cooling, like in cases A3 and A5, results in an order of magnitude denser medium within 1 '' compared to the case with ine ffi cient cooling, i.e. A1. Notice that even higher metallicity in A5 extends the impact the denser region to slightly larger spatial scales. The e ff ect is similar in the temperature profile. In A1, the temperature profile decays with r within the central 1 '' . But when the metallicity is high enough to make cooling e ffi cient the temperature profile can decrease up to two orders of magnitude depending on the exact value. In summary, these experiments allowed us to quantify the impact of radiative cooling by simply decreasing or increasing the radiative cooling influence. However, since the chemical composition in reality is much more complicated and di ff ers from Solar composition significantly these models should be interpreted only as general guides to assess the role of radiative cooling. In the main text of this work, we devoted to explore more physically motivated mixtures that, although depend on more uncertain parameters are more robust when attempting to build a physical model that can be constrasted with observational data.","pages":[17]}],"string":"[\n {\n \"title\": \"ABSTRACT\",\n \"content\": \"Context. The reported discovery of a cold ( ∼ 10 4 K) disc-like structure around the super-massive black hole at the centre of the Milk Way, Sagittarius A* (Sgr A*), has challenged our understanding of the gas dynamics and thermodynamic state of the plasma in its immediate vicinity. State-of-the-art simulations do not agree on whether or not such a disc can indeed be a product of the multiple stellar wind interactions of the mass-losing stars in the region. Aims. This study aims to constrain the conditions for the formation of a cold disc as a natural outcome of the system of the mass-losing stars orbiting around Sgr A*, to investigate if the disc is a transient or long-lasting structure, and to assess the validity of the model through direct comparisons with observations. Methods. We conduct a set of hydrodynamic simulations of the observed Wolf-Rayet (WR) stars feeding Sgr A* using the finitevolume adaptive mesh-refinement code Ramses. We focus, for the first time, on the impact of the chemical composition of the plasma emanating from the WR stars. Results. The simulations show that the chemical composition of the plasma a ff ects the radiative cooling enough to impact the properties of the medium such as density and temperature and, as a consequence, the rate at which the material inflows onto Sgr A*. We demonstrated that the formation of a cold disc from the stellar winds is possible for certain chemical compositions that are consistent with the current observational constraints. However, even in such a case, it is not possible to reproduce the reported properties of the observed disc-like structure, namely its inclination and hydrogen recombination line fluxes. Conclusions. Weconclude that the stellar winds on their own cannot form the cold disc around Sgr A* inferred from the observations. Either relevant ingredients are still missing in the model, or the interpretation of the observed data needs to be revised. Key words. accretion, accretion discs - Galaxy: centre - hydrodynamics - Stars: winds, outflows - Stars: Wolf-Rayet\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"The formation and stability of a cold disc made out of stellar winds in the Galactic Centre\",\n \"content\": \"1,2, 3,4 5 6,7,8 Diego Calderón ⋆ , Jorge Cuadra , Christopher M. P. Russell , Andreas Burkert , Stephan Rosswog 1,9 , and Mayura Balakrishnan 10 Received November 4, 2024; accepted November 4, 2024\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1. Introduction\",\n \"content\": \"The Galactic Centre (GC) hosts the closest super-massive black hole to us, Sagittarius A* (Sgr A*; see Genzel et al. 2010, for a review). Unlike the black holes present in active galactic nuclei, Sgr A* is very underluminous. It does not seem to be currently accreting much material, or to have a standard accretion disc around it (Yuan & Narayan 2014). Murchikova et al. (2019) detected however a disc-like structure around Sgr A*. Using the 1.3-millimetre recombination line H30 α they observed a doublepeaked emission line with full velocity width of ∼ 2200 km s -1 . The centre of the emission coincides with Sgr A* and extends up to 0.11 '' ( ∼ 4.4 × 10 -3 pc) to both the redshifted and blueshifted sides. They interpreted this feature as a rotating disc of mass 10 -5 -10 -4 M ⊙ . Yusef-Zadeh et al. (2020) also reported the presence of broad hydrogen recombination lines, including the double-peaked H30 α line, at the position of Sgr A* using the Very Large Array. However, they interpreted the signatures as a jet emanating from the central region. So far, there has not been any detection in the near infrared, despite many observations. Ciurlo et al. (2021) reports an upper limit for Br γ , which is two orders of magnitude below the extrapolation from the reported H30 α flux. Currently, there is no consensus on how such observations can be reconciled or from where the observed cold material comes from. The gaseous environment around Sgr A* is dominated by the outflows from around 30 Wolf-Rayet (WR) stars, all located within a fraction of a parsec of the black hole (Paumard et al. 2006; Martins et al. 2007). At such close distances, the winds interact strongly in shocks that thermalise them and create a hot and di ff use X-ray emitting plasma (Quataert 2004). However, given the relatively low velocity of some of the outflows (450600 km s -1 ; Martins et al. 2007), the resulting plasma can be prone to radiative cooling and form dense, cold clumps (Cuadra et al. 2005; Calderón et al. 2016), which end up being embedded in the di ff use, hot plasma ( ∼ 10 7 K; Bagano ff et al. 2003). The complex interplay between the stellar winds around Sgr A* has been the subject of hydrodynamic simulations using a variety of numerical techniques. With smoothed particle hydrodynamics (SPH) simulations, Cuadra et al. (2006) showed that, if most of the slow-wind stars are located in a relatively compact stellar disc, many cold clumps form and quickly coalesce in a cold gaseous disc of radius ∼ 1 '' . However, using a stellar distribution closer to the one observed, Cuadra et al. (2008, 2015) showed that no conspicuous disc appears over the 1800 yr time-span of their models. Calderón et al. (2020b) performed finite-volume hydrodynamic simulations on a Cartesian grid of the same system, which are better suited to model shocks, the multi-phase medium, and subsequent cooling, and found that the formation of a cold disc is indeed possible. According to their model, the stellar wind of the star IRS 33E, which is fairly dense ( ∼ 10 -5 M ⊙ yr -1 ) and slow (450 km s -1 ; Martins et al. 2007) interacts with the medium creating a shell that is dense enough to radiate quickly its thermal energy and break into denser and smaller structures. These pieces manage to fall inwards forming a cold ( ∼ 10 4 K) disc with a total mass of ∼ 5 × 10 -3 M ⊙ within a simulation time of 3500 yr. Based on this result, Ciurlo et al. (2021) found that the properties of the modelled disc are consistent with the non-detection in Br γ . Nevertheless, this scenario has not been confirmed by analogous grid-based simulations. Ressler et al. (2020) revisited the same system using the same numerical approach, although a di ff erent code: athena++ (Stone et al. 2020), and found no disc formation even extending their simulation time up to 9000 yr. Building on this model, Solanki et al. (2023) also studied this system through hydrodynamic modelling but encompassing a larger region and evolving it for much longer timescales. They included, for the first time, the presence of the circumnuclear disc (CND), an observed gaseous structure located between 1 . 5-3 . 0 pc (Becklin et al. 1982; Genzel 1989) and with a total mass of 3-4 × 10 4 M ⊙ (Dinh et al. 2021; James et al. 2021). This work showed that the formation of a cold disc is possible on long timescales ( ≳ 300,000 yr) due to the interaction of the innermost boundary of the CND and the WR stellar winds, which results in the transport of CND material to smaller scales. However, the authors warned that the lack of certain physical mechanisms in the model on such timescales (e.g. magnetic fields, supernovae, thermal conduction) might a ff ect the robustness of this result. Thus, the exact formation process of the observed cold disc still remains unknown. Akey factor driving the formation of clumps and a disc in the model of Calderón et al. (2020b) is that radiative cooling allows the gas to get rid of its energy e ffi ciently. The strength of this process depends on the chemical composition of the gas, which is highly unusual in the GC given its origin as winds from evolved massive stars. In this work, we explore for the first time the impact of the radiative cooling through studying specific chemical compositions: Solar with 1 Z ⊙ , 3 Z ⊙ , and 5 Z ⊙ . More importantly, we develop a more realistic chemical setup that considers the atmospheric abundances for the di ff erent WR sub-types present in the region, in order to follow the thermodynamic evolution of the gas in a more appropriate manner. Our results show that the formation of the disc is indeed determined by the composition of the gas. However, the properties of such a disc do not agree with the observations when considering realistic wind compositions compatible with current observational constraints. We also improve the numerical setup in order to better assess the disc orientation and more directly compare our results to the work of Ressler et al. (2018, 2020) as well as with SPH models (Cuadra et al. 2008, 2015; Russell et al. 2017). This article is organised as follows: in Section 2, we present the numerical approach, the setup and the models investigated, Section 3 presents and describes the results of the numerical simulations. In Section 4, we present the synthetic observables obtained through post-processing the models and contrast them with observations. We compare our grid-based models with SPH models in Section 5. In Section 6, we present an analytic analysis of the stability of the observed cold disc. Section 7 discusses our findings and uncertainties in the parameters of our models. Finally, Section 8 presents conclusions and final remarks. Throughout this paper we use the mass and distance to Sgr A* of 4 . 3 × 10 6 M ⊙ and 8 . 33 kpc, respectively (Gillessen et al. 2017; GRAVITY Collaboration et al. 2019, 2021), so that 1 arcsec corresponds to a length of ∼ 0.04 pc ≈ 10 5 RSch, being RSch the Schwarzschild radius.\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"2.1. Equations\",\n \"content\": \"The simulations were performed using the adaptive-mesh refinement (AMR) hydrodynamic code Ramses (Teyssier 2002). This code uses a second-order Godunov method with a shockcapturing scheme to solve the Euler equations in their conservative form, i.e. where ( ρ, u , P ) are the primitive hydrodynamic variables: density, velocity, and pressure, respectively. The set of quantities si correspond to tracer scalar fields that are advected with the fluid whose usage we introduce in Section 2.3. The sink terms on the right-hand side correspond to the e ff ects of the gravitational potential ϕ = ϕ ( x ), assumed to be time independent, and the total radiative loses due to optically-thin radiative cooling Q -tot = Q -tot ( ρ, T , X i), with x the position vector, T the fluid temperature, and X i the chemical composition of the fluid. The total specific energy density e is given by where γ is the adiabatic index that is set to 5 / 3. Additionally, we consider the fluid can be described as an ideal gas so that the temperature can be calculated through P = ( ρ/µ ) k B T , being µ the mean molecular weight and k B the Boltzmann constant.\",\n \"pages\": [\n 2\n ]\n },\n {\n \"title\": \"2.2. Numerical setup\",\n \"content\": \"The model considers the system of WR stars blowing stellar winds while they move on their observed Keplerian orbits around Sgr A*. The setup is analogous to our previous work (Calderón et al. 2020b). The central black hole gravitational field is modelled as a point mass of 4 . 3 × 10 6 M ⊙ (GRAVITY Collaboration et al. 2019, 2021). The stars are simulated as test particles that only feel the gravitational pull of Sgr A*. Their initial Notes. The Solar abundances were taken from Lodders (2003). The abundances for the three WR sub-types correspond to the compilation by Russell et al. (2017) based on previous studies (Herald et al. 2001; Crowther 2007; Onifer et al. 2008). Column 1: chemical mixture. Columns 2-7: hydrogen, helium, carbon, nitrogen, oxygen, and rest of the metals mass fractions, respectively. Column 8: mean molecular weigh assuming full ionisation. Notes. Column 1: simulation ID. Column 2: chemical composition of the winds. Column 3: Riemann solver. Column 4: Slope limiter. Column 5: whether or not the final state of the system shows the presence of a cold disc around Sgr A*. position and velocity vectors are set so that they move on the orbits constrained by observations (Paumard et al. 2006; Cuadra et al. 2008; Gillessen et al. 2019; von Fellenberg et al. 2022). The stellar winds in the simulation are modelled following the procedure by Lemaster et al. (2007), which has been validated and used in our previous work (Calderón et al. 2020a,b). This consists in defining a 'masked region\\\" around each star that is a spherical volume where the hydrodynamic variables are reset in each timestep, in order to reproduce the free wind expansion solution of a spherical wind with certain mass-loss rate ˙ M w, terminal velocity V w, and temperature T w. For these quantities we used the values constrained through modelling of the infrared spectra (Martins et al. 2007; Cuadra et al. 2008). The wind temperature was set to the lowest temperature allowed in the simulation, T w = 10 4 K which is determined by the strong ultraviolet radiation produced by the hundreds of massive stars in the region. The simulations were run in a Cartesian grid in a cubic domain of side 40 '' ( ∼ 1.6 pc) with outflow boundary conditions (zero gradients). We used the exact Riemann solver (e.g. Toro 2009) combined with the MinMod slope limiter, as these choices allow the modelling of hydrodynamic instabilities, which otherwise may be quenched due to numerical di ff usion. We investigated other choices such as the Harten-Lax-van Leer-Contact (HLLC; Toro et al. 1994) and / or the MonCen slope limiter but they produced unwanted numerical artifacts such as spurious oscillations. For a careful analysis of these choices we refer the reader to the Appendix A. Regarding the numerical resolution of our models, the coarse resolution was 64 3 cells, allowing adaptive refinement of four levels, i.e. e ff ectively ∆ x ≈ 0 . 0382 '' . However, the regions around the stars allowed an extra level of refinement ( ∆ x ≈ 0 . 0195 '' ). Additionally, the vicinity around the central black hole has a fixed nested grid with eight refinement levels above the coarse resolution ( ∆ x ≈ 2 . 44 mas). At the location of Sgr A*, we defined a spherical region where the hydrodynamic variables are reset to low values of density and pressure at rest, in order to avoid artificial accumulation of material. The refinement strategy is based on density gradients on top of the geometric criteria previously defined. In order to explore the role of the AMR potentially a ff ecting the results we tested di ff erent values of the smoothing parameter that controls the refinement in transitions between refinement levels. We tested quadrupling the smoothing parameter and the results remained unchanged 1 .\",\n \"pages\": [\n 2,\n 3\n ]\n },\n {\n \"title\": \"2.3. Models\",\n \"content\": \"The main parameter explored in this work is the optically-thin radiative cooling function. This choice is determined by specifying the chemical composition of the fluid. Unfortunately, the metallicity of the young stars in the GC is still poorly known, yet it has been argued that it should be higher than Solar ( Z = 2 -3 Z ⊙ ; Genzel et al. 2010). Past numerical works have considered that the composition of the gas correspond to Solar abundances but with metallicity three times the Solar value, Z = 3 Z ⊙ (Cuadra et al. 2008; Calderón et al. 2016; Ressler et al. 2018, 2020; Solanki et al. 2023). However, more than the 'bulk' metallicity of the stars, the most appropriate composition choice would be the abundances of the WR stellar atmospheres, which normally di ff er significantly from Solar (Herald et al. 2001; Onifer et al. 2008). Although Ressler et al. (2018, 2020) used chemical compositions lacking hydrogen, the rest of the element abundances remained unchanged relative to Solar with 3 Z ⊙ . As a result, the cooling function varied at low temperature ( < 10 5 K) but at higher temperatures it remains unchanged. Moreover, the composition choice not only determines the cooling function but also the ion mean molecular weight as well as the electron to proton number densities n e / n p. These quantities are taken into account in the sink term in the energy equation (see equation 3), and following Schure et al. (2009) can be expressed as follows where the subscript i stands for a given chemical abundance. Bear in mind that di ff erent chemical compositions also correspond to di ff erent temperature Ti , as this is obtained through the ideal gas expression that makes uses of the mean molecular weight of the fluid. In the general case, if we consider a fluid element composed of N mixtures of abundances, the total radiative loses will be given as a summation, i.e. where we have expressed the total radative cooling as a linear combination of the mixtures. Notice that we have introduced the passive scalar fields si , as we used them to quantify the fraction of a given chemical composition. In this work, we introduced three types of compositions that represent the WR subtypes based on their atmosphere abundances. In the region where the winds are generated, the corresponding passive scalar is set to 1 while the rest are set to 0. By doing so, it is possible to identify how much material of a given cell is supplied by which sub-group of WR stars. The mass fractions of each mixture are shown in Table 1. In this work, we investigated five models with di ff erent compositions: Solar composition with metallicities Z ⊙ , 3 Z ⊙ , and 5 Z ⊙ , a mixture of three compositions based on the spectroscopic constraints of the sub-types of WR stars, and a variation of the latter one motivated by the uncertainty in the WR stellar atmospheres and, specifically on the H fraction in them that is set to X H = 40% (see Section 7.2, for a discussion). Figure 1 shows the cooling function Λ = Λ ( T ) as a function of temperature for different chemical mixtures. Each curve was calculated as a linear combination of the abundance of a given element and its contribution to the total radiative cooling. The values of the composition per element are shown in Table 1. The contribution of each element to the total cooling was taken from the plasma models developed by Štofanová et al. (2021). The cooling functions corresponding to WR subtypes compositions: WC8-9, WN5-7, and WN8-9 / Ofpe are shown with dashed blue, dotted orange, and dotted-dashed green lines, respectively. The Solar and Solar with 3 Z ⊙ are represented with thin and thick solid black lines, respectively. Furthermore, we added the cooling function used in previous works by Ressler et al. (2018, 2020) as a dotted-dashed red line. Here it can be observed that a Solar composition with 3 Z ⊙ (the typical used value) is between the range of the di ff erent WR compositions but the subtypes WC89 and WN89 / Ofpe are about a factor two higher. The list of the runs investigated in this work are shown in Table 2. The initial setup was identical to our previous work (Calderón et al. 2020b), i.e. the medium across the whole domain was set to constant and low-enough values of density ρ = 10 -24 g cm -3 and pressure P = γ -1 ρ c 2 s,f with c s,f = 10 km s -1 , so that the stellar winds do not encounter impediments and fill quickly the domain. All models were run for a total of 3,500 yr but setting the starting state of the system in the past, so that the final state of the simulations corresponds to the current state of the system. This is achieved by integrating the current position and velocity vectors of the stars back in time, assuming that they have followed purely Keplerian orbits within this timescale due to the gravitational field of Sgr A* (Cuadra et al. 2008; Ressler et al. 2018; Calderón et al. 2020b).\",\n \"pages\": [\n 3,\n 4\n ]\n },\n {\n \"title\": \"3. Results\",\n \"content\": \"We proceed to describe the evolution and final state of the simulations at t = 0 that corresponds to the present time. The runs with Solar composition, but varying metallicities, were used for understanding the impact of increasing and decreasing the e ff ect of radiative cooling in general. However, since they do not represent realistic compositions we opted not to discuss them in detail here. Instead, we chose to focus on the more physically motivated models, i.e. WR_f07 and WR_f1 which we also refer as fiducial and enhanced , respectively. For completeness, we give a brief description of the models A1, A2, and A3 in Appendix B.\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"3.1. Hydrodynamics\",\n \"content\": \"The simulation evolution of all models is analogous, especially in the initial phases. At t = -3500 yr, the stars begin to orbit around Sgr A* that is located at the centre of the domain, while their winds quickly fill the whole computational domain. After ∼ 500 yr ( t = -3000 yr), the system reaches a quasi-steady state. At this point, the domain is full of di ff use ( ∼ 10 -22 -10 -21 g cm -3 ) and hot plasma ( ∼ 10 7 -10 8 K) due to the shocked stellar winds and their interactions. This is illustrated in Figure 2 that shows density maps across di ff erent simulation times. The maps show projected density fields along the line-of-sight direction ( z axis) weighted by density, i.e. R ρ 2 dz / R ρ dz , in order to highlight the densest regions with the highest resolution. The top panels show two models at t = -700 yr, respectively. In them, it is possible to observe the complex structure developed due to the stellar wind interactions such as bow shocks, instabilities, and dense clumps as a result of condensation through radiative cooling. Left- and right-hand side panels display models with di ff erent chemical compositions WR_f07 and WR_f1, respec- tively. On the left-hand side, it can be seen how mildy e ffi cient cooling a ff ects mostly one of the stellar winds. This star corresponds to IRS 33E and its wind is the slowest ( ∼ 450 km s -1 ). This fact causes its shocked temperature to be in an e ffi cient region of the radiative cooling functions that allows its material to cool down and become denser. Also, some of its material manages to reach the vicinity of Sgr A*, as the map shows an elongated clump being accreted and stretched. The right-hand side also portrays this picture but since the radiative cooling is enhanced more dense clumps can be observed, especially close to IRS 33E and Sgr A*. Finally, the bottom panels of Figure 2 show the models at time t = 0, i.e. the present time. One can clearly see that the amount of dense material that has accumulated around Sgr A* is di ff erent in both maps. Although in the model WR_f07 (see bottom left-hand side panel of Figure 2) some dense material is spiraling towards the black hole overall there are is no clear structure around it. In the case of WR_f1 (see bottom right-hand side panel of Figure 2), the dense material has settled at the centre in a a disc-like structure. This di ff erence is due to the e ffi ciency of the radiative cooling. Since model WR_f1 has enhanced cooling, more dense clumps and filaments form, and some of them manage to fall onto Sgr A*. Overall, the hydrodynamic evolution of the two models is analogous: the winds fill the domain during the first ∼ 500 yr, then the systems reach a quasi-steady state, and in the last 500 yr they diverge as the model WR_f1 creates much more dense, cool material that settles around Sgr A*. A quantitative analysis of the evolution of the properties of system is presented as follows. To quantify the accretion rate at di ff erent spatial scales we calculated the mass flux as a function of both radial distance and time ˙ M ( r , t ) = 4 π r 2 ρ ( r , t ) vr ( r , t ) averaged over a spherical shell. Here, positive and negatives signs in the radial velocity refer to outflow and inflow mass fluxes, respectively. First, we analyse the net mass flux at the innermost radius we can resolve properly, which is equivalent to five times the inner boundary radius, i.e. 5 r in = 5 × 10 -4 pc ≈ 1 . 25 × 10 -2 '' . Figure 3 shows | ˙ M ( r = 5 r in , t ) | as a function of time for models WR_f07 and WR_f1 that are displayed as solid blue and orange lines, respectively. In both cases, at t > -3300 yr the net mass flux reaches levels of ∼ 10 -6 M ⊙ yr -1 with short variability episodes that increase the rate by factors of two to four. During this quasi-steady phase, both simulations display similar behaviour qualitatively, likely determined by the identical stellar wind configuration. This stage lasts until t ≈ -500 yr where the mass flow rates start to deviate from each other. The fiducial model continues to display a variability amplitude of the same order of magnitude. However, the WR_f1 model shows a transition to a strongly gas inflow dominated phase, with inflow rates that are enhanced by a factor four to eight. This stage of the evolution is what we refer to as the disc formation phase, analogously to our previous models reported in Calderón et al. (2020b). Next, we proceed to analyse the time-averaged behaviour of the simulations over the last 500 yr, as this can give us an idea of the general state of the system minimising the e ff ects of the stochastic variability. First, we analyse the (volume-weighted) density and (mass-weighted) temperature radial profiles of the simulations, which are shown on the top and bottom panels in Figure 4, respectively. The dashed blue and solid orange lines represent the simulations WR_f07 and WR_f1, respectively. The orange lines in both panels clearly highlight the presence of the cold disc. The density profile decays with r 2 in both cases at large scales ( ≳ 1 '' ). This is the result of most of the material flowing outwards at these scales following a roughly isotropic spherical wind as seen in previous models (Ressler et al. 2018; Calderón et al. 2020b). However, the profiles di ff er at smaller scales ( < 1 '' ) where the fiducial case transitions to ρ ∝ r -1 , while the model WR_f1 shows a density enhancement due to the presence of the disc. This increase in density is more than one order of magnitude. Regarding the temperature profiles, both models match at larger scales ( ≳ 1 '' ) with a constant temperature of the order of 10 7 K, set by the stellar wind collisions. Again, the profiles differ at smaller scales where the fiducial case follows T ∝ r -1 , and the enhanced cooling case displays a temperature profile about two orders of magnitudes lower on average. The minimum in temperature corresponds to the region where the disc contributes with most of the mass for a given spherical shell. To quantify and characterise the mass inflow and outflow regimes in the simulation domain we calculated them as radial profiles. Figure 5 displays the absolute value of the mass flow rates as a function of distance from Sgr A*. The solid and dashed lines represent the mass inflow and outflow rates, respectively; while the colours follow the same convention as before. Thus, if the solid line is above the dashed line the net mass flow is inwards and vice-versa. Analogous to the model by Ressler et al. (2018), the fiducial model encompasses di ff erent regimes due to the dominant direction of the mass flow at given spatial scales. At r > 3 '' , the outflow component dominates and the inflow is negligible. We find a net mass outflow rate of ∼ 5 × 10 -4 M ⊙ yr -1 . Notice that the enhanced cooling model exhibits exactly the same behaviour at these scales. At smaller scales (1 '' -3 '' ), there is a transition region where | ˙ M in | ∼ | ˙ M out | that corresponds to the location of most mass-losing stars and wind interactions. For model WR_f1, the change is more abrupt due to the presence of the cold disc. In both models, at r < 1 '' the inflow component is larger than the outflow, so the material inflows across these scales and down to the innermost boundary. The net e ff ect makes the mass inflow rate about five times larger which generates the cold disc. The origin of the infalling material can be traced in two ways: analysing the angular momentum of the material close to the inner boundary, and using the scalar tracer fields that we introduced to label the chemical abundances of the di ff erent WR sub-types. Figure 6 shows Hammer projections of the angular momentum direction of the gas enclosed in a sphere of radius 0 . 25 '' ( ∼ 0 . 01 pc) for the fiducial and enhanced cooling runs in the top and bottom panels, respectively. Each point represents a di ff erent simulation time over the last 1000 yr, and they are connected with dashed lines. For reference, we added the angular momentum direction of the orbits of the mass-losing stars as black star symbols, and the location of the clockwise disc based on the orbits we used as a grey shaded circle. This analysis shows that the angular momentum direction of the infalling - - Sun 10 2.0 material varies stochastically but overall tends to align with the orientation of the orbits in the clockwise disc. However, the degree of variability depends on whether or not the model results in the formation of a disc. The fiducial model shows more variability while the enhanced cooling run displays that the angular momentum aligns more consistently with the orientation of the clockwise disc. The fact that the infalling material at this spatial scales comes from these stars agrees with previous numerical models (Cuadra et al. 2008; Ressler et al. 2018; Calderón et al. 2020b). Furthermore, this is not surprising since these stars orbit closer to Sgr A*, and have winds that are relatively slow ( ∼ 600 kms), which results into shocks more prone to radiative cooling and with smaller angular momentum. Overall, the fiducial model displays properties of the medium and gas dynamics consistent with the previous models that do not form a disc (Cuadra et al. 2008; Ressler et al. 2018, 2020; Calderón et al. 2020b, run for 1100 yr). The model with enhanced cooling is consistent with the simulations that show the formation of a cold disc as an outcome of this system (Calderón et al. 2020b). Before discussing if any of the models can be favoured given the current observational constraints we proceed to characterise the cold disc arising in the WR_f1, as its properties will aid us to do so.\",\n \"pages\": [\n 4,\n 5,\n 6,\n 7,\n 8\n ]\n },\n {\n \"title\": \"3.2. Properties of the cold disc\",\n \"content\": \"At the present time ( t = 0), the model WR_f1 shows the presence of a cold disc. Figure 8 shows zoomed maps of the central region of 2 . 5 '' × 2 . 5 '' . The left- and right-hand side panels show projected maps of density and line-of-sight velocity (both weighted by density), respectively. Here it is possible to observe that the projected diameter of the disc is roughly 1 '' . We estimated the mass of the disc by isolating the cold material ( T < 10 5 K) and integrating the cell mass content within a sphere of radius 1.0 '' . We found that the total mass of the disc is 0.005 M ⊙ which is consistent with the value in our previous work (Calderón et al. 2020b). However, the mass accumulated in the disc does not converge and keeps increasing at least for the next hundreds of years in the simulation. On the right-hand side panel of Figure 8, we can see that the line-of-sight velocity peaks at ∼ 2000 km s -1 along both directions (towards and outwards the observed). This coincides with the maximum extension of the observed H30 α line (Murchikova et al. 2019). Additionally, the disc is observed to be ∼ 45 · tilted in projection. This is not in agreement with the observations since, at least in projection the di ff erence is about ∼ 90 · (see Figure 1 of Murchikova et al. 2019).\",\n \"pages\": [\n 8\n ]\n },\n {\n \"title\": \"4. Post-processing of observables\",\n \"content\": \"In order to assess the validity of our models for the Galactic centre we proceed to synthesise observational quantities that we could compare easily with observations. We focus on the recombination lines H30 α and Br γ . Afterwards, we calculate the X-ray spectrum from our models.\",\n \"pages\": [\n 8\n ]\n },\n {\n \"title\": \"4.1. Recombination lines: H30 α and Br γ\",\n \"content\": \"The cold gas at the location of Sgr A* has been observed through the H30 α recombination line (Murchikova et al. 2019; YusefZadeh et al. 2020), and only with an upper limit in the Br γ recombination line (Ciurlo et al. 2021). In order to explore if the models are consistent with these observations we have calculated the expected flux from these emission lines. To compute 1e3 10 10 the emission of the H30 α line we used the coe ffi cients given below fitting a piecewise linear function to them as a function of density and considering a decay ∝ T -1 (see Murchikova et al. 2019, supplementary information), In the case of the Br γ line, its emissivity can be calculated following Schartmann et al. (2015) as Then, in each cell of the domain we computed the respective emissivity values, multiplied them by their n p n e (assuming full ionisation), and then integrated the data cube along the z coordinate which is perpendicular to the line of sight. It is important to remark that in this calculation it is necessary to take into account the hydrogen nucleus n p and electron n e densities appropriately. Since our models consider di ff erent amounts of hydrogen in the plasma we include only the material that contains some, i.e. the material coming from the WN89 / Ofpe stars. Additionally, we only used the hydrogen fraction of that material which is X H = 11 . 5% and X H = 40% for the models WR_f07 and WR_f1. idea of the whole flux of the model we chose a size of p = 1 '' that encloses entirely the disc. The flux values obtained are shown in Table 3 where we also added the upper limit for Br γ from Ciurlo et al. (2021), and the flux of H30 α measured from integrating the spectrum across the velocity channels given by Murchikova et al. (2019). In the case of the Br γ line, we can see that only the model WR_f07 has a flux consistent with the upper limit reported for an aperture of p = 0 . 23 '' . The model WR_f1 has a flux that is ∼ 100 times higher than the upper limit. Also, we see that the disc in our models extends beyond this projected radius, yet its emission is dominated by the inner p = 0 . 23 '' (see Table 3). The situation is more complex for the H30 α line since the observed flux is in both cases higher than the synthetic emission. Although the model that forms a disc is closer it still remains 30% times fainter than the reported value. The model WR_f07 has negligible emission as it is ten order of magnitudes fainter compared to the observation. Thus, it is still not clear how to reconcile the models with both line emission simultaneously as none of them is capable to reproduce the observations. We discuss further the interpretation of these results in Section 7.1.\",\n \"pages\": [\n 8,\n 9\n ]\n },\n {\n \"title\": \"4.2. Spectral X-ray emission\",\n \"content\": \"The X-ray emission from the central parsec is also an observable that has allowed validation of previous numerical models (see Russell et al. 2017; Ressler et al. 2018; Wang et al. 2020). Here we also post-processed our models aiming to obtain the Iron Kalpha spectrum of this region at ∼ 6 . 7 keV as this is the strongest X-ray feature from Sgr A* (e.g. Russell et al. 2017; Corrales et al. 2020). Our approach is based on the procedure outlined in Balakrishnan et al. (2024b) but adapted to our finite-volume grid-based hydrodynamic model, which is briefly described as follows. First, we assigned energy dependent X-ray emissivities j k E = n k e n k i Λ ( E , T k ) that were taken from the vvapec model (Smith et al. 2001), which was obtained with xspec (Arnaud 1996). The densities and temperatures were taken from the last output of our simulation, i.e. at t = 0. Additionally, the tracer fields were used to identify the composition fraction of the parcels of gas according to the WR sub-types. Since the optical depth through the simulation domain is optically thin in the X-ray, the radia- Notes. All fluxes correspond to quantities seen from Earth, i.e. at a distance of 8.33 kpc, and are given in units erg cm -2 s -1 . The observed H30 α flux was calculated integrating the reported flux density across the velocity channels, and taking into consideration the error propagation of the 0.3 mJy reported (see Figure 1 in Murchikova et al. 2019). v th. This function is given as follows tive transfer equation to solve is simply given by the integration along the line of sight of the emissivities of the respective cells. Given that we can trace the fraction of the material that comes from a given WR sub-type this actually corresponds to a linear combination of the tracer field si and the abundance dependent emissivity for each cell, i.e. where ϕ k represents a Gaussian function that it is used to obtain the emission across di ff erent velocity channels, and to take into account the thermal broadening represented with a velocity of We performed calculations through velocity channels spanning line-of-sight velocities v los from -3000 km s -1 to 3000 km s -1 . These were later converted into energy space via Doppler broadening. Thus, for each energy bin from the emissivity tables we obtained 150 line profiles using a fine resolution of 6400 per dex. In order to sum them appropriately we interpolate them into a unique energy range so that we can obtain a single spectrum that includes both the continuum and line contributions. The result of this procedure is shown in Figure 10. This contains the X-ray spectrum emitted from a sky-projected annulus of 0 . 5 '' < p < 1 . 5 '' for models WR_f07 (solid blue line) and WR_f1 (dashed orange line). As a reference, we included the estimation from Balakrishnan et al. (2024b) computed from an analogous model that used the SPH approach, which we discuss in the following section. All spectra show qualitative agreement highlighting the di ff erent features of the Fe complex at 6.7 keV and 6.9 keV. This is also reflected in the broadening of the lines due to the velocity across the line of sight, which is expected since the models are based on the same stellar orbits and wind properties. It is relevant to highlight the level of agreement between the finite-volume and SPH models. The qualitative agreement is reasonable and the level of the emission is of the same order of magnitude. The exact quantitative di ff erence among the models could be mainly attributed to the exact cooling table (composition) used in each simulation, which is the result of the di ff erent density and temperature distributions that can be observed in Figure 4. This is why the WR_f07 shows the highest flux level as it has the least e ffi cient cooling and, as a result it has hotter material. Analogously, the run WR_f1 with an enhanced cooling specified in the composition of its plasma results in a reduction of ∼ 40% of its flux. In case of the SPH model, the quantitative di ff erences could be a ff ected due to the intrinsic ability of the generic SPH to capture shocks which impacts directly the density, velocity structure, and the temperature of the medium. Notice that the flux level of the spectrum of the SPH model by Balakrishnan et al. (2024b) is within the cases explored in this work. Their cooling model is not the same as the ones explored here but this analysis show that it is equivalent to a cooling e ffi -ciency between both of our models. Although not shown here we also compared the X-ray spectra from simulation WR_f1 before and after the disc is formed, and found negligible di ff erences. Finally, we also analysed the slope of the continuum emission within this energy range and ∫ ∫ - 2000 found no significant di ff erences among the models. Thus, in this work the presence of the disc does not imprint a feature in the X-ray spectrum that is resolvable by any current or forthcoming X-ray telescope, in agreement with Balakrishnan et al. (2024b).\",\n \"pages\": [\n 9,\n 10,\n 11\n ]\n },\n {\n \"title\": \"5. Comparison with Lagrangian models\",\n \"content\": \"Hydrodynamic simulations of the feeding of Sgr A* by the WR stellar winds have been conducted with di ff erent numerical tools. In this work, we have presented the results of using a finitevolume approach that solves the hydrodynamic equations in the Eulerian form. However, there has been extensive work on this problem using codes that solve the equations in the Lagrangian form. Specifically, the use of SPH has been the main choice for such a task (Cuadra et al. 2005, 2006, 2008, 2015; Russell et al. 2017; Wang et al. 2020). Despite the limitations of the generic SPH technique (e.g. capturing shocks) the models managed to reproduce the observed accretion rate at the Bondi radius (Cuadra et al. 2008) as well as the X-ray emission (Russell et al. 2017). In this context, we have conducted a comparison of the Eulerian models calculated with Ramses, and the Lagrangian models computed with the SPH code Gadget (Springel 2005). The SPH simulation setup is as similar as it can be to our setup. This corresponds to an updated version from the models in Russell et al. (2017). For more details on the setup of the SPH models, we refer the reader to Balakrishnan et al. (2024a,b). We focus the comparison on the analysis on the final state of the system, specifically in the cold discs formed after 3000 yr. ous structure towards smaller scales while the Lagrangian disc displays a ring-like structure. This feature can be attributed to the spatial resolution and the inner boundary radius. The Eulerian models indeed manage to resolve smaller scales towards the centre, and the radius of the inner boundary is 5 mas while the Lagrangian models resolve ∼ 1 mas and have an inner boundary of 0 . 1 '' . As a result, the Lagrangian model obtained a disc 50 times lighter ( ∼ 10 -4 M ⊙ ) than in the Eulerian simulation. This di ff erence could be attributed to the radiative cooling employed in each simulation: WR sub-type abundance and Solar with 3 Z ⊙ in the Eulerian and the Lagrangian model, respectively. Regarding the projected orientation of the disc both models seem to be in agreement. This can be analysed more carefully on the lineof-sight velocity maps (see right-hand side panels of Figures 8 and 11). Both maps display that the discs indeed match their projected orientation as well as their most blue- and redshifted velocities are roughly -2000 km s -1 and 2000 km s -1 . Overall, the agreement on the outcome a cold disc around Sgr A* whose properties align between two di ff erent approaches indicates that this is a result independent of the numerical technique. Thus, this shows that the determinant factors to obtain this results lies in both the long-enough simulation time ( ≳ 3000 yr ) as well as e ffi cient radiative cooling.\",\n \"pages\": [\n 11\n ]\n },\n {\n \"title\": \"6. Disc stability analysis\",\n \"content\": \"In this section, we present an analysis of the stability of the observed cold disc in the Galactic Centre. Based on its reported properties, the disc should be depleted by accretion after its viscous timescale t ν ≈ 43 kyr, assuming an α = 0 . 1 thin disc (Shakura & Sunyaev 1973), a process much slower than its generation as modelled in our simulations, although faster than its formation out of CND material (Solanki et al. 2023). Nonetheless, there are several external mechanisms that could be capable of destroying the disc faster than through accretion. Among them are the gravitational and hydrodynamic interactions of the disc with the stars, the stellar-wind-disc collisions due to the presence of the S-star cluster in its location, and / or the evaporation due to heat flowing between the hot medium and the cold disc. Although some of these physical scenarios have been studied analytically and numerically, there has not been any direct application to the cold disc discovered in our Galactic Centre.\",\n \"pages\": [\n 11,\n 12\n ]\n },\n {\n \"title\": \"6.1. Stellar cluster perturbing a thin disc\",\n \"content\": \"The stability of a cold disc perturbed by stellar passages has been investigated by Ostriker (1983). Their work presented an analytical approach to calculate how the disc loses angular momentum due to the passages, considering the integrated e ff ect of a stellar cluster in stationary state. Besides this work, most e ff orts have been put on following how the stellar dynamics are a ff ected by the disc presence (e.g., Fabj et al. 2020, and references therein) or on the e ff ect of embedded accreting stars and / or black holes on the disc (e.g., Tagawa et al. 2021) rather than on the consequences to the disc properties. In this Section, we will summarise the approach of Ostriker (1983) and apply it to the Galactic Centre case. Let us start considering a cold, thin accretion disc around a compact object. Within the thin disc we assume that its height H ( r ) is small compared to the radial coordinate r , i.e. H ( r ) / r ≪ 1. Also, we consider that the sound speed of the material in the disc is small compared to the speed of the stars at the same radius. In this scenario, a star passing through the disc will produce a torque that removes angular momentum from the disc. We consider that the star interacts with the disc gravitationally and hydrodynamically The gravitational interaction of a star crossing the disc can be described using the Bondi-Hoyle approach (Bondi & Hoyle 1944), while the hydrodynamic interaction is just a physical collision of a solid sphere with a gaseous disc. Let us consider a system of reference where the x axis is in the direction of rotation and z is in the direction of the rotation pole of the disc. Then, according to Ostriker (1983) the total change of linear momentum along the x direction ∆ px for a single stellar passage is where R ∗ is the radius of the star, Σ d = R disc ρ d H ( r ) is the surface density of the disc, ρ d is the volume density of the disc, q is the hydrodynamic drag parameter that is set to q = 2 assuming a large Mach number collision, v 0 is the local stellar velocity, Λ D ≈ H ( r ) v 2 0 / ( R ∗ v 2 ∗ ) is the Coulomb logarithm, v ∗ is the escape velocity from the surface of the star, η = v ∗ / v 0 is the hardness parameter, v d is the disc velocity, v rel is the relative velocity vector, and ( θ, ϕ ) are the spherical coordinate angles along the polar and azimuthal directions. In order to consider the e ff ect of the complete stellar cluster interacting with the disc we need to integrate over the cluster phase-space volume. First, we integrate over the local stellar velocity distribution. Let us define dN as the number of stars in the velocity range v → v + dv , i.e where f ( v ) is the velocity distribution function, n ∗ is the stellar number density. Then, the number of crossings per unit time and area in the ( θ, ϕ ) direction is Article number, page 12 of 19 Now, integrating over the phase space, the total transfer of momentum to the disc per unit area at r is where µ = cos ϕ . As the local momentum per unit area in the disc is px , d( r ) = Σ ( r ) /τ d( r ) the inverse of the drag timescale to remove the angular momentum of the disc is τ d = -px , d / ˙ px , tot can be written as where the quantities ( I 0 , I 1) depend solely on the stellar distribution function. For instance, in the case of a Maxwellian distribution the values of these parameters can be obtained numerically and are I 0 = 12 . 553 and I 1 = 0 . 474. In order to apply this model to the Galactic Centre we need to calculate the stellar density within the size of the disc. Following Schödel et al. (2007) the stellar mass density at r < 6 arcsec is described by Assuming that the stellar mass density is composed mainly by stars whose radii are R ∗ = 1 R ⊙ and move on their orbits typically at v 0 = 1000 km s -1 we can use Equation 20 to estimate the inverse of the drag timescale, Based on this calculation the angular momentum of the cold disc should be removed after ∼ 185 Myr of interactions of the stars. This is much longer than the formation time-scale found in our simulations. Furthermore, it is even much longer than the age of the Wolf-Rayet stars, which is about 6 Myr (Martins et al. 2007). Then, the gravitational and hydrodynamic interactions of the stars onto the disc should not a ff ect the stability of the disc.\",\n \"pages\": [\n 12\n ]\n },\n {\n \"title\": \"6.2. Stellar winds perturbing a disc\",\n \"content\": \"In the case of stellar irradiation a ff ecting the state of the accretion flow there have been many studies mainly motivated by the recent pericentre passages of the star S2 around Sgr A*. For instance, Cuadra et al. (2003) could rule out the existence of an optically-thick disc based on the lack of its thermallyreprocessed emission as it gets illuminated by S2. Nayakshin et al. (2004) estimated that a star as luminous as S2 could heat up and ionise an inner disc, which could enhance the accretion rate. Giannios & Sironi (2013); Christie et al. (2016) calculated the bremsstrahlung emission as a result of the stellar wind shocking a Radiatively Ine ffi cient Accretion Flow (RIAF) around Sgr A*, finding an X-ray luminosity comparable to quiescent emission of Sgr A*. In fact, no noticeable increase was detected during 2018 (see Table 1 of Andrés et al. 2022). Hosseini et al. (2020) constrained the observed L ' -band variability of S2 to be about 2-3%, based on a model of bow shock of its stellar wind, implying an ambient density < 10 5 cm -3 , which rules out the presence of a standard accretion disc at S2's pericentre, in agreement with the previous studies. It is important to remark however that such a limit marginally allows the existence of both the disc reported by Murchikova et al. (2019), and also the one in our simulation. The stars in the S cluster also have relatively powerful stellar winds. As the stars inhabit the black hole in the vicinity of the cold disc, it is expected that the winds interact with it. By either opening a bubble due to the high kinetic energy carried in the winds or depositing thermal energy they a ff ect the disc properties. We proceed to perform analytical estimates to quantify the impact of these processes.\",\n \"pages\": [\n 12,\n 13\n ]\n },\n {\n \"title\": \"6.2.1. Bubbles in the disc\",\n \"content\": \"Let us consider a star blowing an isotropic stellar wind with a mass-loss rate ˙ M w at a terminal speed v w. Then, its density ρ w at a distance r from the star is given by If such a star is immersed in a medium whose number density is n m at rest at a temperature T m, the radius of the bubble r b opened by the wind can be calculated equating the ram pressure of the wind and the thermal energy of the medium, i.e. where we have used as fiducial values the disc parameters measured by Murchikova et al. (2019) and typical stellar wind properties for B stars (Oskinova et al. 2011; Krtiˇcka 2014). Then, the radius of the bubble is about 0.012 arcsec, which is approximately one tenth of the disc radius. In order to open such a bubble a star would need to be within the disc fort at least the wind crossing time, i.e. r b / v w, which is about ∼ 0 . 5 yr. Let us estimate the duration of a stellar passage through the disc. Along the perpendicular direction a star crossing the disc would take at most 0 . 1 R d / v 0, being H ( r ) / r = 0 . 1. One S star moves typically at v 0 ≈ 1000 km s -1 , then 0 . 1 R d / v 0 ≈ 1 yr. As both timescales are comparable, we conclude that the bubble can be opened. For such a perturbation to last we need to check if either shear or sound waves could eliminate it, i.e. close the bubble. In the case of shear, this is given by the orbital speed of the disc and the size of the bubble: As this expression increases with r , the largest value will be at the outer edge of the disc, then t close < 5 yr. This value should be interpreted as the longest duration of a perturbation of the wind onto the disc. As there are 12 S-stars on orbits whose pericentre distances are shorter than the radius of the disc and their orbital periods are of the order of ∼ 10 yr (Gillessen et al. 2017) we expect a bubble to be opened every year on average. However, as shear would close such bubbles in less than five years there would be at most four bubbles opened on average in a stationary state. As the size of the bubbles is very small compared to the size of the disc (two orders of magnitude in area), four of them would not have an impact on its structure. The sound crossing timescale r b / c s ≈ 30 yr is longer than the shear timescale so we do not expect it to have an impact on erasing perturbations.\",\n \"pages\": [\n 13\n ]\n },\n {\n \"title\": \"6.2.2. Winds injecting thermal energy\",\n \"content\": \"The S stars are of spectral type B with masses of 10 M ⊙ (e.g. Habibi et al. 2017), thus their mass-loss rate must be ∼ 10 -8 M ⊙ yr -1 with terminal velocity of 1000 km s -1 . The supersonic nature of the winds should be enough to compress the shocked material at high temperature. Potentially, this thermal energy could be deposited onto the disc during each stellar passage. If we consider that most of the kinetic energy of the wind is transformed into thermal energy about ∼ 6 . 4 × 10 33 erg would be deposited in the disc during the ∼ 1 yr long passage (see above). In order to check if the wind energy can significantly impact the thermodynamic state of the disc let us estimate its total internal energy U int. where k B is the Boltzmann constant and V d is the disc volume. Replacing the observed disc properties we obtain that U int = 1 . 63 × 10 42 erg. From this calculation it is clear that even if all the energy from the wind in a stellar passage is deposited the contribution is still negligible to modify the state of the disc. Even if we consider the integrated e ff ect of all the S stars passages on a timescale comparable to the viscous timescale of the disc ( ∼ 43 kyr) the contribution is still < 0 . 1% of the total thermal energy of the disc. Thus, the energy supplied by the stellar winds into the disc is not enough to a ff ect its state.\",\n \"pages\": [\n 13\n ]\n },\n {\n \"title\": \"6.3. Thermal conduction\",\n \"content\": \"The vicinity of Sgr A* at the Bondi radius is made out of a diffuse (10 cm -3 ), hot medium (10 7 K; Bagano ff et al. 2003) due to the shocked stellar winds from the Wolf-Rayet stars in the region. At the disc vicinity ( ∼ 10 3 R Sch), the conditions are more extreme as the plasma has a density of about ∼ 5 × 10 3 cm -3 (Gillessen et al. 2019) and temperature of ∼ 10 8 K. As the disc is significantly colder it is expected that heat flows from the medium to the disc. The large temperature gradient should enable thermal conduction to take place and likely evaporate the disc. The problem of evaporating a cold structure sitting in a hot environment due to thermal conduction was first studied by Cowie & McKee (1977). In that work, the authors derived an analytical expression for the mass-loss rate and evaporation timescale for a spherically symmetric gas cloud. Unfortunately, these expressions are not valid for more complex geometries. Meyer & Meyer-Hofmeister (1994) studied specifically the evaporation of a cold thin accretion disc in a hot corona. Although the model considered only one radial zone it was able to calculate the full perpendicular structure of the accretion disc and the corona by means of numerical simulations. Liu et al. (2004) used the model by Meyer & MeyerHofmeister (1994) to analyse which solutions were consistent with the observational data available at the time in case there were a cold disc in the center of the Galaxy. They found that any transient disc would evaporate quickly. With their obtained evaporation rate of ∼ 10 -4 M ⊙ yr -1 , the life-time of a disc with properties such as found by Murchikova et al. (2019) or formed in our models would be ∼ 1 yr. Similar analysis have argued that small clumps ( ∼ M ⊕ ) in the Galactic Centre should also evaporate quickly ( ∼ 1 yr) due to thermal conduction (Burkert et al. 2012; Calderón et al. 2018). According to this, cold gas should not be able to last long if formed, yet we observe many cold gas structures in the central parsec: the many dusty cold clumps detected in the vicinity of IRS 13E (Fritz et al. 2010; Peißker et al. 2023), and a larger gaseous cloud X7 (Ciurlo et al. 2023). Thus, this regime of classical thermal conduction might not be applicable to the Galactic centre environment. Another possibility has been studied by Nayakshin (2004) who focused on the same problem and found solutions in a di ff erent regime, where the electron mean free path is long enough that an accretion disc does not evaporate but rather increases its mass by condensation. A more thorough analysis of the role of thermal conduction is beyond the scope of this work and is deferred to future study. Our findings shows that gravitational and hydrodynamic effects require timescales that are too long to have an impact on the disc stability. On the other hand, we find that thermal conduction in the classical case should evaporate the observed disc in about one year. Given the time-scale for disc formation found in our model, the Nayakshin (2004) condensation regime remains to our knowledge as the only viable physical scenario that would allow its continued existence and potential identification with Murchikova et al. (2019)'s reported disc.\",\n \"pages\": [\n 13,\n 14\n ]\n },\n {\n \"title\": \"7. Discussion\",\n \"content\": \"In this section, we interpret our findings exploring the uncertainties in the abundances of WR stars in general and of the ones in the central parsec. Additionally, we contrast the properties of the simulated and observed cold discs as well as discuss further implications.\",\n \"pages\": [\n 14\n ]\n },\n {\n \"title\": \"7.1. Is the simulated disc the observed disc?\",\n \"content\": \"In order to compare the simulated and the observed discs, first we analyse their physical properties. The simulated disc has a total mass of ∼ 5 × 10 -3 M ⊙ , while the structure observed in H30 α was reported to have a mass of 10 -5 -10 -4 M ⊙ . However, comparing these quantities might not be appropriate since the mass of the observed structure is based on three assumptions: a thin disc with uniform density and temperature, and a masing factor of ∼ 100. All of them minimise the value of the mass inferred from the observations. For instance, dropping the maser assumption the inferred mass increases in a factor ten (see Equations 30-31 of Supplementary Material in Murchikova et al. 2019). Bearing this in mind, we could argue that the mass of the simulated and observed disc might be of the same order provided there was no maser. Although removing that assumption would create tension reconciling both the Br γ and H30 α observed fluxes. Regarding its extension, the simulated disc has a projected radius of ∼ 0 . 5 '' , while the observed disc has a radius of 0 . 23 '' . However, since most of the recombination line emission of the disc is originated from its central region ( p < 0 . 23 '' , see Table 3) the sizes of the simulated and observed discs are in agreement. A more direct comparison can be done through the analysis of the mock observational quantities computed from the simulations. In Section 4.1, we computed the H30 α and Br γ recombination line emission and estimated their fluxes. Here, we have found that both synthetic fluxes are in tension with the observations. First, the simulated Br γ flux is 100 times higher than the upper limit. It is relevant to remark that this limit is already extinction corrected. Second, the simulated H30 α flux is 30% lower than the observed one. These results di ff er from the estimates by Ciurlo et al. (2021) from our previous models due to, again, the uniform disc assumptions. The simulated disc is not thin or possess uniform density or temperature. Although the mean density and temperature of the simulated disc is of the order of ∼ 10 -5 cm -3 and ∼ 10 4 K its geometry is more complex with a larger variance in its properties. As a result, both the synthetic Br γ and H30 α fluxes are much higher than the ones inferred by Ciurlo et al. (2021). Based on the line-of-sight velocity maps shown in Murchikova et al. (2019), the orientation of the observed disc displays the red- and blueshifted sides to be on the East (lefthand) and West (right-hand) sides, respectively. Although our model WR_f1 also shows this configuration the exact inclination of the discs does not match perfectly. Specifically, the discs are inclined in ∼ 90 · among each other, being the simulated disc tilted in the clockwise direction on the sky. In principle, this argues against the WR stars as the main responsible for the formation of the observed disc. But we cannot rule out completely their role in the process, since a combination of the WR stars with other gaseous structures, such as the minispiral or the CND, acting simultaneously might result in a net e ff ect that manages to reproduce the observed angular momentum. Solanki et al. (2023) showed that material from the circumnuclear disc could fall close to Sgr A* though on much longer timescales. Certainly, this scenario would be much more di ffi cult and challenging to simulate with the appropriate chemical abundances as well as with high-enough spatial resolution. It is also relevant to notice that the observed accretion flow orientation at ≲ 10 RSch is consistent with the orientation of the clockwise stellar disc (GRAVITY Collaboration et al. 2020; Event Horizon Telescope Collaboration et al. 2022). Thus, our models reproduce that connection between the dynamics from large ( > 10 5 RSch) to small ( ≲ 10 RSch) scales. However, the observed cold disc does not seem to follow such a connection. Finally, an intriguing aspect of our findings is that the WR atmospheres with more H in them would favour the formation of the disc. At the same time, the disc indeed was detected in the H30 α recombination line, which obviously indicates the presence of H. Thus, it is sensible to ask how much H can be in the winds of WR stars but, specifically in the ones of the sub-type WN89 / Ofpe. This point is discussed in the following section.\",\n \"pages\": [\n 14\n ]\n },\n {\n \"title\": \"7.2. Uncertainties in abundances\",\n \"content\": \"The results of the hydrodynamic models demonstrate that the chemical abundances of the material in the WR winds are the key factor to determining whether or not a cold disc can be formed due to their action. Unfortunately, there are two large uncertainties in this context. First, the metallicity of the young population of the nuclear star cluster is not well constrained (e.g. Genzel et al. 2010). Although there is agreement that it is above Solar its exact value has not been established so far, with most studies quoting the range Z = 2 -3 Z ⊙ . However, even if the metallicity is determined more precisely, the problem would not be entirely solved, since the abundances of the plasma in the region are dominated by the material supplied by the WR star winds whose compositions are much more uncertain. Theoretically, the evolution of these stars is a topic of active research, and improved prescriptions for mass loss are changing the modelled properties of observed WR stars (see e.g., Gormaz-Matamala et al. 2023, and references therein). Simply based on observations, up to now, it has been reported that some of these stars might possess from X H = 0 to X H = 50% for Galactic WR stars of the WN subtype (Hamann et al. 2006, 2019), and even as high as X H = 60% for extragalactic ones (Todt et al. 2015). In the vicinity of Sgr A*, Martins et al. (2007) characterised the wind properties of the sample of WR stars through the analysis of their infrared spectra. They reported abundance H / He ratios in the range 2-5 for these stars which does not allow us to rule out the scenario regarding this aspect. A variation on the hydrogen abundance in the chemical mixture in a factor five might change the net radiative cooling rate by a factor 1 . 1 -1 . 4, especially at lower temperatures. In our models, WR_f07 considered a fiducial value of hydrogen mass fraction of 11.5% (Russell et al. 2017) but the model WR_f1 assumed 40%. Such a change was enough to modify the cooling function by about 30%, which ended up affecting significantly the final result and the state of the plasma. Thus, within our current understanding of the abundances in the WR stellar atmospheres both scenario are plausible. Only better quality spectra and more sophisticated models of them could allow us to constrain the wind properties and their composition more precisely.\",\n \"pages\": [\n 14,\n 15\n ]\n },\n {\n \"title\": \"8. Conclusions\",\n \"content\": \"We have presented a numerical study of the system of masslosing stars feeding Sgr A* focusing on the impact of the chemical composition of the plasma. Through studying simulations with di ff erent abundances we have found that the system can either form or not form a cold disc around the black hole, provided that the model is run for long-enough timescales ( ≳ 3000 yr) to create a dense enough medium in the inner parsec of the Galaxy. As in our past work, we have confirmed that if formed the cold disc can impact significantly the hydrodynamic and thermodynamic state of the plasma within the central parsec. As a result, the inferred mass in flow rate at 5 × 10 -4 pc can reach up to 8 × 10 -6 M ⊙ yr -1 , which is 4 -8 times higher than the value without the presence of such a disc. Our models still point to the mass-losing stars in the clockwise disc being the main source of the accreted material due to their dense and slow winds as previous models have shown (e.g. Cuadra et al. 2008; Ressler et al. 2018; Calderón et al. 2020b). In this work, we have followed more carefully the origin of the inflowing material through the use of passive scalar fields for di ff erent WR subtypes. This allowed us to confirm the origin of the accreted material but, more importantly led us to conclude that this is mostly provided by the winds of the stars of the WR sub-type WN89 / Ofpe. This provides evidence that the infalling material must have a nonnegligible fraction of hydrogen as these WR stars have not lost entirely their atmospheric hydrogen via winds. Our models also showed that the formation of a cold disc depends strictly on the radiative cooling employed, which is determined by the chemical abundances of the plasma. Previous models have only considered Solar abundances with 3 Z ⊙ but had not agreed whether or not a cold disc can be formed. From our results in Appendix B, we speculate that at Z ≈ 3 Z ⊙ there is a transition between the regimes of persistent quasi-steady accretion and disc formation, which would imply that small di ff erences in the cooling function or numerical implementation can result in models reproducing either regime. This may explain the di ff ering results obtained by Calderón et al. (2020b) and Ressler et al. (2020) with similar numerical and physical setups. In the case a cold disc is formed around Sgr A* the structure may resemble the extension and the line-of-sight velocity structure of the observed disc. However, direct observational quantities could not be reproduced. Specifically, the exact skyprojected inclination of the simulated and observed discs di ff ers in ∼ 90 · . The Br γ emission line upper limit is 100 lower than the synthetic flux. The observed H30 α emission line flux is 30% higher than in our model. Furthermore, we have contrasted this work Eulerian (finitevolume grid-based) models with analogous (SPH) Lagrangian models in order to address the potential influence of the chosen approach in the outcome of the simulations. As it has been shown in Balakrishnan et al. (2024b), a cold disc is also formed around Sgr A* if the simulation is run for long timescales. Both cold structures agree on their sizes, their velocity structure, and their sky-projected inclination. Although there are di ff erences in the total mass and specific inner structure of the discs this could be attributed to the resolution employed and the exact assumption to consider the innermost boundary. Despite these di ff erences, the agreement is reasonable among the numerical approaches, especially when analysing the synthetic X-ray emission (see Figure 10). The spectral features agree qualitatively across the models presented in this work and the Lagrangian model. The level of the X-ray emission are all within the same order of magnitude but the exact base level depends on the radiative cooling chosen, which is determined by the chemical abundances in the WR atmospheres. We also have explored the stability and long-term evolution of the putative disc under the e ff ect of perturbing agents: the nuclear star cluster, the wind of all the mass-losing stars (including the S stars) impacting the disc through opening bubbles and / or depositing thermal energy via shocks, and the potential e ff ect of thermal conduction. Our analysis showed that gravitational and hydrodynamic e ff ects would impact the disc on longer timescales then the viscous timescale. However, within the classical thermal conduction framework a cold disc in such a region should not exist. Based on this and the disc formation timescale from our simulations, we speculate that the condensation model of Nayakshin et al. (2004) remains a plausible mechanism consistent with the disc existence. In conclusion, through this work we have been able to reconcile the discrepancy among the numerical simulations of the system of mass-losing stars feeding Sgr A*. The formation of a cold disc on ∼ 3000 yr timescale is indeed possible for certain chemical abundances which are consistent with the current observational constraints. Nonetheless, it is not possible to favour or disfavour this scenario due to the uncertainties in them. Thus, high-resolution spectra of the WR stars and more sophisticated modelling of them could allow us to discern the validity of such scenarios. But we must warn that even in the case that the cold disc is a result of action of the WR stars it is not possible to reproduce all its observed properties: its inclination and recombination line fluxes. More sophisticated models that include infalling material from other sources might be necessary to produce the exact inclination of the structure which could also impact the emission of the structure. Acknowledgements. We thank Prof. Sergei Nayakshin and Prof. Stanley Owocki for very helpful discussions about this project. We also thank Dr. Sean M. Ressler for sharing the radiative cooling function shown in Figure 1. Additionally, we are grateful to Dr. Álex Gormaz-Matamala for helping with references and discussions on the WR wind abundances. DC and JC acknowledge the support of the Kavli Foundation through its summer program at the Max Planck Institute for Astrophysics where fruitful discussions took place. In addition, DC thanks the warm hospitality of the Universidad Adolfo Ibañez in Chile and University of Delaware in the USA where part of this project was developed. We acknowledge the support from ANID in Chile through FONDECYT Regular grant 1211429 and Millennium Science Initiative Program NCN2023_002. The research of DC and SR was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - EXC 2121 - 'Quantum Universe' - 390833306. Since 01.10.2024, DC has been funded by the Alexander von Humboldt Foundation. SR is also supported by the European Research Council (ERC) Advanced Grant INSPIRATION under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 101053985), the Swedish Research Council (VR) under grant number 202005044, and the research environment grant 'Gravitational Radiation and Electromagnetic Astrophysical Transients' (GREAT) funded by the Swedish Research Council (VR) under Dnr 2016-06012, by the Knut and Alice Wallenberg Foundation under grant Dnr. KAW 2019.0112. The numerical simulations of this work were run on the high-performance computing system cobra of the Max Planck Computing and Data Facility. Most of the analysis of the numerical data was carried out using the python package yt (Turk et al. 2011). Furthermore, this work made use of python libraries numpy (Harris et al. 2020) and matplotlib (Hunter 2007), as well as of the NASA's Astrophysics Data System.\",\n \"pages\": [\n 15,\n 16\n ]\n },\n {\n \"title\": \"References\",\n \"content\": \"Andrés, A., van den Eijnden, J., Degenaar, N., et al. 2022, MNRAS, 510, 2851 Arnaud, K. A. 1996, in Astronomical Society of the Pacific Conference Series, Vol. 101, Astronomical Data Analysis Software and Systems V, ed. G. H. Jacoby & J. Barnes, 17 Becklin, E. E., Gatley, I., & Werner, M. W. 1982, ApJ, 258, 135 GRAVITY Collaboration, Abuter, R., Amorim, A., et al. 2021, A&A, 647, A59 GRAVITY Collaboration, Bauböck, M., Dexter, J., et al. 2020, A&A, 635, A143 Hosseini, S. E., Zajaˇcek, M., Eckart, A., Sabha, N. B., & Labadie, L. 2020, A&A, 644, A105 Article number, page 16 of 19 Russell, C. M. P., Wang, Q. D., & Cuadra, J. 2017, MNRAS, 464, 4958 Schartmann, M., Ballone, A., Burkert, A., et al. 2015, ApJ, 811, 155 Practical Introduction (Berlin: Springer) Toro, E. F., Spruce, M., & Speares, W. 1994, Shock Waves, 4, 25 Turk, M. J., Smith, B. D., Oishi, J. S., et al. 2011, ApJS, 192, 9 von Fellenberg, S. D., Gillessen, S., Stadler, J., et al. 2022, ApJ, 932, L6 Štofanová, L., Kaastra, J., Mehdipour, M., & de Plaa, J. 2021, A&A, 655, A2 Wang, Q. D., Li, J., Russell, C. M. P., & Cuadra, J. 2020, MNRAS, 492, 2481 Yelda, S., Ghez, A. M., Lu, J. R., et al. 2014, ApJ, 783, 131 Yuan, F. & Narayan, R. 2014, ARA&A, 52, 529\",\n \"pages\": [\n 16\n ]\n },\n {\n \"title\": \"Appendix A: Numerical setup choices\",\n \"content\": \"Calderón et al. (2020b) simulated the system of WR stars feeding Sgr A* while orbiting around it following an analogous numerical setup compared to the models presented in this work. There, we had found that the resulting cold disc tended to align with the grid after its formation. In order to remedy this numerical artifact we investigated the use of a MinMod slope limiter, instead of MonCen. However, we warned that it was not straightforward to maintain the spherical symmetry of the winds using the MinMod slope limiter. In this work, we investigated more carefully these numerical choices in order to be confident that our results are robust. First, we investigated the choice of the MonCen slope limiter. To do this, we made use of the same setup presented in this work but paying attention to the values of the passive scalar fields. In principle, the values of these fields should be between zero and one. However, this experiment showed that the MonCen slope limiter produced non-physical oscillations that in some cases resulted in negative values of the passive scalar fields. This behaviour was not observed when using the MinMod slope limiter. Thus, we selected this choice throughout this work. To ensure the spherical symmetry of the stellar winds we had to impose a less restrictive refinement criterion based on density gradients as well as a smoother transition between refinement levels through the parameter nexpand . Second, we tested the choice of the Riemann solver employed but using the MinMod slope limiter. Currently, the HLLC Riemann solver (Toro et al. 1994) is widely used in grid-based hydrodynamic simulations. In this work, we have found that this choice results in a cold disc that tends to align with the Cartesian grid as seen in Calderón et al. (2020b). This could indicate problems in conserving the angular momentum which is a known issue with simulations in Cartesian grids when the relevant regions are not well resolved. With the usage of the exact Riemann solver (Toro 2009), though more computationally expensive, it was possible to avoid this artificial alignment. Thus, the only sensible combination was the use of the exact Riemann solver with a MinMod slope limiter but we had to ensure to count with enough resolution and smoothness in the transition of the refinement levels. As a reference of these experiments, Figure A.1 shows density projection (weighted by density) maps of the final state of the system for two runs with Solar composition with 3 Z ⊙ but with di ff erent Riemann solvers. The left- and righthand side panels display the models using the HLLC and the exact Riemann solvers, respectively. Although overall the density structure looks similar there are subtle di ff erences. The most relevant is related to the disc orientation and structure. On the left-hand side, the disc tries to align with one of the Cartesian axes, while on the right-hand side the disc remains consistently with the shown orientation.\",\n \"pages\": [\n 17\n ]\n },\n {\n \"title\": \"Appendix B: Solar composition varying metallicity\",\n \"content\": \"Before investigating the models with the new implementation of the di ff erential cooling we ran some experiments simply modifying the contribution of the metals to the total radiative cooling. We simulated three models varying only the composition used: A1 used Solar composition, A3 used Solar composition with 3 Z ⊙ , and A5 used Solar composition with 5 Z ⊙ . The density projected (weighted by density) maps of the final state of the simulations are shown in Figure B.1. The left-hand side, central, and right-hand side panels display the runs A1, A3, and A5, respectively. A higher metallicity in the plasma increases the radiative cooling directly. As a result, more and denser filaments, clumps, and a denser central disc are obtained with increasing the metallicity. On the contrary, if the metallicity is decreased to a Solar value we observe almost no overdensities, and definitively no disc formation. We speculate that at Z ≈ 3 Z ⊙ there is a transition between the regimes of persistent quasi-steady accretion and disc formation, which would imply that small di ff erences in the cooling function or numerical implementation can result in models reproducing either regime. This may explain the di ff ering results obtained by Calderón et al. (2020b) and Ressler et al. (2020) with similar numerical and physical setups. In order to perform a more quantitative comparison of these results Figure B.2 presents radial profiles of density (weighted by volume) and temperature (weighted by mass) on the top and bottom panels, respectively. The dashed blue, solid orange, and blue dotted-dashed lines represent the models A1, A3, and A5, respectively. The density profiles clearly shows the impact of cooling in the structure of the medium. More e ffi cient cooling, like in cases A3 and A5, results in an order of magnitude denser medium within 1 '' compared to the case with ine ffi cient cooling, i.e. A1. Notice that even higher metallicity in A5 extends the impact the denser region to slightly larger spatial scales. The e ff ect is similar in the temperature profile. In A1, the temperature profile decays with r within the central 1 '' . But when the metallicity is high enough to make cooling e ffi cient the temperature profile can decrease up to two orders of magnitude depending on the exact value. In summary, these experiments allowed us to quantify the impact of radiative cooling by simply decreasing or increasing the radiative cooling influence. However, since the chemical composition in reality is much more complicated and di ff ers from Solar composition significantly these models should be interpreted only as general guides to assess the role of radiative cooling. In the main text of this work, we devoted to explore more physically motivated mixtures that, although depend on more uncertain parameters are more robust when attempting to build a physical model that can be constrasted with observational data.\",\n \"pages\": [\n 17\n ]\n }\n]"}}},{"rowIdx":4970,"cells":{"bibcode":{"kind":"string","value":"2024arXiv241106902L"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/2411.06902.pdf"},"content":{"kind":"string","value":"\nA Systematic Search for Candidate Supermassive Black Hole Binaries Using Periodic Mid-Infrared Light Curves of Active Galactic Nuclei\n\n\n\n1 CAS Key Laboratory for Research in Galaxies and Cosmology, Department of Astronomy, University of Science and Technology of China, Hefei, 230026, China; jnac@ustc.edu.cn\n2 School of Physical Sciences, University of Science and Technology of China, Hefei, 230026, China\n3 School of Astronomy and Space Sciences, University of Science and Technology of China, Hefei, 230026, China\n4 Department of Astronomy, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA\n5\nCenter for Artificial Intelligence Innovation, University of Illinois at Urbana-Champaign, 1205 West Clark Street, Urbana, IL 61801, USA\nNational Center for Supercomputing Applications, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA 6\n(Received 2024 May 24; Revised 2024 November 9; Accepted 2024 November 11)\nSubmitted to ApJ\nABSTRACT\nPeriodic variability in active galactic nuclei (AGNs) is a promising method for studying sub-parsec supermassive black hole binaries (SMBHBs), which are a challenging detection target. While extensive searches have been made in the optical, X-ray and gamma-ray bands, systematic infrared (IR) studies remain limited. Using data from the Wide-field Infrared Survey Explorer (WISE), which provides unique decade-long mid-IR light curves with a six-month cadence, we have conducted the first systematic search for SMBHB candidates based on IR periodicity. Analyzing a parent sample of 48,932 objects selected from about half a million AGNs, we have identified 28 candidate periodic AGNs with periods ranging from 1,268 to 2,437 days (in the observer frame) by fitting their WISE light curves with sinusoidal functions. However, our mock simulation of the parent sample indicates that stochastic variability can actually produce a similar number of periodic sources, underscoring the difficulty in robustly identifying real periodic signals with WISE light curves, given their current sampling. Notably, we found no overlap between our sample and optical periodic sources, which can be explained by a distinct preference for certain periods due to selection bias. By combining archived data from different surveys, we have identified SDSS J140336.43+174136.1 as a candidate exhibiting periodic behavior in both optical and IR bands, a phenomenon that warrants further validation through observational tests. Our results highlight the potential of IR time-domain surveys, including future missions such as the Nancy GraceRoman Space Telescope, for identifying periodic AGNs, but complementary tests are still needed to determine their physical origins such as SMBHBs.\nKeywords: Active galactic nuclei (16); Infrared astronomy(786); Quasars (1319); Supermassive black holes (1663); Time domain astronomy (2109)\n1. INTRODUCTION\nSupermassive black holes (SMBHs) are intriguing celestial objects that reside at the centers of galaxies (Kormendy & Ho 2013). When galaxies merge, it is anticipated that the central black holes will form binary systems known as SMBH binaries (SMBHBs, Begelman et al. 1980). These binary systems offer valuable insights into various astrophysical phenomena, including galaxy formation (Colpi & Dotti 2011), gravitational wave (GW) emission (Hughes 2009), and the evolution of SMBHs (Merritt 2013). More massive binaries are pulsar-timing array (PTA) sources (e.g., Arzoumanian et al. 2018), while less massive binaries are targeted by\nspace-based experiments such as LISA (Klein et al. 2016). They provide a laboratory to directly test strong-field general relativity (Hughes 2009; Centrella et al. 2010).\nHowever, despite their theoretical significance, the direct electromagnetic detection of close SMBHBs below subparsec separation has proven to be a challenging endeavor (see recent reviews such as De Rosa et al. 2019; D'Orazio & Charisi 2023) while numerous SMBH pairs at larger scales have been identified through the presence of dual active galactic nuclei (AGNs) (e.g., Komossa et al. 2003; Zhou et al. 2004; Liu et al. 2010, 2011, 2013; Koss et al. 2011; Liu et al. 2018; Chen et al. 2022). When a binary has exhausted its in-\nteractions with stars but has not approached close enough to emit significant gravitational radiation, its orbit does not have an obvious mechanism for decay. This long-standing challenge presents a major uncertainty in estimating the abundance of SMBHB mergers as GW sources. In theory, the bottleneck may be overcome in gaseous environments (e.g., Gould & Rix 2000; Cuadra et al. 2009; Chapon et al. 2013; del Valle et al. 2015), in triaxial or axisymmetric galaxies (e.g., Khan et al. 2016; Kelley et al. 2017), and/or by interacting with a third BH in hierarchical mergers (e.g., Blaes et al. 2002; Kulkarni & Loeb 2012; Bonetti et al. 2018).\nA recent and promising approach for identifying potential close SMBHB candidates has emerged in the thriving field of time-domain astronomy, which focuses on analyzing the periodic light curves of AGNs. The periodicity observed in these light curves can arise from changes in the accretion rate onto the black holes (Graham et al. 2015b) or from the relativistic Doppler boost resulting from the highly relativistic motion of gas in the mini accretion disk around the smaller black hole in the binary system (D'Orazio et al. 2015). Extensive research has been conducted using optical light curves to search for SMBHBs, and the methods employed in these studies have become well-established (Graham et al. 2015a; Charisi et al. 2016; Zheng et al. 2016; Liu et al. 2019b; Chen et al. 2020; Liao et al. 2021; Chen et al. 2024). However, many of the known candidates have been shown to be subject to false positives due to stochastic quasar variability (e.g., Vaughan et al. 2016). Additionally, recent findings suggest that some AGNs may display periodic variations only at some specific epochs (e.g., Jiang et al. 2022; O'Neill et al. 2022) further enhance the difficulties of periodic searches. Furthermore, previous surveys were only sensitive to the most massive quasars at high redshift ( z ≳ 2) which should have already gone through their major merger process (e.g., Shen 2009). Regardless of the process that imprints periodicity, it is generally accepted that the timescale is primarily determined by the binary orbital period. SMBHBs with masses between 10 6 -10 10 M ⊙ and separations of 0.01 pc possess orbital periods that span from years to several decades, making them possibly detectable by current timedomain surveys. In addition to the optical band, the periodic searches have also been conducted in other wavelength regimes, such as in X-ray (Serafinelli et al. 2020; Liu et al. 2020) and gamma-ray bands (Sandrinelli et al. 2018; Holgado et al. 2018).\nPeriodic infrared (IR) light curves are also expected as a result of reverberation mapping of the circumbinary dusty torus in periodic AGNs. However, the search for SMBHBs using IR data has not been extensively explored, despite the detection of an IR time lag in an individual SMBHB candidate (Jun et al. 2015), which highlights the potential of utilizing periodic IR light curves for SMBHB identifi-\ncation (D'Orazio & Haiman 2017). The IR band not only provides an alternative approach to searching for SMBHBs but also offers distinct advantages. Firstly, IR photons are less susceptible to dust extinction, allowing them to penetrate through obscuring material, which is often abundant in galactic nuclear regions and galaxy merger systems. By extending the search to the IR wavelength range, we can uncover a population of SMBHB candidates that may have been previously overlooked, thereby expanding our knowledge of the prevalence and characteristics of these elusive systems. Even for unobscured AGNs, the variability amplitudes in the mid-IR band are typically larger compared to the optical band due to lower background contamination. Additionally, as discussed in D'Orazio & Haiman (2017), the dust-echo model can assist in determining the origin of central periodicity and place constraints on the physical characteristics of SMBHBs and their dust torus. To address this, we perform the first systematic search for SMBHB candidates via periodic IR light curves. The data we use come from the Widefield Infrared Survey Explorer (WISE, Wright et al. 2010) and its successor NEOWISE (Mainzer et al. 2014), which have provided us with a public dataset from February 2010 to December 2023 with a half-year cadence except for a 2.5year gap for each object. In this study, we primarily focus on sinusoidal periodic signals, which have been commonly adopted in modeling the optical light curves of SMBHB candidates (e.g., D'Orazio et al. 2015), and on modeling the IR light curves using the dust-echo model (D'Orazio & Haiman 2017).\nThe paper is structured as follows. Section 2 introduces the data and methods employed to search for IR periodic AGNs. Section 3 presents the search and corresponding analysis results, where we show compelling evidence demonstrating the highly improbable nature of these periodic light curves being generated by random processes. Section 4 compares selected candidates with normal AGNs and with previous studies that utilized optical light curves. Additionally, we discuss SDSS J140336.43+174136.1 as a candidate exhibiting periodic behavior in both optical and IR bands. Finally, we summarize the main results and suggest directions for future work in Section 5. We assume a cosmology with H 0 = 66 . 88 km s -1 Mpc -1 , Ω m =0 . 32, and Ω Λ =0 . 68, adopted from Planck Collaboration et al. (2020).\n2. DATA AND METHODS\n2.1. IR data\nThe WISE was originally launched to conduct a full-sky survey in four MIR bands centered at 3.4, 4.6, 12, and 22 µ m (labeled W1-W4) from February to August 2010 (Wright et al. 2010). The solid hydrogen cryogen used to cool the W3 and W4 instrumentation was depleted later and the spacecraft was placed in hibernation in 2011 February. However, it was\nreactivated in 2013 October under the new name NEOWISE -R, with only W1 and W2 operational, specifically to search for asteroids that could potentially pose a threat of impact on Earth (Mainzer et al. 2014). The WISE and its new mission NEOWISE operate in the same manner, that is, scanning a specific sky area every six months, with an average of 12 or more individual exposures taken within each epoch, typically spanning a single day. Consequently, each target in the sky has been observed 22-23 times separated by a six-month interval up to December 2023.\nThe single-exposure photometry of WISE and NEOWISE are archived in the AllWISE Multiepoch Photometry Table and NEOWISE-R Single Exposure (L1b) Source Table 1 , respectively. The photometry we adopted is measured by pointspread function (PSF) profile fitting, which is suitable for quasars. We first binned the data within each epoch since the intraday IR variability is negligible except for very radioloud AGNs (Jiang et al. 2012) and the anticipated periods we aim to detect are on much longer timescales. The variance of the binned data was recalculated as shown in Equation 1:\n[ δ mag ( ti )] 2 = 1 ni -1 ni ∑ j =1 [ magj -mag ( ti )] 2 + 1 ni σ 2 s . s . (1)\nwhere ni denotes the number of data points within the i th epoch, mag ( ti ) denotes the mean magnitude within the i th epoch, and σ j denotes the photometric uncertainty of the j th data point. The first part represents the contribution from photometric uncertainty and short-term variability, while the second part represents the contribution from the system stability of WISE. We adopt σ s . s . = 0 . 029 mag for WISE and σ s . s . = 0 . 016 mag for NEOWISE from Lyu et al. (2019).\nPrior to binning, the data were filtered based on the quality flags in the catalog following Jiang et al. (2021). Moreover, outliers that fell outside the 3 σ range of the single data point at each epoch were eliminated, using the mean magnitude and mean photometric uncertainty at that epoch as references. The mean and variance of the single data points were then re-computed, resulting in the binned data used for our analysis. This step was crucial to ensure the accuracy and reliability of the data used in our study.\n2.2. AGN Sample\nThe AGN sample we chose is the widely-used Million Quasar Catalog (v8, 2 August 2023; Flesch 2023). This is a compendium of 907,144 type-I QSOs and AGNs, largely complete from the literature up to 30 June 2023. 66,026 QSO candidates are also included, calculated via radio/X-ray association (including double radio lobes) as being 99% likely\nto be quasars. Blazars and type-II objects are also included, bringing the total count to 1,021,800. To ensure the purity of our candidates, we only selected objects labeled as types 'Q', 'A', 'B', 'K', 'N' in the catalog (see what the legends of these types refer to in the note of Table 2), as these are confirmed AGNs. As a result, the number of objects amounts to 955,744. To enable a reliable long-term variability analysis, we apply a criterion that requires a minimum of 15 detected epochs (about 2/3 of all detected epochs) for each source. This left us with a subset of 576,260 AGNs.\nIn order to rationalize the computational effort in further analysis, we choose to analyze only AGNs with obvious variability. This selection strategy is crucial to ensure that candidates meet the criteria outlined in selection criterion 5 (see Section 2.4). Furthermore, light curves displaying subtle variability tend to pose a challenge in terms of model constraints within the DRW framework (see Section 3.1). To address this issue, we introduce a requirement that, for a given light curve, at least one data point must satisfy mag ( ti ) -2 δ mag ( ti ) > magmean and at least one data point should satisfy mag ( ti ) + 2 δ mag ( ti ) < magmean . This allows us to narrow down our sample to 53,496 AGNs, ensuring that the selected candidates have the necessary characteristics for further analysis. We also require that the light curves be well constrained by the DRW model (see section 3.1), resulting in a parent sample of 48,932 AGNs.\n2.3. Finding Periodicity\nTo identify sources exhibiting periodic variability, we employed a sinusoidal function to fit the light curves of both W1 and W2 bands. The fitting process involved minimizing the following equation using the χ 2 statistics as Equation 2 below (D'Orazio & Haiman 2017):\nχ 2 = n ∑ i =1 [ mag ( ti ) -A sin Ω ( ti -t 0) + B ] 2 [ δ mag ( ti )] 2 (2)\nThe Lomb-Scargle (LS) periodogram (Lomb 1976; Scargle 1982) is widely used for periodicity detection in unevenly sampled data. In our analysis, we utilized the generalized Lomb-Scargle (GLS) periodogram (Zechmeister & Kürster 2009) implemented in the pyastronomy 2 package. In the GLS periodogram, we calculate a series of 'powers' defined as Equation 3:\np = χ 2 0 -χ 2 χ 2 0 (3)\nwhere χ 2 0 is the residual of a model assuming no variability, and χ 2 is equivalent to Equation 2 in the GLS periodogram (Zechmeister & Kürster 2009). Compared to the\n\n\n
Figure 1. Two special examples of candidates. The top panel specifically highlights the highest redshift source at z = 0 . 684 and the bottom panel highlights the candidate showing the most periods. Left: the grey error bars represent the original data (with outlier data points removed). The blue error bars represent the binned data. The orange sinusoids represent the best-fitting sine curves. Right: the black curves show the periodograms of candidates, while other curves show different significance level calculated from 100,000 simulated light curves.
\n\nLS periodogram, the GLS periodogram provides a more accurate frequency prediction by taking into consideration an offset and weights, which allows the calculation of the maximumpower in the GLS periodogram to be equivalent to minimizing Equation 2. The searched period range of the GLS periodogram is from 700 days to T 0 days, where T 0 is the time span of the light curve.\n2.4. Selection Criteria\nWe selected reliable periodic candidates based on five criteria. First, to account for stochastic red noise variability, we adopt a comparative approach by measuring the performance of the candidates against that of simulated light curves, as described in Section 3.1. Rather than applying a single threshold to the normalized periodograms, we require 3 σ significance in both the W1 and W2 bands.\nSecondly, we adopt the signal-to-noise (S/N) ratio defined as Equation 4 (Horne & Baliunas 1986), where A 0 denotes the amplitude and σ 2 r denotes the variance of the residuals. we require that ξ > 2 for both the W1 and W2 bands.\nξ = A 2 0 / (2 σ 2 r ) (4)\nThirdly, we require that the difference between the frequencies of the W1 and W2 light curves must not exceed 5% of their sum, in accordance with Equation 5:\n| f 1 -f 2 | f 1 + f 2 < 0 . 05 (5)\nwhere f 1 and f 2 represent the frequencies of the W1 and W2 light curves. In addition, the phase difference between them must be less than 1/8 cycles. This criterion reflects the requirements of the dust-echo model and guarantees that the\ntwo sinusoids have the same frequency and phase. However, we allow for a margin of error of 1/8 cycles to account for potential minor phase differences arising from low photometric accuracy or intrinsic characteristics of the IR light curves. For instance, the W2 band may receive a greater contribution from dust with lower temperatures located further from the BH, leading to a slight delay in the W2 variability compared to the W1 band (Lyu et al. 2019). If the sources are located at high redshifts, i.e. z > 2, the W1 and W2 emission could mainly come from a relatively inner and outer part of the accretion disk, then a time delay can also be expected. We confirm this small delay for our candidates in Section 3.2.\nFourthly, the available data are required to show more than two cycles in both the W1 and W2 bands. It is important to note that even strictly sinusoidal variations are difficult to distinguish from a simple stochastic process when the number of cycles N cyc is ≲ 5 (Vaughan et al. 2016). Any finite light curve generated by the corresponding variability process represents only a snapshot of that process. The periodogram derived from this light curve will show power distributed around the actual Power Spectral Density (PSD) continuum with some scatter. This scatter can create the appearance of spikes in the periodogram (e.g., see Fig. 1 and Fig. 12), leading the light curve to seemingly exhibit a sinusoidal-like pattern over 2 to 3 cycles, even if the underlying PSD process is purely continuous. Therefore, these patterns can only be reliably distinguished with much longer time series spanning many (e.g., > 4-5) cycles of the putative period. However, most optical searches still adhere to a criterion of > 1.5 cycles due to the limited time span of the data (e.g. Graham et al. 2015a; Charisi et al. 2016; Liu et al. 2019b; Chen et al. 2024). Consequently, false positives would arise in the candidate sample from stochastic red noise variability (Vaughan et al. 2016). Due to the limited number of detected epochs, which amounts to 23, we set the limit to > two cycles. We believe that this is the best approach we can take with the current WISE data, but we note that further improvements should be pursued in future research with more available data.\nFifthly, we check if the maximum power of the periodogram indicates a 'correct' period, since anomalous data points (e.g., an outlier with a small error) could mislead the periodogram calculation. Additionally, this criterion allows us to check if the light curves show strong variability. We estimate the period of a light curve by examining the data points. If all data points within a time segment satisfy mag ( ti ) -2 δ mag ( ti ) > offset (calculated by the periodogram), we classify it as a 'faint segment'. If all data points within a time segment satisfy mag ( ti ) + 2 δ mag ( ti ) < offset, we classify it as a 'bright segment'. Then we conduct a count in chronological order, recording each change from a faint segment to a bright segment or vice versa, and neglecting normal\n\n\n
Fig. 2 illustrates an AGN that meets all the selection criteria except for the selection Criterion 5 in the W2 band. The periodogram analysis reveals a 3 σ significance in the W2 light curve, indicating a period of T max = 1545 . 6 days within a time span of T 0 = 4939 . 9 days. However, the total count for W2 is 3, so that it does not meet Criterion 5. This failure is attributed to the lack of significant variability in the W2 band, as shown in Fig. 2.
\n\nFBQS J0832+2853\nFigure 2. An example for selection Criterion 5. In this example, the total count for W1 is 4, while W2 has 3 counts. Note that the errorbars show 2 δ mag ( ti ), and the purple lines show the best-fit sine curves.\ndata points in this process (see an example shown in Fig. 2). Denoting the total count as n , ideally n / 2 periods would be included in the time span. We require that for both the W1 and W2 bands, the period at the maximum power Tmax and the time span of the light curve T 0 satisfy Equation 6:\nmax { n -2 , 0 } 2 < T 0 Tmax < n + 3 2 (6)\n3. ANALYSIS AND RESULTS\n3.1. Simulating with Stochastic Variability\nAGN variability can roughly be described by a Gaussian first-order continuous autoregressive model (CAR(1), Kelly et al. 2009), also known as a damped random walk (DRW, MacLeod et al. 2010) or Ornstein-Uhlenbeck process. The likelihood function for the DRW process can be represented as:\nL ∝ | C | -1 2 exp -1 2 ∑ i , j ( Xi -q )( C -1 ) i j ( Xj -q ) (7)\nwhere the Xi represents the flux of the data point i , and q is the long-term mean of the light curve. The covariance matrix C is given by:\n\n\n
Figure 3. Distribution of the ∆ BIC value. BIC DRW( c ) is the BIC value of correlated DRW model, BIC DRW( n ) is the BIC value of the uncorrelated DRW model, while BIC DRW( c ) + sin is the BIC value of the sinusoid +correlated DRW model for the candidates.
\n\nCij = δ i j σ 2 i + σ 2 exp ( -| ti -t j | τ ) (8)\nwhere δ i j is the Kronecker delta. In our analysis, we assume a strong correlation between the W1 and W2 data. Subsequently, we have devised a modified form of the covariance matrix that accounts for this correlation, expressed as:\nCai , bj = δ ab δ i j σ 2 i + [ ρ + (1 -ρ ) δ ab ] σ 2 exp ( -| ti -t j | τ ) (9)\nwhere a and b represent the observed band (W1 or W2) of the data points, and ρ is the correlation coefficient.\nRegarding the data quality of WISE, we only use the binned data when estimating DRW parameters. We set a uniform prior for the logarithm of the parameters σ 1, σ 2 and τ , and a uniform prior for ρ W 1 , W 2. We set q as the mean magnitude of the light curve to reduce computational cost and use the Markov Chain Monte Carlo (MCMC) sampler emcee 3 (Foreman-Mackey et al. 2013) to construct the posterior samples of the parameters. However, we find that some light curves lack enough variability but pass the variability criterion (see Section 2.2) due to certain data points with very small errors. Consequently, they are not adequately constrained by the DRW model. To ensure a stable posterior for the MCMC processes, we set a criterion that the 1 σ error of the posterior lg τ sample must be less than 2. This is the final requirement to define our parent sample, resulting in a final number of 48,932 (see Section 2.2). The statistical distributions of σ , τ , and ρ W 1 , W 2 are illustrated in Fig. 4. Upon\ncomparison, the selected candidates exhibit similar values for σ 1, σ 2, and τ compared to the parent sample. Notably, they display a stronger correlation, which aids in meeting selection criterion 4 easier. The distribution of ρ W 1 , W 2 serves as validation for our correlation assumption.\nAs an alternative test, we perform a model selection to test if the correlation parameter is necessary to explain the light curve on top of a uncorrelated stochastic background. We adopt a maximum likelihood approach for the model comparison and parameter estimation. We use the Bayesian information criterion (BIC), which is defined as:\nBIC = -2ln L + k ln N (10)\nwhere L is the likelihood function, k is the number of free model parameters, and N is the number of data points. A lower BIC value indicates a preferable model. Typically, when ∆ BIC < -10, the model with the lower BIC is strongly favored. For the uncorrelated DRW process, we generate posterior parameter samples independently for each band. Subsequently, we integrate these samples into a covariance matrix to compute L with all correlation components set as zero. Our analysis indicates a strong preference for the correlated model over the uncorrelated one, aligning with our earlier analysis (further details provided in Fig. 3).\nWe proceed by randomly selecting a set of parameters to generate two correlated W1 and W2 simulated light curves. These simulated light curves are then compared with the actual observation cadence and measurement errors. This process is repeated 100,000 times to check that the selection criteria are met. In addition, we use the periodogram values of these simulated light curves to determine the significance of our candidates, as described in Criterion 1 in section 2.4. To further validate our simulation methodology, we perform a statistical comparison between the real and simulated light curves. Our analysis shows that they exhibit similar behaviour for each selection criterion (detailed results in Table 1), suggesting that the correlated DRW processes could potentially generate a comparable number of candidates to the real ones.\nNotably, the DRW model is relatively simple and may not fully capture the complexity of actual AGN variability due to degeneracies (Zu et al. 2013). When using the DRW assumption to calculate false alarm probabilities, it might not completely consider the assumptions of the red noise hypothesis. However, using different variability models could lead to some small changes in the results, but usually not significant ones.\n3.2. Candidates of Periodic Sources\nUsing the data and criteria outlined in Section 2, we have identified a total of 28 candidates. The catalog of candidates can be found in Table 2. The light curves and the GLS pe-\n\n\n
Figure 4. The DRW parameters distribution for the selection parent sample (blue) and the candidates (orange). Top: the σ distribution in the W1 band (left) and the W2 band (right). Bottom: the τ distribution (left) and the correlation coefficient distribution (right).
\n\nriodograms of two examples are presented in Fig. 1 and the rest can be found in Fig. 12. The W1 and redshift distributions of both the parent sample and the selected candidates are shown in Fig 5.\nIt is worth noting that the candidates are predominantly situated in the low redshift range (median z=0.126 , with 27/28 at z < 0.4) and bright region (median W1=12.69, 25/28 satisfy W1 < 14), which can be attributed to a selection effect making it easier for them to fulfill the selection criteria.\nIn Section 2.4, we mention the possibility of a slight phase delay between the W2 and W1 light curves. This is confirmed by our candidates, with an average delay of 0.025 periods, as illustrated in Fig. 6. The maximum phase delay is 0.087 period, which remains smaller than the 1/8 period threshold used in our selection process, indicating a reasonable choice.\nAs an alternative test, here we use the BIC method (see details in Section 3.1) to assess if an additional periodic signal is needed to explain the light curve on top of a stochastic\nbackground. The likelihood function for the DRW + sinusoidal model is written as:\nL ∝ | C | -1 2 exp -1 2 ∑ i , j ( Xi -Si )( C -1 ) i j ( Xj -Sj ) (11)\nwhere Si represents the sinusoid signal. Our analysis shows that, for the vast majority of the candidates, BIC DRW + sin -BIC DRW( c ) < -10, indicating robust evidence that the periodic model is significantly favored over the pure stochastic model (refer to Fig. 3 for detailed results). However, this could be explained by the limitation of the DRW model, which might not adequately capture a spiky PSD and may be more suited to smooth continuum-like PSD shapes. Therefore, additional data in the future are necessary to obtain a higher density in frequency sampling for the periodogram and better test the preference for the sinusoid + DRW model.\n
\n\n
Table 1. Statistical comparison between real light curves and simulated ones.NOTE-Data in the 'Mock' column refer to 1 /100,000 of the total from 4,893,200,000 simulated light curves, allowing direct comparison with the real data. '3 σ significance' refers to the selection criterion 1 in Section 2.4. 'correct period' refers to the selection criterion 5 in Section 2.4.
\n
\n\n\n
Figure 5. Distribution of the parent sample (blue), and the candidates (red) in the redshift and W1 magnitude space. We use a 40 × 40 grid to visualize the distribution of the parent sample. The color for each grid cell represents the number of AGNs from the parent sample that fall within that cell.
\n\nIt's worth noting that the sinusoid +DRW model is a commonly used model due to its simplicity and clear physical picture. However, there exist alternative light curve models, such as the bursty model predicted by circumbinary accretion disk simulations (Farris et al. 2014). Despite the potential of this bursty model, we did not simulate it for our candidates as we could not observe 'bursty' characteristics in the WISE light curves.\nHere we compile the derived quantities for our analysis, which are presented in Table 2. The table includes essen-\n\n\n
Figure 6. The distribution of the phase delay of W2 relative to W1 for the candidates (blue), and the corresponding Gaussian fit of the distribution (orange line).
\n\ntial information such as the Milliquas Name, right ascension (R.A.), declination (Decl.), redshift ( z ), and quasar properties as provided by Wu & Shen (2022) and Liu et al. (2019a). Additionally, we have included periodic properties such as offsets of the W1 and W2 bands, and the period of the candidates.\n3.3. False Alarm Probability\nTo estimate the false alarm probability (FAP) for each candidate, we employ the method described in Chen et al. (2020). We estimate the approximate FAP using the effective number of independent frequencies Nef f , which is calculated by dividing the observed frequency window by the expected peak width δ f = 1 / T . The FAP is estimated as (e.g., VanderPlas 2018):\nFAP ∼ 1 -[ Psingle ] Nef f (12)\nWhere Psingle = 1 -e Z , Z is the periodogram value defined in Zechmeister & Kürster (2009). Since candidates are selected based on the light curves of both the W1 band and W2 band, there are two P single and N eff. To simplify the calculation, we can estimate the periodogram FAP as Equation 13:\nFAP ∼ 1 -[ Psingle , W 1] Nef f , W 1 [ Psingle , W 2] Nef f , W 2 (13)\nIt is important to acknowledge that this calculation assumes independent FAPs for the W1 and W2 bands, contrary to the strong correlation observed in the DRW simulation (refer to Section 3.1). Nevertheless, this method provides an upper limit estimate for the FAP, as single-band FAP calculations do not consider this correlation.\n
\n\n
Table 2 . The 28 periodic AGN candidates meeting the selection criteria.
\n
\nNOTE- (1)-(4): Name, RA, DEC, and redshift of the objects. (5)-(6): Median magnitudes in the W1 and W2 bands, respectively. (7): Period (in the observer-frame) in units of days. (8): Legend of type/class from Flesch (2023): Q = QSO, type-I broad-line core-dominated, 860,100 of these. A = AGN, type-I Seyferts/host-dominated, 47,044 of these. B = BL Lac type object, 2,814 of these. (FSRQs are typed as QSOs here) K = NLQSO, type-II narrow-line core-dominated, 6,048 of these. N = NLAGN, type-II Seyferts/host-dominated, 39,768 of these. Incomplete, and includes an unquantified residue of legacy NELGs/ELGs/LINERs, plus some unclear AGN. This is the catch-all category. S = star classified but showing quasar-like photometry and radio/X-ray association, thus included as a quasar candidate; 124 of these. R = radio association displayed. X = X-ray association displayed. 2 = double radio lobes displayed (declared by data-driven algorithm). (9)-(11): Logarithmic bolometric luminosity in units of erg/s, black hole mass in units of solar mass, and Eddington ratio for sources in the SDSS DR16 quasar catalog (Wu & Shen 2022), the DR7 board-line AGN catalog (Liu et al. 2019a).\nAdditionally, we empirically compute the global FAP using the 100,000 simulated light curves for each AGN, following a similar approach as presented by Barth & Stern (2018).\nThe global FAP accounts for all false positives that satisfy all selection criteria within the explored frequency range. The global FAP values for candidates range from 4 . 0 × 10 -4 to\n\n\n
Figure 7. The FAP distribution estimated from periodogram calculation (blue), and the global FAP estimated from 100,000 simulated light curves (orange). The Gaussian fit curves for the 2 distributions are also shown.
\n\n4 . 77 × 10 -3 , with a mean of 2 . 07 × 10 -3 by Gaussian fit. Further details are available in Fig. 7.\nIt is noteworthy that the global FAPs are significantly higher than the periodogram FAPs. This discrepancy can be attributed to the stronger variability exhibited by the simulated light curves compared to real observations, leading to an increase in the global FAP. Moreover, while most false positives exhibit only a 3 σ significance level, certain candidates display notably stronger significance, such as SDSS J162938.09+362452.2 and 6dF J214907.4-175159 as depicted in Fig. 12.\n4. DISCUSSION\n4.1. Do the Candidates Show Distinctive Properties?\nOne might wonder whether our selected periodic sample shows distinct features in other physical properties, such as bolometric luminosity ( L bol), black hole mass ( M BH) and Eddington ratio ( λ edd= L bol / L edd). To investigate this, we try to extract a subsample of candidates with previous measurements in these parameters. We chose to cross-match the candidates with two AGN samples based on Sloan Digital Sky Survey (SDSS), namely the SDSS DR16 quasar catalog (DR16Q) (Wu & Shen 2022) and DR7 broad-line AGN catalog (Liu et al. 2019a). They contained 750,414 and 14,584 sources, concentrated at high and low redshifts, respectively (see Figure 8). We only found a total of 10 candidates that overlapped with the two catalogs.\nSimilar to the occupation of the candidates in the whole parent sample (see Figure 5), they are all in the low redshift and low magnitude space (see Figure 8) due to selection effects. Therefore, we need to construct a control sample for a fair comparison, at least in redshift and IR magnitudes.\n\n\n
Figure 8. Distributions of the selection parent sample (blue), quasars present in the DR16Q catalog that overlap with the parent sample (black), AGNs present in the DR7 broad-line AGN catalog that overlap with the parent sample (green), and the overlapping candidates (red) in the redshift and W1 magnitude space. Enclosed percentiles are marked for each contour level.
\n\nSpecifically, we selected the 10 closest AGNs with a redshift ratio less than 1.01 and with differences in W1 and W2 magnitudes less than 0.1. In cases when no AGN meet the specified criteria, we choose the closest AGN in the catalog as the control sample. After controlling, we then compare the M BH, L bol and λ edd of the periodic sources with normal AGNs. Additionally, we have also considered another parameter dust covering factor ( fd ), defined as the ratio of the W1monochromatic luminosity ( LW 1 = ν W 1 F ν W 1) and the L bol ( fd = L W1 / L bol, e.g., Ma & Wang 2013), to indicate the extent of dust obscuration. However, no significant differences in the distributions of these parameters can be seen visually (see Fig. 9). We have also checked the mean and standard deviation (SD) of their distributions (see Table 3) and confirmed it.\nWe further performed the K-S statistics for the L bol, M BH, λ edd, and fd distributions between the candidates and the control sample, yielding values of 0.375, 0.400, 0.225, and 0.400, respectively. The K-S tests suggest significant differences in these properties with a critical value of 0.480 for a 95% confidence level, so none of them show significant differences. It is worth emphasizing that the systematic uncertainties in the estimates of AGN properties (e.g., singleepoch BH mass, see discussions in Shen 2013) are nonnegligible and would dilute the difference, if there were any. This may be particularly important for the small sample size for comparison here.\n4.2. Comparison with SMBHB Candidates Selected by Optical Periodicity\n\n\n
Figure 10. Redshift and period distributions of SMBHB candidates selected by this work (red), and those reported in previous works by optical search (other colors).
\n\n\n\n
Figure 9. Left: distribution of the control samples (blue), which control for redshift and W1, W2, and the candidates (red) in the log ( M BH) and log ( L bol) space. Right: distribution of the control samples (blue), which control for redshift and L bol, and the candidates (red) in the log ( L bol) and log ( fd ) space (right).The right panel also tests the anti-correlation between L bol and fd .
\n\n
\n\n
Table 3. Statistical comparison of quasar properties between candidates and control samples.NOTE- 'control 1' refers to the control sample that controls z, W1 and W2 magnitude. 'control 2' refers to the control sample that controls z and L bol.
\n
\nAs mentioned in the introduction, previous studies have identified samples of SMBHB candidates based on optical periodic light curves. It is intriguing to investigate whether there are any common sources between our IR-selected sample and these optically-selected samples. Such overlap would provide compelling evidence for the periodicity of the candidates. Additionally, if the periods derived from optical light curves align with those obtained from IR light curves, we can employ the dust-echo model to further explore the properties of the candidates. For this purpose, we compile a collection of optical SMBHB candidates reported in various literature, including Graham et al. (2015a); Charisi et al. (2016); Zheng et al. (2016); Liu et al. (2019b); Chen et al. (2020, 2024).\nUnfortunately, we do not find any overlap between our sample and those samples. The lack of matches is not entirely unexpected, as the two searches prioritize candidates with different periods. Specifically, we find that the periods of the optical candidates predominantly fell below 1700 days,\n\n\n\nz\nwhereas the periods of our selected candidates were mostly longer than 1700 days (see details shown in Fig. 10). Additionally, most of our candidates have low redshifts (z < 0.4) due to the detection ability of WISE, while the redshifts of candidates from optical searches span from 0 to ∼ 3.\n4.3. Optical Light Curves and A Possible Periodic Source\nGiven the abundance of optical photometric archival data, we then examined the optical light curves of the 28 candidates using data from various surveys, such as the Catalina Real Time Transient Survey (CRTS, Drake et al. 2009), the Palomar Transient Factory (PTF, Rau et al. 2009), the Zwicky Transient Facility (ZTF, Masci et al. 2019), and the All-Sky Automated Survey for Supernovae (ASAS-SN, Kochanek\nmag\n\n\n
Figure 11. The W1, W2 and optical light curves of SDSS J140336.43+174136.1. The orange, purple and black lines represent the corresponding best-fitting sinusoids. We assume that the periods of IR light curves are the same as that of the optical light curves.
\n\net al. 2017). Their combined dataset provides optical light curves spanning about two decades. We have normalized the data from different surveys and binned them at 20-day intervals. However, they generally show a small amplitude of optical variability, probably due to either intrinsically weak variability in the optical band or too shallow surveys. Only one candidate, SDSS J140336.43+174136.1, shows significant variability and periodicity in its optical light curve (see Fig. 11).\nThe periodogram analysis of the optical light curve of SDSS J140336.43+174136.1 reveals a period of 2891 ± 22 days, that is slightly longer than the period fitted from the IR light curves, which is 2346 ± 87 days. The errors are given by periodogram peak statistics based on 100,000 perturbed light curves for each real light curve. The discrepancy could be caused by the poor cadence of the WISE surveys, and so in the following analysis we choose the optical period for this particular source.\nWe present the corresponding best-fitting sinusoids for the W1, W2, and optical light curves in Figure 11. The time delay between the W1 (W2) and optical light curves is 630 (727) days (in the rest frame), respectively. Assuming the simplest scenario of a hollow spherical shell of dust with the source located at its center, we can easily estimate that the inner radius of the shell is 0.53 pc. These values are roughly consistent with previous statistical correlations between the\ntime delay and bolometric luminosity of AGNs. For example, using the relations obtained by Lyu et al. (2019), i.e.\n∆ tW 1 / day = 10 2 . 10 ± 0 . 06 ( L bol / 10 11 L ⊙ ) 0 . 47 ± 0 . 06 (14)\nfor the W1 band and\n∆ tW 2 / day = 10 2 . 20 ± 0 . 06 ( L bol / 10 11 L ⊙ ) 0 . 45 ± 0 . 05 (15)\nfor the W2 band, and a given bolometric luminosity of 10 45 . 81 erg s -1 for SDSS J140336.43+174136.1 (see Table 2), the W1 and W2 time delays are estimated to be 476 ± 147 days and 566 ± 158 days, respectively, which is in good agreement with our measurements.\n5. CONCLUSION\nIn this work, we have conducted the first systematic search for SMBHBs in the IR band and identified 28 AGNs exhibiting IR periodic variability using the decade-long WISE and NEOWISE light curves. We performed extensive simulations to evaluate the potential influence of stochastic variability on candidate selection. By counting all the false positives in the simulated light curves, the probability of these periodic light curves being generated by random processes is an average of 0.207% by Gaussian fit. However, the number can be reproduced by our mock simulations with the DRW process of the parent sample. This indicates the challenge of identifying reliable SMBHBs with periodic emission using only WISE light curves, which are constrained by their time span and visit cadence, necessitating a low threshold on the minimum number of cycles ( N cyc > 2). Consequently, the nature of these periodic sources, whether driven by SMBHBs or not, should be carefully tested in future observations. We subsequently investigated whether these IR periodic sources exhibit distinct properties compared to normal AGNs. However, no significant biases were found, and the small sample size prevents us from statistically identifying potential weak differences. On the other hand, we found that there is no overlap between our sample (IRselected) and the SMBHB candidates reported in the literature (optically selected), likely due to differences in their preferred period ranges. Interestingly, we have identified SDSS J140336.43+174136.1 as a candidate displaying periodicity in both optical and IR bands. Further extended and more sensitive observations of these candidates are essential to confirm whether or not they are real periodic sources in both optical and IR bands. As suggested by D'Orazio & Haiman (2017), the periodic sources can also help test the physical mechanisms that produce the observed periodicity, due to relativistic Doppler modulation or accretion rate fluctuations.\nThis study highlights the promising potential of identifying SMBHB candidates through IR time-domain observations, which can significantly complement searches in other bands, although there are challenges in distinguishing genuine sources. It opens up a new avenue for SMBHB searches and can be applied to future surveys. Although NEOWISE unfortunately ended on July 31, 2024, the upcoming NEO Surveyor (Mainzer et al. 2023), the successor to NEOWISE , and the Nancy Grace Roman Space Telescope (Spergel et al. 2015) will ensure a bright future for IR time-domain astronomy, particularly for SMBHB searches (Haiman et al. 2023). These different surveys somewhat form a relay in time and provide us with IR light curves with a long time baseline, which is essential for verifying the periodicity of the candidates with N cyc < 4 and for detailed comparisons against physical models.\nACKNOWLEDGMENTS\nWe sincerely thank the expert referee for very constructive and insightful comments, which helped improve our\nmanuscript greatly. This work is supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB0550200), the National Natural Science Foundation of China (grants 12073025, 12192221, the science research grants from the China Manned Space Project and the Cyrus Chun Ying Tang Foundations. D.L. acknowledges the support from the National Undergraduate Training Program for Innovation and Entrepreneurship. X.L. acknowledges support by NSF grant AST-2206499. This research makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. This research also makes use of data products from NEOWISE-R, which is a project of the Jet Propulsion Laboratory/California Institute of Technology, funded by the Planetary Science Division of the National Aeronautics and Space Administration. This research has made use of the NASA/IPAC Infrared Science Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration.\nREFERENCES\nArzoumanian Z., et al., 2018, ApJ, 859, 47 Barth A. J., Stern D., 2018, ApJ, 859, 10 Begelman M. C., Blandford R. D., Rees M. J., 1980, Nature, 287, 307 Blaes O., Lee M. H., Socrates A., 2002, ApJ, 578, 775 Bonetti M., Sesana A., Barausse E., Haardt F., 2018, MNRAS, 477, 2599 Centrella J., Baker J. G., Kelly B. J., van Meter J. R., 2010, Reviews of Modern Physics, 82, 3069 Chapon D., Mayer L., Teyssier R., 2013, MNRAS, 429, 3114 Charisi M., Bartos I., Haiman Z., Price-Whelan A. M., Graham M. J., Bellm E. C., Laher R. R., Márka S., 2016, MNRAS, 463, 2145 Chen Y.-C., et al., 2020, MNRAS, 499, 2245 Chen Y.-C., Hwang H.-C., Shen Y., Liu X., Zakamska N. L., Yang Q., Li J. I., 2022, ApJ, 925, 162 Chen Y.-J., et al., 2024, MNRAS, 527, 12154 Colpi M., Dotti M., 2011, Advanced Science Letters, 4, 181 Cuadra J., Armitage P. J., Alexander R. D., Begelman M. C., 2009, MNRAS, 393, 1423 D'Orazio D. J., Charisi M., 2023, arXiv e-prints, p. arXiv:2310.16896 D'Orazio D. J., Haiman Z., 2017, MNRAS, 470, 1198 D'Orazio D. J., Haiman Z., Schiminovich D., 2015, Nature, 525, 351\nDe Rosa A., et al., 2019, NewAR, 86, 101525 Drake A. J., et al., 2009, ApJ, 696, 870 Farris B. D., Duffell P., MacFadyen A. I., Haiman Z., 2014, ApJ, 783, 134 Flesch E. W., 2023, The Open Journal of Astrophysics, 6, 49 Foreman-Mackey D., Hogg D. W., Lang D., Goodman J., 2013, PASP, 125, 306 Gould A., Rix H.-W., 2000, ApJL, 532, L29 Graham M. J., et al., 2015a, MNRAS, 453, 1562 Graham M. J., et al., 2015b, Nature, 518, 74 Haiman Z., et al., 2023, arXiv e-prints, p. arXiv:2306.14990 Holgado A. M., Sesana A., Sandrinelli A., Covino S., Treves A., Liu X., Ricker P., 2018, MNRAS, 481, L74 Horne J. H., Baliunas S. L., 1986, ApJ, 302, 757 Hughes S. A., 2009, ARA&A, 47, 107 Jiang N., et al., 2012, ApJL, 759, L31 Jiang N., et al., 2021, ApJS, 252, 32 Jiang N., et al., 2022, arXiv e-prints, p. arXiv:2201.11633 Jun H. D., Stern D., Graham M. J., Djorgovski S. G., Mainzer A., Cutri R. M., Drake A. J., Mahabal A. A., 2015, ApJL, 814, L12 Kelley L. Z., Blecha L., Hernquist L., 2017, MNRAS, 464, 3131 Kelly B. C., Bechtold J., Siemiginowska A., 2009, ApJ, 698, 895 Khan F. M., Fiacconi D., Mayer L., Berczik P., Just A., 2016, ApJ, 828, 73\nKlein A., et al., 2016, PhRvD, 93, 024003\n
\n\n
\n
\n\n
\nAPPENDIX\nA. THE IR LIGHT CURVES OF THE PERIODIC AGN SAMPLE\nWe have shown the light curves of 2 special candidates in Fig. 1. Here we show the light curves of the rest 26 candidates in Fig. 12.\n\n\n
Figure 12. The light curves and periodograms of the rest 26 candidates. The legends correspond to those in Fig. 1.
\n\n\n\n
Figure 12. (Continued).
\n\n\n\n
Figure 12. (Continued).
\n\n\n\n
Figure 12. (Continued).
\n\n\n\n
Figure 12. (Continued).
\n\nDESI 39632956414755490\n\n\n
Figure 12. (Continued).
\n\n\n\n
Figure 12. (Continued).
\n\n\n\n
Figure 12. (Continued).
\n\n\n\n
Figure 12. (Continued).
\n\n"},"sections":{"kind":"list like","value":[{"title":"ABSTRACT","content":"Periodic variability in active galactic nuclei (AGNs) is a promising method for studying sub-parsec supermassive black hole binaries (SMBHBs), which are a challenging detection target. While extensive searches have been made in the optical, X-ray and gamma-ray bands, systematic infrared (IR) studies remain limited. Using data from the Wide-field Infrared Survey Explorer (WISE), which provides unique decade-long mid-IR light curves with a six-month cadence, we have conducted the first systematic search for SMBHB candidates based on IR periodicity. Analyzing a parent sample of 48,932 objects selected from about half a million AGNs, we have identified 28 candidate periodic AGNs with periods ranging from 1,268 to 2,437 days (in the observer frame) by fitting their WISE light curves with sinusoidal functions. However, our mock simulation of the parent sample indicates that stochastic variability can actually produce a similar number of periodic sources, underscoring the difficulty in robustly identifying real periodic signals with WISE light curves, given their current sampling. Notably, we found no overlap between our sample and optical periodic sources, which can be explained by a distinct preference for certain periods due to selection bias. By combining archived data from different surveys, we have identified SDSS J140336.43+174136.1 as a candidate exhibiting periodic behavior in both optical and IR bands, a phenomenon that warrants further validation through observational tests. Our results highlight the potential of IR time-domain surveys, including future missions such as the Nancy GraceRoman Space Telescope, for identifying periodic AGNs, but complementary tests are still needed to determine their physical origins such as SMBHBs. Keywords: Active galactic nuclei (16); Infrared astronomy(786); Quasars (1319); Supermassive black holes (1663); Time domain astronomy (2109)","pages":[1]},{"title":"A Systematic Search for Candidate Supermassive Black Hole Binaries Using Periodic Mid-Infrared Light Curves of Active Galactic Nuclei","content":"1 CAS Key Laboratory for Research in Galaxies and Cosmology, Department of Astronomy, University of Science and Technology of China, Hefei, 230026, China; jnac@ustc.edu.cn 2 School of Physical Sciences, University of Science and Technology of China, Hefei, 230026, China 3 School of Astronomy and Space Sciences, University of Science and Technology of China, Hefei, 230026, China 4 Department of Astronomy, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA 5 Center for Artificial Intelligence Innovation, University of Illinois at Urbana-Champaign, 1205 West Clark Street, Urbana, IL 61801, USA National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA 6 (Received 2024 May 24; Revised 2024 November 9; Accepted 2024 November 11) Submitted to ApJ","pages":[1]},{"title":"1. INTRODUCTION","content":"Supermassive black holes (SMBHs) are intriguing celestial objects that reside at the centers of galaxies (Kormendy & Ho 2013). When galaxies merge, it is anticipated that the central black holes will form binary systems known as SMBH binaries (SMBHBs, Begelman et al. 1980). These binary systems offer valuable insights into various astrophysical phenomena, including galaxy formation (Colpi & Dotti 2011), gravitational wave (GW) emission (Hughes 2009), and the evolution of SMBHs (Merritt 2013). More massive binaries are pulsar-timing array (PTA) sources (e.g., Arzoumanian et al. 2018), while less massive binaries are targeted by space-based experiments such as LISA (Klein et al. 2016). They provide a laboratory to directly test strong-field general relativity (Hughes 2009; Centrella et al. 2010). However, despite their theoretical significance, the direct electromagnetic detection of close SMBHBs below subparsec separation has proven to be a challenging endeavor (see recent reviews such as De Rosa et al. 2019; D'Orazio & Charisi 2023) while numerous SMBH pairs at larger scales have been identified through the presence of dual active galactic nuclei (AGNs) (e.g., Komossa et al. 2003; Zhou et al. 2004; Liu et al. 2010, 2011, 2013; Koss et al. 2011; Liu et al. 2018; Chen et al. 2022). When a binary has exhausted its in- teractions with stars but has not approached close enough to emit significant gravitational radiation, its orbit does not have an obvious mechanism for decay. This long-standing challenge presents a major uncertainty in estimating the abundance of SMBHB mergers as GW sources. In theory, the bottleneck may be overcome in gaseous environments (e.g., Gould & Rix 2000; Cuadra et al. 2009; Chapon et al. 2013; del Valle et al. 2015), in triaxial or axisymmetric galaxies (e.g., Khan et al. 2016; Kelley et al. 2017), and/or by interacting with a third BH in hierarchical mergers (e.g., Blaes et al. 2002; Kulkarni & Loeb 2012; Bonetti et al. 2018). A recent and promising approach for identifying potential close SMBHB candidates has emerged in the thriving field of time-domain astronomy, which focuses on analyzing the periodic light curves of AGNs. The periodicity observed in these light curves can arise from changes in the accretion rate onto the black holes (Graham et al. 2015b) or from the relativistic Doppler boost resulting from the highly relativistic motion of gas in the mini accretion disk around the smaller black hole in the binary system (D'Orazio et al. 2015). Extensive research has been conducted using optical light curves to search for SMBHBs, and the methods employed in these studies have become well-established (Graham et al. 2015a; Charisi et al. 2016; Zheng et al. 2016; Liu et al. 2019b; Chen et al. 2020; Liao et al. 2021; Chen et al. 2024). However, many of the known candidates have been shown to be subject to false positives due to stochastic quasar variability (e.g., Vaughan et al. 2016). Additionally, recent findings suggest that some AGNs may display periodic variations only at some specific epochs (e.g., Jiang et al. 2022; O'Neill et al. 2022) further enhance the difficulties of periodic searches. Furthermore, previous surveys were only sensitive to the most massive quasars at high redshift ( z ≳ 2) which should have already gone through their major merger process (e.g., Shen 2009). Regardless of the process that imprints periodicity, it is generally accepted that the timescale is primarily determined by the binary orbital period. SMBHBs with masses between 10 6 -10 10 M ⊙ and separations of 0.01 pc possess orbital periods that span from years to several decades, making them possibly detectable by current timedomain surveys. In addition to the optical band, the periodic searches have also been conducted in other wavelength regimes, such as in X-ray (Serafinelli et al. 2020; Liu et al. 2020) and gamma-ray bands (Sandrinelli et al. 2018; Holgado et al. 2018). Periodic infrared (IR) light curves are also expected as a result of reverberation mapping of the circumbinary dusty torus in periodic AGNs. However, the search for SMBHBs using IR data has not been extensively explored, despite the detection of an IR time lag in an individual SMBHB candidate (Jun et al. 2015), which highlights the potential of utilizing periodic IR light curves for SMBHB identifi- cation (D'Orazio & Haiman 2017). The IR band not only provides an alternative approach to searching for SMBHBs but also offers distinct advantages. Firstly, IR photons are less susceptible to dust extinction, allowing them to penetrate through obscuring material, which is often abundant in galactic nuclear regions and galaxy merger systems. By extending the search to the IR wavelength range, we can uncover a population of SMBHB candidates that may have been previously overlooked, thereby expanding our knowledge of the prevalence and characteristics of these elusive systems. Even for unobscured AGNs, the variability amplitudes in the mid-IR band are typically larger compared to the optical band due to lower background contamination. Additionally, as discussed in D'Orazio & Haiman (2017), the dust-echo model can assist in determining the origin of central periodicity and place constraints on the physical characteristics of SMBHBs and their dust torus. To address this, we perform the first systematic search for SMBHB candidates via periodic IR light curves. The data we use come from the Widefield Infrared Survey Explorer (WISE, Wright et al. 2010) and its successor NEOWISE (Mainzer et al. 2014), which have provided us with a public dataset from February 2010 to December 2023 with a half-year cadence except for a 2.5year gap for each object. In this study, we primarily focus on sinusoidal periodic signals, which have been commonly adopted in modeling the optical light curves of SMBHB candidates (e.g., D'Orazio et al. 2015), and on modeling the IR light curves using the dust-echo model (D'Orazio & Haiman 2017). The paper is structured as follows. Section 2 introduces the data and methods employed to search for IR periodic AGNs. Section 3 presents the search and corresponding analysis results, where we show compelling evidence demonstrating the highly improbable nature of these periodic light curves being generated by random processes. Section 4 compares selected candidates with normal AGNs and with previous studies that utilized optical light curves. Additionally, we discuss SDSS J140336.43+174136.1 as a candidate exhibiting periodic behavior in both optical and IR bands. Finally, we summarize the main results and suggest directions for future work in Section 5. We assume a cosmology with H 0 = 66 . 88 km s -1 Mpc -1 , Ω m =0 . 32, and Ω Λ =0 . 68, adopted from Planck Collaboration et al. (2020).","pages":[1,2]},{"title":"2.1. IR data","content":"The WISE was originally launched to conduct a full-sky survey in four MIR bands centered at 3.4, 4.6, 12, and 22 µ m (labeled W1-W4) from February to August 2010 (Wright et al. 2010). The solid hydrogen cryogen used to cool the W3 and W4 instrumentation was depleted later and the spacecraft was placed in hibernation in 2011 February. However, it was reactivated in 2013 October under the new name NEOWISE -R, with only W1 and W2 operational, specifically to search for asteroids that could potentially pose a threat of impact on Earth (Mainzer et al. 2014). The WISE and its new mission NEOWISE operate in the same manner, that is, scanning a specific sky area every six months, with an average of 12 or more individual exposures taken within each epoch, typically spanning a single day. Consequently, each target in the sky has been observed 22-23 times separated by a six-month interval up to December 2023. The single-exposure photometry of WISE and NEOWISE are archived in the AllWISE Multiepoch Photometry Table and NEOWISE-R Single Exposure (L1b) Source Table 1 , respectively. The photometry we adopted is measured by pointspread function (PSF) profile fitting, which is suitable for quasars. We first binned the data within each epoch since the intraday IR variability is negligible except for very radioloud AGNs (Jiang et al. 2012) and the anticipated periods we aim to detect are on much longer timescales. The variance of the binned data was recalculated as shown in Equation 1: where ni denotes the number of data points within the i th epoch, mag ( ti ) denotes the mean magnitude within the i th epoch, and σ j denotes the photometric uncertainty of the j th data point. The first part represents the contribution from photometric uncertainty and short-term variability, while the second part represents the contribution from the system stability of WISE. We adopt σ s . s . = 0 . 029 mag for WISE and σ s . s . = 0 . 016 mag for NEOWISE from Lyu et al. (2019). Prior to binning, the data were filtered based on the quality flags in the catalog following Jiang et al. (2021). Moreover, outliers that fell outside the 3 σ range of the single data point at each epoch were eliminated, using the mean magnitude and mean photometric uncertainty at that epoch as references. The mean and variance of the single data points were then re-computed, resulting in the binned data used for our analysis. This step was crucial to ensure the accuracy and reliability of the data used in our study.","pages":[2,3]},{"title":"2.2. AGN Sample","content":"The AGN sample we chose is the widely-used Million Quasar Catalog (v8, 2 August 2023; Flesch 2023). This is a compendium of 907,144 type-I QSOs and AGNs, largely complete from the literature up to 30 June 2023. 66,026 QSO candidates are also included, calculated via radio/X-ray association (including double radio lobes) as being 99% likely to be quasars. Blazars and type-II objects are also included, bringing the total count to 1,021,800. To ensure the purity of our candidates, we only selected objects labeled as types 'Q', 'A', 'B', 'K', 'N' in the catalog (see what the legends of these types refer to in the note of Table 2), as these are confirmed AGNs. As a result, the number of objects amounts to 955,744. To enable a reliable long-term variability analysis, we apply a criterion that requires a minimum of 15 detected epochs (about 2/3 of all detected epochs) for each source. This left us with a subset of 576,260 AGNs. In order to rationalize the computational effort in further analysis, we choose to analyze only AGNs with obvious variability. This selection strategy is crucial to ensure that candidates meet the criteria outlined in selection criterion 5 (see Section 2.4). Furthermore, light curves displaying subtle variability tend to pose a challenge in terms of model constraints within the DRW framework (see Section 3.1). To address this issue, we introduce a requirement that, for a given light curve, at least one data point must satisfy mag ( ti ) -2 δ mag ( ti ) > magmean and at least one data point should satisfy mag ( ti ) + 2 δ mag ( ti ) < magmean . This allows us to narrow down our sample to 53,496 AGNs, ensuring that the selected candidates have the necessary characteristics for further analysis. We also require that the light curves be well constrained by the DRW model (see section 3.1), resulting in a parent sample of 48,932 AGNs.","pages":[3]},{"title":"2.3. Finding Periodicity","content":"To identify sources exhibiting periodic variability, we employed a sinusoidal function to fit the light curves of both W1 and W2 bands. The fitting process involved minimizing the following equation using the χ 2 statistics as Equation 2 below (D'Orazio & Haiman 2017): The Lomb-Scargle (LS) periodogram (Lomb 1976; Scargle 1982) is widely used for periodicity detection in unevenly sampled data. In our analysis, we utilized the generalized Lomb-Scargle (GLS) periodogram (Zechmeister & Kürster 2009) implemented in the pyastronomy 2 package. In the GLS periodogram, we calculate a series of 'powers' defined as Equation 3: where χ 2 0 is the residual of a model assuming no variability, and χ 2 is equivalent to Equation 2 in the GLS periodogram (Zechmeister & Kürster 2009). Compared to the LS periodogram, the GLS periodogram provides a more accurate frequency prediction by taking into consideration an offset and weights, which allows the calculation of the maximumpower in the GLS periodogram to be equivalent to minimizing Equation 2. The searched period range of the GLS periodogram is from 700 days to T 0 days, where T 0 is the time span of the light curve.","pages":[3,4]},{"title":"2.4. Selection Criteria","content":"We selected reliable periodic candidates based on five criteria. First, to account for stochastic red noise variability, we adopt a comparative approach by measuring the performance of the candidates against that of simulated light curves, as described in Section 3.1. Rather than applying a single threshold to the normalized periodograms, we require 3 σ significance in both the W1 and W2 bands. Secondly, we adopt the signal-to-noise (S/N) ratio defined as Equation 4 (Horne & Baliunas 1986), where A 0 denotes the amplitude and σ 2 r denotes the variance of the residuals. we require that ξ > 2 for both the W1 and W2 bands. Thirdly, we require that the difference between the frequencies of the W1 and W2 light curves must not exceed 5% of their sum, in accordance with Equation 5: where f 1 and f 2 represent the frequencies of the W1 and W2 light curves. In addition, the phase difference between them must be less than 1/8 cycles. This criterion reflects the requirements of the dust-echo model and guarantees that the two sinusoids have the same frequency and phase. However, we allow for a margin of error of 1/8 cycles to account for potential minor phase differences arising from low photometric accuracy or intrinsic characteristics of the IR light curves. For instance, the W2 band may receive a greater contribution from dust with lower temperatures located further from the BH, leading to a slight delay in the W2 variability compared to the W1 band (Lyu et al. 2019). If the sources are located at high redshifts, i.e. z > 2, the W1 and W2 emission could mainly come from a relatively inner and outer part of the accretion disk, then a time delay can also be expected. We confirm this small delay for our candidates in Section 3.2. Fourthly, the available data are required to show more than two cycles in both the W1 and W2 bands. It is important to note that even strictly sinusoidal variations are difficult to distinguish from a simple stochastic process when the number of cycles N cyc is ≲ 5 (Vaughan et al. 2016). Any finite light curve generated by the corresponding variability process represents only a snapshot of that process. The periodogram derived from this light curve will show power distributed around the actual Power Spectral Density (PSD) continuum with some scatter. This scatter can create the appearance of spikes in the periodogram (e.g., see Fig. 1 and Fig. 12), leading the light curve to seemingly exhibit a sinusoidal-like pattern over 2 to 3 cycles, even if the underlying PSD process is purely continuous. Therefore, these patterns can only be reliably distinguished with much longer time series spanning many (e.g., > 4-5) cycles of the putative period. However, most optical searches still adhere to a criterion of > 1.5 cycles due to the limited time span of the data (e.g. Graham et al. 2015a; Charisi et al. 2016; Liu et al. 2019b; Chen et al. 2024). Consequently, false positives would arise in the candidate sample from stochastic red noise variability (Vaughan et al. 2016). Due to the limited number of detected epochs, which amounts to 23, we set the limit to > two cycles. We believe that this is the best approach we can take with the current WISE data, but we note that further improvements should be pursued in future research with more available data. Fifthly, we check if the maximum power of the periodogram indicates a 'correct' period, since anomalous data points (e.g., an outlier with a small error) could mislead the periodogram calculation. Additionally, this criterion allows us to check if the light curves show strong variability. We estimate the period of a light curve by examining the data points. If all data points within a time segment satisfy mag ( ti ) -2 δ mag ( ti ) > offset (calculated by the periodogram), we classify it as a 'faint segment'. If all data points within a time segment satisfy mag ( ti ) + 2 δ mag ( ti ) < offset, we classify it as a 'bright segment'. Then we conduct a count in chronological order, recording each change from a faint segment to a bright segment or vice versa, and neglecting normal","pages":[4,5]},{"title":"FBQS J0832+2853","content":"data points in this process (see an example shown in Fig. 2). Denoting the total count as n , ideally n / 2 periods would be included in the time span. We require that for both the W1 and W2 bands, the period at the maximum power Tmax and the time span of the light curve T 0 satisfy Equation 6:","pages":[5]},{"title":"3.1. Simulating with Stochastic Variability","content":"AGN variability can roughly be described by a Gaussian first-order continuous autoregressive model (CAR(1), Kelly et al. 2009), also known as a damped random walk (DRW, MacLeod et al. 2010) or Ornstein-Uhlenbeck process. The likelihood function for the DRW process can be represented as: where the Xi represents the flux of the data point i , and q is the long-term mean of the light curve. The covariance matrix C is given by: where δ i j is the Kronecker delta. In our analysis, we assume a strong correlation between the W1 and W2 data. Subsequently, we have devised a modified form of the covariance matrix that accounts for this correlation, expressed as: where a and b represent the observed band (W1 or W2) of the data points, and ρ is the correlation coefficient. Regarding the data quality of WISE, we only use the binned data when estimating DRW parameters. We set a uniform prior for the logarithm of the parameters σ 1, σ 2 and τ , and a uniform prior for ρ W 1 , W 2. We set q as the mean magnitude of the light curve to reduce computational cost and use the Markov Chain Monte Carlo (MCMC) sampler emcee 3 (Foreman-Mackey et al. 2013) to construct the posterior samples of the parameters. However, we find that some light curves lack enough variability but pass the variability criterion (see Section 2.2) due to certain data points with very small errors. Consequently, they are not adequately constrained by the DRW model. To ensure a stable posterior for the MCMC processes, we set a criterion that the 1 σ error of the posterior lg τ sample must be less than 2. This is the final requirement to define our parent sample, resulting in a final number of 48,932 (see Section 2.2). The statistical distributions of σ , τ , and ρ W 1 , W 2 are illustrated in Fig. 4. Upon comparison, the selected candidates exhibit similar values for σ 1, σ 2, and τ compared to the parent sample. Notably, they display a stronger correlation, which aids in meeting selection criterion 4 easier. The distribution of ρ W 1 , W 2 serves as validation for our correlation assumption. As an alternative test, we perform a model selection to test if the correlation parameter is necessary to explain the light curve on top of a uncorrelated stochastic background. We adopt a maximum likelihood approach for the model comparison and parameter estimation. We use the Bayesian information criterion (BIC), which is defined as: where L is the likelihood function, k is the number of free model parameters, and N is the number of data points. A lower BIC value indicates a preferable model. Typically, when ∆ BIC < -10, the model with the lower BIC is strongly favored. For the uncorrelated DRW process, we generate posterior parameter samples independently for each band. Subsequently, we integrate these samples into a covariance matrix to compute L with all correlation components set as zero. Our analysis indicates a strong preference for the correlated model over the uncorrelated one, aligning with our earlier analysis (further details provided in Fig. 3). We proceed by randomly selecting a set of parameters to generate two correlated W1 and W2 simulated light curves. These simulated light curves are then compared with the actual observation cadence and measurement errors. This process is repeated 100,000 times to check that the selection criteria are met. In addition, we use the periodogram values of these simulated light curves to determine the significance of our candidates, as described in Criterion 1 in section 2.4. To further validate our simulation methodology, we perform a statistical comparison between the real and simulated light curves. Our analysis shows that they exhibit similar behaviour for each selection criterion (detailed results in Table 1), suggesting that the correlated DRW processes could potentially generate a comparable number of candidates to the real ones. Notably, the DRW model is relatively simple and may not fully capture the complexity of actual AGN variability due to degeneracies (Zu et al. 2013). When using the DRW assumption to calculate false alarm probabilities, it might not completely consider the assumptions of the red noise hypothesis. However, using different variability models could lead to some small changes in the results, but usually not significant ones.","pages":[5,6]},{"title":"3.2. Candidates of Periodic Sources","content":"Using the data and criteria outlined in Section 2, we have identified a total of 28 candidates. The catalog of candidates can be found in Table 2. The light curves and the GLS pe- riodograms of two examples are presented in Fig. 1 and the rest can be found in Fig. 12. The W1 and redshift distributions of both the parent sample and the selected candidates are shown in Fig 5. It is worth noting that the candidates are predominantly situated in the low redshift range (median z=0.126 , with 27/28 at z < 0.4) and bright region (median W1=12.69, 25/28 satisfy W1 < 14), which can be attributed to a selection effect making it easier for them to fulfill the selection criteria. In Section 2.4, we mention the possibility of a slight phase delay between the W2 and W1 light curves. This is confirmed by our candidates, with an average delay of 0.025 periods, as illustrated in Fig. 6. The maximum phase delay is 0.087 period, which remains smaller than the 1/8 period threshold used in our selection process, indicating a reasonable choice. As an alternative test, here we use the BIC method (see details in Section 3.1) to assess if an additional periodic signal is needed to explain the light curve on top of a stochastic background. The likelihood function for the DRW + sinusoidal model is written as: where Si represents the sinusoid signal. Our analysis shows that, for the vast majority of the candidates, BIC DRW + sin -BIC DRW( c ) < -10, indicating robust evidence that the periodic model is significantly favored over the pure stochastic model (refer to Fig. 3 for detailed results). However, this could be explained by the limitation of the DRW model, which might not adequately capture a spiky PSD and may be more suited to smooth continuum-like PSD shapes. Therefore, additional data in the future are necessary to obtain a higher density in frequency sampling for the periodogram and better test the preference for the sinusoid + DRW model. It's worth noting that the sinusoid +DRW model is a commonly used model due to its simplicity and clear physical picture. However, there exist alternative light curve models, such as the bursty model predicted by circumbinary accretion disk simulations (Farris et al. 2014). Despite the potential of this bursty model, we did not simulate it for our candidates as we could not observe 'bursty' characteristics in the WISE light curves. Here we compile the derived quantities for our analysis, which are presented in Table 2. The table includes essen- tial information such as the Milliquas Name, right ascension (R.A.), declination (Decl.), redshift ( z ), and quasar properties as provided by Wu & Shen (2022) and Liu et al. (2019a). Additionally, we have included periodic properties such as offsets of the W1 and W2 bands, and the period of the candidates.","pages":[6,7,8]},{"title":"3.3. False Alarm Probability","content":"To estimate the false alarm probability (FAP) for each candidate, we employ the method described in Chen et al. (2020). We estimate the approximate FAP using the effective number of independent frequencies Nef f , which is calculated by dividing the observed frequency window by the expected peak width δ f = 1 / T . The FAP is estimated as (e.g., VanderPlas 2018): Where Psingle = 1 -e Z , Z is the periodogram value defined in Zechmeister & Kürster (2009). Since candidates are selected based on the light curves of both the W1 band and W2 band, there are two P single and N eff. To simplify the calculation, we can estimate the periodogram FAP as Equation 13: It is important to acknowledge that this calculation assumes independent FAPs for the W1 and W2 bands, contrary to the strong correlation observed in the DRW simulation (refer to Section 3.1). Nevertheless, this method provides an upper limit estimate for the FAP, as single-band FAP calculations do not consider this correlation. NOTE- (1)-(4): Name, RA, DEC, and redshift of the objects. (5)-(6): Median magnitudes in the W1 and W2 bands, respectively. (7): Period (in the observer-frame) in units of days. (8): Legend of type/class from Flesch (2023): Q = QSO, type-I broad-line core-dominated, 860,100 of these. A = AGN, type-I Seyferts/host-dominated, 47,044 of these. B = BL Lac type object, 2,814 of these. (FSRQs are typed as QSOs here) K = NLQSO, type-II narrow-line core-dominated, 6,048 of these. N = NLAGN, type-II Seyferts/host-dominated, 39,768 of these. Incomplete, and includes an unquantified residue of legacy NELGs/ELGs/LINERs, plus some unclear AGN. This is the catch-all category. S = star classified but showing quasar-like photometry and radio/X-ray association, thus included as a quasar candidate; 124 of these. R = radio association displayed. X = X-ray association displayed. 2 = double radio lobes displayed (declared by data-driven algorithm). (9)-(11): Logarithmic bolometric luminosity in units of erg/s, black hole mass in units of solar mass, and Eddington ratio for sources in the SDSS DR16 quasar catalog (Wu & Shen 2022), the DR7 board-line AGN catalog (Liu et al. 2019a). Additionally, we empirically compute the global FAP using the 100,000 simulated light curves for each AGN, following a similar approach as presented by Barth & Stern (2018). The global FAP accounts for all false positives that satisfy all selection criteria within the explored frequency range. The global FAP values for candidates range from 4 . 0 × 10 -4 to 4 . 77 × 10 -3 , with a mean of 2 . 07 × 10 -3 by Gaussian fit. Further details are available in Fig. 7. It is noteworthy that the global FAPs are significantly higher than the periodogram FAPs. This discrepancy can be attributed to the stronger variability exhibited by the simulated light curves compared to real observations, leading to an increase in the global FAP. Moreover, while most false positives exhibit only a 3 σ significance level, certain candidates display notably stronger significance, such as SDSS J162938.09+362452.2 and 6dF J214907.4-175159 as depicted in Fig. 12.","pages":[8,9,10]},{"title":"4.1. Do the Candidates Show Distinctive Properties?","content":"One might wonder whether our selected periodic sample shows distinct features in other physical properties, such as bolometric luminosity ( L bol), black hole mass ( M BH) and Eddington ratio ( λ edd= L bol / L edd). To investigate this, we try to extract a subsample of candidates with previous measurements in these parameters. We chose to cross-match the candidates with two AGN samples based on Sloan Digital Sky Survey (SDSS), namely the SDSS DR16 quasar catalog (DR16Q) (Wu & Shen 2022) and DR7 broad-line AGN catalog (Liu et al. 2019a). They contained 750,414 and 14,584 sources, concentrated at high and low redshifts, respectively (see Figure 8). We only found a total of 10 candidates that overlapped with the two catalogs. Similar to the occupation of the candidates in the whole parent sample (see Figure 5), they are all in the low redshift and low magnitude space (see Figure 8) due to selection effects. Therefore, we need to construct a control sample for a fair comparison, at least in redshift and IR magnitudes. Specifically, we selected the 10 closest AGNs with a redshift ratio less than 1.01 and with differences in W1 and W2 magnitudes less than 0.1. In cases when no AGN meet the specified criteria, we choose the closest AGN in the catalog as the control sample. After controlling, we then compare the M BH, L bol and λ edd of the periodic sources with normal AGNs. Additionally, we have also considered another parameter dust covering factor ( fd ), defined as the ratio of the W1monochromatic luminosity ( LW 1 = ν W 1 F ν W 1) and the L bol ( fd = L W1 / L bol, e.g., Ma & Wang 2013), to indicate the extent of dust obscuration. However, no significant differences in the distributions of these parameters can be seen visually (see Fig. 9). We have also checked the mean and standard deviation (SD) of their distributions (see Table 3) and confirmed it. We further performed the K-S statistics for the L bol, M BH, λ edd, and fd distributions between the candidates and the control sample, yielding values of 0.375, 0.400, 0.225, and 0.400, respectively. The K-S tests suggest significant differences in these properties with a critical value of 0.480 for a 95% confidence level, so none of them show significant differences. It is worth emphasizing that the systematic uncertainties in the estimates of AGN properties (e.g., singleepoch BH mass, see discussions in Shen 2013) are nonnegligible and would dilute the difference, if there were any. This may be particularly important for the small sample size for comparison here. 4.2. Comparison with SMBHB Candidates Selected by Optical Periodicity As mentioned in the introduction, previous studies have identified samples of SMBHB candidates based on optical periodic light curves. It is intriguing to investigate whether there are any common sources between our IR-selected sample and these optically-selected samples. Such overlap would provide compelling evidence for the periodicity of the candidates. Additionally, if the periods derived from optical light curves align with those obtained from IR light curves, we can employ the dust-echo model to further explore the properties of the candidates. For this purpose, we compile a collection of optical SMBHB candidates reported in various literature, including Graham et al. (2015a); Charisi et al. (2016); Zheng et al. (2016); Liu et al. (2019b); Chen et al. (2020, 2024). Unfortunately, we do not find any overlap between our sample and those samples. The lack of matches is not entirely unexpected, as the two searches prioritize candidates with different periods. Specifically, we find that the periods of the optical candidates predominantly fell below 1700 days, z whereas the periods of our selected candidates were mostly longer than 1700 days (see details shown in Fig. 10). Additionally, most of our candidates have low redshifts (z < 0.4) due to the detection ability of WISE, while the redshifts of candidates from optical searches span from 0 to ∼ 3.","pages":[10,11]},{"title":"4.3. Optical Light Curves and A Possible Periodic Source","content":"Given the abundance of optical photometric archival data, we then examined the optical light curves of the 28 candidates using data from various surveys, such as the Catalina Real Time Transient Survey (CRTS, Drake et al. 2009), the Palomar Transient Factory (PTF, Rau et al. 2009), the Zwicky Transient Facility (ZTF, Masci et al. 2019), and the All-Sky Automated Survey for Supernovae (ASAS-SN, Kochanek mag et al. 2017). Their combined dataset provides optical light curves spanning about two decades. We have normalized the data from different surveys and binned them at 20-day intervals. However, they generally show a small amplitude of optical variability, probably due to either intrinsically weak variability in the optical band or too shallow surveys. Only one candidate, SDSS J140336.43+174136.1, shows significant variability and periodicity in its optical light curve (see Fig. 11). The periodogram analysis of the optical light curve of SDSS J140336.43+174136.1 reveals a period of 2891 ± 22 days, that is slightly longer than the period fitted from the IR light curves, which is 2346 ± 87 days. The errors are given by periodogram peak statistics based on 100,000 perturbed light curves for each real light curve. The discrepancy could be caused by the poor cadence of the WISE surveys, and so in the following analysis we choose the optical period for this particular source. We present the corresponding best-fitting sinusoids for the W1, W2, and optical light curves in Figure 11. The time delay between the W1 (W2) and optical light curves is 630 (727) days (in the rest frame), respectively. Assuming the simplest scenario of a hollow spherical shell of dust with the source located at its center, we can easily estimate that the inner radius of the shell is 0.53 pc. These values are roughly consistent with previous statistical correlations between the time delay and bolometric luminosity of AGNs. For example, using the relations obtained by Lyu et al. (2019), i.e. for the W1 band and for the W2 band, and a given bolometric luminosity of 10 45 . 81 erg s -1 for SDSS J140336.43+174136.1 (see Table 2), the W1 and W2 time delays are estimated to be 476 ± 147 days and 566 ± 158 days, respectively, which is in good agreement with our measurements.","pages":[11,12]},{"title":"5. CONCLUSION","content":"In this work, we have conducted the first systematic search for SMBHBs in the IR band and identified 28 AGNs exhibiting IR periodic variability using the decade-long WISE and NEOWISE light curves. We performed extensive simulations to evaluate the potential influence of stochastic variability on candidate selection. By counting all the false positives in the simulated light curves, the probability of these periodic light curves being generated by random processes is an average of 0.207% by Gaussian fit. However, the number can be reproduced by our mock simulations with the DRW process of the parent sample. This indicates the challenge of identifying reliable SMBHBs with periodic emission using only WISE light curves, which are constrained by their time span and visit cadence, necessitating a low threshold on the minimum number of cycles ( N cyc > 2). Consequently, the nature of these periodic sources, whether driven by SMBHBs or not, should be carefully tested in future observations. We subsequently investigated whether these IR periodic sources exhibit distinct properties compared to normal AGNs. However, no significant biases were found, and the small sample size prevents us from statistically identifying potential weak differences. On the other hand, we found that there is no overlap between our sample (IRselected) and the SMBHB candidates reported in the literature (optically selected), likely due to differences in their preferred period ranges. Interestingly, we have identified SDSS J140336.43+174136.1 as a candidate displaying periodicity in both optical and IR bands. Further extended and more sensitive observations of these candidates are essential to confirm whether or not they are real periodic sources in both optical and IR bands. As suggested by D'Orazio & Haiman (2017), the periodic sources can also help test the physical mechanisms that produce the observed periodicity, due to relativistic Doppler modulation or accretion rate fluctuations. This study highlights the promising potential of identifying SMBHB candidates through IR time-domain observations, which can significantly complement searches in other bands, although there are challenges in distinguishing genuine sources. It opens up a new avenue for SMBHB searches and can be applied to future surveys. Although NEOWISE unfortunately ended on July 31, 2024, the upcoming NEO Surveyor (Mainzer et al. 2023), the successor to NEOWISE , and the Nancy Grace Roman Space Telescope (Spergel et al. 2015) will ensure a bright future for IR time-domain astronomy, particularly for SMBHB searches (Haiman et al. 2023). These different surveys somewhat form a relay in time and provide us with IR light curves with a long time baseline, which is essential for verifying the periodicity of the candidates with N cyc < 4 and for detailed comparisons against physical models.","pages":[12,13]},{"title":"ACKNOWLEDGMENTS","content":"We sincerely thank the expert referee for very constructive and insightful comments, which helped improve our manuscript greatly. This work is supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB0550200), the National Natural Science Foundation of China (grants 12073025, 12192221, the science research grants from the China Manned Space Project and the Cyrus Chun Ying Tang Foundations. D.L. acknowledges the support from the National Undergraduate Training Program for Innovation and Entrepreneurship. X.L. acknowledges support by NSF grant AST-2206499. This research makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. This research also makes use of data products from NEOWISE-R, which is a project of the Jet Propulsion Laboratory/California Institute of Technology, funded by the Planetary Science Division of the National Aeronautics and Space Administration. This research has made use of the NASA/IPAC Infrared Science Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration.","pages":[13]},{"title":"REFERENCES","content":"Klein A., et al., 2016, PhRvD, 93, 024003","pages":[13]},{"title":"A. THE IR LIGHT CURVES OF THE PERIODIC AGN SAMPLE","content":"We have shown the light curves of 2 special candidates in Fig. 1. Here we show the light curves of the rest 26 candidates in Fig. 12.","pages":[15]}],"string":"[\n {\n \"title\": \"ABSTRACT\",\n \"content\": \"Periodic variability in active galactic nuclei (AGNs) is a promising method for studying sub-parsec supermassive black hole binaries (SMBHBs), which are a challenging detection target. While extensive searches have been made in the optical, X-ray and gamma-ray bands, systematic infrared (IR) studies remain limited. Using data from the Wide-field Infrared Survey Explorer (WISE), which provides unique decade-long mid-IR light curves with a six-month cadence, we have conducted the first systematic search for SMBHB candidates based on IR periodicity. Analyzing a parent sample of 48,932 objects selected from about half a million AGNs, we have identified 28 candidate periodic AGNs with periods ranging from 1,268 to 2,437 days (in the observer frame) by fitting their WISE light curves with sinusoidal functions. However, our mock simulation of the parent sample indicates that stochastic variability can actually produce a similar number of periodic sources, underscoring the difficulty in robustly identifying real periodic signals with WISE light curves, given their current sampling. Notably, we found no overlap between our sample and optical periodic sources, which can be explained by a distinct preference for certain periods due to selection bias. By combining archived data from different surveys, we have identified SDSS J140336.43+174136.1 as a candidate exhibiting periodic behavior in both optical and IR bands, a phenomenon that warrants further validation through observational tests. Our results highlight the potential of IR time-domain surveys, including future missions such as the Nancy GraceRoman Space Telescope, for identifying periodic AGNs, but complementary tests are still needed to determine their physical origins such as SMBHBs. Keywords: Active galactic nuclei (16); Infrared astronomy(786); Quasars (1319); Supermassive black holes (1663); Time domain astronomy (2109)\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"A Systematic Search for Candidate Supermassive Black Hole Binaries Using Periodic Mid-Infrared Light Curves of Active Galactic Nuclei\",\n \"content\": \"1 CAS Key Laboratory for Research in Galaxies and Cosmology, Department of Astronomy, University of Science and Technology of China, Hefei, 230026, China; jnac@ustc.edu.cn 2 School of Physical Sciences, University of Science and Technology of China, Hefei, 230026, China 3 School of Astronomy and Space Sciences, University of Science and Technology of China, Hefei, 230026, China 4 Department of Astronomy, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA 5 Center for Artificial Intelligence Innovation, University of Illinois at Urbana-Champaign, 1205 West Clark Street, Urbana, IL 61801, USA National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA 6 (Received 2024 May 24; Revised 2024 November 9; Accepted 2024 November 11) Submitted to ApJ\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1. INTRODUCTION\",\n \"content\": \"Supermassive black holes (SMBHs) are intriguing celestial objects that reside at the centers of galaxies (Kormendy & Ho 2013). When galaxies merge, it is anticipated that the central black holes will form binary systems known as SMBH binaries (SMBHBs, Begelman et al. 1980). These binary systems offer valuable insights into various astrophysical phenomena, including galaxy formation (Colpi & Dotti 2011), gravitational wave (GW) emission (Hughes 2009), and the evolution of SMBHs (Merritt 2013). More massive binaries are pulsar-timing array (PTA) sources (e.g., Arzoumanian et al. 2018), while less massive binaries are targeted by space-based experiments such as LISA (Klein et al. 2016). They provide a laboratory to directly test strong-field general relativity (Hughes 2009; Centrella et al. 2010). However, despite their theoretical significance, the direct electromagnetic detection of close SMBHBs below subparsec separation has proven to be a challenging endeavor (see recent reviews such as De Rosa et al. 2019; D'Orazio & Charisi 2023) while numerous SMBH pairs at larger scales have been identified through the presence of dual active galactic nuclei (AGNs) (e.g., Komossa et al. 2003; Zhou et al. 2004; Liu et al. 2010, 2011, 2013; Koss et al. 2011; Liu et al. 2018; Chen et al. 2022). When a binary has exhausted its in- teractions with stars but has not approached close enough to emit significant gravitational radiation, its orbit does not have an obvious mechanism for decay. This long-standing challenge presents a major uncertainty in estimating the abundance of SMBHB mergers as GW sources. In theory, the bottleneck may be overcome in gaseous environments (e.g., Gould & Rix 2000; Cuadra et al. 2009; Chapon et al. 2013; del Valle et al. 2015), in triaxial or axisymmetric galaxies (e.g., Khan et al. 2016; Kelley et al. 2017), and/or by interacting with a third BH in hierarchical mergers (e.g., Blaes et al. 2002; Kulkarni & Loeb 2012; Bonetti et al. 2018). A recent and promising approach for identifying potential close SMBHB candidates has emerged in the thriving field of time-domain astronomy, which focuses on analyzing the periodic light curves of AGNs. The periodicity observed in these light curves can arise from changes in the accretion rate onto the black holes (Graham et al. 2015b) or from the relativistic Doppler boost resulting from the highly relativistic motion of gas in the mini accretion disk around the smaller black hole in the binary system (D'Orazio et al. 2015). Extensive research has been conducted using optical light curves to search for SMBHBs, and the methods employed in these studies have become well-established (Graham et al. 2015a; Charisi et al. 2016; Zheng et al. 2016; Liu et al. 2019b; Chen et al. 2020; Liao et al. 2021; Chen et al. 2024). However, many of the known candidates have been shown to be subject to false positives due to stochastic quasar variability (e.g., Vaughan et al. 2016). Additionally, recent findings suggest that some AGNs may display periodic variations only at some specific epochs (e.g., Jiang et al. 2022; O'Neill et al. 2022) further enhance the difficulties of periodic searches. Furthermore, previous surveys were only sensitive to the most massive quasars at high redshift ( z ≳ 2) which should have already gone through their major merger process (e.g., Shen 2009). Regardless of the process that imprints periodicity, it is generally accepted that the timescale is primarily determined by the binary orbital period. SMBHBs with masses between 10 6 -10 10 M ⊙ and separations of 0.01 pc possess orbital periods that span from years to several decades, making them possibly detectable by current timedomain surveys. In addition to the optical band, the periodic searches have also been conducted in other wavelength regimes, such as in X-ray (Serafinelli et al. 2020; Liu et al. 2020) and gamma-ray bands (Sandrinelli et al. 2018; Holgado et al. 2018). Periodic infrared (IR) light curves are also expected as a result of reverberation mapping of the circumbinary dusty torus in periodic AGNs. However, the search for SMBHBs using IR data has not been extensively explored, despite the detection of an IR time lag in an individual SMBHB candidate (Jun et al. 2015), which highlights the potential of utilizing periodic IR light curves for SMBHB identifi- cation (D'Orazio & Haiman 2017). The IR band not only provides an alternative approach to searching for SMBHBs but also offers distinct advantages. Firstly, IR photons are less susceptible to dust extinction, allowing them to penetrate through obscuring material, which is often abundant in galactic nuclear regions and galaxy merger systems. By extending the search to the IR wavelength range, we can uncover a population of SMBHB candidates that may have been previously overlooked, thereby expanding our knowledge of the prevalence and characteristics of these elusive systems. Even for unobscured AGNs, the variability amplitudes in the mid-IR band are typically larger compared to the optical band due to lower background contamination. Additionally, as discussed in D'Orazio & Haiman (2017), the dust-echo model can assist in determining the origin of central periodicity and place constraints on the physical characteristics of SMBHBs and their dust torus. To address this, we perform the first systematic search for SMBHB candidates via periodic IR light curves. The data we use come from the Widefield Infrared Survey Explorer (WISE, Wright et al. 2010) and its successor NEOWISE (Mainzer et al. 2014), which have provided us with a public dataset from February 2010 to December 2023 with a half-year cadence except for a 2.5year gap for each object. In this study, we primarily focus on sinusoidal periodic signals, which have been commonly adopted in modeling the optical light curves of SMBHB candidates (e.g., D'Orazio et al. 2015), and on modeling the IR light curves using the dust-echo model (D'Orazio & Haiman 2017). The paper is structured as follows. Section 2 introduces the data and methods employed to search for IR periodic AGNs. Section 3 presents the search and corresponding analysis results, where we show compelling evidence demonstrating the highly improbable nature of these periodic light curves being generated by random processes. Section 4 compares selected candidates with normal AGNs and with previous studies that utilized optical light curves. Additionally, we discuss SDSS J140336.43+174136.1 as a candidate exhibiting periodic behavior in both optical and IR bands. Finally, we summarize the main results and suggest directions for future work in Section 5. We assume a cosmology with H 0 = 66 . 88 km s -1 Mpc -1 , Ω m =0 . 32, and Ω Λ =0 . 68, adopted from Planck Collaboration et al. (2020).\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"2.1. IR data\",\n \"content\": \"The WISE was originally launched to conduct a full-sky survey in four MIR bands centered at 3.4, 4.6, 12, and 22 µ m (labeled W1-W4) from February to August 2010 (Wright et al. 2010). The solid hydrogen cryogen used to cool the W3 and W4 instrumentation was depleted later and the spacecraft was placed in hibernation in 2011 February. However, it was reactivated in 2013 October under the new name NEOWISE -R, with only W1 and W2 operational, specifically to search for asteroids that could potentially pose a threat of impact on Earth (Mainzer et al. 2014). The WISE and its new mission NEOWISE operate in the same manner, that is, scanning a specific sky area every six months, with an average of 12 or more individual exposures taken within each epoch, typically spanning a single day. Consequently, each target in the sky has been observed 22-23 times separated by a six-month interval up to December 2023. The single-exposure photometry of WISE and NEOWISE are archived in the AllWISE Multiepoch Photometry Table and NEOWISE-R Single Exposure (L1b) Source Table 1 , respectively. The photometry we adopted is measured by pointspread function (PSF) profile fitting, which is suitable for quasars. We first binned the data within each epoch since the intraday IR variability is negligible except for very radioloud AGNs (Jiang et al. 2012) and the anticipated periods we aim to detect are on much longer timescales. The variance of the binned data was recalculated as shown in Equation 1: where ni denotes the number of data points within the i th epoch, mag ( ti ) denotes the mean magnitude within the i th epoch, and σ j denotes the photometric uncertainty of the j th data point. The first part represents the contribution from photometric uncertainty and short-term variability, while the second part represents the contribution from the system stability of WISE. We adopt σ s . s . = 0 . 029 mag for WISE and σ s . s . = 0 . 016 mag for NEOWISE from Lyu et al. (2019). Prior to binning, the data were filtered based on the quality flags in the catalog following Jiang et al. (2021). Moreover, outliers that fell outside the 3 σ range of the single data point at each epoch were eliminated, using the mean magnitude and mean photometric uncertainty at that epoch as references. The mean and variance of the single data points were then re-computed, resulting in the binned data used for our analysis. This step was crucial to ensure the accuracy and reliability of the data used in our study.\",\n \"pages\": [\n 2,\n 3\n ]\n },\n {\n \"title\": \"2.2. AGN Sample\",\n \"content\": \"The AGN sample we chose is the widely-used Million Quasar Catalog (v8, 2 August 2023; Flesch 2023). This is a compendium of 907,144 type-I QSOs and AGNs, largely complete from the literature up to 30 June 2023. 66,026 QSO candidates are also included, calculated via radio/X-ray association (including double radio lobes) as being 99% likely to be quasars. Blazars and type-II objects are also included, bringing the total count to 1,021,800. To ensure the purity of our candidates, we only selected objects labeled as types 'Q', 'A', 'B', 'K', 'N' in the catalog (see what the legends of these types refer to in the note of Table 2), as these are confirmed AGNs. As a result, the number of objects amounts to 955,744. To enable a reliable long-term variability analysis, we apply a criterion that requires a minimum of 15 detected epochs (about 2/3 of all detected epochs) for each source. This left us with a subset of 576,260 AGNs. In order to rationalize the computational effort in further analysis, we choose to analyze only AGNs with obvious variability. This selection strategy is crucial to ensure that candidates meet the criteria outlined in selection criterion 5 (see Section 2.4). Furthermore, light curves displaying subtle variability tend to pose a challenge in terms of model constraints within the DRW framework (see Section 3.1). To address this issue, we introduce a requirement that, for a given light curve, at least one data point must satisfy mag ( ti ) -2 δ mag ( ti ) > magmean and at least one data point should satisfy mag ( ti ) + 2 δ mag ( ti ) < magmean . This allows us to narrow down our sample to 53,496 AGNs, ensuring that the selected candidates have the necessary characteristics for further analysis. We also require that the light curves be well constrained by the DRW model (see section 3.1), resulting in a parent sample of 48,932 AGNs.\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"2.3. Finding Periodicity\",\n \"content\": \"To identify sources exhibiting periodic variability, we employed a sinusoidal function to fit the light curves of both W1 and W2 bands. The fitting process involved minimizing the following equation using the χ 2 statistics as Equation 2 below (D'Orazio & Haiman 2017): The Lomb-Scargle (LS) periodogram (Lomb 1976; Scargle 1982) is widely used for periodicity detection in unevenly sampled data. In our analysis, we utilized the generalized Lomb-Scargle (GLS) periodogram (Zechmeister & Kürster 2009) implemented in the pyastronomy 2 package. In the GLS periodogram, we calculate a series of 'powers' defined as Equation 3: where χ 2 0 is the residual of a model assuming no variability, and χ 2 is equivalent to Equation 2 in the GLS periodogram (Zechmeister & Kürster 2009). Compared to the LS periodogram, the GLS periodogram provides a more accurate frequency prediction by taking into consideration an offset and weights, which allows the calculation of the maximumpower in the GLS periodogram to be equivalent to minimizing Equation 2. The searched period range of the GLS periodogram is from 700 days to T 0 days, where T 0 is the time span of the light curve.\",\n \"pages\": [\n 3,\n 4\n ]\n },\n {\n \"title\": \"2.4. Selection Criteria\",\n \"content\": \"We selected reliable periodic candidates based on five criteria. First, to account for stochastic red noise variability, we adopt a comparative approach by measuring the performance of the candidates against that of simulated light curves, as described in Section 3.1. Rather than applying a single threshold to the normalized periodograms, we require 3 σ significance in both the W1 and W2 bands. Secondly, we adopt the signal-to-noise (S/N) ratio defined as Equation 4 (Horne & Baliunas 1986), where A 0 denotes the amplitude and σ 2 r denotes the variance of the residuals. we require that ξ > 2 for both the W1 and W2 bands. Thirdly, we require that the difference between the frequencies of the W1 and W2 light curves must not exceed 5% of their sum, in accordance with Equation 5: where f 1 and f 2 represent the frequencies of the W1 and W2 light curves. In addition, the phase difference between them must be less than 1/8 cycles. This criterion reflects the requirements of the dust-echo model and guarantees that the two sinusoids have the same frequency and phase. However, we allow for a margin of error of 1/8 cycles to account for potential minor phase differences arising from low photometric accuracy or intrinsic characteristics of the IR light curves. For instance, the W2 band may receive a greater contribution from dust with lower temperatures located further from the BH, leading to a slight delay in the W2 variability compared to the W1 band (Lyu et al. 2019). If the sources are located at high redshifts, i.e. z > 2, the W1 and W2 emission could mainly come from a relatively inner and outer part of the accretion disk, then a time delay can also be expected. We confirm this small delay for our candidates in Section 3.2. Fourthly, the available data are required to show more than two cycles in both the W1 and W2 bands. It is important to note that even strictly sinusoidal variations are difficult to distinguish from a simple stochastic process when the number of cycles N cyc is ≲ 5 (Vaughan et al. 2016). Any finite light curve generated by the corresponding variability process represents only a snapshot of that process. The periodogram derived from this light curve will show power distributed around the actual Power Spectral Density (PSD) continuum with some scatter. This scatter can create the appearance of spikes in the periodogram (e.g., see Fig. 1 and Fig. 12), leading the light curve to seemingly exhibit a sinusoidal-like pattern over 2 to 3 cycles, even if the underlying PSD process is purely continuous. Therefore, these patterns can only be reliably distinguished with much longer time series spanning many (e.g., > 4-5) cycles of the putative period. However, most optical searches still adhere to a criterion of > 1.5 cycles due to the limited time span of the data (e.g. Graham et al. 2015a; Charisi et al. 2016; Liu et al. 2019b; Chen et al. 2024). Consequently, false positives would arise in the candidate sample from stochastic red noise variability (Vaughan et al. 2016). Due to the limited number of detected epochs, which amounts to 23, we set the limit to > two cycles. We believe that this is the best approach we can take with the current WISE data, but we note that further improvements should be pursued in future research with more available data. Fifthly, we check if the maximum power of the periodogram indicates a 'correct' period, since anomalous data points (e.g., an outlier with a small error) could mislead the periodogram calculation. Additionally, this criterion allows us to check if the light curves show strong variability. We estimate the period of a light curve by examining the data points. If all data points within a time segment satisfy mag ( ti ) -2 δ mag ( ti ) > offset (calculated by the periodogram), we classify it as a 'faint segment'. If all data points within a time segment satisfy mag ( ti ) + 2 δ mag ( ti ) < offset, we classify it as a 'bright segment'. Then we conduct a count in chronological order, recording each change from a faint segment to a bright segment or vice versa, and neglecting normal\",\n \"pages\": [\n 4,\n 5\n ]\n },\n {\n \"title\": \"FBQS J0832+2853\",\n \"content\": \"data points in this process (see an example shown in Fig. 2). Denoting the total count as n , ideally n / 2 periods would be included in the time span. We require that for both the W1 and W2 bands, the period at the maximum power Tmax and the time span of the light curve T 0 satisfy Equation 6:\",\n \"pages\": [\n 5\n ]\n },\n {\n \"title\": \"3.1. Simulating with Stochastic Variability\",\n \"content\": \"AGN variability can roughly be described by a Gaussian first-order continuous autoregressive model (CAR(1), Kelly et al. 2009), also known as a damped random walk (DRW, MacLeod et al. 2010) or Ornstein-Uhlenbeck process. The likelihood function for the DRW process can be represented as: where the Xi represents the flux of the data point i , and q is the long-term mean of the light curve. The covariance matrix C is given by: where δ i j is the Kronecker delta. In our analysis, we assume a strong correlation between the W1 and W2 data. Subsequently, we have devised a modified form of the covariance matrix that accounts for this correlation, expressed as: where a and b represent the observed band (W1 or W2) of the data points, and ρ is the correlation coefficient. Regarding the data quality of WISE, we only use the binned data when estimating DRW parameters. We set a uniform prior for the logarithm of the parameters σ 1, σ 2 and τ , and a uniform prior for ρ W 1 , W 2. We set q as the mean magnitude of the light curve to reduce computational cost and use the Markov Chain Monte Carlo (MCMC) sampler emcee 3 (Foreman-Mackey et al. 2013) to construct the posterior samples of the parameters. However, we find that some light curves lack enough variability but pass the variability criterion (see Section 2.2) due to certain data points with very small errors. Consequently, they are not adequately constrained by the DRW model. To ensure a stable posterior for the MCMC processes, we set a criterion that the 1 σ error of the posterior lg τ sample must be less than 2. This is the final requirement to define our parent sample, resulting in a final number of 48,932 (see Section 2.2). The statistical distributions of σ , τ , and ρ W 1 , W 2 are illustrated in Fig. 4. Upon comparison, the selected candidates exhibit similar values for σ 1, σ 2, and τ compared to the parent sample. Notably, they display a stronger correlation, which aids in meeting selection criterion 4 easier. The distribution of ρ W 1 , W 2 serves as validation for our correlation assumption. As an alternative test, we perform a model selection to test if the correlation parameter is necessary to explain the light curve on top of a uncorrelated stochastic background. We adopt a maximum likelihood approach for the model comparison and parameter estimation. We use the Bayesian information criterion (BIC), which is defined as: where L is the likelihood function, k is the number of free model parameters, and N is the number of data points. A lower BIC value indicates a preferable model. Typically, when ∆ BIC < -10, the model with the lower BIC is strongly favored. For the uncorrelated DRW process, we generate posterior parameter samples independently for each band. Subsequently, we integrate these samples into a covariance matrix to compute L with all correlation components set as zero. Our analysis indicates a strong preference for the correlated model over the uncorrelated one, aligning with our earlier analysis (further details provided in Fig. 3). We proceed by randomly selecting a set of parameters to generate two correlated W1 and W2 simulated light curves. These simulated light curves are then compared with the actual observation cadence and measurement errors. This process is repeated 100,000 times to check that the selection criteria are met. In addition, we use the periodogram values of these simulated light curves to determine the significance of our candidates, as described in Criterion 1 in section 2.4. To further validate our simulation methodology, we perform a statistical comparison between the real and simulated light curves. Our analysis shows that they exhibit similar behaviour for each selection criterion (detailed results in Table 1), suggesting that the correlated DRW processes could potentially generate a comparable number of candidates to the real ones. Notably, the DRW model is relatively simple and may not fully capture the complexity of actual AGN variability due to degeneracies (Zu et al. 2013). When using the DRW assumption to calculate false alarm probabilities, it might not completely consider the assumptions of the red noise hypothesis. However, using different variability models could lead to some small changes in the results, but usually not significant ones.\",\n \"pages\": [\n 5,\n 6\n ]\n },\n {\n \"title\": \"3.2. Candidates of Periodic Sources\",\n \"content\": \"Using the data and criteria outlined in Section 2, we have identified a total of 28 candidates. The catalog of candidates can be found in Table 2. The light curves and the GLS pe- riodograms of two examples are presented in Fig. 1 and the rest can be found in Fig. 12. The W1 and redshift distributions of both the parent sample and the selected candidates are shown in Fig 5. It is worth noting that the candidates are predominantly situated in the low redshift range (median z=0.126 , with 27/28 at z < 0.4) and bright region (median W1=12.69, 25/28 satisfy W1 < 14), which can be attributed to a selection effect making it easier for them to fulfill the selection criteria. In Section 2.4, we mention the possibility of a slight phase delay between the W2 and W1 light curves. This is confirmed by our candidates, with an average delay of 0.025 periods, as illustrated in Fig. 6. The maximum phase delay is 0.087 period, which remains smaller than the 1/8 period threshold used in our selection process, indicating a reasonable choice. As an alternative test, here we use the BIC method (see details in Section 3.1) to assess if an additional periodic signal is needed to explain the light curve on top of a stochastic background. The likelihood function for the DRW + sinusoidal model is written as: where Si represents the sinusoid signal. Our analysis shows that, for the vast majority of the candidates, BIC DRW + sin -BIC DRW( c ) < -10, indicating robust evidence that the periodic model is significantly favored over the pure stochastic model (refer to Fig. 3 for detailed results). However, this could be explained by the limitation of the DRW model, which might not adequately capture a spiky PSD and may be more suited to smooth continuum-like PSD shapes. Therefore, additional data in the future are necessary to obtain a higher density in frequency sampling for the periodogram and better test the preference for the sinusoid + DRW model. It's worth noting that the sinusoid +DRW model is a commonly used model due to its simplicity and clear physical picture. However, there exist alternative light curve models, such as the bursty model predicted by circumbinary accretion disk simulations (Farris et al. 2014). Despite the potential of this bursty model, we did not simulate it for our candidates as we could not observe 'bursty' characteristics in the WISE light curves. Here we compile the derived quantities for our analysis, which are presented in Table 2. The table includes essen- tial information such as the Milliquas Name, right ascension (R.A.), declination (Decl.), redshift ( z ), and quasar properties as provided by Wu & Shen (2022) and Liu et al. (2019a). Additionally, we have included periodic properties such as offsets of the W1 and W2 bands, and the period of the candidates.\",\n \"pages\": [\n 6,\n 7,\n 8\n ]\n },\n {\n \"title\": \"3.3. False Alarm Probability\",\n \"content\": \"To estimate the false alarm probability (FAP) for each candidate, we employ the method described in Chen et al. (2020). We estimate the approximate FAP using the effective number of independent frequencies Nef f , which is calculated by dividing the observed frequency window by the expected peak width δ f = 1 / T . The FAP is estimated as (e.g., VanderPlas 2018): Where Psingle = 1 -e Z , Z is the periodogram value defined in Zechmeister & Kürster (2009). Since candidates are selected based on the light curves of both the W1 band and W2 band, there are two P single and N eff. To simplify the calculation, we can estimate the periodogram FAP as Equation 13: It is important to acknowledge that this calculation assumes independent FAPs for the W1 and W2 bands, contrary to the strong correlation observed in the DRW simulation (refer to Section 3.1). Nevertheless, this method provides an upper limit estimate for the FAP, as single-band FAP calculations do not consider this correlation. NOTE- (1)-(4): Name, RA, DEC, and redshift of the objects. (5)-(6): Median magnitudes in the W1 and W2 bands, respectively. (7): Period (in the observer-frame) in units of days. (8): Legend of type/class from Flesch (2023): Q = QSO, type-I broad-line core-dominated, 860,100 of these. A = AGN, type-I Seyferts/host-dominated, 47,044 of these. B = BL Lac type object, 2,814 of these. (FSRQs are typed as QSOs here) K = NLQSO, type-II narrow-line core-dominated, 6,048 of these. N = NLAGN, type-II Seyferts/host-dominated, 39,768 of these. Incomplete, and includes an unquantified residue of legacy NELGs/ELGs/LINERs, plus some unclear AGN. This is the catch-all category. S = star classified but showing quasar-like photometry and radio/X-ray association, thus included as a quasar candidate; 124 of these. R = radio association displayed. X = X-ray association displayed. 2 = double radio lobes displayed (declared by data-driven algorithm). (9)-(11): Logarithmic bolometric luminosity in units of erg/s, black hole mass in units of solar mass, and Eddington ratio for sources in the SDSS DR16 quasar catalog (Wu & Shen 2022), the DR7 board-line AGN catalog (Liu et al. 2019a). Additionally, we empirically compute the global FAP using the 100,000 simulated light curves for each AGN, following a similar approach as presented by Barth & Stern (2018). The global FAP accounts for all false positives that satisfy all selection criteria within the explored frequency range. The global FAP values for candidates range from 4 . 0 × 10 -4 to 4 . 77 × 10 -3 , with a mean of 2 . 07 × 10 -3 by Gaussian fit. Further details are available in Fig. 7. It is noteworthy that the global FAPs are significantly higher than the periodogram FAPs. This discrepancy can be attributed to the stronger variability exhibited by the simulated light curves compared to real observations, leading to an increase in the global FAP. Moreover, while most false positives exhibit only a 3 σ significance level, certain candidates display notably stronger significance, such as SDSS J162938.09+362452.2 and 6dF J214907.4-175159 as depicted in Fig. 12.\",\n \"pages\": [\n 8,\n 9,\n 10\n ]\n },\n {\n \"title\": \"4.1. Do the Candidates Show Distinctive Properties?\",\n \"content\": \"One might wonder whether our selected periodic sample shows distinct features in other physical properties, such as bolometric luminosity ( L bol), black hole mass ( M BH) and Eddington ratio ( λ edd= L bol / L edd). To investigate this, we try to extract a subsample of candidates with previous measurements in these parameters. We chose to cross-match the candidates with two AGN samples based on Sloan Digital Sky Survey (SDSS), namely the SDSS DR16 quasar catalog (DR16Q) (Wu & Shen 2022) and DR7 broad-line AGN catalog (Liu et al. 2019a). They contained 750,414 and 14,584 sources, concentrated at high and low redshifts, respectively (see Figure 8). We only found a total of 10 candidates that overlapped with the two catalogs. Similar to the occupation of the candidates in the whole parent sample (see Figure 5), they are all in the low redshift and low magnitude space (see Figure 8) due to selection effects. Therefore, we need to construct a control sample for a fair comparison, at least in redshift and IR magnitudes. Specifically, we selected the 10 closest AGNs with a redshift ratio less than 1.01 and with differences in W1 and W2 magnitudes less than 0.1. In cases when no AGN meet the specified criteria, we choose the closest AGN in the catalog as the control sample. After controlling, we then compare the M BH, L bol and λ edd of the periodic sources with normal AGNs. Additionally, we have also considered another parameter dust covering factor ( fd ), defined as the ratio of the W1monochromatic luminosity ( LW 1 = ν W 1 F ν W 1) and the L bol ( fd = L W1 / L bol, e.g., Ma & Wang 2013), to indicate the extent of dust obscuration. However, no significant differences in the distributions of these parameters can be seen visually (see Fig. 9). We have also checked the mean and standard deviation (SD) of their distributions (see Table 3) and confirmed it. We further performed the K-S statistics for the L bol, M BH, λ edd, and fd distributions between the candidates and the control sample, yielding values of 0.375, 0.400, 0.225, and 0.400, respectively. The K-S tests suggest significant differences in these properties with a critical value of 0.480 for a 95% confidence level, so none of them show significant differences. It is worth emphasizing that the systematic uncertainties in the estimates of AGN properties (e.g., singleepoch BH mass, see discussions in Shen 2013) are nonnegligible and would dilute the difference, if there were any. This may be particularly important for the small sample size for comparison here. 4.2. Comparison with SMBHB Candidates Selected by Optical Periodicity As mentioned in the introduction, previous studies have identified samples of SMBHB candidates based on optical periodic light curves. It is intriguing to investigate whether there are any common sources between our IR-selected sample and these optically-selected samples. Such overlap would provide compelling evidence for the periodicity of the candidates. Additionally, if the periods derived from optical light curves align with those obtained from IR light curves, we can employ the dust-echo model to further explore the properties of the candidates. For this purpose, we compile a collection of optical SMBHB candidates reported in various literature, including Graham et al. (2015a); Charisi et al. (2016); Zheng et al. (2016); Liu et al. (2019b); Chen et al. (2020, 2024). Unfortunately, we do not find any overlap between our sample and those samples. The lack of matches is not entirely unexpected, as the two searches prioritize candidates with different periods. Specifically, we find that the periods of the optical candidates predominantly fell below 1700 days, z whereas the periods of our selected candidates were mostly longer than 1700 days (see details shown in Fig. 10). Additionally, most of our candidates have low redshifts (z < 0.4) due to the detection ability of WISE, while the redshifts of candidates from optical searches span from 0 to ∼ 3.\",\n \"pages\": [\n 10,\n 11\n ]\n },\n {\n \"title\": \"4.3. Optical Light Curves and A Possible Periodic Source\",\n \"content\": \"Given the abundance of optical photometric archival data, we then examined the optical light curves of the 28 candidates using data from various surveys, such as the Catalina Real Time Transient Survey (CRTS, Drake et al. 2009), the Palomar Transient Factory (PTF, Rau et al. 2009), the Zwicky Transient Facility (ZTF, Masci et al. 2019), and the All-Sky Automated Survey for Supernovae (ASAS-SN, Kochanek mag et al. 2017). Their combined dataset provides optical light curves spanning about two decades. We have normalized the data from different surveys and binned them at 20-day intervals. However, they generally show a small amplitude of optical variability, probably due to either intrinsically weak variability in the optical band or too shallow surveys. Only one candidate, SDSS J140336.43+174136.1, shows significant variability and periodicity in its optical light curve (see Fig. 11). The periodogram analysis of the optical light curve of SDSS J140336.43+174136.1 reveals a period of 2891 ± 22 days, that is slightly longer than the period fitted from the IR light curves, which is 2346 ± 87 days. The errors are given by periodogram peak statistics based on 100,000 perturbed light curves for each real light curve. The discrepancy could be caused by the poor cadence of the WISE surveys, and so in the following analysis we choose the optical period for this particular source. We present the corresponding best-fitting sinusoids for the W1, W2, and optical light curves in Figure 11. The time delay between the W1 (W2) and optical light curves is 630 (727) days (in the rest frame), respectively. Assuming the simplest scenario of a hollow spherical shell of dust with the source located at its center, we can easily estimate that the inner radius of the shell is 0.53 pc. These values are roughly consistent with previous statistical correlations between the time delay and bolometric luminosity of AGNs. For example, using the relations obtained by Lyu et al. (2019), i.e. for the W1 band and for the W2 band, and a given bolometric luminosity of 10 45 . 81 erg s -1 for SDSS J140336.43+174136.1 (see Table 2), the W1 and W2 time delays are estimated to be 476 ± 147 days and 566 ± 158 days, respectively, which is in good agreement with our measurements.\",\n \"pages\": [\n 11,\n 12\n ]\n },\n {\n \"title\": \"5. CONCLUSION\",\n \"content\": \"In this work, we have conducted the first systematic search for SMBHBs in the IR band and identified 28 AGNs exhibiting IR periodic variability using the decade-long WISE and NEOWISE light curves. We performed extensive simulations to evaluate the potential influence of stochastic variability on candidate selection. By counting all the false positives in the simulated light curves, the probability of these periodic light curves being generated by random processes is an average of 0.207% by Gaussian fit. However, the number can be reproduced by our mock simulations with the DRW process of the parent sample. This indicates the challenge of identifying reliable SMBHBs with periodic emission using only WISE light curves, which are constrained by their time span and visit cadence, necessitating a low threshold on the minimum number of cycles ( N cyc > 2). Consequently, the nature of these periodic sources, whether driven by SMBHBs or not, should be carefully tested in future observations. We subsequently investigated whether these IR periodic sources exhibit distinct properties compared to normal AGNs. However, no significant biases were found, and the small sample size prevents us from statistically identifying potential weak differences. On the other hand, we found that there is no overlap between our sample (IRselected) and the SMBHB candidates reported in the literature (optically selected), likely due to differences in their preferred period ranges. Interestingly, we have identified SDSS J140336.43+174136.1 as a candidate displaying periodicity in both optical and IR bands. Further extended and more sensitive observations of these candidates are essential to confirm whether or not they are real periodic sources in both optical and IR bands. As suggested by D'Orazio & Haiman (2017), the periodic sources can also help test the physical mechanisms that produce the observed periodicity, due to relativistic Doppler modulation or accretion rate fluctuations. This study highlights the promising potential of identifying SMBHB candidates through IR time-domain observations, which can significantly complement searches in other bands, although there are challenges in distinguishing genuine sources. It opens up a new avenue for SMBHB searches and can be applied to future surveys. Although NEOWISE unfortunately ended on July 31, 2024, the upcoming NEO Surveyor (Mainzer et al. 2023), the successor to NEOWISE , and the Nancy Grace Roman Space Telescope (Spergel et al. 2015) will ensure a bright future for IR time-domain astronomy, particularly for SMBHB searches (Haiman et al. 2023). These different surveys somewhat form a relay in time and provide us with IR light curves with a long time baseline, which is essential for verifying the periodicity of the candidates with N cyc < 4 and for detailed comparisons against physical models.\",\n \"pages\": [\n 12,\n 13\n ]\n },\n {\n \"title\": \"ACKNOWLEDGMENTS\",\n \"content\": \"We sincerely thank the expert referee for very constructive and insightful comments, which helped improve our manuscript greatly. This work is supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB0550200), the National Natural Science Foundation of China (grants 12073025, 12192221, the science research grants from the China Manned Space Project and the Cyrus Chun Ying Tang Foundations. D.L. acknowledges the support from the National Undergraduate Training Program for Innovation and Entrepreneurship. X.L. acknowledges support by NSF grant AST-2206499. This research makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. This research also makes use of data products from NEOWISE-R, which is a project of the Jet Propulsion Laboratory/California Institute of Technology, funded by the Planetary Science Division of the National Aeronautics and Space Administration. This research has made use of the NASA/IPAC Infrared Science Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration.\",\n \"pages\": [\n 13\n ]\n },\n {\n \"title\": \"REFERENCES\",\n \"content\": \"Klein A., et al., 2016, PhRvD, 93, 024003\",\n \"pages\": [\n 13\n ]\n },\n {\n \"title\": \"A. THE IR LIGHT CURVES OF THE PERIODIC AGN SAMPLE\",\n \"content\": \"We have shown the light curves of 2 special candidates in Fig. 1. Here we show the light curves of the rest 26 candidates in Fig. 12.\",\n \"pages\": [\n 15\n ]\n }\n]"}}},{"rowIdx":4971,"cells":{"bibcode":{"kind":"string","value":"2024arXiv241107305R"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/2411.07305.pdf"},"content":{"kind":"string","value":"\nPICZL : Image-based photometric redshifts for AGN\nWilliam Roster , ⋆ 1 , M. Salvato 1 , 2 , S. Krippendorf 3 , 4 , A. Saxena 5 , R. Shirley 1 , J. Buchner 1 , J. Wolf 2 , 6 , T. Dwelly 1 , F. E. Bauer 7 , 8 , 9 , 10 , J. Aird 11 , C. Ricci 12 , 13 , R. J. Assef 12 , S.F. Anderson 14 , X. Liu 15 , 16 , 17 , A. Merloni 1 , J. Weller 1 , 4\nK. Nandra 1\n(A ffi liations can be found after the references)\nNovember 14, 2024\nABSTRACT\nContext. Computing reliable photometric redshifts (photo-z) for active galactic nuclei (AGN) is a challenging task, primarily due to the complex interplay between the unresolved relative emissions associated with the supermassive black hole and its host galaxy. Spectral energy distribution (SED) fitting methods, while e ff ective for galaxies and AGN in pencil-beam surveys, face limitations in wide or all-sky surveys with fewer bands available, lacking the ability to accurately capture the AGN contribution to the SED, hindering reliable redshift estimation. This limitation is a ff ecting the many 10s of millions of AGN detected in existing datasets, e.g., those AGN clearly singled out and identified by SRG / eROSITA. Aims. Our goal is to enhance photometric redshift performance for AGN in all-sky surveys while simultaneously simplifying the approach by avoiding the need to merge multiple data sets. Instead, we employ readily available data products from the 10th Data Release of the Imaging Legacy Survey for the Dark Energy Spectroscopic Instrument, which covers > 20,000 deg 2 of extragalactic sky with deep imaging and catalogbased photometry in the grizW1-W4 bands. We fully utilize the spatial flux distribution in the vicinity of each source to produce reliable photo-z. Methods. We introduce PICZL, a machine-learning algorithm leveraging an ensemble of convolutional neural networks. Utilizing a cross-channel approach, the algorithm integrates distinct SED features from images with those obtained from catalog-level data. Full probability distributions are achieved via the integration of Gaussian mixture models.\nResults. On a validation sample of 8098 AGN, PICZL achieves an accuracy σ NMAD of 4.5% with an outlier fraction η of 5.6%. These results significantly outperform previous attempts to compute accurate photo-z for AGN using machine learning. We highlight that the model's performance depends on many variables, predominantly the depth of the data and associated photometric error. A thorough evaluation of these dependencies is presented in the paper.\nConclusions. Our streamlined methodology maintains consistent performance across the entire survey area, when accounting for di ff ering data quality. The same approach can be adopted for future deep photometric surveys such as LSST and Euclid, showcasing its potential for wide-scale realization. With this paper, we release updated photo-z (including errors) for the XMM-SERVS W-CDF-S, ELAIS-S1 and LSS fields.\nKey words. Photo-z, AGN, Extragalactic Surveys, Machine Learning\n1. Introduction\nIn recent decades, our understanding of Active Galactic Nuclei (AGN) and their role in galaxy and cosmic evolution has significantly advanced. These luminous celestial powerhouses are thought to be fueled by the accretion of matter onto supermassive black holes (SMBHs) located at the centers of galaxies, exerting intense energetic radiation across the entire electromagnetic spectrum, ranging from radio to γ -rays (Padovani et al. 2017). The close correlation observed between the mass of the central SMBH, whether active or inactive and the properties of its host galaxy's bulge - such as the galaxy's mass and velocity dispersion (e.g., Gebhardt et al. 2000; Ferrarese & Merritt 2000) - suggests a co-evolutionary relationship between galaxies and their central engines (Kormendy & Ho 2013; Heckman & Best 2014). Ongoing research focuses on understanding scaling relations, the evolution of SMBHs within galaxies, and the interconnected rates of star formation (SFR) and black hole accretion (BHAR) over cosmic time (e.g., Madau & Dickinson 2014). To further explore and address these unresolved topics requires diverse AGN samples with reliable redshifts to determine BH demographics and constrain models of galaxy evolution. For all these studies, redshift is an indispensable quantity, with spectroscopic redshifts (spec-z) remaining the preferred es-\nimates for determining precise cosmic distances (Hoyle 2016). However, while multi-objects spectrographs, such as the Sloan Digital Sky Survey (SDSS-V; York et al. 2000; Kollmeier et al. 2019), the Dark Energy Spectroscopic Instrument (DESI; DESICollaboration et al. 2016), the Subaru Prime Focus Spectrograph (PFS; Tamura et al. 2016) or the 4-metre Multi-Object Spectroscopic Telescope (4MOST; De Jong et al. 2019), are set to provide a drastic rise in the number of observed sources over the next several years, we are currently in the situation in which millions of AGN have been detected all-sky by various surveys (e.g., by the Wide-field Infrared Survey Explorer mission (WISE; Wright et al. 2010), and the extended Roentgen Survey with an Imaging Telescope Array (eROSITA; Merloni et al. 2012; Predehl et al. 2021), with only the brightest sources having been observed spectroscopically (Dahlen et al. 2013). The growing disparity between photometric and spectroscopic observations will only widen with upcoming surveys such as the Legacy Survey of Space and Time (LSST; Ivezic et al. 2019) and Euclid (Collaboration et al. 2024), covering unprecedented areas and depths (Newman & Gruen 2022). Thus for the bulk of AGN, we must make use of multiband photometry and rely on photometric redshifts (photo-z).\nFirst implemented by Baum (1957) for inactive galaxies, these low-precision redshift estimates utilize photometric observations to e ff ectively obtain a sparsely sampled spectral energy distribution (SED), trading precision for scalability. They en-\ncompass an array of techniques assuming color-redshift evolution (Connolly et al. 1995; Steidel et al. 1996; Illingworth 1999; Bell et al. 2004), including template-based approaches (e.g., Bolzonella et al. 2000; Ilbert et al. 2006; Salvato et al. 2008; Beck et al. 2017), where redshifted models built on theoretical or empirical SEDs are fitted to observed multi-band photometry. Although a limited number of available bands can introduce uncertainties (see review by Salvato et al. 2018), photo-z methods o ff er an e ffi cient way to estimate distances for all sources in an imaging survey, yielding highly accurate estimates with as few as three bands for passive galaxies (Benitez 2000; Abdalla et al. 2011; Arnouts & Ilbert 2011; Brescia et al. 2014; Desprez et al. 2020). By contrast, reliable photo-z for AGN have historically required highly homogenized photometry across > 20 filters, which was only achievable in pencil-beam surveys (Salvato et al. 2011). As such, this level of detail continues to be unfeasible for wide-area surveys. However, with the 10th data release of the DESI Legacy Imaging Surveys (LS10, Dey et al. 2019), we now have a broad-sky survey that, while lacking NIR coverage, includes a few optical bands supplemented by mid-IR WISE data. This allows us to explore the possibility of generating reliable photo-z for AGN over the full sky, despite having fewer filters compared to the densely sampled pencil-beam surveys.\nSED fitting applied to a broad population of AGN remains particularly challenging due to the uncertainty and di ffi culty of disentangling the relative contributions of the nucleus and respective host to a given band (e.g., Luo et al. 2010; Salvato et al. 2011; Brescia et al. 2019). Since the accretion properties of SMBHs, often characterized as the bolometric luminosity divided by the Eddington limit, or the Eddington ratio, significantly influence the SED of AGN, the intense power-law continuum radiation can either partly (host-dominated) or entirely (quasar-dominated), outshine the respective host, hiding key spectral features that lead to redshift degeneracies (Pierce et al. 2010; Povi'c et al. 2012; Bettoni et al. 2015). Consequently, selecting a limited number of templates can be insufficient for correct redshift determination, while increasing the number of templates raises the degeneracy (see discussion in Salvato et al. 2011; Ananna et al. 2017). In this regime of accounting for AGN contributions to galaxy photo-z, one potential approach involves modeling objects as a combination of quasar and galaxy templates (eg., Cardamone et al. 2010), performed with EAZY (Brammer et al. 2008). In addition, surveys typically estimate fluxes with models that do not account for a mixed contribution from AGN and host galaxy. Ultimately, AGN are also intrinsically variable sources on the timescales explored by the previously mentioned surveys leading to incongruent photometry acquired across di ff erent epochs.\nIn contrast to template-fitting methods, more recent approaches have shifted towards the use of empirical Machine Learning (ML) models, performing regression or classification, to tackle photo-z applied predominantly to inactive galaxies (Collister & Lahav 2004; Laurino et al. 2011; Zhang et al. 2013; Hoyle 2016; D'Isanto & Polsterer 2018; Brescia et al. 2019; Eriksen et al. 2020; Li et al. 2021). Provided with a very large and complete spec-z sample, ML architectures manipulate photometric input features to minimize the divergence between spectroscopic and ML-derived redshifts. Over the years, a plethora of ML architectures, including decision trees (Breiman 2001; Carliles et al. 2010; Li et al. 2022), Gaussian processes (Almosallam et al. 2016) and K-nearest neighbours (Zhang et al. 2013; Luken et al. 2019) have been employed, yielding accurate point predictions and, more interestingly, full probability density functions (PDFs) (Kind & Brunner 2013; Cavuoti et al.\n2016; Rau et al. 2015; Sadeh et al. 2016). The latter grants access to the prediction uncertainty, as otherwise naturally provided by template-fitting approaches, relevant for studies dealing with, e.g. luminosity functions (Aird et al. 2010; Buchner et al. 2015; Georgakakis et al. 2015). However, the limited availability of a sizable training sample of AGN has resulted in only a few attempts to compute photo-z for mostly nucleus-dominated objects with ML-based methods (Mountrichas et al. 2017; Fotopoulou & Paltani 2018; Ruiz et al. 2018; Meshcheryakov et al. 2018; Brescia et al. 2019; Nishizawa et al. 2020).\nMore recently, the conventional approach of manually selecting photometric features for ML has been replaced by bright, well-resolved galaxies at low redshift (Hoyle 2016; Pasquet et al. 2018; Campagne 2020; Hayat et al. 2021). In this regime, integrating galaxy images into deep neural networks inherently captures essential details like flux, morphology, and other features that would typically be extracted from catalogs based on predefined assumptions, leading to a more comprehensive redshift estimation process. This approach is particularly advantageous for addressing current limitations faced by photo-z methods for AGN, as it leverages model-independent fluxes and redshift indicative features, including surface brightness profiles (Stabenau et al. 2008; Jones & Singal 2017; Gomes et al. 2017; Zhou et al. 2021, 2023). Unlike creating a single SED from total flux measurements, projects employing images with independent pixelby-pixel SEDs at identical redshift have demonstrated increased photo-z constraining power, alleviating previous empirical approaches by decreasing the fraction of outliers (Henghes et al. 2022; Schuldt et al. 2021; Lin et al. 2022; Dey et al. 2022a; Newman & Gruen 2022).\nHere, we introduce PICZL (Photometrically Inferred CNN redshift(Z) Likelihoods), an enhanced approach to photoz estimation that builds upon (C ircle Z by Saxena et al. 2024). While the authors demonstrated that redshift degeneracies encountered for AGN, typical in cases of limited photometry, can be broken by integrating aperture photometry alongside traditional total / model fluxes and colors, PICZL instead computes photo-z PDFs for AGN directly from homogenized flux band cutouts by leveraging the more detailed spatial light profile. All inputs are obtained utilizing LS10 exclusively. Similar to Saxena et al. (2024), PICZL can produce reliable photo-z PDFs for all Legacy-detected sources associated with an AGN. However, the model can, in principle, be applied to other extragalactic sources (e.g, inactive galaxies, Götzenberger et al. in prep.) granted that a dedicated training sample is used.\nWe employ an ensemble of the same ML algorithm, notably convolutional neural networks (CNNs), known for their proficiency in learning intricate patterns, as outlined by (Lecun et al. 1998a). Specifically designed for image analysis, CNNs excel at identifying and extracting relevant predictive features directly from images, thereby reducing computational overhead compared to fully connected architectures. Harnessing this more extensive pool of information, these models surpass alternative models based on condensed feature-based input sets.\nThe paper is structured as follows: Sect. 2 introduces the AGN training sample down-selection. Sect. 3 focuses on the photometric data products available within LS10. Sect. 4 details the photometric data preprocessing, followed by Sect. 5, which outlines the model pipeline. Sect. 6 presents and quantifies the redshift results, while Sect. 7 evaluates the photo-z released for the XMM-SERVS (Chen et al. 2018; Ni et al. 2021) fields. Sect. 8 outlines current limitations and discusses how we can achieve further improvements. Sect. 9 explores implications for future surveys, concluding with a summary.\nIn this paper, unless stated di ff erently, we express magnitudes in the AB system and adopt a Λ CDM cosmology with H 0 = 69 . 8 km s -1 Mpc -1 , Ω m = 0 . 28 and Λ = 0 . 72.\n2. AGN training sample selection\nIn X-ray surveys, the identification of AGN has two distinct advantages - i) the reduced impact of moderate obscuration and ii) the lack of host dilution. Due to the inherent brightness of accreting SMBHs compared to their host galaxies, this results in a significantly higher nuclei-to-host emission ratio compared to observations in some of the neighbouring wavelength windows, such as UV-optical-NIR (Padovani et al. 2017). This naturally leads to a larger diversity of AGN observed by an X-ray telescope. That being said, surveys in the more accessible optical and NIR regime can increase the likelihood of detecting higherz , and in the case of MIR more heavily obscured AGN, compared to the soft X-ray bands.\nMLapproaches for photoz estimation in large surveys (e.g., Fotopoulou & Paltani 2018; Duncan 2022) typically classify objects into three broad categories: galaxies, quasars (QSOs), and stars, before computing photo-z. However, this classification is usually based on the optical properties and hence fails for obscured and / or lower-luminosity AGN. Since our goal is to improve on the quality of photo-z estimates for X-ray detected extragalactic sources, including type 2 AGN and low-redshift Seyfert 1 galaxies, generally, our training sample has to replicate this diversity. We achieve this by combining AGN selected across multiple wavelength bands.\nAs a starting point, we include the same X-ray samples used in Saxena et al. (2024), namely the latest version of the XMM catalog, 4XMM, which spans 19 years of observations made with XMM-Newton (Webb et al. 2020) and data from the eROSITA CalPV-phase Final Equatorial-Depth Survey (eFEDS; Brunner et al. 2022), as they provide a reasonably representative and complete set of diverse AGN spanning 5 dex in X-ray flux out to redshift z ≲ 4. However, with just these, some portions of AGNparameter space remain imbalanced, such as highly luminious and / or high-z AGN. Thus, we expand the dataset by adding bright, optical and MIR, selections. We describe each of these samples in more detail below.\nThis approach enhances the completeness of our training sample, which is essential to mitigate covariate shift, i.e., the shift in parameter space between the training and validation samples, so that the model generalizes e ff ectively to new data (Norris et al. 2019). Subsequently, a non-representative training sample may lead to systematically biased outcomes (Newman & Gruen 2022). Accordingly, algorithms will be strongly weighted towards the most densely populated regions of the training space (Duncan 2022).\n2.1. Beyond eFEDS and 4XMM\nWe can enhance the redshift distribution within our sample, particularly towards high ( z ≥ 3) redshift, in this otherwise underrepresented parameter space due to observational selection e ff ects. We recognize the subsequent incorporation of unavoidable selection biases in each survey while restricting the inclusion of sources at low redshift to a minimum. While the balance between dataset quality and size is critical, deep learning algorithms that operate on pixel-level inputs tend to perform optimally only when training datasets contain ≥ 400 000 galaxy images (Schuldt et al. 2021; Dey et al. 2022a; Newman & Gruen\n\n\n
Fig. 1: Flowchart depicting the training sample down-selection pipeline. This includes the sample preprocessing (grey box) and the sample refinement, including redshift extension and duplicate removal below the dashed red line.
\n\n2022). Since our method would benefit from a larger sample (see Table 1), we chose not to apply stringent quality criteria by only considering high-quality data, significantly reducing the number of sources available training. Such a reduction would also prevent the model from learning to handle lower-quality data, limiting its application to only high-quality validation data. By not making an initial down-selection, we retain the flexibility to apply quality cuts to future blind samples by using LS10 flags later.\n2.1.1. Samples from optical selection\nWe include the 2dF QSO Redshift Survey (2QZ, Croom et al. 2004) with ∼ 23k color selected QSOs in the magnitude range 18.25 ≤ bJ ≤ 20.85 at redshifts lower than z ∼ 3 and the QUasars\n
\n\n
Table 1: Overview of the relative fractions of source catalogs used in compiling our AGN training sample.
\n
\nas BRIght beacons for Cosmology in the Southern hemisphere survey (QUBRICS, Boutsia et al. 2020) with 224 bright (i ≤ 18) QSOs at redshifts of z ≥ 2 . 5.\n2.1.2. Samples from optical follow-up of X-ray sources\nAdditionally, we incorporate the SDSS-IV (Blanton et al. 2017) quasar catalog from Data Release 16 (DR16Q, Lyke et al. 2020) with ∼ 150k quasars collected from various subprogrammes including optical / IR selection down to g ≤ 22 in the LS10 footprint after subselecting high quality specz , as well as follow-up of Xray sources from ROSAT (Voges et al. 1999; Boller et al. 2016; Salvato et al. 2018) and XMM (e.g. LaMassa et al. 2019). As successor science programme, we also consider the Black Hole Mapper (BHM) SPectroscopic IDentfication of ERosita Sources (SPIDERS, Anderson et al., in prep, Aydar et al., in prep) from SDSS-V (Kollmeier et al., in prep) Data Release 18 (Dwelly et al. 2017; Co ff ey et al. 2019; Comparat et al. 2020; Almeida et al. 2023).\n2.1.3. Samples of high-z sources\nGiven the strong imbalance above z ∼ 3 . 5, we also include 400 optically / IR selected quasars at redshifts 4.8 ≤ z ≤ 6 . 6 down to g ≤ 24 from the high-redshift quasar survey in the DESI Early Data Release (EDR, Yang et al. 2023) and a compilation of highz quasars at z ≥ 5 . 3 published in literature (Fan et al. 2022).\n2.2. Spectroscopic cross-referencing\nThe parent sample of AGN is annotated with specz , where available. According to Figure 1, we also consider sources, including those from eFEDS and 4XMM, with spatial counterparts from a compilation of public redshifts (Kluge et al. 2024). The procedure by which we match optical counterparts in our combined sample to a compilation of quality criteria down-selected specz , is outlined in Sect. 3.1 of Saxena et al. (2024). Due to overlaps between surveys, we remove duplicates when combining samples. The final sample of sources with specz comprises 40 489 objects, with a breakdown in Table 1. Correspondingly, the (cumulative) histograms illustrating the n( z ) distributions that collectively constitute the PICZL sample are presented in Figure 2.\n\n\n\n0\n\n\n
Fig. 2: Binned redshift histogram (top panel) and cumulative distribution (bottom panel) of the sources utilized in the PICZL AGN sample. Note that these samples are not necessarily rank ordered by importance but for improved readability.
\n\n3. The survey\nTo streamline and simplify our methodology, we have chosen to employ data from LS10 exclusively to mitigate potential complications arising from the heterogeneity of multiple datasets. Crucially, the survey area now extends over 20 000 deg 2 of optical griz and WISE W 1 -W 4 forced photometry, by incorporating the following datasets:\n\n-DECam Legacy Survey observations (DECaLS, Flaugher et al. 2015; Dey et al. 2019), including data from the Dark Energy Survey (DES, Collaboration: et al. 2016), which covers a 5000 deg 2 contiguous area in the South Galactic Cap. In the DES area, the depth reached is higher than elsewhere in the footprint.\n-DECam observations from a range of non-DECaLS surveys, including the full six years of the Dark Energy Survey, publicly released DECam imaging (NOIRLab Archive) from other projects, including the DECam Local Volume Exploration Survey (DELVE, Drlica-Wagner et al. 2021) and the DECam eROSITA survey (DeROSITAs, PI: A. Zenteno, Zenteno et al. in prep).\n\nIn the north ( δ > 32.375 deg), LS10 uses the Beijing-Arizona Sky Survey (BASS, Zou et al. 2017) for g - and r -band coverage, and the Mayall z -band Legacy Survey (MzLS, Silva et al. 2016) for z -band coverage (Kluge et al. 2024).\nCount\n\n\n
Fig. 3: Grid of subplots showing various model inputs for PICZL. In the upper row: the LS10 g-band image (a), along with its 2-D model flux (b), residual (c), and aperture flux map (d). In the bottom row: the original g-r color is shown in (a). The presence of a saturated pixel in the top right corner is visible, indicating the need for pre-processing. The result of the preprocessing is shown in b). The bottom panels c) and d) show the g / r-band and w1 / w2-band aperture flux maps, respectively.
\n\n3.1. Photometric data\nLS10 o ff ers registered, background-subtracted, and photometrically calibrated point spread function (PSF)-forced photometry, including corresponding errors. To extend their wavelength coverage, DR10 catalogs incorporate mid-infrared (mid-IR) forced photometry at wavelengths of 3.4, 4.6, 12, and 22 µ m(referred to as W1, W2, W3, and W4, respectively) for all optically detected sources in the LS10 via the Near-Earth Object Wide-field Infrared Survey Explorer (NEOWISE) (Mainzer et al. 2011; Lang 2014; Meisner et al. 2017). Sources are modeled simultaneously across all optical bands, ensuring consistency in shape and size measurements by fitting a set of light profiles, even for spatially extended sources. Consequently, alongside reliable total and multi-aperture (8 annuli ≤ 7 arcseconds, five annuli ≤ 11 arcseconds for the optical and mid-infrared bands, respectively) flux measurements, the LS10 catalog o ff ers seeing-convolved PSF, de Vaucouleurs, exponential disk, or composite de Vaucouleurs + exponential disk models obtained with the Tractor algorithm (Lang et al. 2016). Additionally, providing fluxes rather than magnitudes, enables considering sources with very low signalto-noise ratios without introducing biases at faint levels. This characteristic also facilitates flux stacking at the catalog level, enhancing the overall versatility and utility of the classification and fitting process within LS10 1 .\n3.2. Imaging data\nIn addition to catalog data, LS10 provides a rich set of imaging products. These include observations, flux model images, and residual maps for all available bands. For instance, the top pan-\nels (a, b, c) of Figure 3 display all image products for a g -band observation, respectively.\n4. Preprocessing\nHere we detail the preprocessing steps taken to prepare our dataset, ensuring that it is clean, normalized, and structured appropriately.\n4.1. Image preprocessing\nBuilding on the approach of Saxena et al. (2024), which demonstrated significant improvements by shifting from total to aperture flux utilizing information on the 2D light distribution, we aim to further refine the spatial characterization of sources. This is achieved by incorporating pixel-level flux resolution through imaging as base input. Images in individual bands or in combination (i.e., colors) reflect the surface brightness, angular size, and sub-component structures of the sources, indirectly providing redshift information (Stabenau et al. 2008). To obtain reliable photo-z directly from images, we utilize flux-calibrated optical cutouts across as many filters as available.\nWith an average seeing FWHM of 1.3 arcseconds under nominal conditions, LS10 provides a pixel resolution of 0.262 arcseconds per pixel for the optical bands, reaching depths between 23 and 24.7 AB, depending on the specific band and region of the sky (see Dey et al. (2019) and Figure 1 in Saxena et al. (2024)). To enhance computational e ffi ciency and mitigate contamination from nearby sources, we restrict our cutout dimensions to 23 × 23 pixel, centered on the AGN coordinates in the four griz LS10 bands. Our cutouts correspond to a field of view (FOV) of approximately 6 arcseconds × 6 arcseconds. We\n\n\n
Fig. 4: Angular size of sources with a fixed physical size of 2040 kpc, as a function of redshift. The orange dashed line depicts a fixed image dimension of 23 pixels, assuming the LS10 spatial resolution of 0.262 arc seconds per pixel, which su ffi ces to cover objects of 30 kpc diameters in size out to redshifts of ∼ 0 . 3 ≤ z ≤ 7 . 8.
\n\nbase our choice of FOV on the angular size-redshift relation by computing the angular diameter distance d Λ via:\nc H 0 1 (1 + z ) Z z 0 d z ' p Ω M(1 + z ' ) 3 + ΩΛ . (1)\nEquation 1 and Figure 4 elucidate the connection between an object's physical size, its angular size, and redshift. Notably, we can e ff ectively map galaxies with a diameter of 30 kpc - representative of main sequence galaxies (Wuyts et al. 2011) within the confines of a 23 × 23 pixel cutout, covering the range of 0 . 5 ≤ z ≤ 7 . 7.\nGiven that the FWHM of the W1, W2, and W3 images is 6 arcseconds, and of 12 arcseconds for W4, WISE band cutouts do not provide meaningful spatial information at this scale (see Figure 5). Therefore, we have opted to use images solely from the optical bands. Problematic sources, exhibiting signs of defects in various ways are flagged in the Legacy Survey by specific bitmasks (Dey et al. 2019).\nWe acknowledge that LS10 images exhibit varying quality due to di ff erences in seeing conditions during observations taken over many years, which impact in particular the measurements of color within source apertures. Despite this, we rely on the model's ability to adapt to these intrinsic variations given the size and diversity of our training sample, as the sources withheld from training represent a shu ffl ed subsample of the main dataset, ensuring robust evaluation. We have verified that the distribution in seeing quality (expressed as the weighted average PSF FWHMofthe images) for the training and validation are comparable (refer to Figure B.1). However, preliminary tests indicate that adding PSF size and PSF depth of the observations-both available in LS10-as additional features enhances model performance (Götzenberger et al., in prep.). While it is not unreasonable to expect the model to implicitly infer the PSF or a related abstract representation thereof from the images themselves, these features will be included by default in future PICZL versions. With ongoing developments, PSF cutouts are expected to\n\n\n\n\n\n
Fig. 5: Example AGN in our sample, as seen by LS10, in the optical RGB image (left) and in the W1 image from NEOWISE7 (right). The spatial resolution is 0.262 arcseconds and 2.75 arcseconds per pixel, respectively. The size of the cutouts is 27.5 arc seconds × 27.5 arc seconds.
\n\nbecome more accessible for integration into the image stack (see Table 2). In the longer term, we anticipate that upcoming surveys like LSST, with their improved consistency in image quality, will further reduce these limitations and boost the precision of pixelbased analyses such as ours.\n4.2. Color images\nIt proves advantageous to provide the network with color images (ratio between images from di ff erent bands) as an additional input. This approach avoids the necessity for the model to learn the significance of colors solely from the flux images, which is inherently a more di ffi cult task. Likewise, rather than processing numerical features separately and merging them with the information extracted from the image cube at a later stage, we find it beneficial to integrate them directly at the pixel level. As a result, whenever possible, we transform catalog features into 2D arrays to align them with the original images in the same thread, enabling smoother integration and more coherent analysis (see e.g. Hayat et al. 2021). We improve the depth of our data cube by converting catalog-based quantities, e.g., flux measurements from apertures across di ff erent bands, into synthetic images, with a respective image size depending on the cutout size (see Figure 3). We expand this approach by generating images for all viable color combinations of aperture fluxes, constrained to those with matching aperture sizes. Additionally, we produce color images for flux cutouts where pixel resolutions are consistent (see lower panels b) and c) of Figure 3). Since the WISEand optical bands di ff er in both aperture size and pixel resolution, cross-wavelength color images are not feasible; instead, color combinations are restricted to within the optical or within the WISE bands (refer to Table 2).\nTo maintain FOV consistency, we integrate WISE data for only the two innermost apertures (see Figure 5), preserving the 23x23 pixel data cube format. Although no additional spatial details are expected at this scale (see Sect. 4.1), the WISE data still captures aperture flux in a format that enables direct crosschannel connections between optical and mid-infrared data at the image level.\nNevertheless, defected images can introduce challenges when generating color images (see bottom panel a) in Figure 3). Given that colors are derived from the ratio of images in di ff erent bands, the occurrence of unphysical negative or abnormally high / low values poses a significant concern. To address this issue, we examine whether the median value of neighbouring pix-\n
\n\n
Table 2: Overview of the various feature formats and dimensions utilized in PICZL's multi-channel approach.
\n
\nNotes. Each column represents a distinct processing channel, detailing its input data type and the respective dimensions. All flux-related inputs have been corrected for reddening e ff ects.\nels is more than three times lower than the value of noisy pixels per band. If so, the central pixel's value is substituted with the median value of the surrounding pixels to smooth out fluctuations. In cases of non-detections or severely corrupted images where the largest value among all pixels in an image is < 0.001, the image is treated as a non-detection and all pixels are set to zero as default. After undergoing preprocessing, the images are utilized to create color images by exploring the six possible color combinations: gr , gi , gz , ri , rz and iz . If either of the two flux images involved in creating a color image is identified as a nondetection, the resulting color image is set to a default value of -99.\nAt the catalog level, spatially invariant features, such as bestfit model classifications and signal-to-noise ratios (S / N), are processed separately in a dedicated channel, as they cannot be converted into image data. Notably, regarding normalization, we adopt a uniform min-max scaling approach to handle columns with cross-dependencies, such as flux bands. This strategy aims to preserve crucial information, such as the original shape of the SED.\nContrary to the conventional approach of stacking all available images into a single input (Hoyle 2016; D'Isanto & Polsterer 2018; Pasquet et al. 2018; Dey et al. 2022a; Treyer et al. 2023), we find it advantageous to separate the color image data cube (23,23,24) from the flux band cutouts (23,23,32). This distinction is necessary because the color images often have di ff ering pixel value scales, including negative values, which require specialized processing in our machine learning application, such as tailored loss functions in the CNN. Non-spatial attributes are then combined with image-based data at a subsequent stage of the model, allowing the model to capture both spatial and nonspatial aspects. By processing data types in parallel channels, we leverage their complementary information by merging the data at a later stage, enhancing the extraction of inter-band correlations and ultimately improving redshift precision (Ma et al. 2015; Ait-Ouahmed et al. 2023). A detailed breakdown of the features integrated into each channel is provided in Table 2.\n5. Neural network\nML embodies an artificial intelligence paradigm where computers learn patterns and relationships from data, enabling them to make predictions or perform tasks without explicit programming. Multilayer perceptrons (MLPs), a feature-based feedforward neural network, draw inspiration from their biological\ncounterparts, namely excitable cells responsible for processing and transmitting information (Rosenblatt 1958; I. Goodfellow 2016; M. Deru 2019). Likewise, each assigned a distinct weight, computational input vectors can be organized into layers, relayed to one or more hidden layers, to compute a scalar output value (Géron 2019). During training, these models learn data mappings by adjusting the weights and biases associated with their connections. The margin of change to the model after every training epoch is dictated by the choice of optimizer and loss function, which e ff ectively calculates the Euclidean distance between the prediction and so-called ground truth in multi-dimensional feature space, thereby significantly impacting model convergence and performance.\nThe current state-of-the-art deep learning (DL) networks, characterized by their many hidden layers, have shown exceptional capabilities in handling complex non-linear tasks.\n5.1. Convolutional model layers\nAmong such architectures, Convolutional Neural Networks (CNNs; Fukushima 1980; LeCun et al. 1989; Lecun et al. 1998b) distinguish themselves by their remarkable e ff ectiveness in handling grid-like data, a prevalent form of which is represented by images. CNNs leverage a model architecture that is particularly e ff ective in tasks like image recognition, object detection, and image segmentation (O'Shea & Nash 2015; Liu et al. 2022). In convolutional layers, neurons establish connections exclusively with pixels within their receptive field, ensuring successive layers are linked only to specific regions of the previous layer. Subsequently, the model extracts low, image-level features in early layers and progressively complex, higher-level features in later layers. This allows the CNN to learn representations of the input data at multiple levels of abstraction.\nThe convolutional operation involves sliding several kernels K , here of sizes 3 × 3 or 5 × 5 pixel, across the images with a fixed stride of size s = 1, compressing each mosaic element into a single scalar. This process entails element-wise multiplication, generating a set of K feature maps. Filters, the learnable parameters of convolutional layers, enable the network to detect and highlight di ff erent aspects of the input data, such as edges or corners. Each convolution is followed by a pooling layer that reduces spatial dimensions and computational load. Pooling involves sliding a kernel, in our case of size 2 × 2 and s = 1, across the feature maps, selecting the maximum value (max pooling) within each kernel window. After flattening the data cube into a single array,\nit is passed through (several) fully connected (FC) layers. Each FC layer contains a layer-specific number of neurons n , tailored to the model's specific needs. The final FC layer's neuron count varies based on the task, with n = 1 referring to regression tasks and n ≥ 2 to other endeavors such as multi-label classification.\n5.2. Gaussian mixture models\nBy default, single output regression models (Schuldt et al. 2021) provide point estimates without quantifying uncertainty. This severely limits their area of application, particularly in scenarios where quantifying the uncertainty is crucial, for example, when performing precision cosmology with Euclid (Bordoloi et al. 2010; Scaramella et al. 2022; Newman & Gruen 2022). By instead employing an architecture that provides PDFs, we can not only retrieve point estimates but also encapsulate the uncertainty associated with the predictions in a concise format (D'Isanto & Polsterer 2018). Given the inherent complexity in determining redshifts from only a few broadband cutouts, our results are expected to often exhibit degeneracy with multi-modal posteriors, making a single Gaussian insu ffi cient for representing the photoz PDFs. Therefore, estimates are computed using Bayesian Gaussian Mixture Models (GMMs, Duda et al. 1973; Viroli & McLachlan 2017; Hatfield et al. 2020). These networks provide a unique probabilistic modeling approach that di ff erentiates them from traditional MLPs. One of the key distinctions lies in the output nature, where GMMs provide a set of variables to compute weighted multi-Gaussian distributions as opposed to single-point estimates. Subsequently, each component is characterized by its mean µ , standard deviation σ , and weight w , allowing GMMs to produce a full PDF for a given set of inputs x . The PDF is expressed as\nP ( x ) = K X k = 1 wk · N ( x | µ k , σ 2 k ) , (2)\nwith, wk representing the weight and N ( x | µ k , σ 2 k ) denoting the Gaussian distribution of the k -th component. As such, we extend our CNN approach by a GMM backend to output PDFs based on the information-rich feature maps produced during the front-end phase of the network. However, we encounter a limiting challenge with inputs of such small dimensions, i.e. 23 × 23, as they are not well-suited for established image-based Deep Learning architectures, such as \"ResNet\" (He et al. 2015), which typically require larger dimension scales, typically exceeding 200 × 200 pixels, to accommodate the large number of pooling layers they employ. Therefore, we developed a custom architecture to fit our data dimensionality. The resulting model architecture features roughly 490,000 trainable parameters, far fewer than found in comparable studies (see Pasquet et al. 2018; Treyer et al. 2023), and is displayed in Figure A.1 of Appendix A.\n5.3. Model refinement\nNumerous hyper-parameters are crucial in shaping the network architecture while influencing training and convergence. Extensive optimization has been conducted across various parameters utilizing the Optuna framework (Akiba et al. 2019). The current configuration accounts for the vast array of potential combinations. The key parameters with the most significant impact are outlined below:\n\n-Batch size [8-2048]: The number of objects utilized in a single training iteration per epoch. Larger batch sizes o ff er com-\n\nputational e ffi ciency, while smaller batches may help generalize better.\n\n-Learning rate [0.1-0.00001]: Controls the size of the model adjustment step during optimization. It influences the convergence speed and the risk of overshooting optimal settings.\n-Number of Gaussians [1-50]: Determines the complexity and flexibility of the GMM, influencing how well the model captures the underlying data distribution.\n-Convolutional & pooling layers [2-10]: the dimension of the kernel influences the size of the receptive field and the learnable features.\n-Number of neurons and layers [10-300]: Determines the depth and complexity of the neural network, impacting its capacity to capture hierarchical features and relationships in the input data.\n-Activation function: Introduces non-linearity to the model, enabling it to learn complex mappings between inputs and outputs, commonly a sigmoid, hyperbolic tangent (tanh), rectified linear unit (ReLu) or versions thereof.\n-Dropout layer [0-0.9]: Refers to the fraction of randomly selected neurons that are temporarily dropped or ignored during training, helping to prevent overfitting by enhancing network robustness and generalization.\n-Max and average pooling: Pooling layers, such as Max Pooling and Average Pooling, are used to downsample the spatial dimensions of the input feature maps, reducing the computational load.\n-Batch normalization: Batch Normalization can improve the training stability and speed of convergence in neural networks by subtracting the batch mean and dividing by the batch standard deviation.\n\n5.4. Loss functions\nWhen utilizing PDFs instead of point estimates, it is crucial to quantify whether a predicted PDF e ff ectively reflects the ground truth, in our supervised ML case, specz (Sánchez et al. 2014; Malz et al. 2018; Schmidt et al. 2020; Treyer et al. 2023). The PDF predicted by a model should be concentrated near the true value. A straight-forward and meaningful proper scoring rule (i.e. one which is lowest when the prediction is at the truth), is the product of probabilities at the true redshifts. The negative log-likelihood (NLL) 2 of which passed to a minimizer\nE ( w ) = -N X n = 1 ln GLYPH<18> k X k = 1 wk ( xn ) N ( y | µ k ( xn ) , σ k ( xn ) 2 ) GLYPH<19> , (3)\nyields the likelihood of observing a data distribution given a specific set of model parameters. This expression coincides with the average Kullback-Leibler divergence (KL, Kullback & Leibler 1951) when going from a delta PDF at specz to the photo-z PDF.\nAn alternative growing in popularity is given by the Continuous Ranked Probability Score (CRPS), initially applied in weather forecasting (Grimit et al. 2006), which serves as a valuable metric for photo-z estimation via PDF quantification (D'Isanto & Polsterer 2018). Computationally, CRPS calculates the integral of the squared di ff erence between the predicted cumulative probability distribution function (CDF) and a Heaviside step-function H ( x ) at the value of the specz ( xz ) as\nCRPS( F , xz ) = Z ∞ -∞ GLYPH<20> Z x -∞ f ( t ) dt -H ( x -xz ) GLYPH<21> 2 d x . (4)\nFor a finite mixture of normal distributions M , the CRPS can be expressed in closed form:\nCRPS = M X i = 1 wiA ( δ i , σ 2 i ) -1 2 M X i = 1 M X j = 1 wiwjA ( µ i -µ j , σ 2 i + σ 2 j ) , (5)\nwith δ i = y -µ i , the uncertainty σ i and weight wi of the i -th component respectively, as well as\nA ( µ, σ 2 ) = µ GLYPH<20> 2 Φ GLYPH<18> µ σ GLYPH<19> -1 GLYPH<21> + 2 σϕ GLYPH<18> µ σ GLYPH<19> , ϕ ( x ) = 1 √ 2 π exp GLYPH<18> -x 2 2 GLYPH<19> , Φ ( x ) = Z x -∞ ϕ ( t )d t . (6)\nBy extending the CRPS loss via normalizing it by (1 + z), we adjust the penalty for prediction errors based on redshift. In doing so, we prioritize accuracy for low and intermediate redshift sources by imposing higher constraints while allowing for more leniency in less critical high-redshift areas. Integrating over the entire range of possible redshift values, the CRPS takes into account both location and spread of the predicted PDF. It, therefore, provides a comprehensive evaluation metric, generating globally well-calibrated PDFs (Dey et al. 2022b).\n5.5. Training & data augmentation\nOur dataset is divided into training and validation sets in an 80:20 ratio. The model is trained over 1000 training epochs utilizing the Adam algorithm (Kingma & Ba 2017) along with a learning rate scheduler to adjust the learning rates during training dynamically. The inclusion of both is a consequence of the Optuna hyperparameter optimization. Typically, the model achieves its lowest validation loss around the 600th epoch when considering both loss functions (see Figure 6). After this point, although the training loss continues to decrease, the validation loss begins to increase, indicating overfitting. This overfitting likely arises due to the limited size of our training sample, allowing the model to memorize specific patterns rather than learning generalizable features. We implement a checkpoint system to mitigate this, saving the model's architecture and parameters when it reaches its lowest validation loss. This approach ensures that we capture the best-performing model configuration while preventing overfitting.\nTo further exploit the advantages of deep learning, particularly its capacity to perform well with extensive datasets, we have tested the incorporation of data augmentation techniques into our methodology. Recognizing the substantial impact of training data amount on model performance, we employ common augmentation data products, such as rotated and mirrored images. Although these transformations significantly increase the apparent size of the dataset, we do not observe an increase in performance when including augmentation techniques (Dey et al. 2022a; Treyer et al. 2023). This is potentially not surprising as the influence of augmentation techniques depends on several factors. While CNNs are inherently equivariant to translations within images, enabling them to recognize objects even if\nthey are shifted or o ff -centered, they and most other machinelearning approaches are not naturally equivariant to rotations and reflections. This lack of rotational equivariance means that the model output e ff ectively depends on the rotation of the input images. However, as most of our inputs are approximately point sources with radial symmetry, we do not explore this aspect further in this study. More recent publications have focused on developing rotationally equivariant CNN architectures, which have shown improved performance despite increased computational costs (Cohen & Welling 2016; Weiler & Cesa 2021; Lines et al. 2024). Incorporating such architectures could be particularly beneficial for future works dealing with large amounts of imaging data.\n5.6. Model ensemble\nEnsemble models capitalize on the diversity of multiple models to improve prediction accuracy. Our approach involves diversifying models by adjusting parameters such as the number of Gaussians per GMM, learning rates, batch sizes, and the choice of loss function. We employ both NLL and CRPS as complementary loss functions to achieve this. While CRPS excels in achieving lower outlier fractions, NLL enhances overall variance. Each loss function trains a set of 144 models, resulting in a diverse pool of 288. In line with Treyer et al. (2023), we achieve superior performance by randomly combining equally weighted CRPS and NLL models from the pool, compared to the best-performing individual model (Dahlen et al. 2013; Newman & Gruen 2022). This finding is also aided by the result that roughly 30% of outliers are model-specific as opposed to 20% of outliers appearing in all models, therefore considered to be genuine. Put di ff erently, 80% of outliers were found to not be an outlier in at least one other model.\nWe additionally recognize that the initial configuration of each model has a minor influence on the performance. Although training additional models could further enhance our results, the computational resources required to generate three to five sets of models would make this approach impractical relative to the potential performance gains. To create an e ff ective ensemble, we evaluate the outlier fraction of all individual models on an un-\n\n\n
Fig. 6: Training (dashed) and Validation (solid) loss curves for a single PICZL model. The solid or dashed lines represent the mean loss values computed over a window size of 20, while the shaded areas denote the 1 σ error range around the mean. The NLL loss is depicted in blue, while the normalized CRPS loss is shown in red.
\n\n\n\n
Fig. 7: Binned Scatter plots of photo-z obtained from our model ZPICZL versus spectroscopic redshift Zspec, for sources classified as type point-like (PSF, left) and extended (EXT, right). Each plot includes identity and two lines denoting the outlier boundary of | z phot -z spec | (1 + z spec) > 0 . 15. Normalized residuals and trend lines and errors are shown in the bottom panels, aiding in the visualization of systematic deviations across the redshift range.
\n\nseen test sample. From this evaluation, we select models based on their relative influence on the model performance. We then observe how the ensemble's performance evolves as more models are continuously incorporated. Our ensemble ultimately comprises 10 models or 84 Gaussians, evenly incorporating models optimized using both NLL and CRPS approaches. To refine the model weights within the ensemble, we employ Optuna for finetuning. Subsequently, the ensemble posterior likelihood distribution P ensemble(x) is given by:\nP ensemble( x ) = N X i = 1 wiP ( i )( x ) , (7)\nwhere the weights wi are normalized such that:\nN X i = 1 wi = 1 , (8)\nto assure that the integral of the ensemble PDF is equal to 1. We obtain a point estimate for our photoz prediction by identifying the dominant mode of the resulting PDF, recognizing that the mean or median could be misleading for highly non-Gaussian and bi- or multi-modal distributions. Additionally, we provide asymmetric 1 and 3 sigma upper and lower errors by evaluating the PDF within the 16th / 84th and 0.15 / 99.85th percentiles, respectively, along with the entire PDF.\n6. Photo-z results\nWe evaluate the performance of PICLZ on both point estimates and PDFs. This comprehensive assessment includes testing the statistical reliability and accuracy of our predictions. We adopt several commonly employed statistical metrics, providing insights into the accuracy, precision, and reliability of the photo-z estimates. In line with the definitions from the literature, we use the following metrics:\n\n-Prediction bias: the mean of the normalised residuals, ⟨ ∆ z ⟩ = ( z spec -z phot) (1 + z spec) .\n-Variance: following Ilbert et al. (2006), we adopt the standard deviation from the normalised median absolute deviation σ NMAD = 1 . 4826 × Median | z spec -z phot | (1 + z spec) .\n-Fraction of outliers η is determined by the proportion of photo-z estimates with absolute normalised residuals exceeding η = | z spec -z phot | (1 + z spec) > 0 . 15.\n-PIT score: a diagnostic tool used to evaluate the calibration of probabilistic forecasts by evaluating the cumulative distribution function (CDF) given the PDF at the (true) specz z corr as CDF( z corr) = R z corr 0 PDF( z ) dz .\n\nWe quantify PICZL's performances on a validation sample of 8098 sources and find an outlier fraction η , of 5.6% with a variance σ NMAD, of 0.045. Since the validation sample is used\nfor feedback in training, it should be mentioned that the results mentioned above are likely overestimating the performance and therefore may not representative of what a future user could achieve with say, a test set previously unknown to the model, leave-one-out, or k-fold cross-validation. To this end, we have estimated our performance on independent, blind, X-rayselected samples (see Sect. 6.2 and 7).\nFigure 7 compares photometric versus spectroscopic redshifts, split by point-like (type = PSF) and extended (type = EXT) morphology. With respect to comparable work (e.g., Figure 13 from Salvato et al. 2022), we observe enhanced performance with a much-reduced fraction of outliers, especially for PSF sources. These objects lack morphological information at redshifts z ≳ 1, hence we attribute this improvement to the model's ability to recognise how the radial extension of the azimuth profile of sources changes with redshift (refer to Figure 4). The scatter, for both PSF and EXT distributions, is tight and symmetrically distributed around the z PICZL = z spec identity line, with outliers appearing randomly scattered, suggesting stable performance across the redshift range and minimal systematic errors (see Figure 8 and Dey et al. (2019) for reference).\n\n\n
Fig. 8: Histogram distribution of the normalised residuals, overlaid with a Gaussian fit (orange). The parameters of the Gaussian distribution are determined by the prediction bias and σ NMAD values obtained from Table 5 for the validation sample.
\n\nGiven that our point estimates are derived from PDFs, we must ensure their global calibration and accuracy. To achieve this, we employ the probability integral transform (PIT) statistic (Dawid 1984; Gneiting et al. 2005), a widely accepted method in the field for assessing the quality of redshift PDFs (Pasquet et al. 2018; Schuldt et al. 2021; Newman & Gruen 2022). An ideal scenario is represented by a uniformly distributed histogram of PIT values. Any deviation from uniformity can indicate issues in PDF calibration. Under-dispersion may suggest overly narrow PDFs, while over-dispersion often results from excessively wide PDFs (D'Isanto & Polsterer 2018). Peaks close to zero or one can be explained by catastrophic outliers, where the true redshift lies so far in the wing of the PDF that it essentially falls outside of it.\nWe employ the PIT histogram (top panel Figure 9) alongside quantile-quantile (QQ) plots (bottom panel Figure 9), to visually assess the statistical properties of our model predictions. For reference, these can be compared to Figure 2 of in Schmidt et al.\n\n\n
Fig. 9: The top panel displays the QQ plot, comparing identity (flat histogram) against the PIT values derived from our redshift PDFs of the validation sample. The bottom panel shows the differences between the QQ plot and identity, highlighting systematic biases or trends for the two morphological classes (PSF and EXT). For all panels, blue refers to results achieved with a single model (see Figure A.1) while orange reflects the results obtained when utilizing ensemble results (refer to Sect. 5.6)
\n\nPIT Score\n.\n(2020); however, those code performances are completely dominated by inactive galaxies, making a direct comparison impractical. The QQ plot compares the CDFs of observed PIT values and identity U (0 , 1), to visualize QQ di ff erences split by morphological type.\nA well-calibrated model will exhibit a QQ plot closely following identity with a flat PIT histogram, indicating accurate and reliable redshift predictions. Our analysis shows that, while asymmetry in the PIT distribution could suggest systematic bias, the QQ plot reveals small residuals overall. Notably, for both the single- (refer to Sect. 5.2 and Figure A.1) and ensemble model (refer to Sect. 5.6), the curves do not deviate significantly from zero, lesser so for PSF-type than EXT-type objects, indicating well-calibrated models. A distinct observation is the shift towards central PIT values for the ensemble model. This shift is not unexpected, given that the ensemble approach integrates multiple redshift solutions, each potentially contributing di ff erent peaks to the resulting ensemble PDF. Consequently, the ensemble PDF exhibits a broader bulk of probability across redshift, with PIT scores accumulating more area under the curve\n
\n\n
Table 3: Overview of wall-clock time τ , ∆ z , σ NMAD and η for sources classified as being extended (EXT) or point-like (PSF) from a single model (as opposed to an ensemble utilized for the results depicted in Figure 7) trained with four input configurations.
\n
\nNotes. The last two setups are an improvement over the previous performances, with the last setup providing comparatively good results, despite ignoring the features provided by catalogs, particularly relevant for potential implementation in e.g., LSST.\n\n( a ) Catalog-based model fluxes only, as traditionally done; ( b ) Catalog-based aperture fluxes and respective colors as done in Saxena et al. (2024);\n\nwhen integrated to the main mode. As a result, rather than yielding a flat PIT distribution, the histogram shifts towards having a concentration of values around 0.5. While this phenomenon aligns with our expectations, further enhancements may be realized by incorporating metrics tailored instead to local calibration accounting for population-specific subgroups (see e.g. Zhao et al. 2021; Dey et al. 2022b).\n6.1. Model performance for different inputs\nImages provide a more comprehensive view of astronomical sources by capturing their full spatial structure and light distribution, o ff ering finer details on morphology, apparent size and extended features that are often lost in the averaging process of aperture photometry. This is pivotal, as SED features crucial for determining redshift solutions and resolving degeneracies mostly reside within the host galaxy (Soo et al. 2017; Wilson et al. 2020). In particular for AGN, images help to spatially separate the pixels corresponding to AGN-dominated emission and those corresponding mostly to the host galaxies. This approach enhances our ability to isolate and analyze individual pixel strings across multiple filters, to e ff ectively construct SEDs for the host galaxies independently. We thereby capture subtle features, neighbors, and patterns that may not be discernible from total flux measures. Critically, the independent photo-z from the AGN and host must agree, thus narrowing the overall source PDF.\nTable 3 presents the fraction of outliers for both EXT and PSF sources using di ff erent configurations of the PICZL algorithm. The first setup replicates the feature-only method, relying solely on total fluxes from the catalog, which, as seen in other studies, results in the fraction of outliers for PSF sources being nearly three times higher than for EXT sources (Salvato et al. 2022). In the second configuration, we adopt the approach of Saxena et al. (2024), incorporating the 2D light distribution and color gradients within annuli. The third configuration, i.e., PICZL as outlined above, enhances this by incorporating images, supplemented with catalog values transformed into image format, which leads to an additional improvement over the method in Saxena et al. (2024).\nSince images inherently capture all features typically extracted numerically and presented in catalogs, we show in the fourth case that PICZL achieves excellent results without feature / catalog information, using solely optical images supplemented by WISE aperture flux maps, even without explicit hyperparameter tuning. This approach is particularly promising for future surveys such as LSST and Euclid, where relying solely on\nmulti-band imaging, including NIR / IR, will eliminate the need for complex source modeling or aperture photometry, regardless of the sources being galaxies or AGN.\n6.2. Blind sample comparison\nTo evaluate the robustness of our approach, we tested PICZL on an independent blind sample from the Chandra Source Catalog 2 (CSC2 3 ). This sample, selected to ensure no overlap with the training data, but yield a comparable X-ray to MIR distribution, is the same one used in the C ircle Z analysis (Saxena et al. 2024), allowing for a direct comparison with their results. After filtering for sources with spectroscopic redshift and removing duplicates with the training dataset, the sample comprises 416 sources within the LS10-South area. For comparison, Saxena et al. (2024) achieve an outlier fraction, η , of 12.3% and a nor-\n\n\n
Fig. 10: PICZL point estimates as a function of PDF width, represented by the di ff erence between upper and lower 1 σ values. Each data point is color-coded according to its r -band magnitude from the LS10. The tilt of the distribution traces the increased uncertainties for PDFs of fainter sources. Horizontal distributions represent areas of large uncertainty in the photo-z, indicating that PICZL errors are realistic.
\n\n7\n\n\n
Figure 10 displays PICZL point estimates as a function of PDF width, represented by the upper 1 σ minus lower 1 σ , colorcoded by the Legacy Survey r -band magnitude, as it o ff ers the most extensive coverage compared to the g , i , or z bands, containing the largest number of sources. We observe wider PDFs correlating with higher redshifts and fainter sources characterized by higher photometric errors or lower S / N, as discussed
\n\n\n\n
Fig. 11: Left: Examples of an inlier (orange) and an outlier (purple) PDF. By definition, the point estimate derived for the two sources from the main mode falls within or outside the grey shaded region. Both sources show a secondary peak in the PDF. We can define the probability of being an inlier (colored area under the PDF which coincides with the grey shaded area). Right: Normalized | ∆ Z | (1 + z spec) as a function of inlier probability, color-coded by number density. The majority of the inliers have a high inlier probability, indicating the stability of our results (see main text for details).
\n\nmalized median absolute deviation, σ NMAD, of 0.055, while our algorithm yields a superior σ NMAD of 0.046 and η of 8.3% (see Table 5 and Figure C.1). This comparison, therefore, demonstrates how utilizing image cubes further improves the already good results achieved using 2D-information from a catalog. For more details on the sample, see Saxena et al. (2024).\n6.3. Prediction uncertainty quantification\nAccurate error estimation is crucial for assessing the reliability of any prediction, particularly in those astrophysical contexts where uncertainties can significantly impact the interpretation of data. To this end, we provide asymmetrical 1 σ and 3 σ errors for every source, together with the photoz PDF, as detailed in Sect. 5.6. These error estimates o ff er a comprehensive understanding of the potential variance in our predictions, accurately reflecting the inherent uncertainties in our data and model. We include asymmetrical errors, as the true distribution of uncertainties is often not symmetrical. This asymmetry arises from factors such as photometric noise and degeneracies in the highly nonlinear color-redshift space, where certain higher or lower redshift values may be more probable than their counterparts, leading to skewed PDFs. Although symmetrical redshift errors are observed for the majority of sources, approximately one-third of the sources exhibit asymmetrical 1 σ errors with deviations in redshift up to | ∆ z asym | ≃ 0 . 25.\nin Sect. 8.4. Additionally, the widths of the PDFs exhibit noticeable horizontal structures, largely influenced by the nature of the LS10 filters. We find that PDFs with wider modes, indicating less certain redshift estimates, often correspond to redshift ranges where key spectral features, such as the Ca break, fall outside the filter coverage (e.g. g & r at z ≃ 0 . 4, r & i at z ≃ 0 . 8 and z ≃ 1.5 or z ≃ 2.2). At redshifts approaching z ≈ 3 and extending below z ≈ 5, the Lymanα break starts to fall within the filter ranges, which contributes to narrower PDFs with increased PDF width only observed for extremely faint sources at even higher redshifts. Lastly, we find that the number of sources with wider PDFs increases as a function of | ∆ Z | (1 + Z spec) , indicating that narrower PDFs correspond to more accurate point predictions.\n6.4. Insights from multimodal PDFs\nTo investigate whether predicted outliers are genuine anomalies or instead stem from machine learning processes, we inquire whether secondary peaks in PDFs are physically meaningful or training artifacts. For each source, we compute the fraction of its PDF, that satisfies the condition | ∆ Z | (1 + z spec) ≥ 0 . 15, to provide the probability of being considered an inlier. The left panel of Figure 11 illustrates this with examples of two PDFs (one for an inlier and one for an outlier), where we highlight the corresponding inlier probabilities. In the right panel of Figure 11, we plot | ∆ Z | (1 + z spec) re-normalised to scale [0,1], such that\n| ∆ z | (1 + z spec) norm = | ∆ z | (1 + z spec) · 0 . 5 0 . 15 if | ∆ z | (1 + z spec) < 0 . 15 | ∆ z | (1 + z spec) · 0 . 5 0 . 15 + 0 . 5 if | ∆ z | (1 + z spec) ≥ 0 . 15\nas a function of inlier probability for all sources. The horizontal dashed black line separates the inliers ( < 0.5) from the outliers ( ≥ 0.5). The majority of inliers are concentrated at high inlier probabilities. This suggests that most inliers provide confident, unimodal PDFs with low errors. Likewise, most outliers have low inlier probability, with only a few outliers having more than 50% inlier probability. While this quantity cannot be computed for sources lacking spectroscopic redshift, it can be used to evaluate the reliability of our point estimates considering their associated PDFs.\nTo evaluate whether ensemble solutions with broad or complex distributions e ff ectively capture the true redshift, we identify sources with multiple peaks and measure the proportion of PDFs exhibiting more than one mode. For these sources, we define a secondary peak as significant if it accounts for at least 10% of the height of the primary peak, which is the minimum prominence threshold used in our analysis. The top panel of Figure 12 shows that only 3% of inliers exhibit secondary peaks as opposed to the roughly 30% of outliers, where occurrences of strong secondary modes decrease with increasing prominence for both cases. While a single mode typically indicates a secure estimate for inliers, the presence of a unique mode alone can subsequently not be used to determine whether a source is an outlier.\nConversely, in the bottom panel of Figure 12, we present the recovery fraction, which denotes how often a secondary peak in PDFs exhibiting multi-modal distributions corresponds to the true redshift. Among the outliers with 30% multi-modal PDF, more than 70% have one of their secondary peaks corresponding to | ∆ Z | (1 + Z spec) ≤ 0 . 15, indicating that there is significant probability that the redshift could consequently be considered as an inlier if the peak heights were reversed. Given the PIT histograms shown\n\n\n
Fig. 12: Top: Fraction of sources having a PDFs presenting secondary peaks for outliers (burgundy) and inliers (blue). Bottom: Corresponding recovery fraction for outliers. Both metrics are plotted as a function of the relative prominence threshold of the secondary peak.
\n\nin Figure 9, it appears likely that the the PDFs are accurately capturing the relative frequency of multi-& bimodal PDFs, as if they were not, there would likely be bias evident in the overall PIT distribution. In other words, using this particular dataset, despite incorporating full multiwavelength image cutouts and cataloglevel information, there persist areas of parameter space that are legitimately degenerate in redshift, and the PDF parameterization looks to be capturing that degeneracy accurately. Consequently, we recommend using the entire PDF rather than point estimates, when possible.\n7. PICZL applied to other surveys\nWe want to investigate how PICZL, utilizing LS10 photometry and imaging, performs in determining photo-z for AGN in a more generic setting. Our focus is set on the LSST deep drilling fields (DDFs), selected to study SMBH growth across the full range of cosmic environments. These fields o ff er deeper, more comprehensive spectroscopic redshift coverage, along with a broader range of high-sensitivity bands, providing an enhanced dataset for photo-z estimation.\n7.1. XMM-SERVS\nThe XMM-Spitzer Extragalactic Representative Volume Survey (XMM-SERVS, Mauduit et al. 2012) encompasses three key fields: the XMM-Large Scale Structure (LSS, Chen et al. 2018), spannning 5.3 deg 2 with a flux limit of 6 . 5 × 10 -15 erg cm -2 s -1 over 90% of the survey area in the 0.5-10keV band (Savi'c et al. 2023); the Wide Chandra Deep Field-South (W-CDF-S, Ni et al. 2021) and the European Large-Area ISO Survey-South 1 (ELAIS-S1, Ni et al. 2021), covering approximately 4.6 deg 2 and 3.2 deg 2 (Brandt et al. 2018; Scolnic et al. 2018), limited to 1 . 3 × 10 -14 erg cm -2 s -1 , respectively, each selected for their exceptional multiwavelength coverage and strategic alignment with the DDFs.\n7.2. Photo-z computation\nConsidering that the original XMM-SERVS photo-z benefit from high S / N photometry and broader wavelength coverage, including u -band and NIR coverage, we would expect significantly better results compared to those achievable by PICZL using LS10 data alone. However, PICZL obtains comparable if not better results when limiting the sample to the depth at which it was trained on (see XMM-SERVS magnitude distribution in Figure E.1). Like any ML model, it struggles to extrapolate to faint sources outside its training distribution. Therefore, a fair comparison requires limiting the analysis to a similar feature space, while the results in Table 4 reflect performance across the entire, diverse XMM-SERVS samples. To facilitate a comparison with PICZL, our analysis begins with the counterpart associations of X-ray emissions as presented in Ni et al. (2021) and Chen et al. (2018). Whenever possible, we limit our samples to sources flagged as AGN (as defined in Sect. 6 of Ni et al. 2021). Matching to objects detected in LS10, due to its more limited depth, we identify approximately one-third of the original sources (see Table 4). This limitation is particularly pronounced in the XMM-LSS survey, where, in addition to deeper X-ray data corresponding to fainter counterparts, Chen et al. (2018) restrict their sample of AGN for which they calculate photo-z, to strictly non-broad-line X-ray sources. We select all sources with spectroscopic redshift and exclude those previously included in the\n
\n\n
Table 4: Summary of the XMM-SERVS samples and their respective photoz metrics.
\n
\nNotes. From left to right, the columns list: Sample field; the original number of X-ray sources; the number of sources in each field that were detected in the LS10 survey; have spectroscopic redshift z > 0 . 002; and were not already present in the training sample of PICZL (i.e. previously unseen); the fraction of failed photo-z in XMM-SERVS zphot = -99; and the bias ⟨ ∆ z ⟩ , variance σ NMAD and outlier fraction η obtained with XMMSERVS (S, excluding zphot = -99) and PICZL (P) photo-z.\n\n\n
Fig. 13: Top: Outlier fraction as a function of LS10 i-band magnitude (left) and LS10 i-band S / N (right), comparing the original photo-z from XMM-SERVS to the photo-z computed with PICZL. The XMM-SERVS photo-z results are shown both with and without the inclusion of sources for which XMM-SERVS photo-z failed (photo-z = -99). The shaded regions represent the range of XMM-SERVS outlier fractions across the ELAIS-S1, W-CDF-S, and LSS fields. Bottom: the range of XMM-SERVS variance in the three fields as a function of LS10 i-band magnitude and S / N. PICZL demonstrates a lower outlier fraction for brighter sources while maintaining comparable accuracy to the original photo-z estimates. Additionally, PICZL successfully computes photo-z for sources that are failures in the original methods.
\n\nPICZL training sample (see Table 4). Unlike the original works, PICZL also successfully computes photo-z for the significant fraction of sources for which the SED fitting adopted in XMMSERVS failed (refer to w / photoz S in Table 4). Figure 13 visualizes the outlier fractions (top) and variances (bottom) for the aggregate XMM-SERVS samples. Blue and burgundy represent the distributions including / excluding catastrophic failures in the XMM-SERVScatalogs. The plot is limited to outlier fractions of 30% and variances of 10%, as XMM-SERVS curves including zphot = -99 solutions continue to rise for fainter magnitudes or lower S / N. This suggests that, with increasing magnitude, a decreasing number of accurate estimates are available, making the subset excluding such solutions increasingly non-representative in terms of outlier fraction.\n\n\n
Fig. 14: Normalized residuals for PICZL photo-z compared to XMM-SERVS photo-z, using data from XMM-SERVS ELAISS1. PICZL outliers are shown as triangles, XMM-SERVS outliers as stars, and sources that are outliers in both methods are depicted as octagons. All sources are color-coded based on their LS10 i-band magnitude.
\n\nHowever, by limiting our comparison to sources with photoz obtained from XMM-SERVS, we find that the accuracy of these photo-z remains comparable up to magnitudes around 23.5 AB, despite the more limited data being used. Additionally, we observe enhanced performance in PICZL photo-z for sources with S / N values ≳ 60, depending on the specific XMM-SERVS sample. Importantly, sources with S / N ≳ 10-20 mark a critical threshold range in LS10 where lower S / N values lead to an exponential rise in photometric errors (see Figure D.1). This trend is particularly evident for magnitudes fainter than 21.5 in the iband (Saxena et al. 2024) and visible in all panels of Figure 13.\nFigure 14 presents a visual comparison of normalized residuals for outliers identified by either PICZL or the original photo-z from Ni et al. (2021), exemplified via the ELAIS-S1 field. The spread of biases, excluding failures from XMM-SERVS, is similar for both methods, falling mostly within the range of [-0.5, 0.5]. The outliers in common appear to be modestly brighter than those that are outliers for a single method only and are evenly distributed between overestimation and underestimation. Notably, the outliers from Ni et al. (2021) are over a narrower range, while PICZL has a few brighter and several more fainter, the majority of which with magnitudes beyond the range covered during training. The observed di ff erence in outlier rates, is\n\n\n
Fig. 15: Prediction bias (top row), fraction of outliers (second row), variance (third row), and number count (fourth row) as a function of magnitude ( r -band from LS10), 0.2-2.3 keV X-ray flux, and specz , categorized by type (extended (EXT) / point-like (PSF)).
\n\n-\nnot primarily due to the use of template-fitting versus machinelearning methods but rather a reflection of the available photometry. XMM-SERVS, with its deeper photometry, is wellsuited for faint objects, though saturation in brighter sources may contribute to some outliers. PICZL, which leverages the LS10 dataset, is optimized for brighter sources due to the shallower photometry. While it performs well in this regime, fainter objects that fall outside the known parameter space are consequently less constrained.\n8. Discussion\nWhile Saxena et al. (2024) made significant strides in overcoming the limitations of relying solely on total or model fluxes for photoz estimation of AGN by utilizing all redshift-correlated features in the LS10 catalog, we have further advanced this approach. By integrating data from both optical and MIR wavelengths, our study highlights the transformative potential of combining imaging with catalog-level data. Despite the state-of-theart advancements in photoz estimation for AGN in this work, further improvements will be possible only by solving issues related to data quality. By overcoming these issues, we can significantly reduce the fraction of catastrophic outliers and achieve\neven greater accuracy in our predictions. In the following, we discuss the limitations that need to addressed in order to apply PICZL to upcoming imaging surveys.\n8.1. Incorrect spec-z\nFirst and foremost, we need to address the reliability of the labels used in our, and generally any supervised machine learning algorithm. While spectroscopic redshift are typically trusted as accurate representations of true redshifts, this assumption does not always hold. As surveys continue to expand in scale, automated pipelines become indispensable for assigning spectroscopic redshift and identifying artifacts / problems for each source, as manual verification by visual inspection is impractical and will remain so for future surveys (Bolton et al. 2012). Importantly, most pipelines are trained on determining the redshift of the bulk of objects (Alexander et al. 2023), which are inactive galaxies and not AGN. This implies that a non-negligible fraction of AGN have wrong redshift estimates, including many labelled with \"no warnings\" (Hewett & Wild 2010; Wu & Shen 2022). Consequently, our performance is constrained by the unknown amount of incorrectly identified spectroscopic redshift\nAbsolute LS10 r band [mag]\nAbsolute LS10 r band [mag]\n\n\n
Fig. 16: Cumulative fraction of outliers as function of absolute magnitudes, to validate their classification based on TYPE. The left panel shows outliers classified as PSF, where most sources have absolute magnitudes consistent with AGN or QSO characteristics. In the right panel, 16% of EXT-classified outliers exhibit absolute magnitudes more typical of QSOs.
\n\nper survey. While this fraction may be smaller for galaxies without nuclear activity, AGN present a range of challenges where pipelines fail. Examples of failures include cases where e.g. MgII is misidentified as Lymanα , leading to ambiguous redshifts around z = 0 . 7 -0 . 9 ⇔ 1 . 5 -2 . 5 or sources with strong featureless continuum typical of Blazars or, finally, sources with intense broad emission lines compared to the background continuum (Chen et al. 2018; Ni et al. 2021), as well as those with weak emission lines and a flat continuum background.\n8.2. Incorrect type classification\nA subset of sources in the training sample, although having redshifts of z ≥ 1, are classified as extended (i.e. non-PSF). This classification is likely incorrect, given the pixel resolution of LS10. The apparent extension is more plausibly a result of poor seeing conditions that were not properly accounted for, or the presence of nearby neighboring sources (see Sect. 5.3 of Hsu et al. 2014). Since the morphological type a ff ects predictions, akin to a prior in template fitting methods, this misclassification explains the increase in outliers within the EXT sample as redshift increases (see middle row in right panel of Figure 15). To verify this, we computed the absolute g magnitudes for outliers to assess whether their values fall within the expected range for galaxies and AGN (Véron-Cetty & Véron 2010). Figure 16 shows that 16% of sources classified as EXT (and thus galaxy dominated) have an absolute magnitude which exceeds the range typical of galaxies MEXT [-16,-24], confirming that their TYPE is wrongly assigned. Interestingly, approximately 96% of PSF sources meet the absolute magnitude requirement of MPSF [-20,31], indicating that their TYPE is generally accurate.\n8.3. Faulty photometry\nIn addition to issues connected to the reliability of labels, predictions also face challenges due to data inconsistencies arising from faulty photometry. Surveys often flag issues of this kind directly, including problems such as hot pixels, saturation, cosmic rays, bleed trails, and other uncategorized anomalies. Detecting and addressing such observational defects is crucial because the accuracy and reliability of photoz predictions cannot be guaranteed for compromised photometric inputs. We experimented with removing various groups of sources a ff ected by observational artefacts but found no improvement when excluding any single group. We therefore assume that, while removing some problematic sources could potentially improve performance, the reduction in training data size o ff sets any gains. Adding this kind of noise to the current training sample could be an approach to decouple these two e ff ects. Consequently, we treat the inclusion of all problematic sources as natural noise, as opposed to artificially degrading images employing methods such as Gaussian noise (Hayat et al. 2021).\n8.4. Survey depth\nWhen splitting the validation sample by S / N across all bands, we find a higher outlier fraction in sources with low S / N (see Table 5). Consequently, to evaluate the reliability of our photoz, we need to consider the photometric depth of LS10 and how increased photometric errors for fainter sources impact redshift estimate uncertainties. Figure 15 presents the prediction bias ⟨ z ⟩ , outlier fraction η , and dispersion σ NMAD as functions of r -band magnitude in LS10, X-ray flux (when available), and specz . As the r -band magnitude increases, as expected, the outlier fraction and scatter rise. This pattern is consistent with the exponential increase in photometric errors for, e.g. i -band magnitudes ≳ 21.5, beyond which the outlier fraction, which otherwise remains well below 10%, and scatter also appears to rise, especially for sources of type EXT (refer to Figure D.1). While faint LS10 sources have unreliable photo-z, future surveys such as LSST and Euclid will probe deeper with improved S / N. As such, our methodology can be adapted to these next-generation surveys, ensuring high accuracy across a broader range of magnitudes and providing robust redshift estimates for the extensive AGN populations these and other surveys will detect.\n8.5. Missing i-band\nFor DR10, the Legacy Survey footprint was extended by incorporating all available DECam data from various contributing surveys (refer to Sect. 3), including coverage in the i -band, though limited to a single pass. As a result, ∼ 10% of sources lack i -band observations. Unsurprisingly, and as shown in Table 5, the accuracy of photo-z increases when this band is also available.\n8.6. Biases\nWhile the photo-z residuals exhibit a symmetric distribution around zero with minimal scatter, they do not consistently center at zero, suggesting a potential systematic bias in the photo-z estimates. Ideally, we would like to achieve normalized residuals that remain consistent irrespective of redshift or selection. Currently, this is not the case, as biases are not corrected for and are mostly influenced by the distribution of our training samples, which are specific to their respective survey and inherently biased.\n
\n\n
Table 5: Comparison of bias, fraction of outliers, and variance between photo-z computed with PICZL for various subsamples.
\n
\nNotes. ( 1 ) all sources; ( 2 ) split by type (PSF vs. EXT); ( 3 ) split by signal-to-noise ratio (S / N) of all available bands; ( 4 ) split by availability of the i -band; ( 5 ) split by selection criteria; ( 6 ) split by the presence of a faint [(mAGN - mNeigh . ) within -3 ≤ ∆ mag < -1 for all available bands] or bright [(mAGN - mNeigh . ) within ∆ mag > -1 for all available bands] neighbor within a 5 arcsecond radius. Additionally, results derived for the CSC2 blind sample for both PICZL and C ircle Z are provided for comparison.\n8.6.1. Selection effects\nWe investigate whether the combination of various ways of selecting AGN entering the training sample a ff ects the quality of the photo-z for specific subsamples. Using the classification outlined in Table 1, we split the validation sample based on observational criteria, creating a binary division between sources selected in optical or via X-ray. While there is only a minor difference in σ NMAD, we observe a higher outlier fraction for Xray-detected sources.\nWhile strong X-ray emission from an AGN implies strong optical and mid-IR from an AGN, which should overpower the galaxy emission in the latter two bands, complicating the process of determining accurate photo-z, local Seyfert 1 galaxies with high X-ray flux have their host remain distinctly visible. As shown in the central panel of Figure 15, however, the outlier fractions and accuracy remain consistent across the full range of X-ray fluxes, aside from small-number statistics for the wings of the distribution.\n8.6.2. Sample characteristics\nUnlike for galaxies, where the redshift distribution n( z ) is wellestablished and can serve as a reliable prior, the AGN n( z ) is not su ffi ciently characterised. Therefore uncertainty surrounds the ML algorithm subliminally adopting a distribution similar to the n( z ) of the training sample.\nAt very low redshifts, AGN have a low surface density, resulting in only a few rare objects that can be considered for training. This phenomenon is intensified by the fact that deep surveys, as opposed to wide-field surveys, usually bypass such bright nearby objects in search of intermediate and high redshift sources. This leads to areas of scarcity in the training sample's specz distribution, with biased predictions towards redshift values where more data points exist. To mitigate this, we nor-\nmalize the CRPS score by (1 + z ), emphasizing the accuracy of low-redshift predictions. Additionally, at extremely low redshifts where the 4000 Å break is barely covered by the g filter, accurate redshift predictions are more di ffi cult to obtain. At very low redshift, the 6 arcsecond × 6 arcsecond cutout may be entirely filled by the galaxy, potentially misleading the algorithm into interpreting it as excess noise.\n8.6.3. Non-representative training samples\nOne method to tackle covariate shift by the imbalance of the bright-end dominated spectroscopic sample, is given by subdividing both the training and validation samples into subsets of n -dimensional feature space of distinguishable properties (Newman & Gruen 2022). Specifically, the prediction accuracy improves if the model used to generate a posterior for a blind source was trained exclusively on training sources residing within the corresponding feature space (Rosenbaum & Rubin 1984; Revsbech et al. 2017; Autenrieth et al. 2023). However, this approach might significantly reduce the training sample size per model making it more applicable to photoz codes dealing with large datasets, such as those for inactive galaxies (Newman & Gruen 2022).\nIn this work we have demonstrated that, in contrast to just a few years ago, AGN-targeted training samples are now su ffi -ciently large to provide reliable photo-z. Overcoming inherent biases still requires either gathering more representative samples that uniformly cover the full redshift range-particularly underrepresented regions-or applying statistical corrections. Thus, the focus has shifted from simply increasing the number of spectra to strategically obtaining spectra that cover specific regions of parameter space (Masters et al. 2015). However, given the substantial gains our model has shown over existing methods, a detailed examination of how these biases a ff ect prac-\ntical applications is beyond the scope of this study. Looking ahead, spectroscopic follow-up campaigns such as DESI, SDSSV / BHM (Kollmeier et al., in prep), and 4MOST (De Jong et al. 2019) will further enhance our capabilities. Nevertheless, predicting photo-z for AGN in deeper Euclid and LSST datasets remains challenging due to the limited availability of spectroscopic data for faint sources. The upcoming Subaru Prime Focus Spectrograph (PFS, Tamura et al. 2016) and Multi Object Optical and Near-infrared Spectrograph for the Very Large Telescope (VLT / MOONS; Cirasuolo et al. 2020) are expected to help address this shortfall by providing much-needed spectra for faint (& obscured / red) objects.\n8.7. Observational constraints: source crowding\nOne factor beyond our control is the local environment or conditions along the line of sight, which can significantly influence the emission characteristics of AGN. Nearby objects can add extra flux, which can a ff ect the accuracy of photometric measurements. We perform a positional cross-match between the validation sample and all LS10 sources within a 5 arcsecond radius of their optical counterparts. For each neighbour meeting this proximity criterion, we calculate apparent magnitudes and flag all sources where the brightest neighbour is no less than 1 magnitude dimmer. Table 5 shows that isolated sources exhibit better overall performance, while those with bright neighbours show drastically reduced quality compared to those with fainter neighbours. The presence of bright neighbours influences the observed flux in two ways: firstly, it complicates the derivation of sourcespecific colors due to convolved fluxes, and secondly, it a ff ects the observed spectra, potentially leading to the detection of two sets of emission lines and incorrect specz identification (Newman & Gruen 2022). A potential solution to this issue could be the implementation of segmentation maps, similar to SExtractor (Source Extractor, Bertin, E. & Arnouts, S. 1996) at image level, as LS10 currently only masks neighbours during their model flux fitting procedure.\nA persistent physical issue that cannot be entirely mitigated is blending. Unlike bright neighbours, which can be masked in principle, blending a ff ects both observed photometry and spectroscopy. While particularly faint neighbours do not substantially a ff ect the predictions negatively, sources identified as blends should be excluded from samples where accurate photoz estimates are required, until we can partially mask sources at the pixel level. The Rubin Observatory expects overlapping (inactive) galaxies to contribute at least 1% of the total flux within their pixels (Sanchez et al. 2021; Newman & Gruen 2022). Blends are also expected to occur in the spectroscopic sample, increasing systematic uncertainty and subsequently the fraction of outliers by up to 5% (Masters et al. 2019). This may be mitigated in future by higher-quality imaging from spacebased data from missions such as Euclid or better tools for deblending (e.g., SCARLET / blendz ; Melchior et al. 2018; Jones & Heavens 2018). Alternatively, data augmentation or standard computer vision techniques could be implemented, such as artificially introducing blending e ff ects in the training sample. However, this approach is not trivial, as LS10 currently lacks a corresponding blending flag.\n8.8. Variability\nAGN are inherently variable sources, meaning that their observed flux can change significantly across di ff erent epochs and\nwavelengths, especially when observations are separated by substantial time gaps. This extends to images that are created by stacking multiple observations taken over extended periods (as in LS10), where the resulting flux represents an average value. Such variability complicates the accurate prediction of AGN redshifts (Simm et al. 2015). However, future time-resolved imaging from LSST will allow us to better account for these variations and potentially use the correlation between AGN variability and their physical properties as a feature to improve photo-z predictions.\n8.9. Scalability in image-based photo-z\nOpen issues remain regarding the feasibility of handling the vast data volumes and computational requirements associated with photo-z estimated from images. Exemplified by the calculation of new estimates for XMM-SERVS concerning storage, processing time, and computational resources, we give an overview in Appendix G.\nA crucial future direction involves exploring ways to utilize imaging data e ffi ciently without necessitating extensive local downloads and computations for all image-based analyses of such data sets, potentially leveraging online platforms for realtime analysis, bringing the compute to the data (Zhang & Zhao 2015).\n9. Summary and outlook\nThis work has been driven by the goal of determining reliable photo-z for X-ray detected AGN in wide-field surveys, such as eROSITA (Merloni et al. 2024). For that reason, we have concentrated on utilizing data from LS10, which provides su ffi ciently homogeneous coverage and depth in 4 optical bands, enriched by the flux measured on NEOWISE7 data for all identified sources (see Sect. 3). LS10 overlaps almost entirely with the eROSITA footprint, simplifying the cross calibration of data, a processing step typically necessary when merging di ff erent surveys.\nWe introduce PICZL, a CNN-based machine learning model designed primarily for AGN redshift estimation, but which holds the potential to reliably measure photoz s for a broad range of extragalactic sources, provided an appropriately constructed training sample is available. In this study, the training sample comprises both X-ray and optically selected AGN. Across our validation set, PICZL demonstrates consistently robust performance, with comparable accuracies and lower outlier fractions particularly for PSF sources, as opposed to the results obtained by Salvato et al. (2022) for similar objects using SED-fitting (see Figure 7 and Table 5). The results show significant improvements for both point-like and extended sources, indicating that the model's performance is primarily driven by the depth and hence photometric error of the training data. Additionally, when tested on a blind sample of X-ray-selected AGN, PICZL maintained comparable results with respect to σ NMAD and η . Notably, on this test set it outperformed (C irclez ; Saxena et al. 2024) by achieving a 20% improvement in accuracy and a 30% reduction in the fraction of outliers (refer to Sect. 6).\nWe also applied PICZL to estimate photo-z for the approximately 30% of XMM-SERVS sources (Chen et al. 2018; Ni et al. 2021), detected in LS10 (see Table 4), demonstrating comparable variance and a substantially improved outlier fraction up to a limiting magnitude of 23.5 AB, using much fewer bands and significantly less sensitive imaging. However, it is important to note that the spectroscopic sample at this faint limit is small\nand likely biased toward sources with higher spectroscopic success rates. Consequently, the most reliable photo-z results are expected at brighter magnitudes (as discussed in Saxena et al. 2024). In response to these findings, we are releasing a new catalog of photo-z, complete with 1 and 3 σ error margins for the XMM-SERVS fields (see Sect. 7).\nPICZL will be used in the next generation of photo-z for sources detected by eROSITA, possibly switching from LS10 to Euclid and, most importantly, LSST, as they are poised to deliver more homogeneous and deeper data with broader wavelength coverage. In particular the availability of NIR data at image level, will improve our ability to determine accurate redshift for faint and high-z sources. Our study has shown that the performance of PICZL, when relying predominantly on images, is already robust (see Table 3). Crucially, we have shown that wellcalibrated images alone can su ffi ce for accurate photo-z estimation, eliminating the need for catalog creation, which is often based on predefined models. This suggests that future surveys could reduce their dependence on catalog-based data for photo-z computation (refer to Sect. 6). Although this study has focused on calculating photo-z for AGN, the approach can be generalised and adapted to other source types, e.g., normal galaxies, with an appropriately constructed training sample (Götzenberger et al. in prep.). Subsequently our findings point to a bright future for all-sky surveys, also thanks to the continue expansion of training sample sizes, supported by initiatives like SDSS-V / BHM, 4MOST and VLT / MOONS, which will enhance the reliability and completeness of photoz estimates (refer to Sect. 8).\nLooking forward, the next step in advancing photo-z estimation for AGN lies in exploring more sophisticated machine learning architectures beyond CNNs. For example, transformers with shared latent space embeddings o ff er a promising avenue. These models have shown success in integrating information from various data sources, such as images and entire spectra, potentially reducing uncertainties associated with traditional specz pipelines by leveraging multi-modal data fusion (DonosoOliva et al. 2023; Parker et al. 2024). Additionally, incorporating other informative features like X-ray flux, when available, holds promise. Integrating these diverse data sources within a unified framework has the potential to refine redshift estimates even further.\n10. Data availability\nWith this paper, we present a new catalog of photo-z derived using PICZL for the ∼ 30% of sources within the XMM-SERVS (ELAIS-S1, W-CDF-S, and LSS) X-ray source catalogs (Chen et al. 2018; Ni et al. 2021) that are detected in the LS10 survey. Hence, it includes updated photo-z for sources with catastrophic failures in the original works. A detailed description of the catalog columns can be found in Appendix F. The full catalog is only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http:// cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/ .\nAcknowledgements. WRand MS are grateful for the constant support of Dustin Lang in handling Legacy Survey-related issues. Part of this work was supported by the German Deutsche Forschungsgemeinschaft, DFG , under Germany's Excellence Strategy - EXC 2094 - 390783311. We gratefully acknowledge funding from FONDECYT Regular - 1231718 (RJA), 1230345 (CR), and 1241005 (FEB), CATA-BASAL - FB210003 (RJA, CR, FEB), and ANID - Millennium Science Initiative - AIM23-0001 (FEB). JA acknowledges support from a UKRI Future Leaders Fellowship (grant code: MR / T020989 / 1) This work is based on data from eROSITA, the soft X-ray instrument aboard SRG, a joint RussianGerman science mission supported by the Russian Space Agency (Roskosmos), in the interests of the Russian Academy of Sciences represented by its Space\nResearch Institute (IKI), and the Deutsches Zentrum für Luft- und Raumfahrt (DLR). The SRG spacecraft was built by Lavochkin Association (NPOL) and its subcontractors and is operated by NPOL with support from the Max Planck Institute for Extraterrestrial Physics (MPE). The development and construction of the eROSITA X-ray instrument were led by MPE, with contributions from the Dr. Karl Remeis Observatory Bamberg & ECAP (FAU Erlangen-Nuernberg), the University of Hamburg Observatory, the Leibniz Institute for Astrophysics Potsdam (AIP), and the Institute for Astronomy and Astrophysics of the University of Tübingen, with the support of DLR and the Max Planck Society. The Argelander Institute for Astronomy of the University of Bonn and the Ludwig Maximilians Universität Munich also participated in the science preparation for eROSITA. The Legacy Surveys consist of three individual and complementary projects: the Dark Energy Camera Legacy Survey (DECaLS; Proposal ID 2014B-0404; PIs: David Schlegel and Arjun Dey), the Beijing-Arizona Sky Survey (BASS; NOAO Prop. ID 2015A-0801; PIs: Zhou Xu and Xiaohui Fan), and the Mayall z-band Legacy Survey (MzLS; Prop. ID 2016A-0453; PI: Arjun Dey). DECaLS, BASS, and MzLS together include data obtained, respectively, at the Blanco telescope, Cerro Tololo Inter-American Observatory, NSF's NOIRLab; the Bok telescope, Steward Observatory, University of Arizona; and the Mayall telescope, Kitt Peak National Observatory, NOIRLab. Pipeline processing and analyses of the data were supported by NOIRLab and the Lawrence Berkeley National Laboratory (LBNL). The Legacy Surveys project is honored to be permitted to conduct astronomical research on Iolkam Du'ag (Kitt Peak), a mountain with particular significance to the Tohono O'odham Nation. NOIRLab is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. LBNL is managed by the Regents of the University of California under contract to the U.S. Department of Energy. This project used data obtained with the Dark Energy Camera (DECam), which was constructed by the Dark Energy Survey (DES) collaboration. Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A & M University, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho de Amparo, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Cientifico e Tecnologico and the Ministerio da Ciencia, Tecnologia e Inovacao, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey. The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Energeticas, Medioambientales y Tecnologicas-Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgenossische Technische Hochschule (ETH) Zurich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ciencies de l'Espai (IEEC / CSIC), the Institut de Fisica d'Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig Maximilians Universitat Munchen and the associated Excellence Cluster Universe, the University of Michigan, NSF's NOIRLab, the University of Nottingham, the Ohio State University, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, and Texas A&M University. BASS is a key project of the Telescope Access Program (TAP), which has been funded by the National Astronomical Observatories of China, the Chinese Academy of Sciences (the Strategic Priority Research Program 'The Emergence of Cosmological Structures' Grant # XDB09000000), and the Special Fund for Astronomy from the Ministry of Finance. The BASS is also supported by the External Cooperation Program of Chinese Academy of Sciences (Grant # 114A11KYSB20160057), and Chinese National Natural Science Foundation (Grant # 12120101003, # 11433005). The Legacy Survey team uses data products from the Near-Earth Object Wide-field Infrared Survey Explorer (NEOWISE), a project of the Jet Propulsion Laboratory / California Institute of Technology. NEOWISE is funded by the National Aeronautics and Space Administration. The Legacy Surveys imaging of the DESI footprint is supported by the Director, O ffi ce of Science, O ffi ce of High Energy Physics of the U.S. Department of Energy under Contract No. DE-AC02-05CH1123, by the National Energy Research Scientific Computing Center, a DOE O ffi ce of Science User Facility under the same contract, and by the U.S. National Science Foundation, Division of Astronomical Sciences under Contract No. AST-0950945 to NOAO. Funding for the Sloan Digital Sky Survey V has been provided by the Alfred P. Sloan Foundation, the Heising-Simons Foundation, the National Science Foundation, and the Participating Institutions. SDSS acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. SDSS telescopes are located at Apache Point Observatory, funded by the Astrophysical Research Consortium and operated by New Mexico State University, and at Las Campanas Observatory, operated by the Carnegie Institution for\nScience. The SDSS web site is www.sdss.org . SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration, including Caltech, The Carnegie Institution for Science, Chilean National Time Allocation Committee (CNTAC) ratified researchers, The Flatiron Institute, the Gotham Participation Group, Harvard University, Heidelberg University, The Johns Hopkins University, L'Ecole polytechnique fédérale de Lausanne (EPFL), Leibniz-Institut für Astrophysik Potsdam (AIP), Max-PlanckInstitut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Extraterrestrische Physik (MPE), Nanjing University, National Astronomical Observatories of China (NAOC), New Mexico State University, The Ohio State University, Pennsylvania State University, Smithsonian Astrophysical Observatory, Space Telescope Science Institute (STScI), the Stellar Astrophysics Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Illinois at Urbana-Champaign, University of Toronto, University of Utah, University of Virginia, Yale University, and Yunnan University.\nReferences\nAbdalla, F. B., Banerji, M., Lahav, O., & Rashkov, V. 2011, MNRAS, 417, 1891 Aird, J., Nandra, K., Laird, E. S., et al. 2010, MNRAS, 401, 2531-2551\n\nAit-Ouahmed, R., Arnouts, S., Pasquet, J., Treyer, M., & Bertin, E. 2023, Multimodality for improved CNN photometric redshifts\nAkiba, T., Sano, S., Yanase, T., Ohta, T., & Koyama, M. 2019, Optuna: A Nextgeneration Hyperparameter Optimization Framework\n\nAlexander, D. M., Davis, T. M., Chaussidon, E., et al. 2023, ApJ, 165, 124\nAlmeida, A., Anderson, S. F., Argudo-Fernández, M., et al. 2023, ApJ Supple-\nment Series, 267, 44\nAlmosallam, I. A., Jarvis, M. J., & Roberts, S. J. 2016, MNRAS, 462, 726\nAnanna, T. T., Salvato, M., LaMassa, S., et al. 2017, ApJ, 850, 66\n\nArnouts, S. & Ilbert, O. 2011, Astrophysics Source Code Library, 08009\nAutenrieth, M., van Dyk, D. A., Trotta, R., & Stenning, D. C. 2023, Stratified Learning: A General-Purpose Statistical Method for Improved Learning under Covariate Shift\n\nBaum, W. 1957, ApJ, 62, 6\n\nBeck, R., Dobos, L., Budavári, T., Szalay, A., & Csabai, I. 2017, Astronomy and Computing, 19, 34\n\nBell, E. F., Wolf, C., Meisenheimer, K., et al. 2004, ApJ, 608, 752-767 Benitez, N. 2000, ApJ, 536, 571\nBertin, E. & Arnouts, S. 1996, Astron. Astrophys. Suppl. Ser., 117, 393\nBettoni, D., Falomo, R., Kotilainen, J. K., Karhunen, K., & Uslenghi, M. 2015,\nMNRAS, 454, 4103\nBlanton, M. R., Bershady, M. A., Abolfathi, B., et al. 2017, ApJ, 154, 28\nBoller, T., Freyberg, M. J., Trümper, J., et al. 2016, A&A, 588, A103\nBolton, A. S., Schlegel, D. J., Aubourg, E., et al. 2012, ApJ, 144, 144\n\nBolzonella, M., Miralles, J.-M., & Pello', R. 2000, Photometric Redshifts based on standard SED fitting procedures\n\nBordoloi, R., Lilly, S. J., & Amara, A. 2010, MNRAS, no\nBoutsia, K., Grazian, A., Calderone, G., et al. 2020, ApJ Supplement Series, 250, 26\n\nBrammer, G. B., van Dokkum, P. G., & Coppi, P. 2008, ApJ, 686, 1503-1513 Brandt, W. N., Ni, Q., Yang, G., et al. 2018, Active Galaxy Science in the LSST Deep-Drilling Fields: Footprints, Cadence Requirements, and Total-Depth Requirements\n\nBreiman, L. 2001, Machine Learning, 545, 5\nBrescia, M., Cavuoti, S., Longo, G., & Stefano, V. D. 2014, A&A, 568, A126\n\nBrescia, M., Salvato, M., Cavuoti, S., et al. 2019, MNRAS, 489, 663\nBrunner, H., Liu, T., Lamer, G., et al. 2022, A&A, 661, A1\nBuchner, J., Georgakakis, A., Nandra, K., et al. 2015, ApJ, 802, 89\nCampagne, J.-E. 2020, Adversarial training applied to Convolutional Neural Network for photometric redshift predictions\nCardamone, C. N., van Dokkum, P. G., Urry, C. M., et al. 2010, ApJS, 189, 270 Carliles, S., Budavári, T., Heinis, S., Priebe, C., & Szalay, A. S. 2010, ApJ, 712, 511\nCavuoti, S., Amaro, V., Brescia, M., et al. 2016, MNRAS, 465, 1959\nChen, C.-T. J., Brandt, W. N., Luo, B., et al. 2018, MNRAS, 478, 2132-2163\nCirasuolo, M., Fairley, A., Rees, P., et al. 2020, Published in The Messenger vol. 180, pp. 10-17, June 2020.\nCo ff ey, D., Salvato, M., Merloni, A., et al. 2019, A&A, 625, A123\nCohen, T. & Welling, M. 2016, in Proceedings of Machine Learning Research, Vol. 48, Proceedings of The 33rd International Conference on Machine Learning, ed. M. F. Balcan & K. Q. Weinberger (New York, New York, USA: PMLR), 2990-2999\nCollaboration:, D. E. S., Abbott, T., Abdalla, F. B., et al. 2016, MNRAS, 460, 1270\nCollaboration, E., Mellier, Y., Abdurro'uf, et al. 2024, Euclid. I. Overview of the Euclid mission\nCollister, A. A. & Lahav, O. 2004, Publications of the Astronomical Society of the Pacific, 116, 345\n\nComparat, J., Merloni, A., Dwelly, T., et al. 2020, A&A, 636, A97 Connolly, A. J., Csabai, I., Szalay, A. S., et al. 1995, ApJ, 110, 2655 Croom, S. M., Smith, R. J., Boyle, B. J., et al. 2004, MNRAS, 349, 1397-1418 Dahlen, T., Mobasher, B., Faber, S. M., et al. 2013, ApJ, 775, 93 Dawid, A. P. 1984, Journal of the Royal Statistical Society. Series A (General),\n\n147, 278\nDeJong, R. S., Agertz, O., Berbel, A. A., et al. 2019, Published in The Messenger vol. 175, pp. 3-11, March 2019.\nDESI-Collaboration, Aghamousa, A., Aguilar, J., et al. 2016, The DESI Experiment Part I: Science,Targeting, and Survey Design\nDesprez, G., Paltani, S., Coupon, J., et al. 2020, A&A, 644, A31\nDey, A., Schlegel, D. J., Lang, D., et al. 2019, ApJ, 157, 168\nDey, B., Andrews, B. H., Newman, J. A., et al. 2022a, MNRAS, 515, 5285-5305\n\nDey, B., Newman, J. A., Andrews, B. H., et al. 2022b, Re-calibrating Photometric\nRedshift Probability Distributions Using Feature-space Regression\nD'Isanto, A. & Polsterer, K. L. 2018, A&A, 609, A111\n\nDonoso-Oliva, C., Becker, I., Protopapas, P., et al. 2023, A&A, 670, A54\nDrlica-Wagner, A., Carlin, J. L., Nidever, D. L., Ferguson, P. S., & Kuropatkin, N. 2021, ApJ Supplement Series, 256, 2\nDuda, R. O., Hart, P. E., & Stork, D. G. 1973, Pattern Classification, 2nd edn. (New York: Wiley)\nDuncan, K. J. 2022, MNRAS, 512, 3662-3683\nDwelly, T., Salvato, M., Merloni, A., et al. 2017, MNRAS, 469, 1065\n\nEriksen, M., Alarcon, A., Cabayol, L., et al. 2020, MNRAS, 497, 4565-4579\n\nFan, X., Banados, E., & Simcoe, R. A. 2022, Quasars and the Intergalactic Medium at Cosmic Dawn\nFerrarese, L. & Merritt, D. 2000, ApJ, 539, L9-L12\nFlaugher, B., Diehl, H. T., Honscheid, K., et al. 2015, ApJ, 150, 150\nFotopoulou, S. & Paltani, S. 2018, A&A, 619, A14\nFukushima, K. 1980, Biological Cybernetics, 36, 193\nGebhardt, K., Bender, R., Bower, G., et al. 2000, ApJ, 539, L13-L16\nGeorgakakis, A., Aird, J., Buchner, J., et al. 2015, MNRAS, 453, 1946-1964\nGneiting, T., Raftery, A., Westveld, A., & Goldman, A. 2005, Monthly Weather Review - MON WEATHER REV, 133\nGomes, Z., Jarvis, M. J., Almosallam, I. A., & Roberts, S. J. 2017, MNRAS, 475, 331\nGrimit, E., Gneiting, T., Berrocal, V., & Johnson, N. 2006, Quarterly Journal of the Royal Meteorological Society, 132, 2925\nGéron, A. 2019, Hands-On Machine Learning with Scikit-Learn, Keras & TensorFlow (O'Reilly, Kiwisoft S.A.S.)\nHatfield, P. W., Almosallam, I. A., Jarvis, M. J., et al. 2020, MNRAS, 498, 5498 Hayat, M. A., Stein, G., Harrington, P., Luki'c , Z., & Mustafa, M. 2021, ApJ Letters, 911, L33\nHe, K., Zhang, X., Ren, S., & Sun, J. 2015, Deep Residual Learning for Image Recognition\n\nHeckman, T. M. & Best, P. N. 2014, Annual Review of A&A, 52, 589\n\nHenghes, B., Thiyagalingam, J., Pettitt, C., Hey, T., & Lahav, O. 2022, MNRAS, 512, 1696\nHewett, P. C. & Wild, V. 2010, MNRAS, no\nHoyle, B. 2016, Measuring photometric redshifts using galaxy images and Deep Neural Networks\nHsu, L.-T., Salvato, M., Nandra, K., et al. 2014, ApJ, 796, 60\nI. Goodfellow, Y. Bengio, A. C. 2016, Deep Learning (Massachusetts Institute of Technology)\nIlbert, O., Arnouts, S., McCracken, H. J., et al. 2006, A&A, 457, 841-856 Illingworth, G. D. 1999, Astrophysics, 269, 165\nIvezic, Z., Kahn, S. M., Tyson, J. A., et al. 2019, ApJ, 873, 111\nJones, D. M. & Heavens, A. F. 2018, MNRAS, 483, 2487-2505\nJones, E. & Singal, J. 2017, A&A, 600, A113\nKind, M. C. & Brunner, R. J. 2013, MNRAS, 432, 1483\nKingma, D. P. & Ba, J. 2017, Adam: A Method for Stochastic Optimization\nKluge, M., Comparat, J., Liu, A., et al. 2024, The First SRG / eROSITA All-Sky Survey: Optical Identification and Properties of Galaxy Clusters and Groups in the Western Galactic Hemisphere\nKollmeier, J., Anderson, S. F., Blanc, G. A., et al. 2019, Bulletin of the AAS, 51, https: // baas.aas.org / pub / 2020n7i274\nKormendy, J. & Ho, L. C. 2013, Annual Review of A&A, 51, 511\nKullback, S. & Leibler, R. A. 1951, The Annals of Mathematical Statistics, 22, 79\nLaMassa, S. M., Georgakakis, A., Vivek, M., et al. 2019, ApJ, 876, 50 Lang, D. 2014, ApJ, 147, 108\nLang, D., Hogg, D. W., & Mykytyn, D. 2016, The Tractor: Probabilistic astronomical source detection and measurement\nLaurino, O., D'Abrusco, R., Longo, G., & Riccio, G. 2011, MNRAS, 418, 2165 LeCun, Y., Boser, B., Denker, J. S., et al. 1989, Neural Computation, 1, 541 Lecun, Y., Bottou, L., Bengio, Y., & Ha ff ner, P. 1998a, Proceedings of the IEEE, 86, 2278\n\nLecun, Y., Bottou, L., Bengio, Y., & Ha ff ner, P. 1998b, Proceedings of the IEEE,\n86, 2278\nLi, C., Zhang, Y., Cui, C., et al. 2021, MNRAS, 509, 2289 Li, C., Zhang, Y., Cui, C., et al. 2022, MNRAS, 518, 513 Lin, Q., Fouchez, D., Pasquet, J., et al. 2022, A&A, 662, A36\nLines, N. E. P., Roset, J. F.-Q., & Scaife, A. M. M. 2024, E(2)-Equivariant Features in Machine Learning for Morphological Classification of Radio Galaxies\nLiu, Z., Mao, H., Wu, C.-Y., et al. 2022, in Proceedings of the IEEE\n/\nCVF Con-\nference on Computer Vision and Pattern Recognition (CVPR), 11976-11986\nLuken, K. J., Norris, R. P., & Park, L. A. F. 2019, Publications of the Astronom-\nical Society of the Pacific, 131, 108003 Luo, B., Brandt, W. N., Xue, Y. Q., et al. 2010, ApJ Supplement Series, 187, 560 Lyke, B. W., Higley, A. N., McLane, J. N., et al. 2020, ApJ Supplement Series, 250, 8\n\nM. Deru, A. N. 2019, Deep Learning mit TensorFlow, Keras und TensorFlow.js. (Rheinwerk Verlag)\n\nMa, L., Lu, Z., Shang, L., & Li, H. 2015, Multimodal Convolutional Neural Networks for Matching Image and Sentence\nMadau, P. & Dickinson, M. 2014, Annual Review of A&A, 52, 415-486\nMainzer, A., Bauer, J., Grav, T., et al. 2011, ApJ, 731, 53\n\nMalz, A. I., Marshall, P. J., DeRose, J., et al. 2018, ApJ, 156, 35\n\nMasters, D., Capak, P., Stern, D., et al. 2015, ApJ, 813, 53\n\nMasters, D. C., Stern, D. K., Cohen, J. G., et al. 2019, ApJ, 877, 81\n\nMauduit, J.-C., Lacy, M., Farrah, D., et al. 2012, Publications of the Astronomical Society of the Pacific, 124, 714-736\nMeisner, A. M., Lang, D., & Schlegel, D. J. 2017, ApJ, 153, 38\n\nMelchior, P., Moolekamp, F., Jerdee, M., et al. 2018, Astronomy and Computing, 24, 129-142\nMerloni, A., Lamer, G., Liu, T., et al. 2024, A&A, 682, A34\n\nMerloni, A., Predehl, P., Becker, W., et al. 2012, eROSITA Science Book: Mapping the Structure of the Energetic Universe\nMeshcheryakov, A., Glazkova, V., Gerasimov, S., & Mashechkin, I. 2018, Astronomy Letters, 44, 735\nMountrichas, G., Corral, A., Masoura, V. A., et al. 2017, A&A, 608, A39 Newman, J. A. & Gruen, D. 2022, Annual Review of A&A, 60\nNi, Q., Brandt, W. N., Chen, C.-T., et al. 2021, ApJ Supplement Series, 256, 21 Nishizawa, A. J., Hsieh, B.-C., Tanaka, M., & Takata, T. 2020, Photometric Redshifts for the Hyper Suprime-Cam Subaru Strategic Program Data Release 2\n\nNorris, R. P., Salvato, M., Longo, G., et al. 2019, Publications of the Astronomical Society of the Pacific, 131, 108004\n\nO'Shea, K. & Nash, R. 2015, An Introduction to Convolutional Neural Networks Padovani, P., Alexander, D. M., Assef, R. J., et al. 2017, The A&A Review, 25 Parker, L., Lanusse, F., Golkar, S., et al. 2024, MNRAS, 531, 4990-5011\n\nPasquet, J., Bertin, E., Treyer, M., Arnouts, S., & Fouchez, D. 2018, A&A, 621, A26\n\nPierce, C. M., Lotz, J. M., Primack, J. R., et al. 2010, MNRAS\nPovi'c , M., Sánchez-Portal, M., García, A. M. P., et al. 2012, A&A, 541, A118 Predehl, P., Andritschke, R., Arefiev, V., et al. 2021, A&A, 647, A1\n\nRau, M. M., Seitz, S., Brimioulle, F., et al. 2015, Accurate photometric redshift probability density estimation - method comparison and application\n\nRevsbech, E. A., Trotta, R., & van Dyk, D. A. 2017, MNRAS, 473, 3969-3986\nRosenbaum, P. & Rubin, D. 1984, Journal of the American Statistical Associa-\ntion, 79\n\nRosenblatt, F. 1958, The perceptron: A probabilistic model for information storage and organization in the brain\nRuiz, A., Corral, A., Mountrichas, G., & Georgantopoulos, I. 2018, A&A, 618, A52\nSadeh, I., Abdalla, F. B., & Lahav, O. 2016, Publications of the Astronomical Society of the Pacific, 128, 104502\nSalvato, M., Hasinger, G., Ilbert, O., et al. 2008, ApJ, 690, 1250\nSalvato, M., Ilbert, O., Hasinger, G., et al. 2011, ApJ, 742, 61\nSalvato, M., Ilbert, O., & Hoyle, B. 2018, The many flavours of photometric redshifts\nSalvato, M., Wolf, J., Dwelly, T., et al. 2022, A&A, 661, A3\nSánchez, C., Kind, M. C., Lin, H., et al. 2014, MNRAS, 445, 1482\nSanchez, J., Mendoza, I., Kirkby, D. P., & Burchat, P. R. 2021, Journal of Cosmology and Astroparticle Physics, 2021, 043\nSavi'c, D. V., Jankov, I., Yu, W., et al. 2023, The LSST AGN Data Challenge: Selection methods\n\nSaxena, A., Salvato, M., Roster, W., et al. 2024, CircleZ: Reliable Photometric redshifts for AGN computed using only photometry from Legacy Survey Imaging for DESI\nScaramella, R., Amiaux, J., Mellier, Y., et al. 2022, A&A, 662, A112\nSchmidt, S. J., Malz, A. I., Soo, J. Y. H., et al. 2020, MNRAS, 499, 1587\nSchuldt, S., Suyu, S. H., Cañ ameras, R., et al. 2021, A&A, 651, A55\n\nScolnic, D. M., Lochner, M., Gris, P., et al. 2018, Optimizing the LSST Observing Strategy for Dark Energy Science: DESC Recommendations for the Deep Drilling Fields and other Special Programs\nSilva, D. R., Blum, R. D., Allen, L., et al. 2016, in American Astronomical Society Meeting Abstracts, Vol. 228, American Astronomical Society Meeting Abstracts #228, 317.02\n\nSimm, T., Saglia, R., Salvato, M., et al. 2015, A&A, 584, A106 Soo, J. Y. H., Moraes, B., Joachimi, B., et al. 2017, MNRAS, 475, 3613-3632 Stabenau, H. F., Connolly, A., & Jain, B. 2008, MNRAS, 387, 1215\n\nSteidel, C. C., Giavalisco, M., Dickinson, M., & Adelberger, K. L. 1996, ApJ, 112, 352\nTamura, N., Takato, N., Shimono, A., et al. 2016, in Ground-based and Airborne Instrumentation for Astronomy VI, ed. C. J. Evans, L. Simard, & H. Takami (SPIE)\nTreyer, M., Ait-Ouahmed, R., Pasquet, J., et al. 2023, CNN photometric redshifts in the SDSS at r ≤ 20\nVéron-Cetty, M. P. & Véron, P. 2010, A&A, 518, A10\nViroli, C. & McLachlan, G. J. 2017, Deep Gaussian Mixture Models\nVoges, W., Aschenbach, B., Boller, T., et al. 1999, The ROSAT All-Sky Survey Bright Source Catalogue\n\nWebb, N. A., Coriat, M., Traulsen, I., et al. 2020, A&A, 641, A136\nWeiler, M. & Cesa, G. 2021, General\nE\n(2)-Equivariant Steerable CNNs\nWilson, D., Nayyeri, H., Cooray, A., & Häußler, B. 2020, ApJ, 888, 83\n\nWright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, ApJ, 140, 1868-1881\n\nWu, Q. & Shen, Y. 2022, ApJ Supplement Series, 263, 42\n\nWuyts, S., Förster Schreiber, N. M., van der Wel, A., et al. 2011, ApJ, 742, 96\n\nYang, J., Fan, X., Gupta, A., et al. 2023, ApJ Supplement Series, 269, 27\nYork, D. G., Adelman, J., John E. Anderson, J., et al. 2000, ApJ, 120, 1579\n\nZhang, Y., Ma, H., Peng, N., Zhao, Y., & bing Wu, X. 2013, ApJ, 146, 22 Zhang, Y. & Zhao, Y. 2015, Data Science Journal\nZhao, D., Dalmasso, N., Izbicki, R., & Lee, A. B. 2021, Diagnostics for Conditional Density Models and Bayesian Inference Algorithms\nZhou, R., Ferraro, S., White, M., et al. 2023, J. Cosmology Astropart. Phys., 2023, 097\nZhou, R., Newman, J. A., Mao, Y.-Y., et al. 2021, MNRAS, 501, 3309 Zou, H., Zhang, T., Zhou, Z., et al. 2017, ApJ, 153\n1 Max-Planck-Institut für extraterrestrische Physik, Giessenbachstr. 1, 85748 Garching, Germany\n2 Exzellenzcluster ORIGINS, Boltzmannstr. 2, D-85748 Garching, Germany\n3 LMU Munich, Arnold Sommerfeld Center for Theoretical Physics, Theresienstr. 37 80333 München Germany\n4 LMU Munich, Universitäts-Sternwarte, Scheinerstr. 1, 81679 München, Germany\n5 Technical University of Munich Department of Computer Science I26 Boltzmannstr. 3 85748 Garching b. München Germany\n6 Max Planck Institut für Astronomie, Königstuhl 17, D-69117, Heidelberg, Germany\n7 Instituto de Astrofísica, Facultad de Física, Pontificia Universidad Católica de Chile, Campus San Joaquín, Av. Vicuña Mackenna 4860, Macul Santiago, Chile, 7820436\n8 Centro de Astroingeniería, Facultad de Física, Pontificia Universidad Católica de Chile, Campus San Joaquín, Av. Vicuña Mackenna 4860, Macul Santiago, Chile, 7820436\n9 Millennium Institute of Astrophysics, Nuncio Monseñor Sótero Sanz 100, Of 104, Providencia, Santiago, Chile\n10 Space Science Institute, 4750 Walnut Street, Suite 205, Boulder, Colorado 80301\n11 Institute for Astronomy, University of Edinburgh, Royal Observatory, Edinburgh EH9 3HJ, UK\n12 Instituto de Estudios Astrofísicos, Universidad Diego Portales, Av. Ejército Libertador 441, Santiago 8370191, Chile\n13 Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China\n14 Department of Astronomy, University of Washington, Box 351580, Seattle, WA, 98195, USA\n15 Department of Astronomy, University of Illinois at UrbanaChampaign, Urbana, IL 61801, USA\n16 National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA\n17 Center for Artificial Intelligence Innovation, University of Illinois at Urbana-Champaign, 1205 West Clark Street, Urbana, IL 61801, USA\n\n02.09.24, 21\n:\n39\nAppendix B: LS10 seeing for different samples\n\n\n
Fig. B.1: Histograms of the weighted average FWHM (seeing) values for the PICZL training, validation, and CSC2 blind samples. The close alignment of these distributions indicates no significant mismatch between the datasets, as they represent all-sky sampling. This similarity supports the applicability of the training sample to blind samples of comparable seeing.
\n\nAppendix C: CSC2 blind sample results\n\n\n
Page 1 of 2 Fig. C.1: Scatter plot of the CSC2 sample color-coded by point density, illustrating the performance of PICZL on previously unseen data, as the majority of sources lie within the defined outlier boundary, demonstrating the prediction robustness. Sources identified as outliers generally scatter close to the boundary, with only a few instances of significant deviations or catastrophic failures.
\n\nAppendix A: CNN model architecture\n\n\n
Fig. A.1: PICZL architecture, showcasing its three-threaded design. Two threads are dedicated to processing image inputs, while the third handles numerical data. The model's output layer generates distinct vectors corresponding to the means, standard deviations, and weights of multiple Gaussian distributions. This design enables the production of full PDFs, allowing the network to capture uncertainties. Question marks as the first dimension per thread input correspond to the (variable) sample size.
\n\nAppendix D: Photometric errors in LS10\nAppendix F: Column description of released photo-z catalog\n\n\n
Fig. D.1: Relationship between S / N and the associated photometric error for the LS10 i -band, considering all sources from Table 1. Each point is color-coded according to the i -band magnitude. A noticeable turning point is observed for sources with S / N ≲ 15, indicating a significant increase in magnitude error for sources fainter than ∼ 22 magnitude.
\n\nAppendix E: XMM-SERVS sources in LS10\n\n\n
Fig. E.1: Histograms of LS10 i -band magnitudes for the three XMM-SERVS subsamples: ELAIS-S1, W-CDF-S, and LSS. The magnitudes are displayed as normalized densities to allow for direct comparison across the fields. These distributions extend to fainter magnitudes than those shown in Figure 15, accounting for the increased η P denoted in Table 4. As the depth reached in PICZL's LS10 training set is limited toward brighter magnitudes, a direct comparison between subsamples is only valid when conducted within a controlled parameter space, as shown in Figure 13.
\n\n\n1. XID: Unique source ID assigned to each X-ray source in the original papers (see Chen et al. 2018; Ni et al. 2021)\n2. SURVEY: Original survey where the source was detected\n3. LS10_FULLID: Unique LS source ID assigned to optical counterpart. It is created by concatenating the LS coloumns RELEASE, BRICKID and OBJID.\n4. CTP_LS10_RA: Right Ascension in degrees of the LS10 optical counterpart\n5. CTP_LS10_DEC: Declination in degrees of the LS10 optical counterpart\n6. SPECZ: specz from original catalog\n7. PHZ_PICZL: Photo-z from PICZL\n8. PHZ_PICZL_l68: PICZL photo-z min at 1 sigma\n9. PHZ_PICZL_u68: PICZL photo-z max at 1 sigma\n10. PHZ_PICZL_l99: PICZL photo-z min at 3 sigma\n11. PHZ_PICZL_u99: PICZL photo-z max at 3 sigma\n\nAppendix G: PICZL run time\nThe computational demands for training and deploying PICZL depend mostly on the hardware configuration and dataset size. For our experiments, we leveraged a set of two Tesla V100PCIE graphics processing units (GPUs) each equipped with 32 GB of ready access memory (RAM), accompanied by a 48 core multi-thread CPU to accelerate the computational workload. This parallel processing capability significantly expedited the model training process, compared to running solely on a central CPU, allowing for faster convergence. Training a single model (refer to Figure A.1) on a dataset of 32 391 sources, corresponding to the 80:20 train-test split (refer to Table 1), each with 108 features (56 of which are images), takes approximately 25 minutes for 600 epochs. The use of an ensemble further scales the processing time by the number of models trained, with ensemble optimization depending on user preferences, making it di ffi -cult to provide a fixed time estimate. After finalizing the model, computing photo-z for e.g. 1393 sources in the XMM-SERVS W-CDF-S field, as displayed via Table 4, takes roughly 17 seconds. The storage requirements for the data used in this sample corresponds to approximately 250 MB (refer to Table 2). Looking ahead, upcoming surveys such as Euclid and LSST will produce higher-resolution images, demanding increased storage and computational resources due to larger data volumes and pixel counts. If the input dimensions were to increase, for instance, from 23x23 pixels to 64x64 pixels, the computational burden would rise significantly. While our current architecture of 32 GB RAM per GPU is e ffi cient for the current input sizes, processing larger cutouts may necessitate either more GPUs or GPUs with higher memory capacity to maintain feasible training times and performance. In scenarios where larger input sizes are anticipated, a thoughtful approach to model architecture and resource allocation will be crucial. Therefore, adapting to the capabilities of the hardware will be key to successfully utilizing the methods in the context of future data from LSST and Euclid, particularly given the much larger sample sizes that we expect from these surveys. Finally, performance will also improve by having PICZL revised by an expert software developer.\n"},"sections":{"kind":"list like","value":[{"title":"ABSTRACT","content":"Context. Computing reliable photometric redshifts (photo-z) for active galactic nuclei (AGN) is a challenging task, primarily due to the complex interplay between the unresolved relative emissions associated with the supermassive black hole and its host galaxy. Spectral energy distribution (SED) fitting methods, while e ff ective for galaxies and AGN in pencil-beam surveys, face limitations in wide or all-sky surveys with fewer bands available, lacking the ability to accurately capture the AGN contribution to the SED, hindering reliable redshift estimation. This limitation is a ff ecting the many 10s of millions of AGN detected in existing datasets, e.g., those AGN clearly singled out and identified by SRG / eROSITA. Aims. Our goal is to enhance photometric redshift performance for AGN in all-sky surveys while simultaneously simplifying the approach by avoiding the need to merge multiple data sets. Instead, we employ readily available data products from the 10th Data Release of the Imaging Legacy Survey for the Dark Energy Spectroscopic Instrument, which covers > 20,000 deg 2 of extragalactic sky with deep imaging and catalogbased photometry in the grizW1-W4 bands. We fully utilize the spatial flux distribution in the vicinity of each source to produce reliable photo-z. Methods. We introduce PICZL, a machine-learning algorithm leveraging an ensemble of convolutional neural networks. Utilizing a cross-channel approach, the algorithm integrates distinct SED features from images with those obtained from catalog-level data. Full probability distributions are achieved via the integration of Gaussian mixture models. Results. On a validation sample of 8098 AGN, PICZL achieves an accuracy σ NMAD of 4.5% with an outlier fraction η of 5.6%. These results significantly outperform previous attempts to compute accurate photo-z for AGN using machine learning. We highlight that the model's performance depends on many variables, predominantly the depth of the data and associated photometric error. A thorough evaluation of these dependencies is presented in the paper. Conclusions. Our streamlined methodology maintains consistent performance across the entire survey area, when accounting for di ff ering data quality. The same approach can be adopted for future deep photometric surveys such as LSST and Euclid, showcasing its potential for wide-scale realization. With this paper, we release updated photo-z (including errors) for the XMM-SERVS W-CDF-S, ELAIS-S1 and LSS fields. Key words. Photo-z, AGN, Extragalactic Surveys, Machine Learning","pages":[1]},{"title":"PICZL : Image-based photometric redshifts for AGN","content":"William Roster , ⋆ 1 , M. Salvato 1 , 2 , S. Krippendorf 3 , 4 , A. Saxena 5 , R. Shirley 1 , J. Buchner 1 , J. Wolf 2 , 6 , T. Dwelly 1 , F. E. Bauer 7 , 8 , 9 , 10 , J. Aird 11 , C. Ricci 12 , 13 , R. J. Assef 12 , S.F. Anderson 14 , X. Liu 15 , 16 , 17 , A. Merloni 1 , J. Weller 1 , 4 K. Nandra 1 (A ffi liations can be found after the references) November 14, 2024","pages":[1]},{"title":"1. Introduction","content":"In recent decades, our understanding of Active Galactic Nuclei (AGN) and their role in galaxy and cosmic evolution has significantly advanced. These luminous celestial powerhouses are thought to be fueled by the accretion of matter onto supermassive black holes (SMBHs) located at the centers of galaxies, exerting intense energetic radiation across the entire electromagnetic spectrum, ranging from radio to γ -rays (Padovani et al. 2017). The close correlation observed between the mass of the central SMBH, whether active or inactive and the properties of its host galaxy's bulge - such as the galaxy's mass and velocity dispersion (e.g., Gebhardt et al. 2000; Ferrarese & Merritt 2000) - suggests a co-evolutionary relationship between galaxies and their central engines (Kormendy & Ho 2013; Heckman & Best 2014). Ongoing research focuses on understanding scaling relations, the evolution of SMBHs within galaxies, and the interconnected rates of star formation (SFR) and black hole accretion (BHAR) over cosmic time (e.g., Madau & Dickinson 2014). To further explore and address these unresolved topics requires diverse AGN samples with reliable redshifts to determine BH demographics and constrain models of galaxy evolution. For all these studies, redshift is an indispensable quantity, with spectroscopic redshifts (spec-z) remaining the preferred es- imates for determining precise cosmic distances (Hoyle 2016). However, while multi-objects spectrographs, such as the Sloan Digital Sky Survey (SDSS-V; York et al. 2000; Kollmeier et al. 2019), the Dark Energy Spectroscopic Instrument (DESI; DESICollaboration et al. 2016), the Subaru Prime Focus Spectrograph (PFS; Tamura et al. 2016) or the 4-metre Multi-Object Spectroscopic Telescope (4MOST; De Jong et al. 2019), are set to provide a drastic rise in the number of observed sources over the next several years, we are currently in the situation in which millions of AGN have been detected all-sky by various surveys (e.g., by the Wide-field Infrared Survey Explorer mission (WISE; Wright et al. 2010), and the extended Roentgen Survey with an Imaging Telescope Array (eROSITA; Merloni et al. 2012; Predehl et al. 2021), with only the brightest sources having been observed spectroscopically (Dahlen et al. 2013). The growing disparity between photometric and spectroscopic observations will only widen with upcoming surveys such as the Legacy Survey of Space and Time (LSST; Ivezic et al. 2019) and Euclid (Collaboration et al. 2024), covering unprecedented areas and depths (Newman & Gruen 2022). Thus for the bulk of AGN, we must make use of multiband photometry and rely on photometric redshifts (photo-z). First implemented by Baum (1957) for inactive galaxies, these low-precision redshift estimates utilize photometric observations to e ff ectively obtain a sparsely sampled spectral energy distribution (SED), trading precision for scalability. They en- compass an array of techniques assuming color-redshift evolution (Connolly et al. 1995; Steidel et al. 1996; Illingworth 1999; Bell et al. 2004), including template-based approaches (e.g., Bolzonella et al. 2000; Ilbert et al. 2006; Salvato et al. 2008; Beck et al. 2017), where redshifted models built on theoretical or empirical SEDs are fitted to observed multi-band photometry. Although a limited number of available bands can introduce uncertainties (see review by Salvato et al. 2018), photo-z methods o ff er an e ffi cient way to estimate distances for all sources in an imaging survey, yielding highly accurate estimates with as few as three bands for passive galaxies (Benitez 2000; Abdalla et al. 2011; Arnouts & Ilbert 2011; Brescia et al. 2014; Desprez et al. 2020). By contrast, reliable photo-z for AGN have historically required highly homogenized photometry across > 20 filters, which was only achievable in pencil-beam surveys (Salvato et al. 2011). As such, this level of detail continues to be unfeasible for wide-area surveys. However, with the 10th data release of the DESI Legacy Imaging Surveys (LS10, Dey et al. 2019), we now have a broad-sky survey that, while lacking NIR coverage, includes a few optical bands supplemented by mid-IR WISE data. This allows us to explore the possibility of generating reliable photo-z for AGN over the full sky, despite having fewer filters compared to the densely sampled pencil-beam surveys. SED fitting applied to a broad population of AGN remains particularly challenging due to the uncertainty and di ffi culty of disentangling the relative contributions of the nucleus and respective host to a given band (e.g., Luo et al. 2010; Salvato et al. 2011; Brescia et al. 2019). Since the accretion properties of SMBHs, often characterized as the bolometric luminosity divided by the Eddington limit, or the Eddington ratio, significantly influence the SED of AGN, the intense power-law continuum radiation can either partly (host-dominated) or entirely (quasar-dominated), outshine the respective host, hiding key spectral features that lead to redshift degeneracies (Pierce et al. 2010; Povi'c et al. 2012; Bettoni et al. 2015). Consequently, selecting a limited number of templates can be insufficient for correct redshift determination, while increasing the number of templates raises the degeneracy (see discussion in Salvato et al. 2011; Ananna et al. 2017). In this regime of accounting for AGN contributions to galaxy photo-z, one potential approach involves modeling objects as a combination of quasar and galaxy templates (eg., Cardamone et al. 2010), performed with EAZY (Brammer et al. 2008). In addition, surveys typically estimate fluxes with models that do not account for a mixed contribution from AGN and host galaxy. Ultimately, AGN are also intrinsically variable sources on the timescales explored by the previously mentioned surveys leading to incongruent photometry acquired across di ff erent epochs. In contrast to template-fitting methods, more recent approaches have shifted towards the use of empirical Machine Learning (ML) models, performing regression or classification, to tackle photo-z applied predominantly to inactive galaxies (Collister & Lahav 2004; Laurino et al. 2011; Zhang et al. 2013; Hoyle 2016; D'Isanto & Polsterer 2018; Brescia et al. 2019; Eriksen et al. 2020; Li et al. 2021). Provided with a very large and complete spec-z sample, ML architectures manipulate photometric input features to minimize the divergence between spectroscopic and ML-derived redshifts. Over the years, a plethora of ML architectures, including decision trees (Breiman 2001; Carliles et al. 2010; Li et al. 2022), Gaussian processes (Almosallam et al. 2016) and K-nearest neighbours (Zhang et al. 2013; Luken et al. 2019) have been employed, yielding accurate point predictions and, more interestingly, full probability density functions (PDFs) (Kind & Brunner 2013; Cavuoti et al. 2016; Rau et al. 2015; Sadeh et al. 2016). The latter grants access to the prediction uncertainty, as otherwise naturally provided by template-fitting approaches, relevant for studies dealing with, e.g. luminosity functions (Aird et al. 2010; Buchner et al. 2015; Georgakakis et al. 2015). However, the limited availability of a sizable training sample of AGN has resulted in only a few attempts to compute photo-z for mostly nucleus-dominated objects with ML-based methods (Mountrichas et al. 2017; Fotopoulou & Paltani 2018; Ruiz et al. 2018; Meshcheryakov et al. 2018; Brescia et al. 2019; Nishizawa et al. 2020). More recently, the conventional approach of manually selecting photometric features for ML has been replaced by bright, well-resolved galaxies at low redshift (Hoyle 2016; Pasquet et al. 2018; Campagne 2020; Hayat et al. 2021). In this regime, integrating galaxy images into deep neural networks inherently captures essential details like flux, morphology, and other features that would typically be extracted from catalogs based on predefined assumptions, leading to a more comprehensive redshift estimation process. This approach is particularly advantageous for addressing current limitations faced by photo-z methods for AGN, as it leverages model-independent fluxes and redshift indicative features, including surface brightness profiles (Stabenau et al. 2008; Jones & Singal 2017; Gomes et al. 2017; Zhou et al. 2021, 2023). Unlike creating a single SED from total flux measurements, projects employing images with independent pixelby-pixel SEDs at identical redshift have demonstrated increased photo-z constraining power, alleviating previous empirical approaches by decreasing the fraction of outliers (Henghes et al. 2022; Schuldt et al. 2021; Lin et al. 2022; Dey et al. 2022a; Newman & Gruen 2022). Here, we introduce PICZL (Photometrically Inferred CNN redshift(Z) Likelihoods), an enhanced approach to photoz estimation that builds upon (C ircle Z by Saxena et al. 2024). While the authors demonstrated that redshift degeneracies encountered for AGN, typical in cases of limited photometry, can be broken by integrating aperture photometry alongside traditional total / model fluxes and colors, PICZL instead computes photo-z PDFs for AGN directly from homogenized flux band cutouts by leveraging the more detailed spatial light profile. All inputs are obtained utilizing LS10 exclusively. Similar to Saxena et al. (2024), PICZL can produce reliable photo-z PDFs for all Legacy-detected sources associated with an AGN. However, the model can, in principle, be applied to other extragalactic sources (e.g, inactive galaxies, Götzenberger et al. in prep.) granted that a dedicated training sample is used. We employ an ensemble of the same ML algorithm, notably convolutional neural networks (CNNs), known for their proficiency in learning intricate patterns, as outlined by (Lecun et al. 1998a). Specifically designed for image analysis, CNNs excel at identifying and extracting relevant predictive features directly from images, thereby reducing computational overhead compared to fully connected architectures. Harnessing this more extensive pool of information, these models surpass alternative models based on condensed feature-based input sets. The paper is structured as follows: Sect. 2 introduces the AGN training sample down-selection. Sect. 3 focuses on the photometric data products available within LS10. Sect. 4 details the photometric data preprocessing, followed by Sect. 5, which outlines the model pipeline. Sect. 6 presents and quantifies the redshift results, while Sect. 7 evaluates the photo-z released for the XMM-SERVS (Chen et al. 2018; Ni et al. 2021) fields. Sect. 8 outlines current limitations and discusses how we can achieve further improvements. Sect. 9 explores implications for future surveys, concluding with a summary. In this paper, unless stated di ff erently, we express magnitudes in the AB system and adopt a Λ CDM cosmology with H 0 = 69 . 8 km s -1 Mpc -1 , Ω m = 0 . 28 and Λ = 0 . 72.","pages":[1,2,3]},{"title":"2. AGN training sample selection","content":"In X-ray surveys, the identification of AGN has two distinct advantages - i) the reduced impact of moderate obscuration and ii) the lack of host dilution. Due to the inherent brightness of accreting SMBHs compared to their host galaxies, this results in a significantly higher nuclei-to-host emission ratio compared to observations in some of the neighbouring wavelength windows, such as UV-optical-NIR (Padovani et al. 2017). This naturally leads to a larger diversity of AGN observed by an X-ray telescope. That being said, surveys in the more accessible optical and NIR regime can increase the likelihood of detecting higherz , and in the case of MIR more heavily obscured AGN, compared to the soft X-ray bands. MLapproaches for photoz estimation in large surveys (e.g., Fotopoulou & Paltani 2018; Duncan 2022) typically classify objects into three broad categories: galaxies, quasars (QSOs), and stars, before computing photo-z. However, this classification is usually based on the optical properties and hence fails for obscured and / or lower-luminosity AGN. Since our goal is to improve on the quality of photo-z estimates for X-ray detected extragalactic sources, including type 2 AGN and low-redshift Seyfert 1 galaxies, generally, our training sample has to replicate this diversity. We achieve this by combining AGN selected across multiple wavelength bands. As a starting point, we include the same X-ray samples used in Saxena et al. (2024), namely the latest version of the XMM catalog, 4XMM, which spans 19 years of observations made with XMM-Newton (Webb et al. 2020) and data from the eROSITA CalPV-phase Final Equatorial-Depth Survey (eFEDS; Brunner et al. 2022), as they provide a reasonably representative and complete set of diverse AGN spanning 5 dex in X-ray flux out to redshift z ≲ 4. However, with just these, some portions of AGNparameter space remain imbalanced, such as highly luminious and / or high-z AGN. Thus, we expand the dataset by adding bright, optical and MIR, selections. We describe each of these samples in more detail below. This approach enhances the completeness of our training sample, which is essential to mitigate covariate shift, i.e., the shift in parameter space between the training and validation samples, so that the model generalizes e ff ectively to new data (Norris et al. 2019). Subsequently, a non-representative training sample may lead to systematically biased outcomes (Newman & Gruen 2022). Accordingly, algorithms will be strongly weighted towards the most densely populated regions of the training space (Duncan 2022).","pages":[3]},{"title":"2.1. Beyond eFEDS and 4XMM","content":"We can enhance the redshift distribution within our sample, particularly towards high ( z ≥ 3) redshift, in this otherwise underrepresented parameter space due to observational selection e ff ects. We recognize the subsequent incorporation of unavoidable selection biases in each survey while restricting the inclusion of sources at low redshift to a minimum. While the balance between dataset quality and size is critical, deep learning algorithms that operate on pixel-level inputs tend to perform optimally only when training datasets contain ≥ 400 000 galaxy images (Schuldt et al. 2021; Dey et al. 2022a; Newman & Gruen 2022). Since our method would benefit from a larger sample (see Table 1), we chose not to apply stringent quality criteria by only considering high-quality data, significantly reducing the number of sources available training. Such a reduction would also prevent the model from learning to handle lower-quality data, limiting its application to only high-quality validation data. By not making an initial down-selection, we retain the flexibility to apply quality cuts to future blind samples by using LS10 flags later.","pages":[3]},{"title":"2.1.1. Samples from optical selection","content":"We include the 2dF QSO Redshift Survey (2QZ, Croom et al. 2004) with ∼ 23k color selected QSOs in the magnitude range 18.25 ≤ bJ ≤ 20.85 at redshifts lower than z ∼ 3 and the QUasars as BRIght beacons for Cosmology in the Southern hemisphere survey (QUBRICS, Boutsia et al. 2020) with 224 bright (i ≤ 18) QSOs at redshifts of z ≥ 2 . 5.","pages":[3,4]},{"title":"2.1.2. Samples from optical follow-up of X-ray sources","content":"Additionally, we incorporate the SDSS-IV (Blanton et al. 2017) quasar catalog from Data Release 16 (DR16Q, Lyke et al. 2020) with ∼ 150k quasars collected from various subprogrammes including optical / IR selection down to g ≤ 22 in the LS10 footprint after subselecting high quality specz , as well as follow-up of Xray sources from ROSAT (Voges et al. 1999; Boller et al. 2016; Salvato et al. 2018) and XMM (e.g. LaMassa et al. 2019). As successor science programme, we also consider the Black Hole Mapper (BHM) SPectroscopic IDentfication of ERosita Sources (SPIDERS, Anderson et al., in prep, Aydar et al., in prep) from SDSS-V (Kollmeier et al., in prep) Data Release 18 (Dwelly et al. 2017; Co ff ey et al. 2019; Comparat et al. 2020; Almeida et al. 2023).","pages":[4]},{"title":"2.1.3. Samples of high-z sources","content":"Given the strong imbalance above z ∼ 3 . 5, we also include 400 optically / IR selected quasars at redshifts 4.8 ≤ z ≤ 6 . 6 down to g ≤ 24 from the high-redshift quasar survey in the DESI Early Data Release (EDR, Yang et al. 2023) and a compilation of highz quasars at z ≥ 5 . 3 published in literature (Fan et al. 2022).","pages":[4]},{"title":"2.2. Spectroscopic cross-referencing","content":"The parent sample of AGN is annotated with specz , where available. According to Figure 1, we also consider sources, including those from eFEDS and 4XMM, with spatial counterparts from a compilation of public redshifts (Kluge et al. 2024). The procedure by which we match optical counterparts in our combined sample to a compilation of quality criteria down-selected specz , is outlined in Sect. 3.1 of Saxena et al. (2024). Due to overlaps between surveys, we remove duplicates when combining samples. The final sample of sources with specz comprises 40 489 objects, with a breakdown in Table 1. Correspondingly, the (cumulative) histograms illustrating the n( z ) distributions that collectively constitute the PICZL sample are presented in Figure 2. 0","pages":[4]},{"title":"3. The survey","content":"To streamline and simplify our methodology, we have chosen to employ data from LS10 exclusively to mitigate potential complications arising from the heterogeneity of multiple datasets. Crucially, the survey area now extends over 20 000 deg 2 of optical griz and WISE W 1 -W 4 forced photometry, by incorporating the following datasets: In the north ( δ > 32.375 deg), LS10 uses the Beijing-Arizona Sky Survey (BASS, Zou et al. 2017) for g - and r -band coverage, and the Mayall z -band Legacy Survey (MzLS, Silva et al. 2016) for z -band coverage (Kluge et al. 2024). Count","pages":[4]},{"title":"3.1. Photometric data","content":"LS10 o ff ers registered, background-subtracted, and photometrically calibrated point spread function (PSF)-forced photometry, including corresponding errors. To extend their wavelength coverage, DR10 catalogs incorporate mid-infrared (mid-IR) forced photometry at wavelengths of 3.4, 4.6, 12, and 22 µ m(referred to as W1, W2, W3, and W4, respectively) for all optically detected sources in the LS10 via the Near-Earth Object Wide-field Infrared Survey Explorer (NEOWISE) (Mainzer et al. 2011; Lang 2014; Meisner et al. 2017). Sources are modeled simultaneously across all optical bands, ensuring consistency in shape and size measurements by fitting a set of light profiles, even for spatially extended sources. Consequently, alongside reliable total and multi-aperture (8 annuli ≤ 7 arcseconds, five annuli ≤ 11 arcseconds for the optical and mid-infrared bands, respectively) flux measurements, the LS10 catalog o ff ers seeing-convolved PSF, de Vaucouleurs, exponential disk, or composite de Vaucouleurs + exponential disk models obtained with the Tractor algorithm (Lang et al. 2016). Additionally, providing fluxes rather than magnitudes, enables considering sources with very low signalto-noise ratios without introducing biases at faint levels. This characteristic also facilitates flux stacking at the catalog level, enhancing the overall versatility and utility of the classification and fitting process within LS10 1 .","pages":[5]},{"title":"3.2. Imaging data","content":"In addition to catalog data, LS10 provides a rich set of imaging products. These include observations, flux model images, and residual maps for all available bands. For instance, the top pan- els (a, b, c) of Figure 3 display all image products for a g -band observation, respectively.","pages":[5]},{"title":"4. Preprocessing","content":"Here we detail the preprocessing steps taken to prepare our dataset, ensuring that it is clean, normalized, and structured appropriately.","pages":[5]},{"title":"4.1. Image preprocessing","content":"Building on the approach of Saxena et al. (2024), which demonstrated significant improvements by shifting from total to aperture flux utilizing information on the 2D light distribution, we aim to further refine the spatial characterization of sources. This is achieved by incorporating pixel-level flux resolution through imaging as base input. Images in individual bands or in combination (i.e., colors) reflect the surface brightness, angular size, and sub-component structures of the sources, indirectly providing redshift information (Stabenau et al. 2008). To obtain reliable photo-z directly from images, we utilize flux-calibrated optical cutouts across as many filters as available. With an average seeing FWHM of 1.3 arcseconds under nominal conditions, LS10 provides a pixel resolution of 0.262 arcseconds per pixel for the optical bands, reaching depths between 23 and 24.7 AB, depending on the specific band and region of the sky (see Dey et al. (2019) and Figure 1 in Saxena et al. (2024)). To enhance computational e ffi ciency and mitigate contamination from nearby sources, we restrict our cutout dimensions to 23 × 23 pixel, centered on the AGN coordinates in the four griz LS10 bands. Our cutouts correspond to a field of view (FOV) of approximately 6 arcseconds × 6 arcseconds. We base our choice of FOV on the angular size-redshift relation by computing the angular diameter distance d Λ via: Equation 1 and Figure 4 elucidate the connection between an object's physical size, its angular size, and redshift. Notably, we can e ff ectively map galaxies with a diameter of 30 kpc - representative of main sequence galaxies (Wuyts et al. 2011) within the confines of a 23 × 23 pixel cutout, covering the range of 0 . 5 ≤ z ≤ 7 . 7. Given that the FWHM of the W1, W2, and W3 images is 6 arcseconds, and of 12 arcseconds for W4, WISE band cutouts do not provide meaningful spatial information at this scale (see Figure 5). Therefore, we have opted to use images solely from the optical bands. Problematic sources, exhibiting signs of defects in various ways are flagged in the Legacy Survey by specific bitmasks (Dey et al. 2019). We acknowledge that LS10 images exhibit varying quality due to di ff erences in seeing conditions during observations taken over many years, which impact in particular the measurements of color within source apertures. Despite this, we rely on the model's ability to adapt to these intrinsic variations given the size and diversity of our training sample, as the sources withheld from training represent a shu ffl ed subsample of the main dataset, ensuring robust evaluation. We have verified that the distribution in seeing quality (expressed as the weighted average PSF FWHMofthe images) for the training and validation are comparable (refer to Figure B.1). However, preliminary tests indicate that adding PSF size and PSF depth of the observations-both available in LS10-as additional features enhances model performance (Götzenberger et al., in prep.). While it is not unreasonable to expect the model to implicitly infer the PSF or a related abstract representation thereof from the images themselves, these features will be included by default in future PICZL versions. With ongoing developments, PSF cutouts are expected to become more accessible for integration into the image stack (see Table 2). In the longer term, we anticipate that upcoming surveys like LSST, with their improved consistency in image quality, will further reduce these limitations and boost the precision of pixelbased analyses such as ours.","pages":[5,6]},{"title":"4.2. Color images","content":"It proves advantageous to provide the network with color images (ratio between images from di ff erent bands) as an additional input. This approach avoids the necessity for the model to learn the significance of colors solely from the flux images, which is inherently a more di ffi cult task. Likewise, rather than processing numerical features separately and merging them with the information extracted from the image cube at a later stage, we find it beneficial to integrate them directly at the pixel level. As a result, whenever possible, we transform catalog features into 2D arrays to align them with the original images in the same thread, enabling smoother integration and more coherent analysis (see e.g. Hayat et al. 2021). We improve the depth of our data cube by converting catalog-based quantities, e.g., flux measurements from apertures across di ff erent bands, into synthetic images, with a respective image size depending on the cutout size (see Figure 3). We expand this approach by generating images for all viable color combinations of aperture fluxes, constrained to those with matching aperture sizes. Additionally, we produce color images for flux cutouts where pixel resolutions are consistent (see lower panels b) and c) of Figure 3). Since the WISEand optical bands di ff er in both aperture size and pixel resolution, cross-wavelength color images are not feasible; instead, color combinations are restricted to within the optical or within the WISE bands (refer to Table 2). To maintain FOV consistency, we integrate WISE data for only the two innermost apertures (see Figure 5), preserving the 23x23 pixel data cube format. Although no additional spatial details are expected at this scale (see Sect. 4.1), the WISE data still captures aperture flux in a format that enables direct crosschannel connections between optical and mid-infrared data at the image level. Nevertheless, defected images can introduce challenges when generating color images (see bottom panel a) in Figure 3). Given that colors are derived from the ratio of images in di ff erent bands, the occurrence of unphysical negative or abnormally high / low values poses a significant concern. To address this issue, we examine whether the median value of neighbouring pix- Notes. Each column represents a distinct processing channel, detailing its input data type and the respective dimensions. All flux-related inputs have been corrected for reddening e ff ects. els is more than three times lower than the value of noisy pixels per band. If so, the central pixel's value is substituted with the median value of the surrounding pixels to smooth out fluctuations. In cases of non-detections or severely corrupted images where the largest value among all pixels in an image is < 0.001, the image is treated as a non-detection and all pixels are set to zero as default. After undergoing preprocessing, the images are utilized to create color images by exploring the six possible color combinations: gr , gi , gz , ri , rz and iz . If either of the two flux images involved in creating a color image is identified as a nondetection, the resulting color image is set to a default value of -99. At the catalog level, spatially invariant features, such as bestfit model classifications and signal-to-noise ratios (S / N), are processed separately in a dedicated channel, as they cannot be converted into image data. Notably, regarding normalization, we adopt a uniform min-max scaling approach to handle columns with cross-dependencies, such as flux bands. This strategy aims to preserve crucial information, such as the original shape of the SED. Contrary to the conventional approach of stacking all available images into a single input (Hoyle 2016; D'Isanto & Polsterer 2018; Pasquet et al. 2018; Dey et al. 2022a; Treyer et al. 2023), we find it advantageous to separate the color image data cube (23,23,24) from the flux band cutouts (23,23,32). This distinction is necessary because the color images often have di ff ering pixel value scales, including negative values, which require specialized processing in our machine learning application, such as tailored loss functions in the CNN. Non-spatial attributes are then combined with image-based data at a subsequent stage of the model, allowing the model to capture both spatial and nonspatial aspects. By processing data types in parallel channels, we leverage their complementary information by merging the data at a later stage, enhancing the extraction of inter-band correlations and ultimately improving redshift precision (Ma et al. 2015; Ait-Ouahmed et al. 2023). A detailed breakdown of the features integrated into each channel is provided in Table 2.","pages":[6,7]},{"title":"5. Neural network","content":"ML embodies an artificial intelligence paradigm where computers learn patterns and relationships from data, enabling them to make predictions or perform tasks without explicit programming. Multilayer perceptrons (MLPs), a feature-based feedforward neural network, draw inspiration from their biological counterparts, namely excitable cells responsible for processing and transmitting information (Rosenblatt 1958; I. Goodfellow 2016; M. Deru 2019). Likewise, each assigned a distinct weight, computational input vectors can be organized into layers, relayed to one or more hidden layers, to compute a scalar output value (Géron 2019). During training, these models learn data mappings by adjusting the weights and biases associated with their connections. The margin of change to the model after every training epoch is dictated by the choice of optimizer and loss function, which e ff ectively calculates the Euclidean distance between the prediction and so-called ground truth in multi-dimensional feature space, thereby significantly impacting model convergence and performance. The current state-of-the-art deep learning (DL) networks, characterized by their many hidden layers, have shown exceptional capabilities in handling complex non-linear tasks.","pages":[7]},{"title":"5.1. Convolutional model layers","content":"Among such architectures, Convolutional Neural Networks (CNNs; Fukushima 1980; LeCun et al. 1989; Lecun et al. 1998b) distinguish themselves by their remarkable e ff ectiveness in handling grid-like data, a prevalent form of which is represented by images. CNNs leverage a model architecture that is particularly e ff ective in tasks like image recognition, object detection, and image segmentation (O'Shea & Nash 2015; Liu et al. 2022). In convolutional layers, neurons establish connections exclusively with pixels within their receptive field, ensuring successive layers are linked only to specific regions of the previous layer. Subsequently, the model extracts low, image-level features in early layers and progressively complex, higher-level features in later layers. This allows the CNN to learn representations of the input data at multiple levels of abstraction. The convolutional operation involves sliding several kernels K , here of sizes 3 × 3 or 5 × 5 pixel, across the images with a fixed stride of size s = 1, compressing each mosaic element into a single scalar. This process entails element-wise multiplication, generating a set of K feature maps. Filters, the learnable parameters of convolutional layers, enable the network to detect and highlight di ff erent aspects of the input data, such as edges or corners. Each convolution is followed by a pooling layer that reduces spatial dimensions and computational load. Pooling involves sliding a kernel, in our case of size 2 × 2 and s = 1, across the feature maps, selecting the maximum value (max pooling) within each kernel window. After flattening the data cube into a single array, it is passed through (several) fully connected (FC) layers. Each FC layer contains a layer-specific number of neurons n , tailored to the model's specific needs. The final FC layer's neuron count varies based on the task, with n = 1 referring to regression tasks and n ≥ 2 to other endeavors such as multi-label classification.","pages":[7,8]},{"title":"5.2. Gaussian mixture models","content":"By default, single output regression models (Schuldt et al. 2021) provide point estimates without quantifying uncertainty. This severely limits their area of application, particularly in scenarios where quantifying the uncertainty is crucial, for example, when performing precision cosmology with Euclid (Bordoloi et al. 2010; Scaramella et al. 2022; Newman & Gruen 2022). By instead employing an architecture that provides PDFs, we can not only retrieve point estimates but also encapsulate the uncertainty associated with the predictions in a concise format (D'Isanto & Polsterer 2018). Given the inherent complexity in determining redshifts from only a few broadband cutouts, our results are expected to often exhibit degeneracy with multi-modal posteriors, making a single Gaussian insu ffi cient for representing the photoz PDFs. Therefore, estimates are computed using Bayesian Gaussian Mixture Models (GMMs, Duda et al. 1973; Viroli & McLachlan 2017; Hatfield et al. 2020). These networks provide a unique probabilistic modeling approach that di ff erentiates them from traditional MLPs. One of the key distinctions lies in the output nature, where GMMs provide a set of variables to compute weighted multi-Gaussian distributions as opposed to single-point estimates. Subsequently, each component is characterized by its mean µ , standard deviation σ , and weight w , allowing GMMs to produce a full PDF for a given set of inputs x . The PDF is expressed as with, wk representing the weight and N ( x | µ k , σ 2 k ) denoting the Gaussian distribution of the k -th component. As such, we extend our CNN approach by a GMM backend to output PDFs based on the information-rich feature maps produced during the front-end phase of the network. However, we encounter a limiting challenge with inputs of such small dimensions, i.e. 23 × 23, as they are not well-suited for established image-based Deep Learning architectures, such as \"ResNet\" (He et al. 2015), which typically require larger dimension scales, typically exceeding 200 × 200 pixels, to accommodate the large number of pooling layers they employ. Therefore, we developed a custom architecture to fit our data dimensionality. The resulting model architecture features roughly 490,000 trainable parameters, far fewer than found in comparable studies (see Pasquet et al. 2018; Treyer et al. 2023), and is displayed in Figure A.1 of Appendix A.","pages":[8]},{"title":"5.3. Model refinement","content":"Numerous hyper-parameters are crucial in shaping the network architecture while influencing training and convergence. Extensive optimization has been conducted across various parameters utilizing the Optuna framework (Akiba et al. 2019). The current configuration accounts for the vast array of potential combinations. The key parameters with the most significant impact are outlined below: putational e ffi ciency, while smaller batches may help generalize better.","pages":[8]},{"title":"5.4. Loss functions","content":"When utilizing PDFs instead of point estimates, it is crucial to quantify whether a predicted PDF e ff ectively reflects the ground truth, in our supervised ML case, specz (Sánchez et al. 2014; Malz et al. 2018; Schmidt et al. 2020; Treyer et al. 2023). The PDF predicted by a model should be concentrated near the true value. A straight-forward and meaningful proper scoring rule (i.e. one which is lowest when the prediction is at the truth), is the product of probabilities at the true redshifts. The negative log-likelihood (NLL) 2 of which passed to a minimizer yields the likelihood of observing a data distribution given a specific set of model parameters. This expression coincides with the average Kullback-Leibler divergence (KL, Kullback & Leibler 1951) when going from a delta PDF at specz to the photo-z PDF. An alternative growing in popularity is given by the Continuous Ranked Probability Score (CRPS), initially applied in weather forecasting (Grimit et al. 2006), which serves as a valuable metric for photo-z estimation via PDF quantification (D'Isanto & Polsterer 2018). Computationally, CRPS calculates the integral of the squared di ff erence between the predicted cumulative probability distribution function (CDF) and a Heaviside step-function H ( x ) at the value of the specz ( xz ) as For a finite mixture of normal distributions M , the CRPS can be expressed in closed form: with δ i = y -µ i , the uncertainty σ i and weight wi of the i -th component respectively, as well as By extending the CRPS loss via normalizing it by (1 + z), we adjust the penalty for prediction errors based on redshift. In doing so, we prioritize accuracy for low and intermediate redshift sources by imposing higher constraints while allowing for more leniency in less critical high-redshift areas. Integrating over the entire range of possible redshift values, the CRPS takes into account both location and spread of the predicted PDF. It, therefore, provides a comprehensive evaluation metric, generating globally well-calibrated PDFs (Dey et al. 2022b).","pages":[8,9]},{"title":"5.5. Training & data augmentation","content":"Our dataset is divided into training and validation sets in an 80:20 ratio. The model is trained over 1000 training epochs utilizing the Adam algorithm (Kingma & Ba 2017) along with a learning rate scheduler to adjust the learning rates during training dynamically. The inclusion of both is a consequence of the Optuna hyperparameter optimization. Typically, the model achieves its lowest validation loss around the 600th epoch when considering both loss functions (see Figure 6). After this point, although the training loss continues to decrease, the validation loss begins to increase, indicating overfitting. This overfitting likely arises due to the limited size of our training sample, allowing the model to memorize specific patterns rather than learning generalizable features. We implement a checkpoint system to mitigate this, saving the model's architecture and parameters when it reaches its lowest validation loss. This approach ensures that we capture the best-performing model configuration while preventing overfitting. To further exploit the advantages of deep learning, particularly its capacity to perform well with extensive datasets, we have tested the incorporation of data augmentation techniques into our methodology. Recognizing the substantial impact of training data amount on model performance, we employ common augmentation data products, such as rotated and mirrored images. Although these transformations significantly increase the apparent size of the dataset, we do not observe an increase in performance when including augmentation techniques (Dey et al. 2022a; Treyer et al. 2023). This is potentially not surprising as the influence of augmentation techniques depends on several factors. While CNNs are inherently equivariant to translations within images, enabling them to recognize objects even if they are shifted or o ff -centered, they and most other machinelearning approaches are not naturally equivariant to rotations and reflections. This lack of rotational equivariance means that the model output e ff ectively depends on the rotation of the input images. However, as most of our inputs are approximately point sources with radial symmetry, we do not explore this aspect further in this study. More recent publications have focused on developing rotationally equivariant CNN architectures, which have shown improved performance despite increased computational costs (Cohen & Welling 2016; Weiler & Cesa 2021; Lines et al. 2024). Incorporating such architectures could be particularly beneficial for future works dealing with large amounts of imaging data.","pages":[9]},{"title":"5.6. Model ensemble","content":"Ensemble models capitalize on the diversity of multiple models to improve prediction accuracy. Our approach involves diversifying models by adjusting parameters such as the number of Gaussians per GMM, learning rates, batch sizes, and the choice of loss function. We employ both NLL and CRPS as complementary loss functions to achieve this. While CRPS excels in achieving lower outlier fractions, NLL enhances overall variance. Each loss function trains a set of 144 models, resulting in a diverse pool of 288. In line with Treyer et al. (2023), we achieve superior performance by randomly combining equally weighted CRPS and NLL models from the pool, compared to the best-performing individual model (Dahlen et al. 2013; Newman & Gruen 2022). This finding is also aided by the result that roughly 30% of outliers are model-specific as opposed to 20% of outliers appearing in all models, therefore considered to be genuine. Put di ff erently, 80% of outliers were found to not be an outlier in at least one other model. We additionally recognize that the initial configuration of each model has a minor influence on the performance. Although training additional models could further enhance our results, the computational resources required to generate three to five sets of models would make this approach impractical relative to the potential performance gains. To create an e ff ective ensemble, we evaluate the outlier fraction of all individual models on an un- seen test sample. From this evaluation, we select models based on their relative influence on the model performance. We then observe how the ensemble's performance evolves as more models are continuously incorporated. Our ensemble ultimately comprises 10 models or 84 Gaussians, evenly incorporating models optimized using both NLL and CRPS approaches. To refine the model weights within the ensemble, we employ Optuna for finetuning. Subsequently, the ensemble posterior likelihood distribution P ensemble(x) is given by: where the weights wi are normalized such that: to assure that the integral of the ensemble PDF is equal to 1. We obtain a point estimate for our photoz prediction by identifying the dominant mode of the resulting PDF, recognizing that the mean or median could be misleading for highly non-Gaussian and bi- or multi-modal distributions. Additionally, we provide asymmetric 1 and 3 sigma upper and lower errors by evaluating the PDF within the 16th / 84th and 0.15 / 99.85th percentiles, respectively, along with the entire PDF.","pages":[9,10]},{"title":"6. Photo-z results","content":"We evaluate the performance of PICLZ on both point estimates and PDFs. This comprehensive assessment includes testing the statistical reliability and accuracy of our predictions. We adopt several commonly employed statistical metrics, providing insights into the accuracy, precision, and reliability of the photo-z estimates. In line with the definitions from the literature, we use the following metrics: We quantify PICZL's performances on a validation sample of 8098 sources and find an outlier fraction η , of 5.6% with a variance σ NMAD, of 0.045. Since the validation sample is used for feedback in training, it should be mentioned that the results mentioned above are likely overestimating the performance and therefore may not representative of what a future user could achieve with say, a test set previously unknown to the model, leave-one-out, or k-fold cross-validation. To this end, we have estimated our performance on independent, blind, X-rayselected samples (see Sect. 6.2 and 7). Figure 7 compares photometric versus spectroscopic redshifts, split by point-like (type = PSF) and extended (type = EXT) morphology. With respect to comparable work (e.g., Figure 13 from Salvato et al. 2022), we observe enhanced performance with a much-reduced fraction of outliers, especially for PSF sources. These objects lack morphological information at redshifts z ≳ 1, hence we attribute this improvement to the model's ability to recognise how the radial extension of the azimuth profile of sources changes with redshift (refer to Figure 4). The scatter, for both PSF and EXT distributions, is tight and symmetrically distributed around the z PICZL = z spec identity line, with outliers appearing randomly scattered, suggesting stable performance across the redshift range and minimal systematic errors (see Figure 8 and Dey et al. (2019) for reference). Given that our point estimates are derived from PDFs, we must ensure their global calibration and accuracy. To achieve this, we employ the probability integral transform (PIT) statistic (Dawid 1984; Gneiting et al. 2005), a widely accepted method in the field for assessing the quality of redshift PDFs (Pasquet et al. 2018; Schuldt et al. 2021; Newman & Gruen 2022). An ideal scenario is represented by a uniformly distributed histogram of PIT values. Any deviation from uniformity can indicate issues in PDF calibration. Under-dispersion may suggest overly narrow PDFs, while over-dispersion often results from excessively wide PDFs (D'Isanto & Polsterer 2018). Peaks close to zero or one can be explained by catastrophic outliers, where the true redshift lies so far in the wing of the PDF that it essentially falls outside of it. We employ the PIT histogram (top panel Figure 9) alongside quantile-quantile (QQ) plots (bottom panel Figure 9), to visually assess the statistical properties of our model predictions. For reference, these can be compared to Figure 2 of in Schmidt et al. PIT Score . (2020); however, those code performances are completely dominated by inactive galaxies, making a direct comparison impractical. The QQ plot compares the CDFs of observed PIT values and identity U (0 , 1), to visualize QQ di ff erences split by morphological type. A well-calibrated model will exhibit a QQ plot closely following identity with a flat PIT histogram, indicating accurate and reliable redshift predictions. Our analysis shows that, while asymmetry in the PIT distribution could suggest systematic bias, the QQ plot reveals small residuals overall. Notably, for both the single- (refer to Sect. 5.2 and Figure A.1) and ensemble model (refer to Sect. 5.6), the curves do not deviate significantly from zero, lesser so for PSF-type than EXT-type objects, indicating well-calibrated models. A distinct observation is the shift towards central PIT values for the ensemble model. This shift is not unexpected, given that the ensemble approach integrates multiple redshift solutions, each potentially contributing di ff erent peaks to the resulting ensemble PDF. Consequently, the ensemble PDF exhibits a broader bulk of probability across redshift, with PIT scores accumulating more area under the curve Notes. The last two setups are an improvement over the previous performances, with the last setup providing comparatively good results, despite ignoring the features provided by catalogs, particularly relevant for potential implementation in e.g., LSST. when integrated to the main mode. As a result, rather than yielding a flat PIT distribution, the histogram shifts towards having a concentration of values around 0.5. While this phenomenon aligns with our expectations, further enhancements may be realized by incorporating metrics tailored instead to local calibration accounting for population-specific subgroups (see e.g. Zhao et al. 2021; Dey et al. 2022b).","pages":[10,11,12]},{"title":"6.1. Model performance for different inputs","content":"Images provide a more comprehensive view of astronomical sources by capturing their full spatial structure and light distribution, o ff ering finer details on morphology, apparent size and extended features that are often lost in the averaging process of aperture photometry. This is pivotal, as SED features crucial for determining redshift solutions and resolving degeneracies mostly reside within the host galaxy (Soo et al. 2017; Wilson et al. 2020). In particular for AGN, images help to spatially separate the pixels corresponding to AGN-dominated emission and those corresponding mostly to the host galaxies. This approach enhances our ability to isolate and analyze individual pixel strings across multiple filters, to e ff ectively construct SEDs for the host galaxies independently. We thereby capture subtle features, neighbors, and patterns that may not be discernible from total flux measures. Critically, the independent photo-z from the AGN and host must agree, thus narrowing the overall source PDF. Table 3 presents the fraction of outliers for both EXT and PSF sources using di ff erent configurations of the PICZL algorithm. The first setup replicates the feature-only method, relying solely on total fluxes from the catalog, which, as seen in other studies, results in the fraction of outliers for PSF sources being nearly three times higher than for EXT sources (Salvato et al. 2022). In the second configuration, we adopt the approach of Saxena et al. (2024), incorporating the 2D light distribution and color gradients within annuli. The third configuration, i.e., PICZL as outlined above, enhances this by incorporating images, supplemented with catalog values transformed into image format, which leads to an additional improvement over the method in Saxena et al. (2024). Since images inherently capture all features typically extracted numerically and presented in catalogs, we show in the fourth case that PICZL achieves excellent results without feature / catalog information, using solely optical images supplemented by WISE aperture flux maps, even without explicit hyperparameter tuning. This approach is particularly promising for future surveys such as LSST and Euclid, where relying solely on multi-band imaging, including NIR / IR, will eliminate the need for complex source modeling or aperture photometry, regardless of the sources being galaxies or AGN.","pages":[12]},{"title":"6.2. Blind sample comparison","content":"To evaluate the robustness of our approach, we tested PICZL on an independent blind sample from the Chandra Source Catalog 2 (CSC2 3 ). This sample, selected to ensure no overlap with the training data, but yield a comparable X-ray to MIR distribution, is the same one used in the C ircle Z analysis (Saxena et al. 2024), allowing for a direct comparison with their results. After filtering for sources with spectroscopic redshift and removing duplicates with the training dataset, the sample comprises 416 sources within the LS10-South area. For comparison, Saxena et al. (2024) achieve an outlier fraction, η , of 12.3% and a nor- 7 malized median absolute deviation, σ NMAD, of 0.055, while our algorithm yields a superior σ NMAD of 0.046 and η of 8.3% (see Table 5 and Figure C.1). This comparison, therefore, demonstrates how utilizing image cubes further improves the already good results achieved using 2D-information from a catalog. For more details on the sample, see Saxena et al. (2024).","pages":[12,13]},{"title":"6.3. Prediction uncertainty quantification","content":"Accurate error estimation is crucial for assessing the reliability of any prediction, particularly in those astrophysical contexts where uncertainties can significantly impact the interpretation of data. To this end, we provide asymmetrical 1 σ and 3 σ errors for every source, together with the photoz PDF, as detailed in Sect. 5.6. These error estimates o ff er a comprehensive understanding of the potential variance in our predictions, accurately reflecting the inherent uncertainties in our data and model. We include asymmetrical errors, as the true distribution of uncertainties is often not symmetrical. This asymmetry arises from factors such as photometric noise and degeneracies in the highly nonlinear color-redshift space, where certain higher or lower redshift values may be more probable than their counterparts, leading to skewed PDFs. Although symmetrical redshift errors are observed for the majority of sources, approximately one-third of the sources exhibit asymmetrical 1 σ errors with deviations in redshift up to | ∆ z asym | ≃ 0 . 25. in Sect. 8.4. Additionally, the widths of the PDFs exhibit noticeable horizontal structures, largely influenced by the nature of the LS10 filters. We find that PDFs with wider modes, indicating less certain redshift estimates, often correspond to redshift ranges where key spectral features, such as the Ca break, fall outside the filter coverage (e.g. g & r at z ≃ 0 . 4, r & i at z ≃ 0 . 8 and z ≃ 1.5 or z ≃ 2.2). At redshifts approaching z ≈ 3 and extending below z ≈ 5, the Lymanα break starts to fall within the filter ranges, which contributes to narrower PDFs with increased PDF width only observed for extremely faint sources at even higher redshifts. Lastly, we find that the number of sources with wider PDFs increases as a function of | ∆ Z | (1 + Z spec) , indicating that narrower PDFs correspond to more accurate point predictions.","pages":[13]},{"title":"6.4. Insights from multimodal PDFs","content":"To investigate whether predicted outliers are genuine anomalies or instead stem from machine learning processes, we inquire whether secondary peaks in PDFs are physically meaningful or training artifacts. For each source, we compute the fraction of its PDF, that satisfies the condition | ∆ Z | (1 + z spec) ≥ 0 . 15, to provide the probability of being considered an inlier. The left panel of Figure 11 illustrates this with examples of two PDFs (one for an inlier and one for an outlier), where we highlight the corresponding inlier probabilities. In the right panel of Figure 11, we plot | ∆ Z | (1 + z spec) re-normalised to scale [0,1], such that as a function of inlier probability for all sources. The horizontal dashed black line separates the inliers ( < 0.5) from the outliers ( ≥ 0.5). The majority of inliers are concentrated at high inlier probabilities. This suggests that most inliers provide confident, unimodal PDFs with low errors. Likewise, most outliers have low inlier probability, with only a few outliers having more than 50% inlier probability. While this quantity cannot be computed for sources lacking spectroscopic redshift, it can be used to evaluate the reliability of our point estimates considering their associated PDFs. To evaluate whether ensemble solutions with broad or complex distributions e ff ectively capture the true redshift, we identify sources with multiple peaks and measure the proportion of PDFs exhibiting more than one mode. For these sources, we define a secondary peak as significant if it accounts for at least 10% of the height of the primary peak, which is the minimum prominence threshold used in our analysis. The top panel of Figure 12 shows that only 3% of inliers exhibit secondary peaks as opposed to the roughly 30% of outliers, where occurrences of strong secondary modes decrease with increasing prominence for both cases. While a single mode typically indicates a secure estimate for inliers, the presence of a unique mode alone can subsequently not be used to determine whether a source is an outlier. Conversely, in the bottom panel of Figure 12, we present the recovery fraction, which denotes how often a secondary peak in PDFs exhibiting multi-modal distributions corresponds to the true redshift. Among the outliers with 30% multi-modal PDF, more than 70% have one of their secondary peaks corresponding to | ∆ Z | (1 + Z spec) ≤ 0 . 15, indicating that there is significant probability that the redshift could consequently be considered as an inlier if the peak heights were reversed. Given the PIT histograms shown in Figure 9, it appears likely that the the PDFs are accurately capturing the relative frequency of multi-& bimodal PDFs, as if they were not, there would likely be bias evident in the overall PIT distribution. In other words, using this particular dataset, despite incorporating full multiwavelength image cutouts and cataloglevel information, there persist areas of parameter space that are legitimately degenerate in redshift, and the PDF parameterization looks to be capturing that degeneracy accurately. Consequently, we recommend using the entire PDF rather than point estimates, when possible.","pages":[13,14]},{"title":"7. PICZL applied to other surveys","content":"We want to investigate how PICZL, utilizing LS10 photometry and imaging, performs in determining photo-z for AGN in a more generic setting. Our focus is set on the LSST deep drilling fields (DDFs), selected to study SMBH growth across the full range of cosmic environments. These fields o ff er deeper, more comprehensive spectroscopic redshift coverage, along with a broader range of high-sensitivity bands, providing an enhanced dataset for photo-z estimation.","pages":[14]},{"title":"7.1. XMM-SERVS","content":"The XMM-Spitzer Extragalactic Representative Volume Survey (XMM-SERVS, Mauduit et al. 2012) encompasses three key fields: the XMM-Large Scale Structure (LSS, Chen et al. 2018), spannning 5.3 deg 2 with a flux limit of 6 . 5 × 10 -15 erg cm -2 s -1 over 90% of the survey area in the 0.5-10keV band (Savi'c et al. 2023); the Wide Chandra Deep Field-South (W-CDF-S, Ni et al. 2021) and the European Large-Area ISO Survey-South 1 (ELAIS-S1, Ni et al. 2021), covering approximately 4.6 deg 2 and 3.2 deg 2 (Brandt et al. 2018; Scolnic et al. 2018), limited to 1 . 3 × 10 -14 erg cm -2 s -1 , respectively, each selected for their exceptional multiwavelength coverage and strategic alignment with the DDFs.","pages":[14]},{"title":"7.2. Photo-z computation","content":"Considering that the original XMM-SERVS photo-z benefit from high S / N photometry and broader wavelength coverage, including u -band and NIR coverage, we would expect significantly better results compared to those achievable by PICZL using LS10 data alone. However, PICZL obtains comparable if not better results when limiting the sample to the depth at which it was trained on (see XMM-SERVS magnitude distribution in Figure E.1). Like any ML model, it struggles to extrapolate to faint sources outside its training distribution. Therefore, a fair comparison requires limiting the analysis to a similar feature space, while the results in Table 4 reflect performance across the entire, diverse XMM-SERVS samples. To facilitate a comparison with PICZL, our analysis begins with the counterpart associations of X-ray emissions as presented in Ni et al. (2021) and Chen et al. (2018). Whenever possible, we limit our samples to sources flagged as AGN (as defined in Sect. 6 of Ni et al. 2021). Matching to objects detected in LS10, due to its more limited depth, we identify approximately one-third of the original sources (see Table 4). This limitation is particularly pronounced in the XMM-LSS survey, where, in addition to deeper X-ray data corresponding to fainter counterparts, Chen et al. (2018) restrict their sample of AGN for which they calculate photo-z, to strictly non-broad-line X-ray sources. We select all sources with spectroscopic redshift and exclude those previously included in the Notes. From left to right, the columns list: Sample field; the original number of X-ray sources; the number of sources in each field that were detected in the LS10 survey; have spectroscopic redshift z > 0 . 002; and were not already present in the training sample of PICZL (i.e. previously unseen); the fraction of failed photo-z in XMM-SERVS zphot = -99; and the bias ⟨ ∆ z ⟩ , variance σ NMAD and outlier fraction η obtained with XMMSERVS (S, excluding zphot = -99) and PICZL (P) photo-z. PICZL training sample (see Table 4). Unlike the original works, PICZL also successfully computes photo-z for the significant fraction of sources for which the SED fitting adopted in XMMSERVS failed (refer to w / photoz S in Table 4). Figure 13 visualizes the outlier fractions (top) and variances (bottom) for the aggregate XMM-SERVS samples. Blue and burgundy represent the distributions including / excluding catastrophic failures in the XMM-SERVScatalogs. The plot is limited to outlier fractions of 30% and variances of 10%, as XMM-SERVS curves including zphot = -99 solutions continue to rise for fainter magnitudes or lower S / N. This suggests that, with increasing magnitude, a decreasing number of accurate estimates are available, making the subset excluding such solutions increasingly non-representative in terms of outlier fraction. However, by limiting our comparison to sources with photoz obtained from XMM-SERVS, we find that the accuracy of these photo-z remains comparable up to magnitudes around 23.5 AB, despite the more limited data being used. Additionally, we observe enhanced performance in PICZL photo-z for sources with S / N values ≳ 60, depending on the specific XMM-SERVS sample. Importantly, sources with S / N ≳ 10-20 mark a critical threshold range in LS10 where lower S / N values lead to an exponential rise in photometric errors (see Figure D.1). This trend is particularly evident for magnitudes fainter than 21.5 in the iband (Saxena et al. 2024) and visible in all panels of Figure 13. Figure 14 presents a visual comparison of normalized residuals for outliers identified by either PICZL or the original photo-z from Ni et al. (2021), exemplified via the ELAIS-S1 field. The spread of biases, excluding failures from XMM-SERVS, is similar for both methods, falling mostly within the range of [-0.5, 0.5]. The outliers in common appear to be modestly brighter than those that are outliers for a single method only and are evenly distributed between overestimation and underestimation. Notably, the outliers from Ni et al. (2021) are over a narrower range, while PICZL has a few brighter and several more fainter, the majority of which with magnitudes beyond the range covered during training. The observed di ff erence in outlier rates, is - not primarily due to the use of template-fitting versus machinelearning methods but rather a reflection of the available photometry. XMM-SERVS, with its deeper photometry, is wellsuited for faint objects, though saturation in brighter sources may contribute to some outliers. PICZL, which leverages the LS10 dataset, is optimized for brighter sources due to the shallower photometry. While it performs well in this regime, fainter objects that fall outside the known parameter space are consequently less constrained.","pages":[14,15,16]},{"title":"8. Discussion","content":"While Saxena et al. (2024) made significant strides in overcoming the limitations of relying solely on total or model fluxes for photoz estimation of AGN by utilizing all redshift-correlated features in the LS10 catalog, we have further advanced this approach. By integrating data from both optical and MIR wavelengths, our study highlights the transformative potential of combining imaging with catalog-level data. Despite the state-of-theart advancements in photoz estimation for AGN in this work, further improvements will be possible only by solving issues related to data quality. By overcoming these issues, we can significantly reduce the fraction of catastrophic outliers and achieve even greater accuracy in our predictions. In the following, we discuss the limitations that need to addressed in order to apply PICZL to upcoming imaging surveys.","pages":[16]},{"title":"8.1. Incorrect spec-z","content":"First and foremost, we need to address the reliability of the labels used in our, and generally any supervised machine learning algorithm. While spectroscopic redshift are typically trusted as accurate representations of true redshifts, this assumption does not always hold. As surveys continue to expand in scale, automated pipelines become indispensable for assigning spectroscopic redshift and identifying artifacts / problems for each source, as manual verification by visual inspection is impractical and will remain so for future surveys (Bolton et al. 2012). Importantly, most pipelines are trained on determining the redshift of the bulk of objects (Alexander et al. 2023), which are inactive galaxies and not AGN. This implies that a non-negligible fraction of AGN have wrong redshift estimates, including many labelled with \"no warnings\" (Hewett & Wild 2010; Wu & Shen 2022). Consequently, our performance is constrained by the unknown amount of incorrectly identified spectroscopic redshift Absolute LS10 r band [mag] Absolute LS10 r band [mag] per survey. While this fraction may be smaller for galaxies without nuclear activity, AGN present a range of challenges where pipelines fail. Examples of failures include cases where e.g. MgII is misidentified as Lymanα , leading to ambiguous redshifts around z = 0 . 7 -0 . 9 ⇔ 1 . 5 -2 . 5 or sources with strong featureless continuum typical of Blazars or, finally, sources with intense broad emission lines compared to the background continuum (Chen et al. 2018; Ni et al. 2021), as well as those with weak emission lines and a flat continuum background.","pages":[16,17]},{"title":"8.2. Incorrect type classification","content":"A subset of sources in the training sample, although having redshifts of z ≥ 1, are classified as extended (i.e. non-PSF). This classification is likely incorrect, given the pixel resolution of LS10. The apparent extension is more plausibly a result of poor seeing conditions that were not properly accounted for, or the presence of nearby neighboring sources (see Sect. 5.3 of Hsu et al. 2014). Since the morphological type a ff ects predictions, akin to a prior in template fitting methods, this misclassification explains the increase in outliers within the EXT sample as redshift increases (see middle row in right panel of Figure 15). To verify this, we computed the absolute g magnitudes for outliers to assess whether their values fall within the expected range for galaxies and AGN (Véron-Cetty & Véron 2010). Figure 16 shows that 16% of sources classified as EXT (and thus galaxy dominated) have an absolute magnitude which exceeds the range typical of galaxies MEXT [-16,-24], confirming that their TYPE is wrongly assigned. Interestingly, approximately 96% of PSF sources meet the absolute magnitude requirement of MPSF [-20,31], indicating that their TYPE is generally accurate.","pages":[17]},{"title":"8.3. Faulty photometry","content":"In addition to issues connected to the reliability of labels, predictions also face challenges due to data inconsistencies arising from faulty photometry. Surveys often flag issues of this kind directly, including problems such as hot pixels, saturation, cosmic rays, bleed trails, and other uncategorized anomalies. Detecting and addressing such observational defects is crucial because the accuracy and reliability of photoz predictions cannot be guaranteed for compromised photometric inputs. We experimented with removing various groups of sources a ff ected by observational artefacts but found no improvement when excluding any single group. We therefore assume that, while removing some problematic sources could potentially improve performance, the reduction in training data size o ff sets any gains. Adding this kind of noise to the current training sample could be an approach to decouple these two e ff ects. Consequently, we treat the inclusion of all problematic sources as natural noise, as opposed to artificially degrading images employing methods such as Gaussian noise (Hayat et al. 2021).","pages":[17]},{"title":"8.4. Survey depth","content":"When splitting the validation sample by S / N across all bands, we find a higher outlier fraction in sources with low S / N (see Table 5). Consequently, to evaluate the reliability of our photoz, we need to consider the photometric depth of LS10 and how increased photometric errors for fainter sources impact redshift estimate uncertainties. Figure 15 presents the prediction bias ⟨ z ⟩ , outlier fraction η , and dispersion σ NMAD as functions of r -band magnitude in LS10, X-ray flux (when available), and specz . As the r -band magnitude increases, as expected, the outlier fraction and scatter rise. This pattern is consistent with the exponential increase in photometric errors for, e.g. i -band magnitudes ≳ 21.5, beyond which the outlier fraction, which otherwise remains well below 10%, and scatter also appears to rise, especially for sources of type EXT (refer to Figure D.1). While faint LS10 sources have unreliable photo-z, future surveys such as LSST and Euclid will probe deeper with improved S / N. As such, our methodology can be adapted to these next-generation surveys, ensuring high accuracy across a broader range of magnitudes and providing robust redshift estimates for the extensive AGN populations these and other surveys will detect.","pages":[17]},{"title":"8.5. Missing i-band","content":"For DR10, the Legacy Survey footprint was extended by incorporating all available DECam data from various contributing surveys (refer to Sect. 3), including coverage in the i -band, though limited to a single pass. As a result, ∼ 10% of sources lack i -band observations. Unsurprisingly, and as shown in Table 5, the accuracy of photo-z increases when this band is also available.","pages":[17]},{"title":"8.6. Biases","content":"While the photo-z residuals exhibit a symmetric distribution around zero with minimal scatter, they do not consistently center at zero, suggesting a potential systematic bias in the photo-z estimates. Ideally, we would like to achieve normalized residuals that remain consistent irrespective of redshift or selection. Currently, this is not the case, as biases are not corrected for and are mostly influenced by the distribution of our training samples, which are specific to their respective survey and inherently biased. Notes. ( 1 ) all sources; ( 2 ) split by type (PSF vs. EXT); ( 3 ) split by signal-to-noise ratio (S / N) of all available bands; ( 4 ) split by availability of the i -band; ( 5 ) split by selection criteria; ( 6 ) split by the presence of a faint [(mAGN - mNeigh . ) within -3 ≤ ∆ mag < -1 for all available bands] or bright [(mAGN - mNeigh . ) within ∆ mag > -1 for all available bands] neighbor within a 5 arcsecond radius. Additionally, results derived for the CSC2 blind sample for both PICZL and C ircle Z are provided for comparison.","pages":[17,18]},{"title":"8.6.1. Selection effects","content":"We investigate whether the combination of various ways of selecting AGN entering the training sample a ff ects the quality of the photo-z for specific subsamples. Using the classification outlined in Table 1, we split the validation sample based on observational criteria, creating a binary division between sources selected in optical or via X-ray. While there is only a minor difference in σ NMAD, we observe a higher outlier fraction for Xray-detected sources. While strong X-ray emission from an AGN implies strong optical and mid-IR from an AGN, which should overpower the galaxy emission in the latter two bands, complicating the process of determining accurate photo-z, local Seyfert 1 galaxies with high X-ray flux have their host remain distinctly visible. As shown in the central panel of Figure 15, however, the outlier fractions and accuracy remain consistent across the full range of X-ray fluxes, aside from small-number statistics for the wings of the distribution.","pages":[18]},{"title":"8.6.2. Sample characteristics","content":"Unlike for galaxies, where the redshift distribution n( z ) is wellestablished and can serve as a reliable prior, the AGN n( z ) is not su ffi ciently characterised. Therefore uncertainty surrounds the ML algorithm subliminally adopting a distribution similar to the n( z ) of the training sample. At very low redshifts, AGN have a low surface density, resulting in only a few rare objects that can be considered for training. This phenomenon is intensified by the fact that deep surveys, as opposed to wide-field surveys, usually bypass such bright nearby objects in search of intermediate and high redshift sources. This leads to areas of scarcity in the training sample's specz distribution, with biased predictions towards redshift values where more data points exist. To mitigate this, we nor- malize the CRPS score by (1 + z ), emphasizing the accuracy of low-redshift predictions. Additionally, at extremely low redshifts where the 4000 Å break is barely covered by the g filter, accurate redshift predictions are more di ffi cult to obtain. At very low redshift, the 6 arcsecond × 6 arcsecond cutout may be entirely filled by the galaxy, potentially misleading the algorithm into interpreting it as excess noise.","pages":[18]},{"title":"8.6.3. Non-representative training samples","content":"One method to tackle covariate shift by the imbalance of the bright-end dominated spectroscopic sample, is given by subdividing both the training and validation samples into subsets of n -dimensional feature space of distinguishable properties (Newman & Gruen 2022). Specifically, the prediction accuracy improves if the model used to generate a posterior for a blind source was trained exclusively on training sources residing within the corresponding feature space (Rosenbaum & Rubin 1984; Revsbech et al. 2017; Autenrieth et al. 2023). However, this approach might significantly reduce the training sample size per model making it more applicable to photoz codes dealing with large datasets, such as those for inactive galaxies (Newman & Gruen 2022). In this work we have demonstrated that, in contrast to just a few years ago, AGN-targeted training samples are now su ffi -ciently large to provide reliable photo-z. Overcoming inherent biases still requires either gathering more representative samples that uniformly cover the full redshift range-particularly underrepresented regions-or applying statistical corrections. Thus, the focus has shifted from simply increasing the number of spectra to strategically obtaining spectra that cover specific regions of parameter space (Masters et al. 2015). However, given the substantial gains our model has shown over existing methods, a detailed examination of how these biases a ff ect prac- tical applications is beyond the scope of this study. Looking ahead, spectroscopic follow-up campaigns such as DESI, SDSSV / BHM (Kollmeier et al., in prep), and 4MOST (De Jong et al. 2019) will further enhance our capabilities. Nevertheless, predicting photo-z for AGN in deeper Euclid and LSST datasets remains challenging due to the limited availability of spectroscopic data for faint sources. The upcoming Subaru Prime Focus Spectrograph (PFS, Tamura et al. 2016) and Multi Object Optical and Near-infrared Spectrograph for the Very Large Telescope (VLT / MOONS; Cirasuolo et al. 2020) are expected to help address this shortfall by providing much-needed spectra for faint (& obscured / red) objects.","pages":[18,19]},{"title":"8.7. Observational constraints: source crowding","content":"One factor beyond our control is the local environment or conditions along the line of sight, which can significantly influence the emission characteristics of AGN. Nearby objects can add extra flux, which can a ff ect the accuracy of photometric measurements. We perform a positional cross-match between the validation sample and all LS10 sources within a 5 arcsecond radius of their optical counterparts. For each neighbour meeting this proximity criterion, we calculate apparent magnitudes and flag all sources where the brightest neighbour is no less than 1 magnitude dimmer. Table 5 shows that isolated sources exhibit better overall performance, while those with bright neighbours show drastically reduced quality compared to those with fainter neighbours. The presence of bright neighbours influences the observed flux in two ways: firstly, it complicates the derivation of sourcespecific colors due to convolved fluxes, and secondly, it a ff ects the observed spectra, potentially leading to the detection of two sets of emission lines and incorrect specz identification (Newman & Gruen 2022). A potential solution to this issue could be the implementation of segmentation maps, similar to SExtractor (Source Extractor, Bertin, E. & Arnouts, S. 1996) at image level, as LS10 currently only masks neighbours during their model flux fitting procedure. A persistent physical issue that cannot be entirely mitigated is blending. Unlike bright neighbours, which can be masked in principle, blending a ff ects both observed photometry and spectroscopy. While particularly faint neighbours do not substantially a ff ect the predictions negatively, sources identified as blends should be excluded from samples where accurate photoz estimates are required, until we can partially mask sources at the pixel level. The Rubin Observatory expects overlapping (inactive) galaxies to contribute at least 1% of the total flux within their pixels (Sanchez et al. 2021; Newman & Gruen 2022). Blends are also expected to occur in the spectroscopic sample, increasing systematic uncertainty and subsequently the fraction of outliers by up to 5% (Masters et al. 2019). This may be mitigated in future by higher-quality imaging from spacebased data from missions such as Euclid or better tools for deblending (e.g., SCARLET / blendz ; Melchior et al. 2018; Jones & Heavens 2018). Alternatively, data augmentation or standard computer vision techniques could be implemented, such as artificially introducing blending e ff ects in the training sample. However, this approach is not trivial, as LS10 currently lacks a corresponding blending flag.","pages":[19]},{"title":"8.8. Variability","content":"AGN are inherently variable sources, meaning that their observed flux can change significantly across di ff erent epochs and wavelengths, especially when observations are separated by substantial time gaps. This extends to images that are created by stacking multiple observations taken over extended periods (as in LS10), where the resulting flux represents an average value. Such variability complicates the accurate prediction of AGN redshifts (Simm et al. 2015). However, future time-resolved imaging from LSST will allow us to better account for these variations and potentially use the correlation between AGN variability and their physical properties as a feature to improve photo-z predictions.","pages":[19]},{"title":"8.9. Scalability in image-based photo-z","content":"Open issues remain regarding the feasibility of handling the vast data volumes and computational requirements associated with photo-z estimated from images. Exemplified by the calculation of new estimates for XMM-SERVS concerning storage, processing time, and computational resources, we give an overview in Appendix G. A crucial future direction involves exploring ways to utilize imaging data e ffi ciently without necessitating extensive local downloads and computations for all image-based analyses of such data sets, potentially leveraging online platforms for realtime analysis, bringing the compute to the data (Zhang & Zhao 2015).","pages":[19]},{"title":"9. Summary and outlook","content":"This work has been driven by the goal of determining reliable photo-z for X-ray detected AGN in wide-field surveys, such as eROSITA (Merloni et al. 2024). For that reason, we have concentrated on utilizing data from LS10, which provides su ffi ciently homogeneous coverage and depth in 4 optical bands, enriched by the flux measured on NEOWISE7 data for all identified sources (see Sect. 3). LS10 overlaps almost entirely with the eROSITA footprint, simplifying the cross calibration of data, a processing step typically necessary when merging di ff erent surveys. We introduce PICZL, a CNN-based machine learning model designed primarily for AGN redshift estimation, but which holds the potential to reliably measure photoz s for a broad range of extragalactic sources, provided an appropriately constructed training sample is available. In this study, the training sample comprises both X-ray and optically selected AGN. Across our validation set, PICZL demonstrates consistently robust performance, with comparable accuracies and lower outlier fractions particularly for PSF sources, as opposed to the results obtained by Salvato et al. (2022) for similar objects using SED-fitting (see Figure 7 and Table 5). The results show significant improvements for both point-like and extended sources, indicating that the model's performance is primarily driven by the depth and hence photometric error of the training data. Additionally, when tested on a blind sample of X-ray-selected AGN, PICZL maintained comparable results with respect to σ NMAD and η . Notably, on this test set it outperformed (C irclez ; Saxena et al. 2024) by achieving a 20% improvement in accuracy and a 30% reduction in the fraction of outliers (refer to Sect. 6). We also applied PICZL to estimate photo-z for the approximately 30% of XMM-SERVS sources (Chen et al. 2018; Ni et al. 2021), detected in LS10 (see Table 4), demonstrating comparable variance and a substantially improved outlier fraction up to a limiting magnitude of 23.5 AB, using much fewer bands and significantly less sensitive imaging. However, it is important to note that the spectroscopic sample at this faint limit is small and likely biased toward sources with higher spectroscopic success rates. Consequently, the most reliable photo-z results are expected at brighter magnitudes (as discussed in Saxena et al. 2024). In response to these findings, we are releasing a new catalog of photo-z, complete with 1 and 3 σ error margins for the XMM-SERVS fields (see Sect. 7). PICZL will be used in the next generation of photo-z for sources detected by eROSITA, possibly switching from LS10 to Euclid and, most importantly, LSST, as they are poised to deliver more homogeneous and deeper data with broader wavelength coverage. In particular the availability of NIR data at image level, will improve our ability to determine accurate redshift for faint and high-z sources. Our study has shown that the performance of PICZL, when relying predominantly on images, is already robust (see Table 3). Crucially, we have shown that wellcalibrated images alone can su ffi ce for accurate photo-z estimation, eliminating the need for catalog creation, which is often based on predefined models. This suggests that future surveys could reduce their dependence on catalog-based data for photo-z computation (refer to Sect. 6). Although this study has focused on calculating photo-z for AGN, the approach can be generalised and adapted to other source types, e.g., normal galaxies, with an appropriately constructed training sample (Götzenberger et al. in prep.). Subsequently our findings point to a bright future for all-sky surveys, also thanks to the continue expansion of training sample sizes, supported by initiatives like SDSS-V / BHM, 4MOST and VLT / MOONS, which will enhance the reliability and completeness of photoz estimates (refer to Sect. 8). Looking forward, the next step in advancing photo-z estimation for AGN lies in exploring more sophisticated machine learning architectures beyond CNNs. For example, transformers with shared latent space embeddings o ff er a promising avenue. These models have shown success in integrating information from various data sources, such as images and entire spectra, potentially reducing uncertainties associated with traditional specz pipelines by leveraging multi-modal data fusion (DonosoOliva et al. 2023; Parker et al. 2024). Additionally, incorporating other informative features like X-ray flux, when available, holds promise. Integrating these diverse data sources within a unified framework has the potential to refine redshift estimates even further.","pages":[19,20]},{"title":"10. Data availability","content":"With this paper, we present a new catalog of photo-z derived using PICZL for the ∼ 30% of sources within the XMM-SERVS (ELAIS-S1, W-CDF-S, and LSS) X-ray source catalogs (Chen et al. 2018; Ni et al. 2021) that are detected in the LS10 survey. Hence, it includes updated photo-z for sources with catastrophic failures in the original works. A detailed description of the catalog columns can be found in Appendix F. The full catalog is only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http:// cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/ . Acknowledgements. WRand MS are grateful for the constant support of Dustin Lang in handling Legacy Survey-related issues. Part of this work was supported by the German Deutsche Forschungsgemeinschaft, DFG , under Germany's Excellence Strategy - EXC 2094 - 390783311. We gratefully acknowledge funding from FONDECYT Regular - 1231718 (RJA), 1230345 (CR), and 1241005 (FEB), CATA-BASAL - FB210003 (RJA, CR, FEB), and ANID - Millennium Science Initiative - AIM23-0001 (FEB). JA acknowledges support from a UKRI Future Leaders Fellowship (grant code: MR / T020989 / 1) This work is based on data from eROSITA, the soft X-ray instrument aboard SRG, a joint RussianGerman science mission supported by the Russian Space Agency (Roskosmos), in the interests of the Russian Academy of Sciences represented by its Space Research Institute (IKI), and the Deutsches Zentrum für Luft- und Raumfahrt (DLR). The SRG spacecraft was built by Lavochkin Association (NPOL) and its subcontractors and is operated by NPOL with support from the Max Planck Institute for Extraterrestrial Physics (MPE). The development and construction of the eROSITA X-ray instrument were led by MPE, with contributions from the Dr. Karl Remeis Observatory Bamberg & ECAP (FAU Erlangen-Nuernberg), the University of Hamburg Observatory, the Leibniz Institute for Astrophysics Potsdam (AIP), and the Institute for Astronomy and Astrophysics of the University of Tübingen, with the support of DLR and the Max Planck Society. The Argelander Institute for Astronomy of the University of Bonn and the Ludwig Maximilians Universität Munich also participated in the science preparation for eROSITA. The Legacy Surveys consist of three individual and complementary projects: the Dark Energy Camera Legacy Survey (DECaLS; Proposal ID 2014B-0404; PIs: David Schlegel and Arjun Dey), the Beijing-Arizona Sky Survey (BASS; NOAO Prop. ID 2015A-0801; PIs: Zhou Xu and Xiaohui Fan), and the Mayall z-band Legacy Survey (MzLS; Prop. ID 2016A-0453; PI: Arjun Dey). DECaLS, BASS, and MzLS together include data obtained, respectively, at the Blanco telescope, Cerro Tololo Inter-American Observatory, NSF's NOIRLab; the Bok telescope, Steward Observatory, University of Arizona; and the Mayall telescope, Kitt Peak National Observatory, NOIRLab. Pipeline processing and analyses of the data were supported by NOIRLab and the Lawrence Berkeley National Laboratory (LBNL). The Legacy Surveys project is honored to be permitted to conduct astronomical research on Iolkam Du'ag (Kitt Peak), a mountain with particular significance to the Tohono O'odham Nation. NOIRLab is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. LBNL is managed by the Regents of the University of California under contract to the U.S. Department of Energy. This project used data obtained with the Dark Energy Camera (DECam), which was constructed by the Dark Energy Survey (DES) collaboration. Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A & M University, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho de Amparo, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Cientifico e Tecnologico and the Ministerio da Ciencia, Tecnologia e Inovacao, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey. The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Energeticas, Medioambientales y Tecnologicas-Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgenossische Technische Hochschule (ETH) Zurich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ciencies de l'Espai (IEEC / CSIC), the Institut de Fisica d'Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig Maximilians Universitat Munchen and the associated Excellence Cluster Universe, the University of Michigan, NSF's NOIRLab, the University of Nottingham, the Ohio State University, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, and Texas A&M University. BASS is a key project of the Telescope Access Program (TAP), which has been funded by the National Astronomical Observatories of China, the Chinese Academy of Sciences (the Strategic Priority Research Program 'The Emergence of Cosmological Structures' Grant # XDB09000000), and the Special Fund for Astronomy from the Ministry of Finance. The BASS is also supported by the External Cooperation Program of Chinese Academy of Sciences (Grant # 114A11KYSB20160057), and Chinese National Natural Science Foundation (Grant # 12120101003, # 11433005). The Legacy Survey team uses data products from the Near-Earth Object Wide-field Infrared Survey Explorer (NEOWISE), a project of the Jet Propulsion Laboratory / California Institute of Technology. NEOWISE is funded by the National Aeronautics and Space Administration. The Legacy Surveys imaging of the DESI footprint is supported by the Director, O ffi ce of Science, O ffi ce of High Energy Physics of the U.S. Department of Energy under Contract No. DE-AC02-05CH1123, by the National Energy Research Scientific Computing Center, a DOE O ffi ce of Science User Facility under the same contract, and by the U.S. National Science Foundation, Division of Astronomical Sciences under Contract No. AST-0950945 to NOAO. Funding for the Sloan Digital Sky Survey V has been provided by the Alfred P. Sloan Foundation, the Heising-Simons Foundation, the National Science Foundation, and the Participating Institutions. SDSS acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. SDSS telescopes are located at Apache Point Observatory, funded by the Astrophysical Research Consortium and operated by New Mexico State University, and at Las Campanas Observatory, operated by the Carnegie Institution for Science. The SDSS web site is www.sdss.org . SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration, including Caltech, The Carnegie Institution for Science, Chilean National Time Allocation Committee (CNTAC) ratified researchers, The Flatiron Institute, the Gotham Participation Group, Harvard University, Heidelberg University, The Johns Hopkins University, L'Ecole polytechnique fédérale de Lausanne (EPFL), Leibniz-Institut für Astrophysik Potsdam (AIP), Max-PlanckInstitut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Extraterrestrische Physik (MPE), Nanjing University, National Astronomical Observatories of China (NAOC), New Mexico State University, The Ohio State University, Pennsylvania State University, Smithsonian Astrophysical Observatory, Space Telescope Science Institute (STScI), the Stellar Astrophysics Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Illinois at Urbana-Champaign, University of Toronto, University of Utah, University of Virginia, Yale University, and Yunnan University.","pages":[20,21]},{"title":"References","content":"Abdalla, F. B., Banerji, M., Lahav, O., & Rashkov, V. 2011, MNRAS, 417, 1891 Aird, J., Nandra, K., Laird, E. S., et al. 2010, MNRAS, 401, 2531-2551 Alexander, D. M., Davis, T. M., Chaussidon, E., et al. 2023, ApJ, 165, 124 Almeida, A., Anderson, S. F., Argudo-Fernández, M., et al. 2023, ApJ Supple- ment Series, 267, 44 Almosallam, I. A., Jarvis, M. J., & Roberts, S. J. 2016, MNRAS, 462, 726 Ananna, T. T., Salvato, M., LaMassa, S., et al. 2017, ApJ, 850, 66 Baum, W. 1957, ApJ, 62, 6 Bell, E. F., Wolf, C., Meisenheimer, K., et al. 2004, ApJ, 608, 752-767 Benitez, N. 2000, ApJ, 536, 571 Bertin, E. & Arnouts, S. 1996, Astron. Astrophys. Suppl. Ser., 117, 393 Bettoni, D., Falomo, R., Kotilainen, J. K., Karhunen, K., & Uslenghi, M. 2015, MNRAS, 454, 4103 Blanton, M. R., Bershady, M. A., Abolfathi, B., et al. 2017, ApJ, 154, 28 Boller, T., Freyberg, M. J., Trümper, J., et al. 2016, A&A, 588, A103 Bolton, A. S., Schlegel, D. J., Aubourg, E., et al. 2012, ApJ, 144, 144 Bordoloi, R., Lilly, S. J., & Amara, A. 2010, MNRAS, no Boutsia, K., Grazian, A., Calderone, G., et al. 2020, ApJ Supplement Series, 250, 26 Breiman, L. 2001, Machine Learning, 545, 5 Brescia, M., Cavuoti, S., Longo, G., & Stefano, V. D. 2014, A&A, 568, A126 Comparat, J., Merloni, A., Dwelly, T., et al. 2020, A&A, 636, A97 Connolly, A. J., Csabai, I., Szalay, A. S., et al. 1995, ApJ, 110, 2655 Croom, S. M., Smith, R. J., Boyle, B. J., et al. 2004, MNRAS, 349, 1397-1418 Dahlen, T., Mobasher, B., Faber, S. M., et al. 2013, ApJ, 775, 93 Dawid, A. P. 1984, Journal of the Royal Statistical Society. Series A (General), Dey, B., Newman, J. A., Andrews, B. H., et al. 2022b, Re-calibrating Photometric Redshift Probability Distributions Using Feature-space Regression D'Isanto, A. & Polsterer, K. L. 2018, A&A, 609, A111 Eriksen, M., Alarcon, A., Cabayol, L., et al. 2020, MNRAS, 497, 4565-4579 Heckman, T. M. & Best, P. N. 2014, Annual Review of A&A, 52, 589 Lecun, Y., Bottou, L., Bengio, Y., & Ha ff ner, P. 1998b, Proceedings of the IEEE, 86, 2278 Li, C., Zhang, Y., Cui, C., et al. 2021, MNRAS, 509, 2289 Li, C., Zhang, Y., Cui, C., et al. 2022, MNRAS, 518, 513 Lin, Q., Fouchez, D., Pasquet, J., et al. 2022, A&A, 662, A36 Lines, N. E. P., Roset, J. F.-Q., & Scaife, A. M. M. 2024, E(2)-Equivariant Features in Machine Learning for Morphological Classification of Radio Galaxies Liu, Z., Mao, H., Wu, C.-Y., et al. 2022, in Proceedings of the IEEE / CVF Con- ference on Computer Vision and Pattern Recognition (CVPR), 11976-11986 Luken, K. J., Norris, R. P., & Park, L. A. F. 2019, Publications of the Astronom- ical Society of the Pacific, 131, 108003 Luo, B., Brandt, W. N., Xue, Y. Q., et al. 2010, ApJ Supplement Series, 187, 560 Lyke, B. W., Higley, A. N., McLane, J. N., et al. 2020, ApJ Supplement Series, 250, 8 Ma, L., Lu, Z., Shang, L., & Li, H. 2015, Multimodal Convolutional Neural Networks for Matching Image and Sentence Madau, P. & Dickinson, M. 2014, Annual Review of A&A, 52, 415-486 Mainzer, A., Bauer, J., Grav, T., et al. 2011, ApJ, 731, 53 Masters, D., Capak, P., Stern, D., et al. 2015, ApJ, 813, 53 Mauduit, J.-C., Lacy, M., Farrah, D., et al. 2012, Publications of the Astronomical Society of the Pacific, 124, 714-736 Meisner, A. M., Lang, D., & Schlegel, D. J. 2017, ApJ, 153, 38 Merloni, A., Predehl, P., Becker, W., et al. 2012, eROSITA Science Book: Mapping the Structure of the Energetic Universe Meshcheryakov, A., Glazkova, V., Gerasimov, S., & Mashechkin, I. 2018, Astronomy Letters, 44, 735 Mountrichas, G., Corral, A., Masoura, V. A., et al. 2017, A&A, 608, A39 Newman, J. A. & Gruen, D. 2022, Annual Review of A&A, 60 Ni, Q., Brandt, W. N., Chen, C.-T., et al. 2021, ApJ Supplement Series, 256, 21 Nishizawa, A. J., Hsieh, B.-C., Tanaka, M., & Takata, T. 2020, Photometric Redshifts for the Hyper Suprime-Cam Subaru Strategic Program Data Release 2 O'Shea, K. & Nash, R. 2015, An Introduction to Convolutional Neural Networks Padovani, P., Alexander, D. M., Assef, R. J., et al. 2017, The A&A Review, 25 Parker, L., Lanusse, F., Golkar, S., et al. 2024, MNRAS, 531, 4990-5011 Pierce, C. M., Lotz, J. M., Primack, J. R., et al. 2010, MNRAS Povi'c , M., Sánchez-Portal, M., García, A. M. P., et al. 2012, A&A, 541, A118 Predehl, P., Andritschke, R., Arefiev, V., et al. 2021, A&A, 647, A1 Revsbech, E. A., Trotta, R., & van Dyk, D. A. 2017, MNRAS, 473, 3969-3986 Rosenbaum, P. & Rubin, D. 1984, Journal of the American Statistical Associa- tion, 79 Saxena, A., Salvato, M., Roster, W., et al. 2024, CircleZ: Reliable Photometric redshifts for AGN computed using only photometry from Legacy Survey Imaging for DESI Scaramella, R., Amiaux, J., Mellier, Y., et al. 2022, A&A, 662, A112 Schmidt, S. J., Malz, A. I., Soo, J. Y. H., et al. 2020, MNRAS, 499, 1587 Schuldt, S., Suyu, S. H., Cañ ameras, R., et al. 2021, A&A, 651, A55 Simm, T., Saglia, R., Salvato, M., et al. 2015, A&A, 584, A106 Soo, J. Y. H., Moraes, B., Joachimi, B., et al. 2017, MNRAS, 475, 3613-3632 Stabenau, H. F., Connolly, A., & Jain, B. 2008, MNRAS, 387, 1215 Webb, N. A., Coriat, M., Traulsen, I., et al. 2020, A&A, 641, A136 Weiler, M. & Cesa, G. 2021, General E (2)-Equivariant Steerable CNNs Wilson, D., Nayyeri, H., Cooray, A., & Häußler, B. 2020, ApJ, 888, 83 Wu, Q. & Shen, Y. 2022, ApJ Supplement Series, 263, 42 Yang, J., Fan, X., Gupta, A., et al. 2023, ApJ Supplement Series, 269, 27 York, D. G., Adelman, J., John E. Anderson, J., et al. 2000, ApJ, 120, 1579 02.09.24, 21 : 39","pages":[21,22,23]},{"title":"Appendix G: PICZL run time","content":"The computational demands for training and deploying PICZL depend mostly on the hardware configuration and dataset size. For our experiments, we leveraged a set of two Tesla V100PCIE graphics processing units (GPUs) each equipped with 32 GB of ready access memory (RAM), accompanied by a 48 core multi-thread CPU to accelerate the computational workload. This parallel processing capability significantly expedited the model training process, compared to running solely on a central CPU, allowing for faster convergence. Training a single model (refer to Figure A.1) on a dataset of 32 391 sources, corresponding to the 80:20 train-test split (refer to Table 1), each with 108 features (56 of which are images), takes approximately 25 minutes for 600 epochs. The use of an ensemble further scales the processing time by the number of models trained, with ensemble optimization depending on user preferences, making it di ffi -cult to provide a fixed time estimate. After finalizing the model, computing photo-z for e.g. 1393 sources in the XMM-SERVS W-CDF-S field, as displayed via Table 4, takes roughly 17 seconds. The storage requirements for the data used in this sample corresponds to approximately 250 MB (refer to Table 2). Looking ahead, upcoming surveys such as Euclid and LSST will produce higher-resolution images, demanding increased storage and computational resources due to larger data volumes and pixel counts. If the input dimensions were to increase, for instance, from 23x23 pixels to 64x64 pixels, the computational burden would rise significantly. While our current architecture of 32 GB RAM per GPU is e ffi cient for the current input sizes, processing larger cutouts may necessitate either more GPUs or GPUs with higher memory capacity to maintain feasible training times and performance. In scenarios where larger input sizes are anticipated, a thoughtful approach to model architecture and resource allocation will be crucial. Therefore, adapting to the capabilities of the hardware will be key to successfully utilizing the methods in the context of future data from LSST and Euclid, particularly given the much larger sample sizes that we expect from these surveys. Finally, performance will also improve by having PICZL revised by an expert software developer.","pages":[24]}],"string":"[\n {\n \"title\": \"ABSTRACT\",\n \"content\": \"Context. Computing reliable photometric redshifts (photo-z) for active galactic nuclei (AGN) is a challenging task, primarily due to the complex interplay between the unresolved relative emissions associated with the supermassive black hole and its host galaxy. Spectral energy distribution (SED) fitting methods, while e ff ective for galaxies and AGN in pencil-beam surveys, face limitations in wide or all-sky surveys with fewer bands available, lacking the ability to accurately capture the AGN contribution to the SED, hindering reliable redshift estimation. This limitation is a ff ecting the many 10s of millions of AGN detected in existing datasets, e.g., those AGN clearly singled out and identified by SRG / eROSITA. Aims. Our goal is to enhance photometric redshift performance for AGN in all-sky surveys while simultaneously simplifying the approach by avoiding the need to merge multiple data sets. Instead, we employ readily available data products from the 10th Data Release of the Imaging Legacy Survey for the Dark Energy Spectroscopic Instrument, which covers > 20,000 deg 2 of extragalactic sky with deep imaging and catalogbased photometry in the grizW1-W4 bands. We fully utilize the spatial flux distribution in the vicinity of each source to produce reliable photo-z. Methods. We introduce PICZL, a machine-learning algorithm leveraging an ensemble of convolutional neural networks. Utilizing a cross-channel approach, the algorithm integrates distinct SED features from images with those obtained from catalog-level data. Full probability distributions are achieved via the integration of Gaussian mixture models. Results. On a validation sample of 8098 AGN, PICZL achieves an accuracy σ NMAD of 4.5% with an outlier fraction η of 5.6%. These results significantly outperform previous attempts to compute accurate photo-z for AGN using machine learning. We highlight that the model's performance depends on many variables, predominantly the depth of the data and associated photometric error. A thorough evaluation of these dependencies is presented in the paper. Conclusions. Our streamlined methodology maintains consistent performance across the entire survey area, when accounting for di ff ering data quality. The same approach can be adopted for future deep photometric surveys such as LSST and Euclid, showcasing its potential for wide-scale realization. With this paper, we release updated photo-z (including errors) for the XMM-SERVS W-CDF-S, ELAIS-S1 and LSS fields. Key words. Photo-z, AGN, Extragalactic Surveys, Machine Learning\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"PICZL : Image-based photometric redshifts for AGN\",\n \"content\": \"William Roster , ⋆ 1 , M. Salvato 1 , 2 , S. Krippendorf 3 , 4 , A. Saxena 5 , R. Shirley 1 , J. Buchner 1 , J. Wolf 2 , 6 , T. Dwelly 1 , F. E. Bauer 7 , 8 , 9 , 10 , J. Aird 11 , C. Ricci 12 , 13 , R. J. Assef 12 , S.F. Anderson 14 , X. Liu 15 , 16 , 17 , A. Merloni 1 , J. Weller 1 , 4 K. Nandra 1 (A ffi liations can be found after the references) November 14, 2024\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1. Introduction\",\n \"content\": \"In recent decades, our understanding of Active Galactic Nuclei (AGN) and their role in galaxy and cosmic evolution has significantly advanced. These luminous celestial powerhouses are thought to be fueled by the accretion of matter onto supermassive black holes (SMBHs) located at the centers of galaxies, exerting intense energetic radiation across the entire electromagnetic spectrum, ranging from radio to γ -rays (Padovani et al. 2017). The close correlation observed between the mass of the central SMBH, whether active or inactive and the properties of its host galaxy's bulge - such as the galaxy's mass and velocity dispersion (e.g., Gebhardt et al. 2000; Ferrarese & Merritt 2000) - suggests a co-evolutionary relationship between galaxies and their central engines (Kormendy & Ho 2013; Heckman & Best 2014). Ongoing research focuses on understanding scaling relations, the evolution of SMBHs within galaxies, and the interconnected rates of star formation (SFR) and black hole accretion (BHAR) over cosmic time (e.g., Madau & Dickinson 2014). To further explore and address these unresolved topics requires diverse AGN samples with reliable redshifts to determine BH demographics and constrain models of galaxy evolution. For all these studies, redshift is an indispensable quantity, with spectroscopic redshifts (spec-z) remaining the preferred es- imates for determining precise cosmic distances (Hoyle 2016). However, while multi-objects spectrographs, such as the Sloan Digital Sky Survey (SDSS-V; York et al. 2000; Kollmeier et al. 2019), the Dark Energy Spectroscopic Instrument (DESI; DESICollaboration et al. 2016), the Subaru Prime Focus Spectrograph (PFS; Tamura et al. 2016) or the 4-metre Multi-Object Spectroscopic Telescope (4MOST; De Jong et al. 2019), are set to provide a drastic rise in the number of observed sources over the next several years, we are currently in the situation in which millions of AGN have been detected all-sky by various surveys (e.g., by the Wide-field Infrared Survey Explorer mission (WISE; Wright et al. 2010), and the extended Roentgen Survey with an Imaging Telescope Array (eROSITA; Merloni et al. 2012; Predehl et al. 2021), with only the brightest sources having been observed spectroscopically (Dahlen et al. 2013). The growing disparity between photometric and spectroscopic observations will only widen with upcoming surveys such as the Legacy Survey of Space and Time (LSST; Ivezic et al. 2019) and Euclid (Collaboration et al. 2024), covering unprecedented areas and depths (Newman & Gruen 2022). Thus for the bulk of AGN, we must make use of multiband photometry and rely on photometric redshifts (photo-z). First implemented by Baum (1957) for inactive galaxies, these low-precision redshift estimates utilize photometric observations to e ff ectively obtain a sparsely sampled spectral energy distribution (SED), trading precision for scalability. They en- compass an array of techniques assuming color-redshift evolution (Connolly et al. 1995; Steidel et al. 1996; Illingworth 1999; Bell et al. 2004), including template-based approaches (e.g., Bolzonella et al. 2000; Ilbert et al. 2006; Salvato et al. 2008; Beck et al. 2017), where redshifted models built on theoretical or empirical SEDs are fitted to observed multi-band photometry. Although a limited number of available bands can introduce uncertainties (see review by Salvato et al. 2018), photo-z methods o ff er an e ffi cient way to estimate distances for all sources in an imaging survey, yielding highly accurate estimates with as few as three bands for passive galaxies (Benitez 2000; Abdalla et al. 2011; Arnouts & Ilbert 2011; Brescia et al. 2014; Desprez et al. 2020). By contrast, reliable photo-z for AGN have historically required highly homogenized photometry across > 20 filters, which was only achievable in pencil-beam surveys (Salvato et al. 2011). As such, this level of detail continues to be unfeasible for wide-area surveys. However, with the 10th data release of the DESI Legacy Imaging Surveys (LS10, Dey et al. 2019), we now have a broad-sky survey that, while lacking NIR coverage, includes a few optical bands supplemented by mid-IR WISE data. This allows us to explore the possibility of generating reliable photo-z for AGN over the full sky, despite having fewer filters compared to the densely sampled pencil-beam surveys. SED fitting applied to a broad population of AGN remains particularly challenging due to the uncertainty and di ffi culty of disentangling the relative contributions of the nucleus and respective host to a given band (e.g., Luo et al. 2010; Salvato et al. 2011; Brescia et al. 2019). Since the accretion properties of SMBHs, often characterized as the bolometric luminosity divided by the Eddington limit, or the Eddington ratio, significantly influence the SED of AGN, the intense power-law continuum radiation can either partly (host-dominated) or entirely (quasar-dominated), outshine the respective host, hiding key spectral features that lead to redshift degeneracies (Pierce et al. 2010; Povi'c et al. 2012; Bettoni et al. 2015). Consequently, selecting a limited number of templates can be insufficient for correct redshift determination, while increasing the number of templates raises the degeneracy (see discussion in Salvato et al. 2011; Ananna et al. 2017). In this regime of accounting for AGN contributions to galaxy photo-z, one potential approach involves modeling objects as a combination of quasar and galaxy templates (eg., Cardamone et al. 2010), performed with EAZY (Brammer et al. 2008). In addition, surveys typically estimate fluxes with models that do not account for a mixed contribution from AGN and host galaxy. Ultimately, AGN are also intrinsically variable sources on the timescales explored by the previously mentioned surveys leading to incongruent photometry acquired across di ff erent epochs. In contrast to template-fitting methods, more recent approaches have shifted towards the use of empirical Machine Learning (ML) models, performing regression or classification, to tackle photo-z applied predominantly to inactive galaxies (Collister & Lahav 2004; Laurino et al. 2011; Zhang et al. 2013; Hoyle 2016; D'Isanto & Polsterer 2018; Brescia et al. 2019; Eriksen et al. 2020; Li et al. 2021). Provided with a very large and complete spec-z sample, ML architectures manipulate photometric input features to minimize the divergence between spectroscopic and ML-derived redshifts. Over the years, a plethora of ML architectures, including decision trees (Breiman 2001; Carliles et al. 2010; Li et al. 2022), Gaussian processes (Almosallam et al. 2016) and K-nearest neighbours (Zhang et al. 2013; Luken et al. 2019) have been employed, yielding accurate point predictions and, more interestingly, full probability density functions (PDFs) (Kind & Brunner 2013; Cavuoti et al. 2016; Rau et al. 2015; Sadeh et al. 2016). The latter grants access to the prediction uncertainty, as otherwise naturally provided by template-fitting approaches, relevant for studies dealing with, e.g. luminosity functions (Aird et al. 2010; Buchner et al. 2015; Georgakakis et al. 2015). However, the limited availability of a sizable training sample of AGN has resulted in only a few attempts to compute photo-z for mostly nucleus-dominated objects with ML-based methods (Mountrichas et al. 2017; Fotopoulou & Paltani 2018; Ruiz et al. 2018; Meshcheryakov et al. 2018; Brescia et al. 2019; Nishizawa et al. 2020). More recently, the conventional approach of manually selecting photometric features for ML has been replaced by bright, well-resolved galaxies at low redshift (Hoyle 2016; Pasquet et al. 2018; Campagne 2020; Hayat et al. 2021). In this regime, integrating galaxy images into deep neural networks inherently captures essential details like flux, morphology, and other features that would typically be extracted from catalogs based on predefined assumptions, leading to a more comprehensive redshift estimation process. This approach is particularly advantageous for addressing current limitations faced by photo-z methods for AGN, as it leverages model-independent fluxes and redshift indicative features, including surface brightness profiles (Stabenau et al. 2008; Jones & Singal 2017; Gomes et al. 2017; Zhou et al. 2021, 2023). Unlike creating a single SED from total flux measurements, projects employing images with independent pixelby-pixel SEDs at identical redshift have demonstrated increased photo-z constraining power, alleviating previous empirical approaches by decreasing the fraction of outliers (Henghes et al. 2022; Schuldt et al. 2021; Lin et al. 2022; Dey et al. 2022a; Newman & Gruen 2022). Here, we introduce PICZL (Photometrically Inferred CNN redshift(Z) Likelihoods), an enhanced approach to photoz estimation that builds upon (C ircle Z by Saxena et al. 2024). While the authors demonstrated that redshift degeneracies encountered for AGN, typical in cases of limited photometry, can be broken by integrating aperture photometry alongside traditional total / model fluxes and colors, PICZL instead computes photo-z PDFs for AGN directly from homogenized flux band cutouts by leveraging the more detailed spatial light profile. All inputs are obtained utilizing LS10 exclusively. Similar to Saxena et al. (2024), PICZL can produce reliable photo-z PDFs for all Legacy-detected sources associated with an AGN. However, the model can, in principle, be applied to other extragalactic sources (e.g, inactive galaxies, Götzenberger et al. in prep.) granted that a dedicated training sample is used. We employ an ensemble of the same ML algorithm, notably convolutional neural networks (CNNs), known for their proficiency in learning intricate patterns, as outlined by (Lecun et al. 1998a). Specifically designed for image analysis, CNNs excel at identifying and extracting relevant predictive features directly from images, thereby reducing computational overhead compared to fully connected architectures. Harnessing this more extensive pool of information, these models surpass alternative models based on condensed feature-based input sets. The paper is structured as follows: Sect. 2 introduces the AGN training sample down-selection. Sect. 3 focuses on the photometric data products available within LS10. Sect. 4 details the photometric data preprocessing, followed by Sect. 5, which outlines the model pipeline. Sect. 6 presents and quantifies the redshift results, while Sect. 7 evaluates the photo-z released for the XMM-SERVS (Chen et al. 2018; Ni et al. 2021) fields. Sect. 8 outlines current limitations and discusses how we can achieve further improvements. Sect. 9 explores implications for future surveys, concluding with a summary. In this paper, unless stated di ff erently, we express magnitudes in the AB system and adopt a Λ CDM cosmology with H 0 = 69 . 8 km s -1 Mpc -1 , Ω m = 0 . 28 and Λ = 0 . 72.\",\n \"pages\": [\n 1,\n 2,\n 3\n ]\n },\n {\n \"title\": \"2. AGN training sample selection\",\n \"content\": \"In X-ray surveys, the identification of AGN has two distinct advantages - i) the reduced impact of moderate obscuration and ii) the lack of host dilution. Due to the inherent brightness of accreting SMBHs compared to their host galaxies, this results in a significantly higher nuclei-to-host emission ratio compared to observations in some of the neighbouring wavelength windows, such as UV-optical-NIR (Padovani et al. 2017). This naturally leads to a larger diversity of AGN observed by an X-ray telescope. That being said, surveys in the more accessible optical and NIR regime can increase the likelihood of detecting higherz , and in the case of MIR more heavily obscured AGN, compared to the soft X-ray bands. MLapproaches for photoz estimation in large surveys (e.g., Fotopoulou & Paltani 2018; Duncan 2022) typically classify objects into three broad categories: galaxies, quasars (QSOs), and stars, before computing photo-z. However, this classification is usually based on the optical properties and hence fails for obscured and / or lower-luminosity AGN. Since our goal is to improve on the quality of photo-z estimates for X-ray detected extragalactic sources, including type 2 AGN and low-redshift Seyfert 1 galaxies, generally, our training sample has to replicate this diversity. We achieve this by combining AGN selected across multiple wavelength bands. As a starting point, we include the same X-ray samples used in Saxena et al. (2024), namely the latest version of the XMM catalog, 4XMM, which spans 19 years of observations made with XMM-Newton (Webb et al. 2020) and data from the eROSITA CalPV-phase Final Equatorial-Depth Survey (eFEDS; Brunner et al. 2022), as they provide a reasonably representative and complete set of diverse AGN spanning 5 dex in X-ray flux out to redshift z ≲ 4. However, with just these, some portions of AGNparameter space remain imbalanced, such as highly luminious and / or high-z AGN. Thus, we expand the dataset by adding bright, optical and MIR, selections. We describe each of these samples in more detail below. This approach enhances the completeness of our training sample, which is essential to mitigate covariate shift, i.e., the shift in parameter space between the training and validation samples, so that the model generalizes e ff ectively to new data (Norris et al. 2019). Subsequently, a non-representative training sample may lead to systematically biased outcomes (Newman & Gruen 2022). Accordingly, algorithms will be strongly weighted towards the most densely populated regions of the training space (Duncan 2022).\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"2.1. Beyond eFEDS and 4XMM\",\n \"content\": \"We can enhance the redshift distribution within our sample, particularly towards high ( z ≥ 3) redshift, in this otherwise underrepresented parameter space due to observational selection e ff ects. We recognize the subsequent incorporation of unavoidable selection biases in each survey while restricting the inclusion of sources at low redshift to a minimum. While the balance between dataset quality and size is critical, deep learning algorithms that operate on pixel-level inputs tend to perform optimally only when training datasets contain ≥ 400 000 galaxy images (Schuldt et al. 2021; Dey et al. 2022a; Newman & Gruen 2022). Since our method would benefit from a larger sample (see Table 1), we chose not to apply stringent quality criteria by only considering high-quality data, significantly reducing the number of sources available training. Such a reduction would also prevent the model from learning to handle lower-quality data, limiting its application to only high-quality validation data. By not making an initial down-selection, we retain the flexibility to apply quality cuts to future blind samples by using LS10 flags later.\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"2.1.1. Samples from optical selection\",\n \"content\": \"We include the 2dF QSO Redshift Survey (2QZ, Croom et al. 2004) with ∼ 23k color selected QSOs in the magnitude range 18.25 ≤ bJ ≤ 20.85 at redshifts lower than z ∼ 3 and the QUasars as BRIght beacons for Cosmology in the Southern hemisphere survey (QUBRICS, Boutsia et al. 2020) with 224 bright (i ≤ 18) QSOs at redshifts of z ≥ 2 . 5.\",\n \"pages\": [\n 3,\n 4\n ]\n },\n {\n \"title\": \"2.1.2. Samples from optical follow-up of X-ray sources\",\n \"content\": \"Additionally, we incorporate the SDSS-IV (Blanton et al. 2017) quasar catalog from Data Release 16 (DR16Q, Lyke et al. 2020) with ∼ 150k quasars collected from various subprogrammes including optical / IR selection down to g ≤ 22 in the LS10 footprint after subselecting high quality specz , as well as follow-up of Xray sources from ROSAT (Voges et al. 1999; Boller et al. 2016; Salvato et al. 2018) and XMM (e.g. LaMassa et al. 2019). As successor science programme, we also consider the Black Hole Mapper (BHM) SPectroscopic IDentfication of ERosita Sources (SPIDERS, Anderson et al., in prep, Aydar et al., in prep) from SDSS-V (Kollmeier et al., in prep) Data Release 18 (Dwelly et al. 2017; Co ff ey et al. 2019; Comparat et al. 2020; Almeida et al. 2023).\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"2.1.3. Samples of high-z sources\",\n \"content\": \"Given the strong imbalance above z ∼ 3 . 5, we also include 400 optically / IR selected quasars at redshifts 4.8 ≤ z ≤ 6 . 6 down to g ≤ 24 from the high-redshift quasar survey in the DESI Early Data Release (EDR, Yang et al. 2023) and a compilation of highz quasars at z ≥ 5 . 3 published in literature (Fan et al. 2022).\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"2.2. Spectroscopic cross-referencing\",\n \"content\": \"The parent sample of AGN is annotated with specz , where available. According to Figure 1, we also consider sources, including those from eFEDS and 4XMM, with spatial counterparts from a compilation of public redshifts (Kluge et al. 2024). The procedure by which we match optical counterparts in our combined sample to a compilation of quality criteria down-selected specz , is outlined in Sect. 3.1 of Saxena et al. (2024). Due to overlaps between surveys, we remove duplicates when combining samples. The final sample of sources with specz comprises 40 489 objects, with a breakdown in Table 1. Correspondingly, the (cumulative) histograms illustrating the n( z ) distributions that collectively constitute the PICZL sample are presented in Figure 2. 0\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"3. The survey\",\n \"content\": \"To streamline and simplify our methodology, we have chosen to employ data from LS10 exclusively to mitigate potential complications arising from the heterogeneity of multiple datasets. Crucially, the survey area now extends over 20 000 deg 2 of optical griz and WISE W 1 -W 4 forced photometry, by incorporating the following datasets: In the north ( δ > 32.375 deg), LS10 uses the Beijing-Arizona Sky Survey (BASS, Zou et al. 2017) for g - and r -band coverage, and the Mayall z -band Legacy Survey (MzLS, Silva et al. 2016) for z -band coverage (Kluge et al. 2024). Count\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"3.1. Photometric data\",\n \"content\": \"LS10 o ff ers registered, background-subtracted, and photometrically calibrated point spread function (PSF)-forced photometry, including corresponding errors. To extend their wavelength coverage, DR10 catalogs incorporate mid-infrared (mid-IR) forced photometry at wavelengths of 3.4, 4.6, 12, and 22 µ m(referred to as W1, W2, W3, and W4, respectively) for all optically detected sources in the LS10 via the Near-Earth Object Wide-field Infrared Survey Explorer (NEOWISE) (Mainzer et al. 2011; Lang 2014; Meisner et al. 2017). Sources are modeled simultaneously across all optical bands, ensuring consistency in shape and size measurements by fitting a set of light profiles, even for spatially extended sources. Consequently, alongside reliable total and multi-aperture (8 annuli ≤ 7 arcseconds, five annuli ≤ 11 arcseconds for the optical and mid-infrared bands, respectively) flux measurements, the LS10 catalog o ff ers seeing-convolved PSF, de Vaucouleurs, exponential disk, or composite de Vaucouleurs + exponential disk models obtained with the Tractor algorithm (Lang et al. 2016). Additionally, providing fluxes rather than magnitudes, enables considering sources with very low signalto-noise ratios without introducing biases at faint levels. This characteristic also facilitates flux stacking at the catalog level, enhancing the overall versatility and utility of the classification and fitting process within LS10 1 .\",\n \"pages\": [\n 5\n ]\n },\n {\n \"title\": \"3.2. Imaging data\",\n \"content\": \"In addition to catalog data, LS10 provides a rich set of imaging products. These include observations, flux model images, and residual maps for all available bands. For instance, the top pan- els (a, b, c) of Figure 3 display all image products for a g -band observation, respectively.\",\n \"pages\": [\n 5\n ]\n },\n {\n \"title\": \"4. Preprocessing\",\n \"content\": \"Here we detail the preprocessing steps taken to prepare our dataset, ensuring that it is clean, normalized, and structured appropriately.\",\n \"pages\": [\n 5\n ]\n },\n {\n \"title\": \"4.1. Image preprocessing\",\n \"content\": \"Building on the approach of Saxena et al. (2024), which demonstrated significant improvements by shifting from total to aperture flux utilizing information on the 2D light distribution, we aim to further refine the spatial characterization of sources. This is achieved by incorporating pixel-level flux resolution through imaging as base input. Images in individual bands or in combination (i.e., colors) reflect the surface brightness, angular size, and sub-component structures of the sources, indirectly providing redshift information (Stabenau et al. 2008). To obtain reliable photo-z directly from images, we utilize flux-calibrated optical cutouts across as many filters as available. With an average seeing FWHM of 1.3 arcseconds under nominal conditions, LS10 provides a pixel resolution of 0.262 arcseconds per pixel for the optical bands, reaching depths between 23 and 24.7 AB, depending on the specific band and region of the sky (see Dey et al. (2019) and Figure 1 in Saxena et al. (2024)). To enhance computational e ffi ciency and mitigate contamination from nearby sources, we restrict our cutout dimensions to 23 × 23 pixel, centered on the AGN coordinates in the four griz LS10 bands. Our cutouts correspond to a field of view (FOV) of approximately 6 arcseconds × 6 arcseconds. We base our choice of FOV on the angular size-redshift relation by computing the angular diameter distance d Λ via: Equation 1 and Figure 4 elucidate the connection between an object's physical size, its angular size, and redshift. Notably, we can e ff ectively map galaxies with a diameter of 30 kpc - representative of main sequence galaxies (Wuyts et al. 2011) within the confines of a 23 × 23 pixel cutout, covering the range of 0 . 5 ≤ z ≤ 7 . 7. Given that the FWHM of the W1, W2, and W3 images is 6 arcseconds, and of 12 arcseconds for W4, WISE band cutouts do not provide meaningful spatial information at this scale (see Figure 5). Therefore, we have opted to use images solely from the optical bands. Problematic sources, exhibiting signs of defects in various ways are flagged in the Legacy Survey by specific bitmasks (Dey et al. 2019). We acknowledge that LS10 images exhibit varying quality due to di ff erences in seeing conditions during observations taken over many years, which impact in particular the measurements of color within source apertures. Despite this, we rely on the model's ability to adapt to these intrinsic variations given the size and diversity of our training sample, as the sources withheld from training represent a shu ffl ed subsample of the main dataset, ensuring robust evaluation. We have verified that the distribution in seeing quality (expressed as the weighted average PSF FWHMofthe images) for the training and validation are comparable (refer to Figure B.1). However, preliminary tests indicate that adding PSF size and PSF depth of the observations-both available in LS10-as additional features enhances model performance (Götzenberger et al., in prep.). While it is not unreasonable to expect the model to implicitly infer the PSF or a related abstract representation thereof from the images themselves, these features will be included by default in future PICZL versions. With ongoing developments, PSF cutouts are expected to become more accessible for integration into the image stack (see Table 2). In the longer term, we anticipate that upcoming surveys like LSST, with their improved consistency in image quality, will further reduce these limitations and boost the precision of pixelbased analyses such as ours.\",\n \"pages\": [\n 5,\n 6\n ]\n },\n {\n \"title\": \"4.2. Color images\",\n \"content\": \"It proves advantageous to provide the network with color images (ratio between images from di ff erent bands) as an additional input. This approach avoids the necessity for the model to learn the significance of colors solely from the flux images, which is inherently a more di ffi cult task. Likewise, rather than processing numerical features separately and merging them with the information extracted from the image cube at a later stage, we find it beneficial to integrate them directly at the pixel level. As a result, whenever possible, we transform catalog features into 2D arrays to align them with the original images in the same thread, enabling smoother integration and more coherent analysis (see e.g. Hayat et al. 2021). We improve the depth of our data cube by converting catalog-based quantities, e.g., flux measurements from apertures across di ff erent bands, into synthetic images, with a respective image size depending on the cutout size (see Figure 3). We expand this approach by generating images for all viable color combinations of aperture fluxes, constrained to those with matching aperture sizes. Additionally, we produce color images for flux cutouts where pixel resolutions are consistent (see lower panels b) and c) of Figure 3). Since the WISEand optical bands di ff er in both aperture size and pixel resolution, cross-wavelength color images are not feasible; instead, color combinations are restricted to within the optical or within the WISE bands (refer to Table 2). To maintain FOV consistency, we integrate WISE data for only the two innermost apertures (see Figure 5), preserving the 23x23 pixel data cube format. Although no additional spatial details are expected at this scale (see Sect. 4.1), the WISE data still captures aperture flux in a format that enables direct crosschannel connections between optical and mid-infrared data at the image level. Nevertheless, defected images can introduce challenges when generating color images (see bottom panel a) in Figure 3). Given that colors are derived from the ratio of images in di ff erent bands, the occurrence of unphysical negative or abnormally high / low values poses a significant concern. To address this issue, we examine whether the median value of neighbouring pix- Notes. Each column represents a distinct processing channel, detailing its input data type and the respective dimensions. All flux-related inputs have been corrected for reddening e ff ects. els is more than three times lower than the value of noisy pixels per band. If so, the central pixel's value is substituted with the median value of the surrounding pixels to smooth out fluctuations. In cases of non-detections or severely corrupted images where the largest value among all pixels in an image is < 0.001, the image is treated as a non-detection and all pixels are set to zero as default. After undergoing preprocessing, the images are utilized to create color images by exploring the six possible color combinations: gr , gi , gz , ri , rz and iz . If either of the two flux images involved in creating a color image is identified as a nondetection, the resulting color image is set to a default value of -99. At the catalog level, spatially invariant features, such as bestfit model classifications and signal-to-noise ratios (S / N), are processed separately in a dedicated channel, as they cannot be converted into image data. Notably, regarding normalization, we adopt a uniform min-max scaling approach to handle columns with cross-dependencies, such as flux bands. This strategy aims to preserve crucial information, such as the original shape of the SED. Contrary to the conventional approach of stacking all available images into a single input (Hoyle 2016; D'Isanto & Polsterer 2018; Pasquet et al. 2018; Dey et al. 2022a; Treyer et al. 2023), we find it advantageous to separate the color image data cube (23,23,24) from the flux band cutouts (23,23,32). This distinction is necessary because the color images often have di ff ering pixel value scales, including negative values, which require specialized processing in our machine learning application, such as tailored loss functions in the CNN. Non-spatial attributes are then combined with image-based data at a subsequent stage of the model, allowing the model to capture both spatial and nonspatial aspects. By processing data types in parallel channels, we leverage their complementary information by merging the data at a later stage, enhancing the extraction of inter-band correlations and ultimately improving redshift precision (Ma et al. 2015; Ait-Ouahmed et al. 2023). A detailed breakdown of the features integrated into each channel is provided in Table 2.\",\n \"pages\": [\n 6,\n 7\n ]\n },\n {\n \"title\": \"5. Neural network\",\n \"content\": \"ML embodies an artificial intelligence paradigm where computers learn patterns and relationships from data, enabling them to make predictions or perform tasks without explicit programming. Multilayer perceptrons (MLPs), a feature-based feedforward neural network, draw inspiration from their biological counterparts, namely excitable cells responsible for processing and transmitting information (Rosenblatt 1958; I. Goodfellow 2016; M. Deru 2019). Likewise, each assigned a distinct weight, computational input vectors can be organized into layers, relayed to one or more hidden layers, to compute a scalar output value (Géron 2019). During training, these models learn data mappings by adjusting the weights and biases associated with their connections. The margin of change to the model after every training epoch is dictated by the choice of optimizer and loss function, which e ff ectively calculates the Euclidean distance between the prediction and so-called ground truth in multi-dimensional feature space, thereby significantly impacting model convergence and performance. The current state-of-the-art deep learning (DL) networks, characterized by their many hidden layers, have shown exceptional capabilities in handling complex non-linear tasks.\",\n \"pages\": [\n 7\n ]\n },\n {\n \"title\": \"5.1. Convolutional model layers\",\n \"content\": \"Among such architectures, Convolutional Neural Networks (CNNs; Fukushima 1980; LeCun et al. 1989; Lecun et al. 1998b) distinguish themselves by their remarkable e ff ectiveness in handling grid-like data, a prevalent form of which is represented by images. CNNs leverage a model architecture that is particularly e ff ective in tasks like image recognition, object detection, and image segmentation (O'Shea & Nash 2015; Liu et al. 2022). In convolutional layers, neurons establish connections exclusively with pixels within their receptive field, ensuring successive layers are linked only to specific regions of the previous layer. Subsequently, the model extracts low, image-level features in early layers and progressively complex, higher-level features in later layers. This allows the CNN to learn representations of the input data at multiple levels of abstraction. The convolutional operation involves sliding several kernels K , here of sizes 3 × 3 or 5 × 5 pixel, across the images with a fixed stride of size s = 1, compressing each mosaic element into a single scalar. This process entails element-wise multiplication, generating a set of K feature maps. Filters, the learnable parameters of convolutional layers, enable the network to detect and highlight di ff erent aspects of the input data, such as edges or corners. Each convolution is followed by a pooling layer that reduces spatial dimensions and computational load. Pooling involves sliding a kernel, in our case of size 2 × 2 and s = 1, across the feature maps, selecting the maximum value (max pooling) within each kernel window. After flattening the data cube into a single array, it is passed through (several) fully connected (FC) layers. Each FC layer contains a layer-specific number of neurons n , tailored to the model's specific needs. The final FC layer's neuron count varies based on the task, with n = 1 referring to regression tasks and n ≥ 2 to other endeavors such as multi-label classification.\",\n \"pages\": [\n 7,\n 8\n ]\n },\n {\n \"title\": \"5.2. Gaussian mixture models\",\n \"content\": \"By default, single output regression models (Schuldt et al. 2021) provide point estimates without quantifying uncertainty. This severely limits their area of application, particularly in scenarios where quantifying the uncertainty is crucial, for example, when performing precision cosmology with Euclid (Bordoloi et al. 2010; Scaramella et al. 2022; Newman & Gruen 2022). By instead employing an architecture that provides PDFs, we can not only retrieve point estimates but also encapsulate the uncertainty associated with the predictions in a concise format (D'Isanto & Polsterer 2018). Given the inherent complexity in determining redshifts from only a few broadband cutouts, our results are expected to often exhibit degeneracy with multi-modal posteriors, making a single Gaussian insu ffi cient for representing the photoz PDFs. Therefore, estimates are computed using Bayesian Gaussian Mixture Models (GMMs, Duda et al. 1973; Viroli & McLachlan 2017; Hatfield et al. 2020). These networks provide a unique probabilistic modeling approach that di ff erentiates them from traditional MLPs. One of the key distinctions lies in the output nature, where GMMs provide a set of variables to compute weighted multi-Gaussian distributions as opposed to single-point estimates. Subsequently, each component is characterized by its mean µ , standard deviation σ , and weight w , allowing GMMs to produce a full PDF for a given set of inputs x . The PDF is expressed as with, wk representing the weight and N ( x | µ k , σ 2 k ) denoting the Gaussian distribution of the k -th component. As such, we extend our CNN approach by a GMM backend to output PDFs based on the information-rich feature maps produced during the front-end phase of the network. However, we encounter a limiting challenge with inputs of such small dimensions, i.e. 23 × 23, as they are not well-suited for established image-based Deep Learning architectures, such as \\\"ResNet\\\" (He et al. 2015), which typically require larger dimension scales, typically exceeding 200 × 200 pixels, to accommodate the large number of pooling layers they employ. Therefore, we developed a custom architecture to fit our data dimensionality. The resulting model architecture features roughly 490,000 trainable parameters, far fewer than found in comparable studies (see Pasquet et al. 2018; Treyer et al. 2023), and is displayed in Figure A.1 of Appendix A.\",\n \"pages\": [\n 8\n ]\n },\n {\n \"title\": \"5.3. Model refinement\",\n \"content\": \"Numerous hyper-parameters are crucial in shaping the network architecture while influencing training and convergence. Extensive optimization has been conducted across various parameters utilizing the Optuna framework (Akiba et al. 2019). The current configuration accounts for the vast array of potential combinations. The key parameters with the most significant impact are outlined below: putational e ffi ciency, while smaller batches may help generalize better.\",\n \"pages\": [\n 8\n ]\n },\n {\n \"title\": \"5.4. Loss functions\",\n \"content\": \"When utilizing PDFs instead of point estimates, it is crucial to quantify whether a predicted PDF e ff ectively reflects the ground truth, in our supervised ML case, specz (Sánchez et al. 2014; Malz et al. 2018; Schmidt et al. 2020; Treyer et al. 2023). The PDF predicted by a model should be concentrated near the true value. A straight-forward and meaningful proper scoring rule (i.e. one which is lowest when the prediction is at the truth), is the product of probabilities at the true redshifts. The negative log-likelihood (NLL) 2 of which passed to a minimizer yields the likelihood of observing a data distribution given a specific set of model parameters. This expression coincides with the average Kullback-Leibler divergence (KL, Kullback & Leibler 1951) when going from a delta PDF at specz to the photo-z PDF. An alternative growing in popularity is given by the Continuous Ranked Probability Score (CRPS), initially applied in weather forecasting (Grimit et al. 2006), which serves as a valuable metric for photo-z estimation via PDF quantification (D'Isanto & Polsterer 2018). Computationally, CRPS calculates the integral of the squared di ff erence between the predicted cumulative probability distribution function (CDF) and a Heaviside step-function H ( x ) at the value of the specz ( xz ) as For a finite mixture of normal distributions M , the CRPS can be expressed in closed form: with δ i = y -µ i , the uncertainty σ i and weight wi of the i -th component respectively, as well as By extending the CRPS loss via normalizing it by (1 + z), we adjust the penalty for prediction errors based on redshift. In doing so, we prioritize accuracy for low and intermediate redshift sources by imposing higher constraints while allowing for more leniency in less critical high-redshift areas. Integrating over the entire range of possible redshift values, the CRPS takes into account both location and spread of the predicted PDF. It, therefore, provides a comprehensive evaluation metric, generating globally well-calibrated PDFs (Dey et al. 2022b).\",\n \"pages\": [\n 8,\n 9\n ]\n },\n {\n \"title\": \"5.5. Training & data augmentation\",\n \"content\": \"Our dataset is divided into training and validation sets in an 80:20 ratio. The model is trained over 1000 training epochs utilizing the Adam algorithm (Kingma & Ba 2017) along with a learning rate scheduler to adjust the learning rates during training dynamically. The inclusion of both is a consequence of the Optuna hyperparameter optimization. Typically, the model achieves its lowest validation loss around the 600th epoch when considering both loss functions (see Figure 6). After this point, although the training loss continues to decrease, the validation loss begins to increase, indicating overfitting. This overfitting likely arises due to the limited size of our training sample, allowing the model to memorize specific patterns rather than learning generalizable features. We implement a checkpoint system to mitigate this, saving the model's architecture and parameters when it reaches its lowest validation loss. This approach ensures that we capture the best-performing model configuration while preventing overfitting. To further exploit the advantages of deep learning, particularly its capacity to perform well with extensive datasets, we have tested the incorporation of data augmentation techniques into our methodology. Recognizing the substantial impact of training data amount on model performance, we employ common augmentation data products, such as rotated and mirrored images. Although these transformations significantly increase the apparent size of the dataset, we do not observe an increase in performance when including augmentation techniques (Dey et al. 2022a; Treyer et al. 2023). This is potentially not surprising as the influence of augmentation techniques depends on several factors. While CNNs are inherently equivariant to translations within images, enabling them to recognize objects even if they are shifted or o ff -centered, they and most other machinelearning approaches are not naturally equivariant to rotations and reflections. This lack of rotational equivariance means that the model output e ff ectively depends on the rotation of the input images. However, as most of our inputs are approximately point sources with radial symmetry, we do not explore this aspect further in this study. More recent publications have focused on developing rotationally equivariant CNN architectures, which have shown improved performance despite increased computational costs (Cohen & Welling 2016; Weiler & Cesa 2021; Lines et al. 2024). Incorporating such architectures could be particularly beneficial for future works dealing with large amounts of imaging data.\",\n \"pages\": [\n 9\n ]\n },\n {\n \"title\": \"5.6. Model ensemble\",\n \"content\": \"Ensemble models capitalize on the diversity of multiple models to improve prediction accuracy. Our approach involves diversifying models by adjusting parameters such as the number of Gaussians per GMM, learning rates, batch sizes, and the choice of loss function. We employ both NLL and CRPS as complementary loss functions to achieve this. While CRPS excels in achieving lower outlier fractions, NLL enhances overall variance. Each loss function trains a set of 144 models, resulting in a diverse pool of 288. In line with Treyer et al. (2023), we achieve superior performance by randomly combining equally weighted CRPS and NLL models from the pool, compared to the best-performing individual model (Dahlen et al. 2013; Newman & Gruen 2022). This finding is also aided by the result that roughly 30% of outliers are model-specific as opposed to 20% of outliers appearing in all models, therefore considered to be genuine. Put di ff erently, 80% of outliers were found to not be an outlier in at least one other model. We additionally recognize that the initial configuration of each model has a minor influence on the performance. Although training additional models could further enhance our results, the computational resources required to generate three to five sets of models would make this approach impractical relative to the potential performance gains. To create an e ff ective ensemble, we evaluate the outlier fraction of all individual models on an un- seen test sample. From this evaluation, we select models based on their relative influence on the model performance. We then observe how the ensemble's performance evolves as more models are continuously incorporated. Our ensemble ultimately comprises 10 models or 84 Gaussians, evenly incorporating models optimized using both NLL and CRPS approaches. To refine the model weights within the ensemble, we employ Optuna for finetuning. Subsequently, the ensemble posterior likelihood distribution P ensemble(x) is given by: where the weights wi are normalized such that: to assure that the integral of the ensemble PDF is equal to 1. We obtain a point estimate for our photoz prediction by identifying the dominant mode of the resulting PDF, recognizing that the mean or median could be misleading for highly non-Gaussian and bi- or multi-modal distributions. Additionally, we provide asymmetric 1 and 3 sigma upper and lower errors by evaluating the PDF within the 16th / 84th and 0.15 / 99.85th percentiles, respectively, along with the entire PDF.\",\n \"pages\": [\n 9,\n 10\n ]\n },\n {\n \"title\": \"6. Photo-z results\",\n \"content\": \"We evaluate the performance of PICLZ on both point estimates and PDFs. This comprehensive assessment includes testing the statistical reliability and accuracy of our predictions. We adopt several commonly employed statistical metrics, providing insights into the accuracy, precision, and reliability of the photo-z estimates. In line with the definitions from the literature, we use the following metrics: We quantify PICZL's performances on a validation sample of 8098 sources and find an outlier fraction η , of 5.6% with a variance σ NMAD, of 0.045. Since the validation sample is used for feedback in training, it should be mentioned that the results mentioned above are likely overestimating the performance and therefore may not representative of what a future user could achieve with say, a test set previously unknown to the model, leave-one-out, or k-fold cross-validation. To this end, we have estimated our performance on independent, blind, X-rayselected samples (see Sect. 6.2 and 7). Figure 7 compares photometric versus spectroscopic redshifts, split by point-like (type = PSF) and extended (type = EXT) morphology. With respect to comparable work (e.g., Figure 13 from Salvato et al. 2022), we observe enhanced performance with a much-reduced fraction of outliers, especially for PSF sources. These objects lack morphological information at redshifts z ≳ 1, hence we attribute this improvement to the model's ability to recognise how the radial extension of the azimuth profile of sources changes with redshift (refer to Figure 4). The scatter, for both PSF and EXT distributions, is tight and symmetrically distributed around the z PICZL = z spec identity line, with outliers appearing randomly scattered, suggesting stable performance across the redshift range and minimal systematic errors (see Figure 8 and Dey et al. (2019) for reference). Given that our point estimates are derived from PDFs, we must ensure their global calibration and accuracy. To achieve this, we employ the probability integral transform (PIT) statistic (Dawid 1984; Gneiting et al. 2005), a widely accepted method in the field for assessing the quality of redshift PDFs (Pasquet et al. 2018; Schuldt et al. 2021; Newman & Gruen 2022). An ideal scenario is represented by a uniformly distributed histogram of PIT values. Any deviation from uniformity can indicate issues in PDF calibration. Under-dispersion may suggest overly narrow PDFs, while over-dispersion often results from excessively wide PDFs (D'Isanto & Polsterer 2018). Peaks close to zero or one can be explained by catastrophic outliers, where the true redshift lies so far in the wing of the PDF that it essentially falls outside of it. We employ the PIT histogram (top panel Figure 9) alongside quantile-quantile (QQ) plots (bottom panel Figure 9), to visually assess the statistical properties of our model predictions. For reference, these can be compared to Figure 2 of in Schmidt et al. PIT Score . (2020); however, those code performances are completely dominated by inactive galaxies, making a direct comparison impractical. The QQ plot compares the CDFs of observed PIT values and identity U (0 , 1), to visualize QQ di ff erences split by morphological type. A well-calibrated model will exhibit a QQ plot closely following identity with a flat PIT histogram, indicating accurate and reliable redshift predictions. Our analysis shows that, while asymmetry in the PIT distribution could suggest systematic bias, the QQ plot reveals small residuals overall. Notably, for both the single- (refer to Sect. 5.2 and Figure A.1) and ensemble model (refer to Sect. 5.6), the curves do not deviate significantly from zero, lesser so for PSF-type than EXT-type objects, indicating well-calibrated models. A distinct observation is the shift towards central PIT values for the ensemble model. This shift is not unexpected, given that the ensemble approach integrates multiple redshift solutions, each potentially contributing di ff erent peaks to the resulting ensemble PDF. Consequently, the ensemble PDF exhibits a broader bulk of probability across redshift, with PIT scores accumulating more area under the curve Notes. The last two setups are an improvement over the previous performances, with the last setup providing comparatively good results, despite ignoring the features provided by catalogs, particularly relevant for potential implementation in e.g., LSST. when integrated to the main mode. As a result, rather than yielding a flat PIT distribution, the histogram shifts towards having a concentration of values around 0.5. While this phenomenon aligns with our expectations, further enhancements may be realized by incorporating metrics tailored instead to local calibration accounting for population-specific subgroups (see e.g. Zhao et al. 2021; Dey et al. 2022b).\",\n \"pages\": [\n 10,\n 11,\n 12\n ]\n },\n {\n \"title\": \"6.1. Model performance for different inputs\",\n \"content\": \"Images provide a more comprehensive view of astronomical sources by capturing their full spatial structure and light distribution, o ff ering finer details on morphology, apparent size and extended features that are often lost in the averaging process of aperture photometry. This is pivotal, as SED features crucial for determining redshift solutions and resolving degeneracies mostly reside within the host galaxy (Soo et al. 2017; Wilson et al. 2020). In particular for AGN, images help to spatially separate the pixels corresponding to AGN-dominated emission and those corresponding mostly to the host galaxies. This approach enhances our ability to isolate and analyze individual pixel strings across multiple filters, to e ff ectively construct SEDs for the host galaxies independently. We thereby capture subtle features, neighbors, and patterns that may not be discernible from total flux measures. Critically, the independent photo-z from the AGN and host must agree, thus narrowing the overall source PDF. Table 3 presents the fraction of outliers for both EXT and PSF sources using di ff erent configurations of the PICZL algorithm. The first setup replicates the feature-only method, relying solely on total fluxes from the catalog, which, as seen in other studies, results in the fraction of outliers for PSF sources being nearly three times higher than for EXT sources (Salvato et al. 2022). In the second configuration, we adopt the approach of Saxena et al. (2024), incorporating the 2D light distribution and color gradients within annuli. The third configuration, i.e., PICZL as outlined above, enhances this by incorporating images, supplemented with catalog values transformed into image format, which leads to an additional improvement over the method in Saxena et al. (2024). Since images inherently capture all features typically extracted numerically and presented in catalogs, we show in the fourth case that PICZL achieves excellent results without feature / catalog information, using solely optical images supplemented by WISE aperture flux maps, even without explicit hyperparameter tuning. This approach is particularly promising for future surveys such as LSST and Euclid, where relying solely on multi-band imaging, including NIR / IR, will eliminate the need for complex source modeling or aperture photometry, regardless of the sources being galaxies or AGN.\",\n \"pages\": [\n 12\n ]\n },\n {\n \"title\": \"6.2. Blind sample comparison\",\n \"content\": \"To evaluate the robustness of our approach, we tested PICZL on an independent blind sample from the Chandra Source Catalog 2 (CSC2 3 ). This sample, selected to ensure no overlap with the training data, but yield a comparable X-ray to MIR distribution, is the same one used in the C ircle Z analysis (Saxena et al. 2024), allowing for a direct comparison with their results. After filtering for sources with spectroscopic redshift and removing duplicates with the training dataset, the sample comprises 416 sources within the LS10-South area. For comparison, Saxena et al. (2024) achieve an outlier fraction, η , of 12.3% and a nor- 7 malized median absolute deviation, σ NMAD, of 0.055, while our algorithm yields a superior σ NMAD of 0.046 and η of 8.3% (see Table 5 and Figure C.1). This comparison, therefore, demonstrates how utilizing image cubes further improves the already good results achieved using 2D-information from a catalog. For more details on the sample, see Saxena et al. (2024).\",\n \"pages\": [\n 12,\n 13\n ]\n },\n {\n \"title\": \"6.3. Prediction uncertainty quantification\",\n \"content\": \"Accurate error estimation is crucial for assessing the reliability of any prediction, particularly in those astrophysical contexts where uncertainties can significantly impact the interpretation of data. To this end, we provide asymmetrical 1 σ and 3 σ errors for every source, together with the photoz PDF, as detailed in Sect. 5.6. These error estimates o ff er a comprehensive understanding of the potential variance in our predictions, accurately reflecting the inherent uncertainties in our data and model. We include asymmetrical errors, as the true distribution of uncertainties is often not symmetrical. This asymmetry arises from factors such as photometric noise and degeneracies in the highly nonlinear color-redshift space, where certain higher or lower redshift values may be more probable than their counterparts, leading to skewed PDFs. Although symmetrical redshift errors are observed for the majority of sources, approximately one-third of the sources exhibit asymmetrical 1 σ errors with deviations in redshift up to | ∆ z asym | ≃ 0 . 25. in Sect. 8.4. Additionally, the widths of the PDFs exhibit noticeable horizontal structures, largely influenced by the nature of the LS10 filters. We find that PDFs with wider modes, indicating less certain redshift estimates, often correspond to redshift ranges where key spectral features, such as the Ca break, fall outside the filter coverage (e.g. g & r at z ≃ 0 . 4, r & i at z ≃ 0 . 8 and z ≃ 1.5 or z ≃ 2.2). At redshifts approaching z ≈ 3 and extending below z ≈ 5, the Lymanα break starts to fall within the filter ranges, which contributes to narrower PDFs with increased PDF width only observed for extremely faint sources at even higher redshifts. Lastly, we find that the number of sources with wider PDFs increases as a function of | ∆ Z | (1 + Z spec) , indicating that narrower PDFs correspond to more accurate point predictions.\",\n \"pages\": [\n 13\n ]\n },\n {\n \"title\": \"6.4. Insights from multimodal PDFs\",\n \"content\": \"To investigate whether predicted outliers are genuine anomalies or instead stem from machine learning processes, we inquire whether secondary peaks in PDFs are physically meaningful or training artifacts. For each source, we compute the fraction of its PDF, that satisfies the condition | ∆ Z | (1 + z spec) ≥ 0 . 15, to provide the probability of being considered an inlier. The left panel of Figure 11 illustrates this with examples of two PDFs (one for an inlier and one for an outlier), where we highlight the corresponding inlier probabilities. In the right panel of Figure 11, we plot | ∆ Z | (1 + z spec) re-normalised to scale [0,1], such that as a function of inlier probability for all sources. The horizontal dashed black line separates the inliers ( < 0.5) from the outliers ( ≥ 0.5). The majority of inliers are concentrated at high inlier probabilities. This suggests that most inliers provide confident, unimodal PDFs with low errors. Likewise, most outliers have low inlier probability, with only a few outliers having more than 50% inlier probability. While this quantity cannot be computed for sources lacking spectroscopic redshift, it can be used to evaluate the reliability of our point estimates considering their associated PDFs. To evaluate whether ensemble solutions with broad or complex distributions e ff ectively capture the true redshift, we identify sources with multiple peaks and measure the proportion of PDFs exhibiting more than one mode. For these sources, we define a secondary peak as significant if it accounts for at least 10% of the height of the primary peak, which is the minimum prominence threshold used in our analysis. The top panel of Figure 12 shows that only 3% of inliers exhibit secondary peaks as opposed to the roughly 30% of outliers, where occurrences of strong secondary modes decrease with increasing prominence for both cases. While a single mode typically indicates a secure estimate for inliers, the presence of a unique mode alone can subsequently not be used to determine whether a source is an outlier. Conversely, in the bottom panel of Figure 12, we present the recovery fraction, which denotes how often a secondary peak in PDFs exhibiting multi-modal distributions corresponds to the true redshift. Among the outliers with 30% multi-modal PDF, more than 70% have one of their secondary peaks corresponding to | ∆ Z | (1 + Z spec) ≤ 0 . 15, indicating that there is significant probability that the redshift could consequently be considered as an inlier if the peak heights were reversed. Given the PIT histograms shown in Figure 9, it appears likely that the the PDFs are accurately capturing the relative frequency of multi-& bimodal PDFs, as if they were not, there would likely be bias evident in the overall PIT distribution. In other words, using this particular dataset, despite incorporating full multiwavelength image cutouts and cataloglevel information, there persist areas of parameter space that are legitimately degenerate in redshift, and the PDF parameterization looks to be capturing that degeneracy accurately. Consequently, we recommend using the entire PDF rather than point estimates, when possible.\",\n \"pages\": [\n 13,\n 14\n ]\n },\n {\n \"title\": \"7. PICZL applied to other surveys\",\n \"content\": \"We want to investigate how PICZL, utilizing LS10 photometry and imaging, performs in determining photo-z for AGN in a more generic setting. Our focus is set on the LSST deep drilling fields (DDFs), selected to study SMBH growth across the full range of cosmic environments. These fields o ff er deeper, more comprehensive spectroscopic redshift coverage, along with a broader range of high-sensitivity bands, providing an enhanced dataset for photo-z estimation.\",\n \"pages\": [\n 14\n ]\n },\n {\n \"title\": \"7.1. XMM-SERVS\",\n \"content\": \"The XMM-Spitzer Extragalactic Representative Volume Survey (XMM-SERVS, Mauduit et al. 2012) encompasses three key fields: the XMM-Large Scale Structure (LSS, Chen et al. 2018), spannning 5.3 deg 2 with a flux limit of 6 . 5 × 10 -15 erg cm -2 s -1 over 90% of the survey area in the 0.5-10keV band (Savi'c et al. 2023); the Wide Chandra Deep Field-South (W-CDF-S, Ni et al. 2021) and the European Large-Area ISO Survey-South 1 (ELAIS-S1, Ni et al. 2021), covering approximately 4.6 deg 2 and 3.2 deg 2 (Brandt et al. 2018; Scolnic et al. 2018), limited to 1 . 3 × 10 -14 erg cm -2 s -1 , respectively, each selected for their exceptional multiwavelength coverage and strategic alignment with the DDFs.\",\n \"pages\": [\n 14\n ]\n },\n {\n \"title\": \"7.2. Photo-z computation\",\n \"content\": \"Considering that the original XMM-SERVS photo-z benefit from high S / N photometry and broader wavelength coverage, including u -band and NIR coverage, we would expect significantly better results compared to those achievable by PICZL using LS10 data alone. However, PICZL obtains comparable if not better results when limiting the sample to the depth at which it was trained on (see XMM-SERVS magnitude distribution in Figure E.1). Like any ML model, it struggles to extrapolate to faint sources outside its training distribution. Therefore, a fair comparison requires limiting the analysis to a similar feature space, while the results in Table 4 reflect performance across the entire, diverse XMM-SERVS samples. To facilitate a comparison with PICZL, our analysis begins with the counterpart associations of X-ray emissions as presented in Ni et al. (2021) and Chen et al. (2018). Whenever possible, we limit our samples to sources flagged as AGN (as defined in Sect. 6 of Ni et al. 2021). Matching to objects detected in LS10, due to its more limited depth, we identify approximately one-third of the original sources (see Table 4). This limitation is particularly pronounced in the XMM-LSS survey, where, in addition to deeper X-ray data corresponding to fainter counterparts, Chen et al. (2018) restrict their sample of AGN for which they calculate photo-z, to strictly non-broad-line X-ray sources. We select all sources with spectroscopic redshift and exclude those previously included in the Notes. From left to right, the columns list: Sample field; the original number of X-ray sources; the number of sources in each field that were detected in the LS10 survey; have spectroscopic redshift z > 0 . 002; and were not already present in the training sample of PICZL (i.e. previously unseen); the fraction of failed photo-z in XMM-SERVS zphot = -99; and the bias ⟨ ∆ z ⟩ , variance σ NMAD and outlier fraction η obtained with XMMSERVS (S, excluding zphot = -99) and PICZL (P) photo-z. PICZL training sample (see Table 4). Unlike the original works, PICZL also successfully computes photo-z for the significant fraction of sources for which the SED fitting adopted in XMMSERVS failed (refer to w / photoz S in Table 4). Figure 13 visualizes the outlier fractions (top) and variances (bottom) for the aggregate XMM-SERVS samples. Blue and burgundy represent the distributions including / excluding catastrophic failures in the XMM-SERVScatalogs. The plot is limited to outlier fractions of 30% and variances of 10%, as XMM-SERVS curves including zphot = -99 solutions continue to rise for fainter magnitudes or lower S / N. This suggests that, with increasing magnitude, a decreasing number of accurate estimates are available, making the subset excluding such solutions increasingly non-representative in terms of outlier fraction. However, by limiting our comparison to sources with photoz obtained from XMM-SERVS, we find that the accuracy of these photo-z remains comparable up to magnitudes around 23.5 AB, despite the more limited data being used. Additionally, we observe enhanced performance in PICZL photo-z for sources with S / N values ≳ 60, depending on the specific XMM-SERVS sample. Importantly, sources with S / N ≳ 10-20 mark a critical threshold range in LS10 where lower S / N values lead to an exponential rise in photometric errors (see Figure D.1). This trend is particularly evident for magnitudes fainter than 21.5 in the iband (Saxena et al. 2024) and visible in all panels of Figure 13. Figure 14 presents a visual comparison of normalized residuals for outliers identified by either PICZL or the original photo-z from Ni et al. (2021), exemplified via the ELAIS-S1 field. The spread of biases, excluding failures from XMM-SERVS, is similar for both methods, falling mostly within the range of [-0.5, 0.5]. The outliers in common appear to be modestly brighter than those that are outliers for a single method only and are evenly distributed between overestimation and underestimation. Notably, the outliers from Ni et al. (2021) are over a narrower range, while PICZL has a few brighter and several more fainter, the majority of which with magnitudes beyond the range covered during training. The observed di ff erence in outlier rates, is - not primarily due to the use of template-fitting versus machinelearning methods but rather a reflection of the available photometry. XMM-SERVS, with its deeper photometry, is wellsuited for faint objects, though saturation in brighter sources may contribute to some outliers. PICZL, which leverages the LS10 dataset, is optimized for brighter sources due to the shallower photometry. While it performs well in this regime, fainter objects that fall outside the known parameter space are consequently less constrained.\",\n \"pages\": [\n 14,\n 15,\n 16\n ]\n },\n {\n \"title\": \"8. Discussion\",\n \"content\": \"While Saxena et al. (2024) made significant strides in overcoming the limitations of relying solely on total or model fluxes for photoz estimation of AGN by utilizing all redshift-correlated features in the LS10 catalog, we have further advanced this approach. By integrating data from both optical and MIR wavelengths, our study highlights the transformative potential of combining imaging with catalog-level data. Despite the state-of-theart advancements in photoz estimation for AGN in this work, further improvements will be possible only by solving issues related to data quality. By overcoming these issues, we can significantly reduce the fraction of catastrophic outliers and achieve even greater accuracy in our predictions. In the following, we discuss the limitations that need to addressed in order to apply PICZL to upcoming imaging surveys.\",\n \"pages\": [\n 16\n ]\n },\n {\n \"title\": \"8.1. Incorrect spec-z\",\n \"content\": \"First and foremost, we need to address the reliability of the labels used in our, and generally any supervised machine learning algorithm. While spectroscopic redshift are typically trusted as accurate representations of true redshifts, this assumption does not always hold. As surveys continue to expand in scale, automated pipelines become indispensable for assigning spectroscopic redshift and identifying artifacts / problems for each source, as manual verification by visual inspection is impractical and will remain so for future surveys (Bolton et al. 2012). Importantly, most pipelines are trained on determining the redshift of the bulk of objects (Alexander et al. 2023), which are inactive galaxies and not AGN. This implies that a non-negligible fraction of AGN have wrong redshift estimates, including many labelled with \\\"no warnings\\\" (Hewett & Wild 2010; Wu & Shen 2022). Consequently, our performance is constrained by the unknown amount of incorrectly identified spectroscopic redshift Absolute LS10 r band [mag] Absolute LS10 r band [mag] per survey. While this fraction may be smaller for galaxies without nuclear activity, AGN present a range of challenges where pipelines fail. Examples of failures include cases where e.g. MgII is misidentified as Lymanα , leading to ambiguous redshifts around z = 0 . 7 -0 . 9 ⇔ 1 . 5 -2 . 5 or sources with strong featureless continuum typical of Blazars or, finally, sources with intense broad emission lines compared to the background continuum (Chen et al. 2018; Ni et al. 2021), as well as those with weak emission lines and a flat continuum background.\",\n \"pages\": [\n 16,\n 17\n ]\n },\n {\n \"title\": \"8.2. Incorrect type classification\",\n \"content\": \"A subset of sources in the training sample, although having redshifts of z ≥ 1, are classified as extended (i.e. non-PSF). This classification is likely incorrect, given the pixel resolution of LS10. The apparent extension is more plausibly a result of poor seeing conditions that were not properly accounted for, or the presence of nearby neighboring sources (see Sect. 5.3 of Hsu et al. 2014). Since the morphological type a ff ects predictions, akin to a prior in template fitting methods, this misclassification explains the increase in outliers within the EXT sample as redshift increases (see middle row in right panel of Figure 15). To verify this, we computed the absolute g magnitudes for outliers to assess whether their values fall within the expected range for galaxies and AGN (Véron-Cetty & Véron 2010). Figure 16 shows that 16% of sources classified as EXT (and thus galaxy dominated) have an absolute magnitude which exceeds the range typical of galaxies MEXT [-16,-24], confirming that their TYPE is wrongly assigned. Interestingly, approximately 96% of PSF sources meet the absolute magnitude requirement of MPSF [-20,31], indicating that their TYPE is generally accurate.\",\n \"pages\": [\n 17\n ]\n },\n {\n \"title\": \"8.3. Faulty photometry\",\n \"content\": \"In addition to issues connected to the reliability of labels, predictions also face challenges due to data inconsistencies arising from faulty photometry. Surveys often flag issues of this kind directly, including problems such as hot pixels, saturation, cosmic rays, bleed trails, and other uncategorized anomalies. Detecting and addressing such observational defects is crucial because the accuracy and reliability of photoz predictions cannot be guaranteed for compromised photometric inputs. We experimented with removing various groups of sources a ff ected by observational artefacts but found no improvement when excluding any single group. We therefore assume that, while removing some problematic sources could potentially improve performance, the reduction in training data size o ff sets any gains. Adding this kind of noise to the current training sample could be an approach to decouple these two e ff ects. Consequently, we treat the inclusion of all problematic sources as natural noise, as opposed to artificially degrading images employing methods such as Gaussian noise (Hayat et al. 2021).\",\n \"pages\": [\n 17\n ]\n },\n {\n \"title\": \"8.4. Survey depth\",\n \"content\": \"When splitting the validation sample by S / N across all bands, we find a higher outlier fraction in sources with low S / N (see Table 5). Consequently, to evaluate the reliability of our photoz, we need to consider the photometric depth of LS10 and how increased photometric errors for fainter sources impact redshift estimate uncertainties. Figure 15 presents the prediction bias ⟨ z ⟩ , outlier fraction η , and dispersion σ NMAD as functions of r -band magnitude in LS10, X-ray flux (when available), and specz . As the r -band magnitude increases, as expected, the outlier fraction and scatter rise. This pattern is consistent with the exponential increase in photometric errors for, e.g. i -band magnitudes ≳ 21.5, beyond which the outlier fraction, which otherwise remains well below 10%, and scatter also appears to rise, especially for sources of type EXT (refer to Figure D.1). While faint LS10 sources have unreliable photo-z, future surveys such as LSST and Euclid will probe deeper with improved S / N. As such, our methodology can be adapted to these next-generation surveys, ensuring high accuracy across a broader range of magnitudes and providing robust redshift estimates for the extensive AGN populations these and other surveys will detect.\",\n \"pages\": [\n 17\n ]\n },\n {\n \"title\": \"8.5. Missing i-band\",\n \"content\": \"For DR10, the Legacy Survey footprint was extended by incorporating all available DECam data from various contributing surveys (refer to Sect. 3), including coverage in the i -band, though limited to a single pass. As a result, ∼ 10% of sources lack i -band observations. Unsurprisingly, and as shown in Table 5, the accuracy of photo-z increases when this band is also available.\",\n \"pages\": [\n 17\n ]\n },\n {\n \"title\": \"8.6. Biases\",\n \"content\": \"While the photo-z residuals exhibit a symmetric distribution around zero with minimal scatter, they do not consistently center at zero, suggesting a potential systematic bias in the photo-z estimates. Ideally, we would like to achieve normalized residuals that remain consistent irrespective of redshift or selection. Currently, this is not the case, as biases are not corrected for and are mostly influenced by the distribution of our training samples, which are specific to their respective survey and inherently biased. Notes. ( 1 ) all sources; ( 2 ) split by type (PSF vs. EXT); ( 3 ) split by signal-to-noise ratio (S / N) of all available bands; ( 4 ) split by availability of the i -band; ( 5 ) split by selection criteria; ( 6 ) split by the presence of a faint [(mAGN - mNeigh . ) within -3 ≤ ∆ mag < -1 for all available bands] or bright [(mAGN - mNeigh . ) within ∆ mag > -1 for all available bands] neighbor within a 5 arcsecond radius. Additionally, results derived for the CSC2 blind sample for both PICZL and C ircle Z are provided for comparison.\",\n \"pages\": [\n 17,\n 18\n ]\n },\n {\n \"title\": \"8.6.1. Selection effects\",\n \"content\": \"We investigate whether the combination of various ways of selecting AGN entering the training sample a ff ects the quality of the photo-z for specific subsamples. Using the classification outlined in Table 1, we split the validation sample based on observational criteria, creating a binary division between sources selected in optical or via X-ray. While there is only a minor difference in σ NMAD, we observe a higher outlier fraction for Xray-detected sources. While strong X-ray emission from an AGN implies strong optical and mid-IR from an AGN, which should overpower the galaxy emission in the latter two bands, complicating the process of determining accurate photo-z, local Seyfert 1 galaxies with high X-ray flux have their host remain distinctly visible. As shown in the central panel of Figure 15, however, the outlier fractions and accuracy remain consistent across the full range of X-ray fluxes, aside from small-number statistics for the wings of the distribution.\",\n \"pages\": [\n 18\n ]\n },\n {\n \"title\": \"8.6.2. Sample characteristics\",\n \"content\": \"Unlike for galaxies, where the redshift distribution n( z ) is wellestablished and can serve as a reliable prior, the AGN n( z ) is not su ffi ciently characterised. Therefore uncertainty surrounds the ML algorithm subliminally adopting a distribution similar to the n( z ) of the training sample. At very low redshifts, AGN have a low surface density, resulting in only a few rare objects that can be considered for training. This phenomenon is intensified by the fact that deep surveys, as opposed to wide-field surveys, usually bypass such bright nearby objects in search of intermediate and high redshift sources. This leads to areas of scarcity in the training sample's specz distribution, with biased predictions towards redshift values where more data points exist. To mitigate this, we nor- malize the CRPS score by (1 + z ), emphasizing the accuracy of low-redshift predictions. Additionally, at extremely low redshifts where the 4000 Å break is barely covered by the g filter, accurate redshift predictions are more di ffi cult to obtain. At very low redshift, the 6 arcsecond × 6 arcsecond cutout may be entirely filled by the galaxy, potentially misleading the algorithm into interpreting it as excess noise.\",\n \"pages\": [\n 18\n ]\n },\n {\n \"title\": \"8.6.3. Non-representative training samples\",\n \"content\": \"One method to tackle covariate shift by the imbalance of the bright-end dominated spectroscopic sample, is given by subdividing both the training and validation samples into subsets of n -dimensional feature space of distinguishable properties (Newman & Gruen 2022). Specifically, the prediction accuracy improves if the model used to generate a posterior for a blind source was trained exclusively on training sources residing within the corresponding feature space (Rosenbaum & Rubin 1984; Revsbech et al. 2017; Autenrieth et al. 2023). However, this approach might significantly reduce the training sample size per model making it more applicable to photoz codes dealing with large datasets, such as those for inactive galaxies (Newman & Gruen 2022). In this work we have demonstrated that, in contrast to just a few years ago, AGN-targeted training samples are now su ffi -ciently large to provide reliable photo-z. Overcoming inherent biases still requires either gathering more representative samples that uniformly cover the full redshift range-particularly underrepresented regions-or applying statistical corrections. Thus, the focus has shifted from simply increasing the number of spectra to strategically obtaining spectra that cover specific regions of parameter space (Masters et al. 2015). However, given the substantial gains our model has shown over existing methods, a detailed examination of how these biases a ff ect prac- tical applications is beyond the scope of this study. Looking ahead, spectroscopic follow-up campaigns such as DESI, SDSSV / BHM (Kollmeier et al., in prep), and 4MOST (De Jong et al. 2019) will further enhance our capabilities. Nevertheless, predicting photo-z for AGN in deeper Euclid and LSST datasets remains challenging due to the limited availability of spectroscopic data for faint sources. The upcoming Subaru Prime Focus Spectrograph (PFS, Tamura et al. 2016) and Multi Object Optical and Near-infrared Spectrograph for the Very Large Telescope (VLT / MOONS; Cirasuolo et al. 2020) are expected to help address this shortfall by providing much-needed spectra for faint (& obscured / red) objects.\",\n \"pages\": [\n 18,\n 19\n ]\n },\n {\n \"title\": \"8.7. Observational constraints: source crowding\",\n \"content\": \"One factor beyond our control is the local environment or conditions along the line of sight, which can significantly influence the emission characteristics of AGN. Nearby objects can add extra flux, which can a ff ect the accuracy of photometric measurements. We perform a positional cross-match between the validation sample and all LS10 sources within a 5 arcsecond radius of their optical counterparts. For each neighbour meeting this proximity criterion, we calculate apparent magnitudes and flag all sources where the brightest neighbour is no less than 1 magnitude dimmer. Table 5 shows that isolated sources exhibit better overall performance, while those with bright neighbours show drastically reduced quality compared to those with fainter neighbours. The presence of bright neighbours influences the observed flux in two ways: firstly, it complicates the derivation of sourcespecific colors due to convolved fluxes, and secondly, it a ff ects the observed spectra, potentially leading to the detection of two sets of emission lines and incorrect specz identification (Newman & Gruen 2022). A potential solution to this issue could be the implementation of segmentation maps, similar to SExtractor (Source Extractor, Bertin, E. & Arnouts, S. 1996) at image level, as LS10 currently only masks neighbours during their model flux fitting procedure. A persistent physical issue that cannot be entirely mitigated is blending. Unlike bright neighbours, which can be masked in principle, blending a ff ects both observed photometry and spectroscopy. While particularly faint neighbours do not substantially a ff ect the predictions negatively, sources identified as blends should be excluded from samples where accurate photoz estimates are required, until we can partially mask sources at the pixel level. The Rubin Observatory expects overlapping (inactive) galaxies to contribute at least 1% of the total flux within their pixels (Sanchez et al. 2021; Newman & Gruen 2022). Blends are also expected to occur in the spectroscopic sample, increasing systematic uncertainty and subsequently the fraction of outliers by up to 5% (Masters et al. 2019). This may be mitigated in future by higher-quality imaging from spacebased data from missions such as Euclid or better tools for deblending (e.g., SCARLET / blendz ; Melchior et al. 2018; Jones & Heavens 2018). Alternatively, data augmentation or standard computer vision techniques could be implemented, such as artificially introducing blending e ff ects in the training sample. However, this approach is not trivial, as LS10 currently lacks a corresponding blending flag.\",\n \"pages\": [\n 19\n ]\n },\n {\n \"title\": \"8.8. Variability\",\n \"content\": \"AGN are inherently variable sources, meaning that their observed flux can change significantly across di ff erent epochs and wavelengths, especially when observations are separated by substantial time gaps. This extends to images that are created by stacking multiple observations taken over extended periods (as in LS10), where the resulting flux represents an average value. Such variability complicates the accurate prediction of AGN redshifts (Simm et al. 2015). However, future time-resolved imaging from LSST will allow us to better account for these variations and potentially use the correlation between AGN variability and their physical properties as a feature to improve photo-z predictions.\",\n \"pages\": [\n 19\n ]\n },\n {\n \"title\": \"8.9. Scalability in image-based photo-z\",\n \"content\": \"Open issues remain regarding the feasibility of handling the vast data volumes and computational requirements associated with photo-z estimated from images. Exemplified by the calculation of new estimates for XMM-SERVS concerning storage, processing time, and computational resources, we give an overview in Appendix G. A crucial future direction involves exploring ways to utilize imaging data e ffi ciently without necessitating extensive local downloads and computations for all image-based analyses of such data sets, potentially leveraging online platforms for realtime analysis, bringing the compute to the data (Zhang & Zhao 2015).\",\n \"pages\": [\n 19\n ]\n },\n {\n \"title\": \"9. Summary and outlook\",\n \"content\": \"This work has been driven by the goal of determining reliable photo-z for X-ray detected AGN in wide-field surveys, such as eROSITA (Merloni et al. 2024). For that reason, we have concentrated on utilizing data from LS10, which provides su ffi ciently homogeneous coverage and depth in 4 optical bands, enriched by the flux measured on NEOWISE7 data for all identified sources (see Sect. 3). LS10 overlaps almost entirely with the eROSITA footprint, simplifying the cross calibration of data, a processing step typically necessary when merging di ff erent surveys. We introduce PICZL, a CNN-based machine learning model designed primarily for AGN redshift estimation, but which holds the potential to reliably measure photoz s for a broad range of extragalactic sources, provided an appropriately constructed training sample is available. In this study, the training sample comprises both X-ray and optically selected AGN. Across our validation set, PICZL demonstrates consistently robust performance, with comparable accuracies and lower outlier fractions particularly for PSF sources, as opposed to the results obtained by Salvato et al. (2022) for similar objects using SED-fitting (see Figure 7 and Table 5). The results show significant improvements for both point-like and extended sources, indicating that the model's performance is primarily driven by the depth and hence photometric error of the training data. Additionally, when tested on a blind sample of X-ray-selected AGN, PICZL maintained comparable results with respect to σ NMAD and η . Notably, on this test set it outperformed (C irclez ; Saxena et al. 2024) by achieving a 20% improvement in accuracy and a 30% reduction in the fraction of outliers (refer to Sect. 6). We also applied PICZL to estimate photo-z for the approximately 30% of XMM-SERVS sources (Chen et al. 2018; Ni et al. 2021), detected in LS10 (see Table 4), demonstrating comparable variance and a substantially improved outlier fraction up to a limiting magnitude of 23.5 AB, using much fewer bands and significantly less sensitive imaging. However, it is important to note that the spectroscopic sample at this faint limit is small and likely biased toward sources with higher spectroscopic success rates. Consequently, the most reliable photo-z results are expected at brighter magnitudes (as discussed in Saxena et al. 2024). In response to these findings, we are releasing a new catalog of photo-z, complete with 1 and 3 σ error margins for the XMM-SERVS fields (see Sect. 7). PICZL will be used in the next generation of photo-z for sources detected by eROSITA, possibly switching from LS10 to Euclid and, most importantly, LSST, as they are poised to deliver more homogeneous and deeper data with broader wavelength coverage. In particular the availability of NIR data at image level, will improve our ability to determine accurate redshift for faint and high-z sources. Our study has shown that the performance of PICZL, when relying predominantly on images, is already robust (see Table 3). Crucially, we have shown that wellcalibrated images alone can su ffi ce for accurate photo-z estimation, eliminating the need for catalog creation, which is often based on predefined models. This suggests that future surveys could reduce their dependence on catalog-based data for photo-z computation (refer to Sect. 6). Although this study has focused on calculating photo-z for AGN, the approach can be generalised and adapted to other source types, e.g., normal galaxies, with an appropriately constructed training sample (Götzenberger et al. in prep.). Subsequently our findings point to a bright future for all-sky surveys, also thanks to the continue expansion of training sample sizes, supported by initiatives like SDSS-V / BHM, 4MOST and VLT / MOONS, which will enhance the reliability and completeness of photoz estimates (refer to Sect. 8). Looking forward, the next step in advancing photo-z estimation for AGN lies in exploring more sophisticated machine learning architectures beyond CNNs. For example, transformers with shared latent space embeddings o ff er a promising avenue. These models have shown success in integrating information from various data sources, such as images and entire spectra, potentially reducing uncertainties associated with traditional specz pipelines by leveraging multi-modal data fusion (DonosoOliva et al. 2023; Parker et al. 2024). Additionally, incorporating other informative features like X-ray flux, when available, holds promise. Integrating these diverse data sources within a unified framework has the potential to refine redshift estimates even further.\",\n \"pages\": [\n 19,\n 20\n ]\n },\n {\n \"title\": \"10. Data availability\",\n \"content\": \"With this paper, we present a new catalog of photo-z derived using PICZL for the ∼ 30% of sources within the XMM-SERVS (ELAIS-S1, W-CDF-S, and LSS) X-ray source catalogs (Chen et al. 2018; Ni et al. 2021) that are detected in the LS10 survey. Hence, it includes updated photo-z for sources with catastrophic failures in the original works. A detailed description of the catalog columns can be found in Appendix F. The full catalog is only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http:// cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/ . Acknowledgements. WRand MS are grateful for the constant support of Dustin Lang in handling Legacy Survey-related issues. Part of this work was supported by the German Deutsche Forschungsgemeinschaft, DFG , under Germany's Excellence Strategy - EXC 2094 - 390783311. We gratefully acknowledge funding from FONDECYT Regular - 1231718 (RJA), 1230345 (CR), and 1241005 (FEB), CATA-BASAL - FB210003 (RJA, CR, FEB), and ANID - Millennium Science Initiative - AIM23-0001 (FEB). JA acknowledges support from a UKRI Future Leaders Fellowship (grant code: MR / T020989 / 1) This work is based on data from eROSITA, the soft X-ray instrument aboard SRG, a joint RussianGerman science mission supported by the Russian Space Agency (Roskosmos), in the interests of the Russian Academy of Sciences represented by its Space Research Institute (IKI), and the Deutsches Zentrum für Luft- und Raumfahrt (DLR). The SRG spacecraft was built by Lavochkin Association (NPOL) and its subcontractors and is operated by NPOL with support from the Max Planck Institute for Extraterrestrial Physics (MPE). The development and construction of the eROSITA X-ray instrument were led by MPE, with contributions from the Dr. Karl Remeis Observatory Bamberg & ECAP (FAU Erlangen-Nuernberg), the University of Hamburg Observatory, the Leibniz Institute for Astrophysics Potsdam (AIP), and the Institute for Astronomy and Astrophysics of the University of Tübingen, with the support of DLR and the Max Planck Society. The Argelander Institute for Astronomy of the University of Bonn and the Ludwig Maximilians Universität Munich also participated in the science preparation for eROSITA. The Legacy Surveys consist of three individual and complementary projects: the Dark Energy Camera Legacy Survey (DECaLS; Proposal ID 2014B-0404; PIs: David Schlegel and Arjun Dey), the Beijing-Arizona Sky Survey (BASS; NOAO Prop. ID 2015A-0801; PIs: Zhou Xu and Xiaohui Fan), and the Mayall z-band Legacy Survey (MzLS; Prop. ID 2016A-0453; PI: Arjun Dey). DECaLS, BASS, and MzLS together include data obtained, respectively, at the Blanco telescope, Cerro Tololo Inter-American Observatory, NSF's NOIRLab; the Bok telescope, Steward Observatory, University of Arizona; and the Mayall telescope, Kitt Peak National Observatory, NOIRLab. Pipeline processing and analyses of the data were supported by NOIRLab and the Lawrence Berkeley National Laboratory (LBNL). The Legacy Surveys project is honored to be permitted to conduct astronomical research on Iolkam Du'ag (Kitt Peak), a mountain with particular significance to the Tohono O'odham Nation. NOIRLab is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. LBNL is managed by the Regents of the University of California under contract to the U.S. Department of Energy. This project used data obtained with the Dark Energy Camera (DECam), which was constructed by the Dark Energy Survey (DES) collaboration. Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A & M University, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho de Amparo, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Cientifico e Tecnologico and the Ministerio da Ciencia, Tecnologia e Inovacao, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey. The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Energeticas, Medioambientales y Tecnologicas-Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgenossische Technische Hochschule (ETH) Zurich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ciencies de l'Espai (IEEC / CSIC), the Institut de Fisica d'Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig Maximilians Universitat Munchen and the associated Excellence Cluster Universe, the University of Michigan, NSF's NOIRLab, the University of Nottingham, the Ohio State University, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, and Texas A&M University. BASS is a key project of the Telescope Access Program (TAP), which has been funded by the National Astronomical Observatories of China, the Chinese Academy of Sciences (the Strategic Priority Research Program 'The Emergence of Cosmological Structures' Grant # XDB09000000), and the Special Fund for Astronomy from the Ministry of Finance. The BASS is also supported by the External Cooperation Program of Chinese Academy of Sciences (Grant # 114A11KYSB20160057), and Chinese National Natural Science Foundation (Grant # 12120101003, # 11433005). The Legacy Survey team uses data products from the Near-Earth Object Wide-field Infrared Survey Explorer (NEOWISE), a project of the Jet Propulsion Laboratory / California Institute of Technology. NEOWISE is funded by the National Aeronautics and Space Administration. The Legacy Surveys imaging of the DESI footprint is supported by the Director, O ffi ce of Science, O ffi ce of High Energy Physics of the U.S. Department of Energy under Contract No. DE-AC02-05CH1123, by the National Energy Research Scientific Computing Center, a DOE O ffi ce of Science User Facility under the same contract, and by the U.S. National Science Foundation, Division of Astronomical Sciences under Contract No. AST-0950945 to NOAO. Funding for the Sloan Digital Sky Survey V has been provided by the Alfred P. Sloan Foundation, the Heising-Simons Foundation, the National Science Foundation, and the Participating Institutions. SDSS acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. SDSS telescopes are located at Apache Point Observatory, funded by the Astrophysical Research Consortium and operated by New Mexico State University, and at Las Campanas Observatory, operated by the Carnegie Institution for Science. The SDSS web site is www.sdss.org . SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration, including Caltech, The Carnegie Institution for Science, Chilean National Time Allocation Committee (CNTAC) ratified researchers, The Flatiron Institute, the Gotham Participation Group, Harvard University, Heidelberg University, The Johns Hopkins University, L'Ecole polytechnique fédérale de Lausanne (EPFL), Leibniz-Institut für Astrophysik Potsdam (AIP), Max-PlanckInstitut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Extraterrestrische Physik (MPE), Nanjing University, National Astronomical Observatories of China (NAOC), New Mexico State University, The Ohio State University, Pennsylvania State University, Smithsonian Astrophysical Observatory, Space Telescope Science Institute (STScI), the Stellar Astrophysics Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Illinois at Urbana-Champaign, University of Toronto, University of Utah, University of Virginia, Yale University, and Yunnan University.\",\n \"pages\": [\n 20,\n 21\n ]\n },\n {\n \"title\": \"References\",\n \"content\": \"Abdalla, F. B., Banerji, M., Lahav, O., & Rashkov, V. 2011, MNRAS, 417, 1891 Aird, J., Nandra, K., Laird, E. S., et al. 2010, MNRAS, 401, 2531-2551 Alexander, D. M., Davis, T. M., Chaussidon, E., et al. 2023, ApJ, 165, 124 Almeida, A., Anderson, S. F., Argudo-Fernández, M., et al. 2023, ApJ Supple- ment Series, 267, 44 Almosallam, I. A., Jarvis, M. J., & Roberts, S. J. 2016, MNRAS, 462, 726 Ananna, T. T., Salvato, M., LaMassa, S., et al. 2017, ApJ, 850, 66 Baum, W. 1957, ApJ, 62, 6 Bell, E. F., Wolf, C., Meisenheimer, K., et al. 2004, ApJ, 608, 752-767 Benitez, N. 2000, ApJ, 536, 571 Bertin, E. & Arnouts, S. 1996, Astron. Astrophys. Suppl. Ser., 117, 393 Bettoni, D., Falomo, R., Kotilainen, J. K., Karhunen, K., & Uslenghi, M. 2015, MNRAS, 454, 4103 Blanton, M. R., Bershady, M. A., Abolfathi, B., et al. 2017, ApJ, 154, 28 Boller, T., Freyberg, M. J., Trümper, J., et al. 2016, A&A, 588, A103 Bolton, A. S., Schlegel, D. J., Aubourg, E., et al. 2012, ApJ, 144, 144 Bordoloi, R., Lilly, S. J., & Amara, A. 2010, MNRAS, no Boutsia, K., Grazian, A., Calderone, G., et al. 2020, ApJ Supplement Series, 250, 26 Breiman, L. 2001, Machine Learning, 545, 5 Brescia, M., Cavuoti, S., Longo, G., & Stefano, V. D. 2014, A&A, 568, A126 Comparat, J., Merloni, A., Dwelly, T., et al. 2020, A&A, 636, A97 Connolly, A. J., Csabai, I., Szalay, A. S., et al. 1995, ApJ, 110, 2655 Croom, S. M., Smith, R. J., Boyle, B. J., et al. 2004, MNRAS, 349, 1397-1418 Dahlen, T., Mobasher, B., Faber, S. M., et al. 2013, ApJ, 775, 93 Dawid, A. P. 1984, Journal of the Royal Statistical Society. Series A (General), Dey, B., Newman, J. A., Andrews, B. H., et al. 2022b, Re-calibrating Photometric Redshift Probability Distributions Using Feature-space Regression D'Isanto, A. & Polsterer, K. L. 2018, A&A, 609, A111 Eriksen, M., Alarcon, A., Cabayol, L., et al. 2020, MNRAS, 497, 4565-4579 Heckman, T. M. & Best, P. N. 2014, Annual Review of A&A, 52, 589 Lecun, Y., Bottou, L., Bengio, Y., & Ha ff ner, P. 1998b, Proceedings of the IEEE, 86, 2278 Li, C., Zhang, Y., Cui, C., et al. 2021, MNRAS, 509, 2289 Li, C., Zhang, Y., Cui, C., et al. 2022, MNRAS, 518, 513 Lin, Q., Fouchez, D., Pasquet, J., et al. 2022, A&A, 662, A36 Lines, N. E. P., Roset, J. F.-Q., & Scaife, A. M. M. 2024, E(2)-Equivariant Features in Machine Learning for Morphological Classification of Radio Galaxies Liu, Z., Mao, H., Wu, C.-Y., et al. 2022, in Proceedings of the IEEE / CVF Con- ference on Computer Vision and Pattern Recognition (CVPR), 11976-11986 Luken, K. J., Norris, R. P., & Park, L. A. F. 2019, Publications of the Astronom- ical Society of the Pacific, 131, 108003 Luo, B., Brandt, W. N., Xue, Y. Q., et al. 2010, ApJ Supplement Series, 187, 560 Lyke, B. W., Higley, A. N., McLane, J. N., et al. 2020, ApJ Supplement Series, 250, 8 Ma, L., Lu, Z., Shang, L., & Li, H. 2015, Multimodal Convolutional Neural Networks for Matching Image and Sentence Madau, P. & Dickinson, M. 2014, Annual Review of A&A, 52, 415-486 Mainzer, A., Bauer, J., Grav, T., et al. 2011, ApJ, 731, 53 Masters, D., Capak, P., Stern, D., et al. 2015, ApJ, 813, 53 Mauduit, J.-C., Lacy, M., Farrah, D., et al. 2012, Publications of the Astronomical Society of the Pacific, 124, 714-736 Meisner, A. M., Lang, D., & Schlegel, D. J. 2017, ApJ, 153, 38 Merloni, A., Predehl, P., Becker, W., et al. 2012, eROSITA Science Book: Mapping the Structure of the Energetic Universe Meshcheryakov, A., Glazkova, V., Gerasimov, S., & Mashechkin, I. 2018, Astronomy Letters, 44, 735 Mountrichas, G., Corral, A., Masoura, V. A., et al. 2017, A&A, 608, A39 Newman, J. A. & Gruen, D. 2022, Annual Review of A&A, 60 Ni, Q., Brandt, W. N., Chen, C.-T., et al. 2021, ApJ Supplement Series, 256, 21 Nishizawa, A. J., Hsieh, B.-C., Tanaka, M., & Takata, T. 2020, Photometric Redshifts for the Hyper Suprime-Cam Subaru Strategic Program Data Release 2 O'Shea, K. & Nash, R. 2015, An Introduction to Convolutional Neural Networks Padovani, P., Alexander, D. M., Assef, R. J., et al. 2017, The A&A Review, 25 Parker, L., Lanusse, F., Golkar, S., et al. 2024, MNRAS, 531, 4990-5011 Pierce, C. M., Lotz, J. M., Primack, J. R., et al. 2010, MNRAS Povi'c , M., Sánchez-Portal, M., García, A. M. P., et al. 2012, A&A, 541, A118 Predehl, P., Andritschke, R., Arefiev, V., et al. 2021, A&A, 647, A1 Revsbech, E. A., Trotta, R., & van Dyk, D. A. 2017, MNRAS, 473, 3969-3986 Rosenbaum, P. & Rubin, D. 1984, Journal of the American Statistical Associa- tion, 79 Saxena, A., Salvato, M., Roster, W., et al. 2024, CircleZ: Reliable Photometric redshifts for AGN computed using only photometry from Legacy Survey Imaging for DESI Scaramella, R., Amiaux, J., Mellier, Y., et al. 2022, A&A, 662, A112 Schmidt, S. J., Malz, A. I., Soo, J. Y. H., et al. 2020, MNRAS, 499, 1587 Schuldt, S., Suyu, S. H., Cañ ameras, R., et al. 2021, A&A, 651, A55 Simm, T., Saglia, R., Salvato, M., et al. 2015, A&A, 584, A106 Soo, J. Y. H., Moraes, B., Joachimi, B., et al. 2017, MNRAS, 475, 3613-3632 Stabenau, H. F., Connolly, A., & Jain, B. 2008, MNRAS, 387, 1215 Webb, N. A., Coriat, M., Traulsen, I., et al. 2020, A&A, 641, A136 Weiler, M. & Cesa, G. 2021, General E (2)-Equivariant Steerable CNNs Wilson, D., Nayyeri, H., Cooray, A., & Häußler, B. 2020, ApJ, 888, 83 Wu, Q. & Shen, Y. 2022, ApJ Supplement Series, 263, 42 Yang, J., Fan, X., Gupta, A., et al. 2023, ApJ Supplement Series, 269, 27 York, D. G., Adelman, J., John E. Anderson, J., et al. 2000, ApJ, 120, 1579 02.09.24, 21 : 39\",\n \"pages\": [\n 21,\n 22,\n 23\n ]\n },\n {\n \"title\": \"Appendix G: PICZL run time\",\n \"content\": \"The computational demands for training and deploying PICZL depend mostly on the hardware configuration and dataset size. For our experiments, we leveraged a set of two Tesla V100PCIE graphics processing units (GPUs) each equipped with 32 GB of ready access memory (RAM), accompanied by a 48 core multi-thread CPU to accelerate the computational workload. This parallel processing capability significantly expedited the model training process, compared to running solely on a central CPU, allowing for faster convergence. Training a single model (refer to Figure A.1) on a dataset of 32 391 sources, corresponding to the 80:20 train-test split (refer to Table 1), each with 108 features (56 of which are images), takes approximately 25 minutes for 600 epochs. The use of an ensemble further scales the processing time by the number of models trained, with ensemble optimization depending on user preferences, making it di ffi -cult to provide a fixed time estimate. After finalizing the model, computing photo-z for e.g. 1393 sources in the XMM-SERVS W-CDF-S field, as displayed via Table 4, takes roughly 17 seconds. The storage requirements for the data used in this sample corresponds to approximately 250 MB (refer to Table 2). Looking ahead, upcoming surveys such as Euclid and LSST will produce higher-resolution images, demanding increased storage and computational resources due to larger data volumes and pixel counts. If the input dimensions were to increase, for instance, from 23x23 pixels to 64x64 pixels, the computational burden would rise significantly. While our current architecture of 32 GB RAM per GPU is e ffi cient for the current input sizes, processing larger cutouts may necessitate either more GPUs or GPUs with higher memory capacity to maintain feasible training times and performance. In scenarios where larger input sizes are anticipated, a thoughtful approach to model architecture and resource allocation will be crucial. Therefore, adapting to the capabilities of the hardware will be key to successfully utilizing the methods in the context of future data from LSST and Euclid, particularly given the much larger sample sizes that we expect from these surveys. Finally, performance will also improve by having PICZL revised by an expert software developer.\",\n \"pages\": [\n 24\n ]\n }\n]"}}},{"rowIdx":4972,"cells":{"bibcode":{"kind":"string","value":"2024arXiv241108310G"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/2411.08310.pdf"},"content":{"kind":"string","value":"\nR esearch in\nA\nstronomy and\nA\nstrophysics\nMeasurements of the solar coronal magnetic field based on coronal seismology with propagating Alfv'enic waves: forward modeling\nYuhang Gao 1 , 2 , Hui Tian 1 , Tom Van Doorsselaere 2 , Zihao Yang 1 , 3 , Mingzhe Guo 2 , 4 and Konstantinos Karampelas 2\n\n1 School of Earth and Space Sciences, Peking University, Beijing, 100871, People's Republic of China; huitian@pku.edu.cn\n2 Centre for mathematical Plasma Astrophysics, Department of Mathematics, KU Leuven, Celestijnenlaan 200B bus 2400, B-3001 Leuven, Belgium\n3 High Altitude Observatory, National Center for Atmospheric Research, Boulder, CO 80307, USA\n4 Shandong Provincial Key Laboratory of Optical Astronomy and Solar-Terrestrial Environment, Institute of Space Sciences, Shandong University, Weihai 264209, China\n\nReceived 20xx month day; accepted 20xx month day\nAbstract Recent observations have demonstrated the capability of mapping the solar coronal magnetic field using the technique of coronal seismology based on the ubiquitous propagating Alfv'enic/kink waves through imaging spectroscopy. We established a magnetohydrodynamic (MHD) model of a gravitationally stratified open magnetic flux tube, exciting kink waves propagating upwards along the tube. Forward modeling was performed to synthesize the Fe XIII 1074.7 and 1079.8 nm spectral line profiles, which were then used to determine the wave phase speed, plasma density, and magnetic field with seismology method. A comparison between the seismologically inferred results and the corresponding input values verifies the reliability of the seismology method. In addition, we also identified some factors that could lead to errors during magnetic field measurements. Our results may serve as a valuable reference for current and future coronal magnetic field measurements based on observations of propagating kink waves.\nKey words: Sun: corona - Sun: magnetic fields - magnetohydrodynamics\n1 INTRODUCTION\nThe magnetic field plays a crucial role in various physical processes in the solar and stellar coronae. The dissipation of magnetic energy is believed to drive solar eruptive events (e.g., flares and coronal mass ejections) and cause heating of the corona. While the magnetic field in the lower solar atmosphere can be reliably measured through spectro-polarimetric observations, direct measurements of the coronal magnetic field have remained challenging for decades. Several methods, including the spectro-polarimetry of coronal infrared lines (Lin et al. 2000, 2004; Schad et al. 2024), coronal radio observations (e.g., Fleishman et al. 2020; Chen et al. 2020; Tan 2022), and magnetic-field-induced transitions of extreme ultraviolet emission lines (e.g., Li et al. 2015, 2016; Landi et al. 2020; Chen et al. 2021, 2023), have been proposed and attempts of measurements have been made. However, these approaches all face limitations and none of them could be used for routine measurements of the global coronal magnetic field.\nAnother technique that could be used to measure the coronal magnetic field is coronal seismology. This technique combines the magneto-hydrodynamic (MHD) wave theory with observed wave parameters (e.g., period, amplitude, propagation speed, damping time) to diagnose various physical properties,\nparticularly the magnetic field. Different wave phenomena in the corona have been used for coronal seismology, including standing kink waves in magnetic loops (e.g. Nakariakov & Ofman 2001; Aschwanden et al. 2002; Van Doorsselaere et al. 2007; Tian et al. 2012; Zhang et al. 2022; Gao et al. 2022, 2024a; Zhong et al. 2023; Li & Long 2023, to name but a few), propagating slow magneto-acoustic waves (e.g., Jess et al. 2016), sausage waves (Chen et al. 2015; Guo et al. 2016; see Li et al. 2020 for a review), propagating kink waves in streamers (Chen et al. 2011; Guo et al. 2022), and torsional oscillations in solar surges (Kohutova et al. 2020). However, these studies provided only one-dimensional (1D) distributions or single values of the magnetic field in specific coronal structures, such as oscillating loops or streamers. To create global 2D magnetic field maps, we need to utilize ubiquitous and continuous wave phenomena. The pervasive propagating disturbances in Dopplergrams (Tomczyk et al. 2007; Tomczyk & McIntosh 2009; Liu et al. 2015; Morton et al. 2015, 2019) observed by the Coronal Multi-channel Polarimeter (CoMP; Tomczyk et al. 2008) are ideal for this purpose. These propagating disturbances are interpreted as kink or Alfv'enic waves (Van Doorsselaere et al. 2008), and their propagation speeds are naturally linked to the local magnetic field. Based on these CoMP observations, Yang et al. (2020b,a) have successfully measured the global distribution of the coronal magnetic field for the first time. Following these successful attempts, a routine (continuous) measurement of the global coronal magnetic field based on similar observations from the Upgraded CoMP (UCoMP; Landi et al. 2016) has been recently achieved, which allows for the construction of coronal synoptic magnetograms (Carrington maps) (Yang et al. 2024).\nThe CoMP and UCoMP instruments can conduct spectroscopic observations of the Fe XIII lines at 1074.7 and 1079.8 nm in the coronal region above the solar limb. From the Dopplergrams of Fe XIII 1074.7 nm, propagating kink waves can be identified throughout the corona. The propagating or phase speed c k of these waves (also named the kink speed) could be obtained by constructing a timedistance map of Doppler velocity, while the coronal density can be inferred from the observed Fe XIII 1079.8-nm/1074.7-nm intensity ratio.\nFor kink waves, we have\nc 2 k = B 2 i + B 2 e µ 0 ( ρ i + ρ e ) , (1)\nwhere µ 0 is the magnetic permeability, B and ρ are the magnetic field and mass density, respectively. The subscripts i and e refer to parameters inside and outside the magnetic flux tubes (the waveguides). Since CoMP and UCoMP likely cannot resolve individual flux tubes, we can only work with an average density ⟨ ρ ⟩ . Furthermore, in the lowβ coronal environment, the internal and external magnetic fields are often assumed to be approximately equal (see e.g., Tomczyk & McIntosh 2009; Morton et al. 2015; Zhong et al. 2023). This leads to the simplified expression for the kink speed:\nc 2 k = B 2 µ 0 ⟨ ρ ⟩ , (2)\nwhich is widely used in estimations of coronal magnetic fields (Long et al. 2017; Yang et al. 2020a,b; Yang et al. 2024).\nGiven the potential of these measurements to provide routine coronal magnetograms on a daily basis, which could play a crucial role in future solar physics research, it is essential to thoroughly assess the reliability and robustness of the methodology used in Yang et al. (2020a,b) and Yang et al. (2024). Magyar & Van Doorsselaere (2018) performed 3D MHD simulations of propagating kink waves under various conditions and conducted forward modeling to evaluate the reliability of this method in deriving the magnetic field strength. Their findings indicated that the magnetic field strengths inferred through seismology closely match the input values, typically with an error less than ∼ 20%. However, there is a limitation in their simulation. They utilized a non-stratified setup that excluded the effects of gravity, resulting in uniform initial density and propagation speed in the vertical direction.\nThe gravitational stratification can play a significant role as it causes the Alfv'en speed and kink speed c k to vary with height. This variation can affect the wave tracking method which typically relies on a linear fit of velocity signals (e.g., see Figure 6 in Tomczyk & McIntosh 2009). Therefore, it\nis important to assess how the gravitational stratification affects the seismological results and estimate the possible error range. So we conducted 3D MHD simulations of propagating kink waves in stratified coronal open flux tubes. Following Magyar & Van Doorsselaere (2018), we employed forward modeling to compare the seismologically derived results with the actual values of physical parameters from our simulation. This paper is organized as follows: Section 2 describes our simulation setup and methodology, Section 3 presents the simulation and forward-modeling results, along with detailed comparisons between seismology results and input values, and Section 4 provides a discussion and summary of our findings.\n2 METHOD\nThe model used in this study is a gravitationally stratified, open magnetic flux tube with a radius of 1 Mm, similar to that in Gao et al. (2024b) (hereafter referred to as Paper I). The main difference is that the initial magnetic field in this study is set at around 4 G, instead of 10 G, to better match the seismologically inferred results in Yang et al. (2020b). As in Paper I, the magnetic field is oriented in the z direction and remains nearly uniform across the simulation domain, with only small spatial gradients to maintain total pressure balance. As a result, the Alfv'en and kink speeds increase with height as the density decreases due to stratification. A relaxation process of 2400 s was conducted to achieve a quasimagnetohydrostatic state, as shown in Figure 1. The figure shows that the post-relaxation magnetic field has some spatial variation but mostly within 0.7 G.\nThe kink wave driver is also similar to that in Paper I, with a velocity amplitude of 8 km s -1 and a period of 300 s. The primary velocity perturbation is along the x direction.\nWe ran the 3D MHD simulation in Cartesian coordinates with the PLUTO code (Mignone et al. 2007). The simulation domain spans [-4, 4] Mm × [-4, 4] Mm × [0, 150] Mm, with a uniform grid of 128 × 128 × 1024 cells, providing spatial resolutions of 62.5 km in the horizontal ( x and y ) directions and 146.5 km in the vertical ( z ) direction. We chose a second-order parabolic spatial scheme and a Roe Riemann solver. The boundary conditions were set to be outflow, except for the lower boundary, where the kink wave driver was introduced by adjusting v x and v y , while other parameters were fixed. As in Paper I, the upper 50 Mm ( z > 100 Mm) was set as a velocity absorption region (VAR) to minimize numerical reflections from the upper boundary (see also Pelouze et al. 2023; Guo et al. 2023; Gao et al. 2023). For the subsequent analysis, we only considered the region below z = 100 Mmwhich can give us physical results.\nOnce the kink wave driver was applied, the propagating waves were rapidly excited, with their properties thoroughly analyzed in Paper I. Here, we focus on assessing the reliability of the seismology method described in Section 1 using the simulation outputs. To do so, we performed forward calculations to synthesize spectroscopic observables of CoMP and UCoMP, namely, the Fe XIII 1074.7 and 1079.8 nm spectral line profiles. Specifically, we used synthesized observation of the 1074.7 nm line to perform the wave tracking and determine the propagation speed, while synthesized observation of the 1079.8 nm line was only employed for the density diagnostic using the intensity ratio method (Yang et al. 2020a,b).\nWe applied the FoMo code (Van Doorsselaere et al. 2016) to synthesize the spectral profiles at all pixels in the yz plane. The photo-excitation (see Young et al. 2003; Yang et al. 2020a) was not considered for simplification, as their effects on the spectral lines are not significant at the lower corona. The line of sight (LOS) was chosen as the x axis. In this way, we can reconstruct 2D maps (in the yz plane) of the intensity and Doppler velocity by fitting a single Gaussian to each spectral profile. The resulting maps consist of 128 pixels in the y direction from -4 Mm to 4 Mm, and 500 pixels in the z direction from 0 Mmto 100 Mm. Since the primary velocity perturbation is along the x direction (or in the LOS plane), the Doppler velocity maps capture the wave propagation signals, while the intensity maps do not show any transverse displacement.\n\n\n
Fig. 1: Model of the open magnetic flux tube. (A) Variation at the x = 0 plane after relaxation. (B)-(C) Horizontal profiles of density and magnetic field along the x axis at three different heights (indicated by different line styles). (D) Vertical profile of density demonstrating gravitational stratification, with solid and dashed lines indicating density inside and outside the flux tube, respectively.
\n\nz (Mm)\n3 RESULTS\n3.1 Before Degrading\nWe first generated time-distance (TD) maps of the Doppler velocity along the z direction, which can be produced at each y position. In Figure 2(A), the TD map along the flux tube's axis (i.e., y = 0 ) is presented, clearly illustrating the propagation of Doppler velocity disturbances. The increasing slope reflects the growing propagation speed with height. However, after 400 s, unusual patterns appear due to wave reflections. Despite efforts to suppress numerical reflections from the upper boundary at z = 150 Mm using a VAR (see Section 2), it is still difficult to eliminate all reflections. Some reflections may come from the layer of z = 100 Mm, which is the lower boundary of the VAR; while others could be attributed to the vertical inhomogeneities of density and phase speed. As noted in previous studies, even smooth phase speed gradients can cause partial wave reflection (e.g., Verdini & Velli 2007; Hahn et al. 2018; Pascoe et al. 2022; Bose et al. 2024). In fact, real CoMP observations of coronal kink waves often reveal both upward and downward propagating components (Tomczyk & McIntosh 2009; Morton et al. 2015). However, when calculating the propagation speed, these reflected waves can introduce significant errors. To reduce this, we applied the Fourier filtering method to separate the upward and downward propagating wave components (see e.g., Tomczyk & McIntosh 2009; Threlfall et al. 2013; Liu et al.\n\n\n
Fig. 2: Time distance maps of the Doppler velocity along the z axis at y = 0 . (A) corresponds to the original forward modeling output, while (B) corresponds to the upward propagating component obtained from Fourier filtering.
\n\n\n\n
Fig. 3: Seismology results for the case with original resolution (before degrading). (A) Spatial distribution of the phase speed. (B) Spatial distribution of the magnetic field ( B seis). (C) Spatial distribution of the relative error of B seis compared with the input magnetic field B los. Blue and red contours correspond to the relative error of -30% and 30%, respectively.
\n\n2014; Tiwari et al. 2019; Yang et al. 2020b). Figure 2(B) shows the TD map for the upward-propagating component, which is used to determine the phase speed.\nBy applying the wave-tracking method to all pixels (for more details, see e.g., Yang et al. 2020b), we obtained a 2D phase speed distribution, as shown in Figure 3(A). For most pixels, the phase speed falls in the range of 200-800 km s -1 (see also Figure 4(B)), with a general trend of increasing with altitude, though some fluctuations are present (see relevant discussions in Section 4).\n\n\n
Fig. 4: (A) 2D histogram comparing the density derived from the Fe XIII intensity ratio and the input density ρ los. (B) 2D histogram comparing the propagation speed obtained from wave tracking and the Alfv'en speed in the model. (C)-(D) Histograms of B seis and the relative error. The Gaussian fit results are overplotted, with the mean value ( µ ) and standard deviation ( σ ) indicated at the top.
\n\nNext, we derived the density by calculating the intensity ratio of the synthesized 1074.7 nm and 1079.8 nm intensity maps. The theoretical relationship between intensity ratio and density can be obtained from the CHIANTI database (Dere et al. 2023). The obtained density values were compared with the input values, as shown in Figure 4(A). The input density corresponds to the emissivity-weighted density along the LOS (Yang et al. 2024):\nρ los = ∫ x ρ ( x ) ϵ ( x )d x ∫ x ϵ ( x )d x , (3)\nwhere ϵ is the Fe XIII 1074.7 nm line emissivity at the corresponding pixel, calculated using the IDL routine emiss calc.pro from the CHIANTI software package. The comparison shows that the density\n\n\n\n\n\n\n\n\n
Fig. 5: Phase speed, density, and magnetic field as a function of height ( z ). In panel (A), the red lines correspond to input values, including the internal Alfv'en speed (dot-dashed line), the external Alfv'en speed (dashed line), and the kink speed (solid line). The solid black line represents the wave propagation speed inferred from the wave tracking. In panel (B), the solid black line corresponds to the density derived from the Fe XIII intensity ratio ( ρ Fe XIII), and the dotted black line corresponds to the input value ( ρ los). The red lines represent the density in the model (similar to Figure 1(D)), including the internal density ρ i (dashed line), the external density ρ e (dotted line), and the average density ⟨ ρ ⟩ = ( ρ i + ρ e ) / 2 (solid line). In panel (C), the solid black line depicts the magnetic field inferred from coronal seismology ( B seis). The dashed and dotted red lines correspond to the internal and external magnetic fields ( B i and B e) in the model, respectively. The solid red line represents the input magnetic field ( B los).
\n\nderived from the intensity ratio is reliable for most pixels, though about 20% pixels show an overestimation ( ≳ 15%). However, this overestimation would not lead to large errors in the seismologically inferred magnetic field, as only the square root of density is needed during the calculation.\nWith the derived phase speed and density, we calculated the magnetic field B seis. Here we chose to treat the phase speed as the local Alfv'en speed (i.e., c ph = B seis / √ µ 0 ρ ), rather than employ Equation(1). Because in our magnetic field configuration, Equation (1) can only provide the phase speed as a function of z ; however, here with a high spatial resolution, we are also interested in the horizontal distribution of phase speed and magnetic field. We note that such a choice may lead to some errors, especially within the flux tube region, which will be further discussed in Section 4. In Figure 4(B), we compared the Alfv'en speed in the model and the wave propagation speed obtained from wave tracking. Despite some noticeable discrepancies, the overall correspondence between the two is reasonably good.\nThe calculated magnetic field is shown in Figure 3(B), and its relative error (compared to the input value, B los, which was calculated in the same manner as ρ los) is presented in Figure 3(C). The contours that correspond to the values of ± 30% indicate that the relative error is less than 30% for most pixels. Figure 4(C) and (D) display histograms of B seis and its relative error. Statistically, there is a slight overestimation of the magnetic field by ∼ 5% (with a standard deviation of 17%). Given that the 5% bias is quite small, the results support the reliability of the coronal seismology technique. Further discussion on the error distributions in Figure 3(C) and the cause of the overestimation can be found in Section 4.\n3.2 After Degrading\nWe then degraded the spatial and temporal resolutions to match those of CoMP observations, with a pixel size of approximately 3.3 Mm and a cadence of 36 s. We focused on the central pixels (from y = -1 . 65 Mm to y = 1 . 65 Mm), which fully encompass the flux tube (diameter ∼ 2 Mm). At this resolution, the tube boundary and finer structures are unresolved, similar to the case in observational studies (Yang et al. 2020a,b).\nWe can now track the physical parameters along the tube axis. Figure 5 shows the phase speed, density, and magnetic field as a function of height z . In panel (A), the phase speed derived using the\nwave tracking method ( c seis) is compared to the characteristic speeds in the model (input values). The c seis closely matches the input kink speed ( c k, input), which lies between the internal and external Alfv'en speeds.\nPanel (B) shows the density derived from the Fe XIII intensity ratio ( ρ Fe XIII), which also falls between the internal density ( ρ i) and external density ( ρ e) in the model. The ρ Fe XIII is slightly higher than the emissivity-weighted density along the LOS ( ρ los), similar to the pre-degradation results shown in Figure 4(A). When calculating the magnetic field with Equation (2), the density value should ideally be the average of internal and external density (i.e., ⟨ ρ ⟩ = ( ρ i + ρ e ) / 2 ), because Equation (2) is derived from Equation (1) when assuming B i = B e. However, in practice, ρ Fe XIII is used, which is slightly lower than ⟨ ρ ⟩ . It means that using ρ Fe XIII as a representative of ⟨ ρ ⟩ can lead to a slight underestimation. The density underestimation is about 12-20% based on Figure 5(B), and the resulting error in B seis would be minimal (less than 5%) since the density is taken the square root of.\nIn fact, the deviation of ρ Fe XIII from ⟨ ρ ⟩ can vary with the filling factor, which represents the fraction of the pixel occupied by the flux tube. Given the current pixel size (3.3 Mm), LOS integration length (8 Mm), and flux tube radius (1 Mm), the filling factor is approximately 12%. If the filling factor is lower, the ρ Fe XIII will be closer to ρ e, increasing the density underestimation. The maximum underestimation depends on the density contrast ζ = ρ i /ρ e, with the deviation factor given by:\nα = ρ Fe XIII ⟨ ρ ⟩ ∼ ρ e ( ρ e + ρ i ) / 2 = 2 1 + ζ . (4)\nIn our simulation, ζ was initially set to be 3, but after relaxation, it decreased to around 2 (see Figure 1(D)). Such a value is comparable with previous observational estimates (Tian et al. 2012; Verwichte et al. 2013; Morton et al. 2021). We tested the case with a reduced filling factor ( 5%), where a density contrast of 2 led to an underestimation of ∼ 30%. Therefore, parameters like the filling factor and density contrast can be crucial when assessing the accuracy of coronal magnetic field measurements.\nFigure 5(C) presents the magnetic field inferred from coronal seismology ( B seis), compared to the internal ( B i), external ( B e), and emissivity-weighted ( B los) magnetic fields. The B seis ranges from 3.9 to 5.1 G, with errors below 15%, indicating that the technique of coronal seismolgy can be used for reliable measurements of the coronal magnetic field strengths.\n4 DISCUSSION AND CONCLUSION\nIn this study, we tested the accuracy of the previously developed seismological techniques based on observations of propagating kink waves for deriving coronal magnetic fields. The results indicate that seismology-based measurements of the magnetic field are accurate and reliable to a large extent.\nIn the high-resolution case, we obtained a 2D distribution of the magnetic field ( B seis) and its relative error. For most regions, the relative error is less than 30%, as shown in Figure 3(C). Statistically, the average magnetic field derived from coronal seismology is about 5% larger than the input value. Although this case has a much higher resolution than the CoMP and UCoMP observations, the pixel size (62.5 km in the z direction and 200 km in the y direction) and cadence (12 s) can be comparable to those of the Cryogenic Near-Infrared Spectro-Polarimeter (Cryo-NIRSP; Fehlmann et al. 2023) and the Diffraction-Limited Near-Infrared Spectropolarimeter (DL-NIRSP; Jaeggli et al. 2022) of the Daniel K. Inouye Solar Telescope (DKIST; Rimmele et al. 2020). The intruments offer high-resolution coronal spectroscopic observations with the Fe XIII lines (Schad et al. 2023), and recently Schad et al. (2024) successfully obtained a coronal LOS magnetogram with Cryo-NIRSP observation based on the Zeeman effect. With rapid repeated raster scans, it is possible that DKIST could also detect propagating transverse waves via Doppler velocity measurements, enabling seismological diagnostics of the plane-of-sky (POS) coronal magnetic field. In this way, DKIST observation can also provide us with maps of the POS magnetic field, but with a much higher spatial resolution compared to CoMP and UCoMP. Our results, particularly Figure 3, can serve as a reference for such diagnostics. Combined with the Stokes-V measurements, DKIST may then be able to achieve measurements of the full magnetic field vector.\nNevertheless, we note that some errors appear in the seismologically inferred magnetic field. First, large errors appeared near the upper and lower boundaries (around z = 0 and z = 100 Mm). This is due to the Fourier filtering method used to subtract downward propagating wave components. As shown in Figure 2(B), the upward propagating components of the Doppler velocity show some artificial velocity amplification near the upper and lower boundaries, especially after 250 s. This could affect the phase speed determination through wave tracking, leading to errors in B seis near the z boundaries. Thus, in observations, it may be useful to exclude boundary signals before calculating phase speeds. Another contributing factor comes from the wave-tracking process itself. When calculating the phase speed, a certain number of data points along the propagating direction is required. Near the lower and upper boundaries, fewer data points will be available to perform cross-correlation, leading to larger uncertainties in the calculated phase speed and consequently in the inferred magnetic field.\nSecond, overestimations were observed inside or along the lateral boundaries of the flux tube. This is likely because we treat the phase velocity as the Alfv'en speed, which applies well to the regions away from the flux tube or cases with spatial averaging (see Yang et al. 2020b and Section 3.2). However, in this case, we modeled kink waves, and the flux tube can be well resolved. Thus, at the flux tube region, particularly the tube boundary, it would be more appropriate to apply Equation (1) since the waves have a dominant kink wave characteristic (e.g., Goossens et al. 2009). Calculating the magnetic field with B seis = c ph √ µ 0 ρ leads to overestimation at high-density regions, explaining the positive errors within the flux tube in Figure 3(C). In future DKIST observations, we would suggest first applying Equation (2) to diagnose the magnetic field outside the flux tube ( B e), then calculate the magnetic field inside the flux tube ( B i) with Equation (1).\nIn addition, there are some other confusing patterns in Figure 3(B) and (C). For instance, B seis and the relative error manifest fluctuations along the z direction. It might be related to longitudinal oscillations excited by kink waves due to some non-linear effects (e.g., Goldstein 1978; Del Zanna et al. 2001; Terradas & Ofman 2004). Further investigations are needed to understand these patterns.\nWhen we degraded the spatial and temporal resolutions to approximately match those of CoMP, we found that the distributions of phase speed, density, and B seis along z all show remarkable similarities to the input values (Figure 5). Again we can notice a deviation near the upper and lower boundaries, particularly the lower one. However, within the height range of 20-80 Mm, the magnetic field error is generally less than 10%. Additionally, the CoMP instrument has recently been upgraded to UCoMP, which has a slightly higher spatial resolution and a larger field of view. Our main conclusions should also apply to the case with the UCoMP observations (Yang et al. 2024). We also tested the case when degrading to UCoMP's pixel size ( ∼ 2.2 Mm), and the results are largely similar to those shown in Figure 5, with a slightly smaller error.\nAnother factor that may impact the phase speed and magnetic field measurements is the magnitude of the input magnetic field and phase speed. A stronger magnetic field and higher phase speed result in steeper slopes in the time-distance maps of Doppler velocity, which can reduce the accuracy of the wave-tracking method, particularly when the spatial and temporal resolutions are degraded. We ran a separate simulation with a background magnetic field of ∼ 10 G, which gives phase speeds of 1-2 Mm s -1 . We found a systematic underestimation of the magnetic field by approximately 30% in this case. Nevertheless, the observed coronal magnetic field around 1 . 05 -1 . 5 R ⊙ is 1-4 G (Lin et al. 2004; Gopalswamy et al. 2012; Kumari et al. 2019; Yang et al. 2020b; Zhong et al. 2023), which is more comparable with the case discussed in Section 3. In conclusion, the coronal seismology technique based on propagating kink waves can provide reliable magnetic field measurements according to our forward modeling.\nWe note that this study only focuses on open coronal structures, including plumes in coronal holes and fan loops at the boundaries of active regions (Morton et al. 2015; Banerjee et al. 2021). Specifically, in our simulation, the magnetic flux tube is perpendicular to the solar surface, with both gravity and magnetic field aligned along the tube axis. However, the propagating kink waves can be detected not only in these structures but also in closed-field regions (e.g., Tomczyk et al. 2007; Yang et al. 2020a,b). Our magnetic configuration can roughly describe one leg of a large-scale closed coronal loop, where the curvature can be neglected. For smaller loops where curvature cannot be ignored, our model is no\nlonger applicable due to differences in gravitational stratification. Nevertheless, for such loops, phase speed along the axis often shows minimal variation McIntosh et al. (2011); Threlfall et al. (2013); Zhong et al. (2023), thus similar to the case in Magyar & Van Doorsselaere (2018), where a model without any vertical gradient in phase speed was used. Therefore, this study complements previous research by highlighting the importance of phase speed variation along the flux tube in coronal seismology.\nFinally, we would like to mention that our model has some limitations. The vertical gradient of the magnetic field and the magnetic expansion are not included. Additionally, we didn't consider the effect of internal flows along the flux tube, which are frequently reported and can affect the apparent wave propagation speed (e.g., Soler et al. 2011; Morton et al. 2015). Moreover, in real observations, there are often multiple flux tubes overlapping along the LOS, introducing further complexities that may affect wave-tracking accuracy and magnetic field measurements. In Figure 5 of Yang et al. (2024), comparisons between seismological results and global coronal MHD models revealed some discrepancies, particularly at higher latitudes where open field lines may dominate. Future work that incorporates more realistic models could offer a more sophisticated evaluation of the magnetic field measurements through coronal seismology and help clarify the discrepancies in Yang et al. (2024).\nAcknowledgements This work is supported by the National Natural Science Foundation of China grant 12425301 and the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB0560000). M.G. acknowledges support from the National Natural Science Foundation of China (NSFC, 12203030). T.V.D was supported by the C1 grant TRACEspace of Internal Funds KU Leuven and a Senior Research Project (G088021N) of the FWO Vlaanderen. Furthermore, TVD received financial support from the Flemish Government under the long-term structural Methusalem funding program, project SOUL: Stellar evolution in full glory, grant METH/24/012 at KU Leuven. The research that led to these results was subsidised by the Belgian Federal Science Policy Office through the contract B2/223/P1/CLOSE-UP. It is also part of the DynaSun project and has thus received funding under the Horizon Europe programme of the European Union under grant agreement (no. 101131534). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union and therefore the European Union cannot be held responsible for them. K.K. acknowledges support by an FWO (Fonds voor Wetenschappelijk Onderzoek - Vlaanderen) postdoctoral fellowship (1273221N).\nReferences\nAschwanden, M. J., De Pontieu, B., Schrijver, C. J., & Title, A. M. 2002, Sol. Phys., 206, 99 2 Banerjee, D., Krishna Prasad, S., Pant, V., et al. 2021, Space Sci. Rev., 217, 76 9 Bose, S., TenBarge, J. M., Carter, T., et al. 2024, ApJ, 971, 72 4 Chen, B., Shen, C., Gary, D. E., et al. 2020, Nature Astronomy, 4, 1140 1 Chen, S.-X., Li, B., Xiong, M., Yu, H., & Guo, M.-Z. 2015, ApJ, 812, 22 2 Chen, Y., Feng, S. W., Li, B., et al. 2011, ApJ, 728, 147 2 Chen, Y., Li, W., Tian, H., et al. 2023, Research in Astronomy and Astrophysics, 23, 022001 1 Chen, Y., Li, W., Tian, H., et al. 2021, ApJ, 920, 116 1 Del Zanna, L., Velli, M., & Londrillo, P. 2001, A&A, 367, 705 9 Dere, K. P., Del Zanna, G., Young, P. R., & Landi, E. 2023, ApJS, 268, 52 6 Fehlmann, A., Kuhn, J. R., Schad, T. A., et al. 2023, Sol. Phys., 298, 5 8 Fleishman, G. D., Gary, D. E., Chen, B., et al. 2020, Science, 367, 278 1 Gao, Y., Guo, M., Van Doorsselaere, T., Tian, H., & Skirvin, S. J. 2023, ApJ, 955, 73 3 Gao, Y., Hou, Z., Van Doorsselaere, T., & Guo, M. 2024a, A&A, 681, L4 2 Gao, Y., Tian, H., Van Doorsselaere, T., & Chen, Y. 2022, ApJ, 930, 55 2 Gao, Y., Van Doorsselaere, T., Tian, H., Guo, M., & Karampelas, K. 2024b, A&A, 689, A195 3 Goldstein, M. L. 1978, ApJ, 219, 700 9 Goossens, M., Terradas, J., Andries, J., Arregui, I., & Ballester, J. L. 2009, A&A, 503, 213 9 Gopalswamy, N., Nitta, N., Akiyama, S., M¨akel¨a, P., & Yashiro, S. 2012, ApJ, 744, 72 9\n
\n\n
\n\nYoung, P. R., Del Zanna, G., Landi, E., et al. 2003, ApJS, 144, 135 3\nZhang, Q. M., Chen, J. L., Li, S. T., Lu, L., & Li, D. 2022, Sol. Phys., 297, 18 2\nZhong, S., Nakariakov, V. M., Miao, Y., Fu, L., & Yuan, D. 2023, Scientific Reports, 13, 12963 2, 9, 10\n"},"sections":{"kind":"list like","value":[{"title":"ABSTRACT","content":"R esearch in A stronomy and A strophysics","pages":[1]},{"title":"Measurements of the solar coronal magnetic field based on coronal seismology with propagating Alfv'enic waves: forward modeling","content":"Yuhang Gao 1 , 2 , Hui Tian 1 , Tom Van Doorsselaere 2 , Zihao Yang 1 , 3 , Mingzhe Guo 2 , 4 and Konstantinos Karampelas 2 Received 20xx month day; accepted 20xx month day Abstract Recent observations have demonstrated the capability of mapping the solar coronal magnetic field using the technique of coronal seismology based on the ubiquitous propagating Alfv'enic/kink waves through imaging spectroscopy. We established a magnetohydrodynamic (MHD) model of a gravitationally stratified open magnetic flux tube, exciting kink waves propagating upwards along the tube. Forward modeling was performed to synthesize the Fe XIII 1074.7 and 1079.8 nm spectral line profiles, which were then used to determine the wave phase speed, plasma density, and magnetic field with seismology method. A comparison between the seismologically inferred results and the corresponding input values verifies the reliability of the seismology method. In addition, we also identified some factors that could lead to errors during magnetic field measurements. Our results may serve as a valuable reference for current and future coronal magnetic field measurements based on observations of propagating kink waves. Key words: Sun: corona - Sun: magnetic fields - magnetohydrodynamics","pages":[1]},{"title":"1 INTRODUCTION","content":"The magnetic field plays a crucial role in various physical processes in the solar and stellar coronae. The dissipation of magnetic energy is believed to drive solar eruptive events (e.g., flares and coronal mass ejections) and cause heating of the corona. While the magnetic field in the lower solar atmosphere can be reliably measured through spectro-polarimetric observations, direct measurements of the coronal magnetic field have remained challenging for decades. Several methods, including the spectro-polarimetry of coronal infrared lines (Lin et al. 2000, 2004; Schad et al. 2024), coronal radio observations (e.g., Fleishman et al. 2020; Chen et al. 2020; Tan 2022), and magnetic-field-induced transitions of extreme ultraviolet emission lines (e.g., Li et al. 2015, 2016; Landi et al. 2020; Chen et al. 2021, 2023), have been proposed and attempts of measurements have been made. However, these approaches all face limitations and none of them could be used for routine measurements of the global coronal magnetic field. Another technique that could be used to measure the coronal magnetic field is coronal seismology. This technique combines the magneto-hydrodynamic (MHD) wave theory with observed wave parameters (e.g., period, amplitude, propagation speed, damping time) to diagnose various physical properties, particularly the magnetic field. Different wave phenomena in the corona have been used for coronal seismology, including standing kink waves in magnetic loops (e.g. Nakariakov & Ofman 2001; Aschwanden et al. 2002; Van Doorsselaere et al. 2007; Tian et al. 2012; Zhang et al. 2022; Gao et al. 2022, 2024a; Zhong et al. 2023; Li & Long 2023, to name but a few), propagating slow magneto-acoustic waves (e.g., Jess et al. 2016), sausage waves (Chen et al. 2015; Guo et al. 2016; see Li et al. 2020 for a review), propagating kink waves in streamers (Chen et al. 2011; Guo et al. 2022), and torsional oscillations in solar surges (Kohutova et al. 2020). However, these studies provided only one-dimensional (1D) distributions or single values of the magnetic field in specific coronal structures, such as oscillating loops or streamers. To create global 2D magnetic field maps, we need to utilize ubiquitous and continuous wave phenomena. The pervasive propagating disturbances in Dopplergrams (Tomczyk et al. 2007; Tomczyk & McIntosh 2009; Liu et al. 2015; Morton et al. 2015, 2019) observed by the Coronal Multi-channel Polarimeter (CoMP; Tomczyk et al. 2008) are ideal for this purpose. These propagating disturbances are interpreted as kink or Alfv'enic waves (Van Doorsselaere et al. 2008), and their propagation speeds are naturally linked to the local magnetic field. Based on these CoMP observations, Yang et al. (2020b,a) have successfully measured the global distribution of the coronal magnetic field for the first time. Following these successful attempts, a routine (continuous) measurement of the global coronal magnetic field based on similar observations from the Upgraded CoMP (UCoMP; Landi et al. 2016) has been recently achieved, which allows for the construction of coronal synoptic magnetograms (Carrington maps) (Yang et al. 2024). The CoMP and UCoMP instruments can conduct spectroscopic observations of the Fe XIII lines at 1074.7 and 1079.8 nm in the coronal region above the solar limb. From the Dopplergrams of Fe XIII 1074.7 nm, propagating kink waves can be identified throughout the corona. The propagating or phase speed c k of these waves (also named the kink speed) could be obtained by constructing a timedistance map of Doppler velocity, while the coronal density can be inferred from the observed Fe XIII 1079.8-nm/1074.7-nm intensity ratio. For kink waves, we have where µ 0 is the magnetic permeability, B and ρ are the magnetic field and mass density, respectively. The subscripts i and e refer to parameters inside and outside the magnetic flux tubes (the waveguides). Since CoMP and UCoMP likely cannot resolve individual flux tubes, we can only work with an average density ⟨ ρ ⟩ . Furthermore, in the lowβ coronal environment, the internal and external magnetic fields are often assumed to be approximately equal (see e.g., Tomczyk & McIntosh 2009; Morton et al. 2015; Zhong et al. 2023). This leads to the simplified expression for the kink speed: which is widely used in estimations of coronal magnetic fields (Long et al. 2017; Yang et al. 2020a,b; Yang et al. 2024). Given the potential of these measurements to provide routine coronal magnetograms on a daily basis, which could play a crucial role in future solar physics research, it is essential to thoroughly assess the reliability and robustness of the methodology used in Yang et al. (2020a,b) and Yang et al. (2024). Magyar & Van Doorsselaere (2018) performed 3D MHD simulations of propagating kink waves under various conditions and conducted forward modeling to evaluate the reliability of this method in deriving the magnetic field strength. Their findings indicated that the magnetic field strengths inferred through seismology closely match the input values, typically with an error less than ∼ 20%. However, there is a limitation in their simulation. They utilized a non-stratified setup that excluded the effects of gravity, resulting in uniform initial density and propagation speed in the vertical direction. The gravitational stratification can play a significant role as it causes the Alfv'en speed and kink speed c k to vary with height. This variation can affect the wave tracking method which typically relies on a linear fit of velocity signals (e.g., see Figure 6 in Tomczyk & McIntosh 2009). Therefore, it is important to assess how the gravitational stratification affects the seismological results and estimate the possible error range. So we conducted 3D MHD simulations of propagating kink waves in stratified coronal open flux tubes. Following Magyar & Van Doorsselaere (2018), we employed forward modeling to compare the seismologically derived results with the actual values of physical parameters from our simulation. This paper is organized as follows: Section 2 describes our simulation setup and methodology, Section 3 presents the simulation and forward-modeling results, along with detailed comparisons between seismology results and input values, and Section 4 provides a discussion and summary of our findings.","pages":[1,2,3]},{"title":"2 METHOD","content":"The model used in this study is a gravitationally stratified, open magnetic flux tube with a radius of 1 Mm, similar to that in Gao et al. (2024b) (hereafter referred to as Paper I). The main difference is that the initial magnetic field in this study is set at around 4 G, instead of 10 G, to better match the seismologically inferred results in Yang et al. (2020b). As in Paper I, the magnetic field is oriented in the z direction and remains nearly uniform across the simulation domain, with only small spatial gradients to maintain total pressure balance. As a result, the Alfv'en and kink speeds increase with height as the density decreases due to stratification. A relaxation process of 2400 s was conducted to achieve a quasimagnetohydrostatic state, as shown in Figure 1. The figure shows that the post-relaxation magnetic field has some spatial variation but mostly within 0.7 G. The kink wave driver is also similar to that in Paper I, with a velocity amplitude of 8 km s -1 and a period of 300 s. The primary velocity perturbation is along the x direction. We ran the 3D MHD simulation in Cartesian coordinates with the PLUTO code (Mignone et al. 2007). The simulation domain spans [-4, 4] Mm × [-4, 4] Mm × [0, 150] Mm, with a uniform grid of 128 × 128 × 1024 cells, providing spatial resolutions of 62.5 km in the horizontal ( x and y ) directions and 146.5 km in the vertical ( z ) direction. We chose a second-order parabolic spatial scheme and a Roe Riemann solver. The boundary conditions were set to be outflow, except for the lower boundary, where the kink wave driver was introduced by adjusting v x and v y , while other parameters were fixed. As in Paper I, the upper 50 Mm ( z > 100 Mm) was set as a velocity absorption region (VAR) to minimize numerical reflections from the upper boundary (see also Pelouze et al. 2023; Guo et al. 2023; Gao et al. 2023). For the subsequent analysis, we only considered the region below z = 100 Mmwhich can give us physical results. Once the kink wave driver was applied, the propagating waves were rapidly excited, with their properties thoroughly analyzed in Paper I. Here, we focus on assessing the reliability of the seismology method described in Section 1 using the simulation outputs. To do so, we performed forward calculations to synthesize spectroscopic observables of CoMP and UCoMP, namely, the Fe XIII 1074.7 and 1079.8 nm spectral line profiles. Specifically, we used synthesized observation of the 1074.7 nm line to perform the wave tracking and determine the propagation speed, while synthesized observation of the 1079.8 nm line was only employed for the density diagnostic using the intensity ratio method (Yang et al. 2020a,b). We applied the FoMo code (Van Doorsselaere et al. 2016) to synthesize the spectral profiles at all pixels in the yz plane. The photo-excitation (see Young et al. 2003; Yang et al. 2020a) was not considered for simplification, as their effects on the spectral lines are not significant at the lower corona. The line of sight (LOS) was chosen as the x axis. In this way, we can reconstruct 2D maps (in the yz plane) of the intensity and Doppler velocity by fitting a single Gaussian to each spectral profile. The resulting maps consist of 128 pixels in the y direction from -4 Mm to 4 Mm, and 500 pixels in the z direction from 0 Mmto 100 Mm. Since the primary velocity perturbation is along the x direction (or in the LOS plane), the Doppler velocity maps capture the wave propagation signals, while the intensity maps do not show any transverse displacement. z (Mm)","pages":[3,4]},{"title":"3.1 Before Degrading","content":"We first generated time-distance (TD) maps of the Doppler velocity along the z direction, which can be produced at each y position. In Figure 2(A), the TD map along the flux tube's axis (i.e., y = 0 ) is presented, clearly illustrating the propagation of Doppler velocity disturbances. The increasing slope reflects the growing propagation speed with height. However, after 400 s, unusual patterns appear due to wave reflections. Despite efforts to suppress numerical reflections from the upper boundary at z = 150 Mm using a VAR (see Section 2), it is still difficult to eliminate all reflections. Some reflections may come from the layer of z = 100 Mm, which is the lower boundary of the VAR; while others could be attributed to the vertical inhomogeneities of density and phase speed. As noted in previous studies, even smooth phase speed gradients can cause partial wave reflection (e.g., Verdini & Velli 2007; Hahn et al. 2018; Pascoe et al. 2022; Bose et al. 2024). In fact, real CoMP observations of coronal kink waves often reveal both upward and downward propagating components (Tomczyk & McIntosh 2009; Morton et al. 2015). However, when calculating the propagation speed, these reflected waves can introduce significant errors. To reduce this, we applied the Fourier filtering method to separate the upward and downward propagating wave components (see e.g., Tomczyk & McIntosh 2009; Threlfall et al. 2013; Liu et al. 2014; Tiwari et al. 2019; Yang et al. 2020b). Figure 2(B) shows the TD map for the upward-propagating component, which is used to determine the phase speed. By applying the wave-tracking method to all pixels (for more details, see e.g., Yang et al. 2020b), we obtained a 2D phase speed distribution, as shown in Figure 3(A). For most pixels, the phase speed falls in the range of 200-800 km s -1 (see also Figure 4(B)), with a general trend of increasing with altitude, though some fluctuations are present (see relevant discussions in Section 4). Next, we derived the density by calculating the intensity ratio of the synthesized 1074.7 nm and 1079.8 nm intensity maps. The theoretical relationship between intensity ratio and density can be obtained from the CHIANTI database (Dere et al. 2023). The obtained density values were compared with the input values, as shown in Figure 4(A). The input density corresponds to the emissivity-weighted density along the LOS (Yang et al. 2024): where ϵ is the Fe XIII 1074.7 nm line emissivity at the corresponding pixel, calculated using the IDL routine emiss calc.pro from the CHIANTI software package. The comparison shows that the density derived from the intensity ratio is reliable for most pixels, though about 20% pixels show an overestimation ( ≳ 15%). However, this overestimation would not lead to large errors in the seismologically inferred magnetic field, as only the square root of density is needed during the calculation. With the derived phase speed and density, we calculated the magnetic field B seis. Here we chose to treat the phase speed as the local Alfv'en speed (i.e., c ph = B seis / √ µ 0 ρ ), rather than employ Equation(1). Because in our magnetic field configuration, Equation (1) can only provide the phase speed as a function of z ; however, here with a high spatial resolution, we are also interested in the horizontal distribution of phase speed and magnetic field. We note that such a choice may lead to some errors, especially within the flux tube region, which will be further discussed in Section 4. In Figure 4(B), we compared the Alfv'en speed in the model and the wave propagation speed obtained from wave tracking. Despite some noticeable discrepancies, the overall correspondence between the two is reasonably good. The calculated magnetic field is shown in Figure 3(B), and its relative error (compared to the input value, B los, which was calculated in the same manner as ρ los) is presented in Figure 3(C). The contours that correspond to the values of ± 30% indicate that the relative error is less than 30% for most pixels. Figure 4(C) and (D) display histograms of B seis and its relative error. Statistically, there is a slight overestimation of the magnetic field by ∼ 5% (with a standard deviation of 17%). Given that the 5% bias is quite small, the results support the reliability of the coronal seismology technique. Further discussion on the error distributions in Figure 3(C) and the cause of the overestimation can be found in Section 4.","pages":[4,5,6,7]},{"title":"3.2 After Degrading","content":"We then degraded the spatial and temporal resolutions to match those of CoMP observations, with a pixel size of approximately 3.3 Mm and a cadence of 36 s. We focused on the central pixels (from y = -1 . 65 Mm to y = 1 . 65 Mm), which fully encompass the flux tube (diameter ∼ 2 Mm). At this resolution, the tube boundary and finer structures are unresolved, similar to the case in observational studies (Yang et al. 2020a,b). We can now track the physical parameters along the tube axis. Figure 5 shows the phase speed, density, and magnetic field as a function of height z . In panel (A), the phase speed derived using the wave tracking method ( c seis) is compared to the characteristic speeds in the model (input values). The c seis closely matches the input kink speed ( c k, input), which lies between the internal and external Alfv'en speeds. Panel (B) shows the density derived from the Fe XIII intensity ratio ( ρ Fe XIII), which also falls between the internal density ( ρ i) and external density ( ρ e) in the model. The ρ Fe XIII is slightly higher than the emissivity-weighted density along the LOS ( ρ los), similar to the pre-degradation results shown in Figure 4(A). When calculating the magnetic field with Equation (2), the density value should ideally be the average of internal and external density (i.e., ⟨ ρ ⟩ = ( ρ i + ρ e ) / 2 ), because Equation (2) is derived from Equation (1) when assuming B i = B e. However, in practice, ρ Fe XIII is used, which is slightly lower than ⟨ ρ ⟩ . It means that using ρ Fe XIII as a representative of ⟨ ρ ⟩ can lead to a slight underestimation. The density underestimation is about 12-20% based on Figure 5(B), and the resulting error in B seis would be minimal (less than 5%) since the density is taken the square root of. In fact, the deviation of ρ Fe XIII from ⟨ ρ ⟩ can vary with the filling factor, which represents the fraction of the pixel occupied by the flux tube. Given the current pixel size (3.3 Mm), LOS integration length (8 Mm), and flux tube radius (1 Mm), the filling factor is approximately 12%. If the filling factor is lower, the ρ Fe XIII will be closer to ρ e, increasing the density underestimation. The maximum underestimation depends on the density contrast ζ = ρ i /ρ e, with the deviation factor given by: In our simulation, ζ was initially set to be 3, but after relaxation, it decreased to around 2 (see Figure 1(D)). Such a value is comparable with previous observational estimates (Tian et al. 2012; Verwichte et al. 2013; Morton et al. 2021). We tested the case with a reduced filling factor ( 5%), where a density contrast of 2 led to an underestimation of ∼ 30%. Therefore, parameters like the filling factor and density contrast can be crucial when assessing the accuracy of coronal magnetic field measurements. Figure 5(C) presents the magnetic field inferred from coronal seismology ( B seis), compared to the internal ( B i), external ( B e), and emissivity-weighted ( B los) magnetic fields. The B seis ranges from 3.9 to 5.1 G, with errors below 15%, indicating that the technique of coronal seismolgy can be used for reliable measurements of the coronal magnetic field strengths.","pages":[7,8]},{"title":"4 DISCUSSION AND CONCLUSION","content":"In this study, we tested the accuracy of the previously developed seismological techniques based on observations of propagating kink waves for deriving coronal magnetic fields. The results indicate that seismology-based measurements of the magnetic field are accurate and reliable to a large extent. In the high-resolution case, we obtained a 2D distribution of the magnetic field ( B seis) and its relative error. For most regions, the relative error is less than 30%, as shown in Figure 3(C). Statistically, the average magnetic field derived from coronal seismology is about 5% larger than the input value. Although this case has a much higher resolution than the CoMP and UCoMP observations, the pixel size (62.5 km in the z direction and 200 km in the y direction) and cadence (12 s) can be comparable to those of the Cryogenic Near-Infrared Spectro-Polarimeter (Cryo-NIRSP; Fehlmann et al. 2023) and the Diffraction-Limited Near-Infrared Spectropolarimeter (DL-NIRSP; Jaeggli et al. 2022) of the Daniel K. Inouye Solar Telescope (DKIST; Rimmele et al. 2020). The intruments offer high-resolution coronal spectroscopic observations with the Fe XIII lines (Schad et al. 2023), and recently Schad et al. (2024) successfully obtained a coronal LOS magnetogram with Cryo-NIRSP observation based on the Zeeman effect. With rapid repeated raster scans, it is possible that DKIST could also detect propagating transverse waves via Doppler velocity measurements, enabling seismological diagnostics of the plane-of-sky (POS) coronal magnetic field. In this way, DKIST observation can also provide us with maps of the POS magnetic field, but with a much higher spatial resolution compared to CoMP and UCoMP. Our results, particularly Figure 3, can serve as a reference for such diagnostics. Combined with the Stokes-V measurements, DKIST may then be able to achieve measurements of the full magnetic field vector. Nevertheless, we note that some errors appear in the seismologically inferred magnetic field. First, large errors appeared near the upper and lower boundaries (around z = 0 and z = 100 Mm). This is due to the Fourier filtering method used to subtract downward propagating wave components. As shown in Figure 2(B), the upward propagating components of the Doppler velocity show some artificial velocity amplification near the upper and lower boundaries, especially after 250 s. This could affect the phase speed determination through wave tracking, leading to errors in B seis near the z boundaries. Thus, in observations, it may be useful to exclude boundary signals before calculating phase speeds. Another contributing factor comes from the wave-tracking process itself. When calculating the phase speed, a certain number of data points along the propagating direction is required. Near the lower and upper boundaries, fewer data points will be available to perform cross-correlation, leading to larger uncertainties in the calculated phase speed and consequently in the inferred magnetic field. Second, overestimations were observed inside or along the lateral boundaries of the flux tube. This is likely because we treat the phase velocity as the Alfv'en speed, which applies well to the regions away from the flux tube or cases with spatial averaging (see Yang et al. 2020b and Section 3.2). However, in this case, we modeled kink waves, and the flux tube can be well resolved. Thus, at the flux tube region, particularly the tube boundary, it would be more appropriate to apply Equation (1) since the waves have a dominant kink wave characteristic (e.g., Goossens et al. 2009). Calculating the magnetic field with B seis = c ph √ µ 0 ρ leads to overestimation at high-density regions, explaining the positive errors within the flux tube in Figure 3(C). In future DKIST observations, we would suggest first applying Equation (2) to diagnose the magnetic field outside the flux tube ( B e), then calculate the magnetic field inside the flux tube ( B i) with Equation (1). In addition, there are some other confusing patterns in Figure 3(B) and (C). For instance, B seis and the relative error manifest fluctuations along the z direction. It might be related to longitudinal oscillations excited by kink waves due to some non-linear effects (e.g., Goldstein 1978; Del Zanna et al. 2001; Terradas & Ofman 2004). Further investigations are needed to understand these patterns. When we degraded the spatial and temporal resolutions to approximately match those of CoMP, we found that the distributions of phase speed, density, and B seis along z all show remarkable similarities to the input values (Figure 5). Again we can notice a deviation near the upper and lower boundaries, particularly the lower one. However, within the height range of 20-80 Mm, the magnetic field error is generally less than 10%. Additionally, the CoMP instrument has recently been upgraded to UCoMP, which has a slightly higher spatial resolution and a larger field of view. Our main conclusions should also apply to the case with the UCoMP observations (Yang et al. 2024). We also tested the case when degrading to UCoMP's pixel size ( ∼ 2.2 Mm), and the results are largely similar to those shown in Figure 5, with a slightly smaller error. Another factor that may impact the phase speed and magnetic field measurements is the magnitude of the input magnetic field and phase speed. A stronger magnetic field and higher phase speed result in steeper slopes in the time-distance maps of Doppler velocity, which can reduce the accuracy of the wave-tracking method, particularly when the spatial and temporal resolutions are degraded. We ran a separate simulation with a background magnetic field of ∼ 10 G, which gives phase speeds of 1-2 Mm s -1 . We found a systematic underestimation of the magnetic field by approximately 30% in this case. Nevertheless, the observed coronal magnetic field around 1 . 05 -1 . 5 R ⊙ is 1-4 G (Lin et al. 2004; Gopalswamy et al. 2012; Kumari et al. 2019; Yang et al. 2020b; Zhong et al. 2023), which is more comparable with the case discussed in Section 3. In conclusion, the coronal seismology technique based on propagating kink waves can provide reliable magnetic field measurements according to our forward modeling. We note that this study only focuses on open coronal structures, including plumes in coronal holes and fan loops at the boundaries of active regions (Morton et al. 2015; Banerjee et al. 2021). Specifically, in our simulation, the magnetic flux tube is perpendicular to the solar surface, with both gravity and magnetic field aligned along the tube axis. However, the propagating kink waves can be detected not only in these structures but also in closed-field regions (e.g., Tomczyk et al. 2007; Yang et al. 2020a,b). Our magnetic configuration can roughly describe one leg of a large-scale closed coronal loop, where the curvature can be neglected. For smaller loops where curvature cannot be ignored, our model is no longer applicable due to differences in gravitational stratification. Nevertheless, for such loops, phase speed along the axis often shows minimal variation McIntosh et al. (2011); Threlfall et al. (2013); Zhong et al. (2023), thus similar to the case in Magyar & Van Doorsselaere (2018), where a model without any vertical gradient in phase speed was used. Therefore, this study complements previous research by highlighting the importance of phase speed variation along the flux tube in coronal seismology. Finally, we would like to mention that our model has some limitations. The vertical gradient of the magnetic field and the magnetic expansion are not included. Additionally, we didn't consider the effect of internal flows along the flux tube, which are frequently reported and can affect the apparent wave propagation speed (e.g., Soler et al. 2011; Morton et al. 2015). Moreover, in real observations, there are often multiple flux tubes overlapping along the LOS, introducing further complexities that may affect wave-tracking accuracy and magnetic field measurements. In Figure 5 of Yang et al. (2024), comparisons between seismological results and global coronal MHD models revealed some discrepancies, particularly at higher latitudes where open field lines may dominate. Future work that incorporates more realistic models could offer a more sophisticated evaluation of the magnetic field measurements through coronal seismology and help clarify the discrepancies in Yang et al. (2024). Acknowledgements This work is supported by the National Natural Science Foundation of China grant 12425301 and the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB0560000). M.G. acknowledges support from the National Natural Science Foundation of China (NSFC, 12203030). T.V.D was supported by the C1 grant TRACEspace of Internal Funds KU Leuven and a Senior Research Project (G088021N) of the FWO Vlaanderen. Furthermore, TVD received financial support from the Flemish Government under the long-term structural Methusalem funding program, project SOUL: Stellar evolution in full glory, grant METH/24/012 at KU Leuven. The research that led to these results was subsidised by the Belgian Federal Science Policy Office through the contract B2/223/P1/CLOSE-UP. It is also part of the DynaSun project and has thus received funding under the Horizon Europe programme of the European Union under grant agreement (no. 101131534). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union and therefore the European Union cannot be held responsible for them. K.K. acknowledges support by an FWO (Fonds voor Wetenschappelijk Onderzoek - Vlaanderen) postdoctoral fellowship (1273221N).","pages":[8,9,10]},{"title":"References","content":"Aschwanden, M. J., De Pontieu, B., Schrijver, C. J., & Title, A. M. 2002, Sol. Phys., 206, 99 2 Banerjee, D., Krishna Prasad, S., Pant, V., et al. 2021, Space Sci. Rev., 217, 76 9 Bose, S., TenBarge, J. M., Carter, T., et al. 2024, ApJ, 971, 72 4 Chen, B., Shen, C., Gary, D. E., et al. 2020, Nature Astronomy, 4, 1140 1 Chen, S.-X., Li, B., Xiong, M., Yu, H., & Guo, M.-Z. 2015, ApJ, 812, 22 2 Chen, Y., Feng, S. W., Li, B., et al. 2011, ApJ, 728, 147 2 Chen, Y., Li, W., Tian, H., et al. 2023, Research in Astronomy and Astrophysics, 23, 022001 1 Chen, Y., Li, W., Tian, H., et al. 2021, ApJ, 920, 116 1 Del Zanna, L., Velli, M., & Londrillo, P. 2001, A&A, 367, 705 9 Dere, K. P., Del Zanna, G., Young, P. R., & Landi, E. 2023, ApJS, 268, 52 6 Fehlmann, A., Kuhn, J. R., Schad, T. A., et al. 2023, Sol. Phys., 298, 5 8 Fleishman, G. D., Gary, D. E., Chen, B., et al. 2020, Science, 367, 278 1 Gao, Y., Guo, M., Van Doorsselaere, T., Tian, H., & Skirvin, S. J. 2023, ApJ, 955, 73 3 Gao, Y., Hou, Z., Van Doorsselaere, T., & Guo, M. 2024a, A&A, 681, L4 2 Gao, Y., Tian, H., Van Doorsselaere, T., & Chen, Y. 2022, ApJ, 930, 55 2 Gao, Y., Van Doorsselaere, T., Tian, H., Guo, M., & Karampelas, K. 2024b, A&A, 689, A195 3 Goldstein, M. L. 1978, ApJ, 219, 700 9 Goossens, M., Terradas, J., Andries, J., Arregui, I., & Ballester, J. L. 2009, A&A, 503, 213 9 Gopalswamy, N., Nitta, N., Akiyama, S., M¨akel¨a, P., & Yashiro, S. 2012, ApJ, 744, 72 9","pages":[10]}],"string":"[\n {\n \"title\": \"ABSTRACT\",\n \"content\": \"R esearch in A stronomy and A strophysics\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"Measurements of the solar coronal magnetic field based on coronal seismology with propagating Alfv'enic waves: forward modeling\",\n \"content\": \"Yuhang Gao 1 , 2 , Hui Tian 1 , Tom Van Doorsselaere 2 , Zihao Yang 1 , 3 , Mingzhe Guo 2 , 4 and Konstantinos Karampelas 2 Received 20xx month day; accepted 20xx month day Abstract Recent observations have demonstrated the capability of mapping the solar coronal magnetic field using the technique of coronal seismology based on the ubiquitous propagating Alfv'enic/kink waves through imaging spectroscopy. We established a magnetohydrodynamic (MHD) model of a gravitationally stratified open magnetic flux tube, exciting kink waves propagating upwards along the tube. Forward modeling was performed to synthesize the Fe XIII 1074.7 and 1079.8 nm spectral line profiles, which were then used to determine the wave phase speed, plasma density, and magnetic field with seismology method. A comparison between the seismologically inferred results and the corresponding input values verifies the reliability of the seismology method. In addition, we also identified some factors that could lead to errors during magnetic field measurements. Our results may serve as a valuable reference for current and future coronal magnetic field measurements based on observations of propagating kink waves. Key words: Sun: corona - Sun: magnetic fields - magnetohydrodynamics\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1 INTRODUCTION\",\n \"content\": \"The magnetic field plays a crucial role in various physical processes in the solar and stellar coronae. The dissipation of magnetic energy is believed to drive solar eruptive events (e.g., flares and coronal mass ejections) and cause heating of the corona. While the magnetic field in the lower solar atmosphere can be reliably measured through spectro-polarimetric observations, direct measurements of the coronal magnetic field have remained challenging for decades. Several methods, including the spectro-polarimetry of coronal infrared lines (Lin et al. 2000, 2004; Schad et al. 2024), coronal radio observations (e.g., Fleishman et al. 2020; Chen et al. 2020; Tan 2022), and magnetic-field-induced transitions of extreme ultraviolet emission lines (e.g., Li et al. 2015, 2016; Landi et al. 2020; Chen et al. 2021, 2023), have been proposed and attempts of measurements have been made. However, these approaches all face limitations and none of them could be used for routine measurements of the global coronal magnetic field. Another technique that could be used to measure the coronal magnetic field is coronal seismology. This technique combines the magneto-hydrodynamic (MHD) wave theory with observed wave parameters (e.g., period, amplitude, propagation speed, damping time) to diagnose various physical properties, particularly the magnetic field. Different wave phenomena in the corona have been used for coronal seismology, including standing kink waves in magnetic loops (e.g. Nakariakov & Ofman 2001; Aschwanden et al. 2002; Van Doorsselaere et al. 2007; Tian et al. 2012; Zhang et al. 2022; Gao et al. 2022, 2024a; Zhong et al. 2023; Li & Long 2023, to name but a few), propagating slow magneto-acoustic waves (e.g., Jess et al. 2016), sausage waves (Chen et al. 2015; Guo et al. 2016; see Li et al. 2020 for a review), propagating kink waves in streamers (Chen et al. 2011; Guo et al. 2022), and torsional oscillations in solar surges (Kohutova et al. 2020). However, these studies provided only one-dimensional (1D) distributions or single values of the magnetic field in specific coronal structures, such as oscillating loops or streamers. To create global 2D magnetic field maps, we need to utilize ubiquitous and continuous wave phenomena. The pervasive propagating disturbances in Dopplergrams (Tomczyk et al. 2007; Tomczyk & McIntosh 2009; Liu et al. 2015; Morton et al. 2015, 2019) observed by the Coronal Multi-channel Polarimeter (CoMP; Tomczyk et al. 2008) are ideal for this purpose. These propagating disturbances are interpreted as kink or Alfv'enic waves (Van Doorsselaere et al. 2008), and their propagation speeds are naturally linked to the local magnetic field. Based on these CoMP observations, Yang et al. (2020b,a) have successfully measured the global distribution of the coronal magnetic field for the first time. Following these successful attempts, a routine (continuous) measurement of the global coronal magnetic field based on similar observations from the Upgraded CoMP (UCoMP; Landi et al. 2016) has been recently achieved, which allows for the construction of coronal synoptic magnetograms (Carrington maps) (Yang et al. 2024). The CoMP and UCoMP instruments can conduct spectroscopic observations of the Fe XIII lines at 1074.7 and 1079.8 nm in the coronal region above the solar limb. From the Dopplergrams of Fe XIII 1074.7 nm, propagating kink waves can be identified throughout the corona. The propagating or phase speed c k of these waves (also named the kink speed) could be obtained by constructing a timedistance map of Doppler velocity, while the coronal density can be inferred from the observed Fe XIII 1079.8-nm/1074.7-nm intensity ratio. For kink waves, we have where µ 0 is the magnetic permeability, B and ρ are the magnetic field and mass density, respectively. The subscripts i and e refer to parameters inside and outside the magnetic flux tubes (the waveguides). Since CoMP and UCoMP likely cannot resolve individual flux tubes, we can only work with an average density ⟨ ρ ⟩ . Furthermore, in the lowβ coronal environment, the internal and external magnetic fields are often assumed to be approximately equal (see e.g., Tomczyk & McIntosh 2009; Morton et al. 2015; Zhong et al. 2023). This leads to the simplified expression for the kink speed: which is widely used in estimations of coronal magnetic fields (Long et al. 2017; Yang et al. 2020a,b; Yang et al. 2024). Given the potential of these measurements to provide routine coronal magnetograms on a daily basis, which could play a crucial role in future solar physics research, it is essential to thoroughly assess the reliability and robustness of the methodology used in Yang et al. (2020a,b) and Yang et al. (2024). Magyar & Van Doorsselaere (2018) performed 3D MHD simulations of propagating kink waves under various conditions and conducted forward modeling to evaluate the reliability of this method in deriving the magnetic field strength. Their findings indicated that the magnetic field strengths inferred through seismology closely match the input values, typically with an error less than ∼ 20%. However, there is a limitation in their simulation. They utilized a non-stratified setup that excluded the effects of gravity, resulting in uniform initial density and propagation speed in the vertical direction. The gravitational stratification can play a significant role as it causes the Alfv'en speed and kink speed c k to vary with height. This variation can affect the wave tracking method which typically relies on a linear fit of velocity signals (e.g., see Figure 6 in Tomczyk & McIntosh 2009). Therefore, it is important to assess how the gravitational stratification affects the seismological results and estimate the possible error range. So we conducted 3D MHD simulations of propagating kink waves in stratified coronal open flux tubes. Following Magyar & Van Doorsselaere (2018), we employed forward modeling to compare the seismologically derived results with the actual values of physical parameters from our simulation. This paper is organized as follows: Section 2 describes our simulation setup and methodology, Section 3 presents the simulation and forward-modeling results, along with detailed comparisons between seismology results and input values, and Section 4 provides a discussion and summary of our findings.\",\n \"pages\": [\n 1,\n 2,\n 3\n ]\n },\n {\n \"title\": \"2 METHOD\",\n \"content\": \"The model used in this study is a gravitationally stratified, open magnetic flux tube with a radius of 1 Mm, similar to that in Gao et al. (2024b) (hereafter referred to as Paper I). The main difference is that the initial magnetic field in this study is set at around 4 G, instead of 10 G, to better match the seismologically inferred results in Yang et al. (2020b). As in Paper I, the magnetic field is oriented in the z direction and remains nearly uniform across the simulation domain, with only small spatial gradients to maintain total pressure balance. As a result, the Alfv'en and kink speeds increase with height as the density decreases due to stratification. A relaxation process of 2400 s was conducted to achieve a quasimagnetohydrostatic state, as shown in Figure 1. The figure shows that the post-relaxation magnetic field has some spatial variation but mostly within 0.7 G. The kink wave driver is also similar to that in Paper I, with a velocity amplitude of 8 km s -1 and a period of 300 s. The primary velocity perturbation is along the x direction. We ran the 3D MHD simulation in Cartesian coordinates with the PLUTO code (Mignone et al. 2007). The simulation domain spans [-4, 4] Mm × [-4, 4] Mm × [0, 150] Mm, with a uniform grid of 128 × 128 × 1024 cells, providing spatial resolutions of 62.5 km in the horizontal ( x and y ) directions and 146.5 km in the vertical ( z ) direction. We chose a second-order parabolic spatial scheme and a Roe Riemann solver. The boundary conditions were set to be outflow, except for the lower boundary, where the kink wave driver was introduced by adjusting v x and v y , while other parameters were fixed. As in Paper I, the upper 50 Mm ( z > 100 Mm) was set as a velocity absorption region (VAR) to minimize numerical reflections from the upper boundary (see also Pelouze et al. 2023; Guo et al. 2023; Gao et al. 2023). For the subsequent analysis, we only considered the region below z = 100 Mmwhich can give us physical results. Once the kink wave driver was applied, the propagating waves were rapidly excited, with their properties thoroughly analyzed in Paper I. Here, we focus on assessing the reliability of the seismology method described in Section 1 using the simulation outputs. To do so, we performed forward calculations to synthesize spectroscopic observables of CoMP and UCoMP, namely, the Fe XIII 1074.7 and 1079.8 nm spectral line profiles. Specifically, we used synthesized observation of the 1074.7 nm line to perform the wave tracking and determine the propagation speed, while synthesized observation of the 1079.8 nm line was only employed for the density diagnostic using the intensity ratio method (Yang et al. 2020a,b). We applied the FoMo code (Van Doorsselaere et al. 2016) to synthesize the spectral profiles at all pixels in the yz plane. The photo-excitation (see Young et al. 2003; Yang et al. 2020a) was not considered for simplification, as their effects on the spectral lines are not significant at the lower corona. The line of sight (LOS) was chosen as the x axis. In this way, we can reconstruct 2D maps (in the yz plane) of the intensity and Doppler velocity by fitting a single Gaussian to each spectral profile. The resulting maps consist of 128 pixels in the y direction from -4 Mm to 4 Mm, and 500 pixels in the z direction from 0 Mmto 100 Mm. Since the primary velocity perturbation is along the x direction (or in the LOS plane), the Doppler velocity maps capture the wave propagation signals, while the intensity maps do not show any transverse displacement. z (Mm)\",\n \"pages\": [\n 3,\n 4\n ]\n },\n {\n \"title\": \"3.1 Before Degrading\",\n \"content\": \"We first generated time-distance (TD) maps of the Doppler velocity along the z direction, which can be produced at each y position. In Figure 2(A), the TD map along the flux tube's axis (i.e., y = 0 ) is presented, clearly illustrating the propagation of Doppler velocity disturbances. The increasing slope reflects the growing propagation speed with height. However, after 400 s, unusual patterns appear due to wave reflections. Despite efforts to suppress numerical reflections from the upper boundary at z = 150 Mm using a VAR (see Section 2), it is still difficult to eliminate all reflections. Some reflections may come from the layer of z = 100 Mm, which is the lower boundary of the VAR; while others could be attributed to the vertical inhomogeneities of density and phase speed. As noted in previous studies, even smooth phase speed gradients can cause partial wave reflection (e.g., Verdini & Velli 2007; Hahn et al. 2018; Pascoe et al. 2022; Bose et al. 2024). In fact, real CoMP observations of coronal kink waves often reveal both upward and downward propagating components (Tomczyk & McIntosh 2009; Morton et al. 2015). However, when calculating the propagation speed, these reflected waves can introduce significant errors. To reduce this, we applied the Fourier filtering method to separate the upward and downward propagating wave components (see e.g., Tomczyk & McIntosh 2009; Threlfall et al. 2013; Liu et al. 2014; Tiwari et al. 2019; Yang et al. 2020b). Figure 2(B) shows the TD map for the upward-propagating component, which is used to determine the phase speed. By applying the wave-tracking method to all pixels (for more details, see e.g., Yang et al. 2020b), we obtained a 2D phase speed distribution, as shown in Figure 3(A). For most pixels, the phase speed falls in the range of 200-800 km s -1 (see also Figure 4(B)), with a general trend of increasing with altitude, though some fluctuations are present (see relevant discussions in Section 4). Next, we derived the density by calculating the intensity ratio of the synthesized 1074.7 nm and 1079.8 nm intensity maps. The theoretical relationship between intensity ratio and density can be obtained from the CHIANTI database (Dere et al. 2023). The obtained density values were compared with the input values, as shown in Figure 4(A). The input density corresponds to the emissivity-weighted density along the LOS (Yang et al. 2024): where ϵ is the Fe XIII 1074.7 nm line emissivity at the corresponding pixel, calculated using the IDL routine emiss calc.pro from the CHIANTI software package. The comparison shows that the density derived from the intensity ratio is reliable for most pixels, though about 20% pixels show an overestimation ( ≳ 15%). However, this overestimation would not lead to large errors in the seismologically inferred magnetic field, as only the square root of density is needed during the calculation. With the derived phase speed and density, we calculated the magnetic field B seis. Here we chose to treat the phase speed as the local Alfv'en speed (i.e., c ph = B seis / √ µ 0 ρ ), rather than employ Equation(1). Because in our magnetic field configuration, Equation (1) can only provide the phase speed as a function of z ; however, here with a high spatial resolution, we are also interested in the horizontal distribution of phase speed and magnetic field. We note that such a choice may lead to some errors, especially within the flux tube region, which will be further discussed in Section 4. In Figure 4(B), we compared the Alfv'en speed in the model and the wave propagation speed obtained from wave tracking. Despite some noticeable discrepancies, the overall correspondence between the two is reasonably good. The calculated magnetic field is shown in Figure 3(B), and its relative error (compared to the input value, B los, which was calculated in the same manner as ρ los) is presented in Figure 3(C). The contours that correspond to the values of ± 30% indicate that the relative error is less than 30% for most pixels. Figure 4(C) and (D) display histograms of B seis and its relative error. Statistically, there is a slight overestimation of the magnetic field by ∼ 5% (with a standard deviation of 17%). Given that the 5% bias is quite small, the results support the reliability of the coronal seismology technique. Further discussion on the error distributions in Figure 3(C) and the cause of the overestimation can be found in Section 4.\",\n \"pages\": [\n 4,\n 5,\n 6,\n 7\n ]\n },\n {\n \"title\": \"3.2 After Degrading\",\n \"content\": \"We then degraded the spatial and temporal resolutions to match those of CoMP observations, with a pixel size of approximately 3.3 Mm and a cadence of 36 s. We focused on the central pixels (from y = -1 . 65 Mm to y = 1 . 65 Mm), which fully encompass the flux tube (diameter ∼ 2 Mm). At this resolution, the tube boundary and finer structures are unresolved, similar to the case in observational studies (Yang et al. 2020a,b). We can now track the physical parameters along the tube axis. Figure 5 shows the phase speed, density, and magnetic field as a function of height z . In panel (A), the phase speed derived using the wave tracking method ( c seis) is compared to the characteristic speeds in the model (input values). The c seis closely matches the input kink speed ( c k, input), which lies between the internal and external Alfv'en speeds. Panel (B) shows the density derived from the Fe XIII intensity ratio ( ρ Fe XIII), which also falls between the internal density ( ρ i) and external density ( ρ e) in the model. The ρ Fe XIII is slightly higher than the emissivity-weighted density along the LOS ( ρ los), similar to the pre-degradation results shown in Figure 4(A). When calculating the magnetic field with Equation (2), the density value should ideally be the average of internal and external density (i.e., ⟨ ρ ⟩ = ( ρ i + ρ e ) / 2 ), because Equation (2) is derived from Equation (1) when assuming B i = B e. However, in practice, ρ Fe XIII is used, which is slightly lower than ⟨ ρ ⟩ . It means that using ρ Fe XIII as a representative of ⟨ ρ ⟩ can lead to a slight underestimation. The density underestimation is about 12-20% based on Figure 5(B), and the resulting error in B seis would be minimal (less than 5%) since the density is taken the square root of. In fact, the deviation of ρ Fe XIII from ⟨ ρ ⟩ can vary with the filling factor, which represents the fraction of the pixel occupied by the flux tube. Given the current pixel size (3.3 Mm), LOS integration length (8 Mm), and flux tube radius (1 Mm), the filling factor is approximately 12%. If the filling factor is lower, the ρ Fe XIII will be closer to ρ e, increasing the density underestimation. The maximum underestimation depends on the density contrast ζ = ρ i /ρ e, with the deviation factor given by: In our simulation, ζ was initially set to be 3, but after relaxation, it decreased to around 2 (see Figure 1(D)). Such a value is comparable with previous observational estimates (Tian et al. 2012; Verwichte et al. 2013; Morton et al. 2021). We tested the case with a reduced filling factor ( 5%), where a density contrast of 2 led to an underestimation of ∼ 30%. Therefore, parameters like the filling factor and density contrast can be crucial when assessing the accuracy of coronal magnetic field measurements. Figure 5(C) presents the magnetic field inferred from coronal seismology ( B seis), compared to the internal ( B i), external ( B e), and emissivity-weighted ( B los) magnetic fields. The B seis ranges from 3.9 to 5.1 G, with errors below 15%, indicating that the technique of coronal seismolgy can be used for reliable measurements of the coronal magnetic field strengths.\",\n \"pages\": [\n 7,\n 8\n ]\n },\n {\n \"title\": \"4 DISCUSSION AND CONCLUSION\",\n \"content\": \"In this study, we tested the accuracy of the previously developed seismological techniques based on observations of propagating kink waves for deriving coronal magnetic fields. The results indicate that seismology-based measurements of the magnetic field are accurate and reliable to a large extent. In the high-resolution case, we obtained a 2D distribution of the magnetic field ( B seis) and its relative error. For most regions, the relative error is less than 30%, as shown in Figure 3(C). Statistically, the average magnetic field derived from coronal seismology is about 5% larger than the input value. Although this case has a much higher resolution than the CoMP and UCoMP observations, the pixel size (62.5 km in the z direction and 200 km in the y direction) and cadence (12 s) can be comparable to those of the Cryogenic Near-Infrared Spectro-Polarimeter (Cryo-NIRSP; Fehlmann et al. 2023) and the Diffraction-Limited Near-Infrared Spectropolarimeter (DL-NIRSP; Jaeggli et al. 2022) of the Daniel K. Inouye Solar Telescope (DKIST; Rimmele et al. 2020). The intruments offer high-resolution coronal spectroscopic observations with the Fe XIII lines (Schad et al. 2023), and recently Schad et al. (2024) successfully obtained a coronal LOS magnetogram with Cryo-NIRSP observation based on the Zeeman effect. With rapid repeated raster scans, it is possible that DKIST could also detect propagating transverse waves via Doppler velocity measurements, enabling seismological diagnostics of the plane-of-sky (POS) coronal magnetic field. In this way, DKIST observation can also provide us with maps of the POS magnetic field, but with a much higher spatial resolution compared to CoMP and UCoMP. Our results, particularly Figure 3, can serve as a reference for such diagnostics. Combined with the Stokes-V measurements, DKIST may then be able to achieve measurements of the full magnetic field vector. Nevertheless, we note that some errors appear in the seismologically inferred magnetic field. First, large errors appeared near the upper and lower boundaries (around z = 0 and z = 100 Mm). This is due to the Fourier filtering method used to subtract downward propagating wave components. As shown in Figure 2(B), the upward propagating components of the Doppler velocity show some artificial velocity amplification near the upper and lower boundaries, especially after 250 s. This could affect the phase speed determination through wave tracking, leading to errors in B seis near the z boundaries. Thus, in observations, it may be useful to exclude boundary signals before calculating phase speeds. Another contributing factor comes from the wave-tracking process itself. When calculating the phase speed, a certain number of data points along the propagating direction is required. Near the lower and upper boundaries, fewer data points will be available to perform cross-correlation, leading to larger uncertainties in the calculated phase speed and consequently in the inferred magnetic field. Second, overestimations were observed inside or along the lateral boundaries of the flux tube. This is likely because we treat the phase velocity as the Alfv'en speed, which applies well to the regions away from the flux tube or cases with spatial averaging (see Yang et al. 2020b and Section 3.2). However, in this case, we modeled kink waves, and the flux tube can be well resolved. Thus, at the flux tube region, particularly the tube boundary, it would be more appropriate to apply Equation (1) since the waves have a dominant kink wave characteristic (e.g., Goossens et al. 2009). Calculating the magnetic field with B seis = c ph √ µ 0 ρ leads to overestimation at high-density regions, explaining the positive errors within the flux tube in Figure 3(C). In future DKIST observations, we would suggest first applying Equation (2) to diagnose the magnetic field outside the flux tube ( B e), then calculate the magnetic field inside the flux tube ( B i) with Equation (1). In addition, there are some other confusing patterns in Figure 3(B) and (C). For instance, B seis and the relative error manifest fluctuations along the z direction. It might be related to longitudinal oscillations excited by kink waves due to some non-linear effects (e.g., Goldstein 1978; Del Zanna et al. 2001; Terradas & Ofman 2004). Further investigations are needed to understand these patterns. When we degraded the spatial and temporal resolutions to approximately match those of CoMP, we found that the distributions of phase speed, density, and B seis along z all show remarkable similarities to the input values (Figure 5). Again we can notice a deviation near the upper and lower boundaries, particularly the lower one. However, within the height range of 20-80 Mm, the magnetic field error is generally less than 10%. Additionally, the CoMP instrument has recently been upgraded to UCoMP, which has a slightly higher spatial resolution and a larger field of view. Our main conclusions should also apply to the case with the UCoMP observations (Yang et al. 2024). We also tested the case when degrading to UCoMP's pixel size ( ∼ 2.2 Mm), and the results are largely similar to those shown in Figure 5, with a slightly smaller error. Another factor that may impact the phase speed and magnetic field measurements is the magnitude of the input magnetic field and phase speed. A stronger magnetic field and higher phase speed result in steeper slopes in the time-distance maps of Doppler velocity, which can reduce the accuracy of the wave-tracking method, particularly when the spatial and temporal resolutions are degraded. We ran a separate simulation with a background magnetic field of ∼ 10 G, which gives phase speeds of 1-2 Mm s -1 . We found a systematic underestimation of the magnetic field by approximately 30% in this case. Nevertheless, the observed coronal magnetic field around 1 . 05 -1 . 5 R ⊙ is 1-4 G (Lin et al. 2004; Gopalswamy et al. 2012; Kumari et al. 2019; Yang et al. 2020b; Zhong et al. 2023), which is more comparable with the case discussed in Section 3. In conclusion, the coronal seismology technique based on propagating kink waves can provide reliable magnetic field measurements according to our forward modeling. We note that this study only focuses on open coronal structures, including plumes in coronal holes and fan loops at the boundaries of active regions (Morton et al. 2015; Banerjee et al. 2021). Specifically, in our simulation, the magnetic flux tube is perpendicular to the solar surface, with both gravity and magnetic field aligned along the tube axis. However, the propagating kink waves can be detected not only in these structures but also in closed-field regions (e.g., Tomczyk et al. 2007; Yang et al. 2020a,b). Our magnetic configuration can roughly describe one leg of a large-scale closed coronal loop, where the curvature can be neglected. For smaller loops where curvature cannot be ignored, our model is no longer applicable due to differences in gravitational stratification. Nevertheless, for such loops, phase speed along the axis often shows minimal variation McIntosh et al. (2011); Threlfall et al. (2013); Zhong et al. (2023), thus similar to the case in Magyar & Van Doorsselaere (2018), where a model without any vertical gradient in phase speed was used. Therefore, this study complements previous research by highlighting the importance of phase speed variation along the flux tube in coronal seismology. Finally, we would like to mention that our model has some limitations. The vertical gradient of the magnetic field and the magnetic expansion are not included. Additionally, we didn't consider the effect of internal flows along the flux tube, which are frequently reported and can affect the apparent wave propagation speed (e.g., Soler et al. 2011; Morton et al. 2015). Moreover, in real observations, there are often multiple flux tubes overlapping along the LOS, introducing further complexities that may affect wave-tracking accuracy and magnetic field measurements. In Figure 5 of Yang et al. (2024), comparisons between seismological results and global coronal MHD models revealed some discrepancies, particularly at higher latitudes where open field lines may dominate. Future work that incorporates more realistic models could offer a more sophisticated evaluation of the magnetic field measurements through coronal seismology and help clarify the discrepancies in Yang et al. (2024). Acknowledgements This work is supported by the National Natural Science Foundation of China grant 12425301 and the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB0560000). M.G. acknowledges support from the National Natural Science Foundation of China (NSFC, 12203030). T.V.D was supported by the C1 grant TRACEspace of Internal Funds KU Leuven and a Senior Research Project (G088021N) of the FWO Vlaanderen. Furthermore, TVD received financial support from the Flemish Government under the long-term structural Methusalem funding program, project SOUL: Stellar evolution in full glory, grant METH/24/012 at KU Leuven. The research that led to these results was subsidised by the Belgian Federal Science Policy Office through the contract B2/223/P1/CLOSE-UP. It is also part of the DynaSun project and has thus received funding under the Horizon Europe programme of the European Union under grant agreement (no. 101131534). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union and therefore the European Union cannot be held responsible for them. K.K. acknowledges support by an FWO (Fonds voor Wetenschappelijk Onderzoek - Vlaanderen) postdoctoral fellowship (1273221N).\",\n \"pages\": [\n 8,\n 9,\n 10\n ]\n },\n {\n \"title\": \"References\",\n \"content\": \"Aschwanden, M. J., De Pontieu, B., Schrijver, C. J., & Title, A. M. 2002, Sol. Phys., 206, 99 2 Banerjee, D., Krishna Prasad, S., Pant, V., et al. 2021, Space Sci. Rev., 217, 76 9 Bose, S., TenBarge, J. M., Carter, T., et al. 2024, ApJ, 971, 72 4 Chen, B., Shen, C., Gary, D. E., et al. 2020, Nature Astronomy, 4, 1140 1 Chen, S.-X., Li, B., Xiong, M., Yu, H., & Guo, M.-Z. 2015, ApJ, 812, 22 2 Chen, Y., Feng, S. W., Li, B., et al. 2011, ApJ, 728, 147 2 Chen, Y., Li, W., Tian, H., et al. 2023, Research in Astronomy and Astrophysics, 23, 022001 1 Chen, Y., Li, W., Tian, H., et al. 2021, ApJ, 920, 116 1 Del Zanna, L., Velli, M., & Londrillo, P. 2001, A&A, 367, 705 9 Dere, K. P., Del Zanna, G., Young, P. R., & Landi, E. 2023, ApJS, 268, 52 6 Fehlmann, A., Kuhn, J. R., Schad, T. A., et al. 2023, Sol. Phys., 298, 5 8 Fleishman, G. D., Gary, D. E., Chen, B., et al. 2020, Science, 367, 278 1 Gao, Y., Guo, M., Van Doorsselaere, T., Tian, H., & Skirvin, S. J. 2023, ApJ, 955, 73 3 Gao, Y., Hou, Z., Van Doorsselaere, T., & Guo, M. 2024a, A&A, 681, L4 2 Gao, Y., Tian, H., Van Doorsselaere, T., & Chen, Y. 2022, ApJ, 930, 55 2 Gao, Y., Van Doorsselaere, T., Tian, H., Guo, M., & Karampelas, K. 2024b, A&A, 689, A195 3 Goldstein, M. L. 1978, ApJ, 219, 700 9 Goossens, M., Terradas, J., Andries, J., Arregui, I., & Ballester, J. L. 2009, A&A, 503, 213 9 Gopalswamy, N., Nitta, N., Akiyama, S., M¨akel¨a, P., & Yashiro, S. 2012, ApJ, 744, 72 9\",\n \"pages\": [\n 10\n ]\n }\n]"}}},{"rowIdx":4973,"cells":{"bibcode":{"kind":"string","value":"2024arXiv241110270B"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/2411.10270.pdf"},"content":{"kind":"string","value":"\nStrong magnetic fields of old white dwarfs are symmetric about the stellar rotation axes\nS. Bagnulo 1 and J.D. Landstreet 1 , 2\n\n1 Armagh Observatory and Planetarium, College Hill, Armagh BT61 9DG, Northern Ireland, UK\n2 Dept. of Physics & Astronomy, University of Western Ontario, London, Ontario N6A 3K7, Canada\n\nReceived July 5, 2024, accepted October 22, 2024\nABSTRACT\nMany magnetic white dwarfs exhibit a polarised spectrum that periodically varies as the star rotates because the magnetic field is not symmetric about the rotation axis. In this work, we report the discovery that while weakly magnetic white dwarfs of all ages with M ≤ 1 M ⊙ show polarimetric variability with a period between hours and several days, the large majority of magnetic white dwarfs in the same mass range with cooling ages older than 2 Gyr and field strengths ≥ 10 MG show little or no polarimetric variability. This could be interpreted as extremely slow rotation, but a lack of known white dwarfs with measured periods longer than two weeks means that we do not see white dwarfs slowing their rotation. We therefore suggest a di ff erent interpretation: old strongly magnetic white dwarfs do not vary because their fields are roughly symmetric about the rotation axes. Symmetry may either be a consequence of field evolution or a physical characteristic intrinsic to the way strong fields are generated in older stars. Specifically, a strong magnetic field could distort the shape of a star, forcing the principal axis of maximum inertia away from the spin axis. Eventually, as a result of energy dissipation, the magnetic axis will align with the angular momentum axis. Alternatively, symmetry could be the hallmark of a dynamo that operates after the beginning of core crystallisation. We also find that the higher-mass strongly magnetised white dwarfs, which are likely the products of the merging of two white dwarfs, may appear as either polarimetrically variable or constant. This may be the symptom of two di ff erent formation channels or the consequence of the fact that a dynamo operating during a merger may produce diverse magnetic configurations. Alternatively, the massive white dwarfs with constant polarisation may be rotating with periods much shorter than the typical exposure times of the observations.\nKey words. polarisation - stars: white dwarfs - stars: magnetic fields\n1. Introduction\nMost white dwarfs do not show any variability, and more than 97% of them can be safely considered as suitable flux standard stars (Hermes et al. 2017). Because of flux stability, it is generally impossible to measure the rotation period of white dwarfs except when they pulsate or when they have a magnetic field.\nIt is well known that magnetic white dwarfs often show periodically variable signals of polarisation and sometimes also periodic photometric variability. In the local 20 pc volume, more than 20% of white dwarfs have a magnetic field (Bagnulo & Landstreet 2021), and about 20% of the magnetic white dwarfs are photometrically variable (Farihi et al. 2024). Polarimetric variability is explained in terms of a magnetic field not symmetric about the rotation axis so that the magnetic configuration seen by the observer (as encoded in the polarisation signal) varies as the star rotates. The causes of photometric variability are not well understood (see, e.g., Bagnulo et al. 2024b), yet photometric variability of non-pulsating white dwarfs is so clearly correlated with the presence of the magnetic field variations that one may hypothesise that all non-pulsating white dwarfs with light variations are magnetic. If a star belongs to the small class of white dwarfs with a well-detected light curve, then the rotational period may usually be deduced from photometry (see Brinkworth et al. 2013; Hernandez et al. 2024; Oliveira da Rosa et al. 2024), but, in general, the time series of polarisation measurements may allow one to recover the rotational period of a larger sample of magnetic white dwarfs.\nInvestigations of magnetic fields and of rotational periods are closely related not only because the presence of a magnetic field enables period measurements but also because one could expect that the occurrence of a magnetic field and its strength is physically related to stellar rotation. For example, should the field be supported by a dynamo, one could predict fields to be stronger in more rapidly rotating stars than in those that are more slowly rotating. Correlations between the presence of a magnetic field and the rotational period of a star exist even when the field has a fossil origin. Most of the chemically peculiar stars of the main sequence (Ap and Bp stars) have a fossil magnetic field, and they all rotate more slowly than the non-magnetic non-chemically peculiar stars in the same region of the Hertzsprung-Russel diagram (e.g. Donati & Landstreet 2009). Some of the magnetic Ap and Bp stars have well measured rotational periods as long as several months, years, or even decades (e.g. γ Equ, Leroy et al. 1994). There is no clear explanation for this well-established correlation between magnetic fields and slow stellar rotation.\nIn the case of white dwarfs, various samples of data have gradually become available in the past few years to constrain possible scenarios for the evolution of their magnetic fields. It now appears that among the normal-mass white dwarfs ( M ≃ 0 . 6 M ⊙ ) that largely descend from single-star evolution, magnetic fields are very rare and weak during the first 1-2 Gyr of cooling but then gradually become much more common and often very strong after about 2-3 Gyr of cooling (Bagnulo & Landstreet 2021). This evolution may occur as a result of gradual relaxation to the surface of fields present in stellar cores during earlier evo-\non and / or as a result of the operation of a dynamo during core crystallisation (Isern et al. 2017; Schreiber et al. 2021; Bagnulo & Landstreet 2021, 2022; Ginzburg et al. 2022). Among normal-mass stars, those with a magnetic field seem to rotate faster than non-magnetic white dwarfs (Hernandez et al. 2024), supporting the idea that crystallisation-driven dynamo may play a role in the formation of their magnetic fields. Many highly massive white dwarfs ( M ≥ 1 . 0 M ⊙ ), which are probably the product of mergers of two white dwarfs, appear magnetic very early in their cooling age, and the origin of the magnetic field may be due to a dynamo operating during the merging (Tout et al. 2008; García-Berro et al. 2012; Briggs et al. 2015; Bagnulo & Landstreet 2022). This idea is supported also by the evidence that some white dwarfs have very short rotation periods (e.g. Feige 7: 131 min, Liebert et al. 1977; Cl Oct: 12.1 min, Barstow et al. 1995; WD2209 + 113: 70 s, Kilic et al. 2021).\nAt the same time, some polarimetric studies have hinted that a few old magnetic white dwarfs with a field strength of the order of tens of to a hundred megagauss have extremely long rotational periods, showing at most quite small variations even on timescales of decades (e.g. Berdyugin & Piirola 1999; Bagnulo &Landstreet 2019b). Traditionally, this lack of obvious variability is explained by the assumption that such non-varying magnetic white dwarfs have very long rotation periods of the order of centuries (Schmidt & Norsworthy 1991). In the absence of clear observational support or contradiction, this assumption has been widely accepted (e.g. Ferrario et al. 2015). The conclusion is basically that the known periods, P , fall into two very di ff erent families: one with P < ∼ 2 weeks, characterising most of the sample, and a few percent with strong fields that have periods of centuries (i.e. no clearly detected rotation). There is no explanation as to why some old white dwarfs would have extremely long rotation periods. Magnetic braking has generally been invoked but without the support of numerical calculations.\nIn this work, we report new polarimetric observations and combine them with data collected from the literature. We analyse the magnetic variability and thus the rotation of a sample of 74 white dwarfs, and we discuss the constraints that these data place on possible evolution paths for magnetic fields in white dwarfs.\n2. Polarimetric observations of magnetic white dwarfs\n2.1. Observations not previously published\nWe present here 49 unpublished spectropolarimetric observations of 13 magnetic white dwarfs. Fourteen spectra of six white dwarfs were recently obtained with the FORS2 instrument (Appenzeller et al. 1998) of the ESO VLT in the course of a spectropolarimetric survey of the solar neighbourhood. Seven polarised spectra of two other white dwarfs were retrieved from the ESO archive: one of the spectra was obtained with FORS1 and the remaining ones with FORS2 (we note that FORS1 and FORS2 are virtually identical instruments). Fourteen new spectra of four white dwarfs were obtained with the ISIS instrument of the William Herschel Telescope (WHT). For all of these data, the observing strategy and data reduction are identical to the procedures described by Bagnulo & Landstreet (2018) for circular polarisation data and by Bagnulo & Landstreet (2019b) for linear polarisation data. For two of the four stars recently observed with ISIS, we also present here three previously unpublished spectra obtained in the 1970s that we used to study the long-term variability of magnetic white dwarfs. These data consist of lowresolution polarised spectra obtained by Angel and Landstreet\nusing the multi-channel spectrophotometer (hereafter MCSP) on the 5-m Palomar telescope. The instrument set-up and data reduction of these observations are described in detail by Angel & Landstreet (1974). Finally, eleven spectra of WD 1105-340 were obtained with the ESPaDOnS instrument of the CanadaFrance-Hawaii Telescope (CFHT). These data were reduced by the automatic CFHT pipeline LibreEsprit.\nThe log of these unpublished observations is given in Table A.1, and results are described in the sections dedicated to individual stars of Appendix B. Apart from their general purpose of assessing polarimetric variability, our new observations confirm that star WD 1116 -470 is magnetic (it was previously considered as suspected to be magnetic by Bagnulo & Landstreet 2021) and bring the number of magnetic white dwarfs in the local 20 pc volume to 34 (see Bagnulo & Landstreet 2021). We have also discovered two new massive magnetic white dwarfs: WD1619 + 054 and WD1754 -550 (both rapidly variable).\n2.2. Literature data\nWe searched the literature and collected data for all magnetic white dwarfs that were observed in polarimetric mode at least twice as well as for all magnetic white dwarfs for which multiple spectroscopic intensity observations revealed magnetic variability. The way literature data and new observations have been used is described in the next section.\n3. Determination of the variability of the magnetic white dwarfs\nWe are interested in establishing whether the magnetic field of a white dwarf appears constant or variable with time to the observer. Magnetic variability is ascribed to changes in the apparent field geometry as a star rotates. It can be detected mainly via circular spectro- or broadband polarimetry, which are both sensitive to the longitudinal component of the magnetic field. Linear polarisation, which is sensitive to the transverse component of the magnetic field, is usually much weaker than circular polarisation, and it is rarely detected in white dwarfs. Intensity spectra (if the star is not featureless) are sensitive to the mean field modulus and may also be used to detect magnetic variability, but they are less sensitive to changes of the apparent magnetic configuration than polarimetry. Magnetic white dwarfs that have been repeatedly observed in spectroscopic mode and that do not show sign of variability have not been considered here because having observed a constant intensity spectrum is not a su ffi ciently strong indication that a star is not magnetically variable. An example is WD2047 + 372, as it shows an almost constant Zeeman triplet in multi-epoch intensity spectra and a variable, sign-reversing circular polarisation spectrum (Landstreet et al. 2017).\nPhotometric variability is observed in magnetic white dwarfs, but in the absence of convincing observational evidence or theoretical arguments, we have chosen not to consider it alone as a proxy for magnetism nor magnetic variability. For example, Brinkworth et al. (2013, see in particular their Tables 1 and 3) detected light variability for a number of magnetic stars. For many of them, however, the light amplitude is quite low, and there are no multi-epoch polarimetric data to confirm magnetic variability. Furthermore, recent analysis of TESS data made by Hernandez et al. (2024) and Oliveira da Rosa et al. (2024) failed to detect a periodic light curve for many of these targets. These stars are not included in our sample. In summary:\n1) Two or more intensity spectra, or polarisation spectra, or\nbroadband polarisation measurements that clearly di ff er from each other are interpreted as being due to magnetic variability.\n\n2) Repeated observations of circular polarisation (either spectropolarimetry or broadband polarimetry) that are consistent among themselves within uncertainties are interpreted as evidence of a lack of magnetic variability (that is, the observer always sees the same magnetic configuration).\n3) No conclusion will be drawn from a series of intensity spectra obtained in di ff erent epochs that appear similar to each other within uncertainties.\n4) Photometric variability that is not associated with observed spectroscopic or polarimetric variability will not be used to decide that a star is magnetically variable.\n\nWegathered reasonable evidence for polarimetric variability, or non-variability, for the sample of 74 magnetic white dwarfs listed in Table 2. Admittedly, for some stars, we found that establishing whether or not two or more observations are consistent among themselves was somewhat subjective. In Sect. 5.3, we argue that our results are not a ff ected by these uncertain cases. Appendix B discusses in detail all individual stars. In this section, we summarise the results and flag special cases.\nAmong the 74 white dwarfs, 44 are magnetically variable (Sect. 3.1), while for 27 there is so far no evidence of variability (Sect. 3.2). The remaining three white dwarfs, which have strong fields and have been observed over several decades, may be further tested for subtle or possibly very slow variability (Sect. 3.3).\n3.1. Stars that show evidence of magnetic variability\nFor 30 white dwarfs among the 44 stars that show polarimetric variability, a rotational period, or at least a good candidate for it, has been established in the literature. This period is given (in days) in the last column of Table 2 and is followed by ':' when only tentative or approximated values are known. Rotational periods are generally of the order of hours; about 25% are of the order of one day or longer, and only two stars have a rotation period longer than one week (the slowest magnetic white dwarf for which the period is known, WD 2316 + 123, has P ≃ 17 . 4 d, Schmidt & Norsworthy 1991). Most of these stars were listed in Table C.1 of Hernandez et al. (2024), and they are shown with blue symbols in Fig. 1 of Sect. 4.\nAmong the other 14 variable stars, we do not have enough data to estimate the rotation period. For ten of them, the evidence of polarimetric variability has been securely established by two or more observations. These ten variable stars are marked in last column of Table 2 with 'var.' and are also represented with a blue symbol in Fig. 1. The remaining four white dwarfs show subtle signs of variability among the observations obtained with the same instrument. They are marked in Table 2 with 'var.:' and are shown with light blue symbols in Fig. 1. For WD 0446-789 and WD1009-184, our conclusion is that the field variations are real but small due to a configuration probably nearly symmetric about the rotation axis. WD 1105-048 was repeatedly observed with FORS1, FORS2, and ESPaDOnS, and a field was detected on only two occasions out of 12 observing epochs. If the star is magnetic, then it is certainly variable with a timescale of the order of days or weeks, but, admittedly, one could suspect that the two detections are in fact spurious. We decided to define the star as a probable variable star. WD 2049-222 shows a signal of polarisation that is marginally higher than instrumental polarisation, but for the reasons explained in Sect. B.68, we decided to consider its subtle variations as real.\n3.2. Stars for which there is no evidence of magnetic variability\nNineteen stars have been observed three or more times in polarimetric mode, and the observations are always consistent among themselves within uncertainties. We consider these stars as established non-variable white dwarfs. They are marked with 'n.v.' in Table 2 and represented with red symbols in Fig. 1. Among these 19 stars, four were observed over a time interval of decades and with di ff erent instruments: WD 0548 -001 = G9937, WD0553 + 053 = G99-47, WD1036 -204 = LP790 -29, and WD1829 + 547 = G227-35. Obviously, it is possible that the other members of this group of 'non-variable stars', for which the observations span a time interval of up to a few months or years, are actually variable on a timescale of decades and that as such they could be stars with (so far undetected) long-term variability.\nEight stars should be considered simply as candidate nonvariable white dwarfs ('n.v.:' in Table 2 and magenta symbols in Fig. 1). For seven of them, the reason for considering them only as candidate non-variable stars is that they have only been observed twice. In particular we mention that for normal-mass WD0236-269 and for the massive WD0330 -000, the nonvariability is inferred based on comments in the original discovery paper by Schmidt et al. (2001), which represents especially weak evidence. Another candidate non-variable star, WD1750 -311, was observed only once (with FORS2). The comparison of spectra obtained within that single observing series shows some change of circular polarisation and certainly a rapidly variable intensity spectrum (timescale of minutes) in the regions of the Balmer lines, particularly around H β . The star also shows photometric variability with a 35-min period (Macfarlane et al. 2017). Our interpretation is that WD 1750-311 is probably a magnetic white dwarf with a field symmetric, or nearly symmetric about the rotation axis, showing strong abundance patches of hydrogen.\n3.3. Stars possibly subtly or slowly variable\nThere are three stars that have been monitored for decades, WD1748 + 708 = G240 + 708, WD1900 + 310 = Grw + 70 · 8247, and WD2010 + 310 = GD229, for which our conclusions are uncertain. These stars, marked with 's.v.:' in Table 2 and shown with black symbols in Fig. 1, may indeed show some low amplitude and possible long-term variability, but there might be room for a di ff erent interpretation. The recent detailed comparison of old and new polarisation spectra of Grw + 70 · 8247 by Bagnulo & Landstreet (2019b) identifies small changes between spectra taken decades apart (see in particular their Fig. 5), but for the two other stars, it is not entirely clear which changes are real and which are due to di ff erences between observational equipment and techniques. Bagnulo & Landstreet (2019b) assumed that timescales of the observed small changes were those corresponding to the large time gaps between isolated data sets. However, because the changes are generally small, one could argue that the subtle changes have timescales as short as weeks and have not have been noticed because the stars were not adequately monitored. For example, as previously discussed, WD 1105 -048 appears constant (and non-magnetic) in ten out of 12 observations, and a magnetic field was detected in only two epochs. In case of WD 1748 + 708, photometry leads to ambiguous results. The star was found photometrically variable with a period between 5 and 48 h by Antonyuk et al. (2016). Brinkworth et al. (2013) did not confirm its short-term photometric variability (nor\ndid Hernandez et al. 2024) but instead claimed detection of light changes over a period of ten months.\n4. Clustering in parameter space\nIn this section, we correlate the variability or non-variability of the stars with their mass, field strength, and stellar temperature. The left panels of Fig. 1 show the field strength versus e ff ective temperature and cooling age for the stars of our sample, with the cooling age on both a linear scale and a logarithmic scale. In these plots, stars are represented with symbols that increase in size with a star's mass, while di ff erent colours are used to mark the characteristics in terms of variability. The right panels of Fig. 1 show the cooling age-mass diagrams for the same sample, with cooling age reported again both with linear and logarithmic scales. The size of the symbols is proportional to the field strength. With the help of these figures and Table 2, we studied the relationships between field strength, mass, age, and variability of the magnetic white dwarfs. We recall that our data do not come from a volume-limited sample of stars and do not reflect the relative density of magnetic white dwarfs in di ff erent regions of the diagrams of Fig. 1. In fact, both young magnetic normalmass white dwarfs and ultra-massive white dwarfs are quite rare objects in space, and they are over-represented in our sample because of biases of the surveys (see Bagnulo & Landstreet 2022). The following analysis is about the relative frequency of variable and non-variable stars in di ff erent regions of the diagrams where we noticed the existence of statistically remarkable di ff erences.\n4.1. Normal-mass and weakly magnetic white dwarfs\nThere are 25 white dwarfs with a field strength of ≤ 1 MG, and all of them but one (WD 2051-208) have M ≤ 1 . 0 M ⊙ . These 'normal-mass' 'weak-field' white dwarfs span an age range up to τ ≈ 6 Gyr. Twenty-one of these 24 stars are variable. Among the three that are non-variable, one is younger than 1 Gyr: WD 1105 -340 ( τ = 0 . 34 Gyr).\n4.2. Normal-mass and strongly magnetic white dwarfs\nThere are 25 magnetic white dwarfs with M ≤ 1 . 0 M ⊙ and field strength of ≥ 10 MG. Among those that are younger than 2 Gyr, six out of eight white dwarfs are variable; however, we recall that young strongly magnetic white dwarfs are rare in space (Bagnulo & Landstreet 2022). Among the magnetic white dwarfs older than 2 Gyr, only one out of the 17 shows polarimetric variability.\n4.3. Massive magnetic white dwarfs\nFifteen stars of our sample have M > 1 . 0 M ⊙ , and all but one are younger than about 1 Gyr. Eight of them are magnetically variable, including the only old ( τ ≃ 4 Gyr) star WD 0756 + 437. Nearly all of these massive magnetically variable white dwarfs have a strong field (tens to hundreds megagauss), except for WD2051-208, which is the only ultra-massive star ( M = 1 . 2 M ⊙ ) with a sub-MG field strength (0.25 MG) in our sample.\nIn contrast, seven massive magnetic white dwarfs show no obvious signs of variability, although none of them may be safely considered as non-variable, either because they were observed only twice (or even just once, i.e. WD 1750-311; see Appendix B.60) or because some subtle sign of variability may have been detected (in WD 1900 + 705 and WD2010 + 310). WD1658 + 440 and WD1750-311 are the only massive non-\n
\n\n
Table 1. Frequency of polarimetrically non-variable stars among normal-mass white dwarfs ( M ≤ 1 M ⊙ ).
\n
\nNotes. The full list includes 59 stars, 23 of which are non-variable. Between brackets, the table also gives the number of non-variable stars N K divided by the number of stars N tot in the subset as defined by the age and field values.\nvariable white dwarfs in our sample with a field as low as ≈ 2 MG. The remaining five non-variable stars have magnetic fields with strengths of the order of hundreds of megagauss.\n4.4. Statistical analysis of normal-mass white dwarfs\nBecause we have only one example of an old massive magnetic white dwarf, we do not know how variability in massive magnetic white dwarfs evolves with time. Vice versa, an evolutionary path is clearly seen in normal-mass ( M < ∼ 1 M ⊙ ) white dwarfs. In our sample, field variability is nearly ubiquitous among weakly magnetic white dwarfs of all ages and in young white dwarfs regardless of the field strength, it is but very rare in old ( τ ≥ 2 Gyr) strongly magnetic ( B ≥ 10 MG) white dwarfs. Next, we assess the statistical significance of this pattern, specifically whether it can be attributed to small number statistics or if it reflects a genuine correlation between cooling age, field strength, and field variability.\nWe first split the sample of stars with M ≤ 1 M ⊙ into two groups: those younger than 2 Gyr and those older than 2 Gyr. Each of these two groups was divided into two subsets: stars with a field strength ⟨| B |⟩ ≤ 1 MG and stars with ⟨| B |⟩ > 10 MG. For each of these four subsets, we considered the ratio between the number of non-variable magnetic stars, N K, and the total number of magnetic stars, N tot. From these numbers, we estimated the probability, Pr , that the sample frequency of non-variable stars is between r and r + d r , normalised to one and obtained assuming that all numbers between zero and one are a priori equally probable, using\nPr = ( N tot + 1)! N K! ( N tot -N K)! r N K (1 -r ) ( N tot -N K) . (1)\nFigure 2 shows the probability density functions for the stars belonging to these sets. It clearly appears that there is little to no overlap between the density functions of the symmetric field in old strongly magnetic white dwarfs and of young white dwarfs. Table 1 reports the points of maximum for these distributions, f = N K / N tot, and their uncertainties\nr f (1 -f ) N tot . (2)\nTable 1 also includes the results for the smaller sets of stars with intermediate strength, 1 ≤ ⟨| B |⟩ ≤ 10 MG (with endpoints overlapping with the other two sets), which could possibly be considered a 'transitioning' field strength range.\nFigure 3 shows the ratio between the number of non-variable magnetic white dwarfs with a field strength lower than a given\n\n\n
Fig. 1. Correlations between field strength, magnetic variability, and other stellar parameters. Left panels: Field strength versus e ff ective temperature and versus cooling age for variable and non-variable stars of Table 2. The size of the symbols is related to the mass of the star as shown in the legend. Red symbols refer to stars that were observed at least three times with no evidence for variability; magenta symbols are for stars that were observed only twice and are tentatively assumed non-variable. Blue symbols refer to stars that show variability. Light blue symbols indicate stars that show marginal but probably real signs of variability, maybe due to a field nearly aligned with the rotation axis. Black symbols are used for stars that show signs of long-term variability that we were not able to interpret. Right panels: Temperature-mass and cooling age-mass diagrams for the same sample of stars. The size of the symbols is related to the field strength as shown in the legend. The meaning of the colour is the same as for the left panels. Black lines represent the onset of crystallisation (solid line for H-thick envelop and dashed line of H-thin envelop) obtained via interpolation of the cooling tables by Bédard et al. (2020). In all panels, the yellow vertical and horizontal lines highlight the position of the box tick marks. We recall that these diagrams do not refer to a volume-limited sample and that both young normal-mass magnetic white dwarfs and highly massive magnetic white dwarfs, which are rare objects in space, are over-represented.
\n\n\n\n
Fig. 3. Cumulative fractions of variable and non-variable magnetic white dwarfs with M ≤ 1 M ⊙ . The blue solid line represents the number of non-variable stars younger than 2 Gyr divided by the total number of stars younger than 2 Gyr with an average field strength lower than the value in the x -axis. The red solid line refers to the same quantity for magnetic white dwarfs older than 2 Gyr. Dotted lines show the estimate of the uncertainties.
\n\n\n\n
Fig. 2. Probability density functions of the non-variable fields in normal-mass white dwarfs with the age and field strength specified in the legend.
\n\nvalue ¯ ⟨| B |⟩ and the total number of magnetic white dwarfs with a field strength lower than that value as a function of ¯ ⟨| B |⟩ for stars with cooling ages τ ≤ 2 Gyr (blue lines) and stars with τ > 2 Gyr (red lines). This plot supports our claim that nonvariable fields are much more common in old, strongly magnetic white dwarfs than in young strongly magnetic white dwarfs, and than in weakly magnetic white dwarfs of all ages. It suggests also that the minimum field strength required for a field to become non-variable is in the range of 5 to 10 MG.\n5. Explanation for the lack of observed variability\nThere are three possible reasons for the observed non-variability of so many magnetic white dwarfs. We examine them in the following sections.\n5.1. Extremely slow rotation\nWe first consider the possibility that the rotation of magnetic white dwarfs slows with age and / or magnetic field strength. Perhaps, as the surface field becomes stronger, the white dwarf loses angular momentum to its environment by electromagnetic dipole radiation (García-Berro et al. 2012), by coupling with gas clouds in the ISM, or by a very weak magnetically coupled wind. Each of these possibilities would shed stellar angular momentum faster from a magnetic white dwarf with a strong field than from one with a weak field, preferentially slowing the magnetic white dwarfs with strong fields. However, all of these mechanisms are expected to exert at most a very weak influence on the rotation of a white dwarf, and it seems quite unlikely that any of them could slow the rotation of a magnetic white dwarf so much that repeated observations even a year apart would not show any rotation. Nevertheless, the idea of extremely slowly rotating white dwarfs has been generally accepted. This hypothesis likely origi-\nates from Sect. 3 of Schmidt & Norsworthy (1991), which says: 'We therefore assign long periods to these stars, with the recognition that other explanations for their polarimetric constancy are possible.' This statement was accompanied by the first plots of field strength versus rotational period, a plot that in updated form has reappeared in numerous reviews (e.g. Ferrario et al. 2015; Kawka 2020; Ferrario et al. 2020) but has never been critically revisited.\nWe note that there is a complete lack of white dwarfs that are known to vary on a timescale between weeks and decades. In this respect, there is a profound di ff erence between candidate longterm variable magnetic white dwarfs and long period magnetic Ap and Bp stars. While we do not know of any white dwarf with a firmly established polarimetric variability with a period longer than approximately two weeks, we are sure that a number of Ap and Bp stars are very slowly rotating stars because they definitely show clearly periodic field variation, even ⟨ Bz ⟩ sign reversals, on a very long but measured timescale that may be months, years, or decades. Furthermore, the distribution of periods is roughly continuous between periods of less than one day and periods of several years (Mathys 2008). If we wanted to interpret the lack of polarimetric variability in white dwarfs as the e ff ect of extremely long rotation periods, we would need to accept that the rotation periods of white dwarfs show an extremely bimodal distribution that peaks around hours and days and around centuries with nothing in between. This appears to be a very unlikely scenario.\n5.2. Extremely fast rotation\nAsecond possibility is that some or all of the non-variable white dwarfs actually are very rapidly rotating stars. If the star has a\nrotation period much shorter than the individual frame exposure time, then the polarimetric variability would be smeared out, and the star would appear constant.\nIn fact, it has been possible to probe variability on the timescale of the exposure time of each individual frame (typically 10 min or less). In very general terms, our FORS2 and ISIS polarimetric observations allowed us to identify a star as variable provided that its rotation period is longer than ≈ 10 min and its magnetic configuration is clearly not symmetric about the rotation axis (for example, WD 1712-590 in Sects. B.57). Isolated white dwarfs that are the product of single-star evolution can hardly have rotation periods shorter than that (Kawaler 2015). This is confirmed by the results of Hernandez et al. (2024), who have shown that rotation periods of young magnetic white dwarfs in the normal mass range are only marginally shorter than the rotation periods of non-magnetic white dwarfs. Merger products, in contrast, can have rotation periods of the order of 1 min (Schwab 2021; Kilic et al. 2021) if most of the binary angular momentum is retained by the merger product. It is therefore possible that the non-variable massive white dwarfs have nonaxisymmetric fields but are rotating very rapidly. Periods down to 1 min or less can be probed via rapid cadence photometry. None of the massive, magnetically non-variable white dwarfs show TESS light variability (Hernandez, priv. comm.; Ramsay, priv. comm.), although of course one cannot rule out that more accurate photometry could reveal some extremely rapid rotators. Provisionally, we rule out very rapid rotation for the massive magnetic white dwarfs that show a constant polarisation.\n5.3. Magnetic fields are symmetric about the stellar rotation axes\nAfter ruling out both extremely slow and extremely fast rotation, we are left with the hypothesis that a non-variable field is approximately axisymmetric about the rotation axis. This means that as the star rotates with a normal period between a fraction of an hour and several days, the observer does not see any significant change in the signal of circular polarisation.\nThis interpretation naturally accounts for some outliers, such as the non-variable weakly magnetic star WD 1105-340. It is reasonable to hypothesise that this one star does not appear variable simply because its rotation axis is tilted at a small angle with respect to the line of sight.\nIn Sect. 3, we highlighted that it may be di ffi cult to firmly assess whether a star is magnetically variable because changes could be too subtle to be detected. However, the risk of classifying as 'non-variable' a star that in fact exhibits subtle but real changes does not weaken our analysis because small changes of the circular polarisation spectrum are still symptoms of a field nearly symmetric about the rotation axis.\n6. Discussion\nHere, we first consider the case of normal-mass white dwarfs (Sect. 6.1) and then that of massive stars (Sect. 6.2). We note that our analysis applies only to isolated white dwarfs; symmetric fields seem uncommon among magnetic white dwarfs accreting material from a companion (Cropper 1988; Reimers et al. 1999; Schmidt et al. 1995).\n6.1. Normal-mass white dwarfs\nIt is clear that most of the weak fields of normal-mass stars of all ages are not symmetric about the stellar rotation axis. Most of the strong fields of normal-mass, young white dwarfs are also not axisymmetric. Most of the strong field of stars older than 2 Gyr are symmetric about the star's rotation axis. We now investigate what are the relationships between these weak and strong fields.\nOur hypothesis is that the weak and non-symmetric magnetic fields of the young normal-mass white dwarfs are due to a relaxation process of a field that was produced in a previous evolutionary stage of the star, a field that was buried below the surface of the newly formed white dwarf, and that it starts to reveal itself with time as the star cools down. Such seed fields could be those found in red-giant stars, in the vicinity of the hydrogenburning shell (e.g. Li et al. 2022, 2023). The question is whether the older, generally stronger fields share essentially the same origin as the weak, younger ones, having just evolved for longer time, or if the strong fields in older stars formed by a totally different mechanism than the weaker, younger ones, for instance by a crystallisation dynamo. Whatever its origin is, we consider first the possibility that the small distortion of the shape of a magnetic white dwarf produced by magnetic forces in outer layers might actually drive the evolution of the global field towards axisymmetry. In Sect. 6.1.1, we discuss one example of a physical e ff ect that might drive such evolution.\n6.1.1. The magnetic field structure of white dwarfs may evolve towards rotational axisymmetry during cooling\nIn the absence of a magnetic field, the rotation of the white dwarf will lead to an equatorial bulge, increasing the component of the principal moment of inertia that is parallel to the rotation axis. In this case, the rotation about the spin axis, which coincides with the largest moment of inertia, is stable. Next, we introduce a dipolar magnetic field inclined to the axis of rotation by, say, 45 · . We suppose that the visible surface field is maintained by a fossil current system deep in the star, which is gradually decaying due to Ohmic losses. As the interior field strength decreases, an azimuthal electric field will be produced by Faraday induction outside the white dwarf's initial current loop. This field will generate a current around the magnetic axis that will in turn interact with the local meridional magnetic field (Landstreet 1987) to produce an outward-directed force, distorting the stellar outer layers into a slightly oblate shape, with the largest radius around the magnetic equator. Because of this e ff ect, the principal moment of inertia of the star will be rotated towards the symmetry axis of the field, which is not aligned with the rotation axis or the angular momentum axis of the star. The white dwarf e ff ectively becomes an unstable free asymmetric rotator (an asymmetric top). If such objects can dissipate some of the energy supporting the asymmetric shape, the field axis will gradually shift to bring the principal moment of inertia closer to the angular momentum axis, which is a stable minimum energy state of an oblate system. Energy dissipation will gradually lead to the alignment of the magnetic axis with the rotation axis, which is the e ff ect we observe in old normal-mass strongly magnetic white dwarfs.\nThis basic physical e ff ect was first identified in the 1970s as possibly operating in main-sequence magnetic Ap stars, which, similar to magnetic white dwarfs, possess global fossil magnetic fields that are usually roughly dipolar and are generally oblique to the rotation axis (Stibbs 1950; Preston 1971). A number of efforts to estimate the timescale of possible evolution of an oblique\nmagnetic field to a state of small or vanishing obliquity have been published, for example, by Mestel & Takhar (1972). Much of this work is summarised in Chapter 9 of Mestel (1999). However, this line of investigation did not lead to a convergence of clear results about possible timescales or about dependence on basic input physics or parameters, such as interior field strength, or details of induced forces in outer layers and their consequences. As the increasing sample of magnetic Ap star models failed to reveal an obliquity distribution suggestive of this e ff ect, possibly because of the relatively short (of order 10 8 yr) mainsequence lifetimes of magnetic Ap stars, theoretical studies of stellar fossil fields moved on to other e ff ects. However, a similar e ff ect may be at work aligning the magnetic fields to the rotation axis in white dwarfs, which have a very di ff erent structure and much longer evolutionary times than Ap stars. This hypothesis could be explored with the aid of numerical modelling of the global (interior and surface) structure and evolution of a rotating white dwarf with an oblique magnetic field. Modelling of comparably complex white dwarf states (e.g. polars, mergers) that include magnetic fields (e.g. Franzon & Schramm 2015; Bisikalo et al. 2021; Zhong et al. 2024) suggest that numerical methods for exploring this problem are already available.\n6.1.2. Crystallisation-driven dynamo and axisymmetric fields\nThe line that marks the beginning of core crystallisation in the age-mass diagram separates variable from non-variable magnetic white dwarfs perhaps in a cleaner way compared to a massindependent age threshold (see Fig. 1). Before core crystallisation begins, magnetic fields are almost always non-symmetric about the rotation axis. From volume-limited surveys, we know that in normal-mass white dwarfs, before crystallisation, strong fields are rare (Bagnulo & Landstreet 2021, 2022), but those that have been discovered and monitored show polarimetric variability. After the beginning of core crystallisation, many strong fields appear, and most of them are symmetric about the rotation axis. White dwarfs with weak and non-symmetric fields continue to appear also after the beginning of core-crystallisation.\nThe strong magnetic fields of old normal-mass white dwarfs could be generated by the crystallisation convective dynamo mechanism (Isern et al. 2017; Schreiber et al. 2021), as almost all have cooling ages longer than the cooling age required for the onset of this dynamo. In this case, the conclusion would be that the crystallisation dynamo is responsible for approximate symmetry about the rotation axis. This situation could have similarities with what has been observed in fully convective late type stars with no di ff erential rotation, which are known to be able to generate strong and simple large-scale, mostly axisymmetric, poloidal fields (Donati et al. 2006; Morin et al. 2008b,a; Kochukhov 2021, Fig. 14). However, how the crystallisation dynamo alone can produce fields stronger than ∼ 1 MG is still unclear (Isern et al. 2017; Castro-Tapia et al. 2024). In fact, Montgomery & Dunlap (2024) have argued that fluid mixing by phase separation is not a viable mechanism to produce the strong fields observed in old white dwarfs. Perhaps the crystallisation dynamo could be more powerful if it were to amplify a pre-existing internal field, such as the same seed field that is seen in some of the white dwarfs prior to the beginning of crystallisation.\nMontgomery & Dunlap (2024) have proposed that corecrystallisation could trigger temporary di ff erential rotation. Therefore, one could suspect that rapid rotation in normal-mass white dwarfs might generate a magnetic field in the stellar core using rotational shear on a seed field left from an earlier point in the star's evolutionary history. In this situation, we might also\nexpect that the field would be roughly axisymmetric, although it is not clear why such fields would always be strong or how they might be related to the weaker oblique fields of stars of similar mass. Furthermore, Spruit (1999) has shown that di ff erential rotation would suppress the non-axisymmetric field component of weak fields, but not in the stronger fields. This is the opposite of what we observed.\nThe origin of strong non-axisymmetric fields of normal-mass white dwarfs that appear before crystallisation could possibly be similar to that of higher-mass white dwarfs (see Sect. 6.2 below).\n6.2. The variability of massive magnetic white dwarfs\nCompared to normal-mass white dwarfs, the magnetic fields of massive white dwarfs present a di ff erent behaviour. Magnetic fields appear when the stars are still very young, and fields may be either symmetric or non-symmetric about the rotation axis. In massive white dwarfs, strong fields (tens or hundreds of megagauss) seem much more common than weaker sub-megagauss fields. It is widely thought that many of these high-mass objects are the result of WD-WD mergers that cause rapid generation of a strong magnetic field (García-Berro et al. 2012; Bagnulo & Landstreet 2022). However, Camisassa et al. (2022) and Blatman & Ginzburg (2024) have shown that at least some of them, depending on their core composition, may have started the process of core crystallisation. Hence, the origin of their field could be linked to the crystallisation dynamo, at least in some cases.\nThe bimodal distribution of the morphologies of the fields of the massive magnetic white dwarfs could indeed reflect two di ff erent channels of formation, one from WD-WD merger (or by a di ff erent binary evolution path), and one by massive singlestar evolution, such as that of normal-mass white dwarfs. Alternatively, the dynamo stimulated by WD-WD merging may be capable of generating both axisymmetric and non-axisymmetric global fields, perhaps depending on the initial angular momentum vectors of the individual merging white dwarfs relative to the orbital angular momentum vector or the mass ratio of the two merging white dwarfs.\nA viable alternative explanation is that the massive white dwarfs that do not show variability are actually rotating with a period much shorter than the typical exposure time of individual polarimetric measurements. In any case, the large mass of stars in this sample and the occurrence of some rotation periods as short as minutes make it very reasonable to suppose that the evolution of the magnetic fields and rotation periods follows a di ff erent course from the evolution of fields in normal-mass magnetic white dwarfs.\nBecause almost all of the massive magnetic white dwarfs of our sample, except one, are very young, we cannot test whether the morphology of older stars is generally axisymmetric. Remarkably, however, the only example we know of a massive star older than 2 Gyr shows a non-axisymmetric field. With an age of more than 4 Gyr, high mass, and a very strong and variable field, the very unusual WD 0756 + 437 may be considered further evidence that the origin of the fields in massive stars is di ff erent than in normal-mass stars.\n7. Conclusions\nSince the earliest discoveries of magnetic white dwarfs (Kemp et al. 1970; Angel & Landstreet 1970, 1971b), two quite di ff erent categories of them have been known to exist. Some magnetic white dwarfs show periodic variations of circular polarisation\n(which probes the longitudinal magnetic field) with timescales ranging from minutes to days. Classically, a signal of circular polarisation constant with time has been interpreted as indicating either a very long rotation period or a lack of rotation.\nUsing both literature data and new observations, we have studied the polarimetric variability of a sample of 74 magnetic white dwarfs. We find that among white dwarfs with M ≤ 1 . 0 M ⊙ ('normal-mass white dwarfs'), nearly all stars with fields weaker than about 1 MG show circular polarisation varying with time. Furthermore, the rare normal-mass white dwarfs younger than ≈ 2 Gyr with strong magnetic fields also show polarimetric variability. In striking contrast, 16 out of the 17 normal-mass stars older than 2 Gyr with fields stronger than about 10 MG in our sample show constant polarisation. Magnetic white dwarfs with M ≥ 1 . 0 M ⊙ ('massive white dwarfs'), many of which are the product of WD-WD merging, show a mixed behaviour. Many of them have a strong magnetic and variable field, but some have a strong and constant magnetic field. In our sample, nearly all the massive magnetic white dwarfs are younger than ≈ 1 Gyr, with the exception of WD 0756 + 437, an old strongly magnetic variable star.\nThe lack of evidence for major variations of circular polarisation on any timescale longer than about two weeks suggests that the interpretation of non-variability arising from extremely long rotational periods is incorrect. We have argued that the nonvariability is due to magnetic field structures roughly symmetric around the star's rotation axis.\nA possible explanation for the observed symmetry could be that most or all of the magnetic fields evolved from pre-existing fields that formed during pre-white dwarf evolution stages (Li et al. 2022, 2023) and gradually relaxed to the stellar surface over a relation time of 1 or 2 Gyr; in fact, very recently, Camisassa et al. (2024) have shown that the magnetic fields of old white dwarfs with M ≥ 0 . 65 M ⊙ may have been generated by a coreconvection dynamo when the star was in the main sequence, and emerged at the stellar surface during the white dwarf cooling phase. As the surface field increases in strength, at some point the Lorentz forces in the outer layers, together with the Coriolis forces acting on rotating convective flows in what are asymmetric rotators, could lead to a gradual relaxation, through energy dissipation, of the global field structure to a form that is symmetric about the rotation axis. The timescale would depend on the field strength, and for a field weaker than several megagauss, it could require a timescale too long to be observed in white dwarfs that still show spectral lines.\nAlternatively, because these normal-mass large-field magnetic white dwarfs mostly occur after the start of crystallisation (Bagnulo & Landstreet 2021, 2022), one could speculate that the crystallisation dynamo (Isern et al. 2017; Schreiber et al. 2021; Ginzburg et al. 2022) may generate some fraction of the observed fields and further that this dynamo produces essentially axisymmetric surface magnetic field structures with extremely strong fields. This kind of origin would have similarities with the axisymmetric fields that are commonly found in fully convective strongly magnetic M-dwarfs and, much weaker, in the planets Earth and Jupiter. However, it has been argued that the instability of the mantle surrounding the core that starts to crystallise cannot produce fields much stronger than 1 MG (Isern et al. 2017; Montgomery & Dunlap 2024; Castro-Tapia et al. 2024). Furthermore, Camisassa et al. (2024) find that, even if the crystallisation-driven dynamo could generate a strong magnetic field, this field would take too long to emerge at the stellar surface. On the other hand, Montgomery & Dunlap (2024) have suggested that crystallisation may still play a role by trig-\ngering a temporary phenomenon of di ff erential rotation, which in turn would generate a magnetic field - the di ff usion timescale of which has not been discussed.\nThe situation for ultra-massive white dwarfs (with M > ∼ 1 . 0 M ⊙ ) is di ff erent, as both strongly magnetic variable and nonvariable white dwarfs are found among them. Some of these massive white dwarfs are the product of WD-WD merging, during which a dynamo could create a strong magnetic field (García-Berro et al. 2012), though it would not necessarily be symmetric about the rotation axis. In addition, some massive magnetic white dwarfs could be the result of single-star evolution and have acquired an axisymmetric field following the same mechanism acting in normal-mass white dwarfs.\nFurther investigation into the variability of magnetic white dwarfs is necessary. Nevertheless, current observations have already imposed significant constraints that any theory explaining the origin and evolution of magnetic fields in degenerate stars must take into account.\n
\n\n
Table 2. Stars used in this work, their main physical parameters, and a note on their magnetic variability.
\n
\nS. Bagnulo and J.D. Landstreet : Strong fields of old white dwarfs are axisymmetric\n
\n\n
Table 2. Continued.
\n
\nNotes. A number in the last column represents the established rotation period (in d) of a star that shows polarimetric variability; tentative periods are followed by the symbol ':'. Other symbols have the following meaning: 'var.' means that the star is certainly polarimetric variable but the period is still unknown; 'var.:' means that hints of subtle variability have been detected over a short timescale; 'n.v.' means that the observed polarisation was constant (within uncertainties) and the star was observed at least three times; 'n.v.:' means that observed polarisation was constant (within uncertainties) but the star was observed only twice; 's.v.' means that polarisation shows some sign of subtle variability over a timescale of a decade or longer. For stars within the local 40 pc volume, the parameters of stellar magnitude, distance, spectral type, atmospheric composition, temperature, mass, and age are from O'Brien et al. (2024); for the remaining stars, we used the catalogue from Gentile Fusillo et al. (2021) with ages interpolated from the tables by Bédard et al. (2020).\nAcknowledgements. The new observations presented in this work were made with the FORS2 instrument at the ESO Telescopes at the La Silla Paranal Observatory under program ID 110.243J.001, 110.23XV.001, 110.23XV.002, and 112.25C9.001, and with the ISIS instrument at the William Herschel Telescope (operated on the island of La Palma by the Isaac Newton Group), under programmes P10 in 19A and P8 in 19B. This research has made use also of additional FORS1 and FORS2 data obtained from the ESO Science Archive Facility: data for WD 1036-204 were obtained under programmes IDs 70.D-0259(A), 087.D-0714(A), 089.D-0612(A), 090.D-0269(A). Data for WD 1105-340 was acquired with ESPaDOnS on the Canada-France-Hawaii Telescope (CFHT) (operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii), under programmes 17AC01, 19AC04, 19BC02, and 21BC02. We thank the anonymous referee for their very constructive criticism. We thank Matthias Schreiber and Antonino Lanza for very useful comments. JDL acknowledges the financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference number 6377-2016.\nReferences\nAchilleos, N. & Wickramasinghe, D. T. 1989, ApJ, 346, 444\n\nAchilleos, N., Wickramasinghe, D. T., Liebert, J., Sa ff er, R. A., & Grauer, A. D. 1992, ApJ, 396, 273\nAngel, J. R. P. 1978, ARA&A, 16, 487\nAngel, J. R. P., Borra, E. F., & Landstreet, J. D. 1981, ApJS, 45, 457\nAngel, J. R. P., Hintzen, P., & Landstreet, J. D. 1975, ApJ, 196, L27\nAngel, J. R. P., Hintzen, P., Strittmatter, P. A., & Martin, P. G. 1974, ApJ, 190, L71\n\nAngel, J. R. P., Illing, R. M. E., & Landstreet, J. D. 1972a, ApJ, 175, L85\n\nAngel, J. R. P., Illing, R. M. E., & Landstreet, J. D. 1972b, ApJ, 175, L85\nAngel, J. R. P. & Landstreet, J. D. 1970, ApJ, 162, L61\nAngel, J. R. P. & Landstreet, J. D. 1971a, ApJ, 164, L15\nAngel, J. R. P. & Landstreet, J. D. 1971b, ApJ, 165, L71\nAngel, J. R. P. & Landstreet, J. D. 1972, ApJ, 178, L21\nAngel, J. R. P. & Landstreet, J. D. 1974, ApJ, 191, 457\nAntonyuk, K. A., Kolesnikov, S. V., Pit, N. V., et al. 2016, Astrophysical Bulletin, 71, 475\nAppenzeller, I., Fricke, K., Fürtig, W., et al. 1998, The Messenger, 94, 1\n\nAznar Cuadrado, R., Jordan, S., Napiwotzki, R., et al. 2004, A&A, 423, 1081\n\nBagnulo, S., Farihi, J., Landstreet, J. D., & Folsom, C. P. 2024a, ApJ, 963, L22\nBagnulo, S. & Landstreet, J. D. 2018, A&A, 618, A113\nBagnulo, S. & Landstreet, J. D. 2019a, A&A, 630, A65\nBagnulo, S. & Landstreet, J. D. 2019b, MNRAS, 486, 4655\nBagnulo, S. & Landstreet, J. D. 2020, A&A, 643, A134\nBagnulo, S. & Landstreet, J. D. 2021, MNRAS, 507, 5902\nBagnulo, S. & Landstreet, J. D. 2022, ApJ, 935, L12\nBagnulo, S., Landstreet, J. D., Farihi, J., et al. 2024b, A&A, 688, L14\nBarstow, M. A., Jordan, S., O'Donoghue, D., et al. 1995, MNRAS, 277, 971\nBédard, A., Bergeron, P., Brassard, P., & Fontaine, G. 2020, ApJ, 901, 93\nBerdyugin, A., Landstreet, J. D., Bagnulo, S., Piirola, V., & Berdyugina, S. V. 2024, A&A, 690, A10\nBerdyugin, A. V. 1995, Astronomy Letters, 21, 695\nBerdyugin, A. V. & Piirola, V. 1999, A&A, 352, 619\nBerdyugin, A. V., Piirola, V., Bagnulo, S., Landstreet, J. D., & Berdyugina, S. V. 2022, A&A, 657, A105\nBerdyugin, A. V., Piirola, V., Bagnulo, S., Landstreet, J. D., & Berdyugina, S. V. 2023, A&A, 670, A2\nBerdyugina, S. V., Berdyugin, A. V., & Piirola, V. 2007, Phys. Rev. Lett., 99, 091101\nBergeron, P., Ruiz, M.-T., & Leggett, S. K. 1992, ApJ, 400, 315\nBeuermann, K. & Reinsch, K. 2002, A&A, 381, 487\nBisikalo, D., Sobolev, A., & Zhilkin, A. 2021, Galaxies, 9, 110\nBlatman, D. & Ginzburg, S. 2024, MNRAS, 533, L13\nBriggs, G. P., Ferrario, L., Tout, C. A., Wickramasinghe, D. T., & Hurley, J. R. 2015, MNRAS, 447, 1713\nBrinkworth, C. S., Burleigh, M. R., Lawrie, K., Marsh, T. R., & Knigge, C. 2013, ApJ, 773, 47\nBurleigh, M. R., Jordan, S., & Schweizer, W. 1999, ApJ, 510, L37\nBychkov, V. D., Fabrika, S. N., & Shtol', V. G. 1991, Pisma v Astronomicheskii Zhurnal, 17, 43\nCamisassa, M., Fuentes, J. R., Schreiber, M. R., et al. 2024, arXiv e-prints, arXiv:2411.02296\nCamisassa, M. E., Raddi, R., Althaus, L. G., et al. 2022, MNRAS, 516, L1\nCastro-Tapia, M., Zhang, S., & Cumming, A. 2024, arXiv e-prints, arXiv:2406.01807\nCohen, M. H., Putney, A., & Goodrich, R. W. 1993, ApJ, 405, L67\nCropper, M. 1988, MNRAS, 231, 597\nDonati, J. F., Howarth, I. D., Jardine, M. M., et al. 2006, MNRAS, 370, 629\nDonati, J.-F. & Landstreet, J. D. 2009, ARA&A, 47, 333\nDownes, R. A. & Margon, B. 1983, PASP, 95, 358\nDufour, P., Bergeron, P., Schmidt, G. D., et al. 2006, ApJ, 651, 1112\nEfimov, I. S. 1981, Izvestiya Ordena Trudovogo Krasnogo Znameni Krymskoj Astrofizicheskoj Observatorii, 63, 118\nEuchner, F., Jordan, S., Beuermann, K., Reinsch, K., & Gänsicke, B. T. 2006, A&A, 451, 671\nEuchner, F., Reinsch, K., Jordan, S., Beuermann, K., & Gänsicke, B. T. 2005, A&A, 442, 651\nFabrika, S. N., Shtol', V. G., Valyavin, G. G., & Bychkov, V. D. 1997, Astronomy Letters, 23, 43\nFarihi, J., Fossati, L., Wheatley, P. J., et al. 2018, MNRAS, 474, 947\nFarihi, J., Robert, A., & Walters, N. 2024, MNRAS, 529, L164\nFerrario, L., de Martino, D., & Gänsicke, B. T. 2015, Space Sci. Rev., 191, 111\nFerrario, L., Vennes, S., Wickramasinghe, D. T., Bailey, J. A., & Christian, D. J. 1997, MNRAS, 292, 205\nFerrario, L., Wickramasinghe, D., & Kawka, A. 2020, Advances in Space Research, 66, 1025\nFranzon, B. & Schramm, S. 2015, Phys. Rev. D, 92, 083006\nFriedrich, S., Östreicher, R., & Ruder, H. 1993, in Astronomische Gesellschaft\n\nArticle number, page 12 of 29\nAbstract Series, Vol. 9, Astronomische Gesellschaft Abstract Series, 145\n\nGänsicke, B. T., Rodríguez-Gil, P., Gentile Fusillo, N. P., et al. 2020, MNRAS, 499, 2564\nGarcía-Berro, E., Lorén-Aguilar, P., Aznar-Siguán, G., et al. 2012, ApJ, 749, 25 Gentile Fusillo, N. P., Tremblay, P. E., Cukanovaite, E., et al. 2021, MNRAS, 508, 3877\nGinzburg, S., Fuller, J., Kawka, A., & Caiazzo, I. 2022, MNRAS, 514, 4111\nHermes, J. J., Gänsicke, B. T., Kawaler, S. D., et al. 2017, ApJS, 232, 23\nHernandez, M. S., Schreiber, M. R., Landstreet, J. D., et al. 2024, MNRAS, 528, 6056\nIsern, J., García-Berro, E., Külebi, B., & Lorén-Aguilar, P. 2017, ApJ, 836, L28 Jordan, S. 2003, in NATO Advanced Study Institute (ASI) Series B, Vol. 105, White Dwarfs, 175\nJordan, S. & Friedrich, S. 2002, A&A, 383, 519\nJordan, S., Schmelcher, P., & Becken, W. 2001, A&A, 376, 614\nJordan, S., Schmelcher, P., Becken, W., & Schweizer, W. 1998, A&A, 336, L33 Kawaler, S. D. 2015, in Astronomical Society of the Pacific Conference Series, Vol. 493, 19th European Workshop on White Dwarfs, ed. P. Dufour, P. Bergeron, & G. Fontaine, 65\nKawka, A. 2020, in White Dwarfs as Probes of Fundamental Physics: Tracers of Planetary, Stellar and Galactic Evolution, ed. M. A. Barstow, S. J. Kleinman, J. L. Provencal, & L. Ferrario, Vol. 357, 60-74\nKawka, A., Briggs, G. P., Vennes, S., et al. 2017, MNRAS, 466, 1127\nKawka, A. & Vennes, S. 2012, MNRAS, 425, 1394\nKawka, A., Vennes, S., & Thorstensen, J. R. 2004, AJ, 127, 1702\nKemp, J. C., Coyne, G. V., Swedlund, S. J. J. B., & Wolstencroft, R. D. 1974, ApJ, 189, L79\nKemp, J. C., Swedlund, J. B., Landstreet, J. D., & Angel, J. R. P. 1970, ApJ, 161, L77\nKilic, M., Kosakowski, A., Moss, A. G., Bergeron, P., & Conly, A. A. 2021, ApJ, 923, L6\nKilic, M., Rolland, B., Bergeron, P., et al. 2019, MNRAS, 489, 3648 Kochukhov, O. 2021, A&A Rev., 29, 1\nKoester, D., Dreizler, S., Weidemann, V., & Allard, N. F. 1998, A&A, 338, 612 Koester, D., Voss, B., Napiwotzki, R., et al. 2009, A&A, 505, 441\nLandi Degl'Innocenti, E. & Landolfi, M., eds. 2004, Astrophysics and Space Science Library, Vol. 307, Polarization in Spectral Lines\nLandstreet, J. D. 1987, MNRAS, 225, 437\nLandstreet, J. D. & Angel, J. R. P. 1971, ApJ, 165, L67\nLandstreet, J. D. & Angel, J. R. P. 1974, ApJ, 190, L25\nLandstreet, J. D. & Bagnulo, S. 2019, A&A, 623, A46\nLandstreet, J. D. & Bagnulo, S. 2020, A&A, 634, L10\nLandstreet, J. D., Bagnulo, S., Martin, A., & Valyavin, G. 2016, A&A, 591, A80\nLandstreet, J. D., Bagnulo, S., Valyavin, G., & Valeev, A. F. 2017, A&A, 607, A92\nLandstreet, J. D., Bagnulo, S., Valyavin, G. G., et al. 2012, A&A, 545, A30\nLandstreet, J. D., Villaver, E., & Bagnulo, S. 2023, ApJ, 952, 129\nLawrie, K. A., Burleigh, M. R., Brinkworth, C. S., et al. 2013, in Astronomical Society of the Pacific Conference Series, Vol. 469, 18th European White Dwarf Workshop., ed. J. Krzesi'nski, G. Stachowski, P. Moskalik, & K. Bajan, 429\nLeroy, J. L., Bagnulo, S., Landolfi, M., & Landi Degl'Innocenti, E. 1994, A&A, 284, 174\nLi, G., Deheuvels, S., Ballot, J., & Lignières, F. 2022, Nature, 610, 43\nLi, G., Deheuvels, S., Li, T., Ballot, J., & Lignières, F. 2023, A&A, 680, A26\nLiebert, J., Angel, J. R. P., & Landstreet, J. D. 1975, ApJ, 202, L139\nLiebert, J., Angel, J. R. P., Stockman, H. S., & Beaver, E. A. 1978, ApJ, 225, 181\nLiebert, J., Angel, J. R. P., Stockman, H. S., Spinrad, H., & Beaver, E. A. 1977, ApJ, 214, 457\n\n\nLiebert, J., Schmidt, G. D., Green, R. F., Stockman, H. S., & McGraw, J. T. 1983, ApJ, 264, 262\n\nLiebert, J., Schmidt, G. D., Sion, E. M., et al. 1985, PASP, 97, 158\nMacfarlane, S. A., Woudt, P. A., Dufour, P., et al. 2017, MNRAS, 470, 732\nManser, C. J., Gänsicke, B. T., Inight, K., et al. 2023, MNRAS, 521, 4976\nMartin, B. & Wickramasinghe, D. T. 1978, MNRAS, 183, 533\nMathys, G. 2008, Contributions of the Astronomical Observatory Skalnate Pleso, 38, 217\nMaxted, P. F. L., Ferrario, L., Marsh, T. R., & Wickramasinghe, D. T. 2000, MNRAS, 315, L41\nMcCook, G. P. & Sion, E. M. 1977, A Catalogue of spectroscopically identified white dwarfs\nMcCook, G. P. & Sion, E. M. 1999, ApJS, 121, 1\nMestel, L. 1999, Stellar magnetism\nMestel, L. & Takhar, H. S. 1972, MNRAS, 156, 419\nMontgomery, M. H. & Dunlap, B. H. 2024, ApJ, 961, 197\nMorin, J., Donati, J. F., Forveille, T., et al. 2008a, MNRAS, 384, 77\nMorin, J., Donati, J. F., Petit, P., et al. 2008b, MNRAS, 390, 567\nMoss, A., Bergeron, P., Kilic, M., et al. 2024, MNRAS, 527, 10111\nO'Brien, M. W., Tremblay, P. E., Gentile Fusillo, N. P., et al. 2023, MNRAS, 518, 3055\nO'Brien, M. W., Tremblay, P. E., Klein, B. L., et al. 2024, MNRAS, 527, 8687 Oliveira da Rosa, G., Kepler, S. O., Soethe, L. T. T., Romero, A. D., & Bell, K. J. 2024, arXiv e-prints, arXiv:2407.05214\nPiirola, V. & Reiz, A. 1992, A&A, 259, 143\nPreston, G. W. 1971, PASP, 83, 571\nPutney, A. 1995, ApJ, 451, L67\nPutney, A. 1997, ApJS, 112, 527\nPutney, A. & Jordan, S. 1995, ApJ, 449, 863\nReding, J. S., Hermes, J. J., Vanderbosch, Z., et al. 2020, ApJ, 894, 19\nReimers, D., Hagen, H. J., & Hopp, U. 1999, A&A, 343, 157\n\nReimers, D., Jordan, S., Koester, D., et al. 1996, A&A, 311, 572\n\nSchmidt, G. D., Bergeron, P., & Fegley, B. 1995, ApJ, 443, 274\n\nSchmidt, G. D., Liebert, J., Harris, H. C., Dahn, C. C., & Leggett, S. K. 1999, ApJ, 512, 916\n\nSchmidt, G. D. & Norsworthy, J. E. 1991, ApJ, 366, 270\nSchmidt, G. D. & Smith, P. S. 1994, ApJ, 423, L63\nSchmidt, G. D. & Smith, P. S. 1995, ApJ, 448, 305\n\nSchmidt, G. D., Vennes, S., Wickramasinghe, D. T., & Ferrario, L. 2001, MNRAS, 328, 203\nSchmidt, G. D., West, S. C., Liebert, J., Green, R. F., & Stockman, H. S. 1986, ApJ, 309, 218\n\nSchreiber, M. R., Belloni, D., Gänsicke, B. T., Parsons, S. G., & Zorotovic, M. 2021, Nature Astronomy [ arXiv:2104.14607 ]\nSchwab, J. 2021, ApJ, 906, 53\nSiebenmorgen, R., Voshchinnikov, N. V., & Bagnulo, S. 2014, A&A, 561, A82\nSion, E. M., Liebert, J., Schmidt, G., & Starrfield, S. G. 1984, in Bulletin of the\nAmerican Astronomical Society, Vol. 16, 725\nSpruit, H. C. 1999, A&A, 349, 189\nStibbs, D. W. N. 1950, MNRAS, 110, 395\nSwedlund, J. B., Wolstencroft, R. D., Michalsky, Jr., J. J., & Kemp, J. C. 1974, ApJ, 187, L121\n\nTout, C. A., Wickramasinghe, D. T., Liebert, J., Ferrario, L., & Pringle, J. E. 2008, MNRAS, 387, 897\n\nValyavin, G., Bagnulo, S., Monin, D., et al. 2005, A&A, 439, 1099\nValyavin, G., Wade, G. A., Bagnulo, S., et al. 2008, ApJ, 683, 466\n\nVennes, S., Kawka, A., Ferrario, L., & Paunzen, E. 2018, Contributions of the Astronomical Observatory Skalnate Pleso, 48, 307\n\nVennes, S., Schmidt, G. D., Ferrario, L., et al. 2003, ApJ, 593, 1040\nVornanen, T., Berdyugina, S. V., Berdyugin, A. V., & Piirola, V. 2010, ApJ, 720,\nL52\nWalters, N., Farihi, J., Marsh, T. R., et al. 2021, MNRAS, 503, 3743\nWegner, G. 1977, Mem. Soc. Astron. Italiana, 48, 27\nWesemael, F., Liebert, J., Schmidt, G. D., et al. 2001, ApJ, 554, 1118\nWest, S. C. 1989a, ApJ, 345, 511\nWest, S. C. 1989b, ApJ, 345, 511\nWickramasinghe, D. T. & Bessell, M. S. 1976, ApJ, 203, L39\nWickramasinghe, D. T. & Cropper, M. 1988, MNRAS, 235, 1451\n\nZhong, Y., Kashiyama, K., Takasao, S., Shigeyama, T., & Fujisawa, K. 2024, ApJ, 963, 26\n\nAppendix A: New observations\n
\nArticle number, page 14 of 29\nS. Bagnulo and J.D. Landstreet : Strong fields of old white dwarfs are axisymmetric\n
\n\n
Table A.1. Continued.
\n
\nAppendix B: Variability of magnetic white dwarfs: Comments on individual stars\nWe preliminary note that to compare observations taken by different authors, we need to know how circular polarisation was defined in di ff erent works. This problem has been highlighted in detail, for example in Sect. 2 of Bagnulo & Landstreet (2020). Here, we have adopted the definition of positive handedness of circular polarisation given by Landi Degl'Innocenti & Landolfi (2004), and converted some of the literature data to it by changing the sign of circular polarisation with respect to the originally published data.\nNumerous examples in the literature (e.g. Bagnulo & Landstreet 2020) and in this section show that many magnetic white dwarfs with very strong fields have a circular polarisation spectrum that varies rapidly with wavelength. The consequence is that the same source observed with even slightly di ff erent broadband filters may result in di ff erences of the measured broadband polarisation signal that are larger than the formal uncertainties. This specific issue is discussed in Sect. 7 of Bagnulo & Landstreet (2019b), where some numerical examples are presented. As a consequence, small di ff erences between broadband polarimetric measurements obtained with di ff erent instruments cannot reliably provide strong evidence that a star is magnetically variable.\nSpectropolarimetry should allow comparisons that are less instrument dependent, provided the observations overlap in wavelength, and that the resolving power of di ff erent instrument is similar. There is no doubt, however, that the safest way to establish variability or constancy of polarisation is to repeatedly observe the star with the same instrument and instrument setting, and to always perform the data reduction using exactly the same technique.\nIn the following, we discuss all stars of our sample (including two stars that did not make it into the final list of Table 2). The titles of the individual subsections contain the star's name in the Villanova system (McCook & Sion 1977, 1999), the main SIMBADidentifier, and our conclusions about the magnetic variability (using the same designation as in Table 2, followed by the rotation period P , if this is known). When the rotation period is obtained only from photometric data, we use the designation P phot.\nAppendix B.1: WD 0004+122 = LP 464-57 (non-variable)\nThis star was discovered to be magnetic, using spectropolarimetry, by Bagnulo & Landstreet (2020), and later observed four times by Berdyugin et al. (2024) in broadband circular polarisation. These measurements are shown in the top left panel of Fig. B.1. None of them deviated more than ≃ 1 σ from the mean value of the measurements obtained in the same filter, except for a ∼ 2 . 5 σ di ff erence between on measurement and the other three in the B filter, which might reflect a real small variability. We reanalysed the FORS2 spectropolarimetry published by Bagnulo & Landstreet (2020) to check for di ff erences between the V / I profiles obtained from the first and the second pair of exposures, sampling an interval in time of about 15 m, without finding any di ff erence (see Fig. B.2). We conclude that the star does not show convincing sign of variability. We estimate a field strength of ≃ 100 MG.\nAppendix B.2: WD 0009+501 = EGGR 381 (variable, P ≃ 8 h)\nThis white dwarf was well monitored and modelled as a magnetic variable with ⟨| B |⟩ between 150 and 250 kG by Valyavin et al. (2005), who presented a model obtained by adopting a rotational period of 0 . 3337 ± 0 . 0031 d. Recent unpublished ESPaDOnS data obtained by us confirm the rotational period.\nAppendix B.3: WD 0011-134 = G 158-45 (variable, P phot ≃ 0 . 73 h)\nDiscovered magnetic (with a field of about 9 MG) by Bergeron et al. (1992), and as a magnetic variable by Putney (1997). From photometry, Lawrie et al. (2013) found a rotational period of 44 ± 0 . 43 min. While there is no guarantee that the photometric period reflects the rotational period of the star, the star is definitely a magnetic variable, and in fact, this very short photometric period is completely consistent with several of the many acceptable periods found from the six widely spaced ⟨ Bz ⟩ data values published by Putney (1997) from spectropolarimetric observations made between 1994 Sep 29 and 1994 Dec 31. This star is one of the magnetic white dwarfs that clearly shows longitudinal field reversal.\nAppendix B.4: WD 0041-102 = Feige 7 (variable, P ≃ 2 h)\nLiebert et al. (1977) showed that the spectrum is rich in faint lines of H and He, and identified a magnetic field of about 18-20 MG from the spectrum. They found, on the basis of broadband circular polarisation measurements, that the field varies with a period of 131.6 min. (Note that the broadband circular polarisation changes sign as the star rotates.) A decentred dipole model of the field structure based on a series of flux spectra was derived by Achilleos et al. (1992); both H and He apparently vary substantially in abundance over the surface. Achilleos et al. (1992) also discovered that the star varies photometrically (in a double wave) with the same period as the spectrum and broad-band polarisation.\nAppendix B.5: WD 0051+117 = PHL 886 (variable, P ≃ 4 . 7 d)\nFrom unpublished spectropolarimetric data (Landstreet & Bagnulo, in prep., hereafter LB25) it is found that the star is periodically variable ( P = 4 . 703 d) with ⟨| B |⟩ ≃ 270 kG. The field ⟨ Bz ⟩ reverses sign as the star rotates. Modelling will be presented in a forthcoming paper.\nAppendix B.6: WD 0058-044 = GD 9 (variable, P ≃ 2 d?)\nFrom our unpublished spectropolarimetric data we found that the star is periodically variable (the period is not determined uniquely from available data, but the most probable period is P = 1 . 96 d), with ⟨| B |⟩ ≃ 300 kG (LB25). The field ⟨ Bz ⟩ varies sinusoidally but does not (quite) reverse sign. Modelling will be presented in a forthcoming paper.\nAppendix B.7: WD 0232+525 = EGGR 314 (variable)\nThis is a well-known star that was repeatedly included in studies of bright white dwarfs and been searched for a magnetic field (Bychkov et al. 1991; Schmidt & Smith 1995; Fabrika et al. 1997) without any detection being reported. Recent particularly\n\n\n
Fig. B.1. Variation in the broadband circular polarisation measurements observed in three filters B ' V ' R ' of the DiPol-UF instrument at the NOT (Berdyugin et al. 2022, 2023, 2024). In each panel, the three dotted lines represent the average of all data obtained in a particular filter: from top to bottom, the R ' filter (red dotted line), the V ' filter (green dotted line) and the B ' filter (blue dotted line). These lines are separated by ∆ V / I = 0 . 3 %. The x -axis represents the time of the observations in days from the first measurement. The solid circles represent the di ff erence between each measurement and the average of all measurements in a given filter (red circles for filter R ' , green circles for filter V ' , and blue circles for filter B ' .
\n\n\n\n
Fig. B.2. WD0004 + 122: Stokes V / I from two pairs of observations obtained within 15 m on 2019-10-07.
\n\nsensitive observations (Bagnulo & Landstreet 2022) succeeded in detecting a weak field. The three published measurements (two with ISIS, one with ESPaDOnS) reveal a very weak, and probably variable longitudinal field (the measurement obtained\nwith ESPaDOnS has the opposite sign as those obtained with ISIS).\nAppendix B.8: WD 0233-242A = LP 830-14 = NLTT 8435A (variable, P phot ≃ 1 . 5 h)\nThe star is discussed by Bagnulo & Landstreet (2021), who suggested the presence of a variable field of the order of 3.8 MG. In fact, ⟨| B |⟩ does not vary between two FORS observations, while ⟨ Bz ⟩ clearly changes sign. According to Vennes et al. (2018) it has a photometric variation period of 95 min.\nAppendix B.9: WD 0236-269 = PHL 4227 (n.v.: - only 2 meas.)\nSchmidt et al. (2001) report two spectropolarimetric detections of circular polarisation at about the 1.2% level. The two observations were obtained three days apart, with no changes detected. Furthermore no significant change in the spectrum-average circular polarisation was observed. We tentatively assume that is not a magnetic variable, but the evidence for non-variability is not compelling (the original paper just reports: \"consistent results were obtained on the two occasions\").\nAppendix B.10: WD 0253+508 = KPD 0253+5052 (variable, P ≃ 4 h)\nThe field was discovered by Downes & Margon (1983), who estimated ⟨| B |⟩ ≈ 13 MG, and modelled by Achilleos & Wickramasinghe (1989) using only flux spectra. The star shows continuum polarisation up to 0.4% that is periodically variable according to Schmidt & Norsworthy (1991) with P = 0 . 170 d. The sign of the broadband continuum circular polarisation briefly reverses during each cycle.\nAppendix B.11: WD 0301+059 = SDSS J030350.63+060748.9 (n.v.: - only 2 meas.)\nThe star was discovered to be magnetic by Landstreet & Bagnulo (2020), who did not find any obvious sign of variability between two observations obtained 1 day apart, nor between individual exposures of the same observing series (exposure times of single frames were 225 s). We tentatively assume that it is not magnetically variable. The star therefore appears as a strongly magnetic, massive, non-variable star.\nAppendix B.12: WD 0313-084 = GALEX J031613.8-081637 (non-variable)\nThe star was discovered to be magnetic by O'Brien et al. (2023). It was observed five times with FORS2 (see Sect. 2.1), and no variability was found within timescales of 1 h, 1 d, and 8 months (see Fig. B.3). The spectrum indicates that ⟨| B |⟩ ≃ 800 kG.\nAppendix B.13: WD 0316-849 = EUVE J0317-855 (variable, P ≃ 0 . 2 h)\nStudied by Ferrario et al. (1997) Burleigh et al. (1999), and Vennes et al. (2003), the star has a rotation period P = 725 s. A detailed model based on UV spectra was obtained by Burleigh et al. (1999), who found that the field varies in strength from about 180 to 800 MG over the surface. Vennes et al. (2003) shown that the star has a light curve with a minimum that corresponds to the maximum of the polarisation. This is one of the very few white dwarfs that show magnetic and photometric variability, for which we have a firmly established phase relationship between photometry and circular polarisation. This star is erroneously missing in the list of such stars compiled by Bagnulo et al. (2024b).\nAppendix B.14: WD 0322-019 = EGGR 566 (variable)\nThe star was discussed by Bagnulo & Landstreet (2021). Two FORS2 measurements by Farihi et al. (2018) obtained during two consecutive nights ( ⟨ Bz ⟩ = -5 . 4 ± 3 . 0 kG and -16 . 5 ± 2 . 3 kG) suggest that the longitudinal field may change on a timescale of a few days.\nAppendix B.15: WD 0330-000 = HE 0330-0002 (only 2 meas., and contradicting data)\nSchmidt et al. (2001) report two spectropolarimetric observations obtained one day apart, and concluded that no significant di ff erences had been detected in the overall spectrum. However, the mean circular polarisation averaged over the observed wavelength range reported in their Table 1 di ff ers by 0.5%, contra-\n\n\n
Fig. B.3. Top and middle panels: WD 0313 -084 observed with FORS2 in five di ff erent epochs. Bottom panels: Stokes V / I from four pairs of observations obtained within 1 h on 2023-09-19.
\n\ndicting the assessment of non-variability. Because of this, this star will not be considered in this paper.\nAppendix B.16: WD 0410-114 = G 160-51 = NLTT 12758 (variable, P ≃ 0 . 3 h)\nA magnetic field of about 1.7 MG was discovered by Kawka & Vennes (2012). Kawka et al. (2017) found that circular polarisation in the H α sigma components varies with P = 22 . 6 min. This is another example of a star for which the phase relationship between light curve and magnetic field are known (Kawka et al. 2017) and it is also missing from the list compiled by Bagnulo et al. (2024b).\nAppendix B.17: WD 0446-789 = WG 47 (var.:)\nBagnulo & Landstreet (2018) show that this is a weak-field star with ⟨ Bz ⟩ ≃ -5 kG; it probably has a dipolar field with axis nearly parallel to the rotation axis, because only very mild variations in ⟨ Bz ⟩ occur on a timescale of days.\n\n\n
Fig. B.4. Upper panel: WD 0548 -001 = G99-37 flux spectra obtained with FORS1 and FORS2 compared. The Stokes I FORS spectra have not been calibrated, and the di ff erences in slopes are almost certainly explained by instrument and atmospheric e ff ects. Lower pane: FORS V / I spectra compared to the low-resolution MCSP spectrum of Angel &Landstreet (1974).
\n\nAppendix B.18: WD 0548-001 = EGGR 248 = G 99-37 (non-variable)\nThe magnetic field was discovered from continuum circular polarisation by Landstreet & Angel (1971). Observations of broadband CP on six nights showed no variations. During one hour, broad-band data were printed every 6 s. These data were Fourier analysed without finding any evidence of short-period variability.\nAngel & Landstreet (1974) obtained an MCSP circular polarisation spectrum in 1972, and the star was re-observed by Vornanen et al. (2010), who remarked that similarity with the previous spectra, reporting also \"no unambiguous variations\" between several polarisation spectra taken in 2003, 2005, 2008. Using FORS1 data obtained in 2005, they also reported no detection of linear polarisation.\nWe retrieved from the ESO FORS1 and FORS2 archives all circular polarisation spectra obtained in 2005, 2008 and 2012. Figure B.4 shows all these spectra compared to the polarisation spectrum of Angel & Landstreet (1974) . We note that there are three spectra obtained in three consecutive nights in November 2012. The spectrum obtained during the last night (2012-1119) shows zero circular polarisation (see green solid line), but a Stokes I flux consistent with that of previous observations. We ascribe this to a non-detected instrument failure (perhaps the retarder waveplate did not move) rather than to stellar variability.\nObservations obtained by Berdyugin et al. (2023) and Berdyugin et al. (2024) show no variability in broadband polarimetry between two measurements obtained 8 months apart (see the top panel of Fig. B.1).\nThis is a nearly unique DQp white dwarf which shows not only the Swan bands in the flux spectrum, but also the CH G band. The magnetic field was modelled by Angel & Landstreet\n(1974) who deduced ⟨ Bz ⟩ ≈ 3 . 6 MG from the strong polarisation signature of the CH G-band. Berdyugina et al. (2007) and Vornanen et al. (2010) modelled FORS1 spectra and found ⟨ Bz ⟩ ≃ 2 . 5 MG.\nAppendix B.19: WD 0553+053 = EGGR 290 = G 99-47 (non-variable)\nThe magnetic field of this star was discovered by Angel & Landstreet (1972), who also made 14 separate BBCP measurements during 10 months, without detecting any significant variation. Low resolution MCSP spectropolarimetry was obtained by Liebert et al. (1975), and 20 years later, a higher resolution polarised spectrum was obtained also by Putney & Jordan (1995). No significant variation has been detected on any timescale up to decades (for details, see Bagnulo & Landstreet 2021). Brinkworth et al. (2013) observed a light curve with 26.8 m period and semi-amplitude = 0.3 %, which suggest that polarimetric observations could have missed short-term variability. However, Angel & Landstreet (1972) had carried out Fourier-analysis of a polarimetric run, and found no significant variability for any period between 11 seconds and 2 h. They claimed that periodic variation with an amplitude of 0.17 % would certainly have been detected. Our conclusion is therefore that the star is non-variable.\nAppendix B.20: WD 0637+478 = GD 77 (variable, P ≃ 1 . 4 d)\nMagnetic variability was reported by Schmidt & Smith (1995). Eight unpublished ISIS polarisation spectra and one ESPaDOnS spectrum show that ⟨ Bz ⟩ varies with P ≃ 1 . 362 d. The value of ⟨| B |⟩ ≈ 1 . 0 MG hardly changes with rotation, but ⟨ Bz ⟩ varies sinusoidally between about + 400 and -250 kG (LB25).\nAppendix B.21: WD 0654+059 = 2MASS J06572938+0550479 (non-variable)\nThree BBCP observations obtained 8 months apart by Berdyugin et al. (2024) show no variability of the polarisation in the three B ' V ' R ' filters (see Fig. B.1). In all three filters, all three observations have V / I ≃ -0 . 3 %. We assume that the star is not variable.\nAppendix B.22: WD 0708-670 = SCR J0708-6706 (n.v.: only 2 meas.)\nTwo spectra published by Bagnulo & Landstreet (2020) show no variability over a 2 month interval. The strength and wavelength dependence of the circular polarisation spectrum suggests that the underlying field could be of the order of 60 to 200 MG. We tentatively assume that the star is magnetically non-variable.\nAppendix B.23: WD0745 + 115 = GALEX J074842.4+112502 (non-variable)\nStrong broadband circular polarisation was detected by Berdyugin et al. (2024). The polarisation changes sign between the B ' and the other bands, and its absolute value is about 1% in all bands. Observed four times, the star does not show variability (see the top right panel of Fig. B.1).\nAppendix B.24: WD 0756+437 = EGGR 428 (variable, P phot ≃ 6 . 5 h )\nThe star was discovered to be magnetic and discussed in detail by Putney (1995). She estimated its magnetic field to be ≃ 200 MG. Recent BBCP observations at NOT (Berdyugin et al. 2024) showed that it is rapidly variable. The first two observations, about 3 h apart, report the largest and smallest polarisation seen in the full data set of five observations; this suggests a rotation period of the order of 6 h. In fact, the photometric study of Brinkworth et al. (2013) found a unique, very large amplitude ( ± 4 %) light variation with a period of P = 6 . 68 h. The similarity of this period with the period range deduced from polarisation measurements strongly confirms that 6.68 h is the rotation period of this white dwarf. A further remarkable fact is the combination of high field (200 MG), high mass (1 . 04 M ⊙ ) with advanced age (4.45 Gyr): this star seems to be a unique example of a very old, still strongly magnetic and rapidly rotating WD-WD merger.\nAppendix B.25: WD 0810-353 = UPM J0812-3529 (non-variable)\nNo variability detected among six polarised spectra obtained over a four year period, as described in a detailed study by Landstreet et al. (2023). According to their modelling, the star shows two regions of di ff erent field strength: one with magnetic field of predominantly 30 MG strength, outward, and one showing a field strength of 45 MG, inward.\nAppendix B.26: WD 0816-310 = SCR J0818-3110 (variable, P ≃ 10 d)\nThis is a DZ white dwarf with strong flux, spectrum and ⟨ Bz ⟩ variations observed in five FORS polarised spectra in 2023 obtained with grism 1200B. It is clear that the surface abundances vary over the surface, and appear to be locked to the surface magnetic field. The rotational period is of the order of 10 d, and ⟨| B |⟩ ≃ 100 kG (Bagnulo et al. 2024a).\nAppendix B.27: WD 0850+192 = LB 8915 (variable, P ∼ hours?)\nThis DBA white dwarf shows very weak, variable H lines, slightly variable He, and variable ⟨ Bz ⟩ (Wesemael et al. 2001). It is found that ⟨| B |⟩ ≃ 850 kG, and the rotation period is relatively short, probably some hours. We note that the correct identification of this star is LB 8915, and not LB 8827 as given in title of the paper by Wesemael et al. (2001).\nAppendix B.28: WD0907+213 = GALEX J091016.5+210555 (variable, P ≃ 10 h)\nMoss et al. (2024) discovered that this is a spectroscopically and magnetically variable DBA star with a rotation period of either 7.7 or 11.3 h (the ambiguity is due to aliasing). They have modelled the field and abundance geometry with a simple model like that used for Feige 7 = WD0041-102. The line splitting in the flux spectra suggests a field of ⟨| B |⟩ ≃ 0 . 5 MG.\n\n\n
Fig. B.5. Circular polarisation spectra of WD 1008 -242 obtained with FORS2 and the 300V grism on 2022-02-10 (top panel) and with the 600B grism on 2022-01-01.
\n\nAppendix B.29: WD 0912+536 = EGGR 250 = G 195-19 (variable, P ≃ 1 . 3 d)\nThis is the second magnetic white dwarf discovered (Angel & Landstreet 1971a), and the first magnetic white dwarf to be discovered to be rotationally variable (Angel & Landstreet 1971b). An improved ephemeris was provided by Angel et al. (1972a). The star shows a large variation of its circular polarisation with a period of 1.33 d (Angel et al. 1972b). Hernandez et al. (2024) measured the period of the photometric variability from TESS data as 1 . 3304 ± 0 . 0054 d (note that high accuracy of period relies on two widely separated TESS observational data sets). Six MCSP V / I spectra roughly uniformly distributed in phase were obtained by Landstreet & Angel, (unpublished). Between 4000 and 5000 Å, V / I reverses sign during rotation; redwards of this, the strong variations retain one sign.\nAppendix B.30: WD 1008-242 = UCAC4 328-061594 (n.v.: only 2 meas.)\nObserved twice with FORS2 in spectropolarimetric mode by Bagnulo & Landstreet (2022), this white dwarf did not show any variation between two spectra obtained 40 d apart, nor within the same observing series (see Fig. B.5). Probably the field is of order 100 MG or more. We consider that it is probably not variable. The star therefore appears a young, strongly magnetic, ultra-massive, non-variable star.\nAppendix B.31: WD 1008+290 = LP 315-42 = LHS 2229 (non-variable)\nThis white dwarf is a cool peculiar DQ star discovered to be magnetic by Schmidt et al. (1999), who tentatively suggested that the field strength is of the order of 100 MG or more. Four observations obtained by Berdyugin et al. (2024) show little to no variability of broadband circular polarisation (see Fig. B.1). We assume that it is not magnetically variable.\nAppendix B.32: WD 1009-184 = WT 1759 (var.:)\nThe star was discovered to be a magnetic white dwarf by Bagnulo & Landstreet (2019a), who published one measurement obtained with FORS2 and one obtained with ISIS, showing that the star has a weak and variable field ( ⟨ Bz ⟩ ≃ 50 kG). Additional unpublished data suggest also weak variability.\nAppendix B.33: WD 1015+014 = PG 1015+014 (variable, P = 98 . 75 m)\nThis is a very strongly polarised white dwarf with V / I ≃ 1 . 5 %. The flux and polarisation spectra are strongly variable with P = 98 . 75 m and a polar field strength of about 120 MG (Angel 1978; Wickramasinghe & Cropper 1988). Schmidt & Norsworthy (1991) report that BBCP also varies with P = 98.75 min, approximately sinusoidally, with extrema of + 1 and -1 % polarisation. Brinkworth et al. (2013) found that the star's light curve has a period consistent, within uncertainties, with that obtained from polarimetry. Euchner et al. (2006) obtained a series of polarised spectra with FORS using the 300V grism. They describe strong I and V spectrum variations, and model the field structure using a multipole field expansion. This white dwarf displays spectral features originating from regions with typical field strengths between about 50 and 90 MG.\nAppendix B.34: WD 1031+234 = Ton 527 = PG 1031+234 (variable, P ≃ 3 . 5 h)\nSchmidt et al. (1986) report strongly variable intensity, circular and linear polarisation spectra with a 3 h 24 min period, and proposed a simple magnetic model with field strength in the range of 200 - 500 MG. Further broadband polarisation observations through the rotation period were obtained by Piirola & Reiz (1992), who measured also a light variation in anti-phase with circular polarisation (that is, the star appears darker when the absolute value of the polarisation is maximum). Brinkworth et al. (2013) found P phot ≃ 3 . 5 h.\nAppendix B.35: WD 1036-204 = LP 790-29 (non-variable)\nThe star was discovered to be magnetic via spectropolarimetry by Liebert et al. (1978), and repeatedly studied (West 1989b; Schmidt et al. 1995, 1999; Beuermann & Reinsch 2002; Jordan & Friedrich 2002). Beuermann & Reinsch (2002) have monitored the star with EFOSC in spectropolarimetric mode to search for short-term variability, without finding any significant variations. Beuermann & Reinsch (2002), however, pointed out that Schmidt et al. (1995) measured a polarisation signal of ≃ -6 % around 6500 Å, a measurement at odds with broadband polarimetric measurements obtained in 1977, 1986, 1994 and 2000, which were all about -9 % (see their Table 1; we recall here that we use the opposite definition for the sign of circular polarisa-\n\n\n\n\n\n
Fig. B.6. Top and middle panels: Intensity and circular polarisation spectra of WD 1036 -204 obtained with FORS1 and FORS2 at four di ff erent epochs; circular polarisation from Schmidt et al. (1995) is also shown with small circles. Bottom panel: Stokes V / I from four pairs of observations obtained within 10 m from each other on 2013-02-21.
\n\nn). Schmidt et al. (1995) therefore suggested the possibility of a very long rotation period of the star. Jordan & Friedrich (2002) carried out a similar study, with similar results regarding short term variation. They also proposed a rotation period in the range of about 24 - 29 yr. We have reduced FORS1 and FORS2 archival polarisation spectra from 2003, 2011, 2012, and 2013, finding absolutely no hint of variability among them. Figure B.6 shows a comparison of archive observations obtained with grism 600B. The flux is not corrected for atmospheric and instrument transmission, and the discrepancies in the slope of the flux measured in 2003 can be explained by the use of a different CCD. A comparison between FORS spectra with those obtained by Schmidt et al. (1995) on May 7 and 8, 1994, (wavelength range 4160-7460 Å) does not show obvious long-term variability, except in the range 5700 -6200 Å. In that range, our data are instead consistent with most of the literature and point to a value of ≃ -9 %. In addition to the polarisation spectra, FORS1 and FORS2 archive contains another two broadband circular polarisation (BBCP) observations in the R filter: a FORS1 BBCP measurement obtained in April 2006, and one obtained with FORS2 in March 2024, using a similar (but not per-fec\nidentical) R filter (program ID 112.25C9.001). We found both measurements consistent with a polarisation signal of about -9 . 4 %. These measurements rule out any significant change in the region around 6500 Å over an interval of 18 years. Finally, Berdyugin et al. (2024) have published the series of BBCP observations obtained in November 2022, November 2023 and February 2024, all consistent among themselves. In the R ' filter they report a polarisation signal of ≃ -9 . 1 %. For the reasons explained at the beginning of this section, it is not possible to accurately compare BBCP measurements obtained with di ff erent instruments, but it is clear that the only deviant point of a series of polarimetric observations obtained in the course of almost half a century is a small portion of a spectrum obtained in 1994. Remarkably, data obtained with the same instrument (FORS) over nearly two decades are fully consistent among themselves, and point strongly to a constant circular polarisation spectrum. Our conclusion is that the star is actually not variable. From the measured signal of circular polarisation of ≃ 10 % we estimate a longitudinal field of the order of 100 MG.\nAppendix B.36: WD 1043-050 = HE 1043-0502 (n.v.: - only 2 meas.)\nThis DBA star was discovered magnetic by Schmidt et al. (2001), who proposed that the field is ≃ 800 MG (although its continuum is not highly polarised). Two observations were taken a few days apart, and Schmidt et al. (2001) state that they did not show variability; the reported wavelength integrated circular polarisation of about 1.5% is also essentially unchanged between the two spectra. So we consider it as candidate nonvariable magnetic white dwarf.\nAppendix B.37: WD 1045-091 = HE 1045-0908 (variable, P ≃ 3 h)\nThis white dwarf was discovered to be magnetic by Reimers et al. (1996) from a flux spectrum. Circular polarisation was confirmed, and shown to be variable by Schmidt et al. (2001). Euchner et al. (2005) obtained a series of I and V spectra and, assuming a rotational period of about 2.7 h, derived a detailed surface field model with a dominant field strength of 16 MG, but local field strength ranging between about 10 and 75 MG.\nAppendix B.38: WD 1105-340 = SCR J1107-3420A (non-variable)\nEleven ESPaDOnS spectra taken between 2018 and 2022, with ten of them during 1 week in 2019 (see Table A.1), show that the star has a weak, non-variable field with ⟨ Bz ⟩ ≈ -22 ± 4 . 5 ˙ kG and ⟨| B |⟩ ≈ 125 ± 5 kG (see Fig. B.7). We also have two FORS spectra, one taken with 1200B and one with 1200R, with lower resolving power but higher signal-to-noise ratio (S / N). Owing to their di ff erence resolution, these spectra were not used in this work. This star is perhaps the best studied non-variable weakfield magnetic white dwarf.\nAppendix B.39: WD 1105-048 = EGGR 76 (ultra-weak field, var.:)\nThe star was repeatedly observed, and a longitudinal field of the order of 1 kG was detected only in two measurements (Bagnulo & Landstreet 2018). The star seems to have a very weak and\n\n\n
Fig. B.7. WD 1105 -340: Stokes I and V / I profiles of H α observed with ESPaDONs at four di ff erent epochs: 2019-03-21 (two observations), 2019-05-31, and 2022-02-22. Only negligible changes are seen between spectra.
\n\nvariable field, but additional measurements should be obtained to confirm the existence of a magnetic field.\nAppendix B.40: WD 1116-470 = SCR J1118-4721 (non-variable)\nThis white dwarf was observed twice with FORS2 by Bagnulo & Landstreet (2021), who flagged it as suspected magnetic star. Both observations show a similar signal of circular polarisation at -0 . 2 %, close to the FORS2 instrumental detection limit. A third observations was obtained in January 2023, and the V / I spectrum is again consistent with that measured previously. The star is definitely magnetic, and most likely not variable. This confirmation (see Table A.1) brings the number of magnetic white dwarfs in the 20 pc volume to 34.\nAppendix B.41: WD 1211-171 = HE 1211-1707 (variable, P ≃ 2 h)\nA magnetic field was suspected in this white dwarf by Reimers et al. (1996), which was confirmed with polarimetry by Schmidt et al. (2001). Both papers show varying flux spectra. Schmidt et al. (2001) estimates P ≃ 100 -120 min and ⟨| B |⟩ ≃ 50 MG. Brinkworth et al. (2013) measured a photometric period of 1.79 h, consistent with the previous estimates from polarimetric data. The star is polarised at a level that varies between 0 and 3% during the rotation cycle. Modelling by Schmidt et al. (2001) strongly suggests a He dominated atmosphere with T e ff ≃ 12000 K, but Reimers et al. (1996), using IUE data, estimates 23000 K. Gentile Fusillo et al. (2021) gives 30000 K and 1 . 20 M ⊙ . We tentatively assume T e ff = 23000 K , M = 1 . 2 M ⊙ , and an He-rich atmosphere.\n\n\n
Fig. B.8. Top panel: WD 1116 -470 observed with FORS2 in three different epochs. Bottom panel: Stokes V / I (re-binned at 410 Å) from individual pairs of exposures as shown in the legend.
\n\nAppendix B.42: WD 1217+475 = SDSS J121929.45+471522.8 (DAHe variable, P phot = 15 . 26 h)\nDAHe with 18.5 MG field strength and a photometric period of about 15.25 h (Gänsicke et al. 2020). Spectroscopy reveals Zeeman components of the Balmer lines varying in strength but with constant splitting. Published data do not demonstrate that the star is magnetically variable but cannot rule out this possibility either. Therefore we have decided not to include this star in our sample.\nAppendix B.43: WD 1249-022 = GALEX J125230.9-023417 (DAHe variable, P phot = 0 . 09 h)\nThis is a DAHe white dwarf that shows variable H β and H α lines that appear sometimes in emission and sometimes in absorption. Field strength has been estimated 5 MG and rotational period = 0.09 h (Reding et al. 2020).\nAppendix B.44: WD 1312+098 = PG 1312+099 - (variable, P ≃ 5 . 4 h)\nVariable continuum circular polarisation was detected in this hot DAH by Schmidt & Norsworthy (1991), who present over 100 measures, and find P = 5 . 43 h. Circular polarisation varies approximately between + 1% and -1%.\nAppendix B.45: WD 1315-781 = LAWD 45 (n.v.: - only 2 meas.)\nNo change between two ⟨| B |⟩ ≃ 5 . 5 MG and ⟨ Bz ⟩ ≃ 0 MG measurements from FORS 300V spectra taken five nights apart\nby Bagnulo & Landstreet (2020). We tentatively assume that it is magnetically non-variable.\nAppendix B.46: WD 1315+222 = LP 378-956 (non-variable)\nTwo observations by Berdyugin et al. (2023) and one by Berdyugin et al. (2024) show no obvious sign of variability (see Fig B.1).\nAppendix B.47: WD 1328+307 = G 165-7 (variable)\nThis star is found to host a magnetic field of ⟨| B |⟩ ≃ 650 kG based on line splitting observed in a good S / NSDSSspectrum (Dufour et al. 2006). These authors have also obtained low-resolution polarised spectra. They state that three 600 sec polarisation spectra were taken on 2005-12-30 at Steward Obs, all yielding essentially the same ⟨ Bz ⟩ ≈ 150 kG, but that the polarisation amplitude in similar Steward polarised spectra from 2006-05-03 (apparently not measured) is at least two times weaker than in 2005 spectrum. We conclude that the star is magnetically variable.\nAppendix B.48: WD 1346+121 = LP 498-66 (non-variable)\nObserved three times by Berdyugin et al. (2023) and Berdyugin et al. (2024), we assume the star, which shows circular polarisation ranging between -1 and 0 % in the three DIPol-UF bands, is magnetically non-variable (see Fig. B.1). The field strength of this white dwarf is probably in the tens of MG.\nAppendix B.49: WD 1350-090 = PG 1350-090 = GJ 3814 (non-variable)\nDiscovered by Schmidt & Smith (1994), who measured ⟨ Bz ⟩ = 85 ± 9 kG. We have obtained four polarised spectra with ESPaDOnS that show ⟨| B |⟩ ≃ 450 -465 kG. There is no strong evidence of field variability, see also Schmidt & Smith (1994). Data will be published in a forthcoming paper (LB25).\nAppendix B.50: WD 1532+129 = G 137-24 (variable)\nOriginally classified as DZ white dwarf by Kawka et al. (2004), the star was discovered to be a magnetic white dwarf by Bagnulo &Landstreet (2019a), who published two FORS2 measurements and one ISIS measurement. The star is variable, with ⟨ Bz ⟩ values from FORS2 measurements of -21 ± 1 kG and -4 ± 1 kG, while ⟨| B |⟩ is not strong enough to split spectral lines, leading to ⟨| B |⟩ < ∼ 300 kG.\nAppendix B.51: WD 1556+044 = PM J15589+0417 (non-variable)\nDiscovered to be magnetic by Berdyugin et al. (2022), the star was re-observed three more times by Berdyugin et al. (2024). The observed circular polarisation, which is detected at the 10 σ level, ranges between -0 . 3 and + 0 . 4% in the three filter bands of DIPol-UF, so the order of magnitude of the field strength is probably some tens of MG. The polarisation does not show any variability (see Fig. B.1).\n\n\n
Fig. B.9. Spectra of WD 1619 + 046 around H β obtained with FORS2 on 2023-06-16.
\n\nAppendix B.52: WD 1615+542 = GALEX J161634.4+541011 (DAHe magnetically variable, P phot = 1 . 59 h)\nDAHe with a variable magnetic field (from 3.5 to 6.5 MG) and a rotation period P = 95 . 29 m estimate via photometry (Manser et al. 2023).\nAppendix B.53: WD 1619+046 = GALEX J162157.7+043219 (variable, P ≃ 40 m?)\nThis white dwarf was discovered to be magnetic and rapidly variable in this work (see Sect. 2.1 and Fig. B.9). From the H β regions, it is clear that the star is a DAH with a rather non-uniform field of ⟨| B |⟩ ≃ 15 MG. We observe clear changes in intensity between spectra obtained a few minutes apart, and also polarisation (which is measured by the combination of two spectra) shows some variation, but without sign reversal. The close similarity between the first pair and the last pair of intensity spectra suggest that the star's rotation period is approximately 40 min.\nAppendix B.54: WD 1639+537 = GD356 (non-variable, but with a light curve and P phot = 1.93 h)\nThis star is the prototype of DAHe white dwarfs. It shows a light curve with low amplitude and P phot ≃ 1 . 93 h (Walters et al. 2021). A time series of circular polarisation spectra obtained during an entire rotational cycle shows now variability while subtle sinusoidal variability is seen in the position of the σ components of H β and H α lines (Walters et al. 2021). Walters et al. (2021) compared spectropolarimetry obtained in 2019, with that\n\n\n\n\n\n
Fig. B.10. Top panels: Comparison of spectra of WD 1658 + 440 obtained in 1980 (Liebert et al. 1983) and in 2019 (this work). Bottom panels: Overplot of eight intensity spectra obtained in sequence with 450s exposure time, and of four circular polarisation spectra obtained from pairs of frames obtained in sequence.
\n\npublished by Ferrario et al. (1997), finding no changes. Its field strength is of the order of 10 MG (Walters et al. 2021). We classify the star as non-variable, but showing very small changes of its apparent magnetic field with a ∼ 2 h rotation period.\nAppendix B.55: WD 1658+440 = PG 1658+441 (n.v.: - only 2 meas., one very old)\nThe star was discovered to be magnetic by Liebert et al. (1983), who measured ⟨ Bz ⟩ ≃ 0 . 7 MG, and ⟨| B |⟩ ≃ 2 . 3 MG from spectropolarimetry of H α , H β and H γ (see their Fig. 5). Their observations were obtained on July 20 1980, with a spectral resolution of 10 Å. The same authors measured a signal of broadband circular polarisation = -0 . 016 ± 0 . 033% and -0 . 044 ± 0 . 019 % in the range 3300-8600 Å and concluded that a 3 σ upper limit of 0.10% semi-amplitude could be set for any presumed sinusoidal variation with a period between 0.5 and 5 h. For possible periods between 4 minutes and 0.5 hours, a less stringent limit of 0.30% polarisation semi-amplitude may be deduced. A comparison with our ISIS spectra obtained on April 21, 2019 (Sect. 2.1) is shown in the top panels of Fig. B.10. We observe a wavelength shift possibly due to bad calibration; most remarkable is the different strength of H α , but perhaps this is instrumental, because the ISIS spectrum was obtained with a setting that puts H α at the edge of the CCD, and no good flat-fielding correction could be applied. Although our comparison between 1980 and 2019 data is not conclusive, we certainly do not see any convincing change of the spectral features that demonstrate long-term variability. A comparison between the four pairs of Stokes V / I profiles obtained with ISIS in 2019 show no di ff erences within the error bars (see the bottom panels of Fig. B.10); therefore we rule out variability on a timescale of 10-15 minutes. This star seems to be a young, strongly magnetic, massive, tentatively classified as non-variable star. Photometric studies lead to inconsistent results: Brinkworth et al. (2013) found that the star is photometrically variable with a period between 6 h and 4 d, while, using TESS photometry, Oliveira da Rosa et al. (2024) derived a period shorter than 1 h. Hernandez et al. (2024), instead, did not detect periodicity in TESS data.\nAppendix B.56: WD 1703-266 = UCAC4 317-104829 (variable)\nThis DA white dwarf was discovered to have a magnetic field by Bagnulo & Landstreet (2020), with ⟨| B |⟩ ≈ 8 MG. They also found that two FORS2 polarised spectra four days apart are significantly di ff erent and yield di ff erent values of ⟨ Bz ⟩ , so the star is variable on a timescale of some days or less.\nAppendix B.57: WD 1712-590 = Gaia DR3 5915797694789556096 (variable)\nDiscovered to be magnetic by O'Brien et al. (2023). We observed this white dwarf in polarimetric mode with grism 1200B three times, twice during the same night. Intensity spectra show very similar splitting of Balmer lines, but the flux distribution within the line cores changes quite significantly, rather similarly to WD2359-434 (Landstreet et al. 2017). The deduced field strength is ⟨| B |⟩ < ∼ 0 . 8 MG. The Stokes V spectra also show that the star is strongly variable within 1-2 h (see Fig. B.11), and that the longitudinal field reverses its polarity during rotation.\nAppendix B.58: WD 1743-521 = L 270-31 = BPM25114 (variable, P = 2 . 84 d)\nWickramasinghe & Bessell (1976) reported a magnetic field of about 35 MG in this southern DA star from close examination of the peculiar flux spectrum. Wegner (1977) found vari-\n\n\n
Fig. B.11. WD1712 -590 observed with FORS2 in the dates shown in the bottom panel. H β detail.
\n\nability of light and of the flux and polarisation spectrum with P ≈ 2 . 84 d, and modelled the spectrum variations, finding results consistent with a magnetic dipole field. Field modelling was carried out in more detail by Martin & Wickramasinghe (1978), who deduced a dipole field strength of 36 MG, corresponding to ⟨| B |⟩ ≃ 18 MG.\nAppendix B.59: WD 1748+708 = EGGR 372 = G 240-72 (s.v.:)\nIntrinsic broadband circular and linear polarisation of this magnetic white dwarf were discovered by Angel et al. (1974), who reported no variations in repeated observations of broad-band circular polarisation over a month, and linear polarisation over two nights. The star was later observed in broadband circular and linear polarimetry by West (1989a), who generally found similar polarisation levels and position angles to earlier work, and in broadband linear polarimetry by Berdyugin & Piirola (1999). Berdyugin & Piirola (1999) found evidence of clear change of the position angle of linear polarisation (mainly rotation of the polarisation angle) on a timescale of 20 years. We have observed the star with ISIS both in circular (twice) and in linear polarisation, and compared these observations with MCSP lowresolution spectra of I , V and P obtained by Angel & Landstreet on September 6 and 7, 1974, that were never published until now. Our ISIS spectra are consistent among themselves. When compared with spectropolarimetry obtained in 1974 and the 1990s, our new ISIS linear polarisation data find small changes in the V and P polarisation amplitude in the blue and visual, whereas the overall position angle behaviour is almost identical to that observed observed in 1974, thus we cannot confirm the changes in the polarisation position angle seen in 1997 by Berdyugin & Piirola (1999). On the other side, Antonyuk et al. (2016) found that the star is photometrically variable with a period between 5 hours and two days. Brinkworth et al. (2013) detected no short period variability, but claim that photometry varied over a ten\n\n\n
\n\nmonth interval. We conclude that the star show some signs of subtle long term variability that should be further investigated.\nAppendix B.60: WD1750 -311 = UCAC4 295-140552 = [MTR2015] OW J175358.85-310728.9 (n.v.: - but P phot ≃ 0 . 5 h)\nAmagnetic field of ⟨| B |⟩ ≈ 2 . 1 MG was identified by Macfarlane et al. (2017) in this hot DQ white dwarf, which has strong lines of neutral C as well as fairly strong Balmer lines. They observed clear light variability of the star with an amplitude of about ± 2 % and a period of 35 min. These authors discuss the origin of the light variations: they argue that the variability is not due to pulsation, as no subsidiary frequencies are observed, and conclude that the variation could be due to rotation. They do not seem to have considered testing this with the spectra that they have collected of the object by looking for spectral variations, although they do test the spectra for radial velocity variations, and find none.\nOur four FORS2 300V spectra, taken with 13 m spacing through the light variation cycle, show virtually no changes in either flux or polarisation except for H β , which has a strongly variable I profile depth but almost constant V / I ( λ ) (see Fig. B.13). Possibly this is due to a magnetic field distribution that is nearly axisymmetric about the rotation axis but a distribution of H that varies strongly around that axis. This might be related to the light variability that has been detected. We have decided to classify the star as candidate magnetically non-variable.\n\n\n
Fig. B.13. WD 1750 -311 observed with FORS2 on 2023-06-12 with grism 300V.
\n\n\n\n
Fig. B.14. FORS2 V / I spectra of WD 1754-550, rebinned at 45 Å steps, from four exposure pairs obtained on 2023-06-12 at the times shown in the legend. The intensity spectrum is featureless.
\n\nAppendix B.61: WD1754 -550 = GALEX J175845.9-550117 (variable)\nThis star was discovered to be magnetic in this work (Sect. 2.1). There is a strong signal of circular polarisation that seems variable on a short timescale between 0 and 4%, suggesting a rotation period of the order of 15 min: see Fig B.14. The inferred field strength is presumably of the order of 100 MG or more. The intensity spectrum appears featureless, so the T e ff ≃ 35 000 K star could be defined a hot DC.\nAppendix B.62: WD 1814+248 = G 183-35 (variable)\nPutney (1997) showed that this white dwarf, formerly classified as a DC, is in fact a cool DAH that shows weak H α and H β , and that both lines are split by a field of about 6.8 MG, roughly the same in two observations. Kilic et al. (2019) observed that the white dwarf shows spectral variations. A series of spectra taken\n\n\n
Fig. B.15. Polarisation spectra of WD 1829 + 547 = G227-35. Top panel: Comparison between MP spectra obtained in 1974 and our ISIS spectra obtained in 2019, after degrading their resolution. Bottom panel: Comparison between ISIS spectra obtained in 2019 and a spectrum obtained by Putney & Jordan (1995) in August 1992.
\n\nover several hours show that the separation of the σ components of H α from the central π component changes rather abruptly between about 90 Å and about 120 Å, equivalent to fairly sudden jumps between ⟨| B |⟩ values of about 4.6 and 6 MG, repeating with a period of about 4 h. The authors suggest that this unusual form of variability may be due to a patchy distribution of H over an He-rich envelope. A plausible model that might explain the observations could be a dipole oblique to the rotation axis, decentred in the direction of one pole so that the polar strengths at the two magnetic poles are unequal, with H-rich patches around both poles, but little or no H in a belt around the magnetic equator.\nAppendix B.63: WD 1829+547 = G 227-35 (non-variable)\nThis white dwarf was discovered to be strongly magnetic by Angel et al. (1975) using both broadband polarimetry and lowresolution spectropolarimetry (on September 9, 1974). Limited tests of broadband variability were negative (Angel et al. 1975, 1981). It was observed again in spectropolarimetric mode by Cohen et al. (1993) and by Putney & Jordan (1995) who estimated a dipolar field strength of 170-180 MG. We observed the same star with ISIS twice in circular polarisation and once in linear polarisation. There are strong circular polarisation features at 6960,\n\n\n
Fig. B.16. MCSP linear and circular polarisation spectra of WD2010 + 310 = GD229.
\n\n7450 and 7930 Å. Figure B.15 shows a comparison between all these spectra. We do not see any obvious sign of variation.\nAppendix B.64: WD 1900+705 = LAWD 73 = Grw + 70 · 8247 (s.v.:)\nThe star has a magnetic field of the order of 180 MG (Jordan 2003), and shows hints of small variability. We refer to Bagnulo & Landstreet (2019b), who present a review and analysis of its polarimetric characteristics, and in particular, a comparison with observations obtained 50 years apart show some discrepancies in both linear and circular polarisation. More recent broadband circular polarisation obtained at NOT (Berdyugin et al. 2022, 2023) show no variability on a timescale of 2 years (see Fig. B.1).\nAppendix B.65: WD 1953-011 = GJ 772 (variable, P ≃ 1 . 5 d)\nMaxted et al. (2000) modelled a series of Stokes I spectra in terms of a global dipolar field with polar field strength of order 100 kG, together with a spot with a field of order 500 kG. This basic modelling was confirmed by the analysis of a series of polarised spectra obtained with FORS1 and on the Russian 6 m telescope, which also revealed a rotation period of 1.448 d (Valyavin et al. 2008).\nAppendix B.66: WD 2010+310 = GD 229 (s.v.:)\nThis star was the fifth circularly polarised white dwarf to be discovered. Swedlund et al. (1974) reported a series of measurements indicating the presence of elevated levels of both circular polarisation and linear polarisation, and strongly suggesting polarisation variability. The claim of variability was then questioned by Kemp et al. (1974), who argued that the initial measurements had been badly contaminated by polarised foreground zodiacal light contamination.\nLinear and / or circular polarisation of GD 229 has been measured by Landstreet & Angel (1974), Efimov (1981), Angel et al. (1981), West (1989a), Berdyugin (1995), and Berdyugin & Piirola (1999).\nAngel & Landstreet (1974) obtained three low-resolution I and V spectra of the star on the nights of 1973 Nov 6-8 using the MCSP. Synthetic broad-band polarimetry derived from these low-resolution spectra revealed no variations over three nights.\nWest (1989b) observed linear polarisation in GD 229 again in 1986 using broadband filters, but found no strong evidence of variation, not even of the position angle of linear polarisation.\nObservations of GD 229 using broadband polarimetry was also carried out for linear polarisation by Berdyugin (1995) and for both circular and linear polarisation, with higher precision, by Berdyugin & Piirola (1999), who compared their results to earlier work. Their results appear to show some quite significant changes in both circular and linear polarimetry compared to earlier observations by other groups. A major di ffi culty of comparing the various measurements is that each group has been made with di ff erent instrumentation. The consequences of using di ff erent filter passbands, di ff erent detector sensitivities as functions of wavelength, and even di ff erent (and possibly incorrect) calibration of instrumental polarimetric e ffi ciency, make it very hard to compare these data. However, Berdyugin & Piirola (1999) do o ff er strong evidence for significant rotation of the angle of linear polarisation, by about 30 · over 20 years, which is di ffi cult to explain by instrumental e ff ects. This is probably the most robust e ff ect that emerges from comparison of the many kinds of polarimetric observations\nLinear and circular polarisation spectra of GD 229 were obtained in 2018 and 2019 using the ISIS spectropolarimeter on the William Herschel Telescope (Sect. 2.1). These new data are shown in Fig. B.16, and compared to the circular polarised spectra by Angel & Landstreet (1974), and to previously unpublished linear spectropolarimetry of Angel and Landstreet (see Table A.1). Remarkably, the angle of linear polarisation appears to have returned to its value during the 1970s. Other di ff erences compared to the spectra of Angel and Landstreet are observed, but some of these are undoubtedly due to greatly di ff erent resolving power, and some may be due to uncertainties in the calibration of the 1970s data, particularly in the UV.\nIt is thus di ffi cult to establish clearly how much variability has occurred in GD 229. There is evidence for variations, but these appear to have relatively small amplitude compared to the overall scale of polarisation, except possibly for position angle rotation of linear polarisation (which however has not been confirmed by our most recent linear spectropolarimetry). Furthermore, because of very limited sampling with a variety of instruments, it is not possible to establish clearly any particular timescale for variations.\nWe note that the spectrum of GD 229 has been modelled as due to He in a field of hundreds of MG by Jordan et al. (1998, 2001) and Jordan (2003).\n\n\n
Fig. B.17. Circular polarisation spectra of WD 2049 -222, rebinned at ≃ 410 Å, and obtained at the time specified in the legend.
\n\nAppendix B.67: WD 2047+372 = EGGR 261 (variable, P ≃ 6 h)\nOriginally, the star was observed polarimetrically by Schmidt & Smith (1995), who did not detect its weak and sign reversing field (their measurements were ⟨ Bz ⟩ = -42 ± 59 kG and -2 . 5 ± 4 . 6 kG).\nThis star was discovered to be magnetic by Landstreet et al. (2016), and monitored and modelled by Landstreet et al. (2017). It is currently the weakest white dwarf field ( ⟨| B |⟩ ≈ 60 kG) that has been modelled in detail, mainly on the basis of a series of 18 ESPaDOnS spectra. The rotation period, determined from the variation of ⟨ Bz ⟩ , is 0.243 d. The observed variations of ⟨ Bz ⟩ and ⟨| B |⟩ are well modelled using a simple dipole model.\nNo rotational variation is detected in TESS photometry (Hernandez et al. 2024).\nAppendix B.68: WD 2049-222 = LP 872-48 (var.:)\nDiscovered to be magnetic by Berdyugin et al. (2022) with BBCP measurements. The star has V / I ≃ + 0 . 1 %, and is one of the weakest polarisation levels securely detected. The inferred ⟨ Bz ⟩ field strength is only of the order of a few MG. Broad-band measurements were repeated in July 2022 (Berdyugin et al. 2024) to check for variability, which was not detected (see Fig. B.1). We also observed the white dwarf three times with FORS2 with grism 300V (Sect. 2.1). The data are barely above the threshold of instrumental polarisation ( ≃ 0 . 1%; see Siebenmorgen et al. 2014), and therefore it is hard to establish whether the hints of variability seen in the spectra are real or not (see Fig. B.17). However, star WD 1116-440 shows a similar level of polarisation, and constant over about 4 year (see Fig. B.8), suggesting that the tiny variability observed in WD 2049-222 may be real and not an instrumental artefact.\nAppendix B.69: WD 2049-253 = UCAC4 325-215293 (non-variable)\nDiscovered to be magnetic by Bagnulo & Landstreet (2020), who observed continuum circular polarisation of order 0.4% and deduced a field strength of order 20 MG. This white dwarf was re-observed in broadband circular polarisation once by Berdyugin et al. (2022) and two more times by Berdyugin et al. (2024), but shows no sign of variability (see Fig. B.1).\nAppendix B.70: WD 2051-208 = BPS CS 22880-0134 (variable, P ≃ 1 . 5 h)\nThe magnetic field of this white dwarf was discovered from the shape of the H α line by Koester et al. (2009).\nWe have two series of five polarised spectra each, one using FORS grism 1200B and one with grism 1200R (LB25). These unpublished data clearly reveal very rapid rotation of this massive magnetic white dwarf. The stellar rotation period is 0.0594 d = 1.425 h. The value of ⟨ Bz ⟩ ranges approximately sinusoidally between about + 50 kG and -30 kG, while the corresponding values of ⟨| B |⟩ increase from about 200 kG to nearly 300 kG, in good agreement with the two values of <| B |> (220 and 290 kG) obtained by Koester et al. (2009) from UVES SPY spectra.\nAppendix B.71: WD 2105-820 = LAWD 83 (variable)\nThis star was suspected to have a weak magnetic field by Koester et al. (1998), who observed that the core of H α was abnormally broad, but they could not decide whether this was due to rapid rotation of v sin i ≈ 65 km / s or to a magnetic field of about 43 kG. Five FORS1 polarised spectra of the star by Landstreet et al. (2012) detected an apparently nearly constant magnetic field of ⟨ Bz ⟩ ≈ 10 kG. Later, FORS2 polarised spectra by Bagnulo & Landstreet (2018) and Farihi et al. (2018) reveal that ⟨ Bz ⟩ sometimes decreases to ⟨ Bz ⟩ ≈ 4 kG, so the star is apparently variable (see Bagnulo & Landstreet 2021).\nAppendix B.72: WD 2138-332 = L 570-26 (variable, P ≃ 6 . 19 h)\nThe DZA star was discovered to be magnetic, with a variable ⟨ Bz ⟩ of the order of 10 kG (Bagnulo & Landstreet 2019a) and a rotational period of P = 6 . 19 h (Hernandez et al. 2024; Farihi et al. 2024). The same period was found from the analysis of the equivalent width and ⟨ Bz ⟩ curves by Bagnulo et al. (2024b), who proposed a magnetic model with a dipolar field with ⟨| B |⟩ ≃ 50 kG. The star shows photometric and ⟨ Bz ⟩ curves with light minimum corresponding to ⟨ Bz ⟩ maximum.\nAppendix B.73: WD 2150+591 = UCAC4 747-070768 (variable, P ≃ 2 . 4 d)\nThe star was discovered to be magnetic by Landstreet & Bagnulo (2019), who reported two ISIS measurements that showed clearly that the field is variable with a period of hours or days. We subsequently monitored the star with one ESPaDOnS observation and several more ISIS spectra, and confirmed variability. These observations and a model of the star's magnetic field will be presented in a forthcoming paper (LB25).\nAppendix B.74: WD 2211+372 = LP 287-35 (non-variable)\nThis DC white dwarf was discovered to be magnetic by Berdyugin et al. (2023), who found circular polarisation in excess of 1% in the blue. It was re-observed by Berdyugin et al. (2024). It does not seem to be variable (see Fig. B.1). The field strength ⟨| B |⟩ is probably of the order of 60 MG or more.\nAppendix B.75: WD 2316+123 = KUV 23162+1220 (variable, P ≃ 18 d)\nDiscovered magnetic by Sion et al. (1984). Schmidt & Norsworthy (1991) reported BBCP of amplitude up to nearly 1% that varies sinusoidally with P = 17.86 d. Both the flux spectrum and the linear and circular polarisation spectra have been modelled repeatedly (Liebert et al. 1985; Achilleos & Wickramasinghe 1989; Friedrich et al. 1993; Putney & Jordan 1995); all agree that the global field strength is of the order of 30 MG. This star has the longest rotational period firmly established for a white dwarf.\nAppendix B.76: WD2359 -434 = LAWD 96 (variable, P ≃ 2 . 7 h)\nWD2359 -434 was suggested to be a magnetic star by Koester et al. (1998) on the basis of the peculiar profile of the H α core, and ⟨ Bz ⟩ was found to be non-zero by Aznar Cuadrado et al. (2004). A series of polarised ESPaDoNS spectra revealed a rotation period of 0.1123 d (Landstreet et al. 2017). The star is also a photometric variable with the same period. The field structure has been modelled approximately, and found to be distinctly more complex than a simple dipole, with ⟨| B |⟩ varying approximately between 50 and 100 kG. This white dwarf o ff ers one of the clearest examples known of a field structure that is substantially more complex than a simple co-linear multipole expansion (Landstreet et al. 2017).\n"},"sections":{"kind":"list like","value":[{"title":"ABSTRACT","content":"Many magnetic white dwarfs exhibit a polarised spectrum that periodically varies as the star rotates because the magnetic field is not symmetric about the rotation axis. In this work, we report the discovery that while weakly magnetic white dwarfs of all ages with M ≤ 1 M ⊙ show polarimetric variability with a period between hours and several days, the large majority of magnetic white dwarfs in the same mass range with cooling ages older than 2 Gyr and field strengths ≥ 10 MG show little or no polarimetric variability. This could be interpreted as extremely slow rotation, but a lack of known white dwarfs with measured periods longer than two weeks means that we do not see white dwarfs slowing their rotation. We therefore suggest a di ff erent interpretation: old strongly magnetic white dwarfs do not vary because their fields are roughly symmetric about the rotation axes. Symmetry may either be a consequence of field evolution or a physical characteristic intrinsic to the way strong fields are generated in older stars. Specifically, a strong magnetic field could distort the shape of a star, forcing the principal axis of maximum inertia away from the spin axis. Eventually, as a result of energy dissipation, the magnetic axis will align with the angular momentum axis. Alternatively, symmetry could be the hallmark of a dynamo that operates after the beginning of core crystallisation. We also find that the higher-mass strongly magnetised white dwarfs, which are likely the products of the merging of two white dwarfs, may appear as either polarimetrically variable or constant. This may be the symptom of two di ff erent formation channels or the consequence of the fact that a dynamo operating during a merger may produce diverse magnetic configurations. Alternatively, the massive white dwarfs with constant polarisation may be rotating with periods much shorter than the typical exposure times of the observations. Key words. polarisation - stars: white dwarfs - stars: magnetic fields","pages":[1]},{"title":"Strong magnetic fields of old white dwarfs are symmetric about the stellar rotation axes","content":"S. Bagnulo 1 and J.D. Landstreet 1 , 2 Received July 5, 2024, accepted October 22, 2024","pages":[1]},{"title":"1. Introduction","content":"Most white dwarfs do not show any variability, and more than 97% of them can be safely considered as suitable flux standard stars (Hermes et al. 2017). Because of flux stability, it is generally impossible to measure the rotation period of white dwarfs except when they pulsate or when they have a magnetic field. It is well known that magnetic white dwarfs often show periodically variable signals of polarisation and sometimes also periodic photometric variability. In the local 20 pc volume, more than 20% of white dwarfs have a magnetic field (Bagnulo & Landstreet 2021), and about 20% of the magnetic white dwarfs are photometrically variable (Farihi et al. 2024). Polarimetric variability is explained in terms of a magnetic field not symmetric about the rotation axis so that the magnetic configuration seen by the observer (as encoded in the polarisation signal) varies as the star rotates. The causes of photometric variability are not well understood (see, e.g., Bagnulo et al. 2024b), yet photometric variability of non-pulsating white dwarfs is so clearly correlated with the presence of the magnetic field variations that one may hypothesise that all non-pulsating white dwarfs with light variations are magnetic. If a star belongs to the small class of white dwarfs with a well-detected light curve, then the rotational period may usually be deduced from photometry (see Brinkworth et al. 2013; Hernandez et al. 2024; Oliveira da Rosa et al. 2024), but, in general, the time series of polarisation measurements may allow one to recover the rotational period of a larger sample of magnetic white dwarfs. Investigations of magnetic fields and of rotational periods are closely related not only because the presence of a magnetic field enables period measurements but also because one could expect that the occurrence of a magnetic field and its strength is physically related to stellar rotation. For example, should the field be supported by a dynamo, one could predict fields to be stronger in more rapidly rotating stars than in those that are more slowly rotating. Correlations between the presence of a magnetic field and the rotational period of a star exist even when the field has a fossil origin. Most of the chemically peculiar stars of the main sequence (Ap and Bp stars) have a fossil magnetic field, and they all rotate more slowly than the non-magnetic non-chemically peculiar stars in the same region of the Hertzsprung-Russel diagram (e.g. Donati & Landstreet 2009). Some of the magnetic Ap and Bp stars have well measured rotational periods as long as several months, years, or even decades (e.g. γ Equ, Leroy et al. 1994). There is no clear explanation for this well-established correlation between magnetic fields and slow stellar rotation. In the case of white dwarfs, various samples of data have gradually become available in the past few years to constrain possible scenarios for the evolution of their magnetic fields. It now appears that among the normal-mass white dwarfs ( M ≃ 0 . 6 M ⊙ ) that largely descend from single-star evolution, magnetic fields are very rare and weak during the first 1-2 Gyr of cooling but then gradually become much more common and often very strong after about 2-3 Gyr of cooling (Bagnulo & Landstreet 2021). This evolution may occur as a result of gradual relaxation to the surface of fields present in stellar cores during earlier evo- on and / or as a result of the operation of a dynamo during core crystallisation (Isern et al. 2017; Schreiber et al. 2021; Bagnulo & Landstreet 2021, 2022; Ginzburg et al. 2022). Among normal-mass stars, those with a magnetic field seem to rotate faster than non-magnetic white dwarfs (Hernandez et al. 2024), supporting the idea that crystallisation-driven dynamo may play a role in the formation of their magnetic fields. Many highly massive white dwarfs ( M ≥ 1 . 0 M ⊙ ), which are probably the product of mergers of two white dwarfs, appear magnetic very early in their cooling age, and the origin of the magnetic field may be due to a dynamo operating during the merging (Tout et al. 2008; García-Berro et al. 2012; Briggs et al. 2015; Bagnulo & Landstreet 2022). This idea is supported also by the evidence that some white dwarfs have very short rotation periods (e.g. Feige 7: 131 min, Liebert et al. 1977; Cl Oct: 12.1 min, Barstow et al. 1995; WD2209 + 113: 70 s, Kilic et al. 2021). At the same time, some polarimetric studies have hinted that a few old magnetic white dwarfs with a field strength of the order of tens of to a hundred megagauss have extremely long rotational periods, showing at most quite small variations even on timescales of decades (e.g. Berdyugin & Piirola 1999; Bagnulo &Landstreet 2019b). Traditionally, this lack of obvious variability is explained by the assumption that such non-varying magnetic white dwarfs have very long rotation periods of the order of centuries (Schmidt & Norsworthy 1991). In the absence of clear observational support or contradiction, this assumption has been widely accepted (e.g. Ferrario et al. 2015). The conclusion is basically that the known periods, P , fall into two very di ff erent families: one with P < ∼ 2 weeks, characterising most of the sample, and a few percent with strong fields that have periods of centuries (i.e. no clearly detected rotation). There is no explanation as to why some old white dwarfs would have extremely long rotation periods. Magnetic braking has generally been invoked but without the support of numerical calculations. In this work, we report new polarimetric observations and combine them with data collected from the literature. We analyse the magnetic variability and thus the rotation of a sample of 74 white dwarfs, and we discuss the constraints that these data place on possible evolution paths for magnetic fields in white dwarfs.","pages":[1,2]},{"title":"2.1. Observations not previously published","content":"We present here 49 unpublished spectropolarimetric observations of 13 magnetic white dwarfs. Fourteen spectra of six white dwarfs were recently obtained with the FORS2 instrument (Appenzeller et al. 1998) of the ESO VLT in the course of a spectropolarimetric survey of the solar neighbourhood. Seven polarised spectra of two other white dwarfs were retrieved from the ESO archive: one of the spectra was obtained with FORS1 and the remaining ones with FORS2 (we note that FORS1 and FORS2 are virtually identical instruments). Fourteen new spectra of four white dwarfs were obtained with the ISIS instrument of the William Herschel Telescope (WHT). For all of these data, the observing strategy and data reduction are identical to the procedures described by Bagnulo & Landstreet (2018) for circular polarisation data and by Bagnulo & Landstreet (2019b) for linear polarisation data. For two of the four stars recently observed with ISIS, we also present here three previously unpublished spectra obtained in the 1970s that we used to study the long-term variability of magnetic white dwarfs. These data consist of lowresolution polarised spectra obtained by Angel and Landstreet using the multi-channel spectrophotometer (hereafter MCSP) on the 5-m Palomar telescope. The instrument set-up and data reduction of these observations are described in detail by Angel & Landstreet (1974). Finally, eleven spectra of WD 1105-340 were obtained with the ESPaDOnS instrument of the CanadaFrance-Hawaii Telescope (CFHT). These data were reduced by the automatic CFHT pipeline LibreEsprit. The log of these unpublished observations is given in Table A.1, and results are described in the sections dedicated to individual stars of Appendix B. Apart from their general purpose of assessing polarimetric variability, our new observations confirm that star WD 1116 -470 is magnetic (it was previously considered as suspected to be magnetic by Bagnulo & Landstreet 2021) and bring the number of magnetic white dwarfs in the local 20 pc volume to 34 (see Bagnulo & Landstreet 2021). We have also discovered two new massive magnetic white dwarfs: WD1619 + 054 and WD1754 -550 (both rapidly variable).","pages":[2]},{"title":"2.2. Literature data","content":"We searched the literature and collected data for all magnetic white dwarfs that were observed in polarimetric mode at least twice as well as for all magnetic white dwarfs for which multiple spectroscopic intensity observations revealed magnetic variability. The way literature data and new observations have been used is described in the next section.","pages":[2]},{"title":"3. Determination of the variability of the magnetic white dwarfs","content":"We are interested in establishing whether the magnetic field of a white dwarf appears constant or variable with time to the observer. Magnetic variability is ascribed to changes in the apparent field geometry as a star rotates. It can be detected mainly via circular spectro- or broadband polarimetry, which are both sensitive to the longitudinal component of the magnetic field. Linear polarisation, which is sensitive to the transverse component of the magnetic field, is usually much weaker than circular polarisation, and it is rarely detected in white dwarfs. Intensity spectra (if the star is not featureless) are sensitive to the mean field modulus and may also be used to detect magnetic variability, but they are less sensitive to changes of the apparent magnetic configuration than polarimetry. Magnetic white dwarfs that have been repeatedly observed in spectroscopic mode and that do not show sign of variability have not been considered here because having observed a constant intensity spectrum is not a su ffi ciently strong indication that a star is not magnetically variable. An example is WD2047 + 372, as it shows an almost constant Zeeman triplet in multi-epoch intensity spectra and a variable, sign-reversing circular polarisation spectrum (Landstreet et al. 2017). Photometric variability is observed in magnetic white dwarfs, but in the absence of convincing observational evidence or theoretical arguments, we have chosen not to consider it alone as a proxy for magnetism nor magnetic variability. For example, Brinkworth et al. (2013, see in particular their Tables 1 and 3) detected light variability for a number of magnetic stars. For many of them, however, the light amplitude is quite low, and there are no multi-epoch polarimetric data to confirm magnetic variability. Furthermore, recent analysis of TESS data made by Hernandez et al. (2024) and Oliveira da Rosa et al. (2024) failed to detect a periodic light curve for many of these targets. These stars are not included in our sample. In summary: 1) Two or more intensity spectra, or polarisation spectra, or broadband polarisation measurements that clearly di ff er from each other are interpreted as being due to magnetic variability. Wegathered reasonable evidence for polarimetric variability, or non-variability, for the sample of 74 magnetic white dwarfs listed in Table 2. Admittedly, for some stars, we found that establishing whether or not two or more observations are consistent among themselves was somewhat subjective. In Sect. 5.3, we argue that our results are not a ff ected by these uncertain cases. Appendix B discusses in detail all individual stars. In this section, we summarise the results and flag special cases. Among the 74 white dwarfs, 44 are magnetically variable (Sect. 3.1), while for 27 there is so far no evidence of variability (Sect. 3.2). The remaining three white dwarfs, which have strong fields and have been observed over several decades, may be further tested for subtle or possibly very slow variability (Sect. 3.3).","pages":[2,3]},{"title":"3.1. Stars that show evidence of magnetic variability","content":"For 30 white dwarfs among the 44 stars that show polarimetric variability, a rotational period, or at least a good candidate for it, has been established in the literature. This period is given (in days) in the last column of Table 2 and is followed by ':' when only tentative or approximated values are known. Rotational periods are generally of the order of hours; about 25% are of the order of one day or longer, and only two stars have a rotation period longer than one week (the slowest magnetic white dwarf for which the period is known, WD 2316 + 123, has P ≃ 17 . 4 d, Schmidt & Norsworthy 1991). Most of these stars were listed in Table C.1 of Hernandez et al. (2024), and they are shown with blue symbols in Fig. 1 of Sect. 4. Among the other 14 variable stars, we do not have enough data to estimate the rotation period. For ten of them, the evidence of polarimetric variability has been securely established by two or more observations. These ten variable stars are marked in last column of Table 2 with 'var.' and are also represented with a blue symbol in Fig. 1. The remaining four white dwarfs show subtle signs of variability among the observations obtained with the same instrument. They are marked in Table 2 with 'var.:' and are shown with light blue symbols in Fig. 1. For WD 0446-789 and WD1009-184, our conclusion is that the field variations are real but small due to a configuration probably nearly symmetric about the rotation axis. WD 1105-048 was repeatedly observed with FORS1, FORS2, and ESPaDOnS, and a field was detected on only two occasions out of 12 observing epochs. If the star is magnetic, then it is certainly variable with a timescale of the order of days or weeks, but, admittedly, one could suspect that the two detections are in fact spurious. We decided to define the star as a probable variable star. WD 2049-222 shows a signal of polarisation that is marginally higher than instrumental polarisation, but for the reasons explained in Sect. B.68, we decided to consider its subtle variations as real.","pages":[3]},{"title":"3.2. Stars for which there is no evidence of magnetic variability","content":"Nineteen stars have been observed three or more times in polarimetric mode, and the observations are always consistent among themselves within uncertainties. We consider these stars as established non-variable white dwarfs. They are marked with 'n.v.' in Table 2 and represented with red symbols in Fig. 1. Among these 19 stars, four were observed over a time interval of decades and with di ff erent instruments: WD 0548 -001 = G9937, WD0553 + 053 = G99-47, WD1036 -204 = LP790 -29, and WD1829 + 547 = G227-35. Obviously, it is possible that the other members of this group of 'non-variable stars', for which the observations span a time interval of up to a few months or years, are actually variable on a timescale of decades and that as such they could be stars with (so far undetected) long-term variability. Eight stars should be considered simply as candidate nonvariable white dwarfs ('n.v.:' in Table 2 and magenta symbols in Fig. 1). For seven of them, the reason for considering them only as candidate non-variable stars is that they have only been observed twice. In particular we mention that for normal-mass WD0236-269 and for the massive WD0330 -000, the nonvariability is inferred based on comments in the original discovery paper by Schmidt et al. (2001), which represents especially weak evidence. Another candidate non-variable star, WD1750 -311, was observed only once (with FORS2). The comparison of spectra obtained within that single observing series shows some change of circular polarisation and certainly a rapidly variable intensity spectrum (timescale of minutes) in the regions of the Balmer lines, particularly around H β . The star also shows photometric variability with a 35-min period (Macfarlane et al. 2017). Our interpretation is that WD 1750-311 is probably a magnetic white dwarf with a field symmetric, or nearly symmetric about the rotation axis, showing strong abundance patches of hydrogen.","pages":[3]},{"title":"3.3. Stars possibly subtly or slowly variable","content":"There are three stars that have been monitored for decades, WD1748 + 708 = G240 + 708, WD1900 + 310 = Grw + 70 · 8247, and WD2010 + 310 = GD229, for which our conclusions are uncertain. These stars, marked with 's.v.:' in Table 2 and shown with black symbols in Fig. 1, may indeed show some low amplitude and possible long-term variability, but there might be room for a di ff erent interpretation. The recent detailed comparison of old and new polarisation spectra of Grw + 70 · 8247 by Bagnulo & Landstreet (2019b) identifies small changes between spectra taken decades apart (see in particular their Fig. 5), but for the two other stars, it is not entirely clear which changes are real and which are due to di ff erences between observational equipment and techniques. Bagnulo & Landstreet (2019b) assumed that timescales of the observed small changes were those corresponding to the large time gaps between isolated data sets. However, because the changes are generally small, one could argue that the subtle changes have timescales as short as weeks and have not have been noticed because the stars were not adequately monitored. For example, as previously discussed, WD 1105 -048 appears constant (and non-magnetic) in ten out of 12 observations, and a magnetic field was detected in only two epochs. In case of WD 1748 + 708, photometry leads to ambiguous results. The star was found photometrically variable with a period between 5 and 48 h by Antonyuk et al. (2016). Brinkworth et al. (2013) did not confirm its short-term photometric variability (nor did Hernandez et al. 2024) but instead claimed detection of light changes over a period of ten months.","pages":[3,4]},{"title":"4. Clustering in parameter space","content":"In this section, we correlate the variability or non-variability of the stars with their mass, field strength, and stellar temperature. The left panels of Fig. 1 show the field strength versus e ff ective temperature and cooling age for the stars of our sample, with the cooling age on both a linear scale and a logarithmic scale. In these plots, stars are represented with symbols that increase in size with a star's mass, while di ff erent colours are used to mark the characteristics in terms of variability. The right panels of Fig. 1 show the cooling age-mass diagrams for the same sample, with cooling age reported again both with linear and logarithmic scales. The size of the symbols is proportional to the field strength. With the help of these figures and Table 2, we studied the relationships between field strength, mass, age, and variability of the magnetic white dwarfs. We recall that our data do not come from a volume-limited sample of stars and do not reflect the relative density of magnetic white dwarfs in di ff erent regions of the diagrams of Fig. 1. In fact, both young magnetic normalmass white dwarfs and ultra-massive white dwarfs are quite rare objects in space, and they are over-represented in our sample because of biases of the surveys (see Bagnulo & Landstreet 2022). The following analysis is about the relative frequency of variable and non-variable stars in di ff erent regions of the diagrams where we noticed the existence of statistically remarkable di ff erences.","pages":[4]},{"title":"4.1. Normal-mass and weakly magnetic white dwarfs","content":"There are 25 white dwarfs with a field strength of ≤ 1 MG, and all of them but one (WD 2051-208) have M ≤ 1 . 0 M ⊙ . These 'normal-mass' 'weak-field' white dwarfs span an age range up to τ ≈ 6 Gyr. Twenty-one of these 24 stars are variable. Among the three that are non-variable, one is younger than 1 Gyr: WD 1105 -340 ( τ = 0 . 34 Gyr).","pages":[4]},{"title":"4.2. Normal-mass and strongly magnetic white dwarfs","content":"There are 25 magnetic white dwarfs with M ≤ 1 . 0 M ⊙ and field strength of ≥ 10 MG. Among those that are younger than 2 Gyr, six out of eight white dwarfs are variable; however, we recall that young strongly magnetic white dwarfs are rare in space (Bagnulo & Landstreet 2022). Among the magnetic white dwarfs older than 2 Gyr, only one out of the 17 shows polarimetric variability.","pages":[4]},{"title":"4.3. Massive magnetic white dwarfs","content":"Fifteen stars of our sample have M > 1 . 0 M ⊙ , and all but one are younger than about 1 Gyr. Eight of them are magnetically variable, including the only old ( τ ≃ 4 Gyr) star WD 0756 + 437. Nearly all of these massive magnetically variable white dwarfs have a strong field (tens to hundreds megagauss), except for WD2051-208, which is the only ultra-massive star ( M = 1 . 2 M ⊙ ) with a sub-MG field strength (0.25 MG) in our sample. In contrast, seven massive magnetic white dwarfs show no obvious signs of variability, although none of them may be safely considered as non-variable, either because they were observed only twice (or even just once, i.e. WD 1750-311; see Appendix B.60) or because some subtle sign of variability may have been detected (in WD 1900 + 705 and WD2010 + 310). WD1658 + 440 and WD1750-311 are the only massive non- Notes. The full list includes 59 stars, 23 of which are non-variable. Between brackets, the table also gives the number of non-variable stars N K divided by the number of stars N tot in the subset as defined by the age and field values. variable white dwarfs in our sample with a field as low as ≈ 2 MG. The remaining five non-variable stars have magnetic fields with strengths of the order of hundreds of megagauss.","pages":[4]},{"title":"4.4. Statistical analysis of normal-mass white dwarfs","content":"Because we have only one example of an old massive magnetic white dwarf, we do not know how variability in massive magnetic white dwarfs evolves with time. Vice versa, an evolutionary path is clearly seen in normal-mass ( M < ∼ 1 M ⊙ ) white dwarfs. In our sample, field variability is nearly ubiquitous among weakly magnetic white dwarfs of all ages and in young white dwarfs regardless of the field strength, it is but very rare in old ( τ ≥ 2 Gyr) strongly magnetic ( B ≥ 10 MG) white dwarfs. Next, we assess the statistical significance of this pattern, specifically whether it can be attributed to small number statistics or if it reflects a genuine correlation between cooling age, field strength, and field variability. We first split the sample of stars with M ≤ 1 M ⊙ into two groups: those younger than 2 Gyr and those older than 2 Gyr. Each of these two groups was divided into two subsets: stars with a field strength ⟨| B |⟩ ≤ 1 MG and stars with ⟨| B |⟩ > 10 MG. For each of these four subsets, we considered the ratio between the number of non-variable magnetic stars, N K, and the total number of magnetic stars, N tot. From these numbers, we estimated the probability, Pr , that the sample frequency of non-variable stars is between r and r + d r , normalised to one and obtained assuming that all numbers between zero and one are a priori equally probable, using Figure 2 shows the probability density functions for the stars belonging to these sets. It clearly appears that there is little to no overlap between the density functions of the symmetric field in old strongly magnetic white dwarfs and of young white dwarfs. Table 1 reports the points of maximum for these distributions, f = N K / N tot, and their uncertainties Table 1 also includes the results for the smaller sets of stars with intermediate strength, 1 ≤ ⟨| B |⟩ ≤ 10 MG (with endpoints overlapping with the other two sets), which could possibly be considered a 'transitioning' field strength range. Figure 3 shows the ratio between the number of non-variable magnetic white dwarfs with a field strength lower than a given value ¯ ⟨| B |⟩ and the total number of magnetic white dwarfs with a field strength lower than that value as a function of ¯ ⟨| B |⟩ for stars with cooling ages τ ≤ 2 Gyr (blue lines) and stars with τ > 2 Gyr (red lines). This plot supports our claim that nonvariable fields are much more common in old, strongly magnetic white dwarfs than in young strongly magnetic white dwarfs, and than in weakly magnetic white dwarfs of all ages. It suggests also that the minimum field strength required for a field to become non-variable is in the range of 5 to 10 MG.","pages":[4,6]},{"title":"5. Explanation for the lack of observed variability","content":"There are three possible reasons for the observed non-variability of so many magnetic white dwarfs. We examine them in the following sections.","pages":[6]},{"title":"5.1. Extremely slow rotation","content":"We first consider the possibility that the rotation of magnetic white dwarfs slows with age and / or magnetic field strength. Perhaps, as the surface field becomes stronger, the white dwarf loses angular momentum to its environment by electromagnetic dipole radiation (García-Berro et al. 2012), by coupling with gas clouds in the ISM, or by a very weak magnetically coupled wind. Each of these possibilities would shed stellar angular momentum faster from a magnetic white dwarf with a strong field than from one with a weak field, preferentially slowing the magnetic white dwarfs with strong fields. However, all of these mechanisms are expected to exert at most a very weak influence on the rotation of a white dwarf, and it seems quite unlikely that any of them could slow the rotation of a magnetic white dwarf so much that repeated observations even a year apart would not show any rotation. Nevertheless, the idea of extremely slowly rotating white dwarfs has been generally accepted. This hypothesis likely origi- ates from Sect. 3 of Schmidt & Norsworthy (1991), which says: 'We therefore assign long periods to these stars, with the recognition that other explanations for their polarimetric constancy are possible.' This statement was accompanied by the first plots of field strength versus rotational period, a plot that in updated form has reappeared in numerous reviews (e.g. Ferrario et al. 2015; Kawka 2020; Ferrario et al. 2020) but has never been critically revisited. We note that there is a complete lack of white dwarfs that are known to vary on a timescale between weeks and decades. In this respect, there is a profound di ff erence between candidate longterm variable magnetic white dwarfs and long period magnetic Ap and Bp stars. While we do not know of any white dwarf with a firmly established polarimetric variability with a period longer than approximately two weeks, we are sure that a number of Ap and Bp stars are very slowly rotating stars because they definitely show clearly periodic field variation, even ⟨ Bz ⟩ sign reversals, on a very long but measured timescale that may be months, years, or decades. Furthermore, the distribution of periods is roughly continuous between periods of less than one day and periods of several years (Mathys 2008). If we wanted to interpret the lack of polarimetric variability in white dwarfs as the e ff ect of extremely long rotation periods, we would need to accept that the rotation periods of white dwarfs show an extremely bimodal distribution that peaks around hours and days and around centuries with nothing in between. This appears to be a very unlikely scenario.","pages":[6]},{"title":"5.2. Extremely fast rotation","content":"Asecond possibility is that some or all of the non-variable white dwarfs actually are very rapidly rotating stars. If the star has a rotation period much shorter than the individual frame exposure time, then the polarimetric variability would be smeared out, and the star would appear constant. In fact, it has been possible to probe variability on the timescale of the exposure time of each individual frame (typically 10 min or less). In very general terms, our FORS2 and ISIS polarimetric observations allowed us to identify a star as variable provided that its rotation period is longer than ≈ 10 min and its magnetic configuration is clearly not symmetric about the rotation axis (for example, WD 1712-590 in Sects. B.57). Isolated white dwarfs that are the product of single-star evolution can hardly have rotation periods shorter than that (Kawaler 2015). This is confirmed by the results of Hernandez et al. (2024), who have shown that rotation periods of young magnetic white dwarfs in the normal mass range are only marginally shorter than the rotation periods of non-magnetic white dwarfs. Merger products, in contrast, can have rotation periods of the order of 1 min (Schwab 2021; Kilic et al. 2021) if most of the binary angular momentum is retained by the merger product. It is therefore possible that the non-variable massive white dwarfs have nonaxisymmetric fields but are rotating very rapidly. Periods down to 1 min or less can be probed via rapid cadence photometry. None of the massive, magnetically non-variable white dwarfs show TESS light variability (Hernandez, priv. comm.; Ramsay, priv. comm.), although of course one cannot rule out that more accurate photometry could reveal some extremely rapid rotators. Provisionally, we rule out very rapid rotation for the massive magnetic white dwarfs that show a constant polarisation.","pages":[6,7]},{"title":"5.3. Magnetic fields are symmetric about the stellar rotation axes","content":"After ruling out both extremely slow and extremely fast rotation, we are left with the hypothesis that a non-variable field is approximately axisymmetric about the rotation axis. This means that as the star rotates with a normal period between a fraction of an hour and several days, the observer does not see any significant change in the signal of circular polarisation. This interpretation naturally accounts for some outliers, such as the non-variable weakly magnetic star WD 1105-340. It is reasonable to hypothesise that this one star does not appear variable simply because its rotation axis is tilted at a small angle with respect to the line of sight. In Sect. 3, we highlighted that it may be di ffi cult to firmly assess whether a star is magnetically variable because changes could be too subtle to be detected. However, the risk of classifying as 'non-variable' a star that in fact exhibits subtle but real changes does not weaken our analysis because small changes of the circular polarisation spectrum are still symptoms of a field nearly symmetric about the rotation axis.","pages":[7]},{"title":"6. Discussion","content":"Here, we first consider the case of normal-mass white dwarfs (Sect. 6.1) and then that of massive stars (Sect. 6.2). We note that our analysis applies only to isolated white dwarfs; symmetric fields seem uncommon among magnetic white dwarfs accreting material from a companion (Cropper 1988; Reimers et al. 1999; Schmidt et al. 1995).","pages":[7]},{"title":"6.1. Normal-mass white dwarfs","content":"It is clear that most of the weak fields of normal-mass stars of all ages are not symmetric about the stellar rotation axis. Most of the strong fields of normal-mass, young white dwarfs are also not axisymmetric. Most of the strong field of stars older than 2 Gyr are symmetric about the star's rotation axis. We now investigate what are the relationships between these weak and strong fields. Our hypothesis is that the weak and non-symmetric magnetic fields of the young normal-mass white dwarfs are due to a relaxation process of a field that was produced in a previous evolutionary stage of the star, a field that was buried below the surface of the newly formed white dwarf, and that it starts to reveal itself with time as the star cools down. Such seed fields could be those found in red-giant stars, in the vicinity of the hydrogenburning shell (e.g. Li et al. 2022, 2023). The question is whether the older, generally stronger fields share essentially the same origin as the weak, younger ones, having just evolved for longer time, or if the strong fields in older stars formed by a totally different mechanism than the weaker, younger ones, for instance by a crystallisation dynamo. Whatever its origin is, we consider first the possibility that the small distortion of the shape of a magnetic white dwarf produced by magnetic forces in outer layers might actually drive the evolution of the global field towards axisymmetry. In Sect. 6.1.1, we discuss one example of a physical e ff ect that might drive such evolution.","pages":[7]},{"title":"6.1.1. The magnetic field structure of white dwarfs may evolve towards rotational axisymmetry during cooling","content":"In the absence of a magnetic field, the rotation of the white dwarf will lead to an equatorial bulge, increasing the component of the principal moment of inertia that is parallel to the rotation axis. In this case, the rotation about the spin axis, which coincides with the largest moment of inertia, is stable. Next, we introduce a dipolar magnetic field inclined to the axis of rotation by, say, 45 · . We suppose that the visible surface field is maintained by a fossil current system deep in the star, which is gradually decaying due to Ohmic losses. As the interior field strength decreases, an azimuthal electric field will be produced by Faraday induction outside the white dwarf's initial current loop. This field will generate a current around the magnetic axis that will in turn interact with the local meridional magnetic field (Landstreet 1987) to produce an outward-directed force, distorting the stellar outer layers into a slightly oblate shape, with the largest radius around the magnetic equator. Because of this e ff ect, the principal moment of inertia of the star will be rotated towards the symmetry axis of the field, which is not aligned with the rotation axis or the angular momentum axis of the star. The white dwarf e ff ectively becomes an unstable free asymmetric rotator (an asymmetric top). If such objects can dissipate some of the energy supporting the asymmetric shape, the field axis will gradually shift to bring the principal moment of inertia closer to the angular momentum axis, which is a stable minimum energy state of an oblate system. Energy dissipation will gradually lead to the alignment of the magnetic axis with the rotation axis, which is the e ff ect we observe in old normal-mass strongly magnetic white dwarfs. This basic physical e ff ect was first identified in the 1970s as possibly operating in main-sequence magnetic Ap stars, which, similar to magnetic white dwarfs, possess global fossil magnetic fields that are usually roughly dipolar and are generally oblique to the rotation axis (Stibbs 1950; Preston 1971). A number of efforts to estimate the timescale of possible evolution of an oblique magnetic field to a state of small or vanishing obliquity have been published, for example, by Mestel & Takhar (1972). Much of this work is summarised in Chapter 9 of Mestel (1999). However, this line of investigation did not lead to a convergence of clear results about possible timescales or about dependence on basic input physics or parameters, such as interior field strength, or details of induced forces in outer layers and their consequences. As the increasing sample of magnetic Ap star models failed to reveal an obliquity distribution suggestive of this e ff ect, possibly because of the relatively short (of order 10 8 yr) mainsequence lifetimes of magnetic Ap stars, theoretical studies of stellar fossil fields moved on to other e ff ects. However, a similar e ff ect may be at work aligning the magnetic fields to the rotation axis in white dwarfs, which have a very di ff erent structure and much longer evolutionary times than Ap stars. This hypothesis could be explored with the aid of numerical modelling of the global (interior and surface) structure and evolution of a rotating white dwarf with an oblique magnetic field. Modelling of comparably complex white dwarf states (e.g. polars, mergers) that include magnetic fields (e.g. Franzon & Schramm 2015; Bisikalo et al. 2021; Zhong et al. 2024) suggest that numerical methods for exploring this problem are already available.","pages":[7,8]},{"title":"6.1.2. Crystallisation-driven dynamo and axisymmetric fields","content":"The line that marks the beginning of core crystallisation in the age-mass diagram separates variable from non-variable magnetic white dwarfs perhaps in a cleaner way compared to a massindependent age threshold (see Fig. 1). Before core crystallisation begins, magnetic fields are almost always non-symmetric about the rotation axis. From volume-limited surveys, we know that in normal-mass white dwarfs, before crystallisation, strong fields are rare (Bagnulo & Landstreet 2021, 2022), but those that have been discovered and monitored show polarimetric variability. After the beginning of core crystallisation, many strong fields appear, and most of them are symmetric about the rotation axis. White dwarfs with weak and non-symmetric fields continue to appear also after the beginning of core-crystallisation. The strong magnetic fields of old normal-mass white dwarfs could be generated by the crystallisation convective dynamo mechanism (Isern et al. 2017; Schreiber et al. 2021), as almost all have cooling ages longer than the cooling age required for the onset of this dynamo. In this case, the conclusion would be that the crystallisation dynamo is responsible for approximate symmetry about the rotation axis. This situation could have similarities with what has been observed in fully convective late type stars with no di ff erential rotation, which are known to be able to generate strong and simple large-scale, mostly axisymmetric, poloidal fields (Donati et al. 2006; Morin et al. 2008b,a; Kochukhov 2021, Fig. 14). However, how the crystallisation dynamo alone can produce fields stronger than ∼ 1 MG is still unclear (Isern et al. 2017; Castro-Tapia et al. 2024). In fact, Montgomery & Dunlap (2024) have argued that fluid mixing by phase separation is not a viable mechanism to produce the strong fields observed in old white dwarfs. Perhaps the crystallisation dynamo could be more powerful if it were to amplify a pre-existing internal field, such as the same seed field that is seen in some of the white dwarfs prior to the beginning of crystallisation. Montgomery & Dunlap (2024) have proposed that corecrystallisation could trigger temporary di ff erential rotation. Therefore, one could suspect that rapid rotation in normal-mass white dwarfs might generate a magnetic field in the stellar core using rotational shear on a seed field left from an earlier point in the star's evolutionary history. In this situation, we might also expect that the field would be roughly axisymmetric, although it is not clear why such fields would always be strong or how they might be related to the weaker oblique fields of stars of similar mass. Furthermore, Spruit (1999) has shown that di ff erential rotation would suppress the non-axisymmetric field component of weak fields, but not in the stronger fields. This is the opposite of what we observed. The origin of strong non-axisymmetric fields of normal-mass white dwarfs that appear before crystallisation could possibly be similar to that of higher-mass white dwarfs (see Sect. 6.2 below).","pages":[8]},{"title":"6.2. The variability of massive magnetic white dwarfs","content":"Compared to normal-mass white dwarfs, the magnetic fields of massive white dwarfs present a di ff erent behaviour. Magnetic fields appear when the stars are still very young, and fields may be either symmetric or non-symmetric about the rotation axis. In massive white dwarfs, strong fields (tens or hundreds of megagauss) seem much more common than weaker sub-megagauss fields. It is widely thought that many of these high-mass objects are the result of WD-WD mergers that cause rapid generation of a strong magnetic field (García-Berro et al. 2012; Bagnulo & Landstreet 2022). However, Camisassa et al. (2022) and Blatman & Ginzburg (2024) have shown that at least some of them, depending on their core composition, may have started the process of core crystallisation. Hence, the origin of their field could be linked to the crystallisation dynamo, at least in some cases. The bimodal distribution of the morphologies of the fields of the massive magnetic white dwarfs could indeed reflect two di ff erent channels of formation, one from WD-WD merger (or by a di ff erent binary evolution path), and one by massive singlestar evolution, such as that of normal-mass white dwarfs. Alternatively, the dynamo stimulated by WD-WD merging may be capable of generating both axisymmetric and non-axisymmetric global fields, perhaps depending on the initial angular momentum vectors of the individual merging white dwarfs relative to the orbital angular momentum vector or the mass ratio of the two merging white dwarfs. A viable alternative explanation is that the massive white dwarfs that do not show variability are actually rotating with a period much shorter than the typical exposure time of individual polarimetric measurements. In any case, the large mass of stars in this sample and the occurrence of some rotation periods as short as minutes make it very reasonable to suppose that the evolution of the magnetic fields and rotation periods follows a di ff erent course from the evolution of fields in normal-mass magnetic white dwarfs. Because almost all of the massive magnetic white dwarfs of our sample, except one, are very young, we cannot test whether the morphology of older stars is generally axisymmetric. Remarkably, however, the only example we know of a massive star older than 2 Gyr shows a non-axisymmetric field. With an age of more than 4 Gyr, high mass, and a very strong and variable field, the very unusual WD 0756 + 437 may be considered further evidence that the origin of the fields in massive stars is di ff erent than in normal-mass stars.","pages":[8]},{"title":"7. Conclusions","content":"Since the earliest discoveries of magnetic white dwarfs (Kemp et al. 1970; Angel & Landstreet 1970, 1971b), two quite di ff erent categories of them have been known to exist. Some magnetic white dwarfs show periodic variations of circular polarisation (which probes the longitudinal magnetic field) with timescales ranging from minutes to days. Classically, a signal of circular polarisation constant with time has been interpreted as indicating either a very long rotation period or a lack of rotation. Using both literature data and new observations, we have studied the polarimetric variability of a sample of 74 magnetic white dwarfs. We find that among white dwarfs with M ≤ 1 . 0 M ⊙ ('normal-mass white dwarfs'), nearly all stars with fields weaker than about 1 MG show circular polarisation varying with time. Furthermore, the rare normal-mass white dwarfs younger than ≈ 2 Gyr with strong magnetic fields also show polarimetric variability. In striking contrast, 16 out of the 17 normal-mass stars older than 2 Gyr with fields stronger than about 10 MG in our sample show constant polarisation. Magnetic white dwarfs with M ≥ 1 . 0 M ⊙ ('massive white dwarfs'), many of which are the product of WD-WD merging, show a mixed behaviour. Many of them have a strong magnetic and variable field, but some have a strong and constant magnetic field. In our sample, nearly all the massive magnetic white dwarfs are younger than ≈ 1 Gyr, with the exception of WD 0756 + 437, an old strongly magnetic variable star. The lack of evidence for major variations of circular polarisation on any timescale longer than about two weeks suggests that the interpretation of non-variability arising from extremely long rotational periods is incorrect. We have argued that the nonvariability is due to magnetic field structures roughly symmetric around the star's rotation axis. A possible explanation for the observed symmetry could be that most or all of the magnetic fields evolved from pre-existing fields that formed during pre-white dwarf evolution stages (Li et al. 2022, 2023) and gradually relaxed to the stellar surface over a relation time of 1 or 2 Gyr; in fact, very recently, Camisassa et al. (2024) have shown that the magnetic fields of old white dwarfs with M ≥ 0 . 65 M ⊙ may have been generated by a coreconvection dynamo when the star was in the main sequence, and emerged at the stellar surface during the white dwarf cooling phase. As the surface field increases in strength, at some point the Lorentz forces in the outer layers, together with the Coriolis forces acting on rotating convective flows in what are asymmetric rotators, could lead to a gradual relaxation, through energy dissipation, of the global field structure to a form that is symmetric about the rotation axis. The timescale would depend on the field strength, and for a field weaker than several megagauss, it could require a timescale too long to be observed in white dwarfs that still show spectral lines. Alternatively, because these normal-mass large-field magnetic white dwarfs mostly occur after the start of crystallisation (Bagnulo & Landstreet 2021, 2022), one could speculate that the crystallisation dynamo (Isern et al. 2017; Schreiber et al. 2021; Ginzburg et al. 2022) may generate some fraction of the observed fields and further that this dynamo produces essentially axisymmetric surface magnetic field structures with extremely strong fields. This kind of origin would have similarities with the axisymmetric fields that are commonly found in fully convective strongly magnetic M-dwarfs and, much weaker, in the planets Earth and Jupiter. However, it has been argued that the instability of the mantle surrounding the core that starts to crystallise cannot produce fields much stronger than 1 MG (Isern et al. 2017; Montgomery & Dunlap 2024; Castro-Tapia et al. 2024). Furthermore, Camisassa et al. (2024) find that, even if the crystallisation-driven dynamo could generate a strong magnetic field, this field would take too long to emerge at the stellar surface. On the other hand, Montgomery & Dunlap (2024) have suggested that crystallisation may still play a role by trig- gering a temporary phenomenon of di ff erential rotation, which in turn would generate a magnetic field - the di ff usion timescale of which has not been discussed. The situation for ultra-massive white dwarfs (with M > ∼ 1 . 0 M ⊙ ) is di ff erent, as both strongly magnetic variable and nonvariable white dwarfs are found among them. Some of these massive white dwarfs are the product of WD-WD merging, during which a dynamo could create a strong magnetic field (García-Berro et al. 2012), though it would not necessarily be symmetric about the rotation axis. In addition, some massive magnetic white dwarfs could be the result of single-star evolution and have acquired an axisymmetric field following the same mechanism acting in normal-mass white dwarfs. Further investigation into the variability of magnetic white dwarfs is necessary. Nevertheless, current observations have already imposed significant constraints that any theory explaining the origin and evolution of magnetic fields in degenerate stars must take into account. Notes. A number in the last column represents the established rotation period (in d) of a star that shows polarimetric variability; tentative periods are followed by the symbol ':'. Other symbols have the following meaning: 'var.' means that the star is certainly polarimetric variable but the period is still unknown; 'var.:' means that hints of subtle variability have been detected over a short timescale; 'n.v.' means that the observed polarisation was constant (within uncertainties) and the star was observed at least three times; 'n.v.:' means that observed polarisation was constant (within uncertainties) but the star was observed only twice; 's.v.' means that polarisation shows some sign of subtle variability over a timescale of a decade or longer. For stars within the local 40 pc volume, the parameters of stellar magnitude, distance, spectral type, atmospheric composition, temperature, mass, and age are from O'Brien et al. (2024); for the remaining stars, we used the catalogue from Gentile Fusillo et al. (2021) with ages interpolated from the tables by Bédard et al. (2020). Acknowledgements. The new observations presented in this work were made with the FORS2 instrument at the ESO Telescopes at the La Silla Paranal Observatory under program ID 110.243J.001, 110.23XV.001, 110.23XV.002, and 112.25C9.001, and with the ISIS instrument at the William Herschel Telescope (operated on the island of La Palma by the Isaac Newton Group), under programmes P10 in 19A and P8 in 19B. This research has made use also of additional FORS1 and FORS2 data obtained from the ESO Science Archive Facility: data for WD 1036-204 were obtained under programmes IDs 70.D-0259(A), 087.D-0714(A), 089.D-0612(A), 090.D-0269(A). Data for WD 1105-340 was acquired with ESPaDOnS on the Canada-France-Hawaii Telescope (CFHT) (operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii), under programmes 17AC01, 19AC04, 19BC02, and 21BC02. We thank the anonymous referee for their very constructive criticism. We thank Matthias Schreiber and Antonino Lanza for very useful comments. JDL acknowledges the financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference number 6377-2016.","pages":[8,9,11,12]},{"title":"References","content":"Achilleos, N. & Wickramasinghe, D. T. 1989, ApJ, 346, 444 Angel, J. R. P., Illing, R. M. E., & Landstreet, J. D. 1972a, ApJ, 175, L85 Aznar Cuadrado, R., Jordan, S., Napiwotzki, R., et al. 2004, A&A, 423, 1081 Article number, page 12 of 29 Abstract Series, Vol. 9, Astronomische Gesellschaft Abstract Series, 145 Liebert, J., Schmidt, G. D., Sion, E. M., et al. 1985, PASP, 97, 158 Macfarlane, S. A., Woudt, P. A., Dufour, P., et al. 2017, MNRAS, 470, 732 Manser, C. J., Gänsicke, B. T., Inight, K., et al. 2023, MNRAS, 521, 4976 Martin, B. & Wickramasinghe, D. T. 1978, MNRAS, 183, 533 Mathys, G. 2008, Contributions of the Astronomical Observatory Skalnate Pleso, 38, 217 Maxted, P. F. L., Ferrario, L., Marsh, T. R., & Wickramasinghe, D. T. 2000, MNRAS, 315, L41 McCook, G. P. & Sion, E. M. 1977, A Catalogue of spectroscopically identified white dwarfs McCook, G. P. & Sion, E. M. 1999, ApJS, 121, 1 Mestel, L. 1999, Stellar magnetism Mestel, L. & Takhar, H. S. 1972, MNRAS, 156, 419 Montgomery, M. H. & Dunlap, B. H. 2024, ApJ, 961, 197 Morin, J., Donati, J. F., Forveille, T., et al. 2008a, MNRAS, 384, 77 Morin, J., Donati, J. F., Petit, P., et al. 2008b, MNRAS, 390, 567 Moss, A., Bergeron, P., Kilic, M., et al. 2024, MNRAS, 527, 10111 O'Brien, M. W., Tremblay, P. E., Gentile Fusillo, N. P., et al. 2023, MNRAS, 518, 3055 O'Brien, M. W., Tremblay, P. E., Klein, B. L., et al. 2024, MNRAS, 527, 8687 Oliveira da Rosa, G., Kepler, S. O., Soethe, L. T. T., Romero, A. D., & Bell, K. J. 2024, arXiv e-prints, arXiv:2407.05214 Piirola, V. & Reiz, A. 1992, A&A, 259, 143 Preston, G. W. 1971, PASP, 83, 571 Putney, A. 1995, ApJ, 451, L67 Putney, A. 1997, ApJS, 112, 527 Putney, A. & Jordan, S. 1995, ApJ, 449, 863 Reding, J. S., Hermes, J. J., Vanderbosch, Z., et al. 2020, ApJ, 894, 19 Reimers, D., Hagen, H. J., & Hopp, U. 1999, A&A, 343, 157 Schmidt, G. D., Bergeron, P., & Fegley, B. 1995, ApJ, 443, 274 Schmidt, G. D. & Norsworthy, J. E. 1991, ApJ, 366, 270 Schmidt, G. D. & Smith, P. S. 1994, ApJ, 423, L63 Schmidt, G. D. & Smith, P. S. 1995, ApJ, 448, 305 Schreiber, M. R., Belloni, D., Gänsicke, B. T., Parsons, S. G., & Zorotovic, M. 2021, Nature Astronomy [ arXiv:2104.14607 ] Schwab, J. 2021, ApJ, 906, 53 Siebenmorgen, R., Voshchinnikov, N. V., & Bagnulo, S. 2014, A&A, 561, A82 Sion, E. M., Liebert, J., Schmidt, G., & Starrfield, S. G. 1984, in Bulletin of the American Astronomical Society, Vol. 16, 725 Spruit, H. C. 1999, A&A, 349, 189 Stibbs, D. W. N. 1950, MNRAS, 110, 395 Swedlund, J. B., Wolstencroft, R. D., Michalsky, Jr., J. J., & Kemp, J. C. 1974, ApJ, 187, L121 Valyavin, G., Bagnulo, S., Monin, D., et al. 2005, A&A, 439, 1099 Valyavin, G., Wade, G. A., Bagnulo, S., et al. 2008, ApJ, 683, 466 Vennes, S., Schmidt, G. D., Ferrario, L., et al. 2003, ApJ, 593, 1040 Vornanen, T., Berdyugina, S. V., Berdyugin, A. V., & Piirola, V. 2010, ApJ, 720, L52 Walters, N., Farihi, J., Marsh, T. R., et al. 2021, MNRAS, 503, 3743 Wegner, G. 1977, Mem. Soc. Astron. Italiana, 48, 27 Wesemael, F., Liebert, J., Schmidt, G. D., et al. 2001, ApJ, 554, 1118 West, S. C. 1989a, ApJ, 345, 511 West, S. C. 1989b, ApJ, 345, 511 Wickramasinghe, D. T. & Bessell, M. S. 1976, ApJ, 203, L39 Wickramasinghe, D. T. & Cropper, M. 1988, MNRAS, 235, 1451","pages":[12,13]},{"title":"Appendix A: New observations","content":"Article number, page 14 of 29","pages":[14]},{"title":"Appendix B: Variability of magnetic white dwarfs: Comments on individual stars","content":"We preliminary note that to compare observations taken by different authors, we need to know how circular polarisation was defined in di ff erent works. This problem has been highlighted in detail, for example in Sect. 2 of Bagnulo & Landstreet (2020). Here, we have adopted the definition of positive handedness of circular polarisation given by Landi Degl'Innocenti & Landolfi (2004), and converted some of the literature data to it by changing the sign of circular polarisation with respect to the originally published data. Numerous examples in the literature (e.g. Bagnulo & Landstreet 2020) and in this section show that many magnetic white dwarfs with very strong fields have a circular polarisation spectrum that varies rapidly with wavelength. The consequence is that the same source observed with even slightly di ff erent broadband filters may result in di ff erences of the measured broadband polarisation signal that are larger than the formal uncertainties. This specific issue is discussed in Sect. 7 of Bagnulo & Landstreet (2019b), where some numerical examples are presented. As a consequence, small di ff erences between broadband polarimetric measurements obtained with di ff erent instruments cannot reliably provide strong evidence that a star is magnetically variable. Spectropolarimetry should allow comparisons that are less instrument dependent, provided the observations overlap in wavelength, and that the resolving power of di ff erent instrument is similar. There is no doubt, however, that the safest way to establish variability or constancy of polarisation is to repeatedly observe the star with the same instrument and instrument setting, and to always perform the data reduction using exactly the same technique. In the following, we discuss all stars of our sample (including two stars that did not make it into the final list of Table 2). The titles of the individual subsections contain the star's name in the Villanova system (McCook & Sion 1977, 1999), the main SIMBADidentifier, and our conclusions about the magnetic variability (using the same designation as in Table 2, followed by the rotation period P , if this is known). When the rotation period is obtained only from photometric data, we use the designation P phot.","pages":[16]},{"title":"Appendix B.1: WD 0004+122 = LP 464-57 (non-variable)","content":"This star was discovered to be magnetic, using spectropolarimetry, by Bagnulo & Landstreet (2020), and later observed four times by Berdyugin et al. (2024) in broadband circular polarisation. These measurements are shown in the top left panel of Fig. B.1. None of them deviated more than ≃ 1 σ from the mean value of the measurements obtained in the same filter, except for a ∼ 2 . 5 σ di ff erence between on measurement and the other three in the B filter, which might reflect a real small variability. We reanalysed the FORS2 spectropolarimetry published by Bagnulo & Landstreet (2020) to check for di ff erences between the V / I profiles obtained from the first and the second pair of exposures, sampling an interval in time of about 15 m, without finding any di ff erence (see Fig. B.2). We conclude that the star does not show convincing sign of variability. We estimate a field strength of ≃ 100 MG.","pages":[16]},{"title":"Appendix B.2: WD 0009+501 = EGGR 381 (variable, P ≃ 8 h)","content":"This white dwarf was well monitored and modelled as a magnetic variable with ⟨| B |⟩ between 150 and 250 kG by Valyavin et al. (2005), who presented a model obtained by adopting a rotational period of 0 . 3337 ± 0 . 0031 d. Recent unpublished ESPaDOnS data obtained by us confirm the rotational period.","pages":[16]},{"title":"Appendix B.3: WD 0011-134 = G 158-45 (variable, P phot ≃ 0 . 73 h)","content":"Discovered magnetic (with a field of about 9 MG) by Bergeron et al. (1992), and as a magnetic variable by Putney (1997). From photometry, Lawrie et al. (2013) found a rotational period of 44 ± 0 . 43 min. While there is no guarantee that the photometric period reflects the rotational period of the star, the star is definitely a magnetic variable, and in fact, this very short photometric period is completely consistent with several of the many acceptable periods found from the six widely spaced ⟨ Bz ⟩ data values published by Putney (1997) from spectropolarimetric observations made between 1994 Sep 29 and 1994 Dec 31. This star is one of the magnetic white dwarfs that clearly shows longitudinal field reversal.","pages":[16]},{"title":"Appendix B.4: WD 0041-102 = Feige 7 (variable, P ≃ 2 h)","content":"Liebert et al. (1977) showed that the spectrum is rich in faint lines of H and He, and identified a magnetic field of about 18-20 MG from the spectrum. They found, on the basis of broadband circular polarisation measurements, that the field varies with a period of 131.6 min. (Note that the broadband circular polarisation changes sign as the star rotates.) A decentred dipole model of the field structure based on a series of flux spectra was derived by Achilleos et al. (1992); both H and He apparently vary substantially in abundance over the surface. Achilleos et al. (1992) also discovered that the star varies photometrically (in a double wave) with the same period as the spectrum and broad-band polarisation.","pages":[16]},{"title":"Appendix B.5: WD 0051+117 = PHL 886 (variable, P ≃ 4 . 7 d)","content":"From unpublished spectropolarimetric data (Landstreet & Bagnulo, in prep., hereafter LB25) it is found that the star is periodically variable ( P = 4 . 703 d) with ⟨| B |⟩ ≃ 270 kG. The field ⟨ Bz ⟩ reverses sign as the star rotates. Modelling will be presented in a forthcoming paper.","pages":[16]},{"title":"Appendix B.6: WD 0058-044 = GD 9 (variable, P ≃ 2 d?)","content":"From our unpublished spectropolarimetric data we found that the star is periodically variable (the period is not determined uniquely from available data, but the most probable period is P = 1 . 96 d), with ⟨| B |⟩ ≃ 300 kG (LB25). The field ⟨ Bz ⟩ varies sinusoidally but does not (quite) reverse sign. Modelling will be presented in a forthcoming paper.","pages":[16]},{"title":"Appendix B.7: WD 0232+525 = EGGR 314 (variable)","content":"This is a well-known star that was repeatedly included in studies of bright white dwarfs and been searched for a magnetic field (Bychkov et al. 1991; Schmidt & Smith 1995; Fabrika et al. 1997) without any detection being reported. Recent particularly sensitive observations (Bagnulo & Landstreet 2022) succeeded in detecting a weak field. The three published measurements (two with ISIS, one with ESPaDOnS) reveal a very weak, and probably variable longitudinal field (the measurement obtained with ESPaDOnS has the opposite sign as those obtained with ISIS).","pages":[16,17]},{"title":"Appendix B.8: WD 0233-242A = LP 830-14 = NLTT 8435A (variable, P phot ≃ 1 . 5 h)","content":"The star is discussed by Bagnulo & Landstreet (2021), who suggested the presence of a variable field of the order of 3.8 MG. In fact, ⟨| B |⟩ does not vary between two FORS observations, while ⟨ Bz ⟩ clearly changes sign. According to Vennes et al. (2018) it has a photometric variation period of 95 min.","pages":[17]},{"title":"Appendix B.9: WD 0236-269 = PHL 4227 (n.v.: - only 2 meas.)","content":"Schmidt et al. (2001) report two spectropolarimetric detections of circular polarisation at about the 1.2% level. The two observations were obtained three days apart, with no changes detected. Furthermore no significant change in the spectrum-average circular polarisation was observed. We tentatively assume that is not a magnetic variable, but the evidence for non-variability is not compelling (the original paper just reports: \"consistent results were obtained on the two occasions\").","pages":[17]},{"title":"Appendix B.10: WD 0253+508 = KPD 0253+5052 (variable, P ≃ 4 h)","content":"The field was discovered by Downes & Margon (1983), who estimated ⟨| B |⟩ ≈ 13 MG, and modelled by Achilleos & Wickramasinghe (1989) using only flux spectra. The star shows continuum polarisation up to 0.4% that is periodically variable according to Schmidt & Norsworthy (1991) with P = 0 . 170 d. The sign of the broadband continuum circular polarisation briefly reverses during each cycle.","pages":[18]},{"title":"Appendix B.11: WD 0301+059 = SDSS J030350.63+060748.9 (n.v.: - only 2 meas.)","content":"The star was discovered to be magnetic by Landstreet & Bagnulo (2020), who did not find any obvious sign of variability between two observations obtained 1 day apart, nor between individual exposures of the same observing series (exposure times of single frames were 225 s). We tentatively assume that it is not magnetically variable. The star therefore appears as a strongly magnetic, massive, non-variable star.","pages":[18]},{"title":"Appendix B.12: WD 0313-084 = GALEX J031613.8-081637 (non-variable)","content":"The star was discovered to be magnetic by O'Brien et al. (2023). It was observed five times with FORS2 (see Sect. 2.1), and no variability was found within timescales of 1 h, 1 d, and 8 months (see Fig. B.3). The spectrum indicates that ⟨| B |⟩ ≃ 800 kG.","pages":[18]},{"title":"Appendix B.13: WD 0316-849 = EUVE J0317-855 (variable, P ≃ 0 . 2 h)","content":"Studied by Ferrario et al. (1997) Burleigh et al. (1999), and Vennes et al. (2003), the star has a rotation period P = 725 s. A detailed model based on UV spectra was obtained by Burleigh et al. (1999), who found that the field varies in strength from about 180 to 800 MG over the surface. Vennes et al. (2003) shown that the star has a light curve with a minimum that corresponds to the maximum of the polarisation. This is one of the very few white dwarfs that show magnetic and photometric variability, for which we have a firmly established phase relationship between photometry and circular polarisation. This star is erroneously missing in the list of such stars compiled by Bagnulo et al. (2024b).","pages":[18]},{"title":"Appendix B.14: WD 0322-019 = EGGR 566 (variable)","content":"The star was discussed by Bagnulo & Landstreet (2021). Two FORS2 measurements by Farihi et al. (2018) obtained during two consecutive nights ( ⟨ Bz ⟩ = -5 . 4 ± 3 . 0 kG and -16 . 5 ± 2 . 3 kG) suggest that the longitudinal field may change on a timescale of a few days.","pages":[18]},{"title":"Appendix B.15: WD 0330-000 = HE 0330-0002 (only 2 meas., and contradicting data)","content":"Schmidt et al. (2001) report two spectropolarimetric observations obtained one day apart, and concluded that no significant di ff erences had been detected in the overall spectrum. However, the mean circular polarisation averaged over the observed wavelength range reported in their Table 1 di ff ers by 0.5%, contra- dicting the assessment of non-variability. Because of this, this star will not be considered in this paper.","pages":[18]},{"title":"Appendix B.16: WD 0410-114 = G 160-51 = NLTT 12758 (variable, P ≃ 0 . 3 h)","content":"A magnetic field of about 1.7 MG was discovered by Kawka & Vennes (2012). Kawka et al. (2017) found that circular polarisation in the H α sigma components varies with P = 22 . 6 min. This is another example of a star for which the phase relationship between light curve and magnetic field are known (Kawka et al. 2017) and it is also missing from the list compiled by Bagnulo et al. (2024b).","pages":[18]},{"title":"Appendix B.17: WD 0446-789 = WG 47 (var.:)","content":"Bagnulo & Landstreet (2018) show that this is a weak-field star with ⟨ Bz ⟩ ≃ -5 kG; it probably has a dipolar field with axis nearly parallel to the rotation axis, because only very mild variations in ⟨ Bz ⟩ occur on a timescale of days.","pages":[18]},{"title":"Appendix B.18: WD 0548-001 = EGGR 248 = G 99-37 (non-variable)","content":"The magnetic field was discovered from continuum circular polarisation by Landstreet & Angel (1971). Observations of broadband CP on six nights showed no variations. During one hour, broad-band data were printed every 6 s. These data were Fourier analysed without finding any evidence of short-period variability. Angel & Landstreet (1974) obtained an MCSP circular polarisation spectrum in 1972, and the star was re-observed by Vornanen et al. (2010), who remarked that similarity with the previous spectra, reporting also \"no unambiguous variations\" between several polarisation spectra taken in 2003, 2005, 2008. Using FORS1 data obtained in 2005, they also reported no detection of linear polarisation. We retrieved from the ESO FORS1 and FORS2 archives all circular polarisation spectra obtained in 2005, 2008 and 2012. Figure B.4 shows all these spectra compared to the polarisation spectrum of Angel & Landstreet (1974) . We note that there are three spectra obtained in three consecutive nights in November 2012. The spectrum obtained during the last night (2012-1119) shows zero circular polarisation (see green solid line), but a Stokes I flux consistent with that of previous observations. We ascribe this to a non-detected instrument failure (perhaps the retarder waveplate did not move) rather than to stellar variability. Observations obtained by Berdyugin et al. (2023) and Berdyugin et al. (2024) show no variability in broadband polarimetry between two measurements obtained 8 months apart (see the top panel of Fig. B.1). This is a nearly unique DQp white dwarf which shows not only the Swan bands in the flux spectrum, but also the CH G band. The magnetic field was modelled by Angel & Landstreet (1974) who deduced ⟨ Bz ⟩ ≈ 3 . 6 MG from the strong polarisation signature of the CH G-band. Berdyugina et al. (2007) and Vornanen et al. (2010) modelled FORS1 spectra and found ⟨ Bz ⟩ ≃ 2 . 5 MG.","pages":[19]},{"title":"Appendix B.19: WD 0553+053 = EGGR 290 = G 99-47 (non-variable)","content":"The magnetic field of this star was discovered by Angel & Landstreet (1972), who also made 14 separate BBCP measurements during 10 months, without detecting any significant variation. Low resolution MCSP spectropolarimetry was obtained by Liebert et al. (1975), and 20 years later, a higher resolution polarised spectrum was obtained also by Putney & Jordan (1995). No significant variation has been detected on any timescale up to decades (for details, see Bagnulo & Landstreet 2021). Brinkworth et al. (2013) observed a light curve with 26.8 m period and semi-amplitude = 0.3 %, which suggest that polarimetric observations could have missed short-term variability. However, Angel & Landstreet (1972) had carried out Fourier-analysis of a polarimetric run, and found no significant variability for any period between 11 seconds and 2 h. They claimed that periodic variation with an amplitude of 0.17 % would certainly have been detected. Our conclusion is therefore that the star is non-variable.","pages":[19]},{"title":"Appendix B.20: WD 0637+478 = GD 77 (variable, P ≃ 1 . 4 d)","content":"Magnetic variability was reported by Schmidt & Smith (1995). Eight unpublished ISIS polarisation spectra and one ESPaDOnS spectrum show that ⟨ Bz ⟩ varies with P ≃ 1 . 362 d. The value of ⟨| B |⟩ ≈ 1 . 0 MG hardly changes with rotation, but ⟨ Bz ⟩ varies sinusoidally between about + 400 and -250 kG (LB25).","pages":[19]},{"title":"Appendix B.21: WD 0654+059 = 2MASS J06572938+0550479 (non-variable)","content":"Three BBCP observations obtained 8 months apart by Berdyugin et al. (2024) show no variability of the polarisation in the three B ' V ' R ' filters (see Fig. B.1). In all three filters, all three observations have V / I ≃ -0 . 3 %. We assume that the star is not variable.","pages":[19]},{"title":"Appendix B.22: WD 0708-670 = SCR J0708-6706 (n.v.: only 2 meas.)","content":"Two spectra published by Bagnulo & Landstreet (2020) show no variability over a 2 month interval. The strength and wavelength dependence of the circular polarisation spectrum suggests that the underlying field could be of the order of 60 to 200 MG. We tentatively assume that the star is magnetically non-variable.","pages":[19]},{"title":"Appendix B.23: WD0745 + 115 = GALEX J074842.4+112502 (non-variable)","content":"Strong broadband circular polarisation was detected by Berdyugin et al. (2024). The polarisation changes sign between the B ' and the other bands, and its absolute value is about 1% in all bands. Observed four times, the star does not show variability (see the top right panel of Fig. B.1).","pages":[19]},{"title":"Appendix B.24: WD 0756+437 = EGGR 428 (variable, P phot ≃ 6 . 5 h )","content":"The star was discovered to be magnetic and discussed in detail by Putney (1995). She estimated its magnetic field to be ≃ 200 MG. Recent BBCP observations at NOT (Berdyugin et al. 2024) showed that it is rapidly variable. The first two observations, about 3 h apart, report the largest and smallest polarisation seen in the full data set of five observations; this suggests a rotation period of the order of 6 h. In fact, the photometric study of Brinkworth et al. (2013) found a unique, very large amplitude ( ± 4 %) light variation with a period of P = 6 . 68 h. The similarity of this period with the period range deduced from polarisation measurements strongly confirms that 6.68 h is the rotation period of this white dwarf. A further remarkable fact is the combination of high field (200 MG), high mass (1 . 04 M ⊙ ) with advanced age (4.45 Gyr): this star seems to be a unique example of a very old, still strongly magnetic and rapidly rotating WD-WD merger.","pages":[20]},{"title":"Appendix B.25: WD 0810-353 = UPM J0812-3529 (non-variable)","content":"No variability detected among six polarised spectra obtained over a four year period, as described in a detailed study by Landstreet et al. (2023). According to their modelling, the star shows two regions of di ff erent field strength: one with magnetic field of predominantly 30 MG strength, outward, and one showing a field strength of 45 MG, inward.","pages":[20]},{"title":"Appendix B.26: WD 0816-310 = SCR J0818-3110 (variable, P ≃ 10 d)","content":"This is a DZ white dwarf with strong flux, spectrum and ⟨ Bz ⟩ variations observed in five FORS polarised spectra in 2023 obtained with grism 1200B. It is clear that the surface abundances vary over the surface, and appear to be locked to the surface magnetic field. The rotational period is of the order of 10 d, and ⟨| B |⟩ ≃ 100 kG (Bagnulo et al. 2024a).","pages":[20]},{"title":"Appendix B.27: WD 0850+192 = LB 8915 (variable, P ∼ hours?)","content":"This DBA white dwarf shows very weak, variable H lines, slightly variable He, and variable ⟨ Bz ⟩ (Wesemael et al. 2001). It is found that ⟨| B |⟩ ≃ 850 kG, and the rotation period is relatively short, probably some hours. We note that the correct identification of this star is LB 8915, and not LB 8827 as given in title of the paper by Wesemael et al. (2001). Appendix B.28: WD0907+213 = GALEX J091016.5+210555 (variable, P ≃ 10 h) Moss et al. (2024) discovered that this is a spectroscopically and magnetically variable DBA star with a rotation period of either 7.7 or 11.3 h (the ambiguity is due to aliasing). They have modelled the field and abundance geometry with a simple model like that used for Feige 7 = WD0041-102. The line splitting in the flux spectra suggests a field of ⟨| B |⟩ ≃ 0 . 5 MG.","pages":[20]},{"title":"Appendix B.29: WD 0912+536 = EGGR 250 = G 195-19 (variable, P ≃ 1 . 3 d)","content":"This is the second magnetic white dwarf discovered (Angel & Landstreet 1971a), and the first magnetic white dwarf to be discovered to be rotationally variable (Angel & Landstreet 1971b). An improved ephemeris was provided by Angel et al. (1972a). The star shows a large variation of its circular polarisation with a period of 1.33 d (Angel et al. 1972b). Hernandez et al. (2024) measured the period of the photometric variability from TESS data as 1 . 3304 ± 0 . 0054 d (note that high accuracy of period relies on two widely separated TESS observational data sets). Six MCSP V / I spectra roughly uniformly distributed in phase were obtained by Landstreet & Angel, (unpublished). Between 4000 and 5000 Å, V / I reverses sign during rotation; redwards of this, the strong variations retain one sign. Appendix B.30: WD 1008-242 = UCAC4 328-061594 (n.v.: only 2 meas.) Observed twice with FORS2 in spectropolarimetric mode by Bagnulo & Landstreet (2022), this white dwarf did not show any variation between two spectra obtained 40 d apart, nor within the same observing series (see Fig. B.5). Probably the field is of order 100 MG or more. We consider that it is probably not variable. The star therefore appears a young, strongly magnetic, ultra-massive, non-variable star.","pages":[20]},{"title":"Appendix B.31: WD 1008+290 = LP 315-42 = LHS 2229 (non-variable)","content":"This white dwarf is a cool peculiar DQ star discovered to be magnetic by Schmidt et al. (1999), who tentatively suggested that the field strength is of the order of 100 MG or more. Four observations obtained by Berdyugin et al. (2024) show little to no variability of broadband circular polarisation (see Fig. B.1). We assume that it is not magnetically variable.","pages":[21]},{"title":"Appendix B.32: WD 1009-184 = WT 1759 (var.:)","content":"The star was discovered to be a magnetic white dwarf by Bagnulo & Landstreet (2019a), who published one measurement obtained with FORS2 and one obtained with ISIS, showing that the star has a weak and variable field ( ⟨ Bz ⟩ ≃ 50 kG). Additional unpublished data suggest also weak variability.","pages":[21]},{"title":"Appendix B.33: WD 1015+014 = PG 1015+014 (variable, P = 98 . 75 m)","content":"This is a very strongly polarised white dwarf with V / I ≃ 1 . 5 %. The flux and polarisation spectra are strongly variable with P = 98 . 75 m and a polar field strength of about 120 MG (Angel 1978; Wickramasinghe & Cropper 1988). Schmidt & Norsworthy (1991) report that BBCP also varies with P = 98.75 min, approximately sinusoidally, with extrema of + 1 and -1 % polarisation. Brinkworth et al. (2013) found that the star's light curve has a period consistent, within uncertainties, with that obtained from polarimetry. Euchner et al. (2006) obtained a series of polarised spectra with FORS using the 300V grism. They describe strong I and V spectrum variations, and model the field structure using a multipole field expansion. This white dwarf displays spectral features originating from regions with typical field strengths between about 50 and 90 MG.","pages":[21]},{"title":"Appendix B.34: WD 1031+234 = Ton 527 = PG 1031+234 (variable, P ≃ 3 . 5 h)","content":"Schmidt et al. (1986) report strongly variable intensity, circular and linear polarisation spectra with a 3 h 24 min period, and proposed a simple magnetic model with field strength in the range of 200 - 500 MG. Further broadband polarisation observations through the rotation period were obtained by Piirola & Reiz (1992), who measured also a light variation in anti-phase with circular polarisation (that is, the star appears darker when the absolute value of the polarisation is maximum). Brinkworth et al. (2013) found P phot ≃ 3 . 5 h.","pages":[21]},{"title":"Appendix B.35: WD 1036-204 = LP 790-29 (non-variable)","content":"The star was discovered to be magnetic via spectropolarimetry by Liebert et al. (1978), and repeatedly studied (West 1989b; Schmidt et al. 1995, 1999; Beuermann & Reinsch 2002; Jordan & Friedrich 2002). Beuermann & Reinsch (2002) have monitored the star with EFOSC in spectropolarimetric mode to search for short-term variability, without finding any significant variations. Beuermann & Reinsch (2002), however, pointed out that Schmidt et al. (1995) measured a polarisation signal of ≃ -6 % around 6500 Å, a measurement at odds with broadband polarimetric measurements obtained in 1977, 1986, 1994 and 2000, which were all about -9 % (see their Table 1; we recall here that we use the opposite definition for the sign of circular polarisa- n). Schmidt et al. (1995) therefore suggested the possibility of a very long rotation period of the star. Jordan & Friedrich (2002) carried out a similar study, with similar results regarding short term variation. They also proposed a rotation period in the range of about 24 - 29 yr. We have reduced FORS1 and FORS2 archival polarisation spectra from 2003, 2011, 2012, and 2013, finding absolutely no hint of variability among them. Figure B.6 shows a comparison of archive observations obtained with grism 600B. The flux is not corrected for atmospheric and instrument transmission, and the discrepancies in the slope of the flux measured in 2003 can be explained by the use of a different CCD. A comparison between FORS spectra with those obtained by Schmidt et al. (1995) on May 7 and 8, 1994, (wavelength range 4160-7460 Å) does not show obvious long-term variability, except in the range 5700 -6200 Å. In that range, our data are instead consistent with most of the literature and point to a value of ≃ -9 %. In addition to the polarisation spectra, FORS1 and FORS2 archive contains another two broadband circular polarisation (BBCP) observations in the R filter: a FORS1 BBCP measurement obtained in April 2006, and one obtained with FORS2 in March 2024, using a similar (but not per-fec identical) R filter (program ID 112.25C9.001). We found both measurements consistent with a polarisation signal of about -9 . 4 %. These measurements rule out any significant change in the region around 6500 Å over an interval of 18 years. Finally, Berdyugin et al. (2024) have published the series of BBCP observations obtained in November 2022, November 2023 and February 2024, all consistent among themselves. In the R ' filter they report a polarisation signal of ≃ -9 . 1 %. For the reasons explained at the beginning of this section, it is not possible to accurately compare BBCP measurements obtained with di ff erent instruments, but it is clear that the only deviant point of a series of polarimetric observations obtained in the course of almost half a century is a small portion of a spectrum obtained in 1994. Remarkably, data obtained with the same instrument (FORS) over nearly two decades are fully consistent among themselves, and point strongly to a constant circular polarisation spectrum. Our conclusion is that the star is actually not variable. From the measured signal of circular polarisation of ≃ 10 % we estimate a longitudinal field of the order of 100 MG.","pages":[21,22]},{"title":"Appendix B.36: WD 1043-050 = HE 1043-0502 (n.v.: - only 2 meas.)","content":"This DBA star was discovered magnetic by Schmidt et al. (2001), who proposed that the field is ≃ 800 MG (although its continuum is not highly polarised). Two observations were taken a few days apart, and Schmidt et al. (2001) state that they did not show variability; the reported wavelength integrated circular polarisation of about 1.5% is also essentially unchanged between the two spectra. So we consider it as candidate nonvariable magnetic white dwarf.","pages":[22]},{"title":"Appendix B.37: WD 1045-091 = HE 1045-0908 (variable, P ≃ 3 h)","content":"This white dwarf was discovered to be magnetic by Reimers et al. (1996) from a flux spectrum. Circular polarisation was confirmed, and shown to be variable by Schmidt et al. (2001). Euchner et al. (2005) obtained a series of I and V spectra and, assuming a rotational period of about 2.7 h, derived a detailed surface field model with a dominant field strength of 16 MG, but local field strength ranging between about 10 and 75 MG.","pages":[22]},{"title":"Appendix B.38: WD 1105-340 = SCR J1107-3420A (non-variable)","content":"Eleven ESPaDOnS spectra taken between 2018 and 2022, with ten of them during 1 week in 2019 (see Table A.1), show that the star has a weak, non-variable field with ⟨ Bz ⟩ ≈ -22 ± 4 . 5 ˙ kG and ⟨| B |⟩ ≈ 125 ± 5 kG (see Fig. B.7). We also have two FORS spectra, one taken with 1200B and one with 1200R, with lower resolving power but higher signal-to-noise ratio (S / N). Owing to their di ff erence resolution, these spectra were not used in this work. This star is perhaps the best studied non-variable weakfield magnetic white dwarf.","pages":[22]},{"title":"Appendix B.39: WD 1105-048 = EGGR 76 (ultra-weak field, var.:)","content":"The star was repeatedly observed, and a longitudinal field of the order of 1 kG was detected only in two measurements (Bagnulo & Landstreet 2018). The star seems to have a very weak and variable field, but additional measurements should be obtained to confirm the existence of a magnetic field.","pages":[22]},{"title":"Appendix B.40: WD 1116-470 = SCR J1118-4721 (non-variable)","content":"This white dwarf was observed twice with FORS2 by Bagnulo & Landstreet (2021), who flagged it as suspected magnetic star. Both observations show a similar signal of circular polarisation at -0 . 2 %, close to the FORS2 instrumental detection limit. A third observations was obtained in January 2023, and the V / I spectrum is again consistent with that measured previously. The star is definitely magnetic, and most likely not variable. This confirmation (see Table A.1) brings the number of magnetic white dwarfs in the 20 pc volume to 34.","pages":[22]},{"title":"Appendix B.41: WD 1211-171 = HE 1211-1707 (variable, P ≃ 2 h)","content":"A magnetic field was suspected in this white dwarf by Reimers et al. (1996), which was confirmed with polarimetry by Schmidt et al. (2001). Both papers show varying flux spectra. Schmidt et al. (2001) estimates P ≃ 100 -120 min and ⟨| B |⟩ ≃ 50 MG. Brinkworth et al. (2013) measured a photometric period of 1.79 h, consistent with the previous estimates from polarimetric data. The star is polarised at a level that varies between 0 and 3% during the rotation cycle. Modelling by Schmidt et al. (2001) strongly suggests a He dominated atmosphere with T e ff ≃ 12000 K, but Reimers et al. (1996), using IUE data, estimates 23000 K. Gentile Fusillo et al. (2021) gives 30000 K and 1 . 20 M ⊙ . We tentatively assume T e ff = 23000 K , M = 1 . 2 M ⊙ , and an He-rich atmosphere.","pages":[22]},{"title":"Appendix B.42: WD 1217+475 = SDSS J121929.45+471522.8 (DAHe variable, P phot = 15 . 26 h)","content":"DAHe with 18.5 MG field strength and a photometric period of about 15.25 h (Gänsicke et al. 2020). Spectroscopy reveals Zeeman components of the Balmer lines varying in strength but with constant splitting. Published data do not demonstrate that the star is magnetically variable but cannot rule out this possibility either. Therefore we have decided not to include this star in our sample.","pages":[23]},{"title":"Appendix B.43: WD 1249-022 = GALEX J125230.9-023417 (DAHe variable, P phot = 0 . 09 h)","content":"This is a DAHe white dwarf that shows variable H β and H α lines that appear sometimes in emission and sometimes in absorption. Field strength has been estimated 5 MG and rotational period = 0.09 h (Reding et al. 2020).","pages":[23]},{"title":"Appendix B.44: WD 1312+098 = PG 1312+099 - (variable, P ≃ 5 . 4 h)","content":"Variable continuum circular polarisation was detected in this hot DAH by Schmidt & Norsworthy (1991), who present over 100 measures, and find P = 5 . 43 h. Circular polarisation varies approximately between + 1% and -1%.","pages":[23]},{"title":"Appendix B.45: WD 1315-781 = LAWD 45 (n.v.: - only 2 meas.)","content":"No change between two ⟨| B |⟩ ≃ 5 . 5 MG and ⟨ Bz ⟩ ≃ 0 MG measurements from FORS 300V spectra taken five nights apart by Bagnulo & Landstreet (2020). We tentatively assume that it is magnetically non-variable.","pages":[23]},{"title":"Appendix B.46: WD 1315+222 = LP 378-956 (non-variable)","content":"Two observations by Berdyugin et al. (2023) and one by Berdyugin et al. (2024) show no obvious sign of variability (see Fig B.1).","pages":[23]},{"title":"Appendix B.47: WD 1328+307 = G 165-7 (variable)","content":"This star is found to host a magnetic field of ⟨| B |⟩ ≃ 650 kG based on line splitting observed in a good S / NSDSSspectrum (Dufour et al. 2006). These authors have also obtained low-resolution polarised spectra. They state that three 600 sec polarisation spectra were taken on 2005-12-30 at Steward Obs, all yielding essentially the same ⟨ Bz ⟩ ≈ 150 kG, but that the polarisation amplitude in similar Steward polarised spectra from 2006-05-03 (apparently not measured) is at least two times weaker than in 2005 spectrum. We conclude that the star is magnetically variable.","pages":[23]},{"title":"Appendix B.48: WD 1346+121 = LP 498-66 (non-variable)","content":"Observed three times by Berdyugin et al. (2023) and Berdyugin et al. (2024), we assume the star, which shows circular polarisation ranging between -1 and 0 % in the three DIPol-UF bands, is magnetically non-variable (see Fig. B.1). The field strength of this white dwarf is probably in the tens of MG.","pages":[23]},{"title":"Appendix B.49: WD 1350-090 = PG 1350-090 = GJ 3814 (non-variable)","content":"Discovered by Schmidt & Smith (1994), who measured ⟨ Bz ⟩ = 85 ± 9 kG. We have obtained four polarised spectra with ESPaDOnS that show ⟨| B |⟩ ≃ 450 -465 kG. There is no strong evidence of field variability, see also Schmidt & Smith (1994). Data will be published in a forthcoming paper (LB25).","pages":[23]},{"title":"Appendix B.50: WD 1532+129 = G 137-24 (variable)","content":"Originally classified as DZ white dwarf by Kawka et al. (2004), the star was discovered to be a magnetic white dwarf by Bagnulo &Landstreet (2019a), who published two FORS2 measurements and one ISIS measurement. The star is variable, with ⟨ Bz ⟩ values from FORS2 measurements of -21 ± 1 kG and -4 ± 1 kG, while ⟨| B |⟩ is not strong enough to split spectral lines, leading to ⟨| B |⟩ < ∼ 300 kG.","pages":[23]},{"title":"Appendix B.51: WD 1556+044 = PM J15589+0417 (non-variable)","content":"Discovered to be magnetic by Berdyugin et al. (2022), the star was re-observed three more times by Berdyugin et al. (2024). The observed circular polarisation, which is detected at the 10 σ level, ranges between -0 . 3 and + 0 . 4% in the three filter bands of DIPol-UF, so the order of magnitude of the field strength is probably some tens of MG. The polarisation does not show any variability (see Fig. B.1).","pages":[23]},{"title":"Appendix B.52: WD 1615+542 = GALEX J161634.4+541011 (DAHe magnetically variable, P phot = 1 . 59 h)","content":"DAHe with a variable magnetic field (from 3.5 to 6.5 MG) and a rotation period P = 95 . 29 m estimate via photometry (Manser et al. 2023).","pages":[24]},{"title":"Appendix B.53: WD 1619+046 = GALEX J162157.7+043219 (variable, P ≃ 40 m?)","content":"This white dwarf was discovered to be magnetic and rapidly variable in this work (see Sect. 2.1 and Fig. B.9). From the H β regions, it is clear that the star is a DAH with a rather non-uniform field of ⟨| B |⟩ ≃ 15 MG. We observe clear changes in intensity between spectra obtained a few minutes apart, and also polarisation (which is measured by the combination of two spectra) shows some variation, but without sign reversal. The close similarity between the first pair and the last pair of intensity spectra suggest that the star's rotation period is approximately 40 min.","pages":[24]},{"title":"Appendix B.54: WD 1639+537 = GD356 (non-variable, but with a light curve and P phot = 1.93 h)","content":"This star is the prototype of DAHe white dwarfs. It shows a light curve with low amplitude and P phot ≃ 1 . 93 h (Walters et al. 2021). A time series of circular polarisation spectra obtained during an entire rotational cycle shows now variability while subtle sinusoidal variability is seen in the position of the σ components of H β and H α lines (Walters et al. 2021). Walters et al. (2021) compared spectropolarimetry obtained in 2019, with that published by Ferrario et al. (1997), finding no changes. Its field strength is of the order of 10 MG (Walters et al. 2021). We classify the star as non-variable, but showing very small changes of its apparent magnetic field with a ∼ 2 h rotation period.","pages":[24]},{"title":"Appendix B.55: WD 1658+440 = PG 1658+441 (n.v.: - only 2 meas., one very old)","content":"The star was discovered to be magnetic by Liebert et al. (1983), who measured ⟨ Bz ⟩ ≃ 0 . 7 MG, and ⟨| B |⟩ ≃ 2 . 3 MG from spectropolarimetry of H α , H β and H γ (see their Fig. 5). Their observations were obtained on July 20 1980, with a spectral resolution of 10 Å. The same authors measured a signal of broadband circular polarisation = -0 . 016 ± 0 . 033% and -0 . 044 ± 0 . 019 % in the range 3300-8600 Å and concluded that a 3 σ upper limit of 0.10% semi-amplitude could be set for any presumed sinusoidal variation with a period between 0.5 and 5 h. For possible periods between 4 minutes and 0.5 hours, a less stringent limit of 0.30% polarisation semi-amplitude may be deduced. A comparison with our ISIS spectra obtained on April 21, 2019 (Sect. 2.1) is shown in the top panels of Fig. B.10. We observe a wavelength shift possibly due to bad calibration; most remarkable is the different strength of H α , but perhaps this is instrumental, because the ISIS spectrum was obtained with a setting that puts H α at the edge of the CCD, and no good flat-fielding correction could be applied. Although our comparison between 1980 and 2019 data is not conclusive, we certainly do not see any convincing change of the spectral features that demonstrate long-term variability. A comparison between the four pairs of Stokes V / I profiles obtained with ISIS in 2019 show no di ff erences within the error bars (see the bottom panels of Fig. B.10); therefore we rule out variability on a timescale of 10-15 minutes. This star seems to be a young, strongly magnetic, massive, tentatively classified as non-variable star. Photometric studies lead to inconsistent results: Brinkworth et al. (2013) found that the star is photometrically variable with a period between 6 h and 4 d, while, using TESS photometry, Oliveira da Rosa et al. (2024) derived a period shorter than 1 h. Hernandez et al. (2024), instead, did not detect periodicity in TESS data.","pages":[25]},{"title":"Appendix B.56: WD 1703-266 = UCAC4 317-104829 (variable)","content":"This DA white dwarf was discovered to have a magnetic field by Bagnulo & Landstreet (2020), with ⟨| B |⟩ ≈ 8 MG. They also found that two FORS2 polarised spectra four days apart are significantly di ff erent and yield di ff erent values of ⟨ Bz ⟩ , so the star is variable on a timescale of some days or less.","pages":[25]},{"title":"Appendix B.57: WD 1712-590 = Gaia DR3 5915797694789556096 (variable)","content":"Discovered to be magnetic by O'Brien et al. (2023). We observed this white dwarf in polarimetric mode with grism 1200B three times, twice during the same night. Intensity spectra show very similar splitting of Balmer lines, but the flux distribution within the line cores changes quite significantly, rather similarly to WD2359-434 (Landstreet et al. 2017). The deduced field strength is ⟨| B |⟩ < ∼ 0 . 8 MG. The Stokes V spectra also show that the star is strongly variable within 1-2 h (see Fig. B.11), and that the longitudinal field reverses its polarity during rotation.","pages":[25]},{"title":"Appendix B.58: WD 1743-521 = L 270-31 = BPM25114 (variable, P = 2 . 84 d)","content":"Wickramasinghe & Bessell (1976) reported a magnetic field of about 35 MG in this southern DA star from close examination of the peculiar flux spectrum. Wegner (1977) found vari- ability of light and of the flux and polarisation spectrum with P ≈ 2 . 84 d, and modelled the spectrum variations, finding results consistent with a magnetic dipole field. Field modelling was carried out in more detail by Martin & Wickramasinghe (1978), who deduced a dipole field strength of 36 MG, corresponding to ⟨| B |⟩ ≃ 18 MG.","pages":[25]},{"title":"Appendix B.59: WD 1748+708 = EGGR 372 = G 240-72 (s.v.:)","content":"Intrinsic broadband circular and linear polarisation of this magnetic white dwarf were discovered by Angel et al. (1974), who reported no variations in repeated observations of broad-band circular polarisation over a month, and linear polarisation over two nights. The star was later observed in broadband circular and linear polarimetry by West (1989a), who generally found similar polarisation levels and position angles to earlier work, and in broadband linear polarimetry by Berdyugin & Piirola (1999). Berdyugin & Piirola (1999) found evidence of clear change of the position angle of linear polarisation (mainly rotation of the polarisation angle) on a timescale of 20 years. We have observed the star with ISIS both in circular (twice) and in linear polarisation, and compared these observations with MCSP lowresolution spectra of I , V and P obtained by Angel & Landstreet on September 6 and 7, 1974, that were never published until now. Our ISIS spectra are consistent among themselves. When compared with spectropolarimetry obtained in 1974 and the 1990s, our new ISIS linear polarisation data find small changes in the V and P polarisation amplitude in the blue and visual, whereas the overall position angle behaviour is almost identical to that observed observed in 1974, thus we cannot confirm the changes in the polarisation position angle seen in 1997 by Berdyugin & Piirola (1999). On the other side, Antonyuk et al. (2016) found that the star is photometrically variable with a period between 5 hours and two days. Brinkworth et al. (2013) detected no short period variability, but claim that photometry varied over a ten month interval. We conclude that the star show some signs of subtle long term variability that should be further investigated. Appendix B.60: WD1750 -311 = UCAC4 295-140552 = [MTR2015] OW J175358.85-310728.9 (n.v.: - but P phot ≃ 0 . 5 h) Amagnetic field of ⟨| B |⟩ ≈ 2 . 1 MG was identified by Macfarlane et al. (2017) in this hot DQ white dwarf, which has strong lines of neutral C as well as fairly strong Balmer lines. They observed clear light variability of the star with an amplitude of about ± 2 % and a period of 35 min. These authors discuss the origin of the light variations: they argue that the variability is not due to pulsation, as no subsidiary frequencies are observed, and conclude that the variation could be due to rotation. They do not seem to have considered testing this with the spectra that they have collected of the object by looking for spectral variations, although they do test the spectra for radial velocity variations, and find none. Our four FORS2 300V spectra, taken with 13 m spacing through the light variation cycle, show virtually no changes in either flux or polarisation except for H β , which has a strongly variable I profile depth but almost constant V / I ( λ ) (see Fig. B.13). Possibly this is due to a magnetic field distribution that is nearly axisymmetric about the rotation axis but a distribution of H that varies strongly around that axis. This might be related to the light variability that has been detected. We have decided to classify the star as candidate magnetically non-variable.","pages":[25,26]},{"title":"Appendix B.61: WD1754 -550 = GALEX J175845.9-550117 (variable)","content":"This star was discovered to be magnetic in this work (Sect. 2.1). There is a strong signal of circular polarisation that seems variable on a short timescale between 0 and 4%, suggesting a rotation period of the order of 15 min: see Fig B.14. The inferred field strength is presumably of the order of 100 MG or more. The intensity spectrum appears featureless, so the T e ff ≃ 35 000 K star could be defined a hot DC.","pages":[26]},{"title":"Appendix B.62: WD 1814+248 = G 183-35 (variable)","content":"Putney (1997) showed that this white dwarf, formerly classified as a DC, is in fact a cool DAH that shows weak H α and H β , and that both lines are split by a field of about 6.8 MG, roughly the same in two observations. Kilic et al. (2019) observed that the white dwarf shows spectral variations. A series of spectra taken over several hours show that the separation of the σ components of H α from the central π component changes rather abruptly between about 90 Å and about 120 Å, equivalent to fairly sudden jumps between ⟨| B |⟩ values of about 4.6 and 6 MG, repeating with a period of about 4 h. The authors suggest that this unusual form of variability may be due to a patchy distribution of H over an He-rich envelope. A plausible model that might explain the observations could be a dipole oblique to the rotation axis, decentred in the direction of one pole so that the polar strengths at the two magnetic poles are unequal, with H-rich patches around both poles, but little or no H in a belt around the magnetic equator.","pages":[26,27]},{"title":"Appendix B.63: WD 1829+547 = G 227-35 (non-variable)","content":"This white dwarf was discovered to be strongly magnetic by Angel et al. (1975) using both broadband polarimetry and lowresolution spectropolarimetry (on September 9, 1974). Limited tests of broadband variability were negative (Angel et al. 1975, 1981). It was observed again in spectropolarimetric mode by Cohen et al. (1993) and by Putney & Jordan (1995) who estimated a dipolar field strength of 170-180 MG. We observed the same star with ISIS twice in circular polarisation and once in linear polarisation. There are strong circular polarisation features at 6960, 7450 and 7930 Å. Figure B.15 shows a comparison between all these spectra. We do not see any obvious sign of variation.","pages":[27]},{"title":"Appendix B.64: WD 1900+705 = LAWD 73 = Grw + 70 · 8247 (s.v.:)","content":"The star has a magnetic field of the order of 180 MG (Jordan 2003), and shows hints of small variability. We refer to Bagnulo & Landstreet (2019b), who present a review and analysis of its polarimetric characteristics, and in particular, a comparison with observations obtained 50 years apart show some discrepancies in both linear and circular polarisation. More recent broadband circular polarisation obtained at NOT (Berdyugin et al. 2022, 2023) show no variability on a timescale of 2 years (see Fig. B.1).","pages":[27]},{"title":"Appendix B.65: WD 1953-011 = GJ 772 (variable, P ≃ 1 . 5 d)","content":"Maxted et al. (2000) modelled a series of Stokes I spectra in terms of a global dipolar field with polar field strength of order 100 kG, together with a spot with a field of order 500 kG. This basic modelling was confirmed by the analysis of a series of polarised spectra obtained with FORS1 and on the Russian 6 m telescope, which also revealed a rotation period of 1.448 d (Valyavin et al. 2008).","pages":[27]},{"title":"Appendix B.66: WD 2010+310 = GD 229 (s.v.:)","content":"This star was the fifth circularly polarised white dwarf to be discovered. Swedlund et al. (1974) reported a series of measurements indicating the presence of elevated levels of both circular polarisation and linear polarisation, and strongly suggesting polarisation variability. The claim of variability was then questioned by Kemp et al. (1974), who argued that the initial measurements had been badly contaminated by polarised foreground zodiacal light contamination. Linear and / or circular polarisation of GD 229 has been measured by Landstreet & Angel (1974), Efimov (1981), Angel et al. (1981), West (1989a), Berdyugin (1995), and Berdyugin & Piirola (1999). Angel & Landstreet (1974) obtained three low-resolution I and V spectra of the star on the nights of 1973 Nov 6-8 using the MCSP. Synthetic broad-band polarimetry derived from these low-resolution spectra revealed no variations over three nights. West (1989b) observed linear polarisation in GD 229 again in 1986 using broadband filters, but found no strong evidence of variation, not even of the position angle of linear polarisation. Observations of GD 229 using broadband polarimetry was also carried out for linear polarisation by Berdyugin (1995) and for both circular and linear polarisation, with higher precision, by Berdyugin & Piirola (1999), who compared their results to earlier work. Their results appear to show some quite significant changes in both circular and linear polarimetry compared to earlier observations by other groups. A major di ffi culty of comparing the various measurements is that each group has been made with di ff erent instrumentation. The consequences of using di ff erent filter passbands, di ff erent detector sensitivities as functions of wavelength, and even di ff erent (and possibly incorrect) calibration of instrumental polarimetric e ffi ciency, make it very hard to compare these data. However, Berdyugin & Piirola (1999) do o ff er strong evidence for significant rotation of the angle of linear polarisation, by about 30 · over 20 years, which is di ffi cult to explain by instrumental e ff ects. This is probably the most robust e ff ect that emerges from comparison of the many kinds of polarimetric observations Linear and circular polarisation spectra of GD 229 were obtained in 2018 and 2019 using the ISIS spectropolarimeter on the William Herschel Telescope (Sect. 2.1). These new data are shown in Fig. B.16, and compared to the circular polarised spectra by Angel & Landstreet (1974), and to previously unpublished linear spectropolarimetry of Angel and Landstreet (see Table A.1). Remarkably, the angle of linear polarisation appears to have returned to its value during the 1970s. Other di ff erences compared to the spectra of Angel and Landstreet are observed, but some of these are undoubtedly due to greatly di ff erent resolving power, and some may be due to uncertainties in the calibration of the 1970s data, particularly in the UV. It is thus di ffi cult to establish clearly how much variability has occurred in GD 229. There is evidence for variations, but these appear to have relatively small amplitude compared to the overall scale of polarisation, except possibly for position angle rotation of linear polarisation (which however has not been confirmed by our most recent linear spectropolarimetry). Furthermore, because of very limited sampling with a variety of instruments, it is not possible to establish clearly any particular timescale for variations. We note that the spectrum of GD 229 has been modelled as due to He in a field of hundreds of MG by Jordan et al. (1998, 2001) and Jordan (2003).","pages":[28]},{"title":"Appendix B.67: WD 2047+372 = EGGR 261 (variable, P ≃ 6 h)","content":"Originally, the star was observed polarimetrically by Schmidt & Smith (1995), who did not detect its weak and sign reversing field (their measurements were ⟨ Bz ⟩ = -42 ± 59 kG and -2 . 5 ± 4 . 6 kG). This star was discovered to be magnetic by Landstreet et al. (2016), and monitored and modelled by Landstreet et al. (2017). It is currently the weakest white dwarf field ( ⟨| B |⟩ ≈ 60 kG) that has been modelled in detail, mainly on the basis of a series of 18 ESPaDOnS spectra. The rotation period, determined from the variation of ⟨ Bz ⟩ , is 0.243 d. The observed variations of ⟨ Bz ⟩ and ⟨| B |⟩ are well modelled using a simple dipole model. No rotational variation is detected in TESS photometry (Hernandez et al. 2024).","pages":[28]},{"title":"Appendix B.68: WD 2049-222 = LP 872-48 (var.:)","content":"Discovered to be magnetic by Berdyugin et al. (2022) with BBCP measurements. The star has V / I ≃ + 0 . 1 %, and is one of the weakest polarisation levels securely detected. The inferred ⟨ Bz ⟩ field strength is only of the order of a few MG. Broad-band measurements were repeated in July 2022 (Berdyugin et al. 2024) to check for variability, which was not detected (see Fig. B.1). We also observed the white dwarf three times with FORS2 with grism 300V (Sect. 2.1). The data are barely above the threshold of instrumental polarisation ( ≃ 0 . 1%; see Siebenmorgen et al. 2014), and therefore it is hard to establish whether the hints of variability seen in the spectra are real or not (see Fig. B.17). However, star WD 1116-440 shows a similar level of polarisation, and constant over about 4 year (see Fig. B.8), suggesting that the tiny variability observed in WD 2049-222 may be real and not an instrumental artefact.","pages":[28]},{"title":"Appendix B.69: WD 2049-253 = UCAC4 325-215293 (non-variable)","content":"Discovered to be magnetic by Bagnulo & Landstreet (2020), who observed continuum circular polarisation of order 0.4% and deduced a field strength of order 20 MG. This white dwarf was re-observed in broadband circular polarisation once by Berdyugin et al. (2022) and two more times by Berdyugin et al. (2024), but shows no sign of variability (see Fig. B.1).","pages":[28]},{"title":"Appendix B.70: WD 2051-208 = BPS CS 22880-0134 (variable, P ≃ 1 . 5 h)","content":"The magnetic field of this white dwarf was discovered from the shape of the H α line by Koester et al. (2009). We have two series of five polarised spectra each, one using FORS grism 1200B and one with grism 1200R (LB25). These unpublished data clearly reveal very rapid rotation of this massive magnetic white dwarf. The stellar rotation period is 0.0594 d = 1.425 h. The value of ⟨ Bz ⟩ ranges approximately sinusoidally between about + 50 kG and -30 kG, while the corresponding values of ⟨| B |⟩ increase from about 200 kG to nearly 300 kG, in good agreement with the two values of <| B |> (220 and 290 kG) obtained by Koester et al. (2009) from UVES SPY spectra.","pages":[29]},{"title":"Appendix B.71: WD 2105-820 = LAWD 83 (variable)","content":"This star was suspected to have a weak magnetic field by Koester et al. (1998), who observed that the core of H α was abnormally broad, but they could not decide whether this was due to rapid rotation of v sin i ≈ 65 km / s or to a magnetic field of about 43 kG. Five FORS1 polarised spectra of the star by Landstreet et al. (2012) detected an apparently nearly constant magnetic field of ⟨ Bz ⟩ ≈ 10 kG. Later, FORS2 polarised spectra by Bagnulo & Landstreet (2018) and Farihi et al. (2018) reveal that ⟨ Bz ⟩ sometimes decreases to ⟨ Bz ⟩ ≈ 4 kG, so the star is apparently variable (see Bagnulo & Landstreet 2021).","pages":[29]},{"title":"Appendix B.72: WD 2138-332 = L 570-26 (variable, P ≃ 6 . 19 h)","content":"The DZA star was discovered to be magnetic, with a variable ⟨ Bz ⟩ of the order of 10 kG (Bagnulo & Landstreet 2019a) and a rotational period of P = 6 . 19 h (Hernandez et al. 2024; Farihi et al. 2024). The same period was found from the analysis of the equivalent width and ⟨ Bz ⟩ curves by Bagnulo et al. (2024b), who proposed a magnetic model with a dipolar field with ⟨| B |⟩ ≃ 50 kG. The star shows photometric and ⟨ Bz ⟩ curves with light minimum corresponding to ⟨ Bz ⟩ maximum.","pages":[29]},{"title":"Appendix B.73: WD 2150+591 = UCAC4 747-070768 (variable, P ≃ 2 . 4 d)","content":"The star was discovered to be magnetic by Landstreet & Bagnulo (2019), who reported two ISIS measurements that showed clearly that the field is variable with a period of hours or days. We subsequently monitored the star with one ESPaDOnS observation and several more ISIS spectra, and confirmed variability. These observations and a model of the star's magnetic field will be presented in a forthcoming paper (LB25).","pages":[29]},{"title":"Appendix B.74: WD 2211+372 = LP 287-35 (non-variable)","content":"This DC white dwarf was discovered to be magnetic by Berdyugin et al. (2023), who found circular polarisation in excess of 1% in the blue. It was re-observed by Berdyugin et al. (2024). It does not seem to be variable (see Fig. B.1). The field strength ⟨| B |⟩ is probably of the order of 60 MG or more.","pages":[29]},{"title":"Appendix B.75: WD 2316+123 = KUV 23162+1220 (variable, P ≃ 18 d)","content":"Discovered magnetic by Sion et al. (1984). Schmidt & Norsworthy (1991) reported BBCP of amplitude up to nearly 1% that varies sinusoidally with P = 17.86 d. Both the flux spectrum and the linear and circular polarisation spectra have been modelled repeatedly (Liebert et al. 1985; Achilleos & Wickramasinghe 1989; Friedrich et al. 1993; Putney & Jordan 1995); all agree that the global field strength is of the order of 30 MG. This star has the longest rotational period firmly established for a white dwarf.","pages":[29]},{"title":"Appendix B.76: WD2359 -434 = LAWD 96 (variable, P ≃ 2 . 7 h)","content":"WD2359 -434 was suggested to be a magnetic star by Koester et al. (1998) on the basis of the peculiar profile of the H α core, and ⟨ Bz ⟩ was found to be non-zero by Aznar Cuadrado et al. (2004). A series of polarised ESPaDoNS spectra revealed a rotation period of 0.1123 d (Landstreet et al. 2017). The star is also a photometric variable with the same period. The field structure has been modelled approximately, and found to be distinctly more complex than a simple dipole, with ⟨| B |⟩ varying approximately between 50 and 100 kG. This white dwarf o ff ers one of the clearest examples known of a field structure that is substantially more complex than a simple co-linear multipole expansion (Landstreet et al. 2017).","pages":[29]}],"string":"[\n {\n \"title\": \"ABSTRACT\",\n \"content\": \"Many magnetic white dwarfs exhibit a polarised spectrum that periodically varies as the star rotates because the magnetic field is not symmetric about the rotation axis. In this work, we report the discovery that while weakly magnetic white dwarfs of all ages with M ≤ 1 M ⊙ show polarimetric variability with a period between hours and several days, the large majority of magnetic white dwarfs in the same mass range with cooling ages older than 2 Gyr and field strengths ≥ 10 MG show little or no polarimetric variability. This could be interpreted as extremely slow rotation, but a lack of known white dwarfs with measured periods longer than two weeks means that we do not see white dwarfs slowing their rotation. We therefore suggest a di ff erent interpretation: old strongly magnetic white dwarfs do not vary because their fields are roughly symmetric about the rotation axes. Symmetry may either be a consequence of field evolution or a physical characteristic intrinsic to the way strong fields are generated in older stars. Specifically, a strong magnetic field could distort the shape of a star, forcing the principal axis of maximum inertia away from the spin axis. Eventually, as a result of energy dissipation, the magnetic axis will align with the angular momentum axis. Alternatively, symmetry could be the hallmark of a dynamo that operates after the beginning of core crystallisation. We also find that the higher-mass strongly magnetised white dwarfs, which are likely the products of the merging of two white dwarfs, may appear as either polarimetrically variable or constant. This may be the symptom of two di ff erent formation channels or the consequence of the fact that a dynamo operating during a merger may produce diverse magnetic configurations. Alternatively, the massive white dwarfs with constant polarisation may be rotating with periods much shorter than the typical exposure times of the observations. Key words. polarisation - stars: white dwarfs - stars: magnetic fields\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"Strong magnetic fields of old white dwarfs are symmetric about the stellar rotation axes\",\n \"content\": \"S. Bagnulo 1 and J.D. Landstreet 1 , 2 Received July 5, 2024, accepted October 22, 2024\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1. Introduction\",\n \"content\": \"Most white dwarfs do not show any variability, and more than 97% of them can be safely considered as suitable flux standard stars (Hermes et al. 2017). Because of flux stability, it is generally impossible to measure the rotation period of white dwarfs except when they pulsate or when they have a magnetic field. It is well known that magnetic white dwarfs often show periodically variable signals of polarisation and sometimes also periodic photometric variability. In the local 20 pc volume, more than 20% of white dwarfs have a magnetic field (Bagnulo & Landstreet 2021), and about 20% of the magnetic white dwarfs are photometrically variable (Farihi et al. 2024). Polarimetric variability is explained in terms of a magnetic field not symmetric about the rotation axis so that the magnetic configuration seen by the observer (as encoded in the polarisation signal) varies as the star rotates. The causes of photometric variability are not well understood (see, e.g., Bagnulo et al. 2024b), yet photometric variability of non-pulsating white dwarfs is so clearly correlated with the presence of the magnetic field variations that one may hypothesise that all non-pulsating white dwarfs with light variations are magnetic. If a star belongs to the small class of white dwarfs with a well-detected light curve, then the rotational period may usually be deduced from photometry (see Brinkworth et al. 2013; Hernandez et al. 2024; Oliveira da Rosa et al. 2024), but, in general, the time series of polarisation measurements may allow one to recover the rotational period of a larger sample of magnetic white dwarfs. Investigations of magnetic fields and of rotational periods are closely related not only because the presence of a magnetic field enables period measurements but also because one could expect that the occurrence of a magnetic field and its strength is physically related to stellar rotation. For example, should the field be supported by a dynamo, one could predict fields to be stronger in more rapidly rotating stars than in those that are more slowly rotating. Correlations between the presence of a magnetic field and the rotational period of a star exist even when the field has a fossil origin. Most of the chemically peculiar stars of the main sequence (Ap and Bp stars) have a fossil magnetic field, and they all rotate more slowly than the non-magnetic non-chemically peculiar stars in the same region of the Hertzsprung-Russel diagram (e.g. Donati & Landstreet 2009). Some of the magnetic Ap and Bp stars have well measured rotational periods as long as several months, years, or even decades (e.g. γ Equ, Leroy et al. 1994). There is no clear explanation for this well-established correlation between magnetic fields and slow stellar rotation. In the case of white dwarfs, various samples of data have gradually become available in the past few years to constrain possible scenarios for the evolution of their magnetic fields. It now appears that among the normal-mass white dwarfs ( M ≃ 0 . 6 M ⊙ ) that largely descend from single-star evolution, magnetic fields are very rare and weak during the first 1-2 Gyr of cooling but then gradually become much more common and often very strong after about 2-3 Gyr of cooling (Bagnulo & Landstreet 2021). This evolution may occur as a result of gradual relaxation to the surface of fields present in stellar cores during earlier evo- on and / or as a result of the operation of a dynamo during core crystallisation (Isern et al. 2017; Schreiber et al. 2021; Bagnulo & Landstreet 2021, 2022; Ginzburg et al. 2022). Among normal-mass stars, those with a magnetic field seem to rotate faster than non-magnetic white dwarfs (Hernandez et al. 2024), supporting the idea that crystallisation-driven dynamo may play a role in the formation of their magnetic fields. Many highly massive white dwarfs ( M ≥ 1 . 0 M ⊙ ), which are probably the product of mergers of two white dwarfs, appear magnetic very early in their cooling age, and the origin of the magnetic field may be due to a dynamo operating during the merging (Tout et al. 2008; García-Berro et al. 2012; Briggs et al. 2015; Bagnulo & Landstreet 2022). This idea is supported also by the evidence that some white dwarfs have very short rotation periods (e.g. Feige 7: 131 min, Liebert et al. 1977; Cl Oct: 12.1 min, Barstow et al. 1995; WD2209 + 113: 70 s, Kilic et al. 2021). At the same time, some polarimetric studies have hinted that a few old magnetic white dwarfs with a field strength of the order of tens of to a hundred megagauss have extremely long rotational periods, showing at most quite small variations even on timescales of decades (e.g. Berdyugin & Piirola 1999; Bagnulo &Landstreet 2019b). Traditionally, this lack of obvious variability is explained by the assumption that such non-varying magnetic white dwarfs have very long rotation periods of the order of centuries (Schmidt & Norsworthy 1991). In the absence of clear observational support or contradiction, this assumption has been widely accepted (e.g. Ferrario et al. 2015). The conclusion is basically that the known periods, P , fall into two very di ff erent families: one with P < ∼ 2 weeks, characterising most of the sample, and a few percent with strong fields that have periods of centuries (i.e. no clearly detected rotation). There is no explanation as to why some old white dwarfs would have extremely long rotation periods. Magnetic braking has generally been invoked but without the support of numerical calculations. In this work, we report new polarimetric observations and combine them with data collected from the literature. We analyse the magnetic variability and thus the rotation of a sample of 74 white dwarfs, and we discuss the constraints that these data place on possible evolution paths for magnetic fields in white dwarfs.\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"2.1. Observations not previously published\",\n \"content\": \"We present here 49 unpublished spectropolarimetric observations of 13 magnetic white dwarfs. Fourteen spectra of six white dwarfs were recently obtained with the FORS2 instrument (Appenzeller et al. 1998) of the ESO VLT in the course of a spectropolarimetric survey of the solar neighbourhood. Seven polarised spectra of two other white dwarfs were retrieved from the ESO archive: one of the spectra was obtained with FORS1 and the remaining ones with FORS2 (we note that FORS1 and FORS2 are virtually identical instruments). Fourteen new spectra of four white dwarfs were obtained with the ISIS instrument of the William Herschel Telescope (WHT). For all of these data, the observing strategy and data reduction are identical to the procedures described by Bagnulo & Landstreet (2018) for circular polarisation data and by Bagnulo & Landstreet (2019b) for linear polarisation data. For two of the four stars recently observed with ISIS, we also present here three previously unpublished spectra obtained in the 1970s that we used to study the long-term variability of magnetic white dwarfs. These data consist of lowresolution polarised spectra obtained by Angel and Landstreet using the multi-channel spectrophotometer (hereafter MCSP) on the 5-m Palomar telescope. The instrument set-up and data reduction of these observations are described in detail by Angel & Landstreet (1974). Finally, eleven spectra of WD 1105-340 were obtained with the ESPaDOnS instrument of the CanadaFrance-Hawaii Telescope (CFHT). These data were reduced by the automatic CFHT pipeline LibreEsprit. The log of these unpublished observations is given in Table A.1, and results are described in the sections dedicated to individual stars of Appendix B. Apart from their general purpose of assessing polarimetric variability, our new observations confirm that star WD 1116 -470 is magnetic (it was previously considered as suspected to be magnetic by Bagnulo & Landstreet 2021) and bring the number of magnetic white dwarfs in the local 20 pc volume to 34 (see Bagnulo & Landstreet 2021). We have also discovered two new massive magnetic white dwarfs: WD1619 + 054 and WD1754 -550 (both rapidly variable).\",\n \"pages\": [\n 2\n ]\n },\n {\n \"title\": \"2.2. Literature data\",\n \"content\": \"We searched the literature and collected data for all magnetic white dwarfs that were observed in polarimetric mode at least twice as well as for all magnetic white dwarfs for which multiple spectroscopic intensity observations revealed magnetic variability. The way literature data and new observations have been used is described in the next section.\",\n \"pages\": [\n 2\n ]\n },\n {\n \"title\": \"3. Determination of the variability of the magnetic white dwarfs\",\n \"content\": \"We are interested in establishing whether the magnetic field of a white dwarf appears constant or variable with time to the observer. Magnetic variability is ascribed to changes in the apparent field geometry as a star rotates. It can be detected mainly via circular spectro- or broadband polarimetry, which are both sensitive to the longitudinal component of the magnetic field. Linear polarisation, which is sensitive to the transverse component of the magnetic field, is usually much weaker than circular polarisation, and it is rarely detected in white dwarfs. Intensity spectra (if the star is not featureless) are sensitive to the mean field modulus and may also be used to detect magnetic variability, but they are less sensitive to changes of the apparent magnetic configuration than polarimetry. Magnetic white dwarfs that have been repeatedly observed in spectroscopic mode and that do not show sign of variability have not been considered here because having observed a constant intensity spectrum is not a su ffi ciently strong indication that a star is not magnetically variable. An example is WD2047 + 372, as it shows an almost constant Zeeman triplet in multi-epoch intensity spectra and a variable, sign-reversing circular polarisation spectrum (Landstreet et al. 2017). Photometric variability is observed in magnetic white dwarfs, but in the absence of convincing observational evidence or theoretical arguments, we have chosen not to consider it alone as a proxy for magnetism nor magnetic variability. For example, Brinkworth et al. (2013, see in particular their Tables 1 and 3) detected light variability for a number of magnetic stars. For many of them, however, the light amplitude is quite low, and there are no multi-epoch polarimetric data to confirm magnetic variability. Furthermore, recent analysis of TESS data made by Hernandez et al. (2024) and Oliveira da Rosa et al. (2024) failed to detect a periodic light curve for many of these targets. These stars are not included in our sample. In summary: 1) Two or more intensity spectra, or polarisation spectra, or broadband polarisation measurements that clearly di ff er from each other are interpreted as being due to magnetic variability. Wegathered reasonable evidence for polarimetric variability, or non-variability, for the sample of 74 magnetic white dwarfs listed in Table 2. Admittedly, for some stars, we found that establishing whether or not two or more observations are consistent among themselves was somewhat subjective. In Sect. 5.3, we argue that our results are not a ff ected by these uncertain cases. Appendix B discusses in detail all individual stars. In this section, we summarise the results and flag special cases. Among the 74 white dwarfs, 44 are magnetically variable (Sect. 3.1), while for 27 there is so far no evidence of variability (Sect. 3.2). The remaining three white dwarfs, which have strong fields and have been observed over several decades, may be further tested for subtle or possibly very slow variability (Sect. 3.3).\",\n \"pages\": [\n 2,\n 3\n ]\n },\n {\n \"title\": \"3.1. Stars that show evidence of magnetic variability\",\n \"content\": \"For 30 white dwarfs among the 44 stars that show polarimetric variability, a rotational period, or at least a good candidate for it, has been established in the literature. This period is given (in days) in the last column of Table 2 and is followed by ':' when only tentative or approximated values are known. Rotational periods are generally of the order of hours; about 25% are of the order of one day or longer, and only two stars have a rotation period longer than one week (the slowest magnetic white dwarf for which the period is known, WD 2316 + 123, has P ≃ 17 . 4 d, Schmidt & Norsworthy 1991). Most of these stars were listed in Table C.1 of Hernandez et al. (2024), and they are shown with blue symbols in Fig. 1 of Sect. 4. Among the other 14 variable stars, we do not have enough data to estimate the rotation period. For ten of them, the evidence of polarimetric variability has been securely established by two or more observations. These ten variable stars are marked in last column of Table 2 with 'var.' and are also represented with a blue symbol in Fig. 1. The remaining four white dwarfs show subtle signs of variability among the observations obtained with the same instrument. They are marked in Table 2 with 'var.:' and are shown with light blue symbols in Fig. 1. For WD 0446-789 and WD1009-184, our conclusion is that the field variations are real but small due to a configuration probably nearly symmetric about the rotation axis. WD 1105-048 was repeatedly observed with FORS1, FORS2, and ESPaDOnS, and a field was detected on only two occasions out of 12 observing epochs. If the star is magnetic, then it is certainly variable with a timescale of the order of days or weeks, but, admittedly, one could suspect that the two detections are in fact spurious. We decided to define the star as a probable variable star. WD 2049-222 shows a signal of polarisation that is marginally higher than instrumental polarisation, but for the reasons explained in Sect. B.68, we decided to consider its subtle variations as real.\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"3.2. Stars for which there is no evidence of magnetic variability\",\n \"content\": \"Nineteen stars have been observed three or more times in polarimetric mode, and the observations are always consistent among themselves within uncertainties. We consider these stars as established non-variable white dwarfs. They are marked with 'n.v.' in Table 2 and represented with red symbols in Fig. 1. Among these 19 stars, four were observed over a time interval of decades and with di ff erent instruments: WD 0548 -001 = G9937, WD0553 + 053 = G99-47, WD1036 -204 = LP790 -29, and WD1829 + 547 = G227-35. Obviously, it is possible that the other members of this group of 'non-variable stars', for which the observations span a time interval of up to a few months or years, are actually variable on a timescale of decades and that as such they could be stars with (so far undetected) long-term variability. Eight stars should be considered simply as candidate nonvariable white dwarfs ('n.v.:' in Table 2 and magenta symbols in Fig. 1). For seven of them, the reason for considering them only as candidate non-variable stars is that they have only been observed twice. In particular we mention that for normal-mass WD0236-269 and for the massive WD0330 -000, the nonvariability is inferred based on comments in the original discovery paper by Schmidt et al. (2001), which represents especially weak evidence. Another candidate non-variable star, WD1750 -311, was observed only once (with FORS2). The comparison of spectra obtained within that single observing series shows some change of circular polarisation and certainly a rapidly variable intensity spectrum (timescale of minutes) in the regions of the Balmer lines, particularly around H β . The star also shows photometric variability with a 35-min period (Macfarlane et al. 2017). Our interpretation is that WD 1750-311 is probably a magnetic white dwarf with a field symmetric, or nearly symmetric about the rotation axis, showing strong abundance patches of hydrogen.\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"3.3. Stars possibly subtly or slowly variable\",\n \"content\": \"There are three stars that have been monitored for decades, WD1748 + 708 = G240 + 708, WD1900 + 310 = Grw + 70 · 8247, and WD2010 + 310 = GD229, for which our conclusions are uncertain. These stars, marked with 's.v.:' in Table 2 and shown with black symbols in Fig. 1, may indeed show some low amplitude and possible long-term variability, but there might be room for a di ff erent interpretation. The recent detailed comparison of old and new polarisation spectra of Grw + 70 · 8247 by Bagnulo & Landstreet (2019b) identifies small changes between spectra taken decades apart (see in particular their Fig. 5), but for the two other stars, it is not entirely clear which changes are real and which are due to di ff erences between observational equipment and techniques. Bagnulo & Landstreet (2019b) assumed that timescales of the observed small changes were those corresponding to the large time gaps between isolated data sets. However, because the changes are generally small, one could argue that the subtle changes have timescales as short as weeks and have not have been noticed because the stars were not adequately monitored. For example, as previously discussed, WD 1105 -048 appears constant (and non-magnetic) in ten out of 12 observations, and a magnetic field was detected in only two epochs. In case of WD 1748 + 708, photometry leads to ambiguous results. The star was found photometrically variable with a period between 5 and 48 h by Antonyuk et al. (2016). Brinkworth et al. (2013) did not confirm its short-term photometric variability (nor did Hernandez et al. 2024) but instead claimed detection of light changes over a period of ten months.\",\n \"pages\": [\n 3,\n 4\n ]\n },\n {\n \"title\": \"4. Clustering in parameter space\",\n \"content\": \"In this section, we correlate the variability or non-variability of the stars with their mass, field strength, and stellar temperature. The left panels of Fig. 1 show the field strength versus e ff ective temperature and cooling age for the stars of our sample, with the cooling age on both a linear scale and a logarithmic scale. In these plots, stars are represented with symbols that increase in size with a star's mass, while di ff erent colours are used to mark the characteristics in terms of variability. The right panels of Fig. 1 show the cooling age-mass diagrams for the same sample, with cooling age reported again both with linear and logarithmic scales. The size of the symbols is proportional to the field strength. With the help of these figures and Table 2, we studied the relationships between field strength, mass, age, and variability of the magnetic white dwarfs. We recall that our data do not come from a volume-limited sample of stars and do not reflect the relative density of magnetic white dwarfs in di ff erent regions of the diagrams of Fig. 1. In fact, both young magnetic normalmass white dwarfs and ultra-massive white dwarfs are quite rare objects in space, and they are over-represented in our sample because of biases of the surveys (see Bagnulo & Landstreet 2022). The following analysis is about the relative frequency of variable and non-variable stars in di ff erent regions of the diagrams where we noticed the existence of statistically remarkable di ff erences.\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"4.1. Normal-mass and weakly magnetic white dwarfs\",\n \"content\": \"There are 25 white dwarfs with a field strength of ≤ 1 MG, and all of them but one (WD 2051-208) have M ≤ 1 . 0 M ⊙ . These 'normal-mass' 'weak-field' white dwarfs span an age range up to τ ≈ 6 Gyr. Twenty-one of these 24 stars are variable. Among the three that are non-variable, one is younger than 1 Gyr: WD 1105 -340 ( τ = 0 . 34 Gyr).\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"4.2. Normal-mass and strongly magnetic white dwarfs\",\n \"content\": \"There are 25 magnetic white dwarfs with M ≤ 1 . 0 M ⊙ and field strength of ≥ 10 MG. Among those that are younger than 2 Gyr, six out of eight white dwarfs are variable; however, we recall that young strongly magnetic white dwarfs are rare in space (Bagnulo & Landstreet 2022). Among the magnetic white dwarfs older than 2 Gyr, only one out of the 17 shows polarimetric variability.\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"4.3. Massive magnetic white dwarfs\",\n \"content\": \"Fifteen stars of our sample have M > 1 . 0 M ⊙ , and all but one are younger than about 1 Gyr. Eight of them are magnetically variable, including the only old ( τ ≃ 4 Gyr) star WD 0756 + 437. Nearly all of these massive magnetically variable white dwarfs have a strong field (tens to hundreds megagauss), except for WD2051-208, which is the only ultra-massive star ( M = 1 . 2 M ⊙ ) with a sub-MG field strength (0.25 MG) in our sample. In contrast, seven massive magnetic white dwarfs show no obvious signs of variability, although none of them may be safely considered as non-variable, either because they were observed only twice (or even just once, i.e. WD 1750-311; see Appendix B.60) or because some subtle sign of variability may have been detected (in WD 1900 + 705 and WD2010 + 310). WD1658 + 440 and WD1750-311 are the only massive non- Notes. The full list includes 59 stars, 23 of which are non-variable. Between brackets, the table also gives the number of non-variable stars N K divided by the number of stars N tot in the subset as defined by the age and field values. variable white dwarfs in our sample with a field as low as ≈ 2 MG. The remaining five non-variable stars have magnetic fields with strengths of the order of hundreds of megagauss.\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"4.4. Statistical analysis of normal-mass white dwarfs\",\n \"content\": \"Because we have only one example of an old massive magnetic white dwarf, we do not know how variability in massive magnetic white dwarfs evolves with time. Vice versa, an evolutionary path is clearly seen in normal-mass ( M < ∼ 1 M ⊙ ) white dwarfs. In our sample, field variability is nearly ubiquitous among weakly magnetic white dwarfs of all ages and in young white dwarfs regardless of the field strength, it is but very rare in old ( τ ≥ 2 Gyr) strongly magnetic ( B ≥ 10 MG) white dwarfs. Next, we assess the statistical significance of this pattern, specifically whether it can be attributed to small number statistics or if it reflects a genuine correlation between cooling age, field strength, and field variability. We first split the sample of stars with M ≤ 1 M ⊙ into two groups: those younger than 2 Gyr and those older than 2 Gyr. Each of these two groups was divided into two subsets: stars with a field strength ⟨| B |⟩ ≤ 1 MG and stars with ⟨| B |⟩ > 10 MG. For each of these four subsets, we considered the ratio between the number of non-variable magnetic stars, N K, and the total number of magnetic stars, N tot. From these numbers, we estimated the probability, Pr , that the sample frequency of non-variable stars is between r and r + d r , normalised to one and obtained assuming that all numbers between zero and one are a priori equally probable, using Figure 2 shows the probability density functions for the stars belonging to these sets. It clearly appears that there is little to no overlap between the density functions of the symmetric field in old strongly magnetic white dwarfs and of young white dwarfs. Table 1 reports the points of maximum for these distributions, f = N K / N tot, and their uncertainties Table 1 also includes the results for the smaller sets of stars with intermediate strength, 1 ≤ ⟨| B |⟩ ≤ 10 MG (with endpoints overlapping with the other two sets), which could possibly be considered a 'transitioning' field strength range. Figure 3 shows the ratio between the number of non-variable magnetic white dwarfs with a field strength lower than a given value ¯ ⟨| B |⟩ and the total number of magnetic white dwarfs with a field strength lower than that value as a function of ¯ ⟨| B |⟩ for stars with cooling ages τ ≤ 2 Gyr (blue lines) and stars with τ > 2 Gyr (red lines). This plot supports our claim that nonvariable fields are much more common in old, strongly magnetic white dwarfs than in young strongly magnetic white dwarfs, and than in weakly magnetic white dwarfs of all ages. It suggests also that the minimum field strength required for a field to become non-variable is in the range of 5 to 10 MG.\",\n \"pages\": [\n 4,\n 6\n ]\n },\n {\n \"title\": \"5. Explanation for the lack of observed variability\",\n \"content\": \"There are three possible reasons for the observed non-variability of so many magnetic white dwarfs. We examine them in the following sections.\",\n \"pages\": [\n 6\n ]\n },\n {\n \"title\": \"5.1. Extremely slow rotation\",\n \"content\": \"We first consider the possibility that the rotation of magnetic white dwarfs slows with age and / or magnetic field strength. Perhaps, as the surface field becomes stronger, the white dwarf loses angular momentum to its environment by electromagnetic dipole radiation (García-Berro et al. 2012), by coupling with gas clouds in the ISM, or by a very weak magnetically coupled wind. Each of these possibilities would shed stellar angular momentum faster from a magnetic white dwarf with a strong field than from one with a weak field, preferentially slowing the magnetic white dwarfs with strong fields. However, all of these mechanisms are expected to exert at most a very weak influence on the rotation of a white dwarf, and it seems quite unlikely that any of them could slow the rotation of a magnetic white dwarf so much that repeated observations even a year apart would not show any rotation. Nevertheless, the idea of extremely slowly rotating white dwarfs has been generally accepted. This hypothesis likely origi- ates from Sect. 3 of Schmidt & Norsworthy (1991), which says: 'We therefore assign long periods to these stars, with the recognition that other explanations for their polarimetric constancy are possible.' This statement was accompanied by the first plots of field strength versus rotational period, a plot that in updated form has reappeared in numerous reviews (e.g. Ferrario et al. 2015; Kawka 2020; Ferrario et al. 2020) but has never been critically revisited. We note that there is a complete lack of white dwarfs that are known to vary on a timescale between weeks and decades. In this respect, there is a profound di ff erence between candidate longterm variable magnetic white dwarfs and long period magnetic Ap and Bp stars. While we do not know of any white dwarf with a firmly established polarimetric variability with a period longer than approximately two weeks, we are sure that a number of Ap and Bp stars are very slowly rotating stars because they definitely show clearly periodic field variation, even ⟨ Bz ⟩ sign reversals, on a very long but measured timescale that may be months, years, or decades. Furthermore, the distribution of periods is roughly continuous between periods of less than one day and periods of several years (Mathys 2008). If we wanted to interpret the lack of polarimetric variability in white dwarfs as the e ff ect of extremely long rotation periods, we would need to accept that the rotation periods of white dwarfs show an extremely bimodal distribution that peaks around hours and days and around centuries with nothing in between. This appears to be a very unlikely scenario.\",\n \"pages\": [\n 6\n ]\n },\n {\n \"title\": \"5.2. Extremely fast rotation\",\n \"content\": \"Asecond possibility is that some or all of the non-variable white dwarfs actually are very rapidly rotating stars. If the star has a rotation period much shorter than the individual frame exposure time, then the polarimetric variability would be smeared out, and the star would appear constant. In fact, it has been possible to probe variability on the timescale of the exposure time of each individual frame (typically 10 min or less). In very general terms, our FORS2 and ISIS polarimetric observations allowed us to identify a star as variable provided that its rotation period is longer than ≈ 10 min and its magnetic configuration is clearly not symmetric about the rotation axis (for example, WD 1712-590 in Sects. B.57). Isolated white dwarfs that are the product of single-star evolution can hardly have rotation periods shorter than that (Kawaler 2015). This is confirmed by the results of Hernandez et al. (2024), who have shown that rotation periods of young magnetic white dwarfs in the normal mass range are only marginally shorter than the rotation periods of non-magnetic white dwarfs. Merger products, in contrast, can have rotation periods of the order of 1 min (Schwab 2021; Kilic et al. 2021) if most of the binary angular momentum is retained by the merger product. It is therefore possible that the non-variable massive white dwarfs have nonaxisymmetric fields but are rotating very rapidly. Periods down to 1 min or less can be probed via rapid cadence photometry. None of the massive, magnetically non-variable white dwarfs show TESS light variability (Hernandez, priv. comm.; Ramsay, priv. comm.), although of course one cannot rule out that more accurate photometry could reveal some extremely rapid rotators. Provisionally, we rule out very rapid rotation for the massive magnetic white dwarfs that show a constant polarisation.\",\n \"pages\": [\n 6,\n 7\n ]\n },\n {\n \"title\": \"5.3. Magnetic fields are symmetric about the stellar rotation axes\",\n \"content\": \"After ruling out both extremely slow and extremely fast rotation, we are left with the hypothesis that a non-variable field is approximately axisymmetric about the rotation axis. This means that as the star rotates with a normal period between a fraction of an hour and several days, the observer does not see any significant change in the signal of circular polarisation. This interpretation naturally accounts for some outliers, such as the non-variable weakly magnetic star WD 1105-340. It is reasonable to hypothesise that this one star does not appear variable simply because its rotation axis is tilted at a small angle with respect to the line of sight. In Sect. 3, we highlighted that it may be di ffi cult to firmly assess whether a star is magnetically variable because changes could be too subtle to be detected. However, the risk of classifying as 'non-variable' a star that in fact exhibits subtle but real changes does not weaken our analysis because small changes of the circular polarisation spectrum are still symptoms of a field nearly symmetric about the rotation axis.\",\n \"pages\": [\n 7\n ]\n },\n {\n \"title\": \"6. Discussion\",\n \"content\": \"Here, we first consider the case of normal-mass white dwarfs (Sect. 6.1) and then that of massive stars (Sect. 6.2). We note that our analysis applies only to isolated white dwarfs; symmetric fields seem uncommon among magnetic white dwarfs accreting material from a companion (Cropper 1988; Reimers et al. 1999; Schmidt et al. 1995).\",\n \"pages\": [\n 7\n ]\n },\n {\n \"title\": \"6.1. Normal-mass white dwarfs\",\n \"content\": \"It is clear that most of the weak fields of normal-mass stars of all ages are not symmetric about the stellar rotation axis. Most of the strong fields of normal-mass, young white dwarfs are also not axisymmetric. Most of the strong field of stars older than 2 Gyr are symmetric about the star's rotation axis. We now investigate what are the relationships between these weak and strong fields. Our hypothesis is that the weak and non-symmetric magnetic fields of the young normal-mass white dwarfs are due to a relaxation process of a field that was produced in a previous evolutionary stage of the star, a field that was buried below the surface of the newly formed white dwarf, and that it starts to reveal itself with time as the star cools down. Such seed fields could be those found in red-giant stars, in the vicinity of the hydrogenburning shell (e.g. Li et al. 2022, 2023). The question is whether the older, generally stronger fields share essentially the same origin as the weak, younger ones, having just evolved for longer time, or if the strong fields in older stars formed by a totally different mechanism than the weaker, younger ones, for instance by a crystallisation dynamo. Whatever its origin is, we consider first the possibility that the small distortion of the shape of a magnetic white dwarf produced by magnetic forces in outer layers might actually drive the evolution of the global field towards axisymmetry. In Sect. 6.1.1, we discuss one example of a physical e ff ect that might drive such evolution.\",\n \"pages\": [\n 7\n ]\n },\n {\n \"title\": \"6.1.1. The magnetic field structure of white dwarfs may evolve towards rotational axisymmetry during cooling\",\n \"content\": \"In the absence of a magnetic field, the rotation of the white dwarf will lead to an equatorial bulge, increasing the component of the principal moment of inertia that is parallel to the rotation axis. In this case, the rotation about the spin axis, which coincides with the largest moment of inertia, is stable. Next, we introduce a dipolar magnetic field inclined to the axis of rotation by, say, 45 · . We suppose that the visible surface field is maintained by a fossil current system deep in the star, which is gradually decaying due to Ohmic losses. As the interior field strength decreases, an azimuthal electric field will be produced by Faraday induction outside the white dwarf's initial current loop. This field will generate a current around the magnetic axis that will in turn interact with the local meridional magnetic field (Landstreet 1987) to produce an outward-directed force, distorting the stellar outer layers into a slightly oblate shape, with the largest radius around the magnetic equator. Because of this e ff ect, the principal moment of inertia of the star will be rotated towards the symmetry axis of the field, which is not aligned with the rotation axis or the angular momentum axis of the star. The white dwarf e ff ectively becomes an unstable free asymmetric rotator (an asymmetric top). If such objects can dissipate some of the energy supporting the asymmetric shape, the field axis will gradually shift to bring the principal moment of inertia closer to the angular momentum axis, which is a stable minimum energy state of an oblate system. Energy dissipation will gradually lead to the alignment of the magnetic axis with the rotation axis, which is the e ff ect we observe in old normal-mass strongly magnetic white dwarfs. This basic physical e ff ect was first identified in the 1970s as possibly operating in main-sequence magnetic Ap stars, which, similar to magnetic white dwarfs, possess global fossil magnetic fields that are usually roughly dipolar and are generally oblique to the rotation axis (Stibbs 1950; Preston 1971). A number of efforts to estimate the timescale of possible evolution of an oblique magnetic field to a state of small or vanishing obliquity have been published, for example, by Mestel & Takhar (1972). Much of this work is summarised in Chapter 9 of Mestel (1999). However, this line of investigation did not lead to a convergence of clear results about possible timescales or about dependence on basic input physics or parameters, such as interior field strength, or details of induced forces in outer layers and their consequences. As the increasing sample of magnetic Ap star models failed to reveal an obliquity distribution suggestive of this e ff ect, possibly because of the relatively short (of order 10 8 yr) mainsequence lifetimes of magnetic Ap stars, theoretical studies of stellar fossil fields moved on to other e ff ects. However, a similar e ff ect may be at work aligning the magnetic fields to the rotation axis in white dwarfs, which have a very di ff erent structure and much longer evolutionary times than Ap stars. This hypothesis could be explored with the aid of numerical modelling of the global (interior and surface) structure and evolution of a rotating white dwarf with an oblique magnetic field. Modelling of comparably complex white dwarf states (e.g. polars, mergers) that include magnetic fields (e.g. Franzon & Schramm 2015; Bisikalo et al. 2021; Zhong et al. 2024) suggest that numerical methods for exploring this problem are already available.\",\n \"pages\": [\n 7,\n 8\n ]\n },\n {\n \"title\": \"6.1.2. Crystallisation-driven dynamo and axisymmetric fields\",\n \"content\": \"The line that marks the beginning of core crystallisation in the age-mass diagram separates variable from non-variable magnetic white dwarfs perhaps in a cleaner way compared to a massindependent age threshold (see Fig. 1). Before core crystallisation begins, magnetic fields are almost always non-symmetric about the rotation axis. From volume-limited surveys, we know that in normal-mass white dwarfs, before crystallisation, strong fields are rare (Bagnulo & Landstreet 2021, 2022), but those that have been discovered and monitored show polarimetric variability. After the beginning of core crystallisation, many strong fields appear, and most of them are symmetric about the rotation axis. White dwarfs with weak and non-symmetric fields continue to appear also after the beginning of core-crystallisation. The strong magnetic fields of old normal-mass white dwarfs could be generated by the crystallisation convective dynamo mechanism (Isern et al. 2017; Schreiber et al. 2021), as almost all have cooling ages longer than the cooling age required for the onset of this dynamo. In this case, the conclusion would be that the crystallisation dynamo is responsible for approximate symmetry about the rotation axis. This situation could have similarities with what has been observed in fully convective late type stars with no di ff erential rotation, which are known to be able to generate strong and simple large-scale, mostly axisymmetric, poloidal fields (Donati et al. 2006; Morin et al. 2008b,a; Kochukhov 2021, Fig. 14). However, how the crystallisation dynamo alone can produce fields stronger than ∼ 1 MG is still unclear (Isern et al. 2017; Castro-Tapia et al. 2024). In fact, Montgomery & Dunlap (2024) have argued that fluid mixing by phase separation is not a viable mechanism to produce the strong fields observed in old white dwarfs. Perhaps the crystallisation dynamo could be more powerful if it were to amplify a pre-existing internal field, such as the same seed field that is seen in some of the white dwarfs prior to the beginning of crystallisation. Montgomery & Dunlap (2024) have proposed that corecrystallisation could trigger temporary di ff erential rotation. Therefore, one could suspect that rapid rotation in normal-mass white dwarfs might generate a magnetic field in the stellar core using rotational shear on a seed field left from an earlier point in the star's evolutionary history. In this situation, we might also expect that the field would be roughly axisymmetric, although it is not clear why such fields would always be strong or how they might be related to the weaker oblique fields of stars of similar mass. Furthermore, Spruit (1999) has shown that di ff erential rotation would suppress the non-axisymmetric field component of weak fields, but not in the stronger fields. This is the opposite of what we observed. The origin of strong non-axisymmetric fields of normal-mass white dwarfs that appear before crystallisation could possibly be similar to that of higher-mass white dwarfs (see Sect. 6.2 below).\",\n \"pages\": [\n 8\n ]\n },\n {\n \"title\": \"6.2. The variability of massive magnetic white dwarfs\",\n \"content\": \"Compared to normal-mass white dwarfs, the magnetic fields of massive white dwarfs present a di ff erent behaviour. Magnetic fields appear when the stars are still very young, and fields may be either symmetric or non-symmetric about the rotation axis. In massive white dwarfs, strong fields (tens or hundreds of megagauss) seem much more common than weaker sub-megagauss fields. It is widely thought that many of these high-mass objects are the result of WD-WD mergers that cause rapid generation of a strong magnetic field (García-Berro et al. 2012; Bagnulo & Landstreet 2022). However, Camisassa et al. (2022) and Blatman & Ginzburg (2024) have shown that at least some of them, depending on their core composition, may have started the process of core crystallisation. Hence, the origin of their field could be linked to the crystallisation dynamo, at least in some cases. The bimodal distribution of the morphologies of the fields of the massive magnetic white dwarfs could indeed reflect two di ff erent channels of formation, one from WD-WD merger (or by a di ff erent binary evolution path), and one by massive singlestar evolution, such as that of normal-mass white dwarfs. Alternatively, the dynamo stimulated by WD-WD merging may be capable of generating both axisymmetric and non-axisymmetric global fields, perhaps depending on the initial angular momentum vectors of the individual merging white dwarfs relative to the orbital angular momentum vector or the mass ratio of the two merging white dwarfs. A viable alternative explanation is that the massive white dwarfs that do not show variability are actually rotating with a period much shorter than the typical exposure time of individual polarimetric measurements. In any case, the large mass of stars in this sample and the occurrence of some rotation periods as short as minutes make it very reasonable to suppose that the evolution of the magnetic fields and rotation periods follows a di ff erent course from the evolution of fields in normal-mass magnetic white dwarfs. Because almost all of the massive magnetic white dwarfs of our sample, except one, are very young, we cannot test whether the morphology of older stars is generally axisymmetric. Remarkably, however, the only example we know of a massive star older than 2 Gyr shows a non-axisymmetric field. With an age of more than 4 Gyr, high mass, and a very strong and variable field, the very unusual WD 0756 + 437 may be considered further evidence that the origin of the fields in massive stars is di ff erent than in normal-mass stars.\",\n \"pages\": [\n 8\n ]\n },\n {\n \"title\": \"7. Conclusions\",\n \"content\": \"Since the earliest discoveries of magnetic white dwarfs (Kemp et al. 1970; Angel & Landstreet 1970, 1971b), two quite di ff erent categories of them have been known to exist. Some magnetic white dwarfs show periodic variations of circular polarisation (which probes the longitudinal magnetic field) with timescales ranging from minutes to days. Classically, a signal of circular polarisation constant with time has been interpreted as indicating either a very long rotation period or a lack of rotation. Using both literature data and new observations, we have studied the polarimetric variability of a sample of 74 magnetic white dwarfs. We find that among white dwarfs with M ≤ 1 . 0 M ⊙ ('normal-mass white dwarfs'), nearly all stars with fields weaker than about 1 MG show circular polarisation varying with time. Furthermore, the rare normal-mass white dwarfs younger than ≈ 2 Gyr with strong magnetic fields also show polarimetric variability. In striking contrast, 16 out of the 17 normal-mass stars older than 2 Gyr with fields stronger than about 10 MG in our sample show constant polarisation. Magnetic white dwarfs with M ≥ 1 . 0 M ⊙ ('massive white dwarfs'), many of which are the product of WD-WD merging, show a mixed behaviour. Many of them have a strong magnetic and variable field, but some have a strong and constant magnetic field. In our sample, nearly all the massive magnetic white dwarfs are younger than ≈ 1 Gyr, with the exception of WD 0756 + 437, an old strongly magnetic variable star. The lack of evidence for major variations of circular polarisation on any timescale longer than about two weeks suggests that the interpretation of non-variability arising from extremely long rotational periods is incorrect. We have argued that the nonvariability is due to magnetic field structures roughly symmetric around the star's rotation axis. A possible explanation for the observed symmetry could be that most or all of the magnetic fields evolved from pre-existing fields that formed during pre-white dwarf evolution stages (Li et al. 2022, 2023) and gradually relaxed to the stellar surface over a relation time of 1 or 2 Gyr; in fact, very recently, Camisassa et al. (2024) have shown that the magnetic fields of old white dwarfs with M ≥ 0 . 65 M ⊙ may have been generated by a coreconvection dynamo when the star was in the main sequence, and emerged at the stellar surface during the white dwarf cooling phase. As the surface field increases in strength, at some point the Lorentz forces in the outer layers, together with the Coriolis forces acting on rotating convective flows in what are asymmetric rotators, could lead to a gradual relaxation, through energy dissipation, of the global field structure to a form that is symmetric about the rotation axis. The timescale would depend on the field strength, and for a field weaker than several megagauss, it could require a timescale too long to be observed in white dwarfs that still show spectral lines. Alternatively, because these normal-mass large-field magnetic white dwarfs mostly occur after the start of crystallisation (Bagnulo & Landstreet 2021, 2022), one could speculate that the crystallisation dynamo (Isern et al. 2017; Schreiber et al. 2021; Ginzburg et al. 2022) may generate some fraction of the observed fields and further that this dynamo produces essentially axisymmetric surface magnetic field structures with extremely strong fields. This kind of origin would have similarities with the axisymmetric fields that are commonly found in fully convective strongly magnetic M-dwarfs and, much weaker, in the planets Earth and Jupiter. However, it has been argued that the instability of the mantle surrounding the core that starts to crystallise cannot produce fields much stronger than 1 MG (Isern et al. 2017; Montgomery & Dunlap 2024; Castro-Tapia et al. 2024). Furthermore, Camisassa et al. (2024) find that, even if the crystallisation-driven dynamo could generate a strong magnetic field, this field would take too long to emerge at the stellar surface. On the other hand, Montgomery & Dunlap (2024) have suggested that crystallisation may still play a role by trig- gering a temporary phenomenon of di ff erential rotation, which in turn would generate a magnetic field - the di ff usion timescale of which has not been discussed. The situation for ultra-massive white dwarfs (with M > ∼ 1 . 0 M ⊙ ) is di ff erent, as both strongly magnetic variable and nonvariable white dwarfs are found among them. Some of these massive white dwarfs are the product of WD-WD merging, during which a dynamo could create a strong magnetic field (García-Berro et al. 2012), though it would not necessarily be symmetric about the rotation axis. In addition, some massive magnetic white dwarfs could be the result of single-star evolution and have acquired an axisymmetric field following the same mechanism acting in normal-mass white dwarfs. Further investigation into the variability of magnetic white dwarfs is necessary. Nevertheless, current observations have already imposed significant constraints that any theory explaining the origin and evolution of magnetic fields in degenerate stars must take into account. Notes. A number in the last column represents the established rotation period (in d) of a star that shows polarimetric variability; tentative periods are followed by the symbol ':'. Other symbols have the following meaning: 'var.' means that the star is certainly polarimetric variable but the period is still unknown; 'var.:' means that hints of subtle variability have been detected over a short timescale; 'n.v.' means that the observed polarisation was constant (within uncertainties) and the star was observed at least three times; 'n.v.:' means that observed polarisation was constant (within uncertainties) but the star was observed only twice; 's.v.' means that polarisation shows some sign of subtle variability over a timescale of a decade or longer. For stars within the local 40 pc volume, the parameters of stellar magnitude, distance, spectral type, atmospheric composition, temperature, mass, and age are from O'Brien et al. (2024); for the remaining stars, we used the catalogue from Gentile Fusillo et al. (2021) with ages interpolated from the tables by Bédard et al. (2020). Acknowledgements. The new observations presented in this work were made with the FORS2 instrument at the ESO Telescopes at the La Silla Paranal Observatory under program ID 110.243J.001, 110.23XV.001, 110.23XV.002, and 112.25C9.001, and with the ISIS instrument at the William Herschel Telescope (operated on the island of La Palma by the Isaac Newton Group), under programmes P10 in 19A and P8 in 19B. This research has made use also of additional FORS1 and FORS2 data obtained from the ESO Science Archive Facility: data for WD 1036-204 were obtained under programmes IDs 70.D-0259(A), 087.D-0714(A), 089.D-0612(A), 090.D-0269(A). Data for WD 1105-340 was acquired with ESPaDOnS on the Canada-France-Hawaii Telescope (CFHT) (operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii), under programmes 17AC01, 19AC04, 19BC02, and 21BC02. We thank the anonymous referee for their very constructive criticism. We thank Matthias Schreiber and Antonino Lanza for very useful comments. JDL acknowledges the financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference number 6377-2016.\",\n \"pages\": [\n 8,\n 9,\n 11,\n 12\n ]\n },\n {\n \"title\": \"References\",\n \"content\": \"Achilleos, N. & Wickramasinghe, D. T. 1989, ApJ, 346, 444 Angel, J. R. P., Illing, R. M. E., & Landstreet, J. D. 1972a, ApJ, 175, L85 Aznar Cuadrado, R., Jordan, S., Napiwotzki, R., et al. 2004, A&A, 423, 1081 Article number, page 12 of 29 Abstract Series, Vol. 9, Astronomische Gesellschaft Abstract Series, 145 Liebert, J., Schmidt, G. D., Sion, E. M., et al. 1985, PASP, 97, 158 Macfarlane, S. A., Woudt, P. A., Dufour, P., et al. 2017, MNRAS, 470, 732 Manser, C. J., Gänsicke, B. T., Inight, K., et al. 2023, MNRAS, 521, 4976 Martin, B. & Wickramasinghe, D. T. 1978, MNRAS, 183, 533 Mathys, G. 2008, Contributions of the Astronomical Observatory Skalnate Pleso, 38, 217 Maxted, P. F. L., Ferrario, L., Marsh, T. R., & Wickramasinghe, D. T. 2000, MNRAS, 315, L41 McCook, G. P. & Sion, E. M. 1977, A Catalogue of spectroscopically identified white dwarfs McCook, G. P. & Sion, E. M. 1999, ApJS, 121, 1 Mestel, L. 1999, Stellar magnetism Mestel, L. & Takhar, H. S. 1972, MNRAS, 156, 419 Montgomery, M. H. & Dunlap, B. H. 2024, ApJ, 961, 197 Morin, J., Donati, J. F., Forveille, T., et al. 2008a, MNRAS, 384, 77 Morin, J., Donati, J. F., Petit, P., et al. 2008b, MNRAS, 390, 567 Moss, A., Bergeron, P., Kilic, M., et al. 2024, MNRAS, 527, 10111 O'Brien, M. W., Tremblay, P. E., Gentile Fusillo, N. P., et al. 2023, MNRAS, 518, 3055 O'Brien, M. W., Tremblay, P. E., Klein, B. L., et al. 2024, MNRAS, 527, 8687 Oliveira da Rosa, G., Kepler, S. O., Soethe, L. T. T., Romero, A. D., & Bell, K. J. 2024, arXiv e-prints, arXiv:2407.05214 Piirola, V. & Reiz, A. 1992, A&A, 259, 143 Preston, G. W. 1971, PASP, 83, 571 Putney, A. 1995, ApJ, 451, L67 Putney, A. 1997, ApJS, 112, 527 Putney, A. & Jordan, S. 1995, ApJ, 449, 863 Reding, J. S., Hermes, J. J., Vanderbosch, Z., et al. 2020, ApJ, 894, 19 Reimers, D., Hagen, H. J., & Hopp, U. 1999, A&A, 343, 157 Schmidt, G. D., Bergeron, P., & Fegley, B. 1995, ApJ, 443, 274 Schmidt, G. D. & Norsworthy, J. E. 1991, ApJ, 366, 270 Schmidt, G. D. & Smith, P. S. 1994, ApJ, 423, L63 Schmidt, G. D. & Smith, P. S. 1995, ApJ, 448, 305 Schreiber, M. R., Belloni, D., Gänsicke, B. T., Parsons, S. G., & Zorotovic, M. 2021, Nature Astronomy [ arXiv:2104.14607 ] Schwab, J. 2021, ApJ, 906, 53 Siebenmorgen, R., Voshchinnikov, N. V., & Bagnulo, S. 2014, A&A, 561, A82 Sion, E. M., Liebert, J., Schmidt, G., & Starrfield, S. G. 1984, in Bulletin of the American Astronomical Society, Vol. 16, 725 Spruit, H. C. 1999, A&A, 349, 189 Stibbs, D. W. N. 1950, MNRAS, 110, 395 Swedlund, J. B., Wolstencroft, R. D., Michalsky, Jr., J. J., & Kemp, J. C. 1974, ApJ, 187, L121 Valyavin, G., Bagnulo, S., Monin, D., et al. 2005, A&A, 439, 1099 Valyavin, G., Wade, G. A., Bagnulo, S., et al. 2008, ApJ, 683, 466 Vennes, S., Schmidt, G. D., Ferrario, L., et al. 2003, ApJ, 593, 1040 Vornanen, T., Berdyugina, S. V., Berdyugin, A. V., & Piirola, V. 2010, ApJ, 720, L52 Walters, N., Farihi, J., Marsh, T. R., et al. 2021, MNRAS, 503, 3743 Wegner, G. 1977, Mem. Soc. Astron. Italiana, 48, 27 Wesemael, F., Liebert, J., Schmidt, G. D., et al. 2001, ApJ, 554, 1118 West, S. C. 1989a, ApJ, 345, 511 West, S. C. 1989b, ApJ, 345, 511 Wickramasinghe, D. T. & Bessell, M. S. 1976, ApJ, 203, L39 Wickramasinghe, D. T. & Cropper, M. 1988, MNRAS, 235, 1451\",\n \"pages\": [\n 12,\n 13\n ]\n },\n {\n \"title\": \"Appendix A: New observations\",\n \"content\": \"Article number, page 14 of 29\",\n \"pages\": [\n 14\n ]\n },\n {\n \"title\": \"Appendix B: Variability of magnetic white dwarfs: Comments on individual stars\",\n \"content\": \"We preliminary note that to compare observations taken by different authors, we need to know how circular polarisation was defined in di ff erent works. This problem has been highlighted in detail, for example in Sect. 2 of Bagnulo & Landstreet (2020). Here, we have adopted the definition of positive handedness of circular polarisation given by Landi Degl'Innocenti & Landolfi (2004), and converted some of the literature data to it by changing the sign of circular polarisation with respect to the originally published data. Numerous examples in the literature (e.g. Bagnulo & Landstreet 2020) and in this section show that many magnetic white dwarfs with very strong fields have a circular polarisation spectrum that varies rapidly with wavelength. The consequence is that the same source observed with even slightly di ff erent broadband filters may result in di ff erences of the measured broadband polarisation signal that are larger than the formal uncertainties. This specific issue is discussed in Sect. 7 of Bagnulo & Landstreet (2019b), where some numerical examples are presented. As a consequence, small di ff erences between broadband polarimetric measurements obtained with di ff erent instruments cannot reliably provide strong evidence that a star is magnetically variable. Spectropolarimetry should allow comparisons that are less instrument dependent, provided the observations overlap in wavelength, and that the resolving power of di ff erent instrument is similar. There is no doubt, however, that the safest way to establish variability or constancy of polarisation is to repeatedly observe the star with the same instrument and instrument setting, and to always perform the data reduction using exactly the same technique. In the following, we discuss all stars of our sample (including two stars that did not make it into the final list of Table 2). The titles of the individual subsections contain the star's name in the Villanova system (McCook & Sion 1977, 1999), the main SIMBADidentifier, and our conclusions about the magnetic variability (using the same designation as in Table 2, followed by the rotation period P , if this is known). When the rotation period is obtained only from photometric data, we use the designation P phot.\",\n \"pages\": [\n 16\n ]\n },\n {\n \"title\": \"Appendix B.1: WD 0004+122 = LP 464-57 (non-variable)\",\n \"content\": \"This star was discovered to be magnetic, using spectropolarimetry, by Bagnulo & Landstreet (2020), and later observed four times by Berdyugin et al. (2024) in broadband circular polarisation. These measurements are shown in the top left panel of Fig. B.1. None of them deviated more than ≃ 1 σ from the mean value of the measurements obtained in the same filter, except for a ∼ 2 . 5 σ di ff erence between on measurement and the other three in the B filter, which might reflect a real small variability. We reanalysed the FORS2 spectropolarimetry published by Bagnulo & Landstreet (2020) to check for di ff erences between the V / I profiles obtained from the first and the second pair of exposures, sampling an interval in time of about 15 m, without finding any di ff erence (see Fig. B.2). We conclude that the star does not show convincing sign of variability. We estimate a field strength of ≃ 100 MG.\",\n \"pages\": [\n 16\n ]\n },\n {\n \"title\": \"Appendix B.2: WD 0009+501 = EGGR 381 (variable, P ≃ 8 h)\",\n \"content\": \"This white dwarf was well monitored and modelled as a magnetic variable with ⟨| B |⟩ between 150 and 250 kG by Valyavin et al. (2005), who presented a model obtained by adopting a rotational period of 0 . 3337 ± 0 . 0031 d. Recent unpublished ESPaDOnS data obtained by us confirm the rotational period.\",\n \"pages\": [\n 16\n ]\n },\n {\n \"title\": \"Appendix B.3: WD 0011-134 = G 158-45 (variable, P phot ≃ 0 . 73 h)\",\n \"content\": \"Discovered magnetic (with a field of about 9 MG) by Bergeron et al. (1992), and as a magnetic variable by Putney (1997). From photometry, Lawrie et al. (2013) found a rotational period of 44 ± 0 . 43 min. While there is no guarantee that the photometric period reflects the rotational period of the star, the star is definitely a magnetic variable, and in fact, this very short photometric period is completely consistent with several of the many acceptable periods found from the six widely spaced ⟨ Bz ⟩ data values published by Putney (1997) from spectropolarimetric observations made between 1994 Sep 29 and 1994 Dec 31. This star is one of the magnetic white dwarfs that clearly shows longitudinal field reversal.\",\n \"pages\": [\n 16\n ]\n },\n {\n \"title\": \"Appendix B.4: WD 0041-102 = Feige 7 (variable, P ≃ 2 h)\",\n \"content\": \"Liebert et al. (1977) showed that the spectrum is rich in faint lines of H and He, and identified a magnetic field of about 18-20 MG from the spectrum. They found, on the basis of broadband circular polarisation measurements, that the field varies with a period of 131.6 min. (Note that the broadband circular polarisation changes sign as the star rotates.) A decentred dipole model of the field structure based on a series of flux spectra was derived by Achilleos et al. (1992); both H and He apparently vary substantially in abundance over the surface. Achilleos et al. (1992) also discovered that the star varies photometrically (in a double wave) with the same period as the spectrum and broad-band polarisation.\",\n \"pages\": [\n 16\n ]\n },\n {\n \"title\": \"Appendix B.5: WD 0051+117 = PHL 886 (variable, P ≃ 4 . 7 d)\",\n \"content\": \"From unpublished spectropolarimetric data (Landstreet & Bagnulo, in prep., hereafter LB25) it is found that the star is periodically variable ( P = 4 . 703 d) with ⟨| B |⟩ ≃ 270 kG. The field ⟨ Bz ⟩ reverses sign as the star rotates. Modelling will be presented in a forthcoming paper.\",\n \"pages\": [\n 16\n ]\n },\n {\n \"title\": \"Appendix B.6: WD 0058-044 = GD 9 (variable, P ≃ 2 d?)\",\n \"content\": \"From our unpublished spectropolarimetric data we found that the star is periodically variable (the period is not determined uniquely from available data, but the most probable period is P = 1 . 96 d), with ⟨| B |⟩ ≃ 300 kG (LB25). The field ⟨ Bz ⟩ varies sinusoidally but does not (quite) reverse sign. Modelling will be presented in a forthcoming paper.\",\n \"pages\": [\n 16\n ]\n },\n {\n \"title\": \"Appendix B.7: WD 0232+525 = EGGR 314 (variable)\",\n \"content\": \"This is a well-known star that was repeatedly included in studies of bright white dwarfs and been searched for a magnetic field (Bychkov et al. 1991; Schmidt & Smith 1995; Fabrika et al. 1997) without any detection being reported. Recent particularly sensitive observations (Bagnulo & Landstreet 2022) succeeded in detecting a weak field. The three published measurements (two with ISIS, one with ESPaDOnS) reveal a very weak, and probably variable longitudinal field (the measurement obtained with ESPaDOnS has the opposite sign as those obtained with ISIS).\",\n \"pages\": [\n 16,\n 17\n ]\n },\n {\n \"title\": \"Appendix B.8: WD 0233-242A = LP 830-14 = NLTT 8435A (variable, P phot ≃ 1 . 5 h)\",\n \"content\": \"The star is discussed by Bagnulo & Landstreet (2021), who suggested the presence of a variable field of the order of 3.8 MG. In fact, ⟨| B |⟩ does not vary between two FORS observations, while ⟨ Bz ⟩ clearly changes sign. According to Vennes et al. (2018) it has a photometric variation period of 95 min.\",\n \"pages\": [\n 17\n ]\n },\n {\n \"title\": \"Appendix B.9: WD 0236-269 = PHL 4227 (n.v.: - only 2 meas.)\",\n \"content\": \"Schmidt et al. (2001) report two spectropolarimetric detections of circular polarisation at about the 1.2% level. The two observations were obtained three days apart, with no changes detected. Furthermore no significant change in the spectrum-average circular polarisation was observed. We tentatively assume that is not a magnetic variable, but the evidence for non-variability is not compelling (the original paper just reports: \\\"consistent results were obtained on the two occasions\\\").\",\n \"pages\": [\n 17\n ]\n },\n {\n \"title\": \"Appendix B.10: WD 0253+508 = KPD 0253+5052 (variable, P ≃ 4 h)\",\n \"content\": \"The field was discovered by Downes & Margon (1983), who estimated ⟨| B |⟩ ≈ 13 MG, and modelled by Achilleos & Wickramasinghe (1989) using only flux spectra. The star shows continuum polarisation up to 0.4% that is periodically variable according to Schmidt & Norsworthy (1991) with P = 0 . 170 d. The sign of the broadband continuum circular polarisation briefly reverses during each cycle.\",\n \"pages\": [\n 18\n ]\n },\n {\n \"title\": \"Appendix B.11: WD 0301+059 = SDSS J030350.63+060748.9 (n.v.: - only 2 meas.)\",\n \"content\": \"The star was discovered to be magnetic by Landstreet & Bagnulo (2020), who did not find any obvious sign of variability between two observations obtained 1 day apart, nor between individual exposures of the same observing series (exposure times of single frames were 225 s). We tentatively assume that it is not magnetically variable. The star therefore appears as a strongly magnetic, massive, non-variable star.\",\n \"pages\": [\n 18\n ]\n },\n {\n \"title\": \"Appendix B.12: WD 0313-084 = GALEX J031613.8-081637 (non-variable)\",\n \"content\": \"The star was discovered to be magnetic by O'Brien et al. (2023). It was observed five times with FORS2 (see Sect. 2.1), and no variability was found within timescales of 1 h, 1 d, and 8 months (see Fig. B.3). The spectrum indicates that ⟨| B |⟩ ≃ 800 kG.\",\n \"pages\": [\n 18\n ]\n },\n {\n \"title\": \"Appendix B.13: WD 0316-849 = EUVE J0317-855 (variable, P ≃ 0 . 2 h)\",\n \"content\": \"Studied by Ferrario et al. (1997) Burleigh et al. (1999), and Vennes et al. (2003), the star has a rotation period P = 725 s. A detailed model based on UV spectra was obtained by Burleigh et al. (1999), who found that the field varies in strength from about 180 to 800 MG over the surface. Vennes et al. (2003) shown that the star has a light curve with a minimum that corresponds to the maximum of the polarisation. This is one of the very few white dwarfs that show magnetic and photometric variability, for which we have a firmly established phase relationship between photometry and circular polarisation. This star is erroneously missing in the list of such stars compiled by Bagnulo et al. (2024b).\",\n \"pages\": [\n 18\n ]\n },\n {\n \"title\": \"Appendix B.14: WD 0322-019 = EGGR 566 (variable)\",\n \"content\": \"The star was discussed by Bagnulo & Landstreet (2021). Two FORS2 measurements by Farihi et al. (2018) obtained during two consecutive nights ( ⟨ Bz ⟩ = -5 . 4 ± 3 . 0 kG and -16 . 5 ± 2 . 3 kG) suggest that the longitudinal field may change on a timescale of a few days.\",\n \"pages\": [\n 18\n ]\n },\n {\n \"title\": \"Appendix B.15: WD 0330-000 = HE 0330-0002 (only 2 meas., and contradicting data)\",\n \"content\": \"Schmidt et al. (2001) report two spectropolarimetric observations obtained one day apart, and concluded that no significant di ff erences had been detected in the overall spectrum. However, the mean circular polarisation averaged over the observed wavelength range reported in their Table 1 di ff ers by 0.5%, contra- dicting the assessment of non-variability. Because of this, this star will not be considered in this paper.\",\n \"pages\": [\n 18\n ]\n },\n {\n \"title\": \"Appendix B.16: WD 0410-114 = G 160-51 = NLTT 12758 (variable, P ≃ 0 . 3 h)\",\n \"content\": \"A magnetic field of about 1.7 MG was discovered by Kawka & Vennes (2012). Kawka et al. (2017) found that circular polarisation in the H α sigma components varies with P = 22 . 6 min. This is another example of a star for which the phase relationship between light curve and magnetic field are known (Kawka et al. 2017) and it is also missing from the list compiled by Bagnulo et al. (2024b).\",\n \"pages\": [\n 18\n ]\n },\n {\n \"title\": \"Appendix B.17: WD 0446-789 = WG 47 (var.:)\",\n \"content\": \"Bagnulo & Landstreet (2018) show that this is a weak-field star with ⟨ Bz ⟩ ≃ -5 kG; it probably has a dipolar field with axis nearly parallel to the rotation axis, because only very mild variations in ⟨ Bz ⟩ occur on a timescale of days.\",\n \"pages\": [\n 18\n ]\n },\n {\n \"title\": \"Appendix B.18: WD 0548-001 = EGGR 248 = G 99-37 (non-variable)\",\n \"content\": \"The magnetic field was discovered from continuum circular polarisation by Landstreet & Angel (1971). Observations of broadband CP on six nights showed no variations. During one hour, broad-band data were printed every 6 s. These data were Fourier analysed without finding any evidence of short-period variability. Angel & Landstreet (1974) obtained an MCSP circular polarisation spectrum in 1972, and the star was re-observed by Vornanen et al. (2010), who remarked that similarity with the previous spectra, reporting also \\\"no unambiguous variations\\\" between several polarisation spectra taken in 2003, 2005, 2008. Using FORS1 data obtained in 2005, they also reported no detection of linear polarisation. We retrieved from the ESO FORS1 and FORS2 archives all circular polarisation spectra obtained in 2005, 2008 and 2012. Figure B.4 shows all these spectra compared to the polarisation spectrum of Angel & Landstreet (1974) . We note that there are three spectra obtained in three consecutive nights in November 2012. The spectrum obtained during the last night (2012-1119) shows zero circular polarisation (see green solid line), but a Stokes I flux consistent with that of previous observations. We ascribe this to a non-detected instrument failure (perhaps the retarder waveplate did not move) rather than to stellar variability. Observations obtained by Berdyugin et al. (2023) and Berdyugin et al. (2024) show no variability in broadband polarimetry between two measurements obtained 8 months apart (see the top panel of Fig. B.1). This is a nearly unique DQp white dwarf which shows not only the Swan bands in the flux spectrum, but also the CH G band. The magnetic field was modelled by Angel & Landstreet (1974) who deduced ⟨ Bz ⟩ ≈ 3 . 6 MG from the strong polarisation signature of the CH G-band. Berdyugina et al. (2007) and Vornanen et al. (2010) modelled FORS1 spectra and found ⟨ Bz ⟩ ≃ 2 . 5 MG.\",\n \"pages\": [\n 19\n ]\n },\n {\n \"title\": \"Appendix B.19: WD 0553+053 = EGGR 290 = G 99-47 (non-variable)\",\n \"content\": \"The magnetic field of this star was discovered by Angel & Landstreet (1972), who also made 14 separate BBCP measurements during 10 months, without detecting any significant variation. Low resolution MCSP spectropolarimetry was obtained by Liebert et al. (1975), and 20 years later, a higher resolution polarised spectrum was obtained also by Putney & Jordan (1995). No significant variation has been detected on any timescale up to decades (for details, see Bagnulo & Landstreet 2021). Brinkworth et al. (2013) observed a light curve with 26.8 m period and semi-amplitude = 0.3 %, which suggest that polarimetric observations could have missed short-term variability. However, Angel & Landstreet (1972) had carried out Fourier-analysis of a polarimetric run, and found no significant variability for any period between 11 seconds and 2 h. They claimed that periodic variation with an amplitude of 0.17 % would certainly have been detected. Our conclusion is therefore that the star is non-variable.\",\n \"pages\": [\n 19\n ]\n },\n {\n \"title\": \"Appendix B.20: WD 0637+478 = GD 77 (variable, P ≃ 1 . 4 d)\",\n \"content\": \"Magnetic variability was reported by Schmidt & Smith (1995). Eight unpublished ISIS polarisation spectra and one ESPaDOnS spectrum show that ⟨ Bz ⟩ varies with P ≃ 1 . 362 d. The value of ⟨| B |⟩ ≈ 1 . 0 MG hardly changes with rotation, but ⟨ Bz ⟩ varies sinusoidally between about + 400 and -250 kG (LB25).\",\n \"pages\": [\n 19\n ]\n },\n {\n \"title\": \"Appendix B.21: WD 0654+059 = 2MASS J06572938+0550479 (non-variable)\",\n \"content\": \"Three BBCP observations obtained 8 months apart by Berdyugin et al. (2024) show no variability of the polarisation in the three B ' V ' R ' filters (see Fig. B.1). In all three filters, all three observations have V / I ≃ -0 . 3 %. We assume that the star is not variable.\",\n \"pages\": [\n 19\n ]\n },\n {\n \"title\": \"Appendix B.22: WD 0708-670 = SCR J0708-6706 (n.v.: only 2 meas.)\",\n \"content\": \"Two spectra published by Bagnulo & Landstreet (2020) show no variability over a 2 month interval. The strength and wavelength dependence of the circular polarisation spectrum suggests that the underlying field could be of the order of 60 to 200 MG. We tentatively assume that the star is magnetically non-variable.\",\n \"pages\": [\n 19\n ]\n },\n {\n \"title\": \"Appendix B.23: WD0745 + 115 = GALEX J074842.4+112502 (non-variable)\",\n \"content\": \"Strong broadband circular polarisation was detected by Berdyugin et al. (2024). The polarisation changes sign between the B ' and the other bands, and its absolute value is about 1% in all bands. Observed four times, the star does not show variability (see the top right panel of Fig. B.1).\",\n \"pages\": [\n 19\n ]\n },\n {\n \"title\": \"Appendix B.24: WD 0756+437 = EGGR 428 (variable, P phot ≃ 6 . 5 h )\",\n \"content\": \"The star was discovered to be magnetic and discussed in detail by Putney (1995). She estimated its magnetic field to be ≃ 200 MG. Recent BBCP observations at NOT (Berdyugin et al. 2024) showed that it is rapidly variable. The first two observations, about 3 h apart, report the largest and smallest polarisation seen in the full data set of five observations; this suggests a rotation period of the order of 6 h. In fact, the photometric study of Brinkworth et al. (2013) found a unique, very large amplitude ( ± 4 %) light variation with a period of P = 6 . 68 h. The similarity of this period with the period range deduced from polarisation measurements strongly confirms that 6.68 h is the rotation period of this white dwarf. A further remarkable fact is the combination of high field (200 MG), high mass (1 . 04 M ⊙ ) with advanced age (4.45 Gyr): this star seems to be a unique example of a very old, still strongly magnetic and rapidly rotating WD-WD merger.\",\n \"pages\": [\n 20\n ]\n },\n {\n \"title\": \"Appendix B.25: WD 0810-353 = UPM J0812-3529 (non-variable)\",\n \"content\": \"No variability detected among six polarised spectra obtained over a four year period, as described in a detailed study by Landstreet et al. (2023). According to their modelling, the star shows two regions of di ff erent field strength: one with magnetic field of predominantly 30 MG strength, outward, and one showing a field strength of 45 MG, inward.\",\n \"pages\": [\n 20\n ]\n },\n {\n \"title\": \"Appendix B.26: WD 0816-310 = SCR J0818-3110 (variable, P ≃ 10 d)\",\n \"content\": \"This is a DZ white dwarf with strong flux, spectrum and ⟨ Bz ⟩ variations observed in five FORS polarised spectra in 2023 obtained with grism 1200B. It is clear that the surface abundances vary over the surface, and appear to be locked to the surface magnetic field. The rotational period is of the order of 10 d, and ⟨| B |⟩ ≃ 100 kG (Bagnulo et al. 2024a).\",\n \"pages\": [\n 20\n ]\n },\n {\n \"title\": \"Appendix B.27: WD 0850+192 = LB 8915 (variable, P ∼ hours?)\",\n \"content\": \"This DBA white dwarf shows very weak, variable H lines, slightly variable He, and variable ⟨ Bz ⟩ (Wesemael et al. 2001). It is found that ⟨| B |⟩ ≃ 850 kG, and the rotation period is relatively short, probably some hours. We note that the correct identification of this star is LB 8915, and not LB 8827 as given in title of the paper by Wesemael et al. (2001). Appendix B.28: WD0907+213 = GALEX J091016.5+210555 (variable, P ≃ 10 h) Moss et al. (2024) discovered that this is a spectroscopically and magnetically variable DBA star with a rotation period of either 7.7 or 11.3 h (the ambiguity is due to aliasing). They have modelled the field and abundance geometry with a simple model like that used for Feige 7 = WD0041-102. The line splitting in the flux spectra suggests a field of ⟨| B |⟩ ≃ 0 . 5 MG.\",\n \"pages\": [\n 20\n ]\n },\n {\n \"title\": \"Appendix B.29: WD 0912+536 = EGGR 250 = G 195-19 (variable, P ≃ 1 . 3 d)\",\n \"content\": \"This is the second magnetic white dwarf discovered (Angel & Landstreet 1971a), and the first magnetic white dwarf to be discovered to be rotationally variable (Angel & Landstreet 1971b). An improved ephemeris was provided by Angel et al. (1972a). The star shows a large variation of its circular polarisation with a period of 1.33 d (Angel et al. 1972b). Hernandez et al. (2024) measured the period of the photometric variability from TESS data as 1 . 3304 ± 0 . 0054 d (note that high accuracy of period relies on two widely separated TESS observational data sets). Six MCSP V / I spectra roughly uniformly distributed in phase were obtained by Landstreet & Angel, (unpublished). Between 4000 and 5000 Å, V / I reverses sign during rotation; redwards of this, the strong variations retain one sign. Appendix B.30: WD 1008-242 = UCAC4 328-061594 (n.v.: only 2 meas.) Observed twice with FORS2 in spectropolarimetric mode by Bagnulo & Landstreet (2022), this white dwarf did not show any variation between two spectra obtained 40 d apart, nor within the same observing series (see Fig. B.5). Probably the field is of order 100 MG or more. We consider that it is probably not variable. The star therefore appears a young, strongly magnetic, ultra-massive, non-variable star.\",\n \"pages\": [\n 20\n ]\n },\n {\n \"title\": \"Appendix B.31: WD 1008+290 = LP 315-42 = LHS 2229 (non-variable)\",\n \"content\": \"This white dwarf is a cool peculiar DQ star discovered to be magnetic by Schmidt et al. (1999), who tentatively suggested that the field strength is of the order of 100 MG or more. Four observations obtained by Berdyugin et al. (2024) show little to no variability of broadband circular polarisation (see Fig. B.1). We assume that it is not magnetically variable.\",\n \"pages\": [\n 21\n ]\n },\n {\n \"title\": \"Appendix B.32: WD 1009-184 = WT 1759 (var.:)\",\n \"content\": \"The star was discovered to be a magnetic white dwarf by Bagnulo & Landstreet (2019a), who published one measurement obtained with FORS2 and one obtained with ISIS, showing that the star has a weak and variable field ( ⟨ Bz ⟩ ≃ 50 kG). Additional unpublished data suggest also weak variability.\",\n \"pages\": [\n 21\n ]\n },\n {\n \"title\": \"Appendix B.33: WD 1015+014 = PG 1015+014 (variable, P = 98 . 75 m)\",\n \"content\": \"This is a very strongly polarised white dwarf with V / I ≃ 1 . 5 %. The flux and polarisation spectra are strongly variable with P = 98 . 75 m and a polar field strength of about 120 MG (Angel 1978; Wickramasinghe & Cropper 1988). Schmidt & Norsworthy (1991) report that BBCP also varies with P = 98.75 min, approximately sinusoidally, with extrema of + 1 and -1 % polarisation. Brinkworth et al. (2013) found that the star's light curve has a period consistent, within uncertainties, with that obtained from polarimetry. Euchner et al. (2006) obtained a series of polarised spectra with FORS using the 300V grism. They describe strong I and V spectrum variations, and model the field structure using a multipole field expansion. This white dwarf displays spectral features originating from regions with typical field strengths between about 50 and 90 MG.\",\n \"pages\": [\n 21\n ]\n },\n {\n \"title\": \"Appendix B.34: WD 1031+234 = Ton 527 = PG 1031+234 (variable, P ≃ 3 . 5 h)\",\n \"content\": \"Schmidt et al. (1986) report strongly variable intensity, circular and linear polarisation spectra with a 3 h 24 min period, and proposed a simple magnetic model with field strength in the range of 200 - 500 MG. Further broadband polarisation observations through the rotation period were obtained by Piirola & Reiz (1992), who measured also a light variation in anti-phase with circular polarisation (that is, the star appears darker when the absolute value of the polarisation is maximum). Brinkworth et al. (2013) found P phot ≃ 3 . 5 h.\",\n \"pages\": [\n 21\n ]\n },\n {\n \"title\": \"Appendix B.35: WD 1036-204 = LP 790-29 (non-variable)\",\n \"content\": \"The star was discovered to be magnetic via spectropolarimetry by Liebert et al. (1978), and repeatedly studied (West 1989b; Schmidt et al. 1995, 1999; Beuermann & Reinsch 2002; Jordan & Friedrich 2002). Beuermann & Reinsch (2002) have monitored the star with EFOSC in spectropolarimetric mode to search for short-term variability, without finding any significant variations. Beuermann & Reinsch (2002), however, pointed out that Schmidt et al. (1995) measured a polarisation signal of ≃ -6 % around 6500 Å, a measurement at odds with broadband polarimetric measurements obtained in 1977, 1986, 1994 and 2000, which were all about -9 % (see their Table 1; we recall here that we use the opposite definition for the sign of circular polarisa- n). Schmidt et al. (1995) therefore suggested the possibility of a very long rotation period of the star. Jordan & Friedrich (2002) carried out a similar study, with similar results regarding short term variation. They also proposed a rotation period in the range of about 24 - 29 yr. We have reduced FORS1 and FORS2 archival polarisation spectra from 2003, 2011, 2012, and 2013, finding absolutely no hint of variability among them. Figure B.6 shows a comparison of archive observations obtained with grism 600B. The flux is not corrected for atmospheric and instrument transmission, and the discrepancies in the slope of the flux measured in 2003 can be explained by the use of a different CCD. A comparison between FORS spectra with those obtained by Schmidt et al. (1995) on May 7 and 8, 1994, (wavelength range 4160-7460 Å) does not show obvious long-term variability, except in the range 5700 -6200 Å. In that range, our data are instead consistent with most of the literature and point to a value of ≃ -9 %. In addition to the polarisation spectra, FORS1 and FORS2 archive contains another two broadband circular polarisation (BBCP) observations in the R filter: a FORS1 BBCP measurement obtained in April 2006, and one obtained with FORS2 in March 2024, using a similar (but not per-fec identical) R filter (program ID 112.25C9.001). We found both measurements consistent with a polarisation signal of about -9 . 4 %. These measurements rule out any significant change in the region around 6500 Å over an interval of 18 years. Finally, Berdyugin et al. (2024) have published the series of BBCP observations obtained in November 2022, November 2023 and February 2024, all consistent among themselves. In the R ' filter they report a polarisation signal of ≃ -9 . 1 %. For the reasons explained at the beginning of this section, it is not possible to accurately compare BBCP measurements obtained with di ff erent instruments, but it is clear that the only deviant point of a series of polarimetric observations obtained in the course of almost half a century is a small portion of a spectrum obtained in 1994. Remarkably, data obtained with the same instrument (FORS) over nearly two decades are fully consistent among themselves, and point strongly to a constant circular polarisation spectrum. Our conclusion is that the star is actually not variable. From the measured signal of circular polarisation of ≃ 10 % we estimate a longitudinal field of the order of 100 MG.\",\n \"pages\": [\n 21,\n 22\n ]\n },\n {\n \"title\": \"Appendix B.36: WD 1043-050 = HE 1043-0502 (n.v.: - only 2 meas.)\",\n \"content\": \"This DBA star was discovered magnetic by Schmidt et al. (2001), who proposed that the field is ≃ 800 MG (although its continuum is not highly polarised). Two observations were taken a few days apart, and Schmidt et al. (2001) state that they did not show variability; the reported wavelength integrated circular polarisation of about 1.5% is also essentially unchanged between the two spectra. So we consider it as candidate nonvariable magnetic white dwarf.\",\n \"pages\": [\n 22\n ]\n },\n {\n \"title\": \"Appendix B.37: WD 1045-091 = HE 1045-0908 (variable, P ≃ 3 h)\",\n \"content\": \"This white dwarf was discovered to be magnetic by Reimers et al. (1996) from a flux spectrum. Circular polarisation was confirmed, and shown to be variable by Schmidt et al. (2001). Euchner et al. (2005) obtained a series of I and V spectra and, assuming a rotational period of about 2.7 h, derived a detailed surface field model with a dominant field strength of 16 MG, but local field strength ranging between about 10 and 75 MG.\",\n \"pages\": [\n 22\n ]\n },\n {\n \"title\": \"Appendix B.38: WD 1105-340 = SCR J1107-3420A (non-variable)\",\n \"content\": \"Eleven ESPaDOnS spectra taken between 2018 and 2022, with ten of them during 1 week in 2019 (see Table A.1), show that the star has a weak, non-variable field with ⟨ Bz ⟩ ≈ -22 ± 4 . 5 ˙ kG and ⟨| B |⟩ ≈ 125 ± 5 kG (see Fig. B.7). We also have two FORS spectra, one taken with 1200B and one with 1200R, with lower resolving power but higher signal-to-noise ratio (S / N). Owing to their di ff erence resolution, these spectra were not used in this work. This star is perhaps the best studied non-variable weakfield magnetic white dwarf.\",\n \"pages\": [\n 22\n ]\n },\n {\n \"title\": \"Appendix B.39: WD 1105-048 = EGGR 76 (ultra-weak field, var.:)\",\n \"content\": \"The star was repeatedly observed, and a longitudinal field of the order of 1 kG was detected only in two measurements (Bagnulo & Landstreet 2018). The star seems to have a very weak and variable field, but additional measurements should be obtained to confirm the existence of a magnetic field.\",\n \"pages\": [\n 22\n ]\n },\n {\n \"title\": \"Appendix B.40: WD 1116-470 = SCR J1118-4721 (non-variable)\",\n \"content\": \"This white dwarf was observed twice with FORS2 by Bagnulo & Landstreet (2021), who flagged it as suspected magnetic star. Both observations show a similar signal of circular polarisation at -0 . 2 %, close to the FORS2 instrumental detection limit. A third observations was obtained in January 2023, and the V / I spectrum is again consistent with that measured previously. The star is definitely magnetic, and most likely not variable. This confirmation (see Table A.1) brings the number of magnetic white dwarfs in the 20 pc volume to 34.\",\n \"pages\": [\n 22\n ]\n },\n {\n \"title\": \"Appendix B.41: WD 1211-171 = HE 1211-1707 (variable, P ≃ 2 h)\",\n \"content\": \"A magnetic field was suspected in this white dwarf by Reimers et al. (1996), which was confirmed with polarimetry by Schmidt et al. (2001). Both papers show varying flux spectra. Schmidt et al. (2001) estimates P ≃ 100 -120 min and ⟨| B |⟩ ≃ 50 MG. Brinkworth et al. (2013) measured a photometric period of 1.79 h, consistent with the previous estimates from polarimetric data. The star is polarised at a level that varies between 0 and 3% during the rotation cycle. Modelling by Schmidt et al. (2001) strongly suggests a He dominated atmosphere with T e ff ≃ 12000 K, but Reimers et al. (1996), using IUE data, estimates 23000 K. Gentile Fusillo et al. (2021) gives 30000 K and 1 . 20 M ⊙ . We tentatively assume T e ff = 23000 K , M = 1 . 2 M ⊙ , and an He-rich atmosphere.\",\n \"pages\": [\n 22\n ]\n },\n {\n \"title\": \"Appendix B.42: WD 1217+475 = SDSS J121929.45+471522.8 (DAHe variable, P phot = 15 . 26 h)\",\n \"content\": \"DAHe with 18.5 MG field strength and a photometric period of about 15.25 h (Gänsicke et al. 2020). Spectroscopy reveals Zeeman components of the Balmer lines varying in strength but with constant splitting. Published data do not demonstrate that the star is magnetically variable but cannot rule out this possibility either. Therefore we have decided not to include this star in our sample.\",\n \"pages\": [\n 23\n ]\n },\n {\n \"title\": \"Appendix B.43: WD 1249-022 = GALEX J125230.9-023417 (DAHe variable, P phot = 0 . 09 h)\",\n \"content\": \"This is a DAHe white dwarf that shows variable H β and H α lines that appear sometimes in emission and sometimes in absorption. Field strength has been estimated 5 MG and rotational period = 0.09 h (Reding et al. 2020).\",\n \"pages\": [\n 23\n ]\n },\n {\n \"title\": \"Appendix B.44: WD 1312+098 = PG 1312+099 - (variable, P ≃ 5 . 4 h)\",\n \"content\": \"Variable continuum circular polarisation was detected in this hot DAH by Schmidt & Norsworthy (1991), who present over 100 measures, and find P = 5 . 43 h. Circular polarisation varies approximately between + 1% and -1%.\",\n \"pages\": [\n 23\n ]\n },\n {\n \"title\": \"Appendix B.45: WD 1315-781 = LAWD 45 (n.v.: - only 2 meas.)\",\n \"content\": \"No change between two ⟨| B |⟩ ≃ 5 . 5 MG and ⟨ Bz ⟩ ≃ 0 MG measurements from FORS 300V spectra taken five nights apart by Bagnulo & Landstreet (2020). We tentatively assume that it is magnetically non-variable.\",\n \"pages\": [\n 23\n ]\n },\n {\n \"title\": \"Appendix B.46: WD 1315+222 = LP 378-956 (non-variable)\",\n \"content\": \"Two observations by Berdyugin et al. (2023) and one by Berdyugin et al. (2024) show no obvious sign of variability (see Fig B.1).\",\n \"pages\": [\n 23\n ]\n },\n {\n \"title\": \"Appendix B.47: WD 1328+307 = G 165-7 (variable)\",\n \"content\": \"This star is found to host a magnetic field of ⟨| B |⟩ ≃ 650 kG based on line splitting observed in a good S / NSDSSspectrum (Dufour et al. 2006). These authors have also obtained low-resolution polarised spectra. They state that three 600 sec polarisation spectra were taken on 2005-12-30 at Steward Obs, all yielding essentially the same ⟨ Bz ⟩ ≈ 150 kG, but that the polarisation amplitude in similar Steward polarised spectra from 2006-05-03 (apparently not measured) is at least two times weaker than in 2005 spectrum. We conclude that the star is magnetically variable.\",\n \"pages\": [\n 23\n ]\n },\n {\n \"title\": \"Appendix B.48: WD 1346+121 = LP 498-66 (non-variable)\",\n \"content\": \"Observed three times by Berdyugin et al. (2023) and Berdyugin et al. (2024), we assume the star, which shows circular polarisation ranging between -1 and 0 % in the three DIPol-UF bands, is magnetically non-variable (see Fig. B.1). The field strength of this white dwarf is probably in the tens of MG.\",\n \"pages\": [\n 23\n ]\n },\n {\n \"title\": \"Appendix B.49: WD 1350-090 = PG 1350-090 = GJ 3814 (non-variable)\",\n \"content\": \"Discovered by Schmidt & Smith (1994), who measured ⟨ Bz ⟩ = 85 ± 9 kG. We have obtained four polarised spectra with ESPaDOnS that show ⟨| B |⟩ ≃ 450 -465 kG. There is no strong evidence of field variability, see also Schmidt & Smith (1994). Data will be published in a forthcoming paper (LB25).\",\n \"pages\": [\n 23\n ]\n },\n {\n \"title\": \"Appendix B.50: WD 1532+129 = G 137-24 (variable)\",\n \"content\": \"Originally classified as DZ white dwarf by Kawka et al. (2004), the star was discovered to be a magnetic white dwarf by Bagnulo &Landstreet (2019a), who published two FORS2 measurements and one ISIS measurement. The star is variable, with ⟨ Bz ⟩ values from FORS2 measurements of -21 ± 1 kG and -4 ± 1 kG, while ⟨| B |⟩ is not strong enough to split spectral lines, leading to ⟨| B |⟩ < ∼ 300 kG.\",\n \"pages\": [\n 23\n ]\n },\n {\n \"title\": \"Appendix B.51: WD 1556+044 = PM J15589+0417 (non-variable)\",\n \"content\": \"Discovered to be magnetic by Berdyugin et al. (2022), the star was re-observed three more times by Berdyugin et al. (2024). The observed circular polarisation, which is detected at the 10 σ level, ranges between -0 . 3 and + 0 . 4% in the three filter bands of DIPol-UF, so the order of magnitude of the field strength is probably some tens of MG. The polarisation does not show any variability (see Fig. B.1).\",\n \"pages\": [\n 23\n ]\n },\n {\n \"title\": \"Appendix B.52: WD 1615+542 = GALEX J161634.4+541011 (DAHe magnetically variable, P phot = 1 . 59 h)\",\n \"content\": \"DAHe with a variable magnetic field (from 3.5 to 6.5 MG) and a rotation period P = 95 . 29 m estimate via photometry (Manser et al. 2023).\",\n \"pages\": [\n 24\n ]\n },\n {\n \"title\": \"Appendix B.53: WD 1619+046 = GALEX J162157.7+043219 (variable, P ≃ 40 m?)\",\n \"content\": \"This white dwarf was discovered to be magnetic and rapidly variable in this work (see Sect. 2.1 and Fig. B.9). From the H β regions, it is clear that the star is a DAH with a rather non-uniform field of ⟨| B |⟩ ≃ 15 MG. We observe clear changes in intensity between spectra obtained a few minutes apart, and also polarisation (which is measured by the combination of two spectra) shows some variation, but without sign reversal. The close similarity between the first pair and the last pair of intensity spectra suggest that the star's rotation period is approximately 40 min.\",\n \"pages\": [\n 24\n ]\n },\n {\n \"title\": \"Appendix B.54: WD 1639+537 = GD356 (non-variable, but with a light curve and P phot = 1.93 h)\",\n \"content\": \"This star is the prototype of DAHe white dwarfs. It shows a light curve with low amplitude and P phot ≃ 1 . 93 h (Walters et al. 2021). A time series of circular polarisation spectra obtained during an entire rotational cycle shows now variability while subtle sinusoidal variability is seen in the position of the σ components of H β and H α lines (Walters et al. 2021). Walters et al. (2021) compared spectropolarimetry obtained in 2019, with that published by Ferrario et al. (1997), finding no changes. Its field strength is of the order of 10 MG (Walters et al. 2021). We classify the star as non-variable, but showing very small changes of its apparent magnetic field with a ∼ 2 h rotation period.\",\n \"pages\": [\n 24\n ]\n },\n {\n \"title\": \"Appendix B.55: WD 1658+440 = PG 1658+441 (n.v.: - only 2 meas., one very old)\",\n \"content\": \"The star was discovered to be magnetic by Liebert et al. (1983), who measured ⟨ Bz ⟩ ≃ 0 . 7 MG, and ⟨| B |⟩ ≃ 2 . 3 MG from spectropolarimetry of H α , H β and H γ (see their Fig. 5). Their observations were obtained on July 20 1980, with a spectral resolution of 10 Å. The same authors measured a signal of broadband circular polarisation = -0 . 016 ± 0 . 033% and -0 . 044 ± 0 . 019 % in the range 3300-8600 Å and concluded that a 3 σ upper limit of 0.10% semi-amplitude could be set for any presumed sinusoidal variation with a period between 0.5 and 5 h. For possible periods between 4 minutes and 0.5 hours, a less stringent limit of 0.30% polarisation semi-amplitude may be deduced. A comparison with our ISIS spectra obtained on April 21, 2019 (Sect. 2.1) is shown in the top panels of Fig. B.10. We observe a wavelength shift possibly due to bad calibration; most remarkable is the different strength of H α , but perhaps this is instrumental, because the ISIS spectrum was obtained with a setting that puts H α at the edge of the CCD, and no good flat-fielding correction could be applied. Although our comparison between 1980 and 2019 data is not conclusive, we certainly do not see any convincing change of the spectral features that demonstrate long-term variability. A comparison between the four pairs of Stokes V / I profiles obtained with ISIS in 2019 show no di ff erences within the error bars (see the bottom panels of Fig. B.10); therefore we rule out variability on a timescale of 10-15 minutes. This star seems to be a young, strongly magnetic, massive, tentatively classified as non-variable star. Photometric studies lead to inconsistent results: Brinkworth et al. (2013) found that the star is photometrically variable with a period between 6 h and 4 d, while, using TESS photometry, Oliveira da Rosa et al. (2024) derived a period shorter than 1 h. Hernandez et al. (2024), instead, did not detect periodicity in TESS data.\",\n \"pages\": [\n 25\n ]\n },\n {\n \"title\": \"Appendix B.56: WD 1703-266 = UCAC4 317-104829 (variable)\",\n \"content\": \"This DA white dwarf was discovered to have a magnetic field by Bagnulo & Landstreet (2020), with ⟨| B |⟩ ≈ 8 MG. They also found that two FORS2 polarised spectra four days apart are significantly di ff erent and yield di ff erent values of ⟨ Bz ⟩ , so the star is variable on a timescale of some days or less.\",\n \"pages\": [\n 25\n ]\n },\n {\n \"title\": \"Appendix B.57: WD 1712-590 = Gaia DR3 5915797694789556096 (variable)\",\n \"content\": \"Discovered to be magnetic by O'Brien et al. (2023). We observed this white dwarf in polarimetric mode with grism 1200B three times, twice during the same night. Intensity spectra show very similar splitting of Balmer lines, but the flux distribution within the line cores changes quite significantly, rather similarly to WD2359-434 (Landstreet et al. 2017). The deduced field strength is ⟨| B |⟩ < ∼ 0 . 8 MG. The Stokes V spectra also show that the star is strongly variable within 1-2 h (see Fig. B.11), and that the longitudinal field reverses its polarity during rotation.\",\n \"pages\": [\n 25\n ]\n },\n {\n \"title\": \"Appendix B.58: WD 1743-521 = L 270-31 = BPM25114 (variable, P = 2 . 84 d)\",\n \"content\": \"Wickramasinghe & Bessell (1976) reported a magnetic field of about 35 MG in this southern DA star from close examination of the peculiar flux spectrum. Wegner (1977) found vari- ability of light and of the flux and polarisation spectrum with P ≈ 2 . 84 d, and modelled the spectrum variations, finding results consistent with a magnetic dipole field. Field modelling was carried out in more detail by Martin & Wickramasinghe (1978), who deduced a dipole field strength of 36 MG, corresponding to ⟨| B |⟩ ≃ 18 MG.\",\n \"pages\": [\n 25\n ]\n },\n {\n \"title\": \"Appendix B.59: WD 1748+708 = EGGR 372 = G 240-72 (s.v.:)\",\n \"content\": \"Intrinsic broadband circular and linear polarisation of this magnetic white dwarf were discovered by Angel et al. (1974), who reported no variations in repeated observations of broad-band circular polarisation over a month, and linear polarisation over two nights. The star was later observed in broadband circular and linear polarimetry by West (1989a), who generally found similar polarisation levels and position angles to earlier work, and in broadband linear polarimetry by Berdyugin & Piirola (1999). Berdyugin & Piirola (1999) found evidence of clear change of the position angle of linear polarisation (mainly rotation of the polarisation angle) on a timescale of 20 years. We have observed the star with ISIS both in circular (twice) and in linear polarisation, and compared these observations with MCSP lowresolution spectra of I , V and P obtained by Angel & Landstreet on September 6 and 7, 1974, that were never published until now. Our ISIS spectra are consistent among themselves. When compared with spectropolarimetry obtained in 1974 and the 1990s, our new ISIS linear polarisation data find small changes in the V and P polarisation amplitude in the blue and visual, whereas the overall position angle behaviour is almost identical to that observed observed in 1974, thus we cannot confirm the changes in the polarisation position angle seen in 1997 by Berdyugin & Piirola (1999). On the other side, Antonyuk et al. (2016) found that the star is photometrically variable with a period between 5 hours and two days. Brinkworth et al. (2013) detected no short period variability, but claim that photometry varied over a ten month interval. We conclude that the star show some signs of subtle long term variability that should be further investigated. Appendix B.60: WD1750 -311 = UCAC4 295-140552 = [MTR2015] OW J175358.85-310728.9 (n.v.: - but P phot ≃ 0 . 5 h) Amagnetic field of ⟨| B |⟩ ≈ 2 . 1 MG was identified by Macfarlane et al. (2017) in this hot DQ white dwarf, which has strong lines of neutral C as well as fairly strong Balmer lines. They observed clear light variability of the star with an amplitude of about ± 2 % and a period of 35 min. These authors discuss the origin of the light variations: they argue that the variability is not due to pulsation, as no subsidiary frequencies are observed, and conclude that the variation could be due to rotation. They do not seem to have considered testing this with the spectra that they have collected of the object by looking for spectral variations, although they do test the spectra for radial velocity variations, and find none. Our four FORS2 300V spectra, taken with 13 m spacing through the light variation cycle, show virtually no changes in either flux or polarisation except for H β , which has a strongly variable I profile depth but almost constant V / I ( λ ) (see Fig. B.13). Possibly this is due to a magnetic field distribution that is nearly axisymmetric about the rotation axis but a distribution of H that varies strongly around that axis. This might be related to the light variability that has been detected. We have decided to classify the star as candidate magnetically non-variable.\",\n \"pages\": [\n 25,\n 26\n ]\n },\n {\n \"title\": \"Appendix B.61: WD1754 -550 = GALEX J175845.9-550117 (variable)\",\n \"content\": \"This star was discovered to be magnetic in this work (Sect. 2.1). There is a strong signal of circular polarisation that seems variable on a short timescale between 0 and 4%, suggesting a rotation period of the order of 15 min: see Fig B.14. The inferred field strength is presumably of the order of 100 MG or more. The intensity spectrum appears featureless, so the T e ff ≃ 35 000 K star could be defined a hot DC.\",\n \"pages\": [\n 26\n ]\n },\n {\n \"title\": \"Appendix B.62: WD 1814+248 = G 183-35 (variable)\",\n \"content\": \"Putney (1997) showed that this white dwarf, formerly classified as a DC, is in fact a cool DAH that shows weak H α and H β , and that both lines are split by a field of about 6.8 MG, roughly the same in two observations. Kilic et al. (2019) observed that the white dwarf shows spectral variations. A series of spectra taken over several hours show that the separation of the σ components of H α from the central π component changes rather abruptly between about 90 Å and about 120 Å, equivalent to fairly sudden jumps between ⟨| B |⟩ values of about 4.6 and 6 MG, repeating with a period of about 4 h. The authors suggest that this unusual form of variability may be due to a patchy distribution of H over an He-rich envelope. A plausible model that might explain the observations could be a dipole oblique to the rotation axis, decentred in the direction of one pole so that the polar strengths at the two magnetic poles are unequal, with H-rich patches around both poles, but little or no H in a belt around the magnetic equator.\",\n \"pages\": [\n 26,\n 27\n ]\n },\n {\n \"title\": \"Appendix B.63: WD 1829+547 = G 227-35 (non-variable)\",\n \"content\": \"This white dwarf was discovered to be strongly magnetic by Angel et al. (1975) using both broadband polarimetry and lowresolution spectropolarimetry (on September 9, 1974). Limited tests of broadband variability were negative (Angel et al. 1975, 1981). It was observed again in spectropolarimetric mode by Cohen et al. (1993) and by Putney & Jordan (1995) who estimated a dipolar field strength of 170-180 MG. We observed the same star with ISIS twice in circular polarisation and once in linear polarisation. There are strong circular polarisation features at 6960, 7450 and 7930 Å. Figure B.15 shows a comparison between all these spectra. We do not see any obvious sign of variation.\",\n \"pages\": [\n 27\n ]\n },\n {\n \"title\": \"Appendix B.64: WD 1900+705 = LAWD 73 = Grw + 70 · 8247 (s.v.:)\",\n \"content\": \"The star has a magnetic field of the order of 180 MG (Jordan 2003), and shows hints of small variability. We refer to Bagnulo & Landstreet (2019b), who present a review and analysis of its polarimetric characteristics, and in particular, a comparison with observations obtained 50 years apart show some discrepancies in both linear and circular polarisation. More recent broadband circular polarisation obtained at NOT (Berdyugin et al. 2022, 2023) show no variability on a timescale of 2 years (see Fig. B.1).\",\n \"pages\": [\n 27\n ]\n },\n {\n \"title\": \"Appendix B.65: WD 1953-011 = GJ 772 (variable, P ≃ 1 . 5 d)\",\n \"content\": \"Maxted et al. (2000) modelled a series of Stokes I spectra in terms of a global dipolar field with polar field strength of order 100 kG, together with a spot with a field of order 500 kG. This basic modelling was confirmed by the analysis of a series of polarised spectra obtained with FORS1 and on the Russian 6 m telescope, which also revealed a rotation period of 1.448 d (Valyavin et al. 2008).\",\n \"pages\": [\n 27\n ]\n },\n {\n \"title\": \"Appendix B.66: WD 2010+310 = GD 229 (s.v.:)\",\n \"content\": \"This star was the fifth circularly polarised white dwarf to be discovered. Swedlund et al. (1974) reported a series of measurements indicating the presence of elevated levels of both circular polarisation and linear polarisation, and strongly suggesting polarisation variability. The claim of variability was then questioned by Kemp et al. (1974), who argued that the initial measurements had been badly contaminated by polarised foreground zodiacal light contamination. Linear and / or circular polarisation of GD 229 has been measured by Landstreet & Angel (1974), Efimov (1981), Angel et al. (1981), West (1989a), Berdyugin (1995), and Berdyugin & Piirola (1999). Angel & Landstreet (1974) obtained three low-resolution I and V spectra of the star on the nights of 1973 Nov 6-8 using the MCSP. Synthetic broad-band polarimetry derived from these low-resolution spectra revealed no variations over three nights. West (1989b) observed linear polarisation in GD 229 again in 1986 using broadband filters, but found no strong evidence of variation, not even of the position angle of linear polarisation. Observations of GD 229 using broadband polarimetry was also carried out for linear polarisation by Berdyugin (1995) and for both circular and linear polarisation, with higher precision, by Berdyugin & Piirola (1999), who compared their results to earlier work. Their results appear to show some quite significant changes in both circular and linear polarimetry compared to earlier observations by other groups. A major di ffi culty of comparing the various measurements is that each group has been made with di ff erent instrumentation. The consequences of using di ff erent filter passbands, di ff erent detector sensitivities as functions of wavelength, and even di ff erent (and possibly incorrect) calibration of instrumental polarimetric e ffi ciency, make it very hard to compare these data. However, Berdyugin & Piirola (1999) do o ff er strong evidence for significant rotation of the angle of linear polarisation, by about 30 · over 20 years, which is di ffi cult to explain by instrumental e ff ects. This is probably the most robust e ff ect that emerges from comparison of the many kinds of polarimetric observations Linear and circular polarisation spectra of GD 229 were obtained in 2018 and 2019 using the ISIS spectropolarimeter on the William Herschel Telescope (Sect. 2.1). These new data are shown in Fig. B.16, and compared to the circular polarised spectra by Angel & Landstreet (1974), and to previously unpublished linear spectropolarimetry of Angel and Landstreet (see Table A.1). Remarkably, the angle of linear polarisation appears to have returned to its value during the 1970s. Other di ff erences compared to the spectra of Angel and Landstreet are observed, but some of these are undoubtedly due to greatly di ff erent resolving power, and some may be due to uncertainties in the calibration of the 1970s data, particularly in the UV. It is thus di ffi cult to establish clearly how much variability has occurred in GD 229. There is evidence for variations, but these appear to have relatively small amplitude compared to the overall scale of polarisation, except possibly for position angle rotation of linear polarisation (which however has not been confirmed by our most recent linear spectropolarimetry). Furthermore, because of very limited sampling with a variety of instruments, it is not possible to establish clearly any particular timescale for variations. We note that the spectrum of GD 229 has been modelled as due to He in a field of hundreds of MG by Jordan et al. (1998, 2001) and Jordan (2003).\",\n \"pages\": [\n 28\n ]\n },\n {\n \"title\": \"Appendix B.67: WD 2047+372 = EGGR 261 (variable, P ≃ 6 h)\",\n \"content\": \"Originally, the star was observed polarimetrically by Schmidt & Smith (1995), who did not detect its weak and sign reversing field (their measurements were ⟨ Bz ⟩ = -42 ± 59 kG and -2 . 5 ± 4 . 6 kG). This star was discovered to be magnetic by Landstreet et al. (2016), and monitored and modelled by Landstreet et al. (2017). It is currently the weakest white dwarf field ( ⟨| B |⟩ ≈ 60 kG) that has been modelled in detail, mainly on the basis of a series of 18 ESPaDOnS spectra. The rotation period, determined from the variation of ⟨ Bz ⟩ , is 0.243 d. The observed variations of ⟨ Bz ⟩ and ⟨| B |⟩ are well modelled using a simple dipole model. No rotational variation is detected in TESS photometry (Hernandez et al. 2024).\",\n \"pages\": [\n 28\n ]\n },\n {\n \"title\": \"Appendix B.68: WD 2049-222 = LP 872-48 (var.:)\",\n \"content\": \"Discovered to be magnetic by Berdyugin et al. (2022) with BBCP measurements. The star has V / I ≃ + 0 . 1 %, and is one of the weakest polarisation levels securely detected. The inferred ⟨ Bz ⟩ field strength is only of the order of a few MG. Broad-band measurements were repeated in July 2022 (Berdyugin et al. 2024) to check for variability, which was not detected (see Fig. B.1). We also observed the white dwarf three times with FORS2 with grism 300V (Sect. 2.1). The data are barely above the threshold of instrumental polarisation ( ≃ 0 . 1%; see Siebenmorgen et al. 2014), and therefore it is hard to establish whether the hints of variability seen in the spectra are real or not (see Fig. B.17). However, star WD 1116-440 shows a similar level of polarisation, and constant over about 4 year (see Fig. B.8), suggesting that the tiny variability observed in WD 2049-222 may be real and not an instrumental artefact.\",\n \"pages\": [\n 28\n ]\n },\n {\n \"title\": \"Appendix B.69: WD 2049-253 = UCAC4 325-215293 (non-variable)\",\n \"content\": \"Discovered to be magnetic by Bagnulo & Landstreet (2020), who observed continuum circular polarisation of order 0.4% and deduced a field strength of order 20 MG. This white dwarf was re-observed in broadband circular polarisation once by Berdyugin et al. (2022) and two more times by Berdyugin et al. (2024), but shows no sign of variability (see Fig. B.1).\",\n \"pages\": [\n 28\n ]\n },\n {\n \"title\": \"Appendix B.70: WD 2051-208 = BPS CS 22880-0134 (variable, P ≃ 1 . 5 h)\",\n \"content\": \"The magnetic field of this white dwarf was discovered from the shape of the H α line by Koester et al. (2009). We have two series of five polarised spectra each, one using FORS grism 1200B and one with grism 1200R (LB25). These unpublished data clearly reveal very rapid rotation of this massive magnetic white dwarf. The stellar rotation period is 0.0594 d = 1.425 h. The value of ⟨ Bz ⟩ ranges approximately sinusoidally between about + 50 kG and -30 kG, while the corresponding values of ⟨| B |⟩ increase from about 200 kG to nearly 300 kG, in good agreement with the two values of <| B |> (220 and 290 kG) obtained by Koester et al. (2009) from UVES SPY spectra.\",\n \"pages\": [\n 29\n ]\n },\n {\n \"title\": \"Appendix B.71: WD 2105-820 = LAWD 83 (variable)\",\n \"content\": \"This star was suspected to have a weak magnetic field by Koester et al. (1998), who observed that the core of H α was abnormally broad, but they could not decide whether this was due to rapid rotation of v sin i ≈ 65 km / s or to a magnetic field of about 43 kG. Five FORS1 polarised spectra of the star by Landstreet et al. (2012) detected an apparently nearly constant magnetic field of ⟨ Bz ⟩ ≈ 10 kG. Later, FORS2 polarised spectra by Bagnulo & Landstreet (2018) and Farihi et al. (2018) reveal that ⟨ Bz ⟩ sometimes decreases to ⟨ Bz ⟩ ≈ 4 kG, so the star is apparently variable (see Bagnulo & Landstreet 2021).\",\n \"pages\": [\n 29\n ]\n },\n {\n \"title\": \"Appendix B.72: WD 2138-332 = L 570-26 (variable, P ≃ 6 . 19 h)\",\n \"content\": \"The DZA star was discovered to be magnetic, with a variable ⟨ Bz ⟩ of the order of 10 kG (Bagnulo & Landstreet 2019a) and a rotational period of P = 6 . 19 h (Hernandez et al. 2024; Farihi et al. 2024). The same period was found from the analysis of the equivalent width and ⟨ Bz ⟩ curves by Bagnulo et al. (2024b), who proposed a magnetic model with a dipolar field with ⟨| B |⟩ ≃ 50 kG. The star shows photometric and ⟨ Bz ⟩ curves with light minimum corresponding to ⟨ Bz ⟩ maximum.\",\n \"pages\": [\n 29\n ]\n },\n {\n \"title\": \"Appendix B.73: WD 2150+591 = UCAC4 747-070768 (variable, P ≃ 2 . 4 d)\",\n \"content\": \"The star was discovered to be magnetic by Landstreet & Bagnulo (2019), who reported two ISIS measurements that showed clearly that the field is variable with a period of hours or days. We subsequently monitored the star with one ESPaDOnS observation and several more ISIS spectra, and confirmed variability. These observations and a model of the star's magnetic field will be presented in a forthcoming paper (LB25).\",\n \"pages\": [\n 29\n ]\n },\n {\n \"title\": \"Appendix B.74: WD 2211+372 = LP 287-35 (non-variable)\",\n \"content\": \"This DC white dwarf was discovered to be magnetic by Berdyugin et al. (2023), who found circular polarisation in excess of 1% in the blue. It was re-observed by Berdyugin et al. (2024). It does not seem to be variable (see Fig. B.1). The field strength ⟨| B |⟩ is probably of the order of 60 MG or more.\",\n \"pages\": [\n 29\n ]\n },\n {\n \"title\": \"Appendix B.75: WD 2316+123 = KUV 23162+1220 (variable, P ≃ 18 d)\",\n \"content\": \"Discovered magnetic by Sion et al. (1984). Schmidt & Norsworthy (1991) reported BBCP of amplitude up to nearly 1% that varies sinusoidally with P = 17.86 d. Both the flux spectrum and the linear and circular polarisation spectra have been modelled repeatedly (Liebert et al. 1985; Achilleos & Wickramasinghe 1989; Friedrich et al. 1993; Putney & Jordan 1995); all agree that the global field strength is of the order of 30 MG. This star has the longest rotational period firmly established for a white dwarf.\",\n \"pages\": [\n 29\n ]\n },\n {\n \"title\": \"Appendix B.76: WD2359 -434 = LAWD 96 (variable, P ≃ 2 . 7 h)\",\n \"content\": \"WD2359 -434 was suggested to be a magnetic star by Koester et al. (1998) on the basis of the peculiar profile of the H α core, and ⟨ Bz ⟩ was found to be non-zero by Aznar Cuadrado et al. (2004). A series of polarised ESPaDoNS spectra revealed a rotation period of 0.1123 d (Landstreet et al. 2017). The star is also a photometric variable with the same period. The field structure has been modelled approximately, and found to be distinctly more complex than a simple dipole, with ⟨| B |⟩ varying approximately between 50 and 100 kG. This white dwarf o ff ers one of the clearest examples known of a field structure that is substantially more complex than a simple co-linear multipole expansion (Landstreet et al. 2017).\",\n \"pages\": [\n 29\n ]\n }\n]"}}},{"rowIdx":4974,"cells":{"bibcode":{"kind":"string","value":"2024arXiv241114911B"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/2411.14911.pdf"},"content":{"kind":"string","value":"\nRadii, masses, and transit-timing variations of the three-planet system orbiting the naked-eye star TOI-396 ⋆\nA. Bonfanti 1 , I. Amateis 2 , 1 , 3 , D. Gandolfi 2 , L. Borsato 4 , J. A. Egger 5 , P. E. Cubillos 1 , 6 , D. Armstrong 7 , 8 , I. C. Leão 9 , M. Fridlund 10 , 11 , B. L. Canto Martins 9 , 12 , S. G. Sousa 13 , J. R. De Medeiros 9 , L. Fossati 1 , V. Adibekyan 13 , A. Collier Cameron 14 , S. Grziwa 15 , K. W. F. Lam 16 , E. Go ff o 17 , L. D. Nielsen 18 , F. Rodler 19 , J. Alarcon 20 , J. Lillo-Box 21 , W. D. Cochran 22 , 23 , R. Luque 24 , S. Redfield 25 , N. C. Santos 13 , 26 , S. C. C. Barros 13 , 26 , D. Bayliss 7 , 8 , X. Dumusque 27 , M. A. F. Keniger 7 , 8 , J. Livingston 28 , 29 , 30 , F. Murgas 31 , 32 , G. Nowak 33 , A. Osborn 34 , H. P. Osborn 5 , 35 , E. Pallé 31 , 32 , C. M. Persson 11 , L. M. Serrano 2 , P. A. Strøm 7 , 8 , S. Udry 27 , and P. J. Wheatley 7 , 8\n(A ffi liations can be found after the references)\nABSTRACT\nContext. TOI-396 is an F6 V bright naked-eye star ( V ≈ 6.4) orbited by three small ( Rp ≈ 2 R ⊕ ) transiting planets discovered thanks to space-based photometry from two TESS sectors. The orbital periods of the two innermost planets, namely TOI-396 b and c, are close to the 5:3 commensurability ( Pb ∼ 3.6 d and Pc ∼ 6.0 d), suggesting that the planets might be trapped in a mean motion resonance (MMR).\nAims. To measure the masses of the three planets, refine their radii, and investigate whether planets b and c are in MMR, we carried out HARPS radial velocity (RV) observations of TOI-396 and retrieved archival high-precision transit photometry from four TESS sectors.\nMethods. We extracted the RVs via a skew-normal fit onto the HARPS cross-correlation functions and performed a Markov chain Monte Carlo joint analysis of the Doppler measurements and transit photometry while employing the breakpoint method to remove stellar activity from the RV time series. We also performed a transit timing variation (TTV) dynamical analysis of the system and simulated the temporal evolution of the TTV amplitudes of the three planets following an N-body numerical integration.\nResults. Our analysis confirms that the three planets have similar sizes ( Rb = 2 . 004 + 0 . 045 -0 . 047 R ⊕ ; Rc = 1 . 979 + 0 . 054 -0 . 051 R ⊕ ; Rd = 2 . 001 + 0 . 063 -0 . 064 R ⊕ ) and is thus in agreement with previous findings. However, our measurements are ∼ 1.4 times more precise thanks to the use of two additional TESS sectors. For the first time, we have determined the RV masses for TOI-396 b and d, finding them to be Mb = 3 . 55 + 0 . 94 -0 . 96 M ⊕ and Md = 7 . 1 ± 1 . 6 M ⊕ , which implies bulk densities of ρ b = 2 . 44 + 0 . 69 -0 . 68 g cm -3 and ρ d = 4 . 9 + 1 . 2 -1 . 1 g cm -3 , respectively. Our results suggest a quite unusual system architecture, with the outermost planet being the densest. Based on a frequency analysis of the HARPS activity indicators and TESS light curves, we find the rotation period of the star to be P rot ,⋆ = 6 . 7 ± 1 . 3 d, in agreement with the value predicted from log R ' HK -based empirical relations. The Doppler reflex motion induced by TOI-396 c remains undetected in our RV time series, likely due to the proximity of the planet's orbital period to the star's rotation period. We also discovered that TOI-396 b and c display significant TTVs. While the TTV dynamical analysis returns a formally precise mass for TOI-396 c of Mc , dyn = 2 . 24 + 0 . 13 -0 . 67 M ⊕ , the result might not be accurate, owing to the poor sampling of the TTV phase. We also conclude that TOI-396 b and c are close to but out of the 5:3 MMR.\nConclusions. A TTV dynamical analysis of additional transit photometry evenly covering the TTV phase and super-period is likely the most e ff ective approach for precisely and accurately determining the mass of TOI-396 c. Our numerical simulation suggests TTV semi-amplitudes of up to 5 hours over a temporal baseline of ∼ 5.2 years, which should be duly taken into account when scheduling future observations of TOI-396.\nKey words. planets and satellites: fundamental parameters - stars: fundamental parameters - techniques: photometric - techniques: radial velocities\n1. Introduction\nMulti-planet systems enable us to place significantly stronger constraints on formation and evolution mechanisms compared to single-planet systems (e.g. Lissauer et al. 2011; Fabrycky et al. 2014; Winn & Fabrycky 2015; Mishra et al. 2023). As a matter of fact, the temporal evolution of the gas content in the proto-planetary disc influences planet migration (e.g. Lin & Papaloizou 1979; Goldreich & Tremaine 1980; Tanaka et al. 2002; D'Angelo & Lubow 2008; Alexander & Armitage 2009), which shapes the orbital architecture of a planetary system. The latter is further expected to correlate with the planet composition (e.g.\nThiabaud et al. 2014, 2015; Walsh et al. 2015; Bergner et al. 2020; Li et al. 2020) that can be inferred once the physical parameters of the planets are known (e.g. Dorn et al. 2017; Otegi et al. 2020b; Leleu et al. 2021; Haldemann et al. 2024).\nIf planets are found in mean motion resonance (MMR; e.g. Lee & Peale 2002; Correia et al. 2018), systems can also shed light on the migration mechanisms during formation, as well as on the impact of tidal e ff ects occurring later on (e.g. Delisle et al. 2012; Izidoro et al. 2017). In addition, planets in, or close to, MMR likely exhibit transit timing variations (TTVs; see e.g. Agol et al. 2005; Agol & Fabrycky 2018; Leleu et al. 2021) that enable one to infer planetary masses without necessarily relying on radial velocity (RV) measurements, which are not always possible (e.g. Hatzes 2016).\nMulti-planet systems also give insights into correlations between physical and orbital parameters of exoplanets. For example, Weiss et al. (2018) noticed that planets in adjacent orbits usually show similar radii, hence the 'peas in a pod' label to describe this scenario. Weiss et al. (2018) further noticed that the outermost planet is the largest in the majority of cases, which agrees with the observed bulk planet density ( ρ p ) trend highlighted by Mishra et al. (2023), where ρ p decreases with the distance from the stellar host as outer planets are expected to be richer in volatiles (thus bigger and less dense).\nThe object TOI-396 represents an interesting laboratory to test these theories as it is the brightest star known so far to host three transiting planets (Vanderburg et al. 2019) after ν 2 Lupi (Delrez et al. 2021). After analysing two sectors from the Transiting Exoplanet Survey Satellite ( TESS ; Ricker et al. 2015), Vanderburg et al. (2019) found that the three planets have radii of ∼ 2 R ⊕ and orbital periods of ∼ 3.6, ∼ 6.0, and ∼ 11.2 d, with TOI-396 c and b showing a period commensurability close to the 5:3 ratio. Following the notation introduced in Mishra et al. (2023), the three planets are 'similar' in terms of radii, and one may wonder whether this architecture class is kept also in the mass-period diagram. Mishra et al. (2023) found a positive and strong correlation of the coe ffi cients of similarity between radii and masses, though exceptions are possible (e.g. Weiss & Marcy 2014; Otegi et al. 2020a, 2022).\nIn this work, we complement the photometric analysis of new TESS light curves (LCs) with RV observations taken with the High Accuracy Radial Velocity Planet Searcher ( HARPS ; Mayor et al. 2003) spectrograph to refine the planet radii and constrain for the first time the planetary masses. Considering the possible 5:3 MMR between TOI-396 c and b, we also simulate the evolution in time of the TTV amplitudes. This paper is organised as follows: Section 2 presents the stellar properties, and Sect. 3 describes the photometric and RV data. After outlining the method to jointly analyse the TESS LCs and the HARPS RV time series in Sect. 4, we present the corresponding results in Sect. 5. We attempt to dynamically model TTV and RV data simultaneously and track the temporal evolution of the TTV signals in Sect. 6, and we study the planets' internal structure in Sect. 7 and explore the prospects for characterising the system with the James Webb Space Telescope (JWST; Gardner et al. 2006) in Sect. 8. Finally Sect. 9 gathers the conclusions.\n2. Host star characterisation\nTOI-396 is an F6 V (Gray et al. 2006) bright naked-eye star with an apparent visual magnitude of V ≈ 6.4 (Perryman et al. 1997). It is located ∼ 31.7 pc away and is visible in the constellation of Fornax in the southern hemisphere. It is member of a visual binary system and its companion HR 858 B is a faint M-dwarf ( G ∼ 16 mag), about 8.4 '' away from the main component.\nWe co-added 78 HARPS spectra (see Sect. 3.2 for further details) and then modelled it with Spectroscopy Made Easy 1 ( SME ; Piskunov & Valenti 2017) version 5.2.2. SME computes synthetic spectra from a grid of well established stellar atmosphere models and adjusts a chosen free parameter based on comparison with the observed spectrum. Here we used the stellar atmosphere grid A tlas 12 (Kurucz 2013) together with atomic line lists from the V ald database (Piskunov et al. 1995) in order to produce the synthetic spectra. We modelled one parameter at a time utilising spectral features sensitive to di ff erent photospheric parameters iterating until convergence of all free parameters. Throughout\nthe modelling, we held the macro- and micro-turbulent velocities, v mac and v mic, fixed to 6 km s -1 (Doyle et al. 2014) and 1.34 km s -1 (Bruntt et al. 2010), respectively. A description of the modelling procedure is detailed in Persson et al. (2018). Finally, we obtained T e ff = 6354 ± 70 K, [Fe / H] = 0 . 025 ± 0 . 050, log g = 4 . 30 ± 0 . 06, and v sin i ⋆ = 7 . 5 ± 0 . 2 kms -1 .\nTo double-check the derived spectroscopy parameters we performed an additional analysis employing ARES + MOOG (Sousa et al. 2021; Sousa 2014; Santos et al. 2013). In detail, we used the latest version of ARES 2 (Sousa et al. 2007, 2015) to consistently measure the equivalent widths (EW) for the list of iron lines presented in Sousa et al. (2008). Following a minimisation process, we then find the ionisation and excitation equilibria to converge for the best set of spectroscopic parameters. This process uses a grid of Kurucz model atmospheres (Kurucz 1993) and the radiative transfer code MOOG (Sneden 1973). We obtained T e ff = 6389 ± 67 K, [Fe / H] = -0 . 014 ± 0 . 045 dex, log g = 4 . 58 ± 0 . 11, and v mic = 1 . 54 ± 0 . 04 kms -1 . In this process we also derived a more accurate trigonometric surface gravity (log g trig = 4 . 34 ± 0 . 02) using recent GAIA data following the same procedure as described in Sousa et al. (2021). In the end ARES + MOOG provides consistent values when compared with the ones derived with SME .\nUsing the SME stellar atmospheric parameters, we determined the abundances of Mg and Si following the classical curve-of-growth analysis method described in Adibekyan et al. (2012, 2015). Similar to the stellar parameter determination, we used ARES to measure the EWs of the spectral lines of these elements and a grid of Kurucz model atmospheres (Kurucz 1993) along with the radiative transfer code MOOG to convert the EWs into abundances, assuming local thermodynamic equilibrium.\nThe stellar radius R ⋆ , mass M ⋆ , and age t ⋆ were derived homogeneously using the isochrone placement algorithm (Bonfanti et al. 2015, 2016) and its capability of interpolating a flexible set of input parameters within pre-computed grids of PARSEC 3 v1.2S (Marigo et al. 2017) isochrones and tracks. For each magnitude listed in Table 1, we performed an isochrone placement run by inserting the spectroscopic parameters derived above, the Gaia parallax π (Gaia Collaboration et al. 2023, o ff set-corrected following Lindegren et al. 2021), and the magnitude value to obtain an estimate for the stellar radius, mass, and age along with their uncertainties. From these results, we built the corresponding Gaussian probability density functions (PDFs) and then we merged (i.e. we summed) the PDFs derived from the di ff erent runs to obtain robust estimates for R ⋆ , M ⋆ , and t ⋆ . The radius R ⋆ derives essentially from T e ff , π , and the stellar magnitude, while M ⋆ and t ⋆ are much more model-dependent; therefore we conservatively inflated their internal uncertainties by 4% and 20%, respectively, following Bonfanti et al. (2021). Our adopted stellar parameters are listed in Table 1.\n3. Observational data\n3.1. TESS photometry\nAs presented in Vanderburg et al. (2019), TOI-396 was photometrically monitored by TESS during the first year of its nominal mission in Sector 3 from 20 September to 17 October 2018 (UT) in CCD 2 of Camera 2, and in Sector 4 from 19 October to 14 November 2018 (UT) in CCD 2 of Camera 1. TOI-396 was later\n\n\n\nTable 1: Stellar properties of TOI-396.\nStar names\nTOI-396\nTIC 178155732\nHR 858\nHD 17926\nHIP 13363\nGaia DR3 5064574724769475968\n
\n\n
\nNotes. RA & DEC are reported as in J2000 reference frame. Values in the bottom half of the table have been derived as part of this paper. ( a ) Zero-point correction from Lindegren et al. (2021) applied.\nre-observed by TESS during the first year of its extended mission in Sector 30 from 23 September to 20 October 2020 (UT) in CCD 2 of Camera 2, and in Sector 31 from 22 October to 16 November 2020 (UT) in CCD 2 of Camera 1. All data were collected at a 2-minute cadence, except for Sector 3 for which TESS only sent down data at 30-minute cadence.\nWe analyse all available TESS time series, including the Sector 3 and 4 data already presented in (Vanderburg et al. 2019). For Sector 3 we used the TESS Asteroseismic Science Opera-\ntion Center (TASOC) photometry (Handberg et al. 2021; Lund et al. 2021), while for the other sectors we analysed the Presearch Data Conditioning Simple Aperture Photometry (PDCSAP) LCs generated by the TESS Science Processing Operation Center (SPOC) pipeline Jenkins et al. (2016), which removes the majority of instrumental artefacts and systematic trends (Smith et al. 2012; Stumpe et al. 2012, 2014). We rejected those data marked with a bad quality flag and we performed a 5 medianabsolute-deviation (MAD) clipping of flux data points. After that, we built our custom LCs by splitting the TESS sectors in temporal windows centred around the transit events keeping ∼ 4 h of out-of-transit data points both before and after the transit for de-trending purposes. We ended up with 41 TESS LCs reporting the epoch of observation ( t ), the flux and its error, and further ancillary vectors available from the TESS science data product, that is mom _ centr 1, mom _ centr 2 (hereafter denoted with x and y , respectively), pos _ corr 1, and pos _ corr 2 (hereafter denoted with pc1 and pc2 , respectively) 4 .\n3.2. HARPS high-resolution spectroscopy\nWe performed the radial velocity (RV) follow-up of TOI-396 with the HARPS spectrograph mounted at the ESO-3.6 m telescope at La Silla Observatory (Chile). We acquired 77 high resolution ( R ≈ 115 000) spectra between 31 January and 27 July 2019 (UT), covering a baseline of ∼ 177 days, as part of the follow-up programs of TESS systems carried out with the HARPS spectrograph (IDs: 1102.C-0923, 1102.C-0249, 0102.C0584; PIs: Gandolfi, Armstrong, De Medeiros). One additional spectrum was acquired during a technical night (ID: 60.A-9700) in February 2019. Following Dumusque et al. (2011b), we averaged out p-mode stellar pulsations by setting the exposure time to 900 s, which led to a median signal-to-noise (S / N) ratio of ∼ 320 per pixel at 550 nm. We used the second fibre of the HARPS spectrograph to simultaneously observe a Fabry-Perot lamp and trace possible instrumental drift down to the sub-metre per second level (Wildi et al. 2010). We reduced the data using the dedicated HARPS data reduction software ( DRS ; Pepe et al. 2002; Lovis, C. & Pepe, F. 2007) and computed the crosscorrelation function (CCF) for each spectrum using a G2 numerical mask (Baranne et al. 1996).\nWe then performed a skew-normal (SN) fit on each CCF (Simola et al. 2019) in orderto extract the stellar radial velocity along with its error, the full width at half maximum (FWHMSN), the contrast ( A ), and the skewness parameter ( γ ) of the CCF. The advantages of using an SN-fit rather than a Normal fit are thoroughly discussed in Simola et al. (2019), while the SN-fitting details are outlined in, for example, Bonfanti et al. (2023); Luque et al. (2023); Fridlund et al. (2024). After the SN-based extraction, we ended up with an RV time series, whose ancillary vectors (FWHMSN, A , γ ) are the activity indicators used to constrain the polynomial basis to model and de-trend the RV component of the stellar activity (see Table A.1).\nAs stellar activity is not stationary, the correlations between the RV observations and the activity indicators are expected to change over time as discussed in Simola et al. (2022), who proposed to apply the breakpoint ( bp ) technique (Bai & Perron 2003) to check whether these correlation changes are statistically significant. If so, the bp algorithm finds the optimal locations of correlation changes, which defines the segmenta-\ntion characterised by the lowest Bayesian Information Criterion (BIC; Schwarz 1978). Indeed, we found one breakpoint at observation 48 (BJD = 2458667.940719; ∆ BIC ≈ -14 with respect to the zero-breakpoint solution). Thus, we split our RV time series in two piecewise stationary segments and de-trended it on a chunk-wise base, rather than performing a global de-trending over the whole time series, similarly to what has already been done in Bonfanti et al. (2023); Luque et al. (2023).\n4. Methods\nWe jointly analysed the TESS LCs and the RV time series within a Markov chain Monte Carlo (MCMC) framework using the MCMCI code (Bonfanti & Gillon 2020). When fitting the TESS LCs extracted from Sector 3 that has a cadence of 30 min, we generated the transit model using a cadence of 2 min (the same as the other TESS sectors) and then rebinned it to 30 min following Kipping (2010), who warns that long-cadence photometry may lead to retrieve erroneous system parameters. We imposed Normal priors on the input stellar parameters (i.e. T e ff , [Fe / H], R ⋆ , and M ⋆ ), as derived in Sect. 2 with a twofold aim: (i) the induced prior on the mean stellar density ρ⋆ (via M ⋆ and R ⋆ ) helps the convergence of the transit model; (ii) limb darkening (LD) coe ffi cients for the TESS filter may be retrieved following interpolation within A tlas 9-based 5 grids that were precomputed using the get_lds.py code 6 by Espinoza & Jordán (2015). We then set Normal priors on the quadratic LD coe ffi -cients using the values coming from the grid interpolation (i.e. u 1 , TESS = 0 . 2318 ± 0 . 0065 and u 2 , TESS = 0 . 3085 ± 0 . 0028) after re-parameterising them following Holman et al. (2006). On the planetary side, we adopted unbounded (except for the physical limits) uniform priors on the transit depth d F , the impact parameter b , the orbital period P , the transit timing T 0, and the RV semi-amplitude K , while we set the eccentricity e = 0 and the argument of periastron ω = 90 · for all planets. We come back to the assumption on the eccentricity below. The specific parameterisations of the jump parameters (aka step parameters) are outlined in Bonfanti & Gillon (2020).\nThe LCs and the RV time series were de-trended simultaneously during the MCMC analysis using polynomials. To assess the polynomial orders to be associated with the di ff erent de-trending parameters for each time series, we first launch several MCMC preliminary runs made of 10 000 steps where we varied only one polynomial order at a time. We then selected the best de-trending baseline (see Table A.2) as the one having the lowest BIC. After that, we performed a preliminary MCMCI run comprising 200 000 steps to evaluate the contribution of both the white and red noise in the LCs following Pont et al. (2006); Bonfanti & Gillon (2020), so to properly rescale the photometric errors and get reliable uncertainties on the output parameters. Finally, three independent MCMCI runs comprising 200 000 steps each (burn-in length equal to 20%) were performed to build the posterior distributions of the output parameters after checking their convergence via the Gelman-Rubin statistic ( ˆ R ; Gelman & Rubin 1992).\nWe also tested the possibility of eccentric orbits by imposing uniform priors on ( √ e cos ω , √ e cos ω ) either bounded to imply e ≲ 0 . 3 or completely unbounded (except for the physical limits). The wider the eccentricity range to be explored by the MCMC scheme, the poorer the parameter convergence, which suggests that the available data are not enough to con-\nhe planetary eccentricities well. Moreover, the MCMCI runs with e , 0 are disfavoured by the ∆ BIC criterion (e.g. Kass & Raftery 1995; Trotta 2007) as well, in fact we obtained ∆ BIC = BIC e , 0 -BIC e = 0 ≳ + 100. This is also in agreement with the simulations performed by Vanderburg et al. (2019), who suggested that the periods' commensurability state of planets b and c is more likely maintained if the system is characterised by low eccentricities. Therefore, we adopted the circular solution as the reference one.\n5. LC and RV data analysis results\n5.1. Joint LC and RV analysis with linear ephemerides\nAs mentioned in Sect. 4, we set P and T 0 as free parameters under the control of a uniform prior, which implies assuming linear ephemerides. With this setup, we improved the transit depth precision of all three planets by a factor ∼ 1.4 if compared with the results of Vanderburg et al. (2019). This improvement level is consistent with having twice the number of data points with respect to the LC analysis performed by Vanderburg et al. (2019), as well as with low TTV amplitudes (see Sect. 5.2).\nBy combining the SN-fit onto the HARPS CCFs along with the bp method, we were able to estimate the masses of TOI-396 b and TOI-396 d to Mb = 3 . 56 + 0 . 92 -0 . 94 M ⊕ and Md = 7 . 2 ± 1 . 6 M ⊕ (detections at the 3.8 and 4.5 σ -level, respectively). Instead, we did not detect any significant Keplerian signal at the orbital period of TOI-396 c within the RV time series. In detail, we obtained a median Kc = 0 . 28 + 0 . 29 -0 . 20 ms -1 (3 σ upper limit K up c = 1 . 2 ms -1 ), which implies Mc = 0 . 92 + 0 . 94 -0 63 M ⊕ ( M up c = 4 . 0 M ⊕ ).\n. When combining the mass and radius values of the three planets, we obtain the following median estimates for the bulk planetary densities: ρ b = 2 . 56 + 0 . 71 -0 . 70 , ρ c = 0 . 67 + 0 . 69 -0 . 46 ( ρ up c = 3 . 1), and ρ d = 5 . 1 + 1 . 3 -1 . 2 g cm -3 . We note that the RV-undetected TOI396 c would be the least dense planet, while the densest planet is the outermost one (i.e. TOI-396 d), which constitutes a quite atypical architecture within the observed exoplanet population (e.g. Ciardi et al. 2013; Weiss et al. 2018; Mishra et al. 2023). However, this conclusion is just tentative, given the uncertainties on the mean planetary densities. Moreover, the detection level of the RV-inferred parameters of TOI-396 c is not statistically significant. We further tested whether including the MINERVAAustralis (Addison et al. 2019) RV data (30 measurements as taken from Vanderburg et al. 2019) can help detect the elusive planet. However, it turned out that their precision level ( ∼ 6 m s -1 ) is not high enough to improve the characterisation of the system. In other words, the RV semi-amplitudes we obtained are consistent and indistinguishable within the statistical fluctuation with what was derived from the more precise HARPS data set. This led us to further check that TOI-396 c indeed belongs to this system (Sect. 5.2) and to investigate the reason for its RV non-detection (Sect. 5.3).\n5.2. Joint LC and RV analysis accounting for TTVs\nAs planet c is undetected in the RV time series, one may wonder whether TOI-396 c is a false positive. However, by using the VESPA tool (Morton 2012, 2015) that accounts for the constraints from the TESS LCs, imaging, and spectroscopy, Vanderburg et al. (2019) already computed that the false-positive probabilities (FPPs) are lower than 10 -3 for all three planets.\nIn line with Vanderburg et al. (2019), we also confirmed that the Pc / Pb period ratio is commensurable and di ff ers from the 5:3 ratio by less than 0.027%. As planets with orbits in, or close to,\n\n\n
Fig. 3 shows the three planets of the system (star symbol) along with other exoplanets whose density is more significant than the 3 σ level 7 (circular marker) in the mass-radius (MR) diagram. The superimposed theoretical MR models as taken from Aguichine et al. (2021, A21) and Haldemann et al. (2024) help guide the eye; however, we warn the reader of possible degeneracies occurring in the MR plane. A thorough internal structure analysis of the well characterised TOI-396 b and d is presented in Sect. 7.
\n\nresonances are likely to show TTVs, we decided to repeat the same MCMC analysis outlined above, but enabling the transit timings of each transit event to vary to then compute the TTV amplitude with respect to the linear ephemerides model derived in Sect. 5.1. All jump parameters converged ( ˆ R ≲ 1 . 01). The medians of the posterior distributions of the most relevant system parameters along with the 68.3% confidence intervals are listed in Table 2. The phase-folded LCs of the three planets are shown in Figure 1, while the phase-folded RV time series are displayed in Figure 2. In particular, the middle panel of Fig. 2 shows that, after subtracting the RV signals of both planets b and d, the RV time series looks flat consistently with the non-detection of TOI396 c.\nThis analysis gives Rb = 2 . 004 + 0 . 045 -0 . 047 R ⊕ , Rc = 1 . 979 + 0 . 054 -0 . 051 R ⊕ , and Rd = 2 . 001 + 0 . 063 -0 . 064 R ⊕ and confirms the RV detection of TOI396 b and TOI-396 d (at the 3.8 and 4.5 σ -level, respectively) with Mb = 3 . 55 + 0 . 94 -0 . 96 M ⊕ and Md = 7 . 1 ± 1 . 6 M ⊕ . These outcomes are consistent with the results obtained from the analysis that assumes linear ephemerides (Sect. 5.1).\nWe found significant TTV signals for planet b and c (see Figure 4, Top and Middle panels) as expected from the commensurability of their periods. The TTV statistical significance of TOI-396 b and c is evident even by eye when comparing the data points location with the shaded regions that represent the 1 σ uncertainties of the linear ephemerides. For each planet, we further quantified the reducedχ 2 ( ˆ χ 2 ) characterising the TTV amplitudes via\nˆ χ 2 j = 1 N tr , j -2 N tr , j X i = 1 TTV j , i σ j , i ! 2 (1)\nwhere N tr , j is the number of transit event of the j -th planet. We obtained ˆ χ 2 b = 4 . 4, ˆ χ 2 c = 4 . 1, and ˆ χ 2 d = 0 . 7 for planets b, c, and d, respectively, which confirms the significance of the TTV amplitudes for TOI-396 b and c.\nFurthermore, the TTV amplitudes of TOI-396 b and c exhibit a clear anti-correlation pattern that we highlight by superimposing the TTV measurements in Figure A.1. This is a typical signature of gravitational interaction between the two planets, which confirms that TOI-396 c belongs to the system despite its elusiveness in the RV time series. For each planet, the timing of each transit event and the corresponding TTV amplitude computed with respect to the linear ephemerides are listed in Table A.3.\n5.3. Discussion regarding why TOI-396 c is not detected in the RV time series\nMagnetic activity combined with stellar rotation induces RV variations that can hide, a ff ect, or even mimic planetary signals (e.g. Queloz et al. 2001; Hatzes et al. 2010; Dumusque et al.\n\n\n
Fig. 1: TESS detrended and phase-folded LCs (blue dots) of TOI396 b ( Top panel ), TOI-396 c ( Middle panel ), and TOI-396 d ( Bottom panel ) with the transit model superimposed in red. The black markers are the binned data points (binning 20 min).
\n\n2011a; Haywood et al. 2014; Suárez Mascareño et al. 2017; Gandolfi et al. 2017). A possible explanation for the non-detection of the Doppler reflex motion induced by TOI-396 c is that stellar activity destructively interferes with the Keplerian signal of the planet. This may happen if the star has a rotation period\n
\n\n
Table 2: System parameters as derived from the joint LC and RV analysis accounting for TTVs.
\n
\nNotes. All jump parameters, but the LD coe ffi cients, were subject to unbounded uniform priors following the parameterisations specified in Bonfanti & Gillon (2020); see text for further details.\n( a ) Uncertainties from the run assuming linear ephemerides (no TTVs). T 0 values are shifted by -2 450 000. ( b ) Assuming zero albedo and full recirculation. ( c ) The 3 σ upper limits on the RV semi-amplitude, the planet mass, and the mean planet density of TOI-396 c are K up c = 1 . 2 ms -1 , M up c = 3 . 8 M ⊕ , and ρ up c = 2 . 9 g cm -3 , respectively.\ncomparable to the orbital period of the planet (e.g. Vanderburg et al. 2016). Disentangling the planetary signal from stellar activity and retrieving the Doppler motion induced by the orbiting planet would then be challenging (e.g. Dragomir et al. 2012; Kossakowski et al. 2022).\nIn order to investigate this hypothesis, we performed a frequency analysis of the line profile variation diagnostics (FWHM, contrast, and skewness) and activity indicators (log R ' HK and H α indexes). Figure 5 displays the time series (left column), along with the respective generalised Lomb-Scargle (GLS, Zechmeister & Kürster 2009) periodograms (right column). We assessed the significance of the peaks detected in the power spectra by estimating their false alarm probability (FAP), that is the probability that noise could produce a peak with power higher than the one we found in the time series. To account for the possible presence of non-Gaussian noise in the data, we estimated the FAP using the bootstrap randomisation method (see, e.g., Murdoch et al. 1993; Kuerster et al. 1997; Hatzes 2019). Briefly, we computed the GLS periodogram of 10 5 mock time series obtained by randomly shu ffl ing the data points along with their error bars, while keeping the timestamps fixed. We defined the FAP as the fraction of those mock periodograms whose highest power exceeds the power of the real data at any frequency. In the present work, we considered a peak to be significant if its false alarm probability is FAP < 0.1 %.\nWe found that the time series of the log R ' HK and H α indexes display long-term trends likely due to the magnetic cycles of the star (Fig. 5, left column, first and third panels). In the Fourier domain, these trends translate into a significant (FAP < 0.1 %) excess of power at frequencies lower than the spectral resolution 8 of our HARPS observations (Fig. 5, right column, first and third panels). We modelled these long-term signals as quadratic trends and subtracted the best-fitting parabolas from the respective time-series. The GLS periodograms of the residuals of the activity indicators display significant peaks between ∼ 6 and 8 d (i.e, between ∼ 0.0125 and 0.167 d -1 ; Fig. 5, yellow area). The peaks are equally spaced by ∼ 0.0068 d -1 , which coincides with a peak found in the periodogram of the window function. Given the current data at our disposal, we are not able to distinguish between true frequencies and aliases. Although not significant, the power spectra of the contrast and skewness show peaks in the same frequency range, suggesting that the rotation period of the star might be ∼ 6-8 d.\nWe note that the periodogram of the FWHM also displays an excess of power at low frequencies. However, the corresponding peak remains below our 0.1 %-FAP significance threshold (see Fig. 5, second to last panel). Yet, if we apply the same procedure\n\n\n\n\n\n
Fig. 2: HARPS detrended and phase-folded RV time series of TOI-396 b ( Top panel ), TOI-396 c ( Middle panel ), and TOI396 d ( Bottom panel ) with the Keplerian model superimposed in red. For each planet, the time series were obtained after subtracting the RV contribution of the other planets. The error bars also account for the jitter contribution (displayed in grey).
\n\ndescribed above and remove this signal by fitting a quadratic trend to the FWHM time series, we find no significant peak in the residuals.\n\n\n
Fig. 3: Mass-radius diagram showing the three planets orbiting TOI-396 (star symbol) along with the exoplanets whose density is more significant than the 3 σ level (circle). All the markers are colour-coded according to the equilibrium temperature ( T eq) of the planets. The thick lines are the theoretical MR BICEPS models as detailed in the legend, except for the light-blue line denoted with A21, which is taken from Aguichine et al. (2021). Earth-like means 32.5% iron + 67.5% silicates, while Mercurylike means 70% iron + 30% silicates. Models were computed for T eq = T eq , b = 1552 K (dashed-dotted lines), T eq = T eq , c = 1309 K (solid lines), and T eq = T eq , d = 1061 K (dashed lines). The di ff erence between the A21 model and its BICEPS counterpart is due to di ff erent assumptions for e.g. pressure-temperature profiles and opacities. The dotted black lines are the iso-density loci of points corresponding to 0.5, 1, 3, 5, 10 g cm -3 (going from top to bottom). We recall that the mass estimate of TOI-396 c (the leftmost star symbol) is not statistically significant and its 3 σ upper limit is M up c ∼ 4 M ⊕ .
\n\nThe projected equatorial velocity of the star ( v sin i ⋆ = 7 . 5 ± 0 . 2 km s -1 ), along with its radius ( R ⋆ = 1 . 258 ± 0 . 019 R ⊙ ), yields an upper limit for the rotation period of P up rot = 8 . 5 ± 0 . 3 d. Using the mean value 9 of log R ' HK = -4 . 926 ± 0 . 014, we inferred a stellar rotation period of 6.7 ± 1.3 d and 6.9 ± 1.3 d from the empirical equations of Noyes et al. (1984) and Mamajek & Hillenbrand (2008), respectively. In addition, by inputting the isochronal age into the gyrochronological relation from Barnes (2010), we computed a stellar rotation period of 7 . 1 + 1 . 0 -1 . 1 d. These results corroborate our interpretations that the peaks between 6 and 8 d significantly detected in the power spectra of the activity indicators originate from stellar rotation.\nTo check whether a quasi-periodic signal compatible with ∼ 7 d is also present in the photometric data, for each TESS sector we extracted custom LCs from pixel data using lightkurve (Lightkurve Collaboration et al. 2018). In detail, we adopted the default quality bitmask and set the aperture to 'all', which corresponds to an aperture larger than the one used by the o ffi cial SAP pipeline. In fact, larger apertures mitigate the e ff ect of slow image drifts that could interfere with slow flux changes, such\nTV [min]\nT\nTV [min]\nT\n40\n30\n20\n10\n0\n10\n20\n60\n50\n40\n30\n20\n10\n0\n10\n20\n8360 8380 8400 8420 8440\n9100 9120 9140 9160 9180 9200\nBJD - 2450000 [d]\n8360 8380 8400 8420 8440\n9100 9120 9140 9160 9180 9200\nBJD - 2450000 [d]\n\n\n
Fig. 4: TTV amplitudes obtained for TOI-396 b ( Top panel ), TOI-396 c ( Middle panel ), and TOI-396 d ( Bottom panel ). The grey shaded region highlights the 1 σ uncertainty region as derived from error propagation of the linear ephemerides.
\n\nas the 6-8 d rotation period signals we aim to detect. After removing the temporal windows containing the transit events, we computed the GLS periodograms of these lightkurve -based LCs. The FAP was computed following the same bootstrap technique outlined above for the RV activity indicators. The four periodograms (Fig. A.2, first column) exhibit very significant peaks at ∼ 7.7, 7.9, 7.5, and 6.8 days for TESS Sectors 3, 4, 30, and 31,\n
\n\n
Table 3: Radial velocity semi-amplitudes K out as retrieved from MCMCIanalyses of the RV time series obtained by adding an artificial Keplerian signal with period P = Pc and semi-amplitudes K in to the original HARPS time series.Notes. Columns ρ p and Mp translate the injected synthetic signals into the corresponding physical parameters of the planet, while the last column quantifies the K out detection level in terms of σ .
\n
\nrespectively. Except for Sector 30, they are not the most prominent peaks; however, they persist even after removing the most significant signals (Fig. A.2, second column). Whereas we acknowledge that the likely rotation period of the star is close to the first harmonic of the orbital period of TESS around the Earth ( P rot ,⋆ ∼ 1 2 P TESS ∼ 14 days), the photometric signal at 6-8 d is significant in all the four TESS sectors and persists after prewhitening the data. This signal is consistent with the rotation period detected in the HARPS activity indicators and inferred from log R ' HK , v sin i ⋆ , and gyrochronology, suggesting it is astrophysical in nature and due to the presence of active regions carried around by stellar rotation. Assuming the log R ' HK -based P rot ,⋆ = 6 . 7 ± 1 . 3 d as our reference estimate, the orbital period of TOI-396 c ( Pc ∼ 6 d) is close to P rot ,⋆ , which may explain the non-detection of planet c within the RV time series.\nIf some kind of destructive interference between the Keplerian signal of planet c and the stellar activity has occurred, any artificial Keplerian signals with period P = Pc added to the observed RV time series should in principle be retrieved. To test this hypothesis, we considered Keplerian signals of the following form\nRV art = -K art sin \" 2 π P ( t -T 0) # (2)\nand generated four di ff erent RV time series by separately adding to the HARPS time series synthetic RV signals following Equation (2) with P = Pc , T 0 = T 0 , c , and K art = K in, where K in are the four di ff erent amplitude values listed in the first column of Table 3. For each RV time series, we then performed an MCMCI analysis to retrieve the RV semi-amplitude of the artificial signal ( K out; see Table 3).\nWe note that the resulting K out ≈ K in + Kc , where Kc = 0 . 28 + 0 . 29 -0 . 19 ms -1 is the RV semi-amplitude of planet c as derived from the analysis on the original RV time series. As we essentially retrieved what we inserted in the HARPS time series, we may conclude that the destructive interference between the RV signals induced by the star and by planet c has already occurred and any further RV signal added to the RV time series is detected.\nAs P rot ,⋆ is not exactly equal to Pc , we then repeated the test outlined above, but this time we injected into the original HARPS time series artificial Keplerian signals with P = P rot ,⋆ . The K out values obtained by the MCMC analyses depending on the di ff erent K in values are reported in Table 4. The K out values are systematically and significantly smaller than the corresponding K in values, which a posteriori supports the conclusion that the stellar rotation period is around 6-8 d. Furthermore, a planet with this\n\n\n\n\n\n
Fig. 5: Time series (left panels) and GLS periodograms (right panels) of the line profile variation diagnostics and activity indicators extracted from TOI-396's HARPS spectra. In the left panels, the red curves in the first four panels mark the quadratic trends and sine functions as obtained from the best fit to the most significant peaks (false alarm probability FAP < 0.1%) identified in the corresponding GLS periodograms. In the right panels, the vertical dashed blue lines mark the orbital frequencies of the tree transiting planets. The yellow area encompasses the peaks likely due to stellar rotation. The horizontal dashed red lines mark the 0.1% false alarm probability.
\n\norbital period would be firmly detected if its RV semi-amplitude were greater than Kd , which is the largest RV semi-amplitude detected for this system. Instead, by injecting a Keplerian signal with K in = Kd and P = P rot ,⋆ , the MCMCI analysis is able to barely detect (at ∼ 2 σ ) a planetary signal whose amplitude is half of that expected. The detection level increases when K in increases; however, we still underestimate K out. These tests prove that it is di ffi cult to retrieve planetary signals with P ∼ P rot ,⋆ .\nIn summary, we conclude that stellar activity is responsible for generating spurious RV signals whose harmonics also include the stellar rotation period. As a consequence, it is hard to reliably detect planets with orbital periods comparable to P rot ,⋆ via the RV technique and Table 4 quantifies the magnitude of this e ff ect. We note that the Kc we obtained from the MCMCI analysis in Sect. 5.2 is comparable to the K out retrieved when inserting an artificial signal having K in = 1 . 0 ms -1 , which let us\n
\n\n
Table 4: Same as Table 3, but this time the Keplerian signals with K = K in have period P = P rot ,⋆ .
\n
\nNotes. Column ∆ K K gives the relative di ff erence (in percentage) between the obtained RV semi-amplitude ( K out) and the expected one ( K in), while the last column ∆ K ≡ K out -K in lists the semi-amplitude di ff erence in terms of the 1 σ uncertainty of K out.\nto speculate that TOI-396 c might have Mc ∼ 3.0 M ⊕ and ρ c ∼ 2.0 g cm -3 .\n6. Joint RV and TTV dynamical analysis\nAs mentioned, the anti-correlation pattern of the observed TTVs (see Fig. A.1) is a typical signal of the dynamical interaction between TOI-396 b and c. However, the data in our hands are not enough to currently derive meaningful planetary masses from the TTVs. Indeed, the photometric observations are clustered in two groups (about two years apart), which results in a partial coverage of the curvature of the TTVs. This prevents us from accurately mapping the full phases and super-periods of the TTV signals. Hence a dynamical fit onto the transit times would lead to a high fraction of low-amplitude TTVs with a short superperiod or to a low fraction of high-amplitude long-period TTV signals.\nDespite this, we attempted to run a dynamical joint fit of the TTV and RV data set with TRADES 10 (Borsato et al. 2014, 2019, 2021), a Fortran-python code developed to model TTVs and RVs simultaneously along with N-body integration. We have taken and fixed the stellar mass and radius values from Table 1. We further fixed the orbital inclinations i to the values in Table 2 and the longitude of the ascending nodes to Ω = 180 · 11 for all the planets. We fitted the planetary masses scaled by the stellar mass ( M b , c , d / M ⋆ ), the orbital periods ( P ), the eccentricities ( e ) and the arguments of the pericentre ( ω ) in the form ( √ e cos ω , √ e sin ω ), and the mean longitudes ( λ 12 ). We also fitted for an RV o ff set ( γ RV) and for an RV jitter term ( σ jitter) by adopting log 2 σ jitter as step parameter, although we used the de-trended RV data set as derived from Sect. 5.2, where a jitter term was already included in the RV errors. All fitting parameters were subject to uniform priors that account for their respective physical boundaries (see Table 5); for the eccentricities we applied a log-penalty (log p e ) based on the half-Gaussian ( e = 0 , σ e = 0 . 083) from Van Eylen et al. (2019).\nThe reference time for the dynamical integration of the orbital parameters was set at T ref = 2 458 379 BJDTDB, that is before all available observations. We combined the quasiglobal di ff erential evolution (Storn & Price 1997) optimisation algorithm implemented in P y DE (Parviainen et al. 2016)\nwith the A ffi ne Invariant MCMC Ensemble sampler (Goodman & Weare 2010) emcee implemented by Foreman-Mackey et al. (2019, 2013). We first ran P y DE and evolved 68 di ff erent initial configurations of parameter sets for 50 000 generations (number of steps for which each parameter is evolved). To perform the dynamical analysis in an MCMC fashion, we then ran emcee , assuming as starting point the outcome obtained with P y DE. We set up 68 chains for 600 000 steps each. To sample the parameter space e ffi ciently, we mixed the DEMove() and DESnookerMove() di ff erential evolution moves 13 in the proportion 80%-20% (Nelson et al. 2014; ter Braak & Vrugt 2008). We repeated the sequence P y DE + emcee twice, with di ff erent seeds for the random number generator. The chains reached convergence according to visual inspection and statistical indicators, such as the Gelman-Rubin ˆ R , the Geweke's statistic (Geweke 1991), and the auto-correlation function 14 . For both runs, we applied a conservative thinning factor of 100 and discarded the first 50% of the chains (burn-in). For each run, we derived the reference outcome (hereinafter also referred to as best-fit), as the maximum-a-posteriori (MAP) parameters' set, that is the set of parameters that maximises the log-probability 15 . The uncertainty of each parameter is quantified by the high-density interval (HDI) at 68.27% 16 of its posterior distribution. For each parameter we computed a Z-score defined as Z-score = | MAP1 -MAP2 | / p max | ERR1 | 2 + max | ERR2 | 2 , where the subscripts denote the two di ff erent runs and ERR = HDI -MAP. It turned out that Z-score < 1 for each parameter, which allows us to merge the posterior distributions deriving from the two runs to finally compute the MAP and the respective HDI from the merged posterior distributions. We further checked the Hill stability of the system (Sundman 1913) when assuming the entire merged posterior distributions by calculating the angular momentum deficit (AMD, Laskar 1997, 2000; Laskar & Petit 2017) criterion (Eq. 26 from Petit et al. 2018). Table 5 lists the parameters returned by TRADES (MAP and HDI) along with their respective priors.\nWe note that the dynamical integration with TRADES allowed for a significant detection ( > 3 σ ) of the mass of TOI-396 c, that is Mc , dyn = 2 . 24 + 0 . 13 -0 . 67 M ⊕ . However, as explained above, TESS data do not allow us to fully map the TTV pattern. Therefore, the present-day estimate of Mc , dyn only provides an indication of the possible mass of the planet that might not be accurate, despite its formal precision. Indeed, to accurately and reliably determine planetary masses via TTVs, it is necessary to monitor the TTV signal by benefiting of a full sampling coverage, as demonstrated for example by the cases of Kepler-9 (Holman et al. 2010) and K2-24 (Petigura et al. 2016), whose orbital parameters and masses were comprehensively revisited by Borsato et al. (2014) and Petigura et al. (2018); Nascimbeni et al. (2024), respectively. In detail, Holman et al. (2010) had reported the masses of Kepler-9 b and c with a precision better than 8%, by performing a TTV analysis on the first three quarters of the Kepler data. Later on, by benefiting of twelve sectors of Kepler data that enabled the full mapping of the TTV phase, Borsato et al. (2014) obtained TTV-based masses di ff ering by a\n\n\n
Figures 10 and 11 show the resulting posteriors of the most important interior structure parameters for TOI-396 b and d, respectively, in comparison with the chosen priors (dotted lines). Tables A.4 and A.5 in the Appendix summarise the median and one sigma error intervals for the full set of internal structure parameters. For both planets, the posterior distributions for the core and mantle mass fractions largely agree with the chosen priors for each of the six models that we ran. Indeed, the only planetary structure parameters for which the observational data contributed to their characterisation were the envelope mass fractions. If we assume that the planets formed outside the iceline, we find envelope mass fractions of 28 ± 10% (A1), 32 + 9 -11 %(A2) and 30 + 11 -13 %(A3) for planet b and 23 + 12 -10 %(A1), 28 ± 11% (A2) and 26 + 12 -13 % (A3) for planet d, with water mass fractions in the envelope of almost 100%. Conversely, if the planets were to have formed inside the iceline, we infer envelope mass fractions of the order of 10 -5 for planet b and 10 -4 for planet d, almost entirely made up of H / He.
\n\nfactor ∼ 2 from the estimate of Holman et al. (2010). Similarly, by accounting on more photometric data, Petigura et al. (2018); Nascimbeni et al. (2024) found that the mass of K2-24 c is lower by almost 2 σ than the estimate of Petigura et al. (2016) who had claimed a detection at the ∼ 4 σ level.\nAfter extracting all the synthetic transit timings T tr (i.e. the 'observed': O) from TRADES' analysis, we computed the 'calculated' (C) counterpart according to the linear ephemerides model based on what listed in Table 2. We plotted the O -C as a function of time as well as the RV best-fit model in Figures 6 and 7.\nAdditionally, we investigated if the best-fit configuration is in or close to an MMR. We integrated the MAP parameters with the N-body code rebound (Rein & Liu 2012) and the symplectic Wisdom-Holman integrator whfast (Rein & Tamayo 2015; Wisdom & Holman 1991) for 10 000 years. We computed the evolution of the critical resonance angles of TOI-396 b and TOI396 c\nϕ b = p λ b -( p + q ) λ c + q ϖ b ϕ c = p λ b -( p + q ) λ c + q ϖ c , (3)\nwhere p = 3 and q = 2 (for a second order 5:3 MMR), while ϖ ≡ ω + Ω is the longitude of the pericentre. We also computed the evolution of ∆ ϖ ≡ ϖ b -ϖ c = ( ϕ b -ϕ c) / q . In case of MMR, we expect that both ϕ b and ϕ c librate (i.e. oscillate) around a fixed value for the entire orbital integration. Instead, if these angles circulate, that is they span the full 0 · -360 · range (or equivalently the -180 · -180 · range), then the planet pair is not in resonance. We found that both ϕ b and ϕ c circulate (see the two upper panels of Fig. 8), which indicates that the system is not in an exact 5:3 MMR. Even if ∆ ϖ seems to oscillate around 0 · , it circulates every ∼ 2 000 years (bottom panel of Fig. 8), which further confirms that the system is not trapped in an MMR state. We also found the same behaviour for the resonant angles of 200 random samples drawn from the posterior distribution (not shown here). Our conclusions are consistent with the simulations performed by Vanderburg et al. (2019), who found that most realisations of the system are not in resonance.\nAs shown in Fig. 6, the MAP parameter set predicts that the TTV super-period is longer than the time spanning the two clustered TESS observations. We decided to track the potential evolution of the TTV signals over a temporal baseline of ∼ 5.2 years. To this end, we ran forward numerical N-body simulations with TRADES, setting the initial conditions of the orbital parameters to be integrated at t = T ref. We ran a first simulation taking all the parameters from the MAP solution. Then, we further ran 200 simulations, where the sets of the system's parameters were randomly drawn from the merged posterior distributions. We computed the synthetic (i.e. observed: O ) transit times T tr and created a simulated O -C plot against time, where the C counterpart of T tr was computed assuming the linear ephemerides of Table 2. The results are displayed in Fig. 9 that emphasises a progressive drift of the O -C values inferred from the MAP parameters (black line) with respect to the zero value. As a consequence, the TTV amplitudes of planets b and c increase with time. In particular, the semi-amplitude of the O -C black curves are about two and five hours for b and c, respectively. The TTV super-period seems to be roughly equal to or larger than the integration time span, that is ∼ 5 years. The variance of the TTV amplitude (shaded area) is inferred from the results of the additional 200 simulations and reflects the widths of the posterior distributions from which the system's parameters were drawn. The remarkable O -C drifts (i.e. the poorly constrained linear\nephemerides) combined with the uncertainty on the TTV amplitudes make challenging to plan future observations of planets b and c, as the actual transit timings might di ff er from the linear ephemerides predictions by ∼ 5 and ∼ 10 hours for TOI-396 b and TOI-396 c, respectively. TESS will not observe the target in the foreseeable future. 17\n7. Internal structure\nUsing the masses and transit depths reported in Table 2, we ran the neural network based internal structure modelling framework plaNETic 18 (Egger et al. 2024) to infer the internal structure of TOI-396 b and d. plaNETic uses a full grid accept-reject sampling method in combination with a deep neural network (DNN) that was trained on the forward model of BICEPS (Haldemann et al. 2024) to infer the internal structure of observed planets. Each planet is modelled as a three-layered structure: an inner iron-dominated core, a silicate mantle and a fully mixed envelope made up of water and H / He. In the case of multi-planet systems, all planets are modelled simultaneously.\nAs modelling the internal structure of exoplanets is a highly degenerate problem, the resulting inferred structure is, at least to a certain extent, dependent on the chosen priors. To mitigate this e ff ect, we ran a total of six models assuming six di ff erent combinations of priors. Most importantly, we use two di ff erent priors for the water content of the modelled planet, one motivated by a formation scenario outside the iceline (case A, water-rich) and one compatible with a formation inside the iceline (case B, water-poor). For both of these water priors, we choose three different options for the planetary Si / Mg / Fe ratios. In a first case, we assume that these match the stellar Si / Mg / Fe ratios exactly Thiabaud et al. (2015). Second, we assume that the planet is enriched in iron compared to its host star by using the fit of Adibekyan et al. (2021). For option 3, we model the planet independent of the stellar Si / Mg / Fe ratios, but just sampling the planetary ratios uniformly from the simplex where the molar Si, Mgand Fe ratios add up to 1, with an upper bound of 0.75 for Fe. These priors are described in more detail in Egger et al. (2024).\n
\n\n
Table 5: Best-fit parameters (MAP and HDI at 68.27%) along with their respective priors as inferred from the dynamical joint modelling of RVs and TTVs with TRADES.Notes. All parameters have been defined at the reference time T ref = BJDTDB -2 450 000 = 8379. U ( X , Y ) means uniform distribution between X and Y values; HG ( µ, σ ) means half-normal distribution with mean µ and standard deviation σ .
\n
\n\n\n\n\n\n
Fig. 6: Observed minus calculated synthetic diagrams derived from the joint RV and TTV dynamical analysis with TRADES for planet b ( left panel ) and c ( right panel ). The O -C for the best-fit (MAP) model is plotted with a black line, while the observed data points are the orange circles. The shaded grey regions displays the confidence intervals at 1, 2, and 3 σ , as inferred from the 200 samples randomly drawn from the merged posterior distributions. Residuals are shown in the lower panels.
\n\n8. JWST characterisation prospects\nAll three planets in the TOI-396 system share similar radii ( ∼ 2 R ⊕ ), but span a wide range of masses (0.9-7.1 M ⊕ ), which\nleaves open the question of whether they have primary or secondary atmospheres. Furthermore, the progression of bulk densities with distance from the host star varies in ways that cannot be described by simple formation and evolution models (e.g. Weiss et al. 2018; Mishra et al. 2023).\nGiven the bright host star and combination of planetary masses, radii, and equilibrium temperatures, the three planets\n\n\n\n\n\n\n\n\n
Fig. 7: Left panel : Same as Fig. 6, but for TOI-396 d. Right panel : Combined RV model of the three planets (black line) superimposed to the de-trended HARPS observations (purple circles). The grey shaded area is determined from the 200 sets of system parameters randomly drawn from the merged posterior distributions as obtained from the joint dynamical analysis with TRADES.
\n\n\n\n
Fig. 8: Temporal evolution of the critical resonance angles ϕ b ( top panel ) and ϕ c ( middle panel ) as well as of ∆ ϖ = ( ϕ b -ϕ c) / q ( bottom panel ) as inferred when assuming the MAP parameters derived by the TRADES dynamical analysis and integrated with rebound+whfast for 10 000 years.
\n\nhave favourable metrics for atmospheric characterisation in both transmission and emission among sub-Neptunes (Kempton et al. 2018, see Fig. 12). This makes the TOI-396 system a highly valuable laboratory to study the formation and evolution of plan-\netary systems. Thus, we explored the prospects for characterisation with JWST. We focused these simulations on emission observations, but we note that transmission and emission have their own advantages and disadvantages in terms of achievable science goals and challenges.\nWe employed the open-source P yrat B ay modelling framework (Cubillos & Blecic 2021) to compute synthetic spectra of the TOI-396 planets. These models consist of 1D cloud-free atmospheres in radiative, thermochemical, and hydrostatic equilibrium (Cubillos et al., in prep.). We varied the models' atmospheric elemental content to explore the wide range of compositions that the planets span. For this comparison we settled on two models to represent a primary- and a secondary-atmosphere scenario: the first is a gas giant with a 5 × solar metallicity, the second is a water world with a 80% H 2 O plus 20% CO 2 composition (based on the C / O ratios seen in the solar system minor bodies, see, e.g., Mumma & Charnley 2011; McKay et al. 2019). For the thermochemical-equilibrium calculations we considered a set of 45 neutral and ionic species, which are the main actors determining the thermal structure. For the radiative-transfer calculation we considered opacities from molecular species for CO, CO 2 , CH 4 , H 2 O, HCN, NH 3 , and C 2 H 2 from hitemp and E xo M ol (Rothman et al. 2010; Tennyson et al. 2016); Na and K resonant lines (Burrows et al. 2000); H, H2, and He Rayleigh (Kurucz 1970); and H2-H2 and H2-He collision-induced absorption (Borysow et al. 2001; Borysow 2002; Richard et al. 2012). We preprocessed the large E xo M ol line lists with the R epack algorithm (Cubillos 2017) to extract the dominant transitions. Figure 13 (top panels) shows the resulting thermal and composition structure for TOI-396 b (planets c and d follow a similar trend).\nThe infrared synthetic emission spectra (Fig. 13, bottom panels) are mainly shaped by H 2 O, CO 2 , and CO features. At most wavelengths the primary- and secondary-atmosphere scenarios roughly di ff er by an o ff set, which would be hard to distinguish unless the energy budget of the planets are known. In contrast,\n\n\n
Fig. 9: Synthetic O -C diagrams obtained after performing forward numerical N-body simulations with TRADES (integration of 5.2 years). The C represents the timings calculated from the linear ephemerides in Table 2. The MAP model is plotted with a black line, while the confidence intervals at 1, 2, and 3 σ are marked as shaded grey regions and come from 200 random samples drawn from the merged posterior distributions derived from the joint RV and TTV dynamical analysis.
\n\nthe 4-5 µ mwindow shows the most distinctive spectral features; here the strong CO 2 absorption band at 4.4 µ m mainly allows one to distinguish primary from secondary atmospheres. Thus, in the following we focus on this region of the spectrum.\nWe simulated JWST observations using the P andeia exposure time calculator Pontoppidan et al. (2016). The brightness of TOI-396 limits the instrument selection to NIRCam (F444W filter) to avoid saturation. We selected the fastest readout and subarray modes, 5 groups per integration, to optimise the S / N. We generated a distribution of (noised up) realisations for each model to estimate how many eclipses are required to distinguish between primary and secondary atmospheres at the 3 σ level. We found that 2, 4, and 8 eclipses (for planets b, c, and d, respectively) would be su ffi cient to di ff erentiate between these two models. Figure 13 shows one of those random realisations when including the required number of eclipses. The decreasing equilibrium temperature of the planets as they are located further away from TOI-396 plays the major role in the decreasing S / N for planets c and d.\n9. Conclusions\nThe object TOI-396 is an F6 V bright naked-eye star orbited by three planets of almost equal size, and the two inner planets are close to but out of a 5:3 MMR. A photometric analysis of the system was already performed by Vanderburg et al. (2019), but by benefiting from two additional TESS sectors, we improved the precision on the planet radii by a factor of ∼ 1.4, obtaining Rb = 2 . 004 + 0 . 045 -0 . 047 R ⊕ , Rc = 1 . 979 + 0 . 054 -0 . 051 R ⊕ , and Rd = 2 . 001 + 0 . 063 -0 . 064 R ⊕ .\nWe determined the masses of the planets by extracting the RV time series from HARPS CCFs using an SN fit followed by a joint LC and RV MCMC analysis, where the RV de-trending uses the breakpoint method. We obtained a firm detection of the RV signals of planets b and d, deriving Mb = 3 . 55 + 0 . 94 -0 . 96 M ⊕ and Md = 7 . 1 ± 1 . 6 M ⊕ , but we can provide only a 3 σ upper limit for the mass of TOI-396 c of M up c = 3 . 8 M ⊕ . This yields the following mean planet densities: ρ b = 2 . 44 + 0 . 69 -0 . 68 , ρ up c = 2 . 9, and ρ d = 4 . 9 + 1 . 2 -1 . 1 g cm -3 , implying a quite unusual system architecture (Mishra et al. 2023) where the mid planet is the least dense and the outermost planet is the densest.\nThe reason for the RV non-detection of any Keplerian signal at P = Pc ∼ 6 d is likely to be ascribed to the vicinity of Pc to the stellar rotation period. As a matter of fact, from the GLS periodograms of both the RV-related activity indices and the TESS raw LCs and from log R ' HK -based empirical relations, we consistently inferred P rot ,⋆ = 6 . 7 ± 1 . 3 d. After injecting synthetic Keplerian signals at P = P rot ,⋆ and di ff erent semi-amplitudes ( K in) into the RV time series, we empirically find that the RV semi-amplitudes output by the MCMC analyses ( K out) are systematically lower than the input ones by almost 3 σ , and they are statistically non-significant as far as K in ≲ Kd . In addition, we find that K out ≈ Kc when considering a planet with Mp ∼ 3 M ⊕ (i.e. ρ p ∼ 2 g cm -3 ), which might correspond to the properties of TOI-396 c. On a more general perspective, these simulations confirm that stellar activity destructively interferes with Keplerian signals having P ∼ P rot ,⋆ (e.g. Vanderburg et al. 2016), and furthermore, they indicate that - even in the case of firm detection - values of K out are significantly underestimated.\nLonger-baseline RV observations may help disentangle coherent signals originated by Keplerian motions from noncoherent signals due to stellar activity, even if degeneracy issues still hold when the planet orbital period is close to the stellar rotation period (Kossakowski et al. 2022). Alternatively, a possible constraint on Mc may come from TTVs, as planets b and c are close to an MMR of the second order. Indeed, the TTV amplitudes of the two planets show a characteristic anti-correlation pattern, as expected; however, the phase coverage given by the available observations is too poor to perform a conclusive TTV dynamical analysis based on the observed transit timings of the planets. We also attempted to fit the TTV and RV simultaneously while integrating the orbits of the system. We found that the masses and densities of planets b and d are consistent with the results from the joint LC and RV analysis. TOI-396 c shows a dynamical mass of Mc , dyn = 2 . 24 + 0 . 13 -0 . 67 M ⊕ , which is greater than that inferred from the joint LC and RV analysis, but it is consistent (Z-score = 1 . 2 σ ); the density is consistent at the 1 . 1 σ level. However, we emphasise that, although formally precise, the Mc , dyn estimate might not be accurate, as the full coverage of the TTV phase is needed to reliably compute TTV-based masses. Therefore, to fully confirm the system architecture, a reliable estimate of the mass of TOI-396 c is still missing.\nWe also checked the evolution of the system over 10 000 years, and the critical resonance angles showed that planets b and c are close to but not in a 5:3 MMR. We further performed forward N-body simulations over a temporal baseline of ∼ 5.2 years in order to track the transit epochs and evaluate the expected TTV amplitudes during time. It turns out that TOI-396 b and TOI-396 c may exhibit TTVs with a super-period of about 5 years and semi-amplitudes of ∼ 2 and ∼ 5 hours, respectively. This translates into a temporal drift of the transit timings that can rise up to ∼ 5 and ∼ 10 hours with respect to the linear ephemerides computed from TESS data.\n\n\n\n\n\n
Fig. 10: Inferred posteriors for the most important internal structure parameters of TOI-396 b. The depicted parameters are the mass fractions of the inner core (wcore), mantle (wmantle) and envelope layers (wenvelope) with respect to the total planet mass, and the mass fraction of water in the envelope (Zenvelope). The top row shows the results when assuming a water prior motivated by a formation of the planet outside the iceline (case A), while the bottom row uses a water prior compatible with a formation inside the iceline (case B). At the same time, we run models with three di ff erent compositional priors for the planetary Si / Mg / Fe ratios: stellar (purple, option 1), iron-enriched compared to the star (pink, option 2) and sampled using a uniform prior (blue, option 3). The dotted lines show the prior distributions, while the dashed vertical lines show the median values of the posteriors.
\n\n\n\n
Fig. 11: Same as Figure 10 but for TOI-396 d.
\n\nStudying the planetary atmospheres with JWST would take advantage of the favourable spectroscopy metrics of the system (Kempton et al. 2018). Therefore, we set up 1D cloud-free atmospheric models, generated the synthetic emission spectra of the three planets, and simulated eclipse observations with JWST. It turns out that 2, 4, and 8 eclipses (for TOI-396 b, c, and d, respectively) would be su ffi cient to distinguish between primary and secondary atmosphere scenarios at the 3 σ level.\nCharacterising the nature of the planetary atmosphere is also key to correctly assessing the planetary bulk densities (in particular for planet c). The potentially high TTVs inferred from our simulations should be duly taken into account when scheduling future observations of the target. This holds not only for JWST, but also for CHEOPS (Benz et al. 2021), which appears espe-\ncially suitable for collecting exquisite photometric data to enable the full characterisation of the system.\nAcknowledgements. We thank the anonymous referee for all the valuable comments that significantly improved the quality of the manuscript. This work has made use of data from the European Space Agency (ESA) mission Gaia ( https://www.cosmos.esa.int/gaia ), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/ gaia/dpac/consortium ). This research made use of Lightkurve, a Python package for Kepler and TESS data analysis (Lightkurve Collaboration et al. 2018). We thank contributors to NumPy (Harris et al. 2020), SciPy (Virtanen et al. 2020), matplotlib (Hunter 2007), astropy (Astropy Collaboration et al. 2013, 2018, 2022), astroquery (Ginsburg et al. 2019), and tesscut (Brasseur et al. 2019). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. We acknowledge financial support from the Agencia Estatal de Investigación of the Ministerio de Ciencia e Innovación MCIN / AEI / 10.13039 / 501100011033 and the ERDF 'A way of making Europe' through project PID2021-125627OB-C32,\n\n\n\n\n\n
Fig. 12: Transit ( top panel ) and eclipse ( bottom panel ) spectroscopic metrics for the TOI-396 planets (see legend). The metrics were calculated using the 2MASS Ks-band magnitude. The grey markers show the metrics for the known sample of transiting exoplanets to date. The blue markers show the metrics for targets with approved JWST programmes.
\n\nand from the Centre of Excellence 'Severo Ochoa' award to the Instituto de Astrofisica de Canarias. This research was funded in part by the UKRI, (Grants ST / X001121 / 1, EP / X027562 / 1). This work was supported by FCT - Fundação para a Ciência e a Tecnologia through national funds and by FEDER through COMPETE2020 - Programa Operacional Competitividade e Internacionalização by these grants: UIDB / 04434 / 2020; UIDP / 04434 / 2020. D.G., A.B., L.F., and L.M.S. gratefully acknowledge the financial support from the grant for internationalization (GAND_GFI_23_01) provided by the University of Turin (Italy). S.G.S acknowledges the support from FCT through Investigador FCT contract nr. CEECIND / 00826 / 2018 and POPH / FSE (EC). P.J.W. acknowledges support from the UK Science and Technology Facilities Council (STFC) through consolidated grants ST / P000495 / 1, ST / T000406 / 1 and ST / X001121 / 1. N.C.S. is funded by the European Union (ERC, FIERCE, 101052347). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. J.L.-B. is funded by the MICIU / AEI / 10.13039 / 501100011033 and NextGenerationEU / PRTR grant PID2019-107061GB-C61 and CNS2023-144309. X.D. acknowledges the support from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement SCORE No 851555) and from the Swiss National Science Foundation under the grant SPECTRE (No 200021_215200). This work has been carried out within the framework of the NCCR PlanetS supported by the Swiss National Science Foundation under grants 51NF40_182901 and 51NF40_205606. G.N. thanks for the research funding from the Ministry of Science and Higher Education programme the \"Excellence Initiative - Research University\" conducted at the Centre of Excellence in Astrophysics and Astrochemistry of the Nicolaus Copernicus University in Toru'n, Poland. Research activities of the Board of Observational and Instrumental Astronomy at the Federal University of Rio Grande do Norte are supported by continuous grants from the Brazilian funding\n\n\n\n\n\n
Fig. 13: Simulations of the atmospheric pressure profiles, eclipse spectra, and JWST observations. Top-left panel : Radiativeequilibrium thermal profile of TOI-396 b assuming a gas-giant atmosphere (5 × solar metallicity) and a secondary atmosphere (80% H 2 O plus 20% CO 2 ). Top-right panels : Volume mixing ratio of TOI-396 b for the most relevant species shaping the infrared spectrum. Bottom panels : Synthetic secondary eclipse spectra of the three TOI-396 planets (solid curves). The round markers with error bars show a realisation of JWST observation with NIRCam / F444W (and their expected uncertainties) when accumulating 2, 4, and 8 observations for planets b, c, and d, respectively. The vertical dashed lines mark the spectral window covered by NIRCam / F444W.
\n\nagencies CNPq. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-Brasil (CAPES)-Finance Code 001 and CAPES-Print program. B.L.C.M., I.C.L., and J.R.M. acknowledge CNPq research fellowships. K.W.F.L. was supported by Deutsche Forschungsgemeinschaft grants RA714 / 14-1, RA714 / 14-2 within the DFG Schwerpunkt SPP 1992, Exploring the Diversity of Extrasolar Planets. L.B. acknowledges support from CHEOPS ASI-INAF agreement n. 2019-29-HH.0. D.G. sincerely thanks Stefano Camera for the inspiring and valuable discussions on the properties of TOI-396.\n\n\n\nReferences\nAddison, B., Wright, D. J., Wittenmyer, R. A., et al. 2019, PASP, 131, 115003 Adibekyan, V., Dorn, C., Sousa, S. G., et al. 2021, Science, 374, 330 Adibekyan, V., Figueira, P., Santos, N. C., et al. 2015, A&A, 583, A94 Adibekyan, V. Z., Sousa, S. G., Santos, N. C., et al. 2012, A&A, 545, A32 Agol, E. & Fabrycky, D. C. 2018, in Handbook of Exoplanets, ed. H. J. Deeg &\n\nJ. A. Belmonte (Cambridge University Press), 7 Agol, E., Ste ff en, J., Sari, R., & Clarkson, W. 2005, MNRAS, 359, 567 Aguichine, A., Mousis, O., Deleuil, M., & Marcq, E. 2021, ApJ, 914, 84 Alexander, R. D. & Armitage, P. J. 2009, ApJ, 704, 989\nAstropy Collaboration, Price-Whelan, A. M., Lim, P. L., et al. 2022, ApJ, 935, 167\nAstropy Collaboration, Price-Whelan, A. M., Sip\"ocz, B. M., et al. 2018, AJ, 156, 123\nAstropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33\nBai, J. & Perron, P. 2003, Journal of Applied Econometrics, 18, 1\n\nBaranne, A., Queloz, D., Mayor, M., et al. 1996, A&AS, 119, 373\nBarnes, S. A. 2010, ApJ, 722, 222\n\nBenz, W., Broeg, C., Fortier, A., et al. 2021, Experimental Astronomy, 51, 109\nBergner, J. B., Öberg, K. I., Bergin, E. A., et al. 2020, ApJ, 898, 97\nBonfanti, A., Delrez, L., Hooton, M. J., et al. 2021, A&A, 646, A157\nBonfanti, A., Gandolfi, D., Egger, J. A., et al. 2023, A&A, 671, L8\nBonfanti, A. & Gillon, M. 2020, A&A, 635, A6\nBonfanti, A., Ortolani, S., & Nascimbeni, V. 2016, A&A, 585, A5\nBonfanti, A., Ortolani, S., Piotto, G., & Nascimbeni, V. 2015, A&A, 575, A18\nBorsato, L., Degen, D., Leleu, A., et al. 2024, A&A, 689, A52\nBorsato, L., Malavolta, L., Piotto, G., et al. 2019, MNRAS, 484, 3233\nBorsato, L., Marzari, F., Nascimbeni, V., et al. 2014, A&A, 571, A38\nBorsato, L., Piotto, G., Gandolfi, D., et al. 2021, MNRAS, 506, 3810 Borysow, A. 2002, A&A, 390, 779\nBorysow, A., Jorgensen, U. G., & Fu, Y. 2001, J. Quant. Spectr. Rad. Transf., 68, 235\nBrasseur, C. E., Phillip, C., Fleming, S. W., Mullally, S. E., & White, R. L. 2019, Astrocut: Tools for creating cutouts of TESS images, Astrophysics Source Code Library, record ascl:1905.007\n\nBruntt, H., Deleuil, M., Fridlund, M., et al. 2010, A&A, 519, A51\nBurrows, A., Marley, M. S., & Sharp, C. M. 2000, ApJ, 531, 438\n\nCiardi, D. R., Fabrycky, D. C., Ford, E. B., et al. 2013, ApJ, 763, 41\nCorreia, A. C. M., Delisle, J.-B., & Laskar, J. 2018, in Handbook of Exoplanets,\ned. H. J. Deeg & J. A. Belmonte (Springer International Publishing AG, part of Springer Nature), 12\nCubillos, P. E. 2017, ApJ, 850, 32\nCubillos, P. E. & Blecic, J. 2021, MNRAS, 505, 2675\nCutri, R. M., Skrutskie, M. F., van Dyk, S., et al. 2003, 2MASS All Sky Catalog of point sources. (University of Massachusetts and Infrared Processing and Analysis Center (IPAC / California Institute of Technology))\n\nD'Angelo, G. & Lubow, S. H. 2008, ApJ, 685, 560\nDelisle, J. B., Laskar, J., Correia, A. C. M., & Boué, G. 2012, A&A, 546, A71\nDelrez, L., Ehrenreich, D., Alibert, Y., et al. 2021, Nature Astronomy, 5, 775\nDorn, C., Venturini, J., Khan, A., et al. 2017, A&A, 597, A37\n\nDoyle, A. P., Davies, G. R., Smalley, B., Chaplin, W. J., & Elsworth, Y. 2014, MNRAS, 444, 3592\nDragomir, D., Kane, S. R., Henry, G. W., et al. 2012, ApJ, 754, 37\nDumusque, X., Santos, N. C., Udry, S., Lovis, C., & Bonfils, X. 2011a, A&A, 527, A82\nDumusque, X., Udry, S., Lovis, C., Santos, N. C., & Monteiro, M. J. P. F. G. 2011b, A&A, 525, A140\n\nEgger, J. A., Osborn, H. P., Kubyshkina, D., et al. 2024, A&A, 688, A223\n\nEspinoza, N. & Jordán, A. 2015, MNRAS, 450, 1879\nFabrycky, D. C., Lissauer, J. J., Ragozzine, D., et al. 2014, ApJ, 790, 146\nForeman-Mackey, D., Farr, W., Sinha, M., et al. 2019, The Journal of Open Source Software, 4, 1864\nForeman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306\nFridlund, M., Georgieva, I. Y., Bonfanti, A., et al. 2024, A&A, 684, A12\nGaia Collaboration, Vallenari, A., Brown, A. G. A., et al. 2023, A&A, 674, A1\nGandolfi, D., Barragán, O., Hatzes, A. P., et al. 2017, AJ, 154, 123\nGardner, J. P., Mather, J. C., Clampin, M., et al. 2006, Space Sci. Rev., 123, 485 Gelman, A. & Rubin, D. B. 1992, Statistical Science, 7, 457\nGeweke, J. F. 1991, Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments, Sta ff Report 148, Federal Reserve Bank of Minneapolis\nGinsburg, A., Sip\"ocz, B. M., Brasseur, C. E., et al. 2019, AJ, 157, 98\nGoldreich, P. & Tremaine, S. 1980, ApJ, 241, 425\nGoodman, J. & Weare, J. 2010, Communications in Applied Mathematics and Computational Science, 5, 65\nGray, R. O., Corbally, C. J., Garrison, R. F., et al. 2006, AJ, 132, 161\nHaldemann, J., Dorn, C., Venturini, J., Alibert, Y., & Benz, W. 2024, A&A, 681, A96\nHandberg, R., Lund, M. N., White, T. R., et al. 2021, AJ, 162, 170\nHarris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357\n\nHatzes, A. P. 2016, in Astrophysics and Space Science Library, Vol. 428, Meth-\n\nods of Detecting Exoplanets: 1st Advanced School on Exoplanetary Science, ed. V. Bozza, L. Mancini, & A. Sozzetti, 3\nHatzes, A. P. 2019, The Doppler Method for the Detection of Exoplanets (Institute of Physics Publishing)\nHatzes, A. P., Dvorak, R., Wuchterl, G., et al. 2010, A&A, 520, A93 Haywood, R. D., Collier Cameron, A., Queloz, D., et al. 2014, MNRAS, 443, 2517\nHøg, E., Fabricius, C., Makarov, V. V., et al. 2000, A&A, 355, L27\n\nHolman, M. J., Fabrycky, D. C., Ragozzine, D., et al. 2010, Science, 330, 51\nHolman, M. J., Winn, J. N., Latham, D. W., et al. 2006, ApJ, 652, 1715\n\nHunter, J. D. 2007, Computing in Science & Engineering, 9, 90\n\nIzidoro, A., Ogihara, M., Raymond, S. N., et al. 2017, MNRAS, 470, 1750\n\nJenkins, J. M., Twicken, J. D., McCauli ff , S., et al. 2016, in Society of PhotoOptical Instrumentation Engineers (SPIE) Conference Series, Vol. 9913, Software and Cyberinfrastructure for Astronomy IV, ed. G. Chiozzi & J. C. Guzman, 99133E\nKass, R. E. & Raftery, A. E. 1995, Journal of the American Statistical Association, 90, 773\nKempton, E. M. R., Bean, J. L., Louie, D. R., et al. 2018, PASP, 130, 114401 Kipping, D. M. 2010, MNRAS, 408, 1758\nKossakowski, D., Kürster, M., Henning, T., et al. 2022, A&A, 666, A143\nKuerster, M., Schmitt, J. H. M. M., Cutispoto, G., & Dennerl, K. 1997, A&A, 320, 831\nKurucz, R. L. 1970, SAO Special Report, 309\nKurucz, R. L. 1993, SYNTHE spectrum synthesis programs and line data (Astrophysics Source Code Library)\nKurucz, R. L. 2013, ATLAS12: Opacity sampling model atmosphere program\nLaskar, J. 1997, A&A, 317, L75\nLaskar, J. 2000, Phys. Rev. Lett., 84, 3240\nLaskar, J. & Petit, A. C. 2017, A&A, 605, A72\nLee, M. H. & Peale, S. J. 2002, arXiv e-prints, astro\nLeleu, A., Alibert, Y., Hara, N. C., et al. 2021, A&A, 649, A26\nLi, M., Huang, S., Petaev, M. I., Zhu, Z., & Ste ff en, J. H. 2020, MNRAS, 495, 2543\nLightkurve Collaboration, Cardoso, J. V. d. M., Hedges, C., et al. 2018, Lightkurve: Kepler and TESS time series analysis in Python, Astrophysics Source Code Library\nLin, D. N. C. & Papaloizou, J. 1979, MNRAS, 186, 799\nLindegren, L., Bastian, U., Biermann, M., et al. 2021, A&A, 649, A4\n\nLissauer, J. J., Ragozzine, D., Fabrycky, D. C., et al. 2011, ApJS, 197, 8\nLovis, C. & Pepe, F. 2007, A&A, 468, 1115\n\nLund, M. N., Handberg, R., Buzasi, D. L., et al. 2021, ApJS, 257, 53\nLuque, R., Osborn, H. P., Leleu, A., et al. 2023, Nature, 623, 932\n\nMamajek, E. E. & Hillenbrand, L. A. 2008, ApJ, 687, 1264\n\nMarigo, P., Girardi, L., Bressan, A., et al. 2017, ApJ, 835, 77\nMayor, M., Pepe, F., Queloz, D., et al. 2003, The Messenger, 114, 20\n\nMcKay, A. J., DiSanti, M. A., Kelley, M. S. P., et al. 2019, AJ, 158, 128\n\nMishra, L., Alibert, Y., Udry, S., & Mordasini, C. 2023, A&A, 670, A68 Morton, T. D. 2012, ApJ, 761, 6\nMorton, T. D. 2015, VESPA: False positive probabilities calculator, Astrophysics Source Code Library, record ascl:1503.011\nMumma, M. J. & Charnley, S. B. 2011, ARA&A, 49, 471\nMurdoch, K. A., Hearnshaw, J. B., & Clark, M. 1993, ApJ, 413, 349\nNascimbeni, V., Borsato, L., Leonardi, P., et al. 2024, A&A, 690, A349\nNascimbeni, V., Borsato, L., Zingales, T., et al. 2023, A&A, 673, A42\nNelson, B., Ford, E. B., & Payne, M. J. 2014, ApJS, 210, 11\nNoyes, R. W., Hartmann, L. W., Baliunas, S. L., Duncan, D. K., & Vaughan, A. H. 1984, ApJ, 279, 763\nOtegi, J. F., Bouchy, F., & Helled, R. 2020a, A&A, 634, A43\nOtegi, J. F., Dorn, C., Helled, R., et al. 2020b, A&A, 640, A135\nOtegi, J. F., Helled, R., & Bouchy, F. 2022, A&A, 658, A107\nParviainen, H., Pallé, E., Nortmann, L., et al. 2016, A&A, 585, A114\nPepe, F., Mayor, M., Galland, F., et al. 2002, A&A, 388, 632\nPerryman, M. A. C., Lindegren, L., Kovalevsky, J., et al. 1997, A&A, 323, L49\nPersson, C. M., Fridlund, M., Barragán, O., et al. 2018, A&A, 618, A33\nPetigura, E. A., Benneke, B., Batygin, K., et al. 2018, AJ, 156, 89\nPetigura, E. A., Howard, A. W., Lopez, E. D., et al. 2016, ApJ, 818, 36\nPetit, A. C., Laskar, J., & Boué, G. 2018, A&A, 617, A93\nPiskunov, N. & Valenti, J. A. 2017, A&A, 597, A16\nPiskunov, N. E., Kupka, F., Ryabchikova, T. A., Weiss, W. W., & Je ff ery, C. S. 1995, A&AS, 112, 525\nPont, F., Zucker, S., & Queloz, D. 2006, MNRAS, 373, 231\n\nPontoppidan, K. M., Pickering, T. E., Laidler, V. G., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9910, Observatory Operations: Strategies, Processes, and Systems VI, ed.\n\nA. B. Peck, R. L. Seaman, & C. R. Benn, 991016\n\nQueloz, D., Henry, G. W., Sivan, J. P., et al. 2001, A&A, 379, 279\nRein, H. & Liu, S. F. 2012, A&A, 537, A128\n\nRein, H. & Tamayo, D. 2015, MNRAS, 452, 376\n\nRichard,\nC.,\nGordon,\nI.\nE.,\nRothman,\nL.\nS.,\net\nal.\n2012,\nJ. Quant. Spectr. Rad. Transf., 113, 1276\n\nRicker, G. R., Winn, J. N., Vanderspek, R., et al. 2015, Journal of Astronomical Telescopes, Instruments, and Systems, 1, 014003\nRothman, L. S., Gordon, I. E., Barber, R. J., et al. 2010, J. Quant. Spectr. Rad. Transf., 111, 2139\nSantos, N. C., Sousa, S. G., Mortier, A., et al. 2013, A&A, 556, A150 Schwarz, G. 1978, Annals of Statistics, 6, 461\nSimola, U., Bonfanti, A., Dumusque, X., et al. 2022, A&A, 664, A127\nSimola, U., Dumusque, X., & Cisewski-Kehe, J. 2019, A&A, 622, A131\nSmith, J. C., Stumpe, M. C., Van Cleve, J. E., et al. 2012, PASP, 124, 1000\n\nSneden, C. A. 1973, PhD thesis, THE UNIVERSITY OF TEXAS AT AUSTIN.\n\nSousa, S. G. 2014, in Determination of Atmospheric Parameters of B-, A-, Fand G-Type Stars., ed. E. Niemczura, B. Smalley, & W. Pych (Springer International Publishing (Cham)), 297-310\nSousa, S. G., Adibekyan, V., Delgado-Mena, E., et al. 2021, A&A, 656, A53 Sousa, S. G., Santos, N. C., Adibekyan, V., Delgado-Mena, E., & Israelian, G. 2015, A&A, 577, A67\nSousa, S. G., Santos, N. C., Israelian, G., Mayor, M., & Monteiro, M. J. P. F. G. 2007, A&A, 469, 783\nSousa, S. G., Santos, N. C., Mayor, M., et al. 2008, A&A, 487, 373\nStorn, R. & Price, K. 1997, Journal of Global Optimization, 11, 341\nStumpe, M. C., Smith, J. C., Catanzarite, J. H., et al. 2014, PASP, 126, 100\nStumpe, M. C., Smith, J. C., Van Cleve, J. E., et al. 2012, PASP, 124, 985\nSuárez Mascareño, A., Rebolo, R., González Hernández, J. I., & Esposito, M. 2017, MNRAS, 468, 4772\nSundman, K. F. 1913, Acta Mathematica, 36, 105\nTanaka, H., Takeuchi, T., & Ward, W. R. 2002, ApJ, 565, 1257\nTennyson, J., Yurchenko, S. N., Al-Refaie, A. F., et al. 2016, Journal of Molecular Spectroscopy, 327, 73\n\nter Braak, C. J. F. & Vrugt, J. A. 2008, Statistics and Computing, 18, 435\nThiabaud, A., Marboeuf, U., Alibert, Y., et al. 2014, A&A, 562, A27\n\nThiabaud, A., Marboeuf, U., Alibert, Y., Leya, I., & Mezger, K. 2015, A&A, 574, A138\nTrotta, R. 2007, MNRAS, 378, 72\nVan Eylen, V., Albrecht, S., Huang, X., et al. 2019, AJ, 157, 61\nVanderburg, A., Huang, C. X., Rodriguez, J. E., et al. 2019, ApJ, 881, L19\nVanderburg, A., Plavchan, P., Johnson, J. A., et al. 2016, MNRAS, 459, 3565\nVirtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261 Walsh, C., Nomura, H., & van Dishoeck, E. 2015, A&A, 582, A88\nWeiss, L. M. & Marcy, G. W. 2014, ApJ, 783, L6\nWeiss, L. M., Marcy, G. W., Petigura, E. A., et al. 2018, AJ, 155, 48\nWildi, F., Pepe, F., Chazelas, B., Lo Curto, G., & Lovis, C. 2010, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7735, Ground-based and Airborne Instrumentation for Astronomy III, ed. I. S. McLean, S. K. Ramsay, & H. Takami, 77354X\nWinn, J. N. 2010, in Exoplanets, ed. S. Seager (University of Arizona Press, Tucson, AZ), 55-77\nWinn, J. N. & Fabrycky, D. C. 2015, ARA&A, 53, 409\n\nWisdom, J. & Holman, M. 1991, AJ, 102, 1528\n\nZechmeister, M. & Kürster, M. 2009, A&A, 496, 577\n1 Austrian Academy of Sciences, Schmiedlstrasse 6, A-8042 Graz, Austria\ne-mail: andrea.bonfanti@oeaw.ac.at\n2 Dipartimento di Fisica, Università degli Studi di Torino, via Pietro Giuria 1, I-10125, Torino, Italy\n3 Department of Physics and Astronomy, Uppsala University, Box 516, Uppsala SE-75120, Sweden\n4 INAF, Osservatorio Astronomico di Padova, Vicolo dell'Osservatorio 5, 35122 Padova, Italy\n5 Weltraumforschung und Planetologie, Physikalisches Institut, University of Bern, Gesellschaftsstrasse 6, 3012 Bern, Switzerland\n6 INAF, Osservatorio Astrofisico di Torino, Via Osservatorio, 20, I10025 Pino Torinese To, Italy\n7 Department of Physics, University of Warwick, Coventry CV4 7AL, UK\n8 Centre for Exoplanets and Habitability, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK\n\n\n9 Departamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, Campus Universitário, Natal, RN, 59072-970, Brazil\n10 Leiden Observatory, University of Leiden, PO Box 9513, 2300 RA Leiden, The Netherlands\n11 Chalmers University of Technology, Department of Space, Earth and Environment, Onsala Space Observatory, SE-439 92 Onsala, Sweden\n12 INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5 Florence, Italy\n13 Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal\n14 School of Physics and Astronomy, Physical Science Building, North Haugh, St Andrews, United Kingdom\n15 Rhenish Institute for Environmental Research, Dep. of Planetary Research, University of Cologne, Aachener Str. 209, 50931 Cologne, Germany\n16 Institute of Planetary Research, German Aerospace Center (DLR), Rutherfordstrasse 2, D-12489 Berlin, Germany\n17 Thüringer Landessternwarte Tautenburg, Sternwarte 5, D-07778 Tautenburg, Germany\n18 University Observatory Munich, Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679 Munich, Germany\n19 European Southern Observatory, Alonso de Cordova 3107, Vitacura, Casilla, 19001, Santiago, Chile\n20 European Southern Observatory, Alonso de Cordova 3107, Vitacura, Santiago de Chile, Chile\n21 Centro de Astrobiología (CAB), CSIC-INTA, Camino Bajo del Castillo s / n, 28692, Villanueva de la Cañada (Madrid), Spain\n22 McDonald Observatory, The University of Texas, Austin Texas USA\n23 Center for Planetary Systems Habitability, The University of Texas, Austin Texas\n24 Department of Astronomy & Astrophysics, University of Chicago, Chicago, IL 60637, USA\n25 Astronomy Department and Van Vleck Observatory, Wesleyan University, Middletown, CT 06459, USA\n26 Departamento de Física e Astronomia, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre, 4169-007 Porto, Portugal\n27 Department of Astronomy of the University of Geneva, chemin Pegasi 51, 1290 Versoix, Switzerland\n28 Astrobiology Center, NINS, 2-21-1 Osawa, Mitaka, Tokyo 1818588, Japan\n29 National Astronomical Observatory of Japan, NINS, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan\n30 Astronomical Science Program, Graduate University for Advanced Studies, SOKENDAI, 2-21-1, Osawa, Mitaka, Tokyo, 181-8588, Japan\n31 Instituto de Astrofísica de Canarias (IAC), calle Vía Láctea s / n, 38205 La Laguna, Tenerife, Spain\n32 Departamento de Astrofísica, Universidad de La Laguna (ULL), 38206 La Laguna, Tenerife, Spain\n33 Institute of Astronomy, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudzia¸dzka 5, 87-100 Toru'n, Poland\n34 Department of Physics and Astronomy, McMaster University, 1280 Main St W, Hamilton, ON, L8S 4L8, Canada\n35 Department of Physics, ETH Zurich, Wolfgang-Pauli-Strasse 2, CH8093 Zurich, Switzerland\n\nA. Bonfanti\n\n\n
Fig. A.1: TTV amplitudes obtained for TOI-396 b (black markers) and TOI-396 c (red markers). This is a superposition of the first two panels of Figure 4 emphasising the anti-correlation pattern.
\n\nAppendix A: Supplementary material\n\n\n
Fig. A.2: Left column : Generalised Lomb-Scargle periodograms of TESS Sector 3 ( first row ), 4 ( second row ), 30 ( third row ), and 31 ( fourth row ) raw photometry. Right column : Generalised Lomb-Scargle periodograms of the residuals after removing the best fitting sinusoidal signals at the most significant frequency identified in the periodograms of the corresponding TESS time series (left column). The long dashed red line marks the FAP at 0.1%. Significant peaks are detected in the 6-8 day range, which we attribute to the presence of active regions co-rotating with the star.
\n\nA. Bonfanti\net al.: Radii, masses, and transit-timing variations of the three-planet system orbiting the naked-eye star TOI-396\n\n\n
Table A.1: Radial velocities RV as extracted from the centred CCFs with their errors σ RV. They are followed by the hyperparameters inferred from the SN fit onto the CCFs (i.e. FWHMSN, A , and γ ) and by the bp -detrended RV values ( RV bp) with their errors which also account for the jitter ( σ RV(bp + jitter)).
\n\n
\n\n
\nA & A proofs: manuscript no. TOI-396\n
\n\n
Table A.1: continued.
\n
\n\n\n\n
\n\n
Table A.2: Polynomial detrending baseline applied to the TESS LCs within the MCMC scheme.
\n
\nNotes. TESS (TE) LCs are identified by a counter based on the chronological order of observation. In particular, LCs from 1 to 11, from 12 to 21, from 22 to 32, and from 33 to 41 were extracted from Sector 3, 4, 30, and 31, respectively. c indicates a normalisation scalar; see text for further details.\n
\n\n
Table A.3: Timing of each transit event T tr and the corresponding TTV amplitude as computed with respect to linear ephemerides. The last column specifies the TESS sector where the transit occurs.
Table A.4: Results of the internal structure modelling for TOI-396 b. The 'w · ' symbol represents the mass fraction with respect to the total planet mass, Zenvelope is the water mass fraction in the planet envelope, while the 'x · ' symbol represents the molar fraction of a given chemical element either in the planet core (x · , core) or in the mantle (x · , mantle).
\n"},"sections":{"kind":"list like","value":[{"title":"ABSTRACT","content":"Context. TOI-396 is an F6 V bright naked-eye star ( V ≈ 6.4) orbited by three small ( Rp ≈ 2 R ⊕ ) transiting planets discovered thanks to space-based photometry from two TESS sectors. The orbital periods of the two innermost planets, namely TOI-396 b and c, are close to the 5:3 commensurability ( Pb ∼ 3.6 d and Pc ∼ 6.0 d), suggesting that the planets might be trapped in a mean motion resonance (MMR). Aims. To measure the masses of the three planets, refine their radii, and investigate whether planets b and c are in MMR, we carried out HARPS radial velocity (RV) observations of TOI-396 and retrieved archival high-precision transit photometry from four TESS sectors. Methods. We extracted the RVs via a skew-normal fit onto the HARPS cross-correlation functions and performed a Markov chain Monte Carlo joint analysis of the Doppler measurements and transit photometry while employing the breakpoint method to remove stellar activity from the RV time series. We also performed a transit timing variation (TTV) dynamical analysis of the system and simulated the temporal evolution of the TTV amplitudes of the three planets following an N-body numerical integration. Results. Our analysis confirms that the three planets have similar sizes ( Rb = 2 . 004 + 0 . 045 -0 . 047 R ⊕ ; Rc = 1 . 979 + 0 . 054 -0 . 051 R ⊕ ; Rd = 2 . 001 + 0 . 063 -0 . 064 R ⊕ ) and is thus in agreement with previous findings. However, our measurements are ∼ 1.4 times more precise thanks to the use of two additional TESS sectors. For the first time, we have determined the RV masses for TOI-396 b and d, finding them to be Mb = 3 . 55 + 0 . 94 -0 . 96 M ⊕ and Md = 7 . 1 ± 1 . 6 M ⊕ , which implies bulk densities of ρ b = 2 . 44 + 0 . 69 -0 . 68 g cm -3 and ρ d = 4 . 9 + 1 . 2 -1 . 1 g cm -3 , respectively. Our results suggest a quite unusual system architecture, with the outermost planet being the densest. Based on a frequency analysis of the HARPS activity indicators and TESS light curves, we find the rotation period of the star to be P rot ,⋆ = 6 . 7 ± 1 . 3 d, in agreement with the value predicted from log R ' HK -based empirical relations. The Doppler reflex motion induced by TOI-396 c remains undetected in our RV time series, likely due to the proximity of the planet's orbital period to the star's rotation period. We also discovered that TOI-396 b and c display significant TTVs. While the TTV dynamical analysis returns a formally precise mass for TOI-396 c of Mc , dyn = 2 . 24 + 0 . 13 -0 . 67 M ⊕ , the result might not be accurate, owing to the poor sampling of the TTV phase. We also conclude that TOI-396 b and c are close to but out of the 5:3 MMR. Conclusions. A TTV dynamical analysis of additional transit photometry evenly covering the TTV phase and super-period is likely the most e ff ective approach for precisely and accurately determining the mass of TOI-396 c. Our numerical simulation suggests TTV semi-amplitudes of up to 5 hours over a temporal baseline of ∼ 5.2 years, which should be duly taken into account when scheduling future observations of TOI-396. Key words. planets and satellites: fundamental parameters - stars: fundamental parameters - techniques: photometric - techniques: radial velocities","pages":[1]},{"title":"Radii, masses, and transit-timing variations of the three-planet system orbiting the naked-eye star TOI-396 ⋆","content":"A. Bonfanti 1 , I. Amateis 2 , 1 , 3 , D. Gandolfi 2 , L. Borsato 4 , J. A. Egger 5 , P. E. Cubillos 1 , 6 , D. Armstrong 7 , 8 , I. C. Leão 9 , M. Fridlund 10 , 11 , B. L. Canto Martins 9 , 12 , S. G. Sousa 13 , J. R. De Medeiros 9 , L. Fossati 1 , V. Adibekyan 13 , A. Collier Cameron 14 , S. Grziwa 15 , K. W. F. Lam 16 , E. Go ff o 17 , L. D. Nielsen 18 , F. Rodler 19 , J. Alarcon 20 , J. Lillo-Box 21 , W. D. Cochran 22 , 23 , R. Luque 24 , S. Redfield 25 , N. C. Santos 13 , 26 , S. C. C. Barros 13 , 26 , D. Bayliss 7 , 8 , X. Dumusque 27 , M. A. F. Keniger 7 , 8 , J. Livingston 28 , 29 , 30 , F. Murgas 31 , 32 , G. Nowak 33 , A. Osborn 34 , H. P. Osborn 5 , 35 , E. Pallé 31 , 32 , C. M. Persson 11 , L. M. Serrano 2 , P. A. Strøm 7 , 8 , S. Udry 27 , and P. J. Wheatley 7 , 8 (A ffi liations can be found after the references)","pages":[1]},{"title":"1. Introduction","content":"Multi-planet systems enable us to place significantly stronger constraints on formation and evolution mechanisms compared to single-planet systems (e.g. Lissauer et al. 2011; Fabrycky et al. 2014; Winn & Fabrycky 2015; Mishra et al. 2023). As a matter of fact, the temporal evolution of the gas content in the proto-planetary disc influences planet migration (e.g. Lin & Papaloizou 1979; Goldreich & Tremaine 1980; Tanaka et al. 2002; D'Angelo & Lubow 2008; Alexander & Armitage 2009), which shapes the orbital architecture of a planetary system. The latter is further expected to correlate with the planet composition (e.g. Thiabaud et al. 2014, 2015; Walsh et al. 2015; Bergner et al. 2020; Li et al. 2020) that can be inferred once the physical parameters of the planets are known (e.g. Dorn et al. 2017; Otegi et al. 2020b; Leleu et al. 2021; Haldemann et al. 2024). If planets are found in mean motion resonance (MMR; e.g. Lee & Peale 2002; Correia et al. 2018), systems can also shed light on the migration mechanisms during formation, as well as on the impact of tidal e ff ects occurring later on (e.g. Delisle et al. 2012; Izidoro et al. 2017). In addition, planets in, or close to, MMR likely exhibit transit timing variations (TTVs; see e.g. Agol et al. 2005; Agol & Fabrycky 2018; Leleu et al. 2021) that enable one to infer planetary masses without necessarily relying on radial velocity (RV) measurements, which are not always possible (e.g. Hatzes 2016). Multi-planet systems also give insights into correlations between physical and orbital parameters of exoplanets. For example, Weiss et al. (2018) noticed that planets in adjacent orbits usually show similar radii, hence the 'peas in a pod' label to describe this scenario. Weiss et al. (2018) further noticed that the outermost planet is the largest in the majority of cases, which agrees with the observed bulk planet density ( ρ p ) trend highlighted by Mishra et al. (2023), where ρ p decreases with the distance from the stellar host as outer planets are expected to be richer in volatiles (thus bigger and less dense). The object TOI-396 represents an interesting laboratory to test these theories as it is the brightest star known so far to host three transiting planets (Vanderburg et al. 2019) after ν 2 Lupi (Delrez et al. 2021). After analysing two sectors from the Transiting Exoplanet Survey Satellite ( TESS ; Ricker et al. 2015), Vanderburg et al. (2019) found that the three planets have radii of ∼ 2 R ⊕ and orbital periods of ∼ 3.6, ∼ 6.0, and ∼ 11.2 d, with TOI-396 c and b showing a period commensurability close to the 5:3 ratio. Following the notation introduced in Mishra et al. (2023), the three planets are 'similar' in terms of radii, and one may wonder whether this architecture class is kept also in the mass-period diagram. Mishra et al. (2023) found a positive and strong correlation of the coe ffi cients of similarity between radii and masses, though exceptions are possible (e.g. Weiss & Marcy 2014; Otegi et al. 2020a, 2022). In this work, we complement the photometric analysis of new TESS light curves (LCs) with RV observations taken with the High Accuracy Radial Velocity Planet Searcher ( HARPS ; Mayor et al. 2003) spectrograph to refine the planet radii and constrain for the first time the planetary masses. Considering the possible 5:3 MMR between TOI-396 c and b, we also simulate the evolution in time of the TTV amplitudes. This paper is organised as follows: Section 2 presents the stellar properties, and Sect. 3 describes the photometric and RV data. After outlining the method to jointly analyse the TESS LCs and the HARPS RV time series in Sect. 4, we present the corresponding results in Sect. 5. We attempt to dynamically model TTV and RV data simultaneously and track the temporal evolution of the TTV signals in Sect. 6, and we study the planets' internal structure in Sect. 7 and explore the prospects for characterising the system with the James Webb Space Telescope (JWST; Gardner et al. 2006) in Sect. 8. Finally Sect. 9 gathers the conclusions.","pages":[1,2]},{"title":"2. Host star characterisation","content":"TOI-396 is an F6 V (Gray et al. 2006) bright naked-eye star with an apparent visual magnitude of V ≈ 6.4 (Perryman et al. 1997). It is located ∼ 31.7 pc away and is visible in the constellation of Fornax in the southern hemisphere. It is member of a visual binary system and its companion HR 858 B is a faint M-dwarf ( G ∼ 16 mag), about 8.4 '' away from the main component. We co-added 78 HARPS spectra (see Sect. 3.2 for further details) and then modelled it with Spectroscopy Made Easy 1 ( SME ; Piskunov & Valenti 2017) version 5.2.2. SME computes synthetic spectra from a grid of well established stellar atmosphere models and adjusts a chosen free parameter based on comparison with the observed spectrum. Here we used the stellar atmosphere grid A tlas 12 (Kurucz 2013) together with atomic line lists from the V ald database (Piskunov et al. 1995) in order to produce the synthetic spectra. We modelled one parameter at a time utilising spectral features sensitive to di ff erent photospheric parameters iterating until convergence of all free parameters. Throughout the modelling, we held the macro- and micro-turbulent velocities, v mac and v mic, fixed to 6 km s -1 (Doyle et al. 2014) and 1.34 km s -1 (Bruntt et al. 2010), respectively. A description of the modelling procedure is detailed in Persson et al. (2018). Finally, we obtained T e ff = 6354 ± 70 K, [Fe / H] = 0 . 025 ± 0 . 050, log g = 4 . 30 ± 0 . 06, and v sin i ⋆ = 7 . 5 ± 0 . 2 kms -1 . To double-check the derived spectroscopy parameters we performed an additional analysis employing ARES + MOOG (Sousa et al. 2021; Sousa 2014; Santos et al. 2013). In detail, we used the latest version of ARES 2 (Sousa et al. 2007, 2015) to consistently measure the equivalent widths (EW) for the list of iron lines presented in Sousa et al. (2008). Following a minimisation process, we then find the ionisation and excitation equilibria to converge for the best set of spectroscopic parameters. This process uses a grid of Kurucz model atmospheres (Kurucz 1993) and the radiative transfer code MOOG (Sneden 1973). We obtained T e ff = 6389 ± 67 K, [Fe / H] = -0 . 014 ± 0 . 045 dex, log g = 4 . 58 ± 0 . 11, and v mic = 1 . 54 ± 0 . 04 kms -1 . In this process we also derived a more accurate trigonometric surface gravity (log g trig = 4 . 34 ± 0 . 02) using recent GAIA data following the same procedure as described in Sousa et al. (2021). In the end ARES + MOOG provides consistent values when compared with the ones derived with SME . Using the SME stellar atmospheric parameters, we determined the abundances of Mg and Si following the classical curve-of-growth analysis method described in Adibekyan et al. (2012, 2015). Similar to the stellar parameter determination, we used ARES to measure the EWs of the spectral lines of these elements and a grid of Kurucz model atmospheres (Kurucz 1993) along with the radiative transfer code MOOG to convert the EWs into abundances, assuming local thermodynamic equilibrium. The stellar radius R ⋆ , mass M ⋆ , and age t ⋆ were derived homogeneously using the isochrone placement algorithm (Bonfanti et al. 2015, 2016) and its capability of interpolating a flexible set of input parameters within pre-computed grids of PARSEC 3 v1.2S (Marigo et al. 2017) isochrones and tracks. For each magnitude listed in Table 1, we performed an isochrone placement run by inserting the spectroscopic parameters derived above, the Gaia parallax π (Gaia Collaboration et al. 2023, o ff set-corrected following Lindegren et al. 2021), and the magnitude value to obtain an estimate for the stellar radius, mass, and age along with their uncertainties. From these results, we built the corresponding Gaussian probability density functions (PDFs) and then we merged (i.e. we summed) the PDFs derived from the di ff erent runs to obtain robust estimates for R ⋆ , M ⋆ , and t ⋆ . The radius R ⋆ derives essentially from T e ff , π , and the stellar magnitude, while M ⋆ and t ⋆ are much more model-dependent; therefore we conservatively inflated their internal uncertainties by 4% and 20%, respectively, following Bonfanti et al. (2021). Our adopted stellar parameters are listed in Table 1.","pages":[2]},{"title":"3.1. TESS photometry","content":"As presented in Vanderburg et al. (2019), TOI-396 was photometrically monitored by TESS during the first year of its nominal mission in Sector 3 from 20 September to 17 October 2018 (UT) in CCD 2 of Camera 2, and in Sector 4 from 19 October to 14 November 2018 (UT) in CCD 2 of Camera 1. TOI-396 was later Table 1: Stellar properties of TOI-396. Star names TOI-396 TIC 178155732 HR 858 HD 17926 HIP 13363 Gaia DR3 5064574724769475968 Notes. RA & DEC are reported as in J2000 reference frame. Values in the bottom half of the table have been derived as part of this paper. ( a ) Zero-point correction from Lindegren et al. (2021) applied. re-observed by TESS during the first year of its extended mission in Sector 30 from 23 September to 20 October 2020 (UT) in CCD 2 of Camera 2, and in Sector 31 from 22 October to 16 November 2020 (UT) in CCD 2 of Camera 1. All data were collected at a 2-minute cadence, except for Sector 3 for which TESS only sent down data at 30-minute cadence. We analyse all available TESS time series, including the Sector 3 and 4 data already presented in (Vanderburg et al. 2019). For Sector 3 we used the TESS Asteroseismic Science Opera- tion Center (TASOC) photometry (Handberg et al. 2021; Lund et al. 2021), while for the other sectors we analysed the Presearch Data Conditioning Simple Aperture Photometry (PDCSAP) LCs generated by the TESS Science Processing Operation Center (SPOC) pipeline Jenkins et al. (2016), which removes the majority of instrumental artefacts and systematic trends (Smith et al. 2012; Stumpe et al. 2012, 2014). We rejected those data marked with a bad quality flag and we performed a 5 medianabsolute-deviation (MAD) clipping of flux data points. After that, we built our custom LCs by splitting the TESS sectors in temporal windows centred around the transit events keeping ∼ 4 h of out-of-transit data points both before and after the transit for de-trending purposes. We ended up with 41 TESS LCs reporting the epoch of observation ( t ), the flux and its error, and further ancillary vectors available from the TESS science data product, that is mom _ centr 1, mom _ centr 2 (hereafter denoted with x and y , respectively), pos _ corr 1, and pos _ corr 2 (hereafter denoted with pc1 and pc2 , respectively) 4 .","pages":[2,3]},{"title":"3.2. HARPS high-resolution spectroscopy","content":"We performed the radial velocity (RV) follow-up of TOI-396 with the HARPS spectrograph mounted at the ESO-3.6 m telescope at La Silla Observatory (Chile). We acquired 77 high resolution ( R ≈ 115 000) spectra between 31 January and 27 July 2019 (UT), covering a baseline of ∼ 177 days, as part of the follow-up programs of TESS systems carried out with the HARPS spectrograph (IDs: 1102.C-0923, 1102.C-0249, 0102.C0584; PIs: Gandolfi, Armstrong, De Medeiros). One additional spectrum was acquired during a technical night (ID: 60.A-9700) in February 2019. Following Dumusque et al. (2011b), we averaged out p-mode stellar pulsations by setting the exposure time to 900 s, which led to a median signal-to-noise (S / N) ratio of ∼ 320 per pixel at 550 nm. We used the second fibre of the HARPS spectrograph to simultaneously observe a Fabry-Perot lamp and trace possible instrumental drift down to the sub-metre per second level (Wildi et al. 2010). We reduced the data using the dedicated HARPS data reduction software ( DRS ; Pepe et al. 2002; Lovis, C. & Pepe, F. 2007) and computed the crosscorrelation function (CCF) for each spectrum using a G2 numerical mask (Baranne et al. 1996). We then performed a skew-normal (SN) fit on each CCF (Simola et al. 2019) in orderto extract the stellar radial velocity along with its error, the full width at half maximum (FWHMSN), the contrast ( A ), and the skewness parameter ( γ ) of the CCF. The advantages of using an SN-fit rather than a Normal fit are thoroughly discussed in Simola et al. (2019), while the SN-fitting details are outlined in, for example, Bonfanti et al. (2023); Luque et al. (2023); Fridlund et al. (2024). After the SN-based extraction, we ended up with an RV time series, whose ancillary vectors (FWHMSN, A , γ ) are the activity indicators used to constrain the polynomial basis to model and de-trend the RV component of the stellar activity (see Table A.1). As stellar activity is not stationary, the correlations between the RV observations and the activity indicators are expected to change over time as discussed in Simola et al. (2022), who proposed to apply the breakpoint ( bp ) technique (Bai & Perron 2003) to check whether these correlation changes are statistically significant. If so, the bp algorithm finds the optimal locations of correlation changes, which defines the segmenta- tion characterised by the lowest Bayesian Information Criterion (BIC; Schwarz 1978). Indeed, we found one breakpoint at observation 48 (BJD = 2458667.940719; ∆ BIC ≈ -14 with respect to the zero-breakpoint solution). Thus, we split our RV time series in two piecewise stationary segments and de-trended it on a chunk-wise base, rather than performing a global de-trending over the whole time series, similarly to what has already been done in Bonfanti et al. (2023); Luque et al. (2023).","pages":[3,4]},{"title":"4. Methods","content":"We jointly analysed the TESS LCs and the RV time series within a Markov chain Monte Carlo (MCMC) framework using the MCMCI code (Bonfanti & Gillon 2020). When fitting the TESS LCs extracted from Sector 3 that has a cadence of 30 min, we generated the transit model using a cadence of 2 min (the same as the other TESS sectors) and then rebinned it to 30 min following Kipping (2010), who warns that long-cadence photometry may lead to retrieve erroneous system parameters. We imposed Normal priors on the input stellar parameters (i.e. T e ff , [Fe / H], R ⋆ , and M ⋆ ), as derived in Sect. 2 with a twofold aim: (i) the induced prior on the mean stellar density ρ⋆ (via M ⋆ and R ⋆ ) helps the convergence of the transit model; (ii) limb darkening (LD) coe ffi cients for the TESS filter may be retrieved following interpolation within A tlas 9-based 5 grids that were precomputed using the get_lds.py code 6 by Espinoza & Jordán (2015). We then set Normal priors on the quadratic LD coe ffi -cients using the values coming from the grid interpolation (i.e. u 1 , TESS = 0 . 2318 ± 0 . 0065 and u 2 , TESS = 0 . 3085 ± 0 . 0028) after re-parameterising them following Holman et al. (2006). On the planetary side, we adopted unbounded (except for the physical limits) uniform priors on the transit depth d F , the impact parameter b , the orbital period P , the transit timing T 0, and the RV semi-amplitude K , while we set the eccentricity e = 0 and the argument of periastron ω = 90 · for all planets. We come back to the assumption on the eccentricity below. The specific parameterisations of the jump parameters (aka step parameters) are outlined in Bonfanti & Gillon (2020). The LCs and the RV time series were de-trended simultaneously during the MCMC analysis using polynomials. To assess the polynomial orders to be associated with the di ff erent de-trending parameters for each time series, we first launch several MCMC preliminary runs made of 10 000 steps where we varied only one polynomial order at a time. We then selected the best de-trending baseline (see Table A.2) as the one having the lowest BIC. After that, we performed a preliminary MCMCI run comprising 200 000 steps to evaluate the contribution of both the white and red noise in the LCs following Pont et al. (2006); Bonfanti & Gillon (2020), so to properly rescale the photometric errors and get reliable uncertainties on the output parameters. Finally, three independent MCMCI runs comprising 200 000 steps each (burn-in length equal to 20%) were performed to build the posterior distributions of the output parameters after checking their convergence via the Gelman-Rubin statistic ( ˆ R ; Gelman & Rubin 1992). We also tested the possibility of eccentric orbits by imposing uniform priors on ( √ e cos ω , √ e cos ω ) either bounded to imply e ≲ 0 . 3 or completely unbounded (except for the physical limits). The wider the eccentricity range to be explored by the MCMC scheme, the poorer the parameter convergence, which suggests that the available data are not enough to con- he planetary eccentricities well. Moreover, the MCMCI runs with e , 0 are disfavoured by the ∆ BIC criterion (e.g. Kass & Raftery 1995; Trotta 2007) as well, in fact we obtained ∆ BIC = BIC e , 0 -BIC e = 0 ≳ + 100. This is also in agreement with the simulations performed by Vanderburg et al. (2019), who suggested that the periods' commensurability state of planets b and c is more likely maintained if the system is characterised by low eccentricities. Therefore, we adopted the circular solution as the reference one.","pages":[4]},{"title":"5.1. Joint LC and RV analysis with linear ephemerides","content":"As mentioned in Sect. 4, we set P and T 0 as free parameters under the control of a uniform prior, which implies assuming linear ephemerides. With this setup, we improved the transit depth precision of all three planets by a factor ∼ 1.4 if compared with the results of Vanderburg et al. (2019). This improvement level is consistent with having twice the number of data points with respect to the LC analysis performed by Vanderburg et al. (2019), as well as with low TTV amplitudes (see Sect. 5.2). By combining the SN-fit onto the HARPS CCFs along with the bp method, we were able to estimate the masses of TOI-396 b and TOI-396 d to Mb = 3 . 56 + 0 . 92 -0 . 94 M ⊕ and Md = 7 . 2 ± 1 . 6 M ⊕ (detections at the 3.8 and 4.5 σ -level, respectively). Instead, we did not detect any significant Keplerian signal at the orbital period of TOI-396 c within the RV time series. In detail, we obtained a median Kc = 0 . 28 + 0 . 29 -0 . 20 ms -1 (3 σ upper limit K up c = 1 . 2 ms -1 ), which implies Mc = 0 . 92 + 0 . 94 -0 63 M ⊕ ( M up c = 4 . 0 M ⊕ ). . When combining the mass and radius values of the three planets, we obtain the following median estimates for the bulk planetary densities: ρ b = 2 . 56 + 0 . 71 -0 . 70 , ρ c = 0 . 67 + 0 . 69 -0 . 46 ( ρ up c = 3 . 1), and ρ d = 5 . 1 + 1 . 3 -1 . 2 g cm -3 . We note that the RV-undetected TOI396 c would be the least dense planet, while the densest planet is the outermost one (i.e. TOI-396 d), which constitutes a quite atypical architecture within the observed exoplanet population (e.g. Ciardi et al. 2013; Weiss et al. 2018; Mishra et al. 2023). However, this conclusion is just tentative, given the uncertainties on the mean planetary densities. Moreover, the detection level of the RV-inferred parameters of TOI-396 c is not statistically significant. We further tested whether including the MINERVAAustralis (Addison et al. 2019) RV data (30 measurements as taken from Vanderburg et al. 2019) can help detect the elusive planet. However, it turned out that their precision level ( ∼ 6 m s -1 ) is not high enough to improve the characterisation of the system. In other words, the RV semi-amplitudes we obtained are consistent and indistinguishable within the statistical fluctuation with what was derived from the more precise HARPS data set. This led us to further check that TOI-396 c indeed belongs to this system (Sect. 5.2) and to investigate the reason for its RV non-detection (Sect. 5.3).","pages":[4]},{"title":"5.2. Joint LC and RV analysis accounting for TTVs","content":"As planet c is undetected in the RV time series, one may wonder whether TOI-396 c is a false positive. However, by using the VESPA tool (Morton 2012, 2015) that accounts for the constraints from the TESS LCs, imaging, and spectroscopy, Vanderburg et al. (2019) already computed that the false-positive probabilities (FPPs) are lower than 10 -3 for all three planets. In line with Vanderburg et al. (2019), we also confirmed that the Pc / Pb period ratio is commensurable and di ff ers from the 5:3 ratio by less than 0.027%. As planets with orbits in, or close to, resonances are likely to show TTVs, we decided to repeat the same MCMC analysis outlined above, but enabling the transit timings of each transit event to vary to then compute the TTV amplitude with respect to the linear ephemerides model derived in Sect. 5.1. All jump parameters converged ( ˆ R ≲ 1 . 01). The medians of the posterior distributions of the most relevant system parameters along with the 68.3% confidence intervals are listed in Table 2. The phase-folded LCs of the three planets are shown in Figure 1, while the phase-folded RV time series are displayed in Figure 2. In particular, the middle panel of Fig. 2 shows that, after subtracting the RV signals of both planets b and d, the RV time series looks flat consistently with the non-detection of TOI396 c. This analysis gives Rb = 2 . 004 + 0 . 045 -0 . 047 R ⊕ , Rc = 1 . 979 + 0 . 054 -0 . 051 R ⊕ , and Rd = 2 . 001 + 0 . 063 -0 . 064 R ⊕ and confirms the RV detection of TOI396 b and TOI-396 d (at the 3.8 and 4.5 σ -level, respectively) with Mb = 3 . 55 + 0 . 94 -0 . 96 M ⊕ and Md = 7 . 1 ± 1 . 6 M ⊕ . These outcomes are consistent with the results obtained from the analysis that assumes linear ephemerides (Sect. 5.1). We found significant TTV signals for planet b and c (see Figure 4, Top and Middle panels) as expected from the commensurability of their periods. The TTV statistical significance of TOI-396 b and c is evident even by eye when comparing the data points location with the shaded regions that represent the 1 σ uncertainties of the linear ephemerides. For each planet, we further quantified the reducedχ 2 ( ˆ χ 2 ) characterising the TTV amplitudes via where N tr , j is the number of transit event of the j -th planet. We obtained ˆ χ 2 b = 4 . 4, ˆ χ 2 c = 4 . 1, and ˆ χ 2 d = 0 . 7 for planets b, c, and d, respectively, which confirms the significance of the TTV amplitudes for TOI-396 b and c. Furthermore, the TTV amplitudes of TOI-396 b and c exhibit a clear anti-correlation pattern that we highlight by superimposing the TTV measurements in Figure A.1. This is a typical signature of gravitational interaction between the two planets, which confirms that TOI-396 c belongs to the system despite its elusiveness in the RV time series. For each planet, the timing of each transit event and the corresponding TTV amplitude computed with respect to the linear ephemerides are listed in Table A.3.","pages":[4,5]},{"title":"5.3. Discussion regarding why TOI-396 c is not detected in the RV time series","content":"Magnetic activity combined with stellar rotation induces RV variations that can hide, a ff ect, or even mimic planetary signals (e.g. Queloz et al. 2001; Hatzes et al. 2010; Dumusque et al. 2011a; Haywood et al. 2014; Suárez Mascareño et al. 2017; Gandolfi et al. 2017). A possible explanation for the non-detection of the Doppler reflex motion induced by TOI-396 c is that stellar activity destructively interferes with the Keplerian signal of the planet. This may happen if the star has a rotation period Notes. All jump parameters, but the LD coe ffi cients, were subject to unbounded uniform priors following the parameterisations specified in Bonfanti & Gillon (2020); see text for further details. ( a ) Uncertainties from the run assuming linear ephemerides (no TTVs). T 0 values are shifted by -2 450 000. ( b ) Assuming zero albedo and full recirculation. ( c ) The 3 σ upper limits on the RV semi-amplitude, the planet mass, and the mean planet density of TOI-396 c are K up c = 1 . 2 ms -1 , M up c = 3 . 8 M ⊕ , and ρ up c = 2 . 9 g cm -3 , respectively. comparable to the orbital period of the planet (e.g. Vanderburg et al. 2016). Disentangling the planetary signal from stellar activity and retrieving the Doppler motion induced by the orbiting planet would then be challenging (e.g. Dragomir et al. 2012; Kossakowski et al. 2022). In order to investigate this hypothesis, we performed a frequency analysis of the line profile variation diagnostics (FWHM, contrast, and skewness) and activity indicators (log R ' HK and H α indexes). Figure 5 displays the time series (left column), along with the respective generalised Lomb-Scargle (GLS, Zechmeister & Kürster 2009) periodograms (right column). We assessed the significance of the peaks detected in the power spectra by estimating their false alarm probability (FAP), that is the probability that noise could produce a peak with power higher than the one we found in the time series. To account for the possible presence of non-Gaussian noise in the data, we estimated the FAP using the bootstrap randomisation method (see, e.g., Murdoch et al. 1993; Kuerster et al. 1997; Hatzes 2019). Briefly, we computed the GLS periodogram of 10 5 mock time series obtained by randomly shu ffl ing the data points along with their error bars, while keeping the timestamps fixed. We defined the FAP as the fraction of those mock periodograms whose highest power exceeds the power of the real data at any frequency. In the present work, we considered a peak to be significant if its false alarm probability is FAP < 0.1 %. We found that the time series of the log R ' HK and H α indexes display long-term trends likely due to the magnetic cycles of the star (Fig. 5, left column, first and third panels). In the Fourier domain, these trends translate into a significant (FAP < 0.1 %) excess of power at frequencies lower than the spectral resolution 8 of our HARPS observations (Fig. 5, right column, first and third panels). We modelled these long-term signals as quadratic trends and subtracted the best-fitting parabolas from the respective time-series. The GLS periodograms of the residuals of the activity indicators display significant peaks between ∼ 6 and 8 d (i.e, between ∼ 0.0125 and 0.167 d -1 ; Fig. 5, yellow area). The peaks are equally spaced by ∼ 0.0068 d -1 , which coincides with a peak found in the periodogram of the window function. Given the current data at our disposal, we are not able to distinguish between true frequencies and aliases. Although not significant, the power spectra of the contrast and skewness show peaks in the same frequency range, suggesting that the rotation period of the star might be ∼ 6-8 d. We note that the periodogram of the FWHM also displays an excess of power at low frequencies. However, the corresponding peak remains below our 0.1 %-FAP significance threshold (see Fig. 5, second to last panel). Yet, if we apply the same procedure described above and remove this signal by fitting a quadratic trend to the FWHM time series, we find no significant peak in the residuals. The projected equatorial velocity of the star ( v sin i ⋆ = 7 . 5 ± 0 . 2 km s -1 ), along with its radius ( R ⋆ = 1 . 258 ± 0 . 019 R ⊙ ), yields an upper limit for the rotation period of P up rot = 8 . 5 ± 0 . 3 d. Using the mean value 9 of log R ' HK = -4 . 926 ± 0 . 014, we inferred a stellar rotation period of 6.7 ± 1.3 d and 6.9 ± 1.3 d from the empirical equations of Noyes et al. (1984) and Mamajek & Hillenbrand (2008), respectively. In addition, by inputting the isochronal age into the gyrochronological relation from Barnes (2010), we computed a stellar rotation period of 7 . 1 + 1 . 0 -1 . 1 d. These results corroborate our interpretations that the peaks between 6 and 8 d significantly detected in the power spectra of the activity indicators originate from stellar rotation. To check whether a quasi-periodic signal compatible with ∼ 7 d is also present in the photometric data, for each TESS sector we extracted custom LCs from pixel data using lightkurve (Lightkurve Collaboration et al. 2018). In detail, we adopted the default quality bitmask and set the aperture to 'all', which corresponds to an aperture larger than the one used by the o ffi cial SAP pipeline. In fact, larger apertures mitigate the e ff ect of slow image drifts that could interfere with slow flux changes, such TV [min] T TV [min] T 40 30 20 10 0 10 20 60 50 40 30 20 10 0 10 20 8360 8380 8400 8420 8440 9100 9120 9140 9160 9180 9200 BJD - 2450000 [d] 8360 8380 8400 8420 8440 9100 9120 9140 9160 9180 9200 BJD - 2450000 [d] as the 6-8 d rotation period signals we aim to detect. After removing the temporal windows containing the transit events, we computed the GLS periodograms of these lightkurve -based LCs. The FAP was computed following the same bootstrap technique outlined above for the RV activity indicators. The four periodograms (Fig. A.2, first column) exhibit very significant peaks at ∼ 7.7, 7.9, 7.5, and 6.8 days for TESS Sectors 3, 4, 30, and 31, respectively. Except for Sector 30, they are not the most prominent peaks; however, they persist even after removing the most significant signals (Fig. A.2, second column). Whereas we acknowledge that the likely rotation period of the star is close to the first harmonic of the orbital period of TESS around the Earth ( P rot ,⋆ ∼ 1 2 P TESS ∼ 14 days), the photometric signal at 6-8 d is significant in all the four TESS sectors and persists after prewhitening the data. This signal is consistent with the rotation period detected in the HARPS activity indicators and inferred from log R ' HK , v sin i ⋆ , and gyrochronology, suggesting it is astrophysical in nature and due to the presence of active regions carried around by stellar rotation. Assuming the log R ' HK -based P rot ,⋆ = 6 . 7 ± 1 . 3 d as our reference estimate, the orbital period of TOI-396 c ( Pc ∼ 6 d) is close to P rot ,⋆ , which may explain the non-detection of planet c within the RV time series. If some kind of destructive interference between the Keplerian signal of planet c and the stellar activity has occurred, any artificial Keplerian signals with period P = Pc added to the observed RV time series should in principle be retrieved. To test this hypothesis, we considered Keplerian signals of the following form and generated four di ff erent RV time series by separately adding to the HARPS time series synthetic RV signals following Equation (2) with P = Pc , T 0 = T 0 , c , and K art = K in, where K in are the four di ff erent amplitude values listed in the first column of Table 3. For each RV time series, we then performed an MCMCI analysis to retrieve the RV semi-amplitude of the artificial signal ( K out; see Table 3). We note that the resulting K out ≈ K in + Kc , where Kc = 0 . 28 + 0 . 29 -0 . 19 ms -1 is the RV semi-amplitude of planet c as derived from the analysis on the original RV time series. As we essentially retrieved what we inserted in the HARPS time series, we may conclude that the destructive interference between the RV signals induced by the star and by planet c has already occurred and any further RV signal added to the RV time series is detected. As P rot ,⋆ is not exactly equal to Pc , we then repeated the test outlined above, but this time we injected into the original HARPS time series artificial Keplerian signals with P = P rot ,⋆ . The K out values obtained by the MCMC analyses depending on the di ff erent K in values are reported in Table 4. The K out values are systematically and significantly smaller than the corresponding K in values, which a posteriori supports the conclusion that the stellar rotation period is around 6-8 d. Furthermore, a planet with this orbital period would be firmly detected if its RV semi-amplitude were greater than Kd , which is the largest RV semi-amplitude detected for this system. Instead, by injecting a Keplerian signal with K in = Kd and P = P rot ,⋆ , the MCMCI analysis is able to barely detect (at ∼ 2 σ ) a planetary signal whose amplitude is half of that expected. The detection level increases when K in increases; however, we still underestimate K out. These tests prove that it is di ffi cult to retrieve planetary signals with P ∼ P rot ,⋆ . In summary, we conclude that stellar activity is responsible for generating spurious RV signals whose harmonics also include the stellar rotation period. As a consequence, it is hard to reliably detect planets with orbital periods comparable to P rot ,⋆ via the RV technique and Table 4 quantifies the magnitude of this e ff ect. We note that the Kc we obtained from the MCMCI analysis in Sect. 5.2 is comparable to the K out retrieved when inserting an artificial signal having K in = 1 . 0 ms -1 , which let us Notes. Column ∆ K K gives the relative di ff erence (in percentage) between the obtained RV semi-amplitude ( K out) and the expected one ( K in), while the last column ∆ K ≡ K out -K in lists the semi-amplitude di ff erence in terms of the 1 σ uncertainty of K out. to speculate that TOI-396 c might have Mc ∼ 3.0 M ⊕ and ρ c ∼ 2.0 g cm -3 .","pages":[5,6,7,8,9,10]},{"title":"6. Joint RV and TTV dynamical analysis","content":"As mentioned, the anti-correlation pattern of the observed TTVs (see Fig. A.1) is a typical signal of the dynamical interaction between TOI-396 b and c. However, the data in our hands are not enough to currently derive meaningful planetary masses from the TTVs. Indeed, the photometric observations are clustered in two groups (about two years apart), which results in a partial coverage of the curvature of the TTVs. This prevents us from accurately mapping the full phases and super-periods of the TTV signals. Hence a dynamical fit onto the transit times would lead to a high fraction of low-amplitude TTVs with a short superperiod or to a low fraction of high-amplitude long-period TTV signals. Despite this, we attempted to run a dynamical joint fit of the TTV and RV data set with TRADES 10 (Borsato et al. 2014, 2019, 2021), a Fortran-python code developed to model TTVs and RVs simultaneously along with N-body integration. We have taken and fixed the stellar mass and radius values from Table 1. We further fixed the orbital inclinations i to the values in Table 2 and the longitude of the ascending nodes to Ω = 180 · 11 for all the planets. We fitted the planetary masses scaled by the stellar mass ( M b , c , d / M ⋆ ), the orbital periods ( P ), the eccentricities ( e ) and the arguments of the pericentre ( ω ) in the form ( √ e cos ω , √ e sin ω ), and the mean longitudes ( λ 12 ). We also fitted for an RV o ff set ( γ RV) and for an RV jitter term ( σ jitter) by adopting log 2 σ jitter as step parameter, although we used the de-trended RV data set as derived from Sect. 5.2, where a jitter term was already included in the RV errors. All fitting parameters were subject to uniform priors that account for their respective physical boundaries (see Table 5); for the eccentricities we applied a log-penalty (log p e ) based on the half-Gaussian ( e = 0 , σ e = 0 . 083) from Van Eylen et al. (2019). The reference time for the dynamical integration of the orbital parameters was set at T ref = 2 458 379 BJDTDB, that is before all available observations. We combined the quasiglobal di ff erential evolution (Storn & Price 1997) optimisation algorithm implemented in P y DE (Parviainen et al. 2016) with the A ffi ne Invariant MCMC Ensemble sampler (Goodman & Weare 2010) emcee implemented by Foreman-Mackey et al. (2019, 2013). We first ran P y DE and evolved 68 di ff erent initial configurations of parameter sets for 50 000 generations (number of steps for which each parameter is evolved). To perform the dynamical analysis in an MCMC fashion, we then ran emcee , assuming as starting point the outcome obtained with P y DE. We set up 68 chains for 600 000 steps each. To sample the parameter space e ffi ciently, we mixed the DEMove() and DESnookerMove() di ff erential evolution moves 13 in the proportion 80%-20% (Nelson et al. 2014; ter Braak & Vrugt 2008). We repeated the sequence P y DE + emcee twice, with di ff erent seeds for the random number generator. The chains reached convergence according to visual inspection and statistical indicators, such as the Gelman-Rubin ˆ R , the Geweke's statistic (Geweke 1991), and the auto-correlation function 14 . For both runs, we applied a conservative thinning factor of 100 and discarded the first 50% of the chains (burn-in). For each run, we derived the reference outcome (hereinafter also referred to as best-fit), as the maximum-a-posteriori (MAP) parameters' set, that is the set of parameters that maximises the log-probability 15 . The uncertainty of each parameter is quantified by the high-density interval (HDI) at 68.27% 16 of its posterior distribution. For each parameter we computed a Z-score defined as Z-score = | MAP1 -MAP2 | / p max | ERR1 | 2 + max | ERR2 | 2 , where the subscripts denote the two di ff erent runs and ERR = HDI -MAP. It turned out that Z-score < 1 for each parameter, which allows us to merge the posterior distributions deriving from the two runs to finally compute the MAP and the respective HDI from the merged posterior distributions. We further checked the Hill stability of the system (Sundman 1913) when assuming the entire merged posterior distributions by calculating the angular momentum deficit (AMD, Laskar 1997, 2000; Laskar & Petit 2017) criterion (Eq. 26 from Petit et al. 2018). Table 5 lists the parameters returned by TRADES (MAP and HDI) along with their respective priors. We note that the dynamical integration with TRADES allowed for a significant detection ( > 3 σ ) of the mass of TOI-396 c, that is Mc , dyn = 2 . 24 + 0 . 13 -0 . 67 M ⊕ . However, as explained above, TESS data do not allow us to fully map the TTV pattern. Therefore, the present-day estimate of Mc , dyn only provides an indication of the possible mass of the planet that might not be accurate, despite its formal precision. Indeed, to accurately and reliably determine planetary masses via TTVs, it is necessary to monitor the TTV signal by benefiting of a full sampling coverage, as demonstrated for example by the cases of Kepler-9 (Holman et al. 2010) and K2-24 (Petigura et al. 2016), whose orbital parameters and masses were comprehensively revisited by Borsato et al. (2014) and Petigura et al. (2018); Nascimbeni et al. (2024), respectively. In detail, Holman et al. (2010) had reported the masses of Kepler-9 b and c with a precision better than 8%, by performing a TTV analysis on the first three quarters of the Kepler data. Later on, by benefiting of twelve sectors of Kepler data that enabled the full mapping of the TTV phase, Borsato et al. (2014) obtained TTV-based masses di ff ering by a factor ∼ 2 from the estimate of Holman et al. (2010). Similarly, by accounting on more photometric data, Petigura et al. (2018); Nascimbeni et al. (2024) found that the mass of K2-24 c is lower by almost 2 σ than the estimate of Petigura et al. (2016) who had claimed a detection at the ∼ 4 σ level. After extracting all the synthetic transit timings T tr (i.e. the 'observed': O) from TRADES' analysis, we computed the 'calculated' (C) counterpart according to the linear ephemerides model based on what listed in Table 2. We plotted the O -C as a function of time as well as the RV best-fit model in Figures 6 and 7. Additionally, we investigated if the best-fit configuration is in or close to an MMR. We integrated the MAP parameters with the N-body code rebound (Rein & Liu 2012) and the symplectic Wisdom-Holman integrator whfast (Rein & Tamayo 2015; Wisdom & Holman 1991) for 10 000 years. We computed the evolution of the critical resonance angles of TOI-396 b and TOI396 c where p = 3 and q = 2 (for a second order 5:3 MMR), while ϖ ≡ ω + Ω is the longitude of the pericentre. We also computed the evolution of ∆ ϖ ≡ ϖ b -ϖ c = ( ϕ b -ϕ c) / q . In case of MMR, we expect that both ϕ b and ϕ c librate (i.e. oscillate) around a fixed value for the entire orbital integration. Instead, if these angles circulate, that is they span the full 0 · -360 · range (or equivalently the -180 · -180 · range), then the planet pair is not in resonance. We found that both ϕ b and ϕ c circulate (see the two upper panels of Fig. 8), which indicates that the system is not in an exact 5:3 MMR. Even if ∆ ϖ seems to oscillate around 0 · , it circulates every ∼ 2 000 years (bottom panel of Fig. 8), which further confirms that the system is not trapped in an MMR state. We also found the same behaviour for the resonant angles of 200 random samples drawn from the posterior distribution (not shown here). Our conclusions are consistent with the simulations performed by Vanderburg et al. (2019), who found that most realisations of the system are not in resonance. As shown in Fig. 6, the MAP parameter set predicts that the TTV super-period is longer than the time spanning the two clustered TESS observations. We decided to track the potential evolution of the TTV signals over a temporal baseline of ∼ 5.2 years. To this end, we ran forward numerical N-body simulations with TRADES, setting the initial conditions of the orbital parameters to be integrated at t = T ref. We ran a first simulation taking all the parameters from the MAP solution. Then, we further ran 200 simulations, where the sets of the system's parameters were randomly drawn from the merged posterior distributions. We computed the synthetic (i.e. observed: O ) transit times T tr and created a simulated O -C plot against time, where the C counterpart of T tr was computed assuming the linear ephemerides of Table 2. The results are displayed in Fig. 9 that emphasises a progressive drift of the O -C values inferred from the MAP parameters (black line) with respect to the zero value. As a consequence, the TTV amplitudes of planets b and c increase with time. In particular, the semi-amplitude of the O -C black curves are about two and five hours for b and c, respectively. The TTV super-period seems to be roughly equal to or larger than the integration time span, that is ∼ 5 years. The variance of the TTV amplitude (shaded area) is inferred from the results of the additional 200 simulations and reflects the widths of the posterior distributions from which the system's parameters were drawn. The remarkable O -C drifts (i.e. the poorly constrained linear ephemerides) combined with the uncertainty on the TTV amplitudes make challenging to plan future observations of planets b and c, as the actual transit timings might di ff er from the linear ephemerides predictions by ∼ 5 and ∼ 10 hours for TOI-396 b and TOI-396 c, respectively. TESS will not observe the target in the foreseeable future. 17","pages":[10,11]},{"title":"7. Internal structure","content":"Using the masses and transit depths reported in Table 2, we ran the neural network based internal structure modelling framework plaNETic 18 (Egger et al. 2024) to infer the internal structure of TOI-396 b and d. plaNETic uses a full grid accept-reject sampling method in combination with a deep neural network (DNN) that was trained on the forward model of BICEPS (Haldemann et al. 2024) to infer the internal structure of observed planets. Each planet is modelled as a three-layered structure: an inner iron-dominated core, a silicate mantle and a fully mixed envelope made up of water and H / He. In the case of multi-planet systems, all planets are modelled simultaneously. As modelling the internal structure of exoplanets is a highly degenerate problem, the resulting inferred structure is, at least to a certain extent, dependent on the chosen priors. To mitigate this e ff ect, we ran a total of six models assuming six di ff erent combinations of priors. Most importantly, we use two di ff erent priors for the water content of the modelled planet, one motivated by a formation scenario outside the iceline (case A, water-rich) and one compatible with a formation inside the iceline (case B, water-poor). For both of these water priors, we choose three different options for the planetary Si / Mg / Fe ratios. In a first case, we assume that these match the stellar Si / Mg / Fe ratios exactly Thiabaud et al. (2015). Second, we assume that the planet is enriched in iron compared to its host star by using the fit of Adibekyan et al. (2021). For option 3, we model the planet independent of the stellar Si / Mg / Fe ratios, but just sampling the planetary ratios uniformly from the simplex where the molar Si, Mgand Fe ratios add up to 1, with an upper bound of 0.75 for Fe. These priors are described in more detail in Egger et al. (2024).","pages":[11]},{"title":"8. JWST characterisation prospects","content":"All three planets in the TOI-396 system share similar radii ( ∼ 2 R ⊕ ), but span a wide range of masses (0.9-7.1 M ⊕ ), which leaves open the question of whether they have primary or secondary atmospheres. Furthermore, the progression of bulk densities with distance from the host star varies in ways that cannot be described by simple formation and evolution models (e.g. Weiss et al. 2018; Mishra et al. 2023). Given the bright host star and combination of planetary masses, radii, and equilibrium temperatures, the three planets have favourable metrics for atmospheric characterisation in both transmission and emission among sub-Neptunes (Kempton et al. 2018, see Fig. 12). This makes the TOI-396 system a highly valuable laboratory to study the formation and evolution of plan- etary systems. Thus, we explored the prospects for characterisation with JWST. We focused these simulations on emission observations, but we note that transmission and emission have their own advantages and disadvantages in terms of achievable science goals and challenges. We employed the open-source P yrat B ay modelling framework (Cubillos & Blecic 2021) to compute synthetic spectra of the TOI-396 planets. These models consist of 1D cloud-free atmospheres in radiative, thermochemical, and hydrostatic equilibrium (Cubillos et al., in prep.). We varied the models' atmospheric elemental content to explore the wide range of compositions that the planets span. For this comparison we settled on two models to represent a primary- and a secondary-atmosphere scenario: the first is a gas giant with a 5 × solar metallicity, the second is a water world with a 80% H 2 O plus 20% CO 2 composition (based on the C / O ratios seen in the solar system minor bodies, see, e.g., Mumma & Charnley 2011; McKay et al. 2019). For the thermochemical-equilibrium calculations we considered a set of 45 neutral and ionic species, which are the main actors determining the thermal structure. For the radiative-transfer calculation we considered opacities from molecular species for CO, CO 2 , CH 4 , H 2 O, HCN, NH 3 , and C 2 H 2 from hitemp and E xo M ol (Rothman et al. 2010; Tennyson et al. 2016); Na and K resonant lines (Burrows et al. 2000); H, H2, and He Rayleigh (Kurucz 1970); and H2-H2 and H2-He collision-induced absorption (Borysow et al. 2001; Borysow 2002; Richard et al. 2012). We preprocessed the large E xo M ol line lists with the R epack algorithm (Cubillos 2017) to extract the dominant transitions. Figure 13 (top panels) shows the resulting thermal and composition structure for TOI-396 b (planets c and d follow a similar trend). The infrared synthetic emission spectra (Fig. 13, bottom panels) are mainly shaped by H 2 O, CO 2 , and CO features. At most wavelengths the primary- and secondary-atmosphere scenarios roughly di ff er by an o ff set, which would be hard to distinguish unless the energy budget of the planets are known. In contrast, the 4-5 µ mwindow shows the most distinctive spectral features; here the strong CO 2 absorption band at 4.4 µ m mainly allows one to distinguish primary from secondary atmospheres. Thus, in the following we focus on this region of the spectrum. We simulated JWST observations using the P andeia exposure time calculator Pontoppidan et al. (2016). The brightness of TOI-396 limits the instrument selection to NIRCam (F444W filter) to avoid saturation. We selected the fastest readout and subarray modes, 5 groups per integration, to optimise the S / N. We generated a distribution of (noised up) realisations for each model to estimate how many eclipses are required to distinguish between primary and secondary atmospheres at the 3 σ level. We found that 2, 4, and 8 eclipses (for planets b, c, and d, respectively) would be su ffi cient to di ff erentiate between these two models. Figure 13 shows one of those random realisations when including the required number of eclipses. The decreasing equilibrium temperature of the planets as they are located further away from TOI-396 plays the major role in the decreasing S / N for planets c and d.","pages":[12,13,14]},{"title":"9. Conclusions","content":"The object TOI-396 is an F6 V bright naked-eye star orbited by three planets of almost equal size, and the two inner planets are close to but out of a 5:3 MMR. A photometric analysis of the system was already performed by Vanderburg et al. (2019), but by benefiting from two additional TESS sectors, we improved the precision on the planet radii by a factor of ∼ 1.4, obtaining Rb = 2 . 004 + 0 . 045 -0 . 047 R ⊕ , Rc = 1 . 979 + 0 . 054 -0 . 051 R ⊕ , and Rd = 2 . 001 + 0 . 063 -0 . 064 R ⊕ . We determined the masses of the planets by extracting the RV time series from HARPS CCFs using an SN fit followed by a joint LC and RV MCMC analysis, where the RV de-trending uses the breakpoint method. We obtained a firm detection of the RV signals of planets b and d, deriving Mb = 3 . 55 + 0 . 94 -0 . 96 M ⊕ and Md = 7 . 1 ± 1 . 6 M ⊕ , but we can provide only a 3 σ upper limit for the mass of TOI-396 c of M up c = 3 . 8 M ⊕ . This yields the following mean planet densities: ρ b = 2 . 44 + 0 . 69 -0 . 68 , ρ up c = 2 . 9, and ρ d = 4 . 9 + 1 . 2 -1 . 1 g cm -3 , implying a quite unusual system architecture (Mishra et al. 2023) where the mid planet is the least dense and the outermost planet is the densest. The reason for the RV non-detection of any Keplerian signal at P = Pc ∼ 6 d is likely to be ascribed to the vicinity of Pc to the stellar rotation period. As a matter of fact, from the GLS periodograms of both the RV-related activity indices and the TESS raw LCs and from log R ' HK -based empirical relations, we consistently inferred P rot ,⋆ = 6 . 7 ± 1 . 3 d. After injecting synthetic Keplerian signals at P = P rot ,⋆ and di ff erent semi-amplitudes ( K in) into the RV time series, we empirically find that the RV semi-amplitudes output by the MCMC analyses ( K out) are systematically lower than the input ones by almost 3 σ , and they are statistically non-significant as far as K in ≲ Kd . In addition, we find that K out ≈ Kc when considering a planet with Mp ∼ 3 M ⊕ (i.e. ρ p ∼ 2 g cm -3 ), which might correspond to the properties of TOI-396 c. On a more general perspective, these simulations confirm that stellar activity destructively interferes with Keplerian signals having P ∼ P rot ,⋆ (e.g. Vanderburg et al. 2016), and furthermore, they indicate that - even in the case of firm detection - values of K out are significantly underestimated. Longer-baseline RV observations may help disentangle coherent signals originated by Keplerian motions from noncoherent signals due to stellar activity, even if degeneracy issues still hold when the planet orbital period is close to the stellar rotation period (Kossakowski et al. 2022). Alternatively, a possible constraint on Mc may come from TTVs, as planets b and c are close to an MMR of the second order. Indeed, the TTV amplitudes of the two planets show a characteristic anti-correlation pattern, as expected; however, the phase coverage given by the available observations is too poor to perform a conclusive TTV dynamical analysis based on the observed transit timings of the planets. We also attempted to fit the TTV and RV simultaneously while integrating the orbits of the system. We found that the masses and densities of planets b and d are consistent with the results from the joint LC and RV analysis. TOI-396 c shows a dynamical mass of Mc , dyn = 2 . 24 + 0 . 13 -0 . 67 M ⊕ , which is greater than that inferred from the joint LC and RV analysis, but it is consistent (Z-score = 1 . 2 σ ); the density is consistent at the 1 . 1 σ level. However, we emphasise that, although formally precise, the Mc , dyn estimate might not be accurate, as the full coverage of the TTV phase is needed to reliably compute TTV-based masses. Therefore, to fully confirm the system architecture, a reliable estimate of the mass of TOI-396 c is still missing. We also checked the evolution of the system over 10 000 years, and the critical resonance angles showed that planets b and c are close to but not in a 5:3 MMR. We further performed forward N-body simulations over a temporal baseline of ∼ 5.2 years in order to track the transit epochs and evaluate the expected TTV amplitudes during time. It turns out that TOI-396 b and TOI-396 c may exhibit TTVs with a super-period of about 5 years and semi-amplitudes of ∼ 2 and ∼ 5 hours, respectively. This translates into a temporal drift of the transit timings that can rise up to ∼ 5 and ∼ 10 hours with respect to the linear ephemerides computed from TESS data. Studying the planetary atmospheres with JWST would take advantage of the favourable spectroscopy metrics of the system (Kempton et al. 2018). Therefore, we set up 1D cloud-free atmospheric models, generated the synthetic emission spectra of the three planets, and simulated eclipse observations with JWST. It turns out that 2, 4, and 8 eclipses (for TOI-396 b, c, and d, respectively) would be su ffi cient to distinguish between primary and secondary atmosphere scenarios at the 3 σ level. Characterising the nature of the planetary atmosphere is also key to correctly assessing the planetary bulk densities (in particular for planet c). The potentially high TTVs inferred from our simulations should be duly taken into account when scheduling future observations of the target. This holds not only for JWST, but also for CHEOPS (Benz et al. 2021), which appears espe- cially suitable for collecting exquisite photometric data to enable the full characterisation of the system. Acknowledgements. We thank the anonymous referee for all the valuable comments that significantly improved the quality of the manuscript. This work has made use of data from the European Space Agency (ESA) mission Gaia ( https://www.cosmos.esa.int/gaia ), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/ gaia/dpac/consortium ). This research made use of Lightkurve, a Python package for Kepler and TESS data analysis (Lightkurve Collaboration et al. 2018). We thank contributors to NumPy (Harris et al. 2020), SciPy (Virtanen et al. 2020), matplotlib (Hunter 2007), astropy (Astropy Collaboration et al. 2013, 2018, 2022), astroquery (Ginsburg et al. 2019), and tesscut (Brasseur et al. 2019). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. We acknowledge financial support from the Agencia Estatal de Investigación of the Ministerio de Ciencia e Innovación MCIN / AEI / 10.13039 / 501100011033 and the ERDF 'A way of making Europe' through project PID2021-125627OB-C32, and from the Centre of Excellence 'Severo Ochoa' award to the Instituto de Astrofisica de Canarias. This research was funded in part by the UKRI, (Grants ST / X001121 / 1, EP / X027562 / 1). This work was supported by FCT - Fundação para a Ciência e a Tecnologia through national funds and by FEDER through COMPETE2020 - Programa Operacional Competitividade e Internacionalização by these grants: UIDB / 04434 / 2020; UIDP / 04434 / 2020. D.G., A.B., L.F., and L.M.S. gratefully acknowledge the financial support from the grant for internationalization (GAND_GFI_23_01) provided by the University of Turin (Italy). S.G.S acknowledges the support from FCT through Investigador FCT contract nr. CEECIND / 00826 / 2018 and POPH / FSE (EC). P.J.W. acknowledges support from the UK Science and Technology Facilities Council (STFC) through consolidated grants ST / P000495 / 1, ST / T000406 / 1 and ST / X001121 / 1. N.C.S. is funded by the European Union (ERC, FIERCE, 101052347). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. J.L.-B. is funded by the MICIU / AEI / 10.13039 / 501100011033 and NextGenerationEU / PRTR grant PID2019-107061GB-C61 and CNS2023-144309. X.D. acknowledges the support from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement SCORE No 851555) and from the Swiss National Science Foundation under the grant SPECTRE (No 200021_215200). This work has been carried out within the framework of the NCCR PlanetS supported by the Swiss National Science Foundation under grants 51NF40_182901 and 51NF40_205606. G.N. thanks for the research funding from the Ministry of Science and Higher Education programme the \"Excellence Initiative - Research University\" conducted at the Centre of Excellence in Astrophysics and Astrochemistry of the Nicolaus Copernicus University in Toru'n, Poland. Research activities of the Board of Observational and Instrumental Astronomy at the Federal University of Rio Grande do Norte are supported by continuous grants from the Brazilian funding agencies CNPq. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-Brasil (CAPES)-Finance Code 001 and CAPES-Print program. B.L.C.M., I.C.L., and J.R.M. acknowledge CNPq research fellowships. K.W.F.L. was supported by Deutsche Forschungsgemeinschaft grants RA714 / 14-1, RA714 / 14-2 within the DFG Schwerpunkt SPP 1992, Exploring the Diversity of Extrasolar Planets. L.B. acknowledges support from CHEOPS ASI-INAF agreement n. 2019-29-HH.0. D.G. sincerely thanks Stefano Camera for the inspiring and valuable discussions on the properties of TOI-396.","pages":[14,15,16]},{"title":"References","content":"Addison, B., Wright, D. J., Wittenmyer, R. A., et al. 2019, PASP, 131, 115003 Adibekyan, V., Dorn, C., Sousa, S. G., et al. 2021, Science, 374, 330 Adibekyan, V., Figueira, P., Santos, N. C., et al. 2015, A&A, 583, A94 Adibekyan, V. Z., Sousa, S. G., Santos, N. C., et al. 2012, A&A, 545, A32 Agol, E. & Fabrycky, D. C. 2018, in Handbook of Exoplanets, ed. H. J. Deeg & Baranne, A., Queloz, D., Mayor, M., et al. 1996, A&AS, 119, 373 Barnes, S. A. 2010, ApJ, 722, 222 Bruntt, H., Deleuil, M., Fridlund, M., et al. 2010, A&A, 519, A51 Burrows, A., Marley, M. S., & Sharp, C. M. 2000, ApJ, 531, 438 D'Angelo, G. & Lubow, S. H. 2008, ApJ, 685, 560 Delisle, J. B., Laskar, J., Correia, A. C. M., & Boué, G. 2012, A&A, 546, A71 Delrez, L., Ehrenreich, D., Alibert, Y., et al. 2021, Nature Astronomy, 5, 775 Dorn, C., Venturini, J., Khan, A., et al. 2017, A&A, 597, A37 Egger, J. A., Osborn, H. P., Kubyshkina, D., et al. 2024, A&A, 688, A223 Hatzes, A. P. 2016, in Astrophysics and Space Science Library, Vol. 428, Meth- Holman, M. J., Fabrycky, D. C., Ragozzine, D., et al. 2010, Science, 330, 51 Holman, M. J., Winn, J. N., Latham, D. W., et al. 2006, ApJ, 652, 1715 Izidoro, A., Ogihara, M., Raymond, S. N., et al. 2017, MNRAS, 470, 1750 Lissauer, J. J., Ragozzine, D., Fabrycky, D. C., et al. 2011, ApJS, 197, 8 Lovis, C. & Pepe, F. 2007, A&A, 468, 1115 Mamajek, E. E. & Hillenbrand, L. A. 2008, ApJ, 687, 1264 McKay, A. J., DiSanti, M. A., Kelley, M. S. P., et al. 2019, AJ, 158, 128 Pontoppidan, K. M., Pickering, T. E., Laidler, V. G., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9910, Observatory Operations: Strategies, Processes, and Systems VI, ed. Queloz, D., Henry, G. W., Sivan, J. P., et al. 2001, A&A, 379, 279 Rein, H. & Liu, S. F. 2012, A&A, 537, A128 Richard, C., Gordon, I. E., Rothman, L. S., et al. 2012, J. Quant. Spectr. Rad. Transf., 113, 1276 Sneden, C. A. 1973, PhD thesis, THE UNIVERSITY OF TEXAS AT AUSTIN. ter Braak, C. J. F. & Vrugt, J. A. 2008, Statistics and Computing, 18, 435 Thiabaud, A., Marboeuf, U., Alibert, Y., et al. 2014, A&A, 562, A27 Wisdom, J. & Holman, M. 1991, AJ, 102, 1528 A. Bonfanti","pages":[17,18,19]},{"title":"Appendix A: Supplementary material","content":"A. Bonfanti et al.: Radii, masses, and transit-timing variations of the three-planet system orbiting the naked-eye star TOI-396 Notes. TESS (TE) LCs are identified by a counter based on the chronological order of observation. In particular, LCs from 1 to 11, from 12 to 21, from 22 to 32, and from 33 to 41 were extracted from Sector 3, 4, 30, and 31, respectively. c indicates a normalisation scalar; see text for further details. T tr [BJD] TTV [min] Sector 8385 . 794 . 8391 . 8397 + 0 - 0 . 010 . + 0 - 0 7447 + 0 - 0 7195 . 8403 . 8415 . 8421 . 8427 . 8433 . 9120 6857 ± 0 + 0 - 0 6337 + 0 - 0 6062 + 0 - 0 5817 + 0 - 0 5482 + 0 - 0 . . . . . . . . . . 5407 . 9126 518 ± 0 . 9132 . 9138 . 9150 . 9156 . 9162 . 9168 . 8387 . 8398 . 8432 . 9117 . 9139 . 9150 . 9162 4916 + 0 - 0 + 0 - 0 4731 + 0 - 0 4169 + 0 - 0 4050 + 0 - 0 3798 + 0 - 0 3562 + 0 - 0 2711 + 0 - 0 5047 + 0 - 0 1919 + 0 - 0 2589 + 0 - 0 7171 + 0 - 0 9371 + 0 - 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 016 . . . . 0068 0063 0046 0047 . 0033 0 0015 0013 0017 0016 0065 0033 0042 0077 0052 0043 . 010 - 8 0032 0020 0057 0040 0085 0090 0029 0044 0027 0030 0068 0084 0036 0034 0056 0051 0050 0034 0076 0068 0047 0025 0072 0037 0038 0031 + 17 + 19 + 23 - 8 + 2 - 3 . . + 0 - 1 + 0 - 10 - 12 . + 43 + 9 + 11 . . + 15 - 22 + 9 - 9 + 6 - 6 . 7 . 0 + 4 - 4 + 2 - 1 0 . . . 4 + 2 - 2 7 + 9 - 4 7 + 6 - 11 + 7 - 6 . . 5 3 5 8 + 14 - 15 + 4 - 2 2 + 8 - 5 8 + 12 - 13 . . 6 + 4 - 6 + 3 - 4 0 + 10 - 12 TOI-396 d - 0 . 4 - 2 + 6 + 2 - 13 + 0 . . 4 + 5 - 4 + 8 - 7 + 7 - 4 0 . 2 + 11 - 10 + 6 - 3 1 + 10 - 5 + 5 - 4 . 5 . 5 3 . . 2 . 9 1 4 . 2 . 9 8 . 6 . . . . . 9 . 2 . 7 . . . . 2 4 8 4 . 8 . 1 . . 7 8 . 2 . 9 . 5 . 3 . 4 . 7 6 8 . 1769 3 3 3 3 4 4 4 4 30 30 30 30 31 31 31 31 3 3 4 30 30 31 31 TOI-396 c","pages":[21,23,24]}],"string":"[\n {\n \"title\": \"ABSTRACT\",\n \"content\": \"Context. TOI-396 is an F6 V bright naked-eye star ( V ≈ 6.4) orbited by three small ( Rp ≈ 2 R ⊕ ) transiting planets discovered thanks to space-based photometry from two TESS sectors. The orbital periods of the two innermost planets, namely TOI-396 b and c, are close to the 5:3 commensurability ( Pb ∼ 3.6 d and Pc ∼ 6.0 d), suggesting that the planets might be trapped in a mean motion resonance (MMR). Aims. To measure the masses of the three planets, refine their radii, and investigate whether planets b and c are in MMR, we carried out HARPS radial velocity (RV) observations of TOI-396 and retrieved archival high-precision transit photometry from four TESS sectors. Methods. We extracted the RVs via a skew-normal fit onto the HARPS cross-correlation functions and performed a Markov chain Monte Carlo joint analysis of the Doppler measurements and transit photometry while employing the breakpoint method to remove stellar activity from the RV time series. We also performed a transit timing variation (TTV) dynamical analysis of the system and simulated the temporal evolution of the TTV amplitudes of the three planets following an N-body numerical integration. Results. Our analysis confirms that the three planets have similar sizes ( Rb = 2 . 004 + 0 . 045 -0 . 047 R ⊕ ; Rc = 1 . 979 + 0 . 054 -0 . 051 R ⊕ ; Rd = 2 . 001 + 0 . 063 -0 . 064 R ⊕ ) and is thus in agreement with previous findings. However, our measurements are ∼ 1.4 times more precise thanks to the use of two additional TESS sectors. For the first time, we have determined the RV masses for TOI-396 b and d, finding them to be Mb = 3 . 55 + 0 . 94 -0 . 96 M ⊕ and Md = 7 . 1 ± 1 . 6 M ⊕ , which implies bulk densities of ρ b = 2 . 44 + 0 . 69 -0 . 68 g cm -3 and ρ d = 4 . 9 + 1 . 2 -1 . 1 g cm -3 , respectively. Our results suggest a quite unusual system architecture, with the outermost planet being the densest. Based on a frequency analysis of the HARPS activity indicators and TESS light curves, we find the rotation period of the star to be P rot ,⋆ = 6 . 7 ± 1 . 3 d, in agreement with the value predicted from log R ' HK -based empirical relations. The Doppler reflex motion induced by TOI-396 c remains undetected in our RV time series, likely due to the proximity of the planet's orbital period to the star's rotation period. We also discovered that TOI-396 b and c display significant TTVs. While the TTV dynamical analysis returns a formally precise mass for TOI-396 c of Mc , dyn = 2 . 24 + 0 . 13 -0 . 67 M ⊕ , the result might not be accurate, owing to the poor sampling of the TTV phase. We also conclude that TOI-396 b and c are close to but out of the 5:3 MMR. Conclusions. A TTV dynamical analysis of additional transit photometry evenly covering the TTV phase and super-period is likely the most e ff ective approach for precisely and accurately determining the mass of TOI-396 c. Our numerical simulation suggests TTV semi-amplitudes of up to 5 hours over a temporal baseline of ∼ 5.2 years, which should be duly taken into account when scheduling future observations of TOI-396. Key words. planets and satellites: fundamental parameters - stars: fundamental parameters - techniques: photometric - techniques: radial velocities\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"Radii, masses, and transit-timing variations of the three-planet system orbiting the naked-eye star TOI-396 ⋆\",\n \"content\": \"A. Bonfanti 1 , I. Amateis 2 , 1 , 3 , D. Gandolfi 2 , L. Borsato 4 , J. A. Egger 5 , P. E. Cubillos 1 , 6 , D. Armstrong 7 , 8 , I. C. Leão 9 , M. Fridlund 10 , 11 , B. L. Canto Martins 9 , 12 , S. G. Sousa 13 , J. R. De Medeiros 9 , L. Fossati 1 , V. Adibekyan 13 , A. Collier Cameron 14 , S. Grziwa 15 , K. W. F. Lam 16 , E. Go ff o 17 , L. D. Nielsen 18 , F. Rodler 19 , J. Alarcon 20 , J. Lillo-Box 21 , W. D. Cochran 22 , 23 , R. Luque 24 , S. Redfield 25 , N. C. Santos 13 , 26 , S. C. C. Barros 13 , 26 , D. Bayliss 7 , 8 , X. Dumusque 27 , M. A. F. Keniger 7 , 8 , J. Livingston 28 , 29 , 30 , F. Murgas 31 , 32 , G. Nowak 33 , A. Osborn 34 , H. P. Osborn 5 , 35 , E. Pallé 31 , 32 , C. M. Persson 11 , L. M. Serrano 2 , P. A. Strøm 7 , 8 , S. Udry 27 , and P. J. Wheatley 7 , 8 (A ffi liations can be found after the references)\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1. Introduction\",\n \"content\": \"Multi-planet systems enable us to place significantly stronger constraints on formation and evolution mechanisms compared to single-planet systems (e.g. Lissauer et al. 2011; Fabrycky et al. 2014; Winn & Fabrycky 2015; Mishra et al. 2023). As a matter of fact, the temporal evolution of the gas content in the proto-planetary disc influences planet migration (e.g. Lin & Papaloizou 1979; Goldreich & Tremaine 1980; Tanaka et al. 2002; D'Angelo & Lubow 2008; Alexander & Armitage 2009), which shapes the orbital architecture of a planetary system. The latter is further expected to correlate with the planet composition (e.g. Thiabaud et al. 2014, 2015; Walsh et al. 2015; Bergner et al. 2020; Li et al. 2020) that can be inferred once the physical parameters of the planets are known (e.g. Dorn et al. 2017; Otegi et al. 2020b; Leleu et al. 2021; Haldemann et al. 2024). If planets are found in mean motion resonance (MMR; e.g. Lee & Peale 2002; Correia et al. 2018), systems can also shed light on the migration mechanisms during formation, as well as on the impact of tidal e ff ects occurring later on (e.g. Delisle et al. 2012; Izidoro et al. 2017). In addition, planets in, or close to, MMR likely exhibit transit timing variations (TTVs; see e.g. Agol et al. 2005; Agol & Fabrycky 2018; Leleu et al. 2021) that enable one to infer planetary masses without necessarily relying on radial velocity (RV) measurements, which are not always possible (e.g. Hatzes 2016). Multi-planet systems also give insights into correlations between physical and orbital parameters of exoplanets. For example, Weiss et al. (2018) noticed that planets in adjacent orbits usually show similar radii, hence the 'peas in a pod' label to describe this scenario. Weiss et al. (2018) further noticed that the outermost planet is the largest in the majority of cases, which agrees with the observed bulk planet density ( ρ p ) trend highlighted by Mishra et al. (2023), where ρ p decreases with the distance from the stellar host as outer planets are expected to be richer in volatiles (thus bigger and less dense). The object TOI-396 represents an interesting laboratory to test these theories as it is the brightest star known so far to host three transiting planets (Vanderburg et al. 2019) after ν 2 Lupi (Delrez et al. 2021). After analysing two sectors from the Transiting Exoplanet Survey Satellite ( TESS ; Ricker et al. 2015), Vanderburg et al. (2019) found that the three planets have radii of ∼ 2 R ⊕ and orbital periods of ∼ 3.6, ∼ 6.0, and ∼ 11.2 d, with TOI-396 c and b showing a period commensurability close to the 5:3 ratio. Following the notation introduced in Mishra et al. (2023), the three planets are 'similar' in terms of radii, and one may wonder whether this architecture class is kept also in the mass-period diagram. Mishra et al. (2023) found a positive and strong correlation of the coe ffi cients of similarity between radii and masses, though exceptions are possible (e.g. Weiss & Marcy 2014; Otegi et al. 2020a, 2022). In this work, we complement the photometric analysis of new TESS light curves (LCs) with RV observations taken with the High Accuracy Radial Velocity Planet Searcher ( HARPS ; Mayor et al. 2003) spectrograph to refine the planet radii and constrain for the first time the planetary masses. Considering the possible 5:3 MMR between TOI-396 c and b, we also simulate the evolution in time of the TTV amplitudes. This paper is organised as follows: Section 2 presents the stellar properties, and Sect. 3 describes the photometric and RV data. After outlining the method to jointly analyse the TESS LCs and the HARPS RV time series in Sect. 4, we present the corresponding results in Sect. 5. We attempt to dynamically model TTV and RV data simultaneously and track the temporal evolution of the TTV signals in Sect. 6, and we study the planets' internal structure in Sect. 7 and explore the prospects for characterising the system with the James Webb Space Telescope (JWST; Gardner et al. 2006) in Sect. 8. Finally Sect. 9 gathers the conclusions.\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"2. Host star characterisation\",\n \"content\": \"TOI-396 is an F6 V (Gray et al. 2006) bright naked-eye star with an apparent visual magnitude of V ≈ 6.4 (Perryman et al. 1997). It is located ∼ 31.7 pc away and is visible in the constellation of Fornax in the southern hemisphere. It is member of a visual binary system and its companion HR 858 B is a faint M-dwarf ( G ∼ 16 mag), about 8.4 '' away from the main component. We co-added 78 HARPS spectra (see Sect. 3.2 for further details) and then modelled it with Spectroscopy Made Easy 1 ( SME ; Piskunov & Valenti 2017) version 5.2.2. SME computes synthetic spectra from a grid of well established stellar atmosphere models and adjusts a chosen free parameter based on comparison with the observed spectrum. Here we used the stellar atmosphere grid A tlas 12 (Kurucz 2013) together with atomic line lists from the V ald database (Piskunov et al. 1995) in order to produce the synthetic spectra. We modelled one parameter at a time utilising spectral features sensitive to di ff erent photospheric parameters iterating until convergence of all free parameters. Throughout the modelling, we held the macro- and micro-turbulent velocities, v mac and v mic, fixed to 6 km s -1 (Doyle et al. 2014) and 1.34 km s -1 (Bruntt et al. 2010), respectively. A description of the modelling procedure is detailed in Persson et al. (2018). Finally, we obtained T e ff = 6354 ± 70 K, [Fe / H] = 0 . 025 ± 0 . 050, log g = 4 . 30 ± 0 . 06, and v sin i ⋆ = 7 . 5 ± 0 . 2 kms -1 . To double-check the derived spectroscopy parameters we performed an additional analysis employing ARES + MOOG (Sousa et al. 2021; Sousa 2014; Santos et al. 2013). In detail, we used the latest version of ARES 2 (Sousa et al. 2007, 2015) to consistently measure the equivalent widths (EW) for the list of iron lines presented in Sousa et al. (2008). Following a minimisation process, we then find the ionisation and excitation equilibria to converge for the best set of spectroscopic parameters. This process uses a grid of Kurucz model atmospheres (Kurucz 1993) and the radiative transfer code MOOG (Sneden 1973). We obtained T e ff = 6389 ± 67 K, [Fe / H] = -0 . 014 ± 0 . 045 dex, log g = 4 . 58 ± 0 . 11, and v mic = 1 . 54 ± 0 . 04 kms -1 . In this process we also derived a more accurate trigonometric surface gravity (log g trig = 4 . 34 ± 0 . 02) using recent GAIA data following the same procedure as described in Sousa et al. (2021). In the end ARES + MOOG provides consistent values when compared with the ones derived with SME . Using the SME stellar atmospheric parameters, we determined the abundances of Mg and Si following the classical curve-of-growth analysis method described in Adibekyan et al. (2012, 2015). Similar to the stellar parameter determination, we used ARES to measure the EWs of the spectral lines of these elements and a grid of Kurucz model atmospheres (Kurucz 1993) along with the radiative transfer code MOOG to convert the EWs into abundances, assuming local thermodynamic equilibrium. The stellar radius R ⋆ , mass M ⋆ , and age t ⋆ were derived homogeneously using the isochrone placement algorithm (Bonfanti et al. 2015, 2016) and its capability of interpolating a flexible set of input parameters within pre-computed grids of PARSEC 3 v1.2S (Marigo et al. 2017) isochrones and tracks. For each magnitude listed in Table 1, we performed an isochrone placement run by inserting the spectroscopic parameters derived above, the Gaia parallax π (Gaia Collaboration et al. 2023, o ff set-corrected following Lindegren et al. 2021), and the magnitude value to obtain an estimate for the stellar radius, mass, and age along with their uncertainties. From these results, we built the corresponding Gaussian probability density functions (PDFs) and then we merged (i.e. we summed) the PDFs derived from the di ff erent runs to obtain robust estimates for R ⋆ , M ⋆ , and t ⋆ . The radius R ⋆ derives essentially from T e ff , π , and the stellar magnitude, while M ⋆ and t ⋆ are much more model-dependent; therefore we conservatively inflated their internal uncertainties by 4% and 20%, respectively, following Bonfanti et al. (2021). Our adopted stellar parameters are listed in Table 1.\",\n \"pages\": [\n 2\n ]\n },\n {\n \"title\": \"3.1. TESS photometry\",\n \"content\": \"As presented in Vanderburg et al. (2019), TOI-396 was photometrically monitored by TESS during the first year of its nominal mission in Sector 3 from 20 September to 17 October 2018 (UT) in CCD 2 of Camera 2, and in Sector 4 from 19 October to 14 November 2018 (UT) in CCD 2 of Camera 1. TOI-396 was later Table 1: Stellar properties of TOI-396. Star names TOI-396 TIC 178155732 HR 858 HD 17926 HIP 13363 Gaia DR3 5064574724769475968 Notes. RA & DEC are reported as in J2000 reference frame. Values in the bottom half of the table have been derived as part of this paper. ( a ) Zero-point correction from Lindegren et al. (2021) applied. re-observed by TESS during the first year of its extended mission in Sector 30 from 23 September to 20 October 2020 (UT) in CCD 2 of Camera 2, and in Sector 31 from 22 October to 16 November 2020 (UT) in CCD 2 of Camera 1. All data were collected at a 2-minute cadence, except for Sector 3 for which TESS only sent down data at 30-minute cadence. We analyse all available TESS time series, including the Sector 3 and 4 data already presented in (Vanderburg et al. 2019). For Sector 3 we used the TESS Asteroseismic Science Opera- tion Center (TASOC) photometry (Handberg et al. 2021; Lund et al. 2021), while for the other sectors we analysed the Presearch Data Conditioning Simple Aperture Photometry (PDCSAP) LCs generated by the TESS Science Processing Operation Center (SPOC) pipeline Jenkins et al. (2016), which removes the majority of instrumental artefacts and systematic trends (Smith et al. 2012; Stumpe et al. 2012, 2014). We rejected those data marked with a bad quality flag and we performed a 5 medianabsolute-deviation (MAD) clipping of flux data points. After that, we built our custom LCs by splitting the TESS sectors in temporal windows centred around the transit events keeping ∼ 4 h of out-of-transit data points both before and after the transit for de-trending purposes. We ended up with 41 TESS LCs reporting the epoch of observation ( t ), the flux and its error, and further ancillary vectors available from the TESS science data product, that is mom _ centr 1, mom _ centr 2 (hereafter denoted with x and y , respectively), pos _ corr 1, and pos _ corr 2 (hereafter denoted with pc1 and pc2 , respectively) 4 .\",\n \"pages\": [\n 2,\n 3\n ]\n },\n {\n \"title\": \"3.2. HARPS high-resolution spectroscopy\",\n \"content\": \"We performed the radial velocity (RV) follow-up of TOI-396 with the HARPS spectrograph mounted at the ESO-3.6 m telescope at La Silla Observatory (Chile). We acquired 77 high resolution ( R ≈ 115 000) spectra between 31 January and 27 July 2019 (UT), covering a baseline of ∼ 177 days, as part of the follow-up programs of TESS systems carried out with the HARPS spectrograph (IDs: 1102.C-0923, 1102.C-0249, 0102.C0584; PIs: Gandolfi, Armstrong, De Medeiros). One additional spectrum was acquired during a technical night (ID: 60.A-9700) in February 2019. Following Dumusque et al. (2011b), we averaged out p-mode stellar pulsations by setting the exposure time to 900 s, which led to a median signal-to-noise (S / N) ratio of ∼ 320 per pixel at 550 nm. We used the second fibre of the HARPS spectrograph to simultaneously observe a Fabry-Perot lamp and trace possible instrumental drift down to the sub-metre per second level (Wildi et al. 2010). We reduced the data using the dedicated HARPS data reduction software ( DRS ; Pepe et al. 2002; Lovis, C. & Pepe, F. 2007) and computed the crosscorrelation function (CCF) for each spectrum using a G2 numerical mask (Baranne et al. 1996). We then performed a skew-normal (SN) fit on each CCF (Simola et al. 2019) in orderto extract the stellar radial velocity along with its error, the full width at half maximum (FWHMSN), the contrast ( A ), and the skewness parameter ( γ ) of the CCF. The advantages of using an SN-fit rather than a Normal fit are thoroughly discussed in Simola et al. (2019), while the SN-fitting details are outlined in, for example, Bonfanti et al. (2023); Luque et al. (2023); Fridlund et al. (2024). After the SN-based extraction, we ended up with an RV time series, whose ancillary vectors (FWHMSN, A , γ ) are the activity indicators used to constrain the polynomial basis to model and de-trend the RV component of the stellar activity (see Table A.1). As stellar activity is not stationary, the correlations between the RV observations and the activity indicators are expected to change over time as discussed in Simola et al. (2022), who proposed to apply the breakpoint ( bp ) technique (Bai & Perron 2003) to check whether these correlation changes are statistically significant. If so, the bp algorithm finds the optimal locations of correlation changes, which defines the segmenta- tion characterised by the lowest Bayesian Information Criterion (BIC; Schwarz 1978). Indeed, we found one breakpoint at observation 48 (BJD = 2458667.940719; ∆ BIC ≈ -14 with respect to the zero-breakpoint solution). Thus, we split our RV time series in two piecewise stationary segments and de-trended it on a chunk-wise base, rather than performing a global de-trending over the whole time series, similarly to what has already been done in Bonfanti et al. (2023); Luque et al. (2023).\",\n \"pages\": [\n 3,\n 4\n ]\n },\n {\n \"title\": \"4. Methods\",\n \"content\": \"We jointly analysed the TESS LCs and the RV time series within a Markov chain Monte Carlo (MCMC) framework using the MCMCI code (Bonfanti & Gillon 2020). When fitting the TESS LCs extracted from Sector 3 that has a cadence of 30 min, we generated the transit model using a cadence of 2 min (the same as the other TESS sectors) and then rebinned it to 30 min following Kipping (2010), who warns that long-cadence photometry may lead to retrieve erroneous system parameters. We imposed Normal priors on the input stellar parameters (i.e. T e ff , [Fe / H], R ⋆ , and M ⋆ ), as derived in Sect. 2 with a twofold aim: (i) the induced prior on the mean stellar density ρ⋆ (via M ⋆ and R ⋆ ) helps the convergence of the transit model; (ii) limb darkening (LD) coe ffi cients for the TESS filter may be retrieved following interpolation within A tlas 9-based 5 grids that were precomputed using the get_lds.py code 6 by Espinoza & Jordán (2015). We then set Normal priors on the quadratic LD coe ffi -cients using the values coming from the grid interpolation (i.e. u 1 , TESS = 0 . 2318 ± 0 . 0065 and u 2 , TESS = 0 . 3085 ± 0 . 0028) after re-parameterising them following Holman et al. (2006). On the planetary side, we adopted unbounded (except for the physical limits) uniform priors on the transit depth d F , the impact parameter b , the orbital period P , the transit timing T 0, and the RV semi-amplitude K , while we set the eccentricity e = 0 and the argument of periastron ω = 90 · for all planets. We come back to the assumption on the eccentricity below. The specific parameterisations of the jump parameters (aka step parameters) are outlined in Bonfanti & Gillon (2020). The LCs and the RV time series were de-trended simultaneously during the MCMC analysis using polynomials. To assess the polynomial orders to be associated with the di ff erent de-trending parameters for each time series, we first launch several MCMC preliminary runs made of 10 000 steps where we varied only one polynomial order at a time. We then selected the best de-trending baseline (see Table A.2) as the one having the lowest BIC. After that, we performed a preliminary MCMCI run comprising 200 000 steps to evaluate the contribution of both the white and red noise in the LCs following Pont et al. (2006); Bonfanti & Gillon (2020), so to properly rescale the photometric errors and get reliable uncertainties on the output parameters. Finally, three independent MCMCI runs comprising 200 000 steps each (burn-in length equal to 20%) were performed to build the posterior distributions of the output parameters after checking their convergence via the Gelman-Rubin statistic ( ˆ R ; Gelman & Rubin 1992). We also tested the possibility of eccentric orbits by imposing uniform priors on ( √ e cos ω , √ e cos ω ) either bounded to imply e ≲ 0 . 3 or completely unbounded (except for the physical limits). The wider the eccentricity range to be explored by the MCMC scheme, the poorer the parameter convergence, which suggests that the available data are not enough to con- he planetary eccentricities well. Moreover, the MCMCI runs with e , 0 are disfavoured by the ∆ BIC criterion (e.g. Kass & Raftery 1995; Trotta 2007) as well, in fact we obtained ∆ BIC = BIC e , 0 -BIC e = 0 ≳ + 100. This is also in agreement with the simulations performed by Vanderburg et al. (2019), who suggested that the periods' commensurability state of planets b and c is more likely maintained if the system is characterised by low eccentricities. Therefore, we adopted the circular solution as the reference one.\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"5.1. Joint LC and RV analysis with linear ephemerides\",\n \"content\": \"As mentioned in Sect. 4, we set P and T 0 as free parameters under the control of a uniform prior, which implies assuming linear ephemerides. With this setup, we improved the transit depth precision of all three planets by a factor ∼ 1.4 if compared with the results of Vanderburg et al. (2019). This improvement level is consistent with having twice the number of data points with respect to the LC analysis performed by Vanderburg et al. (2019), as well as with low TTV amplitudes (see Sect. 5.2). By combining the SN-fit onto the HARPS CCFs along with the bp method, we were able to estimate the masses of TOI-396 b and TOI-396 d to Mb = 3 . 56 + 0 . 92 -0 . 94 M ⊕ and Md = 7 . 2 ± 1 . 6 M ⊕ (detections at the 3.8 and 4.5 σ -level, respectively). Instead, we did not detect any significant Keplerian signal at the orbital period of TOI-396 c within the RV time series. In detail, we obtained a median Kc = 0 . 28 + 0 . 29 -0 . 20 ms -1 (3 σ upper limit K up c = 1 . 2 ms -1 ), which implies Mc = 0 . 92 + 0 . 94 -0 63 M ⊕ ( M up c = 4 . 0 M ⊕ ). . When combining the mass and radius values of the three planets, we obtain the following median estimates for the bulk planetary densities: ρ b = 2 . 56 + 0 . 71 -0 . 70 , ρ c = 0 . 67 + 0 . 69 -0 . 46 ( ρ up c = 3 . 1), and ρ d = 5 . 1 + 1 . 3 -1 . 2 g cm -3 . We note that the RV-undetected TOI396 c would be the least dense planet, while the densest planet is the outermost one (i.e. TOI-396 d), which constitutes a quite atypical architecture within the observed exoplanet population (e.g. Ciardi et al. 2013; Weiss et al. 2018; Mishra et al. 2023). However, this conclusion is just tentative, given the uncertainties on the mean planetary densities. Moreover, the detection level of the RV-inferred parameters of TOI-396 c is not statistically significant. We further tested whether including the MINERVAAustralis (Addison et al. 2019) RV data (30 measurements as taken from Vanderburg et al. 2019) can help detect the elusive planet. However, it turned out that their precision level ( ∼ 6 m s -1 ) is not high enough to improve the characterisation of the system. In other words, the RV semi-amplitudes we obtained are consistent and indistinguishable within the statistical fluctuation with what was derived from the more precise HARPS data set. This led us to further check that TOI-396 c indeed belongs to this system (Sect. 5.2) and to investigate the reason for its RV non-detection (Sect. 5.3).\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"5.2. Joint LC and RV analysis accounting for TTVs\",\n \"content\": \"As planet c is undetected in the RV time series, one may wonder whether TOI-396 c is a false positive. However, by using the VESPA tool (Morton 2012, 2015) that accounts for the constraints from the TESS LCs, imaging, and spectroscopy, Vanderburg et al. (2019) already computed that the false-positive probabilities (FPPs) are lower than 10 -3 for all three planets. In line with Vanderburg et al. (2019), we also confirmed that the Pc / Pb period ratio is commensurable and di ff ers from the 5:3 ratio by less than 0.027%. As planets with orbits in, or close to, resonances are likely to show TTVs, we decided to repeat the same MCMC analysis outlined above, but enabling the transit timings of each transit event to vary to then compute the TTV amplitude with respect to the linear ephemerides model derived in Sect. 5.1. All jump parameters converged ( ˆ R ≲ 1 . 01). The medians of the posterior distributions of the most relevant system parameters along with the 68.3% confidence intervals are listed in Table 2. The phase-folded LCs of the three planets are shown in Figure 1, while the phase-folded RV time series are displayed in Figure 2. In particular, the middle panel of Fig. 2 shows that, after subtracting the RV signals of both planets b and d, the RV time series looks flat consistently with the non-detection of TOI396 c. This analysis gives Rb = 2 . 004 + 0 . 045 -0 . 047 R ⊕ , Rc = 1 . 979 + 0 . 054 -0 . 051 R ⊕ , and Rd = 2 . 001 + 0 . 063 -0 . 064 R ⊕ and confirms the RV detection of TOI396 b and TOI-396 d (at the 3.8 and 4.5 σ -level, respectively) with Mb = 3 . 55 + 0 . 94 -0 . 96 M ⊕ and Md = 7 . 1 ± 1 . 6 M ⊕ . These outcomes are consistent with the results obtained from the analysis that assumes linear ephemerides (Sect. 5.1). We found significant TTV signals for planet b and c (see Figure 4, Top and Middle panels) as expected from the commensurability of their periods. The TTV statistical significance of TOI-396 b and c is evident even by eye when comparing the data points location with the shaded regions that represent the 1 σ uncertainties of the linear ephemerides. For each planet, we further quantified the reducedχ 2 ( ˆ χ 2 ) characterising the TTV amplitudes via where N tr , j is the number of transit event of the j -th planet. We obtained ˆ χ 2 b = 4 . 4, ˆ χ 2 c = 4 . 1, and ˆ χ 2 d = 0 . 7 for planets b, c, and d, respectively, which confirms the significance of the TTV amplitudes for TOI-396 b and c. Furthermore, the TTV amplitudes of TOI-396 b and c exhibit a clear anti-correlation pattern that we highlight by superimposing the TTV measurements in Figure A.1. This is a typical signature of gravitational interaction between the two planets, which confirms that TOI-396 c belongs to the system despite its elusiveness in the RV time series. For each planet, the timing of each transit event and the corresponding TTV amplitude computed with respect to the linear ephemerides are listed in Table A.3.\",\n \"pages\": [\n 4,\n 5\n ]\n },\n {\n \"title\": \"5.3. Discussion regarding why TOI-396 c is not detected in the RV time series\",\n \"content\": \"Magnetic activity combined with stellar rotation induces RV variations that can hide, a ff ect, or even mimic planetary signals (e.g. Queloz et al. 2001; Hatzes et al. 2010; Dumusque et al. 2011a; Haywood et al. 2014; Suárez Mascareño et al. 2017; Gandolfi et al. 2017). A possible explanation for the non-detection of the Doppler reflex motion induced by TOI-396 c is that stellar activity destructively interferes with the Keplerian signal of the planet. This may happen if the star has a rotation period Notes. All jump parameters, but the LD coe ffi cients, were subject to unbounded uniform priors following the parameterisations specified in Bonfanti & Gillon (2020); see text for further details. ( a ) Uncertainties from the run assuming linear ephemerides (no TTVs). T 0 values are shifted by -2 450 000. ( b ) Assuming zero albedo and full recirculation. ( c ) The 3 σ upper limits on the RV semi-amplitude, the planet mass, and the mean planet density of TOI-396 c are K up c = 1 . 2 ms -1 , M up c = 3 . 8 M ⊕ , and ρ up c = 2 . 9 g cm -3 , respectively. comparable to the orbital period of the planet (e.g. Vanderburg et al. 2016). Disentangling the planetary signal from stellar activity and retrieving the Doppler motion induced by the orbiting planet would then be challenging (e.g. Dragomir et al. 2012; Kossakowski et al. 2022). In order to investigate this hypothesis, we performed a frequency analysis of the line profile variation diagnostics (FWHM, contrast, and skewness) and activity indicators (log R ' HK and H α indexes). Figure 5 displays the time series (left column), along with the respective generalised Lomb-Scargle (GLS, Zechmeister & Kürster 2009) periodograms (right column). We assessed the significance of the peaks detected in the power spectra by estimating their false alarm probability (FAP), that is the probability that noise could produce a peak with power higher than the one we found in the time series. To account for the possible presence of non-Gaussian noise in the data, we estimated the FAP using the bootstrap randomisation method (see, e.g., Murdoch et al. 1993; Kuerster et al. 1997; Hatzes 2019). Briefly, we computed the GLS periodogram of 10 5 mock time series obtained by randomly shu ffl ing the data points along with their error bars, while keeping the timestamps fixed. We defined the FAP as the fraction of those mock periodograms whose highest power exceeds the power of the real data at any frequency. In the present work, we considered a peak to be significant if its false alarm probability is FAP < 0.1 %. We found that the time series of the log R ' HK and H α indexes display long-term trends likely due to the magnetic cycles of the star (Fig. 5, left column, first and third panels). In the Fourier domain, these trends translate into a significant (FAP < 0.1 %) excess of power at frequencies lower than the spectral resolution 8 of our HARPS observations (Fig. 5, right column, first and third panels). We modelled these long-term signals as quadratic trends and subtracted the best-fitting parabolas from the respective time-series. The GLS periodograms of the residuals of the activity indicators display significant peaks between ∼ 6 and 8 d (i.e, between ∼ 0.0125 and 0.167 d -1 ; Fig. 5, yellow area). The peaks are equally spaced by ∼ 0.0068 d -1 , which coincides with a peak found in the periodogram of the window function. Given the current data at our disposal, we are not able to distinguish between true frequencies and aliases. Although not significant, the power spectra of the contrast and skewness show peaks in the same frequency range, suggesting that the rotation period of the star might be ∼ 6-8 d. We note that the periodogram of the FWHM also displays an excess of power at low frequencies. However, the corresponding peak remains below our 0.1 %-FAP significance threshold (see Fig. 5, second to last panel). Yet, if we apply the same procedure described above and remove this signal by fitting a quadratic trend to the FWHM time series, we find no significant peak in the residuals. The projected equatorial velocity of the star ( v sin i ⋆ = 7 . 5 ± 0 . 2 km s -1 ), along with its radius ( R ⋆ = 1 . 258 ± 0 . 019 R ⊙ ), yields an upper limit for the rotation period of P up rot = 8 . 5 ± 0 . 3 d. Using the mean value 9 of log R ' HK = -4 . 926 ± 0 . 014, we inferred a stellar rotation period of 6.7 ± 1.3 d and 6.9 ± 1.3 d from the empirical equations of Noyes et al. (1984) and Mamajek & Hillenbrand (2008), respectively. In addition, by inputting the isochronal age into the gyrochronological relation from Barnes (2010), we computed a stellar rotation period of 7 . 1 + 1 . 0 -1 . 1 d. These results corroborate our interpretations that the peaks between 6 and 8 d significantly detected in the power spectra of the activity indicators originate from stellar rotation. To check whether a quasi-periodic signal compatible with ∼ 7 d is also present in the photometric data, for each TESS sector we extracted custom LCs from pixel data using lightkurve (Lightkurve Collaboration et al. 2018). In detail, we adopted the default quality bitmask and set the aperture to 'all', which corresponds to an aperture larger than the one used by the o ffi cial SAP pipeline. In fact, larger apertures mitigate the e ff ect of slow image drifts that could interfere with slow flux changes, such TV [min] T TV [min] T 40 30 20 10 0 10 20 60 50 40 30 20 10 0 10 20 8360 8380 8400 8420 8440 9100 9120 9140 9160 9180 9200 BJD - 2450000 [d] 8360 8380 8400 8420 8440 9100 9120 9140 9160 9180 9200 BJD - 2450000 [d] as the 6-8 d rotation period signals we aim to detect. After removing the temporal windows containing the transit events, we computed the GLS periodograms of these lightkurve -based LCs. The FAP was computed following the same bootstrap technique outlined above for the RV activity indicators. The four periodograms (Fig. A.2, first column) exhibit very significant peaks at ∼ 7.7, 7.9, 7.5, and 6.8 days for TESS Sectors 3, 4, 30, and 31, respectively. Except for Sector 30, they are not the most prominent peaks; however, they persist even after removing the most significant signals (Fig. A.2, second column). Whereas we acknowledge that the likely rotation period of the star is close to the first harmonic of the orbital period of TESS around the Earth ( P rot ,⋆ ∼ 1 2 P TESS ∼ 14 days), the photometric signal at 6-8 d is significant in all the four TESS sectors and persists after prewhitening the data. This signal is consistent with the rotation period detected in the HARPS activity indicators and inferred from log R ' HK , v sin i ⋆ , and gyrochronology, suggesting it is astrophysical in nature and due to the presence of active regions carried around by stellar rotation. Assuming the log R ' HK -based P rot ,⋆ = 6 . 7 ± 1 . 3 d as our reference estimate, the orbital period of TOI-396 c ( Pc ∼ 6 d) is close to P rot ,⋆ , which may explain the non-detection of planet c within the RV time series. If some kind of destructive interference between the Keplerian signal of planet c and the stellar activity has occurred, any artificial Keplerian signals with period P = Pc added to the observed RV time series should in principle be retrieved. To test this hypothesis, we considered Keplerian signals of the following form and generated four di ff erent RV time series by separately adding to the HARPS time series synthetic RV signals following Equation (2) with P = Pc , T 0 = T 0 , c , and K art = K in, where K in are the four di ff erent amplitude values listed in the first column of Table 3. For each RV time series, we then performed an MCMCI analysis to retrieve the RV semi-amplitude of the artificial signal ( K out; see Table 3). We note that the resulting K out ≈ K in + Kc , where Kc = 0 . 28 + 0 . 29 -0 . 19 ms -1 is the RV semi-amplitude of planet c as derived from the analysis on the original RV time series. As we essentially retrieved what we inserted in the HARPS time series, we may conclude that the destructive interference between the RV signals induced by the star and by planet c has already occurred and any further RV signal added to the RV time series is detected. As P rot ,⋆ is not exactly equal to Pc , we then repeated the test outlined above, but this time we injected into the original HARPS time series artificial Keplerian signals with P = P rot ,⋆ . The K out values obtained by the MCMC analyses depending on the di ff erent K in values are reported in Table 4. The K out values are systematically and significantly smaller than the corresponding K in values, which a posteriori supports the conclusion that the stellar rotation period is around 6-8 d. Furthermore, a planet with this orbital period would be firmly detected if its RV semi-amplitude were greater than Kd , which is the largest RV semi-amplitude detected for this system. Instead, by injecting a Keplerian signal with K in = Kd and P = P rot ,⋆ , the MCMCI analysis is able to barely detect (at ∼ 2 σ ) a planetary signal whose amplitude is half of that expected. The detection level increases when K in increases; however, we still underestimate K out. These tests prove that it is di ffi cult to retrieve planetary signals with P ∼ P rot ,⋆ . In summary, we conclude that stellar activity is responsible for generating spurious RV signals whose harmonics also include the stellar rotation period. As a consequence, it is hard to reliably detect planets with orbital periods comparable to P rot ,⋆ via the RV technique and Table 4 quantifies the magnitude of this e ff ect. We note that the Kc we obtained from the MCMCI analysis in Sect. 5.2 is comparable to the K out retrieved when inserting an artificial signal having K in = 1 . 0 ms -1 , which let us Notes. Column ∆ K K gives the relative di ff erence (in percentage) between the obtained RV semi-amplitude ( K out) and the expected one ( K in), while the last column ∆ K ≡ K out -K in lists the semi-amplitude di ff erence in terms of the 1 σ uncertainty of K out. to speculate that TOI-396 c might have Mc ∼ 3.0 M ⊕ and ρ c ∼ 2.0 g cm -3 .\",\n \"pages\": [\n 5,\n 6,\n 7,\n 8,\n 9,\n 10\n ]\n },\n {\n \"title\": \"6. Joint RV and TTV dynamical analysis\",\n \"content\": \"As mentioned, the anti-correlation pattern of the observed TTVs (see Fig. A.1) is a typical signal of the dynamical interaction between TOI-396 b and c. However, the data in our hands are not enough to currently derive meaningful planetary masses from the TTVs. Indeed, the photometric observations are clustered in two groups (about two years apart), which results in a partial coverage of the curvature of the TTVs. This prevents us from accurately mapping the full phases and super-periods of the TTV signals. Hence a dynamical fit onto the transit times would lead to a high fraction of low-amplitude TTVs with a short superperiod or to a low fraction of high-amplitude long-period TTV signals. Despite this, we attempted to run a dynamical joint fit of the TTV and RV data set with TRADES 10 (Borsato et al. 2014, 2019, 2021), a Fortran-python code developed to model TTVs and RVs simultaneously along with N-body integration. We have taken and fixed the stellar mass and radius values from Table 1. We further fixed the orbital inclinations i to the values in Table 2 and the longitude of the ascending nodes to Ω = 180 · 11 for all the planets. We fitted the planetary masses scaled by the stellar mass ( M b , c , d / M ⋆ ), the orbital periods ( P ), the eccentricities ( e ) and the arguments of the pericentre ( ω ) in the form ( √ e cos ω , √ e sin ω ), and the mean longitudes ( λ 12 ). We also fitted for an RV o ff set ( γ RV) and for an RV jitter term ( σ jitter) by adopting log 2 σ jitter as step parameter, although we used the de-trended RV data set as derived from Sect. 5.2, where a jitter term was already included in the RV errors. All fitting parameters were subject to uniform priors that account for their respective physical boundaries (see Table 5); for the eccentricities we applied a log-penalty (log p e ) based on the half-Gaussian ( e = 0 , σ e = 0 . 083) from Van Eylen et al. (2019). The reference time for the dynamical integration of the orbital parameters was set at T ref = 2 458 379 BJDTDB, that is before all available observations. We combined the quasiglobal di ff erential evolution (Storn & Price 1997) optimisation algorithm implemented in P y DE (Parviainen et al. 2016) with the A ffi ne Invariant MCMC Ensemble sampler (Goodman & Weare 2010) emcee implemented by Foreman-Mackey et al. (2019, 2013). We first ran P y DE and evolved 68 di ff erent initial configurations of parameter sets for 50 000 generations (number of steps for which each parameter is evolved). To perform the dynamical analysis in an MCMC fashion, we then ran emcee , assuming as starting point the outcome obtained with P y DE. We set up 68 chains for 600 000 steps each. To sample the parameter space e ffi ciently, we mixed the DEMove() and DESnookerMove() di ff erential evolution moves 13 in the proportion 80%-20% (Nelson et al. 2014; ter Braak & Vrugt 2008). We repeated the sequence P y DE + emcee twice, with di ff erent seeds for the random number generator. The chains reached convergence according to visual inspection and statistical indicators, such as the Gelman-Rubin ˆ R , the Geweke's statistic (Geweke 1991), and the auto-correlation function 14 . For both runs, we applied a conservative thinning factor of 100 and discarded the first 50% of the chains (burn-in). For each run, we derived the reference outcome (hereinafter also referred to as best-fit), as the maximum-a-posteriori (MAP) parameters' set, that is the set of parameters that maximises the log-probability 15 . The uncertainty of each parameter is quantified by the high-density interval (HDI) at 68.27% 16 of its posterior distribution. For each parameter we computed a Z-score defined as Z-score = | MAP1 -MAP2 | / p max | ERR1 | 2 + max | ERR2 | 2 , where the subscripts denote the two di ff erent runs and ERR = HDI -MAP. It turned out that Z-score < 1 for each parameter, which allows us to merge the posterior distributions deriving from the two runs to finally compute the MAP and the respective HDI from the merged posterior distributions. We further checked the Hill stability of the system (Sundman 1913) when assuming the entire merged posterior distributions by calculating the angular momentum deficit (AMD, Laskar 1997, 2000; Laskar & Petit 2017) criterion (Eq. 26 from Petit et al. 2018). Table 5 lists the parameters returned by TRADES (MAP and HDI) along with their respective priors. We note that the dynamical integration with TRADES allowed for a significant detection ( > 3 σ ) of the mass of TOI-396 c, that is Mc , dyn = 2 . 24 + 0 . 13 -0 . 67 M ⊕ . However, as explained above, TESS data do not allow us to fully map the TTV pattern. Therefore, the present-day estimate of Mc , dyn only provides an indication of the possible mass of the planet that might not be accurate, despite its formal precision. Indeed, to accurately and reliably determine planetary masses via TTVs, it is necessary to monitor the TTV signal by benefiting of a full sampling coverage, as demonstrated for example by the cases of Kepler-9 (Holman et al. 2010) and K2-24 (Petigura et al. 2016), whose orbital parameters and masses were comprehensively revisited by Borsato et al. (2014) and Petigura et al. (2018); Nascimbeni et al. (2024), respectively. In detail, Holman et al. (2010) had reported the masses of Kepler-9 b and c with a precision better than 8%, by performing a TTV analysis on the first three quarters of the Kepler data. Later on, by benefiting of twelve sectors of Kepler data that enabled the full mapping of the TTV phase, Borsato et al. (2014) obtained TTV-based masses di ff ering by a factor ∼ 2 from the estimate of Holman et al. (2010). Similarly, by accounting on more photometric data, Petigura et al. (2018); Nascimbeni et al. (2024) found that the mass of K2-24 c is lower by almost 2 σ than the estimate of Petigura et al. (2016) who had claimed a detection at the ∼ 4 σ level. After extracting all the synthetic transit timings T tr (i.e. the 'observed': O) from TRADES' analysis, we computed the 'calculated' (C) counterpart according to the linear ephemerides model based on what listed in Table 2. We plotted the O -C as a function of time as well as the RV best-fit model in Figures 6 and 7. Additionally, we investigated if the best-fit configuration is in or close to an MMR. We integrated the MAP parameters with the N-body code rebound (Rein & Liu 2012) and the symplectic Wisdom-Holman integrator whfast (Rein & Tamayo 2015; Wisdom & Holman 1991) for 10 000 years. We computed the evolution of the critical resonance angles of TOI-396 b and TOI396 c where p = 3 and q = 2 (for a second order 5:3 MMR), while ϖ ≡ ω + Ω is the longitude of the pericentre. We also computed the evolution of ∆ ϖ ≡ ϖ b -ϖ c = ( ϕ b -ϕ c) / q . In case of MMR, we expect that both ϕ b and ϕ c librate (i.e. oscillate) around a fixed value for the entire orbital integration. Instead, if these angles circulate, that is they span the full 0 · -360 · range (or equivalently the -180 · -180 · range), then the planet pair is not in resonance. We found that both ϕ b and ϕ c circulate (see the two upper panels of Fig. 8), which indicates that the system is not in an exact 5:3 MMR. Even if ∆ ϖ seems to oscillate around 0 · , it circulates every ∼ 2 000 years (bottom panel of Fig. 8), which further confirms that the system is not trapped in an MMR state. We also found the same behaviour for the resonant angles of 200 random samples drawn from the posterior distribution (not shown here). Our conclusions are consistent with the simulations performed by Vanderburg et al. (2019), who found that most realisations of the system are not in resonance. As shown in Fig. 6, the MAP parameter set predicts that the TTV super-period is longer than the time spanning the two clustered TESS observations. We decided to track the potential evolution of the TTV signals over a temporal baseline of ∼ 5.2 years. To this end, we ran forward numerical N-body simulations with TRADES, setting the initial conditions of the orbital parameters to be integrated at t = T ref. We ran a first simulation taking all the parameters from the MAP solution. Then, we further ran 200 simulations, where the sets of the system's parameters were randomly drawn from the merged posterior distributions. We computed the synthetic (i.e. observed: O ) transit times T tr and created a simulated O -C plot against time, where the C counterpart of T tr was computed assuming the linear ephemerides of Table 2. The results are displayed in Fig. 9 that emphasises a progressive drift of the O -C values inferred from the MAP parameters (black line) with respect to the zero value. As a consequence, the TTV amplitudes of planets b and c increase with time. In particular, the semi-amplitude of the O -C black curves are about two and five hours for b and c, respectively. The TTV super-period seems to be roughly equal to or larger than the integration time span, that is ∼ 5 years. The variance of the TTV amplitude (shaded area) is inferred from the results of the additional 200 simulations and reflects the widths of the posterior distributions from which the system's parameters were drawn. The remarkable O -C drifts (i.e. the poorly constrained linear ephemerides) combined with the uncertainty on the TTV amplitudes make challenging to plan future observations of planets b and c, as the actual transit timings might di ff er from the linear ephemerides predictions by ∼ 5 and ∼ 10 hours for TOI-396 b and TOI-396 c, respectively. TESS will not observe the target in the foreseeable future. 17\",\n \"pages\": [\n 10,\n 11\n ]\n },\n {\n \"title\": \"7. Internal structure\",\n \"content\": \"Using the masses and transit depths reported in Table 2, we ran the neural network based internal structure modelling framework plaNETic 18 (Egger et al. 2024) to infer the internal structure of TOI-396 b and d. plaNETic uses a full grid accept-reject sampling method in combination with a deep neural network (DNN) that was trained on the forward model of BICEPS (Haldemann et al. 2024) to infer the internal structure of observed planets. Each planet is modelled as a three-layered structure: an inner iron-dominated core, a silicate mantle and a fully mixed envelope made up of water and H / He. In the case of multi-planet systems, all planets are modelled simultaneously. As modelling the internal structure of exoplanets is a highly degenerate problem, the resulting inferred structure is, at least to a certain extent, dependent on the chosen priors. To mitigate this e ff ect, we ran a total of six models assuming six di ff erent combinations of priors. Most importantly, we use two di ff erent priors for the water content of the modelled planet, one motivated by a formation scenario outside the iceline (case A, water-rich) and one compatible with a formation inside the iceline (case B, water-poor). For both of these water priors, we choose three different options for the planetary Si / Mg / Fe ratios. In a first case, we assume that these match the stellar Si / Mg / Fe ratios exactly Thiabaud et al. (2015). Second, we assume that the planet is enriched in iron compared to its host star by using the fit of Adibekyan et al. (2021). For option 3, we model the planet independent of the stellar Si / Mg / Fe ratios, but just sampling the planetary ratios uniformly from the simplex where the molar Si, Mgand Fe ratios add up to 1, with an upper bound of 0.75 for Fe. These priors are described in more detail in Egger et al. (2024).\",\n \"pages\": [\n 11\n ]\n },\n {\n \"title\": \"8. JWST characterisation prospects\",\n \"content\": \"All three planets in the TOI-396 system share similar radii ( ∼ 2 R ⊕ ), but span a wide range of masses (0.9-7.1 M ⊕ ), which leaves open the question of whether they have primary or secondary atmospheres. Furthermore, the progression of bulk densities with distance from the host star varies in ways that cannot be described by simple formation and evolution models (e.g. Weiss et al. 2018; Mishra et al. 2023). Given the bright host star and combination of planetary masses, radii, and equilibrium temperatures, the three planets have favourable metrics for atmospheric characterisation in both transmission and emission among sub-Neptunes (Kempton et al. 2018, see Fig. 12). This makes the TOI-396 system a highly valuable laboratory to study the formation and evolution of plan- etary systems. Thus, we explored the prospects for characterisation with JWST. We focused these simulations on emission observations, but we note that transmission and emission have their own advantages and disadvantages in terms of achievable science goals and challenges. We employed the open-source P yrat B ay modelling framework (Cubillos & Blecic 2021) to compute synthetic spectra of the TOI-396 planets. These models consist of 1D cloud-free atmospheres in radiative, thermochemical, and hydrostatic equilibrium (Cubillos et al., in prep.). We varied the models' atmospheric elemental content to explore the wide range of compositions that the planets span. For this comparison we settled on two models to represent a primary- and a secondary-atmosphere scenario: the first is a gas giant with a 5 × solar metallicity, the second is a water world with a 80% H 2 O plus 20% CO 2 composition (based on the C / O ratios seen in the solar system minor bodies, see, e.g., Mumma & Charnley 2011; McKay et al. 2019). For the thermochemical-equilibrium calculations we considered a set of 45 neutral and ionic species, which are the main actors determining the thermal structure. For the radiative-transfer calculation we considered opacities from molecular species for CO, CO 2 , CH 4 , H 2 O, HCN, NH 3 , and C 2 H 2 from hitemp and E xo M ol (Rothman et al. 2010; Tennyson et al. 2016); Na and K resonant lines (Burrows et al. 2000); H, H2, and He Rayleigh (Kurucz 1970); and H2-H2 and H2-He collision-induced absorption (Borysow et al. 2001; Borysow 2002; Richard et al. 2012). We preprocessed the large E xo M ol line lists with the R epack algorithm (Cubillos 2017) to extract the dominant transitions. Figure 13 (top panels) shows the resulting thermal and composition structure for TOI-396 b (planets c and d follow a similar trend). The infrared synthetic emission spectra (Fig. 13, bottom panels) are mainly shaped by H 2 O, CO 2 , and CO features. At most wavelengths the primary- and secondary-atmosphere scenarios roughly di ff er by an o ff set, which would be hard to distinguish unless the energy budget of the planets are known. In contrast, the 4-5 µ mwindow shows the most distinctive spectral features; here the strong CO 2 absorption band at 4.4 µ m mainly allows one to distinguish primary from secondary atmospheres. Thus, in the following we focus on this region of the spectrum. We simulated JWST observations using the P andeia exposure time calculator Pontoppidan et al. (2016). The brightness of TOI-396 limits the instrument selection to NIRCam (F444W filter) to avoid saturation. We selected the fastest readout and subarray modes, 5 groups per integration, to optimise the S / N. We generated a distribution of (noised up) realisations for each model to estimate how many eclipses are required to distinguish between primary and secondary atmospheres at the 3 σ level. We found that 2, 4, and 8 eclipses (for planets b, c, and d, respectively) would be su ffi cient to di ff erentiate between these two models. Figure 13 shows one of those random realisations when including the required number of eclipses. The decreasing equilibrium temperature of the planets as they are located further away from TOI-396 plays the major role in the decreasing S / N for planets c and d.\",\n \"pages\": [\n 12,\n 13,\n 14\n ]\n },\n {\n \"title\": \"9. Conclusions\",\n \"content\": \"The object TOI-396 is an F6 V bright naked-eye star orbited by three planets of almost equal size, and the two inner planets are close to but out of a 5:3 MMR. A photometric analysis of the system was already performed by Vanderburg et al. (2019), but by benefiting from two additional TESS sectors, we improved the precision on the planet radii by a factor of ∼ 1.4, obtaining Rb = 2 . 004 + 0 . 045 -0 . 047 R ⊕ , Rc = 1 . 979 + 0 . 054 -0 . 051 R ⊕ , and Rd = 2 . 001 + 0 . 063 -0 . 064 R ⊕ . We determined the masses of the planets by extracting the RV time series from HARPS CCFs using an SN fit followed by a joint LC and RV MCMC analysis, where the RV de-trending uses the breakpoint method. We obtained a firm detection of the RV signals of planets b and d, deriving Mb = 3 . 55 + 0 . 94 -0 . 96 M ⊕ and Md = 7 . 1 ± 1 . 6 M ⊕ , but we can provide only a 3 σ upper limit for the mass of TOI-396 c of M up c = 3 . 8 M ⊕ . This yields the following mean planet densities: ρ b = 2 . 44 + 0 . 69 -0 . 68 , ρ up c = 2 . 9, and ρ d = 4 . 9 + 1 . 2 -1 . 1 g cm -3 , implying a quite unusual system architecture (Mishra et al. 2023) where the mid planet is the least dense and the outermost planet is the densest. The reason for the RV non-detection of any Keplerian signal at P = Pc ∼ 6 d is likely to be ascribed to the vicinity of Pc to the stellar rotation period. As a matter of fact, from the GLS periodograms of both the RV-related activity indices and the TESS raw LCs and from log R ' HK -based empirical relations, we consistently inferred P rot ,⋆ = 6 . 7 ± 1 . 3 d. After injecting synthetic Keplerian signals at P = P rot ,⋆ and di ff erent semi-amplitudes ( K in) into the RV time series, we empirically find that the RV semi-amplitudes output by the MCMC analyses ( K out) are systematically lower than the input ones by almost 3 σ , and they are statistically non-significant as far as K in ≲ Kd . In addition, we find that K out ≈ Kc when considering a planet with Mp ∼ 3 M ⊕ (i.e. ρ p ∼ 2 g cm -3 ), which might correspond to the properties of TOI-396 c. On a more general perspective, these simulations confirm that stellar activity destructively interferes with Keplerian signals having P ∼ P rot ,⋆ (e.g. Vanderburg et al. 2016), and furthermore, they indicate that - even in the case of firm detection - values of K out are significantly underestimated. Longer-baseline RV observations may help disentangle coherent signals originated by Keplerian motions from noncoherent signals due to stellar activity, even if degeneracy issues still hold when the planet orbital period is close to the stellar rotation period (Kossakowski et al. 2022). Alternatively, a possible constraint on Mc may come from TTVs, as planets b and c are close to an MMR of the second order. Indeed, the TTV amplitudes of the two planets show a characteristic anti-correlation pattern, as expected; however, the phase coverage given by the available observations is too poor to perform a conclusive TTV dynamical analysis based on the observed transit timings of the planets. We also attempted to fit the TTV and RV simultaneously while integrating the orbits of the system. We found that the masses and densities of planets b and d are consistent with the results from the joint LC and RV analysis. TOI-396 c shows a dynamical mass of Mc , dyn = 2 . 24 + 0 . 13 -0 . 67 M ⊕ , which is greater than that inferred from the joint LC and RV analysis, but it is consistent (Z-score = 1 . 2 σ ); the density is consistent at the 1 . 1 σ level. However, we emphasise that, although formally precise, the Mc , dyn estimate might not be accurate, as the full coverage of the TTV phase is needed to reliably compute TTV-based masses. Therefore, to fully confirm the system architecture, a reliable estimate of the mass of TOI-396 c is still missing. We also checked the evolution of the system over 10 000 years, and the critical resonance angles showed that planets b and c are close to but not in a 5:3 MMR. We further performed forward N-body simulations over a temporal baseline of ∼ 5.2 years in order to track the transit epochs and evaluate the expected TTV amplitudes during time. It turns out that TOI-396 b and TOI-396 c may exhibit TTVs with a super-period of about 5 years and semi-amplitudes of ∼ 2 and ∼ 5 hours, respectively. This translates into a temporal drift of the transit timings that can rise up to ∼ 5 and ∼ 10 hours with respect to the linear ephemerides computed from TESS data. Studying the planetary atmospheres with JWST would take advantage of the favourable spectroscopy metrics of the system (Kempton et al. 2018). Therefore, we set up 1D cloud-free atmospheric models, generated the synthetic emission spectra of the three planets, and simulated eclipse observations with JWST. It turns out that 2, 4, and 8 eclipses (for TOI-396 b, c, and d, respectively) would be su ffi cient to distinguish between primary and secondary atmosphere scenarios at the 3 σ level. Characterising the nature of the planetary atmosphere is also key to correctly assessing the planetary bulk densities (in particular for planet c). The potentially high TTVs inferred from our simulations should be duly taken into account when scheduling future observations of the target. This holds not only for JWST, but also for CHEOPS (Benz et al. 2021), which appears espe- cially suitable for collecting exquisite photometric data to enable the full characterisation of the system. Acknowledgements. We thank the anonymous referee for all the valuable comments that significantly improved the quality of the manuscript. This work has made use of data from the European Space Agency (ESA) mission Gaia ( https://www.cosmos.esa.int/gaia ), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/ gaia/dpac/consortium ). This research made use of Lightkurve, a Python package for Kepler and TESS data analysis (Lightkurve Collaboration et al. 2018). We thank contributors to NumPy (Harris et al. 2020), SciPy (Virtanen et al. 2020), matplotlib (Hunter 2007), astropy (Astropy Collaboration et al. 2013, 2018, 2022), astroquery (Ginsburg et al. 2019), and tesscut (Brasseur et al. 2019). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. We acknowledge financial support from the Agencia Estatal de Investigación of the Ministerio de Ciencia e Innovación MCIN / AEI / 10.13039 / 501100011033 and the ERDF 'A way of making Europe' through project PID2021-125627OB-C32, and from the Centre of Excellence 'Severo Ochoa' award to the Instituto de Astrofisica de Canarias. This research was funded in part by the UKRI, (Grants ST / X001121 / 1, EP / X027562 / 1). This work was supported by FCT - Fundação para a Ciência e a Tecnologia through national funds and by FEDER through COMPETE2020 - Programa Operacional Competitividade e Internacionalização by these grants: UIDB / 04434 / 2020; UIDP / 04434 / 2020. D.G., A.B., L.F., and L.M.S. gratefully acknowledge the financial support from the grant for internationalization (GAND_GFI_23_01) provided by the University of Turin (Italy). S.G.S acknowledges the support from FCT through Investigador FCT contract nr. CEECIND / 00826 / 2018 and POPH / FSE (EC). P.J.W. acknowledges support from the UK Science and Technology Facilities Council (STFC) through consolidated grants ST / P000495 / 1, ST / T000406 / 1 and ST / X001121 / 1. N.C.S. is funded by the European Union (ERC, FIERCE, 101052347). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. J.L.-B. is funded by the MICIU / AEI / 10.13039 / 501100011033 and NextGenerationEU / PRTR grant PID2019-107061GB-C61 and CNS2023-144309. X.D. acknowledges the support from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement SCORE No 851555) and from the Swiss National Science Foundation under the grant SPECTRE (No 200021_215200). This work has been carried out within the framework of the NCCR PlanetS supported by the Swiss National Science Foundation under grants 51NF40_182901 and 51NF40_205606. G.N. thanks for the research funding from the Ministry of Science and Higher Education programme the \\\"Excellence Initiative - Research University\\\" conducted at the Centre of Excellence in Astrophysics and Astrochemistry of the Nicolaus Copernicus University in Toru'n, Poland. Research activities of the Board of Observational and Instrumental Astronomy at the Federal University of Rio Grande do Norte are supported by continuous grants from the Brazilian funding agencies CNPq. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-Brasil (CAPES)-Finance Code 001 and CAPES-Print program. B.L.C.M., I.C.L., and J.R.M. acknowledge CNPq research fellowships. K.W.F.L. was supported by Deutsche Forschungsgemeinschaft grants RA714 / 14-1, RA714 / 14-2 within the DFG Schwerpunkt SPP 1992, Exploring the Diversity of Extrasolar Planets. L.B. acknowledges support from CHEOPS ASI-INAF agreement n. 2019-29-HH.0. D.G. sincerely thanks Stefano Camera for the inspiring and valuable discussions on the properties of TOI-396.\",\n \"pages\": [\n 14,\n 15,\n 16\n ]\n },\n {\n \"title\": \"References\",\n \"content\": \"Addison, B., Wright, D. J., Wittenmyer, R. A., et al. 2019, PASP, 131, 115003 Adibekyan, V., Dorn, C., Sousa, S. G., et al. 2021, Science, 374, 330 Adibekyan, V., Figueira, P., Santos, N. C., et al. 2015, A&A, 583, A94 Adibekyan, V. Z., Sousa, S. G., Santos, N. C., et al. 2012, A&A, 545, A32 Agol, E. & Fabrycky, D. C. 2018, in Handbook of Exoplanets, ed. H. J. Deeg & Baranne, A., Queloz, D., Mayor, M., et al. 1996, A&AS, 119, 373 Barnes, S. A. 2010, ApJ, 722, 222 Bruntt, H., Deleuil, M., Fridlund, M., et al. 2010, A&A, 519, A51 Burrows, A., Marley, M. S., & Sharp, C. M. 2000, ApJ, 531, 438 D'Angelo, G. & Lubow, S. H. 2008, ApJ, 685, 560 Delisle, J. B., Laskar, J., Correia, A. C. M., & Boué, G. 2012, A&A, 546, A71 Delrez, L., Ehrenreich, D., Alibert, Y., et al. 2021, Nature Astronomy, 5, 775 Dorn, C., Venturini, J., Khan, A., et al. 2017, A&A, 597, A37 Egger, J. A., Osborn, H. P., Kubyshkina, D., et al. 2024, A&A, 688, A223 Hatzes, A. P. 2016, in Astrophysics and Space Science Library, Vol. 428, Meth- Holman, M. J., Fabrycky, D. C., Ragozzine, D., et al. 2010, Science, 330, 51 Holman, M. J., Winn, J. N., Latham, D. W., et al. 2006, ApJ, 652, 1715 Izidoro, A., Ogihara, M., Raymond, S. N., et al. 2017, MNRAS, 470, 1750 Lissauer, J. J., Ragozzine, D., Fabrycky, D. C., et al. 2011, ApJS, 197, 8 Lovis, C. & Pepe, F. 2007, A&A, 468, 1115 Mamajek, E. E. & Hillenbrand, L. A. 2008, ApJ, 687, 1264 McKay, A. J., DiSanti, M. A., Kelley, M. S. P., et al. 2019, AJ, 158, 128 Pontoppidan, K. M., Pickering, T. E., Laidler, V. G., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9910, Observatory Operations: Strategies, Processes, and Systems VI, ed. Queloz, D., Henry, G. W., Sivan, J. P., et al. 2001, A&A, 379, 279 Rein, H. & Liu, S. F. 2012, A&A, 537, A128 Richard, C., Gordon, I. E., Rothman, L. S., et al. 2012, J. Quant. Spectr. Rad. Transf., 113, 1276 Sneden, C. A. 1973, PhD thesis, THE UNIVERSITY OF TEXAS AT AUSTIN. ter Braak, C. J. F. & Vrugt, J. A. 2008, Statistics and Computing, 18, 435 Thiabaud, A., Marboeuf, U., Alibert, Y., et al. 2014, A&A, 562, A27 Wisdom, J. & Holman, M. 1991, AJ, 102, 1528 A. Bonfanti\",\n \"pages\": [\n 17,\n 18,\n 19\n ]\n },\n {\n \"title\": \"Appendix A: Supplementary material\",\n \"content\": \"A. Bonfanti et al.: Radii, masses, and transit-timing variations of the three-planet system orbiting the naked-eye star TOI-396 Notes. TESS (TE) LCs are identified by a counter based on the chronological order of observation. In particular, LCs from 1 to 11, from 12 to 21, from 22 to 32, and from 33 to 41 were extracted from Sector 3, 4, 30, and 31, respectively. c indicates a normalisation scalar; see text for further details. T tr [BJD] TTV [min] Sector 8385 . 794 . 8391 . 8397 + 0 - 0 . 010 . + 0 - 0 7447 + 0 - 0 7195 . 8403 . 8415 . 8421 . 8427 . 8433 . 9120 6857 ± 0 + 0 - 0 6337 + 0 - 0 6062 + 0 - 0 5817 + 0 - 0 5482 + 0 - 0 . . . . . . . . . . 5407 . 9126 518 ± 0 . 9132 . 9138 . 9150 . 9156 . 9162 . 9168 . 8387 . 8398 . 8432 . 9117 . 9139 . 9150 . 9162 4916 + 0 - 0 + 0 - 0 4731 + 0 - 0 4169 + 0 - 0 4050 + 0 - 0 3798 + 0 - 0 3562 + 0 - 0 2711 + 0 - 0 5047 + 0 - 0 1919 + 0 - 0 2589 + 0 - 0 7171 + 0 - 0 9371 + 0 - 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 016 . . . . 0068 0063 0046 0047 . 0033 0 0015 0013 0017 0016 0065 0033 0042 0077 0052 0043 . 010 - 8 0032 0020 0057 0040 0085 0090 0029 0044 0027 0030 0068 0084 0036 0034 0056 0051 0050 0034 0076 0068 0047 0025 0072 0037 0038 0031 + 17 + 19 + 23 - 8 + 2 - 3 . . + 0 - 1 + 0 - 10 - 12 . + 43 + 9 + 11 . . + 15 - 22 + 9 - 9 + 6 - 6 . 7 . 0 + 4 - 4 + 2 - 1 0 . . . 4 + 2 - 2 7 + 9 - 4 7 + 6 - 11 + 7 - 6 . . 5 3 5 8 + 14 - 15 + 4 - 2 2 + 8 - 5 8 + 12 - 13 . . 6 + 4 - 6 + 3 - 4 0 + 10 - 12 TOI-396 d - 0 . 4 - 2 + 6 + 2 - 13 + 0 . . 4 + 5 - 4 + 8 - 7 + 7 - 4 0 . 2 + 11 - 10 + 6 - 3 1 + 10 - 5 + 5 - 4 . 5 . 5 3 . . 2 . 9 1 4 . 2 . 9 8 . 6 . . . . . 9 . 2 . 7 . . . . 2 4 8 4 . 8 . 1 . . 7 8 . 2 . 9 . 5 . 3 . 4 . 7 6 8 . 1769 3 3 3 3 4 4 4 4 30 30 30 30 31 31 31 31 3 3 4 30 30 31 31 TOI-396 c\",\n \"pages\": [\n 21,\n 23,\n 24\n ]\n }\n]"}}},{"rowIdx":4975,"cells":{"bibcode":{"kind":"string","value":"2024arXiv241118738F"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/2411.18738.pdf"},"content":{"kind":"string","value":"\nThe Galaxy Activity, Torus, and Outflow Survey (GATOS). VII. The 20-214 𝜇 mimaging atlas of active galactic nuclei using SOFIA\nL/i.pc/n.pc/d.pc/s.pc/a.pc/y.pc F/u.pc/l.pc/l.pc/e.pc/r.pc, 1 E/n.pc/r.pc/i.pc/q.pc/u.pc/e.pc L/o.pc/p.pc/e.pc/z.pc-R/o.pc/d.pc/r.pc/i.pc/g.pc/u.pc/e.pc/z.pc, 2, 3 I/s.pc/m.pc/a.pc/e.pc/l.pc G/a.pc/r.pc/c.pc'/dotlessi.pc/a.pc-B/e.pc/r.pc/n.pc/e.pc/t.pc/e.pc, 4 C/r.pc/i.pc/s.pc/t.pc/i.pc/n.pc/a.pc R/a.pc/m.pc/o.pc/s.pc A/l.pc/m.pc/e.pc/i.pc/d.pc/a.pc, 5, 6 A/l.pc/m.pc/u.pc/d.pc/e.pc/n.pc/a.pc A/l.pc/o.pc/n.pc/s.pc/o.pc-H/e.pc/r.pc/r.pc/e.pc/r.pc/o.pc, 7 C/h.pc/r.pc/i.pc/s.pc P/a.pc/c.pc/k.pc/h.pc/a.pc/m.pc, 1, 8 L/u.pc/l.pc/u.pc Z/h.pc/a.pc/n.pc/g.pc, 1 M/a.pc/s.pc/o.pc/n.pc L/e.pc/i.pc/s.pc/t.pc, 1 N/a.pc/n.pc/c.pc/y.pc L/e.pc/v.pc/e.pc/n.pc/s.pc/o.pc/n.pc, 9 M/a.pc/s.pc/a.pc I/m.pc/a.pc/n.pc/i.pc/s.pc/h.pc/i.pc, 8 S/e.pc/b.pc/a.pc/s.pc/t.pc/i.pc/a.pc/n.pc H/o.pc/e.pc/n.pc/i.pc/g.pc, 10 M/a.pc/r.pc/k.pc/o.pc S/t.pc/a.pc/l.pc/e.pc/v.pc/s.pc/k.pc/i.pc, 11, 12 C/l.pc/a.pc/u.pc/d.pc/i.pc/o.pc R/i.pc/c.pc/c.pc/i.pc, 13, 14 E/r.pc/i.pc/n.pc H/i.pc/c.pc/k.pc/s.pc, 15 E/n.pc/r.pc/i.pc/c.pc/a.pc B/e.pc/l.pc/l.pc/o.pc/c.pc/c.pc/h.pc/i.pc, 16, 17 F/r.pc/a.pc/n.pc/c.pc/o.pc/i.pc/s.pc/e.pc C/o.pc/m.pc/b.pc/e.pc/s.pc, 18 R/i.pc/c.pc D/a.pc/v.pc/i.pc/e.pc/s.pc, 19 S/a.pc/n.pc/t.pc/i.pc/a.pc/g.pc/o.pc G/a.pc/r.pc/c.pc'/dotlessi.pc/a.pc B/u.pc/r.pc/i.pc/l.pc/l.pc/o.pc, 20 O/m.pc/a.pc/i.pc/r.pc/a.pc G/o.pc/n.pc/z.pc '/a.pc/l.pc/e.pc/z.pc M/a.pc/r.pc/t.pc'/dotlessi.pc/n.pc, 21 T/a.pc/k.pc/u.pc/m.pc/a.pc I/z.pc/u.pc/m.pc/i.pc, 8 A/l.pc/v.pc/a.pc/r.pc/o.pc L/a.pc/b.pc/i.pc/a.pc/n.pc/o.pc, 22 M/i.pc/g.pc/u.pc/e.pc/l.pc P/e.pc/r.pc/e.pc/i.pc/r.pc/a.pc S/a.pc/n.pc/t.pc/a.pc/e.pc/l.pc/l.pc/a.pc, 23 D/i.pc/m.pc/i.pc/t.pc/r.pc/a.pc R/i.pc/g.pc/o.pc/p.pc/o.pc/u.pc/l.pc/o.pc/u.pc, 4, 24 D/a.pc/v.pc/i.pc/d.pc R/o.pc/s.pc/a.pc/r.pc/i.pc/o.pc, 25 D/a.pc/n.pc/i.pc/e.pc/l.pc R/o.pc/u.pc/a.pc/n.pc, 26 T/a.pc/r.pc/o.pc S/h.pc/i.pc/m.pc/i.pc/z.pc/u.pc, 19 /a.pc/n.pc/d.pc M/a.pc/r.pc/t.pc/i.pc/n.pc W/a.pc/r.pc/d.pc 27\n1 University of Texas at San Antonio, One UTSA Circle, San Antonio, TX, 78249, USA\n2 Department of Physics & Astronomy, University of South Carolina, Columbia, SC 29208, USA\n3 Kavli Institute for Particle Astrophysics & Cosmology (KIPAC), Stanford University, Stanford, CA 94305, USA\n4 Department of Physics, University of Oxford, Keble Road, Oxford OX1 3RH, UK\n5 Instituto de Astrof'ısica de Canarias, Calle V'ıa L'actea, s/n, E-38205 La Laguna, Tenerife, Spain\n6 Departamento de Astrof'ısica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain\n7 Centro de Astrobiolog'ıa (CAB), CSIC-INTA, Camino Bajo del Castillo s/n, E-28692, Villanueva de la Ca˜nada, Madrid, Spain\n8 National Astronomical Observatory of Japan, National Institutes of Natural Sciences (NINS), 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 9\nSpace Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA\n10 School of Physics & Astronomy, University of Southampton, Southampton SO17 1BJ, UK\n11 Astronomical Observatory, Volgina 7, 11060 Belgrade, Serbia\n12 Sterrenkundig Observatorium, Universiteit Gent, Krijgslaan 281-S9, Gent B-9000, Belgium\n13 N'ucleo de Astronom'ıa de la Facultad de Ingenier'ıa, Universidad Diego Portales, Av. Ej'ercito Libertador 441, Santiago, Chile\n14 Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, People's Republic of China\n15 Department of Physics and Astronomy, University of Alaska Anchorage, Anchorage, AK 99508-4664, USA\n16 Departmento de F'ısica de la Tierra y Astrof'ısica, Fac. de CC F'ısicas, Universidad Complutense de Madrid, E-28040 Madrid, Spain\nInstituto de F'ısica de Part'ıculas y del Cosmos IPARCOS, Fac. CC F'ısicas, Universidad Complutense de Madrid, E-28040 Madrid, Spain\n18 LERMA, Observatoire de Paris, Coll'ege de France, PSL University, CNRS, Sorbonne University, Paris, France\n19 Max Planck Institut fur Extraterrestrische Physik, Giessenbachstrasse 1, D-85748 Garching bei Munchen, Germany\n20 Observatorio de Madrid, OAN-IGN, Alfonso XII, 3, E-28014 Madrid, Spain\n21 Instituto de Radioastronom'ıa y Astrof'ısica (IRyA), Universidad Nacional Aut'onoma de M'exico, Antigua Carretera a P'tzcuaro #8701, ExHda. San Jos'e de la Huerta, Morelia, Michoac'an, C.P. 58089, Mexico\n22 Telespazio UK for the European Space Agency, ESAC, Camino Bajo del Castillo s/n, E-28692 Villanueva de la Ca˜nada, Spain\n23 Instituto de F'ısica Fundamental, CSIC, Calle Serrano 123, E-28006 Madrid, Spain\n24 School of Sciences, European University Cyprus, Diogenes street, Engomi, 1516 Nicosia, Cyprus\n25 School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne NE1 7RU, UK\n26 LESIA, Observatoire de Paris, Universit'e PSL, CNRS, Sorbonne Universit'e, Sorbonne Paris Cite'e, 5 place Jules Janssen, F-92195 Meudon, France\n27 Centre for Extragalactic Astronomy, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK\nABSTRACT\nWepresent a 19 . 7 -214 𝜇 mimaging atlas of local (4 -181 Mpc; median 43 Mpc) active galactic nuclei (AGN) observed with FORCAST and HAWC+ on board the SOFIA telescope with angular resolutions ∼ 3 '' -20 '' . This atlas comprises 22 Seyferts (17 Type 2 and 5 Type 1) with a total of 69 images, 41 of which have not been previously published. The AGN span a range of luminosities of log 10 ( 𝐿 bol [ erg / s ]) = [ 42 , 46 ] with a median of log 10 ( 𝐿 bol [ erg / s ]) = 44 . 1 ± 1 . 0. We provide total fluxes of our sample using aperture photometry for point source objects and a 2-D Gaussian fitting for objects with extended host galaxy emission, which was used to estimate the unresolved nuclear component. Most galaxies in our sample are point-like sources, however, four sources (Centaurus A, Circinus, NGC 1068, and NGC 4388) show extended emission in all wavelengths. The 30 -40 𝜇 m extended emission in NGC 4388 is coincident with the narrow line region at PA ∼ 50 · , while the dusty extension at longer wavelengths arises from the host galaxy at PA ∼ 90 · . Our new observations allow us to construct the best sampled spectral energy distributions (SEDs) available between 30 - 500 𝜇 m for a sample of nearby AGN. We estimate that the average peak wavelength of the nuclear SEDs is ∼ 40 𝜇 m in 𝜈𝐹 𝜈 , which we associate with an unresolved extended dusty region heated by the AGN.\n17\nCO(2-1) emission in NGC 5643 as a nuclear molecular gas component of the torus that is likely collimating the ionization cone. Conditions favorable for launching a cold and molecular wind likely depend on Eddington ratio and nuclear hydrogencolumndensities (e.g., Venanzi et al. 2020; Garc'ıa-Burillo et al. 2021; Alonso-Herrero et al. 2021; Garc'ıa-Bernete et al. 2022a).\nThis pc-scale dusty component is possibly associated with larger scale MIR emission detected out to 100s pc scales. In the case of Circinus, high angular resolution MIR imaging, optical polarimetry and integral field spectra, coupled with state-of-the-art radiative transfer simulations, provide evidence that extended dust emission from pc to tens of pc scales in this object is a result of a hollow dusty cone illuminated by a tilted accretion disk (Stalevski et al. 2017, 2019, 2023; Kakkad et al. 2023). MIR extended emission out to 1' ( ∼ 75 pc) was clearly detected in NGC 1068 by Bock et al. (2000). Later 10 . 8 and 18 . 2 𝜇 m emission extending 3 . '' 5 ( ∼ 200 pc) across NGC 4151 was also attributed to dust in the NLR heated by the central engine (Radomski et al. 2003). Likewise, at similar wavelengths, extended emission in 18 AGN at distances out to hundreds of parsecs was detected (Asmus et al. 2016; Garc'ıa-Bernete et al. 2016; Asmus 2019). Using the 37 . 1 𝜇 m filter on SOFIA/FORCAST and thanks to the increase in angular resolution compared with 𝑆𝑝𝑖𝑡𝑧𝑒𝑟 , extended dust emission in Mrk 3, NGC 4151, and NGC4388 was found on ∼ 100s pc-scales (Fuller et al. 2019) coincident with the NLR and radio axis. This emission may be due to dust along the wall of ionization cones (Mason et al. 2009) or a dusty NLR (Mor et al. 2009; Mor & Netzer 2012).\nIn this manuscript we present an imaging atlas of 22 local (D = 4 -181 Mpc; median 42.8 Mpc) AGN obtained using the FORCAST and HAWC+ instruments on the 2.7-m SOFIA telescope in the wavelength range 20 -214 𝜇 m. Most of these datasets are unpublished or dispersed throughout the literature. We provide a mid- to far-IR imaging atlas at angular scales of ∼ 3 -20 '' . At these scales, contribution from several dust sources is expected and we expect to disentangle the emission sources in a future study. Instead, here we aim to determine whether these objects are extended or not, and at what wavelengths within the resolution of the SOFIA telescope. We also explore the wavelength of turnover in the SED. This atlas is complementary to JWST observations up to ∼ 25 𝜇 mand archival Herschel data (70 -500 𝜇 m).\nThe manuscript is organzed as follows. Section 2 describes the observations and AGN sample definition; Section 3 shows the new IR images; Section 4 contains details of the imaging analysis and Section 5 shows the resulting SEDS; we present results about the data in Section 6.\n2. AGN SAMPLE AND OBSERVATION DATA\n2.1. Sample Selection\nThis imaging atlas was drawn from the ongoing AGN survey performed by the Galactic Activity, Torus, and Out-\nKeywords: galaxies -- active galaxies -- agn\n1. INTRODUCTION\nThere is clear evidence that a considerable amount of dust in the vicinity of supermassive black holes (SMBHs) in active galaxies obscures the central engine (i.e., accretion disk and SMBH) in some lines of sight. Through spectropolarimetric observations of NGC 1068, Antonucci & Miller (1985) showed that its optical polarized spectrum contained broad optical polarized emission lines not originally observed by direct total intensity observations. It was subsequently presumed that an optically and geometrically thick dusty structure ('torus') blocked the central engine in some lines of sight (Antonucci 1993; Urry & Padovani 1995). Under this unified scheme a Type 1 AGN is seen face-on and shows broadened optical lines, while in Type 2 AGN the broadened lines are obscured. This model also predicts that broad silicate features at 10 and 18 𝜇 m will be seen in emission in Type 1 and in absorption in Type 2. However, silicate emission can be seen in emission in some Type 2 AGN, while absorption can be seen in some Type 1 (e.g., Hatziminaoglou et al. 2015). This and other observational features are explained by the inhomogeneous nature of the torus. Clumpy torus models (Nenkova et al. 2008a,b) predict shallower silicate features, more similar infrared spectral energy distributions (SEDs) between Type 1 and Type 2 AGN, etc. (see Ramos Almeida & Ricci 2017, for a review).\nA region of narrow forbidden line emission extends above and below the midplane of the dusty torus structure out to several kpc scales. Recent sub-arcsecond interferometric imaging observations have shown a dust component at pc-scales co-spatial with the base of the narrow line region (NLR; Honig et al. 2012; Tristram et al. 2014; L'opez-Gonzaga et al. 2014, 2016; Burtscher et al. 2013; G'amez Rosas et al. 2022; Isbell et al. 2022). This dusty structure is interpreted as part of a dusty wind launched from the inner hot part of the torus driven by radiation pressure at pc-scales (Honig 2019), but generated by a magnetohydrodynamical wind at sub-pc scales (e.g., Emmering et al. 1992; Lopez-Rodriguez et al. 2015; Takasao et al. 2022; Lopez-Rodriguez et al. 2023). This extended dusty structure has been resolved in a nearby galaxy, ESO 418-G14, using mid-infrared (MIR) images with JWST/MIRI finding that the dust is primarily heated by the AGN and/or radiative jet-induced shocks in the NLR rather than a wind (Haidar et al. 2024).\nALMA observations provide observational support for a dusty torus+outflow scenario. Emission from the nucleus of NGC1068 was mapped with a resolution of ∼ 4 pc, resolving a 7 -10 pc diameter disk interpreted as the sub-mm counterpart of the torus (Garc'ıa-Burillo et al. 2016). Rotation of the compact emission was detected in HCN J=3-2 and HCO+ J=3-2 (Imanishi et al. 2018, 2020). A molecular outflowing wind co-spatial with the dusty and molecular torus was also observed (Garc'ıa-Burillo et al. 2019). Alonso-Herrero et al. (2018) interpreted the measured nuclear (10 -20 pc)\nflow Survey (GATOS; Garc'ıa-Burillo et al. 2021; AlonsoHerrero et al. 2021; Garc'ıa-Bernete et al. 2024). GATOS /one.sup aims to characterize the dynamics and composition of the dusty and molecular torus and multi-phase outflows in AGN. The GATOS parent sample is selected from the 70 Month Swift /BAT AGN catalog, which is flux limited in the ultrahard 14 -195 keV X-rays band (Baumgartner et al. 2013).\nIn the initial study of AGN using SOFIA observations (Fuller et al. 2016), sources from the GATOS survey were selected based on the criteria that the galaxies had been previously studied using C/l.pc/u.pc/m.pc/p.pc/y.pc (Nenkova et al. 2008a,b) torus models and were well-sampled in the 1 -18 𝜇 m regime (Ramos Almeida et al. 2009, 2011; Alonso-Herrero et al. 2011). The study of the 11 objects included the 31 . 5 𝜇 m photometry in the SEDs and found that including the 31.5 𝜇 m photometry reduces the number of C/l.pc/u.pc/m.pc/p.pc/y.pc torus models that are compatible with the data and modifies the model output for the torus outer radius. Fuller et al. (2019) further extended the wavelength range of a subset of 7 AGN SEDs to 37.1 𝜇 m. They subsequently found extended emission in the PSF-subtracted images of Mrk 3, NGC 4151, and NGC 4388 that is coincident with the radio axis and NLR. In a separate study, Lopez-Rodriguez et al. (2018) modeled the torus of NGC 1068 using ∼ 20 -53 𝜇 m FORCAST and HAWC+ observations. They showed that the peak wavelength range of emission from the torus is ∼ 30 - 40 𝜇 m with a characteristic temperature 70 - 100 K. The use of observations ¿ 30 𝜇 m in that study from SOFIA and ALMA highlights the importance of longer wavelength observations to put constraints on MIR emission sources. Based on these results, we extend the wavelength range in objects previously observed, and also expand the number of AGN observed.\nWe present the complete imaging atlas of 22 Seyferts observed by SOFIA in the wavelength range 19 . 7 -214 𝜇 m using FORCAST and HAWC+. The final set of observations presented here was part of a multi-year AGN survey over several observing SOFIA cycles (Proposal IDs: 02 0035, 04 0048, 06 0066, 08 0014; PI: Lopez-Rodriguez; 70 0400 PI: Herter). The SOFIA atlas of AGN in the far-IR (FIR) is a flux-limited sample of nearby, bright, and well-studied AGN. All objects have a point source flux of > 200 mJy at 31 . 5 𝜇 m, which ensures that each band can be observed within 1 hr of on-source time with a signal to noise ratio > 10 using FORCAST/SOFIA. Although the original AGN sample is larger than that presented here, only 22 AGN were observed in total by SOFIA before end of operations in 2022. Note that there are gaps in the 20 -214 𝜇 m wavelength range due to the fact that SOFIA only flies with a single instrument per night. For each SOFIA cycle, we prioritized the objects with observations acquired in a single instrument from the previous cycle.\nThe sample properties are given in Table 1. For most objects, we retrieved redshift data from the NASA Extragalactic Database (NED). However, for nearby objects Centaurus A\n
\n\n
Table 1. SOFIA AGN Sample
\n
\nRedshifts and spectral type were taken from NED. Distances to most sources were obtained using H0 = 70 km s -1 Mpc -1 . Distances to nearby sources Centaurus A and Circinus were taken from Harris et al. (2010) and Tully et al. (2009), respectively. References for log 𝐿 𝑏𝑜𝑙 : a) Borkar et al. (2021) b) Marinucci et al. (2012) c) Alonso-Herrero et al. (2011) d) Ichikawa et al. (2017), e) Leighly et al. (2014) f) Marconi et al. (2004), g) Baumgartner et al. (2013), h) Garc'ıa-Bernete et al. (2015), i) Ramos Almeida et al. (2011), j) Yuan et al. (2002), k) Duras et al. (2020),\nand Circinus, distances were obtained individually (Harris et al. 2010; Tully et al. 2009). The 22 objects in this atlas cover the luminosity range of log 10 ( 𝐿 bol [ erg / s ]) = [ 42 , 46 ] with a median of log 10 ( 𝐿 bol [ erg / s ]) = 44 . 1 ± 1 . 0, and a distance of 4 -181 Mpc with a median of 42.8 Mpc. Figure 1 shows bolometric luminosity plotted against distance, where Seyfert 1 objects are shown as red triangles and Seyfert 2 objects are shown as purple stars.\n2.2. SOFIA Observations and Data Reduction\n2.2.1. FORCAST\nFORCAST is an IR camera and spectrograph sensitive in the wavelength range 5 -40 𝜇 m with a field of view (FOV) of 3.4 ' × 3.2 ' and pixel scale 0.768 '' /pixel. With one exception (NGC 1068 in the 19.7 𝜇 m filter), we only used the Long Wavelength Camera (LWC; 25 -40 𝜇 m) due to the abundance of ground-based images at shorter wavelengths for the objects in our sample. FORCAST observations were made in dual channel mode using the two-position chop-nod (C2N) method with symmetric nod-match-chop (NMC) to remove telescope thermal emission and time variable sky background, and to reduce the effect of 1/ 𝑓 noise from the array. Data were reduced by the SOFIA Science Center using the /f.pc/o.pc/r.pc/c.pc/a.pc/s.pc/t.pc /r.pc/e.pc/d.pc/u.pc/x.pc pipeline following the methods described by Herter et al. (2012). Most of the pipeline changes over the\n\n\n
Figure 1. Luminosity plotted against distance of the 22 AGN in the atlas. Red triangles represent Sy1 and purple stars represent Sy2.
\n\ncycles were to refine the spectroscopic mode of FORCAST with little or no change to the image mode presented here.\nObservations were flux-calibrated using the set of standard stars of the observing run, which provides flux uncertainties of ∼ 10 %. The point spread function (PSF) of the 31.5 𝜇 m observations from Cycle 2 (see Fuller et al. 2016) was the co-average of a set of standard stars from that cycle. Its FWHMwas 3.40 '' , in agreement with the SOFIA Observer's Handbook v3.0.0. The PSFs for Cycle 4 in the 30 - 40 𝜇 mwavelength range were determined by using standard star observations from the individual flights (see Fuller et al. 2019) and averaged at FWHM ∼ 4.33 '' and 4.58 '' in the 31.5 and 37.1 𝜇 m filters, respectively. The FORCAST PSFs for the observations of NGC1068 are detailed in Lopez-Rodriguez et al. (2018).\n2.2.2. HAWC+\nHAWC+isaFIRimaging polarimeter designed to allow total and polarized intensity imaging observations in four broad bands centered at 53, 89, 155, and 214 𝜇 m, corresponding to Bands A, C, D, and E respectively (see Table 2). On-\n-fly mapping (OTFMAP) observing modes were used for both imaging polarimetry and total intensity imaging. Observing modes for the individual observations are given in Table 3. Data taken in polarization mode was reduced using the /h.pc/a.pc/w.pc/c.pc /d.pc/p.pc/r.pc /p.pc/i.pc/p.pc/e.pc/l.pc/i.pc/n.pc/e.pc and the reduction steps presented in Lopez-Rodriguez et al. (2022b). The Comprehensive Reduction Utility for SHARC II (CRUSH; Kov'acs et al. 2006; Kov'acs 2008) was used to obtain the total intensity observations. HAWC+ observations were reduced following the same reduction steps. We quote the pipeline versions or CRUSH versions in Table 3 to differentiate between the observing modes. There are no differences between CRUSH versions to obtain the total intensity images as all the changes in the pipeline were done for the polarimetric mode. Table 3 also shows the pixel scale of each image.\n
\n\n
Table 2. HAWC+ filter suite
\n
\nAs in FORCAST observations, the source of uncertainty in the photometry for HAWC+ results from calibration factors of the standard stars associated with the observation, giving an uncertainty of ∼ 10 % (Lopez-Rodriguez et al. 2022b). HAWC+ PSFs were estimated using standard star observations in 2017. Pallas was observed in Bands A and C on 7 November 2017, while Neptune was observed in Bands D and E on 19 October, 2017. The FWHM of these standards are 5.25 '' , 8.26 '' , 14.74 '' , and 19.65 '' in Bands A, C, D, and E, respectively. The FWHMs from the Observer's Handbook /two.sup are given in Table 2.\n2.3. Observing Data\nTable 3 provides the final AGN sample with information about the wavelength, observing mode, observation and mission details, versions of the separate pipelines, and also the field-of-view (FOV) of the individual images.\nT able 3 .\n
\n\n
T able 3 continued
\n
\n
\n\n
T able 3 (continued)T able 3 continued
\n
\nT able 3 (continued)\nN /o.pc/t.pc/e.pc -Obser v ation Data -Column 1: Object; Column 2: W a v elength; Column 3: Ins tr ument; Column 4: Obser v ation date; Column 5: Obser ving mode; Column 6: Pipeline/CR USH v ershion; Column 7: On-source time; Column 8: Aircraft s tar ting altitude; Column 9: Mission ID; Column 10: Prog ram ID; Column 11: FO V of imag es in Section 3 ; Column 12: Pix el scale\n
\n\n
\n3. IMAGES\nImages of the 22 AGN in the 19 . 7 -214 𝜇 m wavelength range are presented in Figures 2, 3, 4, and 5. The orange scale on the bottom left of the images indicates a scale of 500 pc. The beam size is depicted in white in the top right of the images. In all images, north is up and east is to the left. Complementary Herschel 70 -500 𝜇 m images are shown in Appendix A (Fig. 10, 11, 12, 13). These fully reduced images were obtained through the Herschel Science Archive /three.sup . All objects are presented and analyzed individually.\nCentaurus A . The host galaxy of Centaurus A is clearly visible in the 53 𝜇 m image with a bright compact center. However, the host galaxy becomes more dominant in the 89 𝜇 mimage(seeadetailed analysis of host galaxy dust emission at 89 𝜇 m in Lopez-Rodriguez 2021). The kpc-scale warped dust and gas lane was first observed with Spitzer imaging using IRAC and MIPS (Quillen et al. 2006). On subarcsecond scales, Radomski et al. (2008) observed the nucleus of Centaurus A using the 8.8, 10.4, and 18.3 𝜇 m filters on T-ReCS at Gemini South. They concluded that the mostly likely sources of nuclear MIR emission are an unresolved clumpy dusty torus in the core, and a dusty NLR for the arcsecondscale extended emission (see also Garc'ıa-Bernete et al. 2016).\nCircinus Galaxy . HAWC+ 53 and 89 𝜇 m images of the Circinus Galaxy show a very bright FIR core with extended emission at a PA ∼ 30 · , whereas the 215 𝜇 m image shows a slightly different PA ∼ 55 · . We estimate the FWHM of the extended FIR nuclear emission to be ∼ 6 '' × 6 '' , 13.5 '' × 11.5 '' , and 22.3 '' × 25.6 '' at 53, 89, and 214 𝜇 m, respectively. These are larger than the PSF FWHMs given in Section 2.2.2, which indicates extended emission along the axis of the inner bar of the galaxy. MIR emission was resolved at 8.7 and 18.3 𝜇 m out to 2 '' in an approximate east-west direction, coincident with the ionization cones at PA ∼ 100 · (Packham et al. 2005; Stalevski et al. 2017). However, the elongation seen in the SOFIAimages(andin Herschel imagesinAppendixA)seems to be arising from dust in the host galaxy.\nMCG-5-23-16 . MCG-5-23-16 appears as a point-like source in the 31 . 5 -155 𝜇 mwavelength range whose brightness decreases with increasing wavelength. However, this galaxy appears as an extended source in the MIR using high angular resolution data from VLT (Garc'ıa-Bernete et al. 2016). Likewise, Ferruit et al. (2000) found that this nucleus has an extended optical NLR at a PA of 40 · . They found a dust lane extending 2 '' on either side of the nucleus, parallel to the axis of the galaxy. Their extended dusty emission at 40 · was detected at a 3 𝜎 level up to ∼ 4 '' from the core. This extended structure has no thermal emission counterpart within the 31 . 5 -155 𝜇 mwavelength range.\nMrk3 . Although the FORCAST and HAWC+ images generally appear to be point-like sources, Fuller et al. (2019) found extended emission in the PSF-subtracted 37.1 𝜇 m im-\nof Mrk 3 in the direction of the radio axis (84 · ; Kukula et al. 1993) and the NLR ( ∼ 70 · ; Capetti et al. 1995). Asmus et al. (2013) found an elongated nucleus out to ∼ 170 pc with PA ∼ 70 · in the Si-5 (11.6 𝜇 m) filter using Gemini/MICHELLE. However, the Si-2 (8.7 𝜇 m) image from GTC/Canaricam appears point-like (Alonso-Herrero et al. 2016). The large scale east-west structure in the 53 𝜇 mimage here is a background artifact produced by the data reduction due to the small spatial coverage and short integration time of the observation.\nMrk 231 . The 89 𝜇 mimage shown here is point-like with a FWHM of ∼ 8.5', similar to the standard FWHM of 8.26' (see Section 2.2.2). Mrk 231 is a Type 1 Ultra Luminous Infrared Galaxy (ULIRG) and is the nearest known quasar at a distance of 181 Mpc. It is known for its multi-phase and multi-scale outflows (see Rupke & Veilleux 2011), with a neutral outflow up to 3 kpc in radius (Rupke et al. 2005). Mrk 573 . Although the SNR is very low (3 -4 𝜎 ), both 31.5 and 37.1 𝜇 mimages of Mrk 573 show marginally resolved ∼ 4 . 5 '' elongation in the east-west direction at PA ∼ 110 · . Mrk 573 was previously shown to have a biconical NLR coincident with radio emission 3-4' from the nucleus at a PA ∼ 125 · (Ulvestad & Wilson 1984; Pogge & De Robertis 1995). The marginal detection here may be cold extended dust in the outer layers of the NLR.\nNGC 1068 . The 19 - 53 𝜇 m images of NGC 1068 were published previously in Lopez-Rodriguez et al. (2018), where it was shown that the peak emission from the torus occurs between 30 - 40 𝜇 m with a corresponding temperature of 70 - 100 K. The 89 𝜇 m image was published as a polarimetric observation (Lopez-Rodriguez et al. 2020). Within a scale of about 1 kpc, NGC 1068 shows extended emission in the NE to SW direction at a PA ∼ 45 · , similar to MIR observations using VISIR/VLT (Asmus et al. 2014). Their observations revealed a nuclear structure in the north-south direction and extended structures to the NE and SW. From the N-band spectrum, Mason et al. (2006) concluded that while torus emission dominates NIR wavelengths, large-scale MIR emission is dominated by diffuse dust within the ionization cones.\nNGC 1275 . All 30 - 53 𝜇 m images are dominated by a point-like source. However, the 31.5 𝜇 m image (Fuller et al. 2019) shows 3 𝜎 extended emission along the PA ∼ 140 · . This AGN is known to have a network of H 𝛼 filaments extending out to ∼ 100' (see Conselice et al. 2001) and is possibly the result of a merger (Holtzman et al. 1992). The MIR core shows silicate dust emission in both 10 and 18 𝜇 mbands (see Fuller et al. 2019). Hence, both dust and gas are extended covering several kpc around the core. In the HAWC+ 89 𝜇 mfilter, Lopez-Rodriguez et al. (2023) found extended dust emission at a PA ∼ 125 · out to a 12 kpc radius potentially associated with a magnetized dusty filament along the NW direction (Fabian et al. 2008).\nNGC 2110 . The 30 - 215 𝜇 mimages of NGC 2110 are all point-like. The north-south pattern in the 53 𝜇 mimage is due to background noise and does not represent extended dust. NGC2110is a Type 2 AGN that shows silicate emission at 10 and18 𝜇 mthatis interpreted as a result of a clumpy torus, or as\n\n\n
Figure 2. FORCAST 31.5, and 37.1 𝜇 m images, and HAWC+ 53, 89, 155, and 215 𝜇 m images. Each image has a differing FOV, which can be found in Table 3. For bright objects (Centaurus A, Circinus, and Mrk 231) contours start at 3 𝜎 , then follow log(maximum) from [-1.2 to 0.8], [-1.8 to 0.8] [-1.4,0.8] in steps of 0.2. For MCG-5-23-16 and Mrk 3, the lowest contours are 3 𝜎 and increase in steps of 5 𝜎 . The white transparent circle on the top right indicates the telescope beam size. The orange bar on the bottom left of the images is scaled to 500 pc. For all images, north is up and east is to the left.
\n\ndust within the ionization cones (PA ∼ 160 · ; Mulchaey et al. 1994) in the inner 32 pc of the AGN (Mason et al. 2009). This galaxy appears as an extended source in Gemini/MICHELLE high resolution N-band observations (Garc'ıa-Bernete et al. 2016). However, any structure within the NLR or ionization cones is not resolved by our observations.\nNGC 2273 . The full set of 30 - 215 𝜇 m images of NGC 2273 show a point-like source. The north/south pattern in the 53 𝜇 m image is due to background noise and does not represent extended dust. Within the FWHM of these images\n(see Sections 2.2.1, 2.2.2), there is a known star-forming ring within ∼ 2' of the nucleus (Ferruit et al. 2000; Martini et al. 2003; Sani et al. 2012). GTC/Canaricam observations (Alonso-Herrero et al. 2014, 2016) at 8.7 𝜇 mshowelongation from the north-east to the south-west, likely with contribution from PAH. This structure is consistent with extension seen in the PSF-subtracted 37.1 𝜇 mSOFIAimage(Fulleretal. 2019).\nNGC 2992 . The image of NGC 2992 in the 31.5 𝜇 mfilter is published in Fuller et al. (2016) and appears as a point-like source. Subarcsecond N-band imaging (Garc'ıa-Bernete et al.\n\n\n
Figure 3. FORCAST 31.5 and 37.1 𝜇 m images, and HAWC+ 53, 89, 155, and 215 𝜇 m images. Note that the wavelength range for NGC 1068 starts 19.7 𝜇 m, so its range is shifted. Each image has a differing FOV, which can be found in Table 3. For NGC 1068 FORCAST images, contours begin at 3 𝜎 and follow log(maximum) from [-2.0,0.8] in steps of 0.2, while the HAWC+ images follow the same steps but log (max) ranges [-1.6,0.8]. For all other images, the lowest contours are 3 𝜎 and increase in steps of 5 𝜎 . The white transparent circle on the top right indicates the telescope beam size. The orange bar on the bottom left of the images is scaled to 500 pc. For all images, north is up and east is to the left.
\n\n2015) reveals extended emission along PA ∼ 30 · out to ∼ 3 kpc which is attributed to dust heated by star formation based on corresponding N-band spectroscopy. The FWHM of the SOFIA image is ∼ 3.5' × 3.5' (560 × 560 pc 2 ) so the extension should be resolvable within the SOFIA image. Since we do not see the extension in the image here, we conclude that either the extended dust emission tapers at wavelengths ¿ 20 𝜇 mor SOFIA does not have enough sensitivity to detect it.\nNGC 3081 . The 31.5 𝜇 m image of NGC 3081 was published in Fuller et al. (2016) while the 37.1 𝜇 m image was\npublished in Fuller et al. (2019). The nucleus is known to harbor a region of strong optical emission ∼ 1' from the AGN (Ferruit et al. 2000) likely due to dust or gas heated by the AGN. Fuller et al. (2019) estimated that ∼ 35% of the MIRemission within the central few arcseconds (few hundred parsecs) of the AGN originates in the NLR. High angular resolution N- and Q- band observations show extension towards the north, extending out to ∼ 450 pc from the south-east to the north-west (PA ∼ 160 · ; Garc'ıa-Bernete et al. 2016). On larger scales, optical and NIR observations reveal a series of star\n\n\n
Figure 4. FORCAST 31.5 and 37.1 𝜇 mimages, and HAWC+ 53, 89, 155, and 215 𝜇 mimages. Each image has a differing FOV, which can be found in Table 3. The lowest contours are 3 𝜎 and increase in steps of 5 𝜎 . The white transparent circle on the top right indicates the telescope beam size. The orange bar on the bottom left of the images is scaled to 500 pc. For all images, north is up and east is to the left.
\n\nforming resonance rings at distances of 2.3, 11.0, 26.9 kpc and 33.1 kpc (Buta 1990; Buta et al. 1998, 2004). At longer wavelengths (¿200 𝜇 m), Ramos Almeida et al. (2011) concluded that FIR emission is contaminated by the star-forming ring 2.3 kpc in diameter.\nNGC 3227 . The 31.5 𝜇 m image was published in Fuller et al. (2016) while the 37.1 𝜇 mimage was published in Fuller et al. (2019). These images show a point-like source, although NGC 3227 is known to harbor a nuclear star-forming region (Schinnerer et al. 2001; Davies et al. 2006) with a nuclear cluster within ∼ 70 pc ( ∼ 1') from the core. The 8.7 𝜇 mimage from Alonso-Herrero et al. (2016) shows a slight north/south elongation and the corresponding spectrum shows clear PAH\nin the nucleus (see also Garc'ıa-Bernete et al. 2016). These star forming regions likely contaminate the nuclear MIR emission within the FWHM of our images.\nNGC 3281 . While the 31.5 𝜇 m FORCAST image is published in Fuller et al. (2016), the HAWC+ images at 53, 89, 154, and 214 𝜇 m are presented here for the first time and appear point-like in all filters. The images taken at 53 and 89 𝜇 mappear to have significant noise in their backgrounds. The subarcsecond (0.35') N-band spectrum in Gonz'alez-Mart'ın et al. (2013) shows a deep 10 𝜇 m silicate absorption feature which originates in the inner ∼ 80 pc of the AGN.\nNGC 4151 . The SOFIA images of NGC 4151 appear as a point-like source, with a potential detection of extended\n\n\n
Figure 5. FORCAST 31.5 and 37.1 𝜇 m images, and HAWC+ 53, 89, 155, and 215 𝜇 m images. Each image has a differing FOV, which can be found in Table 3. The white transparent circle on the top right indicates the telescope beam size. The orange bar on the bottom left of the images is scaled to 500 pc. For all images, north is up and east is to the left.
\n\nNGC 7469\n15 5\nNGC 7674\n1\nemission at PA ∼ 120 · at a 3 𝜎 level at 37 . 1 𝜇 m. Fuller et al. (2019) confirmed this elongation in the PSF-subtracted 37.1 𝜇 mimagecoincident with the NLR and radio axes. Radomski et al. (2003) show extended emission in 10.8 and 18.2 𝜇 m images that coincides with the NLR axis at PA ∼ -60 · . For ≥ 37 . 1 𝜇 m, we conclude that any extended emission due to NLR dust is within the FWHM of the SOFIA instruments.\nNGC4258 . NGC 4258 was not detected but we include the data here since it is part of the sample. It has been observed and analyzed in the N-band with Gemini/Michelle by Mason et al. (2012). These authors found a compact nucleus that is marginally resolved at 10 𝜇 m(FWHM ∼ 0 . 5 '' ).\nNGC 4388 . NGC 4388 is an edge-on spiral that shows the most interesting mid- to far-IR morphology in this study. Notably, in the 30 - 40 𝜇 mFORCAST images of NGC 4388, extension can be seen in the NE to SW direction at PA ∼ 40 · (see also Fuller et al. 2019), coincident with the NLR. This emission is seen on smaller scales at shorter wavelengths (Asmus et al. 2016; Garc'ıa-Bernete et al. 2016). The 53 𝜇 m image decreases in intensity and does not show a strong central core of emission as in the 31 . 5 -37 . 1 wavelength range. However, at longer wavelengths (89 -214 𝜇 m), host galaxy emission clearly dominates the images in the east-west direction at PA ∼ 90 · .\nNGC 4941 . NGC 4941 is a low-luminosity AGN that appears here as a faint point-like source in the 31 . 5 𝜇 m and 37 . 1 𝜇 m images, but brighter at 53 and 89 𝜇 m. Subarcsecond resolution N-band imaging on VLT/VISIR (Asmus et al. 2011) showed no significant extended MIR sources outside of the nucleus.\nNGC 5506 NGC 5506 appears as a bright point source in both the 31.5 and 37.1 𝜇 m filters. While the nucleus is unresolved, extended MIR emission has been detected up to a few arcseconds to the northeast at 11.9 𝜇 m (Raban et al. 2008). Extended emission in the north-south direction was detected in the N-band out to ∼ 560 pc, while faint extended emission towards the east in the Q-band was also detected (Garc'ıa-Bernete et al. 2016). However, the PSF-subtracted 12.27 𝜇 m2' × 2' VLT/VISIR image of Alonso-Herrero et al. (2021) shows that the PA of extended emission varies from 30 · in the central ∼ 0.5' to nearly 90 · in the outer regions.\nNGC 7465 . The 31.5 𝜇 m FORCAST image appears faint with a 3 𝜎 upper-limit in the 37.1 𝜇 m image. The 53 and 89 𝜇 m HAWC+ images here appear increasingly brighter, albeit as point-like sources. Cold molecular gas observations (Young et al. 2021) reveal that NGC 7465 is quite gas-rich, possibly from a recent merger.\nNGC 7469 . NGC 7469 appears as a very bright source in the 31.5 𝜇 m image with FWHM ∼ 4.3'. After PSF subtraction, Fuller et al. (2016) found extended emission in the north-south direction. This AGN is known to have a circumnuclear ring of star formation at a radius of ∼ 480 pc ( ∼ 1.4'; Ramos Almeida et al. 2011) in 8.7 and 18.3 𝜇 mimages taken on Gemini/T-ReCS. Recent JWST observations reveal prominent PAH emission, indicative of star formation, in the circumnuclear ring (Garc'ıa-Bernete et al. 2022b; Zhang & Ho 2023).\nNGC 7674 . The previously published (Fuller et al. 2016) FORCAST 31.5 𝜇 m image appears as a point-like source. Asmus et al. (2013) found that the nucleus of NGC 7674 is extended at PA ∼ 125 · at subarcsecond scale resolution, where the extension roughly aligns with the ionization cone.\n4. NUCLEAR FLUX EXTRACTION\nWe aim to construct well-sampled mid- to far-IR SEDs of the nuclear emission of AGN at scales of several arcseconds, depending on the PSF of the observation and possible extended emission. On these scales, we expect multiple dust sources (i.e. torus, star forming regions, dusty outflow), however disentangling these sources is beyond the scope of this imaging atlas. Because the images span a range of observing cycles, instruments, and observing modes, we analyzed each image individually. Of our sample, 17 objects appear visually as point sources. For these sources, we performed aperture photometry where the aperture size was set to be 2 × the FWHM at a given band. For objects that show host galaxy emission, we extract the central PSF to construct the SEDs as described below. We complement our SOFIA data with Herschel imaging data (see Appendix A) and use a similar analysis method to construct the full IR SEDs.\n4.1. Extended Sources: 2D Gaussian Fitting\nFor sources with extended dust emission, we performed a two-component simultaneous fit to accurately model both the central source based on the PSF, and the host galaxy whose fit assumes an elongated 2D Gaussian profile. In order to supplement the SOFIA data for the full mid- to far-IR SEDs, we used a similar methodology with Herschel images.\nFor the SOFIA images, the PSF used was based on the standard stars of the observing runs for each cycle as explained in Section 2.2. However, the same analysis could not be performed on Herschel images due to the threefold lobes associated with the instrument PSFs. To accommodate this, we compared three different PSF models to reproduce and fit the central source. Two of the PSFs were point source images while the third was an approximation of the theoretical instrumental PSF using a Gaussian profile. The fitting routine used four free parameters for the Gaussian profile: (1,2) the position in x and y of the PSF center according to the image center, (3) the amplitude of the PSF, (4) the fourth parameter was dependent on the PSF type used. For archival PSFs, this parameter represents the rotation angle that needs to be applied to the PSF to match the orientation of the image. For the Gaussian PSFs, this fourth parameter represents a scaling factor to the width of the Gaussian compared to its ideal value for a perfect instrument (1 . 22 × 𝜆 / 𝐷 ).\nThe galaxy background is defined by 7 parameters: (1,2) the 2D Gaussian's center position (x0 and y0), (3,4) its width ( 𝜎 𝑥 and 𝜎 𝑦 ) in both directions, (5) its amplitude, and (6) its orientation on the image ( 𝜃 ). To these 6 parameters we added a constant background as a 7th free parameter. We combined these components and fit this simulated intensity map to the observed map using the 11 total free parameters (4 from the PSF and 7 from the 2D gaussian). We thus derived the parameters describing the best central source for our intensity maps, and then studied the properties extracted for the central source and removed it from the initial map to study the host galaxy itself. An example of this procedure is given in the Appendix in Figure 14.\n4.2. AGN and host galaxy contribution\nWhile most SOFIA images were treated as point sources, Centaurus A, Circinus, NGC 1068, and NGC 4388 all had significant host galaxy contamination that needed to be subtracted from at least some of the images. Figure 6 shows the PSF subtracted images of these sources and Table 4 gives the percentage of the contribution of the PSF to the total flux of the object. In several other objects, the shorter wavelength ( ∼ 30 - 100 𝜇 m) images did not show host galaxy contamination, but longer wavelength Herschel images show the colder extended dust. Because this is an atlas of SOFIA images, we include objects with host galaxy contamination only in Herschel images in Appendix C for completeness.\nThe53 𝜇 mimageofCentaurusAcontains ( < 5%)emission from extended sources, so we performed aperture photometry to account for the central emission. At wavelengths ≳ 70 𝜇 m, the host galaxy substantially ( ∼ 56 -70%) contributes to the central AGN emission, so the extended emission was\nsubtracted. The PSF of Centaurus A at these wavelengths contributes ∼ 35 %. This can be interpreted as the nucleus having a relatively constant IR contribution, so the brightness of the nucleus coincides with IR brightness of the host galaxy.\nThe PSF contribution of the Circinus Galaxy decreases between 53 and 160 𝜇 m from 58 % to 35 %. At longer FIR wavelengths, the contribution of the PSF appears to be from the host galaxy and the fitting no longer provides information about the AGN. We interpret this as a decreasing IR contribution from the nucleus compared to the extended emission.\nFor the completeness of the SOFIA Atlas presented here, we used the 19-53 𝜇 m images of NGC 1068 from LopezRodriguez et al. (2018). These datasets were analyzed as described in that study and here we only present the results and images in that wavelength range. The study showed that the fractional contribution from star formation increases from 20 - 50 𝜇 m, while extended emission from 200 K dust decreases. Emission from the torus peaks in this range, a result which is in agreement with Fuller et al. (2016) who found that the turnover in torus emission occurs at wavelengths ¿ 31.5 𝜇 m. Extended emission is observed here at all wavelengths ≳ 70 𝜇 m arising from dust in the host galaxy and star formation regions.\nFor NGC 4388, we show the results of PSF subtraction at all wavelengths, but only use the results in wavelengths ¿ 40 𝜇 m for the SED. In the 30 - 40 𝜇 m range, the NE to SW extension is clear in the PSF-subtracted images. However, almost all of the extended emission lies within the FWHM of the observation; the FWHM of these images are only ∼ 10% greater than the FWHM of the PSF. Thus, while we show the PSF subtracted images of NGC 4388 here, for the SED we use the total 30 - 40 𝜇 mfluxes which encompass the apparent extended emission due to the NLR. The change in the extended emission source and morphology between 40 and 70 𝜇 m is clear in the PSF-subtracted images (Figure 6). The host galaxy clearly dominates the extended emission in the FIR while the NLR region dominates the extended emission in MIR wavelengths. The 53 𝜇 m HAWC+ image appears to show the transition between dominant extended sources. The contribution of the PSF in the images of NGC 4388 is ∼ 60 % in the 30 - 40 𝜇 m range, where the extended emission is in the NE to SW direction. The contribution then decreases drastically to ∼ 20 %. This reflects the turnover in extended emission seen in the images in Figure 5. The contribution of the PSF returns to ∼ 70 % between 70 - 100 𝜇 m, which suggests two separate but significant IR emission sources.\n5. SPECTRAL ENERGY DISTRIBUTIONS\nTables 5 (SOFIA) and 6 ( Herschel ) give the nuclear fluxes of the AGN in our sample along with their associated errors in units of Jy. The sources of uncertainty here are the instrument calibration, sky background, and the 2D gaussian fitting, where applicable. We estimate FORCAST and HAWC+ errors at ∼ 10%. We use PACS instrument errors at 5% /four.sup and\nSPIRE instrument errors as 5.5% /five.sup . The uncertainty due to sky background is determined on an individual basis, but averages ∼ 5%. The average uncertainty due to the 2-D gaussian fitting is ∼ 1.5%. We add these uncertainties in quadrature for the error bar estimation.\nThe nuclear SEDs are shown in Figure 7 in 𝜈 F 𝜈 . The pink diamonds represent SOFIA observations while the black circles represent the complementary Herschel data. We obtained Spitzer /IRS spectra from the Sptizer /CASSIS database (Lebouteiller et al. 2011) for 21 of the 22 objects in our sample (solid black line). There was no spectrum available for NGC 7465. Low-resolution spectra (R ∼ 100) were obtained for 18 of the objects, while moderate resolution spectra (R ∼ 600) were available for Circinus, NGC 1068, and NGC 7674. This dataset provides the most completed SED coverage available between 30 - 500 𝜇 m. Decomposing the SEDs in this sample is outside the scope of this manuscript, as we are presenting an imaging atlas. Here, we provide the main results and features of the SEDs of these objects.\nThe morphological changes seen in the extended emission source in Figure 5 for NGC 4388 are reflected in the SED at 53 𝜇 m, where there is a marked decline in the SED. The drastic decrease seems to be due the change of dominant emitting sources. The extended emission at wavelengths ≲ 40 𝜇 m is due to dust in the direction of the radio axis, and the extended emission at wavelengths ≳ 50 𝜇 mis due to the host galaxy.\nThe wavelength of peak emission can give insight to the primary processes that drive MIR emission. The peak wavelength, determined by the highest flux from photometry and spectroscopy, ranges from 18 to 100 𝜇 m in 𝜈 F 𝜈 with an average of ∼ 40 𝜇 m. This average only includes the peak in continuum values and does not take into account fine structure lines. Most (73%; 11 out of 15) Seyfert 2 have a peak emission at wavelengths ≲ 40 𝜇 m. The SEDs of MCG-5-2316, Mrk3, Mrk 573, NGC 1068, NGC 3081, and NGC 4151 peak at ∼ 18 - 20 𝜇 m. This is in agreement with the correlation peak between the hard-X-rays and the mid-IR for Type 1 AGN in Garc'ıa-Bernete et al. (2017). The peak wavelengths in 𝐹 𝜈 (Jy) range ∼ 20 - 160 𝜇 m, with an average ∼ 93 𝜇 m. NGC 1068 is the only AGN to peak at the same wavelength in both sets of units.\nThe 𝑆𝑝𝑖𝑡𝑧𝑒𝑟 spectrum for NGC 1068 does not align with the SOFIA photometry because of the extensive PSF subtraction that we performed in the photometry that was not accounted for in the spectroscopy. This is the only object that not only has overlapping 20 - 40 𝜇 m 𝑆𝑝𝑖𝑡𝑧𝑒𝑟 and SOFIA data, but also that has had the background emission subtracted at these wavelengths.\n5.1. Luminosity and Peak Wavelength\nTo test whether the peak wavelength is a function of luminosity, we plot L 𝑏𝑜𝑙 vs 𝜆 𝑝𝑒𝑎𝑘 . Figure 8 shows the bolometric luminosities of the AGN plotted against the peak wavelength in the SEDs for both Sy1s, shown as red triangles, and Sy2s,\n\n\n
Figure 6. PSF-subtracted images of host galaxy backgroundsTable 4. Contribution of the PSF to the host galaxy extended emission
\n\n
\n\n
\nshown as purple stars. The correlation coefficient between the luminosity and peak wavelength is | 𝑅 | ∼ 0.63 with statistical significance 𝑝 = 0 . 0015. While it is argued that | 𝑅 | of 0.6 - 0.7 may show moderate to strong correlation (see Section 3.2 in Messenger et al. 2013), a 𝑝 -value ≤ 0.05 is generally accepted as statistically significant. The data suggests that higher luminosity objects have SEDs that peak at shorter wavelengths, which indicates the presence of a hot dust component in the vicinity of the AGN.\n5.2. Mid- to Far-IR Colors\nThe ratio of F 𝜈 (70)/F 𝜈 (160) has been used as a proxy for dust temperature (Mel'endez et al. 2014; Garc'ıa-Gonz'alez et al. 2016), where the ratio is higher for dust heated by the AGN and lower for dust heated by star formation. Here we perform this analysis using the ratio F 𝜈 (31)/F 𝜈 (70) by using the 31.5 𝜇 mSOFIA data in our atlas. For objects that do not have data in the 31.5 𝜇 mfilter, we supplement that with data from the Spitzer /IRS continuum. NGC 7465 did not have 31.5 𝜇 m flux data, nor did it have Spitzer data so we leave that object out of this analysis. Using the fluxes in Table 5, we plot a color-color diagram in F 𝜈 in Figure 9. This figure also visually shows the peak wavelength from the SEDs (in 𝜈𝐹 𝜈 ).\nLonger peak wavelengths tend to cluster at F 𝜈 (70)/F 𝜈 (160) ∼ 1 and F 𝜈 (31)/F 𝜈 (70) between 0.25 - 0.5.\nIn this sample we find an average F 𝜈 (70)/F 𝜈 (160) ratio of 1 . 4 ± 0 . 7. Previous studies (Mel'endez et al. 2014; Garc'ıaGonz'alez et al. 2016) with larger sample sizes (313 and 33, respectively) have found an average ratio of ∼ 0.8, albeit the data was analyzed using independent methods. This suggests a higher amount of AGN heated dust in our sample.\nWe find that 18 objects have F 𝜈 (31)/F 𝜈 (70) ¡1, with an average F 𝜈 (31)/F 𝜈 (70) of 0 . 6 ± 0 . 3. The only object with both F 𝜈 (70)/F 𝜈 (160) and F 𝜈 (31)/F 𝜈 (70) ¿1 is MCG-5-23-16, and an SED that peaks ∼ 20 𝜇 m. This object may be the most AGN dominated source in our sample. The other objects that show a peak at ∼ 20 𝜇 min 𝜈 F 𝜈 still show F 𝜈 (31)/F 𝜈 (70) ¡1. Only one object, NGC 4388, shows F 𝜈 (31)/F 𝜈 (70) ¿1 while F 𝜈 (70)/F 𝜈 (160) ¡1. This reflects the change in extended emission seen in Figure 5.\nHalf (11) of the objects in the sample show ratios F 𝜈 (31)/F 𝜈 (70) ¡1 while F 𝜈 (70)/F 𝜈 (160) ¿1. These objects (Circinus, Mrk 231, Mrk 573, NGC 1275, NGC 2110, NGC 3081, NGC 3227, NGC 3281, NGC 4151, NGC 4941, NGC 5506, NGC 7469) are likely AGN dominated. Six objects show F 𝜈 (31)/F 𝜈 (70) ¡1 and F 𝜈 (70)/F 𝜈 (160) ¡1, meaning that their SEDs peak at longer wavelengths. The emission from\n
\n\n
Table 5. Nuclear fluxes for SOFIA/FORCAST and HAWC+ images.
\n
\nR/e.pc/f.pc/e.pc/r.pc/e.pc/n.pc/c.pc/e.pc/s.pc: a) Fuller et al. (2016), b) Fuller et al. (2019), c) Lopez-Rodriguez et al. (2018), d) Lopez-Rodriguez et al. (2020), e) LopezRodriguez et al. (2022b). *Lopez-Rodriguez et al. (2018) measured the flux of NGC 1068 at 19.7 𝜇 mto be 22.0 ± 1.4.\nthese objects (Centaurus A, NGC 1068, NGC 2273, NGC 2992, NGC 4258, NGC 7674) are likely dominated by star formation.\n6. CONCLUSIONS\nWe have presented a SOFIA atlas of nearby AGN in the 20 - 215 𝜇 m wavelength range using FORCAST and HAWC+. We have released 69 observations of which 41 are newly published and 28 have been previously published (Fuller et al. 2016, 2019; Lopez-Rodriguez et al. 2018, 2022a). From these observations, NGC 4388 shows the most dramatic visual change in emission morphology. The 30 - 40 𝜇 m images show a NE to SW dusty extension associated with the NLR, while the ¿ 50 𝜇 mimages show a East to West dusty emission associated with the plane of the host galaxy. Our observations show that < 10 '' resolution 30 - 70 𝜇 m observations are crucial to disentangle the emitting contribution from AGN and host galaxy.\nWe measured arcsecond scale unresolved nuclear fluxes in order to construct SEDs of the objects in our sample. We included complementary Herschel data to cover up to 500 𝜇 m. For point sources we used aperture photometry to determine\nthe flux. For extended sources we used a 2D gaussian fitting method to extract the central unresolved source(s) of emission from the galaxy background. For this method, the PSF is scaled to represent the central emission while a 2D gaussian represents host galaxy or background emission. Based on the SEDs, we make the following conclusions:\n\n- There is a sharp drop in the SED of NGC 4388 that corresponds to the wavelength where the angle of extended emission transitions from NE/SW (NLR) to E/W (host galaxy).\n- The average peak of the SEDs is 40 𝜇 m in 𝜈 F 𝜈 , spanning a range of [20,100] 𝜇 m.\n- The peak wavelength of the SED appears to be a function of AGN luminosity, where higher luminosity objects peak at shorter wavelengths.\n- MCG-5-23-16 is the only object whose color diagram shows both F 𝜈 (31)/F 𝜈 (70) and F 𝜈 (70)/F 𝜈 (160) ¿1, which may indicate an AGN dominated source.\n- Half of the objects in the sample have flux ratios which suggest that the SED is dominated by AGN heated dust, while six objects show ratios consistent with heating by SF.\n\n
\n\n
Table 6. Nuclear fluxes for Herschel /PACS and SPIRE images.
\n
\nIn future studies, we will combine data from this atlas with incoming data from JWST to update our IR datasets with the latest and highest resolution data available. Newly obtained JWST/MIRI observations will provide new higher angular resolution data for some of the sources in the wavelength range 5 - 28 𝜇 m.\nACKNOWLEDGMENTS\nWe acknowledge Dr. Lucas Grosset for his effort in subtracting the image backgrounds. E.L.-R. is supported by the NASA/DLR Stratospheric Observatory for Infrared Astronomy (SOFIA) under the 08 0012 Program. SOFIA is\njointly operated by the Universities Space Research Association,Inc.(USRA), under NASA contract NNA17BF53C, and the Deutsches SOFIA Institut (DSI) under DLR contract 50OK0901 to the University of Stuttgart. E.L.-R. is supported by the NASA Astrophysics Decadal Survey Precursor Science (ADSPS) Program (NNH22ZDA001NADSPS) with ID 22-ADSPS22-0009 and agreement number 80NSSC23K1585. I.G.B. acknowledges support from STFC through grants ST/S000488/1 and ST/W000903/1. C.R. acknowledges support from Fondecyt Regular grant 1230345 and ANID BASAL project FB210003.\nSoftware: /a.pc/s.pc/t.pc/r.pc/o.pc/p.pc/y.pc (Astropy Collaboration et al. 2022, 2018, 2013); S/c.pc/i.pcP/y.pc Virtanen et al. (2020)\nREFERENCES\nAlonso-Herrero, A., Ramos Almeida, C., Mason, R., et al. 2011,\nApJ, 736, 82, doi: 10.1088/0004-637X/736/2/82\nAlonso-Herrero, A., Ramos Almeida, C., Esquej, P., et al. 2014,\nMNRAS, 443, 2766, doi: 10.1093/mnras/stu1293\nAlonso-Herrero, A., Esquej, P., Roche, P. F., et al. 2016, MNRAS,\n455, 563, doi: 10.1093/mnras/stv2342\nAlonso-Herrero, A., Pereira-Santaella, M., Garc´ıa-Burillo, S., et al. 2018, ApJ, 859, 144, doi: 10.3847/1538-4357/aabe30\nAlonso-Herrero, A., Garc´ıa-Burillo, S., H¨onig, S. F., et al. 2021,\nA&A, 652, A99, doi: 10.1051/0004-6361/202141219\nAntonucci, R. 1993, ARA&A, 31, 473,\ndoi: 10.1146/annurev.aa.31.090193.002353\n\n\n
Figure 7. Mid- to far-IR SEDs of the several-arcsecond-scale nuclear fluxes in our sample of AGN. Pink diamonds represent SOFIA observations while black dots represent complementary Herschel observations. The solid black lines correspond to Spitzer spectra.
\n\n\n\n
Figure 8. Bolometric luminosity vs the peak wavelength of the SED for both Sy1 (red triangles) and Sy2 (purple stars).
\n\nAntonucci, R. R. J., & Miller, J. S. 1985, ApJ, 297, 621,\ndoi: 10.1086/163559\n\n\n
Figure 9. Color diagram of 21 of the 22 AGN in our sample. Sy 1 are represented by triangles while Sy 2 are stars. The scale on the right shows peak wavelength by color.
\n\nAsmus, D. 2019, MNRAS, 489, 2177, doi: 10.1093/mnras/stz2289\n
\n\n
\n
\n\n
\n\nKov'acs, A., Chapman, S. C., Dowell, C. D., et al. 2006, ApJ, 650, 592, doi: 10.1086/506341\nKukula, M. J., Ghosh, T., Pedlar, A., et al. 1993, MNRAS, 264, 893, doi: 10.1093/mnras/264.4.893\nLebouteiller, V., Barry, D. J., Spoon, H. W. W., et al. 2011, ApJS, 196, 8, doi: 10.1088/0067-0049/196/1/8\nLeighly, K. M., Terndrup, D. M., Baron, E., et al. 2014, ApJ, 788, 123, doi: 10.1088/0004-637X/788/2/123\nL'opez-Gonzaga, N., Burtscher, L., Tristram, K. R. W., Meisenheimer, K., & Schartmann, M. 2016, A&A, 591, A47, doi: 10.1051/0004-6361/201527590\nL'opez-Gonzaga, N., Jaffe, W., Burtscher, L., Tristram, K. R. W., & Meisenheimer, K. 2014, A&A, 565, A71, doi: 10.1051/0004-6361/201323002\nLopez-Rodriguez, E. 2021, Nature Astronomy, 5, 604, doi: 10.1038/s41550-021-01329-9\nLopez-Rodriguez, E., Kishimoto, M., Antonucci, R., et al. 2022a, arXiv e-prints, arXiv:2207.09466.\n\nhttps://arxiv.org/abs/2207.09466\n\n-. 2023, ApJ, 951, 31, doi: 10.3847/1538-4357/accb96\nLopez-Rodriguez, E., Packham, C., Jones, T. J., et al. 2015, MNRAS, 452, 1902, doi: 10.1093/mnras/stv1410\nLopez-Rodriguez, E., Fuller, L., Alonso-Herrero, A., et al. 2018, ArXiv e-prints. https://arxiv.org/abs/1804.04134\nLopez-Rodriguez, E., Dowell, C. D., Jones, T. J., et al. 2020, ApJ, 888, 66, doi: 10.3847/1538-4357/ab5849\nLopez-Rodriguez, E., Clarke, M., Shenoy, S., et al. 2022b, ApJ, 936, 65, doi: 10.3847/1538-4357/ac83ac\nMarconi, A., Risaliti, G., Gilli, R., et al. 2004, MNRAS, 351, 169, doi: 10.1111/j.1365-2966.2004.07765.x\nMarinucci, A., Bianchi, S., Nicastro, F., Matt, G., & Goulding, A. D. 2012, ApJ, 748, 130, doi: 10.1088/0004-637X/748/2/130 Martini, P., Regan, M. W., Mulchaey, J. S., & Pogge, R. W. 2003, ApJS, 146, 353, doi: 10.1086/367817\nMason, R. E., Geballe, T. R., Packham, C., et al. 2006, ApJ, 640, 612, doi: 10.1086/500299\nMason, R. E., Levenson, N. A., Shi, Y., et al. 2009, ApJL, 693, L136, doi: 10.1088/0004-637X/693/2/L136\nMason, R. E., Lopez-Rodriguez, E., Packham, C., et al. 2012, AJ, 144, 11, doi: 10.1088/0004-6256/144/1/11\nMel'endez, M., Mushotzky, R. F., Shimizu, T. T., Barger, A. J., & Cowie, L. L. 2014, ApJ, 794, 152,\n\ndoi: 10.1088/0004-637X/794/2/152\n\nMessenger, S. J., Speck, A., & Volk, K. 2013, ApJ, 764, 142, doi: 10.1088/0004-637X/764/2/142\nMor, R., & Netzer, H. 2012, MNRAS, 420, 526, doi: 10.1111/j.1365-2966.2011.20060.x\n\nMor, R., Netzer, H., & Elitzur, M. 2009, ApJ, 705, 298,\n\ndoi: 10.1088/0004-637X/705/1/298\n\n
\n\n
\nVirtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature\nMethods, 17, 261, doi: 10.1038/s41592-019-0686-2\nYoung, L. M., Meier, D. S., Bureau, M., et al. 2021, ApJ, 909, 98, doi: 10.3847/1538-4357/abe126\nYuan, F., Markoff, S., Falcke, H., & Biermann, P. L. 2002, A&A, 391, 139, doi: 10.1051/0004-6361:20020817 Zhang, L., & Ho, L. C. 2023, ApJL, 953, L9, doi: 10.3847/2041-8213/acea73\nF/u.pc/l.pc/l.pc/e.pc/r.pc /e.pc/t.pc /a.pc/l.pc.\n\n\n
Figure 10. Herschel PACS and SPIRE FIR images. The white circle in the top right indicates the beam size of the observation, while the pink line in the bottom left indicates a distance of 1 kpc.
\n\nAPPENDIX\nA. HERSCHEL IMAGES\nWe used images from the Herschel Archive to supplement SOFIA data in FIR wavelengths. Images from the PACS and SPIRE instruments covering the wavelength range 70 - 500 𝜇 m are shown in Figures 10, 11, 12, and 13. The white circle in the upper right indicates the beam size while the pink scale in the bottom left indicates a distance of 1 kpc. To extract nuclear fluxes on scales of several arcseconds, we use the methods described in Section 4. For point sources, we performed aperture photometry. For extended sources, we used the 2D gaussian routine outlined in Section 4.1.\n\n\n
Figure 11. Herschel PACS and SPIRE FIR images. The white circle in the top right indicates the beam size of the observation, while the pink line in the bottom left indicates a distance of 1 kpc.
\n\nB. BACKGROUND FITTING\nFigure 14 shows an example of the 2-D gaussian fitting we used to extract the central flux from the background of the host galaxy in wavelengths 30 - 500 𝜇 m. The object used here is NGC 4388. In the first column, the original observation image is shown. The second column shows a model of the image using a PSF (Column 3) combined with the model background. Column 4 shows a PSF-subtracted image (Column 1 - Column 3) used to make the model background (Column 5). Column 6 shows the PSF- and background- subtracted image of the residuals.\nC. HOST GALAXY BACKGROUNDS HERSCHEL\nHere we show the results of PSF-subtracted images for objects in which only Herschel data showed host galaxy contamination.\nF/u.pc/l.pc/l.pc/e.pc/r.pc /e.pc/t.pc /a.pc/l.pc.\n\n\n\n\n\n\n\n\n\n19'00\n\n\n\n\n\n\n50'00\n\n\n\n52'30\"\n51'30\"\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n50'40\"\n51'00\"\n52'00\"\n24 30\"\n73'30\"\n\n\n\nRA (J2000}\n(2000)\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n19'00\n18'30\n\n\n\n\n\n\n\n\n\n34*50'\n\n\n\n\n\n\n\n\n\n\n\n
Figure 12. Herschel PACS and SPIRE FIR images. The white circle in the top right indicates the beam size of the observation, while the pink line in the bottom left indicates a distance of 1 kpc.
\n\nRA U2000)\n\n\n\n19*53'\n\n\n\n2*48'\n\n\n
Figure 13. Herschel PACS and SPIRE FIR images. The white circle in the top right indicates the beam size of the observation, while the pink line in the bottom left indicates a distance of 1 kpc.
Figure 14. Example of the 2D gaussian fitting routine explained in Section 4.1
\n\nSOFIA A/t.pc/l.pc/a.pc/s.pc\n\n\n
Figure 15. PSF-subtracted host galaxy images from archive Herschel data
\n\n"},"sections":{"kind":"list like","value":[{"title":"ABSTRACT","content":"Wepresent a 19 . 7 -214 𝜇 mimaging atlas of local (4 -181 Mpc; median 43 Mpc) active galactic nuclei (AGN) observed with FORCAST and HAWC+ on board the SOFIA telescope with angular resolutions ∼ 3 '' -20 '' . This atlas comprises 22 Seyferts (17 Type 2 and 5 Type 1) with a total of 69 images, 41 of which have not been previously published. The AGN span a range of luminosities of log 10 ( 𝐿 bol [ erg / s ]) = [ 42 , 46 ] with a median of log 10 ( 𝐿 bol [ erg / s ]) = 44 . 1 ± 1 . 0. We provide total fluxes of our sample using aperture photometry for point source objects and a 2-D Gaussian fitting for objects with extended host galaxy emission, which was used to estimate the unresolved nuclear component. Most galaxies in our sample are point-like sources, however, four sources (Centaurus A, Circinus, NGC 1068, and NGC 4388) show extended emission in all wavelengths. The 30 -40 𝜇 m extended emission in NGC 4388 is coincident with the narrow line region at PA ∼ 50 · , while the dusty extension at longer wavelengths arises from the host galaxy at PA ∼ 90 · . Our new observations allow us to construct the best sampled spectral energy distributions (SEDs) available between 30 - 500 𝜇 m for a sample of nearby AGN. We estimate that the average peak wavelength of the nuclear SEDs is ∼ 40 𝜇 m in 𝜈𝐹 𝜈 , which we associate with an unresolved extended dusty region heated by the AGN. 17 CO(2-1) emission in NGC 5643 as a nuclear molecular gas component of the torus that is likely collimating the ionization cone. Conditions favorable for launching a cold and molecular wind likely depend on Eddington ratio and nuclear hydrogencolumndensities (e.g., Venanzi et al. 2020; Garc'ıa-Burillo et al. 2021; Alonso-Herrero et al. 2021; Garc'ıa-Bernete et al. 2022a). This pc-scale dusty component is possibly associated with larger scale MIR emission detected out to 100s pc scales. In the case of Circinus, high angular resolution MIR imaging, optical polarimetry and integral field spectra, coupled with state-of-the-art radiative transfer simulations, provide evidence that extended dust emission from pc to tens of pc scales in this object is a result of a hollow dusty cone illuminated by a tilted accretion disk (Stalevski et al. 2017, 2019, 2023; Kakkad et al. 2023). MIR extended emission out to 1' ( ∼ 75 pc) was clearly detected in NGC 1068 by Bock et al. (2000). Later 10 . 8 and 18 . 2 𝜇 m emission extending 3 . '' 5 ( ∼ 200 pc) across NGC 4151 was also attributed to dust in the NLR heated by the central engine (Radomski et al. 2003). Likewise, at similar wavelengths, extended emission in 18 AGN at distances out to hundreds of parsecs was detected (Asmus et al. 2016; Garc'ıa-Bernete et al. 2016; Asmus 2019). Using the 37 . 1 𝜇 m filter on SOFIA/FORCAST and thanks to the increase in angular resolution compared with 𝑆𝑝𝑖𝑡𝑧𝑒𝑟 , extended dust emission in Mrk 3, NGC 4151, and NGC4388 was found on ∼ 100s pc-scales (Fuller et al. 2019) coincident with the NLR and radio axis. This emission may be due to dust along the wall of ionization cones (Mason et al. 2009) or a dusty NLR (Mor et al. 2009; Mor & Netzer 2012). In this manuscript we present an imaging atlas of 22 local (D = 4 -181 Mpc; median 42.8 Mpc) AGN obtained using the FORCAST and HAWC+ instruments on the 2.7-m SOFIA telescope in the wavelength range 20 -214 𝜇 m. Most of these datasets are unpublished or dispersed throughout the literature. We provide a mid- to far-IR imaging atlas at angular scales of ∼ 3 -20 '' . At these scales, contribution from several dust sources is expected and we expect to disentangle the emission sources in a future study. Instead, here we aim to determine whether these objects are extended or not, and at what wavelengths within the resolution of the SOFIA telescope. We also explore the wavelength of turnover in the SED. This atlas is complementary to JWST observations up to ∼ 25 𝜇 mand archival Herschel data (70 -500 𝜇 m). The manuscript is organzed as follows. Section 2 describes the observations and AGN sample definition; Section 3 shows the new IR images; Section 4 contains details of the imaging analysis and Section 5 shows the resulting SEDS; we present results about the data in Section 6.","pages":[1,2]},{"title":"The Galaxy Activity, Torus, and Outflow Survey (GATOS). VII. The 20-214 𝜇 mimaging atlas of active galactic nuclei using SOFIA","content":"L/i.pc/n.pc/d.pc/s.pc/a.pc/y.pc F/u.pc/l.pc/l.pc/e.pc/r.pc, 1 E/n.pc/r.pc/i.pc/q.pc/u.pc/e.pc L/o.pc/p.pc/e.pc/z.pc-R/o.pc/d.pc/r.pc/i.pc/g.pc/u.pc/e.pc/z.pc, 2, 3 I/s.pc/m.pc/a.pc/e.pc/l.pc G/a.pc/r.pc/c.pc'/dotlessi.pc/a.pc-B/e.pc/r.pc/n.pc/e.pc/t.pc/e.pc, 4 C/r.pc/i.pc/s.pc/t.pc/i.pc/n.pc/a.pc R/a.pc/m.pc/o.pc/s.pc A/l.pc/m.pc/e.pc/i.pc/d.pc/a.pc, 5, 6 A/l.pc/m.pc/u.pc/d.pc/e.pc/n.pc/a.pc A/l.pc/o.pc/n.pc/s.pc/o.pc-H/e.pc/r.pc/r.pc/e.pc/r.pc/o.pc, 7 C/h.pc/r.pc/i.pc/s.pc P/a.pc/c.pc/k.pc/h.pc/a.pc/m.pc, 1, 8 L/u.pc/l.pc/u.pc Z/h.pc/a.pc/n.pc/g.pc, 1 M/a.pc/s.pc/o.pc/n.pc L/e.pc/i.pc/s.pc/t.pc, 1 N/a.pc/n.pc/c.pc/y.pc L/e.pc/v.pc/e.pc/n.pc/s.pc/o.pc/n.pc, 9 M/a.pc/s.pc/a.pc I/m.pc/a.pc/n.pc/i.pc/s.pc/h.pc/i.pc, 8 S/e.pc/b.pc/a.pc/s.pc/t.pc/i.pc/a.pc/n.pc H/o.pc/e.pc/n.pc/i.pc/g.pc, 10 M/a.pc/r.pc/k.pc/o.pc S/t.pc/a.pc/l.pc/e.pc/v.pc/s.pc/k.pc/i.pc, 11, 12 C/l.pc/a.pc/u.pc/d.pc/i.pc/o.pc R/i.pc/c.pc/c.pc/i.pc, 13, 14 E/r.pc/i.pc/n.pc H/i.pc/c.pc/k.pc/s.pc, 15 E/n.pc/r.pc/i.pc/c.pc/a.pc B/e.pc/l.pc/l.pc/o.pc/c.pc/c.pc/h.pc/i.pc, 16, 17 F/r.pc/a.pc/n.pc/c.pc/o.pc/i.pc/s.pc/e.pc C/o.pc/m.pc/b.pc/e.pc/s.pc, 18 R/i.pc/c.pc D/a.pc/v.pc/i.pc/e.pc/s.pc, 19 S/a.pc/n.pc/t.pc/i.pc/a.pc/g.pc/o.pc G/a.pc/r.pc/c.pc'/dotlessi.pc/a.pc B/u.pc/r.pc/i.pc/l.pc/l.pc/o.pc, 20 O/m.pc/a.pc/i.pc/r.pc/a.pc G/o.pc/n.pc/z.pc '/a.pc/l.pc/e.pc/z.pc M/a.pc/r.pc/t.pc'/dotlessi.pc/n.pc, 21 T/a.pc/k.pc/u.pc/m.pc/a.pc I/z.pc/u.pc/m.pc/i.pc, 8 A/l.pc/v.pc/a.pc/r.pc/o.pc L/a.pc/b.pc/i.pc/a.pc/n.pc/o.pc, 22 M/i.pc/g.pc/u.pc/e.pc/l.pc P/e.pc/r.pc/e.pc/i.pc/r.pc/a.pc S/a.pc/n.pc/t.pc/a.pc/e.pc/l.pc/l.pc/a.pc, 23 D/i.pc/m.pc/i.pc/t.pc/r.pc/a.pc R/i.pc/g.pc/o.pc/p.pc/o.pc/u.pc/l.pc/o.pc/u.pc, 4, 24 D/a.pc/v.pc/i.pc/d.pc R/o.pc/s.pc/a.pc/r.pc/i.pc/o.pc, 25 D/a.pc/n.pc/i.pc/e.pc/l.pc R/o.pc/u.pc/a.pc/n.pc, 26 T/a.pc/r.pc/o.pc S/h.pc/i.pc/m.pc/i.pc/z.pc/u.pc, 19 /a.pc/n.pc/d.pc M/a.pc/r.pc/t.pc/i.pc/n.pc W/a.pc/r.pc/d.pc 27 1 University of Texas at San Antonio, One UTSA Circle, San Antonio, TX, 78249, USA 2 Department of Physics & Astronomy, University of South Carolina, Columbia, SC 29208, USA 3 Kavli Institute for Particle Astrophysics & Cosmology (KIPAC), Stanford University, Stanford, CA 94305, USA 4 Department of Physics, University of Oxford, Keble Road, Oxford OX1 3RH, UK 5 Instituto de Astrof'ısica de Canarias, Calle V'ıa L'actea, s/n, E-38205 La Laguna, Tenerife, Spain 6 Departamento de Astrof'ısica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain 7 Centro de Astrobiolog'ıa (CAB), CSIC-INTA, Camino Bajo del Castillo s/n, E-28692, Villanueva de la Ca˜nada, Madrid, Spain 8 National Astronomical Observatory of Japan, National Institutes of Natural Sciences (NINS), 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 9 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA 10 School of Physics & Astronomy, University of Southampton, Southampton SO17 1BJ, UK 11 Astronomical Observatory, Volgina 7, 11060 Belgrade, Serbia 12 Sterrenkundig Observatorium, Universiteit Gent, Krijgslaan 281-S9, Gent B-9000, Belgium 13 N'ucleo de Astronom'ıa de la Facultad de Ingenier'ıa, Universidad Diego Portales, Av. Ej'ercito Libertador 441, Santiago, Chile 14 Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, People's Republic of China 15 Department of Physics and Astronomy, University of Alaska Anchorage, Anchorage, AK 99508-4664, USA 16 Departmento de F'ısica de la Tierra y Astrof'ısica, Fac. de CC F'ısicas, Universidad Complutense de Madrid, E-28040 Madrid, Spain Instituto de F'ısica de Part'ıculas y del Cosmos IPARCOS, Fac. CC F'ısicas, Universidad Complutense de Madrid, E-28040 Madrid, Spain 18 LERMA, Observatoire de Paris, Coll'ege de France, PSL University, CNRS, Sorbonne University, Paris, France 19 Max Planck Institut fur Extraterrestrische Physik, Giessenbachstrasse 1, D-85748 Garching bei Munchen, Germany 20 Observatorio de Madrid, OAN-IGN, Alfonso XII, 3, E-28014 Madrid, Spain 21 Instituto de Radioastronom'ıa y Astrof'ısica (IRyA), Universidad Nacional Aut'onoma de M'exico, Antigua Carretera a P'tzcuaro #8701, ExHda. San Jos'e de la Huerta, Morelia, Michoac'an, C.P. 58089, Mexico 22 Telespazio UK for the European Space Agency, ESAC, Camino Bajo del Castillo s/n, E-28692 Villanueva de la Ca˜nada, Spain 23 Instituto de F'ısica Fundamental, CSIC, Calle Serrano 123, E-28006 Madrid, Spain 24 School of Sciences, European University Cyprus, Diogenes street, Engomi, 1516 Nicosia, Cyprus 25 School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne NE1 7RU, UK 26 LESIA, Observatoire de Paris, Universit'e PSL, CNRS, Sorbonne Universit'e, Sorbonne Paris Cite'e, 5 place Jules Janssen, F-92195 Meudon, France 27 Centre for Extragalactic Astronomy, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK","pages":[1]},{"title":"2. AGN SAMPLE AND OBSERVATION DATA","content":"2.1. Sample Selection This imaging atlas was drawn from the ongoing AGN survey performed by the Galactic Activity, Torus, and Out- Keywords: galaxies -- active galaxies -- agn","pages":[2]},{"title":"1. INTRODUCTION","content":"There is clear evidence that a considerable amount of dust in the vicinity of supermassive black holes (SMBHs) in active galaxies obscures the central engine (i.e., accretion disk and SMBH) in some lines of sight. Through spectropolarimetric observations of NGC 1068, Antonucci & Miller (1985) showed that its optical polarized spectrum contained broad optical polarized emission lines not originally observed by direct total intensity observations. It was subsequently presumed that an optically and geometrically thick dusty structure ('torus') blocked the central engine in some lines of sight (Antonucci 1993; Urry & Padovani 1995). Under this unified scheme a Type 1 AGN is seen face-on and shows broadened optical lines, while in Type 2 AGN the broadened lines are obscured. This model also predicts that broad silicate features at 10 and 18 𝜇 m will be seen in emission in Type 1 and in absorption in Type 2. However, silicate emission can be seen in emission in some Type 2 AGN, while absorption can be seen in some Type 1 (e.g., Hatziminaoglou et al. 2015). This and other observational features are explained by the inhomogeneous nature of the torus. Clumpy torus models (Nenkova et al. 2008a,b) predict shallower silicate features, more similar infrared spectral energy distributions (SEDs) between Type 1 and Type 2 AGN, etc. (see Ramos Almeida & Ricci 2017, for a review). A region of narrow forbidden line emission extends above and below the midplane of the dusty torus structure out to several kpc scales. Recent sub-arcsecond interferometric imaging observations have shown a dust component at pc-scales co-spatial with the base of the narrow line region (NLR; Honig et al. 2012; Tristram et al. 2014; L'opez-Gonzaga et al. 2014, 2016; Burtscher et al. 2013; G'amez Rosas et al. 2022; Isbell et al. 2022). This dusty structure is interpreted as part of a dusty wind launched from the inner hot part of the torus driven by radiation pressure at pc-scales (Honig 2019), but generated by a magnetohydrodynamical wind at sub-pc scales (e.g., Emmering et al. 1992; Lopez-Rodriguez et al. 2015; Takasao et al. 2022; Lopez-Rodriguez et al. 2023). This extended dusty structure has been resolved in a nearby galaxy, ESO 418-G14, using mid-infrared (MIR) images with JWST/MIRI finding that the dust is primarily heated by the AGN and/or radiative jet-induced shocks in the NLR rather than a wind (Haidar et al. 2024). ALMA observations provide observational support for a dusty torus+outflow scenario. Emission from the nucleus of NGC1068 was mapped with a resolution of ∼ 4 pc, resolving a 7 -10 pc diameter disk interpreted as the sub-mm counterpart of the torus (Garc'ıa-Burillo et al. 2016). Rotation of the compact emission was detected in HCN J=3-2 and HCO+ J=3-2 (Imanishi et al. 2018, 2020). A molecular outflowing wind co-spatial with the dusty and molecular torus was also observed (Garc'ıa-Burillo et al. 2019). Alonso-Herrero et al. (2018) interpreted the measured nuclear (10 -20 pc) flow Survey (GATOS; Garc'ıa-Burillo et al. 2021; AlonsoHerrero et al. 2021; Garc'ıa-Bernete et al. 2024). GATOS /one.sup aims to characterize the dynamics and composition of the dusty and molecular torus and multi-phase outflows in AGN. The GATOS parent sample is selected from the 70 Month Swift /BAT AGN catalog, which is flux limited in the ultrahard 14 -195 keV X-rays band (Baumgartner et al. 2013). In the initial study of AGN using SOFIA observations (Fuller et al. 2016), sources from the GATOS survey were selected based on the criteria that the galaxies had been previously studied using C/l.pc/u.pc/m.pc/p.pc/y.pc (Nenkova et al. 2008a,b) torus models and were well-sampled in the 1 -18 𝜇 m regime (Ramos Almeida et al. 2009, 2011; Alonso-Herrero et al. 2011). The study of the 11 objects included the 31 . 5 𝜇 m photometry in the SEDs and found that including the 31.5 𝜇 m photometry reduces the number of C/l.pc/u.pc/m.pc/p.pc/y.pc torus models that are compatible with the data and modifies the model output for the torus outer radius. Fuller et al. (2019) further extended the wavelength range of a subset of 7 AGN SEDs to 37.1 𝜇 m. They subsequently found extended emission in the PSF-subtracted images of Mrk 3, NGC 4151, and NGC 4388 that is coincident with the radio axis and NLR. In a separate study, Lopez-Rodriguez et al. (2018) modeled the torus of NGC 1068 using ∼ 20 -53 𝜇 m FORCAST and HAWC+ observations. They showed that the peak wavelength range of emission from the torus is ∼ 30 - 40 𝜇 m with a characteristic temperature 70 - 100 K. The use of observations ¿ 30 𝜇 m in that study from SOFIA and ALMA highlights the importance of longer wavelength observations to put constraints on MIR emission sources. Based on these results, we extend the wavelength range in objects previously observed, and also expand the number of AGN observed. We present the complete imaging atlas of 22 Seyferts observed by SOFIA in the wavelength range 19 . 7 -214 𝜇 m using FORCAST and HAWC+. The final set of observations presented here was part of a multi-year AGN survey over several observing SOFIA cycles (Proposal IDs: 02 0035, 04 0048, 06 0066, 08 0014; PI: Lopez-Rodriguez; 70 0400 PI: Herter). The SOFIA atlas of AGN in the far-IR (FIR) is a flux-limited sample of nearby, bright, and well-studied AGN. All objects have a point source flux of > 200 mJy at 31 . 5 𝜇 m, which ensures that each band can be observed within 1 hr of on-source time with a signal to noise ratio > 10 using FORCAST/SOFIA. Although the original AGN sample is larger than that presented here, only 22 AGN were observed in total by SOFIA before end of operations in 2022. Note that there are gaps in the 20 -214 𝜇 m wavelength range due to the fact that SOFIA only flies with a single instrument per night. For each SOFIA cycle, we prioritized the objects with observations acquired in a single instrument from the previous cycle. The sample properties are given in Table 1. For most objects, we retrieved redshift data from the NASA Extragalactic Database (NED). However, for nearby objects Centaurus A Redshifts and spectral type were taken from NED. Distances to most sources were obtained using H0 = 70 km s -1 Mpc -1 . Distances to nearby sources Centaurus A and Circinus were taken from Harris et al. (2010) and Tully et al. (2009), respectively. References for log 𝐿 𝑏𝑜𝑙 : a) Borkar et al. (2021) b) Marinucci et al. (2012) c) Alonso-Herrero et al. (2011) d) Ichikawa et al. (2017), e) Leighly et al. (2014) f) Marconi et al. (2004), g) Baumgartner et al. (2013), h) Garc'ıa-Bernete et al. (2015), i) Ramos Almeida et al. (2011), j) Yuan et al. (2002), k) Duras et al. (2020), and Circinus, distances were obtained individually (Harris et al. 2010; Tully et al. 2009). The 22 objects in this atlas cover the luminosity range of log 10 ( 𝐿 bol [ erg / s ]) = [ 42 , 46 ] with a median of log 10 ( 𝐿 bol [ erg / s ]) = 44 . 1 ± 1 . 0, and a distance of 4 -181 Mpc with a median of 42.8 Mpc. Figure 1 shows bolometric luminosity plotted against distance, where Seyfert 1 objects are shown as red triangles and Seyfert 2 objects are shown as purple stars.","pages":[2,3]},{"title":"2.2.1. FORCAST","content":"FORCAST is an IR camera and spectrograph sensitive in the wavelength range 5 -40 𝜇 m with a field of view (FOV) of 3.4 ' × 3.2 ' and pixel scale 0.768 '' /pixel. With one exception (NGC 1068 in the 19.7 𝜇 m filter), we only used the Long Wavelength Camera (LWC; 25 -40 𝜇 m) due to the abundance of ground-based images at shorter wavelengths for the objects in our sample. FORCAST observations were made in dual channel mode using the two-position chop-nod (C2N) method with symmetric nod-match-chop (NMC) to remove telescope thermal emission and time variable sky background, and to reduce the effect of 1/ 𝑓 noise from the array. Data were reduced by the SOFIA Science Center using the /f.pc/o.pc/r.pc/c.pc/a.pc/s.pc/t.pc /r.pc/e.pc/d.pc/u.pc/x.pc pipeline following the methods described by Herter et al. (2012). Most of the pipeline changes over the cycles were to refine the spectroscopic mode of FORCAST with little or no change to the image mode presented here. Observations were flux-calibrated using the set of standard stars of the observing run, which provides flux uncertainties of ∼ 10 %. The point spread function (PSF) of the 31.5 𝜇 m observations from Cycle 2 (see Fuller et al. 2016) was the co-average of a set of standard stars from that cycle. Its FWHMwas 3.40 '' , in agreement with the SOFIA Observer's Handbook v3.0.0. The PSFs for Cycle 4 in the 30 - 40 𝜇 mwavelength range were determined by using standard star observations from the individual flights (see Fuller et al. 2019) and averaged at FWHM ∼ 4.33 '' and 4.58 '' in the 31.5 and 37.1 𝜇 m filters, respectively. The FORCAST PSFs for the observations of NGC1068 are detailed in Lopez-Rodriguez et al. (2018).","pages":[3,4]},{"title":"2.2.2. HAWC+","content":"HAWC+isaFIRimaging polarimeter designed to allow total and polarized intensity imaging observations in four broad bands centered at 53, 89, 155, and 214 𝜇 m, corresponding to Bands A, C, D, and E respectively (see Table 2). On- -fly mapping (OTFMAP) observing modes were used for both imaging polarimetry and total intensity imaging. Observing modes for the individual observations are given in Table 3. Data taken in polarization mode was reduced using the /h.pc/a.pc/w.pc/c.pc /d.pc/p.pc/r.pc /p.pc/i.pc/p.pc/e.pc/l.pc/i.pc/n.pc/e.pc and the reduction steps presented in Lopez-Rodriguez et al. (2022b). The Comprehensive Reduction Utility for SHARC II (CRUSH; Kov'acs et al. 2006; Kov'acs 2008) was used to obtain the total intensity observations. HAWC+ observations were reduced following the same reduction steps. We quote the pipeline versions or CRUSH versions in Table 3 to differentiate between the observing modes. There are no differences between CRUSH versions to obtain the total intensity images as all the changes in the pipeline were done for the polarimetric mode. Table 3 also shows the pixel scale of each image. As in FORCAST observations, the source of uncertainty in the photometry for HAWC+ results from calibration factors of the standard stars associated with the observation, giving an uncertainty of ∼ 10 % (Lopez-Rodriguez et al. 2022b). HAWC+ PSFs were estimated using standard star observations in 2017. Pallas was observed in Bands A and C on 7 November 2017, while Neptune was observed in Bands D and E on 19 October, 2017. The FWHM of these standards are 5.25 '' , 8.26 '' , 14.74 '' , and 19.65 '' in Bands A, C, D, and E, respectively. The FWHMs from the Observer's Handbook /two.sup are given in Table 2.","pages":[4]},{"title":"2.3. Observing Data","content":"Table 3 provides the final AGN sample with information about the wavelength, observing mode, observation and mission details, versions of the separate pipelines, and also the field-of-view (FOV) of the individual images.","pages":[4]},{"title":"3. IMAGES","content":"Images of the 22 AGN in the 19 . 7 -214 𝜇 m wavelength range are presented in Figures 2, 3, 4, and 5. The orange scale on the bottom left of the images indicates a scale of 500 pc. The beam size is depicted in white in the top right of the images. In all images, north is up and east is to the left. Complementary Herschel 70 -500 𝜇 m images are shown in Appendix A (Fig. 10, 11, 12, 13). These fully reduced images were obtained through the Herschel Science Archive /three.sup . All objects are presented and analyzed individually. Centaurus A . The host galaxy of Centaurus A is clearly visible in the 53 𝜇 m image with a bright compact center. However, the host galaxy becomes more dominant in the 89 𝜇 mimage(seeadetailed analysis of host galaxy dust emission at 89 𝜇 m in Lopez-Rodriguez 2021). The kpc-scale warped dust and gas lane was first observed with Spitzer imaging using IRAC and MIPS (Quillen et al. 2006). On subarcsecond scales, Radomski et al. (2008) observed the nucleus of Centaurus A using the 8.8, 10.4, and 18.3 𝜇 m filters on T-ReCS at Gemini South. They concluded that the mostly likely sources of nuclear MIR emission are an unresolved clumpy dusty torus in the core, and a dusty NLR for the arcsecondscale extended emission (see also Garc'ıa-Bernete et al. 2016). Circinus Galaxy . HAWC+ 53 and 89 𝜇 m images of the Circinus Galaxy show a very bright FIR core with extended emission at a PA ∼ 30 · , whereas the 215 𝜇 m image shows a slightly different PA ∼ 55 · . We estimate the FWHM of the extended FIR nuclear emission to be ∼ 6 '' × 6 '' , 13.5 '' × 11.5 '' , and 22.3 '' × 25.6 '' at 53, 89, and 214 𝜇 m, respectively. These are larger than the PSF FWHMs given in Section 2.2.2, which indicates extended emission along the axis of the inner bar of the galaxy. MIR emission was resolved at 8.7 and 18.3 𝜇 m out to 2 '' in an approximate east-west direction, coincident with the ionization cones at PA ∼ 100 · (Packham et al. 2005; Stalevski et al. 2017). However, the elongation seen in the SOFIAimages(andin Herschel imagesinAppendixA)seems to be arising from dust in the host galaxy. MCG-5-23-16 . MCG-5-23-16 appears as a point-like source in the 31 . 5 -155 𝜇 mwavelength range whose brightness decreases with increasing wavelength. However, this galaxy appears as an extended source in the MIR using high angular resolution data from VLT (Garc'ıa-Bernete et al. 2016). Likewise, Ferruit et al. (2000) found that this nucleus has an extended optical NLR at a PA of 40 · . They found a dust lane extending 2 '' on either side of the nucleus, parallel to the axis of the galaxy. Their extended dusty emission at 40 · was detected at a 3 𝜎 level up to ∼ 4 '' from the core. This extended structure has no thermal emission counterpart within the 31 . 5 -155 𝜇 mwavelength range. Mrk3 . Although the FORCAST and HAWC+ images generally appear to be point-like sources, Fuller et al. (2019) found extended emission in the PSF-subtracted 37.1 𝜇 m im- of Mrk 3 in the direction of the radio axis (84 · ; Kukula et al. 1993) and the NLR ( ∼ 70 · ; Capetti et al. 1995). Asmus et al. (2013) found an elongated nucleus out to ∼ 170 pc with PA ∼ 70 · in the Si-5 (11.6 𝜇 m) filter using Gemini/MICHELLE. However, the Si-2 (8.7 𝜇 m) image from GTC/Canaricam appears point-like (Alonso-Herrero et al. 2016). The large scale east-west structure in the 53 𝜇 mimage here is a background artifact produced by the data reduction due to the small spatial coverage and short integration time of the observation. Mrk 231 . The 89 𝜇 mimage shown here is point-like with a FWHM of ∼ 8.5', similar to the standard FWHM of 8.26' (see Section 2.2.2). Mrk 231 is a Type 1 Ultra Luminous Infrared Galaxy (ULIRG) and is the nearest known quasar at a distance of 181 Mpc. It is known for its multi-phase and multi-scale outflows (see Rupke & Veilleux 2011), with a neutral outflow up to 3 kpc in radius (Rupke et al. 2005). Mrk 573 . Although the SNR is very low (3 -4 𝜎 ), both 31.5 and 37.1 𝜇 mimages of Mrk 573 show marginally resolved ∼ 4 . 5 '' elongation in the east-west direction at PA ∼ 110 · . Mrk 573 was previously shown to have a biconical NLR coincident with radio emission 3-4' from the nucleus at a PA ∼ 125 · (Ulvestad & Wilson 1984; Pogge & De Robertis 1995). The marginal detection here may be cold extended dust in the outer layers of the NLR. NGC 1068 . The 19 - 53 𝜇 m images of NGC 1068 were published previously in Lopez-Rodriguez et al. (2018), where it was shown that the peak emission from the torus occurs between 30 - 40 𝜇 m with a corresponding temperature of 70 - 100 K. The 89 𝜇 m image was published as a polarimetric observation (Lopez-Rodriguez et al. 2020). Within a scale of about 1 kpc, NGC 1068 shows extended emission in the NE to SW direction at a PA ∼ 45 · , similar to MIR observations using VISIR/VLT (Asmus et al. 2014). Their observations revealed a nuclear structure in the north-south direction and extended structures to the NE and SW. From the N-band spectrum, Mason et al. (2006) concluded that while torus emission dominates NIR wavelengths, large-scale MIR emission is dominated by diffuse dust within the ionization cones. NGC 1275 . All 30 - 53 𝜇 m images are dominated by a point-like source. However, the 31.5 𝜇 m image (Fuller et al. 2019) shows 3 𝜎 extended emission along the PA ∼ 140 · . This AGN is known to have a network of H 𝛼 filaments extending out to ∼ 100' (see Conselice et al. 2001) and is possibly the result of a merger (Holtzman et al. 1992). The MIR core shows silicate dust emission in both 10 and 18 𝜇 mbands (see Fuller et al. 2019). Hence, both dust and gas are extended covering several kpc around the core. In the HAWC+ 89 𝜇 mfilter, Lopez-Rodriguez et al. (2023) found extended dust emission at a PA ∼ 125 · out to a 12 kpc radius potentially associated with a magnetized dusty filament along the NW direction (Fabian et al. 2008). NGC 2110 . The 30 - 215 𝜇 mimages of NGC 2110 are all point-like. The north-south pattern in the 53 𝜇 mimage is due to background noise and does not represent extended dust. NGC2110is a Type 2 AGN that shows silicate emission at 10 and18 𝜇 mthatis interpreted as a result of a clumpy torus, or as dust within the ionization cones (PA ∼ 160 · ; Mulchaey et al. 1994) in the inner 32 pc of the AGN (Mason et al. 2009). This galaxy appears as an extended source in Gemini/MICHELLE high resolution N-band observations (Garc'ıa-Bernete et al. 2016). However, any structure within the NLR or ionization cones is not resolved by our observations. NGC 2273 . The full set of 30 - 215 𝜇 m images of NGC 2273 show a point-like source. The north/south pattern in the 53 𝜇 m image is due to background noise and does not represent extended dust. Within the FWHM of these images (see Sections 2.2.1, 2.2.2), there is a known star-forming ring within ∼ 2' of the nucleus (Ferruit et al. 2000; Martini et al. 2003; Sani et al. 2012). GTC/Canaricam observations (Alonso-Herrero et al. 2014, 2016) at 8.7 𝜇 mshowelongation from the north-east to the south-west, likely with contribution from PAH. This structure is consistent with extension seen in the PSF-subtracted 37.1 𝜇 mSOFIAimage(Fulleretal. 2019). NGC 2992 . The image of NGC 2992 in the 31.5 𝜇 mfilter is published in Fuller et al. (2016) and appears as a point-like source. Subarcsecond N-band imaging (Garc'ıa-Bernete et al. 2015) reveals extended emission along PA ∼ 30 · out to ∼ 3 kpc which is attributed to dust heated by star formation based on corresponding N-band spectroscopy. The FWHM of the SOFIA image is ∼ 3.5' × 3.5' (560 × 560 pc 2 ) so the extension should be resolvable within the SOFIA image. Since we do not see the extension in the image here, we conclude that either the extended dust emission tapers at wavelengths ¿ 20 𝜇 mor SOFIA does not have enough sensitivity to detect it. NGC 3081 . The 31.5 𝜇 m image of NGC 3081 was published in Fuller et al. (2016) while the 37.1 𝜇 m image was published in Fuller et al. (2019). The nucleus is known to harbor a region of strong optical emission ∼ 1' from the AGN (Ferruit et al. 2000) likely due to dust or gas heated by the AGN. Fuller et al. (2019) estimated that ∼ 35% of the MIRemission within the central few arcseconds (few hundred parsecs) of the AGN originates in the NLR. High angular resolution N- and Q- band observations show extension towards the north, extending out to ∼ 450 pc from the south-east to the north-west (PA ∼ 160 · ; Garc'ıa-Bernete et al. 2016). On larger scales, optical and NIR observations reveal a series of star forming resonance rings at distances of 2.3, 11.0, 26.9 kpc and 33.1 kpc (Buta 1990; Buta et al. 1998, 2004). At longer wavelengths (¿200 𝜇 m), Ramos Almeida et al. (2011) concluded that FIR emission is contaminated by the star-forming ring 2.3 kpc in diameter. NGC 3227 . The 31.5 𝜇 m image was published in Fuller et al. (2016) while the 37.1 𝜇 mimage was published in Fuller et al. (2019). These images show a point-like source, although NGC 3227 is known to harbor a nuclear star-forming region (Schinnerer et al. 2001; Davies et al. 2006) with a nuclear cluster within ∼ 70 pc ( ∼ 1') from the core. The 8.7 𝜇 mimage from Alonso-Herrero et al. (2016) shows a slight north/south elongation and the corresponding spectrum shows clear PAH in the nucleus (see also Garc'ıa-Bernete et al. 2016). These star forming regions likely contaminate the nuclear MIR emission within the FWHM of our images. NGC 3281 . While the 31.5 𝜇 m FORCAST image is published in Fuller et al. (2016), the HAWC+ images at 53, 89, 154, and 214 𝜇 m are presented here for the first time and appear point-like in all filters. The images taken at 53 and 89 𝜇 mappear to have significant noise in their backgrounds. The subarcsecond (0.35') N-band spectrum in Gonz'alez-Mart'ın et al. (2013) shows a deep 10 𝜇 m silicate absorption feature which originates in the inner ∼ 80 pc of the AGN. NGC 4151 . The SOFIA images of NGC 4151 appear as a point-like source, with a potential detection of extended NGC 7469 15 5 NGC 7674 1 emission at PA ∼ 120 · at a 3 𝜎 level at 37 . 1 𝜇 m. Fuller et al. (2019) confirmed this elongation in the PSF-subtracted 37.1 𝜇 mimagecoincident with the NLR and radio axes. Radomski et al. (2003) show extended emission in 10.8 and 18.2 𝜇 m images that coincides with the NLR axis at PA ∼ -60 · . For ≥ 37 . 1 𝜇 m, we conclude that any extended emission due to NLR dust is within the FWHM of the SOFIA instruments. NGC4258 . NGC 4258 was not detected but we include the data here since it is part of the sample. It has been observed and analyzed in the N-band with Gemini/Michelle by Mason et al. (2012). These authors found a compact nucleus that is marginally resolved at 10 𝜇 m(FWHM ∼ 0 . 5 '' ). NGC 4388 . NGC 4388 is an edge-on spiral that shows the most interesting mid- to far-IR morphology in this study. Notably, in the 30 - 40 𝜇 mFORCAST images of NGC 4388, extension can be seen in the NE to SW direction at PA ∼ 40 · (see also Fuller et al. 2019), coincident with the NLR. This emission is seen on smaller scales at shorter wavelengths (Asmus et al. 2016; Garc'ıa-Bernete et al. 2016). The 53 𝜇 m image decreases in intensity and does not show a strong central core of emission as in the 31 . 5 -37 . 1 wavelength range. However, at longer wavelengths (89 -214 𝜇 m), host galaxy emission clearly dominates the images in the east-west direction at PA ∼ 90 · . NGC 4941 . NGC 4941 is a low-luminosity AGN that appears here as a faint point-like source in the 31 . 5 𝜇 m and 37 . 1 𝜇 m images, but brighter at 53 and 89 𝜇 m. Subarcsecond resolution N-band imaging on VLT/VISIR (Asmus et al. 2011) showed no significant extended MIR sources outside of the nucleus. NGC 5506 NGC 5506 appears as a bright point source in both the 31.5 and 37.1 𝜇 m filters. While the nucleus is unresolved, extended MIR emission has been detected up to a few arcseconds to the northeast at 11.9 𝜇 m (Raban et al. 2008). Extended emission in the north-south direction was detected in the N-band out to ∼ 560 pc, while faint extended emission towards the east in the Q-band was also detected (Garc'ıa-Bernete et al. 2016). However, the PSF-subtracted 12.27 𝜇 m2' × 2' VLT/VISIR image of Alonso-Herrero et al. (2021) shows that the PA of extended emission varies from 30 · in the central ∼ 0.5' to nearly 90 · in the outer regions. NGC 7465 . The 31.5 𝜇 m FORCAST image appears faint with a 3 𝜎 upper-limit in the 37.1 𝜇 m image. The 53 and 89 𝜇 m HAWC+ images here appear increasingly brighter, albeit as point-like sources. Cold molecular gas observations (Young et al. 2021) reveal that NGC 7465 is quite gas-rich, possibly from a recent merger. NGC 7469 . NGC 7469 appears as a very bright source in the 31.5 𝜇 m image with FWHM ∼ 4.3'. After PSF subtraction, Fuller et al. (2016) found extended emission in the north-south direction. This AGN is known to have a circumnuclear ring of star formation at a radius of ∼ 480 pc ( ∼ 1.4'; Ramos Almeida et al. 2011) in 8.7 and 18.3 𝜇 mimages taken on Gemini/T-ReCS. Recent JWST observations reveal prominent PAH emission, indicative of star formation, in the circumnuclear ring (Garc'ıa-Bernete et al. 2022b; Zhang & Ho 2023). NGC 7674 . The previously published (Fuller et al. 2016) FORCAST 31.5 𝜇 m image appears as a point-like source. Asmus et al. (2013) found that the nucleus of NGC 7674 is extended at PA ∼ 125 · at subarcsecond scale resolution, where the extension roughly aligns with the ionization cone.","pages":[8,9,10,11,12,13]},{"title":"4. NUCLEAR FLUX EXTRACTION","content":"We aim to construct well-sampled mid- to far-IR SEDs of the nuclear emission of AGN at scales of several arcseconds, depending on the PSF of the observation and possible extended emission. On these scales, we expect multiple dust sources (i.e. torus, star forming regions, dusty outflow), however disentangling these sources is beyond the scope of this imaging atlas. Because the images span a range of observing cycles, instruments, and observing modes, we analyzed each image individually. Of our sample, 17 objects appear visually as point sources. For these sources, we performed aperture photometry where the aperture size was set to be 2 × the FWHM at a given band. For objects that show host galaxy emission, we extract the central PSF to construct the SEDs as described below. We complement our SOFIA data with Herschel imaging data (see Appendix A) and use a similar analysis method to construct the full IR SEDs.","pages":[13]},{"title":"4.1. Extended Sources: 2D Gaussian Fitting","content":"For sources with extended dust emission, we performed a two-component simultaneous fit to accurately model both the central source based on the PSF, and the host galaxy whose fit assumes an elongated 2D Gaussian profile. In order to supplement the SOFIA data for the full mid- to far-IR SEDs, we used a similar methodology with Herschel images. For the SOFIA images, the PSF used was based on the standard stars of the observing runs for each cycle as explained in Section 2.2. However, the same analysis could not be performed on Herschel images due to the threefold lobes associated with the instrument PSFs. To accommodate this, we compared three different PSF models to reproduce and fit the central source. Two of the PSFs were point source images while the third was an approximation of the theoretical instrumental PSF using a Gaussian profile. The fitting routine used four free parameters for the Gaussian profile: (1,2) the position in x and y of the PSF center according to the image center, (3) the amplitude of the PSF, (4) the fourth parameter was dependent on the PSF type used. For archival PSFs, this parameter represents the rotation angle that needs to be applied to the PSF to match the orientation of the image. For the Gaussian PSFs, this fourth parameter represents a scaling factor to the width of the Gaussian compared to its ideal value for a perfect instrument (1 . 22 × 𝜆 / 𝐷 ). The galaxy background is defined by 7 parameters: (1,2) the 2D Gaussian's center position (x0 and y0), (3,4) its width ( 𝜎 𝑥 and 𝜎 𝑦 ) in both directions, (5) its amplitude, and (6) its orientation on the image ( 𝜃 ). To these 6 parameters we added a constant background as a 7th free parameter. We combined these components and fit this simulated intensity map to the observed map using the 11 total free parameters (4 from the PSF and 7 from the 2D gaussian). We thus derived the parameters describing the best central source for our intensity maps, and then studied the properties extracted for the central source and removed it from the initial map to study the host galaxy itself. An example of this procedure is given in the Appendix in Figure 14.","pages":[13]},{"title":"4.2. AGN and host galaxy contribution","content":"While most SOFIA images were treated as point sources, Centaurus A, Circinus, NGC 1068, and NGC 4388 all had significant host galaxy contamination that needed to be subtracted from at least some of the images. Figure 6 shows the PSF subtracted images of these sources and Table 4 gives the percentage of the contribution of the PSF to the total flux of the object. In several other objects, the shorter wavelength ( ∼ 30 - 100 𝜇 m) images did not show host galaxy contamination, but longer wavelength Herschel images show the colder extended dust. Because this is an atlas of SOFIA images, we include objects with host galaxy contamination only in Herschel images in Appendix C for completeness. The53 𝜇 mimageofCentaurusAcontains ( < 5%)emission from extended sources, so we performed aperture photometry to account for the central emission. At wavelengths ≳ 70 𝜇 m, the host galaxy substantially ( ∼ 56 -70%) contributes to the central AGN emission, so the extended emission was subtracted. The PSF of Centaurus A at these wavelengths contributes ∼ 35 %. This can be interpreted as the nucleus having a relatively constant IR contribution, so the brightness of the nucleus coincides with IR brightness of the host galaxy. The PSF contribution of the Circinus Galaxy decreases between 53 and 160 𝜇 m from 58 % to 35 %. At longer FIR wavelengths, the contribution of the PSF appears to be from the host galaxy and the fitting no longer provides information about the AGN. We interpret this as a decreasing IR contribution from the nucleus compared to the extended emission. For the completeness of the SOFIA Atlas presented here, we used the 19-53 𝜇 m images of NGC 1068 from LopezRodriguez et al. (2018). These datasets were analyzed as described in that study and here we only present the results and images in that wavelength range. The study showed that the fractional contribution from star formation increases from 20 - 50 𝜇 m, while extended emission from 200 K dust decreases. Emission from the torus peaks in this range, a result which is in agreement with Fuller et al. (2016) who found that the turnover in torus emission occurs at wavelengths ¿ 31.5 𝜇 m. Extended emission is observed here at all wavelengths ≳ 70 𝜇 m arising from dust in the host galaxy and star formation regions. For NGC 4388, we show the results of PSF subtraction at all wavelengths, but only use the results in wavelengths ¿ 40 𝜇 m for the SED. In the 30 - 40 𝜇 m range, the NE to SW extension is clear in the PSF-subtracted images. However, almost all of the extended emission lies within the FWHM of the observation; the FWHM of these images are only ∼ 10% greater than the FWHM of the PSF. Thus, while we show the PSF subtracted images of NGC 4388 here, for the SED we use the total 30 - 40 𝜇 mfluxes which encompass the apparent extended emission due to the NLR. The change in the extended emission source and morphology between 40 and 70 𝜇 m is clear in the PSF-subtracted images (Figure 6). The host galaxy clearly dominates the extended emission in the FIR while the NLR region dominates the extended emission in MIR wavelengths. The 53 𝜇 m HAWC+ image appears to show the transition between dominant extended sources. The contribution of the PSF in the images of NGC 4388 is ∼ 60 % in the 30 - 40 𝜇 m range, where the extended emission is in the NE to SW direction. The contribution then decreases drastically to ∼ 20 %. This reflects the turnover in extended emission seen in the images in Figure 5. The contribution of the PSF returns to ∼ 70 % between 70 - 100 𝜇 m, which suggests two separate but significant IR emission sources.","pages":[13,14]},{"title":"5. SPECTRAL ENERGY DISTRIBUTIONS","content":"Tables 5 (SOFIA) and 6 ( Herschel ) give the nuclear fluxes of the AGN in our sample along with their associated errors in units of Jy. The sources of uncertainty here are the instrument calibration, sky background, and the 2D gaussian fitting, where applicable. We estimate FORCAST and HAWC+ errors at ∼ 10%. We use PACS instrument errors at 5% /four.sup and SPIRE instrument errors as 5.5% /five.sup . The uncertainty due to sky background is determined on an individual basis, but averages ∼ 5%. The average uncertainty due to the 2-D gaussian fitting is ∼ 1.5%. We add these uncertainties in quadrature for the error bar estimation. The nuclear SEDs are shown in Figure 7 in 𝜈 F 𝜈 . The pink diamonds represent SOFIA observations while the black circles represent the complementary Herschel data. We obtained Spitzer /IRS spectra from the Sptizer /CASSIS database (Lebouteiller et al. 2011) for 21 of the 22 objects in our sample (solid black line). There was no spectrum available for NGC 7465. Low-resolution spectra (R ∼ 100) were obtained for 18 of the objects, while moderate resolution spectra (R ∼ 600) were available for Circinus, NGC 1068, and NGC 7674. This dataset provides the most completed SED coverage available between 30 - 500 𝜇 m. Decomposing the SEDs in this sample is outside the scope of this manuscript, as we are presenting an imaging atlas. Here, we provide the main results and features of the SEDs of these objects. The morphological changes seen in the extended emission source in Figure 5 for NGC 4388 are reflected in the SED at 53 𝜇 m, where there is a marked decline in the SED. The drastic decrease seems to be due the change of dominant emitting sources. The extended emission at wavelengths ≲ 40 𝜇 m is due to dust in the direction of the radio axis, and the extended emission at wavelengths ≳ 50 𝜇 mis due to the host galaxy. The wavelength of peak emission can give insight to the primary processes that drive MIR emission. The peak wavelength, determined by the highest flux from photometry and spectroscopy, ranges from 18 to 100 𝜇 m in 𝜈 F 𝜈 with an average of ∼ 40 𝜇 m. This average only includes the peak in continuum values and does not take into account fine structure lines. Most (73%; 11 out of 15) Seyfert 2 have a peak emission at wavelengths ≲ 40 𝜇 m. The SEDs of MCG-5-2316, Mrk3, Mrk 573, NGC 1068, NGC 3081, and NGC 4151 peak at ∼ 18 - 20 𝜇 m. This is in agreement with the correlation peak between the hard-X-rays and the mid-IR for Type 1 AGN in Garc'ıa-Bernete et al. (2017). The peak wavelengths in 𝐹 𝜈 (Jy) range ∼ 20 - 160 𝜇 m, with an average ∼ 93 𝜇 m. NGC 1068 is the only AGN to peak at the same wavelength in both sets of units. The 𝑆𝑝𝑖𝑡𝑧𝑒𝑟 spectrum for NGC 1068 does not align with the SOFIA photometry because of the extensive PSF subtraction that we performed in the photometry that was not accounted for in the spectroscopy. This is the only object that not only has overlapping 20 - 40 𝜇 m 𝑆𝑝𝑖𝑡𝑧𝑒𝑟 and SOFIA data, but also that has had the background emission subtracted at these wavelengths.","pages":[14]},{"title":"5.1. Luminosity and Peak Wavelength","content":"To test whether the peak wavelength is a function of luminosity, we plot L 𝑏𝑜𝑙 vs 𝜆 𝑝𝑒𝑎𝑘 . Figure 8 shows the bolometric luminosities of the AGN plotted against the peak wavelength in the SEDs for both Sy1s, shown as red triangles, and Sy2s, shown as purple stars. The correlation coefficient between the luminosity and peak wavelength is | 𝑅 | ∼ 0.63 with statistical significance 𝑝 = 0 . 0015. While it is argued that | 𝑅 | of 0.6 - 0.7 may show moderate to strong correlation (see Section 3.2 in Messenger et al. 2013), a 𝑝 -value ≤ 0.05 is generally accepted as statistically significant. The data suggests that higher luminosity objects have SEDs that peak at shorter wavelengths, which indicates the presence of a hot dust component in the vicinity of the AGN.","pages":[14,15]},{"title":"5.2. Mid- to Far-IR Colors","content":"The ratio of F 𝜈 (70)/F 𝜈 (160) has been used as a proxy for dust temperature (Mel'endez et al. 2014; Garc'ıa-Gonz'alez et al. 2016), where the ratio is higher for dust heated by the AGN and lower for dust heated by star formation. Here we perform this analysis using the ratio F 𝜈 (31)/F 𝜈 (70) by using the 31.5 𝜇 mSOFIA data in our atlas. For objects that do not have data in the 31.5 𝜇 mfilter, we supplement that with data from the Spitzer /IRS continuum. NGC 7465 did not have 31.5 𝜇 m flux data, nor did it have Spitzer data so we leave that object out of this analysis. Using the fluxes in Table 5, we plot a color-color diagram in F 𝜈 in Figure 9. This figure also visually shows the peak wavelength from the SEDs (in 𝜈𝐹 𝜈 ). Longer peak wavelengths tend to cluster at F 𝜈 (70)/F 𝜈 (160) ∼ 1 and F 𝜈 (31)/F 𝜈 (70) between 0.25 - 0.5. In this sample we find an average F 𝜈 (70)/F 𝜈 (160) ratio of 1 . 4 ± 0 . 7. Previous studies (Mel'endez et al. 2014; Garc'ıaGonz'alez et al. 2016) with larger sample sizes (313 and 33, respectively) have found an average ratio of ∼ 0.8, albeit the data was analyzed using independent methods. This suggests a higher amount of AGN heated dust in our sample. We find that 18 objects have F 𝜈 (31)/F 𝜈 (70) ¡1, with an average F 𝜈 (31)/F 𝜈 (70) of 0 . 6 ± 0 . 3. The only object with both F 𝜈 (70)/F 𝜈 (160) and F 𝜈 (31)/F 𝜈 (70) ¿1 is MCG-5-23-16, and an SED that peaks ∼ 20 𝜇 m. This object may be the most AGN dominated source in our sample. The other objects that show a peak at ∼ 20 𝜇 min 𝜈 F 𝜈 still show F 𝜈 (31)/F 𝜈 (70) ¡1. Only one object, NGC 4388, shows F 𝜈 (31)/F 𝜈 (70) ¿1 while F 𝜈 (70)/F 𝜈 (160) ¡1. This reflects the change in extended emission seen in Figure 5. Half (11) of the objects in the sample show ratios F 𝜈 (31)/F 𝜈 (70) ¡1 while F 𝜈 (70)/F 𝜈 (160) ¿1. These objects (Circinus, Mrk 231, Mrk 573, NGC 1275, NGC 2110, NGC 3081, NGC 3227, NGC 3281, NGC 4151, NGC 4941, NGC 5506, NGC 7469) are likely AGN dominated. Six objects show F 𝜈 (31)/F 𝜈 (70) ¡1 and F 𝜈 (70)/F 𝜈 (160) ¡1, meaning that their SEDs peak at longer wavelengths. The emission from R/e.pc/f.pc/e.pc/r.pc/e.pc/n.pc/c.pc/e.pc/s.pc: a) Fuller et al. (2016), b) Fuller et al. (2019), c) Lopez-Rodriguez et al. (2018), d) Lopez-Rodriguez et al. (2020), e) LopezRodriguez et al. (2022b). *Lopez-Rodriguez et al. (2018) measured the flux of NGC 1068 at 19.7 𝜇 mto be 22.0 ± 1.4. these objects (Centaurus A, NGC 1068, NGC 2273, NGC 2992, NGC 4258, NGC 7674) are likely dominated by star formation.","pages":[15,16]},{"title":"6. CONCLUSIONS","content":"We have presented a SOFIA atlas of nearby AGN in the 20 - 215 𝜇 m wavelength range using FORCAST and HAWC+. We have released 69 observations of which 41 are newly published and 28 have been previously published (Fuller et al. 2016, 2019; Lopez-Rodriguez et al. 2018, 2022a). From these observations, NGC 4388 shows the most dramatic visual change in emission morphology. The 30 - 40 𝜇 m images show a NE to SW dusty extension associated with the NLR, while the ¿ 50 𝜇 mimages show a East to West dusty emission associated with the plane of the host galaxy. Our observations show that < 10 '' resolution 30 - 70 𝜇 m observations are crucial to disentangle the emitting contribution from AGN and host galaxy. We measured arcsecond scale unresolved nuclear fluxes in order to construct SEDs of the objects in our sample. We included complementary Herschel data to cover up to 500 𝜇 m. For point sources we used aperture photometry to determine the flux. For extended sources we used a 2D gaussian fitting method to extract the central unresolved source(s) of emission from the galaxy background. For this method, the PSF is scaled to represent the central emission while a 2D gaussian represents host galaxy or background emission. Based on the SEDs, we make the following conclusions: In future studies, we will combine data from this atlas with incoming data from JWST to update our IR datasets with the latest and highest resolution data available. Newly obtained JWST/MIRI observations will provide new higher angular resolution data for some of the sources in the wavelength range 5 - 28 𝜇 m.","pages":[16,17]},{"title":"ACKNOWLEDGMENTS","content":"We acknowledge Dr. Lucas Grosset for his effort in subtracting the image backgrounds. E.L.-R. is supported by the NASA/DLR Stratospheric Observatory for Infrared Astronomy (SOFIA) under the 08 0012 Program. SOFIA is jointly operated by the Universities Space Research Association,Inc.(USRA), under NASA contract NNA17BF53C, and the Deutsches SOFIA Institut (DSI) under DLR contract 50OK0901 to the University of Stuttgart. E.L.-R. is supported by the NASA Astrophysics Decadal Survey Precursor Science (ADSPS) Program (NNH22ZDA001NADSPS) with ID 22-ADSPS22-0009 and agreement number 80NSSC23K1585. I.G.B. acknowledges support from STFC through grants ST/S000488/1 and ST/W000903/1. C.R. acknowledges support from Fondecyt Regular grant 1230345 and ANID BASAL project FB210003. Software: /a.pc/s.pc/t.pc/r.pc/o.pc/p.pc/y.pc (Astropy Collaboration et al. 2022, 2018, 2013); S/c.pc/i.pcP/y.pc Virtanen et al. (2020)","pages":[17]},{"title":"REFERENCES","content":"Alonso-Herrero, A., Ramos Almeida, C., Mason, R., et al. 2011, ApJ, 736, 82, doi: 10.1088/0004-637X/736/2/82 Alonso-Herrero, A., Ramos Almeida, C., Esquej, P., et al. 2014, MNRAS, 443, 2766, doi: 10.1093/mnras/stu1293 Alonso-Herrero, A., Esquej, P., Roche, P. F., et al. 2016, MNRAS, 455, 563, doi: 10.1093/mnras/stv2342 Alonso-Herrero, A., Pereira-Santaella, M., Garc´ıa-Burillo, S., et al. 2018, ApJ, 859, 144, doi: 10.3847/1538-4357/aabe30 Alonso-Herrero, A., Garc´ıa-Burillo, S., H¨onig, S. F., et al. 2021, A&A, 652, A99, doi: 10.1051/0004-6361/202141219 Antonucci, R. 1993, ARA&A, 31, 473, doi: 10.1146/annurev.aa.31.090193.002353 Antonucci, R. R. J., & Miller, J. S. 1985, ApJ, 297, 621, doi: 10.1086/163559 Asmus, D. 2019, MNRAS, 489, 2177, doi: 10.1093/mnras/stz2289 https://arxiv.org/abs/2207.09466 doi: 10.1088/0004-637X/794/2/152 Mor, R., Netzer, H., & Elitzur, M. 2009, ApJ, 705, 298, Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261, doi: 10.1038/s41592-019-0686-2 Young, L. M., Meier, D. S., Bureau, M., et al. 2021, ApJ, 909, 98, doi: 10.3847/1538-4357/abe126 Yuan, F., Markoff, S., Falcke, H., & Biermann, P. L. 2002, A&A, 391, 139, doi: 10.1051/0004-6361:20020817 Zhang, L., & Ho, L. C. 2023, ApJL, 953, L9, doi: 10.3847/2041-8213/acea73","pages":[17,18,20,21]},{"title":"A. HERSCHEL IMAGES","content":"We used images from the Herschel Archive to supplement SOFIA data in FIR wavelengths. Images from the PACS and SPIRE instruments covering the wavelength range 70 - 500 𝜇 m are shown in Figures 10, 11, 12, and 13. The white circle in the upper right indicates the beam size while the pink scale in the bottom left indicates a distance of 1 kpc. To extract nuclear fluxes on scales of several arcseconds, we use the methods described in Section 4. For point sources, we performed aperture photometry. For extended sources, we used the 2D gaussian routine outlined in Section 4.1.","pages":[22]},{"title":"B. BACKGROUND FITTING","content":"Figure 14 shows an example of the 2-D gaussian fitting we used to extract the central flux from the background of the host galaxy in wavelengths 30 - 500 𝜇 m. The object used here is NGC 4388. In the first column, the original observation image is shown. The second column shows a model of the image using a PSF (Column 3) combined with the model background. Column 4 shows a PSF-subtracted image (Column 1 - Column 3) used to make the model background (Column 5). Column 6 shows the PSF- and background- subtracted image of the residuals.","pages":[23]},{"title":"C. HOST GALAXY BACKGROUNDS HERSCHEL","content":"Here we show the results of PSF-subtracted images for objects in which only Herschel data showed host galaxy contamination.","pages":[23]},{"title":"F/u.pc/l.pc/l.pc/e.pc/r.pc /e.pc/t.pc /a.pc/l.pc.","content":"19'00 50'00 52'30\" 51'30\" 50'40\" 51'00\" 52'00\" 24 30\" 73'30\" RA (J2000} (2000) 19'00 18'30 34*50' RA U2000) 19*53' 2*48' RA (J2000) RA (J2000) RA (J2000) 5*32 12'00\" 13'00\" RA (2000) RA (J2000) RA (J2000) RA (J2000) RA (J2000) RA (2000) 53'00 52'40\" 51'40\" 8*54' 8*54' 8*48' 8*48' 8*48 8*54'","pages":[24,25]}],"string":"[\n {\n \"title\": \"ABSTRACT\",\n \"content\": \"Wepresent a 19 . 7 -214 𝜇 mimaging atlas of local (4 -181 Mpc; median 43 Mpc) active galactic nuclei (AGN) observed with FORCAST and HAWC+ on board the SOFIA telescope with angular resolutions ∼ 3 '' -20 '' . This atlas comprises 22 Seyferts (17 Type 2 and 5 Type 1) with a total of 69 images, 41 of which have not been previously published. The AGN span a range of luminosities of log 10 ( 𝐿 bol [ erg / s ]) = [ 42 , 46 ] with a median of log 10 ( 𝐿 bol [ erg / s ]) = 44 . 1 ± 1 . 0. We provide total fluxes of our sample using aperture photometry for point source objects and a 2-D Gaussian fitting for objects with extended host galaxy emission, which was used to estimate the unresolved nuclear component. Most galaxies in our sample are point-like sources, however, four sources (Centaurus A, Circinus, NGC 1068, and NGC 4388) show extended emission in all wavelengths. The 30 -40 𝜇 m extended emission in NGC 4388 is coincident with the narrow line region at PA ∼ 50 · , while the dusty extension at longer wavelengths arises from the host galaxy at PA ∼ 90 · . Our new observations allow us to construct the best sampled spectral energy distributions (SEDs) available between 30 - 500 𝜇 m for a sample of nearby AGN. We estimate that the average peak wavelength of the nuclear SEDs is ∼ 40 𝜇 m in 𝜈𝐹 𝜈 , which we associate with an unresolved extended dusty region heated by the AGN. 17 CO(2-1) emission in NGC 5643 as a nuclear molecular gas component of the torus that is likely collimating the ionization cone. Conditions favorable for launching a cold and molecular wind likely depend on Eddington ratio and nuclear hydrogencolumndensities (e.g., Venanzi et al. 2020; Garc'ıa-Burillo et al. 2021; Alonso-Herrero et al. 2021; Garc'ıa-Bernete et al. 2022a). This pc-scale dusty component is possibly associated with larger scale MIR emission detected out to 100s pc scales. In the case of Circinus, high angular resolution MIR imaging, optical polarimetry and integral field spectra, coupled with state-of-the-art radiative transfer simulations, provide evidence that extended dust emission from pc to tens of pc scales in this object is a result of a hollow dusty cone illuminated by a tilted accretion disk (Stalevski et al. 2017, 2019, 2023; Kakkad et al. 2023). MIR extended emission out to 1' ( ∼ 75 pc) was clearly detected in NGC 1068 by Bock et al. (2000). Later 10 . 8 and 18 . 2 𝜇 m emission extending 3 . '' 5 ( ∼ 200 pc) across NGC 4151 was also attributed to dust in the NLR heated by the central engine (Radomski et al. 2003). Likewise, at similar wavelengths, extended emission in 18 AGN at distances out to hundreds of parsecs was detected (Asmus et al. 2016; Garc'ıa-Bernete et al. 2016; Asmus 2019). Using the 37 . 1 𝜇 m filter on SOFIA/FORCAST and thanks to the increase in angular resolution compared with 𝑆𝑝𝑖𝑡𝑧𝑒𝑟 , extended dust emission in Mrk 3, NGC 4151, and NGC4388 was found on ∼ 100s pc-scales (Fuller et al. 2019) coincident with the NLR and radio axis. This emission may be due to dust along the wall of ionization cones (Mason et al. 2009) or a dusty NLR (Mor et al. 2009; Mor & Netzer 2012). In this manuscript we present an imaging atlas of 22 local (D = 4 -181 Mpc; median 42.8 Mpc) AGN obtained using the FORCAST and HAWC+ instruments on the 2.7-m SOFIA telescope in the wavelength range 20 -214 𝜇 m. Most of these datasets are unpublished or dispersed throughout the literature. We provide a mid- to far-IR imaging atlas at angular scales of ∼ 3 -20 '' . At these scales, contribution from several dust sources is expected and we expect to disentangle the emission sources in a future study. Instead, here we aim to determine whether these objects are extended or not, and at what wavelengths within the resolution of the SOFIA telescope. We also explore the wavelength of turnover in the SED. This atlas is complementary to JWST observations up to ∼ 25 𝜇 mand archival Herschel data (70 -500 𝜇 m). The manuscript is organzed as follows. Section 2 describes the observations and AGN sample definition; Section 3 shows the new IR images; Section 4 contains details of the imaging analysis and Section 5 shows the resulting SEDS; we present results about the data in Section 6.\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"The Galaxy Activity, Torus, and Outflow Survey (GATOS). VII. The 20-214 𝜇 mimaging atlas of active galactic nuclei using SOFIA\",\n \"content\": \"L/i.pc/n.pc/d.pc/s.pc/a.pc/y.pc F/u.pc/l.pc/l.pc/e.pc/r.pc, 1 E/n.pc/r.pc/i.pc/q.pc/u.pc/e.pc L/o.pc/p.pc/e.pc/z.pc-R/o.pc/d.pc/r.pc/i.pc/g.pc/u.pc/e.pc/z.pc, 2, 3 I/s.pc/m.pc/a.pc/e.pc/l.pc G/a.pc/r.pc/c.pc'/dotlessi.pc/a.pc-B/e.pc/r.pc/n.pc/e.pc/t.pc/e.pc, 4 C/r.pc/i.pc/s.pc/t.pc/i.pc/n.pc/a.pc R/a.pc/m.pc/o.pc/s.pc A/l.pc/m.pc/e.pc/i.pc/d.pc/a.pc, 5, 6 A/l.pc/m.pc/u.pc/d.pc/e.pc/n.pc/a.pc A/l.pc/o.pc/n.pc/s.pc/o.pc-H/e.pc/r.pc/r.pc/e.pc/r.pc/o.pc, 7 C/h.pc/r.pc/i.pc/s.pc P/a.pc/c.pc/k.pc/h.pc/a.pc/m.pc, 1, 8 L/u.pc/l.pc/u.pc Z/h.pc/a.pc/n.pc/g.pc, 1 M/a.pc/s.pc/o.pc/n.pc L/e.pc/i.pc/s.pc/t.pc, 1 N/a.pc/n.pc/c.pc/y.pc L/e.pc/v.pc/e.pc/n.pc/s.pc/o.pc/n.pc, 9 M/a.pc/s.pc/a.pc I/m.pc/a.pc/n.pc/i.pc/s.pc/h.pc/i.pc, 8 S/e.pc/b.pc/a.pc/s.pc/t.pc/i.pc/a.pc/n.pc H/o.pc/e.pc/n.pc/i.pc/g.pc, 10 M/a.pc/r.pc/k.pc/o.pc S/t.pc/a.pc/l.pc/e.pc/v.pc/s.pc/k.pc/i.pc, 11, 12 C/l.pc/a.pc/u.pc/d.pc/i.pc/o.pc R/i.pc/c.pc/c.pc/i.pc, 13, 14 E/r.pc/i.pc/n.pc H/i.pc/c.pc/k.pc/s.pc, 15 E/n.pc/r.pc/i.pc/c.pc/a.pc B/e.pc/l.pc/l.pc/o.pc/c.pc/c.pc/h.pc/i.pc, 16, 17 F/r.pc/a.pc/n.pc/c.pc/o.pc/i.pc/s.pc/e.pc C/o.pc/m.pc/b.pc/e.pc/s.pc, 18 R/i.pc/c.pc D/a.pc/v.pc/i.pc/e.pc/s.pc, 19 S/a.pc/n.pc/t.pc/i.pc/a.pc/g.pc/o.pc G/a.pc/r.pc/c.pc'/dotlessi.pc/a.pc B/u.pc/r.pc/i.pc/l.pc/l.pc/o.pc, 20 O/m.pc/a.pc/i.pc/r.pc/a.pc G/o.pc/n.pc/z.pc '/a.pc/l.pc/e.pc/z.pc M/a.pc/r.pc/t.pc'/dotlessi.pc/n.pc, 21 T/a.pc/k.pc/u.pc/m.pc/a.pc I/z.pc/u.pc/m.pc/i.pc, 8 A/l.pc/v.pc/a.pc/r.pc/o.pc L/a.pc/b.pc/i.pc/a.pc/n.pc/o.pc, 22 M/i.pc/g.pc/u.pc/e.pc/l.pc P/e.pc/r.pc/e.pc/i.pc/r.pc/a.pc S/a.pc/n.pc/t.pc/a.pc/e.pc/l.pc/l.pc/a.pc, 23 D/i.pc/m.pc/i.pc/t.pc/r.pc/a.pc R/i.pc/g.pc/o.pc/p.pc/o.pc/u.pc/l.pc/o.pc/u.pc, 4, 24 D/a.pc/v.pc/i.pc/d.pc R/o.pc/s.pc/a.pc/r.pc/i.pc/o.pc, 25 D/a.pc/n.pc/i.pc/e.pc/l.pc R/o.pc/u.pc/a.pc/n.pc, 26 T/a.pc/r.pc/o.pc S/h.pc/i.pc/m.pc/i.pc/z.pc/u.pc, 19 /a.pc/n.pc/d.pc M/a.pc/r.pc/t.pc/i.pc/n.pc W/a.pc/r.pc/d.pc 27 1 University of Texas at San Antonio, One UTSA Circle, San Antonio, TX, 78249, USA 2 Department of Physics & Astronomy, University of South Carolina, Columbia, SC 29208, USA 3 Kavli Institute for Particle Astrophysics & Cosmology (KIPAC), Stanford University, Stanford, CA 94305, USA 4 Department of Physics, University of Oxford, Keble Road, Oxford OX1 3RH, UK 5 Instituto de Astrof'ısica de Canarias, Calle V'ıa L'actea, s/n, E-38205 La Laguna, Tenerife, Spain 6 Departamento de Astrof'ısica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain 7 Centro de Astrobiolog'ıa (CAB), CSIC-INTA, Camino Bajo del Castillo s/n, E-28692, Villanueva de la Ca˜nada, Madrid, Spain 8 National Astronomical Observatory of Japan, National Institutes of Natural Sciences (NINS), 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 9 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA 10 School of Physics & Astronomy, University of Southampton, Southampton SO17 1BJ, UK 11 Astronomical Observatory, Volgina 7, 11060 Belgrade, Serbia 12 Sterrenkundig Observatorium, Universiteit Gent, Krijgslaan 281-S9, Gent B-9000, Belgium 13 N'ucleo de Astronom'ıa de la Facultad de Ingenier'ıa, Universidad Diego Portales, Av. Ej'ercito Libertador 441, Santiago, Chile 14 Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, People's Republic of China 15 Department of Physics and Astronomy, University of Alaska Anchorage, Anchorage, AK 99508-4664, USA 16 Departmento de F'ısica de la Tierra y Astrof'ısica, Fac. de CC F'ısicas, Universidad Complutense de Madrid, E-28040 Madrid, Spain Instituto de F'ısica de Part'ıculas y del Cosmos IPARCOS, Fac. CC F'ısicas, Universidad Complutense de Madrid, E-28040 Madrid, Spain 18 LERMA, Observatoire de Paris, Coll'ege de France, PSL University, CNRS, Sorbonne University, Paris, France 19 Max Planck Institut fur Extraterrestrische Physik, Giessenbachstrasse 1, D-85748 Garching bei Munchen, Germany 20 Observatorio de Madrid, OAN-IGN, Alfonso XII, 3, E-28014 Madrid, Spain 21 Instituto de Radioastronom'ıa y Astrof'ısica (IRyA), Universidad Nacional Aut'onoma de M'exico, Antigua Carretera a P'tzcuaro #8701, ExHda. San Jos'e de la Huerta, Morelia, Michoac'an, C.P. 58089, Mexico 22 Telespazio UK for the European Space Agency, ESAC, Camino Bajo del Castillo s/n, E-28692 Villanueva de la Ca˜nada, Spain 23 Instituto de F'ısica Fundamental, CSIC, Calle Serrano 123, E-28006 Madrid, Spain 24 School of Sciences, European University Cyprus, Diogenes street, Engomi, 1516 Nicosia, Cyprus 25 School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne NE1 7RU, UK 26 LESIA, Observatoire de Paris, Universit'e PSL, CNRS, Sorbonne Universit'e, Sorbonne Paris Cite'e, 5 place Jules Janssen, F-92195 Meudon, France 27 Centre for Extragalactic Astronomy, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"2. AGN SAMPLE AND OBSERVATION DATA\",\n \"content\": \"2.1. Sample Selection This imaging atlas was drawn from the ongoing AGN survey performed by the Galactic Activity, Torus, and Out- Keywords: galaxies -- active galaxies -- agn\",\n \"pages\": [\n 2\n ]\n },\n {\n \"title\": \"1. INTRODUCTION\",\n \"content\": \"There is clear evidence that a considerable amount of dust in the vicinity of supermassive black holes (SMBHs) in active galaxies obscures the central engine (i.e., accretion disk and SMBH) in some lines of sight. Through spectropolarimetric observations of NGC 1068, Antonucci & Miller (1985) showed that its optical polarized spectrum contained broad optical polarized emission lines not originally observed by direct total intensity observations. It was subsequently presumed that an optically and geometrically thick dusty structure ('torus') blocked the central engine in some lines of sight (Antonucci 1993; Urry & Padovani 1995). Under this unified scheme a Type 1 AGN is seen face-on and shows broadened optical lines, while in Type 2 AGN the broadened lines are obscured. This model also predicts that broad silicate features at 10 and 18 𝜇 m will be seen in emission in Type 1 and in absorption in Type 2. However, silicate emission can be seen in emission in some Type 2 AGN, while absorption can be seen in some Type 1 (e.g., Hatziminaoglou et al. 2015). This and other observational features are explained by the inhomogeneous nature of the torus. Clumpy torus models (Nenkova et al. 2008a,b) predict shallower silicate features, more similar infrared spectral energy distributions (SEDs) between Type 1 and Type 2 AGN, etc. (see Ramos Almeida & Ricci 2017, for a review). A region of narrow forbidden line emission extends above and below the midplane of the dusty torus structure out to several kpc scales. Recent sub-arcsecond interferometric imaging observations have shown a dust component at pc-scales co-spatial with the base of the narrow line region (NLR; Honig et al. 2012; Tristram et al. 2014; L'opez-Gonzaga et al. 2014, 2016; Burtscher et al. 2013; G'amez Rosas et al. 2022; Isbell et al. 2022). This dusty structure is interpreted as part of a dusty wind launched from the inner hot part of the torus driven by radiation pressure at pc-scales (Honig 2019), but generated by a magnetohydrodynamical wind at sub-pc scales (e.g., Emmering et al. 1992; Lopez-Rodriguez et al. 2015; Takasao et al. 2022; Lopez-Rodriguez et al. 2023). This extended dusty structure has been resolved in a nearby galaxy, ESO 418-G14, using mid-infrared (MIR) images with JWST/MIRI finding that the dust is primarily heated by the AGN and/or radiative jet-induced shocks in the NLR rather than a wind (Haidar et al. 2024). ALMA observations provide observational support for a dusty torus+outflow scenario. Emission from the nucleus of NGC1068 was mapped with a resolution of ∼ 4 pc, resolving a 7 -10 pc diameter disk interpreted as the sub-mm counterpart of the torus (Garc'ıa-Burillo et al. 2016). Rotation of the compact emission was detected in HCN J=3-2 and HCO+ J=3-2 (Imanishi et al. 2018, 2020). A molecular outflowing wind co-spatial with the dusty and molecular torus was also observed (Garc'ıa-Burillo et al. 2019). Alonso-Herrero et al. (2018) interpreted the measured nuclear (10 -20 pc) flow Survey (GATOS; Garc'ıa-Burillo et al. 2021; AlonsoHerrero et al. 2021; Garc'ıa-Bernete et al. 2024). GATOS /one.sup aims to characterize the dynamics and composition of the dusty and molecular torus and multi-phase outflows in AGN. The GATOS parent sample is selected from the 70 Month Swift /BAT AGN catalog, which is flux limited in the ultrahard 14 -195 keV X-rays band (Baumgartner et al. 2013). In the initial study of AGN using SOFIA observations (Fuller et al. 2016), sources from the GATOS survey were selected based on the criteria that the galaxies had been previously studied using C/l.pc/u.pc/m.pc/p.pc/y.pc (Nenkova et al. 2008a,b) torus models and were well-sampled in the 1 -18 𝜇 m regime (Ramos Almeida et al. 2009, 2011; Alonso-Herrero et al. 2011). The study of the 11 objects included the 31 . 5 𝜇 m photometry in the SEDs and found that including the 31.5 𝜇 m photometry reduces the number of C/l.pc/u.pc/m.pc/p.pc/y.pc torus models that are compatible with the data and modifies the model output for the torus outer radius. Fuller et al. (2019) further extended the wavelength range of a subset of 7 AGN SEDs to 37.1 𝜇 m. They subsequently found extended emission in the PSF-subtracted images of Mrk 3, NGC 4151, and NGC 4388 that is coincident with the radio axis and NLR. In a separate study, Lopez-Rodriguez et al. (2018) modeled the torus of NGC 1068 using ∼ 20 -53 𝜇 m FORCAST and HAWC+ observations. They showed that the peak wavelength range of emission from the torus is ∼ 30 - 40 𝜇 m with a characteristic temperature 70 - 100 K. The use of observations ¿ 30 𝜇 m in that study from SOFIA and ALMA highlights the importance of longer wavelength observations to put constraints on MIR emission sources. Based on these results, we extend the wavelength range in objects previously observed, and also expand the number of AGN observed. We present the complete imaging atlas of 22 Seyferts observed by SOFIA in the wavelength range 19 . 7 -214 𝜇 m using FORCAST and HAWC+. The final set of observations presented here was part of a multi-year AGN survey over several observing SOFIA cycles (Proposal IDs: 02 0035, 04 0048, 06 0066, 08 0014; PI: Lopez-Rodriguez; 70 0400 PI: Herter). The SOFIA atlas of AGN in the far-IR (FIR) is a flux-limited sample of nearby, bright, and well-studied AGN. All objects have a point source flux of > 200 mJy at 31 . 5 𝜇 m, which ensures that each band can be observed within 1 hr of on-source time with a signal to noise ratio > 10 using FORCAST/SOFIA. Although the original AGN sample is larger than that presented here, only 22 AGN were observed in total by SOFIA before end of operations in 2022. Note that there are gaps in the 20 -214 𝜇 m wavelength range due to the fact that SOFIA only flies with a single instrument per night. For each SOFIA cycle, we prioritized the objects with observations acquired in a single instrument from the previous cycle. The sample properties are given in Table 1. For most objects, we retrieved redshift data from the NASA Extragalactic Database (NED). However, for nearby objects Centaurus A Redshifts and spectral type were taken from NED. Distances to most sources were obtained using H0 = 70 km s -1 Mpc -1 . Distances to nearby sources Centaurus A and Circinus were taken from Harris et al. (2010) and Tully et al. (2009), respectively. References for log 𝐿 𝑏𝑜𝑙 : a) Borkar et al. (2021) b) Marinucci et al. (2012) c) Alonso-Herrero et al. (2011) d) Ichikawa et al. (2017), e) Leighly et al. (2014) f) Marconi et al. (2004), g) Baumgartner et al. (2013), h) Garc'ıa-Bernete et al. (2015), i) Ramos Almeida et al. (2011), j) Yuan et al. (2002), k) Duras et al. (2020), and Circinus, distances were obtained individually (Harris et al. 2010; Tully et al. 2009). The 22 objects in this atlas cover the luminosity range of log 10 ( 𝐿 bol [ erg / s ]) = [ 42 , 46 ] with a median of log 10 ( 𝐿 bol [ erg / s ]) = 44 . 1 ± 1 . 0, and a distance of 4 -181 Mpc with a median of 42.8 Mpc. Figure 1 shows bolometric luminosity plotted against distance, where Seyfert 1 objects are shown as red triangles and Seyfert 2 objects are shown as purple stars.\",\n \"pages\": [\n 2,\n 3\n ]\n },\n {\n \"title\": \"2.2.1. FORCAST\",\n \"content\": \"FORCAST is an IR camera and spectrograph sensitive in the wavelength range 5 -40 𝜇 m with a field of view (FOV) of 3.4 ' × 3.2 ' and pixel scale 0.768 '' /pixel. With one exception (NGC 1068 in the 19.7 𝜇 m filter), we only used the Long Wavelength Camera (LWC; 25 -40 𝜇 m) due to the abundance of ground-based images at shorter wavelengths for the objects in our sample. FORCAST observations were made in dual channel mode using the two-position chop-nod (C2N) method with symmetric nod-match-chop (NMC) to remove telescope thermal emission and time variable sky background, and to reduce the effect of 1/ 𝑓 noise from the array. Data were reduced by the SOFIA Science Center using the /f.pc/o.pc/r.pc/c.pc/a.pc/s.pc/t.pc /r.pc/e.pc/d.pc/u.pc/x.pc pipeline following the methods described by Herter et al. (2012). Most of the pipeline changes over the cycles were to refine the spectroscopic mode of FORCAST with little or no change to the image mode presented here. Observations were flux-calibrated using the set of standard stars of the observing run, which provides flux uncertainties of ∼ 10 %. The point spread function (PSF) of the 31.5 𝜇 m observations from Cycle 2 (see Fuller et al. 2016) was the co-average of a set of standard stars from that cycle. Its FWHMwas 3.40 '' , in agreement with the SOFIA Observer's Handbook v3.0.0. The PSFs for Cycle 4 in the 30 - 40 𝜇 mwavelength range were determined by using standard star observations from the individual flights (see Fuller et al. 2019) and averaged at FWHM ∼ 4.33 '' and 4.58 '' in the 31.5 and 37.1 𝜇 m filters, respectively. The FORCAST PSFs for the observations of NGC1068 are detailed in Lopez-Rodriguez et al. (2018).\",\n \"pages\": [\n 3,\n 4\n ]\n },\n {\n \"title\": \"2.2.2. HAWC+\",\n \"content\": \"HAWC+isaFIRimaging polarimeter designed to allow total and polarized intensity imaging observations in four broad bands centered at 53, 89, 155, and 214 𝜇 m, corresponding to Bands A, C, D, and E respectively (see Table 2). On- -fly mapping (OTFMAP) observing modes were used for both imaging polarimetry and total intensity imaging. Observing modes for the individual observations are given in Table 3. Data taken in polarization mode was reduced using the /h.pc/a.pc/w.pc/c.pc /d.pc/p.pc/r.pc /p.pc/i.pc/p.pc/e.pc/l.pc/i.pc/n.pc/e.pc and the reduction steps presented in Lopez-Rodriguez et al. (2022b). The Comprehensive Reduction Utility for SHARC II (CRUSH; Kov'acs et al. 2006; Kov'acs 2008) was used to obtain the total intensity observations. HAWC+ observations were reduced following the same reduction steps. We quote the pipeline versions or CRUSH versions in Table 3 to differentiate between the observing modes. There are no differences between CRUSH versions to obtain the total intensity images as all the changes in the pipeline were done for the polarimetric mode. Table 3 also shows the pixel scale of each image. As in FORCAST observations, the source of uncertainty in the photometry for HAWC+ results from calibration factors of the standard stars associated with the observation, giving an uncertainty of ∼ 10 % (Lopez-Rodriguez et al. 2022b). HAWC+ PSFs were estimated using standard star observations in 2017. Pallas was observed in Bands A and C on 7 November 2017, while Neptune was observed in Bands D and E on 19 October, 2017. The FWHM of these standards are 5.25 '' , 8.26 '' , 14.74 '' , and 19.65 '' in Bands A, C, D, and E, respectively. The FWHMs from the Observer's Handbook /two.sup are given in Table 2.\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"2.3. Observing Data\",\n \"content\": \"Table 3 provides the final AGN sample with information about the wavelength, observing mode, observation and mission details, versions of the separate pipelines, and also the field-of-view (FOV) of the individual images.\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"3. IMAGES\",\n \"content\": \"Images of the 22 AGN in the 19 . 7 -214 𝜇 m wavelength range are presented in Figures 2, 3, 4, and 5. The orange scale on the bottom left of the images indicates a scale of 500 pc. The beam size is depicted in white in the top right of the images. In all images, north is up and east is to the left. Complementary Herschel 70 -500 𝜇 m images are shown in Appendix A (Fig. 10, 11, 12, 13). These fully reduced images were obtained through the Herschel Science Archive /three.sup . All objects are presented and analyzed individually. Centaurus A . The host galaxy of Centaurus A is clearly visible in the 53 𝜇 m image with a bright compact center. However, the host galaxy becomes more dominant in the 89 𝜇 mimage(seeadetailed analysis of host galaxy dust emission at 89 𝜇 m in Lopez-Rodriguez 2021). The kpc-scale warped dust and gas lane was first observed with Spitzer imaging using IRAC and MIPS (Quillen et al. 2006). On subarcsecond scales, Radomski et al. (2008) observed the nucleus of Centaurus A using the 8.8, 10.4, and 18.3 𝜇 m filters on T-ReCS at Gemini South. They concluded that the mostly likely sources of nuclear MIR emission are an unresolved clumpy dusty torus in the core, and a dusty NLR for the arcsecondscale extended emission (see also Garc'ıa-Bernete et al. 2016). Circinus Galaxy . HAWC+ 53 and 89 𝜇 m images of the Circinus Galaxy show a very bright FIR core with extended emission at a PA ∼ 30 · , whereas the 215 𝜇 m image shows a slightly different PA ∼ 55 · . We estimate the FWHM of the extended FIR nuclear emission to be ∼ 6 '' × 6 '' , 13.5 '' × 11.5 '' , and 22.3 '' × 25.6 '' at 53, 89, and 214 𝜇 m, respectively. These are larger than the PSF FWHMs given in Section 2.2.2, which indicates extended emission along the axis of the inner bar of the galaxy. MIR emission was resolved at 8.7 and 18.3 𝜇 m out to 2 '' in an approximate east-west direction, coincident with the ionization cones at PA ∼ 100 · (Packham et al. 2005; Stalevski et al. 2017). However, the elongation seen in the SOFIAimages(andin Herschel imagesinAppendixA)seems to be arising from dust in the host galaxy. MCG-5-23-16 . MCG-5-23-16 appears as a point-like source in the 31 . 5 -155 𝜇 mwavelength range whose brightness decreases with increasing wavelength. However, this galaxy appears as an extended source in the MIR using high angular resolution data from VLT (Garc'ıa-Bernete et al. 2016). Likewise, Ferruit et al. (2000) found that this nucleus has an extended optical NLR at a PA of 40 · . They found a dust lane extending 2 '' on either side of the nucleus, parallel to the axis of the galaxy. Their extended dusty emission at 40 · was detected at a 3 𝜎 level up to ∼ 4 '' from the core. This extended structure has no thermal emission counterpart within the 31 . 5 -155 𝜇 mwavelength range. Mrk3 . Although the FORCAST and HAWC+ images generally appear to be point-like sources, Fuller et al. (2019) found extended emission in the PSF-subtracted 37.1 𝜇 m im- of Mrk 3 in the direction of the radio axis (84 · ; Kukula et al. 1993) and the NLR ( ∼ 70 · ; Capetti et al. 1995). Asmus et al. (2013) found an elongated nucleus out to ∼ 170 pc with PA ∼ 70 · in the Si-5 (11.6 𝜇 m) filter using Gemini/MICHELLE. However, the Si-2 (8.7 𝜇 m) image from GTC/Canaricam appears point-like (Alonso-Herrero et al. 2016). The large scale east-west structure in the 53 𝜇 mimage here is a background artifact produced by the data reduction due to the small spatial coverage and short integration time of the observation. Mrk 231 . The 89 𝜇 mimage shown here is point-like with a FWHM of ∼ 8.5', similar to the standard FWHM of 8.26' (see Section 2.2.2). Mrk 231 is a Type 1 Ultra Luminous Infrared Galaxy (ULIRG) and is the nearest known quasar at a distance of 181 Mpc. It is known for its multi-phase and multi-scale outflows (see Rupke & Veilleux 2011), with a neutral outflow up to 3 kpc in radius (Rupke et al. 2005). Mrk 573 . Although the SNR is very low (3 -4 𝜎 ), both 31.5 and 37.1 𝜇 mimages of Mrk 573 show marginally resolved ∼ 4 . 5 '' elongation in the east-west direction at PA ∼ 110 · . Mrk 573 was previously shown to have a biconical NLR coincident with radio emission 3-4' from the nucleus at a PA ∼ 125 · (Ulvestad & Wilson 1984; Pogge & De Robertis 1995). The marginal detection here may be cold extended dust in the outer layers of the NLR. NGC 1068 . The 19 - 53 𝜇 m images of NGC 1068 were published previously in Lopez-Rodriguez et al. (2018), where it was shown that the peak emission from the torus occurs between 30 - 40 𝜇 m with a corresponding temperature of 70 - 100 K. The 89 𝜇 m image was published as a polarimetric observation (Lopez-Rodriguez et al. 2020). Within a scale of about 1 kpc, NGC 1068 shows extended emission in the NE to SW direction at a PA ∼ 45 · , similar to MIR observations using VISIR/VLT (Asmus et al. 2014). Their observations revealed a nuclear structure in the north-south direction and extended structures to the NE and SW. From the N-band spectrum, Mason et al. (2006) concluded that while torus emission dominates NIR wavelengths, large-scale MIR emission is dominated by diffuse dust within the ionization cones. NGC 1275 . All 30 - 53 𝜇 m images are dominated by a point-like source. However, the 31.5 𝜇 m image (Fuller et al. 2019) shows 3 𝜎 extended emission along the PA ∼ 140 · . This AGN is known to have a network of H 𝛼 filaments extending out to ∼ 100' (see Conselice et al. 2001) and is possibly the result of a merger (Holtzman et al. 1992). The MIR core shows silicate dust emission in both 10 and 18 𝜇 mbands (see Fuller et al. 2019). Hence, both dust and gas are extended covering several kpc around the core. In the HAWC+ 89 𝜇 mfilter, Lopez-Rodriguez et al. (2023) found extended dust emission at a PA ∼ 125 · out to a 12 kpc radius potentially associated with a magnetized dusty filament along the NW direction (Fabian et al. 2008). NGC 2110 . The 30 - 215 𝜇 mimages of NGC 2110 are all point-like. The north-south pattern in the 53 𝜇 mimage is due to background noise and does not represent extended dust. NGC2110is a Type 2 AGN that shows silicate emission at 10 and18 𝜇 mthatis interpreted as a result of a clumpy torus, or as dust within the ionization cones (PA ∼ 160 · ; Mulchaey et al. 1994) in the inner 32 pc of the AGN (Mason et al. 2009). This galaxy appears as an extended source in Gemini/MICHELLE high resolution N-band observations (Garc'ıa-Bernete et al. 2016). However, any structure within the NLR or ionization cones is not resolved by our observations. NGC 2273 . The full set of 30 - 215 𝜇 m images of NGC 2273 show a point-like source. The north/south pattern in the 53 𝜇 m image is due to background noise and does not represent extended dust. Within the FWHM of these images (see Sections 2.2.1, 2.2.2), there is a known star-forming ring within ∼ 2' of the nucleus (Ferruit et al. 2000; Martini et al. 2003; Sani et al. 2012). GTC/Canaricam observations (Alonso-Herrero et al. 2014, 2016) at 8.7 𝜇 mshowelongation from the north-east to the south-west, likely with contribution from PAH. This structure is consistent with extension seen in the PSF-subtracted 37.1 𝜇 mSOFIAimage(Fulleretal. 2019). NGC 2992 . The image of NGC 2992 in the 31.5 𝜇 mfilter is published in Fuller et al. (2016) and appears as a point-like source. Subarcsecond N-band imaging (Garc'ıa-Bernete et al. 2015) reveals extended emission along PA ∼ 30 · out to ∼ 3 kpc which is attributed to dust heated by star formation based on corresponding N-band spectroscopy. The FWHM of the SOFIA image is ∼ 3.5' × 3.5' (560 × 560 pc 2 ) so the extension should be resolvable within the SOFIA image. Since we do not see the extension in the image here, we conclude that either the extended dust emission tapers at wavelengths ¿ 20 𝜇 mor SOFIA does not have enough sensitivity to detect it. NGC 3081 . The 31.5 𝜇 m image of NGC 3081 was published in Fuller et al. (2016) while the 37.1 𝜇 m image was published in Fuller et al. (2019). The nucleus is known to harbor a region of strong optical emission ∼ 1' from the AGN (Ferruit et al. 2000) likely due to dust or gas heated by the AGN. Fuller et al. (2019) estimated that ∼ 35% of the MIRemission within the central few arcseconds (few hundred parsecs) of the AGN originates in the NLR. High angular resolution N- and Q- band observations show extension towards the north, extending out to ∼ 450 pc from the south-east to the north-west (PA ∼ 160 · ; Garc'ıa-Bernete et al. 2016). On larger scales, optical and NIR observations reveal a series of star forming resonance rings at distances of 2.3, 11.0, 26.9 kpc and 33.1 kpc (Buta 1990; Buta et al. 1998, 2004). At longer wavelengths (¿200 𝜇 m), Ramos Almeida et al. (2011) concluded that FIR emission is contaminated by the star-forming ring 2.3 kpc in diameter. NGC 3227 . The 31.5 𝜇 m image was published in Fuller et al. (2016) while the 37.1 𝜇 mimage was published in Fuller et al. (2019). These images show a point-like source, although NGC 3227 is known to harbor a nuclear star-forming region (Schinnerer et al. 2001; Davies et al. 2006) with a nuclear cluster within ∼ 70 pc ( ∼ 1') from the core. The 8.7 𝜇 mimage from Alonso-Herrero et al. (2016) shows a slight north/south elongation and the corresponding spectrum shows clear PAH in the nucleus (see also Garc'ıa-Bernete et al. 2016). These star forming regions likely contaminate the nuclear MIR emission within the FWHM of our images. NGC 3281 . While the 31.5 𝜇 m FORCAST image is published in Fuller et al. (2016), the HAWC+ images at 53, 89, 154, and 214 𝜇 m are presented here for the first time and appear point-like in all filters. The images taken at 53 and 89 𝜇 mappear to have significant noise in their backgrounds. The subarcsecond (0.35') N-band spectrum in Gonz'alez-Mart'ın et al. (2013) shows a deep 10 𝜇 m silicate absorption feature which originates in the inner ∼ 80 pc of the AGN. NGC 4151 . The SOFIA images of NGC 4151 appear as a point-like source, with a potential detection of extended NGC 7469 15 5 NGC 7674 1 emission at PA ∼ 120 · at a 3 𝜎 level at 37 . 1 𝜇 m. Fuller et al. (2019) confirmed this elongation in the PSF-subtracted 37.1 𝜇 mimagecoincident with the NLR and radio axes. Radomski et al. (2003) show extended emission in 10.8 and 18.2 𝜇 m images that coincides with the NLR axis at PA ∼ -60 · . For ≥ 37 . 1 𝜇 m, we conclude that any extended emission due to NLR dust is within the FWHM of the SOFIA instruments. NGC4258 . NGC 4258 was not detected but we include the data here since it is part of the sample. It has been observed and analyzed in the N-band with Gemini/Michelle by Mason et al. (2012). These authors found a compact nucleus that is marginally resolved at 10 𝜇 m(FWHM ∼ 0 . 5 '' ). NGC 4388 . NGC 4388 is an edge-on spiral that shows the most interesting mid- to far-IR morphology in this study. Notably, in the 30 - 40 𝜇 mFORCAST images of NGC 4388, extension can be seen in the NE to SW direction at PA ∼ 40 · (see also Fuller et al. 2019), coincident with the NLR. This emission is seen on smaller scales at shorter wavelengths (Asmus et al. 2016; Garc'ıa-Bernete et al. 2016). The 53 𝜇 m image decreases in intensity and does not show a strong central core of emission as in the 31 . 5 -37 . 1 wavelength range. However, at longer wavelengths (89 -214 𝜇 m), host galaxy emission clearly dominates the images in the east-west direction at PA ∼ 90 · . NGC 4941 . NGC 4941 is a low-luminosity AGN that appears here as a faint point-like source in the 31 . 5 𝜇 m and 37 . 1 𝜇 m images, but brighter at 53 and 89 𝜇 m. Subarcsecond resolution N-band imaging on VLT/VISIR (Asmus et al. 2011) showed no significant extended MIR sources outside of the nucleus. NGC 5506 NGC 5506 appears as a bright point source in both the 31.5 and 37.1 𝜇 m filters. While the nucleus is unresolved, extended MIR emission has been detected up to a few arcseconds to the northeast at 11.9 𝜇 m (Raban et al. 2008). Extended emission in the north-south direction was detected in the N-band out to ∼ 560 pc, while faint extended emission towards the east in the Q-band was also detected (Garc'ıa-Bernete et al. 2016). However, the PSF-subtracted 12.27 𝜇 m2' × 2' VLT/VISIR image of Alonso-Herrero et al. (2021) shows that the PA of extended emission varies from 30 · in the central ∼ 0.5' to nearly 90 · in the outer regions. NGC 7465 . The 31.5 𝜇 m FORCAST image appears faint with a 3 𝜎 upper-limit in the 37.1 𝜇 m image. The 53 and 89 𝜇 m HAWC+ images here appear increasingly brighter, albeit as point-like sources. Cold molecular gas observations (Young et al. 2021) reveal that NGC 7465 is quite gas-rich, possibly from a recent merger. NGC 7469 . NGC 7469 appears as a very bright source in the 31.5 𝜇 m image with FWHM ∼ 4.3'. After PSF subtraction, Fuller et al. (2016) found extended emission in the north-south direction. This AGN is known to have a circumnuclear ring of star formation at a radius of ∼ 480 pc ( ∼ 1.4'; Ramos Almeida et al. 2011) in 8.7 and 18.3 𝜇 mimages taken on Gemini/T-ReCS. Recent JWST observations reveal prominent PAH emission, indicative of star formation, in the circumnuclear ring (Garc'ıa-Bernete et al. 2022b; Zhang & Ho 2023). NGC 7674 . The previously published (Fuller et al. 2016) FORCAST 31.5 𝜇 m image appears as a point-like source. Asmus et al. (2013) found that the nucleus of NGC 7674 is extended at PA ∼ 125 · at subarcsecond scale resolution, where the extension roughly aligns with the ionization cone.\",\n \"pages\": [\n 8,\n 9,\n 10,\n 11,\n 12,\n 13\n ]\n },\n {\n \"title\": \"4. NUCLEAR FLUX EXTRACTION\",\n \"content\": \"We aim to construct well-sampled mid- to far-IR SEDs of the nuclear emission of AGN at scales of several arcseconds, depending on the PSF of the observation and possible extended emission. On these scales, we expect multiple dust sources (i.e. torus, star forming regions, dusty outflow), however disentangling these sources is beyond the scope of this imaging atlas. Because the images span a range of observing cycles, instruments, and observing modes, we analyzed each image individually. Of our sample, 17 objects appear visually as point sources. For these sources, we performed aperture photometry where the aperture size was set to be 2 × the FWHM at a given band. For objects that show host galaxy emission, we extract the central PSF to construct the SEDs as described below. We complement our SOFIA data with Herschel imaging data (see Appendix A) and use a similar analysis method to construct the full IR SEDs.\",\n \"pages\": [\n 13\n ]\n },\n {\n \"title\": \"4.1. Extended Sources: 2D Gaussian Fitting\",\n \"content\": \"For sources with extended dust emission, we performed a two-component simultaneous fit to accurately model both the central source based on the PSF, and the host galaxy whose fit assumes an elongated 2D Gaussian profile. In order to supplement the SOFIA data for the full mid- to far-IR SEDs, we used a similar methodology with Herschel images. For the SOFIA images, the PSF used was based on the standard stars of the observing runs for each cycle as explained in Section 2.2. However, the same analysis could not be performed on Herschel images due to the threefold lobes associated with the instrument PSFs. To accommodate this, we compared three different PSF models to reproduce and fit the central source. Two of the PSFs were point source images while the third was an approximation of the theoretical instrumental PSF using a Gaussian profile. The fitting routine used four free parameters for the Gaussian profile: (1,2) the position in x and y of the PSF center according to the image center, (3) the amplitude of the PSF, (4) the fourth parameter was dependent on the PSF type used. For archival PSFs, this parameter represents the rotation angle that needs to be applied to the PSF to match the orientation of the image. For the Gaussian PSFs, this fourth parameter represents a scaling factor to the width of the Gaussian compared to its ideal value for a perfect instrument (1 . 22 × 𝜆 / 𝐷 ). The galaxy background is defined by 7 parameters: (1,2) the 2D Gaussian's center position (x0 and y0), (3,4) its width ( 𝜎 𝑥 and 𝜎 𝑦 ) in both directions, (5) its amplitude, and (6) its orientation on the image ( 𝜃 ). To these 6 parameters we added a constant background as a 7th free parameter. We combined these components and fit this simulated intensity map to the observed map using the 11 total free parameters (4 from the PSF and 7 from the 2D gaussian). We thus derived the parameters describing the best central source for our intensity maps, and then studied the properties extracted for the central source and removed it from the initial map to study the host galaxy itself. An example of this procedure is given in the Appendix in Figure 14.\",\n \"pages\": [\n 13\n ]\n },\n {\n \"title\": \"4.2. AGN and host galaxy contribution\",\n \"content\": \"While most SOFIA images were treated as point sources, Centaurus A, Circinus, NGC 1068, and NGC 4388 all had significant host galaxy contamination that needed to be subtracted from at least some of the images. Figure 6 shows the PSF subtracted images of these sources and Table 4 gives the percentage of the contribution of the PSF to the total flux of the object. In several other objects, the shorter wavelength ( ∼ 30 - 100 𝜇 m) images did not show host galaxy contamination, but longer wavelength Herschel images show the colder extended dust. Because this is an atlas of SOFIA images, we include objects with host galaxy contamination only in Herschel images in Appendix C for completeness. The53 𝜇 mimageofCentaurusAcontains ( < 5%)emission from extended sources, so we performed aperture photometry to account for the central emission. At wavelengths ≳ 70 𝜇 m, the host galaxy substantially ( ∼ 56 -70%) contributes to the central AGN emission, so the extended emission was subtracted. The PSF of Centaurus A at these wavelengths contributes ∼ 35 %. This can be interpreted as the nucleus having a relatively constant IR contribution, so the brightness of the nucleus coincides with IR brightness of the host galaxy. The PSF contribution of the Circinus Galaxy decreases between 53 and 160 𝜇 m from 58 % to 35 %. At longer FIR wavelengths, the contribution of the PSF appears to be from the host galaxy and the fitting no longer provides information about the AGN. We interpret this as a decreasing IR contribution from the nucleus compared to the extended emission. For the completeness of the SOFIA Atlas presented here, we used the 19-53 𝜇 m images of NGC 1068 from LopezRodriguez et al. (2018). These datasets were analyzed as described in that study and here we only present the results and images in that wavelength range. The study showed that the fractional contribution from star formation increases from 20 - 50 𝜇 m, while extended emission from 200 K dust decreases. Emission from the torus peaks in this range, a result which is in agreement with Fuller et al. (2016) who found that the turnover in torus emission occurs at wavelengths ¿ 31.5 𝜇 m. Extended emission is observed here at all wavelengths ≳ 70 𝜇 m arising from dust in the host galaxy and star formation regions. For NGC 4388, we show the results of PSF subtraction at all wavelengths, but only use the results in wavelengths ¿ 40 𝜇 m for the SED. In the 30 - 40 𝜇 m range, the NE to SW extension is clear in the PSF-subtracted images. However, almost all of the extended emission lies within the FWHM of the observation; the FWHM of these images are only ∼ 10% greater than the FWHM of the PSF. Thus, while we show the PSF subtracted images of NGC 4388 here, for the SED we use the total 30 - 40 𝜇 mfluxes which encompass the apparent extended emission due to the NLR. The change in the extended emission source and morphology between 40 and 70 𝜇 m is clear in the PSF-subtracted images (Figure 6). The host galaxy clearly dominates the extended emission in the FIR while the NLR region dominates the extended emission in MIR wavelengths. The 53 𝜇 m HAWC+ image appears to show the transition between dominant extended sources. The contribution of the PSF in the images of NGC 4388 is ∼ 60 % in the 30 - 40 𝜇 m range, where the extended emission is in the NE to SW direction. The contribution then decreases drastically to ∼ 20 %. This reflects the turnover in extended emission seen in the images in Figure 5. The contribution of the PSF returns to ∼ 70 % between 70 - 100 𝜇 m, which suggests two separate but significant IR emission sources.\",\n \"pages\": [\n 13,\n 14\n ]\n },\n {\n \"title\": \"5. SPECTRAL ENERGY DISTRIBUTIONS\",\n \"content\": \"Tables 5 (SOFIA) and 6 ( Herschel ) give the nuclear fluxes of the AGN in our sample along with their associated errors in units of Jy. The sources of uncertainty here are the instrument calibration, sky background, and the 2D gaussian fitting, where applicable. We estimate FORCAST and HAWC+ errors at ∼ 10%. We use PACS instrument errors at 5% /four.sup and SPIRE instrument errors as 5.5% /five.sup . The uncertainty due to sky background is determined on an individual basis, but averages ∼ 5%. The average uncertainty due to the 2-D gaussian fitting is ∼ 1.5%. We add these uncertainties in quadrature for the error bar estimation. The nuclear SEDs are shown in Figure 7 in 𝜈 F 𝜈 . The pink diamonds represent SOFIA observations while the black circles represent the complementary Herschel data. We obtained Spitzer /IRS spectra from the Sptizer /CASSIS database (Lebouteiller et al. 2011) for 21 of the 22 objects in our sample (solid black line). There was no spectrum available for NGC 7465. Low-resolution spectra (R ∼ 100) were obtained for 18 of the objects, while moderate resolution spectra (R ∼ 600) were available for Circinus, NGC 1068, and NGC 7674. This dataset provides the most completed SED coverage available between 30 - 500 𝜇 m. Decomposing the SEDs in this sample is outside the scope of this manuscript, as we are presenting an imaging atlas. Here, we provide the main results and features of the SEDs of these objects. The morphological changes seen in the extended emission source in Figure 5 for NGC 4388 are reflected in the SED at 53 𝜇 m, where there is a marked decline in the SED. The drastic decrease seems to be due the change of dominant emitting sources. The extended emission at wavelengths ≲ 40 𝜇 m is due to dust in the direction of the radio axis, and the extended emission at wavelengths ≳ 50 𝜇 mis due to the host galaxy. The wavelength of peak emission can give insight to the primary processes that drive MIR emission. The peak wavelength, determined by the highest flux from photometry and spectroscopy, ranges from 18 to 100 𝜇 m in 𝜈 F 𝜈 with an average of ∼ 40 𝜇 m. This average only includes the peak in continuum values and does not take into account fine structure lines. Most (73%; 11 out of 15) Seyfert 2 have a peak emission at wavelengths ≲ 40 𝜇 m. The SEDs of MCG-5-2316, Mrk3, Mrk 573, NGC 1068, NGC 3081, and NGC 4151 peak at ∼ 18 - 20 𝜇 m. This is in agreement with the correlation peak between the hard-X-rays and the mid-IR for Type 1 AGN in Garc'ıa-Bernete et al. (2017). The peak wavelengths in 𝐹 𝜈 (Jy) range ∼ 20 - 160 𝜇 m, with an average ∼ 93 𝜇 m. NGC 1068 is the only AGN to peak at the same wavelength in both sets of units. The 𝑆𝑝𝑖𝑡𝑧𝑒𝑟 spectrum for NGC 1068 does not align with the SOFIA photometry because of the extensive PSF subtraction that we performed in the photometry that was not accounted for in the spectroscopy. This is the only object that not only has overlapping 20 - 40 𝜇 m 𝑆𝑝𝑖𝑡𝑧𝑒𝑟 and SOFIA data, but also that has had the background emission subtracted at these wavelengths.\",\n \"pages\": [\n 14\n ]\n },\n {\n \"title\": \"5.1. Luminosity and Peak Wavelength\",\n \"content\": \"To test whether the peak wavelength is a function of luminosity, we plot L 𝑏𝑜𝑙 vs 𝜆 𝑝𝑒𝑎𝑘 . Figure 8 shows the bolometric luminosities of the AGN plotted against the peak wavelength in the SEDs for both Sy1s, shown as red triangles, and Sy2s, shown as purple stars. The correlation coefficient between the luminosity and peak wavelength is | 𝑅 | ∼ 0.63 with statistical significance 𝑝 = 0 . 0015. While it is argued that | 𝑅 | of 0.6 - 0.7 may show moderate to strong correlation (see Section 3.2 in Messenger et al. 2013), a 𝑝 -value ≤ 0.05 is generally accepted as statistically significant. The data suggests that higher luminosity objects have SEDs that peak at shorter wavelengths, which indicates the presence of a hot dust component in the vicinity of the AGN.\",\n \"pages\": [\n 14,\n 15\n ]\n },\n {\n \"title\": \"5.2. Mid- to Far-IR Colors\",\n \"content\": \"The ratio of F 𝜈 (70)/F 𝜈 (160) has been used as a proxy for dust temperature (Mel'endez et al. 2014; Garc'ıa-Gonz'alez et al. 2016), where the ratio is higher for dust heated by the AGN and lower for dust heated by star formation. Here we perform this analysis using the ratio F 𝜈 (31)/F 𝜈 (70) by using the 31.5 𝜇 mSOFIA data in our atlas. For objects that do not have data in the 31.5 𝜇 mfilter, we supplement that with data from the Spitzer /IRS continuum. NGC 7465 did not have 31.5 𝜇 m flux data, nor did it have Spitzer data so we leave that object out of this analysis. Using the fluxes in Table 5, we plot a color-color diagram in F 𝜈 in Figure 9. This figure also visually shows the peak wavelength from the SEDs (in 𝜈𝐹 𝜈 ). Longer peak wavelengths tend to cluster at F 𝜈 (70)/F 𝜈 (160) ∼ 1 and F 𝜈 (31)/F 𝜈 (70) between 0.25 - 0.5. In this sample we find an average F 𝜈 (70)/F 𝜈 (160) ratio of 1 . 4 ± 0 . 7. Previous studies (Mel'endez et al. 2014; Garc'ıaGonz'alez et al. 2016) with larger sample sizes (313 and 33, respectively) have found an average ratio of ∼ 0.8, albeit the data was analyzed using independent methods. This suggests a higher amount of AGN heated dust in our sample. We find that 18 objects have F 𝜈 (31)/F 𝜈 (70) ¡1, with an average F 𝜈 (31)/F 𝜈 (70) of 0 . 6 ± 0 . 3. The only object with both F 𝜈 (70)/F 𝜈 (160) and F 𝜈 (31)/F 𝜈 (70) ¿1 is MCG-5-23-16, and an SED that peaks ∼ 20 𝜇 m. This object may be the most AGN dominated source in our sample. The other objects that show a peak at ∼ 20 𝜇 min 𝜈 F 𝜈 still show F 𝜈 (31)/F 𝜈 (70) ¡1. Only one object, NGC 4388, shows F 𝜈 (31)/F 𝜈 (70) ¿1 while F 𝜈 (70)/F 𝜈 (160) ¡1. This reflects the change in extended emission seen in Figure 5. Half (11) of the objects in the sample show ratios F 𝜈 (31)/F 𝜈 (70) ¡1 while F 𝜈 (70)/F 𝜈 (160) ¿1. These objects (Circinus, Mrk 231, Mrk 573, NGC 1275, NGC 2110, NGC 3081, NGC 3227, NGC 3281, NGC 4151, NGC 4941, NGC 5506, NGC 7469) are likely AGN dominated. Six objects show F 𝜈 (31)/F 𝜈 (70) ¡1 and F 𝜈 (70)/F 𝜈 (160) ¡1, meaning that their SEDs peak at longer wavelengths. The emission from R/e.pc/f.pc/e.pc/r.pc/e.pc/n.pc/c.pc/e.pc/s.pc: a) Fuller et al. (2016), b) Fuller et al. (2019), c) Lopez-Rodriguez et al. (2018), d) Lopez-Rodriguez et al. (2020), e) LopezRodriguez et al. (2022b). *Lopez-Rodriguez et al. (2018) measured the flux of NGC 1068 at 19.7 𝜇 mto be 22.0 ± 1.4. these objects (Centaurus A, NGC 1068, NGC 2273, NGC 2992, NGC 4258, NGC 7674) are likely dominated by star formation.\",\n \"pages\": [\n 15,\n 16\n ]\n },\n {\n \"title\": \"6. CONCLUSIONS\",\n \"content\": \"We have presented a SOFIA atlas of nearby AGN in the 20 - 215 𝜇 m wavelength range using FORCAST and HAWC+. We have released 69 observations of which 41 are newly published and 28 have been previously published (Fuller et al. 2016, 2019; Lopez-Rodriguez et al. 2018, 2022a). From these observations, NGC 4388 shows the most dramatic visual change in emission morphology. The 30 - 40 𝜇 m images show a NE to SW dusty extension associated with the NLR, while the ¿ 50 𝜇 mimages show a East to West dusty emission associated with the plane of the host galaxy. Our observations show that < 10 '' resolution 30 - 70 𝜇 m observations are crucial to disentangle the emitting contribution from AGN and host galaxy. We measured arcsecond scale unresolved nuclear fluxes in order to construct SEDs of the objects in our sample. We included complementary Herschel data to cover up to 500 𝜇 m. For point sources we used aperture photometry to determine the flux. For extended sources we used a 2D gaussian fitting method to extract the central unresolved source(s) of emission from the galaxy background. For this method, the PSF is scaled to represent the central emission while a 2D gaussian represents host galaxy or background emission. Based on the SEDs, we make the following conclusions: In future studies, we will combine data from this atlas with incoming data from JWST to update our IR datasets with the latest and highest resolution data available. Newly obtained JWST/MIRI observations will provide new higher angular resolution data for some of the sources in the wavelength range 5 - 28 𝜇 m.\",\n \"pages\": [\n 16,\n 17\n ]\n },\n {\n \"title\": \"ACKNOWLEDGMENTS\",\n \"content\": \"We acknowledge Dr. Lucas Grosset for his effort in subtracting the image backgrounds. E.L.-R. is supported by the NASA/DLR Stratospheric Observatory for Infrared Astronomy (SOFIA) under the 08 0012 Program. SOFIA is jointly operated by the Universities Space Research Association,Inc.(USRA), under NASA contract NNA17BF53C, and the Deutsches SOFIA Institut (DSI) under DLR contract 50OK0901 to the University of Stuttgart. E.L.-R. is supported by the NASA Astrophysics Decadal Survey Precursor Science (ADSPS) Program (NNH22ZDA001NADSPS) with ID 22-ADSPS22-0009 and agreement number 80NSSC23K1585. I.G.B. acknowledges support from STFC through grants ST/S000488/1 and ST/W000903/1. C.R. acknowledges support from Fondecyt Regular grant 1230345 and ANID BASAL project FB210003. Software: /a.pc/s.pc/t.pc/r.pc/o.pc/p.pc/y.pc (Astropy Collaboration et al. 2022, 2018, 2013); S/c.pc/i.pcP/y.pc Virtanen et al. (2020)\",\n \"pages\": [\n 17\n ]\n },\n {\n \"title\": \"REFERENCES\",\n \"content\": \"Alonso-Herrero, A., Ramos Almeida, C., Mason, R., et al. 2011, ApJ, 736, 82, doi: 10.1088/0004-637X/736/2/82 Alonso-Herrero, A., Ramos Almeida, C., Esquej, P., et al. 2014, MNRAS, 443, 2766, doi: 10.1093/mnras/stu1293 Alonso-Herrero, A., Esquej, P., Roche, P. F., et al. 2016, MNRAS, 455, 563, doi: 10.1093/mnras/stv2342 Alonso-Herrero, A., Pereira-Santaella, M., Garc´ıa-Burillo, S., et al. 2018, ApJ, 859, 144, doi: 10.3847/1538-4357/aabe30 Alonso-Herrero, A., Garc´ıa-Burillo, S., H¨onig, S. F., et al. 2021, A&A, 652, A99, doi: 10.1051/0004-6361/202141219 Antonucci, R. 1993, ARA&A, 31, 473, doi: 10.1146/annurev.aa.31.090193.002353 Antonucci, R. R. J., & Miller, J. S. 1985, ApJ, 297, 621, doi: 10.1086/163559 Asmus, D. 2019, MNRAS, 489, 2177, doi: 10.1093/mnras/stz2289 https://arxiv.org/abs/2207.09466 doi: 10.1088/0004-637X/794/2/152 Mor, R., Netzer, H., & Elitzur, M. 2009, ApJ, 705, 298, Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261, doi: 10.1038/s41592-019-0686-2 Young, L. M., Meier, D. S., Bureau, M., et al. 2021, ApJ, 909, 98, doi: 10.3847/1538-4357/abe126 Yuan, F., Markoff, S., Falcke, H., & Biermann, P. L. 2002, A&A, 391, 139, doi: 10.1051/0004-6361:20020817 Zhang, L., & Ho, L. C. 2023, ApJL, 953, L9, doi: 10.3847/2041-8213/acea73\",\n \"pages\": [\n 17,\n 18,\n 20,\n 21\n ]\n },\n {\n \"title\": \"A. HERSCHEL IMAGES\",\n \"content\": \"We used images from the Herschel Archive to supplement SOFIA data in FIR wavelengths. Images from the PACS and SPIRE instruments covering the wavelength range 70 - 500 𝜇 m are shown in Figures 10, 11, 12, and 13. The white circle in the upper right indicates the beam size while the pink scale in the bottom left indicates a distance of 1 kpc. To extract nuclear fluxes on scales of several arcseconds, we use the methods described in Section 4. For point sources, we performed aperture photometry. For extended sources, we used the 2D gaussian routine outlined in Section 4.1.\",\n \"pages\": [\n 22\n ]\n },\n {\n \"title\": \"B. BACKGROUND FITTING\",\n \"content\": \"Figure 14 shows an example of the 2-D gaussian fitting we used to extract the central flux from the background of the host galaxy in wavelengths 30 - 500 𝜇 m. The object used here is NGC 4388. In the first column, the original observation image is shown. The second column shows a model of the image using a PSF (Column 3) combined with the model background. Column 4 shows a PSF-subtracted image (Column 1 - Column 3) used to make the model background (Column 5). Column 6 shows the PSF- and background- subtracted image of the residuals.\",\n \"pages\": [\n 23\n ]\n },\n {\n \"title\": \"C. HOST GALAXY BACKGROUNDS HERSCHEL\",\n \"content\": \"Here we show the results of PSF-subtracted images for objects in which only Herschel data showed host galaxy contamination.\",\n \"pages\": [\n 23\n ]\n },\n {\n \"title\": \"F/u.pc/l.pc/l.pc/e.pc/r.pc /e.pc/t.pc /a.pc/l.pc.\",\n \"content\": \"19'00 50'00 52'30\\\" 51'30\\\" 50'40\\\" 51'00\\\" 52'00\\\" 24 30\\\" 73'30\\\" RA (J2000} (2000) 19'00 18'30 34*50' RA U2000) 19*53' 2*48' RA (J2000) RA (J2000) RA (J2000) 5*32 12'00\\\" 13'00\\\" RA (2000) RA (J2000) RA (J2000) RA (J2000) RA (J2000) RA (2000) 53'00 52'40\\\" 51'40\\\" 8*54' 8*54' 8*48' 8*48' 8*48 8*54'\",\n \"pages\": [\n 24,\n 25\n ]\n }\n]"}}},{"rowIdx":4976,"cells":{"bibcode":{"kind":"string","value":"2024arXiv241119306P"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/2411.19306.pdf"},"content":{"kind":"string","value":"\nMethods to Characterise Exoplanet Host Stars from Spectroscopy\nCarina M. Persson\nAbstract A key to understand exoplanets is characterisation of their host stars. One of the most powerful tools to characterise stellar properties like effective temperature, surface gravity and metallicity, is spectroscopy based on observations of stellar atmospheres. This chapter describes the stellar parameters that can be derived from a spectrum with examples of well established methods and theoretical model atmospheres. Combined with photometry and parallax measurements, the outcome of the spectroscopic modelling can be used to derive stellar radii and masses.\nIntroduction\nExoplanets are intimately connected to their host stars through formation and evolution. In addition, detection and characterisation of exoplanets depend on detailed knowledge of their host stars since the current major detection techniques, transit photometry and the radial velocity (RV) method, detect planet sizes and masses relative to their host star (see Chapters by Deeg & Alonso and Wright in Volume 1 of the Handbook of Exoplanets). Uncertainties in a host star's parameters propagate directly to the planets.\nStellar modelling is normally based on two major techniques - photometry and spectroscopy. These methods are model-dependent in contrast to the direct measurements by interferometry, eclipsing binaries, and asteroseismology.\nFor eclipsing binaries (e.g. Andersen 1991; Torres et al. 2010; Serenelli et al. 2021), and for the few large and nearby stars that enable interferometric measurements (e.g. Quirrenbach 2001; Eisenhauer et al. 2023), it is possible to derive the stellar radius with an accuracy of a few per cent. The timing of the duration of the eclipses of eclipsing binaries and their orbital velocities allows accurate estimates\nChalmers University of Technology, department of Space, Earth, and Environment, Onsala space observatory, 439 92 ONSALA, Sweden, e-mail: carina.persson@chalmers.se\nof their sizes. Similarly, stellar masses can be accurately determined for visual binaries from observed separations from the common centre-of-mass with Kepler's third law that relates their masses with observed orbital period and separation. Asteroseismology can also be used to derive stellar mass, radius, and age with a high precision (see Chapter by Lundkvist, Huber, Aguirre & Chaplin in this volume of the Handbook of Exoplanets). Recently, the seismic surface gravity of the star has also been used to obtain the effective temperature and metallicity, in particular with APOGEE (Ahumada et al. 2020, 2022).\nThese methods are, however, currently only possible to apply to a small subset of all stars. Spectroscopic measurements open a window to derive stellar parameters from a larger pool of stars than from direct methods. High-resolution spectroscopy is a powerful tool that provides a wealth of information; the effective temperature, surface gravity, chemical composition, and velocities. The stellar radius, and the luminosity, can then readily be derived from its spectral energy distribution (SED) via the spectroscopic parameters combined with photometry and parallax.\nThe spectroscopic parameters also serve as a base to model characteristics that normally cannot be directly measured, like mass and age, with stellar evolution models and the complementary tool isochrones (e.g. Dotter 2016; Hidalgo et al. 2018). An isochrone is an evolutionary track for a population of stars with different masses on the Hertzsprung-Russell diagram with the same ( iso ) age ( chrone ). The mass and radius can also be obtained from the spectroscopic parameters and empirical calibration equations (e.g. Torres et al. 2010; Enoch et al. 2010; Southworth 2011) albeit often with a higher uncertainty.\nThe downside of high-resolution spectroscopy is that it is expensive; the observations are time-consuming and also requires bright stars or a large collecting area. Modelling of stellar parameters has therefore traditionally been performed by photometry because a much larger number of stars can quickly be observed and analysed. The effective temperature of a star can for instance be derived from color-color diagrams which have been scaled to stars measured with direct methods (e.g. Bell and Gustafsson 1989; Alonso et al. 1996; Nordstrom et al. 2004). However, such models have in general higher uncertainties than based on spectroscopic measurements. In that respect they may only be reliable in a statistical sense and not for individual stars. Thus for exoplanet host stars (for which direct measurements are not applicable), high-resolution spectroscopy is preferred.\nThis chapter begins with a short summary of a few basic requirements in order to obtain spectroscopic measurements of a star. It continues with an overview of the parameters that can be extracted from a high-resolution spectrum and a few examples of well established methods. Some advantages and caveats are highlighted. The chapter ends with a brief summary of how to combine spectroscopic parameters, photometric measurements, and stellar evolution models to obtain stellar radii and masses. Modelling of stellar ages are described by Christensen-Dalsgaard & Aguirre in a Chapter in this volume of the Handbook of Exoplanets.\nInstruments and spectral resolution\nImportant properties of a spectrum is high-resolution, high signal-to-noise (S/N), and wavelength coverage. There are several different types of spectrometers and many textbooks have been written about this topic including advantages and challenges for different types of instruments (e.g. Gray 2008). The most successfull type in observations of exoplanets is echelle spectrographs. The main advantages is the high resolution combined with a wide wavelength coverage obtained in a single exposure. A description of high-precision cross-dispersed echelle spectrographs for exoplanet research (CORAVEL, ELODIE, CORALIE, SOPHIE and HARPS) is found in a Chapter by Fransesco Pepe in this volume of the Handbook of Exoplanets.\nFor the purpose of spectroscopic modelling of host stars, we want high-resolution in order to resolve the spectral lines. The spectral resolution ∆λ at wavelength λ is\n∆λ λ = 1 R , (1)\nwhere R is the resolving power of the spectrograph. It can be translated into a velocity resolution according to\n∆ V = ∆λ λ c = c R . (2)\nDoppler broadening of spectral lines due to thermal and turbulent motions of absorbing species in the atmospheres produce line widths of ∼ 6 kms -1 for latetype stars. This corresponds to a spectral resolution of 50 000. Low-mass, slowly rotating stars can have line widths of only ≈ 1 -2 km s -1 which require R ≳ 300000 to resolve the spectral lines and disentangle blended lines.\nIn terms of high resolution, ultra-high precision, and long-term stability, the High Accuracy Radial Velocity Planet Searcher (HARPS; Mayor et al. 2003) and its decade younger sibling HARPS-North (Cosentino et al. 2012) mounted on the ESO 3.6 m telescope (La Silla observatory, Chile) and the Telescopio Nazionale Galileo (TNG) of Roque de los Muchachos Observatory (La Palma, Spain), respectively, have been the leading instruments in detecting exoplanets over the last two decades. Both HARPS instruments are fiber-fed cross-dispersed high-precision echelle spectrographs covering 380 nm to 690 nm. The spectroscopic resolution is approximately 115000 at visual wavelengths corresponding to a velocity resolution of 2.6 km s -1 . The design is based on experience with the groundbreaking ELODIE and CORALIE instruments where the former was used to detect the first exoplanet 51 Peg b (Mayor and Queloz 1995). Both HARPS spectrographs can be considered as the 'gold standard' when searching for exoplanets in RV data and are also used for characterisation of exoplanet host stars. When searching for exoplanets with the RV method, many measurements are collected, sometimes over a period over many years. The individual spectra can be co-added (after correcting for the peri-\nodic changes in radial velocity) in order to increase the S/N enabling spectroscopic characterisation of the host star.\nIn addition to the HARPS spectrographs, there are many other instruments used for exoplanet detection that can also be used to characterise host stars e.g. ESPRESSO (Pepe et al. 2021), FIES (Telting et al. 2014), HIRES (Vogt et al. 1994), CHIRON (Tokovinin et al. 2013), TULL (Tull et al. 1995) with different resolutions operating at different wavelengths.\nStellar properties from spectroscopy\nOnly a few parameters characterise a stellar atmosphere: the effective temperature ( T eff ), the surface gravity (log g ⋆ ), the overall metal abundance ([M/H]), and atmospheric and rotational velocities.\nThe surface of a star is defined as the location where photons escape from the star which occurs at a characteristic optical depth of 2/3. This occurs within the photosphere, the innermost ≈ 500 km of a star's atmosphere which overlies the opaque interior. The photospheric temperature and density varies with depth and depends on the surface gravity as well as the abundances and opacity of the gases. The Sun's photosphere has a temperature that varies between 4 400 K and 6 600 K with an effective temperature of 5 772 K, while the density is approximately 3 × 10 -4 kg m -3 , increasing with depth into the Sun. Other stars may have hotter or cooler photospheres. Spectral absorption lines originate from different depths and opacities within the photosphere. Weak and optically thin spectral lines, and the line wings of optically thick lines, originate essentially in the same layer as the continuum. In contrast, the cores of optically thick saturated lines develop in higher layers. An example is the core of the hydrogen α line at 6562.81 ˚ A which originates in the hotter and less dense chromosphere which lies on top of the photosphere, where the assumption of local thermal equilibrium (LTE) is no longer valid.\nEffective temperature\nStars are classified according to effective temperature from the hot O-stars, also called early-type stars, to the cool M-stars (late-type stars). If a star is on the main sequence, the effective temperature immediately signals which type of star it is along with typical mass and radius.\nInstead of choosing a particular depth to define a star's surface temperature, the effective temperature is defined in terms of flux. The effective temperature is defined via the Stefan-Bolztmann law in terms of the total power per unit area, radiated by the star (e.g. Gray 2008)\n∫ ∞ 0 F ν d ν = σ T 4 eff . (3)\nHere F ν is the total flux passing through the star's surface and σ is the Boltzmann constant. The effective temperature is thus the temperature of a black body having the same power output per unit area as a star. It is related to the flux we measure at Earth, f ν , via (Gray 2008)\n∫ ∞ 0 f ν d ν = ( R ⋆ d ) 2 ∫ ∞ 0 F ν d ν = ( θ ⋆ 2 ) 2 σ T 4 eff , (4)\nwhere the left integral is the total radiative flux from the star received at the top of the Earth's atmosphere (bolometric flux), R ⋆ is the stellar radius, d is the distance, and θ ⋆ is the angular diameter of the star. The effective temperature can hence be derived by measuring the angular size of the star from interferometry and the received flux at Earth over a wide spectral range. By combining the angular size with distance from parallax measurements, the linear radius of the star can be inferred for a wide range of spectral types nearly model-independent except for the dependence on the adopted limb-darkening coefficients and bolometric correction. Alternatively, if the radius of the star is known from e.g. eclipsing binaries and the distance from parallax measurments, this will give the angular size which then can be used to compute T eff . Since the target stars have to be nearby to measure their angular sizes, interstellar absorption can be neglected. The infrared flux method (IRFM Blackwell and Shallis 1977; Blackwell et al. 1980; Bell and Gustafsson 1989; Blackwell et al. 1990; Casagrande et al. 2006, 2010) is based on observations of the angular size of the star and the measured infrared flux at the top of the Earth's atmosphere. The bolometric flux is derived taking into account the bolometric correction which gives the effective temperature.\nIn addition to the above methods, T eff can also be derived from spectroscopy as described in the following section. However, it is worth highlighting that the temperature derived from spectroscopy is a microscopic value which is close to, but not exactly the same as the effective temperature due to its definition being a macroscopic description.\nSurface gravity\nThe surface gravity is an indication of the luminosity class of a star where V is the main sequence and I - IV is different types of giants. Thus surface gravity contains information of the size and age of a star. A low surface gravity immediately indicates that the star has left the main sequence.\nThe surface gravity of a star is defined by (e.g. Gray 2008)\ng ⋆ = g ⊙ M ⋆ R 2 ⋆ , (5)\nwhere g ⊙ is the surface gravity of the Sun (2 . 740 × 10 4 cm s -2 ) and the radius, R ⋆ , and mass, M ⋆ , of the star are in solar units. The surface gravity is commonly measured in a logarithmic scale, log g ⋆ , where the solar value is 4.44.\nThe surface gravity determines the gas density in the photosphere and is the spectroscopic parameter that has the highest impact on the stellar radius. Unfortunately, log g ⋆ is often poorly constrained by spectral analysis. Since the uncertainties of T eff and metallicity can be strongly correlated with surface gravity this can lead to a significant source of systematic error in some analysis techniques.\nMetallicity\nA stars chemical composition is an outcome of the nucleosynthesis by previous generations of stars. This is important when reconstructing star and planet formation history in our Galaxy. There is a large variation of metallicity, i.e. the abundance of all elements heavier than helium denoted with [M/H], in the Milky Way up to about twice the solar value to hundreds of thousands of times lower than the solar value (e.g. Christlieb et al. 2004; Li et al. 2022; Nepal et al. 2024).\nThe chemical composition of a star is commonly fixed to the overall metallicity of a star relative to the Sun. The Grevesse et al. (2007), Asplund et al. (2009) and Lodders (2003) abundance scales for the Sun are currently the most adopted. However, individual abundances of a star may not follow solar composition and may require modelling of individual elemental abundances. Abundances are generally measured on a logarithmic scale normalised to the Sun where zero equals the Sun's metallicity. In the case of iron we have [Fe/H] ⋆ = log(Fe/H) ⋆ - log(Fe/H) ⊙ . For example, an iron abundance of [Fe/H] = -0 . 5 means that the abundance is 10 -0 . 5 relative to the Sun. Since iron is by far the most abundant species in a stellar atmosphere after hydrogen and helium, measurements of iron has become a proxy for the metallicity.\nStellar abundances is also important when modelling exoplanet interiors in particular rocky super-Earths without significant gaseous envelopes. The degeneracy of interior composition inferred from radius and mass measurements can for this type of planet be reduced assuming an interior structure with a differentiated iron core and a rocky mantle. In these cases, the host star abundances is often used as a proxy of the primary planet-building elements Fe, Mg, and Si, which are expected to be reflected in the planet composition, planet interior, and core mass fraction (Dorn et al. 2015; Acu˜na et al. 2023).\nVelocities\nThermal widths of spectral lines are only a fraction of the observed line widths for dwarf stars hotter than spectral type K0. The line widths are instead mainly governed by Doppler shifts produced by motions of the star's photospheric gases. The radial velocity of the star is only shifting the wavelengths of all spectral lines in the observed spectrum compared to the observations and can easily be corrected for. The velocity that dominates the line shape and width for hot stars is the projected equatorial rotational velocity of the star, V sin i ⋆ , where i ⋆ is the inclination of the stellar rotation axis relative to the line of sight. It can be measured via the full width\nat half maximum (FWHM) of a large number of optically thin and unblended lines not sensitive to pressure broadening. The line shapes are, however, also affected by turbulence from convective motion, granulation, high-order pulsations, stellar activity, and other types of local flows in the photosphere. Turbulence is represented in the models by the macro-turbulent velocity ( V mac) that describes motions on scales larger than the mean free path within the photosphere that induce a change in the line shape; and the micro-turbulent velocity ( V mic ) that describes motions on scales smaller than the mean free path leading to increased line opacity (Gray 2008; Bruntt et al. 2010; Doyle et al. 2014). The latter velocity is a 'fudge' factor originally introduced to reconcile observed and predicted equivalent widths (Mihalas 1978). It includes all remaining types of broadening mechanisms and is at present standard to include in analyses of solar-type stars. Both turbulent velocities depend on temperature and to a lesser extent on surface gravity. The micro-turbulent velocity has a width of the order of 1 km s -1 for dwarfs and several km s -1 for giants. For lowmass stars, V mac and V sin i ⋆ have comparable widths of the order of a few km s -1 . As a reference, the Sun's V sin i ⋆ is 2 km s -1 at the equator while hotter stars have much higher rotational velocities (tens to hundreds of km s -1 ).\nMethods for spectroscopic modelling\nThere are several ways to model a spectrum which can be divided into two main groups. The first is based on spectral synthesis. Here observations are fitted to a synthetic spectrum of stellar atmosphere models by comparison of line profiles. The second is a line-by-line analysis based on measured strengths of observed spectral lines, their equivalent widths (EWs). Detailed description of the physics can be found in many textbooks e.g. Gray (2008). Spectroscopic observations can also be compared to a library of spectra of well-characterised stars via for example interferometric measurements or spectroscopic binaries.\nOne major problem in spectroscopic analysis is to accurately determine the continuum which can introduce large errors. This is particularly difficult for poor spectra with low spectral resolution or low S/N. It also depends on the spectral type of the star and the wavelength region. The number of spectral lines increases towards shorter wavelengths for all types of stars. In addition, late-type stars have a much higher density of spectral lines than early-type stars arising from both atoms and molecules leading to blending and confusion of the continuum location. The higher temperatures of early-type stars ionize a large fraction of their atoms, leading to significantly fewer spectral features than low-mass stars. Differences in rotational velocities also affect the spectral line density. In contrast to late-type stars, the early types have very high rotational velocities which leads to very broad spectral lines that smear out spectral features. Thus solar-type stars (FGK) are the easiest stars to model, while high- and low-mass stars often entails a significantly higher degree of difficulty in the modelling. M-dwarfs have in addition generally a much longer period of high stellar activity than FGK stars, exacerbating the problems (e.g. Mignon\net al. 2023). This is unfortunate since M-dwarfs are popular exoplanet host stars due to their small masses and sizes which increase the exoplanet signals.\nCare must be taken when selecting which spectral lines to model. A large set of narrow, non-blended spectral lines are preferred (unless modelling pressure broadened line wings, see below). If a spectral line becomes optically thick, the abundance of a species stops growing linearly with absorption depth. The characteristics of an optically thick line is a saturated line centre which flattens the bottom and broaden the line wings. Not all optically thin spectral lines may, however, be useful since a large number comes with poorly determined atomic parameters which are needed to compute synthetic spectra. This can be circumvented for solar-type stars if adopting new atomic parameters after comparing the lines from observations of the Sun.\nFitting observations to synthetic spectra\nComputations of a synthetic spectrum requires a model atmosphere based on solutions to the stellar structure equations to synthesize a spectrum. Most stellar atmosphere models are pre-calculated and tabulated on grids describing the profiles of the temperature, surface gravity and abundances as functions of atmospheric depth. Each layer in the model atmosphere is contributing to the formation of absorption line profiles in the final spectrum. Some widely used atmospheric model atmospheres are Atlas12 (Kurucz 2013), Atlas9 (Kurucz 1993a; Heiter et al. 2002), MARCS (Gustafsson et al. 2008) for cool and giant stars, and LL models (Shulyak et al. 2004) for hot main sequence stars. Line lists of atomic and molecular data needed in the computation can be provided by the Vienna Atomic Line Database (VALD3; Piskunov et al. 1995; Ryabchikova et al. 2015).\nA radiative transfer code then computes a synthetic model spectrum of the star for the chosen set of stellar parameters ( T eff , log g ⋆ , [Fe/H], V sin i ⋆ , V mic , V mac) which are matched against the observed spectra based on the spectral line shapes and strengths. The parameters generally affect either the line strength ( T eff , log g ⋆ , abundances, and V mic ) or the line shape ( V mac, V sin i ⋆ , and the instrumental resolution). Degeneracies are stronger within the subsets.\nThe dependence of the line profile on the V mac parameter is broadened line wings and a cusp-shaped core. Unfortunately, disentangling the effect on the line profile from V sin i ⋆ and V mac is difficult, leading to a degeneracy between the two. Prior information of V sin i ⋆ may be obtained from time-resolved photometry, available for the known transiting planets, or from asteroseismology (cf. Doyle et al. 2014). If no prior information is available, calibration equations of both turbulent velocities are often used (e.g. Bruntt et al. 2010; Doyle et al. 2014) which allows modelling of V sin i ⋆ . Note that in order to properly model the different velocities above, it is imperative to take into account how the spectrograph itself broadens the lines. Spectral lines can also be pressure broadened through various mechanisms which further broaden the lines, e.g. the Balmer lines. Details of the various broadening mechanisms can be found in many textbooks such as Gray (2008).\nThere are many softwares that computes synthetic spectra to be used as a model constraint to interpret the observed spectrum. The popular open-source spectroscopic tool iSpec (Blanco-Cuaresma et al. 2014; Blanco-Cuaresma 2019) support several of the most well-known radiative transfer codes such as Spectrocsopy Made Easy ( SME ; Valenti and Piskunov 1996; Piskunov and Valenti 2017; Wehrhahn et al. 2023), SPECTRUM (Gray and Corbally 1994), Turbospectrum (Plez 2012; de Laverny et al. 2012; Gerber et al. 2023), Synthe/WIDTH9 (Kurucz 1993b), and MOOG (Sneden 1973) where the latter is based on the equivalent width method. The codes often adopt LTE and a plane-parallel geometry as default motivated by the fact that for main-sequence stars, the photosphere constitutes ≪ 1 % of the stellar radius. Aspherically symmetric geometry is on the other hand required for giant stars where the atmosphere makes up a substantial portion of its radius (Heiter and Eriksson 2006). However, since complex interaction of gas particles and the non-local radiation fields leads to deviations from LTE in the atmospheres of FGKM-type stars, some of the softwares have also the option of non-local thermodynamic equilibrium (NLTE). For instance, SME includes NLTE departure coefficients for the MARCS and LL models atmospheres (Piskunov and Valenti 2017), and Turbospectrum also have the option of NLTE (Gerber et al. 2023).\nFitting can in several of the softwares be made for one or several parameters at the same time using a χ 2 -minimization algorithm. However, care must be taken when solving for several parameters simultaneously (Torres et al. 2012) due to degeneracies and difficulties in the modelling. Different initial assumptions of one free parameter at a time can therefore be made in the fitting process to iterate to the final solution, thereby mitigating degeneracies. Also, since the surface gravity is often difficult to constrain it is thus advantageous to have external information about the stellar density that can facilitate modelling.\nSpectroscopic modelling can take advantage of the sensitivity of certain spectral lines to specific parameters. For instance, all Balmer lines exhibit pressure (collisionally) broadened line wings which makes their profiles strongly sensitive to temperature (e.g. Fuhrmann et al. 1993, 1994; Barklem et al. 2000, 2002) as shown in Fig. 1. The Balmer line wings are formed in the deepest photospheric layers likely close to the LTE. The metal-line blending increase from H α at 6562.81 ˚ A to H δ at 4101.75 ˚ A which perturb the higher transitions of the Balmer line profiles making H α the best choice. For late F, G, and early K-stars, H α is very insensitive to log g ⋆ and abundance making its line wings an excellent T eff indicator. Two examples of H α line profiles towards a K5V and a G0V host star are shown in Fig. 1. It is obvious that the line wings have almost completely disappeared in the K5V spectrum, while prominent and very broad towards the G-star. For M-dwarfs where the line wings of H α are absent, TiO lines can also be used as an T eff indicator (Valenti et al. 1998).\nFor late F- and G-dwarfs there are several pressure-sensitive lines that can be used to constrain log g ⋆ : the Mg I triplet at 5167.33 ˚ A, 5172.70 ˚ A and 5183.62 ˚ A (e.g. Fuhrmann et al. 1997; Valenti and Fischer 2005) as shown in Fig. 2, and the Ca I lines at 6122.23 ˚ A and 6162.18 ˚ A (e.g. Gray 2008). For lower gravity or higher temperature, the line wings disappear due to lower photospheric density or ionisa-\n\n\n
Fig. 1 Examples of SME modelling of H α using the MARCS (LTE) stellar atmosphere model towards a K5V (HD85512) and a G0V (HD164509) host star observed with HARPS. The observations are plotted in black and the models in blue. The line wings of H α are very sensitive to the effective temperature. Towards the K5V star they have almost disappeared, while they are very broad towards the G0V star.
\n\nhe Mg I lines are, however, very wide which makes normalization difficult. Two of the lines lie very close without continuum between them, and there are numerous overlying narrow metal lines in particular for late-type stars. In these cases, only the 5183.62 ˚ A line can be used reliably. Figure 2 shows the Mg I triplet line profiles towards the same host stars as in Fig. 1. As already seen in Fig. 1, the spectral line density is much higher towards the K5V star than the G0V star. However, the effect is even more pronounciated at shorter wavelengths. In addition to surface gravity, the Ca and Mg pressure broadened line wings are, however, to some degree also sensitive to temperature and abundance. The procedure is therefore to model the temperature first, e.g. via H α , and the abundance via narrow, unblended Ca (e.g. 6156.02 ˚ A, 6166.439 ˚ A, 6169.042 ˚ A, 6455.985 ˚ A) and Mg (e.g. 5711.09 ˚ A) lines and iterate to a solution.\nThe broad line wings of Na I D-lines (5889.97 ˚ A and 5895.94 ˚ A) are sensitive to both T eff , log g ⋆ (and the Na abundance). Hence these lines can be used to check the final model for consistency.\n\n\n
Fig. 2 Examples of SME modelling of the Mg I triplet (5167.33 ˚ A, 5172.70 ˚ A, and 5183.62 ˚ A) towards the same stars as in Fig. 1. The line wings are sensitive to log g ⋆ . Note the difference in spectral line density and line blanketing in HD85512 which introduce severe problems in the spectral modelling.
\n\nMore details and exampels of spectral modelling can for instance be found in Valenti and Fischer (2005), Jofr'e et al. (2014), and Brewer et al. (2016).\nFitting equivalent widths\nIn contrast to synthetic spectral synthesis methods, the equivalent width (EW) method begins with the observed spectrum by measuring the strengths of selected absorption lines which are translated into individual line abundances. The method is based on theoretical atmosphere models and excitation and ionisation equilibrium which determines the population in a certain level of an atom or an ion.\nThe EW of an absorption line is a convenient measurement of its strength. It is defined as the width of a rectangle reaching up to the continuum (with length one) having the same area as the spectral line. Measuring EWs of well-defined weak neutral iron (Fe I) and ionised iron (Fe II) lines is a traditional method to model stellar spectroscopic parameters since there are numerous iron lines in stellar spectra, many\nwith accurate atomic parameters. Metal lines can be very sensitive to temperature although large variations between lines exist. Depending on spectral type, neutral metals are often used as a temperature indicator in solar-type stars, while ionised metal lines are better tracers in early-type stars.\nAssuming LTE, the ratio of population in two levels n and m of an atom (or ion) of a species can be computed with the Bolzmann equation (e.g. Gray 2008; Carroll and Ostlie 2017)\nNn Nm = gn gm e -( χ n -χ m ) / kT , (6)\nwhere gn and gm are the statistical weights of the two levels, χ n and χ m are the corresponding excitation potentials, k is the Boltzmann constant, and T is the temperature. Different line ratios have different sensitivity to temperature.\nAs the temperature increases so will ionisation which occurs quite abruptly once the threshold temperature is reached. In a star's photosphere, elements exist mainly in just two ionisation stages which can be computed with the Saha equation (e.g. Gray 2008; Carroll and Ostlie 2017) between ionisation states i and i + 1 as\nN i + 1 Ni Pe = ( 2 π me ) 3 / 2 ( kT ) 5 / 2 h 3 2 u i + 1 ( T ) ui ( T ) e -I / kT , (7)\nwhere the electron pressure is Pe = NekT (indicator of the surface gravity), me is the electron mass, h is Planck's constant, u ( T ) = ∑ gi e -χ i / kT is the partition function, and I is the ionisation energy.\nThe EW method requires measurements of a large number of equivalent widths either measured by direct integration over the entire line or by fitting a Gaussian profile (or a Lorentzian profile that may fit optically thick lines better). Measuring EWs manually is, however, very time-consuming and therefore several softwares have been developed to automate the measurements for example ARES (Sousa et al. 2007) and DAOSPEC (Stetson and Pancino 2008).\nWeak and optically thin iron lines depend mainly on T eff and the iron abundance and less on log g ⋆ and V mic . In order to avoid abundance dependence with the EW method, it is best to choose two lines from the same element. However, since continuum normalisation is often a large source of error it may sometimes be necessary to use pairs of lines at nearby wavelengths. In these cases, pairs of similar elements like Fe, V, and Ti that normally have similar abundances, are often used. Different sets of lines are chosen for different spectral types since they are useful in different temperature ranges.\nThe effective temperature is constrained in the following modelling by the correlation between the excitation potential and the iron abundance of each individual line, the microturbulence is constrained by the correlation between the abundance of each line and the reduced equivalent width, while the surface gravity is constrained by the ionisation balance. The parameters are adjusted until there are no correlations left and all individual abundances are the same (Sousa 2014).\nAn example of a widely used open radiative transfer code based on the EW method is MOOG (Sneden 1973). This software uses a grid of the ATLAS9 (Ku-\nrucz 1993a) plane-parallel model atmospheres to produce model spectra which is compared to the measured EWs of individual Fe I and Fe II lines. The equilibrium conditions are solved simultaneously to derive T eff , log g ⋆ , [Fe/H], and V mic . A good description of the process is found in Sousa (2014). It has been used by for instance Sousa et al. (2021) together with ARES for a homogeneous spectroscopic characterisation of almost one thousand exoplanet host stars (the SWEETCat online catalogue). Another example of a software based on the EW method is ODUSSEAS (Observing Dwarfs Using Stellar Spectroscopic Energy-Absorption Shapes; Antoniadis-Karnavas et al. 2020). This code uses the machine learning Python package scikit-learn to offer a quick and automatic derivation of T eff and [Fe/H] for M dwarfs from optical spectroscopy.\nWhen comparing the outcome of the EW and models based on synthetic spectra, T eff and log g ⋆ normally agree within the uncertainties and also with results from asteroseismology within 100 K and 0.1 dex, respectively. The methods, however, do not perform equally well for all spectral types. Since the EW method is based on differential analysis with respect to the Sun, it can be applied to FGK stars with T eff ≈ 4500 -6400 K (Sousa et al. 2011). Modelling based on synthetic spectra and the line shapes are also sensitive to spectral type since some of the fundamental traits, e.g. the broad H α line wings, can only be made for solar-type stars since the line wings disappear for colder and hotter stars. In these cases it may still, however, be possible to use other spectral lines (e.g. TiO for low-mass stars). Furthermore, the EWmethod cannot be used on fast-rotator stars because of the severe line blending. For these stars, it may still be possible to use the synthetic method. Another difference is that the EW method is normally much faster than the synthetic method, while synthetic models provide a more complete description of the star.\nEmpirical methods\nInstead of the above methods, it is also possible to compare an observed highresolution spectrum to spectra of well-characterised stars and in this way derive stellar parameters. An example of such software is Spechmatch-emp (Yee et al. 2017) that compares an observed optical spectrum with a dense empirical spectral library thereby deriving R ⋆ (or log g ⋆ ), T eff , and [Fe/H]. The library contains 404 stars observed with the HIRES instrument with R ≈ 60000 at the Keck telescope. The properties of all the library stars have been derived from asteroseismoloy, interferometry, spectrophotometry and LTE spectral synthesis and represents spectral types from approximately M5 to F1. This method performs very well for the difficult late type stars (K4 stars and later) which are challenging for other methods reaching accuracies of 10 % in stellar radius, 70 K in effective temperature, and 0.09 dex in metallicity ([Fe/H]). The software and the library are publicly available.\nStellar radius and mass\nOnce we have obtained the stellar spectroscopic parameters for our host star we now want to find out the radius and the mass of the star.\nRadius from spectroscopy and spectral energy distribution\nWhen spectroscopic parameters and photometric measurements are available it is straightforward to obtain the stellar radius from a fit of the observed magnitudes in different bands to its spectral energy distribution (SED). In addition to magnitudes, the SED also depends on the distance (most accurately computed from the observed parallax), the spectroscopic parameters (derived from spectroscopic modelling), the extinction along the line-of-sight, and the stellar radius which is a free parameter in the fit. Parallax measurements have been performed by the European space missions Hipparchos (van Leeuwen 1997) and Gaia (Gaia Collaboration et al. 2016) launched in 1989 and 2013, respectively. The latter mission have provided the community with astrometric and photometric measurements of almost two billion stars in the Milky Way. Gaia has also delivered excellent measurements of the magnitudes in the Gaia optical band. Observations will end in early 2025. Figure 3 shows an example of a SED fit with the Phoenix (Husser et al. 2013) atmospheric model grid. An example of a publicly available software that automatically fits broadband photometry to six different stellar atmosphere models using Nested Sampling algorithms is the ARIADNE (Vines and Jenkins 2022) software.\nStellar mass\nUnless the star is a binary, the mass cannot usually be determined directly but can be modeled via stellar evolution models. Some of the most popular models are BaSTI (Hidalgo et al. 2018), Padova (Bertelli et al. 2008, 2009), DESP (the Dartmouth Stellar Evolution Database; Dotter et al. 2008), MIST (MESA Isochrones and Stellar Tracks; Choi et al. 2016), PARSEC (the PAdova and TRieste Stellar Evolution Code; Bressan et al. 2012), and Y 2 (Yonsei-Yale; Yi et al. 2003; Demarque et al. 2004). A quick way to obtain a Bayesian estimation of both mass and radius based on PARSEC and MESA (Rodrigues et al. 2017) models are the web interfaces Param1.3 and Param1.5 (da Silva et al. 2006). Another publicly available software that provides a simple interface for MCMC fitting of MIST stellar model grids is isochrones (Morton 2015).\nA complementing method is to use a set of stars with well-known masses and radii, e.g. through interferometry and eclipsing binaries, to derive empirical calibration equations. Such equations can give the mass and radius of a star given a set\n\n\n
Fig. 3 Example of a spectral energy distribution (SED) fit of the host star TOI-2196 discovered by the Transiting Exoplanet Survey Satellite (Persson et al. 2022). The observed photometric measurements are plotted with blue circles and the effective width of the passband are represented with horizontal bars. The Phoenix (Husser et al. 2013) atmosphere model is outlined in black and the magenta diamond symbols are the corresponding model fluxes.
\n\nof stellar atmosphere values (e.g. Torres et al. 2010; Enoch et al. 2010; Southworth 2011).\nFinal checks\nIf a planet is transiting a final important check can be made. The stellar density obtained from the above radius and mass can be checked against the density obtained from transit photometry and Kepler's third law. Assuming a circular orbit we have (Seager and Mall'en-Ornelas 2003)\nρ ⋆ = 3 π GP 2 rot ( a R ⋆ ) 3 , (8)\nwhere G is the gravitational constant, P rot is the orbital period, a is the semi-major axis, and R ⋆ is the stellar radius. If this density does not agree with the above derived value, you need to check your modelling again. Equation 8 can be modified to include the eccentricity if known (e.g. Tingley et al. 2011).\nAnother checkpoint is to compare the stellar mass computed from the spectroscopic log g ⋆ combined with R ⋆ which should be consistent with the final mass from other methods.\nA final remark: if possible, it is best to use several methods to make sure that the derived stellar parameters are robust and consistent.\nReferences\n
\n\n
\n\nCasagrande L, Ram'ırez I, Mel'endez J, Bessell M Asplund M (2010) An absolutely calibrated T e f f scale from the infrared flux method. Dwarfs and subgiants. A&A512:A54\nChoi J, Dotter A, Conroy C et al. (2016) Mesa Isochrones and Stellar Tracks (MIST). I. Solarscaled Models. ApJ823(2):102\nChristlieb N, Gustafsson B, Korn AJ et al. (2004) HE 0107-5240, a Chemically Ancient Star. I. A Detailed Abundance Analysis. ApJ603(2):708-728\nCosentino R, Lovis C, Pepe F et al. (2012) Harps-N: the new planet hunter at TNG. In: Groundbased and Airborne Instrumentation for Astronomy IV, Proc SPIE, vol 8446, p 84461V, DOI 10.1117/12.925738\nda Silva L, Girardi L, Pasquini L et al. (2006) Basic physical parameters of a selected sample of evolved stars. A&A458:609-623\nde Laverny P, Recio-Blanco A, Worley CC Plez B (2012) The AMBRE project: A new synthetic grid of high-resolution FGKM stellar spectra. A&A544:A126\nDemarque P, Woo JH, Kim YC Yi SK (2004) Y 2 Isochrones with an Improved Core Overshoot Treatment. ApJS155(2):667-674\n\nDorn C, Khan A, Heng K et al. (2015) Can we constrain the interior structure of rocky exoplanets from mass and radius measurements? A&A577:A83\nDotter A (2016) MESA Isochrones and Stellar Tracks (MIST) 0: Methods for the Construction of Stellar Isochrones. ApJS222(1):8\nDotter A, Chaboyer B, Jevremovi'c D et al. (2008) The Dartmouth Stellar Evolution Database.\nApJS178:89-101\nDoyle AP, Davies GR, Smalley B, Chaplin WJ Elsworth Y (2014) Determining stellar macroturbulence using asteroseismic rotational velocities from Kepler. MNRAS444:3592-3602\n\nEisenhauer F, Monnier JD Pfuhl O (2023) Advances in Optical/Infrared Interferometry. ARA&A61:237-285\n\nEnoch B, Collier Cameron A, Parley NR Hebb L (2010) An improved method for estimating the\nmasses of stars with transiting planets. A&A516:A33\n\nFuhrmann K, Axer M Gehren T (1993) Balmer lines in cool dwarf stars. 1. Basic influence of atmospheric models. A&A271:451\nFuhrmann K, Axer M Gehren T (1994) Balmer lines in cool dwarf stars II. Effective temperatures and calibration of colour indices. A&A285:585-594\nFuhrmann K, Pfeiffer M, Frank C, Reetz J Gehren T (1997) The surface gravities of cool dwarf stars revisited. A&A323:909-922\nGaia Collaboration, Prusti T, de Bruijne JHJ et al. (2016) The Gaia mission. A&A595:A1\nGerber JM, Magg E, Plez B et al. (2023) Non-LTE radiative transfer with Turbospectrum. A&A669:A43\nGray DF (2008) The Observation and Analysis of Stellar Photospheres. Cambridge University Press\nGray RO Corbally CJ (1994) The Calibration of MK Spectral Classes Using Spectral Synthesis. I. The Effective Temperature Calibration of Dwarf Stars. AJ107:742\nGrevesse N, Asplund M Sauval AJ (2007) The Solar Chemical Composition. Space Sci Rev130(14):105-114\nGustafsson B, Edvardsson B, Eriksson K et al. (2008) A grid of MARCS model atmospheres for late-type stars. I. Methods and general properties. A&A486:951-970\nHeiter U Eriksson K (2006) Geometry of giant star model atmospheres: a consistency test. A&A452(3):1039-1048\nHeiter U, Kupka F, van't Veer-Menneret C et al. (2002) New grids of ATLAS9 atmospheres I: Influence of convection treatments on model structure and on observable quantities. A&A392:619636\nHidalgo SL, Pietrinferni A, Cassisi S et al. (2018) The Updated BaSTI Stellar Evolution Models and Isochrones. I. Solar-scaled Calculations. ApJ856(2):125\nHusser TO, Wende-von Berg S, Dreizler S et al. (2013) A new extensive library of PHOENIX stellar atmospheres and synthetic spectra. A&A553:A6\nJofr'e P, Heiter U, Soubiran C et al. (2014) Gaia FGK benchmark stars: Metallicity. A&A564:A133\n\n\nKurucz R (1993a) ATLAS9 Stellar Atmosphere Programs and 2 km/s grid. ATLAS9 Stellar Atmosphere Programs and 2 km/s grid Kurucz CD-ROM No 13 Cambridge, Mass: Smithsonian Astrophysical Observatory, 1993 13\nKurucz R (1993b) SYNTHE Spectrum Synthesis Programs and Line Data. SYNTHE Spectrum Synthesis Programs and Line Data Kurucz CD-ROM No 18 Cambridge, Mass: Smithsonian Astrophysical Observatory, 1993 18\nKurucz RL (2013) ATLAS12: Opacity sampling model atmosphere program. Astrophysics Source Code Library\nLi H, Aoki W, Matsuno T et al. (2022) Four-hundred Very Metal-poor Stars Studied with LAMOST and Subaru. II. Elemental Abundances. ApJ931(2):147\nLodders K (2003) Solar System Abundances and Condensation Temperatures of the Elements. ApJ591(2):1220-1247\nMayor M Queloz D (1995) A Jupiter-mass companion to a solar-type star. Nature378:355-359 Mayor M, Pepe F, Queloz D et al. (2003) Setting New Standards with HARPS. The Messenger 114:20-24\n\nMignon L, Meunier N, Delfosse X et al. (2023) Characterisation of stellar activity of M dwarfs. I. Long-timescale variability in a large sample and detection of new cycles. A&A675:A168 Mihalas D (1978) Stellar atmospheres\n\nMorton TD (2015) isochrones: Stellar model grid package\nNepal S, Chiappini C, Guiglion G et al. (2024) Insights from super-metal-rich stars: Is the Milky Way bar young? A&A681:L8\n\nNordstrom B, Mayor M, Andersen J et al. (2004) The Geneva-Copenhagen survey of the Solar neighbourhood. Ages, metallicities, and kinematic properties of ∼ 14 000 F and G dwarfs. A&A418:989-1019\nPepe F, Cristiani S, Rebolo R et al. (2021) ESPRESSO at VLT. On-sky performance and first results. A&A645:A96\nPersson CM, Georgieva IY, Gandolfi D et al. (2022) TOI-2196 b: Rare planet in the hot Neptune desert transiting a G-type star. A&A666:A184\nPiskunov N Valenti JA (2017) Spectroscopy Made Easy: Evolution. A&A597:A16\n\nPiskunov NE, Kupka F, Ryabchikova TA, Weiss WW Jeffery CS (1995) VALD: The Vienna Atomic Line Data Base. A&AS112:525\n\nPlez B (2012) Turbospectrum: Code for spectral synthesis. Astrophysics Source Code Library, record ascl:1205.004\nQuirrenbach A (2001) Optical Interferometry. ARA&A39:353-401\nRodrigues TS, Bossini D, Miglio A et al. (2017) Determining stellar parameters of asteroseismic targets: going beyond the use of scaling relations. MNRAS467(2):1433-1448\nRyabchikova T, Piskunov N, Kurucz RL et al. (2015) A major upgrade of the VALD database. Phys Scr90(5):054005\nSeager S Mall'en-Ornelas G (2003) A Unique Solution of Planet and Star Parameters from an Extrasolar Planet Transit Light Curve. ApJ585:1038-1055\n\nSerenelli A, Weiss A, Aerts C et al. (2021) Weighing stars from birth to death: mass determination methods across the HRD. A&A Rev29(1):4\nShulyak D, Tsymbal V, Ryabchikova T, Stutz C Weiss WW (2004) Line-by-line opacity stellar model atmospheres. A&A428:993-1000\nSneden CA (1973) Carbon and Nitrogen Abundances in Metal-Poor Stars. PhD thesis, University of Texas, Austin\nSousa SG (2014) ARES + MOOG: A Practical Overview of an Equivalent Width (EW) Method to Derive Stellar Parameters. In: Determination of Atmospheric Parameters of B, pp 297-310, DOI 10.1007/978-3-319-06956-2 26\n\nSousa SG, Santos NC, Israelian G, Mayor M Monteiro MJPFG (2007) A new code for automatic determination of equivalent widths: Automatic Routine for line Equivalent widths in stellar Spectra (ARES). A&A469:783-791\nSousa SG, Santos NC, Israelian G, Mayor M Udry S (2011) Spectroscopic stellar parameters for 582 FGK stars in the HARPS volume-limited sample. Revising the metallicity-planet correlation. A&A533:A141\nSousa SG, Adibekyan V, Delgado-Mena E et al. (2021) SWEET-Cat 2.0: The Cat just got SWEETer. Higher quality spectra and precise parallaxes from Gaia eDR3. A&A656:A53\nSouthworth J (2011) Homogeneous studies of transiting extrasolar planets - IV. Thirty systems with space-based light curves. MNRAS417:2166-2196\nStetson PB Pancino E (2008) DAOSPEC: An Automatic Code for Measuring Equivalent Widths in High-Resolution Stellar Spectra. PASP120(874):1332\nTelting JH, Avila G, Buchhave L et al. (2014) FIES: The high-resolution Fiber-fed Echelle Spectrograph at the Nordic Optical Telescope. Astronomische Nachrichten 335:41\nTingley B, Bonomo AS Deeg HJ (2011) Using Stellar Densities to Evaluate Transiting Exoplanetary Candidates. ApJ726(2):112\nTokovinin A, Fischer DA, Bonati M et al. (2013) CHIRON-A Fiber Fed Spectrometer for Precise Radial Velocities. PASP125(933):1336\nTorres G, Andersen J Gim'enez A (2010) Accurate masses and radii of normal stars: modern results and applications. A&A Rev18(1-2):67-126\nTorres G, Fischer DA, Sozzetti A et al. (2012) Improved Spectroscopic Parameters for Transiting Planet Hosts. ApJ757(2):161\nTull RG, MacQueen PJ, Sneden C Lambert DL (1995) The high-resolution cross-dispersed echelle white-pupil spectrometer of the McDonald Observatory 2.7-m telescope. PASP107:251-264\nValenti JA Fischer DA (2005) Spectroscopic Properties of Cool Stars (SPOCS). I. 1040 F, G, and K Dwarfs from Keck, Lick, and AAT Planet Search Programs. ApJS159:141-166\nValenti JA Piskunov N (1996) Spectroscopy made easy: A new tool for fitting observations with synthetic spectra. A&AS118:595-603\nValenti JA, Piskunov N Johns-Krull CM (1998) Spectral Synthesis of TiO Lines. ApJ498(2):851862\nvan Leeuwen F (1997) The HIPPARCOS Mission. Space Sci Rev81:201-409\nVines JI Jenkins JS (2022) ARIADNE: measuring accurate and precise stellar parameters through SED fitting. MNRAS513(2):2719-2731\nVogt SS, Allen SL, Bigelow BC et al. (1994) HIRES: the high-resolution echelle spectrometer on the Keck 10-m Telescope. In: Crawford DL Craine ER (eds) Instrumentation in Astronomy VIII, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol 2198, p 362, DOI 10.1117/12.176725\nWehrhahn A, Piskunov N Ryabchikova T (2023) PySME. Spectroscopy Made Easier. A&A671:A171\nYee SW, Petigura EA von Braun K (2017) Precision Stellar Characterization of FGKM Stars using an Empirical Spectral Library. ApJ836:77\n\nYi SK, Kim YC Demarque P (2003) The Y 2 Stellar Evolutionary Tracks. ApJS144(2):259-261\n"},"sections":{"kind":"list like","value":[{"title":"Methods to Characterise Exoplanet Host Stars from Spectroscopy","content":"Carina M. Persson Abstract A key to understand exoplanets is characterisation of their host stars. One of the most powerful tools to characterise stellar properties like effective temperature, surface gravity and metallicity, is spectroscopy based on observations of stellar atmospheres. This chapter describes the stellar parameters that can be derived from a spectrum with examples of well established methods and theoretical model atmospheres. Combined with photometry and parallax measurements, the outcome of the spectroscopic modelling can be used to derive stellar radii and masses.","pages":[1]},{"title":"Introduction","content":"Exoplanets are intimately connected to their host stars through formation and evolution. In addition, detection and characterisation of exoplanets depend on detailed knowledge of their host stars since the current major detection techniques, transit photometry and the radial velocity (RV) method, detect planet sizes and masses relative to their host star (see Chapters by Deeg & Alonso and Wright in Volume 1 of the Handbook of Exoplanets). Uncertainties in a host star's parameters propagate directly to the planets. Stellar modelling is normally based on two major techniques - photometry and spectroscopy. These methods are model-dependent in contrast to the direct measurements by interferometry, eclipsing binaries, and asteroseismology. For eclipsing binaries (e.g. Andersen 1991; Torres et al. 2010; Serenelli et al. 2021), and for the few large and nearby stars that enable interferometric measurements (e.g. Quirrenbach 2001; Eisenhauer et al. 2023), it is possible to derive the stellar radius with an accuracy of a few per cent. The timing of the duration of the eclipses of eclipsing binaries and their orbital velocities allows accurate estimates Chalmers University of Technology, department of Space, Earth, and Environment, Onsala space observatory, 439 92 ONSALA, Sweden, e-mail: carina.persson@chalmers.se of their sizes. Similarly, stellar masses can be accurately determined for visual binaries from observed separations from the common centre-of-mass with Kepler's third law that relates their masses with observed orbital period and separation. Asteroseismology can also be used to derive stellar mass, radius, and age with a high precision (see Chapter by Lundkvist, Huber, Aguirre & Chaplin in this volume of the Handbook of Exoplanets). Recently, the seismic surface gravity of the star has also been used to obtain the effective temperature and metallicity, in particular with APOGEE (Ahumada et al. 2020, 2022). These methods are, however, currently only possible to apply to a small subset of all stars. Spectroscopic measurements open a window to derive stellar parameters from a larger pool of stars than from direct methods. High-resolution spectroscopy is a powerful tool that provides a wealth of information; the effective temperature, surface gravity, chemical composition, and velocities. The stellar radius, and the luminosity, can then readily be derived from its spectral energy distribution (SED) via the spectroscopic parameters combined with photometry and parallax. The spectroscopic parameters also serve as a base to model characteristics that normally cannot be directly measured, like mass and age, with stellar evolution models and the complementary tool isochrones (e.g. Dotter 2016; Hidalgo et al. 2018). An isochrone is an evolutionary track for a population of stars with different masses on the Hertzsprung-Russell diagram with the same ( iso ) age ( chrone ). The mass and radius can also be obtained from the spectroscopic parameters and empirical calibration equations (e.g. Torres et al. 2010; Enoch et al. 2010; Southworth 2011) albeit often with a higher uncertainty. The downside of high-resolution spectroscopy is that it is expensive; the observations are time-consuming and also requires bright stars or a large collecting area. Modelling of stellar parameters has therefore traditionally been performed by photometry because a much larger number of stars can quickly be observed and analysed. The effective temperature of a star can for instance be derived from color-color diagrams which have been scaled to stars measured with direct methods (e.g. Bell and Gustafsson 1989; Alonso et al. 1996; Nordstrom et al. 2004). However, such models have in general higher uncertainties than based on spectroscopic measurements. In that respect they may only be reliable in a statistical sense and not for individual stars. Thus for exoplanet host stars (for which direct measurements are not applicable), high-resolution spectroscopy is preferred. This chapter begins with a short summary of a few basic requirements in order to obtain spectroscopic measurements of a star. It continues with an overview of the parameters that can be extracted from a high-resolution spectrum and a few examples of well established methods. Some advantages and caveats are highlighted. The chapter ends with a brief summary of how to combine spectroscopic parameters, photometric measurements, and stellar evolution models to obtain stellar radii and masses. Modelling of stellar ages are described by Christensen-Dalsgaard & Aguirre in a Chapter in this volume of the Handbook of Exoplanets.","pages":[1,2]},{"title":"Instruments and spectral resolution","content":"Important properties of a spectrum is high-resolution, high signal-to-noise (S/N), and wavelength coverage. There are several different types of spectrometers and many textbooks have been written about this topic including advantages and challenges for different types of instruments (e.g. Gray 2008). The most successfull type in observations of exoplanets is echelle spectrographs. The main advantages is the high resolution combined with a wide wavelength coverage obtained in a single exposure. A description of high-precision cross-dispersed echelle spectrographs for exoplanet research (CORAVEL, ELODIE, CORALIE, SOPHIE and HARPS) is found in a Chapter by Fransesco Pepe in this volume of the Handbook of Exoplanets. For the purpose of spectroscopic modelling of host stars, we want high-resolution in order to resolve the spectral lines. The spectral resolution ∆λ at wavelength λ is where R is the resolving power of the spectrograph. It can be translated into a velocity resolution according to Doppler broadening of spectral lines due to thermal and turbulent motions of absorbing species in the atmospheres produce line widths of ∼ 6 kms -1 for latetype stars. This corresponds to a spectral resolution of 50 000. Low-mass, slowly rotating stars can have line widths of only ≈ 1 -2 km s -1 which require R ≳ 300000 to resolve the spectral lines and disentangle blended lines. In terms of high resolution, ultra-high precision, and long-term stability, the High Accuracy Radial Velocity Planet Searcher (HARPS; Mayor et al. 2003) and its decade younger sibling HARPS-North (Cosentino et al. 2012) mounted on the ESO 3.6 m telescope (La Silla observatory, Chile) and the Telescopio Nazionale Galileo (TNG) of Roque de los Muchachos Observatory (La Palma, Spain), respectively, have been the leading instruments in detecting exoplanets over the last two decades. Both HARPS instruments are fiber-fed cross-dispersed high-precision echelle spectrographs covering 380 nm to 690 nm. The spectroscopic resolution is approximately 115000 at visual wavelengths corresponding to a velocity resolution of 2.6 km s -1 . The design is based on experience with the groundbreaking ELODIE and CORALIE instruments where the former was used to detect the first exoplanet 51 Peg b (Mayor and Queloz 1995). Both HARPS spectrographs can be considered as the 'gold standard' when searching for exoplanets in RV data and are also used for characterisation of exoplanet host stars. When searching for exoplanets with the RV method, many measurements are collected, sometimes over a period over many years. The individual spectra can be co-added (after correcting for the peri- odic changes in radial velocity) in order to increase the S/N enabling spectroscopic characterisation of the host star. In addition to the HARPS spectrographs, there are many other instruments used for exoplanet detection that can also be used to characterise host stars e.g. ESPRESSO (Pepe et al. 2021), FIES (Telting et al. 2014), HIRES (Vogt et al. 1994), CHIRON (Tokovinin et al. 2013), TULL (Tull et al. 1995) with different resolutions operating at different wavelengths.","pages":[3,4]},{"title":"Stellar properties from spectroscopy","content":"Only a few parameters characterise a stellar atmosphere: the effective temperature ( T eff ), the surface gravity (log g ⋆ ), the overall metal abundance ([M/H]), and atmospheric and rotational velocities. The surface of a star is defined as the location where photons escape from the star which occurs at a characteristic optical depth of 2/3. This occurs within the photosphere, the innermost ≈ 500 km of a star's atmosphere which overlies the opaque interior. The photospheric temperature and density varies with depth and depends on the surface gravity as well as the abundances and opacity of the gases. The Sun's photosphere has a temperature that varies between 4 400 K and 6 600 K with an effective temperature of 5 772 K, while the density is approximately 3 × 10 -4 kg m -3 , increasing with depth into the Sun. Other stars may have hotter or cooler photospheres. Spectral absorption lines originate from different depths and opacities within the photosphere. Weak and optically thin spectral lines, and the line wings of optically thick lines, originate essentially in the same layer as the continuum. In contrast, the cores of optically thick saturated lines develop in higher layers. An example is the core of the hydrogen α line at 6562.81 ˚ A which originates in the hotter and less dense chromosphere which lies on top of the photosphere, where the assumption of local thermal equilibrium (LTE) is no longer valid.","pages":[4]},{"title":"Effective temperature","content":"Stars are classified according to effective temperature from the hot O-stars, also called early-type stars, to the cool M-stars (late-type stars). If a star is on the main sequence, the effective temperature immediately signals which type of star it is along with typical mass and radius. Instead of choosing a particular depth to define a star's surface temperature, the effective temperature is defined in terms of flux. The effective temperature is defined via the Stefan-Bolztmann law in terms of the total power per unit area, radiated by the star (e.g. Gray 2008) Here F ν is the total flux passing through the star's surface and σ is the Boltzmann constant. The effective temperature is thus the temperature of a black body having the same power output per unit area as a star. It is related to the flux we measure at Earth, f ν , via (Gray 2008) where the left integral is the total radiative flux from the star received at the top of the Earth's atmosphere (bolometric flux), R ⋆ is the stellar radius, d is the distance, and θ ⋆ is the angular diameter of the star. The effective temperature can hence be derived by measuring the angular size of the star from interferometry and the received flux at Earth over a wide spectral range. By combining the angular size with distance from parallax measurements, the linear radius of the star can be inferred for a wide range of spectral types nearly model-independent except for the dependence on the adopted limb-darkening coefficients and bolometric correction. Alternatively, if the radius of the star is known from e.g. eclipsing binaries and the distance from parallax measurments, this will give the angular size which then can be used to compute T eff . Since the target stars have to be nearby to measure their angular sizes, interstellar absorption can be neglected. The infrared flux method (IRFM Blackwell and Shallis 1977; Blackwell et al. 1980; Bell and Gustafsson 1989; Blackwell et al. 1990; Casagrande et al. 2006, 2010) is based on observations of the angular size of the star and the measured infrared flux at the top of the Earth's atmosphere. The bolometric flux is derived taking into account the bolometric correction which gives the effective temperature. In addition to the above methods, T eff can also be derived from spectroscopy as described in the following section. However, it is worth highlighting that the temperature derived from spectroscopy is a microscopic value which is close to, but not exactly the same as the effective temperature due to its definition being a macroscopic description.","pages":[4,5]},{"title":"Surface gravity","content":"The surface gravity is an indication of the luminosity class of a star where V is the main sequence and I - IV is different types of giants. Thus surface gravity contains information of the size and age of a star. A low surface gravity immediately indicates that the star has left the main sequence. The surface gravity of a star is defined by (e.g. Gray 2008) where g ⊙ is the surface gravity of the Sun (2 . 740 × 10 4 cm s -2 ) and the radius, R ⋆ , and mass, M ⋆ , of the star are in solar units. The surface gravity is commonly measured in a logarithmic scale, log g ⋆ , where the solar value is 4.44. The surface gravity determines the gas density in the photosphere and is the spectroscopic parameter that has the highest impact on the stellar radius. Unfortunately, log g ⋆ is often poorly constrained by spectral analysis. Since the uncertainties of T eff and metallicity can be strongly correlated with surface gravity this can lead to a significant source of systematic error in some analysis techniques.","pages":[5,6]},{"title":"Metallicity","content":"A stars chemical composition is an outcome of the nucleosynthesis by previous generations of stars. This is important when reconstructing star and planet formation history in our Galaxy. There is a large variation of metallicity, i.e. the abundance of all elements heavier than helium denoted with [M/H], in the Milky Way up to about twice the solar value to hundreds of thousands of times lower than the solar value (e.g. Christlieb et al. 2004; Li et al. 2022; Nepal et al. 2024). The chemical composition of a star is commonly fixed to the overall metallicity of a star relative to the Sun. The Grevesse et al. (2007), Asplund et al. (2009) and Lodders (2003) abundance scales for the Sun are currently the most adopted. However, individual abundances of a star may not follow solar composition and may require modelling of individual elemental abundances. Abundances are generally measured on a logarithmic scale normalised to the Sun where zero equals the Sun's metallicity. In the case of iron we have [Fe/H] ⋆ = log(Fe/H) ⋆ - log(Fe/H) ⊙ . For example, an iron abundance of [Fe/H] = -0 . 5 means that the abundance is 10 -0 . 5 relative to the Sun. Since iron is by far the most abundant species in a stellar atmosphere after hydrogen and helium, measurements of iron has become a proxy for the metallicity. Stellar abundances is also important when modelling exoplanet interiors in particular rocky super-Earths without significant gaseous envelopes. The degeneracy of interior composition inferred from radius and mass measurements can for this type of planet be reduced assuming an interior structure with a differentiated iron core and a rocky mantle. In these cases, the host star abundances is often used as a proxy of the primary planet-building elements Fe, Mg, and Si, which are expected to be reflected in the planet composition, planet interior, and core mass fraction (Dorn et al. 2015; Acu˜na et al. 2023).","pages":[6]},{"title":"Velocities","content":"Thermal widths of spectral lines are only a fraction of the observed line widths for dwarf stars hotter than spectral type K0. The line widths are instead mainly governed by Doppler shifts produced by motions of the star's photospheric gases. The radial velocity of the star is only shifting the wavelengths of all spectral lines in the observed spectrum compared to the observations and can easily be corrected for. The velocity that dominates the line shape and width for hot stars is the projected equatorial rotational velocity of the star, V sin i ⋆ , where i ⋆ is the inclination of the stellar rotation axis relative to the line of sight. It can be measured via the full width at half maximum (FWHM) of a large number of optically thin and unblended lines not sensitive to pressure broadening. The line shapes are, however, also affected by turbulence from convective motion, granulation, high-order pulsations, stellar activity, and other types of local flows in the photosphere. Turbulence is represented in the models by the macro-turbulent velocity ( V mac) that describes motions on scales larger than the mean free path within the photosphere that induce a change in the line shape; and the micro-turbulent velocity ( V mic ) that describes motions on scales smaller than the mean free path leading to increased line opacity (Gray 2008; Bruntt et al. 2010; Doyle et al. 2014). The latter velocity is a 'fudge' factor originally introduced to reconcile observed and predicted equivalent widths (Mihalas 1978). It includes all remaining types of broadening mechanisms and is at present standard to include in analyses of solar-type stars. Both turbulent velocities depend on temperature and to a lesser extent on surface gravity. The micro-turbulent velocity has a width of the order of 1 km s -1 for dwarfs and several km s -1 for giants. For lowmass stars, V mac and V sin i ⋆ have comparable widths of the order of a few km s -1 . As a reference, the Sun's V sin i ⋆ is 2 km s -1 at the equator while hotter stars have much higher rotational velocities (tens to hundreds of km s -1 ).","pages":[6,7]},{"title":"Methods for spectroscopic modelling","content":"There are several ways to model a spectrum which can be divided into two main groups. The first is based on spectral synthesis. Here observations are fitted to a synthetic spectrum of stellar atmosphere models by comparison of line profiles. The second is a line-by-line analysis based on measured strengths of observed spectral lines, their equivalent widths (EWs). Detailed description of the physics can be found in many textbooks e.g. Gray (2008). Spectroscopic observations can also be compared to a library of spectra of well-characterised stars via for example interferometric measurements or spectroscopic binaries. One major problem in spectroscopic analysis is to accurately determine the continuum which can introduce large errors. This is particularly difficult for poor spectra with low spectral resolution or low S/N. It also depends on the spectral type of the star and the wavelength region. The number of spectral lines increases towards shorter wavelengths for all types of stars. In addition, late-type stars have a much higher density of spectral lines than early-type stars arising from both atoms and molecules leading to blending and confusion of the continuum location. The higher temperatures of early-type stars ionize a large fraction of their atoms, leading to significantly fewer spectral features than low-mass stars. Differences in rotational velocities also affect the spectral line density. In contrast to late-type stars, the early types have very high rotational velocities which leads to very broad spectral lines that smear out spectral features. Thus solar-type stars (FGK) are the easiest stars to model, while high- and low-mass stars often entails a significantly higher degree of difficulty in the modelling. M-dwarfs have in addition generally a much longer period of high stellar activity than FGK stars, exacerbating the problems (e.g. Mignon et al. 2023). This is unfortunate since M-dwarfs are popular exoplanet host stars due to their small masses and sizes which increase the exoplanet signals. Care must be taken when selecting which spectral lines to model. A large set of narrow, non-blended spectral lines are preferred (unless modelling pressure broadened line wings, see below). If a spectral line becomes optically thick, the abundance of a species stops growing linearly with absorption depth. The characteristics of an optically thick line is a saturated line centre which flattens the bottom and broaden the line wings. Not all optically thin spectral lines may, however, be useful since a large number comes with poorly determined atomic parameters which are needed to compute synthetic spectra. This can be circumvented for solar-type stars if adopting new atomic parameters after comparing the lines from observations of the Sun.","pages":[7,8]},{"title":"Fitting observations to synthetic spectra","content":"Computations of a synthetic spectrum requires a model atmosphere based on solutions to the stellar structure equations to synthesize a spectrum. Most stellar atmosphere models are pre-calculated and tabulated on grids describing the profiles of the temperature, surface gravity and abundances as functions of atmospheric depth. Each layer in the model atmosphere is contributing to the formation of absorption line profiles in the final spectrum. Some widely used atmospheric model atmospheres are Atlas12 (Kurucz 2013), Atlas9 (Kurucz 1993a; Heiter et al. 2002), MARCS (Gustafsson et al. 2008) for cool and giant stars, and LL models (Shulyak et al. 2004) for hot main sequence stars. Line lists of atomic and molecular data needed in the computation can be provided by the Vienna Atomic Line Database (VALD3; Piskunov et al. 1995; Ryabchikova et al. 2015). A radiative transfer code then computes a synthetic model spectrum of the star for the chosen set of stellar parameters ( T eff , log g ⋆ , [Fe/H], V sin i ⋆ , V mic , V mac) which are matched against the observed spectra based on the spectral line shapes and strengths. The parameters generally affect either the line strength ( T eff , log g ⋆ , abundances, and V mic ) or the line shape ( V mac, V sin i ⋆ , and the instrumental resolution). Degeneracies are stronger within the subsets. The dependence of the line profile on the V mac parameter is broadened line wings and a cusp-shaped core. Unfortunately, disentangling the effect on the line profile from V sin i ⋆ and V mac is difficult, leading to a degeneracy between the two. Prior information of V sin i ⋆ may be obtained from time-resolved photometry, available for the known transiting planets, or from asteroseismology (cf. Doyle et al. 2014). If no prior information is available, calibration equations of both turbulent velocities are often used (e.g. Bruntt et al. 2010; Doyle et al. 2014) which allows modelling of V sin i ⋆ . Note that in order to properly model the different velocities above, it is imperative to take into account how the spectrograph itself broadens the lines. Spectral lines can also be pressure broadened through various mechanisms which further broaden the lines, e.g. the Balmer lines. Details of the various broadening mechanisms can be found in many textbooks such as Gray (2008). There are many softwares that computes synthetic spectra to be used as a model constraint to interpret the observed spectrum. The popular open-source spectroscopic tool iSpec (Blanco-Cuaresma et al. 2014; Blanco-Cuaresma 2019) support several of the most well-known radiative transfer codes such as Spectrocsopy Made Easy ( SME ; Valenti and Piskunov 1996; Piskunov and Valenti 2017; Wehrhahn et al. 2023), SPECTRUM (Gray and Corbally 1994), Turbospectrum (Plez 2012; de Laverny et al. 2012; Gerber et al. 2023), Synthe/WIDTH9 (Kurucz 1993b), and MOOG (Sneden 1973) where the latter is based on the equivalent width method. The codes often adopt LTE and a plane-parallel geometry as default motivated by the fact that for main-sequence stars, the photosphere constitutes ≪ 1 % of the stellar radius. Aspherically symmetric geometry is on the other hand required for giant stars where the atmosphere makes up a substantial portion of its radius (Heiter and Eriksson 2006). However, since complex interaction of gas particles and the non-local radiation fields leads to deviations from LTE in the atmospheres of FGKM-type stars, some of the softwares have also the option of non-local thermodynamic equilibrium (NLTE). For instance, SME includes NLTE departure coefficients for the MARCS and LL models atmospheres (Piskunov and Valenti 2017), and Turbospectrum also have the option of NLTE (Gerber et al. 2023). Fitting can in several of the softwares be made for one or several parameters at the same time using a χ 2 -minimization algorithm. However, care must be taken when solving for several parameters simultaneously (Torres et al. 2012) due to degeneracies and difficulties in the modelling. Different initial assumptions of one free parameter at a time can therefore be made in the fitting process to iterate to the final solution, thereby mitigating degeneracies. Also, since the surface gravity is often difficult to constrain it is thus advantageous to have external information about the stellar density that can facilitate modelling. Spectroscopic modelling can take advantage of the sensitivity of certain spectral lines to specific parameters. For instance, all Balmer lines exhibit pressure (collisionally) broadened line wings which makes their profiles strongly sensitive to temperature (e.g. Fuhrmann et al. 1993, 1994; Barklem et al. 2000, 2002) as shown in Fig. 1. The Balmer line wings are formed in the deepest photospheric layers likely close to the LTE. The metal-line blending increase from H α at 6562.81 ˚ A to H δ at 4101.75 ˚ A which perturb the higher transitions of the Balmer line profiles making H α the best choice. For late F, G, and early K-stars, H α is very insensitive to log g ⋆ and abundance making its line wings an excellent T eff indicator. Two examples of H α line profiles towards a K5V and a G0V host star are shown in Fig. 1. It is obvious that the line wings have almost completely disappeared in the K5V spectrum, while prominent and very broad towards the G-star. For M-dwarfs where the line wings of H α are absent, TiO lines can also be used as an T eff indicator (Valenti et al. 1998). For late F- and G-dwarfs there are several pressure-sensitive lines that can be used to constrain log g ⋆ : the Mg I triplet at 5167.33 ˚ A, 5172.70 ˚ A and 5183.62 ˚ A (e.g. Fuhrmann et al. 1997; Valenti and Fischer 2005) as shown in Fig. 2, and the Ca I lines at 6122.23 ˚ A and 6162.18 ˚ A (e.g. Gray 2008). For lower gravity or higher temperature, the line wings disappear due to lower photospheric density or ionisa- he Mg I lines are, however, very wide which makes normalization difficult. Two of the lines lie very close without continuum between them, and there are numerous overlying narrow metal lines in particular for late-type stars. In these cases, only the 5183.62 ˚ A line can be used reliably. Figure 2 shows the Mg I triplet line profiles towards the same host stars as in Fig. 1. As already seen in Fig. 1, the spectral line density is much higher towards the K5V star than the G0V star. However, the effect is even more pronounciated at shorter wavelengths. In addition to surface gravity, the Ca and Mg pressure broadened line wings are, however, to some degree also sensitive to temperature and abundance. The procedure is therefore to model the temperature first, e.g. via H α , and the abundance via narrow, unblended Ca (e.g. 6156.02 ˚ A, 6166.439 ˚ A, 6169.042 ˚ A, 6455.985 ˚ A) and Mg (e.g. 5711.09 ˚ A) lines and iterate to a solution. The broad line wings of Na I D-lines (5889.97 ˚ A and 5895.94 ˚ A) are sensitive to both T eff , log g ⋆ (and the Na abundance). Hence these lines can be used to check the final model for consistency. More details and exampels of spectral modelling can for instance be found in Valenti and Fischer (2005), Jofr'e et al. (2014), and Brewer et al. (2016).","pages":[8,9,10,11]},{"title":"Fitting equivalent widths","content":"In contrast to synthetic spectral synthesis methods, the equivalent width (EW) method begins with the observed spectrum by measuring the strengths of selected absorption lines which are translated into individual line abundances. The method is based on theoretical atmosphere models and excitation and ionisation equilibrium which determines the population in a certain level of an atom or an ion. The EW of an absorption line is a convenient measurement of its strength. It is defined as the width of a rectangle reaching up to the continuum (with length one) having the same area as the spectral line. Measuring EWs of well-defined weak neutral iron (Fe I) and ionised iron (Fe II) lines is a traditional method to model stellar spectroscopic parameters since there are numerous iron lines in stellar spectra, many with accurate atomic parameters. Metal lines can be very sensitive to temperature although large variations between lines exist. Depending on spectral type, neutral metals are often used as a temperature indicator in solar-type stars, while ionised metal lines are better tracers in early-type stars. Assuming LTE, the ratio of population in two levels n and m of an atom (or ion) of a species can be computed with the Bolzmann equation (e.g. Gray 2008; Carroll and Ostlie 2017) where gn and gm are the statistical weights of the two levels, χ n and χ m are the corresponding excitation potentials, k is the Boltzmann constant, and T is the temperature. Different line ratios have different sensitivity to temperature. As the temperature increases so will ionisation which occurs quite abruptly once the threshold temperature is reached. In a star's photosphere, elements exist mainly in just two ionisation stages which can be computed with the Saha equation (e.g. Gray 2008; Carroll and Ostlie 2017) between ionisation states i and i + 1 as where the electron pressure is Pe = NekT (indicator of the surface gravity), me is the electron mass, h is Planck's constant, u ( T ) = ∑ gi e -χ i / kT is the partition function, and I is the ionisation energy. The EW method requires measurements of a large number of equivalent widths either measured by direct integration over the entire line or by fitting a Gaussian profile (or a Lorentzian profile that may fit optically thick lines better). Measuring EWs manually is, however, very time-consuming and therefore several softwares have been developed to automate the measurements for example ARES (Sousa et al. 2007) and DAOSPEC (Stetson and Pancino 2008). Weak and optically thin iron lines depend mainly on T eff and the iron abundance and less on log g ⋆ and V mic . In order to avoid abundance dependence with the EW method, it is best to choose two lines from the same element. However, since continuum normalisation is often a large source of error it may sometimes be necessary to use pairs of lines at nearby wavelengths. In these cases, pairs of similar elements like Fe, V, and Ti that normally have similar abundances, are often used. Different sets of lines are chosen for different spectral types since they are useful in different temperature ranges. The effective temperature is constrained in the following modelling by the correlation between the excitation potential and the iron abundance of each individual line, the microturbulence is constrained by the correlation between the abundance of each line and the reduced equivalent width, while the surface gravity is constrained by the ionisation balance. The parameters are adjusted until there are no correlations left and all individual abundances are the same (Sousa 2014). An example of a widely used open radiative transfer code based on the EW method is MOOG (Sneden 1973). This software uses a grid of the ATLAS9 (Ku- rucz 1993a) plane-parallel model atmospheres to produce model spectra which is compared to the measured EWs of individual Fe I and Fe II lines. The equilibrium conditions are solved simultaneously to derive T eff , log g ⋆ , [Fe/H], and V mic . A good description of the process is found in Sousa (2014). It has been used by for instance Sousa et al. (2021) together with ARES for a homogeneous spectroscopic characterisation of almost one thousand exoplanet host stars (the SWEETCat online catalogue). Another example of a software based on the EW method is ODUSSEAS (Observing Dwarfs Using Stellar Spectroscopic Energy-Absorption Shapes; Antoniadis-Karnavas et al. 2020). This code uses the machine learning Python package scikit-learn to offer a quick and automatic derivation of T eff and [Fe/H] for M dwarfs from optical spectroscopy. When comparing the outcome of the EW and models based on synthetic spectra, T eff and log g ⋆ normally agree within the uncertainties and also with results from asteroseismology within 100 K and 0.1 dex, respectively. The methods, however, do not perform equally well for all spectral types. Since the EW method is based on differential analysis with respect to the Sun, it can be applied to FGK stars with T eff ≈ 4500 -6400 K (Sousa et al. 2011). Modelling based on synthetic spectra and the line shapes are also sensitive to spectral type since some of the fundamental traits, e.g. the broad H α line wings, can only be made for solar-type stars since the line wings disappear for colder and hotter stars. In these cases it may still, however, be possible to use other spectral lines (e.g. TiO for low-mass stars). Furthermore, the EWmethod cannot be used on fast-rotator stars because of the severe line blending. For these stars, it may still be possible to use the synthetic method. Another difference is that the EW method is normally much faster than the synthetic method, while synthetic models provide a more complete description of the star.","pages":[11,12,13]},{"title":"Empirical methods","content":"Instead of the above methods, it is also possible to compare an observed highresolution spectrum to spectra of well-characterised stars and in this way derive stellar parameters. An example of such software is Spechmatch-emp (Yee et al. 2017) that compares an observed optical spectrum with a dense empirical spectral library thereby deriving R ⋆ (or log g ⋆ ), T eff , and [Fe/H]. The library contains 404 stars observed with the HIRES instrument with R ≈ 60000 at the Keck telescope. The properties of all the library stars have been derived from asteroseismoloy, interferometry, spectrophotometry and LTE spectral synthesis and represents spectral types from approximately M5 to F1. This method performs very well for the difficult late type stars (K4 stars and later) which are challenging for other methods reaching accuracies of 10 % in stellar radius, 70 K in effective temperature, and 0.09 dex in metallicity ([Fe/H]). The software and the library are publicly available.","pages":[13]},{"title":"Stellar radius and mass","content":"Once we have obtained the stellar spectroscopic parameters for our host star we now want to find out the radius and the mass of the star.","pages":[14]},{"title":"Radius from spectroscopy and spectral energy distribution","content":"When spectroscopic parameters and photometric measurements are available it is straightforward to obtain the stellar radius from a fit of the observed magnitudes in different bands to its spectral energy distribution (SED). In addition to magnitudes, the SED also depends on the distance (most accurately computed from the observed parallax), the spectroscopic parameters (derived from spectroscopic modelling), the extinction along the line-of-sight, and the stellar radius which is a free parameter in the fit. Parallax measurements have been performed by the European space missions Hipparchos (van Leeuwen 1997) and Gaia (Gaia Collaboration et al. 2016) launched in 1989 and 2013, respectively. The latter mission have provided the community with astrometric and photometric measurements of almost two billion stars in the Milky Way. Gaia has also delivered excellent measurements of the magnitudes in the Gaia optical band. Observations will end in early 2025. Figure 3 shows an example of a SED fit with the Phoenix (Husser et al. 2013) atmospheric model grid. An example of a publicly available software that automatically fits broadband photometry to six different stellar atmosphere models using Nested Sampling algorithms is the ARIADNE (Vines and Jenkins 2022) software.","pages":[14]},{"title":"Stellar mass","content":"Unless the star is a binary, the mass cannot usually be determined directly but can be modeled via stellar evolution models. Some of the most popular models are BaSTI (Hidalgo et al. 2018), Padova (Bertelli et al. 2008, 2009), DESP (the Dartmouth Stellar Evolution Database; Dotter et al. 2008), MIST (MESA Isochrones and Stellar Tracks; Choi et al. 2016), PARSEC (the PAdova and TRieste Stellar Evolution Code; Bressan et al. 2012), and Y 2 (Yonsei-Yale; Yi et al. 2003; Demarque et al. 2004). A quick way to obtain a Bayesian estimation of both mass and radius based on PARSEC and MESA (Rodrigues et al. 2017) models are the web interfaces Param1.3 and Param1.5 (da Silva et al. 2006). Another publicly available software that provides a simple interface for MCMC fitting of MIST stellar model grids is isochrones (Morton 2015). A complementing method is to use a set of stars with well-known masses and radii, e.g. through interferometry and eclipsing binaries, to derive empirical calibration equations. Such equations can give the mass and radius of a star given a set of stellar atmosphere values (e.g. Torres et al. 2010; Enoch et al. 2010; Southworth 2011).","pages":[14,15]},{"title":"Final checks","content":"If a planet is transiting a final important check can be made. The stellar density obtained from the above radius and mass can be checked against the density obtained from transit photometry and Kepler's third law. Assuming a circular orbit we have (Seager and Mall'en-Ornelas 2003) where G is the gravitational constant, P rot is the orbital period, a is the semi-major axis, and R ⋆ is the stellar radius. If this density does not agree with the above derived value, you need to check your modelling again. Equation 8 can be modified to include the eccentricity if known (e.g. Tingley et al. 2011). Another checkpoint is to compare the stellar mass computed from the spectroscopic log g ⋆ combined with R ⋆ which should be consistent with the final mass from other methods. A final remark: if possible, it is best to use several methods to make sure that the derived stellar parameters are robust and consistent.","pages":[15,16]},{"title":"References","content":"Dorn C, Khan A, Heng K et al. (2015) Can we constrain the interior structure of rocky exoplanets from mass and radius measurements? A&A577:A83 Dotter A (2016) MESA Isochrones and Stellar Tracks (MIST) 0: Methods for the Construction of Stellar Isochrones. ApJS222(1):8 Dotter A, Chaboyer B, Jevremovi'c D et al. (2008) The Dartmouth Stellar Evolution Database. ApJS178:89-101 Doyle AP, Davies GR, Smalley B, Chaplin WJ Elsworth Y (2014) Determining stellar macroturbulence using asteroseismic rotational velocities from Kepler. MNRAS444:3592-3602 Enoch B, Collier Cameron A, Parley NR Hebb L (2010) An improved method for estimating the masses of stars with transiting planets. A&A516:A33 Mignon L, Meunier N, Delfosse X et al. (2023) Characterisation of stellar activity of M dwarfs. I. Long-timescale variability in a large sample and detection of new cycles. A&A675:A168 Mihalas D (1978) Stellar atmospheres Nordstrom B, Mayor M, Andersen J et al. (2004) The Geneva-Copenhagen survey of the Solar neighbourhood. Ages, metallicities, and kinematic properties of ∼ 14 000 F and G dwarfs. A&A418:989-1019 Pepe F, Cristiani S, Rebolo R et al. (2021) ESPRESSO at VLT. On-sky performance and first results. A&A645:A96 Persson CM, Georgieva IY, Gandolfi D et al. (2022) TOI-2196 b: Rare planet in the hot Neptune desert transiting a G-type star. A&A666:A184 Piskunov N Valenti JA (2017) Spectroscopy Made Easy: Evolution. A&A597:A16 Plez B (2012) Turbospectrum: Code for spectral synthesis. Astrophysics Source Code Library, record ascl:1205.004 Quirrenbach A (2001) Optical Interferometry. ARA&A39:353-401 Rodrigues TS, Bossini D, Miglio A et al. (2017) Determining stellar parameters of asteroseismic targets: going beyond the use of scaling relations. MNRAS467(2):1433-1448 Ryabchikova T, Piskunov N, Kurucz RL et al. (2015) A major upgrade of the VALD database. Phys Scr90(5):054005 Seager S Mall'en-Ornelas G (2003) A Unique Solution of Planet and Star Parameters from an Extrasolar Planet Transit Light Curve. ApJ585:1038-1055 Sousa SG, Santos NC, Israelian G, Mayor M Monteiro MJPFG (2007) A new code for automatic determination of equivalent widths: Automatic Routine for line Equivalent widths in stellar Spectra (ARES). A&A469:783-791 Sousa SG, Santos NC, Israelian G, Mayor M Udry S (2011) Spectroscopic stellar parameters for 582 FGK stars in the HARPS volume-limited sample. Revising the metallicity-planet correlation. A&A533:A141 Sousa SG, Adibekyan V, Delgado-Mena E et al. (2021) SWEET-Cat 2.0: The Cat just got SWEETer. Higher quality spectra and precise parallaxes from Gaia eDR3. A&A656:A53 Southworth J (2011) Homogeneous studies of transiting extrasolar planets - IV. Thirty systems with space-based light curves. MNRAS417:2166-2196 Stetson PB Pancino E (2008) DAOSPEC: An Automatic Code for Measuring Equivalent Widths in High-Resolution Stellar Spectra. PASP120(874):1332 Telting JH, Avila G, Buchhave L et al. (2014) FIES: The high-resolution Fiber-fed Echelle Spectrograph at the Nordic Optical Telescope. Astronomische Nachrichten 335:41 Tingley B, Bonomo AS Deeg HJ (2011) Using Stellar Densities to Evaluate Transiting Exoplanetary Candidates. ApJ726(2):112 Tokovinin A, Fischer DA, Bonati M et al. (2013) CHIRON-A Fiber Fed Spectrometer for Precise Radial Velocities. PASP125(933):1336 Torres G, Andersen J Gim'enez A (2010) Accurate masses and radii of normal stars: modern results and applications. A&A Rev18(1-2):67-126 Torres G, Fischer DA, Sozzetti A et al. (2012) Improved Spectroscopic Parameters for Transiting Planet Hosts. ApJ757(2):161 Tull RG, MacQueen PJ, Sneden C Lambert DL (1995) The high-resolution cross-dispersed echelle white-pupil spectrometer of the McDonald Observatory 2.7-m telescope. PASP107:251-264 Valenti JA Fischer DA (2005) Spectroscopic Properties of Cool Stars (SPOCS). I. 1040 F, G, and K Dwarfs from Keck, Lick, and AAT Planet Search Programs. ApJS159:141-166 Valenti JA Piskunov N (1996) Spectroscopy made easy: A new tool for fitting observations with synthetic spectra. A&AS118:595-603 Valenti JA, Piskunov N Johns-Krull CM (1998) Spectral Synthesis of TiO Lines. ApJ498(2):851862 van Leeuwen F (1997) The HIPPARCOS Mission. Space Sci Rev81:201-409 Vines JI Jenkins JS (2022) ARIADNE: measuring accurate and precise stellar parameters through SED fitting. MNRAS513(2):2719-2731 Vogt SS, Allen SL, Bigelow BC et al. (1994) HIRES: the high-resolution echelle spectrometer on the Keck 10-m Telescope. In: Crawford DL Craine ER (eds) Instrumentation in Astronomy VIII, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol 2198, p 362, DOI 10.1117/12.176725 Wehrhahn A, Piskunov N Ryabchikova T (2023) PySME. Spectroscopy Made Easier. A&A671:A171 Yee SW, Petigura EA von Braun K (2017) Precision Stellar Characterization of FGKM Stars using an Empirical Spectral Library. ApJ836:77","pages":[17,18,19]}],"string":"[\n {\n \"title\": \"Methods to Characterise Exoplanet Host Stars from Spectroscopy\",\n \"content\": \"Carina M. Persson Abstract A key to understand exoplanets is characterisation of their host stars. One of the most powerful tools to characterise stellar properties like effective temperature, surface gravity and metallicity, is spectroscopy based on observations of stellar atmospheres. This chapter describes the stellar parameters that can be derived from a spectrum with examples of well established methods and theoretical model atmospheres. Combined with photometry and parallax measurements, the outcome of the spectroscopic modelling can be used to derive stellar radii and masses.\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"Introduction\",\n \"content\": \"Exoplanets are intimately connected to their host stars through formation and evolution. In addition, detection and characterisation of exoplanets depend on detailed knowledge of their host stars since the current major detection techniques, transit photometry and the radial velocity (RV) method, detect planet sizes and masses relative to their host star (see Chapters by Deeg & Alonso and Wright in Volume 1 of the Handbook of Exoplanets). Uncertainties in a host star's parameters propagate directly to the planets. Stellar modelling is normally based on two major techniques - photometry and spectroscopy. These methods are model-dependent in contrast to the direct measurements by interferometry, eclipsing binaries, and asteroseismology. For eclipsing binaries (e.g. Andersen 1991; Torres et al. 2010; Serenelli et al. 2021), and for the few large and nearby stars that enable interferometric measurements (e.g. Quirrenbach 2001; Eisenhauer et al. 2023), it is possible to derive the stellar radius with an accuracy of a few per cent. The timing of the duration of the eclipses of eclipsing binaries and their orbital velocities allows accurate estimates Chalmers University of Technology, department of Space, Earth, and Environment, Onsala space observatory, 439 92 ONSALA, Sweden, e-mail: carina.persson@chalmers.se of their sizes. Similarly, stellar masses can be accurately determined for visual binaries from observed separations from the common centre-of-mass with Kepler's third law that relates their masses with observed orbital period and separation. Asteroseismology can also be used to derive stellar mass, radius, and age with a high precision (see Chapter by Lundkvist, Huber, Aguirre & Chaplin in this volume of the Handbook of Exoplanets). Recently, the seismic surface gravity of the star has also been used to obtain the effective temperature and metallicity, in particular with APOGEE (Ahumada et al. 2020, 2022). These methods are, however, currently only possible to apply to a small subset of all stars. Spectroscopic measurements open a window to derive stellar parameters from a larger pool of stars than from direct methods. High-resolution spectroscopy is a powerful tool that provides a wealth of information; the effective temperature, surface gravity, chemical composition, and velocities. The stellar radius, and the luminosity, can then readily be derived from its spectral energy distribution (SED) via the spectroscopic parameters combined with photometry and parallax. The spectroscopic parameters also serve as a base to model characteristics that normally cannot be directly measured, like mass and age, with stellar evolution models and the complementary tool isochrones (e.g. Dotter 2016; Hidalgo et al. 2018). An isochrone is an evolutionary track for a population of stars with different masses on the Hertzsprung-Russell diagram with the same ( iso ) age ( chrone ). The mass and radius can also be obtained from the spectroscopic parameters and empirical calibration equations (e.g. Torres et al. 2010; Enoch et al. 2010; Southworth 2011) albeit often with a higher uncertainty. The downside of high-resolution spectroscopy is that it is expensive; the observations are time-consuming and also requires bright stars or a large collecting area. Modelling of stellar parameters has therefore traditionally been performed by photometry because a much larger number of stars can quickly be observed and analysed. The effective temperature of a star can for instance be derived from color-color diagrams which have been scaled to stars measured with direct methods (e.g. Bell and Gustafsson 1989; Alonso et al. 1996; Nordstrom et al. 2004). However, such models have in general higher uncertainties than based on spectroscopic measurements. In that respect they may only be reliable in a statistical sense and not for individual stars. Thus for exoplanet host stars (for which direct measurements are not applicable), high-resolution spectroscopy is preferred. This chapter begins with a short summary of a few basic requirements in order to obtain spectroscopic measurements of a star. It continues with an overview of the parameters that can be extracted from a high-resolution spectrum and a few examples of well established methods. Some advantages and caveats are highlighted. The chapter ends with a brief summary of how to combine spectroscopic parameters, photometric measurements, and stellar evolution models to obtain stellar radii and masses. Modelling of stellar ages are described by Christensen-Dalsgaard & Aguirre in a Chapter in this volume of the Handbook of Exoplanets.\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"Instruments and spectral resolution\",\n \"content\": \"Important properties of a spectrum is high-resolution, high signal-to-noise (S/N), and wavelength coverage. There are several different types of spectrometers and many textbooks have been written about this topic including advantages and challenges for different types of instruments (e.g. Gray 2008). The most successfull type in observations of exoplanets is echelle spectrographs. The main advantages is the high resolution combined with a wide wavelength coverage obtained in a single exposure. A description of high-precision cross-dispersed echelle spectrographs for exoplanet research (CORAVEL, ELODIE, CORALIE, SOPHIE and HARPS) is found in a Chapter by Fransesco Pepe in this volume of the Handbook of Exoplanets. For the purpose of spectroscopic modelling of host stars, we want high-resolution in order to resolve the spectral lines. The spectral resolution ∆λ at wavelength λ is where R is the resolving power of the spectrograph. It can be translated into a velocity resolution according to Doppler broadening of spectral lines due to thermal and turbulent motions of absorbing species in the atmospheres produce line widths of ∼ 6 kms -1 for latetype stars. This corresponds to a spectral resolution of 50 000. Low-mass, slowly rotating stars can have line widths of only ≈ 1 -2 km s -1 which require R ≳ 300000 to resolve the spectral lines and disentangle blended lines. In terms of high resolution, ultra-high precision, and long-term stability, the High Accuracy Radial Velocity Planet Searcher (HARPS; Mayor et al. 2003) and its decade younger sibling HARPS-North (Cosentino et al. 2012) mounted on the ESO 3.6 m telescope (La Silla observatory, Chile) and the Telescopio Nazionale Galileo (TNG) of Roque de los Muchachos Observatory (La Palma, Spain), respectively, have been the leading instruments in detecting exoplanets over the last two decades. Both HARPS instruments are fiber-fed cross-dispersed high-precision echelle spectrographs covering 380 nm to 690 nm. The spectroscopic resolution is approximately 115000 at visual wavelengths corresponding to a velocity resolution of 2.6 km s -1 . The design is based on experience with the groundbreaking ELODIE and CORALIE instruments where the former was used to detect the first exoplanet 51 Peg b (Mayor and Queloz 1995). Both HARPS spectrographs can be considered as the 'gold standard' when searching for exoplanets in RV data and are also used for characterisation of exoplanet host stars. When searching for exoplanets with the RV method, many measurements are collected, sometimes over a period over many years. The individual spectra can be co-added (after correcting for the peri- odic changes in radial velocity) in order to increase the S/N enabling spectroscopic characterisation of the host star. In addition to the HARPS spectrographs, there are many other instruments used for exoplanet detection that can also be used to characterise host stars e.g. ESPRESSO (Pepe et al. 2021), FIES (Telting et al. 2014), HIRES (Vogt et al. 1994), CHIRON (Tokovinin et al. 2013), TULL (Tull et al. 1995) with different resolutions operating at different wavelengths.\",\n \"pages\": [\n 3,\n 4\n ]\n },\n {\n \"title\": \"Stellar properties from spectroscopy\",\n \"content\": \"Only a few parameters characterise a stellar atmosphere: the effective temperature ( T eff ), the surface gravity (log g ⋆ ), the overall metal abundance ([M/H]), and atmospheric and rotational velocities. The surface of a star is defined as the location where photons escape from the star which occurs at a characteristic optical depth of 2/3. This occurs within the photosphere, the innermost ≈ 500 km of a star's atmosphere which overlies the opaque interior. The photospheric temperature and density varies with depth and depends on the surface gravity as well as the abundances and opacity of the gases. The Sun's photosphere has a temperature that varies between 4 400 K and 6 600 K with an effective temperature of 5 772 K, while the density is approximately 3 × 10 -4 kg m -3 , increasing with depth into the Sun. Other stars may have hotter or cooler photospheres. Spectral absorption lines originate from different depths and opacities within the photosphere. Weak and optically thin spectral lines, and the line wings of optically thick lines, originate essentially in the same layer as the continuum. In contrast, the cores of optically thick saturated lines develop in higher layers. An example is the core of the hydrogen α line at 6562.81 ˚ A which originates in the hotter and less dense chromosphere which lies on top of the photosphere, where the assumption of local thermal equilibrium (LTE) is no longer valid.\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"Effective temperature\",\n \"content\": \"Stars are classified according to effective temperature from the hot O-stars, also called early-type stars, to the cool M-stars (late-type stars). If a star is on the main sequence, the effective temperature immediately signals which type of star it is along with typical mass and radius. Instead of choosing a particular depth to define a star's surface temperature, the effective temperature is defined in terms of flux. The effective temperature is defined via the Stefan-Bolztmann law in terms of the total power per unit area, radiated by the star (e.g. Gray 2008) Here F ν is the total flux passing through the star's surface and σ is the Boltzmann constant. The effective temperature is thus the temperature of a black body having the same power output per unit area as a star. It is related to the flux we measure at Earth, f ν , via (Gray 2008) where the left integral is the total radiative flux from the star received at the top of the Earth's atmosphere (bolometric flux), R ⋆ is the stellar radius, d is the distance, and θ ⋆ is the angular diameter of the star. The effective temperature can hence be derived by measuring the angular size of the star from interferometry and the received flux at Earth over a wide spectral range. By combining the angular size with distance from parallax measurements, the linear radius of the star can be inferred for a wide range of spectral types nearly model-independent except for the dependence on the adopted limb-darkening coefficients and bolometric correction. Alternatively, if the radius of the star is known from e.g. eclipsing binaries and the distance from parallax measurments, this will give the angular size which then can be used to compute T eff . Since the target stars have to be nearby to measure their angular sizes, interstellar absorption can be neglected. The infrared flux method (IRFM Blackwell and Shallis 1977; Blackwell et al. 1980; Bell and Gustafsson 1989; Blackwell et al. 1990; Casagrande et al. 2006, 2010) is based on observations of the angular size of the star and the measured infrared flux at the top of the Earth's atmosphere. The bolometric flux is derived taking into account the bolometric correction which gives the effective temperature. In addition to the above methods, T eff can also be derived from spectroscopy as described in the following section. However, it is worth highlighting that the temperature derived from spectroscopy is a microscopic value which is close to, but not exactly the same as the effective temperature due to its definition being a macroscopic description.\",\n \"pages\": [\n 4,\n 5\n ]\n },\n {\n \"title\": \"Surface gravity\",\n \"content\": \"The surface gravity is an indication of the luminosity class of a star where V is the main sequence and I - IV is different types of giants. Thus surface gravity contains information of the size and age of a star. A low surface gravity immediately indicates that the star has left the main sequence. The surface gravity of a star is defined by (e.g. Gray 2008) where g ⊙ is the surface gravity of the Sun (2 . 740 × 10 4 cm s -2 ) and the radius, R ⋆ , and mass, M ⋆ , of the star are in solar units. The surface gravity is commonly measured in a logarithmic scale, log g ⋆ , where the solar value is 4.44. The surface gravity determines the gas density in the photosphere and is the spectroscopic parameter that has the highest impact on the stellar radius. Unfortunately, log g ⋆ is often poorly constrained by spectral analysis. Since the uncertainties of T eff and metallicity can be strongly correlated with surface gravity this can lead to a significant source of systematic error in some analysis techniques.\",\n \"pages\": [\n 5,\n 6\n ]\n },\n {\n \"title\": \"Metallicity\",\n \"content\": \"A stars chemical composition is an outcome of the nucleosynthesis by previous generations of stars. This is important when reconstructing star and planet formation history in our Galaxy. There is a large variation of metallicity, i.e. the abundance of all elements heavier than helium denoted with [M/H], in the Milky Way up to about twice the solar value to hundreds of thousands of times lower than the solar value (e.g. Christlieb et al. 2004; Li et al. 2022; Nepal et al. 2024). The chemical composition of a star is commonly fixed to the overall metallicity of a star relative to the Sun. The Grevesse et al. (2007), Asplund et al. (2009) and Lodders (2003) abundance scales for the Sun are currently the most adopted. However, individual abundances of a star may not follow solar composition and may require modelling of individual elemental abundances. Abundances are generally measured on a logarithmic scale normalised to the Sun where zero equals the Sun's metallicity. In the case of iron we have [Fe/H] ⋆ = log(Fe/H) ⋆ - log(Fe/H) ⊙ . For example, an iron abundance of [Fe/H] = -0 . 5 means that the abundance is 10 -0 . 5 relative to the Sun. Since iron is by far the most abundant species in a stellar atmosphere after hydrogen and helium, measurements of iron has become a proxy for the metallicity. Stellar abundances is also important when modelling exoplanet interiors in particular rocky super-Earths without significant gaseous envelopes. The degeneracy of interior composition inferred from radius and mass measurements can for this type of planet be reduced assuming an interior structure with a differentiated iron core and a rocky mantle. In these cases, the host star abundances is often used as a proxy of the primary planet-building elements Fe, Mg, and Si, which are expected to be reflected in the planet composition, planet interior, and core mass fraction (Dorn et al. 2015; Acu˜na et al. 2023).\",\n \"pages\": [\n 6\n ]\n },\n {\n \"title\": \"Velocities\",\n \"content\": \"Thermal widths of spectral lines are only a fraction of the observed line widths for dwarf stars hotter than spectral type K0. The line widths are instead mainly governed by Doppler shifts produced by motions of the star's photospheric gases. The radial velocity of the star is only shifting the wavelengths of all spectral lines in the observed spectrum compared to the observations and can easily be corrected for. The velocity that dominates the line shape and width for hot stars is the projected equatorial rotational velocity of the star, V sin i ⋆ , where i ⋆ is the inclination of the stellar rotation axis relative to the line of sight. It can be measured via the full width at half maximum (FWHM) of a large number of optically thin and unblended lines not sensitive to pressure broadening. The line shapes are, however, also affected by turbulence from convective motion, granulation, high-order pulsations, stellar activity, and other types of local flows in the photosphere. Turbulence is represented in the models by the macro-turbulent velocity ( V mac) that describes motions on scales larger than the mean free path within the photosphere that induce a change in the line shape; and the micro-turbulent velocity ( V mic ) that describes motions on scales smaller than the mean free path leading to increased line opacity (Gray 2008; Bruntt et al. 2010; Doyle et al. 2014). The latter velocity is a 'fudge' factor originally introduced to reconcile observed and predicted equivalent widths (Mihalas 1978). It includes all remaining types of broadening mechanisms and is at present standard to include in analyses of solar-type stars. Both turbulent velocities depend on temperature and to a lesser extent on surface gravity. The micro-turbulent velocity has a width of the order of 1 km s -1 for dwarfs and several km s -1 for giants. For lowmass stars, V mac and V sin i ⋆ have comparable widths of the order of a few km s -1 . As a reference, the Sun's V sin i ⋆ is 2 km s -1 at the equator while hotter stars have much higher rotational velocities (tens to hundreds of km s -1 ).\",\n \"pages\": [\n 6,\n 7\n ]\n },\n {\n \"title\": \"Methods for spectroscopic modelling\",\n \"content\": \"There are several ways to model a spectrum which can be divided into two main groups. The first is based on spectral synthesis. Here observations are fitted to a synthetic spectrum of stellar atmosphere models by comparison of line profiles. The second is a line-by-line analysis based on measured strengths of observed spectral lines, their equivalent widths (EWs). Detailed description of the physics can be found in many textbooks e.g. Gray (2008). Spectroscopic observations can also be compared to a library of spectra of well-characterised stars via for example interferometric measurements or spectroscopic binaries. One major problem in spectroscopic analysis is to accurately determine the continuum which can introduce large errors. This is particularly difficult for poor spectra with low spectral resolution or low S/N. It also depends on the spectral type of the star and the wavelength region. The number of spectral lines increases towards shorter wavelengths for all types of stars. In addition, late-type stars have a much higher density of spectral lines than early-type stars arising from both atoms and molecules leading to blending and confusion of the continuum location. The higher temperatures of early-type stars ionize a large fraction of their atoms, leading to significantly fewer spectral features than low-mass stars. Differences in rotational velocities also affect the spectral line density. In contrast to late-type stars, the early types have very high rotational velocities which leads to very broad spectral lines that smear out spectral features. Thus solar-type stars (FGK) are the easiest stars to model, while high- and low-mass stars often entails a significantly higher degree of difficulty in the modelling. M-dwarfs have in addition generally a much longer period of high stellar activity than FGK stars, exacerbating the problems (e.g. Mignon et al. 2023). This is unfortunate since M-dwarfs are popular exoplanet host stars due to their small masses and sizes which increase the exoplanet signals. Care must be taken when selecting which spectral lines to model. A large set of narrow, non-blended spectral lines are preferred (unless modelling pressure broadened line wings, see below). If a spectral line becomes optically thick, the abundance of a species stops growing linearly with absorption depth. The characteristics of an optically thick line is a saturated line centre which flattens the bottom and broaden the line wings. Not all optically thin spectral lines may, however, be useful since a large number comes with poorly determined atomic parameters which are needed to compute synthetic spectra. This can be circumvented for solar-type stars if adopting new atomic parameters after comparing the lines from observations of the Sun.\",\n \"pages\": [\n 7,\n 8\n ]\n },\n {\n \"title\": \"Fitting observations to synthetic spectra\",\n \"content\": \"Computations of a synthetic spectrum requires a model atmosphere based on solutions to the stellar structure equations to synthesize a spectrum. Most stellar atmosphere models are pre-calculated and tabulated on grids describing the profiles of the temperature, surface gravity and abundances as functions of atmospheric depth. Each layer in the model atmosphere is contributing to the formation of absorption line profiles in the final spectrum. Some widely used atmospheric model atmospheres are Atlas12 (Kurucz 2013), Atlas9 (Kurucz 1993a; Heiter et al. 2002), MARCS (Gustafsson et al. 2008) for cool and giant stars, and LL models (Shulyak et al. 2004) for hot main sequence stars. Line lists of atomic and molecular data needed in the computation can be provided by the Vienna Atomic Line Database (VALD3; Piskunov et al. 1995; Ryabchikova et al. 2015). A radiative transfer code then computes a synthetic model spectrum of the star for the chosen set of stellar parameters ( T eff , log g ⋆ , [Fe/H], V sin i ⋆ , V mic , V mac) which are matched against the observed spectra based on the spectral line shapes and strengths. The parameters generally affect either the line strength ( T eff , log g ⋆ , abundances, and V mic ) or the line shape ( V mac, V sin i ⋆ , and the instrumental resolution). Degeneracies are stronger within the subsets. The dependence of the line profile on the V mac parameter is broadened line wings and a cusp-shaped core. Unfortunately, disentangling the effect on the line profile from V sin i ⋆ and V mac is difficult, leading to a degeneracy between the two. Prior information of V sin i ⋆ may be obtained from time-resolved photometry, available for the known transiting planets, or from asteroseismology (cf. Doyle et al. 2014). If no prior information is available, calibration equations of both turbulent velocities are often used (e.g. Bruntt et al. 2010; Doyle et al. 2014) which allows modelling of V sin i ⋆ . Note that in order to properly model the different velocities above, it is imperative to take into account how the spectrograph itself broadens the lines. Spectral lines can also be pressure broadened through various mechanisms which further broaden the lines, e.g. the Balmer lines. Details of the various broadening mechanisms can be found in many textbooks such as Gray (2008). There are many softwares that computes synthetic spectra to be used as a model constraint to interpret the observed spectrum. The popular open-source spectroscopic tool iSpec (Blanco-Cuaresma et al. 2014; Blanco-Cuaresma 2019) support several of the most well-known radiative transfer codes such as Spectrocsopy Made Easy ( SME ; Valenti and Piskunov 1996; Piskunov and Valenti 2017; Wehrhahn et al. 2023), SPECTRUM (Gray and Corbally 1994), Turbospectrum (Plez 2012; de Laverny et al. 2012; Gerber et al. 2023), Synthe/WIDTH9 (Kurucz 1993b), and MOOG (Sneden 1973) where the latter is based on the equivalent width method. The codes often adopt LTE and a plane-parallel geometry as default motivated by the fact that for main-sequence stars, the photosphere constitutes ≪ 1 % of the stellar radius. Aspherically symmetric geometry is on the other hand required for giant stars where the atmosphere makes up a substantial portion of its radius (Heiter and Eriksson 2006). However, since complex interaction of gas particles and the non-local radiation fields leads to deviations from LTE in the atmospheres of FGKM-type stars, some of the softwares have also the option of non-local thermodynamic equilibrium (NLTE). For instance, SME includes NLTE departure coefficients for the MARCS and LL models atmospheres (Piskunov and Valenti 2017), and Turbospectrum also have the option of NLTE (Gerber et al. 2023). Fitting can in several of the softwares be made for one or several parameters at the same time using a χ 2 -minimization algorithm. However, care must be taken when solving for several parameters simultaneously (Torres et al. 2012) due to degeneracies and difficulties in the modelling. Different initial assumptions of one free parameter at a time can therefore be made in the fitting process to iterate to the final solution, thereby mitigating degeneracies. Also, since the surface gravity is often difficult to constrain it is thus advantageous to have external information about the stellar density that can facilitate modelling. Spectroscopic modelling can take advantage of the sensitivity of certain spectral lines to specific parameters. For instance, all Balmer lines exhibit pressure (collisionally) broadened line wings which makes their profiles strongly sensitive to temperature (e.g. Fuhrmann et al. 1993, 1994; Barklem et al. 2000, 2002) as shown in Fig. 1. The Balmer line wings are formed in the deepest photospheric layers likely close to the LTE. The metal-line blending increase from H α at 6562.81 ˚ A to H δ at 4101.75 ˚ A which perturb the higher transitions of the Balmer line profiles making H α the best choice. For late F, G, and early K-stars, H α is very insensitive to log g ⋆ and abundance making its line wings an excellent T eff indicator. Two examples of H α line profiles towards a K5V and a G0V host star are shown in Fig. 1. It is obvious that the line wings have almost completely disappeared in the K5V spectrum, while prominent and very broad towards the G-star. For M-dwarfs where the line wings of H α are absent, TiO lines can also be used as an T eff indicator (Valenti et al. 1998). For late F- and G-dwarfs there are several pressure-sensitive lines that can be used to constrain log g ⋆ : the Mg I triplet at 5167.33 ˚ A, 5172.70 ˚ A and 5183.62 ˚ A (e.g. Fuhrmann et al. 1997; Valenti and Fischer 2005) as shown in Fig. 2, and the Ca I lines at 6122.23 ˚ A and 6162.18 ˚ A (e.g. Gray 2008). For lower gravity or higher temperature, the line wings disappear due to lower photospheric density or ionisa- he Mg I lines are, however, very wide which makes normalization difficult. Two of the lines lie very close without continuum between them, and there are numerous overlying narrow metal lines in particular for late-type stars. In these cases, only the 5183.62 ˚ A line can be used reliably. Figure 2 shows the Mg I triplet line profiles towards the same host stars as in Fig. 1. As already seen in Fig. 1, the spectral line density is much higher towards the K5V star than the G0V star. However, the effect is even more pronounciated at shorter wavelengths. In addition to surface gravity, the Ca and Mg pressure broadened line wings are, however, to some degree also sensitive to temperature and abundance. The procedure is therefore to model the temperature first, e.g. via H α , and the abundance via narrow, unblended Ca (e.g. 6156.02 ˚ A, 6166.439 ˚ A, 6169.042 ˚ A, 6455.985 ˚ A) and Mg (e.g. 5711.09 ˚ A) lines and iterate to a solution. The broad line wings of Na I D-lines (5889.97 ˚ A and 5895.94 ˚ A) are sensitive to both T eff , log g ⋆ (and the Na abundance). Hence these lines can be used to check the final model for consistency. More details and exampels of spectral modelling can for instance be found in Valenti and Fischer (2005), Jofr'e et al. (2014), and Brewer et al. (2016).\",\n \"pages\": [\n 8,\n 9,\n 10,\n 11\n ]\n },\n {\n \"title\": \"Fitting equivalent widths\",\n \"content\": \"In contrast to synthetic spectral synthesis methods, the equivalent width (EW) method begins with the observed spectrum by measuring the strengths of selected absorption lines which are translated into individual line abundances. The method is based on theoretical atmosphere models and excitation and ionisation equilibrium which determines the population in a certain level of an atom or an ion. The EW of an absorption line is a convenient measurement of its strength. It is defined as the width of a rectangle reaching up to the continuum (with length one) having the same area as the spectral line. Measuring EWs of well-defined weak neutral iron (Fe I) and ionised iron (Fe II) lines is a traditional method to model stellar spectroscopic parameters since there are numerous iron lines in stellar spectra, many with accurate atomic parameters. Metal lines can be very sensitive to temperature although large variations between lines exist. Depending on spectral type, neutral metals are often used as a temperature indicator in solar-type stars, while ionised metal lines are better tracers in early-type stars. Assuming LTE, the ratio of population in two levels n and m of an atom (or ion) of a species can be computed with the Bolzmann equation (e.g. Gray 2008; Carroll and Ostlie 2017) where gn and gm are the statistical weights of the two levels, χ n and χ m are the corresponding excitation potentials, k is the Boltzmann constant, and T is the temperature. Different line ratios have different sensitivity to temperature. As the temperature increases so will ionisation which occurs quite abruptly once the threshold temperature is reached. In a star's photosphere, elements exist mainly in just two ionisation stages which can be computed with the Saha equation (e.g. Gray 2008; Carroll and Ostlie 2017) between ionisation states i and i + 1 as where the electron pressure is Pe = NekT (indicator of the surface gravity), me is the electron mass, h is Planck's constant, u ( T ) = ∑ gi e -χ i / kT is the partition function, and I is the ionisation energy. The EW method requires measurements of a large number of equivalent widths either measured by direct integration over the entire line or by fitting a Gaussian profile (or a Lorentzian profile that may fit optically thick lines better). Measuring EWs manually is, however, very time-consuming and therefore several softwares have been developed to automate the measurements for example ARES (Sousa et al. 2007) and DAOSPEC (Stetson and Pancino 2008). Weak and optically thin iron lines depend mainly on T eff and the iron abundance and less on log g ⋆ and V mic . In order to avoid abundance dependence with the EW method, it is best to choose two lines from the same element. However, since continuum normalisation is often a large source of error it may sometimes be necessary to use pairs of lines at nearby wavelengths. In these cases, pairs of similar elements like Fe, V, and Ti that normally have similar abundances, are often used. Different sets of lines are chosen for different spectral types since they are useful in different temperature ranges. The effective temperature is constrained in the following modelling by the correlation between the excitation potential and the iron abundance of each individual line, the microturbulence is constrained by the correlation between the abundance of each line and the reduced equivalent width, while the surface gravity is constrained by the ionisation balance. The parameters are adjusted until there are no correlations left and all individual abundances are the same (Sousa 2014). An example of a widely used open radiative transfer code based on the EW method is MOOG (Sneden 1973). This software uses a grid of the ATLAS9 (Ku- rucz 1993a) plane-parallel model atmospheres to produce model spectra which is compared to the measured EWs of individual Fe I and Fe II lines. The equilibrium conditions are solved simultaneously to derive T eff , log g ⋆ , [Fe/H], and V mic . A good description of the process is found in Sousa (2014). It has been used by for instance Sousa et al. (2021) together with ARES for a homogeneous spectroscopic characterisation of almost one thousand exoplanet host stars (the SWEETCat online catalogue). Another example of a software based on the EW method is ODUSSEAS (Observing Dwarfs Using Stellar Spectroscopic Energy-Absorption Shapes; Antoniadis-Karnavas et al. 2020). This code uses the machine learning Python package scikit-learn to offer a quick and automatic derivation of T eff and [Fe/H] for M dwarfs from optical spectroscopy. When comparing the outcome of the EW and models based on synthetic spectra, T eff and log g ⋆ normally agree within the uncertainties and also with results from asteroseismology within 100 K and 0.1 dex, respectively. The methods, however, do not perform equally well for all spectral types. Since the EW method is based on differential analysis with respect to the Sun, it can be applied to FGK stars with T eff ≈ 4500 -6400 K (Sousa et al. 2011). Modelling based on synthetic spectra and the line shapes are also sensitive to spectral type since some of the fundamental traits, e.g. the broad H α line wings, can only be made for solar-type stars since the line wings disappear for colder and hotter stars. In these cases it may still, however, be possible to use other spectral lines (e.g. TiO for low-mass stars). Furthermore, the EWmethod cannot be used on fast-rotator stars because of the severe line blending. For these stars, it may still be possible to use the synthetic method. Another difference is that the EW method is normally much faster than the synthetic method, while synthetic models provide a more complete description of the star.\",\n \"pages\": [\n 11,\n 12,\n 13\n ]\n },\n {\n \"title\": \"Empirical methods\",\n \"content\": \"Instead of the above methods, it is also possible to compare an observed highresolution spectrum to spectra of well-characterised stars and in this way derive stellar parameters. An example of such software is Spechmatch-emp (Yee et al. 2017) that compares an observed optical spectrum with a dense empirical spectral library thereby deriving R ⋆ (or log g ⋆ ), T eff , and [Fe/H]. The library contains 404 stars observed with the HIRES instrument with R ≈ 60000 at the Keck telescope. The properties of all the library stars have been derived from asteroseismoloy, interferometry, spectrophotometry and LTE spectral synthesis and represents spectral types from approximately M5 to F1. This method performs very well for the difficult late type stars (K4 stars and later) which are challenging for other methods reaching accuracies of 10 % in stellar radius, 70 K in effective temperature, and 0.09 dex in metallicity ([Fe/H]). The software and the library are publicly available.\",\n \"pages\": [\n 13\n ]\n },\n {\n \"title\": \"Stellar radius and mass\",\n \"content\": \"Once we have obtained the stellar spectroscopic parameters for our host star we now want to find out the radius and the mass of the star.\",\n \"pages\": [\n 14\n ]\n },\n {\n \"title\": \"Radius from spectroscopy and spectral energy distribution\",\n \"content\": \"When spectroscopic parameters and photometric measurements are available it is straightforward to obtain the stellar radius from a fit of the observed magnitudes in different bands to its spectral energy distribution (SED). In addition to magnitudes, the SED also depends on the distance (most accurately computed from the observed parallax), the spectroscopic parameters (derived from spectroscopic modelling), the extinction along the line-of-sight, and the stellar radius which is a free parameter in the fit. Parallax measurements have been performed by the European space missions Hipparchos (van Leeuwen 1997) and Gaia (Gaia Collaboration et al. 2016) launched in 1989 and 2013, respectively. The latter mission have provided the community with astrometric and photometric measurements of almost two billion stars in the Milky Way. Gaia has also delivered excellent measurements of the magnitudes in the Gaia optical band. Observations will end in early 2025. Figure 3 shows an example of a SED fit with the Phoenix (Husser et al. 2013) atmospheric model grid. An example of a publicly available software that automatically fits broadband photometry to six different stellar atmosphere models using Nested Sampling algorithms is the ARIADNE (Vines and Jenkins 2022) software.\",\n \"pages\": [\n 14\n ]\n },\n {\n \"title\": \"Stellar mass\",\n \"content\": \"Unless the star is a binary, the mass cannot usually be determined directly but can be modeled via stellar evolution models. Some of the most popular models are BaSTI (Hidalgo et al. 2018), Padova (Bertelli et al. 2008, 2009), DESP (the Dartmouth Stellar Evolution Database; Dotter et al. 2008), MIST (MESA Isochrones and Stellar Tracks; Choi et al. 2016), PARSEC (the PAdova and TRieste Stellar Evolution Code; Bressan et al. 2012), and Y 2 (Yonsei-Yale; Yi et al. 2003; Demarque et al. 2004). A quick way to obtain a Bayesian estimation of both mass and radius based on PARSEC and MESA (Rodrigues et al. 2017) models are the web interfaces Param1.3 and Param1.5 (da Silva et al. 2006). Another publicly available software that provides a simple interface for MCMC fitting of MIST stellar model grids is isochrones (Morton 2015). A complementing method is to use a set of stars with well-known masses and radii, e.g. through interferometry and eclipsing binaries, to derive empirical calibration equations. Such equations can give the mass and radius of a star given a set of stellar atmosphere values (e.g. Torres et al. 2010; Enoch et al. 2010; Southworth 2011).\",\n \"pages\": [\n 14,\n 15\n ]\n },\n {\n \"title\": \"Final checks\",\n \"content\": \"If a planet is transiting a final important check can be made. The stellar density obtained from the above radius and mass can be checked against the density obtained from transit photometry and Kepler's third law. Assuming a circular orbit we have (Seager and Mall'en-Ornelas 2003) where G is the gravitational constant, P rot is the orbital period, a is the semi-major axis, and R ⋆ is the stellar radius. If this density does not agree with the above derived value, you need to check your modelling again. Equation 8 can be modified to include the eccentricity if known (e.g. Tingley et al. 2011). Another checkpoint is to compare the stellar mass computed from the spectroscopic log g ⋆ combined with R ⋆ which should be consistent with the final mass from other methods. A final remark: if possible, it is best to use several methods to make sure that the derived stellar parameters are robust and consistent.\",\n \"pages\": [\n 15,\n 16\n ]\n },\n {\n \"title\": \"References\",\n \"content\": \"Dorn C, Khan A, Heng K et al. (2015) Can we constrain the interior structure of rocky exoplanets from mass and radius measurements? A&A577:A83 Dotter A (2016) MESA Isochrones and Stellar Tracks (MIST) 0: Methods for the Construction of Stellar Isochrones. ApJS222(1):8 Dotter A, Chaboyer B, Jevremovi'c D et al. (2008) The Dartmouth Stellar Evolution Database. ApJS178:89-101 Doyle AP, Davies GR, Smalley B, Chaplin WJ Elsworth Y (2014) Determining stellar macroturbulence using asteroseismic rotational velocities from Kepler. MNRAS444:3592-3602 Enoch B, Collier Cameron A, Parley NR Hebb L (2010) An improved method for estimating the masses of stars with transiting planets. A&A516:A33 Mignon L, Meunier N, Delfosse X et al. (2023) Characterisation of stellar activity of M dwarfs. I. Long-timescale variability in a large sample and detection of new cycles. A&A675:A168 Mihalas D (1978) Stellar atmospheres Nordstrom B, Mayor M, Andersen J et al. (2004) The Geneva-Copenhagen survey of the Solar neighbourhood. Ages, metallicities, and kinematic properties of ∼ 14 000 F and G dwarfs. A&A418:989-1019 Pepe F, Cristiani S, Rebolo R et al. (2021) ESPRESSO at VLT. On-sky performance and first results. A&A645:A96 Persson CM, Georgieva IY, Gandolfi D et al. (2022) TOI-2196 b: Rare planet in the hot Neptune desert transiting a G-type star. A&A666:A184 Piskunov N Valenti JA (2017) Spectroscopy Made Easy: Evolution. A&A597:A16 Plez B (2012) Turbospectrum: Code for spectral synthesis. Astrophysics Source Code Library, record ascl:1205.004 Quirrenbach A (2001) Optical Interferometry. ARA&A39:353-401 Rodrigues TS, Bossini D, Miglio A et al. (2017) Determining stellar parameters of asteroseismic targets: going beyond the use of scaling relations. MNRAS467(2):1433-1448 Ryabchikova T, Piskunov N, Kurucz RL et al. (2015) A major upgrade of the VALD database. Phys Scr90(5):054005 Seager S Mall'en-Ornelas G (2003) A Unique Solution of Planet and Star Parameters from an Extrasolar Planet Transit Light Curve. ApJ585:1038-1055 Sousa SG, Santos NC, Israelian G, Mayor M Monteiro MJPFG (2007) A new code for automatic determination of equivalent widths: Automatic Routine for line Equivalent widths in stellar Spectra (ARES). A&A469:783-791 Sousa SG, Santos NC, Israelian G, Mayor M Udry S (2011) Spectroscopic stellar parameters for 582 FGK stars in the HARPS volume-limited sample. Revising the metallicity-planet correlation. A&A533:A141 Sousa SG, Adibekyan V, Delgado-Mena E et al. (2021) SWEET-Cat 2.0: The Cat just got SWEETer. Higher quality spectra and precise parallaxes from Gaia eDR3. A&A656:A53 Southworth J (2011) Homogeneous studies of transiting extrasolar planets - IV. Thirty systems with space-based light curves. MNRAS417:2166-2196 Stetson PB Pancino E (2008) DAOSPEC: An Automatic Code for Measuring Equivalent Widths in High-Resolution Stellar Spectra. PASP120(874):1332 Telting JH, Avila G, Buchhave L et al. (2014) FIES: The high-resolution Fiber-fed Echelle Spectrograph at the Nordic Optical Telescope. Astronomische Nachrichten 335:41 Tingley B, Bonomo AS Deeg HJ (2011) Using Stellar Densities to Evaluate Transiting Exoplanetary Candidates. ApJ726(2):112 Tokovinin A, Fischer DA, Bonati M et al. (2013) CHIRON-A Fiber Fed Spectrometer for Precise Radial Velocities. PASP125(933):1336 Torres G, Andersen J Gim'enez A (2010) Accurate masses and radii of normal stars: modern results and applications. A&A Rev18(1-2):67-126 Torres G, Fischer DA, Sozzetti A et al. (2012) Improved Spectroscopic Parameters for Transiting Planet Hosts. ApJ757(2):161 Tull RG, MacQueen PJ, Sneden C Lambert DL (1995) The high-resolution cross-dispersed echelle white-pupil spectrometer of the McDonald Observatory 2.7-m telescope. PASP107:251-264 Valenti JA Fischer DA (2005) Spectroscopic Properties of Cool Stars (SPOCS). I. 1040 F, G, and K Dwarfs from Keck, Lick, and AAT Planet Search Programs. ApJS159:141-166 Valenti JA Piskunov N (1996) Spectroscopy made easy: A new tool for fitting observations with synthetic spectra. A&AS118:595-603 Valenti JA, Piskunov N Johns-Krull CM (1998) Spectral Synthesis of TiO Lines. ApJ498(2):851862 van Leeuwen F (1997) The HIPPARCOS Mission. Space Sci Rev81:201-409 Vines JI Jenkins JS (2022) ARIADNE: measuring accurate and precise stellar parameters through SED fitting. MNRAS513(2):2719-2731 Vogt SS, Allen SL, Bigelow BC et al. (1994) HIRES: the high-resolution echelle spectrometer on the Keck 10-m Telescope. In: Crawford DL Craine ER (eds) Instrumentation in Astronomy VIII, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol 2198, p 362, DOI 10.1117/12.176725 Wehrhahn A, Piskunov N Ryabchikova T (2023) PySME. Spectroscopy Made Easier. A&A671:A171 Yee SW, Petigura EA von Braun K (2017) Precision Stellar Characterization of FGKM Stars using an Empirical Spectral Library. ApJ836:77\",\n \"pages\": [\n 17,\n 18,\n 19\n ]\n }\n]"}}},{"rowIdx":4977,"cells":{"bibcode":{"kind":"string","value":"2024arXiv241119893Y"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/2411.19893.pdf"},"content":{"kind":"string","value":"\nA Galaxy with an Extremely Blue UV Slope β = -3 at z = 9 . 25 Identified by JWST Spectroscopy: Evidence for a Weak Nebular Continuum and Efficient Ionizing Photon Escape?\nHiroto Yanagisawa , 1, 2 Masami Ouchi , 3, 1, 4, 5 Kimihiko Nakajima , 3 Yuichi Harikane , 1 Seiji Fujimoto , 6, 7, 8 Yoshiaki Ono , 1 Hiroya Umeda , 1, 2 Minami Nakane , 1, 2 Hidenobu Yajima , 9 Hajime Fukushima , 9 and Yi Xu 1, 10\n1 Institute for Cosmic Ray Research, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan 2 Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan 3 National Astronomical Observatory of Japan, National Institutes of Natural Sciences, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 4 Department of Astronomical Science, SOKENDAI (The Graduate University for Advanced Studies), 2-21-1 Osawa, Mitaka, Tokyo, 181-8588, Japan\n5 Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, Chiba 277-8583, Japan 6 Department of Astronomy, The University of Texas at Austin, Austin, TX 78712, USA 7 Cosmic Dawn Center (DAWN), Denmark\n8 Niels Bohr Institute, University of Copenhagen, Lyngbyvej 2, DK2100 Copenhagen Ø, Denmark\n10\n9 Center for Computational Sciences, University of Tsukuba, Ten-nodai, 1-1-1 Tsukuba, Ibaraki 305-8577, Japan Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan\nABSTRACT\nWe investigate UV continuum slopes β of 974 galaxies at z = 4 -14 using archival JWST/NIRSpec PRISM spectra obtained from major JWST GTO, ERS, and GO programs, including JADES, CEERS, and UNCOVER. Among these galaxies, we identify a remarkable galaxy at z = 9 . 25, dubbed EBG-1, with a significantly blue UV slope β = -2 . 99 ± 0 . 15, unlike the rest of the galaxies that exhibit red continua or ambiguous blue continua hindered by large uncertainties. We confirm that the β value negligibly changes by the data reduction and fitting wavelength ranges for UV emission/absorption line masking. The extreme blue slope, β = -3 . 0, rules out significant contributions from dust extinction or AGN activity. Comparing with stellar and nebular emission models, we find that such a blue UV slope cannot be reproduced solely by stellar models even with very young, metal-poor, or top-heavy contiguous star formation associated with strong nebular continua making the UV slopes red, but with a high ionizing photon escape fraction, f ion esc ≳ 0 . 5, for a weak nebular continuum. While the H β emission line is not detected, likely due to the limited sensitivity of the spectrum, we find moderately weak [O iii ] λ 4959,5007 emission lines for the given star-formation rate (3 M ⊙ yr -1 ) and stellar mass (10 8 . 0 M ⊙ ) that are about three times weaker than the average emission lines, again suggestive of the high ionizing photon escape fraction, f ion esc ∼ 0 . 7 or more. EBG-1 would provide crucial insights into stellar and nebular continuum emission in high-redshift galaxies, serving as an example of the ionizing photon escaping site at the epoch of reionization.\nKeywords: Ultraviolet color; Reionization; Galaxy evolution; Galaxy formation; High-redshift galaxies\n1. INTRODUCTION\nThe first galaxies initiate star formation in the early universe, with young massive stars exhibiting blue UV continua and producing substantial amounts of ionizing photons. Although the ionizing photons are mainly used to ionize the interstellar medium, a fraction of the ionizing photons escape from the galaxy and ionize the intergalactic medium, driving the cosmic reionization.\nThe escape of the hydrogen ionizing photons is characterized by an escape fraction, f ion esc , which is the key quantity to understand how the ionizing photons of the galaxies contribute to the cosmic reionization.\nThe UV continuum slope β ( f ( λ ) ∝ λ β ) is an important indicator of the production and escape of ionizing photons. Young stellar populations produce β < -2, while dust extinction and active galactic nucleus (AGN) provide a red ( β ≳ -2) UV continuum (e.g., Bouwens\net al. 2012; Finkelstein et al. 2012). Typically the β values are larger than ∼ -2 . 6 (Chisholm et al. 2022), because an intrinsically blue UV continuum intensely ionizes the nebula, leading to a significant contribution from a nebular continuum, which has β ≳ -2 (Katz et al. 2024; Cameron et al. 2024).\nHowever, without the nebular continuum, the β values can be as low as -3 . 4 (e.g., Bouwens et al. 2010). This situation is achieved if the escape fraction of the ionizing photon f ion esc is large, because in that case the nebula is less ionized and the nebular continuum has less contribution to the UV spectrum (Zackrisson et al. 2017) . It is thus important to search for galaxies with β < -2 . 6 because such blue galaxies may have extremely high f ion esc , which contribute to the cosmic reionization.\nBecause galaxies with such high f ion esc are probably rare (Leitet et al. 2013; Matthee et al. 2017), it is necessary to search large sample of galaxies. Photometric studies are conducted with JWST data by Topping et al. (2022, 2024); Morales et al. (2024); Cullen et al. (2023, 2024), although deriving β using photometry suffers from contamination by emission lines. The accurate measurement of β requires the high quality spectroscopic data.\nIn this work, we search a large spectroscopic sample of galaxies provided by the DAWN JWST Archive for a galaxy with a blue UV slope, and report a galaxy at z = 9 . 25 that have an extremely blue UV slope. In Section 2, we describe our sample and method of UV slope measurements. In Section 3, we present the results the extremely blue object. We discuss physical origins of the extremely blue UV continuum in Section 4. Section 5 summarizes our results.\n2. SEARCH FOR BLUE UV SLOPE OBJECTS\n2.1. Lower Limit of β for f ion esc = 0\nWe first define a quantitative criterion for a blue UV continuum. We calculate β assuming f ion esc = 0 using Cloudy version 23.01 (Gunasekera et al. 2023) with incident radiations of Kroupa IMF (Kroupa 2001) with a mass range of 0 . 1 -100 M ⊙ provided by BPASS v2.2.1 (Stanway & Eldridge 2018), top-heavy IMF with a mass range of 50 -500 M ⊙ taken from Yggdrasil Pop III.1 model (Zackrisson et al. 2011), and blackbody. We assume a hydrogen density of n e = 10 2 cm -3 , nebular metallicity of log Z neb / Z ⊙ = -2, number of ionizing photon Q (H) = 10 50 s -1 , and inner radius of 10 14 cm. We assume stellar metallicity of log Z star / Z ⊙ = -2 for the Kroupa IMF model, while Z star = 0 is used for top-heavy IMF. Figure 1 shows time evolution of β values. Although the incident radiation of the top-heavy IMF is bluer than that of the Kroupa IMF, the topheavy IMF+nebular continuum model is redder than the\n\n\n
Figure 1. UV slope β as a function of age calculated with Cloudy. The blue, green, and gray lines present the incident radiations of top-heavy IMF (Zackrisson et al. 2011), Kroupa IMF (Kroupa 2001), and blackbody, respectively. The solid lines denote the models with f ion esc = 0 (i.e., the ionizing photons are completely consumed to ionize the nebula), while the dashed lines represent the models with f ion esc = 1 (i.e., the ionizing photons are not used to ionize the nebula). For the blackbody models, the blackbody temperatures are converted into the age using the typical lifetimes of stars having the same temperature.
\n\nKroupa+nebular continuum model. This is because the intrinsically blue incident radiation in top-heavy IMF model intensely ionize the nebula, which leads to a significant contribution from the red nebular continuum. One can also see that the lower limit of β value for f ion esc = 0 is -2 . 6, while the β value reach as low as -3 . 4 if nebular continuum is not included. We thus define a criterion for a blue UV continuum as β = -2 . 6.\n2.2. Our Sample\nWe use 974 PRISM/CLEAR spectra of galaxies provided by the DAWN JWST Archive (DJA) 1 , which compile the major JWST GTO, ERS, and GO programs. 2 We select the galaxies within the redshift range of 4 < z < 14, whose UV continua are covered by NIRSpec. The spectra in the DJA are reduced with msaexp (Brammer 2023; Heintz et al. 2024). Hereafter, these 974 galaxies are referred to as our sample. The redshift distribution of galaxies in our sample is shown in the top panel of Figure 2.\n2.3. UV Slope Measurements\n\n\n
Figure 2. (Top) Redshift distribution of our sample. (Bottom) Fitted β value as a function of redshift. The blue points represent the galaxies in our sample. The β values with unreliably large error ( > 0 . 5) are omitted in this figure. The red point indicates EBG-1. The dotted line denote the lower limit of β if f ion esc = 0 is assumed.
\n\nTo derive β , we fit a galaxy spectra with a function\nf ( λ ) = Aλ β , (1)\nwhere A is a constant for normalization and λ is wavelengths. We employ a Markov Chain Monte Carlo (MCMC) method for the fitting using emcee (ForemanMackey et al. 2013). We minimize a likelihood function\nlog( L ) = -1 2 ∑ λ [ ( f mod ( λ ) -f obs ( λ ) σ ( λ ) ) 2 +log( σ ( λ ) 2 ) ] , (2)\nwhere f mod and f obs are the model and observed flux, respectively, and σ ( λ ) is the 1 σ error of the flux. We use a rest-frame wavelength range of 1268 -2580 ˚ A for the fitting, following the fitting range presented by Calzetti et al. (1994). The measured β values are shown in the bottom panel of Figure 2. Most of the galaxies show β > -2 . 6, which is larger than the lower limit for f ion esc = 0 (Figure 1). Among our sample, we find one galaxy at z = 9 . 25, showing β = -2 . 99 ± 0 . 15, which is smaller than -2 . 6 beyond the 2 σ level. Hereafter, we refer to this object as extremely blue galaxy 1 (EBG-1). EBG-1 is originally identified as MACS0647-z9-20158 at z phot = 9 . 5 by the program GO 1433 (PI: Coe; McLeod et al.\n2024), and then spectroscopically observed by the same program GO 1433 as a filler target for MACS0647-JD. The NIRCam images and spectra of EBG-1 are shown in Figure 3.\nRecently, Saxena et al. (2024) have found six galaxies at 5 . 5 < z < 8 showing β ∼ -3 from the spectroscopic sample obtained from the JADES survey. One out of six galaxies, JADES-GS-210003 at z = 5 . 779, is also included in our sample. Using the same fitting range as Saxena et al. (2024), we obtain β = -2 . 70 ± 0 . 13 for JADES-GS-210003 (Figure 4). There is a ∼ 2 σ difference between our and Saxena et al. (2024) of the β values. This is probably because Saxena et al. (2024) conduct a sigma-clipping method to exclude outlying pixels. JADES-GS-210003 is not selected in our study because we cannot distinguish from β = -2 . 6 at the 2 σ level. On the other hand, EBG-1 is not selected by Saxena et al. (2024), as EBG-1 falls in the MACS0647 lensing field that is not covered in the sample of Saxena et al. (2024).\nWe also compare our β values with those in previous studies in Figure 4. We plot photometric β measurements of Cullen et al. (2024), who conduct β measurements for the sample galaxies taken from NGDEEP, JADES DR1, UNCOVER, and those from McLeod et al. (2024). Cullen et al. (2024) also derive β for EBG-1, which show good agreement with our measurement.\n2.4. Observations and Data of EBG-1\nIn the previous section, we identify EBG-1 out of 974 objects from the spectra reduced by DJA. In this section, we first describe how EBG-1 was observed in Section 2.4.1. We next conduct the photometry for EBG-1 to confirm whether the photometry is consistent with spectrum in Section 2.4.2. We then independently performed data reduction to verify whether the β value changes with different data reduction procedures, as described in Section 2.4.3.\n2.4.1. Observations\nMACS0647 lensing field was observed with JWST/NIRCam in January 8th 2023 in GO 1433 (PI: Coe) targeting MACS0647-JD (Coe et al. 2013; Hsiao et al. 2023). EBG-1 is falling on the footprints of this NIRCam observations, which is photometrically identified at R.A.=06:47:36.95 and Decl.=+70:14:34.69 by McLeod et al. (2024).\nEBG-1 was then spectroscopically observed with JWST/NIRSpec as a filler target for MACS0647-JD in GO 1433 in February 20th, 2023. The observations were performed with PRISM/CLEAR ( R ∼ 100) for a total exposure time of 2200 s. The data were reduced with msaexp by DJA. For details of the reduction, see Heintz\n\n\n
Figure 3. Top and bottom panels show the 2D and 1D NIRSpec spectra of EBG-1 reduced by DJA, respectively. In the bottom panel, the black histogram and gray shaded regions represent the spectrum and its 1 σ uncertainty, respectively. The red line presents the best-fit UV slope derived with the Calzetti et al. (1994) fitting windows. The dark blue shaded regions are the regions that are not used for the UV slope fitting. The light blue shaded regions denote the masks presented by Calzetti et al. (1994) and the mask for the artifact at 2500 ˚ A, which are not used for fitting. The dotted lines present the positions of emission lines. The F150W image and slit position are presented in the inset.
\n\n\n\n
Figure 4. Comparison of β measurements in this work and previous work. The red circle represents the values for EBG-1, whose β values in the previous works are taken from the photometric measurement of Cullen et al. (2024). The crosses denote the β values taken from the photometric measurements of Cullen et al. (2024), while the blue diamond represents those taken from the spectroscopic value of Saxena et al. (2024). The black solid line present the lower limit of β for no ionizing photon escape. The gray dashed line denotes the line of equality.
\n\net al. (2024). The spectrum reduced by DJA is shown in Figure 3.\n2.4.2. Photometry\nCalibrated NIRCam data are collected from DJA. We conduct aperture photometry with 0 . '' 35 aperture size, which are shown in Table 1. The errors are estimated by conducting photometry within a 0 . '' 80 annulus and taking the standard deviations. An apparent magnitude of F200W is derived as 27.7, which is consistent with that derived by McLeod et al. (2024).\n2.4.3. Reanalysis of EBG-1 spectra\nWe performed the data reduction in this work following the method described in Nakajima et al. (2023). Starting from the level 1 products provided by MAST, we executed Spec2 and Spec3 pipelines using Python library jwst (ver. 1.16.1; Bushouse et al. 2024). The reference files stored in the latest pmap file of jwst 1299.pmap were used. The pathloss corrections were conducted by comparing the position of the source and MSA shutter, where we assumed that the source was point-like. We then combined 2D spectra by medianstacking to reduce the effect of hot pixels, with extractions of 3 pixels in the spatial direction. For more details, see Nakajima et al. (2023). In Figure 5 we present\nthe spectrum reduced in this work, which is consistent with both of the spectrum reduced by DJA and the NIRCam photometry.\nThe spectrum of EBG-1 shows [O iii ] λλ 4959, 5007 and tentative C iv λλ 1548, 1550 emission lines. We measure flux values and 3 σ upper limits of emission lines by integrating the flux and error in each wavelength bin (Table 2). The C iv emission might be associated with an AGN, although the extremely blue UV slope disfavors the contribution from an AGN, whose dust content reddens the spectrum (Francis et al. 1991).\n3. RESULTS\n3.1. Confirmation of the Blue UV Slope\nTo check whether the extremely blue UV continuum of EBG-1 is caused by systematics or not, we adopt three fitting methods: 1) conventional Calzetti et al. (1994) windows, 2) using the whole wavelength range without masking, and 3) simply avoiding the possible C iv emission line (note that we mask out at 2480 -2520 ˚ A contaminated by the artifact). For the spectrum reduced by DJA, each fitting method gives β = -3 . 03 ± 0 . 22 , -2 . 99 ± 0 . 15 , and -3 . 08 ± 0 . 15, respectively, all of which are smaller than β = -2 . 6 at the ∼ 2 σ level.\nWe derive β also for the spectrum reduced in this work in the same manner as described above. The fitting methods of 1), 2), and 3) yield β = -2 . 96 ± 0 . 19 , -2 . 92 ± 0 . 13 , and -2 . 79 ± 0 . 14, respectively. Although the significance levels are slightly lower than the β values from the DJA spectrum, the results of the extremely blue UV slope of EBG-1 does not significantly change. We thus conclude that the extremely blue UV slope of EBG-1 is not attributed to observational systematics.\n3.2. SED Fitting\nTo measure stellar properties, we conduct SED fitting to the NIRCam data presented in Table 1 using Prospector (Johnson et al. 2021a). We use the photometry values that are corrected for magnification factor. The Binary Population and Spectral Synthesis (BPASS; Eldridge et al. 2017) model is used for an isochrone library. We apply Calzetti et al. (2000) dust extinction law. We assume a non-parametric star-formation history (SFH) with five bins that are evenly spaced in logarithmic times between 0 Myr and a look-back time corresponding to z = 30. The redshift of the source is fixed at z spec = 9 . 25. We vary the stellar mass M ∗ , metallicity Z , optical depth of dust attenuation at 5500 ˚ A τ dust (5500 ˚ A), and ionization parameter U . The best-fit\n
\n\n
Table 1. Properties of EBG-1
\n
\nNote -a The value is corrected for magnification factor.\n
\n\n
Table 2. Line fluxes and 3 σ upper limits
\n
\nNote -\nparameters and SFH are shown in Table 1 and Figure 6, respectively. The results imply the dust-free condition and the recent starburst, which agree with the extremely blue UV continuum of EBG-1. Because f ion esc = 0 is assumed in our SED fitting, the weak emission line feature of EBG-1 indicates a low metallicity, which compensates for the effect of nonzero f ion esc .\n3.3. Weak Nebular Emission Lines\nThe extremely blue UV continuum in EBG-1 may be explained by high f esc . However, the detection of [O iii ] λλ 4959,5007 emission indicates a certain amount of the ionizing photon is used to ionize the nebulae. To\n\n\n
Figure 5. Comparison of NIRSpec spectrum of EBG-1 reduced in this work and DJA. The yellow histogram and shaded region represent the spectrum reduced in this work and its 1 σ error, respectively. The blue points denote the NIRCam photometry. The other symbols are the same as Figure 3.
\n\n\n\n
Figure 6. Best-fit SFH for EBG-1. The black line and gray shaded region represent the best-fit and 1 σ error of SFH, respectively.
\n\nquantify the escape of ionizing photons in EBG-1, we plot ratios of luminosity of [O iii ] λ 5007 line, L [OIII] , to star-formation rate (SFR) as a function of stellar mass in Figure 7. For comparison, we also plot average values calculated from 126 galaxies at 4 < z < 9 compiled by Nakajima et al. (2023). The SFRs are estimated from UV luminosities of galaxies by using Equation (1) of Kennicutt (1998). The L [OIII] /SFR value of EBG-1 is ∼ 0 . 5 dex smaller than the average value at the same stellar mass, suggesting that [O iii ] emission in EBG-1 is ∼ 3 times weaker than the average. This can be interpreted as a result of the escape of ionizing photon without ionizing the nebulae. If we assume f ion esc = 0 for the\ngalaxies in Nakajima et al. (2023), the [O iii ] emission three times weaker than the average suggests f ion esc ∼ 0 . 7 for EBG-1. However, the f ion esc values of the galaxies in Nakajima et al. (2023) can be larger than zero, in which case our estimate of f ion esc becomes larger. Furthermore, if we assume density-bounded nebulae, weak [O iii ] emission means an excess of O ++ ionizing photons (ionization energy of 35 eV) compared to the amount of the gas, with H + ionizing photon (ionization energy of 13 . 6 eV) being even more abundant. In such a situation, the escape fraction of H + ionizing photon can be larger than that of O ++ ionizing photon.\nHowever, the estimate of f ion esc from the [O iii ] emission is susceptible to metallicity. To estimate f ion esc independently of the assumption of metallicity, we use the β and EW(H β ) values of EBG-1. We derive the upper limit of the H β flux by summing up the error spectrum in quadrature. We then divide the upper limit of H β by the continuum flux density, obtaining log (EW(H β ) / ˚ A) < 2 . 8. In Figure 8, we compare the β and EW(H β ) values with the Cloudy models, which are the same as those in Figure 1 except for varying f ion esc . In all of the models, the very small β values of EBG1 imply f esc ≳ 0 . 5 (although models with extremely metal-poor or metal-free stellar populations reproduce β ∼ -3 even with f ion esc = 0 see Figure 4 of (Bouwens et al. 2010), the detection of [O iii ] emission in EBG-1 disfavors this scenario). However, due to the weak upper limit of EW(H β ), it is difficult to constrain the ionizing radiation and f esc with current data.\n4. DISCUSSION\n\n\n
Figure 7. L [OIII] /SFR as a function of stellar mass. The red circle represents EBG-1. The blue squares denote the average values of galaxies at 4 < z < 9 taken from Nakajima et al. (2023). The ticks at the top of the figure denote metallicity, which is converted from stellar mass using the mass-metallicity relation at z = 4 -10 derived by Nakajima et al. (2023).
\n\nThere are two scenarios that explain the high escape fraction: density-bounded nebulae, or ionizationbounded nebulae with holes (Zackrisson et al. 2013). In density-bounded nebulae, ionizing photons exceeding the gas supply escape the nebula, while in ionizationbounded nebulae with holes, ionizing photons leak through these holes. Both of the scenarios are consistent with our results, and it is difficult to test these two scenarios with the current data. However, the morphology of EBG-1 (Figure 3) may provide a clue to the origin of the high escape fraction. EBG-1 has a small tail extending towards north-west direction from the main component. The slit covers around the tail of the EBG1. The tail may be stripped from EBG-1 by galaxy interactions, where the gas could be lacking. We may be looking at this gas-deficient region, where the nebulae are density-bounded or with many holes. In such a case, the center of EBG-1 can be redder than the tail (see also Schombert et al. (1990), who claim that tidal tails tend to be bluer than primary galaxies from observations). To examine this scenario, it is necessary to conduct spectroscopy by placing a slit along the northwest direction that covers from the center to the tail of the EBG-1 with spatial extent.\nThe fraction of the extremely blue galaxy in this work is low (one out of 974 galaxies) compared to the other studies such as Saxena et al. (2024). This is probably because of the low signal-to-noise ratio of most of the\ngalaxies. Although some galaxies show the best-fit β values smaller than -2 . 6 in Figure 2, the large error bars hinder us from confirming that these galaxies truly have blue UV continua. There may be more galaxies like EBG-1, while deeper spectroscopic observations are required to confirm.\nThe non-detection of H β line and the large uncertainty in [O iii ] emission lines make it difficult to constrain the f ion esc value and type of ionizing source of EBG-1. It is thus necessary to conduct deeper spectroscopy on EBG1 to definitively confirm the high f ion esc and the ionizing source, by which one can utilize EBG-1 as an example to study the process of the significant ionizing photon escape during the epoch of reionization.\n5. SUMMARY\nIn this work, we search the large JWST/NIRSpec spectroscopic sample for a galaxy with an extremely blue UV continuum. Among our sample consisting of the 974 galaxies at 4 < z < 14 taken from the major JWST GTO, ERS, and GO programs, we identify EBG-1, a galaxy at z = 9 . 25 showing β ∼ -3. Our major findings are summarized below:\n\n· By fitting f ( λ ) ∝ λ β to the UV continua for our sample, we find EBG-1 showing β = -2 . 99 ± 0 . 15, which is below the lower limit for no ionizing photon escape, β = -2 . 6, beyond the 2 σ level. This small β value does not change significantly by changing the fitting method and data reduction procedure, which suggest that the extremely blue UV continuum is not caused by systematics.\n· The NIRSpec PRISM spectrum of EBG-1 shows [O iii ] λλ 4959, 5007 emission lines. By estimating the SFR from the UV luminosity, we calculate the L [OIII] / SFR ratio for EBG-1. The comparison with the galaxies at 4 < z < 9 compiled by Nakajima et al. (2023) suggests that the [O iii ] emission is slightly weaker than expected from the UV luminosity if f ion esc = 0 is assumed, indicating f ion esc ∼ 0 . 7 for EBG-1.\n· We compare the observed β and EW(H β ) with our Cloudy modeling. The extremely blue β value imply f ion esc ≳ 0 . 5, which is consistent with the f ion esc value inferred from the [O iii ] emission. However, the weak upper limit of H β emission line prevents us from break the degeneracy between f ion esc and the shape of the ionizing source. It is thus important to conduct deeper spectroscopy on EBG-1.\n\nACKNOWLEDGMENTS\n\n\n
Figure 8. Relation of β and EW(H β ). The blue, green, and gray grids denote the same models in Figure 1. The numbers shown beside the grids represent f ion esc . The red circle presents the current constraint of β = -2 . 99 ± 0 . 15.
\n\nWe are grateful to Steven L. Finkelstein, Moka Nishigaki, and Kuria Watanabe for the valuable discussions. This work is based on observations made with the NASA/ESA/CSA James Webb Space Telescope. The NIRSpec data of EBG-1 were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. The data products presented herein were retrieved from DJA. DJA is an initiative of the Cosmic Dawn Center (DAWN), which is funded by the Danish National Research Foundation under grant DNRF140. We thank DJA for providing the reduced NIRSpec data. The observational data collected from DJA are associated with programs ERS 1345 (CEERS; PI: Finkelstein), DDT 2756 (PI: Chen), DDT 2750 (CEERS; PI: Arrabal Haro), GTO 1180 (PI: Eisenstein), GTO 1181 (PI: Eisenstein), GTO 1210 (PI: Luetzgendorf), GTO 1211 (PI: Isaak), GTO 1286 (PI: Luetzgendorf), DDT 6541 (PI: Egami), GO 1433 (PI: Coe), GO 1747 (PI: Roberts-Borsani), GO 1810 (PI: Belli), GO 1871 (PI: Chisholm), GO 2110 (PI: Kriek), GO 2198 (PI: Barrufet), GO 2561 (UN-\nCOVER; PI: Labbe), GO 2565 (PI: Glazebrook), DDT 2767 (PI: Kelly), ERO 2736 (PI: Pontoppidan), GO 3215 (PI: Eisenstein), GO 4233 (PI: de Graaff) GO 4246 (PI: Abdurro'uf), DDT 4446(PI: Frye), and DDT 4557 (PI: Yan). The authors acknowledge the teams conducting these observations for publicly releasing the data. This publication is based on work supported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, KAKENHI (20H00180, 21H04467, 21H04489, 24K07102) through the Japan Society for the Promotion of Science, and JST FOREST Program (JP-MJFR202Z). This work was supported by the joint research program of the Institute for Cosmic Ray Research (ICRR), University of Tokyo.\nFacilities:\nSoftware: astropy (Astropy Collaboration et al. 2013, 2018, 2022), NumPy (Harris et al. 2020), matplotlib (Hunter 2007), SciPy (Virtanen et al. 2020), corner (Foreman-Mackey 2016) Cloudy (Ferland et al. 2013), emcee (Foreman-Mackey et al. 2013), msaexp (Brammer 2023) Prospector (Johnson et al. 2021b)\nREFERENCES\nAstropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068\nAstropy Collaboration, Price-Whelan, A. M., Sip\"ocz, B. M.,\net al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f\n
\n\n
\nHunter, J. D. 2007, Computing in Science and Engineering, 9, 90, doi: 10.1109/MCSE.2007.55\n
\n\n
\n"},"sections":{"kind":"list like","value":[{"title":"ABSTRACT","content":"We investigate UV continuum slopes β of 974 galaxies at z = 4 -14 using archival JWST/NIRSpec PRISM spectra obtained from major JWST GTO, ERS, and GO programs, including JADES, CEERS, and UNCOVER. Among these galaxies, we identify a remarkable galaxy at z = 9 . 25, dubbed EBG-1, with a significantly blue UV slope β = -2 . 99 ± 0 . 15, unlike the rest of the galaxies that exhibit red continua or ambiguous blue continua hindered by large uncertainties. We confirm that the β value negligibly changes by the data reduction and fitting wavelength ranges for UV emission/absorption line masking. The extreme blue slope, β = -3 . 0, rules out significant contributions from dust extinction or AGN activity. Comparing with stellar and nebular emission models, we find that such a blue UV slope cannot be reproduced solely by stellar models even with very young, metal-poor, or top-heavy contiguous star formation associated with strong nebular continua making the UV slopes red, but with a high ionizing photon escape fraction, f ion esc ≳ 0 . 5, for a weak nebular continuum. While the H β emission line is not detected, likely due to the limited sensitivity of the spectrum, we find moderately weak [O iii ] λ 4959,5007 emission lines for the given star-formation rate (3 M ⊙ yr -1 ) and stellar mass (10 8 . 0 M ⊙ ) that are about three times weaker than the average emission lines, again suggestive of the high ionizing photon escape fraction, f ion esc ∼ 0 . 7 or more. EBG-1 would provide crucial insights into stellar and nebular continuum emission in high-redshift galaxies, serving as an example of the ionizing photon escaping site at the epoch of reionization. Keywords: Ultraviolet color; Reionization; Galaxy evolution; Galaxy formation; High-redshift galaxies","pages":[1]},{"title":"A Galaxy with an Extremely Blue UV Slope β = -3 at z = 9 . 25 Identified by JWST Spectroscopy: Evidence for a Weak Nebular Continuum and Efficient Ionizing Photon Escape?","content":"Hiroto Yanagisawa , 1, 2 Masami Ouchi , 3, 1, 4, 5 Kimihiko Nakajima , 3 Yuichi Harikane , 1 Seiji Fujimoto , 6, 7, 8 Yoshiaki Ono , 1 Hiroya Umeda , 1, 2 Minami Nakane , 1, 2 Hidenobu Yajima , 9 Hajime Fukushima , 9 and Yi Xu 1, 10 1 Institute for Cosmic Ray Research, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan 2 Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan 3 National Astronomical Observatory of Japan, National Institutes of Natural Sciences, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 4 Department of Astronomical Science, SOKENDAI (The Graduate University for Advanced Studies), 2-21-1 Osawa, Mitaka, Tokyo, 181-8588, Japan 5 Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, Chiba 277-8583, Japan 6 Department of Astronomy, The University of Texas at Austin, Austin, TX 78712, USA 7 Cosmic Dawn Center (DAWN), Denmark 8 Niels Bohr Institute, University of Copenhagen, Lyngbyvej 2, DK2100 Copenhagen Ø, Denmark 10 9 Center for Computational Sciences, University of Tsukuba, Ten-nodai, 1-1-1 Tsukuba, Ibaraki 305-8577, Japan Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan","pages":[1]},{"title":"1. INTRODUCTION","content":"The first galaxies initiate star formation in the early universe, with young massive stars exhibiting blue UV continua and producing substantial amounts of ionizing photons. Although the ionizing photons are mainly used to ionize the interstellar medium, a fraction of the ionizing photons escape from the galaxy and ionize the intergalactic medium, driving the cosmic reionization. The escape of the hydrogen ionizing photons is characterized by an escape fraction, f ion esc , which is the key quantity to understand how the ionizing photons of the galaxies contribute to the cosmic reionization. The UV continuum slope β ( f ( λ ) ∝ λ β ) is an important indicator of the production and escape of ionizing photons. Young stellar populations produce β < -2, while dust extinction and active galactic nucleus (AGN) provide a red ( β ≳ -2) UV continuum (e.g., Bouwens et al. 2012; Finkelstein et al. 2012). Typically the β values are larger than ∼ -2 . 6 (Chisholm et al. 2022), because an intrinsically blue UV continuum intensely ionizes the nebula, leading to a significant contribution from a nebular continuum, which has β ≳ -2 (Katz et al. 2024; Cameron et al. 2024). However, without the nebular continuum, the β values can be as low as -3 . 4 (e.g., Bouwens et al. 2010). This situation is achieved if the escape fraction of the ionizing photon f ion esc is large, because in that case the nebula is less ionized and the nebular continuum has less contribution to the UV spectrum (Zackrisson et al. 2017) . It is thus important to search for galaxies with β < -2 . 6 because such blue galaxies may have extremely high f ion esc , which contribute to the cosmic reionization. Because galaxies with such high f ion esc are probably rare (Leitet et al. 2013; Matthee et al. 2017), it is necessary to search large sample of galaxies. Photometric studies are conducted with JWST data by Topping et al. (2022, 2024); Morales et al. (2024); Cullen et al. (2023, 2024), although deriving β using photometry suffers from contamination by emission lines. The accurate measurement of β requires the high quality spectroscopic data. In this work, we search a large spectroscopic sample of galaxies provided by the DAWN JWST Archive for a galaxy with a blue UV slope, and report a galaxy at z = 9 . 25 that have an extremely blue UV slope. In Section 2, we describe our sample and method of UV slope measurements. In Section 3, we present the results the extremely blue object. We discuss physical origins of the extremely blue UV continuum in Section 4. Section 5 summarizes our results.","pages":[1,2]},{"title":"2.1. Lower Limit of β for f ion esc = 0","content":"We first define a quantitative criterion for a blue UV continuum. We calculate β assuming f ion esc = 0 using Cloudy version 23.01 (Gunasekera et al. 2023) with incident radiations of Kroupa IMF (Kroupa 2001) with a mass range of 0 . 1 -100 M ⊙ provided by BPASS v2.2.1 (Stanway & Eldridge 2018), top-heavy IMF with a mass range of 50 -500 M ⊙ taken from Yggdrasil Pop III.1 model (Zackrisson et al. 2011), and blackbody. We assume a hydrogen density of n e = 10 2 cm -3 , nebular metallicity of log Z neb / Z ⊙ = -2, number of ionizing photon Q (H) = 10 50 s -1 , and inner radius of 10 14 cm. We assume stellar metallicity of log Z star / Z ⊙ = -2 for the Kroupa IMF model, while Z star = 0 is used for top-heavy IMF. Figure 1 shows time evolution of β values. Although the incident radiation of the top-heavy IMF is bluer than that of the Kroupa IMF, the topheavy IMF+nebular continuum model is redder than the Kroupa+nebular continuum model. This is because the intrinsically blue incident radiation in top-heavy IMF model intensely ionize the nebula, which leads to a significant contribution from the red nebular continuum. One can also see that the lower limit of β value for f ion esc = 0 is -2 . 6, while the β value reach as low as -3 . 4 if nebular continuum is not included. We thus define a criterion for a blue UV continuum as β = -2 . 6.","pages":[2]},{"title":"2.2. Our Sample","content":"We use 974 PRISM/CLEAR spectra of galaxies provided by the DAWN JWST Archive (DJA) 1 , which compile the major JWST GTO, ERS, and GO programs. 2 We select the galaxies within the redshift range of 4 < z < 14, whose UV continua are covered by NIRSpec. The spectra in the DJA are reduced with msaexp (Brammer 2023; Heintz et al. 2024). Hereafter, these 974 galaxies are referred to as our sample. The redshift distribution of galaxies in our sample is shown in the top panel of Figure 2.","pages":[2]},{"title":"2.3. UV Slope Measurements","content":"To derive β , we fit a galaxy spectra with a function where A is a constant for normalization and λ is wavelengths. We employ a Markov Chain Monte Carlo (MCMC) method for the fitting using emcee (ForemanMackey et al. 2013). We minimize a likelihood function where f mod and f obs are the model and observed flux, respectively, and σ ( λ ) is the 1 σ error of the flux. We use a rest-frame wavelength range of 1268 -2580 ˚ A for the fitting, following the fitting range presented by Calzetti et al. (1994). The measured β values are shown in the bottom panel of Figure 2. Most of the galaxies show β > -2 . 6, which is larger than the lower limit for f ion esc = 0 (Figure 1). Among our sample, we find one galaxy at z = 9 . 25, showing β = -2 . 99 ± 0 . 15, which is smaller than -2 . 6 beyond the 2 σ level. Hereafter, we refer to this object as extremely blue galaxy 1 (EBG-1). EBG-1 is originally identified as MACS0647-z9-20158 at z phot = 9 . 5 by the program GO 1433 (PI: Coe; McLeod et al. 2024), and then spectroscopically observed by the same program GO 1433 as a filler target for MACS0647-JD. The NIRCam images and spectra of EBG-1 are shown in Figure 3. Recently, Saxena et al. (2024) have found six galaxies at 5 . 5 < z < 8 showing β ∼ -3 from the spectroscopic sample obtained from the JADES survey. One out of six galaxies, JADES-GS-210003 at z = 5 . 779, is also included in our sample. Using the same fitting range as Saxena et al. (2024), we obtain β = -2 . 70 ± 0 . 13 for JADES-GS-210003 (Figure 4). There is a ∼ 2 σ difference between our and Saxena et al. (2024) of the β values. This is probably because Saxena et al. (2024) conduct a sigma-clipping method to exclude outlying pixels. JADES-GS-210003 is not selected in our study because we cannot distinguish from β = -2 . 6 at the 2 σ level. On the other hand, EBG-1 is not selected by Saxena et al. (2024), as EBG-1 falls in the MACS0647 lensing field that is not covered in the sample of Saxena et al. (2024). We also compare our β values with those in previous studies in Figure 4. We plot photometric β measurements of Cullen et al. (2024), who conduct β measurements for the sample galaxies taken from NGDEEP, JADES DR1, UNCOVER, and those from McLeod et al. (2024). Cullen et al. (2024) also derive β for EBG-1, which show good agreement with our measurement.","pages":[3]},{"title":"2.4. Observations and Data of EBG-1","content":"In the previous section, we identify EBG-1 out of 974 objects from the spectra reduced by DJA. In this section, we first describe how EBG-1 was observed in Section 2.4.1. We next conduct the photometry for EBG-1 to confirm whether the photometry is consistent with spectrum in Section 2.4.2. We then independently performed data reduction to verify whether the β value changes with different data reduction procedures, as described in Section 2.4.3.","pages":[3]},{"title":"2.4.1. Observations","content":"MACS0647 lensing field was observed with JWST/NIRCam in January 8th 2023 in GO 1433 (PI: Coe) targeting MACS0647-JD (Coe et al. 2013; Hsiao et al. 2023). EBG-1 is falling on the footprints of this NIRCam observations, which is photometrically identified at R.A.=06:47:36.95 and Decl.=+70:14:34.69 by McLeod et al. (2024). EBG-1 was then spectroscopically observed with JWST/NIRSpec as a filler target for MACS0647-JD in GO 1433 in February 20th, 2023. The observations were performed with PRISM/CLEAR ( R ∼ 100) for a total exposure time of 2200 s. The data were reduced with msaexp by DJA. For details of the reduction, see Heintz et al. (2024). The spectrum reduced by DJA is shown in Figure 3.","pages":[3,4]},{"title":"2.4.2. Photometry","content":"Calibrated NIRCam data are collected from DJA. We conduct aperture photometry with 0 . '' 35 aperture size, which are shown in Table 1. The errors are estimated by conducting photometry within a 0 . '' 80 annulus and taking the standard deviations. An apparent magnitude of F200W is derived as 27.7, which is consistent with that derived by McLeod et al. (2024).","pages":[4]},{"title":"2.4.3. Reanalysis of EBG-1 spectra","content":"We performed the data reduction in this work following the method described in Nakajima et al. (2023). Starting from the level 1 products provided by MAST, we executed Spec2 and Spec3 pipelines using Python library jwst (ver. 1.16.1; Bushouse et al. 2024). The reference files stored in the latest pmap file of jwst 1299.pmap were used. The pathloss corrections were conducted by comparing the position of the source and MSA shutter, where we assumed that the source was point-like. We then combined 2D spectra by medianstacking to reduce the effect of hot pixels, with extractions of 3 pixels in the spatial direction. For more details, see Nakajima et al. (2023). In Figure 5 we present the spectrum reduced in this work, which is consistent with both of the spectrum reduced by DJA and the NIRCam photometry. The spectrum of EBG-1 shows [O iii ] λλ 4959, 5007 and tentative C iv λλ 1548, 1550 emission lines. We measure flux values and 3 σ upper limits of emission lines by integrating the flux and error in each wavelength bin (Table 2). The C iv emission might be associated with an AGN, although the extremely blue UV slope disfavors the contribution from an AGN, whose dust content reddens the spectrum (Francis et al. 1991).","pages":[4,5]},{"title":"3.1. Confirmation of the Blue UV Slope","content":"To check whether the extremely blue UV continuum of EBG-1 is caused by systematics or not, we adopt three fitting methods: 1) conventional Calzetti et al. (1994) windows, 2) using the whole wavelength range without masking, and 3) simply avoiding the possible C iv emission line (note that we mask out at 2480 -2520 ˚ A contaminated by the artifact). For the spectrum reduced by DJA, each fitting method gives β = -3 . 03 ± 0 . 22 , -2 . 99 ± 0 . 15 , and -3 . 08 ± 0 . 15, respectively, all of which are smaller than β = -2 . 6 at the ∼ 2 σ level. We derive β also for the spectrum reduced in this work in the same manner as described above. The fitting methods of 1), 2), and 3) yield β = -2 . 96 ± 0 . 19 , -2 . 92 ± 0 . 13 , and -2 . 79 ± 0 . 14, respectively. Although the significance levels are slightly lower than the β values from the DJA spectrum, the results of the extremely blue UV slope of EBG-1 does not significantly change. We thus conclude that the extremely blue UV slope of EBG-1 is not attributed to observational systematics.","pages":[5]},{"title":"3.2. SED Fitting","content":"To measure stellar properties, we conduct SED fitting to the NIRCam data presented in Table 1 using Prospector (Johnson et al. 2021a). We use the photometry values that are corrected for magnification factor. The Binary Population and Spectral Synthesis (BPASS; Eldridge et al. 2017) model is used for an isochrone library. We apply Calzetti et al. (2000) dust extinction law. We assume a non-parametric star-formation history (SFH) with five bins that are evenly spaced in logarithmic times between 0 Myr and a look-back time corresponding to z = 30. The redshift of the source is fixed at z spec = 9 . 25. We vary the stellar mass M ∗ , metallicity Z , optical depth of dust attenuation at 5500 ˚ A τ dust (5500 ˚ A), and ionization parameter U . The best-fit Note -a The value is corrected for magnification factor. Note - parameters and SFH are shown in Table 1 and Figure 6, respectively. The results imply the dust-free condition and the recent starburst, which agree with the extremely blue UV continuum of EBG-1. Because f ion esc = 0 is assumed in our SED fitting, the weak emission line feature of EBG-1 indicates a low metallicity, which compensates for the effect of nonzero f ion esc .","pages":[5]},{"title":"3.3. Weak Nebular Emission Lines","content":"The extremely blue UV continuum in EBG-1 may be explained by high f esc . However, the detection of [O iii ] λλ 4959,5007 emission indicates a certain amount of the ionizing photon is used to ionize the nebulae. To quantify the escape of ionizing photons in EBG-1, we plot ratios of luminosity of [O iii ] λ 5007 line, L [OIII] , to star-formation rate (SFR) as a function of stellar mass in Figure 7. For comparison, we also plot average values calculated from 126 galaxies at 4 < z < 9 compiled by Nakajima et al. (2023). The SFRs are estimated from UV luminosities of galaxies by using Equation (1) of Kennicutt (1998). The L [OIII] /SFR value of EBG-1 is ∼ 0 . 5 dex smaller than the average value at the same stellar mass, suggesting that [O iii ] emission in EBG-1 is ∼ 3 times weaker than the average. This can be interpreted as a result of the escape of ionizing photon without ionizing the nebulae. If we assume f ion esc = 0 for the galaxies in Nakajima et al. (2023), the [O iii ] emission three times weaker than the average suggests f ion esc ∼ 0 . 7 for EBG-1. However, the f ion esc values of the galaxies in Nakajima et al. (2023) can be larger than zero, in which case our estimate of f ion esc becomes larger. Furthermore, if we assume density-bounded nebulae, weak [O iii ] emission means an excess of O ++ ionizing photons (ionization energy of 35 eV) compared to the amount of the gas, with H + ionizing photon (ionization energy of 13 . 6 eV) being even more abundant. In such a situation, the escape fraction of H + ionizing photon can be larger than that of O ++ ionizing photon. However, the estimate of f ion esc from the [O iii ] emission is susceptible to metallicity. To estimate f ion esc independently of the assumption of metallicity, we use the β and EW(H β ) values of EBG-1. We derive the upper limit of the H β flux by summing up the error spectrum in quadrature. We then divide the upper limit of H β by the continuum flux density, obtaining log (EW(H β ) / ˚ A) < 2 . 8. In Figure 8, we compare the β and EW(H β ) values with the Cloudy models, which are the same as those in Figure 1 except for varying f ion esc . In all of the models, the very small β values of EBG1 imply f esc ≳ 0 . 5 (although models with extremely metal-poor or metal-free stellar populations reproduce β ∼ -3 even with f ion esc = 0 see Figure 4 of (Bouwens et al. 2010), the detection of [O iii ] emission in EBG-1 disfavors this scenario). However, due to the weak upper limit of EW(H β ), it is difficult to constrain the ionizing radiation and f esc with current data.","pages":[5,6]},{"title":"4. DISCUSSION","content":"There are two scenarios that explain the high escape fraction: density-bounded nebulae, or ionizationbounded nebulae with holes (Zackrisson et al. 2013). In density-bounded nebulae, ionizing photons exceeding the gas supply escape the nebula, while in ionizationbounded nebulae with holes, ionizing photons leak through these holes. Both of the scenarios are consistent with our results, and it is difficult to test these two scenarios with the current data. However, the morphology of EBG-1 (Figure 3) may provide a clue to the origin of the high escape fraction. EBG-1 has a small tail extending towards north-west direction from the main component. The slit covers around the tail of the EBG1. The tail may be stripped from EBG-1 by galaxy interactions, where the gas could be lacking. We may be looking at this gas-deficient region, where the nebulae are density-bounded or with many holes. In such a case, the center of EBG-1 can be redder than the tail (see also Schombert et al. (1990), who claim that tidal tails tend to be bluer than primary galaxies from observations). To examine this scenario, it is necessary to conduct spectroscopy by placing a slit along the northwest direction that covers from the center to the tail of the EBG-1 with spatial extent. The fraction of the extremely blue galaxy in this work is low (one out of 974 galaxies) compared to the other studies such as Saxena et al. (2024). This is probably because of the low signal-to-noise ratio of most of the galaxies. Although some galaxies show the best-fit β values smaller than -2 . 6 in Figure 2, the large error bars hinder us from confirming that these galaxies truly have blue UV continua. There may be more galaxies like EBG-1, while deeper spectroscopic observations are required to confirm. The non-detection of H β line and the large uncertainty in [O iii ] emission lines make it difficult to constrain the f ion esc value and type of ionizing source of EBG-1. It is thus necessary to conduct deeper spectroscopy on EBG1 to definitively confirm the high f ion esc and the ionizing source, by which one can utilize EBG-1 as an example to study the process of the significant ionizing photon escape during the epoch of reionization.","pages":[7]},{"title":"5. SUMMARY","content":"In this work, we search the large JWST/NIRSpec spectroscopic sample for a galaxy with an extremely blue UV continuum. Among our sample consisting of the 974 galaxies at 4 < z < 14 taken from the major JWST GTO, ERS, and GO programs, we identify EBG-1, a galaxy at z = 9 . 25 showing β ∼ -3. Our major findings are summarized below:","pages":[7]},{"title":"ACKNOWLEDGMENTS","content":"We are grateful to Steven L. Finkelstein, Moka Nishigaki, and Kuria Watanabe for the valuable discussions. This work is based on observations made with the NASA/ESA/CSA James Webb Space Telescope. The NIRSpec data of EBG-1 were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. The data products presented herein were retrieved from DJA. DJA is an initiative of the Cosmic Dawn Center (DAWN), which is funded by the Danish National Research Foundation under grant DNRF140. We thank DJA for providing the reduced NIRSpec data. The observational data collected from DJA are associated with programs ERS 1345 (CEERS; PI: Finkelstein), DDT 2756 (PI: Chen), DDT 2750 (CEERS; PI: Arrabal Haro), GTO 1180 (PI: Eisenstein), GTO 1181 (PI: Eisenstein), GTO 1210 (PI: Luetzgendorf), GTO 1211 (PI: Isaak), GTO 1286 (PI: Luetzgendorf), DDT 6541 (PI: Egami), GO 1433 (PI: Coe), GO 1747 (PI: Roberts-Borsani), GO 1810 (PI: Belli), GO 1871 (PI: Chisholm), GO 2110 (PI: Kriek), GO 2198 (PI: Barrufet), GO 2561 (UN- COVER; PI: Labbe), GO 2565 (PI: Glazebrook), DDT 2767 (PI: Kelly), ERO 2736 (PI: Pontoppidan), GO 3215 (PI: Eisenstein), GO 4233 (PI: de Graaff) GO 4246 (PI: Abdurro'uf), DDT 4446(PI: Frye), and DDT 4557 (PI: Yan). The authors acknowledge the teams conducting these observations for publicly releasing the data. This publication is based on work supported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, KAKENHI (20H00180, 21H04467, 21H04489, 24K07102) through the Japan Society for the Promotion of Science, and JST FOREST Program (JP-MJFR202Z). This work was supported by the joint research program of the Institute for Cosmic Ray Research (ICRR), University of Tokyo.","pages":[8]},{"title":"Facilities:","content":"Software: astropy (Astropy Collaboration et al. 2013, 2018, 2022), NumPy (Harris et al. 2020), matplotlib (Hunter 2007), SciPy (Virtanen et al. 2020), corner (Foreman-Mackey 2016) Cloudy (Ferland et al. 2013), emcee (Foreman-Mackey et al. 2013), msaexp (Brammer 2023) Prospector (Johnson et al. 2021b)","pages":[8]},{"title":"REFERENCES","content":"Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068 Astropy Collaboration, Price-Whelan, A. M., Sip\"ocz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f Hunter, J. D. 2007, Computing in Science and Engineering, 9, 90, doi: 10.1109/MCSE.2007.55","pages":[8,9]}],"string":"[\n {\n \"title\": \"ABSTRACT\",\n \"content\": \"We investigate UV continuum slopes β of 974 galaxies at z = 4 -14 using archival JWST/NIRSpec PRISM spectra obtained from major JWST GTO, ERS, and GO programs, including JADES, CEERS, and UNCOVER. Among these galaxies, we identify a remarkable galaxy at z = 9 . 25, dubbed EBG-1, with a significantly blue UV slope β = -2 . 99 ± 0 . 15, unlike the rest of the galaxies that exhibit red continua or ambiguous blue continua hindered by large uncertainties. We confirm that the β value negligibly changes by the data reduction and fitting wavelength ranges for UV emission/absorption line masking. The extreme blue slope, β = -3 . 0, rules out significant contributions from dust extinction or AGN activity. Comparing with stellar and nebular emission models, we find that such a blue UV slope cannot be reproduced solely by stellar models even with very young, metal-poor, or top-heavy contiguous star formation associated with strong nebular continua making the UV slopes red, but with a high ionizing photon escape fraction, f ion esc ≳ 0 . 5, for a weak nebular continuum. While the H β emission line is not detected, likely due to the limited sensitivity of the spectrum, we find moderately weak [O iii ] λ 4959,5007 emission lines for the given star-formation rate (3 M ⊙ yr -1 ) and stellar mass (10 8 . 0 M ⊙ ) that are about three times weaker than the average emission lines, again suggestive of the high ionizing photon escape fraction, f ion esc ∼ 0 . 7 or more. EBG-1 would provide crucial insights into stellar and nebular continuum emission in high-redshift galaxies, serving as an example of the ionizing photon escaping site at the epoch of reionization. Keywords: Ultraviolet color; Reionization; Galaxy evolution; Galaxy formation; High-redshift galaxies\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"A Galaxy with an Extremely Blue UV Slope β = -3 at z = 9 . 25 Identified by JWST Spectroscopy: Evidence for a Weak Nebular Continuum and Efficient Ionizing Photon Escape?\",\n \"content\": \"Hiroto Yanagisawa , 1, 2 Masami Ouchi , 3, 1, 4, 5 Kimihiko Nakajima , 3 Yuichi Harikane , 1 Seiji Fujimoto , 6, 7, 8 Yoshiaki Ono , 1 Hiroya Umeda , 1, 2 Minami Nakane , 1, 2 Hidenobu Yajima , 9 Hajime Fukushima , 9 and Yi Xu 1, 10 1 Institute for Cosmic Ray Research, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan 2 Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan 3 National Astronomical Observatory of Japan, National Institutes of Natural Sciences, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 4 Department of Astronomical Science, SOKENDAI (The Graduate University for Advanced Studies), 2-21-1 Osawa, Mitaka, Tokyo, 181-8588, Japan 5 Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, Chiba 277-8583, Japan 6 Department of Astronomy, The University of Texas at Austin, Austin, TX 78712, USA 7 Cosmic Dawn Center (DAWN), Denmark 8 Niels Bohr Institute, University of Copenhagen, Lyngbyvej 2, DK2100 Copenhagen Ø, Denmark 10 9 Center for Computational Sciences, University of Tsukuba, Ten-nodai, 1-1-1 Tsukuba, Ibaraki 305-8577, Japan Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1. INTRODUCTION\",\n \"content\": \"The first galaxies initiate star formation in the early universe, with young massive stars exhibiting blue UV continua and producing substantial amounts of ionizing photons. Although the ionizing photons are mainly used to ionize the interstellar medium, a fraction of the ionizing photons escape from the galaxy and ionize the intergalactic medium, driving the cosmic reionization. The escape of the hydrogen ionizing photons is characterized by an escape fraction, f ion esc , which is the key quantity to understand how the ionizing photons of the galaxies contribute to the cosmic reionization. The UV continuum slope β ( f ( λ ) ∝ λ β ) is an important indicator of the production and escape of ionizing photons. Young stellar populations produce β < -2, while dust extinction and active galactic nucleus (AGN) provide a red ( β ≳ -2) UV continuum (e.g., Bouwens et al. 2012; Finkelstein et al. 2012). Typically the β values are larger than ∼ -2 . 6 (Chisholm et al. 2022), because an intrinsically blue UV continuum intensely ionizes the nebula, leading to a significant contribution from a nebular continuum, which has β ≳ -2 (Katz et al. 2024; Cameron et al. 2024). However, without the nebular continuum, the β values can be as low as -3 . 4 (e.g., Bouwens et al. 2010). This situation is achieved if the escape fraction of the ionizing photon f ion esc is large, because in that case the nebula is less ionized and the nebular continuum has less contribution to the UV spectrum (Zackrisson et al. 2017) . It is thus important to search for galaxies with β < -2 . 6 because such blue galaxies may have extremely high f ion esc , which contribute to the cosmic reionization. Because galaxies with such high f ion esc are probably rare (Leitet et al. 2013; Matthee et al. 2017), it is necessary to search large sample of galaxies. Photometric studies are conducted with JWST data by Topping et al. (2022, 2024); Morales et al. (2024); Cullen et al. (2023, 2024), although deriving β using photometry suffers from contamination by emission lines. The accurate measurement of β requires the high quality spectroscopic data. In this work, we search a large spectroscopic sample of galaxies provided by the DAWN JWST Archive for a galaxy with a blue UV slope, and report a galaxy at z = 9 . 25 that have an extremely blue UV slope. In Section 2, we describe our sample and method of UV slope measurements. In Section 3, we present the results the extremely blue object. We discuss physical origins of the extremely blue UV continuum in Section 4. Section 5 summarizes our results.\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"2.1. Lower Limit of β for f ion esc = 0\",\n \"content\": \"We first define a quantitative criterion for a blue UV continuum. We calculate β assuming f ion esc = 0 using Cloudy version 23.01 (Gunasekera et al. 2023) with incident radiations of Kroupa IMF (Kroupa 2001) with a mass range of 0 . 1 -100 M ⊙ provided by BPASS v2.2.1 (Stanway & Eldridge 2018), top-heavy IMF with a mass range of 50 -500 M ⊙ taken from Yggdrasil Pop III.1 model (Zackrisson et al. 2011), and blackbody. We assume a hydrogen density of n e = 10 2 cm -3 , nebular metallicity of log Z neb / Z ⊙ = -2, number of ionizing photon Q (H) = 10 50 s -1 , and inner radius of 10 14 cm. We assume stellar metallicity of log Z star / Z ⊙ = -2 for the Kroupa IMF model, while Z star = 0 is used for top-heavy IMF. Figure 1 shows time evolution of β values. Although the incident radiation of the top-heavy IMF is bluer than that of the Kroupa IMF, the topheavy IMF+nebular continuum model is redder than the Kroupa+nebular continuum model. This is because the intrinsically blue incident radiation in top-heavy IMF model intensely ionize the nebula, which leads to a significant contribution from the red nebular continuum. One can also see that the lower limit of β value for f ion esc = 0 is -2 . 6, while the β value reach as low as -3 . 4 if nebular continuum is not included. We thus define a criterion for a blue UV continuum as β = -2 . 6.\",\n \"pages\": [\n 2\n ]\n },\n {\n \"title\": \"2.2. Our Sample\",\n \"content\": \"We use 974 PRISM/CLEAR spectra of galaxies provided by the DAWN JWST Archive (DJA) 1 , which compile the major JWST GTO, ERS, and GO programs. 2 We select the galaxies within the redshift range of 4 < z < 14, whose UV continua are covered by NIRSpec. The spectra in the DJA are reduced with msaexp (Brammer 2023; Heintz et al. 2024). Hereafter, these 974 galaxies are referred to as our sample. The redshift distribution of galaxies in our sample is shown in the top panel of Figure 2.\",\n \"pages\": [\n 2\n ]\n },\n {\n \"title\": \"2.3. UV Slope Measurements\",\n \"content\": \"To derive β , we fit a galaxy spectra with a function where A is a constant for normalization and λ is wavelengths. We employ a Markov Chain Monte Carlo (MCMC) method for the fitting using emcee (ForemanMackey et al. 2013). We minimize a likelihood function where f mod and f obs are the model and observed flux, respectively, and σ ( λ ) is the 1 σ error of the flux. We use a rest-frame wavelength range of 1268 -2580 ˚ A for the fitting, following the fitting range presented by Calzetti et al. (1994). The measured β values are shown in the bottom panel of Figure 2. Most of the galaxies show β > -2 . 6, which is larger than the lower limit for f ion esc = 0 (Figure 1). Among our sample, we find one galaxy at z = 9 . 25, showing β = -2 . 99 ± 0 . 15, which is smaller than -2 . 6 beyond the 2 σ level. Hereafter, we refer to this object as extremely blue galaxy 1 (EBG-1). EBG-1 is originally identified as MACS0647-z9-20158 at z phot = 9 . 5 by the program GO 1433 (PI: Coe; McLeod et al. 2024), and then spectroscopically observed by the same program GO 1433 as a filler target for MACS0647-JD. The NIRCam images and spectra of EBG-1 are shown in Figure 3. Recently, Saxena et al. (2024) have found six galaxies at 5 . 5 < z < 8 showing β ∼ -3 from the spectroscopic sample obtained from the JADES survey. One out of six galaxies, JADES-GS-210003 at z = 5 . 779, is also included in our sample. Using the same fitting range as Saxena et al. (2024), we obtain β = -2 . 70 ± 0 . 13 for JADES-GS-210003 (Figure 4). There is a ∼ 2 σ difference between our and Saxena et al. (2024) of the β values. This is probably because Saxena et al. (2024) conduct a sigma-clipping method to exclude outlying pixels. JADES-GS-210003 is not selected in our study because we cannot distinguish from β = -2 . 6 at the 2 σ level. On the other hand, EBG-1 is not selected by Saxena et al. (2024), as EBG-1 falls in the MACS0647 lensing field that is not covered in the sample of Saxena et al. (2024). We also compare our β values with those in previous studies in Figure 4. We plot photometric β measurements of Cullen et al. (2024), who conduct β measurements for the sample galaxies taken from NGDEEP, JADES DR1, UNCOVER, and those from McLeod et al. (2024). Cullen et al. (2024) also derive β for EBG-1, which show good agreement with our measurement.\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"2.4. Observations and Data of EBG-1\",\n \"content\": \"In the previous section, we identify EBG-1 out of 974 objects from the spectra reduced by DJA. In this section, we first describe how EBG-1 was observed in Section 2.4.1. We next conduct the photometry for EBG-1 to confirm whether the photometry is consistent with spectrum in Section 2.4.2. We then independently performed data reduction to verify whether the β value changes with different data reduction procedures, as described in Section 2.4.3.\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"2.4.1. Observations\",\n \"content\": \"MACS0647 lensing field was observed with JWST/NIRCam in January 8th 2023 in GO 1433 (PI: Coe) targeting MACS0647-JD (Coe et al. 2013; Hsiao et al. 2023). EBG-1 is falling on the footprints of this NIRCam observations, which is photometrically identified at R.A.=06:47:36.95 and Decl.=+70:14:34.69 by McLeod et al. (2024). EBG-1 was then spectroscopically observed with JWST/NIRSpec as a filler target for MACS0647-JD in GO 1433 in February 20th, 2023. The observations were performed with PRISM/CLEAR ( R ∼ 100) for a total exposure time of 2200 s. The data were reduced with msaexp by DJA. For details of the reduction, see Heintz et al. (2024). The spectrum reduced by DJA is shown in Figure 3.\",\n \"pages\": [\n 3,\n 4\n ]\n },\n {\n \"title\": \"2.4.2. Photometry\",\n \"content\": \"Calibrated NIRCam data are collected from DJA. We conduct aperture photometry with 0 . '' 35 aperture size, which are shown in Table 1. The errors are estimated by conducting photometry within a 0 . '' 80 annulus and taking the standard deviations. An apparent magnitude of F200W is derived as 27.7, which is consistent with that derived by McLeod et al. (2024).\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"2.4.3. Reanalysis of EBG-1 spectra\",\n \"content\": \"We performed the data reduction in this work following the method described in Nakajima et al. (2023). Starting from the level 1 products provided by MAST, we executed Spec2 and Spec3 pipelines using Python library jwst (ver. 1.16.1; Bushouse et al. 2024). The reference files stored in the latest pmap file of jwst 1299.pmap were used. The pathloss corrections were conducted by comparing the position of the source and MSA shutter, where we assumed that the source was point-like. We then combined 2D spectra by medianstacking to reduce the effect of hot pixels, with extractions of 3 pixels in the spatial direction. For more details, see Nakajima et al. (2023). In Figure 5 we present the spectrum reduced in this work, which is consistent with both of the spectrum reduced by DJA and the NIRCam photometry. The spectrum of EBG-1 shows [O iii ] λλ 4959, 5007 and tentative C iv λλ 1548, 1550 emission lines. We measure flux values and 3 σ upper limits of emission lines by integrating the flux and error in each wavelength bin (Table 2). The C iv emission might be associated with an AGN, although the extremely blue UV slope disfavors the contribution from an AGN, whose dust content reddens the spectrum (Francis et al. 1991).\",\n \"pages\": [\n 4,\n 5\n ]\n },\n {\n \"title\": \"3.1. Confirmation of the Blue UV Slope\",\n \"content\": \"To check whether the extremely blue UV continuum of EBG-1 is caused by systematics or not, we adopt three fitting methods: 1) conventional Calzetti et al. (1994) windows, 2) using the whole wavelength range without masking, and 3) simply avoiding the possible C iv emission line (note that we mask out at 2480 -2520 ˚ A contaminated by the artifact). For the spectrum reduced by DJA, each fitting method gives β = -3 . 03 ± 0 . 22 , -2 . 99 ± 0 . 15 , and -3 . 08 ± 0 . 15, respectively, all of which are smaller than β = -2 . 6 at the ∼ 2 σ level. We derive β also for the spectrum reduced in this work in the same manner as described above. The fitting methods of 1), 2), and 3) yield β = -2 . 96 ± 0 . 19 , -2 . 92 ± 0 . 13 , and -2 . 79 ± 0 . 14, respectively. Although the significance levels are slightly lower than the β values from the DJA spectrum, the results of the extremely blue UV slope of EBG-1 does not significantly change. We thus conclude that the extremely blue UV slope of EBG-1 is not attributed to observational systematics.\",\n \"pages\": [\n 5\n ]\n },\n {\n \"title\": \"3.2. SED Fitting\",\n \"content\": \"To measure stellar properties, we conduct SED fitting to the NIRCam data presented in Table 1 using Prospector (Johnson et al. 2021a). We use the photometry values that are corrected for magnification factor. The Binary Population and Spectral Synthesis (BPASS; Eldridge et al. 2017) model is used for an isochrone library. We apply Calzetti et al. (2000) dust extinction law. We assume a non-parametric star-formation history (SFH) with five bins that are evenly spaced in logarithmic times between 0 Myr and a look-back time corresponding to z = 30. The redshift of the source is fixed at z spec = 9 . 25. We vary the stellar mass M ∗ , metallicity Z , optical depth of dust attenuation at 5500 ˚ A τ dust (5500 ˚ A), and ionization parameter U . The best-fit Note -a The value is corrected for magnification factor. Note - parameters and SFH are shown in Table 1 and Figure 6, respectively. The results imply the dust-free condition and the recent starburst, which agree with the extremely blue UV continuum of EBG-1. Because f ion esc = 0 is assumed in our SED fitting, the weak emission line feature of EBG-1 indicates a low metallicity, which compensates for the effect of nonzero f ion esc .\",\n \"pages\": [\n 5\n ]\n },\n {\n \"title\": \"3.3. Weak Nebular Emission Lines\",\n \"content\": \"The extremely blue UV continuum in EBG-1 may be explained by high f esc . However, the detection of [O iii ] λλ 4959,5007 emission indicates a certain amount of the ionizing photon is used to ionize the nebulae. To quantify the escape of ionizing photons in EBG-1, we plot ratios of luminosity of [O iii ] λ 5007 line, L [OIII] , to star-formation rate (SFR) as a function of stellar mass in Figure 7. For comparison, we also plot average values calculated from 126 galaxies at 4 < z < 9 compiled by Nakajima et al. (2023). The SFRs are estimated from UV luminosities of galaxies by using Equation (1) of Kennicutt (1998). The L [OIII] /SFR value of EBG-1 is ∼ 0 . 5 dex smaller than the average value at the same stellar mass, suggesting that [O iii ] emission in EBG-1 is ∼ 3 times weaker than the average. This can be interpreted as a result of the escape of ionizing photon without ionizing the nebulae. If we assume f ion esc = 0 for the galaxies in Nakajima et al. (2023), the [O iii ] emission three times weaker than the average suggests f ion esc ∼ 0 . 7 for EBG-1. However, the f ion esc values of the galaxies in Nakajima et al. (2023) can be larger than zero, in which case our estimate of f ion esc becomes larger. Furthermore, if we assume density-bounded nebulae, weak [O iii ] emission means an excess of O ++ ionizing photons (ionization energy of 35 eV) compared to the amount of the gas, with H + ionizing photon (ionization energy of 13 . 6 eV) being even more abundant. In such a situation, the escape fraction of H + ionizing photon can be larger than that of O ++ ionizing photon. However, the estimate of f ion esc from the [O iii ] emission is susceptible to metallicity. To estimate f ion esc independently of the assumption of metallicity, we use the β and EW(H β ) values of EBG-1. We derive the upper limit of the H β flux by summing up the error spectrum in quadrature. We then divide the upper limit of H β by the continuum flux density, obtaining log (EW(H β ) / ˚ A) < 2 . 8. In Figure 8, we compare the β and EW(H β ) values with the Cloudy models, which are the same as those in Figure 1 except for varying f ion esc . In all of the models, the very small β values of EBG1 imply f esc ≳ 0 . 5 (although models with extremely metal-poor or metal-free stellar populations reproduce β ∼ -3 even with f ion esc = 0 see Figure 4 of (Bouwens et al. 2010), the detection of [O iii ] emission in EBG-1 disfavors this scenario). However, due to the weak upper limit of EW(H β ), it is difficult to constrain the ionizing radiation and f esc with current data.\",\n \"pages\": [\n 5,\n 6\n ]\n },\n {\n \"title\": \"4. DISCUSSION\",\n \"content\": \"There are two scenarios that explain the high escape fraction: density-bounded nebulae, or ionizationbounded nebulae with holes (Zackrisson et al. 2013). In density-bounded nebulae, ionizing photons exceeding the gas supply escape the nebula, while in ionizationbounded nebulae with holes, ionizing photons leak through these holes. Both of the scenarios are consistent with our results, and it is difficult to test these two scenarios with the current data. However, the morphology of EBG-1 (Figure 3) may provide a clue to the origin of the high escape fraction. EBG-1 has a small tail extending towards north-west direction from the main component. The slit covers around the tail of the EBG1. The tail may be stripped from EBG-1 by galaxy interactions, where the gas could be lacking. We may be looking at this gas-deficient region, where the nebulae are density-bounded or with many holes. In such a case, the center of EBG-1 can be redder than the tail (see also Schombert et al. (1990), who claim that tidal tails tend to be bluer than primary galaxies from observations). To examine this scenario, it is necessary to conduct spectroscopy by placing a slit along the northwest direction that covers from the center to the tail of the EBG-1 with spatial extent. The fraction of the extremely blue galaxy in this work is low (one out of 974 galaxies) compared to the other studies such as Saxena et al. (2024). This is probably because of the low signal-to-noise ratio of most of the galaxies. Although some galaxies show the best-fit β values smaller than -2 . 6 in Figure 2, the large error bars hinder us from confirming that these galaxies truly have blue UV continua. There may be more galaxies like EBG-1, while deeper spectroscopic observations are required to confirm. The non-detection of H β line and the large uncertainty in [O iii ] emission lines make it difficult to constrain the f ion esc value and type of ionizing source of EBG-1. It is thus necessary to conduct deeper spectroscopy on EBG1 to definitively confirm the high f ion esc and the ionizing source, by which one can utilize EBG-1 as an example to study the process of the significant ionizing photon escape during the epoch of reionization.\",\n \"pages\": [\n 7\n ]\n },\n {\n \"title\": \"5. SUMMARY\",\n \"content\": \"In this work, we search the large JWST/NIRSpec spectroscopic sample for a galaxy with an extremely blue UV continuum. Among our sample consisting of the 974 galaxies at 4 < z < 14 taken from the major JWST GTO, ERS, and GO programs, we identify EBG-1, a galaxy at z = 9 . 25 showing β ∼ -3. Our major findings are summarized below:\",\n \"pages\": [\n 7\n ]\n },\n {\n \"title\": \"ACKNOWLEDGMENTS\",\n \"content\": \"We are grateful to Steven L. Finkelstein, Moka Nishigaki, and Kuria Watanabe for the valuable discussions. This work is based on observations made with the NASA/ESA/CSA James Webb Space Telescope. The NIRSpec data of EBG-1 were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. The data products presented herein were retrieved from DJA. DJA is an initiative of the Cosmic Dawn Center (DAWN), which is funded by the Danish National Research Foundation under grant DNRF140. We thank DJA for providing the reduced NIRSpec data. The observational data collected from DJA are associated with programs ERS 1345 (CEERS; PI: Finkelstein), DDT 2756 (PI: Chen), DDT 2750 (CEERS; PI: Arrabal Haro), GTO 1180 (PI: Eisenstein), GTO 1181 (PI: Eisenstein), GTO 1210 (PI: Luetzgendorf), GTO 1211 (PI: Isaak), GTO 1286 (PI: Luetzgendorf), DDT 6541 (PI: Egami), GO 1433 (PI: Coe), GO 1747 (PI: Roberts-Borsani), GO 1810 (PI: Belli), GO 1871 (PI: Chisholm), GO 2110 (PI: Kriek), GO 2198 (PI: Barrufet), GO 2561 (UN- COVER; PI: Labbe), GO 2565 (PI: Glazebrook), DDT 2767 (PI: Kelly), ERO 2736 (PI: Pontoppidan), GO 3215 (PI: Eisenstein), GO 4233 (PI: de Graaff) GO 4246 (PI: Abdurro'uf), DDT 4446(PI: Frye), and DDT 4557 (PI: Yan). The authors acknowledge the teams conducting these observations for publicly releasing the data. This publication is based on work supported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, KAKENHI (20H00180, 21H04467, 21H04489, 24K07102) through the Japan Society for the Promotion of Science, and JST FOREST Program (JP-MJFR202Z). This work was supported by the joint research program of the Institute for Cosmic Ray Research (ICRR), University of Tokyo.\",\n \"pages\": [\n 8\n ]\n },\n {\n \"title\": \"Facilities:\",\n \"content\": \"Software: astropy (Astropy Collaboration et al. 2013, 2018, 2022), NumPy (Harris et al. 2020), matplotlib (Hunter 2007), SciPy (Virtanen et al. 2020), corner (Foreman-Mackey 2016) Cloudy (Ferland et al. 2013), emcee (Foreman-Mackey et al. 2013), msaexp (Brammer 2023) Prospector (Johnson et al. 2021b)\",\n \"pages\": [\n 8\n ]\n },\n {\n \"title\": \"REFERENCES\",\n \"content\": \"Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068 Astropy Collaboration, Price-Whelan, A. M., Sip\\\"ocz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f Hunter, J. D. 2007, Computing in Science and Engineering, 9, 90, doi: 10.1109/MCSE.2007.55\",\n \"pages\": [\n 8,\n 9\n ]\n }\n]"}}},{"rowIdx":4978,"cells":{"bibcode":{"kind":"string","value":"2024arXiv241200191E"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/2412.00191.pdf"},"content":{"kind":"string","value":"\nEuclid preparation\nThe impact of line-of-sight projections on the covariance between galaxy cluster multi-wavelength observable properties - insights from hydrodynamic simulations\nEuclid Collaboration: A. Ragagnin ⋆ 1 , 2 , 3 , 4 , A. Saro 5 , 2 , 6 , 7 , 4 , S. Andreon 8 , A. Biviano 6 , 2 , K. Dolag 9 , S. Ettori 1 , 10 , C. Giocoli 1 , 11 , A. M. C. Le Brun 12 , G. A. Mamon 13 , 14 , B. J. Maughan 15 , M. Meneghetti 1 , 16 , L. Moscardini 3 , 1 , 16 , F. Pacaud 17 , G. W. Pratt 18 , M. Sereno 1 , 16 , S. Borgani 5 , 2 , 6 , 7 , F. Calura 1 , G. Castignani 1 , M. De Petris 19 , D. Eckert 20 , G. F. Lesci 3 , 1 , J. Macias-Perez 21 , M. Maturi 22 , 23 , A. Amara 24 , N. Auricchio 1 , C. Baccigalupi 2 , 6 , 7 , 25 , M. Baldi 26 , 1 , 16 , S. Bardelli 1 , D. Bonino 27 , E. Branchini 28 , 29 , 8 , M. Brescia 30 , 31 , 32 , J. Brinchmann 33 , S. Camera 34 , 35 , 27 , V. Capobianco 27 , C. Carbone 36 , J. Carretero 37 , 38 , S. Casas 39 , M. Castellano 40 , S. Cavuoti 31 , 32 , A. Cimatti 41 , C. Colodro-Conde 42 , G. Congedo 43 , C. J. Conselice 44 , L. Conversi 45 , 46 , Y. Copin 47 , F. Courbin 48 , H. M. Courtois 49 , A. Da Silva 50 , 51 , H. Degaudenzi 20 , G. De Lucia 6 , J. Dinis 50 , 51 , F. Dubath 20 , X. Dupac 46 , M. Farina 52 , S. Farrens 18 , S. Ferriol 47 , M. Frailis 6 , E. Franceschi 1 , M. Fumana 36 , K. George 9 , B. Gillis 43 , A. Grazian 53 , F. Grupp 54 , 9 , S. V. H. Haugan 55 , W. Holmes 56 , I. Hook 57 , F. Hormuth 58 , A. Hornstrup 59 , 60 , K. Jahnke 61 , E. Keihänen 62 , S. Kermiche 63 , A. Kiessling 56 , M. Kilbinger 18 , B. Kubik 47 , M. Kümmel 9 , M. Kunz 64 , H. Kurki-Suonio 65 , 66 , S. Ligori 27 , P. B. Lilje 55 , V. Lindholm 65 , 66 , I. Lloro 67 , D. Maino 68 , 36 , 69 , E. Maiorano 1 , O. Mansutti 6 , O. Marggraf 17 , K. Markovic 56 , M. Martinelli 40 , 70 , N. Martinet 71 , F. Marulli 3 , 1 , 16 , R. Massey 72 , S. Maurogordato 73 , E. Medinaceli 1 , S. Mei 74 , Y. Mellier 13 , 14 , G. Meylan 48 , M. Moresco 3 , 1 , E. Munari 6 , 2 , C. Neissner 75 , 38 , S.-M. Niemi 76 , J. W. Nightingale 77 , 72 , C. Padilla 75 , S. Paltani 20 , F. Pasian 6 , K. Pedersen 78 , V. Pettorino 76 , G. Polenta 79 , M. Poncet 80 , L. A. Popa 81 , L. Pozzetti 1 , F. Raison 54 , A. Renzi 82 , 83 , J. Rhodes 56 , G. Riccio 31 , E. Romelli 6 , M. Roncarelli 1 , E. Rossetti 26 , R. Saglia 9 , 54 , Z. Sakr 22 , 84 , 85 , A. G. Sánchez 54 , D. Sapone 86 , B. Sartoris 9 , 6 , R. Scaramella 40 , 70 , P. Schneider 17 , T. Schrabback 87 , A. Secroun 63 , E. Sefusatti 6 , 2 , 7 , G. Seidel 61 , S. Serrano 88 , 89 , 90 , C. Sirignano 82 , 83 , G. Sirri 16 , L. Stanco 83 , J. Steinwagner 54 , P. Tallada-Crespí 37 , 38 , I. Tereno 50 , 91 , R. Toledo-Moreo 92 , F. Torradeflot 38 , 37 , I. Tutusaus 84 , L. Valenziano 1 , 10 , T. Vassallo 9 , 6 , G. Verdoes Kleijn 93 , A. Veropalumbo 8 , 29 , Y. Wang 94 , J. Weller 9 , 54 , G. Zamorani 1 , E. Zucca 1 , M. Bolzonella 1 , A. Boucaud 74 , E. Bozzo 20 , C. Burigana 95 , 10 , M. Calabrese 96 , 36 , D. Di Ferdinando 16 , J. A. Escartin Vigo 54 , R. Farinelli 1 , J. Gracia-Carpio 54 , N. Mauri 41 , 16 , V. Scottez 13 , 97 , M. Tenti 16 , M. Viel 2 , 6 , 25 , 7 , 4 , M. Wiesmann 55 , Y. Akrami 98 , 99 , V. Allevato 31 , S. Anselmi 83 , 82 , 12 , M. Ballardini 100 , 1 , 101 , P. Bergamini 68 , 1 , A. Blanchard 84 , L. Blot 102 , 12 , S. Bruton 103 , R. Cabanac 84 , A. Calabro 40 , G. Canas-Herrera 76 , 104 , A. Cappi 1 , 73 , C. S. Carvalho 91 , T. Castro 6 , 7 , 2 , 4 , K. C. Chambers 105 , S. Contarini 54 , 3 , A. R. Cooray 106 , M. Costanzi 5 , 6 , 2 , B. De Caro 83 , 82 , S. de la Torre 71 , G. Desprez 107 , A. Díaz-Sánchez 108 , S. Di Domizio 28 , 29 , H. Dole 109 , S. Esco ffi er 63 , A. G. Ferrari 41 , 16 , P. G. Ferreira 110 , I. Ferrero 55 , F. Finelli 1 , 10 , F. Fornari 10 , L. Gabarra 110 , K. Ganga 74 , J. García-Bellido 98 , E. Gaztanaga 89 , 88 , 111 , F. Giacomini 16 , G. Gozaliasl 112 , 65 , A. Hall 43 , H. Hildebrandt 113 , J. Hjorth 114 , A. Jimenez Muñoz 21 , J. J. E. Kajava 115 , 116 , V. Kansal 117 , 118 , D. Karagiannis 119 , 120 , C. C. Kirkpatrick 62 , L. Legrand 121 , G. Libet 80 , A. Loureiro 122 , 123 , G. Maggio 6 , M. Magliocchetti 52 , F. Mannucci 124 , R. Maoli 19 , 40 , C. J. A. P. Martins 125 , 33 , S. Matthew 43 , L. Maurin 109 , R. B. Metcalf 3 , 1 , P. Monaco 5 , 6 , 7 , 2 , C. Moretti 25 , 4 , 6 , 2 , 7 , G. Morgante 1 , Nicholas A. Walton 126 , L. Patrizii 16 , A. Pezzotta 54 , M. Pöntinen 65 , V. Popa 81 , C. Porciani 17 , D. Potter 127 , I. Risso 128 , P.-F. Rocci 109 , M. Sahlén 129 , A. Schneider 127 , M. Schultheis 73 , P. Simon 17 , A. Spurio Mancini 130 , 131 , C. Tao 63 , G. Testera 29 , R. Teyssier 132 , S. Toft 60 , 133 , 134 , S. Tosi 28 , 29 , 8 , A. Troja 82 , 83 , M. Tucci 20 , C. Valieri 16 , J. Valiviita 65 , 66 , D. Vergani 1 , and G. Verza 135 , 136 arXiv:2412.00191v1 [astro-ph.CO] 29 Nov 2024\n(A ffi liations can be found after the references)\nJune 16, 2023\nABSTRACT\nContext. Cluster cosmology can benefit from combining multi-wavelength studies, which themselves can benefit from a characterisation of the correlation coe ffi cients between di ff erent mass-observable relations.\nAims. In this work, we aim to provide information on the scatter, the skewness, and the covariance of various mass-observable relations in galaxy clusters in cosmological hydrodynamic simulations. This information will help future analyses to better tackle accretion histories and projection e ff ects and model mass observable relations for cosmology studies.\nMethods. Weidentify galaxy clusters in Magneticum Box2b simulations with mass M 200c > 10 14 M ⊙ at redshift z = 0 . 24 and z = 0 . 90. Our analysis includes Euclid -derived properties such as richness, stellar mass, lensing mass, and concentration. Additionally, we investigate complementary multi-wavelength data, including X-ray luminosity, integrated Comptony parameter, gas mass, and temperature. The impact of projection e ff ects on mass-observable residuals and correlations is then examined.\nResults. We find that at intermediate redshift ( z = 0 . 24) projection e ff ects impact lensing concentration, richness, and gas mass the most in terms of scatter and skewness of log-residuals of scaling relations. The contribution of projection e ff ects can be significant enough to boost a spurious hot- vs. cold-baryons correlation and consequently hide underlying correlations due to halo accretion histories. At high redshift ( z = 0 . 9), the richness has a much lower scatter (of log-residuals), and the quantity that is most impacted by projection e ff ects is the lensing mass. Lensing concentration reconstruction, in particular, is a ff ected by deviations of the reduced-shear profile shape from the one derived by an NFW profile rather than interlopers in the line of sight.\nKey words. Galaxies: clusters: general - Cosmology: cosmological parameters - galaxies: abundances - methods: numerical\n1. Introduction\nGalaxy clusters are the largest gravitationally bound, collapsed, and virialised structures in our Universe and represent unique laboratories for testing cosmological models, galaxy evolution, and thermodynamics of the intracluster medium (ICM, see Kravtsov & Borgani 2012, for a review on galaxy clusters). Regarding galaxy cluster cosmology studies (see, e.g., Rozo et al. 2010; Bocquet et al. 2019), an accurate characterisation of the selection function and of the mass-observable scaling relations represent the dominant systematic uncertainties (see the review on the cluster mass scale in Pratt et al. 2019).\nCluster masses cannot be observed directly, and their reconstruction requires both a number of assumptions and highquality data (see, e.g., Meneghetti et al. 2010). This means that precise estimates are rare (Okabe et al. 2010; Hoekstra et al. 2012; Melchior et al. 2015; van Uitert et al. 2016; Stern et al. 2019; Sugiyama et al. 2023; Bocquet et al. 2024). Once a set of highly accurate mass determinations are available, together with other mass proxies recovered from multi-band observations, well-calibrated mass-observable relations (for instance, the mass-richness relation or the mass-temperature relation) can be established and used to estimate galaxy cluster masses for larger samples with known observable properties. For this purpose, it is important to calibrate accurately the mass-observable relations (Giodini et al. 2013; Allen et al. 2011; Schrabback et al. 2021), including proper modelling of their associated scatter (Lima & Hu 2005; Bocquet et al. 2019).\nThis process is complicated by the fact that studies at di ff erent wavelengths are biased by various astrophysical processes and projection e ff ects to various degrees. For instance, X-ray surveys tend to favour the selection of clusters with centrally peaked gas distributions (Pacaud et al. 2007; Hudson et al. 2010; Andreon & Moretti 2011; Andreon et al. 2016; Xu et al. 2018) and su ff er from AGN contamination (see, e.g., Bhargava et al. 2023), while projection e ff ects are known to strongly impact weak lensing mass reconstructions (Meneghetti et al. 2014; Euclid Collaboration: Giocoli et al. 2024) and richness evaluations (e.g., Castignani & Benoist 2016). This is particularly relevant for cluster cosmology studies, where the aim is to reduce uncertainty by combining constraints on di ff erent massobservable relations. For Euclid (Euclid Collaboration: Mellier et al. 2024), this will include quantities such as richness, stellar mass, and properties of stacked weak lensing signals (Pires et al. 2020), of the cluster samples detected using tools such as AMICO (Bellagamba et al. 2018a; Maturi et al. 2019) or PZWav (Euclid Collaboration: Adam et al. 2019), possibly to-\ngether with other multi-wavelength observations Allen et al. (2011). These properties are known to be biased by projection e ff ects (Meneghetti et al. 2014), accretion histories (Ragagnin et al. 2022), mis-centring (Sommer et al. 2022, 2023), and the fit procedure (Sereno et al. 2016). Projection e ff ects, especially, are expected to generate some covariance between the richness and weak lensing signal, the uncertainty of which may significantly a ff ect the performance of the mission for cluster population analyses. This e ff ect is one the major sources of systematics for current optical cluster surveys (Costanzi et al. 2019; Abbott et al. 2020), and thus is expected to play an even more critical role for the Euclid cluster sample.\nNumerical simulations are thus a critical tool to mitigate the impact of the aforementioned biases on cosmological cluster studies. Indeed, the power of observations to constrain them is limited, thus increasing the final uncertainty budget. However, scatter and covariance parameters are also prime sources of uncertainty when aiming to combine information originating from di ff erent wavelengths. For instance, various observational works hint towards di ff erent directions for the hot- vs. cold-baryon covariance (Farahi et al. 2019; Puddu & Andreon 2022; Ragagnin et al. 2022), as di ff erent formation times are related with satellite accretion history (Giocoli et al. 2008).\nIn this context, numerical simulations have proven to be a very powerful tool for helping observational studies in modelling mass-observable relations, which are strongly a ff ected by galaxy cluster accretion histories (Ludlow et al. 2012; Bose et al. 2019; Davies et al. 2020; Anbajagane et al. 2020; Ragagnin et al. 2022), projection e ff ects (Meneghetti et al. 2014), and deviations (see, e.g., Ragagnin et al. 2021) from the Navarro-FrenkWhite density profile (NFW, Navarro et al. 1997), which is often adopted in weak lensing studies. Thus, simulations can suggest the most suitable functional forms of scaling relations for cosmological studies (as in the works of Costanzi et al. 2019; Bocquet et al. 2016, 2019; Ghirardini et al. 2024). They can provide informative priors on their correlation coe ffi cients, which are among the most di ffi cult parameters to be constrained directly from observed quantities, guiding the forward modelling setup of cluster cosmology studies.\nThere are various works in the literature that study how simulations can help disentangle physical models (see e.g., Cui et al. 2022; Angelinelli et al. 2023a) cosmological models (see e.g., Bocquet et al. 2020; Angulo et al. 2021; Villaescusa-Navarro et al. 2022) or dark matter types (see e.g., Ragagnin et al. 2024; Fischer et al. 2024; Contreras-Santos et al. 2024), and study observable cross-correlations (see e.g., Stanek et al. 2010; Anbajagane et al. 2020).\nIn this work, besides focusing on correlations between observable properties of interests for multi-wave length studies, we also study the impact of projection e ff ects. The impact of uncorrelated large-scale structure on the covariance between observable properties can be modelled analytically (Hoekstra 2003; McClintock et al. 2019; Costanzi et al. 2019), but the covariance of di ff erent observable properties below a few tens of Mpc requires dedicated simulations. At these scales, numerical hydrodynamic simulations, with their self-consistent depiction of the ICM, emerge as an ideal tool for exploring multiwavelength observable properties since they incorporate the effects of large-scale structures within which clusters are situated. Indeed, baryon feedback influences the ICM not only within cluster virial radii but also beyond (see, e.g., Angelinelli et al. 2022, 2023b).\nWhile it is true that cosmological simulations are influenced by the underlying sub-grid prescriptions, and while it is true that these simulations may diverge on small scales, they generally exhibit agreement on quantities integrated up to the sizes of galaxy groups and clusters (see, e.g., Anbajagane et al. 2020). At the same time, di ff erent cosmological parameters can a ff ect galaxy cluster properties, such as their masses (Ragagnin et al. 2021), satellite abundance (van den Bosch et al. 2005), and massobservable relations (Singh et al. 2020). On the other hand, the qualitative significance of covariances and projection e ff ects on observable properties is not expected to significantly hinge on cosmological parameters (Bocquet et al. 2019), and possible deviations from this expectation could be estimated using emulators (see, e.g., Bocquet et al. 2020; Ragagnin et al. 2021; Angulo et al. 2021; Ragagnin et al. 2023).\nWe will study the impact of projection e ff ects using hydrodynamic simulations in order to gain insight into which fraction of the scatter and skewness of scaling relations originates from projection e ff ects (i.e., alignment with filaments and objects) or di ff erent accretion histories.\nIn Sect. 2, we present how we set up our Euclid -like observable properties and the others coming from the other wavelengths. In Sect. 3, we study how projection e ff ects impact the scatter and skewness of log-residuals of scaling relations and discuss the impact of projection e ff ects on observable covariance. In Sect. 4, we focus on the mass-concentration relation and how it is a ff ected by projection e ff ects and deviations from the functional form of profiles and the radial ranges of the fits. In Sect. 5, we focus on the covariance between observable properties and study how di ff erent accretion histories and projection e ff ects impact them. Finally, we draw our conclusions in Sect. 6.\n2. Numerical Setup\nWe will conduct our study by analysing clusters obtained from the Magneticum 1 hydrodynamic cosmological simulations (Bi ffi et al. 2013; Saro et al. 2014; Steinborn et al. 2015; Dolag et al. 2016, 2015; Teklu et al. 2015; Steinborn et al. 2016; Bocquet et al. 2016; Ragagnin et al. 2019). They are based on the N -body code Gadget3 , which is built upon Gadget2 (Springel et al. 2005b; Springel 2005; Boylan-Kolchin et al. 2009) with an improved smoothed particle hydrodynamics (SPH) solver from Beck et al. (2016). Magneticum initial conditions are generated using a standard Λ CDM cosmology with Wilkinson Microwave Anisotropy Probe 7 (Komatsu et al. 2011) cosmological parameters. The large-scale structure evolution in Magneticum simulations includes a treatment of radiative cooling, heating\nfrom a uniform redshift-dependent ultraviolet (UV) background, star formation, and stellar feedback processes as in Springel et al. (2005a). The stellar feedback is then connected to a detailed chemical evolution and enrichment model as in Tornatore et al. (2007), which follows 11 chemical elements (H, He, C, N, O, Ne, Mg, Si, S, Ca, and Fe, with cooling tables from Wiersma et al. 2009) which are produced with the CLOUDY photo-ionisation code (Ferland et al. 1998). Fabjan et al. (2010) and Hirschmann et al. (2014) described prescriptions for black hole growth and feedback from AGNs. Haloes that host galaxy clusters and groups are identified using the friends-of-friends halo finder (Davis et al. 1985), and subhaloes together with their associated galaxies are identified with an improved version of SUBFIND (Springel et al. 2001), which takes into account the presence of baryons (Dolag et al. 2009).\nWe define r ∆ c as the radius that encloses an average density of ∆ c ρ cr , where ρ cr is the critical density of the Universe at a given redshift,\nM ∆ c = 4 3 π r 3 ∆ c ∆ c ρ cr . (1)\nThroughout this paper, when we omit ∆ c from masses and radii we imply the usage of ∆ c = 200 (i.e., M = M 200c) .\nTo disentangle the scatter of the mass-observable relation from projection e ff ects, we compute quantities within a sphere of radius r 200c and as integrated into a cylinder. Projected quantities will be denoted with the superscript 2D (for instance, the total mass inside the cylinder is denoted as M 2D ). We opted to employ a random projection plane for each cluster. Additionally, we set an integration depth of 20 comoving h -1 Mpc, corresponding to approximately 23 Mpc at z = 0 . 24 and 15 Mpc at z = 0 . 9 (with h = 0 . 704). This cylinder depth is smaller than Euclid 's galaxy cluster photoz equivalent uncertainty (Euclid Collaboration: Desprez et al. 2020) and, while we exclude some uncorrelated projection e ff ects, they are known not to play an important role (Sunayama et al. 2020; Wu et al. 2022). Thus, we ensure that we do not overestimate any projection e ff ect that could be mitigated using photoz . Consequently, all projection e ff ects examined in this paper hold relevance for interpreting forthcoming Euclid -based catalogues.\nThis study is based on the results from Box2b / hr (Hirschmann et al. 2014) Magneticum simulation, which covers a length of 900 comoving Mpc, with dark matter particle masses m DM = 9 . 8 × 10 8 M ⊙ , gas initial particle masses of m gas = 2 × 10 8 M ⊙ , and a gravitational softening of both gas and dark matter of ϵ = 3 . 75 comoving kpc. Euclid is expected to detect clusters with masses M > 10 14 M ⊙ up to a redshift of approximately z ≈ 2 (Sartoris et al. 2016; Euclid Collaboration: Adam et al. 2019), where the bulk of the cluster population which will be used for mass-calibration will lie below redshift z ≈ 1 . Furthermore, the number of haloes contained in this Magneticum simulation drops significantly beyond the same redshift value. Therefore, we decided to extract haloes at two representative redshift slices: at an intermediate redshift of approximately z ≈ 0 . 24, yielding 4300 objects, and at a higher redshift of approximately z ≈ 0 . 9, yielding 1300 objects. These extractions were performed using the web portal 2 introduced in Ragagnin et al. (2017). We focus most of the analyses on the qualitative e ff ect of projection e ff ects at our intermediate redshift slice because of the larger statistics of clusters to help us determine projection e ff ects. We stress that this mass threshold is high enough so that all of our galaxy clusters have at least\n10 4 particles and, therefore, can be considered well resolved in terms of density profile fitting (Navarro et al. 1997).\n2.1. Observable properties\nWe now discuss the properties that we compute for each cluster and report a summary in Table 1. We compute the total stellar masses M ⋆ and M 2D ⋆ as the sum of all stellar particles within the respective volumes. We compute the richness n with a cut of satellites of log 10 ( M ⋆/ M ⊙ ) > 10 . 65 , and the projected version n 2D that includes Euclid -like corrections for projection effects where similarly to Andreon et al. (2016). In particular, we compute the average projected richness between 3 . 5 and 8 Mpc radii from the cluster centre, divided the annulus in 8 slices of equal angles, excluded the two least dense and most dense slices and removed the average projected number density of the 4 remaining octants from the projected richness within r 200c; 3 We compute the X-ray luminosities LX , and L 2D X , in the [0 . 5 , 2] keV energy band computed using the APEC model (Smith et al. 2001), using SPH particle temperatures together with the XSPEC package 4 (Arnaud 1996), which considers the emission of a collisionally-ionized, chemically-enriched plasma implemented with metallicity values taken from the simulated particles 5 . We compute the temperature T and T 2D as weighted by the X-ray emissivity of gas particles. We compute the hot gas mass M g and M 3D g , computed as the sum of the mass of SPH particles with cold gas fraction greater than 0 . 1 and T > 3 × 10 4 K in order to filter out cold gas.\nNote that the projected gas mass is not to be confused with the one inferred from X-ray observations, as X-ray observational works typically perform a de-projection of the surface brightness ∝ n 2 e , which provides a gas-mass estimate that is closer to the spherical M g , with the addition of some possible alignment e ff ects coming from the central region of clusters. Moreover, observational works have the capability to mask possible bright substructures, thus minimising the presence of interlopers. Consequently, we can conceptualise the observed projected gas mass as an intermediate value between our M g and M 2D g .\nWe estimate the integrated Comptony parameter produced by thermal Sunyaev-Zeldovich (SZ, Sunyaev & Zeldovich 1972). The Comptony parameter is defined as\ny = k B σ T m e c 2 Z T n e d l , (2)\nwhere T is the temperature, n e the number density of the electrons, k B the Boltzmann constant, σ T the Thomson cross-section, c the speed of light, and m e the electron rest mass. We compute the integrated Comptony parameter Y = R y d Ω , both within the volume of sphere of ( Y ) and a cylinder ( Y 2D ). We estimate the integral in Eq. (2) as\nZ T n e d l ≈ 1 π R 2 X i Ti f e , i mi m p , (3)\nwhere the sum runs over all SPH particles, mi is the i -th SPH particle mass, Ti its temperature and f e , i is its electron fraction, expressed as local electron number density normalised to the hydrogen number density, and m p is the proton mass.\nFor each halo, we also perform fits of the NFW profile ρ NFW, defined as\nρ NFW ( r ) = ρ 0 r / r s (1 + r / r s) 2 , (4)\nwhere the scaling density ρ 0 and the scale radius r s (that is the radius where the density log-slope equals -2) are free parameters. We perform this fit on the total matter density profile on 100 log-spaced radial bins between 75 ckpc (which corresponds to 60 kpc at z = 0 . 24 , and to 40 kpc at z = 0 . 9; as it is enough to exclude the deep central potential of baryons) and r 200c , and define the corresponding NFW masses and concentration parameters as M NFW and c NFW respectively, and the concentration as c NFW = r NFW / r s , where r NFW is obtained from M NFW via the Eq. (1).\nThe projected version of the mass and concentrations are obtained by mimicking a lensing reconstruction procedure by fitting the corresponding reduced shear. We define the derived masses and concentrations as M 2D NFW and c 2D NFW , where c 2D NFW = R 2D NFW / r s , where R 2D NFW is obtained from M 2D NFW via the Eq. (1). Note that the \"2D\" here, as for the other quantities, means that the quantity is computed in projection, however, a correct fit of the mass from reduced shear NFW profile (namely, our M 2D NFW ) should provide an estimate of the same NFW halo mass M NFW that would be recovered from a 3D fit.\nThe fit is computed within the cylinder described above, with a projected radial range of [300 , 3000] kpc at z = 0 . 2 . We performed the analyses at z = 0 . 9 by rescaling that range with H -2 / 3 ( z ) , where H ( z ) is the Hubble parameter, in order to retain the same fractional distances from the virial radius (at fixed mass), which resulted in a range of [234 , 2300] kpc . Note that in this work we are not interested in estimating the contribution of the uncorrelated large-scale structure in the reduced shear reconstruction, therefore we limit our density projection to a cylinder of the depth of 20 cMpc , (see, e.g., Euclid Collaboration: Giocoli et al. 2024; Becker & Kravtsov 2011).\nThe signal from the source-averaged excess surface mass density ∆Σ gt , averaged over circular radii R and a population of sources distributed in redshift, can be written as\nD ∆Σ gt E ( R ) ≃ ⟨ ∆Σ t ⟩ ( R ) 1 -D Σ -1 cr E ⟨ Σ ⟩ ( R ) . (5)\nHere Σ denotes the surface mass density. The symbol ⟨ ... ⟩ denotes an average over radial bins and redshift lens sources, where we used a redshift distribution as proposed in Euclid Collaboration: Ajani et al. (2023), and Euclid Collaboration: Giocoli et al. (2024). The quantity ⟨ ∆Σ t ⟩ is the excess of surface mass density, averaged over polar coordinates and defined as\n⟨ ∆Σ t ⟩ ( R ) = 1 π R 2 Z R 0 2 π r ⟨ Σ ⟩ ( r ) d r - ⟨ Σ ⟩ ( R ) . (6)\nThe symbol Σ cr in Eq. (5) is the critical surface mass density, that for a given redshift source equals to\nΣ cr = c 2 D s 4 π GD ds D d , (7)\nwhere G the universal gravity constant, D s the angular diameter distance to the source, D d the angular distance to the lens, and\n
\n\n
Table 1: List of observable properties used in this work and presented in Sect. 2.1
\n
\nD ds the angular distance between the source and the lens. Similarly to Euclid Collaboration: Giocoli et al. (2024), we define the error associated with each radial bin of the profile in Eq. (5) as\nδ D ∆Σ gt E = ⟨ Σ cr ⟩ σϵ q π n g GLYPH<16> R 2 2 -R 2 1 GLYPH<17> , (8)\nwhere σϵ = 0 . 3 (Hoekstra et al. 2012; Euclid Collaboration: Blanchard et al. 2020) is the dispersion of the total intrinsic ellipticity ϵ = (1 -q ) / (1 + q ) , where q is the axis ratio, R 1 and R 2 represent the inner and outer radius of a bin, and n g is the number density of galaxies. For our redshift source distribution (we assume the same as in Euclid Collaboration: Giocoli et al. 2024), we find that n g ≈ 28 arcmin -2 for lenses at redshift z = 0 . 24 and n g ≈ 14 arcmin -2 for lenses at redshift z = 0 . 9 .\n2.2. Scaling relations\nIn Fig. 1 we show the observable properties vs. true mass M of clusters, derived from Magneticum Box2b / hr simulation for properties that could be derived by Euclid -like catalogues, such as lensing concentration (first row from top), lensing mass (second row), projected richness (third row), and projected stellar mass (last row), as presented in Sect. 2.1. For each property, we fit a scaling relation performed using a linear regression in the log-log space. We utilise a log-log linear regression because a single power law proves to be e ff ective in modelling our scaling relations.\nIn the right panel of Fig. 1, we show the log-residual distribution for both low-mass haloes ( M < 2 × 10 14 M ⊙ ), highmass haloes ( M > 2 × 10 14 M ⊙ ), and for the complete sample of the log-residuals σ ln , i , defined as the logarithmic ratio between the i -th cluster property and the corresponding scaling relation value at its mass. In the second column, we also report the logscatter σ ln defined here as the corresponding standard deviation of the log-residual, namely\nσ ln = E h GLYPH<0> σ ln , i -E GLYPH<2> σ ln , i GLYPH<3>GLYPH<1> 2 i 1 / 2 , (9)\nwhere E is the expectation operator that averages over our catalogue data. We note that the concentration has a scatter of 0 . 45 which is higher than theoretical expectations (see, e.g., Child et al. 2018). Throughout this paper, we will show that this is due to projection e ff ects; in fact, the 3D concentration has a scatter of ≈ 0 . 33 .\nNote that our scatter in temperature exceeds that reported in the theoretical work by Truong et al. (2018). We verified that, if\n\n\n
Fig. 1: Magneticum mass-observable relation for Euclid -like derived quantities. The left column shows scaling relations, relative fit (solid grey line), and a corridor corresponding to one standard deviation (dashed grey line). The right column shows the residual PDF and scatter of log-residuals σ ln . We report the following properties: lensing concentration c 2D NFW (first row), lensing mass M 2D NFW (second row), projected richness n 2D (third row), and projected stellar mass M 2D ⋆ (fourth row). The histogram of residuals for haloes with M < 2 × 10 14 M ⊙ is in blue dotted lines, for haloes with M > 2 × 10 14 M ⊙ is in orange dashed lines, and for the complete sample is in solid grey lines. Note that the three histograms almost overlap. Each distribution panel reports the value of the natural log scatter σ ln for the complete sample.
\n\nwe compute mass-weighted temperature, which is known to behave very well in a power law scaling relation, reveals a log scatter of 0 . 07 , in agreement with their work. The additional scatter that we see may be due to di ff erent X-ray temperature computa-\n\n\n
Fig. 2: As Fig. 1, but for the following multi-wavelength observable properties: projected integrated Comptony parameter Y 2D (first row), the 3D gas mass M g , 500c (second row), the projected X-ray luminosity in the soft band (in range [0 . 5 , 2] keV) L 2D X , 500c (third row), and the projected temperature T 2D 500c (fourth row). The values of X-ray luminosity, gas mass, and temperatures are reported within an overdensity of r 500c as this definition is a typical choice in X-ray-based observations.
\n\ntions (they use core-excised temperature while we take the contribution of the core into account).\nIn Fig. 2 we show the mass-observable relations of quantities that could potentially be obtained from various multi-wavelength observations that could enrich studies based on Euclid -like data products: the integrated Comptony parameter, the gas mass M g , 500c, the X-ray luminosity L 2D X , 500c converted in the soft band [0 . 5 , 2] keV , and the temperature T 500c . Wedecided to plot the Xray luminosity, gas mass, and temperature within r 500c because this radius is typically used in various X-ray observations (see, e.g., Vikhlinin et al. 2006; Sun et al. 2009).\nThe typical Euclid cluster cosmology analysis will therefore rely on mass-observable relations calibrated within r 200 c (e.g., richness, weak-lensing mass), and follow-up observations calibrated within r 500 c (e.g., X-ray and SZ mass-proxies). We will thus need to take into account the covariance between observable properties extracted at di ff erent radii. . Finally, we note that generally, X-ray observations are expected to align more closely with the 3D mass rather than the projected one (Ettori et al. 2013), although several details adopted to analyse the Xray observations (e.g., including masking of substructures, deprojection procedures, etc.) can significantly impact the final result. These choices critically depend on the quality of the observations themselves. Dedicated mocks will thus be required to properly take into account all these e ff ects and are, therefore,\nbeyond the purpose of this work. For simplicity, in this work, we consider an X-ray-derived gas mass as close to M g , while an SZ-derived gas mass closer to M 2D g .\n3. Projection effects\nThe main objective of this work is to disentangle the amount of scatter and skewness in scaling relations that is purely due to projection e ff ects. Note that observational data are also a ff ected by measurement errors that we do not tackle in this work (as, for instance, the Poisson error of the limited number of galaxies used to infer the richness). In this Section, we discuss the scatter and skewness of mass-observable relation qualitatively and limit the discussion for the data at redshift z = 0 . 24 , as we have a larger sample of galaxy clusters, and the results are qualitatively similar to the ones at z = 0 . 9. We stress that we leave the quantitative discussion of the scatter and correlation coe ffi cients for both redshifts on Sect. 5.\nTo assess the role of projection e ff ects, Fig. 3 reports the 3D and 2D scatter of our cluster properties. In the left panel of Fig. 3 we show the value of the scatter (see Eq. 9) of log-residuals σ ln for all our mass-observable relations. In the shaded region, we also report the values computed within r 500c for the X-ray luminosity, gas mass, and temperature values because this is the characteristic overdensity used in X-ray analyses.\nFor each observable, in Fig. 3, we report (with di ff erent symbols) both the scatter of the complete sample as well as the one of two separate mass range of 10 14 < M < 2 × 10 14 M ⊙ and M > 2 × 10 14 M ⊙ respectively. We note that the lower-mass bin ( M < 2 × 10 14 M ⊙ ) is the one with the largest scatter because, for a given external object in the line-of-sight (LoS), the profile of a small cluster will be more perturbed with respect to a cluster.\nIn the left panel of Fig. 3, we see that some quantities have a low scatter in the 3D space and do gain a large amount of scatter once they are seen in projection. To better quantify what is the actual impact of projection e ff ects, in the right panel of Fig. 3 we present the metric q σ 2 ln , 2D -σ 2 ln , 3D . This metric shows that the quantities that are most a ff ected by projection e ff ects are the weak lensing concentration, integrated Comptony parameter, gas mass, and NFW profile lensing mass. This is expected as all these observable properties (except for weak lensing concentration and X-ray luminosity) scale linearly with the respective observable mass. Further, we note that the scatter in the richness agrees to the theoretical predictions from Castignani & Benoist (2016).\nWe observe that X-ray luminosity and temperature are the least a ff ected by projection e ff ects. This is attributed to the fact that X-ray luminosity is contingent upon the square of gas density, thereby being primarily influenced by the most bright regions within an image. Similarly, the temperature is predominantly influenced by the innermost regions of a cluster. Note that we lack the value of projection e ff ects for L X , 500c because the 2D scatter is slightly smaller than the 3D one. This happened because projection e ff ects impact under-luminous haloes more strongly than overly luminous haloes (at a fixed mass bin), with the consequence of the projected X-ray luminosity having a higher normalisation and a lower scatter (see Fig. B.1).\nSome mass-observable relations have a large skewness, to aid observational works in modelling these relations, we will estimate their skewness. Therefore, we also quantify deviations of residuals from a symmetrical distribution by means of the Fisher-Pearson coe ffi cient of skewness m 3 / m 3 / 2 2 , where mk is the\n\n\n
Fig. 3: Scatter of our mass-observable relations, paired with their projected version: concentration, integrated Comptony parameter, gas mass, NFW mass, richness, stellar mass, X-ray luminosity, and temperature, within a radius of r 200c; the grey band reports the gas mass, X-ray luminosity, and temperature within r 500c . The left panel reports the fractional scatter of both 3D and 2D quantities. Each dotted vertical line separates the regions that report a given quantity and its 2D version. The right panel reports the contribution of projection e ff ects. Points are coloured by their mass range as in Fig. 1, blue down-triangles represent the low mass bin, orange up-triangles represent the high mass bin, and grey crosses represent the complete sample. Note that we lack the value of projection e ff ects for L X , 500c (see discussion). Points are ordered according to the value of the second panel. Error bars are computed using the jackknife method.
\n\n\n\n
Fig. 4: Skewness parameters for our cluster properties. Data, line styles and colours are as in Fig. 3. Error bars are computed using the jackknife method.
\n\nsample k th central moment. 6 We report its dependency on projection e ff ects in Fig. 4, showing that the properties whose skewness is most impacted by projection e ff ects are the integrated Comptony parameter, the gas mass, and the lensing NFW mass. We also notice that some scaling relation residuals move from having a negative skewness (for the NFW concentration, for instance, due to un-relaxed and merging clusters) to a positive one once projection e ff ects are taken into account (namely, from an asymmetry towards negative residuals towards an asymmetry towards positive residuals).\n\n\n
Fig. 5: Probability density distribution of M 2D / M at di ff erent mass bins: for haloes with M < 2 × 10 14 M ⊙ as a blue dotted line, for haloes with M > 2 × 10 14 M ⊙ as a dashed orange line and for the complete sample as a grey solid line. For each mass bin, we report a vertical line with the median values (note that the three lines are very close together).
\n\nTo assess the impact of projection e ff ects, we introduce a variable to quantify the amount of additional matter in the line of sight that can skew our observable properties. We define it as the ratio of the mass within the cylinder and the mass within a sphere M 2D / M , both of radius r 200c , where the length of the cylinder is described in Sect. 2. We present the distribution of M 2D / M in Fig. 5, where we can see that this quantity is strongly skewed and its median value is M 2D / M ≈ 1 . 26 (note that for a NFW profile with c = 4 , the corresponding analytical cylinder vs. spherical mass is 1 . 25). Although this quantity is not directly observable, we will use it to assess the contribution of LoS objects in the scatter of scaling relations. Note that besides objects in the LoS, di ff erent fitting procedures may impact the scatter of projection e ff ects, as we will see in Sect. 4.\n2D\n\n\n
Fig. 6: Projected maps along a cylinder of length 23 Mpc , with radius r 200c and centred on a random sample of our galaxy clusters, ordered by their M 2D / M (over-plotted above each map) values decreasing from left to right. The pixel red, green, and blue channels are used as follows: the red channel maps the gas projected mass, the green channel maps the dark matter projected mass, and the blue channel maps the stellar projected mass. Columns widths are proportional to the cluster radii.
\n\n\n\n
Fig. 7: Impact of LoS contamination in scaling relations. We show halo properties as a function of halo mass M in the left column, and colour-coded by the fractional amount of mass in a cylinder ( M 2D / M ), and the residuals PDFs in the right column (grey shaded histogram). Rows correspond to richness, the integrated Comptony parameter, lensing mass, and lensing concentration. We also show the residuals of a subset of haloes with low LoS contamination (in particular M 2D / M < 1 . 26 , dashed line histogram). Each panel reports the scatter of the residuals σ and the mean µ of the low LoS residuals.
\n\n\n\n
Fig. 8: Integrated Comptony parameter vs halo mass, colourcoded by a stellar mass fraction. The top panel shows quantities computed within a sphere of radius r 200c, while on the bottom panel, they are computed within a cylinder (of radius r 200c and length 23 Mpc as already presented in Sect. 2). We limit the plot in the mass range M ∈ [1 , 2] 10 14 M ⊙ .
\n\nIn Fig. 6, we show a random selection of clusters, ordered by decreasing M 2D / M from left to right. Objects with high M 2D / M (the objects in the left-most panels) include clusters that are merging, elongated, or in the LoS. In the rest of this paper, we will refer to the objects having M 2D / M greater than the median of the distribution 1 . 26 as having LoS excess.\nTo study the impact of projection e ff ects in scaling relations, in Fig. 7 we show the scaling relations of the following projected quantities: richness, integrated Comptony parameter, lensing mass, and concentrations, that are the ones that are most a ff ected by projection e ff ects. We colour-code these points by M 2D / M and focus on a narrow mass range of M ∈ [1 , 2] × 10 14 M ⊙ in order to visualise better how LoS excess impacts these scaling relations. On the left column, we can visually see that, except for concentration, they strongly correlate with M 2D / M , as the upper points of the scatter plot tend to have higher values of M 2D / M .\nWe quantify this finding in the right column by comparing the residual distributions (of the power-law fit over the complete mass range presented in Sect. 2.2) of the complete sample with the distribution of objects with low LoS contamination only (we adopt the criteria of M 2D / M < 1 . 26 being the median of the M 2D / M distribution, as shown in Fig. 5), and report the respective scatter σ and average value µ of the residual distributions (for the complete sample we have that µ = 0). Except for the concentration, residuals of haloes with low M 2D / M (see dashed histogram) significantly shift towards negative values of µ and σ. For instance, when we consider only objects with a low LoS excess, the scatter of Y 2D decreases from 0 . 35 down to 0 . 11 , and the M 2D NFW scatter goes from 0 . 23 to 0 . 9. The lensing mass is, in fact, well known to be a ff ected by projection e ff ects (Meneghetti et al. 2014; Euclid Collaboration: Giocoli et al. 2024).\nIn this paper, we refer to projection e ff ects as all e ff ects that take place when going from 3D to projection; they include both LoS e ff ects and model uncertainties. This definition becomes relevant when dealing with concentration, which is not impacted\n\n\n
Fig. 9: Probability density distribution of the parameters γ (inner slope, upper panel) and β (outer slope, right panel) of Eq. (10) of the successful gNFW profile fits. The central panel shows the scatter plot between the two parameters colour-coded by M . The dotted lines show the NFW parameters γ = 1 and β = 3 .
\n\nonly by LoS objects but also by the NFW profile fitting procedure. We stress that in Fig. 3 we proved that our projected concentration is actually highly impacted by projection e ff ects, yet only weakly a ff ected by LoS e ff ects. We show in the next Section the reason for concentration being strongly a ff ected by projection e ff ects is that their reduced shear profile deviates strongly from the one produced by NFW profile (see, e.g., Ragagnin et al. 2021), which is used to reconstruct the reduced shear profile.\nTo conclude this section, we now study how correlations between cluster observable properties can be a ff ected by projection e ff ects. To this end, we take the case of a possible hot- vs. coldbaryon correlation by studying the stellar mass vs. integrated Comptony parameter (as the latter should strongly correlate with the gas mass). In Fig. 8, we show the integrated Comptony parameter scaling relation, colour-coded by stellar mass for the 3D quantities (top panel) and projected quantities (bottom panel).\nExamining the correlations at a constant halo mass among the computed quantities within spheres (as depicted in the top panel of Fig. 8), we find no discernible weak anti-correlation between stellar mass and the integrated Comptony parameter (which is defined in Sect. 5). Conversely, when investigating the properties in the projected space, a more pronounced correlation becomes evident. This implies that projection e ff ects can strongly impact the correlation between observable properties. While this analysis is purely qualitative, we will quantify the impact of these projection e ff ects in Sect. 5, where we will compute the correlation coe ffi cients for both 3D quantities and 2D quantities.\n4. Projection effects on lensing concentration\nAs we found in the previous section, projection e ff ects significantly increase the scatter and skewness in the scaling of lensing concentration with mass. However, this scatter increase is not related to external objects along the LoS. Now, we assess if the high scatter of lensing concentration is due to deviations of the reduced shear profile from the one induced by an NFW profile.\nIn this work, we will not delve into the origin of this deviation as it falls beyond the scope of this paper. Such deviation\n\n\n
Fig. 10: Ratio between 2D NFW profile fit parameters and 3D parameters for haloes with a successful 3D gNFW profile fit. Upper panel: the ratio between concentrations; lower panel: the ratio between halo masses. Points are colour-coded by the external log-slope β of the 3D fit of gNFW.
\n\nmay arise due to halo elongations, suggesting that alternative profiles such as truncated NFW profiles may better suit galaxy clusters (Oguri & Hamana 2011). Alternatively, it could stem from the expectation that the NFW profile is intended to describe stacked haloes rather than individual objects. Our focus in this paper is to understand the impact of assuming an NFW profile for each of our haloes. We emphasize that these NFW deviations only a ff ect weak lensing signal reconstruction, as the NFW profile is highly e ff ective in recovering halo mass in 3D.\nTo study deviations from the NFW profile of haloes we fit a generalised NFW profile (Nagai et al. 2007), hereafter gNFW, in spherical coordinates over the same radial range as our previous NFW profile (described in Sect. 2.1), where the density profile ρ gNFW ( r ) is defined as\nρ gNFW ( r ) = ρ 0 ( r / r s) γ (1 + r / r s) β -γ , (10)\nwhere γ and β are respectively the internal and external logslopes of the total matter density profiles. The case γ = 1 and β = 3 produces the NFW profile as in Eq. (4). Note that the Nagai et al. (2007) gNFW profile also depends on the parameter α that we fix to α = 1 in this work in order to explore internal and external log-slope variations only.\nWe present the PDF for the gNFW profile parameters γ and β in Fig. 9, where the fit was performed in 3D with a flat priors\n\n\n
Fig. 11: Residuals of lensing concentrations with respect to the power-law fit. As in Fig. 7, the dashed line histogram indicates the residuals for objects with low LoS e ff ects (low value of M 2D / M ). The solid line histogram contains the additional constraints of haloes with β and γ parameters close to the ones of an NFW profile (2 . 8 < β < 3 . 2 and 0 . 8 < γ < 1 . 2). Each histogram label reports the scatter σ of the residuals.
\n\n\n\n
Fig. 12: Scatter plot of ratio between 2D concentration and 3D concentration against β, namely the outer slope of Eq. (10), of well behaving 3D gNFW profile fits in a narrow mass range of M ∈ [1 , 2] × 10 14 M ⊙ . We also report the correlation coe ffi cient.
\n\nfor γ ∈ [0 , 3] and β ∈ [0 , 6]. The data points are colour-coded according to the variable M , revealing no discernible strong trend with respect to the fitted parameters. For 19% of the objects, the resulting best-fit parameters hit the boundaries of hard-cut priors. Upon visual inspection, these objects are characterised by a very steep matter density profile at large cluster-centric distances, possibly suggesting that a truncated NFW profile might be a better model choice. As our objective is to examine the effects of deviations from the generalised NFW profile, we omit these objects from the subsequent analysis in this Section. Given\nthe substantial deviation of these objects from NFW profile, they could potentially o ff er additional insights for our analyses. However, incorporating them would necessitate the use of a profile more general than Eq. 10. Therefore, we excluded them in order to make our analysis clearer.\nWe observe that the external logslope of Magneticum profiles appears to be slightly flatter than -3 . While we emphasize that this discrepancy does not a ff ect the accurate recovery of mass and concentration parameters in 3D NFW fits (such fits can still yield precise estimates of halo mass and concentrations). However, these deviations in the NFW profiles may a ff ect the reduced shear fit, particularly when observed over large radii (remember that in this work, we use 3 Mpc).\nFurthermore, we observe a degeneracy between the β and γ parameters, indicating that our profiles deviating from NFW profile tend to exhibit a flatter profile compared to NFW profile (as illustrated in Fig. A.1). However, investigating this discrepancy is beyond the scope of this paper, as the internal log slope of clusters is not currently captured by existing weak lensing studies.\nIn Fig. 10, we plot the values of concentration and mass obtained from reduced shear fit, divided by the corresponding 3D quantities and colour-coded by the external 3D gNFW slope β for our intermediate redshift haloes. As we can see, haloes with large values of β have a projected concentration that is significantly higher than the 3D one (see upper panel). In Appendix A we report the example of a simulated halo with low LoS excess (see Fig. A.1) and an analytical one (see Fig. A.2), both with a flat external log-slope, and we show how the under-estimation of the concentration is caused by the fact that the NFW profile fit on the reduced shear is weighting too much the external part of the profile, that deviates from an NFW profile.\nIn Fig. 11, we show the concentration residual distribution and report their scatter. We note that the projected concentration scatter is not a ff ected by external material along the LoS (dashed line and shaded histograms match). However, if one restricts our sample to objects having NFW-like profile log-slopes (we used criteria of 2 . 8 < β < 3 . 2 and 0 . 8 < γ < 1 . 2), then the scatter distribution changes drastically. The concentration residuals decrease from 0 . 43 to 0 . 38, and the residuals shift towards higher values, suggesting that these objects are more relaxed. Such e ff ect is well known, as studied for instance in Macciò et al. (2007).\nWe also show how the external log-slope of the halo profile a ff ects the lensing reconstruction by plotting the ratio between the projected and 3D concentration (i.e., c 2D NFW / c NFW) value versus the 3D log-slope β in the narrow mass bin of M ∈ [1 , 2] × 10 14 M ⊙ in Fig. 12, where we find a positive correlation coe ffi cient of ≈ 0 . 28 , in agreement with a shift of residuals we showed in Fig. 11.\n5. Correlations between properties\nWhile in the last sections, we investigated the origin of the impact of projection e ff ects in the scatter and skewness of observable properties, we will now quantify how projection e ff ects impact the correlation between observable properties. To this end, we quantify the Pearson correlation coe ffi cients between their log-residuals (as defined in Sect. 2.2). We adopt the standard error associated with the Pearson coe ffi cient ρ as derived from two normal distributions, given by σρ = p 1 -ρ 2 / √ N -2 (see Eq. 12-93 in Pugh & Winslow 1966), where N represents the number of objects. This corresponds to a maximum error of 0 . 015\n(for ρ = 0) for the sample size at z = 0 . 24 and a maximum error of 0 . 028 for the sample size at z = 0 . 90. It is worth noting that in the correlation coe ffi cient matrices generated in subsequent analyses, we will only colour values with correlation coe ffi cients | ρ | > 0 . 3, aiming to highlight strongly correlating properties. We define mild correlation as 0 . 2 < | ρ | < 0 . 3, as we choose to exercise caution. Correlation coe ffi cients with | ρ | < 0 . 1 are disregarded.\n5.1. Analysis at z = 0 . 24\nIn this Section we focus on the haloes at intermediate redshift z = 0 . 24 . In Fig. 13 we show the correlation coe ffi cient matrix between log-residuals at fixed halo mass of our projected observable both from Euclid -like data (lensing concentration, lensing mass, richness, and stellar mass, respectively) and possible outcomes from multi-wavelength observations (integrated Comptony parameter, gas mass, X-ray luminosity, and temperature) for intermediate-redshift objects.\nIn the lower triangle, we present scatter plots alongside the slope derived from the correlation coe ffi cient. This visualization allows for the identification of instances where the correlation coe ffi cient slope accurately captures the trend of the residuals. Typically, this alignment occurs for quantities that exhibit strong correlation coe ffi cients. For example, our data points show a robust correlation between certain hot baryon tracers ( M 2D g and Y 2D ), some cold baryon components ( n 2D and M 2D ⋆ ), and weak lensing mass M 2D NFW . The underlying reason for these strong correlations lies in projection e ff ects: the greater the amount of matter along the line of sight, the higher the observed values. We will further demonstrate this in the subsequent section by presenting the correlation coe ffi cient matrix for 3D quantities, where many of these correlations diminish. This can be anticipated by observing that L 2D x and c 2D NFW do not exhibit this positive trend of correlations.\nNotably, we observe that the correlation between richness and stellar mass ( ρ = 0 . 48) is not exceptionally high. Moreover, the stellar mass appears to be more influenced by projection e ff ects compared to richness (evident in their correlations with M 2D NFW , where they exhibit ρ = 0 . 69 and ρ = 0 . 42, respectively). We can speculate on two potential causes: firstly, unlike stellar mass, our richness computation incorporates some observationally-motivated background subtraction; alternatively, since stellar mass encompasses all stellar particles (while richness involves a luminosity-motivated galaxy stellar-mass cut), it is plausible that small subhaloes are influencing the projected stellar mass.\nWe note that concentration anti-correlates with gas-mass (and integrated Comptony parameter). This is in agreement with recent analyses of simulations. In fact, richness at fixed mass anti-correlates with concentration (Bose et al. 2019); low concentration is an index of the system being perturbed (Ludlow et al. 2012); and un-relaxed systems tend to be gas-rich (Davies et al. 2020). We refer to Ragagnin et al. (2022) for a more comprehensive study on low luminous groups. Moreover, at fixed halo mass, the lensing mass correlates strongly with total projected stellar mass ( ρ = 0 . 69) and projected gas mass ( ρ = 0 . 59), which may be due to the fact that both correlate strongly with LoS contamination. The same holds for the correlation among richness, gas mass, and stellar mass. This is due to projection e ff ects, where LoS excess amplifies all these quantities, as discussed in Section 2.2. We note that the 2D lensing mass and projected X-ray luminosity have a slight positive ( ρ = 0 . 20) cor-\nation, in agreement with the observational work of Sereno et al. (2020).\nIn Fig. 14 we show the covariance matrix of non-projected quantities for intermediate redshift objects. We see that as opposed to Fig. 13, the 3D covariance matrix shows a mild yet negative covariance between gas mass and stellar mass ( ρ = -0 . 24), and a positive correlation between richness and gas mass ( ρ = 0 . 23) because most of their correlations in the projection are due to Line of sights excess, which significantly increases the values of the gas mass, the richness, and the stellar mass. In Fig. 15 we report the correlation matrix as in Fig. 13 where we present X-ray luminosity, gas mass, and temperature, as computed within r 500c , which shows an anti-correlation between the gas mass and the concentration residuals ( ρ = -0 . 14) that is significantly lower than the one found in Fig. 13 and Fig. 14 ( ρ equals to -0 . 26 and -0 . 34 respectively). One possibility is that this change in sign of the correlation is caused by the fact that mixing overdensities (concentration is within ∆ c = 200 and gasmass is within ∆ c = 500) does introduce an additional correlation with the sparsity (Balmès et al. 2014; Corasaniti et al. 2022) that itself correlates with the concentration (see Appendix B).\nFor completeness, we report the correlation coe ffi cient matrix and the scatter of log-residuals of all quantities in Table B.1. There we also added the core-excised projected X-ray luminosity L 2D X , ce500c , as it is typically used in X-ray-based observational studies, where we can see that the scatter and most of the correlation coe ffi cients are smaller than L 2D ce500c , while the correlations with the concentration and gas mass increase. Note that we do not report the 3D NFW mass ( M NFW) because it has an extremely low intrinsic scatter σ ln ( M NFW) ≈ 0 . 01 and its correlation coefficients are not meaningful.\n5.2. Analysis at z = 0 . 9\nIn this Section, we focus on observational property covariance matrixes of our haloes z = 0 . 9. At this redshift, we computed projected quantities within a cylinder depth of 35 Mpc in order to retain the same relative ratio as the photoz uncertainty of the low-redshift analysis (it scales with 1 + z ). For what concerns the cylinder used to integrate ∆Σ gt, we rescaled so as to keep it constant in comoving units with the low-redshift analyses. We rescaled the 3D NFW profile minimum radius to 40 kpc while we kept the maximum radius at r 200c . For what concerns the radial range of the lensing fit, we rescaled it with H -2 / 3 ( z ), therefore performing it in the range of [234 , 2300] kpc .\nWe stress that we do not model observational uncertainty. Therefore, the decrease in background source count with redshift does not impact our best fits. However, it still impacts the fact that we weigh external radial bins more than internal ones. We report the values of the scatter and the projection contribution at z = 0 . 9 in Fig. 16, while we report the log-residuals and the skewness for each property in Fig. 17.\nIn particular, the quantities most a ff ected by projection effects are the lensing mass and concentration, whereas the temperature is the lowest. These results are qualitatively similar to the low redshift analyses, with the ComptonY parameter and gas mass being slightly less a ff ected by projection e ff ects. Note that since the virial radius is smaller at higher redshift values, our radial range of the reduced shear is closer to the NFW scale radius; therefore, the weak lensing reconstruction is more e ff ective in capturing the scale radius and more sensitive to deviations from an NFW profile. As a consequence, we found that the in-\ncrease of scatter going from c NFW to c 2D NFW compared to the low redshift analyses.\nWe report the correlation coe ffi cient matrix and the scatter log-residuals of the quantities at z = 0 . 9 in Table B.2. As for the case at z = 0 . 24 , note that we do not report the 3D NFW mass ( M NFW) because it has an extremely small intrinsic scatter of 0 . 01 , and thus its correlation coe ffi cients have no impact in our study.\n6. Conclusions\nIn this work, we analysed a number of galaxy clusters from Magneticum hydrodynamic simulation Box2b / hr. We did so in a mass range, tailored for Euclid -like data products (see Sartoris et al. 2016; Euclid Collaboration: Adam et al. 2019), namely with a mass of M 200c > 10 14 M ⊙ . To this end, we computed properties that could come from Euclid catalogues of galaxy clusters, such as richness, stellar mass, and lensing masses and concentration, and possible properties coming from multi-wavelength studies such as X-ray luminosity, integrated Comptony parameter, gas mass, and temperature. All these properties were computed both within a sphere and within a cylinder (both with radius r 200c) to account for projection e ff ects. Our study considers the remarkable capabilities of Euclid photoz measurements in identifying interlopers. However, their importance decreases significantly at scales as small as a few tens Mpc. This depth is still long enough to contain multiple haloes along the LoS. Hence, we studied the projection e ff ects on a scale that is significantly smaller than the Euclid photoz uncertainty.\nWe then studied how the scatter and skewness change when one measures quantities in 3D space or in projection. Below, we summarise our findings:\n\n-The properties that are most a ff ected by projection e ff ects are the mass and concentration from lensing, the integrated Comptony parameter, and the gas mass. In contrast, temperature and X-ray luminosity are the quantities least a ff ected by projection e ff ects.\n-In both redshift slices ( z = 0 . 24 and z = 0 . 9), the influence of LoS e ff ects is substantial and potentially leads to a spurious correlation between gas and stellar masses. These projection e ff ects have the capacity to markedly enhance correlations between gas and stellar mass (they go from a negative value of -0 . 24 to a significantly high value of 0 . 57), e ff ectively masking the intrinsic correlation (for instance, driven by distinct accretion histories) beneath.\n-The lensing concentration, on the other hand, is mainly affected by the fact that the profile outskirt of reduced shear deviates from the one coming from an NFW profile (which is the profile typically used in WL analyses). We found that deviations from an ideal NFW profile increase the skewness from 0 . 6 to 2 . 5 and increase the scatter of log-residuals from 0 . 33 (in agreement with theoretical works) up to 0 . 46 .\n\nThe analysis presented here has been carried out using a single suite of hydrodynamic simulations. Regarding weak lensing masses and concentration, since in this work, we did not consider the profile noise due to the finite number of background galaxies, future studies are needed to improve our estimations.\nSome works show that both scatter and correlation coefficients vary between cosmological simulations with di ff erent cosmologies (Ragagnin et al. 2023), the presence of feedback schemes (Stanek et al. 2010), and di ff erent cosmological simulation suite in the market (see Fig. 7 in Anbajagane et al. 2020).\nSo, while simulations can provide directions on how to model correlation coe ffi cients, it is possible that when using this kind of data, one needs to allow for variation due to the di ff erent baryon physics.\nFurthermore, when striving for even more precise results, it is important to acknowledge that mass-observable relations are not exact power laws. Therefore, employing more generic fitting techniques, such as a running median, could yield improvements. Additionally, there is room for enhancement in how we compute correlation coe ffi cients in future studies. One potential approach could involve simultaneously fitting both the massobservable relation scatter and the correlation coe ffi cients by maximizing multivariate likelihoods. We anticipate that future studies combining Euclid data with multi-wavelength observations may encounter challenges in shedding light on currently puzzling residual correlations, primarily dominated by projection e ff ects.\nAcknowledgements. We thanks the anonymous referee for the useful comments. The Magneticum Pathfinder simulations were partially performed at the LeibnizRechenzentrum with CPU time assigned to the Project 'pr86re'. AR and LM acknowledge support from the grant PRIN-MIUR 2017 WSCC32 and acknowledges the usage of the INAF-OATs IT framework (Ta ff oni et al. 2020; Bertocco et al. 2020), and the space filling curve improvement on Gadget3 (Ragagnin et al. 2016). Antonio Ragagnin thanks Veronica Bi ffi and Elena Rasia for the X-ray computation routines and tables. KD acknowledges support by the COMPLEX project from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program grant agreement ERC-2019-AdG 882679. LM acknowledges the financial contribution from the grant PRIN-MUR 2022 20227RNLY3 'The concordance cosmological model: stress-tests with galaxy clusters' supported by Next Generation EU. CG and LM acknowledge support from the grant ASI n.2018-23-HH.0. AR and CG acknowledge funding from INAF theory Grant 2022: Illuminating Dark Matter using Weak Lensing by Cluster Satellites, PI: Carlo Giocoli. SB acknowledges partial financial support from the INFN InDark grant. AMCLB was supported by a fellowship of PSL University hosted by the Paris Observatory. We used the package colossus (see Diemer 2018) for computing Σ and ∆Σ as expected from NFW profiles. AR and FC acknowledge co-funding by the European Union - NextGenerationEU within PRIN 2022 project n.20229YBSAN - Globular clusters in cosmological simulations and in lensed fields: from their birth to the present epoch. The Euclid Consortium acknowledges the European Space Agency and a number of agencies and institutes that have supported the development of Euclid , in particular the Academy of Finland, the Agenzia Spaziale Italiana, the Belgian Science Policy, the Canadian Euclid Consortium, the French Centre National d'Etudes Spatiales, the Deutsches Zentrum für Luft- und Raumfahrt, the Danish Space Research Institute, the Fundação para a Ciência e a Tecnologia, the Ministerio de Ciencia e Innovación, the National Aeronautics and Space Administration, the National Astronomical Observatory of Japan, the Netherlandse Onderzoekschool Voor Astronomie, the Norwegian Space Agency, the Romanian Space Agency, the State Secretariat for Education, Research and Innovation (SERI) at the Swiss Space O ffi ce (SSO), and the United Kingdom Space Agency. A complete and detailed list is available on the Euclid web site ( http://www.euclid-ec.org ).\nData Availability\nRaw simulation data were generated at C 2 PAP / LRZ cosmology simulation web portal https://c2papcosmosim.uc. lrz.de/ . Derived data supporting the findings of this study are available from the corresponding author AR on request.\nReferences\n\nAbbott, T. M. C., Aguena, M., Alarcon, A., et al. 2020, Phys. Rev. D, 102, 023509\nAllen, S. W., Evrard, A. E., & Mantz, A. B. 2011, ARA&A, 49, 409\nAnbajagane, D., Evrard, A. E., Farahi, A., et al. 2020, MNRAS, 495, 686\nAndreon, S., Dong, H., & Raichoor, A. 2016, A&A, 593, A2\nAndreon, S. & Moretti, A. 2011, A&A, 536, A37\nAngelinelli, M., Ettori, S., Dolag, K., Vazza, F., & Ragagnin, A. 2022, A&A, 663, L6\nAngelinelli, M., Ettori, S., Dolag, K., Vazza, F., & Ragagnin, A. 2023a, A&A, 675, A188\nAngelinelli, M., Ettori, S., Dolag, K., Vazza, F., & Ragagnin, A. 2023b, A&A, 675, A188\nAngulo, R. E., Zennaro, M., Contreras, S., et al. 2021, MNRAS, 507, 5869\nArnaud, K. A. 1996, in Astronomical Society of the Pacific Conference Series, Vol. 101, Astronomical Data Analysis Software and Systems V, ed. G. H. Jacoby & J. Barnes, 17\nBalmès, I., Rasera, Y., Corasaniti, P. S., & Alimi, J. M. 2014, MNRAS, 437, 2328\nBeck, A. M., Murante, G., Arth, A., et al. 2016, MNRAS, 455, 2110\nBecker, M. R. & Kravtsov, A. V. 2011, ApJ, 740, 25\nBellagamba, F., Roncarelli, M., Maturi, M., & Moscardini, L. 2018a, MNRAS, 473, 5221\nBellagamba, F., Roncarelli, M., Maturi, M., & Moscardini, L. 2018b, MNRAS, 473, 5221\nBertocco, S., Goz, D., Tornatore, L., et al. 2020, in Astronomical Society of the Pacific Conference Series, Vol. 527, Astronomical Society of the Pacific Conference Series, ed. R. Pizzo, E. R. Deul, J. D. Mol, J. de Plaa, & H. Verkouter, 303\nBhargava, S., Garrel, C., Koulouridis, E., et al. 2023, A&A, 673, A92 Bi ffi , V., Dolag, K., & Böhringer, H. 2013, MNRAS, 428, 1395 Bi ffi , V., Planelles, S., Borgani, S., et al. 2017, MNRAS, 468, 531 Bocquet, S., Dietrich, J. P., Schrabback, T., et al. 2019, ApJ, 878, 55 Bocquet, S., Grandis, S., Bleem, L. E., et al. 2024, Phys. Rev. D, 110, 083510 Bocquet, S., Heitmann, K., Habib, S., et al. 2020, ApJ, 901, 5 Bocquet, S., Saro, A., Dolag, K., & Mohr, J. J. 2016, MNRAS, 456, 2361 Bose, S., Eisenstein, D. J., Hernquist, L., et al. 2019, MNRAS, 490, 5693 Boylan-Kolchin, M., Springel, V., White, S. D. M., Jenkins, A., & Lemson, G.\n2009, MNRAS, 398, 1150\nCastignani, G. & Benoist, C. 2016, A&A, 595, A111\nChild, H. L., Habib, S., Heitmann, K., et al. 2018, ApJ, 859, 55\nContreras-Santos, A., Buitrago, F., Knebe, A., et al. 2024, A&A, 690, A109\nCorasaniti, P. S., Le Brun, A. M. C., Richardson, T. R. G., et al. 2022, MNRAS, 516, 437\nCostanzi, M., Rozo, E., Simet, M., et al. 2019, MNRAS, 488, 4779\nCui, W., Dave, R., Knebe, A., et al. 2022, MNRAS, 514, 977\nDavies, J. J., Crain, R. A., Oppenheimer, B. D., & Schaye, J. 2020, MNRAS, 491, 4462\nDavis, M., Efstathiou, G., Frenk, C. S., & White, S. D. M. 1985, ApJ, 292, 371 Diemer, B. 2018, ApJS, 239, 35\nDolag, K., Borgani, S., Murante, G., & Springel, V. 2009, MNRAS, 399, 497\nDolag, K., Gaensler, B. M., Beck, A. M., & Beck, M. C. 2015, MNRAS, 451, 4277\nDolag, K., Komatsu, E., & Sunyaev, R. 2016, MNRAS, 463, 1797\nEttori, S., Donnarumma, A., Pointecouteau, E., et al. 2013, Space Sci. Rev., 177, 119\nEuclid Collaboration: Adam, R., Vannier, M., Maurogordato, S., et al. 2019, A&A, 627, A23\nEuclid Collaboration: Ajani, V., Baldi, M., Barthelemy, A., et al. 2023, A&A, 675, A120\nEuclid Collaboration: Blanchard, A., Camera, S., Carbone, C., et al. 2020, A&A, 642, A191\nEuclid Collaboration: Desprez, G., Paltani, S., Coupon, J., et al. 2020, A&A, 644, A31\nEuclid Collaboration: Giocoli, C., Meneghetti, M., Rasia, E., et al. 2024, A&A, 681, A67\nEuclid Collaboration: Mellier, Y., Abdurro'uf, Acevedo Barroso, J., Achúcarro, A., et al. 2024, A&A, submitted, arXiv:2405.13491\nFabjan, D., Borgani, S., Tornatore, L., et al. 2010, MNRAS, 401, 1670\nFarahi, A., Mulroy, S. L., Evrard, A. E., et al. 2019, Nature Communications, 10, 2504\nFerland, G. J., Korista, K. T., Verner, D. A., et al. 1998, PASP, 110, 761\nFischer, M. S., Kasselmann, L., Brüggen, M., et al. 2024, MNRAS, 529, 2327 Ghirardini, V., Bulbul, E., Artis, E., et al. 2024, arXiv e-prints, arXiv:2402.08458 Giocoli, C., Tormen, G., & van den Bosch, F. C. 2008, MNRAS, 386, 2135 Giodini, S., Lovisari, L., Pointecouteau, E., et al. 2013, Space Sci. Rev., 177, 247 Hirschmann, M., Dolag, K., Saro, A., et al. 2014, MNRAS, 442, 2304 Hoekstra, H. 2003, MNRAS, 339, 1155\nHoekstra, H., Mahdavi, A., Babul, A., & Bildfell, C. 2012, MNRAS, 427, 1298 Hudson, D. S., Mittal, R., Reiprich, T. H., et al. 2010, A&A, 513, A37\nKomatsu, E., Smith, K. M., Dunkley, J., et al. 2011, ApJS, 192, 18\n\nKravtsov, A. V. & Borgani, S. 2012, ARA&A, 50, 353\n\nLima, M. & Hu, W. 2005, Phys. Rev. D, 72, 043006\nLudlow, A. D., Navarro, J. F., Li, M., et al. 2012, MNRAS, 427, 1322\nMacciò, A. V., Dutton, A. A., van den Bosch, F. C., et al. 2007, MNRAS, 378, 55\nMaturi, M., Bellagamba, F., Radovich, M., et al. 2019, MNRAS, 485, 498 McClintock, T., Varga, T. N., Gruen, D., et al. 2019, MNRAS, 482, 1352 Melchior, P., Suchyta, E., Hu ff , E., et al. 2015, MNRAS, 449, 2219\n\n\nMeneghetti, M., Rasia, E., Merten, J., et al. 2010, A&A, 514, A93 Meneghetti, M., Rasia, E., Vega, J., et al. 2014, ApJ, 797, 34 Nagai, D., Kravtsov, A. V., & Vikhlinin, A. 2007, ApJ, 668, 1 Navarro, J. F., Frenk, C. S., & White, S. D. M. 1997, ApJ, 490, 493 Oguri, M. & Hamana, T. 2011, MNRAS, 414, 1851 Okabe, N., Zhang, Y. Y., Finoguenov, A., et al. 2010, ApJ, 721, 875 Pacaud, F., Pierre, M., Adami, C., et al. 2007, MNRAS, 382, 1289 Pires, S., Vandenbussche, V., Kansal, V., et al. 2020, A&A, 638, A141 Pratt, G. W., Arnaud, M., Biviano, A., et al. 2019, Space Sci. Rev., 215, 25 Puddu, E. & Andreon, S. 2022, MNRAS, 511, 2968 Pugh, E. & Winslow, G. 1966, The Analysis of Physical Measurements (By > Emerson M. Pugh (And > George H. Winslow, Addison-Wesley Series in Physics\n\nRagagnin, A., Andreon, S., & Puddu, E. 2022, A&A, 666, A22\n\nRagagnin, A., Dolag, K., Bi ffi , V., et al. 2017, Astronomy and Computing, 20, 52\nRagagnin, A., Dolag, K., Moscardini, L., Biviano, A., & D'Onofrio, M. 2019, MNRAS, 486, 4001\n\nRagagnin, A., Fumagalli, A., Castro, T., et al. 2023, A&A, 675, A77\n\nRagagnin, A., Meneghetti, M., Calura, F., et al. 2024, A&A, 687, A270\nRagagnin, A., Saro, A., Singh, P., & Dolag, K. 2021, MNRAS, 500, 5056\nRagagnin, A., Tchipev, N., Bader, M., Dolag, K., & Hammer, N. J. 2016, in Advances in Parallel Computing, Volume 27: Parallel Computing: On the Road to Exascale, Edited by Gerhard R. Joubert, Hugh Leather, Mark Parsons, Frans Peters, Mark Sawyer. IOP Ebook, ISBN: 978-1-61499-621-7, pages 411-420\nRozo, E., Wechsler, R. H., Ryko ff , E. S., et al. 2010, ApJ, 708, 645 Saro, A., Liu, J., Mohr, J. J., et al. 2014, MNRAS, 440, 2610 Sartoris, B., Biviano, A., Fedeli, C., et al. 2016, MNRAS, 459, 1764 Schrabback, T., Bocquet, S., Sommer, M., et al. 2021, MNRAS, 505, 3923 Sereno, M., Fedeli, C., & Moscardini, L. 2016, J. Cosmology Astropart. Phys., 2016, 042\n\nSereno, M., Umetsu, K., Ettori, S., et al. 2020, MNRAS, 492, 4528\n\nSingh, P., Saro, A., Costanzi, M., & Dolag, K. 2020, MNRAS, 494, 3728\nSmith, R. K., Brickhouse, N. S., Liedahl, D. A., & Raymond, J. C. 2001, ApJ, 556, L91\nSommer, M. W., Schrabback, T., Applegate, D. E., et al. 2022, MNRAS, 509, 1127\nSommer, M. W., Schrabback, T., Ragagnin, A., & Rockenfeller, R. 2023, arXiv, arXiv:2306.13187\nSpringel, V. 2005, MNRAS, 364, 1105\nSpringel, V., Di Matteo, T., & Hernquist, L. 2005a, MNRAS, 361, 776\nSpringel, V., White, S. D. M., Jenkins, A., et al. 2005b, Nature, 435, 629\nSpringel, V., White, S. D. M., Tormen, G., & Kau ff mann, G. 2001, MNRAS, 328, 726\nStanek, R., Rasia, E., Evrard, A. E., Pearce, F., & Gazzola, L. 2010, ApJ, 715, 1508\nSteinborn, L. K., Dolag, K., Comerford, J. M., et al. 2016, MNRAS, 458, 1013 Steinborn, L. K., Dolag, K., Hirschmann, M., Prieto, M. A., & Remus, R.-S. 2015, MNRAS, 448, 1504\nStern, C., Dietrich, J. P., Bocquet, S., et al. 2019, MNRAS, 485, 69\n\nSugiyama, S., Miyatake, H., More, S., et al. 2023, Phys. Rev. D, 108, 123521\nSun, M., Voit, G. M., Donahue, M., et al. 2009, ApJ, 693, 1142\n\nSunayama, T., Park, Y., Takada, M., et al. 2020, MNRAS, 496, 4468\nSunyaev, R. A. & Zeldovich, Y. B. 1972, Comments on Astrophysics and Space Physics, 4, 173\nTa ff oni, G., Becciani, U., Garilli, B., et al. 2020, in Astronomical Society of the Pacific Conference Series, Vol. 527, Astronomical Society of the Pacific Conference Series, ed. R. Pizzo, E. R. Deul, J. D. Mol, J. de Plaa, & H. Verkouter, 307\n\nTeklu, A. F., Remus, R.-S., Dolag, K., et al. 2015, ApJ, 812, 29\nTornatore, L., Borgani, S., Dolag, K., & Matteucci, F. 2007, MNRAS, 382, 1050 Truong, N., Rasia, E., Mazzotta, P., et al. 2018, MNRAS, 474, 4089\n\nvan den Bosch, F. C., Yang, X., Mo, H. J., & Norberg, P. 2005, MNRAS, 356, 1233\n\nvan Uitert, E., Cacciato, M., Hoekstra, H., et al. 2016, MNRAS, 459, 3251\n\nVikhlinin, A., Kravtsov, A., Forman, W., et al. 2006, ApJ, 640, 691\nVillaescusa-Navarro, F., Ding, J., Genel, S., et al. 2022, ApJ, 929, 132\nWiersma, R. P. C., Schaye, J., Theuns, T., Dalla Vecchia, C., & Tornatore, L. 2009, MNRAS, 399, 574\nWu, H.-Y., Costanzi, M., To, C.-H., et al. 2022, MNRAS, 515, 4471\nXu, W., Ramos-Ceja, M. E., Pacaud, F., Reiprich, T. H., & Erben, T. 2018, A&A, 619, A162\n1 INAF-Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via Piero Gobetti 93 / 3, 40129 Bologna, Italy\n2 IFPU, Institute for Fundamental Physics of the Universe, via Beirut 2, 34151 Trieste, Italy\n3 Dipartimento di Fisica e Astronomia \"Augusto Righi\" - Alma Mater Studiorum Università di Bologna, via Piero Gobetti 93 / 2, 40129 Bologna, Italy\n4 ICSC - Centro Nazionale di Ricerca in High Performance Computing, Big Data e Quantum Computing, Via Magnanelli 2, Bologna, Italy\n5 Dipartimento di Fisica - Sezione di Astronomia, Università di Trieste, Via Tiepolo 11, 34131 Trieste, Italy\n6 INAF-Osservatorio Astronomico di Trieste, Via G. B. Tiepolo 11, 34143 Trieste, Italy\n7 INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste TS, Italy\n8 INAF-Osservatorio Astronomico di Brera, Via Brera 28, 20122 Milano, Italy\n9 Universitäts-Sternwarte München, Fakultät für Physik, LudwigMaximilians-Universität München, Scheinerstrasse 1, 81679 München, Germany\n10 INFN-Bologna, Via Irnerio 46, 40126 Bologna, Italy\n11 Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Via Irnerio 46, 40126 Bologna, Italy\n12 Laboratoire Univers et Théorie, Observatoire de Paris, Université PSL, Université Paris Cité, CNRS, 92190 Meudon, France\n13 Institut d'Astrophysique de Paris, 98bis Boulevard Arago, 75014, Paris, France\n14 Institut d'Astrophysique de Paris, UMR 7095, CNRS, and Sorbonne Université, 98 bis boulevard Arago, 75014 Paris, France\n15 School of Physics, HH Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, UK\n16 INFN-Sezione di Bologna, Viale Berti Pichat 6 / 2, 40127 Bologna, Italy\n17 Universität Bonn, Argelander-Institut für Astronomie, Auf dem Hügel 71, 53121 Bonn, Germany\n\nArticle number, page 14 of 26\n18\nUniversité Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM,\n91191, Gif-sur-Yvette, France\n\n19 Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy\n20 Department of Astronomy, University of Geneva, ch. d'Ecogia 16, 1290 Versoix, Switzerland\n21 Univ. Grenoble Alpes, CNRS, Grenoble INP, LPSC-IN2P3, 53, Avenue des Martyrs, 38000, Grenoble, France\n22 Institut für Theoretische Physik, University of Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany\n\n23\nZentrum für Astronomie, Universität Heidelberg, Philosophenweg\n12, 69120 Heidelberg, Germany\n\n24 School of Mathematics and Physics, University of Surrey, Guildford, Surrey, GU2 7XH, UK\n25 SISSA, International School for Advanced Studies, Via Bonomea 265, 34136 Trieste TS, Italy\n26 Dipartimento di Fisica e Astronomia, Università di Bologna, Via Gobetti 93 / 2, 40129 Bologna, Italy\n27 INAF-Osservatorio Astrofisico di Torino, Via Osservatorio 20, 10025 Pino Torinese (TO), Italy\n28 Dipartimento di Fisica, Università di Genova, Via Dodecaneso 33, 16146, Genova, Italy\n29 INFN-Sezione di Genova, Via Dodecaneso 33, 16146, Genova, Italy\n30 Department of Physics \"E. Pancini\", University Federico II, Via Cinthia 6, 80126, Napoli, Italy\n31 INAF-Osservatorio Astronomico di Capodimonte, Via Moiariello 16, 80131 Napoli, Italy\n32 INFN section of Naples, Via Cinthia 6, 80126, Napoli, Italy\n33 Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, PT4150-762 Porto, Portugal\n34 Dipartimento di Fisica, Università degli Studi di Torino, Via P. Giuria 1, 10125 Torino, Italy\n35 INFN-Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy\n\n\n36 INAF-IASF Milano, Via Alfonso Corti 12, 20133 Milano, Italy\n37 Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Avenida Complutense 40, 28040 Madrid, Spain\n38 Port d'Informació Científica, Campus UAB, C. Albareda s / n, 08193 Bellaterra (Barcelona), Spain\n39 Institute for Theoretical Particle Physics and Cosmology (TTK), RWTH Aachen University, 52056 Aachen, Germany\n40 INAF-Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monteporzio Catone, Italy\n41 Dipartimento di Fisica e Astronomia \"Augusto Righi\" - Alma Mater Studiorum Università di Bologna, Viale Berti Pichat 6 / 2, 40127 Bologna, Italy\n42 Instituto de Astrofísica de Canarias, Calle Vía Láctea s / n, 38204, San Cristóbal de La Laguna, Tenerife, Spain\n43 Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK\n44 Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, UK\n45 European Space Agency / ESRIN, Largo Galileo Galilei 1, 00044 Frascati, Roma, Italy\n46 ESAC / ESA, Camino Bajo del Castillo, s / n., Urb. Villafranca del Castillo, 28692 Villanueva de la Cañada, Madrid, Spain\n47 Université Claude Bernard Lyon 1, CNRS / IN2P3, IP2I Lyon, UMR 5822, Villeurbanne, F-69100, France\n48 Institute of Physics, Laboratory of Astrophysics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland\n49 UCB Lyon 1, CNRS / IN2P3, IUF, IP2I Lyon, 4 rue Enrico Fermi, 69622 Villeurbanne, France\n50 Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Edifício C8, Campo Grande, PT1749-016 Lisboa, Portugal\n51 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal\n52 INAF-Istituto di Astrofisica e Planetologia Spaziali, via del Fosso del Cavaliere, 100, 00100 Roma, Italy\n53 INAF-Osservatorio Astronomico di Padova, Via dell'Osservatorio 5, 35122 Padova, Italy\n54 Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1, 85748 Garching, Germany\n55 Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, 0315 Oslo, Norway\n56 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA, 91109, USA\n57 Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK\n58 Felix Hormuth Engineering, Goethestr. 17, 69181 Leimen, Germany\n59 Technical University of Denmark, Elektrovej 327, 2800 Kgs. Lyngby, Denmark\n60 Cosmic Dawn Center (DAWN), Denmark\n61 Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany\n62 Department of Physics and Helsinki Institute of Physics, Gustaf Hällströmin katu 2, 00014 University of Helsinki, Finland\n63 Aix-Marseille Université, CNRS / IN2P3, CPPM, Marseille, France\n64 Université de Genève, Département de Physique Théorique and Centre for Astroparticle Physics, 24 quai Ernest-Ansermet, CH1211 Genève 4, Switzerland\n65 Department of Physics, P.O. Box 64, 00014 University of Helsinki, Finland\n66 Helsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, Finland\n67 NOVA optical infrared instrumentation group at ASTRON, Oude Hoogeveensedijk 4, 7991PD, Dwingeloo, The Netherlands\n68 Dipartimento di Fisica \"Aldo Pontremoli\", Università degli Studi di Milano, Via Celoria 16, 20133 Milano, Italy\n69 INFN-Sezione di Milano, Via Celoria 16, 20133 Milano, Italy\n70 INFN-Sezione di Roma, Piazzale Aldo Moro, 2 - c / o Dipartimento di Fisica, Edificio G. Marconi, 00185 Roma, Italy\n71 Aix-Marseille Université, CNRS, CNES, LAM, Marseille, France\n72 Department of Physics, Institute for Computational Cosmology, Durham University, South Road, Durham, DH1 3LE, UK\n73 Université Côte d'Azur, Observatoire de la Côte d'Azur, CNRS, Laboratoire Lagrange, Bd de l'Observatoire, CS 34229, 06304 Nice cedex 4, France\n74 Université Paris Cité, CNRS, Astroparticule et Cosmologie, 75013 Paris, France\n75 Institut de Física d'Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona), Spain\n76 European Space Agency / ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands\n77 School of Mathematics, Statistics and Physics, Newcastle University, Herschel Building, Newcastle-upon-Tyne, NE1 7RU, UK\n78 Department of Physics and Astronomy, University of Aarhus, Ny Munkegade 120, DK-8000 Aarhus C, Denmark\n79 Space Science Data Center, Italian Space Agency, via del Politecnico snc, 00133 Roma, Italy\n80 Centre National d'Etudes Spatiales - Centre spatial de Toulouse, 18 avenue Edouard Belin, 31401 Toulouse Cedex 9, France\n81 Institute of Space Science, Str. Atomistilor, nr. 409 M˘agurele, Ilfov, 077125, Romania\n82 Dipartimento di Fisica e Astronomia \"G. Galilei\", Università di Padova, Via Marzolo 8, 35131 Padova, Italy\n83 INFN-Padova, Via Marzolo 8, 35131 Padova, Italy\n84 Institut de Recherche en Astrophysique et Planétologie (IRAP), Université de Toulouse, CNRS, UPS, CNES, 14 Av. Edouard Belin, 31400 Toulouse, France\n85 Université St Joseph; Faculty of Sciences, Beirut, Lebanon\n86 Departamento de Física, FCFM, Universidad de Chile, Blanco Encalada 2008, Santiago, Chile\n87 Universität Innsbruck, Institut für Astro- und Teilchenphysik, Technikerstr. 25 / 8, 6020 Innsbruck, Austria\n88 Institut d'Estudis Espacials de Catalunya (IEEC), Edifici RDIT, Campus UPC, 08860 Castelldefels, Barcelona, Spain\n89 Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, s / n, 08193 Barcelona, Spain\n90 Satlantis, University Science Park, Sede Bld 48940, Leioa-Bilbao, Spain\n91 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, Tapada da Ajuda, 1349-018 Lisboa, Portugal\n92 Universidad Politécnica de Cartagena, Departamento de Electrónica y Tecnología de Computadoras, Plaza del Hospital 1, 30202 Cartagena, Spain\n93 Kapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands\n94 Infrared Processing and Analysis Center, California Institute of Technology, Pasadena, CA 91125, USA\n95 INAF, Istituto di Radioastronomia, Via Piero Gobetti 101, 40129 Bologna, Italy\n96 Astronomical Observatory of the Autonomous Region of the Aosta Valley (OAVdA), Loc. Lignan 39, I-11020, Nus (Aosta Valley), Italy\n97 ICL, Junia, Université Catholique de Lille, LITL, 59000 Lille, France\n98 Instituto de Física Teórica UAM-CSIC, Campus de Cantoblanco, 28049 Madrid, Spain\n99 CERCA / ISO, Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106, USA\n100 Dipartimento di Fisica e Scienze della Terra, Università degli Studi di Ferrara, Via Giuseppe Saragat 1, 44122 Ferrara, Italy\n101 Istituto Nazionale di Fisica Nucleare, Sezione di Ferrara, Via Giuseppe Saragat 1, 44122 Ferrara, Italy\n102 Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, Chiba 277-8583, Japan\n\n\n103 Minnesota Institute for Astrophysics, University of Minnesota, 116 Church St SE, Minneapolis, MN 55455, USA\n104 Institute Lorentz, Leiden University, Niels Bohrweg 2, 2333 CA Leiden, The Netherlands\n105 Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, USA\n106 Department of Physics & Astronomy, University of California Irvine, Irvine CA 92697, USA\n107 Department of Astronomy & Physics and Institute for Computational Astrophysics, Saint Mary's University, 923 Robie Street, Halifax, Nova Scotia, B3H 3C3, Canada\n108 Departamento Física Aplicada, Universidad Politécnica de Cartagena, Campus Muralla del Mar, 30202 Cartagena, Murcia, Spain\n109 Université Paris-Saclay, CNRS, Institut d'astrophysique spatiale, 91405, Orsay, France\n110 Department of Physics, Oxford University, Keble Road, Oxford OX1 3RH, UK\n111 Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth PO1 3FX, UK\n112 Department of Computer Science, Aalto University, PO Box 15400, Espoo, FI-00 076, Finland\n113 Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), German Centre for Cosmological Lensing (GCCL), 44780 Bochum, Germany\n114 DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 155, 2200 Copenhagen, Denmark\n115 Department of Physics and Astronomy, Vesilinnantie 5, 20014 University of Turku, Finland\n116 Serco for European Space Agency (ESA), Camino bajo del Castillo, s / n, Urbanizacion Villafranca del Castillo, Villanueva de la Cañada, 28692 Madrid, Spain\n117 ARC Centre of Excellence for Dark Matter Particle Physics, Melbourne, Australia\n118 Centre for Astrophysics & Supercomputing, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia\n119 School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London E1 4NS, UK\n120 Department of Physics and Astronomy, University of the Western Cape, Bellville, Cape Town, 7535, South Africa\n121 ICTP South American Institute for Fundamental Research, Instituto de Física Teórica, Universidade Estadual Paulista, São Paulo, Brazil\n122 Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, Stockholm, SE-106 91, Sweden\n123 Astrophysics Group, Blackett Laboratory, Imperial College London, London SW7 2AZ, UK\n124 INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125, Firenze, Italy\n125 Centro de Astrofísica da Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal\n126 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK\n127 Department of Astrophysics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland\n128 Dipartimento di Fisica, Università degli studi di Genova, and INFN-Sezione di Genova, via Dodecaneso 33, 16146, Genova, Italy\n129 Theoretical astrophysics, Department of Physics and Astronomy, Uppsala University, Box 515, 751 20 Uppsala, Sweden\n130 Department of Physics, Royal Holloway, University of London, TW20 0EX, UK\n131 Mullard Space Science Laboratory, University College London, Holmbury St Mary, Dorking, Surrey RH5 6NT, UK\n132 Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA\n133 Cosmic Dawn Center (DAWN)\n134 Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 Copenhagen, Denmark\n135 Center for Cosmology and Particle Physics, Department of Physics, New York University, New York, NY 10003, USA\n\n\n136 Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, 10010, New York, NY, USA\n\n\n\n
Fig. 13: Correlation coe ffi cients matrix (upper-right triangle) and scatter plot (bottom-left triangle) of power-law log-residuals of Euclid -data (lensing concentration, lensing mass, richness, and stellar mass, respectively) and possible outcomes from multiwavelength observations (integrated Comptony parameter, gas mass, X-ray luminosity, and temperature, respectively). Cell colouring goes from blue (negative correlation coe ffi cients) to red (positive correlation coe ffi cients) and is white in the interval [ -0 . 35 , 0 . 35] in order to enhance the visibility of the most significant coe ffi cients.
\n\n\n\n
Fig. 14: As Fig. 13, here we show the quantities computed in the 3D space.
\n\n\n\n
Fig. 15: As Fig. 13, but we show the projected X-ray luminosity, the projected gas mass, and the projected temperature computed within the overdensity of ∆ c = 500 instead of the respective quantities within ∆ c = 200 .
\n\n\n\n
Fig. 16: Same as Fig. 3 for the data at z = 0 . 9 .
\n\n\n\n
Fig. 17: Same as Fig. 4 for the data at z = 0 . 9 . We do not report the value for the concentration because in our fit radial range the lensing one does not correlate with the 3D one.
\n\nAppendix A: Fit of gNFW\nIn Fig. A.1, we present the density profile of a halo that deviates from NFW profile and has no LoS contamination. In particular, it has β ≈ 1 . 8 and γ = 1 . 5 . We show its NFW profile fit profile on the 3D density in the upper panel of Fig. A.1, where we can see that 3D NFW profile (performed on radial bins in a sphere) is capable of capturing the shape of the halo and to estimate its mass with high accuracy (within ≈ 5%). In the central panel, we show the reduced shear profile and best fits, where we can see that the fit performed on the reduced shear underestimated the concentration and is not able to capture the more internal part of the shear profile, as it is done by the profile that was fit in 3D.\nWe first exclude this mismatch as being due to projection effects by showing that both the reduced shear from the particle data (orange line) matches the one recovered by performing an analytical projection of the 3D profile (blue solid line). In particular we project the density profile ρ ( r ) and derive the surface mass density Σ conv ., as follows\nΣ conv . ( R ) = Z + ∞ -∞ ρ GLYPH<18> p R 2 + z 2 GLYPH<19> d z = 2 Z ∞ R ρ ( r ) r √ r 2 -R 2 d r . (A.1)\nWhat we find is that the shear obtained with the aid of an analytical projection Σ conv . matches very well the real one (i.e., orange and blue lines do match). This hints that for this cluster there are no strong LoS e ff ects.\nTo understand why the fit on the shear is not able to capture the concentration of the original halo, we zoom our fit in the bottom panel of Fig. A.1, where it looks like the fit is very good in capturing the final part of the profile and not able to capture the internal. It is crucial to emphasise that we did not include observational uncertainties in these analyses. Therefore, the uncertainty outlined in Eq. (8) a ff ects the fit by assigning more weight to external radial bins compared to internal ones. It is worth noting that the proportionality factors in Eq. (8) will not a ff ect our best fit.\nTo validate this point in Fig. A.2 we study the bias on fitting an NFW profile on a mock gNFW profile that has β = 1 . 8 and γ = 1 . 5 , a mass of 3 × 10 14 M ⊙ and a concentration c = 2 . 4 (as the halo presented in Fig. A.1). We see that the 3D NFW profile is capable of estimating both its mass and its gNFW concentration with high accuracy (see top panel match between blue and dashed black lines). On the other hand the fit of the shear (we report in the bottom panel of Fig. A.2) has the same problems as the one on the cluster in Fig. A.1: it recovers a low concentration (with a value of 1 . 5). This may be because, at outer radii, the model fits the data. It is possible that the under-estimation of concentration at low radii is caused by the combination of two factors: the fit under-estimates the shear at lower radii (with the result of under-estimating the lensing concentration), or the fact that γ is di ff erent than 3 induces an NFW profile fit with a low concentration.\nWe then performed the experiment of fitting the analytical profile with constant (yet unrealistic) error bars. While the fit was able to capture the shape of the profile, it recovered a concentration of 1 . 6 , implying that there is indeed a degeneracy between the shear of low-concentrated NFW profiles and steeper-NFW profiles.\n\n\n\n\n\n\n\n\n
Fig. A.1: Density profiles of a simulated halo and the corresponding NFW profile fit. Upper panel: total matter density profile (blue solid line) and the respective NFW profile fit profile (black dashed line). Central panel: reduced shear from simulated particles (orange solid line), and from the analytical projection of the density profile Σ conv . presented in Eq. (A.1) and performed in the radial range [60 , 3000] kpc, in the blue solid line. The dashed vertical line indicates the minimum radius of the shear fit, and the fit profiles (black lines) are extrapolated down to 10 kpc to enhance the central densities predicted by the two fits. The bottom panel shows the same as the central panel but focuses on the radial range of the fit. The error bars indicate the uncertainty for each radial bin, as defined in Eq. (8).
\n\nAppendix B: Correlations with different overdensities\nIn this appendix, we discuss the di ff erences between scaling relation scatters and covariance values at di ff erent overdensities. First of all, we tackle the fact that when we compute X-ray luminosity within r 500c (instead of r 200c), we find that the scatter of the scaling relation of the projected quantity is larger than the 3D one.\nTo investigate this feature, we will focus on the bolometric X-ray luminosity. We report the 3D and projected bolometric X-\n\n\n\n\n\n
Fig. A.2: Density profiles of a mock halo that deviates from NFW and the corresponding NFW profile fit. The mock halo mass, concentration parameter and gNFW log-slopes are chosen to match the ones of the simulated halo presented in Fig. A.1. The upper panel reports the total matter density profile of the mock halo (solid blue line) and the profile from the corresponding NFW profile fit (dashed black line). The bottom panel shows the reduced shear and the profile from the corresponding NFW profile fit (dotted black line). The error bars indicate the uncertainty for each radial bin, as defined in Eq. (8).
\n\nray luminosity in Fig. B.1 (top panel), where it is visually clear that the projected X-ray luminosity is (as expected) always larger than the 3D one. One can also notice that the increase in X-ray luminosity depends on the fact that a halo is over-luminous or not: the increase of luminosity growing from the 3D to 2D is larger for under-luminous haloes than for over-luminous haloes. We prove this point in the bottom panel of Fig. B.1 where we show the ratio between the 2D and 3D luminosity as a function of their residual of the 3D scaling relation (the higher the value of the x axis, the more over-luminous is the object for its mass bin), where we can see a strong anti-correlation: overly luminous objects (for a given mass bin) are not going to be a ff ected much by the fact that their luminosity is computed in 3D or 2D. The possible cause is that an interloper in the LoS will not a ff ect much an overly luminous object.\nFor completeness, in Fig. B.2 we show the correlation coefficients between the gas mass and stellar mass computed within both r 500c and r 200c and the concentration. Here we can see a change of sign between M ⋆, 500cMg , 500c correlations and M ⋆, 500cM g correlations and a change in the sign between c NFWM g correlations and c NFWMg , 500c correlations.\nFinally, in Fig. B.3 we report the scatter of observable properties at fixed mass for both Euclid -like quantities (lensing mass, richness, and projected stellar mass), and possible multiwavelength properties (integrated Comptony parameter, X-ray luminosity, and temperature), where we compute X-ray lumi-\n\n\n\n\n\n
Fig. B.1: Comparison between 3D and projected X-ray luminosities. Top panel shows a scatter plot of the two mass-observable relations, while the bottom panel shows their ratio as a function of the 3D X-ray scaling relation residuals.
\n\nnosity and temperature within r 500c as they are typically derived within this overdensity. The upper panel of Fig. B.3 shows the residuals of the log-log linear regression where we see that in terms of 2D scatter, the properties with the lowest scatter are the stellar mass and the temperature. The bottom panel shows the data points used to perform the fit (in black) where we used a visually-inspected cut on the halo mass values in order to ensure that mass values are complete for a given observable value.\n
\n\n
Table B.1: Scatter and correlation coe ffi cient matrix between z = 0 . 24 log-residuals of scaling relations.
\n
\nNotes. Diagonal terms report the scatter of the log-residuals of each quantity, namely σ ln of Eq. (9), while the o ff -diagonal terms report the correlation coe ffi cient between the log-residuals. We do not report values of the correlation coe ffi cient below 0 . 20 because they are not significant. We do not report the values for the 3D NFW mass M 200c because it has a very low scatter of log-residuals ( ≈ 0 . 01) and its correlation coe ffi cients are not meaningful. Note that in this table we also added the core-excised X-ray luminosity.\n
\n\n
Table B.2: Scatter and correlation coe ffi cient matrix between z = 0 . 9 log-residuals of scaling relations.
\n
\nNotes. Rows and columns are as in Table B.1.\n\n\n
Fig. B.2: We report the correlation coe ffi cient between concentration, gas mass and stellar mass computed at both r 500c and r 200c .
\n\n\n\n
Fig. B.3: Halo masses at fixed observable properties. We report the lensing mass, lensing richness, projected stellar mass, projected integrated Comptony , projected X-ray luminosity, and projected temperature in each column, respectively. The top panel shows residuals of the observable-mass relations and respective scatter of log-residuals σ ln , and its axes are on the upper part of the plot. The bottom panel shows the scaling relation fit (blue solid line); the data used to perform the fit (black data points) over-plotted on top of the total sample (grey data points) of the mass M as a function of the observable properties.
\n\n"},"sections":{"kind":"list like","value":[{"title":"ABSTRACT","content":"Context. Cluster cosmology can benefit from combining multi-wavelength studies, which themselves can benefit from a characterisation of the correlation coe ffi cients between di ff erent mass-observable relations. Aims. In this work, we aim to provide information on the scatter, the skewness, and the covariance of various mass-observable relations in galaxy clusters in cosmological hydrodynamic simulations. This information will help future analyses to better tackle accretion histories and projection e ff ects and model mass observable relations for cosmology studies. Methods. Weidentify galaxy clusters in Magneticum Box2b simulations with mass M 200c > 10 14 M ⊙ at redshift z = 0 . 24 and z = 0 . 90. Our analysis includes Euclid -derived properties such as richness, stellar mass, lensing mass, and concentration. Additionally, we investigate complementary multi-wavelength data, including X-ray luminosity, integrated Comptony parameter, gas mass, and temperature. The impact of projection e ff ects on mass-observable residuals and correlations is then examined. Results. We find that at intermediate redshift ( z = 0 . 24) projection e ff ects impact lensing concentration, richness, and gas mass the most in terms of scatter and skewness of log-residuals of scaling relations. The contribution of projection e ff ects can be significant enough to boost a spurious hot- vs. cold-baryons correlation and consequently hide underlying correlations due to halo accretion histories. At high redshift ( z = 0 . 9), the richness has a much lower scatter (of log-residuals), and the quantity that is most impacted by projection e ff ects is the lensing mass. Lensing concentration reconstruction, in particular, is a ff ected by deviations of the reduced-shear profile shape from the one derived by an NFW profile rather than interlopers in the line of sight.","pages":[1,2]},{"title":"The impact of line-of-sight projections on the covariance between galaxy cluster multi-wavelength observable properties - insights from hydrodynamic simulations","content":"Euclid Collaboration: A. Ragagnin ⋆ 1 , 2 , 3 , 4 , A. Saro 5 , 2 , 6 , 7 , 4 , S. Andreon 8 , A. Biviano 6 , 2 , K. Dolag 9 , S. Ettori 1 , 10 , C. Giocoli 1 , 11 , A. M. C. Le Brun 12 , G. A. Mamon 13 , 14 , B. J. Maughan 15 , M. Meneghetti 1 , 16 , L. Moscardini 3 , 1 , 16 , F. Pacaud 17 , G. W. Pratt 18 , M. Sereno 1 , 16 , S. Borgani 5 , 2 , 6 , 7 , F. Calura 1 , G. Castignani 1 , M. De Petris 19 , D. Eckert 20 , G. F. Lesci 3 , 1 , J. Macias-Perez 21 , M. Maturi 22 , 23 , A. Amara 24 , N. Auricchio 1 , C. Baccigalupi 2 , 6 , 7 , 25 , M. Baldi 26 , 1 , 16 , S. Bardelli 1 , D. Bonino 27 , E. Branchini 28 , 29 , 8 , M. Brescia 30 , 31 , 32 , J. Brinchmann 33 , S. Camera 34 , 35 , 27 , V. Capobianco 27 , C. Carbone 36 , J. Carretero 37 , 38 , S. Casas 39 , M. Castellano 40 , S. Cavuoti 31 , 32 , A. Cimatti 41 , C. Colodro-Conde 42 , G. Congedo 43 , C. J. Conselice 44 , L. Conversi 45 , 46 , Y. Copin 47 , F. Courbin 48 , H. M. Courtois 49 , A. Da Silva 50 , 51 , H. Degaudenzi 20 , G. De Lucia 6 , J. Dinis 50 , 51 , F. Dubath 20 , X. Dupac 46 , M. Farina 52 , S. Farrens 18 , S. Ferriol 47 , M. Frailis 6 , E. Franceschi 1 , M. Fumana 36 , K. George 9 , B. Gillis 43 , A. Grazian 53 , F. Grupp 54 , 9 , S. V. H. Haugan 55 , W. Holmes 56 , I. Hook 57 , F. Hormuth 58 , A. Hornstrup 59 , 60 , K. Jahnke 61 , E. Keihänen 62 , S. Kermiche 63 , A. Kiessling 56 , M. Kilbinger 18 , B. Kubik 47 , M. Kümmel 9 , M. Kunz 64 , H. Kurki-Suonio 65 , 66 , S. Ligori 27 , P. B. Lilje 55 , V. Lindholm 65 , 66 , I. Lloro 67 , D. Maino 68 , 36 , 69 , E. Maiorano 1 , O. Mansutti 6 , O. Marggraf 17 , K. Markovic 56 , M. Martinelli 40 , 70 , N. Martinet 71 , F. Marulli 3 , 1 , 16 , R. Massey 72 , S. Maurogordato 73 , E. Medinaceli 1 , S. Mei 74 , Y. Mellier 13 , 14 , G. Meylan 48 , M. Moresco 3 , 1 , E. Munari 6 , 2 , C. Neissner 75 , 38 , S.-M. Niemi 76 , J. W. Nightingale 77 , 72 , C. Padilla 75 , S. Paltani 20 , F. Pasian 6 , K. Pedersen 78 , V. Pettorino 76 , G. Polenta 79 , M. Poncet 80 , L. A. Popa 81 , L. Pozzetti 1 , F. Raison 54 , A. Renzi 82 , 83 , J. Rhodes 56 , G. Riccio 31 , E. Romelli 6 , M. Roncarelli 1 , E. Rossetti 26 , R. Saglia 9 , 54 , Z. Sakr 22 , 84 , 85 , A. G. Sánchez 54 , D. Sapone 86 , B. Sartoris 9 , 6 , R. Scaramella 40 , 70 , P. Schneider 17 , T. Schrabback 87 , A. Secroun 63 , E. Sefusatti 6 , 2 , 7 , G. Seidel 61 , S. Serrano 88 , 89 , 90 , C. Sirignano 82 , 83 , G. Sirri 16 , L. Stanco 83 , J. Steinwagner 54 , P. Tallada-Crespí 37 , 38 , I. Tereno 50 , 91 , R. Toledo-Moreo 92 , F. Torradeflot 38 , 37 , I. Tutusaus 84 , L. Valenziano 1 , 10 , T. Vassallo 9 , 6 , G. Verdoes Kleijn 93 , A. Veropalumbo 8 , 29 , Y. Wang 94 , J. Weller 9 , 54 , G. Zamorani 1 , E. Zucca 1 , M. Bolzonella 1 , A. Boucaud 74 , E. Bozzo 20 , C. Burigana 95 , 10 , M. Calabrese 96 , 36 , D. Di Ferdinando 16 , J. A. Escartin Vigo 54 , R. Farinelli 1 , J. Gracia-Carpio 54 , N. Mauri 41 , 16 , V. Scottez 13 , 97 , M. Tenti 16 , M. Viel 2 , 6 , 25 , 7 , 4 , M. Wiesmann 55 , Y. Akrami 98 , 99 , V. Allevato 31 , S. Anselmi 83 , 82 , 12 , M. Ballardini 100 , 1 , 101 , P. Bergamini 68 , 1 , A. Blanchard 84 , L. Blot 102 , 12 , S. Bruton 103 , R. Cabanac 84 , A. Calabro 40 , G. Canas-Herrera 76 , 104 , A. Cappi 1 , 73 , C. S. Carvalho 91 , T. Castro 6 , 7 , 2 , 4 , K. C. Chambers 105 , S. Contarini 54 , 3 , A. R. Cooray 106 , M. Costanzi 5 , 6 , 2 , B. De Caro 83 , 82 , S. de la Torre 71 , G. Desprez 107 , A. Díaz-Sánchez 108 , S. Di Domizio 28 , 29 , H. Dole 109 , S. Esco ffi er 63 , A. G. Ferrari 41 , 16 , P. G. Ferreira 110 , I. Ferrero 55 , F. Finelli 1 , 10 , F. Fornari 10 , L. Gabarra 110 , K. Ganga 74 , J. García-Bellido 98 , E. Gaztanaga 89 , 88 , 111 , F. Giacomini 16 , G. Gozaliasl 112 , 65 , A. Hall 43 , H. Hildebrandt 113 , J. Hjorth 114 , A. Jimenez Muñoz 21 , J. J. E. Kajava 115 , 116 , V. Kansal 117 , 118 , D. Karagiannis 119 , 120 , C. C. Kirkpatrick 62 , L. Legrand 121 , G. Libet 80 , A. Loureiro 122 , 123 , G. Maggio 6 , M. Magliocchetti 52 , F. Mannucci 124 , R. Maoli 19 , 40 , C. J. A. P. Martins 125 , 33 , S. Matthew 43 , L. Maurin 109 , R. B. Metcalf 3 , 1 , P. Monaco 5 , 6 , 7 , 2 , C. Moretti 25 , 4 , 6 , 2 , 7 , G. Morgante 1 , Nicholas A. Walton 126 , L. Patrizii 16 , A. Pezzotta 54 , M. Pöntinen 65 , V. Popa 81 , C. Porciani 17 , D. Potter 127 , I. Risso 128 , P.-F. Rocci 109 , M. Sahlén 129 , A. Schneider 127 , M. Schultheis 73 , P. Simon 17 , A. Spurio Mancini 130 , 131 , C. Tao 63 , G. Testera 29 , R. Teyssier 132 , S. Toft 60 , 133 , 134 , S. Tosi 28 , 29 , 8 , A. Troja 82 , 83 , M. Tucci 20 , C. Valieri 16 , J. Valiviita 65 , 66 , D. Vergani 1 , and G. Verza 135 , 136 arXiv:2412.00191v1 [astro-ph.CO] 29 Nov 2024 (A ffi liations can be found after the references) June 16, 2023","pages":[1]},{"title":"1. Introduction","content":"Galaxy clusters are the largest gravitationally bound, collapsed, and virialised structures in our Universe and represent unique laboratories for testing cosmological models, galaxy evolution, and thermodynamics of the intracluster medium (ICM, see Kravtsov & Borgani 2012, for a review on galaxy clusters). Regarding galaxy cluster cosmology studies (see, e.g., Rozo et al. 2010; Bocquet et al. 2019), an accurate characterisation of the selection function and of the mass-observable scaling relations represent the dominant systematic uncertainties (see the review on the cluster mass scale in Pratt et al. 2019). Cluster masses cannot be observed directly, and their reconstruction requires both a number of assumptions and highquality data (see, e.g., Meneghetti et al. 2010). This means that precise estimates are rare (Okabe et al. 2010; Hoekstra et al. 2012; Melchior et al. 2015; van Uitert et al. 2016; Stern et al. 2019; Sugiyama et al. 2023; Bocquet et al. 2024). Once a set of highly accurate mass determinations are available, together with other mass proxies recovered from multi-band observations, well-calibrated mass-observable relations (for instance, the mass-richness relation or the mass-temperature relation) can be established and used to estimate galaxy cluster masses for larger samples with known observable properties. For this purpose, it is important to calibrate accurately the mass-observable relations (Giodini et al. 2013; Allen et al. 2011; Schrabback et al. 2021), including proper modelling of their associated scatter (Lima & Hu 2005; Bocquet et al. 2019). This process is complicated by the fact that studies at di ff erent wavelengths are biased by various astrophysical processes and projection e ff ects to various degrees. For instance, X-ray surveys tend to favour the selection of clusters with centrally peaked gas distributions (Pacaud et al. 2007; Hudson et al. 2010; Andreon & Moretti 2011; Andreon et al. 2016; Xu et al. 2018) and su ff er from AGN contamination (see, e.g., Bhargava et al. 2023), while projection e ff ects are known to strongly impact weak lensing mass reconstructions (Meneghetti et al. 2014; Euclid Collaboration: Giocoli et al. 2024) and richness evaluations (e.g., Castignani & Benoist 2016). This is particularly relevant for cluster cosmology studies, where the aim is to reduce uncertainty by combining constraints on di ff erent massobservable relations. For Euclid (Euclid Collaboration: Mellier et al. 2024), this will include quantities such as richness, stellar mass, and properties of stacked weak lensing signals (Pires et al. 2020), of the cluster samples detected using tools such as AMICO (Bellagamba et al. 2018a; Maturi et al. 2019) or PZWav (Euclid Collaboration: Adam et al. 2019), possibly to- gether with other multi-wavelength observations Allen et al. (2011). These properties are known to be biased by projection e ff ects (Meneghetti et al. 2014), accretion histories (Ragagnin et al. 2022), mis-centring (Sommer et al. 2022, 2023), and the fit procedure (Sereno et al. 2016). Projection e ff ects, especially, are expected to generate some covariance between the richness and weak lensing signal, the uncertainty of which may significantly a ff ect the performance of the mission for cluster population analyses. This e ff ect is one the major sources of systematics for current optical cluster surveys (Costanzi et al. 2019; Abbott et al. 2020), and thus is expected to play an even more critical role for the Euclid cluster sample. Numerical simulations are thus a critical tool to mitigate the impact of the aforementioned biases on cosmological cluster studies. Indeed, the power of observations to constrain them is limited, thus increasing the final uncertainty budget. However, scatter and covariance parameters are also prime sources of uncertainty when aiming to combine information originating from di ff erent wavelengths. For instance, various observational works hint towards di ff erent directions for the hot- vs. cold-baryon covariance (Farahi et al. 2019; Puddu & Andreon 2022; Ragagnin et al. 2022), as di ff erent formation times are related with satellite accretion history (Giocoli et al. 2008). In this context, numerical simulations have proven to be a very powerful tool for helping observational studies in modelling mass-observable relations, which are strongly a ff ected by galaxy cluster accretion histories (Ludlow et al. 2012; Bose et al. 2019; Davies et al. 2020; Anbajagane et al. 2020; Ragagnin et al. 2022), projection e ff ects (Meneghetti et al. 2014), and deviations (see, e.g., Ragagnin et al. 2021) from the Navarro-FrenkWhite density profile (NFW, Navarro et al. 1997), which is often adopted in weak lensing studies. Thus, simulations can suggest the most suitable functional forms of scaling relations for cosmological studies (as in the works of Costanzi et al. 2019; Bocquet et al. 2016, 2019; Ghirardini et al. 2024). They can provide informative priors on their correlation coe ffi cients, which are among the most di ffi cult parameters to be constrained directly from observed quantities, guiding the forward modelling setup of cluster cosmology studies. There are various works in the literature that study how simulations can help disentangle physical models (see e.g., Cui et al. 2022; Angelinelli et al. 2023a) cosmological models (see e.g., Bocquet et al. 2020; Angulo et al. 2021; Villaescusa-Navarro et al. 2022) or dark matter types (see e.g., Ragagnin et al. 2024; Fischer et al. 2024; Contreras-Santos et al. 2024), and study observable cross-correlations (see e.g., Stanek et al. 2010; Anbajagane et al. 2020). In this work, besides focusing on correlations between observable properties of interests for multi-wave length studies, we also study the impact of projection e ff ects. The impact of uncorrelated large-scale structure on the covariance between observable properties can be modelled analytically (Hoekstra 2003; McClintock et al. 2019; Costanzi et al. 2019), but the covariance of di ff erent observable properties below a few tens of Mpc requires dedicated simulations. At these scales, numerical hydrodynamic simulations, with their self-consistent depiction of the ICM, emerge as an ideal tool for exploring multiwavelength observable properties since they incorporate the effects of large-scale structures within which clusters are situated. Indeed, baryon feedback influences the ICM not only within cluster virial radii but also beyond (see, e.g., Angelinelli et al. 2022, 2023b). While it is true that cosmological simulations are influenced by the underlying sub-grid prescriptions, and while it is true that these simulations may diverge on small scales, they generally exhibit agreement on quantities integrated up to the sizes of galaxy groups and clusters (see, e.g., Anbajagane et al. 2020). At the same time, di ff erent cosmological parameters can a ff ect galaxy cluster properties, such as their masses (Ragagnin et al. 2021), satellite abundance (van den Bosch et al. 2005), and massobservable relations (Singh et al. 2020). On the other hand, the qualitative significance of covariances and projection e ff ects on observable properties is not expected to significantly hinge on cosmological parameters (Bocquet et al. 2019), and possible deviations from this expectation could be estimated using emulators (see, e.g., Bocquet et al. 2020; Ragagnin et al. 2021; Angulo et al. 2021; Ragagnin et al. 2023). We will study the impact of projection e ff ects using hydrodynamic simulations in order to gain insight into which fraction of the scatter and skewness of scaling relations originates from projection e ff ects (i.e., alignment with filaments and objects) or di ff erent accretion histories. In Sect. 2, we present how we set up our Euclid -like observable properties and the others coming from the other wavelengths. In Sect. 3, we study how projection e ff ects impact the scatter and skewness of log-residuals of scaling relations and discuss the impact of projection e ff ects on observable covariance. In Sect. 4, we focus on the mass-concentration relation and how it is a ff ected by projection e ff ects and deviations from the functional form of profiles and the radial ranges of the fits. In Sect. 5, we focus on the covariance between observable properties and study how di ff erent accretion histories and projection e ff ects impact them. Finally, we draw our conclusions in Sect. 6.","pages":[2,3]},{"title":"2. Numerical Setup","content":"We will conduct our study by analysing clusters obtained from the Magneticum 1 hydrodynamic cosmological simulations (Bi ffi et al. 2013; Saro et al. 2014; Steinborn et al. 2015; Dolag et al. 2016, 2015; Teklu et al. 2015; Steinborn et al. 2016; Bocquet et al. 2016; Ragagnin et al. 2019). They are based on the N -body code Gadget3 , which is built upon Gadget2 (Springel et al. 2005b; Springel 2005; Boylan-Kolchin et al. 2009) with an improved smoothed particle hydrodynamics (SPH) solver from Beck et al. (2016). Magneticum initial conditions are generated using a standard Λ CDM cosmology with Wilkinson Microwave Anisotropy Probe 7 (Komatsu et al. 2011) cosmological parameters. The large-scale structure evolution in Magneticum simulations includes a treatment of radiative cooling, heating from a uniform redshift-dependent ultraviolet (UV) background, star formation, and stellar feedback processes as in Springel et al. (2005a). The stellar feedback is then connected to a detailed chemical evolution and enrichment model as in Tornatore et al. (2007), which follows 11 chemical elements (H, He, C, N, O, Ne, Mg, Si, S, Ca, and Fe, with cooling tables from Wiersma et al. 2009) which are produced with the CLOUDY photo-ionisation code (Ferland et al. 1998). Fabjan et al. (2010) and Hirschmann et al. (2014) described prescriptions for black hole growth and feedback from AGNs. Haloes that host galaxy clusters and groups are identified using the friends-of-friends halo finder (Davis et al. 1985), and subhaloes together with their associated galaxies are identified with an improved version of SUBFIND (Springel et al. 2001), which takes into account the presence of baryons (Dolag et al. 2009). We define r ∆ c as the radius that encloses an average density of ∆ c ρ cr , where ρ cr is the critical density of the Universe at a given redshift, Throughout this paper, when we omit ∆ c from masses and radii we imply the usage of ∆ c = 200 (i.e., M = M 200c) . To disentangle the scatter of the mass-observable relation from projection e ff ects, we compute quantities within a sphere of radius r 200c and as integrated into a cylinder. Projected quantities will be denoted with the superscript 2D (for instance, the total mass inside the cylinder is denoted as M 2D ). We opted to employ a random projection plane for each cluster. Additionally, we set an integration depth of 20 comoving h -1 Mpc, corresponding to approximately 23 Mpc at z = 0 . 24 and 15 Mpc at z = 0 . 9 (with h = 0 . 704). This cylinder depth is smaller than Euclid 's galaxy cluster photoz equivalent uncertainty (Euclid Collaboration: Desprez et al. 2020) and, while we exclude some uncorrelated projection e ff ects, they are known not to play an important role (Sunayama et al. 2020; Wu et al. 2022). Thus, we ensure that we do not overestimate any projection e ff ect that could be mitigated using photoz . Consequently, all projection e ff ects examined in this paper hold relevance for interpreting forthcoming Euclid -based catalogues. This study is based on the results from Box2b / hr (Hirschmann et al. 2014) Magneticum simulation, which covers a length of 900 comoving Mpc, with dark matter particle masses m DM = 9 . 8 × 10 8 M ⊙ , gas initial particle masses of m gas = 2 × 10 8 M ⊙ , and a gravitational softening of both gas and dark matter of ϵ = 3 . 75 comoving kpc. Euclid is expected to detect clusters with masses M > 10 14 M ⊙ up to a redshift of approximately z ≈ 2 (Sartoris et al. 2016; Euclid Collaboration: Adam et al. 2019), where the bulk of the cluster population which will be used for mass-calibration will lie below redshift z ≈ 1 . Furthermore, the number of haloes contained in this Magneticum simulation drops significantly beyond the same redshift value. Therefore, we decided to extract haloes at two representative redshift slices: at an intermediate redshift of approximately z ≈ 0 . 24, yielding 4300 objects, and at a higher redshift of approximately z ≈ 0 . 9, yielding 1300 objects. These extractions were performed using the web portal 2 introduced in Ragagnin et al. (2017). We focus most of the analyses on the qualitative e ff ect of projection e ff ects at our intermediate redshift slice because of the larger statistics of clusters to help us determine projection e ff ects. We stress that this mass threshold is high enough so that all of our galaxy clusters have at least 10 4 particles and, therefore, can be considered well resolved in terms of density profile fitting (Navarro et al. 1997).","pages":[3,4]},{"title":"2.1. Observable properties","content":"We now discuss the properties that we compute for each cluster and report a summary in Table 1. We compute the total stellar masses M ⋆ and M 2D ⋆ as the sum of all stellar particles within the respective volumes. We compute the richness n with a cut of satellites of log 10 ( M ⋆/ M ⊙ ) > 10 . 65 , and the projected version n 2D that includes Euclid -like corrections for projection effects where similarly to Andreon et al. (2016). In particular, we compute the average projected richness between 3 . 5 and 8 Mpc radii from the cluster centre, divided the annulus in 8 slices of equal angles, excluded the two least dense and most dense slices and removed the average projected number density of the 4 remaining octants from the projected richness within r 200c; 3 We compute the X-ray luminosities LX , and L 2D X , in the [0 . 5 , 2] keV energy band computed using the APEC model (Smith et al. 2001), using SPH particle temperatures together with the XSPEC package 4 (Arnaud 1996), which considers the emission of a collisionally-ionized, chemically-enriched plasma implemented with metallicity values taken from the simulated particles 5 . We compute the temperature T and T 2D as weighted by the X-ray emissivity of gas particles. We compute the hot gas mass M g and M 3D g , computed as the sum of the mass of SPH particles with cold gas fraction greater than 0 . 1 and T > 3 × 10 4 K in order to filter out cold gas. Note that the projected gas mass is not to be confused with the one inferred from X-ray observations, as X-ray observational works typically perform a de-projection of the surface brightness ∝ n 2 e , which provides a gas-mass estimate that is closer to the spherical M g , with the addition of some possible alignment e ff ects coming from the central region of clusters. Moreover, observational works have the capability to mask possible bright substructures, thus minimising the presence of interlopers. Consequently, we can conceptualise the observed projected gas mass as an intermediate value between our M g and M 2D g . We estimate the integrated Comptony parameter produced by thermal Sunyaev-Zeldovich (SZ, Sunyaev & Zeldovich 1972). The Comptony parameter is defined as where T is the temperature, n e the number density of the electrons, k B the Boltzmann constant, σ T the Thomson cross-section, c the speed of light, and m e the electron rest mass. We compute the integrated Comptony parameter Y = R y d Ω , both within the volume of sphere of ( Y ) and a cylinder ( Y 2D ). We estimate the integral in Eq. (2) as where the sum runs over all SPH particles, mi is the i -th SPH particle mass, Ti its temperature and f e , i is its electron fraction, expressed as local electron number density normalised to the hydrogen number density, and m p is the proton mass. For each halo, we also perform fits of the NFW profile ρ NFW, defined as where the scaling density ρ 0 and the scale radius r s (that is the radius where the density log-slope equals -2) are free parameters. We perform this fit on the total matter density profile on 100 log-spaced radial bins between 75 ckpc (which corresponds to 60 kpc at z = 0 . 24 , and to 40 kpc at z = 0 . 9; as it is enough to exclude the deep central potential of baryons) and r 200c , and define the corresponding NFW masses and concentration parameters as M NFW and c NFW respectively, and the concentration as c NFW = r NFW / r s , where r NFW is obtained from M NFW via the Eq. (1). The projected version of the mass and concentrations are obtained by mimicking a lensing reconstruction procedure by fitting the corresponding reduced shear. We define the derived masses and concentrations as M 2D NFW and c 2D NFW , where c 2D NFW = R 2D NFW / r s , where R 2D NFW is obtained from M 2D NFW via the Eq. (1). Note that the \"2D\" here, as for the other quantities, means that the quantity is computed in projection, however, a correct fit of the mass from reduced shear NFW profile (namely, our M 2D NFW ) should provide an estimate of the same NFW halo mass M NFW that would be recovered from a 3D fit. The fit is computed within the cylinder described above, with a projected radial range of [300 , 3000] kpc at z = 0 . 2 . We performed the analyses at z = 0 . 9 by rescaling that range with H -2 / 3 ( z ) , where H ( z ) is the Hubble parameter, in order to retain the same fractional distances from the virial radius (at fixed mass), which resulted in a range of [234 , 2300] kpc . Note that in this work we are not interested in estimating the contribution of the uncorrelated large-scale structure in the reduced shear reconstruction, therefore we limit our density projection to a cylinder of the depth of 20 cMpc , (see, e.g., Euclid Collaboration: Giocoli et al. 2024; Becker & Kravtsov 2011). The signal from the source-averaged excess surface mass density ∆Σ gt , averaged over circular radii R and a population of sources distributed in redshift, can be written as Here Σ denotes the surface mass density. The symbol ⟨ ... ⟩ denotes an average over radial bins and redshift lens sources, where we used a redshift distribution as proposed in Euclid Collaboration: Ajani et al. (2023), and Euclid Collaboration: Giocoli et al. (2024). The quantity ⟨ ∆Σ t ⟩ is the excess of surface mass density, averaged over polar coordinates and defined as The symbol Σ cr in Eq. (5) is the critical surface mass density, that for a given redshift source equals to where G the universal gravity constant, D s the angular diameter distance to the source, D d the angular distance to the lens, and D ds the angular distance between the source and the lens. Similarly to Euclid Collaboration: Giocoli et al. (2024), we define the error associated with each radial bin of the profile in Eq. (5) as where σϵ = 0 . 3 (Hoekstra et al. 2012; Euclid Collaboration: Blanchard et al. 2020) is the dispersion of the total intrinsic ellipticity ϵ = (1 -q ) / (1 + q ) , where q is the axis ratio, R 1 and R 2 represent the inner and outer radius of a bin, and n g is the number density of galaxies. For our redshift source distribution (we assume the same as in Euclid Collaboration: Giocoli et al. 2024), we find that n g ≈ 28 arcmin -2 for lenses at redshift z = 0 . 24 and n g ≈ 14 arcmin -2 for lenses at redshift z = 0 . 9 .","pages":[4,5]},{"title":"2.2. Scaling relations","content":"In Fig. 1 we show the observable properties vs. true mass M of clusters, derived from Magneticum Box2b / hr simulation for properties that could be derived by Euclid -like catalogues, such as lensing concentration (first row from top), lensing mass (second row), projected richness (third row), and projected stellar mass (last row), as presented in Sect. 2.1. For each property, we fit a scaling relation performed using a linear regression in the log-log space. We utilise a log-log linear regression because a single power law proves to be e ff ective in modelling our scaling relations. In the right panel of Fig. 1, we show the log-residual distribution for both low-mass haloes ( M < 2 × 10 14 M ⊙ ), highmass haloes ( M > 2 × 10 14 M ⊙ ), and for the complete sample of the log-residuals σ ln , i , defined as the logarithmic ratio between the i -th cluster property and the corresponding scaling relation value at its mass. In the second column, we also report the logscatter σ ln defined here as the corresponding standard deviation of the log-residual, namely where E is the expectation operator that averages over our catalogue data. We note that the concentration has a scatter of 0 . 45 which is higher than theoretical expectations (see, e.g., Child et al. 2018). Throughout this paper, we will show that this is due to projection e ff ects; in fact, the 3D concentration has a scatter of ≈ 0 . 33 . Note that our scatter in temperature exceeds that reported in the theoretical work by Truong et al. (2018). We verified that, if we compute mass-weighted temperature, which is known to behave very well in a power law scaling relation, reveals a log scatter of 0 . 07 , in agreement with their work. The additional scatter that we see may be due to di ff erent X-ray temperature computa- tions (they use core-excised temperature while we take the contribution of the core into account). In Fig. 2 we show the mass-observable relations of quantities that could potentially be obtained from various multi-wavelength observations that could enrich studies based on Euclid -like data products: the integrated Comptony parameter, the gas mass M g , 500c, the X-ray luminosity L 2D X , 500c converted in the soft band [0 . 5 , 2] keV , and the temperature T 500c . Wedecided to plot the Xray luminosity, gas mass, and temperature within r 500c because this radius is typically used in various X-ray observations (see, e.g., Vikhlinin et al. 2006; Sun et al. 2009). The typical Euclid cluster cosmology analysis will therefore rely on mass-observable relations calibrated within r 200 c (e.g., richness, weak-lensing mass), and follow-up observations calibrated within r 500 c (e.g., X-ray and SZ mass-proxies). We will thus need to take into account the covariance between observable properties extracted at di ff erent radii. . Finally, we note that generally, X-ray observations are expected to align more closely with the 3D mass rather than the projected one (Ettori et al. 2013), although several details adopted to analyse the Xray observations (e.g., including masking of substructures, deprojection procedures, etc.) can significantly impact the final result. These choices critically depend on the quality of the observations themselves. Dedicated mocks will thus be required to properly take into account all these e ff ects and are, therefore, beyond the purpose of this work. For simplicity, in this work, we consider an X-ray-derived gas mass as close to M g , while an SZ-derived gas mass closer to M 2D g .","pages":[5,6]},{"title":"3. Projection effects","content":"The main objective of this work is to disentangle the amount of scatter and skewness in scaling relations that is purely due to projection e ff ects. Note that observational data are also a ff ected by measurement errors that we do not tackle in this work (as, for instance, the Poisson error of the limited number of galaxies used to infer the richness). In this Section, we discuss the scatter and skewness of mass-observable relation qualitatively and limit the discussion for the data at redshift z = 0 . 24 , as we have a larger sample of galaxy clusters, and the results are qualitatively similar to the ones at z = 0 . 9. We stress that we leave the quantitative discussion of the scatter and correlation coe ffi cients for both redshifts on Sect. 5. To assess the role of projection e ff ects, Fig. 3 reports the 3D and 2D scatter of our cluster properties. In the left panel of Fig. 3 we show the value of the scatter (see Eq. 9) of log-residuals σ ln for all our mass-observable relations. In the shaded region, we also report the values computed within r 500c for the X-ray luminosity, gas mass, and temperature values because this is the characteristic overdensity used in X-ray analyses. For each observable, in Fig. 3, we report (with di ff erent symbols) both the scatter of the complete sample as well as the one of two separate mass range of 10 14 < M < 2 × 10 14 M ⊙ and M > 2 × 10 14 M ⊙ respectively. We note that the lower-mass bin ( M < 2 × 10 14 M ⊙ ) is the one with the largest scatter because, for a given external object in the line-of-sight (LoS), the profile of a small cluster will be more perturbed with respect to a cluster. In the left panel of Fig. 3, we see that some quantities have a low scatter in the 3D space and do gain a large amount of scatter once they are seen in projection. To better quantify what is the actual impact of projection e ff ects, in the right panel of Fig. 3 we present the metric q σ 2 ln , 2D -σ 2 ln , 3D . This metric shows that the quantities that are most a ff ected by projection e ff ects are the weak lensing concentration, integrated Comptony parameter, gas mass, and NFW profile lensing mass. This is expected as all these observable properties (except for weak lensing concentration and X-ray luminosity) scale linearly with the respective observable mass. Further, we note that the scatter in the richness agrees to the theoretical predictions from Castignani & Benoist (2016). We observe that X-ray luminosity and temperature are the least a ff ected by projection e ff ects. This is attributed to the fact that X-ray luminosity is contingent upon the square of gas density, thereby being primarily influenced by the most bright regions within an image. Similarly, the temperature is predominantly influenced by the innermost regions of a cluster. Note that we lack the value of projection e ff ects for L X , 500c because the 2D scatter is slightly smaller than the 3D one. This happened because projection e ff ects impact under-luminous haloes more strongly than overly luminous haloes (at a fixed mass bin), with the consequence of the projected X-ray luminosity having a higher normalisation and a lower scatter (see Fig. B.1). Some mass-observable relations have a large skewness, to aid observational works in modelling these relations, we will estimate their skewness. Therefore, we also quantify deviations of residuals from a symmetrical distribution by means of the Fisher-Pearson coe ffi cient of skewness m 3 / m 3 / 2 2 , where mk is the sample k th central moment. 6 We report its dependency on projection e ff ects in Fig. 4, showing that the properties whose skewness is most impacted by projection e ff ects are the integrated Comptony parameter, the gas mass, and the lensing NFW mass. We also notice that some scaling relation residuals move from having a negative skewness (for the NFW concentration, for instance, due to un-relaxed and merging clusters) to a positive one once projection e ff ects are taken into account (namely, from an asymmetry towards negative residuals towards an asymmetry towards positive residuals). To assess the impact of projection e ff ects, we introduce a variable to quantify the amount of additional matter in the line of sight that can skew our observable properties. We define it as the ratio of the mass within the cylinder and the mass within a sphere M 2D / M , both of radius r 200c , where the length of the cylinder is described in Sect. 2. We present the distribution of M 2D / M in Fig. 5, where we can see that this quantity is strongly skewed and its median value is M 2D / M ≈ 1 . 26 (note that for a NFW profile with c = 4 , the corresponding analytical cylinder vs. spherical mass is 1 . 25). Although this quantity is not directly observable, we will use it to assess the contribution of LoS objects in the scatter of scaling relations. Note that besides objects in the LoS, di ff erent fitting procedures may impact the scatter of projection e ff ects, as we will see in Sect. 4. 2D In Fig. 6, we show a random selection of clusters, ordered by decreasing M 2D / M from left to right. Objects with high M 2D / M (the objects in the left-most panels) include clusters that are merging, elongated, or in the LoS. In the rest of this paper, we will refer to the objects having M 2D / M greater than the median of the distribution 1 . 26 as having LoS excess. To study the impact of projection e ff ects in scaling relations, in Fig. 7 we show the scaling relations of the following projected quantities: richness, integrated Comptony parameter, lensing mass, and concentrations, that are the ones that are most a ff ected by projection e ff ects. We colour-code these points by M 2D / M and focus on a narrow mass range of M ∈ [1 , 2] × 10 14 M ⊙ in order to visualise better how LoS excess impacts these scaling relations. On the left column, we can visually see that, except for concentration, they strongly correlate with M 2D / M , as the upper points of the scatter plot tend to have higher values of M 2D / M . We quantify this finding in the right column by comparing the residual distributions (of the power-law fit over the complete mass range presented in Sect. 2.2) of the complete sample with the distribution of objects with low LoS contamination only (we adopt the criteria of M 2D / M < 1 . 26 being the median of the M 2D / M distribution, as shown in Fig. 5), and report the respective scatter σ and average value µ of the residual distributions (for the complete sample we have that µ = 0). Except for the concentration, residuals of haloes with low M 2D / M (see dashed histogram) significantly shift towards negative values of µ and σ. For instance, when we consider only objects with a low LoS excess, the scatter of Y 2D decreases from 0 . 35 down to 0 . 11 , and the M 2D NFW scatter goes from 0 . 23 to 0 . 9. The lensing mass is, in fact, well known to be a ff ected by projection e ff ects (Meneghetti et al. 2014; Euclid Collaboration: Giocoli et al. 2024). In this paper, we refer to projection e ff ects as all e ff ects that take place when going from 3D to projection; they include both LoS e ff ects and model uncertainties. This definition becomes relevant when dealing with concentration, which is not impacted only by LoS objects but also by the NFW profile fitting procedure. We stress that in Fig. 3 we proved that our projected concentration is actually highly impacted by projection e ff ects, yet only weakly a ff ected by LoS e ff ects. We show in the next Section the reason for concentration being strongly a ff ected by projection e ff ects is that their reduced shear profile deviates strongly from the one produced by NFW profile (see, e.g., Ragagnin et al. 2021), which is used to reconstruct the reduced shear profile. To conclude this section, we now study how correlations between cluster observable properties can be a ff ected by projection e ff ects. To this end, we take the case of a possible hot- vs. coldbaryon correlation by studying the stellar mass vs. integrated Comptony parameter (as the latter should strongly correlate with the gas mass). In Fig. 8, we show the integrated Comptony parameter scaling relation, colour-coded by stellar mass for the 3D quantities (top panel) and projected quantities (bottom panel). Examining the correlations at a constant halo mass among the computed quantities within spheres (as depicted in the top panel of Fig. 8), we find no discernible weak anti-correlation between stellar mass and the integrated Comptony parameter (which is defined in Sect. 5). Conversely, when investigating the properties in the projected space, a more pronounced correlation becomes evident. This implies that projection e ff ects can strongly impact the correlation between observable properties. While this analysis is purely qualitative, we will quantify the impact of these projection e ff ects in Sect. 5, where we will compute the correlation coe ffi cients for both 3D quantities and 2D quantities.","pages":[6,7,8,9]},{"title":"4. Projection effects on lensing concentration","content":"As we found in the previous section, projection e ff ects significantly increase the scatter and skewness in the scaling of lensing concentration with mass. However, this scatter increase is not related to external objects along the LoS. Now, we assess if the high scatter of lensing concentration is due to deviations of the reduced shear profile from the one induced by an NFW profile. In this work, we will not delve into the origin of this deviation as it falls beyond the scope of this paper. Such deviation may arise due to halo elongations, suggesting that alternative profiles such as truncated NFW profiles may better suit galaxy clusters (Oguri & Hamana 2011). Alternatively, it could stem from the expectation that the NFW profile is intended to describe stacked haloes rather than individual objects. Our focus in this paper is to understand the impact of assuming an NFW profile for each of our haloes. We emphasize that these NFW deviations only a ff ect weak lensing signal reconstruction, as the NFW profile is highly e ff ective in recovering halo mass in 3D. To study deviations from the NFW profile of haloes we fit a generalised NFW profile (Nagai et al. 2007), hereafter gNFW, in spherical coordinates over the same radial range as our previous NFW profile (described in Sect. 2.1), where the density profile ρ gNFW ( r ) is defined as where γ and β are respectively the internal and external logslopes of the total matter density profiles. The case γ = 1 and β = 3 produces the NFW profile as in Eq. (4). Note that the Nagai et al. (2007) gNFW profile also depends on the parameter α that we fix to α = 1 in this work in order to explore internal and external log-slope variations only. We present the PDF for the gNFW profile parameters γ and β in Fig. 9, where the fit was performed in 3D with a flat priors for γ ∈ [0 , 3] and β ∈ [0 , 6]. The data points are colour-coded according to the variable M , revealing no discernible strong trend with respect to the fitted parameters. For 19% of the objects, the resulting best-fit parameters hit the boundaries of hard-cut priors. Upon visual inspection, these objects are characterised by a very steep matter density profile at large cluster-centric distances, possibly suggesting that a truncated NFW profile might be a better model choice. As our objective is to examine the effects of deviations from the generalised NFW profile, we omit these objects from the subsequent analysis in this Section. Given the substantial deviation of these objects from NFW profile, they could potentially o ff er additional insights for our analyses. However, incorporating them would necessitate the use of a profile more general than Eq. 10. Therefore, we excluded them in order to make our analysis clearer. We observe that the external logslope of Magneticum profiles appears to be slightly flatter than -3 . While we emphasize that this discrepancy does not a ff ect the accurate recovery of mass and concentration parameters in 3D NFW fits (such fits can still yield precise estimates of halo mass and concentrations). However, these deviations in the NFW profiles may a ff ect the reduced shear fit, particularly when observed over large radii (remember that in this work, we use 3 Mpc). Furthermore, we observe a degeneracy between the β and γ parameters, indicating that our profiles deviating from NFW profile tend to exhibit a flatter profile compared to NFW profile (as illustrated in Fig. A.1). However, investigating this discrepancy is beyond the scope of this paper, as the internal log slope of clusters is not currently captured by existing weak lensing studies. In Fig. 10, we plot the values of concentration and mass obtained from reduced shear fit, divided by the corresponding 3D quantities and colour-coded by the external 3D gNFW slope β for our intermediate redshift haloes. As we can see, haloes with large values of β have a projected concentration that is significantly higher than the 3D one (see upper panel). In Appendix A we report the example of a simulated halo with low LoS excess (see Fig. A.1) and an analytical one (see Fig. A.2), both with a flat external log-slope, and we show how the under-estimation of the concentration is caused by the fact that the NFW profile fit on the reduced shear is weighting too much the external part of the profile, that deviates from an NFW profile. In Fig. 11, we show the concentration residual distribution and report their scatter. We note that the projected concentration scatter is not a ff ected by external material along the LoS (dashed line and shaded histograms match). However, if one restricts our sample to objects having NFW-like profile log-slopes (we used criteria of 2 . 8 < β < 3 . 2 and 0 . 8 < γ < 1 . 2), then the scatter distribution changes drastically. The concentration residuals decrease from 0 . 43 to 0 . 38, and the residuals shift towards higher values, suggesting that these objects are more relaxed. Such e ff ect is well known, as studied for instance in Macciò et al. (2007). We also show how the external log-slope of the halo profile a ff ects the lensing reconstruction by plotting the ratio between the projected and 3D concentration (i.e., c 2D NFW / c NFW) value versus the 3D log-slope β in the narrow mass bin of M ∈ [1 , 2] × 10 14 M ⊙ in Fig. 12, where we find a positive correlation coe ffi cient of ≈ 0 . 28 , in agreement with a shift of residuals we showed in Fig. 11.","pages":[9,10,11]},{"title":"5. Correlations between properties","content":"While in the last sections, we investigated the origin of the impact of projection e ff ects in the scatter and skewness of observable properties, we will now quantify how projection e ff ects impact the correlation between observable properties. To this end, we quantify the Pearson correlation coe ffi cients between their log-residuals (as defined in Sect. 2.2). We adopt the standard error associated with the Pearson coe ffi cient ρ as derived from two normal distributions, given by σρ = p 1 -ρ 2 / √ N -2 (see Eq. 12-93 in Pugh & Winslow 1966), where N represents the number of objects. This corresponds to a maximum error of 0 . 015 (for ρ = 0) for the sample size at z = 0 . 24 and a maximum error of 0 . 028 for the sample size at z = 0 . 90. It is worth noting that in the correlation coe ffi cient matrices generated in subsequent analyses, we will only colour values with correlation coe ffi cients | ρ | > 0 . 3, aiming to highlight strongly correlating properties. We define mild correlation as 0 . 2 < | ρ | < 0 . 3, as we choose to exercise caution. Correlation coe ffi cients with | ρ | < 0 . 1 are disregarded.","pages":[11]},{"title":"5.1. Analysis at z = 0 . 24","content":"In this Section we focus on the haloes at intermediate redshift z = 0 . 24 . In Fig. 13 we show the correlation coe ffi cient matrix between log-residuals at fixed halo mass of our projected observable both from Euclid -like data (lensing concentration, lensing mass, richness, and stellar mass, respectively) and possible outcomes from multi-wavelength observations (integrated Comptony parameter, gas mass, X-ray luminosity, and temperature) for intermediate-redshift objects. In the lower triangle, we present scatter plots alongside the slope derived from the correlation coe ffi cient. This visualization allows for the identification of instances where the correlation coe ffi cient slope accurately captures the trend of the residuals. Typically, this alignment occurs for quantities that exhibit strong correlation coe ffi cients. For example, our data points show a robust correlation between certain hot baryon tracers ( M 2D g and Y 2D ), some cold baryon components ( n 2D and M 2D ⋆ ), and weak lensing mass M 2D NFW . The underlying reason for these strong correlations lies in projection e ff ects: the greater the amount of matter along the line of sight, the higher the observed values. We will further demonstrate this in the subsequent section by presenting the correlation coe ffi cient matrix for 3D quantities, where many of these correlations diminish. This can be anticipated by observing that L 2D x and c 2D NFW do not exhibit this positive trend of correlations. Notably, we observe that the correlation between richness and stellar mass ( ρ = 0 . 48) is not exceptionally high. Moreover, the stellar mass appears to be more influenced by projection e ff ects compared to richness (evident in their correlations with M 2D NFW , where they exhibit ρ = 0 . 69 and ρ = 0 . 42, respectively). We can speculate on two potential causes: firstly, unlike stellar mass, our richness computation incorporates some observationally-motivated background subtraction; alternatively, since stellar mass encompasses all stellar particles (while richness involves a luminosity-motivated galaxy stellar-mass cut), it is plausible that small subhaloes are influencing the projected stellar mass. We note that concentration anti-correlates with gas-mass (and integrated Comptony parameter). This is in agreement with recent analyses of simulations. In fact, richness at fixed mass anti-correlates with concentration (Bose et al. 2019); low concentration is an index of the system being perturbed (Ludlow et al. 2012); and un-relaxed systems tend to be gas-rich (Davies et al. 2020). We refer to Ragagnin et al. (2022) for a more comprehensive study on low luminous groups. Moreover, at fixed halo mass, the lensing mass correlates strongly with total projected stellar mass ( ρ = 0 . 69) and projected gas mass ( ρ = 0 . 59), which may be due to the fact that both correlate strongly with LoS contamination. The same holds for the correlation among richness, gas mass, and stellar mass. This is due to projection e ff ects, where LoS excess amplifies all these quantities, as discussed in Section 2.2. We note that the 2D lensing mass and projected X-ray luminosity have a slight positive ( ρ = 0 . 20) cor- ation, in agreement with the observational work of Sereno et al. (2020). In Fig. 14 we show the covariance matrix of non-projected quantities for intermediate redshift objects. We see that as opposed to Fig. 13, the 3D covariance matrix shows a mild yet negative covariance between gas mass and stellar mass ( ρ = -0 . 24), and a positive correlation between richness and gas mass ( ρ = 0 . 23) because most of their correlations in the projection are due to Line of sights excess, which significantly increases the values of the gas mass, the richness, and the stellar mass. In Fig. 15 we report the correlation matrix as in Fig. 13 where we present X-ray luminosity, gas mass, and temperature, as computed within r 500c , which shows an anti-correlation between the gas mass and the concentration residuals ( ρ = -0 . 14) that is significantly lower than the one found in Fig. 13 and Fig. 14 ( ρ equals to -0 . 26 and -0 . 34 respectively). One possibility is that this change in sign of the correlation is caused by the fact that mixing overdensities (concentration is within ∆ c = 200 and gasmass is within ∆ c = 500) does introduce an additional correlation with the sparsity (Balmès et al. 2014; Corasaniti et al. 2022) that itself correlates with the concentration (see Appendix B). For completeness, we report the correlation coe ffi cient matrix and the scatter of log-residuals of all quantities in Table B.1. There we also added the core-excised projected X-ray luminosity L 2D X , ce500c , as it is typically used in X-ray-based observational studies, where we can see that the scatter and most of the correlation coe ffi cients are smaller than L 2D ce500c , while the correlations with the concentration and gas mass increase. Note that we do not report the 3D NFW mass ( M NFW) because it has an extremely low intrinsic scatter σ ln ( M NFW) ≈ 0 . 01 and its correlation coefficients are not meaningful.","pages":[11,12]},{"title":"5.2. Analysis at z = 0 . 9","content":"In this Section, we focus on observational property covariance matrixes of our haloes z = 0 . 9. At this redshift, we computed projected quantities within a cylinder depth of 35 Mpc in order to retain the same relative ratio as the photoz uncertainty of the low-redshift analysis (it scales with 1 + z ). For what concerns the cylinder used to integrate ∆Σ gt, we rescaled so as to keep it constant in comoving units with the low-redshift analyses. We rescaled the 3D NFW profile minimum radius to 40 kpc while we kept the maximum radius at r 200c . For what concerns the radial range of the lensing fit, we rescaled it with H -2 / 3 ( z ), therefore performing it in the range of [234 , 2300] kpc . We stress that we do not model observational uncertainty. Therefore, the decrease in background source count with redshift does not impact our best fits. However, it still impacts the fact that we weigh external radial bins more than internal ones. We report the values of the scatter and the projection contribution at z = 0 . 9 in Fig. 16, while we report the log-residuals and the skewness for each property in Fig. 17. In particular, the quantities most a ff ected by projection effects are the lensing mass and concentration, whereas the temperature is the lowest. These results are qualitatively similar to the low redshift analyses, with the ComptonY parameter and gas mass being slightly less a ff ected by projection e ff ects. Note that since the virial radius is smaller at higher redshift values, our radial range of the reduced shear is closer to the NFW scale radius; therefore, the weak lensing reconstruction is more e ff ective in capturing the scale radius and more sensitive to deviations from an NFW profile. As a consequence, we found that the in- crease of scatter going from c NFW to c 2D NFW compared to the low redshift analyses. We report the correlation coe ffi cient matrix and the scatter log-residuals of the quantities at z = 0 . 9 in Table B.2. As for the case at z = 0 . 24 , note that we do not report the 3D NFW mass ( M NFW) because it has an extremely small intrinsic scatter of 0 . 01 , and thus its correlation coe ffi cients have no impact in our study.","pages":[12]},{"title":"6. Conclusions","content":"In this work, we analysed a number of galaxy clusters from Magneticum hydrodynamic simulation Box2b / hr. We did so in a mass range, tailored for Euclid -like data products (see Sartoris et al. 2016; Euclid Collaboration: Adam et al. 2019), namely with a mass of M 200c > 10 14 M ⊙ . To this end, we computed properties that could come from Euclid catalogues of galaxy clusters, such as richness, stellar mass, and lensing masses and concentration, and possible properties coming from multi-wavelength studies such as X-ray luminosity, integrated Comptony parameter, gas mass, and temperature. All these properties were computed both within a sphere and within a cylinder (both with radius r 200c) to account for projection e ff ects. Our study considers the remarkable capabilities of Euclid photoz measurements in identifying interlopers. However, their importance decreases significantly at scales as small as a few tens Mpc. This depth is still long enough to contain multiple haloes along the LoS. Hence, we studied the projection e ff ects on a scale that is significantly smaller than the Euclid photoz uncertainty. We then studied how the scatter and skewness change when one measures quantities in 3D space or in projection. Below, we summarise our findings: The analysis presented here has been carried out using a single suite of hydrodynamic simulations. Regarding weak lensing masses and concentration, since in this work, we did not consider the profile noise due to the finite number of background galaxies, future studies are needed to improve our estimations. Some works show that both scatter and correlation coefficients vary between cosmological simulations with di ff erent cosmologies (Ragagnin et al. 2023), the presence of feedback schemes (Stanek et al. 2010), and di ff erent cosmological simulation suite in the market (see Fig. 7 in Anbajagane et al. 2020). So, while simulations can provide directions on how to model correlation coe ffi cients, it is possible that when using this kind of data, one needs to allow for variation due to the di ff erent baryon physics. Furthermore, when striving for even more precise results, it is important to acknowledge that mass-observable relations are not exact power laws. Therefore, employing more generic fitting techniques, such as a running median, could yield improvements. Additionally, there is room for enhancement in how we compute correlation coe ffi cients in future studies. One potential approach could involve simultaneously fitting both the massobservable relation scatter and the correlation coe ffi cients by maximizing multivariate likelihoods. We anticipate that future studies combining Euclid data with multi-wavelength observations may encounter challenges in shedding light on currently puzzling residual correlations, primarily dominated by projection e ff ects. Acknowledgements. We thanks the anonymous referee for the useful comments. The Magneticum Pathfinder simulations were partially performed at the LeibnizRechenzentrum with CPU time assigned to the Project 'pr86re'. AR and LM acknowledge support from the grant PRIN-MIUR 2017 WSCC32 and acknowledges the usage of the INAF-OATs IT framework (Ta ff oni et al. 2020; Bertocco et al. 2020), and the space filling curve improvement on Gadget3 (Ragagnin et al. 2016). Antonio Ragagnin thanks Veronica Bi ffi and Elena Rasia for the X-ray computation routines and tables. KD acknowledges support by the COMPLEX project from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program grant agreement ERC-2019-AdG 882679. LM acknowledges the financial contribution from the grant PRIN-MUR 2022 20227RNLY3 'The concordance cosmological model: stress-tests with galaxy clusters' supported by Next Generation EU. CG and LM acknowledge support from the grant ASI n.2018-23-HH.0. AR and CG acknowledge funding from INAF theory Grant 2022: Illuminating Dark Matter using Weak Lensing by Cluster Satellites, PI: Carlo Giocoli. SB acknowledges partial financial support from the INFN InDark grant. AMCLB was supported by a fellowship of PSL University hosted by the Paris Observatory. We used the package colossus (see Diemer 2018) for computing Σ and ∆Σ as expected from NFW profiles. AR and FC acknowledge co-funding by the European Union - NextGenerationEU within PRIN 2022 project n.20229YBSAN - Globular clusters in cosmological simulations and in lensed fields: from their birth to the present epoch. The Euclid Consortium acknowledges the European Space Agency and a number of agencies and institutes that have supported the development of Euclid , in particular the Academy of Finland, the Agenzia Spaziale Italiana, the Belgian Science Policy, the Canadian Euclid Consortium, the French Centre National d'Etudes Spatiales, the Deutsches Zentrum für Luft- und Raumfahrt, the Danish Space Research Institute, the Fundação para a Ciência e a Tecnologia, the Ministerio de Ciencia e Innovación, the National Aeronautics and Space Administration, the National Astronomical Observatory of Japan, the Netherlandse Onderzoekschool Voor Astronomie, the Norwegian Space Agency, the Romanian Space Agency, the State Secretariat for Education, Research and Innovation (SERI) at the Swiss Space O ffi ce (SSO), and the United Kingdom Space Agency. A complete and detailed list is available on the Euclid web site ( http://www.euclid-ec.org ).","pages":[12,13]},{"title":"Data Availability","content":"Raw simulation data were generated at C 2 PAP / LRZ cosmology simulation web portal https://c2papcosmosim.uc. lrz.de/ . Derived data supporting the findings of this study are available from the corresponding author AR on request.","pages":[13]},{"title":"References","content":"Kravtsov, A. V. & Borgani, S. 2012, ARA&A, 50, 353 Ragagnin, A., Andreon, S., & Puddu, E. 2022, A&A, 666, A22 Ragagnin, A., Fumagalli, A., Castro, T., et al. 2023, A&A, 675, A77 Sereno, M., Umetsu, K., Ettori, S., et al. 2020, MNRAS, 492, 4528 Sugiyama, S., Miyatake, H., More, S., et al. 2023, Phys. Rev. D, 108, 123521 Sun, M., Voit, G. M., Donahue, M., et al. 2009, ApJ, 693, 1142 Teklu, A. F., Remus, R.-S., Dolag, K., et al. 2015, ApJ, 812, 29 Tornatore, L., Borgani, S., Dolag, K., & Matteucci, F. 2007, MNRAS, 382, 1050 Truong, N., Rasia, E., Mazzotta, P., et al. 2018, MNRAS, 474, 4089 van Uitert, E., Cacciato, M., Hoekstra, H., et al. 2016, MNRAS, 459, 3251 Article number, page 14 of 26 18 Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France 23 Zentrum für Astronomie, Universität Heidelberg, Philosophenweg 12, 69120 Heidelberg, Germany","pages":[13,14]},{"title":"Appendix A: Fit of gNFW","content":"In Fig. A.1, we present the density profile of a halo that deviates from NFW profile and has no LoS contamination. In particular, it has β ≈ 1 . 8 and γ = 1 . 5 . We show its NFW profile fit profile on the 3D density in the upper panel of Fig. A.1, where we can see that 3D NFW profile (performed on radial bins in a sphere) is capable of capturing the shape of the halo and to estimate its mass with high accuracy (within ≈ 5%). In the central panel, we show the reduced shear profile and best fits, where we can see that the fit performed on the reduced shear underestimated the concentration and is not able to capture the more internal part of the shear profile, as it is done by the profile that was fit in 3D. We first exclude this mismatch as being due to projection effects by showing that both the reduced shear from the particle data (orange line) matches the one recovered by performing an analytical projection of the 3D profile (blue solid line). In particular we project the density profile ρ ( r ) and derive the surface mass density Σ conv ., as follows What we find is that the shear obtained with the aid of an analytical projection Σ conv . matches very well the real one (i.e., orange and blue lines do match). This hints that for this cluster there are no strong LoS e ff ects. To understand why the fit on the shear is not able to capture the concentration of the original halo, we zoom our fit in the bottom panel of Fig. A.1, where it looks like the fit is very good in capturing the final part of the profile and not able to capture the internal. It is crucial to emphasise that we did not include observational uncertainties in these analyses. Therefore, the uncertainty outlined in Eq. (8) a ff ects the fit by assigning more weight to external radial bins compared to internal ones. It is worth noting that the proportionality factors in Eq. (8) will not a ff ect our best fit. To validate this point in Fig. A.2 we study the bias on fitting an NFW profile on a mock gNFW profile that has β = 1 . 8 and γ = 1 . 5 , a mass of 3 × 10 14 M ⊙ and a concentration c = 2 . 4 (as the halo presented in Fig. A.1). We see that the 3D NFW profile is capable of estimating both its mass and its gNFW concentration with high accuracy (see top panel match between blue and dashed black lines). On the other hand the fit of the shear (we report in the bottom panel of Fig. A.2) has the same problems as the one on the cluster in Fig. A.1: it recovers a low concentration (with a value of 1 . 5). This may be because, at outer radii, the model fits the data. It is possible that the under-estimation of concentration at low radii is caused by the combination of two factors: the fit under-estimates the shear at lower radii (with the result of under-estimating the lensing concentration), or the fact that γ is di ff erent than 3 induces an NFW profile fit with a low concentration. We then performed the experiment of fitting the analytical profile with constant (yet unrealistic) error bars. While the fit was able to capture the shape of the profile, it recovered a concentration of 1 . 6 , implying that there is indeed a degeneracy between the shear of low-concentrated NFW profiles and steeper-NFW profiles.","pages":[22]},{"title":"Appendix B: Correlations with different overdensities","content":"In this appendix, we discuss the di ff erences between scaling relation scatters and covariance values at di ff erent overdensities. First of all, we tackle the fact that when we compute X-ray luminosity within r 500c (instead of r 200c), we find that the scatter of the scaling relation of the projected quantity is larger than the 3D one. To investigate this feature, we will focus on the bolometric X-ray luminosity. We report the 3D and projected bolometric X- ray luminosity in Fig. B.1 (top panel), where it is visually clear that the projected X-ray luminosity is (as expected) always larger than the 3D one. One can also notice that the increase in X-ray luminosity depends on the fact that a halo is over-luminous or not: the increase of luminosity growing from the 3D to 2D is larger for under-luminous haloes than for over-luminous haloes. We prove this point in the bottom panel of Fig. B.1 where we show the ratio between the 2D and 3D luminosity as a function of their residual of the 3D scaling relation (the higher the value of the x axis, the more over-luminous is the object for its mass bin), where we can see a strong anti-correlation: overly luminous objects (for a given mass bin) are not going to be a ff ected much by the fact that their luminosity is computed in 3D or 2D. The possible cause is that an interloper in the LoS will not a ff ect much an overly luminous object. For completeness, in Fig. B.2 we show the correlation coefficients between the gas mass and stellar mass computed within both r 500c and r 200c and the concentration. Here we can see a change of sign between M ⋆, 500cMg , 500c correlations and M ⋆, 500cM g correlations and a change in the sign between c NFWM g correlations and c NFWMg , 500c correlations. Finally, in Fig. B.3 we report the scatter of observable properties at fixed mass for both Euclid -like quantities (lensing mass, richness, and projected stellar mass), and possible multiwavelength properties (integrated Comptony parameter, X-ray luminosity, and temperature), where we compute X-ray lumi- nosity and temperature within r 500c as they are typically derived within this overdensity. The upper panel of Fig. B.3 shows the residuals of the log-log linear regression where we see that in terms of 2D scatter, the properties with the lowest scatter are the stellar mass and the temperature. The bottom panel shows the data points used to perform the fit (in black) where we used a visually-inspected cut on the halo mass values in order to ensure that mass values are complete for a given observable value. Notes. Diagonal terms report the scatter of the log-residuals of each quantity, namely σ ln of Eq. (9), while the o ff -diagonal terms report the correlation coe ffi cient between the log-residuals. We do not report values of the correlation coe ffi cient below 0 . 20 because they are not significant. We do not report the values for the 3D NFW mass M 200c because it has a very low scatter of log-residuals ( ≈ 0 . 01) and its correlation coe ffi cients are not meaningful. Note that in this table we also added the core-excised X-ray luminosity. Notes. Rows and columns are as in Table B.1.","pages":[22,23,24]}],"string":"[\n {\n \"title\": \"ABSTRACT\",\n \"content\": \"Context. Cluster cosmology can benefit from combining multi-wavelength studies, which themselves can benefit from a characterisation of the correlation coe ffi cients between di ff erent mass-observable relations. Aims. In this work, we aim to provide information on the scatter, the skewness, and the covariance of various mass-observable relations in galaxy clusters in cosmological hydrodynamic simulations. This information will help future analyses to better tackle accretion histories and projection e ff ects and model mass observable relations for cosmology studies. Methods. Weidentify galaxy clusters in Magneticum Box2b simulations with mass M 200c > 10 14 M ⊙ at redshift z = 0 . 24 and z = 0 . 90. Our analysis includes Euclid -derived properties such as richness, stellar mass, lensing mass, and concentration. Additionally, we investigate complementary multi-wavelength data, including X-ray luminosity, integrated Comptony parameter, gas mass, and temperature. The impact of projection e ff ects on mass-observable residuals and correlations is then examined. Results. We find that at intermediate redshift ( z = 0 . 24) projection e ff ects impact lensing concentration, richness, and gas mass the most in terms of scatter and skewness of log-residuals of scaling relations. The contribution of projection e ff ects can be significant enough to boost a spurious hot- vs. cold-baryons correlation and consequently hide underlying correlations due to halo accretion histories. At high redshift ( z = 0 . 9), the richness has a much lower scatter (of log-residuals), and the quantity that is most impacted by projection e ff ects is the lensing mass. Lensing concentration reconstruction, in particular, is a ff ected by deviations of the reduced-shear profile shape from the one derived by an NFW profile rather than interlopers in the line of sight.\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"The impact of line-of-sight projections on the covariance between galaxy cluster multi-wavelength observable properties - insights from hydrodynamic simulations\",\n \"content\": \"Euclid Collaboration: A. Ragagnin ⋆ 1 , 2 , 3 , 4 , A. Saro 5 , 2 , 6 , 7 , 4 , S. Andreon 8 , A. Biviano 6 , 2 , K. Dolag 9 , S. Ettori 1 , 10 , C. Giocoli 1 , 11 , A. M. C. Le Brun 12 , G. A. Mamon 13 , 14 , B. J. Maughan 15 , M. Meneghetti 1 , 16 , L. Moscardini 3 , 1 , 16 , F. Pacaud 17 , G. W. Pratt 18 , M. Sereno 1 , 16 , S. Borgani 5 , 2 , 6 , 7 , F. Calura 1 , G. Castignani 1 , M. De Petris 19 , D. Eckert 20 , G. F. Lesci 3 , 1 , J. Macias-Perez 21 , M. Maturi 22 , 23 , A. Amara 24 , N. Auricchio 1 , C. Baccigalupi 2 , 6 , 7 , 25 , M. Baldi 26 , 1 , 16 , S. Bardelli 1 , D. Bonino 27 , E. Branchini 28 , 29 , 8 , M. Brescia 30 , 31 , 32 , J. Brinchmann 33 , S. Camera 34 , 35 , 27 , V. Capobianco 27 , C. Carbone 36 , J. Carretero 37 , 38 , S. Casas 39 , M. Castellano 40 , S. Cavuoti 31 , 32 , A. Cimatti 41 , C. Colodro-Conde 42 , G. Congedo 43 , C. J. Conselice 44 , L. Conversi 45 , 46 , Y. Copin 47 , F. Courbin 48 , H. M. Courtois 49 , A. Da Silva 50 , 51 , H. Degaudenzi 20 , G. De Lucia 6 , J. Dinis 50 , 51 , F. Dubath 20 , X. Dupac 46 , M. Farina 52 , S. Farrens 18 , S. Ferriol 47 , M. Frailis 6 , E. Franceschi 1 , M. Fumana 36 , K. George 9 , B. Gillis 43 , A. Grazian 53 , F. Grupp 54 , 9 , S. V. H. Haugan 55 , W. Holmes 56 , I. Hook 57 , F. Hormuth 58 , A. Hornstrup 59 , 60 , K. Jahnke 61 , E. Keihänen 62 , S. Kermiche 63 , A. Kiessling 56 , M. Kilbinger 18 , B. Kubik 47 , M. Kümmel 9 , M. Kunz 64 , H. Kurki-Suonio 65 , 66 , S. Ligori 27 , P. B. Lilje 55 , V. Lindholm 65 , 66 , I. Lloro 67 , D. Maino 68 , 36 , 69 , E. Maiorano 1 , O. Mansutti 6 , O. Marggraf 17 , K. Markovic 56 , M. Martinelli 40 , 70 , N. Martinet 71 , F. Marulli 3 , 1 , 16 , R. Massey 72 , S. Maurogordato 73 , E. Medinaceli 1 , S. Mei 74 , Y. Mellier 13 , 14 , G. Meylan 48 , M. Moresco 3 , 1 , E. Munari 6 , 2 , C. Neissner 75 , 38 , S.-M. Niemi 76 , J. W. Nightingale 77 , 72 , C. Padilla 75 , S. Paltani 20 , F. Pasian 6 , K. Pedersen 78 , V. Pettorino 76 , G. Polenta 79 , M. Poncet 80 , L. A. Popa 81 , L. Pozzetti 1 , F. Raison 54 , A. Renzi 82 , 83 , J. Rhodes 56 , G. Riccio 31 , E. Romelli 6 , M. Roncarelli 1 , E. Rossetti 26 , R. Saglia 9 , 54 , Z. Sakr 22 , 84 , 85 , A. G. Sánchez 54 , D. Sapone 86 , B. Sartoris 9 , 6 , R. Scaramella 40 , 70 , P. Schneider 17 , T. Schrabback 87 , A. Secroun 63 , E. Sefusatti 6 , 2 , 7 , G. Seidel 61 , S. Serrano 88 , 89 , 90 , C. Sirignano 82 , 83 , G. Sirri 16 , L. Stanco 83 , J. Steinwagner 54 , P. Tallada-Crespí 37 , 38 , I. Tereno 50 , 91 , R. Toledo-Moreo 92 , F. Torradeflot 38 , 37 , I. Tutusaus 84 , L. Valenziano 1 , 10 , T. Vassallo 9 , 6 , G. Verdoes Kleijn 93 , A. Veropalumbo 8 , 29 , Y. Wang 94 , J. Weller 9 , 54 , G. Zamorani 1 , E. Zucca 1 , M. Bolzonella 1 , A. Boucaud 74 , E. Bozzo 20 , C. Burigana 95 , 10 , M. Calabrese 96 , 36 , D. Di Ferdinando 16 , J. A. Escartin Vigo 54 , R. Farinelli 1 , J. Gracia-Carpio 54 , N. Mauri 41 , 16 , V. Scottez 13 , 97 , M. Tenti 16 , M. Viel 2 , 6 , 25 , 7 , 4 , M. Wiesmann 55 , Y. Akrami 98 , 99 , V. Allevato 31 , S. Anselmi 83 , 82 , 12 , M. Ballardini 100 , 1 , 101 , P. Bergamini 68 , 1 , A. Blanchard 84 , L. Blot 102 , 12 , S. Bruton 103 , R. Cabanac 84 , A. Calabro 40 , G. Canas-Herrera 76 , 104 , A. Cappi 1 , 73 , C. S. Carvalho 91 , T. Castro 6 , 7 , 2 , 4 , K. C. Chambers 105 , S. Contarini 54 , 3 , A. R. Cooray 106 , M. Costanzi 5 , 6 , 2 , B. De Caro 83 , 82 , S. de la Torre 71 , G. Desprez 107 , A. Díaz-Sánchez 108 , S. Di Domizio 28 , 29 , H. Dole 109 , S. Esco ffi er 63 , A. G. Ferrari 41 , 16 , P. G. Ferreira 110 , I. Ferrero 55 , F. Finelli 1 , 10 , F. Fornari 10 , L. Gabarra 110 , K. Ganga 74 , J. García-Bellido 98 , E. Gaztanaga 89 , 88 , 111 , F. Giacomini 16 , G. Gozaliasl 112 , 65 , A. Hall 43 , H. Hildebrandt 113 , J. Hjorth 114 , A. Jimenez Muñoz 21 , J. J. E. Kajava 115 , 116 , V. Kansal 117 , 118 , D. Karagiannis 119 , 120 , C. C. Kirkpatrick 62 , L. Legrand 121 , G. Libet 80 , A. Loureiro 122 , 123 , G. Maggio 6 , M. Magliocchetti 52 , F. Mannucci 124 , R. Maoli 19 , 40 , C. J. A. P. Martins 125 , 33 , S. Matthew 43 , L. Maurin 109 , R. B. Metcalf 3 , 1 , P. Monaco 5 , 6 , 7 , 2 , C. Moretti 25 , 4 , 6 , 2 , 7 , G. Morgante 1 , Nicholas A. Walton 126 , L. Patrizii 16 , A. Pezzotta 54 , M. Pöntinen 65 , V. Popa 81 , C. Porciani 17 , D. Potter 127 , I. Risso 128 , P.-F. Rocci 109 , M. Sahlén 129 , A. Schneider 127 , M. Schultheis 73 , P. Simon 17 , A. Spurio Mancini 130 , 131 , C. Tao 63 , G. Testera 29 , R. Teyssier 132 , S. Toft 60 , 133 , 134 , S. Tosi 28 , 29 , 8 , A. Troja 82 , 83 , M. Tucci 20 , C. Valieri 16 , J. Valiviita 65 , 66 , D. Vergani 1 , and G. Verza 135 , 136 arXiv:2412.00191v1 [astro-ph.CO] 29 Nov 2024 (A ffi liations can be found after the references) June 16, 2023\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1. Introduction\",\n \"content\": \"Galaxy clusters are the largest gravitationally bound, collapsed, and virialised structures in our Universe and represent unique laboratories for testing cosmological models, galaxy evolution, and thermodynamics of the intracluster medium (ICM, see Kravtsov & Borgani 2012, for a review on galaxy clusters). Regarding galaxy cluster cosmology studies (see, e.g., Rozo et al. 2010; Bocquet et al. 2019), an accurate characterisation of the selection function and of the mass-observable scaling relations represent the dominant systematic uncertainties (see the review on the cluster mass scale in Pratt et al. 2019). Cluster masses cannot be observed directly, and their reconstruction requires both a number of assumptions and highquality data (see, e.g., Meneghetti et al. 2010). This means that precise estimates are rare (Okabe et al. 2010; Hoekstra et al. 2012; Melchior et al. 2015; van Uitert et al. 2016; Stern et al. 2019; Sugiyama et al. 2023; Bocquet et al. 2024). Once a set of highly accurate mass determinations are available, together with other mass proxies recovered from multi-band observations, well-calibrated mass-observable relations (for instance, the mass-richness relation or the mass-temperature relation) can be established and used to estimate galaxy cluster masses for larger samples with known observable properties. For this purpose, it is important to calibrate accurately the mass-observable relations (Giodini et al. 2013; Allen et al. 2011; Schrabback et al. 2021), including proper modelling of their associated scatter (Lima & Hu 2005; Bocquet et al. 2019). This process is complicated by the fact that studies at di ff erent wavelengths are biased by various astrophysical processes and projection e ff ects to various degrees. For instance, X-ray surveys tend to favour the selection of clusters with centrally peaked gas distributions (Pacaud et al. 2007; Hudson et al. 2010; Andreon & Moretti 2011; Andreon et al. 2016; Xu et al. 2018) and su ff er from AGN contamination (see, e.g., Bhargava et al. 2023), while projection e ff ects are known to strongly impact weak lensing mass reconstructions (Meneghetti et al. 2014; Euclid Collaboration: Giocoli et al. 2024) and richness evaluations (e.g., Castignani & Benoist 2016). This is particularly relevant for cluster cosmology studies, where the aim is to reduce uncertainty by combining constraints on di ff erent massobservable relations. For Euclid (Euclid Collaboration: Mellier et al. 2024), this will include quantities such as richness, stellar mass, and properties of stacked weak lensing signals (Pires et al. 2020), of the cluster samples detected using tools such as AMICO (Bellagamba et al. 2018a; Maturi et al. 2019) or PZWav (Euclid Collaboration: Adam et al. 2019), possibly to- gether with other multi-wavelength observations Allen et al. (2011). These properties are known to be biased by projection e ff ects (Meneghetti et al. 2014), accretion histories (Ragagnin et al. 2022), mis-centring (Sommer et al. 2022, 2023), and the fit procedure (Sereno et al. 2016). Projection e ff ects, especially, are expected to generate some covariance between the richness and weak lensing signal, the uncertainty of which may significantly a ff ect the performance of the mission for cluster population analyses. This e ff ect is one the major sources of systematics for current optical cluster surveys (Costanzi et al. 2019; Abbott et al. 2020), and thus is expected to play an even more critical role for the Euclid cluster sample. Numerical simulations are thus a critical tool to mitigate the impact of the aforementioned biases on cosmological cluster studies. Indeed, the power of observations to constrain them is limited, thus increasing the final uncertainty budget. However, scatter and covariance parameters are also prime sources of uncertainty when aiming to combine information originating from di ff erent wavelengths. For instance, various observational works hint towards di ff erent directions for the hot- vs. cold-baryon covariance (Farahi et al. 2019; Puddu & Andreon 2022; Ragagnin et al. 2022), as di ff erent formation times are related with satellite accretion history (Giocoli et al. 2008). In this context, numerical simulations have proven to be a very powerful tool for helping observational studies in modelling mass-observable relations, which are strongly a ff ected by galaxy cluster accretion histories (Ludlow et al. 2012; Bose et al. 2019; Davies et al. 2020; Anbajagane et al. 2020; Ragagnin et al. 2022), projection e ff ects (Meneghetti et al. 2014), and deviations (see, e.g., Ragagnin et al. 2021) from the Navarro-FrenkWhite density profile (NFW, Navarro et al. 1997), which is often adopted in weak lensing studies. Thus, simulations can suggest the most suitable functional forms of scaling relations for cosmological studies (as in the works of Costanzi et al. 2019; Bocquet et al. 2016, 2019; Ghirardini et al. 2024). They can provide informative priors on their correlation coe ffi cients, which are among the most di ffi cult parameters to be constrained directly from observed quantities, guiding the forward modelling setup of cluster cosmology studies. There are various works in the literature that study how simulations can help disentangle physical models (see e.g., Cui et al. 2022; Angelinelli et al. 2023a) cosmological models (see e.g., Bocquet et al. 2020; Angulo et al. 2021; Villaescusa-Navarro et al. 2022) or dark matter types (see e.g., Ragagnin et al. 2024; Fischer et al. 2024; Contreras-Santos et al. 2024), and study observable cross-correlations (see e.g., Stanek et al. 2010; Anbajagane et al. 2020). In this work, besides focusing on correlations between observable properties of interests for multi-wave length studies, we also study the impact of projection e ff ects. The impact of uncorrelated large-scale structure on the covariance between observable properties can be modelled analytically (Hoekstra 2003; McClintock et al. 2019; Costanzi et al. 2019), but the covariance of di ff erent observable properties below a few tens of Mpc requires dedicated simulations. At these scales, numerical hydrodynamic simulations, with their self-consistent depiction of the ICM, emerge as an ideal tool for exploring multiwavelength observable properties since they incorporate the effects of large-scale structures within which clusters are situated. Indeed, baryon feedback influences the ICM not only within cluster virial radii but also beyond (see, e.g., Angelinelli et al. 2022, 2023b). While it is true that cosmological simulations are influenced by the underlying sub-grid prescriptions, and while it is true that these simulations may diverge on small scales, they generally exhibit agreement on quantities integrated up to the sizes of galaxy groups and clusters (see, e.g., Anbajagane et al. 2020). At the same time, di ff erent cosmological parameters can a ff ect galaxy cluster properties, such as their masses (Ragagnin et al. 2021), satellite abundance (van den Bosch et al. 2005), and massobservable relations (Singh et al. 2020). On the other hand, the qualitative significance of covariances and projection e ff ects on observable properties is not expected to significantly hinge on cosmological parameters (Bocquet et al. 2019), and possible deviations from this expectation could be estimated using emulators (see, e.g., Bocquet et al. 2020; Ragagnin et al. 2021; Angulo et al. 2021; Ragagnin et al. 2023). We will study the impact of projection e ff ects using hydrodynamic simulations in order to gain insight into which fraction of the scatter and skewness of scaling relations originates from projection e ff ects (i.e., alignment with filaments and objects) or di ff erent accretion histories. In Sect. 2, we present how we set up our Euclid -like observable properties and the others coming from the other wavelengths. In Sect. 3, we study how projection e ff ects impact the scatter and skewness of log-residuals of scaling relations and discuss the impact of projection e ff ects on observable covariance. In Sect. 4, we focus on the mass-concentration relation and how it is a ff ected by projection e ff ects and deviations from the functional form of profiles and the radial ranges of the fits. In Sect. 5, we focus on the covariance between observable properties and study how di ff erent accretion histories and projection e ff ects impact them. Finally, we draw our conclusions in Sect. 6.\",\n \"pages\": [\n 2,\n 3\n ]\n },\n {\n \"title\": \"2. Numerical Setup\",\n \"content\": \"We will conduct our study by analysing clusters obtained from the Magneticum 1 hydrodynamic cosmological simulations (Bi ffi et al. 2013; Saro et al. 2014; Steinborn et al. 2015; Dolag et al. 2016, 2015; Teklu et al. 2015; Steinborn et al. 2016; Bocquet et al. 2016; Ragagnin et al. 2019). They are based on the N -body code Gadget3 , which is built upon Gadget2 (Springel et al. 2005b; Springel 2005; Boylan-Kolchin et al. 2009) with an improved smoothed particle hydrodynamics (SPH) solver from Beck et al. (2016). Magneticum initial conditions are generated using a standard Λ CDM cosmology with Wilkinson Microwave Anisotropy Probe 7 (Komatsu et al. 2011) cosmological parameters. The large-scale structure evolution in Magneticum simulations includes a treatment of radiative cooling, heating from a uniform redshift-dependent ultraviolet (UV) background, star formation, and stellar feedback processes as in Springel et al. (2005a). The stellar feedback is then connected to a detailed chemical evolution and enrichment model as in Tornatore et al. (2007), which follows 11 chemical elements (H, He, C, N, O, Ne, Mg, Si, S, Ca, and Fe, with cooling tables from Wiersma et al. 2009) which are produced with the CLOUDY photo-ionisation code (Ferland et al. 1998). Fabjan et al. (2010) and Hirschmann et al. (2014) described prescriptions for black hole growth and feedback from AGNs. Haloes that host galaxy clusters and groups are identified using the friends-of-friends halo finder (Davis et al. 1985), and subhaloes together with their associated galaxies are identified with an improved version of SUBFIND (Springel et al. 2001), which takes into account the presence of baryons (Dolag et al. 2009). We define r ∆ c as the radius that encloses an average density of ∆ c ρ cr , where ρ cr is the critical density of the Universe at a given redshift, Throughout this paper, when we omit ∆ c from masses and radii we imply the usage of ∆ c = 200 (i.e., M = M 200c) . To disentangle the scatter of the mass-observable relation from projection e ff ects, we compute quantities within a sphere of radius r 200c and as integrated into a cylinder. Projected quantities will be denoted with the superscript 2D (for instance, the total mass inside the cylinder is denoted as M 2D ). We opted to employ a random projection plane for each cluster. Additionally, we set an integration depth of 20 comoving h -1 Mpc, corresponding to approximately 23 Mpc at z = 0 . 24 and 15 Mpc at z = 0 . 9 (with h = 0 . 704). This cylinder depth is smaller than Euclid 's galaxy cluster photoz equivalent uncertainty (Euclid Collaboration: Desprez et al. 2020) and, while we exclude some uncorrelated projection e ff ects, they are known not to play an important role (Sunayama et al. 2020; Wu et al. 2022). Thus, we ensure that we do not overestimate any projection e ff ect that could be mitigated using photoz . Consequently, all projection e ff ects examined in this paper hold relevance for interpreting forthcoming Euclid -based catalogues. This study is based on the results from Box2b / hr (Hirschmann et al. 2014) Magneticum simulation, which covers a length of 900 comoving Mpc, with dark matter particle masses m DM = 9 . 8 × 10 8 M ⊙ , gas initial particle masses of m gas = 2 × 10 8 M ⊙ , and a gravitational softening of both gas and dark matter of ϵ = 3 . 75 comoving kpc. Euclid is expected to detect clusters with masses M > 10 14 M ⊙ up to a redshift of approximately z ≈ 2 (Sartoris et al. 2016; Euclid Collaboration: Adam et al. 2019), where the bulk of the cluster population which will be used for mass-calibration will lie below redshift z ≈ 1 . Furthermore, the number of haloes contained in this Magneticum simulation drops significantly beyond the same redshift value. Therefore, we decided to extract haloes at two representative redshift slices: at an intermediate redshift of approximately z ≈ 0 . 24, yielding 4300 objects, and at a higher redshift of approximately z ≈ 0 . 9, yielding 1300 objects. These extractions were performed using the web portal 2 introduced in Ragagnin et al. (2017). We focus most of the analyses on the qualitative e ff ect of projection e ff ects at our intermediate redshift slice because of the larger statistics of clusters to help us determine projection e ff ects. We stress that this mass threshold is high enough so that all of our galaxy clusters have at least 10 4 particles and, therefore, can be considered well resolved in terms of density profile fitting (Navarro et al. 1997).\",\n \"pages\": [\n 3,\n 4\n ]\n },\n {\n \"title\": \"2.1. Observable properties\",\n \"content\": \"We now discuss the properties that we compute for each cluster and report a summary in Table 1. We compute the total stellar masses M ⋆ and M 2D ⋆ as the sum of all stellar particles within the respective volumes. We compute the richness n with a cut of satellites of log 10 ( M ⋆/ M ⊙ ) > 10 . 65 , and the projected version n 2D that includes Euclid -like corrections for projection effects where similarly to Andreon et al. (2016). In particular, we compute the average projected richness between 3 . 5 and 8 Mpc radii from the cluster centre, divided the annulus in 8 slices of equal angles, excluded the two least dense and most dense slices and removed the average projected number density of the 4 remaining octants from the projected richness within r 200c; 3 We compute the X-ray luminosities LX , and L 2D X , in the [0 . 5 , 2] keV energy band computed using the APEC model (Smith et al. 2001), using SPH particle temperatures together with the XSPEC package 4 (Arnaud 1996), which considers the emission of a collisionally-ionized, chemically-enriched plasma implemented with metallicity values taken from the simulated particles 5 . We compute the temperature T and T 2D as weighted by the X-ray emissivity of gas particles. We compute the hot gas mass M g and M 3D g , computed as the sum of the mass of SPH particles with cold gas fraction greater than 0 . 1 and T > 3 × 10 4 K in order to filter out cold gas. Note that the projected gas mass is not to be confused with the one inferred from X-ray observations, as X-ray observational works typically perform a de-projection of the surface brightness ∝ n 2 e , which provides a gas-mass estimate that is closer to the spherical M g , with the addition of some possible alignment e ff ects coming from the central region of clusters. Moreover, observational works have the capability to mask possible bright substructures, thus minimising the presence of interlopers. Consequently, we can conceptualise the observed projected gas mass as an intermediate value between our M g and M 2D g . We estimate the integrated Comptony parameter produced by thermal Sunyaev-Zeldovich (SZ, Sunyaev & Zeldovich 1972). The Comptony parameter is defined as where T is the temperature, n e the number density of the electrons, k B the Boltzmann constant, σ T the Thomson cross-section, c the speed of light, and m e the electron rest mass. We compute the integrated Comptony parameter Y = R y d Ω , both within the volume of sphere of ( Y ) and a cylinder ( Y 2D ). We estimate the integral in Eq. (2) as where the sum runs over all SPH particles, mi is the i -th SPH particle mass, Ti its temperature and f e , i is its electron fraction, expressed as local electron number density normalised to the hydrogen number density, and m p is the proton mass. For each halo, we also perform fits of the NFW profile ρ NFW, defined as where the scaling density ρ 0 and the scale radius r s (that is the radius where the density log-slope equals -2) are free parameters. We perform this fit on the total matter density profile on 100 log-spaced radial bins between 75 ckpc (which corresponds to 60 kpc at z = 0 . 24 , and to 40 kpc at z = 0 . 9; as it is enough to exclude the deep central potential of baryons) and r 200c , and define the corresponding NFW masses and concentration parameters as M NFW and c NFW respectively, and the concentration as c NFW = r NFW / r s , where r NFW is obtained from M NFW via the Eq. (1). The projected version of the mass and concentrations are obtained by mimicking a lensing reconstruction procedure by fitting the corresponding reduced shear. We define the derived masses and concentrations as M 2D NFW and c 2D NFW , where c 2D NFW = R 2D NFW / r s , where R 2D NFW is obtained from M 2D NFW via the Eq. (1). Note that the \\\"2D\\\" here, as for the other quantities, means that the quantity is computed in projection, however, a correct fit of the mass from reduced shear NFW profile (namely, our M 2D NFW ) should provide an estimate of the same NFW halo mass M NFW that would be recovered from a 3D fit. The fit is computed within the cylinder described above, with a projected radial range of [300 , 3000] kpc at z = 0 . 2 . We performed the analyses at z = 0 . 9 by rescaling that range with H -2 / 3 ( z ) , where H ( z ) is the Hubble parameter, in order to retain the same fractional distances from the virial radius (at fixed mass), which resulted in a range of [234 , 2300] kpc . Note that in this work we are not interested in estimating the contribution of the uncorrelated large-scale structure in the reduced shear reconstruction, therefore we limit our density projection to a cylinder of the depth of 20 cMpc , (see, e.g., Euclid Collaboration: Giocoli et al. 2024; Becker & Kravtsov 2011). The signal from the source-averaged excess surface mass density ∆Σ gt , averaged over circular radii R and a population of sources distributed in redshift, can be written as Here Σ denotes the surface mass density. The symbol ⟨ ... ⟩ denotes an average over radial bins and redshift lens sources, where we used a redshift distribution as proposed in Euclid Collaboration: Ajani et al. (2023), and Euclid Collaboration: Giocoli et al. (2024). The quantity ⟨ ∆Σ t ⟩ is the excess of surface mass density, averaged over polar coordinates and defined as The symbol Σ cr in Eq. (5) is the critical surface mass density, that for a given redshift source equals to where G the universal gravity constant, D s the angular diameter distance to the source, D d the angular distance to the lens, and D ds the angular distance between the source and the lens. Similarly to Euclid Collaboration: Giocoli et al. (2024), we define the error associated with each radial bin of the profile in Eq. (5) as where σϵ = 0 . 3 (Hoekstra et al. 2012; Euclid Collaboration: Blanchard et al. 2020) is the dispersion of the total intrinsic ellipticity ϵ = (1 -q ) / (1 + q ) , where q is the axis ratio, R 1 and R 2 represent the inner and outer radius of a bin, and n g is the number density of galaxies. For our redshift source distribution (we assume the same as in Euclid Collaboration: Giocoli et al. 2024), we find that n g ≈ 28 arcmin -2 for lenses at redshift z = 0 . 24 and n g ≈ 14 arcmin -2 for lenses at redshift z = 0 . 9 .\",\n \"pages\": [\n 4,\n 5\n ]\n },\n {\n \"title\": \"2.2. Scaling relations\",\n \"content\": \"In Fig. 1 we show the observable properties vs. true mass M of clusters, derived from Magneticum Box2b / hr simulation for properties that could be derived by Euclid -like catalogues, such as lensing concentration (first row from top), lensing mass (second row), projected richness (third row), and projected stellar mass (last row), as presented in Sect. 2.1. For each property, we fit a scaling relation performed using a linear regression in the log-log space. We utilise a log-log linear regression because a single power law proves to be e ff ective in modelling our scaling relations. In the right panel of Fig. 1, we show the log-residual distribution for both low-mass haloes ( M < 2 × 10 14 M ⊙ ), highmass haloes ( M > 2 × 10 14 M ⊙ ), and for the complete sample of the log-residuals σ ln , i , defined as the logarithmic ratio between the i -th cluster property and the corresponding scaling relation value at its mass. In the second column, we also report the logscatter σ ln defined here as the corresponding standard deviation of the log-residual, namely where E is the expectation operator that averages over our catalogue data. We note that the concentration has a scatter of 0 . 45 which is higher than theoretical expectations (see, e.g., Child et al. 2018). Throughout this paper, we will show that this is due to projection e ff ects; in fact, the 3D concentration has a scatter of ≈ 0 . 33 . Note that our scatter in temperature exceeds that reported in the theoretical work by Truong et al. (2018). We verified that, if we compute mass-weighted temperature, which is known to behave very well in a power law scaling relation, reveals a log scatter of 0 . 07 , in agreement with their work. The additional scatter that we see may be due to di ff erent X-ray temperature computa- tions (they use core-excised temperature while we take the contribution of the core into account). In Fig. 2 we show the mass-observable relations of quantities that could potentially be obtained from various multi-wavelength observations that could enrich studies based on Euclid -like data products: the integrated Comptony parameter, the gas mass M g , 500c, the X-ray luminosity L 2D X , 500c converted in the soft band [0 . 5 , 2] keV , and the temperature T 500c . Wedecided to plot the Xray luminosity, gas mass, and temperature within r 500c because this radius is typically used in various X-ray observations (see, e.g., Vikhlinin et al. 2006; Sun et al. 2009). The typical Euclid cluster cosmology analysis will therefore rely on mass-observable relations calibrated within r 200 c (e.g., richness, weak-lensing mass), and follow-up observations calibrated within r 500 c (e.g., X-ray and SZ mass-proxies). We will thus need to take into account the covariance between observable properties extracted at di ff erent radii. . Finally, we note that generally, X-ray observations are expected to align more closely with the 3D mass rather than the projected one (Ettori et al. 2013), although several details adopted to analyse the Xray observations (e.g., including masking of substructures, deprojection procedures, etc.) can significantly impact the final result. These choices critically depend on the quality of the observations themselves. Dedicated mocks will thus be required to properly take into account all these e ff ects and are, therefore, beyond the purpose of this work. For simplicity, in this work, we consider an X-ray-derived gas mass as close to M g , while an SZ-derived gas mass closer to M 2D g .\",\n \"pages\": [\n 5,\n 6\n ]\n },\n {\n \"title\": \"3. Projection effects\",\n \"content\": \"The main objective of this work is to disentangle the amount of scatter and skewness in scaling relations that is purely due to projection e ff ects. Note that observational data are also a ff ected by measurement errors that we do not tackle in this work (as, for instance, the Poisson error of the limited number of galaxies used to infer the richness). In this Section, we discuss the scatter and skewness of mass-observable relation qualitatively and limit the discussion for the data at redshift z = 0 . 24 , as we have a larger sample of galaxy clusters, and the results are qualitatively similar to the ones at z = 0 . 9. We stress that we leave the quantitative discussion of the scatter and correlation coe ffi cients for both redshifts on Sect. 5. To assess the role of projection e ff ects, Fig. 3 reports the 3D and 2D scatter of our cluster properties. In the left panel of Fig. 3 we show the value of the scatter (see Eq. 9) of log-residuals σ ln for all our mass-observable relations. In the shaded region, we also report the values computed within r 500c for the X-ray luminosity, gas mass, and temperature values because this is the characteristic overdensity used in X-ray analyses. For each observable, in Fig. 3, we report (with di ff erent symbols) both the scatter of the complete sample as well as the one of two separate mass range of 10 14 < M < 2 × 10 14 M ⊙ and M > 2 × 10 14 M ⊙ respectively. We note that the lower-mass bin ( M < 2 × 10 14 M ⊙ ) is the one with the largest scatter because, for a given external object in the line-of-sight (LoS), the profile of a small cluster will be more perturbed with respect to a cluster. In the left panel of Fig. 3, we see that some quantities have a low scatter in the 3D space and do gain a large amount of scatter once they are seen in projection. To better quantify what is the actual impact of projection e ff ects, in the right panel of Fig. 3 we present the metric q σ 2 ln , 2D -σ 2 ln , 3D . This metric shows that the quantities that are most a ff ected by projection e ff ects are the weak lensing concentration, integrated Comptony parameter, gas mass, and NFW profile lensing mass. This is expected as all these observable properties (except for weak lensing concentration and X-ray luminosity) scale linearly with the respective observable mass. Further, we note that the scatter in the richness agrees to the theoretical predictions from Castignani & Benoist (2016). We observe that X-ray luminosity and temperature are the least a ff ected by projection e ff ects. This is attributed to the fact that X-ray luminosity is contingent upon the square of gas density, thereby being primarily influenced by the most bright regions within an image. Similarly, the temperature is predominantly influenced by the innermost regions of a cluster. Note that we lack the value of projection e ff ects for L X , 500c because the 2D scatter is slightly smaller than the 3D one. This happened because projection e ff ects impact under-luminous haloes more strongly than overly luminous haloes (at a fixed mass bin), with the consequence of the projected X-ray luminosity having a higher normalisation and a lower scatter (see Fig. B.1). Some mass-observable relations have a large skewness, to aid observational works in modelling these relations, we will estimate their skewness. Therefore, we also quantify deviations of residuals from a symmetrical distribution by means of the Fisher-Pearson coe ffi cient of skewness m 3 / m 3 / 2 2 , where mk is the sample k th central moment. 6 We report its dependency on projection e ff ects in Fig. 4, showing that the properties whose skewness is most impacted by projection e ff ects are the integrated Comptony parameter, the gas mass, and the lensing NFW mass. We also notice that some scaling relation residuals move from having a negative skewness (for the NFW concentration, for instance, due to un-relaxed and merging clusters) to a positive one once projection e ff ects are taken into account (namely, from an asymmetry towards negative residuals towards an asymmetry towards positive residuals). To assess the impact of projection e ff ects, we introduce a variable to quantify the amount of additional matter in the line of sight that can skew our observable properties. We define it as the ratio of the mass within the cylinder and the mass within a sphere M 2D / M , both of radius r 200c , where the length of the cylinder is described in Sect. 2. We present the distribution of M 2D / M in Fig. 5, where we can see that this quantity is strongly skewed and its median value is M 2D / M ≈ 1 . 26 (note that for a NFW profile with c = 4 , the corresponding analytical cylinder vs. spherical mass is 1 . 25). Although this quantity is not directly observable, we will use it to assess the contribution of LoS objects in the scatter of scaling relations. Note that besides objects in the LoS, di ff erent fitting procedures may impact the scatter of projection e ff ects, as we will see in Sect. 4. 2D In Fig. 6, we show a random selection of clusters, ordered by decreasing M 2D / M from left to right. Objects with high M 2D / M (the objects in the left-most panels) include clusters that are merging, elongated, or in the LoS. In the rest of this paper, we will refer to the objects having M 2D / M greater than the median of the distribution 1 . 26 as having LoS excess. To study the impact of projection e ff ects in scaling relations, in Fig. 7 we show the scaling relations of the following projected quantities: richness, integrated Comptony parameter, lensing mass, and concentrations, that are the ones that are most a ff ected by projection e ff ects. We colour-code these points by M 2D / M and focus on a narrow mass range of M ∈ [1 , 2] × 10 14 M ⊙ in order to visualise better how LoS excess impacts these scaling relations. On the left column, we can visually see that, except for concentration, they strongly correlate with M 2D / M , as the upper points of the scatter plot tend to have higher values of M 2D / M . We quantify this finding in the right column by comparing the residual distributions (of the power-law fit over the complete mass range presented in Sect. 2.2) of the complete sample with the distribution of objects with low LoS contamination only (we adopt the criteria of M 2D / M < 1 . 26 being the median of the M 2D / M distribution, as shown in Fig. 5), and report the respective scatter σ and average value µ of the residual distributions (for the complete sample we have that µ = 0). Except for the concentration, residuals of haloes with low M 2D / M (see dashed histogram) significantly shift towards negative values of µ and σ. For instance, when we consider only objects with a low LoS excess, the scatter of Y 2D decreases from 0 . 35 down to 0 . 11 , and the M 2D NFW scatter goes from 0 . 23 to 0 . 9. The lensing mass is, in fact, well known to be a ff ected by projection e ff ects (Meneghetti et al. 2014; Euclid Collaboration: Giocoli et al. 2024). In this paper, we refer to projection e ff ects as all e ff ects that take place when going from 3D to projection; they include both LoS e ff ects and model uncertainties. This definition becomes relevant when dealing with concentration, which is not impacted only by LoS objects but also by the NFW profile fitting procedure. We stress that in Fig. 3 we proved that our projected concentration is actually highly impacted by projection e ff ects, yet only weakly a ff ected by LoS e ff ects. We show in the next Section the reason for concentration being strongly a ff ected by projection e ff ects is that their reduced shear profile deviates strongly from the one produced by NFW profile (see, e.g., Ragagnin et al. 2021), which is used to reconstruct the reduced shear profile. To conclude this section, we now study how correlations between cluster observable properties can be a ff ected by projection e ff ects. To this end, we take the case of a possible hot- vs. coldbaryon correlation by studying the stellar mass vs. integrated Comptony parameter (as the latter should strongly correlate with the gas mass). In Fig. 8, we show the integrated Comptony parameter scaling relation, colour-coded by stellar mass for the 3D quantities (top panel) and projected quantities (bottom panel). Examining the correlations at a constant halo mass among the computed quantities within spheres (as depicted in the top panel of Fig. 8), we find no discernible weak anti-correlation between stellar mass and the integrated Comptony parameter (which is defined in Sect. 5). Conversely, when investigating the properties in the projected space, a more pronounced correlation becomes evident. This implies that projection e ff ects can strongly impact the correlation between observable properties. While this analysis is purely qualitative, we will quantify the impact of these projection e ff ects in Sect. 5, where we will compute the correlation coe ffi cients for both 3D quantities and 2D quantities.\",\n \"pages\": [\n 6,\n 7,\n 8,\n 9\n ]\n },\n {\n \"title\": \"4. Projection effects on lensing concentration\",\n \"content\": \"As we found in the previous section, projection e ff ects significantly increase the scatter and skewness in the scaling of lensing concentration with mass. However, this scatter increase is not related to external objects along the LoS. Now, we assess if the high scatter of lensing concentration is due to deviations of the reduced shear profile from the one induced by an NFW profile. In this work, we will not delve into the origin of this deviation as it falls beyond the scope of this paper. Such deviation may arise due to halo elongations, suggesting that alternative profiles such as truncated NFW profiles may better suit galaxy clusters (Oguri & Hamana 2011). Alternatively, it could stem from the expectation that the NFW profile is intended to describe stacked haloes rather than individual objects. Our focus in this paper is to understand the impact of assuming an NFW profile for each of our haloes. We emphasize that these NFW deviations only a ff ect weak lensing signal reconstruction, as the NFW profile is highly e ff ective in recovering halo mass in 3D. To study deviations from the NFW profile of haloes we fit a generalised NFW profile (Nagai et al. 2007), hereafter gNFW, in spherical coordinates over the same radial range as our previous NFW profile (described in Sect. 2.1), where the density profile ρ gNFW ( r ) is defined as where γ and β are respectively the internal and external logslopes of the total matter density profiles. The case γ = 1 and β = 3 produces the NFW profile as in Eq. (4). Note that the Nagai et al. (2007) gNFW profile also depends on the parameter α that we fix to α = 1 in this work in order to explore internal and external log-slope variations only. We present the PDF for the gNFW profile parameters γ and β in Fig. 9, where the fit was performed in 3D with a flat priors for γ ∈ [0 , 3] and β ∈ [0 , 6]. The data points are colour-coded according to the variable M , revealing no discernible strong trend with respect to the fitted parameters. For 19% of the objects, the resulting best-fit parameters hit the boundaries of hard-cut priors. Upon visual inspection, these objects are characterised by a very steep matter density profile at large cluster-centric distances, possibly suggesting that a truncated NFW profile might be a better model choice. As our objective is to examine the effects of deviations from the generalised NFW profile, we omit these objects from the subsequent analysis in this Section. Given the substantial deviation of these objects from NFW profile, they could potentially o ff er additional insights for our analyses. However, incorporating them would necessitate the use of a profile more general than Eq. 10. Therefore, we excluded them in order to make our analysis clearer. We observe that the external logslope of Magneticum profiles appears to be slightly flatter than -3 . While we emphasize that this discrepancy does not a ff ect the accurate recovery of mass and concentration parameters in 3D NFW fits (such fits can still yield precise estimates of halo mass and concentrations). However, these deviations in the NFW profiles may a ff ect the reduced shear fit, particularly when observed over large radii (remember that in this work, we use 3 Mpc). Furthermore, we observe a degeneracy between the β and γ parameters, indicating that our profiles deviating from NFW profile tend to exhibit a flatter profile compared to NFW profile (as illustrated in Fig. A.1). However, investigating this discrepancy is beyond the scope of this paper, as the internal log slope of clusters is not currently captured by existing weak lensing studies. In Fig. 10, we plot the values of concentration and mass obtained from reduced shear fit, divided by the corresponding 3D quantities and colour-coded by the external 3D gNFW slope β for our intermediate redshift haloes. As we can see, haloes with large values of β have a projected concentration that is significantly higher than the 3D one (see upper panel). In Appendix A we report the example of a simulated halo with low LoS excess (see Fig. A.1) and an analytical one (see Fig. A.2), both with a flat external log-slope, and we show how the under-estimation of the concentration is caused by the fact that the NFW profile fit on the reduced shear is weighting too much the external part of the profile, that deviates from an NFW profile. In Fig. 11, we show the concentration residual distribution and report their scatter. We note that the projected concentration scatter is not a ff ected by external material along the LoS (dashed line and shaded histograms match). However, if one restricts our sample to objects having NFW-like profile log-slopes (we used criteria of 2 . 8 < β < 3 . 2 and 0 . 8 < γ < 1 . 2), then the scatter distribution changes drastically. The concentration residuals decrease from 0 . 43 to 0 . 38, and the residuals shift towards higher values, suggesting that these objects are more relaxed. Such e ff ect is well known, as studied for instance in Macciò et al. (2007). We also show how the external log-slope of the halo profile a ff ects the lensing reconstruction by plotting the ratio between the projected and 3D concentration (i.e., c 2D NFW / c NFW) value versus the 3D log-slope β in the narrow mass bin of M ∈ [1 , 2] × 10 14 M ⊙ in Fig. 12, where we find a positive correlation coe ffi cient of ≈ 0 . 28 , in agreement with a shift of residuals we showed in Fig. 11.\",\n \"pages\": [\n 9,\n 10,\n 11\n ]\n },\n {\n \"title\": \"5. Correlations between properties\",\n \"content\": \"While in the last sections, we investigated the origin of the impact of projection e ff ects in the scatter and skewness of observable properties, we will now quantify how projection e ff ects impact the correlation between observable properties. To this end, we quantify the Pearson correlation coe ffi cients between their log-residuals (as defined in Sect. 2.2). We adopt the standard error associated with the Pearson coe ffi cient ρ as derived from two normal distributions, given by σρ = p 1 -ρ 2 / √ N -2 (see Eq. 12-93 in Pugh & Winslow 1966), where N represents the number of objects. This corresponds to a maximum error of 0 . 015 (for ρ = 0) for the sample size at z = 0 . 24 and a maximum error of 0 . 028 for the sample size at z = 0 . 90. It is worth noting that in the correlation coe ffi cient matrices generated in subsequent analyses, we will only colour values with correlation coe ffi cients | ρ | > 0 . 3, aiming to highlight strongly correlating properties. We define mild correlation as 0 . 2 < | ρ | < 0 . 3, as we choose to exercise caution. Correlation coe ffi cients with | ρ | < 0 . 1 are disregarded.\",\n \"pages\": [\n 11\n ]\n },\n {\n \"title\": \"5.1. Analysis at z = 0 . 24\",\n \"content\": \"In this Section we focus on the haloes at intermediate redshift z = 0 . 24 . In Fig. 13 we show the correlation coe ffi cient matrix between log-residuals at fixed halo mass of our projected observable both from Euclid -like data (lensing concentration, lensing mass, richness, and stellar mass, respectively) and possible outcomes from multi-wavelength observations (integrated Comptony parameter, gas mass, X-ray luminosity, and temperature) for intermediate-redshift objects. In the lower triangle, we present scatter plots alongside the slope derived from the correlation coe ffi cient. This visualization allows for the identification of instances where the correlation coe ffi cient slope accurately captures the trend of the residuals. Typically, this alignment occurs for quantities that exhibit strong correlation coe ffi cients. For example, our data points show a robust correlation between certain hot baryon tracers ( M 2D g and Y 2D ), some cold baryon components ( n 2D and M 2D ⋆ ), and weak lensing mass M 2D NFW . The underlying reason for these strong correlations lies in projection e ff ects: the greater the amount of matter along the line of sight, the higher the observed values. We will further demonstrate this in the subsequent section by presenting the correlation coe ffi cient matrix for 3D quantities, where many of these correlations diminish. This can be anticipated by observing that L 2D x and c 2D NFW do not exhibit this positive trend of correlations. Notably, we observe that the correlation between richness and stellar mass ( ρ = 0 . 48) is not exceptionally high. Moreover, the stellar mass appears to be more influenced by projection e ff ects compared to richness (evident in their correlations with M 2D NFW , where they exhibit ρ = 0 . 69 and ρ = 0 . 42, respectively). We can speculate on two potential causes: firstly, unlike stellar mass, our richness computation incorporates some observationally-motivated background subtraction; alternatively, since stellar mass encompasses all stellar particles (while richness involves a luminosity-motivated galaxy stellar-mass cut), it is plausible that small subhaloes are influencing the projected stellar mass. We note that concentration anti-correlates with gas-mass (and integrated Comptony parameter). This is in agreement with recent analyses of simulations. In fact, richness at fixed mass anti-correlates with concentration (Bose et al. 2019); low concentration is an index of the system being perturbed (Ludlow et al. 2012); and un-relaxed systems tend to be gas-rich (Davies et al. 2020). We refer to Ragagnin et al. (2022) for a more comprehensive study on low luminous groups. Moreover, at fixed halo mass, the lensing mass correlates strongly with total projected stellar mass ( ρ = 0 . 69) and projected gas mass ( ρ = 0 . 59), which may be due to the fact that both correlate strongly with LoS contamination. The same holds for the correlation among richness, gas mass, and stellar mass. This is due to projection e ff ects, where LoS excess amplifies all these quantities, as discussed in Section 2.2. We note that the 2D lensing mass and projected X-ray luminosity have a slight positive ( ρ = 0 . 20) cor- ation, in agreement with the observational work of Sereno et al. (2020). In Fig. 14 we show the covariance matrix of non-projected quantities for intermediate redshift objects. We see that as opposed to Fig. 13, the 3D covariance matrix shows a mild yet negative covariance between gas mass and stellar mass ( ρ = -0 . 24), and a positive correlation between richness and gas mass ( ρ = 0 . 23) because most of their correlations in the projection are due to Line of sights excess, which significantly increases the values of the gas mass, the richness, and the stellar mass. In Fig. 15 we report the correlation matrix as in Fig. 13 where we present X-ray luminosity, gas mass, and temperature, as computed within r 500c , which shows an anti-correlation between the gas mass and the concentration residuals ( ρ = -0 . 14) that is significantly lower than the one found in Fig. 13 and Fig. 14 ( ρ equals to -0 . 26 and -0 . 34 respectively). One possibility is that this change in sign of the correlation is caused by the fact that mixing overdensities (concentration is within ∆ c = 200 and gasmass is within ∆ c = 500) does introduce an additional correlation with the sparsity (Balmès et al. 2014; Corasaniti et al. 2022) that itself correlates with the concentration (see Appendix B). For completeness, we report the correlation coe ffi cient matrix and the scatter of log-residuals of all quantities in Table B.1. There we also added the core-excised projected X-ray luminosity L 2D X , ce500c , as it is typically used in X-ray-based observational studies, where we can see that the scatter and most of the correlation coe ffi cients are smaller than L 2D ce500c , while the correlations with the concentration and gas mass increase. Note that we do not report the 3D NFW mass ( M NFW) because it has an extremely low intrinsic scatter σ ln ( M NFW) ≈ 0 . 01 and its correlation coefficients are not meaningful.\",\n \"pages\": [\n 11,\n 12\n ]\n },\n {\n \"title\": \"5.2. Analysis at z = 0 . 9\",\n \"content\": \"In this Section, we focus on observational property covariance matrixes of our haloes z = 0 . 9. At this redshift, we computed projected quantities within a cylinder depth of 35 Mpc in order to retain the same relative ratio as the photoz uncertainty of the low-redshift analysis (it scales with 1 + z ). For what concerns the cylinder used to integrate ∆Σ gt, we rescaled so as to keep it constant in comoving units with the low-redshift analyses. We rescaled the 3D NFW profile minimum radius to 40 kpc while we kept the maximum radius at r 200c . For what concerns the radial range of the lensing fit, we rescaled it with H -2 / 3 ( z ), therefore performing it in the range of [234 , 2300] kpc . We stress that we do not model observational uncertainty. Therefore, the decrease in background source count with redshift does not impact our best fits. However, it still impacts the fact that we weigh external radial bins more than internal ones. We report the values of the scatter and the projection contribution at z = 0 . 9 in Fig. 16, while we report the log-residuals and the skewness for each property in Fig. 17. In particular, the quantities most a ff ected by projection effects are the lensing mass and concentration, whereas the temperature is the lowest. These results are qualitatively similar to the low redshift analyses, with the ComptonY parameter and gas mass being slightly less a ff ected by projection e ff ects. Note that since the virial radius is smaller at higher redshift values, our radial range of the reduced shear is closer to the NFW scale radius; therefore, the weak lensing reconstruction is more e ff ective in capturing the scale radius and more sensitive to deviations from an NFW profile. As a consequence, we found that the in- crease of scatter going from c NFW to c 2D NFW compared to the low redshift analyses. We report the correlation coe ffi cient matrix and the scatter log-residuals of the quantities at z = 0 . 9 in Table B.2. As for the case at z = 0 . 24 , note that we do not report the 3D NFW mass ( M NFW) because it has an extremely small intrinsic scatter of 0 . 01 , and thus its correlation coe ffi cients have no impact in our study.\",\n \"pages\": [\n 12\n ]\n },\n {\n \"title\": \"6. Conclusions\",\n \"content\": \"In this work, we analysed a number of galaxy clusters from Magneticum hydrodynamic simulation Box2b / hr. We did so in a mass range, tailored for Euclid -like data products (see Sartoris et al. 2016; Euclid Collaboration: Adam et al. 2019), namely with a mass of M 200c > 10 14 M ⊙ . To this end, we computed properties that could come from Euclid catalogues of galaxy clusters, such as richness, stellar mass, and lensing masses and concentration, and possible properties coming from multi-wavelength studies such as X-ray luminosity, integrated Comptony parameter, gas mass, and temperature. All these properties were computed both within a sphere and within a cylinder (both with radius r 200c) to account for projection e ff ects. Our study considers the remarkable capabilities of Euclid photoz measurements in identifying interlopers. However, their importance decreases significantly at scales as small as a few tens Mpc. This depth is still long enough to contain multiple haloes along the LoS. Hence, we studied the projection e ff ects on a scale that is significantly smaller than the Euclid photoz uncertainty. We then studied how the scatter and skewness change when one measures quantities in 3D space or in projection. Below, we summarise our findings: The analysis presented here has been carried out using a single suite of hydrodynamic simulations. Regarding weak lensing masses and concentration, since in this work, we did not consider the profile noise due to the finite number of background galaxies, future studies are needed to improve our estimations. Some works show that both scatter and correlation coefficients vary between cosmological simulations with di ff erent cosmologies (Ragagnin et al. 2023), the presence of feedback schemes (Stanek et al. 2010), and di ff erent cosmological simulation suite in the market (see Fig. 7 in Anbajagane et al. 2020). So, while simulations can provide directions on how to model correlation coe ffi cients, it is possible that when using this kind of data, one needs to allow for variation due to the di ff erent baryon physics. Furthermore, when striving for even more precise results, it is important to acknowledge that mass-observable relations are not exact power laws. Therefore, employing more generic fitting techniques, such as a running median, could yield improvements. Additionally, there is room for enhancement in how we compute correlation coe ffi cients in future studies. One potential approach could involve simultaneously fitting both the massobservable relation scatter and the correlation coe ffi cients by maximizing multivariate likelihoods. We anticipate that future studies combining Euclid data with multi-wavelength observations may encounter challenges in shedding light on currently puzzling residual correlations, primarily dominated by projection e ff ects. Acknowledgements. We thanks the anonymous referee for the useful comments. The Magneticum Pathfinder simulations were partially performed at the LeibnizRechenzentrum with CPU time assigned to the Project 'pr86re'. AR and LM acknowledge support from the grant PRIN-MIUR 2017 WSCC32 and acknowledges the usage of the INAF-OATs IT framework (Ta ff oni et al. 2020; Bertocco et al. 2020), and the space filling curve improvement on Gadget3 (Ragagnin et al. 2016). Antonio Ragagnin thanks Veronica Bi ffi and Elena Rasia for the X-ray computation routines and tables. KD acknowledges support by the COMPLEX project from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program grant agreement ERC-2019-AdG 882679. LM acknowledges the financial contribution from the grant PRIN-MUR 2022 20227RNLY3 'The concordance cosmological model: stress-tests with galaxy clusters' supported by Next Generation EU. CG and LM acknowledge support from the grant ASI n.2018-23-HH.0. AR and CG acknowledge funding from INAF theory Grant 2022: Illuminating Dark Matter using Weak Lensing by Cluster Satellites, PI: Carlo Giocoli. SB acknowledges partial financial support from the INFN InDark grant. AMCLB was supported by a fellowship of PSL University hosted by the Paris Observatory. We used the package colossus (see Diemer 2018) for computing Σ and ∆Σ as expected from NFW profiles. AR and FC acknowledge co-funding by the European Union - NextGenerationEU within PRIN 2022 project n.20229YBSAN - Globular clusters in cosmological simulations and in lensed fields: from their birth to the present epoch. The Euclid Consortium acknowledges the European Space Agency and a number of agencies and institutes that have supported the development of Euclid , in particular the Academy of Finland, the Agenzia Spaziale Italiana, the Belgian Science Policy, the Canadian Euclid Consortium, the French Centre National d'Etudes Spatiales, the Deutsches Zentrum für Luft- und Raumfahrt, the Danish Space Research Institute, the Fundação para a Ciência e a Tecnologia, the Ministerio de Ciencia e Innovación, the National Aeronautics and Space Administration, the National Astronomical Observatory of Japan, the Netherlandse Onderzoekschool Voor Astronomie, the Norwegian Space Agency, the Romanian Space Agency, the State Secretariat for Education, Research and Innovation (SERI) at the Swiss Space O ffi ce (SSO), and the United Kingdom Space Agency. A complete and detailed list is available on the Euclid web site ( http://www.euclid-ec.org ).\",\n \"pages\": [\n 12,\n 13\n ]\n },\n {\n \"title\": \"Data Availability\",\n \"content\": \"Raw simulation data were generated at C 2 PAP / LRZ cosmology simulation web portal https://c2papcosmosim.uc. lrz.de/ . Derived data supporting the findings of this study are available from the corresponding author AR on request.\",\n \"pages\": [\n 13\n ]\n },\n {\n \"title\": \"References\",\n \"content\": \"Kravtsov, A. V. & Borgani, S. 2012, ARA&A, 50, 353 Ragagnin, A., Andreon, S., & Puddu, E. 2022, A&A, 666, A22 Ragagnin, A., Fumagalli, A., Castro, T., et al. 2023, A&A, 675, A77 Sereno, M., Umetsu, K., Ettori, S., et al. 2020, MNRAS, 492, 4528 Sugiyama, S., Miyatake, H., More, S., et al. 2023, Phys. Rev. D, 108, 123521 Sun, M., Voit, G. M., Donahue, M., et al. 2009, ApJ, 693, 1142 Teklu, A. F., Remus, R.-S., Dolag, K., et al. 2015, ApJ, 812, 29 Tornatore, L., Borgani, S., Dolag, K., & Matteucci, F. 2007, MNRAS, 382, 1050 Truong, N., Rasia, E., Mazzotta, P., et al. 2018, MNRAS, 474, 4089 van Uitert, E., Cacciato, M., Hoekstra, H., et al. 2016, MNRAS, 459, 3251 Article number, page 14 of 26 18 Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France 23 Zentrum für Astronomie, Universität Heidelberg, Philosophenweg 12, 69120 Heidelberg, Germany\",\n \"pages\": [\n 13,\n 14\n ]\n },\n {\n \"title\": \"Appendix A: Fit of gNFW\",\n \"content\": \"In Fig. A.1, we present the density profile of a halo that deviates from NFW profile and has no LoS contamination. In particular, it has β ≈ 1 . 8 and γ = 1 . 5 . We show its NFW profile fit profile on the 3D density in the upper panel of Fig. A.1, where we can see that 3D NFW profile (performed on radial bins in a sphere) is capable of capturing the shape of the halo and to estimate its mass with high accuracy (within ≈ 5%). In the central panel, we show the reduced shear profile and best fits, where we can see that the fit performed on the reduced shear underestimated the concentration and is not able to capture the more internal part of the shear profile, as it is done by the profile that was fit in 3D. We first exclude this mismatch as being due to projection effects by showing that both the reduced shear from the particle data (orange line) matches the one recovered by performing an analytical projection of the 3D profile (blue solid line). In particular we project the density profile ρ ( r ) and derive the surface mass density Σ conv ., as follows What we find is that the shear obtained with the aid of an analytical projection Σ conv . matches very well the real one (i.e., orange and blue lines do match). This hints that for this cluster there are no strong LoS e ff ects. To understand why the fit on the shear is not able to capture the concentration of the original halo, we zoom our fit in the bottom panel of Fig. A.1, where it looks like the fit is very good in capturing the final part of the profile and not able to capture the internal. It is crucial to emphasise that we did not include observational uncertainties in these analyses. Therefore, the uncertainty outlined in Eq. (8) a ff ects the fit by assigning more weight to external radial bins compared to internal ones. It is worth noting that the proportionality factors in Eq. (8) will not a ff ect our best fit. To validate this point in Fig. A.2 we study the bias on fitting an NFW profile on a mock gNFW profile that has β = 1 . 8 and γ = 1 . 5 , a mass of 3 × 10 14 M ⊙ and a concentration c = 2 . 4 (as the halo presented in Fig. A.1). We see that the 3D NFW profile is capable of estimating both its mass and its gNFW concentration with high accuracy (see top panel match between blue and dashed black lines). On the other hand the fit of the shear (we report in the bottom panel of Fig. A.2) has the same problems as the one on the cluster in Fig. A.1: it recovers a low concentration (with a value of 1 . 5). This may be because, at outer radii, the model fits the data. It is possible that the under-estimation of concentration at low radii is caused by the combination of two factors: the fit under-estimates the shear at lower radii (with the result of under-estimating the lensing concentration), or the fact that γ is di ff erent than 3 induces an NFW profile fit with a low concentration. We then performed the experiment of fitting the analytical profile with constant (yet unrealistic) error bars. While the fit was able to capture the shape of the profile, it recovered a concentration of 1 . 6 , implying that there is indeed a degeneracy between the shear of low-concentrated NFW profiles and steeper-NFW profiles.\",\n \"pages\": [\n 22\n ]\n },\n {\n \"title\": \"Appendix B: Correlations with different overdensities\",\n \"content\": \"In this appendix, we discuss the di ff erences between scaling relation scatters and covariance values at di ff erent overdensities. First of all, we tackle the fact that when we compute X-ray luminosity within r 500c (instead of r 200c), we find that the scatter of the scaling relation of the projected quantity is larger than the 3D one. To investigate this feature, we will focus on the bolometric X-ray luminosity. We report the 3D and projected bolometric X- ray luminosity in Fig. B.1 (top panel), where it is visually clear that the projected X-ray luminosity is (as expected) always larger than the 3D one. One can also notice that the increase in X-ray luminosity depends on the fact that a halo is over-luminous or not: the increase of luminosity growing from the 3D to 2D is larger for under-luminous haloes than for over-luminous haloes. We prove this point in the bottom panel of Fig. B.1 where we show the ratio between the 2D and 3D luminosity as a function of their residual of the 3D scaling relation (the higher the value of the x axis, the more over-luminous is the object for its mass bin), where we can see a strong anti-correlation: overly luminous objects (for a given mass bin) are not going to be a ff ected much by the fact that their luminosity is computed in 3D or 2D. The possible cause is that an interloper in the LoS will not a ff ect much an overly luminous object. For completeness, in Fig. B.2 we show the correlation coefficients between the gas mass and stellar mass computed within both r 500c and r 200c and the concentration. Here we can see a change of sign between M ⋆, 500cMg , 500c correlations and M ⋆, 500cM g correlations and a change in the sign between c NFWM g correlations and c NFWMg , 500c correlations. Finally, in Fig. B.3 we report the scatter of observable properties at fixed mass for both Euclid -like quantities (lensing mass, richness, and projected stellar mass), and possible multiwavelength properties (integrated Comptony parameter, X-ray luminosity, and temperature), where we compute X-ray lumi- nosity and temperature within r 500c as they are typically derived within this overdensity. The upper panel of Fig. B.3 shows the residuals of the log-log linear regression where we see that in terms of 2D scatter, the properties with the lowest scatter are the stellar mass and the temperature. The bottom panel shows the data points used to perform the fit (in black) where we used a visually-inspected cut on the halo mass values in order to ensure that mass values are complete for a given observable value. Notes. Diagonal terms report the scatter of the log-residuals of each quantity, namely σ ln of Eq. (9), while the o ff -diagonal terms report the correlation coe ffi cient between the log-residuals. We do not report values of the correlation coe ffi cient below 0 . 20 because they are not significant. We do not report the values for the 3D NFW mass M 200c because it has a very low scatter of log-residuals ( ≈ 0 . 01) and its correlation coe ffi cients are not meaningful. Note that in this table we also added the core-excised X-ray luminosity. Notes. Rows and columns are as in Table B.1.\",\n \"pages\": [\n 22,\n 23,\n 24\n ]\n }\n]"}}},{"rowIdx":4979,"cells":{"bibcode":{"kind":"string","value":"2024arXiv241200975R"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/2412.00975.pdf"},"content":{"kind":"string","value":"\nOn the Orbital Effects of Stellar Collisions in Galactic Nuclei: Tidal Disruption Events and Ejected Stars\n\n\n\n1 Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA), Northwestern University, 1800 Sherman Ave, Evanston, IL 60201, USA\n2 The Observatories of the Carnegie Institution for Science, Pasadena, CA 91101, USA\nABSTRACT\nDense stellar clusters surround the supermassive black holes (SMBH) in galactic nuclei. Interactions within the cluster can alter the stellar orbits, occasionally driving a star into the SMBH's tidal radius where it becomes ruptured. This proof-of-concept study examines the orbital effects of stellar collisions using a semianalytic model. Both low and high speed collisions occur in the SMBH's sphere of influence. Our model treats stars in low speed collisions as sticky spheres. For high-speed collisions, we develop a simple prescription based on the limiting case of a hyperbolic encounter. We test a range of collision treatments and cluster conditions. We find that collisions can place stars on nearly radial orbits. Depositing stars within the tidal radius, collisions may drive the disruption of stars with unusual masses and structures: depending on the nature of the collision, the star could be the product of a recent merger, or it could have lost its outer layers in a high speed impact, appearing as a stripped star. We also find that high speed collisions near the periapsis of an eccentric orbit can unbind stars from the SMBH. However, dissipation during these high-speed collisions can substantially reduce the number of unbound stars achieved in our simulations. We conclude that TDEs and ejected stars, even in the hypervelocity regime, are plausible outcomes of stellar collisions, though their frequency in a three-dimensional nuclear star cluster are uncertain. Future work will address the rates and properties of these events.\nKeywords: Stellar dynamics; Galactic center; Star clusters; Stellar mergers\n1. INTRODUCTION\nAsupermassive black hole resides at the center of most galaxies, where it is surrounded by a dense stellar cluster (e.g. Ferrarese & Ford 2005; Kormendy & Ho 2013; Schodel et al. 2003; Ghez et al. 2005, 2008; Gillessen et al. 2009, 2017; Neumayer et al. 2020). A tidal disruption event (TDE) occurs when a star from the cluster passes within a critical distance from the SMBH and becomes ruptured by tidal forces (e.g., Hills 1975; Rees 1988; Alexander 1999; Magorrian & Tremaine 1999; Wang & Merritt 2004; MacLeod et al. 2012). The SMBH then accretes the stellar material, producing an electromagnetic signature (e.g., Guillochon & Ramirez-Ruiz 2013). Spectra of these events encode valuable information about the mass, structure, and composition of the ruptured star (e.g., Kochanek 2016a,b; Yang et al. 2017; Mockler et al. 2022; Miller et al. 2023). Observations of\nCorresponding author: Sanaea C. Rose\nsanaea.rose@northwestern.edu\nTDEs represent a powerful way to probe the stellar populations in galactic nuclei and the processes that shape them.\nAmongst these physical processes are direct collisions. Collisions can occur within the sphere of influence of the SMBH due to the high densities and velocity dispersion (e.g., Freitag & Benz 2002; Dale et al. 2009; Dale & Davies 2006; Rubin & Loeb 2011; Balberg et al. 2013; Balberg 2024; Mastrobuono-Battisti et al. 2014; Rose & MacLeod 2024). These events may produce electromagnetic signatures, both from the collisions themselves and from interactions between liberated material and the SMBH (e.g., Rosswog et al. 2009; Lee et al. 2010; Balberg et al. 2013; Dessart et al. 2024; Ryu et al. 2024b,a; Brutman et al. 2024), though it is difficult to ignite a main-sequence star with compression (Guillochon et al. 2009). They can also shape the stellar population. Both low and high speed collisions can alter the mass of a star - the latter can destroy stars completely and the former can give rise to blue stragglers in dense stellar systems (e.g., Lai et al. 1993; Rauch 1999; Sills et al. 1997, 2001; Lombardi et al. 2002; MacLeod et al. 2013; Leigh et al.\n2016; Rose et al. 2023). Recently, Gibson et al. (2024) have shown that high speed collisions can also produce stripped stars similar to what might be seen through binary evolution.\nCollisions that produce stellar mergers or stripped stars may explain recent unexpected TDE observations. For example, detections of high nitrogen-tocarbon (N/C) ratios in TDEs point to the disruption of more stars that burn on the CNO cycle (as first proposed by Kochanek 2016a,b), and that are ≳ 1 -2 M ⊙ (Kochanek 2016a; Yang et al. 2017; Gallegos-Garcia et al. 2018; Mockler et al. 2023) than is predicted by the host galaxies' stellar populations (Mockler et al. 2023). One particular TDE has such an extremely high N/C abundance ratio that it is difficult to explain with single stellar evolution alone (Miller et al. 2023), but could be the result of the disruption of a stripped star that has lost its nitrogen-poor envelope (Mockler et al. 2024). Additionally, a recently discovered population of extremely bright nuclear transients has also been suggested to originate from the disruption of high mass stars (e.g. ≳ 10 M ⊙ , Subrayan et al. 2023; Hinkle et al. 2024). Because stars that end in disruptions are expected to be drawn approximately at random from the stellar mass function (with small adjustments for stellar type, MacLeod et al. 2012), this preference for higher mass stars may imply that the mass function in galactic nuclei is more top-heavy (and/or bottom-light) than in the rest of the galaxy (see e.g., Lu et al. 2013; Hosek et al. 2019).\nThe potential link between TDEs and collisions is intriguing: collisions can simultaneously affect both the properties of the stars and their orbits about the SMBH. MacLeod et al. (2012) first considered collisions, particularly destructive ones, in the context of the TDE rate. Changes to a star's trajectory from collisions have also been considered in general dynamical models of dense stellar systems (e.g., Sanders 1970; Rauch 1999; Freitag & Benz 2002; Kremer et al. 2020; Gonz'alez et al. 2021; Rodriguez et al. 2022). In this proof-of-concept study, we study the orbital effects of stellar collisions in galactic nuclei. We assess whether the collisions that produce unusual stars could plausibly deposit those same stars onto TDE-producing orbits. Our models leverage simple, intuitive treatments for collision outcomes and fitting formulae from previous studies (e.g. Lai et al. 1993; Rauch 1999). We test a range of initial conditions and nuclear star cluster properties. Our paper is organized as follows:\nIn Section 2, we discuss our general approach to modeling the Milky Way's nuclear star cluster. Section 3 describes the treatment of various physics in our code, with\nSection 3.2 in particular outlining our methodology for updating the stellar orbits post collision. These orbital changes represent the most major addition to the code as compared to previous iterations (Rose et al. 2022, 2023). Section 4 presents simulated results for direct collisions. Sections 4.1, 4.2, and 4.3 discuss the implications for TDEs, orbital properties of collision-affected stars, and unbound and hypervelocity stars. We then incorporate relaxation into our simulations in addition to stellar collisions and present results in Section 5. Lastly, we summarize the scope and findings of our study in Section 6.\n2. MODEL NUCLEAR STAR CLUSTER\nWe leverage semi-analytic models to study the effects of collisions on the nuclear star cluster. Our fiducial model uses the conditions and properties of the Milky Way's GN, whose proximity makes it the best studied galactic center. We follow a sample of stars embedded in a fixed, unevolving cluster. For simplicity, both the evolving sample and the surrounding cluster are composed of 1 M ⊙ stars. The cluster can be understood in two key properties, density and velocity dispersion, which govern the dynamical processes unfolding within it.\nThe stellar density sets the the frequency with which stars interact. We describe the stellar density as a function of distance from the SMBH using a power law:\nρ ( r · ) = ρ 0 ( r · r 0 ) -α , (1)\nwhere α is the slope and r · , distance from the SMBH. Based on observations of the cluster within the sphere of influence, this equation is normalized using ρ 0 = 1 . 35 × 10 6 M ⊙ / pc 3 at r 0 = 0 . 25 pc (Genzel et al. 2010). Our fiducial model uses a slope of 1 . 75, the expectation for a single-mass population (Bahcall & Wolf 1976), consistent with the fact that our simple model cluster has only solar mass stars. However, in order to capture the range of theoretical predictions and observational constraints on the stellar cusp (e.g., Bahcall & Wolf 1976; Gallego-Cano et al. 2018a; Linial & Sari 2022), we also test a few simulations with α = 1 . 25, shown in Appendix A. We assume that the slope of the stellar cusp is roughly constant, or varying slowly, over the timescales of interest in our simulations.\nThe velocity dispersion within the cluster also influences the frequency and nature of stellar interactions. It decreases with distance from the SMBH:\nσ ( r · ) = √ GM · r · (1 + α ) , (2)\nwhere α is the slope of the density profile and M · is the mass of the SMBH (Alexander 1999; Alexander & Pfuhl 2014). We take M · to be 4 × 10 6 M ⊙ , like the Milky Way's SMBH (e.g., Ghez et al. 2003). For a uniform mass cluster of 1 M ⊙ stars, the number density n is simply ρ ( r · ) 1 M ⊙ .\n3. SEMIANALYTIC MODEL\nWe follow a sample of 1 M ⊙ tracer stars embedded in our model cluster. We draw their orbital eccentricities from a thermal distribution. We select their semimajor axes so that they lie on a cusp with slope α , matching the background cluster. These stars are allowed to evolve under the influence of two main dynamical processes, direct collisions and two-body relaxation, using a model first developed by Rose et al. (2022, 2023).\n3.1. Collisions\nDirect collisions occur over a characteristic timescale t -1 coll = nσA , where A is the cross-section of interaction, n is the number density, and σ is the velocity dispersion. For an impact to occur, A is the physical crosssection enhanced by gravitational focusing. The collision timescale also depends weakly on the star's orbital eccentricity and can be written as:\nt -1 coll = πn ( a · ) σ ( a · ) × ( f 1 ( e · ) r 2 c + f 2 ( e · ) r c 2 G ( M ⊙ + M ) σ ( a · ) 2 ) . (3)\nwhere f 1 ( e · ) and f 2 ( e · ) are equations 20 and 21 from Rose et al. (2020), G is the gravitational constant, and a · is the star's semimajor axis. r c is the sum of the radii of the colliding stars, or 2 R ⊙ for a uniform population of solar mass stars. We plot this timescale in red in the upper panel of Figure 1. We consider a range of slopes for the stellar density profile, spanning α = 1 . 25 (dashed line) to α = 1 . 75 (solid line). The horizontal grey line shows the total simulation time, included to guide the eye. Where the collision timescale is less than the simulation time, within 0 . 1 pc of the SMBH, collisions become important to understanding the evolution of the cluster (e.g., Rose & MacLeod 2024).\nWe treat stellar collisions using a statistical approach. We begin by computing the probability that a star in our sample will experience a collision. Over a timestep ∆ t , this probability equals ∆ t/t coll . ∆ t is taken to be 10 6 years so that the probability ∆ t/t coll is always less than one; t coll ≳ 10 7 years for the parameter space we consider. The code then draws a random number between 0 and 1, which, if less than or equal to the collision probability ∆ t/t coll , means a collision has occurred. We repeat this prescription until the desired simulation\nruntime or the star's main-sequence lifetime has been reached, whichever is shorter.\nIf a collision has occurred over a given timestep, we update the mass and age of the star using a prescription detailed in Rose et al. (2023). A full discussion of our approach can be found in previous papers (Rose et al. 2023; Rose & MacLeod 2024). In brief, we combine fitting formulae from hydrodynamics simulations of stellar collisions (Rauch 1999; Lai et al. 1993) with heuristic arguments to (1) determine if a given collision will result in a merger and (2) determine the amount of mass lost from either the merger product or the unbound individual stars. Since speeds in galactic nuclei often exceed hundreds of kilometers per second, collisions can eject anywhere from a percent to all of the star's mass, effectively blowing it up (Spitzer & Saslaw 1966; Lai et al. 1993; Balberg et al. 2013; Balberg 2024; Brutman et al. 2024). Mergers with minimal mass loss are most likely to occur outside of about 0 . 01 pc, where the relative speeds tend to be less than the escape speed from the stars. Within 0 . 01 pc, however, velocities exceed the escape speed from the stars, and collisions can result in peculiar, low-mass 'stripped stars' (Rose et al. 2023; Rose & MacLeod 2024; Gibson et al. 2024).\nIn practice, the outcome of a collision is more complex to determine, depending on properties such as the impact parameter and stellar structure (e.g., Freitag & Benz 2005; Gibson et al. 2024, Rose et al. in prep.). For the purposes of this study, which focuses on the general implications of collisions for stellar orbits in galactic nuclei, the specific mass loss and merger prescriptions should not qualitatively change our results. Unless otherwise specified, all simulations shown herein use fitting formulae from Rauch (1999) for the mass loss and escape speed arguments to determine whether or not a collision results in a merger (Rose et al. 2023).\nIn previous iterations of this code, the collision speed was taken to be simply the velocity dispersion at the star's distance from the supermassive black hole, given by Eq. (2) (Rose et al. 2022, 2023). However, this approach is insufficient for studying the effects of collisions on stellar orbits. In this study, we draw the orbit of the second colliding star, using a procedure detailed below.\n3.2. Orbital Dynamics\nIf a star in our sample with semimajor axis a · and eccentricity e · collides over timestep ∆ t , we find a plausible orbit for the second colliding star. As we operate under the assumption that the cluster is composed of a uniform stellar population, we always take the second star to have m collider = 1 M ⊙ . We then draw semimajor axes and eccentricities from the cluster's property distri-\n\n\n
Figure 1. Upper Panel: We plot the relevant timescales as a function of distance from the SMBH for a range of stellar density profiles, α = 1 . 25 (solid line) to α = 1 . 75 (dashed line). The collision and relaxation timescales are in red and green, respectively, while the grey line marks the total simulation time of 10 Gyr. The vertical red line emphasizes the radius at which the collision timescale equals the simulation time; within this radius, collisions are common and shape the evolution of the stars and cluster. We also mark the radius at which the velocity dispersion is approximately the escape speed from the surface of a Sun-like star using the green vertical line. External to this radius, we expect collisions to result in mergers. Lower Panel: We consider a hyperbolic encounter between two stars in the cluster. This plot shows the impact parameter in solar radii for a 90 · deflection as a function of the relative speed between the stars. Note that the x-axis is inverted to reflect the dependence of velocity dispersion on distance from the SMBH; it parallels the x-axis of the plot above, with the largest velocities occurring near the SMBH and decreasing further out. We have marked the speeds that correspond to the inner and outer allowed initial semimajor axes of the stars in our sample, 0 . 001 and 1 pc, respectively. Additionally, we mark where b 90 becomes equal to the star's radius. This point coincides well with the speed at which we would expect the collision outcome to transition from mergers to the sorts of high-speed encounters that might strip or destroy stars, but leave them unbound from each other.
\n\n\n\n\nas described in Section 2, until we find an orbit that intersects with that of our tracer star. Our code assumes that the cluster is spherically symmetric and only tracks the semimajor axis and eccentricity of each tracer star. In this set up, our tracer star always has argument of periapsis equal to zero, however the second colliding star's periapsis need not be aligned with it. The angle between them is drawn from a uniform distribution. The orbits are always assumed to be coplanar.\nObservations suggest that a subset of the stars in the Milky Way's galactic center reside in a disk, while others have an isotropic distribution (e.g., Levin & Beloborodov 2003; Ghez et al. 2003, 2005; Gillessen et al. 2009; Paumard et al. 2006; Lu et al. 2009; Bartko et al. 2009; Yelda et al. 2014). In our co-planar physical picture, we have two options: either both of the colliding stars can orbit in the same direction about the supermassive black hole, or they can be equally likely to be prograde or retrograde. We test two extremes to capture the effects of different orbital orientations. Our first case, 'disk-like', has the two colliding stars orbiting the SMBH in the same direction. In the second case, 'isotropic-like', we assume that fifty percent of the time the two stars orbit the SMBH in the same direction and fifty percent of the time in opposite directions. The latter orientation means that the orbital angular momentum vectors are anti-parallel to one another.\nOnce we have found an intersecting orbit for our colliding tracer star, we determine the intersection point and compute the velocity vectors of the two stars at that point. The relative speed tells us whether or not a merger occurs and the degree of fractional mass loss from the system (see above section). From here, determining the final orbit(s) will depend on the type of outcome and therefore the relative speed.\n3.2.1. Final Orbit in Merger Case\nA stellar merger is the natural outcome of a low-speed collision. We calculate the final mass, age, and now trajectory of the product assuming the stars act as 'sticky spheres' (e.g., Rodriguez et al. 2022; Rose et al. 2023). The two stars approach each other with velocity vectors v · and v collider as determined by the intersection point of their orbit. Momentum conservation demands that the final velocity of the merger product, v final , equals ( M v · + M ⊙ v collider ) / ( M + M ⊙ ). Based on hydrodynamics studies of stellar mergers, the mass loss in these collisions should be low (e.g.., Lai et al. 1993; Rauch 1999, Rose et al. in prep.). With the final velocity, mass, and location of the collision - the intersection point of the original orbits - we can calculate the new orbit of the merger product about the SMBH.\n3.2.2. Final Orbits Following a High-Speed Collision\nOrbital changes from high-speed collisions are much more difficult to determine in the absence of hydrodynamics simulations. However, we present a framework for treating these interactions. In order to understand these collisions, we start at the limit where the two stars barely graze. This interaction should unfold as a hyperbolic encounter. In the center of mass frame of the two stars, their speeds remain constant, but their velocity vectors are deflected by angle θ hyp . θ hyp can be found analytically:\nθ hyp = 2arctan( b 90 /b ) , (4)\nwhere b is the impact parameter and b 90 is defined as the impact parameter needed for a 90 · deflection. b 90 equals 2 G/v 2 rel , where v rel is the relative speed between the two stars (e.g., Binney & Tremaine 2008).\nWe plot b 90 as a function of the relative speed in the bottom panel of Figure 1. In the nuclear star cluster, the velocity dispersion can be understood as the characteristic relative speed between stars at a given distance from the SMBH (see Eq. 2). We have therefore inverted the x-axis of the bottom plot to parallel that of the upper plot, distance from the SMBH, and marked the velocity dispersion at key distances using vertical dashed lines. In both the upper and lower plots, we also indicate the regions in which we expect mergers versus high-speed collisions, which leave the stars unbound from each other. Interestingly, for most of the parameter space where these high-speed, non-sticky sphere collisions occur, b 90 is less than the star's radius. Another way of interpreting this statement is that at high speeds, physical collisions are required for a strong-angle deflection.\nIf the stars were point particles, the impact parameter could be arbitrarily small and the interaction would still unfold as a hyperbolic encounter. Two Sun-like stars begin to touch when b = 2 R ⊙ . With b < 2 R ⊙ , the stars physically impede each other as they interact, leading to a smaller deflection angle than the one given in Eq. (4). Furthermore, if the stars approach each other perfectly head-on with b = 0, the center of mass velocity is 0 and there is no angular momentum. In this case, there would be no deflection. Heuristically, then, we expect the deflection angle to be given by Eq. (4) for grazing encounters, and some fraction of this angle for encounters with b < 2 R . That fraction should decrease with impact parameter until they are both zero. The precise dependence is impossible to determine in the absence of hydro simulations, but we define a collision deflection\nangle that meets the two limiting criterion:\nθ coll = 2 b r c arctan( b 90 /b ) , (5)\nwhere r c is the sum of the radii of the two colliding stars.\nIt is reasonable to expect that the collision affects the stars' speeds as well. Even more so than the deflection angle, changes in speed can only be understood through hydrodynamic simulations. In our models, we simply test three cases: one in which the speed is not affected at all, one in which the speed in the center of mass frame is always reduced by 10%, and one in which it is always reduced by a factor of 2.\nWe treat these collisions as follows: at the intersection point of the orbits of the two colliding stars, we calculate their center of mass velocity. We transform to the center of mass frame. Only then do we rotate and scale their velocity vectors. While we only track semimajor axes and eccentricities, we do allow the deflections to be three dimensional, with components out of the plane. The final velocity can be any vector, chosen randomly, along a cone defined by angle θ coll with respect to the original velocity. Isotropizing the deflection is important because otherwise the colliding stars would always be scattered exactly towards or away from the SMBH, leading to an gross overestimate of TDE rates. We then transform back to the frame of the SMBH and calculate the new orbits given the velocity and position vectors.\nWe note that our calculation assumes that the center of mass is not moving in the frame of the supermassive black hole. In actuality, the center of mass of the two stars would orbit the supermassive black hole, but the effects will be negligable due to the high collision speeds.\n3.3. Two-Body Relaxation\nIn addition to collisions, stars in the cluster experience the weak gravitational effects of nearby neighbors. The effects of these interactions can accumulate, eventually changing the star's orbital energy and angular momentum by order of itself. The original orbit is 'erased' over a characteristic timescale:\nt rlx = 0 . 34 σ 3 G 2 ρ ⟨ M ∗ ⟩ ln Λ rlx , (6)\nwhere ⟨ M ∗ ⟩ is the average star's mass, here taken to be 1 M ⊙ , and ln Λ rlx is the coulomb logarithm (e.g., Binney & Tremaine 2008; Merritt 2013). Figure 1 shows the relaxation timescale as a function of distance from the SMBH in blue for a range of stellar density profiles.\nWe account for relaxation by allowing the orbital eccentricity and semimajor axis of each of our tracer stars to slowly evolve. Once per orbit, we apply a small in-\nstantaneous change in velocity to each star (e.g., Bradnick et al. 2017; Lu & Naoz 2019; Rose et al. 2022, 2023; Naoz et al. 2022, see the latter for the full set of equations). The kick is calibrated so that ∆ v/v ∼ √ ∆ t/t rlx , and if ∆ t = t rlx , ∆ v is of order of the velocity. This prescription simulates the diffusion of the orbital orbital parameters over time from interactions with other stars in the cluster. Previously, it has been used in studies of TDEs and extreme mass ratio inspirals of stellar mass black holes into the SMBH (Naoz et al. 2022; Melchor et al. 2024).\n3.4. Orbital Stopping Conditions\nAs noted in Section 3.1, we terminate the simulation when the desired runtime of 10 billion years is reached or when the time elapsed has exceeded the star's mainsequence lifetime, whichever comes first. However, orbital changes from collisions or relaxation can also send stars into the tidal radius, where they will be destroyed. We trigger a stopping condition if the star's periapsis a · (1 -e · ) becomes less than twice the tidal radius from the SMBH, R star ( M · /M star ) 1 / 3 (e.g., Guillochon & Ramirez-Ruiz 2013; Mockler et al. 2023). Tidal disruption events can be characterized by impact parameter β = R tidal / ( a · (1 -e · )). Our stopping condition corresponds to β = 0 . 5, allowing us to capture partial as well as full disruptions.\nBecause both direct collisions and relaxation processes can place a star onto a nearly radial orbit, we log whether the critical orbit was reached through a direct collision or our relaxation prescription. A star that becomes a TDE due to relaxation processes can still have experienced one or more collision previously in its life. However, a star that collides and becomes a TDE due to the collision itself may still be inflated from the impact when it reaches its periapsis. In this case, the stopping condition noted above may be conservative; the R star is simply the radius expected using a mass-radius relation for a main-sequence star of mass m star .\nHigh-speed collisions, unlike those that result in mergers, can also cause stars to be ejected from the nuclear star cluster. Consider two stars that collide on elliptical orbits that intersect near periapsis. As the eccentricity approaches unity, the speed at periapsis approaches the escape speed, just shy of becoming an unbound, parabolic orbit. In the limit of a hyperbolic encounter, the speeds of the stars are unchanged in the center of mass frame, but they undergo a deflection. In the frame of the SMBH, this change can boost one star's speed enough to unbind it from the SMBH, while the other star ends up on a more tightly bound orbit. Close encounters have previously been shown to eject stars from\ndense stellar systems (e.g., Henon 1969; Lin & Tremaine 1980). Additionally, high-speed collisions often lead to mass loss, which can unbind the orbit not unlike a supernova (e.g., Lu & Naoz 2019). Therefore, we include a stopping condition for orbital energy ≥ 0 and eccentricity ≥ 1. We do not consider stars that become unbound after losing mass in a tidal disruption, as proposed by Manukian et al. (2013).\n4. NUMERICAL RESULTS WITHOUT RELAXATION\nWe run large sets of 10000 tracer stars and let them evolve over 10 billion years. We begin by presenting simulated results without our relaxation prescription. These simulations allows us to isolate the orbital effects of collisions unobscured by relaxation. As in a nuclear star cluster, most of our tracer stars reside near ∼ 1 pc, the edge of the sphere of influence. About 500 of our tracer stars, or 5% of the sample, lie within 0 . 1 pc, where collisions are most common (see Figure 1). Sampling in this way gives a more complete picture of the relative rates of each outcome.\nFigure 2 juxtaposes results for slightly different treatments of the collision physics, labeled as follows:\nType A: In this simulation, two colliding stars have a 50% chance of orbiting the SMBH in the same direction and a 50% chance of orbiting in opposite directions. The final velocities after a high-speed collision are calculated using Eq. 5. We assume that these high speed collisions do not impact the speed of the stars in their center of mass frame.\nType B: Simulations with type B conditions differ from type A by assuming that all stars orbit the SMBH in the same direction.\nType C: These simulations are identical to type A, except high speed collisions decrease the speeds in the stars' center of mass frame by 10%. In other words, we scale the velocity vectors of the stars by 0 . 9 before tranforming back to the frame of the SMBH.\nType D: These conditions are identical to types A and C except high speed collisions drecrease the speed in the center of mass frame by 50%.\nThe left column of Figure 2 shows the final masses and semimajor axes of the tracer stars. Red open circles represent stars at the end of the simulation or their mainsequence lifetime, whichever came first. Grey points represent the initial conditions. Blue diamonds indicate stars that were placed on unbound orbits by high speed collisions. Note that unbound orbits have negative semimajor axes, so we plot the absolute value. We provide the total number of stars that escape on unbound orbits in the upper right corner of each plot. In addition to\n\n\n
Figure 2. Left Column: We show the final masses of our tracer stars versus their final distances from the SMBH for four select simulations without relaxation. Turning off relaxation allows us to isolate the effect of collisions. The final mass and distance are either the stars properties when it evolved off the main-sequence or at the end of the 10 Gyr integration time, whichever is shorter. The grey points are the initial conditions, while the red points are the final tracer stars that remained in the cluster. Blue diamonds denote stars which escaped from the cluster, and black and lime stars represent TDEs from different types of collisions, mergers and high-speed collision deflections, respectively. The crosses represents stars which were placed onto such radial orbits that they plunged right into the Schwarschild radius of the SMBH. In the upper left hand corner, we indicate the number of stars that were placed onto unbound orbits. Since unbound orbits are defined to have negative semimajor axis, here we are plotting the absolute value for their final orbits. Right Column: These plots show the orbital period of the stars when they become TDEs versus β , the ratio of the tidal radius to the orbit's periapsis. The grey dashed horizontal line marks 30 yr, approximately the threshold at which we would expect repeating TDEs because the orbital period is short enough to observe multiple passages. Rows: Each row corresponds to slightly different initial conditions or collision treatment. A uses Eq. 5 to calculate the post-high speed collision deflection and assumes our so-called 'isotropic' conditions for the relative velocities between the colliding stars. B is identical to the above except the collision conditions are 'disky', meaning stars are always assumed to orbit the SMBH in the same direction. C uses isotropic conditions and reduces the post-collision speed in the center of mass frame by 10% for high-speed collisions. D scales the post-collision speed by 50% in center of mass frame following high-speed, destructive collision cases.
\n\nthese unbound orbits, collisions can also place stars on orbits with semimajor axes outside the inner pc.\nTDEs from direct collisions are represented by the star shaped symbols. The color indicates the type of collision responsible for the radial orbit: low-speed collisions, which we refer to as merger TDEs, are shown in black, while lime represents high speed collisions. Some of these orbits are so extreme that the periapsis distance lies within the Schwarzschild radius. We mark these plunging stars using an 'x' symbol.\nThe right column shows the properties of the TDEs and plunges for each simulation. The y-axis shows the orbital period of the final orbit, which carries them into the tidal radius. On the x-axis, we plot the parameter β , which quantifies how deeply the orbit penetrates the tidal radius. 0 . 5 < β < 1 indicate a partial disruption. To guide the eye, grey dashed line marks where the orbital period equals 30 yr. TDEs with periods less than this could conceivably be observed as a repeating TDE. We discuss the implications for TDEs below.\n4.1. Tidal Disruption Events\nCollision-induced mergers can result in TDEs under conditions A, C, and D, but not B. These merged stars are placed on nearly radial orbits when the colliding stars are orbiting in opposite directions. Their angular momenta vector are anti-aligned, giving a low angular momentum to the final orbit of the merger product. Under type B collision conditions, in contrast, there are no merger TDEs. The orbital angular momenta are never in a position to cancel each other out.\nFewer TDEs result from high speed collisions. The relative abundances reflect the relative frequency of each type of collision. High speed collisions tend to occur near the SMBH where speeds are sufficiently high, but fewer stars reside. These high speed TDEs are relatively rarer because this type of collision is relatively rarer. Collisions in general tend to become common within the inner 0 . 1 pc of the cluster. However, only about 500 stars from our sample lie in this region. We find that when we sample this region in greater detail, both high and low speed collision TDEs become more common. One high speed collision even produced a repeating TDE (see additional simulations in Section A.2). Others were close to our repeating TDE threshold with 30-50 yr periods.\nThere are some initial conditions that consistently fail to produce high speed collision TDEs, notably Type D conditions. A shallower density profile will also lead to a lower collision rate overall because the timescale is longer (see Figure 1), in turn reducing the rate of collision TDEs (see Appendix A.2). Additionally, we note that our prescription for determining whether or\nnot a merger results from a collision may over-predict the number of mergers. We use fitting formulae from Rauch (1999) to determine the mass loss in a collision and heuristic arguments, similar to the aforementioned study, to determine if a merger occurs. Previous studies with this code (Rose et al. 2023; Rose & MacLeod 2024) have also tested simulations using fitting formulae from Lai et al. (1993), who include a fitting formula for the merger capture radius. Their formula leads to fewer mergers compared to our first approach. However, collision-induced mergers would still be viable as a mechanism to create TDEs; they would simply occur at a much lower rate. As with other additional simulations, we show examples using the Lai et al. (1993) fitting formulae in the Appendix (Figure 8) to avoid overcrowding the main text.\n4.2. Orbital Changes from Collisions\nWe consider general trends in the effects of collisions on the orbital properties of the stars. We do this by examining sub-populations of the tracer stars based on their final mass. We stress that this classification does not necessarily give the full picture of a star's collision history. While all stars with M final > 1 M ⊙ must have undergone a merger - recall that all stars in are cluster are initially 1 M ⊙ - and stars with M final < 1 M ⊙ must have lost mass in a high speed collision, these two populations are not mutually exclusive. A single star can experience both types of collisions over its lifetime depending on where it is in its orbit and the orbit of the second colliding star. However, final mass still represents the best 'observable' probe of a star's collision history. We compare the semimajor axis versus 1 -e of the stars in Figure 3, the latter being proportional to the periapsis. Grey dots show the initial conditions for all the stars. Green circles represent stars with M final > 1 M ⊙ , while orange dots represent stars with M final < 1 M ⊙ . TDEs and unbound stars are represented using the same symbols as in previous figures. We confirm that the TDEs in our simulation do indeed come from nearly radial orbits.\nMergers and high speed collisions differ in terms of their orbital outcomes. Conservation of energy demands that mergers always shrink the semimajor axis, while high speed collisions can place stars on both wider or smaller orbits by facilitating an energy exchange between the two stars. These trends are visible in the Figure. The green open circles are mostly confined to the inner cluster, while orange dots have an extended distribution. A few merged stars lie outside our initial cluster. These merged stars experienced a high speed collision that deflected them onto wider orbits.\n\n\n\n\n\n\n\n\n
Figure 3. This figure shows orbital parameters of our tracer stars from the simulations in the first three rows of Figure 2. We plot semimajor axis versus 1 -e , a quantity proportional to the periapsis distance. Systems with e f > 1, that is, stars placed on unbound orbits, are ommitted from this plot. Grey dots represent the initial population. Stars that experienced a post-collision merger over their lifetime are represented by green open circles. These stars are defined as having M > 1 M ⊙ . Orange dots, or stripped stars, are defined as having M < 1 M ⊙ , the result of a high speed collision. TDEs and plunges are marked with stars and crosses, respectively. As can be seen in the figure, these stars were placed on nearly radial orbits with extreme eccentricity values. We omit the final properties of stars that have not experienced a collision. These simulations have no relaxation, so un-collided stars' parameters remain unchanged.
\n\nIn simulations A and B, a roughly equal number of stripped stars moved to more (less) tightly bound orbits. We count unbound stars in the less tightly bound category because their final orbital energies are larger than the initial values. Additionally, about 10 stars are placed on orbits with semimajor axes outside of 1 pc. This result suggests that there may be collision-affected stripped stars masquerading outside of the sphere of influence. Including a treatment for dissipation during a high speed collision, simulation C presents a different story. The orbits of stripped stars are more likely to shrink and become more circular. About 75% become more tightly bound. Additionally, a population of short period stripped stars emerges. These stars likely experienced multiple high speed collisions, giving them multiple opportunities to shrink their semimajor axes. The effects are even more pronounced for simulation D (not shown). Under type D conditions, stripped stars are overwhelmingly on less eccentric orbits, explaining why the high speed collision TDE rate for these conditions are so low.\nEccentricity trends can also be gleaned from Figure 3. For type B conditions, mergers always act to make the orbit more circular. In addition to energy, angular momentum must be conserved. For stars orbiting in the same direction, their angular momentum vectors are always aligned. Since the semimajor axis shrinks after a collision, the eccentricity must decrease to conserve angular momentum. This effect can be seen in vertical extent of the green circles in B compared to A. The most eccentric orbits for merged stars in B have been largely removed. We reiterate that the results are somewhat muddied by the occurrence of high speed collisions in a subset of this population. Under type A conditions, low speed collisions both decrease and increase the orbital eccentricity. The most eccentric final orbits become TDEs or plunges, removing the stars from the population. We note that in actuality, the orbital properties of the stars can be modified by resonant and non-resonant relaxation processes, not accounted for in these results (e.g., Rauch & Tremaine 1996; Hopman & Alexander 2006; Kocsis & Tremaine 2011).\n4.3. Unbound Stars\nHigh speed collisions can unbind stars from the SMBH. These interactions, treated similarly to a hyperbolic encounter, can place one star on a more tightly bound orbit while giving the other star a positive orbital energy. We show properties of the unbound stars from simulations A and B in Figure 4. The initial orbits tend to be eccentric. We also confirmed by inspection of specific cases that the collisions tend to occur near periapsis.\nFor the most part, the final orbits have eccentricity just above unity. We calculate the speed at infinity, v inf , for these stars based on their final orbital energy and color code the points in Figure 4 based on its value. Typical speeds range from ∼ 100 to 600 km/s, but occasionally one star will have a v inf above 1000 km/s, as was the case in simulation B. For simulation C, which includes some form of dissipation during high speed collisions, there were fewer unbound stars overall, but their distribution was similar. The maximum v inf was 642 km/s. The single unbound star from the fourth row of Figure 2 (simulation D) had v inf equal to 377 km/s.\nThese maximum values suggest that high speed collisions may represent another mechanism to launch hypervelocity stars from galactic nuclei. The most famous of these mechanisms is the Hills Mechanism, in which a binary is disrupted by the SMBH such that one star is ejected at high speed while the other is retained on a tightly bound orbit (e.g., Hills 1988; Generozov & Madigan 2020), though other mechanisms exist (e.g., Yu & Tremaine 2003; Perets 2009a). Close encounters between single stars are known to eject stars from dense stellar clusters (Henon 1969; Lin & Tremaine 1980), and in fact Yu & Tremaine (2003) consider such interactions as a means of producing hypervelocity stars. Omitting collisions, they find a low ejection rate at speeds ≳ 300 km/s. We consider collisions exclusively and treat them as modified close encounters. While our largest speed is consistent with those generated by the Hill's Mechanism ( ≳ 1000 km/s, Hills 1988), about 30% have v inf > 300 km/s. Observations of hypervelocity stars with origins pointing to the Galactic center exhibit a range of speeds, from the hundreds to thousands of km/s (e.g., Brown et al. 2005, 2018; Koposov et al. 2020; Generozov & Perets 2022, to quote from the latter, stars need ejection speed ≳ 750 km/s from the Galactic center to have 275 km/s at 20 kpc). The v inf of our stars shown in Figure 4 do not account for any additional terms in the potential beyond the SMBH. We reserve a detailed comparison of our results to observations for future work.\nThe mass loss fitting formulae from Rauch (1999) give very low mass loss for larger impact parameter collisions ( b ≳ 0 . 6 r c ) at moderately high speeds ( ∼ 1000 km/s); generally, the fractional mass loss in these cases is less than a percent, and unbound stars in Figure 2 appear as if they have not lost any mass at all. However, fitting formulae from high speed collisions can be unreliable (Freitag & Benz 2005). We therefore refrain from commenting on the masses of the unbound stars in detail except to state that some of them may look like stripped stars (Gibson et al. 2024).\nAs can be seen in Figure 2, a maximum of ∼ 1% of the stars in the inner parsec region may become unbound from the SMBH. This number far exceeds those that become TDEs through high speed collisions. Furthermore, this outcome represents a surprisingly high fraction of stars that experience such collisions: about a fifth of stars that experience high speed collisions become unbound. This high number may owe to a few conspiring conditions:\nFor reasons discussed in Section 3.4 and as supported by Figure 4, unbound stars tend come from high speed collisions near the periapsis of eccentric orbits. Near periapsis, a small boost in the frame of the SMBH can tip the star's speed over the escape speed. However, the merger-to-high speed collision boundary also lies around 800 km/s. The velocity dispersion only exceeds this value within about 0 . 02 pc, and the vast majority of stars reside outside this distance. In consequence, most high speed collisions can only occur near periapsis, where speeds are high enough to no longer be in the merger regime and also where unbinding the star from the SMBH becomes a more favorable outcome. Our thermal initial eccentricity distribution ensures that plenty of stars begin on very eccentric orbits (see Appendix A.1 for the impact of the eccentricity distribution).\nDissipation during the collision can reduce the number of unbound stars. As discussed in Section 3.2.2, it is possible that high speed direct collisions reduce the speeds of the stars in their center of mass frame. The physical impact of the collision may reduce the number of stars that get enough of a boost to escape the inner parsec. Type C and D collision conditions test the limits of this mechanism in producing unbound stars. Notably, if high speed collisions reduce the post-collision speed by 50% in the center of mass frame, only a few tenths of a percent of the cluster stars escape.\n5. NUMERICAL RESULTS WITH RELAXATION\nNow that we have built a physical picture of how collisions shape the stellar cluster, we turn on our prescription for two-body relaxation in the code. These simulations take longer to run, so the sample size of the tracer stars presented herein number at 4000 instead of 10000. We present results in Figure 5. The symbols are all the same as in previous figures, with one addition: stars that became TDEs through two body relaxation are shown in dark green. The orbital evolution of these stars where still affected by collisions, as can be seen by their masses. Interestingly, the stripped star that became a TDE via relaxation had experienced a collision within the same timestep. The collision had placed it on\nUnbound Orbits Properties\n\n\n
Figure 4. We plot the final versus initial eccentricities for the stars on unbound orbits from simulations A and B (first and second row in Figure 2). Both the initial and final eccentricities are clustered near 1, the initial eccentricities slightly under this value and the final eccentricities above. The diamonds are colorcoded by the speed at infinity of the unbound stars. We note that this calculation does not take into account any additional contribution to the potential, such as the stellar cluster, beyond the mass of the SMBH. Typical v inf are order 100 km/s, though some exceed 1000 km/s.
\n\nan eccentric orbit just shy of the tidal radius, periapsis of 1 . 135 AU versus 0 . 956 AU needed for partial disruption. Had the star been inflated at all from the collision, material would have been siphoned off by the SMBH. As the star's parameters did not trigger the stopping condition, our relaxation prescription tipped the star into the tidal radius.\nWe place a caveat on the TDE inside 0 . 001 pc. The closest known star to the Milky Way's SMBH has a periapsis just under 0 . 001 pc (e.g., Gillessen et al. 2017). It is questionable whether our two-body relaxation prescription applies within this radius. However, this star too experienced a collision in the previous timestep which, had the star been inflated, would have led to a disruption. Altogether, our results suggest that collisions represent a viable mechanism to preferentially deposit stars that are (1) more massive than the surrounding population due to a merger or (2) stripped stars into the\ntidal radius. Some of these TDEs could be observed as repeating TDEs. Additionally, we may be missing disruptions from stars that are inflated post collision, leading to qualitatively different electromagnetic signatures (e.g., MacLeod et al. 2013; Gibson et al. 2024).\nThe unbound stars present something of a puzzle. Even accounting for the difference in sample size, we find much fewer compared to Figure 2 (no relaxation). There are a few possible explanations. First, the relaxation prescription changes the orbital energy by roughly order of itself over the relaxation timescale, which is less than 10 Gyr in the model cluster. Stars at larger radii can therefore wander towards the SMBH, where destructive collisions become possible. Once in this 'danger zone', they can experience enough high-speed collisions to become destroyed. Roughly 500 stars were destroyed over this simulation. While this process is slow (see e.g., Rose & MacLeod 2024), it does drain stars from the sample population. In the simulation without relaxation, the majority of stars began and remained outside of this region. Only ∼ 50 stars were destroyed, a much smaller fraction of the sample, and most unbound stars had initial semimajor axes outside of 0 . 1 pc (see Appendix A.4).\nThe reduction of unbound stars may also be related to the interplay between the two dynamical processes. We speculate on why this second option might be true. Collisions are discrete events. Relaxation, in contrast, is a diffusion process that slowly alters a star's orbital parameters. This timescale is less than the collision timescale for most of the stars in our sample (see Figure 1), particularly where the unbound stars originate in our earlier simulations. The implication is that when a collision does occur, it may be less likely to catch the orbit in a state favorable for unbinding the star. The last possibility is that there is something in our relaxation prescription which we have yet to detect that is artificially suppressing the unbound stars. 1 Regardless, the interplay of relaxation and collisions merit a more thorough examination. We reserve a detailed study for future work.\n6. CONCLUSIONS\nCollisions between main-sequence stars occur in the nuclear star cluster and become common within the inner 0 . 1 pc (e.g., Rose & MacLeod 2024). In this proof-ofconcept study, we examine the orbital changes that result from these collisions. Specifically, we assess whether\n\n\n
Figure 5. We show a simulation with type A collision conditions that includes relaxation. The symbols are consistent with those in our other figures, with the addition of the dark green stars to represent TDEs from our relaxation prescription. We place a caveat on the dark green star (relaxation TDE) with final semimajor axis within 0 . 001 pc, also the TDE with the shortest orbital period. The applicability of our relaxation prescription within this region is uncertain.
\n\nphysical collisions can contribute to the production of TDEs and ejected stars. Collisions in galactic nuclei can be understood in two types.\nIn the first type, the relative speed between the stars is low and the collision results in a merger. Little mass loss occurs (e.g., Lai et al. 1993) and we can treat the stars as sticky spheres (e.g., Kremer et al. 2020; Gonz'alez et al. 2021). Conservation of momentum allows us to calculate the final orbit of the merged star. The second type of collision occurs at speeds that exceed the escape velocity from the stars. While the high speeds ensure that the stars remain unbound from each other, they can also drive mass loss from the stars, in some cases producing a stripped star (e.g., Lai et al. 1993; Rauch 1999; Freitag & Benz 2005; Gibson et al. 2024). We use fitting formulae from Rauch (1999) to calculate the mass loss. However, we refrain from commenting in detail on the final masses of these stars because fitting formulae are not always accurate for collisions in galactic nuclei (Freitag & Benz 2005). We are examining these collisions further using sph simulations in forthcoming work (Rose et al. in prep.).\nWe use limiting cases and heuristic arguments to determine the final orbits from high speed collisions. If the stars were point masses, these interactions would unfold as a hyperbolic encounter in the center of mass frame of the stars. As an upper limit, we could calculate the deflection angle using Eq. 4. However, while a grazing collision approaches the limit given by Eq. 4, a head-on collision should not result in any deflection at all. We therefore adopt Eq. 5, which meets both criteria. We also test the role of dissipation by reducing the speeds of the stars post-collision in their center of mass frame.\nOur simulations follow a sample of tracer stars embedded in a fixed, uniform cluster. We test a variety of cluster conditions and find the following:\n\n1. Tidal disruption events: We find that both high and low speed collisions can produce TDEs by placing stars on highly eccentric orbits. Certain conditions, however, can be prohibitive to this mechanism. If stars all orbit the SMBH in the same direction, for example, collision-induced mergers cannot produce TDEs because the orbital angular momentum vectors of the colliding stars are aligned. If high speed collisions are highly dissipative, then TDEs from this class of collisions are similarly suppressed. The overall rates of collision TDEs remain to be seen. Generally, however, our results mean that collision-affected stars can be delivered to the tidal radius by the collisions themselves, in addition to standard two-body relaxation that affects the population at large. The TDEs immediately post-collision versus after the star has relaxed may look quite difficult (see discussion of imminent versus eventual TDEs in Gibson et al. 2024), adding to the richness of the observations that this dynamical process can yield.\n2. Unbound stars: High speed collisions can place stars on unbound orbits, especially if the collisions occur near the periapsis of an eccentric orbit. As evidenced by the low mass loss in Figure 2, these collisions tend to have larger impact parameter. These grazing collisions approach the limit given by Eq. 4. Of course, in a physical collision, stars impede each other's motion. We test simple cases in which the star is only deflected a fraction of the angle and its speed is reduced post-collision.\n\nDespite these changes, the unbound stars persist, albeit in much lower numbers. We reserve a precise examination of their rates and properties for future work, though some may look like stripped stars (Gibson et al. 2024). In the optimistic case, stellar collisions represent a mechanism to launch hypervelocity stars from galactic nuclei, joining the list of existing mechanisms (e.g., Hills 1988; Yu & Tremaine 2003; Ginsburg & Loeb 2007; Perets 2009a,b; Generozov & Madigan 2020). The Hills Mechanism elegantly explains both the origins of the S-star cluster, young-seeming massive stars in the vicinity of the SMBH, and hypervelocity stars using the same dynamical process (e.g., Hills 1988; Ghez et al. 2003; Ginsburg & Loeb 2007; Perets et al. 2007; Madigan et al. 2009; Lockmann et al. 2009; Generozov & Madigan 2020; Generozov 2021). Collisions present another possibility, where S-stars are the high-mass tail of the merger products created by successive low-speed collisions (see e.g., Rose et al. 2023), and hypervelocity stars\ntrace to high speed collisions near periapsis. However, the viability of this ejection mechanism still faces tests in the form of dissipation during collisions, a stellar mass function, interplay with other dynamical processes such as relaxation, and overall rates in a three-dimensional cluster.\nWe thank Fred Rasio, Enrico Ramirez-Ruiz, Jamie Lombardi, Fulya Kiroglu, Charles Gibson, and Claire Ye for invaluable discussion and input. SR thanks the CIERA Lindheimer Fellowship for support. BM thanks the Carnegie CTAC postdoctoral fellowship for support. This project began while SCR and BM were at the Aspen Center for Physics, which is supported by NSF grant PHY-2210452. This research was supported in part through the computational resources and staff contributions provided for the Quest high performance computing facility at Northwestern University which is jointly supported by the Office of the Provost, the Office for Research, and Northwestern University Information Technology.\nAPPENDIX\nA. SUPPLEMENTAL FIGURES\nA.1. Role of the Initial Eccentricity Distribution\nLess eccentric orbits on average should result in fewer unbound stars. We test this hypothesis by drawing the eccentricities of tracer stars from a uniform distribution (average e = 0 . 5) instead of a thermal one (average e = 0 . 67). We compare the results in the first row of Figure 2, which uses a thermal initial eccentricity distribution, to that shown in Figure 6. Despite both having type A initial conditions, changing the eccentricity distribution to uniform reduces the number of unbound stars by about a factor of 2. There were also fewer high speed collisions overall, a ∼ 35% reduction. These results come from reducing the number of stars with speeds at periapsis that place them in the high speed collision regime.\nA.2. Sampling the Inner Collision-dominated Region\nMost collisions occur in the inner 0 . 1 parsec of the nuclear star cluster, where the collision timescale becomes shorter than 10 Gyr. In order to probe this region in greater detail, we run a similar set of simulations to those shown in Figure 2, but with the stars sampled from 0 . 001 to 0 . 1 pc. We note that a cusp with slope α = 1 . 75 results in a very high density in this region. The actual cusp in the Galactic center may be closer to 1 . 4 (Gallego-Cano et al. 2018b). Additionally, collisions themselves can deplete the cusp in the inner 0 . 1 pc. A shallower cusp would lead to a lower collision rate. We therefore test α = 1 . 25 with the idea that reality may lie somewhere in the middle. These simulations are marked with an asterisk in Figure 7. The shallower profile means that the collision timescale is longer and that fewer collisions occur within the cluster over the integration time. However, as can be seen in the Figure, the collisions that do occur can stil produce TDEs. Additionally, type B conditions, which prohibit merger TDEs, can still have collision-driven TDEs of a different kind.\nA.3. Alternative Fitting Formulae\nAs mentioned in Section 3.1, we leverage fitting formulae and intuitive toy models to calculate the mass loss from a collision and determine whether a merger occurs. Simulations shown in the main text and in Figure 7 use fitting formulae from Rauch (1999) and heuristic arguments to determine if a merger occurs. In this section, we present\n\n\n
Figure 6. We show a simulation with type A collision conditions that uses a uniform eccentricity distribution for the stars instead of a thermal one. The presentation of the figure and symbols are the same as those of Figure 2. Having less eccentric orbits on average reduces the number of unbound stars compared to the thermal case.
\n\nsimulations with type A collision conditions that use fitting formulae from Lai et al. (1993). Lai et al. (1993) include fitting formula for the capture radius, a function of the relative speed between the stars and their impact parameter. This formula leads to less mergers than the simple prescription that we use in the main text of this paper, which is also similar to the prescription used in Rauch (1999) (see for a full discussion Rose et al. 2023). We present the results of two simulations using the Lai et al. (1993) formulae in Figure 8. The upper row samples tracer stars on a cusp from 0 . 001 to 1 pc, while the second row focuses on the inner 0 . 1 pc. As expected, with the overall reduction in mergers - they instead are treated as high speed collisions - there are fewer merger TDEs. However, it is still possible to get TDEs of all types and unbound stars.\nA.4. Orbital Parameters\nWe expand upon Section 4.2 in this appendix. We compare the final semimajor axes versus the initial semimajor axes of the stars in Figure 9. As is our convention in the main text, green circles represent stars with M final > 1 M ⊙ , while orange dots have M final < 1 M ⊙ . TDEs and unbound stars are represented using stars and diamonds. The black dashed line shows where the initial and final semimajor axes are equal.\nMergers work to shrink the semimajor axis, while high speed collisions can increase or decrease it, as is visible in the Figure. For the most part, green circles lie below the black dashed lines. A few lie above the black line, mostly within 0 . 1 pc, because they also experienced a high speed collision. They initially resided within ∼ 0 . 1 pc, where high speed collisions become more likely.\nWe examine the final eccentricities for merger-affected stars in Figure 10. We compare the final eccentricity distribution to the initial thermal distribution. For reasons outlined in Section 4.2, type B mergers circularize orbits. This effect can be seen in Figure 10, where there is a dearth of eccentric orbits.\nThe effects on eccentricity in type A conditions are harder to tease out. 50% of the time the orbits are aligned and the eccentricity decreases. The other 50% of the time, the angular momenta vector are anti-aligned, so even though the semimajor axis still shrinks, the eccentricity can increase. The most eccentric merged stars are removed from the population because they become TDEs or plunges. The result is an eccentricity distribution that flattens at high ( e > 0 . 5) eccentricities. However, the actual number of merged star orbits that became more (less) eccentric are similar.\nREFERENCES\nAlexander, T. 1999, ApJ, 527, 835, doi: 10.1086/308129\nAlexander, T., & Pfuhl, O. 2014, ApJ, 780, 148,\ndoi: 10.1088/0004-637X/780/2/148\nBahcall, J. N., & Wolf, R. A. 1976, ApJ, 209, 214,\ndoi: 10.1086/154711\nBalberg, S. 2024, ApJ, 962, 150,\ndoi: 10.3847/1538-4357/ad1690\nSampling Tracer Stars from the Inner 0.1 pc\n\n\n
Figure 7. This Figure has the same form as Figure 2, except the 10000 tracer stars have been sampled on a cusp from 0 . 001 to 0 . 1 pc. In this region, the collision timescale becomes comparable to the lifetime of a solar mass star, and collision become common. The simulations conditions are denoted by the letter on the left and correspond to those with the same letter in Figure 2. Simulations with an asterisk use a shallower slope for the stellar density profile, α = 1 . 25 instead of 1 . 75.
\n\n*uses a = 1.25 instead of 1.75\n\n\n
Figure 8. Again with the same form as Figure 2, here we present results using fitting formulae from Lai et al. (1993). The upper row has tracer stars sampled on a cusp from 0 . 001 to 1 pc, while the outer limit in the bottom row for the initial semimajor axes is 0 . 1.
\n\n\n\n
Figure 9. We compare the final semimajor axes of interesting tracer stars to their initial values. We note that since there was no relaxation in any of these simulations, all changes are due to direct collisions. The black dashed line shows where the initial semimajor axes equals the final. Stars that experienced a post-collision merger over their lifetime are represented by green open circles. These stars are defined as having M > 1 M ⊙ . Orange dots, or stripped stars, are defined as having M < 1 M ⊙ , the result of a high speed collision. Blue diamonds represent unbound stars. We plot the absolute value of their semimajor axes. Black (lime) stars represent TDEs from merger (high speed) collisions. Each initial condition case are qualitatively similar, with mergers producing more tightly bound orbits and high speed collisions moving stars to both tighter and wider orbits. The exception is the simulation with type C initial conditions. This prescription slows stars down by 10% following a high-speed collision in the stars' center of mass frame. As a result, a population ot stripped stars that have sunk towards the SMBH emerges.
\n\n\n\n\n\n\n\n\n\n
Figure 10. We compare the final eccentricity distribution of stars that have experienced a merger with the initial (thermal) eccentricity distribution. The first column uses type A inital conditions ('isotropic'), in which the colliding stars can orbit the SMBH in opposite direction, while the second column uses type B initial conditions ('disky'), in which stars always orbit the SMBH in the same direction. Mergers seem to circularize the stellar orbits, completely erasing the peak at high eccentricities of a thermal distribution. We note that these simulations do not include relaxation; all orbital changes are due to direct collisions.
\n\nBalberg, S., Sari, R., & Loeb, A. 2013, MNRAS, 434, L26, doi: 10.1093/mnrasl/slt071 Bartko, H., Martins, F., Fritz, T. K., et al. 2009, ApJ, 697, 1741, doi: 10.1088/0004-637X/697/2/1741 Binney, J., & Tremaine, S. 2008, Galactic Dynamics: Second Edition Bradnick, B., Mandel, I., & Levin, Y. 2017, MNRAS, 469, 2042, doi: 10.1093/mnras/stx1007 Brown, W. R., Geller, M. J., Kenyon, S. J., & Kurtz, M. J. 2005, ApJL, 622, L33, doi: 10.1086/429378 Brown, W. R., Lattanzi, M. G., Kenyon, S. J., & Geller, M. J. 2018, ApJ, 866, 39, doi: 10.3847/1538-4357/aadb8e Brutman, Y., Steinberg, E., & Balberg, S. 2024, arXiv e-prints, arXiv:2408.16383, doi: 10.48550/arXiv.2408.16383 Dale, J. E., & Davies, M. B. 2006, MNRAS, 366, 1424, doi: 10.1111/j.1365-2966.2005.09937.x Dale, J. E., Davies, M. B., Church, R. P., & Freitag, M. 2009, MNRAS, 393, 1016, doi: 10.1111/j.1365-2966.2008.14254.x Dessart, L., Ryu, T., Amaro Seoane, P., & Taylor, A. M. 2024, A&A, 682, A58, doi: 10.1051/0004-6361/202348228 Ferrarese, L., & Ford, H. 2005, SSRv, 116, 523, doi: 10.1007/s11214-005-3947-6 Freitag, M., & Benz, W. 2002, A&A, 394, 345, doi: 10.1051/0004-6361:20021142 -. 2005, MNRAS, 358, 1133,\ndoi: 10.1111/j.1365-2966.2005.08770.x\nGallego-Cano, E., Sch¨odel, R., Dong, H., et al. 2018a, A&A, 609, A26, doi: 10.1051/0004-6361/201730451 -. 2018b, A&A, 609, A26, doi: 10.1051/0004-6361/201730451 Gallegos-Garcia, M., Law-Smith, J., & Ramirez-Ruiz, E. 2018, ApJ, 857, 109, doi: 10.3847/1538-4357/aab5b8 Generozov, A. 2021, MNRAS, 501, 3088, doi: 10.1093/mnras/staa3851 Generozov, A., & Madigan, A.-M. 2020, ApJ, 896, 137, doi: 10.3847/1538-4357/ab94bc Generozov, A., & Perets, H. B. 2022, MNRAS, 513, 4257, doi: 10.1093/mnras/stac1108 Genzel, R., Eisenhauer, F., & Gillessen, S. 2010, Reviews of Modern Physics, 82, 3121, doi: 10.1103/RevModPhys.82.3121 Ghez, A. M., Salim, S., Hornstein, S. D., et al. 2005, ApJ, 620, 744, doi: 10.1086/427175 Ghez, A. M., Duchˆene, G., Matthews, K., et al. 2003, ApJL, 586, L127, doi: 10.1086/374804 Ghez, A. M., Salim, S., Weinberg, N. N., et al. 2008, ApJ, 689, 1044, doi: 10.1086/592738 Gibson, C., Kıro˘glu, F., Lombardi, J. C., J., et al. 2024, arXiv e-prints, arXiv:2410.02146, doi: 10.48550/arXiv.2410.02146 Gillessen, S., Eisenhauer, F., Trippe, S., et al. 2009, ApJ, 692, 1075, doi: 10.1088/0004-637X/692/2/1075\nGillessen, S., Plewa, P. M., Eisenhauer, F., et al. 2017, ApJ, 837, 30, doi: 10.3847/1538-4357/aa5c41\n
\n\n
\n
\n\n
\n
\n\n
\n
\n\n
\n"},"sections":{"kind":"list like","value":[{"title":"ABSTRACT","content":"Dense stellar clusters surround the supermassive black holes (SMBH) in galactic nuclei. Interactions within the cluster can alter the stellar orbits, occasionally driving a star into the SMBH's tidal radius where it becomes ruptured. This proof-of-concept study examines the orbital effects of stellar collisions using a semianalytic model. Both low and high speed collisions occur in the SMBH's sphere of influence. Our model treats stars in low speed collisions as sticky spheres. For high-speed collisions, we develop a simple prescription based on the limiting case of a hyperbolic encounter. We test a range of collision treatments and cluster conditions. We find that collisions can place stars on nearly radial orbits. Depositing stars within the tidal radius, collisions may drive the disruption of stars with unusual masses and structures: depending on the nature of the collision, the star could be the product of a recent merger, or it could have lost its outer layers in a high speed impact, appearing as a stripped star. We also find that high speed collisions near the periapsis of an eccentric orbit can unbind stars from the SMBH. However, dissipation during these high-speed collisions can substantially reduce the number of unbound stars achieved in our simulations. We conclude that TDEs and ejected stars, even in the hypervelocity regime, are plausible outcomes of stellar collisions, though their frequency in a three-dimensional nuclear star cluster are uncertain. Future work will address the rates and properties of these events. Keywords: Stellar dynamics; Galactic center; Star clusters; Stellar mergers","pages":[1]},{"title":"On the Orbital Effects of Stellar Collisions in Galactic Nuclei: Tidal Disruption Events and Ejected Stars","content":"1 Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA), Northwestern University, 1800 Sherman Ave, Evanston, IL 60201, USA 2 The Observatories of the Carnegie Institution for Science, Pasadena, CA 91101, USA","pages":[1]},{"title":"1. INTRODUCTION","content":"Asupermassive black hole resides at the center of most galaxies, where it is surrounded by a dense stellar cluster (e.g. Ferrarese & Ford 2005; Kormendy & Ho 2013; Schodel et al. 2003; Ghez et al. 2005, 2008; Gillessen et al. 2009, 2017; Neumayer et al. 2020). A tidal disruption event (TDE) occurs when a star from the cluster passes within a critical distance from the SMBH and becomes ruptured by tidal forces (e.g., Hills 1975; Rees 1988; Alexander 1999; Magorrian & Tremaine 1999; Wang & Merritt 2004; MacLeod et al. 2012). The SMBH then accretes the stellar material, producing an electromagnetic signature (e.g., Guillochon & Ramirez-Ruiz 2013). Spectra of these events encode valuable information about the mass, structure, and composition of the ruptured star (e.g., Kochanek 2016a,b; Yang et al. 2017; Mockler et al. 2022; Miller et al. 2023). Observations of Corresponding author: Sanaea C. Rose sanaea.rose@northwestern.edu TDEs represent a powerful way to probe the stellar populations in galactic nuclei and the processes that shape them. Amongst these physical processes are direct collisions. Collisions can occur within the sphere of influence of the SMBH due to the high densities and velocity dispersion (e.g., Freitag & Benz 2002; Dale et al. 2009; Dale & Davies 2006; Rubin & Loeb 2011; Balberg et al. 2013; Balberg 2024; Mastrobuono-Battisti et al. 2014; Rose & MacLeod 2024). These events may produce electromagnetic signatures, both from the collisions themselves and from interactions between liberated material and the SMBH (e.g., Rosswog et al. 2009; Lee et al. 2010; Balberg et al. 2013; Dessart et al. 2024; Ryu et al. 2024b,a; Brutman et al. 2024), though it is difficult to ignite a main-sequence star with compression (Guillochon et al. 2009). They can also shape the stellar population. Both low and high speed collisions can alter the mass of a star - the latter can destroy stars completely and the former can give rise to blue stragglers in dense stellar systems (e.g., Lai et al. 1993; Rauch 1999; Sills et al. 1997, 2001; Lombardi et al. 2002; MacLeod et al. 2013; Leigh et al. 2016; Rose et al. 2023). Recently, Gibson et al. (2024) have shown that high speed collisions can also produce stripped stars similar to what might be seen through binary evolution. Collisions that produce stellar mergers or stripped stars may explain recent unexpected TDE observations. For example, detections of high nitrogen-tocarbon (N/C) ratios in TDEs point to the disruption of more stars that burn on the CNO cycle (as first proposed by Kochanek 2016a,b), and that are ≳ 1 -2 M ⊙ (Kochanek 2016a; Yang et al. 2017; Gallegos-Garcia et al. 2018; Mockler et al. 2023) than is predicted by the host galaxies' stellar populations (Mockler et al. 2023). One particular TDE has such an extremely high N/C abundance ratio that it is difficult to explain with single stellar evolution alone (Miller et al. 2023), but could be the result of the disruption of a stripped star that has lost its nitrogen-poor envelope (Mockler et al. 2024). Additionally, a recently discovered population of extremely bright nuclear transients has also been suggested to originate from the disruption of high mass stars (e.g. ≳ 10 M ⊙ , Subrayan et al. 2023; Hinkle et al. 2024). Because stars that end in disruptions are expected to be drawn approximately at random from the stellar mass function (with small adjustments for stellar type, MacLeod et al. 2012), this preference for higher mass stars may imply that the mass function in galactic nuclei is more top-heavy (and/or bottom-light) than in the rest of the galaxy (see e.g., Lu et al. 2013; Hosek et al. 2019). The potential link between TDEs and collisions is intriguing: collisions can simultaneously affect both the properties of the stars and their orbits about the SMBH. MacLeod et al. (2012) first considered collisions, particularly destructive ones, in the context of the TDE rate. Changes to a star's trajectory from collisions have also been considered in general dynamical models of dense stellar systems (e.g., Sanders 1970; Rauch 1999; Freitag & Benz 2002; Kremer et al. 2020; Gonz'alez et al. 2021; Rodriguez et al. 2022). In this proof-of-concept study, we study the orbital effects of stellar collisions in galactic nuclei. We assess whether the collisions that produce unusual stars could plausibly deposit those same stars onto TDE-producing orbits. Our models leverage simple, intuitive treatments for collision outcomes and fitting formulae from previous studies (e.g. Lai et al. 1993; Rauch 1999). We test a range of initial conditions and nuclear star cluster properties. Our paper is organized as follows: In Section 2, we discuss our general approach to modeling the Milky Way's nuclear star cluster. Section 3 describes the treatment of various physics in our code, with Section 3.2 in particular outlining our methodology for updating the stellar orbits post collision. These orbital changes represent the most major addition to the code as compared to previous iterations (Rose et al. 2022, 2023). Section 4 presents simulated results for direct collisions. Sections 4.1, 4.2, and 4.3 discuss the implications for TDEs, orbital properties of collision-affected stars, and unbound and hypervelocity stars. We then incorporate relaxation into our simulations in addition to stellar collisions and present results in Section 5. Lastly, we summarize the scope and findings of our study in Section 6.","pages":[1,2]},{"title":"2. MODEL NUCLEAR STAR CLUSTER","content":"We leverage semi-analytic models to study the effects of collisions on the nuclear star cluster. Our fiducial model uses the conditions and properties of the Milky Way's GN, whose proximity makes it the best studied galactic center. We follow a sample of stars embedded in a fixed, unevolving cluster. For simplicity, both the evolving sample and the surrounding cluster are composed of 1 M ⊙ stars. The cluster can be understood in two key properties, density and velocity dispersion, which govern the dynamical processes unfolding within it. The stellar density sets the the frequency with which stars interact. We describe the stellar density as a function of distance from the SMBH using a power law: where α is the slope and r · , distance from the SMBH. Based on observations of the cluster within the sphere of influence, this equation is normalized using ρ 0 = 1 . 35 × 10 6 M ⊙ / pc 3 at r 0 = 0 . 25 pc (Genzel et al. 2010). Our fiducial model uses a slope of 1 . 75, the expectation for a single-mass population (Bahcall & Wolf 1976), consistent with the fact that our simple model cluster has only solar mass stars. However, in order to capture the range of theoretical predictions and observational constraints on the stellar cusp (e.g., Bahcall & Wolf 1976; Gallego-Cano et al. 2018a; Linial & Sari 2022), we also test a few simulations with α = 1 . 25, shown in Appendix A. We assume that the slope of the stellar cusp is roughly constant, or varying slowly, over the timescales of interest in our simulations. The velocity dispersion within the cluster also influences the frequency and nature of stellar interactions. It decreases with distance from the SMBH: where α is the slope of the density profile and M · is the mass of the SMBH (Alexander 1999; Alexander & Pfuhl 2014). We take M · to be 4 × 10 6 M ⊙ , like the Milky Way's SMBH (e.g., Ghez et al. 2003). For a uniform mass cluster of 1 M ⊙ stars, the number density n is simply ρ ( r · ) 1 M ⊙ .","pages":[2,3]},{"title":"3. SEMIANALYTIC MODEL","content":"We follow a sample of 1 M ⊙ tracer stars embedded in our model cluster. We draw their orbital eccentricities from a thermal distribution. We select their semimajor axes so that they lie on a cusp with slope α , matching the background cluster. These stars are allowed to evolve under the influence of two main dynamical processes, direct collisions and two-body relaxation, using a model first developed by Rose et al. (2022, 2023).","pages":[3]},{"title":"3.1. Collisions","content":"Direct collisions occur over a characteristic timescale t -1 coll = nσA , where A is the cross-section of interaction, n is the number density, and σ is the velocity dispersion. For an impact to occur, A is the physical crosssection enhanced by gravitational focusing. The collision timescale also depends weakly on the star's orbital eccentricity and can be written as: where f 1 ( e · ) and f 2 ( e · ) are equations 20 and 21 from Rose et al. (2020), G is the gravitational constant, and a · is the star's semimajor axis. r c is the sum of the radii of the colliding stars, or 2 R ⊙ for a uniform population of solar mass stars. We plot this timescale in red in the upper panel of Figure 1. We consider a range of slopes for the stellar density profile, spanning α = 1 . 25 (dashed line) to α = 1 . 75 (solid line). The horizontal grey line shows the total simulation time, included to guide the eye. Where the collision timescale is less than the simulation time, within 0 . 1 pc of the SMBH, collisions become important to understanding the evolution of the cluster (e.g., Rose & MacLeod 2024). We treat stellar collisions using a statistical approach. We begin by computing the probability that a star in our sample will experience a collision. Over a timestep ∆ t , this probability equals ∆ t/t coll . ∆ t is taken to be 10 6 years so that the probability ∆ t/t coll is always less than one; t coll ≳ 10 7 years for the parameter space we consider. The code then draws a random number between 0 and 1, which, if less than or equal to the collision probability ∆ t/t coll , means a collision has occurred. We repeat this prescription until the desired simulation runtime or the star's main-sequence lifetime has been reached, whichever is shorter. If a collision has occurred over a given timestep, we update the mass and age of the star using a prescription detailed in Rose et al. (2023). A full discussion of our approach can be found in previous papers (Rose et al. 2023; Rose & MacLeod 2024). In brief, we combine fitting formulae from hydrodynamics simulations of stellar collisions (Rauch 1999; Lai et al. 1993) with heuristic arguments to (1) determine if a given collision will result in a merger and (2) determine the amount of mass lost from either the merger product or the unbound individual stars. Since speeds in galactic nuclei often exceed hundreds of kilometers per second, collisions can eject anywhere from a percent to all of the star's mass, effectively blowing it up (Spitzer & Saslaw 1966; Lai et al. 1993; Balberg et al. 2013; Balberg 2024; Brutman et al. 2024). Mergers with minimal mass loss are most likely to occur outside of about 0 . 01 pc, where the relative speeds tend to be less than the escape speed from the stars. Within 0 . 01 pc, however, velocities exceed the escape speed from the stars, and collisions can result in peculiar, low-mass 'stripped stars' (Rose et al. 2023; Rose & MacLeod 2024; Gibson et al. 2024). In practice, the outcome of a collision is more complex to determine, depending on properties such as the impact parameter and stellar structure (e.g., Freitag & Benz 2005; Gibson et al. 2024, Rose et al. in prep.). For the purposes of this study, which focuses on the general implications of collisions for stellar orbits in galactic nuclei, the specific mass loss and merger prescriptions should not qualitatively change our results. Unless otherwise specified, all simulations shown herein use fitting formulae from Rauch (1999) for the mass loss and escape speed arguments to determine whether or not a collision results in a merger (Rose et al. 2023). In previous iterations of this code, the collision speed was taken to be simply the velocity dispersion at the star's distance from the supermassive black hole, given by Eq. (2) (Rose et al. 2022, 2023). However, this approach is insufficient for studying the effects of collisions on stellar orbits. In this study, we draw the orbit of the second colliding star, using a procedure detailed below.","pages":[3]},{"title":"3.2. Orbital Dynamics","content":"If a star in our sample with semimajor axis a · and eccentricity e · collides over timestep ∆ t , we find a plausible orbit for the second colliding star. As we operate under the assumption that the cluster is composed of a uniform stellar population, we always take the second star to have m collider = 1 M ⊙ . We then draw semimajor axes and eccentricities from the cluster's property distri- as described in Section 2, until we find an orbit that intersects with that of our tracer star. Our code assumes that the cluster is spherically symmetric and only tracks the semimajor axis and eccentricity of each tracer star. In this set up, our tracer star always has argument of periapsis equal to zero, however the second colliding star's periapsis need not be aligned with it. The angle between them is drawn from a uniform distribution. The orbits are always assumed to be coplanar. Observations suggest that a subset of the stars in the Milky Way's galactic center reside in a disk, while others have an isotropic distribution (e.g., Levin & Beloborodov 2003; Ghez et al. 2003, 2005; Gillessen et al. 2009; Paumard et al. 2006; Lu et al. 2009; Bartko et al. 2009; Yelda et al. 2014). In our co-planar physical picture, we have two options: either both of the colliding stars can orbit in the same direction about the supermassive black hole, or they can be equally likely to be prograde or retrograde. We test two extremes to capture the effects of different orbital orientations. Our first case, 'disk-like', has the two colliding stars orbiting the SMBH in the same direction. In the second case, 'isotropic-like', we assume that fifty percent of the time the two stars orbit the SMBH in the same direction and fifty percent of the time in opposite directions. The latter orientation means that the orbital angular momentum vectors are anti-parallel to one another. Once we have found an intersecting orbit for our colliding tracer star, we determine the intersection point and compute the velocity vectors of the two stars at that point. The relative speed tells us whether or not a merger occurs and the degree of fractional mass loss from the system (see above section). From here, determining the final orbit(s) will depend on the type of outcome and therefore the relative speed.","pages":[3,4]},{"title":"3.2.1. Final Orbit in Merger Case","content":"A stellar merger is the natural outcome of a low-speed collision. We calculate the final mass, age, and now trajectory of the product assuming the stars act as 'sticky spheres' (e.g., Rodriguez et al. 2022; Rose et al. 2023). The two stars approach each other with velocity vectors v · and v collider as determined by the intersection point of their orbit. Momentum conservation demands that the final velocity of the merger product, v final , equals ( M v · + M ⊙ v collider ) / ( M + M ⊙ ). Based on hydrodynamics studies of stellar mergers, the mass loss in these collisions should be low (e.g.., Lai et al. 1993; Rauch 1999, Rose et al. in prep.). With the final velocity, mass, and location of the collision - the intersection point of the original orbits - we can calculate the new orbit of the merger product about the SMBH.","pages":[4]},{"title":"3.2.2. Final Orbits Following a High-Speed Collision","content":"Orbital changes from high-speed collisions are much more difficult to determine in the absence of hydrodynamics simulations. However, we present a framework for treating these interactions. In order to understand these collisions, we start at the limit where the two stars barely graze. This interaction should unfold as a hyperbolic encounter. In the center of mass frame of the two stars, their speeds remain constant, but their velocity vectors are deflected by angle θ hyp . θ hyp can be found analytically: where b is the impact parameter and b 90 is defined as the impact parameter needed for a 90 · deflection. b 90 equals 2 G/v 2 rel , where v rel is the relative speed between the two stars (e.g., Binney & Tremaine 2008). We plot b 90 as a function of the relative speed in the bottom panel of Figure 1. In the nuclear star cluster, the velocity dispersion can be understood as the characteristic relative speed between stars at a given distance from the SMBH (see Eq. 2). We have therefore inverted the x-axis of the bottom plot to parallel that of the upper plot, distance from the SMBH, and marked the velocity dispersion at key distances using vertical dashed lines. In both the upper and lower plots, we also indicate the regions in which we expect mergers versus high-speed collisions, which leave the stars unbound from each other. Interestingly, for most of the parameter space where these high-speed, non-sticky sphere collisions occur, b 90 is less than the star's radius. Another way of interpreting this statement is that at high speeds, physical collisions are required for a strong-angle deflection. If the stars were point particles, the impact parameter could be arbitrarily small and the interaction would still unfold as a hyperbolic encounter. Two Sun-like stars begin to touch when b = 2 R ⊙ . With b < 2 R ⊙ , the stars physically impede each other as they interact, leading to a smaller deflection angle than the one given in Eq. (4). Furthermore, if the stars approach each other perfectly head-on with b = 0, the center of mass velocity is 0 and there is no angular momentum. In this case, there would be no deflection. Heuristically, then, we expect the deflection angle to be given by Eq. (4) for grazing encounters, and some fraction of this angle for encounters with b < 2 R . That fraction should decrease with impact parameter until they are both zero. The precise dependence is impossible to determine in the absence of hydro simulations, but we define a collision deflection angle that meets the two limiting criterion: where r c is the sum of the radii of the two colliding stars. It is reasonable to expect that the collision affects the stars' speeds as well. Even more so than the deflection angle, changes in speed can only be understood through hydrodynamic simulations. In our models, we simply test three cases: one in which the speed is not affected at all, one in which the speed in the center of mass frame is always reduced by 10%, and one in which it is always reduced by a factor of 2. We treat these collisions as follows: at the intersection point of the orbits of the two colliding stars, we calculate their center of mass velocity. We transform to the center of mass frame. Only then do we rotate and scale their velocity vectors. While we only track semimajor axes and eccentricities, we do allow the deflections to be three dimensional, with components out of the plane. The final velocity can be any vector, chosen randomly, along a cone defined by angle θ coll with respect to the original velocity. Isotropizing the deflection is important because otherwise the colliding stars would always be scattered exactly towards or away from the SMBH, leading to an gross overestimate of TDE rates. We then transform back to the frame of the SMBH and calculate the new orbits given the velocity and position vectors. We note that our calculation assumes that the center of mass is not moving in the frame of the supermassive black hole. In actuality, the center of mass of the two stars would orbit the supermassive black hole, but the effects will be negligable due to the high collision speeds.","pages":[5]},{"title":"3.3. Two-Body Relaxation","content":"In addition to collisions, stars in the cluster experience the weak gravitational effects of nearby neighbors. The effects of these interactions can accumulate, eventually changing the star's orbital energy and angular momentum by order of itself. The original orbit is 'erased' over a characteristic timescale: where ⟨ M ∗ ⟩ is the average star's mass, here taken to be 1 M ⊙ , and ln Λ rlx is the coulomb logarithm (e.g., Binney & Tremaine 2008; Merritt 2013). Figure 1 shows the relaxation timescale as a function of distance from the SMBH in blue for a range of stellar density profiles. We account for relaxation by allowing the orbital eccentricity and semimajor axis of each of our tracer stars to slowly evolve. Once per orbit, we apply a small in- stantaneous change in velocity to each star (e.g., Bradnick et al. 2017; Lu & Naoz 2019; Rose et al. 2022, 2023; Naoz et al. 2022, see the latter for the full set of equations). The kick is calibrated so that ∆ v/v ∼ √ ∆ t/t rlx , and if ∆ t = t rlx , ∆ v is of order of the velocity. This prescription simulates the diffusion of the orbital orbital parameters over time from interactions with other stars in the cluster. Previously, it has been used in studies of TDEs and extreme mass ratio inspirals of stellar mass black holes into the SMBH (Naoz et al. 2022; Melchor et al. 2024).","pages":[5,6]},{"title":"3.4. Orbital Stopping Conditions","content":"As noted in Section 3.1, we terminate the simulation when the desired runtime of 10 billion years is reached or when the time elapsed has exceeded the star's mainsequence lifetime, whichever comes first. However, orbital changes from collisions or relaxation can also send stars into the tidal radius, where they will be destroyed. We trigger a stopping condition if the star's periapsis a · (1 -e · ) becomes less than twice the tidal radius from the SMBH, R star ( M · /M star ) 1 / 3 (e.g., Guillochon & Ramirez-Ruiz 2013; Mockler et al. 2023). Tidal disruption events can be characterized by impact parameter β = R tidal / ( a · (1 -e · )). Our stopping condition corresponds to β = 0 . 5, allowing us to capture partial as well as full disruptions. Because both direct collisions and relaxation processes can place a star onto a nearly radial orbit, we log whether the critical orbit was reached through a direct collision or our relaxation prescription. A star that becomes a TDE due to relaxation processes can still have experienced one or more collision previously in its life. However, a star that collides and becomes a TDE due to the collision itself may still be inflated from the impact when it reaches its periapsis. In this case, the stopping condition noted above may be conservative; the R star is simply the radius expected using a mass-radius relation for a main-sequence star of mass m star . High-speed collisions, unlike those that result in mergers, can also cause stars to be ejected from the nuclear star cluster. Consider two stars that collide on elliptical orbits that intersect near periapsis. As the eccentricity approaches unity, the speed at periapsis approaches the escape speed, just shy of becoming an unbound, parabolic orbit. In the limit of a hyperbolic encounter, the speeds of the stars are unchanged in the center of mass frame, but they undergo a deflection. In the frame of the SMBH, this change can boost one star's speed enough to unbind it from the SMBH, while the other star ends up on a more tightly bound orbit. Close encounters have previously been shown to eject stars from dense stellar systems (e.g., Henon 1969; Lin & Tremaine 1980). Additionally, high-speed collisions often lead to mass loss, which can unbind the orbit not unlike a supernova (e.g., Lu & Naoz 2019). Therefore, we include a stopping condition for orbital energy ≥ 0 and eccentricity ≥ 1. We do not consider stars that become unbound after losing mass in a tidal disruption, as proposed by Manukian et al. (2013).","pages":[6]},{"title":"4. NUMERICAL RESULTS WITHOUT RELAXATION","content":"We run large sets of 10000 tracer stars and let them evolve over 10 billion years. We begin by presenting simulated results without our relaxation prescription. These simulations allows us to isolate the orbital effects of collisions unobscured by relaxation. As in a nuclear star cluster, most of our tracer stars reside near ∼ 1 pc, the edge of the sphere of influence. About 500 of our tracer stars, or 5% of the sample, lie within 0 . 1 pc, where collisions are most common (see Figure 1). Sampling in this way gives a more complete picture of the relative rates of each outcome. Figure 2 juxtaposes results for slightly different treatments of the collision physics, labeled as follows: Type A: In this simulation, two colliding stars have a 50% chance of orbiting the SMBH in the same direction and a 50% chance of orbiting in opposite directions. The final velocities after a high-speed collision are calculated using Eq. 5. We assume that these high speed collisions do not impact the speed of the stars in their center of mass frame. Type B: Simulations with type B conditions differ from type A by assuming that all stars orbit the SMBH in the same direction. Type C: These simulations are identical to type A, except high speed collisions decrease the speeds in the stars' center of mass frame by 10%. In other words, we scale the velocity vectors of the stars by 0 . 9 before tranforming back to the frame of the SMBH. Type D: These conditions are identical to types A and C except high speed collisions drecrease the speed in the center of mass frame by 50%. The left column of Figure 2 shows the final masses and semimajor axes of the tracer stars. Red open circles represent stars at the end of the simulation or their mainsequence lifetime, whichever came first. Grey points represent the initial conditions. Blue diamonds indicate stars that were placed on unbound orbits by high speed collisions. Note that unbound orbits have negative semimajor axes, so we plot the absolute value. We provide the total number of stars that escape on unbound orbits in the upper right corner of each plot. In addition to these unbound orbits, collisions can also place stars on orbits with semimajor axes outside the inner pc. TDEs from direct collisions are represented by the star shaped symbols. The color indicates the type of collision responsible for the radial orbit: low-speed collisions, which we refer to as merger TDEs, are shown in black, while lime represents high speed collisions. Some of these orbits are so extreme that the periapsis distance lies within the Schwarzschild radius. We mark these plunging stars using an 'x' symbol. The right column shows the properties of the TDEs and plunges for each simulation. The y-axis shows the orbital period of the final orbit, which carries them into the tidal radius. On the x-axis, we plot the parameter β , which quantifies how deeply the orbit penetrates the tidal radius. 0 . 5 < β < 1 indicate a partial disruption. To guide the eye, grey dashed line marks where the orbital period equals 30 yr. TDEs with periods less than this could conceivably be observed as a repeating TDE. We discuss the implications for TDEs below.","pages":[6,8]},{"title":"4.1. Tidal Disruption Events","content":"Collision-induced mergers can result in TDEs under conditions A, C, and D, but not B. These merged stars are placed on nearly radial orbits when the colliding stars are orbiting in opposite directions. Their angular momenta vector are anti-aligned, giving a low angular momentum to the final orbit of the merger product. Under type B collision conditions, in contrast, there are no merger TDEs. The orbital angular momenta are never in a position to cancel each other out. Fewer TDEs result from high speed collisions. The relative abundances reflect the relative frequency of each type of collision. High speed collisions tend to occur near the SMBH where speeds are sufficiently high, but fewer stars reside. These high speed TDEs are relatively rarer because this type of collision is relatively rarer. Collisions in general tend to become common within the inner 0 . 1 pc of the cluster. However, only about 500 stars from our sample lie in this region. We find that when we sample this region in greater detail, both high and low speed collision TDEs become more common. One high speed collision even produced a repeating TDE (see additional simulations in Section A.2). Others were close to our repeating TDE threshold with 30-50 yr periods. There are some initial conditions that consistently fail to produce high speed collision TDEs, notably Type D conditions. A shallower density profile will also lead to a lower collision rate overall because the timescale is longer (see Figure 1), in turn reducing the rate of collision TDEs (see Appendix A.2). Additionally, we note that our prescription for determining whether or not a merger results from a collision may over-predict the number of mergers. We use fitting formulae from Rauch (1999) to determine the mass loss in a collision and heuristic arguments, similar to the aforementioned study, to determine if a merger occurs. Previous studies with this code (Rose et al. 2023; Rose & MacLeod 2024) have also tested simulations using fitting formulae from Lai et al. (1993), who include a fitting formula for the merger capture radius. Their formula leads to fewer mergers compared to our first approach. However, collision-induced mergers would still be viable as a mechanism to create TDEs; they would simply occur at a much lower rate. As with other additional simulations, we show examples using the Lai et al. (1993) fitting formulae in the Appendix (Figure 8) to avoid overcrowding the main text.","pages":[8]},{"title":"4.2. Orbital Changes from Collisions","content":"We consider general trends in the effects of collisions on the orbital properties of the stars. We do this by examining sub-populations of the tracer stars based on their final mass. We stress that this classification does not necessarily give the full picture of a star's collision history. While all stars with M final > 1 M ⊙ must have undergone a merger - recall that all stars in are cluster are initially 1 M ⊙ - and stars with M final < 1 M ⊙ must have lost mass in a high speed collision, these two populations are not mutually exclusive. A single star can experience both types of collisions over its lifetime depending on where it is in its orbit and the orbit of the second colliding star. However, final mass still represents the best 'observable' probe of a star's collision history. We compare the semimajor axis versus 1 -e of the stars in Figure 3, the latter being proportional to the periapsis. Grey dots show the initial conditions for all the stars. Green circles represent stars with M final > 1 M ⊙ , while orange dots represent stars with M final < 1 M ⊙ . TDEs and unbound stars are represented using the same symbols as in previous figures. We confirm that the TDEs in our simulation do indeed come from nearly radial orbits. Mergers and high speed collisions differ in terms of their orbital outcomes. Conservation of energy demands that mergers always shrink the semimajor axis, while high speed collisions can place stars on both wider or smaller orbits by facilitating an energy exchange between the two stars. These trends are visible in the Figure. The green open circles are mostly confined to the inner cluster, while orange dots have an extended distribution. A few merged stars lie outside our initial cluster. These merged stars experienced a high speed collision that deflected them onto wider orbits. In simulations A and B, a roughly equal number of stripped stars moved to more (less) tightly bound orbits. We count unbound stars in the less tightly bound category because their final orbital energies are larger than the initial values. Additionally, about 10 stars are placed on orbits with semimajor axes outside of 1 pc. This result suggests that there may be collision-affected stripped stars masquerading outside of the sphere of influence. Including a treatment for dissipation during a high speed collision, simulation C presents a different story. The orbits of stripped stars are more likely to shrink and become more circular. About 75% become more tightly bound. Additionally, a population of short period stripped stars emerges. These stars likely experienced multiple high speed collisions, giving them multiple opportunities to shrink their semimajor axes. The effects are even more pronounced for simulation D (not shown). Under type D conditions, stripped stars are overwhelmingly on less eccentric orbits, explaining why the high speed collision TDE rate for these conditions are so low. Eccentricity trends can also be gleaned from Figure 3. For type B conditions, mergers always act to make the orbit more circular. In addition to energy, angular momentum must be conserved. For stars orbiting in the same direction, their angular momentum vectors are always aligned. Since the semimajor axis shrinks after a collision, the eccentricity must decrease to conserve angular momentum. This effect can be seen in vertical extent of the green circles in B compared to A. The most eccentric orbits for merged stars in B have been largely removed. We reiterate that the results are somewhat muddied by the occurrence of high speed collisions in a subset of this population. Under type A conditions, low speed collisions both decrease and increase the orbital eccentricity. The most eccentric final orbits become TDEs or plunges, removing the stars from the population. We note that in actuality, the orbital properties of the stars can be modified by resonant and non-resonant relaxation processes, not accounted for in these results (e.g., Rauch & Tremaine 1996; Hopman & Alexander 2006; Kocsis & Tremaine 2011).","pages":[8,9]},{"title":"4.3. Unbound Stars","content":"High speed collisions can unbind stars from the SMBH. These interactions, treated similarly to a hyperbolic encounter, can place one star on a more tightly bound orbit while giving the other star a positive orbital energy. We show properties of the unbound stars from simulations A and B in Figure 4. The initial orbits tend to be eccentric. We also confirmed by inspection of specific cases that the collisions tend to occur near periapsis. For the most part, the final orbits have eccentricity just above unity. We calculate the speed at infinity, v inf , for these stars based on their final orbital energy and color code the points in Figure 4 based on its value. Typical speeds range from ∼ 100 to 600 km/s, but occasionally one star will have a v inf above 1000 km/s, as was the case in simulation B. For simulation C, which includes some form of dissipation during high speed collisions, there were fewer unbound stars overall, but their distribution was similar. The maximum v inf was 642 km/s. The single unbound star from the fourth row of Figure 2 (simulation D) had v inf equal to 377 km/s. These maximum values suggest that high speed collisions may represent another mechanism to launch hypervelocity stars from galactic nuclei. The most famous of these mechanisms is the Hills Mechanism, in which a binary is disrupted by the SMBH such that one star is ejected at high speed while the other is retained on a tightly bound orbit (e.g., Hills 1988; Generozov & Madigan 2020), though other mechanisms exist (e.g., Yu & Tremaine 2003; Perets 2009a). Close encounters between single stars are known to eject stars from dense stellar clusters (Henon 1969; Lin & Tremaine 1980), and in fact Yu & Tremaine (2003) consider such interactions as a means of producing hypervelocity stars. Omitting collisions, they find a low ejection rate at speeds ≳ 300 km/s. We consider collisions exclusively and treat them as modified close encounters. While our largest speed is consistent with those generated by the Hill's Mechanism ( ≳ 1000 km/s, Hills 1988), about 30% have v inf > 300 km/s. Observations of hypervelocity stars with origins pointing to the Galactic center exhibit a range of speeds, from the hundreds to thousands of km/s (e.g., Brown et al. 2005, 2018; Koposov et al. 2020; Generozov & Perets 2022, to quote from the latter, stars need ejection speed ≳ 750 km/s from the Galactic center to have 275 km/s at 20 kpc). The v inf of our stars shown in Figure 4 do not account for any additional terms in the potential beyond the SMBH. We reserve a detailed comparison of our results to observations for future work. The mass loss fitting formulae from Rauch (1999) give very low mass loss for larger impact parameter collisions ( b ≳ 0 . 6 r c ) at moderately high speeds ( ∼ 1000 km/s); generally, the fractional mass loss in these cases is less than a percent, and unbound stars in Figure 2 appear as if they have not lost any mass at all. However, fitting formulae from high speed collisions can be unreliable (Freitag & Benz 2005). We therefore refrain from commenting on the masses of the unbound stars in detail except to state that some of them may look like stripped stars (Gibson et al. 2024). As can be seen in Figure 2, a maximum of ∼ 1% of the stars in the inner parsec region may become unbound from the SMBH. This number far exceeds those that become TDEs through high speed collisions. Furthermore, this outcome represents a surprisingly high fraction of stars that experience such collisions: about a fifth of stars that experience high speed collisions become unbound. This high number may owe to a few conspiring conditions: For reasons discussed in Section 3.4 and as supported by Figure 4, unbound stars tend come from high speed collisions near the periapsis of eccentric orbits. Near periapsis, a small boost in the frame of the SMBH can tip the star's speed over the escape speed. However, the merger-to-high speed collision boundary also lies around 800 km/s. The velocity dispersion only exceeds this value within about 0 . 02 pc, and the vast majority of stars reside outside this distance. In consequence, most high speed collisions can only occur near periapsis, where speeds are high enough to no longer be in the merger regime and also where unbinding the star from the SMBH becomes a more favorable outcome. Our thermal initial eccentricity distribution ensures that plenty of stars begin on very eccentric orbits (see Appendix A.1 for the impact of the eccentricity distribution). Dissipation during the collision can reduce the number of unbound stars. As discussed in Section 3.2.2, it is possible that high speed direct collisions reduce the speeds of the stars in their center of mass frame. The physical impact of the collision may reduce the number of stars that get enough of a boost to escape the inner parsec. Type C and D collision conditions test the limits of this mechanism in producing unbound stars. Notably, if high speed collisions reduce the post-collision speed by 50% in the center of mass frame, only a few tenths of a percent of the cluster stars escape.","pages":[9,10]},{"title":"5. NUMERICAL RESULTS WITH RELAXATION","content":"Now that we have built a physical picture of how collisions shape the stellar cluster, we turn on our prescription for two-body relaxation in the code. These simulations take longer to run, so the sample size of the tracer stars presented herein number at 4000 instead of 10000. We present results in Figure 5. The symbols are all the same as in previous figures, with one addition: stars that became TDEs through two body relaxation are shown in dark green. The orbital evolution of these stars where still affected by collisions, as can be seen by their masses. Interestingly, the stripped star that became a TDE via relaxation had experienced a collision within the same timestep. The collision had placed it on","pages":[10]},{"title":"Unbound Orbits Properties","content":"an eccentric orbit just shy of the tidal radius, periapsis of 1 . 135 AU versus 0 . 956 AU needed for partial disruption. Had the star been inflated at all from the collision, material would have been siphoned off by the SMBH. As the star's parameters did not trigger the stopping condition, our relaxation prescription tipped the star into the tidal radius. We place a caveat on the TDE inside 0 . 001 pc. The closest known star to the Milky Way's SMBH has a periapsis just under 0 . 001 pc (e.g., Gillessen et al. 2017). It is questionable whether our two-body relaxation prescription applies within this radius. However, this star too experienced a collision in the previous timestep which, had the star been inflated, would have led to a disruption. Altogether, our results suggest that collisions represent a viable mechanism to preferentially deposit stars that are (1) more massive than the surrounding population due to a merger or (2) stripped stars into the tidal radius. Some of these TDEs could be observed as repeating TDEs. Additionally, we may be missing disruptions from stars that are inflated post collision, leading to qualitatively different electromagnetic signatures (e.g., MacLeod et al. 2013; Gibson et al. 2024). The unbound stars present something of a puzzle. Even accounting for the difference in sample size, we find much fewer compared to Figure 2 (no relaxation). There are a few possible explanations. First, the relaxation prescription changes the orbital energy by roughly order of itself over the relaxation timescale, which is less than 10 Gyr in the model cluster. Stars at larger radii can therefore wander towards the SMBH, where destructive collisions become possible. Once in this 'danger zone', they can experience enough high-speed collisions to become destroyed. Roughly 500 stars were destroyed over this simulation. While this process is slow (see e.g., Rose & MacLeod 2024), it does drain stars from the sample population. In the simulation without relaxation, the majority of stars began and remained outside of this region. Only ∼ 50 stars were destroyed, a much smaller fraction of the sample, and most unbound stars had initial semimajor axes outside of 0 . 1 pc (see Appendix A.4). The reduction of unbound stars may also be related to the interplay between the two dynamical processes. We speculate on why this second option might be true. Collisions are discrete events. Relaxation, in contrast, is a diffusion process that slowly alters a star's orbital parameters. This timescale is less than the collision timescale for most of the stars in our sample (see Figure 1), particularly where the unbound stars originate in our earlier simulations. The implication is that when a collision does occur, it may be less likely to catch the orbit in a state favorable for unbinding the star. The last possibility is that there is something in our relaxation prescription which we have yet to detect that is artificially suppressing the unbound stars. 1 Regardless, the interplay of relaxation and collisions merit a more thorough examination. We reserve a detailed study for future work.","pages":[11]},{"title":"6. CONCLUSIONS","content":"Collisions between main-sequence stars occur in the nuclear star cluster and become common within the inner 0 . 1 pc (e.g., Rose & MacLeod 2024). In this proof-ofconcept study, we examine the orbital changes that result from these collisions. Specifically, we assess whether physical collisions can contribute to the production of TDEs and ejected stars. Collisions in galactic nuclei can be understood in two types. In the first type, the relative speed between the stars is low and the collision results in a merger. Little mass loss occurs (e.g., Lai et al. 1993) and we can treat the stars as sticky spheres (e.g., Kremer et al. 2020; Gonz'alez et al. 2021). Conservation of momentum allows us to calculate the final orbit of the merged star. The second type of collision occurs at speeds that exceed the escape velocity from the stars. While the high speeds ensure that the stars remain unbound from each other, they can also drive mass loss from the stars, in some cases producing a stripped star (e.g., Lai et al. 1993; Rauch 1999; Freitag & Benz 2005; Gibson et al. 2024). We use fitting formulae from Rauch (1999) to calculate the mass loss. However, we refrain from commenting in detail on the final masses of these stars because fitting formulae are not always accurate for collisions in galactic nuclei (Freitag & Benz 2005). We are examining these collisions further using sph simulations in forthcoming work (Rose et al. in prep.). We use limiting cases and heuristic arguments to determine the final orbits from high speed collisions. If the stars were point masses, these interactions would unfold as a hyperbolic encounter in the center of mass frame of the stars. As an upper limit, we could calculate the deflection angle using Eq. 4. However, while a grazing collision approaches the limit given by Eq. 4, a head-on collision should not result in any deflection at all. We therefore adopt Eq. 5, which meets both criteria. We also test the role of dissipation by reducing the speeds of the stars post-collision in their center of mass frame. Our simulations follow a sample of tracer stars embedded in a fixed, uniform cluster. We test a variety of cluster conditions and find the following: Despite these changes, the unbound stars persist, albeit in much lower numbers. We reserve a precise examination of their rates and properties for future work, though some may look like stripped stars (Gibson et al. 2024). In the optimistic case, stellar collisions represent a mechanism to launch hypervelocity stars from galactic nuclei, joining the list of existing mechanisms (e.g., Hills 1988; Yu & Tremaine 2003; Ginsburg & Loeb 2007; Perets 2009a,b; Generozov & Madigan 2020). The Hills Mechanism elegantly explains both the origins of the S-star cluster, young-seeming massive stars in the vicinity of the SMBH, and hypervelocity stars using the same dynamical process (e.g., Hills 1988; Ghez et al. 2003; Ginsburg & Loeb 2007; Perets et al. 2007; Madigan et al. 2009; Lockmann et al. 2009; Generozov & Madigan 2020; Generozov 2021). Collisions present another possibility, where S-stars are the high-mass tail of the merger products created by successive low-speed collisions (see e.g., Rose et al. 2023), and hypervelocity stars trace to high speed collisions near periapsis. However, the viability of this ejection mechanism still faces tests in the form of dissipation during collisions, a stellar mass function, interplay with other dynamical processes such as relaxation, and overall rates in a three-dimensional cluster. We thank Fred Rasio, Enrico Ramirez-Ruiz, Jamie Lombardi, Fulya Kiroglu, Charles Gibson, and Claire Ye for invaluable discussion and input. SR thanks the CIERA Lindheimer Fellowship for support. BM thanks the Carnegie CTAC postdoctoral fellowship for support. This project began while SCR and BM were at the Aspen Center for Physics, which is supported by NSF grant PHY-2210452. This research was supported in part through the computational resources and staff contributions provided for the Quest high performance computing facility at Northwestern University which is jointly supported by the Office of the Provost, the Office for Research, and Northwestern University Information Technology.","pages":[11,12,13]},{"title":"A.1. Role of the Initial Eccentricity Distribution","content":"Less eccentric orbits on average should result in fewer unbound stars. We test this hypothesis by drawing the eccentricities of tracer stars from a uniform distribution (average e = 0 . 5) instead of a thermal one (average e = 0 . 67). We compare the results in the first row of Figure 2, which uses a thermal initial eccentricity distribution, to that shown in Figure 6. Despite both having type A initial conditions, changing the eccentricity distribution to uniform reduces the number of unbound stars by about a factor of 2. There were also fewer high speed collisions overall, a ∼ 35% reduction. These results come from reducing the number of stars with speeds at periapsis that place them in the high speed collision regime.","pages":[13]},{"title":"A.2. Sampling the Inner Collision-dominated Region","content":"Most collisions occur in the inner 0 . 1 parsec of the nuclear star cluster, where the collision timescale becomes shorter than 10 Gyr. In order to probe this region in greater detail, we run a similar set of simulations to those shown in Figure 2, but with the stars sampled from 0 . 001 to 0 . 1 pc. We note that a cusp with slope α = 1 . 75 results in a very high density in this region. The actual cusp in the Galactic center may be closer to 1 . 4 (Gallego-Cano et al. 2018b). Additionally, collisions themselves can deplete the cusp in the inner 0 . 1 pc. A shallower cusp would lead to a lower collision rate. We therefore test α = 1 . 25 with the idea that reality may lie somewhere in the middle. These simulations are marked with an asterisk in Figure 7. The shallower profile means that the collision timescale is longer and that fewer collisions occur within the cluster over the integration time. However, as can be seen in the Figure, the collisions that do occur can stil produce TDEs. Additionally, type B conditions, which prohibit merger TDEs, can still have collision-driven TDEs of a different kind.","pages":[13]},{"title":"A.3. Alternative Fitting Formulae","content":"As mentioned in Section 3.1, we leverage fitting formulae and intuitive toy models to calculate the mass loss from a collision and determine whether a merger occurs. Simulations shown in the main text and in Figure 7 use fitting formulae from Rauch (1999) and heuristic arguments to determine if a merger occurs. In this section, we present simulations with type A collision conditions that use fitting formulae from Lai et al. (1993). Lai et al. (1993) include fitting formula for the capture radius, a function of the relative speed between the stars and their impact parameter. This formula leads to less mergers than the simple prescription that we use in the main text of this paper, which is also similar to the prescription used in Rauch (1999) (see for a full discussion Rose et al. 2023). We present the results of two simulations using the Lai et al. (1993) formulae in Figure 8. The upper row samples tracer stars on a cusp from 0 . 001 to 1 pc, while the second row focuses on the inner 0 . 1 pc. As expected, with the overall reduction in mergers - they instead are treated as high speed collisions - there are fewer merger TDEs. However, it is still possible to get TDEs of all types and unbound stars.","pages":[13,14]},{"title":"A.4. Orbital Parameters","content":"We expand upon Section 4.2 in this appendix. We compare the final semimajor axes versus the initial semimajor axes of the stars in Figure 9. As is our convention in the main text, green circles represent stars with M final > 1 M ⊙ , while orange dots have M final < 1 M ⊙ . TDEs and unbound stars are represented using stars and diamonds. The black dashed line shows where the initial and final semimajor axes are equal. Mergers work to shrink the semimajor axis, while high speed collisions can increase or decrease it, as is visible in the Figure. For the most part, green circles lie below the black dashed lines. A few lie above the black line, mostly within 0 . 1 pc, because they also experienced a high speed collision. They initially resided within ∼ 0 . 1 pc, where high speed collisions become more likely. We examine the final eccentricities for merger-affected stars in Figure 10. We compare the final eccentricity distribution to the initial thermal distribution. For reasons outlined in Section 4.2, type B mergers circularize orbits. This effect can be seen in Figure 10, where there is a dearth of eccentric orbits. The effects on eccentricity in type A conditions are harder to tease out. 50% of the time the orbits are aligned and the eccentricity decreases. The other 50% of the time, the angular momenta vector are anti-aligned, so even though the semimajor axis still shrinks, the eccentricity can increase. The most eccentric merged stars are removed from the population because they become TDEs or plunges. The result is an eccentricity distribution that flattens at high ( e > 0 . 5) eccentricities. However, the actual number of merged star orbits that became more (less) eccentric are similar.","pages":[14]},{"title":"REFERENCES","content":"Alexander, T. 1999, ApJ, 527, 835, doi: 10.1086/308129 Alexander, T., & Pfuhl, O. 2014, ApJ, 780, 148, doi: 10.1088/0004-637X/780/2/148 Bahcall, J. N., & Wolf, R. A. 1976, ApJ, 209, 214, doi: 10.1086/154711 Balberg, S. 2024, ApJ, 962, 150, doi: 10.3847/1538-4357/ad1690","pages":[14]},{"title":"Sampling Tracer Stars from the Inner 0.1 pc","content":"*uses a = 1.25 instead of 1.75 Balberg, S., Sari, R., & Loeb, A. 2013, MNRAS, 434, L26, doi: 10.1093/mnrasl/slt071 Bartko, H., Martins, F., Fritz, T. K., et al. 2009, ApJ, 697, 1741, doi: 10.1088/0004-637X/697/2/1741 Binney, J., & Tremaine, S. 2008, Galactic Dynamics: Second Edition Bradnick, B., Mandel, I., & Levin, Y. 2017, MNRAS, 469, 2042, doi: 10.1093/mnras/stx1007 Brown, W. R., Geller, M. J., Kenyon, S. J., & Kurtz, M. J. 2005, ApJL, 622, L33, doi: 10.1086/429378 Brown, W. R., Lattanzi, M. G., Kenyon, S. J., & Geller, M. J. 2018, ApJ, 866, 39, doi: 10.3847/1538-4357/aadb8e Brutman, Y., Steinberg, E., & Balberg, S. 2024, arXiv e-prints, arXiv:2408.16383, doi: 10.48550/arXiv.2408.16383 Dale, J. E., & Davies, M. B. 2006, MNRAS, 366, 1424, doi: 10.1111/j.1365-2966.2005.09937.x Dale, J. E., Davies, M. B., Church, R. P., & Freitag, M. 2009, MNRAS, 393, 1016, doi: 10.1111/j.1365-2966.2008.14254.x Dessart, L., Ryu, T., Amaro Seoane, P., & Taylor, A. M. 2024, A&A, 682, A58, doi: 10.1051/0004-6361/202348228 Ferrarese, L., & Ford, H. 2005, SSRv, 116, 523, doi: 10.1007/s11214-005-3947-6 Freitag, M., & Benz, W. 2002, A&A, 394, 345, doi: 10.1051/0004-6361:20021142 -. 2005, MNRAS, 358, 1133, doi: 10.1111/j.1365-2966.2005.08770.x Gallego-Cano, E., Sch¨odel, R., Dong, H., et al. 2018a, A&A, 609, A26, doi: 10.1051/0004-6361/201730451 -. 2018b, A&A, 609, A26, doi: 10.1051/0004-6361/201730451 Gallegos-Garcia, M., Law-Smith, J., & Ramirez-Ruiz, E. 2018, ApJ, 857, 109, doi: 10.3847/1538-4357/aab5b8 Generozov, A. 2021, MNRAS, 501, 3088, doi: 10.1093/mnras/staa3851 Generozov, A., & Madigan, A.-M. 2020, ApJ, 896, 137, doi: 10.3847/1538-4357/ab94bc Generozov, A., & Perets, H. B. 2022, MNRAS, 513, 4257, doi: 10.1093/mnras/stac1108 Genzel, R., Eisenhauer, F., & Gillessen, S. 2010, Reviews of Modern Physics, 82, 3121, doi: 10.1103/RevModPhys.82.3121 Ghez, A. M., Salim, S., Hornstein, S. D., et al. 2005, ApJ, 620, 744, doi: 10.1086/427175 Ghez, A. M., Duchˆene, G., Matthews, K., et al. 2003, ApJL, 586, L127, doi: 10.1086/374804 Ghez, A. M., Salim, S., Weinberg, N. N., et al. 2008, ApJ, 689, 1044, doi: 10.1086/592738 Gibson, C., Kıro˘glu, F., Lombardi, J. C., J., et al. 2024, arXiv e-prints, arXiv:2410.02146, doi: 10.48550/arXiv.2410.02146 Gillessen, S., Eisenhauer, F., Trippe, S., et al. 2009, ApJ, 692, 1075, doi: 10.1088/0004-637X/692/2/1075 Gillessen, S., Plewa, P. M., Eisenhauer, F., et al. 2017, ApJ, 837, 30, doi: 10.3847/1538-4357/aa5c41","pages":[15,17]}],"string":"[\n {\n \"title\": \"ABSTRACT\",\n \"content\": \"Dense stellar clusters surround the supermassive black holes (SMBH) in galactic nuclei. Interactions within the cluster can alter the stellar orbits, occasionally driving a star into the SMBH's tidal radius where it becomes ruptured. This proof-of-concept study examines the orbital effects of stellar collisions using a semianalytic model. Both low and high speed collisions occur in the SMBH's sphere of influence. Our model treats stars in low speed collisions as sticky spheres. For high-speed collisions, we develop a simple prescription based on the limiting case of a hyperbolic encounter. We test a range of collision treatments and cluster conditions. We find that collisions can place stars on nearly radial orbits. Depositing stars within the tidal radius, collisions may drive the disruption of stars with unusual masses and structures: depending on the nature of the collision, the star could be the product of a recent merger, or it could have lost its outer layers in a high speed impact, appearing as a stripped star. We also find that high speed collisions near the periapsis of an eccentric orbit can unbind stars from the SMBH. However, dissipation during these high-speed collisions can substantially reduce the number of unbound stars achieved in our simulations. We conclude that TDEs and ejected stars, even in the hypervelocity regime, are plausible outcomes of stellar collisions, though their frequency in a three-dimensional nuclear star cluster are uncertain. Future work will address the rates and properties of these events. Keywords: Stellar dynamics; Galactic center; Star clusters; Stellar mergers\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"On the Orbital Effects of Stellar Collisions in Galactic Nuclei: Tidal Disruption Events and Ejected Stars\",\n \"content\": \"1 Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA), Northwestern University, 1800 Sherman Ave, Evanston, IL 60201, USA 2 The Observatories of the Carnegie Institution for Science, Pasadena, CA 91101, USA\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1. INTRODUCTION\",\n \"content\": \"Asupermassive black hole resides at the center of most galaxies, where it is surrounded by a dense stellar cluster (e.g. Ferrarese & Ford 2005; Kormendy & Ho 2013; Schodel et al. 2003; Ghez et al. 2005, 2008; Gillessen et al. 2009, 2017; Neumayer et al. 2020). A tidal disruption event (TDE) occurs when a star from the cluster passes within a critical distance from the SMBH and becomes ruptured by tidal forces (e.g., Hills 1975; Rees 1988; Alexander 1999; Magorrian & Tremaine 1999; Wang & Merritt 2004; MacLeod et al. 2012). The SMBH then accretes the stellar material, producing an electromagnetic signature (e.g., Guillochon & Ramirez-Ruiz 2013). Spectra of these events encode valuable information about the mass, structure, and composition of the ruptured star (e.g., Kochanek 2016a,b; Yang et al. 2017; Mockler et al. 2022; Miller et al. 2023). Observations of Corresponding author: Sanaea C. Rose sanaea.rose@northwestern.edu TDEs represent a powerful way to probe the stellar populations in galactic nuclei and the processes that shape them. Amongst these physical processes are direct collisions. Collisions can occur within the sphere of influence of the SMBH due to the high densities and velocity dispersion (e.g., Freitag & Benz 2002; Dale et al. 2009; Dale & Davies 2006; Rubin & Loeb 2011; Balberg et al. 2013; Balberg 2024; Mastrobuono-Battisti et al. 2014; Rose & MacLeod 2024). These events may produce electromagnetic signatures, both from the collisions themselves and from interactions between liberated material and the SMBH (e.g., Rosswog et al. 2009; Lee et al. 2010; Balberg et al. 2013; Dessart et al. 2024; Ryu et al. 2024b,a; Brutman et al. 2024), though it is difficult to ignite a main-sequence star with compression (Guillochon et al. 2009). They can also shape the stellar population. Both low and high speed collisions can alter the mass of a star - the latter can destroy stars completely and the former can give rise to blue stragglers in dense stellar systems (e.g., Lai et al. 1993; Rauch 1999; Sills et al. 1997, 2001; Lombardi et al. 2002; MacLeod et al. 2013; Leigh et al. 2016; Rose et al. 2023). Recently, Gibson et al. (2024) have shown that high speed collisions can also produce stripped stars similar to what might be seen through binary evolution. Collisions that produce stellar mergers or stripped stars may explain recent unexpected TDE observations. For example, detections of high nitrogen-tocarbon (N/C) ratios in TDEs point to the disruption of more stars that burn on the CNO cycle (as first proposed by Kochanek 2016a,b), and that are ≳ 1 -2 M ⊙ (Kochanek 2016a; Yang et al. 2017; Gallegos-Garcia et al. 2018; Mockler et al. 2023) than is predicted by the host galaxies' stellar populations (Mockler et al. 2023). One particular TDE has such an extremely high N/C abundance ratio that it is difficult to explain with single stellar evolution alone (Miller et al. 2023), but could be the result of the disruption of a stripped star that has lost its nitrogen-poor envelope (Mockler et al. 2024). Additionally, a recently discovered population of extremely bright nuclear transients has also been suggested to originate from the disruption of high mass stars (e.g. ≳ 10 M ⊙ , Subrayan et al. 2023; Hinkle et al. 2024). Because stars that end in disruptions are expected to be drawn approximately at random from the stellar mass function (with small adjustments for stellar type, MacLeod et al. 2012), this preference for higher mass stars may imply that the mass function in galactic nuclei is more top-heavy (and/or bottom-light) than in the rest of the galaxy (see e.g., Lu et al. 2013; Hosek et al. 2019). The potential link between TDEs and collisions is intriguing: collisions can simultaneously affect both the properties of the stars and their orbits about the SMBH. MacLeod et al. (2012) first considered collisions, particularly destructive ones, in the context of the TDE rate. Changes to a star's trajectory from collisions have also been considered in general dynamical models of dense stellar systems (e.g., Sanders 1970; Rauch 1999; Freitag & Benz 2002; Kremer et al. 2020; Gonz'alez et al. 2021; Rodriguez et al. 2022). In this proof-of-concept study, we study the orbital effects of stellar collisions in galactic nuclei. We assess whether the collisions that produce unusual stars could plausibly deposit those same stars onto TDE-producing orbits. Our models leverage simple, intuitive treatments for collision outcomes and fitting formulae from previous studies (e.g. Lai et al. 1993; Rauch 1999). We test a range of initial conditions and nuclear star cluster properties. Our paper is organized as follows: In Section 2, we discuss our general approach to modeling the Milky Way's nuclear star cluster. Section 3 describes the treatment of various physics in our code, with Section 3.2 in particular outlining our methodology for updating the stellar orbits post collision. These orbital changes represent the most major addition to the code as compared to previous iterations (Rose et al. 2022, 2023). Section 4 presents simulated results for direct collisions. Sections 4.1, 4.2, and 4.3 discuss the implications for TDEs, orbital properties of collision-affected stars, and unbound and hypervelocity stars. We then incorporate relaxation into our simulations in addition to stellar collisions and present results in Section 5. Lastly, we summarize the scope and findings of our study in Section 6.\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"2. MODEL NUCLEAR STAR CLUSTER\",\n \"content\": \"We leverage semi-analytic models to study the effects of collisions on the nuclear star cluster. Our fiducial model uses the conditions and properties of the Milky Way's GN, whose proximity makes it the best studied galactic center. We follow a sample of stars embedded in a fixed, unevolving cluster. For simplicity, both the evolving sample and the surrounding cluster are composed of 1 M ⊙ stars. The cluster can be understood in two key properties, density and velocity dispersion, which govern the dynamical processes unfolding within it. The stellar density sets the the frequency with which stars interact. We describe the stellar density as a function of distance from the SMBH using a power law: where α is the slope and r · , distance from the SMBH. Based on observations of the cluster within the sphere of influence, this equation is normalized using ρ 0 = 1 . 35 × 10 6 M ⊙ / pc 3 at r 0 = 0 . 25 pc (Genzel et al. 2010). Our fiducial model uses a slope of 1 . 75, the expectation for a single-mass population (Bahcall & Wolf 1976), consistent with the fact that our simple model cluster has only solar mass stars. However, in order to capture the range of theoretical predictions and observational constraints on the stellar cusp (e.g., Bahcall & Wolf 1976; Gallego-Cano et al. 2018a; Linial & Sari 2022), we also test a few simulations with α = 1 . 25, shown in Appendix A. We assume that the slope of the stellar cusp is roughly constant, or varying slowly, over the timescales of interest in our simulations. The velocity dispersion within the cluster also influences the frequency and nature of stellar interactions. It decreases with distance from the SMBH: where α is the slope of the density profile and M · is the mass of the SMBH (Alexander 1999; Alexander & Pfuhl 2014). We take M · to be 4 × 10 6 M ⊙ , like the Milky Way's SMBH (e.g., Ghez et al. 2003). For a uniform mass cluster of 1 M ⊙ stars, the number density n is simply ρ ( r · ) 1 M ⊙ .\",\n \"pages\": [\n 2,\n 3\n ]\n },\n {\n \"title\": \"3. SEMIANALYTIC MODEL\",\n \"content\": \"We follow a sample of 1 M ⊙ tracer stars embedded in our model cluster. We draw their orbital eccentricities from a thermal distribution. We select their semimajor axes so that they lie on a cusp with slope α , matching the background cluster. These stars are allowed to evolve under the influence of two main dynamical processes, direct collisions and two-body relaxation, using a model first developed by Rose et al. (2022, 2023).\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"3.1. Collisions\",\n \"content\": \"Direct collisions occur over a characteristic timescale t -1 coll = nσA , where A is the cross-section of interaction, n is the number density, and σ is the velocity dispersion. For an impact to occur, A is the physical crosssection enhanced by gravitational focusing. The collision timescale also depends weakly on the star's orbital eccentricity and can be written as: where f 1 ( e · ) and f 2 ( e · ) are equations 20 and 21 from Rose et al. (2020), G is the gravitational constant, and a · is the star's semimajor axis. r c is the sum of the radii of the colliding stars, or 2 R ⊙ for a uniform population of solar mass stars. We plot this timescale in red in the upper panel of Figure 1. We consider a range of slopes for the stellar density profile, spanning α = 1 . 25 (dashed line) to α = 1 . 75 (solid line). The horizontal grey line shows the total simulation time, included to guide the eye. Where the collision timescale is less than the simulation time, within 0 . 1 pc of the SMBH, collisions become important to understanding the evolution of the cluster (e.g., Rose & MacLeod 2024). We treat stellar collisions using a statistical approach. We begin by computing the probability that a star in our sample will experience a collision. Over a timestep ∆ t , this probability equals ∆ t/t coll . ∆ t is taken to be 10 6 years so that the probability ∆ t/t coll is always less than one; t coll ≳ 10 7 years for the parameter space we consider. The code then draws a random number between 0 and 1, which, if less than or equal to the collision probability ∆ t/t coll , means a collision has occurred. We repeat this prescription until the desired simulation runtime or the star's main-sequence lifetime has been reached, whichever is shorter. If a collision has occurred over a given timestep, we update the mass and age of the star using a prescription detailed in Rose et al. (2023). A full discussion of our approach can be found in previous papers (Rose et al. 2023; Rose & MacLeod 2024). In brief, we combine fitting formulae from hydrodynamics simulations of stellar collisions (Rauch 1999; Lai et al. 1993) with heuristic arguments to (1) determine if a given collision will result in a merger and (2) determine the amount of mass lost from either the merger product or the unbound individual stars. Since speeds in galactic nuclei often exceed hundreds of kilometers per second, collisions can eject anywhere from a percent to all of the star's mass, effectively blowing it up (Spitzer & Saslaw 1966; Lai et al. 1993; Balberg et al. 2013; Balberg 2024; Brutman et al. 2024). Mergers with minimal mass loss are most likely to occur outside of about 0 . 01 pc, where the relative speeds tend to be less than the escape speed from the stars. Within 0 . 01 pc, however, velocities exceed the escape speed from the stars, and collisions can result in peculiar, low-mass 'stripped stars' (Rose et al. 2023; Rose & MacLeod 2024; Gibson et al. 2024). In practice, the outcome of a collision is more complex to determine, depending on properties such as the impact parameter and stellar structure (e.g., Freitag & Benz 2005; Gibson et al. 2024, Rose et al. in prep.). For the purposes of this study, which focuses on the general implications of collisions for stellar orbits in galactic nuclei, the specific mass loss and merger prescriptions should not qualitatively change our results. Unless otherwise specified, all simulations shown herein use fitting formulae from Rauch (1999) for the mass loss and escape speed arguments to determine whether or not a collision results in a merger (Rose et al. 2023). In previous iterations of this code, the collision speed was taken to be simply the velocity dispersion at the star's distance from the supermassive black hole, given by Eq. (2) (Rose et al. 2022, 2023). However, this approach is insufficient for studying the effects of collisions on stellar orbits. In this study, we draw the orbit of the second colliding star, using a procedure detailed below.\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"3.2. Orbital Dynamics\",\n \"content\": \"If a star in our sample with semimajor axis a · and eccentricity e · collides over timestep ∆ t , we find a plausible orbit for the second colliding star. As we operate under the assumption that the cluster is composed of a uniform stellar population, we always take the second star to have m collider = 1 M ⊙ . We then draw semimajor axes and eccentricities from the cluster's property distri- as described in Section 2, until we find an orbit that intersects with that of our tracer star. Our code assumes that the cluster is spherically symmetric and only tracks the semimajor axis and eccentricity of each tracer star. In this set up, our tracer star always has argument of periapsis equal to zero, however the second colliding star's periapsis need not be aligned with it. The angle between them is drawn from a uniform distribution. The orbits are always assumed to be coplanar. Observations suggest that a subset of the stars in the Milky Way's galactic center reside in a disk, while others have an isotropic distribution (e.g., Levin & Beloborodov 2003; Ghez et al. 2003, 2005; Gillessen et al. 2009; Paumard et al. 2006; Lu et al. 2009; Bartko et al. 2009; Yelda et al. 2014). In our co-planar physical picture, we have two options: either both of the colliding stars can orbit in the same direction about the supermassive black hole, or they can be equally likely to be prograde or retrograde. We test two extremes to capture the effects of different orbital orientations. Our first case, 'disk-like', has the two colliding stars orbiting the SMBH in the same direction. In the second case, 'isotropic-like', we assume that fifty percent of the time the two stars orbit the SMBH in the same direction and fifty percent of the time in opposite directions. The latter orientation means that the orbital angular momentum vectors are anti-parallel to one another. Once we have found an intersecting orbit for our colliding tracer star, we determine the intersection point and compute the velocity vectors of the two stars at that point. The relative speed tells us whether or not a merger occurs and the degree of fractional mass loss from the system (see above section). From here, determining the final orbit(s) will depend on the type of outcome and therefore the relative speed.\",\n \"pages\": [\n 3,\n 4\n ]\n },\n {\n \"title\": \"3.2.1. Final Orbit in Merger Case\",\n \"content\": \"A stellar merger is the natural outcome of a low-speed collision. We calculate the final mass, age, and now trajectory of the product assuming the stars act as 'sticky spheres' (e.g., Rodriguez et al. 2022; Rose et al. 2023). The two stars approach each other with velocity vectors v · and v collider as determined by the intersection point of their orbit. Momentum conservation demands that the final velocity of the merger product, v final , equals ( M v · + M ⊙ v collider ) / ( M + M ⊙ ). Based on hydrodynamics studies of stellar mergers, the mass loss in these collisions should be low (e.g.., Lai et al. 1993; Rauch 1999, Rose et al. in prep.). With the final velocity, mass, and location of the collision - the intersection point of the original orbits - we can calculate the new orbit of the merger product about the SMBH.\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"3.2.2. Final Orbits Following a High-Speed Collision\",\n \"content\": \"Orbital changes from high-speed collisions are much more difficult to determine in the absence of hydrodynamics simulations. However, we present a framework for treating these interactions. In order to understand these collisions, we start at the limit where the two stars barely graze. This interaction should unfold as a hyperbolic encounter. In the center of mass frame of the two stars, their speeds remain constant, but their velocity vectors are deflected by angle θ hyp . θ hyp can be found analytically: where b is the impact parameter and b 90 is defined as the impact parameter needed for a 90 · deflection. b 90 equals 2 G/v 2 rel , where v rel is the relative speed between the two stars (e.g., Binney & Tremaine 2008). We plot b 90 as a function of the relative speed in the bottom panel of Figure 1. In the nuclear star cluster, the velocity dispersion can be understood as the characteristic relative speed between stars at a given distance from the SMBH (see Eq. 2). We have therefore inverted the x-axis of the bottom plot to parallel that of the upper plot, distance from the SMBH, and marked the velocity dispersion at key distances using vertical dashed lines. In both the upper and lower plots, we also indicate the regions in which we expect mergers versus high-speed collisions, which leave the stars unbound from each other. Interestingly, for most of the parameter space where these high-speed, non-sticky sphere collisions occur, b 90 is less than the star's radius. Another way of interpreting this statement is that at high speeds, physical collisions are required for a strong-angle deflection. If the stars were point particles, the impact parameter could be arbitrarily small and the interaction would still unfold as a hyperbolic encounter. Two Sun-like stars begin to touch when b = 2 R ⊙ . With b < 2 R ⊙ , the stars physically impede each other as they interact, leading to a smaller deflection angle than the one given in Eq. (4). Furthermore, if the stars approach each other perfectly head-on with b = 0, the center of mass velocity is 0 and there is no angular momentum. In this case, there would be no deflection. Heuristically, then, we expect the deflection angle to be given by Eq. (4) for grazing encounters, and some fraction of this angle for encounters with b < 2 R . That fraction should decrease with impact parameter until they are both zero. The precise dependence is impossible to determine in the absence of hydro simulations, but we define a collision deflection angle that meets the two limiting criterion: where r c is the sum of the radii of the two colliding stars. It is reasonable to expect that the collision affects the stars' speeds as well. Even more so than the deflection angle, changes in speed can only be understood through hydrodynamic simulations. In our models, we simply test three cases: one in which the speed is not affected at all, one in which the speed in the center of mass frame is always reduced by 10%, and one in which it is always reduced by a factor of 2. We treat these collisions as follows: at the intersection point of the orbits of the two colliding stars, we calculate their center of mass velocity. We transform to the center of mass frame. Only then do we rotate and scale their velocity vectors. While we only track semimajor axes and eccentricities, we do allow the deflections to be three dimensional, with components out of the plane. The final velocity can be any vector, chosen randomly, along a cone defined by angle θ coll with respect to the original velocity. Isotropizing the deflection is important because otherwise the colliding stars would always be scattered exactly towards or away from the SMBH, leading to an gross overestimate of TDE rates. We then transform back to the frame of the SMBH and calculate the new orbits given the velocity and position vectors. We note that our calculation assumes that the center of mass is not moving in the frame of the supermassive black hole. In actuality, the center of mass of the two stars would orbit the supermassive black hole, but the effects will be negligable due to the high collision speeds.\",\n \"pages\": [\n 5\n ]\n },\n {\n \"title\": \"3.3. Two-Body Relaxation\",\n \"content\": \"In addition to collisions, stars in the cluster experience the weak gravitational effects of nearby neighbors. The effects of these interactions can accumulate, eventually changing the star's orbital energy and angular momentum by order of itself. The original orbit is 'erased' over a characteristic timescale: where ⟨ M ∗ ⟩ is the average star's mass, here taken to be 1 M ⊙ , and ln Λ rlx is the coulomb logarithm (e.g., Binney & Tremaine 2008; Merritt 2013). Figure 1 shows the relaxation timescale as a function of distance from the SMBH in blue for a range of stellar density profiles. We account for relaxation by allowing the orbital eccentricity and semimajor axis of each of our tracer stars to slowly evolve. Once per orbit, we apply a small in- stantaneous change in velocity to each star (e.g., Bradnick et al. 2017; Lu & Naoz 2019; Rose et al. 2022, 2023; Naoz et al. 2022, see the latter for the full set of equations). The kick is calibrated so that ∆ v/v ∼ √ ∆ t/t rlx , and if ∆ t = t rlx , ∆ v is of order of the velocity. This prescription simulates the diffusion of the orbital orbital parameters over time from interactions with other stars in the cluster. Previously, it has been used in studies of TDEs and extreme mass ratio inspirals of stellar mass black holes into the SMBH (Naoz et al. 2022; Melchor et al. 2024).\",\n \"pages\": [\n 5,\n 6\n ]\n },\n {\n \"title\": \"3.4. Orbital Stopping Conditions\",\n \"content\": \"As noted in Section 3.1, we terminate the simulation when the desired runtime of 10 billion years is reached or when the time elapsed has exceeded the star's mainsequence lifetime, whichever comes first. However, orbital changes from collisions or relaxation can also send stars into the tidal radius, where they will be destroyed. We trigger a stopping condition if the star's periapsis a · (1 -e · ) becomes less than twice the tidal radius from the SMBH, R star ( M · /M star ) 1 / 3 (e.g., Guillochon & Ramirez-Ruiz 2013; Mockler et al. 2023). Tidal disruption events can be characterized by impact parameter β = R tidal / ( a · (1 -e · )). Our stopping condition corresponds to β = 0 . 5, allowing us to capture partial as well as full disruptions. Because both direct collisions and relaxation processes can place a star onto a nearly radial orbit, we log whether the critical orbit was reached through a direct collision or our relaxation prescription. A star that becomes a TDE due to relaxation processes can still have experienced one or more collision previously in its life. However, a star that collides and becomes a TDE due to the collision itself may still be inflated from the impact when it reaches its periapsis. In this case, the stopping condition noted above may be conservative; the R star is simply the radius expected using a mass-radius relation for a main-sequence star of mass m star . High-speed collisions, unlike those that result in mergers, can also cause stars to be ejected from the nuclear star cluster. Consider two stars that collide on elliptical orbits that intersect near periapsis. As the eccentricity approaches unity, the speed at periapsis approaches the escape speed, just shy of becoming an unbound, parabolic orbit. In the limit of a hyperbolic encounter, the speeds of the stars are unchanged in the center of mass frame, but they undergo a deflection. In the frame of the SMBH, this change can boost one star's speed enough to unbind it from the SMBH, while the other star ends up on a more tightly bound orbit. Close encounters have previously been shown to eject stars from dense stellar systems (e.g., Henon 1969; Lin & Tremaine 1980). Additionally, high-speed collisions often lead to mass loss, which can unbind the orbit not unlike a supernova (e.g., Lu & Naoz 2019). Therefore, we include a stopping condition for orbital energy ≥ 0 and eccentricity ≥ 1. We do not consider stars that become unbound after losing mass in a tidal disruption, as proposed by Manukian et al. (2013).\",\n \"pages\": [\n 6\n ]\n },\n {\n \"title\": \"4. NUMERICAL RESULTS WITHOUT RELAXATION\",\n \"content\": \"We run large sets of 10000 tracer stars and let them evolve over 10 billion years. We begin by presenting simulated results without our relaxation prescription. These simulations allows us to isolate the orbital effects of collisions unobscured by relaxation. As in a nuclear star cluster, most of our tracer stars reside near ∼ 1 pc, the edge of the sphere of influence. About 500 of our tracer stars, or 5% of the sample, lie within 0 . 1 pc, where collisions are most common (see Figure 1). Sampling in this way gives a more complete picture of the relative rates of each outcome. Figure 2 juxtaposes results for slightly different treatments of the collision physics, labeled as follows: Type A: In this simulation, two colliding stars have a 50% chance of orbiting the SMBH in the same direction and a 50% chance of orbiting in opposite directions. The final velocities after a high-speed collision are calculated using Eq. 5. We assume that these high speed collisions do not impact the speed of the stars in their center of mass frame. Type B: Simulations with type B conditions differ from type A by assuming that all stars orbit the SMBH in the same direction. Type C: These simulations are identical to type A, except high speed collisions decrease the speeds in the stars' center of mass frame by 10%. In other words, we scale the velocity vectors of the stars by 0 . 9 before tranforming back to the frame of the SMBH. Type D: These conditions are identical to types A and C except high speed collisions drecrease the speed in the center of mass frame by 50%. The left column of Figure 2 shows the final masses and semimajor axes of the tracer stars. Red open circles represent stars at the end of the simulation or their mainsequence lifetime, whichever came first. Grey points represent the initial conditions. Blue diamonds indicate stars that were placed on unbound orbits by high speed collisions. Note that unbound orbits have negative semimajor axes, so we plot the absolute value. We provide the total number of stars that escape on unbound orbits in the upper right corner of each plot. In addition to these unbound orbits, collisions can also place stars on orbits with semimajor axes outside the inner pc. TDEs from direct collisions are represented by the star shaped symbols. The color indicates the type of collision responsible for the radial orbit: low-speed collisions, which we refer to as merger TDEs, are shown in black, while lime represents high speed collisions. Some of these orbits are so extreme that the periapsis distance lies within the Schwarzschild radius. We mark these plunging stars using an 'x' symbol. The right column shows the properties of the TDEs and plunges for each simulation. The y-axis shows the orbital period of the final orbit, which carries them into the tidal radius. On the x-axis, we plot the parameter β , which quantifies how deeply the orbit penetrates the tidal radius. 0 . 5 < β < 1 indicate a partial disruption. To guide the eye, grey dashed line marks where the orbital period equals 30 yr. TDEs with periods less than this could conceivably be observed as a repeating TDE. We discuss the implications for TDEs below.\",\n \"pages\": [\n 6,\n 8\n ]\n },\n {\n \"title\": \"4.1. Tidal Disruption Events\",\n \"content\": \"Collision-induced mergers can result in TDEs under conditions A, C, and D, but not B. These merged stars are placed on nearly radial orbits when the colliding stars are orbiting in opposite directions. Their angular momenta vector are anti-aligned, giving a low angular momentum to the final orbit of the merger product. Under type B collision conditions, in contrast, there are no merger TDEs. The orbital angular momenta are never in a position to cancel each other out. Fewer TDEs result from high speed collisions. The relative abundances reflect the relative frequency of each type of collision. High speed collisions tend to occur near the SMBH where speeds are sufficiently high, but fewer stars reside. These high speed TDEs are relatively rarer because this type of collision is relatively rarer. Collisions in general tend to become common within the inner 0 . 1 pc of the cluster. However, only about 500 stars from our sample lie in this region. We find that when we sample this region in greater detail, both high and low speed collision TDEs become more common. One high speed collision even produced a repeating TDE (see additional simulations in Section A.2). Others were close to our repeating TDE threshold with 30-50 yr periods. There are some initial conditions that consistently fail to produce high speed collision TDEs, notably Type D conditions. A shallower density profile will also lead to a lower collision rate overall because the timescale is longer (see Figure 1), in turn reducing the rate of collision TDEs (see Appendix A.2). Additionally, we note that our prescription for determining whether or not a merger results from a collision may over-predict the number of mergers. We use fitting formulae from Rauch (1999) to determine the mass loss in a collision and heuristic arguments, similar to the aforementioned study, to determine if a merger occurs. Previous studies with this code (Rose et al. 2023; Rose & MacLeod 2024) have also tested simulations using fitting formulae from Lai et al. (1993), who include a fitting formula for the merger capture radius. Their formula leads to fewer mergers compared to our first approach. However, collision-induced mergers would still be viable as a mechanism to create TDEs; they would simply occur at a much lower rate. As with other additional simulations, we show examples using the Lai et al. (1993) fitting formulae in the Appendix (Figure 8) to avoid overcrowding the main text.\",\n \"pages\": [\n 8\n ]\n },\n {\n \"title\": \"4.2. Orbital Changes from Collisions\",\n \"content\": \"We consider general trends in the effects of collisions on the orbital properties of the stars. We do this by examining sub-populations of the tracer stars based on their final mass. We stress that this classification does not necessarily give the full picture of a star's collision history. While all stars with M final > 1 M ⊙ must have undergone a merger - recall that all stars in are cluster are initially 1 M ⊙ - and stars with M final < 1 M ⊙ must have lost mass in a high speed collision, these two populations are not mutually exclusive. A single star can experience both types of collisions over its lifetime depending on where it is in its orbit and the orbit of the second colliding star. However, final mass still represents the best 'observable' probe of a star's collision history. We compare the semimajor axis versus 1 -e of the stars in Figure 3, the latter being proportional to the periapsis. Grey dots show the initial conditions for all the stars. Green circles represent stars with M final > 1 M ⊙ , while orange dots represent stars with M final < 1 M ⊙ . TDEs and unbound stars are represented using the same symbols as in previous figures. We confirm that the TDEs in our simulation do indeed come from nearly radial orbits. Mergers and high speed collisions differ in terms of their orbital outcomes. Conservation of energy demands that mergers always shrink the semimajor axis, while high speed collisions can place stars on both wider or smaller orbits by facilitating an energy exchange between the two stars. These trends are visible in the Figure. The green open circles are mostly confined to the inner cluster, while orange dots have an extended distribution. A few merged stars lie outside our initial cluster. These merged stars experienced a high speed collision that deflected them onto wider orbits. In simulations A and B, a roughly equal number of stripped stars moved to more (less) tightly bound orbits. We count unbound stars in the less tightly bound category because their final orbital energies are larger than the initial values. Additionally, about 10 stars are placed on orbits with semimajor axes outside of 1 pc. This result suggests that there may be collision-affected stripped stars masquerading outside of the sphere of influence. Including a treatment for dissipation during a high speed collision, simulation C presents a different story. The orbits of stripped stars are more likely to shrink and become more circular. About 75% become more tightly bound. Additionally, a population of short period stripped stars emerges. These stars likely experienced multiple high speed collisions, giving them multiple opportunities to shrink their semimajor axes. The effects are even more pronounced for simulation D (not shown). Under type D conditions, stripped stars are overwhelmingly on less eccentric orbits, explaining why the high speed collision TDE rate for these conditions are so low. Eccentricity trends can also be gleaned from Figure 3. For type B conditions, mergers always act to make the orbit more circular. In addition to energy, angular momentum must be conserved. For stars orbiting in the same direction, their angular momentum vectors are always aligned. Since the semimajor axis shrinks after a collision, the eccentricity must decrease to conserve angular momentum. This effect can be seen in vertical extent of the green circles in B compared to A. The most eccentric orbits for merged stars in B have been largely removed. We reiterate that the results are somewhat muddied by the occurrence of high speed collisions in a subset of this population. Under type A conditions, low speed collisions both decrease and increase the orbital eccentricity. The most eccentric final orbits become TDEs or plunges, removing the stars from the population. We note that in actuality, the orbital properties of the stars can be modified by resonant and non-resonant relaxation processes, not accounted for in these results (e.g., Rauch & Tremaine 1996; Hopman & Alexander 2006; Kocsis & Tremaine 2011).\",\n \"pages\": [\n 8,\n 9\n ]\n },\n {\n \"title\": \"4.3. Unbound Stars\",\n \"content\": \"High speed collisions can unbind stars from the SMBH. These interactions, treated similarly to a hyperbolic encounter, can place one star on a more tightly bound orbit while giving the other star a positive orbital energy. We show properties of the unbound stars from simulations A and B in Figure 4. The initial orbits tend to be eccentric. We also confirmed by inspection of specific cases that the collisions tend to occur near periapsis. For the most part, the final orbits have eccentricity just above unity. We calculate the speed at infinity, v inf , for these stars based on their final orbital energy and color code the points in Figure 4 based on its value. Typical speeds range from ∼ 100 to 600 km/s, but occasionally one star will have a v inf above 1000 km/s, as was the case in simulation B. For simulation C, which includes some form of dissipation during high speed collisions, there were fewer unbound stars overall, but their distribution was similar. The maximum v inf was 642 km/s. The single unbound star from the fourth row of Figure 2 (simulation D) had v inf equal to 377 km/s. These maximum values suggest that high speed collisions may represent another mechanism to launch hypervelocity stars from galactic nuclei. The most famous of these mechanisms is the Hills Mechanism, in which a binary is disrupted by the SMBH such that one star is ejected at high speed while the other is retained on a tightly bound orbit (e.g., Hills 1988; Generozov & Madigan 2020), though other mechanisms exist (e.g., Yu & Tremaine 2003; Perets 2009a). Close encounters between single stars are known to eject stars from dense stellar clusters (Henon 1969; Lin & Tremaine 1980), and in fact Yu & Tremaine (2003) consider such interactions as a means of producing hypervelocity stars. Omitting collisions, they find a low ejection rate at speeds ≳ 300 km/s. We consider collisions exclusively and treat them as modified close encounters. While our largest speed is consistent with those generated by the Hill's Mechanism ( ≳ 1000 km/s, Hills 1988), about 30% have v inf > 300 km/s. Observations of hypervelocity stars with origins pointing to the Galactic center exhibit a range of speeds, from the hundreds to thousands of km/s (e.g., Brown et al. 2005, 2018; Koposov et al. 2020; Generozov & Perets 2022, to quote from the latter, stars need ejection speed ≳ 750 km/s from the Galactic center to have 275 km/s at 20 kpc). The v inf of our stars shown in Figure 4 do not account for any additional terms in the potential beyond the SMBH. We reserve a detailed comparison of our results to observations for future work. The mass loss fitting formulae from Rauch (1999) give very low mass loss for larger impact parameter collisions ( b ≳ 0 . 6 r c ) at moderately high speeds ( ∼ 1000 km/s); generally, the fractional mass loss in these cases is less than a percent, and unbound stars in Figure 2 appear as if they have not lost any mass at all. However, fitting formulae from high speed collisions can be unreliable (Freitag & Benz 2005). We therefore refrain from commenting on the masses of the unbound stars in detail except to state that some of them may look like stripped stars (Gibson et al. 2024). As can be seen in Figure 2, a maximum of ∼ 1% of the stars in the inner parsec region may become unbound from the SMBH. This number far exceeds those that become TDEs through high speed collisions. Furthermore, this outcome represents a surprisingly high fraction of stars that experience such collisions: about a fifth of stars that experience high speed collisions become unbound. This high number may owe to a few conspiring conditions: For reasons discussed in Section 3.4 and as supported by Figure 4, unbound stars tend come from high speed collisions near the periapsis of eccentric orbits. Near periapsis, a small boost in the frame of the SMBH can tip the star's speed over the escape speed. However, the merger-to-high speed collision boundary also lies around 800 km/s. The velocity dispersion only exceeds this value within about 0 . 02 pc, and the vast majority of stars reside outside this distance. In consequence, most high speed collisions can only occur near periapsis, where speeds are high enough to no longer be in the merger regime and also where unbinding the star from the SMBH becomes a more favorable outcome. Our thermal initial eccentricity distribution ensures that plenty of stars begin on very eccentric orbits (see Appendix A.1 for the impact of the eccentricity distribution). Dissipation during the collision can reduce the number of unbound stars. As discussed in Section 3.2.2, it is possible that high speed direct collisions reduce the speeds of the stars in their center of mass frame. The physical impact of the collision may reduce the number of stars that get enough of a boost to escape the inner parsec. Type C and D collision conditions test the limits of this mechanism in producing unbound stars. Notably, if high speed collisions reduce the post-collision speed by 50% in the center of mass frame, only a few tenths of a percent of the cluster stars escape.\",\n \"pages\": [\n 9,\n 10\n ]\n },\n {\n \"title\": \"5. NUMERICAL RESULTS WITH RELAXATION\",\n \"content\": \"Now that we have built a physical picture of how collisions shape the stellar cluster, we turn on our prescription for two-body relaxation in the code. These simulations take longer to run, so the sample size of the tracer stars presented herein number at 4000 instead of 10000. We present results in Figure 5. The symbols are all the same as in previous figures, with one addition: stars that became TDEs through two body relaxation are shown in dark green. The orbital evolution of these stars where still affected by collisions, as can be seen by their masses. Interestingly, the stripped star that became a TDE via relaxation had experienced a collision within the same timestep. The collision had placed it on\",\n \"pages\": [\n 10\n ]\n },\n {\n \"title\": \"Unbound Orbits Properties\",\n \"content\": \"an eccentric orbit just shy of the tidal radius, periapsis of 1 . 135 AU versus 0 . 956 AU needed for partial disruption. Had the star been inflated at all from the collision, material would have been siphoned off by the SMBH. As the star's parameters did not trigger the stopping condition, our relaxation prescription tipped the star into the tidal radius. We place a caveat on the TDE inside 0 . 001 pc. The closest known star to the Milky Way's SMBH has a periapsis just under 0 . 001 pc (e.g., Gillessen et al. 2017). It is questionable whether our two-body relaxation prescription applies within this radius. However, this star too experienced a collision in the previous timestep which, had the star been inflated, would have led to a disruption. Altogether, our results suggest that collisions represent a viable mechanism to preferentially deposit stars that are (1) more massive than the surrounding population due to a merger or (2) stripped stars into the tidal radius. Some of these TDEs could be observed as repeating TDEs. Additionally, we may be missing disruptions from stars that are inflated post collision, leading to qualitatively different electromagnetic signatures (e.g., MacLeod et al. 2013; Gibson et al. 2024). The unbound stars present something of a puzzle. Even accounting for the difference in sample size, we find much fewer compared to Figure 2 (no relaxation). There are a few possible explanations. First, the relaxation prescription changes the orbital energy by roughly order of itself over the relaxation timescale, which is less than 10 Gyr in the model cluster. Stars at larger radii can therefore wander towards the SMBH, where destructive collisions become possible. Once in this 'danger zone', they can experience enough high-speed collisions to become destroyed. Roughly 500 stars were destroyed over this simulation. While this process is slow (see e.g., Rose & MacLeod 2024), it does drain stars from the sample population. In the simulation without relaxation, the majority of stars began and remained outside of this region. Only ∼ 50 stars were destroyed, a much smaller fraction of the sample, and most unbound stars had initial semimajor axes outside of 0 . 1 pc (see Appendix A.4). The reduction of unbound stars may also be related to the interplay between the two dynamical processes. We speculate on why this second option might be true. Collisions are discrete events. Relaxation, in contrast, is a diffusion process that slowly alters a star's orbital parameters. This timescale is less than the collision timescale for most of the stars in our sample (see Figure 1), particularly where the unbound stars originate in our earlier simulations. The implication is that when a collision does occur, it may be less likely to catch the orbit in a state favorable for unbinding the star. The last possibility is that there is something in our relaxation prescription which we have yet to detect that is artificially suppressing the unbound stars. 1 Regardless, the interplay of relaxation and collisions merit a more thorough examination. We reserve a detailed study for future work.\",\n \"pages\": [\n 11\n ]\n },\n {\n \"title\": \"6. CONCLUSIONS\",\n \"content\": \"Collisions between main-sequence stars occur in the nuclear star cluster and become common within the inner 0 . 1 pc (e.g., Rose & MacLeod 2024). In this proof-ofconcept study, we examine the orbital changes that result from these collisions. Specifically, we assess whether physical collisions can contribute to the production of TDEs and ejected stars. Collisions in galactic nuclei can be understood in two types. In the first type, the relative speed between the stars is low and the collision results in a merger. Little mass loss occurs (e.g., Lai et al. 1993) and we can treat the stars as sticky spheres (e.g., Kremer et al. 2020; Gonz'alez et al. 2021). Conservation of momentum allows us to calculate the final orbit of the merged star. The second type of collision occurs at speeds that exceed the escape velocity from the stars. While the high speeds ensure that the stars remain unbound from each other, they can also drive mass loss from the stars, in some cases producing a stripped star (e.g., Lai et al. 1993; Rauch 1999; Freitag & Benz 2005; Gibson et al. 2024). We use fitting formulae from Rauch (1999) to calculate the mass loss. However, we refrain from commenting in detail on the final masses of these stars because fitting formulae are not always accurate for collisions in galactic nuclei (Freitag & Benz 2005). We are examining these collisions further using sph simulations in forthcoming work (Rose et al. in prep.). We use limiting cases and heuristic arguments to determine the final orbits from high speed collisions. If the stars were point masses, these interactions would unfold as a hyperbolic encounter in the center of mass frame of the stars. As an upper limit, we could calculate the deflection angle using Eq. 4. However, while a grazing collision approaches the limit given by Eq. 4, a head-on collision should not result in any deflection at all. We therefore adopt Eq. 5, which meets both criteria. We also test the role of dissipation by reducing the speeds of the stars post-collision in their center of mass frame. Our simulations follow a sample of tracer stars embedded in a fixed, uniform cluster. We test a variety of cluster conditions and find the following: Despite these changes, the unbound stars persist, albeit in much lower numbers. We reserve a precise examination of their rates and properties for future work, though some may look like stripped stars (Gibson et al. 2024). In the optimistic case, stellar collisions represent a mechanism to launch hypervelocity stars from galactic nuclei, joining the list of existing mechanisms (e.g., Hills 1988; Yu & Tremaine 2003; Ginsburg & Loeb 2007; Perets 2009a,b; Generozov & Madigan 2020). The Hills Mechanism elegantly explains both the origins of the S-star cluster, young-seeming massive stars in the vicinity of the SMBH, and hypervelocity stars using the same dynamical process (e.g., Hills 1988; Ghez et al. 2003; Ginsburg & Loeb 2007; Perets et al. 2007; Madigan et al. 2009; Lockmann et al. 2009; Generozov & Madigan 2020; Generozov 2021). Collisions present another possibility, where S-stars are the high-mass tail of the merger products created by successive low-speed collisions (see e.g., Rose et al. 2023), and hypervelocity stars trace to high speed collisions near periapsis. However, the viability of this ejection mechanism still faces tests in the form of dissipation during collisions, a stellar mass function, interplay with other dynamical processes such as relaxation, and overall rates in a three-dimensional cluster. We thank Fred Rasio, Enrico Ramirez-Ruiz, Jamie Lombardi, Fulya Kiroglu, Charles Gibson, and Claire Ye for invaluable discussion and input. SR thanks the CIERA Lindheimer Fellowship for support. BM thanks the Carnegie CTAC postdoctoral fellowship for support. This project began while SCR and BM were at the Aspen Center for Physics, which is supported by NSF grant PHY-2210452. This research was supported in part through the computational resources and staff contributions provided for the Quest high performance computing facility at Northwestern University which is jointly supported by the Office of the Provost, the Office for Research, and Northwestern University Information Technology.\",\n \"pages\": [\n 11,\n 12,\n 13\n ]\n },\n {\n \"title\": \"A.1. Role of the Initial Eccentricity Distribution\",\n \"content\": \"Less eccentric orbits on average should result in fewer unbound stars. We test this hypothesis by drawing the eccentricities of tracer stars from a uniform distribution (average e = 0 . 5) instead of a thermal one (average e = 0 . 67). We compare the results in the first row of Figure 2, which uses a thermal initial eccentricity distribution, to that shown in Figure 6. Despite both having type A initial conditions, changing the eccentricity distribution to uniform reduces the number of unbound stars by about a factor of 2. There were also fewer high speed collisions overall, a ∼ 35% reduction. These results come from reducing the number of stars with speeds at periapsis that place them in the high speed collision regime.\",\n \"pages\": [\n 13\n ]\n },\n {\n \"title\": \"A.2. Sampling the Inner Collision-dominated Region\",\n \"content\": \"Most collisions occur in the inner 0 . 1 parsec of the nuclear star cluster, where the collision timescale becomes shorter than 10 Gyr. In order to probe this region in greater detail, we run a similar set of simulations to those shown in Figure 2, but with the stars sampled from 0 . 001 to 0 . 1 pc. We note that a cusp with slope α = 1 . 75 results in a very high density in this region. The actual cusp in the Galactic center may be closer to 1 . 4 (Gallego-Cano et al. 2018b). Additionally, collisions themselves can deplete the cusp in the inner 0 . 1 pc. A shallower cusp would lead to a lower collision rate. We therefore test α = 1 . 25 with the idea that reality may lie somewhere in the middle. These simulations are marked with an asterisk in Figure 7. The shallower profile means that the collision timescale is longer and that fewer collisions occur within the cluster over the integration time. However, as can be seen in the Figure, the collisions that do occur can stil produce TDEs. Additionally, type B conditions, which prohibit merger TDEs, can still have collision-driven TDEs of a different kind.\",\n \"pages\": [\n 13\n ]\n },\n {\n \"title\": \"A.3. Alternative Fitting Formulae\",\n \"content\": \"As mentioned in Section 3.1, we leverage fitting formulae and intuitive toy models to calculate the mass loss from a collision and determine whether a merger occurs. Simulations shown in the main text and in Figure 7 use fitting formulae from Rauch (1999) and heuristic arguments to determine if a merger occurs. In this section, we present simulations with type A collision conditions that use fitting formulae from Lai et al. (1993). Lai et al. (1993) include fitting formula for the capture radius, a function of the relative speed between the stars and their impact parameter. This formula leads to less mergers than the simple prescription that we use in the main text of this paper, which is also similar to the prescription used in Rauch (1999) (see for a full discussion Rose et al. 2023). We present the results of two simulations using the Lai et al. (1993) formulae in Figure 8. The upper row samples tracer stars on a cusp from 0 . 001 to 1 pc, while the second row focuses on the inner 0 . 1 pc. As expected, with the overall reduction in mergers - they instead are treated as high speed collisions - there are fewer merger TDEs. However, it is still possible to get TDEs of all types and unbound stars.\",\n \"pages\": [\n 13,\n 14\n ]\n },\n {\n \"title\": \"A.4. Orbital Parameters\",\n \"content\": \"We expand upon Section 4.2 in this appendix. We compare the final semimajor axes versus the initial semimajor axes of the stars in Figure 9. As is our convention in the main text, green circles represent stars with M final > 1 M ⊙ , while orange dots have M final < 1 M ⊙ . TDEs and unbound stars are represented using stars and diamonds. The black dashed line shows where the initial and final semimajor axes are equal. Mergers work to shrink the semimajor axis, while high speed collisions can increase or decrease it, as is visible in the Figure. For the most part, green circles lie below the black dashed lines. A few lie above the black line, mostly within 0 . 1 pc, because they also experienced a high speed collision. They initially resided within ∼ 0 . 1 pc, where high speed collisions become more likely. We examine the final eccentricities for merger-affected stars in Figure 10. We compare the final eccentricity distribution to the initial thermal distribution. For reasons outlined in Section 4.2, type B mergers circularize orbits. This effect can be seen in Figure 10, where there is a dearth of eccentric orbits. The effects on eccentricity in type A conditions are harder to tease out. 50% of the time the orbits are aligned and the eccentricity decreases. The other 50% of the time, the angular momenta vector are anti-aligned, so even though the semimajor axis still shrinks, the eccentricity can increase. The most eccentric merged stars are removed from the population because they become TDEs or plunges. The result is an eccentricity distribution that flattens at high ( e > 0 . 5) eccentricities. However, the actual number of merged star orbits that became more (less) eccentric are similar.\",\n \"pages\": [\n 14\n ]\n },\n {\n \"title\": \"REFERENCES\",\n \"content\": \"Alexander, T. 1999, ApJ, 527, 835, doi: 10.1086/308129 Alexander, T., & Pfuhl, O. 2014, ApJ, 780, 148, doi: 10.1088/0004-637X/780/2/148 Bahcall, J. N., & Wolf, R. A. 1976, ApJ, 209, 214, doi: 10.1086/154711 Balberg, S. 2024, ApJ, 962, 150, doi: 10.3847/1538-4357/ad1690\",\n \"pages\": [\n 14\n ]\n },\n {\n \"title\": \"Sampling Tracer Stars from the Inner 0.1 pc\",\n \"content\": \"*uses a = 1.25 instead of 1.75 Balberg, S., Sari, R., & Loeb, A. 2013, MNRAS, 434, L26, doi: 10.1093/mnrasl/slt071 Bartko, H., Martins, F., Fritz, T. K., et al. 2009, ApJ, 697, 1741, doi: 10.1088/0004-637X/697/2/1741 Binney, J., & Tremaine, S. 2008, Galactic Dynamics: Second Edition Bradnick, B., Mandel, I., & Levin, Y. 2017, MNRAS, 469, 2042, doi: 10.1093/mnras/stx1007 Brown, W. R., Geller, M. J., Kenyon, S. J., & Kurtz, M. J. 2005, ApJL, 622, L33, doi: 10.1086/429378 Brown, W. R., Lattanzi, M. G., Kenyon, S. J., & Geller, M. J. 2018, ApJ, 866, 39, doi: 10.3847/1538-4357/aadb8e Brutman, Y., Steinberg, E., & Balberg, S. 2024, arXiv e-prints, arXiv:2408.16383, doi: 10.48550/arXiv.2408.16383 Dale, J. E., & Davies, M. B. 2006, MNRAS, 366, 1424, doi: 10.1111/j.1365-2966.2005.09937.x Dale, J. E., Davies, M. B., Church, R. P., & Freitag, M. 2009, MNRAS, 393, 1016, doi: 10.1111/j.1365-2966.2008.14254.x Dessart, L., Ryu, T., Amaro Seoane, P., & Taylor, A. M. 2024, A&A, 682, A58, doi: 10.1051/0004-6361/202348228 Ferrarese, L., & Ford, H. 2005, SSRv, 116, 523, doi: 10.1007/s11214-005-3947-6 Freitag, M., & Benz, W. 2002, A&A, 394, 345, doi: 10.1051/0004-6361:20021142 -. 2005, MNRAS, 358, 1133, doi: 10.1111/j.1365-2966.2005.08770.x Gallego-Cano, E., Sch¨odel, R., Dong, H., et al. 2018a, A&A, 609, A26, doi: 10.1051/0004-6361/201730451 -. 2018b, A&A, 609, A26, doi: 10.1051/0004-6361/201730451 Gallegos-Garcia, M., Law-Smith, J., & Ramirez-Ruiz, E. 2018, ApJ, 857, 109, doi: 10.3847/1538-4357/aab5b8 Generozov, A. 2021, MNRAS, 501, 3088, doi: 10.1093/mnras/staa3851 Generozov, A., & Madigan, A.-M. 2020, ApJ, 896, 137, doi: 10.3847/1538-4357/ab94bc Generozov, A., & Perets, H. B. 2022, MNRAS, 513, 4257, doi: 10.1093/mnras/stac1108 Genzel, R., Eisenhauer, F., & Gillessen, S. 2010, Reviews of Modern Physics, 82, 3121, doi: 10.1103/RevModPhys.82.3121 Ghez, A. M., Salim, S., Hornstein, S. D., et al. 2005, ApJ, 620, 744, doi: 10.1086/427175 Ghez, A. M., Duchˆene, G., Matthews, K., et al. 2003, ApJL, 586, L127, doi: 10.1086/374804 Ghez, A. M., Salim, S., Weinberg, N. N., et al. 2008, ApJ, 689, 1044, doi: 10.1086/592738 Gibson, C., Kıro˘glu, F., Lombardi, J. C., J., et al. 2024, arXiv e-prints, arXiv:2410.02146, doi: 10.48550/arXiv.2410.02146 Gillessen, S., Eisenhauer, F., Trippe, S., et al. 2009, ApJ, 692, 1075, doi: 10.1088/0004-637X/692/2/1075 Gillessen, S., Plewa, P. M., Eisenhauer, F., et al. 2017, ApJ, 837, 30, doi: 10.3847/1538-4357/aa5c41\",\n \"pages\": [\n 15,\n 17\n ]\n }\n]"}}},{"rowIdx":4980,"cells":{"bibcode":{"kind":"string","value":"2024arXiv241201150D"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/2412.01150.pdf"},"content":{"kind":"string","value":"\n, 1-23 (2024)\nRepresentation Learning for Time-Domain High-Energy Astrophysics: Discovery of Extragalactic Fast X-ray Transient XRT 200515\nSteven Dillmann 1 , 2 ★ † , Rafael Martínez-Galarza 3 , Roberto Soria 4 , 5 , Rosanne Di Stefano 3 and Vinay L. Kashyap 3\n\n1 Stanford University, Institute for Computational and Mathematical Engineering, Stanford, CA 94305, USA\n2 University of Cambridge, Department of Physics, Cavendish Laboratory, Cambridge, CB3 0HE, UK\n3 Center for Astrophysics | Harvard & Smithsonian, Cambridge, MA 02138, USA\n4 INAF-Osservatorio Astrofisico di Torino, Strada Osservatorio 20, I-10025 Pino Torinese, Italy\n5 Sydney Institute for Astronomy, School of Physics A28, The University of Sydney, Sydney, NSW 2006, Australia\n\nAccepted XXX. Received YYY; in original form ZZZ\nABSTRACT\nWe present a novel representation learning method for downstream tasks such as anomaly detection and unsupervised transient classification in high-energy datasets. This approach enabled the discovery of a new fast X-ray transient (FXT) in the Chandra archive, XRT 200515, a needle-in-the-haystack event and the first Chandra FXT of its kind. Recent serendipitous breakthroughs in X-ray astronomy, including FXTs from binary neutron star mergers and an extragalactic planetary transit candidate, highlight the need for systematic transient searches in X-ray archives. We introduce new event file representations, 𝐸 -𝑡 Maps and 𝐸 -𝑡 -𝑑𝑡 Cubes, designed to capture both temporal and spectral information, effectively addressing the challenges posed by variable-length event file time series in machine learning applications. Our pipeline extracts low-dimensional, informative features from these representations using principal component analysis or sparse autoencoders, followed by clustering in the embedding space with DBSCAN. New transients are identified within transient-dominant clusters or through nearest-neighbor searches around known transients, producing a catalog of 3,539 candidates (3,427 flares and 112 dips). XRT 200515 exhibits unique temporal and spectral variability, including an intense, hard < 10 s initial burst followed by spectral softening in an ∼ 800 s oscillating tail. We interpret XRT 200515 as either the first giant magnetar flare observed at low X-ray energies or the first extragalactic Type I X-ray burst from a faint LMXB in the LMC. Our method extends to datasets from other observatories such as XMM-Newton , Swift-XRT , eROSITA , Einstein Probe , and upcoming missions like AXIS .\nKey words: software: machine learning, methods: data analysis, X-rays: bursts, stars: magnetars, transients: gamma-ray bursts, stars: peculiar\n1 INTRODUCTION\nRecent serendipitous discoveries, such as extragalactic fast X-ray transients (FXTs) linked to neutron star merger candidates as electromagnetic counterparts to gravitational wave events (Lin et al. 2022) and an X-ray dip associated with the first extragalactic planet candidate (Di Stefano et al. 2021), underscore the challenges of identifying such rare events within large X-ray catalogs. Beyond magnetarpowered FXTs as the aftermath of binary neutron star mergers (Dai et al. 2006; Metzger et al. 2008; Zhang 2013; Sun et al. 2017; Bauer et al. 2017; Xue et al. 2019), other interesting origins of extragalactic FXTs include supernova shock breakouts (SBOs) (Soderberg et al. 2008; Modjaz et al. 2009; Alp & Larsson 2020; Novara et al. 2020), tidal disruption events (TDEs) (Jonker et al. 2013) including quasiperiodic eruptions (QPEs) (Arcodia et al. 2021; Chakraborty et al. 2021), thermonuclear (Type I) X-ray bursts from accreting neutron stars (in't Zand et al. 2013), or binary self-lensing events (D'Orazio\n\n★ E-mail: stevendi@stanford.edu\n† Present address: 450 Jane Stanford Way, Stanford, CA 94305, USA\n\n&DiStefano 2018, 2020; Hu et al. 2020). Both because of their very stochastic nature, and because narrow field X-ray missions such as the Chandra X-ray Observatory ( Chandra ) (Weisskopf et al. 2000), XMM-Newton (Jansen et al. 2001) and Swift-XRT (Burrows et al. 2005) are not designed as wide time-domain surveys, X-ray transient discoveries are often serendipitous. They can be found in observations that were originally proposed for a completely unrelated science objective and are rarely the target of the observation. In many cases serendipitously found X-ray sources do not get characterized or classified, since their transient nature is not immediately obvious. Instead, observations with X-ray transients often get stored in large data archives and remain unnoticed. This raises the need for a systematic search for short-duration phenomena in high-energy catalogs. New missions such as eROSITA (Predehl et al. 2021), Einstein Probe (Yuan et al. 2022) and the upcoming AXIS Observatory (Reynolds et al. 2024) target X-ray transients more directly, thus the development of novel transient detection methods is becoming even more relevant. The temporary, unpredictable and 'unusual' nature of X-ray transients distinguishes them from 'normal' X-ray source emissions. From a data science perspective, they can be understood\nas 'anomalies' within a large dataset. Existing methods for identifying X-ray transients primarily rely on statistical tests of variability (Yang et al. 2019; Pastor-Marazuela et al. 2020; Quirola-Vásquez et al. 2022; Quirola-Vásquez et al. 2023). While effective within specific constraints, these approaches are inherently limited by their underlying assumptions, which may not capture the diverse nature of transient phenomena. In contrast, machine learning offers a more flexible, expressive, and scalable framework, making it particularly well-suited for anomaly detection in large, high-dimensional datasets with diverse transient types. While optical time-domain surveys are at the forefront of leveraging extensive observational programs, like ZTF (Bellm et al. 2019) or the upcoming LSST survey (Ivezić et al. 2019), and neural network-based anomaly detection tools to identify rare sources among countless ordinary objects (Villar et al. 2021; Muthukrishna et al. 2022), the X-ray astronomy community has only recently begun exploring the potential of machine learning to classify sources (Yang et al. 2022; Pérez-Díaz et al. 2024) or to search for needle-in-a-haystack events in large X-ray datasets and archives (Kovačević et al. 2022; Dillmann & Martínez-Galarza 2023). The effectiveness of machine learning methods largely depends on the algorithm's ability to learn useful representations from the data.\nRepresentation learning (Bengio et al. 2013) is an increasingly popular technique in astronomy used in supervised, semi-supervised, self-supervised and unsupervised frameworks (Naul et al. 2018; Hayat et al. 2021; Walmsley et al. 2022; Slijepcevic et al. 2024; Mohale & Lochner 2024). It involves creating or learning meaningful representations for specific modalities of scientific data, which can then be used for downstream tasks such as regression, classification, or, as in this work, anomaly detection. The compressed representations live in a low-dimensional embedding space, in which anomalous data samples are well-separated from more ordinary ones.\nWe propose a new unsupervised representation learning method to perform a large-scale search for X-ray transients in the Chandra archive. High-energy catalogs include individual X-ray source observations in the form of event files. The variable length of these time series poses a challenge in creating consistent representations suitable for transient searches with machine learning. Most deep learning algorithms take a fixed-length input for all data samples. In order to effectively represent event files over a broad range of lengths, we introduce novel fixed-length event file representations, which take into account both their time-domain and energy-domain information. Applying feature extraction and dimensionality reduction techniques, for example with sparse autoencoders, we create a representation space that encodes scientifically meaningful information, such as the spectral and variability properties of the astrophysical sources. Previously identified X-ray transients occupy distinct, well-isolated clusters in the embedding space. Using clustering techniques and nearest neighbor searches allows us to effectively explore these transient-dominant clusters to discover new X-ray transients. We collect the identified X-ray flare and dip candidates in a publicly available catalog, serving as a fertile ground for new discoveries in time-domain high-energy astrophysics.\nAmong these candidates, we identify an intriguing extragalactic FXT, XRT 200515, which exhibits unique temporal and spectral characteristics distinct from any previously reported Chandra FXTs. The transient's initial hard < 10 s burst shows a sharp rise exceeding 4 orders of magnitude, followed by spectral softening in an ∼ 800 s oscillating tail. This transient is likely related to either a giant magnetar flare (GMF) from a distant soft gamma repeater (SGR) behind the Large Magellanic Cloud (LMC) or an extragalactic Type I X-ray burst from a faint LMXB in the LMC. Each of these interpretations presents its own set of challenges. Alternatively, XRT 200515 could\nbe a new type of astronomical phenomenon found by our anomaly detection method using machine learning.\nOur method is the first representation learning approach for anomaly detection in high-energy astrophysics. It is applicable to datasets from high-energy catalogs like Chandra , XMM-Newton , Swift-XRT , eROSITA , and Einstein Probe . We created semantically meaningful representations that can be aligned with other data modalities, such as optical images or infrared spectra to design multi-modal models (Parker et al. 2024; Mishra-Sharma et al. 2024; Zhang et al. 2024; Rizhko & Bloom 2024) using contrastive learning (Radford et al. 2021), that can improve on current state-of-the-art algorithms used to characterize the physics of the associated objects. Ultimately, this work and other representation and contrastive learning approaches lay the groundwork for developing generalized foundation model in astronomy.\nThe paper is organized as follows: In § 2, we provide information on the dataset of Chandra event files used in this analysis. In § 3, we describe in detail the implementation of our novel transient detection approach leveraging representation learning. In § 4, we present and discuss the results in form of the semantically meaningful representation space of the event files, the catalog of X-ray transient candidates and the discovery of the new Chandra transient XRT 200515. Finally, we highlight our contributions to time-domain high-energy astrophysics and outline potential directions for extending this work in the future in § 5.\nThe relevant code, a demonstration of the pipeline, and an interactive embedding selection, transient search and lightcurve plotting tool are available online at the GitHub repository https://github.com/StevenDillmann/ml-xraytransients-mnras.\n2 DATASET\nWe use data from the Chandra Source Catalog (CSC) version 2.1 (Evans et al. 2024), which includes all publicly available X-ray sources detected by Chandra as of December 2021. For this study, we focus specifically on observations from the Advanced CCD Imaging Spectrometer (ACIS). CSC 2.1 had not been fully released at the time our analysis was performed, but catalog data was available for sources that had completed processing in the Current Database View 1 , a snapshot of which we took on 11 April 2023. CSC 2.1 performs source detection on stacked observations, and catalog properties are provided both for these stack-level detections, and for each of observation-level detection that contribute to a stack detection. Because we are interested in short-time variability that happens within a single observation of a source, we use the catalog products for the observation-level detections in our analysis. For a given X-ray detection, two types of products are provided in the CSC: (i) database tables with source properties, such as fluxes in the different X-ray energy bands, hardness ratios, variability indices, etc., and (ii) filebased data products for each detection of a source, such as the detect regions, the Chandra PSF at that location, etc. The following observation-level catalog properties are relevant for our analysis:\n\n· var_prob_b : The probability that a source detection is variable in time for the broad energy band (0.5-7 keV), as estimated using the Gregory-Loredo algorithm (Gregory & Loredo 1992). In this paper we call this quantity 𝑝 𝑏 var .\n· var_index_b : The variability index in the broad band, which indicates the level of confidence for time variability. A variability\n\nindex of 6 or larger indicates variability at a confidence of at least 2 𝜎 . In this paper we call this quantity 𝐼 𝑏 var .\n\n· hard_ : The hardness ratios, which quantify the relative fraction of photons detected in two given bands chosen between the soft (0.5-1.2 keV), medium (1.2-2 keV), and hard (27 keV) bands for a source detection. For example, a value of the hard-to-soft hardness ratio close to 1 indicates that most of the photons detected are in the hard energy band, whereas a value close to -1 indicates that most photons are detected in the soft band. In this paper we call these quantities 𝐻𝑅 hs , 𝐻𝑅 ms , and 𝐻𝑅 hm .\n\nFrom the catalog data products available for observation-level Xray detections, we are interested in the region event file. This event file consists of a list of all individual photon events detected in a small bounding box around a source detection, listing their energies, arrival times, and detector coordinates. These event files are the basis for the characterization of an X-ray source: lightcurves, spectra, images, coordinates, and other properties are derived from the distribution of the listed quantities. In this analysis, we directly use these event files as our primary data products. The values of the catalog properties listed above serve as summary statistics for the detection associated with a given region event file. We only include event files with more than 5 events and a signal-to-noise ratio above 5 to minimize spurious signals from low number statistics in faint sources. We also exclude detections that are flagged for pile-up 2 , i.e., those with a pileup fraction larger than 5%, which corresponds to a maximum pileup warning of 0.1 in CSC 2.1. For the resulting detections, we filter the event files to include only events contained within the detection region for each source. These detection regions are also provided as data products in CSC 2.1, and consist of the ellipse that includes the 90% encircled counts fraction of the PSF at the source location. Due to the low background level in Chandra observations, the majority of events selected after this spatial filtering are expected to be events associated with the X-ray source, not the background. In the selected event files, we only include photon events within good time intervals (GTIs), which are time periods of valid, high-quality data. No other pre-processing is required. The final dataset consists of 95,473 filtered event files from 58,932 sources, resulting in an average of 1 . 62 observations per source. This includes 9,003 new sources that have been added as part of the CSC 2.1 release, in addition to the sources from the previous release.\n3 METHODS\nIn this work, we introduce a novel representation learning based anomaly detection method to systematically search for X-ray transients in high-energy archives. We begin with an overview of the method here and provide detailed explanations of each step in individual subsections. The full pipeline is illustrated in Figure 1. Starting with the event files described in § 2, we (i) build two novel and uniform event file representations by binning their arrival times and energies into 𝐸 -𝑡 Maps (Event File Representation I) or 𝐸 -𝑡 -𝑑𝑡 Cubes (Event File Representation II); (ii) use principal component analysis (Feature Extraction I) or sparse autoencoders (Feature Extraction II) to extract informative features from the event\n
\n\n
Table 1. Naming of the different embedding result cases based on the event file representation and feature extraction method.
\n
\nfile representations; (iii) apply dimensionality reduction to the extracted features to create a low-dimensional embedding space; (iv) use density-based clustering to create embedding clusters that group event files with similar characteristics, for example transient behavior or certain spectral features. Previously identified transients like the extragalactic magnetar-powered flare candidate reported by Lin et al. (2022) and the extragalactic planet candidate dip reported by Di Stefano et al. (2021), shown in Figure 2, occupy well-isolated clusters in the embedding space. Exploring these clusters and conducting nearest-neighbor searches enables us to effectively find analogs to bona-fide time-domain anomalies, while at the same time grouping them according to their spectral properties. We compile the identified transient candidates in a catalog. While our approach is designed and tested using Chandra data, it is applicable to any dataset consisting of event lists, like those from other high-energy telescopes. The described transient detection approach is applied to both types of event file representations with both feature extraction methods, resulting in four different embeddings. We denote the different cases as described in Table 1.\n3.1 Event File Representation\nThe different event files in the dataset are variable in length 𝑁 and duration 𝑇 , as shown in Appendix A. The large variation in the number of events and duration highlights the challenge in producing uniform data representations that preserve relevant information on time variability and spectral properties. While there exist machine learning architectures that take variable length inputs, the significant differences in the number of events from object to object make standardization of the inputs challenging, even when these architectures are used (Martínez-Galarza & Makinen 2022). As a first step in our analysis, we introduce 2-dimensional and 3-dimensional fixed-length representations based on an informed binning strategy for the event files, similar to the DMDT maps for optical lightcurves introduced by Mahabal et al. (2017).\n3.1.1 2D Histogram Representation ( 𝐸 -𝑡 Maps)\nAssume an event file with 𝑁 photons and a photon arrival time column 𝒕 with entries { 𝑡 𝑘 } 𝑁 𝑘 = 1 and energy column 𝑬 with entries { 𝐸 𝑘 } 𝑁 𝑘 = 1 . The event file duration is given by 𝑇 = 𝑡 𝑁 -𝑡 1 . The energy column entries take values in the broad energy band of Chandra 's ACIS instrument, i.e. 𝐸 𝑘 ∈ [ 𝐸 𝑚𝑖𝑛 , 𝐸 𝑚𝑎𝑥 ] , where 𝐸 𝑚𝑖𝑛 = 0 . 5 keV and 𝐸 𝑚𝑎𝑥 = 7 keV comes from considering appropriate boundaries for the energy response of Chandra 's ACIS instrument. Beyond these boundaries, the telescope's aperture effective area is low for the majority of detected sources. First, we obtain the normalized time column, given by 𝝉 = 𝒕 -𝑡 1 𝑇 , and the logarithm of the energy column, given by 𝝐 = log 𝑬 . The resulting boundaries for normalized time column are 𝝉 ∈ [ 𝜏 𝑚𝑖𝑛 , 𝜏 𝑚𝑎𝑥 ] , where 𝜏 𝑚𝑖𝑛 = 0 and 𝜏 𝑚𝑎𝑥 = 1.\n\n\n
Figure 1. Flowchart of the proposed representation learning approach for anomaly detection in time-domain high-energy astrophysics, enabling the systematic detection of transients in high-energy archives. The first step is to create uniform event file representations by binning photon arrival times and energies in the event files into into 𝐸 -𝑡 Maps (Event File Representation I) or 𝐸 -𝑡 -𝑑𝑡 Cubes (Event File Representation II). The second step involves extracting informative features from these event file representations via principal component analysis (Feature Extraction I) or sparse autoencoders (Feature Extraction II). The third step is to apply dimensionality reduction to the extracted features and to create a low-dimensional embedding space, which is clustered in the fourth step using density-based clustering. Previously identified transients occupy well-isolated clusters on the edges of the embedding space, thus new transients can be identified by exploring these edge clusters and performing nearest-neighbor searches around known bona-fide flares and dips. Finally, we compile these search results in a publicly available catalog of transient candidates, serving as a fertile ground for the discovery of new X-ray transients.
\n\n\n\n
Figure 2. Left panel: Lightcurve of the first extragalactic planet candidate dip reported by Di Stefano et al. (2021) detected in the Chandra observation ObsID 13814. Right panel: Lightcurve of the magnetar-powered X-ray flare candidate reported by Lin et al. (2022) detected in the Chandra observation ObsID 4062.
\n\n\n\n
Figure 3. The distribution of the optimal number of bins for the energy dimension 𝑛 𝜖 (left), time dimension 𝑛 𝜏 (middle), time difference dimension 𝑛 𝛿𝜏 (right). The distribution of 𝑛 𝜏 only includes event files for which 𝑝 𝑏 var > 0 . 9. The vertical lines indicate the number of bins chosen for the event file representations.
\n\nThe range for the log-energy column is 𝝐 ∈ [ 𝜖 𝑚𝑖𝑛 , 𝜖 𝑚𝑎𝑥 ] , where 𝜖 𝑚𝑖𝑛 = log 0 . 5 keV and 𝜖 𝑚𝑎𝑥 = log 7 keV.\nNext, we determine the dimensionality of our representations. For a each event file, we determine the optimal number of bins in the energy dimension, 𝑛 𝜖 , with the Freedman-Diaconis rule (Freedman & Diaconis 1981), a widely used method that balances the trade-off between too noisy histograms (too many bins) and not informative enough histograms (too few bins). The optimal bin width 𝑏 𝜖 according to this rule is calculated in the following way:\n𝑏 𝜖 = 2 𝐼𝑄𝑅 ( 𝝐 ) 𝑁 1 3 , (1)\nwhere 𝐼𝑄𝑅 ( 𝜖 ) represents the interquartile range of the 𝜖 values for a given event file of length 𝑁 . Subsequently, we obtain the optimal number of energy bins 𝑛 𝜖 with:\n𝑛 𝜖 = 𝜖 𝑚𝑎𝑥 -𝜖 𝑚𝑖𝑛 𝑏 𝜖 . (2)\nFor each event file, we determine the optimal number of bins in the time dimension, 𝑛 𝜏 , with the help of the Bayesian Blocks algorithm, which was specifically developed for time series analysis in astronomy (Scargle et al. 2013). This algorithm partitions the time series into adaptive width bins or blocks that are statistically distinct from neighboring blocks; that is, within a given time-ordered Bayesian block, events grouped in that block are consistent with having a similar event arrival rate. We use the default Astropy implementation of Bayesian blocks, and set the false alarm probability parameter to 𝑝 0 = 0 . 01 (Astropy Collaboration et al. 2013), which implies a 1% probability of declaring a change of rate when there is none. For each event file, we define the optimal uniform bin width 𝑏 𝜏 as the minimum bin width calculated by the Bayesian Blocks algorithm, and then find the optimal number of time bins 𝑛 𝜏 with:\n𝑛 𝜏 = 𝜏 𝑚𝑎𝑥 -𝜏 𝑚𝑖𝑛 𝑏 𝜏 . (3)\nThe optimal number of bins is different for each event file, due to their different lengths 𝑁 and durations 𝑇 . To select a bin size that can be applied to all event files, we consider the distributions of these optimal bin sizes, which are shown in Figure 3. For the distribution of 𝑛 𝜏 values we only use those event files for which 𝑝 𝑏 var > 0 . 9. The intent of this is to effectively capture variability timescales that are associated with short time-domain events, such as flares and dips.\nWe choose the 90th percentile value of each distribution to set the final number of bins in each dimension. That is, only 10% of the event files will have an optimal number of bins that is larger than the chosen values 𝑛 𝜖 = 16 and 𝑛 𝜏 = 24. The choice of the 90th percentile, rather than the mean or mode, is motivated by the need to capture sufficient statistical detail even for long event files, while\nkeeping the size of the resulting representations computationally tractable. Choosing a lower resolution would risk losing significant details in the representation, particularly short-duration events such as flares and dips within longer event files. The 𝐸 -𝑡 Maps are the 2D histogram representations with size ( 𝑛 𝜏 , 𝑛 𝜖 ) = ( 24 , 16 ) that result from binning the events according to the optimized number of bins.\nFigure 4 shows the 𝐸 -𝑡 Maps for the known extragalactic dip reported by Di Stefano et al. (2021) and known extragalactic flare reported by Lin et al. (2022).\n3.1.2 3D Histogram Representation ( 𝐸 -𝑡 -𝑑𝑡 Cubes)\nWenowintroduce the 𝐸 -𝑡 -𝑑𝑡 Cubes, which extend the 𝐸 -𝑡 Maps by a third dimension that serves as a proxy for the photon arrival rate. For an event file of length 𝑁 , consider the array of time differences between consecutive photon arrivals 𝚫 𝒕 with entries Δ 𝑡 𝑘 = 𝑡 𝑘 + 1 -𝑡 𝑘 for 𝑘 = 1 , 2 , . . . , 𝑁 -1. We again scale and normalize the obtained values, so that they adopt values between 0 and 1, using in each case the minimum value Δ 𝑡 𝑚𝑖𝑛 and maximum value Δ 𝑡 𝑚𝑎𝑥 . This provides the third dimension 𝜹𝝉 :\n𝜹𝝉 = 𝚫 𝒕 -Δ 𝑡 𝑚𝑖𝑛 Δ 𝑡 𝑚𝑎𝑥 -Δ 𝑡 𝑚𝑖𝑛 . (4)\nThe additional dimension is intended to better isolate short-duration features in time variability by capturing high photon arrival rates, which are typical of flares, as well as very low photon arrival rates, which are typical of dips. The boundaries of our histogram representations in this dimension are 𝜹𝝉 ∈ [ 𝛿𝜏 𝑚𝑖𝑛 , 𝛿𝜏 𝑚𝑎𝑥 ] , where 𝛿𝜏 𝑚𝑖𝑛 = 0 and 𝛿𝜏 𝑚𝑎𝑥 = 1. We determine the optimal number of bins in the 𝜹𝝉 dimension, 𝑛 𝛿𝜏 , again by computing the optimal bin width 𝑏 𝛿𝜏 with the Freedman-Diaconis rule and dividing the range for 𝜹𝝉 by 𝑏 𝛿𝜏 :\n𝑏 𝛿𝜏 = 2 𝐼𝑄𝑅 ( 𝜹𝝉 ) 𝑁 1 3 , (5)\n𝑛 𝛿𝜏 = 𝛿𝜏 𝑚𝑎𝑥 -𝛿𝜏 𝑚𝑖𝑛 𝑏 𝛿𝜏 . (6)\nThe distribution of 𝑛 𝛿𝜏 across the event files is shown in Figure 3. Most of the relevant time-domain information is already captured by 𝝉 , but adding 𝜹𝝉 provides an additional marker for dips and flares that can be shorter than the timescales probed by our chosen binning of 𝝉 .\nUnlike in the other two dimensions, we choose the 75th percentile value of the distribution as our final choice of common binning, which results in 𝑛 𝛿𝜏 = 16. This is because in order to identify short transients, we need to capture strong deviations in 𝜹𝝉 only. Choosing a lower value for 𝑛 𝛿𝜏 reduces noise an improves computational tractability. Having both 𝝉 and 𝜹𝝉 represented also breaks\n\n\n
Figure 4 shows the 𝐸 -𝑡 -𝑑𝑡 Cubes for the known extragalactic dip reported by Di Stefano et al. (2021) and known extragalactic flare reported by Lin et al. (2022).
\n\n\n\n\n\n\n\n\n\n
Figure 4. Upper panel: 2D histogram representations ( 𝐸 -𝑡 Maps) for the extragalactic dip in (Di Stefano et al. 2021) (left) and extragalactic flare in (Lin et al. 2022) (right). Lower panel: 3D histogram representations ( 𝐸 -𝑡 -𝑑𝑡 Cubes) for the same events. Darker bins indicate higher counts, while lighter bins indicate lower counts.
\n\nany assumption of stationarity, in that we can be sensitive to transient events happening at any time during the observation of the source, and break degeneracies between periodic and non-periodic features in the representations presented by Martínez-Galarza & Makinen (2022). The 𝐸 -𝑡 -𝑑𝑡 Cubes are the resulting 3D histogram event file representations with size ( 𝑛 𝜏 , 𝑛 𝜖 , 𝑛 𝛿𝜏 ) = ( 24 , 16 , 16 ) .\n3.1.3 Feature Notation\nThe event file representations can now be used as inputs for various statistical learning and machine learning algorithms. For the 𝑖 𝑡 ℎ event file in the dataset of length 𝑚 = 95,473, we denote the corresponding feature vector as fi 𝑥 𝑖 = [ 𝑥 1 , 𝑥 2 , . . . , 𝑥 𝑛 ] 𝑖 , where 𝑛 = 𝑛 𝜏 · 𝑛 𝜖 = 384 for the 𝐸 -𝑡 Maps and 𝑛 = 𝑛 𝜏 · 𝑛 𝜖 · 𝑛 𝛿𝜏 = 6 , 144 for the 𝐸 -𝑡 -𝑑𝑡 Cubes. The set of all feature vectors is denoted as X = [fi 𝑥 1 , fi 𝑥 2 , . . . , fi 𝑥 𝑚 ] ⊤ with size ( 𝑚, 𝑛 ) .\n3.2 Feature Extraction I: Principal Component Analysis\nWe use Principal Component Analysis (PCA) (Pearson 1901) provided by scikit-learn (Pedregosa et al. 2011) as our first feature extraction method. The extracted principal components should encode relevant time-domain and spectral information of the event file they represent. PCA involves transforming a dataset into a new coordinate system by finding the principal components of the data that capture most of the variance in the data. By projecting the dataset onto principal components, PCA reduces the dimensionality of the data while retaining the most important information, which increases the interpretability of high-dimensional data.\n\n\n
Figure 5. Scree plot for the principal components of the 𝐸 -𝑡 Maps (top) and 𝐸 -𝑡 -𝑑𝑡 Cubes (bottom). The scree plots show the amount of variance explained by each individual principal component including the knee point in the cumulative variance.
\n\n3.2.1 PCA Algorithm\nWe start with the feature vector set X of size ( 𝑚, 𝑛 ) representing our dataset with 𝑚 samples and 𝑛 dimensions. PCA aims to find a new coordinate system defined by a set of orthogonal axes, i.e. the principal components, that captures the maximum amount of variance in the data. The PCA result is a transformed dataset Xpc obtained by projecting X onto the principal components:\nXpc = XW , (7)\nwhere W is matrix of size ( 𝑛, 𝑛 𝑝𝑐 ) containing the first 𝑛 𝑝𝑐 principal components to be retained as its columns and Xpc is of size ( 𝑚 , 𝑛 𝑝𝑐 ) with a reduced dimensionality of 𝑛 𝑝𝑐 . For a more detailed explanation of the algorithm, we refer the reader to Jolliffe (2002).\n3.2.2 Principal Components Retained\nThe main PCA hyperparameter is the number of principal components 𝑛 𝑝𝑐 to retain. Figure 5 shows two scree plots illustrating the amount of variance explained by each principal component in descending order and the cumulative proportion of variance explained by the principal components for both 𝐸 -𝑡 Mapsand 𝐸 -𝑡 -𝑑𝑡 Cubes. Acommon approach to determine the optimal value of 𝑛 𝑝𝑐 is to find the knee point in the cumulative scree plot of the principal components. This balances the objective of minimizing the dimensionality while retaining as much information as possible. Defining the knee point as the point beyond which adding additional principal components increases the amount of variance by less than 0 . 1% gives 𝑛 𝑝𝑐 = 15 for 𝐸 -𝑡 Maps and 𝑛 𝑝𝑐 = 22 for 𝐸 -𝑡 -𝑑𝑡 Cubes as indicated in Figure 5. These capture 94 . 1% and 89 . 9% of the variance respectively.\n\n\n\n\n\n
Figure 6. Encoder architecture of the convolutional autoencoder used for the 𝐸 -𝑡 Maps (top) and of the fully connected autoencoder used for the 𝐸 -𝑡 -𝑑𝑡 Cubes (bottom). The decoder architecture is simply a mirror image of the encoder.
\n\n3.3 Feature Extraction II: Sparse Autoencoder Neural Network\nAs an alternative to PCA, we now build Autoencoder (Hinton & Salakhutdinov 2006) models with TensorFlow (Abadi et al. 2015) to learn a set of latent features from the 𝐸 -𝑡 Maps and 𝐸 -𝑡 -𝑑𝑡 Cubes that can be used to isolate transients and encode specific spectral properties. An autoencoder is composed of two neural networks, an encoder and a decoder, which work together to learn a compressed representation of the input data. The encoder network takes the input data and maps it to a lower-dimensional representation, often called 'latent space' or 'bottleneck'. The number of neurons in the bottleneck determines the dimensionality of the learned representation. The decoder network then aims to reconstruct the original input from this compressed representation. The decoder is typically a mirrored version of the encoder gradually upsampling the latent space until the output matches the dimensions of the original input. By minimizing the reconstruction error between input and output during training, the model learns a low-dimensional representation of the input. The bottleneck forces the encoder to capture the most important information necessary for accurate reconstruction, effectively compressing the input and learning to extract informative features in an unsupervised manner. Once the autoencoder is trained, the encoder network can be used as a standalone feature extractor to obtain a compressed representation of the input data, which can be used for downstream tasks such as clustering or anomaly detection. As opposed to PCA, which is a linear technique that works well for linearly correlated data but fails to capture complex non-linear relationships, an autoencoder is able to learn complex non-linear relationships. We design two different autoencoders to process the 𝐸 -𝑡 Maps and 𝐸 -𝑡 -𝑑𝑡 Cubes.\n
\n\n
Table 2. Summary of the encoder architecture of the convolutional autoencoder used to extract informative features from the 𝐸 -𝑡 Maps. Note that each layer has a Leaky ReLU activation function and that each standard convolutional layer is followed by batch normalization with momentum 0.9.
\n
\n3.3.1 Convolutional Autoencoder\nIn a convolutional autoencoder (Masci et al. 2011), both the encoder and decoder network consist of convolutional layers (LeCun et al. 1998), which perform convolutions over the input using a filter. These filters are small matrix kernels with learnable weights that slide across the input, allowing the network to capture high-level features while preserving important spatial hierarchies and relationships, which is why they are often used for image-like data. This makes this architecture particularly well-suited to recognize spatial patterns such as dips or flares in our 𝐸 -𝑡 Maps. To gradually reduce the dimension of the input while it is being passed through the encoder network, we use stride convolution layers (Simonyan & Zisserman 2014) with a stride value of 2 for downsampling. This means that the learnable filter jumps two pixels at a time as it slides over the input. The output of the convolutional layers is a feature map, which is then flattened to a feature vector and passed through a series of fully connected layers, where every neuron in the previous layer is connected to every neuron in the next layer. These fully connected layers are responsible for mapping the learned features to a lower-dimensional latent representation in the bottleneck and perform non-linear transformations while downsampling through the use of non-linear activation functions. The final latent space has 𝑛 𝑎𝑒 = 12 elements, representing the most essential features of the input data, which can now be used for further downstream tasks. Figure 6 shows a diagram of the encoder part of the model and Table 2 summarizes its architecture.\n3.3.2 Fully Connected Autoencoder\nOur 𝐸 -𝑡 -𝑑𝑡 Cubes introduce an additional dimension resulting in sparse 3D input data. Convolutional layers assume regular grid-like data, making them less effective for handling sparse data. Moreover, very expensive 3D convolutional operations would substantially increase complexity of the model. Therefore, we use a simple fully connected autoencoder for the 𝐸 -𝑡 -𝑑𝑡 Cubes. Its encoder network consists of a series of fully connected layers, which gradually map the original input data to a latent space with 𝑛 𝑎𝑒 = 24 elements. Figure 6 shows a diagram of the encoder part of the model and Table 3 summarizes its architecture.\n3.3.3 Activation Functions\nNeural networks are able to learn and represent complex non-linear relationships due to the introduction of non-linear activation func-\n
\n\n
Table 3. Summary of the encoder architecture of the fully connected autoencoder used to extract informative features from the 𝐸 -𝑡 -𝑑𝑡 Cubes. Note that each layer has a Leaky ReLU activation function and that each standard fully-connected layer is followed by batch normalization with momentum 0.9.
\n
\ntions within their layers. An activation function is a mathematical function used in a neural network to determine whether a neuron should be activated or not, based on its input. It essentially decides how much of the input signal should pass through the neuron, producing an output that can either be passed to the next layer or used to make predictions. The popular Rectified Linear Unit (ReLU) activation function 𝑅𝑒𝐿𝑈 ( 𝑥 ) = max ( 0 , 𝑥 ) (Nair & Hinton 2010) is simple and computationally efficient. To mitigate any potential encounters of the 'dying the ReLU problem', where neurons become non-responsive during training, we choose an extended version called Leaky ReLU (Maas et al. 2013):\n𝐿𝑒𝑎𝑘𝑦𝑅𝑒𝐿𝑈 ( 𝑥 ) = max ( 𝛼𝑥, 𝑥 ) , (8)\nwhere 𝛼 = 0 . 1 is a hyperparameter that defines the slope of the function for negative input values. ReLU sets all negative values in the input to zero, while Leaky ReLU allows a small negative slope for negative inputs, which can help prevent neurons from dying. As for the output layer, we want any values to be mapped to a range between 0 and 1, which is achieved by using the sigmoid activation function:\n𝑠𝑖𝑔𝑚𝑜𝑖𝑑 ( 𝑥 ) = 1 1 + 𝑒 -𝑥 . (9)\n3.3.4 Loss Function and Sparsity Regularization\nIn order to encourage the autoencoder to generate reconstructions close to the original inputs, we use the mean squared error ( 𝑀𝑆𝐸 ) as as a measure of the reconstruction quality given by:\n𝑀𝑆𝐸 = 1 𝑚 𝑚 ∑︁ 𝑖 = 1 ( 𝑥 𝑖 -ˆ 𝑥 𝑖 ) 2 , (10)\nwhere 𝑥 𝑖 is the 𝑖 𝑡 ℎ element of the input vector and ˆ 𝑥 𝑖 is the corresponding is reconstructed output. The 𝑀𝑆𝐸 is a straightforward measure of reconstruction error, and its differentiability allows efficient gradient computation for updating model weights via gradient-based optimization.\nOur neural networks are so called sparse autoencoders (Ng et al. 2011), which promote sparsity in the learned representation, meaning only a small subset of the neurons in the network are active at any given time. Sparse representations are valuable for our work because they help extract highly informative features from the input, while disregarding irrelevant or noisy information. To encourage sparsity in the latent space, we introduce a L1 regularization term in the objective, resulting in the following loss function:\n𝐿 = 𝑀𝑆𝐸 + 𝜆 · 𝑛 𝑤 ∑︁ 𝑗 = 1 | 𝑤 𝑗 | = 1 𝑚 𝑚 ∑︁ 𝑖 = 1 ( 𝑥 𝑖 -ˆ 𝑥 𝑖 ) 2 + 𝜆 · 𝑛 𝑤 ∑︁ 𝑗 = 1 | 𝑤 𝑗 | , (11)\nwhere 𝜆 = 0 . 1 is the regularization strength and 𝑤 𝑗 are the individual bottleneck weight values of which there are 𝑛 𝑤 in total. L1 regularization pushes small weights to zero and thus helps the model prioritize the most significant features of the input data, leading to a semantically meaningful latent space.\n3.3.5 Training\nStarting with the original dataset with a 𝑚 = 95,473 samples and using a test split of 0 . 1 gives us a training and validation set of length 85,925 and a test set of length 9,548. Further using a validation split of 0 . 2, gives 68,740 samples for training and 17,185 for validation. We run the training process for a maximum of 200 epochs with a batch size of 1,024 samples. The initial learning rate was set to 0 . 01 along with an on plateau learning rate scheduler, which dynamically reduces the learning rate by a factor of 0 . 1 if the validation loss plateaus for longer than 10 epochs. Reducing the learning rate when a plateau is detected can help escape local minima in the loss surface and converge to a more optimal solution in the parameter space. This scheduler is used in combination with the Adaptive Moment Estimation (Adam) optimizer (Kingma & Ba 2014), which is a stochastic gradient descent algorithm combining the benefits of both adaptive learning rates (Duchi et al. 2011) and momentum-based optimization techniques (Sutskever et al. 2013). Finally, we use an early stopping callback to monitor the validation loss. It automatically interrupts the training process if the validation loss does not improve for 25 epochs and restores the weights of the model to the best observed weights during training. The training process for both autoencoder models is shown in Appendix B. Once the autoencoder is trained, we can use the encoder to transform the original dataset X to the feature vector space Xae of size ( 𝑚 , 𝑛 𝑎𝑒 ) with a reduced dimensionality of 𝑛 𝑎𝑒 features.\n3.4 Dimensionality Reduction\nUsing t-SNE (Maaten & Hinton 2008), short for t-Distributed Stochastic Neighbor Embedding, we create two-dimensional embeddings of the informative features previously extracted from the event file representations using PCA or sparse autoencoders. The t-SNE algorithm is a method used to map the input data onto a low-dimensional embedding space, and is particularly useful for the visualization of clusters and patterns in high-dimensional datasets. Each high-dimensional sample is transformed into a low-dimensional embedding in such a way that similar object are nearby points, while dissimilar objects are distant points in the embedding space. Essentially, it aims to capture the local structure of the data by preserving the pairwise similarities between objects while mapping them to a lower-dimensional embedding space.\n3.4.1 Algorithm\nWe use our informative features, X if = Xpc or X if = Xae , as input to the t-SNE algorithm to reduce the data to a two-dimensional embedding, denoted as Z . First, t-SNE creates a probability distribution 𝑃 for pairs of high-dimensional data points in X if , assigning higher probabilities to similar pairs and lower probabilities to dissimilar ones. This is done by modeling pairwise similarities using a Gaussian kernel with a specific perplexity parameter, which controls the effective number of neighbors considered for each point. Next, t-SNE defines a similar probability distribution 𝑄 for the pairwise\n
\n\n
Table 4. Chosen t-SNE hyperparameters for different embedding cases.
\n
\nsimilarities in the low-dimensional space Z , modeled using a Student's t-distribution. The goal of t-SNE is to minimize the difference between 𝑃 and 𝑄 using gradient descent, with the Kullback-Leibler (KL) divergence (Kullback & Leibler 1951) as the cost function:\n𝐷 𝐾𝐿 ( 𝑃 | 𝑄 ) = ∑︁ 𝑖 ≠ 𝑗 𝑃 𝑖 𝑗 log 𝑃 𝑖 𝑗 𝑄 𝑖 𝑗 , (12)\nwhere 𝑃 𝑖 𝑗 and 𝑄 𝑖 𝑗 represent pairwise similarities in the high- and low-dimensional spaces, respectively. The algorithm iteratively adjusts the low-dimensional embedding Z to minimize the KL divergence, often requiring hundreds to thousands of iterations for convergence. The result of this optimization is a two-dimensional representation Z of size ( 𝑚, 2 ) , where similar points in the high-dimensional space are clustered closely together.\n3.4.2 Hyperparameter Optimization\nThe t-SNE algorithm has a number of important hyperparameters to be tuned. The two most important parameters are the perplexity and the learning_rate . The perplexity parameter controls the balance between capturing the local versus global structure in the data, while the learning_rate controls the step size at each iteration of the optimization process. The n_iter parameter is the number of iterations. To ensure reproducibility, we set a fixed random_state . Our t-SNE hyperparameter optimization approach is detailed in Appendix C. A summary of the final t-SNE hyperparameters is provided in Table 4.\n3.5 Clustering\nThe next step is the identification of individual clusters in the embedding space using DBSCAN (Hartigan & Wong 1979), short for Density-Based Spatial Clustering of Applications with Noise. Unlike traditional clustering algorithms such as k-means, DBSCAN does not require the number of clusters to be specified, as it identifies dense regions in the data space based on a density criterion.\n3.5.1 Algorithm\nWe use our t-SNE embedding space Z as input to the DBSCAN algorithm, which segments the embedding space into multiple clusters. The DBSCAN algorithm has two main hyperparameters. The eps parameter defines the radius of the neighborhood surrounding each point in the dataset, while the minPts parameter specifies the minimum number of points required within this neighborhood for a data point to be classified as a core point. A border point is defined as a point that is in the vicinity of at least one core point but has fewer than minPts within its neighborhood. All other points are considered to be noise points. Clusters are then created from the aggregation of core points and their associated border points, with noise points being categorized as outliers. Figure 7 visualizes the clustering method.\n\n\n
Figure 7. Illustration of the DBSCAN clustering algorithm, showing core points as densely connected regions, border points along cluster edges, and noise points as outliers. Adapted from Slipski et al. (2024) with permission.
\n\n
\n\n
Table 5. Chosen DBSCAN hyperparameters for different embedding cases.
\n
\n3.5.2 Hyperparameter Optimization\nOur DBSCAN hyperparameter optimization approach is detailed in Appendix C. A summary of the final t-SNE hyperparameters is provided in Table 5.\n3.6 Previously Reported Transients\nWe highlight the embeddings of previously reported bona-fide transients, listed in Table 6, in our low-dimensional representation space to identify transient-dominant clusters. The flares include extragalactic FXTs reported by Jonker et al. (2013), Glennie et al. (2015), Yang et al. (2019), Lin et al. (2021), Lin et al. (2022), Quirola-Vásquez et al. (2022) and a set of stellar flares found in the dataset by manual inspection. The dips include the extragalactic planet candidate in M51 reported by Di Stefano et al. (2021), the ultraluminous X-ray source (ULX) 2E 1402 . 4+5440 in NGC 5457 (Colbert & Ptak 2002; Swartz et al. 2004) and the well-studied eclipsing and bursting lowmass X-ray binary (LMXB) EXO 0748 -676 (Parmar et al. 1986; D'Aì et al. 2014). These transients occupy well-isolated clusters. Exploring transient-dominant clusters and performing nearest-neighbor searches around known transients allows us to find new transients.\n3.7 Candidate Selection\nNewtransients are identified in embedding clusters containing previously reported transients. For well-isolated clusters containing known discovered transients, we use the entire cluster to define new transient candidates. The well-isolated transient-dominant clusters used for candidate selection are listed in Appendix E. However, in a few cases known discovered transients reside within larger poorly separated clusters. Selecting the entire cluster would result in a high number of false positives. To address this, we instead use the k-nearest\n
\n\n
Table 6. Previously reported flares and dips used to identify transient-dominant clusters.
\n
\nneighbors ( kNN ) algorithm (Cover & Hart 1967), identifying the 50 nearest neighbors for each known transient residing in a poorly separated cluster to define additional transient candidates.\n3.8 Cross Matching\nWe use an existing cross-match table (Green et al. 2023) between CSC2.1 and five other catalogs - Gaia DR3 (Gaia Collaboration et al. 2021), DESI Legacy Survey DR10 (Dey et al. 2019), PanSTARRS-1 (Chamberset al. 2016), 2MASS (Skrutskie et al. 2006), and the SDSS DR17 catalog - to complement the X-ray properties derived from the CSC with additional multi-wavelength observations. This includes catalog identifiers, positions, magnitudes, source type classifications and other columns. We cross-matched our transient candidates with the SIMBAD database (Wenger et al. 2000) by associating each candidate with the nearest SIMBAD object, provided the object is located within a 5 arcsec radius of the candidate's coordinates listed\nin the CSC. The multi-wavelength observations of the transient candidates provide valuable information for their characterization and classification.\n4 RESULTS AND DISCUSSION\nWe now present the results of applying the methods in § 3 to the set of representations of X-ray event files in the dataset from § 2.\n4.1 Representation Embedding Space and Clusters\nFigure 8 shows the t-SNE embedding space for the 3D-PCA and 3DAE cases color-coded by the hardness ratio 𝐻𝑅 hs . The embedding space for the other two cases, 2D-PCA and 2D-AE , are shown in Appendix D. The observed hardness ratio gradients in all embedding spaces indicate that the learned representations effectively encode\n\n\n
Figure 8. Embedding representations color-coded by 𝐻𝑅 hs for the 3D-PCA case (left) and 3D-AE case (right).
\n\n\n\n
Figure 9. Embedding representations color-coded by 𝐼 𝑏 var for 3D-PCA (left) and 3D-AE (right). Known transients and XRT 200515 are highlighted.
\n\n\n\n
Figure 10. Embedding clusters for the 3D-PCA case (left) and 3D-AE case (right).
\n\n
\n\n
Table 7. The first 5 samples of our transient candidates catalog showing a subset of selected columns.
\n
\n\n\n
Figure 11. Lightcurves in the 0.5-7 kev energy range for different examples of dips (blue), flares (red) and pulsating or quasi-periodic sources (green) in the transient candidates catalog. The shown pulsating or quasi-periodic sources are part of the flare candidates.
\n\nspectral information, in particular at the level of individual clusters, allowing for the identification of X-ray sources with specific spectral signatures. For the 2D-PCA and 2D-AE cases, these gradients are more uniform across the embedding space, because the temporal and spectral information of event files are captured by one axis each in the 𝐸 -𝑡 Maps. Moreover, some clusters consist exclusively of soft or hard sources, demonstrating that our representations can be leveraged not only to identify transients but also to find analogs to sources with specific spectral characteristics.\nFigure 9 shows the 3D-PCA and 3D-AE embedding spaces, now color-coded by the variability index 𝐼 𝑏 index with the other two cases shown in Appendix D. The learned embeddings also encode the temporal behavior of the sources, with some clusters being dominated by X-ray detections with significant variability, including transient behavior. To demonstrate this, we also highlight the embeddings of the bona-fide flares and dips listed in Table 6. Note that these occupy very well-defined clusters on the edges of the representation space, allowing for queries of analog transient behavior. In the 2D-PCA and 2D-AE cases, transient sources are distributed across multiple small clusters on the edges of the embedding spaces. In contrast, the 3D-PCA and 3D-AE embedding spaces achieve a significantly more compact clustering of bona-fide transients because temporal features in the event files are given a higher importance by the introduction of an additional time-related axis in the 𝐸 -𝑡 -𝑑𝑡 Cubes.\nFigure 10 shows the clusters identified by the DBSCAN algorithm in the 3D-PCA and 3D-AE cases. The clusters for the other two cases, 2D-PCA and 2D-AE , are shown in Appendix D. The largest cluster in all cases (Cluster 1) corresponds to observations that are not 'anomalous', for example non-variable sources or noisy detections in the low-count regime. We also see multiple smaller clusters on the edges of the embedding space clearly separated from this main cluster. Of special interest are clusters that contain known discovered transients, as these likely host other interesting transients that have not yet been discovered. Some of the edge clusters group observations with similar temporal and spectral behavior. For example, Cluster 4 in the 3D-PCA case only contains flares with high hardness ratios. Other clusters instead group observations primarily by similar temporal behavior, but then show a within-cluster grouping of similar spectral behaviors. For example, Cluster 4 in the 3D-AE case contains many dipping sources, but show a hardness ratio gradient within the cluster. When comparing the results of different feature extraction methods, we observe that in the 3D-AE embedding space, nearly all previously identified extragalactic FXTs live within a single, wellisolated cluster (Cluster 8). In contrast, the 3D-PCA embedding space distributes these extragalactic FXTs across multiple clusters. All of these points underline the effectiveness of our method and that the created representation space is highly informative.\n4.2 Catalog of X-ray Flare and Dip Candidates\nWe identify new transient candidates within clusters that are occupied by previously reported transients and by conducting nearestneighbor searches around these known transients. We compile these in a catalog of X-ray transient candidates, which includes both flares and dips. Table E1 lists the selected clusters used to define the new flare and dip candidates in addition to the 50 nearest neighbors of each bona-fide transient. From each selected cluster, we only include X-ray detections for which the variability index 𝐼 𝑏 var ≥ 5. This threshold corresponds to detections for which the Gregory-Loredo algorithm yields a confidence in the variability of at least 90%, and allows us to discard flare and dip-like behaviors that are not statistically significant. We also manually exclude a small fraction of\n\n\n
Figure 12. Distribution SIMBAD object types in the dip candidates catalog (blue) and flare candidates catalog (red). There are 11 dip candidates and 897 flare candidates, for which no SIMBAD match is found.
\n\nfalse positives identified by visual inspection of the lightcurves for both flare and dip candidates. The resulting catalog contains a total of 3539 detections, and the columns included are described in Appendix F. Table 7 shows the first 5 samples in our catalog for a subset of selected columns. Figure 11 shows a number of example lightcurves of the dips and flares in our catalog. The dip selection shows dips from LMXBs, a low-mass X-ray binary (HMXB), an ULX, an eclipsing binary, a cataclysmic binary, and a quasar. The flare selection shows flares from an eruptive variable, a pulsar, an AGN, a HMXB, a cataclysmic variables and young stars. We also find a number of pulsating or quasi-periodic lightcurves from pulsars, magnetic cataclysmic variables and SGRs. Figure 12 shows the distribution of SIMBAD object types in our transient catalog. About 25% of the transient candidates do not have a SIMBAD match, making them particularly interesting sources for new transient discoveries. Our dip candidates include 6 Chandra observations with prominent dips from the known source CXOGlb J002400.9 -720453 in the globular cluster NGC 104 (47 Tuc). The catalog identifiers for these are CATALOG_ID: 2737_139, 16527_79, 15747_79, 16529_79, 15748_79, 16528_14 . Our flare candidates include a newly discovered extragalactic FXT, which is characterized and discussed in detail in § 4.3. Its catalog identifier is CATALOG_ID: 23022_122 . We recommend using our catalog to identify a diverse range of flares and dips. While this work is primarily motivated by the discovery of new extragalactic transients, we intentionally did not exclude galactic stellar flares to enable systematic follow-up studies to study flare incidence rates and the rotational evolution of stars. Users interested exclusively in extragalactic transients can filter out galactic sources using metadata from the CSC and the cross-match columns in the catalog.\n4.3 XRT 200515: A New Extragalactic Fast X-ray Transient\nAmong the flare candidates in our catalog, we discovered an intriguing new extragalactic Chandra FXT in an observation of the supernova remnant SNR 0509 -67.5 in the LMC on May 15, 2020 (Guest et al. 2022). What made this transient stand out from thousands of other flares discovered in this work is the unique temporal variability in its lightcurve, which exhibits no detectable pre-flare X-ray emission, a sharp rise of at least 4 orders of magnitude in the count rate to peak intensity followed by a sharp fall, all in a matter of a < 10 s, down to ∼ 800 s long oscillating tail. There is also notable spectral variability during the flare, characterized by an initially hard spectrum at the peak, followed by spectral softening in the tail. The combination of these temporal and spectral properties establishes this transient as the first of its kind within the sample of discovered Chandra FXTs. We designate this newly discovered FXT as XRT 200515 and present a detailed study and discussion of its potential origins.\n4.3.1 X-Ray Detection by Chandra\nThe transient XRT 200515 was detected in Chandra ObsID 23022. The target of the observation was the supernova remnant SNR 0509 -67.5 in the LMC, which is shown in Figure 13 alongside the newly discovered FXT event. Table 8 summarizes the properties of XRT 200515 and its associated Chandra source 2CXO J051117.2 -672556 in ObdID 23022. The transient was captured by the ACIS camera in the S4 chip, and is located significantly off-axis in this observation, at an angular distance of 11.75 arcmin from the aimpoint in the S3 chip. This leads to an elongated and relatively large PSF, which, in this case, is advantageous as it substantially reduces photon pile-up in the initial spike, by spreading the counts over many pixels. We processed the data of Chandra observation ObsID 23022 with the Chandra Interactive Analysis of Observations (/c.pc/i.pc/a.pc/o.pc) version 4.15 (Fruscione et al. 2006), with calibration data base version 4.9.8. In particular, we created a new level-2 event file with the /c.pc/i.pc/a.pc/o.pc task chandra_repro and filter it in energy and time with dmcopy . We obtained the sky position in Table 8 using the /c.pc/i.pc/a.pc/o.pc tool wavdetect . To reduce background noise and improve the determination of the source centroid, we applied wavdetect on an image filtered to include only the time interval from the beginning of the flare ( 𝑡 0 ) until a time 𝑡 0 + 920 s. The 90% uncertainty radius of 2.0 arcsec is the combination of the uncertainty in the source centroid position reported by wavdetect , and the absolute astrometry uncertainty in a typical ACIS observation for off-axis sources 3 .\nThe field was previously covered by four other Chandra observations (ObsIDs 776, 7635, 8554, and 23023) with no source detections at the location of 2CXO J051117.2 -672556. We estimated modelindependent upper limits to the source flux and luminosity with /c.pc/i.pc/a.pc/o.pc tool srcflux . In the pre-flare part of ObsID 23022, we obtained a 90% confidence limit of 𝐿 X < 1 . 0 × 10 34 erg / s in the 0.3-7 keV band at the LMC distance of 50 kpc. Stacking the data from all the ObsIDs with non-detections, including the pre-flare part of ObsID 23022, results in a total observed exposure of approximately ∼ 150 ks, and yields a 90% confidence upper limit on the X-ray luminosity is 𝐿 X < 3 × 10 33 erg / s.\n
\n\n
Table 8. Properties of the Chandra observation ObsID 23022 and source 2CXO J051117.2 -672556 associated with XRT 200515.
\n
\n\n\n
Figure 13. ACIS-S image for Chandra observation ObsID 23022 showing the target, SNR 0509 -67.5, on the bottom right and the transient event, XRT 200515, on the top left. For this image, the event file was filtered to include only the time interval 𝑡 0 + 920 s around the flare. Red counts correspond to photons in the 0.3-1.2 keV band, yellow counts correspond to photons in the 1.2-2.4 keV band, and blue photons correspond to photons in the 2.4-7 keV band. The inset image is a 1.0 arcmin × 1.0 arcmin zoomed-in view. The dashed ellipse has semi-minor and semi-major axes of 15 arcsec × 20 arcsec, and is the source region used for spectral extraction.
\n\n4.3.2 X-ray Temporal Analysis\nWe used the /c.pc/i.pc/a.pc/o.pc tool dmextract to extract background-subtracted lightcurves in several energy bands, from the reprocessed event file of Chandra ObsID 23022. We defined an elliptical source extraction region, with semi-minor and semi-major axes of 15 arcsec and 20 arcsec (matching the point-source PSF at the source location); the local background region was chosen in the same ACIS chip, with an area approximately eight times larger.\nFigure 14 shows the 0.3-7 keV background-subtracted lightcurve of XRT 200515 with a time resolution of 20 s. The lightcurve is consistent with no source detection at the location of the transient, before the start of the flare at around 23.5 ks into the observation. The few pre-flare counts are consistent with background noise. The lightcurve exhibits a strong initial spike with a sharp rise of at least 4 orders of magnitude in < 10 s, containing 44 out of all ∼ 180 flare counts. This initial burst is followed by a sudden drop to a ∼ 800 s long pulsating and decaying tail. We estimate a 𝑇 90 ∼ 580-740 s for the photons observed in the 0.3-7 keV band 4 , depending on the definition of total flare counts.\n\n\n
Figure 14. Background-subtracted lightcurve of XRT 200515 in the 0.37 keV energy range with a bin size of 20 s. The zero start time is taken as the start of the Chandra observation ObsID 23022. The inset shows the last ∼ 2 ks of the observations, and captures the initial burst and tail of the transient event. The bin size is chosen to better visualize the oscillatory decay of the tail. The actual duration of the initial burst peak is < 10 s. The presence of a few negative counts arises from the process of background subtraction.
\n\nFigure 15 shows the lightcurve of XRT 200515 at a resolution matching the ACIS frame time of 3.2 s, the hardness ratio, and the energy evolution for the time interval 𝑡 0 + 920 s. The lightcurve exhibits a spike in the count rate across only 3 bins (with a total of 4, 31 and 9 counts, respectively), hence the burst duration of < 10 s. The rise and fall times of the burst are both between 3.2 s and 6.4 s. The maximum count rate at the Chandra frame time resolution is ∼ 9.7 counts/s, acting as the lower bound for the peak count rate of the burst. Those counts are spatially spread over a PSF area of ∼ 3000 pixels; therefore, pile-up is not an issue. We evaluated the hardness ratio evolution during the flare with the Bayesian estimation method BEHR (Park et al. 2006). Here, the hardness ratio is defined as:\n𝐻𝑅 = ℎ -𝑚 -𝑠 ℎ + 𝑚 + 𝑠 , (13)\nwhere 𝑠 is the number of soft photons (0.3-1.2 keV), 𝑚 is the number of medium photons (1.2-2 keV), and ℎ is the number of hard photons (2-7 keV) in each bin. We also track the running average of the photon energies during the flare with a moving window of ± 10 counts. The hardness ratio and energy evolution indicate spectral softening during the flare, with the hardness ratio starting at 1 during the hard burst peak and decreasing to a range of 0.4 to 0.6 in the tail, highlighting the notable spectral variability of XRT 200515.\n4.3.3 X-ray Spectral Analysis\nWe used the /c.pc/i.pc/a.pc/o.pc tool specextract to extract the spectrum and the associated response and ancillary response files from the reprocessed event file of Chandra ObsID 23022. We used the same source and background extraction regions defined for the lightcurve extraction. To improve the signal-to-noise ratio of the source, we extracted the spectrum only from the time interval 𝑡 0 + 920 s. We binned the spectrum to a minimum of 1 count per bin with the grppha task within the /f.pc/t.pc/o.pc/o.pc/l.pc/s.pc package suite (Blackburn 1995) from NASA's High Energy Astrophysics Science Archive Research Center (HEASARC) 5 . For all spectral modelling and flux estimates, we used the /x.pc/s.pc/p.pc/e.pc/c.pc software version 12.13.0 (Arnaud 1996). With only 179 net counts, we\n\n\n
Figure 15. Upperpanel: Background-subtracted count rate lightcurve of XRT 200515 in the 0.3-7 keV energy range at the full Chandra detector resolution of 3.2 s and hardness ratio evolution during the flare obtained for a minimum of 20 counts per bin. The zero start time is taken as the flare start time 𝑡 0 of XRT 200515. The initial peak is < 10 s long and is very hard, while the ∼ 800 s long oscillatory tail is significantly softer. Bottom panel: The energy evolution during the flare obtained from the running average energy with a moving window of ± 10 counts, showing significant spectral variability and softening during the flare. The scatter points represent the time and energy of individual photons from XRT 200515 in the event file associated with Chandra observation ObsID 23022.
\n\nare unable to fit complex spectral models; thus, we limit our analysis to the simplest one-component models representative of opposite scenarios: a power law ( powerlaw ) and a blackbody model ( bbody ), both modified by photo-electric absorption ( tbabs ). In both cases, we adopted the Tuebingen-Boulder absorption model with Wilms abundances (Wilms et al. 2000). We minimized the Cash statistic (Cash 1979), as we do not have enough counts for 𝜒 2 fitting.\nThe best-fitting power-law model (Table 9 and Figure 16) has a photon index of Γ = 0 . 5 ± 0 . 3. The fit statistics yield a null hypothesis probability of 3 . 5 × 10 -3 , with a Cstat value of 132.7 for 137 degrees of freedom. For the blackbody model, the best-fitting temperature is 𝑘𝑇 bb = 1 . 8 ± 0 . 3 keV (Table 9). The fit statistics yield a null hypothesis probability of 1 . 2 × 10 -2 , with a Cstat value of 129.6 for 137 degrees of freedom. The reason this blackbody spectrum may appear hard in the Chandra band, resembling a Γ ∼ 0 . 5 power law, is that at a temperature of 𝑘𝑇 bb ∼ 2 keV, the ACIS detector samples only the peak and the Rayleigh-Jeans (rising) portion of the blackbody emission.\nWe can use either model to determine an average conversion between the count rate and luminosity. This will then enable us to estimate the peak luminosity in the initial spike, for which we have previously estimated a peak count rate of ≳ 10 counts/s. The best-fitting power law model implies a peak flux of 𝐹 p ≳ 5 . 6 × 10 -10 erg/s/cm 2 , a total flare fluence of 𝐸 f ≳ 1 . 1 × 10 -8 erg/cm 2 , and a peak unabsorbed 0.3-10 keV luminosity of 𝐿 X ≳ 1 . 7 × 10 38 erg/s at the LMC distance of 50 kpc. For the best-fitting blackbody model, the peak flux and flare fluence would be 𝐹 p ≳ 4 . 0 × 10 -10 erg/s/cm 2 and 𝐸 f ≳ 0 . 8 × 10 -8 erg/cm 2 respectively. The peak unabsorbed 0.3-10 keV luminosity would be 𝐿 X ≳ 1 . 2 × 10 38 erg/s and the peak\n
\n\n
Table 9. Best-fitting parameters of the Chandra /ACIS-S spectrum of XRT 200515, fitted with the Cash statistics, for an absorbed power law and an absorbed blackbody model. Because of the relatively low number of counts, parameter uncertainties are reported at the confidence interval Δ 𝐶 = ± 1 . 0: this is asymptotically equivalent to the 68% confidence interval (1 𝜎 ) in the 𝜒 2 statistics.
\n
\n\n\n
Figure 16. Upper panel: Observed X-ray spectral energy distribution from ∼ 920 s around the flare XRT 200515 and the best-fit absorbed power law model. The data have been rebinned to a minimum of 10 counts per bin for plotting purposes only; a binning of 1 count per bin was instead used for the fitting (Cash statistics). Lower panel: Residuals between the data and the best-fit model.
\n\nbolometric luminosity would be 𝐿 bol ≳ 1 . 5 × 10 38 erg/s. These values should be considered conservative lower limits for two reasons: (i) the peak count rate provides only a lower bound estimate, as it is constrained by the Chandra frame time resolution of the observations, potentially underestimating the true peak count rate; and (ii) the conversion factor applied is derived from the average spectrum over the entire flare, even though the spectrum of the initial spike is significantly harder compared to the tail, as shown in Figure 15.\nWe searched for potential detections of XRT 200515 by other highenergy facilities. However, no significant X-ray or 𝛾 -ray events in the field around the X-ray source coordinates and flare start time 𝑡 0 reported in Table 8 were detected by the Fermi Gamma-ray Space Telescope ( Fermi ), the Burst Alert Telescope ( BAT ) on the Neil Gehrels Swift Observatory ( Swift ), the International Gamma-Ray Astrophysics Laboratory ( INTEGRAL ), or the Monitor of All-sky X-ray Image MAXI . LIGO was not operational during the time of the FXT, hence no gravitational wave signal could have been detected if the origin of XRT 200515 was a compact object merger.\n4.3.5 Optical Counterpart Search\nWe used the X-ray source coordinates reported in Table 8 to search for optical and infrared counterparts to XRT 200515. The field of XRT 200515 was covered by the Survey of Magellanic Stellar History ( SMASH ) (Nidever et al. 2017), a deep optical survey in the ugriz bands with the Dark Energy Camera (DECam) mounted on the Víctor M. Blanco Telescope at the Cerro Tololo Inter-American Observatory (CTIO) in Chile. We used the Astro Data Lab Jupyter Notebook server (Nikutta et al. 2020; Juneau et al. 2021) to access and visualize the SMASH catalog 6 . Figure 17 shows a color image of the field created from the deepest available stacked images in the u , g and i bands; the 5 𝜎 detection limits in these bands are 23.9 mag, 24.8 mag and 24.2 mag, respectively. The images were taken on December 7, 2015 with exposure times of 1,179 s, 981 s and 1,179 s respectively. The astrometry of the SMASH images is calibrated on the Gaia DR3 reference frame, thus their positional uncertainty is negligible compared to the X-ray source position uncertainty. Within the Chandra position error circle in Figure 17, there is no obvious optical counterpart that stands out in brightness or color from the surrounding stellar population. We performed relative photometry on the sources inside the error circle, comparing them to several nearby sources with known positions and brightnesses listed in the Gaia DR3 catalog. We used the SMASH g band as the closest approximation to Gaia's G band. We estimate the brightest optical source within the error circle to have a Vega magnitude of 𝑔 = 22 . 7 ± 0 . 1 mag, corresponding to an absolute magnitude of 𝑀 𝑔 ≈ 4 . 2, assuming it is in the LMC. Additionally, three other point-like sources are detected with 𝑔 band magnitudes in the range of 23-24 mag. All four sources appear pointlike, consistent with the seeing conditions of the SMASH survey, with no evidence of any spatially extended background galaxies. The three brightest stars visible in Figure 17 within ∼ 12 arcsec of the Chandra source are solar-mass stars on the red giant branch, indicative of an old stellar population.\nThe lack of bright optical counterparts and the short burst duration of < 10 s rules out a stellar flare from a foreground Galactic low-mass star (Güdel 2004; Reale 2007; Reale & Landi 2012; Pye et al. 2015; Kuznetsov & Kolotkov 2021). A flare from a Be/X-ray binary or any other HMXB in the LMC is also excluded by the lack of a bright optical counterpart (Ducci et al. 2019, 2022). The temporal and spectral properties of XRT 200515, combined with the absence of an optical counterpart, suggests three possibilities: (i) a relativistic jet phenomenon, such as a 𝛾 -ray burst (GRB); (ii) a rapid, high-energy process linked to extreme magnetic fields, such as a giant magnetar flare (GMF); or (iii) a thermonuclear Type I X-ray burst caused by surface nuclear burning on a neutron star.\n\n\n
Figure 17. SMASH survey color image of the field of XRT 200515 created from the deepest available stacked images in the u , g an i bands. Red corresponds to the i band, green to the g band, and blue to the u band. The dashed circle has a radius of 2 arcsec, and is the 90% position uncertainty of the Chandra source.
\n\n4.3.6 Gamma Ray Burst from a Compact Object Merger?\nEvidence in favor or against the association of at least some Chandra FXTs with low-luminosity long-GRBs or off-axis short-GRBs (see Berger 2014 for a review), at moderate or high redshifts, is extensively discussed in Quirola-Vásquez et al. (2022), Quirola-Vásquez et al. (2023), and Wichern et al. (2024). A detailed re-investigation of this issue is beyond the scope of this work. Here, we simply point out that XRT 200515, like the other Chandra FXTs in the literature, does not have any 𝛾 -ray detection. On the other hand, XRT 200515 has a significantly harder spectrum ( Γ = 0 . 5 ± 0 . 3) in the Chandra band than the rest of the FXT sample, all of which have photon indices of Γ > 1 (Jonker et al. 2013; Glennie et al. 2015; Bauer et al. 2017; Xue et al. 2019; Lin et al. 2022; Quirola-Vásquez et al. 2022; Quirola-Vásquez et al. 2023; Eappachen et al. 2023). A photon index of Γ ∼ 0 . 5 below 10 keV is indeed expected and observed from both core-collapse GRBs and compact-merger GRBs (Ghirlanda et al. 2009; Bromberg et al. 2013; Oganesyan et al. 2018; Ravasio et al. 2019; Toffano et al. 2021). This might support the association of the initial spike of XRT 200515 with a GRB. However, the presence and properties of the ∼ 800 s tail (candidate GRB afterglow) is puzzling. The 𝑇 90 ∼ 580-740 s value for XRT 200515 is significantly shorter than in most other Chandra FXTs (Quirola-Vásquez et al. 2022; Lin et al. 2022; Quirola-Vásquez et al. 2023), which have 𝑇 90 values on the order of several ks and are already pushing the limit for a GRB afterglow detection (Wichern et al. 2024). Moreover, XRT 200515's initial burst duration ( < 10 s), its short rise and fall times (3.2-6.4 s), and the lack of a peak plateau are inconsistent with the lightcurves of Chandra FXTs interpreted as magnetar-powered GRBs as the aftermath of a binary neutron star merger, such as CDF-S XT1 (Bauer et al. 2017), CDF-S XT2 (Xue et al. 2019) and the sample in Lin et al. (2022). Finally, the lack of any optical evidence for a host galaxy is another element disfavoring the high-redshift GRB interpretation.\n4.3.7 Giant Magnetar Flare from a Soft Gamma Repeater?\nBased on its temporal and spectral variability, it is tempting to interpret XRT 200515 as a rare GMF from a SGR (Mereghetti 2008; Turolla et al. 2015) in the LMC or behind it, which can easily explain the burst's strong increase of at least 4 orders of magnitude in < 10 s (Coti Zelati et al. 2018). Similar to XRT 200515, GMFs are characterized by a short and hard initial spike and a longer and softer, pulsating tail. GMFs are extremely rare, with only a select few ever discovered. Well-studied examples are SGR 0526 -66 in the LMC (Mazets et al. 1979), and the Galactic sources SGR 1900 + 14 (Hurley et al. 1999) and SGR 1806 -20 (Hurley et al. 2005; Palmer et al. 2005; Israel et al. 2005). More recently, GMFs have been identified in M 31 (Mazets et al. 2008a), NGC 253 (Fermi-LAT Collaboration et al. 2021; Svinkin et al. 2021; Roberts et al. 2021; Trigg et al. 2024) and M82 (Mereghetti et al. 2024). All of these have been observed by high time resolution instruments in the hard X-rays and soft 𝛾 -rays with luminosities above 10 46 erg/s for a fraction of a second in the initial spike. The tails of GMFs are often modulated by magnetar spin periods of 2-12 s, leading to quasi-periodic oscillations (QPOs). For XRT 200515, there is no hard X-ray or 𝛾 -ray detection, despite the LMC direction being in good visibility for most of the previously mentioned high-energy facilities. We were unable to identify any significant periodicities in the tail of XRT 200515 through periodogram analysis, which is unsurprising given the low time resolution of Chandra observations. No X-ray activity has been observed by Chandra or other X-ray telescopes in the years before or after XRT 200515, which may be because SGRs are very faint when they are not bursting. The strongest argument against a magnetar in the LMC as the origin of XRT 200515 is that magnetars are short-lived objects ( ≲ 10 5 yr) associated to young stellar populations (Olausen & Kaspi 2014; Nakano et al. 2015; Mondal 2021). Even allowing for the persistence of magnetar-like activity in ordinary radio pulsars as old as ∼ 10 7 yr (Rea et al. 2010), this scenario is still inconsistent with the old stellar population (several Gyr) in the LMC field shown in Figure 17. The nearest star-forming regions in the LMC are ∼ 10 arcmin ( ∼ 150 pc) away. If (in a very contrived scenario), we assume that XRT 200515 is powered by a young neutron star ejected from one of those regions, we estimate a characteristic time of 1 Myr to travel that distance at a speed of 150 km/s. Therefore, if XRT 200515 is a GMF, it must be located behind the LMC, in a low-redshift galaxy (Hurley et al. 2005; Tanvir et al. 2005). Since GMFs have been observed only a few times and never at soft X-ray energies, their properties in the soft X-ray band detectable by Chandra remain largely unexplored. XRT 200515 could indeed be the first GMF detected at soft X-ray energies. Distinguishing distant short GRBs from GMFs has historically been difficult and there are multiple studies suggesting that a subset of short GRBs are actually extragalactic GMFs (Hurley et al. 2005; Palmer et al. 2005; Tanvir et al. 2005; Ofek et al. 2006; Mazets et al. 2008b; Hurley 2011; Yang et al. 2020; Svinkin et al. 2021; Negro & Burns 2023). Just as for the distant GRB interpretation, the non-detection of any optical counterpart remains puzzling for a distant GMF scenario, unless we are dealing with a very distant and exceptionally luminous GMF.\n4.3.8 Thermonuclear X-ray Burst from a quiet LMXB in the LMC?\nIf XRT 200515 is in the LMC, a peak luminosity near the Eddington luminosity 𝐿 Edd ∼ 10 38 erg/s and sharp rise time of the flare suggests a Type I X-ray burst interpretation, which is a thermonuclear explosion on the surface of a weakly magnetized, accreting neutron star (Lewin et al. 1993; Strohmayer & Bildsten 2003; Gal-\nloway et al. 2008, 2020; Galloway & Keek 2021; Alizai et al. 2023).\nThe old stellar population in the field of XRT 200515 is consistent with the presence of neutron star LMXBs. Following the definition of burst timescale 𝜏 = 𝐸 f / 𝐹 p in Galloway et al. (2008), we estimate 𝜏 ∼ 20 s for XRT 200515, which is consistent with Type I X-ray bursts (Galloway & Keek 2021; Alizai et al. 2023). The fitted temperature 𝑘𝑇 bb ∼ 2 keV when the average spectrum is fitted with a simple blackbody, and the softening of the spectrum (temperature decrease) in the tail is also typical of Type I X-ray bursts (Galloway et al. 2008, 2020; Güver et al. 2012). On the other hand, several observed properties of XRT 200515 are unusual for Type I X-ray bursts. In particular, most Type I X-ray bursts occur when the persistent luminosity (proportional to the accretion rate) of a LMXB is 𝐿 X > 10 -4 𝐿 Edd (and, in most cases, 𝐿 X > 10 -3 𝐿 Edd ) (Galloway et al. 2008). Instead, in the initial part of ObsID 23022, the upper limit on the X-ray luminosity at the position of XRT 200515 is 𝐿 X < 10 -4 𝐿 Edd , so that the X-ray flux increased by at least 4 orders of magnitudes. On another note, the sharp decline after the initial burst of XRT 200515 would be unusual for Type I X-ray bursts, which typically exhibit a gradual and exponential decay. However, note that most Type I X-ray bursters were observed by the Rossi X-Ray Timing Explorer ( RXTE ) (Jahoda et al. 1996), which has a high time resolution. The low time resolution of Chandra may have obscured such a decay for XRT 200515. Moreover, most Type I bursts tend to repeat every few hours (Galloway et al. 2008); instead, XRT 200515 is the only event detected at that location over a total observed time of ∼ 150 ks. No LMXB has ever been noted at that position before or after the event. The time interval between bursts is related to an index 𝛼 defined as the ratio between the integrated persistent fluence between subsequent bursts and the burst fluence; from a comparison of the energy released by accretion (contributing to the persistent fluence) and by thermonuclear burning (burst fluence), we expect 𝛼 ≳ 40, in agreement with the observations of Type I bursts (Galloway et al. 2008). If we apply the same criterion ( 𝛼 ≳ 40) to the persistent and flare fluences of XRT 200515, we would have to wait > 10 7 s (4 months) to observe another similar event, assuming the persistent flux level upper limit in ObsID 23022 before the transient event. This waiting time extends to at least one year if we assume the persistent flux upper limit derived from the stacked ∼ 150 ks Chandra observations. Only a few one-off bursts from Galactic neutron stars at a very low persistent luminosity ( 𝐿 X ∼ 10 32 -10 33 erg/s) were found by Cornelisse et al. (2002a,b) with estimated recurrence times of tens of years. The vast majority of Type I X-ray bursts are Galactic, due to their lower flux at large distances. Only a handful of extragalactic Type I X-ray bursts are documented, for example in M 31 (Pastor-Marazuela et al. 2020) and the Magellanic Bridge (Haberl et al. 2023). If XRT 200515 is a Type I X-ray burst, it is the first extragalactic Type I X-ray burster in the LMC and represents the tip of the iceberg for a vast population of faint LMXBs in nearby galaxies, too dim to be detected by Chandra or XMM-Newton , but which may occasionally reveal themselves via thermonuclear bursts with a long duty cycle.\n4.3.9 Concluding Remarks and Outlook for XRT 200515\nXRT 200515 is a unique and intriguing extragalactic Chandra FXT. The combination of its temporal and spectral properties is unlike any of the other Chandra FXT samples. Based on our analysis, the two most likely scenarios for XRT 200515 are: (i) a distant GMF from a SGR behind the LMC; the first observed in the low X-ray energy band, missed by any other high-energy facilities, or (ii) an unusual Type I X-ray burst from a previously unknown faint LMXB; the first extragalactic X-ray burster in the LMC. Nevertheless, both\n\n\n
Figure 18. 𝐸 -𝑡 Mapevent file representation (left) and 𝐸 -𝑡 -𝑑𝑡 Cube event file representation (right) of XRT 200515 in Chandra observation ObsID 23022. The catalog identifier for XRT 200515 is CATALOG_ID: 23022_122 .
\n\nof these interpretations come with their own unique challenges. XRT 200515 could, in fact, represent an entirely new type of astronomical phenomenon. After all, the primary objective of our work was to use machine learning to find rare, needle-in-the-haystack anomalies hidden within vast astronomical datasets. We invite further detailed studies of XRT 200515 to evaluate our interpretations and explore alternative scenarios, such as potential associations with a fast radio burst (FRB) or a SBO. We highly recommend follow-up multi-band observations at the source coordinates of XRT 200515 to better constrain its nature. Lastly, we note that XRT 200515 and the second transient discovered by Glennie et al. (2015), XRT 120830, have remarkably similar temporal evolutions in their lightcurves (J. Irwin, personal communication, November 2024), however with very different spectral properties ( Γ ∼ 2 . 5 for XRT 120830 versus Γ ∼ 0 . 5 for XRT 200515). We leave a detailed comparative analysis of these transients for future work.\nFigure 18 shows the 𝐸 -𝑡 Map and 𝐸 -𝑡 -𝑑𝑡 Cube event file representations for XRT 200515. These exhibit high counts at high energies in a narrow time window, which is in line with the hard spectrum and transient nature of XRT 200515.\n4.4 Technical Caveats\nThe main technical caveat of our approach is related to the representation of event files. While our new event file representations enable a simple, yet powerful representation learning approach to find new and rare X-ray transients, any simplification of raw event files, like the fixed number of time bins we use across all event files, is associated with a loss of information. This could lead to us missing a small amount of transients. To minimize this, we have implemented a rigorous approach to justify the resolution of the event file representations in § 3.1. Moreover, flares, in particular known extragalactic FXTs, cluster notably well in our representation spaces. This is because their distinctive features are less dependent on the temporal binning resolution in the 𝐸 -𝑡 Maps and 𝐸 -𝑡 -𝑑𝑡 Cubes. To improve the effectiveness of dip searches with our proposed method, we suggest using higher resolution event file representations. Nevertheless, our comprehensive transient candidate catalog includes numerous newly identified transients that were previously overlooked by other X-ray transient searches in the Chandra archive. Among these is the remarkable needle-in-the-haystack event XRT 200515 discovered in this work, underscoring the effectiveness of our method. A followup representation learning algorithm will learn informative features from raw and unbinned event files while accounting for the Poisson nature of X-ray observations (Song et al., 2025, in preparation).\n5 CONCLUSION\nWe have introduced a novel representation learning method, the first of its kind applied to X-ray event files, enabling downstream tasks such as unsupervised classification and anomaly detection in highenergy astrophysics. We have used the learned representation to investigate time-domain properties of sources in the Chandra archive, with a particular emphasis on the discovery of X-ray transients. As a result, we have compiled the identified X-ray flares and dips in a comprehensive catalog of transient candidates. Notably, our method led to the discovery of XRT 200515; a previously unidentified extragalactic FXT with unique temporal and spectral properties, representing a genuine needle-in-the-haystack discovery. Our key results are as follows:\n\n(i) Weintroduce novel event file representations, the E-t Mapsand E-t-dt Cubes, which capture both temporal and spectral information.\n(ii) We apply two feature extraction methods to the event file representations, PCA and sparse autoencoder neural networks, to extract or learn informative features that can be utilized for downstream tasks, such as unsupervised classification or anomaly detection.\n(iii) We project the learned features to two-dimensional embedding spaces, enabling interpretable queries of analogs to objects of interest based on their temporal and spectral properties.\n(iv) We cluster the embedding spaces with DBSCAN, successfully isolating previously identified X-ray transients. We identify new transient candidates within specific transient-dominant clusters or through nearest-neighbor searches using kNN.\n(v) We compile a catalog of the X-ray transient candidates, including 3,427 flares and 112 dips, and make it openly accessible to the community and the broader scientific audience.\n(vi) We report the discovery of XRT 200515, a rare extragalactic FXT characterized by unique temporal and spectral features. We explore its potential origins and suggest that it may be associated with one of the following scenarios, presented in no particular order:\n· A rare GMF from an SGR behind the LMC, marking the first GMF detected in the low X-ray energy range covered by telescopes like Chandra , XMM-Newton , Swift-XRT , eROSITA , or Einstein Probe .\n· A rare extragalactic Type I X-ray burst from a faint LMXB in the LMC, representing the first such detection in the LMC.\n· A new type of astronomical phenomenon and a genuine anomaly, previously hidden in the vast Chandra archive.\n\nXRT200515wasonlydetectedby Chandra , with no identified optical counterparts. We strongly encourage a multi-wavelength search for additional signals from the source associated with XRT 200515 to better understand its origin and nature.\nOurworkadvancestime-domainhigh-energy astrophysics by making the Chandra transient candidates catalog publicly available and open-sourcing the representation learning based transient search pipeline 7 . The catalog enables queries to identify new Chandra transients. Future work involves applying the detection pipeline to additional high-energy archives and adapting it to a variety of other scientific datasets, paving the way for further machine learning driven discoveries of rare transients and other scientific anomalies.\nACKNOWLEDGEMENTS\nThis research has made use of data obtained from the Chandra Source Catalog, and software provided by the Chandra X-ray Center (CXC) in the /c.pc/i.pc/a.pc/o.pc application package.\nSD's work was partially funded by the UK government's Turing Scheme and mainly carried out at the Center for Astrophysics | Harvard & Smithsonian as part of the SAO Predoctoral Program, with the support of AstroAI. SD acknowledges hospitality at the Institute of Astronomy at the University of Cambridge and at the Stanford Center for Decoding the Universe (Stanford Data Science) during the later parts of this project. RS's work was performed in part at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-2210452. RS acknowledges support and hospitality at the National Astronomical Observatories of China (NAOC) in Beijing, during part of this project.\nWe thank Edo Berger, Massimiliano De Pasquale, Ken Ebisawa, Duncan Galloway, Jimmy Irwin, Peter Jonker, Daniel Kocevski, Amy Lien, Sandro Meregetthi, Daniel Muthukrishna, Nicola Omodei, Jonathan Quirola-Vásquez, Shivam Raval, Ashley Villar, and Silvia Zane for their fruitful discussions.\nDATA AVAILABILITY\nThe data used in this paper, composed of X-ray event files and source detection regions, was obtained from the publicly available CSC, using their public interfaces (https://cxc.cfa.harvard.edu/csc/). The catalog of transient candidates and the clustered embedding spaces generated using our unsupervised representation learning method can be accessed in the supplementary material. All intermediate data products, i.e. as 𝐸 -𝑡 Maps, 𝐸 -𝑡 -𝑑𝑡 Cubes, principal components and latent features, feature embeddings and embedding clusters can be produced using the code provided in the GitHub repository https://github.com/StevenDillmann/ml-xraytransients-mnras.\nREFERENCES\n\nAbadi M., et al., 2015, TensorFlow: Large-Scale Machine Learning on Heterogeneous Systems, https://www.tensorflow.org/ Alizai K., et al., 2023, Monthly Notices of the RAS, 521, 3608 Alp D., Larsson J., 2020, The Astrophysical Journal, 896, 39 Arcodia R., et al., 2021, Nature, 592, 704 Arnaud K., 1996, in ASP Conf.. Astropy Collaboration et al., 2013, Astronomy and Astrophysics, 558, A33 Bauer F. E., et al., 2017, Monthly Notices of the RAS, 467, 4841 Bellm E. C., et al., 2019, PASP, 131, 018002 Bengio Y., Courville A., Vincent P., 2013, IEEE transactions on pattern analysis and machine intelligence, 35, 1798 Berger E., 2014, Annual Review of Astronomy and Astrophysics, 52, 43 Blackburn J. K., 1995, in Shaw R. A., Payne H. E., Hayes J. J. E., eds, Astronomical Society of the Pacific Conference Series Vol. 77, Astronomical Data Analysis Software and Systems IV. p. 367 Bromberg O., Nakar E., Piran T., Sari R., 2013, The Astrophysical Journal, 764, 179 Burrows D. N., et al., 2005, SSR, 120, 165 Caliński T., Harabasz J., 1974, Communications in Statistics-theory and Methods, 3, 1 Cash W., 1979, The Astrophysical Journal, 228, 939 Chakraborty J., Kara E., Masterson M., Giustini M., Miniutti G., Saxton R., 2021, The Astrophysical Journal, Letters, 921, L40 Chambers K. C., et al., 2016, arXiv e-prints, p. arXiv:1612.05560 Colbert E. J. M., Ptak A. F., 2002, The Astrophysical Journal, Supplement,\n\n
\n\n
\n
\n\n
\n
\n\n
\nWalmsley M., et al., 2022, Monthly Notices of the RAS, 513, 1581\nWeisskopf M. C., Tananbaum H. D., Van Speybroeck L. P., O'Dell S. L., 2000, in Truemper J. E., Aschenbach B., eds, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Vol. 4012, X-Ray Optics, Instruments, and Missions III. pp 2-16 ( arXiv:astro-ph/0004127 ), doi:10.1117/12.391545\n\nWenger M., et al., 2000, Astronomy and Astrophysics, Supplement, 143, 9\n\nWichern H. C. I., Ravasio M. E., Jonker P. G., Quirola-Vásquez J. A., Levan\n\nA. J., Bauer F. E., Kann D. A., 2024, Astronomy and Astrophysics, 690, A101\nWilms J., Allen A., McCray R., 2000, The Astrophysical Journal, 542, 914 Xue Y. Q., et al., 2019, Nature, 568, 198\nYang G., Brandt W. N., Zhu S. F., Bauer F. E., Luo B., Xue Y. Q., Zheng\n\nX. C., 2019, Monthly Notices of the RAS, 487, 4721\n\nYang J., et al., 2020, The Astrophysical Journal, 899, 106\nYang H., Hare J., Kargaltsev O., Volkov I., Chen S., Rangelov B., 2022, The Astrophysical Journal, 941, 104\nYuan W., Zhang C., Chen Y., Ling Z., 2022, in Bambi C., Sangangelo A., eds, , Handbook of X-ray and Gamma-ray Astrophysics. p. 86, doi:10.1007/978-981-16-4544-0_151-1\nZhang B., 2013, The Astrophysical Journal, Letters, 763, L22\nZhang G., Helfer T., Gagliano A. T., Mishra-Sharma S., Villar V. A., 2024, arXiv e-prints, p. arXiv:2408.16829\nin't Zand J. J. M., et al., 2013, Astronomy and Astrophysics, 553, A83\n\n\n\n
Figure B1. Training process for the convolutional autoencoder applied to the 𝐸 -𝑡 Maps (top) and fully connected autoencoder applied to the 𝐸 -𝑡 -𝑑𝑡 Cubes (bottom). The plots show the evolution of the training and validation loss with the number of epochs including the early stopping point and epoch of the restored weights.
\n\n\n\n
Figure A1. Distribution of Chandra event file lengths 𝑁 (top) and durations 𝑇 (bottom) in the dataset used in this work.
\n\nAPPENDIX A: DISTRIBUTION OF EVENT FILE LENGTHS AND DURATIONS\nFigure A1 shows the distribution of the length 𝑁 and duration 𝑇 of event files in the dataset used in this work.\nAPPENDIX B: AUTOENCODER TRAINING PROCESS\nFigure B1 shows the training process of the autoencoders used in this work.\nAPPENDIX C: HYPERPARAMETER OPTIMIZATION\nBelow, we summarize the optimization strategy for the t-SNE and DBSCAN hyperparameters. For even more details on this approach, please refer to Dillmann & Martínez-Galarza (2023).\nC1 t-SNE Hyperparameters\nThe choice of the perplexity and learning_rate can have a large impact on the resulting t-SNE embedding space. Ideally, we want the two-dimensional embedding space to effectively capture both energy information (in form of the hardness ratio 𝐻𝑅 ) and variability information (in form of the variability probability 𝑝 var ). That means that event files with similar values for 𝐻𝑅 and 𝑝 var should live close to each other in the final embedding space. We can use this information to define a performance metric for different t-SNE hyperparameter inputs. First, we compute the pairwise distance matrix D Z of size ( 𝑚, 𝑚 ) , where the distance 𝐷 𝑍 𝑖 𝑗 between points 𝑖 and 𝑗 is computed using a Euclidean distance metric. Next, we define the property vector Y , which includes 7 CSC properties (hardness ratios 𝐻𝑅 hm , 𝐻𝑅 hs , 𝐻𝑅 ms and variability probabilities 𝑝 b var , 𝑝 h var , 𝑝 m var , 𝑝 s var ) for\neach event file and thus each t-SNE point. As a measure of similarity between the labels of different points, we can again compute a pairwise similarity matrix D Y of size ( 𝑚, 𝑚 ) . To compute the similarity distance 𝐷 𝑌 𝑖 𝑗 between sample 𝑖 and 𝑗 , we use the Mahalanobis distance metric (Mahalanobis 1936). Unlike the Euclidean distance metric, the Mahalanobis distance metric accounts for the correlation between different labels by taking into account the covariance structure of the data. Note that our hardness ratios are correlated with each other, and that the same holds for the variability probabilities. Accounting for these correlations provides a more accurate measure of the similarity distance between different samples. Having computed D Z and D Y , we can define a performance metric that allows us to compare the performance of different t-SNE hyperparameters. The smaller the distance 𝐷 𝑍 𝑖 𝑗 between two points 𝑖 and 𝑗 in the t-SNE embedding, the smaller should be difference in their associated labels as measured by the distance 𝐷 𝑌 𝑖 𝑗 . We can thus define a performance metric based on the statistical correlation of D Z and D Y using the Spearman's rank correlation coefficient 𝜌 𝑍𝑌 (Spearman 1904). The higher 𝜌 𝑍𝑌 , the higher is the positive correlation between D Z and D Y and the better the performance of the t-SNE embedding. The hyperparameter space is given by the ranges learning_rate ∈ ( 20 , 200 ) with a step size of 20 and perplexity ∈ ( 10 , 100 ) with a step size of 10. This optimization process is performed using a reduced dataset of 15,353 samples for 2,000 iterations per hyperparameter combination due to computational constraints. While subsampling, the overall structure of the data was preserved by selecting the same distributions between any combinations of hard, medium, soft, variable and non-variable samples. This ensures that the sample set is\nrepresentative of the original data. We choose the hyperparameter combination that produces the highest value of 𝜌 𝑍𝑌 .\nC2 DBSCAN Hyperparameters\nDifferent hyperparameter combinations of eps and minPts can have a large impact on the resulting DBSCAN clusters. We use a combination of the Davies-Bouldin index 𝐷𝐵 (Davies & Bouldin 1979) and Calinski-Harabasz index 𝐶𝐻 (Caliński & Harabasz 1974) as a performance metric to find the optimal DBSCAN hyperparameter inputs. The 𝐷𝐵 index is a measure of the average similarity between each cluster and its most similar cluster, relative to the average distance between points within each cluster. The 𝐷𝐵 index is given by the following formula:\n𝐷𝐵 = 1 𝑛 𝑐 𝑛 𝑐 ∑︁ 𝑖 = 1 max 𝑗 ≠ 𝑖 GLYPH<18> 𝑊 𝑖 + 𝑊 𝑗 𝑑 ( 𝑐 𝑖 , 𝑐 𝑗 ) GLYPH<19> , (C1)\nwhere 𝑛 𝑐 is the number of clusters, 𝑊 𝑖 and 𝑊 𝑗 are the within-cluster sum of squares for cluster 𝑖 and 𝑗 , and 𝑑 ( 𝑐 𝑖 , 𝑐 𝑗 ) is the distance between the centroids of clusters 𝑖 and 𝑗 . On the other hand, the 𝐶𝐻 index is based on the concept that good clusters should have high intra-cluster similarity (cohesion) measured by the between-cluster dispersion 𝐵 and low inter-cluster similarity (separation) measured by the within-cluster dispersion 𝑊 . 𝐵 is the sum of the pairwise distances between cluster centroids, and 𝑊 is the sum of the pairwise distances between points within each cluster. The 𝐶𝐻 index is given by the following formula:\n𝐶𝐻 = 𝐵 𝑊 × 𝑚 -𝑛 𝑐 𝑛 𝑐 -1 , (C2)\nwhere the scaling factor 𝑚 -𝑛 𝑐 𝑛 𝑐 -1 accounts for the total number of data points 𝑚 and the number of clusters 𝑛 𝑐 . A lower 𝐷𝐵 index and higher 𝐶𝐻 index indicate that the clustering algorithm is more effective in grouping similar data points together and separating different data points into distinct clusters. We thus define the performance metric 𝜌 𝐷𝐶 as the ratio of the normalized indices 𝐷𝐵 𝑛 = 𝐷𝐵 max ( 𝐷𝐵 ) and 𝐶𝐻 𝑛 = 𝐶𝐻 max ( 𝐶𝐻 ) in the hyperparameter space given by eps ∈ ( 1 . 0 , 3 . 0 ) with a step size of 0 . 1 and minPts ∈ ( 10 , 30 ) with a step size of 1:\n𝜌 𝐷𝐵𝑆𝐶𝐴𝑁 = 𝐶𝐻 𝑛 𝐷𝐵 𝑛 . (C3)\nWe choose the hyperparameter combination that produces the highest value of 𝜌 𝐷𝐵𝑆𝐶𝐴𝑁 .\nAPPENDIX D: EMBEDDINGS\nFigures D1, D2 and D3 show the 2D-PCA and 2D-AE embeddedings.\nAPPENDIX E: TRANSIENT-DOMINANT EMBEDDING CLUSTERS\nTable E1 lists the transient-dominant clusters in the different embedding spaces used for the selection of transient candidates.\nAPPENDIX F: CATALOG COLUMNS\nTable F1 shows the of X-ray transient candidate catalog column descriptions.\n
\n\n
Table E1. Transient-dominant clusters used to find new transient candidates.
\n
\nThis paper has been typeset from a T E X/L A T E X file prepared by the author.\n\n\n
Figure D1. Embedding representations color-coded by 𝐻𝑅 hs for the 2D-PCA case (left) and 2D-AE case (right).
\n\n\n\n
Figure D2. Embedding representations color-coded by 𝐼 𝑏 var for 2D-PCA (left) and 2D-AE (right). The bona-fide transients and XRT 200515 are highlighted.
\n\n\n\n
Figure D3. Embedding clusters for the 2D-PCA case (left) and 2D-AE case (right).
\n\n
\n\n
Table F1. Column descriptions of the catalog of X-ray transient candidates found in this work.
\n
\n"},"sections":{"kind":"list like","value":[{"title":"ABSTRACT","content":"We present a novel representation learning method for downstream tasks such as anomaly detection and unsupervised transient classification in high-energy datasets. This approach enabled the discovery of a new fast X-ray transient (FXT) in the Chandra archive, XRT 200515, a needle-in-the-haystack event and the first Chandra FXT of its kind. Recent serendipitous breakthroughs in X-ray astronomy, including FXTs from binary neutron star mergers and an extragalactic planetary transit candidate, highlight the need for systematic transient searches in X-ray archives. We introduce new event file representations, 𝐸 -𝑡 Maps and 𝐸 -𝑡 -𝑑𝑡 Cubes, designed to capture both temporal and spectral information, effectively addressing the challenges posed by variable-length event file time series in machine learning applications. Our pipeline extracts low-dimensional, informative features from these representations using principal component analysis or sparse autoencoders, followed by clustering in the embedding space with DBSCAN. New transients are identified within transient-dominant clusters or through nearest-neighbor searches around known transients, producing a catalog of 3,539 candidates (3,427 flares and 112 dips). XRT 200515 exhibits unique temporal and spectral variability, including an intense, hard < 10 s initial burst followed by spectral softening in an ∼ 800 s oscillating tail. We interpret XRT 200515 as either the first giant magnetar flare observed at low X-ray energies or the first extragalactic Type I X-ray burst from a faint LMXB in the LMC. Our method extends to datasets from other observatories such as XMM-Newton , Swift-XRT , eROSITA , Einstein Probe , and upcoming missions like AXIS . Key words: software: machine learning, methods: data analysis, X-rays: bursts, stars: magnetars, transients: gamma-ray bursts, stars: peculiar","pages":[1]},{"title":"Representation Learning for Time-Domain High-Energy Astrophysics: Discovery of Extragalactic Fast X-ray Transient XRT 200515","content":"Steven Dillmann 1 , 2 ★ † , Rafael Martínez-Galarza 3 , Roberto Soria 4 , 5 , Rosanne Di Stefano 3 and Vinay L. Kashyap 3 Accepted XXX. Received YYY; in original form ZZZ","pages":[1]},{"title":"1 INTRODUCTION","content":"Recent serendipitous discoveries, such as extragalactic fast X-ray transients (FXTs) linked to neutron star merger candidates as electromagnetic counterparts to gravitational wave events (Lin et al. 2022) and an X-ray dip associated with the first extragalactic planet candidate (Di Stefano et al. 2021), underscore the challenges of identifying such rare events within large X-ray catalogs. Beyond magnetarpowered FXTs as the aftermath of binary neutron star mergers (Dai et al. 2006; Metzger et al. 2008; Zhang 2013; Sun et al. 2017; Bauer et al. 2017; Xue et al. 2019), other interesting origins of extragalactic FXTs include supernova shock breakouts (SBOs) (Soderberg et al. 2008; Modjaz et al. 2009; Alp & Larsson 2020; Novara et al. 2020), tidal disruption events (TDEs) (Jonker et al. 2013) including quasiperiodic eruptions (QPEs) (Arcodia et al. 2021; Chakraborty et al. 2021), thermonuclear (Type I) X-ray bursts from accreting neutron stars (in't Zand et al. 2013), or binary self-lensing events (D'Orazio &DiStefano 2018, 2020; Hu et al. 2020). Both because of their very stochastic nature, and because narrow field X-ray missions such as the Chandra X-ray Observatory ( Chandra ) (Weisskopf et al. 2000), XMM-Newton (Jansen et al. 2001) and Swift-XRT (Burrows et al. 2005) are not designed as wide time-domain surveys, X-ray transient discoveries are often serendipitous. They can be found in observations that were originally proposed for a completely unrelated science objective and are rarely the target of the observation. In many cases serendipitously found X-ray sources do not get characterized or classified, since their transient nature is not immediately obvious. Instead, observations with X-ray transients often get stored in large data archives and remain unnoticed. This raises the need for a systematic search for short-duration phenomena in high-energy catalogs. New missions such as eROSITA (Predehl et al. 2021), Einstein Probe (Yuan et al. 2022) and the upcoming AXIS Observatory (Reynolds et al. 2024) target X-ray transients more directly, thus the development of novel transient detection methods is becoming even more relevant. The temporary, unpredictable and 'unusual' nature of X-ray transients distinguishes them from 'normal' X-ray source emissions. From a data science perspective, they can be understood as 'anomalies' within a large dataset. Existing methods for identifying X-ray transients primarily rely on statistical tests of variability (Yang et al. 2019; Pastor-Marazuela et al. 2020; Quirola-Vásquez et al. 2022; Quirola-Vásquez et al. 2023). While effective within specific constraints, these approaches are inherently limited by their underlying assumptions, which may not capture the diverse nature of transient phenomena. In contrast, machine learning offers a more flexible, expressive, and scalable framework, making it particularly well-suited for anomaly detection in large, high-dimensional datasets with diverse transient types. While optical time-domain surveys are at the forefront of leveraging extensive observational programs, like ZTF (Bellm et al. 2019) or the upcoming LSST survey (Ivezić et al. 2019), and neural network-based anomaly detection tools to identify rare sources among countless ordinary objects (Villar et al. 2021; Muthukrishna et al. 2022), the X-ray astronomy community has only recently begun exploring the potential of machine learning to classify sources (Yang et al. 2022; Pérez-Díaz et al. 2024) or to search for needle-in-a-haystack events in large X-ray datasets and archives (Kovačević et al. 2022; Dillmann & Martínez-Galarza 2023). The effectiveness of machine learning methods largely depends on the algorithm's ability to learn useful representations from the data. Representation learning (Bengio et al. 2013) is an increasingly popular technique in astronomy used in supervised, semi-supervised, self-supervised and unsupervised frameworks (Naul et al. 2018; Hayat et al. 2021; Walmsley et al. 2022; Slijepcevic et al. 2024; Mohale & Lochner 2024). It involves creating or learning meaningful representations for specific modalities of scientific data, which can then be used for downstream tasks such as regression, classification, or, as in this work, anomaly detection. The compressed representations live in a low-dimensional embedding space, in which anomalous data samples are well-separated from more ordinary ones. We propose a new unsupervised representation learning method to perform a large-scale search for X-ray transients in the Chandra archive. High-energy catalogs include individual X-ray source observations in the form of event files. The variable length of these time series poses a challenge in creating consistent representations suitable for transient searches with machine learning. Most deep learning algorithms take a fixed-length input for all data samples. In order to effectively represent event files over a broad range of lengths, we introduce novel fixed-length event file representations, which take into account both their time-domain and energy-domain information. Applying feature extraction and dimensionality reduction techniques, for example with sparse autoencoders, we create a representation space that encodes scientifically meaningful information, such as the spectral and variability properties of the astrophysical sources. Previously identified X-ray transients occupy distinct, well-isolated clusters in the embedding space. Using clustering techniques and nearest neighbor searches allows us to effectively explore these transient-dominant clusters to discover new X-ray transients. We collect the identified X-ray flare and dip candidates in a publicly available catalog, serving as a fertile ground for new discoveries in time-domain high-energy astrophysics. Among these candidates, we identify an intriguing extragalactic FXT, XRT 200515, which exhibits unique temporal and spectral characteristics distinct from any previously reported Chandra FXTs. The transient's initial hard < 10 s burst shows a sharp rise exceeding 4 orders of magnitude, followed by spectral softening in an ∼ 800 s oscillating tail. This transient is likely related to either a giant magnetar flare (GMF) from a distant soft gamma repeater (SGR) behind the Large Magellanic Cloud (LMC) or an extragalactic Type I X-ray burst from a faint LMXB in the LMC. Each of these interpretations presents its own set of challenges. Alternatively, XRT 200515 could be a new type of astronomical phenomenon found by our anomaly detection method using machine learning. Our method is the first representation learning approach for anomaly detection in high-energy astrophysics. It is applicable to datasets from high-energy catalogs like Chandra , XMM-Newton , Swift-XRT , eROSITA , and Einstein Probe . We created semantically meaningful representations that can be aligned with other data modalities, such as optical images or infrared spectra to design multi-modal models (Parker et al. 2024; Mishra-Sharma et al. 2024; Zhang et al. 2024; Rizhko & Bloom 2024) using contrastive learning (Radford et al. 2021), that can improve on current state-of-the-art algorithms used to characterize the physics of the associated objects. Ultimately, this work and other representation and contrastive learning approaches lay the groundwork for developing generalized foundation model in astronomy. The paper is organized as follows: In § 2, we provide information on the dataset of Chandra event files used in this analysis. In § 3, we describe in detail the implementation of our novel transient detection approach leveraging representation learning. In § 4, we present and discuss the results in form of the semantically meaningful representation space of the event files, the catalog of X-ray transient candidates and the discovery of the new Chandra transient XRT 200515. Finally, we highlight our contributions to time-domain high-energy astrophysics and outline potential directions for extending this work in the future in § 5. The relevant code, a demonstration of the pipeline, and an interactive embedding selection, transient search and lightcurve plotting tool are available online at the GitHub repository https://github.com/StevenDillmann/ml-xraytransients-mnras.","pages":[1,2]},{"title":"2 DATASET","content":"We use data from the Chandra Source Catalog (CSC) version 2.1 (Evans et al. 2024), which includes all publicly available X-ray sources detected by Chandra as of December 2021. For this study, we focus specifically on observations from the Advanced CCD Imaging Spectrometer (ACIS). CSC 2.1 had not been fully released at the time our analysis was performed, but catalog data was available for sources that had completed processing in the Current Database View 1 , a snapshot of which we took on 11 April 2023. CSC 2.1 performs source detection on stacked observations, and catalog properties are provided both for these stack-level detections, and for each of observation-level detection that contribute to a stack detection. Because we are interested in short-time variability that happens within a single observation of a source, we use the catalog products for the observation-level detections in our analysis. For a given X-ray detection, two types of products are provided in the CSC: (i) database tables with source properties, such as fluxes in the different X-ray energy bands, hardness ratios, variability indices, etc., and (ii) filebased data products for each detection of a source, such as the detect regions, the Chandra PSF at that location, etc. The following observation-level catalog properties are relevant for our analysis: index of 6 or larger indicates variability at a confidence of at least 2 𝜎 . In this paper we call this quantity 𝐼 𝑏 var . From the catalog data products available for observation-level Xray detections, we are interested in the region event file. This event file consists of a list of all individual photon events detected in a small bounding box around a source detection, listing their energies, arrival times, and detector coordinates. These event files are the basis for the characterization of an X-ray source: lightcurves, spectra, images, coordinates, and other properties are derived from the distribution of the listed quantities. In this analysis, we directly use these event files as our primary data products. The values of the catalog properties listed above serve as summary statistics for the detection associated with a given region event file. We only include event files with more than 5 events and a signal-to-noise ratio above 5 to minimize spurious signals from low number statistics in faint sources. We also exclude detections that are flagged for pile-up 2 , i.e., those with a pileup fraction larger than 5%, which corresponds to a maximum pileup warning of 0.1 in CSC 2.1. For the resulting detections, we filter the event files to include only events contained within the detection region for each source. These detection regions are also provided as data products in CSC 2.1, and consist of the ellipse that includes the 90% encircled counts fraction of the PSF at the source location. Due to the low background level in Chandra observations, the majority of events selected after this spatial filtering are expected to be events associated with the X-ray source, not the background. In the selected event files, we only include photon events within good time intervals (GTIs), which are time periods of valid, high-quality data. No other pre-processing is required. The final dataset consists of 95,473 filtered event files from 58,932 sources, resulting in an average of 1 . 62 observations per source. This includes 9,003 new sources that have been added as part of the CSC 2.1 release, in addition to the sources from the previous release.","pages":[2,3]},{"title":"3 METHODS","content":"In this work, we introduce a novel representation learning based anomaly detection method to systematically search for X-ray transients in high-energy archives. We begin with an overview of the method here and provide detailed explanations of each step in individual subsections. The full pipeline is illustrated in Figure 1. Starting with the event files described in § 2, we (i) build two novel and uniform event file representations by binning their arrival times and energies into 𝐸 -𝑡 Maps (Event File Representation I) or 𝐸 -𝑡 -𝑑𝑡 Cubes (Event File Representation II); (ii) use principal component analysis (Feature Extraction I) or sparse autoencoders (Feature Extraction II) to extract informative features from the event file representations; (iii) apply dimensionality reduction to the extracted features to create a low-dimensional embedding space; (iv) use density-based clustering to create embedding clusters that group event files with similar characteristics, for example transient behavior or certain spectral features. Previously identified transients like the extragalactic magnetar-powered flare candidate reported by Lin et al. (2022) and the extragalactic planet candidate dip reported by Di Stefano et al. (2021), shown in Figure 2, occupy well-isolated clusters in the embedding space. Exploring these clusters and conducting nearest-neighbor searches enables us to effectively find analogs to bona-fide time-domain anomalies, while at the same time grouping them according to their spectral properties. We compile the identified transient candidates in a catalog. While our approach is designed and tested using Chandra data, it is applicable to any dataset consisting of event lists, like those from other high-energy telescopes. The described transient detection approach is applied to both types of event file representations with both feature extraction methods, resulting in four different embeddings. We denote the different cases as described in Table 1.","pages":[3]},{"title":"3.1 Event File Representation","content":"The different event files in the dataset are variable in length 𝑁 and duration 𝑇 , as shown in Appendix A. The large variation in the number of events and duration highlights the challenge in producing uniform data representations that preserve relevant information on time variability and spectral properties. While there exist machine learning architectures that take variable length inputs, the significant differences in the number of events from object to object make standardization of the inputs challenging, even when these architectures are used (Martínez-Galarza & Makinen 2022). As a first step in our analysis, we introduce 2-dimensional and 3-dimensional fixed-length representations based on an informed binning strategy for the event files, similar to the DMDT maps for optical lightcurves introduced by Mahabal et al. (2017).","pages":[3]},{"title":"3.1.1 2D Histogram Representation ( 𝐸 -𝑡 Maps)","content":"Assume an event file with 𝑁 photons and a photon arrival time column 𝒕 with entries { 𝑡 𝑘 } 𝑁 𝑘 = 1 and energy column 𝑬 with entries { 𝐸 𝑘 } 𝑁 𝑘 = 1 . The event file duration is given by 𝑇 = 𝑡 𝑁 -𝑡 1 . The energy column entries take values in the broad energy band of Chandra 's ACIS instrument, i.e. 𝐸 𝑘 ∈ [ 𝐸 𝑚𝑖𝑛 , 𝐸 𝑚𝑎𝑥 ] , where 𝐸 𝑚𝑖𝑛 = 0 . 5 keV and 𝐸 𝑚𝑎𝑥 = 7 keV comes from considering appropriate boundaries for the energy response of Chandra 's ACIS instrument. Beyond these boundaries, the telescope's aperture effective area is low for the majority of detected sources. First, we obtain the normalized time column, given by 𝝉 = 𝒕 -𝑡 1 𝑇 , and the logarithm of the energy column, given by 𝝐 = log 𝑬 . The resulting boundaries for normalized time column are 𝝉 ∈ [ 𝜏 𝑚𝑖𝑛 , 𝜏 𝑚𝑎𝑥 ] , where 𝜏 𝑚𝑖𝑛 = 0 and 𝜏 𝑚𝑎𝑥 = 1. The range for the log-energy column is 𝝐 ∈ [ 𝜖 𝑚𝑖𝑛 , 𝜖 𝑚𝑎𝑥 ] , where 𝜖 𝑚𝑖𝑛 = log 0 . 5 keV and 𝜖 𝑚𝑎𝑥 = log 7 keV. Next, we determine the dimensionality of our representations. For a each event file, we determine the optimal number of bins in the energy dimension, 𝑛 𝜖 , with the Freedman-Diaconis rule (Freedman & Diaconis 1981), a widely used method that balances the trade-off between too noisy histograms (too many bins) and not informative enough histograms (too few bins). The optimal bin width 𝑏 𝜖 according to this rule is calculated in the following way: where 𝐼𝑄𝑅 ( 𝜖 ) represents the interquartile range of the 𝜖 values for a given event file of length 𝑁 . Subsequently, we obtain the optimal number of energy bins 𝑛 𝜖 with: For each event file, we determine the optimal number of bins in the time dimension, 𝑛 𝜏 , with the help of the Bayesian Blocks algorithm, which was specifically developed for time series analysis in astronomy (Scargle et al. 2013). This algorithm partitions the time series into adaptive width bins or blocks that are statistically distinct from neighboring blocks; that is, within a given time-ordered Bayesian block, events grouped in that block are consistent with having a similar event arrival rate. We use the default Astropy implementation of Bayesian blocks, and set the false alarm probability parameter to 𝑝 0 = 0 . 01 (Astropy Collaboration et al. 2013), which implies a 1% probability of declaring a change of rate when there is none. For each event file, we define the optimal uniform bin width 𝑏 𝜏 as the minimum bin width calculated by the Bayesian Blocks algorithm, and then find the optimal number of time bins 𝑛 𝜏 with: The optimal number of bins is different for each event file, due to their different lengths 𝑁 and durations 𝑇 . To select a bin size that can be applied to all event files, we consider the distributions of these optimal bin sizes, which are shown in Figure 3. For the distribution of 𝑛 𝜏 values we only use those event files for which 𝑝 𝑏 var > 0 . 9. The intent of this is to effectively capture variability timescales that are associated with short time-domain events, such as flares and dips. We choose the 90th percentile value of each distribution to set the final number of bins in each dimension. That is, only 10% of the event files will have an optimal number of bins that is larger than the chosen values 𝑛 𝜖 = 16 and 𝑛 𝜏 = 24. The choice of the 90th percentile, rather than the mean or mode, is motivated by the need to capture sufficient statistical detail even for long event files, while keeping the size of the resulting representations computationally tractable. Choosing a lower resolution would risk losing significant details in the representation, particularly short-duration events such as flares and dips within longer event files. The 𝐸 -𝑡 Maps are the 2D histogram representations with size ( 𝑛 𝜏 , 𝑛 𝜖 ) = ( 24 , 16 ) that result from binning the events according to the optimized number of bins. Figure 4 shows the 𝐸 -𝑡 Maps for the known extragalactic dip reported by Di Stefano et al. (2021) and known extragalactic flare reported by Lin et al. (2022).","pages":[3,5]},{"title":"3.1.2 3D Histogram Representation ( 𝐸 -𝑡 -𝑑𝑡 Cubes)","content":"Wenowintroduce the 𝐸 -𝑡 -𝑑𝑡 Cubes, which extend the 𝐸 -𝑡 Maps by a third dimension that serves as a proxy for the photon arrival rate. For an event file of length 𝑁 , consider the array of time differences between consecutive photon arrivals 𝚫 𝒕 with entries Δ 𝑡 𝑘 = 𝑡 𝑘 + 1 -𝑡 𝑘 for 𝑘 = 1 , 2 , . . . , 𝑁 -1. We again scale and normalize the obtained values, so that they adopt values between 0 and 1, using in each case the minimum value Δ 𝑡 𝑚𝑖𝑛 and maximum value Δ 𝑡 𝑚𝑎𝑥 . This provides the third dimension 𝜹𝝉 : The additional dimension is intended to better isolate short-duration features in time variability by capturing high photon arrival rates, which are typical of flares, as well as very low photon arrival rates, which are typical of dips. The boundaries of our histogram representations in this dimension are 𝜹𝝉 ∈ [ 𝛿𝜏 𝑚𝑖𝑛 , 𝛿𝜏 𝑚𝑎𝑥 ] , where 𝛿𝜏 𝑚𝑖𝑛 = 0 and 𝛿𝜏 𝑚𝑎𝑥 = 1. We determine the optimal number of bins in the 𝜹𝝉 dimension, 𝑛 𝛿𝜏 , again by computing the optimal bin width 𝑏 𝛿𝜏 with the Freedman-Diaconis rule and dividing the range for 𝜹𝝉 by 𝑏 𝛿𝜏 : The distribution of 𝑛 𝛿𝜏 across the event files is shown in Figure 3. Most of the relevant time-domain information is already captured by 𝝉 , but adding 𝜹𝝉 provides an additional marker for dips and flares that can be shorter than the timescales probed by our chosen binning of 𝝉 . Unlike in the other two dimensions, we choose the 75th percentile value of the distribution as our final choice of common binning, which results in 𝑛 𝛿𝜏 = 16. This is because in order to identify short transients, we need to capture strong deviations in 𝜹𝝉 only. Choosing a lower value for 𝑛 𝛿𝜏 reduces noise an improves computational tractability. Having both 𝝉 and 𝜹𝝉 represented also breaks any assumption of stationarity, in that we can be sensitive to transient events happening at any time during the observation of the source, and break degeneracies between periodic and non-periodic features in the representations presented by Martínez-Galarza & Makinen (2022). The 𝐸 -𝑡 -𝑑𝑡 Cubes are the resulting 3D histogram event file representations with size ( 𝑛 𝜏 , 𝑛 𝜖 , 𝑛 𝛿𝜏 ) = ( 24 , 16 , 16 ) .","pages":[5,6]},{"title":"3.1.3 Feature Notation","content":"The event file representations can now be used as inputs for various statistical learning and machine learning algorithms. For the 𝑖 𝑡 ℎ event file in the dataset of length 𝑚 = 95,473, we denote the corresponding feature vector as fi 𝑥 𝑖 = [ 𝑥 1 , 𝑥 2 , . . . , 𝑥 𝑛 ] 𝑖 , where 𝑛 = 𝑛 𝜏 · 𝑛 𝜖 = 384 for the 𝐸 -𝑡 Maps and 𝑛 = 𝑛 𝜏 · 𝑛 𝜖 · 𝑛 𝛿𝜏 = 6 , 144 for the 𝐸 -𝑡 -𝑑𝑡 Cubes. The set of all feature vectors is denoted as X = [fi 𝑥 1 , fi 𝑥 2 , . . . , fi 𝑥 𝑚 ] ⊤ with size ( 𝑚, 𝑛 ) .","pages":[6]},{"title":"3.2 Feature Extraction I: Principal Component Analysis","content":"We use Principal Component Analysis (PCA) (Pearson 1901) provided by scikit-learn (Pedregosa et al. 2011) as our first feature extraction method. The extracted principal components should encode relevant time-domain and spectral information of the event file they represent. PCA involves transforming a dataset into a new coordinate system by finding the principal components of the data that capture most of the variance in the data. By projecting the dataset onto principal components, PCA reduces the dimensionality of the data while retaining the most important information, which increases the interpretability of high-dimensional data.","pages":[6]},{"title":"3.2.1 PCA Algorithm","content":"We start with the feature vector set X of size ( 𝑚, 𝑛 ) representing our dataset with 𝑚 samples and 𝑛 dimensions. PCA aims to find a new coordinate system defined by a set of orthogonal axes, i.e. the principal components, that captures the maximum amount of variance in the data. The PCA result is a transformed dataset Xpc obtained by projecting X onto the principal components: where W is matrix of size ( 𝑛, 𝑛 𝑝𝑐 ) containing the first 𝑛 𝑝𝑐 principal components to be retained as its columns and Xpc is of size ( 𝑚 , 𝑛 𝑝𝑐 ) with a reduced dimensionality of 𝑛 𝑝𝑐 . For a more detailed explanation of the algorithm, we refer the reader to Jolliffe (2002).","pages":[6]},{"title":"3.2.2 Principal Components Retained","content":"The main PCA hyperparameter is the number of principal components 𝑛 𝑝𝑐 to retain. Figure 5 shows two scree plots illustrating the amount of variance explained by each principal component in descending order and the cumulative proportion of variance explained by the principal components for both 𝐸 -𝑡 Mapsand 𝐸 -𝑡 -𝑑𝑡 Cubes. Acommon approach to determine the optimal value of 𝑛 𝑝𝑐 is to find the knee point in the cumulative scree plot of the principal components. This balances the objective of minimizing the dimensionality while retaining as much information as possible. Defining the knee point as the point beyond which adding additional principal components increases the amount of variance by less than 0 . 1% gives 𝑛 𝑝𝑐 = 15 for 𝐸 -𝑡 Maps and 𝑛 𝑝𝑐 = 22 for 𝐸 -𝑡 -𝑑𝑡 Cubes as indicated in Figure 5. These capture 94 . 1% and 89 . 9% of the variance respectively.","pages":[6]},{"title":"3.3 Feature Extraction II: Sparse Autoencoder Neural Network","content":"As an alternative to PCA, we now build Autoencoder (Hinton & Salakhutdinov 2006) models with TensorFlow (Abadi et al. 2015) to learn a set of latent features from the 𝐸 -𝑡 Maps and 𝐸 -𝑡 -𝑑𝑡 Cubes that can be used to isolate transients and encode specific spectral properties. An autoencoder is composed of two neural networks, an encoder and a decoder, which work together to learn a compressed representation of the input data. The encoder network takes the input data and maps it to a lower-dimensional representation, often called 'latent space' or 'bottleneck'. The number of neurons in the bottleneck determines the dimensionality of the learned representation. The decoder network then aims to reconstruct the original input from this compressed representation. The decoder is typically a mirrored version of the encoder gradually upsampling the latent space until the output matches the dimensions of the original input. By minimizing the reconstruction error between input and output during training, the model learns a low-dimensional representation of the input. The bottleneck forces the encoder to capture the most important information necessary for accurate reconstruction, effectively compressing the input and learning to extract informative features in an unsupervised manner. Once the autoencoder is trained, the encoder network can be used as a standalone feature extractor to obtain a compressed representation of the input data, which can be used for downstream tasks such as clustering or anomaly detection. As opposed to PCA, which is a linear technique that works well for linearly correlated data but fails to capture complex non-linear relationships, an autoencoder is able to learn complex non-linear relationships. We design two different autoencoders to process the 𝐸 -𝑡 Maps and 𝐸 -𝑡 -𝑑𝑡 Cubes.","pages":[7]},{"title":"3.3.1 Convolutional Autoencoder","content":"In a convolutional autoencoder (Masci et al. 2011), both the encoder and decoder network consist of convolutional layers (LeCun et al. 1998), which perform convolutions over the input using a filter. These filters are small matrix kernels with learnable weights that slide across the input, allowing the network to capture high-level features while preserving important spatial hierarchies and relationships, which is why they are often used for image-like data. This makes this architecture particularly well-suited to recognize spatial patterns such as dips or flares in our 𝐸 -𝑡 Maps. To gradually reduce the dimension of the input while it is being passed through the encoder network, we use stride convolution layers (Simonyan & Zisserman 2014) with a stride value of 2 for downsampling. This means that the learnable filter jumps two pixels at a time as it slides over the input. The output of the convolutional layers is a feature map, which is then flattened to a feature vector and passed through a series of fully connected layers, where every neuron in the previous layer is connected to every neuron in the next layer. These fully connected layers are responsible for mapping the learned features to a lower-dimensional latent representation in the bottleneck and perform non-linear transformations while downsampling through the use of non-linear activation functions. The final latent space has 𝑛 𝑎𝑒 = 12 elements, representing the most essential features of the input data, which can now be used for further downstream tasks. Figure 6 shows a diagram of the encoder part of the model and Table 2 summarizes its architecture.","pages":[7]},{"title":"3.3.2 Fully Connected Autoencoder","content":"Our 𝐸 -𝑡 -𝑑𝑡 Cubes introduce an additional dimension resulting in sparse 3D input data. Convolutional layers assume regular grid-like data, making them less effective for handling sparse data. Moreover, very expensive 3D convolutional operations would substantially increase complexity of the model. Therefore, we use a simple fully connected autoencoder for the 𝐸 -𝑡 -𝑑𝑡 Cubes. Its encoder network consists of a series of fully connected layers, which gradually map the original input data to a latent space with 𝑛 𝑎𝑒 = 24 elements. Figure 6 shows a diagram of the encoder part of the model and Table 3 summarizes its architecture.","pages":[7]},{"title":"3.3.3 Activation Functions","content":"Neural networks are able to learn and represent complex non-linear relationships due to the introduction of non-linear activation func- tions within their layers. An activation function is a mathematical function used in a neural network to determine whether a neuron should be activated or not, based on its input. It essentially decides how much of the input signal should pass through the neuron, producing an output that can either be passed to the next layer or used to make predictions. The popular Rectified Linear Unit (ReLU) activation function 𝑅𝑒𝐿𝑈 ( 𝑥 ) = max ( 0 , 𝑥 ) (Nair & Hinton 2010) is simple and computationally efficient. To mitigate any potential encounters of the 'dying the ReLU problem', where neurons become non-responsive during training, we choose an extended version called Leaky ReLU (Maas et al. 2013): where 𝛼 = 0 . 1 is a hyperparameter that defines the slope of the function for negative input values. ReLU sets all negative values in the input to zero, while Leaky ReLU allows a small negative slope for negative inputs, which can help prevent neurons from dying. As for the output layer, we want any values to be mapped to a range between 0 and 1, which is achieved by using the sigmoid activation function:","pages":[7,8]},{"title":"3.3.4 Loss Function and Sparsity Regularization","content":"In order to encourage the autoencoder to generate reconstructions close to the original inputs, we use the mean squared error ( 𝑀𝑆𝐸 ) as as a measure of the reconstruction quality given by: where 𝑥 𝑖 is the 𝑖 𝑡 ℎ element of the input vector and ˆ 𝑥 𝑖 is the corresponding is reconstructed output. The 𝑀𝑆𝐸 is a straightforward measure of reconstruction error, and its differentiability allows efficient gradient computation for updating model weights via gradient-based optimization. Our neural networks are so called sparse autoencoders (Ng et al. 2011), which promote sparsity in the learned representation, meaning only a small subset of the neurons in the network are active at any given time. Sparse representations are valuable for our work because they help extract highly informative features from the input, while disregarding irrelevant or noisy information. To encourage sparsity in the latent space, we introduce a L1 regularization term in the objective, resulting in the following loss function: where 𝜆 = 0 . 1 is the regularization strength and 𝑤 𝑗 are the individual bottleneck weight values of which there are 𝑛 𝑤 in total. L1 regularization pushes small weights to zero and thus helps the model prioritize the most significant features of the input data, leading to a semantically meaningful latent space.","pages":[8]},{"title":"3.3.5 Training","content":"Starting with the original dataset with a 𝑚 = 95,473 samples and using a test split of 0 . 1 gives us a training and validation set of length 85,925 and a test set of length 9,548. Further using a validation split of 0 . 2, gives 68,740 samples for training and 17,185 for validation. We run the training process for a maximum of 200 epochs with a batch size of 1,024 samples. The initial learning rate was set to 0 . 01 along with an on plateau learning rate scheduler, which dynamically reduces the learning rate by a factor of 0 . 1 if the validation loss plateaus for longer than 10 epochs. Reducing the learning rate when a plateau is detected can help escape local minima in the loss surface and converge to a more optimal solution in the parameter space. This scheduler is used in combination with the Adaptive Moment Estimation (Adam) optimizer (Kingma & Ba 2014), which is a stochastic gradient descent algorithm combining the benefits of both adaptive learning rates (Duchi et al. 2011) and momentum-based optimization techniques (Sutskever et al. 2013). Finally, we use an early stopping callback to monitor the validation loss. It automatically interrupts the training process if the validation loss does not improve for 25 epochs and restores the weights of the model to the best observed weights during training. The training process for both autoencoder models is shown in Appendix B. Once the autoencoder is trained, we can use the encoder to transform the original dataset X to the feature vector space Xae of size ( 𝑚 , 𝑛 𝑎𝑒 ) with a reduced dimensionality of 𝑛 𝑎𝑒 features.","pages":[8]},{"title":"3.4 Dimensionality Reduction","content":"Using t-SNE (Maaten & Hinton 2008), short for t-Distributed Stochastic Neighbor Embedding, we create two-dimensional embeddings of the informative features previously extracted from the event file representations using PCA or sparse autoencoders. The t-SNE algorithm is a method used to map the input data onto a low-dimensional embedding space, and is particularly useful for the visualization of clusters and patterns in high-dimensional datasets. Each high-dimensional sample is transformed into a low-dimensional embedding in such a way that similar object are nearby points, while dissimilar objects are distant points in the embedding space. Essentially, it aims to capture the local structure of the data by preserving the pairwise similarities between objects while mapping them to a lower-dimensional embedding space.","pages":[8]},{"title":"3.4.1 Algorithm","content":"We use our informative features, X if = Xpc or X if = Xae , as input to the t-SNE algorithm to reduce the data to a two-dimensional embedding, denoted as Z . First, t-SNE creates a probability distribution 𝑃 for pairs of high-dimensional data points in X if , assigning higher probabilities to similar pairs and lower probabilities to dissimilar ones. This is done by modeling pairwise similarities using a Gaussian kernel with a specific perplexity parameter, which controls the effective number of neighbors considered for each point. Next, t-SNE defines a similar probability distribution 𝑄 for the pairwise similarities in the low-dimensional space Z , modeled using a Student's t-distribution. The goal of t-SNE is to minimize the difference between 𝑃 and 𝑄 using gradient descent, with the Kullback-Leibler (KL) divergence (Kullback & Leibler 1951) as the cost function: where 𝑃 𝑖 𝑗 and 𝑄 𝑖 𝑗 represent pairwise similarities in the high- and low-dimensional spaces, respectively. The algorithm iteratively adjusts the low-dimensional embedding Z to minimize the KL divergence, often requiring hundreds to thousands of iterations for convergence. The result of this optimization is a two-dimensional representation Z of size ( 𝑚, 2 ) , where similar points in the high-dimensional space are clustered closely together.","pages":[8,9]},{"title":"3.4.2 Hyperparameter Optimization","content":"The t-SNE algorithm has a number of important hyperparameters to be tuned. The two most important parameters are the perplexity and the learning_rate . The perplexity parameter controls the balance between capturing the local versus global structure in the data, while the learning_rate controls the step size at each iteration of the optimization process. The n_iter parameter is the number of iterations. To ensure reproducibility, we set a fixed random_state . Our t-SNE hyperparameter optimization approach is detailed in Appendix C. A summary of the final t-SNE hyperparameters is provided in Table 4.","pages":[9]},{"title":"3.5 Clustering","content":"The next step is the identification of individual clusters in the embedding space using DBSCAN (Hartigan & Wong 1979), short for Density-Based Spatial Clustering of Applications with Noise. Unlike traditional clustering algorithms such as k-means, DBSCAN does not require the number of clusters to be specified, as it identifies dense regions in the data space based on a density criterion.","pages":[9]},{"title":"3.5.1 Algorithm","content":"We use our t-SNE embedding space Z as input to the DBSCAN algorithm, which segments the embedding space into multiple clusters. The DBSCAN algorithm has two main hyperparameters. The eps parameter defines the radius of the neighborhood surrounding each point in the dataset, while the minPts parameter specifies the minimum number of points required within this neighborhood for a data point to be classified as a core point. A border point is defined as a point that is in the vicinity of at least one core point but has fewer than minPts within its neighborhood. All other points are considered to be noise points. Clusters are then created from the aggregation of core points and their associated border points, with noise points being categorized as outliers. Figure 7 visualizes the clustering method.","pages":[9]},{"title":"3.5.2 Hyperparameter Optimization","content":"Our DBSCAN hyperparameter optimization approach is detailed in Appendix C. A summary of the final t-SNE hyperparameters is provided in Table 5.","pages":[9]},{"title":"3.6 Previously Reported Transients","content":"We highlight the embeddings of previously reported bona-fide transients, listed in Table 6, in our low-dimensional representation space to identify transient-dominant clusters. The flares include extragalactic FXTs reported by Jonker et al. (2013), Glennie et al. (2015), Yang et al. (2019), Lin et al. (2021), Lin et al. (2022), Quirola-Vásquez et al. (2022) and a set of stellar flares found in the dataset by manual inspection. The dips include the extragalactic planet candidate in M51 reported by Di Stefano et al. (2021), the ultraluminous X-ray source (ULX) 2E 1402 . 4+5440 in NGC 5457 (Colbert & Ptak 2002; Swartz et al. 2004) and the well-studied eclipsing and bursting lowmass X-ray binary (LMXB) EXO 0748 -676 (Parmar et al. 1986; D'Aì et al. 2014). These transients occupy well-isolated clusters. Exploring transient-dominant clusters and performing nearest-neighbor searches around known transients allows us to find new transients.","pages":[9]},{"title":"3.7 Candidate Selection","content":"Newtransients are identified in embedding clusters containing previously reported transients. For well-isolated clusters containing known discovered transients, we use the entire cluster to define new transient candidates. The well-isolated transient-dominant clusters used for candidate selection are listed in Appendix E. However, in a few cases known discovered transients reside within larger poorly separated clusters. Selecting the entire cluster would result in a high number of false positives. To address this, we instead use the k-nearest neighbors ( kNN ) algorithm (Cover & Hart 1967), identifying the 50 nearest neighbors for each known transient residing in a poorly separated cluster to define additional transient candidates.","pages":[9,10]},{"title":"3.8 Cross Matching","content":"We use an existing cross-match table (Green et al. 2023) between CSC2.1 and five other catalogs - Gaia DR3 (Gaia Collaboration et al. 2021), DESI Legacy Survey DR10 (Dey et al. 2019), PanSTARRS-1 (Chamberset al. 2016), 2MASS (Skrutskie et al. 2006), and the SDSS DR17 catalog - to complement the X-ray properties derived from the CSC with additional multi-wavelength observations. This includes catalog identifiers, positions, magnitudes, source type classifications and other columns. We cross-matched our transient candidates with the SIMBAD database (Wenger et al. 2000) by associating each candidate with the nearest SIMBAD object, provided the object is located within a 5 arcsec radius of the candidate's coordinates listed in the CSC. The multi-wavelength observations of the transient candidates provide valuable information for their characterization and classification.","pages":[10]},{"title":"4 RESULTS AND DISCUSSION","content":"We now present the results of applying the methods in § 3 to the set of representations of X-ray event files in the dataset from § 2.","pages":[10]},{"title":"4.1 Representation Embedding Space and Clusters","content":"Figure 8 shows the t-SNE embedding space for the 3D-PCA and 3DAE cases color-coded by the hardness ratio 𝐻𝑅 hs . The embedding space for the other two cases, 2D-PCA and 2D-AE , are shown in Appendix D. The observed hardness ratio gradients in all embedding spaces indicate that the learned representations effectively encode spectral information, in particular at the level of individual clusters, allowing for the identification of X-ray sources with specific spectral signatures. For the 2D-PCA and 2D-AE cases, these gradients are more uniform across the embedding space, because the temporal and spectral information of event files are captured by one axis each in the 𝐸 -𝑡 Maps. Moreover, some clusters consist exclusively of soft or hard sources, demonstrating that our representations can be leveraged not only to identify transients but also to find analogs to sources with specific spectral characteristics. Figure 9 shows the 3D-PCA and 3D-AE embedding spaces, now color-coded by the variability index 𝐼 𝑏 index with the other two cases shown in Appendix D. The learned embeddings also encode the temporal behavior of the sources, with some clusters being dominated by X-ray detections with significant variability, including transient behavior. To demonstrate this, we also highlight the embeddings of the bona-fide flares and dips listed in Table 6. Note that these occupy very well-defined clusters on the edges of the representation space, allowing for queries of analog transient behavior. In the 2D-PCA and 2D-AE cases, transient sources are distributed across multiple small clusters on the edges of the embedding spaces. In contrast, the 3D-PCA and 3D-AE embedding spaces achieve a significantly more compact clustering of bona-fide transients because temporal features in the event files are given a higher importance by the introduction of an additional time-related axis in the 𝐸 -𝑡 -𝑑𝑡 Cubes. Figure 10 shows the clusters identified by the DBSCAN algorithm in the 3D-PCA and 3D-AE cases. The clusters for the other two cases, 2D-PCA and 2D-AE , are shown in Appendix D. The largest cluster in all cases (Cluster 1) corresponds to observations that are not 'anomalous', for example non-variable sources or noisy detections in the low-count regime. We also see multiple smaller clusters on the edges of the embedding space clearly separated from this main cluster. Of special interest are clusters that contain known discovered transients, as these likely host other interesting transients that have not yet been discovered. Some of the edge clusters group observations with similar temporal and spectral behavior. For example, Cluster 4 in the 3D-PCA case only contains flares with high hardness ratios. Other clusters instead group observations primarily by similar temporal behavior, but then show a within-cluster grouping of similar spectral behaviors. For example, Cluster 4 in the 3D-AE case contains many dipping sources, but show a hardness ratio gradient within the cluster. When comparing the results of different feature extraction methods, we observe that in the 3D-AE embedding space, nearly all previously identified extragalactic FXTs live within a single, wellisolated cluster (Cluster 8). In contrast, the 3D-PCA embedding space distributes these extragalactic FXTs across multiple clusters. All of these points underline the effectiveness of our method and that the created representation space is highly informative.","pages":[10,13]},{"title":"4.2 Catalog of X-ray Flare and Dip Candidates","content":"We identify new transient candidates within clusters that are occupied by previously reported transients and by conducting nearestneighbor searches around these known transients. We compile these in a catalog of X-ray transient candidates, which includes both flares and dips. Table E1 lists the selected clusters used to define the new flare and dip candidates in addition to the 50 nearest neighbors of each bona-fide transient. From each selected cluster, we only include X-ray detections for which the variability index 𝐼 𝑏 var ≥ 5. This threshold corresponds to detections for which the Gregory-Loredo algorithm yields a confidence in the variability of at least 90%, and allows us to discard flare and dip-like behaviors that are not statistically significant. We also manually exclude a small fraction of false positives identified by visual inspection of the lightcurves for both flare and dip candidates. The resulting catalog contains a total of 3539 detections, and the columns included are described in Appendix F. Table 7 shows the first 5 samples in our catalog for a subset of selected columns. Figure 11 shows a number of example lightcurves of the dips and flares in our catalog. The dip selection shows dips from LMXBs, a low-mass X-ray binary (HMXB), an ULX, an eclipsing binary, a cataclysmic binary, and a quasar. The flare selection shows flares from an eruptive variable, a pulsar, an AGN, a HMXB, a cataclysmic variables and young stars. We also find a number of pulsating or quasi-periodic lightcurves from pulsars, magnetic cataclysmic variables and SGRs. Figure 12 shows the distribution of SIMBAD object types in our transient catalog. About 25% of the transient candidates do not have a SIMBAD match, making them particularly interesting sources for new transient discoveries. Our dip candidates include 6 Chandra observations with prominent dips from the known source CXOGlb J002400.9 -720453 in the globular cluster NGC 104 (47 Tuc). The catalog identifiers for these are CATALOG_ID: 2737_139, 16527_79, 15747_79, 16529_79, 15748_79, 16528_14 . Our flare candidates include a newly discovered extragalactic FXT, which is characterized and discussed in detail in § 4.3. Its catalog identifier is CATALOG_ID: 23022_122 . We recommend using our catalog to identify a diverse range of flares and dips. While this work is primarily motivated by the discovery of new extragalactic transients, we intentionally did not exclude galactic stellar flares to enable systematic follow-up studies to study flare incidence rates and the rotational evolution of stars. Users interested exclusively in extragalactic transients can filter out galactic sources using metadata from the CSC and the cross-match columns in the catalog.","pages":[13]},{"title":"4.3 XRT 200515: A New Extragalactic Fast X-ray Transient","content":"Among the flare candidates in our catalog, we discovered an intriguing new extragalactic Chandra FXT in an observation of the supernova remnant SNR 0509 -67.5 in the LMC on May 15, 2020 (Guest et al. 2022). What made this transient stand out from thousands of other flares discovered in this work is the unique temporal variability in its lightcurve, which exhibits no detectable pre-flare X-ray emission, a sharp rise of at least 4 orders of magnitude in the count rate to peak intensity followed by a sharp fall, all in a matter of a < 10 s, down to ∼ 800 s long oscillating tail. There is also notable spectral variability during the flare, characterized by an initially hard spectrum at the peak, followed by spectral softening in the tail. The combination of these temporal and spectral properties establishes this transient as the first of its kind within the sample of discovered Chandra FXTs. We designate this newly discovered FXT as XRT 200515 and present a detailed study and discussion of its potential origins.","pages":[14]},{"title":"4.3.1 X-Ray Detection by Chandra","content":"The transient XRT 200515 was detected in Chandra ObsID 23022. The target of the observation was the supernova remnant SNR 0509 -67.5 in the LMC, which is shown in Figure 13 alongside the newly discovered FXT event. Table 8 summarizes the properties of XRT 200515 and its associated Chandra source 2CXO J051117.2 -672556 in ObdID 23022. The transient was captured by the ACIS camera in the S4 chip, and is located significantly off-axis in this observation, at an angular distance of 11.75 arcmin from the aimpoint in the S3 chip. This leads to an elongated and relatively large PSF, which, in this case, is advantageous as it substantially reduces photon pile-up in the initial spike, by spreading the counts over many pixels. We processed the data of Chandra observation ObsID 23022 with the Chandra Interactive Analysis of Observations (/c.pc/i.pc/a.pc/o.pc) version 4.15 (Fruscione et al. 2006), with calibration data base version 4.9.8. In particular, we created a new level-2 event file with the /c.pc/i.pc/a.pc/o.pc task chandra_repro and filter it in energy and time with dmcopy . We obtained the sky position in Table 8 using the /c.pc/i.pc/a.pc/o.pc tool wavdetect . To reduce background noise and improve the determination of the source centroid, we applied wavdetect on an image filtered to include only the time interval from the beginning of the flare ( 𝑡 0 ) until a time 𝑡 0 + 920 s. The 90% uncertainty radius of 2.0 arcsec is the combination of the uncertainty in the source centroid position reported by wavdetect , and the absolute astrometry uncertainty in a typical ACIS observation for off-axis sources 3 . The field was previously covered by four other Chandra observations (ObsIDs 776, 7635, 8554, and 23023) with no source detections at the location of 2CXO J051117.2 -672556. We estimated modelindependent upper limits to the source flux and luminosity with /c.pc/i.pc/a.pc/o.pc tool srcflux . In the pre-flare part of ObsID 23022, we obtained a 90% confidence limit of 𝐿 X < 1 . 0 × 10 34 erg / s in the 0.3-7 keV band at the LMC distance of 50 kpc. Stacking the data from all the ObsIDs with non-detections, including the pre-flare part of ObsID 23022, results in a total observed exposure of approximately ∼ 150 ks, and yields a 90% confidence upper limit on the X-ray luminosity is 𝐿 X < 3 × 10 33 erg / s.","pages":[14]},{"title":"4.3.2 X-ray Temporal Analysis","content":"We used the /c.pc/i.pc/a.pc/o.pc tool dmextract to extract background-subtracted lightcurves in several energy bands, from the reprocessed event file of Chandra ObsID 23022. We defined an elliptical source extraction region, with semi-minor and semi-major axes of 15 arcsec and 20 arcsec (matching the point-source PSF at the source location); the local background region was chosen in the same ACIS chip, with an area approximately eight times larger. Figure 14 shows the 0.3-7 keV background-subtracted lightcurve of XRT 200515 with a time resolution of 20 s. The lightcurve is consistent with no source detection at the location of the transient, before the start of the flare at around 23.5 ks into the observation. The few pre-flare counts are consistent with background noise. The lightcurve exhibits a strong initial spike with a sharp rise of at least 4 orders of magnitude in < 10 s, containing 44 out of all ∼ 180 flare counts. This initial burst is followed by a sudden drop to a ∼ 800 s long pulsating and decaying tail. We estimate a 𝑇 90 ∼ 580-740 s for the photons observed in the 0.3-7 keV band 4 , depending on the definition of total flare counts. Figure 15 shows the lightcurve of XRT 200515 at a resolution matching the ACIS frame time of 3.2 s, the hardness ratio, and the energy evolution for the time interval 𝑡 0 + 920 s. The lightcurve exhibits a spike in the count rate across only 3 bins (with a total of 4, 31 and 9 counts, respectively), hence the burst duration of < 10 s. The rise and fall times of the burst are both between 3.2 s and 6.4 s. The maximum count rate at the Chandra frame time resolution is ∼ 9.7 counts/s, acting as the lower bound for the peak count rate of the burst. Those counts are spatially spread over a PSF area of ∼ 3000 pixels; therefore, pile-up is not an issue. We evaluated the hardness ratio evolution during the flare with the Bayesian estimation method BEHR (Park et al. 2006). Here, the hardness ratio is defined as: where 𝑠 is the number of soft photons (0.3-1.2 keV), 𝑚 is the number of medium photons (1.2-2 keV), and ℎ is the number of hard photons (2-7 keV) in each bin. We also track the running average of the photon energies during the flare with a moving window of ± 10 counts. The hardness ratio and energy evolution indicate spectral softening during the flare, with the hardness ratio starting at 1 during the hard burst peak and decreasing to a range of 0.4 to 0.6 in the tail, highlighting the notable spectral variability of XRT 200515.","pages":[14,15]},{"title":"4.3.3 X-ray Spectral Analysis","content":"We used the /c.pc/i.pc/a.pc/o.pc tool specextract to extract the spectrum and the associated response and ancillary response files from the reprocessed event file of Chandra ObsID 23022. We used the same source and background extraction regions defined for the lightcurve extraction. To improve the signal-to-noise ratio of the source, we extracted the spectrum only from the time interval 𝑡 0 + 920 s. We binned the spectrum to a minimum of 1 count per bin with the grppha task within the /f.pc/t.pc/o.pc/o.pc/l.pc/s.pc package suite (Blackburn 1995) from NASA's High Energy Astrophysics Science Archive Research Center (HEASARC) 5 . For all spectral modelling and flux estimates, we used the /x.pc/s.pc/p.pc/e.pc/c.pc software version 12.13.0 (Arnaud 1996). With only 179 net counts, we are unable to fit complex spectral models; thus, we limit our analysis to the simplest one-component models representative of opposite scenarios: a power law ( powerlaw ) and a blackbody model ( bbody ), both modified by photo-electric absorption ( tbabs ). In both cases, we adopted the Tuebingen-Boulder absorption model with Wilms abundances (Wilms et al. 2000). We minimized the Cash statistic (Cash 1979), as we do not have enough counts for 𝜒 2 fitting. The best-fitting power-law model (Table 9 and Figure 16) has a photon index of Γ = 0 . 5 ± 0 . 3. The fit statistics yield a null hypothesis probability of 3 . 5 × 10 -3 , with a Cstat value of 132.7 for 137 degrees of freedom. For the blackbody model, the best-fitting temperature is 𝑘𝑇 bb = 1 . 8 ± 0 . 3 keV (Table 9). The fit statistics yield a null hypothesis probability of 1 . 2 × 10 -2 , with a Cstat value of 129.6 for 137 degrees of freedom. The reason this blackbody spectrum may appear hard in the Chandra band, resembling a Γ ∼ 0 . 5 power law, is that at a temperature of 𝑘𝑇 bb ∼ 2 keV, the ACIS detector samples only the peak and the Rayleigh-Jeans (rising) portion of the blackbody emission. We can use either model to determine an average conversion between the count rate and luminosity. This will then enable us to estimate the peak luminosity in the initial spike, for which we have previously estimated a peak count rate of ≳ 10 counts/s. The best-fitting power law model implies a peak flux of 𝐹 p ≳ 5 . 6 × 10 -10 erg/s/cm 2 , a total flare fluence of 𝐸 f ≳ 1 . 1 × 10 -8 erg/cm 2 , and a peak unabsorbed 0.3-10 keV luminosity of 𝐿 X ≳ 1 . 7 × 10 38 erg/s at the LMC distance of 50 kpc. For the best-fitting blackbody model, the peak flux and flare fluence would be 𝐹 p ≳ 4 . 0 × 10 -10 erg/s/cm 2 and 𝐸 f ≳ 0 . 8 × 10 -8 erg/cm 2 respectively. The peak unabsorbed 0.3-10 keV luminosity would be 𝐿 X ≳ 1 . 2 × 10 38 erg/s and the peak bolometric luminosity would be 𝐿 bol ≳ 1 . 5 × 10 38 erg/s. These values should be considered conservative lower limits for two reasons: (i) the peak count rate provides only a lower bound estimate, as it is constrained by the Chandra frame time resolution of the observations, potentially underestimating the true peak count rate; and (ii) the conversion factor applied is derived from the average spectrum over the entire flare, even though the spectrum of the initial spike is significantly harder compared to the tail, as shown in Figure 15. We searched for potential detections of XRT 200515 by other highenergy facilities. However, no significant X-ray or 𝛾 -ray events in the field around the X-ray source coordinates and flare start time 𝑡 0 reported in Table 8 were detected by the Fermi Gamma-ray Space Telescope ( Fermi ), the Burst Alert Telescope ( BAT ) on the Neil Gehrels Swift Observatory ( Swift ), the International Gamma-Ray Astrophysics Laboratory ( INTEGRAL ), or the Monitor of All-sky X-ray Image MAXI . LIGO was not operational during the time of the FXT, hence no gravitational wave signal could have been detected if the origin of XRT 200515 was a compact object merger.","pages":[15,16]},{"title":"4.3.5 Optical Counterpart Search","content":"We used the X-ray source coordinates reported in Table 8 to search for optical and infrared counterparts to XRT 200515. The field of XRT 200515 was covered by the Survey of Magellanic Stellar History ( SMASH ) (Nidever et al. 2017), a deep optical survey in the ugriz bands with the Dark Energy Camera (DECam) mounted on the Víctor M. Blanco Telescope at the Cerro Tololo Inter-American Observatory (CTIO) in Chile. We used the Astro Data Lab Jupyter Notebook server (Nikutta et al. 2020; Juneau et al. 2021) to access and visualize the SMASH catalog 6 . Figure 17 shows a color image of the field created from the deepest available stacked images in the u , g and i bands; the 5 𝜎 detection limits in these bands are 23.9 mag, 24.8 mag and 24.2 mag, respectively. The images were taken on December 7, 2015 with exposure times of 1,179 s, 981 s and 1,179 s respectively. The astrometry of the SMASH images is calibrated on the Gaia DR3 reference frame, thus their positional uncertainty is negligible compared to the X-ray source position uncertainty. Within the Chandra position error circle in Figure 17, there is no obvious optical counterpart that stands out in brightness or color from the surrounding stellar population. We performed relative photometry on the sources inside the error circle, comparing them to several nearby sources with known positions and brightnesses listed in the Gaia DR3 catalog. We used the SMASH g band as the closest approximation to Gaia's G band. We estimate the brightest optical source within the error circle to have a Vega magnitude of 𝑔 = 22 . 7 ± 0 . 1 mag, corresponding to an absolute magnitude of 𝑀 𝑔 ≈ 4 . 2, assuming it is in the LMC. Additionally, three other point-like sources are detected with 𝑔 band magnitudes in the range of 23-24 mag. All four sources appear pointlike, consistent with the seeing conditions of the SMASH survey, with no evidence of any spatially extended background galaxies. The three brightest stars visible in Figure 17 within ∼ 12 arcsec of the Chandra source are solar-mass stars on the red giant branch, indicative of an old stellar population. The lack of bright optical counterparts and the short burst duration of < 10 s rules out a stellar flare from a foreground Galactic low-mass star (Güdel 2004; Reale 2007; Reale & Landi 2012; Pye et al. 2015; Kuznetsov & Kolotkov 2021). A flare from a Be/X-ray binary or any other HMXB in the LMC is also excluded by the lack of a bright optical counterpart (Ducci et al. 2019, 2022). The temporal and spectral properties of XRT 200515, combined with the absence of an optical counterpart, suggests three possibilities: (i) a relativistic jet phenomenon, such as a 𝛾 -ray burst (GRB); (ii) a rapid, high-energy process linked to extreme magnetic fields, such as a giant magnetar flare (GMF); or (iii) a thermonuclear Type I X-ray burst caused by surface nuclear burning on a neutron star.","pages":[16]},{"title":"4.3.6 Gamma Ray Burst from a Compact Object Merger?","content":"Evidence in favor or against the association of at least some Chandra FXTs with low-luminosity long-GRBs or off-axis short-GRBs (see Berger 2014 for a review), at moderate or high redshifts, is extensively discussed in Quirola-Vásquez et al. (2022), Quirola-Vásquez et al. (2023), and Wichern et al. (2024). A detailed re-investigation of this issue is beyond the scope of this work. Here, we simply point out that XRT 200515, like the other Chandra FXTs in the literature, does not have any 𝛾 -ray detection. On the other hand, XRT 200515 has a significantly harder spectrum ( Γ = 0 . 5 ± 0 . 3) in the Chandra band than the rest of the FXT sample, all of which have photon indices of Γ > 1 (Jonker et al. 2013; Glennie et al. 2015; Bauer et al. 2017; Xue et al. 2019; Lin et al. 2022; Quirola-Vásquez et al. 2022; Quirola-Vásquez et al. 2023; Eappachen et al. 2023). A photon index of Γ ∼ 0 . 5 below 10 keV is indeed expected and observed from both core-collapse GRBs and compact-merger GRBs (Ghirlanda et al. 2009; Bromberg et al. 2013; Oganesyan et al. 2018; Ravasio et al. 2019; Toffano et al. 2021). This might support the association of the initial spike of XRT 200515 with a GRB. However, the presence and properties of the ∼ 800 s tail (candidate GRB afterglow) is puzzling. The 𝑇 90 ∼ 580-740 s value for XRT 200515 is significantly shorter than in most other Chandra FXTs (Quirola-Vásquez et al. 2022; Lin et al. 2022; Quirola-Vásquez et al. 2023), which have 𝑇 90 values on the order of several ks and are already pushing the limit for a GRB afterglow detection (Wichern et al. 2024). Moreover, XRT 200515's initial burst duration ( < 10 s), its short rise and fall times (3.2-6.4 s), and the lack of a peak plateau are inconsistent with the lightcurves of Chandra FXTs interpreted as magnetar-powered GRBs as the aftermath of a binary neutron star merger, such as CDF-S XT1 (Bauer et al. 2017), CDF-S XT2 (Xue et al. 2019) and the sample in Lin et al. (2022). Finally, the lack of any optical evidence for a host galaxy is another element disfavoring the high-redshift GRB interpretation. 4.3.7 Giant Magnetar Flare from a Soft Gamma Repeater? Based on its temporal and spectral variability, it is tempting to interpret XRT 200515 as a rare GMF from a SGR (Mereghetti 2008; Turolla et al. 2015) in the LMC or behind it, which can easily explain the burst's strong increase of at least 4 orders of magnitude in < 10 s (Coti Zelati et al. 2018). Similar to XRT 200515, GMFs are characterized by a short and hard initial spike and a longer and softer, pulsating tail. GMFs are extremely rare, with only a select few ever discovered. Well-studied examples are SGR 0526 -66 in the LMC (Mazets et al. 1979), and the Galactic sources SGR 1900 + 14 (Hurley et al. 1999) and SGR 1806 -20 (Hurley et al. 2005; Palmer et al. 2005; Israel et al. 2005). More recently, GMFs have been identified in M 31 (Mazets et al. 2008a), NGC 253 (Fermi-LAT Collaboration et al. 2021; Svinkin et al. 2021; Roberts et al. 2021; Trigg et al. 2024) and M82 (Mereghetti et al. 2024). All of these have been observed by high time resolution instruments in the hard X-rays and soft 𝛾 -rays with luminosities above 10 46 erg/s for a fraction of a second in the initial spike. The tails of GMFs are often modulated by magnetar spin periods of 2-12 s, leading to quasi-periodic oscillations (QPOs). For XRT 200515, there is no hard X-ray or 𝛾 -ray detection, despite the LMC direction being in good visibility for most of the previously mentioned high-energy facilities. We were unable to identify any significant periodicities in the tail of XRT 200515 through periodogram analysis, which is unsurprising given the low time resolution of Chandra observations. No X-ray activity has been observed by Chandra or other X-ray telescopes in the years before or after XRT 200515, which may be because SGRs are very faint when they are not bursting. The strongest argument against a magnetar in the LMC as the origin of XRT 200515 is that magnetars are short-lived objects ( ≲ 10 5 yr) associated to young stellar populations (Olausen & Kaspi 2014; Nakano et al. 2015; Mondal 2021). Even allowing for the persistence of magnetar-like activity in ordinary radio pulsars as old as ∼ 10 7 yr (Rea et al. 2010), this scenario is still inconsistent with the old stellar population (several Gyr) in the LMC field shown in Figure 17. The nearest star-forming regions in the LMC are ∼ 10 arcmin ( ∼ 150 pc) away. If (in a very contrived scenario), we assume that XRT 200515 is powered by a young neutron star ejected from one of those regions, we estimate a characteristic time of 1 Myr to travel that distance at a speed of 150 km/s. Therefore, if XRT 200515 is a GMF, it must be located behind the LMC, in a low-redshift galaxy (Hurley et al. 2005; Tanvir et al. 2005). Since GMFs have been observed only a few times and never at soft X-ray energies, their properties in the soft X-ray band detectable by Chandra remain largely unexplored. XRT 200515 could indeed be the first GMF detected at soft X-ray energies. Distinguishing distant short GRBs from GMFs has historically been difficult and there are multiple studies suggesting that a subset of short GRBs are actually extragalactic GMFs (Hurley et al. 2005; Palmer et al. 2005; Tanvir et al. 2005; Ofek et al. 2006; Mazets et al. 2008b; Hurley 2011; Yang et al. 2020; Svinkin et al. 2021; Negro & Burns 2023). Just as for the distant GRB interpretation, the non-detection of any optical counterpart remains puzzling for a distant GMF scenario, unless we are dealing with a very distant and exceptionally luminous GMF.","pages":[17]},{"title":"4.3.8 Thermonuclear X-ray Burst from a quiet LMXB in the LMC?","content":"If XRT 200515 is in the LMC, a peak luminosity near the Eddington luminosity 𝐿 Edd ∼ 10 38 erg/s and sharp rise time of the flare suggests a Type I X-ray burst interpretation, which is a thermonuclear explosion on the surface of a weakly magnetized, accreting neutron star (Lewin et al. 1993; Strohmayer & Bildsten 2003; Gal-","pages":[17]},{"title":"loway et al. 2008, 2020; Galloway & Keek 2021; Alizai et al. 2023).","content":"The old stellar population in the field of XRT 200515 is consistent with the presence of neutron star LMXBs. Following the definition of burst timescale 𝜏 = 𝐸 f / 𝐹 p in Galloway et al. (2008), we estimate 𝜏 ∼ 20 s for XRT 200515, which is consistent with Type I X-ray bursts (Galloway & Keek 2021; Alizai et al. 2023). The fitted temperature 𝑘𝑇 bb ∼ 2 keV when the average spectrum is fitted with a simple blackbody, and the softening of the spectrum (temperature decrease) in the tail is also typical of Type I X-ray bursts (Galloway et al. 2008, 2020; Güver et al. 2012). On the other hand, several observed properties of XRT 200515 are unusual for Type I X-ray bursts. In particular, most Type I X-ray bursts occur when the persistent luminosity (proportional to the accretion rate) of a LMXB is 𝐿 X > 10 -4 𝐿 Edd (and, in most cases, 𝐿 X > 10 -3 𝐿 Edd ) (Galloway et al. 2008). Instead, in the initial part of ObsID 23022, the upper limit on the X-ray luminosity at the position of XRT 200515 is 𝐿 X < 10 -4 𝐿 Edd , so that the X-ray flux increased by at least 4 orders of magnitudes. On another note, the sharp decline after the initial burst of XRT 200515 would be unusual for Type I X-ray bursts, which typically exhibit a gradual and exponential decay. However, note that most Type I X-ray bursters were observed by the Rossi X-Ray Timing Explorer ( RXTE ) (Jahoda et al. 1996), which has a high time resolution. The low time resolution of Chandra may have obscured such a decay for XRT 200515. Moreover, most Type I bursts tend to repeat every few hours (Galloway et al. 2008); instead, XRT 200515 is the only event detected at that location over a total observed time of ∼ 150 ks. No LMXB has ever been noted at that position before or after the event. The time interval between bursts is related to an index 𝛼 defined as the ratio between the integrated persistent fluence between subsequent bursts and the burst fluence; from a comparison of the energy released by accretion (contributing to the persistent fluence) and by thermonuclear burning (burst fluence), we expect 𝛼 ≳ 40, in agreement with the observations of Type I bursts (Galloway et al. 2008). If we apply the same criterion ( 𝛼 ≳ 40) to the persistent and flare fluences of XRT 200515, we would have to wait > 10 7 s (4 months) to observe another similar event, assuming the persistent flux level upper limit in ObsID 23022 before the transient event. This waiting time extends to at least one year if we assume the persistent flux upper limit derived from the stacked ∼ 150 ks Chandra observations. Only a few one-off bursts from Galactic neutron stars at a very low persistent luminosity ( 𝐿 X ∼ 10 32 -10 33 erg/s) were found by Cornelisse et al. (2002a,b) with estimated recurrence times of tens of years. The vast majority of Type I X-ray bursts are Galactic, due to their lower flux at large distances. Only a handful of extragalactic Type I X-ray bursts are documented, for example in M 31 (Pastor-Marazuela et al. 2020) and the Magellanic Bridge (Haberl et al. 2023). If XRT 200515 is a Type I X-ray burst, it is the first extragalactic Type I X-ray burster in the LMC and represents the tip of the iceberg for a vast population of faint LMXBs in nearby galaxies, too dim to be detected by Chandra or XMM-Newton , but which may occasionally reveal themselves via thermonuclear bursts with a long duty cycle.","pages":[18]},{"title":"4.3.9 Concluding Remarks and Outlook for XRT 200515","content":"XRT 200515 is a unique and intriguing extragalactic Chandra FXT. The combination of its temporal and spectral properties is unlike any of the other Chandra FXT samples. Based on our analysis, the two most likely scenarios for XRT 200515 are: (i) a distant GMF from a SGR behind the LMC; the first observed in the low X-ray energy band, missed by any other high-energy facilities, or (ii) an unusual Type I X-ray burst from a previously unknown faint LMXB; the first extragalactic X-ray burster in the LMC. Nevertheless, both of these interpretations come with their own unique challenges. XRT 200515 could, in fact, represent an entirely new type of astronomical phenomenon. After all, the primary objective of our work was to use machine learning to find rare, needle-in-the-haystack anomalies hidden within vast astronomical datasets. We invite further detailed studies of XRT 200515 to evaluate our interpretations and explore alternative scenarios, such as potential associations with a fast radio burst (FRB) or a SBO. We highly recommend follow-up multi-band observations at the source coordinates of XRT 200515 to better constrain its nature. Lastly, we note that XRT 200515 and the second transient discovered by Glennie et al. (2015), XRT 120830, have remarkably similar temporal evolutions in their lightcurves (J. Irwin, personal communication, November 2024), however with very different spectral properties ( Γ ∼ 2 . 5 for XRT 120830 versus Γ ∼ 0 . 5 for XRT 200515). We leave a detailed comparative analysis of these transients for future work. Figure 18 shows the 𝐸 -𝑡 Map and 𝐸 -𝑡 -𝑑𝑡 Cube event file representations for XRT 200515. These exhibit high counts at high energies in a narrow time window, which is in line with the hard spectrum and transient nature of XRT 200515.","pages":[18]},{"title":"4.4 Technical Caveats","content":"The main technical caveat of our approach is related to the representation of event files. While our new event file representations enable a simple, yet powerful representation learning approach to find new and rare X-ray transients, any simplification of raw event files, like the fixed number of time bins we use across all event files, is associated with a loss of information. This could lead to us missing a small amount of transients. To minimize this, we have implemented a rigorous approach to justify the resolution of the event file representations in § 3.1. Moreover, flares, in particular known extragalactic FXTs, cluster notably well in our representation spaces. This is because their distinctive features are less dependent on the temporal binning resolution in the 𝐸 -𝑡 Maps and 𝐸 -𝑡 -𝑑𝑡 Cubes. To improve the effectiveness of dip searches with our proposed method, we suggest using higher resolution event file representations. Nevertheless, our comprehensive transient candidate catalog includes numerous newly identified transients that were previously overlooked by other X-ray transient searches in the Chandra archive. Among these is the remarkable needle-in-the-haystack event XRT 200515 discovered in this work, underscoring the effectiveness of our method. A followup representation learning algorithm will learn informative features from raw and unbinned event files while accounting for the Poisson nature of X-ray observations (Song et al., 2025, in preparation).","pages":[18]},{"title":"5 CONCLUSION","content":"We have introduced a novel representation learning method, the first of its kind applied to X-ray event files, enabling downstream tasks such as unsupervised classification and anomaly detection in highenergy astrophysics. We have used the learned representation to investigate time-domain properties of sources in the Chandra archive, with a particular emphasis on the discovery of X-ray transients. As a result, we have compiled the identified X-ray flares and dips in a comprehensive catalog of transient candidates. Notably, our method led to the discovery of XRT 200515; a previously unidentified extragalactic FXT with unique temporal and spectral properties, representing a genuine needle-in-the-haystack discovery. Our key results are as follows: XRT200515wasonlydetectedby Chandra , with no identified optical counterparts. We strongly encourage a multi-wavelength search for additional signals from the source associated with XRT 200515 to better understand its origin and nature. Ourworkadvancestime-domainhigh-energy astrophysics by making the Chandra transient candidates catalog publicly available and open-sourcing the representation learning based transient search pipeline 7 . The catalog enables queries to identify new Chandra transients. Future work involves applying the detection pipeline to additional high-energy archives and adapting it to a variety of other scientific datasets, paving the way for further machine learning driven discoveries of rare transients and other scientific anomalies.","pages":[19]},{"title":"ACKNOWLEDGEMENTS","content":"This research has made use of data obtained from the Chandra Source Catalog, and software provided by the Chandra X-ray Center (CXC) in the /c.pc/i.pc/a.pc/o.pc application package. SD's work was partially funded by the UK government's Turing Scheme and mainly carried out at the Center for Astrophysics | Harvard & Smithsonian as part of the SAO Predoctoral Program, with the support of AstroAI. SD acknowledges hospitality at the Institute of Astronomy at the University of Cambridge and at the Stanford Center for Decoding the Universe (Stanford Data Science) during the later parts of this project. RS's work was performed in part at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-2210452. RS acknowledges support and hospitality at the National Astronomical Observatories of China (NAOC) in Beijing, during part of this project. We thank Edo Berger, Massimiliano De Pasquale, Ken Ebisawa, Duncan Galloway, Jimmy Irwin, Peter Jonker, Daniel Kocevski, Amy Lien, Sandro Meregetthi, Daniel Muthukrishna, Nicola Omodei, Jonathan Quirola-Vásquez, Shivam Raval, Ashley Villar, and Silvia Zane for their fruitful discussions.","pages":[19]},{"title":"DATA AVAILABILITY","content":"The data used in this paper, composed of X-ray event files and source detection regions, was obtained from the publicly available CSC, using their public interfaces (https://cxc.cfa.harvard.edu/csc/). The catalog of transient candidates and the clustered embedding spaces generated using our unsupervised representation learning method can be accessed in the supplementary material. All intermediate data products, i.e. as 𝐸 -𝑡 Maps, 𝐸 -𝑡 -𝑑𝑡 Cubes, principal components and latent features, feature embeddings and embedding clusters can be produced using the code provided in the GitHub repository https://github.com/StevenDillmann/ml-xraytransients-mnras.","pages":[19]},{"title":"REFERENCES","content":"Walmsley M., et al., 2022, Monthly Notices of the RAS, 513, 1581 Weisskopf M. C., Tananbaum H. D., Van Speybroeck L. P., O'Dell S. L., 2000, in Truemper J. E., Aschenbach B., eds, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Vol. 4012, X-Ray Optics, Instruments, and Missions III. pp 2-16 ( arXiv:astro-ph/0004127 ), doi:10.1117/12.391545 Wichern H. C. I., Ravasio M. E., Jonker P. G., Quirola-Vásquez J. A., Levan X. C., 2019, Monthly Notices of the RAS, 487, 4721","pages":[21]},{"title":"APPENDIX A: DISTRIBUTION OF EVENT FILE LENGTHS AND DURATIONS","content":"Figure A1 shows the distribution of the length 𝑁 and duration 𝑇 of event files in the dataset used in this work.","pages":[22]},{"title":"APPENDIX B: AUTOENCODER TRAINING PROCESS","content":"Figure B1 shows the training process of the autoencoders used in this work.","pages":[22]},{"title":"APPENDIX C: HYPERPARAMETER OPTIMIZATION","content":"Below, we summarize the optimization strategy for the t-SNE and DBSCAN hyperparameters. For even more details on this approach, please refer to Dillmann & Martínez-Galarza (2023).","pages":[22]},{"title":"C1 t-SNE Hyperparameters","content":"The choice of the perplexity and learning_rate can have a large impact on the resulting t-SNE embedding space. Ideally, we want the two-dimensional embedding space to effectively capture both energy information (in form of the hardness ratio 𝐻𝑅 ) and variability information (in form of the variability probability 𝑝 var ). That means that event files with similar values for 𝐻𝑅 and 𝑝 var should live close to each other in the final embedding space. We can use this information to define a performance metric for different t-SNE hyperparameter inputs. First, we compute the pairwise distance matrix D Z of size ( 𝑚, 𝑚 ) , where the distance 𝐷 𝑍 𝑖 𝑗 between points 𝑖 and 𝑗 is computed using a Euclidean distance metric. Next, we define the property vector Y , which includes 7 CSC properties (hardness ratios 𝐻𝑅 hm , 𝐻𝑅 hs , 𝐻𝑅 ms and variability probabilities 𝑝 b var , 𝑝 h var , 𝑝 m var , 𝑝 s var ) for each event file and thus each t-SNE point. As a measure of similarity between the labels of different points, we can again compute a pairwise similarity matrix D Y of size ( 𝑚, 𝑚 ) . To compute the similarity distance 𝐷 𝑌 𝑖 𝑗 between sample 𝑖 and 𝑗 , we use the Mahalanobis distance metric (Mahalanobis 1936). Unlike the Euclidean distance metric, the Mahalanobis distance metric accounts for the correlation between different labels by taking into account the covariance structure of the data. Note that our hardness ratios are correlated with each other, and that the same holds for the variability probabilities. Accounting for these correlations provides a more accurate measure of the similarity distance between different samples. Having computed D Z and D Y , we can define a performance metric that allows us to compare the performance of different t-SNE hyperparameters. The smaller the distance 𝐷 𝑍 𝑖 𝑗 between two points 𝑖 and 𝑗 in the t-SNE embedding, the smaller should be difference in their associated labels as measured by the distance 𝐷 𝑌 𝑖 𝑗 . We can thus define a performance metric based on the statistical correlation of D Z and D Y using the Spearman's rank correlation coefficient 𝜌 𝑍𝑌 (Spearman 1904). The higher 𝜌 𝑍𝑌 , the higher is the positive correlation between D Z and D Y and the better the performance of the t-SNE embedding. The hyperparameter space is given by the ranges learning_rate ∈ ( 20 , 200 ) with a step size of 20 and perplexity ∈ ( 10 , 100 ) with a step size of 10. This optimization process is performed using a reduced dataset of 15,353 samples for 2,000 iterations per hyperparameter combination due to computational constraints. While subsampling, the overall structure of the data was preserved by selecting the same distributions between any combinations of hard, medium, soft, variable and non-variable samples. This ensures that the sample set is representative of the original data. We choose the hyperparameter combination that produces the highest value of 𝜌 𝑍𝑌 .","pages":[22,23]},{"title":"C2 DBSCAN Hyperparameters","content":"Different hyperparameter combinations of eps and minPts can have a large impact on the resulting DBSCAN clusters. We use a combination of the Davies-Bouldin index 𝐷𝐵 (Davies & Bouldin 1979) and Calinski-Harabasz index 𝐶𝐻 (Caliński & Harabasz 1974) as a performance metric to find the optimal DBSCAN hyperparameter inputs. The 𝐷𝐵 index is a measure of the average similarity between each cluster and its most similar cluster, relative to the average distance between points within each cluster. The 𝐷𝐵 index is given by the following formula: where 𝑛 𝑐 is the number of clusters, 𝑊 𝑖 and 𝑊 𝑗 are the within-cluster sum of squares for cluster 𝑖 and 𝑗 , and 𝑑 ( 𝑐 𝑖 , 𝑐 𝑗 ) is the distance between the centroids of clusters 𝑖 and 𝑗 . On the other hand, the 𝐶𝐻 index is based on the concept that good clusters should have high intra-cluster similarity (cohesion) measured by the between-cluster dispersion 𝐵 and low inter-cluster similarity (separation) measured by the within-cluster dispersion 𝑊 . 𝐵 is the sum of the pairwise distances between cluster centroids, and 𝑊 is the sum of the pairwise distances between points within each cluster. The 𝐶𝐻 index is given by the following formula: where the scaling factor 𝑚 -𝑛 𝑐 𝑛 𝑐 -1 accounts for the total number of data points 𝑚 and the number of clusters 𝑛 𝑐 . A lower 𝐷𝐵 index and higher 𝐶𝐻 index indicate that the clustering algorithm is more effective in grouping similar data points together and separating different data points into distinct clusters. We thus define the performance metric 𝜌 𝐷𝐶 as the ratio of the normalized indices 𝐷𝐵 𝑛 = 𝐷𝐵 max ( 𝐷𝐵 ) and 𝐶𝐻 𝑛 = 𝐶𝐻 max ( 𝐶𝐻 ) in the hyperparameter space given by eps ∈ ( 1 . 0 , 3 . 0 ) with a step size of 0 . 1 and minPts ∈ ( 10 , 30 ) with a step size of 1: We choose the hyperparameter combination that produces the highest value of 𝜌 𝐷𝐵𝑆𝐶𝐴𝑁 .","pages":[23]},{"title":"APPENDIX D: EMBEDDINGS","content":"Figures D1, D2 and D3 show the 2D-PCA and 2D-AE embeddedings.","pages":[23]},{"title":"APPENDIX E: TRANSIENT-DOMINANT EMBEDDING CLUSTERS","content":"Table E1 lists the transient-dominant clusters in the different embedding spaces used for the selection of transient candidates.","pages":[23]},{"title":"APPENDIX F: CATALOG COLUMNS","content":"Table F1 shows the of X-ray transient candidate catalog column descriptions. This paper has been typeset from a T E X/L A T E X file prepared by the author.","pages":[23]}],"string":"[\n {\n \"title\": \"ABSTRACT\",\n \"content\": \"We present a novel representation learning method for downstream tasks such as anomaly detection and unsupervised transient classification in high-energy datasets. This approach enabled the discovery of a new fast X-ray transient (FXT) in the Chandra archive, XRT 200515, a needle-in-the-haystack event and the first Chandra FXT of its kind. Recent serendipitous breakthroughs in X-ray astronomy, including FXTs from binary neutron star mergers and an extragalactic planetary transit candidate, highlight the need for systematic transient searches in X-ray archives. We introduce new event file representations, 𝐸 -𝑡 Maps and 𝐸 -𝑡 -𝑑𝑡 Cubes, designed to capture both temporal and spectral information, effectively addressing the challenges posed by variable-length event file time series in machine learning applications. Our pipeline extracts low-dimensional, informative features from these representations using principal component analysis or sparse autoencoders, followed by clustering in the embedding space with DBSCAN. New transients are identified within transient-dominant clusters or through nearest-neighbor searches around known transients, producing a catalog of 3,539 candidates (3,427 flares and 112 dips). XRT 200515 exhibits unique temporal and spectral variability, including an intense, hard < 10 s initial burst followed by spectral softening in an ∼ 800 s oscillating tail. We interpret XRT 200515 as either the first giant magnetar flare observed at low X-ray energies or the first extragalactic Type I X-ray burst from a faint LMXB in the LMC. Our method extends to datasets from other observatories such as XMM-Newton , Swift-XRT , eROSITA , Einstein Probe , and upcoming missions like AXIS . Key words: software: machine learning, methods: data analysis, X-rays: bursts, stars: magnetars, transients: gamma-ray bursts, stars: peculiar\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"Representation Learning for Time-Domain High-Energy Astrophysics: Discovery of Extragalactic Fast X-ray Transient XRT 200515\",\n \"content\": \"Steven Dillmann 1 , 2 ★ † , Rafael Martínez-Galarza 3 , Roberto Soria 4 , 5 , Rosanne Di Stefano 3 and Vinay L. Kashyap 3 Accepted XXX. Received YYY; in original form ZZZ\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1 INTRODUCTION\",\n \"content\": \"Recent serendipitous discoveries, such as extragalactic fast X-ray transients (FXTs) linked to neutron star merger candidates as electromagnetic counterparts to gravitational wave events (Lin et al. 2022) and an X-ray dip associated with the first extragalactic planet candidate (Di Stefano et al. 2021), underscore the challenges of identifying such rare events within large X-ray catalogs. Beyond magnetarpowered FXTs as the aftermath of binary neutron star mergers (Dai et al. 2006; Metzger et al. 2008; Zhang 2013; Sun et al. 2017; Bauer et al. 2017; Xue et al. 2019), other interesting origins of extragalactic FXTs include supernova shock breakouts (SBOs) (Soderberg et al. 2008; Modjaz et al. 2009; Alp & Larsson 2020; Novara et al. 2020), tidal disruption events (TDEs) (Jonker et al. 2013) including quasiperiodic eruptions (QPEs) (Arcodia et al. 2021; Chakraborty et al. 2021), thermonuclear (Type I) X-ray bursts from accreting neutron stars (in't Zand et al. 2013), or binary self-lensing events (D'Orazio &DiStefano 2018, 2020; Hu et al. 2020). Both because of their very stochastic nature, and because narrow field X-ray missions such as the Chandra X-ray Observatory ( Chandra ) (Weisskopf et al. 2000), XMM-Newton (Jansen et al. 2001) and Swift-XRT (Burrows et al. 2005) are not designed as wide time-domain surveys, X-ray transient discoveries are often serendipitous. They can be found in observations that were originally proposed for a completely unrelated science objective and are rarely the target of the observation. In many cases serendipitously found X-ray sources do not get characterized or classified, since their transient nature is not immediately obvious. Instead, observations with X-ray transients often get stored in large data archives and remain unnoticed. This raises the need for a systematic search for short-duration phenomena in high-energy catalogs. New missions such as eROSITA (Predehl et al. 2021), Einstein Probe (Yuan et al. 2022) and the upcoming AXIS Observatory (Reynolds et al. 2024) target X-ray transients more directly, thus the development of novel transient detection methods is becoming even more relevant. The temporary, unpredictable and 'unusual' nature of X-ray transients distinguishes them from 'normal' X-ray source emissions. From a data science perspective, they can be understood as 'anomalies' within a large dataset. Existing methods for identifying X-ray transients primarily rely on statistical tests of variability (Yang et al. 2019; Pastor-Marazuela et al. 2020; Quirola-Vásquez et al. 2022; Quirola-Vásquez et al. 2023). While effective within specific constraints, these approaches are inherently limited by their underlying assumptions, which may not capture the diverse nature of transient phenomena. In contrast, machine learning offers a more flexible, expressive, and scalable framework, making it particularly well-suited for anomaly detection in large, high-dimensional datasets with diverse transient types. While optical time-domain surveys are at the forefront of leveraging extensive observational programs, like ZTF (Bellm et al. 2019) or the upcoming LSST survey (Ivezić et al. 2019), and neural network-based anomaly detection tools to identify rare sources among countless ordinary objects (Villar et al. 2021; Muthukrishna et al. 2022), the X-ray astronomy community has only recently begun exploring the potential of machine learning to classify sources (Yang et al. 2022; Pérez-Díaz et al. 2024) or to search for needle-in-a-haystack events in large X-ray datasets and archives (Kovačević et al. 2022; Dillmann & Martínez-Galarza 2023). The effectiveness of machine learning methods largely depends on the algorithm's ability to learn useful representations from the data. Representation learning (Bengio et al. 2013) is an increasingly popular technique in astronomy used in supervised, semi-supervised, self-supervised and unsupervised frameworks (Naul et al. 2018; Hayat et al. 2021; Walmsley et al. 2022; Slijepcevic et al. 2024; Mohale & Lochner 2024). It involves creating or learning meaningful representations for specific modalities of scientific data, which can then be used for downstream tasks such as regression, classification, or, as in this work, anomaly detection. The compressed representations live in a low-dimensional embedding space, in which anomalous data samples are well-separated from more ordinary ones. We propose a new unsupervised representation learning method to perform a large-scale search for X-ray transients in the Chandra archive. High-energy catalogs include individual X-ray source observations in the form of event files. The variable length of these time series poses a challenge in creating consistent representations suitable for transient searches with machine learning. Most deep learning algorithms take a fixed-length input for all data samples. In order to effectively represent event files over a broad range of lengths, we introduce novel fixed-length event file representations, which take into account both their time-domain and energy-domain information. Applying feature extraction and dimensionality reduction techniques, for example with sparse autoencoders, we create a representation space that encodes scientifically meaningful information, such as the spectral and variability properties of the astrophysical sources. Previously identified X-ray transients occupy distinct, well-isolated clusters in the embedding space. Using clustering techniques and nearest neighbor searches allows us to effectively explore these transient-dominant clusters to discover new X-ray transients. We collect the identified X-ray flare and dip candidates in a publicly available catalog, serving as a fertile ground for new discoveries in time-domain high-energy astrophysics. Among these candidates, we identify an intriguing extragalactic FXT, XRT 200515, which exhibits unique temporal and spectral characteristics distinct from any previously reported Chandra FXTs. The transient's initial hard < 10 s burst shows a sharp rise exceeding 4 orders of magnitude, followed by spectral softening in an ∼ 800 s oscillating tail. This transient is likely related to either a giant magnetar flare (GMF) from a distant soft gamma repeater (SGR) behind the Large Magellanic Cloud (LMC) or an extragalactic Type I X-ray burst from a faint LMXB in the LMC. Each of these interpretations presents its own set of challenges. Alternatively, XRT 200515 could be a new type of astronomical phenomenon found by our anomaly detection method using machine learning. Our method is the first representation learning approach for anomaly detection in high-energy astrophysics. It is applicable to datasets from high-energy catalogs like Chandra , XMM-Newton , Swift-XRT , eROSITA , and Einstein Probe . We created semantically meaningful representations that can be aligned with other data modalities, such as optical images or infrared spectra to design multi-modal models (Parker et al. 2024; Mishra-Sharma et al. 2024; Zhang et al. 2024; Rizhko & Bloom 2024) using contrastive learning (Radford et al. 2021), that can improve on current state-of-the-art algorithms used to characterize the physics of the associated objects. Ultimately, this work and other representation and contrastive learning approaches lay the groundwork for developing generalized foundation model in astronomy. The paper is organized as follows: In § 2, we provide information on the dataset of Chandra event files used in this analysis. In § 3, we describe in detail the implementation of our novel transient detection approach leveraging representation learning. In § 4, we present and discuss the results in form of the semantically meaningful representation space of the event files, the catalog of X-ray transient candidates and the discovery of the new Chandra transient XRT 200515. Finally, we highlight our contributions to time-domain high-energy astrophysics and outline potential directions for extending this work in the future in § 5. The relevant code, a demonstration of the pipeline, and an interactive embedding selection, transient search and lightcurve plotting tool are available online at the GitHub repository https://github.com/StevenDillmann/ml-xraytransients-mnras.\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"2 DATASET\",\n \"content\": \"We use data from the Chandra Source Catalog (CSC) version 2.1 (Evans et al. 2024), which includes all publicly available X-ray sources detected by Chandra as of December 2021. For this study, we focus specifically on observations from the Advanced CCD Imaging Spectrometer (ACIS). CSC 2.1 had not been fully released at the time our analysis was performed, but catalog data was available for sources that had completed processing in the Current Database View 1 , a snapshot of which we took on 11 April 2023. CSC 2.1 performs source detection on stacked observations, and catalog properties are provided both for these stack-level detections, and for each of observation-level detection that contribute to a stack detection. Because we are interested in short-time variability that happens within a single observation of a source, we use the catalog products for the observation-level detections in our analysis. For a given X-ray detection, two types of products are provided in the CSC: (i) database tables with source properties, such as fluxes in the different X-ray energy bands, hardness ratios, variability indices, etc., and (ii) filebased data products for each detection of a source, such as the detect regions, the Chandra PSF at that location, etc. The following observation-level catalog properties are relevant for our analysis: index of 6 or larger indicates variability at a confidence of at least 2 𝜎 . In this paper we call this quantity 𝐼 𝑏 var . From the catalog data products available for observation-level Xray detections, we are interested in the region event file. This event file consists of a list of all individual photon events detected in a small bounding box around a source detection, listing their energies, arrival times, and detector coordinates. These event files are the basis for the characterization of an X-ray source: lightcurves, spectra, images, coordinates, and other properties are derived from the distribution of the listed quantities. In this analysis, we directly use these event files as our primary data products. The values of the catalog properties listed above serve as summary statistics for the detection associated with a given region event file. We only include event files with more than 5 events and a signal-to-noise ratio above 5 to minimize spurious signals from low number statistics in faint sources. We also exclude detections that are flagged for pile-up 2 , i.e., those with a pileup fraction larger than 5%, which corresponds to a maximum pileup warning of 0.1 in CSC 2.1. For the resulting detections, we filter the event files to include only events contained within the detection region for each source. These detection regions are also provided as data products in CSC 2.1, and consist of the ellipse that includes the 90% encircled counts fraction of the PSF at the source location. Due to the low background level in Chandra observations, the majority of events selected after this spatial filtering are expected to be events associated with the X-ray source, not the background. In the selected event files, we only include photon events within good time intervals (GTIs), which are time periods of valid, high-quality data. No other pre-processing is required. The final dataset consists of 95,473 filtered event files from 58,932 sources, resulting in an average of 1 . 62 observations per source. This includes 9,003 new sources that have been added as part of the CSC 2.1 release, in addition to the sources from the previous release.\",\n \"pages\": [\n 2,\n 3\n ]\n },\n {\n \"title\": \"3 METHODS\",\n \"content\": \"In this work, we introduce a novel representation learning based anomaly detection method to systematically search for X-ray transients in high-energy archives. We begin with an overview of the method here and provide detailed explanations of each step in individual subsections. The full pipeline is illustrated in Figure 1. Starting with the event files described in § 2, we (i) build two novel and uniform event file representations by binning their arrival times and energies into 𝐸 -𝑡 Maps (Event File Representation I) or 𝐸 -𝑡 -𝑑𝑡 Cubes (Event File Representation II); (ii) use principal component analysis (Feature Extraction I) or sparse autoencoders (Feature Extraction II) to extract informative features from the event file representations; (iii) apply dimensionality reduction to the extracted features to create a low-dimensional embedding space; (iv) use density-based clustering to create embedding clusters that group event files with similar characteristics, for example transient behavior or certain spectral features. Previously identified transients like the extragalactic magnetar-powered flare candidate reported by Lin et al. (2022) and the extragalactic planet candidate dip reported by Di Stefano et al. (2021), shown in Figure 2, occupy well-isolated clusters in the embedding space. Exploring these clusters and conducting nearest-neighbor searches enables us to effectively find analogs to bona-fide time-domain anomalies, while at the same time grouping them according to their spectral properties. We compile the identified transient candidates in a catalog. While our approach is designed and tested using Chandra data, it is applicable to any dataset consisting of event lists, like those from other high-energy telescopes. The described transient detection approach is applied to both types of event file representations with both feature extraction methods, resulting in four different embeddings. We denote the different cases as described in Table 1.\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"3.1 Event File Representation\",\n \"content\": \"The different event files in the dataset are variable in length 𝑁 and duration 𝑇 , as shown in Appendix A. The large variation in the number of events and duration highlights the challenge in producing uniform data representations that preserve relevant information on time variability and spectral properties. While there exist machine learning architectures that take variable length inputs, the significant differences in the number of events from object to object make standardization of the inputs challenging, even when these architectures are used (Martínez-Galarza & Makinen 2022). As a first step in our analysis, we introduce 2-dimensional and 3-dimensional fixed-length representations based on an informed binning strategy for the event files, similar to the DMDT maps for optical lightcurves introduced by Mahabal et al. (2017).\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"3.1.1 2D Histogram Representation ( 𝐸 -𝑡 Maps)\",\n \"content\": \"Assume an event file with 𝑁 photons and a photon arrival time column 𝒕 with entries { 𝑡 𝑘 } 𝑁 𝑘 = 1 and energy column 𝑬 with entries { 𝐸 𝑘 } 𝑁 𝑘 = 1 . The event file duration is given by 𝑇 = 𝑡 𝑁 -𝑡 1 . The energy column entries take values in the broad energy band of Chandra 's ACIS instrument, i.e. 𝐸 𝑘 ∈ [ 𝐸 𝑚𝑖𝑛 , 𝐸 𝑚𝑎𝑥 ] , where 𝐸 𝑚𝑖𝑛 = 0 . 5 keV and 𝐸 𝑚𝑎𝑥 = 7 keV comes from considering appropriate boundaries for the energy response of Chandra 's ACIS instrument. Beyond these boundaries, the telescope's aperture effective area is low for the majority of detected sources. First, we obtain the normalized time column, given by 𝝉 = 𝒕 -𝑡 1 𝑇 , and the logarithm of the energy column, given by 𝝐 = log 𝑬 . The resulting boundaries for normalized time column are 𝝉 ∈ [ 𝜏 𝑚𝑖𝑛 , 𝜏 𝑚𝑎𝑥 ] , where 𝜏 𝑚𝑖𝑛 = 0 and 𝜏 𝑚𝑎𝑥 = 1. The range for the log-energy column is 𝝐 ∈ [ 𝜖 𝑚𝑖𝑛 , 𝜖 𝑚𝑎𝑥 ] , where 𝜖 𝑚𝑖𝑛 = log 0 . 5 keV and 𝜖 𝑚𝑎𝑥 = log 7 keV. Next, we determine the dimensionality of our representations. For a each event file, we determine the optimal number of bins in the energy dimension, 𝑛 𝜖 , with the Freedman-Diaconis rule (Freedman & Diaconis 1981), a widely used method that balances the trade-off between too noisy histograms (too many bins) and not informative enough histograms (too few bins). The optimal bin width 𝑏 𝜖 according to this rule is calculated in the following way: where 𝐼𝑄𝑅 ( 𝜖 ) represents the interquartile range of the 𝜖 values for a given event file of length 𝑁 . Subsequently, we obtain the optimal number of energy bins 𝑛 𝜖 with: For each event file, we determine the optimal number of bins in the time dimension, 𝑛 𝜏 , with the help of the Bayesian Blocks algorithm, which was specifically developed for time series analysis in astronomy (Scargle et al. 2013). This algorithm partitions the time series into adaptive width bins or blocks that are statistically distinct from neighboring blocks; that is, within a given time-ordered Bayesian block, events grouped in that block are consistent with having a similar event arrival rate. We use the default Astropy implementation of Bayesian blocks, and set the false alarm probability parameter to 𝑝 0 = 0 . 01 (Astropy Collaboration et al. 2013), which implies a 1% probability of declaring a change of rate when there is none. For each event file, we define the optimal uniform bin width 𝑏 𝜏 as the minimum bin width calculated by the Bayesian Blocks algorithm, and then find the optimal number of time bins 𝑛 𝜏 with: The optimal number of bins is different for each event file, due to their different lengths 𝑁 and durations 𝑇 . To select a bin size that can be applied to all event files, we consider the distributions of these optimal bin sizes, which are shown in Figure 3. For the distribution of 𝑛 𝜏 values we only use those event files for which 𝑝 𝑏 var > 0 . 9. The intent of this is to effectively capture variability timescales that are associated with short time-domain events, such as flares and dips. We choose the 90th percentile value of each distribution to set the final number of bins in each dimension. That is, only 10% of the event files will have an optimal number of bins that is larger than the chosen values 𝑛 𝜖 = 16 and 𝑛 𝜏 = 24. The choice of the 90th percentile, rather than the mean or mode, is motivated by the need to capture sufficient statistical detail even for long event files, while keeping the size of the resulting representations computationally tractable. Choosing a lower resolution would risk losing significant details in the representation, particularly short-duration events such as flares and dips within longer event files. The 𝐸 -𝑡 Maps are the 2D histogram representations with size ( 𝑛 𝜏 , 𝑛 𝜖 ) = ( 24 , 16 ) that result from binning the events according to the optimized number of bins. Figure 4 shows the 𝐸 -𝑡 Maps for the known extragalactic dip reported by Di Stefano et al. (2021) and known extragalactic flare reported by Lin et al. (2022).\",\n \"pages\": [\n 3,\n 5\n ]\n },\n {\n \"title\": \"3.1.2 3D Histogram Representation ( 𝐸 -𝑡 -𝑑𝑡 Cubes)\",\n \"content\": \"Wenowintroduce the 𝐸 -𝑡 -𝑑𝑡 Cubes, which extend the 𝐸 -𝑡 Maps by a third dimension that serves as a proxy for the photon arrival rate. For an event file of length 𝑁 , consider the array of time differences between consecutive photon arrivals 𝚫 𝒕 with entries Δ 𝑡 𝑘 = 𝑡 𝑘 + 1 -𝑡 𝑘 for 𝑘 = 1 , 2 , . . . , 𝑁 -1. We again scale and normalize the obtained values, so that they adopt values between 0 and 1, using in each case the minimum value Δ 𝑡 𝑚𝑖𝑛 and maximum value Δ 𝑡 𝑚𝑎𝑥 . This provides the third dimension 𝜹𝝉 : The additional dimension is intended to better isolate short-duration features in time variability by capturing high photon arrival rates, which are typical of flares, as well as very low photon arrival rates, which are typical of dips. The boundaries of our histogram representations in this dimension are 𝜹𝝉 ∈ [ 𝛿𝜏 𝑚𝑖𝑛 , 𝛿𝜏 𝑚𝑎𝑥 ] , where 𝛿𝜏 𝑚𝑖𝑛 = 0 and 𝛿𝜏 𝑚𝑎𝑥 = 1. We determine the optimal number of bins in the 𝜹𝝉 dimension, 𝑛 𝛿𝜏 , again by computing the optimal bin width 𝑏 𝛿𝜏 with the Freedman-Diaconis rule and dividing the range for 𝜹𝝉 by 𝑏 𝛿𝜏 : The distribution of 𝑛 𝛿𝜏 across the event files is shown in Figure 3. Most of the relevant time-domain information is already captured by 𝝉 , but adding 𝜹𝝉 provides an additional marker for dips and flares that can be shorter than the timescales probed by our chosen binning of 𝝉 . Unlike in the other two dimensions, we choose the 75th percentile value of the distribution as our final choice of common binning, which results in 𝑛 𝛿𝜏 = 16. This is because in order to identify short transients, we need to capture strong deviations in 𝜹𝝉 only. Choosing a lower value for 𝑛 𝛿𝜏 reduces noise an improves computational tractability. Having both 𝝉 and 𝜹𝝉 represented also breaks any assumption of stationarity, in that we can be sensitive to transient events happening at any time during the observation of the source, and break degeneracies between periodic and non-periodic features in the representations presented by Martínez-Galarza & Makinen (2022). The 𝐸 -𝑡 -𝑑𝑡 Cubes are the resulting 3D histogram event file representations with size ( 𝑛 𝜏 , 𝑛 𝜖 , 𝑛 𝛿𝜏 ) = ( 24 , 16 , 16 ) .\",\n \"pages\": [\n 5,\n 6\n ]\n },\n {\n \"title\": \"3.1.3 Feature Notation\",\n \"content\": \"The event file representations can now be used as inputs for various statistical learning and machine learning algorithms. For the 𝑖 𝑡 ℎ event file in the dataset of length 𝑚 = 95,473, we denote the corresponding feature vector as fi 𝑥 𝑖 = [ 𝑥 1 , 𝑥 2 , . . . , 𝑥 𝑛 ] 𝑖 , where 𝑛 = 𝑛 𝜏 · 𝑛 𝜖 = 384 for the 𝐸 -𝑡 Maps and 𝑛 = 𝑛 𝜏 · 𝑛 𝜖 · 𝑛 𝛿𝜏 = 6 , 144 for the 𝐸 -𝑡 -𝑑𝑡 Cubes. The set of all feature vectors is denoted as X = [fi 𝑥 1 , fi 𝑥 2 , . . . , fi 𝑥 𝑚 ] ⊤ with size ( 𝑚, 𝑛 ) .\",\n \"pages\": [\n 6\n ]\n },\n {\n \"title\": \"3.2 Feature Extraction I: Principal Component Analysis\",\n \"content\": \"We use Principal Component Analysis (PCA) (Pearson 1901) provided by scikit-learn (Pedregosa et al. 2011) as our first feature extraction method. The extracted principal components should encode relevant time-domain and spectral information of the event file they represent. PCA involves transforming a dataset into a new coordinate system by finding the principal components of the data that capture most of the variance in the data. By projecting the dataset onto principal components, PCA reduces the dimensionality of the data while retaining the most important information, which increases the interpretability of high-dimensional data.\",\n \"pages\": [\n 6\n ]\n },\n {\n \"title\": \"3.2.1 PCA Algorithm\",\n \"content\": \"We start with the feature vector set X of size ( 𝑚, 𝑛 ) representing our dataset with 𝑚 samples and 𝑛 dimensions. PCA aims to find a new coordinate system defined by a set of orthogonal axes, i.e. the principal components, that captures the maximum amount of variance in the data. The PCA result is a transformed dataset Xpc obtained by projecting X onto the principal components: where W is matrix of size ( 𝑛, 𝑛 𝑝𝑐 ) containing the first 𝑛 𝑝𝑐 principal components to be retained as its columns and Xpc is of size ( 𝑚 , 𝑛 𝑝𝑐 ) with a reduced dimensionality of 𝑛 𝑝𝑐 . For a more detailed explanation of the algorithm, we refer the reader to Jolliffe (2002).\",\n \"pages\": [\n 6\n ]\n },\n {\n \"title\": \"3.2.2 Principal Components Retained\",\n \"content\": \"The main PCA hyperparameter is the number of principal components 𝑛 𝑝𝑐 to retain. Figure 5 shows two scree plots illustrating the amount of variance explained by each principal component in descending order and the cumulative proportion of variance explained by the principal components for both 𝐸 -𝑡 Mapsand 𝐸 -𝑡 -𝑑𝑡 Cubes. Acommon approach to determine the optimal value of 𝑛 𝑝𝑐 is to find the knee point in the cumulative scree plot of the principal components. This balances the objective of minimizing the dimensionality while retaining as much information as possible. Defining the knee point as the point beyond which adding additional principal components increases the amount of variance by less than 0 . 1% gives 𝑛 𝑝𝑐 = 15 for 𝐸 -𝑡 Maps and 𝑛 𝑝𝑐 = 22 for 𝐸 -𝑡 -𝑑𝑡 Cubes as indicated in Figure 5. These capture 94 . 1% and 89 . 9% of the variance respectively.\",\n \"pages\": [\n 6\n ]\n },\n {\n \"title\": \"3.3 Feature Extraction II: Sparse Autoencoder Neural Network\",\n \"content\": \"As an alternative to PCA, we now build Autoencoder (Hinton & Salakhutdinov 2006) models with TensorFlow (Abadi et al. 2015) to learn a set of latent features from the 𝐸 -𝑡 Maps and 𝐸 -𝑡 -𝑑𝑡 Cubes that can be used to isolate transients and encode specific spectral properties. An autoencoder is composed of two neural networks, an encoder and a decoder, which work together to learn a compressed representation of the input data. The encoder network takes the input data and maps it to a lower-dimensional representation, often called 'latent space' or 'bottleneck'. The number of neurons in the bottleneck determines the dimensionality of the learned representation. The decoder network then aims to reconstruct the original input from this compressed representation. The decoder is typically a mirrored version of the encoder gradually upsampling the latent space until the output matches the dimensions of the original input. By minimizing the reconstruction error between input and output during training, the model learns a low-dimensional representation of the input. The bottleneck forces the encoder to capture the most important information necessary for accurate reconstruction, effectively compressing the input and learning to extract informative features in an unsupervised manner. Once the autoencoder is trained, the encoder network can be used as a standalone feature extractor to obtain a compressed representation of the input data, which can be used for downstream tasks such as clustering or anomaly detection. As opposed to PCA, which is a linear technique that works well for linearly correlated data but fails to capture complex non-linear relationships, an autoencoder is able to learn complex non-linear relationships. We design two different autoencoders to process the 𝐸 -𝑡 Maps and 𝐸 -𝑡 -𝑑𝑡 Cubes.\",\n \"pages\": [\n 7\n ]\n },\n {\n \"title\": \"3.3.1 Convolutional Autoencoder\",\n \"content\": \"In a convolutional autoencoder (Masci et al. 2011), both the encoder and decoder network consist of convolutional layers (LeCun et al. 1998), which perform convolutions over the input using a filter. These filters are small matrix kernels with learnable weights that slide across the input, allowing the network to capture high-level features while preserving important spatial hierarchies and relationships, which is why they are often used for image-like data. This makes this architecture particularly well-suited to recognize spatial patterns such as dips or flares in our 𝐸 -𝑡 Maps. To gradually reduce the dimension of the input while it is being passed through the encoder network, we use stride convolution layers (Simonyan & Zisserman 2014) with a stride value of 2 for downsampling. This means that the learnable filter jumps two pixels at a time as it slides over the input. The output of the convolutional layers is a feature map, which is then flattened to a feature vector and passed through a series of fully connected layers, where every neuron in the previous layer is connected to every neuron in the next layer. These fully connected layers are responsible for mapping the learned features to a lower-dimensional latent representation in the bottleneck and perform non-linear transformations while downsampling through the use of non-linear activation functions. The final latent space has 𝑛 𝑎𝑒 = 12 elements, representing the most essential features of the input data, which can now be used for further downstream tasks. Figure 6 shows a diagram of the encoder part of the model and Table 2 summarizes its architecture.\",\n \"pages\": [\n 7\n ]\n },\n {\n \"title\": \"3.3.2 Fully Connected Autoencoder\",\n \"content\": \"Our 𝐸 -𝑡 -𝑑𝑡 Cubes introduce an additional dimension resulting in sparse 3D input data. Convolutional layers assume regular grid-like data, making them less effective for handling sparse data. Moreover, very expensive 3D convolutional operations would substantially increase complexity of the model. Therefore, we use a simple fully connected autoencoder for the 𝐸 -𝑡 -𝑑𝑡 Cubes. Its encoder network consists of a series of fully connected layers, which gradually map the original input data to a latent space with 𝑛 𝑎𝑒 = 24 elements. Figure 6 shows a diagram of the encoder part of the model and Table 3 summarizes its architecture.\",\n \"pages\": [\n 7\n ]\n },\n {\n \"title\": \"3.3.3 Activation Functions\",\n \"content\": \"Neural networks are able to learn and represent complex non-linear relationships due to the introduction of non-linear activation func- tions within their layers. An activation function is a mathematical function used in a neural network to determine whether a neuron should be activated or not, based on its input. It essentially decides how much of the input signal should pass through the neuron, producing an output that can either be passed to the next layer or used to make predictions. The popular Rectified Linear Unit (ReLU) activation function 𝑅𝑒𝐿𝑈 ( 𝑥 ) = max ( 0 , 𝑥 ) (Nair & Hinton 2010) is simple and computationally efficient. To mitigate any potential encounters of the 'dying the ReLU problem', where neurons become non-responsive during training, we choose an extended version called Leaky ReLU (Maas et al. 2013): where 𝛼 = 0 . 1 is a hyperparameter that defines the slope of the function for negative input values. ReLU sets all negative values in the input to zero, while Leaky ReLU allows a small negative slope for negative inputs, which can help prevent neurons from dying. As for the output layer, we want any values to be mapped to a range between 0 and 1, which is achieved by using the sigmoid activation function:\",\n \"pages\": [\n 7,\n 8\n ]\n },\n {\n \"title\": \"3.3.4 Loss Function and Sparsity Regularization\",\n \"content\": \"In order to encourage the autoencoder to generate reconstructions close to the original inputs, we use the mean squared error ( 𝑀𝑆𝐸 ) as as a measure of the reconstruction quality given by: where 𝑥 𝑖 is the 𝑖 𝑡 ℎ element of the input vector and ˆ 𝑥 𝑖 is the corresponding is reconstructed output. The 𝑀𝑆𝐸 is a straightforward measure of reconstruction error, and its differentiability allows efficient gradient computation for updating model weights via gradient-based optimization. Our neural networks are so called sparse autoencoders (Ng et al. 2011), which promote sparsity in the learned representation, meaning only a small subset of the neurons in the network are active at any given time. Sparse representations are valuable for our work because they help extract highly informative features from the input, while disregarding irrelevant or noisy information. To encourage sparsity in the latent space, we introduce a L1 regularization term in the objective, resulting in the following loss function: where 𝜆 = 0 . 1 is the regularization strength and 𝑤 𝑗 are the individual bottleneck weight values of which there are 𝑛 𝑤 in total. L1 regularization pushes small weights to zero and thus helps the model prioritize the most significant features of the input data, leading to a semantically meaningful latent space.\",\n \"pages\": [\n 8\n ]\n },\n {\n \"title\": \"3.3.5 Training\",\n \"content\": \"Starting with the original dataset with a 𝑚 = 95,473 samples and using a test split of 0 . 1 gives us a training and validation set of length 85,925 and a test set of length 9,548. Further using a validation split of 0 . 2, gives 68,740 samples for training and 17,185 for validation. We run the training process for a maximum of 200 epochs with a batch size of 1,024 samples. The initial learning rate was set to 0 . 01 along with an on plateau learning rate scheduler, which dynamically reduces the learning rate by a factor of 0 . 1 if the validation loss plateaus for longer than 10 epochs. Reducing the learning rate when a plateau is detected can help escape local minima in the loss surface and converge to a more optimal solution in the parameter space. This scheduler is used in combination with the Adaptive Moment Estimation (Adam) optimizer (Kingma & Ba 2014), which is a stochastic gradient descent algorithm combining the benefits of both adaptive learning rates (Duchi et al. 2011) and momentum-based optimization techniques (Sutskever et al. 2013). Finally, we use an early stopping callback to monitor the validation loss. It automatically interrupts the training process if the validation loss does not improve for 25 epochs and restores the weights of the model to the best observed weights during training. The training process for both autoencoder models is shown in Appendix B. Once the autoencoder is trained, we can use the encoder to transform the original dataset X to the feature vector space Xae of size ( 𝑚 , 𝑛 𝑎𝑒 ) with a reduced dimensionality of 𝑛 𝑎𝑒 features.\",\n \"pages\": [\n 8\n ]\n },\n {\n \"title\": \"3.4 Dimensionality Reduction\",\n \"content\": \"Using t-SNE (Maaten & Hinton 2008), short for t-Distributed Stochastic Neighbor Embedding, we create two-dimensional embeddings of the informative features previously extracted from the event file representations using PCA or sparse autoencoders. The t-SNE algorithm is a method used to map the input data onto a low-dimensional embedding space, and is particularly useful for the visualization of clusters and patterns in high-dimensional datasets. Each high-dimensional sample is transformed into a low-dimensional embedding in such a way that similar object are nearby points, while dissimilar objects are distant points in the embedding space. Essentially, it aims to capture the local structure of the data by preserving the pairwise similarities between objects while mapping them to a lower-dimensional embedding space.\",\n \"pages\": [\n 8\n ]\n },\n {\n \"title\": \"3.4.1 Algorithm\",\n \"content\": \"We use our informative features, X if = Xpc or X if = Xae , as input to the t-SNE algorithm to reduce the data to a two-dimensional embedding, denoted as Z . First, t-SNE creates a probability distribution 𝑃 for pairs of high-dimensional data points in X if , assigning higher probabilities to similar pairs and lower probabilities to dissimilar ones. This is done by modeling pairwise similarities using a Gaussian kernel with a specific perplexity parameter, which controls the effective number of neighbors considered for each point. Next, t-SNE defines a similar probability distribution 𝑄 for the pairwise similarities in the low-dimensional space Z , modeled using a Student's t-distribution. The goal of t-SNE is to minimize the difference between 𝑃 and 𝑄 using gradient descent, with the Kullback-Leibler (KL) divergence (Kullback & Leibler 1951) as the cost function: where 𝑃 𝑖 𝑗 and 𝑄 𝑖 𝑗 represent pairwise similarities in the high- and low-dimensional spaces, respectively. The algorithm iteratively adjusts the low-dimensional embedding Z to minimize the KL divergence, often requiring hundreds to thousands of iterations for convergence. The result of this optimization is a two-dimensional representation Z of size ( 𝑚, 2 ) , where similar points in the high-dimensional space are clustered closely together.\",\n \"pages\": [\n 8,\n 9\n ]\n },\n {\n \"title\": \"3.4.2 Hyperparameter Optimization\",\n \"content\": \"The t-SNE algorithm has a number of important hyperparameters to be tuned. The two most important parameters are the perplexity and the learning_rate . The perplexity parameter controls the balance between capturing the local versus global structure in the data, while the learning_rate controls the step size at each iteration of the optimization process. The n_iter parameter is the number of iterations. To ensure reproducibility, we set a fixed random_state . Our t-SNE hyperparameter optimization approach is detailed in Appendix C. A summary of the final t-SNE hyperparameters is provided in Table 4.\",\n \"pages\": [\n 9\n ]\n },\n {\n \"title\": \"3.5 Clustering\",\n \"content\": \"The next step is the identification of individual clusters in the embedding space using DBSCAN (Hartigan & Wong 1979), short for Density-Based Spatial Clustering of Applications with Noise. Unlike traditional clustering algorithms such as k-means, DBSCAN does not require the number of clusters to be specified, as it identifies dense regions in the data space based on a density criterion.\",\n \"pages\": [\n 9\n ]\n },\n {\n \"title\": \"3.5.1 Algorithm\",\n \"content\": \"We use our t-SNE embedding space Z as input to the DBSCAN algorithm, which segments the embedding space into multiple clusters. The DBSCAN algorithm has two main hyperparameters. The eps parameter defines the radius of the neighborhood surrounding each point in the dataset, while the minPts parameter specifies the minimum number of points required within this neighborhood for a data point to be classified as a core point. A border point is defined as a point that is in the vicinity of at least one core point but has fewer than minPts within its neighborhood. All other points are considered to be noise points. Clusters are then created from the aggregation of core points and their associated border points, with noise points being categorized as outliers. Figure 7 visualizes the clustering method.\",\n \"pages\": [\n 9\n ]\n },\n {\n \"title\": \"3.5.2 Hyperparameter Optimization\",\n \"content\": \"Our DBSCAN hyperparameter optimization approach is detailed in Appendix C. A summary of the final t-SNE hyperparameters is provided in Table 5.\",\n \"pages\": [\n 9\n ]\n },\n {\n \"title\": \"3.6 Previously Reported Transients\",\n \"content\": \"We highlight the embeddings of previously reported bona-fide transients, listed in Table 6, in our low-dimensional representation space to identify transient-dominant clusters. The flares include extragalactic FXTs reported by Jonker et al. (2013), Glennie et al. (2015), Yang et al. (2019), Lin et al. (2021), Lin et al. (2022), Quirola-Vásquez et al. (2022) and a set of stellar flares found in the dataset by manual inspection. The dips include the extragalactic planet candidate in M51 reported by Di Stefano et al. (2021), the ultraluminous X-ray source (ULX) 2E 1402 . 4+5440 in NGC 5457 (Colbert & Ptak 2002; Swartz et al. 2004) and the well-studied eclipsing and bursting lowmass X-ray binary (LMXB) EXO 0748 -676 (Parmar et al. 1986; D'Aì et al. 2014). These transients occupy well-isolated clusters. Exploring transient-dominant clusters and performing nearest-neighbor searches around known transients allows us to find new transients.\",\n \"pages\": [\n 9\n ]\n },\n {\n \"title\": \"3.7 Candidate Selection\",\n \"content\": \"Newtransients are identified in embedding clusters containing previously reported transients. For well-isolated clusters containing known discovered transients, we use the entire cluster to define new transient candidates. The well-isolated transient-dominant clusters used for candidate selection are listed in Appendix E. However, in a few cases known discovered transients reside within larger poorly separated clusters. Selecting the entire cluster would result in a high number of false positives. To address this, we instead use the k-nearest neighbors ( kNN ) algorithm (Cover & Hart 1967), identifying the 50 nearest neighbors for each known transient residing in a poorly separated cluster to define additional transient candidates.\",\n \"pages\": [\n 9,\n 10\n ]\n },\n {\n \"title\": \"3.8 Cross Matching\",\n \"content\": \"We use an existing cross-match table (Green et al. 2023) between CSC2.1 and five other catalogs - Gaia DR3 (Gaia Collaboration et al. 2021), DESI Legacy Survey DR10 (Dey et al. 2019), PanSTARRS-1 (Chamberset al. 2016), 2MASS (Skrutskie et al. 2006), and the SDSS DR17 catalog - to complement the X-ray properties derived from the CSC with additional multi-wavelength observations. This includes catalog identifiers, positions, magnitudes, source type classifications and other columns. We cross-matched our transient candidates with the SIMBAD database (Wenger et al. 2000) by associating each candidate with the nearest SIMBAD object, provided the object is located within a 5 arcsec radius of the candidate's coordinates listed in the CSC. The multi-wavelength observations of the transient candidates provide valuable information for their characterization and classification.\",\n \"pages\": [\n 10\n ]\n },\n {\n \"title\": \"4 RESULTS AND DISCUSSION\",\n \"content\": \"We now present the results of applying the methods in § 3 to the set of representations of X-ray event files in the dataset from § 2.\",\n \"pages\": [\n 10\n ]\n },\n {\n \"title\": \"4.1 Representation Embedding Space and Clusters\",\n \"content\": \"Figure 8 shows the t-SNE embedding space for the 3D-PCA and 3DAE cases color-coded by the hardness ratio 𝐻𝑅 hs . The embedding space for the other two cases, 2D-PCA and 2D-AE , are shown in Appendix D. The observed hardness ratio gradients in all embedding spaces indicate that the learned representations effectively encode spectral information, in particular at the level of individual clusters, allowing for the identification of X-ray sources with specific spectral signatures. For the 2D-PCA and 2D-AE cases, these gradients are more uniform across the embedding space, because the temporal and spectral information of event files are captured by one axis each in the 𝐸 -𝑡 Maps. Moreover, some clusters consist exclusively of soft or hard sources, demonstrating that our representations can be leveraged not only to identify transients but also to find analogs to sources with specific spectral characteristics. Figure 9 shows the 3D-PCA and 3D-AE embedding spaces, now color-coded by the variability index 𝐼 𝑏 index with the other two cases shown in Appendix D. The learned embeddings also encode the temporal behavior of the sources, with some clusters being dominated by X-ray detections with significant variability, including transient behavior. To demonstrate this, we also highlight the embeddings of the bona-fide flares and dips listed in Table 6. Note that these occupy very well-defined clusters on the edges of the representation space, allowing for queries of analog transient behavior. In the 2D-PCA and 2D-AE cases, transient sources are distributed across multiple small clusters on the edges of the embedding spaces. In contrast, the 3D-PCA and 3D-AE embedding spaces achieve a significantly more compact clustering of bona-fide transients because temporal features in the event files are given a higher importance by the introduction of an additional time-related axis in the 𝐸 -𝑡 -𝑑𝑡 Cubes. Figure 10 shows the clusters identified by the DBSCAN algorithm in the 3D-PCA and 3D-AE cases. The clusters for the other two cases, 2D-PCA and 2D-AE , are shown in Appendix D. The largest cluster in all cases (Cluster 1) corresponds to observations that are not 'anomalous', for example non-variable sources or noisy detections in the low-count regime. We also see multiple smaller clusters on the edges of the embedding space clearly separated from this main cluster. Of special interest are clusters that contain known discovered transients, as these likely host other interesting transients that have not yet been discovered. Some of the edge clusters group observations with similar temporal and spectral behavior. For example, Cluster 4 in the 3D-PCA case only contains flares with high hardness ratios. Other clusters instead group observations primarily by similar temporal behavior, but then show a within-cluster grouping of similar spectral behaviors. For example, Cluster 4 in the 3D-AE case contains many dipping sources, but show a hardness ratio gradient within the cluster. When comparing the results of different feature extraction methods, we observe that in the 3D-AE embedding space, nearly all previously identified extragalactic FXTs live within a single, wellisolated cluster (Cluster 8). In contrast, the 3D-PCA embedding space distributes these extragalactic FXTs across multiple clusters. All of these points underline the effectiveness of our method and that the created representation space is highly informative.\",\n \"pages\": [\n 10,\n 13\n ]\n },\n {\n \"title\": \"4.2 Catalog of X-ray Flare and Dip Candidates\",\n \"content\": \"We identify new transient candidates within clusters that are occupied by previously reported transients and by conducting nearestneighbor searches around these known transients. We compile these in a catalog of X-ray transient candidates, which includes both flares and dips. Table E1 lists the selected clusters used to define the new flare and dip candidates in addition to the 50 nearest neighbors of each bona-fide transient. From each selected cluster, we only include X-ray detections for which the variability index 𝐼 𝑏 var ≥ 5. This threshold corresponds to detections for which the Gregory-Loredo algorithm yields a confidence in the variability of at least 90%, and allows us to discard flare and dip-like behaviors that are not statistically significant. We also manually exclude a small fraction of false positives identified by visual inspection of the lightcurves for both flare and dip candidates. The resulting catalog contains a total of 3539 detections, and the columns included are described in Appendix F. Table 7 shows the first 5 samples in our catalog for a subset of selected columns. Figure 11 shows a number of example lightcurves of the dips and flares in our catalog. The dip selection shows dips from LMXBs, a low-mass X-ray binary (HMXB), an ULX, an eclipsing binary, a cataclysmic binary, and a quasar. The flare selection shows flares from an eruptive variable, a pulsar, an AGN, a HMXB, a cataclysmic variables and young stars. We also find a number of pulsating or quasi-periodic lightcurves from pulsars, magnetic cataclysmic variables and SGRs. Figure 12 shows the distribution of SIMBAD object types in our transient catalog. About 25% of the transient candidates do not have a SIMBAD match, making them particularly interesting sources for new transient discoveries. Our dip candidates include 6 Chandra observations with prominent dips from the known source CXOGlb J002400.9 -720453 in the globular cluster NGC 104 (47 Tuc). The catalog identifiers for these are CATALOG_ID: 2737_139, 16527_79, 15747_79, 16529_79, 15748_79, 16528_14 . Our flare candidates include a newly discovered extragalactic FXT, which is characterized and discussed in detail in § 4.3. Its catalog identifier is CATALOG_ID: 23022_122 . We recommend using our catalog to identify a diverse range of flares and dips. While this work is primarily motivated by the discovery of new extragalactic transients, we intentionally did not exclude galactic stellar flares to enable systematic follow-up studies to study flare incidence rates and the rotational evolution of stars. Users interested exclusively in extragalactic transients can filter out galactic sources using metadata from the CSC and the cross-match columns in the catalog.\",\n \"pages\": [\n 13\n ]\n },\n {\n \"title\": \"4.3 XRT 200515: A New Extragalactic Fast X-ray Transient\",\n \"content\": \"Among the flare candidates in our catalog, we discovered an intriguing new extragalactic Chandra FXT in an observation of the supernova remnant SNR 0509 -67.5 in the LMC on May 15, 2020 (Guest et al. 2022). What made this transient stand out from thousands of other flares discovered in this work is the unique temporal variability in its lightcurve, which exhibits no detectable pre-flare X-ray emission, a sharp rise of at least 4 orders of magnitude in the count rate to peak intensity followed by a sharp fall, all in a matter of a < 10 s, down to ∼ 800 s long oscillating tail. There is also notable spectral variability during the flare, characterized by an initially hard spectrum at the peak, followed by spectral softening in the tail. The combination of these temporal and spectral properties establishes this transient as the first of its kind within the sample of discovered Chandra FXTs. We designate this newly discovered FXT as XRT 200515 and present a detailed study and discussion of its potential origins.\",\n \"pages\": [\n 14\n ]\n },\n {\n \"title\": \"4.3.1 X-Ray Detection by Chandra\",\n \"content\": \"The transient XRT 200515 was detected in Chandra ObsID 23022. The target of the observation was the supernova remnant SNR 0509 -67.5 in the LMC, which is shown in Figure 13 alongside the newly discovered FXT event. Table 8 summarizes the properties of XRT 200515 and its associated Chandra source 2CXO J051117.2 -672556 in ObdID 23022. The transient was captured by the ACIS camera in the S4 chip, and is located significantly off-axis in this observation, at an angular distance of 11.75 arcmin from the aimpoint in the S3 chip. This leads to an elongated and relatively large PSF, which, in this case, is advantageous as it substantially reduces photon pile-up in the initial spike, by spreading the counts over many pixels. We processed the data of Chandra observation ObsID 23022 with the Chandra Interactive Analysis of Observations (/c.pc/i.pc/a.pc/o.pc) version 4.15 (Fruscione et al. 2006), with calibration data base version 4.9.8. In particular, we created a new level-2 event file with the /c.pc/i.pc/a.pc/o.pc task chandra_repro and filter it in energy and time with dmcopy . We obtained the sky position in Table 8 using the /c.pc/i.pc/a.pc/o.pc tool wavdetect . To reduce background noise and improve the determination of the source centroid, we applied wavdetect on an image filtered to include only the time interval from the beginning of the flare ( 𝑡 0 ) until a time 𝑡 0 + 920 s. The 90% uncertainty radius of 2.0 arcsec is the combination of the uncertainty in the source centroid position reported by wavdetect , and the absolute astrometry uncertainty in a typical ACIS observation for off-axis sources 3 . The field was previously covered by four other Chandra observations (ObsIDs 776, 7635, 8554, and 23023) with no source detections at the location of 2CXO J051117.2 -672556. We estimated modelindependent upper limits to the source flux and luminosity with /c.pc/i.pc/a.pc/o.pc tool srcflux . In the pre-flare part of ObsID 23022, we obtained a 90% confidence limit of 𝐿 X < 1 . 0 × 10 34 erg / s in the 0.3-7 keV band at the LMC distance of 50 kpc. Stacking the data from all the ObsIDs with non-detections, including the pre-flare part of ObsID 23022, results in a total observed exposure of approximately ∼ 150 ks, and yields a 90% confidence upper limit on the X-ray luminosity is 𝐿 X < 3 × 10 33 erg / s.\",\n \"pages\": [\n 14\n ]\n },\n {\n \"title\": \"4.3.2 X-ray Temporal Analysis\",\n \"content\": \"We used the /c.pc/i.pc/a.pc/o.pc tool dmextract to extract background-subtracted lightcurves in several energy bands, from the reprocessed event file of Chandra ObsID 23022. We defined an elliptical source extraction region, with semi-minor and semi-major axes of 15 arcsec and 20 arcsec (matching the point-source PSF at the source location); the local background region was chosen in the same ACIS chip, with an area approximately eight times larger. Figure 14 shows the 0.3-7 keV background-subtracted lightcurve of XRT 200515 with a time resolution of 20 s. The lightcurve is consistent with no source detection at the location of the transient, before the start of the flare at around 23.5 ks into the observation. The few pre-flare counts are consistent with background noise. The lightcurve exhibits a strong initial spike with a sharp rise of at least 4 orders of magnitude in < 10 s, containing 44 out of all ∼ 180 flare counts. This initial burst is followed by a sudden drop to a ∼ 800 s long pulsating and decaying tail. We estimate a 𝑇 90 ∼ 580-740 s for the photons observed in the 0.3-7 keV band 4 , depending on the definition of total flare counts. Figure 15 shows the lightcurve of XRT 200515 at a resolution matching the ACIS frame time of 3.2 s, the hardness ratio, and the energy evolution for the time interval 𝑡 0 + 920 s. The lightcurve exhibits a spike in the count rate across only 3 bins (with a total of 4, 31 and 9 counts, respectively), hence the burst duration of < 10 s. The rise and fall times of the burst are both between 3.2 s and 6.4 s. The maximum count rate at the Chandra frame time resolution is ∼ 9.7 counts/s, acting as the lower bound for the peak count rate of the burst. Those counts are spatially spread over a PSF area of ∼ 3000 pixels; therefore, pile-up is not an issue. We evaluated the hardness ratio evolution during the flare with the Bayesian estimation method BEHR (Park et al. 2006). Here, the hardness ratio is defined as: where 𝑠 is the number of soft photons (0.3-1.2 keV), 𝑚 is the number of medium photons (1.2-2 keV), and ℎ is the number of hard photons (2-7 keV) in each bin. We also track the running average of the photon energies during the flare with a moving window of ± 10 counts. The hardness ratio and energy evolution indicate spectral softening during the flare, with the hardness ratio starting at 1 during the hard burst peak and decreasing to a range of 0.4 to 0.6 in the tail, highlighting the notable spectral variability of XRT 200515.\",\n \"pages\": [\n 14,\n 15\n ]\n },\n {\n \"title\": \"4.3.3 X-ray Spectral Analysis\",\n \"content\": \"We used the /c.pc/i.pc/a.pc/o.pc tool specextract to extract the spectrum and the associated response and ancillary response files from the reprocessed event file of Chandra ObsID 23022. We used the same source and background extraction regions defined for the lightcurve extraction. To improve the signal-to-noise ratio of the source, we extracted the spectrum only from the time interval 𝑡 0 + 920 s. We binned the spectrum to a minimum of 1 count per bin with the grppha task within the /f.pc/t.pc/o.pc/o.pc/l.pc/s.pc package suite (Blackburn 1995) from NASA's High Energy Astrophysics Science Archive Research Center (HEASARC) 5 . For all spectral modelling and flux estimates, we used the /x.pc/s.pc/p.pc/e.pc/c.pc software version 12.13.0 (Arnaud 1996). With only 179 net counts, we are unable to fit complex spectral models; thus, we limit our analysis to the simplest one-component models representative of opposite scenarios: a power law ( powerlaw ) and a blackbody model ( bbody ), both modified by photo-electric absorption ( tbabs ). In both cases, we adopted the Tuebingen-Boulder absorption model with Wilms abundances (Wilms et al. 2000). We minimized the Cash statistic (Cash 1979), as we do not have enough counts for 𝜒 2 fitting. The best-fitting power-law model (Table 9 and Figure 16) has a photon index of Γ = 0 . 5 ± 0 . 3. The fit statistics yield a null hypothesis probability of 3 . 5 × 10 -3 , with a Cstat value of 132.7 for 137 degrees of freedom. For the blackbody model, the best-fitting temperature is 𝑘𝑇 bb = 1 . 8 ± 0 . 3 keV (Table 9). The fit statistics yield a null hypothesis probability of 1 . 2 × 10 -2 , with a Cstat value of 129.6 for 137 degrees of freedom. The reason this blackbody spectrum may appear hard in the Chandra band, resembling a Γ ∼ 0 . 5 power law, is that at a temperature of 𝑘𝑇 bb ∼ 2 keV, the ACIS detector samples only the peak and the Rayleigh-Jeans (rising) portion of the blackbody emission. We can use either model to determine an average conversion between the count rate and luminosity. This will then enable us to estimate the peak luminosity in the initial spike, for which we have previously estimated a peak count rate of ≳ 10 counts/s. The best-fitting power law model implies a peak flux of 𝐹 p ≳ 5 . 6 × 10 -10 erg/s/cm 2 , a total flare fluence of 𝐸 f ≳ 1 . 1 × 10 -8 erg/cm 2 , and a peak unabsorbed 0.3-10 keV luminosity of 𝐿 X ≳ 1 . 7 × 10 38 erg/s at the LMC distance of 50 kpc. For the best-fitting blackbody model, the peak flux and flare fluence would be 𝐹 p ≳ 4 . 0 × 10 -10 erg/s/cm 2 and 𝐸 f ≳ 0 . 8 × 10 -8 erg/cm 2 respectively. The peak unabsorbed 0.3-10 keV luminosity would be 𝐿 X ≳ 1 . 2 × 10 38 erg/s and the peak bolometric luminosity would be 𝐿 bol ≳ 1 . 5 × 10 38 erg/s. These values should be considered conservative lower limits for two reasons: (i) the peak count rate provides only a lower bound estimate, as it is constrained by the Chandra frame time resolution of the observations, potentially underestimating the true peak count rate; and (ii) the conversion factor applied is derived from the average spectrum over the entire flare, even though the spectrum of the initial spike is significantly harder compared to the tail, as shown in Figure 15. We searched for potential detections of XRT 200515 by other highenergy facilities. However, no significant X-ray or 𝛾 -ray events in the field around the X-ray source coordinates and flare start time 𝑡 0 reported in Table 8 were detected by the Fermi Gamma-ray Space Telescope ( Fermi ), the Burst Alert Telescope ( BAT ) on the Neil Gehrels Swift Observatory ( Swift ), the International Gamma-Ray Astrophysics Laboratory ( INTEGRAL ), or the Monitor of All-sky X-ray Image MAXI . LIGO was not operational during the time of the FXT, hence no gravitational wave signal could have been detected if the origin of XRT 200515 was a compact object merger.\",\n \"pages\": [\n 15,\n 16\n ]\n },\n {\n \"title\": \"4.3.5 Optical Counterpart Search\",\n \"content\": \"We used the X-ray source coordinates reported in Table 8 to search for optical and infrared counterparts to XRT 200515. The field of XRT 200515 was covered by the Survey of Magellanic Stellar History ( SMASH ) (Nidever et al. 2017), a deep optical survey in the ugriz bands with the Dark Energy Camera (DECam) mounted on the Víctor M. Blanco Telescope at the Cerro Tololo Inter-American Observatory (CTIO) in Chile. We used the Astro Data Lab Jupyter Notebook server (Nikutta et al. 2020; Juneau et al. 2021) to access and visualize the SMASH catalog 6 . Figure 17 shows a color image of the field created from the deepest available stacked images in the u , g and i bands; the 5 𝜎 detection limits in these bands are 23.9 mag, 24.8 mag and 24.2 mag, respectively. The images were taken on December 7, 2015 with exposure times of 1,179 s, 981 s and 1,179 s respectively. The astrometry of the SMASH images is calibrated on the Gaia DR3 reference frame, thus their positional uncertainty is negligible compared to the X-ray source position uncertainty. Within the Chandra position error circle in Figure 17, there is no obvious optical counterpart that stands out in brightness or color from the surrounding stellar population. We performed relative photometry on the sources inside the error circle, comparing them to several nearby sources with known positions and brightnesses listed in the Gaia DR3 catalog. We used the SMASH g band as the closest approximation to Gaia's G band. We estimate the brightest optical source within the error circle to have a Vega magnitude of 𝑔 = 22 . 7 ± 0 . 1 mag, corresponding to an absolute magnitude of 𝑀 𝑔 ≈ 4 . 2, assuming it is in the LMC. Additionally, three other point-like sources are detected with 𝑔 band magnitudes in the range of 23-24 mag. All four sources appear pointlike, consistent with the seeing conditions of the SMASH survey, with no evidence of any spatially extended background galaxies. The three brightest stars visible in Figure 17 within ∼ 12 arcsec of the Chandra source are solar-mass stars on the red giant branch, indicative of an old stellar population. The lack of bright optical counterparts and the short burst duration of < 10 s rules out a stellar flare from a foreground Galactic low-mass star (Güdel 2004; Reale 2007; Reale & Landi 2012; Pye et al. 2015; Kuznetsov & Kolotkov 2021). A flare from a Be/X-ray binary or any other HMXB in the LMC is also excluded by the lack of a bright optical counterpart (Ducci et al. 2019, 2022). The temporal and spectral properties of XRT 200515, combined with the absence of an optical counterpart, suggests three possibilities: (i) a relativistic jet phenomenon, such as a 𝛾 -ray burst (GRB); (ii) a rapid, high-energy process linked to extreme magnetic fields, such as a giant magnetar flare (GMF); or (iii) a thermonuclear Type I X-ray burst caused by surface nuclear burning on a neutron star.\",\n \"pages\": [\n 16\n ]\n },\n {\n \"title\": \"4.3.6 Gamma Ray Burst from a Compact Object Merger?\",\n \"content\": \"Evidence in favor or against the association of at least some Chandra FXTs with low-luminosity long-GRBs or off-axis short-GRBs (see Berger 2014 for a review), at moderate or high redshifts, is extensively discussed in Quirola-Vásquez et al. (2022), Quirola-Vásquez et al. (2023), and Wichern et al. (2024). A detailed re-investigation of this issue is beyond the scope of this work. Here, we simply point out that XRT 200515, like the other Chandra FXTs in the literature, does not have any 𝛾 -ray detection. On the other hand, XRT 200515 has a significantly harder spectrum ( Γ = 0 . 5 ± 0 . 3) in the Chandra band than the rest of the FXT sample, all of which have photon indices of Γ > 1 (Jonker et al. 2013; Glennie et al. 2015; Bauer et al. 2017; Xue et al. 2019; Lin et al. 2022; Quirola-Vásquez et al. 2022; Quirola-Vásquez et al. 2023; Eappachen et al. 2023). A photon index of Γ ∼ 0 . 5 below 10 keV is indeed expected and observed from both core-collapse GRBs and compact-merger GRBs (Ghirlanda et al. 2009; Bromberg et al. 2013; Oganesyan et al. 2018; Ravasio et al. 2019; Toffano et al. 2021). This might support the association of the initial spike of XRT 200515 with a GRB. However, the presence and properties of the ∼ 800 s tail (candidate GRB afterglow) is puzzling. The 𝑇 90 ∼ 580-740 s value for XRT 200515 is significantly shorter than in most other Chandra FXTs (Quirola-Vásquez et al. 2022; Lin et al. 2022; Quirola-Vásquez et al. 2023), which have 𝑇 90 values on the order of several ks and are already pushing the limit for a GRB afterglow detection (Wichern et al. 2024). Moreover, XRT 200515's initial burst duration ( < 10 s), its short rise and fall times (3.2-6.4 s), and the lack of a peak plateau are inconsistent with the lightcurves of Chandra FXTs interpreted as magnetar-powered GRBs as the aftermath of a binary neutron star merger, such as CDF-S XT1 (Bauer et al. 2017), CDF-S XT2 (Xue et al. 2019) and the sample in Lin et al. (2022). Finally, the lack of any optical evidence for a host galaxy is another element disfavoring the high-redshift GRB interpretation. 4.3.7 Giant Magnetar Flare from a Soft Gamma Repeater? Based on its temporal and spectral variability, it is tempting to interpret XRT 200515 as a rare GMF from a SGR (Mereghetti 2008; Turolla et al. 2015) in the LMC or behind it, which can easily explain the burst's strong increase of at least 4 orders of magnitude in < 10 s (Coti Zelati et al. 2018). Similar to XRT 200515, GMFs are characterized by a short and hard initial spike and a longer and softer, pulsating tail. GMFs are extremely rare, with only a select few ever discovered. Well-studied examples are SGR 0526 -66 in the LMC (Mazets et al. 1979), and the Galactic sources SGR 1900 + 14 (Hurley et al. 1999) and SGR 1806 -20 (Hurley et al. 2005; Palmer et al. 2005; Israel et al. 2005). More recently, GMFs have been identified in M 31 (Mazets et al. 2008a), NGC 253 (Fermi-LAT Collaboration et al. 2021; Svinkin et al. 2021; Roberts et al. 2021; Trigg et al. 2024) and M82 (Mereghetti et al. 2024). All of these have been observed by high time resolution instruments in the hard X-rays and soft 𝛾 -rays with luminosities above 10 46 erg/s for a fraction of a second in the initial spike. The tails of GMFs are often modulated by magnetar spin periods of 2-12 s, leading to quasi-periodic oscillations (QPOs). For XRT 200515, there is no hard X-ray or 𝛾 -ray detection, despite the LMC direction being in good visibility for most of the previously mentioned high-energy facilities. We were unable to identify any significant periodicities in the tail of XRT 200515 through periodogram analysis, which is unsurprising given the low time resolution of Chandra observations. No X-ray activity has been observed by Chandra or other X-ray telescopes in the years before or after XRT 200515, which may be because SGRs are very faint when they are not bursting. The strongest argument against a magnetar in the LMC as the origin of XRT 200515 is that magnetars are short-lived objects ( ≲ 10 5 yr) associated to young stellar populations (Olausen & Kaspi 2014; Nakano et al. 2015; Mondal 2021). Even allowing for the persistence of magnetar-like activity in ordinary radio pulsars as old as ∼ 10 7 yr (Rea et al. 2010), this scenario is still inconsistent with the old stellar population (several Gyr) in the LMC field shown in Figure 17. The nearest star-forming regions in the LMC are ∼ 10 arcmin ( ∼ 150 pc) away. If (in a very contrived scenario), we assume that XRT 200515 is powered by a young neutron star ejected from one of those regions, we estimate a characteristic time of 1 Myr to travel that distance at a speed of 150 km/s. Therefore, if XRT 200515 is a GMF, it must be located behind the LMC, in a low-redshift galaxy (Hurley et al. 2005; Tanvir et al. 2005). Since GMFs have been observed only a few times and never at soft X-ray energies, their properties in the soft X-ray band detectable by Chandra remain largely unexplored. XRT 200515 could indeed be the first GMF detected at soft X-ray energies. Distinguishing distant short GRBs from GMFs has historically been difficult and there are multiple studies suggesting that a subset of short GRBs are actually extragalactic GMFs (Hurley et al. 2005; Palmer et al. 2005; Tanvir et al. 2005; Ofek et al. 2006; Mazets et al. 2008b; Hurley 2011; Yang et al. 2020; Svinkin et al. 2021; Negro & Burns 2023). Just as for the distant GRB interpretation, the non-detection of any optical counterpart remains puzzling for a distant GMF scenario, unless we are dealing with a very distant and exceptionally luminous GMF.\",\n \"pages\": [\n 17\n ]\n },\n {\n \"title\": \"4.3.8 Thermonuclear X-ray Burst from a quiet LMXB in the LMC?\",\n \"content\": \"If XRT 200515 is in the LMC, a peak luminosity near the Eddington luminosity 𝐿 Edd ∼ 10 38 erg/s and sharp rise time of the flare suggests a Type I X-ray burst interpretation, which is a thermonuclear explosion on the surface of a weakly magnetized, accreting neutron star (Lewin et al. 1993; Strohmayer & Bildsten 2003; Gal-\",\n \"pages\": [\n 17\n ]\n },\n {\n \"title\": \"loway et al. 2008, 2020; Galloway & Keek 2021; Alizai et al. 2023).\",\n \"content\": \"The old stellar population in the field of XRT 200515 is consistent with the presence of neutron star LMXBs. Following the definition of burst timescale 𝜏 = 𝐸 f / 𝐹 p in Galloway et al. (2008), we estimate 𝜏 ∼ 20 s for XRT 200515, which is consistent with Type I X-ray bursts (Galloway & Keek 2021; Alizai et al. 2023). The fitted temperature 𝑘𝑇 bb ∼ 2 keV when the average spectrum is fitted with a simple blackbody, and the softening of the spectrum (temperature decrease) in the tail is also typical of Type I X-ray bursts (Galloway et al. 2008, 2020; Güver et al. 2012). On the other hand, several observed properties of XRT 200515 are unusual for Type I X-ray bursts. In particular, most Type I X-ray bursts occur when the persistent luminosity (proportional to the accretion rate) of a LMXB is 𝐿 X > 10 -4 𝐿 Edd (and, in most cases, 𝐿 X > 10 -3 𝐿 Edd ) (Galloway et al. 2008). Instead, in the initial part of ObsID 23022, the upper limit on the X-ray luminosity at the position of XRT 200515 is 𝐿 X < 10 -4 𝐿 Edd , so that the X-ray flux increased by at least 4 orders of magnitudes. On another note, the sharp decline after the initial burst of XRT 200515 would be unusual for Type I X-ray bursts, which typically exhibit a gradual and exponential decay. However, note that most Type I X-ray bursters were observed by the Rossi X-Ray Timing Explorer ( RXTE ) (Jahoda et al. 1996), which has a high time resolution. The low time resolution of Chandra may have obscured such a decay for XRT 200515. Moreover, most Type I bursts tend to repeat every few hours (Galloway et al. 2008); instead, XRT 200515 is the only event detected at that location over a total observed time of ∼ 150 ks. No LMXB has ever been noted at that position before or after the event. The time interval between bursts is related to an index 𝛼 defined as the ratio between the integrated persistent fluence between subsequent bursts and the burst fluence; from a comparison of the energy released by accretion (contributing to the persistent fluence) and by thermonuclear burning (burst fluence), we expect 𝛼 ≳ 40, in agreement with the observations of Type I bursts (Galloway et al. 2008). If we apply the same criterion ( 𝛼 ≳ 40) to the persistent and flare fluences of XRT 200515, we would have to wait > 10 7 s (4 months) to observe another similar event, assuming the persistent flux level upper limit in ObsID 23022 before the transient event. This waiting time extends to at least one year if we assume the persistent flux upper limit derived from the stacked ∼ 150 ks Chandra observations. Only a few one-off bursts from Galactic neutron stars at a very low persistent luminosity ( 𝐿 X ∼ 10 32 -10 33 erg/s) were found by Cornelisse et al. (2002a,b) with estimated recurrence times of tens of years. The vast majority of Type I X-ray bursts are Galactic, due to their lower flux at large distances. Only a handful of extragalactic Type I X-ray bursts are documented, for example in M 31 (Pastor-Marazuela et al. 2020) and the Magellanic Bridge (Haberl et al. 2023). If XRT 200515 is a Type I X-ray burst, it is the first extragalactic Type I X-ray burster in the LMC and represents the tip of the iceberg for a vast population of faint LMXBs in nearby galaxies, too dim to be detected by Chandra or XMM-Newton , but which may occasionally reveal themselves via thermonuclear bursts with a long duty cycle.\",\n \"pages\": [\n 18\n ]\n },\n {\n \"title\": \"4.3.9 Concluding Remarks and Outlook for XRT 200515\",\n \"content\": \"XRT 200515 is a unique and intriguing extragalactic Chandra FXT. The combination of its temporal and spectral properties is unlike any of the other Chandra FXT samples. Based on our analysis, the two most likely scenarios for XRT 200515 are: (i) a distant GMF from a SGR behind the LMC; the first observed in the low X-ray energy band, missed by any other high-energy facilities, or (ii) an unusual Type I X-ray burst from a previously unknown faint LMXB; the first extragalactic X-ray burster in the LMC. Nevertheless, both of these interpretations come with their own unique challenges. XRT 200515 could, in fact, represent an entirely new type of astronomical phenomenon. After all, the primary objective of our work was to use machine learning to find rare, needle-in-the-haystack anomalies hidden within vast astronomical datasets. We invite further detailed studies of XRT 200515 to evaluate our interpretations and explore alternative scenarios, such as potential associations with a fast radio burst (FRB) or a SBO. We highly recommend follow-up multi-band observations at the source coordinates of XRT 200515 to better constrain its nature. Lastly, we note that XRT 200515 and the second transient discovered by Glennie et al. (2015), XRT 120830, have remarkably similar temporal evolutions in their lightcurves (J. Irwin, personal communication, November 2024), however with very different spectral properties ( Γ ∼ 2 . 5 for XRT 120830 versus Γ ∼ 0 . 5 for XRT 200515). We leave a detailed comparative analysis of these transients for future work. Figure 18 shows the 𝐸 -𝑡 Map and 𝐸 -𝑡 -𝑑𝑡 Cube event file representations for XRT 200515. These exhibit high counts at high energies in a narrow time window, which is in line with the hard spectrum and transient nature of XRT 200515.\",\n \"pages\": [\n 18\n ]\n },\n {\n \"title\": \"4.4 Technical Caveats\",\n \"content\": \"The main technical caveat of our approach is related to the representation of event files. While our new event file representations enable a simple, yet powerful representation learning approach to find new and rare X-ray transients, any simplification of raw event files, like the fixed number of time bins we use across all event files, is associated with a loss of information. This could lead to us missing a small amount of transients. To minimize this, we have implemented a rigorous approach to justify the resolution of the event file representations in § 3.1. Moreover, flares, in particular known extragalactic FXTs, cluster notably well in our representation spaces. This is because their distinctive features are less dependent on the temporal binning resolution in the 𝐸 -𝑡 Maps and 𝐸 -𝑡 -𝑑𝑡 Cubes. To improve the effectiveness of dip searches with our proposed method, we suggest using higher resolution event file representations. Nevertheless, our comprehensive transient candidate catalog includes numerous newly identified transients that were previously overlooked by other X-ray transient searches in the Chandra archive. Among these is the remarkable needle-in-the-haystack event XRT 200515 discovered in this work, underscoring the effectiveness of our method. A followup representation learning algorithm will learn informative features from raw and unbinned event files while accounting for the Poisson nature of X-ray observations (Song et al., 2025, in preparation).\",\n \"pages\": [\n 18\n ]\n },\n {\n \"title\": \"5 CONCLUSION\",\n \"content\": \"We have introduced a novel representation learning method, the first of its kind applied to X-ray event files, enabling downstream tasks such as unsupervised classification and anomaly detection in highenergy astrophysics. We have used the learned representation to investigate time-domain properties of sources in the Chandra archive, with a particular emphasis on the discovery of X-ray transients. As a result, we have compiled the identified X-ray flares and dips in a comprehensive catalog of transient candidates. Notably, our method led to the discovery of XRT 200515; a previously unidentified extragalactic FXT with unique temporal and spectral properties, representing a genuine needle-in-the-haystack discovery. Our key results are as follows: XRT200515wasonlydetectedby Chandra , with no identified optical counterparts. We strongly encourage a multi-wavelength search for additional signals from the source associated with XRT 200515 to better understand its origin and nature. Ourworkadvancestime-domainhigh-energy astrophysics by making the Chandra transient candidates catalog publicly available and open-sourcing the representation learning based transient search pipeline 7 . The catalog enables queries to identify new Chandra transients. Future work involves applying the detection pipeline to additional high-energy archives and adapting it to a variety of other scientific datasets, paving the way for further machine learning driven discoveries of rare transients and other scientific anomalies.\",\n \"pages\": [\n 19\n ]\n },\n {\n \"title\": \"ACKNOWLEDGEMENTS\",\n \"content\": \"This research has made use of data obtained from the Chandra Source Catalog, and software provided by the Chandra X-ray Center (CXC) in the /c.pc/i.pc/a.pc/o.pc application package. SD's work was partially funded by the UK government's Turing Scheme and mainly carried out at the Center for Astrophysics | Harvard & Smithsonian as part of the SAO Predoctoral Program, with the support of AstroAI. SD acknowledges hospitality at the Institute of Astronomy at the University of Cambridge and at the Stanford Center for Decoding the Universe (Stanford Data Science) during the later parts of this project. RS's work was performed in part at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-2210452. RS acknowledges support and hospitality at the National Astronomical Observatories of China (NAOC) in Beijing, during part of this project. We thank Edo Berger, Massimiliano De Pasquale, Ken Ebisawa, Duncan Galloway, Jimmy Irwin, Peter Jonker, Daniel Kocevski, Amy Lien, Sandro Meregetthi, Daniel Muthukrishna, Nicola Omodei, Jonathan Quirola-Vásquez, Shivam Raval, Ashley Villar, and Silvia Zane for their fruitful discussions.\",\n \"pages\": [\n 19\n ]\n },\n {\n \"title\": \"DATA AVAILABILITY\",\n \"content\": \"The data used in this paper, composed of X-ray event files and source detection regions, was obtained from the publicly available CSC, using their public interfaces (https://cxc.cfa.harvard.edu/csc/). The catalog of transient candidates and the clustered embedding spaces generated using our unsupervised representation learning method can be accessed in the supplementary material. All intermediate data products, i.e. as 𝐸 -𝑡 Maps, 𝐸 -𝑡 -𝑑𝑡 Cubes, principal components and latent features, feature embeddings and embedding clusters can be produced using the code provided in the GitHub repository https://github.com/StevenDillmann/ml-xraytransients-mnras.\",\n \"pages\": [\n 19\n ]\n },\n {\n \"title\": \"REFERENCES\",\n \"content\": \"Walmsley M., et al., 2022, Monthly Notices of the RAS, 513, 1581 Weisskopf M. C., Tananbaum H. D., Van Speybroeck L. P., O'Dell S. L., 2000, in Truemper J. E., Aschenbach B., eds, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Vol. 4012, X-Ray Optics, Instruments, and Missions III. pp 2-16 ( arXiv:astro-ph/0004127 ), doi:10.1117/12.391545 Wichern H. C. I., Ravasio M. E., Jonker P. G., Quirola-Vásquez J. A., Levan X. C., 2019, Monthly Notices of the RAS, 487, 4721\",\n \"pages\": [\n 21\n ]\n },\n {\n \"title\": \"APPENDIX A: DISTRIBUTION OF EVENT FILE LENGTHS AND DURATIONS\",\n \"content\": \"Figure A1 shows the distribution of the length 𝑁 and duration 𝑇 of event files in the dataset used in this work.\",\n \"pages\": [\n 22\n ]\n },\n {\n \"title\": \"APPENDIX B: AUTOENCODER TRAINING PROCESS\",\n \"content\": \"Figure B1 shows the training process of the autoencoders used in this work.\",\n \"pages\": [\n 22\n ]\n },\n {\n \"title\": \"APPENDIX C: HYPERPARAMETER OPTIMIZATION\",\n \"content\": \"Below, we summarize the optimization strategy for the t-SNE and DBSCAN hyperparameters. For even more details on this approach, please refer to Dillmann & Martínez-Galarza (2023).\",\n \"pages\": [\n 22\n ]\n },\n {\n \"title\": \"C1 t-SNE Hyperparameters\",\n \"content\": \"The choice of the perplexity and learning_rate can have a large impact on the resulting t-SNE embedding space. Ideally, we want the two-dimensional embedding space to effectively capture both energy information (in form of the hardness ratio 𝐻𝑅 ) and variability information (in form of the variability probability 𝑝 var ). That means that event files with similar values for 𝐻𝑅 and 𝑝 var should live close to each other in the final embedding space. We can use this information to define a performance metric for different t-SNE hyperparameter inputs. First, we compute the pairwise distance matrix D Z of size ( 𝑚, 𝑚 ) , where the distance 𝐷 𝑍 𝑖 𝑗 between points 𝑖 and 𝑗 is computed using a Euclidean distance metric. Next, we define the property vector Y , which includes 7 CSC properties (hardness ratios 𝐻𝑅 hm , 𝐻𝑅 hs , 𝐻𝑅 ms and variability probabilities 𝑝 b var , 𝑝 h var , 𝑝 m var , 𝑝 s var ) for each event file and thus each t-SNE point. As a measure of similarity between the labels of different points, we can again compute a pairwise similarity matrix D Y of size ( 𝑚, 𝑚 ) . To compute the similarity distance 𝐷 𝑌 𝑖 𝑗 between sample 𝑖 and 𝑗 , we use the Mahalanobis distance metric (Mahalanobis 1936). Unlike the Euclidean distance metric, the Mahalanobis distance metric accounts for the correlation between different labels by taking into account the covariance structure of the data. Note that our hardness ratios are correlated with each other, and that the same holds for the variability probabilities. Accounting for these correlations provides a more accurate measure of the similarity distance between different samples. Having computed D Z and D Y , we can define a performance metric that allows us to compare the performance of different t-SNE hyperparameters. The smaller the distance 𝐷 𝑍 𝑖 𝑗 between two points 𝑖 and 𝑗 in the t-SNE embedding, the smaller should be difference in their associated labels as measured by the distance 𝐷 𝑌 𝑖 𝑗 . We can thus define a performance metric based on the statistical correlation of D Z and D Y using the Spearman's rank correlation coefficient 𝜌 𝑍𝑌 (Spearman 1904). The higher 𝜌 𝑍𝑌 , the higher is the positive correlation between D Z and D Y and the better the performance of the t-SNE embedding. The hyperparameter space is given by the ranges learning_rate ∈ ( 20 , 200 ) with a step size of 20 and perplexity ∈ ( 10 , 100 ) with a step size of 10. This optimization process is performed using a reduced dataset of 15,353 samples for 2,000 iterations per hyperparameter combination due to computational constraints. While subsampling, the overall structure of the data was preserved by selecting the same distributions between any combinations of hard, medium, soft, variable and non-variable samples. This ensures that the sample set is representative of the original data. We choose the hyperparameter combination that produces the highest value of 𝜌 𝑍𝑌 .\",\n \"pages\": [\n 22,\n 23\n ]\n },\n {\n \"title\": \"C2 DBSCAN Hyperparameters\",\n \"content\": \"Different hyperparameter combinations of eps and minPts can have a large impact on the resulting DBSCAN clusters. We use a combination of the Davies-Bouldin index 𝐷𝐵 (Davies & Bouldin 1979) and Calinski-Harabasz index 𝐶𝐻 (Caliński & Harabasz 1974) as a performance metric to find the optimal DBSCAN hyperparameter inputs. The 𝐷𝐵 index is a measure of the average similarity between each cluster and its most similar cluster, relative to the average distance between points within each cluster. The 𝐷𝐵 index is given by the following formula: where 𝑛 𝑐 is the number of clusters, 𝑊 𝑖 and 𝑊 𝑗 are the within-cluster sum of squares for cluster 𝑖 and 𝑗 , and 𝑑 ( 𝑐 𝑖 , 𝑐 𝑗 ) is the distance between the centroids of clusters 𝑖 and 𝑗 . On the other hand, the 𝐶𝐻 index is based on the concept that good clusters should have high intra-cluster similarity (cohesion) measured by the between-cluster dispersion 𝐵 and low inter-cluster similarity (separation) measured by the within-cluster dispersion 𝑊 . 𝐵 is the sum of the pairwise distances between cluster centroids, and 𝑊 is the sum of the pairwise distances between points within each cluster. The 𝐶𝐻 index is given by the following formula: where the scaling factor 𝑚 -𝑛 𝑐 𝑛 𝑐 -1 accounts for the total number of data points 𝑚 and the number of clusters 𝑛 𝑐 . A lower 𝐷𝐵 index and higher 𝐶𝐻 index indicate that the clustering algorithm is more effective in grouping similar data points together and separating different data points into distinct clusters. We thus define the performance metric 𝜌 𝐷𝐶 as the ratio of the normalized indices 𝐷𝐵 𝑛 = 𝐷𝐵 max ( 𝐷𝐵 ) and 𝐶𝐻 𝑛 = 𝐶𝐻 max ( 𝐶𝐻 ) in the hyperparameter space given by eps ∈ ( 1 . 0 , 3 . 0 ) with a step size of 0 . 1 and minPts ∈ ( 10 , 30 ) with a step size of 1: We choose the hyperparameter combination that produces the highest value of 𝜌 𝐷𝐵𝑆𝐶𝐴𝑁 .\",\n \"pages\": [\n 23\n ]\n },\n {\n \"title\": \"APPENDIX D: EMBEDDINGS\",\n \"content\": \"Figures D1, D2 and D3 show the 2D-PCA and 2D-AE embeddedings.\",\n \"pages\": [\n 23\n ]\n },\n {\n \"title\": \"APPENDIX E: TRANSIENT-DOMINANT EMBEDDING CLUSTERS\",\n \"content\": \"Table E1 lists the transient-dominant clusters in the different embedding spaces used for the selection of transient candidates.\",\n \"pages\": [\n 23\n ]\n },\n {\n \"title\": \"APPENDIX F: CATALOG COLUMNS\",\n \"content\": \"Table F1 shows the of X-ray transient candidate catalog column descriptions. This paper has been typeset from a T E X/L A T E X file prepared by the author.\",\n \"pages\": [\n 23\n ]\n }\n]"}}},{"rowIdx":4981,"cells":{"bibcode":{"kind":"string","value":"2024arXiv241201157B"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/2412.01157.pdf"},"content":{"kind":"string","value":"\nLOCAL VARIATIONS OF THE RADIAL METALLICITY GRADIENT IN A SIMULATED NIHAO-UHD MILKY WAY ANALOGUE AND THEIR IMPLICATIONS FOR (EXTRA-)GALACTIC STUDIES\nS/v.pc/e.pc/n.pc B/u.pc/d.pc/e.pc/r.pc 1 , 2 , ∗ , T/o.pc/b.pc/i.pc/a.pc/s.pc B/u.pc/c.pc/k.pc 3 , 4 , Q/i.pc/a.pc/n.pc-H/u.pc/i.pc C/h.pc/e.pc/n.pc ( 陈 千 惠 ) 1 , 2 , /a.pc/n.pc/d.pc K/a.pc/t.pc/h.pc/r.pc/y.pc/n.pc G/r.pc/a.pc/s.pc/h.pc/a.pc 1 , 2 , ∗\n1 Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT 2611, Australia\n2 ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Australia\n3 Universität Heidelberg, Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, Im Neuenheimer Feld 205, D-69120 Heidelberg, Germany and\n4 Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische Astrophysik, Albert-Ueberle-Strae 2, D-69120 Heidelberg, Germany Version December 3, 2024\nABSTRACT\nRadial metallicity gradients are fundamental to understanding galaxy formation and evolution. In our high-resolution simulation of a NIHAO-UHD Milky Way analogue, we analyze the linearity, scatter, spatial coherence, and age-related variations of metallicity gradients using young stars and gas. While a global linear model generally captures the gradient, it ever so slightly overestimates metallicity in the inner galaxy and underestimates it in the outer regions of our simulated galaxy. Both a quadratic model, showing an initially steeper gradient that smoothly flattens outward, and a piecewise linear model with a break radius at 10 kpc (2.5 effective radii) fit the data equally better. The spread of [Fe/H] of young stars in the simulation increases by tenfold from the innermost to the outer galaxy at a radius of 20 kpc. We find that stars born at similar times along radial spirals drive this spread in the outer galaxy, with a chemical under- and over-enhancement of up to 0.1 dex at leading and trailing regions of such spirals, respectively. This localised chemical variance highlights the need to examine radial and azimuthal selection effects for both Galactic and extragalactic observational studies. The arguably idealised but volume-complete simulations suggest that future studies should not only test linear and piecewise linear gradients, but also non-linear functions such as quadratic ones to test for a smooth gradient rather than one with a break radius. Either finding would help to determine the importance of different enrichment or mixing pathways and thus our understanding of galaxy formation and evolution scenarios.\nSubject headings: Galaxy: structure - Galaxy: abundances - galaxies: structure - galaxies: abundances\n1. INTRODUCTION\nUnderstanding the radial metallicity gradient, defined as the change in heavy element abundance with galactocentric radius, in galaxies provides critical insights into their formation and evolutionary processes, such as inside-out formation, gas accretion, outflows, and radial migration (e.g. Quirk & Tinsley 1973; Tinsley 1980; Lacey & Fall 1985; Wyse & Silk 1989; Kauffmann 1996; Chiappini et al. 1997; Schönrich & Binney 2009; Moran et al. 2012; Bird et al. 2013). The decrease in metallicity with increasing distance from the Galactic centre is well-established both theoretically (Larson 1976; Tinsley 1980; Chiosi 1980) and observationally in the Milky Way (Searle 1971; Janes 1979; Twarog et al. 1997) and other massive spiral galaxies (e.g. Tinsley 1980; Zaritsky et al. 1994; Bresolin et al. 2012). The Milky Way, being the only galaxy where we can resolve millions of stars, provides a unique opportunity to study these gradients and deviations from them in detail. Early evidence by Janes (1979) suggested a linear gradient on the order of d [ Fe / H ]/ d 𝑅 = -0 . 05 ± 0 . 01 dex kpc -1 for the Milky Way which aligns very well with more recent measurements (Anders et al. 2017; Hayden et al. 2015). However, these gradients are accompanied by a significant spread in [Fe/H] of 0 . 1 -0 . 15 dex, as noted by Twarog (1980), which may imply a fine structure of the metallicity gradient (see Genovali et al. 2014).\nWith increasing sample size and measurement precision, the specific shape and characteristics of this gradient remain somewhat unclear (Chiappini 2002). Previous studies have\n∗ Australian Research Council DECRA Fellow\nfor example claimed more intricate non-linear trends, bends, and flattening in the gradient of the Milky Way (e.g. Donor et al. 2020) and other galaxies (e.g. Pilyugin 2003; Sánchez et al. 2014) or even sequences of shapes (Pilyugin et al. 2017; Pilyugin & Tautvaišien˙e 2024), which were fitted with different models (Rosales-Ortega et al. 2011; Bresolin et al. 2012), such as piecewise linear ones (e.g. Sánchez-Menguiano et al. 2016) or non-linear ones (e.g. Scarano & Lépine 2013).\nVariations in the metal distribution, including breaks of the gradient at specific radii, give rise to a plethora of possible physical explanations, such as star formation efficiency variations and localised star formation bursts (Sánchez et al. 2014; Sánchez-Blázquez et al. 2014; Ho et al. 2015), gas accretion and dilution at different rates (Bresolin et al. 2012; Sánchez et al. 2013; Belfiore et al. 2016; Sánchez-Menguiano et al. 2016), gas outflows and feedback (Lilly et al. 2013; Ma et al. 2017a), as well as disk instabilities or local overdensities (Grand et al. 2016; Ho et al. 2017). In particular, Scarano & Lépine (2013) suggested that gradient break radii coincided with the corotation radii of spiral arms.\nRecent advancements in both computations and observations have significantly expanded our capabilities. For example, in terms of observational data in the Milky Way, the Gaia mission (Gaia Collaboration et al. 2016) enables more detailed studies of these gradients. New suites of large-scale simulations now allow us to gain insights into radial metallicity gradients across a range of simulated galaxies, including Milky Way analogues. This presents opportunities to revisit outstanding challenges of the detailed shape of the radial metallicity gradient. For instance, Hogg et al. (2019) created an extensive metallicity map of the Milky Way using APOGEE and Gaia\ndata, while Poggio et al. (2022) mapped young stars and found metallicity variations around spiral arms (see also Zari et al. 2018, 2021; Poggio et al. 2021; Hackshaw et al. 2024). Similarly, Imig et al. (2023, among others) traced gradients across different stellar populations and ages, emphasizing the importance of considering radial migration effects (Binney & Tremaine 2008; Frankel et al. 2018, 2020).\nHistorically, radial metallicity gradients have been measured using various stellar populations and gas tracers. Estimated gradients seem to be broadly consistent across different stellar tracers, such as young open clusters (e.g. Yong et al. 2012; Cunha et al. 2016; Magrini et al. 2017; Casamiquela et al. 2019; Donor et al. 2020; Spina et al. 2021; Myers et al. 2022), young hot (OB-type) stars (Zari et al. 2018, 2021; Poggio et al. 2021, 2022), field stars close to the Galactic plane (e.g. Bergemann et al. 2014) or Cepheids (Andrievsky et al. 2002a,b; Lemasle et al. 2007, 2013).\nDespite extensive observational efforts, several challenges persist for studies in the Milky Way. The completeness (or patchiness) of observed datasets remains a fundamental issue (Bergemann et al. 2014). The robustness of fits to the incomplete data is still contentious, including the need to actually fit two linear gradients with a break radius at corotation radius (Bresolin et al. 2012, and references therein) or further out (Yong et al. 2012; Donor et al. 2020) - or even more complicated functions (see e.g. Chiappini et al. 2001; Kubryk et al. 2015). Furthermore, methodologies for fitting linear models to scattered data need re-evaluation (Metha et al. 2021). Different samples yield varying gradients, potentially due to biases in data or the inclusion of older stars (e.g. Allende Prieto et al. 2006; Hayden et al. 2014; Anders et al. 2014; Vickers et al. 2021; Willett et al. 2023). The impact of spiral arm structures (Poggio et al. 2021), the Galactic warp (Lemasle et al. 2022) or bar-driven mixing (Di Matteo et al. 2013) on metallicity gradients is not fully understood.\nUnderstanding these gradients in the Milky Way is also crucial for extragalactic studies, where spatial resolution limits observations in different ways. In extragalactic systems, metals are mainly traced via gas, because it provides a more direct measure of the ongoing enrichment processes, unlike stars, which primarily reflect the integrated chemical history of the past. Observationally, gas emission lines are typically brighter and more accessible across large distances than stellar absorption lines, allowing for broader spatial coverage, especially in distant galaxies. Consequently, extragalactic studies often focus on gas-phase metallicity as traced by oxygen, A ( O ) = 12 + log ( O / H ) , while Galactic studies typically use stellar iron abundance [ Fe / H ] = A ( Fe ) -A ( Fe ) ⊙ as a metallicity tracer (e.g. Nicholls et al. 2017; Fraser-McKelvie et al. 2022).\nNew instruments like the MUSE integral field spectrograph have enabled a plethora of extragalactic studies to contrast the Milky Way and techniques like the spectroscopy of H /i.pc/i.pc regions and planetary nebulae have helped to infer gas metallicity distributions in external galaxies (Shaver et al. 1983; Vilchez & Esteban 1996; Rolleston et al. 2000; Bresolin et al. 2012). Recent examples include Sánchez et al. (2014) with CALIFA galaxy observations as well as Mun et al. (2024) and Chen et al. (2024a) who use MAGPI observations to probe for example the effects of spiral arms. Notable is also the scatter that Chen et al. (2023) found for the gas metallicity across galactic radii with TYPHOON observations (see their. Figs. 4-6). Grasha et al. (2022) found that the gas metallicity gradient plateaus at a lower limit in their TYPHOON galaxies at the outermost\nradii - an observation replicated by IllustrisTNG simulations (Hemler et al. 2021; Garcia et al. 2023).\nFrom a modelling perspective, galactic chemical evolution models can both test understanding of radial metallicity gradients and make predictions beyond the limited volumes and tracers tested by Milky Way and extragalactic studies. Such galactic chemical evolution models include Chiappini et al. (2001); Matteucci & Recchi (2001); Minchev et al. (2014); Kubryk et al. (2015); Stanghellini et al. (2015); Rybizki et al. (2017); Spitoni et al. (2023); Johnson et al. (2024). Sharda et al. (2021) even presented a model for gas phase metallicity gradients in galaxies and their evolution from first principles (see also Krumholz & Ting 2018).\nIn exploring radial metallicity gradients through simulations, we have better understood how different processes influence these gradients across galactic models and temporal scales. Studies such as Pilkington et al. (2012) in RaDES simulations reveal that gradients are typically established via inside-out galaxy formation. Khoperskov et al. (2023) quantified the scatter of gas metallicity to ≈ 0 . 04 -0 . 06 dex at a given galactocentric distance in their simulations. Meanwhile, the EAGLE simulations used by Tissera et al. (2019) provide insights into how these gradients vary with galaxy characteristics like stellar mass and merger history, emphasizing the dynamic nature of metallicity distributions. The plethora of simulations such as AURIGA (Grand et al. 2016), FIRE (Ma et al. 2017b, see their Fig. 6) or VINTERGATAN (see their Fig. 9; Agertz et al. 2021) also allow us to explore the gradient evolution of galactic timescales. Buck et al. (2023), for example, found a link of major accretion events with periods of unexpected steepening in the metallicity gradient within NIHAO-UHD simulations - closely resembling findings for the Milky Way by Lu et al. (2022) and Ratcliffe et al. (2023). FIRE simulations, examined by Bellardini et al. (2021), Bellardini et al. (2022), and Graf et al. (2024), extend these findings by comparing radial metallicity gradients and their azimuthal scatter across gas and stellar components and illustrate the complex interplay between galactic structure and metal enrichment processes. Similarly, Grand et al. (2015, 2016) highlight temporal changes in metallicity gradients already within 120 Myr, or roughly one galactic rotation. Such rapid changes underscore the impact of transient galactic events on the metal distribution, linking them to star formation patterns along spiral arms and the broader evolutionary history of the galaxy.\nIn this study, we analyze a high-resolution NIHAO-UHD simulation of a Milky Way analogue to bridge the observational gap between detailed studies of our galaxy and broader extragalactic surveys. We aim to reveal subtle features of the radial metallicity gradient, which may be obscured by observational constraints in both the Milky Way and distant galaxies, by testing the following properties within the observationally probed inner 𝑅 Gal . ≤ 20 kpc:\n\n1. Linearity of the gradient: Assess the extent to which the radial metallicity gradient of young stars is linear.\n2. Scatter in the gradient: Quantify the expected scatter in the radial metallicity gradient of young stars.\n3. Coherence of the gradient with position: Investigate the gradient's variation with radial coverage and azimuth.\n4. Coherence of the gradient with age: Test the reliability of stars as tracers of the gas disk over different ages.\n\n\n\n\nXGal.\n\n\n
F/i.pc/g.pc. 1.- Logarithmic spatial density distribution of stars (upper panels) and gas (lower panels) within 𝑅 < 20 kpc ∼ 5 Re of the NIHAO-UHD Milky Way analogue g8.26e11 in galactocentric cartesian and cylindrical coordinates. Panel c) shows the influence of selecting only young stars with ages below 0 . 5 Gyr.F/i.pc/g.pc. 2.- Face-on view of average simulated metallicity (left panels), a linear radial fit to it (middle panels, see Section 3.1 and Eq. 1) and the fit residuals (right panels) in bins of 0 . 5 kpc. Shown are metallicity as traced by young star iron abundances (top panels) and gas phase metallicity (bottom) panels.
\n\nThe paper is structured as follows: Section 2 describes the data of our Milky Way analogue NIHAO-UHD simulation. Section 3 analyses the linearity of the radial metallicity gradient of the simulation, the first of our four objectives. Section 4 then analyses both the scatter and local deviations from the gradient as well as the coherence of the gradient with vertical and azimuthal position as well as age (the remaining three objectives). Section 5 discusses them individually. We note that in this research we are mainly interested in the specific shape of the radial metallicity gradient for radii relevant to Galactic observations. However, we also discuss our results in the context of extragalactic results that probe beyond the inner 20 kpc of a galaxy. Section 6 bundles our results into an overarching conclusion.\n2. DATA: A NIHAO-UHD MILKY WAY ANALOGUE SIMULATION\nFor this project, we use a cosmological zoom-in simulation of a Milky Way analogue ( g8.26e11 ) from the Numerical Investigation of a Hundred Astronomical Objects (NIHAO, Wang et al. 2015) suite. This model galaxy was calculated as part of the NIHAO-UHD project (Buck et al. 2020) and has previously been used in various works studying Milky Way satellites (Buck et al. 2019), Milky Way's dark halo spin (Obreja et al. 2022), inferring birth properties of stars with abundance clustering (Ratcliffe et al. 2022), as well as the evolution of the interstellar medium's radial metallicity gradient since redshift three (Ratcliffe et al. 2024).\nSimulations were carried out with the smoothed particle hydrodynamics code Gasoline2 (Wadsley et al. 2017), including sub-grid turbulent mixing, using cosmological parameters from Planck Collaboration et al. (2014) with initial conditions and energetic feedback descriptions from the NIHAO project (Wang et al. 2015). Zoom-in simulations were then performed as described in detail by Buck et al. (2021) with star formation following Stinson et al. (2006) and energetic feedback following Stinson et al. (2013). We note that this is a slightly different rerun of the same simulation than the one studied by Buder et al. (2024); in this work, we use a higher resolution version and updated chemical yields.\nBecause computational resources still limit the mass resolution of simulations, we are relying on tracer particles that represent simple stellar populations (SSPs) with the same age, overall metallicity and discrete initial mass function (IMF). Buck et al. (2021) have implemented the flexible chemical evolution code /c.pc/h.pc/e.pc/m.pc/p.pc/y.pc (Rybizki et al. 2017) to calculate the chemical yields for the SSPs. In particular, we use the alternative ( alt ) setup of /c.pc/h.pc/e.pc/m.pc/p.pc/y.pc that assumes a Chabrier (2003) IMF with high-mass slope of 𝛼 IMF = -2 . 3 over a mass range of 0 . 1 -100 M ⊙ for SSPs across a metallicity range of 𝑍 / 𝑍 ⊙ ∈ [ 10 -5 , 2 ] . The code calculates the contribution from asymptotic giant branch (AGB) stars, CCSN across a mass range of 8 -40 M ⊙ , and SNIa with an exponential function with exponent -1 . 12, a delay time of 40 Myr, and a normalization of the SNIa rate of log 10 ( NIa ) = -2 . 9. For each of these nucleosynthetic channels, yields from the following studies are used: Chieffi & Limongi (2004) for CCSN, Seitenzahl et al. (2013) for SNIa, and Karakas & Lugaro (2016) for AGB stars ( new_fit model in Buck et al. 2021). Contrary to a previous study by Buder et al. (2024), we take the elemental abundances at face value and do not apply any shifts.\nWe limit the simulation data to the main halo by applying /p.pc/y.pc/n.pc/b.pc/o.pc/d.pc/y.pc's implementation of the Amiga Halo Finder (Knollmann & Knebe 2009) and then reposition and rotate this\nmain halo to be face-on based on the angular momentum with /p.pc/y.pc/n.pc/b.pc/o.pc/d.pc/y.pc's /a.pc/n.pc/a.pc/l.pc/y.pc/s.pc/i.pc/s.pc./a.pc/n.pc/g.pc/m.pc/o.pc/m.pc./f.pc/a.pc/c.pc/e.pc/o.pc/n.pc module (Pontzen et al. 2013). We then further transform the resulting galactocentric Cartesian coordinate ( 𝑋,𝑌, 𝑍 ) and velocities ( 𝑉 𝑋 , 𝑉 𝑌 , 𝑉 𝑍 ) to Cylindrical ones as done in a previous study of this main halo by Buder et al. (2024).\nThe model galaxy has a virial radius of 𝑅 vir = 𝑅 200 = 206 kpc and a total mass (gas, stars and dark matter) inside 𝑅 vir of 9 . 1 · 10 11 M ⊙ . At redshift zero, it contains 8 . 2 · 10 11 M ⊙ dark matter, 6 . 4 · 10 10 M ⊙ gas mass and 2 . 3 · 10 10 M ⊙ stellar mass with a stellar mass resolution of around 7 . 5 · 10 3 M ⊙ . When using a fifth of the virial radius as a reference to calculate total luminosity /one.sup and mass, we estimate a half-light radius, that is, effective radius of 𝑅 𝑒 = 3 . 79 kpc and a half-stellar-mass radius of 2 . 97 kpc.\nTo achieve a roughly similar selection as the observational data of the Milky Way (Genovali et al. 2014) and other galaxies (e.g. Chen et al. 2023), we restrict the simulation data to a galactocentric radius of 𝑅 Gal ≤ 20 kpc and | 𝑧 | ≤ 10 kpc, as shown in Fig. 1. Similar to the Milky Way (Poggio et al. 2018; Lemasle et al. 2022), we note a warp of the stellar and gaseous disk (see Figs. 1b and 1e, respectively).\nTo avoid too strong effects of radial migration (Binney & Tremaine 2008; Frankel et al. 2018; Grand et al. 2016; Minchev et al. 2018) while maintaining a sufficiently large sample size we further enforce stars to be younger than 0 . 5 Gyr, corresponding to roughly the time of four galactic rotations, and being half the value found by Minchev et al. (2018) for very limited migration in the Milky Way. This selection defacto limits the vertical range of 99% of stars to | 𝑧 | = 1 . 4 kpc. The strong influence of this age cut on the vertical distribution of stars in the Milky Way analogue can best be appreciated from the difference of vertical density distributions of stars in Figs. 1b and 1c. We are applying these cuts for all following analyses of the radial metallicity gradient in Section 3.\n3. THE LINEARITY OF THE RADIAL METALLICITY GRADIENT IN NIHAO-UHD\nIn this section, we analyse the functional shape of the radial metallicity gradient. To get a first impression of possible shapes, we show the face-on view of the decreasing radial metallicity gradient of the simulation in Figs. 2a (for young stars) and Fig. 2d (for gas). Foreshadowing the later parts of this work on local variations, we also show a linear radial fit to either distribution in Fig. 2b and 2e and show the fit residuals in Figs. 2c and 2f.\nAt the moment, however, we focus on the linearity and thus the logarithmic density distribution of star particle iron abundances [Fe/H] across different galactocentric radii 𝑅 Gal . . This distribution is shown in Fig. 3a and strongly suggests that the gradient is predominantly linear, similar to findings for the Milky Way. More complex shapes, such as piecewise linear ones have been suggested based on incomplete and limited data in the literature. We are thus also analysing these shapes with the complete and better-sampled data points of the NIHAO-UHD simulation. We firstly test different global fits in Section 3.1, before testing the influence of binning and coverage in Sections 3.2 and 3.3, respectively.\n
\n3.1. Global gradient fits\nWe fit three different functional forms to the global data: a linear function (used for Fig. 2b)\n𝑓 lin ( 𝑅 Gal . ) = 𝑐 1 · 𝑅 Gal . + 𝑐 2 , (1)\na piecewise linear with a break radius 𝑅 break\n𝑓 piece ( 𝑅 Gal . ) = GLYPH<26> 𝑐 1 · 𝑅 Gal . + 𝑐 2 if 𝑅 Gal . ≤ 𝑅 break 𝑐 3 · 𝑅 Gal . + 𝑐 4 if 𝑅 Gal . > 𝑅 break , (2)\nand a quadratic function\n𝑓 quad ( 𝑅 Gal . ) = 𝑐 1 · 𝑅 2 Gal . + 𝑐 2 · 𝑅 Gal . + 𝑐 3 . (3)\nThe coefficients of the functions are fitted with the /s.pc/c.pc/i.pc/p.pc/y.pc./o.pc/p.pc/t.pc/i.pc/m.pc/i.pc/z.pc/e.pc function /c.pc/u.pc/r.pc/v.pc/e.pc_/f.pc/i.pc/t.pc (Virtanen et al. 2020) and listed in Table 2. To estimate the uncertainty of the break radius 𝑅 break, we use the profile likelihood method to identify the radii at which the residual sum of squares (RSS) values are increased by 1 𝜎 from the best RSS radius in steps of Δ 𝑅 break = 0 . 1 kpc and 0 . 5 kpc for the full and binned data set, respectively. We compute the RSS for each model 𝑓 𝑖 (see Eqs. 1-3) based on the 𝑁 data points as\nRSS 𝑖 = 𝑁 ∑︁ 𝑛 = 1 GLYPH<0> [ Fe / H ] 𝑛 -𝑓 𝑖 ( 𝑅 Gal .,𝑛 ) GLYPH<1> 2 . (4)\nWe have confirmed the robustness of our fits by applying other fitting routines as outlined in Table 1. After fitting three different functional forms, we use a combination of parameters to determine which model provides the best fit.\nIn Table 2, the RSS value is the smallest (although only by a small margin) for the quadratic function. When assuming 𝜎 2 = 𝑅𝑆𝑆 / 𝑁 , we can also define a logarithmic likelihood\nln 𝐿 = -𝑁 2 ln ( 2 𝜋 ) -𝑁 2 ln 𝑅𝑆𝑆 𝑁 -𝑁 2 (5)\nfor the 𝑁 data points. For 𝑘 free parameters, we then calculate the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) as\n𝐴𝐼𝐶 = 2 𝑘 -2 ln 𝐿 𝐵𝐼𝐶 = 𝑘 ln 𝑁 -2 ln 𝐿. (6)\nFor these criteria, the quadratic function performs slightly better than the linear or piecewise linear functions (see Table 2).\nWe show the fit residuals in Fig. 3b as density distribution as well as in Fig. 3c as percentile distributions in radial bins of Δ 𝑅 Gal = 0 . 5 kpc. While the density distribution shows substructure, which we investigate later in Section 4, we note an increase in the median residuals of the linear fit in Fig. 3c towards the inner and outer radii, especially for 𝑅 Gal . > 17 kpc. A quadratic fit (see orange lines in Fig. 3) results in a slightly steeper linear component of the gradient (from -0 . 0411 to -0 . 0497 dex kpc -1 ), which is counteracted by the quadratic flattening term of + 0 . 0005 dex kpc -2 . The latter leads to an\n\n\n\nRGal.\nF/i.pc/g.pc. 3.- Global fits and deviation to the radial metallicity gradient 𝑅 -[ Fe / H ] . Functional forms of the linear (red) and quadratic (orange) lines are shown in the legend. Panel a) shows the underlying data of all data points as logarithmic density and the global fit to them as red dashed line. Panel b) shows the deviation of data from a linear gradient as a logarithmic density plot, whereas panel c) shows the 16th and 84th percentile around the median deviation as error bars in Δ 𝑅 Gal = 0 . 5 kpc bins.\neffective flattening of -0 . 172 + 0 . 200 = 0 . 028 dex (linear vs. quadratic terms) at 𝑅 Gal . = 20 dex. While seemingly only a nuisance correction across the large extent of [Fe/H] and 𝑅 Gal . , this quadratic function outperforms the linear fit. This is most apparent in Fig. 3c, where the orange line better traces the median residuals from the linear function across all radii. This suggests that non-linear functions, such as piecewise linear or quadratic ones describe the gradient better.\nDistinguishing between the two latter functional forms is challenging. The quantitative performance indicators-RSS, AIC, and BIC-show very similar values for both forms, and a closer examination of the residuals in Fig. 4 reveals no clear visual advantage for either the piecewise linear or quadratic model.\nTAKE-AWAY: Both piecewise linear and quadratic functions provide a better fit to the radial metallicity relation than a simple linear model. However, based on our assessments, there is no clear preference between the piecewise and quadratic functions.\n3.2. The influence of binning\nIn this section, we test the influence of fitting a function to all points of the distribution or binned data in steps of Δ 𝑅 Gal . = 0 . 5 kpc, using median values as data points and standard deviations /two.sup as uncertainty (see also Hemler et al. 2021, who fitted functions to radially binned IllustrisTNG data). The results are shown in Fig. 4. Given that more than half of the young star particles of the galaxy are within\n/two.sup Wenote that this 𝜎 is not equivalent to observational uncertainty and can thus not be directly applied onto observational analyses.\nTABLE 2\n
F/i.pc/g.pc. 4.- Deviation of different radial metallicity gradient functions from the global linear fit. Shown are the different functions (linear, quadratic, and piecewise linear) estimated from the full distribution (solid lines) or medians and standard deviations (error bars) in Δ 𝑅 Gal . = 0 . 5 kpc bins (dashed lines).
\n\n𝑅 Gal . < 4 kpc, this binning - although counteracted by the smaller spread of [Fe/H] in the inner galaxy - weighs the distribution of the inner galaxy significantly less than when weighing all particles equally (20 vs. 34 000 data points). The parameters fitted to the binned data exhibit a larger uncertainty due to our use of the spread of [Fe/H] per bin as absolute uncertainty 𝜎 , but the fitted parameters agree well within the fitting uncertainties.\nTAKE-AWAY: While the specific slopes differ when fitting all points or binned data, they agree within the small fitting uncertainties.\n3.3. The influence of radial coverage on linear fits\nAlthough we have gained useful insight into the global function, observational data will rarely cover the full extent of the stellar disk. Milky Way studies have previously been limited to the range of around 5 -15 kpc. There are often similar limitations and even gaps in extragalactic data. Using smaller ranges, observational studies have found hints of piece-wise linear gradients with a break radius in them based on limited radial coverage (e.g. Andrievsky et al. 2002a; Yong et al. 2012; Boeche et al. 2013; Hayden et al. 2014; Anders et al. 2017; Donor et al. 2020; Chen et al. 2023). These results are intriguing, since a quadratic function can, to first order, be approximated by two linear functions with a break radius. We therefore want to use our simulation to test if the radial coverage may indeed delude us into identifying broken linear gradients.\nWetest how smaller coverage and piecewise linear fits could mimic a complex global gradient by fitting in piecewise lin-\n\n\n
F/i.pc/g.pc. 5.- Impact of different coverage in galactocentric radius when fitting a linear radial metallicity gradient to young stars. Each horizontal segment uses a different running radial fitting range between 0.25 and 15 kpc as outlined on the right. For better contrast, the global linear fit is subtracted from the local gradient estimates and each line is colored by the gradient slope with a color scale centered around the global fit slope. Additionally, the slope of each line segment highlights the difference between global and local slopes. Thus, if the local fit exactly matches the global fit, we display it as a flat line (zero slope difference) on top of the gray dashed line (zero offset deviation) and give it a gray color signaling the same gradient value as the global fit.
\n\near radial ranges of 0.25, 0.5, 1, 2, 5, 8, 10, 15, and 20 kpc. We show their difference with respect to a global linear fit in Fig. 5, with color-coding indicating the slope of the local gradient. A horizontal dashed line indicates the same slope as the global fit, whereas the offset of a line from the said horizontal dashed line indicates the local deviation from the global gradient intercept. Differences in line slopes are visualising the difference in gradient slopes between the global and local fits. We see that all ranges suggest more or less significant deviations from a global linear fit. The innermost fit suggests a significantly different gradient than the outermost fit. We also note increasing slope differences towards the smallest scales, hinting at local deviations from a global pattern. We follow these up in Section 4, but for now, focus on the larger-scale trends.\nWhen directly comparing an inner and outer radius fit, such as between 𝑅 Gal . = 5 -10 kpc (thick grey line in Fig. 5) and 𝑅 Gal . = 10 -15 kpc (thick black line in Fig. 5), we note a significant change, similar to previous estimates of the Milky Way (e.g. Yong et al. 2012; Lemasle et al. 2008). In our case, the gradient estimate changes from [ Fe / H ] ( 𝑅 Gal . ) = 0 . 471 -0 . 044 · 𝑅 Gal . to [ Fe / H ] ( 𝑅 Gal . ) = 0 . 375 -0 . 034 · 𝑅 Gal . . When looking at linear gradient fits across the radial coverage of Δ 𝑅 Gal . = 5 -15 kpc in Fig. 5, the gradient is steeper (bluer color) for smaller radii and flatter (redder) for larger radii. Indeed, a piecewise linear function can well mimic a complex global gradient.\nWe note that in the simulated data, we see local deviations that become traceable below Δ 𝑅 Gal . ≤ 2 kpc ∼ 0 . 5 Re (bottom part of Fig. 5). This might indicate the spatial resolution required to see local effects, such as spiral arms, for extragalactic studies (see also Krumholz & Ting 2018; Li et al. 2024). We pursue this observation in the following Section 4.\nTAKE-AWAY: We find that a piecewise linear function can well mimicaquadraticfunction across the scales used in Milky Way and extragalactic studies. Local deviations become traceable below are spatial resolution of Δ 𝑅 Gal . ≤ 2 kpc (or Δ 𝑅 Gal . ≤ 0 . 5 Re).\n4. SCATTER AND LOCAL DEVIATIONS FROM THE GRADIENT\nNow that we are sufficiently satisfied that our flattening gradient function reproduces the overall shape of the radial metallicity gradient, we are concerned with both the scatter and local slope deviations across the galactocentric radii in this section. In detail, we analyse the scatter (Section 4.2), vertical variations (Section 4.2), azimuthal variations (Section 4.3, particularly motivated by the localised, spiral-shaped fit residuals of Figs. 2c and 2f) and deviations across different ages (Section 4.4).\n4.1. Scatter\nWhen investigating the change in scatter from the innermost radii to the outermost (see Fig. 3c), we see a steady increase in 1 -𝜎 spread. This spread increases from\n𝜎 [ Fe / H ] = 0 . 01 dex at 𝑅 Gal . = 0 . 25 ± 0 . 25 kpc to\n𝜎 [ Fe / H ] = 0 . 06 dex at 𝑅 Gal . = 8 . 25 ± 0 . 25 kpc and reaches\n𝜎 [ Fe / H ] = 0 . 10 dex at 𝑅 Gal . = 19 . 75 ± 0 . 25 kpc.\nWhen we recall the observed significant spread in metallicities of young open clusters at the solar radius beyond observational uncertainty (e.g. Donor et al. 2020; Spina et al. 2021) and our selection of only young ( < 0 . 5 Gyr) stars from the simulation, a strong impact of this scatter by radial migration should be excluded. At this point, we can imagine that this chemical diversity might be caused by less well-mixed gas or non-radial effects (such as vertical or azimuthal ones), which we investigate subsequently.\n4.2. Vertical deviations\nIn this section, we now look at deviations with respect to the vertical dimension, that is, 𝑅 -𝑧 . In Fig. 6 we show the previously identified local gradient deviations (lines following the left axis label) on top of the vertical density distribution ( 𝑅 -𝑧 ) of young stars (Fig. 6a) and gas (Fig. 6b) between -3 < 𝑧 < 3 kpc. Although the quickly decreasing number of young stars (Fig. 6a) at outer radii does not show substructure in the density plots for reasonable bin sizes, we see more substructure for the gaseous component in Fig. 6b). In particular,\n\n\n\n\n\n
F/i.pc/g.pc. 6.- Local gradient deviations (red-blue lines) similar to the second lowest row of Fig. 5 for radial gradients in 0 . 5 kpc steps (but compared to a global quadratic function) overlapped on top of the logarithmic density distribution in 𝑅 -𝑧 for | 𝑧 | < 3 kpc of gas (panel a) and stars (panel b). We see no strong correlation between local gradient slopes (red-blue lines) and star or gas density in this projection.
\n\nwe see rather minor deviations at small radii (where most stars and gas are close to the plane). At increasing radii, we notice an increase in both the vertical distribution of stars, and increasing local gradient deviations. In particular, we note a significant deviation of the slope around 𝑅 Gal . ∼ 15 kpc, where the gradient deviation line is steep and blue (indicating a much steeper gradient at this radius), and we notice a significant overdensity of gas around 𝑧 ∼ 1 kpc. Overall, however, we do not see strong correlations in this particular plane.\nThis could, however, be caused by a super-position effect of the up- and downturn at larger radii due to the galactic warp (see Figs. 1b and 1e). Although the warp of the stellar disk in Fig. 1b is not as clear, we confirm that both the gas disk and the youngest stars below 0 . 5 Gyr are tracing each other across the simulation in both galactocentric radius 𝑅 Gal . and height 𝑧 Gal . for different sectors in the azimuthal direction. We note that the superposition in Fig. 6 could smear out local correlations of slope changes with gas overdensities, for example, by spiral arms. Although such an edge-on view of the galaxy may indeed be the only observable one for extragalactic targets, for example, of the GECKOS survey of edge-on galaxies (van de Sande et al. 2023), we have the luxury of being able to analyse the azimuthal direction of our simulated galaxy, too.\nTAKE-AWAY: We see no strong correlations of deviations in the vertical direction throughout the simulation. Such correlations could, however, be blurred by azimuthal effects, like the galactic warp, which needs to be disentangled in the azimuthal dimension.\n4.3. Azimuthal deviations\nTo analyse the deviations from a global gradient across different azimuthal viewing angles, we divide the galaxy into 8 sectors with Δ 𝜛 Gal . = 45 · (see Fig. 7a). This allows us to study the positions around the upturn and downturn of the\n\n\n
F/i.pc/g.pc. 7.- Stellar density variation across 8 different sectors (with color-code visualised in panel a) of the radial metallicity gradient 𝑅 - [ Fe / H ] across 8 different azimuth ranges (panels b-i). A rotating lighthouse-like GIF animation of the median age and median density of the 𝑅 - [ Fe / H ] -relation across different azimuths is freely available on a repository.
\n\n\n\n
F/i.pc/g.pc. 8.- Same as Fig. 7, but colored by median age per bin. We identify 3 groups with boxes in panels e, f, and h. A rotating lighthouse-like GIF animation of the median age and median density of the 𝑅 - [ Fe / H ] -relation across different azimuths is freely available a repository.
\n\ngalactic warp with the median azimuth of young star particles below and above the plane being 𝜑 Gal . ∼ 183 · and 𝜑 Gal . ∼ 4 · , respectively (see Fig. 1e), while maintaining a reasonable sample size.\nAt face value, the distribution of 𝑅 Gal . - [ Fe / H ] for each sector follows a similar, rather linear shape with most stars being born in the inner 5 kpc of the galaxy. However, we find significant deviations in different sectors of the galaxy (Fig. 7). On the one hand, we find non-linear deviations as bumps with slightly increased or decreased iron abundance - up to 0 . 1 -0 . 2 dex - in Figs. 7c at 𝑅 Gal ∼ 18 kpc, 7d at 𝑅 Gal ∼ 10 kpc, 7f at 𝑅 Gal ∼ 14 kpc, and 7g at 𝑅 Gal ∼ 17 kpc. On the other hand, we find significant gaps in the distribution at similar [Fe/H], most strikingly at the upturn of the galactic warp in Fig. 7e ( 𝜛 Gal . = 135 -180 · ) at [ Fe / H ] ∼ 0 dex and 𝑅 Gal . ∼ 8 -14 kpc. We note that the sector e) with the gap is surrounded by two sectors (d and f) with significant overabundance at the same radius. This could be indicative of stars having formed as a result of gas moving from sector e towards either azimuthal direction, causing a gas overdensity which could in turn lead to higher star formation activity. To establish this observation, we take a closer look at the timedomain, that is, stellar age as well as the spatial domain of 𝑅 Gal . -𝜑 Gal . in the next section.\nTAKE-AWAY: We find various deviations from the global trend in the azimuthal direction, including gaps and isolated streaks of stars with similar [Fe/H] throughout Δ 𝑅 Gal . = 2 -6 kpc. These can introduce local over- and under-enhancement of up to ± 0 . 2 dex in [Fe/H] at a given radius. In the next section, we analyse whether the stars of these streaks have been born at the same or different time.\n4.4. Deviations with time and age\nIn this section, we examine the radial metallicity gradient in a small age range less than 0 . 5 Gyr. To do so, we color Fig. 7 by the median stellar age rather than the logarithmic density in Fig. 8. We find an overall significant scatter across time, suggesting a good mix of star formation across all sectors for stars born within less than 0 . 5 Gyr. For stars within this restricted age range, we do not see a strong correlation with radius, such as older stars being born further inside, but a larger amount of stars being born closer to the galactic centre. Wenote that stars with similar [Fe/H] in each sector tend to be formed at similar times (within 50 Myr), that is, as flat lines with the same color (age) in Figs. 8b-i. To guide the eye, we have identified Group 1 in Fig. 8e (around 𝑅 Gal . ∼ 14 kpc at 𝜛 Gal . = 180 -225 · ). We further note that the enriched bumps identified earlier are born at similar times, see, for example, Group 2 in Fig. 8f. The coloring by age also reveals that stars with lower [Fe/H] than expected (see Group 3 in Fig. 8h) are born at similar times. In some cases, these extend to Δ 𝑅 Gal . = 2 -6 kpc, see Groups 1, 2, and 3 in Fig. 8. At a given radius 𝑅 Gal . , these streaks cause a significant spread in local [Fe/H] of up to ± 0 . 2 dex (see Fig. 8).\nFrom the analysis of azimuthal sectors, the impression arose that the star formation in this simulated Milky Way analogue is - as expected for a spiral galaxy - rather patchy and localised on the smallest timescales. This is confirmed by looking at the spatial distribution of azimuth 𝜑 Gal . and radius 𝑅 Gal . in Fig. 9. Already when looking at the density distribution of all stars born within less than 0 . 5 Gyr in Fig. 9a, multiple streams are visible, stars on spiral patterns (see also Kreckel et al. 2019; Chen et al. 2024b). When following up the previously identi-\n\n\n\n\n\n
F/i.pc/g.pc. 9.- Density distribution (panel a) and age distribution (panel b) of young stars in the azimuthal and radial direction 𝜑 Gal . -𝑅 Gal . . In panel a), we also show the groups previously identified in Fig. 8.
\n\nroups 1, 2, and 3, we recover them on said spiral patterns (Groups 1 and 3) or a local overdensity (Group 2). Although one could imagine that radial migration might induce such a spiral-like shape for the stars of groups 1 and 3, their low age of less than 250 Myr would require a significant migration effect of several kpc, while having no influence on the older stars of group 2. When tracing the position of significant overdensities from Fig. 9a in the same projection colored by age in Fig. 9b, we note that for radii above 𝑅 Gal . > 5 kpc these overdensities are colored in red, that is, containing indeed young stars with ages below 200 Myr and being consistent with the most recent star formation along these spiral patterns in the outer galaxy.\nTAKE-AWAY: We find significant scatter across the radial metallicity distribution caused by streaks of stars born with similar [Fe/H] at similar times (within 50 Myr) across either very local or radially extended spiral-shaped regions of the galaxy, suggesting local enhancement patterns in small overdensities or along spiral arms.\n5. DISCUSSION\nHaving presented the analysis, we now put our results into the context of other work in terms of our initial aims: to analyse the shape (Section 5.1), scatter (Section 5.2), local deviations (Section 5.3), and time-dependence (Section 5.4) of the radial metallicity gradient. These initial discussions inform our thoughts on the implications of this work for Milky Way studies in Section 5.5 and the studies of other galaxies in Section 5.6.\n5.1. Linearity of the radial metallicity gradient\nThe radial metallicity gradient of our simulated NIHAOUHDMilky Way analogue showed an overall decreasing, predominantly linear shape, as established in Section 3. Motivated by previous works by Sánchez-Menguiano et al. (2016), among others, we also fitted piecewise linear and quadratic\n\n\n
F/i.pc/g.pc. 10.- Comparison of the Milky Way's radial metallicity trend as traced by Cepheids (black triangles, compiled from literature by Genovali et al. 2014, G+14) as well as young ( < 0 . 5 Gyr) open cluster of the Milky Way as traced by the literature compilation from Genovali et al. (2014, G+14 as squares), APOGEE DR17 from Myers et al. (2022, M+22 as crosses), and GALAH DR3 from Spina et al. (2021, S+21 as circles). The latter two are compiled based on the membership and age catalogue by Cantat-Gaudin & Anders (2020, CG+20).
\n\nfunctions to the data in Section 3.1. Both forms perform better than a linear trend. The very similar fitting performances indicate no significant preference between either piecewise linear or quadratic function. Due to both functions' rather good overall fit, we have not tried more exotic non-linear functions as done by Scarano & Lépine (2013). Increasing the flexibility of the gradient function could, however, improve the fit at the innermost kpc, where a flattening is predicted by our simulation, but chemical enrichment is also harder to simulate (see also Minchev et al. 2013; Sun et al. 2024). We have found no significant influence of binning for our gradient estimates (Section 3.2), but have found that a limited radial coverage - as is the case for the Milky Way - could mimic a truly quadratic function with two piecewise linear fits (Section 3.3). This is important, as it has significant implications for the conclusions we draw from the incomplete data of our Milky Way, as we will discuss in more detail in Section 5.5.\nThe balance between a quadratic and piecewise linear radial metallicity gradient teeters at the breaking radius. If present, our analysis of the Milky Way analogue would place it at 𝑅 break ∼ 10 ± 0 . 5 kpc. This radius is strikingly close to the radius of 9 kpc found by Hemler et al. (2021) for a TNG50 galaxy simulation with a stellar mass of log ( 𝑀 ★ / M ⊙ ) = 10 . 72, that is, close to the Milky Way's (see their Fig. 2). In terms of physical reasons for a breaking radius at this location, a direct and secular influence of a stellar bar with non-symmetric effects around the corotation radius (Di Matteo et al. 2013; Scarano & Lépine 2013) should be minor for our specific scenario due to the low ages of the stars considered in our analysis. In particular, our identified break radius is significantly larger than the corotation radius of the Milky Way bar at 4 . 5 -7 . 0 kpc (Bland-Hawthorn & Gerhard 2016, and references therein) anyway. We are intrigued, however, by the proposition by Garcia et al. (2023) of galactic discs consisting of a star-forming inner disc with a steep gradient and a mixing-dominated outer disc with a flat gradient, with the break radius marking the region of transition between them. In Illustris TNG50-1 data, they found such a transition and break radius to be situated much further out at 30 kpc for Milky Way mass galaxies (10 . 1 ≤ log ( 𝑀 ★ / M ⊙ ) ≤ 10 . 6). While our bestfitting break radius - if present - is inconsistent with theirs, we will follow this up in more detail in Section 5.6, where we also discuss the implications for extragalactic studies in general.\n\n\n
F/i.pc/g.pc. 11.- Same as Fig. 3, but for gas.
\n\n5.2. Scatter of the radial metallicity gradient\nIn Section 4.1 we found an increasing scatter from 𝜎 [ Fe / H ] = 0 . 01 dex in the inner galaxy to 𝜎 [ Fe / H ] = 0 . 10 dex around 𝑅 Gal . ∼ 20 kpc. Comparing these values with simulations other than TNG50 with a similar metallicity spread (see Fig. 2 by Hemler et al. 2021) is rather difficult, as the literature focuses on the shape and density distribution (see e.g. Minchev et al. 2014, their Fig. 10). When comparing with Milky Way studies (e.g. Anders et al. 2017), the scatter in the simulation is smaller than the observed spread of [Fe/H]. This can be visually appreciated by comparing the combinations of different measurements in the Milky Way (Genovali et al. 2014; Spina et al. 2021; Myers et al. 2022) in Fig. 10a and our simulation in Fig. 10b and c. We discuss the implications of this on studies of the Milky Way's gradient in Section 5.5.\nWhen assuming that young star and gas phase abundances\n\n\n
F/i.pc/g.pc. 12.- Tracing young stars and gas across galactocentric radii 𝑅 Gal . and height 𝑧 Gal . across the whole galaxy (panel a) and different azimuthal ranges/sectors (panels b-i). Small rectangles with cool-warm colors along the horizontal axis indicate the local gradient slopes as in Fig. 6.
\n\nare similar, we find comparable scatter of abundances for example with respect to TYPHOON observations by Chen et al. (2023). To test this assumption, we also show the gas phase metallicity in Fig. 11, for which we find a similar shape and scatter of the gradient, but systematically less scatter or spread than observed gas phase abundance, thus urging us to treat the absolute values of abundances and abundance scatters as well as spreads with caution. We furthermore note that the spread of abundance in observations does only increase for some but not all of the observed (and thus observationally limited) galaxies by Chen et al. (2023). This potentially limits the range of galaxies to which our conclusions may apply. While the simulated abundance scatter is consistent with the predictions by the theoretical forced-diffusion model by Krumholz & Ting (2018), that is, a scatter of ∼ 0 . 1 dex over timescales of ∼ 100 -300 Myr, our simulations suggest that the scatter is driven by the radial structure and large-scale spiral arms, which were not included in their model.\n5.3. Localised vertical and azimuthal deviations and their correlation with gas\nIn Sections 4.2 and 4.3 we established that local deviations contribute significantly to the spread of the global metallicity gradient above 𝑅 Gal . > 8 kpc ∼ 2 Re. We noted in particular a void of stars where we found an upturning warp of the galaxy around 𝜑 Gal . ∼ 180 · spatially close to regions of the galaxy (Groups 1 and 2) that deviated most significantly from the overall trend in Fig. 8 and Fig. 9.\nThese stellar voids pose the question if they are also void of gas thus suggesting the gas has shifted to the more enhanced regions. In Fig. 12 we are thus tracing both the spatial distribution of gas as colored density distribution and stars as grey contour lines and gas. We find that while stars and gas trace each other in the vertical direction, we do not always see a match between the two tracers in the radial direction. In particular, we do find a significant amount of gas around the stellar void of 𝜑 Gal . ∼ 180 · and 𝑅 Gal . ∼ 8 -11 kpc in Figs. 12e and 12f. This gas seems to be more tightly concentrated though for example in the tight wave around 𝑅 Gal . ∼ 8 -11 kpc in Fig. 12e. We also note that significant gas overdensities, for example around 𝜑 Gal . ∼ 0 -45 · and 𝑅 Gal . ∼ 7 kpc in Fig. 12a do not seem to correlate with significant overenhancement in\niron abundance (compare to Fig. 7a). While we see a hint of a coinciding deviation of Δ [ Fe / H ] for larger deviations from the galactic plane Δ 𝑧 in the upturning outer region of Fig. 12f, this does not seem to be the case for the downturning outer region of Figs. 12b and 12i.\nAs the edge-on projection is not providing conclusive insights, we are now looking into the phase-on projection in Fig. 13. We have chosen a region of the simulated galaxy whose gas density at solar radius (Fig. 13c) matches with the recently measured distribution of young stars in the Milky Way at face value (Fig. 13a) by Poggio et al. (2021). Comparing simulated gas and observed young stars is preferable in this case, as the density of simulated stars is too low to easily identify overdensities (Fig. 13b). The region and its gas spiral structures appear to be representative, as these structures exist throughout the whole galaxy (see Fig. 1d).\nIn the different panels of Fig. 14, we thus show this representative region of the galaxy, but color each spatial bin by stellar metallicity (Fig. 14a), the deviation from the global linear trend (Fig. 14b) as well as the gas metallicity (Fig. 14c) and its deviation from the global linear trend (Fig. 14d). In all cases, we also overlay the density contours of the significant gas overdensities (red regions in Fig. 13c). As expected, we see that the metallicity color map of the stars in Fig. 14a shows a decreasing trend from right to left (inner to outer galaxy) and an increasing scatter (more blue and red points towards the left) in the residual plot of Fig. 14b. We cannot identify a strong correlation between gas overdensities and stellar metallicity or residuals in either plot - possibly caused by the low number density. In Figs. 14c and 14d, however, the radial gas metallicity gradient shows significant local variations, that is, a trend from left to right that is not very smooth. In particular, we find significant deviations of up to + 0 . 15 dex in [Fe/H] behind the outer gas spiral (lower left of Fig. 14d) and -0 . 1 dex in [Fe/H] in front of the inner gas spiral (upper center of Fig. 14d) with a steep edge consistent with the gas spiral edge. We have identified the same patterns in both [Fe/H] and A(O) as both elements trace each other rather well in the simulation of young stars (see Fig. A1).\nTentatively, we even see a slight enhancement of A(O) at the trailing edge of the inner spiral arm (top of Fig. 14d). We convince ourselves of the step-like behaviour by selecting a\n\n\n
F/i.pc/g.pc. 13.- Comparison of the density distribution of young stars and gas in the Milky Way and the NIHAO Milky Way analogue simulation. Panel a) shows the measurements of the Solar vicinity within 5 kpc by Poggio et al. (2021). Panels b) and c) show young stars and gas NIHAO, respectively, for a selected region similar to panel a). Black and white contour lines in panel b) trace overdensities in the gas distribution of panel c).
\n\n\n\n
F/i.pc/g.pc. 14.- Comparison of density distribution of young stars and gas in the NIHAO-UHD Milky Way analogue simulation for the same regions as Figs. 13b and 13c. Panels a) and c) trace median young star Fe and gas O abundances, respectively. Panels b) and d) plot the metallicity residuals of stars and gas, respectively, when correcting with a radial metallicity gradient fit. Black and white contour lines in each panel trace overdensities in the gas distribution of Fig. 13c).
\n\nsmall slit-like region of 𝜑 Gal . ∼ 0 · and -2 < 𝑌 Gal . / kpc < -1 and tracing the gas metallicity and gas density as a function of radius in Fig. 15. We indeed find steps and confirm that they coincide with the location of significant gas overdensities. These step-patterns have also been found by Grand et al. (2015) in another simulation and observationally by Ho et al. (2017). In Fig. 15, we note an extended flat region just beyond 𝑅 Gal . > 10 kpc, the best fitting 𝑅 break of an assumed piecewise linear fit.\nOur analyses suggest that the correlation of void and overdensities with chemical enrichment of gas and young stars is more complicated and should better be followed up by tracing these structures over simulation look-back time in a dedicated follow-up analysis to unravel the physical mechanisms of star formation feedback cycles . This could also involve the tracing of star formation bursts and disk instabilities (Sánchez et al. 2014; Sánchez-Blázquez et al. 2014; Ho et al. 2015) as well as tracing how much self-enrichment as well as mixing and dilution takes place around the gas spirals (Ho et al. 2017). Rather than going back in simulation time, the present simulation data of a single snapshot in time already allows us to look back in terms of stellar lifetime - similar to Milky Way studies, as we discuss subsequently.\n\n\n
F/i.pc/g.pc. 15.- Radial gas metallicity gradient of a slit-like region ( -2 < 𝑌 Gal . / kpc < -1) from Fig. 14. The plot extends towards larger and smaller radii and shows the step-like distribution of individual gas particle metallicities colored by their deviation from a global fit. A running median along 1000 particles is shown as a black line. The gray histogram indicates the gas density along the radius with prominent overdensities coinciding with step edges.
\n\n\n\n
F/i.pc/g.pc. 16.- Stellar density distribution and spread of [Fe/H] across different galactocentric radii with respect to a global linear radial metallicity gradient across different age ranges. Panels a-j) show young stars and exhibit a rather similar trend, whereas the scatter increases significantly for stars above 0 . 5 Gyr in panels k) and l).
\n\n5.4. The impact of time and age: mixing and migration\nAlthough we have chosen a rather small stellar age window of 0 . 5 Gyr to trace the radial metallicity gradient without the expected significant impact of radial mixing and migration, we are testing and discussing this particular choice in this section in two ways. Firstly, we test the deviation of the radial metallicity gradient from the same global shape as well as the abundance spread across smaller age bins of 100 Myr between 0 -1000 Myr in Fig. 16. Secondly, we trace the distribution of stellar metallicity across galactic radii for increasing age bins from 50 Myr up to the maximum stellar age of 13 . 8 Gyr in Fig. 17.\nOur first test in Fig. 16 shows that the deviation from a global trend remains similar in functional form. We find that the spread of iron abundance does indeed scatter significantly, but the distributions stay within the same overall shape across the ten age bins. We note though, that the smallest age bin of 0 -100 Myr shows the least abundance scatter.\nThis is consistent with the picture from our second test of increasing age ranges in Fig. 17. Here we find the first significant deviation from a tighter and already slightly quadratic relation for an age of 100 -150 Myr in Fig. 17c - our previously identified Group 3. As expected from previous simulations and observations, we see an increase in the scatter as we include more and more older stars. We note a still similar albeit more scattered shape for stars below 4 Gyr in Fig. 17h, before we start to see a more metal-poor population of stars in the inner\n\n\n
F/i.pc/g.pc. 17.- Radial metallicity gradients and quadratic fits for different maximum age ranges. The quadratic fit to stars below 0 . 5 Gyr is shown as a dashed red line for reference and the quadratic fit to each shown distribution is overlaid as a solid red line with the functional form given as inset text. At 𝑅 Gal . = 8 . 21 kpc, spread increases from 𝜎 [ Fe / H ] = 0 . 05 for youngest stars to 0.09 and 0.11 for stars below 4 and 8 Gyr, respectively.
\n\ngalaxy appear between 4 -8 Gyr in Fig. 17i. These also begin to significantly impact the quadratic fit to the radial metallicity distribution, shown as a solid red line, in contrast to our reference fit, represented by a dashed red line. The significant amount of metal-poor stars in the inner galaxy then completely tilts the distribution when also including stars between 8 -13 . 8 Gyr in Fig. 17j (see also Johnson et al. 2024). Similar to the Milky Way (Bland-Hawthorn & Gerhard 2016), these oldest stars are those of the relatively more metal-poor thick disk that are confined to the inner disk with a shorter scale length.\nWhile we cannot exclude radial migration playing a role for change of radius for the youngest stars of the simulation, since Frankel et al. (2018) predicted significant shifts even for ages below 0 . 5 Gyr (see their Fig. 10), the larger scatter for older stars is certainly suggesting a larger (re-)distribution of stars along the radial axis, as found in previous simulations (Minchev & Famaey 2010; Grand et al. 2015).\n5.5. Implications for Milky Way studies\nOur analysis of the radial metallicity gradient in a simulated NIHAO-UHD galaxy offers several insights that are directly applicable to understanding the Milky Way's gradient.\nFirst, the nature of the gradient - whether it is linear or better described by more complex functional forms - remains a critical question. Previous studies, such as those by Lépine et al. (2011) and Donor et al. (2020), have suggested the po-\nential for a break radius, possibly at the corotation radius or further out (Scarano & Lépine 2013), which could indicate two distinct linear regimes. In our analysis, we find evidence that the gradient is at least is not purely linear, but could also be smoothly flattening. Applying a smooth quadratic function on observational data (Yong et al. 2012; Andrievsky et al. 2004; Genovali et al. 2014), might provide a better or at least consistent fit for the Milky Way data without the need for a break radius. However, even this may not fully capture the nuances observed in our simulations. Chemical evolution models propose a more sophisticated behaviour (e.g. Chiappini et al. 2001; Kubryk et al. 2015; Palla et al. 2024), reflecting varying influences of galactic processes at different radii. Understanding this structure in the simulated galaxy provides a framework for interpreting similar complexities in the Milky Way.\nGiven these complexities, it is also essential to consider how local sampling biases might affect our understanding of the Milky Way's metallicity gradient. For instance, incomplete samples that omit low [Fe/H] clusters or stars could skew gradient estimates, as suggested by our comparisons in Figures 10a and 10b. Our results indicate that young clusters with lower (or higher) [Fe/H] than expected at a given radius could indicate the previous presence of a spiral arm (see our identified Groups in Figs. 8 and 9). Furthermore, we caution that localised effects - both intrinsic and in terms of selection function - could also mimic non-linear shapes and more spatial coverage is needed in the Milky Way. Our results also indicate that older clusters, which have been found more frequently at larger distances than young clusters - are likely influenced by radial migration - and thus complicate the interpretation of these radial metallicity gradients (Magrini et al. 2009; Lépine et al. 2011).\nCosmological zoom-in simulations like NIHAO-UHD are approaching the resolution needed to examine regions analogous to the solar vicinity, though the star particle numbers and mass resolution remain a limiting factor. Nonetheless, we observe distinct patterns in the distribution of young stars and gas, including lower [Fe/H] and A(O) in the leading edges of gas overdensities and higher [Fe/H] and A(O) in the trailing edges, consistent with findings by Grand et al. (2016), Ho et al. (2017), and Kreckel et al. (2019). These trends suggest that local metallicity variations, driven by gas dynamics, may also play a significant role in shaping the observed gradients in the Milky Way.\nAdditionally, our study hints at the potential for more nuanced variations in [Fe/H] across different regions of the galaxy. In particular, the gas shows a step-like behavior of A(O)and[Fe/H] changes around the edges of gas overdensities (Fig. 15), with significant deviations from the global gradient in specific regions. We have also found a larger stellar void around -12 < 𝑅 Gal < -10kpc. Although further investigation is needed, these findings could have important implications for understanding localized star formation events and their impact on the overall metallicity distribution in the Milky Way (Sánchez et al. 2014; Sánchez-Blázquez et al. 2014; Ho et al. 2015). It will certainly be exciting to see how much more insights (Poggio et al. 2021; Hackshaw et al. 2024) we will get from the more extended data of future data releases of Gaia and spectroscopic surveys.\nWecannot directly link spiral arms to bar resonances or bardriven mixing in our simulation, because of a negligible bar strength in our galaxy /three.sup (but see Minchev & Famaey 2010; Di\nMatteo et al. 2013). However, the influence of a galactic bar on the spiral arms and, by extension, on the radial metallicity gradient, remains a possibility (see again Chen et al. 2023). Disk instabilities and warps might further complicate the interpretation of these gradients and progress will likely rely on the detailed disentangling of these effects from both cosmological simulations as well as idealised simulations and models (Minchev et al. 2013; Grand et al. 2015, 2016; Krumholz et al. 2018; Sharda et al. 2021; Bland-Hawthorn et al. 2024; Tepper-Garcia et al. 2024).\n5.6. Implications for extragalactic studies\nThe insights gained from our analysis of the radial metallicity gradient in a simulated NIHAO-UHD galaxy extend beyond the Milky Way, offering valuable implications for the study of extragalactic systems.\nOne key observation is that deviations from a purely linear metallicity gradient, as seen in our Milky Way analogue, are common in other galaxies as well. When fitting a piecewise linear fit to our data, we found a break radius at 𝑅 Gal . = 10 . 0 ± 0 . 5 kpc. Converted to effective radii Re or radii R25 covering the 25 mag arcsec -2 isophote /four.sup , this corresponds to 𝑅 break ∼ 2 . 5 Re ≡ 0 . 7 R25 for our simulation. This would be consistent with the observational results by Sánchez et al. (2014) who found that breaks in metallicity gradients are common in both spiral and barred galaxies, with flattening of the abundance being evident beyond ∼ 2 Re (compare also to Belfiore et al. 2017). Similar to our suggestion for Milky Way studies, we suggest to also test a smooth function, such as a quadratic one, on extragalactic observational data (e.g. Bresolin et al. 2012; Chen et al. 2023) to test the preference of a distinct break radius.\nAlthough the focus of this research lies on the observable region of the Milky Way ( 𝑅 Gal . < 20 kpc) and most other galaxies ( 𝑅 Gal . < 2 . 5 Re), the finding of significant gradient changes in the outskirts of galaxies by Garcia et al. (2023), suggests to also test this region of our Milky Way analogue. Garcia et al. (2023, see their Fig. 4) found a metallicity floor in IllustrisTNG galaxies. When using their sample to identify a metallicity floor radius for a Milky Way mass galaxy with log ( 𝑀 ★ / M ⊙ ) = 10 . 7 (Bland-Hawthorn & Gerhard 2016) at redshift 𝑧 ∼ 0, we would expect to find it around 25 -30 kpc. We therefore extend the analysed radius to 𝑅 Gal . ≤ 100 kpc (see Fig. 18) and indeed find a similar abundance floor of [ Fe / H ] ≥ -0 . 64 for young stars and A ( O ) ≥ 8 . 12 for the majority of gas (see Fig. 19 at a similar radius. We note that another galaxy without gas in this figure is a sufficiently large distance of 92 kpc, that is, ( 𝑌,𝑌, 𝑍 ) = (-50 , -75 , 20 ) kpc.\nThese lowest abundances remind us of two observational results. Firstly, the iron abundance floor is consistent with the lower end of the Milky Way thin - and coincidentally outer disk of [ Fe / H ] ∼ -0 . 7 (Bensby et al. 2014; Buder et al. 2019). Secondly this oxygen abundance floor is consistent with the results by Grasha et al. (2022) from TYPHOON galaxy observations. Grasha et al. (2022) suggested this could be caused by changes in the ratio of supernovae II and AGB reflected by a changing ratio of nitrogen to oxygen abundance N/O which also flattens towards a lower plateau below metallicities of A ( O ) ∼ 8 . 0 (Nicholls et al. 2017). While we cannot follow this observation up with the present simulation, a similar simulation used by Buder et al. (2024) has traced the relative\n\n\n\nXGal.\n\n\n\n\n\n\nXGal.\nXGal.\n\n\n
F/i.pc/g.pc. 18.- Same as Fig. 1, but for an extended 𝑅 Gal . ≤ 100 kpc and | 𝑧 Gal . | ≤ 50 kpc.F/i.pc/g.pc. 19.- Radial metallicity functions for all stars (panel a), young stars (panel b), and gas (panels c and d for iron and oxygen as metallicity tracers) out to 𝑅 Gal . ≤ 100 kpc. Panels b and c are comparable to Figs. 3a and 11a for a smaller radial coverage.
\n\ncontribution of both supernovae II and AGB and should be used to test this hypothesis in the future.\nIt is important to note that the chemical evolution model in the NIHAO-UHD simulations is constrained by the current, incomplete understanding of evolutionary pathways and yields (Buck et al. 2021), as well as by limitations in resolution and the imperfect physics inherent to cosmological zoom-in simulations (Buck 2020). Both could contribute to the identified differences in absolute and relative abundances across different scales - including a different scatter of abundances for example of the gas phase metallicity between NIHAO-UHD of up to 0 . 1 dex and the low scatter of 0 . 03 -0 . 05 dex (and even lower on local scales) found by PHANGS-MUSE faceon observations (Kreckel et al. 2020). Extending our analysis to other simulations and further improving the resolution and physics of the simulations will be key in uniting the observational and theoretical insights into galactic chemical evolution on small and large scales.\nSimilar to more resolved and higher quality observations in the Milky Way, we also expect more, better, and diverse faceon and edge-on observations and analyses across a range of wavelengths by the PHANGS and GECKOS teams (Kreckel et al. 2019, 2020; van de Sande et al. 2023) as well as the SDSS-V and MAGPI collaborations (Kollmeier et al. 2017; Foster et al. 2021; Mun et al. 2024; Chen et al. 2024a), among many other ongoing efforts.\n6. CONCLUSIONS\nTo conclude our study, we first iterate the main take-away of our research in Section 6.1 before giving suggestions for future research in Section 6.2.\n\n\n\n\n\n\n\n\n\n6.1. Take-Away\nWehaveanalysed the radial metallicity distribution of young stars and gas in the inner 20 kpc of a NIHAO-UHD Milky Way analogue (Fig. 1), finding a predominantly linear decrease (Figs. 2 and 3). Although our analysis of a single spiral galaxy simulation has limited applicability to the entire population of diverse spiral galaxies, it reveals several intriguing findings about the shape and local metallicity variations. The results we find in this work hold relevance for both the Milky Way and extragalactic research communities:\n\n· Looking into the shape in detail, we find that piecewise linear and quadratic functions both perform better than a linear fit to the radial metallicity relation. However, we see no significant preference between piecewise and quadratic functions based on our assessments (Fig. 4). While the specific slopes differ when fitting all points or binned data, they agree within the still rather small fitting uncertainties.\n· We find that a piecewise linear function can effectively approximate a quadratic function across scales commonly applied in Milky Way and extragalactic studies. Local deviations become traceable below a spatial resolution of Δ 𝑅 Gal . ≤ 2 kpc (Fig. 5).\n· We see no strong correlations of deviations in the vertical direction across the whole simulation (Fig. 6). Such correlations could, however, be blurred by azimuthal effects, like the galactic warp, which needs to be disentangled in the azimuthal dimension.\n· We find various deviations from the global trend in azimuthal direction, including gaps as well as isolated streaks of stars with similar [Fe/H] across Δ 𝑅 Gal . = 2 -6 kpc (Fig. 7). These can introduce significant local over-/underenhancement of up to ± 0 . 2 dex in [Fe/H] at a given radius.\n· We find significant scatter across the radial metallicity distribution caused by streaks of stars born with similar [Fe/H] at similar times and similar but slightly extended regions of the galaxy (Figs. 8 and 9).\n· Our results imply the need for more careful consideration of local intrinsic effects and selection effects on radial metallicity gradient and scatter studies in the Milky Way (Fig. 10).\n· Expanding our work to the gas phase metallicity gradient (Fig. 11), we perform a preliminary comparison of observed and simulated young stars as well as simulated gas distribution and chemistry (Figs. 12 and 13), finding significant step-like changes in the gas chemistry at the leading and trailing edges of gas spirals, with lower and higher enhancement respectively (Figs. 14 and 15).\n· We have further identified that the abundance scatter, which increases towards larger radii, is as large as 0 . 1 dex and already present at the youngest ages of 100 Myr (Fig. 16). While not the focus of our analysis, we have also confirmed that the scatter significantly increases towards larger ages (Fig. 17).\n· We have discussed the implications of our findings for studies of the Milky Way (Section 5.5) as well as external galaxies (Section. 5.6). Here, we firstly suggest to explore the spread of abundances across different radii in\n\nmore detail. Secondly, we suggest approaching the fitting of gradients in external galaxies in a more agnostic way to the shape. This will be particularly interesting when we can observe the outermost regions of galaxies, where simulations predict an abundance floor (Figs. 18 and 19).\n6.2. Future Research\nIn our study, we have focused on the present-day snapshot of the NIHAO-UHD Milky Way analogue simulation - similar to present-day observations that are possible in our local Universe. Given that the simulation is tracing particles and gas over time, a detailed follow-up study should trace the evolution and coherence of spatial and chemical over- and underdensities over time and different elements (see also Zhang et al. 2024). This would in particular allow us to quantify the change of abundance in the leading and trailing edges of spiral arms and subsequently track the mixing and blurring of these over time. Certainly, more studies are needed to establish a link to a physical mechanism and further quantify its importance. More results are expected as we extend the reach, number, and quality of stellar measurements in our Galaxy (e.g. Barbillon et al. 2024) and beyond. These improvements will allow us to move beyond a one-dimensional analysis of gradients and better incorporate and model local variations.\nWhile previous works are showing that relative trends for several elemental abundances do not strongly disagree from observations (Buck et al. 2021; Buder et al. 2024), we are still missing several details on the origin of elements, such as a complete picture of the synthesis sites, environments, and yields for elements. Not least because of these imperfections of absolute chemical enrichment predictions, we are refraining from quantitatively comparing the shape of our Milky Way analogue with the actual Milky Way. We have previously also mentioned the limitations in mass resolution of stars and gas, which may introduce unrealistic effects and could for example drive deviations from actual chemical enrichment at the smallest scales. We also note that the results of our simulation may not apply to the actual Milky Way due to different galaxy properties. These could be different due to different galaxy formation pathways, such as the amount and importance of mergers (Buck et al. 2023; Buder et al. 2024). In our discussion, we have already eluded to the weak bar in this simulation. Motivated by the analysis by Tuntipong et al. (2024), we have also investigated the bulge to total stellar mass ratio 𝐵 / 𝑇 . We find a strong bulge with 𝐵 / 𝑇 = 0 . 48 when selecting bulge stars with orbit circularity 𝑗 𝑧 / 𝑗 𝑐 < 0 . 5 and disk stars with 𝑗 𝑧 / 𝑗 𝑐 > 0 . 7 based on actions 𝑗 , consistent with values found by Obreja et al. (2019) of a lower resolution simulation of 8.26e11 . Our simulated galaxy has both a very weak bar and a smoothly changing radial metallicity gradient. This is in line with the findings by Chen et al. (2023) for five strongly and weakly barred spiral galaxies, where bars seem to drive the dominance of break radii in radial metallicity gradients. Future work should certainly look at a variety of galaxies to establish the causality of bar strength with the smoothness or abrupt change at a break radius for the radial metallicity gradient. Expanding our study to more and other simulations such as the VINTERGATAN suite (Renaud et al. 2024) and subsequently comparing to observations of galaxies with varying parameters like mass, formation history, bar strength or environment would allow us to quantify their effect on metallicity gradients and further disentangle the influence of different enrichment mechanisms on the chemical evolution of galaxies.\nSOFTWARE\nThe research for this publication was coded in /p.pc/y.pc/t.pc/h.pc/o.pc/n.pc (version 3.7.4) and included its packages /a.pc/s.pc/t.pc/r.pc/o.pc/p.pc/y.pc (v. 3.2.2; Astropy Collaboration et al. 2013, 2018), IP/y.pc/t.pc/h.pc/o.pc/n.pc (v. 7.8.0; Pérez & Granger 2007), /m.pc/a.pc/t.pc/p.pc/l.pc/o.pc/t.pc/l.pc/i.pc/b.pc (v. 3.1.3; Hunter 2007), N/u.pc/m.pcP/y.pc (v. 1.17.2; Walt et al. 2011), /p.pc/y.pc/n.pc/b.pc/o.pc/d.pc/y.pc (v. 1.1.0; Pontzen et al. 2013), /s.pc/c.pc/i.pc/p.pc/y.pc (v. 1.3.1; Virtanen et al. 2020), /s.pc/k.pc/l.pc/e.pc/a.pc/r.pc/n.pc (v. 1.5.1 Pedregosa et al. 2011) /s.pc/t.pc/a.pc/t.pc/s.pc/m.pc/o.pc/d.pc/e.pc/l.pc/s.pc (v. 0.14.2 Perktold et al. 2024) We further made use of /t.pc/o.pc/p.pc/c.pc/a.pc/t.pc (version 4.7; Taylor 2005);\nDATA AVAILABILITY\nAll code to reproduce the analysis and figures can be publicly accessed via https://github.com/svenbuder/ nihao_radial_metallicity_gradients . The used simulationsnapshot can be accessed as FITS file via https: //github.com/svenbuder/preparing_NIHAO . Original data, more snapshots and other galaxies can be found at https://tobias-buck.de/#sim_data . We encourage interested readers to get in contact with the authors for full data access and advice for use and cite Buck et al. (2020, 2021).\nACKNOWLEDGMENTS\nWeacknowledge the traditional owners of the land on which the ANU stands, the Ngunnawal and Ngambri people. We pay our respects to elders past, and present and are proud to continue their tradition of surveying the night sky and its mysteries to better understand our Universe.\nThis work was supported by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013. SB and KGacknowledge support from the Australian Research Council under grant numbers DE240100150 and DE220100766, respectively. TB acknowledges funding from the Carl Zeiss Stiftung and support from the European Research Council under ERC-CoG grant CRAGSMAN-646955. We gratefully acknowledge the Gauss Centre for Supercomputing e.V. ( www.gauss-centre.eu ) for funding this project by providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre ( www.lrz.de ). Simulations were partially computed with High Performance Computing resources at New York University, Abu Dhabi.\nREFERENCES\n\nAgertz O., et al., 2021, MNRAS, 503, 5826\nCunha K., et al., 2016, Astronomische Nachrichten, 337, 922\nAllende Prieto C., Beers T. C., Wilhelm R., Newberg H. J., Rockosi C. M.,\n\nYanny B., Lee Y. S., 2006, ApJ, 636, 804\n\nAnders F., et al., 2014, A&A, 564, A115\nAnders F., et al., 2017, A&A, 600, A70\nAndrievsky S. M., et al., 2002a, A&A, 381, 32\nAndrievsky S. M., Bersier D., Kovtyukh V. V., Luck R. E., Maciel W. J.,\nLépine J. R. D., Beletsky Y. V., 2002b, A&A, 384, 140\nAndrievsky S. M., Luck R. E., Martin P., Lépine J. R. D., 2004, A&A, 413, 159\nAstropy Collaboration et al., 2013, A&A, 558, A33\nAstropy Collaboration et al., 2018, AJ, 156, 123\nBarbillon M., Recio-Blanco A., Poggio E., Palicio P. A., Spitoni E., de\nLaverny P., Cescutti G., 2024, arXiv e-prints, p. arXiv:2411.10007\nBelfiore F., et al., 2016, MNRAS, 461, 3111\nBelfiore F., et al., 2017, MNRAS, 469, 151\nBellardini M. A., Wetzel A., Loebman S. R., Faucher-Giguère C.-A., Ma X., Feldmann R., 2021, MNRAS, 505, 4586\nBellardini M. A., Wetzel A., Loebman S. R., Bailin J., 2022, MNRAS, 514, 4270\nBensby T., Feltzing S., Oey M. S., 2014, A&A, 562, A71\nBergemann M., et al., 2014, A&A, 565, A89\nBinney J., Tremaine S., 2008, Galactic Dynamics: Second Edition. Princeton University Press\nBird J. C., Kazantzidis S., Weinberg D. H., Guedes J., Callegari S., Mayer L., Madau P., 2013, ApJ, 773, 43\nBland-Hawthorn J., Gerhard O., 2016, ARA&A, 54, 529\nBland-Hawthorn J., Tepper-Garcia T., Agertz O., Federrath C., 2024, ApJ, 968, 86\nBoeche C., et al., 2013, A&A, 559, A59\nBresolin F., Kennicutt R. C., Ryan-Weber E., 2012, ApJ, 750, 122\nBuck T., 2020, MNRAS, 491, 5435\nBuck T., Macciò A. V., Dutton A. A., Obreja A., Frings J., 2019, MNRAS, 483, 1314\nBuck T., Obreja A., Macciò A. V., Minchev I., Dutton A. A., Ostriker J. P., 2020, MNRAS, 491, 3461\nBuck T., Rybizki J., Buder S., Obreja A., Macciò A. V., Pfrommer C., Steinmetz M., Ness M., 2021, MNRAS, 508, 3365\nBuck T., Obreja A., Ratcliffe B., Lu Y., Minchev I., Macciò A. V., 2023,\nMNRAS, 523, 1565\nBuder S., et al., 2019, A&A, 624, A19\nBuder S., Mijnarends L., Buck T., 2024, MNRAS, 532, 1010\nCantat-Gaudin T., Anders F., 2020, A&A, 633, A99\nCasamiquela L., et al., 2019, MNRAS, 490, 1821\nChabrier G., 2003, PASP, 115, 763\nChen Q.-H., Grasha K., Battisti A. J., Kewley L. J., Madore B. F., Seibert M., Rich J. A., Beaton R. L., 2023, MNRAS, 519, 4801\nChen Q.-H., et al., 2024a, MNRAS, 527, 2991\nChen Q.-H., et al., 2024b, MNRAS, 534, 883\nChiappini C., 2002, Ap&SS, 281, 253\nChiappini C., Matteucci F., Gratton R., 1997, ApJ, 477, 765\nChiappini C., Matteucci F., Romano D., 2001, ApJ, 554, 1044\nChieffi A., Limongi M., 2004, ApJ, 608, 405\nChiosi C., 1980, A&A, 83, 206\nDi Matteo P., Haywood M., Combes F., Semelin B., Snaith O. N., 2013, A&A, 553, A102\nDonor J., et al., 2020, AJ, 159, 199\nFoster C., et al., 2021, PASA, 38, e031\nFrankel N., Rix H.-W., Ting Y.-S., Ness M., Hogg D. W., 2018, ApJ, 865, 96\nFrankel N., Sanders J., Ting Y.-S., Rix H.-W., 2020, ApJ, 896, 15\nFraser-McKelvie A., et al., 2022, MNRAS, 510, 320\nGaia Collaboration et al., 2016, A&A, 595, A1\nGarcia A. M., et al., 2023, MNRAS, 519, 4716\nGenovali K., et al., 2014, A&A, 566, A37\nGraf R. L., Wetzel A., Bellardini M. A., Bailin J., 2024, arXiv e-prints, p. arXiv:2402.15614\nGrand R. J. J., Kawata D., Cropper M., 2015, MNRAS, 447, 4018\nGrand R. J. J., et al., 2016, MNRAS, 460, L94\nGrasha K., et al., 2022, ApJ, 929, 118\nHackshaw Z., Hawkins K., Filion C., Horta D., Laporte C. F. P., Carr C.,\nPrice-Whelan A. M., 2024, arXiv e-prints, p. arXiv:2405.18120\nHayden M. R., et al., 2014, AJ, 147, 116\nHayden M. R., et al., 2015, ApJ, 808, 132\nHemler Z. S., et al., 2021, MNRAS, 506, 3024\nHo I. T., Kudritzki R.-P., Kewley L. J., Zahid H. J., Dopita M. A., Bresolin\nF., Rupke D. S. N., 2015, MNRAS, 448, 2030\nHo I. T., et al., 2017, ApJ, 846, 39\nHogg D. W., Eilers A.-C., Rix H.-W., 2019, AJ, 158, 147\nHunter J. D., 2007, Comput Sci Eng, 9, 90\nImig J., et al., 2023, ApJ, 954, 124\nJanes K. A., 1979, ApJS, 39, 135\nJohnson J. W., et al., 2024, arXiv e-prints, p. arXiv:2410.13256\nKarakas A. I., Lugaro M., 2016, ApJ, 825, 26\nKauffmann G., 1996, MNRAS, 281, 475\nKhoperskov S., Sivkova E., Saburova A., Vasiliev E., Shustov B., Minchev I., Walcher C. J., 2023, A&A, 671, A56\nKnollmann S. R., Knebe A., 2009, ApJS, 182, 608\nKollmeier J. A., et al., 2017, arXiv e-prints, p. arXiv:1711.03234\nKreckel K., et al., 2019, ApJ, 887, 80\nKreckel K., et al., 2020, MNRAS, 499, 193\nKrumholz M. R., Ting Y.-S., 2018, MNRAS, 475, 2236\nKrumholz M. R., Burkhart B., Forbes J. C., Crocker R. M., 2018, MNRAS, 477, 2716\nKubryk M., Prantzos N., Athanassoula E., 2015, A&A, 580, A126\nLacey C. G., Fall S. M., 1985, ApJ, 290, 154\nLarson R. B., 1976, MNRAS, 176, 31\nLemasle B., François P., Bono G., Mottini M., Primas F., Romaniello M., 2007, A&A, 467, 283\nLemasle B., François P., Piersimoni A., Pedicelli S., Bono G., Laney C. D., Primas F., Romaniello M., 2008, A&A, 490, 613\nLemasle B., et al., 2013, A&A, 558, A31\nLemasle B., et al., 2022, A&A, 668, A40\nLépine J. R. D., et al., 2011, MNRAS, 417, 698\nLi Z., et al., 2024, MNRAS, 528, 7103\nLilly S. J., Carollo C. M., Pipino A., Renzini A., Peng Y., 2013, ApJ, 772, 119\nLu Y., Buck T., Minchev I., Ness M. K., 2022, MNRAS, 515, L34\n\n\nMa X., Hopkins P. F., Feldmann R., Torrey P., Faucher-Giguère C.-A., Kereš D., 2017a, MNRAS, 466, 4780\nMa X., Hopkins P. F., Wetzel A. R., Kirby E. N., Anglés-Alcázar D.,\nFaucher-Giguère C.-A., Kereš D., Quataert E., 2017b, MNRAS, 467, 2430\nMagrini L., Sestito P., Randich S., Galli D., 2009, A&A, 494, 95\nMagrini L., et al., 2017, A&A, 603, A2\nMatteucci F., Recchi S., 2001, ApJ, 558, 351\nMetha B., Trenti M., Chu T., 2021, MNRAS, 508, 489\nMinchev I., Famaey B., 2010, ApJ, 722, 112\nMinchev I., Chiappini C., Martig M., 2013, A&A, 558, A9\nMinchev I., Chiappini C., Martig M., 2014, A&A, 572, A92\nMinchev I., et al., 2018, MNRAS, 481, 1645\nMoran S. M., et al., 2012, ApJ, 745, 66\nMun M., et al., 2024, MNRAS, 530, 5072\nMyers N., et al., 2022, AJ, 164, 85\nNicholls D. C., Sutherland R. S., Dopita M. A., Kewley L. J., Groves B. A., 2017, MNRAS, 466, 4403\nObreja A., et al., 2019, MNRAS, 487, 4424\nObreja A., Buck T., Macciò A. V., 2022, A&A, 657, A15\nPalla M., Magrini L., Spitoni E., Matteucci F., Viscasillas Vázquez C.,\n\nFranchini M., Molero M., Randich S., 2024, A&A, 690, A334\nPedregosa F., et al., 2011, J Mach Learn Res, 12, 2825\n\nPérez F., Granger B. E., 2007, Comput Sci Eng, 9, 21\n\nPerktold J., et al., 2024, statsmodels/statsmodels: Release 0.14.2,\ndoi:10.5281/zenodo.593847\n\nPilkington K., et al., 2012, A&A, 540, A56\nPilyugin L. S., 2003, A&A, 397, 109\nPilyugin L. S., Tautvaišien˙e G., 2024, A&A, 682, A41\nPilyugin L. S., Grebel E. K., Zinchenko I. A., Nefedyev Y. A., Vílchez J. M., 2017, A&A, 608, A127\nPlanck Collaboration et al., 2014, A&A, 571, A16\nPoggio E., et al., 2018, MNRAS, 481, L21\nPoggio E., et al., 2021, A&A, 651, A104\nPoggio E., et al., 2022, A&A, 666, L4\nPontzen A., Roškar R., Stinson G. S., Woods R., 2013, pynbody:\nAstrophysics Simulation Analysis for Python, Astrophysics Source Code Library, record ascl:1305.002\nQuirk W. J., Tinsley B. M., 1973, ApJ, 179, 69\n\nRatcliffe B. L., Ness M. K., Buck T., Johnston K. V., Sen B., Beraldo e Silva\n\nL., Debattista V. P., 2022, ApJ, 924, 60\nRatcliffe B., et al., 2023, MNRAS, 525, 2208\nRatcliffe B., Khoperskov S., Minchev I., Lee N. D., Buck T., Marques L., Lu L., Steinmetz M., 2024, arXiv e-prints, p. arXiv:2410.17326\nRenaud F., Ratcliffe B., Minchev I., Haywood M., Di Matteo P., Agertz O., Romeo A. B., 2024, arXiv e-prints, p. arXiv:2409.10598\nRolleston W. R. J., Smartt S. J., Dufton P. L., Ryans R. S. I., 2000, A&A, 363, 537\nRosales-Ortega F. F., Díaz A. I., Kennicutt R. C., Sánchez S. F., 2011, MNRAS, 415, 2439\n\nRybizki J., Just A., Rix H.-W., 2017, A&A, 605, A59\n\n\n
F/i.pc/g.pc. A1.- Comparison of gas abundances for oxygen and iron. Shown are absolute oxygen abundances A(O) and the comparison of relative iron and oxygen abundances [Fe/H] - [O/H]. The top panel shows values at face values, whereas the bottom panel shows the comparison for a linear approximation of [Fe/H] from [O/H].
\n\n\nSánchez-Blázquez P., et al., 2014, A&A, 570, A6\nSánchez-Menguiano L., et al., 2016, A&A, 587, A70\nSánchez S. F., et al., 2013, A&A, 554, A58\nSánchez S. F., et al., 2014, A&A, 563, A49\nScarano S., Lépine J. R. D., 2013, MNRAS, 428, 625\n\nSchönrich R., Binney J., 2009, MNRAS, 396, 203\n\nSearle L., 1971, ApJ, 168, 327\nSeitenzahl I. R., et al., 2013, MNRAS, 429, 1156\n\nSharda P., Krumholz M. R., Wisnioski E., Forbes J. C., Federrath C.,\nAcharyya A., 2021, MNRAS, 502, 5935\n\nShaver P. A., McGee R. X., Newton L. M., Danks A. C., Pottasch S. R., 1983, MNRAS, 204, 53\nSpina L., et al., 2021, MNRAS, 503, 3279\nSpitoni E., et al., 2023, A&A, 680, A85\nStanghellini L., Magrini L., Casasola V., 2015, ApJ, 812, 39\nStinson G., Seth A., Katz N., Wadsley J., Governato F., Quinn T., 2006, MNRAS, 373, 1074\nStinson G. S., Brook C., Macciò A. V., Wadsley J., Quinn T. R., Couchman H. M. P., 2013, MNRAS, 428, 129\nSun X., et al., 2024, arXiv e-prints, p. arXiv:2409.09290\n\nTaylor M. B., 2005, ASPC, 347, 29\n\nTepper-Garcia T., Bland-Hawthorn J., Vasiliev E., Agertz O., Teyssier R.,\n\nFederrath C., 2024, arXiv e-prints, p. arXiv:2406.00342\n\nTinsley B. M., 1980, Fund. Cosmic Phys., 5, 287\nTissera P. B., Rosas-Guevara Y., Bower R. G., Crain R. A., del P Lagos C.,\n\nSchaller M., Schaye J., Theuns T., 2019, MNRAS, 482, 2208\n\nTuntipong S., et al., 2024, arXiv e-prints, p. arXiv:2408.12223\nTwarog B. A., 1980, ApJ, 242, 242\nTwarog B. A., Ashman K. M., Anthony-Twarog B. J., 1997, AJ, 114, 2556\nVickers J. J., Shen J., Li Z.-Y., 2021, ApJ, 922, 189\nVilchez J. M., Esteban C., 1996, MNRAS, 280, 720\nVirtanen P., et al., 2020, Nature Methods, 17, 261\nWadsley J. W., Keller B. W., Quinn T. R., 2017, MNRAS, 471, 2357\n\nWalt S. v. d., Colbert S. C., Varoquaux G., 2011, Comput Sci Eng, 13, 22\nWang L., Dutton A. A., Stinson G. S., Macciò A. V., Penzo C., Kang X.,\nKeller B. W., Wadsley J., 2015, MNRAS, 454, 83\nWillett E., et al., 2023, MNRAS, 526, 2141\nWilliams M. J., Bureau M., Cappellari M., 2009, MNRAS, 400, 1665\n\nWyse R. F. G., Silk J., 1989, ApJ, 339, 700\nYong D., Carney B. W., Friel E. D., 2012, AJ, 144, 95\nZari E., Hashemi H., Brown A. G. A., Jardine K., de Zeeuw P. T., 2018, A&A, 620, A172\nZari E., Rix H. W., Frankel N., Xiang M., Poggio E., Drimmel R., Tkachenko A., 2021, A&A, 650, A112\n\nZaritsky D., Kennicutt Robert C. J., Huchra J. P., 1994, ApJ, 420, 87\nZhang C., Li Z., Hu Z., Krumholz M. R., 2024, arXiv e-prints, p.\narXiv:2411.01518\nvan de Sande J., Fraser-McKelvie A., Fisher D. B., Martig M., Hayden M. R., the GECKOS Survey collaboration 2023, arXiv e-prints, p. arXiv:2306.00059\nAPPENDIX\nA. ADDITIONAL FIGURES\nFig. A1 demonstrates the tight correlation of gas phase iron and oxygen abundance and how to approximate them linearly.\nThis paper was built using the Open Journal of Astrophysics L A T E X template. The OJA is a journal which provides fast and easy peer review for new papers in the astro-ph section of the arXiv, making the reviewing process simpler for authors and referees alike. Learn more at http://astro.theoj.org .\n"},"sections":{"kind":"list like","value":[{"title":"ABSTRACT","content":"Radial metallicity gradients are fundamental to understanding galaxy formation and evolution. In our high-resolution simulation of a NIHAO-UHD Milky Way analogue, we analyze the linearity, scatter, spatial coherence, and age-related variations of metallicity gradients using young stars and gas. While a global linear model generally captures the gradient, it ever so slightly overestimates metallicity in the inner galaxy and underestimates it in the outer regions of our simulated galaxy. Both a quadratic model, showing an initially steeper gradient that smoothly flattens outward, and a piecewise linear model with a break radius at 10 kpc (2.5 effective radii) fit the data equally better. The spread of [Fe/H] of young stars in the simulation increases by tenfold from the innermost to the outer galaxy at a radius of 20 kpc. We find that stars born at similar times along radial spirals drive this spread in the outer galaxy, with a chemical under- and over-enhancement of up to 0.1 dex at leading and trailing regions of such spirals, respectively. This localised chemical variance highlights the need to examine radial and azimuthal selection effects for both Galactic and extragalactic observational studies. The arguably idealised but volume-complete simulations suggest that future studies should not only test linear and piecewise linear gradients, but also non-linear functions such as quadratic ones to test for a smooth gradient rather than one with a break radius. Either finding would help to determine the importance of different enrichment or mixing pathways and thus our understanding of galaxy formation and evolution scenarios. Subject headings: Galaxy: structure - Galaxy: abundances - galaxies: structure - galaxies: abundances","pages":[1]},{"title":"LOCAL VARIATIONS OF THE RADIAL METALLICITY GRADIENT IN A SIMULATED NIHAO-UHD MILKY WAY ANALOGUE AND THEIR IMPLICATIONS FOR (EXTRA-)GALACTIC STUDIES","content":"S/v.pc/e.pc/n.pc B/u.pc/d.pc/e.pc/r.pc 1 , 2 , ∗ , T/o.pc/b.pc/i.pc/a.pc/s.pc B/u.pc/c.pc/k.pc 3 , 4 , Q/i.pc/a.pc/n.pc-H/u.pc/i.pc C/h.pc/e.pc/n.pc ( 陈 千 惠 ) 1 , 2 , /a.pc/n.pc/d.pc K/a.pc/t.pc/h.pc/r.pc/y.pc/n.pc G/r.pc/a.pc/s.pc/h.pc/a.pc 1 , 2 , ∗ 1 Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT 2611, Australia 2 ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Australia 3 Universität Heidelberg, Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, Im Neuenheimer Feld 205, D-69120 Heidelberg, Germany and 4 Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische Astrophysik, Albert-Ueberle-Strae 2, D-69120 Heidelberg, Germany Version December 3, 2024","pages":[1]},{"title":"1. INTRODUCTION","content":"Understanding the radial metallicity gradient, defined as the change in heavy element abundance with galactocentric radius, in galaxies provides critical insights into their formation and evolutionary processes, such as inside-out formation, gas accretion, outflows, and radial migration (e.g. Quirk & Tinsley 1973; Tinsley 1980; Lacey & Fall 1985; Wyse & Silk 1989; Kauffmann 1996; Chiappini et al. 1997; Schönrich & Binney 2009; Moran et al. 2012; Bird et al. 2013). The decrease in metallicity with increasing distance from the Galactic centre is well-established both theoretically (Larson 1976; Tinsley 1980; Chiosi 1980) and observationally in the Milky Way (Searle 1971; Janes 1979; Twarog et al. 1997) and other massive spiral galaxies (e.g. Tinsley 1980; Zaritsky et al. 1994; Bresolin et al. 2012). The Milky Way, being the only galaxy where we can resolve millions of stars, provides a unique opportunity to study these gradients and deviations from them in detail. Early evidence by Janes (1979) suggested a linear gradient on the order of d [ Fe / H ]/ d 𝑅 = -0 . 05 ± 0 . 01 dex kpc -1 for the Milky Way which aligns very well with more recent measurements (Anders et al. 2017; Hayden et al. 2015). However, these gradients are accompanied by a significant spread in [Fe/H] of 0 . 1 -0 . 15 dex, as noted by Twarog (1980), which may imply a fine structure of the metallicity gradient (see Genovali et al. 2014). With increasing sample size and measurement precision, the specific shape and characteristics of this gradient remain somewhat unclear (Chiappini 2002). Previous studies have ∗ Australian Research Council DECRA Fellow for example claimed more intricate non-linear trends, bends, and flattening in the gradient of the Milky Way (e.g. Donor et al. 2020) and other galaxies (e.g. Pilyugin 2003; Sánchez et al. 2014) or even sequences of shapes (Pilyugin et al. 2017; Pilyugin & Tautvaišien˙e 2024), which were fitted with different models (Rosales-Ortega et al. 2011; Bresolin et al. 2012), such as piecewise linear ones (e.g. Sánchez-Menguiano et al. 2016) or non-linear ones (e.g. Scarano & Lépine 2013). Variations in the metal distribution, including breaks of the gradient at specific radii, give rise to a plethora of possible physical explanations, such as star formation efficiency variations and localised star formation bursts (Sánchez et al. 2014; Sánchez-Blázquez et al. 2014; Ho et al. 2015), gas accretion and dilution at different rates (Bresolin et al. 2012; Sánchez et al. 2013; Belfiore et al. 2016; Sánchez-Menguiano et al. 2016), gas outflows and feedback (Lilly et al. 2013; Ma et al. 2017a), as well as disk instabilities or local overdensities (Grand et al. 2016; Ho et al. 2017). In particular, Scarano & Lépine (2013) suggested that gradient break radii coincided with the corotation radii of spiral arms. Recent advancements in both computations and observations have significantly expanded our capabilities. For example, in terms of observational data in the Milky Way, the Gaia mission (Gaia Collaboration et al. 2016) enables more detailed studies of these gradients. New suites of large-scale simulations now allow us to gain insights into radial metallicity gradients across a range of simulated galaxies, including Milky Way analogues. This presents opportunities to revisit outstanding challenges of the detailed shape of the radial metallicity gradient. For instance, Hogg et al. (2019) created an extensive metallicity map of the Milky Way using APOGEE and Gaia data, while Poggio et al. (2022) mapped young stars and found metallicity variations around spiral arms (see also Zari et al. 2018, 2021; Poggio et al. 2021; Hackshaw et al. 2024). Similarly, Imig et al. (2023, among others) traced gradients across different stellar populations and ages, emphasizing the importance of considering radial migration effects (Binney & Tremaine 2008; Frankel et al. 2018, 2020). Historically, radial metallicity gradients have been measured using various stellar populations and gas tracers. Estimated gradients seem to be broadly consistent across different stellar tracers, such as young open clusters (e.g. Yong et al. 2012; Cunha et al. 2016; Magrini et al. 2017; Casamiquela et al. 2019; Donor et al. 2020; Spina et al. 2021; Myers et al. 2022), young hot (OB-type) stars (Zari et al. 2018, 2021; Poggio et al. 2021, 2022), field stars close to the Galactic plane (e.g. Bergemann et al. 2014) or Cepheids (Andrievsky et al. 2002a,b; Lemasle et al. 2007, 2013). Despite extensive observational efforts, several challenges persist for studies in the Milky Way. The completeness (or patchiness) of observed datasets remains a fundamental issue (Bergemann et al. 2014). The robustness of fits to the incomplete data is still contentious, including the need to actually fit two linear gradients with a break radius at corotation radius (Bresolin et al. 2012, and references therein) or further out (Yong et al. 2012; Donor et al. 2020) - or even more complicated functions (see e.g. Chiappini et al. 2001; Kubryk et al. 2015). Furthermore, methodologies for fitting linear models to scattered data need re-evaluation (Metha et al. 2021). Different samples yield varying gradients, potentially due to biases in data or the inclusion of older stars (e.g. Allende Prieto et al. 2006; Hayden et al. 2014; Anders et al. 2014; Vickers et al. 2021; Willett et al. 2023). The impact of spiral arm structures (Poggio et al. 2021), the Galactic warp (Lemasle et al. 2022) or bar-driven mixing (Di Matteo et al. 2013) on metallicity gradients is not fully understood. Understanding these gradients in the Milky Way is also crucial for extragalactic studies, where spatial resolution limits observations in different ways. In extragalactic systems, metals are mainly traced via gas, because it provides a more direct measure of the ongoing enrichment processes, unlike stars, which primarily reflect the integrated chemical history of the past. Observationally, gas emission lines are typically brighter and more accessible across large distances than stellar absorption lines, allowing for broader spatial coverage, especially in distant galaxies. Consequently, extragalactic studies often focus on gas-phase metallicity as traced by oxygen, A ( O ) = 12 + log ( O / H ) , while Galactic studies typically use stellar iron abundance [ Fe / H ] = A ( Fe ) -A ( Fe ) ⊙ as a metallicity tracer (e.g. Nicholls et al. 2017; Fraser-McKelvie et al. 2022). New instruments like the MUSE integral field spectrograph have enabled a plethora of extragalactic studies to contrast the Milky Way and techniques like the spectroscopy of H /i.pc/i.pc regions and planetary nebulae have helped to infer gas metallicity distributions in external galaxies (Shaver et al. 1983; Vilchez & Esteban 1996; Rolleston et al. 2000; Bresolin et al. 2012). Recent examples include Sánchez et al. (2014) with CALIFA galaxy observations as well as Mun et al. (2024) and Chen et al. (2024a) who use MAGPI observations to probe for example the effects of spiral arms. Notable is also the scatter that Chen et al. (2023) found for the gas metallicity across galactic radii with TYPHOON observations (see their. Figs. 4-6). Grasha et al. (2022) found that the gas metallicity gradient plateaus at a lower limit in their TYPHOON galaxies at the outermost radii - an observation replicated by IllustrisTNG simulations (Hemler et al. 2021; Garcia et al. 2023). From a modelling perspective, galactic chemical evolution models can both test understanding of radial metallicity gradients and make predictions beyond the limited volumes and tracers tested by Milky Way and extragalactic studies. Such galactic chemical evolution models include Chiappini et al. (2001); Matteucci & Recchi (2001); Minchev et al. (2014); Kubryk et al. (2015); Stanghellini et al. (2015); Rybizki et al. (2017); Spitoni et al. (2023); Johnson et al. (2024). Sharda et al. (2021) even presented a model for gas phase metallicity gradients in galaxies and their evolution from first principles (see also Krumholz & Ting 2018). In exploring radial metallicity gradients through simulations, we have better understood how different processes influence these gradients across galactic models and temporal scales. Studies such as Pilkington et al. (2012) in RaDES simulations reveal that gradients are typically established via inside-out galaxy formation. Khoperskov et al. (2023) quantified the scatter of gas metallicity to ≈ 0 . 04 -0 . 06 dex at a given galactocentric distance in their simulations. Meanwhile, the EAGLE simulations used by Tissera et al. (2019) provide insights into how these gradients vary with galaxy characteristics like stellar mass and merger history, emphasizing the dynamic nature of metallicity distributions. The plethora of simulations such as AURIGA (Grand et al. 2016), FIRE (Ma et al. 2017b, see their Fig. 6) or VINTERGATAN (see their Fig. 9; Agertz et al. 2021) also allow us to explore the gradient evolution of galactic timescales. Buck et al. (2023), for example, found a link of major accretion events with periods of unexpected steepening in the metallicity gradient within NIHAO-UHD simulations - closely resembling findings for the Milky Way by Lu et al. (2022) and Ratcliffe et al. (2023). FIRE simulations, examined by Bellardini et al. (2021), Bellardini et al. (2022), and Graf et al. (2024), extend these findings by comparing radial metallicity gradients and their azimuthal scatter across gas and stellar components and illustrate the complex interplay between galactic structure and metal enrichment processes. Similarly, Grand et al. (2015, 2016) highlight temporal changes in metallicity gradients already within 120 Myr, or roughly one galactic rotation. Such rapid changes underscore the impact of transient galactic events on the metal distribution, linking them to star formation patterns along spiral arms and the broader evolutionary history of the galaxy. In this study, we analyze a high-resolution NIHAO-UHD simulation of a Milky Way analogue to bridge the observational gap between detailed studies of our galaxy and broader extragalactic surveys. We aim to reveal subtle features of the radial metallicity gradient, which may be obscured by observational constraints in both the Milky Way and distant galaxies, by testing the following properties within the observationally probed inner 𝑅 Gal . ≤ 20 kpc: XGal. The paper is structured as follows: Section 2 describes the data of our Milky Way analogue NIHAO-UHD simulation. Section 3 analyses the linearity of the radial metallicity gradient of the simulation, the first of our four objectives. Section 4 then analyses both the scatter and local deviations from the gradient as well as the coherence of the gradient with vertical and azimuthal position as well as age (the remaining three objectives). Section 5 discusses them individually. We note that in this research we are mainly interested in the specific shape of the radial metallicity gradient for radii relevant to Galactic observations. However, we also discuss our results in the context of extragalactic results that probe beyond the inner 20 kpc of a galaxy. Section 6 bundles our results into an overarching conclusion.","pages":[1,2,3,4]},{"title":"2. DATA: A NIHAO-UHD MILKY WAY ANALOGUE SIMULATION","content":"For this project, we use a cosmological zoom-in simulation of a Milky Way analogue ( g8.26e11 ) from the Numerical Investigation of a Hundred Astronomical Objects (NIHAO, Wang et al. 2015) suite. This model galaxy was calculated as part of the NIHAO-UHD project (Buck et al. 2020) and has previously been used in various works studying Milky Way satellites (Buck et al. 2019), Milky Way's dark halo spin (Obreja et al. 2022), inferring birth properties of stars with abundance clustering (Ratcliffe et al. 2022), as well as the evolution of the interstellar medium's radial metallicity gradient since redshift three (Ratcliffe et al. 2024). Simulations were carried out with the smoothed particle hydrodynamics code Gasoline2 (Wadsley et al. 2017), including sub-grid turbulent mixing, using cosmological parameters from Planck Collaboration et al. (2014) with initial conditions and energetic feedback descriptions from the NIHAO project (Wang et al. 2015). Zoom-in simulations were then performed as described in detail by Buck et al. (2021) with star formation following Stinson et al. (2006) and energetic feedback following Stinson et al. (2013). We note that this is a slightly different rerun of the same simulation than the one studied by Buder et al. (2024); in this work, we use a higher resolution version and updated chemical yields. Because computational resources still limit the mass resolution of simulations, we are relying on tracer particles that represent simple stellar populations (SSPs) with the same age, overall metallicity and discrete initial mass function (IMF). Buck et al. (2021) have implemented the flexible chemical evolution code /c.pc/h.pc/e.pc/m.pc/p.pc/y.pc (Rybizki et al. 2017) to calculate the chemical yields for the SSPs. In particular, we use the alternative ( alt ) setup of /c.pc/h.pc/e.pc/m.pc/p.pc/y.pc that assumes a Chabrier (2003) IMF with high-mass slope of 𝛼 IMF = -2 . 3 over a mass range of 0 . 1 -100 M ⊙ for SSPs across a metallicity range of 𝑍 / 𝑍 ⊙ ∈ [ 10 -5 , 2 ] . The code calculates the contribution from asymptotic giant branch (AGB) stars, CCSN across a mass range of 8 -40 M ⊙ , and SNIa with an exponential function with exponent -1 . 12, a delay time of 40 Myr, and a normalization of the SNIa rate of log 10 ( NIa ) = -2 . 9. For each of these nucleosynthetic channels, yields from the following studies are used: Chieffi & Limongi (2004) for CCSN, Seitenzahl et al. (2013) for SNIa, and Karakas & Lugaro (2016) for AGB stars ( new_fit model in Buck et al. 2021). Contrary to a previous study by Buder et al. (2024), we take the elemental abundances at face value and do not apply any shifts. We limit the simulation data to the main halo by applying /p.pc/y.pc/n.pc/b.pc/o.pc/d.pc/y.pc's implementation of the Amiga Halo Finder (Knollmann & Knebe 2009) and then reposition and rotate this main halo to be face-on based on the angular momentum with /p.pc/y.pc/n.pc/b.pc/o.pc/d.pc/y.pc's /a.pc/n.pc/a.pc/l.pc/y.pc/s.pc/i.pc/s.pc./a.pc/n.pc/g.pc/m.pc/o.pc/m.pc./f.pc/a.pc/c.pc/e.pc/o.pc/n.pc module (Pontzen et al. 2013). We then further transform the resulting galactocentric Cartesian coordinate ( 𝑋,𝑌, 𝑍 ) and velocities ( 𝑉 𝑋 , 𝑉 𝑌 , 𝑉 𝑍 ) to Cylindrical ones as done in a previous study of this main halo by Buder et al. (2024). The model galaxy has a virial radius of 𝑅 vir = 𝑅 200 = 206 kpc and a total mass (gas, stars and dark matter) inside 𝑅 vir of 9 . 1 · 10 11 M ⊙ . At redshift zero, it contains 8 . 2 · 10 11 M ⊙ dark matter, 6 . 4 · 10 10 M ⊙ gas mass and 2 . 3 · 10 10 M ⊙ stellar mass with a stellar mass resolution of around 7 . 5 · 10 3 M ⊙ . When using a fifth of the virial radius as a reference to calculate total luminosity /one.sup and mass, we estimate a half-light radius, that is, effective radius of 𝑅 𝑒 = 3 . 79 kpc and a half-stellar-mass radius of 2 . 97 kpc. To achieve a roughly similar selection as the observational data of the Milky Way (Genovali et al. 2014) and other galaxies (e.g. Chen et al. 2023), we restrict the simulation data to a galactocentric radius of 𝑅 Gal ≤ 20 kpc and | 𝑧 | ≤ 10 kpc, as shown in Fig. 1. Similar to the Milky Way (Poggio et al. 2018; Lemasle et al. 2022), we note a warp of the stellar and gaseous disk (see Figs. 1b and 1e, respectively). To avoid too strong effects of radial migration (Binney & Tremaine 2008; Frankel et al. 2018; Grand et al. 2016; Minchev et al. 2018) while maintaining a sufficiently large sample size we further enforce stars to be younger than 0 . 5 Gyr, corresponding to roughly the time of four galactic rotations, and being half the value found by Minchev et al. (2018) for very limited migration in the Milky Way. This selection defacto limits the vertical range of 99% of stars to | 𝑧 | = 1 . 4 kpc. The strong influence of this age cut on the vertical distribution of stars in the Milky Way analogue can best be appreciated from the difference of vertical density distributions of stars in Figs. 1b and 1c. We are applying these cuts for all following analyses of the radial metallicity gradient in Section 3.","pages":[4]},{"title":"3. THE LINEARITY OF THE RADIAL METALLICITY GRADIENT IN NIHAO-UHD","content":"In this section, we analyse the functional shape of the radial metallicity gradient. To get a first impression of possible shapes, we show the face-on view of the decreasing radial metallicity gradient of the simulation in Figs. 2a (for young stars) and Fig. 2d (for gas). Foreshadowing the later parts of this work on local variations, we also show a linear radial fit to either distribution in Fig. 2b and 2e and show the fit residuals in Figs. 2c and 2f. At the moment, however, we focus on the linearity and thus the logarithmic density distribution of star particle iron abundances [Fe/H] across different galactocentric radii 𝑅 Gal . . This distribution is shown in Fig. 3a and strongly suggests that the gradient is predominantly linear, similar to findings for the Milky Way. More complex shapes, such as piecewise linear ones have been suggested based on incomplete and limited data in the literature. We are thus also analysing these shapes with the complete and better-sampled data points of the NIHAO-UHD simulation. We firstly test different global fits in Section 3.1, before testing the influence of binning and coverage in Sections 3.2 and 3.3, respectively.","pages":[4]},{"title":"3.1. Global gradient fits","content":"We fit three different functional forms to the global data: a linear function (used for Fig. 2b) a piecewise linear with a break radius 𝑅 break and a quadratic function The coefficients of the functions are fitted with the /s.pc/c.pc/i.pc/p.pc/y.pc./o.pc/p.pc/t.pc/i.pc/m.pc/i.pc/z.pc/e.pc function /c.pc/u.pc/r.pc/v.pc/e.pc_/f.pc/i.pc/t.pc (Virtanen et al. 2020) and listed in Table 2. To estimate the uncertainty of the break radius 𝑅 break, we use the profile likelihood method to identify the radii at which the residual sum of squares (RSS) values are increased by 1 𝜎 from the best RSS radius in steps of Δ 𝑅 break = 0 . 1 kpc and 0 . 5 kpc for the full and binned data set, respectively. We compute the RSS for each model 𝑓 𝑖 (see Eqs. 1-3) based on the 𝑁 data points as We have confirmed the robustness of our fits by applying other fitting routines as outlined in Table 1. After fitting three different functional forms, we use a combination of parameters to determine which model provides the best fit. In Table 2, the RSS value is the smallest (although only by a small margin) for the quadratic function. When assuming 𝜎 2 = 𝑅𝑆𝑆 / 𝑁 , we can also define a logarithmic likelihood for the 𝑁 data points. For 𝑘 free parameters, we then calculate the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) as For these criteria, the quadratic function performs slightly better than the linear or piecewise linear functions (see Table 2). We show the fit residuals in Fig. 3b as density distribution as well as in Fig. 3c as percentile distributions in radial bins of Δ 𝑅 Gal = 0 . 5 kpc. While the density distribution shows substructure, which we investigate later in Section 4, we note an increase in the median residuals of the linear fit in Fig. 3c towards the inner and outer radii, especially for 𝑅 Gal . > 17 kpc. A quadratic fit (see orange lines in Fig. 3) results in a slightly steeper linear component of the gradient (from -0 . 0411 to -0 . 0497 dex kpc -1 ), which is counteracted by the quadratic flattening term of + 0 . 0005 dex kpc -2 . The latter leads to an RGal. effective flattening of -0 . 172 + 0 . 200 = 0 . 028 dex (linear vs. quadratic terms) at 𝑅 Gal . = 20 dex. While seemingly only a nuisance correction across the large extent of [Fe/H] and 𝑅 Gal . , this quadratic function outperforms the linear fit. This is most apparent in Fig. 3c, where the orange line better traces the median residuals from the linear function across all radii. This suggests that non-linear functions, such as piecewise linear or quadratic ones describe the gradient better. Distinguishing between the two latter functional forms is challenging. The quantitative performance indicators-RSS, AIC, and BIC-show very similar values for both forms, and a closer examination of the residuals in Fig. 4 reveals no clear visual advantage for either the piecewise linear or quadratic model. TAKE-AWAY: Both piecewise linear and quadratic functions provide a better fit to the radial metallicity relation than a simple linear model. However, based on our assessments, there is no clear preference between the piecewise and quadratic functions.","pages":[5]},{"title":"3.2. The influence of binning","content":"In this section, we test the influence of fitting a function to all points of the distribution or binned data in steps of Δ 𝑅 Gal . = 0 . 5 kpc, using median values as data points and standard deviations /two.sup as uncertainty (see also Hemler et al. 2021, who fitted functions to radially binned IllustrisTNG data). The results are shown in Fig. 4. Given that more than half of the young star particles of the galaxy are within /two.sup Wenote that this 𝜎 is not equivalent to observational uncertainty and can thus not be directly applied onto observational analyses. 𝑅 Gal . < 4 kpc, this binning - although counteracted by the smaller spread of [Fe/H] in the inner galaxy - weighs the distribution of the inner galaxy significantly less than when weighing all particles equally (20 vs. 34 000 data points). The parameters fitted to the binned data exhibit a larger uncertainty due to our use of the spread of [Fe/H] per bin as absolute uncertainty 𝜎 , but the fitted parameters agree well within the fitting uncertainties. TAKE-AWAY: While the specific slopes differ when fitting all points or binned data, they agree within the small fitting uncertainties.","pages":[5,6]},{"title":"3.3. The influence of radial coverage on linear fits","content":"Although we have gained useful insight into the global function, observational data will rarely cover the full extent of the stellar disk. Milky Way studies have previously been limited to the range of around 5 -15 kpc. There are often similar limitations and even gaps in extragalactic data. Using smaller ranges, observational studies have found hints of piece-wise linear gradients with a break radius in them based on limited radial coverage (e.g. Andrievsky et al. 2002a; Yong et al. 2012; Boeche et al. 2013; Hayden et al. 2014; Anders et al. 2017; Donor et al. 2020; Chen et al. 2023). These results are intriguing, since a quadratic function can, to first order, be approximated by two linear functions with a break radius. We therefore want to use our simulation to test if the radial coverage may indeed delude us into identifying broken linear gradients. Wetest how smaller coverage and piecewise linear fits could mimic a complex global gradient by fitting in piecewise lin- ear radial ranges of 0.25, 0.5, 1, 2, 5, 8, 10, 15, and 20 kpc. We show their difference with respect to a global linear fit in Fig. 5, with color-coding indicating the slope of the local gradient. A horizontal dashed line indicates the same slope as the global fit, whereas the offset of a line from the said horizontal dashed line indicates the local deviation from the global gradient intercept. Differences in line slopes are visualising the difference in gradient slopes between the global and local fits. We see that all ranges suggest more or less significant deviations from a global linear fit. The innermost fit suggests a significantly different gradient than the outermost fit. We also note increasing slope differences towards the smallest scales, hinting at local deviations from a global pattern. We follow these up in Section 4, but for now, focus on the larger-scale trends. When directly comparing an inner and outer radius fit, such as between 𝑅 Gal . = 5 -10 kpc (thick grey line in Fig. 5) and 𝑅 Gal . = 10 -15 kpc (thick black line in Fig. 5), we note a significant change, similar to previous estimates of the Milky Way (e.g. Yong et al. 2012; Lemasle et al. 2008). In our case, the gradient estimate changes from [ Fe / H ] ( 𝑅 Gal . ) = 0 . 471 -0 . 044 · 𝑅 Gal . to [ Fe / H ] ( 𝑅 Gal . ) = 0 . 375 -0 . 034 · 𝑅 Gal . . When looking at linear gradient fits across the radial coverage of Δ 𝑅 Gal . = 5 -15 kpc in Fig. 5, the gradient is steeper (bluer color) for smaller radii and flatter (redder) for larger radii. Indeed, a piecewise linear function can well mimic a complex global gradient. We note that in the simulated data, we see local deviations that become traceable below Δ 𝑅 Gal . ≤ 2 kpc ∼ 0 . 5 Re (bottom part of Fig. 5). This might indicate the spatial resolution required to see local effects, such as spiral arms, for extragalactic studies (see also Krumholz & Ting 2018; Li et al. 2024). We pursue this observation in the following Section 4. TAKE-AWAY: We find that a piecewise linear function can well mimicaquadraticfunction across the scales used in Milky Way and extragalactic studies. Local deviations become traceable below are spatial resolution of Δ 𝑅 Gal . ≤ 2 kpc (or Δ 𝑅 Gal . ≤ 0 . 5 Re).","pages":[6,7]},{"title":"4. SCATTER AND LOCAL DEVIATIONS FROM THE GRADIENT","content":"Now that we are sufficiently satisfied that our flattening gradient function reproduces the overall shape of the radial metallicity gradient, we are concerned with both the scatter and local slope deviations across the galactocentric radii in this section. In detail, we analyse the scatter (Section 4.2), vertical variations (Section 4.2), azimuthal variations (Section 4.3, particularly motivated by the localised, spiral-shaped fit residuals of Figs. 2c and 2f) and deviations across different ages (Section 4.4).","pages":[7]},{"title":"4.1. Scatter","content":"When investigating the change in scatter from the innermost radii to the outermost (see Fig. 3c), we see a steady increase in 1 -𝜎 spread. This spread increases from 𝜎 [ Fe / H ] = 0 . 01 dex at 𝑅 Gal . = 0 . 25 ± 0 . 25 kpc to 𝜎 [ Fe / H ] = 0 . 06 dex at 𝑅 Gal . = 8 . 25 ± 0 . 25 kpc and reaches 𝜎 [ Fe / H ] = 0 . 10 dex at 𝑅 Gal . = 19 . 75 ± 0 . 25 kpc. When we recall the observed significant spread in metallicities of young open clusters at the solar radius beyond observational uncertainty (e.g. Donor et al. 2020; Spina et al. 2021) and our selection of only young ( < 0 . 5 Gyr) stars from the simulation, a strong impact of this scatter by radial migration should be excluded. At this point, we can imagine that this chemical diversity might be caused by less well-mixed gas or non-radial effects (such as vertical or azimuthal ones), which we investigate subsequently.","pages":[7]},{"title":"4.2. Vertical deviations","content":"In this section, we now look at deviations with respect to the vertical dimension, that is, 𝑅 -𝑧 . In Fig. 6 we show the previously identified local gradient deviations (lines following the left axis label) on top of the vertical density distribution ( 𝑅 -𝑧 ) of young stars (Fig. 6a) and gas (Fig. 6b) between -3 < 𝑧 < 3 kpc. Although the quickly decreasing number of young stars (Fig. 6a) at outer radii does not show substructure in the density plots for reasonable bin sizes, we see more substructure for the gaseous component in Fig. 6b). In particular, we see rather minor deviations at small radii (where most stars and gas are close to the plane). At increasing radii, we notice an increase in both the vertical distribution of stars, and increasing local gradient deviations. In particular, we note a significant deviation of the slope around 𝑅 Gal . ∼ 15 kpc, where the gradient deviation line is steep and blue (indicating a much steeper gradient at this radius), and we notice a significant overdensity of gas around 𝑧 ∼ 1 kpc. Overall, however, we do not see strong correlations in this particular plane. This could, however, be caused by a super-position effect of the up- and downturn at larger radii due to the galactic warp (see Figs. 1b and 1e). Although the warp of the stellar disk in Fig. 1b is not as clear, we confirm that both the gas disk and the youngest stars below 0 . 5 Gyr are tracing each other across the simulation in both galactocentric radius 𝑅 Gal . and height 𝑧 Gal . for different sectors in the azimuthal direction. We note that the superposition in Fig. 6 could smear out local correlations of slope changes with gas overdensities, for example, by spiral arms. Although such an edge-on view of the galaxy may indeed be the only observable one for extragalactic targets, for example, of the GECKOS survey of edge-on galaxies (van de Sande et al. 2023), we have the luxury of being able to analyse the azimuthal direction of our simulated galaxy, too. TAKE-AWAY: We see no strong correlations of deviations in the vertical direction throughout the simulation. Such correlations could, however, be blurred by azimuthal effects, like the galactic warp, which needs to be disentangled in the azimuthal dimension.","pages":[7]},{"title":"4.3. Azimuthal deviations","content":"To analyse the deviations from a global gradient across different azimuthal viewing angles, we divide the galaxy into 8 sectors with Δ 𝜛 Gal . = 45 · (see Fig. 7a). This allows us to study the positions around the upturn and downturn of the galactic warp with the median azimuth of young star particles below and above the plane being 𝜑 Gal . ∼ 183 · and 𝜑 Gal . ∼ 4 · , respectively (see Fig. 1e), while maintaining a reasonable sample size. At face value, the distribution of 𝑅 Gal . - [ Fe / H ] for each sector follows a similar, rather linear shape with most stars being born in the inner 5 kpc of the galaxy. However, we find significant deviations in different sectors of the galaxy (Fig. 7). On the one hand, we find non-linear deviations as bumps with slightly increased or decreased iron abundance - up to 0 . 1 -0 . 2 dex - in Figs. 7c at 𝑅 Gal ∼ 18 kpc, 7d at 𝑅 Gal ∼ 10 kpc, 7f at 𝑅 Gal ∼ 14 kpc, and 7g at 𝑅 Gal ∼ 17 kpc. On the other hand, we find significant gaps in the distribution at similar [Fe/H], most strikingly at the upturn of the galactic warp in Fig. 7e ( 𝜛 Gal . = 135 -180 · ) at [ Fe / H ] ∼ 0 dex and 𝑅 Gal . ∼ 8 -14 kpc. We note that the sector e) with the gap is surrounded by two sectors (d and f) with significant overabundance at the same radius. This could be indicative of stars having formed as a result of gas moving from sector e towards either azimuthal direction, causing a gas overdensity which could in turn lead to higher star formation activity. To establish this observation, we take a closer look at the timedomain, that is, stellar age as well as the spatial domain of 𝑅 Gal . -𝜑 Gal . in the next section. TAKE-AWAY: We find various deviations from the global trend in the azimuthal direction, including gaps and isolated streaks of stars with similar [Fe/H] throughout Δ 𝑅 Gal . = 2 -6 kpc. These can introduce local over- and under-enhancement of up to ± 0 . 2 dex in [Fe/H] at a given radius. In the next section, we analyse whether the stars of these streaks have been born at the same or different time.","pages":[7,9]},{"title":"4.4. Deviations with time and age","content":"In this section, we examine the radial metallicity gradient in a small age range less than 0 . 5 Gyr. To do so, we color Fig. 7 by the median stellar age rather than the logarithmic density in Fig. 8. We find an overall significant scatter across time, suggesting a good mix of star formation across all sectors for stars born within less than 0 . 5 Gyr. For stars within this restricted age range, we do not see a strong correlation with radius, such as older stars being born further inside, but a larger amount of stars being born closer to the galactic centre. Wenote that stars with similar [Fe/H] in each sector tend to be formed at similar times (within 50 Myr), that is, as flat lines with the same color (age) in Figs. 8b-i. To guide the eye, we have identified Group 1 in Fig. 8e (around 𝑅 Gal . ∼ 14 kpc at 𝜛 Gal . = 180 -225 · ). We further note that the enriched bumps identified earlier are born at similar times, see, for example, Group 2 in Fig. 8f. The coloring by age also reveals that stars with lower [Fe/H] than expected (see Group 3 in Fig. 8h) are born at similar times. In some cases, these extend to Δ 𝑅 Gal . = 2 -6 kpc, see Groups 1, 2, and 3 in Fig. 8. At a given radius 𝑅 Gal . , these streaks cause a significant spread in local [Fe/H] of up to ± 0 . 2 dex (see Fig. 8). From the analysis of azimuthal sectors, the impression arose that the star formation in this simulated Milky Way analogue is - as expected for a spiral galaxy - rather patchy and localised on the smallest timescales. This is confirmed by looking at the spatial distribution of azimuth 𝜑 Gal . and radius 𝑅 Gal . in Fig. 9. Already when looking at the density distribution of all stars born within less than 0 . 5 Gyr in Fig. 9a, multiple streams are visible, stars on spiral patterns (see also Kreckel et al. 2019; Chen et al. 2024b). When following up the previously identi- roups 1, 2, and 3, we recover them on said spiral patterns (Groups 1 and 3) or a local overdensity (Group 2). Although one could imagine that radial migration might induce such a spiral-like shape for the stars of groups 1 and 3, their low age of less than 250 Myr would require a significant migration effect of several kpc, while having no influence on the older stars of group 2. When tracing the position of significant overdensities from Fig. 9a in the same projection colored by age in Fig. 9b, we note that for radii above 𝑅 Gal . > 5 kpc these overdensities are colored in red, that is, containing indeed young stars with ages below 200 Myr and being consistent with the most recent star formation along these spiral patterns in the outer galaxy. TAKE-AWAY: We find significant scatter across the radial metallicity distribution caused by streaks of stars born with similar [Fe/H] at similar times (within 50 Myr) across either very local or radially extended spiral-shaped regions of the galaxy, suggesting local enhancement patterns in small overdensities or along spiral arms.","pages":[9]},{"title":"5. DISCUSSION","content":"Having presented the analysis, we now put our results into the context of other work in terms of our initial aims: to analyse the shape (Section 5.1), scatter (Section 5.2), local deviations (Section 5.3), and time-dependence (Section 5.4) of the radial metallicity gradient. These initial discussions inform our thoughts on the implications of this work for Milky Way studies in Section 5.5 and the studies of other galaxies in Section 5.6.","pages":[9]},{"title":"5.1. Linearity of the radial metallicity gradient","content":"The radial metallicity gradient of our simulated NIHAOUHDMilky Way analogue showed an overall decreasing, predominantly linear shape, as established in Section 3. Motivated by previous works by Sánchez-Menguiano et al. (2016), among others, we also fitted piecewise linear and quadratic functions to the data in Section 3.1. Both forms perform better than a linear trend. The very similar fitting performances indicate no significant preference between either piecewise linear or quadratic function. Due to both functions' rather good overall fit, we have not tried more exotic non-linear functions as done by Scarano & Lépine (2013). Increasing the flexibility of the gradient function could, however, improve the fit at the innermost kpc, where a flattening is predicted by our simulation, but chemical enrichment is also harder to simulate (see also Minchev et al. 2013; Sun et al. 2024). We have found no significant influence of binning for our gradient estimates (Section 3.2), but have found that a limited radial coverage - as is the case for the Milky Way - could mimic a truly quadratic function with two piecewise linear fits (Section 3.3). This is important, as it has significant implications for the conclusions we draw from the incomplete data of our Milky Way, as we will discuss in more detail in Section 5.5. The balance between a quadratic and piecewise linear radial metallicity gradient teeters at the breaking radius. If present, our analysis of the Milky Way analogue would place it at 𝑅 break ∼ 10 ± 0 . 5 kpc. This radius is strikingly close to the radius of 9 kpc found by Hemler et al. (2021) for a TNG50 galaxy simulation with a stellar mass of log ( 𝑀 ★ / M ⊙ ) = 10 . 72, that is, close to the Milky Way's (see their Fig. 2). In terms of physical reasons for a breaking radius at this location, a direct and secular influence of a stellar bar with non-symmetric effects around the corotation radius (Di Matteo et al. 2013; Scarano & Lépine 2013) should be minor for our specific scenario due to the low ages of the stars considered in our analysis. In particular, our identified break radius is significantly larger than the corotation radius of the Milky Way bar at 4 . 5 -7 . 0 kpc (Bland-Hawthorn & Gerhard 2016, and references therein) anyway. We are intrigued, however, by the proposition by Garcia et al. (2023) of galactic discs consisting of a star-forming inner disc with a steep gradient and a mixing-dominated outer disc with a flat gradient, with the break radius marking the region of transition between them. In Illustris TNG50-1 data, they found such a transition and break radius to be situated much further out at 30 kpc for Milky Way mass galaxies (10 . 1 ≤ log ( 𝑀 ★ / M ⊙ ) ≤ 10 . 6). While our bestfitting break radius - if present - is inconsistent with theirs, we will follow this up in more detail in Section 5.6, where we also discuss the implications for extragalactic studies in general.","pages":[9,10]},{"title":"5.2. Scatter of the radial metallicity gradient","content":"In Section 4.1 we found an increasing scatter from 𝜎 [ Fe / H ] = 0 . 01 dex in the inner galaxy to 𝜎 [ Fe / H ] = 0 . 10 dex around 𝑅 Gal . ∼ 20 kpc. Comparing these values with simulations other than TNG50 with a similar metallicity spread (see Fig. 2 by Hemler et al. 2021) is rather difficult, as the literature focuses on the shape and density distribution (see e.g. Minchev et al. 2014, their Fig. 10). When comparing with Milky Way studies (e.g. Anders et al. 2017), the scatter in the simulation is smaller than the observed spread of [Fe/H]. This can be visually appreciated by comparing the combinations of different measurements in the Milky Way (Genovali et al. 2014; Spina et al. 2021; Myers et al. 2022) in Fig. 10a and our simulation in Fig. 10b and c. We discuss the implications of this on studies of the Milky Way's gradient in Section 5.5. When assuming that young star and gas phase abundances are similar, we find comparable scatter of abundances for example with respect to TYPHOON observations by Chen et al. (2023). To test this assumption, we also show the gas phase metallicity in Fig. 11, for which we find a similar shape and scatter of the gradient, but systematically less scatter or spread than observed gas phase abundance, thus urging us to treat the absolute values of abundances and abundance scatters as well as spreads with caution. We furthermore note that the spread of abundance in observations does only increase for some but not all of the observed (and thus observationally limited) galaxies by Chen et al. (2023). This potentially limits the range of galaxies to which our conclusions may apply. While the simulated abundance scatter is consistent with the predictions by the theoretical forced-diffusion model by Krumholz & Ting (2018), that is, a scatter of ∼ 0 . 1 dex over timescales of ∼ 100 -300 Myr, our simulations suggest that the scatter is driven by the radial structure and large-scale spiral arms, which were not included in their model.","pages":[10,11]},{"title":"5.3. Localised vertical and azimuthal deviations and their correlation with gas","content":"In Sections 4.2 and 4.3 we established that local deviations contribute significantly to the spread of the global metallicity gradient above 𝑅 Gal . > 8 kpc ∼ 2 Re. We noted in particular a void of stars where we found an upturning warp of the galaxy around 𝜑 Gal . ∼ 180 · spatially close to regions of the galaxy (Groups 1 and 2) that deviated most significantly from the overall trend in Fig. 8 and Fig. 9. These stellar voids pose the question if they are also void of gas thus suggesting the gas has shifted to the more enhanced regions. In Fig. 12 we are thus tracing both the spatial distribution of gas as colored density distribution and stars as grey contour lines and gas. We find that while stars and gas trace each other in the vertical direction, we do not always see a match between the two tracers in the radial direction. In particular, we do find a significant amount of gas around the stellar void of 𝜑 Gal . ∼ 180 · and 𝑅 Gal . ∼ 8 -11 kpc in Figs. 12e and 12f. This gas seems to be more tightly concentrated though for example in the tight wave around 𝑅 Gal . ∼ 8 -11 kpc in Fig. 12e. We also note that significant gas overdensities, for example around 𝜑 Gal . ∼ 0 -45 · and 𝑅 Gal . ∼ 7 kpc in Fig. 12a do not seem to correlate with significant overenhancement in iron abundance (compare to Fig. 7a). While we see a hint of a coinciding deviation of Δ [ Fe / H ] for larger deviations from the galactic plane Δ 𝑧 in the upturning outer region of Fig. 12f, this does not seem to be the case for the downturning outer region of Figs. 12b and 12i. As the edge-on projection is not providing conclusive insights, we are now looking into the phase-on projection in Fig. 13. We have chosen a region of the simulated galaxy whose gas density at solar radius (Fig. 13c) matches with the recently measured distribution of young stars in the Milky Way at face value (Fig. 13a) by Poggio et al. (2021). Comparing simulated gas and observed young stars is preferable in this case, as the density of simulated stars is too low to easily identify overdensities (Fig. 13b). The region and its gas spiral structures appear to be representative, as these structures exist throughout the whole galaxy (see Fig. 1d). In the different panels of Fig. 14, we thus show this representative region of the galaxy, but color each spatial bin by stellar metallicity (Fig. 14a), the deviation from the global linear trend (Fig. 14b) as well as the gas metallicity (Fig. 14c) and its deviation from the global linear trend (Fig. 14d). In all cases, we also overlay the density contours of the significant gas overdensities (red regions in Fig. 13c). As expected, we see that the metallicity color map of the stars in Fig. 14a shows a decreasing trend from right to left (inner to outer galaxy) and an increasing scatter (more blue and red points towards the left) in the residual plot of Fig. 14b. We cannot identify a strong correlation between gas overdensities and stellar metallicity or residuals in either plot - possibly caused by the low number density. In Figs. 14c and 14d, however, the radial gas metallicity gradient shows significant local variations, that is, a trend from left to right that is not very smooth. In particular, we find significant deviations of up to + 0 . 15 dex in [Fe/H] behind the outer gas spiral (lower left of Fig. 14d) and -0 . 1 dex in [Fe/H] in front of the inner gas spiral (upper center of Fig. 14d) with a steep edge consistent with the gas spiral edge. We have identified the same patterns in both [Fe/H] and A(O) as both elements trace each other rather well in the simulation of young stars (see Fig. A1). Tentatively, we even see a slight enhancement of A(O) at the trailing edge of the inner spiral arm (top of Fig. 14d). We convince ourselves of the step-like behaviour by selecting a small slit-like region of 𝜑 Gal . ∼ 0 · and -2 < 𝑌 Gal . / kpc < -1 and tracing the gas metallicity and gas density as a function of radius in Fig. 15. We indeed find steps and confirm that they coincide with the location of significant gas overdensities. These step-patterns have also been found by Grand et al. (2015) in another simulation and observationally by Ho et al. (2017). In Fig. 15, we note an extended flat region just beyond 𝑅 Gal . > 10 kpc, the best fitting 𝑅 break of an assumed piecewise linear fit. Our analyses suggest that the correlation of void and overdensities with chemical enrichment of gas and young stars is more complicated and should better be followed up by tracing these structures over simulation look-back time in a dedicated follow-up analysis to unravel the physical mechanisms of star formation feedback cycles . This could also involve the tracing of star formation bursts and disk instabilities (Sánchez et al. 2014; Sánchez-Blázquez et al. 2014; Ho et al. 2015) as well as tracing how much self-enrichment as well as mixing and dilution takes place around the gas spirals (Ho et al. 2017). Rather than going back in simulation time, the present simulation data of a single snapshot in time already allows us to look back in terms of stellar lifetime - similar to Milky Way studies, as we discuss subsequently.","pages":[11,12]},{"title":"5.4. The impact of time and age: mixing and migration","content":"Although we have chosen a rather small stellar age window of 0 . 5 Gyr to trace the radial metallicity gradient without the expected significant impact of radial mixing and migration, we are testing and discussing this particular choice in this section in two ways. Firstly, we test the deviation of the radial metallicity gradient from the same global shape as well as the abundance spread across smaller age bins of 100 Myr between 0 -1000 Myr in Fig. 16. Secondly, we trace the distribution of stellar metallicity across galactic radii for increasing age bins from 50 Myr up to the maximum stellar age of 13 . 8 Gyr in Fig. 17. Our first test in Fig. 16 shows that the deviation from a global trend remains similar in functional form. We find that the spread of iron abundance does indeed scatter significantly, but the distributions stay within the same overall shape across the ten age bins. We note though, that the smallest age bin of 0 -100 Myr shows the least abundance scatter. This is consistent with the picture from our second test of increasing age ranges in Fig. 17. Here we find the first significant deviation from a tighter and already slightly quadratic relation for an age of 100 -150 Myr in Fig. 17c - our previously identified Group 3. As expected from previous simulations and observations, we see an increase in the scatter as we include more and more older stars. We note a still similar albeit more scattered shape for stars below 4 Gyr in Fig. 17h, before we start to see a more metal-poor population of stars in the inner galaxy appear between 4 -8 Gyr in Fig. 17i. These also begin to significantly impact the quadratic fit to the radial metallicity distribution, shown as a solid red line, in contrast to our reference fit, represented by a dashed red line. The significant amount of metal-poor stars in the inner galaxy then completely tilts the distribution when also including stars between 8 -13 . 8 Gyr in Fig. 17j (see also Johnson et al. 2024). Similar to the Milky Way (Bland-Hawthorn & Gerhard 2016), these oldest stars are those of the relatively more metal-poor thick disk that are confined to the inner disk with a shorter scale length. While we cannot exclude radial migration playing a role for change of radius for the youngest stars of the simulation, since Frankel et al. (2018) predicted significant shifts even for ages below 0 . 5 Gyr (see their Fig. 10), the larger scatter for older stars is certainly suggesting a larger (re-)distribution of stars along the radial axis, as found in previous simulations (Minchev & Famaey 2010; Grand et al. 2015).","pages":[13]},{"title":"5.5. Implications for Milky Way studies","content":"Our analysis of the radial metallicity gradient in a simulated NIHAO-UHD galaxy offers several insights that are directly applicable to understanding the Milky Way's gradient. First, the nature of the gradient - whether it is linear or better described by more complex functional forms - remains a critical question. Previous studies, such as those by Lépine et al. (2011) and Donor et al. (2020), have suggested the po- ential for a break radius, possibly at the corotation radius or further out (Scarano & Lépine 2013), which could indicate two distinct linear regimes. In our analysis, we find evidence that the gradient is at least is not purely linear, but could also be smoothly flattening. Applying a smooth quadratic function on observational data (Yong et al. 2012; Andrievsky et al. 2004; Genovali et al. 2014), might provide a better or at least consistent fit for the Milky Way data without the need for a break radius. However, even this may not fully capture the nuances observed in our simulations. Chemical evolution models propose a more sophisticated behaviour (e.g. Chiappini et al. 2001; Kubryk et al. 2015; Palla et al. 2024), reflecting varying influences of galactic processes at different radii. Understanding this structure in the simulated galaxy provides a framework for interpreting similar complexities in the Milky Way. Given these complexities, it is also essential to consider how local sampling biases might affect our understanding of the Milky Way's metallicity gradient. For instance, incomplete samples that omit low [Fe/H] clusters or stars could skew gradient estimates, as suggested by our comparisons in Figures 10a and 10b. Our results indicate that young clusters with lower (or higher) [Fe/H] than expected at a given radius could indicate the previous presence of a spiral arm (see our identified Groups in Figs. 8 and 9). Furthermore, we caution that localised effects - both intrinsic and in terms of selection function - could also mimic non-linear shapes and more spatial coverage is needed in the Milky Way. Our results also indicate that older clusters, which have been found more frequently at larger distances than young clusters - are likely influenced by radial migration - and thus complicate the interpretation of these radial metallicity gradients (Magrini et al. 2009; Lépine et al. 2011). Cosmological zoom-in simulations like NIHAO-UHD are approaching the resolution needed to examine regions analogous to the solar vicinity, though the star particle numbers and mass resolution remain a limiting factor. Nonetheless, we observe distinct patterns in the distribution of young stars and gas, including lower [Fe/H] and A(O) in the leading edges of gas overdensities and higher [Fe/H] and A(O) in the trailing edges, consistent with findings by Grand et al. (2016), Ho et al. (2017), and Kreckel et al. (2019). These trends suggest that local metallicity variations, driven by gas dynamics, may also play a significant role in shaping the observed gradients in the Milky Way. Additionally, our study hints at the potential for more nuanced variations in [Fe/H] across different regions of the galaxy. In particular, the gas shows a step-like behavior of A(O)and[Fe/H] changes around the edges of gas overdensities (Fig. 15), with significant deviations from the global gradient in specific regions. We have also found a larger stellar void around -12 < 𝑅 Gal < -10kpc. Although further investigation is needed, these findings could have important implications for understanding localized star formation events and their impact on the overall metallicity distribution in the Milky Way (Sánchez et al. 2014; Sánchez-Blázquez et al. 2014; Ho et al. 2015). It will certainly be exciting to see how much more insights (Poggio et al. 2021; Hackshaw et al. 2024) we will get from the more extended data of future data releases of Gaia and spectroscopic surveys. Wecannot directly link spiral arms to bar resonances or bardriven mixing in our simulation, because of a negligible bar strength in our galaxy /three.sup (but see Minchev & Famaey 2010; Di Matteo et al. 2013). However, the influence of a galactic bar on the spiral arms and, by extension, on the radial metallicity gradient, remains a possibility (see again Chen et al. 2023). Disk instabilities and warps might further complicate the interpretation of these gradients and progress will likely rely on the detailed disentangling of these effects from both cosmological simulations as well as idealised simulations and models (Minchev et al. 2013; Grand et al. 2015, 2016; Krumholz et al. 2018; Sharda et al. 2021; Bland-Hawthorn et al. 2024; Tepper-Garcia et al. 2024).","pages":[13,14]},{"title":"5.6. Implications for extragalactic studies","content":"The insights gained from our analysis of the radial metallicity gradient in a simulated NIHAO-UHD galaxy extend beyond the Milky Way, offering valuable implications for the study of extragalactic systems. One key observation is that deviations from a purely linear metallicity gradient, as seen in our Milky Way analogue, are common in other galaxies as well. When fitting a piecewise linear fit to our data, we found a break radius at 𝑅 Gal . = 10 . 0 ± 0 . 5 kpc. Converted to effective radii Re or radii R25 covering the 25 mag arcsec -2 isophote /four.sup , this corresponds to 𝑅 break ∼ 2 . 5 Re ≡ 0 . 7 R25 for our simulation. This would be consistent with the observational results by Sánchez et al. (2014) who found that breaks in metallicity gradients are common in both spiral and barred galaxies, with flattening of the abundance being evident beyond ∼ 2 Re (compare also to Belfiore et al. 2017). Similar to our suggestion for Milky Way studies, we suggest to also test a smooth function, such as a quadratic one, on extragalactic observational data (e.g. Bresolin et al. 2012; Chen et al. 2023) to test the preference of a distinct break radius. Although the focus of this research lies on the observable region of the Milky Way ( 𝑅 Gal . < 20 kpc) and most other galaxies ( 𝑅 Gal . < 2 . 5 Re), the finding of significant gradient changes in the outskirts of galaxies by Garcia et al. (2023), suggests to also test this region of our Milky Way analogue. Garcia et al. (2023, see their Fig. 4) found a metallicity floor in IllustrisTNG galaxies. When using their sample to identify a metallicity floor radius for a Milky Way mass galaxy with log ( 𝑀 ★ / M ⊙ ) = 10 . 7 (Bland-Hawthorn & Gerhard 2016) at redshift 𝑧 ∼ 0, we would expect to find it around 25 -30 kpc. We therefore extend the analysed radius to 𝑅 Gal . ≤ 100 kpc (see Fig. 18) and indeed find a similar abundance floor of [ Fe / H ] ≥ -0 . 64 for young stars and A ( O ) ≥ 8 . 12 for the majority of gas (see Fig. 19 at a similar radius. We note that another galaxy without gas in this figure is a sufficiently large distance of 92 kpc, that is, ( 𝑌,𝑌, 𝑍 ) = (-50 , -75 , 20 ) kpc. These lowest abundances remind us of two observational results. Firstly, the iron abundance floor is consistent with the lower end of the Milky Way thin - and coincidentally outer disk of [ Fe / H ] ∼ -0 . 7 (Bensby et al. 2014; Buder et al. 2019). Secondly this oxygen abundance floor is consistent with the results by Grasha et al. (2022) from TYPHOON galaxy observations. Grasha et al. (2022) suggested this could be caused by changes in the ratio of supernovae II and AGB reflected by a changing ratio of nitrogen to oxygen abundance N/O which also flattens towards a lower plateau below metallicities of A ( O ) ∼ 8 . 0 (Nicholls et al. 2017). While we cannot follow this observation up with the present simulation, a similar simulation used by Buder et al. (2024) has traced the relative XGal. XGal. XGal. contribution of both supernovae II and AGB and should be used to test this hypothesis in the future. It is important to note that the chemical evolution model in the NIHAO-UHD simulations is constrained by the current, incomplete understanding of evolutionary pathways and yields (Buck et al. 2021), as well as by limitations in resolution and the imperfect physics inherent to cosmological zoom-in simulations (Buck 2020). Both could contribute to the identified differences in absolute and relative abundances across different scales - including a different scatter of abundances for example of the gas phase metallicity between NIHAO-UHD of up to 0 . 1 dex and the low scatter of 0 . 03 -0 . 05 dex (and even lower on local scales) found by PHANGS-MUSE faceon observations (Kreckel et al. 2020). Extending our analysis to other simulations and further improving the resolution and physics of the simulations will be key in uniting the observational and theoretical insights into galactic chemical evolution on small and large scales. Similar to more resolved and higher quality observations in the Milky Way, we also expect more, better, and diverse faceon and edge-on observations and analyses across a range of wavelengths by the PHANGS and GECKOS teams (Kreckel et al. 2019, 2020; van de Sande et al. 2023) as well as the SDSS-V and MAGPI collaborations (Kollmeier et al. 2017; Foster et al. 2021; Mun et al. 2024; Chen et al. 2024a), among many other ongoing efforts.","pages":[14,15]},{"title":"6. CONCLUSIONS","content":"To conclude our study, we first iterate the main take-away of our research in Section 6.1 before giving suggestions for future research in Section 6.2.","pages":[15]},{"title":"6.1. Take-Away","content":"Wehaveanalysed the radial metallicity distribution of young stars and gas in the inner 20 kpc of a NIHAO-UHD Milky Way analogue (Fig. 1), finding a predominantly linear decrease (Figs. 2 and 3). Although our analysis of a single spiral galaxy simulation has limited applicability to the entire population of diverse spiral galaxies, it reveals several intriguing findings about the shape and local metallicity variations. The results we find in this work hold relevance for both the Milky Way and extragalactic research communities: more detail. Secondly, we suggest approaching the fitting of gradients in external galaxies in a more agnostic way to the shape. This will be particularly interesting when we can observe the outermost regions of galaxies, where simulations predict an abundance floor (Figs. 18 and 19).","pages":[16]},{"title":"6.2. Future Research","content":"In our study, we have focused on the present-day snapshot of the NIHAO-UHD Milky Way analogue simulation - similar to present-day observations that are possible in our local Universe. Given that the simulation is tracing particles and gas over time, a detailed follow-up study should trace the evolution and coherence of spatial and chemical over- and underdensities over time and different elements (see also Zhang et al. 2024). This would in particular allow us to quantify the change of abundance in the leading and trailing edges of spiral arms and subsequently track the mixing and blurring of these over time. Certainly, more studies are needed to establish a link to a physical mechanism and further quantify its importance. More results are expected as we extend the reach, number, and quality of stellar measurements in our Galaxy (e.g. Barbillon et al. 2024) and beyond. These improvements will allow us to move beyond a one-dimensional analysis of gradients and better incorporate and model local variations. While previous works are showing that relative trends for several elemental abundances do not strongly disagree from observations (Buck et al. 2021; Buder et al. 2024), we are still missing several details on the origin of elements, such as a complete picture of the synthesis sites, environments, and yields for elements. Not least because of these imperfections of absolute chemical enrichment predictions, we are refraining from quantitatively comparing the shape of our Milky Way analogue with the actual Milky Way. We have previously also mentioned the limitations in mass resolution of stars and gas, which may introduce unrealistic effects and could for example drive deviations from actual chemical enrichment at the smallest scales. We also note that the results of our simulation may not apply to the actual Milky Way due to different galaxy properties. These could be different due to different galaxy formation pathways, such as the amount and importance of mergers (Buck et al. 2023; Buder et al. 2024). In our discussion, we have already eluded to the weak bar in this simulation. Motivated by the analysis by Tuntipong et al. (2024), we have also investigated the bulge to total stellar mass ratio 𝐵 / 𝑇 . We find a strong bulge with 𝐵 / 𝑇 = 0 . 48 when selecting bulge stars with orbit circularity 𝑗 𝑧 / 𝑗 𝑐 < 0 . 5 and disk stars with 𝑗 𝑧 / 𝑗 𝑐 > 0 . 7 based on actions 𝑗 , consistent with values found by Obreja et al. (2019) of a lower resolution simulation of 8.26e11 . Our simulated galaxy has both a very weak bar and a smoothly changing radial metallicity gradient. This is in line with the findings by Chen et al. (2023) for five strongly and weakly barred spiral galaxies, where bars seem to drive the dominance of break radii in radial metallicity gradients. Future work should certainly look at a variety of galaxies to establish the causality of bar strength with the smoothness or abrupt change at a break radius for the radial metallicity gradient. Expanding our study to more and other simulations such as the VINTERGATAN suite (Renaud et al. 2024) and subsequently comparing to observations of galaxies with varying parameters like mass, formation history, bar strength or environment would allow us to quantify their effect on metallicity gradients and further disentangle the influence of different enrichment mechanisms on the chemical evolution of galaxies.","pages":[16]},{"title":"SOFTWARE","content":"The research for this publication was coded in /p.pc/y.pc/t.pc/h.pc/o.pc/n.pc (version 3.7.4) and included its packages /a.pc/s.pc/t.pc/r.pc/o.pc/p.pc/y.pc (v. 3.2.2; Astropy Collaboration et al. 2013, 2018), IP/y.pc/t.pc/h.pc/o.pc/n.pc (v. 7.8.0; Pérez & Granger 2007), /m.pc/a.pc/t.pc/p.pc/l.pc/o.pc/t.pc/l.pc/i.pc/b.pc (v. 3.1.3; Hunter 2007), N/u.pc/m.pcP/y.pc (v. 1.17.2; Walt et al. 2011), /p.pc/y.pc/n.pc/b.pc/o.pc/d.pc/y.pc (v. 1.1.0; Pontzen et al. 2013), /s.pc/c.pc/i.pc/p.pc/y.pc (v. 1.3.1; Virtanen et al. 2020), /s.pc/k.pc/l.pc/e.pc/a.pc/r.pc/n.pc (v. 1.5.1 Pedregosa et al. 2011) /s.pc/t.pc/a.pc/t.pc/s.pc/m.pc/o.pc/d.pc/e.pc/l.pc/s.pc (v. 0.14.2 Perktold et al. 2024) We further made use of /t.pc/o.pc/p.pc/c.pc/a.pc/t.pc (version 4.7; Taylor 2005);","pages":[17]},{"title":"DATA AVAILABILITY","content":"All code to reproduce the analysis and figures can be publicly accessed via https://github.com/svenbuder/ nihao_radial_metallicity_gradients . The used simulationsnapshot can be accessed as FITS file via https: //github.com/svenbuder/preparing_NIHAO . Original data, more snapshots and other galaxies can be found at https://tobias-buck.de/#sim_data . We encourage interested readers to get in contact with the authors for full data access and advice for use and cite Buck et al. (2020, 2021).","pages":[17]},{"title":"ACKNOWLEDGMENTS","content":"Weacknowledge the traditional owners of the land on which the ANU stands, the Ngunnawal and Ngambri people. We pay our respects to elders past, and present and are proud to continue their tradition of surveying the night sky and its mysteries to better understand our Universe. This work was supported by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013. SB and KGacknowledge support from the Australian Research Council under grant numbers DE240100150 and DE220100766, respectively. TB acknowledges funding from the Carl Zeiss Stiftung and support from the European Research Council under ERC-CoG grant CRAGSMAN-646955. We gratefully acknowledge the Gauss Centre for Supercomputing e.V. ( www.gauss-centre.eu ) for funding this project by providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre ( www.lrz.de ). Simulations were partially computed with High Performance Computing resources at New York University, Abu Dhabi.","pages":[17]},{"title":"REFERENCES","content":"Yanny B., Lee Y. S., 2006, ApJ, 636, 804 Franchini M., Molero M., Randich S., 2024, A&A, 690, A334 Pedregosa F., et al., 2011, J Mach Learn Res, 12, 2825 Perktold J., et al., 2024, statsmodels/statsmodels: Release 0.14.2, doi:10.5281/zenodo.593847 Ratcliffe B. L., Ness M. K., Buck T., Johnston K. V., Sen B., Beraldo e Silva Rybizki J., Just A., Rix H.-W., 2017, A&A, 605, A59 Schönrich R., Binney J., 2009, MNRAS, 396, 203 Sharda P., Krumholz M. R., Wisnioski E., Forbes J. C., Federrath C., Acharyya A., 2021, MNRAS, 502, 5935 Taylor M. B., 2005, ASPC, 347, 29 Federrath C., 2024, arXiv e-prints, p. arXiv:2406.00342 Schaller M., Schaye J., Theuns T., 2019, MNRAS, 482, 2208 Walt S. v. d., Colbert S. C., Varoquaux G., 2011, Comput Sci Eng, 13, 22 Wang L., Dutton A. A., Stinson G. S., Macciò A. V., Penzo C., Kang X., Keller B. W., Wadsley J., 2015, MNRAS, 454, 83 Willett E., et al., 2023, MNRAS, 526, 2141 Williams M. J., Bureau M., Cappellari M., 2009, MNRAS, 400, 1665 Zaritsky D., Kennicutt Robert C. J., Huchra J. P., 1994, ApJ, 420, 87 Zhang C., Li Z., Hu Z., Krumholz M. R., 2024, arXiv e-prints, p. arXiv:2411.01518 van de Sande J., Fraser-McKelvie A., Fisher D. B., Martig M., Hayden M. R., the GECKOS Survey collaboration 2023, arXiv e-prints, p. arXiv:2306.00059","pages":[17,18]},{"title":"A. ADDITIONAL FIGURES","content":"Fig. A1 demonstrates the tight correlation of gas phase iron and oxygen abundance and how to approximate them linearly. This paper was built using the Open Journal of Astrophysics L A T E X template. The OJA is a journal which provides fast and easy peer review for new papers in the astro-ph section of the arXiv, making the reviewing process simpler for authors and referees alike. Learn more at http://astro.theoj.org .","pages":[18]}],"string":"[\n {\n \"title\": \"ABSTRACT\",\n \"content\": \"Radial metallicity gradients are fundamental to understanding galaxy formation and evolution. In our high-resolution simulation of a NIHAO-UHD Milky Way analogue, we analyze the linearity, scatter, spatial coherence, and age-related variations of metallicity gradients using young stars and gas. While a global linear model generally captures the gradient, it ever so slightly overestimates metallicity in the inner galaxy and underestimates it in the outer regions of our simulated galaxy. Both a quadratic model, showing an initially steeper gradient that smoothly flattens outward, and a piecewise linear model with a break radius at 10 kpc (2.5 effective radii) fit the data equally better. The spread of [Fe/H] of young stars in the simulation increases by tenfold from the innermost to the outer galaxy at a radius of 20 kpc. We find that stars born at similar times along radial spirals drive this spread in the outer galaxy, with a chemical under- and over-enhancement of up to 0.1 dex at leading and trailing regions of such spirals, respectively. This localised chemical variance highlights the need to examine radial and azimuthal selection effects for both Galactic and extragalactic observational studies. The arguably idealised but volume-complete simulations suggest that future studies should not only test linear and piecewise linear gradients, but also non-linear functions such as quadratic ones to test for a smooth gradient rather than one with a break radius. Either finding would help to determine the importance of different enrichment or mixing pathways and thus our understanding of galaxy formation and evolution scenarios. Subject headings: Galaxy: structure - Galaxy: abundances - galaxies: structure - galaxies: abundances\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"LOCAL VARIATIONS OF THE RADIAL METALLICITY GRADIENT IN A SIMULATED NIHAO-UHD MILKY WAY ANALOGUE AND THEIR IMPLICATIONS FOR (EXTRA-)GALACTIC STUDIES\",\n \"content\": \"S/v.pc/e.pc/n.pc B/u.pc/d.pc/e.pc/r.pc 1 , 2 , ∗ , T/o.pc/b.pc/i.pc/a.pc/s.pc B/u.pc/c.pc/k.pc 3 , 4 , Q/i.pc/a.pc/n.pc-H/u.pc/i.pc C/h.pc/e.pc/n.pc ( 陈 千 惠 ) 1 , 2 , /a.pc/n.pc/d.pc K/a.pc/t.pc/h.pc/r.pc/y.pc/n.pc G/r.pc/a.pc/s.pc/h.pc/a.pc 1 , 2 , ∗ 1 Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT 2611, Australia 2 ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Australia 3 Universität Heidelberg, Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, Im Neuenheimer Feld 205, D-69120 Heidelberg, Germany and 4 Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische Astrophysik, Albert-Ueberle-Strae 2, D-69120 Heidelberg, Germany Version December 3, 2024\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1. INTRODUCTION\",\n \"content\": \"Understanding the radial metallicity gradient, defined as the change in heavy element abundance with galactocentric radius, in galaxies provides critical insights into their formation and evolutionary processes, such as inside-out formation, gas accretion, outflows, and radial migration (e.g. Quirk & Tinsley 1973; Tinsley 1980; Lacey & Fall 1985; Wyse & Silk 1989; Kauffmann 1996; Chiappini et al. 1997; Schönrich & Binney 2009; Moran et al. 2012; Bird et al. 2013). The decrease in metallicity with increasing distance from the Galactic centre is well-established both theoretically (Larson 1976; Tinsley 1980; Chiosi 1980) and observationally in the Milky Way (Searle 1971; Janes 1979; Twarog et al. 1997) and other massive spiral galaxies (e.g. Tinsley 1980; Zaritsky et al. 1994; Bresolin et al. 2012). The Milky Way, being the only galaxy where we can resolve millions of stars, provides a unique opportunity to study these gradients and deviations from them in detail. Early evidence by Janes (1979) suggested a linear gradient on the order of d [ Fe / H ]/ d 𝑅 = -0 . 05 ± 0 . 01 dex kpc -1 for the Milky Way which aligns very well with more recent measurements (Anders et al. 2017; Hayden et al. 2015). However, these gradients are accompanied by a significant spread in [Fe/H] of 0 . 1 -0 . 15 dex, as noted by Twarog (1980), which may imply a fine structure of the metallicity gradient (see Genovali et al. 2014). With increasing sample size and measurement precision, the specific shape and characteristics of this gradient remain somewhat unclear (Chiappini 2002). Previous studies have ∗ Australian Research Council DECRA Fellow for example claimed more intricate non-linear trends, bends, and flattening in the gradient of the Milky Way (e.g. Donor et al. 2020) and other galaxies (e.g. Pilyugin 2003; Sánchez et al. 2014) or even sequences of shapes (Pilyugin et al. 2017; Pilyugin & Tautvaišien˙e 2024), which were fitted with different models (Rosales-Ortega et al. 2011; Bresolin et al. 2012), such as piecewise linear ones (e.g. Sánchez-Menguiano et al. 2016) or non-linear ones (e.g. Scarano & Lépine 2013). Variations in the metal distribution, including breaks of the gradient at specific radii, give rise to a plethora of possible physical explanations, such as star formation efficiency variations and localised star formation bursts (Sánchez et al. 2014; Sánchez-Blázquez et al. 2014; Ho et al. 2015), gas accretion and dilution at different rates (Bresolin et al. 2012; Sánchez et al. 2013; Belfiore et al. 2016; Sánchez-Menguiano et al. 2016), gas outflows and feedback (Lilly et al. 2013; Ma et al. 2017a), as well as disk instabilities or local overdensities (Grand et al. 2016; Ho et al. 2017). In particular, Scarano & Lépine (2013) suggested that gradient break radii coincided with the corotation radii of spiral arms. Recent advancements in both computations and observations have significantly expanded our capabilities. For example, in terms of observational data in the Milky Way, the Gaia mission (Gaia Collaboration et al. 2016) enables more detailed studies of these gradients. New suites of large-scale simulations now allow us to gain insights into radial metallicity gradients across a range of simulated galaxies, including Milky Way analogues. This presents opportunities to revisit outstanding challenges of the detailed shape of the radial metallicity gradient. For instance, Hogg et al. (2019) created an extensive metallicity map of the Milky Way using APOGEE and Gaia data, while Poggio et al. (2022) mapped young stars and found metallicity variations around spiral arms (see also Zari et al. 2018, 2021; Poggio et al. 2021; Hackshaw et al. 2024). Similarly, Imig et al. (2023, among others) traced gradients across different stellar populations and ages, emphasizing the importance of considering radial migration effects (Binney & Tremaine 2008; Frankel et al. 2018, 2020). Historically, radial metallicity gradients have been measured using various stellar populations and gas tracers. Estimated gradients seem to be broadly consistent across different stellar tracers, such as young open clusters (e.g. Yong et al. 2012; Cunha et al. 2016; Magrini et al. 2017; Casamiquela et al. 2019; Donor et al. 2020; Spina et al. 2021; Myers et al. 2022), young hot (OB-type) stars (Zari et al. 2018, 2021; Poggio et al. 2021, 2022), field stars close to the Galactic plane (e.g. Bergemann et al. 2014) or Cepheids (Andrievsky et al. 2002a,b; Lemasle et al. 2007, 2013). Despite extensive observational efforts, several challenges persist for studies in the Milky Way. The completeness (or patchiness) of observed datasets remains a fundamental issue (Bergemann et al. 2014). The robustness of fits to the incomplete data is still contentious, including the need to actually fit two linear gradients with a break radius at corotation radius (Bresolin et al. 2012, and references therein) or further out (Yong et al. 2012; Donor et al. 2020) - or even more complicated functions (see e.g. Chiappini et al. 2001; Kubryk et al. 2015). Furthermore, methodologies for fitting linear models to scattered data need re-evaluation (Metha et al. 2021). Different samples yield varying gradients, potentially due to biases in data or the inclusion of older stars (e.g. Allende Prieto et al. 2006; Hayden et al. 2014; Anders et al. 2014; Vickers et al. 2021; Willett et al. 2023). The impact of spiral arm structures (Poggio et al. 2021), the Galactic warp (Lemasle et al. 2022) or bar-driven mixing (Di Matteo et al. 2013) on metallicity gradients is not fully understood. Understanding these gradients in the Milky Way is also crucial for extragalactic studies, where spatial resolution limits observations in different ways. In extragalactic systems, metals are mainly traced via gas, because it provides a more direct measure of the ongoing enrichment processes, unlike stars, which primarily reflect the integrated chemical history of the past. Observationally, gas emission lines are typically brighter and more accessible across large distances than stellar absorption lines, allowing for broader spatial coverage, especially in distant galaxies. Consequently, extragalactic studies often focus on gas-phase metallicity as traced by oxygen, A ( O ) = 12 + log ( O / H ) , while Galactic studies typically use stellar iron abundance [ Fe / H ] = A ( Fe ) -A ( Fe ) ⊙ as a metallicity tracer (e.g. Nicholls et al. 2017; Fraser-McKelvie et al. 2022). New instruments like the MUSE integral field spectrograph have enabled a plethora of extragalactic studies to contrast the Milky Way and techniques like the spectroscopy of H /i.pc/i.pc regions and planetary nebulae have helped to infer gas metallicity distributions in external galaxies (Shaver et al. 1983; Vilchez & Esteban 1996; Rolleston et al. 2000; Bresolin et al. 2012). Recent examples include Sánchez et al. (2014) with CALIFA galaxy observations as well as Mun et al. (2024) and Chen et al. (2024a) who use MAGPI observations to probe for example the effects of spiral arms. Notable is also the scatter that Chen et al. (2023) found for the gas metallicity across galactic radii with TYPHOON observations (see their. Figs. 4-6). Grasha et al. (2022) found that the gas metallicity gradient plateaus at a lower limit in their TYPHOON galaxies at the outermost radii - an observation replicated by IllustrisTNG simulations (Hemler et al. 2021; Garcia et al. 2023). From a modelling perspective, galactic chemical evolution models can both test understanding of radial metallicity gradients and make predictions beyond the limited volumes and tracers tested by Milky Way and extragalactic studies. Such galactic chemical evolution models include Chiappini et al. (2001); Matteucci & Recchi (2001); Minchev et al. (2014); Kubryk et al. (2015); Stanghellini et al. (2015); Rybizki et al. (2017); Spitoni et al. (2023); Johnson et al. (2024). Sharda et al. (2021) even presented a model for gas phase metallicity gradients in galaxies and their evolution from first principles (see also Krumholz & Ting 2018). In exploring radial metallicity gradients through simulations, we have better understood how different processes influence these gradients across galactic models and temporal scales. Studies such as Pilkington et al. (2012) in RaDES simulations reveal that gradients are typically established via inside-out galaxy formation. Khoperskov et al. (2023) quantified the scatter of gas metallicity to ≈ 0 . 04 -0 . 06 dex at a given galactocentric distance in their simulations. Meanwhile, the EAGLE simulations used by Tissera et al. (2019) provide insights into how these gradients vary with galaxy characteristics like stellar mass and merger history, emphasizing the dynamic nature of metallicity distributions. The plethora of simulations such as AURIGA (Grand et al. 2016), FIRE (Ma et al. 2017b, see their Fig. 6) or VINTERGATAN (see their Fig. 9; Agertz et al. 2021) also allow us to explore the gradient evolution of galactic timescales. Buck et al. (2023), for example, found a link of major accretion events with periods of unexpected steepening in the metallicity gradient within NIHAO-UHD simulations - closely resembling findings for the Milky Way by Lu et al. (2022) and Ratcliffe et al. (2023). FIRE simulations, examined by Bellardini et al. (2021), Bellardini et al. (2022), and Graf et al. (2024), extend these findings by comparing radial metallicity gradients and their azimuthal scatter across gas and stellar components and illustrate the complex interplay between galactic structure and metal enrichment processes. Similarly, Grand et al. (2015, 2016) highlight temporal changes in metallicity gradients already within 120 Myr, or roughly one galactic rotation. Such rapid changes underscore the impact of transient galactic events on the metal distribution, linking them to star formation patterns along spiral arms and the broader evolutionary history of the galaxy. In this study, we analyze a high-resolution NIHAO-UHD simulation of a Milky Way analogue to bridge the observational gap between detailed studies of our galaxy and broader extragalactic surveys. We aim to reveal subtle features of the radial metallicity gradient, which may be obscured by observational constraints in both the Milky Way and distant galaxies, by testing the following properties within the observationally probed inner 𝑅 Gal . ≤ 20 kpc: XGal. The paper is structured as follows: Section 2 describes the data of our Milky Way analogue NIHAO-UHD simulation. Section 3 analyses the linearity of the radial metallicity gradient of the simulation, the first of our four objectives. Section 4 then analyses both the scatter and local deviations from the gradient as well as the coherence of the gradient with vertical and azimuthal position as well as age (the remaining three objectives). Section 5 discusses them individually. We note that in this research we are mainly interested in the specific shape of the radial metallicity gradient for radii relevant to Galactic observations. However, we also discuss our results in the context of extragalactic results that probe beyond the inner 20 kpc of a galaxy. Section 6 bundles our results into an overarching conclusion.\",\n \"pages\": [\n 1,\n 2,\n 3,\n 4\n ]\n },\n {\n \"title\": \"2. DATA: A NIHAO-UHD MILKY WAY ANALOGUE SIMULATION\",\n \"content\": \"For this project, we use a cosmological zoom-in simulation of a Milky Way analogue ( g8.26e11 ) from the Numerical Investigation of a Hundred Astronomical Objects (NIHAO, Wang et al. 2015) suite. This model galaxy was calculated as part of the NIHAO-UHD project (Buck et al. 2020) and has previously been used in various works studying Milky Way satellites (Buck et al. 2019), Milky Way's dark halo spin (Obreja et al. 2022), inferring birth properties of stars with abundance clustering (Ratcliffe et al. 2022), as well as the evolution of the interstellar medium's radial metallicity gradient since redshift three (Ratcliffe et al. 2024). Simulations were carried out with the smoothed particle hydrodynamics code Gasoline2 (Wadsley et al. 2017), including sub-grid turbulent mixing, using cosmological parameters from Planck Collaboration et al. (2014) with initial conditions and energetic feedback descriptions from the NIHAO project (Wang et al. 2015). Zoom-in simulations were then performed as described in detail by Buck et al. (2021) with star formation following Stinson et al. (2006) and energetic feedback following Stinson et al. (2013). We note that this is a slightly different rerun of the same simulation than the one studied by Buder et al. (2024); in this work, we use a higher resolution version and updated chemical yields. Because computational resources still limit the mass resolution of simulations, we are relying on tracer particles that represent simple stellar populations (SSPs) with the same age, overall metallicity and discrete initial mass function (IMF). Buck et al. (2021) have implemented the flexible chemical evolution code /c.pc/h.pc/e.pc/m.pc/p.pc/y.pc (Rybizki et al. 2017) to calculate the chemical yields for the SSPs. In particular, we use the alternative ( alt ) setup of /c.pc/h.pc/e.pc/m.pc/p.pc/y.pc that assumes a Chabrier (2003) IMF with high-mass slope of 𝛼 IMF = -2 . 3 over a mass range of 0 . 1 -100 M ⊙ for SSPs across a metallicity range of 𝑍 / 𝑍 ⊙ ∈ [ 10 -5 , 2 ] . The code calculates the contribution from asymptotic giant branch (AGB) stars, CCSN across a mass range of 8 -40 M ⊙ , and SNIa with an exponential function with exponent -1 . 12, a delay time of 40 Myr, and a normalization of the SNIa rate of log 10 ( NIa ) = -2 . 9. For each of these nucleosynthetic channels, yields from the following studies are used: Chieffi & Limongi (2004) for CCSN, Seitenzahl et al. (2013) for SNIa, and Karakas & Lugaro (2016) for AGB stars ( new_fit model in Buck et al. 2021). Contrary to a previous study by Buder et al. (2024), we take the elemental abundances at face value and do not apply any shifts. We limit the simulation data to the main halo by applying /p.pc/y.pc/n.pc/b.pc/o.pc/d.pc/y.pc's implementation of the Amiga Halo Finder (Knollmann & Knebe 2009) and then reposition and rotate this main halo to be face-on based on the angular momentum with /p.pc/y.pc/n.pc/b.pc/o.pc/d.pc/y.pc's /a.pc/n.pc/a.pc/l.pc/y.pc/s.pc/i.pc/s.pc./a.pc/n.pc/g.pc/m.pc/o.pc/m.pc./f.pc/a.pc/c.pc/e.pc/o.pc/n.pc module (Pontzen et al. 2013). We then further transform the resulting galactocentric Cartesian coordinate ( 𝑋,𝑌, 𝑍 ) and velocities ( 𝑉 𝑋 , 𝑉 𝑌 , 𝑉 𝑍 ) to Cylindrical ones as done in a previous study of this main halo by Buder et al. (2024). The model galaxy has a virial radius of 𝑅 vir = 𝑅 200 = 206 kpc and a total mass (gas, stars and dark matter) inside 𝑅 vir of 9 . 1 · 10 11 M ⊙ . At redshift zero, it contains 8 . 2 · 10 11 M ⊙ dark matter, 6 . 4 · 10 10 M ⊙ gas mass and 2 . 3 · 10 10 M ⊙ stellar mass with a stellar mass resolution of around 7 . 5 · 10 3 M ⊙ . When using a fifth of the virial radius as a reference to calculate total luminosity /one.sup and mass, we estimate a half-light radius, that is, effective radius of 𝑅 𝑒 = 3 . 79 kpc and a half-stellar-mass radius of 2 . 97 kpc. To achieve a roughly similar selection as the observational data of the Milky Way (Genovali et al. 2014) and other galaxies (e.g. Chen et al. 2023), we restrict the simulation data to a galactocentric radius of 𝑅 Gal ≤ 20 kpc and | 𝑧 | ≤ 10 kpc, as shown in Fig. 1. Similar to the Milky Way (Poggio et al. 2018; Lemasle et al. 2022), we note a warp of the stellar and gaseous disk (see Figs. 1b and 1e, respectively). To avoid too strong effects of radial migration (Binney & Tremaine 2008; Frankel et al. 2018; Grand et al. 2016; Minchev et al. 2018) while maintaining a sufficiently large sample size we further enforce stars to be younger than 0 . 5 Gyr, corresponding to roughly the time of four galactic rotations, and being half the value found by Minchev et al. (2018) for very limited migration in the Milky Way. This selection defacto limits the vertical range of 99% of stars to | 𝑧 | = 1 . 4 kpc. The strong influence of this age cut on the vertical distribution of stars in the Milky Way analogue can best be appreciated from the difference of vertical density distributions of stars in Figs. 1b and 1c. We are applying these cuts for all following analyses of the radial metallicity gradient in Section 3.\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"3. THE LINEARITY OF THE RADIAL METALLICITY GRADIENT IN NIHAO-UHD\",\n \"content\": \"In this section, we analyse the functional shape of the radial metallicity gradient. To get a first impression of possible shapes, we show the face-on view of the decreasing radial metallicity gradient of the simulation in Figs. 2a (for young stars) and Fig. 2d (for gas). Foreshadowing the later parts of this work on local variations, we also show a linear radial fit to either distribution in Fig. 2b and 2e and show the fit residuals in Figs. 2c and 2f. At the moment, however, we focus on the linearity and thus the logarithmic density distribution of star particle iron abundances [Fe/H] across different galactocentric radii 𝑅 Gal . . This distribution is shown in Fig. 3a and strongly suggests that the gradient is predominantly linear, similar to findings for the Milky Way. More complex shapes, such as piecewise linear ones have been suggested based on incomplete and limited data in the literature. We are thus also analysing these shapes with the complete and better-sampled data points of the NIHAO-UHD simulation. We firstly test different global fits in Section 3.1, before testing the influence of binning and coverage in Sections 3.2 and 3.3, respectively.\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"3.1. Global gradient fits\",\n \"content\": \"We fit three different functional forms to the global data: a linear function (used for Fig. 2b) a piecewise linear with a break radius 𝑅 break and a quadratic function The coefficients of the functions are fitted with the /s.pc/c.pc/i.pc/p.pc/y.pc./o.pc/p.pc/t.pc/i.pc/m.pc/i.pc/z.pc/e.pc function /c.pc/u.pc/r.pc/v.pc/e.pc_/f.pc/i.pc/t.pc (Virtanen et al. 2020) and listed in Table 2. To estimate the uncertainty of the break radius 𝑅 break, we use the profile likelihood method to identify the radii at which the residual sum of squares (RSS) values are increased by 1 𝜎 from the best RSS radius in steps of Δ 𝑅 break = 0 . 1 kpc and 0 . 5 kpc for the full and binned data set, respectively. We compute the RSS for each model 𝑓 𝑖 (see Eqs. 1-3) based on the 𝑁 data points as We have confirmed the robustness of our fits by applying other fitting routines as outlined in Table 1. After fitting three different functional forms, we use a combination of parameters to determine which model provides the best fit. In Table 2, the RSS value is the smallest (although only by a small margin) for the quadratic function. When assuming 𝜎 2 = 𝑅𝑆𝑆 / 𝑁 , we can also define a logarithmic likelihood for the 𝑁 data points. For 𝑘 free parameters, we then calculate the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) as For these criteria, the quadratic function performs slightly better than the linear or piecewise linear functions (see Table 2). We show the fit residuals in Fig. 3b as density distribution as well as in Fig. 3c as percentile distributions in radial bins of Δ 𝑅 Gal = 0 . 5 kpc. While the density distribution shows substructure, which we investigate later in Section 4, we note an increase in the median residuals of the linear fit in Fig. 3c towards the inner and outer radii, especially for 𝑅 Gal . > 17 kpc. A quadratic fit (see orange lines in Fig. 3) results in a slightly steeper linear component of the gradient (from -0 . 0411 to -0 . 0497 dex kpc -1 ), which is counteracted by the quadratic flattening term of + 0 . 0005 dex kpc -2 . The latter leads to an RGal. effective flattening of -0 . 172 + 0 . 200 = 0 . 028 dex (linear vs. quadratic terms) at 𝑅 Gal . = 20 dex. While seemingly only a nuisance correction across the large extent of [Fe/H] and 𝑅 Gal . , this quadratic function outperforms the linear fit. This is most apparent in Fig. 3c, where the orange line better traces the median residuals from the linear function across all radii. This suggests that non-linear functions, such as piecewise linear or quadratic ones describe the gradient better. Distinguishing between the two latter functional forms is challenging. The quantitative performance indicators-RSS, AIC, and BIC-show very similar values for both forms, and a closer examination of the residuals in Fig. 4 reveals no clear visual advantage for either the piecewise linear or quadratic model. TAKE-AWAY: Both piecewise linear and quadratic functions provide a better fit to the radial metallicity relation than a simple linear model. However, based on our assessments, there is no clear preference between the piecewise and quadratic functions.\",\n \"pages\": [\n 5\n ]\n },\n {\n \"title\": \"3.2. The influence of binning\",\n \"content\": \"In this section, we test the influence of fitting a function to all points of the distribution or binned data in steps of Δ 𝑅 Gal . = 0 . 5 kpc, using median values as data points and standard deviations /two.sup as uncertainty (see also Hemler et al. 2021, who fitted functions to radially binned IllustrisTNG data). The results are shown in Fig. 4. Given that more than half of the young star particles of the galaxy are within /two.sup Wenote that this 𝜎 is not equivalent to observational uncertainty and can thus not be directly applied onto observational analyses. 𝑅 Gal . < 4 kpc, this binning - although counteracted by the smaller spread of [Fe/H] in the inner galaxy - weighs the distribution of the inner galaxy significantly less than when weighing all particles equally (20 vs. 34 000 data points). The parameters fitted to the binned data exhibit a larger uncertainty due to our use of the spread of [Fe/H] per bin as absolute uncertainty 𝜎 , but the fitted parameters agree well within the fitting uncertainties. TAKE-AWAY: While the specific slopes differ when fitting all points or binned data, they agree within the small fitting uncertainties.\",\n \"pages\": [\n 5,\n 6\n ]\n },\n {\n \"title\": \"3.3. The influence of radial coverage on linear fits\",\n \"content\": \"Although we have gained useful insight into the global function, observational data will rarely cover the full extent of the stellar disk. Milky Way studies have previously been limited to the range of around 5 -15 kpc. There are often similar limitations and even gaps in extragalactic data. Using smaller ranges, observational studies have found hints of piece-wise linear gradients with a break radius in them based on limited radial coverage (e.g. Andrievsky et al. 2002a; Yong et al. 2012; Boeche et al. 2013; Hayden et al. 2014; Anders et al. 2017; Donor et al. 2020; Chen et al. 2023). These results are intriguing, since a quadratic function can, to first order, be approximated by two linear functions with a break radius. We therefore want to use our simulation to test if the radial coverage may indeed delude us into identifying broken linear gradients. Wetest how smaller coverage and piecewise linear fits could mimic a complex global gradient by fitting in piecewise lin- ear radial ranges of 0.25, 0.5, 1, 2, 5, 8, 10, 15, and 20 kpc. We show their difference with respect to a global linear fit in Fig. 5, with color-coding indicating the slope of the local gradient. A horizontal dashed line indicates the same slope as the global fit, whereas the offset of a line from the said horizontal dashed line indicates the local deviation from the global gradient intercept. Differences in line slopes are visualising the difference in gradient slopes between the global and local fits. We see that all ranges suggest more or less significant deviations from a global linear fit. The innermost fit suggests a significantly different gradient than the outermost fit. We also note increasing slope differences towards the smallest scales, hinting at local deviations from a global pattern. We follow these up in Section 4, but for now, focus on the larger-scale trends. When directly comparing an inner and outer radius fit, such as between 𝑅 Gal . = 5 -10 kpc (thick grey line in Fig. 5) and 𝑅 Gal . = 10 -15 kpc (thick black line in Fig. 5), we note a significant change, similar to previous estimates of the Milky Way (e.g. Yong et al. 2012; Lemasle et al. 2008). In our case, the gradient estimate changes from [ Fe / H ] ( 𝑅 Gal . ) = 0 . 471 -0 . 044 · 𝑅 Gal . to [ Fe / H ] ( 𝑅 Gal . ) = 0 . 375 -0 . 034 · 𝑅 Gal . . When looking at linear gradient fits across the radial coverage of Δ 𝑅 Gal . = 5 -15 kpc in Fig. 5, the gradient is steeper (bluer color) for smaller radii and flatter (redder) for larger radii. Indeed, a piecewise linear function can well mimic a complex global gradient. We note that in the simulated data, we see local deviations that become traceable below Δ 𝑅 Gal . ≤ 2 kpc ∼ 0 . 5 Re (bottom part of Fig. 5). This might indicate the spatial resolution required to see local effects, such as spiral arms, for extragalactic studies (see also Krumholz & Ting 2018; Li et al. 2024). We pursue this observation in the following Section 4. TAKE-AWAY: We find that a piecewise linear function can well mimicaquadraticfunction across the scales used in Milky Way and extragalactic studies. Local deviations become traceable below are spatial resolution of Δ 𝑅 Gal . ≤ 2 kpc (or Δ 𝑅 Gal . ≤ 0 . 5 Re).\",\n \"pages\": [\n 6,\n 7\n ]\n },\n {\n \"title\": \"4. SCATTER AND LOCAL DEVIATIONS FROM THE GRADIENT\",\n \"content\": \"Now that we are sufficiently satisfied that our flattening gradient function reproduces the overall shape of the radial metallicity gradient, we are concerned with both the scatter and local slope deviations across the galactocentric radii in this section. In detail, we analyse the scatter (Section 4.2), vertical variations (Section 4.2), azimuthal variations (Section 4.3, particularly motivated by the localised, spiral-shaped fit residuals of Figs. 2c and 2f) and deviations across different ages (Section 4.4).\",\n \"pages\": [\n 7\n ]\n },\n {\n \"title\": \"4.1. Scatter\",\n \"content\": \"When investigating the change in scatter from the innermost radii to the outermost (see Fig. 3c), we see a steady increase in 1 -𝜎 spread. This spread increases from 𝜎 [ Fe / H ] = 0 . 01 dex at 𝑅 Gal . = 0 . 25 ± 0 . 25 kpc to 𝜎 [ Fe / H ] = 0 . 06 dex at 𝑅 Gal . = 8 . 25 ± 0 . 25 kpc and reaches 𝜎 [ Fe / H ] = 0 . 10 dex at 𝑅 Gal . = 19 . 75 ± 0 . 25 kpc. When we recall the observed significant spread in metallicities of young open clusters at the solar radius beyond observational uncertainty (e.g. Donor et al. 2020; Spina et al. 2021) and our selection of only young ( < 0 . 5 Gyr) stars from the simulation, a strong impact of this scatter by radial migration should be excluded. At this point, we can imagine that this chemical diversity might be caused by less well-mixed gas or non-radial effects (such as vertical or azimuthal ones), which we investigate subsequently.\",\n \"pages\": [\n 7\n ]\n },\n {\n \"title\": \"4.2. Vertical deviations\",\n \"content\": \"In this section, we now look at deviations with respect to the vertical dimension, that is, 𝑅 -𝑧 . In Fig. 6 we show the previously identified local gradient deviations (lines following the left axis label) on top of the vertical density distribution ( 𝑅 -𝑧 ) of young stars (Fig. 6a) and gas (Fig. 6b) between -3 < 𝑧 < 3 kpc. Although the quickly decreasing number of young stars (Fig. 6a) at outer radii does not show substructure in the density plots for reasonable bin sizes, we see more substructure for the gaseous component in Fig. 6b). In particular, we see rather minor deviations at small radii (where most stars and gas are close to the plane). At increasing radii, we notice an increase in both the vertical distribution of stars, and increasing local gradient deviations. In particular, we note a significant deviation of the slope around 𝑅 Gal . ∼ 15 kpc, where the gradient deviation line is steep and blue (indicating a much steeper gradient at this radius), and we notice a significant overdensity of gas around 𝑧 ∼ 1 kpc. Overall, however, we do not see strong correlations in this particular plane. This could, however, be caused by a super-position effect of the up- and downturn at larger radii due to the galactic warp (see Figs. 1b and 1e). Although the warp of the stellar disk in Fig. 1b is not as clear, we confirm that both the gas disk and the youngest stars below 0 . 5 Gyr are tracing each other across the simulation in both galactocentric radius 𝑅 Gal . and height 𝑧 Gal . for different sectors in the azimuthal direction. We note that the superposition in Fig. 6 could smear out local correlations of slope changes with gas overdensities, for example, by spiral arms. Although such an edge-on view of the galaxy may indeed be the only observable one for extragalactic targets, for example, of the GECKOS survey of edge-on galaxies (van de Sande et al. 2023), we have the luxury of being able to analyse the azimuthal direction of our simulated galaxy, too. TAKE-AWAY: We see no strong correlations of deviations in the vertical direction throughout the simulation. Such correlations could, however, be blurred by azimuthal effects, like the galactic warp, which needs to be disentangled in the azimuthal dimension.\",\n \"pages\": [\n 7\n ]\n },\n {\n \"title\": \"4.3. Azimuthal deviations\",\n \"content\": \"To analyse the deviations from a global gradient across different azimuthal viewing angles, we divide the galaxy into 8 sectors with Δ 𝜛 Gal . = 45 · (see Fig. 7a). This allows us to study the positions around the upturn and downturn of the galactic warp with the median azimuth of young star particles below and above the plane being 𝜑 Gal . ∼ 183 · and 𝜑 Gal . ∼ 4 · , respectively (see Fig. 1e), while maintaining a reasonable sample size. At face value, the distribution of 𝑅 Gal . - [ Fe / H ] for each sector follows a similar, rather linear shape with most stars being born in the inner 5 kpc of the galaxy. However, we find significant deviations in different sectors of the galaxy (Fig. 7). On the one hand, we find non-linear deviations as bumps with slightly increased or decreased iron abundance - up to 0 . 1 -0 . 2 dex - in Figs. 7c at 𝑅 Gal ∼ 18 kpc, 7d at 𝑅 Gal ∼ 10 kpc, 7f at 𝑅 Gal ∼ 14 kpc, and 7g at 𝑅 Gal ∼ 17 kpc. On the other hand, we find significant gaps in the distribution at similar [Fe/H], most strikingly at the upturn of the galactic warp in Fig. 7e ( 𝜛 Gal . = 135 -180 · ) at [ Fe / H ] ∼ 0 dex and 𝑅 Gal . ∼ 8 -14 kpc. We note that the sector e) with the gap is surrounded by two sectors (d and f) with significant overabundance at the same radius. This could be indicative of stars having formed as a result of gas moving from sector e towards either azimuthal direction, causing a gas overdensity which could in turn lead to higher star formation activity. To establish this observation, we take a closer look at the timedomain, that is, stellar age as well as the spatial domain of 𝑅 Gal . -𝜑 Gal . in the next section. TAKE-AWAY: We find various deviations from the global trend in the azimuthal direction, including gaps and isolated streaks of stars with similar [Fe/H] throughout Δ 𝑅 Gal . = 2 -6 kpc. These can introduce local over- and under-enhancement of up to ± 0 . 2 dex in [Fe/H] at a given radius. In the next section, we analyse whether the stars of these streaks have been born at the same or different time.\",\n \"pages\": [\n 7,\n 9\n ]\n },\n {\n \"title\": \"4.4. Deviations with time and age\",\n \"content\": \"In this section, we examine the radial metallicity gradient in a small age range less than 0 . 5 Gyr. To do so, we color Fig. 7 by the median stellar age rather than the logarithmic density in Fig. 8. We find an overall significant scatter across time, suggesting a good mix of star formation across all sectors for stars born within less than 0 . 5 Gyr. For stars within this restricted age range, we do not see a strong correlation with radius, such as older stars being born further inside, but a larger amount of stars being born closer to the galactic centre. Wenote that stars with similar [Fe/H] in each sector tend to be formed at similar times (within 50 Myr), that is, as flat lines with the same color (age) in Figs. 8b-i. To guide the eye, we have identified Group 1 in Fig. 8e (around 𝑅 Gal . ∼ 14 kpc at 𝜛 Gal . = 180 -225 · ). We further note that the enriched bumps identified earlier are born at similar times, see, for example, Group 2 in Fig. 8f. The coloring by age also reveals that stars with lower [Fe/H] than expected (see Group 3 in Fig. 8h) are born at similar times. In some cases, these extend to Δ 𝑅 Gal . = 2 -6 kpc, see Groups 1, 2, and 3 in Fig. 8. At a given radius 𝑅 Gal . , these streaks cause a significant spread in local [Fe/H] of up to ± 0 . 2 dex (see Fig. 8). From the analysis of azimuthal sectors, the impression arose that the star formation in this simulated Milky Way analogue is - as expected for a spiral galaxy - rather patchy and localised on the smallest timescales. This is confirmed by looking at the spatial distribution of azimuth 𝜑 Gal . and radius 𝑅 Gal . in Fig. 9. Already when looking at the density distribution of all stars born within less than 0 . 5 Gyr in Fig. 9a, multiple streams are visible, stars on spiral patterns (see also Kreckel et al. 2019; Chen et al. 2024b). When following up the previously identi- roups 1, 2, and 3, we recover them on said spiral patterns (Groups 1 and 3) or a local overdensity (Group 2). Although one could imagine that radial migration might induce such a spiral-like shape for the stars of groups 1 and 3, their low age of less than 250 Myr would require a significant migration effect of several kpc, while having no influence on the older stars of group 2. When tracing the position of significant overdensities from Fig. 9a in the same projection colored by age in Fig. 9b, we note that for radii above 𝑅 Gal . > 5 kpc these overdensities are colored in red, that is, containing indeed young stars with ages below 200 Myr and being consistent with the most recent star formation along these spiral patterns in the outer galaxy. TAKE-AWAY: We find significant scatter across the radial metallicity distribution caused by streaks of stars born with similar [Fe/H] at similar times (within 50 Myr) across either very local or radially extended spiral-shaped regions of the galaxy, suggesting local enhancement patterns in small overdensities or along spiral arms.\",\n \"pages\": [\n 9\n ]\n },\n {\n \"title\": \"5. DISCUSSION\",\n \"content\": \"Having presented the analysis, we now put our results into the context of other work in terms of our initial aims: to analyse the shape (Section 5.1), scatter (Section 5.2), local deviations (Section 5.3), and time-dependence (Section 5.4) of the radial metallicity gradient. These initial discussions inform our thoughts on the implications of this work for Milky Way studies in Section 5.5 and the studies of other galaxies in Section 5.6.\",\n \"pages\": [\n 9\n ]\n },\n {\n \"title\": \"5.1. Linearity of the radial metallicity gradient\",\n \"content\": \"The radial metallicity gradient of our simulated NIHAOUHDMilky Way analogue showed an overall decreasing, predominantly linear shape, as established in Section 3. Motivated by previous works by Sánchez-Menguiano et al. (2016), among others, we also fitted piecewise linear and quadratic functions to the data in Section 3.1. Both forms perform better than a linear trend. The very similar fitting performances indicate no significant preference between either piecewise linear or quadratic function. Due to both functions' rather good overall fit, we have not tried more exotic non-linear functions as done by Scarano & Lépine (2013). Increasing the flexibility of the gradient function could, however, improve the fit at the innermost kpc, where a flattening is predicted by our simulation, but chemical enrichment is also harder to simulate (see also Minchev et al. 2013; Sun et al. 2024). We have found no significant influence of binning for our gradient estimates (Section 3.2), but have found that a limited radial coverage - as is the case for the Milky Way - could mimic a truly quadratic function with two piecewise linear fits (Section 3.3). This is important, as it has significant implications for the conclusions we draw from the incomplete data of our Milky Way, as we will discuss in more detail in Section 5.5. The balance between a quadratic and piecewise linear radial metallicity gradient teeters at the breaking radius. If present, our analysis of the Milky Way analogue would place it at 𝑅 break ∼ 10 ± 0 . 5 kpc. This radius is strikingly close to the radius of 9 kpc found by Hemler et al. (2021) for a TNG50 galaxy simulation with a stellar mass of log ( 𝑀 ★ / M ⊙ ) = 10 . 72, that is, close to the Milky Way's (see their Fig. 2). In terms of physical reasons for a breaking radius at this location, a direct and secular influence of a stellar bar with non-symmetric effects around the corotation radius (Di Matteo et al. 2013; Scarano & Lépine 2013) should be minor for our specific scenario due to the low ages of the stars considered in our analysis. In particular, our identified break radius is significantly larger than the corotation radius of the Milky Way bar at 4 . 5 -7 . 0 kpc (Bland-Hawthorn & Gerhard 2016, and references therein) anyway. We are intrigued, however, by the proposition by Garcia et al. (2023) of galactic discs consisting of a star-forming inner disc with a steep gradient and a mixing-dominated outer disc with a flat gradient, with the break radius marking the region of transition between them. In Illustris TNG50-1 data, they found such a transition and break radius to be situated much further out at 30 kpc for Milky Way mass galaxies (10 . 1 ≤ log ( 𝑀 ★ / M ⊙ ) ≤ 10 . 6). While our bestfitting break radius - if present - is inconsistent with theirs, we will follow this up in more detail in Section 5.6, where we also discuss the implications for extragalactic studies in general.\",\n \"pages\": [\n 9,\n 10\n ]\n },\n {\n \"title\": \"5.2. Scatter of the radial metallicity gradient\",\n \"content\": \"In Section 4.1 we found an increasing scatter from 𝜎 [ Fe / H ] = 0 . 01 dex in the inner galaxy to 𝜎 [ Fe / H ] = 0 . 10 dex around 𝑅 Gal . ∼ 20 kpc. Comparing these values with simulations other than TNG50 with a similar metallicity spread (see Fig. 2 by Hemler et al. 2021) is rather difficult, as the literature focuses on the shape and density distribution (see e.g. Minchev et al. 2014, their Fig. 10). When comparing with Milky Way studies (e.g. Anders et al. 2017), the scatter in the simulation is smaller than the observed spread of [Fe/H]. This can be visually appreciated by comparing the combinations of different measurements in the Milky Way (Genovali et al. 2014; Spina et al. 2021; Myers et al. 2022) in Fig. 10a and our simulation in Fig. 10b and c. We discuss the implications of this on studies of the Milky Way's gradient in Section 5.5. When assuming that young star and gas phase abundances are similar, we find comparable scatter of abundances for example with respect to TYPHOON observations by Chen et al. (2023). To test this assumption, we also show the gas phase metallicity in Fig. 11, for which we find a similar shape and scatter of the gradient, but systematically less scatter or spread than observed gas phase abundance, thus urging us to treat the absolute values of abundances and abundance scatters as well as spreads with caution. We furthermore note that the spread of abundance in observations does only increase for some but not all of the observed (and thus observationally limited) galaxies by Chen et al. (2023). This potentially limits the range of galaxies to which our conclusions may apply. While the simulated abundance scatter is consistent with the predictions by the theoretical forced-diffusion model by Krumholz & Ting (2018), that is, a scatter of ∼ 0 . 1 dex over timescales of ∼ 100 -300 Myr, our simulations suggest that the scatter is driven by the radial structure and large-scale spiral arms, which were not included in their model.\",\n \"pages\": [\n 10,\n 11\n ]\n },\n {\n \"title\": \"5.3. Localised vertical and azimuthal deviations and their correlation with gas\",\n \"content\": \"In Sections 4.2 and 4.3 we established that local deviations contribute significantly to the spread of the global metallicity gradient above 𝑅 Gal . > 8 kpc ∼ 2 Re. We noted in particular a void of stars where we found an upturning warp of the galaxy around 𝜑 Gal . ∼ 180 · spatially close to regions of the galaxy (Groups 1 and 2) that deviated most significantly from the overall trend in Fig. 8 and Fig. 9. These stellar voids pose the question if they are also void of gas thus suggesting the gas has shifted to the more enhanced regions. In Fig. 12 we are thus tracing both the spatial distribution of gas as colored density distribution and stars as grey contour lines and gas. We find that while stars and gas trace each other in the vertical direction, we do not always see a match between the two tracers in the radial direction. In particular, we do find a significant amount of gas around the stellar void of 𝜑 Gal . ∼ 180 · and 𝑅 Gal . ∼ 8 -11 kpc in Figs. 12e and 12f. This gas seems to be more tightly concentrated though for example in the tight wave around 𝑅 Gal . ∼ 8 -11 kpc in Fig. 12e. We also note that significant gas overdensities, for example around 𝜑 Gal . ∼ 0 -45 · and 𝑅 Gal . ∼ 7 kpc in Fig. 12a do not seem to correlate with significant overenhancement in iron abundance (compare to Fig. 7a). While we see a hint of a coinciding deviation of Δ [ Fe / H ] for larger deviations from the galactic plane Δ 𝑧 in the upturning outer region of Fig. 12f, this does not seem to be the case for the downturning outer region of Figs. 12b and 12i. As the edge-on projection is not providing conclusive insights, we are now looking into the phase-on projection in Fig. 13. We have chosen a region of the simulated galaxy whose gas density at solar radius (Fig. 13c) matches with the recently measured distribution of young stars in the Milky Way at face value (Fig. 13a) by Poggio et al. (2021). Comparing simulated gas and observed young stars is preferable in this case, as the density of simulated stars is too low to easily identify overdensities (Fig. 13b). The region and its gas spiral structures appear to be representative, as these structures exist throughout the whole galaxy (see Fig. 1d). In the different panels of Fig. 14, we thus show this representative region of the galaxy, but color each spatial bin by stellar metallicity (Fig. 14a), the deviation from the global linear trend (Fig. 14b) as well as the gas metallicity (Fig. 14c) and its deviation from the global linear trend (Fig. 14d). In all cases, we also overlay the density contours of the significant gas overdensities (red regions in Fig. 13c). As expected, we see that the metallicity color map of the stars in Fig. 14a shows a decreasing trend from right to left (inner to outer galaxy) and an increasing scatter (more blue and red points towards the left) in the residual plot of Fig. 14b. We cannot identify a strong correlation between gas overdensities and stellar metallicity or residuals in either plot - possibly caused by the low number density. In Figs. 14c and 14d, however, the radial gas metallicity gradient shows significant local variations, that is, a trend from left to right that is not very smooth. In particular, we find significant deviations of up to + 0 . 15 dex in [Fe/H] behind the outer gas spiral (lower left of Fig. 14d) and -0 . 1 dex in [Fe/H] in front of the inner gas spiral (upper center of Fig. 14d) with a steep edge consistent with the gas spiral edge. We have identified the same patterns in both [Fe/H] and A(O) as both elements trace each other rather well in the simulation of young stars (see Fig. A1). Tentatively, we even see a slight enhancement of A(O) at the trailing edge of the inner spiral arm (top of Fig. 14d). We convince ourselves of the step-like behaviour by selecting a small slit-like region of 𝜑 Gal . ∼ 0 · and -2 < 𝑌 Gal . / kpc < -1 and tracing the gas metallicity and gas density as a function of radius in Fig. 15. We indeed find steps and confirm that they coincide with the location of significant gas overdensities. These step-patterns have also been found by Grand et al. (2015) in another simulation and observationally by Ho et al. (2017). In Fig. 15, we note an extended flat region just beyond 𝑅 Gal . > 10 kpc, the best fitting 𝑅 break of an assumed piecewise linear fit. Our analyses suggest that the correlation of void and overdensities with chemical enrichment of gas and young stars is more complicated and should better be followed up by tracing these structures over simulation look-back time in a dedicated follow-up analysis to unravel the physical mechanisms of star formation feedback cycles . This could also involve the tracing of star formation bursts and disk instabilities (Sánchez et al. 2014; Sánchez-Blázquez et al. 2014; Ho et al. 2015) as well as tracing how much self-enrichment as well as mixing and dilution takes place around the gas spirals (Ho et al. 2017). Rather than going back in simulation time, the present simulation data of a single snapshot in time already allows us to look back in terms of stellar lifetime - similar to Milky Way studies, as we discuss subsequently.\",\n \"pages\": [\n 11,\n 12\n ]\n },\n {\n \"title\": \"5.4. The impact of time and age: mixing and migration\",\n \"content\": \"Although we have chosen a rather small stellar age window of 0 . 5 Gyr to trace the radial metallicity gradient without the expected significant impact of radial mixing and migration, we are testing and discussing this particular choice in this section in two ways. Firstly, we test the deviation of the radial metallicity gradient from the same global shape as well as the abundance spread across smaller age bins of 100 Myr between 0 -1000 Myr in Fig. 16. Secondly, we trace the distribution of stellar metallicity across galactic radii for increasing age bins from 50 Myr up to the maximum stellar age of 13 . 8 Gyr in Fig. 17. Our first test in Fig. 16 shows that the deviation from a global trend remains similar in functional form. We find that the spread of iron abundance does indeed scatter significantly, but the distributions stay within the same overall shape across the ten age bins. We note though, that the smallest age bin of 0 -100 Myr shows the least abundance scatter. This is consistent with the picture from our second test of increasing age ranges in Fig. 17. Here we find the first significant deviation from a tighter and already slightly quadratic relation for an age of 100 -150 Myr in Fig. 17c - our previously identified Group 3. As expected from previous simulations and observations, we see an increase in the scatter as we include more and more older stars. We note a still similar albeit more scattered shape for stars below 4 Gyr in Fig. 17h, before we start to see a more metal-poor population of stars in the inner galaxy appear between 4 -8 Gyr in Fig. 17i. These also begin to significantly impact the quadratic fit to the radial metallicity distribution, shown as a solid red line, in contrast to our reference fit, represented by a dashed red line. The significant amount of metal-poor stars in the inner galaxy then completely tilts the distribution when also including stars between 8 -13 . 8 Gyr in Fig. 17j (see also Johnson et al. 2024). Similar to the Milky Way (Bland-Hawthorn & Gerhard 2016), these oldest stars are those of the relatively more metal-poor thick disk that are confined to the inner disk with a shorter scale length. While we cannot exclude radial migration playing a role for change of radius for the youngest stars of the simulation, since Frankel et al. (2018) predicted significant shifts even for ages below 0 . 5 Gyr (see their Fig. 10), the larger scatter for older stars is certainly suggesting a larger (re-)distribution of stars along the radial axis, as found in previous simulations (Minchev & Famaey 2010; Grand et al. 2015).\",\n \"pages\": [\n 13\n ]\n },\n {\n \"title\": \"5.5. Implications for Milky Way studies\",\n \"content\": \"Our analysis of the radial metallicity gradient in a simulated NIHAO-UHD galaxy offers several insights that are directly applicable to understanding the Milky Way's gradient. First, the nature of the gradient - whether it is linear or better described by more complex functional forms - remains a critical question. Previous studies, such as those by Lépine et al. (2011) and Donor et al. (2020), have suggested the po- ential for a break radius, possibly at the corotation radius or further out (Scarano & Lépine 2013), which could indicate two distinct linear regimes. In our analysis, we find evidence that the gradient is at least is not purely linear, but could also be smoothly flattening. Applying a smooth quadratic function on observational data (Yong et al. 2012; Andrievsky et al. 2004; Genovali et al. 2014), might provide a better or at least consistent fit for the Milky Way data without the need for a break radius. However, even this may not fully capture the nuances observed in our simulations. Chemical evolution models propose a more sophisticated behaviour (e.g. Chiappini et al. 2001; Kubryk et al. 2015; Palla et al. 2024), reflecting varying influences of galactic processes at different radii. Understanding this structure in the simulated galaxy provides a framework for interpreting similar complexities in the Milky Way. Given these complexities, it is also essential to consider how local sampling biases might affect our understanding of the Milky Way's metallicity gradient. For instance, incomplete samples that omit low [Fe/H] clusters or stars could skew gradient estimates, as suggested by our comparisons in Figures 10a and 10b. Our results indicate that young clusters with lower (or higher) [Fe/H] than expected at a given radius could indicate the previous presence of a spiral arm (see our identified Groups in Figs. 8 and 9). Furthermore, we caution that localised effects - both intrinsic and in terms of selection function - could also mimic non-linear shapes and more spatial coverage is needed in the Milky Way. Our results also indicate that older clusters, which have been found more frequently at larger distances than young clusters - are likely influenced by radial migration - and thus complicate the interpretation of these radial metallicity gradients (Magrini et al. 2009; Lépine et al. 2011). Cosmological zoom-in simulations like NIHAO-UHD are approaching the resolution needed to examine regions analogous to the solar vicinity, though the star particle numbers and mass resolution remain a limiting factor. Nonetheless, we observe distinct patterns in the distribution of young stars and gas, including lower [Fe/H] and A(O) in the leading edges of gas overdensities and higher [Fe/H] and A(O) in the trailing edges, consistent with findings by Grand et al. (2016), Ho et al. (2017), and Kreckel et al. (2019). These trends suggest that local metallicity variations, driven by gas dynamics, may also play a significant role in shaping the observed gradients in the Milky Way. Additionally, our study hints at the potential for more nuanced variations in [Fe/H] across different regions of the galaxy. In particular, the gas shows a step-like behavior of A(O)and[Fe/H] changes around the edges of gas overdensities (Fig. 15), with significant deviations from the global gradient in specific regions. We have also found a larger stellar void around -12 < 𝑅 Gal < -10kpc. Although further investigation is needed, these findings could have important implications for understanding localized star formation events and their impact on the overall metallicity distribution in the Milky Way (Sánchez et al. 2014; Sánchez-Blázquez et al. 2014; Ho et al. 2015). It will certainly be exciting to see how much more insights (Poggio et al. 2021; Hackshaw et al. 2024) we will get from the more extended data of future data releases of Gaia and spectroscopic surveys. Wecannot directly link spiral arms to bar resonances or bardriven mixing in our simulation, because of a negligible bar strength in our galaxy /three.sup (but see Minchev & Famaey 2010; Di Matteo et al. 2013). However, the influence of a galactic bar on the spiral arms and, by extension, on the radial metallicity gradient, remains a possibility (see again Chen et al. 2023). Disk instabilities and warps might further complicate the interpretation of these gradients and progress will likely rely on the detailed disentangling of these effects from both cosmological simulations as well as idealised simulations and models (Minchev et al. 2013; Grand et al. 2015, 2016; Krumholz et al. 2018; Sharda et al. 2021; Bland-Hawthorn et al. 2024; Tepper-Garcia et al. 2024).\",\n \"pages\": [\n 13,\n 14\n ]\n },\n {\n \"title\": \"5.6. Implications for extragalactic studies\",\n \"content\": \"The insights gained from our analysis of the radial metallicity gradient in a simulated NIHAO-UHD galaxy extend beyond the Milky Way, offering valuable implications for the study of extragalactic systems. One key observation is that deviations from a purely linear metallicity gradient, as seen in our Milky Way analogue, are common in other galaxies as well. When fitting a piecewise linear fit to our data, we found a break radius at 𝑅 Gal . = 10 . 0 ± 0 . 5 kpc. Converted to effective radii Re or radii R25 covering the 25 mag arcsec -2 isophote /four.sup , this corresponds to 𝑅 break ∼ 2 . 5 Re ≡ 0 . 7 R25 for our simulation. This would be consistent with the observational results by Sánchez et al. (2014) who found that breaks in metallicity gradients are common in both spiral and barred galaxies, with flattening of the abundance being evident beyond ∼ 2 Re (compare also to Belfiore et al. 2017). Similar to our suggestion for Milky Way studies, we suggest to also test a smooth function, such as a quadratic one, on extragalactic observational data (e.g. Bresolin et al. 2012; Chen et al. 2023) to test the preference of a distinct break radius. Although the focus of this research lies on the observable region of the Milky Way ( 𝑅 Gal . < 20 kpc) and most other galaxies ( 𝑅 Gal . < 2 . 5 Re), the finding of significant gradient changes in the outskirts of galaxies by Garcia et al. (2023), suggests to also test this region of our Milky Way analogue. Garcia et al. (2023, see their Fig. 4) found a metallicity floor in IllustrisTNG galaxies. When using their sample to identify a metallicity floor radius for a Milky Way mass galaxy with log ( 𝑀 ★ / M ⊙ ) = 10 . 7 (Bland-Hawthorn & Gerhard 2016) at redshift 𝑧 ∼ 0, we would expect to find it around 25 -30 kpc. We therefore extend the analysed radius to 𝑅 Gal . ≤ 100 kpc (see Fig. 18) and indeed find a similar abundance floor of [ Fe / H ] ≥ -0 . 64 for young stars and A ( O ) ≥ 8 . 12 for the majority of gas (see Fig. 19 at a similar radius. We note that another galaxy without gas in this figure is a sufficiently large distance of 92 kpc, that is, ( 𝑌,𝑌, 𝑍 ) = (-50 , -75 , 20 ) kpc. These lowest abundances remind us of two observational results. Firstly, the iron abundance floor is consistent with the lower end of the Milky Way thin - and coincidentally outer disk of [ Fe / H ] ∼ -0 . 7 (Bensby et al. 2014; Buder et al. 2019). Secondly this oxygen abundance floor is consistent with the results by Grasha et al. (2022) from TYPHOON galaxy observations. Grasha et al. (2022) suggested this could be caused by changes in the ratio of supernovae II and AGB reflected by a changing ratio of nitrogen to oxygen abundance N/O which also flattens towards a lower plateau below metallicities of A ( O ) ∼ 8 . 0 (Nicholls et al. 2017). While we cannot follow this observation up with the present simulation, a similar simulation used by Buder et al. (2024) has traced the relative XGal. XGal. XGal. contribution of both supernovae II and AGB and should be used to test this hypothesis in the future. It is important to note that the chemical evolution model in the NIHAO-UHD simulations is constrained by the current, incomplete understanding of evolutionary pathways and yields (Buck et al. 2021), as well as by limitations in resolution and the imperfect physics inherent to cosmological zoom-in simulations (Buck 2020). Both could contribute to the identified differences in absolute and relative abundances across different scales - including a different scatter of abundances for example of the gas phase metallicity between NIHAO-UHD of up to 0 . 1 dex and the low scatter of 0 . 03 -0 . 05 dex (and even lower on local scales) found by PHANGS-MUSE faceon observations (Kreckel et al. 2020). Extending our analysis to other simulations and further improving the resolution and physics of the simulations will be key in uniting the observational and theoretical insights into galactic chemical evolution on small and large scales. Similar to more resolved and higher quality observations in the Milky Way, we also expect more, better, and diverse faceon and edge-on observations and analyses across a range of wavelengths by the PHANGS and GECKOS teams (Kreckel et al. 2019, 2020; van de Sande et al. 2023) as well as the SDSS-V and MAGPI collaborations (Kollmeier et al. 2017; Foster et al. 2021; Mun et al. 2024; Chen et al. 2024a), among many other ongoing efforts.\",\n \"pages\": [\n 14,\n 15\n ]\n },\n {\n \"title\": \"6. CONCLUSIONS\",\n \"content\": \"To conclude our study, we first iterate the main take-away of our research in Section 6.1 before giving suggestions for future research in Section 6.2.\",\n \"pages\": [\n 15\n ]\n },\n {\n \"title\": \"6.1. Take-Away\",\n \"content\": \"Wehaveanalysed the radial metallicity distribution of young stars and gas in the inner 20 kpc of a NIHAO-UHD Milky Way analogue (Fig. 1), finding a predominantly linear decrease (Figs. 2 and 3). Although our analysis of a single spiral galaxy simulation has limited applicability to the entire population of diverse spiral galaxies, it reveals several intriguing findings about the shape and local metallicity variations. The results we find in this work hold relevance for both the Milky Way and extragalactic research communities: more detail. Secondly, we suggest approaching the fitting of gradients in external galaxies in a more agnostic way to the shape. This will be particularly interesting when we can observe the outermost regions of galaxies, where simulations predict an abundance floor (Figs. 18 and 19).\",\n \"pages\": [\n 16\n ]\n },\n {\n \"title\": \"6.2. Future Research\",\n \"content\": \"In our study, we have focused on the present-day snapshot of the NIHAO-UHD Milky Way analogue simulation - similar to present-day observations that are possible in our local Universe. Given that the simulation is tracing particles and gas over time, a detailed follow-up study should trace the evolution and coherence of spatial and chemical over- and underdensities over time and different elements (see also Zhang et al. 2024). This would in particular allow us to quantify the change of abundance in the leading and trailing edges of spiral arms and subsequently track the mixing and blurring of these over time. Certainly, more studies are needed to establish a link to a physical mechanism and further quantify its importance. More results are expected as we extend the reach, number, and quality of stellar measurements in our Galaxy (e.g. Barbillon et al. 2024) and beyond. These improvements will allow us to move beyond a one-dimensional analysis of gradients and better incorporate and model local variations. While previous works are showing that relative trends for several elemental abundances do not strongly disagree from observations (Buck et al. 2021; Buder et al. 2024), we are still missing several details on the origin of elements, such as a complete picture of the synthesis sites, environments, and yields for elements. Not least because of these imperfections of absolute chemical enrichment predictions, we are refraining from quantitatively comparing the shape of our Milky Way analogue with the actual Milky Way. We have previously also mentioned the limitations in mass resolution of stars and gas, which may introduce unrealistic effects and could for example drive deviations from actual chemical enrichment at the smallest scales. We also note that the results of our simulation may not apply to the actual Milky Way due to different galaxy properties. These could be different due to different galaxy formation pathways, such as the amount and importance of mergers (Buck et al. 2023; Buder et al. 2024). In our discussion, we have already eluded to the weak bar in this simulation. Motivated by the analysis by Tuntipong et al. (2024), we have also investigated the bulge to total stellar mass ratio 𝐵 / 𝑇 . We find a strong bulge with 𝐵 / 𝑇 = 0 . 48 when selecting bulge stars with orbit circularity 𝑗 𝑧 / 𝑗 𝑐 < 0 . 5 and disk stars with 𝑗 𝑧 / 𝑗 𝑐 > 0 . 7 based on actions 𝑗 , consistent with values found by Obreja et al. (2019) of a lower resolution simulation of 8.26e11 . Our simulated galaxy has both a very weak bar and a smoothly changing radial metallicity gradient. This is in line with the findings by Chen et al. (2023) for five strongly and weakly barred spiral galaxies, where bars seem to drive the dominance of break radii in radial metallicity gradients. Future work should certainly look at a variety of galaxies to establish the causality of bar strength with the smoothness or abrupt change at a break radius for the radial metallicity gradient. Expanding our study to more and other simulations such as the VINTERGATAN suite (Renaud et al. 2024) and subsequently comparing to observations of galaxies with varying parameters like mass, formation history, bar strength or environment would allow us to quantify their effect on metallicity gradients and further disentangle the influence of different enrichment mechanisms on the chemical evolution of galaxies.\",\n \"pages\": [\n 16\n ]\n },\n {\n \"title\": \"SOFTWARE\",\n \"content\": \"The research for this publication was coded in /p.pc/y.pc/t.pc/h.pc/o.pc/n.pc (version 3.7.4) and included its packages /a.pc/s.pc/t.pc/r.pc/o.pc/p.pc/y.pc (v. 3.2.2; Astropy Collaboration et al. 2013, 2018), IP/y.pc/t.pc/h.pc/o.pc/n.pc (v. 7.8.0; Pérez & Granger 2007), /m.pc/a.pc/t.pc/p.pc/l.pc/o.pc/t.pc/l.pc/i.pc/b.pc (v. 3.1.3; Hunter 2007), N/u.pc/m.pcP/y.pc (v. 1.17.2; Walt et al. 2011), /p.pc/y.pc/n.pc/b.pc/o.pc/d.pc/y.pc (v. 1.1.0; Pontzen et al. 2013), /s.pc/c.pc/i.pc/p.pc/y.pc (v. 1.3.1; Virtanen et al. 2020), /s.pc/k.pc/l.pc/e.pc/a.pc/r.pc/n.pc (v. 1.5.1 Pedregosa et al. 2011) /s.pc/t.pc/a.pc/t.pc/s.pc/m.pc/o.pc/d.pc/e.pc/l.pc/s.pc (v. 0.14.2 Perktold et al. 2024) We further made use of /t.pc/o.pc/p.pc/c.pc/a.pc/t.pc (version 4.7; Taylor 2005);\",\n \"pages\": [\n 17\n ]\n },\n {\n \"title\": \"DATA AVAILABILITY\",\n \"content\": \"All code to reproduce the analysis and figures can be publicly accessed via https://github.com/svenbuder/ nihao_radial_metallicity_gradients . The used simulationsnapshot can be accessed as FITS file via https: //github.com/svenbuder/preparing_NIHAO . Original data, more snapshots and other galaxies can be found at https://tobias-buck.de/#sim_data . We encourage interested readers to get in contact with the authors for full data access and advice for use and cite Buck et al. (2020, 2021).\",\n \"pages\": [\n 17\n ]\n },\n {\n \"title\": \"ACKNOWLEDGMENTS\",\n \"content\": \"Weacknowledge the traditional owners of the land on which the ANU stands, the Ngunnawal and Ngambri people. We pay our respects to elders past, and present and are proud to continue their tradition of surveying the night sky and its mysteries to better understand our Universe. This work was supported by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013. SB and KGacknowledge support from the Australian Research Council under grant numbers DE240100150 and DE220100766, respectively. TB acknowledges funding from the Carl Zeiss Stiftung and support from the European Research Council under ERC-CoG grant CRAGSMAN-646955. We gratefully acknowledge the Gauss Centre for Supercomputing e.V. ( www.gauss-centre.eu ) for funding this project by providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre ( www.lrz.de ). Simulations were partially computed with High Performance Computing resources at New York University, Abu Dhabi.\",\n \"pages\": [\n 17\n ]\n },\n {\n \"title\": \"REFERENCES\",\n \"content\": \"Yanny B., Lee Y. S., 2006, ApJ, 636, 804 Franchini M., Molero M., Randich S., 2024, A&A, 690, A334 Pedregosa F., et al., 2011, J Mach Learn Res, 12, 2825 Perktold J., et al., 2024, statsmodels/statsmodels: Release 0.14.2, doi:10.5281/zenodo.593847 Ratcliffe B. L., Ness M. K., Buck T., Johnston K. V., Sen B., Beraldo e Silva Rybizki J., Just A., Rix H.-W., 2017, A&A, 605, A59 Schönrich R., Binney J., 2009, MNRAS, 396, 203 Sharda P., Krumholz M. R., Wisnioski E., Forbes J. C., Federrath C., Acharyya A., 2021, MNRAS, 502, 5935 Taylor M. B., 2005, ASPC, 347, 29 Federrath C., 2024, arXiv e-prints, p. arXiv:2406.00342 Schaller M., Schaye J., Theuns T., 2019, MNRAS, 482, 2208 Walt S. v. d., Colbert S. C., Varoquaux G., 2011, Comput Sci Eng, 13, 22 Wang L., Dutton A. A., Stinson G. S., Macciò A. V., Penzo C., Kang X., Keller B. W., Wadsley J., 2015, MNRAS, 454, 83 Willett E., et al., 2023, MNRAS, 526, 2141 Williams M. J., Bureau M., Cappellari M., 2009, MNRAS, 400, 1665 Zaritsky D., Kennicutt Robert C. J., Huchra J. P., 1994, ApJ, 420, 87 Zhang C., Li Z., Hu Z., Krumholz M. R., 2024, arXiv e-prints, p. arXiv:2411.01518 van de Sande J., Fraser-McKelvie A., Fisher D. B., Martig M., Hayden M. R., the GECKOS Survey collaboration 2023, arXiv e-prints, p. arXiv:2306.00059\",\n \"pages\": [\n 17,\n 18\n ]\n },\n {\n \"title\": \"A. ADDITIONAL FIGURES\",\n \"content\": \"Fig. A1 demonstrates the tight correlation of gas phase iron and oxygen abundance and how to approximate them linearly. This paper was built using the Open Journal of Astrophysics L A T E X template. The OJA is a journal which provides fast and easy peer review for new papers in the astro-ph section of the arXiv, making the reviewing process simpler for authors and referees alike. Learn more at http://astro.theoj.org .\",\n \"pages\": [\n 18\n ]\n }\n]"}}},{"rowIdx":4982,"cells":{"bibcode":{"kind":"string","value":"2024arXiv241201746M"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/2412.01746.pdf"},"content":{"kind":"string","value":"\nStrong gravitational lensing with upcoming wide-field radio surveys\nSamuel McCarty 1 , 2 , ★ Liam Connor 3\n1 Department of Astronomy, University of Washington, Seattle, WA 98195-1580, USA\n2 Cahill Center for Astronomy and Astrophysics, MC 249-17, California Institute of Technology, Pasadena CA 91125, USA\n3 Center for Astrophysics | Harvard & Smithsonian, Cambridge, MA 02138-1516, USA\nAccepted XXX. Received YYY; in original form ZZZ\nABSTRACT\nThe number of strong lensing systems will soon increase by orders of magnitude thanks to sensitive, wide-field optical and infrared imaging surveys such as Euclid, Rubin-LSST, and Roman. A dramatic increase in strong lenses will also occur at radio wavelengths. The 2000-antenna Deep Synoptic Array (DSA-2000) will detect over 10 9 continuum sources in the Northern Hemisphere with a high mean redshift ( ⟨ 𝑧 𝑠 ⟩ ≈ 2) and the Square Kilometer Array (SKA) will observe a large sample of extragalactic sources in the South with sub-arcsecond resolution. We forecast lensing rates, finding that the DSA-2000 will discover O( 10 5 ) strongly lensed systems, many of which will be galaxy group and cluster lenses. We propose strategies for strong lensing discovery in the limit where the Einstein radii are comparable to the PSF angular scale, taking advantage of modern computer vision techniques and multi-survey data. We also forecast synergies with optical and infrared surveys, which will provide redshifts as well as multiwavelength information about the lens systems. Finally, we describe applications of radio strong lensing systems, including time-delay cosmography with transient and variable sources. We find that ∼ 100 time-variable flat-spectrum AGN discovered by the DSA-2000 could be used to constrain 𝐻 0 at the percent level with the appropriate follow-up.\nKey words: gravitational lensing: strong - radio continuum: general\n1 INTRODUCTION\nStrong gravitational lensing has a multitude of applications in astrophysics and cosmology (Treu 2010). Previously theoretical ideas have been put into practice in recent decades as the number of known lensed systems has increased and observational data have improved. For example, strongly lensed time-variable and transient sources can be used to constrain the Hubble constant, 𝐻 0 , because the time delay of a multiply imaged source depends on the geometry of the Universe (Refsdal 1964). The technique is known as time-delay cosmography. With just six lensed quasar systems, the The 𝐻 0 Lenses in COSMOGRAIL's Wellspring (H0LiCOW) collaboration has reported 2.4 % precision on their 𝐻 0 measurement, which is independent of the distance ladder and the CMB (Wong et al. 2019).\nIn addition to the Universe's large-scale geometry, lensing observables are sensitive to the total mass of the deflector galaxy or cluster, allowing one to measure the spatial distribution of matter and test different dark matter models (Massey et al. 2010; Vegetti et al. 2024). Lensing magnification allows astronomers to observe objects in the distant Universe, as was pointed out at the field's inception (Zwicky 1937). Dramatic examples have come from the James Webb Space Telescope (JWST), including a red supergiant star at 𝑧 ≈ 2 . 2 that appears to be magnified by a factor of several thousand due to its proximity to caustics in a cluster lens (Diego et al. 2023).\nNearly all of these applications would benefit from a larger sample of strong lensing systems. To date, roughly 10 3 strong lensing systems have been discovered, most of which were identified at optical\nand infrared wavelengths (O/IR). Fortunately, upcoming wide-field imaging surveys such as Euclid and The Vera C. Rubin Observatory's Legacy Survey of Space and Time (Rubin-LSST) are each expected to detect as many as ∼ 10 5 strong lenses (Collett 2015). An early release of a 0.7 deg 2 field from Euclid has recently affirmed these forecasts (Barroso et al. 2024). The Nancy Grace Roman Space Telescope's 2000 square degree survey could find of order 20,000 strong lenses (Weiner et al. 2020), and many more if the proposed multi-epoch 4 𝜋 sr survey is carried out (Han et al. 2023). An increase in the total number of lenses by two-orders of magnitude will usher in a new era of strong lensing science.\nThe first strongly lensed system ever discovered was co-detected at radio wavelengths (Walsh et al. 1979). The first survey for lenses, the Mit-Green Bank survey, was also in the radio (Bennett et al. 1986). Despite this, fewer than ∼ 100 radio lensing systems have been detected to date. This is due to the relatively small total number of known radio sources ( ∼ 10 7 ) and the lack of wide-field imaging surveys with ∼ arcsecond resolution. Both of those limitations will soon be overcome with the advent of next-generation radio survey telescopes.\nThe 2000-antenna Deep Synoptic Array (DSA-2000) will detect over one billion radio sources with a deep redshift distribution, most of which will be star-forming radio galaxies (SFRG) or active galactic nuclei (AGN) (Hallinan et al. 2019). The DSA-2000 is expected to see first light in 2027 with key surveys running between 2028 and 2033. Its point-spread function (PSF) will be roughly 2' at the top of the 0.7-2 GHz radio band. A fifty-fold increase in the total radio source catalog is made possible by the DSA-2000's high survey speed, driven by a large field-of-view ( ∼ 10 deg 2 ) and high sensitivity (the expected\nsystem-equivalent flux density or 'SEFD' is just 2.5 Jy). Its cadenced all-sky survey will map out the 3 𝜋 sr above declination -30 · down to 500 nJy/beam root-mean square noise (Hallinan et al. 2019). While the nominal cadence of the continuum survey is four months, certain fields may be visited more regularly, presenting an opportunity to measure lensing time delays with better temporal sampling.\nThe mid-frequency telescope for the square kilometre array (SKAMid) will consist of 197 fully steerable 13.5 m dishes (including the existing MeerKAT radio telescope), operating between 350 MHz and 15.4 GHz with a field-of-view of roughly 1 deg 2 at 1400 MHz. The end of its contruction is anticipated to be July 2029 1 . Although the SKA's lower survey speed will result in fewer sources than the DSA-2000, the wide frequency range and long maximum baseline (150 km) will enable high-resolution imaging, an asset to strong lensing studies (McKean et al. 2015). The Next Generation Very Large Array (ngVLA) is a planned interferometer with extraordinary sensitivity covering a wide range of frequencies (1.2-116 GHz) (Selina et al. 2018a). It will be able to resolve features at milliarcseconds scales. While its broad science goals did not require optimizing the instrument for mapping speed (Selina et al. 2018b), the ngVLA will be a world-class instrument for strong lensing science.\nRadio strong lensing offers distinct advantages to studies at shorter wavelengths and will complement O/IR imaging surveys. Starforming radio galaxies and AGN can be detected to great distances, boosting the mean optical depth of radio continuum sources. Obscuration by dust in lensing galaxies is not an issue at radio wavelengths, nor is the variable 'seeing' that impacts ground-based O/IR telescopes. Relatedly, the point-spread function (PSF) of a radio interferometer is directly determined by observing frequency and array configuration, allowing us to accurately forward model the instrument's response. In the limit of a large number of antennas and a filled aperture (a 'radio camera'), the deterministic PSF allows us to be more ambitious in image-plane deconvolution, enabling techniques such as superresolution (Connor et al. 2022). Radio telescopes also measure full polarization information, which is conserved under gravitational lensing (Greenfield et al. 1985; Dyer & Shaver 1992). The rotation measure (RM) of a lensed source allows one to measure the magnetic field and ionized gas properties of intervening halos (Mao et al. 2017). Finally, the larger emission regions of radio AGN render them less susceptible to microlensing by substructure in the lens, and may provide cleaner modelling of the deflector mass distribution (Birrer et al. 2024). This is critical for measuring 𝐻 0 via time-delay cosmography.\nThere are of course several drawbacks to strong lensing studies in radio surveys, often precluding the use of standalone radio observations. For example, continuum sources will not contain redshift information, and multi-wavelength datasets or follow-up will be necessary for the majority of lensed systems at radio frequencies. Secondly, the morphology of radio lobes can mimic lensing arcs, exacerbating the already-challenging lens identification problem (consider, for example, head-tail galaxies 3C 465 or 3C 129 at redshift 2). In the case of the DSA-2000, the angular resolution is such that 75-90% of galaxy-galaxy lensing systems will be unresolved in the absence of superresolution.\nIn this work we seek to study radio strong lensing in upcoming interferometric surveys, and develop strategies that alleviate the drawbacks of radio lensing. We focus on the DSA-2000, first by forecasting its strong lensing rates for different source classes. We compare this with a forecast for strong lensing discovery on the SKA. Next, due to\n\n\n
Figure 1. The theoretical distribution of strongly lensed image separations (2 𝜃 𝐸 ) from our model for galaxies (solid), galaxy groups (dashed), and galaxy clusters (dotted), assuming a DSA-2000 source redshift distribution. The portion that will be discoverable by the DSA-2000 is shown in the blue-shaded region, while the portion that could be discoverable with superresolution is shown in the green-shaded region. The galaxy group distribution is flat at the top because the density profile is constant over the entire mass range in our model.
\n\nthe 2-3' PSF of the DSA-2000, we discuss methods for superresolving lensing candidates with modern computer vision techniques in order to increase the yield of strongly lensed systems. We then consider lensed time-variable and transient sources that the DSA-2000 will find and forecast constraints on 𝐻 0 that can be achieved with the appropriate follow-up, as well as other applications of strong lensing.\n2 OPTICAL DEPTH CALCULATION & EXPECTED RATES\nFirst, we offer a simple empirical forecast based on observational data and cosmological simulations, and then offer a more detailed forward model that accounts for different source classes and the deflector population.\n2.1 Empirical forecast\nThe Cosmic Lens All Sky Survey (CLASS) (Myers et al. 2003) sought to study a statistical sample of radio-loud gravitationally lensed systems. Despite significant advances in both radio instruments and lens-finding algorithms in the two decades since CLASS, the number of known radio lenses has not increased dramatically. We can use CLASS for a rough estimate of the number of lenses the DSA-2000 will find. CLASS found that a complete sample of 8,958 flat-spectrum radio point sources brighter than 30 mJy had a mean lensing optical depth of 1 . 5 + 0 . 5 -0 . 3 × 10 -3 (Browne et al. 2003). By targeting compact sources with flat spectra, many of their objects are quasars at high redshifts, comparable to typical redshifts of DSA-discovered sources, despite the difference in flux scales. Of the\nconfirmed radio lenses, approximately 20% had angular separations larger than 2 arcseconds, which could be detected by the DSA-2000. Thus, a rough estimate indicates that for every 10,000 sources detected by the DSA-2000, several could be identified as strong lenses. If we assume a detection threshold of 10 𝜎 , the number of extragalactic DSA-2000 sources becomes roughly 5 × 10 8 (Section 2.3). This would result in O( 10 5 ) new galaxy-scale radio lenses, depending on the practical signal-to-noise threshold for candidate systems.\nGalaxy group and cluster scale lenses have been neglected in previous investigations of lensing statistics for upcoming surveys because they make up a smaller, but still significant, fraction of the total lenses. Additionally, lens modeling is more complicated in this regime. The dark matter halos of groups, in particular, are complex and not well-understood structures. From Oguri (2006), the number of group and cluster lenses are around 11% and 3% of the total galaxy lenses respectively (excluding sub-halo lensing). However, these systems make up a considerable portion of lens systems with angular separation of order 1', and almost all of the lenses at ≥ 10'. Because the PSF of the DSA-2000 is relatively large (2' at 2 GHz and 3.3' at 1.4 GHz), these lenses will make up a large fraction of the discoverable lensing systems and so must be accounted for in our investigation.\nA radio group/cluster lens survey as complete as CLASS does not exist, but we can make a simple empirical estimate from the literature. The total number of galaxy scale lenses should be O( 10 6 ) assuming 10 9 sources and the CLASS lensing optical depth. Taking the percentages of group/cluster lenses from Oguri (2006), we should see O( 10 5 ) groupscale lenses and O( 10 4 ) cluster scale lenses, almost all of which will have angular separation large enough to be detected in the DSA-2000. The first and largest survey for group-scale lenses was the Strong Lensing Legacy Survey (SL2S) (Cabanac, R. A. et al. 2007). SL2S found 13 strong lens systems in the mass range of groups in ∼ 100 deg 2 of the Canada France Hawaii Telescope Legacy Survey (CFHTLS), giving a rate of ∼ 0.13 deg -2 (Limousin, M. et al. 2009). A later publication identified at least 54 promising group lenses in ∼ 150 deg 2 of the CFHTLS (More et al. 2012). However, the mean redshift of the CFHTLS sources used in SL2S is ⟨ 𝑧 𝑠 ⟩ < 1, and we expect ⟨ 𝑧 𝑠 ⟩ ≈ 2 for the DSA-2000 (see section 2.3, figure 3 below) (Coupon, J. et al. 2009). At these redshifts, we expect the lensing optical depth to be roughly ∝ 𝑧 2 (see equation 31 of Oguri (2019); figure 2 below). This means that we would expect to see ∼ 10 + 6 -6 times more group lenses in the DSA-2000, for a total of ∼ 1 . 1 + 0 . 6 -0 . 9 × 10 5 . Similarly, the Red Sequence Cluster Survey found 8 cluster lenses in ∼ 90 deg 2 with shallower CFHT imaging, indicating ∼ 2 . 6 + 1 . 6 -1 . 6 × 10 4 cluster lenses in the DSA-2000 (Gladders et al. 2003). These estimates are of the same order of magnitude as expected from Oguri (2006).\nAlternatively, if we take the cross section for giant arcs from the simulations of Puchwein & Hilbert (2009), Fedeli, C. et al. (2010), and Mahdi et al. (2014), assume typical values of 𝑀 = 1 × 10 14 M ⊙ , 𝑧 𝑑 = 0 . 5, and 𝑧 𝑠 = 2, we can calculate a rough estimate of the cluster lensing optical depth at 𝑧 𝑠 = 2. The cross sections at these values are all on the order of 10 arcsec 2 . From Böhringer, Hans et al. (2017), the number density at 𝑀 = 1 × 10 14 M ⊙ is approximately ≳ 1 × 10 -6 Mpc -3 , and after multiplying by the differential coming volumeat 𝑧 𝑠 = 2andskycoverageoftheDSA-2000weget ≈ 5 × 10 -5 for all three simulations. When taking into account that this is the cross-section for giant arcs only, this means we should see several tens of thousands of cluster lenses in the DSA-2000, in good agreement with the previous estimates.\nWealso make an empirical estimate of the rate of lensed transients\nin the DSA-2000. These objects are of particular interest for 𝐻 0 measurements but are much harder to forward model because of the lack of observational data. We can use data from the Very Large Array Sky Survey (VLASS), which is currently the most comparable survey to the DSA-2000, for a simple estimate (Lacy et al. 2020). The determined log rates of supernovae (SNe) and tidal disruption events (TDEs) in deg -2 yr -1 are -1.91 + 0 . 15 -0 . 16 and -2.85 + 0 . 28 -0 . 38 respectively (Dong et al. 2024 in prep). If we assume that the total number of observable sources scales as 𝑆 -1 . 5 min , the value for a Euclidean universe, and that the flux limit of VLASS is 0.7 mJy (10 𝜎 from Lacy et al. (2020)), the DSA-2000 should find ∼ 200 more sources of each type above 20 𝜇 Jy, which is a 10 𝜎 detection in a single epoch of the DSA-2000. We can also assume a lensing optical depth of 1 × 10 -4 and a magnification bias of 2, which are typical for the low redshifts at which we expect to detect transients. Because typical radio rise times of TDEs are on the order of 10 3 days the cadence of the DSA-2000, ∼ 1 / 3 yr, is fast enough to catch most TDEs (Cendes et al. 2023). Scaling for the sky coverage of the DSA-2000, and the cadence for SNe, gives a total yield of 5 + 2 -1 yr -1 and 0.6 + 0 . 5 -0 . 3 yr -1 lensed sources per year for SNe and TDEs respectively. Applying a similar estimate of the lensed rate of GRB afterglows using the predictions of Ghirlanda et al. (2013, 2014) gives a rate much less than 1 per year, and so we ignore them in the rest of this investigation.\nFurther, Yao et al. (2023) find a volumetric rate of optically selected TDEs of 290 + 60 -130 Gpc -3 yr -1 . Cendes et al. (2023) find that ≈ 50% of optically selected TDEs emit in the radio on longer timescales. Becausethere are likely TDEs that emit in radio but not the optical, we take 50% of the optically selected rate to be a conservative estimate. The luminosities of the TDEs in the sample from Cendes et al. (2023) are ∼ 10 37 -10 39 ergs/s, but they note that many of these are likely a lower limit because most of the TDEs still had rising emission at the time of observation. Given this, the DSA-2000 should be able to detect this rate of TDEs out to 𝑧 ≈ 0 . 5, which, when combined with the same optical depth and magnification bias as above, gives 0 . 8 + 0 . 2 -0 . 4 yr -1 lensed TDEs. This number is in good agreement with the previous estimate, so in general, we expect to see about 1 lensed TDE per year in the DSA-2000. However, as noted before, this is a conservative estimate, and there is reason to believe that the actual rate of total and lensed TDEs in the DSA-2000 might be significantly higher. Additionally, because many of the TDEs will be bright in multiple epochs, the DSA-2000 could detect them well below the 20 𝜇 Jy limit, which would increase the rate by a large factor.\nThe DSA-2000 will also discover tens of thousands of distant fast radio bursts (FRBs) (Petroff et al. 2019; Cordes & Chatterjee 2019), some of which will be strongly lensed (Connor & Ravi 2023). The key advantage to using FRBs for time-delay lensing is that their short duration and coherence allows for exceptional precision on the gravitational lensing time delay (Wucknitz et al. 2021). Radio telescopes can preserve phase information about the electromagnetic waveform at nanosecond sampling, which means microlensing signals can be searched for at ultrashort timescales (Leung et al. 2022; Kader et al. 2022). However, for cosmological lensing time delays longer than a pointing (i.e. deflectors more massive than ∼ 10 8 M ⊙ ), one needs to catch the lensed images by pointing at the same patch of sky when it arrives. We do not forecast lensed FRB rates here and point the reader to previous estimates (Connor & Ravi 2023).\n2.2 Lens Model\nNext, we build a forward model based on the lens and source distributions. Following Yue et al. (2022b), the lensing optical depth for a\n
\n\n
Table 1. Total sources above 10 𝜎 𝑛 and expected number of discoverable lensing events by deflector type in the DSA-2000, organized by source classes. Blazars are both FSRQs and BLLacs and transients here include ccSNe and TDEs.
\n
\nsingular isothermal sphere (SIS) is\n𝜏 ( 𝑧 ) = ∫ 𝑧 𝑠 0 𝑑𝑧 𝑑 ∫ 𝑑𝜎 Φ ( 𝜎, 𝑧 𝑑 ) 𝑑 2 𝑉 𝑐 𝑑 Ω 𝑑𝑧 𝑑 𝜋𝜃 𝐸 ( 𝜎, 𝑧 𝑑 , 𝑧 𝑠 ) 2 𝐷 2 𝑑 , (1)\nwhere 𝑧 𝑑 is the redshift of the deflector, 𝑧 𝑠 is the redshift of the source, 𝜎 is the 1D velocity dispersion of the deflector, Φ ( 𝜎, 𝑧 𝑑 ) is the velocity dispersion function (VDF) of the deflectors, 𝑑 2 𝑉 𝑐 𝑑 Ω 𝑑𝑧 𝑑 = ( 1 + 𝑧 𝑑 ) 3 𝑐 𝑑𝑡 𝑑𝑧 𝑑 is the differential comoving volume, 𝜃 𝐸 ( 𝜎, 𝑧 𝑑 , 𝑧 𝑠 ) is the Einstein radius, and 𝐷 𝑑 is the angular diameter distance at the deflector redshift. The SIS model (or its elliptical generalization) has been shown to replicate the properties of early-type galaxies, which make up the majority of galaxy lenses (e.g. Gavazzi et al. (2007); Koopmans et al. (2009); Li et al. (2018)), and is a widely used model for galaxy strong lensing statistics (e.g. Oguri & Marshall (2010); Collett (2015); Yue et al. (2022a). For a SIS, the Eintein radius becomes\n𝜃 𝐸 = 4 𝜋 GLYPH<16> 𝜎 𝑐 GLYPH<17> 2 𝐷 𝑑𝑠 𝐷 𝑠 , (2)\nwhere 𝐷 𝑑𝑠 is the angular diameter distance from the deflector to the source and 𝐷 𝑠 is the angular diameter distance from the observer to the source. We use the analytical VDF from Yue et al. (2022b) to model the galaxy deflector population, which they show to be in good agreement with observations. We include a sharp exponential cutoff on the VDF at 300 kms -1 which corresponds roughly to the transition between galaxies and galaxy groups.\nAn important ingredient in the calculation is the magnification bias, which increases the rate of lensed sources by about a factor of two at low redshifts, or by orders of magnitude at high redshifts. The general magnification bias is\n𝐵 = ∫ +∞ 𝜇 min 𝑑𝜇𝑝 ( 𝜇 ) 𝑁 ( > 𝐿 min / 𝜇 ) 𝑁 ( > 𝐿 min ) , (3)\nwhere 𝜇 is the magnification of a lensed source, 𝑝 ( 𝜇 ) is the probability distribution of the magnification, 𝐿 min is the smallest observable luminosity, and 𝑁 ( > 𝐿 min ) is the number of sources brighter than 𝐿 min . For SIS, 𝜇 min = 2 for the total magnification of the multiple images. A SIS will produce only two multiple images; if the separation of the multiple lens images is large enough to be resolved then 𝑝 ( 𝜇 ) = 2 ( 𝜇 ± 1 ) 3 describes the magnification of the fainter (+) and brighter (-) image (Wyithe et al. 2001). If the lens is not resolved then 𝑝 ( 𝜇 ) = 8 𝜇 3 is the total magnification of both images. Combining 𝐵 and 𝜏 for the total fraction of lensed sources:\n𝐹 lensed = 𝐵𝜏 𝐵𝜏 + 𝐵 ' ( 1 -𝜏 ) , (4)\nwhere 𝐵 ' is the magnification bias of sources that are not lensed, which is assumed to be unity.\nWhile the SIS model describes deflector galaxies well, the mass distributions of galaxy groups and clusters behave differently. It has been traditionally thought that in the limit of large mass and large image separation, i.e. clusters, the distribution will be dominated by the dark matter halo, generally following the Navarro-Frenk-White (NFW) profile (Navarro et al. 1997). Recent works suggest that NFW profiles may not be the most accurate representation of large dark matter halos (e.g. Klypin et al. (2016)), but we use them here because they are a decent approximation and their lensing properties are well known. Regardless, for the mass range of groups, it is clear that some intermediate model between SIS and NFW is necessary (Williams et al. 1999; Oguri 2006; More et al. 2012). In a SIS the density 𝜌 ∝ 𝑟 -2 , while the NFW has 𝜌 ∝ 𝑟 -1 on small scales and 𝜌 ∝ 𝑟 -3 on larger scales. The shallower NFW profile has a significantly smaller cross-section than SIS. Oguri (2006) include the effects of baryon cooling and the large elliptical galaxies found at the center of most halos in their model, which will steepen the central density profile and increase the cross-section of halos (especially for groups). Without introducing these complexities, we can steepen the inner profile of halos by modeling them as Generalized NFW (GNFW) profiles. The GNFW is:\n𝜌 ( 𝑟 ) = 𝜌 𝑠 𝑟 3 𝑠 𝑟 𝛼 ( 𝑟 + 𝑟 𝑠 ) 3 -𝛼 , (5)\nwhere 𝑟 𝑠 is the scale radius and 𝜌 𝑠 is the scale density (Li & Ostriker 2002). When 𝛼 = 1 we have the standard NFW. When 𝛼 = 2 the profile resembles an SIS below the scale radius and when 1 < 𝛼 < 2 the inner profile is somewhere in between. Using weak lensing and stellar kinematics, Wang et al. (2023) find the inner dark matter density profiles of group (10 13 𝑀 ⊙ ≤ 𝑀 < 10 14 𝑀 ⊙ ) and cluster ( 𝑀 ≥ 10 14 𝑀 ⊙ ) halos are 1 . 82 + 0 . 15 -0 . 25 and 1 . 48 + 0 . 2 -0 . 41 respectively. Similarly, Mandelbaum et al. (2006) find that the total inner profiles of groups with mass ∼ 2 . 5 × 10 13 𝑀 ⊙ are decently described by a power-law of slope 𝛾 tot = 1 . 85. However, at ∼ 7 × 10 13 𝑀 ⊙ they are less steep than 1.85 and closer to the traditional NFW. Newman et al. (2013) find 𝛾 tot = 1 . 16 + 0 . 10 -0 . 12 for the inner profile of clusters with 𝑀 = 0 . 4 -2 × 10 15 𝑀 ⊙ , and Newman et al. (2015) find the total inner profile slope 𝛾 tot ≈ 1 . 7 while the dark matter slope 𝛼 ≈ 1 . 35 for galaxy groups with ⟨ 𝑀 ⟩ ≈ 10 14 𝑀 ⊙ . In this work, we use the GNFW profile with 𝛼 = 1 . 6 to model groups and 𝛼 = 1 . 2 to model clusters. The resulting GNFW profiles will capture the inner profile well, which is crucial for lensing, and also converge to the standard NFW profile at large radii where we expect the density to be dominated by dark matter (e.g. figures 3 and 4 of Wang et al. (2023)). The GNFW is completely parameterized by 𝛼 , its mass 𝑀 , and the concentration parameter, 𝑐 , from which 𝑟 𝑠 and 𝜌 𝑠 can be determined as in Li & Ostriker (2002). The mass and 𝑐 are correlated with some scatter,\nwe use the relationship presented by Dutton & Macciò (2014) for 𝑐 200 . The scatter in 𝑐 is lognormal with 𝜎 log 𝑐 = 0 . 11. We include an additional factor of ( 2 -𝛼 ) as in Oguri et al. (2001) to correct for the generalized form of the NFW which ensures that the radius at which the logarithmic density slope becomes -2 is the same as in the standard NFW for all 𝛼 . The lensing power of the GNFW halo is very sensitive to 𝑐 and its scatter, so having an accurate relationship is important; early investigations tend to overestimate these values.\nThe lens equation relates a position on the lens plane, 𝑥 , to a position on the source plane, 𝑦 , which for a GNFW profile is\n𝑦 = 𝑥 -𝜇 𝑠 𝑔 ( 𝑥, 𝛼 ) 𝑥 , (6)\nwhere\n𝜇 𝑠 = 4 𝑝 𝑠 𝑟 𝑠 Σ 𝑐𝑟 , Σ 𝑐𝑟 = 𝑐 2 4 𝜋𝐺 𝐷 𝑠 𝐷 𝑑 𝐷 𝑑𝑠 , (7)\nand 𝑔 ( 𝑥, 𝛼 ) is the same as in Li & Ostriker (2002). The lens equation has three solutions inside the radial caustic, 𝑦 𝑐𝑟 , which are the multiple images produced by the GNFW profile. The cross-section for lensing is the area in the source plane in which multiple images will be produced, i.e. the area inside the radial caustic. This is approximated as:\n𝜎 ( 𝑀, 𝑧 ) ≈ 𝜋𝑦 2 𝑐𝑟 𝑟 2 𝑠 (8)\nwhere 𝑦 𝑐𝑟 = -𝑦 ( 𝑥 𝑐𝑟 ) and 𝑥 𝑐𝑟 is the location of the minimum of equation 6 (Li & Ostriker 2002). The image separation is given by the separation between the outer two images, which can be approximated as\nΔ 𝜃 ≈ 2 𝑥 0 𝑟 𝑠 𝐷 𝑑 (9)\nwhere 𝑥 0 is the position of the tangential critical curve and can be found as the root of equation 6 (Li & Ostriker 2002). The Einstein radius is just half of the image separation, 𝜃 𝐸 = Δ 𝜃 / 2.\nIn general, because our GNFW profiles are shallower than a SIS, they will be worse at producing multiple images but significantly better at magnifying. This is particularly sensitive to 𝑐 , which defines the shallowness of the GNFW profile. For our concentration parameters, especially at high redshift, we expect the magnifications from the GNFWs to be several times larger than from SIS (Wyithe et al. 2001). To calculate the magnification bias for a GNFW profile, we first determine the minimum magnification as (Li & Ostriker 2002):\n𝜇 min ≈ 2 𝑥 0 𝑦 𝑐𝑟 . (10)\nWith 𝜇 min wecalculate B according to equation 3 with (Li & Ostriker 2002):\n𝑝 nfw ( 𝜇 ) = 2 𝜇 2 min 𝜇 3 . (11)\nThe minimum magnification for both the first and second brightest image can be approximated as half of the total minimum magnification (Oguri et al. 2002). Substituting 𝜇 min = 2 into equation 11 recovers the total magnification distribution for SIS.\nTo model the populations of galaxy groups and clusters we determine Mass Functions (MF) from the CosmoDC2 synthetic catalogue Korytov et al. (2019). For simplicity, we define a galaxy group as a\nsystem with mass 10 13 𝑀 ⊙ ≤ 𝑀 < 10 14 𝑀 ⊙ and a galaxy cluster as 𝑀 ≥ 10 14 𝑀 ⊙ . We find all dark matter halos in a redshift interval where the mass of the halo and its constituent galaxies fall in these ranges. We fit Schechter functions to the binned results. The fitted functions match the results of Böhringer, Hans et al. (2017) well at 𝑧 = 0 and show a realistic decline in the mass function towards higher redshift. To calculate the optical depth, we replace 𝜋𝜃 𝐸 ( 𝜎, 𝑧 𝑑 , 𝑧 𝑠 ) 2 𝐷 2 𝑑 in equation 1 with equation 8, Φ ( 𝜎, 𝑧 𝑑 ) with the CosmoDC2 MF, and integrate over 𝑀 instead of 𝜎 .\nWhile equation 4 gives the total fraction of lensed sightlines, we are more interested in the number of discoverable lenses. We define a lens as discoverable if its image separation Δ 𝜃 = 2 𝜃 𝐸 is larger than 2/3rds of the FWHM of the PSF of the survey instrument (as in Oguri & Marshall (2010); Yue et al. (2022a)) so that multiple images can be resolved. Further, for point-like sources (i.e. flat-spectrum sources and transients), we require the second faintest image to be detectable at 10 𝜎 confidence so that the lens can be identified. To account for this we integrate equation 1 from 𝜎 min (or equivalently 𝑀 min for groups and clusters) corresponding to 𝜃 𝐸 min and use the magnification bias of the fainter image. The second image from a SIS will often be demagnified, so the magnification bias of the fainter image is typically a bit less than 1, while the magnification bias of the fainter image from an NFW will be about half of the total. For SFRGs and SS-AGN, which in general have extended emission regions, the lensed images will be stretched into arcs and rings, and so for this case we require the brightest image to be detectable at 10 𝜎 confidence (using the magnification bias of the brightest image) because only one arc is needed to identify a lens.\nThe image separation distribution predicted by our model and the lensing optical depths for galaxies, groups, and clusters are shown in Figures 1 and 2 respectively. We see that the optical depth increases rapidly at small redshift as expected, and approximately matches the estimates given in Section 2.1. The galaxy lens optical depth is smaller than the approximation from Oguri (2019). The discoverable optical depth, limited by the PSF of the DSA-2000, is significantly smaller than the total optical depth for galaxies but similar to the total for groups and clusters. The image separation distribution matches the results of Oguri (2006) well despite the simpler model. As expected, galaxies lie mostly at Δ 𝜃 ≈ 1', while groups and clusters correspond to Δ 𝜃 ≈ 10' and 10' ≲ Δ 𝜃 ≲ 100' respectively. The largest uncertainty is in the model of groups, as our simple model does not capture many of the complexities of halo profiles and is constant over the whole mass range. Further, varying 𝛼 between 1 and 2 creates a significant change in the calculated group optical depth. On the other hand, there is inherent uncertainty in all three models from the assumption of spherical symmetry. Using SIS instead of Singular Isothermal Ellipsoids (SIE) for galaxies will overestimate the galaxy lens population by ∼ 10%, which is corrected for in our final numbers (Yue et al. 2022b; Ferrami & Wyithe 2024). Introducing ellipticity will similarly reduce the cross-section of NFW profiles.\nWe do not account for effects such as sub-halo lensing or line-ofsight structures and lens environments. It is shown in Oguri (2006) that a significant fraction of lenses, roughly half for groups and clusters and 10% for galaxies, come from lensing by sub-halos. The environment around these sub-halos boosts their lens capabilities and image separations. At least one of the CLASS lenses is caused by a galaxy within a galaxy group (Auger et al. 2007). In the context of cluster lensing, this is termed galaxy-galaxy strong lensing (GGSL) and it is reported that observational numbers of GGSL exceed expectations of the Λ CDM model (Meneghetti, Massimo et al. 2022; Tokayer et al. 2024). This means that we are likely underestimating the number of lenses in the image separation range of galaxy scale\n\n\n
Figure 2. Lensing optical depth vs. redshift without magnification bias. Solid lines are the total optical depth while dashed lines are the optical depth for lenses discoverable by the DSA-2000.
\n\nlenses. However, most of these will be below the discoverable limit of the DSA-2000. Further, it is well known that intervening masses along the line of sight between the deflector and the observer could increase the probability of multiple imaging by a significant fraction, especially at high source redshifts, which we also neglect in this study (Fleury et al. 2021). Because these higher-order aspects of the model are likely to increase the number of lenses, for the purpose of this paper it is safe to ignore them.\n2.3 Source Populations\nThe other important piece of the model is the population of the sources. The DSA-2000 will map ∼ 30,000 deg 2 of the sky to a combined 𝜎 𝑛 = 500 nJy/beam rms noise (Hallinan et al. 2019). Matthews et al. (2021) presents detailed radio source counts from the MeerKat DEEP2 image, including direct source counts above 10 𝜇 Jy and statistical counts extrapolating below 10 𝜇 Jy. They find that for sources <10 𝜇 Jy, the differential source count is significantly flatter than Euclidean. Scaling for the 30,000 deg 2 footprint of DSA2000's continuum survey and integrating down to a minimum flux density 𝑆 = 2 . 5 𝜇 Jy for a 5 𝜎 𝑛 combined detection gives an expected total source count of 1 . 45 + 0 . 25 -0 . 10 × 10 9 , while for 𝑆 ≥ 5 𝜇 Jy = 10 𝜎 𝑛 that number is 8 . 6 + 1 . 1 -0 . 4 × 10 8 .\nTo get redshift distributions for different source classes, we model the Star-Forming Radio Galaxy (SFRG) and Active Galactic Nucleus (AGN) populations using the luminosity functions (LF) from the Tiered Radio Extra-galactic Continuum Simulation (T-RECS) (Bonaldi et al. 2018). These have been shown to match observations out to high redshift. Following Bonaldi et al. (2018), we divide the AGN into three sub populations: Flat Spectrum Radio Quasars (FSRQ), BL-Laceratae (BLLac), and Steep Spectrum AGN (SSAGN). We expect SFRGs to make up roughly 95% of all of the sources in any synoptic radio survey, and BLLacs to be the rarest source.\nFrom the LF we calculate the number of sources in a redshift interval as\n𝑛 ( 𝑧 ) 𝑑𝑧 = ∫ 𝐿 min ( 𝑧 ) Φ ( 𝐿, 𝑧 ) 𝑑 log 𝐿 × 𝑑 2 𝑉 𝑐 𝑑 Ω 𝑑𝑧 𝑑 × 𝑑𝑧 × Ω survey (12)\nwhere Ω survey is the sky coverage of the survey in steradians ( ≈ 3 𝜋 for the DSA-2000) and Φ ( 𝐿, 𝑧 ) is the LF of the source population. 𝐿 min ( 𝑧 ) = 4 𝜋𝐷 2 𝐿 ( 1 + 𝑧 ) 1 + 𝛼 𝑆 min is the minimum observable luminosity at a redshift 𝑧 , where 𝐷 𝐿 is the luminosity distance, 𝛼 is the spectral index of the source population, 𝑆 min is the 10 𝜎 𝑛 flux sensitivity of the DSA-2000, and the factor ( 1 + 𝑧 ) -( 1 + 𝛼 ) is the standard cosmological radio K-correction. We scaled the total number of sources in the LF model to match the observational prediction from the DEEP2 image given above.\nIn addition to SFRGs and AGN, we build a model for radio SNe. To date, only ∼ 100 radio core-collapse supernovae (ccSNe) have been detected and the first Type Ia radio supernova was detected last year (Bietenholz et al. 2021; Kool et al. 2023). To model the expected rate of radio ccSNe we follow Lien et al. (2011). Because ccSNe are shortlived, the rate of ccSNe is closely related to the cosmic Star Formation Rate (SFR), which has been observational measured to high redshift and accuracy. We use the ccSNe radio luminosity distribution of Bietenholz et al. (2021), which is more complete than the one in Lien et al. (2011). The best-fit distribution is log-normal and incorporates all known radio ccSNe as well as radio non-detections of supernovae. Bietenholz et al. (2021) note that their data is likely biased towards radio detections, finding that 30% of all ccSNe are detected in the radio. We divide the total number of radio ccSNe in the model by 3 to get a more conservative 10% as in Lien et al. (2011). From the luminosity distribution, we determine the number of radio ccSNe at each redshift above the flux limit of the DSA-2000. Because the SNe observations used in Bietenholz et al. (2021) have frequencies ranging from 2-10 GHz, there is uncertainty when converting to 1.4 GHz. We correct 𝐿 min 6 GHz ( 𝑧 ) = 𝐿 min 1.4 GHz GLYPH<16> 6 1 . 4 GLYPH<17> 𝛼 , where we assume 𝛼 = -0 . 7 for the synchrotron emission of radio ccSNe. Further, because radio ccSNe have a mean rise time of log 10 ( 𝑡 rise ) = 1 . 7 days (Bietenholz et al. 2021), we use the single epoch flux limit of the DSA-2000 and account for the DSA-2000's cadence. The rms noise for a single epoch is 𝜎 𝑛 = 2 𝜇 Jy/beam, so a 10 𝜎 𝑛 detection is 20 𝜇 Jy. The cadence will be ∼ 4 months, so conservatively we will detect 1/3 of the radio ccSNe per year. None of the known radio ccSNe have luminosities above 10 29 erg s -1 Hz -1 , but given the wide spread we expect that the highest luminosities could be several times larger. Because of this we set an exponential cutoff at 10 30 erg s -1 Hz -1 , which we can expect to see out to about 𝑧 ≈ 1 . 5 in a single epoch of the DSA-2000.\nThe predicted redshift distributions of SFRGs, AGN, and ccSNe from our model are shown in Figure 3 and compared to the expected distribution of Rubin-LSST sources. As expected, the total source count is dominated by SFRGs until high redshift. Blazars, meaning both FSRQs and BLLacs, are the AGN that will be most useful for time delay measurements, which is explored in Section 5.1. These are the most common sources at very high redshift. We expect the DSA2000 to probe much higher redshifts than the Rubin-LSST because radio emission is much less susceptible to intervening gas or seeing. This is one of the reasons that the DSA-2000 will be effective at lens finding. The total number of expected sources of each type in a DSA-2000 all-sky survey with a 10 𝜎 𝑛 detection are listed in the first row of Table 1.\n2.4 Expected rates\nThe final results of our model are shown in Table 1. The results agree quite well with the empirical estimates given in Section 2.1. As expected from CLASS there are roughly 10 6 total galaxy scale lenses contained in the DSA-2000 all-sky survey, less than 10% of\n\n\n
Figure 3. Expected source redshift distribution for the DSA-2000 from the model of Section 2.3. Blazars refers to both FSRQs and BLLacs. The distribution for Rubin-LSST is taken from Alonso & Ferreira (2015).
\n\nwhich will be discoverable. The total number of group and cluster lenses are about 10% and 2% of the galaxy number respectively, which is consistent with Oguri (2006). Roughly half of the group and cluster lenses will be discoverable, leading to them making up about a third of the total discoverable lenses, a significant fraction. Yue et al. (2022a) find that the Rubin-LSST will discover about 2000 lensed QSOs; our number is similar. This is a result of the competing mechanisms of the DSA-2000 detecting many more at higher redshifts and simultaneously being able to discover a smaller fraction because of the PSF size. The main purpose of Yue et al. (2022b) is to investigate the discrepancy between the fraction of lensed high redshift ( 𝑧 𝑠 ≳ 5) quasars in observations ( ∼ 0 . 2%) and previous predictions ( ∼ 4%).Yueetal.(2022b)find ∼ 0 . 4 -0 . 8%with their model, and we have ∼ 0 . 3%, which is consistent. The number of discoverable lensed ccSNe per year in the DSA-2000 from the full model is about 2, which is of the same order of magnitude as the empirical VLASS estimate. Oguri & Marshall (2010) conclude that Rubin-LSST will find roughly 8 SNe per year. We expect there to be less lensed SNe at radio wavelengths because not all SNe emit in the radio and the radio emission is weaker than in other wavelengths. The redshift distribution of the deflectors and sources is shown in Figure 4. Almost all deflectors are at 𝑧 𝑑 < 3 and most are at 𝑧 𝑑 ≈ 1, consistent with Oguri & Marshall (2010); Collett (2015); Yue et al. (2022a,b). This justifies the use of the CosmoDC2 catalog, which is limited to 𝑧 < 3, to determine the MF of groups and clusters.\nWe also run the simulation for the survey specs of the SKA-mid and VLASS. The main differences are the sensitivity and resolution. The expected rms noise of the SKA-mid at 1.4 GHz will be 𝜎 𝑛 = 2 𝜇 Jy/beam, and the PSF size will be about 0.4' (Braun et al. 2019). From the DEEP2 source counts, the SKA-mid should find about 3 × 10 8 sources above 10 𝜎 𝑛 . With this, our model predicts that the SKA-mid should see 1.8 ± 0.7 × 10 5 , 1.8 ± 1.2 × 10 4 , and 2.7 ± 1.1 × 10 3 galaxy, group, and cluster lenses, given the same discoverability limit defined in Section 2.2. Despite less depth leading to around a third of the total galaxy lenses, the PSF of the SKA-mid is such that it will resolve over half of them, resulting in about twice as many discoverable galaxy lenses as in the DSA-2000. The SKA-mid will find fewer group and cluster scale lenses than the DSA-2000 because sensitivity is more important for discovering these systems\n\n\n
Figure 4. Redshift distribution of the deflectors (blue) and sources (orange) for galaxies (solid), groups (dashed), and clusters (dotted) expected in the DSA-2000 with magnification bias.
\n\nthan angular resolution. In addition, we expect about 14 lensed radio ccSNe per year in an SKA-mid all-sky survey. McKean et al. (2015) estimate ∼ 3 × 10 5 lenses in an SKA-mid all-sky survey with 𝜎 𝑛 = 3 𝜇 Jy/beam and a lens detection limit of 15 𝜎 𝑛 and >0.3'. This is slightly more optimistic than our forecast, but broadly consistent. VLASS has 𝜎 𝑛 = 70 𝜇 Jy/beam combined and an angular resolution of 2.5' (Lacy et al. 2020). The DEEP2 source counts indicate that we should expect ∼ 5 × 10 6 sources above 10 𝜎 𝑛 in VLASS (Lacy et al. (2020) estimate 5.3 × 10 6 above 5 𝜎 𝑛 ). With these numbers, we estimate that there should be about 300 discoverable lenses in the full sky survey. As VLASS enters its epoch 3, many of these lenses are likely already in the data (several have recently been identified (Martinez et al. 2024)).\n3 LENS DISCOVERY\n3.1 Superresolution\nIn the past decade, tremendous progress has been made by the computer vision community with respect to the classical ill-posed inverse problems. These include deblurring (Zhang et al. 2022), deconvolution and superresolution (Alzubaidi et al. 2021), and image inpainting (Yu et al. 2018b). Nearly all of these advances were borne out of the deep learning revolution, as efficient neural network architectures and training strategies have enabled powerful learning-based tools for machine vision.\nAstronomy naturally lends itself to these tools because sparse sampling and ill-posedness arise in many astronomical imaging and reconstruction contexts, especially in interferometry. Several groups have begun developing machine learning methods for interferometric image reconstruction (Connor et al. 2022; Aghabiglou et al. 2024; Mars et al. 2024). One reason this approach is suitable for radio astronomy is that the PSF of an array is given deterministically by the spatial distribution of antennas and the observing frequency: The on-sky response of an interferometer is the 2D Fourier Transform of its sampling in UV-space (the aperture plane). Therefore, prior physical knowledge of the PSF can be incorporated in the model either via training data (Connor et al. 2022) or fed directly to the network itself (Mars et al. 2024). This is not the case in ground-\n\n\n
Figure 5. An example of superresolution image reconstruction on radio strong lenses. The simulated sky model is a mix of star-forming radio galaxies and AGN point sources. We randomly select 5% of sources to be strongly lensed. The artificially high value ensures that we have several lensing examples to reconstruct. The left panel shows a 3.3'x3.3' region observed (i.e. the 'dirty image') with the DSA-2000 full-band PSF (size shown with mauve circle) without any deconvolution. The middle panel is a reconstruction of that field with the POLISH algorithm. The rightmost panel is the true sky. The dirty images have been gamma encoded with 𝐹𝑙𝑢𝑥 0 . 75 with a value range of the inset figures chosen to highlight structure. In this toy example, strong lensing systems with Einstein radii below the PSF scale can be identified.
\n\nbased optical astronomy where 'seeing' and complex optics mean the PSF is not known a priori.\nAsasimple demonstration, we use the super-resolution and imageplane deconvolution method POLISH to show how strong lenses with image separations below the PSF scale can be recovered with machine learning. POLISH is a supervised machine learning model that learns the mapping between the true sky and observed images (the 'dirty image' in radio parlance) (Connor et al. 2022). It uses the Wide Activation for Efficient and Accurate Image Super-Resolution (WDSR) architecture (Yu et al. 2018a), but any super-resolution neural network could be swapped in (e.g., a standard U-Net or the Efficient Super-Resolution Transformer (Lim et al. 2017)).\nWehave trained a POLISH network on forward-modelled synthetic data using a DSA-2000 PSF averaged over the whole band, resulting in an angular resolution of ∼ 3.3'. This is worse resolution than the top of the band, where many strong lenses could be found, but we offer this as a toy example. The sky model is described in Connor et al. (2022), with the addition of strongly lensed galaxies in both the training and validation set. In Figure 5 we show an example validation image that contains multiple lenses. Lensed systems that are undetected in dirty image, including both arcs (left inset panels) and Einstein rings (right inset panels), are identifiable in the POLISH reconstruction. In the future, we plan to develop super-resolution methods explicitly for lens finding, but we take this as a promising sign that the DSA-2000 will be able to find systems with 𝜃 𝐸 ≳ 0 . 5 '' .\n3.2 Identification\nThe first major hurdle for strong lensing science with DSA-2000 and other surveys is identification. The large number of sources produced by these surveys makes visual inspection by experts impossible. Despite much work on automated lens discovery (see Lemon et al.\n(2023) for a review), little effort has been spent on identification in the radio. On the one hand, radio lens images are made cleaner by the lack of radio emission from the massive quiescent galaxies that make up the deflector population. However, the large, extended lobes of radio galaxies can easily be misidentified as lensing features. Further, color and photometric redshift information from UV and optical surveys have traditionally played a role in lens identification, despite being a foreground to the morphology of the lensed source.\nCatalog-level searches, where large cuts are made based on certain features before a smaller subset is visually inspected, have been successful in the past. CLASS selected for flat spectrum radio sources ( 𝛼 > -0 . 5) (Myers et al. 2003). This had the advantage of picking out only blazars, whose radio emission is dominated by a small compact core, thus eliminating any confusion with intrinsic structure. A similar strategy would likely be effective in the DSA-2000. However, dedicated follow up of all of these sources, as in the ∼ 11,000 blazars in the initial CLASS sample, will be impossible. Similarly, a visual inspection of even a small subset of the O( 10 7 ) blazars expected in the DSA-2000 is ambitious (hence the need for intermediate automated steps in the identification pipeline). Still, targeting flat spectrum sources could increase the yield of any search, and spectral information from the whole DSA-2000 band will be useful for determining whether components are different sources or multiple images. Jackson & Browne (2006) show that combining astrometric data from radio and optical surveys can significantly improve the efficiency of lens searches. If the optical emission is dominated by the deflector galaxy then there will exist an offset between the centroid position of the optical and radio sources in a lens system. This offset was exploited to find lenses with separation down to ∼ 1' even with the poor 5' resolution of the Faint Images of the Radio Sky at Twenty-one centimeters (FIRST) survey. They predict that this effect will become more efficient at lower radio fluxes because the optical\nflux is more likely to be dominated by the defelctor galaxy. Given the sensitivity of the DSA-2000 and its significant overlap with current and planned large surveys in other wavelengths, this could prove to be a very effective way of overcoming the limitations of its PSF. Regardless, exploiting the extra information from overlap with other surveys will be crucial for any lens search in the radio.\nMachine learning models have been successfully used to discover many new candidate lens systems. Convolutional Neural Networks (CNNs) in particular are effective at classifying astronomical images. The main disadvantage with CNNs and other supervised learners is the need for large realistic training sets. Using CNNs on simulated data for the International LOFAR Telescope (ILT) at 150 MHz, Rezaei et al. (2022) are able to recover over 90% of galaxy-size lenses with a false-positive rate of only 0.008%. They find that a strong 20 𝜎 detection and large separation 𝜃 𝐸 ≥ 3 / 2 beam size (stricter than the discoverable limit imposed in Section 2.2) are necessary for reliable identification with the CNN. Incorporating superresolution models into the lens-finding routine would significantly increase the yield of such a CNN, especially at high S/N. Resolving lenses down to 𝜃 𝐸 ≈ 0 . 5' with would enable the same level of accuracy as in Rezaei et al. (2022). Simulating accurate DSA-2000 lenses and training models for identifying them is a goal of our future work. The SKA-mid will already have sufficient resolution for these purposes, making CNNs an attractive option for lens identification in its all-sky survey.\nThe main limitation of any ML-based approach to group/cluster lens finding is the difficulty of creating realistic training sets. The irregularity of group/cluster lensing potentials may require building a forward-modelled training set from ray-tracing in large cosmological magnetohydrodynamical simulations such as TNG-Cluster, which should have a heterogenous population of massive clusters (Nelson et al. 2024). More generally, a realistic training set of strong lenses across scales could be produced with radio source samples from T-RECS and a deflector population drawn from cosmological simulations.\n4 MULTIWAVELENGTH SYNERGIES\nAcquiring redshifts for lensed galaxies and deflectors is critical for the application of strong lensing to cosmology and astrophysics. We consider here how well we could do in the absence of targeted followup observations, capitalizing on the suite of large survey telescopes that will be operating in the mid- to late-2020s.\nDSA-2000 spectroscopic HI survey In addition to the cadenced allsky survey, the DSA-2000 will undertake a large spectroscopic HI survey over the full visible sky (Dec > -30 · ) (Hallinan et al. 2019). This is expected to produce ∼ 10 6 HI galaxies at 𝑧 < 1. Most deflectors will be massive elliptical galaxies with limited 21 cm emission, but a sub-set of early-type galaxies with 𝑛 𝐻 ≥ 10 20 cm -2 𝑧 -2 𝑑 will be detected. Some late-type deflector galaxies will have enough cold gas to obtain a redshift. Additionally, 21 cm absorption in the inner CGM of deflector galaxies may be present in the lensed source's continuum spectrum, providing a measurement of cold gas along different sightlines of the same CGM (see Rudie et al. (2019) for a similar application in lensed quasar systems). Thus, some deflector galaxies will have a spectroscopic redshift without any external O/IR data. Of the ∼ 10 6 HI galaxies detected by the DSA-2000 in emission, we find that only a handful will be strongly lensed, due to their shallow redshift distribution. Spectroscopic radio observations will not provide a significant portion of redshifts required for our purposes, but may be a minority of interesting deflectors.\nThe Dark Energy Spectroscopic Instrument (DESI) will produce\nredshifts of ∼ 40 million galaxies and quasars (DESI Collaboration et al. 2016; Dey et al. 2019). Its 3 yr public data release is expected before the DSA-2000's first light. DESI's Bright Galaxy Survey (BGS) targets galaxies with 𝑟 -band magnitudes brighter than 19.5. For the luminous red galaxies (LRGs), emission line galaxies (ELGs), and quasars, the depth will be 𝑟 ≈ 23. Each class of target will have a different redshift distribution, but together they will provide a rich catalog of spectroscopic redshifts for deflectors in the lensing systems discovered by DSA-2000. The BGS and LRG samples will be over-represented as deflector galaxies, whereas the ELG and QSO samples may make up a small fraction of lensed DSA-2000 sources. ELGs, by definition, have strong emission lines indicative of star formation, which produce synchrotron radiation detectable by radio surveys. Some of these will be strongly lensed star-forming radio galaxies with 𝑆 𝜈 above 5 𝜇 Jy. Still, we do not expect more than a small minority of the radio continuum lenses to be detected in DESI due to the relative number of total sources.\nSPHEREx is a near-infrared space mission expected to launch before Spring 2025. It will map the full 4 𝜋 sr sky with 6 arcsecond pixels and 96 color bands, producing a catalog of hundreds of millions of galaxies with 'spectro-photometric redshifts' (Doré et al. 2018). Although its PSF it too large to discover strong lenses directly, the galaxy catalog will be valuable for obtaining reliable redshifts of both deflectors and lensed galaxies.\nThe Rubin Observatory will find billions of galaxies in its Legacy Survey of Space and Time (LSST) (Ivezić et al. 2019). Most of these galaxies will be at 𝑧 𝑠 ≲ 1 . 5 (Collaboration et al. 2021), providing an excellent deflector catalog for the DSA-2000 lens candidates. In the DSA-2000/LSST overlapping footprint between -30 and +30 Declination, we find that a significant fraction of deflectors and several thousand lensed sources in the DSA-2000 sample will have photometric redshifts from the Rubin Observatory.\nRoman The High Latitude Spectroscopic Survey (HLSS) is expected to detect 10 million galaxies in 𝐻𝛼 between 1 < 𝑧 < 2and ∼ 2 million [OIII] galaxies at 𝑧 = 2 -3 using the near-IR grism and wide-field camera (Wang et al. 2022). Roman's 2,000 deg 2 footprint will be fully mapped by DSA-2000. With a dust-attenuated 𝐻𝛼 flux higher than 10 -16 erg s -1 cm -2 , several million of these galaxies will be detectable at GHz frequencies at > 5 𝜇 Jy (Murphy et al. 2011). Assuming a mean lensing optical depth of 5 × 10 -4 , this suggests that O( 10 3 ) will be strongly lensed and detected by both the DSA-2000 and Roman , the latter providing 0.1' resolution and a source redshift.\nEuclid is a visible and near-infrared space telescope that launched in July 2023 (Collaboration et al. 2024). Its large photometric survey will find over one billion galaxies, with a spectroscopic survey that will obtain accurate redshifts for ∼ 30 million galaxies mostly at 0 . 9 < 𝑧 < 1 . 8. Euclid and the DSA-2000 share over 10,000 deg 2 of sky, providing an excellent sample of photometric and spectroscopic redshifts of lens systems.\n5 STRONG LENSING APPLICATIONS\nA key limitation of strong lensing science is the small number of known systems, especially at radio wavelengths. We discuss the impact that the large expected number of radio lenses and their multiwavelength counterparts will have on several important applications of strong lensing.\n5.1 Time-delay cosmography & 𝐻 0\nLight from multiple images in a gravitational lens will reach the observer at different times. For continuum sources, we do not observe the difference in arrival times, but transients or time-variable sources allow us to measure the time delay. This time delay is due to a difference in path lengths and gravitational time dilation near the deflector. As such, the time delay encodes information about the geometry of the universe, and is inversely proportional to 𝐻 0 (Oguri 2019). Much work has been dedicated to measuring these time delays and using them to constrain 𝐻 0 (Treu & Marshall 2016; Birrer et al. 2024). The H0LiCOW project has measured 𝐻 0 to 2.4%, which is independent of and competitive with state-of-the-art 𝐻 0 constraints (Wong et al. 2019). A large increase in the number of measurable time-delay systems will allow even tighter constraints and could help settle the current Hubble tension.\nWe estimate the number of lensed variable AGN that will be useful for measuring 𝐻 0 in the DSA-2000. To first order, all blazars (FSRQs and BLLacs) are inherently variable, ∼ 10 3 of which will be discoverable as lenses in DSA-2000 data (Table 1). However, several factors complicate our ability to use them for time delay measurements; we would like to know how many will be in a \"gold sample\" for which the time-delay can be reliably determined. To date, the most extensive study of radio AGN variability is the OVRO 40 m blazar monitoring campaign. Since 2007 this program has monitored around 1500 blazars with a cadence of two weeks (Richards et al. 2011). Richards et al. (2011) provides distributions of the intrinsic modulation index, a measure of the relative intrinsic variability of the AGN, for FSRQs and BLLacs. The DSA-2000 will measure flux variability at the level of a few percent, but to be conservative we consider candidates with ≥ 10% variability. We also want sources with a strong detection, so we chose sources with flux ≥ 20 𝜎 𝑛 = 10 𝜇 Jy. The OVRO blazars are monitored at 15 GHz but we expect lower variability at 1.4 GHz. Fan, J. H. et al. (2007) find that blazars are about 20% less variable at 4.8 GHz than 14.5 GHz. Sotnikova et al. (2024) find that radio AGN at 22.3 GHz are about 35% more variable than at 2.3 GHz, although there is little difference between 11.2 GHz and 2.3 GHz. We assume that the mean modulation index drops by 40% from the OVRO 40 m data to 1.4 GHz for the DSA-2000. Further, for multiple images, we need at least two of the images to be bright enough to measure the time delay so we use the magnification bias of the fainter image. With these restrictions, the OVRO data, and our lens model, we estimate that there will be 68 ± 30 lensed FSRQs and 23 ± 10 lensed BLLacs with sufficient brightness, variability, and image separation to measure time delays. The same analysis for the SKA-mid gives 120 ± 50 and 43 ± 16 lensed FSRQs and BLLacs.\nOf the lensed AGN, the most useful systems for time delay analysis will be those that have four or more images because they allow for multiple constraints on the time delay. A SIS is only capable of creating two images and a spherical NFW can make three, but factors such as ellipticity, irregularities in the lens potential, and external environments of real halos will allow a number of these systems to make four or more images. The fraction of quad galaxy-QSO lenses predicted in Rubin-LSST is ∼ 10-15% (Oguri & Marshall 2010; Yue et al. 2022a). However, of the statistically complete CLASS sample, 5 out of 13 lensed blazars were quadruply imaged and one had six images (B1933+503 has two sets of quadruply imaged sources Sykes et al. (1998)). Therefore we can expect 100-400 quadruply-lensed blazars in the DSA-2000 all-sky survey and about twice as many in the SKA-mid, roughly 10% of which may be used in practice for measuring 𝐻 0 . We note that quad systems often have higher external shear due to their preponderance of group or cluster environments,\nwhich results in an added systematic in lens modeling (Holder & Schechter 2003).\nWong et al. (2019) reach a 2.4% measurement of 𝐻 0 with a sample of 6 lensed quasars, and predict that around 40 lenses are needed to constrain 𝐻 0 to within 1%. Jee et al. (2016) argue that with a well-defined sample of quadruply imaged quasars, and combining time-delay distances and velocity dispersion measurements to estimate angular diameter distances within 5%, 𝐻 0 can be measured to the same precision as Planck with as few as 10 systems. Napier et al. (2023) reach a 10% measurement with three galaxy clusters, and estimate that roughly 50 will be needed for a sub 1% constraint, assuming that modeling uncertainties can be reduced in coming years. While we expect uncertainty to decrease significantly with a larger sample of lensed quasars, there are several systematics that must be kept under control for an accurate measurement. Namely, an accurate redshift measurement is necessary, ideally of both the source and deflector, as well as precise measurements of the time delay, complete sampling over many cycles of the time delay to negate contamination from micro-lensing, accurate lens models, and constraints on any line of sight structures. If we assume that all of the lenses in our gold sample can be modeled to the accuracy of the H0LiCOW sample - a roughly 6% combined uncertainty on the time delay, mass model, and line of sight contribution for each lens - then a sample of ∼ 100 lensed quasars would give about a 0.6% constraint on 𝐻 0 . This is of course not possible with the DSA-2000 or SKA-mid alone but will require dedicated follow-up at multiple wavelengths. As discussed in Section 4, we expect that a significant minority of lenses should have redshifts available in other concurrent public surveys. Interesting subsamples of lenses will require targeted follow-up for both mass modeling and spectroscopic redshifts.\nThe DSA-2000's cadenced all-sky survey will image each field above -30 Dec once every 4 months. Typical time delays for galaxy, group, and cluster lenses in our model are a month or less, several months, and a year or more respectively (Oguri et al. 2002). Special systems could therefore be visited with a higher cadence, similar to how pulsar fields will be visited regularly for timing experiments.\nSo far we have only considered lensed blazars for time delay measurements, which are desirable because of their characteristic variability and compact emission regions. Lensed transients will also be useful for time delay measurements, provided that they are detected early and monitored closely. The variability of radio emission from TDEs is too slow to make them useful for this application (on-axis jetted TDEs can have shorter rise times, but they are significantly rarer such that we do not expect any lensed jetted TDEs in upcoming radio surveys), but SNe have faster rise times. We expect O( 1 ) lensed radio ccSNe per year in the DSA-2000 and O( 10 ) in the SKA-mid that can be used for 𝐻 0 measurements.\n5.2 Dark Matter Structure\nGravitational lensing has been very useful for determining the structure of galaxies, groups, and clusters at cosmological distances and distinguishing between dark matter models (see Vegetti et al. (2024); Natarajan et al. (2024) for a review). The best of these studies use high-resolution HST images, but radio observations also played a pivotal early role. Cohn et al. (2001) used the 10-image CLASS lens B1933+503, which provides many more constraints than usual, to determine the density profile of the deflecting galaxy, finding it close to isothermal. Wucknitz et al. (2004) find the same for B0218+357. Dalal & Kochanek (2002) used radio observations of flux ratio anomalies between multiple images to constrain the mass fraction of dark matter substructure to 2%. Radio observations are especially\nsuited for this type of study because they are less susceptible to micro-lensing (Koopmans et al. 2003; Kochanek & Dalal 2004). Despite this, dark matter studies with radio lenses have been limited for many years by the small sample size (Vegetti et al. 2024). Because resolution is a key factor in the ability to accurately model mass distributions or detect substructure, the DSA-2000 alone will not be able to use lens systems to study dark matter. Its main utility will come from identifying huge numbers of systems for detailed followup. The extraordinary resolution of the ngVLA on a large sample of lenses selected by the DSA-2000 will revolutionize this field. In the more immediate future, the SKA will also provide enough resolution for these studies in their overlapping sky coverage.\nAnother exciting application of lensing is using a central image to study small scales near the centers of galaxies (Treu 2010; McKean et al. 2015; Shajib et al. 2024). A SIS lens only forms two images, but for realistic galaxy profiles a highly demagnified image is expected to form at the center of the image configuration. This image will be sensitive to the mass contained within very small distances of the deflector's center, and so can be used to study supermassive black holes at cosmological distances. This is especially suited for radio wavelengths because in many cases the deflector will be radio-quiet allowing the center image to be detected. Because the center image is highly demagnified, only sensitive radio surveys such as the DSA2000 will be able to reliably identify them. At least one such central image has been detected in the radio with brightness ∼ 0 . 8 mJy at 8.46 GHz (Winn et al. 2004), so we can likely expect O( 10 2 ) or more with the DSA-2000. A main challenge will be disentangling the faint emission of the central image from the other images. For groups and clusters, whose inner density profiles are shallower than isothermal, central images are more common and often less demagnified. The very inner profiles of groups/clusters can thus be studied, which are hard to constrain with images near the Einstein radius alone. These central images will be easier to identify because the other images will be farther away (although many clusters have large central elliptical galaxies emitting in the radio that will obscure the image).\n5.3 Polarization\nMeasuring full Stokes information of gravitationally lensed signals offers insight into the source plane because polarization is not affected by gravitational potentials. This is an advantage of radio surveys, where polarization information is often stored and Faraday rotation measure (RM) can be measured. This enables us to study propagation effects along different sightlines for lensed objects. For example, Mao et al. (2017) used the VLA to measure the polarization properties of two lensed images of CLASS B1152+199, finding a large difference in RM ( + 9 . 7 ± 0 . 5 rad m -2 for image ' 𝐴 ' and + 517 ± 3 rad m -2 for image ' 𝐵 '), which is due to the magnetized plasma in the lens galaxy's interstellar medium.\nThe DSA-2000 will have the ability to measure full polarization information for any source it detects, with a maximum |RM| of roughly 10 4 rad m -2 . Compact sources such as Blazars (and especially BL Lac objects) can show significant polarization fractions at 1 GHz. Assuming a 50 𝜎 detection threshold in total power, we estimate that the DSA-2000 could detect Stokes Q and U for 5-10 million AGN in its 5 year continuum survey. Using the empirical CLASS lensing optical depth and image separation cut, we take 𝜏 𝐴𝐺𝑁 ≈ 5 × 10 -4 . We estimate that the DSA-2000 could find O( 10 3 ) strong lenses for which polarization properties could be used to model the lens distribution and study magnetic fields at cosmological distances.\n5.4 Other applications\nIn addition to constraining 𝐻 0 , time-delay measurements are weakly sensitive to other cosmological parameters, such as Ω 𝑚 , Ω Λ , and the dark energy equation of state parameter 𝑤 (Natarajan et al. 2024). Cluster lenses are used for studying dark energy as well (Macciò 2005; Gilmore & Natarajan 2009; Jullo et al. 2010). A large influx in measured time delays and observed cluster lenses with the DSA2000 will allow better constraints on these cosmological parameters. Lenses are also commonly used as 'nature's telescopes'. The magnification of background sources allows very distant galaxies to be studied even by smaller telescopes (Jackson 2011). This is especially true for group/cluster lenses, where we expect the magnifications to be larger (Robertson et al. 2020). We expect to see O( 10 4 ) lenses with 𝑧 𝑠 > 5 in the DSA-2000 (figure 4), many of which will be group and cluster lenses due to their high magnifications. These high redshift sources will provide insight into the population of radio sources in the early universe. Lensing statistics can also be used to test cosmological models. The abundance of giant arcs in cluster lenses was a topic of much debate in the past few decades (Meneghetti et al. 2013). Similarly, the rate of GGSL events in clusters may be in tension with the Λ CDM(Meneghetti et al. 2020; Meneghetti, Massimo et al. 2022; Tokayer et al. 2024). These studies require a large sample of lenses for accurate statistics, which is only now becoming possible with large surveys such as the DSA-2000.\n6 CONCLUSIONS\nIn this paper, we forecast expected strong lensing rates in the upcoming DSA-2000 and SKA-mid wide-field radio surveys. We first provide empirical estimates based on previous surveys and simulations, and then develop a detailed forward model that accounts for the source and deflector populations, finding them to be in good agreement. Notably, we model the expected number of galaxy group and cluster scale lenses because these systems will be easily discovered due to their wide angular separations. We find that both the DSA-2000 and the SKA-mid will discover roughly 10 5 strong lens systems. We discuss strategies for identifying these lenses in the data, which will all benefit from emerging superresolution techniques. Finally, we discuss the scientific application of the huge numbers of lenses that will be discovered by these surveys. One of the most exciting applications is 𝐻 0 cosmography with variable and transient sources. The DSA-2000 and SKA-mid will discover about 100 and 200 lensed flat spectrum AGN with >10% variability respectively, as well as about O( 1 ) and O( 10 ) lensed transients per year. With dedicated multi-wavelength follow up these systems could be used to constrain 𝐻 0 to within 1%. The new lens systems will also be useful for studying the distribution of dark matter at cosmological distances, among other applications.\nACKNOWLEDGEMENTS\nWeare grateful to Schmidt Sciences for supporting Samuel McCarty as a Summer Undergraduate Research Fellow at Caltech. We thank Kim-Vy Tran, Tony Readhead, and Tommaso Treu for helpful conversations on strong lensing, as well as Paul Schechter for insights into quad systems.\nDATA AVAILABILITY\nWe have placed a reproduction package on the public GitHub repository available at https://github.com/smmccrty/ radiolensing .\nREFERENCES\n
\n\n
\n
\n\n
\nR., Motta V., 2012, The Astrophysical Journal, 749, 38\n
\n\n
\n\nYao Y., et al., 2023, The Astrophysical Journal Letters, 955, L6\nYu J., Fan Y., Yang J., Xu N., Wang Z., Wang X., Huang T., 2018a, Wide Activation for Efficient and Accurate Image Super-Resolution ( arXiv:1808.08718 ), https://arxiv.org/abs/1808.08718\nYu J., Lin Z., Yang J., Shen X., Lu X., Huang T. S., 2018b, in Proceedings of the IEEE conference on computer vision and pattern recognition. pp 5505-5514\nYue M., Fan X., Yang J., Wang F., 2022a, The Astronomical Journal, 163, 139\nYue M., Fan X., Yang J., Wang F., 2022b, The Astrophysical Journal, 925, 169\nZhang K., Ren W., Luo W., Lai W.-S., Stenger B., Yang M.-H., Li H., 2022, International Journal of Computer Vision, 130, 2103\nZwicky F., 1937, ApJ, 86, 217\n\nThis paper has been typeset from a T E X/L A T E X file prepared by the author.\n"},"sections":{"kind":"list like","value":[{"title":"ABSTRACT","content":"The number of strong lensing systems will soon increase by orders of magnitude thanks to sensitive, wide-field optical and infrared imaging surveys such as Euclid, Rubin-LSST, and Roman. A dramatic increase in strong lenses will also occur at radio wavelengths. The 2000-antenna Deep Synoptic Array (DSA-2000) will detect over 10 9 continuum sources in the Northern Hemisphere with a high mean redshift ( ⟨ 𝑧 𝑠 ⟩ ≈ 2) and the Square Kilometer Array (SKA) will observe a large sample of extragalactic sources in the South with sub-arcsecond resolution. We forecast lensing rates, finding that the DSA-2000 will discover O( 10 5 ) strongly lensed systems, many of which will be galaxy group and cluster lenses. We propose strategies for strong lensing discovery in the limit where the Einstein radii are comparable to the PSF angular scale, taking advantage of modern computer vision techniques and multi-survey data. We also forecast synergies with optical and infrared surveys, which will provide redshifts as well as multiwavelength information about the lens systems. Finally, we describe applications of radio strong lensing systems, including time-delay cosmography with transient and variable sources. We find that ∼ 100 time-variable flat-spectrum AGN discovered by the DSA-2000 could be used to constrain 𝐻 0 at the percent level with the appropriate follow-up. Key words: gravitational lensing: strong - radio continuum: general","pages":[1]},{"title":"Strong gravitational lensing with upcoming wide-field radio surveys","content":"Samuel McCarty 1 , 2 , ★ Liam Connor 3 1 Department of Astronomy, University of Washington, Seattle, WA 98195-1580, USA 2 Cahill Center for Astronomy and Astrophysics, MC 249-17, California Institute of Technology, Pasadena CA 91125, USA 3 Center for Astrophysics | Harvard & Smithsonian, Cambridge, MA 02138-1516, USA Accepted XXX. Received YYY; in original form ZZZ","pages":[1]},{"title":"1 INTRODUCTION","content":"Strong gravitational lensing has a multitude of applications in astrophysics and cosmology (Treu 2010). Previously theoretical ideas have been put into practice in recent decades as the number of known lensed systems has increased and observational data have improved. For example, strongly lensed time-variable and transient sources can be used to constrain the Hubble constant, 𝐻 0 , because the time delay of a multiply imaged source depends on the geometry of the Universe (Refsdal 1964). The technique is known as time-delay cosmography. With just six lensed quasar systems, the The 𝐻 0 Lenses in COSMOGRAIL's Wellspring (H0LiCOW) collaboration has reported 2.4 % precision on their 𝐻 0 measurement, which is independent of the distance ladder and the CMB (Wong et al. 2019). In addition to the Universe's large-scale geometry, lensing observables are sensitive to the total mass of the deflector galaxy or cluster, allowing one to measure the spatial distribution of matter and test different dark matter models (Massey et al. 2010; Vegetti et al. 2024). Lensing magnification allows astronomers to observe objects in the distant Universe, as was pointed out at the field's inception (Zwicky 1937). Dramatic examples have come from the James Webb Space Telescope (JWST), including a red supergiant star at 𝑧 ≈ 2 . 2 that appears to be magnified by a factor of several thousand due to its proximity to caustics in a cluster lens (Diego et al. 2023). Nearly all of these applications would benefit from a larger sample of strong lensing systems. To date, roughly 10 3 strong lensing systems have been discovered, most of which were identified at optical and infrared wavelengths (O/IR). Fortunately, upcoming wide-field imaging surveys such as Euclid and The Vera C. Rubin Observatory's Legacy Survey of Space and Time (Rubin-LSST) are each expected to detect as many as ∼ 10 5 strong lenses (Collett 2015). An early release of a 0.7 deg 2 field from Euclid has recently affirmed these forecasts (Barroso et al. 2024). The Nancy Grace Roman Space Telescope's 2000 square degree survey could find of order 20,000 strong lenses (Weiner et al. 2020), and many more if the proposed multi-epoch 4 𝜋 sr survey is carried out (Han et al. 2023). An increase in the total number of lenses by two-orders of magnitude will usher in a new era of strong lensing science. The first strongly lensed system ever discovered was co-detected at radio wavelengths (Walsh et al. 1979). The first survey for lenses, the Mit-Green Bank survey, was also in the radio (Bennett et al. 1986). Despite this, fewer than ∼ 100 radio lensing systems have been detected to date. This is due to the relatively small total number of known radio sources ( ∼ 10 7 ) and the lack of wide-field imaging surveys with ∼ arcsecond resolution. Both of those limitations will soon be overcome with the advent of next-generation radio survey telescopes. The 2000-antenna Deep Synoptic Array (DSA-2000) will detect over one billion radio sources with a deep redshift distribution, most of which will be star-forming radio galaxies (SFRG) or active galactic nuclei (AGN) (Hallinan et al. 2019). The DSA-2000 is expected to see first light in 2027 with key surveys running between 2028 and 2033. Its point-spread function (PSF) will be roughly 2' at the top of the 0.7-2 GHz radio band. A fifty-fold increase in the total radio source catalog is made possible by the DSA-2000's high survey speed, driven by a large field-of-view ( ∼ 10 deg 2 ) and high sensitivity (the expected system-equivalent flux density or 'SEFD' is just 2.5 Jy). Its cadenced all-sky survey will map out the 3 𝜋 sr above declination -30 · down to 500 nJy/beam root-mean square noise (Hallinan et al. 2019). While the nominal cadence of the continuum survey is four months, certain fields may be visited more regularly, presenting an opportunity to measure lensing time delays with better temporal sampling. The mid-frequency telescope for the square kilometre array (SKAMid) will consist of 197 fully steerable 13.5 m dishes (including the existing MeerKAT radio telescope), operating between 350 MHz and 15.4 GHz with a field-of-view of roughly 1 deg 2 at 1400 MHz. The end of its contruction is anticipated to be July 2029 1 . Although the SKA's lower survey speed will result in fewer sources than the DSA-2000, the wide frequency range and long maximum baseline (150 km) will enable high-resolution imaging, an asset to strong lensing studies (McKean et al. 2015). The Next Generation Very Large Array (ngVLA) is a planned interferometer with extraordinary sensitivity covering a wide range of frequencies (1.2-116 GHz) (Selina et al. 2018a). It will be able to resolve features at milliarcseconds scales. While its broad science goals did not require optimizing the instrument for mapping speed (Selina et al. 2018b), the ngVLA will be a world-class instrument for strong lensing science. Radio strong lensing offers distinct advantages to studies at shorter wavelengths and will complement O/IR imaging surveys. Starforming radio galaxies and AGN can be detected to great distances, boosting the mean optical depth of radio continuum sources. Obscuration by dust in lensing galaxies is not an issue at radio wavelengths, nor is the variable 'seeing' that impacts ground-based O/IR telescopes. Relatedly, the point-spread function (PSF) of a radio interferometer is directly determined by observing frequency and array configuration, allowing us to accurately forward model the instrument's response. In the limit of a large number of antennas and a filled aperture (a 'radio camera'), the deterministic PSF allows us to be more ambitious in image-plane deconvolution, enabling techniques such as superresolution (Connor et al. 2022). Radio telescopes also measure full polarization information, which is conserved under gravitational lensing (Greenfield et al. 1985; Dyer & Shaver 1992). The rotation measure (RM) of a lensed source allows one to measure the magnetic field and ionized gas properties of intervening halos (Mao et al. 2017). Finally, the larger emission regions of radio AGN render them less susceptible to microlensing by substructure in the lens, and may provide cleaner modelling of the deflector mass distribution (Birrer et al. 2024). This is critical for measuring 𝐻 0 via time-delay cosmography. There are of course several drawbacks to strong lensing studies in radio surveys, often precluding the use of standalone radio observations. For example, continuum sources will not contain redshift information, and multi-wavelength datasets or follow-up will be necessary for the majority of lensed systems at radio frequencies. Secondly, the morphology of radio lobes can mimic lensing arcs, exacerbating the already-challenging lens identification problem (consider, for example, head-tail galaxies 3C 465 or 3C 129 at redshift 2). In the case of the DSA-2000, the angular resolution is such that 75-90% of galaxy-galaxy lensing systems will be unresolved in the absence of superresolution. In this work we seek to study radio strong lensing in upcoming interferometric surveys, and develop strategies that alleviate the drawbacks of radio lensing. We focus on the DSA-2000, first by forecasting its strong lensing rates for different source classes. We compare this with a forecast for strong lensing discovery on the SKA. Next, due to the 2-3' PSF of the DSA-2000, we discuss methods for superresolving lensing candidates with modern computer vision techniques in order to increase the yield of strongly lensed systems. We then consider lensed time-variable and transient sources that the DSA-2000 will find and forecast constraints on 𝐻 0 that can be achieved with the appropriate follow-up, as well as other applications of strong lensing.","pages":[1,2]},{"title":"2 OPTICAL DEPTH CALCULATION & EXPECTED RATES","content":"First, we offer a simple empirical forecast based on observational data and cosmological simulations, and then offer a more detailed forward model that accounts for different source classes and the deflector population.","pages":[2]},{"title":"2.1 Empirical forecast","content":"The Cosmic Lens All Sky Survey (CLASS) (Myers et al. 2003) sought to study a statistical sample of radio-loud gravitationally lensed systems. Despite significant advances in both radio instruments and lens-finding algorithms in the two decades since CLASS, the number of known radio lenses has not increased dramatically. We can use CLASS for a rough estimate of the number of lenses the DSA-2000 will find. CLASS found that a complete sample of 8,958 flat-spectrum radio point sources brighter than 30 mJy had a mean lensing optical depth of 1 . 5 + 0 . 5 -0 . 3 × 10 -3 (Browne et al. 2003). By targeting compact sources with flat spectra, many of their objects are quasars at high redshifts, comparable to typical redshifts of DSA-discovered sources, despite the difference in flux scales. Of the confirmed radio lenses, approximately 20% had angular separations larger than 2 arcseconds, which could be detected by the DSA-2000. Thus, a rough estimate indicates that for every 10,000 sources detected by the DSA-2000, several could be identified as strong lenses. If we assume a detection threshold of 10 𝜎 , the number of extragalactic DSA-2000 sources becomes roughly 5 × 10 8 (Section 2.3). This would result in O( 10 5 ) new galaxy-scale radio lenses, depending on the practical signal-to-noise threshold for candidate systems. Galaxy group and cluster scale lenses have been neglected in previous investigations of lensing statistics for upcoming surveys because they make up a smaller, but still significant, fraction of the total lenses. Additionally, lens modeling is more complicated in this regime. The dark matter halos of groups, in particular, are complex and not well-understood structures. From Oguri (2006), the number of group and cluster lenses are around 11% and 3% of the total galaxy lenses respectively (excluding sub-halo lensing). However, these systems make up a considerable portion of lens systems with angular separation of order 1', and almost all of the lenses at ≥ 10'. Because the PSF of the DSA-2000 is relatively large (2' at 2 GHz and 3.3' at 1.4 GHz), these lenses will make up a large fraction of the discoverable lensing systems and so must be accounted for in our investigation. A radio group/cluster lens survey as complete as CLASS does not exist, but we can make a simple empirical estimate from the literature. The total number of galaxy scale lenses should be O( 10 6 ) assuming 10 9 sources and the CLASS lensing optical depth. Taking the percentages of group/cluster lenses from Oguri (2006), we should see O( 10 5 ) groupscale lenses and O( 10 4 ) cluster scale lenses, almost all of which will have angular separation large enough to be detected in the DSA-2000. The first and largest survey for group-scale lenses was the Strong Lensing Legacy Survey (SL2S) (Cabanac, R. A. et al. 2007). SL2S found 13 strong lens systems in the mass range of groups in ∼ 100 deg 2 of the Canada France Hawaii Telescope Legacy Survey (CFHTLS), giving a rate of ∼ 0.13 deg -2 (Limousin, M. et al. 2009). A later publication identified at least 54 promising group lenses in ∼ 150 deg 2 of the CFHTLS (More et al. 2012). However, the mean redshift of the CFHTLS sources used in SL2S is ⟨ 𝑧 𝑠 ⟩ < 1, and we expect ⟨ 𝑧 𝑠 ⟩ ≈ 2 for the DSA-2000 (see section 2.3, figure 3 below) (Coupon, J. et al. 2009). At these redshifts, we expect the lensing optical depth to be roughly ∝ 𝑧 2 (see equation 31 of Oguri (2019); figure 2 below). This means that we would expect to see ∼ 10 + 6 -6 times more group lenses in the DSA-2000, for a total of ∼ 1 . 1 + 0 . 6 -0 . 9 × 10 5 . Similarly, the Red Sequence Cluster Survey found 8 cluster lenses in ∼ 90 deg 2 with shallower CFHT imaging, indicating ∼ 2 . 6 + 1 . 6 -1 . 6 × 10 4 cluster lenses in the DSA-2000 (Gladders et al. 2003). These estimates are of the same order of magnitude as expected from Oguri (2006). Alternatively, if we take the cross section for giant arcs from the simulations of Puchwein & Hilbert (2009), Fedeli, C. et al. (2010), and Mahdi et al. (2014), assume typical values of 𝑀 = 1 × 10 14 M ⊙ , 𝑧 𝑑 = 0 . 5, and 𝑧 𝑠 = 2, we can calculate a rough estimate of the cluster lensing optical depth at 𝑧 𝑠 = 2. The cross sections at these values are all on the order of 10 arcsec 2 . From Böhringer, Hans et al. (2017), the number density at 𝑀 = 1 × 10 14 M ⊙ is approximately ≳ 1 × 10 -6 Mpc -3 , and after multiplying by the differential coming volumeat 𝑧 𝑠 = 2andskycoverageoftheDSA-2000weget ≈ 5 × 10 -5 for all three simulations. When taking into account that this is the cross-section for giant arcs only, this means we should see several tens of thousands of cluster lenses in the DSA-2000, in good agreement with the previous estimates. Wealso make an empirical estimate of the rate of lensed transients in the DSA-2000. These objects are of particular interest for 𝐻 0 measurements but are much harder to forward model because of the lack of observational data. We can use data from the Very Large Array Sky Survey (VLASS), which is currently the most comparable survey to the DSA-2000, for a simple estimate (Lacy et al. 2020). The determined log rates of supernovae (SNe) and tidal disruption events (TDEs) in deg -2 yr -1 are -1.91 + 0 . 15 -0 . 16 and -2.85 + 0 . 28 -0 . 38 respectively (Dong et al. 2024 in prep). If we assume that the total number of observable sources scales as 𝑆 -1 . 5 min , the value for a Euclidean universe, and that the flux limit of VLASS is 0.7 mJy (10 𝜎 from Lacy et al. (2020)), the DSA-2000 should find ∼ 200 more sources of each type above 20 𝜇 Jy, which is a 10 𝜎 detection in a single epoch of the DSA-2000. We can also assume a lensing optical depth of 1 × 10 -4 and a magnification bias of 2, which are typical for the low redshifts at which we expect to detect transients. Because typical radio rise times of TDEs are on the order of 10 3 days the cadence of the DSA-2000, ∼ 1 / 3 yr, is fast enough to catch most TDEs (Cendes et al. 2023). Scaling for the sky coverage of the DSA-2000, and the cadence for SNe, gives a total yield of 5 + 2 -1 yr -1 and 0.6 + 0 . 5 -0 . 3 yr -1 lensed sources per year for SNe and TDEs respectively. Applying a similar estimate of the lensed rate of GRB afterglows using the predictions of Ghirlanda et al. (2013, 2014) gives a rate much less than 1 per year, and so we ignore them in the rest of this investigation. Further, Yao et al. (2023) find a volumetric rate of optically selected TDEs of 290 + 60 -130 Gpc -3 yr -1 . Cendes et al. (2023) find that ≈ 50% of optically selected TDEs emit in the radio on longer timescales. Becausethere are likely TDEs that emit in radio but not the optical, we take 50% of the optically selected rate to be a conservative estimate. The luminosities of the TDEs in the sample from Cendes et al. (2023) are ∼ 10 37 -10 39 ergs/s, but they note that many of these are likely a lower limit because most of the TDEs still had rising emission at the time of observation. Given this, the DSA-2000 should be able to detect this rate of TDEs out to 𝑧 ≈ 0 . 5, which, when combined with the same optical depth and magnification bias as above, gives 0 . 8 + 0 . 2 -0 . 4 yr -1 lensed TDEs. This number is in good agreement with the previous estimate, so in general, we expect to see about 1 lensed TDE per year in the DSA-2000. However, as noted before, this is a conservative estimate, and there is reason to believe that the actual rate of total and lensed TDEs in the DSA-2000 might be significantly higher. Additionally, because many of the TDEs will be bright in multiple epochs, the DSA-2000 could detect them well below the 20 𝜇 Jy limit, which would increase the rate by a large factor. The DSA-2000 will also discover tens of thousands of distant fast radio bursts (FRBs) (Petroff et al. 2019; Cordes & Chatterjee 2019), some of which will be strongly lensed (Connor & Ravi 2023). The key advantage to using FRBs for time-delay lensing is that their short duration and coherence allows for exceptional precision on the gravitational lensing time delay (Wucknitz et al. 2021). Radio telescopes can preserve phase information about the electromagnetic waveform at nanosecond sampling, which means microlensing signals can be searched for at ultrashort timescales (Leung et al. 2022; Kader et al. 2022). However, for cosmological lensing time delays longer than a pointing (i.e. deflectors more massive than ∼ 10 8 M ⊙ ), one needs to catch the lensed images by pointing at the same patch of sky when it arrives. We do not forecast lensed FRB rates here and point the reader to previous estimates (Connor & Ravi 2023).","pages":[2,3]},{"title":"2.2 Lens Model","content":"Next, we build a forward model based on the lens and source distributions. Following Yue et al. (2022b), the lensing optical depth for a singular isothermal sphere (SIS) is where 𝑧 𝑑 is the redshift of the deflector, 𝑧 𝑠 is the redshift of the source, 𝜎 is the 1D velocity dispersion of the deflector, Φ ( 𝜎, 𝑧 𝑑 ) is the velocity dispersion function (VDF) of the deflectors, 𝑑 2 𝑉 𝑐 𝑑 Ω 𝑑𝑧 𝑑 = ( 1 + 𝑧 𝑑 ) 3 𝑐 𝑑𝑡 𝑑𝑧 𝑑 is the differential comoving volume, 𝜃 𝐸 ( 𝜎, 𝑧 𝑑 , 𝑧 𝑠 ) is the Einstein radius, and 𝐷 𝑑 is the angular diameter distance at the deflector redshift. The SIS model (or its elliptical generalization) has been shown to replicate the properties of early-type galaxies, which make up the majority of galaxy lenses (e.g. Gavazzi et al. (2007); Koopmans et al. (2009); Li et al. (2018)), and is a widely used model for galaxy strong lensing statistics (e.g. Oguri & Marshall (2010); Collett (2015); Yue et al. (2022a). For a SIS, the Eintein radius becomes where 𝐷 𝑑𝑠 is the angular diameter distance from the deflector to the source and 𝐷 𝑠 is the angular diameter distance from the observer to the source. We use the analytical VDF from Yue et al. (2022b) to model the galaxy deflector population, which they show to be in good agreement with observations. We include a sharp exponential cutoff on the VDF at 300 kms -1 which corresponds roughly to the transition between galaxies and galaxy groups. An important ingredient in the calculation is the magnification bias, which increases the rate of lensed sources by about a factor of two at low redshifts, or by orders of magnitude at high redshifts. The general magnification bias is where 𝜇 is the magnification of a lensed source, 𝑝 ( 𝜇 ) is the probability distribution of the magnification, 𝐿 min is the smallest observable luminosity, and 𝑁 ( > 𝐿 min ) is the number of sources brighter than 𝐿 min . For SIS, 𝜇 min = 2 for the total magnification of the multiple images. A SIS will produce only two multiple images; if the separation of the multiple lens images is large enough to be resolved then 𝑝 ( 𝜇 ) = 2 ( 𝜇 ± 1 ) 3 describes the magnification of the fainter (+) and brighter (-) image (Wyithe et al. 2001). If the lens is not resolved then 𝑝 ( 𝜇 ) = 8 𝜇 3 is the total magnification of both images. Combining 𝐵 and 𝜏 for the total fraction of lensed sources: where 𝐵 ' is the magnification bias of sources that are not lensed, which is assumed to be unity. While the SIS model describes deflector galaxies well, the mass distributions of galaxy groups and clusters behave differently. It has been traditionally thought that in the limit of large mass and large image separation, i.e. clusters, the distribution will be dominated by the dark matter halo, generally following the Navarro-Frenk-White (NFW) profile (Navarro et al. 1997). Recent works suggest that NFW profiles may not be the most accurate representation of large dark matter halos (e.g. Klypin et al. (2016)), but we use them here because they are a decent approximation and their lensing properties are well known. Regardless, for the mass range of groups, it is clear that some intermediate model between SIS and NFW is necessary (Williams et al. 1999; Oguri 2006; More et al. 2012). In a SIS the density 𝜌 ∝ 𝑟 -2 , while the NFW has 𝜌 ∝ 𝑟 -1 on small scales and 𝜌 ∝ 𝑟 -3 on larger scales. The shallower NFW profile has a significantly smaller cross-section than SIS. Oguri (2006) include the effects of baryon cooling and the large elliptical galaxies found at the center of most halos in their model, which will steepen the central density profile and increase the cross-section of halos (especially for groups). Without introducing these complexities, we can steepen the inner profile of halos by modeling them as Generalized NFW (GNFW) profiles. The GNFW is: where 𝑟 𝑠 is the scale radius and 𝜌 𝑠 is the scale density (Li & Ostriker 2002). When 𝛼 = 1 we have the standard NFW. When 𝛼 = 2 the profile resembles an SIS below the scale radius and when 1 < 𝛼 < 2 the inner profile is somewhere in between. Using weak lensing and stellar kinematics, Wang et al. (2023) find the inner dark matter density profiles of group (10 13 𝑀 ⊙ ≤ 𝑀 < 10 14 𝑀 ⊙ ) and cluster ( 𝑀 ≥ 10 14 𝑀 ⊙ ) halos are 1 . 82 + 0 . 15 -0 . 25 and 1 . 48 + 0 . 2 -0 . 41 respectively. Similarly, Mandelbaum et al. (2006) find that the total inner profiles of groups with mass ∼ 2 . 5 × 10 13 𝑀 ⊙ are decently described by a power-law of slope 𝛾 tot = 1 . 85. However, at ∼ 7 × 10 13 𝑀 ⊙ they are less steep than 1.85 and closer to the traditional NFW. Newman et al. (2013) find 𝛾 tot = 1 . 16 + 0 . 10 -0 . 12 for the inner profile of clusters with 𝑀 = 0 . 4 -2 × 10 15 𝑀 ⊙ , and Newman et al. (2015) find the total inner profile slope 𝛾 tot ≈ 1 . 7 while the dark matter slope 𝛼 ≈ 1 . 35 for galaxy groups with ⟨ 𝑀 ⟩ ≈ 10 14 𝑀 ⊙ . In this work, we use the GNFW profile with 𝛼 = 1 . 6 to model groups and 𝛼 = 1 . 2 to model clusters. The resulting GNFW profiles will capture the inner profile well, which is crucial for lensing, and also converge to the standard NFW profile at large radii where we expect the density to be dominated by dark matter (e.g. figures 3 and 4 of Wang et al. (2023)). The GNFW is completely parameterized by 𝛼 , its mass 𝑀 , and the concentration parameter, 𝑐 , from which 𝑟 𝑠 and 𝜌 𝑠 can be determined as in Li & Ostriker (2002). The mass and 𝑐 are correlated with some scatter, we use the relationship presented by Dutton & Macciò (2014) for 𝑐 200 . The scatter in 𝑐 is lognormal with 𝜎 log 𝑐 = 0 . 11. We include an additional factor of ( 2 -𝛼 ) as in Oguri et al. (2001) to correct for the generalized form of the NFW which ensures that the radius at which the logarithmic density slope becomes -2 is the same as in the standard NFW for all 𝛼 . The lensing power of the GNFW halo is very sensitive to 𝑐 and its scatter, so having an accurate relationship is important; early investigations tend to overestimate these values. The lens equation relates a position on the lens plane, 𝑥 , to a position on the source plane, 𝑦 , which for a GNFW profile is where and 𝑔 ( 𝑥, 𝛼 ) is the same as in Li & Ostriker (2002). The lens equation has three solutions inside the radial caustic, 𝑦 𝑐𝑟 , which are the multiple images produced by the GNFW profile. The cross-section for lensing is the area in the source plane in which multiple images will be produced, i.e. the area inside the radial caustic. This is approximated as: where 𝑦 𝑐𝑟 = -𝑦 ( 𝑥 𝑐𝑟 ) and 𝑥 𝑐𝑟 is the location of the minimum of equation 6 (Li & Ostriker 2002). The image separation is given by the separation between the outer two images, which can be approximated as where 𝑥 0 is the position of the tangential critical curve and can be found as the root of equation 6 (Li & Ostriker 2002). The Einstein radius is just half of the image separation, 𝜃 𝐸 = Δ 𝜃 / 2. In general, because our GNFW profiles are shallower than a SIS, they will be worse at producing multiple images but significantly better at magnifying. This is particularly sensitive to 𝑐 , which defines the shallowness of the GNFW profile. For our concentration parameters, especially at high redshift, we expect the magnifications from the GNFWs to be several times larger than from SIS (Wyithe et al. 2001). To calculate the magnification bias for a GNFW profile, we first determine the minimum magnification as (Li & Ostriker 2002): With 𝜇 min wecalculate B according to equation 3 with (Li & Ostriker 2002): The minimum magnification for both the first and second brightest image can be approximated as half of the total minimum magnification (Oguri et al. 2002). Substituting 𝜇 min = 2 into equation 11 recovers the total magnification distribution for SIS. To model the populations of galaxy groups and clusters we determine Mass Functions (MF) from the CosmoDC2 synthetic catalogue Korytov et al. (2019). For simplicity, we define a galaxy group as a system with mass 10 13 𝑀 ⊙ ≤ 𝑀 < 10 14 𝑀 ⊙ and a galaxy cluster as 𝑀 ≥ 10 14 𝑀 ⊙ . We find all dark matter halos in a redshift interval where the mass of the halo and its constituent galaxies fall in these ranges. We fit Schechter functions to the binned results. The fitted functions match the results of Böhringer, Hans et al. (2017) well at 𝑧 = 0 and show a realistic decline in the mass function towards higher redshift. To calculate the optical depth, we replace 𝜋𝜃 𝐸 ( 𝜎, 𝑧 𝑑 , 𝑧 𝑠 ) 2 𝐷 2 𝑑 in equation 1 with equation 8, Φ ( 𝜎, 𝑧 𝑑 ) with the CosmoDC2 MF, and integrate over 𝑀 instead of 𝜎 . While equation 4 gives the total fraction of lensed sightlines, we are more interested in the number of discoverable lenses. We define a lens as discoverable if its image separation Δ 𝜃 = 2 𝜃 𝐸 is larger than 2/3rds of the FWHM of the PSF of the survey instrument (as in Oguri & Marshall (2010); Yue et al. (2022a)) so that multiple images can be resolved. Further, for point-like sources (i.e. flat-spectrum sources and transients), we require the second faintest image to be detectable at 10 𝜎 confidence so that the lens can be identified. To account for this we integrate equation 1 from 𝜎 min (or equivalently 𝑀 min for groups and clusters) corresponding to 𝜃 𝐸 min and use the magnification bias of the fainter image. The second image from a SIS will often be demagnified, so the magnification bias of the fainter image is typically a bit less than 1, while the magnification bias of the fainter image from an NFW will be about half of the total. For SFRGs and SS-AGN, which in general have extended emission regions, the lensed images will be stretched into arcs and rings, and so for this case we require the brightest image to be detectable at 10 𝜎 confidence (using the magnification bias of the brightest image) because only one arc is needed to identify a lens. The image separation distribution predicted by our model and the lensing optical depths for galaxies, groups, and clusters are shown in Figures 1 and 2 respectively. We see that the optical depth increases rapidly at small redshift as expected, and approximately matches the estimates given in Section 2.1. The galaxy lens optical depth is smaller than the approximation from Oguri (2019). The discoverable optical depth, limited by the PSF of the DSA-2000, is significantly smaller than the total optical depth for galaxies but similar to the total for groups and clusters. The image separation distribution matches the results of Oguri (2006) well despite the simpler model. As expected, galaxies lie mostly at Δ 𝜃 ≈ 1', while groups and clusters correspond to Δ 𝜃 ≈ 10' and 10' ≲ Δ 𝜃 ≲ 100' respectively. The largest uncertainty is in the model of groups, as our simple model does not capture many of the complexities of halo profiles and is constant over the whole mass range. Further, varying 𝛼 between 1 and 2 creates a significant change in the calculated group optical depth. On the other hand, there is inherent uncertainty in all three models from the assumption of spherical symmetry. Using SIS instead of Singular Isothermal Ellipsoids (SIE) for galaxies will overestimate the galaxy lens population by ∼ 10%, which is corrected for in our final numbers (Yue et al. 2022b; Ferrami & Wyithe 2024). Introducing ellipticity will similarly reduce the cross-section of NFW profiles. We do not account for effects such as sub-halo lensing or line-ofsight structures and lens environments. It is shown in Oguri (2006) that a significant fraction of lenses, roughly half for groups and clusters and 10% for galaxies, come from lensing by sub-halos. The environment around these sub-halos boosts their lens capabilities and image separations. At least one of the CLASS lenses is caused by a galaxy within a galaxy group (Auger et al. 2007). In the context of cluster lensing, this is termed galaxy-galaxy strong lensing (GGSL) and it is reported that observational numbers of GGSL exceed expectations of the Λ CDM model (Meneghetti, Massimo et al. 2022; Tokayer et al. 2024). This means that we are likely underestimating the number of lenses in the image separation range of galaxy scale lenses. However, most of these will be below the discoverable limit of the DSA-2000. Further, it is well known that intervening masses along the line of sight between the deflector and the observer could increase the probability of multiple imaging by a significant fraction, especially at high source redshifts, which we also neglect in this study (Fleury et al. 2021). Because these higher-order aspects of the model are likely to increase the number of lenses, for the purpose of this paper it is safe to ignore them.","pages":[3,4,5,6]},{"title":"2.3 Source Populations","content":"The other important piece of the model is the population of the sources. The DSA-2000 will map ∼ 30,000 deg 2 of the sky to a combined 𝜎 𝑛 = 500 nJy/beam rms noise (Hallinan et al. 2019). Matthews et al. (2021) presents detailed radio source counts from the MeerKat DEEP2 image, including direct source counts above 10 𝜇 Jy and statistical counts extrapolating below 10 𝜇 Jy. They find that for sources <10 𝜇 Jy, the differential source count is significantly flatter than Euclidean. Scaling for the 30,000 deg 2 footprint of DSA2000's continuum survey and integrating down to a minimum flux density 𝑆 = 2 . 5 𝜇 Jy for a 5 𝜎 𝑛 combined detection gives an expected total source count of 1 . 45 + 0 . 25 -0 . 10 × 10 9 , while for 𝑆 ≥ 5 𝜇 Jy = 10 𝜎 𝑛 that number is 8 . 6 + 1 . 1 -0 . 4 × 10 8 . To get redshift distributions for different source classes, we model the Star-Forming Radio Galaxy (SFRG) and Active Galactic Nucleus (AGN) populations using the luminosity functions (LF) from the Tiered Radio Extra-galactic Continuum Simulation (T-RECS) (Bonaldi et al. 2018). These have been shown to match observations out to high redshift. Following Bonaldi et al. (2018), we divide the AGN into three sub populations: Flat Spectrum Radio Quasars (FSRQ), BL-Laceratae (BLLac), and Steep Spectrum AGN (SSAGN). We expect SFRGs to make up roughly 95% of all of the sources in any synoptic radio survey, and BLLacs to be the rarest source. From the LF we calculate the number of sources in a redshift interval as where Ω survey is the sky coverage of the survey in steradians ( ≈ 3 𝜋 for the DSA-2000) and Φ ( 𝐿, 𝑧 ) is the LF of the source population. 𝐿 min ( 𝑧 ) = 4 𝜋𝐷 2 𝐿 ( 1 + 𝑧 ) 1 + 𝛼 𝑆 min is the minimum observable luminosity at a redshift 𝑧 , where 𝐷 𝐿 is the luminosity distance, 𝛼 is the spectral index of the source population, 𝑆 min is the 10 𝜎 𝑛 flux sensitivity of the DSA-2000, and the factor ( 1 + 𝑧 ) -( 1 + 𝛼 ) is the standard cosmological radio K-correction. We scaled the total number of sources in the LF model to match the observational prediction from the DEEP2 image given above. In addition to SFRGs and AGN, we build a model for radio SNe. To date, only ∼ 100 radio core-collapse supernovae (ccSNe) have been detected and the first Type Ia radio supernova was detected last year (Bietenholz et al. 2021; Kool et al. 2023). To model the expected rate of radio ccSNe we follow Lien et al. (2011). Because ccSNe are shortlived, the rate of ccSNe is closely related to the cosmic Star Formation Rate (SFR), which has been observational measured to high redshift and accuracy. We use the ccSNe radio luminosity distribution of Bietenholz et al. (2021), which is more complete than the one in Lien et al. (2011). The best-fit distribution is log-normal and incorporates all known radio ccSNe as well as radio non-detections of supernovae. Bietenholz et al. (2021) note that their data is likely biased towards radio detections, finding that 30% of all ccSNe are detected in the radio. We divide the total number of radio ccSNe in the model by 3 to get a more conservative 10% as in Lien et al. (2011). From the luminosity distribution, we determine the number of radio ccSNe at each redshift above the flux limit of the DSA-2000. Because the SNe observations used in Bietenholz et al. (2021) have frequencies ranging from 2-10 GHz, there is uncertainty when converting to 1.4 GHz. We correct 𝐿 min 6 GHz ( 𝑧 ) = 𝐿 min 1.4 GHz GLYPH<16> 6 1 . 4 GLYPH<17> 𝛼 , where we assume 𝛼 = -0 . 7 for the synchrotron emission of radio ccSNe. Further, because radio ccSNe have a mean rise time of log 10 ( 𝑡 rise ) = 1 . 7 days (Bietenholz et al. 2021), we use the single epoch flux limit of the DSA-2000 and account for the DSA-2000's cadence. The rms noise for a single epoch is 𝜎 𝑛 = 2 𝜇 Jy/beam, so a 10 𝜎 𝑛 detection is 20 𝜇 Jy. The cadence will be ∼ 4 months, so conservatively we will detect 1/3 of the radio ccSNe per year. None of the known radio ccSNe have luminosities above 10 29 erg s -1 Hz -1 , but given the wide spread we expect that the highest luminosities could be several times larger. Because of this we set an exponential cutoff at 10 30 erg s -1 Hz -1 , which we can expect to see out to about 𝑧 ≈ 1 . 5 in a single epoch of the DSA-2000. The predicted redshift distributions of SFRGs, AGN, and ccSNe from our model are shown in Figure 3 and compared to the expected distribution of Rubin-LSST sources. As expected, the total source count is dominated by SFRGs until high redshift. Blazars, meaning both FSRQs and BLLacs, are the AGN that will be most useful for time delay measurements, which is explored in Section 5.1. These are the most common sources at very high redshift. We expect the DSA2000 to probe much higher redshifts than the Rubin-LSST because radio emission is much less susceptible to intervening gas or seeing. This is one of the reasons that the DSA-2000 will be effective at lens finding. The total number of expected sources of each type in a DSA-2000 all-sky survey with a 10 𝜎 𝑛 detection are listed in the first row of Table 1.","pages":[6]},{"title":"2.4 Expected rates","content":"The final results of our model are shown in Table 1. The results agree quite well with the empirical estimates given in Section 2.1. As expected from CLASS there are roughly 10 6 total galaxy scale lenses contained in the DSA-2000 all-sky survey, less than 10% of which will be discoverable. The total number of group and cluster lenses are about 10% and 2% of the galaxy number respectively, which is consistent with Oguri (2006). Roughly half of the group and cluster lenses will be discoverable, leading to them making up about a third of the total discoverable lenses, a significant fraction. Yue et al. (2022a) find that the Rubin-LSST will discover about 2000 lensed QSOs; our number is similar. This is a result of the competing mechanisms of the DSA-2000 detecting many more at higher redshifts and simultaneously being able to discover a smaller fraction because of the PSF size. The main purpose of Yue et al. (2022b) is to investigate the discrepancy between the fraction of lensed high redshift ( 𝑧 𝑠 ≳ 5) quasars in observations ( ∼ 0 . 2%) and previous predictions ( ∼ 4%).Yueetal.(2022b)find ∼ 0 . 4 -0 . 8%with their model, and we have ∼ 0 . 3%, which is consistent. The number of discoverable lensed ccSNe per year in the DSA-2000 from the full model is about 2, which is of the same order of magnitude as the empirical VLASS estimate. Oguri & Marshall (2010) conclude that Rubin-LSST will find roughly 8 SNe per year. We expect there to be less lensed SNe at radio wavelengths because not all SNe emit in the radio and the radio emission is weaker than in other wavelengths. The redshift distribution of the deflectors and sources is shown in Figure 4. Almost all deflectors are at 𝑧 𝑑 < 3 and most are at 𝑧 𝑑 ≈ 1, consistent with Oguri & Marshall (2010); Collett (2015); Yue et al. (2022a,b). This justifies the use of the CosmoDC2 catalog, which is limited to 𝑧 < 3, to determine the MF of groups and clusters. We also run the simulation for the survey specs of the SKA-mid and VLASS. The main differences are the sensitivity and resolution. The expected rms noise of the SKA-mid at 1.4 GHz will be 𝜎 𝑛 = 2 𝜇 Jy/beam, and the PSF size will be about 0.4' (Braun et al. 2019). From the DEEP2 source counts, the SKA-mid should find about 3 × 10 8 sources above 10 𝜎 𝑛 . With this, our model predicts that the SKA-mid should see 1.8 ± 0.7 × 10 5 , 1.8 ± 1.2 × 10 4 , and 2.7 ± 1.1 × 10 3 galaxy, group, and cluster lenses, given the same discoverability limit defined in Section 2.2. Despite less depth leading to around a third of the total galaxy lenses, the PSF of the SKA-mid is such that it will resolve over half of them, resulting in about twice as many discoverable galaxy lenses as in the DSA-2000. The SKA-mid will find fewer group and cluster scale lenses than the DSA-2000 because sensitivity is more important for discovering these systems than angular resolution. In addition, we expect about 14 lensed radio ccSNe per year in an SKA-mid all-sky survey. McKean et al. (2015) estimate ∼ 3 × 10 5 lenses in an SKA-mid all-sky survey with 𝜎 𝑛 = 3 𝜇 Jy/beam and a lens detection limit of 15 𝜎 𝑛 and >0.3'. This is slightly more optimistic than our forecast, but broadly consistent. VLASS has 𝜎 𝑛 = 70 𝜇 Jy/beam combined and an angular resolution of 2.5' (Lacy et al. 2020). The DEEP2 source counts indicate that we should expect ∼ 5 × 10 6 sources above 10 𝜎 𝑛 in VLASS (Lacy et al. (2020) estimate 5.3 × 10 6 above 5 𝜎 𝑛 ). With these numbers, we estimate that there should be about 300 discoverable lenses in the full sky survey. As VLASS enters its epoch 3, many of these lenses are likely already in the data (several have recently been identified (Martinez et al. 2024)).","pages":[6,7]},{"title":"3.1 Superresolution","content":"In the past decade, tremendous progress has been made by the computer vision community with respect to the classical ill-posed inverse problems. These include deblurring (Zhang et al. 2022), deconvolution and superresolution (Alzubaidi et al. 2021), and image inpainting (Yu et al. 2018b). Nearly all of these advances were borne out of the deep learning revolution, as efficient neural network architectures and training strategies have enabled powerful learning-based tools for machine vision. Astronomy naturally lends itself to these tools because sparse sampling and ill-posedness arise in many astronomical imaging and reconstruction contexts, especially in interferometry. Several groups have begun developing machine learning methods for interferometric image reconstruction (Connor et al. 2022; Aghabiglou et al. 2024; Mars et al. 2024). One reason this approach is suitable for radio astronomy is that the PSF of an array is given deterministically by the spatial distribution of antennas and the observing frequency: The on-sky response of an interferometer is the 2D Fourier Transform of its sampling in UV-space (the aperture plane). Therefore, prior physical knowledge of the PSF can be incorporated in the model either via training data (Connor et al. 2022) or fed directly to the network itself (Mars et al. 2024). This is not the case in ground- based optical astronomy where 'seeing' and complex optics mean the PSF is not known a priori. Asasimple demonstration, we use the super-resolution and imageplane deconvolution method POLISH to show how strong lenses with image separations below the PSF scale can be recovered with machine learning. POLISH is a supervised machine learning model that learns the mapping between the true sky and observed images (the 'dirty image' in radio parlance) (Connor et al. 2022). It uses the Wide Activation for Efficient and Accurate Image Super-Resolution (WDSR) architecture (Yu et al. 2018a), but any super-resolution neural network could be swapped in (e.g., a standard U-Net or the Efficient Super-Resolution Transformer (Lim et al. 2017)). Wehave trained a POLISH network on forward-modelled synthetic data using a DSA-2000 PSF averaged over the whole band, resulting in an angular resolution of ∼ 3.3'. This is worse resolution than the top of the band, where many strong lenses could be found, but we offer this as a toy example. The sky model is described in Connor et al. (2022), with the addition of strongly lensed galaxies in both the training and validation set. In Figure 5 we show an example validation image that contains multiple lenses. Lensed systems that are undetected in dirty image, including both arcs (left inset panels) and Einstein rings (right inset panels), are identifiable in the POLISH reconstruction. In the future, we plan to develop super-resolution methods explicitly for lens finding, but we take this as a promising sign that the DSA-2000 will be able to find systems with 𝜃 𝐸 ≳ 0 . 5 '' .","pages":[7,8]},{"title":"3.2 Identification","content":"The first major hurdle for strong lensing science with DSA-2000 and other surveys is identification. The large number of sources produced by these surveys makes visual inspection by experts impossible. Despite much work on automated lens discovery (see Lemon et al. (2023) for a review), little effort has been spent on identification in the radio. On the one hand, radio lens images are made cleaner by the lack of radio emission from the massive quiescent galaxies that make up the deflector population. However, the large, extended lobes of radio galaxies can easily be misidentified as lensing features. Further, color and photometric redshift information from UV and optical surveys have traditionally played a role in lens identification, despite being a foreground to the morphology of the lensed source. Catalog-level searches, where large cuts are made based on certain features before a smaller subset is visually inspected, have been successful in the past. CLASS selected for flat spectrum radio sources ( 𝛼 > -0 . 5) (Myers et al. 2003). This had the advantage of picking out only blazars, whose radio emission is dominated by a small compact core, thus eliminating any confusion with intrinsic structure. A similar strategy would likely be effective in the DSA-2000. However, dedicated follow up of all of these sources, as in the ∼ 11,000 blazars in the initial CLASS sample, will be impossible. Similarly, a visual inspection of even a small subset of the O( 10 7 ) blazars expected in the DSA-2000 is ambitious (hence the need for intermediate automated steps in the identification pipeline). Still, targeting flat spectrum sources could increase the yield of any search, and spectral information from the whole DSA-2000 band will be useful for determining whether components are different sources or multiple images. Jackson & Browne (2006) show that combining astrometric data from radio and optical surveys can significantly improve the efficiency of lens searches. If the optical emission is dominated by the deflector galaxy then there will exist an offset between the centroid position of the optical and radio sources in a lens system. This offset was exploited to find lenses with separation down to ∼ 1' even with the poor 5' resolution of the Faint Images of the Radio Sky at Twenty-one centimeters (FIRST) survey. They predict that this effect will become more efficient at lower radio fluxes because the optical flux is more likely to be dominated by the defelctor galaxy. Given the sensitivity of the DSA-2000 and its significant overlap with current and planned large surveys in other wavelengths, this could prove to be a very effective way of overcoming the limitations of its PSF. Regardless, exploiting the extra information from overlap with other surveys will be crucial for any lens search in the radio. Machine learning models have been successfully used to discover many new candidate lens systems. Convolutional Neural Networks (CNNs) in particular are effective at classifying astronomical images. The main disadvantage with CNNs and other supervised learners is the need for large realistic training sets. Using CNNs on simulated data for the International LOFAR Telescope (ILT) at 150 MHz, Rezaei et al. (2022) are able to recover over 90% of galaxy-size lenses with a false-positive rate of only 0.008%. They find that a strong 20 𝜎 detection and large separation 𝜃 𝐸 ≥ 3 / 2 beam size (stricter than the discoverable limit imposed in Section 2.2) are necessary for reliable identification with the CNN. Incorporating superresolution models into the lens-finding routine would significantly increase the yield of such a CNN, especially at high S/N. Resolving lenses down to 𝜃 𝐸 ≈ 0 . 5' with would enable the same level of accuracy as in Rezaei et al. (2022). Simulating accurate DSA-2000 lenses and training models for identifying them is a goal of our future work. The SKA-mid will already have sufficient resolution for these purposes, making CNNs an attractive option for lens identification in its all-sky survey. The main limitation of any ML-based approach to group/cluster lens finding is the difficulty of creating realistic training sets. The irregularity of group/cluster lensing potentials may require building a forward-modelled training set from ray-tracing in large cosmological magnetohydrodynamical simulations such as TNG-Cluster, which should have a heterogenous population of massive clusters (Nelson et al. 2024). More generally, a realistic training set of strong lenses across scales could be produced with radio source samples from T-RECS and a deflector population drawn from cosmological simulations.","pages":[8,9]},{"title":"4 MULTIWAVELENGTH SYNERGIES","content":"Acquiring redshifts for lensed galaxies and deflectors is critical for the application of strong lensing to cosmology and astrophysics. We consider here how well we could do in the absence of targeted followup observations, capitalizing on the suite of large survey telescopes that will be operating in the mid- to late-2020s. DSA-2000 spectroscopic HI survey In addition to the cadenced allsky survey, the DSA-2000 will undertake a large spectroscopic HI survey over the full visible sky (Dec > -30 · ) (Hallinan et al. 2019). This is expected to produce ∼ 10 6 HI galaxies at 𝑧 < 1. Most deflectors will be massive elliptical galaxies with limited 21 cm emission, but a sub-set of early-type galaxies with 𝑛 𝐻 ≥ 10 20 cm -2 𝑧 -2 𝑑 will be detected. Some late-type deflector galaxies will have enough cold gas to obtain a redshift. Additionally, 21 cm absorption in the inner CGM of deflector galaxies may be present in the lensed source's continuum spectrum, providing a measurement of cold gas along different sightlines of the same CGM (see Rudie et al. (2019) for a similar application in lensed quasar systems). Thus, some deflector galaxies will have a spectroscopic redshift without any external O/IR data. Of the ∼ 10 6 HI galaxies detected by the DSA-2000 in emission, we find that only a handful will be strongly lensed, due to their shallow redshift distribution. Spectroscopic radio observations will not provide a significant portion of redshifts required for our purposes, but may be a minority of interesting deflectors. The Dark Energy Spectroscopic Instrument (DESI) will produce redshifts of ∼ 40 million galaxies and quasars (DESI Collaboration et al. 2016; Dey et al. 2019). Its 3 yr public data release is expected before the DSA-2000's first light. DESI's Bright Galaxy Survey (BGS) targets galaxies with 𝑟 -band magnitudes brighter than 19.5. For the luminous red galaxies (LRGs), emission line galaxies (ELGs), and quasars, the depth will be 𝑟 ≈ 23. Each class of target will have a different redshift distribution, but together they will provide a rich catalog of spectroscopic redshifts for deflectors in the lensing systems discovered by DSA-2000. The BGS and LRG samples will be over-represented as deflector galaxies, whereas the ELG and QSO samples may make up a small fraction of lensed DSA-2000 sources. ELGs, by definition, have strong emission lines indicative of star formation, which produce synchrotron radiation detectable by radio surveys. Some of these will be strongly lensed star-forming radio galaxies with 𝑆 𝜈 above 5 𝜇 Jy. Still, we do not expect more than a small minority of the radio continuum lenses to be detected in DESI due to the relative number of total sources. SPHEREx is a near-infrared space mission expected to launch before Spring 2025. It will map the full 4 𝜋 sr sky with 6 arcsecond pixels and 96 color bands, producing a catalog of hundreds of millions of galaxies with 'spectro-photometric redshifts' (Doré et al. 2018). Although its PSF it too large to discover strong lenses directly, the galaxy catalog will be valuable for obtaining reliable redshifts of both deflectors and lensed galaxies. The Rubin Observatory will find billions of galaxies in its Legacy Survey of Space and Time (LSST) (Ivezić et al. 2019). Most of these galaxies will be at 𝑧 𝑠 ≲ 1 . 5 (Collaboration et al. 2021), providing an excellent deflector catalog for the DSA-2000 lens candidates. In the DSA-2000/LSST overlapping footprint between -30 and +30 Declination, we find that a significant fraction of deflectors and several thousand lensed sources in the DSA-2000 sample will have photometric redshifts from the Rubin Observatory. Roman The High Latitude Spectroscopic Survey (HLSS) is expected to detect 10 million galaxies in 𝐻𝛼 between 1 < 𝑧 < 2and ∼ 2 million [OIII] galaxies at 𝑧 = 2 -3 using the near-IR grism and wide-field camera (Wang et al. 2022). Roman's 2,000 deg 2 footprint will be fully mapped by DSA-2000. With a dust-attenuated 𝐻𝛼 flux higher than 10 -16 erg s -1 cm -2 , several million of these galaxies will be detectable at GHz frequencies at > 5 𝜇 Jy (Murphy et al. 2011). Assuming a mean lensing optical depth of 5 × 10 -4 , this suggests that O( 10 3 ) will be strongly lensed and detected by both the DSA-2000 and Roman , the latter providing 0.1' resolution and a source redshift. Euclid is a visible and near-infrared space telescope that launched in July 2023 (Collaboration et al. 2024). Its large photometric survey will find over one billion galaxies, with a spectroscopic survey that will obtain accurate redshifts for ∼ 30 million galaxies mostly at 0 . 9 < 𝑧 < 1 . 8. Euclid and the DSA-2000 share over 10,000 deg 2 of sky, providing an excellent sample of photometric and spectroscopic redshifts of lens systems.","pages":[9]},{"title":"5 STRONG LENSING APPLICATIONS","content":"A key limitation of strong lensing science is the small number of known systems, especially at radio wavelengths. We discuss the impact that the large expected number of radio lenses and their multiwavelength counterparts will have on several important applications of strong lensing.","pages":[9]},{"title":"5.1 Time-delay cosmography & 𝐻 0","content":"Light from multiple images in a gravitational lens will reach the observer at different times. For continuum sources, we do not observe the difference in arrival times, but transients or time-variable sources allow us to measure the time delay. This time delay is due to a difference in path lengths and gravitational time dilation near the deflector. As such, the time delay encodes information about the geometry of the universe, and is inversely proportional to 𝐻 0 (Oguri 2019). Much work has been dedicated to measuring these time delays and using them to constrain 𝐻 0 (Treu & Marshall 2016; Birrer et al. 2024). The H0LiCOW project has measured 𝐻 0 to 2.4%, which is independent of and competitive with state-of-the-art 𝐻 0 constraints (Wong et al. 2019). A large increase in the number of measurable time-delay systems will allow even tighter constraints and could help settle the current Hubble tension. We estimate the number of lensed variable AGN that will be useful for measuring 𝐻 0 in the DSA-2000. To first order, all blazars (FSRQs and BLLacs) are inherently variable, ∼ 10 3 of which will be discoverable as lenses in DSA-2000 data (Table 1). However, several factors complicate our ability to use them for time delay measurements; we would like to know how many will be in a \"gold sample\" for which the time-delay can be reliably determined. To date, the most extensive study of radio AGN variability is the OVRO 40 m blazar monitoring campaign. Since 2007 this program has monitored around 1500 blazars with a cadence of two weeks (Richards et al. 2011). Richards et al. (2011) provides distributions of the intrinsic modulation index, a measure of the relative intrinsic variability of the AGN, for FSRQs and BLLacs. The DSA-2000 will measure flux variability at the level of a few percent, but to be conservative we consider candidates with ≥ 10% variability. We also want sources with a strong detection, so we chose sources with flux ≥ 20 𝜎 𝑛 = 10 𝜇 Jy. The OVRO blazars are monitored at 15 GHz but we expect lower variability at 1.4 GHz. Fan, J. H. et al. (2007) find that blazars are about 20% less variable at 4.8 GHz than 14.5 GHz. Sotnikova et al. (2024) find that radio AGN at 22.3 GHz are about 35% more variable than at 2.3 GHz, although there is little difference between 11.2 GHz and 2.3 GHz. We assume that the mean modulation index drops by 40% from the OVRO 40 m data to 1.4 GHz for the DSA-2000. Further, for multiple images, we need at least two of the images to be bright enough to measure the time delay so we use the magnification bias of the fainter image. With these restrictions, the OVRO data, and our lens model, we estimate that there will be 68 ± 30 lensed FSRQs and 23 ± 10 lensed BLLacs with sufficient brightness, variability, and image separation to measure time delays. The same analysis for the SKA-mid gives 120 ± 50 and 43 ± 16 lensed FSRQs and BLLacs. Of the lensed AGN, the most useful systems for time delay analysis will be those that have four or more images because they allow for multiple constraints on the time delay. A SIS is only capable of creating two images and a spherical NFW can make three, but factors such as ellipticity, irregularities in the lens potential, and external environments of real halos will allow a number of these systems to make four or more images. The fraction of quad galaxy-QSO lenses predicted in Rubin-LSST is ∼ 10-15% (Oguri & Marshall 2010; Yue et al. 2022a). However, of the statistically complete CLASS sample, 5 out of 13 lensed blazars were quadruply imaged and one had six images (B1933+503 has two sets of quadruply imaged sources Sykes et al. (1998)). Therefore we can expect 100-400 quadruply-lensed blazars in the DSA-2000 all-sky survey and about twice as many in the SKA-mid, roughly 10% of which may be used in practice for measuring 𝐻 0 . We note that quad systems often have higher external shear due to their preponderance of group or cluster environments, which results in an added systematic in lens modeling (Holder & Schechter 2003). Wong et al. (2019) reach a 2.4% measurement of 𝐻 0 with a sample of 6 lensed quasars, and predict that around 40 lenses are needed to constrain 𝐻 0 to within 1%. Jee et al. (2016) argue that with a well-defined sample of quadruply imaged quasars, and combining time-delay distances and velocity dispersion measurements to estimate angular diameter distances within 5%, 𝐻 0 can be measured to the same precision as Planck with as few as 10 systems. Napier et al. (2023) reach a 10% measurement with three galaxy clusters, and estimate that roughly 50 will be needed for a sub 1% constraint, assuming that modeling uncertainties can be reduced in coming years. While we expect uncertainty to decrease significantly with a larger sample of lensed quasars, there are several systematics that must be kept under control for an accurate measurement. Namely, an accurate redshift measurement is necessary, ideally of both the source and deflector, as well as precise measurements of the time delay, complete sampling over many cycles of the time delay to negate contamination from micro-lensing, accurate lens models, and constraints on any line of sight structures. If we assume that all of the lenses in our gold sample can be modeled to the accuracy of the H0LiCOW sample - a roughly 6% combined uncertainty on the time delay, mass model, and line of sight contribution for each lens - then a sample of ∼ 100 lensed quasars would give about a 0.6% constraint on 𝐻 0 . This is of course not possible with the DSA-2000 or SKA-mid alone but will require dedicated follow-up at multiple wavelengths. As discussed in Section 4, we expect that a significant minority of lenses should have redshifts available in other concurrent public surveys. Interesting subsamples of lenses will require targeted follow-up for both mass modeling and spectroscopic redshifts. The DSA-2000's cadenced all-sky survey will image each field above -30 Dec once every 4 months. Typical time delays for galaxy, group, and cluster lenses in our model are a month or less, several months, and a year or more respectively (Oguri et al. 2002). Special systems could therefore be visited with a higher cadence, similar to how pulsar fields will be visited regularly for timing experiments. So far we have only considered lensed blazars for time delay measurements, which are desirable because of their characteristic variability and compact emission regions. Lensed transients will also be useful for time delay measurements, provided that they are detected early and monitored closely. The variability of radio emission from TDEs is too slow to make them useful for this application (on-axis jetted TDEs can have shorter rise times, but they are significantly rarer such that we do not expect any lensed jetted TDEs in upcoming radio surveys), but SNe have faster rise times. We expect O( 1 ) lensed radio ccSNe per year in the DSA-2000 and O( 10 ) in the SKA-mid that can be used for 𝐻 0 measurements.","pages":[10]},{"title":"5.2 Dark Matter Structure","content":"Gravitational lensing has been very useful for determining the structure of galaxies, groups, and clusters at cosmological distances and distinguishing between dark matter models (see Vegetti et al. (2024); Natarajan et al. (2024) for a review). The best of these studies use high-resolution HST images, but radio observations also played a pivotal early role. Cohn et al. (2001) used the 10-image CLASS lens B1933+503, which provides many more constraints than usual, to determine the density profile of the deflecting galaxy, finding it close to isothermal. Wucknitz et al. (2004) find the same for B0218+357. Dalal & Kochanek (2002) used radio observations of flux ratio anomalies between multiple images to constrain the mass fraction of dark matter substructure to 2%. Radio observations are especially suited for this type of study because they are less susceptible to micro-lensing (Koopmans et al. 2003; Kochanek & Dalal 2004). Despite this, dark matter studies with radio lenses have been limited for many years by the small sample size (Vegetti et al. 2024). Because resolution is a key factor in the ability to accurately model mass distributions or detect substructure, the DSA-2000 alone will not be able to use lens systems to study dark matter. Its main utility will come from identifying huge numbers of systems for detailed followup. The extraordinary resolution of the ngVLA on a large sample of lenses selected by the DSA-2000 will revolutionize this field. In the more immediate future, the SKA will also provide enough resolution for these studies in their overlapping sky coverage. Another exciting application of lensing is using a central image to study small scales near the centers of galaxies (Treu 2010; McKean et al. 2015; Shajib et al. 2024). A SIS lens only forms two images, but for realistic galaxy profiles a highly demagnified image is expected to form at the center of the image configuration. This image will be sensitive to the mass contained within very small distances of the deflector's center, and so can be used to study supermassive black holes at cosmological distances. This is especially suited for radio wavelengths because in many cases the deflector will be radio-quiet allowing the center image to be detected. Because the center image is highly demagnified, only sensitive radio surveys such as the DSA2000 will be able to reliably identify them. At least one such central image has been detected in the radio with brightness ∼ 0 . 8 mJy at 8.46 GHz (Winn et al. 2004), so we can likely expect O( 10 2 ) or more with the DSA-2000. A main challenge will be disentangling the faint emission of the central image from the other images. For groups and clusters, whose inner density profiles are shallower than isothermal, central images are more common and often less demagnified. The very inner profiles of groups/clusters can thus be studied, which are hard to constrain with images near the Einstein radius alone. These central images will be easier to identify because the other images will be farther away (although many clusters have large central elliptical galaxies emitting in the radio that will obscure the image).","pages":[10,11]},{"title":"5.3 Polarization","content":"Measuring full Stokes information of gravitationally lensed signals offers insight into the source plane because polarization is not affected by gravitational potentials. This is an advantage of radio surveys, where polarization information is often stored and Faraday rotation measure (RM) can be measured. This enables us to study propagation effects along different sightlines for lensed objects. For example, Mao et al. (2017) used the VLA to measure the polarization properties of two lensed images of CLASS B1152+199, finding a large difference in RM ( + 9 . 7 ± 0 . 5 rad m -2 for image ' 𝐴 ' and + 517 ± 3 rad m -2 for image ' 𝐵 '), which is due to the magnetized plasma in the lens galaxy's interstellar medium. The DSA-2000 will have the ability to measure full polarization information for any source it detects, with a maximum |RM| of roughly 10 4 rad m -2 . Compact sources such as Blazars (and especially BL Lac objects) can show significant polarization fractions at 1 GHz. Assuming a 50 𝜎 detection threshold in total power, we estimate that the DSA-2000 could detect Stokes Q and U for 5-10 million AGN in its 5 year continuum survey. Using the empirical CLASS lensing optical depth and image separation cut, we take 𝜏 𝐴𝐺𝑁 ≈ 5 × 10 -4 . We estimate that the DSA-2000 could find O( 10 3 ) strong lenses for which polarization properties could be used to model the lens distribution and study magnetic fields at cosmological distances.","pages":[11]},{"title":"5.4 Other applications","content":"In addition to constraining 𝐻 0 , time-delay measurements are weakly sensitive to other cosmological parameters, such as Ω 𝑚 , Ω Λ , and the dark energy equation of state parameter 𝑤 (Natarajan et al. 2024). Cluster lenses are used for studying dark energy as well (Macciò 2005; Gilmore & Natarajan 2009; Jullo et al. 2010). A large influx in measured time delays and observed cluster lenses with the DSA2000 will allow better constraints on these cosmological parameters. Lenses are also commonly used as 'nature's telescopes'. The magnification of background sources allows very distant galaxies to be studied even by smaller telescopes (Jackson 2011). This is especially true for group/cluster lenses, where we expect the magnifications to be larger (Robertson et al. 2020). We expect to see O( 10 4 ) lenses with 𝑧 𝑠 > 5 in the DSA-2000 (figure 4), many of which will be group and cluster lenses due to their high magnifications. These high redshift sources will provide insight into the population of radio sources in the early universe. Lensing statistics can also be used to test cosmological models. The abundance of giant arcs in cluster lenses was a topic of much debate in the past few decades (Meneghetti et al. 2013). Similarly, the rate of GGSL events in clusters may be in tension with the Λ CDM(Meneghetti et al. 2020; Meneghetti, Massimo et al. 2022; Tokayer et al. 2024). These studies require a large sample of lenses for accurate statistics, which is only now becoming possible with large surveys such as the DSA-2000.","pages":[11]},{"title":"6 CONCLUSIONS","content":"In this paper, we forecast expected strong lensing rates in the upcoming DSA-2000 and SKA-mid wide-field radio surveys. We first provide empirical estimates based on previous surveys and simulations, and then develop a detailed forward model that accounts for the source and deflector populations, finding them to be in good agreement. Notably, we model the expected number of galaxy group and cluster scale lenses because these systems will be easily discovered due to their wide angular separations. We find that both the DSA-2000 and the SKA-mid will discover roughly 10 5 strong lens systems. We discuss strategies for identifying these lenses in the data, which will all benefit from emerging superresolution techniques. Finally, we discuss the scientific application of the huge numbers of lenses that will be discovered by these surveys. One of the most exciting applications is 𝐻 0 cosmography with variable and transient sources. The DSA-2000 and SKA-mid will discover about 100 and 200 lensed flat spectrum AGN with >10% variability respectively, as well as about O( 1 ) and O( 10 ) lensed transients per year. With dedicated multi-wavelength follow up these systems could be used to constrain 𝐻 0 to within 1%. The new lens systems will also be useful for studying the distribution of dark matter at cosmological distances, among other applications.","pages":[11]},{"title":"ACKNOWLEDGEMENTS","content":"Weare grateful to Schmidt Sciences for supporting Samuel McCarty as a Summer Undergraduate Research Fellow at Caltech. We thank Kim-Vy Tran, Tony Readhead, and Tommaso Treu for helpful conversations on strong lensing, as well as Paul Schechter for insights into quad systems.","pages":[11]},{"title":"DATA AVAILABILITY","content":"We have placed a reproduction package on the public GitHub repository available at https://github.com/smmccrty/ radiolensing .","pages":[12]},{"title":"REFERENCES","content":"R., Motta V., 2012, The Astrophysical Journal, 749, 38 This paper has been typeset from a T E X/L A T E X file prepared by the author.","pages":[12,13]}],"string":"[\n {\n \"title\": \"ABSTRACT\",\n \"content\": \"The number of strong lensing systems will soon increase by orders of magnitude thanks to sensitive, wide-field optical and infrared imaging surveys such as Euclid, Rubin-LSST, and Roman. A dramatic increase in strong lenses will also occur at radio wavelengths. The 2000-antenna Deep Synoptic Array (DSA-2000) will detect over 10 9 continuum sources in the Northern Hemisphere with a high mean redshift ( ⟨ 𝑧 𝑠 ⟩ ≈ 2) and the Square Kilometer Array (SKA) will observe a large sample of extragalactic sources in the South with sub-arcsecond resolution. We forecast lensing rates, finding that the DSA-2000 will discover O( 10 5 ) strongly lensed systems, many of which will be galaxy group and cluster lenses. We propose strategies for strong lensing discovery in the limit where the Einstein radii are comparable to the PSF angular scale, taking advantage of modern computer vision techniques and multi-survey data. We also forecast synergies with optical and infrared surveys, which will provide redshifts as well as multiwavelength information about the lens systems. Finally, we describe applications of radio strong lensing systems, including time-delay cosmography with transient and variable sources. We find that ∼ 100 time-variable flat-spectrum AGN discovered by the DSA-2000 could be used to constrain 𝐻 0 at the percent level with the appropriate follow-up. Key words: gravitational lensing: strong - radio continuum: general\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"Strong gravitational lensing with upcoming wide-field radio surveys\",\n \"content\": \"Samuel McCarty 1 , 2 , ★ Liam Connor 3 1 Department of Astronomy, University of Washington, Seattle, WA 98195-1580, USA 2 Cahill Center for Astronomy and Astrophysics, MC 249-17, California Institute of Technology, Pasadena CA 91125, USA 3 Center for Astrophysics | Harvard & Smithsonian, Cambridge, MA 02138-1516, USA Accepted XXX. Received YYY; in original form ZZZ\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1 INTRODUCTION\",\n \"content\": \"Strong gravitational lensing has a multitude of applications in astrophysics and cosmology (Treu 2010). Previously theoretical ideas have been put into practice in recent decades as the number of known lensed systems has increased and observational data have improved. For example, strongly lensed time-variable and transient sources can be used to constrain the Hubble constant, 𝐻 0 , because the time delay of a multiply imaged source depends on the geometry of the Universe (Refsdal 1964). The technique is known as time-delay cosmography. With just six lensed quasar systems, the The 𝐻 0 Lenses in COSMOGRAIL's Wellspring (H0LiCOW) collaboration has reported 2.4 % precision on their 𝐻 0 measurement, which is independent of the distance ladder and the CMB (Wong et al. 2019). In addition to the Universe's large-scale geometry, lensing observables are sensitive to the total mass of the deflector galaxy or cluster, allowing one to measure the spatial distribution of matter and test different dark matter models (Massey et al. 2010; Vegetti et al. 2024). Lensing magnification allows astronomers to observe objects in the distant Universe, as was pointed out at the field's inception (Zwicky 1937). Dramatic examples have come from the James Webb Space Telescope (JWST), including a red supergiant star at 𝑧 ≈ 2 . 2 that appears to be magnified by a factor of several thousand due to its proximity to caustics in a cluster lens (Diego et al. 2023). Nearly all of these applications would benefit from a larger sample of strong lensing systems. To date, roughly 10 3 strong lensing systems have been discovered, most of which were identified at optical and infrared wavelengths (O/IR). Fortunately, upcoming wide-field imaging surveys such as Euclid and The Vera C. Rubin Observatory's Legacy Survey of Space and Time (Rubin-LSST) are each expected to detect as many as ∼ 10 5 strong lenses (Collett 2015). An early release of a 0.7 deg 2 field from Euclid has recently affirmed these forecasts (Barroso et al. 2024). The Nancy Grace Roman Space Telescope's 2000 square degree survey could find of order 20,000 strong lenses (Weiner et al. 2020), and many more if the proposed multi-epoch 4 𝜋 sr survey is carried out (Han et al. 2023). An increase in the total number of lenses by two-orders of magnitude will usher in a new era of strong lensing science. The first strongly lensed system ever discovered was co-detected at radio wavelengths (Walsh et al. 1979). The first survey for lenses, the Mit-Green Bank survey, was also in the radio (Bennett et al. 1986). Despite this, fewer than ∼ 100 radio lensing systems have been detected to date. This is due to the relatively small total number of known radio sources ( ∼ 10 7 ) and the lack of wide-field imaging surveys with ∼ arcsecond resolution. Both of those limitations will soon be overcome with the advent of next-generation radio survey telescopes. The 2000-antenna Deep Synoptic Array (DSA-2000) will detect over one billion radio sources with a deep redshift distribution, most of which will be star-forming radio galaxies (SFRG) or active galactic nuclei (AGN) (Hallinan et al. 2019). The DSA-2000 is expected to see first light in 2027 with key surveys running between 2028 and 2033. Its point-spread function (PSF) will be roughly 2' at the top of the 0.7-2 GHz radio band. A fifty-fold increase in the total radio source catalog is made possible by the DSA-2000's high survey speed, driven by a large field-of-view ( ∼ 10 deg 2 ) and high sensitivity (the expected system-equivalent flux density or 'SEFD' is just 2.5 Jy). Its cadenced all-sky survey will map out the 3 𝜋 sr above declination -30 · down to 500 nJy/beam root-mean square noise (Hallinan et al. 2019). While the nominal cadence of the continuum survey is four months, certain fields may be visited more regularly, presenting an opportunity to measure lensing time delays with better temporal sampling. The mid-frequency telescope for the square kilometre array (SKAMid) will consist of 197 fully steerable 13.5 m dishes (including the existing MeerKAT radio telescope), operating between 350 MHz and 15.4 GHz with a field-of-view of roughly 1 deg 2 at 1400 MHz. The end of its contruction is anticipated to be July 2029 1 . Although the SKA's lower survey speed will result in fewer sources than the DSA-2000, the wide frequency range and long maximum baseline (150 km) will enable high-resolution imaging, an asset to strong lensing studies (McKean et al. 2015). The Next Generation Very Large Array (ngVLA) is a planned interferometer with extraordinary sensitivity covering a wide range of frequencies (1.2-116 GHz) (Selina et al. 2018a). It will be able to resolve features at milliarcseconds scales. While its broad science goals did not require optimizing the instrument for mapping speed (Selina et al. 2018b), the ngVLA will be a world-class instrument for strong lensing science. Radio strong lensing offers distinct advantages to studies at shorter wavelengths and will complement O/IR imaging surveys. Starforming radio galaxies and AGN can be detected to great distances, boosting the mean optical depth of radio continuum sources. Obscuration by dust in lensing galaxies is not an issue at radio wavelengths, nor is the variable 'seeing' that impacts ground-based O/IR telescopes. Relatedly, the point-spread function (PSF) of a radio interferometer is directly determined by observing frequency and array configuration, allowing us to accurately forward model the instrument's response. In the limit of a large number of antennas and a filled aperture (a 'radio camera'), the deterministic PSF allows us to be more ambitious in image-plane deconvolution, enabling techniques such as superresolution (Connor et al. 2022). Radio telescopes also measure full polarization information, which is conserved under gravitational lensing (Greenfield et al. 1985; Dyer & Shaver 1992). The rotation measure (RM) of a lensed source allows one to measure the magnetic field and ionized gas properties of intervening halos (Mao et al. 2017). Finally, the larger emission regions of radio AGN render them less susceptible to microlensing by substructure in the lens, and may provide cleaner modelling of the deflector mass distribution (Birrer et al. 2024). This is critical for measuring 𝐻 0 via time-delay cosmography. There are of course several drawbacks to strong lensing studies in radio surveys, often precluding the use of standalone radio observations. For example, continuum sources will not contain redshift information, and multi-wavelength datasets or follow-up will be necessary for the majority of lensed systems at radio frequencies. Secondly, the morphology of radio lobes can mimic lensing arcs, exacerbating the already-challenging lens identification problem (consider, for example, head-tail galaxies 3C 465 or 3C 129 at redshift 2). In the case of the DSA-2000, the angular resolution is such that 75-90% of galaxy-galaxy lensing systems will be unresolved in the absence of superresolution. In this work we seek to study radio strong lensing in upcoming interferometric surveys, and develop strategies that alleviate the drawbacks of radio lensing. We focus on the DSA-2000, first by forecasting its strong lensing rates for different source classes. We compare this with a forecast for strong lensing discovery on the SKA. Next, due to the 2-3' PSF of the DSA-2000, we discuss methods for superresolving lensing candidates with modern computer vision techniques in order to increase the yield of strongly lensed systems. We then consider lensed time-variable and transient sources that the DSA-2000 will find and forecast constraints on 𝐻 0 that can be achieved with the appropriate follow-up, as well as other applications of strong lensing.\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"2 OPTICAL DEPTH CALCULATION & EXPECTED RATES\",\n \"content\": \"First, we offer a simple empirical forecast based on observational data and cosmological simulations, and then offer a more detailed forward model that accounts for different source classes and the deflector population.\",\n \"pages\": [\n 2\n ]\n },\n {\n \"title\": \"2.1 Empirical forecast\",\n \"content\": \"The Cosmic Lens All Sky Survey (CLASS) (Myers et al. 2003) sought to study a statistical sample of radio-loud gravitationally lensed systems. Despite significant advances in both radio instruments and lens-finding algorithms in the two decades since CLASS, the number of known radio lenses has not increased dramatically. We can use CLASS for a rough estimate of the number of lenses the DSA-2000 will find. CLASS found that a complete sample of 8,958 flat-spectrum radio point sources brighter than 30 mJy had a mean lensing optical depth of 1 . 5 + 0 . 5 -0 . 3 × 10 -3 (Browne et al. 2003). By targeting compact sources with flat spectra, many of their objects are quasars at high redshifts, comparable to typical redshifts of DSA-discovered sources, despite the difference in flux scales. Of the confirmed radio lenses, approximately 20% had angular separations larger than 2 arcseconds, which could be detected by the DSA-2000. Thus, a rough estimate indicates that for every 10,000 sources detected by the DSA-2000, several could be identified as strong lenses. If we assume a detection threshold of 10 𝜎 , the number of extragalactic DSA-2000 sources becomes roughly 5 × 10 8 (Section 2.3). This would result in O( 10 5 ) new galaxy-scale radio lenses, depending on the practical signal-to-noise threshold for candidate systems. Galaxy group and cluster scale lenses have been neglected in previous investigations of lensing statistics for upcoming surveys because they make up a smaller, but still significant, fraction of the total lenses. Additionally, lens modeling is more complicated in this regime. The dark matter halos of groups, in particular, are complex and not well-understood structures. From Oguri (2006), the number of group and cluster lenses are around 11% and 3% of the total galaxy lenses respectively (excluding sub-halo lensing). However, these systems make up a considerable portion of lens systems with angular separation of order 1', and almost all of the lenses at ≥ 10'. Because the PSF of the DSA-2000 is relatively large (2' at 2 GHz and 3.3' at 1.4 GHz), these lenses will make up a large fraction of the discoverable lensing systems and so must be accounted for in our investigation. A radio group/cluster lens survey as complete as CLASS does not exist, but we can make a simple empirical estimate from the literature. The total number of galaxy scale lenses should be O( 10 6 ) assuming 10 9 sources and the CLASS lensing optical depth. Taking the percentages of group/cluster lenses from Oguri (2006), we should see O( 10 5 ) groupscale lenses and O( 10 4 ) cluster scale lenses, almost all of which will have angular separation large enough to be detected in the DSA-2000. The first and largest survey for group-scale lenses was the Strong Lensing Legacy Survey (SL2S) (Cabanac, R. A. et al. 2007). SL2S found 13 strong lens systems in the mass range of groups in ∼ 100 deg 2 of the Canada France Hawaii Telescope Legacy Survey (CFHTLS), giving a rate of ∼ 0.13 deg -2 (Limousin, M. et al. 2009). A later publication identified at least 54 promising group lenses in ∼ 150 deg 2 of the CFHTLS (More et al. 2012). However, the mean redshift of the CFHTLS sources used in SL2S is ⟨ 𝑧 𝑠 ⟩ < 1, and we expect ⟨ 𝑧 𝑠 ⟩ ≈ 2 for the DSA-2000 (see section 2.3, figure 3 below) (Coupon, J. et al. 2009). At these redshifts, we expect the lensing optical depth to be roughly ∝ 𝑧 2 (see equation 31 of Oguri (2019); figure 2 below). This means that we would expect to see ∼ 10 + 6 -6 times more group lenses in the DSA-2000, for a total of ∼ 1 . 1 + 0 . 6 -0 . 9 × 10 5 . Similarly, the Red Sequence Cluster Survey found 8 cluster lenses in ∼ 90 deg 2 with shallower CFHT imaging, indicating ∼ 2 . 6 + 1 . 6 -1 . 6 × 10 4 cluster lenses in the DSA-2000 (Gladders et al. 2003). These estimates are of the same order of magnitude as expected from Oguri (2006). Alternatively, if we take the cross section for giant arcs from the simulations of Puchwein & Hilbert (2009), Fedeli, C. et al. (2010), and Mahdi et al. (2014), assume typical values of 𝑀 = 1 × 10 14 M ⊙ , 𝑧 𝑑 = 0 . 5, and 𝑧 𝑠 = 2, we can calculate a rough estimate of the cluster lensing optical depth at 𝑧 𝑠 = 2. The cross sections at these values are all on the order of 10 arcsec 2 . From Böhringer, Hans et al. (2017), the number density at 𝑀 = 1 × 10 14 M ⊙ is approximately ≳ 1 × 10 -6 Mpc -3 , and after multiplying by the differential coming volumeat 𝑧 𝑠 = 2andskycoverageoftheDSA-2000weget ≈ 5 × 10 -5 for all three simulations. When taking into account that this is the cross-section for giant arcs only, this means we should see several tens of thousands of cluster lenses in the DSA-2000, in good agreement with the previous estimates. Wealso make an empirical estimate of the rate of lensed transients in the DSA-2000. These objects are of particular interest for 𝐻 0 measurements but are much harder to forward model because of the lack of observational data. We can use data from the Very Large Array Sky Survey (VLASS), which is currently the most comparable survey to the DSA-2000, for a simple estimate (Lacy et al. 2020). The determined log rates of supernovae (SNe) and tidal disruption events (TDEs) in deg -2 yr -1 are -1.91 + 0 . 15 -0 . 16 and -2.85 + 0 . 28 -0 . 38 respectively (Dong et al. 2024 in prep). If we assume that the total number of observable sources scales as 𝑆 -1 . 5 min , the value for a Euclidean universe, and that the flux limit of VLASS is 0.7 mJy (10 𝜎 from Lacy et al. (2020)), the DSA-2000 should find ∼ 200 more sources of each type above 20 𝜇 Jy, which is a 10 𝜎 detection in a single epoch of the DSA-2000. We can also assume a lensing optical depth of 1 × 10 -4 and a magnification bias of 2, which are typical for the low redshifts at which we expect to detect transients. Because typical radio rise times of TDEs are on the order of 10 3 days the cadence of the DSA-2000, ∼ 1 / 3 yr, is fast enough to catch most TDEs (Cendes et al. 2023). Scaling for the sky coverage of the DSA-2000, and the cadence for SNe, gives a total yield of 5 + 2 -1 yr -1 and 0.6 + 0 . 5 -0 . 3 yr -1 lensed sources per year for SNe and TDEs respectively. Applying a similar estimate of the lensed rate of GRB afterglows using the predictions of Ghirlanda et al. (2013, 2014) gives a rate much less than 1 per year, and so we ignore them in the rest of this investigation. Further, Yao et al. (2023) find a volumetric rate of optically selected TDEs of 290 + 60 -130 Gpc -3 yr -1 . Cendes et al. (2023) find that ≈ 50% of optically selected TDEs emit in the radio on longer timescales. Becausethere are likely TDEs that emit in radio but not the optical, we take 50% of the optically selected rate to be a conservative estimate. The luminosities of the TDEs in the sample from Cendes et al. (2023) are ∼ 10 37 -10 39 ergs/s, but they note that many of these are likely a lower limit because most of the TDEs still had rising emission at the time of observation. Given this, the DSA-2000 should be able to detect this rate of TDEs out to 𝑧 ≈ 0 . 5, which, when combined with the same optical depth and magnification bias as above, gives 0 . 8 + 0 . 2 -0 . 4 yr -1 lensed TDEs. This number is in good agreement with the previous estimate, so in general, we expect to see about 1 lensed TDE per year in the DSA-2000. However, as noted before, this is a conservative estimate, and there is reason to believe that the actual rate of total and lensed TDEs in the DSA-2000 might be significantly higher. Additionally, because many of the TDEs will be bright in multiple epochs, the DSA-2000 could detect them well below the 20 𝜇 Jy limit, which would increase the rate by a large factor. The DSA-2000 will also discover tens of thousands of distant fast radio bursts (FRBs) (Petroff et al. 2019; Cordes & Chatterjee 2019), some of which will be strongly lensed (Connor & Ravi 2023). The key advantage to using FRBs for time-delay lensing is that their short duration and coherence allows for exceptional precision on the gravitational lensing time delay (Wucknitz et al. 2021). Radio telescopes can preserve phase information about the electromagnetic waveform at nanosecond sampling, which means microlensing signals can be searched for at ultrashort timescales (Leung et al. 2022; Kader et al. 2022). However, for cosmological lensing time delays longer than a pointing (i.e. deflectors more massive than ∼ 10 8 M ⊙ ), one needs to catch the lensed images by pointing at the same patch of sky when it arrives. We do not forecast lensed FRB rates here and point the reader to previous estimates (Connor & Ravi 2023).\",\n \"pages\": [\n 2,\n 3\n ]\n },\n {\n \"title\": \"2.2 Lens Model\",\n \"content\": \"Next, we build a forward model based on the lens and source distributions. Following Yue et al. (2022b), the lensing optical depth for a singular isothermal sphere (SIS) is where 𝑧 𝑑 is the redshift of the deflector, 𝑧 𝑠 is the redshift of the source, 𝜎 is the 1D velocity dispersion of the deflector, Φ ( 𝜎, 𝑧 𝑑 ) is the velocity dispersion function (VDF) of the deflectors, 𝑑 2 𝑉 𝑐 𝑑 Ω 𝑑𝑧 𝑑 = ( 1 + 𝑧 𝑑 ) 3 𝑐 𝑑𝑡 𝑑𝑧 𝑑 is the differential comoving volume, 𝜃 𝐸 ( 𝜎, 𝑧 𝑑 , 𝑧 𝑠 ) is the Einstein radius, and 𝐷 𝑑 is the angular diameter distance at the deflector redshift. The SIS model (or its elliptical generalization) has been shown to replicate the properties of early-type galaxies, which make up the majority of galaxy lenses (e.g. Gavazzi et al. (2007); Koopmans et al. (2009); Li et al. (2018)), and is a widely used model for galaxy strong lensing statistics (e.g. Oguri & Marshall (2010); Collett (2015); Yue et al. (2022a). For a SIS, the Eintein radius becomes where 𝐷 𝑑𝑠 is the angular diameter distance from the deflector to the source and 𝐷 𝑠 is the angular diameter distance from the observer to the source. We use the analytical VDF from Yue et al. (2022b) to model the galaxy deflector population, which they show to be in good agreement with observations. We include a sharp exponential cutoff on the VDF at 300 kms -1 which corresponds roughly to the transition between galaxies and galaxy groups. An important ingredient in the calculation is the magnification bias, which increases the rate of lensed sources by about a factor of two at low redshifts, or by orders of magnitude at high redshifts. The general magnification bias is where 𝜇 is the magnification of a lensed source, 𝑝 ( 𝜇 ) is the probability distribution of the magnification, 𝐿 min is the smallest observable luminosity, and 𝑁 ( > 𝐿 min ) is the number of sources brighter than 𝐿 min . For SIS, 𝜇 min = 2 for the total magnification of the multiple images. A SIS will produce only two multiple images; if the separation of the multiple lens images is large enough to be resolved then 𝑝 ( 𝜇 ) = 2 ( 𝜇 ± 1 ) 3 describes the magnification of the fainter (+) and brighter (-) image (Wyithe et al. 2001). If the lens is not resolved then 𝑝 ( 𝜇 ) = 8 𝜇 3 is the total magnification of both images. Combining 𝐵 and 𝜏 for the total fraction of lensed sources: where 𝐵 ' is the magnification bias of sources that are not lensed, which is assumed to be unity. While the SIS model describes deflector galaxies well, the mass distributions of galaxy groups and clusters behave differently. It has been traditionally thought that in the limit of large mass and large image separation, i.e. clusters, the distribution will be dominated by the dark matter halo, generally following the Navarro-Frenk-White (NFW) profile (Navarro et al. 1997). Recent works suggest that NFW profiles may not be the most accurate representation of large dark matter halos (e.g. Klypin et al. (2016)), but we use them here because they are a decent approximation and their lensing properties are well known. Regardless, for the mass range of groups, it is clear that some intermediate model between SIS and NFW is necessary (Williams et al. 1999; Oguri 2006; More et al. 2012). In a SIS the density 𝜌 ∝ 𝑟 -2 , while the NFW has 𝜌 ∝ 𝑟 -1 on small scales and 𝜌 ∝ 𝑟 -3 on larger scales. The shallower NFW profile has a significantly smaller cross-section than SIS. Oguri (2006) include the effects of baryon cooling and the large elliptical galaxies found at the center of most halos in their model, which will steepen the central density profile and increase the cross-section of halos (especially for groups). Without introducing these complexities, we can steepen the inner profile of halos by modeling them as Generalized NFW (GNFW) profiles. The GNFW is: where 𝑟 𝑠 is the scale radius and 𝜌 𝑠 is the scale density (Li & Ostriker 2002). When 𝛼 = 1 we have the standard NFW. When 𝛼 = 2 the profile resembles an SIS below the scale radius and when 1 < 𝛼 < 2 the inner profile is somewhere in between. Using weak lensing and stellar kinematics, Wang et al. (2023) find the inner dark matter density profiles of group (10 13 𝑀 ⊙ ≤ 𝑀 < 10 14 𝑀 ⊙ ) and cluster ( 𝑀 ≥ 10 14 𝑀 ⊙ ) halos are 1 . 82 + 0 . 15 -0 . 25 and 1 . 48 + 0 . 2 -0 . 41 respectively. Similarly, Mandelbaum et al. (2006) find that the total inner profiles of groups with mass ∼ 2 . 5 × 10 13 𝑀 ⊙ are decently described by a power-law of slope 𝛾 tot = 1 . 85. However, at ∼ 7 × 10 13 𝑀 ⊙ they are less steep than 1.85 and closer to the traditional NFW. Newman et al. (2013) find 𝛾 tot = 1 . 16 + 0 . 10 -0 . 12 for the inner profile of clusters with 𝑀 = 0 . 4 -2 × 10 15 𝑀 ⊙ , and Newman et al. (2015) find the total inner profile slope 𝛾 tot ≈ 1 . 7 while the dark matter slope 𝛼 ≈ 1 . 35 for galaxy groups with ⟨ 𝑀 ⟩ ≈ 10 14 𝑀 ⊙ . In this work, we use the GNFW profile with 𝛼 = 1 . 6 to model groups and 𝛼 = 1 . 2 to model clusters. The resulting GNFW profiles will capture the inner profile well, which is crucial for lensing, and also converge to the standard NFW profile at large radii where we expect the density to be dominated by dark matter (e.g. figures 3 and 4 of Wang et al. (2023)). The GNFW is completely parameterized by 𝛼 , its mass 𝑀 , and the concentration parameter, 𝑐 , from which 𝑟 𝑠 and 𝜌 𝑠 can be determined as in Li & Ostriker (2002). The mass and 𝑐 are correlated with some scatter, we use the relationship presented by Dutton & Macciò (2014) for 𝑐 200 . The scatter in 𝑐 is lognormal with 𝜎 log 𝑐 = 0 . 11. We include an additional factor of ( 2 -𝛼 ) as in Oguri et al. (2001) to correct for the generalized form of the NFW which ensures that the radius at which the logarithmic density slope becomes -2 is the same as in the standard NFW for all 𝛼 . The lensing power of the GNFW halo is very sensitive to 𝑐 and its scatter, so having an accurate relationship is important; early investigations tend to overestimate these values. The lens equation relates a position on the lens plane, 𝑥 , to a position on the source plane, 𝑦 , which for a GNFW profile is where and 𝑔 ( 𝑥, 𝛼 ) is the same as in Li & Ostriker (2002). The lens equation has three solutions inside the radial caustic, 𝑦 𝑐𝑟 , which are the multiple images produced by the GNFW profile. The cross-section for lensing is the area in the source plane in which multiple images will be produced, i.e. the area inside the radial caustic. This is approximated as: where 𝑦 𝑐𝑟 = -𝑦 ( 𝑥 𝑐𝑟 ) and 𝑥 𝑐𝑟 is the location of the minimum of equation 6 (Li & Ostriker 2002). The image separation is given by the separation between the outer two images, which can be approximated as where 𝑥 0 is the position of the tangential critical curve and can be found as the root of equation 6 (Li & Ostriker 2002). The Einstein radius is just half of the image separation, 𝜃 𝐸 = Δ 𝜃 / 2. In general, because our GNFW profiles are shallower than a SIS, they will be worse at producing multiple images but significantly better at magnifying. This is particularly sensitive to 𝑐 , which defines the shallowness of the GNFW profile. For our concentration parameters, especially at high redshift, we expect the magnifications from the GNFWs to be several times larger than from SIS (Wyithe et al. 2001). To calculate the magnification bias for a GNFW profile, we first determine the minimum magnification as (Li & Ostriker 2002): With 𝜇 min wecalculate B according to equation 3 with (Li & Ostriker 2002): The minimum magnification for both the first and second brightest image can be approximated as half of the total minimum magnification (Oguri et al. 2002). Substituting 𝜇 min = 2 into equation 11 recovers the total magnification distribution for SIS. To model the populations of galaxy groups and clusters we determine Mass Functions (MF) from the CosmoDC2 synthetic catalogue Korytov et al. (2019). For simplicity, we define a galaxy group as a system with mass 10 13 𝑀 ⊙ ≤ 𝑀 < 10 14 𝑀 ⊙ and a galaxy cluster as 𝑀 ≥ 10 14 𝑀 ⊙ . We find all dark matter halos in a redshift interval where the mass of the halo and its constituent galaxies fall in these ranges. We fit Schechter functions to the binned results. The fitted functions match the results of Böhringer, Hans et al. (2017) well at 𝑧 = 0 and show a realistic decline in the mass function towards higher redshift. To calculate the optical depth, we replace 𝜋𝜃 𝐸 ( 𝜎, 𝑧 𝑑 , 𝑧 𝑠 ) 2 𝐷 2 𝑑 in equation 1 with equation 8, Φ ( 𝜎, 𝑧 𝑑 ) with the CosmoDC2 MF, and integrate over 𝑀 instead of 𝜎 . While equation 4 gives the total fraction of lensed sightlines, we are more interested in the number of discoverable lenses. We define a lens as discoverable if its image separation Δ 𝜃 = 2 𝜃 𝐸 is larger than 2/3rds of the FWHM of the PSF of the survey instrument (as in Oguri & Marshall (2010); Yue et al. (2022a)) so that multiple images can be resolved. Further, for point-like sources (i.e. flat-spectrum sources and transients), we require the second faintest image to be detectable at 10 𝜎 confidence so that the lens can be identified. To account for this we integrate equation 1 from 𝜎 min (or equivalently 𝑀 min for groups and clusters) corresponding to 𝜃 𝐸 min and use the magnification bias of the fainter image. The second image from a SIS will often be demagnified, so the magnification bias of the fainter image is typically a bit less than 1, while the magnification bias of the fainter image from an NFW will be about half of the total. For SFRGs and SS-AGN, which in general have extended emission regions, the lensed images will be stretched into arcs and rings, and so for this case we require the brightest image to be detectable at 10 𝜎 confidence (using the magnification bias of the brightest image) because only one arc is needed to identify a lens. The image separation distribution predicted by our model and the lensing optical depths for galaxies, groups, and clusters are shown in Figures 1 and 2 respectively. We see that the optical depth increases rapidly at small redshift as expected, and approximately matches the estimates given in Section 2.1. The galaxy lens optical depth is smaller than the approximation from Oguri (2019). The discoverable optical depth, limited by the PSF of the DSA-2000, is significantly smaller than the total optical depth for galaxies but similar to the total for groups and clusters. The image separation distribution matches the results of Oguri (2006) well despite the simpler model. As expected, galaxies lie mostly at Δ 𝜃 ≈ 1', while groups and clusters correspond to Δ 𝜃 ≈ 10' and 10' ≲ Δ 𝜃 ≲ 100' respectively. The largest uncertainty is in the model of groups, as our simple model does not capture many of the complexities of halo profiles and is constant over the whole mass range. Further, varying 𝛼 between 1 and 2 creates a significant change in the calculated group optical depth. On the other hand, there is inherent uncertainty in all three models from the assumption of spherical symmetry. Using SIS instead of Singular Isothermal Ellipsoids (SIE) for galaxies will overestimate the galaxy lens population by ∼ 10%, which is corrected for in our final numbers (Yue et al. 2022b; Ferrami & Wyithe 2024). Introducing ellipticity will similarly reduce the cross-section of NFW profiles. We do not account for effects such as sub-halo lensing or line-ofsight structures and lens environments. It is shown in Oguri (2006) that a significant fraction of lenses, roughly half for groups and clusters and 10% for galaxies, come from lensing by sub-halos. The environment around these sub-halos boosts their lens capabilities and image separations. At least one of the CLASS lenses is caused by a galaxy within a galaxy group (Auger et al. 2007). In the context of cluster lensing, this is termed galaxy-galaxy strong lensing (GGSL) and it is reported that observational numbers of GGSL exceed expectations of the Λ CDM model (Meneghetti, Massimo et al. 2022; Tokayer et al. 2024). This means that we are likely underestimating the number of lenses in the image separation range of galaxy scale lenses. However, most of these will be below the discoverable limit of the DSA-2000. Further, it is well known that intervening masses along the line of sight between the deflector and the observer could increase the probability of multiple imaging by a significant fraction, especially at high source redshifts, which we also neglect in this study (Fleury et al. 2021). Because these higher-order aspects of the model are likely to increase the number of lenses, for the purpose of this paper it is safe to ignore them.\",\n \"pages\": [\n 3,\n 4,\n 5,\n 6\n ]\n },\n {\n \"title\": \"2.3 Source Populations\",\n \"content\": \"The other important piece of the model is the population of the sources. The DSA-2000 will map ∼ 30,000 deg 2 of the sky to a combined 𝜎 𝑛 = 500 nJy/beam rms noise (Hallinan et al. 2019). Matthews et al. (2021) presents detailed radio source counts from the MeerKat DEEP2 image, including direct source counts above 10 𝜇 Jy and statistical counts extrapolating below 10 𝜇 Jy. They find that for sources <10 𝜇 Jy, the differential source count is significantly flatter than Euclidean. Scaling for the 30,000 deg 2 footprint of DSA2000's continuum survey and integrating down to a minimum flux density 𝑆 = 2 . 5 𝜇 Jy for a 5 𝜎 𝑛 combined detection gives an expected total source count of 1 . 45 + 0 . 25 -0 . 10 × 10 9 , while for 𝑆 ≥ 5 𝜇 Jy = 10 𝜎 𝑛 that number is 8 . 6 + 1 . 1 -0 . 4 × 10 8 . To get redshift distributions for different source classes, we model the Star-Forming Radio Galaxy (SFRG) and Active Galactic Nucleus (AGN) populations using the luminosity functions (LF) from the Tiered Radio Extra-galactic Continuum Simulation (T-RECS) (Bonaldi et al. 2018). These have been shown to match observations out to high redshift. Following Bonaldi et al. (2018), we divide the AGN into three sub populations: Flat Spectrum Radio Quasars (FSRQ), BL-Laceratae (BLLac), and Steep Spectrum AGN (SSAGN). We expect SFRGs to make up roughly 95% of all of the sources in any synoptic radio survey, and BLLacs to be the rarest source. From the LF we calculate the number of sources in a redshift interval as where Ω survey is the sky coverage of the survey in steradians ( ≈ 3 𝜋 for the DSA-2000) and Φ ( 𝐿, 𝑧 ) is the LF of the source population. 𝐿 min ( 𝑧 ) = 4 𝜋𝐷 2 𝐿 ( 1 + 𝑧 ) 1 + 𝛼 𝑆 min is the minimum observable luminosity at a redshift 𝑧 , where 𝐷 𝐿 is the luminosity distance, 𝛼 is the spectral index of the source population, 𝑆 min is the 10 𝜎 𝑛 flux sensitivity of the DSA-2000, and the factor ( 1 + 𝑧 ) -( 1 + 𝛼 ) is the standard cosmological radio K-correction. We scaled the total number of sources in the LF model to match the observational prediction from the DEEP2 image given above. In addition to SFRGs and AGN, we build a model for radio SNe. To date, only ∼ 100 radio core-collapse supernovae (ccSNe) have been detected and the first Type Ia radio supernova was detected last year (Bietenholz et al. 2021; Kool et al. 2023). To model the expected rate of radio ccSNe we follow Lien et al. (2011). Because ccSNe are shortlived, the rate of ccSNe is closely related to the cosmic Star Formation Rate (SFR), which has been observational measured to high redshift and accuracy. We use the ccSNe radio luminosity distribution of Bietenholz et al. (2021), which is more complete than the one in Lien et al. (2011). The best-fit distribution is log-normal and incorporates all known radio ccSNe as well as radio non-detections of supernovae. Bietenholz et al. (2021) note that their data is likely biased towards radio detections, finding that 30% of all ccSNe are detected in the radio. We divide the total number of radio ccSNe in the model by 3 to get a more conservative 10% as in Lien et al. (2011). From the luminosity distribution, we determine the number of radio ccSNe at each redshift above the flux limit of the DSA-2000. Because the SNe observations used in Bietenholz et al. (2021) have frequencies ranging from 2-10 GHz, there is uncertainty when converting to 1.4 GHz. We correct 𝐿 min 6 GHz ( 𝑧 ) = 𝐿 min 1.4 GHz GLYPH<16> 6 1 . 4 GLYPH<17> 𝛼 , where we assume 𝛼 = -0 . 7 for the synchrotron emission of radio ccSNe. Further, because radio ccSNe have a mean rise time of log 10 ( 𝑡 rise ) = 1 . 7 days (Bietenholz et al. 2021), we use the single epoch flux limit of the DSA-2000 and account for the DSA-2000's cadence. The rms noise for a single epoch is 𝜎 𝑛 = 2 𝜇 Jy/beam, so a 10 𝜎 𝑛 detection is 20 𝜇 Jy. The cadence will be ∼ 4 months, so conservatively we will detect 1/3 of the radio ccSNe per year. None of the known radio ccSNe have luminosities above 10 29 erg s -1 Hz -1 , but given the wide spread we expect that the highest luminosities could be several times larger. Because of this we set an exponential cutoff at 10 30 erg s -1 Hz -1 , which we can expect to see out to about 𝑧 ≈ 1 . 5 in a single epoch of the DSA-2000. The predicted redshift distributions of SFRGs, AGN, and ccSNe from our model are shown in Figure 3 and compared to the expected distribution of Rubin-LSST sources. As expected, the total source count is dominated by SFRGs until high redshift. Blazars, meaning both FSRQs and BLLacs, are the AGN that will be most useful for time delay measurements, which is explored in Section 5.1. These are the most common sources at very high redshift. We expect the DSA2000 to probe much higher redshifts than the Rubin-LSST because radio emission is much less susceptible to intervening gas or seeing. This is one of the reasons that the DSA-2000 will be effective at lens finding. The total number of expected sources of each type in a DSA-2000 all-sky survey with a 10 𝜎 𝑛 detection are listed in the first row of Table 1.\",\n \"pages\": [\n 6\n ]\n },\n {\n \"title\": \"2.4 Expected rates\",\n \"content\": \"The final results of our model are shown in Table 1. The results agree quite well with the empirical estimates given in Section 2.1. As expected from CLASS there are roughly 10 6 total galaxy scale lenses contained in the DSA-2000 all-sky survey, less than 10% of which will be discoverable. The total number of group and cluster lenses are about 10% and 2% of the galaxy number respectively, which is consistent with Oguri (2006). Roughly half of the group and cluster lenses will be discoverable, leading to them making up about a third of the total discoverable lenses, a significant fraction. Yue et al. (2022a) find that the Rubin-LSST will discover about 2000 lensed QSOs; our number is similar. This is a result of the competing mechanisms of the DSA-2000 detecting many more at higher redshifts and simultaneously being able to discover a smaller fraction because of the PSF size. The main purpose of Yue et al. (2022b) is to investigate the discrepancy between the fraction of lensed high redshift ( 𝑧 𝑠 ≳ 5) quasars in observations ( ∼ 0 . 2%) and previous predictions ( ∼ 4%).Yueetal.(2022b)find ∼ 0 . 4 -0 . 8%with their model, and we have ∼ 0 . 3%, which is consistent. The number of discoverable lensed ccSNe per year in the DSA-2000 from the full model is about 2, which is of the same order of magnitude as the empirical VLASS estimate. Oguri & Marshall (2010) conclude that Rubin-LSST will find roughly 8 SNe per year. We expect there to be less lensed SNe at radio wavelengths because not all SNe emit in the radio and the radio emission is weaker than in other wavelengths. The redshift distribution of the deflectors and sources is shown in Figure 4. Almost all deflectors are at 𝑧 𝑑 < 3 and most are at 𝑧 𝑑 ≈ 1, consistent with Oguri & Marshall (2010); Collett (2015); Yue et al. (2022a,b). This justifies the use of the CosmoDC2 catalog, which is limited to 𝑧 < 3, to determine the MF of groups and clusters. We also run the simulation for the survey specs of the SKA-mid and VLASS. The main differences are the sensitivity and resolution. The expected rms noise of the SKA-mid at 1.4 GHz will be 𝜎 𝑛 = 2 𝜇 Jy/beam, and the PSF size will be about 0.4' (Braun et al. 2019). From the DEEP2 source counts, the SKA-mid should find about 3 × 10 8 sources above 10 𝜎 𝑛 . With this, our model predicts that the SKA-mid should see 1.8 ± 0.7 × 10 5 , 1.8 ± 1.2 × 10 4 , and 2.7 ± 1.1 × 10 3 galaxy, group, and cluster lenses, given the same discoverability limit defined in Section 2.2. Despite less depth leading to around a third of the total galaxy lenses, the PSF of the SKA-mid is such that it will resolve over half of them, resulting in about twice as many discoverable galaxy lenses as in the DSA-2000. The SKA-mid will find fewer group and cluster scale lenses than the DSA-2000 because sensitivity is more important for discovering these systems than angular resolution. In addition, we expect about 14 lensed radio ccSNe per year in an SKA-mid all-sky survey. McKean et al. (2015) estimate ∼ 3 × 10 5 lenses in an SKA-mid all-sky survey with 𝜎 𝑛 = 3 𝜇 Jy/beam and a lens detection limit of 15 𝜎 𝑛 and >0.3'. This is slightly more optimistic than our forecast, but broadly consistent. VLASS has 𝜎 𝑛 = 70 𝜇 Jy/beam combined and an angular resolution of 2.5' (Lacy et al. 2020). The DEEP2 source counts indicate that we should expect ∼ 5 × 10 6 sources above 10 𝜎 𝑛 in VLASS (Lacy et al. (2020) estimate 5.3 × 10 6 above 5 𝜎 𝑛 ). With these numbers, we estimate that there should be about 300 discoverable lenses in the full sky survey. As VLASS enters its epoch 3, many of these lenses are likely already in the data (several have recently been identified (Martinez et al. 2024)).\",\n \"pages\": [\n 6,\n 7\n ]\n },\n {\n \"title\": \"3.1 Superresolution\",\n \"content\": \"In the past decade, tremendous progress has been made by the computer vision community with respect to the classical ill-posed inverse problems. These include deblurring (Zhang et al. 2022), deconvolution and superresolution (Alzubaidi et al. 2021), and image inpainting (Yu et al. 2018b). Nearly all of these advances were borne out of the deep learning revolution, as efficient neural network architectures and training strategies have enabled powerful learning-based tools for machine vision. Astronomy naturally lends itself to these tools because sparse sampling and ill-posedness arise in many astronomical imaging and reconstruction contexts, especially in interferometry. Several groups have begun developing machine learning methods for interferometric image reconstruction (Connor et al. 2022; Aghabiglou et al. 2024; Mars et al. 2024). One reason this approach is suitable for radio astronomy is that the PSF of an array is given deterministically by the spatial distribution of antennas and the observing frequency: The on-sky response of an interferometer is the 2D Fourier Transform of its sampling in UV-space (the aperture plane). Therefore, prior physical knowledge of the PSF can be incorporated in the model either via training data (Connor et al. 2022) or fed directly to the network itself (Mars et al. 2024). This is not the case in ground- based optical astronomy where 'seeing' and complex optics mean the PSF is not known a priori. Asasimple demonstration, we use the super-resolution and imageplane deconvolution method POLISH to show how strong lenses with image separations below the PSF scale can be recovered with machine learning. POLISH is a supervised machine learning model that learns the mapping between the true sky and observed images (the 'dirty image' in radio parlance) (Connor et al. 2022). It uses the Wide Activation for Efficient and Accurate Image Super-Resolution (WDSR) architecture (Yu et al. 2018a), but any super-resolution neural network could be swapped in (e.g., a standard U-Net or the Efficient Super-Resolution Transformer (Lim et al. 2017)). Wehave trained a POLISH network on forward-modelled synthetic data using a DSA-2000 PSF averaged over the whole band, resulting in an angular resolution of ∼ 3.3'. This is worse resolution than the top of the band, where many strong lenses could be found, but we offer this as a toy example. The sky model is described in Connor et al. (2022), with the addition of strongly lensed galaxies in both the training and validation set. In Figure 5 we show an example validation image that contains multiple lenses. Lensed systems that are undetected in dirty image, including both arcs (left inset panels) and Einstein rings (right inset panels), are identifiable in the POLISH reconstruction. In the future, we plan to develop super-resolution methods explicitly for lens finding, but we take this as a promising sign that the DSA-2000 will be able to find systems with 𝜃 𝐸 ≳ 0 . 5 '' .\",\n \"pages\": [\n 7,\n 8\n ]\n },\n {\n \"title\": \"3.2 Identification\",\n \"content\": \"The first major hurdle for strong lensing science with DSA-2000 and other surveys is identification. The large number of sources produced by these surveys makes visual inspection by experts impossible. Despite much work on automated lens discovery (see Lemon et al. (2023) for a review), little effort has been spent on identification in the radio. On the one hand, radio lens images are made cleaner by the lack of radio emission from the massive quiescent galaxies that make up the deflector population. However, the large, extended lobes of radio galaxies can easily be misidentified as lensing features. Further, color and photometric redshift information from UV and optical surveys have traditionally played a role in lens identification, despite being a foreground to the morphology of the lensed source. Catalog-level searches, where large cuts are made based on certain features before a smaller subset is visually inspected, have been successful in the past. CLASS selected for flat spectrum radio sources ( 𝛼 > -0 . 5) (Myers et al. 2003). This had the advantage of picking out only blazars, whose radio emission is dominated by a small compact core, thus eliminating any confusion with intrinsic structure. A similar strategy would likely be effective in the DSA-2000. However, dedicated follow up of all of these sources, as in the ∼ 11,000 blazars in the initial CLASS sample, will be impossible. Similarly, a visual inspection of even a small subset of the O( 10 7 ) blazars expected in the DSA-2000 is ambitious (hence the need for intermediate automated steps in the identification pipeline). Still, targeting flat spectrum sources could increase the yield of any search, and spectral information from the whole DSA-2000 band will be useful for determining whether components are different sources or multiple images. Jackson & Browne (2006) show that combining astrometric data from radio and optical surveys can significantly improve the efficiency of lens searches. If the optical emission is dominated by the deflector galaxy then there will exist an offset between the centroid position of the optical and radio sources in a lens system. This offset was exploited to find lenses with separation down to ∼ 1' even with the poor 5' resolution of the Faint Images of the Radio Sky at Twenty-one centimeters (FIRST) survey. They predict that this effect will become more efficient at lower radio fluxes because the optical flux is more likely to be dominated by the defelctor galaxy. Given the sensitivity of the DSA-2000 and its significant overlap with current and planned large surveys in other wavelengths, this could prove to be a very effective way of overcoming the limitations of its PSF. Regardless, exploiting the extra information from overlap with other surveys will be crucial for any lens search in the radio. Machine learning models have been successfully used to discover many new candidate lens systems. Convolutional Neural Networks (CNNs) in particular are effective at classifying astronomical images. The main disadvantage with CNNs and other supervised learners is the need for large realistic training sets. Using CNNs on simulated data for the International LOFAR Telescope (ILT) at 150 MHz, Rezaei et al. (2022) are able to recover over 90% of galaxy-size lenses with a false-positive rate of only 0.008%. They find that a strong 20 𝜎 detection and large separation 𝜃 𝐸 ≥ 3 / 2 beam size (stricter than the discoverable limit imposed in Section 2.2) are necessary for reliable identification with the CNN. Incorporating superresolution models into the lens-finding routine would significantly increase the yield of such a CNN, especially at high S/N. Resolving lenses down to 𝜃 𝐸 ≈ 0 . 5' with would enable the same level of accuracy as in Rezaei et al. (2022). Simulating accurate DSA-2000 lenses and training models for identifying them is a goal of our future work. The SKA-mid will already have sufficient resolution for these purposes, making CNNs an attractive option for lens identification in its all-sky survey. The main limitation of any ML-based approach to group/cluster lens finding is the difficulty of creating realistic training sets. The irregularity of group/cluster lensing potentials may require building a forward-modelled training set from ray-tracing in large cosmological magnetohydrodynamical simulations such as TNG-Cluster, which should have a heterogenous population of massive clusters (Nelson et al. 2024). More generally, a realistic training set of strong lenses across scales could be produced with radio source samples from T-RECS and a deflector population drawn from cosmological simulations.\",\n \"pages\": [\n 8,\n 9\n ]\n },\n {\n \"title\": \"4 MULTIWAVELENGTH SYNERGIES\",\n \"content\": \"Acquiring redshifts for lensed galaxies and deflectors is critical for the application of strong lensing to cosmology and astrophysics. We consider here how well we could do in the absence of targeted followup observations, capitalizing on the suite of large survey telescopes that will be operating in the mid- to late-2020s. DSA-2000 spectroscopic HI survey In addition to the cadenced allsky survey, the DSA-2000 will undertake a large spectroscopic HI survey over the full visible sky (Dec > -30 · ) (Hallinan et al. 2019). This is expected to produce ∼ 10 6 HI galaxies at 𝑧 < 1. Most deflectors will be massive elliptical galaxies with limited 21 cm emission, but a sub-set of early-type galaxies with 𝑛 𝐻 ≥ 10 20 cm -2 𝑧 -2 𝑑 will be detected. Some late-type deflector galaxies will have enough cold gas to obtain a redshift. Additionally, 21 cm absorption in the inner CGM of deflector galaxies may be present in the lensed source's continuum spectrum, providing a measurement of cold gas along different sightlines of the same CGM (see Rudie et al. (2019) for a similar application in lensed quasar systems). Thus, some deflector galaxies will have a spectroscopic redshift without any external O/IR data. Of the ∼ 10 6 HI galaxies detected by the DSA-2000 in emission, we find that only a handful will be strongly lensed, due to their shallow redshift distribution. Spectroscopic radio observations will not provide a significant portion of redshifts required for our purposes, but may be a minority of interesting deflectors. The Dark Energy Spectroscopic Instrument (DESI) will produce redshifts of ∼ 40 million galaxies and quasars (DESI Collaboration et al. 2016; Dey et al. 2019). Its 3 yr public data release is expected before the DSA-2000's first light. DESI's Bright Galaxy Survey (BGS) targets galaxies with 𝑟 -band magnitudes brighter than 19.5. For the luminous red galaxies (LRGs), emission line galaxies (ELGs), and quasars, the depth will be 𝑟 ≈ 23. Each class of target will have a different redshift distribution, but together they will provide a rich catalog of spectroscopic redshifts for deflectors in the lensing systems discovered by DSA-2000. The BGS and LRG samples will be over-represented as deflector galaxies, whereas the ELG and QSO samples may make up a small fraction of lensed DSA-2000 sources. ELGs, by definition, have strong emission lines indicative of star formation, which produce synchrotron radiation detectable by radio surveys. Some of these will be strongly lensed star-forming radio galaxies with 𝑆 𝜈 above 5 𝜇 Jy. Still, we do not expect more than a small minority of the radio continuum lenses to be detected in DESI due to the relative number of total sources. SPHEREx is a near-infrared space mission expected to launch before Spring 2025. It will map the full 4 𝜋 sr sky with 6 arcsecond pixels and 96 color bands, producing a catalog of hundreds of millions of galaxies with 'spectro-photometric redshifts' (Doré et al. 2018). Although its PSF it too large to discover strong lenses directly, the galaxy catalog will be valuable for obtaining reliable redshifts of both deflectors and lensed galaxies. The Rubin Observatory will find billions of galaxies in its Legacy Survey of Space and Time (LSST) (Ivezić et al. 2019). Most of these galaxies will be at 𝑧 𝑠 ≲ 1 . 5 (Collaboration et al. 2021), providing an excellent deflector catalog for the DSA-2000 lens candidates. In the DSA-2000/LSST overlapping footprint between -30 and +30 Declination, we find that a significant fraction of deflectors and several thousand lensed sources in the DSA-2000 sample will have photometric redshifts from the Rubin Observatory. Roman The High Latitude Spectroscopic Survey (HLSS) is expected to detect 10 million galaxies in 𝐻𝛼 between 1 < 𝑧 < 2and ∼ 2 million [OIII] galaxies at 𝑧 = 2 -3 using the near-IR grism and wide-field camera (Wang et al. 2022). Roman's 2,000 deg 2 footprint will be fully mapped by DSA-2000. With a dust-attenuated 𝐻𝛼 flux higher than 10 -16 erg s -1 cm -2 , several million of these galaxies will be detectable at GHz frequencies at > 5 𝜇 Jy (Murphy et al. 2011). Assuming a mean lensing optical depth of 5 × 10 -4 , this suggests that O( 10 3 ) will be strongly lensed and detected by both the DSA-2000 and Roman , the latter providing 0.1' resolution and a source redshift. Euclid is a visible and near-infrared space telescope that launched in July 2023 (Collaboration et al. 2024). Its large photometric survey will find over one billion galaxies, with a spectroscopic survey that will obtain accurate redshifts for ∼ 30 million galaxies mostly at 0 . 9 < 𝑧 < 1 . 8. Euclid and the DSA-2000 share over 10,000 deg 2 of sky, providing an excellent sample of photometric and spectroscopic redshifts of lens systems.\",\n \"pages\": [\n 9\n ]\n },\n {\n \"title\": \"5 STRONG LENSING APPLICATIONS\",\n \"content\": \"A key limitation of strong lensing science is the small number of known systems, especially at radio wavelengths. We discuss the impact that the large expected number of radio lenses and their multiwavelength counterparts will have on several important applications of strong lensing.\",\n \"pages\": [\n 9\n ]\n },\n {\n \"title\": \"5.1 Time-delay cosmography & 𝐻 0\",\n \"content\": \"Light from multiple images in a gravitational lens will reach the observer at different times. For continuum sources, we do not observe the difference in arrival times, but transients or time-variable sources allow us to measure the time delay. This time delay is due to a difference in path lengths and gravitational time dilation near the deflector. As such, the time delay encodes information about the geometry of the universe, and is inversely proportional to 𝐻 0 (Oguri 2019). Much work has been dedicated to measuring these time delays and using them to constrain 𝐻 0 (Treu & Marshall 2016; Birrer et al. 2024). The H0LiCOW project has measured 𝐻 0 to 2.4%, which is independent of and competitive with state-of-the-art 𝐻 0 constraints (Wong et al. 2019). A large increase in the number of measurable time-delay systems will allow even tighter constraints and could help settle the current Hubble tension. We estimate the number of lensed variable AGN that will be useful for measuring 𝐻 0 in the DSA-2000. To first order, all blazars (FSRQs and BLLacs) are inherently variable, ∼ 10 3 of which will be discoverable as lenses in DSA-2000 data (Table 1). However, several factors complicate our ability to use them for time delay measurements; we would like to know how many will be in a \\\"gold sample\\\" for which the time-delay can be reliably determined. To date, the most extensive study of radio AGN variability is the OVRO 40 m blazar monitoring campaign. Since 2007 this program has monitored around 1500 blazars with a cadence of two weeks (Richards et al. 2011). Richards et al. (2011) provides distributions of the intrinsic modulation index, a measure of the relative intrinsic variability of the AGN, for FSRQs and BLLacs. The DSA-2000 will measure flux variability at the level of a few percent, but to be conservative we consider candidates with ≥ 10% variability. We also want sources with a strong detection, so we chose sources with flux ≥ 20 𝜎 𝑛 = 10 𝜇 Jy. The OVRO blazars are monitored at 15 GHz but we expect lower variability at 1.4 GHz. Fan, J. H. et al. (2007) find that blazars are about 20% less variable at 4.8 GHz than 14.5 GHz. Sotnikova et al. (2024) find that radio AGN at 22.3 GHz are about 35% more variable than at 2.3 GHz, although there is little difference between 11.2 GHz and 2.3 GHz. We assume that the mean modulation index drops by 40% from the OVRO 40 m data to 1.4 GHz for the DSA-2000. Further, for multiple images, we need at least two of the images to be bright enough to measure the time delay so we use the magnification bias of the fainter image. With these restrictions, the OVRO data, and our lens model, we estimate that there will be 68 ± 30 lensed FSRQs and 23 ± 10 lensed BLLacs with sufficient brightness, variability, and image separation to measure time delays. The same analysis for the SKA-mid gives 120 ± 50 and 43 ± 16 lensed FSRQs and BLLacs. Of the lensed AGN, the most useful systems for time delay analysis will be those that have four or more images because they allow for multiple constraints on the time delay. A SIS is only capable of creating two images and a spherical NFW can make three, but factors such as ellipticity, irregularities in the lens potential, and external environments of real halos will allow a number of these systems to make four or more images. The fraction of quad galaxy-QSO lenses predicted in Rubin-LSST is ∼ 10-15% (Oguri & Marshall 2010; Yue et al. 2022a). However, of the statistically complete CLASS sample, 5 out of 13 lensed blazars were quadruply imaged and one had six images (B1933+503 has two sets of quadruply imaged sources Sykes et al. (1998)). Therefore we can expect 100-400 quadruply-lensed blazars in the DSA-2000 all-sky survey and about twice as many in the SKA-mid, roughly 10% of which may be used in practice for measuring 𝐻 0 . We note that quad systems often have higher external shear due to their preponderance of group or cluster environments, which results in an added systematic in lens modeling (Holder & Schechter 2003). Wong et al. (2019) reach a 2.4% measurement of 𝐻 0 with a sample of 6 lensed quasars, and predict that around 40 lenses are needed to constrain 𝐻 0 to within 1%. Jee et al. (2016) argue that with a well-defined sample of quadruply imaged quasars, and combining time-delay distances and velocity dispersion measurements to estimate angular diameter distances within 5%, 𝐻 0 can be measured to the same precision as Planck with as few as 10 systems. Napier et al. (2023) reach a 10% measurement with three galaxy clusters, and estimate that roughly 50 will be needed for a sub 1% constraint, assuming that modeling uncertainties can be reduced in coming years. While we expect uncertainty to decrease significantly with a larger sample of lensed quasars, there are several systematics that must be kept under control for an accurate measurement. Namely, an accurate redshift measurement is necessary, ideally of both the source and deflector, as well as precise measurements of the time delay, complete sampling over many cycles of the time delay to negate contamination from micro-lensing, accurate lens models, and constraints on any line of sight structures. If we assume that all of the lenses in our gold sample can be modeled to the accuracy of the H0LiCOW sample - a roughly 6% combined uncertainty on the time delay, mass model, and line of sight contribution for each lens - then a sample of ∼ 100 lensed quasars would give about a 0.6% constraint on 𝐻 0 . This is of course not possible with the DSA-2000 or SKA-mid alone but will require dedicated follow-up at multiple wavelengths. As discussed in Section 4, we expect that a significant minority of lenses should have redshifts available in other concurrent public surveys. Interesting subsamples of lenses will require targeted follow-up for both mass modeling and spectroscopic redshifts. The DSA-2000's cadenced all-sky survey will image each field above -30 Dec once every 4 months. Typical time delays for galaxy, group, and cluster lenses in our model are a month or less, several months, and a year or more respectively (Oguri et al. 2002). Special systems could therefore be visited with a higher cadence, similar to how pulsar fields will be visited regularly for timing experiments. So far we have only considered lensed blazars for time delay measurements, which are desirable because of their characteristic variability and compact emission regions. Lensed transients will also be useful for time delay measurements, provided that they are detected early and monitored closely. The variability of radio emission from TDEs is too slow to make them useful for this application (on-axis jetted TDEs can have shorter rise times, but they are significantly rarer such that we do not expect any lensed jetted TDEs in upcoming radio surveys), but SNe have faster rise times. We expect O( 1 ) lensed radio ccSNe per year in the DSA-2000 and O( 10 ) in the SKA-mid that can be used for 𝐻 0 measurements.\",\n \"pages\": [\n 10\n ]\n },\n {\n \"title\": \"5.2 Dark Matter Structure\",\n \"content\": \"Gravitational lensing has been very useful for determining the structure of galaxies, groups, and clusters at cosmological distances and distinguishing between dark matter models (see Vegetti et al. (2024); Natarajan et al. (2024) for a review). The best of these studies use high-resolution HST images, but radio observations also played a pivotal early role. Cohn et al. (2001) used the 10-image CLASS lens B1933+503, which provides many more constraints than usual, to determine the density profile of the deflecting galaxy, finding it close to isothermal. Wucknitz et al. (2004) find the same for B0218+357. Dalal & Kochanek (2002) used radio observations of flux ratio anomalies between multiple images to constrain the mass fraction of dark matter substructure to 2%. Radio observations are especially suited for this type of study because they are less susceptible to micro-lensing (Koopmans et al. 2003; Kochanek & Dalal 2004). Despite this, dark matter studies with radio lenses have been limited for many years by the small sample size (Vegetti et al. 2024). Because resolution is a key factor in the ability to accurately model mass distributions or detect substructure, the DSA-2000 alone will not be able to use lens systems to study dark matter. Its main utility will come from identifying huge numbers of systems for detailed followup. The extraordinary resolution of the ngVLA on a large sample of lenses selected by the DSA-2000 will revolutionize this field. In the more immediate future, the SKA will also provide enough resolution for these studies in their overlapping sky coverage. Another exciting application of lensing is using a central image to study small scales near the centers of galaxies (Treu 2010; McKean et al. 2015; Shajib et al. 2024). A SIS lens only forms two images, but for realistic galaxy profiles a highly demagnified image is expected to form at the center of the image configuration. This image will be sensitive to the mass contained within very small distances of the deflector's center, and so can be used to study supermassive black holes at cosmological distances. This is especially suited for radio wavelengths because in many cases the deflector will be radio-quiet allowing the center image to be detected. Because the center image is highly demagnified, only sensitive radio surveys such as the DSA2000 will be able to reliably identify them. At least one such central image has been detected in the radio with brightness ∼ 0 . 8 mJy at 8.46 GHz (Winn et al. 2004), so we can likely expect O( 10 2 ) or more with the DSA-2000. A main challenge will be disentangling the faint emission of the central image from the other images. For groups and clusters, whose inner density profiles are shallower than isothermal, central images are more common and often less demagnified. The very inner profiles of groups/clusters can thus be studied, which are hard to constrain with images near the Einstein radius alone. These central images will be easier to identify because the other images will be farther away (although many clusters have large central elliptical galaxies emitting in the radio that will obscure the image).\",\n \"pages\": [\n 10,\n 11\n ]\n },\n {\n \"title\": \"5.3 Polarization\",\n \"content\": \"Measuring full Stokes information of gravitationally lensed signals offers insight into the source plane because polarization is not affected by gravitational potentials. This is an advantage of radio surveys, where polarization information is often stored and Faraday rotation measure (RM) can be measured. This enables us to study propagation effects along different sightlines for lensed objects. For example, Mao et al. (2017) used the VLA to measure the polarization properties of two lensed images of CLASS B1152+199, finding a large difference in RM ( + 9 . 7 ± 0 . 5 rad m -2 for image ' 𝐴 ' and + 517 ± 3 rad m -2 for image ' 𝐵 '), which is due to the magnetized plasma in the lens galaxy's interstellar medium. The DSA-2000 will have the ability to measure full polarization information for any source it detects, with a maximum |RM| of roughly 10 4 rad m -2 . Compact sources such as Blazars (and especially BL Lac objects) can show significant polarization fractions at 1 GHz. Assuming a 50 𝜎 detection threshold in total power, we estimate that the DSA-2000 could detect Stokes Q and U for 5-10 million AGN in its 5 year continuum survey. Using the empirical CLASS lensing optical depth and image separation cut, we take 𝜏 𝐴𝐺𝑁 ≈ 5 × 10 -4 . We estimate that the DSA-2000 could find O( 10 3 ) strong lenses for which polarization properties could be used to model the lens distribution and study magnetic fields at cosmological distances.\",\n \"pages\": [\n 11\n ]\n },\n {\n \"title\": \"5.4 Other applications\",\n \"content\": \"In addition to constraining 𝐻 0 , time-delay measurements are weakly sensitive to other cosmological parameters, such as Ω 𝑚 , Ω Λ , and the dark energy equation of state parameter 𝑤 (Natarajan et al. 2024). Cluster lenses are used for studying dark energy as well (Macciò 2005; Gilmore & Natarajan 2009; Jullo et al. 2010). A large influx in measured time delays and observed cluster lenses with the DSA2000 will allow better constraints on these cosmological parameters. Lenses are also commonly used as 'nature's telescopes'. The magnification of background sources allows very distant galaxies to be studied even by smaller telescopes (Jackson 2011). This is especially true for group/cluster lenses, where we expect the magnifications to be larger (Robertson et al. 2020). We expect to see O( 10 4 ) lenses with 𝑧 𝑠 > 5 in the DSA-2000 (figure 4), many of which will be group and cluster lenses due to their high magnifications. These high redshift sources will provide insight into the population of radio sources in the early universe. Lensing statistics can also be used to test cosmological models. The abundance of giant arcs in cluster lenses was a topic of much debate in the past few decades (Meneghetti et al. 2013). Similarly, the rate of GGSL events in clusters may be in tension with the Λ CDM(Meneghetti et al. 2020; Meneghetti, Massimo et al. 2022; Tokayer et al. 2024). These studies require a large sample of lenses for accurate statistics, which is only now becoming possible with large surveys such as the DSA-2000.\",\n \"pages\": [\n 11\n ]\n },\n {\n \"title\": \"6 CONCLUSIONS\",\n \"content\": \"In this paper, we forecast expected strong lensing rates in the upcoming DSA-2000 and SKA-mid wide-field radio surveys. We first provide empirical estimates based on previous surveys and simulations, and then develop a detailed forward model that accounts for the source and deflector populations, finding them to be in good agreement. Notably, we model the expected number of galaxy group and cluster scale lenses because these systems will be easily discovered due to their wide angular separations. We find that both the DSA-2000 and the SKA-mid will discover roughly 10 5 strong lens systems. We discuss strategies for identifying these lenses in the data, which will all benefit from emerging superresolution techniques. Finally, we discuss the scientific application of the huge numbers of lenses that will be discovered by these surveys. One of the most exciting applications is 𝐻 0 cosmography with variable and transient sources. The DSA-2000 and SKA-mid will discover about 100 and 200 lensed flat spectrum AGN with >10% variability respectively, as well as about O( 1 ) and O( 10 ) lensed transients per year. With dedicated multi-wavelength follow up these systems could be used to constrain 𝐻 0 to within 1%. The new lens systems will also be useful for studying the distribution of dark matter at cosmological distances, among other applications.\",\n \"pages\": [\n 11\n ]\n },\n {\n \"title\": \"ACKNOWLEDGEMENTS\",\n \"content\": \"Weare grateful to Schmidt Sciences for supporting Samuel McCarty as a Summer Undergraduate Research Fellow at Caltech. We thank Kim-Vy Tran, Tony Readhead, and Tommaso Treu for helpful conversations on strong lensing, as well as Paul Schechter for insights into quad systems.\",\n \"pages\": [\n 11\n ]\n },\n {\n \"title\": \"DATA AVAILABILITY\",\n \"content\": \"We have placed a reproduction package on the public GitHub repository available at https://github.com/smmccrty/ radiolensing .\",\n \"pages\": [\n 12\n ]\n },\n {\n \"title\": \"REFERENCES\",\n \"content\": \"R., Motta V., 2012, The Astrophysical Journal, 749, 38 This paper has been typeset from a T E X/L A T E X file prepared by the author.\",\n \"pages\": [\n 12,\n 13\n ]\n }\n]"}}},{"rowIdx":4983,"cells":{"bibcode":{"kind":"string","value":"2024arXiv241204059S"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/2412.04059.pdf"},"content":{"kind":"string","value":"\nThe puzzling long GRB 191019A: Evidence for Kilonova Light\nG. Stratta , 1, 2, 3, 4 A. M. Nicuesa Guelbenzu , 5 S. Klose , 5 A. Rossi , 6 P. Singh, 1, 6 E. Palazzi , 6 C. Guidorzi, 7, 8, 6 A. Camisasca, 9 S. Bernuzzi, 10 A. Rau, 11 M. Bulla, 7, 8, 12 F. Ragosta, 13, 14 E. Maiorano, 6 and 15\nD. Paris\n1 Institut fur Theoretische Physik, Goethe Universitat, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany 2 INAF - Osservatorio di Astrofisica e Scienza dello Spazio, via Piero Gobetti 93/3, 40129 Bologna, Italy 3 Istituto di Astrofisica e Planetologia Spaziali, via Fosso del Cavaliere 100, I-00133 Roma, Italy 4 Istituto Nazionale di Fisica Nucleare-Roma 1, Piazzale Aldo Moro 2, I-00185 Roma, Italy 5 Thuringer Landessternwarte Tautenburg, Sternwarte 5, 07778 Tautenburg, Germany 6 INAF - Osservatorio di Astrofisica e Scienza dello Spazio, via Piero Gobetti 93/3, 40129 Bologna, Italy 7 Department of Physics and Earth Science, University of Ferrara, Via Saragat 1, 44122 Ferrara, Italy 8 INFN - Sezione di Ferrara, Via Saragat 1, 44122 Ferrara, Italy 9 Astronomical Observatory of the Autonomous Region of the Aosta Valley (OAVdA), Loc. Lignan 39, I-11020, Nus (Aosta Valley), Italy 10 Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universitat Jena, 07743, Jena, Germany 11 Max-Planck-Institut fur Extraterrestrische Physik, Giessenbachstraße 1, 85748 Garching, Germany 12 INAF, Osservatorio Astronomico d'Abruzzo, via Mentore Maggini snc, 64100 Teramo, Italy 13 Dipartimento di Fisica 'Ettore Pancini', Universit'a di Napoli Federico II, Via Cinthia 9, 80126 Naples, Italy 14 INAF - Osservatorio Astronomico di Capodimonte, Via Moiariello 16, I-80131 Naples, Italy 15 INAF - Osservatorio Astronomico di Roma, via Frascati 33, 00040 Monte Porzio Catone, Italy\nABSTRACT\nGRB 191019A was a long Gamma-ray burst (GRB) lasting ∼ 65 s and, as such, originally thought to be linked to a core-collapse supernova. However, even though follow-up observations identified the optical counterpart close to the bright nucleus of a nearby ancient galaxy ( z =0.248), no associated supernova was found. This led to the suggestion that the burst was caused by the merger of two compact stellar objects, likely in a dense circumnuclear environment. By using a recently developed diagnostic tool based on prompt emission temporal properties, we noticed that GRB 191019A falls among those long GRBs which are associated with compact mergers and with evidence of kilonova light. We thus re-analyzed unpublished GROND multi-color ( g ' r ' i ' z ' JHK s ) data obtained between 0.4 and 15 days post trigger. Image subtraction confirmed the optical counterpart in all four optical bands, with GROND tracking its fading until 1.5 days post-burst. Incorporating publicly available Swift -XRT data, a joint fit of an afterglow plus a kilonova model revealed a better match than an afterglow-only scenario. The resulting kilonova properties resemble those of AT2017gfo associated with the binary neutron star merger GW170817, with a total ejected mass of ∼ 0 . 06 M ⊙ . Contrary to previous findings inferring a high-density circumburst environment ( n 0 ∼ 10 7 -8 cm -3 ), our analysis finds standard conditions ( n 0 ∼ 1 cm -3 ), suggesting the long duration of GRB 191019A was intrinsic rather than due to jet interaction with a dense external medium.\nKeywords: Gamma-ray bursts - Compact objects\n1. INTRODUCTION\nThe empirical classification of gamma-ray bursts (GRBs) into two classes according to the bimodal burst duration and hardness distribution (see, e.g., Kouveliotou et al. 1993), has been interpreted as the evidence of two different progenitor channels. Indeed, it was found that the vast majority of sufficiently nearby long GRBs are found to be spatially and temporally coincident with core-collapse (CC) supernovae (SNe) and hosted in star-forming galaxies. On the other hand, short GRBs lack any associated SN and are typically located in the outer regions of early-type galaxies, suggesting a progenitor belonging to an old stellar population, formally consistent with compact binary mergers. This\npicture was confirmed by the association of the short GRB 170817A with the gravitational wave event GW170817 that was compatible with a binary neutron star (NS-NS) merger (e.g., Abbott et al. 2017). From the very same event and thanks to its spatial proximity (40 Mpc), the first robust evidence of a kilonova (AT2017gfo) was obtained (e.g., Coulter et al. 2017), confirming the predicted thermal emission from matter released and heated during a NS-NS merger (e.g., Li & Paczy'nski 1998). In the last decade, kilonova signatures were observed in a number of past short GRBs that were sufficiently nearby, as for instance GRB 130603B ( z =0.3565, Tanvir et al. 2013), GRB 150101B ( z =0.134, Troja et al. 2018), GRB 160821B ( z =0.1613, Troja et al. 2019), though with much lower significance than AT2017gfo due to the larger distances of these events (see, e.g., Troja 2023 for a review).\nIn recent years, the standard long and short-GRB progenitor paradigm has been blurred by mounting evidence of the existence of long GRBs with no associated CC-SN but with evidence of a kilonova component, more consistent with a compact binary merger progenitor. Two striking examples are the long GRB 211211A at redshift z = 0 . 076 (350 Mpc; e.g., Troja et al. 2018; Mei et al. 2022; Rastinejad et al. 2022; Yang et al. 2022; Gompertz et al. 2023) and GRB 230307A at z = 0 . 0646 (300 Mpc; e.g., Levan et al. 2023), for which a kilonova was clearly identified in the optical afterglow. In addition, evidence of short GRBs showing massive star collapse properties were also found (e.g., Ahumada et al. 2021; Rossi et al. 2022) further contributed to dismantle the standard GRB classification scheme. These results suggest that some of the past GRB progenitor classifications based on the burst duration and hardness only, have to be revisited. This holds in particular for GRB 191019A.\nGRB 191019A was discovered (Simpson et al. 2019) with the Burst Alert Telescope (BAT; Barthelmy et al. 2005) on board the Neil Gehrels Swift Observatory (Gehrels et al. 2004). It showed a complex burst light curve with multiple peaks with a duration T 90 = 64 . 6 ± 4 . 5 s. Its 15-150 keV spectrum could be well modeled with a power law with a photon index of 2 . 25 ± 0 . 05 and a fluence of 10 -5 erg cm -2 (Krimm et al. 2019) putting this source among the soft/long GRBs. Follow-up observations with the Swift X-ray telescope (XRT; Burrows et al. 2005) robustly identified an uncatalogued X-ray source (Evans et al. 2019). Rapid follow-up observations with the Zeiss-1000 1-m telescope of Tien Shan Astronomical Observatory revealed only one optical source within the XRT error circle with R = 18 . 37 ± 0 . 03 mag at T 0 +14 min (Reva et al. 2019). The position of this source was coincident with a catalogued object quoted to be fainter than the observed one, leading to the interpretation this is the host galaxy of GRB 191019A with the afterglow on top of it (Reva et al. 2019). The Zadko telescope (Coward et al. 2017) observed GRB 191019A field at similar epochs and confirmed the object reported by Reva et al. (2019) with R =18.92 mag, with no significant flux variation within 1.5 hours (Gendre et al. 2019). Similar claims of a lack of flux variations were provided by other teams performing early-epoch observations (Fynbo et al. 2019, Nicuesa Guelbenzu 2019, Perley et al. 2019a, Zhu et al. 2019). However, late observations performed with the Nordic Optical Telescope (NOT) at T 0 +3.25 days were analyzed with image subtraction methods and revealed that the source was fading, confirming the optical transient plus host identification (Perley et al. 2019b).\nSpectroscopic analysis of the host galaxy showed several absorption lines at redshift z = 0 . 248 and a spectral energy distribution consistent with a galaxy dominated by an old stellar population ( > 1 Gyr), with a star formation rate (SFR) of 0 . 06 ± 0 . 03 M ⊙ yr -1 , a stellar mass of 3 × 10 10 M ⊙ , and small dust extinction of A V = 0 . 19 ± 0 . 08 mag (Levan et al. 2023). The measured SFR is at odds with the typical high values inferred in long GRB hosts (but see Rossi et al. 2014). Further hints of anomalies for the long GRB 191019A came from the absence of an associated SN, despite the low redshift. Deep limits in g, r and z obtained between 2 and 73 days with the NOT and optical imaging with the Hubble Space Telescope at 30 and 184 days, put strong constraints on any SN emission up to > 20 times less luminous than SN1998bw (Levan et al. 2023). All these properties are at odds with a massive-star origin of GRB 191019A, while formally consistent with a compact object binary merger progenitor. Though, the deep limits obtained with the NOT Telescope at > 2 days could not confirm the presence of an early kilonova emission (Levan et al. 2023).\nAn interesting feature characterizing GRB 191019A is its projected distance from the host galaxy center of ≲ 100 pc, the closest distance among all known short GRBs to their corresponding galactic nucleus (Levan et al. 2023). Indeed, compact binary progenitors formed in stellar binary systems are thought to have large kick velocities acquired during the CC-SN phase of the binary components, and for this reason short GRBs are typically found in the outskirts of their host galaxies, with large offsets from the center (e.g., Berger 2014; Fong et al. 2022; O'Connor et al. 2022). As noted by Levan et al. (2023), the proximity of GRB 191019A to the host galaxy center suggests that the possible progenitor compact binary system could have formed through dynamical encounters, which are thought to be favored in the dense gaseous environment of supermassive black hole surrounding disks through kinetic energy dissipation ('gas-capture' binary formation channel, Tagawa et al. 2020). An exciting consequence of this scenario is that dense environments\ncould also alter the prompt emission properties of a short GRB, making it longer and softer (Lazzati et al. 2023). This possibility was explored for GRB 191019A and it was found compatible with an environmental density of the order of 10 7 to 10 8 cm -3 (Lazzati et al. 2023). Starting from Lazzati et al.'s conclusions, Wang et al. (2024) investigated the interaction of a possible kilonova ejecta from GRB 191019A with a dense circumstellar medium (CSM) that could be detected years after the merger. Their model predicts that in a very dense environment, the smaller the kilonova ejected mass, the fainter is the radioactively-powered luminosity but the higher is the contribution from kilonova-CSM interaction at late times. No evidence of kilonova ejecta-CSM interaction was found for GRB 191019A so far, and a possible detection in the future would require an ejected mass less than 2 × 10 -5 M ⊙ (Wang et al. 2024).\nHere we further investigate the properties of GRB 1901910A and the nature of the optical transient that followed the burst, with the goal to find arguments in favour or against its possible compact merger origin. We use the recently developed GRB prompt emission minimum variability (MVT) criterion (Camisasca et al. 2023) to explore the properties of the burst, and perform a complete reanalysis of the light curve of the optical/X-ray transient with particular emphasis on so far not analyzed GROND (Greiner et al. 2008) observations starting 10.3 hours post burst (Nicuesa Guelbenzu 2019). We use the sophisticated Nuclear - Multi-Messenger Astronomy (NMMA, Dietrich et al. 2020; Pang et al. 2023) software package which allows for afterglow and kilonova joint Bayesian inference, to explore if the multi-color light curve of the optical transient can be understood as powered by afterglow emission only or if a kilonova component was present too.\nThroughout this paper, we adopt a flat cosmological model with H 0 =67.4 km s -1 Mpc -1 , Ω M =0.315, and Ω Λ =0.685 (Planck Collaboration et al. 2020). For these parameters a redshift of z =0.248 (Fynbo et al. 2019) corresponds to a luminosity distance of d L = 1.29 Gpc, 1 arcsec corresponds to 4.03 kpc projected distance, and distance modulus is m -M = 40 . 55 mag. The Milky Way reddening E ( B -V ) along the line of sight towards the source is between 0.03 and 0.04 mag (Schlegel et al. 1998; Schlafly & Finkbeiner 2011). We present the observations and data reduction in Sect. 2, and our results in Sect. 3. In our data analysis we corrected the apparent magnitudes for the Galactic reddening and adopted a host-galaxy visual extinction along the line of sight of A host V = 0.06 mag (Levan et al. 2023). Discussion and conclusions are presented in Sect. 4.\n2. OBSERVATIONS AND DATA REDUCTION\n2.1. GROND early-time Observations\nWith a duration of about 65 seconds, GRB 191019A was a classical long GRB. At a redshift of z = 0 . 248, a rising SN 1998bw component was thus expected to be detectable with GROND within about two weeks after the GRB trigger. Therefore, following earlier efforts with GROND to explore GRB-SN light curves (Olivares E. et al. 2012a, 2015; Greiner et al. 2015; Kann et al. 2019a; Klose et al. 2019) we performed follow-up observations of the field during several epochs between 0.4 (Nicuesa Guelbenzu 2019) and 15.4 days post GRB trigger (Table 1). Observations were then terminated since no evidence for SN light was found.\nGROND data were reduced in a standard fashion (bias subtraction, flat fielding, co-adding; Kruhler et al. 2008; Yolda¸s et al. 2008), based on numerical routines in IRAF (Image Reduction and Analysis Facility; Tody 1993).\n2.2. LBT late-time Observations\nLate-time observations of the field were performed with the Large Binocular Telescope (LBT) using the two twin LBC instruments equipped with the Sloan filters g ' r ' i ' z ' on October 20, 2023. These data were reduced using the data reduction pipeline developed at INAF - Osservatorio Astronomico di Roma (Fontana et al. 2014) which includes bias subtraction and flat-fielding, bad pixel and cosmic ray masking, astrometric calibration, and coaddition.\nOn the LBT images, we measure the AB magnitudes for the host given in Table 4 (see Appendix A). Within the errors, these values are consistent with those measured with GROND in epoch 4 ( T -T 0 = 7 . 4756 days). Based on these data we conclude that 7 days post burst the transient did not contribute anymore to the observed flux in g ' r ' i ' z ' in a measurable quantity.\n2.3. Image subtraction and photometry\nIn order to search for a transient hidden by the relatively bright host galaxy, we performed image subtraction on the GROND images using HOTPANTS (Becker 2015). HOTPANTS convolves the template and the source images to the same\nStratta et al.\n
\n\n
Table 1. Measured AB magnitudes and upper limits of the transient.
\n
\nNote -Column (2) provides the midtime of the observation, in units of days from burst trigger. Magnitudes are not corrected for Galactic foreground extinction. The photometry for epochs 1a to 3 was obtained after image subtraction against epoch 4 (see § 2.3 for details). On the contrary, the 3 σ upper limits for epoch 4 to 6 have been measured directly, without image subtraction.\npoint spread function (PSF) and photometric scale. As a template image we used the GROND 4th-epoch observations (Table 1) since these images have the best seeing ( ∼ 1 . '' 0). We aligned the template and the source images by means of wcsremap 1 .\nThe Gaussian input parameters in HOTPANTS were calculated following Becker (2015) considering the case in which the template FWHM is smaller than the source FWHM. In this way, the template images were smoothed to the seeing of the corresponding source images. In addition, we fixed several other parameters for HOTPANTS following Hu et al. (2022), but let free to vary the number of each region's stamps, the convolution kernel half width, and the size of Gaussians which compose the kernel. In doing so, we selected those input parameters that minimized the background noise on the residual image in a star-free region close to the position of the host galaxy. All resulting residual images were then photometrically calibrated with respect to the source images (see below). In the case of a non-detection of the transient the upper limit we used was measured locally on the subtracted image (Table 1). We have confirmed our results using also the 5th GROND epoch as template (seeing ∼ 1 . '' 3), although we obtained a lower SNR.\nWe double checked the detections using the late LBT images (about 4 years or 1460 days after the burst trigger, see Sect. 2.2 and Table 4 in Appendix A). The g ' i ' z ' detections are confirmed for epochs 1a and 2, but with lower SNR likely due to the additional plate scaling, and a general worse PSF shape due to not perfect collimation. We have not checked the r ' -band since the above mentioned problem was worse in this filter, although it did not compromise the aperture photometry in Table 4. For this filter we have instead used (as template) a stack of the best Gemini-S images 2 obtained at 58-63 days after the burst (see Levan et al. 2023), and we obtained consistent results. In other words, the results obtained with GROND could not be improved. Finally, to better constrain the late transient decay, we performed image subtraction on the Gemini r -band image at 11.4 days, using the later Gemini images as template (see above), and obtained for the optical transient an upper limit of r > 25 . 8 AB mag.\nApparent magnitudes were obtained by performing aperture photometry using DAOPHOT and APPHOT under PyRAF/IRAF with increasing apertures up to 4 times the FWHM and by PSF photometry with the DAOPHOT and ALLSTAR tasks of IRAF. PSF-fitting was used to measure the magnitudes of the transient after image subtraction. In the optical bands the data were calibrated using Pan-STARRS (Chambers et al. 2016), in the NIR bands using 2MASS (Skrutskie et al. 2006). Central wavelengths of the GROND filter bands are listed in Rossi et al. (2011).\n3. RESULTS\n3.1. Application of the prompt emission minimum variability criterion\nAccording to Camisasca et al. (2023), GRB prompt emission minimum variability timescale (MVT) represents a diagnostic to infer the progenitor nature of a GRB. By estimating MVT from the full width at half maximum of the shortest, statistically significant peak in the prompt emission light curve (FWHM min ) of hundreds of GRBs, it was found that high variability (low FWHM min values) typically belongs to compact binary mergers. Interestingly, this result was found to be independent of the burst duration and for this reason it is an important probe for the nature of peculiar long GRBs with properties more similar to those of short GRBs.\nBy analyzing the results from the Swift GRB sample analyzed by Camisasca et al. (2023), we checked which long GRBs sufficiently nearby for a possible kilonova detection, i.e. at z < 0 . 5, and without any associated SN, had an FWHM min compatible with compact binary merger values, and we found that GRB 191019A was satisfying our criteria. Specifically, for this GRB an MVT of FWHM min = 0 . 196 +0 . 068 -0 . 05 s has been measured, which is compatible with the values typically found for GRBs associated with compact binary mergers (Fig. 1). In particular, given the long burst duration of GRB 191019A ( T 90 = 64 . 35 ± 4 . 35 s), its position in the FWHM min vs T 90 diagram lies among the short GRBs showing a soft 'Extended Emission' (SEE, e.g., Norris & Bonnell 2006; Minaev et al. 2010). Interestingly, two famous long GRBs for which a kilonova component was found in the optical afterglow, namely GRB 211211A (Rastinejad et al. 2022; Troja et al. 2022; Yang et al. 2022; Gompertz et al. 2023) and GRB 230307A (e.g., Levan et al. 2023), lie in the same region of the diagram of GRB 191019A (Fig. 1).\nThe SEE distribution in the T 90 -FWHM min plane clearly overlaps with the distribution of long GRBs associated with SNe, but at the same time the centroids of the two distributions are well separated: interestingly, GRB 191019A lies in the middle of the SEE region and in the outskirts of the SN-associated cases. This property alone cannot be taken as compelling evidence that GRB 191019A behaves like a SEE, but suggests a probable SEE-like nature than a collapsar one. These findings further support past interpretation that also GRB 191019A originated from a compact binary merger (e.g., Levan et al. 2023), and address the possible presence of a kilonova component, similarly to GRB 211211A and GRB 230307A.\n3.2. Re-analysis of the light curve of the Transient that followed the burst\n3.2.1. X-rays\nAdopting a power-law spectral energy distribution (SED), the best fit of the Swift -XRT 0.3-10.0 keV spectrum, computed by the automatic algorithm of the UK Swift Science Data Center (UKSSDC) Swift -XRT GRB Spectrum Repository 3 in the 3 . 2 -32 . 4 ks temporal window, provides a photon index of Γ = 2 . 0 +0 . 4 -0 . 3 and an intrinsic equivalent hydrogen column density of N H , z = 1 . 2 +1 . 6 -1 . 2 × 10 21 cm -2 in addition to a Galactic N H = 3 . 28 × 10 20 cm -2 . We obtained from the UKSSDC Burst Analyzer tool (Evans et al. 2010), the absorption-corrected X-ray fluxes in the energy range 0.3-10.0 keV, and we converted them into 1 keV flux densities by assuming a power law spectrum and the corresponding photon index provided by the Burst Analyzer in each temporal bin of the light curve. The flux density evolution in the X-ray band is compatible with a power-law decay 4 with α X = 1 . 44 ± 0 . 14.\n3.2.2. Optical bands\nWe detected the optical transient during the first three GROND observations of the field, up to 1.5 days post burst (Table 1, Fig. 2). The transient has coordinates R.A., decl. (J2000) = 22:40:05.861, -17:19:42.77 ( ± 0.3 arcsec), measured on the combined g ' r ' i ' residual images of the 2nd-epoch observations (Fig. 3). Within the error, these coordinates agree with what has been reported by Perley et al. (2019b).\nBased on a joint fit of the griz -band magnitudes provided by Levan et al. (2023) and GROND's first visit of the field (epoch 1a in Table 1), we find that between both observing runs the optical transient was fading with a decay slope α opt = 1 . 43 ± 0 . 07. Within the errors, α X ∼ α opt up to T 0 +0 . 4 days (epoch 1a in Table 1), suggesting a common cooling regime of the electrons radiating optical and X-ray synchrotron light (see Fig. 4). This finding is compatible with results by (Levan et al. 2023) who showed that a single power-law SED from X-rays to the optical bands can describe the observations at T 0 +0 . 21 days.\n\n\n
Figure 1. Burst duration versus prompt emission minimum temporal variability of 1291 GRBs detected with Swift -BAT (from January 2005 to July 2022) and the corresponding marginal distributions. The sample includes 78 Short GRBs (blue) and 24 short GRBs with 'Extended Emission' (SEE, green). Gold points are Long GRBs with an associated SN (adapted from Camisasca et al. 2023). The location of GRB 191019A is indicated by a green star in the top-left side of the diagram where SEE are located, together with two long GRBs better identified with being originated by compact binary coalescence progenitors (GRB 211211A and GRB 230307A).
\n\nHowever, GROND's following observing runs 1b and 2 (Table 1) clearly show a significant flattening of the light curve, with a new decay slope of α opt ∼ 0 . 5 ± 0 . 1 (Fig. 4). The rescaled X-ray power-law model underpredicts the flux in the optical bands, with a > 3 σ deviation observed at T 0 +1 . 5 days. Even by assuming a shallower optical decay index (i.e., α opt = α X -0 . 25), as for the case where the synchrotron cooling frequency lies between the X-ray and the optical regime (e.g., Sari et al. 1998), the discrepancy at late times ( > 1 day) persists (Fig. 4).\n\n\n
Figure 2. Multi-band light curve of the transient that followed GRB 191019A. Optical and NIR data are from Levan et al. (2023) (from their table 4 of the supplementary material, where flux densities corrected for dust extinction and host galaxy substracted are quoted: we consider here only those with relative error ≤ 30%) and from this work (larger markers) where GROND fluxes have been computed after image subtraction of the host galaxy observed at epoch 4 with GROND, and corrected for both Milky Way ( A V =0.10 mag) and host-galaxy dust extinction ( A V =0.06 mag). The vertical dashed lines mark the first three observation epochs quoted in Table 1. X-ray data were taken from the Swift -XRT GRB light curve Repository (Evans et al. 2007, 2009) as unabsorbed 0.3-10 keV fluxes computed with the Burst Analyzer (see Evans et al. 2010), and converted to 1 keV flux densities by assuming a power law spectrum and the corresponding photon index provided in each temporal bin of the light curve.
\n\n\n\n
Figure 3. From left to right, GROND r ' -band observations obtained during epochs #2 at 1.5 days, #4 at 7.5 days and the residual of the image subtraction (epoch #2 minus epoch #4) using HOTPANTS . The optical transient is indicated by the circle.
\n\nTo quantify the null hypothesis that a power law model cannot fit the data and test for chromaticity of flux evolution, we considered the early data by Levan et al. (2023) and our GROND detections (i.e. from ∼ 0.2 to 1.5 days) for the two filters with best temporal sampling (i.e. r ' and g ' ): a simple power law model does not provide an acceptable fit, with probability of having the observed data set, given the null hypothesis (p-values), of 1 . 7 × 10 -5 and 4 × 10 -4 , respectively. By assuming a broken power law, with a steep-to-shallow behavior and a temporal break at 0.44 days, we find a significant improvement of the fit, with p-values of 0 . 79 and 0 . 73 in the r ' and g ' bands, respectively. The best\nfit broken power laws show that the flux evolution after the break is slightly shallower in the redder filter, suggesting a chromatic behaviour.\nWe note that this peculiar steep-to-shallow optical light curve morphology is not unique, it was already observed in a small subset of GRBs in the early years of the Swift era (see, e.g., Kann et al. 2010), although with light curve breaks on average much earlier than what we observe for GRB 191019A. For instance, the optical light curve of GRB 090102 (Gendre et al. 2010) is similar to GRB 191019A but the break time was ∼ 0 . 1 hr (rest frame) after the burst.\nIn the standard fireball model, which fairly accounts for late afterglow phenomenology as forward-shock radiation, a steep-to-shallow light curve is not envisioned (e.g., Sari et al. 1998). Such morphology could be produced by the presence of a reverse shock component dominating the forward shock at early times. However, reverse shock is expected on timescales of the order of minutes (e.g., Nakar & Piran 2004), in contrast with what we observe for GRB 191019A. Alternatively, energy injection into the forward shock could originate a light curve flattening. The spin-down radiation from a newly born magnetar (e.g., Zhang & M'esz'aros 2001) or a slightly off-axis structured jet (e.g., Beniamini et al. 2020) are plausible energy sources that are often invoked to explain the X-ray afterglow 'plateaus' observed in GRBs, and that might have an optical counterpart. In the case of GRB 191019A the lack of any evidence of a shallow phase in X-rays cast some doubts against the energy injection scenario, however.\n3.2.3. Constraints on Supernova Light\nIf GRB 191019A had been a classical long burst, then for a redshift of z =0.248 a supernova (SN) component was expected to be detectable with GROND. Observational constraints on SN light were first reported by Levan et al. (2023) based on HST observations 30 and 184 days post GRB trigger. No evidence for transient emission was found ( g > 24 , r > 23 . 5, and z > 22 mag).\nFigure 5 shows the upper limits we can set on any SN that followed GRB 191019A in comparison to the r ' -band light curves of 13 GRB-SNe that have been observed with GROND between 2007 and 2014 (see Table 2 in Klose et al. 2019). These events cover the whole GRB-SN luminosity distribution of well sampled GRB-SNe, from GRB 100316D/SN 2010bh (Olivares E. et al. 2012a), one of the faintest GRB-SN ever detected (e.g., Melandri et al. 2014; Cano et al. 2017; Dainotti et al. 2022), to GRB 111209A / SN 2011kl (Kann et al. 2019a), the most luminous GRB-SN ever observed.\nA visual inspection of Figure 5 shows that all light curves fall into a strip with a width of about 2-3 mag, which is limited by the very faint GRB 100316D/SN 2010bh and the very bright GRB 111209/SN 2011kl. The strongest limit we can set stamps from the time span between 7 and 12 days post burst and is well below the identified GRB-SN magnitude strip: if a CC-SN followed GRB 191019A, in r ' it was at least 3 mag less luminous than SN 1998bw and 2 mag fainter than SN 2010bh. According to the archived Gemini data (Sect. 2.3), the constraint on the luminosity is even stronger, > 4 mag in r at 11.4 days post burst.\nIn conclusion, even if we take into account that the time evolution of GRB-SNe light curves show a certain parameter range (characterized by stretch factor), the SN limit we can set provides a constraint on the entire SN light curves. In this respect, any SN related to GRB 191019A must have been less luminous than each GRB-SN (see Appendix B).\n3.3. Joint Afterglow and Kilonova modelling\nSince the flattening observed in the optical light curve at late times (epochs 1b and 2 in Table 1) cannot be explained by the presence of a slowly rising SN (Sect. 3.2.3), and probably not by a reverse shock or energy injection either (Sect. 3.2.2), we explore now the possibility that the source of this additional radiation component was kilonova light.\nGiven the relatively small redshift of GRB 191019A ( z = 0 . 248), a kilonova with a luminosity similar to AT2017gfo is expected to lie within the discovery space of a 2-m telescope, if not hidden by an intrinsic low luminosity. Therefore, following the procedure described in Rossi et al. (2020), we compared our data at T 0 +1 . 5 days in each filter, with the flux of the kilonova AT2017gfo (Coulter et al. 2017) associated with the NS-NS merger GW170817 (Abbott et al. 2017), shifted to the redshift of GRB 191019A. We find that an emission component similar to AT2017gfo but ∼ 4 times more luminous could reproduce the observations in the i ' and r ' bands, while our g ' -band flux seems to be brighter (Fig. 4). Apparently, the presence of a kilonova associated with GRB 191019A is overall compatible with the general properties of the observed optical-NIR emission at 0.4 and 1.5 days, although with slightly different brightness than AT2017gfo 5 .\n\n\n
Figure 4. X-ray versus optical light curve, where larger circles indicate GROND data from this work and the other data is taken from the literature (as in Figure 2). The blue dashed line shows the X-ray flux best fit power-law model with a decay index α X = 1 . 44 ± 0 . 14. This model is then scaled to match the optical data (dashed lines) together with a shallower power-law decay with α X -0 . 25 (dotted lines). The two power-law models are expected if the optical emission is produced by electrons in the same cooling regime as those emitting in X-rays (dashed line) or in a different cooling regime (dotted line). The solid lines indicate the flux of a AT2017gfo-like kilonova if it were at the redshift of GRB 191019A and ∼ 4 times more luminous. For visual purposes only, the r ' , i ' , and z ' -band data have been rescaled by a factor of 3, 5, and 7 respectively.
\n\nMotivated by the results obtained with our previous simplistic approach, we then explored the possible presence of a kilonova by comparing our multi-band dataset ( g ' , r ' , i ' , z ' , and X-ray) with a much more sophisticated model that takes into account simultaneously the presence of an afterglow and a kilonova component through a joint fit. For this purpose, we exploited the Nuclear - Multi-Messenger Astronomy (NMMA v0.2.0) framework that allows to estimate best-fit parameters with a bayesian inference method (Dietrich et al. 2020; Pang et al. 2023).\nFor the afterglow modelling, NMMA uses the Afterglowpy python module (Ryan et al. 2020). Afterglowpy allows to model GRB afterglow light curves and spectra by taking into account the possible effects due to a complex jet structure and an off-axis observer. We here assumed a Gaussian profile for the jet structure, and that the whole\n\n\n
Figure 5. The r ' -band light curves of all 13 GRB-SNe observed by GROND between 2007 and 2014 which are listed in Table 2 in Klose et al. (2019). All light curves have been calculated based on the luminosity and stretch factors found for these SNe and have been corrected for Galactic and host galaxy extinction along the line of sight. They have been red-shifted to z=0.248 taking into account the appropriate cosmological corrections following the procedure described in Zeh et al. (2004). Also shown is the r ' -band light curve of SN1998bw (in blue) shifted to z=0.248 taking into account the appropriate cosmological corrections. Overplotted are the upper limits we can set based on the GROND data (Table 1; r ' -band) and the late Gemini observations (Sect. 2.3; dark green).
\n\n( ξ N = 1) electron population is shock-accelerated. With the exception of the viewing angle ι , which has a sine function, to all parameters we assigned a uniform function to model the prior probability (for details see Table 2).\nFor the kilonova emission, NMMA allows to fit and simulate data using several numerical and analytical models. Assuming a binary neutron star merger as a progenitor system, we adopted here the kilonova modelling resulting from the time dependent 3D Monte Carlo code POSSIS for modelling radiation transport (Bulla 2019, 2023). We used the kilonova model grid published in Dietrich et al. (2020) based on the first version of POSSIS (Bulla 2019) where the kilonova ejecta is represented with two components: 1) a high-velocity (0 . 08 < v dyn /c < 0 . 3) dynamical ejecta of mass M dyn rj with a lanthanide-rich composition distributed about the equatorial plane with half-opening angle Φ, and lanthanide-poor composition at higher latitudes, and 2) a slower (0 . 025 < v wind /c < 0 . 08) wind (or 'post-merger') component, which is a spherical ejecta released from the merger remnant and debris disk, with an intermediate\nlanthanide content and mass M wind ej (see Dietrich et al. 2020). In the fit we fixed the GRB 191019A luminosity distance at 1289.3 Mpc, as obtained from the measured redshift ( z = 0 . 248) and assuming a flat cosmological model (see Sect. 1). We also considered an additional error budget (in magnitudes), which takes into account the uncertainties on the model predictions as well as possible systematic errors ( em syserr in the corner plot), to which we assigned a uniform prior with range from 0 to 2 mag. The resulting best-fit light curves computed in each band are plotted in Figure 6, the fit corner plot is in Figure 7, while the 90% confidence interval and median values of each parameter , as well as the adopted prior functions, are quoted in Table 2.\nBy simply inspecting the resulting light curves, it is evident that the afterglow component is constrained mostly by the X-ray data and it is dominant during the early epochs in the optical bands. The afterglow best fit jet core semiaperture angle is about ∼ 10 deg (9 +4 -3 deg), and the observer viewing angle is ∼ 4 deg (4 +3 -2 deg), thus within the jet core. The fireball isotropic kinetic energy E 0 (1 . 4 +1 . 8 -1 . 0 × 10 52 erg) is nicely compatible with the observed prompt emission equivalent isotropic energy E iso , by assuming a reasonable efficiency of about η ∼ 10%, where η = E iso E iso + E 0 . Indeed, the 15-150 keV fluence measured by Swift -BAT is (1 . 00 ± 0 . 03) × 10 -5 erg cm -2 (Krimm et al. 2019), corresponding to a radiated energy E iso = (1 . 70 ± 0 . 05) × 10 51 erg. The latter was computed by assuming a power-law spectrum in the BAT bandpass: this assumption is reasonable given the softness of the spectrum, which is best fit with a photon index Γ = 2 . 25 ± 0 . 05 (Krimm et al. 2019), suggesting that the prompt emission peak energy lies below the BAT bandpass.\nFigure 6 also clearly shows that at later epochs ( > 1 day), the observed flux lies above the predicted levels of the afterglow component, and the presence of a kilonova provides a better match. Indeed, by assuming only the afterglow model, i.e. by removing the kilonova component from our initial model, we obtained a worse fit, with Bayesian evidence ln( Z ) = -21 . 1. By considering the ratio with the Bayesian evidences of the joint afterglow and kilonova model, for which we obtain ln( Z 0 ) = -14 . 1, the resulting Bayes factor (ln( B ) =ln( Z/Z 0 ) = -7 . 0) indicates a strong preference for the model which includes the kilonova component (see, e.g., Kunert et al. 2024 and references therein).\nThe kilonova properties we find from our fit are compatible with a dynamical ejecta mass of M dyn ej ∼ 0 . 02 M ⊙ and a wind mass M wind ej ∼ 0 . 04 M ⊙ , though with large uncertainties (relative errors ≥ 60%). These values are slightly higher (yet consistent within the uncertainties) with those found for AT2017gfo by assuming the same kilonova modelling (Dietrich et al. 2020). This is in line with the need of a brighter kilonova (of about a factor 4, see Sect. 3.3 and Fig. 4) by simply superposing an AT2017gfo-like light curve on the data.\nWe stress here that the real picture is likely much more complicated than a two-component scenario, and the dynamical/wind masses inferred should rather be interpreted as belonging to some high/low velocity components of a multi-component scenario with multiple ejecta episodes (e.g., Bernuzzi 2020; Nedora et al. 2021). At the same time, the kilonova model we assumed is among the most sophisticated ones publicly available, and the assumption of a kilonova from a NS-NS merger progenitor is the most reasonable choice given that, so far, the only GRB with kilonova emission for which we were able to infer the progenitor nature was GRB170817/AT2017gfo, wich we know from gravitational wave data analysis being originated from a binary neutron star merger. Nevertheless, we explored also other kilonova models available within the NMMA framework, with different levels of sophistication, and which have different assumptions on the progenitor nature and on the kilonova ejecta properties (see Appendix C). We find that the dataset for GRB 191019A did not allow us to confidently distinguish among different kilonova models, apart disfavouring the most simplistic ones. However, in all cases, we find that a joint afterglow plus kilonova fit is preferred with respect to an afterglow-only model.\nInterestingly, both the afterglow-only and afterglow plus kilonova model, provide as a best-fit for the circumburst environment particle density a value n 0 < 1 cm -3 , which is typically deduced for short-GRB environments (e.g., Berger 2014; Fong et al. 2015). This result is in stark contrast with the interpretation by Lazzati et al. (2023) that invokes the presence of a very high density environment with n 0 ∼ 10 7 -10 8 cm -3 (see also Levan et al. 2023).\nT able 2 . The 90% confidence in terv al and median v alues of eac h parameter inferred from the sim ultaneous afterglo w ( Ry an et al. 202 0 ) and kilono v a ( Dietric h et al. 2020 ) mo dell ing p erformed b y using the co de NMMA. F or the afterglo w comp onen t, w e assumed a Gaussian jet profi le and fiducial v alue s for the microph ysical parameters (see Sect. 3.3 an d Fig. 6 and Figure 7 ). The b ottom part of the table quotes the assumed p r i ors in the Ba y esian inference analysis, where 'U' stands for Uniform function, with minim um and maxim um v alue s quoted in the brac k ets.\n
\n\n
\nNote -E 0 = kinetic fireball energy; n = particle n um b er densit y in the circum burst en vironmen t; θ c =half-op ening angle of the jet core; θ w =half-op ening a ngle of the jet truncated-wings; ι =viewing angle with resp ect to jet axis; p = electron energy distribution p o w er-la w index; ϵ e = sho c k e nergy fraction that go es in to th e electrons; ϵ B = sho c k energy fraction that go es in to the magne tic energy densit y; M dyn ej = dynamical ejecta mass; M wej = wind ejecta mass; Φ = half-op ening angle of lan thanide-ric h equatorial ejecta.\n
\n\n
Table 3. Expected and observed SN AB magnitudes.
\n
\nNote -All observed (obs) AB magnitudes in the r ' , i ' and z ' filters are 3 σ upper limits (Fig. 5). Columns (2), (5), and (8) quote the expected (exp) magnitudes. All the corresponding differences (observed minus expected; columns 4, 7, 10) are lower limits.\n4. DISCUSSION AND CONCLUSIONS\nRecent analysis of an increasing number of GRBs which were initially classified as long bursts, and therefore thought to be originating from collapsing massive stars, suggest a better compatibility of the observational data with compact binary merger progenitors. By ignoring the standard burst duration classification, typical features of GRBs associated with compact binary mergers are (1) the absence of an associated CC-SN, (2) the presence of an optical/NIR rebrightening compatible with a kilonova, (3) an early-type host galaxy, and (4) a distant GRB explosion site with respect to the corresponding galaxy center. GRB 191019A was suggested to belong to this sample of misclassified long GRBs, given the non detection of any SN component (Levan et al. 2023) despite its close cosmological distance ( z =0.248), and the evidence of a host galaxy dominated by an old stellar population. However, the proximity of the GRB 191019A explosion site to the photometric center of its host galaxy ( ≲ 100 pc) is at odds with typical short-GRB galactocentric distances, and could suggest a compact binary formed in the dense circumnuclear disk of an AGN. In this scenario (Lazzati et al. 2022) a very high-density environment ( n 0 > 10 7 cm -3 ) could be the origin of the long duration of the prompt emission (see Lazzati et al. 2023). Given the low distance of this burst, a kilonova ligh is expected to be seen. However, the temporal coverage and limiting magnitudes of previous data sets, did not allow to address this question yet.\nIn this work, we further investigate the burst properties of GRB 1901910A and the nature of the optical transient that followed the burst, with the goal to find arguments in favour or against its possible compact merger origin. We found an independent support on the compact binary merger progenitor nature from the high-energy prompt emission temporal variability, which has been recently found to be an additional potential diagnostic to infer the progenitor nature, where high variability values suggest a compact binary merger progenitor (Camisasca et al. 2023). We noticed that, for GRB 191019A, the obtained MVT value of ∼ 20 ms locates this burst close to the centroid of the distribution of those GRBs associated with compact binary merger progenitors (Camisasca et al. 2023 and Fig. 1). The same variability properties were found also for two other long GRBs with evidence of kilonova (GRB 111210A and GRB 230307A). These results independently support past interpretations on the progenitor nature of GRB 191019A and address to the possible presence of a kilonova component in its optical transient.\nAn optical transient following GRB 191019A was at first reported by Perley et al. (2019b) but its nature could not be clearly pinned down. Using GROND multi-color data obtained between 0.4 and 15 days post burst, we argued here that the temporal evolution of the transient's brightness disfavours a pure afterglow emission (Fig. 4). While the GROND data confirm the absence of a SN component (Fig. 5), we found that the luminosity evolution of the optical transient is in agreement with an afterglow plus a kilonova signal compatible with AT2017gfo redshifted to the distance of the GRB host galaxy, though slightly brighter by a factor of a few.\n\n\n
Figure 6. Joint fit of an afterglow plus a kilonova model, performed with NMMA (see Table 2). The dashed lines show the median light curve, while the shaded areas show the 95% interval. Red circles and black triangles mark the detections and upper limits, respectively, in AB mag.
\n\n\n\n
Figure 7. Corner plot obtained with NMMA from a joint fit assuming a model that incorporates both an afterglow and a kilonova component (see Table 2 and Fig. 6). Different shadings mark the 68%, 95%, and 99% confidence intervals. For the 1D posterior probability distributions, the 90% confidence interval (dashed lines) and median values above each panel are indicated.
\n\nBy modelling the afterglow and the kilonova component, we were able to estimate a kilonova ejecta mass which is slightly higher but still consistent, within the large uncertainties, with the one measured for AT2017gfo with similar assumptions on the progenitor system (a binary neutron star) and on the kilonova modelling (e.g., Dietrich et al. 2020). By assuming other kilonova models with different levels of complexity and different progenitor assumptions, the data did not allow to discriminate among them, apart disfavouring the most simplistic one. However, in all cases, results strongly favour the presence of a kilonova with respect to an afterglow-only model (Table 5).\nOur findings strongly suggest that GRB 191019A might belong to the increasing list of long GRBs with an associated kilonova, beside GRB 211211A and GRB 230307A, and other cases with less robust evidence, as for instance GRB 060614 (e.g., Jin et al. 2015). Another interesting findings from our analysis, is the evidence of a low circumburst density (Table 2). This result is in stark constrast with the one obtained by Lazzati et al. (2023) from prompt emission light curve modelling, where extreme values of n 0 > 10 7 cm -3 were found. These high density values were considered consistent with the apparent position of the GRB, very close to the center of an AGN-like host galaxy (though no direct proofs on the existence of an AGN have been provided yet, see Levan et al. 2023). The interaction between a jet and an AGN dense circumnuclear environment, however, may choke or strongly suppress GRB relativistic jets, unless particularly bright and long, as also supported by recent studies (e.g., Zhang et al. 2024). GRB 191019A is classified as a normal burst, with a well detected optical counterpart (see, e.g., Zhang et al. 2024), and with low dust extinction along the line-of-sight within the host ( A host V in the range from 0.06 to 0.10 mag, Levan et al. 2023), further supported by the UV-band detections (LaPorte et al. 2019). These properties are more consistent with a low density environment, as the one we obtained from a joint fitting of an afterglow and a kilonova component model on the X-ray and optical data of GRB 191019A. More in general, if GRB 191019A is a disguised short GRB with a compact binary merger origin, as both past studies and this work suggest, a low density environment is in agreement with the typical density values found in short GRBs and with the environments expected for compact binary mergers.\nIn a low density scenario, the very small offset from the host galaxy of GRB 191019A, can be explained with a negligible transverse projection of a GRB located far off centre. In this scenario, the long duration of the prompt emission, which Lazzati et al. (2023) explained as caused by the extremely dense circumburst environment, may have instead an intrinsic origin. An interesting hypothesis that was proposed in the past to explain short GRBs with soft extended emission (e.g., Barkov & Pozanenko 2011; Kisaka & Ioka 2015) and which has been recently studied in much greater details (e.g., Musolino et al. 2024), invokes fall-back accretion onto the remnant compact object. Whether this is the case for GRB 191019A goes beyond the scope of this work and will be addressed in another study.\nWe thank the anonymous referee for useful comments and suggestions which improved our work. We are grateful to P. Pang for his precious support on NMMA and useful discussions. ANG acknowledges logistic support by the Thuringer Landessternwarte Tautenburg, Germany. G.S. and P.S acknowledge the support by the State of Hesse within the Research Cluster ELEMENTS (Project ID 500/10.006). A.R. and E.P. acknowledge support by PRIN-MIUR 2017 (grant 20179ZF5KS). Part of the funding for GROND (both hardware and personnel) was generously granted by the Leibniz-Prize to G. Hasinger (DFG grant HA 1850/28-1) and by the Thuringer Landessternwarte Tautenburg. S.B. aknowledges funding from the EU Horizon under ERC Consolidator Grant, no. InspiReM-101043372. A.E.C. is partially supported by the 2023/24 'Research and Education' grant from Fondazione CRT. The OAVdA is managed by the Fondazione Cl'ement Fillietroz-ONLUS, which is supported by the Regional Government of the Aosta Valley, the Town Municipality of Nus and the Unit'e des Communes valdotaines Mont-Emilius. The LBT is an international collaboration of the University of Arizona, Italy (INAF: Istituto Nazionale di Astrofisica), Germany (LBTB: LBT Beteiligungsgesellschaft), The Ohio State University, representing also the University of Minnesota, the University of Virginia, and the University of Notre Dame. This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester.\nFacilities: GROND, Swift (BAT, XRT and UVOT), LBT\nSoftware: Afterglowpy (Ryan et al. 2020), NMMA (Pang et al. 2023), astropy (Astropy Collaboration et al. 2013, 2018), HOTPANTS (Becker 2015), PyRAF (Science Software Branch at STScI 2012), DRAGONS (Labrie et al. 2019), WCSTools (Mink 2019).\nAPPENDIX\nA. HOST GALAXY\n
\n\n
Table 4. Measured AB magnitudes of the host.
\n
\nNote -Magnitudes are measured within an aperture with radius of 4 × FWHM and are not corrected for Galactic foreground extinction.\nTable 4 provides the magnitude of the GRB host galaxy based on GROND's 4th-epoch observations and LBT's visit of the field four years after the burst (see Sect. 2.2).\nB. GRB-SN PEAK LUMINOSITY RANGE\nWe further investigated on possible biases towards the faint end of our GRB-SN sample by comparing with SN Ic peak luminosity properties observed so far. The r-band absolute peak-magnitude ( M r,peak ) distribution of broad line (BL) Type Ic SNe spans at most 2.5 mags (between -17 . 7 and -20 . 8 mag, with an average of M r,peak = -18 . 7 ± 0 . 7 mag, Taddia et al. 2019; Barbarino et al. 2021; Gomez et al. 2022). GRB-SNe observed to date with good data sampling, are at least ∼ 1 mag brighter and trace the high-luminosity end of the above Type Ic BL distribution (e.g., Hjorth & Bloom 2012; Hjorth 2013; Cano et al. 2017; Klose et al. 2019), but the width of their peak luminosity distribution (Hjorth & Bloom 2012; Cano et al. 2017; Kann et al. 2019b; Klose et al. 2019) is not substantially different to the width of the corresponding peak luminosity distribution of type Ic and BL-SNe (see also Soderberg et al. 2006). By considering the GRB-SNe with the best follow-up in multiple bands (including spectroscopy), on the high end site of the peak luminosity distribution remains GRB 111209A/SN 2011kl (e.g., Nakauchi et al. 2013; Kann et al. 2019b) while on the low-end site there are GRB 100316D/SN 2010bh (e.g., Starling et al. 2011; Olivares E. et al. 2012b) and GRB 060218/SN 2006aj (e.g., Pian et al. 2006; Ferrero et al. 2006): the r -band peak magnitudes of these extreme GRB-SNe have a difference that lies between 2 and 3 mags, similar to the broad line (BL) Type Ic SNe. In principle, the long bursts GRB 990712 (Bjornsson et al. 2001), GRB 021211 (Della Valle et al. 2003; Zeh et al. 2004), GRB 040924 (Soderberg et al. 2006; Wiersema et al. 2008), and GRB 060904B (Cano et al. 2017) could have been followed by even slightly fainter SNe than SN 2010bh, but in these cases the data base is comparably poor and no strong conclusions could be made. While, e.g., any SN that was associated with the long bursts GRB 060605 and GRB 060614 must have had a luminosity < 1% of the luminosity of the prototypical SN 1998bw (Fynbo et al. 2006), here and in similar cases (e.g., GRB 111005A; Michaglyph[suppress]lowskI et al. 2018; Tanga et al. 2018) there is no evidence for SN light.\nIn conclusion, if substantially less luminous GRB-SNe do exist formally remains an open question, at least from the obervational point of view, and the peak luminosity range of GRB-SNe sample we have used to infer the luminosity of any SN associated with GRB 191019A is the most robust so far.\nC. AFTERGLOW AND KILONOVA JOINT FIT COMPARISON\nIn Table 5 we present the results obtained for different kilonova models within the NMMA framework, in addition to the kilonova model presented in Sect. 3.3.\nAt first, we have tested a model from Kasen et al. (2017) (Kasen17-Jet in Table 5) which assumes only one ejecta component and has three parameters: the ejecta mass, the ejecta velocity and the lanthanide mass fraction ( χ lan ). Then we made a different assumption on the progenitor by considering a kilonova emission from a neutron star black hole binary system as predicted with POSSIS (Bu19-NSBH-Jet in Table 5), which has three parameters: the dynamical ejecta mass, the wind mass and the orbital plane inclination, which is linked to the viewing angle of the jet assumed in the afterglow modelling. These models were finally compared with the results obtained by assuming the simple analytical model described in Metzger (2017) (Metzger17-Jet in Table 5), which assumes one component and has four parameters: the ejecta mass, the ejecta velocity, the power-law index β of the ejecta mass distribution expressed as a function of its velocity (the faster ejecta matter lies ahead of slower matter and the distribution of mass with velocity greater than a value v 0 can be approximated with a power-law M ( > v 0 ) = M ( v/v 0 ) -β , and the opacity k r .\nThe afterglow emission was modelled within the Afterglowpy framework (Afterglow in Table 5), as described in Sect. 3.3. However, contrary to our previous analysis, the microphyiscal parameters ϵ e and ϵ B are now fixed to 0 . 5 and 0 . 01, respectively. In doing so, we aim at reproduce similar assumptions to Lazzati et al. (2023) and further investigate on the circumburst density estimates.\nThe highest Bayesian evidence ( Z ) is obtained for the Bu19-NSBH-Jet model. Following Kunert et al. (2024) and references therein, by considering Bu19-NSBH-Jet as the reference model, we find -1 . 10 < ln( Z/Z Bu 19 -NSBH -Jet ) < 0 for the Kasen17-Jet and Bu19-BNS-Jet models, while ln( Z/Z Bu 19 -NSBH -Jet ) < -4 . 5 for Metzger17-Jet and Afterglow models, indicating a strong evidence against the latter two models, while no preference could be set among Bu19-BNSJet, Bu19-NSBH-Jet and Kasen17-Jet models.\nREFERENCES\nAbbott, B. P., et al. 2017, ApJ, 848, L13,\ndoi: 10.3847/2041-8213/aa920c\nAhumada, T., Singer, L. P., Anand, S., et al. 2021, Nature Astronomy, 5, 917, doi: 10.1038/s41550-021-01428-7 Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33,\ndoi: 10.1051/0004-6361/201322068\nAstropy Collaboration, Price-Whelan, A. M., Sip\"ocz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f Barbarino, C., Sollerman, J., Taddia, F., et al. 2021, A&A,\n651, A81, doi: 10.1051/0004-6361/202038890\nBarkov, M. V., & Pozanenko, A. S. 2011, MNRAS, 417, 2161, doi: 10.1111/j.1365-2966.2011.19398.x\nBarthelmy, S. D., Barbier, L. M., Cummings, J. R., et al. 2005, SSRv, 120, 143, doi: 10.1007/s11214-005-5096-3\nBecker, A. 2015, HOTPANTS: High Order Transform of PSF ANd Template Subtraction, Astrophysics Source Code Library, record ascl:1504.004.\nhttp://ascl.net/1504.004\nBeniamini, P., Duque, R., Daigne, F., & Mochkovitch, R. 2020, MNRAS, 492, 2847, doi: 10.1093/mnras/staa070 Berger, E. 2014, ARA&A, 52, 43,\ndoi: 10.1146/annurev-astro-081913-035926\nBernuzzi, S. 2020, General Relativity and Gravitation, 52, 108, doi: 10.1007/s10714-020-02752-5\nBjornsson, G., Hjorth, J., Jakobsson, P., Christensen, L., & Holland, S. 2001, ApJL, 552, L121, doi: 10.1086/320328 Bulla, M. 2019, MNRAS, 489, 5037,\ndoi: 10.1093/mnras/stz2495\n-. 2023, MNRAS, 520, 2558, doi: 10.1093/mnras/stad232 Burrows, D. N., Hill, J. E., Nousek, J. A., et al. 2005, SSRv, 120, 165, doi: 10.1007/s11214-005-5097-2\nCamisasca, A. E., Guidorzi, C., Amati, L., et al. 2023, A&A, 671, A112, doi: 10.1051/0004-6361/202245657\nCano, Z., Wang, S.-Q., Dai, Z.-G., & Wu, X.-F. 2017, Advances in Astronomy, 2017, 8929054, doi: 10.1155/2017/8929054\nChambers, K. C., Magnier, E. A., Metcalfe, N., et al. 2016, arXiv e-prints, arXiv:1612.05560,\ndoi: 10.48550/arXiv.1612.05560\nCoulter, D. A., Foley, R. J., Kilpatrick, C. D., et al. 2017, Science, 358, 1556, doi: 10.1126/science.aap9811\nCoward, D. M., Gendre, B., Tanga, P., et al. 2017, PASA, 34, e005, doi: 10.1017/pasa.2016.61\nDainotti, M. G., De Simone, B., Islam, K. M., et al. 2022,\nApJ, 938, 41, doi: 10.3847/1538-4357/ac8b77\nDella Valle, M., Malesani, D., Benetti, S., et al. 2003, A&A, 406, L33, doi: 10.1051/0004-6361:20030855\nDietrich, T., Coughlin, M. W., Pang, P. T. H., et al. 2020, Science, 370, 1450, doi: 10.1126/science.abb4317\n
\n\n
Table 5. The 90% confidence interval and median values for each parameter inferred from joint afterglow and kilonova and afterglow only jet modelling performed by using the code NMMA.
\n
\nNote -E 0 = kinetic fireball energy; n = particle number density in the circumburst environment; θ c =half-opening angle of the jet core; ι =viewing angle with respect to jet axis; p = electron energy distribution power-law index; M dyn ej = dynamical ejecta mass; M wind ej = wind ejecta mass; M ej = total ejecta mass; v ej = ejecta velocity; Φ = half-opening angle of lanthanide-rich equatorial ejecta; ϵ e = shock energy fraction that goes into the electrons; ϵ B = shock energy fraction that goes into the magnetic energy density; χ lan = lanthanide mass fraction (Kasen et al. 2017) ; κ r = opacity; β = power-law index of the ejecta mass distribution as a function of its velocity (Metzger 2017); ln( Z ) = natural logarithm of the bayes evidence. We note that the Bu19-BNS-Jet model is the same as the one presented in Sect. 3.3 where now ϵ e and ϵ B are fixed.\nEvans, P. A., Osborne, J. P., Burrows, D. N., et al. 2019,\nGRB Coordinates Network, 26034, 1 Evans, P. A., Beardmore, A. P., Page, K. L., et al. 2007, A&A, 469, 379, doi: 10.1051/0004-6361:20077530 -. 2009, MNRAS, 397, 1177, doi: 10.1111/j.1365-2966.2009.14913.x Evans, P. A., Willingale, R., Osborne, J. P., et al. 2010, A&A, 519, A102, doi: 10.1051/0004-6361/201014819 Ferrero, P., Kann, D. A., Zeh, A., et al. 2006, A&A, 457, 857, doi: 10.1051/0004-6361:20065530 Fong, W., Berger, E., Margutti, R., & Zauderer, B. A. 2015, ApJ, 815, 102, doi: 10.1088/0004-637X/815/2/102 Fong, W.-f., Nugent, A. E., Dong, Y., et al. 2022, ApJ, 940, 56, doi: 10.3847/1538-4357/ac91d0 Fontana, A., Dunlop, J. S., Paris, D., et al. 2014, A&A, 570, A11, doi: 10.1051/0004-6361/201423543 Fynbo, J. P. U., Perley, D. A., de Ugarte Postigo, A., et al.\n2019, GRB Coordinates Network, 26041, 1\nFynbo, J. P. U., Watson, D., Thone, C. C., et al. 2006, Nature, 444, 1047, doi: 10.1038/nature05375 Gehrels, N., Chincarini, G., Giommi, P., et al. 2004, ApJ, 611, 1005, doi: 10.1086/422091 Gendre, B., Crisp, H., Coward, D., et al. 2019, GRB Coordinates Network, 26098, 1 Gomez, S., Berger, E., Nicholl, M., Blanchard, P. K., & Hosseinzadeh, G. 2022, ApJ, 941, 107, doi: 10.3847/1538-4357/ac9842 Gompertz, B. P., Ravasio, M. E., Nicholl, M., et al. 2023, Nature Astronomy, 7, 67, doi: 10.1038/s41550-022-01819-4 Greiner, J., Bornemann, W., Clemens, C., et al. 2008, PASP, 120, 405, doi: 10.1086/587032 Greiner, J., Mazzali, P. A., Kann, D. A., et al. 2015, Nature, 523, 189, doi: 10.1038/nature14579 Hjorth, J. 2013, Philosophical Transactions of the Royal Society of London Series A, 371, 20120275,\ndoi: 10.1098/rsta.2012.0275\n
\n\n
\n
\n\n
\n
\n\n
\n\nTody, D. 1993, ASP Conf. Ser., 52, 173\n\nTroja, E. 2023, Universe, 9, 245,\ndoi: 10.3390/universe9060245\n\nTroja, E., Ryan, G., Piro, L., et al. 2018, Nature Communications, 9, 4089, doi: 10.1038/s41467-018-06558-7\nTroja, E., Castro-Tirado, A. J., Becerra Gonz'alez, J., et al. 2019, MNRAS, 489, 2104, doi: 10.1093/mnras/stz2255\nTroja, E., Fryer, C. L., O'Connor, B., et al. 2022, Nature, 612, 228, doi: 10.1038/s41586-022-05327-3\nWang, S.-N., Lu, H.-J., Yuan, Y., et al. 2024, ApJ, 963, 156, doi: 10.3847/1538-4357/ad2205\nWiersema, K., van der Horst, A. J., Kann, D. A., et al. 2008, A&A, 481, 319, doi: 10.1051/0004-6361:20078050\nYang, J., Ai, S., Zhang, B.-B., et al. 2022, Nature, 612, 232, doi: 10.1038/s41586-022-05403-8\n\nYolda¸s, A. K., Kruhler, T., Greiner, J., et al. 2008, in American Institute of Physics Conference Series, Vol. 1000, American Institute of Physics Conference Series, ed. M. Galassi, D. Palmer, & E. Fenimore, 227-231, doi: 10.1063/1.2943450\n\nZeh, A., Klose, S., & Hartmann, D. H. 2004, ApJ, 609, 952, doi: 10.1086/421100\nZhang, B., & M'esz'aros, P. 2001, ApJL, 552, L35, doi: 10.1086/320255\nZhang, H.-H., Zhu, J.-P., & Yu, Y.-W. 2024, arXiv e-prints, arXiv:2406.10904, doi: 10.48550/arXiv.2406.10904\nZhu, Z. P., Yu, B. Y., Xu, D., & Gao, X. 2019, GRB Coordinates Network, 26059, 1\n\n"},"sections":{"kind":"list like","value":[{"title":"ABSTRACT","content":"GRB 191019A was a long Gamma-ray burst (GRB) lasting ∼ 65 s and, as such, originally thought to be linked to a core-collapse supernova. However, even though follow-up observations identified the optical counterpart close to the bright nucleus of a nearby ancient galaxy ( z =0.248), no associated supernova was found. This led to the suggestion that the burst was caused by the merger of two compact stellar objects, likely in a dense circumnuclear environment. By using a recently developed diagnostic tool based on prompt emission temporal properties, we noticed that GRB 191019A falls among those long GRBs which are associated with compact mergers and with evidence of kilonova light. We thus re-analyzed unpublished GROND multi-color ( g ' r ' i ' z ' JHK s ) data obtained between 0.4 and 15 days post trigger. Image subtraction confirmed the optical counterpart in all four optical bands, with GROND tracking its fading until 1.5 days post-burst. Incorporating publicly available Swift -XRT data, a joint fit of an afterglow plus a kilonova model revealed a better match than an afterglow-only scenario. The resulting kilonova properties resemble those of AT2017gfo associated with the binary neutron star merger GW170817, with a total ejected mass of ∼ 0 . 06 M ⊙ . Contrary to previous findings inferring a high-density circumburst environment ( n 0 ∼ 10 7 -8 cm -3 ), our analysis finds standard conditions ( n 0 ∼ 1 cm -3 ), suggesting the long duration of GRB 191019A was intrinsic rather than due to jet interaction with a dense external medium. Keywords: Gamma-ray bursts - Compact objects","pages":[1]},{"title":"The puzzling long GRB 191019A: Evidence for Kilonova Light","content":"G. Stratta , 1, 2, 3, 4 A. M. Nicuesa Guelbenzu , 5 S. Klose , 5 A. Rossi , 6 P. Singh, 1, 6 E. Palazzi , 6 C. Guidorzi, 7, 8, 6 A. Camisasca, 9 S. Bernuzzi, 10 A. Rau, 11 M. Bulla, 7, 8, 12 F. Ragosta, 13, 14 E. Maiorano, 6 and 15 D. Paris 1 Institut fur Theoretische Physik, Goethe Universitat, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany 2 INAF - Osservatorio di Astrofisica e Scienza dello Spazio, via Piero Gobetti 93/3, 40129 Bologna, Italy 3 Istituto di Astrofisica e Planetologia Spaziali, via Fosso del Cavaliere 100, I-00133 Roma, Italy 4 Istituto Nazionale di Fisica Nucleare-Roma 1, Piazzale Aldo Moro 2, I-00185 Roma, Italy 5 Thuringer Landessternwarte Tautenburg, Sternwarte 5, 07778 Tautenburg, Germany 6 INAF - Osservatorio di Astrofisica e Scienza dello Spazio, via Piero Gobetti 93/3, 40129 Bologna, Italy 7 Department of Physics and Earth Science, University of Ferrara, Via Saragat 1, 44122 Ferrara, Italy 8 INFN - Sezione di Ferrara, Via Saragat 1, 44122 Ferrara, Italy 9 Astronomical Observatory of the Autonomous Region of the Aosta Valley (OAVdA), Loc. Lignan 39, I-11020, Nus (Aosta Valley), Italy 10 Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universitat Jena, 07743, Jena, Germany 11 Max-Planck-Institut fur Extraterrestrische Physik, Giessenbachstraße 1, 85748 Garching, Germany 12 INAF, Osservatorio Astronomico d'Abruzzo, via Mentore Maggini snc, 64100 Teramo, Italy 13 Dipartimento di Fisica 'Ettore Pancini', Universit'a di Napoli Federico II, Via Cinthia 9, 80126 Naples, Italy 14 INAF - Osservatorio Astronomico di Capodimonte, Via Moiariello 16, I-80131 Naples, Italy 15 INAF - Osservatorio Astronomico di Roma, via Frascati 33, 00040 Monte Porzio Catone, Italy","pages":[1]},{"title":"1. INTRODUCTION","content":"The empirical classification of gamma-ray bursts (GRBs) into two classes according to the bimodal burst duration and hardness distribution (see, e.g., Kouveliotou et al. 1993), has been interpreted as the evidence of two different progenitor channels. Indeed, it was found that the vast majority of sufficiently nearby long GRBs are found to be spatially and temporally coincident with core-collapse (CC) supernovae (SNe) and hosted in star-forming galaxies. On the other hand, short GRBs lack any associated SN and are typically located in the outer regions of early-type galaxies, suggesting a progenitor belonging to an old stellar population, formally consistent with compact binary mergers. This picture was confirmed by the association of the short GRB 170817A with the gravitational wave event GW170817 that was compatible with a binary neutron star (NS-NS) merger (e.g., Abbott et al. 2017). From the very same event and thanks to its spatial proximity (40 Mpc), the first robust evidence of a kilonova (AT2017gfo) was obtained (e.g., Coulter et al. 2017), confirming the predicted thermal emission from matter released and heated during a NS-NS merger (e.g., Li & Paczy'nski 1998). In the last decade, kilonova signatures were observed in a number of past short GRBs that were sufficiently nearby, as for instance GRB 130603B ( z =0.3565, Tanvir et al. 2013), GRB 150101B ( z =0.134, Troja et al. 2018), GRB 160821B ( z =0.1613, Troja et al. 2019), though with much lower significance than AT2017gfo due to the larger distances of these events (see, e.g., Troja 2023 for a review). In recent years, the standard long and short-GRB progenitor paradigm has been blurred by mounting evidence of the existence of long GRBs with no associated CC-SN but with evidence of a kilonova component, more consistent with a compact binary merger progenitor. Two striking examples are the long GRB 211211A at redshift z = 0 . 076 (350 Mpc; e.g., Troja et al. 2018; Mei et al. 2022; Rastinejad et al. 2022; Yang et al. 2022; Gompertz et al. 2023) and GRB 230307A at z = 0 . 0646 (300 Mpc; e.g., Levan et al. 2023), for which a kilonova was clearly identified in the optical afterglow. In addition, evidence of short GRBs showing massive star collapse properties were also found (e.g., Ahumada et al. 2021; Rossi et al. 2022) further contributed to dismantle the standard GRB classification scheme. These results suggest that some of the past GRB progenitor classifications based on the burst duration and hardness only, have to be revisited. This holds in particular for GRB 191019A. GRB 191019A was discovered (Simpson et al. 2019) with the Burst Alert Telescope (BAT; Barthelmy et al. 2005) on board the Neil Gehrels Swift Observatory (Gehrels et al. 2004). It showed a complex burst light curve with multiple peaks with a duration T 90 = 64 . 6 ± 4 . 5 s. Its 15-150 keV spectrum could be well modeled with a power law with a photon index of 2 . 25 ± 0 . 05 and a fluence of 10 -5 erg cm -2 (Krimm et al. 2019) putting this source among the soft/long GRBs. Follow-up observations with the Swift X-ray telescope (XRT; Burrows et al. 2005) robustly identified an uncatalogued X-ray source (Evans et al. 2019). Rapid follow-up observations with the Zeiss-1000 1-m telescope of Tien Shan Astronomical Observatory revealed only one optical source within the XRT error circle with R = 18 . 37 ± 0 . 03 mag at T 0 +14 min (Reva et al. 2019). The position of this source was coincident with a catalogued object quoted to be fainter than the observed one, leading to the interpretation this is the host galaxy of GRB 191019A with the afterglow on top of it (Reva et al. 2019). The Zadko telescope (Coward et al. 2017) observed GRB 191019A field at similar epochs and confirmed the object reported by Reva et al. (2019) with R =18.92 mag, with no significant flux variation within 1.5 hours (Gendre et al. 2019). Similar claims of a lack of flux variations were provided by other teams performing early-epoch observations (Fynbo et al. 2019, Nicuesa Guelbenzu 2019, Perley et al. 2019a, Zhu et al. 2019). However, late observations performed with the Nordic Optical Telescope (NOT) at T 0 +3.25 days were analyzed with image subtraction methods and revealed that the source was fading, confirming the optical transient plus host identification (Perley et al. 2019b). Spectroscopic analysis of the host galaxy showed several absorption lines at redshift z = 0 . 248 and a spectral energy distribution consistent with a galaxy dominated by an old stellar population ( > 1 Gyr), with a star formation rate (SFR) of 0 . 06 ± 0 . 03 M ⊙ yr -1 , a stellar mass of 3 × 10 10 M ⊙ , and small dust extinction of A V = 0 . 19 ± 0 . 08 mag (Levan et al. 2023). The measured SFR is at odds with the typical high values inferred in long GRB hosts (but see Rossi et al. 2014). Further hints of anomalies for the long GRB 191019A came from the absence of an associated SN, despite the low redshift. Deep limits in g, r and z obtained between 2 and 73 days with the NOT and optical imaging with the Hubble Space Telescope at 30 and 184 days, put strong constraints on any SN emission up to > 20 times less luminous than SN1998bw (Levan et al. 2023). All these properties are at odds with a massive-star origin of GRB 191019A, while formally consistent with a compact object binary merger progenitor. Though, the deep limits obtained with the NOT Telescope at > 2 days could not confirm the presence of an early kilonova emission (Levan et al. 2023). An interesting feature characterizing GRB 191019A is its projected distance from the host galaxy center of ≲ 100 pc, the closest distance among all known short GRBs to their corresponding galactic nucleus (Levan et al. 2023). Indeed, compact binary progenitors formed in stellar binary systems are thought to have large kick velocities acquired during the CC-SN phase of the binary components, and for this reason short GRBs are typically found in the outskirts of their host galaxies, with large offsets from the center (e.g., Berger 2014; Fong et al. 2022; O'Connor et al. 2022). As noted by Levan et al. (2023), the proximity of GRB 191019A to the host galaxy center suggests that the possible progenitor compact binary system could have formed through dynamical encounters, which are thought to be favored in the dense gaseous environment of supermassive black hole surrounding disks through kinetic energy dissipation ('gas-capture' binary formation channel, Tagawa et al. 2020). An exciting consequence of this scenario is that dense environments could also alter the prompt emission properties of a short GRB, making it longer and softer (Lazzati et al. 2023). This possibility was explored for GRB 191019A and it was found compatible with an environmental density of the order of 10 7 to 10 8 cm -3 (Lazzati et al. 2023). Starting from Lazzati et al.'s conclusions, Wang et al. (2024) investigated the interaction of a possible kilonova ejecta from GRB 191019A with a dense circumstellar medium (CSM) that could be detected years after the merger. Their model predicts that in a very dense environment, the smaller the kilonova ejected mass, the fainter is the radioactively-powered luminosity but the higher is the contribution from kilonova-CSM interaction at late times. No evidence of kilonova ejecta-CSM interaction was found for GRB 191019A so far, and a possible detection in the future would require an ejected mass less than 2 × 10 -5 M ⊙ (Wang et al. 2024). Here we further investigate the properties of GRB 1901910A and the nature of the optical transient that followed the burst, with the goal to find arguments in favour or against its possible compact merger origin. We use the recently developed GRB prompt emission minimum variability (MVT) criterion (Camisasca et al. 2023) to explore the properties of the burst, and perform a complete reanalysis of the light curve of the optical/X-ray transient with particular emphasis on so far not analyzed GROND (Greiner et al. 2008) observations starting 10.3 hours post burst (Nicuesa Guelbenzu 2019). We use the sophisticated Nuclear - Multi-Messenger Astronomy (NMMA, Dietrich et al. 2020; Pang et al. 2023) software package which allows for afterglow and kilonova joint Bayesian inference, to explore if the multi-color light curve of the optical transient can be understood as powered by afterglow emission only or if a kilonova component was present too. Throughout this paper, we adopt a flat cosmological model with H 0 =67.4 km s -1 Mpc -1 , Ω M =0.315, and Ω Λ =0.685 (Planck Collaboration et al. 2020). For these parameters a redshift of z =0.248 (Fynbo et al. 2019) corresponds to a luminosity distance of d L = 1.29 Gpc, 1 arcsec corresponds to 4.03 kpc projected distance, and distance modulus is m -M = 40 . 55 mag. The Milky Way reddening E ( B -V ) along the line of sight towards the source is between 0.03 and 0.04 mag (Schlegel et al. 1998; Schlafly & Finkbeiner 2011). We present the observations and data reduction in Sect. 2, and our results in Sect. 3. In our data analysis we corrected the apparent magnitudes for the Galactic reddening and adopted a host-galaxy visual extinction along the line of sight of A host V = 0.06 mag (Levan et al. 2023). Discussion and conclusions are presented in Sect. 4.","pages":[1,2,3]},{"title":"2.1. GROND early-time Observations","content":"With a duration of about 65 seconds, GRB 191019A was a classical long GRB. At a redshift of z = 0 . 248, a rising SN 1998bw component was thus expected to be detectable with GROND within about two weeks after the GRB trigger. Therefore, following earlier efforts with GROND to explore GRB-SN light curves (Olivares E. et al. 2012a, 2015; Greiner et al. 2015; Kann et al. 2019a; Klose et al. 2019) we performed follow-up observations of the field during several epochs between 0.4 (Nicuesa Guelbenzu 2019) and 15.4 days post GRB trigger (Table 1). Observations were then terminated since no evidence for SN light was found. GROND data were reduced in a standard fashion (bias subtraction, flat fielding, co-adding; Kruhler et al. 2008; Yolda¸s et al. 2008), based on numerical routines in IRAF (Image Reduction and Analysis Facility; Tody 1993).","pages":[3]},{"title":"2.2. LBT late-time Observations","content":"Late-time observations of the field were performed with the Large Binocular Telescope (LBT) using the two twin LBC instruments equipped with the Sloan filters g ' r ' i ' z ' on October 20, 2023. These data were reduced using the data reduction pipeline developed at INAF - Osservatorio Astronomico di Roma (Fontana et al. 2014) which includes bias subtraction and flat-fielding, bad pixel and cosmic ray masking, astrometric calibration, and coaddition. On the LBT images, we measure the AB magnitudes for the host given in Table 4 (see Appendix A). Within the errors, these values are consistent with those measured with GROND in epoch 4 ( T -T 0 = 7 . 4756 days). Based on these data we conclude that 7 days post burst the transient did not contribute anymore to the observed flux in g ' r ' i ' z ' in a measurable quantity.","pages":[3]},{"title":"2.3. Image subtraction and photometry","content":"In order to search for a transient hidden by the relatively bright host galaxy, we performed image subtraction on the GROND images using HOTPANTS (Becker 2015). HOTPANTS convolves the template and the source images to the same","pages":[3]},{"title":"Stratta et al.","content":"Note -Column (2) provides the midtime of the observation, in units of days from burst trigger. Magnitudes are not corrected for Galactic foreground extinction. The photometry for epochs 1a to 3 was obtained after image subtraction against epoch 4 (see § 2.3 for details). On the contrary, the 3 σ upper limits for epoch 4 to 6 have been measured directly, without image subtraction. point spread function (PSF) and photometric scale. As a template image we used the GROND 4th-epoch observations (Table 1) since these images have the best seeing ( ∼ 1 . '' 0). We aligned the template and the source images by means of wcsremap 1 . The Gaussian input parameters in HOTPANTS were calculated following Becker (2015) considering the case in which the template FWHM is smaller than the source FWHM. In this way, the template images were smoothed to the seeing of the corresponding source images. In addition, we fixed several other parameters for HOTPANTS following Hu et al. (2022), but let free to vary the number of each region's stamps, the convolution kernel half width, and the size of Gaussians which compose the kernel. In doing so, we selected those input parameters that minimized the background noise on the residual image in a star-free region close to the position of the host galaxy. All resulting residual images were then photometrically calibrated with respect to the source images (see below). In the case of a non-detection of the transient the upper limit we used was measured locally on the subtracted image (Table 1). We have confirmed our results using also the 5th GROND epoch as template (seeing ∼ 1 . '' 3), although we obtained a lower SNR. We double checked the detections using the late LBT images (about 4 years or 1460 days after the burst trigger, see Sect. 2.2 and Table 4 in Appendix A). The g ' i ' z ' detections are confirmed for epochs 1a and 2, but with lower SNR likely due to the additional plate scaling, and a general worse PSF shape due to not perfect collimation. We have not checked the r ' -band since the above mentioned problem was worse in this filter, although it did not compromise the aperture photometry in Table 4. For this filter we have instead used (as template) a stack of the best Gemini-S images 2 obtained at 58-63 days after the burst (see Levan et al. 2023), and we obtained consistent results. In other words, the results obtained with GROND could not be improved. Finally, to better constrain the late transient decay, we performed image subtraction on the Gemini r -band image at 11.4 days, using the later Gemini images as template (see above), and obtained for the optical transient an upper limit of r > 25 . 8 AB mag. Apparent magnitudes were obtained by performing aperture photometry using DAOPHOT and APPHOT under PyRAF/IRAF with increasing apertures up to 4 times the FWHM and by PSF photometry with the DAOPHOT and ALLSTAR tasks of IRAF. PSF-fitting was used to measure the magnitudes of the transient after image subtraction. In the optical bands the data were calibrated using Pan-STARRS (Chambers et al. 2016), in the NIR bands using 2MASS (Skrutskie et al. 2006). Central wavelengths of the GROND filter bands are listed in Rossi et al. (2011).","pages":[4]},{"title":"3.1. Application of the prompt emission minimum variability criterion","content":"According to Camisasca et al. (2023), GRB prompt emission minimum variability timescale (MVT) represents a diagnostic to infer the progenitor nature of a GRB. By estimating MVT from the full width at half maximum of the shortest, statistically significant peak in the prompt emission light curve (FWHM min ) of hundreds of GRBs, it was found that high variability (low FWHM min values) typically belongs to compact binary mergers. Interestingly, this result was found to be independent of the burst duration and for this reason it is an important probe for the nature of peculiar long GRBs with properties more similar to those of short GRBs. By analyzing the results from the Swift GRB sample analyzed by Camisasca et al. (2023), we checked which long GRBs sufficiently nearby for a possible kilonova detection, i.e. at z < 0 . 5, and without any associated SN, had an FWHM min compatible with compact binary merger values, and we found that GRB 191019A was satisfying our criteria. Specifically, for this GRB an MVT of FWHM min = 0 . 196 +0 . 068 -0 . 05 s has been measured, which is compatible with the values typically found for GRBs associated with compact binary mergers (Fig. 1). In particular, given the long burst duration of GRB 191019A ( T 90 = 64 . 35 ± 4 . 35 s), its position in the FWHM min vs T 90 diagram lies among the short GRBs showing a soft 'Extended Emission' (SEE, e.g., Norris & Bonnell 2006; Minaev et al. 2010). Interestingly, two famous long GRBs for which a kilonova component was found in the optical afterglow, namely GRB 211211A (Rastinejad et al. 2022; Troja et al. 2022; Yang et al. 2022; Gompertz et al. 2023) and GRB 230307A (e.g., Levan et al. 2023), lie in the same region of the diagram of GRB 191019A (Fig. 1). The SEE distribution in the T 90 -FWHM min plane clearly overlaps with the distribution of long GRBs associated with SNe, but at the same time the centroids of the two distributions are well separated: interestingly, GRB 191019A lies in the middle of the SEE region and in the outskirts of the SN-associated cases. This property alone cannot be taken as compelling evidence that GRB 191019A behaves like a SEE, but suggests a probable SEE-like nature than a collapsar one. These findings further support past interpretation that also GRB 191019A originated from a compact binary merger (e.g., Levan et al. 2023), and address the possible presence of a kilonova component, similarly to GRB 211211A and GRB 230307A.","pages":[5]},{"title":"3.2. Re-analysis of the light curve of the Transient that followed the burst","content":"3.2.1. X-rays Adopting a power-law spectral energy distribution (SED), the best fit of the Swift -XRT 0.3-10.0 keV spectrum, computed by the automatic algorithm of the UK Swift Science Data Center (UKSSDC) Swift -XRT GRB Spectrum Repository 3 in the 3 . 2 -32 . 4 ks temporal window, provides a photon index of Γ = 2 . 0 +0 . 4 -0 . 3 and an intrinsic equivalent hydrogen column density of N H , z = 1 . 2 +1 . 6 -1 . 2 × 10 21 cm -2 in addition to a Galactic N H = 3 . 28 × 10 20 cm -2 . We obtained from the UKSSDC Burst Analyzer tool (Evans et al. 2010), the absorption-corrected X-ray fluxes in the energy range 0.3-10.0 keV, and we converted them into 1 keV flux densities by assuming a power law spectrum and the corresponding photon index provided by the Burst Analyzer in each temporal bin of the light curve. The flux density evolution in the X-ray band is compatible with a power-law decay 4 with α X = 1 . 44 ± 0 . 14.","pages":[5]},{"title":"3.2.2. Optical bands","content":"We detected the optical transient during the first three GROND observations of the field, up to 1.5 days post burst (Table 1, Fig. 2). The transient has coordinates R.A., decl. (J2000) = 22:40:05.861, -17:19:42.77 ( ± 0.3 arcsec), measured on the combined g ' r ' i ' residual images of the 2nd-epoch observations (Fig. 3). Within the error, these coordinates agree with what has been reported by Perley et al. (2019b). Based on a joint fit of the griz -band magnitudes provided by Levan et al. (2023) and GROND's first visit of the field (epoch 1a in Table 1), we find that between both observing runs the optical transient was fading with a decay slope α opt = 1 . 43 ± 0 . 07. Within the errors, α X ∼ α opt up to T 0 +0 . 4 days (epoch 1a in Table 1), suggesting a common cooling regime of the electrons radiating optical and X-ray synchrotron light (see Fig. 4). This finding is compatible with results by (Levan et al. 2023) who showed that a single power-law SED from X-rays to the optical bands can describe the observations at T 0 +0 . 21 days. However, GROND's following observing runs 1b and 2 (Table 1) clearly show a significant flattening of the light curve, with a new decay slope of α opt ∼ 0 . 5 ± 0 . 1 (Fig. 4). The rescaled X-ray power-law model underpredicts the flux in the optical bands, with a > 3 σ deviation observed at T 0 +1 . 5 days. Even by assuming a shallower optical decay index (i.e., α opt = α X -0 . 25), as for the case where the synchrotron cooling frequency lies between the X-ray and the optical regime (e.g., Sari et al. 1998), the discrepancy at late times ( > 1 day) persists (Fig. 4). To quantify the null hypothesis that a power law model cannot fit the data and test for chromaticity of flux evolution, we considered the early data by Levan et al. (2023) and our GROND detections (i.e. from ∼ 0.2 to 1.5 days) for the two filters with best temporal sampling (i.e. r ' and g ' ): a simple power law model does not provide an acceptable fit, with probability of having the observed data set, given the null hypothesis (p-values), of 1 . 7 × 10 -5 and 4 × 10 -4 , respectively. By assuming a broken power law, with a steep-to-shallow behavior and a temporal break at 0.44 days, we find a significant improvement of the fit, with p-values of 0 . 79 and 0 . 73 in the r ' and g ' bands, respectively. The best fit broken power laws show that the flux evolution after the break is slightly shallower in the redder filter, suggesting a chromatic behaviour. We note that this peculiar steep-to-shallow optical light curve morphology is not unique, it was already observed in a small subset of GRBs in the early years of the Swift era (see, e.g., Kann et al. 2010), although with light curve breaks on average much earlier than what we observe for GRB 191019A. For instance, the optical light curve of GRB 090102 (Gendre et al. 2010) is similar to GRB 191019A but the break time was ∼ 0 . 1 hr (rest frame) after the burst. In the standard fireball model, which fairly accounts for late afterglow phenomenology as forward-shock radiation, a steep-to-shallow light curve is not envisioned (e.g., Sari et al. 1998). Such morphology could be produced by the presence of a reverse shock component dominating the forward shock at early times. However, reverse shock is expected on timescales of the order of minutes (e.g., Nakar & Piran 2004), in contrast with what we observe for GRB 191019A. Alternatively, energy injection into the forward shock could originate a light curve flattening. The spin-down radiation from a newly born magnetar (e.g., Zhang & M'esz'aros 2001) or a slightly off-axis structured jet (e.g., Beniamini et al. 2020) are plausible energy sources that are often invoked to explain the X-ray afterglow 'plateaus' observed in GRBs, and that might have an optical counterpart. In the case of GRB 191019A the lack of any evidence of a shallow phase in X-rays cast some doubts against the energy injection scenario, however.","pages":[5,6,7,8]},{"title":"3.2.3. Constraints on Supernova Light","content":"If GRB 191019A had been a classical long burst, then for a redshift of z =0.248 a supernova (SN) component was expected to be detectable with GROND. Observational constraints on SN light were first reported by Levan et al. (2023) based on HST observations 30 and 184 days post GRB trigger. No evidence for transient emission was found ( g > 24 , r > 23 . 5, and z > 22 mag). Figure 5 shows the upper limits we can set on any SN that followed GRB 191019A in comparison to the r ' -band light curves of 13 GRB-SNe that have been observed with GROND between 2007 and 2014 (see Table 2 in Klose et al. 2019). These events cover the whole GRB-SN luminosity distribution of well sampled GRB-SNe, from GRB 100316D/SN 2010bh (Olivares E. et al. 2012a), one of the faintest GRB-SN ever detected (e.g., Melandri et al. 2014; Cano et al. 2017; Dainotti et al. 2022), to GRB 111209A / SN 2011kl (Kann et al. 2019a), the most luminous GRB-SN ever observed. A visual inspection of Figure 5 shows that all light curves fall into a strip with a width of about 2-3 mag, which is limited by the very faint GRB 100316D/SN 2010bh and the very bright GRB 111209/SN 2011kl. The strongest limit we can set stamps from the time span between 7 and 12 days post burst and is well below the identified GRB-SN magnitude strip: if a CC-SN followed GRB 191019A, in r ' it was at least 3 mag less luminous than SN 1998bw and 2 mag fainter than SN 2010bh. According to the archived Gemini data (Sect. 2.3), the constraint on the luminosity is even stronger, > 4 mag in r at 11.4 days post burst. In conclusion, even if we take into account that the time evolution of GRB-SNe light curves show a certain parameter range (characterized by stretch factor), the SN limit we can set provides a constraint on the entire SN light curves. In this respect, any SN related to GRB 191019A must have been less luminous than each GRB-SN (see Appendix B).","pages":[8]},{"title":"3.3. Joint Afterglow and Kilonova modelling","content":"Since the flattening observed in the optical light curve at late times (epochs 1b and 2 in Table 1) cannot be explained by the presence of a slowly rising SN (Sect. 3.2.3), and probably not by a reverse shock or energy injection either (Sect. 3.2.2), we explore now the possibility that the source of this additional radiation component was kilonova light. Given the relatively small redshift of GRB 191019A ( z = 0 . 248), a kilonova with a luminosity similar to AT2017gfo is expected to lie within the discovery space of a 2-m telescope, if not hidden by an intrinsic low luminosity. Therefore, following the procedure described in Rossi et al. (2020), we compared our data at T 0 +1 . 5 days in each filter, with the flux of the kilonova AT2017gfo (Coulter et al. 2017) associated with the NS-NS merger GW170817 (Abbott et al. 2017), shifted to the redshift of GRB 191019A. We find that an emission component similar to AT2017gfo but ∼ 4 times more luminous could reproduce the observations in the i ' and r ' bands, while our g ' -band flux seems to be brighter (Fig. 4). Apparently, the presence of a kilonova associated with GRB 191019A is overall compatible with the general properties of the observed optical-NIR emission at 0.4 and 1.5 days, although with slightly different brightness than AT2017gfo 5 . Motivated by the results obtained with our previous simplistic approach, we then explored the possible presence of a kilonova by comparing our multi-band dataset ( g ' , r ' , i ' , z ' , and X-ray) with a much more sophisticated model that takes into account simultaneously the presence of an afterglow and a kilonova component through a joint fit. For this purpose, we exploited the Nuclear - Multi-Messenger Astronomy (NMMA v0.2.0) framework that allows to estimate best-fit parameters with a bayesian inference method (Dietrich et al. 2020; Pang et al. 2023). For the afterglow modelling, NMMA uses the Afterglowpy python module (Ryan et al. 2020). Afterglowpy allows to model GRB afterglow light curves and spectra by taking into account the possible effects due to a complex jet structure and an off-axis observer. We here assumed a Gaussian profile for the jet structure, and that the whole ( ξ N = 1) electron population is shock-accelerated. With the exception of the viewing angle ι , which has a sine function, to all parameters we assigned a uniform function to model the prior probability (for details see Table 2). For the kilonova emission, NMMA allows to fit and simulate data using several numerical and analytical models. Assuming a binary neutron star merger as a progenitor system, we adopted here the kilonova modelling resulting from the time dependent 3D Monte Carlo code POSSIS for modelling radiation transport (Bulla 2019, 2023). We used the kilonova model grid published in Dietrich et al. (2020) based on the first version of POSSIS (Bulla 2019) where the kilonova ejecta is represented with two components: 1) a high-velocity (0 . 08 < v dyn /c < 0 . 3) dynamical ejecta of mass M dyn rj with a lanthanide-rich composition distributed about the equatorial plane with half-opening angle Φ, and lanthanide-poor composition at higher latitudes, and 2) a slower (0 . 025 < v wind /c < 0 . 08) wind (or 'post-merger') component, which is a spherical ejecta released from the merger remnant and debris disk, with an intermediate lanthanide content and mass M wind ej (see Dietrich et al. 2020). In the fit we fixed the GRB 191019A luminosity distance at 1289.3 Mpc, as obtained from the measured redshift ( z = 0 . 248) and assuming a flat cosmological model (see Sect. 1). We also considered an additional error budget (in magnitudes), which takes into account the uncertainties on the model predictions as well as possible systematic errors ( em syserr in the corner plot), to which we assigned a uniform prior with range from 0 to 2 mag. The resulting best-fit light curves computed in each band are plotted in Figure 6, the fit corner plot is in Figure 7, while the 90% confidence interval and median values of each parameter , as well as the adopted prior functions, are quoted in Table 2. By simply inspecting the resulting light curves, it is evident that the afterglow component is constrained mostly by the X-ray data and it is dominant during the early epochs in the optical bands. The afterglow best fit jet core semiaperture angle is about ∼ 10 deg (9 +4 -3 deg), and the observer viewing angle is ∼ 4 deg (4 +3 -2 deg), thus within the jet core. The fireball isotropic kinetic energy E 0 (1 . 4 +1 . 8 -1 . 0 × 10 52 erg) is nicely compatible with the observed prompt emission equivalent isotropic energy E iso , by assuming a reasonable efficiency of about η ∼ 10%, where η = E iso E iso + E 0 . Indeed, the 15-150 keV fluence measured by Swift -BAT is (1 . 00 ± 0 . 03) × 10 -5 erg cm -2 (Krimm et al. 2019), corresponding to a radiated energy E iso = (1 . 70 ± 0 . 05) × 10 51 erg. The latter was computed by assuming a power-law spectrum in the BAT bandpass: this assumption is reasonable given the softness of the spectrum, which is best fit with a photon index Γ = 2 . 25 ± 0 . 05 (Krimm et al. 2019), suggesting that the prompt emission peak energy lies below the BAT bandpass. Figure 6 also clearly shows that at later epochs ( > 1 day), the observed flux lies above the predicted levels of the afterglow component, and the presence of a kilonova provides a better match. Indeed, by assuming only the afterglow model, i.e. by removing the kilonova component from our initial model, we obtained a worse fit, with Bayesian evidence ln( Z ) = -21 . 1. By considering the ratio with the Bayesian evidences of the joint afterglow and kilonova model, for which we obtain ln( Z 0 ) = -14 . 1, the resulting Bayes factor (ln( B ) =ln( Z/Z 0 ) = -7 . 0) indicates a strong preference for the model which includes the kilonova component (see, e.g., Kunert et al. 2024 and references therein). The kilonova properties we find from our fit are compatible with a dynamical ejecta mass of M dyn ej ∼ 0 . 02 M ⊙ and a wind mass M wind ej ∼ 0 . 04 M ⊙ , though with large uncertainties (relative errors ≥ 60%). These values are slightly higher (yet consistent within the uncertainties) with those found for AT2017gfo by assuming the same kilonova modelling (Dietrich et al. 2020). This is in line with the need of a brighter kilonova (of about a factor 4, see Sect. 3.3 and Fig. 4) by simply superposing an AT2017gfo-like light curve on the data. We stress here that the real picture is likely much more complicated than a two-component scenario, and the dynamical/wind masses inferred should rather be interpreted as belonging to some high/low velocity components of a multi-component scenario with multiple ejecta episodes (e.g., Bernuzzi 2020; Nedora et al. 2021). At the same time, the kilonova model we assumed is among the most sophisticated ones publicly available, and the assumption of a kilonova from a NS-NS merger progenitor is the most reasonable choice given that, so far, the only GRB with kilonova emission for which we were able to infer the progenitor nature was GRB170817/AT2017gfo, wich we know from gravitational wave data analysis being originated from a binary neutron star merger. Nevertheless, we explored also other kilonova models available within the NMMA framework, with different levels of sophistication, and which have different assumptions on the progenitor nature and on the kilonova ejecta properties (see Appendix C). We find that the dataset for GRB 191019A did not allow us to confidently distinguish among different kilonova models, apart disfavouring the most simplistic ones. However, in all cases, we find that a joint afterglow plus kilonova fit is preferred with respect to an afterglow-only model. Interestingly, both the afterglow-only and afterglow plus kilonova model, provide as a best-fit for the circumburst environment particle density a value n 0 < 1 cm -3 , which is typically deduced for short-GRB environments (e.g., Berger 2014; Fong et al. 2015). This result is in stark contrast with the interpretation by Lazzati et al. (2023) that invokes the presence of a very high density environment with n 0 ∼ 10 7 -10 8 cm -3 (see also Levan et al. 2023). T able 2 . The 90% confidence in terv al and median v alues of eac h parameter inferred from the sim ultaneous afterglo w ( Ry an et al. 202 0 ) and kilono v a ( Dietric h et al. 2020 ) mo dell ing p erformed b y using the co de NMMA. F or the afterglo w comp onen t, w e assumed a Gaussian jet profi le and fiducial v alue s for the microph ysical parameters (see Sect. 3.3 an d Fig. 6 and Figure 7 ). The b ottom part of the table quotes the assumed p r i ors in the Ba y esian inference analysis, where 'U' stands for Uniform function, with minim um and maxim um v alue s quoted in the brac k ets. Note -E 0 = kinetic fireball energy; n = particle n um b er densit y in the circum burst en vironmen t; θ c =half-op ening angle of the jet core; θ w =half-op ening a ngle of the jet truncated-wings; ι =viewing angle with resp ect to jet axis; p = electron energy distribution p o w er-la w index; ϵ e = sho c k e nergy fraction that go es in to th e electrons; ϵ B = sho c k energy fraction that go es in to the magne tic energy densit y; M dyn ej = dynamical ejecta mass; M wej = wind ejecta mass; Φ = half-op ening angle of lan thanide-ric h equatorial ejecta. Note -All observed (obs) AB magnitudes in the r ' , i ' and z ' filters are 3 σ upper limits (Fig. 5). Columns (2), (5), and (8) quote the expected (exp) magnitudes. All the corresponding differences (observed minus expected; columns 4, 7, 10) are lower limits.","pages":[8,9,10,11,12,13]},{"title":"4. DISCUSSION AND CONCLUSIONS","content":"Recent analysis of an increasing number of GRBs which were initially classified as long bursts, and therefore thought to be originating from collapsing massive stars, suggest a better compatibility of the observational data with compact binary merger progenitors. By ignoring the standard burst duration classification, typical features of GRBs associated with compact binary mergers are (1) the absence of an associated CC-SN, (2) the presence of an optical/NIR rebrightening compatible with a kilonova, (3) an early-type host galaxy, and (4) a distant GRB explosion site with respect to the corresponding galaxy center. GRB 191019A was suggested to belong to this sample of misclassified long GRBs, given the non detection of any SN component (Levan et al. 2023) despite its close cosmological distance ( z =0.248), and the evidence of a host galaxy dominated by an old stellar population. However, the proximity of the GRB 191019A explosion site to the photometric center of its host galaxy ( ≲ 100 pc) is at odds with typical short-GRB galactocentric distances, and could suggest a compact binary formed in the dense circumnuclear disk of an AGN. In this scenario (Lazzati et al. 2022) a very high-density environment ( n 0 > 10 7 cm -3 ) could be the origin of the long duration of the prompt emission (see Lazzati et al. 2023). Given the low distance of this burst, a kilonova ligh is expected to be seen. However, the temporal coverage and limiting magnitudes of previous data sets, did not allow to address this question yet. In this work, we further investigate the burst properties of GRB 1901910A and the nature of the optical transient that followed the burst, with the goal to find arguments in favour or against its possible compact merger origin. We found an independent support on the compact binary merger progenitor nature from the high-energy prompt emission temporal variability, which has been recently found to be an additional potential diagnostic to infer the progenitor nature, where high variability values suggest a compact binary merger progenitor (Camisasca et al. 2023). We noticed that, for GRB 191019A, the obtained MVT value of ∼ 20 ms locates this burst close to the centroid of the distribution of those GRBs associated with compact binary merger progenitors (Camisasca et al. 2023 and Fig. 1). The same variability properties were found also for two other long GRBs with evidence of kilonova (GRB 111210A and GRB 230307A). These results independently support past interpretations on the progenitor nature of GRB 191019A and address to the possible presence of a kilonova component in its optical transient. An optical transient following GRB 191019A was at first reported by Perley et al. (2019b) but its nature could not be clearly pinned down. Using GROND multi-color data obtained between 0.4 and 15 days post burst, we argued here that the temporal evolution of the transient's brightness disfavours a pure afterglow emission (Fig. 4). While the GROND data confirm the absence of a SN component (Fig. 5), we found that the luminosity evolution of the optical transient is in agreement with an afterglow plus a kilonova signal compatible with AT2017gfo redshifted to the distance of the GRB host galaxy, though slightly brighter by a factor of a few. By modelling the afterglow and the kilonova component, we were able to estimate a kilonova ejecta mass which is slightly higher but still consistent, within the large uncertainties, with the one measured for AT2017gfo with similar assumptions on the progenitor system (a binary neutron star) and on the kilonova modelling (e.g., Dietrich et al. 2020). By assuming other kilonova models with different levels of complexity and different progenitor assumptions, the data did not allow to discriminate among them, apart disfavouring the most simplistic one. However, in all cases, results strongly favour the presence of a kilonova with respect to an afterglow-only model (Table 5). Our findings strongly suggest that GRB 191019A might belong to the increasing list of long GRBs with an associated kilonova, beside GRB 211211A and GRB 230307A, and other cases with less robust evidence, as for instance GRB 060614 (e.g., Jin et al. 2015). Another interesting findings from our analysis, is the evidence of a low circumburst density (Table 2). This result is in stark constrast with the one obtained by Lazzati et al. (2023) from prompt emission light curve modelling, where extreme values of n 0 > 10 7 cm -3 were found. These high density values were considered consistent with the apparent position of the GRB, very close to the center of an AGN-like host galaxy (though no direct proofs on the existence of an AGN have been provided yet, see Levan et al. 2023). The interaction between a jet and an AGN dense circumnuclear environment, however, may choke or strongly suppress GRB relativistic jets, unless particularly bright and long, as also supported by recent studies (e.g., Zhang et al. 2024). GRB 191019A is classified as a normal burst, with a well detected optical counterpart (see, e.g., Zhang et al. 2024), and with low dust extinction along the line-of-sight within the host ( A host V in the range from 0.06 to 0.10 mag, Levan et al. 2023), further supported by the UV-band detections (LaPorte et al. 2019). These properties are more consistent with a low density environment, as the one we obtained from a joint fitting of an afterglow and a kilonova component model on the X-ray and optical data of GRB 191019A. More in general, if GRB 191019A is a disguised short GRB with a compact binary merger origin, as both past studies and this work suggest, a low density environment is in agreement with the typical density values found in short GRBs and with the environments expected for compact binary mergers. In a low density scenario, the very small offset from the host galaxy of GRB 191019A, can be explained with a negligible transverse projection of a GRB located far off centre. In this scenario, the long duration of the prompt emission, which Lazzati et al. (2023) explained as caused by the extremely dense circumburst environment, may have instead an intrinsic origin. An interesting hypothesis that was proposed in the past to explain short GRBs with soft extended emission (e.g., Barkov & Pozanenko 2011; Kisaka & Ioka 2015) and which has been recently studied in much greater details (e.g., Musolino et al. 2024), invokes fall-back accretion onto the remnant compact object. Whether this is the case for GRB 191019A goes beyond the scope of this work and will be addressed in another study. We thank the anonymous referee for useful comments and suggestions which improved our work. We are grateful to P. Pang for his precious support on NMMA and useful discussions. ANG acknowledges logistic support by the Thuringer Landessternwarte Tautenburg, Germany. G.S. and P.S acknowledge the support by the State of Hesse within the Research Cluster ELEMENTS (Project ID 500/10.006). A.R. and E.P. acknowledge support by PRIN-MIUR 2017 (grant 20179ZF5KS). Part of the funding for GROND (both hardware and personnel) was generously granted by the Leibniz-Prize to G. Hasinger (DFG grant HA 1850/28-1) and by the Thuringer Landessternwarte Tautenburg. S.B. aknowledges funding from the EU Horizon under ERC Consolidator Grant, no. InspiReM-101043372. A.E.C. is partially supported by the 2023/24 'Research and Education' grant from Fondazione CRT. The OAVdA is managed by the Fondazione Cl'ement Fillietroz-ONLUS, which is supported by the Regional Government of the Aosta Valley, the Town Municipality of Nus and the Unit'e des Communes valdotaines Mont-Emilius. The LBT is an international collaboration of the University of Arizona, Italy (INAF: Istituto Nazionale di Astrofisica), Germany (LBTB: LBT Beteiligungsgesellschaft), The Ohio State University, representing also the University of Minnesota, the University of Virginia, and the University of Notre Dame. This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester.","pages":[13,15,16]},{"title":"Facilities: GROND, Swift (BAT, XRT and UVOT), LBT","content":"Software: Afterglowpy (Ryan et al. 2020), NMMA (Pang et al. 2023), astropy (Astropy Collaboration et al. 2013, 2018), HOTPANTS (Becker 2015), PyRAF (Science Software Branch at STScI 2012), DRAGONS (Labrie et al. 2019), WCSTools (Mink 2019).","pages":[16]},{"title":"A. HOST GALAXY","content":"Note -Magnitudes are measured within an aperture with radius of 4 × FWHM and are not corrected for Galactic foreground extinction. Table 4 provides the magnitude of the GRB host galaxy based on GROND's 4th-epoch observations and LBT's visit of the field four years after the burst (see Sect. 2.2).","pages":[17]},{"title":"B. GRB-SN PEAK LUMINOSITY RANGE","content":"We further investigated on possible biases towards the faint end of our GRB-SN sample by comparing with SN Ic peak luminosity properties observed so far. The r-band absolute peak-magnitude ( M r,peak ) distribution of broad line (BL) Type Ic SNe spans at most 2.5 mags (between -17 . 7 and -20 . 8 mag, with an average of M r,peak = -18 . 7 ± 0 . 7 mag, Taddia et al. 2019; Barbarino et al. 2021; Gomez et al. 2022). GRB-SNe observed to date with good data sampling, are at least ∼ 1 mag brighter and trace the high-luminosity end of the above Type Ic BL distribution (e.g., Hjorth & Bloom 2012; Hjorth 2013; Cano et al. 2017; Klose et al. 2019), but the width of their peak luminosity distribution (Hjorth & Bloom 2012; Cano et al. 2017; Kann et al. 2019b; Klose et al. 2019) is not substantially different to the width of the corresponding peak luminosity distribution of type Ic and BL-SNe (see also Soderberg et al. 2006). By considering the GRB-SNe with the best follow-up in multiple bands (including spectroscopy), on the high end site of the peak luminosity distribution remains GRB 111209A/SN 2011kl (e.g., Nakauchi et al. 2013; Kann et al. 2019b) while on the low-end site there are GRB 100316D/SN 2010bh (e.g., Starling et al. 2011; Olivares E. et al. 2012b) and GRB 060218/SN 2006aj (e.g., Pian et al. 2006; Ferrero et al. 2006): the r -band peak magnitudes of these extreme GRB-SNe have a difference that lies between 2 and 3 mags, similar to the broad line (BL) Type Ic SNe. In principle, the long bursts GRB 990712 (Bjornsson et al. 2001), GRB 021211 (Della Valle et al. 2003; Zeh et al. 2004), GRB 040924 (Soderberg et al. 2006; Wiersema et al. 2008), and GRB 060904B (Cano et al. 2017) could have been followed by even slightly fainter SNe than SN 2010bh, but in these cases the data base is comparably poor and no strong conclusions could be made. While, e.g., any SN that was associated with the long bursts GRB 060605 and GRB 060614 must have had a luminosity < 1% of the luminosity of the prototypical SN 1998bw (Fynbo et al. 2006), here and in similar cases (e.g., GRB 111005A; Michaglyph[suppress]lowskI et al. 2018; Tanga et al. 2018) there is no evidence for SN light. In conclusion, if substantially less luminous GRB-SNe do exist formally remains an open question, at least from the obervational point of view, and the peak luminosity range of GRB-SNe sample we have used to infer the luminosity of any SN associated with GRB 191019A is the most robust so far.","pages":[17]},{"title":"C. AFTERGLOW AND KILONOVA JOINT FIT COMPARISON","content":"In Table 5 we present the results obtained for different kilonova models within the NMMA framework, in addition to the kilonova model presented in Sect. 3.3. At first, we have tested a model from Kasen et al. (2017) (Kasen17-Jet in Table 5) which assumes only one ejecta component and has three parameters: the ejecta mass, the ejecta velocity and the lanthanide mass fraction ( χ lan ). Then we made a different assumption on the progenitor by considering a kilonova emission from a neutron star black hole binary system as predicted with POSSIS (Bu19-NSBH-Jet in Table 5), which has three parameters: the dynamical ejecta mass, the wind mass and the orbital plane inclination, which is linked to the viewing angle of the jet assumed in the afterglow modelling. These models were finally compared with the results obtained by assuming the simple analytical model described in Metzger (2017) (Metzger17-Jet in Table 5), which assumes one component and has four parameters: the ejecta mass, the ejecta velocity, the power-law index β of the ejecta mass distribution expressed as a function of its velocity (the faster ejecta matter lies ahead of slower matter and the distribution of mass with velocity greater than a value v 0 can be approximated with a power-law M ( > v 0 ) = M ( v/v 0 ) -β , and the opacity k r . The afterglow emission was modelled within the Afterglowpy framework (Afterglow in Table 5), as described in Sect. 3.3. However, contrary to our previous analysis, the microphyiscal parameters ϵ e and ϵ B are now fixed to 0 . 5 and 0 . 01, respectively. In doing so, we aim at reproduce similar assumptions to Lazzati et al. (2023) and further investigate on the circumburst density estimates. The highest Bayesian evidence ( Z ) is obtained for the Bu19-NSBH-Jet model. Following Kunert et al. (2024) and references therein, by considering Bu19-NSBH-Jet as the reference model, we find -1 . 10 < ln( Z/Z Bu 19 -NSBH -Jet ) < 0 for the Kasen17-Jet and Bu19-BNS-Jet models, while ln( Z/Z Bu 19 -NSBH -Jet ) < -4 . 5 for Metzger17-Jet and Afterglow models, indicating a strong evidence against the latter two models, while no preference could be set among Bu19-BNSJet, Bu19-NSBH-Jet and Kasen17-Jet models.","pages":[17,18]},{"title":"REFERENCES","content":"Abbott, B. P., et al. 2017, ApJ, 848, L13, doi: 10.3847/2041-8213/aa920c Ahumada, T., Singer, L. P., Anand, S., et al. 2021, Nature Astronomy, 5, 917, doi: 10.1038/s41550-021-01428-7 Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068 Astropy Collaboration, Price-Whelan, A. M., Sip\"ocz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f Barbarino, C., Sollerman, J., Taddia, F., et al. 2021, A&A, 651, A81, doi: 10.1051/0004-6361/202038890 Barkov, M. V., & Pozanenko, A. S. 2011, MNRAS, 417, 2161, doi: 10.1111/j.1365-2966.2011.19398.x Barthelmy, S. D., Barbier, L. M., Cummings, J. R., et al. 2005, SSRv, 120, 143, doi: 10.1007/s11214-005-5096-3 Becker, A. 2015, HOTPANTS: High Order Transform of PSF ANd Template Subtraction, Astrophysics Source Code Library, record ascl:1504.004. http://ascl.net/1504.004 Beniamini, P., Duque, R., Daigne, F., & Mochkovitch, R. 2020, MNRAS, 492, 2847, doi: 10.1093/mnras/staa070 Berger, E. 2014, ARA&A, 52, 43, doi: 10.1146/annurev-astro-081913-035926 Bernuzzi, S. 2020, General Relativity and Gravitation, 52, 108, doi: 10.1007/s10714-020-02752-5 Bjornsson, G., Hjorth, J., Jakobsson, P., Christensen, L., & Holland, S. 2001, ApJL, 552, L121, doi: 10.1086/320328 Bulla, M. 2019, MNRAS, 489, 5037, doi: 10.1093/mnras/stz2495 -. 2023, MNRAS, 520, 2558, doi: 10.1093/mnras/stad232 Burrows, D. N., Hill, J. E., Nousek, J. A., et al. 2005, SSRv, 120, 165, doi: 10.1007/s11214-005-5097-2 Camisasca, A. E., Guidorzi, C., Amati, L., et al. 2023, A&A, 671, A112, doi: 10.1051/0004-6361/202245657 Cano, Z., Wang, S.-Q., Dai, Z.-G., & Wu, X.-F. 2017, Advances in Astronomy, 2017, 8929054, doi: 10.1155/2017/8929054 Chambers, K. C., Magnier, E. A., Metcalfe, N., et al. 2016, arXiv e-prints, arXiv:1612.05560, doi: 10.48550/arXiv.1612.05560 Coulter, D. A., Foley, R. J., Kilpatrick, C. D., et al. 2017, Science, 358, 1556, doi: 10.1126/science.aap9811 Coward, D. M., Gendre, B., Tanga, P., et al. 2017, PASA, 34, e005, doi: 10.1017/pasa.2016.61 Dainotti, M. G., De Simone, B., Islam, K. M., et al. 2022, ApJ, 938, 41, doi: 10.3847/1538-4357/ac8b77 Della Valle, M., Malesani, D., Benetti, S., et al. 2003, A&A, 406, L33, doi: 10.1051/0004-6361:20030855 Dietrich, T., Coughlin, M. W., Pang, P. T. H., et al. 2020, Science, 370, 1450, doi: 10.1126/science.abb4317 Note -E 0 = kinetic fireball energy; n = particle number density in the circumburst environment; θ c =half-opening angle of the jet core; ι =viewing angle with respect to jet axis; p = electron energy distribution power-law index; M dyn ej = dynamical ejecta mass; M wind ej = wind ejecta mass; M ej = total ejecta mass; v ej = ejecta velocity; Φ = half-opening angle of lanthanide-rich equatorial ejecta; ϵ e = shock energy fraction that goes into the electrons; ϵ B = shock energy fraction that goes into the magnetic energy density; χ lan = lanthanide mass fraction (Kasen et al. 2017) ; κ r = opacity; β = power-law index of the ejecta mass distribution as a function of its velocity (Metzger 2017); ln( Z ) = natural logarithm of the bayes evidence. We note that the Bu19-BNS-Jet model is the same as the one presented in Sect. 3.3 where now ϵ e and ϵ B are fixed. Evans, P. A., Osborne, J. P., Burrows, D. N., et al. 2019, GRB Coordinates Network, 26034, 1 Evans, P. A., Beardmore, A. P., Page, K. L., et al. 2007, A&A, 469, 379, doi: 10.1051/0004-6361:20077530 -. 2009, MNRAS, 397, 1177, doi: 10.1111/j.1365-2966.2009.14913.x Evans, P. A., Willingale, R., Osborne, J. P., et al. 2010, A&A, 519, A102, doi: 10.1051/0004-6361/201014819 Ferrero, P., Kann, D. A., Zeh, A., et al. 2006, A&A, 457, 857, doi: 10.1051/0004-6361:20065530 Fong, W., Berger, E., Margutti, R., & Zauderer, B. A. 2015, ApJ, 815, 102, doi: 10.1088/0004-637X/815/2/102 Fong, W.-f., Nugent, A. E., Dong, Y., et al. 2022, ApJ, 940, 56, doi: 10.3847/1538-4357/ac91d0 Fontana, A., Dunlop, J. S., Paris, D., et al. 2014, A&A, 570, A11, doi: 10.1051/0004-6361/201423543 Fynbo, J. P. U., Perley, D. A., de Ugarte Postigo, A., et al. 2019, GRB Coordinates Network, 26041, 1 Fynbo, J. P. U., Watson, D., Thone, C. C., et al. 2006, Nature, 444, 1047, doi: 10.1038/nature05375 Gehrels, N., Chincarini, G., Giommi, P., et al. 2004, ApJ, 611, 1005, doi: 10.1086/422091 Gendre, B., Crisp, H., Coward, D., et al. 2019, GRB Coordinates Network, 26098, 1 Gomez, S., Berger, E., Nicholl, M., Blanchard, P. K., & Hosseinzadeh, G. 2022, ApJ, 941, 107, doi: 10.3847/1538-4357/ac9842 Gompertz, B. P., Ravasio, M. E., Nicholl, M., et al. 2023, Nature Astronomy, 7, 67, doi: 10.1038/s41550-022-01819-4 Greiner, J., Bornemann, W., Clemens, C., et al. 2008, PASP, 120, 405, doi: 10.1086/587032 Greiner, J., Mazzali, P. A., Kann, D. A., et al. 2015, Nature, 523, 189, doi: 10.1038/nature14579 Hjorth, J. 2013, Philosophical Transactions of the Royal Society of London Series A, 371, 20120275, doi: 10.1098/rsta.2012.0275 Troja, E. 2023, Universe, 9, 245, doi: 10.3390/universe9060245 Yolda¸s, A. K., Kruhler, T., Greiner, J., et al. 2008, in American Institute of Physics Conference Series, Vol. 1000, American Institute of Physics Conference Series, ed. M. Galassi, D. Palmer, & E. Fenimore, 227-231, doi: 10.1063/1.2943450","pages":[18,19,21]}],"string":"[\n {\n \"title\": \"ABSTRACT\",\n \"content\": \"GRB 191019A was a long Gamma-ray burst (GRB) lasting ∼ 65 s and, as such, originally thought to be linked to a core-collapse supernova. However, even though follow-up observations identified the optical counterpart close to the bright nucleus of a nearby ancient galaxy ( z =0.248), no associated supernova was found. This led to the suggestion that the burst was caused by the merger of two compact stellar objects, likely in a dense circumnuclear environment. By using a recently developed diagnostic tool based on prompt emission temporal properties, we noticed that GRB 191019A falls among those long GRBs which are associated with compact mergers and with evidence of kilonova light. We thus re-analyzed unpublished GROND multi-color ( g ' r ' i ' z ' JHK s ) data obtained between 0.4 and 15 days post trigger. Image subtraction confirmed the optical counterpart in all four optical bands, with GROND tracking its fading until 1.5 days post-burst. Incorporating publicly available Swift -XRT data, a joint fit of an afterglow plus a kilonova model revealed a better match than an afterglow-only scenario. The resulting kilonova properties resemble those of AT2017gfo associated with the binary neutron star merger GW170817, with a total ejected mass of ∼ 0 . 06 M ⊙ . Contrary to previous findings inferring a high-density circumburst environment ( n 0 ∼ 10 7 -8 cm -3 ), our analysis finds standard conditions ( n 0 ∼ 1 cm -3 ), suggesting the long duration of GRB 191019A was intrinsic rather than due to jet interaction with a dense external medium. Keywords: Gamma-ray bursts - Compact objects\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"The puzzling long GRB 191019A: Evidence for Kilonova Light\",\n \"content\": \"G. Stratta , 1, 2, 3, 4 A. M. Nicuesa Guelbenzu , 5 S. Klose , 5 A. Rossi , 6 P. Singh, 1, 6 E. Palazzi , 6 C. Guidorzi, 7, 8, 6 A. Camisasca, 9 S. Bernuzzi, 10 A. Rau, 11 M. Bulla, 7, 8, 12 F. Ragosta, 13, 14 E. Maiorano, 6 and 15 D. Paris 1 Institut fur Theoretische Physik, Goethe Universitat, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany 2 INAF - Osservatorio di Astrofisica e Scienza dello Spazio, via Piero Gobetti 93/3, 40129 Bologna, Italy 3 Istituto di Astrofisica e Planetologia Spaziali, via Fosso del Cavaliere 100, I-00133 Roma, Italy 4 Istituto Nazionale di Fisica Nucleare-Roma 1, Piazzale Aldo Moro 2, I-00185 Roma, Italy 5 Thuringer Landessternwarte Tautenburg, Sternwarte 5, 07778 Tautenburg, Germany 6 INAF - Osservatorio di Astrofisica e Scienza dello Spazio, via Piero Gobetti 93/3, 40129 Bologna, Italy 7 Department of Physics and Earth Science, University of Ferrara, Via Saragat 1, 44122 Ferrara, Italy 8 INFN - Sezione di Ferrara, Via Saragat 1, 44122 Ferrara, Italy 9 Astronomical Observatory of the Autonomous Region of the Aosta Valley (OAVdA), Loc. Lignan 39, I-11020, Nus (Aosta Valley), Italy 10 Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universitat Jena, 07743, Jena, Germany 11 Max-Planck-Institut fur Extraterrestrische Physik, Giessenbachstraße 1, 85748 Garching, Germany 12 INAF, Osservatorio Astronomico d'Abruzzo, via Mentore Maggini snc, 64100 Teramo, Italy 13 Dipartimento di Fisica 'Ettore Pancini', Universit'a di Napoli Federico II, Via Cinthia 9, 80126 Naples, Italy 14 INAF - Osservatorio Astronomico di Capodimonte, Via Moiariello 16, I-80131 Naples, Italy 15 INAF - Osservatorio Astronomico di Roma, via Frascati 33, 00040 Monte Porzio Catone, Italy\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1. INTRODUCTION\",\n \"content\": \"The empirical classification of gamma-ray bursts (GRBs) into two classes according to the bimodal burst duration and hardness distribution (see, e.g., Kouveliotou et al. 1993), has been interpreted as the evidence of two different progenitor channels. Indeed, it was found that the vast majority of sufficiently nearby long GRBs are found to be spatially and temporally coincident with core-collapse (CC) supernovae (SNe) and hosted in star-forming galaxies. On the other hand, short GRBs lack any associated SN and are typically located in the outer regions of early-type galaxies, suggesting a progenitor belonging to an old stellar population, formally consistent with compact binary mergers. This picture was confirmed by the association of the short GRB 170817A with the gravitational wave event GW170817 that was compatible with a binary neutron star (NS-NS) merger (e.g., Abbott et al. 2017). From the very same event and thanks to its spatial proximity (40 Mpc), the first robust evidence of a kilonova (AT2017gfo) was obtained (e.g., Coulter et al. 2017), confirming the predicted thermal emission from matter released and heated during a NS-NS merger (e.g., Li & Paczy'nski 1998). In the last decade, kilonova signatures were observed in a number of past short GRBs that were sufficiently nearby, as for instance GRB 130603B ( z =0.3565, Tanvir et al. 2013), GRB 150101B ( z =0.134, Troja et al. 2018), GRB 160821B ( z =0.1613, Troja et al. 2019), though with much lower significance than AT2017gfo due to the larger distances of these events (see, e.g., Troja 2023 for a review). In recent years, the standard long and short-GRB progenitor paradigm has been blurred by mounting evidence of the existence of long GRBs with no associated CC-SN but with evidence of a kilonova component, more consistent with a compact binary merger progenitor. Two striking examples are the long GRB 211211A at redshift z = 0 . 076 (350 Mpc; e.g., Troja et al. 2018; Mei et al. 2022; Rastinejad et al. 2022; Yang et al. 2022; Gompertz et al. 2023) and GRB 230307A at z = 0 . 0646 (300 Mpc; e.g., Levan et al. 2023), for which a kilonova was clearly identified in the optical afterglow. In addition, evidence of short GRBs showing massive star collapse properties were also found (e.g., Ahumada et al. 2021; Rossi et al. 2022) further contributed to dismantle the standard GRB classification scheme. These results suggest that some of the past GRB progenitor classifications based on the burst duration and hardness only, have to be revisited. This holds in particular for GRB 191019A. GRB 191019A was discovered (Simpson et al. 2019) with the Burst Alert Telescope (BAT; Barthelmy et al. 2005) on board the Neil Gehrels Swift Observatory (Gehrels et al. 2004). It showed a complex burst light curve with multiple peaks with a duration T 90 = 64 . 6 ± 4 . 5 s. Its 15-150 keV spectrum could be well modeled with a power law with a photon index of 2 . 25 ± 0 . 05 and a fluence of 10 -5 erg cm -2 (Krimm et al. 2019) putting this source among the soft/long GRBs. Follow-up observations with the Swift X-ray telescope (XRT; Burrows et al. 2005) robustly identified an uncatalogued X-ray source (Evans et al. 2019). Rapid follow-up observations with the Zeiss-1000 1-m telescope of Tien Shan Astronomical Observatory revealed only one optical source within the XRT error circle with R = 18 . 37 ± 0 . 03 mag at T 0 +14 min (Reva et al. 2019). The position of this source was coincident with a catalogued object quoted to be fainter than the observed one, leading to the interpretation this is the host galaxy of GRB 191019A with the afterglow on top of it (Reva et al. 2019). The Zadko telescope (Coward et al. 2017) observed GRB 191019A field at similar epochs and confirmed the object reported by Reva et al. (2019) with R =18.92 mag, with no significant flux variation within 1.5 hours (Gendre et al. 2019). Similar claims of a lack of flux variations were provided by other teams performing early-epoch observations (Fynbo et al. 2019, Nicuesa Guelbenzu 2019, Perley et al. 2019a, Zhu et al. 2019). However, late observations performed with the Nordic Optical Telescope (NOT) at T 0 +3.25 days were analyzed with image subtraction methods and revealed that the source was fading, confirming the optical transient plus host identification (Perley et al. 2019b). Spectroscopic analysis of the host galaxy showed several absorption lines at redshift z = 0 . 248 and a spectral energy distribution consistent with a galaxy dominated by an old stellar population ( > 1 Gyr), with a star formation rate (SFR) of 0 . 06 ± 0 . 03 M ⊙ yr -1 , a stellar mass of 3 × 10 10 M ⊙ , and small dust extinction of A V = 0 . 19 ± 0 . 08 mag (Levan et al. 2023). The measured SFR is at odds with the typical high values inferred in long GRB hosts (but see Rossi et al. 2014). Further hints of anomalies for the long GRB 191019A came from the absence of an associated SN, despite the low redshift. Deep limits in g, r and z obtained between 2 and 73 days with the NOT and optical imaging with the Hubble Space Telescope at 30 and 184 days, put strong constraints on any SN emission up to > 20 times less luminous than SN1998bw (Levan et al. 2023). All these properties are at odds with a massive-star origin of GRB 191019A, while formally consistent with a compact object binary merger progenitor. Though, the deep limits obtained with the NOT Telescope at > 2 days could not confirm the presence of an early kilonova emission (Levan et al. 2023). An interesting feature characterizing GRB 191019A is its projected distance from the host galaxy center of ≲ 100 pc, the closest distance among all known short GRBs to their corresponding galactic nucleus (Levan et al. 2023). Indeed, compact binary progenitors formed in stellar binary systems are thought to have large kick velocities acquired during the CC-SN phase of the binary components, and for this reason short GRBs are typically found in the outskirts of their host galaxies, with large offsets from the center (e.g., Berger 2014; Fong et al. 2022; O'Connor et al. 2022). As noted by Levan et al. (2023), the proximity of GRB 191019A to the host galaxy center suggests that the possible progenitor compact binary system could have formed through dynamical encounters, which are thought to be favored in the dense gaseous environment of supermassive black hole surrounding disks through kinetic energy dissipation ('gas-capture' binary formation channel, Tagawa et al. 2020). An exciting consequence of this scenario is that dense environments could also alter the prompt emission properties of a short GRB, making it longer and softer (Lazzati et al. 2023). This possibility was explored for GRB 191019A and it was found compatible with an environmental density of the order of 10 7 to 10 8 cm -3 (Lazzati et al. 2023). Starting from Lazzati et al.'s conclusions, Wang et al. (2024) investigated the interaction of a possible kilonova ejecta from GRB 191019A with a dense circumstellar medium (CSM) that could be detected years after the merger. Their model predicts that in a very dense environment, the smaller the kilonova ejected mass, the fainter is the radioactively-powered luminosity but the higher is the contribution from kilonova-CSM interaction at late times. No evidence of kilonova ejecta-CSM interaction was found for GRB 191019A so far, and a possible detection in the future would require an ejected mass less than 2 × 10 -5 M ⊙ (Wang et al. 2024). Here we further investigate the properties of GRB 1901910A and the nature of the optical transient that followed the burst, with the goal to find arguments in favour or against its possible compact merger origin. We use the recently developed GRB prompt emission minimum variability (MVT) criterion (Camisasca et al. 2023) to explore the properties of the burst, and perform a complete reanalysis of the light curve of the optical/X-ray transient with particular emphasis on so far not analyzed GROND (Greiner et al. 2008) observations starting 10.3 hours post burst (Nicuesa Guelbenzu 2019). We use the sophisticated Nuclear - Multi-Messenger Astronomy (NMMA, Dietrich et al. 2020; Pang et al. 2023) software package which allows for afterglow and kilonova joint Bayesian inference, to explore if the multi-color light curve of the optical transient can be understood as powered by afterglow emission only or if a kilonova component was present too. Throughout this paper, we adopt a flat cosmological model with H 0 =67.4 km s -1 Mpc -1 , Ω M =0.315, and Ω Λ =0.685 (Planck Collaboration et al. 2020). For these parameters a redshift of z =0.248 (Fynbo et al. 2019) corresponds to a luminosity distance of d L = 1.29 Gpc, 1 arcsec corresponds to 4.03 kpc projected distance, and distance modulus is m -M = 40 . 55 mag. The Milky Way reddening E ( B -V ) along the line of sight towards the source is between 0.03 and 0.04 mag (Schlegel et al. 1998; Schlafly & Finkbeiner 2011). We present the observations and data reduction in Sect. 2, and our results in Sect. 3. In our data analysis we corrected the apparent magnitudes for the Galactic reddening and adopted a host-galaxy visual extinction along the line of sight of A host V = 0.06 mag (Levan et al. 2023). Discussion and conclusions are presented in Sect. 4.\",\n \"pages\": [\n 1,\n 2,\n 3\n ]\n },\n {\n \"title\": \"2.1. GROND early-time Observations\",\n \"content\": \"With a duration of about 65 seconds, GRB 191019A was a classical long GRB. At a redshift of z = 0 . 248, a rising SN 1998bw component was thus expected to be detectable with GROND within about two weeks after the GRB trigger. Therefore, following earlier efforts with GROND to explore GRB-SN light curves (Olivares E. et al. 2012a, 2015; Greiner et al. 2015; Kann et al. 2019a; Klose et al. 2019) we performed follow-up observations of the field during several epochs between 0.4 (Nicuesa Guelbenzu 2019) and 15.4 days post GRB trigger (Table 1). Observations were then terminated since no evidence for SN light was found. GROND data were reduced in a standard fashion (bias subtraction, flat fielding, co-adding; Kruhler et al. 2008; Yolda¸s et al. 2008), based on numerical routines in IRAF (Image Reduction and Analysis Facility; Tody 1993).\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"2.2. LBT late-time Observations\",\n \"content\": \"Late-time observations of the field were performed with the Large Binocular Telescope (LBT) using the two twin LBC instruments equipped with the Sloan filters g ' r ' i ' z ' on October 20, 2023. These data were reduced using the data reduction pipeline developed at INAF - Osservatorio Astronomico di Roma (Fontana et al. 2014) which includes bias subtraction and flat-fielding, bad pixel and cosmic ray masking, astrometric calibration, and coaddition. On the LBT images, we measure the AB magnitudes for the host given in Table 4 (see Appendix A). Within the errors, these values are consistent with those measured with GROND in epoch 4 ( T -T 0 = 7 . 4756 days). Based on these data we conclude that 7 days post burst the transient did not contribute anymore to the observed flux in g ' r ' i ' z ' in a measurable quantity.\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"2.3. Image subtraction and photometry\",\n \"content\": \"In order to search for a transient hidden by the relatively bright host galaxy, we performed image subtraction on the GROND images using HOTPANTS (Becker 2015). HOTPANTS convolves the template and the source images to the same\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"Stratta et al.\",\n \"content\": \"Note -Column (2) provides the midtime of the observation, in units of days from burst trigger. Magnitudes are not corrected for Galactic foreground extinction. The photometry for epochs 1a to 3 was obtained after image subtraction against epoch 4 (see § 2.3 for details). On the contrary, the 3 σ upper limits for epoch 4 to 6 have been measured directly, without image subtraction. point spread function (PSF) and photometric scale. As a template image we used the GROND 4th-epoch observations (Table 1) since these images have the best seeing ( ∼ 1 . '' 0). We aligned the template and the source images by means of wcsremap 1 . The Gaussian input parameters in HOTPANTS were calculated following Becker (2015) considering the case in which the template FWHM is smaller than the source FWHM. In this way, the template images were smoothed to the seeing of the corresponding source images. In addition, we fixed several other parameters for HOTPANTS following Hu et al. (2022), but let free to vary the number of each region's stamps, the convolution kernel half width, and the size of Gaussians which compose the kernel. In doing so, we selected those input parameters that minimized the background noise on the residual image in a star-free region close to the position of the host galaxy. All resulting residual images were then photometrically calibrated with respect to the source images (see below). In the case of a non-detection of the transient the upper limit we used was measured locally on the subtracted image (Table 1). We have confirmed our results using also the 5th GROND epoch as template (seeing ∼ 1 . '' 3), although we obtained a lower SNR. We double checked the detections using the late LBT images (about 4 years or 1460 days after the burst trigger, see Sect. 2.2 and Table 4 in Appendix A). The g ' i ' z ' detections are confirmed for epochs 1a and 2, but with lower SNR likely due to the additional plate scaling, and a general worse PSF shape due to not perfect collimation. We have not checked the r ' -band since the above mentioned problem was worse in this filter, although it did not compromise the aperture photometry in Table 4. For this filter we have instead used (as template) a stack of the best Gemini-S images 2 obtained at 58-63 days after the burst (see Levan et al. 2023), and we obtained consistent results. In other words, the results obtained with GROND could not be improved. Finally, to better constrain the late transient decay, we performed image subtraction on the Gemini r -band image at 11.4 days, using the later Gemini images as template (see above), and obtained for the optical transient an upper limit of r > 25 . 8 AB mag. Apparent magnitudes were obtained by performing aperture photometry using DAOPHOT and APPHOT under PyRAF/IRAF with increasing apertures up to 4 times the FWHM and by PSF photometry with the DAOPHOT and ALLSTAR tasks of IRAF. PSF-fitting was used to measure the magnitudes of the transient after image subtraction. In the optical bands the data were calibrated using Pan-STARRS (Chambers et al. 2016), in the NIR bands using 2MASS (Skrutskie et al. 2006). Central wavelengths of the GROND filter bands are listed in Rossi et al. (2011).\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"3.1. Application of the prompt emission minimum variability criterion\",\n \"content\": \"According to Camisasca et al. (2023), GRB prompt emission minimum variability timescale (MVT) represents a diagnostic to infer the progenitor nature of a GRB. By estimating MVT from the full width at half maximum of the shortest, statistically significant peak in the prompt emission light curve (FWHM min ) of hundreds of GRBs, it was found that high variability (low FWHM min values) typically belongs to compact binary mergers. Interestingly, this result was found to be independent of the burst duration and for this reason it is an important probe for the nature of peculiar long GRBs with properties more similar to those of short GRBs. By analyzing the results from the Swift GRB sample analyzed by Camisasca et al. (2023), we checked which long GRBs sufficiently nearby for a possible kilonova detection, i.e. at z < 0 . 5, and without any associated SN, had an FWHM min compatible with compact binary merger values, and we found that GRB 191019A was satisfying our criteria. Specifically, for this GRB an MVT of FWHM min = 0 . 196 +0 . 068 -0 . 05 s has been measured, which is compatible with the values typically found for GRBs associated with compact binary mergers (Fig. 1). In particular, given the long burst duration of GRB 191019A ( T 90 = 64 . 35 ± 4 . 35 s), its position in the FWHM min vs T 90 diagram lies among the short GRBs showing a soft 'Extended Emission' (SEE, e.g., Norris & Bonnell 2006; Minaev et al. 2010). Interestingly, two famous long GRBs for which a kilonova component was found in the optical afterglow, namely GRB 211211A (Rastinejad et al. 2022; Troja et al. 2022; Yang et al. 2022; Gompertz et al. 2023) and GRB 230307A (e.g., Levan et al. 2023), lie in the same region of the diagram of GRB 191019A (Fig. 1). The SEE distribution in the T 90 -FWHM min plane clearly overlaps with the distribution of long GRBs associated with SNe, but at the same time the centroids of the two distributions are well separated: interestingly, GRB 191019A lies in the middle of the SEE region and in the outskirts of the SN-associated cases. This property alone cannot be taken as compelling evidence that GRB 191019A behaves like a SEE, but suggests a probable SEE-like nature than a collapsar one. These findings further support past interpretation that also GRB 191019A originated from a compact binary merger (e.g., Levan et al. 2023), and address the possible presence of a kilonova component, similarly to GRB 211211A and GRB 230307A.\",\n \"pages\": [\n 5\n ]\n },\n {\n \"title\": \"3.2. Re-analysis of the light curve of the Transient that followed the burst\",\n \"content\": \"3.2.1. X-rays Adopting a power-law spectral energy distribution (SED), the best fit of the Swift -XRT 0.3-10.0 keV spectrum, computed by the automatic algorithm of the UK Swift Science Data Center (UKSSDC) Swift -XRT GRB Spectrum Repository 3 in the 3 . 2 -32 . 4 ks temporal window, provides a photon index of Γ = 2 . 0 +0 . 4 -0 . 3 and an intrinsic equivalent hydrogen column density of N H , z = 1 . 2 +1 . 6 -1 . 2 × 10 21 cm -2 in addition to a Galactic N H = 3 . 28 × 10 20 cm -2 . We obtained from the UKSSDC Burst Analyzer tool (Evans et al. 2010), the absorption-corrected X-ray fluxes in the energy range 0.3-10.0 keV, and we converted them into 1 keV flux densities by assuming a power law spectrum and the corresponding photon index provided by the Burst Analyzer in each temporal bin of the light curve. The flux density evolution in the X-ray band is compatible with a power-law decay 4 with α X = 1 . 44 ± 0 . 14.\",\n \"pages\": [\n 5\n ]\n },\n {\n \"title\": \"3.2.2. Optical bands\",\n \"content\": \"We detected the optical transient during the first three GROND observations of the field, up to 1.5 days post burst (Table 1, Fig. 2). The transient has coordinates R.A., decl. (J2000) = 22:40:05.861, -17:19:42.77 ( ± 0.3 arcsec), measured on the combined g ' r ' i ' residual images of the 2nd-epoch observations (Fig. 3). Within the error, these coordinates agree with what has been reported by Perley et al. (2019b). Based on a joint fit of the griz -band magnitudes provided by Levan et al. (2023) and GROND's first visit of the field (epoch 1a in Table 1), we find that between both observing runs the optical transient was fading with a decay slope α opt = 1 . 43 ± 0 . 07. Within the errors, α X ∼ α opt up to T 0 +0 . 4 days (epoch 1a in Table 1), suggesting a common cooling regime of the electrons radiating optical and X-ray synchrotron light (see Fig. 4). This finding is compatible with results by (Levan et al. 2023) who showed that a single power-law SED from X-rays to the optical bands can describe the observations at T 0 +0 . 21 days. However, GROND's following observing runs 1b and 2 (Table 1) clearly show a significant flattening of the light curve, with a new decay slope of α opt ∼ 0 . 5 ± 0 . 1 (Fig. 4). The rescaled X-ray power-law model underpredicts the flux in the optical bands, with a > 3 σ deviation observed at T 0 +1 . 5 days. Even by assuming a shallower optical decay index (i.e., α opt = α X -0 . 25), as for the case where the synchrotron cooling frequency lies between the X-ray and the optical regime (e.g., Sari et al. 1998), the discrepancy at late times ( > 1 day) persists (Fig. 4). To quantify the null hypothesis that a power law model cannot fit the data and test for chromaticity of flux evolution, we considered the early data by Levan et al. (2023) and our GROND detections (i.e. from ∼ 0.2 to 1.5 days) for the two filters with best temporal sampling (i.e. r ' and g ' ): a simple power law model does not provide an acceptable fit, with probability of having the observed data set, given the null hypothesis (p-values), of 1 . 7 × 10 -5 and 4 × 10 -4 , respectively. By assuming a broken power law, with a steep-to-shallow behavior and a temporal break at 0.44 days, we find a significant improvement of the fit, with p-values of 0 . 79 and 0 . 73 in the r ' and g ' bands, respectively. The best fit broken power laws show that the flux evolution after the break is slightly shallower in the redder filter, suggesting a chromatic behaviour. We note that this peculiar steep-to-shallow optical light curve morphology is not unique, it was already observed in a small subset of GRBs in the early years of the Swift era (see, e.g., Kann et al. 2010), although with light curve breaks on average much earlier than what we observe for GRB 191019A. For instance, the optical light curve of GRB 090102 (Gendre et al. 2010) is similar to GRB 191019A but the break time was ∼ 0 . 1 hr (rest frame) after the burst. In the standard fireball model, which fairly accounts for late afterglow phenomenology as forward-shock radiation, a steep-to-shallow light curve is not envisioned (e.g., Sari et al. 1998). Such morphology could be produced by the presence of a reverse shock component dominating the forward shock at early times. However, reverse shock is expected on timescales of the order of minutes (e.g., Nakar & Piran 2004), in contrast with what we observe for GRB 191019A. Alternatively, energy injection into the forward shock could originate a light curve flattening. The spin-down radiation from a newly born magnetar (e.g., Zhang & M'esz'aros 2001) or a slightly off-axis structured jet (e.g., Beniamini et al. 2020) are plausible energy sources that are often invoked to explain the X-ray afterglow 'plateaus' observed in GRBs, and that might have an optical counterpart. In the case of GRB 191019A the lack of any evidence of a shallow phase in X-rays cast some doubts against the energy injection scenario, however.\",\n \"pages\": [\n 5,\n 6,\n 7,\n 8\n ]\n },\n {\n \"title\": \"3.2.3. Constraints on Supernova Light\",\n \"content\": \"If GRB 191019A had been a classical long burst, then for a redshift of z =0.248 a supernova (SN) component was expected to be detectable with GROND. Observational constraints on SN light were first reported by Levan et al. (2023) based on HST observations 30 and 184 days post GRB trigger. No evidence for transient emission was found ( g > 24 , r > 23 . 5, and z > 22 mag). Figure 5 shows the upper limits we can set on any SN that followed GRB 191019A in comparison to the r ' -band light curves of 13 GRB-SNe that have been observed with GROND between 2007 and 2014 (see Table 2 in Klose et al. 2019). These events cover the whole GRB-SN luminosity distribution of well sampled GRB-SNe, from GRB 100316D/SN 2010bh (Olivares E. et al. 2012a), one of the faintest GRB-SN ever detected (e.g., Melandri et al. 2014; Cano et al. 2017; Dainotti et al. 2022), to GRB 111209A / SN 2011kl (Kann et al. 2019a), the most luminous GRB-SN ever observed. A visual inspection of Figure 5 shows that all light curves fall into a strip with a width of about 2-3 mag, which is limited by the very faint GRB 100316D/SN 2010bh and the very bright GRB 111209/SN 2011kl. The strongest limit we can set stamps from the time span between 7 and 12 days post burst and is well below the identified GRB-SN magnitude strip: if a CC-SN followed GRB 191019A, in r ' it was at least 3 mag less luminous than SN 1998bw and 2 mag fainter than SN 2010bh. According to the archived Gemini data (Sect. 2.3), the constraint on the luminosity is even stronger, > 4 mag in r at 11.4 days post burst. In conclusion, even if we take into account that the time evolution of GRB-SNe light curves show a certain parameter range (characterized by stretch factor), the SN limit we can set provides a constraint on the entire SN light curves. In this respect, any SN related to GRB 191019A must have been less luminous than each GRB-SN (see Appendix B).\",\n \"pages\": [\n 8\n ]\n },\n {\n \"title\": \"3.3. Joint Afterglow and Kilonova modelling\",\n \"content\": \"Since the flattening observed in the optical light curve at late times (epochs 1b and 2 in Table 1) cannot be explained by the presence of a slowly rising SN (Sect. 3.2.3), and probably not by a reverse shock or energy injection either (Sect. 3.2.2), we explore now the possibility that the source of this additional radiation component was kilonova light. Given the relatively small redshift of GRB 191019A ( z = 0 . 248), a kilonova with a luminosity similar to AT2017gfo is expected to lie within the discovery space of a 2-m telescope, if not hidden by an intrinsic low luminosity. Therefore, following the procedure described in Rossi et al. (2020), we compared our data at T 0 +1 . 5 days in each filter, with the flux of the kilonova AT2017gfo (Coulter et al. 2017) associated with the NS-NS merger GW170817 (Abbott et al. 2017), shifted to the redshift of GRB 191019A. We find that an emission component similar to AT2017gfo but ∼ 4 times more luminous could reproduce the observations in the i ' and r ' bands, while our g ' -band flux seems to be brighter (Fig. 4). Apparently, the presence of a kilonova associated with GRB 191019A is overall compatible with the general properties of the observed optical-NIR emission at 0.4 and 1.5 days, although with slightly different brightness than AT2017gfo 5 . Motivated by the results obtained with our previous simplistic approach, we then explored the possible presence of a kilonova by comparing our multi-band dataset ( g ' , r ' , i ' , z ' , and X-ray) with a much more sophisticated model that takes into account simultaneously the presence of an afterglow and a kilonova component through a joint fit. For this purpose, we exploited the Nuclear - Multi-Messenger Astronomy (NMMA v0.2.0) framework that allows to estimate best-fit parameters with a bayesian inference method (Dietrich et al. 2020; Pang et al. 2023). For the afterglow modelling, NMMA uses the Afterglowpy python module (Ryan et al. 2020). Afterglowpy allows to model GRB afterglow light curves and spectra by taking into account the possible effects due to a complex jet structure and an off-axis observer. We here assumed a Gaussian profile for the jet structure, and that the whole ( ξ N = 1) electron population is shock-accelerated. With the exception of the viewing angle ι , which has a sine function, to all parameters we assigned a uniform function to model the prior probability (for details see Table 2). For the kilonova emission, NMMA allows to fit and simulate data using several numerical and analytical models. Assuming a binary neutron star merger as a progenitor system, we adopted here the kilonova modelling resulting from the time dependent 3D Monte Carlo code POSSIS for modelling radiation transport (Bulla 2019, 2023). We used the kilonova model grid published in Dietrich et al. (2020) based on the first version of POSSIS (Bulla 2019) where the kilonova ejecta is represented with two components: 1) a high-velocity (0 . 08 < v dyn /c < 0 . 3) dynamical ejecta of mass M dyn rj with a lanthanide-rich composition distributed about the equatorial plane with half-opening angle Φ, and lanthanide-poor composition at higher latitudes, and 2) a slower (0 . 025 < v wind /c < 0 . 08) wind (or 'post-merger') component, which is a spherical ejecta released from the merger remnant and debris disk, with an intermediate lanthanide content and mass M wind ej (see Dietrich et al. 2020). In the fit we fixed the GRB 191019A luminosity distance at 1289.3 Mpc, as obtained from the measured redshift ( z = 0 . 248) and assuming a flat cosmological model (see Sect. 1). We also considered an additional error budget (in magnitudes), which takes into account the uncertainties on the model predictions as well as possible systematic errors ( em syserr in the corner plot), to which we assigned a uniform prior with range from 0 to 2 mag. The resulting best-fit light curves computed in each band are plotted in Figure 6, the fit corner plot is in Figure 7, while the 90% confidence interval and median values of each parameter , as well as the adopted prior functions, are quoted in Table 2. By simply inspecting the resulting light curves, it is evident that the afterglow component is constrained mostly by the X-ray data and it is dominant during the early epochs in the optical bands. The afterglow best fit jet core semiaperture angle is about ∼ 10 deg (9 +4 -3 deg), and the observer viewing angle is ∼ 4 deg (4 +3 -2 deg), thus within the jet core. The fireball isotropic kinetic energy E 0 (1 . 4 +1 . 8 -1 . 0 × 10 52 erg) is nicely compatible with the observed prompt emission equivalent isotropic energy E iso , by assuming a reasonable efficiency of about η ∼ 10%, where η = E iso E iso + E 0 . Indeed, the 15-150 keV fluence measured by Swift -BAT is (1 . 00 ± 0 . 03) × 10 -5 erg cm -2 (Krimm et al. 2019), corresponding to a radiated energy E iso = (1 . 70 ± 0 . 05) × 10 51 erg. The latter was computed by assuming a power-law spectrum in the BAT bandpass: this assumption is reasonable given the softness of the spectrum, which is best fit with a photon index Γ = 2 . 25 ± 0 . 05 (Krimm et al. 2019), suggesting that the prompt emission peak energy lies below the BAT bandpass. Figure 6 also clearly shows that at later epochs ( > 1 day), the observed flux lies above the predicted levels of the afterglow component, and the presence of a kilonova provides a better match. Indeed, by assuming only the afterglow model, i.e. by removing the kilonova component from our initial model, we obtained a worse fit, with Bayesian evidence ln( Z ) = -21 . 1. By considering the ratio with the Bayesian evidences of the joint afterglow and kilonova model, for which we obtain ln( Z 0 ) = -14 . 1, the resulting Bayes factor (ln( B ) =ln( Z/Z 0 ) = -7 . 0) indicates a strong preference for the model which includes the kilonova component (see, e.g., Kunert et al. 2024 and references therein). The kilonova properties we find from our fit are compatible with a dynamical ejecta mass of M dyn ej ∼ 0 . 02 M ⊙ and a wind mass M wind ej ∼ 0 . 04 M ⊙ , though with large uncertainties (relative errors ≥ 60%). These values are slightly higher (yet consistent within the uncertainties) with those found for AT2017gfo by assuming the same kilonova modelling (Dietrich et al. 2020). This is in line with the need of a brighter kilonova (of about a factor 4, see Sect. 3.3 and Fig. 4) by simply superposing an AT2017gfo-like light curve on the data. We stress here that the real picture is likely much more complicated than a two-component scenario, and the dynamical/wind masses inferred should rather be interpreted as belonging to some high/low velocity components of a multi-component scenario with multiple ejecta episodes (e.g., Bernuzzi 2020; Nedora et al. 2021). At the same time, the kilonova model we assumed is among the most sophisticated ones publicly available, and the assumption of a kilonova from a NS-NS merger progenitor is the most reasonable choice given that, so far, the only GRB with kilonova emission for which we were able to infer the progenitor nature was GRB170817/AT2017gfo, wich we know from gravitational wave data analysis being originated from a binary neutron star merger. Nevertheless, we explored also other kilonova models available within the NMMA framework, with different levels of sophistication, and which have different assumptions on the progenitor nature and on the kilonova ejecta properties (see Appendix C). We find that the dataset for GRB 191019A did not allow us to confidently distinguish among different kilonova models, apart disfavouring the most simplistic ones. However, in all cases, we find that a joint afterglow plus kilonova fit is preferred with respect to an afterglow-only model. Interestingly, both the afterglow-only and afterglow plus kilonova model, provide as a best-fit for the circumburst environment particle density a value n 0 < 1 cm -3 , which is typically deduced for short-GRB environments (e.g., Berger 2014; Fong et al. 2015). This result is in stark contrast with the interpretation by Lazzati et al. (2023) that invokes the presence of a very high density environment with n 0 ∼ 10 7 -10 8 cm -3 (see also Levan et al. 2023). T able 2 . The 90% confidence in terv al and median v alues of eac h parameter inferred from the sim ultaneous afterglo w ( Ry an et al. 202 0 ) and kilono v a ( Dietric h et al. 2020 ) mo dell ing p erformed b y using the co de NMMA. F or the afterglo w comp onen t, w e assumed a Gaussian jet profi le and fiducial v alue s for the microph ysical parameters (see Sect. 3.3 an d Fig. 6 and Figure 7 ). The b ottom part of the table quotes the assumed p r i ors in the Ba y esian inference analysis, where 'U' stands for Uniform function, with minim um and maxim um v alue s quoted in the brac k ets. Note -E 0 = kinetic fireball energy; n = particle n um b er densit y in the circum burst en vironmen t; θ c =half-op ening angle of the jet core; θ w =half-op ening a ngle of the jet truncated-wings; ι =viewing angle with resp ect to jet axis; p = electron energy distribution p o w er-la w index; ϵ e = sho c k e nergy fraction that go es in to th e electrons; ϵ B = sho c k energy fraction that go es in to the magne tic energy densit y; M dyn ej = dynamical ejecta mass; M wej = wind ejecta mass; Φ = half-op ening angle of lan thanide-ric h equatorial ejecta. Note -All observed (obs) AB magnitudes in the r ' , i ' and z ' filters are 3 σ upper limits (Fig. 5). Columns (2), (5), and (8) quote the expected (exp) magnitudes. All the corresponding differences (observed minus expected; columns 4, 7, 10) are lower limits.\",\n \"pages\": [\n 8,\n 9,\n 10,\n 11,\n 12,\n 13\n ]\n },\n {\n \"title\": \"4. DISCUSSION AND CONCLUSIONS\",\n \"content\": \"Recent analysis of an increasing number of GRBs which were initially classified as long bursts, and therefore thought to be originating from collapsing massive stars, suggest a better compatibility of the observational data with compact binary merger progenitors. By ignoring the standard burst duration classification, typical features of GRBs associated with compact binary mergers are (1) the absence of an associated CC-SN, (2) the presence of an optical/NIR rebrightening compatible with a kilonova, (3) an early-type host galaxy, and (4) a distant GRB explosion site with respect to the corresponding galaxy center. GRB 191019A was suggested to belong to this sample of misclassified long GRBs, given the non detection of any SN component (Levan et al. 2023) despite its close cosmological distance ( z =0.248), and the evidence of a host galaxy dominated by an old stellar population. However, the proximity of the GRB 191019A explosion site to the photometric center of its host galaxy ( ≲ 100 pc) is at odds with typical short-GRB galactocentric distances, and could suggest a compact binary formed in the dense circumnuclear disk of an AGN. In this scenario (Lazzati et al. 2022) a very high-density environment ( n 0 > 10 7 cm -3 ) could be the origin of the long duration of the prompt emission (see Lazzati et al. 2023). Given the low distance of this burst, a kilonova ligh is expected to be seen. However, the temporal coverage and limiting magnitudes of previous data sets, did not allow to address this question yet. In this work, we further investigate the burst properties of GRB 1901910A and the nature of the optical transient that followed the burst, with the goal to find arguments in favour or against its possible compact merger origin. We found an independent support on the compact binary merger progenitor nature from the high-energy prompt emission temporal variability, which has been recently found to be an additional potential diagnostic to infer the progenitor nature, where high variability values suggest a compact binary merger progenitor (Camisasca et al. 2023). We noticed that, for GRB 191019A, the obtained MVT value of ∼ 20 ms locates this burst close to the centroid of the distribution of those GRBs associated with compact binary merger progenitors (Camisasca et al. 2023 and Fig. 1). The same variability properties were found also for two other long GRBs with evidence of kilonova (GRB 111210A and GRB 230307A). These results independently support past interpretations on the progenitor nature of GRB 191019A and address to the possible presence of a kilonova component in its optical transient. An optical transient following GRB 191019A was at first reported by Perley et al. (2019b) but its nature could not be clearly pinned down. Using GROND multi-color data obtained between 0.4 and 15 days post burst, we argued here that the temporal evolution of the transient's brightness disfavours a pure afterglow emission (Fig. 4). While the GROND data confirm the absence of a SN component (Fig. 5), we found that the luminosity evolution of the optical transient is in agreement with an afterglow plus a kilonova signal compatible with AT2017gfo redshifted to the distance of the GRB host galaxy, though slightly brighter by a factor of a few. By modelling the afterglow and the kilonova component, we were able to estimate a kilonova ejecta mass which is slightly higher but still consistent, within the large uncertainties, with the one measured for AT2017gfo with similar assumptions on the progenitor system (a binary neutron star) and on the kilonova modelling (e.g., Dietrich et al. 2020). By assuming other kilonova models with different levels of complexity and different progenitor assumptions, the data did not allow to discriminate among them, apart disfavouring the most simplistic one. However, in all cases, results strongly favour the presence of a kilonova with respect to an afterglow-only model (Table 5). Our findings strongly suggest that GRB 191019A might belong to the increasing list of long GRBs with an associated kilonova, beside GRB 211211A and GRB 230307A, and other cases with less robust evidence, as for instance GRB 060614 (e.g., Jin et al. 2015). Another interesting findings from our analysis, is the evidence of a low circumburst density (Table 2). This result is in stark constrast with the one obtained by Lazzati et al. (2023) from prompt emission light curve modelling, where extreme values of n 0 > 10 7 cm -3 were found. These high density values were considered consistent with the apparent position of the GRB, very close to the center of an AGN-like host galaxy (though no direct proofs on the existence of an AGN have been provided yet, see Levan et al. 2023). The interaction between a jet and an AGN dense circumnuclear environment, however, may choke or strongly suppress GRB relativistic jets, unless particularly bright and long, as also supported by recent studies (e.g., Zhang et al. 2024). GRB 191019A is classified as a normal burst, with a well detected optical counterpart (see, e.g., Zhang et al. 2024), and with low dust extinction along the line-of-sight within the host ( A host V in the range from 0.06 to 0.10 mag, Levan et al. 2023), further supported by the UV-band detections (LaPorte et al. 2019). These properties are more consistent with a low density environment, as the one we obtained from a joint fitting of an afterglow and a kilonova component model on the X-ray and optical data of GRB 191019A. More in general, if GRB 191019A is a disguised short GRB with a compact binary merger origin, as both past studies and this work suggest, a low density environment is in agreement with the typical density values found in short GRBs and with the environments expected for compact binary mergers. In a low density scenario, the very small offset from the host galaxy of GRB 191019A, can be explained with a negligible transverse projection of a GRB located far off centre. In this scenario, the long duration of the prompt emission, which Lazzati et al. (2023) explained as caused by the extremely dense circumburst environment, may have instead an intrinsic origin. An interesting hypothesis that was proposed in the past to explain short GRBs with soft extended emission (e.g., Barkov & Pozanenko 2011; Kisaka & Ioka 2015) and which has been recently studied in much greater details (e.g., Musolino et al. 2024), invokes fall-back accretion onto the remnant compact object. Whether this is the case for GRB 191019A goes beyond the scope of this work and will be addressed in another study. We thank the anonymous referee for useful comments and suggestions which improved our work. We are grateful to P. Pang for his precious support on NMMA and useful discussions. ANG acknowledges logistic support by the Thuringer Landessternwarte Tautenburg, Germany. G.S. and P.S acknowledge the support by the State of Hesse within the Research Cluster ELEMENTS (Project ID 500/10.006). A.R. and E.P. acknowledge support by PRIN-MIUR 2017 (grant 20179ZF5KS). Part of the funding for GROND (both hardware and personnel) was generously granted by the Leibniz-Prize to G. Hasinger (DFG grant HA 1850/28-1) and by the Thuringer Landessternwarte Tautenburg. S.B. aknowledges funding from the EU Horizon under ERC Consolidator Grant, no. InspiReM-101043372. A.E.C. is partially supported by the 2023/24 'Research and Education' grant from Fondazione CRT. The OAVdA is managed by the Fondazione Cl'ement Fillietroz-ONLUS, which is supported by the Regional Government of the Aosta Valley, the Town Municipality of Nus and the Unit'e des Communes valdotaines Mont-Emilius. The LBT is an international collaboration of the University of Arizona, Italy (INAF: Istituto Nazionale di Astrofisica), Germany (LBTB: LBT Beteiligungsgesellschaft), The Ohio State University, representing also the University of Minnesota, the University of Virginia, and the University of Notre Dame. This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester.\",\n \"pages\": [\n 13,\n 15,\n 16\n ]\n },\n {\n \"title\": \"Facilities: GROND, Swift (BAT, XRT and UVOT), LBT\",\n \"content\": \"Software: Afterglowpy (Ryan et al. 2020), NMMA (Pang et al. 2023), astropy (Astropy Collaboration et al. 2013, 2018), HOTPANTS (Becker 2015), PyRAF (Science Software Branch at STScI 2012), DRAGONS (Labrie et al. 2019), WCSTools (Mink 2019).\",\n \"pages\": [\n 16\n ]\n },\n {\n \"title\": \"A. HOST GALAXY\",\n \"content\": \"Note -Magnitudes are measured within an aperture with radius of 4 × FWHM and are not corrected for Galactic foreground extinction. Table 4 provides the magnitude of the GRB host galaxy based on GROND's 4th-epoch observations and LBT's visit of the field four years after the burst (see Sect. 2.2).\",\n \"pages\": [\n 17\n ]\n },\n {\n \"title\": \"B. GRB-SN PEAK LUMINOSITY RANGE\",\n \"content\": \"We further investigated on possible biases towards the faint end of our GRB-SN sample by comparing with SN Ic peak luminosity properties observed so far. The r-band absolute peak-magnitude ( M r,peak ) distribution of broad line (BL) Type Ic SNe spans at most 2.5 mags (between -17 . 7 and -20 . 8 mag, with an average of M r,peak = -18 . 7 ± 0 . 7 mag, Taddia et al. 2019; Barbarino et al. 2021; Gomez et al. 2022). GRB-SNe observed to date with good data sampling, are at least ∼ 1 mag brighter and trace the high-luminosity end of the above Type Ic BL distribution (e.g., Hjorth & Bloom 2012; Hjorth 2013; Cano et al. 2017; Klose et al. 2019), but the width of their peak luminosity distribution (Hjorth & Bloom 2012; Cano et al. 2017; Kann et al. 2019b; Klose et al. 2019) is not substantially different to the width of the corresponding peak luminosity distribution of type Ic and BL-SNe (see also Soderberg et al. 2006). By considering the GRB-SNe with the best follow-up in multiple bands (including spectroscopy), on the high end site of the peak luminosity distribution remains GRB 111209A/SN 2011kl (e.g., Nakauchi et al. 2013; Kann et al. 2019b) while on the low-end site there are GRB 100316D/SN 2010bh (e.g., Starling et al. 2011; Olivares E. et al. 2012b) and GRB 060218/SN 2006aj (e.g., Pian et al. 2006; Ferrero et al. 2006): the r -band peak magnitudes of these extreme GRB-SNe have a difference that lies between 2 and 3 mags, similar to the broad line (BL) Type Ic SNe. In principle, the long bursts GRB 990712 (Bjornsson et al. 2001), GRB 021211 (Della Valle et al. 2003; Zeh et al. 2004), GRB 040924 (Soderberg et al. 2006; Wiersema et al. 2008), and GRB 060904B (Cano et al. 2017) could have been followed by even slightly fainter SNe than SN 2010bh, but in these cases the data base is comparably poor and no strong conclusions could be made. While, e.g., any SN that was associated with the long bursts GRB 060605 and GRB 060614 must have had a luminosity < 1% of the luminosity of the prototypical SN 1998bw (Fynbo et al. 2006), here and in similar cases (e.g., GRB 111005A; Michaglyph[suppress]lowskI et al. 2018; Tanga et al. 2018) there is no evidence for SN light. In conclusion, if substantially less luminous GRB-SNe do exist formally remains an open question, at least from the obervational point of view, and the peak luminosity range of GRB-SNe sample we have used to infer the luminosity of any SN associated with GRB 191019A is the most robust so far.\",\n \"pages\": [\n 17\n ]\n },\n {\n \"title\": \"C. AFTERGLOW AND KILONOVA JOINT FIT COMPARISON\",\n \"content\": \"In Table 5 we present the results obtained for different kilonova models within the NMMA framework, in addition to the kilonova model presented in Sect. 3.3. At first, we have tested a model from Kasen et al. (2017) (Kasen17-Jet in Table 5) which assumes only one ejecta component and has three parameters: the ejecta mass, the ejecta velocity and the lanthanide mass fraction ( χ lan ). Then we made a different assumption on the progenitor by considering a kilonova emission from a neutron star black hole binary system as predicted with POSSIS (Bu19-NSBH-Jet in Table 5), which has three parameters: the dynamical ejecta mass, the wind mass and the orbital plane inclination, which is linked to the viewing angle of the jet assumed in the afterglow modelling. These models were finally compared with the results obtained by assuming the simple analytical model described in Metzger (2017) (Metzger17-Jet in Table 5), which assumes one component and has four parameters: the ejecta mass, the ejecta velocity, the power-law index β of the ejecta mass distribution expressed as a function of its velocity (the faster ejecta matter lies ahead of slower matter and the distribution of mass with velocity greater than a value v 0 can be approximated with a power-law M ( > v 0 ) = M ( v/v 0 ) -β , and the opacity k r . The afterglow emission was modelled within the Afterglowpy framework (Afterglow in Table 5), as described in Sect. 3.3. However, contrary to our previous analysis, the microphyiscal parameters ϵ e and ϵ B are now fixed to 0 . 5 and 0 . 01, respectively. In doing so, we aim at reproduce similar assumptions to Lazzati et al. (2023) and further investigate on the circumburst density estimates. The highest Bayesian evidence ( Z ) is obtained for the Bu19-NSBH-Jet model. Following Kunert et al. (2024) and references therein, by considering Bu19-NSBH-Jet as the reference model, we find -1 . 10 < ln( Z/Z Bu 19 -NSBH -Jet ) < 0 for the Kasen17-Jet and Bu19-BNS-Jet models, while ln( Z/Z Bu 19 -NSBH -Jet ) < -4 . 5 for Metzger17-Jet and Afterglow models, indicating a strong evidence against the latter two models, while no preference could be set among Bu19-BNSJet, Bu19-NSBH-Jet and Kasen17-Jet models.\",\n \"pages\": [\n 17,\n 18\n ]\n },\n {\n \"title\": \"REFERENCES\",\n \"content\": \"Abbott, B. P., et al. 2017, ApJ, 848, L13, doi: 10.3847/2041-8213/aa920c Ahumada, T., Singer, L. P., Anand, S., et al. 2021, Nature Astronomy, 5, 917, doi: 10.1038/s41550-021-01428-7 Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068 Astropy Collaboration, Price-Whelan, A. M., Sip\\\"ocz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f Barbarino, C., Sollerman, J., Taddia, F., et al. 2021, A&A, 651, A81, doi: 10.1051/0004-6361/202038890 Barkov, M. V., & Pozanenko, A. S. 2011, MNRAS, 417, 2161, doi: 10.1111/j.1365-2966.2011.19398.x Barthelmy, S. D., Barbier, L. M., Cummings, J. R., et al. 2005, SSRv, 120, 143, doi: 10.1007/s11214-005-5096-3 Becker, A. 2015, HOTPANTS: High Order Transform of PSF ANd Template Subtraction, Astrophysics Source Code Library, record ascl:1504.004. http://ascl.net/1504.004 Beniamini, P., Duque, R., Daigne, F., & Mochkovitch, R. 2020, MNRAS, 492, 2847, doi: 10.1093/mnras/staa070 Berger, E. 2014, ARA&A, 52, 43, doi: 10.1146/annurev-astro-081913-035926 Bernuzzi, S. 2020, General Relativity and Gravitation, 52, 108, doi: 10.1007/s10714-020-02752-5 Bjornsson, G., Hjorth, J., Jakobsson, P., Christensen, L., & Holland, S. 2001, ApJL, 552, L121, doi: 10.1086/320328 Bulla, M. 2019, MNRAS, 489, 5037, doi: 10.1093/mnras/stz2495 -. 2023, MNRAS, 520, 2558, doi: 10.1093/mnras/stad232 Burrows, D. N., Hill, J. E., Nousek, J. A., et al. 2005, SSRv, 120, 165, doi: 10.1007/s11214-005-5097-2 Camisasca, A. E., Guidorzi, C., Amati, L., et al. 2023, A&A, 671, A112, doi: 10.1051/0004-6361/202245657 Cano, Z., Wang, S.-Q., Dai, Z.-G., & Wu, X.-F. 2017, Advances in Astronomy, 2017, 8929054, doi: 10.1155/2017/8929054 Chambers, K. C., Magnier, E. A., Metcalfe, N., et al. 2016, arXiv e-prints, arXiv:1612.05560, doi: 10.48550/arXiv.1612.05560 Coulter, D. A., Foley, R. J., Kilpatrick, C. D., et al. 2017, Science, 358, 1556, doi: 10.1126/science.aap9811 Coward, D. M., Gendre, B., Tanga, P., et al. 2017, PASA, 34, e005, doi: 10.1017/pasa.2016.61 Dainotti, M. G., De Simone, B., Islam, K. M., et al. 2022, ApJ, 938, 41, doi: 10.3847/1538-4357/ac8b77 Della Valle, M., Malesani, D., Benetti, S., et al. 2003, A&A, 406, L33, doi: 10.1051/0004-6361:20030855 Dietrich, T., Coughlin, M. W., Pang, P. T. H., et al. 2020, Science, 370, 1450, doi: 10.1126/science.abb4317 Note -E 0 = kinetic fireball energy; n = particle number density in the circumburst environment; θ c =half-opening angle of the jet core; ι =viewing angle with respect to jet axis; p = electron energy distribution power-law index; M dyn ej = dynamical ejecta mass; M wind ej = wind ejecta mass; M ej = total ejecta mass; v ej = ejecta velocity; Φ = half-opening angle of lanthanide-rich equatorial ejecta; ϵ e = shock energy fraction that goes into the electrons; ϵ B = shock energy fraction that goes into the magnetic energy density; χ lan = lanthanide mass fraction (Kasen et al. 2017) ; κ r = opacity; β = power-law index of the ejecta mass distribution as a function of its velocity (Metzger 2017); ln( Z ) = natural logarithm of the bayes evidence. We note that the Bu19-BNS-Jet model is the same as the one presented in Sect. 3.3 where now ϵ e and ϵ B are fixed. Evans, P. A., Osborne, J. P., Burrows, D. N., et al. 2019, GRB Coordinates Network, 26034, 1 Evans, P. A., Beardmore, A. P., Page, K. L., et al. 2007, A&A, 469, 379, doi: 10.1051/0004-6361:20077530 -. 2009, MNRAS, 397, 1177, doi: 10.1111/j.1365-2966.2009.14913.x Evans, P. A., Willingale, R., Osborne, J. P., et al. 2010, A&A, 519, A102, doi: 10.1051/0004-6361/201014819 Ferrero, P., Kann, D. A., Zeh, A., et al. 2006, A&A, 457, 857, doi: 10.1051/0004-6361:20065530 Fong, W., Berger, E., Margutti, R., & Zauderer, B. A. 2015, ApJ, 815, 102, doi: 10.1088/0004-637X/815/2/102 Fong, W.-f., Nugent, A. E., Dong, Y., et al. 2022, ApJ, 940, 56, doi: 10.3847/1538-4357/ac91d0 Fontana, A., Dunlop, J. S., Paris, D., et al. 2014, A&A, 570, A11, doi: 10.1051/0004-6361/201423543 Fynbo, J. P. U., Perley, D. A., de Ugarte Postigo, A., et al. 2019, GRB Coordinates Network, 26041, 1 Fynbo, J. P. U., Watson, D., Thone, C. C., et al. 2006, Nature, 444, 1047, doi: 10.1038/nature05375 Gehrels, N., Chincarini, G., Giommi, P., et al. 2004, ApJ, 611, 1005, doi: 10.1086/422091 Gendre, B., Crisp, H., Coward, D., et al. 2019, GRB Coordinates Network, 26098, 1 Gomez, S., Berger, E., Nicholl, M., Blanchard, P. K., & Hosseinzadeh, G. 2022, ApJ, 941, 107, doi: 10.3847/1538-4357/ac9842 Gompertz, B. P., Ravasio, M. E., Nicholl, M., et al. 2023, Nature Astronomy, 7, 67, doi: 10.1038/s41550-022-01819-4 Greiner, J., Bornemann, W., Clemens, C., et al. 2008, PASP, 120, 405, doi: 10.1086/587032 Greiner, J., Mazzali, P. A., Kann, D. A., et al. 2015, Nature, 523, 189, doi: 10.1038/nature14579 Hjorth, J. 2013, Philosophical Transactions of the Royal Society of London Series A, 371, 20120275, doi: 10.1098/rsta.2012.0275 Troja, E. 2023, Universe, 9, 245, doi: 10.3390/universe9060245 Yolda¸s, A. K., Kruhler, T., Greiner, J., et al. 2008, in American Institute of Physics Conference Series, Vol. 1000, American Institute of Physics Conference Series, ed. M. Galassi, D. Palmer, & E. Fenimore, 227-231, doi: 10.1063/1.2943450\",\n \"pages\": [\n 18,\n 19,\n 21\n ]\n }\n]"}}},{"rowIdx":4984,"cells":{"bibcode":{"kind":"string","value":"2024arXiv241205099M"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/2412.05099.pdf"},"content":{"kind":"string","value":"\n\n\n\nARTICLE TYPE\nC/O ratios in self-gravitating protoplanetary discs with dust evolution\nTamara Molyarova, 1,2 Eduard Vorobyov, 1,2 and Vitaly Akimkin 1\n1 Institute of Astronomy, Russian Academy of Sciences, 48 Pyatnitskaya St., Moscow, 119017, Russia 2 Research Institute of Physics, Southern Federal University, Stachki Ave. 194, Rostov-on-Don 344090, Russia Author for correspondence: Tamara Molyarova, Email: moliarova@sfedu.ru.\nAbstract\nElemental abundances, particularly the C/O ratio, are seen as a way to connect the composition of planetary atmospheres with planet formation scenario and the disc chemical environment. We model the chemical composition of gas and ices in a self-gravitating disc on timescales of 0.5 Myr since its formation to study the evolution of C/O ratio due to dust dynamics and growth, and phase transitions of the volatile species. We use the thin-disc hydrodynamic code FEOSAD, which includes disc self-gravity, thermal balance, dust evolution and turbulent diffusion, and treats dust as a dynamically different and evolving component interacting with the gas. It also describes freeze-out, sublimation and advection of four most abundant volatile species: H2O, CO2, CH4 and CO. We demonstrate the effect of gas and dust substructures such as spirals and rings on the distribution of volatiles and C/O ratios, including the formation of multiple snowlines of one species, and point out the anticorrelation between dust-to-gas ratio and total C/O ratio emerging due to the contribution of oxygen-rich ice mantles. We identify time and spatial locations where two distinct trigger mechanisms for planet formation are operating and differentiate them by C/O ratio range: wide range of the C/O ratios of 0 - 1.4 for streaming instability, and a much narrower range 0.3 - 0.6 for gravitational instability (with the initial value of 0.34). This conclusion is corroborated by observations, showing that transiting exoplanets, which possibly experienced migration through a variety of disc conditions, have significantly larger spread of C/O in comparison with directly imaged exoplanets likely formed in gravitationally unstable outer disk regions. We show that the ice-phase C/O ≈ 0.2 - 0.3 between the CO, CO2 and CH4 snowlines corresponds to the composition of the Solar system comets, that represent primordial planetesimals.\nKeywords: protoplanetary disc, volatiles, dust evolution\n1. Introduction\nThe protoplanetary disc matter can be roughly divided into three component: gaseous chemical species, solid dust particles, and icy mantles covering the surface of dust grains. Gas and solid particles become dynamically decoupled, as evolving dust grows and acquires relative velocities leading to the redistribution of elements in the disc and between the phases, and creating the premises for different chemical environments. When planets start to form, their properties, including chemical composition of the atmosphere, are inevitably affected by the location and the mechanism of their formation. This suggests that the origin of (exo)planets might be investigated using their observed chemical composition, and makes understanding the disc chemical evolution vital for creating a consistent planet formation theory.\nOne of the key parameters that govern the chemical setup of a planetary atmosphere is the relation between the abundances of carbon and oxygen, often referred to as carbon-tooxygen ratio (hereafter C/O ratio). The variations of C/O ratio in the ice and gas phases at the snowlines of main disc volatiles (CO, CO 2 , and H 2 O) and the prospects of connecting them to planet formation were discussed in Öberg, Murray-Clay, and Bergin (2011) within a qualitative freeze-out model. Since then, C/O ratio received a lot of attention in this context. It was thoroughly investigated in modelling (see, e.g., Booth et al. 2017; Eistrup, Walsh, and van Dishoeck 2018; Cridland, Eistrup, and van Dishoeck 2019; Cridland et al. 2019; Crid-\nand, Bosman, and van Dishoeck 2020; Cridland et al. 2020; Krijt et al. 2020; Turrini et al. 2021; Schneider and Bitsch 2021). The connection of disc chemical composition with C/O in exoplanetary atmospheres was modelled using core accretion model (Thiabaud et al. 2015) and 'chain' planet population synthesis model (Mordasini et al. 2016). Paul Mollière et al. (2022) considered a simple formation retrieval pipeline and found that this task requires careful consideration of the model assumptions.\nThe measurements of molecular abundances in the atmospheres of giant exoplanets obtained by a variety of modern facilities, such as HST, Spitzer, VLTI, JWST, Gemini, indicate a diversity of C/O ratios: from low C/O ratio values ( ≈ 0.4, below the solar value of 0.54; Benneke et al. 2019; GRAVITY Collaboration et al. 2020; Worthen et al. 2024; Xue et al. 2024) to stellar ( ≈ 0.5, close to solar; P. Mollière et al. 2020; Zhang et al. 2021; Smith et al. 2024) and close to or above unity (Swain et al. 2009; Madhusudhan et al. 2011). A variety of solar and super-solar C/O ratios is observed in four planets within the HR8799 system (Nasedkin et al. 2024). Chemical composition of the atmospheres of many hot Jupiters indicates high C/O>1 of the forming material (Moses et al. 2013). There is an observational evidence of young planets in PDS 70 disc accreting the material with C/O>1 (Facchini et al. 2021). The number of exoplanets with constrained atmospheric C/O ratios grows with large studies of multiple planets such as Changeat et al. (2022), which allows us to make some statistical conclusions. The\npopulation study of C/O ratios in exoplanetary atmospheres reveals that there are two populations with different elemental ratio, which are likely formed in different mechanisms (Hoch et al. 2023). Khorshid, Min, and Désert (2023) were able to restrict the formation scenario for WASP-77b based on the measured C/O ratio of the planet (Line et al. 2021) and the modelling of planet formation and migration.\nElemental abundances in protoplanetary discs can be constrained from observations (Fedele and Favre 2020), and C/O ratio in the gas can be estimated. Spatially resolved observations can help distinguish between different C/O ratios spectroscopically (Matter, Pignatale, and Lopez 2020). Cleeves et al. (2018) report C/O ≈ 0.8 in the molecular layer of IM Lup disc. ALMA observations of hydrocarbons and sulphur-bearing species indicate C/O > 1 in the upper disc layers and in the outer disc in TW Hya and DM Tau (Dutrey et al. 2011; Bergin et al. 2016; Semenov et al. 2018) and for a population of discs in Lupus (Miotello et al. 2019). For the nearby discs the solar elemental composition with C/O ≈ 0.54 is usually expected, thus the observed higher values confirm redistribution of carbon and oxygen in discs. In addition to high C/O in disc atmospheres, there is evidence of both carbon and oxygen depletion from gas (Kama et al. 2016; Miotello et al. 2019). However, some of heavy oxygen carriers might not be observable, leading to overestimated C/O in disc observations (Walsh, Nomura, and van Dishoeck 2015).\nThe volatile composition is also used to constrain the origin of bodies in the Solar System. Fraction of CO and CO 2 ices relative to water in cometary comae indicate their formation between the CO and CO 2 snowlines or exterior to the CO snowline (A'Hearn et al. 2012; Seligman et al. 2022). Abundances of CO and N 2 ices were used to analyse the original location of Pluto and Triton (Mousis, Anderson, et al. 2024). Observed elemental abundances were used to constrain the Jupiter formation scenario (Lodders 2004), relying also on abundances of nitrogen (Öberg and Wordsworth 2019; Bosman, Cridland, and Miguel 2019) and chemically inactive species like Ar. However, the model assumptions can lead to different interpretation of the observations: while Öberg and Wordsworth (2019) and Bosman, Cridland, and Miguel (2019) suggest that Jupiter formed outside N 2 snowline (at > 30 au), Ohno and Ueda (2021) consider the concept disc shadow, which allows Jupiter to form near its current location.\nChemical processes other than freeze-out and sublimation at the snowlines can alter the composition of ice and gas as well. Due to gas-phase and surface reactions, snowlines can become important for the redistribution of elements. More detailed chemical modelling shows that the C/O ratio in the gas and in the ice depends also on the initial chemical setup and ionisation by cosmic rays and radioactive nuclei (Eistrup, Walsh, and van Dishoeck 2016, 2018). It directly affects the interpretation of observations. Another essential chemical process is CO depletion from the gas, resulting in its transformation to CO 2 ice (Bosman, Tielens, and van Dishoeck 2018). For example, stellar C/O ratio in the atmosphere of HR 8799e indicates that the planet accreted its material beyond CO snowline ( ≈ 45 au), but chemical modelling suggests that due to CO depletion,\nthe C/O in the ice already approaches the stellar ratio beyond CO 2 snowline ( ≈ 20 au), which is closer to the star (P. Mollière et al. 2020).\nAnother key process affecting the elemental ratio is dust drift, which leads to spatial segregation between the chemical constituents of the gas and the grains covered with ice. The distribution of CO in the gas and ice phases was studied within dynamical models of dust evolution (Stammler et al. 2017; Krijt et al. 2018). Even without chemical processes, dust evolution and dynamics can substantially alter C/O ratio in the atmospheres of forming planets (Booth et al. 2017). Some models combine chemical reactions treatment with dust evolution and transport, usually within 1D viscous models. Dust transport can have a strong effect on the abundances of volatiles in the inner disc regions (Bosman, Tielens, and van Dishoeck 2018). However, for discs with low turbulence and high cosmic ray ionisation rate, C/O ratio is rather defined by chemical evolution (Booth and Ilee 2019).\nIn our previous work (Molyarova et al. 2021) we showed that the volatile species tend to concentrate around their snowlines both in the gas and more notably on the dust surface. This accumulation was found to be caused by effective transport of volatiles through the snowlines by azimuthal variations in the gas and dust radial and angular velocity, an effect that cannot be captured in 1D viscous disc models. Such accumulation should immediately affect the local C/O ratio, which suggests the connection between the snowlines of various volatiles and the formation of planets with altered C/O in their atmospheres. In this work, we follow the distribution of the main volatile species in the disc to investigate the distribution of C/O ratio in gas and ice in a 2D thin-disc hydrodynamic model. We study the effect of dust growth and dynamics on the elemental ratios and consider the role of the initial mass of the collapsing core on the distribution of volatiles.\nThe paper is organised as follows. The main features of the used FEOSAD model are described in Section 2, with the details of the treatment of the volatiles given in Section 2.3. In Section 3 we describe the results of the simulations, focusing on distribution of the volatiles in Section 3.2, the C/O ratios in Section 3.3, and their evolution in Section 3.4. In Section 4, we discuss the implications of our results in the context of planet formation via different mechanisms. The main conclusions are listed in Section 5.\n2. Model\nWe use the global model of protoplanetary disc formation FEOSAD (Vorobyov et al. 2018), which includes disc selfgravity, dust evolution and interaction with gas (including backreaction of dust on gas), turbulent viscosity, adiabatic and radiative cooling and heating. It describes the formation of a protostar and a protoplanetary disc from a collapsing cloud in a 2D thin-disc approach. The model also includes freeze-out of main volatile species as in Molyarova et al. (2021), with the feedback from ice mantles on dust evolution via fragmentation velocity. Here we summarise the key characteristics of the model, more details can be found in the previous works (Vorobyov\net al. 2018; Molyarova et al. 2021; Kadam, Vorobyov, and Basu 2022). The main difference from our previous study in Molyarova et al. (2021) is that here we consider the formation of dead zones via variable α -parameter of Shakura and Sunyaev and also include turbulent diffusion.\n2.1 Gas evolution\nFor the gas component, the hydrodynamic equations for mass, momentum, and internal energy conservation are the following\n∂Σ g ∂ t + ∇· ( Σ g v ) = 0, (1)\n∂ ∂ t ( Σ g v ) + [ ∇· ( Σ g v ⊗ v ) ] = -∇P + Σ g g + + ∇· Π -Σ d,gr f , (2)\n∂ e ∂ t + ∇· ( e v ) = -P ( ∇· v ) -Λ + Γ + ∇ v : Π , (3)\nwhere subscripts p and p ' denote the planar components ( r , ϕ ) in polar coordinates, Σ g is the gas mass surface density, e is the internal energy per surface area, P is the vertically integrated gas pressure calculated via the ideal equation of state as P = ( γ - 1) e with γ = 7/5, f is the friction force between gas and dust, v = vr ˆ r + v ϕ ˆ ϕ is the gas velocity in the disc plane, and ∇ = ˆ r ∂ / ∂ r + ˆ ϕ r -1 ∂ / ∂ϕ is the gradient along the planar coordinates of the disc. The gravitational acceleration in the disc plane, g = gr ˆ r + g ϕ ˆ ϕ , includes the gravity of the central protostar when formed and takes into account disc self-gravity of both gas and dust found by solving the Poisson integral (Binney and Tremaine 1987).\nTheconsideration of time-dependent energy balance (Eq. (3)) allows us to accurately calculate the midplane temperature T mp and is particularly important to describe the phase state of the volatiles and the level of turbulent viscosity. The terms Λ and Γ describe the rates of dust cooling and heating, respectively, by stellar and background irradiation. They are calculated based on the analytical solution of the radiation transfer equations in the vertical direction (Dong et al. 2016; Vorobyov et al. 2018)\nΛ = 8 τ P σ T 4 mp 1 + 2 τ P + 3 2 τ P τ R , Γ = 8 τ P σ T 4 irr 1 + 2 τ P + 3 2 τ P τ R . (4)\nHere, σ is the Stefan-Boltzmann constant, τ P and τ R are the Planck and Rosseland mean optical depths to the disc midplane, calculated as τ = κΣ dust from Planck and Rosseland mean opacities κ P and κ R (Semenov et al. 2003) and total dust surface density Σ dust . Gas and dust temperatures are assumed to be equal, and the midplane temperature is linked with gas pressure as T mp = P µ / R Σ g, where µ = 2.3 is the mean molecular weight of the gas, and R is the universal gas constant. The irradiation temperature at the disc surface T irr is determined by both stellar and background irradiation. Stellar irradiation includes the luminosity from the photosphere of the protostar\nand accretion luminosity. The background radiation is assumed as a black body with the temperature of 15 K. For more details on the irradiation we refer to Vorobyov et al. (2018).\nTurbulent viscosity is described using the common α -parameter approach of Shakura and Sunyaev (1973). It is taken into account via the viscous stress tensor Π (see Vorobyov and Basu 2010, for explicit expressions for the components of the terms with Π ). The magnitude of kinematic viscosity is ν = α c s H , where c s is the sound speed and H is the vertical scale height of the gas disc calculated using an assumption of local hydrostatic equilibrium of a self-gravitating disc (see Vorobyov and Basu 2010, Appendix A). Here, we use the adaptive α approach implying accretion through a layered disc (Gammie 1996; Armitage, Livio, and Pringle 2001; Kadam, Vorobyov, and Basu 2022). Turbulence is assumed to be generated by magneto-rotational instability (MRI) which only develops in layers of the disc where the ionisation level is high enough. The MRI-active layer is characterised by its surface density Σ MRI and relatively high value of turbulent viscosity α MRI = 10 -3 . As thermal and photo-ionisation are not efficient enough for the relatively cold and dense matter in the disc at >0.5 au, the main process determining the thickness of the MRI-active layer is ionisation by cosmic rays. It is assumed to be constant Σ MRI = 100g cm -2 , which is the typical depth of Galactic cosmic rays penetration in the ISM (Umebayashi and Nakano 1981) and in protoplanetary discs (Padovani, Galli, and Glassgold 2009). The dead zone is characterised by surface density from the midplane Σ dz = Σ g/2Σ MRI with residual turbulence α dz . The turbulence in this layer is only hydrodynamic turbulence driven by the Maxwell stress in the active layer, and small value α dz = 10 -5 is adopted (Okuzumi and Hirose 2011). However, if local temperature exceeds the critical value of 1300 K, thermal ionisation becomes possible, the MRI develops and the dead zone is no longer dead, in which case α dz = 10 -1 (Zhu et al. 2010; Kadam et al. 2019). This value is higher than α MRI in the outer disc due to the different ionisation processes and local conditions. In the outer disc, the MRI can be suppressed by non-ideal magneto-hydrodynamic (MHD) effects such as ambipolar diffusion and Ohmic resistivity (Bai and Stone 2013; Gressel et al. 2015). In the dead zone in the inner disc, α can reach higher values when the MRI is triggered by thermal ionisation, as shown by 3D MHD simulations (Zhu, Jiang, and Stone 2020). The adopted parameterization makes use of an effective parameter α eff , which at any given location is calculated as\nα eff = Σ MRI α MRI + Σ dz α dz Σ MRI + Σ dz , α dz = { 10 -5 , if T mp < 1300K; 10 -1 , if T mp ≥ 1300K. (5)\n2.2 Dust evolution\nDust is described as consisting of two components: small grains that are dynamically coupled to the gas, with the mass surface density Σ d,sm , and grown grains with the mass surface density Σ d,gr that can move relative to the gas and change in size. The total dust surface density necessary for the calculation\nof the optical depths for Eq. (4) is Σ dust = ( Σ d,gr + Σ d,sm )/2. The factor 1/2 appears as the optical depths is calculated to the midplane. Each dust population has a power-law size distribution f ( a ) = dN / da = Ca -p with a normalisation constant C and a fixed exponent p = 3.5. Small dust has sizes between a min = 5 × 10 -7 cm and a ∗ = 10 -4 cm, grown dust has sizes between a ∗ and a max, which can vary due to dust coagulation and fragmentation. Dynamics of these dust components follows the continuity and momentum equations\n∂Σ d,sm ∂ t + ∇· ( Σ d,sm v ) = -S ( a max) + ∇· ( D Σ g ∇ ( Σ d,sm Σ g )) , (6)\n∂Σ d,gr ∂ t + ∇· ( Σ d,gr u ) = S ( a max)+ ∇· ( D Σ g ∇ ( Σ d,gr Σ g )) , (7)\n∂ ∂ t ( Σ d,gr u ) + [ ∇· ( Σ d,gr u ⊗ u )] = Σ d,gr g + + Σ d,gr f + S ( a max) v , (8)\nwhere u is the grown dust velocity. The term S ( a max) is responsible for the exchange of matter between the dust populations, as dust is converted from the small to grown component due to coagulation and back due to fragmentation. The details of the dust evolution model are presented in Vorobyov et al. (2018). The last term in Eqs. (6) and (7) is responsible for dust turbulent diffusion, similar to Vorobyov, Elbakyan, et al. (2020). The coefficient of turbulent diffusion D is related to the kinematic viscosity ν as D = ν /(1 + St 2 ) (Birnstiel 2023). Diffusion affects dust grains along with their ice mantles, as well as the gas-phase species (see Section 2.3).\nThe innermost regions of the disc are challenging to simulate explicitly due to the Courant criterion: in the highly dynamic inner regions (fraction of au) the timescales are so short that the code demands very small time step in order to preserve stability. Therefore, the inner regions are represented by a sink cell, with a carefully chosen inflow-outflow boundary condition at the sink cell and a parametric description of the accretion onto the star (see Vorobyov et al. 2018, for details). In the simulations presented below, the radius of the sink cell is 0.62 au.\nWe consider two disc models with different initial mass of the collapsing cloud, 0.66 and 1 M ⊙ . We note that about 10% of the gas mass that crosses the sink cell is assumed to be evacuated by jets and outflows, and the other 90% lands on to the star. A small amount of mass remains in the envelope by the end of simulations. In both models, the initial gas temperature T init = 15 K and the ratio of rotational to gravitational energy β = 0.28%. The simulations start from the collapse of a molecular cloud, with only small dust grains. The simulation continues until the age of the system becomes equal to 0.5 Myr. Masses of the central protostar and the disc by the end of simulation are M ⋆ = 0.4 M ⊙ and M disc = 0.22 M ⊙ in model M1 and M ⋆ = 0.58 M ⊙ and M disc = 0.35 M ⊙ in model M2. The disc masses are around 0.5 stellar masses, which makes them essentially self-gravitating.\nA number of recent studies develop the idea that the mass infall from the ambient ISM continues during the lifetime of the disc, including the Class II stage (Padoan et al. 2024; Pelkonen et al. 2024; Winter, Benisty, and Andrews 2024). Such models describe a Bondi-Hoyle accretion regime and are in good agreement with the observed properties of the disc population, such as accretion rates, masses and sizes. This input of matter can play an important role in disc evolution and planet formation process (Vorobyov, Lin, and Guedel 2015; Vorobyov et al. 2016). In the FEOSAD model, the mass infall to the disc can be accounted for (Vorobyov, Lin, and Guedel 2015; Vorobyov et al. 2016), but in the present simulation this effect is not included. The mass infall from the envelope continues until the cloud is depleted of matter. Because of the thin-disc geometry, the gravitational contraction of the cloud proceeds in the plane of the disc. The matter is landing on the disc outer edge and is transported towards the star by the combined action of gravitational and viscous torques. The infall on the disc inner regions is therefore neglected. This is a reasonable approximation, considering that most of the matter and angular momentum in a three-dimensional cloud is located at relatively large polar angles and a flared outer edge of the disc intercepts most of them (see, e.g., Visser et al. 2009, Figure 1). In our modelling, we do not consider the continuous Bondi-Hoyle accretion and concentrate on the internal disc processes.\n2.3 Evolution of volatiles\nWe follow the evolution of four main volatile species: H 2 O, CO 2 , CO and CH 4 . These are the most abundant carbon- and oxygen-bearing ices observed in protostellar cores (Karin I. Öberg et al. 2011). In the model, each of these species can be present in three states: in the gas, in the ice on the surface of small dust, and in the ice on the surface of grown dust. Each species s is described by its surface density in the gas Σ gas s , on small dust Σ sm s , and on grown dust Σ gr s . Their distributions in the disc can change through three main processes: advection together with the corresponding component (gas or small/grown dust); exchange of mantles between dust populations due to grain collisions; phase transitions, including adsorption from gas to dust, and thermal and photo-desorption. Initially, all ices are on small grains and no volatiles present in the gas. The treatment of volatiles is adopted from Molyarova et al. (2021), who describe the models in more details. Here we recap main features of the chemical model.\nOur chemical model only describes phase transitions, i.e. adsorption and desorption, which includes thermal desorption and photo-desorption by interstellar UV radiation. These reactions were shown to be he most important for gas-phase abundances of most species (Ilee et al. 2011). Due to high computational costs, no other chemical processes, either gasphase or surface reactions are included, although they also may have significant effect on the composition of both ice and gas (Semenov and Wiebe 2011). The chemical evolution of the surface densities of volatile species is calculated from the\nsystem of equations\n∂Σ\ngas s ∂ t + ∇· ( Σ gas s v ) -∇· ( D Σ g ∇ ( Σ gas s Σ g )) = -λ s Σ gas s + η sm s + η gr s , (9) ∂Σ sm s ∂ t + ∇· ( Σ sm s v ) -∇· ( D Σ g ∇ ( Σ sm s Σ g )) = λ sm s Σ gas s -η sm s , (10) ∂Σ gr s ∂ t + ∇· ( Σ gr s u ) -∇· ( D Σ g ∇ ( Σ gr s Σ g )) = λ gr s Σ gas s -η gr s , (11)\nwhere the mass rate coefficients per disc unit area of adsorption λ s and desorption η s for the species s are calculated for local conditions at every hydrodynamic step, separately for small and grown dust populations. Eqs. (9)-(11) are solved in two steps: first, the right-hand side is considered without the advection term. The case of pure adsorption/desorption represented by the right-hand side of the equation can be solved analytically (see Appendix A in Molyarova et al. 2021). This is done at every hydrodynamic step, before the dust growth step, when ices on small and grown grains are redistributed proportionally to mass exchange between the dust populations. This is followed by a transport step, when the surface densities of the volatiles are changed according to the fluxes of their respective gas or dust components between the cells. Restricting chemical processes to only adsorption and desorption allows the chemical step to be calculated fast, which is very important for computationally demanding hydrodynamic simulations.\nFor each dust population, the desorption rate is a sum of thermal desorption and photo-desorption η = η td + η pd (the indices 'sm' and 'gr' are omitted for convenience). We operate under the assumption of zeroth-order desorption, i.e. the desorption rate does not depend on the present amount of ice ( Σ sm s or Σ gr s ). It implies that only the upper layers of the ice mantle are able to sublimate, which is a more appropriate approach for thick mantles. This assumption better describes desorption of CO and H 2 Oin temperature programmed desorption (TPD) experiments (Fraser et al. 2001; K. I. Öberg et al. 2005; Bisschop et al. 2006) than first-order desorption, which is more suitable for thin mantles of several monolayers. The rate of thermal desorption is calculated as\nη td = ˜ σ tot N ss µ s m p √ 2 N ss E b k B π 2 µ s m p exp ( -E b T mp ) , (12)\nwhere N ss = 10 15 cm -2 is the surface density of binding sites (Cuppen et al. 2017), E b (K) is the binding energy of the species to the surface, µ s is the species molecular mass, m p is atomic mass unit, k B is the Boltzmann constant. We follow Hasegawa and Herbst (1993) and use binding energy to calculate the pre-exponential factor in Eq. (12), the same way it was done in Molyarova et al. (2021). However, this approach was recently demonstrated by Minissale et al. (2022) to underestimate the pre-exponential factor by a few orders of magnitude,\nas it does not account for the rotational part of the partition functions of desorbing molecules. Total surface area of dust grains (small or grown) per disc unit area ˜ σ tot (cm 2 cm -2 ) is calculated for the adopted power-law size distribution with p = 3.5 as\n˜ σ sm tot = 3 Σ d,sm ρ s √ a min a ∗ , (13)\n˜ σ gr tot = 3 Σ d,gr ρ s √ a ∗ a max . (14)\nHere, ρ s = 3 g cm -3 is the material density of silicate cores of the dust grains.\nThe model includes photodesorption of volatiles by interstellar irradiation, which is mostly relevant in the outer disc regions with lower optical depth. We do not consider the UV radiation field produced by the star and the accretion region as a source of photodesorption, assuming that they do not reach disc midplane due to high optical depth. The photo-desorption rate is calculated as\nη pd = ˜ σ tot µ sp m p YF UV . (15)\nwhere F UV = F UV 0 G UV (photons cm -2 s -1 ) is the UV photon flux expressed in the units of standard UV field and Y = 3.5 × 10 -3 + 0.13 exp ( -336K/ T mp ) (mol photon -1 ) is the photodesorption yield adopted from Westley et al. (1995) for water ice. The intensity of the interstellar UV radiation field with G 0 = 1 is F UV 0 = 4.63 × 10 7 photon cm -2 s -1 (Draine 1978). We assume that the disc situated in a star-forming region is illuminated by a slightly elevated unattenuated UV field with G env = 5.5 G 0 . For the disc midplane, which is illuminated from above and below, this field scales with the UV optical depth τ UV towards the disc midplane as\nG UV = 0.5 G env e -τ UV . (16)\nThe optical depth can be calculated as τ UV = 0.5( κ sm Σ d,sm + κ gr Σ d,gr ), where κ sm = 10 4 cm 2 g -1 , κ gr = 2 × 10 2 cm 2 g -1 are typical values of absorption coefficients in the UV for small and grown grains (Pavlyuchenkov et al. 2019, Fig. 1).\nWe calculate the adsorption rate λ following Brown and Charnley (1990). It is proportional to the total cross-section of dust grains per unit volume, which can be obtained from the total surface area of dust grains per unit disc surface ˜ σ tot. To change the normalisation to the 2D case, ˜ σ tot needs to be multiplied by 1/2 H . Another factor of 1/4 follows from the difference between cross-section and surface area of a sphere. As a result, the adsorption rate is calculated as\nλ = ˜ σ tot 8 H √ 8 k B T mp πµ sp m p . (17)\nA more detailed derivation of rate coefficients for adsorption and desorption is presented in Section 2.3 of Molyarova et al. (2021).\n
\n\n
Table 1. Binding energies, molecular weights, and initial abundances for the considered volatiles adopted in the modelling. Initial abundances of the species f s are shown relative to number density of water molecules in ice phase, and Σ sm s / Σ init g is the corresponding initial mass fraction of the ices relative to gas surface density.
\n
\nTable 1 summarises the molecular parameters used in both models in this work. Binding energies for H 2 O, CO 2 , and CO are based on the experimental data from Cuppen et al. (2017) for desorption from crystalline water ice. The binding energy for methane is taken from Aikawa et al. (1996). The values of the initial abundances relative to water f s are based on the observations of ices in protostellar cores around the low-mass protostars (Karin I. Öberg et al. 2011). They are transformed to the mass fraction relative to gas assuming the water abundance of 5 × 10 -5 relative to gas number density. Total initial mass of ices comprises ≈ 8.5% of the total mass of refractory grain cores. This is relatively low compared to the estimates suggesting comparable masses of ices and refractories in the discs (Pontoppidan et al. 2014). However, the lower fraction of ices is more suitable in our approach, that suggests that ice mantles do not change mass and radius of dust grains (Molyarova et al. 2021). As we are mostly interested in the elemental ratios and relative abundances of the considered ices, lower ice fraction is an appropriate simplification. However, we note that dust dynamics can lead to accumulation of ices and ice mantles exceeding masses of silicate cores in some disc regions, as was shown previously in Molyarova et al. (2021).\nIce mantles also provide feedback on dust evolution. The model includes the effect of ices on fragmentation velocity v frag , which is the the maximum collision velocity leading to sticking instead of fragmentation. According to laboratory experiments, icy grains have higher v frag than bare silicate grains by an order of magnitude (Wada et al. 2009; Gundlach and Blum 2015). In Molyarova et al. (2021) we used the values of fragmentation velocity v frag = 1.5 and 15 m s -1 for bare and icy grains, correspondingly. Here we follow Okuzumi and Tazaki (2019) and adopt lower values of v frag = 0.5 and 5 m s -1 , which are more relevant for grains consisting of µ m-sized monomers. These lower values of v frag will lead to higher importance of fragmentation compared to Molyarova et al. (2021). To determine if a dust grain should be considered icy or bare, we compare the local total surface density of all ices on grown dust divided by Σ d,gr with the threshold value K , which is calculated as\nK = 3 a ml ρ ice √ a ∗ a max ρ s , (18)\ni.e., an icy grain must have at least one monolayer of ice. Here, a ml is the thickness of the ice monolayer estimated as the size\nof a water molecule 3 × 10 -8 cm. The material density of ice ρ ice = 1 g cm -3 and the mean radius of a grown grain is calculated as √ a ∗ a max for the power-law distribution with p = 3.5.\n3. Results\nTo understand the distribution of the species, we first need to consider the global evolution of the disc and its structure. The distribution of volatiles and the C/O ratio is very sensitive to gas and dust substructures appearing in the disc. Variations in temperature and pressure lead to the complex shape and temporal evolution of the snowlines. The dependence of dust fragmentation velocity on the presence of ice mantles implies the feedback from the volatiles on dust and (through backreaction) on gas.\nOur modelling starts with the gravitational collapse of a flattened, slowly rotating molecular cloud. The protoplanetary disc is formed after the formation of the protostar, when the in-spiraling layers of the contracting cloud hit the centrifugal barrier near the inner edge of the sink cell, at a time instance depending on the initial core mass. The disc and the protostar are formed ≈ 53 kyr after the beginning of the cloud collapse in model M1 and at ≈ 78kyr in model M2. If not stated otherwise, times are specified counting from the beginning of the simulation, e.g. the 100 kyr time instance for model M1 refers to a ≈ 50kyr old disc, as the first stage includes cloud collapse as well.\nAn important characteristic of young stellar objects is their variable accretion rate. Our modelling produces accretion bursts with the magnitude of ∼ 100 L ⊙ occurring every ∼ 10 4 years during the first hundred thousands years of disc evolution. These burst parameters are in line with the episodic accretion scenario (Hartmann and Kenyon 1985) and resemble the observed phenomenon of FU Ori type stars (see Audard et al. 2014, for a review). The luminosity outbursts heat up the disc and typically shift the snowlines further away from the star. Although this effect is temporary, it can be reflected in the distributions of the volatiles, and leave long-term imprints in the observed dust properties (Vorobyov et al. 2022). The detailed analysis of the effect of such outbursts on the distribution of the elements is worthy of a separate study and lies beyond the scope of this paper.\n3.1 Dust and gas structures\nDuring the first hundred thousands years of evolution, protoplanetary discs change from highly asymmetric and dynamic objects to nearly axisymmetric and slowly evolving structures. Figure 1 shows the examples of different gas and dust substructures that are present in the disc at different stages. The snapshots are shown for model M1, they include a young disc (160 kyr), an intermediate state (350 kyr), and a more evolved and axisymmetric disc (490 kyr). Each of the selected time instances represent some characteristic morphological features addressed below. In model M2, similar structures appear, although sometimes at different evolution times. In this subsection, we consider model M1 as an example and discuss\n\n\n
Figure 1. Surface density of gas and grown dust, Toomre Q -parameter, maximum dust radius, temperature and viscous α -parameter in model M1 at selected time moments: 160 kyr, 80 × 80 au; 350 kyr, 35 × 35 au; 490 kyr, 9 × 9 au. The contours indicate the position of the water snowline. Note that at the panels with multiple water snowlines, water is frozen outside the outer line and inside an inner dust ring at 1 - 2 au.
\n\nthese features, highlighting the properties of dust and gas substructures that are most relevant for the distribution of the volatiles.\nThe earliest phases of disc evolution are characterised by a large-scale spiral structure in both dust and gas, as well as episodic appearance of clumps. They are the result of gravitational instability (GI) in a massive disc, as in our modelling, the disc mass comprises more than 0.1 of the stellar mass, which is roughly a threshold of the disc stability against GI (Vorobyov 2013; Kratter and Lodato 2016). This is illustrated by the Toomre Q -parameter (Toomre 1964) in the third column of Figure 1: inside the spirals and the clump Q < 1, which indicates the dominance of self-gravity over Keplerian sheer and gas density. The spiral structures become less prominent with time as the disc looses mass and the Q -parameter increases. However, spirals persist in the model throughout the disc evolution up to 0.5 Myr. For example, at 350 kyr, a very tight spiral is present in the gas, starting at the gas and dust ring at ≈ 10au. At 490 kyr, the spiral pattern is weak and exists at > 10 au distances, which are not displayed in Figure 1.\nOne- or two-armed grand design spirals are common 100 - 200 kyr after the disc formation in the models. The analogues of such spirals in the observed young protoplanetary discs around low-mass stars are found, for example, in Elias 227 or WaOph 6 (Pérez et al. 2016; Huang, Andrews, Pérez, et al. 2018). It is not yet clear if the observed spirals are the result of GI or caused by a perturbation from a companion\nplanet or a (sub)stellar object (Meru et al. 2017; Brown-Sevilla et al. 2021). A spiral with multiple clumps produced by GI was recently observed in the disc around a FUor V960 Mon (Weber et al. 2023).\nAnother important feature of gas and dust spatial distributions is ring-like structures at various scales. The system of rings starts to form as early as 100 kyr, and develops to the high-contrast multiple ring structure (1 - 3 orders of magnitude difference between surface densities in rings and gaps), which is evident in the middle and bottom rows of Figure 1. While gas rings are also common, the annual structures are more prominent in the dust surface density, as well as in dust size. Some of the dust rings correspond spatially to the gas rings, while others are barely reflected in the gas distribution. Overall, the difference between the gas and dust structures develops with time, as the result of dust growth and drift (Testi et al. 2014). Only some particular substructures, such as the ring between 1 - 2 au, are present in the distributions of both gas and dust components.\nAnother location where prominent rings form in both dust and gas is the water snowline. Here, the water snowline is defined as the location in the disc midplane where the amount of water in the gas equals to its amount in the ice (on both dust populations). There are multiple location in the disc where this happens. Water is frozen in most of the disc beyond 5 - 10 au, and we will refer to the furthest snowline dividing these outer frozen region from the inner one with the gas-phase water\nas a primary snowline. Generally, the primary snowline is roughly circular, but it can have asymmetries due to the spiral structure and an additional snowline may appear, e.g., around a gravitationally bound clump (upper rows in Figure 1). Besides, at ≈ 260kyr, another region rich in water ice appears in the inner disc, creating a pattern of double or triple water snowline at later times (middle and lower rows in Figure 1). We will refer to these inner additional snowlines as the secondary snowlines. They circumcise a cold and dense gas-dust ring that forms at 1 - 2 au. As the disc cools down with time, the primary snowline position moves from ≈ 10au distance at 160 kyr, to ≈ 6au at 350 kyr and ≈ 4au at 490 kyr.\nSnowlines are known to be associated with the enhancement of dust and volatiles (Stevenson and Lunine 1988; Cuzzi and Zahnle 2004; Drążkowska and Alibert 2017; Molyarova et al. 2021). Dust enhancement was also shown to affect the distribution of gas and its accretion rate through the disc (Gárate et al. 2020) by means of dust back reaction, which is also accounted for in our modelling. In our models, an about 2 au wide dust ring is formed at the inner edge of the water snowline (at ≈ 10au) as soon as 50 kyr after the disc formation. Grown dust grains drifting towards the star through the snowline lose their mantles, their fragmentation velocity drops, rendering them more vulnerable to fragmentation. Consequently, the grain maximum size a max decreases, their drift towards the star slows down, which leads to the accumulation of grown dust, as well as small dust as a product of fragmentation. Increase of total dust density also leads to less efficient cooling and results in higher temperature inside the snowline (see the right panels in Figure 1). Later, at times > 200 kyr, several additional rings form outside the water snowline at distances up to 100 au, the most notable one being 1 - 2 au exterior to the primary snowline. Dust is trapped inside the gas pressure maxima in these rings, which is a self-supporting phenomenon as the temperature also increases inside the dust ring due to high optical depth.\nThese rings are worthy of attention in the context of possible planet formation. Dust surface density and size are higher in the rings, with a max reaching centimetres, dust-to-gas ratio up to 0.1 - 0.2, and Stokes number up to 0.05 - 0.1. This could ease the development of the streaming instability, which typically requires pebble-size grains with St ≳ 0.01 and dustto-gas ratio ≳ 0.02 (Carrera, Johansen, and Davies 2015). Multiple ring-like structures are commonly observed in protoplanetary discs at a range of ages and display a variety of examples, with different widths, contrasts and numbers of rings (Huang, Andrews, Dullemond, et al. 2018, and many others). However, to directly compare the ring structures in the simulated dust surface density with the observed dust emission, require radiative transfer modelling is needed. Some of the observed ring structures could be produced by radial variation in dust size even in the absence of gaps in dust surface density (Akimkin et al. 2020).\nThe most prominent dust rings are located in the vicinity of the water snowline: the ring outside the primary snowline at 5 - 8 au (depending on the time) and the ring at 1 - 2 au, inside the primary snowline, which at later times also contains water\nice and additional snowlines. The main effect of the snowlines is the change in fragmentation velocity between mantled and non-mantled grains: dust size sharply decreases by ≈ 2 orders of magnitude for the latter. Immediately outside the water snowline, the midplane temperature is lower, due to lower dust opacity and thus more efficient cooling. Turbulent α is on the contrary, higher, providing more efficient radial transport of matter. It leads to lowering the gas surface density in this gap, which in turn increases α (see Eq. (5)), creating the positive feedback and further deepening the gap. One of the possible mechanisms to create the initial decrease in the gas surface density is dust back-reaction, which can affect the inward flow of gas at the snowline. This effect was investigated by Gárate et al. (2020) for different initial dust-to-gas ratios. The radial variation of α itself lead to the appearance of gas substructures (Tong, Alexander, and Rosotti 2024). The combined effect of lower temperature and density creates a pressure minimum, which dust tends to avoid.\nThe dead zone, where α -parameter values are lower than 10 -3 , includes the ice-free inner disc and has an approximately two times larger radial span than the iceless region. The distribution of α -parameter is shown in the right column of Figure 1. Inside the primary water snowline, the values of α are the lowest due to high surface density of gas. The dead zone is not axisymmetric and reflects the spiral structures of the gas distribution, as it is sensitive to Σ g (see Eq. (5)). The spiral arms of the dead zone span to 15 - 25 au at 160 kyr, to 10 - 18 au at 350 kyr and to 10 - 14 au at 490 kyr. The comparison with the temperature distribution in Figure 1 indicates that α and T mp are anticorrelated, as higher viscosity provides faster accretion, hence lower surface densities and more efficient cooling. Similarly, a lot of dust accumulates in the dead zone, increasing opacity, which hampers cooling.\nThe icy dust ring at 1 - 2 au is especially interesting in the context of planet formation. In this ring, dust-to-gas ratio exceeds 0.1, and surface densities of both gas and dust are increased by more than an order of magnitude compared to the adjacent regions. Due to the ice mantles that protect dust from fragmentation, the dust size reaches several cm, close to the values behind the water snowline. When the ring is established, it is self-supporting, in a sense that without external perturbation (e.g. sharp change in accretion flow from the outer disc), it can be stable for a long time, over tens of kyr.\nThe presence of water ice inside the dead zone was investigated by Vallet et al. (2023). They showed that in the discs around lower-mass stars, the turbulent heating in the inner disc can be low enough to allow the existence of ices. In our modelling the cooling of the inner region is assisted by the inner dust ring. The freeze-out has a positive feedback on dust growth due to higher v frag , which helps dust grow larger and further accumulate toward the pressure maximum in the ring. So in our modelling ices coexist with a lot of grown dust in the dead zone. Such an icy inner region appears to be a promising place for the formation of the volatile rich planets.\n\n\n
Figure 2. Radial distribution of azimuthally averaged surface densities of the volatiles in the gas and in the ice at various time instances in M1 ( M core = 0.66 M ⊙ ). Pale lines indicate the total surface density of species. The upper panels show surface densities of gas, small dust and grown dust, and the midplane temperature.
\n\n\n\n
Figure 3. Same as Figure 3 but for model M2 ( M core = 1 M ⊙ ).
\n\n3.2 Distribution of volatiles\nEven in the absence of chemical reactions, over the years of disc evolution the distribution of volatiles significantly changes compared to the initial one. This is the result of both phase transitions and dust growth and advection. Dust drift brings ices from the outer disc, enriching the inner disc with volatiles. Snowlines provide the conditions favouring accumulation of ices and gas-phase species. The formation of the established disc structures, such as dust and gas rings and spirals (see Section 3.1) leads to a complex pattern of intermittent snowlines. In this Section, we describe the main features of the molecular distributions and their implications for the composition of dust and gas at early stages of protoplanetary disc evolution.\nFigures 2 and 3 show the azimuthally averaged radial profiles of the volatile species in models M1 and M2, respectively. As mentioned earlier, we define the snowline position as the location in the disc midplane where the amount of species in the gas equals to its amount in the ice (on both dust populations). This definition allows the snowline to have complex shape, characterised by different positions for each azimuthal direction. In the azimuthally averaged distributions shown in Figures 2 and 3, the position would only be approximate. The slope of the species distribution in the vicinity of the snowline reflects the degree of the axial asymmetry in the disc. Flatter distributions appear as the contributions sum up from the snowlines, the radial position of which vary at different azimuthal angles ϕ .\nIn our model setup, the volatiles can either have no snowline in the disc, or have two or more snowlines depending on the local conditions. Snowlines are absent for more volatile species (CO and CH 4 ) at earlier times or during bright luminosity outbursts, when the disc is too hot for them to freeze out. When there are two snowlines, the inner snowline in the disc is the one determined by thermal desorption. We refer to it as the primary snowline. The outer snowline is determined by photo-desorption, it lies in the embedding envelope outside of the body of the disc where optical depth is low. It must be noted that the gas-phase species outside this photo-snowline are vulnerable to photo-dissociation by the UV radiation. This process is not explicitly included in our chemical model, but can be assessed as in Molyarova et al. (2021).\nMore than two snowlines appear when the disc physical structure becomes more complex, mainly due to the presence of the ring-like structures. Species can freeze inside a cold dense dust ring, creating additional secondary snowlines, also governed by thermal desorption. The concepts of primary and secondary snowline is necessary for H 2 O and CO 2 , which have multiple snowlines at the later stages of disc evolution. For water, the inner icy region appears at ≈ 1au at 490 kyr (see right column of Figure 3) inside a dense dust ring. For CO 2 , the ring that appears at 7 au, outside the primary water snowline at 4 au, creates the inverted thermal structure in the region with T mp close to the typical CO2 sublimation temperature of 70 - 90 K. Increase in T mp in these ring is caused by higher optical depth and consequently lower cooling on the viscously heated midplane.\nApart from the snowlines, the distributions of volatiles in Figures 2 and 3 present local radial variations in all of the species components, including total abundance of the species. The initial total abundance is kept only in the envelope. As the matter is being redistributed, abundances of all volatiles in the inner disc grow. Particularly, the process responsible for this is dust drift. It brings the grown ice-covered grains to the inner disc regions, where their ice mantles are sublimated and no longer move with the drifting silicate dust cores. The effect is stronger for less volatile species H 2 Oand CO 2 : their abundance in the gas grows by a factor of a few inside their primary snowlines. For CO and CH 4 the effect is weaker, because there is less grown dust outside their snowlines, and those snowlines are not very much established at earlier times. Thus their abundances inside the snowline only grow by a factor of unity, except for the immediate vicinity of the snowline. Moreover, both CO and CH 4 have a bump in the gas-phase distribution just outside the dust ring at 1 au, that is absent in H 2 Oand CO 2 . At the inner disc edge the abundances of CO and CH 4 in the gas are lower than the initial value.\nThe abundance enhancements appear most notably at the snowlines, with local bumps in all three phases at later times (after ≈ 350kyr). They are produced by the combination of the dust radial drift and the azimuthal oscillations of dust and gas radial velocity, described in more detail in Molyarova et al. (2021) and Molyarova et al., in prep. By 500 kyr, the surface density of gas-phase H 2 Oexceeds the initial value by a factor of 5, of CO 2 - by a factor of 7, of CH 4 - by 3.5 and of CO by 2.5. Similar accumulation powered by turbulent diffusion was previously studied for CO molecule in axially symmetric model setup (Stammler et al. 2017; Krijt et al. 2018). Their results indicated similar enhancement in the gas phase by a factor of a few. In our non-axisymmetric approach, diffusion is not a necessary requirement, and the necessary transport is rather provided by azimuthal variations of radial velocity induced by disc self-gravity (Molyarova et al., in prep).\nDistributions of ices on small and grown dust are different as they are affected by dust growth and drift. In general, there is more grown dust than small by mass, and in most of the disc there is more ices on grown dust, particularly at later times. However, in the outer disc and at the earlier times, ices on small grains dominate. As initially all ices are on small dust, it seems inevitable that they will gradually move to the grown grains as small dust coagulates and turns into grown grains. However, near all the snowlines, amount of ices on small dust increases due to the effect of the spiral pattern. Ice abundances are determined by episodic sublimation and freeze-out in the warm spiral arms of the complex density and temperature pattern (see Section 3.3.2. in Molyarova et al. 2021). As adsorption preferably happens to the smaller grains due to their larger total surface area, there is much more ices on small dust in the wake of the spiral arms. This concerns the photo-desorption snowlines as well.\n\n\n
Figure 4 shows radial profiles of the C/O ratios averaged over the azimuthal angle ϕ by the end of the simulations, at 490kyr. The key values of C/O ratio relevant in the context of planet formation are the initial value of 0.34, motivated by the comparison with the initial abundances, and 1.0, which is a boarder between carbon- and oxygen-dominated chemical regimes in the ISM and (exo)planetary atmospheres (see, e.g., Tielens 2005; Seager et al. 2005; Madhusudhan 2012).
\n\n\n\n
Figure 4. Radial profiles of the C/O ratio at 490 kyr in models M1 (le/f_t) and M2 (right). The plots show C/O in total (black), in the gas (red), and in the ice (blue). The C/O ratio in the ice (gas) is only shown for radial distances where the mass of the volatiles in the ice (gas) is larger than 0.1% of the total mass of the volatiles in the gas (ice). The grey horizontal line indicates the baseline C/O = 0.34 . Positions of the snowlines are highlighted by vertical dashed lines. The regions where water (thus all other species) is in the gas are shaded with purple.
\n\n3.3 C/O ratios\nHere we consider the relative amount of carbon and oxygen in the gas, on the surface of small and grown grains, and in total for all for all phases and disc locations. C/O ratio is seen as a perspective tracer of planet formation mechanisms (see Öberg, Murray-Clay, and Bergin 2011; Thiabaud et al. 2015). Therefore, we are especially interested in the regions of the disc and the volatile phase component where the C/O ratio declines from the total initial value (see below). Particularly, weare interested in C/O ratio noticeably above the initial value, as it was observed in atmospheres of exoplanets and suggests the formation of these planets in similarly carbon-enriched environments (Swain et al. 2009; Madhusudhan et al. 2011; Moses et al. 2013; Facchini et al. 2021).\nThe C/O ratio is calculated as the total number of carbon atoms contained in the molecules, divided by the total number of oxygen atoms in the molecules. This calculation can be done separately for the species contained in the gas phase, species on dust surface, or for the species in all phases. Thus, we calculate the C/O ratio in the gas, in the ice, and total C/O, respectively. For the ice species, we include both ices on grown and small dust grains.In some disc regions, the amount of carbon and oxygen in in a particular phase is very low, e.g. in the ice phase inside the water snowline, where all the molecules are in the gas. For such cases, C/O ratio would not be representative of the chemical composition, so we exclude the corresponding computational cells from the consideration. We only display the C/O ratio if the mass of the volatiles in the phase is higher than 10 -3 of the total mass of volatiles at a given location.\nFor the adopted molecular abundances based on Karin I. Öberg et al. (2011) and also used by Eistrup, Walsh, and van Dishoeck (2018), the baseline value is C/O = 0.34, which is in line with the gas-phase abundances in the ISM ( ≈ 0.36, Przybilla, Nieva, and Butler 2008). This value is different from the typical solar value of ≈ 0.5 (Przybilla, Nieva, and Butler\n2008), which is commonly used as a reference (e.g., Öberg, Murray-Clay, and Bergin 2011; Semenov et al. 2018). First, local galactic abundances changed since the Solar system formation 4.6 billion years ago (see, e.g., Appendix A in Bergin et al. 2024, for the comparison). Second, the difference also stems from inclusion or non-inclusion of the dust component. The stellar atmospheric abundances comprise all the elements present in the medium where the star formed, while the elements available for the volatiles are only a fraction of that. The values of the elemental abundances can be determined separately for the volatile (gas and ices) and the refractory (rocky dust grain cores) components of the ISM (Hensley and Draine 2021). Here, we do not take into account carbon and oxygen from the rocky cores of dust grains. Due to the model assumptions, the initial ice-to-rock mass ratio in our model is quite low (0.08 compared to 2 - 4 in Pontoppidan et al. 2014). Therefore, adding the elements from solid dust grains to those contained in ices would be misleading, as the former would dominate in the resulting C/O ratio. In this work, we concentrate on considering the C/O ratios of the volatile component (ice and gas), and compare them with the initial value of 0.34.\nBy the age of 490 yr, the C/O ratio in both models is significantly different from the initial value. This concerns the total C/O ratio, as well as the C/O ratio in the gas and in the ice. Let us analyse the distributions of C/O ratios that we can see from Figure 4, moving inwards from the envelope to the centre. The main features of the C/O distributions are the\nfollowing:\n\n· the C/O ratio in the envelope is close to the initial value 0.34 in the gas and in total, and it grows up to unity from the envelope to the CO snowline in the disc;\n· around the CO snowline, the C/O approaches unity;\n· between CO 2 and CH 4 snowlines the C/O in the gas is > 1, and the C/O in the ice is below the initial;\n· at the distances of tens of au, there are variations in the total C/O ratio that are not connected with any snowlines, or variations in gas- or ice-phase C/O;\n· ice-phase C/O ratios peak around all snowlines except water;\n· inside the primary water snowline is the region rich in volatiles, with both gas- and ice-phase C/O are the lowest in the disc.\n\nAs expected, the key changes in the C/O ratio distribution are associated with the positions of the snowlines. There are two main processes. First, the freeze-out and desorption at the snowlines transfer the elements between phases, altering the C/O in the gas and in the ice. Second, the snowlines favour the accumulation of the respective volatiles both in the gas and in the ice, as was discussed in Section 3.2, pumping up the amount of both components and altering their proportion. Besides the snowlines, radial drift of grown grains (Weidenschilling 1977) transports the volatiles inwards. We discuss below which processes are responsible for the formation of the listed features.\nC/O in the outer disc. In the surrounding envelope, outside the disc, all volatiles are in the gas due to photo-desorption, and C/O in the gas is close to the initial value of 0.34. Around the COsnowline and beyond, the C/O in the gas is close to unity, and in the gas C/O is higher that the initial, which requires explanation. The region beyond the CO snowline is usually described as the place where all species are frozen, thus having a stellar C/O ratio in the solid phase and practically no carbon or oxygen in the gas (e.g., Öberg, Murray-Clay, and Bergin 2011; Öberg and Wordsworth 2019; P. Mollière et al. 2020). In our simulations, only in model M2 there is a region where less than 10 -3 of C and O is in the gas, and in model M1 such region is absent. Due to the asymmetric spiral structure that persists even at 490 kyr, even though most of CO is frozen beyond the snowline, there is a significant (> 10 -3 ) fraction of it in the gas. Additionally, the position of CO snowline itself is significantly affected by the dust drift, as it declines from the equilibrium between adsorption and desorption.The indistinctness of the CO snowline also helps CO to persist in the outer regions: all other species are ultimately frozen, so they are efficiently carried away to the inner disc via radial drift, while this works worse for only partially frozen CO. It creates relative overabundance of CO in the outer disc, elevating the total C/O ratio and later the ice-phase C/O ratio, when the preserved CO freezes out. Between the CO ice line and the envelope, there is a gradient of C/O in the gas due to photodesorption of the ices. The last molecule to be photo-desorbed is water, which returns oxygen to the gas phase at the farthest radial distance.\nC / O ≈ 1 around the CO snowline. Beyond the CH 4 snowline, CO dominates the composition of volatiles in the gas phase, leading to the gas-phase C/O close to unity, the value characteristic of the CO molecule. At the CO snowline, the CO dominates the ice-phase composition as well. While dust drift substantially lowered the abundances of CO 2 and H 2 Oby 490 kyr in these regions, CO ice accumulated at the snowline. This leads to the ice-phase C/O closer to 1, too, making the vicinity of the CO snowline a region where total amounts of C and O are similar.\nHigh C/O in the gas, low C/O in the ice. In the disc regions beyond the primary CO 2 snowline, only CO and CH 4 are in the gas, meaning the dominance of carbon and C/O > 1. Consequently, the C/O in the ice is generally lower, around 0.2, as the ices are mainly H 2 O and CO 2 rich in oxygen. This is consistent with the classical step-like picture of Öberg, Murray-Clay, and Bergin (2011), with the addition of carbonrich methane allowing the C/O > 1. The midplane C/O ratios we simulate are difficult to directly compare with observations, which mostly trace the molecular layer above the disc midplane. High C/O ratios in the gas are indeed observed in many protoplanetary discs (e.g. Miotello et al. 2019), but they are considered to be a natural consequence of dust settling (Krijt et al. 2018; Krijt et al. 2020). Dust inward drift could also enhance this effect in the outer disc regions, creating the radial gradient of total C/O ratio in the disc. Observations of CS and SO emission coming from close to the midplane layers potentially indicate the presence of such radial gradient of the gas-phase C/O in the PDS 70 disc (Rampinelli et al. 2024).\nVariations of total C/O. Throughout the disc, there are sharp changes in total C/O ratio not connected with any snowline. Most noticeable are the variations between CO 2 and CH 4 snowlines, where gas-phase and ice-phase C/O ratios are stable. These variations are associated with the disc substructures, particularly with the dense dust rings described in Section 3.1. The total C/O changes due to radial variations in dust-to-gas ratio: when it is higher, the total C/O is closer to the ice-phase C/O, and vice versa. Inside the dust-rich rings, H 2 Oand CO 2 ices are abundant, due to high surface density of dust relative to gas. At the same time, CO and CH 4 in the gas phase have similar surface densities inside dust rings and between them. Thus, in the dust rings, CO 2 and water are overabundant, leading to lower total C/O ratio. We consider this effect in more detail in Section 3.5. Variations of the total C/O ratio due to dust substructures are also present in the cold ring at ≈ 1au.\nC/O peaks at the snowlines. There are peaks of the C/O ratio in the ice right outside the snowlines of CO 2 , CH 4 and CO, produced by the accumulation of the respective ices (see Figures 2 and 3). In M2 model, the amount of CH 4 and CO ices at their respective snowlines becomes comparable with or even larger than those of CO 2 and H 2 O(compare the right panels of Figure 3), leading to C/O in the ice ≈ 0.6 - 0.9. In model M1, the accumulation is more prominent, so the C/O ratio in the ice approaches unity at CO snowline and > 1 at the methane snowline. These peaks distort the pattern of generally\nlow C/O ratio in the ice and preset additional regions where carbon-rich planetesimals could be formed, and carbon-rich pebbles could be accreted onto forming protoplanets. At the snowlines of H 2 Oand CO 2 , the C/O in the ice approaches the C/O ratios of these molecules, 0 and 0.5, respectively.\nLower C/O inside the water snowline. Inside the primary water snowline, the C/O ratio is generally lower than outside of it, close to the initial 0.34. Contrary to the outer regions depleted of ices due to the radial drift, the disc parts inside and around the water snowline are enriched in volatiles, and particularly of oxygen-rich water and CO 2 . Total and gas-phase C/O ratios vary from ≈ 0.2 to 0.6, as the secondary water snowlines add more substructure to the C/O distribution. The regions where the only ice is water and the ice-phase C/O = 0 are the inner cold dust ring at 1 au and the narrow annuli between water and CO 2 snowlines. The enrichment of the inner disc regions with oxygen as a result of dust radial drift is suggested by the resolved observations of molecules in protoplanetary discs (Banzatti et al. 2020).\nThese characteristic features of the C/O distributions are similar in the two presented models. The main difference is the radial distances where the borders between the zones are located; they are closer to the star in the less massive and thus colder model M1. This is mainly due to slightly different masses of the central star accumulated throughout 490 kyr of non-identical protostellar accretion history, which lead to different luminosity and thermal structure (see upper panels of Figure 5). Particularly, the stellar masses and luminosities at this time instance are: 1.07 L ⊙ and 0.34 M ⊙ (for M1), 1.89 L ⊙ and 0.58 M ⊙ (for M2). The less massive model M1 demonstrates overall higher C/O ratio in both phases.\n3.4 Evolution of the snowlines and the C/O ratios\nThe positions of the snowlines are crucial for the values of C/O ratios, both because of the direct change through the phase transitions and the associated accumulation of volatiles. They depend on the local gas and dust properties, particularly on temperature. They can also be shifted inward due to dust drift (Piso et al. 2015). Snowlines evolve as the disc structure changes with time. In this Section, we consider the co-evolution of the snowlines and the C/O ratios and discuss the mechanisms of species redistribution over the disc.\nThe temporal evolution of the azimuthally averaged C/O ratios and the equilibrium positions of the snowlines is shown in Figure 5. It is evident from Figure 5 that the C/O ratio indeed follows the snowlines, particularly inside ≈ 10 au. The C/O structure changes throughout the disc evolution, and some key features only appear at later times.\nOne of the key factors affecting the disc thermal structure is the luminosity of the central source. In the upper panels of Figure 5, we show the evolution of total luminosity, which directly affects the positions of the snowlines. The luminosity is the sum of stellar and accretion components, coming from the protorstar itself and gravitational potential energy of the accreted matter. The stellar luminosity gradually decreases as the protostar becomes more compact. Accretion luminosity\ndepends on the accretion rate, which is highly variable as a result of magnetorotational and gravitational instabilities in the disc (Kadam et al. 2019, 2020; Vorobyov, Khaibrakhmanov, et al. 2020). The simulated episodic luminosity outbursts are similar to those occurring in the observed YSOs (Connelley and Reipurth 2018), with their amplitudes of tens and hundreds L ⊙ . In the more massive model M2, the outbursts are more frequent, brighter and occur until later times. The reasons behind this difference is the massive disc being more prone to both MRI and GI, which needs to be investigated in more detail in a separate study.\nSnowlines of the least volatile of the considered species, H 2 Oand CO 2 , exist in the model since the earliest phases of the disc formation. Even during bright luminosity outbursts ( ∼ 100 L ⊙ ) they do not disappear, but move farther away from the star. During the first ≈ 50kyr the disc is spreading out, it is highly asymmetric and dynamic, so the snowline positions oscillate. Later on, the disc generally cools down, and the snowlines gradually move toward the star (except during the outbursts). In model M2, between 130 and 500 kyr, the primary water snowline moves from 12 to 4.5 au; the primary CO 2 snowline moves from 23 to 7 au. In model M1, the water snowline moves from 9 to 4 au, and the CO 2 snowline moves from 17 to 4.5 au.\nDespite a factor of two different binding energies of H 2 O and CO 2 (5770 and 2360 K, respectively), the locations of their primary snowlines do not differ much due to the steep radial temperature gradient around these distances (see upper panels of Figures 3 and 2). Inside 10 - 20 au, T mp is determined by heating mechanisms other than external irradiation: viscous heating, gas work (PdV heating), heating by shocks and energy transport with advection. This also means that the water snowline is less sensitive to the level of irradiative heating, thus only slightly affected by the luminosity outbursts. The temperature change is particularly sharp at the water snowline(s), with the absolute value of the approximated power-law slope of 1.5 - 6. High surface density of dust in water-ice-free regions makes cooling less efficient and leads to locking up the produced heat and consequently higher temperatures. Besides, both these species have multiple snowlines due to the formation of ring-like substructures with the conditions close to the borderline between their frozen and gaseous state. These additional snowlines also affect the C/O distributions.\nMethane and CO are the more volatile species in our model. They either have zero or two snowlines in the disc. The snowlines are absent during the outbursts brighter than ≈ 100 L ⊙ for methane and ≈ 200 L ⊙ for CO. Until approximately 200 - 250 kyr, there is no established CO snowline inside the disc. For example, at 160 kyr (see lower left panel in Figures 3 and 2) there is CO ice on both small and grown dust, but their amount is an order of magnitude lower than that of the gas. Additionally, most of this ice is located at the outer disc edge, where gas and dust surface densities sharply drop. Similar distribution appears for CO and CH 4 during bright outbursts: their ices are present in some disc regions, but due to non-axisymmetric disc structure, they do not dominate in the averaged profiles, and there is no common snowline for\n\n\n
Figure 5. Evolution of central source luminosity and C/O ratio in models M1 ( M core = 0.66 M ⊙ , le/f_t) and M2 ( M core = 1 M ⊙ , right). The upper panels show stellar and accretion luminosity depending on time. Below are, successively, total C/O ratio, C/O in the gas, and C/O in the ice, depending on time. The C/O values above and below the initial value of 0.34 are coloured in shades of red and blue, respectively. The regions with low abundances of both carbon and oxygen, either in the gas or ice phases, are shown in white. Coloured contours correspond to the positions of the snowlines. Photo-dissociation snowlines are not shown.
\n\nthe whole disc. There are no secondary snowlines of CO or CH 4 , because there is no prominent gas and dust structures in the outer disc where these species are frozen.\nSnowlines divide the disc into several zones with different characteristic C/O ratios. However, the chemical composition and C/O ratios in these zones change with time. One of the distinct zones is the region where water is not frozen, shaded with purple in Figure 4 and circumscribed by the dark purple dotted line in Figure 5. In this zone, all the species are in the gas phase, so the C/O ratio is initially close to 0.34. However, as the disc evolves, dust brings more volatiles from the outer disc. Particularly, the abundance of water grows most significantly, making C/O in the gas decrease with time. This happens because the main mechanism of the transport of the volatiles is dust radial drift, which works best in the regions where grown grains can sustain their mantles. Banzatti et al. (2020) suggest that dust growth and drift can be responsible for the observed anticorrelation between disc radius and H 2 O emission, implying that the inner regions of small discs with are enriched in water brought by efficiently drifting grains. Water is the least volatile species in our model, it is frozen in the largest part of the disc, thus its distribution is most strongly affected by the dust drift. At later times, this zone is divided into two, when a cold dense dust ring forms at 1 - 2 au, which happens at ≈ 460 kyr in model M2 and at ≈ 270 kyr in model M1. Interior to the ring, water abundance in the gas is around an order of magnitude lower than outside of it in both models. This happens because the inward flow of gasphase H 2 Ois 'blocked' by the cold ring where it freezes and accumulates with the grown grains in the pressure maximum. So the C/O ratio in the gas at r ⪅ 2au is determined by CO 2 and thus close to 0.5.\nThe C/O ratio in the ice is not defined in the envelope, where all ices are photo-desorbed, and in the warmest inner disc, where even water is in the gas. In disc regions where only water is frozen, the C/O ratio in the ice is zero. Before 200 - 250 kyr, when the disc cools down enough for CO to freeze out, the C/O ratio in the ice in the rest of the disc is close to 0.2. It is determined by CO 2 and H 2 O ices, with a small contribution from the low-abundance CH 4 . After that, when CO freezes out and CO snowline appears, the C/O ratio in the outer disc region becomes close to the initial value both in the ice and in total. This region beyond the CO snowline is frequently referred to in relation to giant exoplanets with stellar C/O ratios (e.g., P. Mollière et al. 2020; Öberg and Wordsworth 2019; Ohno and Ueda 2021). This region would be a perfect location for planets to accrete pebbles covered with icy mantles with the primordial elemental ratio, which would directly become part of the planetary atmosphere. However, pebbles are not initially present in the disc, and their existence in these outer regions is not guaranteed. Pebbles are dust grains large enough to move relative to gas e.g., Lambrechts and Johansen 2012; Lenz, Klahr, and Birnstiel 2019, and a fraction of the grown dust in our modelling can be classified as pebbles. The properties of pebbles and composition of their ice mantles in the same setting of the FEOSAD model were studied by Topchieva et al. (2024). They show that pebbles\nappear in the disc as early as 50 kyr after its formation, and exist in a wide region of the disc. This partially includes the region beyond the CO snowline, but only the area of CO enhancement, where CO dominates in the ice composition and thus the C/O ratio in the ice is close to unity (see lower panels in their Figure 3).\nAs shown by Topchieva et al. (2024), ices on pebbles are dominated by H 2 O and CO 2 . In this case, relatively high values of the C/O in the ice ( ≈ 0.5) correspond to the regions where there is more CO 2 , i.e. around the CO 2 snowlines. This is also the region where a prominent dust ring forms under the influence of the primary water snowline. It is characterised by accumulation of CO 2 and to lesser degree H 2 Oice, as well as vapours, and relatively high amount of grown dust in the ring. The dust ring is situated between the two regions with C/O ≈ 0.34 in the ice. It presents another favourable location for accreting the ice content with close-to-initial C/O ratio.\nThe total C/O ratio in the disc also changed significantly from the initial value due to dust drift that redistributes the ices. The matter becomes more carbon-rich as the grains bring CO and CH 4 from the outer disc parts. This effect was previously investigated by Stammler et al. (2017) and Krijt et al. (2018) in their modelling of CO dynamics and dust evolution. However, as was shown by Krijt et al. (2018) and Krijt et al. (2020), vertical settling of grown grains towards the midplane is responsible for depleting the upper layers in the outer disc of gas-phase oxygen, which cannot be captured within our thin-disc modelling. The panels in the second row of Figure 5 demonstrate strong enhancement of total C/O ratio in the intermediate disc regions. In model M2, the total C/O ratio between 10 - 100 au becomes ≈ 0.7 after ∼ 300 kyr, which is two times higher than the initial value. At the snowlines of CH 4 and CO 2 it approaches unity, mostly due to the accumulation of these species in the gas. In model M1, this process begins ≈ 200kyr earlier and consequently leads to even higher total C/O ratio. At 400 - 500 kyr, most of the disc between 5 and 100 au has total C/O ≳ 1 in model M2, which demonstrates the powerful impact of dust drift.\nA distinctive feature of the produced C/O distributions is the peaks of total and ice-phase C/O ratios around the CO and CH 4 snowlines. In model M2, the increase of the C/O ratios becomes noticeable only after ≈ 450kyr, while in model M1 it starts to form around ≈ 300 kyr. Initial abundances of CH 4 and CO are lower than that of CO 2 and H 2 O. As these species accumulate at their snowlines, the abundances become comparable, leading to C/O in the ice ≈ 0.6 - 0.8 in model M2 and up to 1 in model M1. Similar accumulation is seen around CO 2 snowline, but unlike CO and CH 4 snowlines, it is also connected to the interaction with the dust ring structure and is considered in more detail in Section 3.5.\n3.5 Two-dimentional structure\nThe disc is not axisymmetric even at later stages of its evolution (see Section 3.1 and Figure 1), which is also reflected in the distributions of volatiles and C/O ratios. Examples of 2D distributions of C/O in model M1 at two time instances are\n\n\n
Figure 6. Distributions of C/O ratios and gas/dust surface densities. Model M1 160 kyr (upper le/f_t) and 300 kyr (upper right); model M2 250 kyr (lower le/f_t) and 490 kyr (lower right). Dotted lines mark the positions of the snowlines for H 2 O(dark purple), CO 2 (magenta), CH 4 (green), and CO (yellow).
\n\n\n\n
Figure 7. Averaged radial profiles of dust-to-gas ratio, the total C/O ratio, and CO 2 and H 2 Oin the gas and in the ice in model M2 at 490 kyr. The horizontal lines in the upper panel show the reference values for the C/O ratio (0.34) and dust-to-gas ratio (0.01).
\n\n\n\n
Figure 8. Dependence between total C/O ratio and dust-to-gas mass ratio in models M1 (upper panel) and M2 (lower panel). Three time instances are shown. The dashed line shows the fitted log-linear dependence for 490 kyr.
\n\nshown in upper panels of Figure 6. The left-hand side group of panels shows the structure that is characteristic of the earlier phases, the same time instance as the upper row in Figure 1. The prominent spiral structure and a clump both have their reflection in the C/O ratios. The snowlines of CO 2 and CH 4 have clearly non-regular shape affected by the spiral pattern of the gas, with blobs of frozen methane inside the main snowline. Inside the clump, the temperature is higher (see Figure 1), and both CO 2 and water are in the gas. The separation between gas-phase and ice-phase C/O ratios is clear, but the total C/O ratio in the clump is similar to the surroundings and is only slightly above the primordial value. The gas-phase C/O ratio inside the clump is around 0.7, lower than in the surrounding gas at the same radial distances. Similar decrease of C/O ratio indicated by lowered CS/CO ratio was observed in the disc around DR Tau (Huang et al. 2024). Lifetime of the clump (several orbital periods) is too short for significant differentiation between gas- and ice-phase composition to develop, mainly because of insufficient numerical resolution. Focused studies with higher resolution are needed to explore the C/O ratio in the clumps as precursor of giant planets formed via disc fragmentation.\nThe upper right-hand side group of panels in Figure 6 features a later stage of the disc evolution in model M1. By 300kyr, the accretion rate and the average positions of the snowlines are stabilised (see Figure 5). The spiral structure in the gas is much weaker but still present at r > 10 au. The CO snowline is established at ≈ 80au. The 2D shape of the snowlines is more circular at 300 kyr, particularly for the less volatile CO 2 and H 2 O. The effect of spirals is still evident in the contours of the CO and CH 4 snowlines. At the outer side of these snowlines, at approximately 40 and 80 au, respectively, the C/O in the gas has local maxima, which are also seen in Figure 4. The one at 40 au also has a clearly spiral shape, repeating the pattern of the gas distribution. The radial span of these peaks is of the same scale as the dispersion of the respective snowline distance from the star. For example, at the CH 4 snowline the C/O peak is ≈ 10au wide, and the snowline is at 28 to 37 au distances. Similar accumulation powered by diffusion of water vapour was shown, e.g., by Drążkowska and Alibert (2017). In addition to diffusion, in our modelling, the species are delivered outward from the snowline due to the dynamic shape of the snowline and two-dimensional movement of gas and dust (see Molyarova et al. 2021, and Molyarova et al., in prep).\nIn more massive model M2, the shape of the snowlines remains complex for longer times. Examples of C/O distributions in M2 are shown in the lower panels of Figure 6. The radial scale is chosen so that CO 2 and H 2 O snowlines are seen in more detail. At 250 kyr, even the grown dust distribution still has spiral pattern, and CO 2 snowline is following its complex multi-armed shape. The accumulation of CO 2 at the snowline is also very efficient, but it does not strongly affect the C/O ratios, as it drives them to the value 0.5 of the CO 2 molecule, which is close to the intrinsic value. The complex-shaped region between these snowlines has the ice-phase C/O ≈ 0.7. This pattern moves and changes its shape, thus affecting all\nradial distances between 10 and 20 au. The total C/O ratio is slightly above the ambient value, approaching 0.5, because CO 2 in both phases begins to accumulate in this region.\nAt later times, the C/O distribution becomes more complex due to the presence of disc substructures, particularly the dust rings. The lower right-hand side group of panels in Figure 6 show this in more details. First, the temperature and density variations across the dust rings lead to the formation of multiple CO 2 and H 2 Osnowlines. An additional annulus of icy CO 2 appears in a relatively cold region between the dense dust rings at 6 - 7 and 8 - 13 au. The dust rings are warmer due to higher optical depth and active heating by disk internal sources, and the inner edge of the 6 au ring is warm enough to sustain gas-phase CO 2 . Second, the accumulation of icy dust grains in the rings alters the total C/O ratio. The total C/O ratio anticorrelates with the distribution of grown dust grains: inside the dense rings it is close to the initial value of 0.34, while between the rings, it is higher and reaches 0.8 - 0.9. At the same time, neither ice-phase, nor gas-phase C/O ratio displays any noticeable variations at 12-25 au, but they do have variations at r < 12 au, following the dust ring pattern.\nThe total C/O ratio is defined by the combination of the ice- and the gas-phase component. Their relative contribution is proportional to the dust-to-gas mass ratio. This is illustrated in Figure 7. Within the rings, the total C/O is dominated by ices of oxygen-rich species CO 2 and H 2 O, particularly on grown dust accumulated in the pressure maxima. For example, there is a lot of water ice at 6 - 7 au, and its contribution to the total C/O is weighted with high dust-to-gas ratio of almost 0.1, so the resulting total C/O is lower than the initial value. At the same time, at ≈ 5au, dust-to-gas ratio is around 10 -4 . The dominant species there are in the gas, ice surface densities are 1 - 2 orders of magnitude lower, so the total C/O ratio goes up. In the wide dust ring at 9-13 au, both H 2 Oand CO 2 contribute, although at the warmer inner edge of the dust ring CO 2 is sublimated. The value of total C/O is approaches 0.34, also elevated by the presence of gas-phase CO and CH 4 . Beyond ≈ 10au, both H 2 O and CO 2 are frozen, and the variations of the total C/O ratio are clearly anticorrelated with the dust-to-gas ratio and the position of the rings. Between the dust rings, where contribution of ices is low, the C/O is determined by CO and CH 4 gases, and reaches values of ≈ 1.\nThe anticorrelation between the total C/O ratio and dustto-gas mass ratio is an interesting finding. It is illustrated in Figure 8 for both models. Only the points within the disc are shown, with Σ gas > 0.1 g cm -2 .The pattern obviously changes with time, but the anticorrelation persists. Model M1 demonstrates wider variety of C/O and dust-to-gas ratios. The disc points are grouped in tangled curved \"branches\", some of them steeper than others. The 'width' (or spread) of these branches is determined by the azimuthal substructures. In the axially symmetric parts of the disc the values of dust-to-gas ratio and the total C/O are similar at a given radial distance. Different branches, which can be closer to vertical or horizontal orientation, are the result of the radial variations in the ice-phase C/O ratio and in ice fraction relative to the dust silicate cores. Horizontal branches correspond to weak or absent anticorrelation.\nFor example, near the snowlines of carbon-rich species, C/O in the ice is high and close to that of the gas, which decreases the effect of dust mass fraction on the total C/O. In areas with no ices, the anticorrelation is also irrelevant, because the total C/O is determined entirely by the gas phase. Vertical branches, on the contrary, correspond to the strongest anticorrelation effect, which is expected in the regions between the snowlines, where ice-phase C/O is the lowest. The identified anticorrelation in Figure 8 is similar to the results of chemical population synthesis modelling (Cridland et al. 2019, 2020) showing that the more solids a planet accreted in the disc, the lower the C/O ratio is in its atmosphere.\nWe fit the data for t = 490kyr with a linear law (taking the logarithm of dust-to-gas ratio) and obtain the following fits: C/O = -0.056 - 0.25 log 10 ( ξ ) for model M1, and C/O = -0.18 - 0.27 log 10 ( ξ ) for model M2. Here, ξ is the dust-to-gas mass ratio. The correlation coefficients are -0.54 for M1 and -0.57 for M2. At the shown earlier times (160 and 350 kyr), the correlation coefficient changes between approximately -0.9 and -0.5, which indicates noticeable anticorrelation throughout the disc evolution.\nWe note that these C/O ratios only include the volatile component, without the contribution from the refractory material. Although the refractory material is typically considered as silicates, which are rich oxygen, it contains a significant amount of carbon, with the resulting C/O ≈ 0.5 (see Table 2 in Hensley and Draine 2021). This solid carbon can be subject to carbon grain destruction (Lee, Bergin, and Nomura 2010; Gail and Trieloff 2017; Wei et al. 2019), but this process should be treated separately, as it also affects the gas-phase carbon abundance. Taking refractory cores into account should affect the dependence between total C/O ratio and dust-to-gas ratio, as adding more rock would make the C/O ratio closer to 0.5. Thus the degree of the anticorrelation must be affected by the composition of rocky cores, but the anticorrelation itself should remain even when refractories are included, because the C/O ≈ 0.5 is still lower than typical C/O of the gas in most of the disc ( ⪆ 1).\nDust rings are detected in the majority of the observed protoplanetary discs (Long et al. 2018; Huang, Andrews, Dullemond, et al. 2018). They are considered as a plausible sites of planet formation (Carrera, Johansen, and Davies 2015; Yang, Johansen, and Carrera 2017; Li and Youdin 2021; Lee, Fuentes, and Hopkins 2022; Jiang and Ormel 2023). The anticorrelation between the total C/O ratio of the volatiles and dust-to-gas mass ratio that we point out is a logical consequence of ices having typically lower C/O ratios and being attached to dust grains. If planets are formed in the dust rings with high dustto-gas ratios (> 10 -2 ), either exclusively from solids, or with the inclusion of the dust component, this would imply that their material initially has lower C/O ratio of ≈ 0.5 and below. To reach higher C/O ratios up to unity and above, which are observed in many exoplanets, these planets would need to migrate and accrete carbon-rich gas from regions other than their immediate formation sites inside the dust rings. In case if planet formation occurs independently of the dust rings, e.g. in the GI, their material is not determined by this anticorrelation.\n4. DISCUSSION\nOur simulations present a wide range of C/O ratios in the disc in different phases evolving with time. For the atmospheres of giant exoplanets, a variety of C/O ratios were retrieved, too. Here we can compare them to identify the disc regions and times where the chemical and physical conditions for planet formation are consistent. Most of the exoplanets for which the atmospheric composition was retrieved have super-stellar C/O ratios (Hoch et al. 2023; Weiner Mansfield et al. 2024), which draws more attention to carbon-enriched areas. They are suggested to form by core accretion, and accreting mostly the gas, which is typically more carbon rich (beyond water snowline). A lot of planets are observed to have stellar or slightly super-stellar C/O (e.g., P. Mollière et al. 2020; Zhang et al. 2021; Smith et al. 2024; Sing et al. 2024; Nortmann et al. 2024, and many others). One way to form such planets is gravitational instability, which includes solids and gas together, thus undifferentiated matter is suitable for producing planets with stellar C/O. Disc fragmentation to clumps due to GI requires particular conditions (Meru and Bate 2010; Vorobyov 2013), and the direct collapse of gravitationally unstable clumps tends to produce rather massive objects (e.g. ≈ 5 M J planets and ≈ 60 - 70 M J brown dwarfs, see Figure 4 in Vorobyov, Zakhozhay, and Dunham 2013) at larger radial distances (> 10 - 100 au, see Vorobyov, Zakhozhay, and Dunham 2013; Kratter and Lodato 2016). GI can also assist the assemblage of planetary cores (Nayakshin 2010a, 2010b; Nayakshin, Helled, and Boley 2014; Vorobyov and Elbakyan 2019). A planet formed through core accretion can also accrete planetesimals, which can be covered with ice, and enrich the atmosphere with oxygen, making the C/O ratio close to the initial stellar value. There are particular exoplanets, where lower than stellar C/O ratio is observed in the atmosphere, such as β Pic b (GRAVITY Collaboration et al. 2020; Worthen et al. 2024), HD 209458 b (Xue et al. 2024), or HD 189733b (Fu et al. 2024). Such planets need even more enrichment in ices with low C/O, which makes the regions with low C/O in the ice also more attractive sites for planet formation.\nGravitational instability implies that the planet forms from a mix of gas and dust (Bodenheimer 1974), this is why it is suitable to explain the formation of planets with solar, or unaltered C/O ratios. In our modelling, GI would be associated with the total C/O ratio, which we find to be significantly variable, too. For GI to form a planet with a primordial C/O ratio, it has to occur during the first 100 kyr after the disc formation. At later times, the total C/O ratio changes, and the only region with the primordial C/O ratio is the very outer disc parts, at > 100 au, which is the part of a protoplanetary disc, where conditions for GI are the most consistent with the observed properties of these objects (Rafikov 2005). Planet formation through GI is indeed more likely at earlier evolutionary stages, when gas surface density is higher (Armitage 2010). We can highlight the areas where planet formation via GI is possible in our modelling as the regions where Q Toomre ≤ 1. They are shown in the upper panel of Figure 9 for model M1. These regions appear between ≈ 10-100au before ≈ 300 kyr.\nHowever, at later times, clumps could appear in the disc as a result of an external perturbation, such as a stellar flyby (Thies et al. 2010). In this case, the planet would be formed from the material with altered C/O ratio, most probably with elevated amount of carbon, as the regions outside 5 - 10 au are typically more gravitationally unstable. This means that GI can produce planets with super-solar C/O ratios, if it is induced by external influence at later stages of disc evolution.\nCore accretion is another most widely discussed scenario of giant planet formation. Accretion of gas should produce atmospheres with the C/O ratio close to the one in the gas phase of protoplanetary disc. However, dust grains are also accreted, so pebble and planetesimal accretion can enrich the atmosphere in volatile components (Mordasini et al. 2016; Danti, Bitsch, and Mah 2023). This makes atmospheric C/O ratio closer to the ice-phase C/O, but in case of gas giants, the amount of the solids needed to compensate the prevalence of carbon in the gas should be quite high, up to hundreds of Earth masses (GRAVITY Collaboration et al. 2020). The C/O ratios in exoplanetary atmospheres are often interpreted in terms of pebble accretion, so planets with stellar C/O ratios are assumed to form in the environments where solid phase C/O is unprocessed and thus close to the initial value. One of such locations is beyond CO snowline, where most of the carbon- and oxygen- bearing material is in the ice, e.g. for Jupiter (e.g. Öberg and Wordsworth 2019; Ohno and Ueda 2021). In our models, this is rather the vicinity of the CO 2 snowlines, and the pebbles beyond the CO snowline are mostly covered with carbon-rich CO ice (Topchieva et al. 2024). Additionally, the CO snowline is typically very far from the star (> 40 au), so such scenarios must rely on planet migration to obtain their current location. Interpretations relying on chemical modelling extend this region to include the area beyond CO 2 snowline due to additional chemical processing of CO in this region, e.g. HR 8799e (P. Mollière et al. 2020). This puts milder constraints on the original distance from the star where pebbles should be accreted and requires less migration. Modelling of planet formation and migration including pebble and gas accretion puts the formation location of planets with super-solar C/O ratios beyond water and CO 2 snowlines (Bitsch, Schneider, and Kreidberg 2022).\nIn our modelling, the C/O in the ice is close to initial value in the regions beyond CO snowline, excluding the area of CO accumulation. Between CO and CO 2 snowlines, it is lower, as our model does not include chemical processes apart from adsorption and desorption. However, there is another region with C/O in the ice close to initial. The vicinity of primary water snowline and the ring induced outside of it has values of C/O in the ice only slightly above the initial value of 0.34. It is surrounded by the snowlines of CO 2 . This region could be another favourable location for forming planets with the stellar C/O. As it is situated closer to the star, it would imply less migration.\nRare planets with lower than stellar C/O ratio, such as β Pic b (GRAVITY Collaboration et al. 2020; Reggiani et al. 2024), HD 209458 b (Xue et al. 2024), HD 189733b (Fu et al. 2024), or KELT-1 b, Kepler-13A b and WASP-79 b\n(less precisely determined, see Hoch et al. 2023), need to have accreted a lot of oxygen-rich ice. Therefore, they are more likely to accrete solid material in the regions with the lowest ice-phase C/O ratios. The most suitable region would be at the distances between H 2 O and CO 2 snowlines, where ice mantles are made of pure water. However, in our modelling results, this region is very small, typically only a few au wide, as the snowlines are close to each other. This is because of steep temperature profile in this region, which is a result of the significant contribution of non-irradiation heating mechanisms, particularly viscous heating. Beyond CO 2 snowline, there are also regions with relatively low (0.2 - 0.3) C/O in the ice, but much more solids need to be accreted in such areas to compensate for the excess of carbon from the gas.\nLet us summarise the above constraints on planet formation locations and mechanisms implied by our simulated C/O ratios. Core accretion is suitable for forming planets with high C/O ( ≈ 1) in the atmosphere around the snowlines of CO, CH 4 and CO 2 , or anywhere beyond CO 2 snowline if they did not accrete much solids. Planets with stellar or slightly superstellar C/O ratio need to accrete (a lot of ) oxygen-rich solids to compensate their initially high C/O inherited from the gas. The locations where this is possible is between CO 2 and CH 4 and between CO and CH 4 snowlines. Planets with low C/O ratio could accrete ices between H 2 O and CO 2 snowlines. Snowlines are favourable planetesimal formation sites, so the planets that accreted planetesimals/pebbles there can have altered C/O ratios. The values will be lower than the initial if they form at the water snowline, and higher if they form at the snowlines of carbon-rich species. At the same time, to obtain planets with stellar C/O formed at the snowlines, these planets would need to migrate and accrete matter in different regions of the disc to make their C/O ratio close to the initial stellar value. Alternatively, planets with stellar C/O ratio can form via disc fragmentation through GI at earlier stages. Dedicated modelling of planet formation accounting for evolution of dust and volatiles is necessary to put more particular constraints on planet formation scenarios.\nPlanetesimals play an important role in delivering the icephase elements to planetary atmospheres. To form planetesimals, additional physical process is needed, such as streaming instability (SI, see Youdin and Goodman 2005), which is not explicitly included in our modelling because of insufficient numerical resolution and simplified vertical disc structure. However, we can post-process the simulation results to check if the conditions for SI are fulfilled in some regions of the disc where dust-to-gas ratio and dust size are enhanced, following Vorobyov et al. (2024). Dense dust rings forming at later stages (see Section 3.1) seem to be an ideal location for triggering the SI, which would ultimately lead to formation of planetesimals and then planets in the disc. Triggering the SI requires specific relations between local dust-to-gas ratio and Stokes number (Yang, Johansen, and Carrera 2017). The criteria vary depending on the model, we adopt them from Li and Youdin (2021). Another criteria would be the requirement of volume density of dust to exceed that of gas in the midplane (Youdin and Goodman 2005). We do not apply it, as in our modelling,\n\n\n
Figure 9. The disc regions where the conditions for GI and SI are fulfilled in model M1. The regions and times where there is no instability are shaded in white. In the upper panel, the colour indicates the minimum value of Q Toomre at a given radius, if Q Toomre ≤ 1 . In the lower panel, the colour indicates the fraction of mass at a given radius where SI can be triggered according to Li and Youdin (2021) criterion. Positions of the snowlines are shown for reference in dashed lines.
\n\n\n\n
Figure 10. Distribution of C/O ratios in the regions where gravitational and streaming instabilities are triggered. For GI, total C/O ratio is shown, for SI, the C/O ratios in the ice and in the gas. Black and grey points show the observed C/O ratios in two populations of exoplanets, the data is adopted from Hoch et al. (2023).
\n\nthe residual value of α is 10 -3 which makes this condition unreachable outside of the dead zone. The regions in model M1 where the conditions of Li and Youdin (2021) are satisfied are shown in the lower panel of Figure 9. Most of the suitable regions are in the inner disc ( r < 20 au) inside the dust rings, and appear after 200 kyr. However, there are some suitable regions between 10 - 100 au at earlier times, where SI could be triggered in the spirals.\nThe regions where GI and SI are possible shown in Figure 9 are separated in space and time, and they have different characteristic C/O ratios. We can sum up all the volatiles in these regions (throughout the disc lifetime) to assess typical C/O ratios of the planet-forming material. For GI, we exclude the pre-disc phase ( t < 53 kyr) and consider the total C/O ratio, assuming both gas and solids are included in the forming planet. For SI, we separate the gas- and ice-phase C/O ratios. The formed planetesimals would only include the ices, however, if they form the planetary cores, these cores would also accrete gas. We note that the composition of the rocks, which are typically carbon-rich, is not included in our assessment. The resulting distributions of the C/O ratios in planet-forming regions are shown in Figure 10. For GI regions, the C/O distribution has a distinct and relatively narrow peak around 0.5. It is slightly higher than the initial value of 0.34. For SI regions, the ice-phase C/O is below 0.5, with major peaks at 0 and ≈ 0.2, and the gas-phase C/O has a broad distribution with multiple peaks between ≈ 0.2 - 1.4. Distributions of C/O ratios in the regions where GI and SI can be triggered are noticeably different.\nIt was shown by Hoch et al. (2023) that there are two different populations of C/O ratios observed in giant exoplanets. They find that directly imaged exoplanets have C/O ≈ 0.5-0.8, while transiting hot Jupiters have a wider variety of C/O ratios ( ≈ 0.3-1.7, see Figures 12 and 13 in Hoch et al. 2023), and suggest that these two populations could have different formation pathways. We add the C/O data of exoplanetary atmospheres compiled in Table 3 of Hoch et al. (2023) to Figure 10 (with arbitrary position at the y -axis). The narrow distribution of C/O ratios in the regions with GI is in step with the distribution of directly imaged exoplanets, albeit with a slightly shifted value due to our assumed initial conditions, while the wide range of C/O values in the regions of SI matches the variety of C/O ratios in transiting exoplanets. This could suggest that directly imaged exoplanets could form as a result of gravitational instability, which is also in line with their typically higher masses and orbital separations. At the same time, the transiting hot Jupiters could have experienced a lot of migration (Lin, Bodenheimer, and Richardson 1996; Dawson and Johnson 2018), during which they accrete material with a variety of C/O ratios both from the gas and solid phase. It suggests that they could also form in the core accretion scenario. The origin of wide separation planets was also investigated by Bergin et al. (2024), based on the comparison with the observed C/O > 1 in protoplanetary discs (including the full composition of solids). They conclude that both core accretion and gravitational instability can work as the formation mechanism of these planets.\nApart from (exo)planets, the C/O ratios can be measured\nfor the comets, which present the best preserved sample of the primordial composition of the ices in the Solar System. Spectroscopic measurements of molecular composition in the comae suggest that the C/O ratio of cometary ice is quite low, typically below 0.1 due to the dominance of water ice (A'Hearn et al. 2012; Seligman et al. 2022; Harrington Pinto et al. 2022). Although most comets are carbon-depleted, there are individual measurements of C/O in comets above 0.5, for example in C/2006 W3 Christensen and 29P/SchwassmannWachmann (Ootsubo et al. 2012; Seligman et al. 2022), or even close to 1 in C/2016 R2 (PanSTARRS) (Wierzchos and Womack 2018; McKay et al. 2019). Additionally, high value of C/O ≈ 1 was observed in the interstellar object 2I/Borisov (Bodewits et al. 2020). Our modelling results show the icephase C/O = 0 in the vicinity of the water snowline, as well as low values between the CO 2 and CH 4 snowlines ( ≈ 0.2) and between the CH 4 and CO snowlines ( ≈ 0.3). These are the locations where comets could originate from. However, in the vicinity of the CO 2 , CH 4 and CO ice lines themselves, the C/O ratio in the ice phase is much higher. The fact that carbon-rich cometary ices are extremely rare in the Solar System may indicate that planetesimals formed on ice lines from carbon-rich volatiles do not persist throughout the evolution of a planetary system. This means that they are likely to be included in larger bodies, which favours the snowline-aided planet formation scenarios (Drążkowska and Alibert 2017; Hyodo et al. 2021). This is also consistent with the abundance of exoplanets with high C/O (Weiner Mansfield et al. 2024), which could be formed around the snowlines of carbon-rich species. In the giant planets of the Solar System, the C/O ratios are not well constrained (Mousis, Cavalié, et al. 2024). However, the existing data suggest rather super-solar values for all giant planets except for Neptune (Cavalié et al. 2024); for Jupiter, the C/O ratio is assessed as ≈ 0.9 (Wong et al. 2004; Li et al. 2024).\nOur model only considers four most abundant chemical species. However, there can be other more complex molecules in protoplanetary discs, which could affect the balance of carbon and oxygen. The most obvious candidate is methanol CH 3 OH, which has the abundance similar to methane in the protostellar cores (Karin I. Öberg et al. 2011). It was also observed in a protoplanetary disc around an erupting star V883 Ori (Lee et al. 2019). We do not consider it in the model as its binding energy is close to that of water, thus the snowlines would have similar positions, but the abundance is an order of magnitude lower. However, it could somewhat increase the local C/O ratio in the inner regions where there are no other carbon-bearing species, such as the ice in the ring at 1 au. Including methanol would alter the distribution of the C/O ratio. Interactions between the ices considered in the model could also affect the results. As was recently shown by Ligterink, Kipfer, and Gavino (2024), trapping of volatile species inside the mantles of less volatile ices could have a significant impact on the distribution of C/O ratios.\nAnother important process missing in our modelling is gas-phase and surface chemical reactions. They could significantly affect the distribution of C/O ratio in the gas and in\nthe ice, particularly with high level of cosmic ray ionisation (Eistrup, Walsh, and van Dishoeck 2016) or if carbon grain destruction is considered (Cridland, Eistrup, and van Dishoeck 2019). One particular mechanism is the transformation of CO to CO 2 on the surface of dust grains, which can lead to the depletion of CO from both gas and ice phases between CO and CO 2 snowlines (Molyarova et al. 2017; Bosman, Tielens, and van Dishoeck 2018). Considering this mechanism can change the conclusions about planet formation location (P. Mollière et al. 2020). Nevertheless, radial variations of the C/O ratio are necessary to explain molecular emission of discs with gaps (Leemker et al. 2024), and they can only be result of dust dynamics. In order to more consistently describe the distribution of molecules and elements in the disc, the models combining dust evolution and dynamics with more complex chemistry treatment are necessary.\nOur simulations adopt the thin-disc approximation and focus on the midplane of the protoplanetary discs, in order to capture the essential physics of self-gravity, thermal balance, and dust evolution in a global modelling within reasonable computational times. This means that some relevant processes connected with the vertical structure are inevitably excluded. For example, vertical mixing and dust settling affect the C/O ratio in the upper layers of the disc (Krijt et al. 2018; Krijt et al. 2020). Dust settling is implicitly included in our modelling through separate scale heights of drown dust and gas (as well as small dust), affecting dust number density in the midplane. However, this approach does not allow to reproduce vertical stratification in dust properties and chemical composition, which is particularly relevant for the interpretation of molecular observations. Vertical structure is also relevant for the accretion of matter on forming giant planets, which should proceed in 3D manner through meridional flows (Morbidelli et al. 2014). Cridland, Bosman, and van Dishoeck (2020) showed that the C/O ratio in the atmospheres of giant planets is rather affected by the composition of the molecular layer than that of the midplane.\n5. Conclusions\nIn this work, we studied the distribution of volatiles in a viscous self-gravitating protoplanetary disc with dust evolution using a thin-disc hydrodynamic code FEOSAD (Vorobyov et al. 2018; Molyarova et al. 2021). We calculated the C/O elemental ratio in the gas, in the ice, and in total, identified the key properties of the distribution of elements over 500 kyr of disc evolution and considered their implications for planet formation theory. Our main findings can be summarised as follows.\n\n· The simulated C/O ratios in the regions where GI and SI conditions are fulfilled are consistent with the C/O ratios in two populations of exoplanets possibly formed in different mechanisms pointed out by Hoch et al. (2023). We show that narrow C/O distribution of directly imaged planets is consistent with their formation via gravitational instability, while a variety of C/O in transiting hot Jupiters is in line with their migration through varying C/O conditions after the formation via either core accretion or GI.\n· The lower C/O ratio in the ice between the CO 2 , CH 4 and COsnowlines is consistent with the typical composition of Solar System comets, while the higher value of C/O ≈ 0.51 at these snowlines corresponds to the composition of rare carbon-rich comets. This may indicate that matter from the snowlines is hardly preserved during the evolution of the disc and planetary system, possibly due to the inclusion in to planets.\n· The distribution of volatiles is affected by the disc substructures, such as rings and spirals, as well as by dust radial drift. Variations of physical conditions create multiple snowlines of CO 2 and H 2 O inside 10 au. Dust drift of icy grains brings the volatiles from the outer to the inner disc, enriching the inner disc with both C and O. It also creates a radial gradient of the total C/O ratio: its value is around 0.2 where water is not frozen, and 0.6 - 0.9 where it is icy, compared to the initial value of 0.34.\n· Volatiles accumulate at their snowlines in both ice and gas phases due to the combined effect of dust drift and azimuthal variations of gas and dust radial velocities in a self-gravitating, non-axisymmetric disc. The species with low initial abundances, such as CH 4 (or methanol not considered here), can significantly affect C/O ratio, as their accumulation at the snowline creates a bump in C/O in all phases above 1.0. The total mass of the model affects the timescales and the magnitude of the accumulation by a factor of two.\n· Forming planets can accrete gas with C/O > 1 beyond CO 2 snowline, ices with C/O ≈ 0.5-1 at the CO, CH 4 and CO 2 snowlines, ices with C/O ≈ 0.2 - 0.3 between these snowlines and ices with C/O = 0 between H 2 Oand CO 2 snowlines. Planets with stellar C/O would need to migrate through these regions to acquire necessary composition or form via GI at earlier stages from the mixture of gas and dust with unaltered C/O ratio.\n· Dust-to-gas mass ratio and the total C/O ratio are systematically anticorrelated, because in dust-rich regions the volatile composition is close to that of the ice (which is lower), and in dust-poor regions, gas determines the C/O ratio.\n\nThe connection between protoplanetary disc components and exoplanets based on their composition should be more thoroughly investigated in the models focused on the planet formation process. We emphasise that these models should also take into account the effect of dust evolution and dynamics on the distribution of the elements in the planet-forming material. Inclusion of chemical processes and more accurate consideration of the bulk composition of dust grains could also affect the C/O ratios of the planet-forming environment.\nAcknowledgement\nWeare thankful to the anonymous referee for useful comments that helped to improve the manuscript. The computational results presented have been achieved using the Vienna Scientific Cluster (VSC) and the local computing facility of the Southern Federal University.\nFunding Statement The work is supported by Russian Science Foundation grant 22-72-10029, https://rscf.ru/project/2272-10029/\nCompeting Interests None\nData Availability Statement The data underlying this article will be shared on reasonable request to the corresponding author.\nReferences\nA'Hearn, Michael F., Lori M. Feaga, H. Uwe Keller, Hideyo Kawakita, Donald L. Hampton, Jochen Kissel, Kenneth P. Klaasen, et al. 2012. Cometary Volatiles and the Origin of Comets. ApJ 758, no. 1 (October): 29. https: //doi.org/10.1088/0004-637X/758/1/29.\nAikawa, Yuri, Shoken M. Miyama, Takenori Nakano, and Toyoharu Umebayashi. 1996. Evolution of Molecular Abundance in Gaseous Disks around Young Stars: Depletion of CO Molecules. ApJ 467 (August): 684. https://doi.org/10.1086/177644.\nAkimkin, Vitaly, Eduard Vorobyov, Yaroslav Pavlyuchenkov, and Olga Stoyanovskaya. 2020. Gravitoviscous protoplanetary discs with a dust component - IV. Disc outer edges, spectral indices, and opacity gaps. MNRAS 499, no. 4 (December): 5578-5597. https://doi.org/10.1093/mnras/ staa3134. arXiv: 2010.06566 [astro-ph.EP] .\nArmitage, Philip J. 2010. Astrophysics of Planet Formation.\nArmitage, Philip J., Mario Livio, and J. E. Pringle. 2001. Episodic accretion in magnetically layered protoplanetary discs. MNRAS 324, no. 3 (June): 705-711. https://doi.org/10.1046/j.1365-8711.2001.04356.x. arXiv: astro-ph/0101253 [astro-ph] .\nAudard, M., P. Ábrahám, M. M. Dunham, J. D. Green, N. Grosso, K. Hamaguchi, J. H. Kastner, et al. 2014. Episodic Accretion in Young Stars. In Protostars and planets vi, edited by Henrik Beuther, Ralf S. Klessen, Cornelis P. Dullemond, and Thomas Henning, 387-410. January. https: / / doi . org / 10 . 2458 / azu _ uapress _ 9780816531240 - ch017. arXiv: 1401.3368 [astro-ph.SR] .\nBai, Xue-Ning, and James M. Stone. 2013. Wind-driven Accretion in Protoplanetary Disks. I. Suppression of the Magnetorotational Instability and Launching of the Magnetocentrifugal Wind. ApJ 769, no. 1 (May): 76. https://doi.org/10.1088/0004-637X/769/1/76. arXiv: 1301.0318 [astro-ph.EP] .\nBanzatti, Andrea, Ilaria Pascucci, Arthur D. Bosman, Paola Pinilla, Colette Salyk, Gregory J. Herczeg, Klaus M. Pontoppidan, et al. 2020. Hints for Icy Pebble Migration Feeding an Oxygen-rich Chemistry in the Inner Planet-forming Region of Disks. ApJ 903, no. 2 (November): 124. https://doi.org/10.3847/1538-4357/abbc1a. arXiv: 2009.13525 [astro-ph.EP] .\nBenneke, Björn, Heather A. Knutson, Joshua Lothringer, Ian J. M. Crossfield, Julianne I. Moses, Caroline Morley, Laura Kreidberg, et al. 2019. A sub-Neptune exoplanet with a low-metallicity methane-depleted atmosphere and Mie-scattering clouds. Nature Astronomy 3 (July): 813-821. https://doi.org/10.1038/s41550- 019- 0800- 5. arXiv: 1907.00449 [astro-ph.EP] .\nBergin, Edwin A., Richard A. Booth, Maria Jose Colmenares, and John D. Ilee. 2024. C/O Ratios and the formation of wide separation exoplanets. arXiv e-prints (June): arXiv:2406.12037. https://doi.org/10.48550/arXiv. 2406.12037. arXiv: 2406.12037 [astro-ph.EP] .\nBergin, Edwin A., Fujun Du, L. Ilsedore Cleeves, G. A. Blake, K. Schwarz, R. Visser, and K. Zhang. 2016. Hydrocarbon Emission Rings in Protoplanetary Disks Induced by Dust Evolution. ApJ 831, no. 1 (November): 101. https://doi.org/10.3847/0004-637X/831/1/101. arXiv: 1609.06337 [astro-ph.EP] .\nBinney, James, and Scott Tremaine. 1987. Galactic dynamics.\nBirnstiel, Tilman. 2023. Dust growth and evolution in protoplanetary disks. arXiv e-prints (December): arXiv:2312.13287. https://doi.org/10.48550/ arXiv.2312.13287. arXiv: 2312.13287 [astro-ph.EP] .\nBisschop, S. E., H. J. Fraser, K. I. Öberg, E. F. van Dishoeck, and S. Schlemmer. 2006. Desorption rates and sticking coefficients for CO and N2 interstellar ices. A&A 449, no. 3 (April): 1297-1309. https://doi.org/10. 1051/0004-6361:20054051. arXiv: astro-ph/0601082 [astro-ph] .\nBitsch, Bertram, Aaron David Schneider, and Laura Kreidberg. 2022. How drifting and evaporating pebbles shape giant planets. III. The formation of WASP-77A b and τ Boötis b. A&A 665 (September): A138. http s : / / doi . org / 10 . 1051 / 0004 - 6361 / 202243345. arXiv: 2207 . 06077 [astro-ph.EP] .\nBodenheimer, P. 1974. Calculations of the Early Evolution of Jupiter. Icarus 23, no. 3 (November): 319-325. https : / / doi . org / 10 . 1016 / 0019 1035(74)90050-5.\nBodewits, D., J. W. Noonan, P. D. Feldman, M. T. Bannister, D. Farnocchia, W. M. Harris, J. -Y. Li, K. E. Mandt, J. Wm. Parker, and Z. -X. Xing. 2020. The carbon monoxide-rich interstellar comet 2I/Borisov. Nature Astronomy 4 (April): 867-871. https://doi.org/10.1038/s41550-0201095-2. arXiv: 2004.08972 [astro-ph.EP] .\nBooth, R. A., and J. D. Ilee. 2019. Planet-forming material in a protoplanetary disc: the interplay between chemical evolution and pebble drift. MNRAS 487, no. 3 (August): 3998-4011. https://doi.org/10.1093/mnras/stz1488. arXiv: 1905.12639 [astro-ph.EP] .\nBooth, Richard A., Cathie J. Clarke, Nikku Madhusudhan, and John D. Ilee. 2017. Chemical enrichment of giant planets and discs due to pebble drift. MNRAS 469, no. 4 (August): 3994-4011. https://doi.org/10.1093/ mnras/stx1103. arXiv: 1705.03305 [astro-ph.EP] .\nBosman, A. D., A. J. Cridland, and Y. Miguel. 2019. Jupiter formed as a pebble pile around the N2 ice line. A&A 632 (December): L11. https://doi.org/ 10.1051/0004-6361/201936827. arXiv: 1911.11154 [astro-ph.EP] .\nBosman, Arthur D., Alexander G. G. M. Tielens, and Ewine F. van Dishoeck. 2018. Efficiency of radial transport of ices in protoplanetary disks probed with infrared observations: the case of CO2. A&A 611 (April): A80. https://doi.org/10.1051/0004-6361/201732056. arXiv: 1712.03989 [astro-ph.EP] .\nBrown, Paul D., and S. B. Charnley. 1990. Chemical models of interstellar gas-grain processes. I. Modelling and the effect of accretion on gas abundances and mantle composition in dense clouds. MNRAS 244 (June): 432.\nBrown-Sevilla, S. B., M. Keppler, M. Barraza-Alfaro, J. D. Melon Fuksman, N. Kurtovic, P. Pinilla, M. Feldt, et al. 2021. A multiwavelength analysis of the spiral arms in the protoplanetary disk around WaOph 6. A&A 654 (October): A35. https://doi.org/10.1051/0004-6361/202140783. arXiv: 2107.13560 [astro-ph.EP] .\nCarrera, Daniel, Anders Johansen, and Melvyn B. Davies. 2015. How to form planetesimals from mm-sized chondrules and chondrule aggregates. A&A 579 (July): A43. https://doi.org/10.1051/0004-6361/201425120. arXiv: 1501.05314 [astro-ph.EP] .\nCavalié, Thibault, Jonathan Lunine, Olivier Mousis, and Ricardo Hueso. 2024. The Deep Oxygen Abundance in Solar System Giant Planets, with a New Derivation for Saturn. Space Sci. Rev. 220, no. 1 (January): 8. https://doi.org/10.1007/s11214-024-01045-6. arXiv: 2407.07515 [astro-ph.EP] .\nChangeat, Q., B. Edwards, A. F. Al-Refaie, A. Tsiaras, J. W. Skinner, J. Y. K. Cho, K. H. Yip, et al. 2022. Five Key Exoplanet Questions Answered via the Analysis of 25 Hot-Jupiter Atmospheres in Eclipse. ApJS 260, no. 1 (May): 3. https://doi.org/10.3847/1538-4365/ac5cc2. arXiv: 2204.11729 [astro-ph.EP] .\nCleeves, L. Ilsedore, Karin I. Öberg, David J. Wilner, Jane Huang, Ryan A. Loomis, Sean M. Andrews, and V. V. Guzman. 2018. Constraining Gasphase Carbon, Oxygen, and Nitrogen in the IM Lup Protoplanetary Disk. ApJ 865, no. 2 (October): 155. https://doi.org/10.3847/15384357/aade96. arXiv: 1808.10682 [astro-ph.SR] .\nConnelley, Michael S., and Bo Reipurth. 2018. A Near-infrared Spectroscopic Survey of FU Orionis Objects. ApJ 861, no. 2 (July): 145. https://doi. org/10.3847/1538-4357/aaba7b. arXiv: 1806.08880 [astro-ph.SR] .\nCridland, Alex J., Ewine F. van Dishoeck, Matthew Alessi, and Ralph E. Pudritz. 2019. Connecting planet formation and astrochemistry. A main sequence for C/O in hot exoplanetary atmospheres. A&A 632 (December): A63. https://doi.org/10.1051/0004- 6361/201936105. arXiv: 1910.13171 [astro-ph.EP] .\n. 2020. Connecting planet formation and astrochemistry. C/Os and N/Os of warm giant planets and Jupiter analogues. A&A 642 (October): A229. https://doi.org/10.1051/0004-6361/202038767. arXiv: 2009. 02907 [astro-ph.EP] .\nCridland, Alexander J., Arthur D. Bosman, and Ewine F. van Dishoeck. 2020. Impact of vertical gas accretion on the carbon-to-oxygen ratio of gas giant atmospheres. A&A 635 (March): A68. https://doi.org/10.1051/ 0004-6361/201936858. arXiv: 2001.05808 [astro-ph.EP] .\nCridland, Alexander J., Christian Eistrup, and Ewine F. van Dishoeck. 2019. Connecting planet formation and astrochemistry. Refractory carbon depletion leading to super-stellar C/O in giant planetary atmospheres. A&A 627 (July): A127. https://doi.org/10.1051/0004-6361/201834378. arXiv: 1901.08896 [astro-ph.EP] .\nCuppen, H. M., C. Walsh, T. Lamberts, D. Semenov, R. T. Garrod, E. M. Penteado, and S. Ioppolo. 2017. Grain Surface Models and Data for Astrochemistry. Space Sci. Rev. 212, nos. 1-2 (October): 1-58. https: //doi.org/10.1007/s11214-016-0319-3.\nCuzzi, Jeffrey N., and Kevin J. Zahnle. 2004. Material Enhancement in Protoplanetary Nebulae by Particle Drift through Evaporation Fronts. ApJ 614, no. 1 (October): 490-496. https://doi.org/10.1086/423611. arXiv: astro-ph/0409276 [astro-ph] .\nDanti, C., B. Bitsch, and J. Mah. 2023. Composition of giant planets: The roles of pebbles and planetesimals. A&A 679 (November): L7. https://doi.org/ 10.1051/0004-6361/202347501. arXiv: 2310.02886 [astro-ph.EP] .\nDawson, Rebekah I., and John Asher Johnson. 2018. Origins of Hot Jupiters. ARA&A 56 (September): 175-221. https://doi.org/10.1146/annurevastro-081817-051853. arXiv: 1801.06117 [astro-ph.EP] .\nDong, Ruobing, Eduard Vorobyov, Yaroslav Pavlyuchenkov, Eugene Chiang, and Hauyu Baobab Liu. 2016. Signatures of Gravitational Instability in Resolved Images of Protostellar Disks. ApJ 823, no. 2 (June): 141. https://doi.org/10.3847/0004-637X/823/2/141. arXiv: 1603.01618 [astro-ph.SR] .\nDraine, B. T. 1978. Photoelectric heating of interstellar gas. ApJS 36 (April): 595-619. https://doi.org/10.1086/190513.\nDrążkowska, J., and Y. Alibert. 2017. Planetesimal formation starts at the snow line. A&A 608 (December): A92. https://doi.org/10.1051/00046361/201731491. arXiv: 1710.00009 [astro-ph.EP] .\nDutrey, A., V. Wakelam, Y. Boehler, S. Guilloteau, F. Hersant, D. Semenov, E. Chapillon, et al. 2011. Chemistry in disks. V. Sulfur-bearing molecules in the protoplanetary disks surrounding LkCa15, MWC480, DM Tauri, and GO Tauri. A&A 535 (November): A104. https://doi.org/10.1051/ 0004-6361/201116931. arXiv: 1109.5870 [astro-ph.SR] .\nEistrup, Christian, Catherine Walsh, and Ewine F. van Dishoeck. 2016. Setting the volatile composition of (exo)planet-building material. Does chemical evolution in disk midplanes matter? A&A 595 (November): A83. htt ps://doi.org/10.1051/0004- 6361/201628509. arXiv: 1607.06710 [astro-ph.EP] .\n. 2018. Molecular abundances and C/O ratios in chemically evolving planet-forming disk midplanes. A&A 613 (May): A14. https://doi.org/ 10.1051/0004-6361/201731302. arXiv: 1709.07863 [astro-ph.EP] .\nFacchini, Stefano, Richard Teague, Jaehan Bae, Myriam Benisty, Miriam Keppler, and Andrea Isella. 2021. The Chemical Inventory of the Planethosting Disk PDS 70. AJ 162, no. 3 (September): 99. https://doi.org/10. 3847/1538-3881/abf0a4. arXiv: 2101.08369 [astro-ph.EP] .\nFedele, D., and C. Favre. 2020. Measuring elemental abundance ratios in protoplanetary disks at millimeter wavelengths. A&A 638 (June): A110. https://doi.org/10.1051/0004-6361/202037927. arXiv: 2005.03891 [astro-ph.SR] .\nFraser, Helen J., Mark P. Collings, Martin R. S. McCoustra, and David A. Williams. 2001. Thermal desorption of water ice in the interstellar medium. MNRAS 327, no. 4 (November): 1165-1172. https://doi. org/10.1046/j.1365-8711.2001.04835.x. arXiv: astro-ph/0107487 [astro-ph] .\nFu, Guangwei, Luis Welbanks, Drake Deming, Julie Inglis, Michael Zhang, Joshua Lothringer, Jegug Ih, et al. 2024. Hydrogen sulfide and metalenriched atmosphere for a Jupiter-mass exoplanet. arXiv e-prints (July): arXiv:2407.06163. https://doi.org/10.48550/arXiv.2407.06163. arXiv: 2407.06163 [astro-ph.EP] .\nGail, Hans-Peter, and Mario Trieloff. 2017. Spatial distribution of carbon dust in the early solar nebula and the carbon content of planetesimals. A&A 606 (September): A16. https://doi.org/10.1051/0004-6361/201730480. arXiv: 1707.07611 [astro-ph.EP] .\nGammie, Charles F. 1996. Layered Accretion in T Tauri Disks. ApJ 457 (January): 355. https://doi.org/10.1086/176735.\nGárate, Matias, Til Birnstiel, Joanna Drążkowska, and Sebastian Markus Stammler. 2020. Gas accretion damped by dust back-reaction at the snow line. A&A 635 (March): A149. https://doi.org/10.1051/00046361/201936067. arXiv: 1906.07708 [astro-ph.EP] .\nGRAVITY Collaboration, M. Nowak, S. Lacour, P. Mollière, J. Wang, B. Charnay, E. F. van Dishoeck, et al. 2020. Peering into the formation history of β Pictoris b with VLTI/GRAVITY long-baseline interferometry. A&A 633 (January): A110. https://doi.org/10.1051/00046361/201936898. arXiv: 1912.04651 [astro-ph.EP] .\nGressel, Oliver, Neal J. Turner, Richard P. Nelson, and Colin P. McNally. 2015. Global Simulations of Protoplanetary Disks With Ohmic Resistivity and Ambipolar Diffusion. ApJ 801, no. 2 (March): 84. https://doi.org/ 10.1088/0004-637X/801/2/84. arXiv: 1501.05431 [astro-ph.EP] .\nGundlach, B., and J. Blum. 2015. The Stickiness of Micrometer-sized Waterice Particles. ApJ 798, no. 1 (January): 34. https://doi.org/10.1088/0004637X/798/1/34. arXiv: 1410.7199 [astro-ph.EP] .\nHarrington Pinto, Olga, Maria Womack, Yanga Fernandez, and James Bauer. 2022. A Survey of CO, CO2, and H2O in Comets and Centaurs. PSJ 3, no. 11 (November): 247. https://doi.org/10.3847/PSJ/ac960d. arXiv: 2209.09985 [astro-ph.EP] .\nHartmann, L., and S. J. Kenyon. 1985. On the nature of FU Orionis objects. ApJ 299 (December): 462-478. https://doi.org/10.1086/163713.\nHasegawa, T. I., and E. Herbst. 1993. Three-Phase Chemical Models of Dense Interstellar Clouds - Gas Dust Particle Mantles and Dust Particle Surfaces. MNRAS 263 (August): 589. https://doi.org/10.1093/mnras/263.3.589.\nHensley, Brandon S., and B. T. Draine. 2021. Observational Constraints on the Physical Properties of Interstellar Dust in the Post-Planck Era. ApJ 906, no. 2 (January): 73. https://doi.org/10.3847/1538-4357/abc8f1. arXiv: 2009.00018 [astro-ph.GA] .\nHoch, Kielan K. W., Quinn M. Konopacky, Christopher A. Theissen, JeanBaptiste Ruffio, Travis S. Barman, Emily L. Rickman, Marshall D. Perrin, Bruce Macintosh, and Christian Marois. 2023. Assessing the C/O Ratio Formation Diagnostic: A Potential Trend with Companion Mass. AJ 166, no. 3 (September): 85. https://doi.org/10.3847/1538-3881/ace442. arXiv: 2212.04557 [astro-ph.EP] .\nHuang, Jane, Sean M. Andrews, Cornelis P. Dullemond, Andrea Isella, Laura M. Pérez, Viviana V. Guzmán, Karin I. Öberg, et al. 2018. The Disk Substructures at High Angular Resolution Project (DSHARP). II. Characteristics of Annular Substructures. ApJ 869, no. 2 (December): L42. https : / / doi . org / 10 . 3847 / 2041 - 8213 / aaf 740. arXiv: 1812 . 04041 [astro-ph.EP] .\nHuang, Jane, Sean M. Andrews, Laura M. Pérez, Zhaohuan Zhu, Cornelis P. Dullemond, Andrea Isella, Myriam Benisty, et al. 2018. The Disk Substructures at High Angular Resolution Project (DSHARP). III. Spiral Structures in the Millimeter Continuum of the Elias 27, IM Lup, and WaOph 6 Disks. ApJ 869, no. 2 (December): L43. https://doi.org/10. 3847/2041-8213/aaf7a0. arXiv: 1812.04193 [astro-ph.SR] .\nHuang, Jane, Edwin A. Bergin, Romane Le Gal, Sean M. Andrews, Jaehan Bae, Luke Keyte, and J. A. Sturm. 2024. Constraints on the gas-phase C/O ratio of DR Tau's outer disk from CS, SO, and C2H observations. arXiv e-prints (July): arXiv:2407.01679. https://doi.org/10.48550/arXiv. 2407.01679. arXiv: 2407.01679 [astro-ph.EP] .\nHyodo, Ryuki, Tristan Guillot, Shigeru Ida, Satoshi Okuzumi, and Andrew N. Youdin. 2021. Planetesimal formation around the snow line. II. Dust or pebbles? A&A 646 (February): A14. https://doi.org/10.1051/00046361/202039894. arXiv: 2012.06700 [astro-ph.EP] .\nIlee, J. D., A. C. Boley, P. Caselli, R. H. Durisen, T. W. Hartquist, and J. M. C. Rawlings. 2011. Chemistry in a gravitationally unstable protoplanetary disc. MNRAS 417, no. 4 (November): 2950-2961. https://doi.org/10. 1111/j.1365-2966.2011.19455.x. arXiv: 1107.3041 [astro-ph.GA] .\nJiang, Haochang, and Chris W. Ormel. 2023. Efficient planet formation by pebble accretion in ALMA rings. MNRAS 518, no. 3 (January): 38773900. https://doi.org/10.1093/mnras/stac3275. arXiv: 2207.13002 [astro-ph.EP] .\nKadam, Kundan, Eduard Vorobyov, and Shantanu Basu. 2022. Primordial dusty rings and episodic outbursts in protoplanetary discs. MNRAS 516, no. 3 (November): 4448-4468. https://doi.org/10.1093/mnras/stac2455. arXiv: 2208.12105 [astro-ph.EP] .\nKadam, Kundan, Eduard Vorobyov, Zsolt Regály, Ágnes Kóspál, and Péter Ábrahám. 2019. Dynamical Gaseous Rings in Global Simulations of Protoplanetary Disk Formation. ApJ 882, no. 2 (September): 96. h ttps : / / doi . org / 10 . 3847 / 1538 - 4357 / ab378a. arXiv: 1908 . 02515 [astro-ph.SR] .\n. 2020. Outbursts in Global Protoplanetary Disk Simulations. ApJ 895, no. 1 (May): 41. https://doi.org/10.3847/1538-4357/ab8bd8. arXiv: 2005.03578 [astro-ph.SR] .\nKama, M., S. Bruderer, E. F. van Dishoeck, M. Hogerheijde, C. P. Folsom, A. Miotello, D. Fedele, A. Belloche, R. Güsten, and F. Wyrowski. 2016. Volatile-carbon locking and release in protoplanetary disks. A study of TWHya and HD 100546. A&A 592 (August): A83. https://doi.org/10. 1051/0004-6361/201526991. arXiv: 1605.05093 [astro-ph.EP] .\nKhorshid, N., M. Min, and J. M. Désert. 2023. Retrieving planet formation parameters of WASP-77Ab using SimAb. A&A 675 (July): A95. htt ps://doi.org/10.1051/0004- 6361/202245469. arXiv: 2311.15702 [astro-ph.EP] .\nKratter, Kaitlin, and Giuseppe Lodato. 2016. Gravitational Instabilities in Circumstellar Disks. ARA&A 54 (September): 271-311. https://doi. org/10.1146/annurev- astro- 081915- 023307. arXiv: 1603.01280 [astro-ph.SR] .\nKrijt, Sebastiaan, Arthur D. Bosman, Ke Zhang, Kamber R. Schwarz, Fred J. Ciesla, and Edwin A. Bergin. 2020. CO Depletion in Protoplanetary Disks: A Unified Picture Combining Physical Sequestration and Chemical Processing. ApJ 899, no. 2 (August): 134. https://doi.org/10.3847/ 1538-4357/aba75d. arXiv: 2007.09517 [astro-ph.SR] .\nKrijt, Sebastiaan, Kamber R. Schwarz, Edwin A. Bergin, and Fred J. Ciesla. 2018. Transport of CO in Protoplanetary Disks: Consequences of Pebble Formation, Settling, and Radial Drift. ApJ 864, no. 1 (September): 78. https : / / doi . org / 10 . 3847 / 1538 - 4357 / aad69b. arXiv: 1808 . 01840 [astro-ph.EP] .\nLambrechts, M., and A. Johansen. 2012. Rapid growth of gas-giant cores by pebble accretion. A&A 544 (August): A32. https://doi.org/10.1051/00046361/201219127. arXiv: 1205.3030 [astro-ph.EP] .\nLee, Eve J., J. R. Fuentes, and Philip F. Hopkins. 2022. Establishing Dust Rings and Forming Planets within Them. ApJ 937, no. 2 (October): 95. https://doi.org/10.3847/1538-4357/ac8cfe. arXiv: 2206.01219 [astro-ph.EP] .\nLee, Jeong-Eun, Edwin A. Bergin, and Hideko Nomura. 2010. The Solar Nebula on Fire: A Solution to the Carbon Deficit in the Inner Solar System. ApJ 710, no. 1 (February): L21-L25. https://doi.org/10.1088/ 2041-8205/710/1/L21. arXiv: 1001.0818 [astro-ph.GA] .\nLee, Jeong-Eun, Seokho Lee, Giseon Baek, Yuri Aikawa, Lucas Cieza, SungYong Yoon, Gregory Herczeg, Doug Johnstone, and Simon Casassus. 2019. The ice composition in the disk around V883 Ori revealed by its stellar outburst. Nature Astronomy 3 (February): 314-319. https://doi. org/10.1038/s41550-018-0680-0. arXiv: 1809.00353 [astro-ph.SR] .\nLeemker, M., A. S. Booth, E. F. van Dishoeck, L. Wölfer, and B. Dent. 2024. Chemistry across dust and gas gaps in protoplanetary disks: modelling the co-spatial molecular rings in the HD 100546 disk. arXiv e-prints (May): arXiv:2405.10361. https://doi.org/10.48550/arXiv.2405.10361. arXiv: 2405.10361 [astro-ph.EP] .\nLenz, Christian T., Hubert Klahr, and Tilman Birnstiel. 2019. Planetesimal Population Synthesis: Pebble Flux-regulated Planetesimal Formation. ApJ 874, no. 1 (March): 36. https://doi.org/10.3847/1538-4357/ab05d9. arXiv: 1902.07089 [astro-ph.EP] .\nLi, Cheng, Michael Allison, Sushil Atreya, Shawn Brueshaber, Leigh N. Fletcher, Tristan Guillot, Liming Li, et al. 2024. Super-adiabatic temperature gradient at Jupiter's equatorial zone and implications for the water abundance. Icarus 414 (May): 116028. https://doi.org/10.1016/j. icarus.2024.116028. arXiv: 2403.05363 [astro-ph.EP] .\nLi, Rixin, and Andrew N. Youdin. 2021. Thresholds for Particle Clumping by the Streaming Instability. ApJ 919, no. 2 (October): 107. https://doi. org/10.3847/1538-4357/ac0e9f. arXiv: 2105.06042 [astro-ph.EP] .\nLigterink, N. F. W., K. A. Kipfer, and S. Gavino. 2024. Mind the Trap: Non-negligible effect of volatile trapping in ice on C/O ratios in protoplanetary disks and exoplanetary atmospheres. arXiv e-prints (June): arXiv:2406.16029. https://doi.org/10.48550/arXiv.2406.16029. arXiv: 2406.16029 [astro-ph.EP] .\nLin, D. N. C., P. Bodenheimer, and D. C. Richardson. 1996. Orbital migration of the planetary companion of 51 Pegasi to its present location. Nature 380, no. 6575 (April): 606-607. https://doi.org/10.1038/380606a0.\nLine, Michael R., Matteo Brogi, Jacob L. Bean, Siddharth Gandhi, Joseph Zalesky, Vivien Parmentier, Peter Smith, et al. 2021. A solar C/O and sub-solar metallicity in a hot Jupiter atmosphere. Nature 598, no. 7882 (October): 580-584. https://doi.org/10.1038/s41586-021-03912-6. arXiv: 2110.14821 [astro-ph.EP] .\nLodders, Katharina. 2004. Jupiter Formed with More Tar than Ice. ApJ 611, no. 1 (August): 587-597. https://doi.org/10.1086/421970.\nLong, Feng, Paola Pinilla, Gregory J. Herczeg, Daniel Harsono, Giovanni Dipierro, Ilaria Pascucci, Nathan Hendler, et al. 2018. Gaps and Rings in an ALMA Survey of Disks in the Taurus Star-forming Region. ApJ 869, no. 1 (December): 17. https://doi.org/10.3847/1538-4357/aae8e1. arXiv: 1810.06044 [astro-ph.SR] .\nMadhusudhan, Nikku. 2012. C/O Ratio as a Dimension for Characterizing Exoplanetary Atmospheres. ApJ 758, no. 1 (October): 36. https://doi. org/10.1088/0004-637X/758/1/36. arXiv: 1209.2412 [astro-ph.EP] .\nMadhusudhan, Nikku, Joseph Harrington, Kevin B. Stevenson, Sarah Nymeyer, Christopher J. Campo, Peter J. Wheatley, Drake Deming, et al. 2011. A high C/O ratio and weak thermal inversion in the atmosphere of exoplanet WASP-12b. Nature 469, no. 7328 (January): 64-67. https: //doi.org/10.1038/nature09602. arXiv: 1012.1603 [astro-ph.EP] .\nMatter, A., F. C. Pignatale, and B. Lopez. 2020. Spatially resolving the chemical composition of the planet building blocks. MNRAS 497, no. 3 (September): 2540-2552. https://doi.org/10.1093/mnras/staa2137. arXiv: 2007.09385 [astro-ph.EP] .\nMcKay, Adam J., Michael A. DiSanti, Michael S. P. Kelley, Matthew M. Knight, Maria Womack, Kacper Wierzchos, Olga Harrington Pinto, et al. 2019. The Peculiar Volatile Composition of CO-dominated Comet C/2016 R2 (PanSTARRS). AJ 158, no. 3 (September): 128. https://doi. org/10.3847/1538-3881/ab32e4. arXiv: 1907.07208 [astro-ph.EP] .\nMeru, Farzana, and Matthew R. Bate. 2010. Exploring the conditions required to form giant planets via gravitational instability in massive protoplanetary discs. MNRAS 406, no. 4 (August): 2279-2288. https: //doi.org/10.1111/j.1365- 2966.2010.16867.x. arXiv: 1004.3766 [astro-ph.EP] .\nMeru, Farzana, Attila Juhász, John D. Ilee, Cathie J. Clarke, Giovanni P. Rosotti, and Richard A. Booth. 2017. On the Origin of the Spiral Morphology in the Elias 2-27 Circumstellar Disk. ApJ 839, no. 2 (April): L24. https://doi.org/10.3847/2041-8213/aa6837. arXiv: 1703.05338 [astro-ph.EP] .\nMinissale, Marco, Yuri Aikawa, Edwin Bergin, Mathieu Bertin, Wendy A. Brown, Stephanie Cazaux, Steven B. Charnley, et al. 2022. Thermal Desorption of Interstellar Ices: A Review on the Controlling Parameters and Their Implications from Snowlines to Chemical Complexity. ACS Earth and Space Chemistry 6, no. 3 (March): 597-630. https://doi.org/10. 1021/acsearthspacechem.1c00357. arXiv: 2201.07512 [astro-ph.GA] .\nMiotello, A., S. Facchini, E. F. van Dishoeck, P. Cazzoletti, L. Testi, J. P. Williams, M. Ansdell, S. van Terwisga, and N. van der Marel. 2019. Bright C2H emission in protoplanetary discs in Lupus: high volatile C/O > 1 ratios. A&A 631 (November): A69. https://doi.org/10.1051/00046361/201935441. arXiv: 1909.04477 [astro-ph.SR] .\nMollière, P., T. Stolker, S. Lacour, G. P. P. L. Otten, J. Shangguan, B. Charnay, T. Molyarova, et al. 2020. Retrieving scattering clouds and disequilibrium chemistry in the atmosphere of HR 8799e. A&A 640 (August): A131. https://doi.org/10.1051/0004-6361/202038325. arXiv: 2006.09394 [astro-ph.EP] .\nMollière, Paul, Tamara Molyarova, Bertram Bitsch, Thomas Henning, Aaron Schneider, Laura Kreidberg, Christian Eistrup, et al. 2022. Interpreting the Atmospheric Composition of Exoplanets: Sensitivity to Planet Formation Assumptions. ApJ 934, no. 1 (July): 74. https://doi.org/10.3847/ 1538-4357/ac6a56. arXiv: 2204.13714 [astro-ph.EP] .\nMolyarova, Tamara, Vitaly Akimkin, Dmitry Semenov, Thomas Henning, Anton Vasyunin, and Dmitri Wiebe. 2017. Gas Mass Tracers in Protoplanetary Disks: CO is Still the Best. ApJ 849, no. 2 (November): 130. https://doi.org/10.3847/1538-4357/aa9227. arXiv: 1710.02993 [astro-ph.EP] .\nMolyarova, Tamara, Eduard I. Vorobyov, Vitaly Akimkin, Aleksandr Skliarevskii, Dmitri Wiebe, and Manuel Güdel. 2021. Gravitoviscous Protoplanetary Disks with a Dust Component. V. The Dynamic Model for Freeze-out and Sublimation of Volatiles. ApJ 910, no. 2 (April): 153. https://doi.org/ 10.3847/1538-4357/abe2b0. arXiv: 2103.06045 [astro-ph.EP] .\nMorbidelli, A., J. Szulágyi, A. Crida, E. Lega, B. Bitsch, T. Tanigawa, and K. Kanagawa. 2014. Meridional circulation of gas into gaps opened by giant planets in three-dimensional low-viscosity disks. Icarus 232 (April): 266-270. https://doi.org/10.1016/j.icarus.2014.01.010. arXiv: 1401.2925 [astro-ph.EP] .\nMordasini, C., R. van Boekel, P. Mollière, Th. Henning, and Björn Benneke. 2016. The Imprint of Exoplanet Formation History on Observable Present-day Spectra of Hot Jupiters. ApJ 832, no. 1 (November): 41. https://doi.org/10.3847/0004-637X/832/1/41. arXiv: 1609.03019 [astro-ph.EP] .\nMoses, J. I., N. Madhusudhan, C. Visscher, and R. S. Freedman. 2013. Chemical Consequences of the C/O Ratio on Hot Jupiters: Examples from WASP-12b, CoRoT-2b, XO-1b, and HD 189733b. ApJ 763, no. 1 (January): 25. https://doi.org/10.1088/0004-637X/763/1/25. arXiv: 1211.2996 [astro-ph.EP] .\nMousis, Olivier, Sarah E. Anderson, Adrienn Luspay-Kuti, Kathleen E. Mandt, and Pierre Vernazza. 2024. Triton and Pluto: same origin but separated at birth. arXiv e-prints (June): arXiv:2406.03815. https://doi.org/10. 48550/arXiv.2406.03815. arXiv: 2406.03815 [astro-ph.EP] .\nMousis, Olivier, Thibault Cavalié, Jonathan I. Lunine, Kathleen E. Mandt, Ricardo Hueso, Artyom Aguichine, Antoine Schneeberger, et al. 2024. Recipes for Forming a Carbon-Rich Giant Planet. Space Sci. Rev. 220, no. 4 (June): 44. https://doi.org/10.1007/s11214-024-01071-4. arXiv: 2405.19748 [astro-ph.EP] .\nNasedkin, E., P. Mollière, S. Lacour, M. Nowak, L. Kreidberg, T. Stolker, J. J. Wang, et al. 2024. Four-of-a-kind? Comprehensive atmospheric characterisation of the HR 8799 planets with VLTI/GRAVITY. arXiv e-prints (April): arXiv:2404.03776. https://doi.org/10.48550/arXiv.2404. 03776. arXiv: 2404.03776 [astro-ph.EP] .\nNayakshin, Sergei. 2010a. Formation of planets by tidal downsizing of giant planet embryos. MNRAS 408, no. 1 (October): L36-L40. https://doi.org/ 10.1111/j.1745-3933.2010.00923.x. arXiv: 1007.4159 [astro-ph.EP] .\n. 2010b. Grain sedimentation inside giant planet embryos. MNRAS 408, no. 4 (November): 2381-2396. https://doi.org/10.1111/j.13652966.2010.17289.x. arXiv: 1007.4162 [astro-ph.EP] .\nNayakshin, Sergei, Ravit Helled, and Aaron C. Boley. 2014. Core-assisted gas capture instability: a new mode of giant planet formation by gravitationally unstable discs. MNRAS 440, no. 4 (June): 3797-3808. https: //doi.org/10.1093/mnras/stu473. arXiv: 1403.1813 [astro-ph.EP] .\nNortmann, L., F. Lesjak, F. Yan, D. Cont, S. Czesla, A. Lavail, A. D. Rains, et al. 2024. CRIRES + transmission spectroscopy of WASP-127b. Detection of the resolved signatures of a supersonic equatorial jet and cool poles in a hot planet. arXiv e-prints (April): arXiv:2404.12363. https://doi.org/ 10.48550/arXiv.2404.12363. arXiv: 2404.12363 [astro-ph.EP] .\nÖberg, K. I., F. van Broekhuizen, H. J. Fraser, S. E. Bisschop, E. F. van Dishoeck, and S. Schlemmer. 2005. Competition between CO and N2 Desorption from Interstellar Ices. ApJ 621, no. 1 (March): L33-L36. https://doi.org/10.1086/428901.\nÖberg, Karin I., A. C. Adwin Boogert, Klaus M. Pontoppidan, Saskia van den Broek, Ewine F. van Dishoeck, Sandrine Bottinelli, Geoffrey A. Blake, and II Evans Neal J. 2011. The Spitzer Ice Legacy: Ice Evolution from Cores to Protostars. ApJ 740, no. 2 (October): 109. https://doi.org/10. 1088/0004-637X/740/2/109. arXiv: 1107.5825 [astro-ph.GA] .\nÖberg, Karin I., Ruth Murray-Clay, and Edwin A. Bergin. 2011. The Effects of Snowlines on C/O in Planetary Atmospheres. ApJ 743, no. 1 (December): L16. https://doi.org/10.1088/2041-8205/743/1/L16. arXiv: 1110.5567 [astro-ph.GA] .\nÖberg, Karin I., and Robin Wordsworth. 2019. Jupiter's Composition Suggests its Core Assembled Exterior to the N2 Snowline. AJ 158, no. 5 (November): 194. https://doi.org/10.3847/1538-3881/ab46a8. arXiv: 1909.11246 [astro-ph.EP] .\nOhno, Kazumasa, and Takahiro Ueda. 2021. Jupiter's 'cold' formation in the protosolar disk shadow. An explanation for the planet's uniformly enriched atmosphere. A&A 651 (July): L2. https://doi.org/10.1051/00046361/202141169. arXiv: 2106.09084 [astro-ph.EP] .\nOkuzumi, Satoshi, and Shigenobu Hirose. 2011. Modeling Magnetorotational Turbulence in Protoplanetary Disks with Dead Zones. ApJ 742, no. 2 (December): 65. https://doi.org/10.1088/0004-637X/742/2/65. arXiv: 1108.4892 [astro-ph.EP] .\nOkuzumi, Satoshi, and Ryo Tazaki. 2019. Nonsticky Ice at the Origin of the Uniformly Polarized Submillimeter Emission from the HL Tau Disk. ApJ 878, no. 2 (June): 132. https://doi.org/10.3847/1538-4357/ab204d. arXiv: 1904.03869 [astro-ph.EP] .\nOotsubo, Takafumi, Hideyo Kawakita, Saki Hamada, Hitomi Kobayashi, Mitsuru Yamaguchi, Fumihiko Usui, Takao Nakagawa, et al. 2012. AKARI Near-infrared Spectroscopic Survey for CO2 in 18 Comets. ApJ 752, no. 1 (June): 15. https://doi.org/10.1088/0004-637X/752/1/15.\nPadoan, Paolo, Liubin Pan, Veli-Matti Pelkonen, Troels Haugboelle, and AAke Nordlund. 2024. Protoplanetary Disks from Pre-Main Sequence BondiHoyle Accretion. arXiv e-prints (May): arXiv:2405.07334. https://doi. org/10.48550/arXiv.2405.07334. arXiv: 2405.07334 [astro-ph.GA] .\nPadovani, M., D. Galli, and A. E. Glassgold. 2009. Cosmic-ray ionization of molecular clouds. A&A 501, no. 2 (July): 619-631. https://doi.org/10. 1051/0004-6361/200911794. arXiv: 0904.4149 [astro-ph.SR] .\nPavlyuchenkov, Yaroslav, Vitaly Akimkin, Dmitri Wiebe, and Eduard Vorobyov. 2019. Revealing dust segregation in protoplanetary discs with the help of multifrequency spectral index maps. MNRAS 486, no. 3 (July): 39073914. https://doi.org/10.1093/mnras/stz1046. arXiv: 1904.05251 [astro-ph.IM] .\nPelkonen, Veli-Matti, Paolo Padoan, Mika Juvela, Troels Haugbølle, and Åke Nordlund. 2024. Origin and Evolution of Angular Momentum of Class II Disks. arXiv e-prints (May): arXiv:2405.06520. https://doi.org/10. 48550/arXiv.2405.06520. arXiv: 2405.06520 [astro-ph.SR] .\nPérez, Laura M., John M. Carpenter, Sean M. Andrews, Luca Ricci, Andrea Isella, Hendrik Linz, Anneila I. Sargent, et al. 2016. Spiral density waves in a young protoplanetary disk. Science 353, no. 6307 (September): 15191521. https://doi.org/10.1126/science.aaf8296. arXiv: 1610.05139 [astro-ph.GA] .\nPiso, Ana-Maria A., Karin I. Öberg, Tilman Birnstiel, and Ruth A. MurrayClay. 2015. C/O and Snowline Locations in Protoplanetary Disks: The Effect of Radial Drift and Viscous Gas Accretion. ApJ 815, no. 2 (December): 109. https://doi.org/10.1088/0004-637X/815/2/109. arXiv: 1511.05563 [astro-ph.EP] .\nPontoppidan, K. M., C. Salyk, E. A. Bergin, S. Brittain, B. Marty, O. Mousis, and K. I. Öberg. 2014. Volatiles in Protoplanetary Disks. In Protostars and planets vi, edited by Henrik Beuther, Ralf S. Klessen, Cornelis P. Dullemond, and Thomas Henning, 363-385. January. https : / / doi . org/10.2458/azu_uapress_9780816531240-ch016. arXiv: 1401.2423 [astro-ph.EP] .\nPrzybilla, Norbert, Maria-Fernanda Nieva, and Keith Butler. 2008. A Cosmic Abundance Standard: Chemical Homogeneity of the Solar Neighborhood and the ISM Dust-Phase Composition. ApJ 688, no. 2 (December): L103. https://doi.org/10.1086/595618. arXiv: 0809.2403 [astro-ph] .\nRafikov, Roman R. 2005. Can Giant Planets Form by Direct Gravitational Instability? ApJ 621, no. 1 (March): L69-L72. https://doi.org/10.1086/ 428899. arXiv: astro-ph/0406469 [astro-ph] .\nRampinelli, L., S. Facchini, M. Leemker, J. Bae, M. Benisty, R. Teague, C. J. Law, K. I. Öberg, B. Portilla-Revelo, and A. J. Cridland. 2024. ALMA high-resolution observations unveil planet formation shaping molecular emission in the PDS 70 disk. arXiv e-prints (July): arXiv:2407.06272. https://doi.org/10.48550/arXiv.2407.06272. arXiv: 2407.06272 [astro-ph.EP] .\nReggiani, Henrique, Jhon Yana Galarza, Kevin C. Schlaufman, David K. Sing, Brian F. Healy, Andrew McWilliam, Joshua D. Lothringer, and Laurent Pueyo. 2024. Insight into the Formation of β Pic b through the Composition of Its Parent Protoplanetary Disk as Revealed by the β Pic Moving Group Member HD 181327. AJ 167, no. 1 (January): 45. https://doi.org/10.3847/1538-3881/ad0f93. arXiv: 2311.12210 [astro-ph.SR] .\nSchneider, Aaron David, and Bertram Bitsch. 2021. How drifting and evaporating pebbles shape giant planets. I. Heavy element content and atmospheric C/O. A&A 654 (October): A71. https://doi.org/10.1051/00046361/202039640. arXiv: 2105.13267 [astro-ph.EP] .\nSeager, S., L. J. Richardson, B. M. S. Hansen, K. Menou, J. Y. -K. Cho, and D. Deming. 2005. On the Dayside Thermal Emission of Hot Jupiters. ApJ 632, no. 2 (October): 1122-1131. https://doi.org/10.1086/444411. arXiv: astro-ph/0504212 [astro-ph] .\nSeligman, Darryl Z., Leslie A. Rogers, Samuel H. C. Cabot, John W. Noonan, Theodore Kareta, Kathleen E. Mandt, Fred Ciesla, et al. 2022. The Volatile Carbon-to-oxygen Ratio as a Tracer for the Formation Locations of Interstellar Comets. PSJ 3, no. 7 (July): 150. https://doi.org/10. 3847/PSJ/ac75b5. arXiv: 2204.13211 [astro-ph.EP] .\nSemenov, D., C. Favre, D. Fedele, S. Guilloteau, R. Teague, Th. Henning, A. Dutrey, E. Chapillon, F. Hersant, and V. Piétu. 2018. Chemistry in disks. XI. Sulfur-bearing species as tracers of protoplanetary disk physics and chemistry: the DM Tau case. A&A 617 (September): A28. https://doi.org/10.1051/0004-6361/201832980. arXiv: 1806.07707 [astro-ph.GA] .\nSemenov, D., Th. Henning, Ch. Helling, M. Ilgner, and E. Sedlmayr. 2003. Rosseland and Planck mean opacities for protoplanetary discs. A&A 410 (November): 611-621. https://doi.org/10.1051/0004-6361:20031279. arXiv: astro-ph/0308344 [astro-ph] .\nSemenov, D., and D. Wiebe. 2011. Chemical Evolution of Turbulent Protoplanetary Disks and the Solar Nebula. ApJS 196, no. 2 (October): 25. https://doi.org/10.1088/0067- 0049/196/2/25. arXiv: 1104.4358 [astro-ph.GA] .\nShakura, N. I., and R. A. Sunyaev. 1973. Black holes in binary systems. Observational appearance. A&A 24 (January): 337-355.\nSing, David K., Zafar Rustamkulov, Daniel P. Thorngren, Joanna K. Barstow, Pascal Tremblin, Catarina Alves de Oliveira, Tracy L. Beck, et al. 2024. A warm Neptune's methane reveals core mass and vigorous atmospheric mixing. arXiv e-prints (May): arXiv:2405.11027. https://doi.org/10. 48550/arXiv.2405.11027. arXiv: 2405.11027 [astro-ph.EP] .\nSmith, Peter C. B., Michael R. Line, Jacob L. Bean, Matteo Brogi, Prune August, Luis Welbanks, Jean-Michel Desert, et al. 2024. A Combined Ground-based and JWST Atmospheric Retrieval Analysis: Both IGRINS and NIRSpec Agree that the Atmosphere of WASP-77A b Is Metal-poor. AJ 167, no. 3 (March): 110. https://doi.org/10.3847/1538-3881/ad17bf. arXiv: 2312.13069 [astro-ph.EP] .\nStammler, Sebastian Markus, Tilman Birnstiel, Olja Panić, Cornelis Petrus Dullemond, and Carsten Dominik. 2017. Redistribution of CO at the location of the CO ice line in evolving gas and dust disks. A&A 600 (April): A140. https://doi.org/10.1051/0004-6361/201629041. arXiv: 1701.02385 [astro-ph.EP] .\nStevenson, David J., and Jonathan I. Lunine. 1988. Rapid formation of Jupiter by diffusive redistribution of water vapor in the solar nebula. Icarus 75, no. 1 (July): 146-155. https://doi.org/10.1016/0019-1035(88)90133-9.\nSwain, M. R., G. Tinetti, G. Vasisht, P. Deroo, C. Griffith, J. Bouwman, Pin Chen, et al. 2009. Water, Methane, and Carbon Dioxide Present in the Dayside Spectrum of the Exoplanet HD 209458b. ApJ 704, no. 2 (October): 1616-1621. https://doi.org/10.1088/0004-637X/704/2/1616. arXiv: 0908.4010 [astro-ph.EP] .\nTesti, L., T. Birnstiel, L. Ricci, S. Andrews, J. Blum, J. Carpenter, C. Dominik, et al. 2014. Dust Evolution in Protoplanetary Disks. In Protostars and planets vi, edited by Henrik Beuther, Ralf S. Klessen, Cornelis P. Dullemond, and Thomas Henning, 339-361. January. https://doi.org/10.2458/azu_ uapress_9780816531240-ch015. arXiv: 1402.1354 [astro-ph.SR] .\nThiabaud, A., U. Marboeuf, Y. Alibert, I. Leya, and K. Mezger. 2015. Gas composition of the main volatile elements in protoplanetary discs and its implication for planet formation. A&A 574 (February): A138. https: //doi.org/10.1051/0004-6361/201424868.\nThies, Ingo, Pavel Kroupa, Simon P. Goodwin, Dimitrios Stamatellos, and Anthony P. Whitworth. 2010. Tidally Induced Brown Dwarf and Planet Formation in Circumstellar Disks. ApJ 717, no. 1 (July): 577-585. h ttps://doi.org/10.1088/0004- 637X/717/1/577. arXiv: 1005.3017 [astro-ph.SR] .\nTielens, A. G. G. M. 2005. The Physics and Chemistry of the Interstellar Medium.\nTong, Simin, Richard Alexander, and Giovanni Rosotti. 2024. A question of personalities: evolution of viscous and wind-driven protoplanetary discs in the presence of dead zones. MNRAS (July). https://doi.org/10.1093/ mnras/stae1748. arXiv: 2407.12209 [astro-ph.EP] .\nToomre, A. 1964. On the gravitational stability of a disk of stars. ApJ 139 (May): 1217-1238. https://doi.org/10.1086/147861.\nTopchieva, A., T. Molyarova, V. Akimkin, L. Maksimova, and E. Vorobyov. 2024. Ices on pebbles in protoplanetary discs. MNRAS 530, no. 3 (May): 2731-2748. https://doi.org/10.1093/mnras/stae597. arXiv: 2403.02895 [astro-ph.EP] .\nTurrini, D., E. Schisano, S. Fonte, S. Molinari, R. Politi, D. Fedele, O. Panić, M. Kama, Q. Changeat, and G. Tinetti. 2021. Tracing the Formation History of Giant Planets in Protoplanetary Disks with Carbon, Oxygen, Nitrogen, and Sulfur. ApJ 909, no. 1 (March): 40. https://doi.org/10. 3847/1538-4357/abd6e5. arXiv: 2012.14315 [astro-ph.EP] .\nUmebayashi, T., and T. Nakano. 1981. Fluxes of Energetic Particles and the Ionization Rate in Very Dense Interstellar Clouds. PASJ 33 (January): 617.\nVallet, David, Anna C. Childs, Rebecca G. Martin, Mario Livio, and Stephen Lepp. 2023. Formation of super-Earths in icy dead zones around lowmass stars. MNRAS 519, no. 1 (February): L10-L14. https://doi.org/10. 1093/mnrasl/slac144. arXiv: 2211.07759 [astro-ph.EP] .\nVisser, R., E. F. van Dishoeck, S. D. Doty, and C. P. Dullemond. 2009. The chemical history of molecules in circumstellar disks. I. Ices. A&A 495, no. 3 (March): 881-897. https://doi.org/10.1051/0004-6361/200810846. arXiv: 0901.1313 [astro-ph.SR] .\nVorobyov, E. I. 2013. Formation of giant planets and brown dwarfs on wide orbits. A&A 552 (April): A129. https://doi.org/10.1051/0004-6361/ 201220601. arXiv: 1302.1892 [astro-ph.EP] .\nVorobyov, Eduard I., Vitaly Akimkin, Olga Stoyanovskaya, Yaroslav Pavlyuchenkov, and Hauyu Baobab Liu. 2018. Early evolution of viscous and selfgravitating circumstellar disks with a dust component. A&A 614 (June): A98. https://doi.org/10.1051/0004-6361/201731690. arXiv: 1801.06898 [astro-ph.EP] .\nVorobyov, Eduard I., and Shantanu Basu. 2010. The Burst Mode of Accretion and Disk Fragmentation in the Early Embedded Stages of Star Formation. ApJ 719, no. 2 (August): 1896-1911. https://doi.org/10.1088/0004637X/719/2/1896. arXiv: 1007.2993 [astro-ph.SR] .\nVorobyov, Eduard I., and Vardan G. Elbakyan. 2019. Gravitoviscous protoplanetary disks with a dust component. II. Spatial distribution and growth of dust in a clumpy disk. A&A 631 (November): A1. https: / / doi . org / 10 . 1051 / 0004 - 6361 / 201936132. arXiv: 1908 . 10589 [astro-ph.SR] .\nVorobyov, Eduard I., Vardan G. Elbakyan, Michihiro Takami, and Hauyu B. Liu. 2020. Effect of luminosity outbursts on protoplanetary disk dynamics. A&A 643 (November): A13. https://doi.org/10.1051/00046361/202038122. arXiv: 2009.01888 [astro-ph.SR] .\nVorobyov, Eduard I., Sergey Khaibrakhmanov, Shantanu Basu, and Marc Audard. 2020. Accretion bursts in magnetized gas-dust protoplanetary disks. A&A 644 (December): A74. https://doi.org/10.1051/00046361/202039081. arXiv: 2011.00951 [astro-ph.SR] .\nVorobyov, Eduard I., D. N. C. Lin, and Manuel Guedel. 2015. The effect of external environment on the evolution of protostellar disks. A&A 573 (January): A5. https://doi.org/10.1051/0004-6361/201424583. arXiv: 1410.1743 [astro-ph.SR] .\nVorobyov, Eduard I., Zsolt Regaly, Manuel Guedel, and Doug N. C. Lin. 2016. An alternative model for the origin of gaps in circumstellar disks. A&A 587 (March): A146. https://doi.org/10.1051/0004-6361/201527701. arXiv: 1601.08089 [astro-ph.SR] .\nVorobyov, Eduard I., Aleksandr M. Skliarevskii, Manuel Guedel, and Tamara Molyarova. 2024. Primordial dust rings, hidden dust mass, and the first generation of planetesimals in gravitationally unstable protoplanetary disks. arXiv e-prints (April): arXiv:2404.16151. https://doi.org/10.48550/ arXiv.2404.16151. arXiv: 2404.16151 [astro-ph.EP] .\nVorobyov, Eduard I., Aleksandr M. Skliarevskii, Tamara Molyarova, Vitaly Akimkin, Yaroslav Pavlyuchenkov, Ágnes Kóspál, Hauyu Baobab Liu, Michihiro Takami, and Anastasiia Topchieva. 2022. Evolution of dust in protoplanetary disks of eruptive stars. A&A 658 (February): A191. https://doi.org/10.1051/0004-6361/202141932. arXiv: 2112.06004 [astro-ph.EP] .\nVorobyov, Eduard I., Olga V. Zakhozhay, and Michael M. Dunham. 2013. Fragmenting protostellar discs: properties and observational signatures. MNRAS 433, no. 4 (August): 3256-3273. https://doi.org/10.1093/ mnras/stt970. arXiv: 1306.4074 [astro-ph.SR] .\nWada, Koji, Hidekazu Tanaka, Toru Suyama, Hiroshi Kimura, and Tetsuo Yamamoto. 2009. Collisional Growth Conditions for Dust Aggregates. ApJ 702, no. 2 (September): 1490-1501. https://doi.org/10.1088/0004637X/702/2/1490.\nWalsh, Catherine, Hideko Nomura, and Ewine van Dishoeck. 2015. The molecular composition of the planet-forming regions of protoplanetary disks across the luminosity regime. A&A 582 (October): A88. http s : / / doi . org / 10 . 1051 / 0004 - 6361 / 201526751. arXiv: 1507 . 08544 [astro-ph.EP] .\nWeber, Philipp, Sebastián Pérez, Alice Zurlo, James Miley, Antonio Hales, Lucas Cieza, David Principe, et al. 2023. Spirals and Clumps in V960 Mon: Signs of Planet Formation via Gravitational Instability around an FU Ori Star? ApJ 952, no. 1 (July): L17. https://doi.org/10.3847/20418213/ace186. arXiv: 2307.13433 [astro-ph.EP] .\nWei, Chen-En, Hideko Nomura, Jeong-Eun Lee, Wing-Huen Ip, Catherine Walsh, and T. J. Millar. 2019. The Effect of Carbon Grain Destruction on the Chemical Structure of Protoplanetary Disks. ApJ 870, no. 2 (January): 129. https://doi.org/10.3847/1538- 4357/aaf390. arXiv: 1811.10194 [astro-ph.EP] .\nWeidenschilling, S. J. 1977. Aerodynamics of solid bodies in the solar nebula. MNRAS 180 (July): 57-70. https://doi.org/10.1093/mnras/180.2.57.\nWeiner Mansfield, Megan, Michael R. Line, Joost P. Wardenier, Matteo Brogi, Jacob L. Bean, Hayley Beltz, Peter Smith, et al. 2024. The metallicity and carbon-to-oxygen ratio of the ultra-hot Jupiter WASP-76b from Gemini-S/IGRINS. arXiv e-prints (May): arXiv:2405.09769. https://doi. org/10.48550/arXiv.2405.09769. arXiv: 2405.09769 [astro-ph.EP] .\nWestley, M. S., R. A. Baragiola, R. E. Johnson, and G. A. Baratta. 1995. Photodesorption from low-temperature water ice in interstellar and circumsolar grains. Nature 373, no. 6513 (February): 405-407. https: //doi.org/10.1038/373405a0.\nWierzchos, K., and M. Womack. 2018. C/2016 R2 (PANSTARRS): A Comet Rich in CO and Depleted in HCN. AJ 156, no. 1 (July): 34. https://doi. org/10.3847/1538-3881/aac6bc. arXiv: 1805.06918 [astro-ph.EP] .\nWinter, Andrew J., Myriam Benisty, and Sean M. Andrews. 2024. Planet formation regulated by galactic-scale interstellar turbulence. arXiv eprints (May): arXiv:2405.08451. arXiv: 2405.08451 [astro-ph.EP] .\nWong, Michael H., Paul R. Mahaffy, Sushil K. Atreya, Hasso B. Niemann, and Tobias C. Owen. 2004. Updated Galileo probe mass spectrometer measurements of carbon, oxygen, nitrogen, and sulfur on Jupiter. Icarus 171, no. 1 (September): 153-170. https://doi.org/10.1016/j.icarus.2004. 04.010.\nWorthen, Kadin, Christine H. Chen, David R. Law, Cicero X. Lu, Kielan Hoch, Yiwei Chai, G. C. Sloan, et al. 2024. MIRI MRS Observations of β Pictoris. I. The Inner Dust, the Planet, and the Gas. ApJ 964, no. 2 (April): 168. https://doi.org/10.3847/1538-4357/ad2354.\nXue, Qiao, Jacob L. Bean, Michael Zhang, Luis Welbanks, Jonathan Lunine, and Prune August. 2024. JWST Transmission Spectroscopy of HD 209458b: A Supersolar Metallicity, a Very Low C/O, and No Evidence of CH4, HCN, or C2H2. ApJ 963, no. 1 (March): L5. https://doi.org/ 10.3847/2041-8213/ad2682. arXiv: 2310.03245 [astro-ph.EP] .\nYang, Chao-Chin, Anders Johansen, and Daniel Carrera. 2017. Concentrating small particles in protoplanetary disks through the streaming instability. A&A 606 (October): A80. https : / / doi . org / 10 . 1051 / 0004 - 6361 / 201630106. arXiv: 1611.07014 [astro-ph.EP] .\nYoudin, Andrew N., and Jeremy Goodman. 2005. Streaming Instabilities in Protoplanetary Disks. ApJ 620, no. 1 (February): 459-469. https: //doi.org/10.1086/426895. arXiv: astro-ph/0409263 [astro-ph] .\nZhang, Yapeng, Ignas A. G. Snellen, Alexander J. Bohn, Paul Mollière, Christian Ginski, H. Jens Hoeijmakers, Matthew A. Kenworthy, et al. 2021. The 13 CO-rich atmosphere of a young accreting super-Jupiter. Nature 595, no. 7867 (July): 370-372. https://doi.org/10.1038/s41586-02103616-x. arXiv: 2107.06297 [astro-ph.EP] .\nZhu, Zhaohuan, Lee Hartmann, Charles F. Gammie, Laura G. Book, Jacob B. Simon, and Eric Engelhard. 2010. Long-term Evolution of Protostellar and Protoplanetary Disks. I. Outbursts. ApJ 713, no. 2 (April): 11341142. https : / / doi . org / 10 . 1088 / 0004 - 637X / 713 / 2 / 1134. arXiv: 1003.1759 [astro-ph.SR] .\nZhu, Zhaohuan, Yan-Fei Jiang, and James M. Stone. 2020. Global 3D radiation magnetohydrodynamic simulations for FU Ori's accretion disc and observational signatures of magnetic fields. MNRAS 495, no. 3 (January): 3494-3514. https://doi.org/10.1093/mnras/staa952. arXiv: 1912.01632 [astro-ph.EP] .\n"},"sections":{"kind":"list like","value":[{"title":"C/O ratios in self-gravitating protoplanetary discs with dust evolution","content":"Tamara Molyarova, 1,2 Eduard Vorobyov, 1,2 and Vitaly Akimkin 1 1 Institute of Astronomy, Russian Academy of Sciences, 48 Pyatnitskaya St., Moscow, 119017, Russia 2 Research Institute of Physics, Southern Federal University, Stachki Ave. 194, Rostov-on-Don 344090, Russia Author for correspondence: Tamara Molyarova, Email: moliarova@sfedu.ru.","pages":[1]},{"title":"Abstract","content":"Elemental abundances, particularly the C/O ratio, are seen as a way to connect the composition of planetary atmospheres with planet formation scenario and the disc chemical environment. We model the chemical composition of gas and ices in a self-gravitating disc on timescales of 0.5 Myr since its formation to study the evolution of C/O ratio due to dust dynamics and growth, and phase transitions of the volatile species. We use the thin-disc hydrodynamic code FEOSAD, which includes disc self-gravity, thermal balance, dust evolution and turbulent diffusion, and treats dust as a dynamically different and evolving component interacting with the gas. It also describes freeze-out, sublimation and advection of four most abundant volatile species: H2O, CO2, CH4 and CO. We demonstrate the effect of gas and dust substructures such as spirals and rings on the distribution of volatiles and C/O ratios, including the formation of multiple snowlines of one species, and point out the anticorrelation between dust-to-gas ratio and total C/O ratio emerging due to the contribution of oxygen-rich ice mantles. We identify time and spatial locations where two distinct trigger mechanisms for planet formation are operating and differentiate them by C/O ratio range: wide range of the C/O ratios of 0 - 1.4 for streaming instability, and a much narrower range 0.3 - 0.6 for gravitational instability (with the initial value of 0.34). This conclusion is corroborated by observations, showing that transiting exoplanets, which possibly experienced migration through a variety of disc conditions, have significantly larger spread of C/O in comparison with directly imaged exoplanets likely formed in gravitationally unstable outer disk regions. We show that the ice-phase C/O ≈ 0.2 - 0.3 between the CO, CO2 and CH4 snowlines corresponds to the composition of the Solar system comets, that represent primordial planetesimals. Keywords: protoplanetary disc, volatiles, dust evolution","pages":[1]},{"title":"1. Introduction","content":"The protoplanetary disc matter can be roughly divided into three component: gaseous chemical species, solid dust particles, and icy mantles covering the surface of dust grains. Gas and solid particles become dynamically decoupled, as evolving dust grows and acquires relative velocities leading to the redistribution of elements in the disc and between the phases, and creating the premises for different chemical environments. When planets start to form, their properties, including chemical composition of the atmosphere, are inevitably affected by the location and the mechanism of their formation. This suggests that the origin of (exo)planets might be investigated using their observed chemical composition, and makes understanding the disc chemical evolution vital for creating a consistent planet formation theory. One of the key parameters that govern the chemical setup of a planetary atmosphere is the relation between the abundances of carbon and oxygen, often referred to as carbon-tooxygen ratio (hereafter C/O ratio). The variations of C/O ratio in the ice and gas phases at the snowlines of main disc volatiles (CO, CO 2 , and H 2 O) and the prospects of connecting them to planet formation were discussed in Öberg, Murray-Clay, and Bergin (2011) within a qualitative freeze-out model. Since then, C/O ratio received a lot of attention in this context. It was thoroughly investigated in modelling (see, e.g., Booth et al. 2017; Eistrup, Walsh, and van Dishoeck 2018; Cridland, Eistrup, and van Dishoeck 2019; Cridland et al. 2019; Crid- and, Bosman, and van Dishoeck 2020; Cridland et al. 2020; Krijt et al. 2020; Turrini et al. 2021; Schneider and Bitsch 2021). The connection of disc chemical composition with C/O in exoplanetary atmospheres was modelled using core accretion model (Thiabaud et al. 2015) and 'chain' planet population synthesis model (Mordasini et al. 2016). Paul Mollière et al. (2022) considered a simple formation retrieval pipeline and found that this task requires careful consideration of the model assumptions. The measurements of molecular abundances in the atmospheres of giant exoplanets obtained by a variety of modern facilities, such as HST, Spitzer, VLTI, JWST, Gemini, indicate a diversity of C/O ratios: from low C/O ratio values ( ≈ 0.4, below the solar value of 0.54; Benneke et al. 2019; GRAVITY Collaboration et al. 2020; Worthen et al. 2024; Xue et al. 2024) to stellar ( ≈ 0.5, close to solar; P. Mollière et al. 2020; Zhang et al. 2021; Smith et al. 2024) and close to or above unity (Swain et al. 2009; Madhusudhan et al. 2011). A variety of solar and super-solar C/O ratios is observed in four planets within the HR8799 system (Nasedkin et al. 2024). Chemical composition of the atmospheres of many hot Jupiters indicates high C/O>1 of the forming material (Moses et al. 2013). There is an observational evidence of young planets in PDS 70 disc accreting the material with C/O>1 (Facchini et al. 2021). The number of exoplanets with constrained atmospheric C/O ratios grows with large studies of multiple planets such as Changeat et al. (2022), which allows us to make some statistical conclusions. The population study of C/O ratios in exoplanetary atmospheres reveals that there are two populations with different elemental ratio, which are likely formed in different mechanisms (Hoch et al. 2023). Khorshid, Min, and Désert (2023) were able to restrict the formation scenario for WASP-77b based on the measured C/O ratio of the planet (Line et al. 2021) and the modelling of planet formation and migration. Elemental abundances in protoplanetary discs can be constrained from observations (Fedele and Favre 2020), and C/O ratio in the gas can be estimated. Spatially resolved observations can help distinguish between different C/O ratios spectroscopically (Matter, Pignatale, and Lopez 2020). Cleeves et al. (2018) report C/O ≈ 0.8 in the molecular layer of IM Lup disc. ALMA observations of hydrocarbons and sulphur-bearing species indicate C/O > 1 in the upper disc layers and in the outer disc in TW Hya and DM Tau (Dutrey et al. 2011; Bergin et al. 2016; Semenov et al. 2018) and for a population of discs in Lupus (Miotello et al. 2019). For the nearby discs the solar elemental composition with C/O ≈ 0.54 is usually expected, thus the observed higher values confirm redistribution of carbon and oxygen in discs. In addition to high C/O in disc atmospheres, there is evidence of both carbon and oxygen depletion from gas (Kama et al. 2016; Miotello et al. 2019). However, some of heavy oxygen carriers might not be observable, leading to overestimated C/O in disc observations (Walsh, Nomura, and van Dishoeck 2015). The volatile composition is also used to constrain the origin of bodies in the Solar System. Fraction of CO and CO 2 ices relative to water in cometary comae indicate their formation between the CO and CO 2 snowlines or exterior to the CO snowline (A'Hearn et al. 2012; Seligman et al. 2022). Abundances of CO and N 2 ices were used to analyse the original location of Pluto and Triton (Mousis, Anderson, et al. 2024). Observed elemental abundances were used to constrain the Jupiter formation scenario (Lodders 2004), relying also on abundances of nitrogen (Öberg and Wordsworth 2019; Bosman, Cridland, and Miguel 2019) and chemically inactive species like Ar. However, the model assumptions can lead to different interpretation of the observations: while Öberg and Wordsworth (2019) and Bosman, Cridland, and Miguel (2019) suggest that Jupiter formed outside N 2 snowline (at > 30 au), Ohno and Ueda (2021) consider the concept disc shadow, which allows Jupiter to form near its current location. Chemical processes other than freeze-out and sublimation at the snowlines can alter the composition of ice and gas as well. Due to gas-phase and surface reactions, snowlines can become important for the redistribution of elements. More detailed chemical modelling shows that the C/O ratio in the gas and in the ice depends also on the initial chemical setup and ionisation by cosmic rays and radioactive nuclei (Eistrup, Walsh, and van Dishoeck 2016, 2018). It directly affects the interpretation of observations. Another essential chemical process is CO depletion from the gas, resulting in its transformation to CO 2 ice (Bosman, Tielens, and van Dishoeck 2018). For example, stellar C/O ratio in the atmosphere of HR 8799e indicates that the planet accreted its material beyond CO snowline ( ≈ 45 au), but chemical modelling suggests that due to CO depletion, the C/O in the ice already approaches the stellar ratio beyond CO 2 snowline ( ≈ 20 au), which is closer to the star (P. Mollière et al. 2020). Another key process affecting the elemental ratio is dust drift, which leads to spatial segregation between the chemical constituents of the gas and the grains covered with ice. The distribution of CO in the gas and ice phases was studied within dynamical models of dust evolution (Stammler et al. 2017; Krijt et al. 2018). Even without chemical processes, dust evolution and dynamics can substantially alter C/O ratio in the atmospheres of forming planets (Booth et al. 2017). Some models combine chemical reactions treatment with dust evolution and transport, usually within 1D viscous models. Dust transport can have a strong effect on the abundances of volatiles in the inner disc regions (Bosman, Tielens, and van Dishoeck 2018). However, for discs with low turbulence and high cosmic ray ionisation rate, C/O ratio is rather defined by chemical evolution (Booth and Ilee 2019). In our previous work (Molyarova et al. 2021) we showed that the volatile species tend to concentrate around their snowlines both in the gas and more notably on the dust surface. This accumulation was found to be caused by effective transport of volatiles through the snowlines by azimuthal variations in the gas and dust radial and angular velocity, an effect that cannot be captured in 1D viscous disc models. Such accumulation should immediately affect the local C/O ratio, which suggests the connection between the snowlines of various volatiles and the formation of planets with altered C/O in their atmospheres. In this work, we follow the distribution of the main volatile species in the disc to investigate the distribution of C/O ratio in gas and ice in a 2D thin-disc hydrodynamic model. We study the effect of dust growth and dynamics on the elemental ratios and consider the role of the initial mass of the collapsing core on the distribution of volatiles. The paper is organised as follows. The main features of the used FEOSAD model are described in Section 2, with the details of the treatment of the volatiles given in Section 2.3. In Section 3 we describe the results of the simulations, focusing on distribution of the volatiles in Section 3.2, the C/O ratios in Section 3.3, and their evolution in Section 3.4. In Section 4, we discuss the implications of our results in the context of planet formation via different mechanisms. The main conclusions are listed in Section 5.","pages":[1,2]},{"title":"2. Model","content":"We use the global model of protoplanetary disc formation FEOSAD (Vorobyov et al. 2018), which includes disc selfgravity, dust evolution and interaction with gas (including backreaction of dust on gas), turbulent viscosity, adiabatic and radiative cooling and heating. It describes the formation of a protostar and a protoplanetary disc from a collapsing cloud in a 2D thin-disc approach. The model also includes freeze-out of main volatile species as in Molyarova et al. (2021), with the feedback from ice mantles on dust evolution via fragmentation velocity. Here we summarise the key characteristics of the model, more details can be found in the previous works (Vorobyov et al. 2018; Molyarova et al. 2021; Kadam, Vorobyov, and Basu 2022). The main difference from our previous study in Molyarova et al. (2021) is that here we consider the formation of dead zones via variable α -parameter of Shakura and Sunyaev and also include turbulent diffusion.","pages":[2,3]},{"title":"2.1 Gas evolution","content":"For the gas component, the hydrodynamic equations for mass, momentum, and internal energy conservation are the following where subscripts p and p ' denote the planar components ( r , ϕ ) in polar coordinates, Σ g is the gas mass surface density, e is the internal energy per surface area, P is the vertically integrated gas pressure calculated via the ideal equation of state as P = ( γ - 1) e with γ = 7/5, f is the friction force between gas and dust, v = vr ˆ r + v ϕ ˆ ϕ is the gas velocity in the disc plane, and ∇ = ˆ r ∂ / ∂ r + ˆ ϕ r -1 ∂ / ∂ϕ is the gradient along the planar coordinates of the disc. The gravitational acceleration in the disc plane, g = gr ˆ r + g ϕ ˆ ϕ , includes the gravity of the central protostar when formed and takes into account disc self-gravity of both gas and dust found by solving the Poisson integral (Binney and Tremaine 1987). Theconsideration of time-dependent energy balance (Eq. (3)) allows us to accurately calculate the midplane temperature T mp and is particularly important to describe the phase state of the volatiles and the level of turbulent viscosity. The terms Λ and Γ describe the rates of dust cooling and heating, respectively, by stellar and background irradiation. They are calculated based on the analytical solution of the radiation transfer equations in the vertical direction (Dong et al. 2016; Vorobyov et al. 2018) Here, σ is the Stefan-Boltzmann constant, τ P and τ R are the Planck and Rosseland mean optical depths to the disc midplane, calculated as τ = κΣ dust from Planck and Rosseland mean opacities κ P and κ R (Semenov et al. 2003) and total dust surface density Σ dust . Gas and dust temperatures are assumed to be equal, and the midplane temperature is linked with gas pressure as T mp = P µ / R Σ g, where µ = 2.3 is the mean molecular weight of the gas, and R is the universal gas constant. The irradiation temperature at the disc surface T irr is determined by both stellar and background irradiation. Stellar irradiation includes the luminosity from the photosphere of the protostar and accretion luminosity. The background radiation is assumed as a black body with the temperature of 15 K. For more details on the irradiation we refer to Vorobyov et al. (2018). Turbulent viscosity is described using the common α -parameter approach of Shakura and Sunyaev (1973). It is taken into account via the viscous stress tensor Π (see Vorobyov and Basu 2010, for explicit expressions for the components of the terms with Π ). The magnitude of kinematic viscosity is ν = α c s H , where c s is the sound speed and H is the vertical scale height of the gas disc calculated using an assumption of local hydrostatic equilibrium of a self-gravitating disc (see Vorobyov and Basu 2010, Appendix A). Here, we use the adaptive α approach implying accretion through a layered disc (Gammie 1996; Armitage, Livio, and Pringle 2001; Kadam, Vorobyov, and Basu 2022). Turbulence is assumed to be generated by magneto-rotational instability (MRI) which only develops in layers of the disc where the ionisation level is high enough. The MRI-active layer is characterised by its surface density Σ MRI and relatively high value of turbulent viscosity α MRI = 10 -3 . As thermal and photo-ionisation are not efficient enough for the relatively cold and dense matter in the disc at >0.5 au, the main process determining the thickness of the MRI-active layer is ionisation by cosmic rays. It is assumed to be constant Σ MRI = 100g cm -2 , which is the typical depth of Galactic cosmic rays penetration in the ISM (Umebayashi and Nakano 1981) and in protoplanetary discs (Padovani, Galli, and Glassgold 2009). The dead zone is characterised by surface density from the midplane Σ dz = Σ g/2Σ MRI with residual turbulence α dz . The turbulence in this layer is only hydrodynamic turbulence driven by the Maxwell stress in the active layer, and small value α dz = 10 -5 is adopted (Okuzumi and Hirose 2011). However, if local temperature exceeds the critical value of 1300 K, thermal ionisation becomes possible, the MRI develops and the dead zone is no longer dead, in which case α dz = 10 -1 (Zhu et al. 2010; Kadam et al. 2019). This value is higher than α MRI in the outer disc due to the different ionisation processes and local conditions. In the outer disc, the MRI can be suppressed by non-ideal magneto-hydrodynamic (MHD) effects such as ambipolar diffusion and Ohmic resistivity (Bai and Stone 2013; Gressel et al. 2015). In the dead zone in the inner disc, α can reach higher values when the MRI is triggered by thermal ionisation, as shown by 3D MHD simulations (Zhu, Jiang, and Stone 2020). The adopted parameterization makes use of an effective parameter α eff , which at any given location is calculated as","pages":[3]},{"title":"2.2 Dust evolution","content":"Dust is described as consisting of two components: small grains that are dynamically coupled to the gas, with the mass surface density Σ d,sm , and grown grains with the mass surface density Σ d,gr that can move relative to the gas and change in size. The total dust surface density necessary for the calculation of the optical depths for Eq. (4) is Σ dust = ( Σ d,gr + Σ d,sm )/2. The factor 1/2 appears as the optical depths is calculated to the midplane. Each dust population has a power-law size distribution f ( a ) = dN / da = Ca -p with a normalisation constant C and a fixed exponent p = 3.5. Small dust has sizes between a min = 5 × 10 -7 cm and a ∗ = 10 -4 cm, grown dust has sizes between a ∗ and a max, which can vary due to dust coagulation and fragmentation. Dynamics of these dust components follows the continuity and momentum equations where u is the grown dust velocity. The term S ( a max) is responsible for the exchange of matter between the dust populations, as dust is converted from the small to grown component due to coagulation and back due to fragmentation. The details of the dust evolution model are presented in Vorobyov et al. (2018). The last term in Eqs. (6) and (7) is responsible for dust turbulent diffusion, similar to Vorobyov, Elbakyan, et al. (2020). The coefficient of turbulent diffusion D is related to the kinematic viscosity ν as D = ν /(1 + St 2 ) (Birnstiel 2023). Diffusion affects dust grains along with their ice mantles, as well as the gas-phase species (see Section 2.3). The innermost regions of the disc are challenging to simulate explicitly due to the Courant criterion: in the highly dynamic inner regions (fraction of au) the timescales are so short that the code demands very small time step in order to preserve stability. Therefore, the inner regions are represented by a sink cell, with a carefully chosen inflow-outflow boundary condition at the sink cell and a parametric description of the accretion onto the star (see Vorobyov et al. 2018, for details). In the simulations presented below, the radius of the sink cell is 0.62 au. We consider two disc models with different initial mass of the collapsing cloud, 0.66 and 1 M ⊙ . We note that about 10% of the gas mass that crosses the sink cell is assumed to be evacuated by jets and outflows, and the other 90% lands on to the star. A small amount of mass remains in the envelope by the end of simulations. In both models, the initial gas temperature T init = 15 K and the ratio of rotational to gravitational energy β = 0.28%. The simulations start from the collapse of a molecular cloud, with only small dust grains. The simulation continues until the age of the system becomes equal to 0.5 Myr. Masses of the central protostar and the disc by the end of simulation are M ⋆ = 0.4 M ⊙ and M disc = 0.22 M ⊙ in model M1 and M ⋆ = 0.58 M ⊙ and M disc = 0.35 M ⊙ in model M2. The disc masses are around 0.5 stellar masses, which makes them essentially self-gravitating. A number of recent studies develop the idea that the mass infall from the ambient ISM continues during the lifetime of the disc, including the Class II stage (Padoan et al. 2024; Pelkonen et al. 2024; Winter, Benisty, and Andrews 2024). Such models describe a Bondi-Hoyle accretion regime and are in good agreement with the observed properties of the disc population, such as accretion rates, masses and sizes. This input of matter can play an important role in disc evolution and planet formation process (Vorobyov, Lin, and Guedel 2015; Vorobyov et al. 2016). In the FEOSAD model, the mass infall to the disc can be accounted for (Vorobyov, Lin, and Guedel 2015; Vorobyov et al. 2016), but in the present simulation this effect is not included. The mass infall from the envelope continues until the cloud is depleted of matter. Because of the thin-disc geometry, the gravitational contraction of the cloud proceeds in the plane of the disc. The matter is landing on the disc outer edge and is transported towards the star by the combined action of gravitational and viscous torques. The infall on the disc inner regions is therefore neglected. This is a reasonable approximation, considering that most of the matter and angular momentum in a three-dimensional cloud is located at relatively large polar angles and a flared outer edge of the disc intercepts most of them (see, e.g., Visser et al. 2009, Figure 1). In our modelling, we do not consider the continuous Bondi-Hoyle accretion and concentrate on the internal disc processes.","pages":[3,4]},{"title":"2.3 Evolution of volatiles","content":"We follow the evolution of four main volatile species: H 2 O, CO 2 , CO and CH 4 . These are the most abundant carbon- and oxygen-bearing ices observed in protostellar cores (Karin I. Öberg et al. 2011). In the model, each of these species can be present in three states: in the gas, in the ice on the surface of small dust, and in the ice on the surface of grown dust. Each species s is described by its surface density in the gas Σ gas s , on small dust Σ sm s , and on grown dust Σ gr s . Their distributions in the disc can change through three main processes: advection together with the corresponding component (gas or small/grown dust); exchange of mantles between dust populations due to grain collisions; phase transitions, including adsorption from gas to dust, and thermal and photo-desorption. Initially, all ices are on small grains and no volatiles present in the gas. The treatment of volatiles is adopted from Molyarova et al. (2021), who describe the models in more details. Here we recap main features of the chemical model. Our chemical model only describes phase transitions, i.e. adsorption and desorption, which includes thermal desorption and photo-desorption by interstellar UV radiation. These reactions were shown to be he most important for gas-phase abundances of most species (Ilee et al. 2011). Due to high computational costs, no other chemical processes, either gasphase or surface reactions are included, although they also may have significant effect on the composition of both ice and gas (Semenov and Wiebe 2011). The chemical evolution of the surface densities of volatile species is calculated from the system of equations where the mass rate coefficients per disc unit area of adsorption λ s and desorption η s for the species s are calculated for local conditions at every hydrodynamic step, separately for small and grown dust populations. Eqs. (9)-(11) are solved in two steps: first, the right-hand side is considered without the advection term. The case of pure adsorption/desorption represented by the right-hand side of the equation can be solved analytically (see Appendix A in Molyarova et al. 2021). This is done at every hydrodynamic step, before the dust growth step, when ices on small and grown grains are redistributed proportionally to mass exchange between the dust populations. This is followed by a transport step, when the surface densities of the volatiles are changed according to the fluxes of their respective gas or dust components between the cells. Restricting chemical processes to only adsorption and desorption allows the chemical step to be calculated fast, which is very important for computationally demanding hydrodynamic simulations. For each dust population, the desorption rate is a sum of thermal desorption and photo-desorption η = η td + η pd (the indices 'sm' and 'gr' are omitted for convenience). We operate under the assumption of zeroth-order desorption, i.e. the desorption rate does not depend on the present amount of ice ( Σ sm s or Σ gr s ). It implies that only the upper layers of the ice mantle are able to sublimate, which is a more appropriate approach for thick mantles. This assumption better describes desorption of CO and H 2 Oin temperature programmed desorption (TPD) experiments (Fraser et al. 2001; K. I. Öberg et al. 2005; Bisschop et al. 2006) than first-order desorption, which is more suitable for thin mantles of several monolayers. The rate of thermal desorption is calculated as where N ss = 10 15 cm -2 is the surface density of binding sites (Cuppen et al. 2017), E b (K) is the binding energy of the species to the surface, µ s is the species molecular mass, m p is atomic mass unit, k B is the Boltzmann constant. We follow Hasegawa and Herbst (1993) and use binding energy to calculate the pre-exponential factor in Eq. (12), the same way it was done in Molyarova et al. (2021). However, this approach was recently demonstrated by Minissale et al. (2022) to underestimate the pre-exponential factor by a few orders of magnitude, as it does not account for the rotational part of the partition functions of desorbing molecules. Total surface area of dust grains (small or grown) per disc unit area ˜ σ tot (cm 2 cm -2 ) is calculated for the adopted power-law size distribution with p = 3.5 as Here, ρ s = 3 g cm -3 is the material density of silicate cores of the dust grains. The model includes photodesorption of volatiles by interstellar irradiation, which is mostly relevant in the outer disc regions with lower optical depth. We do not consider the UV radiation field produced by the star and the accretion region as a source of photodesorption, assuming that they do not reach disc midplane due to high optical depth. The photo-desorption rate is calculated as where F UV = F UV 0 G UV (photons cm -2 s -1 ) is the UV photon flux expressed in the units of standard UV field and Y = 3.5 × 10 -3 + 0.13 exp ( -336K/ T mp ) (mol photon -1 ) is the photodesorption yield adopted from Westley et al. (1995) for water ice. The intensity of the interstellar UV radiation field with G 0 = 1 is F UV 0 = 4.63 × 10 7 photon cm -2 s -1 (Draine 1978). We assume that the disc situated in a star-forming region is illuminated by a slightly elevated unattenuated UV field with G env = 5.5 G 0 . For the disc midplane, which is illuminated from above and below, this field scales with the UV optical depth τ UV towards the disc midplane as The optical depth can be calculated as τ UV = 0.5( κ sm Σ d,sm + κ gr Σ d,gr ), where κ sm = 10 4 cm 2 g -1 , κ gr = 2 × 10 2 cm 2 g -1 are typical values of absorption coefficients in the UV for small and grown grains (Pavlyuchenkov et al. 2019, Fig. 1). We calculate the adsorption rate λ following Brown and Charnley (1990). It is proportional to the total cross-section of dust grains per unit volume, which can be obtained from the total surface area of dust grains per unit disc surface ˜ σ tot. To change the normalisation to the 2D case, ˜ σ tot needs to be multiplied by 1/2 H . Another factor of 1/4 follows from the difference between cross-section and surface area of a sphere. As a result, the adsorption rate is calculated as A more detailed derivation of rate coefficients for adsorption and desorption is presented in Section 2.3 of Molyarova et al. (2021). Table 1 summarises the molecular parameters used in both models in this work. Binding energies for H 2 O, CO 2 , and CO are based on the experimental data from Cuppen et al. (2017) for desorption from crystalline water ice. The binding energy for methane is taken from Aikawa et al. (1996). The values of the initial abundances relative to water f s are based on the observations of ices in protostellar cores around the low-mass protostars (Karin I. Öberg et al. 2011). They are transformed to the mass fraction relative to gas assuming the water abundance of 5 × 10 -5 relative to gas number density. Total initial mass of ices comprises ≈ 8.5% of the total mass of refractory grain cores. This is relatively low compared to the estimates suggesting comparable masses of ices and refractories in the discs (Pontoppidan et al. 2014). However, the lower fraction of ices is more suitable in our approach, that suggests that ice mantles do not change mass and radius of dust grains (Molyarova et al. 2021). As we are mostly interested in the elemental ratios and relative abundances of the considered ices, lower ice fraction is an appropriate simplification. However, we note that dust dynamics can lead to accumulation of ices and ice mantles exceeding masses of silicate cores in some disc regions, as was shown previously in Molyarova et al. (2021). Ice mantles also provide feedback on dust evolution. The model includes the effect of ices on fragmentation velocity v frag , which is the the maximum collision velocity leading to sticking instead of fragmentation. According to laboratory experiments, icy grains have higher v frag than bare silicate grains by an order of magnitude (Wada et al. 2009; Gundlach and Blum 2015). In Molyarova et al. (2021) we used the values of fragmentation velocity v frag = 1.5 and 15 m s -1 for bare and icy grains, correspondingly. Here we follow Okuzumi and Tazaki (2019) and adopt lower values of v frag = 0.5 and 5 m s -1 , which are more relevant for grains consisting of µ m-sized monomers. These lower values of v frag will lead to higher importance of fragmentation compared to Molyarova et al. (2021). To determine if a dust grain should be considered icy or bare, we compare the local total surface density of all ices on grown dust divided by Σ d,gr with the threshold value K , which is calculated as i.e., an icy grain must have at least one monolayer of ice. Here, a ml is the thickness of the ice monolayer estimated as the size of a water molecule 3 × 10 -8 cm. The material density of ice ρ ice = 1 g cm -3 and the mean radius of a grown grain is calculated as √ a ∗ a max for the power-law distribution with p = 3.5.","pages":[4,5,6]},{"title":"3. Results","content":"To understand the distribution of the species, we first need to consider the global evolution of the disc and its structure. The distribution of volatiles and the C/O ratio is very sensitive to gas and dust substructures appearing in the disc. Variations in temperature and pressure lead to the complex shape and temporal evolution of the snowlines. The dependence of dust fragmentation velocity on the presence of ice mantles implies the feedback from the volatiles on dust and (through backreaction) on gas. Our modelling starts with the gravitational collapse of a flattened, slowly rotating molecular cloud. The protoplanetary disc is formed after the formation of the protostar, when the in-spiraling layers of the contracting cloud hit the centrifugal barrier near the inner edge of the sink cell, at a time instance depending on the initial core mass. The disc and the protostar are formed ≈ 53 kyr after the beginning of the cloud collapse in model M1 and at ≈ 78kyr in model M2. If not stated otherwise, times are specified counting from the beginning of the simulation, e.g. the 100 kyr time instance for model M1 refers to a ≈ 50kyr old disc, as the first stage includes cloud collapse as well. An important characteristic of young stellar objects is their variable accretion rate. Our modelling produces accretion bursts with the magnitude of ∼ 100 L ⊙ occurring every ∼ 10 4 years during the first hundred thousands years of disc evolution. These burst parameters are in line with the episodic accretion scenario (Hartmann and Kenyon 1985) and resemble the observed phenomenon of FU Ori type stars (see Audard et al. 2014, for a review). The luminosity outbursts heat up the disc and typically shift the snowlines further away from the star. Although this effect is temporary, it can be reflected in the distributions of the volatiles, and leave long-term imprints in the observed dust properties (Vorobyov et al. 2022). The detailed analysis of the effect of such outbursts on the distribution of the elements is worthy of a separate study and lies beyond the scope of this paper.","pages":[6]},{"title":"3.1 Dust and gas structures","content":"During the first hundred thousands years of evolution, protoplanetary discs change from highly asymmetric and dynamic objects to nearly axisymmetric and slowly evolving structures. Figure 1 shows the examples of different gas and dust substructures that are present in the disc at different stages. The snapshots are shown for model M1, they include a young disc (160 kyr), an intermediate state (350 kyr), and a more evolved and axisymmetric disc (490 kyr). Each of the selected time instances represent some characteristic morphological features addressed below. In model M2, similar structures appear, although sometimes at different evolution times. In this subsection, we consider model M1 as an example and discuss these features, highlighting the properties of dust and gas substructures that are most relevant for the distribution of the volatiles. The earliest phases of disc evolution are characterised by a large-scale spiral structure in both dust and gas, as well as episodic appearance of clumps. They are the result of gravitational instability (GI) in a massive disc, as in our modelling, the disc mass comprises more than 0.1 of the stellar mass, which is roughly a threshold of the disc stability against GI (Vorobyov 2013; Kratter and Lodato 2016). This is illustrated by the Toomre Q -parameter (Toomre 1964) in the third column of Figure 1: inside the spirals and the clump Q < 1, which indicates the dominance of self-gravity over Keplerian sheer and gas density. The spiral structures become less prominent with time as the disc looses mass and the Q -parameter increases. However, spirals persist in the model throughout the disc evolution up to 0.5 Myr. For example, at 350 kyr, a very tight spiral is present in the gas, starting at the gas and dust ring at ≈ 10au. At 490 kyr, the spiral pattern is weak and exists at > 10 au distances, which are not displayed in Figure 1. One- or two-armed grand design spirals are common 100 - 200 kyr after the disc formation in the models. The analogues of such spirals in the observed young protoplanetary discs around low-mass stars are found, for example, in Elias 227 or WaOph 6 (Pérez et al. 2016; Huang, Andrews, Pérez, et al. 2018). It is not yet clear if the observed spirals are the result of GI or caused by a perturbation from a companion planet or a (sub)stellar object (Meru et al. 2017; Brown-Sevilla et al. 2021). A spiral with multiple clumps produced by GI was recently observed in the disc around a FUor V960 Mon (Weber et al. 2023). Another important feature of gas and dust spatial distributions is ring-like structures at various scales. The system of rings starts to form as early as 100 kyr, and develops to the high-contrast multiple ring structure (1 - 3 orders of magnitude difference between surface densities in rings and gaps), which is evident in the middle and bottom rows of Figure 1. While gas rings are also common, the annual structures are more prominent in the dust surface density, as well as in dust size. Some of the dust rings correspond spatially to the gas rings, while others are barely reflected in the gas distribution. Overall, the difference between the gas and dust structures develops with time, as the result of dust growth and drift (Testi et al. 2014). Only some particular substructures, such as the ring between 1 - 2 au, are present in the distributions of both gas and dust components. Another location where prominent rings form in both dust and gas is the water snowline. Here, the water snowline is defined as the location in the disc midplane where the amount of water in the gas equals to its amount in the ice (on both dust populations). There are multiple location in the disc where this happens. Water is frozen in most of the disc beyond 5 - 10 au, and we will refer to the furthest snowline dividing these outer frozen region from the inner one with the gas-phase water as a primary snowline. Generally, the primary snowline is roughly circular, but it can have asymmetries due to the spiral structure and an additional snowline may appear, e.g., around a gravitationally bound clump (upper rows in Figure 1). Besides, at ≈ 260kyr, another region rich in water ice appears in the inner disc, creating a pattern of double or triple water snowline at later times (middle and lower rows in Figure 1). We will refer to these inner additional snowlines as the secondary snowlines. They circumcise a cold and dense gas-dust ring that forms at 1 - 2 au. As the disc cools down with time, the primary snowline position moves from ≈ 10au distance at 160 kyr, to ≈ 6au at 350 kyr and ≈ 4au at 490 kyr. Snowlines are known to be associated with the enhancement of dust and volatiles (Stevenson and Lunine 1988; Cuzzi and Zahnle 2004; Drążkowska and Alibert 2017; Molyarova et al. 2021). Dust enhancement was also shown to affect the distribution of gas and its accretion rate through the disc (Gárate et al. 2020) by means of dust back reaction, which is also accounted for in our modelling. In our models, an about 2 au wide dust ring is formed at the inner edge of the water snowline (at ≈ 10au) as soon as 50 kyr after the disc formation. Grown dust grains drifting towards the star through the snowline lose their mantles, their fragmentation velocity drops, rendering them more vulnerable to fragmentation. Consequently, the grain maximum size a max decreases, their drift towards the star slows down, which leads to the accumulation of grown dust, as well as small dust as a product of fragmentation. Increase of total dust density also leads to less efficient cooling and results in higher temperature inside the snowline (see the right panels in Figure 1). Later, at times > 200 kyr, several additional rings form outside the water snowline at distances up to 100 au, the most notable one being 1 - 2 au exterior to the primary snowline. Dust is trapped inside the gas pressure maxima in these rings, which is a self-supporting phenomenon as the temperature also increases inside the dust ring due to high optical depth. These rings are worthy of attention in the context of possible planet formation. Dust surface density and size are higher in the rings, with a max reaching centimetres, dust-to-gas ratio up to 0.1 - 0.2, and Stokes number up to 0.05 - 0.1. This could ease the development of the streaming instability, which typically requires pebble-size grains with St ≳ 0.01 and dustto-gas ratio ≳ 0.02 (Carrera, Johansen, and Davies 2015). Multiple ring-like structures are commonly observed in protoplanetary discs at a range of ages and display a variety of examples, with different widths, contrasts and numbers of rings (Huang, Andrews, Dullemond, et al. 2018, and many others). However, to directly compare the ring structures in the simulated dust surface density with the observed dust emission, require radiative transfer modelling is needed. Some of the observed ring structures could be produced by radial variation in dust size even in the absence of gaps in dust surface density (Akimkin et al. 2020). The most prominent dust rings are located in the vicinity of the water snowline: the ring outside the primary snowline at 5 - 8 au (depending on the time) and the ring at 1 - 2 au, inside the primary snowline, which at later times also contains water ice and additional snowlines. The main effect of the snowlines is the change in fragmentation velocity between mantled and non-mantled grains: dust size sharply decreases by ≈ 2 orders of magnitude for the latter. Immediately outside the water snowline, the midplane temperature is lower, due to lower dust opacity and thus more efficient cooling. Turbulent α is on the contrary, higher, providing more efficient radial transport of matter. It leads to lowering the gas surface density in this gap, which in turn increases α (see Eq. (5)), creating the positive feedback and further deepening the gap. One of the possible mechanisms to create the initial decrease in the gas surface density is dust back-reaction, which can affect the inward flow of gas at the snowline. This effect was investigated by Gárate et al. (2020) for different initial dust-to-gas ratios. The radial variation of α itself lead to the appearance of gas substructures (Tong, Alexander, and Rosotti 2024). The combined effect of lower temperature and density creates a pressure minimum, which dust tends to avoid. The dead zone, where α -parameter values are lower than 10 -3 , includes the ice-free inner disc and has an approximately two times larger radial span than the iceless region. The distribution of α -parameter is shown in the right column of Figure 1. Inside the primary water snowline, the values of α are the lowest due to high surface density of gas. The dead zone is not axisymmetric and reflects the spiral structures of the gas distribution, as it is sensitive to Σ g (see Eq. (5)). The spiral arms of the dead zone span to 15 - 25 au at 160 kyr, to 10 - 18 au at 350 kyr and to 10 - 14 au at 490 kyr. The comparison with the temperature distribution in Figure 1 indicates that α and T mp are anticorrelated, as higher viscosity provides faster accretion, hence lower surface densities and more efficient cooling. Similarly, a lot of dust accumulates in the dead zone, increasing opacity, which hampers cooling. The icy dust ring at 1 - 2 au is especially interesting in the context of planet formation. In this ring, dust-to-gas ratio exceeds 0.1, and surface densities of both gas and dust are increased by more than an order of magnitude compared to the adjacent regions. Due to the ice mantles that protect dust from fragmentation, the dust size reaches several cm, close to the values behind the water snowline. When the ring is established, it is self-supporting, in a sense that without external perturbation (e.g. sharp change in accretion flow from the outer disc), it can be stable for a long time, over tens of kyr. The presence of water ice inside the dead zone was investigated by Vallet et al. (2023). They showed that in the discs around lower-mass stars, the turbulent heating in the inner disc can be low enough to allow the existence of ices. In our modelling the cooling of the inner region is assisted by the inner dust ring. The freeze-out has a positive feedback on dust growth due to higher v frag , which helps dust grow larger and further accumulate toward the pressure maximum in the ring. So in our modelling ices coexist with a lot of grown dust in the dead zone. Such an icy inner region appears to be a promising place for the formation of the volatile rich planets.","pages":[6,7,8]},{"title":"3.2 Distribution of volatiles","content":"Even in the absence of chemical reactions, over the years of disc evolution the distribution of volatiles significantly changes compared to the initial one. This is the result of both phase transitions and dust growth and advection. Dust drift brings ices from the outer disc, enriching the inner disc with volatiles. Snowlines provide the conditions favouring accumulation of ices and gas-phase species. The formation of the established disc structures, such as dust and gas rings and spirals (see Section 3.1) leads to a complex pattern of intermittent snowlines. In this Section, we describe the main features of the molecular distributions and their implications for the composition of dust and gas at early stages of protoplanetary disc evolution. Figures 2 and 3 show the azimuthally averaged radial profiles of the volatile species in models M1 and M2, respectively. As mentioned earlier, we define the snowline position as the location in the disc midplane where the amount of species in the gas equals to its amount in the ice (on both dust populations). This definition allows the snowline to have complex shape, characterised by different positions for each azimuthal direction. In the azimuthally averaged distributions shown in Figures 2 and 3, the position would only be approximate. The slope of the species distribution in the vicinity of the snowline reflects the degree of the axial asymmetry in the disc. Flatter distributions appear as the contributions sum up from the snowlines, the radial position of which vary at different azimuthal angles ϕ . In our model setup, the volatiles can either have no snowline in the disc, or have two or more snowlines depending on the local conditions. Snowlines are absent for more volatile species (CO and CH 4 ) at earlier times or during bright luminosity outbursts, when the disc is too hot for them to freeze out. When there are two snowlines, the inner snowline in the disc is the one determined by thermal desorption. We refer to it as the primary snowline. The outer snowline is determined by photo-desorption, it lies in the embedding envelope outside of the body of the disc where optical depth is low. It must be noted that the gas-phase species outside this photo-snowline are vulnerable to photo-dissociation by the UV radiation. This process is not explicitly included in our chemical model, but can be assessed as in Molyarova et al. (2021). More than two snowlines appear when the disc physical structure becomes more complex, mainly due to the presence of the ring-like structures. Species can freeze inside a cold dense dust ring, creating additional secondary snowlines, also governed by thermal desorption. The concepts of primary and secondary snowline is necessary for H 2 O and CO 2 , which have multiple snowlines at the later stages of disc evolution. For water, the inner icy region appears at ≈ 1au at 490 kyr (see right column of Figure 3) inside a dense dust ring. For CO 2 , the ring that appears at 7 au, outside the primary water snowline at 4 au, creates the inverted thermal structure in the region with T mp close to the typical CO2 sublimation temperature of 70 - 90 K. Increase in T mp in these ring is caused by higher optical depth and consequently lower cooling on the viscously heated midplane. Apart from the snowlines, the distributions of volatiles in Figures 2 and 3 present local radial variations in all of the species components, including total abundance of the species. The initial total abundance is kept only in the envelope. As the matter is being redistributed, abundances of all volatiles in the inner disc grow. Particularly, the process responsible for this is dust drift. It brings the grown ice-covered grains to the inner disc regions, where their ice mantles are sublimated and no longer move with the drifting silicate dust cores. The effect is stronger for less volatile species H 2 Oand CO 2 : their abundance in the gas grows by a factor of a few inside their primary snowlines. For CO and CH 4 the effect is weaker, because there is less grown dust outside their snowlines, and those snowlines are not very much established at earlier times. Thus their abundances inside the snowline only grow by a factor of unity, except for the immediate vicinity of the snowline. Moreover, both CO and CH 4 have a bump in the gas-phase distribution just outside the dust ring at 1 au, that is absent in H 2 Oand CO 2 . At the inner disc edge the abundances of CO and CH 4 in the gas are lower than the initial value. The abundance enhancements appear most notably at the snowlines, with local bumps in all three phases at later times (after ≈ 350kyr). They are produced by the combination of the dust radial drift and the azimuthal oscillations of dust and gas radial velocity, described in more detail in Molyarova et al. (2021) and Molyarova et al., in prep. By 500 kyr, the surface density of gas-phase H 2 Oexceeds the initial value by a factor of 5, of CO 2 - by a factor of 7, of CH 4 - by 3.5 and of CO by 2.5. Similar accumulation powered by turbulent diffusion was previously studied for CO molecule in axially symmetric model setup (Stammler et al. 2017; Krijt et al. 2018). Their results indicated similar enhancement in the gas phase by a factor of a few. In our non-axisymmetric approach, diffusion is not a necessary requirement, and the necessary transport is rather provided by azimuthal variations of radial velocity induced by disc self-gravity (Molyarova et al., in prep). Distributions of ices on small and grown dust are different as they are affected by dust growth and drift. In general, there is more grown dust than small by mass, and in most of the disc there is more ices on grown dust, particularly at later times. However, in the outer disc and at the earlier times, ices on small grains dominate. As initially all ices are on small dust, it seems inevitable that they will gradually move to the grown grains as small dust coagulates and turns into grown grains. However, near all the snowlines, amount of ices on small dust increases due to the effect of the spiral pattern. Ice abundances are determined by episodic sublimation and freeze-out in the warm spiral arms of the complex density and temperature pattern (see Section 3.3.2. in Molyarova et al. 2021). As adsorption preferably happens to the smaller grains due to their larger total surface area, there is much more ices on small dust in the wake of the spiral arms. This concerns the photo-desorption snowlines as well.","pages":[10]},{"title":"3.3 C/O ratios","content":"Here we consider the relative amount of carbon and oxygen in the gas, on the surface of small and grown grains, and in total for all for all phases and disc locations. C/O ratio is seen as a perspective tracer of planet formation mechanisms (see Öberg, Murray-Clay, and Bergin 2011; Thiabaud et al. 2015). Therefore, we are especially interested in the regions of the disc and the volatile phase component where the C/O ratio declines from the total initial value (see below). Particularly, weare interested in C/O ratio noticeably above the initial value, as it was observed in atmospheres of exoplanets and suggests the formation of these planets in similarly carbon-enriched environments (Swain et al. 2009; Madhusudhan et al. 2011; Moses et al. 2013; Facchini et al. 2021). The C/O ratio is calculated as the total number of carbon atoms contained in the molecules, divided by the total number of oxygen atoms in the molecules. This calculation can be done separately for the species contained in the gas phase, species on dust surface, or for the species in all phases. Thus, we calculate the C/O ratio in the gas, in the ice, and total C/O, respectively. For the ice species, we include both ices on grown and small dust grains.In some disc regions, the amount of carbon and oxygen in in a particular phase is very low, e.g. in the ice phase inside the water snowline, where all the molecules are in the gas. For such cases, C/O ratio would not be representative of the chemical composition, so we exclude the corresponding computational cells from the consideration. We only display the C/O ratio if the mass of the volatiles in the phase is higher than 10 -3 of the total mass of volatiles at a given location. For the adopted molecular abundances based on Karin I. Öberg et al. (2011) and also used by Eistrup, Walsh, and van Dishoeck (2018), the baseline value is C/O = 0.34, which is in line with the gas-phase abundances in the ISM ( ≈ 0.36, Przybilla, Nieva, and Butler 2008). This value is different from the typical solar value of ≈ 0.5 (Przybilla, Nieva, and Butler 2008), which is commonly used as a reference (e.g., Öberg, Murray-Clay, and Bergin 2011; Semenov et al. 2018). First, local galactic abundances changed since the Solar system formation 4.6 billion years ago (see, e.g., Appendix A in Bergin et al. 2024, for the comparison). Second, the difference also stems from inclusion or non-inclusion of the dust component. The stellar atmospheric abundances comprise all the elements present in the medium where the star formed, while the elements available for the volatiles are only a fraction of that. The values of the elemental abundances can be determined separately for the volatile (gas and ices) and the refractory (rocky dust grain cores) components of the ISM (Hensley and Draine 2021). Here, we do not take into account carbon and oxygen from the rocky cores of dust grains. Due to the model assumptions, the initial ice-to-rock mass ratio in our model is quite low (0.08 compared to 2 - 4 in Pontoppidan et al. 2014). Therefore, adding the elements from solid dust grains to those contained in ices would be misleading, as the former would dominate in the resulting C/O ratio. In this work, we concentrate on considering the C/O ratios of the volatile component (ice and gas), and compare them with the initial value of 0.34. By the age of 490 yr, the C/O ratio in both models is significantly different from the initial value. This concerns the total C/O ratio, as well as the C/O ratio in the gas and in the ice. Let us analyse the distributions of C/O ratios that we can see from Figure 4, moving inwards from the envelope to the centre. The main features of the C/O distributions are the following: As expected, the key changes in the C/O ratio distribution are associated with the positions of the snowlines. There are two main processes. First, the freeze-out and desorption at the snowlines transfer the elements between phases, altering the C/O in the gas and in the ice. Second, the snowlines favour the accumulation of the respective volatiles both in the gas and in the ice, as was discussed in Section 3.2, pumping up the amount of both components and altering their proportion. Besides the snowlines, radial drift of grown grains (Weidenschilling 1977) transports the volatiles inwards. We discuss below which processes are responsible for the formation of the listed features. C/O in the outer disc. In the surrounding envelope, outside the disc, all volatiles are in the gas due to photo-desorption, and C/O in the gas is close to the initial value of 0.34. Around the COsnowline and beyond, the C/O in the gas is close to unity, and in the gas C/O is higher that the initial, which requires explanation. The region beyond the CO snowline is usually described as the place where all species are frozen, thus having a stellar C/O ratio in the solid phase and practically no carbon or oxygen in the gas (e.g., Öberg, Murray-Clay, and Bergin 2011; Öberg and Wordsworth 2019; P. Mollière et al. 2020). In our simulations, only in model M2 there is a region where less than 10 -3 of C and O is in the gas, and in model M1 such region is absent. Due to the asymmetric spiral structure that persists even at 490 kyr, even though most of CO is frozen beyond the snowline, there is a significant (> 10 -3 ) fraction of it in the gas. Additionally, the position of CO snowline itself is significantly affected by the dust drift, as it declines from the equilibrium between adsorption and desorption.The indistinctness of the CO snowline also helps CO to persist in the outer regions: all other species are ultimately frozen, so they are efficiently carried away to the inner disc via radial drift, while this works worse for only partially frozen CO. It creates relative overabundance of CO in the outer disc, elevating the total C/O ratio and later the ice-phase C/O ratio, when the preserved CO freezes out. Between the CO ice line and the envelope, there is a gradient of C/O in the gas due to photodesorption of the ices. The last molecule to be photo-desorbed is water, which returns oxygen to the gas phase at the farthest radial distance. C / O ≈ 1 around the CO snowline. Beyond the CH 4 snowline, CO dominates the composition of volatiles in the gas phase, leading to the gas-phase C/O close to unity, the value characteristic of the CO molecule. At the CO snowline, the CO dominates the ice-phase composition as well. While dust drift substantially lowered the abundances of CO 2 and H 2 Oby 490 kyr in these regions, CO ice accumulated at the snowline. This leads to the ice-phase C/O closer to 1, too, making the vicinity of the CO snowline a region where total amounts of C and O are similar. High C/O in the gas, low C/O in the ice. In the disc regions beyond the primary CO 2 snowline, only CO and CH 4 are in the gas, meaning the dominance of carbon and C/O > 1. Consequently, the C/O in the ice is generally lower, around 0.2, as the ices are mainly H 2 O and CO 2 rich in oxygen. This is consistent with the classical step-like picture of Öberg, Murray-Clay, and Bergin (2011), with the addition of carbonrich methane allowing the C/O > 1. The midplane C/O ratios we simulate are difficult to directly compare with observations, which mostly trace the molecular layer above the disc midplane. High C/O ratios in the gas are indeed observed in many protoplanetary discs (e.g. Miotello et al. 2019), but they are considered to be a natural consequence of dust settling (Krijt et al. 2018; Krijt et al. 2020). Dust inward drift could also enhance this effect in the outer disc regions, creating the radial gradient of total C/O ratio in the disc. Observations of CS and SO emission coming from close to the midplane layers potentially indicate the presence of such radial gradient of the gas-phase C/O in the PDS 70 disc (Rampinelli et al. 2024). Variations of total C/O. Throughout the disc, there are sharp changes in total C/O ratio not connected with any snowline. Most noticeable are the variations between CO 2 and CH 4 snowlines, where gas-phase and ice-phase C/O ratios are stable. These variations are associated with the disc substructures, particularly with the dense dust rings described in Section 3.1. The total C/O changes due to radial variations in dust-to-gas ratio: when it is higher, the total C/O is closer to the ice-phase C/O, and vice versa. Inside the dust-rich rings, H 2 Oand CO 2 ices are abundant, due to high surface density of dust relative to gas. At the same time, CO and CH 4 in the gas phase have similar surface densities inside dust rings and between them. Thus, in the dust rings, CO 2 and water are overabundant, leading to lower total C/O ratio. We consider this effect in more detail in Section 3.5. Variations of the total C/O ratio due to dust substructures are also present in the cold ring at ≈ 1au. C/O peaks at the snowlines. There are peaks of the C/O ratio in the ice right outside the snowlines of CO 2 , CH 4 and CO, produced by the accumulation of the respective ices (see Figures 2 and 3). In M2 model, the amount of CH 4 and CO ices at their respective snowlines becomes comparable with or even larger than those of CO 2 and H 2 O(compare the right panels of Figure 3), leading to C/O in the ice ≈ 0.6 - 0.9. In model M1, the accumulation is more prominent, so the C/O ratio in the ice approaches unity at CO snowline and > 1 at the methane snowline. These peaks distort the pattern of generally low C/O ratio in the ice and preset additional regions where carbon-rich planetesimals could be formed, and carbon-rich pebbles could be accreted onto forming protoplanets. At the snowlines of H 2 Oand CO 2 , the C/O in the ice approaches the C/O ratios of these molecules, 0 and 0.5, respectively. Lower C/O inside the water snowline. Inside the primary water snowline, the C/O ratio is generally lower than outside of it, close to the initial 0.34. Contrary to the outer regions depleted of ices due to the radial drift, the disc parts inside and around the water snowline are enriched in volatiles, and particularly of oxygen-rich water and CO 2 . Total and gas-phase C/O ratios vary from ≈ 0.2 to 0.6, as the secondary water snowlines add more substructure to the C/O distribution. The regions where the only ice is water and the ice-phase C/O = 0 are the inner cold dust ring at 1 au and the narrow annuli between water and CO 2 snowlines. The enrichment of the inner disc regions with oxygen as a result of dust radial drift is suggested by the resolved observations of molecules in protoplanetary discs (Banzatti et al. 2020). These characteristic features of the C/O distributions are similar in the two presented models. The main difference is the radial distances where the borders between the zones are located; they are closer to the star in the less massive and thus colder model M1. This is mainly due to slightly different masses of the central star accumulated throughout 490 kyr of non-identical protostellar accretion history, which lead to different luminosity and thermal structure (see upper panels of Figure 5). Particularly, the stellar masses and luminosities at this time instance are: 1.07 L ⊙ and 0.34 M ⊙ (for M1), 1.89 L ⊙ and 0.58 M ⊙ (for M2). The less massive model M1 demonstrates overall higher C/O ratio in both phases.","pages":[11,12,13]},{"title":"3.4 Evolution of the snowlines and the C/O ratios","content":"The positions of the snowlines are crucial for the values of C/O ratios, both because of the direct change through the phase transitions and the associated accumulation of volatiles. They depend on the local gas and dust properties, particularly on temperature. They can also be shifted inward due to dust drift (Piso et al. 2015). Snowlines evolve as the disc structure changes with time. In this Section, we consider the co-evolution of the snowlines and the C/O ratios and discuss the mechanisms of species redistribution over the disc. The temporal evolution of the azimuthally averaged C/O ratios and the equilibrium positions of the snowlines is shown in Figure 5. It is evident from Figure 5 that the C/O ratio indeed follows the snowlines, particularly inside ≈ 10 au. The C/O structure changes throughout the disc evolution, and some key features only appear at later times. One of the key factors affecting the disc thermal structure is the luminosity of the central source. In the upper panels of Figure 5, we show the evolution of total luminosity, which directly affects the positions of the snowlines. The luminosity is the sum of stellar and accretion components, coming from the protorstar itself and gravitational potential energy of the accreted matter. The stellar luminosity gradually decreases as the protostar becomes more compact. Accretion luminosity depends on the accretion rate, which is highly variable as a result of magnetorotational and gravitational instabilities in the disc (Kadam et al. 2019, 2020; Vorobyov, Khaibrakhmanov, et al. 2020). The simulated episodic luminosity outbursts are similar to those occurring in the observed YSOs (Connelley and Reipurth 2018), with their amplitudes of tens and hundreds L ⊙ . In the more massive model M2, the outbursts are more frequent, brighter and occur until later times. The reasons behind this difference is the massive disc being more prone to both MRI and GI, which needs to be investigated in more detail in a separate study. Snowlines of the least volatile of the considered species, H 2 Oand CO 2 , exist in the model since the earliest phases of the disc formation. Even during bright luminosity outbursts ( ∼ 100 L ⊙ ) they do not disappear, but move farther away from the star. During the first ≈ 50kyr the disc is spreading out, it is highly asymmetric and dynamic, so the snowline positions oscillate. Later on, the disc generally cools down, and the snowlines gradually move toward the star (except during the outbursts). In model M2, between 130 and 500 kyr, the primary water snowline moves from 12 to 4.5 au; the primary CO 2 snowline moves from 23 to 7 au. In model M1, the water snowline moves from 9 to 4 au, and the CO 2 snowline moves from 17 to 4.5 au. Despite a factor of two different binding energies of H 2 O and CO 2 (5770 and 2360 K, respectively), the locations of their primary snowlines do not differ much due to the steep radial temperature gradient around these distances (see upper panels of Figures 3 and 2). Inside 10 - 20 au, T mp is determined by heating mechanisms other than external irradiation: viscous heating, gas work (PdV heating), heating by shocks and energy transport with advection. This also means that the water snowline is less sensitive to the level of irradiative heating, thus only slightly affected by the luminosity outbursts. The temperature change is particularly sharp at the water snowline(s), with the absolute value of the approximated power-law slope of 1.5 - 6. High surface density of dust in water-ice-free regions makes cooling less efficient and leads to locking up the produced heat and consequently higher temperatures. Besides, both these species have multiple snowlines due to the formation of ring-like substructures with the conditions close to the borderline between their frozen and gaseous state. These additional snowlines also affect the C/O distributions. Methane and CO are the more volatile species in our model. They either have zero or two snowlines in the disc. The snowlines are absent during the outbursts brighter than ≈ 100 L ⊙ for methane and ≈ 200 L ⊙ for CO. Until approximately 200 - 250 kyr, there is no established CO snowline inside the disc. For example, at 160 kyr (see lower left panel in Figures 3 and 2) there is CO ice on both small and grown dust, but their amount is an order of magnitude lower than that of the gas. Additionally, most of this ice is located at the outer disc edge, where gas and dust surface densities sharply drop. Similar distribution appears for CO and CH 4 during bright outbursts: their ices are present in some disc regions, but due to non-axisymmetric disc structure, they do not dominate in the averaged profiles, and there is no common snowline for the whole disc. There are no secondary snowlines of CO or CH 4 , because there is no prominent gas and dust structures in the outer disc where these species are frozen. Snowlines divide the disc into several zones with different characteristic C/O ratios. However, the chemical composition and C/O ratios in these zones change with time. One of the distinct zones is the region where water is not frozen, shaded with purple in Figure 4 and circumscribed by the dark purple dotted line in Figure 5. In this zone, all the species are in the gas phase, so the C/O ratio is initially close to 0.34. However, as the disc evolves, dust brings more volatiles from the outer disc. Particularly, the abundance of water grows most significantly, making C/O in the gas decrease with time. This happens because the main mechanism of the transport of the volatiles is dust radial drift, which works best in the regions where grown grains can sustain their mantles. Banzatti et al. (2020) suggest that dust growth and drift can be responsible for the observed anticorrelation between disc radius and H 2 O emission, implying that the inner regions of small discs with are enriched in water brought by efficiently drifting grains. Water is the least volatile species in our model, it is frozen in the largest part of the disc, thus its distribution is most strongly affected by the dust drift. At later times, this zone is divided into two, when a cold dense dust ring forms at 1 - 2 au, which happens at ≈ 460 kyr in model M2 and at ≈ 270 kyr in model M1. Interior to the ring, water abundance in the gas is around an order of magnitude lower than outside of it in both models. This happens because the inward flow of gasphase H 2 Ois 'blocked' by the cold ring where it freezes and accumulates with the grown grains in the pressure maximum. So the C/O ratio in the gas at r ⪅ 2au is determined by CO 2 and thus close to 0.5. The C/O ratio in the ice is not defined in the envelope, where all ices are photo-desorbed, and in the warmest inner disc, where even water is in the gas. In disc regions where only water is frozen, the C/O ratio in the ice is zero. Before 200 - 250 kyr, when the disc cools down enough for CO to freeze out, the C/O ratio in the ice in the rest of the disc is close to 0.2. It is determined by CO 2 and H 2 O ices, with a small contribution from the low-abundance CH 4 . After that, when CO freezes out and CO snowline appears, the C/O ratio in the outer disc region becomes close to the initial value both in the ice and in total. This region beyond the CO snowline is frequently referred to in relation to giant exoplanets with stellar C/O ratios (e.g., P. Mollière et al. 2020; Öberg and Wordsworth 2019; Ohno and Ueda 2021). This region would be a perfect location for planets to accrete pebbles covered with icy mantles with the primordial elemental ratio, which would directly become part of the planetary atmosphere. However, pebbles are not initially present in the disc, and their existence in these outer regions is not guaranteed. Pebbles are dust grains large enough to move relative to gas e.g., Lambrechts and Johansen 2012; Lenz, Klahr, and Birnstiel 2019, and a fraction of the grown dust in our modelling can be classified as pebbles. The properties of pebbles and composition of their ice mantles in the same setting of the FEOSAD model were studied by Topchieva et al. (2024). They show that pebbles appear in the disc as early as 50 kyr after its formation, and exist in a wide region of the disc. This partially includes the region beyond the CO snowline, but only the area of CO enhancement, where CO dominates in the ice composition and thus the C/O ratio in the ice is close to unity (see lower panels in their Figure 3). As shown by Topchieva et al. (2024), ices on pebbles are dominated by H 2 O and CO 2 . In this case, relatively high values of the C/O in the ice ( ≈ 0.5) correspond to the regions where there is more CO 2 , i.e. around the CO 2 snowlines. This is also the region where a prominent dust ring forms under the influence of the primary water snowline. It is characterised by accumulation of CO 2 and to lesser degree H 2 Oice, as well as vapours, and relatively high amount of grown dust in the ring. The dust ring is situated between the two regions with C/O ≈ 0.34 in the ice. It presents another favourable location for accreting the ice content with close-to-initial C/O ratio. The total C/O ratio in the disc also changed significantly from the initial value due to dust drift that redistributes the ices. The matter becomes more carbon-rich as the grains bring CO and CH 4 from the outer disc parts. This effect was previously investigated by Stammler et al. (2017) and Krijt et al. (2018) in their modelling of CO dynamics and dust evolution. However, as was shown by Krijt et al. (2018) and Krijt et al. (2020), vertical settling of grown grains towards the midplane is responsible for depleting the upper layers in the outer disc of gas-phase oxygen, which cannot be captured within our thin-disc modelling. The panels in the second row of Figure 5 demonstrate strong enhancement of total C/O ratio in the intermediate disc regions. In model M2, the total C/O ratio between 10 - 100 au becomes ≈ 0.7 after ∼ 300 kyr, which is two times higher than the initial value. At the snowlines of CH 4 and CO 2 it approaches unity, mostly due to the accumulation of these species in the gas. In model M1, this process begins ≈ 200kyr earlier and consequently leads to even higher total C/O ratio. At 400 - 500 kyr, most of the disc between 5 and 100 au has total C/O ≳ 1 in model M2, which demonstrates the powerful impact of dust drift. A distinctive feature of the produced C/O distributions is the peaks of total and ice-phase C/O ratios around the CO and CH 4 snowlines. In model M2, the increase of the C/O ratios becomes noticeable only after ≈ 450kyr, while in model M1 it starts to form around ≈ 300 kyr. Initial abundances of CH 4 and CO are lower than that of CO 2 and H 2 O. As these species accumulate at their snowlines, the abundances become comparable, leading to C/O in the ice ≈ 0.6 - 0.8 in model M2 and up to 1 in model M1. Similar accumulation is seen around CO 2 snowline, but unlike CO and CH 4 snowlines, it is also connected to the interaction with the dust ring structure and is considered in more detail in Section 3.5.","pages":[13,15]},{"title":"3.5 Two-dimentional structure","content":"The disc is not axisymmetric even at later stages of its evolution (see Section 3.1 and Figure 1), which is also reflected in the distributions of volatiles and C/O ratios. Examples of 2D distributions of C/O in model M1 at two time instances are shown in upper panels of Figure 6. The left-hand side group of panels shows the structure that is characteristic of the earlier phases, the same time instance as the upper row in Figure 1. The prominent spiral structure and a clump both have their reflection in the C/O ratios. The snowlines of CO 2 and CH 4 have clearly non-regular shape affected by the spiral pattern of the gas, with blobs of frozen methane inside the main snowline. Inside the clump, the temperature is higher (see Figure 1), and both CO 2 and water are in the gas. The separation between gas-phase and ice-phase C/O ratios is clear, but the total C/O ratio in the clump is similar to the surroundings and is only slightly above the primordial value. The gas-phase C/O ratio inside the clump is around 0.7, lower than in the surrounding gas at the same radial distances. Similar decrease of C/O ratio indicated by lowered CS/CO ratio was observed in the disc around DR Tau (Huang et al. 2024). Lifetime of the clump (several orbital periods) is too short for significant differentiation between gas- and ice-phase composition to develop, mainly because of insufficient numerical resolution. Focused studies with higher resolution are needed to explore the C/O ratio in the clumps as precursor of giant planets formed via disc fragmentation. The upper right-hand side group of panels in Figure 6 features a later stage of the disc evolution in model M1. By 300kyr, the accretion rate and the average positions of the snowlines are stabilised (see Figure 5). The spiral structure in the gas is much weaker but still present at r > 10 au. The CO snowline is established at ≈ 80au. The 2D shape of the snowlines is more circular at 300 kyr, particularly for the less volatile CO 2 and H 2 O. The effect of spirals is still evident in the contours of the CO and CH 4 snowlines. At the outer side of these snowlines, at approximately 40 and 80 au, respectively, the C/O in the gas has local maxima, which are also seen in Figure 4. The one at 40 au also has a clearly spiral shape, repeating the pattern of the gas distribution. The radial span of these peaks is of the same scale as the dispersion of the respective snowline distance from the star. For example, at the CH 4 snowline the C/O peak is ≈ 10au wide, and the snowline is at 28 to 37 au distances. Similar accumulation powered by diffusion of water vapour was shown, e.g., by Drążkowska and Alibert (2017). In addition to diffusion, in our modelling, the species are delivered outward from the snowline due to the dynamic shape of the snowline and two-dimensional movement of gas and dust (see Molyarova et al. 2021, and Molyarova et al., in prep). In more massive model M2, the shape of the snowlines remains complex for longer times. Examples of C/O distributions in M2 are shown in the lower panels of Figure 6. The radial scale is chosen so that CO 2 and H 2 O snowlines are seen in more detail. At 250 kyr, even the grown dust distribution still has spiral pattern, and CO 2 snowline is following its complex multi-armed shape. The accumulation of CO 2 at the snowline is also very efficient, but it does not strongly affect the C/O ratios, as it drives them to the value 0.5 of the CO 2 molecule, which is close to the intrinsic value. The complex-shaped region between these snowlines has the ice-phase C/O ≈ 0.7. This pattern moves and changes its shape, thus affecting all radial distances between 10 and 20 au. The total C/O ratio is slightly above the ambient value, approaching 0.5, because CO 2 in both phases begins to accumulate in this region. At later times, the C/O distribution becomes more complex due to the presence of disc substructures, particularly the dust rings. The lower right-hand side group of panels in Figure 6 show this in more details. First, the temperature and density variations across the dust rings lead to the formation of multiple CO 2 and H 2 Osnowlines. An additional annulus of icy CO 2 appears in a relatively cold region between the dense dust rings at 6 - 7 and 8 - 13 au. The dust rings are warmer due to higher optical depth and active heating by disk internal sources, and the inner edge of the 6 au ring is warm enough to sustain gas-phase CO 2 . Second, the accumulation of icy dust grains in the rings alters the total C/O ratio. The total C/O ratio anticorrelates with the distribution of grown dust grains: inside the dense rings it is close to the initial value of 0.34, while between the rings, it is higher and reaches 0.8 - 0.9. At the same time, neither ice-phase, nor gas-phase C/O ratio displays any noticeable variations at 12-25 au, but they do have variations at r < 12 au, following the dust ring pattern. The total C/O ratio is defined by the combination of the ice- and the gas-phase component. Their relative contribution is proportional to the dust-to-gas mass ratio. This is illustrated in Figure 7. Within the rings, the total C/O is dominated by ices of oxygen-rich species CO 2 and H 2 O, particularly on grown dust accumulated in the pressure maxima. For example, there is a lot of water ice at 6 - 7 au, and its contribution to the total C/O is weighted with high dust-to-gas ratio of almost 0.1, so the resulting total C/O is lower than the initial value. At the same time, at ≈ 5au, dust-to-gas ratio is around 10 -4 . The dominant species there are in the gas, ice surface densities are 1 - 2 orders of magnitude lower, so the total C/O ratio goes up. In the wide dust ring at 9-13 au, both H 2 Oand CO 2 contribute, although at the warmer inner edge of the dust ring CO 2 is sublimated. The value of total C/O is approaches 0.34, also elevated by the presence of gas-phase CO and CH 4 . Beyond ≈ 10au, both H 2 O and CO 2 are frozen, and the variations of the total C/O ratio are clearly anticorrelated with the dust-to-gas ratio and the position of the rings. Between the dust rings, where contribution of ices is low, the C/O is determined by CO and CH 4 gases, and reaches values of ≈ 1. The anticorrelation between the total C/O ratio and dustto-gas mass ratio is an interesting finding. It is illustrated in Figure 8 for both models. Only the points within the disc are shown, with Σ gas > 0.1 g cm -2 .The pattern obviously changes with time, but the anticorrelation persists. Model M1 demonstrates wider variety of C/O and dust-to-gas ratios. The disc points are grouped in tangled curved \"branches\", some of them steeper than others. The 'width' (or spread) of these branches is determined by the azimuthal substructures. In the axially symmetric parts of the disc the values of dust-to-gas ratio and the total C/O are similar at a given radial distance. Different branches, which can be closer to vertical or horizontal orientation, are the result of the radial variations in the ice-phase C/O ratio and in ice fraction relative to the dust silicate cores. Horizontal branches correspond to weak or absent anticorrelation. For example, near the snowlines of carbon-rich species, C/O in the ice is high and close to that of the gas, which decreases the effect of dust mass fraction on the total C/O. In areas with no ices, the anticorrelation is also irrelevant, because the total C/O is determined entirely by the gas phase. Vertical branches, on the contrary, correspond to the strongest anticorrelation effect, which is expected in the regions between the snowlines, where ice-phase C/O is the lowest. The identified anticorrelation in Figure 8 is similar to the results of chemical population synthesis modelling (Cridland et al. 2019, 2020) showing that the more solids a planet accreted in the disc, the lower the C/O ratio is in its atmosphere. We fit the data for t = 490kyr with a linear law (taking the logarithm of dust-to-gas ratio) and obtain the following fits: C/O = -0.056 - 0.25 log 10 ( ξ ) for model M1, and C/O = -0.18 - 0.27 log 10 ( ξ ) for model M2. Here, ξ is the dust-to-gas mass ratio. The correlation coefficients are -0.54 for M1 and -0.57 for M2. At the shown earlier times (160 and 350 kyr), the correlation coefficient changes between approximately -0.9 and -0.5, which indicates noticeable anticorrelation throughout the disc evolution. We note that these C/O ratios only include the volatile component, without the contribution from the refractory material. Although the refractory material is typically considered as silicates, which are rich oxygen, it contains a significant amount of carbon, with the resulting C/O ≈ 0.5 (see Table 2 in Hensley and Draine 2021). This solid carbon can be subject to carbon grain destruction (Lee, Bergin, and Nomura 2010; Gail and Trieloff 2017; Wei et al. 2019), but this process should be treated separately, as it also affects the gas-phase carbon abundance. Taking refractory cores into account should affect the dependence between total C/O ratio and dust-to-gas ratio, as adding more rock would make the C/O ratio closer to 0.5. Thus the degree of the anticorrelation must be affected by the composition of rocky cores, but the anticorrelation itself should remain even when refractories are included, because the C/O ≈ 0.5 is still lower than typical C/O of the gas in most of the disc ( ⪆ 1). Dust rings are detected in the majority of the observed protoplanetary discs (Long et al. 2018; Huang, Andrews, Dullemond, et al. 2018). They are considered as a plausible sites of planet formation (Carrera, Johansen, and Davies 2015; Yang, Johansen, and Carrera 2017; Li and Youdin 2021; Lee, Fuentes, and Hopkins 2022; Jiang and Ormel 2023). The anticorrelation between the total C/O ratio of the volatiles and dust-to-gas mass ratio that we point out is a logical consequence of ices having typically lower C/O ratios and being attached to dust grains. If planets are formed in the dust rings with high dustto-gas ratios (> 10 -2 ), either exclusively from solids, or with the inclusion of the dust component, this would imply that their material initially has lower C/O ratio of ≈ 0.5 and below. To reach higher C/O ratios up to unity and above, which are observed in many exoplanets, these planets would need to migrate and accrete carbon-rich gas from regions other than their immediate formation sites inside the dust rings. In case if planet formation occurs independently of the dust rings, e.g. in the GI, their material is not determined by this anticorrelation.","pages":[15,17,18]},{"title":"4. DISCUSSION","content":"Our simulations present a wide range of C/O ratios in the disc in different phases evolving with time. For the atmospheres of giant exoplanets, a variety of C/O ratios were retrieved, too. Here we can compare them to identify the disc regions and times where the chemical and physical conditions for planet formation are consistent. Most of the exoplanets for which the atmospheric composition was retrieved have super-stellar C/O ratios (Hoch et al. 2023; Weiner Mansfield et al. 2024), which draws more attention to carbon-enriched areas. They are suggested to form by core accretion, and accreting mostly the gas, which is typically more carbon rich (beyond water snowline). A lot of planets are observed to have stellar or slightly super-stellar C/O (e.g., P. Mollière et al. 2020; Zhang et al. 2021; Smith et al. 2024; Sing et al. 2024; Nortmann et al. 2024, and many others). One way to form such planets is gravitational instability, which includes solids and gas together, thus undifferentiated matter is suitable for producing planets with stellar C/O. Disc fragmentation to clumps due to GI requires particular conditions (Meru and Bate 2010; Vorobyov 2013), and the direct collapse of gravitationally unstable clumps tends to produce rather massive objects (e.g. ≈ 5 M J planets and ≈ 60 - 70 M J brown dwarfs, see Figure 4 in Vorobyov, Zakhozhay, and Dunham 2013) at larger radial distances (> 10 - 100 au, see Vorobyov, Zakhozhay, and Dunham 2013; Kratter and Lodato 2016). GI can also assist the assemblage of planetary cores (Nayakshin 2010a, 2010b; Nayakshin, Helled, and Boley 2014; Vorobyov and Elbakyan 2019). A planet formed through core accretion can also accrete planetesimals, which can be covered with ice, and enrich the atmosphere with oxygen, making the C/O ratio close to the initial stellar value. There are particular exoplanets, where lower than stellar C/O ratio is observed in the atmosphere, such as β Pic b (GRAVITY Collaboration et al. 2020; Worthen et al. 2024), HD 209458 b (Xue et al. 2024), or HD 189733b (Fu et al. 2024). Such planets need even more enrichment in ices with low C/O, which makes the regions with low C/O in the ice also more attractive sites for planet formation. Gravitational instability implies that the planet forms from a mix of gas and dust (Bodenheimer 1974), this is why it is suitable to explain the formation of planets with solar, or unaltered C/O ratios. In our modelling, GI would be associated with the total C/O ratio, which we find to be significantly variable, too. For GI to form a planet with a primordial C/O ratio, it has to occur during the first 100 kyr after the disc formation. At later times, the total C/O ratio changes, and the only region with the primordial C/O ratio is the very outer disc parts, at > 100 au, which is the part of a protoplanetary disc, where conditions for GI are the most consistent with the observed properties of these objects (Rafikov 2005). Planet formation through GI is indeed more likely at earlier evolutionary stages, when gas surface density is higher (Armitage 2010). We can highlight the areas where planet formation via GI is possible in our modelling as the regions where Q Toomre ≤ 1. They are shown in the upper panel of Figure 9 for model M1. These regions appear between ≈ 10-100au before ≈ 300 kyr. However, at later times, clumps could appear in the disc as a result of an external perturbation, such as a stellar flyby (Thies et al. 2010). In this case, the planet would be formed from the material with altered C/O ratio, most probably with elevated amount of carbon, as the regions outside 5 - 10 au are typically more gravitationally unstable. This means that GI can produce planets with super-solar C/O ratios, if it is induced by external influence at later stages of disc evolution. Core accretion is another most widely discussed scenario of giant planet formation. Accretion of gas should produce atmospheres with the C/O ratio close to the one in the gas phase of protoplanetary disc. However, dust grains are also accreted, so pebble and planetesimal accretion can enrich the atmosphere in volatile components (Mordasini et al. 2016; Danti, Bitsch, and Mah 2023). This makes atmospheric C/O ratio closer to the ice-phase C/O, but in case of gas giants, the amount of the solids needed to compensate the prevalence of carbon in the gas should be quite high, up to hundreds of Earth masses (GRAVITY Collaboration et al. 2020). The C/O ratios in exoplanetary atmospheres are often interpreted in terms of pebble accretion, so planets with stellar C/O ratios are assumed to form in the environments where solid phase C/O is unprocessed and thus close to the initial value. One of such locations is beyond CO snowline, where most of the carbon- and oxygen- bearing material is in the ice, e.g. for Jupiter (e.g. Öberg and Wordsworth 2019; Ohno and Ueda 2021). In our models, this is rather the vicinity of the CO 2 snowlines, and the pebbles beyond the CO snowline are mostly covered with carbon-rich CO ice (Topchieva et al. 2024). Additionally, the CO snowline is typically very far from the star (> 40 au), so such scenarios must rely on planet migration to obtain their current location. Interpretations relying on chemical modelling extend this region to include the area beyond CO 2 snowline due to additional chemical processing of CO in this region, e.g. HR 8799e (P. Mollière et al. 2020). This puts milder constraints on the original distance from the star where pebbles should be accreted and requires less migration. Modelling of planet formation and migration including pebble and gas accretion puts the formation location of planets with super-solar C/O ratios beyond water and CO 2 snowlines (Bitsch, Schneider, and Kreidberg 2022). In our modelling, the C/O in the ice is close to initial value in the regions beyond CO snowline, excluding the area of CO accumulation. Between CO and CO 2 snowlines, it is lower, as our model does not include chemical processes apart from adsorption and desorption. However, there is another region with C/O in the ice close to initial. The vicinity of primary water snowline and the ring induced outside of it has values of C/O in the ice only slightly above the initial value of 0.34. It is surrounded by the snowlines of CO 2 . This region could be another favourable location for forming planets with the stellar C/O. As it is situated closer to the star, it would imply less migration. Rare planets with lower than stellar C/O ratio, such as β Pic b (GRAVITY Collaboration et al. 2020; Reggiani et al. 2024), HD 209458 b (Xue et al. 2024), HD 189733b (Fu et al. 2024), or KELT-1 b, Kepler-13A b and WASP-79 b (less precisely determined, see Hoch et al. 2023), need to have accreted a lot of oxygen-rich ice. Therefore, they are more likely to accrete solid material in the regions with the lowest ice-phase C/O ratios. The most suitable region would be at the distances between H 2 O and CO 2 snowlines, where ice mantles are made of pure water. However, in our modelling results, this region is very small, typically only a few au wide, as the snowlines are close to each other. This is because of steep temperature profile in this region, which is a result of the significant contribution of non-irradiation heating mechanisms, particularly viscous heating. Beyond CO 2 snowline, there are also regions with relatively low (0.2 - 0.3) C/O in the ice, but much more solids need to be accreted in such areas to compensate for the excess of carbon from the gas. Let us summarise the above constraints on planet formation locations and mechanisms implied by our simulated C/O ratios. Core accretion is suitable for forming planets with high C/O ( ≈ 1) in the atmosphere around the snowlines of CO, CH 4 and CO 2 , or anywhere beyond CO 2 snowline if they did not accrete much solids. Planets with stellar or slightly superstellar C/O ratio need to accrete (a lot of ) oxygen-rich solids to compensate their initially high C/O inherited from the gas. The locations where this is possible is between CO 2 and CH 4 and between CO and CH 4 snowlines. Planets with low C/O ratio could accrete ices between H 2 O and CO 2 snowlines. Snowlines are favourable planetesimal formation sites, so the planets that accreted planetesimals/pebbles there can have altered C/O ratios. The values will be lower than the initial if they form at the water snowline, and higher if they form at the snowlines of carbon-rich species. At the same time, to obtain planets with stellar C/O formed at the snowlines, these planets would need to migrate and accrete matter in different regions of the disc to make their C/O ratio close to the initial stellar value. Alternatively, planets with stellar C/O ratio can form via disc fragmentation through GI at earlier stages. Dedicated modelling of planet formation accounting for evolution of dust and volatiles is necessary to put more particular constraints on planet formation scenarios. Planetesimals play an important role in delivering the icephase elements to planetary atmospheres. To form planetesimals, additional physical process is needed, such as streaming instability (SI, see Youdin and Goodman 2005), which is not explicitly included in our modelling because of insufficient numerical resolution and simplified vertical disc structure. However, we can post-process the simulation results to check if the conditions for SI are fulfilled in some regions of the disc where dust-to-gas ratio and dust size are enhanced, following Vorobyov et al. (2024). Dense dust rings forming at later stages (see Section 3.1) seem to be an ideal location for triggering the SI, which would ultimately lead to formation of planetesimals and then planets in the disc. Triggering the SI requires specific relations between local dust-to-gas ratio and Stokes number (Yang, Johansen, and Carrera 2017). The criteria vary depending on the model, we adopt them from Li and Youdin (2021). Another criteria would be the requirement of volume density of dust to exceed that of gas in the midplane (Youdin and Goodman 2005). We do not apply it, as in our modelling, the residual value of α is 10 -3 which makes this condition unreachable outside of the dead zone. The regions in model M1 where the conditions of Li and Youdin (2021) are satisfied are shown in the lower panel of Figure 9. Most of the suitable regions are in the inner disc ( r < 20 au) inside the dust rings, and appear after 200 kyr. However, there are some suitable regions between 10 - 100 au at earlier times, where SI could be triggered in the spirals. The regions where GI and SI are possible shown in Figure 9 are separated in space and time, and they have different characteristic C/O ratios. We can sum up all the volatiles in these regions (throughout the disc lifetime) to assess typical C/O ratios of the planet-forming material. For GI, we exclude the pre-disc phase ( t < 53 kyr) and consider the total C/O ratio, assuming both gas and solids are included in the forming planet. For SI, we separate the gas- and ice-phase C/O ratios. The formed planetesimals would only include the ices, however, if they form the planetary cores, these cores would also accrete gas. We note that the composition of the rocks, which are typically carbon-rich, is not included in our assessment. The resulting distributions of the C/O ratios in planet-forming regions are shown in Figure 10. For GI regions, the C/O distribution has a distinct and relatively narrow peak around 0.5. It is slightly higher than the initial value of 0.34. For SI regions, the ice-phase C/O is below 0.5, with major peaks at 0 and ≈ 0.2, and the gas-phase C/O has a broad distribution with multiple peaks between ≈ 0.2 - 1.4. Distributions of C/O ratios in the regions where GI and SI can be triggered are noticeably different. It was shown by Hoch et al. (2023) that there are two different populations of C/O ratios observed in giant exoplanets. They find that directly imaged exoplanets have C/O ≈ 0.5-0.8, while transiting hot Jupiters have a wider variety of C/O ratios ( ≈ 0.3-1.7, see Figures 12 and 13 in Hoch et al. 2023), and suggest that these two populations could have different formation pathways. We add the C/O data of exoplanetary atmospheres compiled in Table 3 of Hoch et al. (2023) to Figure 10 (with arbitrary position at the y -axis). The narrow distribution of C/O ratios in the regions with GI is in step with the distribution of directly imaged exoplanets, albeit with a slightly shifted value due to our assumed initial conditions, while the wide range of C/O values in the regions of SI matches the variety of C/O ratios in transiting exoplanets. This could suggest that directly imaged exoplanets could form as a result of gravitational instability, which is also in line with their typically higher masses and orbital separations. At the same time, the transiting hot Jupiters could have experienced a lot of migration (Lin, Bodenheimer, and Richardson 1996; Dawson and Johnson 2018), during which they accrete material with a variety of C/O ratios both from the gas and solid phase. It suggests that they could also form in the core accretion scenario. The origin of wide separation planets was also investigated by Bergin et al. (2024), based on the comparison with the observed C/O > 1 in protoplanetary discs (including the full composition of solids). They conclude that both core accretion and gravitational instability can work as the formation mechanism of these planets. Apart from (exo)planets, the C/O ratios can be measured for the comets, which present the best preserved sample of the primordial composition of the ices in the Solar System. Spectroscopic measurements of molecular composition in the comae suggest that the C/O ratio of cometary ice is quite low, typically below 0.1 due to the dominance of water ice (A'Hearn et al. 2012; Seligman et al. 2022; Harrington Pinto et al. 2022). Although most comets are carbon-depleted, there are individual measurements of C/O in comets above 0.5, for example in C/2006 W3 Christensen and 29P/SchwassmannWachmann (Ootsubo et al. 2012; Seligman et al. 2022), or even close to 1 in C/2016 R2 (PanSTARRS) (Wierzchos and Womack 2018; McKay et al. 2019). Additionally, high value of C/O ≈ 1 was observed in the interstellar object 2I/Borisov (Bodewits et al. 2020). Our modelling results show the icephase C/O = 0 in the vicinity of the water snowline, as well as low values between the CO 2 and CH 4 snowlines ( ≈ 0.2) and between the CH 4 and CO snowlines ( ≈ 0.3). These are the locations where comets could originate from. However, in the vicinity of the CO 2 , CH 4 and CO ice lines themselves, the C/O ratio in the ice phase is much higher. The fact that carbon-rich cometary ices are extremely rare in the Solar System may indicate that planetesimals formed on ice lines from carbon-rich volatiles do not persist throughout the evolution of a planetary system. This means that they are likely to be included in larger bodies, which favours the snowline-aided planet formation scenarios (Drążkowska and Alibert 2017; Hyodo et al. 2021). This is also consistent with the abundance of exoplanets with high C/O (Weiner Mansfield et al. 2024), which could be formed around the snowlines of carbon-rich species. In the giant planets of the Solar System, the C/O ratios are not well constrained (Mousis, Cavalié, et al. 2024). However, the existing data suggest rather super-solar values for all giant planets except for Neptune (Cavalié et al. 2024); for Jupiter, the C/O ratio is assessed as ≈ 0.9 (Wong et al. 2004; Li et al. 2024). Our model only considers four most abundant chemical species. However, there can be other more complex molecules in protoplanetary discs, which could affect the balance of carbon and oxygen. The most obvious candidate is methanol CH 3 OH, which has the abundance similar to methane in the protostellar cores (Karin I. Öberg et al. 2011). It was also observed in a protoplanetary disc around an erupting star V883 Ori (Lee et al. 2019). We do not consider it in the model as its binding energy is close to that of water, thus the snowlines would have similar positions, but the abundance is an order of magnitude lower. However, it could somewhat increase the local C/O ratio in the inner regions where there are no other carbon-bearing species, such as the ice in the ring at 1 au. Including methanol would alter the distribution of the C/O ratio. Interactions between the ices considered in the model could also affect the results. As was recently shown by Ligterink, Kipfer, and Gavino (2024), trapping of volatile species inside the mantles of less volatile ices could have a significant impact on the distribution of C/O ratios. Another important process missing in our modelling is gas-phase and surface chemical reactions. They could significantly affect the distribution of C/O ratio in the gas and in the ice, particularly with high level of cosmic ray ionisation (Eistrup, Walsh, and van Dishoeck 2016) or if carbon grain destruction is considered (Cridland, Eistrup, and van Dishoeck 2019). One particular mechanism is the transformation of CO to CO 2 on the surface of dust grains, which can lead to the depletion of CO from both gas and ice phases between CO and CO 2 snowlines (Molyarova et al. 2017; Bosman, Tielens, and van Dishoeck 2018). Considering this mechanism can change the conclusions about planet formation location (P. Mollière et al. 2020). Nevertheless, radial variations of the C/O ratio are necessary to explain molecular emission of discs with gaps (Leemker et al. 2024), and they can only be result of dust dynamics. In order to more consistently describe the distribution of molecules and elements in the disc, the models combining dust evolution and dynamics with more complex chemistry treatment are necessary. Our simulations adopt the thin-disc approximation and focus on the midplane of the protoplanetary discs, in order to capture the essential physics of self-gravity, thermal balance, and dust evolution in a global modelling within reasonable computational times. This means that some relevant processes connected with the vertical structure are inevitably excluded. For example, vertical mixing and dust settling affect the C/O ratio in the upper layers of the disc (Krijt et al. 2018; Krijt et al. 2020). Dust settling is implicitly included in our modelling through separate scale heights of drown dust and gas (as well as small dust), affecting dust number density in the midplane. However, this approach does not allow to reproduce vertical stratification in dust properties and chemical composition, which is particularly relevant for the interpretation of molecular observations. Vertical structure is also relevant for the accretion of matter on forming giant planets, which should proceed in 3D manner through meridional flows (Morbidelli et al. 2014). Cridland, Bosman, and van Dishoeck (2020) showed that the C/O ratio in the atmospheres of giant planets is rather affected by the composition of the molecular layer than that of the midplane.","pages":[19,20,21,22]},{"title":"5. Conclusions","content":"In this work, we studied the distribution of volatiles in a viscous self-gravitating protoplanetary disc with dust evolution using a thin-disc hydrodynamic code FEOSAD (Vorobyov et al. 2018; Molyarova et al. 2021). We calculated the C/O elemental ratio in the gas, in the ice, and in total, identified the key properties of the distribution of elements over 500 kyr of disc evolution and considered their implications for planet formation theory. Our main findings can be summarised as follows. The connection between protoplanetary disc components and exoplanets based on their composition should be more thoroughly investigated in the models focused on the planet formation process. We emphasise that these models should also take into account the effect of dust evolution and dynamics on the distribution of the elements in the planet-forming material. Inclusion of chemical processes and more accurate consideration of the bulk composition of dust grains could also affect the C/O ratios of the planet-forming environment.","pages":[22]},{"title":"Acknowledgement","content":"Weare thankful to the anonymous referee for useful comments that helped to improve the manuscript. The computational results presented have been achieved using the Vienna Scientific Cluster (VSC) and the local computing facility of the Southern Federal University. Funding Statement The work is supported by Russian Science Foundation grant 22-72-10029, https://rscf.ru/project/2272-10029/ Competing Interests None Data Availability Statement The data underlying this article will be shared on reasonable request to the corresponding author.","pages":[22,23]},{"title":"References","content":"A'Hearn, Michael F., Lori M. Feaga, H. Uwe Keller, Hideyo Kawakita, Donald L. Hampton, Jochen Kissel, Kenneth P. Klaasen, et al. 2012. Cometary Volatiles and the Origin of Comets. ApJ 758, no. 1 (October): 29. https: //doi.org/10.1088/0004-637X/758/1/29. Aikawa, Yuri, Shoken M. Miyama, Takenori Nakano, and Toyoharu Umebayashi. 1996. Evolution of Molecular Abundance in Gaseous Disks around Young Stars: Depletion of CO Molecules. ApJ 467 (August): 684. https://doi.org/10.1086/177644. Akimkin, Vitaly, Eduard Vorobyov, Yaroslav Pavlyuchenkov, and Olga Stoyanovskaya. 2020. Gravitoviscous protoplanetary discs with a dust component - IV. Disc outer edges, spectral indices, and opacity gaps. MNRAS 499, no. 4 (December): 5578-5597. https://doi.org/10.1093/mnras/ staa3134. arXiv: 2010.06566 [astro-ph.EP] . Armitage, Philip J. 2010. Astrophysics of Planet Formation. Armitage, Philip J., Mario Livio, and J. E. Pringle. 2001. Episodic accretion in magnetically layered protoplanetary discs. MNRAS 324, no. 3 (June): 705-711. https://doi.org/10.1046/j.1365-8711.2001.04356.x. arXiv: astro-ph/0101253 [astro-ph] . Audard, M., P. Ábrahám, M. M. Dunham, J. D. Green, N. Grosso, K. Hamaguchi, J. H. Kastner, et al. 2014. Episodic Accretion in Young Stars. In Protostars and planets vi, edited by Henrik Beuther, Ralf S. Klessen, Cornelis P. Dullemond, and Thomas Henning, 387-410. January. https: / / doi . org / 10 . 2458 / azu _ uapress _ 9780816531240 - ch017. arXiv: 1401.3368 [astro-ph.SR] . Bai, Xue-Ning, and James M. Stone. 2013. Wind-driven Accretion in Protoplanetary Disks. I. Suppression of the Magnetorotational Instability and Launching of the Magnetocentrifugal Wind. ApJ 769, no. 1 (May): 76. https://doi.org/10.1088/0004-637X/769/1/76. arXiv: 1301.0318 [astro-ph.EP] . Banzatti, Andrea, Ilaria Pascucci, Arthur D. Bosman, Paola Pinilla, Colette Salyk, Gregory J. Herczeg, Klaus M. Pontoppidan, et al. 2020. Hints for Icy Pebble Migration Feeding an Oxygen-rich Chemistry in the Inner Planet-forming Region of Disks. ApJ 903, no. 2 (November): 124. https://doi.org/10.3847/1538-4357/abbc1a. arXiv: 2009.13525 [astro-ph.EP] . Benneke, Björn, Heather A. Knutson, Joshua Lothringer, Ian J. M. Crossfield, Julianne I. Moses, Caroline Morley, Laura Kreidberg, et al. 2019. A sub-Neptune exoplanet with a low-metallicity methane-depleted atmosphere and Mie-scattering clouds. Nature Astronomy 3 (July): 813-821. https://doi.org/10.1038/s41550- 019- 0800- 5. arXiv: 1907.00449 [astro-ph.EP] . Bergin, Edwin A., Richard A. Booth, Maria Jose Colmenares, and John D. Ilee. 2024. C/O Ratios and the formation of wide separation exoplanets. arXiv e-prints (June): arXiv:2406.12037. https://doi.org/10.48550/arXiv. 2406.12037. arXiv: 2406.12037 [astro-ph.EP] . Bergin, Edwin A., Fujun Du, L. Ilsedore Cleeves, G. A. Blake, K. Schwarz, R. Visser, and K. Zhang. 2016. Hydrocarbon Emission Rings in Protoplanetary Disks Induced by Dust Evolution. ApJ 831, no. 1 (November): 101. https://doi.org/10.3847/0004-637X/831/1/101. arXiv: 1609.06337 [astro-ph.EP] . Binney, James, and Scott Tremaine. 1987. Galactic dynamics. Birnstiel, Tilman. 2023. Dust growth and evolution in protoplanetary disks. arXiv e-prints (December): arXiv:2312.13287. https://doi.org/10.48550/ arXiv.2312.13287. arXiv: 2312.13287 [astro-ph.EP] . Bisschop, S. E., H. J. Fraser, K. I. Öberg, E. F. van Dishoeck, and S. Schlemmer. 2006. Desorption rates and sticking coefficients for CO and N2 interstellar ices. A&A 449, no. 3 (April): 1297-1309. https://doi.org/10. 1051/0004-6361:20054051. arXiv: astro-ph/0601082 [astro-ph] . Bitsch, Bertram, Aaron David Schneider, and Laura Kreidberg. 2022. How drifting and evaporating pebbles shape giant planets. III. The formation of WASP-77A b and τ Boötis b. A&A 665 (September): A138. http s : / / doi . org / 10 . 1051 / 0004 - 6361 / 202243345. arXiv: 2207 . 06077 [astro-ph.EP] . Bodenheimer, P. 1974. Calculations of the Early Evolution of Jupiter. Icarus 23, no. 3 (November): 319-325. https : / / doi . org / 10 . 1016 / 0019 1035(74)90050-5. Bodewits, D., J. W. Noonan, P. D. Feldman, M. T. Bannister, D. Farnocchia, W. M. Harris, J. -Y. Li, K. E. Mandt, J. Wm. Parker, and Z. -X. Xing. 2020. The carbon monoxide-rich interstellar comet 2I/Borisov. Nature Astronomy 4 (April): 867-871. https://doi.org/10.1038/s41550-0201095-2. arXiv: 2004.08972 [astro-ph.EP] . Booth, R. A., and J. D. Ilee. 2019. Planet-forming material in a protoplanetary disc: the interplay between chemical evolution and pebble drift. MNRAS 487, no. 3 (August): 3998-4011. https://doi.org/10.1093/mnras/stz1488. arXiv: 1905.12639 [astro-ph.EP] . Booth, Richard A., Cathie J. Clarke, Nikku Madhusudhan, and John D. Ilee. 2017. Chemical enrichment of giant planets and discs due to pebble drift. MNRAS 469, no. 4 (August): 3994-4011. https://doi.org/10.1093/ mnras/stx1103. arXiv: 1705.03305 [astro-ph.EP] . Bosman, A. D., A. J. Cridland, and Y. Miguel. 2019. Jupiter formed as a pebble pile around the N2 ice line. A&A 632 (December): L11. https://doi.org/ 10.1051/0004-6361/201936827. arXiv: 1911.11154 [astro-ph.EP] . Bosman, Arthur D., Alexander G. G. M. Tielens, and Ewine F. van Dishoeck. 2018. Efficiency of radial transport of ices in protoplanetary disks probed with infrared observations: the case of CO2. A&A 611 (April): A80. https://doi.org/10.1051/0004-6361/201732056. arXiv: 1712.03989 [astro-ph.EP] . Brown, Paul D., and S. B. Charnley. 1990. Chemical models of interstellar gas-grain processes. I. Modelling and the effect of accretion on gas abundances and mantle composition in dense clouds. MNRAS 244 (June): 432. Brown-Sevilla, S. B., M. Keppler, M. Barraza-Alfaro, J. D. Melon Fuksman, N. Kurtovic, P. Pinilla, M. Feldt, et al. 2021. A multiwavelength analysis of the spiral arms in the protoplanetary disk around WaOph 6. A&A 654 (October): A35. https://doi.org/10.1051/0004-6361/202140783. arXiv: 2107.13560 [astro-ph.EP] . Carrera, Daniel, Anders Johansen, and Melvyn B. Davies. 2015. How to form planetesimals from mm-sized chondrules and chondrule aggregates. A&A 579 (July): A43. https://doi.org/10.1051/0004-6361/201425120. arXiv: 1501.05314 [astro-ph.EP] . Cavalié, Thibault, Jonathan Lunine, Olivier Mousis, and Ricardo Hueso. 2024. The Deep Oxygen Abundance in Solar System Giant Planets, with a New Derivation for Saturn. Space Sci. Rev. 220, no. 1 (January): 8. https://doi.org/10.1007/s11214-024-01045-6. arXiv: 2407.07515 [astro-ph.EP] . Changeat, Q., B. Edwards, A. F. Al-Refaie, A. Tsiaras, J. W. Skinner, J. Y. K. Cho, K. H. Yip, et al. 2022. Five Key Exoplanet Questions Answered via the Analysis of 25 Hot-Jupiter Atmospheres in Eclipse. ApJS 260, no. 1 (May): 3. https://doi.org/10.3847/1538-4365/ac5cc2. arXiv: 2204.11729 [astro-ph.EP] . Cleeves, L. Ilsedore, Karin I. Öberg, David J. Wilner, Jane Huang, Ryan A. Loomis, Sean M. Andrews, and V. V. Guzman. 2018. Constraining Gasphase Carbon, Oxygen, and Nitrogen in the IM Lup Protoplanetary Disk. ApJ 865, no. 2 (October): 155. https://doi.org/10.3847/15384357/aade96. arXiv: 1808.10682 [astro-ph.SR] . Connelley, Michael S., and Bo Reipurth. 2018. A Near-infrared Spectroscopic Survey of FU Orionis Objects. ApJ 861, no. 2 (July): 145. https://doi. org/10.3847/1538-4357/aaba7b. arXiv: 1806.08880 [astro-ph.SR] . Cridland, Alex J., Ewine F. van Dishoeck, Matthew Alessi, and Ralph E. Pudritz. 2019. Connecting planet formation and astrochemistry. A main sequence for C/O in hot exoplanetary atmospheres. A&A 632 (December): A63. https://doi.org/10.1051/0004- 6361/201936105. arXiv: 1910.13171 [astro-ph.EP] . . 2020. Connecting planet formation and astrochemistry. C/Os and N/Os of warm giant planets and Jupiter analogues. A&A 642 (October): A229. https://doi.org/10.1051/0004-6361/202038767. arXiv: 2009. 02907 [astro-ph.EP] . Cridland, Alexander J., Arthur D. Bosman, and Ewine F. van Dishoeck. 2020. Impact of vertical gas accretion on the carbon-to-oxygen ratio of gas giant atmospheres. A&A 635 (March): A68. https://doi.org/10.1051/ 0004-6361/201936858. arXiv: 2001.05808 [astro-ph.EP] . Cridland, Alexander J., Christian Eistrup, and Ewine F. van Dishoeck. 2019. Connecting planet formation and astrochemistry. Refractory carbon depletion leading to super-stellar C/O in giant planetary atmospheres. A&A 627 (July): A127. https://doi.org/10.1051/0004-6361/201834378. arXiv: 1901.08896 [astro-ph.EP] . Cuppen, H. M., C. Walsh, T. Lamberts, D. Semenov, R. T. Garrod, E. M. Penteado, and S. Ioppolo. 2017. Grain Surface Models and Data for Astrochemistry. Space Sci. Rev. 212, nos. 1-2 (October): 1-58. https: //doi.org/10.1007/s11214-016-0319-3. Cuzzi, Jeffrey N., and Kevin J. Zahnle. 2004. Material Enhancement in Protoplanetary Nebulae by Particle Drift through Evaporation Fronts. ApJ 614, no. 1 (October): 490-496. https://doi.org/10.1086/423611. arXiv: astro-ph/0409276 [astro-ph] . Danti, C., B. Bitsch, and J. Mah. 2023. Composition of giant planets: The roles of pebbles and planetesimals. A&A 679 (November): L7. https://doi.org/ 10.1051/0004-6361/202347501. arXiv: 2310.02886 [astro-ph.EP] . Dawson, Rebekah I., and John Asher Johnson. 2018. Origins of Hot Jupiters. ARA&A 56 (September): 175-221. https://doi.org/10.1146/annurevastro-081817-051853. arXiv: 1801.06117 [astro-ph.EP] . Dong, Ruobing, Eduard Vorobyov, Yaroslav Pavlyuchenkov, Eugene Chiang, and Hauyu Baobab Liu. 2016. Signatures of Gravitational Instability in Resolved Images of Protostellar Disks. ApJ 823, no. 2 (June): 141. https://doi.org/10.3847/0004-637X/823/2/141. arXiv: 1603.01618 [astro-ph.SR] . Draine, B. T. 1978. Photoelectric heating of interstellar gas. ApJS 36 (April): 595-619. https://doi.org/10.1086/190513. Drążkowska, J., and Y. Alibert. 2017. Planetesimal formation starts at the snow line. A&A 608 (December): A92. https://doi.org/10.1051/00046361/201731491. arXiv: 1710.00009 [astro-ph.EP] . Dutrey, A., V. Wakelam, Y. Boehler, S. Guilloteau, F. Hersant, D. Semenov, E. Chapillon, et al. 2011. Chemistry in disks. V. Sulfur-bearing molecules in the protoplanetary disks surrounding LkCa15, MWC480, DM Tauri, and GO Tauri. A&A 535 (November): A104. https://doi.org/10.1051/ 0004-6361/201116931. arXiv: 1109.5870 [astro-ph.SR] . Eistrup, Christian, Catherine Walsh, and Ewine F. van Dishoeck. 2016. Setting the volatile composition of (exo)planet-building material. Does chemical evolution in disk midplanes matter? A&A 595 (November): A83. htt ps://doi.org/10.1051/0004- 6361/201628509. arXiv: 1607.06710 [astro-ph.EP] . . 2018. Molecular abundances and C/O ratios in chemically evolving planet-forming disk midplanes. A&A 613 (May): A14. https://doi.org/ 10.1051/0004-6361/201731302. arXiv: 1709.07863 [astro-ph.EP] . Facchini, Stefano, Richard Teague, Jaehan Bae, Myriam Benisty, Miriam Keppler, and Andrea Isella. 2021. The Chemical Inventory of the Planethosting Disk PDS 70. AJ 162, no. 3 (September): 99. https://doi.org/10. 3847/1538-3881/abf0a4. arXiv: 2101.08369 [astro-ph.EP] . Fedele, D., and C. Favre. 2020. Measuring elemental abundance ratios in protoplanetary disks at millimeter wavelengths. A&A 638 (June): A110. https://doi.org/10.1051/0004-6361/202037927. arXiv: 2005.03891 [astro-ph.SR] . Fraser, Helen J., Mark P. Collings, Martin R. S. McCoustra, and David A. Williams. 2001. Thermal desorption of water ice in the interstellar medium. MNRAS 327, no. 4 (November): 1165-1172. https://doi. org/10.1046/j.1365-8711.2001.04835.x. arXiv: astro-ph/0107487 [astro-ph] . Fu, Guangwei, Luis Welbanks, Drake Deming, Julie Inglis, Michael Zhang, Joshua Lothringer, Jegug Ih, et al. 2024. Hydrogen sulfide and metalenriched atmosphere for a Jupiter-mass exoplanet. arXiv e-prints (July): arXiv:2407.06163. https://doi.org/10.48550/arXiv.2407.06163. arXiv: 2407.06163 [astro-ph.EP] . Gail, Hans-Peter, and Mario Trieloff. 2017. Spatial distribution of carbon dust in the early solar nebula and the carbon content of planetesimals. A&A 606 (September): A16. https://doi.org/10.1051/0004-6361/201730480. arXiv: 1707.07611 [astro-ph.EP] . Gammie, Charles F. 1996. Layered Accretion in T Tauri Disks. ApJ 457 (January): 355. https://doi.org/10.1086/176735. Gárate, Matias, Til Birnstiel, Joanna Drążkowska, and Sebastian Markus Stammler. 2020. Gas accretion damped by dust back-reaction at the snow line. A&A 635 (March): A149. https://doi.org/10.1051/00046361/201936067. arXiv: 1906.07708 [astro-ph.EP] . GRAVITY Collaboration, M. Nowak, S. Lacour, P. Mollière, J. Wang, B. Charnay, E. F. van Dishoeck, et al. 2020. Peering into the formation history of β Pictoris b with VLTI/GRAVITY long-baseline interferometry. A&A 633 (January): A110. https://doi.org/10.1051/00046361/201936898. arXiv: 1912.04651 [astro-ph.EP] . Gressel, Oliver, Neal J. Turner, Richard P. Nelson, and Colin P. McNally. 2015. Global Simulations of Protoplanetary Disks With Ohmic Resistivity and Ambipolar Diffusion. ApJ 801, no. 2 (March): 84. https://doi.org/ 10.1088/0004-637X/801/2/84. arXiv: 1501.05431 [astro-ph.EP] . Gundlach, B., and J. Blum. 2015. The Stickiness of Micrometer-sized Waterice Particles. ApJ 798, no. 1 (January): 34. https://doi.org/10.1088/0004637X/798/1/34. arXiv: 1410.7199 [astro-ph.EP] . Harrington Pinto, Olga, Maria Womack, Yanga Fernandez, and James Bauer. 2022. A Survey of CO, CO2, and H2O in Comets and Centaurs. PSJ 3, no. 11 (November): 247. https://doi.org/10.3847/PSJ/ac960d. arXiv: 2209.09985 [astro-ph.EP] . Hartmann, L., and S. J. Kenyon. 1985. On the nature of FU Orionis objects. ApJ 299 (December): 462-478. https://doi.org/10.1086/163713. Hasegawa, T. I., and E. Herbst. 1993. Three-Phase Chemical Models of Dense Interstellar Clouds - Gas Dust Particle Mantles and Dust Particle Surfaces. MNRAS 263 (August): 589. https://doi.org/10.1093/mnras/263.3.589. Hensley, Brandon S., and B. T. Draine. 2021. Observational Constraints on the Physical Properties of Interstellar Dust in the Post-Planck Era. ApJ 906, no. 2 (January): 73. https://doi.org/10.3847/1538-4357/abc8f1. arXiv: 2009.00018 [astro-ph.GA] . Hoch, Kielan K. W., Quinn M. Konopacky, Christopher A. Theissen, JeanBaptiste Ruffio, Travis S. Barman, Emily L. Rickman, Marshall D. Perrin, Bruce Macintosh, and Christian Marois. 2023. Assessing the C/O Ratio Formation Diagnostic: A Potential Trend with Companion Mass. AJ 166, no. 3 (September): 85. https://doi.org/10.3847/1538-3881/ace442. arXiv: 2212.04557 [astro-ph.EP] . Huang, Jane, Sean M. Andrews, Cornelis P. Dullemond, Andrea Isella, Laura M. Pérez, Viviana V. Guzmán, Karin I. Öberg, et al. 2018. The Disk Substructures at High Angular Resolution Project (DSHARP). II. Characteristics of Annular Substructures. ApJ 869, no. 2 (December): L42. https : / / doi . org / 10 . 3847 / 2041 - 8213 / aaf 740. arXiv: 1812 . 04041 [astro-ph.EP] . Huang, Jane, Sean M. Andrews, Laura M. Pérez, Zhaohuan Zhu, Cornelis P. Dullemond, Andrea Isella, Myriam Benisty, et al. 2018. The Disk Substructures at High Angular Resolution Project (DSHARP). III. Spiral Structures in the Millimeter Continuum of the Elias 27, IM Lup, and WaOph 6 Disks. ApJ 869, no. 2 (December): L43. https://doi.org/10. 3847/2041-8213/aaf7a0. arXiv: 1812.04193 [astro-ph.SR] . Huang, Jane, Edwin A. Bergin, Romane Le Gal, Sean M. Andrews, Jaehan Bae, Luke Keyte, and J. A. Sturm. 2024. Constraints on the gas-phase C/O ratio of DR Tau's outer disk from CS, SO, and C2H observations. arXiv e-prints (July): arXiv:2407.01679. https://doi.org/10.48550/arXiv. 2407.01679. arXiv: 2407.01679 [astro-ph.EP] . Hyodo, Ryuki, Tristan Guillot, Shigeru Ida, Satoshi Okuzumi, and Andrew N. Youdin. 2021. Planetesimal formation around the snow line. II. Dust or pebbles? A&A 646 (February): A14. https://doi.org/10.1051/00046361/202039894. arXiv: 2012.06700 [astro-ph.EP] . Ilee, J. D., A. C. Boley, P. Caselli, R. H. Durisen, T. W. Hartquist, and J. M. C. Rawlings. 2011. Chemistry in a gravitationally unstable protoplanetary disc. MNRAS 417, no. 4 (November): 2950-2961. https://doi.org/10. 1111/j.1365-2966.2011.19455.x. arXiv: 1107.3041 [astro-ph.GA] . Jiang, Haochang, and Chris W. Ormel. 2023. Efficient planet formation by pebble accretion in ALMA rings. MNRAS 518, no. 3 (January): 38773900. https://doi.org/10.1093/mnras/stac3275. arXiv: 2207.13002 [astro-ph.EP] . Kadam, Kundan, Eduard Vorobyov, and Shantanu Basu. 2022. Primordial dusty rings and episodic outbursts in protoplanetary discs. MNRAS 516, no. 3 (November): 4448-4468. https://doi.org/10.1093/mnras/stac2455. arXiv: 2208.12105 [astro-ph.EP] . Kadam, Kundan, Eduard Vorobyov, Zsolt Regály, Ágnes Kóspál, and Péter Ábrahám. 2019. Dynamical Gaseous Rings in Global Simulations of Protoplanetary Disk Formation. ApJ 882, no. 2 (September): 96. h ttps : / / doi . org / 10 . 3847 / 1538 - 4357 / ab378a. arXiv: 1908 . 02515 [astro-ph.SR] . . 2020. Outbursts in Global Protoplanetary Disk Simulations. ApJ 895, no. 1 (May): 41. https://doi.org/10.3847/1538-4357/ab8bd8. arXiv: 2005.03578 [astro-ph.SR] . Kama, M., S. Bruderer, E. F. van Dishoeck, M. Hogerheijde, C. P. Folsom, A. Miotello, D. Fedele, A. Belloche, R. Güsten, and F. Wyrowski. 2016. Volatile-carbon locking and release in protoplanetary disks. A study of TWHya and HD 100546. A&A 592 (August): A83. https://doi.org/10. 1051/0004-6361/201526991. arXiv: 1605.05093 [astro-ph.EP] . Khorshid, N., M. Min, and J. M. Désert. 2023. Retrieving planet formation parameters of WASP-77Ab using SimAb. A&A 675 (July): A95. htt ps://doi.org/10.1051/0004- 6361/202245469. arXiv: 2311.15702 [astro-ph.EP] . Kratter, Kaitlin, and Giuseppe Lodato. 2016. Gravitational Instabilities in Circumstellar Disks. ARA&A 54 (September): 271-311. https://doi. org/10.1146/annurev- astro- 081915- 023307. arXiv: 1603.01280 [astro-ph.SR] . Krijt, Sebastiaan, Arthur D. Bosman, Ke Zhang, Kamber R. Schwarz, Fred J. Ciesla, and Edwin A. Bergin. 2020. CO Depletion in Protoplanetary Disks: A Unified Picture Combining Physical Sequestration and Chemical Processing. ApJ 899, no. 2 (August): 134. https://doi.org/10.3847/ 1538-4357/aba75d. arXiv: 2007.09517 [astro-ph.SR] . Krijt, Sebastiaan, Kamber R. Schwarz, Edwin A. Bergin, and Fred J. Ciesla. 2018. Transport of CO in Protoplanetary Disks: Consequences of Pebble Formation, Settling, and Radial Drift. ApJ 864, no. 1 (September): 78. https : / / doi . org / 10 . 3847 / 1538 - 4357 / aad69b. arXiv: 1808 . 01840 [astro-ph.EP] . Lambrechts, M., and A. Johansen. 2012. Rapid growth of gas-giant cores by pebble accretion. A&A 544 (August): A32. https://doi.org/10.1051/00046361/201219127. arXiv: 1205.3030 [astro-ph.EP] . Lee, Eve J., J. R. Fuentes, and Philip F. Hopkins. 2022. Establishing Dust Rings and Forming Planets within Them. ApJ 937, no. 2 (October): 95. https://doi.org/10.3847/1538-4357/ac8cfe. arXiv: 2206.01219 [astro-ph.EP] . Lee, Jeong-Eun, Edwin A. Bergin, and Hideko Nomura. 2010. The Solar Nebula on Fire: A Solution to the Carbon Deficit in the Inner Solar System. ApJ 710, no. 1 (February): L21-L25. https://doi.org/10.1088/ 2041-8205/710/1/L21. arXiv: 1001.0818 [astro-ph.GA] . Lee, Jeong-Eun, Seokho Lee, Giseon Baek, Yuri Aikawa, Lucas Cieza, SungYong Yoon, Gregory Herczeg, Doug Johnstone, and Simon Casassus. 2019. The ice composition in the disk around V883 Ori revealed by its stellar outburst. Nature Astronomy 3 (February): 314-319. https://doi. org/10.1038/s41550-018-0680-0. arXiv: 1809.00353 [astro-ph.SR] . Leemker, M., A. S. Booth, E. F. van Dishoeck, L. Wölfer, and B. Dent. 2024. Chemistry across dust and gas gaps in protoplanetary disks: modelling the co-spatial molecular rings in the HD 100546 disk. arXiv e-prints (May): arXiv:2405.10361. https://doi.org/10.48550/arXiv.2405.10361. arXiv: 2405.10361 [astro-ph.EP] . Lenz, Christian T., Hubert Klahr, and Tilman Birnstiel. 2019. Planetesimal Population Synthesis: Pebble Flux-regulated Planetesimal Formation. ApJ 874, no. 1 (March): 36. https://doi.org/10.3847/1538-4357/ab05d9. arXiv: 1902.07089 [astro-ph.EP] . Li, Cheng, Michael Allison, Sushil Atreya, Shawn Brueshaber, Leigh N. Fletcher, Tristan Guillot, Liming Li, et al. 2024. Super-adiabatic temperature gradient at Jupiter's equatorial zone and implications for the water abundance. Icarus 414 (May): 116028. https://doi.org/10.1016/j. icarus.2024.116028. arXiv: 2403.05363 [astro-ph.EP] . Li, Rixin, and Andrew N. Youdin. 2021. Thresholds for Particle Clumping by the Streaming Instability. ApJ 919, no. 2 (October): 107. https://doi. org/10.3847/1538-4357/ac0e9f. arXiv: 2105.06042 [astro-ph.EP] . Ligterink, N. F. W., K. A. Kipfer, and S. Gavino. 2024. Mind the Trap: Non-negligible effect of volatile trapping in ice on C/O ratios in protoplanetary disks and exoplanetary atmospheres. arXiv e-prints (June): arXiv:2406.16029. https://doi.org/10.48550/arXiv.2406.16029. arXiv: 2406.16029 [astro-ph.EP] . Lin, D. N. C., P. Bodenheimer, and D. C. Richardson. 1996. Orbital migration of the planetary companion of 51 Pegasi to its present location. Nature 380, no. 6575 (April): 606-607. https://doi.org/10.1038/380606a0. Line, Michael R., Matteo Brogi, Jacob L. Bean, Siddharth Gandhi, Joseph Zalesky, Vivien Parmentier, Peter Smith, et al. 2021. A solar C/O and sub-solar metallicity in a hot Jupiter atmosphere. Nature 598, no. 7882 (October): 580-584. https://doi.org/10.1038/s41586-021-03912-6. arXiv: 2110.14821 [astro-ph.EP] . Lodders, Katharina. 2004. Jupiter Formed with More Tar than Ice. ApJ 611, no. 1 (August): 587-597. https://doi.org/10.1086/421970. Long, Feng, Paola Pinilla, Gregory J. Herczeg, Daniel Harsono, Giovanni Dipierro, Ilaria Pascucci, Nathan Hendler, et al. 2018. Gaps and Rings in an ALMA Survey of Disks in the Taurus Star-forming Region. ApJ 869, no. 1 (December): 17. https://doi.org/10.3847/1538-4357/aae8e1. arXiv: 1810.06044 [astro-ph.SR] . Madhusudhan, Nikku. 2012. C/O Ratio as a Dimension for Characterizing Exoplanetary Atmospheres. ApJ 758, no. 1 (October): 36. https://doi. org/10.1088/0004-637X/758/1/36. arXiv: 1209.2412 [astro-ph.EP] . Madhusudhan, Nikku, Joseph Harrington, Kevin B. Stevenson, Sarah Nymeyer, Christopher J. Campo, Peter J. Wheatley, Drake Deming, et al. 2011. A high C/O ratio and weak thermal inversion in the atmosphere of exoplanet WASP-12b. Nature 469, no. 7328 (January): 64-67. https: //doi.org/10.1038/nature09602. arXiv: 1012.1603 [astro-ph.EP] . Matter, A., F. C. Pignatale, and B. Lopez. 2020. Spatially resolving the chemical composition of the planet building blocks. MNRAS 497, no. 3 (September): 2540-2552. https://doi.org/10.1093/mnras/staa2137. arXiv: 2007.09385 [astro-ph.EP] . McKay, Adam J., Michael A. DiSanti, Michael S. P. Kelley, Matthew M. Knight, Maria Womack, Kacper Wierzchos, Olga Harrington Pinto, et al. 2019. The Peculiar Volatile Composition of CO-dominated Comet C/2016 R2 (PanSTARRS). AJ 158, no. 3 (September): 128. https://doi. org/10.3847/1538-3881/ab32e4. arXiv: 1907.07208 [astro-ph.EP] . Meru, Farzana, and Matthew R. Bate. 2010. Exploring the conditions required to form giant planets via gravitational instability in massive protoplanetary discs. MNRAS 406, no. 4 (August): 2279-2288. https: //doi.org/10.1111/j.1365- 2966.2010.16867.x. arXiv: 1004.3766 [astro-ph.EP] . Meru, Farzana, Attila Juhász, John D. Ilee, Cathie J. Clarke, Giovanni P. Rosotti, and Richard A. Booth. 2017. On the Origin of the Spiral Morphology in the Elias 2-27 Circumstellar Disk. ApJ 839, no. 2 (April): L24. https://doi.org/10.3847/2041-8213/aa6837. arXiv: 1703.05338 [astro-ph.EP] . Minissale, Marco, Yuri Aikawa, Edwin Bergin, Mathieu Bertin, Wendy A. Brown, Stephanie Cazaux, Steven B. Charnley, et al. 2022. Thermal Desorption of Interstellar Ices: A Review on the Controlling Parameters and Their Implications from Snowlines to Chemical Complexity. ACS Earth and Space Chemistry 6, no. 3 (March): 597-630. https://doi.org/10. 1021/acsearthspacechem.1c00357. arXiv: 2201.07512 [astro-ph.GA] . Miotello, A., S. Facchini, E. F. van Dishoeck, P. Cazzoletti, L. Testi, J. P. Williams, M. Ansdell, S. van Terwisga, and N. van der Marel. 2019. Bright C2H emission in protoplanetary discs in Lupus: high volatile C/O > 1 ratios. A&A 631 (November): A69. https://doi.org/10.1051/00046361/201935441. arXiv: 1909.04477 [astro-ph.SR] . Mollière, P., T. Stolker, S. Lacour, G. P. P. L. Otten, J. Shangguan, B. Charnay, T. Molyarova, et al. 2020. Retrieving scattering clouds and disequilibrium chemistry in the atmosphere of HR 8799e. A&A 640 (August): A131. https://doi.org/10.1051/0004-6361/202038325. arXiv: 2006.09394 [astro-ph.EP] . Mollière, Paul, Tamara Molyarova, Bertram Bitsch, Thomas Henning, Aaron Schneider, Laura Kreidberg, Christian Eistrup, et al. 2022. Interpreting the Atmospheric Composition of Exoplanets: Sensitivity to Planet Formation Assumptions. ApJ 934, no. 1 (July): 74. https://doi.org/10.3847/ 1538-4357/ac6a56. arXiv: 2204.13714 [astro-ph.EP] . Molyarova, Tamara, Vitaly Akimkin, Dmitry Semenov, Thomas Henning, Anton Vasyunin, and Dmitri Wiebe. 2017. Gas Mass Tracers in Protoplanetary Disks: CO is Still the Best. ApJ 849, no. 2 (November): 130. https://doi.org/10.3847/1538-4357/aa9227. arXiv: 1710.02993 [astro-ph.EP] . Molyarova, Tamara, Eduard I. Vorobyov, Vitaly Akimkin, Aleksandr Skliarevskii, Dmitri Wiebe, and Manuel Güdel. 2021. Gravitoviscous Protoplanetary Disks with a Dust Component. V. The Dynamic Model for Freeze-out and Sublimation of Volatiles. ApJ 910, no. 2 (April): 153. https://doi.org/ 10.3847/1538-4357/abe2b0. arXiv: 2103.06045 [astro-ph.EP] . Morbidelli, A., J. Szulágyi, A. Crida, E. Lega, B. Bitsch, T. Tanigawa, and K. Kanagawa. 2014. Meridional circulation of gas into gaps opened by giant planets in three-dimensional low-viscosity disks. Icarus 232 (April): 266-270. https://doi.org/10.1016/j.icarus.2014.01.010. arXiv: 1401.2925 [astro-ph.EP] . Mordasini, C., R. van Boekel, P. Mollière, Th. Henning, and Björn Benneke. 2016. The Imprint of Exoplanet Formation History on Observable Present-day Spectra of Hot Jupiters. ApJ 832, no. 1 (November): 41. https://doi.org/10.3847/0004-637X/832/1/41. arXiv: 1609.03019 [astro-ph.EP] . Moses, J. I., N. Madhusudhan, C. Visscher, and R. S. Freedman. 2013. Chemical Consequences of the C/O Ratio on Hot Jupiters: Examples from WASP-12b, CoRoT-2b, XO-1b, and HD 189733b. ApJ 763, no. 1 (January): 25. https://doi.org/10.1088/0004-637X/763/1/25. arXiv: 1211.2996 [astro-ph.EP] . Mousis, Olivier, Sarah E. Anderson, Adrienn Luspay-Kuti, Kathleen E. Mandt, and Pierre Vernazza. 2024. Triton and Pluto: same origin but separated at birth. arXiv e-prints (June): arXiv:2406.03815. https://doi.org/10. 48550/arXiv.2406.03815. arXiv: 2406.03815 [astro-ph.EP] . Mousis, Olivier, Thibault Cavalié, Jonathan I. Lunine, Kathleen E. Mandt, Ricardo Hueso, Artyom Aguichine, Antoine Schneeberger, et al. 2024. Recipes for Forming a Carbon-Rich Giant Planet. Space Sci. Rev. 220, no. 4 (June): 44. https://doi.org/10.1007/s11214-024-01071-4. arXiv: 2405.19748 [astro-ph.EP] . Nasedkin, E., P. Mollière, S. Lacour, M. Nowak, L. Kreidberg, T. Stolker, J. J. Wang, et al. 2024. Four-of-a-kind? Comprehensive atmospheric characterisation of the HR 8799 planets with VLTI/GRAVITY. arXiv e-prints (April): arXiv:2404.03776. https://doi.org/10.48550/arXiv.2404. 03776. arXiv: 2404.03776 [astro-ph.EP] . Nayakshin, Sergei. 2010a. Formation of planets by tidal downsizing of giant planet embryos. MNRAS 408, no. 1 (October): L36-L40. https://doi.org/ 10.1111/j.1745-3933.2010.00923.x. arXiv: 1007.4159 [astro-ph.EP] . . 2010b. Grain sedimentation inside giant planet embryos. MNRAS 408, no. 4 (November): 2381-2396. https://doi.org/10.1111/j.13652966.2010.17289.x. arXiv: 1007.4162 [astro-ph.EP] . Nayakshin, Sergei, Ravit Helled, and Aaron C. Boley. 2014. Core-assisted gas capture instability: a new mode of giant planet formation by gravitationally unstable discs. MNRAS 440, no. 4 (June): 3797-3808. https: //doi.org/10.1093/mnras/stu473. arXiv: 1403.1813 [astro-ph.EP] . Nortmann, L., F. Lesjak, F. Yan, D. Cont, S. Czesla, A. Lavail, A. D. Rains, et al. 2024. CRIRES + transmission spectroscopy of WASP-127b. Detection of the resolved signatures of a supersonic equatorial jet and cool poles in a hot planet. arXiv e-prints (April): arXiv:2404.12363. https://doi.org/ 10.48550/arXiv.2404.12363. arXiv: 2404.12363 [astro-ph.EP] . Öberg, K. I., F. van Broekhuizen, H. J. Fraser, S. E. Bisschop, E. F. van Dishoeck, and S. Schlemmer. 2005. Competition between CO and N2 Desorption from Interstellar Ices. ApJ 621, no. 1 (March): L33-L36. https://doi.org/10.1086/428901. Öberg, Karin I., A. C. Adwin Boogert, Klaus M. Pontoppidan, Saskia van den Broek, Ewine F. van Dishoeck, Sandrine Bottinelli, Geoffrey A. Blake, and II Evans Neal J. 2011. The Spitzer Ice Legacy: Ice Evolution from Cores to Protostars. ApJ 740, no. 2 (October): 109. https://doi.org/10. 1088/0004-637X/740/2/109. arXiv: 1107.5825 [astro-ph.GA] . Öberg, Karin I., Ruth Murray-Clay, and Edwin A. Bergin. 2011. The Effects of Snowlines on C/O in Planetary Atmospheres. ApJ 743, no. 1 (December): L16. https://doi.org/10.1088/2041-8205/743/1/L16. arXiv: 1110.5567 [astro-ph.GA] . Öberg, Karin I., and Robin Wordsworth. 2019. Jupiter's Composition Suggests its Core Assembled Exterior to the N2 Snowline. AJ 158, no. 5 (November): 194. https://doi.org/10.3847/1538-3881/ab46a8. arXiv: 1909.11246 [astro-ph.EP] . Ohno, Kazumasa, and Takahiro Ueda. 2021. Jupiter's 'cold' formation in the protosolar disk shadow. An explanation for the planet's uniformly enriched atmosphere. A&A 651 (July): L2. https://doi.org/10.1051/00046361/202141169. arXiv: 2106.09084 [astro-ph.EP] . Okuzumi, Satoshi, and Shigenobu Hirose. 2011. Modeling Magnetorotational Turbulence in Protoplanetary Disks with Dead Zones. ApJ 742, no. 2 (December): 65. https://doi.org/10.1088/0004-637X/742/2/65. arXiv: 1108.4892 [astro-ph.EP] . Okuzumi, Satoshi, and Ryo Tazaki. 2019. Nonsticky Ice at the Origin of the Uniformly Polarized Submillimeter Emission from the HL Tau Disk. ApJ 878, no. 2 (June): 132. https://doi.org/10.3847/1538-4357/ab204d. arXiv: 1904.03869 [astro-ph.EP] . Ootsubo, Takafumi, Hideyo Kawakita, Saki Hamada, Hitomi Kobayashi, Mitsuru Yamaguchi, Fumihiko Usui, Takao Nakagawa, et al. 2012. AKARI Near-infrared Spectroscopic Survey for CO2 in 18 Comets. ApJ 752, no. 1 (June): 15. https://doi.org/10.1088/0004-637X/752/1/15. Padoan, Paolo, Liubin Pan, Veli-Matti Pelkonen, Troels Haugboelle, and AAke Nordlund. 2024. Protoplanetary Disks from Pre-Main Sequence BondiHoyle Accretion. arXiv e-prints (May): arXiv:2405.07334. https://doi. org/10.48550/arXiv.2405.07334. arXiv: 2405.07334 [astro-ph.GA] . Padovani, M., D. Galli, and A. E. Glassgold. 2009. Cosmic-ray ionization of molecular clouds. A&A 501, no. 2 (July): 619-631. https://doi.org/10. 1051/0004-6361/200911794. arXiv: 0904.4149 [astro-ph.SR] . Pavlyuchenkov, Yaroslav, Vitaly Akimkin, Dmitri Wiebe, and Eduard Vorobyov. 2019. Revealing dust segregation in protoplanetary discs with the help of multifrequency spectral index maps. MNRAS 486, no. 3 (July): 39073914. https://doi.org/10.1093/mnras/stz1046. arXiv: 1904.05251 [astro-ph.IM] . Pelkonen, Veli-Matti, Paolo Padoan, Mika Juvela, Troels Haugbølle, and Åke Nordlund. 2024. Origin and Evolution of Angular Momentum of Class II Disks. arXiv e-prints (May): arXiv:2405.06520. https://doi.org/10. 48550/arXiv.2405.06520. arXiv: 2405.06520 [astro-ph.SR] . Pérez, Laura M., John M. Carpenter, Sean M. Andrews, Luca Ricci, Andrea Isella, Hendrik Linz, Anneila I. Sargent, et al. 2016. Spiral density waves in a young protoplanetary disk. Science 353, no. 6307 (September): 15191521. https://doi.org/10.1126/science.aaf8296. arXiv: 1610.05139 [astro-ph.GA] . Piso, Ana-Maria A., Karin I. Öberg, Tilman Birnstiel, and Ruth A. MurrayClay. 2015. C/O and Snowline Locations in Protoplanetary Disks: The Effect of Radial Drift and Viscous Gas Accretion. ApJ 815, no. 2 (December): 109. https://doi.org/10.1088/0004-637X/815/2/109. arXiv: 1511.05563 [astro-ph.EP] . Pontoppidan, K. M., C. Salyk, E. A. Bergin, S. Brittain, B. Marty, O. Mousis, and K. I. Öberg. 2014. Volatiles in Protoplanetary Disks. In Protostars and planets vi, edited by Henrik Beuther, Ralf S. Klessen, Cornelis P. Dullemond, and Thomas Henning, 363-385. January. https : / / doi . org/10.2458/azu_uapress_9780816531240-ch016. arXiv: 1401.2423 [astro-ph.EP] . Przybilla, Norbert, Maria-Fernanda Nieva, and Keith Butler. 2008. A Cosmic Abundance Standard: Chemical Homogeneity of the Solar Neighborhood and the ISM Dust-Phase Composition. ApJ 688, no. 2 (December): L103. https://doi.org/10.1086/595618. arXiv: 0809.2403 [astro-ph] . Rafikov, Roman R. 2005. Can Giant Planets Form by Direct Gravitational Instability? ApJ 621, no. 1 (March): L69-L72. https://doi.org/10.1086/ 428899. arXiv: astro-ph/0406469 [astro-ph] . Rampinelli, L., S. Facchini, M. Leemker, J. Bae, M. Benisty, R. Teague, C. J. Law, K. I. Öberg, B. Portilla-Revelo, and A. J. Cridland. 2024. ALMA high-resolution observations unveil planet formation shaping molecular emission in the PDS 70 disk. arXiv e-prints (July): arXiv:2407.06272. https://doi.org/10.48550/arXiv.2407.06272. arXiv: 2407.06272 [astro-ph.EP] . Reggiani, Henrique, Jhon Yana Galarza, Kevin C. Schlaufman, David K. Sing, Brian F. Healy, Andrew McWilliam, Joshua D. Lothringer, and Laurent Pueyo. 2024. Insight into the Formation of β Pic b through the Composition of Its Parent Protoplanetary Disk as Revealed by the β Pic Moving Group Member HD 181327. AJ 167, no. 1 (January): 45. https://doi.org/10.3847/1538-3881/ad0f93. arXiv: 2311.12210 [astro-ph.SR] . Schneider, Aaron David, and Bertram Bitsch. 2021. How drifting and evaporating pebbles shape giant planets. I. Heavy element content and atmospheric C/O. A&A 654 (October): A71. https://doi.org/10.1051/00046361/202039640. arXiv: 2105.13267 [astro-ph.EP] . Seager, S., L. J. Richardson, B. M. S. Hansen, K. Menou, J. Y. -K. Cho, and D. Deming. 2005. On the Dayside Thermal Emission of Hot Jupiters. ApJ 632, no. 2 (October): 1122-1131. https://doi.org/10.1086/444411. arXiv: astro-ph/0504212 [astro-ph] . Seligman, Darryl Z., Leslie A. Rogers, Samuel H. C. Cabot, John W. Noonan, Theodore Kareta, Kathleen E. Mandt, Fred Ciesla, et al. 2022. The Volatile Carbon-to-oxygen Ratio as a Tracer for the Formation Locations of Interstellar Comets. PSJ 3, no. 7 (July): 150. https://doi.org/10. 3847/PSJ/ac75b5. arXiv: 2204.13211 [astro-ph.EP] . Semenov, D., C. Favre, D. Fedele, S. Guilloteau, R. Teague, Th. Henning, A. Dutrey, E. Chapillon, F. Hersant, and V. Piétu. 2018. Chemistry in disks. XI. Sulfur-bearing species as tracers of protoplanetary disk physics and chemistry: the DM Tau case. A&A 617 (September): A28. https://doi.org/10.1051/0004-6361/201832980. arXiv: 1806.07707 [astro-ph.GA] . Semenov, D., Th. Henning, Ch. Helling, M. Ilgner, and E. Sedlmayr. 2003. Rosseland and Planck mean opacities for protoplanetary discs. A&A 410 (November): 611-621. https://doi.org/10.1051/0004-6361:20031279. arXiv: astro-ph/0308344 [astro-ph] . Semenov, D., and D. Wiebe. 2011. Chemical Evolution of Turbulent Protoplanetary Disks and the Solar Nebula. ApJS 196, no. 2 (October): 25. https://doi.org/10.1088/0067- 0049/196/2/25. arXiv: 1104.4358 [astro-ph.GA] . Shakura, N. I., and R. A. Sunyaev. 1973. Black holes in binary systems. Observational appearance. A&A 24 (January): 337-355. Sing, David K., Zafar Rustamkulov, Daniel P. Thorngren, Joanna K. Barstow, Pascal Tremblin, Catarina Alves de Oliveira, Tracy L. Beck, et al. 2024. A warm Neptune's methane reveals core mass and vigorous atmospheric mixing. arXiv e-prints (May): arXiv:2405.11027. https://doi.org/10. 48550/arXiv.2405.11027. arXiv: 2405.11027 [astro-ph.EP] . Smith, Peter C. B., Michael R. Line, Jacob L. Bean, Matteo Brogi, Prune August, Luis Welbanks, Jean-Michel Desert, et al. 2024. A Combined Ground-based and JWST Atmospheric Retrieval Analysis: Both IGRINS and NIRSpec Agree that the Atmosphere of WASP-77A b Is Metal-poor. AJ 167, no. 3 (March): 110. https://doi.org/10.3847/1538-3881/ad17bf. arXiv: 2312.13069 [astro-ph.EP] . Stammler, Sebastian Markus, Tilman Birnstiel, Olja Panić, Cornelis Petrus Dullemond, and Carsten Dominik. 2017. Redistribution of CO at the location of the CO ice line in evolving gas and dust disks. A&A 600 (April): A140. https://doi.org/10.1051/0004-6361/201629041. arXiv: 1701.02385 [astro-ph.EP] . Stevenson, David J., and Jonathan I. Lunine. 1988. Rapid formation of Jupiter by diffusive redistribution of water vapor in the solar nebula. Icarus 75, no. 1 (July): 146-155. https://doi.org/10.1016/0019-1035(88)90133-9. Swain, M. R., G. Tinetti, G. Vasisht, P. Deroo, C. Griffith, J. Bouwman, Pin Chen, et al. 2009. Water, Methane, and Carbon Dioxide Present in the Dayside Spectrum of the Exoplanet HD 209458b. ApJ 704, no. 2 (October): 1616-1621. https://doi.org/10.1088/0004-637X/704/2/1616. arXiv: 0908.4010 [astro-ph.EP] . Testi, L., T. Birnstiel, L. Ricci, S. Andrews, J. Blum, J. Carpenter, C. Dominik, et al. 2014. Dust Evolution in Protoplanetary Disks. In Protostars and planets vi, edited by Henrik Beuther, Ralf S. Klessen, Cornelis P. Dullemond, and Thomas Henning, 339-361. January. https://doi.org/10.2458/azu_ uapress_9780816531240-ch015. arXiv: 1402.1354 [astro-ph.SR] . Thiabaud, A., U. Marboeuf, Y. Alibert, I. Leya, and K. Mezger. 2015. Gas composition of the main volatile elements in protoplanetary discs and its implication for planet formation. A&A 574 (February): A138. https: //doi.org/10.1051/0004-6361/201424868. Thies, Ingo, Pavel Kroupa, Simon P. Goodwin, Dimitrios Stamatellos, and Anthony P. Whitworth. 2010. Tidally Induced Brown Dwarf and Planet Formation in Circumstellar Disks. ApJ 717, no. 1 (July): 577-585. h ttps://doi.org/10.1088/0004- 637X/717/1/577. arXiv: 1005.3017 [astro-ph.SR] . Tielens, A. G. G. M. 2005. The Physics and Chemistry of the Interstellar Medium. Tong, Simin, Richard Alexander, and Giovanni Rosotti. 2024. A question of personalities: evolution of viscous and wind-driven protoplanetary discs in the presence of dead zones. MNRAS (July). https://doi.org/10.1093/ mnras/stae1748. arXiv: 2407.12209 [astro-ph.EP] . Toomre, A. 1964. On the gravitational stability of a disk of stars. ApJ 139 (May): 1217-1238. https://doi.org/10.1086/147861. Topchieva, A., T. Molyarova, V. Akimkin, L. Maksimova, and E. Vorobyov. 2024. Ices on pebbles in protoplanetary discs. MNRAS 530, no. 3 (May): 2731-2748. https://doi.org/10.1093/mnras/stae597. arXiv: 2403.02895 [astro-ph.EP] . Turrini, D., E. Schisano, S. Fonte, S. Molinari, R. Politi, D. Fedele, O. Panić, M. Kama, Q. Changeat, and G. Tinetti. 2021. Tracing the Formation History of Giant Planets in Protoplanetary Disks with Carbon, Oxygen, Nitrogen, and Sulfur. ApJ 909, no. 1 (March): 40. https://doi.org/10. 3847/1538-4357/abd6e5. arXiv: 2012.14315 [astro-ph.EP] . Umebayashi, T., and T. Nakano. 1981. Fluxes of Energetic Particles and the Ionization Rate in Very Dense Interstellar Clouds. PASJ 33 (January): 617. Vallet, David, Anna C. Childs, Rebecca G. Martin, Mario Livio, and Stephen Lepp. 2023. Formation of super-Earths in icy dead zones around lowmass stars. MNRAS 519, no. 1 (February): L10-L14. https://doi.org/10. 1093/mnrasl/slac144. arXiv: 2211.07759 [astro-ph.EP] . Visser, R., E. F. van Dishoeck, S. D. Doty, and C. P. Dullemond. 2009. The chemical history of molecules in circumstellar disks. I. Ices. A&A 495, no. 3 (March): 881-897. https://doi.org/10.1051/0004-6361/200810846. arXiv: 0901.1313 [astro-ph.SR] . Vorobyov, E. I. 2013. Formation of giant planets and brown dwarfs on wide orbits. A&A 552 (April): A129. https://doi.org/10.1051/0004-6361/ 201220601. arXiv: 1302.1892 [astro-ph.EP] . Vorobyov, Eduard I., Vitaly Akimkin, Olga Stoyanovskaya, Yaroslav Pavlyuchenkov, and Hauyu Baobab Liu. 2018. Early evolution of viscous and selfgravitating circumstellar disks with a dust component. A&A 614 (June): A98. https://doi.org/10.1051/0004-6361/201731690. arXiv: 1801.06898 [astro-ph.EP] . Vorobyov, Eduard I., and Shantanu Basu. 2010. The Burst Mode of Accretion and Disk Fragmentation in the Early Embedded Stages of Star Formation. ApJ 719, no. 2 (August): 1896-1911. https://doi.org/10.1088/0004637X/719/2/1896. arXiv: 1007.2993 [astro-ph.SR] . Vorobyov, Eduard I., and Vardan G. Elbakyan. 2019. Gravitoviscous protoplanetary disks with a dust component. II. Spatial distribution and growth of dust in a clumpy disk. A&A 631 (November): A1. https: / / doi . org / 10 . 1051 / 0004 - 6361 / 201936132. arXiv: 1908 . 10589 [astro-ph.SR] . Vorobyov, Eduard I., Vardan G. Elbakyan, Michihiro Takami, and Hauyu B. Liu. 2020. Effect of luminosity outbursts on protoplanetary disk dynamics. A&A 643 (November): A13. https://doi.org/10.1051/00046361/202038122. arXiv: 2009.01888 [astro-ph.SR] . Vorobyov, Eduard I., Sergey Khaibrakhmanov, Shantanu Basu, and Marc Audard. 2020. Accretion bursts in magnetized gas-dust protoplanetary disks. A&A 644 (December): A74. https://doi.org/10.1051/00046361/202039081. arXiv: 2011.00951 [astro-ph.SR] . Vorobyov, Eduard I., D. N. C. Lin, and Manuel Guedel. 2015. The effect of external environment on the evolution of protostellar disks. A&A 573 (January): A5. https://doi.org/10.1051/0004-6361/201424583. arXiv: 1410.1743 [astro-ph.SR] . Vorobyov, Eduard I., Zsolt Regaly, Manuel Guedel, and Doug N. C. Lin. 2016. An alternative model for the origin of gaps in circumstellar disks. A&A 587 (March): A146. https://doi.org/10.1051/0004-6361/201527701. arXiv: 1601.08089 [astro-ph.SR] . Vorobyov, Eduard I., Aleksandr M. Skliarevskii, Manuel Guedel, and Tamara Molyarova. 2024. Primordial dust rings, hidden dust mass, and the first generation of planetesimals in gravitationally unstable protoplanetary disks. arXiv e-prints (April): arXiv:2404.16151. https://doi.org/10.48550/ arXiv.2404.16151. arXiv: 2404.16151 [astro-ph.EP] . Vorobyov, Eduard I., Aleksandr M. Skliarevskii, Tamara Molyarova, Vitaly Akimkin, Yaroslav Pavlyuchenkov, Ágnes Kóspál, Hauyu Baobab Liu, Michihiro Takami, and Anastasiia Topchieva. 2022. Evolution of dust in protoplanetary disks of eruptive stars. A&A 658 (February): A191. https://doi.org/10.1051/0004-6361/202141932. arXiv: 2112.06004 [astro-ph.EP] . Vorobyov, Eduard I., Olga V. Zakhozhay, and Michael M. Dunham. 2013. Fragmenting protostellar discs: properties and observational signatures. MNRAS 433, no. 4 (August): 3256-3273. https://doi.org/10.1093/ mnras/stt970. arXiv: 1306.4074 [astro-ph.SR] . Wada, Koji, Hidekazu Tanaka, Toru Suyama, Hiroshi Kimura, and Tetsuo Yamamoto. 2009. Collisional Growth Conditions for Dust Aggregates. ApJ 702, no. 2 (September): 1490-1501. https://doi.org/10.1088/0004637X/702/2/1490. Walsh, Catherine, Hideko Nomura, and Ewine van Dishoeck. 2015. The molecular composition of the planet-forming regions of protoplanetary disks across the luminosity regime. A&A 582 (October): A88. http s : / / doi . org / 10 . 1051 / 0004 - 6361 / 201526751. arXiv: 1507 . 08544 [astro-ph.EP] . Weber, Philipp, Sebastián Pérez, Alice Zurlo, James Miley, Antonio Hales, Lucas Cieza, David Principe, et al. 2023. Spirals and Clumps in V960 Mon: Signs of Planet Formation via Gravitational Instability around an FU Ori Star? ApJ 952, no. 1 (July): L17. https://doi.org/10.3847/20418213/ace186. arXiv: 2307.13433 [astro-ph.EP] . Wei, Chen-En, Hideko Nomura, Jeong-Eun Lee, Wing-Huen Ip, Catherine Walsh, and T. J. Millar. 2019. The Effect of Carbon Grain Destruction on the Chemical Structure of Protoplanetary Disks. ApJ 870, no. 2 (January): 129. https://doi.org/10.3847/1538- 4357/aaf390. arXiv: 1811.10194 [astro-ph.EP] . Weidenschilling, S. J. 1977. Aerodynamics of solid bodies in the solar nebula. MNRAS 180 (July): 57-70. https://doi.org/10.1093/mnras/180.2.57. Weiner Mansfield, Megan, Michael R. Line, Joost P. Wardenier, Matteo Brogi, Jacob L. Bean, Hayley Beltz, Peter Smith, et al. 2024. The metallicity and carbon-to-oxygen ratio of the ultra-hot Jupiter WASP-76b from Gemini-S/IGRINS. arXiv e-prints (May): arXiv:2405.09769. https://doi. org/10.48550/arXiv.2405.09769. arXiv: 2405.09769 [astro-ph.EP] . Westley, M. S., R. A. Baragiola, R. E. Johnson, and G. A. Baratta. 1995. Photodesorption from low-temperature water ice in interstellar and circumsolar grains. Nature 373, no. 6513 (February): 405-407. https: //doi.org/10.1038/373405a0. Wierzchos, K., and M. Womack. 2018. C/2016 R2 (PANSTARRS): A Comet Rich in CO and Depleted in HCN. AJ 156, no. 1 (July): 34. https://doi. org/10.3847/1538-3881/aac6bc. arXiv: 1805.06918 [astro-ph.EP] . Winter, Andrew J., Myriam Benisty, and Sean M. Andrews. 2024. Planet formation regulated by galactic-scale interstellar turbulence. arXiv eprints (May): arXiv:2405.08451. arXiv: 2405.08451 [astro-ph.EP] . Wong, Michael H., Paul R. Mahaffy, Sushil K. Atreya, Hasso B. Niemann, and Tobias C. Owen. 2004. Updated Galileo probe mass spectrometer measurements of carbon, oxygen, nitrogen, and sulfur on Jupiter. Icarus 171, no. 1 (September): 153-170. https://doi.org/10.1016/j.icarus.2004. 04.010. Worthen, Kadin, Christine H. Chen, David R. Law, Cicero X. Lu, Kielan Hoch, Yiwei Chai, G. C. Sloan, et al. 2024. MIRI MRS Observations of β Pictoris. I. The Inner Dust, the Planet, and the Gas. ApJ 964, no. 2 (April): 168. https://doi.org/10.3847/1538-4357/ad2354. Xue, Qiao, Jacob L. Bean, Michael Zhang, Luis Welbanks, Jonathan Lunine, and Prune August. 2024. JWST Transmission Spectroscopy of HD 209458b: A Supersolar Metallicity, a Very Low C/O, and No Evidence of CH4, HCN, or C2H2. ApJ 963, no. 1 (March): L5. https://doi.org/ 10.3847/2041-8213/ad2682. arXiv: 2310.03245 [astro-ph.EP] . Yang, Chao-Chin, Anders Johansen, and Daniel Carrera. 2017. Concentrating small particles in protoplanetary disks through the streaming instability. A&A 606 (October): A80. https : / / doi . org / 10 . 1051 / 0004 - 6361 / 201630106. arXiv: 1611.07014 [astro-ph.EP] . Youdin, Andrew N., and Jeremy Goodman. 2005. Streaming Instabilities in Protoplanetary Disks. ApJ 620, no. 1 (February): 459-469. https: //doi.org/10.1086/426895. arXiv: astro-ph/0409263 [astro-ph] . Zhang, Yapeng, Ignas A. G. Snellen, Alexander J. Bohn, Paul Mollière, Christian Ginski, H. Jens Hoeijmakers, Matthew A. Kenworthy, et al. 2021. The 13 CO-rich atmosphere of a young accreting super-Jupiter. Nature 595, no. 7867 (July): 370-372. https://doi.org/10.1038/s41586-02103616-x. arXiv: 2107.06297 [astro-ph.EP] . Zhu, Zhaohuan, Lee Hartmann, Charles F. Gammie, Laura G. Book, Jacob B. Simon, and Eric Engelhard. 2010. Long-term Evolution of Protostellar and Protoplanetary Disks. I. Outbursts. ApJ 713, no. 2 (April): 11341142. https : / / doi . org / 10 . 1088 / 0004 - 637X / 713 / 2 / 1134. arXiv: 1003.1759 [astro-ph.SR] . Zhu, Zhaohuan, Yan-Fei Jiang, and James M. Stone. 2020. Global 3D radiation magnetohydrodynamic simulations for FU Ori's accretion disc and observational signatures of magnetic fields. MNRAS 495, no. 3 (January): 3494-3514. https://doi.org/10.1093/mnras/staa952. arXiv: 1912.01632 [astro-ph.EP] .","pages":[23,24,25,26,27,28,29]}],"string":"[\n {\n \"title\": \"C/O ratios in self-gravitating protoplanetary discs with dust evolution\",\n \"content\": \"Tamara Molyarova, 1,2 Eduard Vorobyov, 1,2 and Vitaly Akimkin 1 1 Institute of Astronomy, Russian Academy of Sciences, 48 Pyatnitskaya St., Moscow, 119017, Russia 2 Research Institute of Physics, Southern Federal University, Stachki Ave. 194, Rostov-on-Don 344090, Russia Author for correspondence: Tamara Molyarova, Email: moliarova@sfedu.ru.\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"Abstract\",\n \"content\": \"Elemental abundances, particularly the C/O ratio, are seen as a way to connect the composition of planetary atmospheres with planet formation scenario and the disc chemical environment. We model the chemical composition of gas and ices in a self-gravitating disc on timescales of 0.5 Myr since its formation to study the evolution of C/O ratio due to dust dynamics and growth, and phase transitions of the volatile species. We use the thin-disc hydrodynamic code FEOSAD, which includes disc self-gravity, thermal balance, dust evolution and turbulent diffusion, and treats dust as a dynamically different and evolving component interacting with the gas. It also describes freeze-out, sublimation and advection of four most abundant volatile species: H2O, CO2, CH4 and CO. We demonstrate the effect of gas and dust substructures such as spirals and rings on the distribution of volatiles and C/O ratios, including the formation of multiple snowlines of one species, and point out the anticorrelation between dust-to-gas ratio and total C/O ratio emerging due to the contribution of oxygen-rich ice mantles. We identify time and spatial locations where two distinct trigger mechanisms for planet formation are operating and differentiate them by C/O ratio range: wide range of the C/O ratios of 0 - 1.4 for streaming instability, and a much narrower range 0.3 - 0.6 for gravitational instability (with the initial value of 0.34). This conclusion is corroborated by observations, showing that transiting exoplanets, which possibly experienced migration through a variety of disc conditions, have significantly larger spread of C/O in comparison with directly imaged exoplanets likely formed in gravitationally unstable outer disk regions. We show that the ice-phase C/O ≈ 0.2 - 0.3 between the CO, CO2 and CH4 snowlines corresponds to the composition of the Solar system comets, that represent primordial planetesimals. Keywords: protoplanetary disc, volatiles, dust evolution\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1. Introduction\",\n \"content\": \"The protoplanetary disc matter can be roughly divided into three component: gaseous chemical species, solid dust particles, and icy mantles covering the surface of dust grains. Gas and solid particles become dynamically decoupled, as evolving dust grows and acquires relative velocities leading to the redistribution of elements in the disc and between the phases, and creating the premises for different chemical environments. When planets start to form, their properties, including chemical composition of the atmosphere, are inevitably affected by the location and the mechanism of their formation. This suggests that the origin of (exo)planets might be investigated using their observed chemical composition, and makes understanding the disc chemical evolution vital for creating a consistent planet formation theory. One of the key parameters that govern the chemical setup of a planetary atmosphere is the relation between the abundances of carbon and oxygen, often referred to as carbon-tooxygen ratio (hereafter C/O ratio). The variations of C/O ratio in the ice and gas phases at the snowlines of main disc volatiles (CO, CO 2 , and H 2 O) and the prospects of connecting them to planet formation were discussed in Öberg, Murray-Clay, and Bergin (2011) within a qualitative freeze-out model. Since then, C/O ratio received a lot of attention in this context. It was thoroughly investigated in modelling (see, e.g., Booth et al. 2017; Eistrup, Walsh, and van Dishoeck 2018; Cridland, Eistrup, and van Dishoeck 2019; Cridland et al. 2019; Crid- and, Bosman, and van Dishoeck 2020; Cridland et al. 2020; Krijt et al. 2020; Turrini et al. 2021; Schneider and Bitsch 2021). The connection of disc chemical composition with C/O in exoplanetary atmospheres was modelled using core accretion model (Thiabaud et al. 2015) and 'chain' planet population synthesis model (Mordasini et al. 2016). Paul Mollière et al. (2022) considered a simple formation retrieval pipeline and found that this task requires careful consideration of the model assumptions. The measurements of molecular abundances in the atmospheres of giant exoplanets obtained by a variety of modern facilities, such as HST, Spitzer, VLTI, JWST, Gemini, indicate a diversity of C/O ratios: from low C/O ratio values ( ≈ 0.4, below the solar value of 0.54; Benneke et al. 2019; GRAVITY Collaboration et al. 2020; Worthen et al. 2024; Xue et al. 2024) to stellar ( ≈ 0.5, close to solar; P. Mollière et al. 2020; Zhang et al. 2021; Smith et al. 2024) and close to or above unity (Swain et al. 2009; Madhusudhan et al. 2011). A variety of solar and super-solar C/O ratios is observed in four planets within the HR8799 system (Nasedkin et al. 2024). Chemical composition of the atmospheres of many hot Jupiters indicates high C/O>1 of the forming material (Moses et al. 2013). There is an observational evidence of young planets in PDS 70 disc accreting the material with C/O>1 (Facchini et al. 2021). The number of exoplanets with constrained atmospheric C/O ratios grows with large studies of multiple planets such as Changeat et al. (2022), which allows us to make some statistical conclusions. The population study of C/O ratios in exoplanetary atmospheres reveals that there are two populations with different elemental ratio, which are likely formed in different mechanisms (Hoch et al. 2023). Khorshid, Min, and Désert (2023) were able to restrict the formation scenario for WASP-77b based on the measured C/O ratio of the planet (Line et al. 2021) and the modelling of planet formation and migration. Elemental abundances in protoplanetary discs can be constrained from observations (Fedele and Favre 2020), and C/O ratio in the gas can be estimated. Spatially resolved observations can help distinguish between different C/O ratios spectroscopically (Matter, Pignatale, and Lopez 2020). Cleeves et al. (2018) report C/O ≈ 0.8 in the molecular layer of IM Lup disc. ALMA observations of hydrocarbons and sulphur-bearing species indicate C/O > 1 in the upper disc layers and in the outer disc in TW Hya and DM Tau (Dutrey et al. 2011; Bergin et al. 2016; Semenov et al. 2018) and for a population of discs in Lupus (Miotello et al. 2019). For the nearby discs the solar elemental composition with C/O ≈ 0.54 is usually expected, thus the observed higher values confirm redistribution of carbon and oxygen in discs. In addition to high C/O in disc atmospheres, there is evidence of both carbon and oxygen depletion from gas (Kama et al. 2016; Miotello et al. 2019). However, some of heavy oxygen carriers might not be observable, leading to overestimated C/O in disc observations (Walsh, Nomura, and van Dishoeck 2015). The volatile composition is also used to constrain the origin of bodies in the Solar System. Fraction of CO and CO 2 ices relative to water in cometary comae indicate their formation between the CO and CO 2 snowlines or exterior to the CO snowline (A'Hearn et al. 2012; Seligman et al. 2022). Abundances of CO and N 2 ices were used to analyse the original location of Pluto and Triton (Mousis, Anderson, et al. 2024). Observed elemental abundances were used to constrain the Jupiter formation scenario (Lodders 2004), relying also on abundances of nitrogen (Öberg and Wordsworth 2019; Bosman, Cridland, and Miguel 2019) and chemically inactive species like Ar. However, the model assumptions can lead to different interpretation of the observations: while Öberg and Wordsworth (2019) and Bosman, Cridland, and Miguel (2019) suggest that Jupiter formed outside N 2 snowline (at > 30 au), Ohno and Ueda (2021) consider the concept disc shadow, which allows Jupiter to form near its current location. Chemical processes other than freeze-out and sublimation at the snowlines can alter the composition of ice and gas as well. Due to gas-phase and surface reactions, snowlines can become important for the redistribution of elements. More detailed chemical modelling shows that the C/O ratio in the gas and in the ice depends also on the initial chemical setup and ionisation by cosmic rays and radioactive nuclei (Eistrup, Walsh, and van Dishoeck 2016, 2018). It directly affects the interpretation of observations. Another essential chemical process is CO depletion from the gas, resulting in its transformation to CO 2 ice (Bosman, Tielens, and van Dishoeck 2018). For example, stellar C/O ratio in the atmosphere of HR 8799e indicates that the planet accreted its material beyond CO snowline ( ≈ 45 au), but chemical modelling suggests that due to CO depletion, the C/O in the ice already approaches the stellar ratio beyond CO 2 snowline ( ≈ 20 au), which is closer to the star (P. Mollière et al. 2020). Another key process affecting the elemental ratio is dust drift, which leads to spatial segregation between the chemical constituents of the gas and the grains covered with ice. The distribution of CO in the gas and ice phases was studied within dynamical models of dust evolution (Stammler et al. 2017; Krijt et al. 2018). Even without chemical processes, dust evolution and dynamics can substantially alter C/O ratio in the atmospheres of forming planets (Booth et al. 2017). Some models combine chemical reactions treatment with dust evolution and transport, usually within 1D viscous models. Dust transport can have a strong effect on the abundances of volatiles in the inner disc regions (Bosman, Tielens, and van Dishoeck 2018). However, for discs with low turbulence and high cosmic ray ionisation rate, C/O ratio is rather defined by chemical evolution (Booth and Ilee 2019). In our previous work (Molyarova et al. 2021) we showed that the volatile species tend to concentrate around their snowlines both in the gas and more notably on the dust surface. This accumulation was found to be caused by effective transport of volatiles through the snowlines by azimuthal variations in the gas and dust radial and angular velocity, an effect that cannot be captured in 1D viscous disc models. Such accumulation should immediately affect the local C/O ratio, which suggests the connection between the snowlines of various volatiles and the formation of planets with altered C/O in their atmospheres. In this work, we follow the distribution of the main volatile species in the disc to investigate the distribution of C/O ratio in gas and ice in a 2D thin-disc hydrodynamic model. We study the effect of dust growth and dynamics on the elemental ratios and consider the role of the initial mass of the collapsing core on the distribution of volatiles. The paper is organised as follows. The main features of the used FEOSAD model are described in Section 2, with the details of the treatment of the volatiles given in Section 2.3. In Section 3 we describe the results of the simulations, focusing on distribution of the volatiles in Section 3.2, the C/O ratios in Section 3.3, and their evolution in Section 3.4. In Section 4, we discuss the implications of our results in the context of planet formation via different mechanisms. The main conclusions are listed in Section 5.\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"2. Model\",\n \"content\": \"We use the global model of protoplanetary disc formation FEOSAD (Vorobyov et al. 2018), which includes disc selfgravity, dust evolution and interaction with gas (including backreaction of dust on gas), turbulent viscosity, adiabatic and radiative cooling and heating. It describes the formation of a protostar and a protoplanetary disc from a collapsing cloud in a 2D thin-disc approach. The model also includes freeze-out of main volatile species as in Molyarova et al. (2021), with the feedback from ice mantles on dust evolution via fragmentation velocity. Here we summarise the key characteristics of the model, more details can be found in the previous works (Vorobyov et al. 2018; Molyarova et al. 2021; Kadam, Vorobyov, and Basu 2022). The main difference from our previous study in Molyarova et al. (2021) is that here we consider the formation of dead zones via variable α -parameter of Shakura and Sunyaev and also include turbulent diffusion.\",\n \"pages\": [\n 2,\n 3\n ]\n },\n {\n \"title\": \"2.1 Gas evolution\",\n \"content\": \"For the gas component, the hydrodynamic equations for mass, momentum, and internal energy conservation are the following where subscripts p and p ' denote the planar components ( r , ϕ ) in polar coordinates, Σ g is the gas mass surface density, e is the internal energy per surface area, P is the vertically integrated gas pressure calculated via the ideal equation of state as P = ( γ - 1) e with γ = 7/5, f is the friction force between gas and dust, v = vr ˆ r + v ϕ ˆ ϕ is the gas velocity in the disc plane, and ∇ = ˆ r ∂ / ∂ r + ˆ ϕ r -1 ∂ / ∂ϕ is the gradient along the planar coordinates of the disc. The gravitational acceleration in the disc plane, g = gr ˆ r + g ϕ ˆ ϕ , includes the gravity of the central protostar when formed and takes into account disc self-gravity of both gas and dust found by solving the Poisson integral (Binney and Tremaine 1987). Theconsideration of time-dependent energy balance (Eq. (3)) allows us to accurately calculate the midplane temperature T mp and is particularly important to describe the phase state of the volatiles and the level of turbulent viscosity. The terms Λ and Γ describe the rates of dust cooling and heating, respectively, by stellar and background irradiation. They are calculated based on the analytical solution of the radiation transfer equations in the vertical direction (Dong et al. 2016; Vorobyov et al. 2018) Here, σ is the Stefan-Boltzmann constant, τ P and τ R are the Planck and Rosseland mean optical depths to the disc midplane, calculated as τ = κΣ dust from Planck and Rosseland mean opacities κ P and κ R (Semenov et al. 2003) and total dust surface density Σ dust . Gas and dust temperatures are assumed to be equal, and the midplane temperature is linked with gas pressure as T mp = P µ / R Σ g, where µ = 2.3 is the mean molecular weight of the gas, and R is the universal gas constant. The irradiation temperature at the disc surface T irr is determined by both stellar and background irradiation. Stellar irradiation includes the luminosity from the photosphere of the protostar and accretion luminosity. The background radiation is assumed as a black body with the temperature of 15 K. For more details on the irradiation we refer to Vorobyov et al. (2018). Turbulent viscosity is described using the common α -parameter approach of Shakura and Sunyaev (1973). It is taken into account via the viscous stress tensor Π (see Vorobyov and Basu 2010, for explicit expressions for the components of the terms with Π ). The magnitude of kinematic viscosity is ν = α c s H , where c s is the sound speed and H is the vertical scale height of the gas disc calculated using an assumption of local hydrostatic equilibrium of a self-gravitating disc (see Vorobyov and Basu 2010, Appendix A). Here, we use the adaptive α approach implying accretion through a layered disc (Gammie 1996; Armitage, Livio, and Pringle 2001; Kadam, Vorobyov, and Basu 2022). Turbulence is assumed to be generated by magneto-rotational instability (MRI) which only develops in layers of the disc where the ionisation level is high enough. The MRI-active layer is characterised by its surface density Σ MRI and relatively high value of turbulent viscosity α MRI = 10 -3 . As thermal and photo-ionisation are not efficient enough for the relatively cold and dense matter in the disc at >0.5 au, the main process determining the thickness of the MRI-active layer is ionisation by cosmic rays. It is assumed to be constant Σ MRI = 100g cm -2 , which is the typical depth of Galactic cosmic rays penetration in the ISM (Umebayashi and Nakano 1981) and in protoplanetary discs (Padovani, Galli, and Glassgold 2009). The dead zone is characterised by surface density from the midplane Σ dz = Σ g/2Σ MRI with residual turbulence α dz . The turbulence in this layer is only hydrodynamic turbulence driven by the Maxwell stress in the active layer, and small value α dz = 10 -5 is adopted (Okuzumi and Hirose 2011). However, if local temperature exceeds the critical value of 1300 K, thermal ionisation becomes possible, the MRI develops and the dead zone is no longer dead, in which case α dz = 10 -1 (Zhu et al. 2010; Kadam et al. 2019). This value is higher than α MRI in the outer disc due to the different ionisation processes and local conditions. In the outer disc, the MRI can be suppressed by non-ideal magneto-hydrodynamic (MHD) effects such as ambipolar diffusion and Ohmic resistivity (Bai and Stone 2013; Gressel et al. 2015). In the dead zone in the inner disc, α can reach higher values when the MRI is triggered by thermal ionisation, as shown by 3D MHD simulations (Zhu, Jiang, and Stone 2020). The adopted parameterization makes use of an effective parameter α eff , which at any given location is calculated as\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"2.2 Dust evolution\",\n \"content\": \"Dust is described as consisting of two components: small grains that are dynamically coupled to the gas, with the mass surface density Σ d,sm , and grown grains with the mass surface density Σ d,gr that can move relative to the gas and change in size. The total dust surface density necessary for the calculation of the optical depths for Eq. (4) is Σ dust = ( Σ d,gr + Σ d,sm )/2. The factor 1/2 appears as the optical depths is calculated to the midplane. Each dust population has a power-law size distribution f ( a ) = dN / da = Ca -p with a normalisation constant C and a fixed exponent p = 3.5. Small dust has sizes between a min = 5 × 10 -7 cm and a ∗ = 10 -4 cm, grown dust has sizes between a ∗ and a max, which can vary due to dust coagulation and fragmentation. Dynamics of these dust components follows the continuity and momentum equations where u is the grown dust velocity. The term S ( a max) is responsible for the exchange of matter between the dust populations, as dust is converted from the small to grown component due to coagulation and back due to fragmentation. The details of the dust evolution model are presented in Vorobyov et al. (2018). The last term in Eqs. (6) and (7) is responsible for dust turbulent diffusion, similar to Vorobyov, Elbakyan, et al. (2020). The coefficient of turbulent diffusion D is related to the kinematic viscosity ν as D = ν /(1 + St 2 ) (Birnstiel 2023). Diffusion affects dust grains along with their ice mantles, as well as the gas-phase species (see Section 2.3). The innermost regions of the disc are challenging to simulate explicitly due to the Courant criterion: in the highly dynamic inner regions (fraction of au) the timescales are so short that the code demands very small time step in order to preserve stability. Therefore, the inner regions are represented by a sink cell, with a carefully chosen inflow-outflow boundary condition at the sink cell and a parametric description of the accretion onto the star (see Vorobyov et al. 2018, for details). In the simulations presented below, the radius of the sink cell is 0.62 au. We consider two disc models with different initial mass of the collapsing cloud, 0.66 and 1 M ⊙ . We note that about 10% of the gas mass that crosses the sink cell is assumed to be evacuated by jets and outflows, and the other 90% lands on to the star. A small amount of mass remains in the envelope by the end of simulations. In both models, the initial gas temperature T init = 15 K and the ratio of rotational to gravitational energy β = 0.28%. The simulations start from the collapse of a molecular cloud, with only small dust grains. The simulation continues until the age of the system becomes equal to 0.5 Myr. Masses of the central protostar and the disc by the end of simulation are M ⋆ = 0.4 M ⊙ and M disc = 0.22 M ⊙ in model M1 and M ⋆ = 0.58 M ⊙ and M disc = 0.35 M ⊙ in model M2. The disc masses are around 0.5 stellar masses, which makes them essentially self-gravitating. A number of recent studies develop the idea that the mass infall from the ambient ISM continues during the lifetime of the disc, including the Class II stage (Padoan et al. 2024; Pelkonen et al. 2024; Winter, Benisty, and Andrews 2024). Such models describe a Bondi-Hoyle accretion regime and are in good agreement with the observed properties of the disc population, such as accretion rates, masses and sizes. This input of matter can play an important role in disc evolution and planet formation process (Vorobyov, Lin, and Guedel 2015; Vorobyov et al. 2016). In the FEOSAD model, the mass infall to the disc can be accounted for (Vorobyov, Lin, and Guedel 2015; Vorobyov et al. 2016), but in the present simulation this effect is not included. The mass infall from the envelope continues until the cloud is depleted of matter. Because of the thin-disc geometry, the gravitational contraction of the cloud proceeds in the plane of the disc. The matter is landing on the disc outer edge and is transported towards the star by the combined action of gravitational and viscous torques. The infall on the disc inner regions is therefore neglected. This is a reasonable approximation, considering that most of the matter and angular momentum in a three-dimensional cloud is located at relatively large polar angles and a flared outer edge of the disc intercepts most of them (see, e.g., Visser et al. 2009, Figure 1). In our modelling, we do not consider the continuous Bondi-Hoyle accretion and concentrate on the internal disc processes.\",\n \"pages\": [\n 3,\n 4\n ]\n },\n {\n \"title\": \"2.3 Evolution of volatiles\",\n \"content\": \"We follow the evolution of four main volatile species: H 2 O, CO 2 , CO and CH 4 . These are the most abundant carbon- and oxygen-bearing ices observed in protostellar cores (Karin I. Öberg et al. 2011). In the model, each of these species can be present in three states: in the gas, in the ice on the surface of small dust, and in the ice on the surface of grown dust. Each species s is described by its surface density in the gas Σ gas s , on small dust Σ sm s , and on grown dust Σ gr s . Their distributions in the disc can change through three main processes: advection together with the corresponding component (gas or small/grown dust); exchange of mantles between dust populations due to grain collisions; phase transitions, including adsorption from gas to dust, and thermal and photo-desorption. Initially, all ices are on small grains and no volatiles present in the gas. The treatment of volatiles is adopted from Molyarova et al. (2021), who describe the models in more details. Here we recap main features of the chemical model. Our chemical model only describes phase transitions, i.e. adsorption and desorption, which includes thermal desorption and photo-desorption by interstellar UV radiation. These reactions were shown to be he most important for gas-phase abundances of most species (Ilee et al. 2011). Due to high computational costs, no other chemical processes, either gasphase or surface reactions are included, although they also may have significant effect on the composition of both ice and gas (Semenov and Wiebe 2011). The chemical evolution of the surface densities of volatile species is calculated from the system of equations where the mass rate coefficients per disc unit area of adsorption λ s and desorption η s for the species s are calculated for local conditions at every hydrodynamic step, separately for small and grown dust populations. Eqs. (9)-(11) are solved in two steps: first, the right-hand side is considered without the advection term. The case of pure adsorption/desorption represented by the right-hand side of the equation can be solved analytically (see Appendix A in Molyarova et al. 2021). This is done at every hydrodynamic step, before the dust growth step, when ices on small and grown grains are redistributed proportionally to mass exchange between the dust populations. This is followed by a transport step, when the surface densities of the volatiles are changed according to the fluxes of their respective gas or dust components between the cells. Restricting chemical processes to only adsorption and desorption allows the chemical step to be calculated fast, which is very important for computationally demanding hydrodynamic simulations. For each dust population, the desorption rate is a sum of thermal desorption and photo-desorption η = η td + η pd (the indices 'sm' and 'gr' are omitted for convenience). We operate under the assumption of zeroth-order desorption, i.e. the desorption rate does not depend on the present amount of ice ( Σ sm s or Σ gr s ). It implies that only the upper layers of the ice mantle are able to sublimate, which is a more appropriate approach for thick mantles. This assumption better describes desorption of CO and H 2 Oin temperature programmed desorption (TPD) experiments (Fraser et al. 2001; K. I. Öberg et al. 2005; Bisschop et al. 2006) than first-order desorption, which is more suitable for thin mantles of several monolayers. The rate of thermal desorption is calculated as where N ss = 10 15 cm -2 is the surface density of binding sites (Cuppen et al. 2017), E b (K) is the binding energy of the species to the surface, µ s is the species molecular mass, m p is atomic mass unit, k B is the Boltzmann constant. We follow Hasegawa and Herbst (1993) and use binding energy to calculate the pre-exponential factor in Eq. (12), the same way it was done in Molyarova et al. (2021). However, this approach was recently demonstrated by Minissale et al. (2022) to underestimate the pre-exponential factor by a few orders of magnitude, as it does not account for the rotational part of the partition functions of desorbing molecules. Total surface area of dust grains (small or grown) per disc unit area ˜ σ tot (cm 2 cm -2 ) is calculated for the adopted power-law size distribution with p = 3.5 as Here, ρ s = 3 g cm -3 is the material density of silicate cores of the dust grains. The model includes photodesorption of volatiles by interstellar irradiation, which is mostly relevant in the outer disc regions with lower optical depth. We do not consider the UV radiation field produced by the star and the accretion region as a source of photodesorption, assuming that they do not reach disc midplane due to high optical depth. The photo-desorption rate is calculated as where F UV = F UV 0 G UV (photons cm -2 s -1 ) is the UV photon flux expressed in the units of standard UV field and Y = 3.5 × 10 -3 + 0.13 exp ( -336K/ T mp ) (mol photon -1 ) is the photodesorption yield adopted from Westley et al. (1995) for water ice. The intensity of the interstellar UV radiation field with G 0 = 1 is F UV 0 = 4.63 × 10 7 photon cm -2 s -1 (Draine 1978). We assume that the disc situated in a star-forming region is illuminated by a slightly elevated unattenuated UV field with G env = 5.5 G 0 . For the disc midplane, which is illuminated from above and below, this field scales with the UV optical depth τ UV towards the disc midplane as The optical depth can be calculated as τ UV = 0.5( κ sm Σ d,sm + κ gr Σ d,gr ), where κ sm = 10 4 cm 2 g -1 , κ gr = 2 × 10 2 cm 2 g -1 are typical values of absorption coefficients in the UV for small and grown grains (Pavlyuchenkov et al. 2019, Fig. 1). We calculate the adsorption rate λ following Brown and Charnley (1990). It is proportional to the total cross-section of dust grains per unit volume, which can be obtained from the total surface area of dust grains per unit disc surface ˜ σ tot. To change the normalisation to the 2D case, ˜ σ tot needs to be multiplied by 1/2 H . Another factor of 1/4 follows from the difference between cross-section and surface area of a sphere. As a result, the adsorption rate is calculated as A more detailed derivation of rate coefficients for adsorption and desorption is presented in Section 2.3 of Molyarova et al. (2021). Table 1 summarises the molecular parameters used in both models in this work. Binding energies for H 2 O, CO 2 , and CO are based on the experimental data from Cuppen et al. (2017) for desorption from crystalline water ice. The binding energy for methane is taken from Aikawa et al. (1996). The values of the initial abundances relative to water f s are based on the observations of ices in protostellar cores around the low-mass protostars (Karin I. Öberg et al. 2011). They are transformed to the mass fraction relative to gas assuming the water abundance of 5 × 10 -5 relative to gas number density. Total initial mass of ices comprises ≈ 8.5% of the total mass of refractory grain cores. This is relatively low compared to the estimates suggesting comparable masses of ices and refractories in the discs (Pontoppidan et al. 2014). However, the lower fraction of ices is more suitable in our approach, that suggests that ice mantles do not change mass and radius of dust grains (Molyarova et al. 2021). As we are mostly interested in the elemental ratios and relative abundances of the considered ices, lower ice fraction is an appropriate simplification. However, we note that dust dynamics can lead to accumulation of ices and ice mantles exceeding masses of silicate cores in some disc regions, as was shown previously in Molyarova et al. (2021). Ice mantles also provide feedback on dust evolution. The model includes the effect of ices on fragmentation velocity v frag , which is the the maximum collision velocity leading to sticking instead of fragmentation. According to laboratory experiments, icy grains have higher v frag than bare silicate grains by an order of magnitude (Wada et al. 2009; Gundlach and Blum 2015). In Molyarova et al. (2021) we used the values of fragmentation velocity v frag = 1.5 and 15 m s -1 for bare and icy grains, correspondingly. Here we follow Okuzumi and Tazaki (2019) and adopt lower values of v frag = 0.5 and 5 m s -1 , which are more relevant for grains consisting of µ m-sized monomers. These lower values of v frag will lead to higher importance of fragmentation compared to Molyarova et al. (2021). To determine if a dust grain should be considered icy or bare, we compare the local total surface density of all ices on grown dust divided by Σ d,gr with the threshold value K , which is calculated as i.e., an icy grain must have at least one monolayer of ice. Here, a ml is the thickness of the ice monolayer estimated as the size of a water molecule 3 × 10 -8 cm. The material density of ice ρ ice = 1 g cm -3 and the mean radius of a grown grain is calculated as √ a ∗ a max for the power-law distribution with p = 3.5.\",\n \"pages\": [\n 4,\n 5,\n 6\n ]\n },\n {\n \"title\": \"3. Results\",\n \"content\": \"To understand the distribution of the species, we first need to consider the global evolution of the disc and its structure. The distribution of volatiles and the C/O ratio is very sensitive to gas and dust substructures appearing in the disc. Variations in temperature and pressure lead to the complex shape and temporal evolution of the snowlines. The dependence of dust fragmentation velocity on the presence of ice mantles implies the feedback from the volatiles on dust and (through backreaction) on gas. Our modelling starts with the gravitational collapse of a flattened, slowly rotating molecular cloud. The protoplanetary disc is formed after the formation of the protostar, when the in-spiraling layers of the contracting cloud hit the centrifugal barrier near the inner edge of the sink cell, at a time instance depending on the initial core mass. The disc and the protostar are formed ≈ 53 kyr after the beginning of the cloud collapse in model M1 and at ≈ 78kyr in model M2. If not stated otherwise, times are specified counting from the beginning of the simulation, e.g. the 100 kyr time instance for model M1 refers to a ≈ 50kyr old disc, as the first stage includes cloud collapse as well. An important characteristic of young stellar objects is their variable accretion rate. Our modelling produces accretion bursts with the magnitude of ∼ 100 L ⊙ occurring every ∼ 10 4 years during the first hundred thousands years of disc evolution. These burst parameters are in line with the episodic accretion scenario (Hartmann and Kenyon 1985) and resemble the observed phenomenon of FU Ori type stars (see Audard et al. 2014, for a review). The luminosity outbursts heat up the disc and typically shift the snowlines further away from the star. Although this effect is temporary, it can be reflected in the distributions of the volatiles, and leave long-term imprints in the observed dust properties (Vorobyov et al. 2022). The detailed analysis of the effect of such outbursts on the distribution of the elements is worthy of a separate study and lies beyond the scope of this paper.\",\n \"pages\": [\n 6\n ]\n },\n {\n \"title\": \"3.1 Dust and gas structures\",\n \"content\": \"During the first hundred thousands years of evolution, protoplanetary discs change from highly asymmetric and dynamic objects to nearly axisymmetric and slowly evolving structures. Figure 1 shows the examples of different gas and dust substructures that are present in the disc at different stages. The snapshots are shown for model M1, they include a young disc (160 kyr), an intermediate state (350 kyr), and a more evolved and axisymmetric disc (490 kyr). Each of the selected time instances represent some characteristic morphological features addressed below. In model M2, similar structures appear, although sometimes at different evolution times. In this subsection, we consider model M1 as an example and discuss these features, highlighting the properties of dust and gas substructures that are most relevant for the distribution of the volatiles. The earliest phases of disc evolution are characterised by a large-scale spiral structure in both dust and gas, as well as episodic appearance of clumps. They are the result of gravitational instability (GI) in a massive disc, as in our modelling, the disc mass comprises more than 0.1 of the stellar mass, which is roughly a threshold of the disc stability against GI (Vorobyov 2013; Kratter and Lodato 2016). This is illustrated by the Toomre Q -parameter (Toomre 1964) in the third column of Figure 1: inside the spirals and the clump Q < 1, which indicates the dominance of self-gravity over Keplerian sheer and gas density. The spiral structures become less prominent with time as the disc looses mass and the Q -parameter increases. However, spirals persist in the model throughout the disc evolution up to 0.5 Myr. For example, at 350 kyr, a very tight spiral is present in the gas, starting at the gas and dust ring at ≈ 10au. At 490 kyr, the spiral pattern is weak and exists at > 10 au distances, which are not displayed in Figure 1. One- or two-armed grand design spirals are common 100 - 200 kyr after the disc formation in the models. The analogues of such spirals in the observed young protoplanetary discs around low-mass stars are found, for example, in Elias 227 or WaOph 6 (Pérez et al. 2016; Huang, Andrews, Pérez, et al. 2018). It is not yet clear if the observed spirals are the result of GI or caused by a perturbation from a companion planet or a (sub)stellar object (Meru et al. 2017; Brown-Sevilla et al. 2021). A spiral with multiple clumps produced by GI was recently observed in the disc around a FUor V960 Mon (Weber et al. 2023). Another important feature of gas and dust spatial distributions is ring-like structures at various scales. The system of rings starts to form as early as 100 kyr, and develops to the high-contrast multiple ring structure (1 - 3 orders of magnitude difference between surface densities in rings and gaps), which is evident in the middle and bottom rows of Figure 1. While gas rings are also common, the annual structures are more prominent in the dust surface density, as well as in dust size. Some of the dust rings correspond spatially to the gas rings, while others are barely reflected in the gas distribution. Overall, the difference between the gas and dust structures develops with time, as the result of dust growth and drift (Testi et al. 2014). Only some particular substructures, such as the ring between 1 - 2 au, are present in the distributions of both gas and dust components. Another location where prominent rings form in both dust and gas is the water snowline. Here, the water snowline is defined as the location in the disc midplane where the amount of water in the gas equals to its amount in the ice (on both dust populations). There are multiple location in the disc where this happens. Water is frozen in most of the disc beyond 5 - 10 au, and we will refer to the furthest snowline dividing these outer frozen region from the inner one with the gas-phase water as a primary snowline. Generally, the primary snowline is roughly circular, but it can have asymmetries due to the spiral structure and an additional snowline may appear, e.g., around a gravitationally bound clump (upper rows in Figure 1). Besides, at ≈ 260kyr, another region rich in water ice appears in the inner disc, creating a pattern of double or triple water snowline at later times (middle and lower rows in Figure 1). We will refer to these inner additional snowlines as the secondary snowlines. They circumcise a cold and dense gas-dust ring that forms at 1 - 2 au. As the disc cools down with time, the primary snowline position moves from ≈ 10au distance at 160 kyr, to ≈ 6au at 350 kyr and ≈ 4au at 490 kyr. Snowlines are known to be associated with the enhancement of dust and volatiles (Stevenson and Lunine 1988; Cuzzi and Zahnle 2004; Drążkowska and Alibert 2017; Molyarova et al. 2021). Dust enhancement was also shown to affect the distribution of gas and its accretion rate through the disc (Gárate et al. 2020) by means of dust back reaction, which is also accounted for in our modelling. In our models, an about 2 au wide dust ring is formed at the inner edge of the water snowline (at ≈ 10au) as soon as 50 kyr after the disc formation. Grown dust grains drifting towards the star through the snowline lose their mantles, their fragmentation velocity drops, rendering them more vulnerable to fragmentation. Consequently, the grain maximum size a max decreases, their drift towards the star slows down, which leads to the accumulation of grown dust, as well as small dust as a product of fragmentation. Increase of total dust density also leads to less efficient cooling and results in higher temperature inside the snowline (see the right panels in Figure 1). Later, at times > 200 kyr, several additional rings form outside the water snowline at distances up to 100 au, the most notable one being 1 - 2 au exterior to the primary snowline. Dust is trapped inside the gas pressure maxima in these rings, which is a self-supporting phenomenon as the temperature also increases inside the dust ring due to high optical depth. These rings are worthy of attention in the context of possible planet formation. Dust surface density and size are higher in the rings, with a max reaching centimetres, dust-to-gas ratio up to 0.1 - 0.2, and Stokes number up to 0.05 - 0.1. This could ease the development of the streaming instability, which typically requires pebble-size grains with St ≳ 0.01 and dustto-gas ratio ≳ 0.02 (Carrera, Johansen, and Davies 2015). Multiple ring-like structures are commonly observed in protoplanetary discs at a range of ages and display a variety of examples, with different widths, contrasts and numbers of rings (Huang, Andrews, Dullemond, et al. 2018, and many others). However, to directly compare the ring structures in the simulated dust surface density with the observed dust emission, require radiative transfer modelling is needed. Some of the observed ring structures could be produced by radial variation in dust size even in the absence of gaps in dust surface density (Akimkin et al. 2020). The most prominent dust rings are located in the vicinity of the water snowline: the ring outside the primary snowline at 5 - 8 au (depending on the time) and the ring at 1 - 2 au, inside the primary snowline, which at later times also contains water ice and additional snowlines. The main effect of the snowlines is the change in fragmentation velocity between mantled and non-mantled grains: dust size sharply decreases by ≈ 2 orders of magnitude for the latter. Immediately outside the water snowline, the midplane temperature is lower, due to lower dust opacity and thus more efficient cooling. Turbulent α is on the contrary, higher, providing more efficient radial transport of matter. It leads to lowering the gas surface density in this gap, which in turn increases α (see Eq. (5)), creating the positive feedback and further deepening the gap. One of the possible mechanisms to create the initial decrease in the gas surface density is dust back-reaction, which can affect the inward flow of gas at the snowline. This effect was investigated by Gárate et al. (2020) for different initial dust-to-gas ratios. The radial variation of α itself lead to the appearance of gas substructures (Tong, Alexander, and Rosotti 2024). The combined effect of lower temperature and density creates a pressure minimum, which dust tends to avoid. The dead zone, where α -parameter values are lower than 10 -3 , includes the ice-free inner disc and has an approximately two times larger radial span than the iceless region. The distribution of α -parameter is shown in the right column of Figure 1. Inside the primary water snowline, the values of α are the lowest due to high surface density of gas. The dead zone is not axisymmetric and reflects the spiral structures of the gas distribution, as it is sensitive to Σ g (see Eq. (5)). The spiral arms of the dead zone span to 15 - 25 au at 160 kyr, to 10 - 18 au at 350 kyr and to 10 - 14 au at 490 kyr. The comparison with the temperature distribution in Figure 1 indicates that α and T mp are anticorrelated, as higher viscosity provides faster accretion, hence lower surface densities and more efficient cooling. Similarly, a lot of dust accumulates in the dead zone, increasing opacity, which hampers cooling. The icy dust ring at 1 - 2 au is especially interesting in the context of planet formation. In this ring, dust-to-gas ratio exceeds 0.1, and surface densities of both gas and dust are increased by more than an order of magnitude compared to the adjacent regions. Due to the ice mantles that protect dust from fragmentation, the dust size reaches several cm, close to the values behind the water snowline. When the ring is established, it is self-supporting, in a sense that without external perturbation (e.g. sharp change in accretion flow from the outer disc), it can be stable for a long time, over tens of kyr. The presence of water ice inside the dead zone was investigated by Vallet et al. (2023). They showed that in the discs around lower-mass stars, the turbulent heating in the inner disc can be low enough to allow the existence of ices. In our modelling the cooling of the inner region is assisted by the inner dust ring. The freeze-out has a positive feedback on dust growth due to higher v frag , which helps dust grow larger and further accumulate toward the pressure maximum in the ring. So in our modelling ices coexist with a lot of grown dust in the dead zone. Such an icy inner region appears to be a promising place for the formation of the volatile rich planets.\",\n \"pages\": [\n 6,\n 7,\n 8\n ]\n },\n {\n \"title\": \"3.2 Distribution of volatiles\",\n \"content\": \"Even in the absence of chemical reactions, over the years of disc evolution the distribution of volatiles significantly changes compared to the initial one. This is the result of both phase transitions and dust growth and advection. Dust drift brings ices from the outer disc, enriching the inner disc with volatiles. Snowlines provide the conditions favouring accumulation of ices and gas-phase species. The formation of the established disc structures, such as dust and gas rings and spirals (see Section 3.1) leads to a complex pattern of intermittent snowlines. In this Section, we describe the main features of the molecular distributions and their implications for the composition of dust and gas at early stages of protoplanetary disc evolution. Figures 2 and 3 show the azimuthally averaged radial profiles of the volatile species in models M1 and M2, respectively. As mentioned earlier, we define the snowline position as the location in the disc midplane where the amount of species in the gas equals to its amount in the ice (on both dust populations). This definition allows the snowline to have complex shape, characterised by different positions for each azimuthal direction. In the azimuthally averaged distributions shown in Figures 2 and 3, the position would only be approximate. The slope of the species distribution in the vicinity of the snowline reflects the degree of the axial asymmetry in the disc. Flatter distributions appear as the contributions sum up from the snowlines, the radial position of which vary at different azimuthal angles ϕ . In our model setup, the volatiles can either have no snowline in the disc, or have two or more snowlines depending on the local conditions. Snowlines are absent for more volatile species (CO and CH 4 ) at earlier times or during bright luminosity outbursts, when the disc is too hot for them to freeze out. When there are two snowlines, the inner snowline in the disc is the one determined by thermal desorption. We refer to it as the primary snowline. The outer snowline is determined by photo-desorption, it lies in the embedding envelope outside of the body of the disc where optical depth is low. It must be noted that the gas-phase species outside this photo-snowline are vulnerable to photo-dissociation by the UV radiation. This process is not explicitly included in our chemical model, but can be assessed as in Molyarova et al. (2021). More than two snowlines appear when the disc physical structure becomes more complex, mainly due to the presence of the ring-like structures. Species can freeze inside a cold dense dust ring, creating additional secondary snowlines, also governed by thermal desorption. The concepts of primary and secondary snowline is necessary for H 2 O and CO 2 , which have multiple snowlines at the later stages of disc evolution. For water, the inner icy region appears at ≈ 1au at 490 kyr (see right column of Figure 3) inside a dense dust ring. For CO 2 , the ring that appears at 7 au, outside the primary water snowline at 4 au, creates the inverted thermal structure in the region with T mp close to the typical CO2 sublimation temperature of 70 - 90 K. Increase in T mp in these ring is caused by higher optical depth and consequently lower cooling on the viscously heated midplane. Apart from the snowlines, the distributions of volatiles in Figures 2 and 3 present local radial variations in all of the species components, including total abundance of the species. The initial total abundance is kept only in the envelope. As the matter is being redistributed, abundances of all volatiles in the inner disc grow. Particularly, the process responsible for this is dust drift. It brings the grown ice-covered grains to the inner disc regions, where their ice mantles are sublimated and no longer move with the drifting silicate dust cores. The effect is stronger for less volatile species H 2 Oand CO 2 : their abundance in the gas grows by a factor of a few inside their primary snowlines. For CO and CH 4 the effect is weaker, because there is less grown dust outside their snowlines, and those snowlines are not very much established at earlier times. Thus their abundances inside the snowline only grow by a factor of unity, except for the immediate vicinity of the snowline. Moreover, both CO and CH 4 have a bump in the gas-phase distribution just outside the dust ring at 1 au, that is absent in H 2 Oand CO 2 . At the inner disc edge the abundances of CO and CH 4 in the gas are lower than the initial value. The abundance enhancements appear most notably at the snowlines, with local bumps in all three phases at later times (after ≈ 350kyr). They are produced by the combination of the dust radial drift and the azimuthal oscillations of dust and gas radial velocity, described in more detail in Molyarova et al. (2021) and Molyarova et al., in prep. By 500 kyr, the surface density of gas-phase H 2 Oexceeds the initial value by a factor of 5, of CO 2 - by a factor of 7, of CH 4 - by 3.5 and of CO by 2.5. Similar accumulation powered by turbulent diffusion was previously studied for CO molecule in axially symmetric model setup (Stammler et al. 2017; Krijt et al. 2018). Their results indicated similar enhancement in the gas phase by a factor of a few. In our non-axisymmetric approach, diffusion is not a necessary requirement, and the necessary transport is rather provided by azimuthal variations of radial velocity induced by disc self-gravity (Molyarova et al., in prep). Distributions of ices on small and grown dust are different as they are affected by dust growth and drift. In general, there is more grown dust than small by mass, and in most of the disc there is more ices on grown dust, particularly at later times. However, in the outer disc and at the earlier times, ices on small grains dominate. As initially all ices are on small dust, it seems inevitable that they will gradually move to the grown grains as small dust coagulates and turns into grown grains. However, near all the snowlines, amount of ices on small dust increases due to the effect of the spiral pattern. Ice abundances are determined by episodic sublimation and freeze-out in the warm spiral arms of the complex density and temperature pattern (see Section 3.3.2. in Molyarova et al. 2021). As adsorption preferably happens to the smaller grains due to their larger total surface area, there is much more ices on small dust in the wake of the spiral arms. This concerns the photo-desorption snowlines as well.\",\n \"pages\": [\n 10\n ]\n },\n {\n \"title\": \"3.3 C/O ratios\",\n \"content\": \"Here we consider the relative amount of carbon and oxygen in the gas, on the surface of small and grown grains, and in total for all for all phases and disc locations. C/O ratio is seen as a perspective tracer of planet formation mechanisms (see Öberg, Murray-Clay, and Bergin 2011; Thiabaud et al. 2015). Therefore, we are especially interested in the regions of the disc and the volatile phase component where the C/O ratio declines from the total initial value (see below). Particularly, weare interested in C/O ratio noticeably above the initial value, as it was observed in atmospheres of exoplanets and suggests the formation of these planets in similarly carbon-enriched environments (Swain et al. 2009; Madhusudhan et al. 2011; Moses et al. 2013; Facchini et al. 2021). The C/O ratio is calculated as the total number of carbon atoms contained in the molecules, divided by the total number of oxygen atoms in the molecules. This calculation can be done separately for the species contained in the gas phase, species on dust surface, or for the species in all phases. Thus, we calculate the C/O ratio in the gas, in the ice, and total C/O, respectively. For the ice species, we include both ices on grown and small dust grains.In some disc regions, the amount of carbon and oxygen in in a particular phase is very low, e.g. in the ice phase inside the water snowline, where all the molecules are in the gas. For such cases, C/O ratio would not be representative of the chemical composition, so we exclude the corresponding computational cells from the consideration. We only display the C/O ratio if the mass of the volatiles in the phase is higher than 10 -3 of the total mass of volatiles at a given location. For the adopted molecular abundances based on Karin I. Öberg et al. (2011) and also used by Eistrup, Walsh, and van Dishoeck (2018), the baseline value is C/O = 0.34, which is in line with the gas-phase abundances in the ISM ( ≈ 0.36, Przybilla, Nieva, and Butler 2008). This value is different from the typical solar value of ≈ 0.5 (Przybilla, Nieva, and Butler 2008), which is commonly used as a reference (e.g., Öberg, Murray-Clay, and Bergin 2011; Semenov et al. 2018). First, local galactic abundances changed since the Solar system formation 4.6 billion years ago (see, e.g., Appendix A in Bergin et al. 2024, for the comparison). Second, the difference also stems from inclusion or non-inclusion of the dust component. The stellar atmospheric abundances comprise all the elements present in the medium where the star formed, while the elements available for the volatiles are only a fraction of that. The values of the elemental abundances can be determined separately for the volatile (gas and ices) and the refractory (rocky dust grain cores) components of the ISM (Hensley and Draine 2021). Here, we do not take into account carbon and oxygen from the rocky cores of dust grains. Due to the model assumptions, the initial ice-to-rock mass ratio in our model is quite low (0.08 compared to 2 - 4 in Pontoppidan et al. 2014). Therefore, adding the elements from solid dust grains to those contained in ices would be misleading, as the former would dominate in the resulting C/O ratio. In this work, we concentrate on considering the C/O ratios of the volatile component (ice and gas), and compare them with the initial value of 0.34. By the age of 490 yr, the C/O ratio in both models is significantly different from the initial value. This concerns the total C/O ratio, as well as the C/O ratio in the gas and in the ice. Let us analyse the distributions of C/O ratios that we can see from Figure 4, moving inwards from the envelope to the centre. The main features of the C/O distributions are the following: As expected, the key changes in the C/O ratio distribution are associated with the positions of the snowlines. There are two main processes. First, the freeze-out and desorption at the snowlines transfer the elements between phases, altering the C/O in the gas and in the ice. Second, the snowlines favour the accumulation of the respective volatiles both in the gas and in the ice, as was discussed in Section 3.2, pumping up the amount of both components and altering their proportion. Besides the snowlines, radial drift of grown grains (Weidenschilling 1977) transports the volatiles inwards. We discuss below which processes are responsible for the formation of the listed features. C/O in the outer disc. In the surrounding envelope, outside the disc, all volatiles are in the gas due to photo-desorption, and C/O in the gas is close to the initial value of 0.34. Around the COsnowline and beyond, the C/O in the gas is close to unity, and in the gas C/O is higher that the initial, which requires explanation. The region beyond the CO snowline is usually described as the place where all species are frozen, thus having a stellar C/O ratio in the solid phase and practically no carbon or oxygen in the gas (e.g., Öberg, Murray-Clay, and Bergin 2011; Öberg and Wordsworth 2019; P. Mollière et al. 2020). In our simulations, only in model M2 there is a region where less than 10 -3 of C and O is in the gas, and in model M1 such region is absent. Due to the asymmetric spiral structure that persists even at 490 kyr, even though most of CO is frozen beyond the snowline, there is a significant (> 10 -3 ) fraction of it in the gas. Additionally, the position of CO snowline itself is significantly affected by the dust drift, as it declines from the equilibrium between adsorption and desorption.The indistinctness of the CO snowline also helps CO to persist in the outer regions: all other species are ultimately frozen, so they are efficiently carried away to the inner disc via radial drift, while this works worse for only partially frozen CO. It creates relative overabundance of CO in the outer disc, elevating the total C/O ratio and later the ice-phase C/O ratio, when the preserved CO freezes out. Between the CO ice line and the envelope, there is a gradient of C/O in the gas due to photodesorption of the ices. The last molecule to be photo-desorbed is water, which returns oxygen to the gas phase at the farthest radial distance. C / O ≈ 1 around the CO snowline. Beyond the CH 4 snowline, CO dominates the composition of volatiles in the gas phase, leading to the gas-phase C/O close to unity, the value characteristic of the CO molecule. At the CO snowline, the CO dominates the ice-phase composition as well. While dust drift substantially lowered the abundances of CO 2 and H 2 Oby 490 kyr in these regions, CO ice accumulated at the snowline. This leads to the ice-phase C/O closer to 1, too, making the vicinity of the CO snowline a region where total amounts of C and O are similar. High C/O in the gas, low C/O in the ice. In the disc regions beyond the primary CO 2 snowline, only CO and CH 4 are in the gas, meaning the dominance of carbon and C/O > 1. Consequently, the C/O in the ice is generally lower, around 0.2, as the ices are mainly H 2 O and CO 2 rich in oxygen. This is consistent with the classical step-like picture of Öberg, Murray-Clay, and Bergin (2011), with the addition of carbonrich methane allowing the C/O > 1. The midplane C/O ratios we simulate are difficult to directly compare with observations, which mostly trace the molecular layer above the disc midplane. High C/O ratios in the gas are indeed observed in many protoplanetary discs (e.g. Miotello et al. 2019), but they are considered to be a natural consequence of dust settling (Krijt et al. 2018; Krijt et al. 2020). Dust inward drift could also enhance this effect in the outer disc regions, creating the radial gradient of total C/O ratio in the disc. Observations of CS and SO emission coming from close to the midplane layers potentially indicate the presence of such radial gradient of the gas-phase C/O in the PDS 70 disc (Rampinelli et al. 2024). Variations of total C/O. Throughout the disc, there are sharp changes in total C/O ratio not connected with any snowline. Most noticeable are the variations between CO 2 and CH 4 snowlines, where gas-phase and ice-phase C/O ratios are stable. These variations are associated with the disc substructures, particularly with the dense dust rings described in Section 3.1. The total C/O changes due to radial variations in dust-to-gas ratio: when it is higher, the total C/O is closer to the ice-phase C/O, and vice versa. Inside the dust-rich rings, H 2 Oand CO 2 ices are abundant, due to high surface density of dust relative to gas. At the same time, CO and CH 4 in the gas phase have similar surface densities inside dust rings and between them. Thus, in the dust rings, CO 2 and water are overabundant, leading to lower total C/O ratio. We consider this effect in more detail in Section 3.5. Variations of the total C/O ratio due to dust substructures are also present in the cold ring at ≈ 1au. C/O peaks at the snowlines. There are peaks of the C/O ratio in the ice right outside the snowlines of CO 2 , CH 4 and CO, produced by the accumulation of the respective ices (see Figures 2 and 3). In M2 model, the amount of CH 4 and CO ices at their respective snowlines becomes comparable with or even larger than those of CO 2 and H 2 O(compare the right panels of Figure 3), leading to C/O in the ice ≈ 0.6 - 0.9. In model M1, the accumulation is more prominent, so the C/O ratio in the ice approaches unity at CO snowline and > 1 at the methane snowline. These peaks distort the pattern of generally low C/O ratio in the ice and preset additional regions where carbon-rich planetesimals could be formed, and carbon-rich pebbles could be accreted onto forming protoplanets. At the snowlines of H 2 Oand CO 2 , the C/O in the ice approaches the C/O ratios of these molecules, 0 and 0.5, respectively. Lower C/O inside the water snowline. Inside the primary water snowline, the C/O ratio is generally lower than outside of it, close to the initial 0.34. Contrary to the outer regions depleted of ices due to the radial drift, the disc parts inside and around the water snowline are enriched in volatiles, and particularly of oxygen-rich water and CO 2 . Total and gas-phase C/O ratios vary from ≈ 0.2 to 0.6, as the secondary water snowlines add more substructure to the C/O distribution. The regions where the only ice is water and the ice-phase C/O = 0 are the inner cold dust ring at 1 au and the narrow annuli between water and CO 2 snowlines. The enrichment of the inner disc regions with oxygen as a result of dust radial drift is suggested by the resolved observations of molecules in protoplanetary discs (Banzatti et al. 2020). These characteristic features of the C/O distributions are similar in the two presented models. The main difference is the radial distances where the borders between the zones are located; they are closer to the star in the less massive and thus colder model M1. This is mainly due to slightly different masses of the central star accumulated throughout 490 kyr of non-identical protostellar accretion history, which lead to different luminosity and thermal structure (see upper panels of Figure 5). Particularly, the stellar masses and luminosities at this time instance are: 1.07 L ⊙ and 0.34 M ⊙ (for M1), 1.89 L ⊙ and 0.58 M ⊙ (for M2). The less massive model M1 demonstrates overall higher C/O ratio in both phases.\",\n \"pages\": [\n 11,\n 12,\n 13\n ]\n },\n {\n \"title\": \"3.4 Evolution of the snowlines and the C/O ratios\",\n \"content\": \"The positions of the snowlines are crucial for the values of C/O ratios, both because of the direct change through the phase transitions and the associated accumulation of volatiles. They depend on the local gas and dust properties, particularly on temperature. They can also be shifted inward due to dust drift (Piso et al. 2015). Snowlines evolve as the disc structure changes with time. In this Section, we consider the co-evolution of the snowlines and the C/O ratios and discuss the mechanisms of species redistribution over the disc. The temporal evolution of the azimuthally averaged C/O ratios and the equilibrium positions of the snowlines is shown in Figure 5. It is evident from Figure 5 that the C/O ratio indeed follows the snowlines, particularly inside ≈ 10 au. The C/O structure changes throughout the disc evolution, and some key features only appear at later times. One of the key factors affecting the disc thermal structure is the luminosity of the central source. In the upper panels of Figure 5, we show the evolution of total luminosity, which directly affects the positions of the snowlines. The luminosity is the sum of stellar and accretion components, coming from the protorstar itself and gravitational potential energy of the accreted matter. The stellar luminosity gradually decreases as the protostar becomes more compact. Accretion luminosity depends on the accretion rate, which is highly variable as a result of magnetorotational and gravitational instabilities in the disc (Kadam et al. 2019, 2020; Vorobyov, Khaibrakhmanov, et al. 2020). The simulated episodic luminosity outbursts are similar to those occurring in the observed YSOs (Connelley and Reipurth 2018), with their amplitudes of tens and hundreds L ⊙ . In the more massive model M2, the outbursts are more frequent, brighter and occur until later times. The reasons behind this difference is the massive disc being more prone to both MRI and GI, which needs to be investigated in more detail in a separate study. Snowlines of the least volatile of the considered species, H 2 Oand CO 2 , exist in the model since the earliest phases of the disc formation. Even during bright luminosity outbursts ( ∼ 100 L ⊙ ) they do not disappear, but move farther away from the star. During the first ≈ 50kyr the disc is spreading out, it is highly asymmetric and dynamic, so the snowline positions oscillate. Later on, the disc generally cools down, and the snowlines gradually move toward the star (except during the outbursts). In model M2, between 130 and 500 kyr, the primary water snowline moves from 12 to 4.5 au; the primary CO 2 snowline moves from 23 to 7 au. In model M1, the water snowline moves from 9 to 4 au, and the CO 2 snowline moves from 17 to 4.5 au. Despite a factor of two different binding energies of H 2 O and CO 2 (5770 and 2360 K, respectively), the locations of their primary snowlines do not differ much due to the steep radial temperature gradient around these distances (see upper panels of Figures 3 and 2). Inside 10 - 20 au, T mp is determined by heating mechanisms other than external irradiation: viscous heating, gas work (PdV heating), heating by shocks and energy transport with advection. This also means that the water snowline is less sensitive to the level of irradiative heating, thus only slightly affected by the luminosity outbursts. The temperature change is particularly sharp at the water snowline(s), with the absolute value of the approximated power-law slope of 1.5 - 6. High surface density of dust in water-ice-free regions makes cooling less efficient and leads to locking up the produced heat and consequently higher temperatures. Besides, both these species have multiple snowlines due to the formation of ring-like substructures with the conditions close to the borderline between their frozen and gaseous state. These additional snowlines also affect the C/O distributions. Methane and CO are the more volatile species in our model. They either have zero or two snowlines in the disc. The snowlines are absent during the outbursts brighter than ≈ 100 L ⊙ for methane and ≈ 200 L ⊙ for CO. Until approximately 200 - 250 kyr, there is no established CO snowline inside the disc. For example, at 160 kyr (see lower left panel in Figures 3 and 2) there is CO ice on both small and grown dust, but their amount is an order of magnitude lower than that of the gas. Additionally, most of this ice is located at the outer disc edge, where gas and dust surface densities sharply drop. Similar distribution appears for CO and CH 4 during bright outbursts: their ices are present in some disc regions, but due to non-axisymmetric disc structure, they do not dominate in the averaged profiles, and there is no common snowline for the whole disc. There are no secondary snowlines of CO or CH 4 , because there is no prominent gas and dust structures in the outer disc where these species are frozen. Snowlines divide the disc into several zones with different characteristic C/O ratios. However, the chemical composition and C/O ratios in these zones change with time. One of the distinct zones is the region where water is not frozen, shaded with purple in Figure 4 and circumscribed by the dark purple dotted line in Figure 5. In this zone, all the species are in the gas phase, so the C/O ratio is initially close to 0.34. However, as the disc evolves, dust brings more volatiles from the outer disc. Particularly, the abundance of water grows most significantly, making C/O in the gas decrease with time. This happens because the main mechanism of the transport of the volatiles is dust radial drift, which works best in the regions where grown grains can sustain their mantles. Banzatti et al. (2020) suggest that dust growth and drift can be responsible for the observed anticorrelation between disc radius and H 2 O emission, implying that the inner regions of small discs with are enriched in water brought by efficiently drifting grains. Water is the least volatile species in our model, it is frozen in the largest part of the disc, thus its distribution is most strongly affected by the dust drift. At later times, this zone is divided into two, when a cold dense dust ring forms at 1 - 2 au, which happens at ≈ 460 kyr in model M2 and at ≈ 270 kyr in model M1. Interior to the ring, water abundance in the gas is around an order of magnitude lower than outside of it in both models. This happens because the inward flow of gasphase H 2 Ois 'blocked' by the cold ring where it freezes and accumulates with the grown grains in the pressure maximum. So the C/O ratio in the gas at r ⪅ 2au is determined by CO 2 and thus close to 0.5. The C/O ratio in the ice is not defined in the envelope, where all ices are photo-desorbed, and in the warmest inner disc, where even water is in the gas. In disc regions where only water is frozen, the C/O ratio in the ice is zero. Before 200 - 250 kyr, when the disc cools down enough for CO to freeze out, the C/O ratio in the ice in the rest of the disc is close to 0.2. It is determined by CO 2 and H 2 O ices, with a small contribution from the low-abundance CH 4 . After that, when CO freezes out and CO snowline appears, the C/O ratio in the outer disc region becomes close to the initial value both in the ice and in total. This region beyond the CO snowline is frequently referred to in relation to giant exoplanets with stellar C/O ratios (e.g., P. Mollière et al. 2020; Öberg and Wordsworth 2019; Ohno and Ueda 2021). This region would be a perfect location for planets to accrete pebbles covered with icy mantles with the primordial elemental ratio, which would directly become part of the planetary atmosphere. However, pebbles are not initially present in the disc, and their existence in these outer regions is not guaranteed. Pebbles are dust grains large enough to move relative to gas e.g., Lambrechts and Johansen 2012; Lenz, Klahr, and Birnstiel 2019, and a fraction of the grown dust in our modelling can be classified as pebbles. The properties of pebbles and composition of their ice mantles in the same setting of the FEOSAD model were studied by Topchieva et al. (2024). They show that pebbles appear in the disc as early as 50 kyr after its formation, and exist in a wide region of the disc. This partially includes the region beyond the CO snowline, but only the area of CO enhancement, where CO dominates in the ice composition and thus the C/O ratio in the ice is close to unity (see lower panels in their Figure 3). As shown by Topchieva et al. (2024), ices on pebbles are dominated by H 2 O and CO 2 . In this case, relatively high values of the C/O in the ice ( ≈ 0.5) correspond to the regions where there is more CO 2 , i.e. around the CO 2 snowlines. This is also the region where a prominent dust ring forms under the influence of the primary water snowline. It is characterised by accumulation of CO 2 and to lesser degree H 2 Oice, as well as vapours, and relatively high amount of grown dust in the ring. The dust ring is situated between the two regions with C/O ≈ 0.34 in the ice. It presents another favourable location for accreting the ice content with close-to-initial C/O ratio. The total C/O ratio in the disc also changed significantly from the initial value due to dust drift that redistributes the ices. The matter becomes more carbon-rich as the grains bring CO and CH 4 from the outer disc parts. This effect was previously investigated by Stammler et al. (2017) and Krijt et al. (2018) in their modelling of CO dynamics and dust evolution. However, as was shown by Krijt et al. (2018) and Krijt et al. (2020), vertical settling of grown grains towards the midplane is responsible for depleting the upper layers in the outer disc of gas-phase oxygen, which cannot be captured within our thin-disc modelling. The panels in the second row of Figure 5 demonstrate strong enhancement of total C/O ratio in the intermediate disc regions. In model M2, the total C/O ratio between 10 - 100 au becomes ≈ 0.7 after ∼ 300 kyr, which is two times higher than the initial value. At the snowlines of CH 4 and CO 2 it approaches unity, mostly due to the accumulation of these species in the gas. In model M1, this process begins ≈ 200kyr earlier and consequently leads to even higher total C/O ratio. At 400 - 500 kyr, most of the disc between 5 and 100 au has total C/O ≳ 1 in model M2, which demonstrates the powerful impact of dust drift. A distinctive feature of the produced C/O distributions is the peaks of total and ice-phase C/O ratios around the CO and CH 4 snowlines. In model M2, the increase of the C/O ratios becomes noticeable only after ≈ 450kyr, while in model M1 it starts to form around ≈ 300 kyr. Initial abundances of CH 4 and CO are lower than that of CO 2 and H 2 O. As these species accumulate at their snowlines, the abundances become comparable, leading to C/O in the ice ≈ 0.6 - 0.8 in model M2 and up to 1 in model M1. Similar accumulation is seen around CO 2 snowline, but unlike CO and CH 4 snowlines, it is also connected to the interaction with the dust ring structure and is considered in more detail in Section 3.5.\",\n \"pages\": [\n 13,\n 15\n ]\n },\n {\n \"title\": \"3.5 Two-dimentional structure\",\n \"content\": \"The disc is not axisymmetric even at later stages of its evolution (see Section 3.1 and Figure 1), which is also reflected in the distributions of volatiles and C/O ratios. Examples of 2D distributions of C/O in model M1 at two time instances are shown in upper panels of Figure 6. The left-hand side group of panels shows the structure that is characteristic of the earlier phases, the same time instance as the upper row in Figure 1. The prominent spiral structure and a clump both have their reflection in the C/O ratios. The snowlines of CO 2 and CH 4 have clearly non-regular shape affected by the spiral pattern of the gas, with blobs of frozen methane inside the main snowline. Inside the clump, the temperature is higher (see Figure 1), and both CO 2 and water are in the gas. The separation between gas-phase and ice-phase C/O ratios is clear, but the total C/O ratio in the clump is similar to the surroundings and is only slightly above the primordial value. The gas-phase C/O ratio inside the clump is around 0.7, lower than in the surrounding gas at the same radial distances. Similar decrease of C/O ratio indicated by lowered CS/CO ratio was observed in the disc around DR Tau (Huang et al. 2024). Lifetime of the clump (several orbital periods) is too short for significant differentiation between gas- and ice-phase composition to develop, mainly because of insufficient numerical resolution. Focused studies with higher resolution are needed to explore the C/O ratio in the clumps as precursor of giant planets formed via disc fragmentation. The upper right-hand side group of panels in Figure 6 features a later stage of the disc evolution in model M1. By 300kyr, the accretion rate and the average positions of the snowlines are stabilised (see Figure 5). The spiral structure in the gas is much weaker but still present at r > 10 au. The CO snowline is established at ≈ 80au. The 2D shape of the snowlines is more circular at 300 kyr, particularly for the less volatile CO 2 and H 2 O. The effect of spirals is still evident in the contours of the CO and CH 4 snowlines. At the outer side of these snowlines, at approximately 40 and 80 au, respectively, the C/O in the gas has local maxima, which are also seen in Figure 4. The one at 40 au also has a clearly spiral shape, repeating the pattern of the gas distribution. The radial span of these peaks is of the same scale as the dispersion of the respective snowline distance from the star. For example, at the CH 4 snowline the C/O peak is ≈ 10au wide, and the snowline is at 28 to 37 au distances. Similar accumulation powered by diffusion of water vapour was shown, e.g., by Drążkowska and Alibert (2017). In addition to diffusion, in our modelling, the species are delivered outward from the snowline due to the dynamic shape of the snowline and two-dimensional movement of gas and dust (see Molyarova et al. 2021, and Molyarova et al., in prep). In more massive model M2, the shape of the snowlines remains complex for longer times. Examples of C/O distributions in M2 are shown in the lower panels of Figure 6. The radial scale is chosen so that CO 2 and H 2 O snowlines are seen in more detail. At 250 kyr, even the grown dust distribution still has spiral pattern, and CO 2 snowline is following its complex multi-armed shape. The accumulation of CO 2 at the snowline is also very efficient, but it does not strongly affect the C/O ratios, as it drives them to the value 0.5 of the CO 2 molecule, which is close to the intrinsic value. The complex-shaped region between these snowlines has the ice-phase C/O ≈ 0.7. This pattern moves and changes its shape, thus affecting all radial distances between 10 and 20 au. The total C/O ratio is slightly above the ambient value, approaching 0.5, because CO 2 in both phases begins to accumulate in this region. At later times, the C/O distribution becomes more complex due to the presence of disc substructures, particularly the dust rings. The lower right-hand side group of panels in Figure 6 show this in more details. First, the temperature and density variations across the dust rings lead to the formation of multiple CO 2 and H 2 Osnowlines. An additional annulus of icy CO 2 appears in a relatively cold region between the dense dust rings at 6 - 7 and 8 - 13 au. The dust rings are warmer due to higher optical depth and active heating by disk internal sources, and the inner edge of the 6 au ring is warm enough to sustain gas-phase CO 2 . Second, the accumulation of icy dust grains in the rings alters the total C/O ratio. The total C/O ratio anticorrelates with the distribution of grown dust grains: inside the dense rings it is close to the initial value of 0.34, while between the rings, it is higher and reaches 0.8 - 0.9. At the same time, neither ice-phase, nor gas-phase C/O ratio displays any noticeable variations at 12-25 au, but they do have variations at r < 12 au, following the dust ring pattern. The total C/O ratio is defined by the combination of the ice- and the gas-phase component. Their relative contribution is proportional to the dust-to-gas mass ratio. This is illustrated in Figure 7. Within the rings, the total C/O is dominated by ices of oxygen-rich species CO 2 and H 2 O, particularly on grown dust accumulated in the pressure maxima. For example, there is a lot of water ice at 6 - 7 au, and its contribution to the total C/O is weighted with high dust-to-gas ratio of almost 0.1, so the resulting total C/O is lower than the initial value. At the same time, at ≈ 5au, dust-to-gas ratio is around 10 -4 . The dominant species there are in the gas, ice surface densities are 1 - 2 orders of magnitude lower, so the total C/O ratio goes up. In the wide dust ring at 9-13 au, both H 2 Oand CO 2 contribute, although at the warmer inner edge of the dust ring CO 2 is sublimated. The value of total C/O is approaches 0.34, also elevated by the presence of gas-phase CO and CH 4 . Beyond ≈ 10au, both H 2 O and CO 2 are frozen, and the variations of the total C/O ratio are clearly anticorrelated with the dust-to-gas ratio and the position of the rings. Between the dust rings, where contribution of ices is low, the C/O is determined by CO and CH 4 gases, and reaches values of ≈ 1. The anticorrelation between the total C/O ratio and dustto-gas mass ratio is an interesting finding. It is illustrated in Figure 8 for both models. Only the points within the disc are shown, with Σ gas > 0.1 g cm -2 .The pattern obviously changes with time, but the anticorrelation persists. Model M1 demonstrates wider variety of C/O and dust-to-gas ratios. The disc points are grouped in tangled curved \\\"branches\\\", some of them steeper than others. The 'width' (or spread) of these branches is determined by the azimuthal substructures. In the axially symmetric parts of the disc the values of dust-to-gas ratio and the total C/O are similar at a given radial distance. Different branches, which can be closer to vertical or horizontal orientation, are the result of the radial variations in the ice-phase C/O ratio and in ice fraction relative to the dust silicate cores. Horizontal branches correspond to weak or absent anticorrelation. For example, near the snowlines of carbon-rich species, C/O in the ice is high and close to that of the gas, which decreases the effect of dust mass fraction on the total C/O. In areas with no ices, the anticorrelation is also irrelevant, because the total C/O is determined entirely by the gas phase. Vertical branches, on the contrary, correspond to the strongest anticorrelation effect, which is expected in the regions between the snowlines, where ice-phase C/O is the lowest. The identified anticorrelation in Figure 8 is similar to the results of chemical population synthesis modelling (Cridland et al. 2019, 2020) showing that the more solids a planet accreted in the disc, the lower the C/O ratio is in its atmosphere. We fit the data for t = 490kyr with a linear law (taking the logarithm of dust-to-gas ratio) and obtain the following fits: C/O = -0.056 - 0.25 log 10 ( ξ ) for model M1, and C/O = -0.18 - 0.27 log 10 ( ξ ) for model M2. Here, ξ is the dust-to-gas mass ratio. The correlation coefficients are -0.54 for M1 and -0.57 for M2. At the shown earlier times (160 and 350 kyr), the correlation coefficient changes between approximately -0.9 and -0.5, which indicates noticeable anticorrelation throughout the disc evolution. We note that these C/O ratios only include the volatile component, without the contribution from the refractory material. Although the refractory material is typically considered as silicates, which are rich oxygen, it contains a significant amount of carbon, with the resulting C/O ≈ 0.5 (see Table 2 in Hensley and Draine 2021). This solid carbon can be subject to carbon grain destruction (Lee, Bergin, and Nomura 2010; Gail and Trieloff 2017; Wei et al. 2019), but this process should be treated separately, as it also affects the gas-phase carbon abundance. Taking refractory cores into account should affect the dependence between total C/O ratio and dust-to-gas ratio, as adding more rock would make the C/O ratio closer to 0.5. Thus the degree of the anticorrelation must be affected by the composition of rocky cores, but the anticorrelation itself should remain even when refractories are included, because the C/O ≈ 0.5 is still lower than typical C/O of the gas in most of the disc ( ⪆ 1). Dust rings are detected in the majority of the observed protoplanetary discs (Long et al. 2018; Huang, Andrews, Dullemond, et al. 2018). They are considered as a plausible sites of planet formation (Carrera, Johansen, and Davies 2015; Yang, Johansen, and Carrera 2017; Li and Youdin 2021; Lee, Fuentes, and Hopkins 2022; Jiang and Ormel 2023). The anticorrelation between the total C/O ratio of the volatiles and dust-to-gas mass ratio that we point out is a logical consequence of ices having typically lower C/O ratios and being attached to dust grains. If planets are formed in the dust rings with high dustto-gas ratios (> 10 -2 ), either exclusively from solids, or with the inclusion of the dust component, this would imply that their material initially has lower C/O ratio of ≈ 0.5 and below. To reach higher C/O ratios up to unity and above, which are observed in many exoplanets, these planets would need to migrate and accrete carbon-rich gas from regions other than their immediate formation sites inside the dust rings. In case if planet formation occurs independently of the dust rings, e.g. in the GI, their material is not determined by this anticorrelation.\",\n \"pages\": [\n 15,\n 17,\n 18\n ]\n },\n {\n \"title\": \"4. DISCUSSION\",\n \"content\": \"Our simulations present a wide range of C/O ratios in the disc in different phases evolving with time. For the atmospheres of giant exoplanets, a variety of C/O ratios were retrieved, too. Here we can compare them to identify the disc regions and times where the chemical and physical conditions for planet formation are consistent. Most of the exoplanets for which the atmospheric composition was retrieved have super-stellar C/O ratios (Hoch et al. 2023; Weiner Mansfield et al. 2024), which draws more attention to carbon-enriched areas. They are suggested to form by core accretion, and accreting mostly the gas, which is typically more carbon rich (beyond water snowline). A lot of planets are observed to have stellar or slightly super-stellar C/O (e.g., P. Mollière et al. 2020; Zhang et al. 2021; Smith et al. 2024; Sing et al. 2024; Nortmann et al. 2024, and many others). One way to form such planets is gravitational instability, which includes solids and gas together, thus undifferentiated matter is suitable for producing planets with stellar C/O. Disc fragmentation to clumps due to GI requires particular conditions (Meru and Bate 2010; Vorobyov 2013), and the direct collapse of gravitationally unstable clumps tends to produce rather massive objects (e.g. ≈ 5 M J planets and ≈ 60 - 70 M J brown dwarfs, see Figure 4 in Vorobyov, Zakhozhay, and Dunham 2013) at larger radial distances (> 10 - 100 au, see Vorobyov, Zakhozhay, and Dunham 2013; Kratter and Lodato 2016). GI can also assist the assemblage of planetary cores (Nayakshin 2010a, 2010b; Nayakshin, Helled, and Boley 2014; Vorobyov and Elbakyan 2019). A planet formed through core accretion can also accrete planetesimals, which can be covered with ice, and enrich the atmosphere with oxygen, making the C/O ratio close to the initial stellar value. There are particular exoplanets, where lower than stellar C/O ratio is observed in the atmosphere, such as β Pic b (GRAVITY Collaboration et al. 2020; Worthen et al. 2024), HD 209458 b (Xue et al. 2024), or HD 189733b (Fu et al. 2024). Such planets need even more enrichment in ices with low C/O, which makes the regions with low C/O in the ice also more attractive sites for planet formation. Gravitational instability implies that the planet forms from a mix of gas and dust (Bodenheimer 1974), this is why it is suitable to explain the formation of planets with solar, or unaltered C/O ratios. In our modelling, GI would be associated with the total C/O ratio, which we find to be significantly variable, too. For GI to form a planet with a primordial C/O ratio, it has to occur during the first 100 kyr after the disc formation. At later times, the total C/O ratio changes, and the only region with the primordial C/O ratio is the very outer disc parts, at > 100 au, which is the part of a protoplanetary disc, where conditions for GI are the most consistent with the observed properties of these objects (Rafikov 2005). Planet formation through GI is indeed more likely at earlier evolutionary stages, when gas surface density is higher (Armitage 2010). We can highlight the areas where planet formation via GI is possible in our modelling as the regions where Q Toomre ≤ 1. They are shown in the upper panel of Figure 9 for model M1. These regions appear between ≈ 10-100au before ≈ 300 kyr. However, at later times, clumps could appear in the disc as a result of an external perturbation, such as a stellar flyby (Thies et al. 2010). In this case, the planet would be formed from the material with altered C/O ratio, most probably with elevated amount of carbon, as the regions outside 5 - 10 au are typically more gravitationally unstable. This means that GI can produce planets with super-solar C/O ratios, if it is induced by external influence at later stages of disc evolution. Core accretion is another most widely discussed scenario of giant planet formation. Accretion of gas should produce atmospheres with the C/O ratio close to the one in the gas phase of protoplanetary disc. However, dust grains are also accreted, so pebble and planetesimal accretion can enrich the atmosphere in volatile components (Mordasini et al. 2016; Danti, Bitsch, and Mah 2023). This makes atmospheric C/O ratio closer to the ice-phase C/O, but in case of gas giants, the amount of the solids needed to compensate the prevalence of carbon in the gas should be quite high, up to hundreds of Earth masses (GRAVITY Collaboration et al. 2020). The C/O ratios in exoplanetary atmospheres are often interpreted in terms of pebble accretion, so planets with stellar C/O ratios are assumed to form in the environments where solid phase C/O is unprocessed and thus close to the initial value. One of such locations is beyond CO snowline, where most of the carbon- and oxygen- bearing material is in the ice, e.g. for Jupiter (e.g. Öberg and Wordsworth 2019; Ohno and Ueda 2021). In our models, this is rather the vicinity of the CO 2 snowlines, and the pebbles beyond the CO snowline are mostly covered with carbon-rich CO ice (Topchieva et al. 2024). Additionally, the CO snowline is typically very far from the star (> 40 au), so such scenarios must rely on planet migration to obtain their current location. Interpretations relying on chemical modelling extend this region to include the area beyond CO 2 snowline due to additional chemical processing of CO in this region, e.g. HR 8799e (P. Mollière et al. 2020). This puts milder constraints on the original distance from the star where pebbles should be accreted and requires less migration. Modelling of planet formation and migration including pebble and gas accretion puts the formation location of planets with super-solar C/O ratios beyond water and CO 2 snowlines (Bitsch, Schneider, and Kreidberg 2022). In our modelling, the C/O in the ice is close to initial value in the regions beyond CO snowline, excluding the area of CO accumulation. Between CO and CO 2 snowlines, it is lower, as our model does not include chemical processes apart from adsorption and desorption. However, there is another region with C/O in the ice close to initial. The vicinity of primary water snowline and the ring induced outside of it has values of C/O in the ice only slightly above the initial value of 0.34. It is surrounded by the snowlines of CO 2 . This region could be another favourable location for forming planets with the stellar C/O. As it is situated closer to the star, it would imply less migration. Rare planets with lower than stellar C/O ratio, such as β Pic b (GRAVITY Collaboration et al. 2020; Reggiani et al. 2024), HD 209458 b (Xue et al. 2024), HD 189733b (Fu et al. 2024), or KELT-1 b, Kepler-13A b and WASP-79 b (less precisely determined, see Hoch et al. 2023), need to have accreted a lot of oxygen-rich ice. Therefore, they are more likely to accrete solid material in the regions with the lowest ice-phase C/O ratios. The most suitable region would be at the distances between H 2 O and CO 2 snowlines, where ice mantles are made of pure water. However, in our modelling results, this region is very small, typically only a few au wide, as the snowlines are close to each other. This is because of steep temperature profile in this region, which is a result of the significant contribution of non-irradiation heating mechanisms, particularly viscous heating. Beyond CO 2 snowline, there are also regions with relatively low (0.2 - 0.3) C/O in the ice, but much more solids need to be accreted in such areas to compensate for the excess of carbon from the gas. Let us summarise the above constraints on planet formation locations and mechanisms implied by our simulated C/O ratios. Core accretion is suitable for forming planets with high C/O ( ≈ 1) in the atmosphere around the snowlines of CO, CH 4 and CO 2 , or anywhere beyond CO 2 snowline if they did not accrete much solids. Planets with stellar or slightly superstellar C/O ratio need to accrete (a lot of ) oxygen-rich solids to compensate their initially high C/O inherited from the gas. The locations where this is possible is between CO 2 and CH 4 and between CO and CH 4 snowlines. Planets with low C/O ratio could accrete ices between H 2 O and CO 2 snowlines. Snowlines are favourable planetesimal formation sites, so the planets that accreted planetesimals/pebbles there can have altered C/O ratios. The values will be lower than the initial if they form at the water snowline, and higher if they form at the snowlines of carbon-rich species. At the same time, to obtain planets with stellar C/O formed at the snowlines, these planets would need to migrate and accrete matter in different regions of the disc to make their C/O ratio close to the initial stellar value. Alternatively, planets with stellar C/O ratio can form via disc fragmentation through GI at earlier stages. Dedicated modelling of planet formation accounting for evolution of dust and volatiles is necessary to put more particular constraints on planet formation scenarios. Planetesimals play an important role in delivering the icephase elements to planetary atmospheres. To form planetesimals, additional physical process is needed, such as streaming instability (SI, see Youdin and Goodman 2005), which is not explicitly included in our modelling because of insufficient numerical resolution and simplified vertical disc structure. However, we can post-process the simulation results to check if the conditions for SI are fulfilled in some regions of the disc where dust-to-gas ratio and dust size are enhanced, following Vorobyov et al. (2024). Dense dust rings forming at later stages (see Section 3.1) seem to be an ideal location for triggering the SI, which would ultimately lead to formation of planetesimals and then planets in the disc. Triggering the SI requires specific relations between local dust-to-gas ratio and Stokes number (Yang, Johansen, and Carrera 2017). The criteria vary depending on the model, we adopt them from Li and Youdin (2021). Another criteria would be the requirement of volume density of dust to exceed that of gas in the midplane (Youdin and Goodman 2005). We do not apply it, as in our modelling, the residual value of α is 10 -3 which makes this condition unreachable outside of the dead zone. The regions in model M1 where the conditions of Li and Youdin (2021) are satisfied are shown in the lower panel of Figure 9. Most of the suitable regions are in the inner disc ( r < 20 au) inside the dust rings, and appear after 200 kyr. However, there are some suitable regions between 10 - 100 au at earlier times, where SI could be triggered in the spirals. The regions where GI and SI are possible shown in Figure 9 are separated in space and time, and they have different characteristic C/O ratios. We can sum up all the volatiles in these regions (throughout the disc lifetime) to assess typical C/O ratios of the planet-forming material. For GI, we exclude the pre-disc phase ( t < 53 kyr) and consider the total C/O ratio, assuming both gas and solids are included in the forming planet. For SI, we separate the gas- and ice-phase C/O ratios. The formed planetesimals would only include the ices, however, if they form the planetary cores, these cores would also accrete gas. We note that the composition of the rocks, which are typically carbon-rich, is not included in our assessment. The resulting distributions of the C/O ratios in planet-forming regions are shown in Figure 10. For GI regions, the C/O distribution has a distinct and relatively narrow peak around 0.5. It is slightly higher than the initial value of 0.34. For SI regions, the ice-phase C/O is below 0.5, with major peaks at 0 and ≈ 0.2, and the gas-phase C/O has a broad distribution with multiple peaks between ≈ 0.2 - 1.4. Distributions of C/O ratios in the regions where GI and SI can be triggered are noticeably different. It was shown by Hoch et al. (2023) that there are two different populations of C/O ratios observed in giant exoplanets. They find that directly imaged exoplanets have C/O ≈ 0.5-0.8, while transiting hot Jupiters have a wider variety of C/O ratios ( ≈ 0.3-1.7, see Figures 12 and 13 in Hoch et al. 2023), and suggest that these two populations could have different formation pathways. We add the C/O data of exoplanetary atmospheres compiled in Table 3 of Hoch et al. (2023) to Figure 10 (with arbitrary position at the y -axis). The narrow distribution of C/O ratios in the regions with GI is in step with the distribution of directly imaged exoplanets, albeit with a slightly shifted value due to our assumed initial conditions, while the wide range of C/O values in the regions of SI matches the variety of C/O ratios in transiting exoplanets. This could suggest that directly imaged exoplanets could form as a result of gravitational instability, which is also in line with their typically higher masses and orbital separations. At the same time, the transiting hot Jupiters could have experienced a lot of migration (Lin, Bodenheimer, and Richardson 1996; Dawson and Johnson 2018), during which they accrete material with a variety of C/O ratios both from the gas and solid phase. It suggests that they could also form in the core accretion scenario. The origin of wide separation planets was also investigated by Bergin et al. (2024), based on the comparison with the observed C/O > 1 in protoplanetary discs (including the full composition of solids). They conclude that both core accretion and gravitational instability can work as the formation mechanism of these planets. Apart from (exo)planets, the C/O ratios can be measured for the comets, which present the best preserved sample of the primordial composition of the ices in the Solar System. Spectroscopic measurements of molecular composition in the comae suggest that the C/O ratio of cometary ice is quite low, typically below 0.1 due to the dominance of water ice (A'Hearn et al. 2012; Seligman et al. 2022; Harrington Pinto et al. 2022). Although most comets are carbon-depleted, there are individual measurements of C/O in comets above 0.5, for example in C/2006 W3 Christensen and 29P/SchwassmannWachmann (Ootsubo et al. 2012; Seligman et al. 2022), or even close to 1 in C/2016 R2 (PanSTARRS) (Wierzchos and Womack 2018; McKay et al. 2019). Additionally, high value of C/O ≈ 1 was observed in the interstellar object 2I/Borisov (Bodewits et al. 2020). Our modelling results show the icephase C/O = 0 in the vicinity of the water snowline, as well as low values between the CO 2 and CH 4 snowlines ( ≈ 0.2) and between the CH 4 and CO snowlines ( ≈ 0.3). These are the locations where comets could originate from. However, in the vicinity of the CO 2 , CH 4 and CO ice lines themselves, the C/O ratio in the ice phase is much higher. The fact that carbon-rich cometary ices are extremely rare in the Solar System may indicate that planetesimals formed on ice lines from carbon-rich volatiles do not persist throughout the evolution of a planetary system. This means that they are likely to be included in larger bodies, which favours the snowline-aided planet formation scenarios (Drążkowska and Alibert 2017; Hyodo et al. 2021). This is also consistent with the abundance of exoplanets with high C/O (Weiner Mansfield et al. 2024), which could be formed around the snowlines of carbon-rich species. In the giant planets of the Solar System, the C/O ratios are not well constrained (Mousis, Cavalié, et al. 2024). However, the existing data suggest rather super-solar values for all giant planets except for Neptune (Cavalié et al. 2024); for Jupiter, the C/O ratio is assessed as ≈ 0.9 (Wong et al. 2004; Li et al. 2024). Our model only considers four most abundant chemical species. However, there can be other more complex molecules in protoplanetary discs, which could affect the balance of carbon and oxygen. The most obvious candidate is methanol CH 3 OH, which has the abundance similar to methane in the protostellar cores (Karin I. Öberg et al. 2011). It was also observed in a protoplanetary disc around an erupting star V883 Ori (Lee et al. 2019). We do not consider it in the model as its binding energy is close to that of water, thus the snowlines would have similar positions, but the abundance is an order of magnitude lower. However, it could somewhat increase the local C/O ratio in the inner regions where there are no other carbon-bearing species, such as the ice in the ring at 1 au. Including methanol would alter the distribution of the C/O ratio. Interactions between the ices considered in the model could also affect the results. As was recently shown by Ligterink, Kipfer, and Gavino (2024), trapping of volatile species inside the mantles of less volatile ices could have a significant impact on the distribution of C/O ratios. Another important process missing in our modelling is gas-phase and surface chemical reactions. They could significantly affect the distribution of C/O ratio in the gas and in the ice, particularly with high level of cosmic ray ionisation (Eistrup, Walsh, and van Dishoeck 2016) or if carbon grain destruction is considered (Cridland, Eistrup, and van Dishoeck 2019). One particular mechanism is the transformation of CO to CO 2 on the surface of dust grains, which can lead to the depletion of CO from both gas and ice phases between CO and CO 2 snowlines (Molyarova et al. 2017; Bosman, Tielens, and van Dishoeck 2018). Considering this mechanism can change the conclusions about planet formation location (P. Mollière et al. 2020). Nevertheless, radial variations of the C/O ratio are necessary to explain molecular emission of discs with gaps (Leemker et al. 2024), and they can only be result of dust dynamics. In order to more consistently describe the distribution of molecules and elements in the disc, the models combining dust evolution and dynamics with more complex chemistry treatment are necessary. Our simulations adopt the thin-disc approximation and focus on the midplane of the protoplanetary discs, in order to capture the essential physics of self-gravity, thermal balance, and dust evolution in a global modelling within reasonable computational times. This means that some relevant processes connected with the vertical structure are inevitably excluded. For example, vertical mixing and dust settling affect the C/O ratio in the upper layers of the disc (Krijt et al. 2018; Krijt et al. 2020). Dust settling is implicitly included in our modelling through separate scale heights of drown dust and gas (as well as small dust), affecting dust number density in the midplane. However, this approach does not allow to reproduce vertical stratification in dust properties and chemical composition, which is particularly relevant for the interpretation of molecular observations. Vertical structure is also relevant for the accretion of matter on forming giant planets, which should proceed in 3D manner through meridional flows (Morbidelli et al. 2014). Cridland, Bosman, and van Dishoeck (2020) showed that the C/O ratio in the atmospheres of giant planets is rather affected by the composition of the molecular layer than that of the midplane.\",\n \"pages\": [\n 19,\n 20,\n 21,\n 22\n ]\n },\n {\n \"title\": \"5. Conclusions\",\n \"content\": \"In this work, we studied the distribution of volatiles in a viscous self-gravitating protoplanetary disc with dust evolution using a thin-disc hydrodynamic code FEOSAD (Vorobyov et al. 2018; Molyarova et al. 2021). We calculated the C/O elemental ratio in the gas, in the ice, and in total, identified the key properties of the distribution of elements over 500 kyr of disc evolution and considered their implications for planet formation theory. Our main findings can be summarised as follows. The connection between protoplanetary disc components and exoplanets based on their composition should be more thoroughly investigated in the models focused on the planet formation process. We emphasise that these models should also take into account the effect of dust evolution and dynamics on the distribution of the elements in the planet-forming material. Inclusion of chemical processes and more accurate consideration of the bulk composition of dust grains could also affect the C/O ratios of the planet-forming environment.\",\n \"pages\": [\n 22\n ]\n },\n {\n \"title\": \"Acknowledgement\",\n \"content\": \"Weare thankful to the anonymous referee for useful comments that helped to improve the manuscript. The computational results presented have been achieved using the Vienna Scientific Cluster (VSC) and the local computing facility of the Southern Federal University. Funding Statement The work is supported by Russian Science Foundation grant 22-72-10029, https://rscf.ru/project/2272-10029/ Competing Interests None Data Availability Statement The data underlying this article will be shared on reasonable request to the corresponding author.\",\n \"pages\": [\n 22,\n 23\n ]\n },\n {\n \"title\": \"References\",\n \"content\": \"A'Hearn, Michael F., Lori M. Feaga, H. Uwe Keller, Hideyo Kawakita, Donald L. Hampton, Jochen Kissel, Kenneth P. Klaasen, et al. 2012. Cometary Volatiles and the Origin of Comets. ApJ 758, no. 1 (October): 29. https: //doi.org/10.1088/0004-637X/758/1/29. Aikawa, Yuri, Shoken M. Miyama, Takenori Nakano, and Toyoharu Umebayashi. 1996. Evolution of Molecular Abundance in Gaseous Disks around Young Stars: Depletion of CO Molecules. ApJ 467 (August): 684. https://doi.org/10.1086/177644. Akimkin, Vitaly, Eduard Vorobyov, Yaroslav Pavlyuchenkov, and Olga Stoyanovskaya. 2020. Gravitoviscous protoplanetary discs with a dust component - IV. Disc outer edges, spectral indices, and opacity gaps. MNRAS 499, no. 4 (December): 5578-5597. https://doi.org/10.1093/mnras/ staa3134. arXiv: 2010.06566 [astro-ph.EP] . Armitage, Philip J. 2010. Astrophysics of Planet Formation. Armitage, Philip J., Mario Livio, and J. E. Pringle. 2001. Episodic accretion in magnetically layered protoplanetary discs. MNRAS 324, no. 3 (June): 705-711. https://doi.org/10.1046/j.1365-8711.2001.04356.x. arXiv: astro-ph/0101253 [astro-ph] . Audard, M., P. Ábrahám, M. M. Dunham, J. D. Green, N. Grosso, K. Hamaguchi, J. H. Kastner, et al. 2014. Episodic Accretion in Young Stars. In Protostars and planets vi, edited by Henrik Beuther, Ralf S. Klessen, Cornelis P. Dullemond, and Thomas Henning, 387-410. January. https: / / doi . org / 10 . 2458 / azu _ uapress _ 9780816531240 - ch017. arXiv: 1401.3368 [astro-ph.SR] . Bai, Xue-Ning, and James M. Stone. 2013. Wind-driven Accretion in Protoplanetary Disks. I. Suppression of the Magnetorotational Instability and Launching of the Magnetocentrifugal Wind. ApJ 769, no. 1 (May): 76. https://doi.org/10.1088/0004-637X/769/1/76. arXiv: 1301.0318 [astro-ph.EP] . Banzatti, Andrea, Ilaria Pascucci, Arthur D. Bosman, Paola Pinilla, Colette Salyk, Gregory J. Herczeg, Klaus M. Pontoppidan, et al. 2020. Hints for Icy Pebble Migration Feeding an Oxygen-rich Chemistry in the Inner Planet-forming Region of Disks. ApJ 903, no. 2 (November): 124. https://doi.org/10.3847/1538-4357/abbc1a. arXiv: 2009.13525 [astro-ph.EP] . Benneke, Björn, Heather A. Knutson, Joshua Lothringer, Ian J. M. Crossfield, Julianne I. Moses, Caroline Morley, Laura Kreidberg, et al. 2019. A sub-Neptune exoplanet with a low-metallicity methane-depleted atmosphere and Mie-scattering clouds. Nature Astronomy 3 (July): 813-821. https://doi.org/10.1038/s41550- 019- 0800- 5. arXiv: 1907.00449 [astro-ph.EP] . Bergin, Edwin A., Richard A. Booth, Maria Jose Colmenares, and John D. Ilee. 2024. C/O Ratios and the formation of wide separation exoplanets. arXiv e-prints (June): arXiv:2406.12037. https://doi.org/10.48550/arXiv. 2406.12037. arXiv: 2406.12037 [astro-ph.EP] . Bergin, Edwin A., Fujun Du, L. Ilsedore Cleeves, G. A. Blake, K. Schwarz, R. Visser, and K. Zhang. 2016. Hydrocarbon Emission Rings in Protoplanetary Disks Induced by Dust Evolution. ApJ 831, no. 1 (November): 101. https://doi.org/10.3847/0004-637X/831/1/101. arXiv: 1609.06337 [astro-ph.EP] . Binney, James, and Scott Tremaine. 1987. Galactic dynamics. Birnstiel, Tilman. 2023. Dust growth and evolution in protoplanetary disks. arXiv e-prints (December): arXiv:2312.13287. https://doi.org/10.48550/ arXiv.2312.13287. arXiv: 2312.13287 [astro-ph.EP] . Bisschop, S. E., H. J. Fraser, K. I. Öberg, E. F. van Dishoeck, and S. Schlemmer. 2006. Desorption rates and sticking coefficients for CO and N2 interstellar ices. A&A 449, no. 3 (April): 1297-1309. https://doi.org/10. 1051/0004-6361:20054051. arXiv: astro-ph/0601082 [astro-ph] . Bitsch, Bertram, Aaron David Schneider, and Laura Kreidberg. 2022. How drifting and evaporating pebbles shape giant planets. III. The formation of WASP-77A b and τ Boötis b. A&A 665 (September): A138. http s : / / doi . org / 10 . 1051 / 0004 - 6361 / 202243345. arXiv: 2207 . 06077 [astro-ph.EP] . Bodenheimer, P. 1974. Calculations of the Early Evolution of Jupiter. Icarus 23, no. 3 (November): 319-325. https : / / doi . org / 10 . 1016 / 0019 1035(74)90050-5. Bodewits, D., J. W. Noonan, P. D. Feldman, M. T. Bannister, D. Farnocchia, W. M. Harris, J. -Y. Li, K. E. Mandt, J. Wm. Parker, and Z. -X. Xing. 2020. The carbon monoxide-rich interstellar comet 2I/Borisov. Nature Astronomy 4 (April): 867-871. https://doi.org/10.1038/s41550-0201095-2. arXiv: 2004.08972 [astro-ph.EP] . Booth, R. A., and J. D. Ilee. 2019. Planet-forming material in a protoplanetary disc: the interplay between chemical evolution and pebble drift. MNRAS 487, no. 3 (August): 3998-4011. https://doi.org/10.1093/mnras/stz1488. arXiv: 1905.12639 [astro-ph.EP] . Booth, Richard A., Cathie J. Clarke, Nikku Madhusudhan, and John D. Ilee. 2017. Chemical enrichment of giant planets and discs due to pebble drift. MNRAS 469, no. 4 (August): 3994-4011. https://doi.org/10.1093/ mnras/stx1103. arXiv: 1705.03305 [astro-ph.EP] . Bosman, A. D., A. J. Cridland, and Y. Miguel. 2019. Jupiter formed as a pebble pile around the N2 ice line. A&A 632 (December): L11. https://doi.org/ 10.1051/0004-6361/201936827. arXiv: 1911.11154 [astro-ph.EP] . Bosman, Arthur D., Alexander G. G. M. Tielens, and Ewine F. van Dishoeck. 2018. Efficiency of radial transport of ices in protoplanetary disks probed with infrared observations: the case of CO2. A&A 611 (April): A80. https://doi.org/10.1051/0004-6361/201732056. arXiv: 1712.03989 [astro-ph.EP] . Brown, Paul D., and S. B. Charnley. 1990. Chemical models of interstellar gas-grain processes. I. Modelling and the effect of accretion on gas abundances and mantle composition in dense clouds. MNRAS 244 (June): 432. Brown-Sevilla, S. B., M. Keppler, M. Barraza-Alfaro, J. D. Melon Fuksman, N. Kurtovic, P. Pinilla, M. Feldt, et al. 2021. A multiwavelength analysis of the spiral arms in the protoplanetary disk around WaOph 6. A&A 654 (October): A35. https://doi.org/10.1051/0004-6361/202140783. arXiv: 2107.13560 [astro-ph.EP] . Carrera, Daniel, Anders Johansen, and Melvyn B. Davies. 2015. How to form planetesimals from mm-sized chondrules and chondrule aggregates. A&A 579 (July): A43. https://doi.org/10.1051/0004-6361/201425120. arXiv: 1501.05314 [astro-ph.EP] . Cavalié, Thibault, Jonathan Lunine, Olivier Mousis, and Ricardo Hueso. 2024. The Deep Oxygen Abundance in Solar System Giant Planets, with a New Derivation for Saturn. Space Sci. Rev. 220, no. 1 (January): 8. https://doi.org/10.1007/s11214-024-01045-6. arXiv: 2407.07515 [astro-ph.EP] . Changeat, Q., B. Edwards, A. F. Al-Refaie, A. Tsiaras, J. W. Skinner, J. Y. K. Cho, K. H. Yip, et al. 2022. Five Key Exoplanet Questions Answered via the Analysis of 25 Hot-Jupiter Atmospheres in Eclipse. ApJS 260, no. 1 (May): 3. https://doi.org/10.3847/1538-4365/ac5cc2. arXiv: 2204.11729 [astro-ph.EP] . Cleeves, L. Ilsedore, Karin I. Öberg, David J. Wilner, Jane Huang, Ryan A. Loomis, Sean M. Andrews, and V. V. Guzman. 2018. Constraining Gasphase Carbon, Oxygen, and Nitrogen in the IM Lup Protoplanetary Disk. ApJ 865, no. 2 (October): 155. https://doi.org/10.3847/15384357/aade96. arXiv: 1808.10682 [astro-ph.SR] . Connelley, Michael S., and Bo Reipurth. 2018. A Near-infrared Spectroscopic Survey of FU Orionis Objects. ApJ 861, no. 2 (July): 145. https://doi. org/10.3847/1538-4357/aaba7b. arXiv: 1806.08880 [astro-ph.SR] . Cridland, Alex J., Ewine F. van Dishoeck, Matthew Alessi, and Ralph E. Pudritz. 2019. Connecting planet formation and astrochemistry. A main sequence for C/O in hot exoplanetary atmospheres. A&A 632 (December): A63. https://doi.org/10.1051/0004- 6361/201936105. arXiv: 1910.13171 [astro-ph.EP] . . 2020. Connecting planet formation and astrochemistry. C/Os and N/Os of warm giant planets and Jupiter analogues. A&A 642 (October): A229. https://doi.org/10.1051/0004-6361/202038767. arXiv: 2009. 02907 [astro-ph.EP] . Cridland, Alexander J., Arthur D. Bosman, and Ewine F. van Dishoeck. 2020. Impact of vertical gas accretion on the carbon-to-oxygen ratio of gas giant atmospheres. A&A 635 (March): A68. https://doi.org/10.1051/ 0004-6361/201936858. arXiv: 2001.05808 [astro-ph.EP] . Cridland, Alexander J., Christian Eistrup, and Ewine F. van Dishoeck. 2019. Connecting planet formation and astrochemistry. Refractory carbon depletion leading to super-stellar C/O in giant planetary atmospheres. A&A 627 (July): A127. https://doi.org/10.1051/0004-6361/201834378. arXiv: 1901.08896 [astro-ph.EP] . Cuppen, H. M., C. Walsh, T. Lamberts, D. Semenov, R. T. Garrod, E. M. Penteado, and S. Ioppolo. 2017. Grain Surface Models and Data for Astrochemistry. Space Sci. Rev. 212, nos. 1-2 (October): 1-58. https: //doi.org/10.1007/s11214-016-0319-3. Cuzzi, Jeffrey N., and Kevin J. Zahnle. 2004. Material Enhancement in Protoplanetary Nebulae by Particle Drift through Evaporation Fronts. ApJ 614, no. 1 (October): 490-496. https://doi.org/10.1086/423611. arXiv: astro-ph/0409276 [astro-ph] . Danti, C., B. Bitsch, and J. Mah. 2023. Composition of giant planets: The roles of pebbles and planetesimals. A&A 679 (November): L7. https://doi.org/ 10.1051/0004-6361/202347501. arXiv: 2310.02886 [astro-ph.EP] . Dawson, Rebekah I., and John Asher Johnson. 2018. Origins of Hot Jupiters. ARA&A 56 (September): 175-221. https://doi.org/10.1146/annurevastro-081817-051853. arXiv: 1801.06117 [astro-ph.EP] . Dong, Ruobing, Eduard Vorobyov, Yaroslav Pavlyuchenkov, Eugene Chiang, and Hauyu Baobab Liu. 2016. Signatures of Gravitational Instability in Resolved Images of Protostellar Disks. ApJ 823, no. 2 (June): 141. https://doi.org/10.3847/0004-637X/823/2/141. arXiv: 1603.01618 [astro-ph.SR] . Draine, B. T. 1978. Photoelectric heating of interstellar gas. ApJS 36 (April): 595-619. https://doi.org/10.1086/190513. Drążkowska, J., and Y. Alibert. 2017. Planetesimal formation starts at the snow line. A&A 608 (December): A92. https://doi.org/10.1051/00046361/201731491. arXiv: 1710.00009 [astro-ph.EP] . Dutrey, A., V. Wakelam, Y. Boehler, S. Guilloteau, F. Hersant, D. Semenov, E. Chapillon, et al. 2011. Chemistry in disks. V. Sulfur-bearing molecules in the protoplanetary disks surrounding LkCa15, MWC480, DM Tauri, and GO Tauri. A&A 535 (November): A104. https://doi.org/10.1051/ 0004-6361/201116931. arXiv: 1109.5870 [astro-ph.SR] . Eistrup, Christian, Catherine Walsh, and Ewine F. van Dishoeck. 2016. Setting the volatile composition of (exo)planet-building material. Does chemical evolution in disk midplanes matter? A&A 595 (November): A83. htt ps://doi.org/10.1051/0004- 6361/201628509. arXiv: 1607.06710 [astro-ph.EP] . . 2018. Molecular abundances and C/O ratios in chemically evolving planet-forming disk midplanes. A&A 613 (May): A14. https://doi.org/ 10.1051/0004-6361/201731302. arXiv: 1709.07863 [astro-ph.EP] . Facchini, Stefano, Richard Teague, Jaehan Bae, Myriam Benisty, Miriam Keppler, and Andrea Isella. 2021. The Chemical Inventory of the Planethosting Disk PDS 70. AJ 162, no. 3 (September): 99. https://doi.org/10. 3847/1538-3881/abf0a4. arXiv: 2101.08369 [astro-ph.EP] . Fedele, D., and C. Favre. 2020. Measuring elemental abundance ratios in protoplanetary disks at millimeter wavelengths. A&A 638 (June): A110. https://doi.org/10.1051/0004-6361/202037927. arXiv: 2005.03891 [astro-ph.SR] . Fraser, Helen J., Mark P. Collings, Martin R. S. McCoustra, and David A. Williams. 2001. Thermal desorption of water ice in the interstellar medium. MNRAS 327, no. 4 (November): 1165-1172. https://doi. org/10.1046/j.1365-8711.2001.04835.x. arXiv: astro-ph/0107487 [astro-ph] . Fu, Guangwei, Luis Welbanks, Drake Deming, Julie Inglis, Michael Zhang, Joshua Lothringer, Jegug Ih, et al. 2024. Hydrogen sulfide and metalenriched atmosphere for a Jupiter-mass exoplanet. arXiv e-prints (July): arXiv:2407.06163. https://doi.org/10.48550/arXiv.2407.06163. arXiv: 2407.06163 [astro-ph.EP] . Gail, Hans-Peter, and Mario Trieloff. 2017. Spatial distribution of carbon dust in the early solar nebula and the carbon content of planetesimals. A&A 606 (September): A16. https://doi.org/10.1051/0004-6361/201730480. arXiv: 1707.07611 [astro-ph.EP] . Gammie, Charles F. 1996. Layered Accretion in T Tauri Disks. ApJ 457 (January): 355. https://doi.org/10.1086/176735. Gárate, Matias, Til Birnstiel, Joanna Drążkowska, and Sebastian Markus Stammler. 2020. Gas accretion damped by dust back-reaction at the snow line. A&A 635 (March): A149. https://doi.org/10.1051/00046361/201936067. arXiv: 1906.07708 [astro-ph.EP] . GRAVITY Collaboration, M. Nowak, S. Lacour, P. Mollière, J. Wang, B. Charnay, E. F. van Dishoeck, et al. 2020. Peering into the formation history of β Pictoris b with VLTI/GRAVITY long-baseline interferometry. A&A 633 (January): A110. https://doi.org/10.1051/00046361/201936898. arXiv: 1912.04651 [astro-ph.EP] . Gressel, Oliver, Neal J. Turner, Richard P. Nelson, and Colin P. McNally. 2015. Global Simulations of Protoplanetary Disks With Ohmic Resistivity and Ambipolar Diffusion. ApJ 801, no. 2 (March): 84. https://doi.org/ 10.1088/0004-637X/801/2/84. arXiv: 1501.05431 [astro-ph.EP] . Gundlach, B., and J. Blum. 2015. The Stickiness of Micrometer-sized Waterice Particles. ApJ 798, no. 1 (January): 34. https://doi.org/10.1088/0004637X/798/1/34. arXiv: 1410.7199 [astro-ph.EP] . Harrington Pinto, Olga, Maria Womack, Yanga Fernandez, and James Bauer. 2022. A Survey of CO, CO2, and H2O in Comets and Centaurs. PSJ 3, no. 11 (November): 247. https://doi.org/10.3847/PSJ/ac960d. arXiv: 2209.09985 [astro-ph.EP] . Hartmann, L., and S. J. Kenyon. 1985. On the nature of FU Orionis objects. ApJ 299 (December): 462-478. https://doi.org/10.1086/163713. Hasegawa, T. I., and E. Herbst. 1993. Three-Phase Chemical Models of Dense Interstellar Clouds - Gas Dust Particle Mantles and Dust Particle Surfaces. MNRAS 263 (August): 589. https://doi.org/10.1093/mnras/263.3.589. Hensley, Brandon S., and B. T. Draine. 2021. Observational Constraints on the Physical Properties of Interstellar Dust in the Post-Planck Era. ApJ 906, no. 2 (January): 73. https://doi.org/10.3847/1538-4357/abc8f1. arXiv: 2009.00018 [astro-ph.GA] . Hoch, Kielan K. W., Quinn M. Konopacky, Christopher A. Theissen, JeanBaptiste Ruffio, Travis S. Barman, Emily L. Rickman, Marshall D. Perrin, Bruce Macintosh, and Christian Marois. 2023. Assessing the C/O Ratio Formation Diagnostic: A Potential Trend with Companion Mass. AJ 166, no. 3 (September): 85. https://doi.org/10.3847/1538-3881/ace442. arXiv: 2212.04557 [astro-ph.EP] . Huang, Jane, Sean M. Andrews, Cornelis P. Dullemond, Andrea Isella, Laura M. Pérez, Viviana V. Guzmán, Karin I. Öberg, et al. 2018. The Disk Substructures at High Angular Resolution Project (DSHARP). II. Characteristics of Annular Substructures. ApJ 869, no. 2 (December): L42. https : / / doi . org / 10 . 3847 / 2041 - 8213 / aaf 740. arXiv: 1812 . 04041 [astro-ph.EP] . Huang, Jane, Sean M. Andrews, Laura M. Pérez, Zhaohuan Zhu, Cornelis P. Dullemond, Andrea Isella, Myriam Benisty, et al. 2018. The Disk Substructures at High Angular Resolution Project (DSHARP). III. Spiral Structures in the Millimeter Continuum of the Elias 27, IM Lup, and WaOph 6 Disks. ApJ 869, no. 2 (December): L43. https://doi.org/10. 3847/2041-8213/aaf7a0. arXiv: 1812.04193 [astro-ph.SR] . Huang, Jane, Edwin A. Bergin, Romane Le Gal, Sean M. Andrews, Jaehan Bae, Luke Keyte, and J. A. Sturm. 2024. Constraints on the gas-phase C/O ratio of DR Tau's outer disk from CS, SO, and C2H observations. arXiv e-prints (July): arXiv:2407.01679. https://doi.org/10.48550/arXiv. 2407.01679. arXiv: 2407.01679 [astro-ph.EP] . Hyodo, Ryuki, Tristan Guillot, Shigeru Ida, Satoshi Okuzumi, and Andrew N. Youdin. 2021. Planetesimal formation around the snow line. II. Dust or pebbles? A&A 646 (February): A14. https://doi.org/10.1051/00046361/202039894. arXiv: 2012.06700 [astro-ph.EP] . Ilee, J. D., A. C. Boley, P. Caselli, R. H. Durisen, T. W. Hartquist, and J. M. C. Rawlings. 2011. Chemistry in a gravitationally unstable protoplanetary disc. MNRAS 417, no. 4 (November): 2950-2961. https://doi.org/10. 1111/j.1365-2966.2011.19455.x. arXiv: 1107.3041 [astro-ph.GA] . Jiang, Haochang, and Chris W. Ormel. 2023. Efficient planet formation by pebble accretion in ALMA rings. MNRAS 518, no. 3 (January): 38773900. https://doi.org/10.1093/mnras/stac3275. arXiv: 2207.13002 [astro-ph.EP] . Kadam, Kundan, Eduard Vorobyov, and Shantanu Basu. 2022. Primordial dusty rings and episodic outbursts in protoplanetary discs. MNRAS 516, no. 3 (November): 4448-4468. https://doi.org/10.1093/mnras/stac2455. arXiv: 2208.12105 [astro-ph.EP] . Kadam, Kundan, Eduard Vorobyov, Zsolt Regály, Ágnes Kóspál, and Péter Ábrahám. 2019. Dynamical Gaseous Rings in Global Simulations of Protoplanetary Disk Formation. ApJ 882, no. 2 (September): 96. h ttps : / / doi . org / 10 . 3847 / 1538 - 4357 / ab378a. arXiv: 1908 . 02515 [astro-ph.SR] . . 2020. Outbursts in Global Protoplanetary Disk Simulations. ApJ 895, no. 1 (May): 41. https://doi.org/10.3847/1538-4357/ab8bd8. arXiv: 2005.03578 [astro-ph.SR] . Kama, M., S. Bruderer, E. F. van Dishoeck, M. Hogerheijde, C. P. Folsom, A. Miotello, D. Fedele, A. Belloche, R. Güsten, and F. Wyrowski. 2016. Volatile-carbon locking and release in protoplanetary disks. A study of TWHya and HD 100546. A&A 592 (August): A83. https://doi.org/10. 1051/0004-6361/201526991. arXiv: 1605.05093 [astro-ph.EP] . Khorshid, N., M. Min, and J. M. Désert. 2023. Retrieving planet formation parameters of WASP-77Ab using SimAb. A&A 675 (July): A95. htt ps://doi.org/10.1051/0004- 6361/202245469. arXiv: 2311.15702 [astro-ph.EP] . Kratter, Kaitlin, and Giuseppe Lodato. 2016. Gravitational Instabilities in Circumstellar Disks. ARA&A 54 (September): 271-311. https://doi. org/10.1146/annurev- astro- 081915- 023307. arXiv: 1603.01280 [astro-ph.SR] . Krijt, Sebastiaan, Arthur D. Bosman, Ke Zhang, Kamber R. Schwarz, Fred J. Ciesla, and Edwin A. Bergin. 2020. CO Depletion in Protoplanetary Disks: A Unified Picture Combining Physical Sequestration and Chemical Processing. ApJ 899, no. 2 (August): 134. https://doi.org/10.3847/ 1538-4357/aba75d. arXiv: 2007.09517 [astro-ph.SR] . Krijt, Sebastiaan, Kamber R. Schwarz, Edwin A. Bergin, and Fred J. Ciesla. 2018. Transport of CO in Protoplanetary Disks: Consequences of Pebble Formation, Settling, and Radial Drift. ApJ 864, no. 1 (September): 78. https : / / doi . org / 10 . 3847 / 1538 - 4357 / aad69b. arXiv: 1808 . 01840 [astro-ph.EP] . Lambrechts, M., and A. Johansen. 2012. Rapid growth of gas-giant cores by pebble accretion. A&A 544 (August): A32. https://doi.org/10.1051/00046361/201219127. arXiv: 1205.3030 [astro-ph.EP] . Lee, Eve J., J. R. Fuentes, and Philip F. Hopkins. 2022. Establishing Dust Rings and Forming Planets within Them. ApJ 937, no. 2 (October): 95. https://doi.org/10.3847/1538-4357/ac8cfe. arXiv: 2206.01219 [astro-ph.EP] . Lee, Jeong-Eun, Edwin A. Bergin, and Hideko Nomura. 2010. The Solar Nebula on Fire: A Solution to the Carbon Deficit in the Inner Solar System. ApJ 710, no. 1 (February): L21-L25. https://doi.org/10.1088/ 2041-8205/710/1/L21. arXiv: 1001.0818 [astro-ph.GA] . Lee, Jeong-Eun, Seokho Lee, Giseon Baek, Yuri Aikawa, Lucas Cieza, SungYong Yoon, Gregory Herczeg, Doug Johnstone, and Simon Casassus. 2019. The ice composition in the disk around V883 Ori revealed by its stellar outburst. Nature Astronomy 3 (February): 314-319. https://doi. org/10.1038/s41550-018-0680-0. arXiv: 1809.00353 [astro-ph.SR] . Leemker, M., A. S. Booth, E. F. van Dishoeck, L. Wölfer, and B. Dent. 2024. Chemistry across dust and gas gaps in protoplanetary disks: modelling the co-spatial molecular rings in the HD 100546 disk. arXiv e-prints (May): arXiv:2405.10361. https://doi.org/10.48550/arXiv.2405.10361. arXiv: 2405.10361 [astro-ph.EP] . Lenz, Christian T., Hubert Klahr, and Tilman Birnstiel. 2019. Planetesimal Population Synthesis: Pebble Flux-regulated Planetesimal Formation. ApJ 874, no. 1 (March): 36. https://doi.org/10.3847/1538-4357/ab05d9. arXiv: 1902.07089 [astro-ph.EP] . Li, Cheng, Michael Allison, Sushil Atreya, Shawn Brueshaber, Leigh N. Fletcher, Tristan Guillot, Liming Li, et al. 2024. Super-adiabatic temperature gradient at Jupiter's equatorial zone and implications for the water abundance. Icarus 414 (May): 116028. https://doi.org/10.1016/j. icarus.2024.116028. arXiv: 2403.05363 [astro-ph.EP] . Li, Rixin, and Andrew N. Youdin. 2021. Thresholds for Particle Clumping by the Streaming Instability. ApJ 919, no. 2 (October): 107. https://doi. org/10.3847/1538-4357/ac0e9f. arXiv: 2105.06042 [astro-ph.EP] . Ligterink, N. F. W., K. A. Kipfer, and S. Gavino. 2024. Mind the Trap: Non-negligible effect of volatile trapping in ice on C/O ratios in protoplanetary disks and exoplanetary atmospheres. arXiv e-prints (June): arXiv:2406.16029. https://doi.org/10.48550/arXiv.2406.16029. arXiv: 2406.16029 [astro-ph.EP] . Lin, D. N. C., P. Bodenheimer, and D. C. Richardson. 1996. Orbital migration of the planetary companion of 51 Pegasi to its present location. Nature 380, no. 6575 (April): 606-607. https://doi.org/10.1038/380606a0. Line, Michael R., Matteo Brogi, Jacob L. Bean, Siddharth Gandhi, Joseph Zalesky, Vivien Parmentier, Peter Smith, et al. 2021. A solar C/O and sub-solar metallicity in a hot Jupiter atmosphere. Nature 598, no. 7882 (October): 580-584. https://doi.org/10.1038/s41586-021-03912-6. arXiv: 2110.14821 [astro-ph.EP] . Lodders, Katharina. 2004. Jupiter Formed with More Tar than Ice. ApJ 611, no. 1 (August): 587-597. https://doi.org/10.1086/421970. Long, Feng, Paola Pinilla, Gregory J. Herczeg, Daniel Harsono, Giovanni Dipierro, Ilaria Pascucci, Nathan Hendler, et al. 2018. Gaps and Rings in an ALMA Survey of Disks in the Taurus Star-forming Region. ApJ 869, no. 1 (December): 17. https://doi.org/10.3847/1538-4357/aae8e1. arXiv: 1810.06044 [astro-ph.SR] . Madhusudhan, Nikku. 2012. C/O Ratio as a Dimension for Characterizing Exoplanetary Atmospheres. ApJ 758, no. 1 (October): 36. https://doi. org/10.1088/0004-637X/758/1/36. arXiv: 1209.2412 [astro-ph.EP] . Madhusudhan, Nikku, Joseph Harrington, Kevin B. Stevenson, Sarah Nymeyer, Christopher J. Campo, Peter J. Wheatley, Drake Deming, et al. 2011. A high C/O ratio and weak thermal inversion in the atmosphere of exoplanet WASP-12b. Nature 469, no. 7328 (January): 64-67. https: //doi.org/10.1038/nature09602. arXiv: 1012.1603 [astro-ph.EP] . Matter, A., F. C. Pignatale, and B. Lopez. 2020. Spatially resolving the chemical composition of the planet building blocks. MNRAS 497, no. 3 (September): 2540-2552. https://doi.org/10.1093/mnras/staa2137. arXiv: 2007.09385 [astro-ph.EP] . McKay, Adam J., Michael A. DiSanti, Michael S. P. Kelley, Matthew M. Knight, Maria Womack, Kacper Wierzchos, Olga Harrington Pinto, et al. 2019. The Peculiar Volatile Composition of CO-dominated Comet C/2016 R2 (PanSTARRS). AJ 158, no. 3 (September): 128. https://doi. org/10.3847/1538-3881/ab32e4. arXiv: 1907.07208 [astro-ph.EP] . Meru, Farzana, and Matthew R. Bate. 2010. Exploring the conditions required to form giant planets via gravitational instability in massive protoplanetary discs. MNRAS 406, no. 4 (August): 2279-2288. https: //doi.org/10.1111/j.1365- 2966.2010.16867.x. arXiv: 1004.3766 [astro-ph.EP] . Meru, Farzana, Attila Juhász, John D. Ilee, Cathie J. Clarke, Giovanni P. Rosotti, and Richard A. Booth. 2017. On the Origin of the Spiral Morphology in the Elias 2-27 Circumstellar Disk. ApJ 839, no. 2 (April): L24. https://doi.org/10.3847/2041-8213/aa6837. arXiv: 1703.05338 [astro-ph.EP] . Minissale, Marco, Yuri Aikawa, Edwin Bergin, Mathieu Bertin, Wendy A. Brown, Stephanie Cazaux, Steven B. Charnley, et al. 2022. Thermal Desorption of Interstellar Ices: A Review on the Controlling Parameters and Their Implications from Snowlines to Chemical Complexity. ACS Earth and Space Chemistry 6, no. 3 (March): 597-630. https://doi.org/10. 1021/acsearthspacechem.1c00357. arXiv: 2201.07512 [astro-ph.GA] . Miotello, A., S. Facchini, E. F. van Dishoeck, P. Cazzoletti, L. Testi, J. P. Williams, M. Ansdell, S. van Terwisga, and N. van der Marel. 2019. Bright C2H emission in protoplanetary discs in Lupus: high volatile C/O > 1 ratios. A&A 631 (November): A69. https://doi.org/10.1051/00046361/201935441. arXiv: 1909.04477 [astro-ph.SR] . Mollière, P., T. Stolker, S. Lacour, G. P. P. L. Otten, J. Shangguan, B. Charnay, T. Molyarova, et al. 2020. Retrieving scattering clouds and disequilibrium chemistry in the atmosphere of HR 8799e. A&A 640 (August): A131. https://doi.org/10.1051/0004-6361/202038325. arXiv: 2006.09394 [astro-ph.EP] . Mollière, Paul, Tamara Molyarova, Bertram Bitsch, Thomas Henning, Aaron Schneider, Laura Kreidberg, Christian Eistrup, et al. 2022. Interpreting the Atmospheric Composition of Exoplanets: Sensitivity to Planet Formation Assumptions. ApJ 934, no. 1 (July): 74. https://doi.org/10.3847/ 1538-4357/ac6a56. arXiv: 2204.13714 [astro-ph.EP] . Molyarova, Tamara, Vitaly Akimkin, Dmitry Semenov, Thomas Henning, Anton Vasyunin, and Dmitri Wiebe. 2017. Gas Mass Tracers in Protoplanetary Disks: CO is Still the Best. ApJ 849, no. 2 (November): 130. https://doi.org/10.3847/1538-4357/aa9227. arXiv: 1710.02993 [astro-ph.EP] . Molyarova, Tamara, Eduard I. Vorobyov, Vitaly Akimkin, Aleksandr Skliarevskii, Dmitri Wiebe, and Manuel Güdel. 2021. Gravitoviscous Protoplanetary Disks with a Dust Component. V. The Dynamic Model for Freeze-out and Sublimation of Volatiles. ApJ 910, no. 2 (April): 153. https://doi.org/ 10.3847/1538-4357/abe2b0. arXiv: 2103.06045 [astro-ph.EP] . Morbidelli, A., J. Szulágyi, A. Crida, E. Lega, B. Bitsch, T. Tanigawa, and K. Kanagawa. 2014. Meridional circulation of gas into gaps opened by giant planets in three-dimensional low-viscosity disks. Icarus 232 (April): 266-270. https://doi.org/10.1016/j.icarus.2014.01.010. arXiv: 1401.2925 [astro-ph.EP] . Mordasini, C., R. van Boekel, P. Mollière, Th. Henning, and Björn Benneke. 2016. The Imprint of Exoplanet Formation History on Observable Present-day Spectra of Hot Jupiters. ApJ 832, no. 1 (November): 41. https://doi.org/10.3847/0004-637X/832/1/41. arXiv: 1609.03019 [astro-ph.EP] . Moses, J. I., N. Madhusudhan, C. Visscher, and R. S. Freedman. 2013. Chemical Consequences of the C/O Ratio on Hot Jupiters: Examples from WASP-12b, CoRoT-2b, XO-1b, and HD 189733b. ApJ 763, no. 1 (January): 25. https://doi.org/10.1088/0004-637X/763/1/25. arXiv: 1211.2996 [astro-ph.EP] . Mousis, Olivier, Sarah E. Anderson, Adrienn Luspay-Kuti, Kathleen E. Mandt, and Pierre Vernazza. 2024. Triton and Pluto: same origin but separated at birth. arXiv e-prints (June): arXiv:2406.03815. https://doi.org/10. 48550/arXiv.2406.03815. arXiv: 2406.03815 [astro-ph.EP] . Mousis, Olivier, Thibault Cavalié, Jonathan I. Lunine, Kathleen E. Mandt, Ricardo Hueso, Artyom Aguichine, Antoine Schneeberger, et al. 2024. Recipes for Forming a Carbon-Rich Giant Planet. Space Sci. Rev. 220, no. 4 (June): 44. https://doi.org/10.1007/s11214-024-01071-4. arXiv: 2405.19748 [astro-ph.EP] . Nasedkin, E., P. Mollière, S. Lacour, M. Nowak, L. Kreidberg, T. Stolker, J. J. Wang, et al. 2024. Four-of-a-kind? Comprehensive atmospheric characterisation of the HR 8799 planets with VLTI/GRAVITY. arXiv e-prints (April): arXiv:2404.03776. https://doi.org/10.48550/arXiv.2404. 03776. arXiv: 2404.03776 [astro-ph.EP] . Nayakshin, Sergei. 2010a. Formation of planets by tidal downsizing of giant planet embryos. MNRAS 408, no. 1 (October): L36-L40. https://doi.org/ 10.1111/j.1745-3933.2010.00923.x. arXiv: 1007.4159 [astro-ph.EP] . . 2010b. Grain sedimentation inside giant planet embryos. MNRAS 408, no. 4 (November): 2381-2396. https://doi.org/10.1111/j.13652966.2010.17289.x. arXiv: 1007.4162 [astro-ph.EP] . Nayakshin, Sergei, Ravit Helled, and Aaron C. Boley. 2014. Core-assisted gas capture instability: a new mode of giant planet formation by gravitationally unstable discs. MNRAS 440, no. 4 (June): 3797-3808. https: //doi.org/10.1093/mnras/stu473. arXiv: 1403.1813 [astro-ph.EP] . Nortmann, L., F. Lesjak, F. Yan, D. Cont, S. Czesla, A. Lavail, A. D. Rains, et al. 2024. CRIRES + transmission spectroscopy of WASP-127b. Detection of the resolved signatures of a supersonic equatorial jet and cool poles in a hot planet. arXiv e-prints (April): arXiv:2404.12363. https://doi.org/ 10.48550/arXiv.2404.12363. arXiv: 2404.12363 [astro-ph.EP] . Öberg, K. I., F. van Broekhuizen, H. J. Fraser, S. E. Bisschop, E. F. van Dishoeck, and S. Schlemmer. 2005. Competition between CO and N2 Desorption from Interstellar Ices. ApJ 621, no. 1 (March): L33-L36. https://doi.org/10.1086/428901. Öberg, Karin I., A. C. Adwin Boogert, Klaus M. Pontoppidan, Saskia van den Broek, Ewine F. van Dishoeck, Sandrine Bottinelli, Geoffrey A. Blake, and II Evans Neal J. 2011. The Spitzer Ice Legacy: Ice Evolution from Cores to Protostars. ApJ 740, no. 2 (October): 109. https://doi.org/10. 1088/0004-637X/740/2/109. arXiv: 1107.5825 [astro-ph.GA] . Öberg, Karin I., Ruth Murray-Clay, and Edwin A. Bergin. 2011. The Effects of Snowlines on C/O in Planetary Atmospheres. ApJ 743, no. 1 (December): L16. https://doi.org/10.1088/2041-8205/743/1/L16. arXiv: 1110.5567 [astro-ph.GA] . Öberg, Karin I., and Robin Wordsworth. 2019. Jupiter's Composition Suggests its Core Assembled Exterior to the N2 Snowline. AJ 158, no. 5 (November): 194. https://doi.org/10.3847/1538-3881/ab46a8. arXiv: 1909.11246 [astro-ph.EP] . Ohno, Kazumasa, and Takahiro Ueda. 2021. Jupiter's 'cold' formation in the protosolar disk shadow. An explanation for the planet's uniformly enriched atmosphere. A&A 651 (July): L2. https://doi.org/10.1051/00046361/202141169. arXiv: 2106.09084 [astro-ph.EP] . Okuzumi, Satoshi, and Shigenobu Hirose. 2011. Modeling Magnetorotational Turbulence in Protoplanetary Disks with Dead Zones. ApJ 742, no. 2 (December): 65. https://doi.org/10.1088/0004-637X/742/2/65. arXiv: 1108.4892 [astro-ph.EP] . Okuzumi, Satoshi, and Ryo Tazaki. 2019. Nonsticky Ice at the Origin of the Uniformly Polarized Submillimeter Emission from the HL Tau Disk. ApJ 878, no. 2 (June): 132. https://doi.org/10.3847/1538-4357/ab204d. arXiv: 1904.03869 [astro-ph.EP] . Ootsubo, Takafumi, Hideyo Kawakita, Saki Hamada, Hitomi Kobayashi, Mitsuru Yamaguchi, Fumihiko Usui, Takao Nakagawa, et al. 2012. AKARI Near-infrared Spectroscopic Survey for CO2 in 18 Comets. ApJ 752, no. 1 (June): 15. https://doi.org/10.1088/0004-637X/752/1/15. Padoan, Paolo, Liubin Pan, Veli-Matti Pelkonen, Troels Haugboelle, and AAke Nordlund. 2024. Protoplanetary Disks from Pre-Main Sequence BondiHoyle Accretion. arXiv e-prints (May): arXiv:2405.07334. https://doi. org/10.48550/arXiv.2405.07334. arXiv: 2405.07334 [astro-ph.GA] . Padovani, M., D. Galli, and A. E. Glassgold. 2009. Cosmic-ray ionization of molecular clouds. A&A 501, no. 2 (July): 619-631. https://doi.org/10. 1051/0004-6361/200911794. arXiv: 0904.4149 [astro-ph.SR] . Pavlyuchenkov, Yaroslav, Vitaly Akimkin, Dmitri Wiebe, and Eduard Vorobyov. 2019. Revealing dust segregation in protoplanetary discs with the help of multifrequency spectral index maps. MNRAS 486, no. 3 (July): 39073914. https://doi.org/10.1093/mnras/stz1046. arXiv: 1904.05251 [astro-ph.IM] . Pelkonen, Veli-Matti, Paolo Padoan, Mika Juvela, Troels Haugbølle, and Åke Nordlund. 2024. Origin and Evolution of Angular Momentum of Class II Disks. arXiv e-prints (May): arXiv:2405.06520. https://doi.org/10. 48550/arXiv.2405.06520. arXiv: 2405.06520 [astro-ph.SR] . Pérez, Laura M., John M. Carpenter, Sean M. Andrews, Luca Ricci, Andrea Isella, Hendrik Linz, Anneila I. Sargent, et al. 2016. Spiral density waves in a young protoplanetary disk. Science 353, no. 6307 (September): 15191521. https://doi.org/10.1126/science.aaf8296. arXiv: 1610.05139 [astro-ph.GA] . Piso, Ana-Maria A., Karin I. Öberg, Tilman Birnstiel, and Ruth A. MurrayClay. 2015. C/O and Snowline Locations in Protoplanetary Disks: The Effect of Radial Drift and Viscous Gas Accretion. ApJ 815, no. 2 (December): 109. https://doi.org/10.1088/0004-637X/815/2/109. arXiv: 1511.05563 [astro-ph.EP] . Pontoppidan, K. M., C. Salyk, E. A. Bergin, S. Brittain, B. Marty, O. Mousis, and K. I. Öberg. 2014. Volatiles in Protoplanetary Disks. In Protostars and planets vi, edited by Henrik Beuther, Ralf S. Klessen, Cornelis P. Dullemond, and Thomas Henning, 363-385. January. https : / / doi . org/10.2458/azu_uapress_9780816531240-ch016. arXiv: 1401.2423 [astro-ph.EP] . Przybilla, Norbert, Maria-Fernanda Nieva, and Keith Butler. 2008. A Cosmic Abundance Standard: Chemical Homogeneity of the Solar Neighborhood and the ISM Dust-Phase Composition. ApJ 688, no. 2 (December): L103. https://doi.org/10.1086/595618. arXiv: 0809.2403 [astro-ph] . Rafikov, Roman R. 2005. Can Giant Planets Form by Direct Gravitational Instability? ApJ 621, no. 1 (March): L69-L72. https://doi.org/10.1086/ 428899. arXiv: astro-ph/0406469 [astro-ph] . Rampinelli, L., S. Facchini, M. Leemker, J. Bae, M. Benisty, R. Teague, C. J. Law, K. I. Öberg, B. Portilla-Revelo, and A. J. Cridland. 2024. ALMA high-resolution observations unveil planet formation shaping molecular emission in the PDS 70 disk. arXiv e-prints (July): arXiv:2407.06272. https://doi.org/10.48550/arXiv.2407.06272. arXiv: 2407.06272 [astro-ph.EP] . Reggiani, Henrique, Jhon Yana Galarza, Kevin C. Schlaufman, David K. Sing, Brian F. Healy, Andrew McWilliam, Joshua D. Lothringer, and Laurent Pueyo. 2024. Insight into the Formation of β Pic b through the Composition of Its Parent Protoplanetary Disk as Revealed by the β Pic Moving Group Member HD 181327. AJ 167, no. 1 (January): 45. https://doi.org/10.3847/1538-3881/ad0f93. arXiv: 2311.12210 [astro-ph.SR] . Schneider, Aaron David, and Bertram Bitsch. 2021. How drifting and evaporating pebbles shape giant planets. I. Heavy element content and atmospheric C/O. A&A 654 (October): A71. https://doi.org/10.1051/00046361/202039640. arXiv: 2105.13267 [astro-ph.EP] . Seager, S., L. J. Richardson, B. M. S. Hansen, K. Menou, J. Y. -K. Cho, and D. Deming. 2005. On the Dayside Thermal Emission of Hot Jupiters. ApJ 632, no. 2 (October): 1122-1131. https://doi.org/10.1086/444411. arXiv: astro-ph/0504212 [astro-ph] . Seligman, Darryl Z., Leslie A. Rogers, Samuel H. C. Cabot, John W. Noonan, Theodore Kareta, Kathleen E. Mandt, Fred Ciesla, et al. 2022. The Volatile Carbon-to-oxygen Ratio as a Tracer for the Formation Locations of Interstellar Comets. PSJ 3, no. 7 (July): 150. https://doi.org/10. 3847/PSJ/ac75b5. arXiv: 2204.13211 [astro-ph.EP] . Semenov, D., C. Favre, D. Fedele, S. Guilloteau, R. Teague, Th. Henning, A. Dutrey, E. Chapillon, F. Hersant, and V. Piétu. 2018. Chemistry in disks. XI. Sulfur-bearing species as tracers of protoplanetary disk physics and chemistry: the DM Tau case. A&A 617 (September): A28. https://doi.org/10.1051/0004-6361/201832980. arXiv: 1806.07707 [astro-ph.GA] . Semenov, D., Th. Henning, Ch. Helling, M. Ilgner, and E. Sedlmayr. 2003. Rosseland and Planck mean opacities for protoplanetary discs. A&A 410 (November): 611-621. https://doi.org/10.1051/0004-6361:20031279. arXiv: astro-ph/0308344 [astro-ph] . Semenov, D., and D. Wiebe. 2011. Chemical Evolution of Turbulent Protoplanetary Disks and the Solar Nebula. ApJS 196, no. 2 (October): 25. https://doi.org/10.1088/0067- 0049/196/2/25. arXiv: 1104.4358 [astro-ph.GA] . Shakura, N. I., and R. A. Sunyaev. 1973. Black holes in binary systems. Observational appearance. A&A 24 (January): 337-355. Sing, David K., Zafar Rustamkulov, Daniel P. Thorngren, Joanna K. Barstow, Pascal Tremblin, Catarina Alves de Oliveira, Tracy L. Beck, et al. 2024. A warm Neptune's methane reveals core mass and vigorous atmospheric mixing. arXiv e-prints (May): arXiv:2405.11027. https://doi.org/10. 48550/arXiv.2405.11027. arXiv: 2405.11027 [astro-ph.EP] . Smith, Peter C. B., Michael R. Line, Jacob L. Bean, Matteo Brogi, Prune August, Luis Welbanks, Jean-Michel Desert, et al. 2024. A Combined Ground-based and JWST Atmospheric Retrieval Analysis: Both IGRINS and NIRSpec Agree that the Atmosphere of WASP-77A b Is Metal-poor. AJ 167, no. 3 (March): 110. https://doi.org/10.3847/1538-3881/ad17bf. arXiv: 2312.13069 [astro-ph.EP] . Stammler, Sebastian Markus, Tilman Birnstiel, Olja Panić, Cornelis Petrus Dullemond, and Carsten Dominik. 2017. Redistribution of CO at the location of the CO ice line in evolving gas and dust disks. A&A 600 (April): A140. https://doi.org/10.1051/0004-6361/201629041. arXiv: 1701.02385 [astro-ph.EP] . Stevenson, David J., and Jonathan I. Lunine. 1988. Rapid formation of Jupiter by diffusive redistribution of water vapor in the solar nebula. Icarus 75, no. 1 (July): 146-155. https://doi.org/10.1016/0019-1035(88)90133-9. Swain, M. R., G. Tinetti, G. Vasisht, P. Deroo, C. Griffith, J. Bouwman, Pin Chen, et al. 2009. Water, Methane, and Carbon Dioxide Present in the Dayside Spectrum of the Exoplanet HD 209458b. ApJ 704, no. 2 (October): 1616-1621. https://doi.org/10.1088/0004-637X/704/2/1616. arXiv: 0908.4010 [astro-ph.EP] . Testi, L., T. Birnstiel, L. Ricci, S. Andrews, J. Blum, J. Carpenter, C. Dominik, et al. 2014. Dust Evolution in Protoplanetary Disks. In Protostars and planets vi, edited by Henrik Beuther, Ralf S. Klessen, Cornelis P. Dullemond, and Thomas Henning, 339-361. January. https://doi.org/10.2458/azu_ uapress_9780816531240-ch015. arXiv: 1402.1354 [astro-ph.SR] . Thiabaud, A., U. Marboeuf, Y. Alibert, I. Leya, and K. Mezger. 2015. Gas composition of the main volatile elements in protoplanetary discs and its implication for planet formation. A&A 574 (February): A138. https: //doi.org/10.1051/0004-6361/201424868. Thies, Ingo, Pavel Kroupa, Simon P. Goodwin, Dimitrios Stamatellos, and Anthony P. Whitworth. 2010. Tidally Induced Brown Dwarf and Planet Formation in Circumstellar Disks. ApJ 717, no. 1 (July): 577-585. h ttps://doi.org/10.1088/0004- 637X/717/1/577. arXiv: 1005.3017 [astro-ph.SR] . Tielens, A. G. G. M. 2005. The Physics and Chemistry of the Interstellar Medium. Tong, Simin, Richard Alexander, and Giovanni Rosotti. 2024. A question of personalities: evolution of viscous and wind-driven protoplanetary discs in the presence of dead zones. MNRAS (July). https://doi.org/10.1093/ mnras/stae1748. arXiv: 2407.12209 [astro-ph.EP] . Toomre, A. 1964. On the gravitational stability of a disk of stars. ApJ 139 (May): 1217-1238. https://doi.org/10.1086/147861. Topchieva, A., T. Molyarova, V. Akimkin, L. Maksimova, and E. Vorobyov. 2024. Ices on pebbles in protoplanetary discs. MNRAS 530, no. 3 (May): 2731-2748. https://doi.org/10.1093/mnras/stae597. arXiv: 2403.02895 [astro-ph.EP] . Turrini, D., E. Schisano, S. Fonte, S. Molinari, R. Politi, D. Fedele, O. Panić, M. Kama, Q. Changeat, and G. Tinetti. 2021. Tracing the Formation History of Giant Planets in Protoplanetary Disks with Carbon, Oxygen, Nitrogen, and Sulfur. ApJ 909, no. 1 (March): 40. https://doi.org/10. 3847/1538-4357/abd6e5. arXiv: 2012.14315 [astro-ph.EP] . Umebayashi, T., and T. Nakano. 1981. Fluxes of Energetic Particles and the Ionization Rate in Very Dense Interstellar Clouds. PASJ 33 (January): 617. Vallet, David, Anna C. Childs, Rebecca G. Martin, Mario Livio, and Stephen Lepp. 2023. Formation of super-Earths in icy dead zones around lowmass stars. MNRAS 519, no. 1 (February): L10-L14. https://doi.org/10. 1093/mnrasl/slac144. arXiv: 2211.07759 [astro-ph.EP] . Visser, R., E. F. van Dishoeck, S. D. Doty, and C. P. Dullemond. 2009. The chemical history of molecules in circumstellar disks. I. Ices. A&A 495, no. 3 (March): 881-897. https://doi.org/10.1051/0004-6361/200810846. arXiv: 0901.1313 [astro-ph.SR] . Vorobyov, E. I. 2013. Formation of giant planets and brown dwarfs on wide orbits. A&A 552 (April): A129. https://doi.org/10.1051/0004-6361/ 201220601. arXiv: 1302.1892 [astro-ph.EP] . Vorobyov, Eduard I., Vitaly Akimkin, Olga Stoyanovskaya, Yaroslav Pavlyuchenkov, and Hauyu Baobab Liu. 2018. Early evolution of viscous and selfgravitating circumstellar disks with a dust component. A&A 614 (June): A98. https://doi.org/10.1051/0004-6361/201731690. arXiv: 1801.06898 [astro-ph.EP] . Vorobyov, Eduard I., and Shantanu Basu. 2010. The Burst Mode of Accretion and Disk Fragmentation in the Early Embedded Stages of Star Formation. ApJ 719, no. 2 (August): 1896-1911. https://doi.org/10.1088/0004637X/719/2/1896. arXiv: 1007.2993 [astro-ph.SR] . Vorobyov, Eduard I., and Vardan G. Elbakyan. 2019. Gravitoviscous protoplanetary disks with a dust component. II. Spatial distribution and growth of dust in a clumpy disk. A&A 631 (November): A1. https: / / doi . org / 10 . 1051 / 0004 - 6361 / 201936132. arXiv: 1908 . 10589 [astro-ph.SR] . Vorobyov, Eduard I., Vardan G. Elbakyan, Michihiro Takami, and Hauyu B. Liu. 2020. Effect of luminosity outbursts on protoplanetary disk dynamics. A&A 643 (November): A13. https://doi.org/10.1051/00046361/202038122. arXiv: 2009.01888 [astro-ph.SR] . Vorobyov, Eduard I., Sergey Khaibrakhmanov, Shantanu Basu, and Marc Audard. 2020. Accretion bursts in magnetized gas-dust protoplanetary disks. A&A 644 (December): A74. https://doi.org/10.1051/00046361/202039081. arXiv: 2011.00951 [astro-ph.SR] . Vorobyov, Eduard I., D. N. C. Lin, and Manuel Guedel. 2015. The effect of external environment on the evolution of protostellar disks. A&A 573 (January): A5. https://doi.org/10.1051/0004-6361/201424583. arXiv: 1410.1743 [astro-ph.SR] . Vorobyov, Eduard I., Zsolt Regaly, Manuel Guedel, and Doug N. C. Lin. 2016. An alternative model for the origin of gaps in circumstellar disks. A&A 587 (March): A146. https://doi.org/10.1051/0004-6361/201527701. arXiv: 1601.08089 [astro-ph.SR] . Vorobyov, Eduard I., Aleksandr M. Skliarevskii, Manuel Guedel, and Tamara Molyarova. 2024. Primordial dust rings, hidden dust mass, and the first generation of planetesimals in gravitationally unstable protoplanetary disks. arXiv e-prints (April): arXiv:2404.16151. https://doi.org/10.48550/ arXiv.2404.16151. arXiv: 2404.16151 [astro-ph.EP] . Vorobyov, Eduard I., Aleksandr M. Skliarevskii, Tamara Molyarova, Vitaly Akimkin, Yaroslav Pavlyuchenkov, Ágnes Kóspál, Hauyu Baobab Liu, Michihiro Takami, and Anastasiia Topchieva. 2022. Evolution of dust in protoplanetary disks of eruptive stars. A&A 658 (February): A191. https://doi.org/10.1051/0004-6361/202141932. arXiv: 2112.06004 [astro-ph.EP] . Vorobyov, Eduard I., Olga V. Zakhozhay, and Michael M. Dunham. 2013. Fragmenting protostellar discs: properties and observational signatures. MNRAS 433, no. 4 (August): 3256-3273. https://doi.org/10.1093/ mnras/stt970. arXiv: 1306.4074 [astro-ph.SR] . Wada, Koji, Hidekazu Tanaka, Toru Suyama, Hiroshi Kimura, and Tetsuo Yamamoto. 2009. Collisional Growth Conditions for Dust Aggregates. ApJ 702, no. 2 (September): 1490-1501. https://doi.org/10.1088/0004637X/702/2/1490. Walsh, Catherine, Hideko Nomura, and Ewine van Dishoeck. 2015. The molecular composition of the planet-forming regions of protoplanetary disks across the luminosity regime. A&A 582 (October): A88. http s : / / doi . org / 10 . 1051 / 0004 - 6361 / 201526751. arXiv: 1507 . 08544 [astro-ph.EP] . Weber, Philipp, Sebastián Pérez, Alice Zurlo, James Miley, Antonio Hales, Lucas Cieza, David Principe, et al. 2023. Spirals and Clumps in V960 Mon: Signs of Planet Formation via Gravitational Instability around an FU Ori Star? ApJ 952, no. 1 (July): L17. https://doi.org/10.3847/20418213/ace186. arXiv: 2307.13433 [astro-ph.EP] . Wei, Chen-En, Hideko Nomura, Jeong-Eun Lee, Wing-Huen Ip, Catherine Walsh, and T. J. Millar. 2019. The Effect of Carbon Grain Destruction on the Chemical Structure of Protoplanetary Disks. ApJ 870, no. 2 (January): 129. https://doi.org/10.3847/1538- 4357/aaf390. arXiv: 1811.10194 [astro-ph.EP] . Weidenschilling, S. J. 1977. Aerodynamics of solid bodies in the solar nebula. MNRAS 180 (July): 57-70. https://doi.org/10.1093/mnras/180.2.57. Weiner Mansfield, Megan, Michael R. Line, Joost P. Wardenier, Matteo Brogi, Jacob L. Bean, Hayley Beltz, Peter Smith, et al. 2024. The metallicity and carbon-to-oxygen ratio of the ultra-hot Jupiter WASP-76b from Gemini-S/IGRINS. arXiv e-prints (May): arXiv:2405.09769. https://doi. org/10.48550/arXiv.2405.09769. arXiv: 2405.09769 [astro-ph.EP] . Westley, M. S., R. A. Baragiola, R. E. Johnson, and G. A. Baratta. 1995. Photodesorption from low-temperature water ice in interstellar and circumsolar grains. Nature 373, no. 6513 (February): 405-407. https: //doi.org/10.1038/373405a0. Wierzchos, K., and M. Womack. 2018. C/2016 R2 (PANSTARRS): A Comet Rich in CO and Depleted in HCN. AJ 156, no. 1 (July): 34. https://doi. org/10.3847/1538-3881/aac6bc. arXiv: 1805.06918 [astro-ph.EP] . Winter, Andrew J., Myriam Benisty, and Sean M. Andrews. 2024. Planet formation regulated by galactic-scale interstellar turbulence. arXiv eprints (May): arXiv:2405.08451. arXiv: 2405.08451 [astro-ph.EP] . Wong, Michael H., Paul R. Mahaffy, Sushil K. Atreya, Hasso B. Niemann, and Tobias C. Owen. 2004. Updated Galileo probe mass spectrometer measurements of carbon, oxygen, nitrogen, and sulfur on Jupiter. Icarus 171, no. 1 (September): 153-170. https://doi.org/10.1016/j.icarus.2004. 04.010. Worthen, Kadin, Christine H. Chen, David R. Law, Cicero X. Lu, Kielan Hoch, Yiwei Chai, G. C. Sloan, et al. 2024. MIRI MRS Observations of β Pictoris. I. The Inner Dust, the Planet, and the Gas. ApJ 964, no. 2 (April): 168. https://doi.org/10.3847/1538-4357/ad2354. Xue, Qiao, Jacob L. Bean, Michael Zhang, Luis Welbanks, Jonathan Lunine, and Prune August. 2024. JWST Transmission Spectroscopy of HD 209458b: A Supersolar Metallicity, a Very Low C/O, and No Evidence of CH4, HCN, or C2H2. ApJ 963, no. 1 (March): L5. https://doi.org/ 10.3847/2041-8213/ad2682. arXiv: 2310.03245 [astro-ph.EP] . Yang, Chao-Chin, Anders Johansen, and Daniel Carrera. 2017. Concentrating small particles in protoplanetary disks through the streaming instability. A&A 606 (October): A80. https : / / doi . org / 10 . 1051 / 0004 - 6361 / 201630106. arXiv: 1611.07014 [astro-ph.EP] . Youdin, Andrew N., and Jeremy Goodman. 2005. Streaming Instabilities in Protoplanetary Disks. ApJ 620, no. 1 (February): 459-469. https: //doi.org/10.1086/426895. arXiv: astro-ph/0409263 [astro-ph] . Zhang, Yapeng, Ignas A. G. Snellen, Alexander J. Bohn, Paul Mollière, Christian Ginski, H. Jens Hoeijmakers, Matthew A. Kenworthy, et al. 2021. The 13 CO-rich atmosphere of a young accreting super-Jupiter. Nature 595, no. 7867 (July): 370-372. https://doi.org/10.1038/s41586-02103616-x. arXiv: 2107.06297 [astro-ph.EP] . Zhu, Zhaohuan, Lee Hartmann, Charles F. Gammie, Laura G. Book, Jacob B. Simon, and Eric Engelhard. 2010. Long-term Evolution of Protostellar and Protoplanetary Disks. I. Outbursts. ApJ 713, no. 2 (April): 11341142. https : / / doi . org / 10 . 1088 / 0004 - 637X / 713 / 2 / 1134. arXiv: 1003.1759 [astro-ph.SR] . Zhu, Zhaohuan, Yan-Fei Jiang, and James M. Stone. 2020. Global 3D radiation magnetohydrodynamic simulations for FU Ori's accretion disc and observational signatures of magnetic fields. MNRAS 495, no. 3 (January): 3494-3514. https://doi.org/10.1093/mnras/staa952. arXiv: 1912.01632 [astro-ph.EP] .\",\n \"pages\": [\n 23,\n 24,\n 25,\n 26,\n 27,\n 28,\n 29\n ]\n }\n]"}}},{"rowIdx":4985,"cells":{"bibcode":{"kind":"string","value":"2024arXiv241205772M"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/2412.05772.pdf"},"content":{"kind":"string","value":"\nWOLF-RAYET STARS - WHAT WE KNOW AND WHAT WE DON'T\nO. V. Maryeva a *\na Astronomical Institute of the Czech Academy of Sciences, Fričova 298, 25165 Ondřejov, Czech Republic\nToday, we have a sufficiently complete picture of what the Wolf-Rayet (WR) stars are. Predictions of stellar evolution theory are in a good agreement with their parameters, estimated from observational data using stellar atmospheres codes; predictions of population synthesis also agree well with number of known WR stars. This article provides an overview of the main historical milestones in the studies of WR stars, showing how we came to this understanding, and what questions are still unanswered.\nKeywords: Stars: Wolf-Rayet - Stars: evolution - Stars: atmospheres - General: history and philosophy of astronomy\n1. INTRODUCTION\nWolf-Rayet (WR) stars are a class of objects identified on the basis of their spectral features. WR spectra show strong emission lines of helium, nitrogen, carbon and oxygen in different stages of ionization. The width of these lines reaches tens of angstroms, the central intensities are sometimes 10-20 times the intensity of the continuum spectrum. A prominent feature of WRs are two bumps at 4650 1) and 5808 Å, which are groups of closely located lines of N II , N III , C III and C IV , which are visible even in low-resolution spectra. The lines are formed in an extended atmosphere - in the stellar wind, expelling at velocities of 10 2 -10 3 kms -1 . WR stars are characterized by a high mass-loss rate of several 10 -5 M ⊙ yr -1 and by high temperatures ( T eff ≳ 30000 K). WR are the final stage in the evolution of massive stars (stars with an initial mass ≳ 25 M ⊙ ) before the core-collapse supernova explosion.\nWith accumulation of observational data it became clear that emission spectra and WR bumps are intrinsic to objects of different natures. Therefore nowadays there is a clear distinction between WR stars and WR phenomenon , which occurs when fast-moving, hot plasma is expanding around a hot star (Gräfener et al., 2011; Vink, 2015). WR phenomenon may occur in the low mass stars after ejecting their outer layers in the planetary nebula phase and exposing their hot cores prior to the white-dwarf phase (Marcolino et al., 2007; Todt, 2009; van der Hucht et al., 1981). Central stars of planetary nebulae showing WR phenomenon in their spectra are denoted as [WR] (van der Hucht et al., 1981). Rare and still unique case of low mass object with WR phenomenon and the absence of a clearly detected circumstellar nebula is LAMOST J040901.83+323955.6 (Maryeva et al., 2024). It was selected as a WR star by Škoda et al. (2020), but Maryeva et al. (2024) showed that this object is 0 . 9 M ⊙ star caught in a rare transitional phase from post-AGB to CSPN. Another unique object with WR phenomenon is IRAS 00500+6713, formed as a result of a merger of two white dwarfs (Gvaramadze et al., 2019). The WR phenomenon is also observed in young supernovae (GalYam et al., 2014).\nAs we know now, massive stars showing WR spectra are also not a homogeneous class. They are split into two groups: very massive stars and classical WR stars . Very massive stars ( M ini ≥ 100M ⊙ ) experience strong stellar winds and show WR spectra already during core H-burning phase, when they are located on main sequence. All of them show signs of hydrogen on their surface and are classified as WNh objects (Crowther et al., 2010; Martins et al., 2023; Smith et al., 1996). Classical WR stars (usually just named WR stars ) stars are hydrogen-depleted objects that have evolved off the main sequence and suffered intense mass loss (Crowther, 2007; Shenar, 2024). Classical WR stars will be the subject of this review article.\nThe field of contemporary studies of WR stars is enormous and cannot be fully covered in a single small review like this one, so I'd like to recommend two other detailed reviews - Crowther (2007) and Shenar (2024) - for better understanding what WR stars are. On the other hand, here I will concentrate more on the historical aspects of WR studies, on how we have arrived to our present day understanding of them. Therefore Section 2 will cover the history of studies of WR stars, while Section 3 will highlight several topical problems and open questions about them.\n2. FROM DISCOVERY TO UNDERSTANDING\nIn 1867 year astronomers from Paris observatory Charles Wolf and Georges Rayet discovered three stars in Cygnus constellation those spectra significantly\ndiffer from other stars. In contrast with other stars, where usually absorption lines are dominant in spectra, discovered objects showed strong emission lines (Wolf & Rayet, 1867). The class of objects received its name in honor of the discoverers and gradually began to be replenished with new members. One hundred years after the discovery of WRs, by the end of 1960-s the number of known WRs in the Galaxy had reached over a hundred, totalling 127 (Roberts, 1962; Smith, 1968), while now is ∼ 700 (Rosslowe & Crowther, 2015). At the same time, in the late 1950-s middle 1960-s, work began to search for extragalactic WRs - WRs in the Large Magellanic Cloud (Westerlund & Rodgers, 1959; Westerlund & Smith, 1964).\nSignificant contribution to the understanding of the physics of WR stars was given by Carlyle S. Beals (1929). Beals found that some lines in the spectra of WR stars have a P Cygni profile and based on comparison with P Cygni profile lines in spectra of novae he suggested that WRs are surrounded by expanding envelopes (Beals, 1929). Beals also found a difference between WR stars and novae: the P Cygni profiles in the WR stars do not change over time. This allowed to propose that an outflow from WR stars occurs continuously (Beals, 1929). This was confirmed by Chandrasekhar (1934), who developed a solid footing for interpreting P Cygni profiles as arising in expanding atmospheres. Kosirev (1934) used the diagnostics developed by Chandrasekhar to estimate the mass loss and maximum outflow velocity of a WR star and found, respectively, ∼ 10 -5 M ⊙ yr -1 and ∼ 1000 kms -1 .\nAt the beginning of the 20th century, stellar spectroscopy developed in close cooperation with atomic physics. Investigating spectra of the star ζ Puppis Edward Pickering paid his attention to previously unknown lines 5411, 4541, 4200, 4100, 4026, 3924, 3858, 3813, 3782 Å (Pickering, 1897). He interpreted them as another series of hydrogen (besides Balmer) (Pickering, 1901). Line 4686 Å, first discovered during a solar eclipse and often found in WR spectra, was also considered to be a hydrogen line (Fowler, 1912). Although Fowler (1912) was able to obtain these spectral lines in the laboratory in 1912, during an experiment with a helium-filled tube, he considered their appearance to be a contribution from hydrogen impurity. It was not until 1913 that Niels Bohr's work (Bohr, 1913) explained the nature of these lines as ionized helium He II , confirming the atomic model (see Robotti (1983) for historical review).\nIn 1890 Pickering drew attention to the similarity of the spectra of WR stars and planetary nebula (Pickering, 1891). Due to this, in papers devoted to emission spectra of nebulae it is possible to find descriptions of spectra of WR stars. The lines at 4647, 4650, 5696, 5801 and 5812 Å were identified with the carbon by Wright (1918). Ira S. Bowen, who found an interpretation of the 'nebulium' lines as [O III ], gave a detailed list of emission lines observed in nebulae, which\nincludes an identification of almost all lines visible in the WRs (Bowen, 1928). In a 1934 paper Bowen (1934) suggests a significant role of the fluorescence for formation of the N III 4634, 46340 lines belonging to the WR bump.\nAlready the spectra of the first discovered WR stars showed a difference: the broad emission bands are located in different places (Huggins & Huggins, 1890; Wolf & Rayet, 1867). After the identification of all the main spectral lines, Beals (1933) proposed the splitting of WR into two groups. This classification was approved in 1938, the International Astronomical Union (IAU) divided the spectra of WR stars into types WN and WC, depending on whether the spectrum was dominated by lines of nitrogen or carbon-oxygen respectively (Beals, 1933; Swings, 1942). WN stars are believed to show the hydrogen burning products via the CNO cycle, while WC stars reveal the helium burning products via the tripleα cycle. Based on the strength of the emission lines and line ratios, WN stars can be further classified into the spectral subtypes WN2 to WN11, and WC stars into the spectral subtypes WC4 to WC9 (Crowther et al., 1998; Smith, 1968; Smith et al., 1990, 1994, 1996). Also, there are transition types from WN to WC, which are called WN/C, whose spectra show strong emission lines of carbon and nitrogen simultaneously (Conti & Massey, 1989).\nIn 1933 Victor A. Ambartsumian (1933) estimated He/H ratio for WR stars using intensities of H β and He II 4686 lines and found He/H ≳ 1 . 8 . Subsequent studies (Rublev (1972a) and references therein) confirmed excess of helium. Gamow (1943) suggested that the anomalous composition of WR stars was the result of nuclear processed material being visible on their surfaces. Despite these arguments in favor of far evolved status of WR stars, debates about it continued until the mid-1970s. There was alternative hypothesis, that WR are young and more massive than T Tau objects, before main sequence (Sahade, 1958; Underhill, 1968). It is important to mention the studies of Sergej V. Rublev, in which he developed methods for determining the temperatures and luminosities of WR stars and estimated the hydrogen abundances (Rublev, 1965, 1970, 1972b, 1975). These works played a significant role in the formation of the modern understanding of WR stars as far evolved massive objects.\n2.1. Development of numerical methods\nContemporary progress in our understanding of hot stars and especially WR stars has been significantly defined by the progress of numerical modelling of stellar atmospheres, in turn enabled by the development of both physical approximations and numerical methods.\nVictor V. Sobolev developed the theory of radiative transfer in moving media and applied it to the determination of physical conditions in the envelopes\nof WR, Be, novae, etc. type stars (Sobolev, 1947, 1960). He showed that due to the presence of a velocity gradient, photons of spectral lines are able to leave the deep layers of the atmosphere directly, avoiding the usual diffusion process; this significantly simplifies the study of the radiative equilibrium of moving atmospheres. This theory became the basis for the transition to modern quantitative calculations of the stellar wind.\nWRstars are essentially hot stellar cores surrounded by extended, low-density atmospheres. Due to their high luminosity the radiation in the stellar wind dominates over collisional processes, and the models of WR's atmospheres should be calculated taking into account the deviations from local thermodynamic equilibrium (non-LTE). Straightforward iterative procedure for the solution of radiative transfer equations Λ iteration - fails to converge in typical non-LTE conditions, as in scattering dominated media of large optical thickness, its convergence is extremely slow (see overview by Atanackovic-Vukmanovic (2004) and references therein).\nSeveral approaches have been proposed in early 1970-s as a way to numerically solve radiative transfer (RT) problems in non-LTE case. The first fully self-consistent solution of RT problems for non-LTE case was achieved by Auer & Mihalas (1969) who developed the complete linearization method based on the Newton-Raphson scheme to solve the set of discretized structural equations in a robust manner that opened a way for significant progress in the modelling of stellar atmospheres.\nA different approach is the 'core saturation method', introduced by Rybicki (1972) who implemented a modification of the Λ iteration method that made it practically usable for solving the RT and statistical equilibrium equations. In order to eliminate the cause of the slow convergence of simple Λ iteration (poor conditioning of the equations arising from a large number of scatterings), Rybicki proposed the elimination of scattering events in the line core as they do not contribute much to the transfer process.\nAnother approach is 'perturbation technique' proposed by Cannon (1973a,b) who was the first to use the idea of 'operator splitting' in radiative transfer computations. He replaced the full (exact) Λ operator by a simplified one (so called approximate Lambda operator, or ALO) and computed a small error term made by this approximation by using a perturbation technique. Cannon's operator perturbation technique was an efficient way to combine computational advantages of approximate solutions with the accuracy of exact methods.\nIn 1980-s computational approximations to accelerate the Λ iteration started to be used together with the operator perturbation technique introduced by Cannon (1973a,b) to simplify the direct solution, making the basis for the class of methods known as Accelerated Lambda Iteration. Wolf-Rainer Hamann (1985,\n1986) adopted perturbation technique to approximate lambda operator, based on idea of 'core saturation' and presented test calculations for a typical WR star atmosphere. This effort became the foundation for the PoWR code (Hamann & Gräfener, 2003) that is still being actively developed and successfully applied for numerical modelling of various types of astrophysical objects. Independently, D. John Hillier (1990) presented a method based on tridiagonal Newton-Raphson operator, complete linearization scheme and perturbation technique for calculations of WN and WC atmosphere models. Over time, this method has grown into CMFGEN code (Hillier & Miller, 1998).\nSince the mid-1990s, CMFGEN and PoWR have become a reliable instruments for studying hot stars with high mass-loss rates. Systematic study of Galactic WR stars started with a series of works by Paul Crowther titled 'Fundamental parameters of WR stars' and devoted to the analysis of properties of nitrogen-rich WR stars (WN) based on optical and ultraviolet (UV) data and numerical modelling using CMFGEN code (Crowther et al., 1995a,b,c,d,e). Advances in infrared (IR) spectroscopy also enabled studies of WR stars near the Galactic center (Liermann et al., 2010; Martins et al., 2007). Modeling of large sample of Galctic WN (Hamann et al., 2006) and WC (Sander et al., 2012) stars ( ≈ 126 objects in total) allowed to obtain a homogeneous set of stellar and atmospheric parameters for them, to determine the correct location of WR stars in the Hertzsprung-Russell (HR) diagram.\nPoWR code was widely used for studying extragalactic objects. Among WR stars in nearby galaxies, the ones in Magellanic Clouds are most studied (Bestenlehner et al., 2014; Hainich et al., 2014, 2015). Sample of 74 WR stars belonged to M33 galaxy was analysed by Pritzkuleit (2020) (Pritzkuleit, 2020). Sander et al. (2014) investigated late-type WN stars in M31 galaxy (Sander et al., 2014). These studies extended our understanding of properties of WR stars in different galactic environments and metallicity-dependence of massive star winds.\n2.2. Evolution\nModern view on WRs as products of evolution of massive stars was finally formed only in 1980s. Although by the 1980s WR stars were already recognised as descendants of massive OB stars, their exact evolutionary status with respect to the main sequence and other evolved massive stars was still an open question. WR stars were being proposed as possible progenitors of supernovae, and particularly the newly-discovered type Ib supernovae that lack hydrogen but are apparently associated with young massive stars.\nThe start of systematic X-ray observations with space-borne X-ray telescopes in 1960s significantly advanced the theory of close binary stars and their interac-\ntions, and it led Bohdan Paczyński (1967) to the suggestion that WR stars are the products of binary evolution - namely, mass exchange in a binary system. In his scenario, a more massive star in binary system evolves faster and, upon exhaustion of hydrogen in the core, fills the Roche lobe and a rapidly loses its hydrogen envelope that outflows towards the companion. As a result, when the envelope is fully transferred, its hot helium core remains as a WR star with little hydrogen in the outermost layers (Paczyński, 1973).\nThe hypothesis of Paczyński was well accepted and it naturally explained the anomalous chemical composition of WR stars as naked stellar cores enriched with products of nuclear reactions. On the other hand, based on the similarity of Of and WN stars (Figure 1) and on the first estimates of the mass loss rate for hot massive stars based on UV observations Peter Conti (1975) suggested that the stellar wind of O stars is strong enough to remove the outer layers and uncover the stellar core, thus proposing a single star mass loss channel to form WR stars. In addition, brightest WR star, γ Vel, was inconsistent with Paczyński's hypothesis as, while being a binary, its orbit was clearly elliptical and its second component lacks any signs of previous mass exchange, both suggesting that there was no interaction between the components (Conti, 2015).\nIn 1984 Peter Conti (1984) introduced a new class of objects. In his talk at the IAU symposium, Peter Conti grouped S Doradus variables, Hubble-Sandage variables, η Carinae, P Cygni, and other similar stars together under the term 'luminous blue variables' (LBV). LBV are characterized by strong mass loss (around 10 -5 M ⊙ yr -1 ) and occasional giant eruptive events (Humphreys & Davidson, 1994). Conti suggested that massive O stars transit to WR stars through this short-duration LBV phase. This idea is consolidated as the 'Conti scenario' and becomes the main channel for WR star formation. Modern stellar evolution calculations are consistent with the 'Conti scenario' (Groh et al., 2014; Sander et al., 2019). According to numerical calculations the stars with initial mass of 60 M ⊙ spend 2 × 10 5 yr in the LBV phase - only 5% of their lifetime (Groh et al., 2014). Recently observed transition from LBV to WN8 spectral type in Romano' star is also a direct confirmation of 'Conti scenario' (Maryeva et al., 2019; Polcaro et al., 2016).\nSimilar scenario for WR formation was also proposed by Bisnovatyi-Kogan & Nadyozhin (1972), with significant mass loss on the red supergiant phase. They show that large inverse density gradient appears in the outer layers of a 30 M ⊙ star in this phase due to supercritical luminosity, resulting in powerful and rapid mass loss and formation of a stripped helium core.\nIn the last decade, Paczyński's hypothesis has again been discussed as a mechanism for the formation of WR stars. Analysing the spatial positions of LBV and WR stars, Nathan Smith (Smith, 2016, 2019; Smith & Tombleson, 2015) con-\n\n\n
Fig. 1: Comparison of spectra of J013340.19+303134.5 (WC4), J013352.43+304351.7 (WN8) and HD14947 (O4.5If) as illustration of significant differences in intensities of spectral lines for O stars and different sub-types of WR stars.
\n\nluded that both LBVs and WRs could be the products of binary evolution, LBVs being mass-gainers, and WRs - mass-donors, or stripped components. This idea was critically discussed by Humphreys et al. (2016) and Davidson et al. (2016). Now we may confidently conclude only that we do not have enough data for statistical analysis. Modern evolutionary codes allow detailed calculations of binary system evolution (see for example Götberg et al. (2018); Laplace et al. (2021)), thus making it possible to compare theoretical predictions for stripped stars with properties of real WR stars (Götberg et al., 2018). Our current understanding is that binary interaction scenario cannot properly describe all observed WR stars, but most probably takes place in a part of them (see Shenar (2024) and references therein).\nImproving the statistics of binarity among WR stars is a crucial task. Among the first researchers who started to work on it was Virpi S. Niemelä, who determined the stellar masses in binary systems and discovered and analyzed the spectroscopic orbits of many binary systems with WR components (see for ex-\nmple her works Niemela (1995, 2001)). Nowadays this work is being continued, both through spectroscopy (Chené et al., 2022) and using high-resolution imaging (Deshmukh et al., 2024).\n3. CURRENT PROBLEMS\n3.1. Search for new WR stars\nIn our Galaxy there are 679 WR stars currently discovered 2) (Rosslowe & Crowther, 2015), while the theory based on current Milky Way star formation rate and duration of WR phase predicts the numbers of WR stars ∼ 1200 (Rosslowe & Crowther, 2015). It means that significant part of WR stars are still not discovered. WR stars in the Galaxy align with spiral arms and the regions of star formation. Thus, the discovery of new WR stars through optical observations has been significantly limited due to the presence of dust extinction. Although the probability of finding a WR star during optical spectroscopic surveys remains (Maíz Apellániz et al., 2016; Zhang et al., 2020), the majority of discoveries of new WR stars now happens through the observations in IR range. For example, Mauerhan et al. (2011) identified 60 Galactic WR stars: candidates were selected using the photometry from Spitzer and Two Micron All Sky Survey (2MASS) surveys and confirmed by near-IR spectroscopy. Shara et al. (2012) expanded the list by adding 71 more stars, also preliminary selected from J and K band 2MASS photometry.\nMoreover, the search for new WR stars is complicated by the effect of spectral mimicry - low mass [WR] objects masquerading as classical WR stars. Thanks to the results of Gaia mission (Gaia Collaboration et al., 2016) we now have reliable estimations of distances, and it helps to quickly understand the nature of individual objects and separate low-mass evolved stars from massive stars, and, moreover, to find the objects of unusual origin among low mass stars. For example, based on Gaia distance estimation Gvaramadze et al. (2019) demonstrated that IRAS00500+6713 is a product of merging of two white dwarfs, while the star has WO-type spectrum. Counterexample is PMR 5 star classified as [WN6] by Morgan et al. (2003) that was recently reclassified as a classical WN.\nSystematic search for WR stars in nearby galaxies is ongoing since 1970-s. In 1972 first catalog of WR candidate stars located in M33 galaxy was published (Wray & Corso, 1972). Twenty-five objects were selected based on photometry in three narrow-band filters (one filter is centered on He II 4686 - the strongest emission line in WN-type WRs, one is centered on C III -IV ≈ 4650 - the strongest\n\n\n
Fig. 2: HR diagram and evolutionary tracks for the low metallicity massive stars ( Z = 0 . 002 ) from Georgy et al. (2013) (left panel) and Grasha et al. (2021) (right panel) evolutionary models. Geneva tracks do not produce massive WR stars at such low metallicities, while Stromlo tracks show rather different evolutionary path and produce WR stars with M ∗ > 70M ⊙ .
\n\nline in WC-type WRs, and one is centered on continuum). This technique of selection of WR candidates for following spectroscopy has proven itself successfully and is still widely used (Massey & Conti, 1983; Massey & Johnson, 1998; Neugent & Massey, 2011; Neugent et al., 2012). As of today, 211 WR are known in M33 galaxy (Massey et al., 2016) and 173 are in M31 (Neugent & Massey, 2023). Search for new WR and their following investigations are important for understanding the impact of metallicity on mass-loss rate.\n3.2. Wolf-Rayet stars in low metallicity environment\nAs the opacity of stellar wind depends on the metallicity, mass-loss rates of massive stars decrease with Z . It means that in low metallicity environment single massive stars are unable to lose enough material through the stellar wind to become WR star. Thus, the mass loss on pre-WR stages is of fundamental importance for the final fate of massive stars, and may influence e.g. the formation of long-duration Gamma Ray Bursts (GRBs) or the yields from early stellar generations.\nMetallicity impacts overall evolutionary behaviour of massive stars in a complicated way. Georgy et al. (2013) calculated evolutionary tracks for stars in a wide range of initial masses at low metallicity. According to their calculations,\nmetal-poor stars become colder after the end of hydrogen burning in the core and move along the track to the right side of the diagram. Unlike solar metallicity stars, metal-poor ones no longer return to the left side of the diagram (see Figure 2), even when the effects of stellar rotation are taken into account in the models. On the other hand, evolutionary tracks behave differently if the models implement Galactic Concordance abundances instead of solar, scaled-solar, or alpha-element enhanced abundances (Grasha et al., 2021) - the stars with the initial mass of M ∗ > 70M ⊙ with rotation V/V crit = 0 . 2 after the main sequence move to the right and then return to the left to the WR stars region, in contrast to predictions of Georgy et al. (2013).\nYarovova et al. (2023) used evolutionary tracks from Georgy et al. (2013) and Grasha et al. (2021) for the study of emission-line stellar-like object in the nearby low-metallicity ( Z ∼ 0 . 1 Z ⊙ ) dwarf galaxy NGC 4068. They found the best agreement between the modelled and observed spectra for the model assuming ionization by low-metallicity WR star of M ∗ ≈ 80M ⊙ mass, ionizing the nebula through the strong wind and enriching the interstellar medium with nitrogen. Thus, parameters of the object favor Grasha et al. (2021) predictions. Therefore, understanding of mass loss rate requires complex approach that includes both studies of massive stars in different metallicites, and studies of present-day chemical abundances.\n3.3. Wolf-Rayet stars and red supergiants\nIsolated massive stars with initial masses in 8 M ⊙ ≲ M ∗ ≲ 20 M ⊙ interval undergo the transition to Red Supergiants (RSG) after hydrogen in the core is exhausted. For them RSG is the final stage before Type-II supernova (SNII) explosion. More massive stars ( 40 M ⊙ ≲ M ∗ < 90 M ⊙ ) evolve through LBV phase before reaching WR phase, and finally explode as SN Type Ib/c. The evolution of stars with 25 M ⊙ ≲ M ∗ ≲ 35 M ⊙ initial masses is more complicated. Observed population of supernovae Type II progentiors in the local Universe shows that they are produced by the stars with initial masses M ∗ ≲ 18M ⊙ (Smartt, 2015), and it suggests that more massive stars should either directly collapse to black holes, or evolve leftwards in HR diagram, thus becoming Wolf-Rayet stars. It is supported by the luminosity distributions of cool supergiants in Large and Small Magellanic Clouds (Davies et al., 2018) that overlaps with those of apparently-single WR stars in both galaxies, thus suggesting a changing evolutionary sequence of massive stars with increasing initial mass. At the same time, between the RSG and WR regions in HR diagram lies the region of low luminosity LBVs and yellow hypergiants (YHGs) (Figure 3). Therefore, the evolution of 25 M ⊙ ≲ M ∗ ≲ 35 M ⊙\n\n\n
Fig. 3: Hertzsprung-Russell (HR) diagram (luminosities versus effective temperatures). Color horizontal lines show evolutionary tracks, with solid parts showing the phase when hydrogen burns in the stellar core. The tracks are taken from Ekström et al. (2012) (Ekström et al., 2012). ZAMS is zero age Main Sequence.
\n\ninitial mass may look like that:\nO → RSG → Y HG (? ) → low LBV → WR ( for M ∗ ≈ 25 -35 M ⊙ )\nIt means that the star with initial mass about 30 M ⊙ should pass through all four stages - do such objects really exist?\nGroh et al. (2013) demonstrated that the stars with 20 -25 M ⊙ initial masses are unstable on low luminosity LBV stage, and explode as core-collapse supernovae (SN Type IIL/b). Prior to the explosion they show the spectrum similar to S Doradus during its minimum of brightness (WN11h spectral type). Maryeva et al. (2020) have recently found a Galactic object, Wray 15-906, with characteristics typical of a low luminosity LBV and the spectrum looking like WN11h. It was identified via the detection of its infrared circumstellar shell (of ≈ 2 pc in diameter) with the Wide-field Infrared Survey Explorer (WISE) and the Herschel Space Observatory. Maryeva et al. (2020) found that Wray 15-906 is a relatively low-luminosity, log( L ∗ / L ⊙ ) ≈ 5 . 4 , star with a temperature of 25 ± 2 kK, with position in HR diagram corresponding to post-red supergiant with initial mass\nof ≈ 25M ⊙ . Its spectrum and properties are consistent with theoretical predictions of Groh et al. (2013). Thus, the monitoring of this object, and the search for similar ones, may improve our understanding of the link between WRs and RSGs.\n4. CONCLUSION\nWolf-Rayet stars were discovered more than 150 years ago, and since then they were targets of numerous studies that combined astrophysical insights with advancements in atomic physics, development of numerical methods, and instrumental progress. Nowadays we understand quite well what Wolf-Rayet stars are, and they are no longer a hot topic in astrophysics. However they still are keystones for many directions of research, from theory of stellar atmospheres up to stellar populations in other galaxies and gravitational waves. Through the existence of the Wolf-Rayet phenomenon WR stars are linked, besides massive stars, with low mass evolved stars.\nWe expect new progress in understanding the Wolf-Rayet stars and massloss of massive stars in general from the upcoming advances in the instrumentation. For example, SITELLE imaging Fourier transform spectrometer at CanadaFrance-Hawaii telescope (Drissen et al., 2019) may provide spatially resolved line ratios and kinematics of WR nebulae thus hinting us on ionizing flux from the stars, and their mass loss history. James Webb Space Telescope will allow studying dust shells around WR stars (Lau et al., 2022), and to assess their role in enrichment of interstellar medium with organic compounds and carbonaceous dust. Spectropolarimetric surveys of WR and monitoring of variability of linear polarization are also still topical.\nAcknowledgements I would like to thank the organizers of the conference for their kind invitation to present this overview. This research received funding from the European Union's Framework Programme for Research and Innovation Horizon 2020 (2014-2020) under the Marie Skłodowska-Curie Grant Agreement No. 823734 (POEMS project). The Astronomical Institute in Ondřejov is supported by the project RVO:67985815.\nREFERENCES\nAmbartsumian V. A., 1933, Izvestiya Glavnoj Astronomicheskoj Observatorii v Pulkove, 13\nAtanackovic-Vukmanovic O., 2004, Serbian Astronomical Journal, 169, 1\nAuer L. H., Mihalas D., 1969, ApJ, 158, 641 Beals C. S., 1929, MNRAS, 90, 202 Beals C. S., 1933, The Observatory, 56, 196 Bestenlehner J. M., et al., 2014, A&A, 570, A38 Bisnovatyi-Kogan G. S., Nadyozhin D. K., 1972, Ap&SS, 15, 353 Bohr N., 1913, Nature, 92, 231 Bowen I. S., 1928, ApJ, 67, 1 Bowen I. S., 1934, PASP, 46, 146 Cannon C. J., 1973a, J. Quant. Spec. Radiat. Transf., 13, 627 Cannon C. J., 1973b, ApJ, 185, 621 Chandrasekhar S., 1934, MNRAS, 94, 522 Chené A.-N., Mahy L., Gosset E., St-Louis N., Dsilva K., Manick R., 2022, MNRAS, 516, 1022 Conti P. S., 1975, Memoires of the Societe Royale des Sciences de Liege, 9, 193 Conti P. S., 1984, in Maeder A., Renzini A., eds, IAU Symposium Vol. 105, Observational Tests of the Stellar Evolution Theory. p. 233 Conti P. S., 2015, in Hamann W.-R., Sander A., Todt H., eds, Wolf-Rayet Stars. pp 347-350 Conti P. S., Massey P., 1989, ApJ, 337, 251 Crowther P. A., 2007, ARA&A, 45, 177 Crowther P. A., Hillier D. J., Smith L. J., 1995a, A&A, 293, 172 Crowther P. A., Hillier D. J., Smith L. J., 1995b, A&A, 293, 403 Crowther P. A., Smith L. J., Hillier D. J., Schmutz W., 1995c, A&A, 293, 427 Crowther P. A., Smith L. J., Hillier D. J., 1995d, A&A, 302, 457 Crowther P. A., Smith L. J., Willis A. J., 1995e, A&A, 304, 269 Crowther P. A., De Marco O., Barlow M. J., 1998, MNRAS, 296, 367\n
\n\n
\nHamann W. R., Gräfener G., 2003, A&A, 410, 993\nHamann W. R., Gräfener G., Liermann A., 2006, A&A, 457, 1015 Hillier D. J., 1990, A&A, 231, 116 Hillier D. J., Miller D. L., 1998, ApJ, 496, 407 Huggins W., Huggins M., 1890, Proceedings of the Royal Society of London Series I, 49, 33 Humphreys R. M., Davidson K., 1994, PASP, 106, 1025 Humphreys R. M., Weis K., Davidson K., Gordon M. S., 2016, ApJ, 825, 64 Kosirev N. A., 1934, MNRAS, 94, 430 Laplace E., Justham S., Renzo M., Götberg Y., Farmer R., Vartanyan D., de Mink S. E., 2021, A&A, 656, A58 Lau R. M., et al., 2022, Nature Astronomy, 6, 1308 Liermann A., Hamann W. R., Oskinova L. M., Todt H., Butler K., 2010, A&A, 524, A82 Maíz Apellániz J., et al., 2016, ApJS, 224, 4 Marcolino W. L. F., Hillier D. J., de Araujo F. X., Pereira C. B., 2007, ApJ, 654, 1068 Martins F., Genzel R., Hillier D. J., Eisenhauer F., Paumard T., Gillessen S., Ott T., Trippe S., 2007, A&A, 468, 233 Martins F., Schaerer D., Marques-Chaves R., Upadhyaya A., 2023, A&A, 678, A159 Maryeva O., Viotti R. F., Koenigsberger G., Calabresi M., Rossi C., Gualandi R., 2019, Galaxies, 7, 79 Maryeva O. V., Gvaramadze V. V., Kniazev A. Y., Berdnikov L. N., 2020, MNRAS, 498, 5093 Maryeva O., Abdulkarimova A., Karpov S., Moiseev A., Oparin D., 2024, MNRAS, 527, 11925\nMassey P., Conti P. S., 1983, ApJ, 273, 576\nMassey P., Johnson O., 1998, ApJ, 505, 793 Massey P., Neugent K. F., Smart B. M., 2016, AJ, 152, 62 Mauerhan J. C., Van Dyk S. D., Morris P. W., 2011, AJ, 142, 40 Morgan D. H., Parker Q. A., Cohen M., 2003, MNRAS, 346, 719 Neugent K. F., Massey P., 2011, ApJ, 733, 123 Neugent K. F., Massey P., 2023, AJ, 166, 68 Neugent K. F., Massey P., Georgy C., 2012, ApJ, 759, 11 Niemela V. S., 1995, in van der Hucht K. A., Williams P. M., eds, IAU Symposium Vol. 163, Wolf-Rayet Stars: Binaries; Colliding Winds; Evolution. p. 223 Niemela V., 2001, in Revista Mexicana de Astronomia y Astrofisica Conference Series. pp 23-26 Paczyński B., 1967, Acta Astron., 17, 355 Paczyński B., 1973, in Bappu M. K. V., Sahade J., eds, IAU Symposium Vol. 49, Wolf-Rayet and High-Temperature Stars. p. 143 Pickering E. C., 1891, Astronomische Nachrichten, 127, 1 Pickering E. C., 1897, ApJ, 5, 92 Pickering E. C., 1901, Harvard College Observatory Circular, 55, 1 Polcaro V. F., et al., 2016, AJ, 151, 149 Pritzkuleit M., 2020, Master's thesis, University of Potsdam, Germany Roberts M. S., 1962, AJ, 67, 79 Robotti N., 1983, Historical Studies in the Physical Sciences, 14, 123 Rosslowe C. K., Crowther P. A., 2015, MNRAS, 447, 2322 Rublev S. V., 1965, Soviet Ast., 9, 274 Rublev S. V., 1970, Astrofizicheskie Issledovaniia Izvestiya Spetsial'noj As-\ntrofizicheskoj Observatorii, 1, 25\nRublev S. V., 1972a, Astrofizicheskie Issledovaniia Izvestiya Spetsial'noj Astrofizicheskoj Observatorii, 4, 3\n
\n\n
\nUnderhill A. B., 1968, ARA&A, 6, 39\nVink J. S., 2015, in Hamann W.-R., Sander A., Todt H., eds, Wolf-Rayet Stars. pp 133-138 ( arXiv:1510.00227 ), doi:10.48550/arXiv.1510.00227 Westerlund B. E., Rodgers A. W., 1959, The Observatory, 79, 132 Westerlund B. E., Smith L. F., 1964, MNRAS, 128, 311 Wolf C. J. E., Rayet G., 1867, Academie des Sciences Paris Comptes Rendus, 65, 292 Wray J. D., Corso G. J., 1972, ApJ, 172, 577 Wright W. H., 1918, Publications of Lick Observatory, 13, 191 Yarovova A. D., Egorov O. V., Moiseev A. V., Maryeva O. V., 2023, MNRAS, 518, 2256 Zhang W., et al., 2020, ApJ, 902, 62 Škoda P., Podsztavek O., Tvrdík P., 2020, A&A, 643, A122 1981,\nvan der Hucht K. A., Conti P. S., Lundstrom I., Stenholm B., Space Sci. Rev., 28, 227\n"},"sections":{"kind":"list like","value":[{"title":"WOLF-RAYET STARS - WHAT WE KNOW AND WHAT WE DON'T","content":"O. V. Maryeva a * a Astronomical Institute of the Czech Academy of Sciences, Fričova 298, 25165 Ondřejov, Czech Republic Today, we have a sufficiently complete picture of what the Wolf-Rayet (WR) stars are. Predictions of stellar evolution theory are in a good agreement with their parameters, estimated from observational data using stellar atmospheres codes; predictions of population synthesis also agree well with number of known WR stars. This article provides an overview of the main historical milestones in the studies of WR stars, showing how we came to this understanding, and what questions are still unanswered. Keywords: Stars: Wolf-Rayet - Stars: evolution - Stars: atmospheres - General: history and philosophy of astronomy","pages":[1]},{"title":"1. INTRODUCTION","content":"Wolf-Rayet (WR) stars are a class of objects identified on the basis of their spectral features. WR spectra show strong emission lines of helium, nitrogen, carbon and oxygen in different stages of ionization. The width of these lines reaches tens of angstroms, the central intensities are sometimes 10-20 times the intensity of the continuum spectrum. A prominent feature of WRs are two bumps at 4650 1) and 5808 Å, which are groups of closely located lines of N II , N III , C III and C IV , which are visible even in low-resolution spectra. The lines are formed in an extended atmosphere - in the stellar wind, expelling at velocities of 10 2 -10 3 kms -1 . WR stars are characterized by a high mass-loss rate of several 10 -5 M ⊙ yr -1 and by high temperatures ( T eff ≳ 30000 K). WR are the final stage in the evolution of massive stars (stars with an initial mass ≳ 25 M ⊙ ) before the core-collapse supernova explosion. With accumulation of observational data it became clear that emission spectra and WR bumps are intrinsic to objects of different natures. Therefore nowadays there is a clear distinction between WR stars and WR phenomenon , which occurs when fast-moving, hot plasma is expanding around a hot star (Gräfener et al., 2011; Vink, 2015). WR phenomenon may occur in the low mass stars after ejecting their outer layers in the planetary nebula phase and exposing their hot cores prior to the white-dwarf phase (Marcolino et al., 2007; Todt, 2009; van der Hucht et al., 1981). Central stars of planetary nebulae showing WR phenomenon in their spectra are denoted as [WR] (van der Hucht et al., 1981). Rare and still unique case of low mass object with WR phenomenon and the absence of a clearly detected circumstellar nebula is LAMOST J040901.83+323955.6 (Maryeva et al., 2024). It was selected as a WR star by Škoda et al. (2020), but Maryeva et al. (2024) showed that this object is 0 . 9 M ⊙ star caught in a rare transitional phase from post-AGB to CSPN. Another unique object with WR phenomenon is IRAS 00500+6713, formed as a result of a merger of two white dwarfs (Gvaramadze et al., 2019). The WR phenomenon is also observed in young supernovae (GalYam et al., 2014). As we know now, massive stars showing WR spectra are also not a homogeneous class. They are split into two groups: very massive stars and classical WR stars . Very massive stars ( M ini ≥ 100M ⊙ ) experience strong stellar winds and show WR spectra already during core H-burning phase, when they are located on main sequence. All of them show signs of hydrogen on their surface and are classified as WNh objects (Crowther et al., 2010; Martins et al., 2023; Smith et al., 1996). Classical WR stars (usually just named WR stars ) stars are hydrogen-depleted objects that have evolved off the main sequence and suffered intense mass loss (Crowther, 2007; Shenar, 2024). Classical WR stars will be the subject of this review article. The field of contemporary studies of WR stars is enormous and cannot be fully covered in a single small review like this one, so I'd like to recommend two other detailed reviews - Crowther (2007) and Shenar (2024) - for better understanding what WR stars are. On the other hand, here I will concentrate more on the historical aspects of WR studies, on how we have arrived to our present day understanding of them. Therefore Section 2 will cover the history of studies of WR stars, while Section 3 will highlight several topical problems and open questions about them.","pages":[1,2]},{"title":"2. FROM DISCOVERY TO UNDERSTANDING","content":"In 1867 year astronomers from Paris observatory Charles Wolf and Georges Rayet discovered three stars in Cygnus constellation those spectra significantly differ from other stars. In contrast with other stars, where usually absorption lines are dominant in spectra, discovered objects showed strong emission lines (Wolf & Rayet, 1867). The class of objects received its name in honor of the discoverers and gradually began to be replenished with new members. One hundred years after the discovery of WRs, by the end of 1960-s the number of known WRs in the Galaxy had reached over a hundred, totalling 127 (Roberts, 1962; Smith, 1968), while now is ∼ 700 (Rosslowe & Crowther, 2015). At the same time, in the late 1950-s middle 1960-s, work began to search for extragalactic WRs - WRs in the Large Magellanic Cloud (Westerlund & Rodgers, 1959; Westerlund & Smith, 1964). Significant contribution to the understanding of the physics of WR stars was given by Carlyle S. Beals (1929). Beals found that some lines in the spectra of WR stars have a P Cygni profile and based on comparison with P Cygni profile lines in spectra of novae he suggested that WRs are surrounded by expanding envelopes (Beals, 1929). Beals also found a difference between WR stars and novae: the P Cygni profiles in the WR stars do not change over time. This allowed to propose that an outflow from WR stars occurs continuously (Beals, 1929). This was confirmed by Chandrasekhar (1934), who developed a solid footing for interpreting P Cygni profiles as arising in expanding atmospheres. Kosirev (1934) used the diagnostics developed by Chandrasekhar to estimate the mass loss and maximum outflow velocity of a WR star and found, respectively, ∼ 10 -5 M ⊙ yr -1 and ∼ 1000 kms -1 . At the beginning of the 20th century, stellar spectroscopy developed in close cooperation with atomic physics. Investigating spectra of the star ζ Puppis Edward Pickering paid his attention to previously unknown lines 5411, 4541, 4200, 4100, 4026, 3924, 3858, 3813, 3782 Å (Pickering, 1897). He interpreted them as another series of hydrogen (besides Balmer) (Pickering, 1901). Line 4686 Å, first discovered during a solar eclipse and often found in WR spectra, was also considered to be a hydrogen line (Fowler, 1912). Although Fowler (1912) was able to obtain these spectral lines in the laboratory in 1912, during an experiment with a helium-filled tube, he considered their appearance to be a contribution from hydrogen impurity. It was not until 1913 that Niels Bohr's work (Bohr, 1913) explained the nature of these lines as ionized helium He II , confirming the atomic model (see Robotti (1983) for historical review). In 1890 Pickering drew attention to the similarity of the spectra of WR stars and planetary nebula (Pickering, 1891). Due to this, in papers devoted to emission spectra of nebulae it is possible to find descriptions of spectra of WR stars. The lines at 4647, 4650, 5696, 5801 and 5812 Å were identified with the carbon by Wright (1918). Ira S. Bowen, who found an interpretation of the 'nebulium' lines as [O III ], gave a detailed list of emission lines observed in nebulae, which includes an identification of almost all lines visible in the WRs (Bowen, 1928). In a 1934 paper Bowen (1934) suggests a significant role of the fluorescence for formation of the N III 4634, 46340 lines belonging to the WR bump. Already the spectra of the first discovered WR stars showed a difference: the broad emission bands are located in different places (Huggins & Huggins, 1890; Wolf & Rayet, 1867). After the identification of all the main spectral lines, Beals (1933) proposed the splitting of WR into two groups. This classification was approved in 1938, the International Astronomical Union (IAU) divided the spectra of WR stars into types WN and WC, depending on whether the spectrum was dominated by lines of nitrogen or carbon-oxygen respectively (Beals, 1933; Swings, 1942). WN stars are believed to show the hydrogen burning products via the CNO cycle, while WC stars reveal the helium burning products via the tripleα cycle. Based on the strength of the emission lines and line ratios, WN stars can be further classified into the spectral subtypes WN2 to WN11, and WC stars into the spectral subtypes WC4 to WC9 (Crowther et al., 1998; Smith, 1968; Smith et al., 1990, 1994, 1996). Also, there are transition types from WN to WC, which are called WN/C, whose spectra show strong emission lines of carbon and nitrogen simultaneously (Conti & Massey, 1989). In 1933 Victor A. Ambartsumian (1933) estimated He/H ratio for WR stars using intensities of H β and He II 4686 lines and found He/H ≳ 1 . 8 . Subsequent studies (Rublev (1972a) and references therein) confirmed excess of helium. Gamow (1943) suggested that the anomalous composition of WR stars was the result of nuclear processed material being visible on their surfaces. Despite these arguments in favor of far evolved status of WR stars, debates about it continued until the mid-1970s. There was alternative hypothesis, that WR are young and more massive than T Tau objects, before main sequence (Sahade, 1958; Underhill, 1968). It is important to mention the studies of Sergej V. Rublev, in which he developed methods for determining the temperatures and luminosities of WR stars and estimated the hydrogen abundances (Rublev, 1965, 1970, 1972b, 1975). These works played a significant role in the formation of the modern understanding of WR stars as far evolved massive objects.","pages":[2,3,4]},{"title":"2.1. Development of numerical methods","content":"Contemporary progress in our understanding of hot stars and especially WR stars has been significantly defined by the progress of numerical modelling of stellar atmospheres, in turn enabled by the development of both physical approximations and numerical methods. Victor V. Sobolev developed the theory of radiative transfer in moving media and applied it to the determination of physical conditions in the envelopes of WR, Be, novae, etc. type stars (Sobolev, 1947, 1960). He showed that due to the presence of a velocity gradient, photons of spectral lines are able to leave the deep layers of the atmosphere directly, avoiding the usual diffusion process; this significantly simplifies the study of the radiative equilibrium of moving atmospheres. This theory became the basis for the transition to modern quantitative calculations of the stellar wind. WRstars are essentially hot stellar cores surrounded by extended, low-density atmospheres. Due to their high luminosity the radiation in the stellar wind dominates over collisional processes, and the models of WR's atmospheres should be calculated taking into account the deviations from local thermodynamic equilibrium (non-LTE). Straightforward iterative procedure for the solution of radiative transfer equations Λ iteration - fails to converge in typical non-LTE conditions, as in scattering dominated media of large optical thickness, its convergence is extremely slow (see overview by Atanackovic-Vukmanovic (2004) and references therein). Several approaches have been proposed in early 1970-s as a way to numerically solve radiative transfer (RT) problems in non-LTE case. The first fully self-consistent solution of RT problems for non-LTE case was achieved by Auer & Mihalas (1969) who developed the complete linearization method based on the Newton-Raphson scheme to solve the set of discretized structural equations in a robust manner that opened a way for significant progress in the modelling of stellar atmospheres. A different approach is the 'core saturation method', introduced by Rybicki (1972) who implemented a modification of the Λ iteration method that made it practically usable for solving the RT and statistical equilibrium equations. In order to eliminate the cause of the slow convergence of simple Λ iteration (poor conditioning of the equations arising from a large number of scatterings), Rybicki proposed the elimination of scattering events in the line core as they do not contribute much to the transfer process. Another approach is 'perturbation technique' proposed by Cannon (1973a,b) who was the first to use the idea of 'operator splitting' in radiative transfer computations. He replaced the full (exact) Λ operator by a simplified one (so called approximate Lambda operator, or ALO) and computed a small error term made by this approximation by using a perturbation technique. Cannon's operator perturbation technique was an efficient way to combine computational advantages of approximate solutions with the accuracy of exact methods. In 1980-s computational approximations to accelerate the Λ iteration started to be used together with the operator perturbation technique introduced by Cannon (1973a,b) to simplify the direct solution, making the basis for the class of methods known as Accelerated Lambda Iteration. Wolf-Rainer Hamann (1985, 1986) adopted perturbation technique to approximate lambda operator, based on idea of 'core saturation' and presented test calculations for a typical WR star atmosphere. This effort became the foundation for the PoWR code (Hamann & Gräfener, 2003) that is still being actively developed and successfully applied for numerical modelling of various types of astrophysical objects. Independently, D. John Hillier (1990) presented a method based on tridiagonal Newton-Raphson operator, complete linearization scheme and perturbation technique for calculations of WN and WC atmosphere models. Over time, this method has grown into CMFGEN code (Hillier & Miller, 1998). Since the mid-1990s, CMFGEN and PoWR have become a reliable instruments for studying hot stars with high mass-loss rates. Systematic study of Galactic WR stars started with a series of works by Paul Crowther titled 'Fundamental parameters of WR stars' and devoted to the analysis of properties of nitrogen-rich WR stars (WN) based on optical and ultraviolet (UV) data and numerical modelling using CMFGEN code (Crowther et al., 1995a,b,c,d,e). Advances in infrared (IR) spectroscopy also enabled studies of WR stars near the Galactic center (Liermann et al., 2010; Martins et al., 2007). Modeling of large sample of Galctic WN (Hamann et al., 2006) and WC (Sander et al., 2012) stars ( ≈ 126 objects in total) allowed to obtain a homogeneous set of stellar and atmospheric parameters for them, to determine the correct location of WR stars in the Hertzsprung-Russell (HR) diagram. PoWR code was widely used for studying extragalactic objects. Among WR stars in nearby galaxies, the ones in Magellanic Clouds are most studied (Bestenlehner et al., 2014; Hainich et al., 2014, 2015). Sample of 74 WR stars belonged to M33 galaxy was analysed by Pritzkuleit (2020) (Pritzkuleit, 2020). Sander et al. (2014) investigated late-type WN stars in M31 galaxy (Sander et al., 2014). These studies extended our understanding of properties of WR stars in different galactic environments and metallicity-dependence of massive star winds.","pages":[4,5,6]},{"title":"2.2. Evolution","content":"Modern view on WRs as products of evolution of massive stars was finally formed only in 1980s. Although by the 1980s WR stars were already recognised as descendants of massive OB stars, their exact evolutionary status with respect to the main sequence and other evolved massive stars was still an open question. WR stars were being proposed as possible progenitors of supernovae, and particularly the newly-discovered type Ib supernovae that lack hydrogen but are apparently associated with young massive stars. The start of systematic X-ray observations with space-borne X-ray telescopes in 1960s significantly advanced the theory of close binary stars and their interac- tions, and it led Bohdan Paczyński (1967) to the suggestion that WR stars are the products of binary evolution - namely, mass exchange in a binary system. In his scenario, a more massive star in binary system evolves faster and, upon exhaustion of hydrogen in the core, fills the Roche lobe and a rapidly loses its hydrogen envelope that outflows towards the companion. As a result, when the envelope is fully transferred, its hot helium core remains as a WR star with little hydrogen in the outermost layers (Paczyński, 1973). The hypothesis of Paczyński was well accepted and it naturally explained the anomalous chemical composition of WR stars as naked stellar cores enriched with products of nuclear reactions. On the other hand, based on the similarity of Of and WN stars (Figure 1) and on the first estimates of the mass loss rate for hot massive stars based on UV observations Peter Conti (1975) suggested that the stellar wind of O stars is strong enough to remove the outer layers and uncover the stellar core, thus proposing a single star mass loss channel to form WR stars. In addition, brightest WR star, γ Vel, was inconsistent with Paczyński's hypothesis as, while being a binary, its orbit was clearly elliptical and its second component lacks any signs of previous mass exchange, both suggesting that there was no interaction between the components (Conti, 2015). In 1984 Peter Conti (1984) introduced a new class of objects. In his talk at the IAU symposium, Peter Conti grouped S Doradus variables, Hubble-Sandage variables, η Carinae, P Cygni, and other similar stars together under the term 'luminous blue variables' (LBV). LBV are characterized by strong mass loss (around 10 -5 M ⊙ yr -1 ) and occasional giant eruptive events (Humphreys & Davidson, 1994). Conti suggested that massive O stars transit to WR stars through this short-duration LBV phase. This idea is consolidated as the 'Conti scenario' and becomes the main channel for WR star formation. Modern stellar evolution calculations are consistent with the 'Conti scenario' (Groh et al., 2014; Sander et al., 2019). According to numerical calculations the stars with initial mass of 60 M ⊙ spend 2 × 10 5 yr in the LBV phase - only 5% of their lifetime (Groh et al., 2014). Recently observed transition from LBV to WN8 spectral type in Romano' star is also a direct confirmation of 'Conti scenario' (Maryeva et al., 2019; Polcaro et al., 2016). Similar scenario for WR formation was also proposed by Bisnovatyi-Kogan & Nadyozhin (1972), with significant mass loss on the red supergiant phase. They show that large inverse density gradient appears in the outer layers of a 30 M ⊙ star in this phase due to supercritical luminosity, resulting in powerful and rapid mass loss and formation of a stripped helium core. In the last decade, Paczyński's hypothesis has again been discussed as a mechanism for the formation of WR stars. Analysing the spatial positions of LBV and WR stars, Nathan Smith (Smith, 2016, 2019; Smith & Tombleson, 2015) con- luded that both LBVs and WRs could be the products of binary evolution, LBVs being mass-gainers, and WRs - mass-donors, or stripped components. This idea was critically discussed by Humphreys et al. (2016) and Davidson et al. (2016). Now we may confidently conclude only that we do not have enough data for statistical analysis. Modern evolutionary codes allow detailed calculations of binary system evolution (see for example Götberg et al. (2018); Laplace et al. (2021)), thus making it possible to compare theoretical predictions for stripped stars with properties of real WR stars (Götberg et al., 2018). Our current understanding is that binary interaction scenario cannot properly describe all observed WR stars, but most probably takes place in a part of them (see Shenar (2024) and references therein). Improving the statistics of binarity among WR stars is a crucial task. Among the first researchers who started to work on it was Virpi S. Niemelä, who determined the stellar masses in binary systems and discovered and analyzed the spectroscopic orbits of many binary systems with WR components (see for ex- mple her works Niemela (1995, 2001)). Nowadays this work is being continued, both through spectroscopy (Chené et al., 2022) and using high-resolution imaging (Deshmukh et al., 2024).","pages":[6,7,8,9]},{"title":"3.1. Search for new WR stars","content":"In our Galaxy there are 679 WR stars currently discovered 2) (Rosslowe & Crowther, 2015), while the theory based on current Milky Way star formation rate and duration of WR phase predicts the numbers of WR stars ∼ 1200 (Rosslowe & Crowther, 2015). It means that significant part of WR stars are still not discovered. WR stars in the Galaxy align with spiral arms and the regions of star formation. Thus, the discovery of new WR stars through optical observations has been significantly limited due to the presence of dust extinction. Although the probability of finding a WR star during optical spectroscopic surveys remains (Maíz Apellániz et al., 2016; Zhang et al., 2020), the majority of discoveries of new WR stars now happens through the observations in IR range. For example, Mauerhan et al. (2011) identified 60 Galactic WR stars: candidates were selected using the photometry from Spitzer and Two Micron All Sky Survey (2MASS) surveys and confirmed by near-IR spectroscopy. Shara et al. (2012) expanded the list by adding 71 more stars, also preliminary selected from J and K band 2MASS photometry. Moreover, the search for new WR stars is complicated by the effect of spectral mimicry - low mass [WR] objects masquerading as classical WR stars. Thanks to the results of Gaia mission (Gaia Collaboration et al., 2016) we now have reliable estimations of distances, and it helps to quickly understand the nature of individual objects and separate low-mass evolved stars from massive stars, and, moreover, to find the objects of unusual origin among low mass stars. For example, based on Gaia distance estimation Gvaramadze et al. (2019) demonstrated that IRAS00500+6713 is a product of merging of two white dwarfs, while the star has WO-type spectrum. Counterexample is PMR 5 star classified as [WN6] by Morgan et al. (2003) that was recently reclassified as a classical WN. Systematic search for WR stars in nearby galaxies is ongoing since 1970-s. In 1972 first catalog of WR candidate stars located in M33 galaxy was published (Wray & Corso, 1972). Twenty-five objects were selected based on photometry in three narrow-band filters (one filter is centered on He II 4686 - the strongest emission line in WN-type WRs, one is centered on C III -IV ≈ 4650 - the strongest line in WC-type WRs, and one is centered on continuum). This technique of selection of WR candidates for following spectroscopy has proven itself successfully and is still widely used (Massey & Conti, 1983; Massey & Johnson, 1998; Neugent & Massey, 2011; Neugent et al., 2012). As of today, 211 WR are known in M33 galaxy (Massey et al., 2016) and 173 are in M31 (Neugent & Massey, 2023). Search for new WR and their following investigations are important for understanding the impact of metallicity on mass-loss rate.","pages":[9,10]},{"title":"3.2. Wolf-Rayet stars in low metallicity environment","content":"As the opacity of stellar wind depends on the metallicity, mass-loss rates of massive stars decrease with Z . It means that in low metallicity environment single massive stars are unable to lose enough material through the stellar wind to become WR star. Thus, the mass loss on pre-WR stages is of fundamental importance for the final fate of massive stars, and may influence e.g. the formation of long-duration Gamma Ray Bursts (GRBs) or the yields from early stellar generations. Metallicity impacts overall evolutionary behaviour of massive stars in a complicated way. Georgy et al. (2013) calculated evolutionary tracks for stars in a wide range of initial masses at low metallicity. According to their calculations, metal-poor stars become colder after the end of hydrogen burning in the core and move along the track to the right side of the diagram. Unlike solar metallicity stars, metal-poor ones no longer return to the left side of the diagram (see Figure 2), even when the effects of stellar rotation are taken into account in the models. On the other hand, evolutionary tracks behave differently if the models implement Galactic Concordance abundances instead of solar, scaled-solar, or alpha-element enhanced abundances (Grasha et al., 2021) - the stars with the initial mass of M ∗ > 70M ⊙ with rotation V/V crit = 0 . 2 after the main sequence move to the right and then return to the left to the WR stars region, in contrast to predictions of Georgy et al. (2013). Yarovova et al. (2023) used evolutionary tracks from Georgy et al. (2013) and Grasha et al. (2021) for the study of emission-line stellar-like object in the nearby low-metallicity ( Z ∼ 0 . 1 Z ⊙ ) dwarf galaxy NGC 4068. They found the best agreement between the modelled and observed spectra for the model assuming ionization by low-metallicity WR star of M ∗ ≈ 80M ⊙ mass, ionizing the nebula through the strong wind and enriching the interstellar medium with nitrogen. Thus, parameters of the object favor Grasha et al. (2021) predictions. Therefore, understanding of mass loss rate requires complex approach that includes both studies of massive stars in different metallicites, and studies of present-day chemical abundances.","pages":[10,11]},{"title":"3.3. Wolf-Rayet stars and red supergiants","content":"Isolated massive stars with initial masses in 8 M ⊙ ≲ M ∗ ≲ 20 M ⊙ interval undergo the transition to Red Supergiants (RSG) after hydrogen in the core is exhausted. For them RSG is the final stage before Type-II supernova (SNII) explosion. More massive stars ( 40 M ⊙ ≲ M ∗ < 90 M ⊙ ) evolve through LBV phase before reaching WR phase, and finally explode as SN Type Ib/c. The evolution of stars with 25 M ⊙ ≲ M ∗ ≲ 35 M ⊙ initial masses is more complicated. Observed population of supernovae Type II progentiors in the local Universe shows that they are produced by the stars with initial masses M ∗ ≲ 18M ⊙ (Smartt, 2015), and it suggests that more massive stars should either directly collapse to black holes, or evolve leftwards in HR diagram, thus becoming Wolf-Rayet stars. It is supported by the luminosity distributions of cool supergiants in Large and Small Magellanic Clouds (Davies et al., 2018) that overlaps with those of apparently-single WR stars in both galaxies, thus suggesting a changing evolutionary sequence of massive stars with increasing initial mass. At the same time, between the RSG and WR regions in HR diagram lies the region of low luminosity LBVs and yellow hypergiants (YHGs) (Figure 3). Therefore, the evolution of 25 M ⊙ ≲ M ∗ ≲ 35 M ⊙ initial mass may look like that: It means that the star with initial mass about 30 M ⊙ should pass through all four stages - do such objects really exist? Groh et al. (2013) demonstrated that the stars with 20 -25 M ⊙ initial masses are unstable on low luminosity LBV stage, and explode as core-collapse supernovae (SN Type IIL/b). Prior to the explosion they show the spectrum similar to S Doradus during its minimum of brightness (WN11h spectral type). Maryeva et al. (2020) have recently found a Galactic object, Wray 15-906, with characteristics typical of a low luminosity LBV and the spectrum looking like WN11h. It was identified via the detection of its infrared circumstellar shell (of ≈ 2 pc in diameter) with the Wide-field Infrared Survey Explorer (WISE) and the Herschel Space Observatory. Maryeva et al. (2020) found that Wray 15-906 is a relatively low-luminosity, log( L ∗ / L ⊙ ) ≈ 5 . 4 , star with a temperature of 25 ± 2 kK, with position in HR diagram corresponding to post-red supergiant with initial mass of ≈ 25M ⊙ . Its spectrum and properties are consistent with theoretical predictions of Groh et al. (2013). Thus, the monitoring of this object, and the search for similar ones, may improve our understanding of the link between WRs and RSGs.","pages":[11,12,13]},{"title":"4. CONCLUSION","content":"Wolf-Rayet stars were discovered more than 150 years ago, and since then they were targets of numerous studies that combined astrophysical insights with advancements in atomic physics, development of numerical methods, and instrumental progress. Nowadays we understand quite well what Wolf-Rayet stars are, and they are no longer a hot topic in astrophysics. However they still are keystones for many directions of research, from theory of stellar atmospheres up to stellar populations in other galaxies and gravitational waves. Through the existence of the Wolf-Rayet phenomenon WR stars are linked, besides massive stars, with low mass evolved stars. We expect new progress in understanding the Wolf-Rayet stars and massloss of massive stars in general from the upcoming advances in the instrumentation. For example, SITELLE imaging Fourier transform spectrometer at CanadaFrance-Hawaii telescope (Drissen et al., 2019) may provide spatially resolved line ratios and kinematics of WR nebulae thus hinting us on ionizing flux from the stars, and their mass loss history. James Webb Space Telescope will allow studying dust shells around WR stars (Lau et al., 2022), and to assess their role in enrichment of interstellar medium with organic compounds and carbonaceous dust. Spectropolarimetric surveys of WR and monitoring of variability of linear polarization are also still topical. Acknowledgements I would like to thank the organizers of the conference for their kind invitation to present this overview. This research received funding from the European Union's Framework Programme for Research and Innovation Horizon 2020 (2014-2020) under the Marie Skłodowska-Curie Grant Agreement No. 823734 (POEMS project). The Astronomical Institute in Ondřejov is supported by the project RVO:67985815.","pages":[13]},{"title":"REFERENCES","content":"Ambartsumian V. A., 1933, Izvestiya Glavnoj Astronomicheskoj Observatorii v Pulkove, 13 Atanackovic-Vukmanovic O., 2004, Serbian Astronomical Journal, 169, 1 Auer L. H., Mihalas D., 1969, ApJ, 158, 641 Beals C. S., 1929, MNRAS, 90, 202 Beals C. S., 1933, The Observatory, 56, 196 Bestenlehner J. M., et al., 2014, A&A, 570, A38 Bisnovatyi-Kogan G. S., Nadyozhin D. K., 1972, Ap&SS, 15, 353 Bohr N., 1913, Nature, 92, 231 Bowen I. S., 1928, ApJ, 67, 1 Bowen I. S., 1934, PASP, 46, 146 Cannon C. J., 1973a, J. Quant. Spec. Radiat. Transf., 13, 627 Cannon C. J., 1973b, ApJ, 185, 621 Chandrasekhar S., 1934, MNRAS, 94, 522 Chené A.-N., Mahy L., Gosset E., St-Louis N., Dsilva K., Manick R., 2022, MNRAS, 516, 1022 Conti P. S., 1975, Memoires of the Societe Royale des Sciences de Liege, 9, 193 Conti P. S., 1984, in Maeder A., Renzini A., eds, IAU Symposium Vol. 105, Observational Tests of the Stellar Evolution Theory. p. 233 Conti P. S., 2015, in Hamann W.-R., Sander A., Todt H., eds, Wolf-Rayet Stars. pp 347-350 Conti P. S., Massey P., 1989, ApJ, 337, 251 Crowther P. A., 2007, ARA&A, 45, 177 Crowther P. A., Hillier D. J., Smith L. J., 1995a, A&A, 293, 172 Crowther P. A., Hillier D. J., Smith L. J., 1995b, A&A, 293, 403 Crowther P. A., Smith L. J., Hillier D. J., Schmutz W., 1995c, A&A, 293, 427 Crowther P. A., Smith L. J., Hillier D. J., 1995d, A&A, 302, 457 Crowther P. A., Smith L. J., Willis A. J., 1995e, A&A, 304, 269 Crowther P. A., De Marco O., Barlow M. J., 1998, MNRAS, 296, 367 Hamann W. R., Gräfener G., 2003, A&A, 410, 993 Hamann W. R., Gräfener G., Liermann A., 2006, A&A, 457, 1015 Hillier D. J., 1990, A&A, 231, 116 Hillier D. J., Miller D. L., 1998, ApJ, 496, 407 Huggins W., Huggins M., 1890, Proceedings of the Royal Society of London Series I, 49, 33 Humphreys R. M., Davidson K., 1994, PASP, 106, 1025 Humphreys R. M., Weis K., Davidson K., Gordon M. S., 2016, ApJ, 825, 64 Kosirev N. A., 1934, MNRAS, 94, 430 Laplace E., Justham S., Renzo M., Götberg Y., Farmer R., Vartanyan D., de Mink S. E., 2021, A&A, 656, A58 Lau R. M., et al., 2022, Nature Astronomy, 6, 1308 Liermann A., Hamann W. R., Oskinova L. M., Todt H., Butler K., 2010, A&A, 524, A82 Maíz Apellániz J., et al., 2016, ApJS, 224, 4 Marcolino W. L. F., Hillier D. J., de Araujo F. X., Pereira C. B., 2007, ApJ, 654, 1068 Martins F., Genzel R., Hillier D. J., Eisenhauer F., Paumard T., Gillessen S., Ott T., Trippe S., 2007, A&A, 468, 233 Martins F., Schaerer D., Marques-Chaves R., Upadhyaya A., 2023, A&A, 678, A159 Maryeva O., Viotti R. F., Koenigsberger G., Calabresi M., Rossi C., Gualandi R., 2019, Galaxies, 7, 79 Maryeva O. V., Gvaramadze V. V., Kniazev A. Y., Berdnikov L. N., 2020, MNRAS, 498, 5093 Maryeva O., Abdulkarimova A., Karpov S., Moiseev A., Oparin D., 2024, MNRAS, 527, 11925 Massey P., Conti P. S., 1983, ApJ, 273, 576 Massey P., Johnson O., 1998, ApJ, 505, 793 Massey P., Neugent K. F., Smart B. M., 2016, AJ, 152, 62 Mauerhan J. C., Van Dyk S. D., Morris P. W., 2011, AJ, 142, 40 Morgan D. H., Parker Q. A., Cohen M., 2003, MNRAS, 346, 719 Neugent K. F., Massey P., 2011, ApJ, 733, 123 Neugent K. F., Massey P., 2023, AJ, 166, 68 Neugent K. F., Massey P., Georgy C., 2012, ApJ, 759, 11 Niemela V. S., 1995, in van der Hucht K. A., Williams P. M., eds, IAU Symposium Vol. 163, Wolf-Rayet Stars: Binaries; Colliding Winds; Evolution. p. 223 Niemela V., 2001, in Revista Mexicana de Astronomia y Astrofisica Conference Series. pp 23-26 Paczyński B., 1967, Acta Astron., 17, 355 Paczyński B., 1973, in Bappu M. K. V., Sahade J., eds, IAU Symposium Vol. 49, Wolf-Rayet and High-Temperature Stars. p. 143 Pickering E. C., 1891, Astronomische Nachrichten, 127, 1 Pickering E. C., 1897, ApJ, 5, 92 Pickering E. C., 1901, Harvard College Observatory Circular, 55, 1 Polcaro V. F., et al., 2016, AJ, 151, 149 Pritzkuleit M., 2020, Master's thesis, University of Potsdam, Germany Roberts M. S., 1962, AJ, 67, 79 Robotti N., 1983, Historical Studies in the Physical Sciences, 14, 123 Rosslowe C. K., Crowther P. A., 2015, MNRAS, 447, 2322 Rublev S. V., 1965, Soviet Ast., 9, 274 Rublev S. V., 1970, Astrofizicheskie Issledovaniia Izvestiya Spetsial'noj As- trofizicheskoj Observatorii, 1, 25 Rublev S. V., 1972a, Astrofizicheskie Issledovaniia Izvestiya Spetsial'noj Astrofizicheskoj Observatorii, 4, 3 Underhill A. B., 1968, ARA&A, 6, 39 Vink J. S., 2015, in Hamann W.-R., Sander A., Todt H., eds, Wolf-Rayet Stars. pp 133-138 ( arXiv:1510.00227 ), doi:10.48550/arXiv.1510.00227 Westerlund B. E., Rodgers A. W., 1959, The Observatory, 79, 132 Westerlund B. E., Smith L. F., 1964, MNRAS, 128, 311 Wolf C. J. E., Rayet G., 1867, Academie des Sciences Paris Comptes Rendus, 65, 292 Wray J. D., Corso G. J., 1972, ApJ, 172, 577 Wright W. H., 1918, Publications of Lick Observatory, 13, 191 Yarovova A. D., Egorov O. V., Moiseev A. V., Maryeva O. V., 2023, MNRAS, 518, 2256 Zhang W., et al., 2020, ApJ, 902, 62 Škoda P., Podsztavek O., Tvrdík P., 2020, A&A, 643, A122 1981, van der Hucht K. A., Conti P. S., Lundstrom I., Stenholm B., Space Sci. Rev., 28, 227","pages":[13,14,16,17,19]}],"string":"[\n {\n \"title\": \"WOLF-RAYET STARS - WHAT WE KNOW AND WHAT WE DON'T\",\n \"content\": \"O. V. Maryeva a * a Astronomical Institute of the Czech Academy of Sciences, Fričova 298, 25165 Ondřejov, Czech Republic Today, we have a sufficiently complete picture of what the Wolf-Rayet (WR) stars are. Predictions of stellar evolution theory are in a good agreement with their parameters, estimated from observational data using stellar atmospheres codes; predictions of population synthesis also agree well with number of known WR stars. This article provides an overview of the main historical milestones in the studies of WR stars, showing how we came to this understanding, and what questions are still unanswered. Keywords: Stars: Wolf-Rayet - Stars: evolution - Stars: atmospheres - General: history and philosophy of astronomy\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1. INTRODUCTION\",\n \"content\": \"Wolf-Rayet (WR) stars are a class of objects identified on the basis of their spectral features. WR spectra show strong emission lines of helium, nitrogen, carbon and oxygen in different stages of ionization. The width of these lines reaches tens of angstroms, the central intensities are sometimes 10-20 times the intensity of the continuum spectrum. A prominent feature of WRs are two bumps at 4650 1) and 5808 Å, which are groups of closely located lines of N II , N III , C III and C IV , which are visible even in low-resolution spectra. The lines are formed in an extended atmosphere - in the stellar wind, expelling at velocities of 10 2 -10 3 kms -1 . WR stars are characterized by a high mass-loss rate of several 10 -5 M ⊙ yr -1 and by high temperatures ( T eff ≳ 30000 K). WR are the final stage in the evolution of massive stars (stars with an initial mass ≳ 25 M ⊙ ) before the core-collapse supernova explosion. With accumulation of observational data it became clear that emission spectra and WR bumps are intrinsic to objects of different natures. Therefore nowadays there is a clear distinction between WR stars and WR phenomenon , which occurs when fast-moving, hot plasma is expanding around a hot star (Gräfener et al., 2011; Vink, 2015). WR phenomenon may occur in the low mass stars after ejecting their outer layers in the planetary nebula phase and exposing their hot cores prior to the white-dwarf phase (Marcolino et al., 2007; Todt, 2009; van der Hucht et al., 1981). Central stars of planetary nebulae showing WR phenomenon in their spectra are denoted as [WR] (van der Hucht et al., 1981). Rare and still unique case of low mass object with WR phenomenon and the absence of a clearly detected circumstellar nebula is LAMOST J040901.83+323955.6 (Maryeva et al., 2024). It was selected as a WR star by Škoda et al. (2020), but Maryeva et al. (2024) showed that this object is 0 . 9 M ⊙ star caught in a rare transitional phase from post-AGB to CSPN. Another unique object with WR phenomenon is IRAS 00500+6713, formed as a result of a merger of two white dwarfs (Gvaramadze et al., 2019). The WR phenomenon is also observed in young supernovae (GalYam et al., 2014). As we know now, massive stars showing WR spectra are also not a homogeneous class. They are split into two groups: very massive stars and classical WR stars . Very massive stars ( M ini ≥ 100M ⊙ ) experience strong stellar winds and show WR spectra already during core H-burning phase, when they are located on main sequence. All of them show signs of hydrogen on their surface and are classified as WNh objects (Crowther et al., 2010; Martins et al., 2023; Smith et al., 1996). Classical WR stars (usually just named WR stars ) stars are hydrogen-depleted objects that have evolved off the main sequence and suffered intense mass loss (Crowther, 2007; Shenar, 2024). Classical WR stars will be the subject of this review article. The field of contemporary studies of WR stars is enormous and cannot be fully covered in a single small review like this one, so I'd like to recommend two other detailed reviews - Crowther (2007) and Shenar (2024) - for better understanding what WR stars are. On the other hand, here I will concentrate more on the historical aspects of WR studies, on how we have arrived to our present day understanding of them. Therefore Section 2 will cover the history of studies of WR stars, while Section 3 will highlight several topical problems and open questions about them.\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"2. FROM DISCOVERY TO UNDERSTANDING\",\n \"content\": \"In 1867 year astronomers from Paris observatory Charles Wolf and Georges Rayet discovered three stars in Cygnus constellation those spectra significantly differ from other stars. In contrast with other stars, where usually absorption lines are dominant in spectra, discovered objects showed strong emission lines (Wolf & Rayet, 1867). The class of objects received its name in honor of the discoverers and gradually began to be replenished with new members. One hundred years after the discovery of WRs, by the end of 1960-s the number of known WRs in the Galaxy had reached over a hundred, totalling 127 (Roberts, 1962; Smith, 1968), while now is ∼ 700 (Rosslowe & Crowther, 2015). At the same time, in the late 1950-s middle 1960-s, work began to search for extragalactic WRs - WRs in the Large Magellanic Cloud (Westerlund & Rodgers, 1959; Westerlund & Smith, 1964). Significant contribution to the understanding of the physics of WR stars was given by Carlyle S. Beals (1929). Beals found that some lines in the spectra of WR stars have a P Cygni profile and based on comparison with P Cygni profile lines in spectra of novae he suggested that WRs are surrounded by expanding envelopes (Beals, 1929). Beals also found a difference between WR stars and novae: the P Cygni profiles in the WR stars do not change over time. This allowed to propose that an outflow from WR stars occurs continuously (Beals, 1929). This was confirmed by Chandrasekhar (1934), who developed a solid footing for interpreting P Cygni profiles as arising in expanding atmospheres. Kosirev (1934) used the diagnostics developed by Chandrasekhar to estimate the mass loss and maximum outflow velocity of a WR star and found, respectively, ∼ 10 -5 M ⊙ yr -1 and ∼ 1000 kms -1 . At the beginning of the 20th century, stellar spectroscopy developed in close cooperation with atomic physics. Investigating spectra of the star ζ Puppis Edward Pickering paid his attention to previously unknown lines 5411, 4541, 4200, 4100, 4026, 3924, 3858, 3813, 3782 Å (Pickering, 1897). He interpreted them as another series of hydrogen (besides Balmer) (Pickering, 1901). Line 4686 Å, first discovered during a solar eclipse and often found in WR spectra, was also considered to be a hydrogen line (Fowler, 1912). Although Fowler (1912) was able to obtain these spectral lines in the laboratory in 1912, during an experiment with a helium-filled tube, he considered their appearance to be a contribution from hydrogen impurity. It was not until 1913 that Niels Bohr's work (Bohr, 1913) explained the nature of these lines as ionized helium He II , confirming the atomic model (see Robotti (1983) for historical review). In 1890 Pickering drew attention to the similarity of the spectra of WR stars and planetary nebula (Pickering, 1891). Due to this, in papers devoted to emission spectra of nebulae it is possible to find descriptions of spectra of WR stars. The lines at 4647, 4650, 5696, 5801 and 5812 Å were identified with the carbon by Wright (1918). Ira S. Bowen, who found an interpretation of the 'nebulium' lines as [O III ], gave a detailed list of emission lines observed in nebulae, which includes an identification of almost all lines visible in the WRs (Bowen, 1928). In a 1934 paper Bowen (1934) suggests a significant role of the fluorescence for formation of the N III 4634, 46340 lines belonging to the WR bump. Already the spectra of the first discovered WR stars showed a difference: the broad emission bands are located in different places (Huggins & Huggins, 1890; Wolf & Rayet, 1867). After the identification of all the main spectral lines, Beals (1933) proposed the splitting of WR into two groups. This classification was approved in 1938, the International Astronomical Union (IAU) divided the spectra of WR stars into types WN and WC, depending on whether the spectrum was dominated by lines of nitrogen or carbon-oxygen respectively (Beals, 1933; Swings, 1942). WN stars are believed to show the hydrogen burning products via the CNO cycle, while WC stars reveal the helium burning products via the tripleα cycle. Based on the strength of the emission lines and line ratios, WN stars can be further classified into the spectral subtypes WN2 to WN11, and WC stars into the spectral subtypes WC4 to WC9 (Crowther et al., 1998; Smith, 1968; Smith et al., 1990, 1994, 1996). Also, there are transition types from WN to WC, which are called WN/C, whose spectra show strong emission lines of carbon and nitrogen simultaneously (Conti & Massey, 1989). In 1933 Victor A. Ambartsumian (1933) estimated He/H ratio for WR stars using intensities of H β and He II 4686 lines and found He/H ≳ 1 . 8 . Subsequent studies (Rublev (1972a) and references therein) confirmed excess of helium. Gamow (1943) suggested that the anomalous composition of WR stars was the result of nuclear processed material being visible on their surfaces. Despite these arguments in favor of far evolved status of WR stars, debates about it continued until the mid-1970s. There was alternative hypothesis, that WR are young and more massive than T Tau objects, before main sequence (Sahade, 1958; Underhill, 1968). It is important to mention the studies of Sergej V. Rublev, in which he developed methods for determining the temperatures and luminosities of WR stars and estimated the hydrogen abundances (Rublev, 1965, 1970, 1972b, 1975). These works played a significant role in the formation of the modern understanding of WR stars as far evolved massive objects.\",\n \"pages\": [\n 2,\n 3,\n 4\n ]\n },\n {\n \"title\": \"2.1. Development of numerical methods\",\n \"content\": \"Contemporary progress in our understanding of hot stars and especially WR stars has been significantly defined by the progress of numerical modelling of stellar atmospheres, in turn enabled by the development of both physical approximations and numerical methods. Victor V. Sobolev developed the theory of radiative transfer in moving media and applied it to the determination of physical conditions in the envelopes of WR, Be, novae, etc. type stars (Sobolev, 1947, 1960). He showed that due to the presence of a velocity gradient, photons of spectral lines are able to leave the deep layers of the atmosphere directly, avoiding the usual diffusion process; this significantly simplifies the study of the radiative equilibrium of moving atmospheres. This theory became the basis for the transition to modern quantitative calculations of the stellar wind. WRstars are essentially hot stellar cores surrounded by extended, low-density atmospheres. Due to their high luminosity the radiation in the stellar wind dominates over collisional processes, and the models of WR's atmospheres should be calculated taking into account the deviations from local thermodynamic equilibrium (non-LTE). Straightforward iterative procedure for the solution of radiative transfer equations Λ iteration - fails to converge in typical non-LTE conditions, as in scattering dominated media of large optical thickness, its convergence is extremely slow (see overview by Atanackovic-Vukmanovic (2004) and references therein). Several approaches have been proposed in early 1970-s as a way to numerically solve radiative transfer (RT) problems in non-LTE case. The first fully self-consistent solution of RT problems for non-LTE case was achieved by Auer & Mihalas (1969) who developed the complete linearization method based on the Newton-Raphson scheme to solve the set of discretized structural equations in a robust manner that opened a way for significant progress in the modelling of stellar atmospheres. A different approach is the 'core saturation method', introduced by Rybicki (1972) who implemented a modification of the Λ iteration method that made it practically usable for solving the RT and statistical equilibrium equations. In order to eliminate the cause of the slow convergence of simple Λ iteration (poor conditioning of the equations arising from a large number of scatterings), Rybicki proposed the elimination of scattering events in the line core as they do not contribute much to the transfer process. Another approach is 'perturbation technique' proposed by Cannon (1973a,b) who was the first to use the idea of 'operator splitting' in radiative transfer computations. He replaced the full (exact) Λ operator by a simplified one (so called approximate Lambda operator, or ALO) and computed a small error term made by this approximation by using a perturbation technique. Cannon's operator perturbation technique was an efficient way to combine computational advantages of approximate solutions with the accuracy of exact methods. In 1980-s computational approximations to accelerate the Λ iteration started to be used together with the operator perturbation technique introduced by Cannon (1973a,b) to simplify the direct solution, making the basis for the class of methods known as Accelerated Lambda Iteration. Wolf-Rainer Hamann (1985, 1986) adopted perturbation technique to approximate lambda operator, based on idea of 'core saturation' and presented test calculations for a typical WR star atmosphere. This effort became the foundation for the PoWR code (Hamann & Gräfener, 2003) that is still being actively developed and successfully applied for numerical modelling of various types of astrophysical objects. Independently, D. John Hillier (1990) presented a method based on tridiagonal Newton-Raphson operator, complete linearization scheme and perturbation technique for calculations of WN and WC atmosphere models. Over time, this method has grown into CMFGEN code (Hillier & Miller, 1998). Since the mid-1990s, CMFGEN and PoWR have become a reliable instruments for studying hot stars with high mass-loss rates. Systematic study of Galactic WR stars started with a series of works by Paul Crowther titled 'Fundamental parameters of WR stars' and devoted to the analysis of properties of nitrogen-rich WR stars (WN) based on optical and ultraviolet (UV) data and numerical modelling using CMFGEN code (Crowther et al., 1995a,b,c,d,e). Advances in infrared (IR) spectroscopy also enabled studies of WR stars near the Galactic center (Liermann et al., 2010; Martins et al., 2007). Modeling of large sample of Galctic WN (Hamann et al., 2006) and WC (Sander et al., 2012) stars ( ≈ 126 objects in total) allowed to obtain a homogeneous set of stellar and atmospheric parameters for them, to determine the correct location of WR stars in the Hertzsprung-Russell (HR) diagram. PoWR code was widely used for studying extragalactic objects. Among WR stars in nearby galaxies, the ones in Magellanic Clouds are most studied (Bestenlehner et al., 2014; Hainich et al., 2014, 2015). Sample of 74 WR stars belonged to M33 galaxy was analysed by Pritzkuleit (2020) (Pritzkuleit, 2020). Sander et al. (2014) investigated late-type WN stars in M31 galaxy (Sander et al., 2014). These studies extended our understanding of properties of WR stars in different galactic environments and metallicity-dependence of massive star winds.\",\n \"pages\": [\n 4,\n 5,\n 6\n ]\n },\n {\n \"title\": \"2.2. Evolution\",\n \"content\": \"Modern view on WRs as products of evolution of massive stars was finally formed only in 1980s. Although by the 1980s WR stars were already recognised as descendants of massive OB stars, their exact evolutionary status with respect to the main sequence and other evolved massive stars was still an open question. WR stars were being proposed as possible progenitors of supernovae, and particularly the newly-discovered type Ib supernovae that lack hydrogen but are apparently associated with young massive stars. The start of systematic X-ray observations with space-borne X-ray telescopes in 1960s significantly advanced the theory of close binary stars and their interac- tions, and it led Bohdan Paczyński (1967) to the suggestion that WR stars are the products of binary evolution - namely, mass exchange in a binary system. In his scenario, a more massive star in binary system evolves faster and, upon exhaustion of hydrogen in the core, fills the Roche lobe and a rapidly loses its hydrogen envelope that outflows towards the companion. As a result, when the envelope is fully transferred, its hot helium core remains as a WR star with little hydrogen in the outermost layers (Paczyński, 1973). The hypothesis of Paczyński was well accepted and it naturally explained the anomalous chemical composition of WR stars as naked stellar cores enriched with products of nuclear reactions. On the other hand, based on the similarity of Of and WN stars (Figure 1) and on the first estimates of the mass loss rate for hot massive stars based on UV observations Peter Conti (1975) suggested that the stellar wind of O stars is strong enough to remove the outer layers and uncover the stellar core, thus proposing a single star mass loss channel to form WR stars. In addition, brightest WR star, γ Vel, was inconsistent with Paczyński's hypothesis as, while being a binary, its orbit was clearly elliptical and its second component lacks any signs of previous mass exchange, both suggesting that there was no interaction between the components (Conti, 2015). In 1984 Peter Conti (1984) introduced a new class of objects. In his talk at the IAU symposium, Peter Conti grouped S Doradus variables, Hubble-Sandage variables, η Carinae, P Cygni, and other similar stars together under the term 'luminous blue variables' (LBV). LBV are characterized by strong mass loss (around 10 -5 M ⊙ yr -1 ) and occasional giant eruptive events (Humphreys & Davidson, 1994). Conti suggested that massive O stars transit to WR stars through this short-duration LBV phase. This idea is consolidated as the 'Conti scenario' and becomes the main channel for WR star formation. Modern stellar evolution calculations are consistent with the 'Conti scenario' (Groh et al., 2014; Sander et al., 2019). According to numerical calculations the stars with initial mass of 60 M ⊙ spend 2 × 10 5 yr in the LBV phase - only 5% of their lifetime (Groh et al., 2014). Recently observed transition from LBV to WN8 spectral type in Romano' star is also a direct confirmation of 'Conti scenario' (Maryeva et al., 2019; Polcaro et al., 2016). Similar scenario for WR formation was also proposed by Bisnovatyi-Kogan & Nadyozhin (1972), with significant mass loss on the red supergiant phase. They show that large inverse density gradient appears in the outer layers of a 30 M ⊙ star in this phase due to supercritical luminosity, resulting in powerful and rapid mass loss and formation of a stripped helium core. In the last decade, Paczyński's hypothesis has again been discussed as a mechanism for the formation of WR stars. Analysing the spatial positions of LBV and WR stars, Nathan Smith (Smith, 2016, 2019; Smith & Tombleson, 2015) con- luded that both LBVs and WRs could be the products of binary evolution, LBVs being mass-gainers, and WRs - mass-donors, or stripped components. This idea was critically discussed by Humphreys et al. (2016) and Davidson et al. (2016). Now we may confidently conclude only that we do not have enough data for statistical analysis. Modern evolutionary codes allow detailed calculations of binary system evolution (see for example Götberg et al. (2018); Laplace et al. (2021)), thus making it possible to compare theoretical predictions for stripped stars with properties of real WR stars (Götberg et al., 2018). Our current understanding is that binary interaction scenario cannot properly describe all observed WR stars, but most probably takes place in a part of them (see Shenar (2024) and references therein). Improving the statistics of binarity among WR stars is a crucial task. Among the first researchers who started to work on it was Virpi S. Niemelä, who determined the stellar masses in binary systems and discovered and analyzed the spectroscopic orbits of many binary systems with WR components (see for ex- mple her works Niemela (1995, 2001)). Nowadays this work is being continued, both through spectroscopy (Chené et al., 2022) and using high-resolution imaging (Deshmukh et al., 2024).\",\n \"pages\": [\n 6,\n 7,\n 8,\n 9\n ]\n },\n {\n \"title\": \"3.1. Search for new WR stars\",\n \"content\": \"In our Galaxy there are 679 WR stars currently discovered 2) (Rosslowe & Crowther, 2015), while the theory based on current Milky Way star formation rate and duration of WR phase predicts the numbers of WR stars ∼ 1200 (Rosslowe & Crowther, 2015). It means that significant part of WR stars are still not discovered. WR stars in the Galaxy align with spiral arms and the regions of star formation. Thus, the discovery of new WR stars through optical observations has been significantly limited due to the presence of dust extinction. Although the probability of finding a WR star during optical spectroscopic surveys remains (Maíz Apellániz et al., 2016; Zhang et al., 2020), the majority of discoveries of new WR stars now happens through the observations in IR range. For example, Mauerhan et al. (2011) identified 60 Galactic WR stars: candidates were selected using the photometry from Spitzer and Two Micron All Sky Survey (2MASS) surveys and confirmed by near-IR spectroscopy. Shara et al. (2012) expanded the list by adding 71 more stars, also preliminary selected from J and K band 2MASS photometry. Moreover, the search for new WR stars is complicated by the effect of spectral mimicry - low mass [WR] objects masquerading as classical WR stars. Thanks to the results of Gaia mission (Gaia Collaboration et al., 2016) we now have reliable estimations of distances, and it helps to quickly understand the nature of individual objects and separate low-mass evolved stars from massive stars, and, moreover, to find the objects of unusual origin among low mass stars. For example, based on Gaia distance estimation Gvaramadze et al. (2019) demonstrated that IRAS00500+6713 is a product of merging of two white dwarfs, while the star has WO-type spectrum. Counterexample is PMR 5 star classified as [WN6] by Morgan et al. (2003) that was recently reclassified as a classical WN. Systematic search for WR stars in nearby galaxies is ongoing since 1970-s. In 1972 first catalog of WR candidate stars located in M33 galaxy was published (Wray & Corso, 1972). Twenty-five objects were selected based on photometry in three narrow-band filters (one filter is centered on He II 4686 - the strongest emission line in WN-type WRs, one is centered on C III -IV ≈ 4650 - the strongest line in WC-type WRs, and one is centered on continuum). This technique of selection of WR candidates for following spectroscopy has proven itself successfully and is still widely used (Massey & Conti, 1983; Massey & Johnson, 1998; Neugent & Massey, 2011; Neugent et al., 2012). As of today, 211 WR are known in M33 galaxy (Massey et al., 2016) and 173 are in M31 (Neugent & Massey, 2023). Search for new WR and their following investigations are important for understanding the impact of metallicity on mass-loss rate.\",\n \"pages\": [\n 9,\n 10\n ]\n },\n {\n \"title\": \"3.2. Wolf-Rayet stars in low metallicity environment\",\n \"content\": \"As the opacity of stellar wind depends on the metallicity, mass-loss rates of massive stars decrease with Z . It means that in low metallicity environment single massive stars are unable to lose enough material through the stellar wind to become WR star. Thus, the mass loss on pre-WR stages is of fundamental importance for the final fate of massive stars, and may influence e.g. the formation of long-duration Gamma Ray Bursts (GRBs) or the yields from early stellar generations. Metallicity impacts overall evolutionary behaviour of massive stars in a complicated way. Georgy et al. (2013) calculated evolutionary tracks for stars in a wide range of initial masses at low metallicity. According to their calculations, metal-poor stars become colder after the end of hydrogen burning in the core and move along the track to the right side of the diagram. Unlike solar metallicity stars, metal-poor ones no longer return to the left side of the diagram (see Figure 2), even when the effects of stellar rotation are taken into account in the models. On the other hand, evolutionary tracks behave differently if the models implement Galactic Concordance abundances instead of solar, scaled-solar, or alpha-element enhanced abundances (Grasha et al., 2021) - the stars with the initial mass of M ∗ > 70M ⊙ with rotation V/V crit = 0 . 2 after the main sequence move to the right and then return to the left to the WR stars region, in contrast to predictions of Georgy et al. (2013). Yarovova et al. (2023) used evolutionary tracks from Georgy et al. (2013) and Grasha et al. (2021) for the study of emission-line stellar-like object in the nearby low-metallicity ( Z ∼ 0 . 1 Z ⊙ ) dwarf galaxy NGC 4068. They found the best agreement between the modelled and observed spectra for the model assuming ionization by low-metallicity WR star of M ∗ ≈ 80M ⊙ mass, ionizing the nebula through the strong wind and enriching the interstellar medium with nitrogen. Thus, parameters of the object favor Grasha et al. (2021) predictions. Therefore, understanding of mass loss rate requires complex approach that includes both studies of massive stars in different metallicites, and studies of present-day chemical abundances.\",\n \"pages\": [\n 10,\n 11\n ]\n },\n {\n \"title\": \"3.3. Wolf-Rayet stars and red supergiants\",\n \"content\": \"Isolated massive stars with initial masses in 8 M ⊙ ≲ M ∗ ≲ 20 M ⊙ interval undergo the transition to Red Supergiants (RSG) after hydrogen in the core is exhausted. For them RSG is the final stage before Type-II supernova (SNII) explosion. More massive stars ( 40 M ⊙ ≲ M ∗ < 90 M ⊙ ) evolve through LBV phase before reaching WR phase, and finally explode as SN Type Ib/c. The evolution of stars with 25 M ⊙ ≲ M ∗ ≲ 35 M ⊙ initial masses is more complicated. Observed population of supernovae Type II progentiors in the local Universe shows that they are produced by the stars with initial masses M ∗ ≲ 18M ⊙ (Smartt, 2015), and it suggests that more massive stars should either directly collapse to black holes, or evolve leftwards in HR diagram, thus becoming Wolf-Rayet stars. It is supported by the luminosity distributions of cool supergiants in Large and Small Magellanic Clouds (Davies et al., 2018) that overlaps with those of apparently-single WR stars in both galaxies, thus suggesting a changing evolutionary sequence of massive stars with increasing initial mass. At the same time, between the RSG and WR regions in HR diagram lies the region of low luminosity LBVs and yellow hypergiants (YHGs) (Figure 3). Therefore, the evolution of 25 M ⊙ ≲ M ∗ ≲ 35 M ⊙ initial mass may look like that: It means that the star with initial mass about 30 M ⊙ should pass through all four stages - do such objects really exist? Groh et al. (2013) demonstrated that the stars with 20 -25 M ⊙ initial masses are unstable on low luminosity LBV stage, and explode as core-collapse supernovae (SN Type IIL/b). Prior to the explosion they show the spectrum similar to S Doradus during its minimum of brightness (WN11h spectral type). Maryeva et al. (2020) have recently found a Galactic object, Wray 15-906, with characteristics typical of a low luminosity LBV and the spectrum looking like WN11h. It was identified via the detection of its infrared circumstellar shell (of ≈ 2 pc in diameter) with the Wide-field Infrared Survey Explorer (WISE) and the Herschel Space Observatory. Maryeva et al. (2020) found that Wray 15-906 is a relatively low-luminosity, log( L ∗ / L ⊙ ) ≈ 5 . 4 , star with a temperature of 25 ± 2 kK, with position in HR diagram corresponding to post-red supergiant with initial mass of ≈ 25M ⊙ . Its spectrum and properties are consistent with theoretical predictions of Groh et al. (2013). Thus, the monitoring of this object, and the search for similar ones, may improve our understanding of the link between WRs and RSGs.\",\n \"pages\": [\n 11,\n 12,\n 13\n ]\n },\n {\n \"title\": \"4. CONCLUSION\",\n \"content\": \"Wolf-Rayet stars were discovered more than 150 years ago, and since then they were targets of numerous studies that combined astrophysical insights with advancements in atomic physics, development of numerical methods, and instrumental progress. Nowadays we understand quite well what Wolf-Rayet stars are, and they are no longer a hot topic in astrophysics. However they still are keystones for many directions of research, from theory of stellar atmospheres up to stellar populations in other galaxies and gravitational waves. Through the existence of the Wolf-Rayet phenomenon WR stars are linked, besides massive stars, with low mass evolved stars. We expect new progress in understanding the Wolf-Rayet stars and massloss of massive stars in general from the upcoming advances in the instrumentation. For example, SITELLE imaging Fourier transform spectrometer at CanadaFrance-Hawaii telescope (Drissen et al., 2019) may provide spatially resolved line ratios and kinematics of WR nebulae thus hinting us on ionizing flux from the stars, and their mass loss history. James Webb Space Telescope will allow studying dust shells around WR stars (Lau et al., 2022), and to assess their role in enrichment of interstellar medium with organic compounds and carbonaceous dust. Spectropolarimetric surveys of WR and monitoring of variability of linear polarization are also still topical. Acknowledgements I would like to thank the organizers of the conference for their kind invitation to present this overview. This research received funding from the European Union's Framework Programme for Research and Innovation Horizon 2020 (2014-2020) under the Marie Skłodowska-Curie Grant Agreement No. 823734 (POEMS project). The Astronomical Institute in Ondřejov is supported by the project RVO:67985815.\",\n \"pages\": [\n 13\n ]\n },\n {\n \"title\": \"REFERENCES\",\n \"content\": \"Ambartsumian V. A., 1933, Izvestiya Glavnoj Astronomicheskoj Observatorii v Pulkove, 13 Atanackovic-Vukmanovic O., 2004, Serbian Astronomical Journal, 169, 1 Auer L. H., Mihalas D., 1969, ApJ, 158, 641 Beals C. S., 1929, MNRAS, 90, 202 Beals C. S., 1933, The Observatory, 56, 196 Bestenlehner J. M., et al., 2014, A&A, 570, A38 Bisnovatyi-Kogan G. S., Nadyozhin D. K., 1972, Ap&SS, 15, 353 Bohr N., 1913, Nature, 92, 231 Bowen I. S., 1928, ApJ, 67, 1 Bowen I. S., 1934, PASP, 46, 146 Cannon C. J., 1973a, J. Quant. Spec. Radiat. Transf., 13, 627 Cannon C. J., 1973b, ApJ, 185, 621 Chandrasekhar S., 1934, MNRAS, 94, 522 Chené A.-N., Mahy L., Gosset E., St-Louis N., Dsilva K., Manick R., 2022, MNRAS, 516, 1022 Conti P. S., 1975, Memoires of the Societe Royale des Sciences de Liege, 9, 193 Conti P. S., 1984, in Maeder A., Renzini A., eds, IAU Symposium Vol. 105, Observational Tests of the Stellar Evolution Theory. p. 233 Conti P. S., 2015, in Hamann W.-R., Sander A., Todt H., eds, Wolf-Rayet Stars. pp 347-350 Conti P. S., Massey P., 1989, ApJ, 337, 251 Crowther P. A., 2007, ARA&A, 45, 177 Crowther P. A., Hillier D. J., Smith L. J., 1995a, A&A, 293, 172 Crowther P. A., Hillier D. J., Smith L. J., 1995b, A&A, 293, 403 Crowther P. A., Smith L. J., Hillier D. J., Schmutz W., 1995c, A&A, 293, 427 Crowther P. A., Smith L. J., Hillier D. J., 1995d, A&A, 302, 457 Crowther P. A., Smith L. J., Willis A. J., 1995e, A&A, 304, 269 Crowther P. A., De Marco O., Barlow M. J., 1998, MNRAS, 296, 367 Hamann W. R., Gräfener G., 2003, A&A, 410, 993 Hamann W. R., Gräfener G., Liermann A., 2006, A&A, 457, 1015 Hillier D. J., 1990, A&A, 231, 116 Hillier D. J., Miller D. L., 1998, ApJ, 496, 407 Huggins W., Huggins M., 1890, Proceedings of the Royal Society of London Series I, 49, 33 Humphreys R. M., Davidson K., 1994, PASP, 106, 1025 Humphreys R. M., Weis K., Davidson K., Gordon M. S., 2016, ApJ, 825, 64 Kosirev N. A., 1934, MNRAS, 94, 430 Laplace E., Justham S., Renzo M., Götberg Y., Farmer R., Vartanyan D., de Mink S. E., 2021, A&A, 656, A58 Lau R. M., et al., 2022, Nature Astronomy, 6, 1308 Liermann A., Hamann W. R., Oskinova L. M., Todt H., Butler K., 2010, A&A, 524, A82 Maíz Apellániz J., et al., 2016, ApJS, 224, 4 Marcolino W. L. F., Hillier D. J., de Araujo F. X., Pereira C. B., 2007, ApJ, 654, 1068 Martins F., Genzel R., Hillier D. J., Eisenhauer F., Paumard T., Gillessen S., Ott T., Trippe S., 2007, A&A, 468, 233 Martins F., Schaerer D., Marques-Chaves R., Upadhyaya A., 2023, A&A, 678, A159 Maryeva O., Viotti R. F., Koenigsberger G., Calabresi M., Rossi C., Gualandi R., 2019, Galaxies, 7, 79 Maryeva O. V., Gvaramadze V. V., Kniazev A. Y., Berdnikov L. N., 2020, MNRAS, 498, 5093 Maryeva O., Abdulkarimova A., Karpov S., Moiseev A., Oparin D., 2024, MNRAS, 527, 11925 Massey P., Conti P. S., 1983, ApJ, 273, 576 Massey P., Johnson O., 1998, ApJ, 505, 793 Massey P., Neugent K. F., Smart B. M., 2016, AJ, 152, 62 Mauerhan J. C., Van Dyk S. D., Morris P. W., 2011, AJ, 142, 40 Morgan D. H., Parker Q. A., Cohen M., 2003, MNRAS, 346, 719 Neugent K. F., Massey P., 2011, ApJ, 733, 123 Neugent K. F., Massey P., 2023, AJ, 166, 68 Neugent K. F., Massey P., Georgy C., 2012, ApJ, 759, 11 Niemela V. S., 1995, in van der Hucht K. A., Williams P. M., eds, IAU Symposium Vol. 163, Wolf-Rayet Stars: Binaries; Colliding Winds; Evolution. p. 223 Niemela V., 2001, in Revista Mexicana de Astronomia y Astrofisica Conference Series. pp 23-26 Paczyński B., 1967, Acta Astron., 17, 355 Paczyński B., 1973, in Bappu M. K. V., Sahade J., eds, IAU Symposium Vol. 49, Wolf-Rayet and High-Temperature Stars. p. 143 Pickering E. C., 1891, Astronomische Nachrichten, 127, 1 Pickering E. C., 1897, ApJ, 5, 92 Pickering E. C., 1901, Harvard College Observatory Circular, 55, 1 Polcaro V. F., et al., 2016, AJ, 151, 149 Pritzkuleit M., 2020, Master's thesis, University of Potsdam, Germany Roberts M. S., 1962, AJ, 67, 79 Robotti N., 1983, Historical Studies in the Physical Sciences, 14, 123 Rosslowe C. K., Crowther P. A., 2015, MNRAS, 447, 2322 Rublev S. V., 1965, Soviet Ast., 9, 274 Rublev S. V., 1970, Astrofizicheskie Issledovaniia Izvestiya Spetsial'noj As- trofizicheskoj Observatorii, 1, 25 Rublev S. V., 1972a, Astrofizicheskie Issledovaniia Izvestiya Spetsial'noj Astrofizicheskoj Observatorii, 4, 3 Underhill A. B., 1968, ARA&A, 6, 39 Vink J. S., 2015, in Hamann W.-R., Sander A., Todt H., eds, Wolf-Rayet Stars. pp 133-138 ( arXiv:1510.00227 ), doi:10.48550/arXiv.1510.00227 Westerlund B. E., Rodgers A. W., 1959, The Observatory, 79, 132 Westerlund B. E., Smith L. F., 1964, MNRAS, 128, 311 Wolf C. J. E., Rayet G., 1867, Academie des Sciences Paris Comptes Rendus, 65, 292 Wray J. D., Corso G. J., 1972, ApJ, 172, 577 Wright W. H., 1918, Publications of Lick Observatory, 13, 191 Yarovova A. D., Egorov O. V., Moiseev A. V., Maryeva O. V., 2023, MNRAS, 518, 2256 Zhang W., et al., 2020, ApJ, 902, 62 Škoda P., Podsztavek O., Tvrdík P., 2020, A&A, 643, A122 1981, van der Hucht K. A., Conti P. S., Lundstrom I., Stenholm B., Space Sci. Rev., 28, 227\",\n \"pages\": [\n 13,\n 14,\n 16,\n 17,\n 19\n ]\n }\n]"}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":49,"numItemsPerPage":100,"numTotalItems":4986,"offset":4900,"length":100}},"jwt":"eyJhbGciOiJFZERTQSJ9.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTczOTY5NTM2OCwic3ViIjoiL2RhdGFzZXRzL2FzaGlzaGtncGlhbi9hZHNwYXBlcnNfNzBfNzUiLCJleHAiOjE3Mzk2OTg5NjgsImlzcyI6Imh0dHBzOi8vaHVnZ2luZ2ZhY2UuY28ifQ.Tw5loLDlFgxvsV5xeohvRs2KkRNvs66_532VvZM5PBZWY0TU7HWyvGWRHEzs7L27QbkdThkVF1SeBYFLQK-XCQ","displayUrls":true},"discussionsStats":{"closed":0,"open":1,"total":1},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false}">
<document>
<section_header_level_1><location><page_1><loc_15><loc_89><loc_84><loc_91></location>Echoes from charged black holes influenced by quintessence</section_header_level_1>
<text><location><page_1><loc_35><loc_85><loc_65><loc_87></location>Siyuan Hui a,b ∗ and Benrong Mu a,b †</text>
<text><location><page_1><loc_15><loc_74><loc_85><loc_84></location>a Center for Joint Quantum Studies, College of Medical Technology, Chengdu University of Traditional Chinese Medicine, Chengdu, 611137, PR China b Center for Theoretical Physics, College of Physics, Sichuan University, Chengdu, 610064, PR China</text>
<section_header_level_1><location><page_1><loc_45><loc_71><loc_54><loc_72></location>Abstract</section_header_level_1>
<text><location><page_1><loc_12><loc_52><loc_88><loc_69></location>In this paper, we investigate the effective potential and echoes from the dyonic black hole with quintessence. For a dyonic black hole, the quasi-topological electromagnetism provides proper matter energy-momentum tensor to curve the spacetime, and quintessence strengthens this force. We find that when the effect of quintessence becomes stronger, the black hole potential transforms between single-peak and double-peak, which will influence the existence of black hole echoes. In particular, we find that observer will receive a sudden vanishment of high-frequency echoes when quintessence remains a relatively strong effect.</text>
<section_header_level_1><location><page_2><loc_14><loc_89><loc_22><loc_91></location>Contents</section_header_level_1>
<table>
<location><page_2><loc_12><loc_44><loc_89><loc_86></location>
</table>
<section_header_level_1><location><page_2><loc_12><loc_39><loc_32><loc_40></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_2><loc_12><loc_11><loc_88><loc_36></location>LIGO and Virgo first successfully detected gravitational waves from a binary black hole merger[1]. Later, the Event Horizon Telescope photographed the first image of a supermassive black hole at the center of galaxy M87[2-7]. This discovery launches an extraordinary new area in black hole physics. The ringdown waveforms of gravitational waves from two massive objects merging include important information, which is worth studying. The ringdown waveforms are characterized by quasinormal modes (QNMs) of final black holes[8-10]. Specially, the ringdown waveforms will no longer depend on the initial disturbance after the initial wave burst of the perturbation. Therefore, the ringdown waveform measurement provide a great chance to reveal properties of the black hole geometry characterized by an event horizon[11, 12].</text>
<text><location><page_2><loc_14><loc_8><loc_88><loc_9></location>However, QNMs and ringdown signals are in fact related to the photon rings of the massive</text>
<text><location><page_3><loc_12><loc_76><loc_88><loc_91></location>objects, which may be not necessarily black holes[10]. Thus, it is hard to distinguish black holes from many other horizonless massive objects just via QNMs and ringdown signals[1315]. In fact, many horizonless exotic compact objects (ECOs) can behave like black holes, such as boson stars, gravastars, wormholes and firewalls[16-28]. Specially, the late-time ringdown signals together with echoes from the binary black hole merger can be similarly constructed from various ECOs' theoretical models[29-36].</text>
<text><location><page_3><loc_12><loc_37><loc_88><loc_75></location>Meanwhile, data analysis suggests that the recent LIGO/Virgo observation may offer potential evidence for echoes in waveforms of gravitational wave from the binary black hole merger[37, 38]. To gain a deeper insight of echoes, an extra reflecting boundary was laid in a black hole spacetime. It showed that this boundary plays a core role in producing extra multiple time-delay ringdown signals, overlaying to form a series of complete echo waveform which can be received by a distant observer[14]. Several black hole and wormhole models have been studied and confirmed to have echoes, such as Kerr-like wormholes, hairy black holes, quantum black holes, braneworld black holes, and even a singularity[29, 30, 35, 36, 3952]. Specially, echoes have been found in Einstein -nonlinear electrodynamic spacetimes. One is the dyonic black hole in Einstein-Maxwell gravity extended with quasi-topological electromagnetism, which hold two photon spheres and double-peak effective potentials[53, 54]. Another is the multiple-horizon black hole model, which hold effective potentials with three or more peaks[55, 56]. Due to the theoretical and observational importance of echoes, it is significant to look for more black hole spacetimes that can hold multiple effective potentials to produce echo signals.</text>
<text><location><page_3><loc_12><loc_15><loc_88><loc_35></location>Currently, there is no research about the connection between black hole echoes and quintessence. As recent astronomical observations display, the universe is now in an accelerating expansion[57-59], implying a state of negative pressure. Quintessence dark energy is one of the candidates to interpret the negative pressure, whose dynamic may affect the black hole[60, 61]. In this model, the state equation of quintessence is constrained by the pressure p = ρ q ω q , where ρ represents the energy density and ω q represents the state parameter satisfying -1 < ω q < -1 3 . Thus, it is natural for us to think about the effect of quintessence acting to black hole echoes.</text>
<text><location><page_3><loc_12><loc_7><loc_88><loc_14></location>The remaining of this paper is organized as follows. In Section II, we consider a massless time-dependent scalar field perturbation and offer a numerical method for solving wave function. In Section III, we give out the metric of a dyonic black hole with quintessence and</text>
<text><location><page_4><loc_12><loc_84><loc_88><loc_91></location>analyze its parameters. In Section IV, we detailly discuss the various transitions of black hole effective potential and echoes under effect of quintessence. In Section V, the complete conclusion is given.</text>
<section_header_level_1><location><page_4><loc_12><loc_79><loc_24><loc_80></location>II. SET UP</section_header_level_1>
<section_header_level_1><location><page_4><loc_14><loc_74><loc_69><loc_75></location>A. Massless scalar field perturbation and effective potential</section_header_level_1>
<text><location><page_4><loc_12><loc_64><loc_88><loc_71></location>In this paper, we only consider the static and spherically symmetric black hole. In this section, we give out the most general metric form of the black hole and its effective potential under massless scalar perturbation. The general metric of the black hole takes the form[54]</text>
<formula><location><page_4><loc_30><loc_58><loc_88><loc_61></location>ds 2 = -h ( r ) dt 2 + 1 f ( r ) dr 2 + r 2 ( dθ 2 +sin 2 θdφ 2 ) , (1)</formula>
<text><location><page_4><loc_12><loc_50><loc_88><loc_57></location>where we assume that h ( r + ) = f ( r + ) = 0 gives out the event horizon r + of the black hole. To study the echoes due to the black hole metric only, we consider the time-dependent massless scalar field perturbation, whose equation is</text>
<formula><location><page_4><loc_34><loc_45><loc_88><loc_49></location>glyph[square] Ψ ≡ 1 √ -g ∂ µ ( √ -gg µν ∂ ν Ψ( t, r, θ, φ )) . (2)</formula>
<text><location><page_4><loc_12><loc_37><loc_88><loc_43></location>For the static and spherically symmetric black hole metric, the time-dependent scalar field perturbation Ψ( t, r, θ, φ ) can be decomposed by separation of variables in terms of spherical harmonics</text>
<formula><location><page_4><loc_35><loc_32><loc_88><loc_36></location>Ψ( t, r, θ, φ ) = ∑ l,m Φ( t, r ) r Y l,m ( θ, φ ) . (3)</formula>
<text><location><page_4><loc_12><loc_30><loc_60><loc_31></location>Thus, the scalar field perturbation equation(1) reduces to</text>
<formula><location><page_4><loc_18><loc_25><loc_88><loc_28></location>-∂ 2 Φ( t, r ) ∂t 2 + h ( r ) f ( r ) ∂ 2 Φ( t, r ) ∂r 2 + 1 2 ∂ ∂r ( h ( r ) f ( r )) ∂ Φ( t, r ) ∂r -V ( r )Φ( t, r ) = 0 , (4)</formula>
<text><location><page_4><loc_12><loc_22><loc_41><loc_23></location>with the original effective potential</text>
<formula><location><page_4><loc_33><loc_16><loc_88><loc_20></location>V ( r ) = l ( l +1) r 2 h ( r ) + 1 2 r ∂ ∂r ( h ( r ) f ( r )) . (5)</formula>
<text><location><page_4><loc_12><loc_11><loc_88><loc_15></location>In order to map the radial region to ( -∞ , + ∞ ), the definition of the tortoise coordinate x is needed,</text>
<formula><location><page_4><loc_42><loc_7><loc_88><loc_10></location>dx = 1 √ h ( r ) f ( r ) dr. (6)</formula>
<text><location><page_5><loc_12><loc_89><loc_51><loc_91></location>Therefore, the equation(4) can be rewritten as</text>
<formula><location><page_5><loc_34><loc_84><loc_88><loc_88></location>( -∂ 2 ∂t 2 + ∂ 2 ∂x 2 -V eff ( x ) ) ψ ( t, x ) = 0 , (7)</formula>
<text><location><page_5><loc_12><loc_82><loc_84><loc_83></location>with the time-dependent scalar field perturbation and effective potential in terms of x</text>
<formula><location><page_5><loc_35><loc_77><loc_88><loc_79></location>Φ( t, r ) ⇒ ψ ( t, x ) , V ( r ) ⇒ V eff ( x ) . (8)</formula>
<section_header_level_1><location><page_5><loc_14><loc_72><loc_35><loc_73></location>B. Numerical method</section_header_level_1>
<text><location><page_5><loc_12><loc_65><loc_88><loc_69></location>To solve the partial differential equation(7), we can consider a time-dependent Green's function G ( t, x, x ' ), which satisfies[8, 52]</text>
<formula><location><page_5><loc_28><loc_60><loc_88><loc_64></location>( -∂ 2 ∂t 2 + ∂ 2 ∂x 2 -V eff ( x ) ) G ( t, x, x ' ) = δ ( t ) δ ( x -x ' ) . (9)</formula>
<text><location><page_5><loc_12><loc_57><loc_73><loc_59></location>Thus, the solution of equation(7) can be rewritten in terms of G ( t, x, x ' ),</text>
<formula><location><page_5><loc_23><loc_52><loc_88><loc_56></location>ψ ( t, x ) = -∫ + ∞ -∞ ( G ( t, x, x ' ) ∂ t ψ (0 , x ' ) + ∂ t G ( t, x, x ' ) ψ (0 , x ' )) dx ' . (10)</formula>
<text><location><page_5><loc_12><loc_49><loc_63><loc_50></location>With Fourier transform, the solution(10) can be expressed as</text>
<formula><location><page_5><loc_30><loc_44><loc_88><loc_48></location>ψ ( t, x ) = 1 2 π ∫ + ∞ -∞ ˆ G ( ω, x, x ' ) ˆ S ( ω, x ) e -iωt dx ' dω, (11)</formula>
<text><location><page_5><loc_12><loc_41><loc_52><loc_42></location>where ˆ S ( ω, x ) is depend on the initial condition,</text>
<formula><location><page_5><loc_35><loc_35><loc_88><loc_39></location>ˆ S ( ω, x ) = -∂ψ ( t, x ) ∂t ∣ ∣ ∣ ∣ t =0 + iωψ (0 , x ) , (12)</formula>
<text><location><page_5><loc_12><loc_30><loc_88><loc_34></location>Two linearly independent solutions ˆ ψ + ( ω, x ) and ˆ ψ -( ω, x ) are necessary to construct the time-independent Green's function G ( ω, x, x ' ), with the homogeneous differential equation</text>
<formula><location><page_5><loc_35><loc_25><loc_88><loc_28></location>( ∂ 2 ∂x 2 + ω 2 -V eff ( x ) ) ˆ ψ ( ω, x ) = 0 , (13)</formula>
<text><location><page_5><loc_12><loc_22><loc_33><loc_23></location>and boundary conditions</text>
<formula><location><page_5><loc_36><loc_15><loc_88><loc_21></location> ˆ ψ + ( ω, x ) = e iωt , x → + ∞ ˆ ψ -( ω, x ) = e -iωt , x →-∞ (14)</formula>
<text><location><page_5><loc_12><loc_12><loc_72><loc_14></location>Therefore, the time-independent Green's function G ( ω, x, x ' ) is given by</text>
<formula><location><page_5><loc_29><loc_7><loc_88><loc_11></location>ˆ G ( ω, x, x ' ) = ˆ ψ + ( ω, max( x, x ' )) ˆ ψ -( ω, min( x, x ' )) W ( ω ) , (15)</formula>
<text><location><page_6><loc_12><loc_89><loc_46><loc_91></location>where the Wronskian W ( ω ) is defined as</text>
<formula><location><page_6><loc_29><loc_85><loc_88><loc_87></location>W ( ω ) = ˆ ψ -( ω, x ) ∂ x ˆ ψ + ( ω, x ) -∂ x ˆ ψ -( ω, x ) ˆ ψ + ( ω, x ) , (16)</formula>
<text><location><page_6><loc_12><loc_73><loc_88><loc_83></location>In this paper, we investigate different waveforms ψ ( t, x ) caused by different peak structure of effective potential in the black hole, which are all detected by a distant observer. To numerically solve the partial differential equation(7) from a certain initial wave packet, we consider the initial condition to be a Gaussian wave packet,</text>
<formula><location><page_6><loc_28><loc_68><loc_88><loc_72></location>ψ (0 , x ) = 0 , ∂ψ ( t, x ) ∂t ∣ ∣ ∣ ∣ t =0 = A exp ( -( x -x 0 ) 2 2∆ 2 ) , (17)</formula>
<text><location><page_6><loc_12><loc_42><loc_88><loc_67></location>where the parameter x 0 is the center of the initial Gaussian wave packet position, the width ∆ and the amplitude A are also chosen to adapt to the specific case. In this paper, we consider two inequivalent situations. One is that the initial Gaussian wave packet is located outside the double peaks. Another is that it is in the potential well between the two peaks. In [54], it has been found that the black hole echoes have the same characteristics whether it is close to the event horizon or far away from the outer peak. But when the initial location x 0 is closer to the bottom of the potential well, the echo frequency roughly doubles the one associated with the wave packet outside the peaks, which makes the echoes more distinguishable. In this paper, we also compare the variation of echo frequency with respect to quintessence parameter in these two cases, which will be discussed in Section IV.</text>
<section_header_level_1><location><page_6><loc_12><loc_33><loc_88><loc_37></location>III. QUASI-TOPOLOGICAL ELECTROMAGNETISM AND DYONIC BLACK HOLE WITH QUINTESSENCE</section_header_level_1>
<text><location><page_6><loc_12><loc_26><loc_88><loc_30></location>In [53, 54], the quasi-topological electromagnetism is defined to be the squared norm of the topological 4-form F ∧ F . The Lagrangian is</text>
<formula><location><page_6><loc_26><loc_22><loc_88><loc_25></location>L = √ -g ( R -F µν F µν -α ( ( F µν F µν ) 2 -2 F µ ν F ν ρ F ρ σ F σ µ )) . (18)</formula>
<text><location><page_6><loc_12><loc_13><loc_88><loc_20></location>This theory admits the exact solution of the dyonic black hole, which is both static and spherically symmetric. The Einstein field equation and the Maxwell equation of motion can be expressed as[53]</text>
<formula><location><page_6><loc_32><loc_7><loc_88><loc_13></location>R µν -1 2 Rg µν = T µν + ˜ T µν , ∇ µ ( 4 F µν +8 α ( F 2 F µν -2 F µρ F σ rho F ν σ ) ) = 0 , (19)</formula>
<text><location><page_7><loc_12><loc_89><loc_60><loc_91></location>where the energy-momentum tensor T µν can be written as</text>
<formula><location><page_7><loc_14><loc_83><loc_88><loc_88></location>T µν = 2 F µρ F ρ ν -1 2 F 2 g µν + α ( 4 F 2 F µρ F ρ ν -8 F µρ F ρ σ F σ λ F λ ν -1 2 ( ( F 2 ) 2 -2 F µ ν F ν ρ F ρ σ F σ µ ) ) , (20)</formula>
<text><location><page_7><loc_12><loc_80><loc_53><loc_81></location>and ˜ T µν denotes the quintessence matter with[60]</text>
<formula><location><page_7><loc_31><loc_75><loc_88><loc_78></location>˜ T t t = ˜ T r r = -ρ q , ˜ T θ θ = ˜ T φ φ = -1 2 ρ q (3 ω q +1) . (21)</formula>
<text><location><page_7><loc_12><loc_69><loc_88><loc_73></location>Thus, with the above functions, the metric of the black hole with quintessence can be written as</text>
<formula><location><page_7><loc_30><loc_65><loc_88><loc_69></location>ds 2 = -f ( r ) dt 2 + 1 f ( r ) dr 2 + r 2 ( dθ 2 +sin 2 θdφ 2 ) , (22)</formula>
<text><location><page_7><loc_12><loc_63><loc_17><loc_64></location>where</text>
<formula><location><page_7><loc_26><loc_59><loc_88><loc_63></location>f ( r ) = 1 -2 M r + p 2 r 2 + q 2 r 2 2 F 1 ( 1 4 , 1; 5 4 ; -4 αp 2 r 4 ) -a r 3 ω q +1 . (23)</formula>
<text><location><page_7><loc_12><loc_57><loc_80><loc_58></location>With the black hole metric, the original effective potential equation(5) reduces to</text>
<formula><location><page_7><loc_12><loc_48><loc_89><loc_55></location>V ( r ) = l ( l +1) r 2 f ( r ) + 1 r f ( r ) ∂f ( r ) ∂r (24) = f ( r ) r 2 ( l ( l +1) + 2 M r -2 p 2 r 2 -q 2 r 2 4 αp 2 + r 4 -q 2 r 2 2 F 1 ( 1 4 , 1; 5 4 ; -4 p 2 α r 4 ) + a (3 ω q +1) r 3 ω q +1 ) .</formula>
<text><location><page_7><loc_12><loc_34><loc_88><loc_46></location>Here, we can see that the effective potential V ( r ) contains several parameters. M , q and p are the mass, electric and magnetic charges of the black hole, α is the coupling constant, ω q and a are the state parameter and normalization factor of quintessence. When α = 0 and p = 0, the metric(23) reduces to the usual RN black hole metric. When q = 0, it reduces to the original Schwarzschild black hole metric.</text>
<text><location><page_7><loc_12><loc_16><loc_88><loc_33></location>For a dyonic black hole with quintessence, set f ( r ) = 0 and generally we can get three solution. Two of them are black hole horizon, with the larger one becomes the event horizon r + . And the solution far away beyond the event horizon becomes the cosmological horizon r c . The region between the two horizons is called the domain of outer communication, since any two observers in this region may communicate with each other without being hindered by a horizon[62-64]. Under tortoise coordinates, region ( r + , r c ) turns into ( x + , x c ) = ( -∞ , + ∞ ), which also makes it suitable to consider the cosmological horizon as an observer at infinity.</text>
<text><location><page_7><loc_12><loc_8><loc_88><loc_14></location>Moreover, the dyonic black hole has an unusual feature that the metric function f ( r ) is not monotonous in radius region. It has multiple wiggle such that the gravity force can vanish or even become repulsive in a certain finite region outside the horizon. It also leads</text>
<text><location><page_8><loc_12><loc_81><loc_88><loc_91></location>to the black hole solution with four or five horizons for some suitable parameters. As this feature has been discussed detailly in [54], we expect that quintessence will give rise to various changes of the black hole horizons and effective potential, which will be analyzed in the following sections.</text>
<section_header_level_1><location><page_8><loc_12><loc_76><loc_68><loc_77></location>IV. EFFECTIVE POTENTIAL AND BLACK HOLE ECHO</section_header_level_1>
<text><location><page_8><loc_12><loc_56><loc_88><loc_73></location>In this section, we discuss different changes of the black hole due to various parameters. After fixing other parameters, we analyze how quintessence influence the number of local maximum and local minimum. It will reduces to the transition of effective potential between single-peak and double-peak, which will have different effect to the echoes of the black hole. Moreover, different values of spherical harmonics l also have different effects on the effective potential. In this paper, we focus on the situation l = 2 since they play a dominant role in the ringdown gravitational waves after binary black holes merge.</text>
<section_header_level_1><location><page_8><loc_14><loc_50><loc_57><loc_51></location>A. Transition from single-peak to double-peak</section_header_level_1>
<figure>
<location><page_8><loc_13><loc_28><loc_87><loc_46></location>
<caption>FIG. 1: Changes of black hole metric and effective potential under quintessence. Here we take ( M,p,q,α ) = (6 . 44 , 1 . 16 , 6 . 92 , 150).</caption>
</figure>
<text><location><page_8><loc_12><loc_8><loc_88><loc_17></location>For an initial dyonic black hole without quintessence, the coupling constant α is not large enough, meaning that the quasi-topological electromagnetism cannot support strong matter energy-momentum tensor to curve the spacetime. Thus, the effective potential only displays a single-peak, which cannot provide a suitable constructure to generate echoes. The first</text>
<figure>
<location><page_9><loc_17><loc_14><loc_83><loc_91></location>
<caption>FIG. 2: Changes of black hole echoes with ω q . The upper attached figure gives out the positions and relative size of inner-peak and outer-peak. The lower attached figure gives out echoes from different initial positons of wave package.</caption>
</figure>
<text><location><page_10><loc_12><loc_74><loc_88><loc_91></location>imiage in Figure 2 shows that the reflection from the single-peak potential forms an observed burst after the initial wave traveling from the vicinity of the peak to the observer. At late times, the wave signal shows an exponentially damped sinusoid. Due to the absence of the outer peak, we cannot observe any echo after the burst is received. Note that waves propagating on a black hole spacetime usually develop asymptotically late-time tails, which follow exponentially damped sinusoids and decay as an inverse power of time due to scattering from large radius in the black hole geometry.</text>
<text><location><page_10><loc_12><loc_44><loc_88><loc_72></location>After being influenced by quintessence, the effective potential of the black hole reduces continuously and obviously. Meanwhile, as the value of ω q becomes small, the outer peak appears and rapidly rises to a size similar to that of the inner peak. Because of the appearance of double-peak, the perturbation can be reflected off by the inner potential barriers and bounce back and forth between the two peaks. When the perturbation successively tunnels through the outer potential barrier, a series of echoes can be received by a distant observer. From Figure 2, we can see that under the effect of quintessence, the relative height of the outer and inner peaks is decreasing, together with the decreasing of the potential well. This means that there will be more wave signals tunneling through the barriers and superposing to form black hole echoes. Thus, the frequency of echoes will increase exponentially under a smaller ω q , which means a stronger effect from quintessence.</text>
<section_header_level_1><location><page_10><loc_14><loc_39><loc_75><loc_40></location>B. Transition of potential size between inner-peak and outer-peak</section_header_level_1>
<figure>
<location><page_10><loc_13><loc_17><loc_87><loc_35></location>
<caption>FIG. 3: Transition of inner-peak and outer-peak with black hole metric under quintessence. Here we take ( M,p,q,α ) = (6 . 44 , 1 . 185 , 6 . 92 , 150).</caption>
</figure>
<figure>
<location><page_11><loc_17><loc_40><loc_83><loc_91></location>
<caption>FIG. 4: Changes of black hole echoes during the transition of inner-peak and outer-peak with ω q .</caption>
</figure>
<text><location><page_11><loc_12><loc_10><loc_88><loc_32></location>In the subsection above, we have known that the inner potential peak and potential well will decrease under the effect of quintessence. Therefore, it is natural for us to consider how echoes change when the effective potential changes from an inner-large doublepeak to an outer-larger double-peak. When the quasi-topological electromagnetism supports enough matter energy-momentum tensor to curve the spacetime, a dyonic black hole without quintessence can form an effective potential with double-peak. In this case, the distance observer can also receive black hole echoes. And because the inner peak is bigger than the outer peak, the waves reflected by the inner peak can easily tunnel through the outer barrier. This leads to the distinguishable echoes with a high frequency.</text>
<text><location><page_11><loc_14><loc_7><loc_88><loc_8></location>However, under the effect of quintessence, the inner peak turns to be lower than the outer</text>
<text><location><page_12><loc_12><loc_66><loc_88><loc_91></location>one, which influences the behavior of echoes. It should be emphasized that although both the inner potential and potential well are decreasing, the former decreases faster than the latter. As shown in Figure 4, for one thing, as the inner peak decreases, the number of waves that can be reflected by it also decreases accordingly. For another, due to the larger distance between the two peaks, the scattering of a perturbation off one peak is lessly affected by the other peak. Therefore, the black hole echoes become distinguishable but with a low frequency. Meanwhile, waves that has been bounced back need to face a thicker potential barrier, which reduces the probability of tunneling through the barrier. Thus, compared to the initial burst waveform, the amplitude of the echo signal becomes very small, requiring more precise observation instruments.</text>
<section_header_level_1><location><page_12><loc_14><loc_60><loc_75><loc_61></location>C. Smooth and sudden transition from double-peak to single-peak</section_header_level_1>
<figure>
<location><page_12><loc_13><loc_38><loc_50><loc_56></location>
<caption>FIG. 5: Smooth disappearance of inner-peak with the monotonicity change of black hole metric under quintessence. Here we take ( M,p,q,α ) = (6 . 44 , 1 . 23 , 6 . 92 , 150).</caption>
</figure>
<figure>
<location><page_12><loc_51><loc_38><loc_87><loc_56></location>
</figure>
<text><location><page_12><loc_50><loc_48><loc_51><loc_48></location>)</text>
<text><location><page_12><loc_50><loc_48><loc_51><loc_48></location>r</text>
<text><location><page_12><loc_50><loc_48><loc_51><loc_48></location>(</text>
<text><location><page_12><loc_50><loc_47><loc_51><loc_48></location>V</text>
<text><location><page_12><loc_12><loc_7><loc_88><loc_27></location>When the effect of quintessence becomes stronger, there will be two situations where the effective potential transforms from double-peak to single-peak. From Figure 5, one situation is that the black hole still maintains three horizons (event horizon r + , cosmology horizon r c and one horizon inside r + ). In this case, because the inner potential decreases faster than the potential well, we can see that the inner peak decreases and finially disappears. Thus, the potential transition is smooth, with the number and frequency of echoes also slowing down gradually from high-frequency and easily resolved echo signal to low-frequency, no-echo waveform signal, which is shown in Figure 6.</text>
<figure>
<location><page_13><loc_12><loc_60><loc_85><loc_89></location>
<caption>FIG. 6: Changes of black hole echoes during the smooth disappearance of inner-peak with ω q .</caption>
</figure>
<figure>
<location><page_13><loc_13><loc_34><loc_87><loc_52></location>
<caption>FIG. 7: Sudden vanishment of inner-peak with the monotonicity change of black hole metric under quintessence. Here we take ( M,p,q,α ) = (6 . 48 , 1 . 0 , 6 . 8 , 150).</caption>
</figure>
<text><location><page_13><loc_12><loc_9><loc_88><loc_24></location>From Figure 7, another situation is when the inner potential is large enough, the black hole will have four or five horizons ( r + , r c , and two or three horizons inside r + ). In this case, the inner peak will be wrapped by event horizon r + before it disappears. Therefore, as shown in Figure 8, the effective potential will experience a sudden transition from doublepeak to single-peak. And the distance observer will also receive a sudden change of the black hole echoes. This change is sudden and rapid. At this moment, distinguishable and</text>
<figure>
<location><page_14><loc_17><loc_40><loc_83><loc_91></location>
<caption>FIG. 8: Changes of black hole echoes during the sudden vanishment of inner-peak with ω q .</caption>
</figure>
<text><location><page_14><loc_12><loc_28><loc_88><loc_32></location>high-frequency echoes exist. In the next moment, echoes suddenly disappear, leaving only a single-peak potential.</text>
<section_header_level_1><location><page_14><loc_12><loc_23><loc_49><loc_24></location>V. DISCUSSION AND CONCLUTION</section_header_level_1>
<text><location><page_14><loc_12><loc_8><loc_88><loc_20></location>In this paper, we first reviewed the dyonic black hole in Einstein-Maxwell gravity extended with quasi-topological electromagnetism which is defined to be the squared norm of the topological 4-form F ∧ F [53, 54]. Then we studied various effects of quintessence acting to the black hole effective potential and echoes. Several black hole and wormhole models with double effective potentials have been studied[29, 30, 35, 36, 39-52]. And our motivation was</text>
<text><location><page_15><loc_12><loc_79><loc_88><loc_91></location>to look for whether quintessence can influence the spacetime structure of black holes. And our results shown that effective potential indeed varies between single peak and double peaks in a dyonic black hole with quintessence. We also studied echoes with quintessence from different values and distance of inner and outer potential peaks, including wormhole-like potential.</text>
<text><location><page_15><loc_12><loc_55><loc_88><loc_77></location>For suitable parameters, the effect of quintessence acting on the dyonic black hole will cause potential transition between single-peak and double-peak, which can be approximately divided into two cases. One is that an outer peak appears, the original peak becomes the inner one, which decreases and finally generally disappears together with potential well. In this case, the distance observer will receive smooth transition of waveform between exponentially decaying sinusoid and high frequency echoes. Another is that the black hole will exist multiple horizons to wrap the inner peak, which leads to the sudden disappearance of the inner peak. In this case, it will be strange to observe a distinguishable and high-frequency echoes vanish, leaving only an exponentially decaying sinusoid waveform.</text>
<text><location><page_15><loc_12><loc_34><loc_88><loc_54></location>It should be emphasized that quintessence only cannot lead to the multiple potential peaks and echoes in Schwarzschild or RN black holes. In fact, the dyonic black hole itself already has double-peak potential under certain conditions. This is caused by Einstein-Maxwell gravity extended with quasi-topological electromagnetism. Under a large coupling constant α , the quasi-topological electromagnetism provides proper matter energy-momentum tensor to curve the spacetime[53, 54]. And quintessence plays no particular role other than strengthen this force. Thus, it will be interesting if our analysis of quintessence can be extended beyond spherical symmetry to more general black hole spacetimes.</text>
<section_header_level_1><location><page_15><loc_14><loc_28><loc_30><loc_30></location>Acknowledgement</section_header_level_1>
<text><location><page_15><loc_12><loc_19><loc_88><loc_25></location>We are grateful to Haitang Yang, Jun Tao and Yuhang Lu for useful discussions. This work is supported in part by NSFC (Grant No. 11747171), Xinglin Scholars Project of Chengdu University of Traditional Chinese Medicine (Grant no.QNXZ2018050).</text>
<text><location><page_16><loc_16><loc_89><loc_65><loc_91></location>doi:10.1103/PhysRevLett.116.061102 [arXiv:1602.03837 [gr-qc]].</text>
<unordered_list>
<list_item><location><page_16><loc_13><loc_81><loc_88><loc_88></location>[2] K. Akiyama et al. [Event Horizon Telescope], First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole, Astrophys. J. Lett. 875 , L1 (2019) doi:10.3847/2041-8213/ab0ec7 [arXiv:1906.11238 [astro-ph.GA]].</list_item>
<list_item><location><page_16><loc_13><loc_73><loc_88><loc_80></location>[3] K. Akiyama et al. [Event Horizon Telescope], First M87 Event Horizon Telescope Results. II. Array and Instrumentation, Astrophys. J. Lett. 875 , no.1, L2 (2019) doi:10.3847/20418213/ab0c96 [arXiv:1906.11239 [astro-ph.IM]].</list_item>
<list_item><location><page_16><loc_13><loc_65><loc_88><loc_71></location>[4] K. Akiyama et al. [Event Horizon Telescope], First M87 Event Horizon Telescope Results. III. Data Processing and Calibration, Astrophys. J. Lett. 875 , no.1, L3 (2019) doi:10.3847/20418213/ab0c57 [arXiv:1906.11240 [astro-ph.GA]].</list_item>
<list_item><location><page_16><loc_13><loc_57><loc_88><loc_63></location>[5] K. Akiyama et al. [Event Horizon Telescope], First M87 Event Horizon Telescope Results. IV. Imaging the Central Supermassive Black Hole, Astrophys. J. Lett. 875 , no.1, L4 (2019) doi:10.3847/2041-8213/ab0e85 [arXiv:1906.11241 [astro-ph.GA]].</list_item>
<list_item><location><page_16><loc_13><loc_48><loc_88><loc_55></location>[6] K. Akiyama et al. [Event Horizon Telescope], First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring, Astrophys. J. Lett. 875 , no.1, L5 (2019) doi:10.3847/2041-8213/ab0f43 [arXiv:1906.11242 [astro-ph.GA]].</list_item>
<list_item><location><page_16><loc_13><loc_40><loc_88><loc_47></location>[7] K. Akiyama et al. [Event Horizon Telescope], First M87 Event Horizon Telescope Results. VI. The Shadow and Mass of the Central Black Hole, Astrophys. J. Lett. 875 , no.1, L6 (2019) doi:10.3847/2041-8213/ab1141 [arXiv:1906.11243 [astro-ph.GA]].</list_item>
<list_item><location><page_16><loc_13><loc_32><loc_88><loc_39></location>[8] H. P. Nollert, TOPICAL REVIEW: Quasinormal modes: the characteristic 'sound' of black holes and neutron stars, Class. Quant. Grav. 16 , R159-R216 (1999) doi:10.1088/02649381/16/12/201</list_item>
<list_item><location><page_16><loc_13><loc_24><loc_88><loc_30></location>[9] E. Berti, V. Cardoso, J. A. Gonzalez and U. Sperhake, Mining information from binary black hole mergers: A Comparison of estimation methods for complex exponentials in noise, Phys. Rev. D 75 , 124017 (2007) doi:10.1103/PhysRevD.75.124017 [arXiv:gr-qc/0701086 [gr-qc]].</list_item>
<list_item><location><page_16><loc_12><loc_16><loc_88><loc_22></location>[10] V. Cardoso, E. Franzin and P. Pani, Is the gravitational-wave ringdown a probe of the event horizon?, Phys. Rev. Lett. 116 , no.17, 171101 (2016) [erratum: Phys. Rev. Lett. 117 , no.8, 089902 (2016)] doi:10.1103/PhysRevLett.116.171101 [arXiv:1602.07309 [gr-qc]].</list_item>
<list_item><location><page_16><loc_12><loc_10><loc_88><loc_14></location>[11] R. H. Price and G. Khanna, Gravitational wave sources: reflections and echoes, Class. Quant. Grav. 34 , no.22, 225005 (2017) doi:10.1088/1361-6382/aa8f29 [arXiv:1702.04833 [gr-qc]].</list_item>
<list_item><location><page_16><loc_12><loc_7><loc_88><loc_9></location>[12] M. Giesler, M. Isi, M. A. Scheel and S. Teukolsky, Black Hole Ringdown: The Impor-</list_item>
</unordered_list>
<text><location><page_17><loc_16><loc_87><loc_88><loc_91></location>tance of Overtones, Phys. Rev. X 9 , no.4, 041060 (2019) doi:10.1103/PhysRevX.9.041060 [arXiv:1903.08284 [gr-qc]].</text>
<unordered_list>
<list_item><location><page_17><loc_12><loc_78><loc_88><loc_85></location>[13] E. Berti, V. Cardoso and A. O. Starinets, Quasinormal modes of black holes and black branes, Class. Quant. Grav. 26 , 163001 (2009) doi:10.1088/0264-9381/26/16/163001 [arXiv:0905.2975 [gr-qc]].</list_item>
<list_item><location><page_17><loc_12><loc_70><loc_88><loc_77></location>[14] Z. Mark, A. Zimmerman, S. M. Du and Y. Chen, A recipe for echoes from exotic compact objects, Phys. Rev. D 96 , no.8, 084002 (2017) doi:10.1103/PhysRevD.96.084002 [arXiv:1706.06155 [gr-qc]].</list_item>
<list_item><location><page_17><loc_12><loc_62><loc_88><loc_69></location>[15] R. A. Konoplya and A. Zhidenko, Quasinormal modes of black holes: From astrophysics to string theory, Rev. Mod. Phys. 83 , 793-836 (2011) doi:10.1103/RevModPhys.83.793 [arXiv:1102.4014 [gr-qc]].</list_item>
<list_item><location><page_17><loc_12><loc_57><loc_88><loc_61></location>[16] J. P. S. Lemos and O. B. Zaslavskii, Black hole mimickers: Regular versus singular behavior, Phys. Rev. D 78 , 024040 (2008) doi:10.1103/PhysRevD.78.024040 [arXiv:0806.0845 [gr-qc]].</list_item>
<list_item><location><page_17><loc_12><loc_48><loc_88><loc_55></location>[17] P. V. P. Cunha, J. A. Font, C. Herdeiro, E. Radu, N. Sanchis-Gual and M. Zilh˜ao, Lensing and dynamics of ultracompact bosonic stars, Phys. Rev. D 96 , no.10, 104040 (2017) doi:10.1103/PhysRevD.96.104040 [arXiv:1709.06118 [gr-qc]].</list_item>
<list_item><location><page_17><loc_12><loc_40><loc_88><loc_47></location>[18] P. V. P. Cunha and C. A. R. Herdeiro, Shadows and strong gravitational lensing: a brief review, Gen. Rel. Grav. 50 , no.4, 42 (2018) doi:10.1007/s10714-018-2361-9 [arXiv:1801.00860 [gr-qc]].</list_item>
<list_item><location><page_17><loc_12><loc_32><loc_88><loc_39></location>[19] R. Shaikh, P. Banerjee, S. Paul and T. Sarkar, A novel gravitational lensing feature by wormholes, Phys. Lett. B 789 , 270-275 (2019) [erratum: Phys. Lett. B 791 , 422-423 (2019)] doi:10.1016/j.physletb.2018.12.030 [arXiv:1811.08245 [gr-qc]].</list_item>
<list_item><location><page_17><loc_12><loc_26><loc_88><loc_30></location>[20] D. C. Dai and D. Stojkovic, Observing a Wormhole, Phys. Rev. D 100 , no.8, 083513 (2019) doi:10.1103/PhysRevD.100.083513 [arXiv:1910.00429 [gr-qc]].</list_item>
<list_item><location><page_17><loc_12><loc_21><loc_88><loc_25></location>[21] H. Huang and J. Yang, Charged Ellis Wormhole and Black Bounce, Phys. Rev. D 100 , no.12, 124063 (2019) doi:10.1103/PhysRevD.100.124063 [arXiv:1909.04603 [gr-qc]].</list_item>
<list_item><location><page_17><loc_12><loc_13><loc_88><loc_19></location>[22] J. H. Simonetti, M. J. Kavic, D. Minic, D. Stojkovic and D. C. Dai, Sensitive searches for wormholes, Phys. Rev. D 104 , no.8, L081502 (2021) doi:10.1103/PhysRevD.104.L081502 [arXiv:2007.12184 [gr-qc]].</list_item>
<list_item><location><page_17><loc_12><loc_7><loc_88><loc_11></location>[23] M. Wielgus, J. Horak, F. Vincent and M. Abramowicz, Reflection-asymmetric wormholes and their double shadows, Phys. Rev. D 102 , no.8, 084044 (2020)</list_item>
</unordered_list>
<text><location><page_18><loc_16><loc_89><loc_63><loc_91></location>doi:10.1103/PhysRevD.102.084044 [arXiv:2008.10130 [gr-qc]].</text>
<unordered_list>
<list_item><location><page_18><loc_12><loc_81><loc_88><loc_88></location>[24] J. Yang and H. Huang, Trapping horizons of the evolving charged wormhole and black bounce, Phys. Rev. D 104 , no.8, 084005 (2021) doi:10.1103/PhysRevD.104.084005 [arXiv:2104.11134 [gr-qc]].</list_item>
<list_item><location><page_18><loc_12><loc_76><loc_88><loc_80></location>[25] C. Bambi and D. Stojkovic, Astrophysical Wormholes, Universe 7 , no.5, 136 (2021) doi:10.3390/universe7050136 [arXiv:2105.00881 [gr-qc]].</list_item>
<list_item><location><page_18><loc_12><loc_67><loc_88><loc_74></location>[26] J. Peng, M. Guo and X. H. Feng, Observational signature and additional photon rings of an asymmetric thin-shell wormhole, Phys. Rev. D 104 , no.12, 124010 (2021) doi:10.1103/PhysRevD.104.124010 [arXiv:2102.05488 [gr-qc]].</list_item>
<list_item><location><page_18><loc_12><loc_62><loc_88><loc_66></location>[27] A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black Holes: Complementarity or Firewalls?, JHEP 02 , 062 (2013) doi:10.1007/JHEP02(2013)062 [arXiv:1207.3123 [hep-th]].</list_item>
<list_item><location><page_18><loc_12><loc_57><loc_88><loc_61></location>[28] D. E. Kaplan and S. Rajendran, Firewalls in General Relativity, Phys. Rev. D 99 , no.4, 044033 (2019) doi:10.1103/PhysRevD.99.044033 [arXiv:1812.00536 [hep-th]].</list_item>
<list_item><location><page_18><loc_12><loc_48><loc_88><loc_55></location>[29] P. Bueno, P. A. Cano, F. Goelen, T. Hertog and B. Vercnocke, Echoes of Kerr-like wormholes, Phys. Rev. D 97 , no.2, 024040 (2018) doi:10.1103/PhysRevD.97.024040 [arXiv:1711.00391 [gr-qc]].</list_item>
<list_item><location><page_18><loc_12><loc_40><loc_88><loc_47></location>[30] R. A. Konoplya, Z. Stuchl'ık and A. Zhidenko, Echoes of compact objects: new physics near the surface and matter at a distance, Phys. Rev. D 99 , no.2, 024007 (2019) doi:10.1103/PhysRevD.99.024007 [arXiv:1810.01295 [gr-qc]].</list_item>
<list_item><location><page_18><loc_12><loc_35><loc_88><loc_39></location>[31] Y. T. Wang, J. Zhang and Y. S. Piao, Primordial gravastar from inflation, Phys. Lett. B 795 , 314-318 (2019) doi:10.1016/j.physletb.2019.06.036 [arXiv:1810.04885 [gr-qc]].</list_item>
<list_item><location><page_18><loc_12><loc_26><loc_88><loc_33></location>[32] Y. T. Wang, Z. P. Li, J. Zhang, S. Y. Zhou and Y. S. Piao, Are gravitational wave ringdown echoes always equal-interval?, Eur. Phys. J. C 78 , no.6, 482 (2018) doi:10.1140/epjc/s10052018-5974-y [arXiv:1802.02003 [gr-qc]].</list_item>
<list_item><location><page_18><loc_12><loc_21><loc_88><loc_25></location>[33] V. Cardoso and P. Pani, Testing the nature of dark compact objects: a status report, Living Rev. Rel. 22 , no.1, 4 (2019) doi:10.1007/s41114-019-0020-4 [arXiv:1904.05363 [gr-qc]].</list_item>
<list_item><location><page_18><loc_12><loc_13><loc_88><loc_19></location>[34] J. T. G'alvez Ghersi, A. V. Frolov and D. A. Dobre, Echoes from the scattering of wavepackets on wormholes, Class. Quant. Grav. 36 , no.13, 135006 (2019) doi:10.1088/1361-6382/ab23c8 [arXiv:1901.06625 [gr-qc]].</list_item>
<list_item><location><page_18><loc_12><loc_7><loc_88><loc_11></location>[35] H. Liu, P. Liu, Y. Liu, B. Wang and J. P. Wu, Echoes from phantom wormholes, Phys. Rev. D 103 , no.2, 024006 (2021) doi:10.1103/PhysRevD.103.024006 [arXiv:2007.09078 [gr-qc]].</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_19><loc_12><loc_84><loc_88><loc_91></location>[36] M. Y. Ou, M. Y. Lai and H. Huang, Echoes from asymmetric wormholes and black bounce, Eur. Phys. J. C 82 , no.5, 452 (2022) doi:10.1140/epjc/s10052-022-10421-x [arXiv:2111.13890 [gr-qc]].</list_item>
<list_item><location><page_19><loc_12><loc_76><loc_88><loc_82></location>[37] J. Abedi, H. Dykaar and N. Afshordi, Echoes from the Abyss: Tentative evidence for Planck-scale structure at black hole horizons, Phys. Rev. D 96 , no.8, 082004 (2017) doi:10.1103/PhysRevD.96.082004 [arXiv:1612.00266 [gr-qc]].</list_item>
<list_item><location><page_19><loc_12><loc_70><loc_88><loc_74></location>[38] J. Abedi, H. Dykaar and N. Afshordi, Echoes from the Abyss: The Holiday Edition!, [arXiv:1701.03485 [gr-qc]].</list_item>
<list_item><location><page_19><loc_12><loc_62><loc_88><loc_69></location>[39] K. A. Bronnikov and R. A. Konoplya, Echoes in brane worlds: ringing at a black hole-wormhole transition, Phys. Rev. D 101 , no.6, 064004 (2020) doi:10.1103/PhysRevD.101.064004 [arXiv:1912.05315 [gr-qc]].</list_item>
<list_item><location><page_19><loc_12><loc_54><loc_88><loc_61></location>[40] M. S. Churilova and Z. Stuchlik, Ringing of the regular black-hole/wormhole transition, Class. Quant. Grav. 37 , no.7, 075014 (2020) doi:10.1088/1361-6382/ab7717 [arXiv:1911.11823 [grqc]].</list_item>
<list_item><location><page_19><loc_12><loc_48><loc_88><loc_52></location>[41] D. Kartini and A. Sulaksono, Gravitational wave echoes from quark stars, J. Phys. Conf. Ser. 1572 , 012034 (2020) doi:10.1088/1742-6596/1572/1/012034</list_item>
<list_item><location><page_19><loc_12><loc_43><loc_88><loc_47></location>[42] R. Dey, S. Chakraborty and N. Afshordi, Echoes from braneworld black holes, Phys. Rev. D 101 , no.10, 104014 (2020) doi:10.1103/PhysRevD.101.104014 [arXiv:2001.01301 [gr-qc]].</list_item>
<list_item><location><page_19><loc_12><loc_35><loc_88><loc_41></location>[43] P. Pani and V. Ferrari, On gravitational-wave echoes from neutron-star binary coalescences, Class. Quant. Grav. 35 , no.15, 15LT01 (2018) doi:10.1088/1361-6382/aacb8f [arXiv:1804.01444 [gr-qc]].</list_item>
<list_item><location><page_19><loc_12><loc_29><loc_88><loc_33></location>[44] A. Urbano and H. Veermae, On gravitational echoes from ultracompact exotic stars, JCAP 04 , 011 (2019) doi:10.1088/1475-7516/2019/04/011 [arXiv:1810.07137 [gr-qc]].</list_item>
<list_item><location><page_19><loc_12><loc_24><loc_88><loc_28></location>[45] A. Chowdhury and N. Banerjee, Echoes from a singularity, Phys. Rev. D 102 , no.12, 124051 (2020) doi:10.1103/PhysRevD.102.124051 [arXiv:2006.16522 [gr-qc]].</list_item>
<list_item><location><page_19><loc_12><loc_18><loc_88><loc_22></location>[46] Q. Wang, N. Oshita and N. Afshordi, Echoes from Quantum Black Holes, Phys. Rev. D 101 , no.2, 024031 (2020) doi:10.1103/PhysRevD.101.024031 [arXiv:1905.00446 [gr-qc]].</list_item>
<list_item><location><page_19><loc_12><loc_13><loc_88><loc_17></location>[47] K. Saraswat and N. Afshordi, Quantum Nature of Black Holes: Fast Scrambling versus Echoes, JHEP 04 , 136 (2020) doi:10.1007/JHEP04(2020)136 [arXiv:1906.02653 [hep-th]].</list_item>
<list_item><location><page_19><loc_12><loc_7><loc_88><loc_11></location>[48] N. Oshita, D. Tsuna and N. Afshordi, Quantum Black Hole Seismology I: Echoes, Ergospheres, and Spectra, Phys. Rev. D 102 , no.2, 024045 (2020) doi:10.1103/PhysRevD.102.024045</list_item>
</unordered_list>
<text><location><page_20><loc_16><loc_89><loc_36><loc_91></location>[arXiv:2001.11642 [gr-qc]].</text>
<unordered_list>
<list_item><location><page_20><loc_12><loc_84><loc_88><loc_88></location>[49] S. K. Manikandan and K. Rajeev, New kind of echo from quantum black holes, Phys. Rev. D 105 , no.6, 064024 (2022) doi:10.1103/PhysRevD.105.064024 [arXiv:2112.08773 [gr-qc]].</list_item>
<list_item><location><page_20><loc_12><loc_76><loc_88><loc_82></location>[50] H. Liu, W. L. Qian, Y. Liu, J. P. Wu, B. Wang and R. H. Yue, Alternative mechanism for black hole echoes, Phys. Rev. D 104 , no.4, 044012 (2021) doi:10.1103/PhysRevD.104.044012 [arXiv:2104.11912 [gr-qc]].</list_item>
<list_item><location><page_20><loc_12><loc_70><loc_88><loc_74></location>[51] G. D'Amico and N. Kaloper, Black hole echoes, Phys. Rev. D 102 , no.4, 044001 (2020) doi:10.1103/PhysRevD.102.044001 [arXiv:1912.05584 [gr-qc]].</list_item>
<list_item><location><page_20><loc_12><loc_65><loc_88><loc_69></location>[52] G. Guo, P. Wang, H. Wu and H. Yang, Echoes from hairy black holes, JHEP 06 , 073 (2022) doi:10.1007/JHEP06(2022)073 [arXiv:2204.00982 [gr-qc]].</list_item>
<list_item><location><page_20><loc_12><loc_54><loc_88><loc_63></location>[53] H. S. Liu, Z. F. Mai, Y. Z. Li and H. Lu, Quasi-topological Electromagnetism: Dark Energy, Dyonic Black Holes, Stable Photon Spheres and Hidden Electromagnetic Duality, Sci. China Phys. Mech. Astron. 63 , 240411 (2020) doi:10.1007/s11433-019-1446-1 [arXiv:1907.10876 [hepth]].</list_item>
<list_item><location><page_20><loc_12><loc_48><loc_88><loc_52></location>[54] H. Huang, M. Y. Ou, M. Y. Lai and H. Lu, Echoes from classical black holes, Phys. Rev. D 105 , no.10, 104049 (2022) doi:10.1103/PhysRevD.105.104049 [arXiv:2112.14780 [hep-th]].</list_item>
<list_item><location><page_20><loc_12><loc_43><loc_88><loc_47></location>[55] C. Gao, Black holes with many horizons in the theories of nonlinear electrodynamics, Phys. Rev. D 104 , no.6, 064038 (2021) doi:10.1103/PhysRevD.104.064038 [arXiv:2106.13486 [gr-qc]].</list_item>
<list_item><location><page_20><loc_12><loc_35><loc_88><loc_41></location>[56] A. Sang, M. Zhang, S. W. Wei and J. Jiang, Echoes of black holes in Einstein-nonlinear electrodynamic theories, Eur. Phys. J. C 83 , no.4, 291 (2023) doi:10.1140/epjc/s10052-02311448-4 [arXiv:2209.02559 [gr-qc]].</list_item>
<list_item><location><page_20><loc_12><loc_26><loc_88><loc_33></location>[57] S. Perlmutter et al. [Supernova Cosmology Project], Measurements of Ω and Λ from 42 high redshift supernovae, Astrophys. J. 517 , 565-586 (1999) doi:10.1086/307221 [arXiv:astroph/9812133 [astro-ph]].</list_item>
<list_item><location><page_20><loc_12><loc_18><loc_88><loc_25></location>[58] A. G. Riess et al. [Supernova Search Team], Observational evidence from supernovae for an accelerating universe and a cosmological constant, Astron. J. 116 , 1009-1038 (1998) doi:10.1086/300499 [arXiv:astro-ph/9805201 [astro-ph]].</list_item>
<list_item><location><page_20><loc_12><loc_13><loc_88><loc_17></location>[59] P. M. Garnavich et al. [Supernova Search Team], Supernova limits on the cosmic equation of state, Astrophys. J. 509 , 74-79 (1998) doi:10.1086/306495 [arXiv:astro-ph/9806396 [astro-ph]].</list_item>
<list_item><location><page_20><loc_12><loc_7><loc_88><loc_11></location>[60] V. V. Kiselev, Quintessence and black holes, Class. Quant. Grav. 20 , 1187-1198 (2003) doi:10.1088/0264-9381/20/6/310 [arXiv:gr-qc/0210040 [gr-qc]].</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_21><loc_12><loc_84><loc_88><loc_91></location>[61] S. Vagnozzi, C. Bambi and L. Visinelli, Concerns regarding the use of black hole shadows as standard rulers, Class. Quant. Grav. 37 , no.8, 087001 (2020) doi:10.1088/1361-6382/ab7965 [arXiv:2001.02986 [gr-qc]].</list_item>
<list_item><location><page_21><loc_12><loc_76><loc_88><loc_82></location>[62] J. L. Friedman, K. Schleich and D. M. Witt, Topological censorship, Phys. Rev. Lett. 71 , 14861489 (1993) [erratum: Phys. Rev. Lett. 75 , 1872 (1995)] doi:10.1103/PhysRevLett.71.1486 [arXiv:gr-qc/9305017 [gr-qc]].</list_item>
<list_item><location><page_21><loc_12><loc_67><loc_88><loc_74></location>[63] X. X. Zeng and H. Q. Zhang, Influence of quintessence dark energy on the shadow of black hole, Eur. Phys. J. C 80 , no.11, 1058 (2020) doi:10.1140/epjc/s10052-020-08656-7 [arXiv:2007.06333 [gr-qc]].</list_item>
<list_item><location><page_21><loc_12><loc_59><loc_88><loc_66></location>[64] A. He, J. Tao, Y. Xue and L. Zhang, Shadow and photon sphere of black hole in clouds of strings and quintessence *, Chin. Phys. C 46 , no.6, 065102 (2022) doi:10.1088/16741137/ac56cf [arXiv:2109.13807 [gr-qc]].</list_item>
</unordered_list>
</document>
[
{
"title": "Echoes from charged black holes influenced by quintessence",
"content": "Siyuan Hui a,b ∗ and Benrong Mu a,b † a Center for Joint Quantum Studies, College of Medical Technology, Chengdu University of Traditional Chinese Medicine, Chengdu, 611137, PR China b Center for Theoretical Physics, College of Physics, Sichuan University, Chengdu, 610064, PR China",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "In this paper, we investigate the effective potential and echoes from the dyonic black hole with quintessence. For a dyonic black hole, the quasi-topological electromagnetism provides proper matter energy-momentum tensor to curve the spacetime, and quintessence strengthens this force. We find that when the effect of quintessence becomes stronger, the black hole potential transforms between single-peak and double-peak, which will influence the existence of black hole echoes. In particular, we find that observer will receive a sudden vanishment of high-frequency echoes when quintessence remains a relatively strong effect.",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "LIGO and Virgo first successfully detected gravitational waves from a binary black hole merger[1]. Later, the Event Horizon Telescope photographed the first image of a supermassive black hole at the center of galaxy M87[2-7]. This discovery launches an extraordinary new area in black hole physics. The ringdown waveforms of gravitational waves from two massive objects merging include important information, which is worth studying. The ringdown waveforms are characterized by quasinormal modes (QNMs) of final black holes[8-10]. Specially, the ringdown waveforms will no longer depend on the initial disturbance after the initial wave burst of the perturbation. Therefore, the ringdown waveform measurement provide a great chance to reveal properties of the black hole geometry characterized by an event horizon[11, 12]. However, QNMs and ringdown signals are in fact related to the photon rings of the massive objects, which may be not necessarily black holes[10]. Thus, it is hard to distinguish black holes from many other horizonless massive objects just via QNMs and ringdown signals[1315]. In fact, many horizonless exotic compact objects (ECOs) can behave like black holes, such as boson stars, gravastars, wormholes and firewalls[16-28]. Specially, the late-time ringdown signals together with echoes from the binary black hole merger can be similarly constructed from various ECOs' theoretical models[29-36]. Meanwhile, data analysis suggests that the recent LIGO/Virgo observation may offer potential evidence for echoes in waveforms of gravitational wave from the binary black hole merger[37, 38]. To gain a deeper insight of echoes, an extra reflecting boundary was laid in a black hole spacetime. It showed that this boundary plays a core role in producing extra multiple time-delay ringdown signals, overlaying to form a series of complete echo waveform which can be received by a distant observer[14]. Several black hole and wormhole models have been studied and confirmed to have echoes, such as Kerr-like wormholes, hairy black holes, quantum black holes, braneworld black holes, and even a singularity[29, 30, 35, 36, 3952]. Specially, echoes have been found in Einstein -nonlinear electrodynamic spacetimes. One is the dyonic black hole in Einstein-Maxwell gravity extended with quasi-topological electromagnetism, which hold two photon spheres and double-peak effective potentials[53, 54]. Another is the multiple-horizon black hole model, which hold effective potentials with three or more peaks[55, 56]. Due to the theoretical and observational importance of echoes, it is significant to look for more black hole spacetimes that can hold multiple effective potentials to produce echo signals. Currently, there is no research about the connection between black hole echoes and quintessence. As recent astronomical observations display, the universe is now in an accelerating expansion[57-59], implying a state of negative pressure. Quintessence dark energy is one of the candidates to interpret the negative pressure, whose dynamic may affect the black hole[60, 61]. In this model, the state equation of quintessence is constrained by the pressure p = ρ q ω q , where ρ represents the energy density and ω q represents the state parameter satisfying -1 < ω q < -1 3 . Thus, it is natural for us to think about the effect of quintessence acting to black hole echoes. The remaining of this paper is organized as follows. In Section II, we consider a massless time-dependent scalar field perturbation and offer a numerical method for solving wave function. In Section III, we give out the metric of a dyonic black hole with quintessence and analyze its parameters. In Section IV, we detailly discuss the various transitions of black hole effective potential and echoes under effect of quintessence. In Section V, the complete conclusion is given.",
"pages": [
2,
3,
4
]
},
{
"title": "A. Massless scalar field perturbation and effective potential",
"content": "In this paper, we only consider the static and spherically symmetric black hole. In this section, we give out the most general metric form of the black hole and its effective potential under massless scalar perturbation. The general metric of the black hole takes the form[54] where we assume that h ( r + ) = f ( r + ) = 0 gives out the event horizon r + of the black hole. To study the echoes due to the black hole metric only, we consider the time-dependent massless scalar field perturbation, whose equation is For the static and spherically symmetric black hole metric, the time-dependent scalar field perturbation Ψ( t, r, θ, φ ) can be decomposed by separation of variables in terms of spherical harmonics Thus, the scalar field perturbation equation(1) reduces to with the original effective potential In order to map the radial region to ( -∞ , + ∞ ), the definition of the tortoise coordinate x is needed, Therefore, the equation(4) can be rewritten as with the time-dependent scalar field perturbation and effective potential in terms of x",
"pages": [
4,
5
]
},
{
"title": "B. Numerical method",
"content": "To solve the partial differential equation(7), we can consider a time-dependent Green's function G ( t, x, x ' ), which satisfies[8, 52] Thus, the solution of equation(7) can be rewritten in terms of G ( t, x, x ' ), With Fourier transform, the solution(10) can be expressed as where ˆ S ( ω, x ) is depend on the initial condition, Two linearly independent solutions ˆ ψ + ( ω, x ) and ˆ ψ -( ω, x ) are necessary to construct the time-independent Green's function G ( ω, x, x ' ), with the homogeneous differential equation and boundary conditions Therefore, the time-independent Green's function G ( ω, x, x ' ) is given by where the Wronskian W ( ω ) is defined as In this paper, we investigate different waveforms ψ ( t, x ) caused by different peak structure of effective potential in the black hole, which are all detected by a distant observer. To numerically solve the partial differential equation(7) from a certain initial wave packet, we consider the initial condition to be a Gaussian wave packet, where the parameter x 0 is the center of the initial Gaussian wave packet position, the width ∆ and the amplitude A are also chosen to adapt to the specific case. In this paper, we consider two inequivalent situations. One is that the initial Gaussian wave packet is located outside the double peaks. Another is that it is in the potential well between the two peaks. In [54], it has been found that the black hole echoes have the same characteristics whether it is close to the event horizon or far away from the outer peak. But when the initial location x 0 is closer to the bottom of the potential well, the echo frequency roughly doubles the one associated with the wave packet outside the peaks, which makes the echoes more distinguishable. In this paper, we also compare the variation of echo frequency with respect to quintessence parameter in these two cases, which will be discussed in Section IV.",
"pages": [
5,
6
]
},
{
"title": "III. QUASI-TOPOLOGICAL ELECTROMAGNETISM AND DYONIC BLACK HOLE WITH QUINTESSENCE",
"content": "In [53, 54], the quasi-topological electromagnetism is defined to be the squared norm of the topological 4-form F ∧ F . The Lagrangian is This theory admits the exact solution of the dyonic black hole, which is both static and spherically symmetric. The Einstein field equation and the Maxwell equation of motion can be expressed as[53] where the energy-momentum tensor T µν can be written as and ˜ T µν denotes the quintessence matter with[60] Thus, with the above functions, the metric of the black hole with quintessence can be written as where With the black hole metric, the original effective potential equation(5) reduces to Here, we can see that the effective potential V ( r ) contains several parameters. M , q and p are the mass, electric and magnetic charges of the black hole, α is the coupling constant, ω q and a are the state parameter and normalization factor of quintessence. When α = 0 and p = 0, the metric(23) reduces to the usual RN black hole metric. When q = 0, it reduces to the original Schwarzschild black hole metric. For a dyonic black hole with quintessence, set f ( r ) = 0 and generally we can get three solution. Two of them are black hole horizon, with the larger one becomes the event horizon r + . And the solution far away beyond the event horizon becomes the cosmological horizon r c . The region between the two horizons is called the domain of outer communication, since any two observers in this region may communicate with each other without being hindered by a horizon[62-64]. Under tortoise coordinates, region ( r + , r c ) turns into ( x + , x c ) = ( -∞ , + ∞ ), which also makes it suitable to consider the cosmological horizon as an observer at infinity. Moreover, the dyonic black hole has an unusual feature that the metric function f ( r ) is not monotonous in radius region. It has multiple wiggle such that the gravity force can vanish or even become repulsive in a certain finite region outside the horizon. It also leads to the black hole solution with four or five horizons for some suitable parameters. As this feature has been discussed detailly in [54], we expect that quintessence will give rise to various changes of the black hole horizons and effective potential, which will be analyzed in the following sections.",
"pages": [
6,
7,
8
]
},
{
"title": "IV. EFFECTIVE POTENTIAL AND BLACK HOLE ECHO",
"content": "In this section, we discuss different changes of the black hole due to various parameters. After fixing other parameters, we analyze how quintessence influence the number of local maximum and local minimum. It will reduces to the transition of effective potential between single-peak and double-peak, which will have different effect to the echoes of the black hole. Moreover, different values of spherical harmonics l also have different effects on the effective potential. In this paper, we focus on the situation l = 2 since they play a dominant role in the ringdown gravitational waves after binary black holes merge.",
"pages": [
8
]
},
{
"title": "A. Transition from single-peak to double-peak",
"content": "For an initial dyonic black hole without quintessence, the coupling constant α is not large enough, meaning that the quasi-topological electromagnetism cannot support strong matter energy-momentum tensor to curve the spacetime. Thus, the effective potential only displays a single-peak, which cannot provide a suitable constructure to generate echoes. The first imiage in Figure 2 shows that the reflection from the single-peak potential forms an observed burst after the initial wave traveling from the vicinity of the peak to the observer. At late times, the wave signal shows an exponentially damped sinusoid. Due to the absence of the outer peak, we cannot observe any echo after the burst is received. Note that waves propagating on a black hole spacetime usually develop asymptotically late-time tails, which follow exponentially damped sinusoids and decay as an inverse power of time due to scattering from large radius in the black hole geometry. After being influenced by quintessence, the effective potential of the black hole reduces continuously and obviously. Meanwhile, as the value of ω q becomes small, the outer peak appears and rapidly rises to a size similar to that of the inner peak. Because of the appearance of double-peak, the perturbation can be reflected off by the inner potential barriers and bounce back and forth between the two peaks. When the perturbation successively tunnels through the outer potential barrier, a series of echoes can be received by a distant observer. From Figure 2, we can see that under the effect of quintessence, the relative height of the outer and inner peaks is decreasing, together with the decreasing of the potential well. This means that there will be more wave signals tunneling through the barriers and superposing to form black hole echoes. Thus, the frequency of echoes will increase exponentially under a smaller ω q , which means a stronger effect from quintessence.",
"pages": [
8,
10
]
},
{
"title": "B. Transition of potential size between inner-peak and outer-peak",
"content": "In the subsection above, we have known that the inner potential peak and potential well will decrease under the effect of quintessence. Therefore, it is natural for us to consider how echoes change when the effective potential changes from an inner-large doublepeak to an outer-larger double-peak. When the quasi-topological electromagnetism supports enough matter energy-momentum tensor to curve the spacetime, a dyonic black hole without quintessence can form an effective potential with double-peak. In this case, the distance observer can also receive black hole echoes. And because the inner peak is bigger than the outer peak, the waves reflected by the inner peak can easily tunnel through the outer barrier. This leads to the distinguishable echoes with a high frequency. However, under the effect of quintessence, the inner peak turns to be lower than the outer one, which influences the behavior of echoes. It should be emphasized that although both the inner potential and potential well are decreasing, the former decreases faster than the latter. As shown in Figure 4, for one thing, as the inner peak decreases, the number of waves that can be reflected by it also decreases accordingly. For another, due to the larger distance between the two peaks, the scattering of a perturbation off one peak is lessly affected by the other peak. Therefore, the black hole echoes become distinguishable but with a low frequency. Meanwhile, waves that has been bounced back need to face a thicker potential barrier, which reduces the probability of tunneling through the barrier. Thus, compared to the initial burst waveform, the amplitude of the echo signal becomes very small, requiring more precise observation instruments.",
"pages": [
11,
12
]
},
{
"title": "C. Smooth and sudden transition from double-peak to single-peak",
"content": ") r ( V When the effect of quintessence becomes stronger, there will be two situations where the effective potential transforms from double-peak to single-peak. From Figure 5, one situation is that the black hole still maintains three horizons (event horizon r + , cosmology horizon r c and one horizon inside r + ). In this case, because the inner potential decreases faster than the potential well, we can see that the inner peak decreases and finially disappears. Thus, the potential transition is smooth, with the number and frequency of echoes also slowing down gradually from high-frequency and easily resolved echo signal to low-frequency, no-echo waveform signal, which is shown in Figure 6. From Figure 7, another situation is when the inner potential is large enough, the black hole will have four or five horizons ( r + , r c , and two or three horizons inside r + ). In this case, the inner peak will be wrapped by event horizon r + before it disappears. Therefore, as shown in Figure 8, the effective potential will experience a sudden transition from doublepeak to single-peak. And the distance observer will also receive a sudden change of the black hole echoes. This change is sudden and rapid. At this moment, distinguishable and high-frequency echoes exist. In the next moment, echoes suddenly disappear, leaving only a single-peak potential.",
"pages": [
12,
13,
14
]
},
{
"title": "V. DISCUSSION AND CONCLUTION",
"content": "In this paper, we first reviewed the dyonic black hole in Einstein-Maxwell gravity extended with quasi-topological electromagnetism which is defined to be the squared norm of the topological 4-form F ∧ F [53, 54]. Then we studied various effects of quintessence acting to the black hole effective potential and echoes. Several black hole and wormhole models with double effective potentials have been studied[29, 30, 35, 36, 39-52]. And our motivation was to look for whether quintessence can influence the spacetime structure of black holes. And our results shown that effective potential indeed varies between single peak and double peaks in a dyonic black hole with quintessence. We also studied echoes with quintessence from different values and distance of inner and outer potential peaks, including wormhole-like potential. For suitable parameters, the effect of quintessence acting on the dyonic black hole will cause potential transition between single-peak and double-peak, which can be approximately divided into two cases. One is that an outer peak appears, the original peak becomes the inner one, which decreases and finally generally disappears together with potential well. In this case, the distance observer will receive smooth transition of waveform between exponentially decaying sinusoid and high frequency echoes. Another is that the black hole will exist multiple horizons to wrap the inner peak, which leads to the sudden disappearance of the inner peak. In this case, it will be strange to observe a distinguishable and high-frequency echoes vanish, leaving only an exponentially decaying sinusoid waveform. It should be emphasized that quintessence only cannot lead to the multiple potential peaks and echoes in Schwarzschild or RN black holes. In fact, the dyonic black hole itself already has double-peak potential under certain conditions. This is caused by Einstein-Maxwell gravity extended with quasi-topological electromagnetism. Under a large coupling constant α , the quasi-topological electromagnetism provides proper matter energy-momentum tensor to curve the spacetime[53, 54]. And quintessence plays no particular role other than strengthen this force. Thus, it will be interesting if our analysis of quintessence can be extended beyond spherical symmetry to more general black hole spacetimes.",
"pages": [
14,
15
]
},
{
"title": "Acknowledgement",
"content": "We are grateful to Haitang Yang, Jun Tao and Yuhang Lu for useful discussions. This work is supported in part by NSFC (Grant No. 11747171), Xinglin Scholars Project of Chengdu University of Traditional Chinese Medicine (Grant no.QNXZ2018050). doi:10.1103/PhysRevLett.116.061102 [arXiv:1602.03837 [gr-qc]]. tance of Overtones, Phys. Rev. X 9 , no.4, 041060 (2019) doi:10.1103/PhysRevX.9.041060 [arXiv:1903.08284 [gr-qc]]. doi:10.1103/PhysRevD.102.084044 [arXiv:2008.10130 [gr-qc]]. [arXiv:2001.11642 [gr-qc]].",
"pages": [
15,
16,
17,
18,
20
]
}
]
2024PDU....4301404T
https://arxiv.org/pdf/2308.10786.pdf
<document>
<section_header_level_1><location><page_1><loc_7><loc_82><loc_85><loc_84></location>Testing of κ ( R , T ) -gravity through gravastar configurations</section_header_level_1>
<text><location><page_1><loc_7><loc_77><loc_68><loc_80></location>G R P Teruel a,1 , Ksh. Newton Singh b,2 , Tanmoy Chowdhury c,3 , Farook Rahaman d,3 , Monimala Mondal e,3</text>
<unordered_list>
<list_item><location><page_1><loc_7><loc_75><loc_57><loc_76></location>1 Departamento de Matematicas, IES Carrus, Elche 03205, Alicante, Spain.</list_item>
<list_item><location><page_1><loc_7><loc_74><loc_65><loc_75></location>2 Department of Physics, National Defence Academy, Khadakwasla, Pune-411023, India.</list_item>
<list_item><location><page_1><loc_7><loc_73><loc_65><loc_74></location>3 Department of Mathematics, Jadavpur University, Kolkata-700032, West Bengal, India</list_item>
</unordered_list>
<text><location><page_1><loc_7><loc_66><loc_28><loc_67></location>Received: date / Accepted: date</text>
<text><location><page_1><loc_7><loc_36><loc_47><loc_63></location>Abstract In this article, we are reporting for the first time the existence of gravastar configurations in the framework of κ ( R , T ) -gravity, which can be treated as an alternative to a black hole (Mazur and Mottola). This strengthens how much this new gravity theory may be physically demanding to the gravity community in the near future. We first develop the gravastar field equations for a generic κ ( R , T ) functional and then we study four different models within this theory. We find that the solutions for the interior region are regular everywhere regardless of the exact form of the κ ( R , T ) functional. The solutions for the shell region indicate that two of the four models subjected to the study are physically feasible. In addition, the junction conditions are considered at each interface by using the Lanczos equations that yield the surface density and pressure at the thin shell. We investigate various characteristics of the gravastar structure such as the proper length, energy, and entropy of the spherical distribution.</text>
<text><location><page_1><loc_7><loc_32><loc_46><loc_35></location>Keywords Gravastar · κ ( R , T ) -gravity · Junction conditions</text>
<section_header_level_1><location><page_1><loc_7><loc_28><loc_19><loc_29></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_7><loc_18><loc_47><loc_26></location>The hierarchy of principles in physics is an old debate that should be revived from time to time to ask ourselves if we are going in the right direction. On the one hand, there are foundational physical principles, such as the principle of inertia in classical mechanics, the uncertainty principle in quantum mechanics, or the principle</text>
<text><location><page_1><loc_50><loc_37><loc_90><loc_63></location>of relativity that directly founded or contributed to establishing new scientific theories. On the other hand, there are the auxiliary principles, which are not as fundamental as the first ones, but help to strengthen our theories once they have been established by the foundational principles. It is natural to wonder: To which of these two categories does the principle of least action belong? At first glance, one would answer that the least action principle should be an undisputed part of the first group, i.e., a foundational one due to its ubiquitous presence in our current theories. However, let's carefully examine the history of physics. We will appreciate that the variational principles, despite their enormous success and importance, were not founding principles, in the sense that they did not establish new scientific paradigms, although they turned out to be versatile and flexible enough to know how to adapt to every paradigm shift in theoretical physics.</text>
<text><location><page_1><loc_50><loc_10><loc_90><loc_35></location>In particular, if we examine the birth of the two major revolutions of the twentieth century, the theories of relativity and quantum mechanics, we will see that the least action principle had a secondary role [1,2]. In fact, other principles were the foundational building blocks of these theories [3,4]. The action principle was adapted later to them and arrived to enrich their formalism. Analogously, neither of the two great classical field theories was discovered by means of the least action principle. Indeed, Classical Electrodynamics (CE) was fulfilled with the inclusion of a source term directly into the field equations (the Maxwell displacement current). General relativity (GR) was also first derived in this way, after adding the key trace term to the field equations [5,6]. In both cases, the least action principle came to strengthen the formalism after other more fundamental principles had established the theories.</text>
<text><location><page_2><loc_7><loc_84><loc_47><loc_89></location>These historical lessons persuade us that the search for possible Non-Lagrangian theories, that is, theories not based (at least initially) on the variational method, should not be ruled out a priori.</text>
<text><location><page_2><loc_7><loc_64><loc_47><loc_83></location>In 2001, Mazur and Mottola [7,8] proposed a very interesting and encouraging stellar model, named gravitational vacuum star (or gravastar) as an alternating black hole structure. In recent days, the study on gravastar has become a considerably interesting topic as an alternating black hole solution. Mazur and Mottola [7,8] first introduced a novel solution for the extreme point of gravitational collapse of cold and dark matter, a neutral system, and compact stars. Therefore, this is an extended form of Bose-Einstein condensation whose radius is similar to the Schwarzschild radius. The total design of a gravastar can be defined by means of 3 different zones with distinct EoS:</text>
<formula><location><page_2><loc_10><loc_61><loc_35><loc_62></location>I. Interior: 0 < r < r 1 ( p = -ρ ),</formula>
<formula><location><page_2><loc_10><loc_58><loc_32><loc_59></location>II. Shell: r 1 < r < r 2 ( p = ρ ),</formula>
<formula><location><page_2><loc_10><loc_55><loc_35><loc_56></location>III. Exterior: r 2 < r ( p = ρ = 0).</formula>
<text><location><page_2><loc_7><loc_43><loc_47><loc_53></location>Several researchers have already studied gravastar from different perspectives and scooped out a new field as an alternative to the black hole paradigm. [7,8,9,10, 11,12,13,14,15,16,17]. Lack of observational evidence in Einstein's GR on the oscillating model of the universe alongside the presence of dark matter has forced a conceptual challenge to this theory [18,19,20,21].</text>
<text><location><page_2><loc_7><loc_28><loc_47><loc_43></location>The repulsive nature of dark energy in GR requires considering an exotic fluid with a negative pressure to explain the cosmic speed-up. Many modified Lagrangian theories of gravity have been proposed to account for the accelerating phase of the universe. These theories are based on the generalization of the EinsteinHilbert action viz f ( T , T ) [22], f ( T ) [23], f ( R , T ) [24], EGB [25], f ( G ) [26], f ( G, T ) [27], f ( Q, T ) [28] gravity theories, etc, which earned wide attention in recent years.</text>
<text><location><page_2><loc_7><loc_10><loc_47><loc_28></location>A major role of modified gravity theories is to test the validity of the gravastar model by examining various realistic properties of these compact objects. Barzegar et al. [29] explored the 3D AdS gravitational vacuum stars in the framework of gravity's rainbow theory. Ghosh et al. [30] investigated gravastars in f ( T , T ) gravity and enlarged its research in Rastall gravity as well [31]. Das et al. [32], investigated the formation of gravastars in f ( R , T ) gravity and inspect its theoretical efficiency. Enlarge the analysis Yousaf et al., recently explored gravastars in f ( R , T , R µν T µν ) gravity [33]. Bhatti et al. [34] reviewed charged gravastar for</text>
<text><location><page_2><loc_50><loc_69><loc_91><loc_89></location>spherically and cylindrically symmetric spacetimes and in f ( R , G ) gravity [35]. Extending it, Yousaf [36] constructed charged cylindrical gravastar-like structures in f ( R , T ) gravity. Lobo and Garattini [37] analyzed gravastar solutions in the frame of non-invariant geometry with their physical properties and characteristics. Bhatti et al. [38] investigated the stability of this thin shell together with the thermodynamical stability of locally isotropic gravastars with cylindrical space-time. Bhatti et al. [39] talked over the symmetric gravastar model within the framework of the modification of GaussBonnet gravity. Das et al. [40] discussed the physically interesting and valid features within the framework of the f ( T ) theory of gravity.</text>
<text><location><page_2><loc_50><loc_28><loc_90><loc_68></location>Horvet et al. [41] used dominant force conditions to initialize the stability of the gravastar after combining a sufficient external vacuum solution with an internal geometry and meeting some possible constraints. Also, they examined the formation of this type of compact object under the influence of electromagnetism. Rahaman and others [42] analyzed the electrically charged gravastar model considering the (2 + 1) -dimensional geometry. For a stable model of a spherically symmetric gravastar, they also explored some viable properties as well as energy components, length, and entropy of the gravastar. Ghosh et al. [43] investigated some new diagrams in the formation of these matters with and without electric charge in high-order manifolds. Ghosh et al. [44] discussed gravastar layout in the framework of the Kuchowicz metric potential. Ghosh et al. [45] figure out the gravastar model under Einstein's GR in (3+1) dimensions by incorporating the Karmarkar condition. Sumita et al. [46] studied gravastars under Finslerian spacetime geometry. Sengupta et al. [47] investigated a gravastar configuration in RS Brane gravity, while Ray et al. [48] reviewed a very interesting paper on generalizations of gravastars in GR, including studies of higher and lower dimensional GR with physical insights like the Randall-Sundrum theory, and several modified gravity models such as f ( R , T ) theory or Rastall-Rainbow gravity.</text>
<text><location><page_2><loc_50><loc_10><loc_90><loc_28></location>Recently, a modified theory was invented [50] as κ ( R , T ) -gravity. The conceptual framework of this theory is not based on the standard modified gravity program. Rather, its insight is inspired by Maxwell's and Einstein's ideas of including new possible potential terms in the field equations. In this sense, we must recall Maxwell's addition to Ampere's law of the displacement current term to complete the Electromagnetic field equations, and the introduction of the key term, 1 2 R g µν by Einstein to consummate the GR field equations [5,6]. Despite the fact that the variational principle is often regarded as our main tool to formulate</text>
<text><location><page_3><loc_7><loc_54><loc_47><loc_89></location>a new physical theory (and its generalizations), ought not to have been placed at the same fundamental level as other truly first principles, like the equivalence principle and the principle of general covariance, that are the two core principles of GR. In spite of this, the variational method has reached the pinnacle in our hierarchy of physical principles, becoming dominant to the extent of being the starting point for any modification of GR. Indeed, the vast majority of the modified gravity theories that are available in the market, like the examples mentioned above, are Lagrangian in the sense that they arise from the generalization of the Einstein-Hilbert action. The overpopulation of alternatives suggests that some other fundamental principle (beyond the equivalence principle and the principle of general covariance) is required to guide us trough the deep jungle of possible Lagrangian theories. The fact is, we have no reason to assure that general symmetries and general conservation laws can always exist (in their current form) in the final theories of nature, which moves us to think that the Non-Lagrangian proposals should also be explored. In fact, according to Hojman [64], the use of Lagrangian and Hamiltonian functions is not an essential requirement to derive conservation theorems.</text>
<text><location><page_3><loc_7><loc_21><loc_47><loc_53></location>κ ( R , T ) -gravity is a recent proposal, and so its implications have not been explored well enough. However, for the past few years, some works are dedicated to studying its cosmic aspects. In particular, Pradhan and Ahmed [51] showed that the theory can deal with the current scenario of an accelerating universe, claiming that the theory requires a very small value of the cosmological constant, which agrees with observations. Pradhan et al. [52] further extended this study to include a more complete cosmological scenario, while Dixit et al. [53] studied the thermodynamics of the expansion of the cosmos in the context of this theory. Sarkar et al. [54], explored a model of wormhole in κ ( R , T ) -gravity, while Teruel et al. [55] introduced the first internal solutions representing compact stars in κ ( R , T )-gravity by solving the field equations in isotropic coordinates. Also very recently, Tas . er and Dogru [56] investigated whether the Krori-Barua model produces valid results in this theory. All these papers have one remarkable common feature: they were devoted to studying only a particular case of the theory with κ ( R , T ) ≡ κ ( T ) = 8 π -λ T .</text>
<text><location><page_3><loc_7><loc_10><loc_47><loc_20></location>Up to now, nobody studied gravastar solutions in κ ( R , T ) -gravity, it would therefore be interesting to investigate and discuss whether this theory can support a consistent gravastar configuration from a physical point of view. In this sense, four different models, i.e., four different choices of the running κ ( R , T ) -gravitational constant are analyzed in this investiga-</text>
<text><location><page_3><loc_50><loc_82><loc_90><loc_89></location>on. Therefore, we don't restrict ourselves only to the κ ( T ) choice like all the previous works, but extend the study to the more general possible case. Indeed, once derived the equations for a generic κ ( R , T ) functional, we see how the theory would look for specific cases.</text>
<text><location><page_3><loc_50><loc_61><loc_90><loc_82></location>The structure of the paper is as follows. We start in section 2 with the basic framework of the theory. Section 3 deals with the field equations derived from the Morris-Thorne metric for the general κ ( R , T ) case. The solutions for the interior region of the gravastar are discussed in section 4. The shell region of the gravastar is analyzed in section 5, where we study the physical acceptability of 4 different models within the theory. Entropy within the shell is addressed in section 6. The junction interface and surface stresses are the subjects of section 7. The brief section 8 deals with the surface redshift within the thin shell. Finally, in section 9, we summarize the findings and discuss the conclusions of the work.</text>
<section_header_level_1><location><page_3><loc_50><loc_57><loc_78><loc_58></location>2 Framework of κ ( R , T ) -gravity</section_header_level_1>
<text><location><page_3><loc_50><loc_52><loc_90><loc_55></location>The structure of the theory depends on the following field equations</text>
<formula><location><page_3><loc_50><loc_48><loc_90><loc_51></location>R µν -1 2 R g µν -Λg µν = κ ( R , T ) T µν , (1)</formula>
<text><location><page_3><loc_50><loc_28><loc_90><loc_47></location>where R µν is the Ricci tensor, g µν is the space-time metric, Λ corresponds to the cosmological constant, T µν is the stress-energy tensor of the material content, while κ ( R , T ) is a generalization of Einstein's gravitational constant, that we extend to the status of a function of the traces T ≡ g µν T µν , and R≡ g µν R µν . The introduction of the functional κ ( R , T ), means that we inspect the possibility of a running gravitational constant, although not at the level of the variational method. The possibility of considering a variable Einstein's gravitational constant in the action was studied many years ago by Brans and Dicke [57,58], and the resulting theory is quite different from (1).</text>
<text><location><page_3><loc_50><loc_22><loc_90><loc_27></location>The field equations (1) imply the non-covariant conservation of T µν . Indeed, since the divergence of the left-hand side of Einstein's field equation vanishes, we have</text>
<formula><location><page_3><loc_50><loc_19><loc_90><loc_21></location>∇ ν [ κ ( R , T ) T µν ] = 0 . (2)</formula>
<text><location><page_3><loc_50><loc_14><loc_90><loc_17></location>Then, this non-conservation of the T µν can be expressed, for κ ( R , T ) = 0, as</text>
<text><location><page_3><loc_83><loc_11><loc_83><loc_12></location≯</text>
<text><location><page_3><loc_59><loc_14><loc_59><loc_16></location≯</text>
<formula><location><page_3><loc_50><loc_10><loc_90><loc_13></location>∇ ν T µν = -∇ ν κ ( R , T ) κ ( R , T ) T µν , ∀ κ ( R , T ) = 0 . (3)</formula>
<text><location><page_4><loc_7><loc_78><loc_47><loc_89></location>When κ ( R , T ) = 0, the right-hand side of Einstein's field equations vanishes. This interesting case probably may happen in some specific models in the early universe for sufficiently high densities, and it will imply an exponential expansion driven by a cosmological constant. Of course, the theory can be recast into a conservative form by defining a new (effective) stress-energy tensor</text>
<formula><location><page_4><loc_7><loc_75><loc_47><loc_76></location>S µν = κ ( R , T ) T µν . (4)</formula>
<text><location><page_4><loc_7><loc_72><loc_35><loc_74></location>The field equations then take the form</text>
<formula><location><page_4><loc_7><loc_69><loc_47><loc_71></location>R µν -1 2 R g µν -Λg µν = S µν , (5)</formula>
<text><location><page_4><loc_7><loc_67><loc_42><loc_68></location>and the Bianchi identities imply that Eq. (2) is</text>
<formula><location><page_4><loc_7><loc_64><loc_47><loc_65></location>∇ ν S µν = 0 . (6)</formula>
<text><location><page_4><loc_7><loc_55><loc_48><loc_62></location>Then, this theory has the same formal structure as GR, with a non-trivial modification of the material content. Examples of famous non-conservative gravitational theories are Rastall's gravity [59], or the Lagrangian f ( R , T ) theory by Harko et al. [60].</text>
<text><location><page_4><loc_7><loc_26><loc_48><loc_55></location>Lindblom and Hisock [61], and also Visser [62] more recently, criticized some non-conservative gravity theories (in particular the Rastall case), claiming that they are formally identical to Einstein's gravity and that one can always build a conserved effective stress-energy tensor T eff µν , which is constructed solely from the matter sources. Furthermore, they underline the fact that, if the effective stress-energy tensor of the matter sources does not depend on the space-time curvature, (such is the case of Rastall gravity), then these theories will not represent truly alternative theories of gravity, reducing the question of the non-conservation of the stressenergy tensor to the domain of special relativity. Nevertheless, since the effective stress-energy tensor of κ ( R , T ) gravity is not in general determined by the matter sources solely (it can have a dependence on the space-time curvature as well via the trace R ), we conclude that κ ( R , T ) theory represents a true generalization of GR and not only a redefinition of its matter sector. Some relevant features of the theory are the following:</text>
<unordered_list>
<list_item><location><page_4><loc_8><loc_15><loc_47><loc_25></location>-The modification procedure involves the right-hand side of Einstein's field equations, that is, only the material content sector is generalized. Therefore, the equations will be second order in the metric coefficients, and the theory will be free of typical instabilities that plagued many of the higher-order gravitational theories.</list_item>
<list_item><location><page_4><loc_8><loc_10><loc_47><loc_14></location>-Since the pure geometrical sector is the same as GR, the theory boils down to GR in the absence of matter sources.</list_item>
<list_item><location><page_4><loc_51><loc_78><loc_90><loc_89></location>-The dependence on T means that, for particular cases of the type κ ( T ) = 8 π -λ T , or for more general functional 1 , that directly couples the R and T traces, the theory would predict the same physics as GR when coupled to standard (traceless) electromagnetic fields. In such cases, only significant departures are expected for non-linear electrodynamics, where T ̸ = 0.</list_item>
</unordered_list>
<section_header_level_1><location><page_4><loc_50><loc_73><loc_82><loc_74></location>3 Field Equations in κ ( R , T ) -gravity</section_header_level_1>
<text><location><page_4><loc_50><loc_69><loc_90><loc_72></location>The interior of the gravastar is taken to be as MorrisThorne Spacetime given by</text>
<formula><location><page_4><loc_50><loc_66><loc_90><loc_69></location>ds 2 = -e 2 f dt 2 + [ 1 -b r ] -1 dr 2 + r 2 ( dθ 2 +sin 2 θ dϕ 2 ) . (7)</formula>
<text><location><page_4><loc_50><loc_63><loc_90><loc_65></location>For Λ = 0, the Einstein field equations for a general κ ( R , T ) model will be</text>
<formula><location><page_4><loc_50><loc_59><loc_90><loc_62></location>ρ ( r ) κ ( R , T ) = b ' ( r ) r 2 , (8)</formula>
<formula><location><page_4><loc_50><loc_56><loc_90><loc_59></location>p ( r ) κ ( R , T ) = 2 f ' ( r ) r ( 1 -b ( r ) r ) -b ( r ) r 3 . (9)</formula>
<text><location><page_4><loc_50><loc_51><loc_90><loc_55></location>On the other hand, the non-covariant conservation of the stress-energy tensor given by Eq. (2) acquires the form</text>
<formula><location><page_4><loc_50><loc_46><loc_90><loc_50></location>p ( r ) dκ ( R , T ) dr + κ ( R , T ) ( df ( r ) dr { ρ ( r ) + p ( r ) } + dp ( r ) dr ) = 0 . (10)</formula>
<section_header_level_1><location><page_4><loc_50><loc_40><loc_85><loc_43></location>4 Regular Interior Solutions in κ ( R , T ) -gravity with p = -ρ</section_header_level_1>
<text><location><page_4><loc_50><loc_30><loc_91><loc_38></location>In this section, we show that the interior solutions for a gravastar configuration are regular everywhere in κ ( R , T ) theory, regardless of the specific choice of the running gravitational constant. To see how this interesting property of the theory arises, notice that for p ( r ) = -ρ ( r ), Eq. (10) reduces to</text>
<formula><location><page_4><loc_50><loc_26><loc_90><loc_29></location>ρ ( r ) dκ ( R , T ) dr + κ ( R , T ) dρ ( r ) dr = 0 , (11)</formula>
<text><location><page_4><loc_50><loc_24><loc_71><loc_25></location>which further takes the form</text>
<formula><location><page_4><loc_50><loc_20><loc_90><loc_23></location>d dr [ κ ( R , T ) ρ ( r ) ] = 0 . (12)</formula>
<text><location><page_4><loc_50><loc_18><loc_68><loc_19></location>This implies the relation</text>
<formula><location><page_4><loc_50><loc_15><loc_90><loc_16></location>κ ( R , T ) ρ ( r ) = C, (13)</formula>
<text><location><page_5><loc_7><loc_86><loc_47><loc_89></location>where C , the constant of integration. Therefore, field equations reduce to</text>
<formula><location><page_5><loc_8><loc_83><loc_47><loc_86></location>C = b ' ( r ) r 2 , (14)</formula>
<formula><location><page_5><loc_7><loc_80><loc_47><loc_83></location>-C = 2 f ' ( r ) r ( 1 -b ( r ) r ) -b ( r ) r 3 , (15)</formula>
<text><location><page_5><loc_7><loc_78><loc_38><loc_79></location>from which we can obtain b ( r ) and f ( r ) as</text>
<formula><location><page_5><loc_7><loc_75><loc_47><loc_78></location>b ( r ) = C 3 r 3 + D, (16)</formula>
<formula><location><page_5><loc_7><loc_72><loc_47><loc_75></location>f ( r ) = 1 2 ln ( 1 -C 3 r 2 -D r ) . (17)</formula>
<text><location><page_5><loc_7><loc_60><loc_47><loc_72></location>Regularity at r = 0 implies that we set D = 0. At r = 0, the core of the gravastar, we have that f ( r ) = 0 and the metric functions become g tt = -1, and g rr = 1. The density should also be constant and finite at the gravastar's interior. At first we observed that the curvature scalar is constant and non-divergent everywhere at the interior region. After a cumbersome calculation, we have arrived at</text>
<formula><location><page_5><loc_8><loc_54><loc_47><loc_60></location>R = -2 ( 1 -b ( r ) r )[ f '' ( r ) + 2 f ' ( r ) r + f ' ( r ) 2 ] + 2 b ' ( r ) r 2 + d dr ( b ( r ) r ) f ' ( r ) = 4 C. (18)</formula>
<text><location><page_5><loc_7><loc_48><loc_47><loc_54></location>Therefore, an important conclusion of our results is that the interior solution for the gravastar will be regular and well-behaved everywhere, regardless of the specific functional forms of κ ( R , T ).</text>
<formula><location><page_5><loc_7><loc_44><loc_33><loc_45></location>4.1 Model κ ( R , T ) = 8 π + β Rα T</formula>
<text><location><page_5><loc_7><loc_37><loc_47><loc_42></location>Let us consider the choice, κ ( R , T ) = 8 π + β Rα T , where the pre-multiplier are the free parameters. For this particular case we have, using that ρ ( r ) = -p ( r ) and the results above, that Eq. (13) has the form</text>
<formula><location><page_5><loc_7><loc_34><loc_47><loc_36></location>[ 8 π +4 βC -4 αρ ( r ) ] ρ ( r ) = C. (19)</formula>
<text><location><page_5><loc_39><loc_29><loc_39><loc_30></location≯</text>
<text><location><page_5><loc_7><loc_27><loc_47><loc_33></location>This algebraic equation implies, as we mentioned before, that the density (and pressure) for the interior layer of the gravastar is uniform, and for α = 0, it has the following possible values</text>
<formula><location><page_5><loc_7><loc_24><loc_47><loc_27></location>ρ = 1 2 α ( ω ± √ ω 2 -αC ) = ρ 1 (20)</formula>
<formula><location><page_5><loc_7><loc_21><loc_47><loc_24></location>p = -1 2 α ( ω ± √ ω 2 -αC ) = p 1 , (21)</formula>
<text><location><page_5><loc_7><loc_18><loc_47><loc_21></location>where ω is a constant given by ω = 2 π + βC and ω ≥ √ αC .</text>
<text><location><page_5><loc_7><loc_15><loc_47><loc_18></location>The gravitational mass at the interior can is found from</text>
<formula><location><page_5><loc_7><loc_12><loc_47><loc_15></location>M ( C 1 ) = ∫ R 1 = C 1 0 4 πr 2 ρ dr = 4 3 πC 3 1 ρ 1 . (22)</formula>
<formula><location><page_5><loc_50><loc_88><loc_77><loc_89></location>4.2 Model κ ( R , T ) ≡ κ ( T ) = 8 π -α T</formula>
<text><location><page_5><loc_50><loc_81><loc_90><loc_86></location>The density and pressure for this κ ( R , T ) choice can be computed by setting β = 0 in the previous case. Taking into account that for this choice, the constant ω = 2 π + βC reduces to ω = 2 π , we obtain</text>
<formula><location><page_5><loc_50><loc_77><loc_90><loc_80></location>ρ = 1 2 α ( 2 π ± √ 4 π 2 -αC ) = ρ 2 (23)</formula>
<formula><location><page_5><loc_50><loc_75><loc_90><loc_77></location>p = -1 2 α ( 2 π ± √ 4 π 2 -αC ) = p 2 . (24)</formula>
<text><location><page_5><loc_50><loc_72><loc_90><loc_74></location>Thus, the active gravitational follows from the direct computation</text>
<formula><location><page_5><loc_50><loc_68><loc_90><loc_71></location>M ( C 1 ) = ∫ R 1 = C 1 0 4 πr 2 ρdr = 4 3 πC 3 1 ρ 2 . (25)</formula>
<formula><location><page_5><loc_50><loc_62><loc_78><loc_64></location>4.3 Model κ ( R , T ) ≡ κ ( R ) = 8 π + β R</formula>
<text><location><page_5><loc_50><loc_57><loc_90><loc_61></location>The density and pressure for this specific choice can be computed by setting α = 0 in the case A. Then, Eq. (19) reduces to</text>
<formula><location><page_5><loc_50><loc_54><loc_90><loc_56></location>( 8 π +4 βC ) ρ = C. (26)</formula>
<text><location><page_5><loc_50><loc_52><loc_57><loc_53></location>Therefore,</text>
<formula><location><page_5><loc_50><loc_48><loc_90><loc_51></location>ρ = 1 4( 2 π C + β ) = ρ 3 (27)</formula>
<formula><location><page_5><loc_50><loc_45><loc_90><loc_48></location>p = -1 4( 2 π C + β ) = p 3 . (28)</formula>
<text><location><page_5><loc_50><loc_40><loc_90><loc_44></location>The mass is obtained in the same fashion as models (4.1) and (4.2), i.e, by direct integration of the density in the range r = 0 and r = R 1</text>
<formula><location><page_5><loc_50><loc_36><loc_90><loc_39></location>M ( C 1 ) = ∫ R 1 = C 1 0 4 πr 2 ρdr = 4 3 πC 3 1 ρ 3 . (29)</formula>
<formula><location><page_5><loc_50><loc_33><loc_73><loc_34></location>4.4 Model κ ( R , T ) = 8 π -γ RT</formula>
<text><location><page_5><loc_50><loc_27><loc_90><loc_31></location>For this direct coupling among matter and curvature terms, we have that the density, pressure and active gravitational mass are given by</text>
<formula><location><page_5><loc_50><loc_23><loc_90><loc_26></location>ρ = π 4 Cγ ( 1 ± √ 1 -C 2 γ π 2 ) = ρ 4 (30)</formula>
<formula><location><page_5><loc_50><loc_20><loc_90><loc_23></location>p = -π 4 Cγ ( 1 ± √ 1 -C 2 γ π 2 ) = p 4 , (31)</formula>
<formula><location><page_5><loc_50><loc_15><loc_90><loc_18></location>M ( C 1 ) = ∫ R 1 = C 1 0 4 πr 2 ρdr = 4 3 πC 3 1 ρ 4 . (32)</formula>
<text><location><page_5><loc_50><loc_10><loc_90><loc_14></location>where, γ is a parameter of the model that should satisfy the constraint γ ≤ ( π/C ) 2 in order to get positive density in the core.</text>
<section_header_level_1><location><page_6><loc_7><loc_88><loc_24><loc_89></location>5 Shell Region p = ρ</section_header_level_1>
<text><location><page_6><loc_7><loc_78><loc_47><loc_86></location>This region of the gravastar supposedly contains a stiff fluid that obeys EoS p = ρ or v = √ dp/dρ = 1, which is the most stiff EoS known as Zeldovich fluid . This condition implies that the non-covariant conservation of the stress-energy tensor given by Eq. (10) acquires the form</text>
<formula><location><page_6><loc_7><loc_76><loc_47><loc_77></location>ρ ( r ) κ ' ( R , T ) + κ ( R , T ) [ 2 f ' ( r ) ρ ( r ) + ρ ' ( r ) ] = 0 . (33)</formula>
<text><location><page_6><loc_7><loc_73><loc_46><loc_74></location>On the other hand, Einstein's field equations become</text>
<formula><location><page_6><loc_7><loc_70><loc_47><loc_73></location>ρ ( r ) κ ( R , T ) = b ' ( r ) r 2 , (34)</formula>
<formula><location><page_6><loc_7><loc_67><loc_47><loc_70></location>ρ ( r ) κ ( R , T ) = 2 f ' ( r ) r ( 1 -b ( r ) r ) -b ( r ) r 3 . (35)</formula>
<text><location><page_6><loc_7><loc_58><loc_47><loc_66></location>Since, in general, the running gravitational constant κ ( R , T ) depends on the scalars R , T , which in turn are functions of b ( r ), f ( r ), ρ ( r ). To solve the unknown functions b ( r ), f ( r ) and ρ ( r ) we need three equations, two from field equations and one conservation equation. From the last two equations we obtain</text>
<formula><location><page_6><loc_7><loc_54><loc_47><loc_57></location>b ' ( r ) r 2 = 2 f ' ( r ) r ( 1 -b ( r ) r ) -b ( r ) r 3 . (36)</formula>
<text><location><page_6><loc_7><loc_49><loc_41><loc_51></location>5.1 The Shell Region Differential Equations in κ ( R , T ) -gravity</text>
<text><location><page_6><loc_46><loc_43><loc_46><loc_44></location≯</text>
<text><location><page_6><loc_7><loc_42><loc_47><loc_47></location>To find the general equations for the shell region in κ ( R , T ) gravity we can proceed in the following way: using (34), we eliminate κ ( R , T ), i.e. we have for ρ ( r ) = 0, can always write</text>
<formula><location><page_6><loc_7><loc_38><loc_47><loc_41></location>κ ( R , T ) = 1 ρ ( r ) b ' ( r ) r 2 . (37)</formula>
<text><location><page_6><loc_7><loc_36><loc_45><loc_37></location>Substituting this into Eq. (33), we obtain the result</text>
<formula><location><page_6><loc_7><loc_33><loc_47><loc_36></location>d dr [ b ' r 2 ] -b ' r 2 ρ ' ρ + b ' r 2 · 2 f ' + b ' r 2 ρ ' ρ ( r ) = 0 (38)</formula>
<formula><location><page_6><loc_14><loc_29><loc_47><loc_32></location>Or d dr ( b ' r 2 ) + b ' r 2 · 2 f ' ( r ) = 0 . (39)</formula>
<text><location><page_6><loc_7><loc_28><loc_47><loc_29></location>This differential equation can be integrated to provide</text>
<formula><location><page_6><loc_7><loc_24><loc_47><loc_27></location>f ( r ) = -1 2 ln ( b ' ( r ) r 2 ) . (40)</formula>
<text><location><page_6><loc_7><loc_19><loc_47><loc_23></location>This general relation between the metric functions b ( r ), f ( r ) should be satisfied for any κ ( R , T ). Inserting now this outcome into Eq. (36), we obtain a differential equation for b ( r ) as</text>
<formula><location><page_6><loc_7><loc_15><loc_47><loc_18></location>[ b ' ( r ) r 2 ] 2 = -1 r d dr ( b ' ( r ) r 2 )( 1 -b ( r ) r ) -b ( r ) b ' ( r ) r 5 , (41)</formula>
<text><location><page_6><loc_7><loc_10><loc_47><loc_14></location>a non-linear second order differential equation very difficult to solve. To go further, we consider the so-called thin shell approximation.</text>
<section_header_level_1><location><page_6><loc_50><loc_88><loc_75><loc_89></location>5.2 The Thin Shell Approximation</section_header_level_1>
<text><location><page_6><loc_50><loc_78><loc_90><loc_86></location>For this EoS, it is not an easy task to obtain exact analytic solutions. One successful strategy usually employed in the literature is the so-called thin shell approximation, 0 < 1 -b ( r ) /r ≡ h << 1. This approximation allow us to set h ≈ 0 so that one can consider only the lower order terms and we get from (41)</text>
<formula><location><page_6><loc_50><loc_74><loc_90><loc_77></location>b ' ( r ) r 2 ( b ' ( r ) r 2 + b ( r ) r 3 ) = 0 . (42)</formula>
<text><location><page_6><loc_50><loc_72><loc_90><loc_73></location>The non-trivial solution for b ( r ) arises by imposing that</text>
<formula><location><page_6><loc_50><loc_67><loc_90><loc_70></location>b ' ( r ) r 2 + b ( r ) r 3 = 0 . (43)</formula>
<text><location><page_6><loc_50><loc_61><loc_90><loc_66></location>The last equation can also be obtained more directly by applying the thin shell approximation to Eq. (36). Solving for b ( r ), and using (40), we obtain that the metric potentials are given by</text>
<formula><location><page_6><loc_50><loc_57><loc_90><loc_60></location>b ( r ) = D r , f ( r ) = ln( C 0 r 2 ) , (44)</formula>
<text><location><page_6><loc_50><loc_56><loc_70><loc_57></location>where D , C 0 are constants.</text>
<text><location><page_6><loc_50><loc_53><loc_90><loc_55></location>In the thin shell approximation, the curvature scalar is given by</text>
<formula><location><page_6><loc_50><loc_49><loc_90><loc_52></location>R = 2 b ' ( r ) r 2 + d dr ( b ( r ) r ) f ' ( r ) = -6 D r 4 . (45)</formula>
<text><location><page_6><loc_50><loc_45><loc_90><loc_48></location>Regarding the proper length of the thin shell, e λ assumes the following form as</text>
<formula><location><page_6><loc_50><loc_42><loc_90><loc_45></location>e λ ( r ) = ( 1 -b ( r ) r ) -1 = ( 1 -D r 2 ) -1 . (46)</formula>
<text><location><page_6><loc_50><loc_39><loc_77><loc_40></location>we get the value of the proper length</text>
<formula><location><page_6><loc_50><loc_30><loc_90><loc_39></location>l = ∫ R + ϵ R dr e λ/ 2 = ∫ R + ϵ R dr √ 1 -D/r 2 = ∫ R + ϵ R √ r 2 r 2 -D dr = √ r 2 -D ∣ ∣ ∣ ∣ R + ϵ R = √ ( R + ϵ ) 2 -D -√ R 2 -D. (47)</formula>
<text><location><page_6><loc_50><loc_28><loc_75><loc_29></location>This can be rewritten in the form</text>
<formula><location><page_6><loc_50><loc_24><loc_90><loc_27></location>l = ( R + ϵ ) √ 1 -D ( R + ϵ ) 2 -R √ 1 -D R 2 . (48)</formula>
<text><location><page_6><loc_50><loc_20><loc_90><loc_23></location>If the constant D is small compared to the radius R , i.e, D << R we can approximate the proper length as</text>
<formula><location><page_6><loc_50><loc_16><loc_90><loc_19></location>l ≈ ϵ + D 2 ( 1 R -1 R + ϵ ) , (49)</formula>
<text><location><page_6><loc_50><loc_13><loc_90><loc_16></location>this further can approximated if ϵ << R , we finally obtain</text>
<formula><location><page_6><loc_50><loc_10><loc_90><loc_12></location>l ≈ ϵ ( 1 + D 2 R 2 ) . (50)</formula>
<figure>
<location><page_7><loc_7><loc_73><loc_47><loc_89></location>
<caption>Fig. 1 Variation of proper length as a function of the thickness within the thin shell.</caption>
</figure>
<text><location><page_7><loc_7><loc_55><loc_47><loc_67></location>This result means that the proper length of the thin shell is proportional to the thickness ϵ of the shell. The variation of the proper length as a function of the thickness can be seen in Fig. 1. Another important conclusion is that the proper length of the thin shell does not depend on the particular choice of the κ ( R , T ) running gravitational constant. Now, we discuss the solutions for some specific models.</text>
<text><location><page_7><loc_7><loc_48><loc_41><loc_50></location>5.3 Study of the Thin Shell Solutions for Some Specific Models</text>
<formula><location><page_7><loc_7><loc_44><loc_35><loc_46></location>5.3.1 Model κ ( R , T ) = 8 π + β Rα T</formula>
<text><location><page_7><loc_7><loc_37><loc_47><loc_43></location>The density and pressure in the shell for this particular model can be found analytically. Indeed, substituting κ ( R , T ) = 8 π + β Rα T in (34), we find after straightforward manipulations the result</text>
<formula><location><page_7><loc_7><loc_29><loc_47><loc_34></location>p ( r ) = ρ ( r ) = 1 4 a [ 6 bD r 4 -1 + √ ( 1 -6 bD r 4 ) 2 -8 ad r 4 ] , (51)</formula>
<text><location><page_7><loc_7><loc_10><loc_47><loc_26></location>where we have defined the constants a = α/ 8 π , b = β/ 8 π and d = D/ 8 π . In order to represent a feasible physical gravastar, namely, to get positive density in the shell, we have to severely constrain the parameters of the model, i.e, the condition (1 -6 bD/r 4 ) 2 > aD/πr 4 has to be satisfied, possibly together with 6 bD/r 4 > 1, We consider that with these constraints, the model will become a restrictive an unrealistic one. Therefore, we choose not to pursue a further study of this particular choice, and proceed to discuss now other possible forms of κ ( R , T ).</text>
<formula><location><page_7><loc_50><loc_88><loc_71><loc_89></location>5.3.2 Model κ ( T ) = 8 π -α T</formula>
<text><location><page_7><loc_50><loc_85><loc_82><loc_86></location>Setting β = 0 in the former case, we obtain</text>
<formula><location><page_7><loc_50><loc_81><loc_90><loc_84></location>p ( r ) = ρ ( r ) = 1 4 a ( √ 1 -8 ad r 4 -1 ) . (52)</formula>
<text><location><page_7><loc_50><loc_62><loc_90><loc_79></location>This solution seems problematic, i.e., since a is a positive constant, ρ seems to be only positive for negative values of d , namely, negative values of the constant D , and this is not consistent with the thin shell approximation. Indeed, notice that substituting the result b ( r ) = D/r in the approximation 1 -b ( r ) /r << 1, it is obtained that 1 ≈ D/r 2 . Hence, if we denote R as the thin shell radius, the last relation implies that R is of order R ≈ √ D , being D necessarily a positive constant. Thus, we conclude that the κ ( T ) model cannot support a gravastar, at least as long as such approximation is valid.</text>
<formula><location><page_7><loc_50><loc_59><loc_71><loc_60></location>5.3.3 Model κ ( R ) = 8 π + β R</formula>
<text><location><page_7><loc_50><loc_56><loc_82><loc_57></location>The matter density in the model is given by</text>
<formula><location><page_7><loc_50><loc_52><loc_90><loc_55></location>p ( r ) = ρ ( r ) = d 6 bD -r 4 . (53)</formula>
<text><location><page_7><loc_50><loc_47><loc_90><loc_51></location>The trend of the p = ρ with respect to r is shown in Fig. 2 (left). To be a feasible physical model, i.e, to have positive density, we have to impose the constraint</text>
<formula><location><page_7><loc_50><loc_45><loc_90><loc_46></location>6 bD -r 4 > 0 . (54)</formula>
<text><location><page_7><loc_50><loc_41><loc_90><loc_43></location>Then, if we denote R as the radius of the thin shell, we have the following upper bound for the thin shell radius</text>
<formula><location><page_7><loc_50><loc_37><loc_90><loc_39></location>R < 4 √ 6 bD. (55)</formula>
<text><location><page_7><loc_50><loc_33><loc_90><loc_36></location>The energy within the shell for this model is therefore given by</text>
<formula><location><page_7><loc_50><loc_24><loc_90><loc_32></location>E ( r ) = ∫ R + ϵ R 4 πρr 2 dr = 2 πd 4 √ 6 bD [ tanh -1 ( r 4 √ 6 bD ) -tan -1 ( r 4 √ 6 bD ) ] R + ϵ R . (56)</formula>
<text><location><page_7><loc_50><loc_22><loc_87><loc_23></location>The variation of energy in shown in Fig. 2 (right).</text>
<formula><location><page_7><loc_50><loc_19><loc_74><loc_20></location>5.3.4 Model κ ( R , T ) = 8 π -γ RT</formula>
<text><location><page_7><loc_50><loc_13><loc_90><loc_17></location>The matter density and pressure for this specific model that directly couples the matter and curvature trace terms are given by the following expression</text>
<formula><location><page_7><loc_50><loc_10><loc_90><loc_12></location>p ( r ) = ρ ( r ) = ar 4 + 1 2 √ (2 ar 4 ) 2 + b. (57)</formula>
<figure>
<location><page_8><loc_9><loc_73><loc_48><loc_89></location>
</figure>
<text><location><page_8><loc_29><loc_72><loc_30><loc_74></location>R</text>
<figure>
<location><page_8><loc_49><loc_73><loc_88><loc_89></location>
<caption>Fig. 2 Variations of p = ρ and energy within the thin shell for κ ( R , T ) = 8 π + β R by choosing D = 2.</caption>
</figure>
<text><location><page_8><loc_8><loc_60><loc_10><loc_61></location>p</text>
<figure>
<location><page_8><loc_9><loc_53><loc_48><loc_69></location>
</figure>
<figure>
<location><page_8><loc_49><loc_53><loc_88><loc_69></location>
<caption>Fig. 3 Variations of p = ρ and energy within the thin shell for κ ( R , T ) = 8 π -γ RT by choosing D = 0 . 9.</caption>
</figure>
<text><location><page_8><loc_7><loc_38><loc_47><loc_48></location>Where we have defined the constants a = π/ 3 Dγ , b = 1 / 3 γ . Thus, the model κ ( R , T ) = 8 π -γ RT , admits positive density without the need to constrain any free parameter. Again, the variation of p = ρ with respect to r in Fig. 3 (left). On the other hand, the energy within the shell for this model can be computed by means of the integral</text>
<formula><location><page_8><loc_7><loc_29><loc_47><loc_38></location>E ( r ) = ∫ R + ϵ R 4 πρr 2 dr = 2 π 3 γD [ 2 πr 7 7 + √ γ Dr 3 √ 3 2 F 1 ( -1 2 , 3 8 ; 11 8 ; -4 π 2 r 8 3 D 2 γ ) ] R + ϵ R . (58)</formula>
<text><location><page_8><loc_7><loc_25><loc_48><loc_29></location>where 2 F 1 is the ordinary hypergeometric function. The variations of energy for κ ( R , T ) = 8 π + β R and κ ( R , T ) = 8 π -γ RT are shown in Fig. 3 (right).</text>
<section_header_level_1><location><page_8><loc_7><loc_21><loc_38><loc_22></location>6 ENTROPY WITHIN THE SHELL</section_header_level_1>
<text><location><page_8><loc_7><loc_14><loc_47><loc_20></location>The core region of a gravastar has vanishing entropy density [7,8]. However, entropy within the shell is generally not vanishing. For a non-collapsing gravatar, the entropy at the shell is defined as</text>
<formula><location><page_8><loc_7><loc_10><loc_47><loc_13></location>S = ∫ R + ϵ R 4 πr 2 s ( r ) dr √ 1 -b/r with s ( r ) = χ √ p 2 π , (59)</formula>
<text><location><page_8><loc_50><loc_43><loc_90><loc_48></location>in the unit ℏ = κ B = 1 and χ , a dimensionless parameter. The thickness of the shell is ϵ . Since (59) is nonintegrable, we can approximate as follows: consider the primitive integral of (59) as</text>
<formula><location><page_8><loc_50><loc_38><loc_90><loc_42></location>S = ∫ R + ϵ R z ' ( r ) dr = z ( r ) ∣ ∣ ∣ R + ϵ R = z ( R + ϵ ) -z ( R ) ≈ ϵ z ' ( R ) (60)</formula>
<text><location><page_8><loc_50><loc_35><loc_79><loc_36></location>at ϵ → 0. Hence, (59) can be written as</text>
<formula><location><page_8><loc_50><loc_27><loc_90><loc_34></location>S = ∫ R + ϵ R 4 πr 2 s ( r ) dr √ 1 -b/r ≈ 4 πR 2 ϵ s ( R ) √ 1 -b ( R ) /R = 4 πR 2 ϵ α √ 1 -b ( R ) /R √ p ( R ) 2 π . (61)</formula>
<text><location><page_8><loc_50><loc_21><loc_90><loc_26></location>The variations of entropy for κ ( R , T ) = 8 π + β R and κ ( R , T ) = 8 π -γ RT can be seen in Figs. 4. Entropy in both cases increases linearly with the thickness of the shell ϵ .</text>
<section_header_level_1><location><page_8><loc_50><loc_15><loc_89><loc_17></location>7 JUNCTION INTERFACE AND SURFACE STRESSES</section_header_level_1>
<text><location><page_8><loc_50><loc_10><loc_90><loc_13></location>Gravastar has three regions, the interior ( p = -ρ ) filled with dark energy, the shell filled with Zeldovich fluid</text>
<figure>
<location><page_9><loc_8><loc_73><loc_48><loc_89></location>
</figure>
<text><location><page_9><loc_29><loc_73><loc_30><loc_73></location>ϵ</text>
<figure>
<location><page_9><loc_49><loc_73><loc_88><loc_89></location>
<caption>Fig. 4 Variations of entropy within the thin shell for the two cases ( D = 2 and D = 0 . 9 respectively).</caption>
</figure>
<figure>
<location><page_9><loc_9><loc_53><loc_48><loc_69></location>
</figure>
<figure>
<location><page_9><loc_49><loc_52><loc_88><loc_69></location>
<caption>Fig. 5 Variations of surface pressure and energy density.</caption>
</figure>
<text><location><page_9><loc_7><loc_43><loc_47><loc_48></location>( p = ρ ), and an exterior where both pressure and density vanishes ( p = ρ = 0) i.e. a vacuum. The exterior solution is generally accepted as the Schwarzschild solution given by</text>
<formula><location><page_9><loc_7><loc_37><loc_47><loc_41></location>ds 2 = [ 1 -2 M r ] dt 2 -dr 2 1 -2 M/r -r 2 ( dθ 2 +sin θ dϕ 2 ) . (62)</formula>
<text><location><page_9><loc_7><loc_30><loc_47><loc_35></location>At the junctions r = R , the interior and the exterior connect smoothly. However, due to the slight mismatch in their derivatives, there arises stress at the junction. The stress tensor is given by the Lanczos equation as</text>
<formula><location><page_9><loc_7><loc_26><loc_47><loc_28></location>S i j = -1 8 π ( K i j -δ i j K q q ) , (63)</formula>
<text><location><page_9><loc_7><loc_18><loc_47><loc_25></location>where K ij = K + ij -K -ij is the discontinuity in the second fundamental form. Here the signs '+' and ' -' correspond to the exterior and the interior regions respectively. The second fundamental at the junction is given by</text>
<formula><location><page_9><loc_7><loc_14><loc_47><loc_17></location>K ± ij = -n ± ν [ ∂ 2 x ν ∂ξ i ∂ξ j + Γ ν αβ ∂x α ∂ξ i ∂x β ∂ξ i ] Σ . (64)</formula>
<text><location><page_9><loc_7><loc_10><loc_47><loc_13></location>Here, ξ i is the intrinsic coordinates on the shell, and n ± ν is the unit normals to the surface Σ . For the exterior</text>
<text><location><page_9><loc_50><loc_46><loc_90><loc_48></location>spacetime (62), the unit normal to the surface is given by</text>
<formula><location><page_9><loc_50><loc_41><loc_90><loc_44></location>n ± ν = ± ∣ ∣ ∣ ∣ g αβ ∂F ∂x α ∂F ∂x β ∣ ∣ ∣ ∣ -1 / 2 ∂F ∂x ν , ∀ n µ n µ = 1 . (65)</formula>
<text><location><page_9><loc_50><loc_35><loc_90><loc_40></location>and F = 1 -2 M/R . If the surface stress-energy tensor is taken as S i j = diag( σ, -P , -P , -P ), where σ is the surface energy density and P , the surface pressure, they can be determined as</text>
<formula><location><page_9><loc_50><loc_31><loc_90><loc_35></location>σ = -√ F 4 πR ∣ ∣ ∣ ∣ + -, P = -σ 2 + 1 16 π F ' √ F ∣ ∣ ∣ ∣ + -. (66)</formula>
<text><location><page_9><loc_50><loc_27><loc_90><loc_30></location>For the shell, the surface energy density and surface pressure take the form</text>
<formula><location><page_9><loc_50><loc_23><loc_90><loc_27></location>σ = -1 4 πR [ √ 1 -2 M R -C 0 R 2 ] , (67)</formula>
<formula><location><page_9><loc_50><loc_19><loc_90><loc_22></location>P = 2 -2 M/R -3 C 0 R 2 √ 1 -2 M/R 16 πR √ 1 -2 M/R . (68)</formula>
<text><location><page_9><loc_50><loc_14><loc_90><loc_18></location>The trends of surface energy and surface pressure are shown in Fig. 5. Now, we can determine the mass of the shell ( m s ) as</text>
<formula><location><page_9><loc_50><loc_10><loc_90><loc_14></location>m s = 4 πR 2 σ = R [ C 0 R 2 -√ 1 -2 M R ] , (69)</formula>
<figure>
<location><page_10><loc_9><loc_73><loc_48><loc_89></location>
</figure>
<figure>
<location><page_10><loc_49><loc_72><loc_88><loc_89></location>
<caption>Fig. 6 Variations of equation of state parameter and surface redshift C 0 = 0 . 895 and M = 0 . 1.</caption>
</figure>
<figure>
<location><page_10><loc_8><loc_52><loc_47><loc_68></location>
<caption>Fig. 7 Variations of shell mass with respect to shell thickness C 0 = 0 . 895 and M = 0 . 1.</caption>
</figure>
<text><location><page_10><loc_7><loc_44><loc_47><loc_46></location>from which we can determine the total mass of the gravastar as</text>
<formula><location><page_10><loc_7><loc_40><loc_47><loc_43></location>M = 2 C 0 R 3 m s -C 2 0 R 6 -m 2 s + R 2 2 R . (70)</formula>
<text><location><page_10><loc_7><loc_37><loc_47><loc_40></location>Further, one can also determine the equation of state parameter at the interface as</text>
<formula><location><page_10><loc_7><loc_34><loc_47><loc_37></location>ω = P σ = 3 C 0 R 3 √ 1 -2 M/R +2 M -2 R -4 C 0 R 3 √ 1 -2 M/R -8 M +4 R , (71)</formula>
<text><location><page_10><loc_24><loc_30><loc_24><loc_32></location≯</text>
<text><location><page_10><loc_7><loc_30><loc_47><loc_33></location>which must have a real value, requires 2 M/R < 1 and to avoid singularity C 0 = √ 1 -2 M/R/R 2 .</text>
<text><location><page_10><loc_7><loc_26><loc_47><loc_30></location>The variations of the equation of state parameter ω , and shell mass m s can be seen in Fig. 6 (left), and Fig. 7.</text>
<section_header_level_1><location><page_10><loc_7><loc_21><loc_42><loc_23></location>8 SURFACE REDSHIFT WITHIN THE THIN SHELL</section_header_level_1>
<text><location><page_10><loc_7><loc_10><loc_48><loc_19></location>One of the most important physical parameters in gravastar structure analysis is a ratio of wavelengths known as surface redshift Z s . The study of this dimensionless quantity provides valuable information corresponding to the stability and detection of these compact objects. The surface gravitational redshift is defined as</text>
<text><location><page_10><loc_50><loc_60><loc_90><loc_68></location>Z s = ∆λ λ e = λ 0 λ e , where ∆ represents the fractional change wavelength among the emitted λ e and received signal λ 0 . For static isotropic matter distribution, Buchdahl [63] stated that the value of redshift parameter Z s should not exceed 2, i.e. Z s < 2. The expression of surface redshift is presented by</text>
<formula><location><page_10><loc_50><loc_57><loc_90><loc_58></location>Z s = | g tt | -1 / 2 -1 , (72)</formula>
<text><location><page_10><loc_50><loc_51><loc_90><loc_55></location>inserting into (72) the value of the metric potential g tt , for the shell region given by Eq. (44), we get the surface redshift as</text>
<formula><location><page_10><loc_50><loc_47><loc_90><loc_50></location>Z s = 1 C 0 r 2 -1 with R ≤ r ≤ R + ϵ. (73)</formula>
<text><location><page_10><loc_50><loc_44><loc_90><loc_46></location>The variation of surface redshift Z s as a function of the thickness of the shell is shown in Fig. 6 (right).</text>
<section_header_level_1><location><page_10><loc_50><loc_39><loc_78><loc_40></location>9 RESULTS AND DISCUSSION</section_header_level_1>
<text><location><page_10><loc_50><loc_26><loc_90><loc_37></location>In this current work, we have derived and explored the Einstein field equations describing a gravastar for a generic κ ( R , T ) functional, and then we discussed several special cases corresponding to specific choices of this functional, such as κ ( R , T ) = 8 π + β Rα T , κ ( R , T ) ≡ κ ( T ) = 8 π -α T , κ ( R , T ) ≡ κ ( R ) = 8 π + β R and κ ( R , T ) = 8 π -γ RT . A summary of our findings is the following:</text>
<unordered_list>
<list_item><location><page_10><loc_51><loc_19><loc_90><loc_25></location>-In the interior region with EoS ρ = -p , the solutions are regular everywhere and, in particular, in the center regardless of the specific form of the κ ( R , T ) functional.</list_item>
<list_item><location><page_10><loc_51><loc_10><loc_90><loc_19></location>-For the shell region with EoS ρ = p , the specific models, κ ( R , T ) = 8 π + β R α T , and κ ( T ) = 8 π -α T provide negative density, therefore they are unphysical solutions and cannot support a gravastar configuration. The viability of the model κ ( R ) = 8 π + β R requires an upper bound for the thin shell</list_item>
</unordered_list>
<text><location><page_11><loc_10><loc_82><loc_47><loc_89></location>radius to get positive density. Nevertheless, the nonlinear model that directly couples the R and T traces, i.e. κ ( R , T ) = 8 π -γ RT supports a gravastar configuration without resorting to any constraint or fine-tuning of the free parameters.</text>
<unordered_list>
<list_item><location><page_11><loc_8><loc_70><loc_47><loc_82></location>-Inside the shell, the p = ρ and energy are slightly increasing linearly outward with respect to the shell thickness in the case of κ ( R , T ) = 8 π + β R . Both these parameters decrease when the κ ( R ) -coupling parameter β increases. On the other hand, p = ρ and energy for the case κ ( R , T ) = 8 π -γ RT are constants throughout the shell and they too decrease when κ ( R , T ) -coupling strength γ increases.</list_item>
<list_item><location><page_11><loc_8><loc_63><loc_47><loc_70></location>-The entropy within the shell increases when the coupling strength β increases in κ ( R ) -gravity while the same parameter decreases when the coupling strength increases between geometry and matter in nonlinear κ ( R , T ) -gravity.</list_item>
<list_item><location><page_11><loc_8><loc_60><loc_47><loc_62></location>-The surface redshift Z s is less than 2, and hence physically inspired.</list_item>
</unordered_list>
<text><location><page_11><loc_7><loc_47><loc_47><loc_59></location>At the end, it can be concluded that the existence of gravastar configurations in κ ( R , T ) -gravity strongly depends on the chosen form of the κ -function. Similar to the parallel competing modified theories of gravity, κ ( R , T ) had already given the wormhole solutions [54], compact star configurations [55,56] and now gravastar. Hence, this new theory is becoming a promising new theory of gravity.</text>
<section_header_level_1><location><page_11><loc_7><loc_44><loc_22><loc_45></location>Acknowledgments</section_header_level_1>
<text><location><page_11><loc_7><loc_38><loc_48><loc_42></location>FR, KNS would like to thank the authorities of the Inter-University Centre for Astronomy and Astrophysics, Pune, India for providing research facilities.</text>
<section_header_level_1><location><page_11><loc_7><loc_34><loc_16><loc_35></location>References</section_header_level_1>
<unordered_list>
<list_item><location><page_11><loc_8><loc_30><loc_47><loc_32></location>1. J. Reignier. The birth of special relativity. arXiv:physics/0008229</list_item>
<list_item><location><page_11><loc_8><loc_29><loc_47><loc_30></location>2. W. Heisenberg, Zeitschrift fur Physik 33 , 879-893 (1925)</list_item>
<list_item><location><page_11><loc_8><loc_28><loc_38><loc_29></location>3. A. Einstein, Ann. der Phys. 17 , 891 (1905)</list_item>
<list_item><location><page_11><loc_8><loc_27><loc_47><loc_28></location>4. W. Heisenberg, Zeitschrift fur Physik 43 , 172-198 (1927)</list_item>
<list_item><location><page_11><loc_8><loc_22><loc_47><loc_26></location>5. D. Hilbert, Die Grundlagen der Physik.(Erste Mitteilung.). Nachrichten von der Gesellschaft der Wissenschaften zu Gottingen, Mathematisch-Physikalische Klasse 1915 , 395-408 (1915)</list_item>
<list_item><location><page_11><loc_8><loc_19><loc_47><loc_22></location>6. J. Renn (Ed.). (2007). The genesis of general relativity: Sources and interpretations (Vol. 250). Springer Science and Business Media.</list_item>
<list_item><location><page_11><loc_8><loc_17><loc_41><loc_18></location>7. P. Mazur, E. Mottola, arXiv:gr-qc/0109035</list_item>
<list_item><location><page_11><loc_8><loc_15><loc_47><loc_17></location>8. P. Mazur, E. Mottola, Proc. Natl. Acad. Sci. USA, 101 , 9545 (2004)</list_item>
<list_item><location><page_11><loc_8><loc_13><loc_47><loc_15></location>9. M. Visser, D. L. Wiltshire, Class. Quan. Grav. 21 , 1135 (2004)</list_item>
<list_item><location><page_11><loc_7><loc_10><loc_47><loc_12></location>10. C. Cattoen, T. Faber, M. Visser, Class. Quan. Grav. 22 , 4189 (2005)</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_11><loc_50><loc_87><loc_90><loc_89></location>11. N. Bilic, G. B. Tupper, R. D. Viollier, J. Cosmol. Astropart. Phys. 02 , 013 (2006)</list_item>
<list_item><location><page_11><loc_50><loc_84><loc_90><loc_87></location>12. F. S. N. Lobo, and A. V. B. Arellano, Class. Quan. Grav. 24 , 1069 (2007)</list_item>
<list_item><location><page_11><loc_50><loc_82><loc_90><loc_84></location>13. P. Rocha, R. Chan, M. F A. da Silva, A. Wang, J. Cosmol. Astropart. Phys. 11 , 010 (2008)</list_item>
<list_item><location><page_11><loc_50><loc_81><loc_89><loc_82></location>14. D. Horvat, S. Iliji'c, Class. Quan. Grav. 24 , 5637 (2007)</list_item>
<list_item><location><page_11><loc_50><loc_78><loc_90><loc_81></location>15. K. K. Nandi, Y. Z. Zhang, R. G. Cai, A. Panchenko, Phys. Rev. D 79 , 024011 (2009)</list_item>
<list_item><location><page_11><loc_50><loc_75><loc_90><loc_78></location>16. A. A. Usmani, F. Rahaman, S. Ray, K. K. Nandi, P. K. F. Kuhfittig, S. A. Rakib, Z. Hasan, Phys. Lett. B 701 , 388-392 (2011)</list_item>
<list_item><location><page_11><loc_50><loc_72><loc_90><loc_75></location>17. F. Rahaman, S. Chakraborty, S. Ray, A. A. Usmani, S. Islam, Int. J. Theor. Phys. 54 , 50-61 (2015)</list_item>
<list_item><location><page_11><loc_50><loc_70><loc_90><loc_72></location>18. R. Chan, M. F. A. da Silva, P. Rocha, Gen. Rel. Grav. 43 , 2223-2235 (2011)</list_item>
<list_item><location><page_11><loc_50><loc_68><loc_90><loc_70></location>19. C.F.C. Brandt, R. Chan, M.F.A. da Silva, P. Rocha, J. Mod. Phys. 6 , 879 (2013)</list_item>
<list_item><location><page_11><loc_50><loc_65><loc_90><loc_67></location>20. R. Chan, M.F.A. da Silva, P. Rocha, A. Wang, J. Cosmo. Astropart. Phys. 03 , 010 (2009)</list_item>
<list_item><location><page_11><loc_50><loc_63><loc_90><loc_65></location>21. D. J. C. Lombardo, C. D. Vigh, Int. J. Mod. Phys. D 28 , 1950108 (2019)</list_item>
<list_item><location><page_11><loc_50><loc_61><loc_90><loc_63></location>22. T. Harko, F. S.N. Lobo, G. Otalora, E. N. Saridakis, J. Cosmo. Astropart. Phys. 12 , 021 (2014)</list_item>
<list_item><location><page_11><loc_50><loc_58><loc_90><loc_60></location>23. V. C. de Andrade, L. C. T. Guillen, J. G. Pereira arXiv: gr-qc/0011087</list_item>
<list_item><location><page_11><loc_50><loc_56><loc_90><loc_58></location>24. T. Harko, F. S.N. Lobo, S. Nojiri, S. D. Odintsov, Phys. Rev. D 84 , 024020 (2011)</list_item>
<list_item><location><page_11><loc_50><loc_55><loc_84><loc_55></location>25. D. Lovelock, J. Math. Phys. 12 , 498-501 (1971)</list_item>
<list_item><location><page_11><loc_50><loc_52><loc_90><loc_54></location>26. N. M. Garc'ıa, F. S N Lobo, J. P. Mimoso, T. Harko, J. Phys.: Conf. Ser. 314 , 012056 (2011)</list_item>
<list_item><location><page_11><loc_50><loc_51><loc_87><loc_52></location>27. M. Sharif, A. Ikram, Eur. Phys. J. C 76 , 640 (2016)</list_item>
<list_item><location><page_11><loc_50><loc_49><loc_90><loc_51></location>28. Yixin Xu, T. Harko, S. Shahidi, S.-D. Liang, Eur. Phys. J. C 80 , 449 (2020)</list_item>
<list_item><location><page_11><loc_50><loc_46><loc_90><loc_48></location>29. H. Barzegar, M. Bigdeli, G. H. Bordbar, B. E. Panah, Eur. Phys. J. C. 83 , 151 (2023)</list_item>
<list_item><location><page_11><loc_50><loc_44><loc_90><loc_46></location>30. S. Ghosh, A. D. Kanfon, A. Das, M. J. S. Houndjo, I. G. Salako, S. Ray, Int. J. Mod. Phys. A 35 , 2050017 (2020)</list_item>
<list_item><location><page_11><loc_50><loc_41><loc_90><loc_44></location>31. S. Ghosh, S. Dey, A. Das, A. Chanda, B. C. Paul, J. Cosmo. Astropart. Phys. 07 , 004 (2021)</list_item>
<list_item><location><page_11><loc_50><loc_39><loc_90><loc_41></location>32. A. Das, S. Ghosh, B. K. Guha, S. Das, F. Rahaman, S. Ray, Phys. Rev. D 95 , 124011 (2017)</list_item>
<list_item><location><page_11><loc_50><loc_37><loc_90><loc_39></location>33. Z. Yousaf, M.Z. Bhatti, H. Asad, Phys. Dark Uni. 28 , 100501 (2020)</list_item>
<list_item><location><page_11><loc_50><loc_35><loc_88><loc_36></location>34. M. Z. Bhatti, Mod. Phys. Lett. A 35 , 2050069 (2020)</list_item>
<list_item><location><page_11><loc_50><loc_33><loc_90><loc_35></location>35. M. Z. Bhatti, Z. Yousaf, A. Rahaman, Phys. Dark Univ. 29 , 100561 (2020)</list_item>
<list_item><location><page_11><loc_50><loc_32><loc_84><loc_33></location>36. Z. Yousaf, Phys. Dark Univ. 28 , 100509 (2020)</list_item>
<list_item><location><page_11><loc_50><loc_29><loc_90><loc_32></location>37. F. S. N. Lobo, R. Garattini, J. High Ener. Phys. 12 , 065 (2013)</list_item>
<list_item><location><page_11><loc_50><loc_27><loc_90><loc_29></location>38. M. Z. Bhatti, Z. Yousaf, M. Ajmal, Int. J. Mod. Phys. D 28 , 1950123 (2019)</list_item>
<list_item><location><page_11><loc_50><loc_25><loc_90><loc_27></location>39. M.Z. Bhatti, Z. Yousaf, T. Ashraf, Chin. J. Phys. 73 , 167-178 (2021)</list_item>
<list_item><location><page_11><loc_50><loc_22><loc_90><loc_24></location>40. A. Das, S. Ghosh, D. Deb, F. Rahaman, S. Ray, Nucl. Phys. B 954 , 114986 (2020)</list_item>
<list_item><location><page_11><loc_50><loc_20><loc_90><loc_22></location>41. D. Horvat, S. Iliji'c, A. Marunovi'c, Class. Quan. Grav. 26 , 025003 (2009)</list_item>
<list_item><location><page_11><loc_50><loc_18><loc_90><loc_20></location>42. F. Rahaman, A.A. Usmani, S. Ray, S. Islam, Phys. Lett. B 717 , 1-5 (2012)</list_item>
<list_item><location><page_11><loc_50><loc_15><loc_90><loc_17></location>43. S. Ghosh, S. Ray, F. Rahaman, B.K. Guha, Ann. Phys 394 , 230-243 (2018)</list_item>
<list_item><location><page_11><loc_50><loc_13><loc_90><loc_15></location>44. S. Ghosh, D. Shee, S. Ray, F. Rahaman, B.K. Guha, Res. Phys. 14 , 102473 (2019)</list_item>
<list_item><location><page_11><loc_50><loc_10><loc_90><loc_12></location>45. S. Ghosh , S. Biswas , F. Rahaman , B.K. Guha, S. Ray, Ann. Phys. 411 , 167968 (2019)</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_12><loc_7><loc_87><loc_47><loc_89></location>46. S. Banerjee, S. Ghosh, N. Paul, F. Rahaman, Eur. Phys. J. Plus, 135 , 185 (2020)</list_item>
<list_item><location><page_12><loc_7><loc_84><loc_47><loc_87></location>47. R. Sengupta, S. Ghosh, S. Ray, B. Mishra, S. K. Tripathy, Phys. Rev. D 102 , 024037 (2020)</list_item>
<list_item><location><page_12><loc_7><loc_82><loc_47><loc_84></location>48. S. Ray, R. Sengupta, H. Nimesh, Int. J. Mod. Phys. D 29 , 2030004 (2020)</list_item>
<list_item><location><page_12><loc_7><loc_80><loc_47><loc_82></location>49. A. Das, F. Rahaman, B. K. Guha, S. Ray, Eur. Phys. J. C 76 , 654 (2016)</list_item>
<list_item><location><page_12><loc_7><loc_79><loc_41><loc_79></location>50. G. R. P. Teruel, Eur. Phys. J. C 78 , 660 (2018)</list_item>
<list_item><location><page_12><loc_7><loc_77><loc_47><loc_78></location>51. N. Ahmed, A. Pradhan, Ind. J. Phys. 96 , 301-307 (2022)</list_item>
<list_item><location><page_12><loc_7><loc_75><loc_47><loc_77></location>52. A. Dixit, A. Pradhan, R. Chaubey, Int. J. Geom. Math. Mod. Phys. 19 , 2250013 (2022).</list_item>
<list_item><location><page_12><loc_7><loc_73><loc_47><loc_75></location>53. A. Dixit, S. Gupta, A. Pradhan, A. Beesham, Symmetry 15 , 549 (2023)</list_item>
<list_item><location><page_12><loc_7><loc_70><loc_47><loc_72></location>54. S. Sarkar, N. Sarkar, F. Rahaman, Y. Aditya, To Phys. J. 2 , 7 (2019)</list_item>
<list_item><location><page_12><loc_7><loc_68><loc_47><loc_70></location>55. G. R. P. Teruel, K. N. Singh, F. Rahaman, T. Chowdhury, Int. J. Mod. Phys. A 37 , 2250194 (2022)</list_item>
<list_item><location><page_12><loc_7><loc_66><loc_47><loc_68></location>56. D. Ta¸ser, S.S. Do˘gru, Astrophys. Space Sci. 368 , 49 (2023)</list_item>
<list_item><location><page_12><loc_7><loc_63><loc_47><loc_65></location>57. C.H. Brans, R.H. Dicke, Phys. Rev. 124(3) , 925-935 (1961)</list_item>
<list_item><location><page_12><loc_7><loc_62><loc_34><loc_63></location>58. C.H. Brans, arXiv:gr-qc/0506063</list_item>
<list_item><location><page_12><loc_7><loc_61><loc_40><loc_62></location>59. P. Rastall, Phys. Rev. D 6 , 3357-3359 (1972)</list_item>
<list_item><location><page_12><loc_7><loc_58><loc_47><loc_60></location>60. T. Harko, F.S.N. Lobo, S. Nojiri, S.D. Odintsov, arXiv:1104.2669</list_item>
<list_item><location><page_12><loc_7><loc_56><loc_47><loc_58></location>61. L. Lindblom, W.A. Hiscock, J. Phys. A: Math. Gen. 15 , 1827 (1982)</list_item>
<list_item><location><page_12><loc_7><loc_55><loc_36><loc_56></location>62. M. Visser, Phys. Lett. B 782 , 83 (2018)</list_item>
<list_item><location><page_12><loc_7><loc_54><loc_40><loc_55></location>63. H.A. Buchdahl, Phys. Rev. 116 , 1027 (1959)</list_item>
<list_item><location><page_12><loc_7><loc_52><loc_46><loc_53></location>64. S. A. Hojman, J. Phys. A: Math. Gen 25 , L291 (1992)</list_item>
</document>
[
{
"title": "Testing of κ ( R , T ) -gravity through gravastar configurations",
"content": "G R P Teruel a,1 , Ksh. Newton Singh b,2 , Tanmoy Chowdhury c,3 , Farook Rahaman d,3 , Monimala Mondal e,3 Received: date / Accepted: date Abstract In this article, we are reporting for the first time the existence of gravastar configurations in the framework of κ ( R , T ) -gravity, which can be treated as an alternative to a black hole (Mazur and Mottola). This strengthens how much this new gravity theory may be physically demanding to the gravity community in the near future. We first develop the gravastar field equations for a generic κ ( R , T ) functional and then we study four different models within this theory. We find that the solutions for the interior region are regular everywhere regardless of the exact form of the κ ( R , T ) functional. The solutions for the shell region indicate that two of the four models subjected to the study are physically feasible. In addition, the junction conditions are considered at each interface by using the Lanczos equations that yield the surface density and pressure at the thin shell. We investigate various characteristics of the gravastar structure such as the proper length, energy, and entropy of the spherical distribution. Keywords Gravastar · κ ( R , T ) -gravity · Junction conditions",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "The hierarchy of principles in physics is an old debate that should be revived from time to time to ask ourselves if we are going in the right direction. On the one hand, there are foundational physical principles, such as the principle of inertia in classical mechanics, the uncertainty principle in quantum mechanics, or the principle of relativity that directly founded or contributed to establishing new scientific theories. On the other hand, there are the auxiliary principles, which are not as fundamental as the first ones, but help to strengthen our theories once they have been established by the foundational principles. It is natural to wonder: To which of these two categories does the principle of least action belong? At first glance, one would answer that the least action principle should be an undisputed part of the first group, i.e., a foundational one due to its ubiquitous presence in our current theories. However, let's carefully examine the history of physics. We will appreciate that the variational principles, despite their enormous success and importance, were not founding principles, in the sense that they did not establish new scientific paradigms, although they turned out to be versatile and flexible enough to know how to adapt to every paradigm shift in theoretical physics. In particular, if we examine the birth of the two major revolutions of the twentieth century, the theories of relativity and quantum mechanics, we will see that the least action principle had a secondary role [1,2]. In fact, other principles were the foundational building blocks of these theories [3,4]. The action principle was adapted later to them and arrived to enrich their formalism. Analogously, neither of the two great classical field theories was discovered by means of the least action principle. Indeed, Classical Electrodynamics (CE) was fulfilled with the inclusion of a source term directly into the field equations (the Maxwell displacement current). General relativity (GR) was also first derived in this way, after adding the key trace term to the field equations [5,6]. In both cases, the least action principle came to strengthen the formalism after other more fundamental principles had established the theories. These historical lessons persuade us that the search for possible Non-Lagrangian theories, that is, theories not based (at least initially) on the variational method, should not be ruled out a priori. In 2001, Mazur and Mottola [7,8] proposed a very interesting and encouraging stellar model, named gravitational vacuum star (or gravastar) as an alternating black hole structure. In recent days, the study on gravastar has become a considerably interesting topic as an alternating black hole solution. Mazur and Mottola [7,8] first introduced a novel solution for the extreme point of gravitational collapse of cold and dark matter, a neutral system, and compact stars. Therefore, this is an extended form of Bose-Einstein condensation whose radius is similar to the Schwarzschild radius. The total design of a gravastar can be defined by means of 3 different zones with distinct EoS: Several researchers have already studied gravastar from different perspectives and scooped out a new field as an alternative to the black hole paradigm. [7,8,9,10, 11,12,13,14,15,16,17]. Lack of observational evidence in Einstein's GR on the oscillating model of the universe alongside the presence of dark matter has forced a conceptual challenge to this theory [18,19,20,21]. The repulsive nature of dark energy in GR requires considering an exotic fluid with a negative pressure to explain the cosmic speed-up. Many modified Lagrangian theories of gravity have been proposed to account for the accelerating phase of the universe. These theories are based on the generalization of the EinsteinHilbert action viz f ( T , T ) [22], f ( T ) [23], f ( R , T ) [24], EGB [25], f ( G ) [26], f ( G, T ) [27], f ( Q, T ) [28] gravity theories, etc, which earned wide attention in recent years. A major role of modified gravity theories is to test the validity of the gravastar model by examining various realistic properties of these compact objects. Barzegar et al. [29] explored the 3D AdS gravitational vacuum stars in the framework of gravity's rainbow theory. Ghosh et al. [30] investigated gravastars in f ( T , T ) gravity and enlarged its research in Rastall gravity as well [31]. Das et al. [32], investigated the formation of gravastars in f ( R , T ) gravity and inspect its theoretical efficiency. Enlarge the analysis Yousaf et al., recently explored gravastars in f ( R , T , R µν T µν ) gravity [33]. Bhatti et al. [34] reviewed charged gravastar for spherically and cylindrically symmetric spacetimes and in f ( R , G ) gravity [35]. Extending it, Yousaf [36] constructed charged cylindrical gravastar-like structures in f ( R , T ) gravity. Lobo and Garattini [37] analyzed gravastar solutions in the frame of non-invariant geometry with their physical properties and characteristics. Bhatti et al. [38] investigated the stability of this thin shell together with the thermodynamical stability of locally isotropic gravastars with cylindrical space-time. Bhatti et al. [39] talked over the symmetric gravastar model within the framework of the modification of GaussBonnet gravity. Das et al. [40] discussed the physically interesting and valid features within the framework of the f ( T ) theory of gravity. Horvet et al. [41] used dominant force conditions to initialize the stability of the gravastar after combining a sufficient external vacuum solution with an internal geometry and meeting some possible constraints. Also, they examined the formation of this type of compact object under the influence of electromagnetism. Rahaman and others [42] analyzed the electrically charged gravastar model considering the (2 + 1) -dimensional geometry. For a stable model of a spherically symmetric gravastar, they also explored some viable properties as well as energy components, length, and entropy of the gravastar. Ghosh et al. [43] investigated some new diagrams in the formation of these matters with and without electric charge in high-order manifolds. Ghosh et al. [44] discussed gravastar layout in the framework of the Kuchowicz metric potential. Ghosh et al. [45] figure out the gravastar model under Einstein's GR in (3+1) dimensions by incorporating the Karmarkar condition. Sumita et al. [46] studied gravastars under Finslerian spacetime geometry. Sengupta et al. [47] investigated a gravastar configuration in RS Brane gravity, while Ray et al. [48] reviewed a very interesting paper on generalizations of gravastars in GR, including studies of higher and lower dimensional GR with physical insights like the Randall-Sundrum theory, and several modified gravity models such as f ( R , T ) theory or Rastall-Rainbow gravity. Recently, a modified theory was invented [50] as κ ( R , T ) -gravity. The conceptual framework of this theory is not based on the standard modified gravity program. Rather, its insight is inspired by Maxwell's and Einstein's ideas of including new possible potential terms in the field equations. In this sense, we must recall Maxwell's addition to Ampere's law of the displacement current term to complete the Electromagnetic field equations, and the introduction of the key term, 1 2 R g µν by Einstein to consummate the GR field equations [5,6]. Despite the fact that the variational principle is often regarded as our main tool to formulate a new physical theory (and its generalizations), ought not to have been placed at the same fundamental level as other truly first principles, like the equivalence principle and the principle of general covariance, that are the two core principles of GR. In spite of this, the variational method has reached the pinnacle in our hierarchy of physical principles, becoming dominant to the extent of being the starting point for any modification of GR. Indeed, the vast majority of the modified gravity theories that are available in the market, like the examples mentioned above, are Lagrangian in the sense that they arise from the generalization of the Einstein-Hilbert action. The overpopulation of alternatives suggests that some other fundamental principle (beyond the equivalence principle and the principle of general covariance) is required to guide us trough the deep jungle of possible Lagrangian theories. The fact is, we have no reason to assure that general symmetries and general conservation laws can always exist (in their current form) in the final theories of nature, which moves us to think that the Non-Lagrangian proposals should also be explored. In fact, according to Hojman [64], the use of Lagrangian and Hamiltonian functions is not an essential requirement to derive conservation theorems. κ ( R , T ) -gravity is a recent proposal, and so its implications have not been explored well enough. However, for the past few years, some works are dedicated to studying its cosmic aspects. In particular, Pradhan and Ahmed [51] showed that the theory can deal with the current scenario of an accelerating universe, claiming that the theory requires a very small value of the cosmological constant, which agrees with observations. Pradhan et al. [52] further extended this study to include a more complete cosmological scenario, while Dixit et al. [53] studied the thermodynamics of the expansion of the cosmos in the context of this theory. Sarkar et al. [54], explored a model of wormhole in κ ( R , T ) -gravity, while Teruel et al. [55] introduced the first internal solutions representing compact stars in κ ( R , T )-gravity by solving the field equations in isotropic coordinates. Also very recently, Tas . er and Dogru [56] investigated whether the Krori-Barua model produces valid results in this theory. All these papers have one remarkable common feature: they were devoted to studying only a particular case of the theory with κ ( R , T ) ≡ κ ( T ) = 8 π -λ T . Up to now, nobody studied gravastar solutions in κ ( R , T ) -gravity, it would therefore be interesting to investigate and discuss whether this theory can support a consistent gravastar configuration from a physical point of view. In this sense, four different models, i.e., four different choices of the running κ ( R , T ) -gravitational constant are analyzed in this investiga- on. Therefore, we don't restrict ourselves only to the κ ( T ) choice like all the previous works, but extend the study to the more general possible case. Indeed, once derived the equations for a generic κ ( R , T ) functional, we see how the theory would look for specific cases. The structure of the paper is as follows. We start in section 2 with the basic framework of the theory. Section 3 deals with the field equations derived from the Morris-Thorne metric for the general κ ( R , T ) case. The solutions for the interior region of the gravastar are discussed in section 4. The shell region of the gravastar is analyzed in section 5, where we study the physical acceptability of 4 different models within the theory. Entropy within the shell is addressed in section 6. The junction interface and surface stresses are the subjects of section 7. The brief section 8 deals with the surface redshift within the thin shell. Finally, in section 9, we summarize the findings and discuss the conclusions of the work.",
"pages": [
1,
2,
3
]
},
{
"title": "2 Framework of κ ( R , T ) -gravity",
"content": "The structure of the theory depends on the following field equations where R µν is the Ricci tensor, g µν is the space-time metric, Λ corresponds to the cosmological constant, T µν is the stress-energy tensor of the material content, while κ ( R , T ) is a generalization of Einstein's gravitational constant, that we extend to the status of a function of the traces T ≡ g µν T µν , and R≡ g µν R µν . The introduction of the functional κ ( R , T ), means that we inspect the possibility of a running gravitational constant, although not at the level of the variational method. The possibility of considering a variable Einstein's gravitational constant in the action was studied many years ago by Brans and Dicke [57,58], and the resulting theory is quite different from (1). The field equations (1) imply the non-covariant conservation of T µν . Indeed, since the divergence of the left-hand side of Einstein's field equation vanishes, we have Then, this non-conservation of the T µν can be expressed, for κ ( R , T ) = 0, as ̸ ̸ When κ ( R , T ) = 0, the right-hand side of Einstein's field equations vanishes. This interesting case probably may happen in some specific models in the early universe for sufficiently high densities, and it will imply an exponential expansion driven by a cosmological constant. Of course, the theory can be recast into a conservative form by defining a new (effective) stress-energy tensor The field equations then take the form and the Bianchi identities imply that Eq. (2) is Then, this theory has the same formal structure as GR, with a non-trivial modification of the material content. Examples of famous non-conservative gravitational theories are Rastall's gravity [59], or the Lagrangian f ( R , T ) theory by Harko et al. [60]. Lindblom and Hisock [61], and also Visser [62] more recently, criticized some non-conservative gravity theories (in particular the Rastall case), claiming that they are formally identical to Einstein's gravity and that one can always build a conserved effective stress-energy tensor T eff µν , which is constructed solely from the matter sources. Furthermore, they underline the fact that, if the effective stress-energy tensor of the matter sources does not depend on the space-time curvature, (such is the case of Rastall gravity), then these theories will not represent truly alternative theories of gravity, reducing the question of the non-conservation of the stressenergy tensor to the domain of special relativity. Nevertheless, since the effective stress-energy tensor of κ ( R , T ) gravity is not in general determined by the matter sources solely (it can have a dependence on the space-time curvature as well via the trace R ), we conclude that κ ( R , T ) theory represents a true generalization of GR and not only a redefinition of its matter sector. Some relevant features of the theory are the following:",
"pages": [
3,
4
]
},
{
"title": "3 Field Equations in κ ( R , T ) -gravity",
"content": "The interior of the gravastar is taken to be as MorrisThorne Spacetime given by For Λ = 0, the Einstein field equations for a general κ ( R , T ) model will be On the other hand, the non-covariant conservation of the stress-energy tensor given by Eq. (2) acquires the form",
"pages": [
4
]
},
{
"title": "4 Regular Interior Solutions in κ ( R , T ) -gravity with p = -ρ",
"content": "In this section, we show that the interior solutions for a gravastar configuration are regular everywhere in κ ( R , T ) theory, regardless of the specific choice of the running gravitational constant. To see how this interesting property of the theory arises, notice that for p ( r ) = -ρ ( r ), Eq. (10) reduces to which further takes the form This implies the relation where C , the constant of integration. Therefore, field equations reduce to from which we can obtain b ( r ) and f ( r ) as Regularity at r = 0 implies that we set D = 0. At r = 0, the core of the gravastar, we have that f ( r ) = 0 and the metric functions become g tt = -1, and g rr = 1. The density should also be constant and finite at the gravastar's interior. At first we observed that the curvature scalar is constant and non-divergent everywhere at the interior region. After a cumbersome calculation, we have arrived at Therefore, an important conclusion of our results is that the interior solution for the gravastar will be regular and well-behaved everywhere, regardless of the specific functional forms of κ ( R , T ). Let us consider the choice, κ ( R , T ) = 8 π + β Rα T , where the pre-multiplier are the free parameters. For this particular case we have, using that ρ ( r ) = -p ( r ) and the results above, that Eq. (13) has the form ̸ This algebraic equation implies, as we mentioned before, that the density (and pressure) for the interior layer of the gravastar is uniform, and for α = 0, it has the following possible values where ω is a constant given by ω = 2 π + βC and ω ≥ √ αC . The gravitational mass at the interior can is found from The density and pressure for this κ ( R , T ) choice can be computed by setting β = 0 in the previous case. Taking into account that for this choice, the constant ω = 2 π + βC reduces to ω = 2 π , we obtain Thus, the active gravitational follows from the direct computation The density and pressure for this specific choice can be computed by setting α = 0 in the case A. Then, Eq. (19) reduces to Therefore, The mass is obtained in the same fashion as models (4.1) and (4.2), i.e, by direct integration of the density in the range r = 0 and r = R 1 For this direct coupling among matter and curvature terms, we have that the density, pressure and active gravitational mass are given by where, γ is a parameter of the model that should satisfy the constraint γ ≤ ( π/C ) 2 in order to get positive density in the core.",
"pages": [
4,
5
]
},
{
"title": "5 Shell Region p = ρ",
"content": "This region of the gravastar supposedly contains a stiff fluid that obeys EoS p = ρ or v = √ dp/dρ = 1, which is the most stiff EoS known as Zeldovich fluid . This condition implies that the non-covariant conservation of the stress-energy tensor given by Eq. (10) acquires the form On the other hand, Einstein's field equations become Since, in general, the running gravitational constant κ ( R , T ) depends on the scalars R , T , which in turn are functions of b ( r ), f ( r ), ρ ( r ). To solve the unknown functions b ( r ), f ( r ) and ρ ( r ) we need three equations, two from field equations and one conservation equation. From the last two equations we obtain 5.1 The Shell Region Differential Equations in κ ( R , T ) -gravity ̸ To find the general equations for the shell region in κ ( R , T ) gravity we can proceed in the following way: using (34), we eliminate κ ( R , T ), i.e. we have for ρ ( r ) = 0, can always write Substituting this into Eq. (33), we obtain the result This differential equation can be integrated to provide This general relation between the metric functions b ( r ), f ( r ) should be satisfied for any κ ( R , T ). Inserting now this outcome into Eq. (36), we obtain a differential equation for b ( r ) as a non-linear second order differential equation very difficult to solve. To go further, we consider the so-called thin shell approximation.",
"pages": [
6
]
},
{
"title": "5.2 The Thin Shell Approximation",
"content": "For this EoS, it is not an easy task to obtain exact analytic solutions. One successful strategy usually employed in the literature is the so-called thin shell approximation, 0 < 1 -b ( r ) /r ≡ h << 1. This approximation allow us to set h ≈ 0 so that one can consider only the lower order terms and we get from (41) The non-trivial solution for b ( r ) arises by imposing that The last equation can also be obtained more directly by applying the thin shell approximation to Eq. (36). Solving for b ( r ), and using (40), we obtain that the metric potentials are given by where D , C 0 are constants. In the thin shell approximation, the curvature scalar is given by Regarding the proper length of the thin shell, e λ assumes the following form as we get the value of the proper length This can be rewritten in the form If the constant D is small compared to the radius R , i.e, D << R we can approximate the proper length as this further can approximated if ϵ << R , we finally obtain This result means that the proper length of the thin shell is proportional to the thickness ϵ of the shell. The variation of the proper length as a function of the thickness can be seen in Fig. 1. Another important conclusion is that the proper length of the thin shell does not depend on the particular choice of the κ ( R , T ) running gravitational constant. Now, we discuss the solutions for some specific models. 5.3 Study of the Thin Shell Solutions for Some Specific Models The density and pressure in the shell for this particular model can be found analytically. Indeed, substituting κ ( R , T ) = 8 π + β Rα T in (34), we find after straightforward manipulations the result where we have defined the constants a = α/ 8 π , b = β/ 8 π and d = D/ 8 π . In order to represent a feasible physical gravastar, namely, to get positive density in the shell, we have to severely constrain the parameters of the model, i.e, the condition (1 -6 bD/r 4 ) 2 > aD/πr 4 has to be satisfied, possibly together with 6 bD/r 4 > 1, We consider that with these constraints, the model will become a restrictive an unrealistic one. Therefore, we choose not to pursue a further study of this particular choice, and proceed to discuss now other possible forms of κ ( R , T ). Setting β = 0 in the former case, we obtain This solution seems problematic, i.e., since a is a positive constant, ρ seems to be only positive for negative values of d , namely, negative values of the constant D , and this is not consistent with the thin shell approximation. Indeed, notice that substituting the result b ( r ) = D/r in the approximation 1 -b ( r ) /r << 1, it is obtained that 1 ≈ D/r 2 . Hence, if we denote R as the thin shell radius, the last relation implies that R is of order R ≈ √ D , being D necessarily a positive constant. Thus, we conclude that the κ ( T ) model cannot support a gravastar, at least as long as such approximation is valid. The matter density in the model is given by The trend of the p = ρ with respect to r is shown in Fig. 2 (left). To be a feasible physical model, i.e, to have positive density, we have to impose the constraint Then, if we denote R as the radius of the thin shell, we have the following upper bound for the thin shell radius The energy within the shell for this model is therefore given by The variation of energy in shown in Fig. 2 (right). The matter density and pressure for this specific model that directly couples the matter and curvature trace terms are given by the following expression R p Where we have defined the constants a = π/ 3 Dγ , b = 1 / 3 γ . Thus, the model κ ( R , T ) = 8 π -γ RT , admits positive density without the need to constrain any free parameter. Again, the variation of p = ρ with respect to r in Fig. 3 (left). On the other hand, the energy within the shell for this model can be computed by means of the integral where 2 F 1 is the ordinary hypergeometric function. The variations of energy for κ ( R , T ) = 8 π + β R and κ ( R , T ) = 8 π -γ RT are shown in Fig. 3 (right).",
"pages": [
6,
7,
8
]
},
{
"title": "6 ENTROPY WITHIN THE SHELL",
"content": "The core region of a gravastar has vanishing entropy density [7,8]. However, entropy within the shell is generally not vanishing. For a non-collapsing gravatar, the entropy at the shell is defined as in the unit ℏ = κ B = 1 and χ , a dimensionless parameter. The thickness of the shell is ϵ . Since (59) is nonintegrable, we can approximate as follows: consider the primitive integral of (59) as at ϵ → 0. Hence, (59) can be written as The variations of entropy for κ ( R , T ) = 8 π + β R and κ ( R , T ) = 8 π -γ RT can be seen in Figs. 4. Entropy in both cases increases linearly with the thickness of the shell ϵ .",
"pages": [
8
]
},
{
"title": "7 JUNCTION INTERFACE AND SURFACE STRESSES",
"content": "Gravastar has three regions, the interior ( p = -ρ ) filled with dark energy, the shell filled with Zeldovich fluid ϵ ( p = ρ ), and an exterior where both pressure and density vanishes ( p = ρ = 0) i.e. a vacuum. The exterior solution is generally accepted as the Schwarzschild solution given by At the junctions r = R , the interior and the exterior connect smoothly. However, due to the slight mismatch in their derivatives, there arises stress at the junction. The stress tensor is given by the Lanczos equation as where K ij = K + ij -K -ij is the discontinuity in the second fundamental form. Here the signs '+' and ' -' correspond to the exterior and the interior regions respectively. The second fundamental at the junction is given by Here, ξ i is the intrinsic coordinates on the shell, and n ± ν is the unit normals to the surface Σ . For the exterior spacetime (62), the unit normal to the surface is given by and F = 1 -2 M/R . If the surface stress-energy tensor is taken as S i j = diag( σ, -P , -P , -P ), where σ is the surface energy density and P , the surface pressure, they can be determined as For the shell, the surface energy density and surface pressure take the form The trends of surface energy and surface pressure are shown in Fig. 5. Now, we can determine the mass of the shell ( m s ) as from which we can determine the total mass of the gravastar as Further, one can also determine the equation of state parameter at the interface as ̸ which must have a real value, requires 2 M/R < 1 and to avoid singularity C 0 = √ 1 -2 M/R/R 2 . The variations of the equation of state parameter ω , and shell mass m s can be seen in Fig. 6 (left), and Fig. 7.",
"pages": [
8,
9,
10
]
},
{
"title": "8 SURFACE REDSHIFT WITHIN THE THIN SHELL",
"content": "One of the most important physical parameters in gravastar structure analysis is a ratio of wavelengths known as surface redshift Z s . The study of this dimensionless quantity provides valuable information corresponding to the stability and detection of these compact objects. The surface gravitational redshift is defined as Z s = ∆λ λ e = λ 0 λ e , where ∆ represents the fractional change wavelength among the emitted λ e and received signal λ 0 . For static isotropic matter distribution, Buchdahl [63] stated that the value of redshift parameter Z s should not exceed 2, i.e. Z s < 2. The expression of surface redshift is presented by inserting into (72) the value of the metric potential g tt , for the shell region given by Eq. (44), we get the surface redshift as The variation of surface redshift Z s as a function of the thickness of the shell is shown in Fig. 6 (right).",
"pages": [
10
]
},
{
"title": "9 RESULTS AND DISCUSSION",
"content": "In this current work, we have derived and explored the Einstein field equations describing a gravastar for a generic κ ( R , T ) functional, and then we discussed several special cases corresponding to specific choices of this functional, such as κ ( R , T ) = 8 π + β Rα T , κ ( R , T ) ≡ κ ( T ) = 8 π -α T , κ ( R , T ) ≡ κ ( R ) = 8 π + β R and κ ( R , T ) = 8 π -γ RT . A summary of our findings is the following: radius to get positive density. Nevertheless, the nonlinear model that directly couples the R and T traces, i.e. κ ( R , T ) = 8 π -γ RT supports a gravastar configuration without resorting to any constraint or fine-tuning of the free parameters. At the end, it can be concluded that the existence of gravastar configurations in κ ( R , T ) -gravity strongly depends on the chosen form of the κ -function. Similar to the parallel competing modified theories of gravity, κ ( R , T ) had already given the wormhole solutions [54], compact star configurations [55,56] and now gravastar. Hence, this new theory is becoming a promising new theory of gravity.",
"pages": [
10,
11
]
},
{
"title": "Acknowledgments",
"content": "FR, KNS would like to thank the authorities of the Inter-University Centre for Astronomy and Astrophysics, Pune, India for providing research facilities.",
"pages": [
11
]
}
]
2024PDU....4501523A
https://arxiv.org/pdf/2307.16458.pdf
<document>
<section_header_level_1><location><page_1><loc_20><loc_87><loc_80><loc_91></location>Cosmological First-Order Vacuum Phase Transitions in an Expanding Anisotropic Universe</section_header_level_1>
<text><location><page_1><loc_31><loc_81><loc_69><loc_83></location>A. Sava¸s Arapo˘glu ∗ and A. Emrah Y¨ukselci †</text>
<text><location><page_1><loc_21><loc_75><loc_78><loc_79></location>Istanbul Technical University, Faculty of Science and Letters, Physics Engineering Department, 34469, Maslak, Istanbul, Turkey</text>
<section_header_level_1><location><page_1><loc_46><loc_65><loc_54><loc_67></location>Abstract</section_header_level_1>
<text><location><page_1><loc_12><loc_41><loc_88><loc_63></location>We examine the anisotropy originated from a first-order vacuum phase transitions through threedimensional numerical simulations. We apply Bianchi Type-I metric to our model that has one scalar field minimally coupled to the gravity. We calculate the time evolution of the energy density for the shear scalar and the directional Hubble parameters as well as the power spectra for the scalar field and the gravitational radiation although there are a number of caveats for the tensor perturbations in Bianchi Type-I universe. We run simulations with different mass scales of the scalar field, therefore, in addition to investigation of anisotropy via the shear scalar, we also determine at which mass scale the phase transition completes successfully, hence, neglecting the expansion of the Universe does not significantly affect the results. Finally, we showed that such an event may contribute to the total anisotropy depending on the mass scale of the scalar field and the initial population of nucleated bubbles.</text>
<section_header_level_1><location><page_2><loc_40><loc_89><loc_60><loc_91></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_2><loc_12><loc_71><loc_88><loc_87></location>The first direct detection of the gravitational waves (GWs) [1] has opened a new era in observation of the Universe since they can carry the information related to very source phenomenon thanks to their weakly-interacting character. Although recent results of the observation with low-frequency GWs may point out an astrophysical origin [2] as is the first one, yet this is another important step towards mapping the stochastic GW background that may have contributions originated from the events of the early stages in the Universe as well. The imprints of such GWs may be detected through space-based GW detectors that are planned to be built in the future [3-8] .</text>
<text><location><page_2><loc_12><loc_49><loc_88><loc_71></location>Cosmological first-order phase transitions (PTs), which may have possibly occurred in the early Universe, are one type of phenomenon that can create GWs as an outcome [9, 10] . In spite of the fact that the well-known examples are the electroweak and the quark-hadron PTs, they may have taken place at any scale in the early Universe between QCD [11] and GUT [12] scales, respectively. However, the standard model of particle physics does not predict first-order phase transitions [13, 14] , yet there are many extensions of it to allow that (see e.g. [9, 10, 15] and references therein). This type of event could take place through the bubble nucleation mechanism, theory of which was studied at zero [16, 17] and finite [18, 19] temperatures in flat space-time whereas the gravitational effects were discussed later in Ref. [20] . It is known that the most probable initial profile for the bubbles has a O (4) symmetric form [21] although there is no such proof for a curved space-time yet.</text>
<text><location><page_2><loc_12><loc_21><loc_88><loc_48></location>The GWs originated from first-order PTs are created by the shear stress caused by the deformation of the symmetric structure of colliding bubbles. The theoretical approach to examine such an event was studied in Ref. [22] where collided portions of the bubbles are not taken into account. However, it was shown through the numerical simulations [23-25] that those parts should be considered since the scalar field oscillates around its true vacuum and give rise to another peak in the gravitational radiation power spectrum related to its mass in addition to the maximum value associated with the mean separation between bubbles. This has the potential to determine the parameters of a model and/or even the model itself. Recently, in the context of the scalar-tensor theories, the scalar field non-minimally coupled to the gravity has also been studied through the numerical simulations [26] . Another example is the study of the two-step phase transition related to electroweak symmetry breaking [27] . In addition to numerical approaches, it has been shown that it is also possible to analytically calculate the power spectra to some extent for the gravitational radiation formed during the bubble collision phase [28-30] .</text>
<text><location><page_2><loc_12><loc_10><loc_88><loc_20></location>In order to investigate the anisotropy we implement Bianchi Type-I metric where each direction has a different scale factors unlike the Friedmann-Lemaˆıtre-Robertson-Walker metric which is, indeed, a particular type of Bianchi Type-I model in this manner. This model has been widely used in the cosmological context (see e.g. [31] and references therein) since it is one of the simplest extensions for isotropic space-time and it may even offer some solutions to</text>
<text><location><page_3><loc_12><loc_69><loc_88><loc_91></location>the well-known problems such as H 0 tension [32] . Moreover, on the GW front with another aspect of the anisotropy, a possible detection of it in the stochastic GW background may even enable to distinguish between superimposed sources [33] . On the other hand, in light of recent observations, it has been reported that any sort of anisotropy is not encountered in the data [34] . However, the picture may change after the inclusion of more data into the analysis, and this may lead to understand the birthplace of an observed GW signal, in other words, whether it has an astrophysical or cosmological origins. Although there is no concrete evidence yet to determine the origins of the signal [35, 36] , analysis in Ref. [37] on the NanoGRAV data [2] indicates that some cosmological sources, e.g. strong first-order PTs, can provide comparable results with the astrophysical ones such as supermassive black hole binaries.</text>
<text><location><page_3><loc_12><loc_59><loc_88><loc_69></location>The paper is structured as follows: In Section (2) , we describe the main equations of the model; in Section (3) we provide the modified equations in accordance with the numerical scheme; in Section (4) we define the quantities to be followed during the simulations; in Section (5) we present the outcomes of the simulations and discuss the results in Section (6) .</text>
<section_header_level_1><location><page_3><loc_44><loc_54><loc_56><loc_55></location>2. SET-UP</section_header_level_1>
<text><location><page_3><loc_12><loc_48><loc_88><loc_52></location>In this section we provide the main equations that will be used throughout the paper. To this end, we start with the Einstein field equations given as</text>
<formula><location><page_3><loc_39><loc_44><loc_88><loc_47></location>R µν -1 2 R g µν = M -2 Pl T µν (1)</formula>
<text><location><page_3><loc_12><loc_35><loc_88><loc_43></location>where the Planck mass is defined via M -2 Pl ≡ 8 πG and is equal to M Pl = 2 . 435 × 10 18 GeV in natural units, i.e. c = ℏ = 1, which is adopted in this work. In the presence of only one scalar field minimally coupled to the gravity the energy-momentum tensor is given by the following expression</text>
<formula><location><page_3><loc_31><loc_30><loc_88><loc_34></location>T µν = ∇ µ ϕ ∇ ν ϕ -1 2 g µν ∇ σ ϕ ∇ σ ϕ -g µν V ( ϕ ) . (2)</formula>
<text><location><page_3><loc_12><loc_28><loc_75><loc_30></location>On the other hand, the equation of motion for the scalar field is obtained as</text>
<formula><location><page_3><loc_41><loc_23><loc_88><loc_27></location>∇ σ ∇ σ ϕ -∂V ( ϕ ) ∂ϕ = 0 (3)</formula>
<text><location><page_3><loc_12><loc_21><loc_44><loc_22></location>and we use the potential in the form of</text>
<formula><location><page_3><loc_34><loc_16><loc_88><loc_20></location>V ( ϕ ) = 1 2 M 2 ϕ 2 + 1 3 δϕ 3 + 1 4 λϕ 4 -V c (4)</formula>
<text><location><page_3><loc_12><loc_14><loc_42><loc_15></location>where M , δ , λ , and V c are constants.</text>
<text><location><page_3><loc_14><loc_12><loc_76><loc_13></location>For the background evolution, we implement the Bianchi Type-I metric as</text>
<formula><location><page_3><loc_31><loc_7><loc_88><loc_10></location>d s 2 = -d t 2 + a 2 1 ( t ) d x 2 + a 2 2 ( t ) d y 2 + a 2 3 ( t ) d z 2 (5)</formula>
<text><location><page_4><loc_12><loc_87><loc_88><loc_91></location>where a 1 ( t ), a 2 ( t ), and a 3 ( t ) are the scale factors in x , y , and z directions and they are the functions of time only. For future convenience, we also define</text>
<formula><location><page_4><loc_28><loc_83><loc_88><loc_86></location>H i ≡ ˙ a i a i ( i = 1 , 2 , 3) , H ≡ 1 3 ( H 1 + H 2 + H 3 ) (6)</formula>
<text><location><page_4><loc_12><loc_80><loc_69><loc_82></location>where H i is the directional and H is the average Hubble parameters.</text>
<text><location><page_4><loc_12><loc_76><loc_88><loc_80></location>The metric given in Eq. (5) yields tt , xx , yy , and zz components of Eq. (1) , respectively, in the following forms</text>
<formula><location><page_4><loc_14><loc_71><loc_88><loc_75></location>˙ a 1 a 1 ˙ a 2 a 2 + ˙ a 1 a 1 ˙ a 3 a 3 + ˙ a 2 a 2 ˙ a 3 a 3 = M -2 Pl [ 1 2 ( ⟨ ˙ ϕ 2 ⟩ + ⟨ ϕ 2 x ⟩ a 2 1 + ⟨ ϕ 2 y ⟩ a 2 2 + ⟨ ϕ 2 z ⟩ a 2 3 ) + ⟨ V ( ϕ ) ⟩ ] (7)</formula>
<formula><location><page_4><loc_20><loc_66><loc_88><loc_70></location>a 2 a 2 + a 3 a 3 + ˙ a 2 a 2 ˙ a 3 a 3 = M -2 Pl [ 1 2 ( -⟨ ˙ ϕ 2 ⟩ -⟨ ϕ 2 x ⟩ a 2 1 + ⟨ ϕ 2 y ⟩ a 2 2 + ⟨ ϕ 2 z ⟩ a 2 3 ) + ⟨ V ( ϕ ) ⟩ ] (8)</formula>
<formula><location><page_4><loc_20><loc_61><loc_88><loc_65></location>a 1 a 1 + a 3 a 3 + ˙ a 1 a 1 ˙ a 3 a 3 = M -2 Pl [ 1 2 ( -⟨ ˙ ϕ 2 ⟩ + ⟨ ϕ 2 x ⟩ a 2 1 -⟨ ϕ 2 y ⟩ a 2 2 + ⟨ ϕ 2 z ⟩ a 2 3 ) + ⟨ V ( ϕ ) ⟩ ] (9)</formula>
<formula><location><page_4><loc_20><loc_56><loc_88><loc_60></location>a 1 a 1 + a 2 a 2 + ˙ a 1 a 1 ˙ a 2 a 2 = M -2 Pl [ 1 2 ( -⟨ ˙ ϕ 2 ⟩ + ⟨ ϕ 2 x ⟩ a 2 1 + ⟨ ϕ 2 y ⟩ a 2 2 -⟨ ϕ 2 z ⟩ a 2 3 ) + ⟨ V ( ϕ ) ⟩ ] (10)</formula>
<text><location><page_4><loc_12><loc_44><loc_88><loc_56></location>where the angle brackets denotes the spatial average over all simulation box, the dot represents the derivative with respect to t , and the subscript letters x, y, z stand for the spatial derivatives in the corresponding directions. Moreover, for the sake of simplification we eliminate the time derivative of the scalar field from Eqs. (8) , (9) , (10) with the help of the constraint equation, i.e. Eq. (7) , and obtain the final form of the equations of motion for the scale factors as follows</text>
<formula><location><page_4><loc_30><loc_39><loc_88><loc_43></location>a 1 a 1 + ˙ a 1 a 1 ( ˙ a 2 a 2 + ˙ a 3 a 3 ) = M -2 Pl [ ⟨ ϕ 2 x ⟩ a 2 1 + ⟨ V ( ϕ ) ⟩ ] (11)</formula>
<formula><location><page_4><loc_30><loc_34><loc_88><loc_38></location>a 2 a 2 + ˙ a 2 a 2 ( ˙ a 1 a 1 + ˙ a 3 a 3 ) = M -2 Pl [ ⟨ ϕ 2 y ⟩ a 2 2 + ⟨ V ( ϕ ) ⟩ ] (12)</formula>
<formula><location><page_4><loc_30><loc_30><loc_88><loc_33></location>a 3 a 3 + ˙ a 3 a 3 ( ˙ a 1 a 1 + ˙ a 2 a 2 ) = M -2 Pl [ ⟨ ϕ 2 z ⟩ a 2 3 + ⟨ V ( ϕ ) ⟩ ] . (13)</formula>
<text><location><page_4><loc_14><loc_27><loc_81><loc_29></location>On the other hand, the equation of motion for the scalar field, Eq. (3) , becomes</text>
<formula><location><page_4><loc_36><loc_21><loc_88><loc_26></location>¨ ϕ +3 H ˙ ϕ -3 ∑ k =1 ∂ 2 k ϕ a 2 k + ∂V ( ϕ ) ∂ϕ = 0 (14)</formula>
<text><location><page_4><loc_12><loc_17><loc_88><loc_20></location>where the average Hubble parameter, H , is defined in Eq. (6) and the potential for the scalar field is given in Eq. (4) .</text>
<text><location><page_4><loc_12><loc_11><loc_88><loc_16></location>Regarding the gravitational waves, the transverse-traceless (TT) part of the tensor perturbations, h ij , can be related to an auxiliary tensor, u ij , by defining a projection operator [38] as</text>
<formula><location><page_4><loc_38><loc_9><loc_88><loc_10></location>h ij ( t, k ) = Λ ij,lm ( ˆ k ) u lm ( t, k ) (15)</formula>
<text><location><page_5><loc_12><loc_89><loc_88><loc_91></location>where u lm ( t, k ) is the Fourier transform of u ij ( t, x ) and the projection operator is defined as</text>
<formula><location><page_5><loc_21><loc_84><loc_88><loc_88></location>Λ ij,lm ( ˆ k ) = P im ( ˆ k ) P jl ( ˆ k ) -1 2 P ij ( ˆ k ) P lm ( ˆ k ) , P ij ( ˆ k ) = δ ij -k i k j k 2 . (16)</formula>
<text><location><page_5><loc_12><loc_82><loc_70><loc_83></location>Then, this method yields the equation of motion in the following form</text>
<formula><location><page_5><loc_32><loc_75><loc_88><loc_80></location>u ij +3 H ˙ u ij -3 ∑ k =1 ∂ 2 k u ij a 2 k = M -2 Pl ∂ i ϕ∂ j ϕ a i a j . (17)</formula>
<text><location><page_5><loc_12><loc_64><loc_88><loc_74></location>Although we use TT gauge for the tensor perturbations here, this not need to be entirely true since there is a gauge fixing problem in Bianchi Type-I model [39, 40] . Therefore, we should note that the results related to the tensor perturbations, i.e. the gravitational waves, represented in this work are valid up to a gauge transformation. We will make some more comments on this issue in the following sections.</text>
<section_header_level_1><location><page_5><loc_34><loc_56><loc_65><loc_57></location>3. NUMERICAL PROCEDURE</section_header_level_1>
<text><location><page_5><loc_12><loc_44><loc_88><loc_53></location>The code which we used for this work is the same with that of Ref. [26] . It has been written in Python programming language with the help of Cython [41] extension for the intensive iterations. Parallel processing has been realized by the pencil decomposition and the communication between processes has been ensured by mpi4py [42] package. We constructed similar algorithms given in Ref. [43] for the Fourier transforms.</text>
<text><location><page_5><loc_12><loc_34><loc_88><loc_43></location>We implement the staggered leapfrog algorithm for advance in time with 7-point stencil for the Laplacian operator. In accordance with this scheme, it is necessary to eliminate the first time derivatives of the variables in the equations of motion to achieve the stability and the consistency for the numerical calculations. To this end, we define the new variables and the constants as</text>
<formula><location><page_5><loc_16><loc_29><loc_88><loc_32></location>ψ ≡ √ a 1 a 2 a 3 ϕ ϕ t , v ij ≡ M 2 Pl ϕ 2 t √ a 1 a 2 a 3 u ij , U ≡ 1 ϕ 2 t M 2 ( a 1 a 2 a 3 ) V , (18)</formula>
<formula><location><page_5><loc_16><loc_25><loc_88><loc_27></location>dτ ≡ Mdt , r → M r , m Pl ≡ M Pl /M , α ≡ δ/M , β ≡ ϕ t /M . (19)</formula>
<text><location><page_5><loc_12><loc_18><loc_88><loc_24></location>We will denote the derivative with respect to τ by the prime symbol in the following sections whereas we keep the same notation for the spatial derivatives, in other words, the subscript x should be understood as the derivative with respect to Mx .</text>
<text><location><page_5><loc_12><loc_14><loc_88><loc_18></location>In addition to that, we use fixed spatial resolution with dx = 0 . 44 in general and the value of the Courant factor is taken c = 0 . 4 for all simulations.</text>
<section_header_level_1><location><page_6><loc_12><loc_89><loc_38><loc_91></location>3.1. Numerical Equation Set</section_header_level_1>
<text><location><page_6><loc_12><loc_83><loc_88><loc_87></location>With definition of the new variables the equation of motion for the scalar field given in Eq. (3) takes the following form</text>
<formula><location><page_6><loc_35><loc_77><loc_88><loc_82></location>ψ '' + Kψ -3 ∑ k =1 ∂ 2 k ψ a 2 k + ∂U ( ψ ) ∂ψ = 0 (20)</formula>
<text><location><page_6><loc_12><loc_74><loc_16><loc_76></location>where</text>
<formula><location><page_6><loc_15><loc_69><loc_88><loc_73></location>K ≡ 1 4 [ a ' 2 1 a 1 + a ' 2 2 a 2 + a ' 2 3 a 3 ] -1 2 [ a '' 1 a 1 + a '' 2 a 2 + a '' 3 a 3 + a ' 1 a 1 a ' 2 a 2 + a ' 1 a 1 a ' 3 a 3 + a ' 2 a 2 a ' 3 a 3 ] (21)</formula>
<text><location><page_6><loc_12><loc_66><loc_44><loc_68></location>and the redefined potential is given by</text>
<formula><location><page_6><loc_25><loc_61><loc_88><loc_65></location>U ( ψ ) = 1 2 ψ 2 + 1 3 αβ √ a 1 a 2 a 3 ψ 3 + 1 4 λβ 2 a 1 a 2 a 3 ψ 4 -a 1 a 2 a 3 β 2 V c (22)</formula>
<text><location><page_6><loc_12><loc_54><loc_88><loc_60></location>where we set V c such that V ( ϕ t ) = 0. One may also choose a small constant instead of zero potential value as the cosmological constant. However, this is not in the scope of this paper and it needs to be considered with more realistic setups for long-time simulations.</text>
<text><location><page_6><loc_14><loc_52><loc_66><loc_54></location>On the other hand, the equations for the scale factors become</text>
<formula><location><page_6><loc_25><loc_47><loc_88><loc_51></location>a '' 1 a 1 + a ' 1 a 1 ( a ' 2 a 2 + a ' 3 a 3 ) = β 2 m 2 Pl ( a 1 a 2 a 3 ) -1 [ ⟨ ψ 2 x ⟩ a 2 1 + ⟨ U ( ψ ) ⟩ ] (23)</formula>
<formula><location><page_6><loc_25><loc_42><loc_88><loc_46></location>a '' 2 a 2 + a ' 2 a 2 ( a ' 1 a 1 + a ' 3 a 3 ) = β 2 m 2 Pl ( a 1 a 2 a 3 ) -1 [ ⟨ ψ 2 y ⟩ a 2 2 + ⟨ U ( ψ ) ⟩ ] (24)</formula>
<formula><location><page_6><loc_25><loc_37><loc_88><loc_41></location>a '' 3 a 3 + a ' 3 a 3 ( a ' 1 a 1 + a ' 2 a 2 ) = β 2 m 2 Pl ( a 1 a 2 a 3 ) -1 [ ⟨ ψ 2 z ⟩ a 2 3 + ⟨ U ( ψ ) ⟩ ] . (25)</formula>
<text><location><page_6><loc_12><loc_34><loc_74><loc_35></location>Finally, the equation of motion for the tensor perturbations is obtained as</text>
<formula><location><page_6><loc_30><loc_28><loc_88><loc_32></location>v '' ij + Kv ij -3 ∑ k =1 ∂ 2 k v ij a 2 k = ( a 1 a 2 a 3 ) -1 / 2 ∂ i ψ∂ j ψ a i a j (26)</formula>
<text><location><page_6><loc_12><loc_25><loc_38><loc_26></location>where K is defined in Eq. (21) .</text>
<text><location><page_6><loc_12><loc_9><loc_88><loc_24></location>These are the equations that will be solved numerically. The structure of the equations for the scalar field and the tensor perturbations are already in a suitable form for the leapfrog algorithm. However, the equations for the scale factors need a modification since the first time derivatives are one half step behind the corresponding variable at each step. In order to synchronize the variables and their first time derivatives we also keep their values from the previous step, meaning that we calculate the derivatives for this particular purpose as a ' 1 ( t ) ≈ [ a ' 1 ( t +∆ t/ 2) + a 1 ( t -∆ t/ 2)] / 2 where ∆ t is the time step. We use those values to calculate the expression given in Eq. (21) as well.</text>
<section_header_level_1><location><page_7><loc_12><loc_89><loc_33><loc_91></location>3.2. Initial Conditions</section_header_level_1>
<text><location><page_7><loc_12><loc_83><loc_88><loc_87></location>In order to start the simulations we use the thin-wall approximation [16] to determine the initial profile of the scalar field, that is, we implement</text>
<formula><location><page_7><loc_33><loc_79><loc_88><loc_82></location>ψ ( t = 0 , r ) = 1 2 [ 1 -tanh ( r -R c M l 0 M )] (27)</formula>
<text><location><page_7><loc_12><loc_74><loc_88><loc_77></location>where R c and l 0 are the critical radius and the bubble wall length, respectively, which can be found from the following expressions [23]</text>
<formula><location><page_7><loc_21><loc_69><loc_88><loc_73></location>ψ ( R c ) = 1 2 , ψ ( r ± ) = 1 2 [ 1 -tanh ( ± 1 2 )] , l 0 M = r + -r -. (28)</formula>
<text><location><page_7><loc_12><loc_62><loc_88><loc_68></location>Furthermore, the time derivative of the scalar field is taken initially to be zero, i.e. ψ ' ( t = 0) = 0. On the other hand, the bubble nucleation points in the lattice are randomly determined and the bubbles are nucleated simultaneously at the beginning of the simulations.</text>
<text><location><page_7><loc_14><loc_60><loc_42><loc_62></location>As for the scale factors we choose</text>
<formula><location><page_7><loc_34><loc_57><loc_88><loc_59></location>a 1 ( t = 0) = a 2 ( t = 0) = a 3 ( t = 0) = 1 (29)</formula>
<text><location><page_7><loc_12><loc_52><loc_88><loc_55></location>and in order to determine the initial values for the derivatives of the scale factors we use Eq. (7) written with the new variables as</text>
<formula><location><page_7><loc_13><loc_47><loc_88><loc_51></location>a ' 1 a ' 2 + a ' 1 a ' 3 + a ' 2 a ' 3 = β 2 m 2 Pl [ 1 2 ( ⟨ ψ 2 x ⟩ + ⟨ ψ 2 y ⟩ + ⟨ ψ 2 z ⟩ ) + 1 4 ( a ' 1 + a ' 2 + a ' 3 ) 2 ⟨ ψ 2 ⟩ + ⟨ U ( ψ ) ⟩ ] (30)</formula>
<text><location><page_7><loc_12><loc_44><loc_79><loc_46></location>with the help of Eq. (29) . Moreover, assuming that a ' 1 = a ' 2 = a ' 3 initially we get</text>
<formula><location><page_7><loc_24><loc_38><loc_88><loc_43></location>a ' 1 ( t = 0) = a ' 2 ( t = 0) = a ' 3 ( t = 0) = ± 1 3 I 2 -1 √ 1 3 I 1 (1 -3 I 2 ) (31)</formula>
<text><location><page_7><loc_12><loc_37><loc_16><loc_38></location>where</text>
<formula><location><page_7><loc_21><loc_32><loc_88><loc_36></location>I 1 = β 2 m 2 Pl [ 1 2 ( ⟨ ψ 2 x ⟩ + ⟨ ψ 2 y ⟩ + ⟨ ψ 2 z ⟩ ) + ⟨ U ( ψ ) ⟩ ] , I 2 = β 2 m 2 Pl ⟨ ψ 2 ⟩ . (32)</formula>
<text><location><page_7><loc_12><loc_29><loc_70><loc_31></location>Finally, we set v ij ( t = 0) = v ' ij ( t = 0) = 0 for the tensor perturbations.</text>
<section_header_level_1><location><page_7><loc_21><loc_24><loc_79><loc_26></location>4. DENSITIES, POWER SPECTRA, AND SHEAR SCALAR</section_header_level_1>
<text><location><page_7><loc_12><loc_14><loc_88><loc_22></location>Here we give the definitions for the densities, the power spectra for both the scalar field and the gravitational waves, and the shear scalar which will show the amount of anisotropy in simulations with different configurations. Starting with the densities for the scalar field we have</text>
<formula><location><page_7><loc_22><loc_9><loc_88><loc_13></location>¯ ρ K ≡ 1 2 〈 ˙ ϕ 2 〉 , ¯ ρ G ≡ 1 2 〈 3 ∑ k =1 ∂ 2 k ϕ a 2 k 〉 , ¯ ρ V ≡ 〈 V ( ϕ ) -V ( ϕ t ) 〉 (33)</formula>
<text><location><page_8><loc_12><loc_87><loc_88><loc_91></location>which are the kinetic, the gradient, and the potential energies, respectively. On the other hand, for the energy density of the gravitational waves, the following expression is calculated</text>
<formula><location><page_8><loc_23><loc_82><loc_88><loc_86></location>¯ ρ gw ( x , t ) = 1 8 M 2 Pl ∑ i,j 〈 ˙ h ij ( x , t ) ˙ h ij ( x , t ) + ∇ h ij ( x , t ) ∇ h ij ( x , t ) 〉 . (34)</formula>
<text><location><page_8><loc_12><loc_77><loc_88><loc_80></location>Although we keep the gradient terms explicitly, we need to emphasize that they almost have no effect on the results.</text>
<text><location><page_8><loc_14><loc_75><loc_60><loc_76></location>The power spectrum for the scalar field is expressed by</text>
<formula><location><page_8><loc_37><loc_70><loc_88><loc_73></location>P ϕ ( k , t ) = k 3 2 π 2 〈 ϕ ( k , t ) ϕ ∗ ( k , t ) 〉 (35)</formula>
<text><location><page_8><loc_12><loc_67><loc_44><loc_69></location>and for the gravitational waves we use</text>
<formula><location><page_8><loc_32><loc_62><loc_88><loc_66></location>d Ω gw d ln k = 1 3 H 2 k 3 16 π 2 ( P ˙ h ( k , t ) + k 2 P h ( k , t ) ) . (36)</formula>
<text><location><page_8><loc_12><loc_59><loc_51><loc_61></location>We also implement the following normalization</text>
<formula><location><page_8><loc_37><loc_54><loc_88><loc_58></location>d Ω gw d ln k -→ 1 ( H ∗ R ∗ Ω vac ) 2 d Ω gw d ln k (37)</formula>
<text><location><page_8><loc_12><loc_47><loc_88><loc_53></location>where H ∗ is the average Hubble parameter value at the time of the transition, R ∗ is the mean bubble separation equals to ( V /N b ) 1 / 3 in which V and N b are the physical volume of the simulation box and the number of bubbles, respectively.</text>
<text><location><page_8><loc_12><loc_37><loc_88><loc_47></location>We should emphasize that the above definitions for the power spectra are not entirely correct in the anisotropic case. However, we will represent them regardless and left those calculations to future studies. Because, in addition to investigating the mass scale for the scalar field, our main focus in this paper is to demonstrate the time evolution of the shear scalar defined as</text>
<formula><location><page_8><loc_29><loc_33><loc_88><loc_36></location>σ 2 = 1 6 [ ( H 1 -H 2 ) 2 +( H 1 -H 3 ) 2 +( H 2 -H 3 ) 2 ] (38)</formula>
<text><location><page_8><loc_12><loc_28><loc_88><loc_31></location>in order to quantify the anisotropy in the background. The Friedmann equation can then be written in the following form</text>
<formula><location><page_8><loc_26><loc_22><loc_88><loc_26></location>H 2 = σ 2 3 + 1 3 M -2 Pl [ 1 2 ( ⟨ ˙ ϕ 2 ⟩ + 3 ∑ k =1 ⟨ ( ∂ i ϕ ) 2 ⟩ a 2 i ) + ⟨ V ( ϕ ) ⟩ ] (39)</formula>
<text><location><page_8><loc_12><loc_17><loc_88><loc_20></location>or in terms of energy density parameters 1 = Ω σ 2 + Ω ϕ where we have defined the energy density parameter for the shear scalar as follows</text>
<formula><location><page_8><loc_45><loc_12><loc_88><loc_15></location>Ω σ 2 ≡ σ 2 3 H 2 (40)</formula>
<text><location><page_8><loc_12><loc_9><loc_78><loc_11></location>and put all the terms of the scalar field into Ω ϕ since we will not use it further.</text>
<table>
<location><page_9><loc_14><loc_75><loc_86><loc_91></location>
<caption>TABLE 1: Parameter values used to create different configurations for the simulations. N and N b are the number of grid points and the number of bubbles, respectively. From left to right, first five constants are the free parameters that have been chosen to test the dependency of the results and in accordance with the previous works of Refs. [23, 26] . The mass ( M t /M ) and the scalar field value ( ϕ t /M ) in true vacuum, vacuum energy density ( ρ vac /M 4 ), the critical radius ( R c M ), and bubble wall thickness ( l 0 M ) are calculated as explained in the text. For N = 640 ( N = 1280) we take dx = 0 . 44 ( dx = 0 . 22).</caption>
</table>
<figure>
<location><page_9><loc_11><loc_22><loc_87><loc_52></location>
<caption>FIG. 1: Two-dimensional slices of the simulations with M Pl /M = 1 , 10 , 100 from left to right, respectively, for two different times. For all simulations, we take N = 640 and N b = 320. Note that the physical scales are different in each slice due to the difference in expansion rate.</caption>
</figure>
<text><location><page_10><loc_34><loc_18><loc_35><loc_19></location>∗</text>
<section_header_level_1><location><page_10><loc_37><loc_89><loc_63><loc_91></location>5. SIMULATION RESULTS</section_header_level_1>
<text><location><page_10><loc_12><loc_69><loc_88><loc_87></location>We have simulated several configurations of the model with different parameter values as listed in Table (1) . In order to distinguish their effects on the results we have changed the number of nucleated bubbles (simulations 3-6) and the number of grid points (simulations 6, 7) as well as the mass scale (simulations 1, 2, 3) which is one of the main concern of this study in the context of free parameters. The values of the other constants that depend on the free parameters have also been calculated and given in the same table. The outcomes that will be reported and discussed in detail below are the Hubble parameters, densities, power spectra, and the shear scalar whose time evolution is another main interest of our work as it is one of the key indicators of anisotropy.</text>
<text><location><page_10><loc_12><loc_35><loc_88><loc_69></location>First of all, we have run simulations to see the mass scale at which the phase transition can be completed successfully. We see from two-dimensional slices of the simulations represented in Fig. (1) that the phase transition does not complete for the configurations with M Pl /M = 1 , 10. This is due to the fact that the expansion rate is higher for relatively small values of M Pl /M as one can deduce this result from the right-hand side of Eqs. (25) . The high expansion rate causes that universe to outgrow with a speed much more than the enlargement of the bubbles, therefore, the bubble collision phase either can not be completed entirely or does not happen at all. For M Pl /M = 10 the bubbles expand for a while at the start of the simulation and the ones close to each other collide partially, but then, the expansion rate of the universe eventually dominates the dynamics, whereas the bubble collision does not even occur in the case of M Pl /M = 1. On the other hand, for M Pl /M = 100 we notice that this kind of effect does not take place and the bubble collision phase is completed successfully in a time less than H -1 ∗ . At this point it is necessary to emphasize that this mass value is beyond even the GUT scale let alone the EW phase transition epoch, which are around 10 15 GeV and 100 GeV, respectively. Therefore, the completion of a phase transition at either GUT or EW scale is not affected by the expansion of the Universe. On the other hand, this inference can also be supported by the time evolution of the average scale factors and the</text>
<figure>
<location><page_10><loc_12><loc_18><loc_48><loc_32></location>
<caption>FIG. 2: Time evolution of the average Hubble parameters (left) and the scale factors (right) for the configurations with different mass scales. Here N = 640, N b = 320 for all simulations.</caption>
</figure>
<figure>
<location><page_10><loc_52><loc_18><loc_88><loc_32></location>
</figure>
<text><location><page_10><loc_73><loc_18><loc_73><loc_19></location>∗</text>
<text><location><page_10><loc_51><loc_25><loc_53><loc_26></location>a</text>
<figure>
<location><page_11><loc_11><loc_77><loc_47><loc_91></location>
<caption>FIG. 3: Time evolution of the absolute differences between the directional Hubble parameters for the configurations with different values of number of bubbles as depicted in the figures. For all simulations we take N = 640 and M Pl /M = 100.</caption>
</figure>
<text><location><page_11><loc_15><loc_74><loc_16><loc_76></location>×</text>
<text><location><page_11><loc_16><loc_75><loc_18><loc_76></location>10</text>
<text><location><page_11><loc_18><loc_75><loc_19><loc_76></location>-</text>
<text><location><page_11><loc_16><loc_65><loc_17><loc_66></location>0</text>
<text><location><page_11><loc_19><loc_65><loc_20><loc_66></location>1</text>
<text><location><page_11><loc_22><loc_65><loc_23><loc_66></location>2</text>
<text><location><page_11><loc_25><loc_65><loc_26><loc_66></location>3</text>
<text><location><page_11><loc_28><loc_65><loc_28><loc_66></location>4</text>
<text><location><page_11><loc_31><loc_65><loc_31><loc_66></location>5</text>
<text><location><page_11><loc_33><loc_65><loc_34><loc_66></location>6</text>
<text><location><page_11><loc_36><loc_65><loc_37><loc_66></location>7</text>
<text><location><page_11><loc_39><loc_65><loc_40><loc_66></location>8</text>
<text><location><page_11><loc_42><loc_65><loc_43><loc_66></location>9</text>
<text><location><page_11><loc_45><loc_65><loc_46><loc_66></location>10</text>
<text><location><page_11><loc_30><loc_63><loc_32><loc_64></location>tH</text>
<text><location><page_11><loc_32><loc_62><loc_32><loc_64></location>∗</text>
<text><location><page_11><loc_55><loc_88><loc_57><loc_91></location>×</text>
<text><location><page_11><loc_57><loc_89><loc_58><loc_90></location>10</text>
<text><location><page_11><loc_58><loc_89><loc_59><loc_91></location>-</text>
<text><location><page_11><loc_56><loc_79><loc_57><loc_80></location>0</text>
<text><location><page_11><loc_59><loc_79><loc_60><loc_80></location>1</text>
<text><location><page_11><loc_62><loc_79><loc_63><loc_80></location>2</text>
<text><location><page_11><loc_65><loc_79><loc_66><loc_80></location>3</text>
<text><location><page_11><loc_68><loc_79><loc_69><loc_80></location>4</text>
<text><location><page_11><loc_71><loc_79><loc_72><loc_80></location>5</text>
<text><location><page_11><loc_74><loc_79><loc_74><loc_80></location>6</text>
<text><location><page_11><loc_77><loc_79><loc_77><loc_80></location>7</text>
<text><location><page_11><loc_79><loc_79><loc_80><loc_80></location>8</text>
<text><location><page_11><loc_82><loc_79><loc_83><loc_80></location>9</text>
<text><location><page_11><loc_85><loc_79><loc_86><loc_80></location>10</text>
<text><location><page_11><loc_55><loc_74><loc_57><loc_76></location>×</text>
<text><location><page_11><loc_57><loc_75><loc_58><loc_76></location>10</text>
<text><location><page_11><loc_58><loc_75><loc_59><loc_76></location>-</text>
<text><location><page_11><loc_56><loc_65><loc_57><loc_66></location>0</text>
<text><location><page_11><loc_59><loc_65><loc_60><loc_66></location>1</text>
<text><location><page_11><loc_62><loc_65><loc_63><loc_66></location>2</text>
<text><location><page_11><loc_65><loc_65><loc_66><loc_66></location>3</text>
<text><location><page_11><loc_68><loc_65><loc_69><loc_66></location>4</text>
<text><location><page_11><loc_71><loc_65><loc_72><loc_66></location>5</text>
<text><location><page_11><loc_74><loc_65><loc_74><loc_66></location>6</text>
<text><location><page_11><loc_77><loc_65><loc_77><loc_66></location>7</text>
<text><location><page_11><loc_79><loc_65><loc_80><loc_66></location>8</text>
<text><location><page_11><loc_82><loc_65><loc_83><loc_66></location>9</text>
<text><location><page_11><loc_85><loc_65><loc_86><loc_66></location>10</text>
<text><location><page_11><loc_70><loc_63><loc_72><loc_64></location>tH</text>
<text><location><page_11><loc_72><loc_62><loc_72><loc_64></location>∗</text>
<text><location><page_11><loc_12><loc_37><loc_88><loc_52></location>average Hubble parameters represented in Fig. (2) . Higher mass ratios correspond to lower expansion rates in comparison at the same time scale. However, we should note that the scale factor of the case with M Pl /M = 10 is less than that of M Pl /M = 1 at the initial stages of their time evolution although it becomes slightly larger afterwards as can be seen in the figure as well. Nevertheless, the time evolution of those two curves are very similar and they differ from that of M Pl /M = 100 almost two orders of magnitude. However, the cases with M Pl /M = 1 , 10 are the scenarios we did not take into account further as they do not fulfill the requirements to complete the phase transition.</text>
<text><location><page_11><loc_12><loc_10><loc_88><loc_36></location>Since we have confirmed that it is required to choose M Pl /M ≳ 100 roughly in order the transition to complete, now we will investigate the effect of the other parameters on the outcomes such as the number of initiated bubbles, N b , and the number of grid points, N . We examine the impact of N b through four different simulations with N b = 10 , 40 , 320 , 600 fixing M Pl /M = 100 and N = 640 for all runs. The results are shown in Fig. (3) for the differences in the directional Hubble parameters, in Fig. (4) for a sample of the gradient energy densities together with the differences in the directional components, and in Fig. (5) for the average Hubble parameters and the energy density parameter of the shear scalar. As seen from Fig. (3) although there is only one order of magnitude between them, the maximum value in the differences decreases with increasing number of bubbles except for the case of N b = 10 for which we have found that the transition does not complete and gives results similar to the case with M Pl /M = 10 given in Fig. (1) . Regarding the difference in the gradient energies, we have found that they are in the same order of magnitude for all four different simulations</text>
<text><location><page_11><loc_59><loc_76><loc_59><loc_76></location>6</text>
<text><location><page_11><loc_68><loc_73><loc_69><loc_74></location>N</text>
<text><location><page_11><loc_70><loc_77><loc_72><loc_78></location>tH</text>
<text><location><page_11><loc_69><loc_73><loc_70><loc_74></location>b</text>
<text><location><page_11><loc_72><loc_76><loc_72><loc_78></location>∗</text>
<text><location><page_11><loc_70><loc_73><loc_74><loc_74></location>= 600</text>
<text><location><page_11><loc_82><loc_73><loc_84><loc_74></location>i, j</text>
<text><location><page_11><loc_84><loc_72><loc_84><loc_73></location>1</text>
<text><location><page_11><loc_84><loc_72><loc_85><loc_73></location>,</text>
<text><location><page_11><loc_85><loc_72><loc_86><loc_73></location>2</text>
<text><location><page_11><loc_84><loc_70><loc_84><loc_71></location>1</text>
<text><location><page_11><loc_84><loc_70><loc_85><loc_71></location>,</text>
<text><location><page_11><loc_85><loc_70><loc_86><loc_71></location>3</text>
<text><location><page_11><loc_84><loc_69><loc_84><loc_70></location>2</text>
<text><location><page_11><loc_84><loc_69><loc_85><loc_70></location>,</text>
<text><location><page_11><loc_85><loc_69><loc_86><loc_70></location>3</text>
<text><location><page_11><loc_19><loc_76><loc_19><loc_76></location>5</text>
<text><location><page_11><loc_11><loc_73><loc_13><loc_74></location>M</text>
<text><location><page_11><loc_11><loc_72><loc_13><loc_73></location>/</text>
<text><location><page_11><loc_11><loc_72><loc_14><loc_72></location>|</text>
<text><location><page_11><loc_11><loc_72><loc_13><loc_72></location>j</text>
<text><location><page_11><loc_11><loc_71><loc_13><loc_72></location>H</text>
<text><location><page_11><loc_11><loc_69><loc_14><loc_70></location>-</text>
<text><location><page_11><loc_11><loc_68><loc_13><loc_69></location>i</text>
<text><location><page_11><loc_11><loc_67><loc_13><loc_68></location>H</text>
<text><location><page_11><loc_11><loc_67><loc_14><loc_67></location>|</text>
<text><location><page_11><loc_14><loc_66><loc_14><loc_67></location>0</text>
<text><location><page_11><loc_14><loc_73><loc_14><loc_75></location>1</text>
<text><location><page_11><loc_28><loc_73><loc_29><loc_74></location>N</text>
<text><location><page_11><loc_29><loc_73><loc_30><loc_74></location>b</text>
<text><location><page_11><loc_32><loc_76><loc_32><loc_78></location>∗</text>
<text><location><page_11><loc_30><loc_73><loc_34><loc_74></location>= 320</text>
<text><location><page_11><loc_42><loc_73><loc_44><loc_74></location>i, j</text>
<text><location><page_11><loc_43><loc_72><loc_44><loc_73></location>1</text>
<text><location><page_11><loc_44><loc_72><loc_44><loc_73></location>,</text>
<text><location><page_11><loc_45><loc_72><loc_45><loc_73></location>2</text>
<text><location><page_11><loc_43><loc_70><loc_44><loc_71></location>1</text>
<text><location><page_11><loc_44><loc_70><loc_44><loc_71></location>,</text>
<text><location><page_11><loc_45><loc_70><loc_45><loc_71></location>3</text>
<text><location><page_11><loc_43><loc_69><loc_44><loc_70></location>2</text>
<text><location><page_11><loc_44><loc_69><loc_44><loc_70></location>,</text>
<text><location><page_11><loc_45><loc_69><loc_45><loc_70></location>3</text>
<text><location><page_11><loc_51><loc_87><loc_53><loc_89></location>M</text>
<text><location><page_11><loc_51><loc_87><loc_53><loc_87></location>/</text>
<text><location><page_11><loc_51><loc_86><loc_54><loc_87></location>|</text>
<text><location><page_11><loc_52><loc_86><loc_53><loc_86></location>j</text>
<text><location><page_11><loc_51><loc_85><loc_53><loc_86></location>H</text>
<text><location><page_11><loc_51><loc_83><loc_54><loc_84></location>-</text>
<text><location><page_11><loc_54><loc_84><loc_55><loc_85></location>1</text>
<text><location><page_11><loc_52><loc_83><loc_53><loc_83></location>i</text>
<text><location><page_11><loc_51><loc_82><loc_53><loc_83></location>H</text>
<text><location><page_11><loc_51><loc_81><loc_54><loc_82></location>|</text>
<text><location><page_11><loc_51><loc_73><loc_53><loc_74></location>M</text>
<text><location><page_11><loc_51><loc_72><loc_53><loc_73></location>/</text>
<text><location><page_11><loc_51><loc_72><loc_54><loc_72></location>|</text>
<text><location><page_11><loc_52><loc_72><loc_53><loc_72></location>j</text>
<text><location><page_11><loc_51><loc_71><loc_53><loc_72></location>H</text>
<text><location><page_11><loc_54><loc_71><loc_55><loc_72></location>2</text>
<text><location><page_11><loc_51><loc_69><loc_54><loc_70></location>-</text>
<text><location><page_11><loc_52><loc_68><loc_53><loc_69></location>i</text>
<text><location><page_11><loc_51><loc_67><loc_53><loc_68></location>H</text>
<text><location><page_11><loc_51><loc_67><loc_54><loc_67></location>|</text>
<text><location><page_11><loc_54><loc_66><loc_55><loc_67></location>0</text>
<text><location><page_11><loc_54><loc_80><loc_55><loc_81></location>0</text>
<text><location><page_11><loc_54><loc_88><loc_55><loc_89></location>2</text>
<text><location><page_11><loc_59><loc_90><loc_59><loc_91></location>5</text>
<text><location><page_11><loc_69><loc_87><loc_70><loc_88></location>N</text>
<text><location><page_11><loc_70><loc_87><loc_70><loc_88></location>b</text>
<text><location><page_11><loc_71><loc_87><loc_74><loc_88></location>= 40</text>
<text><location><page_11><loc_82><loc_87><loc_84><loc_88></location>i, j</text>
<text><location><page_11><loc_84><loc_86><loc_84><loc_87></location>1</text>
<text><location><page_11><loc_84><loc_86><loc_85><loc_87></location>,</text>
<text><location><page_11><loc_85><loc_86><loc_86><loc_87></location>2</text>
<text><location><page_11><loc_84><loc_84><loc_84><loc_86></location>1</text>
<text><location><page_11><loc_84><loc_84><loc_85><loc_86></location>,</text>
<text><location><page_11><loc_85><loc_84><loc_86><loc_86></location>3</text>
<text><location><page_11><loc_84><loc_83><loc_84><loc_84></location>2</text>
<text><location><page_11><loc_84><loc_83><loc_85><loc_84></location>,</text>
<text><location><page_11><loc_85><loc_83><loc_86><loc_84></location>3</text>
<figure>
<location><page_12><loc_11><loc_77><loc_47><loc_91></location>
<caption>FIG. 5: Time evolution of the Hubble parameters (left) and the shear scalar (right) for the simulations with different values of number of bubbles and grid points. Here we take M Pl /M = 100 for all simulations. For the solid lines, N = 640, and dx = 0 . 44 whereas N = 1280, dx = 0 . 22, and N b = 320 for the dashed line.</caption>
</figure>
<text><location><page_12><loc_32><loc_76><loc_32><loc_78></location>∗</text>
<text><location><page_12><loc_55><loc_88><loc_57><loc_91></location>×</text>
<text><location><page_12><loc_57><loc_89><loc_58><loc_90></location>10</text>
<text><location><page_12><loc_58><loc_89><loc_59><loc_91></location>-</text>
<text><location><page_12><loc_56><loc_79><loc_57><loc_80></location>0</text>
<text><location><page_12><loc_59><loc_79><loc_60><loc_80></location>1</text>
<text><location><page_12><loc_62><loc_79><loc_63><loc_80></location>2</text>
<text><location><page_12><loc_65><loc_79><loc_66><loc_80></location>3</text>
<text><location><page_12><loc_68><loc_79><loc_69><loc_80></location>4</text>
<text><location><page_12><loc_71><loc_79><loc_72><loc_80></location>5</text>
<text><location><page_12><loc_74><loc_79><loc_74><loc_80></location>6</text>
<text><location><page_12><loc_77><loc_79><loc_77><loc_80></location>7</text>
<text><location><page_12><loc_79><loc_79><loc_80><loc_80></location>8</text>
<text><location><page_12><loc_82><loc_79><loc_83><loc_80></location>9</text>
<text><location><page_12><loc_85><loc_79><loc_86><loc_80></location>10</text>
<text><location><page_12><loc_70><loc_77><loc_72><loc_78></location>tH</text>
<text><location><page_12><loc_72><loc_76><loc_73><loc_78></location>∗</text>
<paragraph><location><page_12><loc_12><loc_72><loc_88><loc_75></location>FIG. 4: Time evolution of in the directional average gradient energies (left) and the absolute differences (right) for a sample configuration with N b = 320, N = 640, and M Pl /M = 100.</paragraph>
<text><location><page_12><loc_12><loc_62><loc_88><loc_68></location>and we have represented one example of them in Fig. (4) . As seen from the figure, the directional quantities peak at early stages of the simulations corresponding to the bubble collision phase and then decreases smoothly throughout the run.</text>
<text><location><page_12><loc_12><loc_40><loc_88><loc_62></location>Together with the corresponding Hubble parameters the results for the shear scalar defined in Eq. (38) are represented in Fig. (5) in terms of its energy density parameter given in Eq. (40) . The curves show that the value of the shear scalar increases to some extent with decreasing number of bubbles and then starts to get smaller after some value in accordance with discussion about the difference between the components of the directional Hubble parameters in the previous paragraph and we should recall that for N b = 10 the transition is not accomplished. The shear scalar has almost the same shape throughout its time evolution in different simulations as if it was shifted depending on the number of bubbles. Nevertheless, the maximum values occur around 10 -8 -10 -10 right after the completion of the bubble collision phase and then within our time scale for the simulations it reaches 10 -11 -10 -12 decreasing gradually. Moreover, we have also provided a result drawn with a dashed line on</text>
<figure>
<location><page_12><loc_11><loc_20><loc_48><loc_35></location>
</figure>
<text><location><page_12><loc_33><loc_19><loc_34><loc_20></location>∗</text>
<figure>
<location><page_12><loc_51><loc_20><loc_88><loc_35></location>
</figure>
<text><location><page_12><loc_74><loc_19><loc_74><loc_20></location>∗</text>
<text><location><page_12><loc_52><loc_89><loc_53><loc_90></location>ac</text>
<text><location><page_12><loc_52><loc_89><loc_53><loc_89></location>v</text>
<text><location><page_12><loc_51><loc_87><loc_53><loc_89></location>/ρ</text>
<text><location><page_12><loc_51><loc_87><loc_54><loc_87></location>|</text>
<text><location><page_12><loc_52><loc_85><loc_53><loc_87></location>G,j</text>
<text><location><page_12><loc_51><loc_84><loc_53><loc_85></location>¯</text>
<text><location><page_12><loc_51><loc_84><loc_53><loc_85></location>ρ</text>
<text><location><page_12><loc_51><loc_83><loc_54><loc_84></location>-</text>
<text><location><page_12><loc_52><loc_81><loc_53><loc_82></location>G,i</text>
<text><location><page_12><loc_51><loc_80><loc_53><loc_81></location>¯</text>
<text><location><page_12><loc_51><loc_80><loc_53><loc_81></location>ρ</text>
<text><location><page_12><loc_51><loc_80><loc_54><loc_80></location>|</text>
<text><location><page_12><loc_54><loc_80><loc_55><loc_81></location>0</text>
<text><location><page_12><loc_54><loc_86><loc_55><loc_87></location>2</text>
<text><location><page_12><loc_59><loc_90><loc_60><loc_91></location>6</text>
<text><location><page_12><loc_68><loc_87><loc_69><loc_88></location>N</text>
<text><location><page_12><loc_69><loc_87><loc_70><loc_88></location>b</text>
<text><location><page_12><loc_71><loc_87><loc_74><loc_88></location>= 320</text>
<text><location><page_12><loc_82><loc_87><loc_84><loc_88></location>i, j</text>
<text><location><page_12><loc_84><loc_86><loc_84><loc_87></location>1</text>
<text><location><page_12><loc_84><loc_86><loc_85><loc_87></location>,</text>
<text><location><page_12><loc_85><loc_86><loc_86><loc_87></location>2</text>
<text><location><page_12><loc_84><loc_85><loc_84><loc_86></location>1</text>
<text><location><page_12><loc_84><loc_85><loc_85><loc_86></location>,</text>
<text><location><page_12><loc_85><loc_85><loc_86><loc_86></location>3</text>
<text><location><page_12><loc_84><loc_83><loc_84><loc_84></location>2</text>
<text><location><page_12><loc_84><loc_83><loc_85><loc_84></location>,</text>
<text><location><page_12><loc_85><loc_83><loc_86><loc_84></location>3</text>
<figure>
<location><page_13><loc_29><loc_75><loc_70><loc_91></location>
<caption>FIG. 6: Time evolution of the shear scalar of a simulation with M Pl /M = 100, N = 640, and N b = 320 for a longer run in comparison with the ones given in Fig. (5) .</caption>
</figure>
<text><location><page_13><loc_54><loc_74><loc_55><loc_76></location>∗</text>
<text><location><page_13><loc_12><loc_43><loc_88><loc_67></location>the right panel of Fig. (5) in order to check the effect of resolution of the simulation box. We see that the shear scalar gets slightly smaller for higher resolution with the same number of bubbles. Nevertheless, for all cases the shear scalar increases during the bubble collision phase and then it decreases as the scalar field oscillates around its true vacuum. Since we do not expect an anisotropic structure to develop in this configuration at late times, we can conclude that the maximum value for the shear scalar energy density parameter that we have found is around 10 -8 . Additionally, we have also provided the result of a longer run for a simulation with N b = 320, N = 640, and M Pl /M = 100 in Fig. (6) . We see that the energy density parameter of the shear scalar reaches values around 10 -14 at the end of that simulation. We need to note that this value already matches one of the most stringent constraint on today's value for the energy density parameter of the shear scalar [32] , that is, in the order of 10 -15 , and, moreover, it continues to decrease.</text>
<text><location><page_13><loc_12><loc_15><loc_88><loc_43></location>The results for the power spectra of the scalar field and the GW energy density, defined in Eqs. (35) and (36) respectively, are represented in Fig. (7) . We have shown only one example for the case of N b = 320 due to the fact that change in number of bubbles does not effect the shape of the spectrum neither for the scalar field nor for the GW energy density. We understand from the figures that the bubble collision phase is completed successfully before tH ∗ = 1 since the scalar field already oscillates around its true vacuum corresponding to a peak of its power spectrum near the mass value M t and the power spectrum of the GW energy density develops secondary peak there as well. The overall magnitude in both power spectra decreases due to the expansion while keeping the same shape. Therefore, the characteristic shapes for both power spectra are the same with the results of previous works [23, 26] . However, as we have mentioned before the tensor perturbations should be investigated in detail for Bianchi Type-I model and, in accordance with the spirit of the model, possible anisotropies in the power spectra with compatible definitions are needed to take into consideration which we left for future studies.</text>
<figure>
<location><page_14><loc_11><loc_72><loc_48><loc_91></location>
<caption>FIG. 7: Power spectrum of the scalar field (left) and the GW energy density (right) for a configuration with N b = 320. Here we take N = 640 and M Pl /M = 100.</caption>
</figure>
<text><location><page_14><loc_33><loc_71><loc_34><loc_73></location>∗</text>
<figure>
<location><page_14><loc_51><loc_72><loc_87><loc_91></location>
</figure>
<text><location><page_14><loc_73><loc_71><loc_74><loc_73></location>∗</text>
<section_header_level_1><location><page_14><loc_41><loc_62><loc_59><loc_64></location>6. CONCLUSION</section_header_level_1>
<text><location><page_14><loc_12><loc_28><loc_88><loc_60></location>In this paper, we have examined the cosmological first-order vacuum phase transitions in an anisotropic expanding universe modeled by Bianchi Type-I metric. To do this we have used a model with a scalar field that is minimally coupled to the gravity and has a typical potential for the first-order phase transitions. After representing the main equations in their analytical forms, we have put them into numerical set in accordance with the leapfrog algorithm. Then, we have integrated the equations of motion for the scalar field and for the directional scale factors as well as for the tensor perturbations, the results of which are valid up to a gauge transformation due to the fact that in Bianchi Type-I model the TT gauge should be modified [39, 40] . In addition to that it is also important to check the anisotropy in the GW power spectrum to either validate or eliminate a model or the source of the signal through possible upcoming observations even by taking the periodicity of the simulation box into account [44] . Nevertheless, main purpose of this work was to find out the mass scale at which the bubble collision phase is accomplished and, additionally, to track the anisotropy by determining the behavior of the shear scalar defined in Eq. (38) , in other words, to consider the anisotropy in the background evolution due to the scalar field responsible from the transition.</text>
<text><location><page_14><loc_12><loc_10><loc_88><loc_27></location>We have run several simulations with different number of initiated bubbles which determines the initial conditions and correspondingly has the major impact for the time evolution of all variables. In addition to that due to the computational costs we have simulated only one configuration with higher resolution and the one with a longer run in comparison with the others. Before investigating the shear scalar, we have represented the results for three simulations with different mass scales, namely M Pl /M = 1 , 10 , 100, which have shown that the phase transition does not complete for the runs roughly M Pl /M ≲ 100. In those cases either the bubbles expand for a while and then the expansion of the universe prevents them to coalesce entirely or they do not find a chance to collide at all because of the expansion</text>
<text><location><page_15><loc_12><loc_79><loc_88><loc_91></location>of the universe. We have given the results of examples for those two cases with the mass scales of M Pl /M = 10 and M Pl /M = 1, respectively, in Fig. (1) together with the case of M Pl /M = 100 that was adopted for the rest of the simulations. We did not use mass scales greater than that because of the computational costs and, moreover, this value is enough to examine the anisotropy in first place due to the fact that higher rates for M Pl /M suppress the expansion of the Universe more and more.</text>
<text><location><page_15><loc_12><loc_61><loc_88><loc_79></location>After determining the order of minimum mass scale, that is M Pl /M ≈ 100, at which the phase transition can be completed successfully, we have run simulations to determine the time evolution of the energy density parameter for the shear scalar by examining the effect of different initial conditions created through different number of initiated bubbles. But before this we have shown in Fig. (3) that absolute differences in the directional Hubble parameters are in the order of 10 -5 for N b = 10 , 40 , 320 while it is around 10 -6 for N b = 600 at most. Additionally, we have also provided the directional gradient energies and their differences in Fig. (4) for the same configurations with the number of bubbles mentioned and have shown that their differences are in the order of 10 -6 .</text>
<text><location><page_15><loc_12><loc_27><loc_88><loc_61></location>In Fig. (5) we have presented the results for the Hubble parameters and the shear scalars. Moreover, we have also given the outcomes for a longer run of a specific configuration in Fig. (6) . As indicated before from the results of the directional Hubble parameters it seems that the relatively small number of bubbles give rise to high values for the shear scalar except for the case of N b = 10 which seems to be a counter example for this conclusion at first glance, but the bubble collision phase is not completed for that simulation. Therefore, as one may guess before, in addition to the mass scale, the proportion of the number of initiated bubbles to whole simulation box is another quantity that also determines whether a phase transition can be completed or not. This can also be seen from Fig. (5) for the Hubble parameters where the case of N b = 10 is different from the others at the beginning of the simulation. Nevertheless, we have found that before decreasing smoothly, the energy density parameter for the shear scalar gains a peak between 10 -8 -10 -10 which occurs at bubble collision phase. With the aforementioned longer run we have shown that Ω σ 2 becomes close to one of the constraints obtained for its today's value [32] . Additionally, it seems that the expansion of the Universe does not effect the phase transition for a typical mass scales of M Pl /M ≳ 100 with a fairly distributed number of initiated bubbles, since hereby we have tested impact of the expansion itself as well besides the anisotropy.</text>
<section_header_level_1><location><page_15><loc_41><loc_22><loc_59><loc_23></location>Acknowledgements</section_header_level_1>
<text><location><page_15><loc_12><loc_10><loc_88><loc_19></location>This work is supported by The Scientific and Technological Research Council of Turkiye (T UB ˙ ITAK) through grant number 121F066. Computing resources used in this work were provided by the National Center for High Performance Computing of Turkiye (UHeM) under grant number 5013072022 and the simulations were partially performed at TUBITAK ULAKBIM, High Performance and Grid Computing Center (TRUBA resources).</text>
<unordered_list>
<list_item><location><page_16><loc_12><loc_85><loc_87><loc_86></location>[1] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. Lett. 116 , 061102 (2016), arXiv:1602.03837.</list_item>
<list_item><location><page_16><loc_12><loc_83><loc_76><loc_84></location>[2] G. Agazie et al. (NANOGrav), Astrophys. J. Lett. 951 , L8 (2023), arXiv:2306.16213.</list_item>
<list_item><location><page_16><loc_12><loc_81><loc_45><loc_82></location>[3] P. A.-S. et al., (2017), arXiv:1702.00786.</list_item>
<list_item><location><page_16><loc_12><loc_79><loc_52><loc_80></location>[4] C. Caprini et al. , JCAP 03 , 024, arXiv:1910.13125.</list_item>
<list_item><location><page_16><loc_12><loc_77><loc_88><loc_78></location>[5] N. Seto, S. Kawamura, and T. Nakamura, Phys. Rev. Lett. 87 , 221103 (2001), arXiv:astro-ph/0108011.</list_item>
<list_item><location><page_16><loc_12><loc_72><loc_88><loc_76></location>[6] W.-H. Ruan, Z.-K. Guo, R.-G. Cai, and Y.-Z. Zhang, Int. J. Mod. Phys. A 35 , 2050075 (2020), arXiv:1807.09495.</list_item>
<list_item><location><page_16><loc_12><loc_70><loc_74><loc_71></location>[7] J. Luo et al. (TianQin), Class. Quant. Grav. 33 , 035010 (2016), arXiv:1512.02076.</list_item>
<list_item><location><page_16><loc_12><loc_68><loc_78><loc_69></location>[8] V. Corbin and N. J. Cornish, Class. Quant. Grav. 23 , 2435 (2006), arXiv:gr-qc/0512039.</list_item>
<list_item><location><page_16><loc_12><loc_66><loc_76><loc_67></location>[9] A. Mazumdar and G. White, Rept. Prog. Phys. 82 , 076901 (2019), arXiv:1811.01948.</list_item>
<list_item><location><page_16><loc_12><loc_62><loc_88><loc_65></location>[10] M. B. Hindmarsh, M. Luben, J. Lumma, and M. Pauly, SciPost Phys. Lect. Notes 24 , 1 (2021), arXiv:2008.09136.</list_item>
<list_item><location><page_16><loc_12><loc_60><loc_62><loc_61></location>[11] K. Rajagopal and F. Wilczek, Nuclear Physics B 399 , 395 (1993).</list_item>
<list_item><location><page_16><loc_12><loc_58><loc_62><loc_59></location>[12] S. Dimopoulos and H. Georgi, Nuclear Physics B 193 , 150 (1981).</list_item>
<list_item><location><page_16><loc_12><loc_54><loc_88><loc_57></location>[13] K. Kajantie, M. Laine, K. Rummukainen, and M. E. Shaposhnikov, Phys. Rev. Lett. 77 , 2887 (1996), arXiv:hep-ph/9605288.</list_item>
<list_item><location><page_16><loc_12><loc_49><loc_88><loc_53></location>[14] K. Kajantie, M. Laine, K. Rummukainen, and M. E. Shaposhnikov, Nucl. Phys. B 493 , 413 (1997), arXiv:hep-lat/9612006.</list_item>
<list_item><location><page_16><loc_12><loc_47><loc_75><loc_48></location>[15] D. J. Weir, Phil. Trans. Roy. Soc. Lond. A 376 , 20170126 (2018), arXiv:1705.01783.</list_item>
<list_item><location><page_16><loc_12><loc_45><loc_77><loc_46></location>[16] S. R. Coleman, Phys. Rev. D 15 , 2929 (1977), [Erratum: Phys.Rev.D 16, 1248 (1977)].</list_item>
<list_item><location><page_16><loc_12><loc_43><loc_64><loc_44></location>[17] C. G. Callan, Jr. and S. R. Coleman, Phys. Rev. D 16 , 1762 (1977).</list_item>
<list_item><location><page_16><loc_12><loc_41><loc_46><loc_42></location>[18] A. D. Linde, Phys. Lett. B 100 , 37 (1981).</list_item>
<list_item><location><page_16><loc_12><loc_39><loc_76><loc_40></location>[19] A. D. Linde, Nucl. Phys. B 216 , 421 (1983), [Erratum: Nucl.Phys.B 223, 544 (1983)].</list_item>
<list_item><location><page_16><loc_12><loc_37><loc_61><loc_38></location>[20] S. R. Coleman and F. De Luccia, Phys. Rev. D 21 , 3305 (1980).</list_item>
<list_item><location><page_16><loc_12><loc_35><loc_73><loc_36></location>[21] S. R. Coleman, V. Glaser, and A. Martin, Commun. Math. Phys. 58 , 211 (1978).</list_item>
<list_item><location><page_16><loc_12><loc_33><loc_78><loc_34></location>[22] A. Kosowsky and M. S. Turner, Phys. Rev. D 47 , 4372 (1993), arXiv:astro-ph/9211004.</list_item>
<list_item><location><page_16><loc_12><loc_31><loc_83><loc_32></location>[23] D. Cutting, M. Hindmarsh, and D. J. Weir, Phys. Rev. D 97 , 123513 (2018), arXiv:1802.05712.</list_item>
<list_item><location><page_16><loc_12><loc_28><loc_86><loc_30></location>[24] D. Cutting, M. Hindmarsh, and D. J. Weir, Phys. Rev. Lett. 125 , 021302 (2020), arXiv:1906.00480.</list_item>
<list_item><location><page_16><loc_12><loc_24><loc_88><loc_27></location>[25] D. Cutting, E. G. Escartin, M. Hindmarsh, and D. J. Weir, Phys. Rev. D 103 , 023531 (2021), arXiv:2005.13537.</list_item>
<list_item><location><page_16><loc_12><loc_22><loc_80><loc_23></location>[26] A. S. Arapo˘glu and A. E. Yukselci, Phys. Dark Univ. 40 , 101176 (2023), arXiv:2210.16699.</list_item>
<list_item><location><page_16><loc_12><loc_20><loc_63><loc_21></location>[27] Z. Zhao, Y. Di, L. Bian, and R.-G. Cai, (2022), arXiv:2204.04427.</list_item>
<list_item><location><page_16><loc_12><loc_18><loc_79><loc_19></location>[28] C. Caprini, R. Durrer, and G. Servant, Phys. Rev. D 77 , 124015 (2008), arXiv:0711.2593.</list_item>
<list_item><location><page_16><loc_12><loc_16><loc_59><loc_17></location>[29] R. Jinno and M. Takimoto, JCAP 01 , 060, arXiv:1707.03111.</list_item>
<list_item><location><page_16><loc_12><loc_14><loc_76><loc_15></location>[30] H. Zhong, B. Gong, and T. Qiu 10.1007/JHEP02(2022)077 (2021), arXiv:2107.01845.</list_item>
<list_item><location><page_16><loc_12><loc_12><loc_85><loc_13></location>[31] T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis, Phys. Rept. 513 , 1 (2012), arXiv:1106.2476.</list_item>
<list_item><location><page_16><loc_12><loc_10><loc_88><loc_11></location>[32] O. Akarsu, S. Kumar, S. Sharma, and L. Tedesco, Phys. Rev. D 100 , 023532 (2019), arXiv:1905.06949.</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_17><loc_12><loc_90><loc_77><loc_91></location>[33] N. Bartolo et al. (LISA Cosmology Working Group), JCAP 11 , 009, arXiv:2201.08782.</list_item>
<list_item><location><page_17><loc_12><loc_87><loc_56><loc_89></location>[34] G. Agazie et al. (NANOGrav), (2023), arXiv:2306.16221.</list_item>
<list_item><location><page_17><loc_12><loc_85><loc_76><loc_86></location>[35] L. Bian, S. Ge, J. Shu, B. Wang, X.-Y. Yang, and J. Zong, (2023), arXiv:2307.02376.</list_item>
<list_item><location><page_17><loc_12><loc_83><loc_81><loc_84></location>[36] D. G. Figueroa, M. Pieroni, A. Ricciardone, and P. Simakachorn, (2023), arXiv:2307.02399.</list_item>
<list_item><location><page_17><loc_12><loc_81><loc_64><loc_82></location>[37] Y.-M. Wu, Z.-C. Chen, and Q.-G. Huang, (2023), arXiv:2307.03141.</list_item>
<list_item><location><page_17><loc_12><loc_79><loc_86><loc_80></location>[38] J. Garcia-Bellido, D. G. Figueroa, and A. Sastre, Phys. Rev. D 77 , 043517 (2008), arXiv:0707.0839.</list_item>
<list_item><location><page_17><loc_12><loc_77><loc_66><loc_78></location>[39] P. G. Miedema and W. A. van Leeuwen, Phys. Rev. D 47 , 3151 (1993).</list_item>
<list_item><location><page_17><loc_12><loc_75><loc_79><loc_76></location>[40] H. T. Cho and A. D. Speliotopoulos, Phys. Rev. D 52 , 5445 (1995), arXiv:gr-qc/9504046.</list_item>
<list_item><location><page_17><loc_12><loc_71><loc_88><loc_74></location>[41] S. Behnel, R. Bradshaw, C. Citro, L. Dalcin, D. Seljebotn, and K. Smith, Computing in Science Engineering 13 , 31 (2011).</list_item>
<list_item><location><page_17><loc_12><loc_69><loc_85><loc_70></location>[42] L. Dalc'ın, R. Paz, and M. Storti, Journal of Parallel and Distributed Computing 65 , 1108 (2005).</list_item>
<list_item><location><page_17><loc_12><loc_66><loc_85><loc_68></location>[43] M. Mortensen and H. P. Langtangen, Comput. Phys. Commun. 203 , 53 (2016), arXiv:1602.03638.</list_item>
<list_item><location><page_17><loc_12><loc_62><loc_88><loc_65></location>[44] G. R'acz, I. Szapudi, I. Csabai, and L. Dobos, Mon. Not. Roy. Astron. Soc. 503 , 5638 (2021), arXiv:2006.10399.</list_item>
</unordered_list>
</document>
[
{
"title": "Cosmological First-Order Vacuum Phase Transitions in an Expanding Anisotropic Universe",
"content": "A. Sava¸s Arapo˘glu ∗ and A. Emrah Y¨ukselci † Istanbul Technical University, Faculty of Science and Letters, Physics Engineering Department, 34469, Maslak, Istanbul, Turkey",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "We examine the anisotropy originated from a first-order vacuum phase transitions through threedimensional numerical simulations. We apply Bianchi Type-I metric to our model that has one scalar field minimally coupled to the gravity. We calculate the time evolution of the energy density for the shear scalar and the directional Hubble parameters as well as the power spectra for the scalar field and the gravitational radiation although there are a number of caveats for the tensor perturbations in Bianchi Type-I universe. We run simulations with different mass scales of the scalar field, therefore, in addition to investigation of anisotropy via the shear scalar, we also determine at which mass scale the phase transition completes successfully, hence, neglecting the expansion of the Universe does not significantly affect the results. Finally, we showed that such an event may contribute to the total anisotropy depending on the mass scale of the scalar field and the initial population of nucleated bubbles.",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "The first direct detection of the gravitational waves (GWs) [1] has opened a new era in observation of the Universe since they can carry the information related to very source phenomenon thanks to their weakly-interacting character. Although recent results of the observation with low-frequency GWs may point out an astrophysical origin [2] as is the first one, yet this is another important step towards mapping the stochastic GW background that may have contributions originated from the events of the early stages in the Universe as well. The imprints of such GWs may be detected through space-based GW detectors that are planned to be built in the future [3-8] . Cosmological first-order phase transitions (PTs), which may have possibly occurred in the early Universe, are one type of phenomenon that can create GWs as an outcome [9, 10] . In spite of the fact that the well-known examples are the electroweak and the quark-hadron PTs, they may have taken place at any scale in the early Universe between QCD [11] and GUT [12] scales, respectively. However, the standard model of particle physics does not predict first-order phase transitions [13, 14] , yet there are many extensions of it to allow that (see e.g. [9, 10, 15] and references therein). This type of event could take place through the bubble nucleation mechanism, theory of which was studied at zero [16, 17] and finite [18, 19] temperatures in flat space-time whereas the gravitational effects were discussed later in Ref. [20] . It is known that the most probable initial profile for the bubbles has a O (4) symmetric form [21] although there is no such proof for a curved space-time yet. The GWs originated from first-order PTs are created by the shear stress caused by the deformation of the symmetric structure of colliding bubbles. The theoretical approach to examine such an event was studied in Ref. [22] where collided portions of the bubbles are not taken into account. However, it was shown through the numerical simulations [23-25] that those parts should be considered since the scalar field oscillates around its true vacuum and give rise to another peak in the gravitational radiation power spectrum related to its mass in addition to the maximum value associated with the mean separation between bubbles. This has the potential to determine the parameters of a model and/or even the model itself. Recently, in the context of the scalar-tensor theories, the scalar field non-minimally coupled to the gravity has also been studied through the numerical simulations [26] . Another example is the study of the two-step phase transition related to electroweak symmetry breaking [27] . In addition to numerical approaches, it has been shown that it is also possible to analytically calculate the power spectra to some extent for the gravitational radiation formed during the bubble collision phase [28-30] . In order to investigate the anisotropy we implement Bianchi Type-I metric where each direction has a different scale factors unlike the Friedmann-Lemaˆıtre-Robertson-Walker metric which is, indeed, a particular type of Bianchi Type-I model in this manner. This model has been widely used in the cosmological context (see e.g. [31] and references therein) since it is one of the simplest extensions for isotropic space-time and it may even offer some solutions to the well-known problems such as H 0 tension [32] . Moreover, on the GW front with another aspect of the anisotropy, a possible detection of it in the stochastic GW background may even enable to distinguish between superimposed sources [33] . On the other hand, in light of recent observations, it has been reported that any sort of anisotropy is not encountered in the data [34] . However, the picture may change after the inclusion of more data into the analysis, and this may lead to understand the birthplace of an observed GW signal, in other words, whether it has an astrophysical or cosmological origins. Although there is no concrete evidence yet to determine the origins of the signal [35, 36] , analysis in Ref. [37] on the NanoGRAV data [2] indicates that some cosmological sources, e.g. strong first-order PTs, can provide comparable results with the astrophysical ones such as supermassive black hole binaries. The paper is structured as follows: In Section (2) , we describe the main equations of the model; in Section (3) we provide the modified equations in accordance with the numerical scheme; in Section (4) we define the quantities to be followed during the simulations; in Section (5) we present the outcomes of the simulations and discuss the results in Section (6) .",
"pages": [
2,
3
]
},
{
"title": "2. SET-UP",
"content": "In this section we provide the main equations that will be used throughout the paper. To this end, we start with the Einstein field equations given as where the Planck mass is defined via M -2 Pl ≡ 8 πG and is equal to M Pl = 2 . 435 × 10 18 GeV in natural units, i.e. c = ℏ = 1, which is adopted in this work. In the presence of only one scalar field minimally coupled to the gravity the energy-momentum tensor is given by the following expression On the other hand, the equation of motion for the scalar field is obtained as and we use the potential in the form of where M , δ , λ , and V c are constants. For the background evolution, we implement the Bianchi Type-I metric as where a 1 ( t ), a 2 ( t ), and a 3 ( t ) are the scale factors in x , y , and z directions and they are the functions of time only. For future convenience, we also define where H i is the directional and H is the average Hubble parameters. The metric given in Eq. (5) yields tt , xx , yy , and zz components of Eq. (1) , respectively, in the following forms where the angle brackets denotes the spatial average over all simulation box, the dot represents the derivative with respect to t , and the subscript letters x, y, z stand for the spatial derivatives in the corresponding directions. Moreover, for the sake of simplification we eliminate the time derivative of the scalar field from Eqs. (8) , (9) , (10) with the help of the constraint equation, i.e. Eq. (7) , and obtain the final form of the equations of motion for the scale factors as follows On the other hand, the equation of motion for the scalar field, Eq. (3) , becomes where the average Hubble parameter, H , is defined in Eq. (6) and the potential for the scalar field is given in Eq. (4) . Regarding the gravitational waves, the transverse-traceless (TT) part of the tensor perturbations, h ij , can be related to an auxiliary tensor, u ij , by defining a projection operator [38] as where u lm ( t, k ) is the Fourier transform of u ij ( t, x ) and the projection operator is defined as Then, this method yields the equation of motion in the following form Although we use TT gauge for the tensor perturbations here, this not need to be entirely true since there is a gauge fixing problem in Bianchi Type-I model [39, 40] . Therefore, we should note that the results related to the tensor perturbations, i.e. the gravitational waves, represented in this work are valid up to a gauge transformation. We will make some more comments on this issue in the following sections.",
"pages": [
3,
4,
5
]
},
{
"title": "3. NUMERICAL PROCEDURE",
"content": "The code which we used for this work is the same with that of Ref. [26] . It has been written in Python programming language with the help of Cython [41] extension for the intensive iterations. Parallel processing has been realized by the pencil decomposition and the communication between processes has been ensured by mpi4py [42] package. We constructed similar algorithms given in Ref. [43] for the Fourier transforms. We implement the staggered leapfrog algorithm for advance in time with 7-point stencil for the Laplacian operator. In accordance with this scheme, it is necessary to eliminate the first time derivatives of the variables in the equations of motion to achieve the stability and the consistency for the numerical calculations. To this end, we define the new variables and the constants as We will denote the derivative with respect to τ by the prime symbol in the following sections whereas we keep the same notation for the spatial derivatives, in other words, the subscript x should be understood as the derivative with respect to Mx . In addition to that, we use fixed spatial resolution with dx = 0 . 44 in general and the value of the Courant factor is taken c = 0 . 4 for all simulations.",
"pages": [
5
]
},
{
"title": "3.1. Numerical Equation Set",
"content": "With definition of the new variables the equation of motion for the scalar field given in Eq. (3) takes the following form where and the redefined potential is given by where we set V c such that V ( ϕ t ) = 0. One may also choose a small constant instead of zero potential value as the cosmological constant. However, this is not in the scope of this paper and it needs to be considered with more realistic setups for long-time simulations. On the other hand, the equations for the scale factors become Finally, the equation of motion for the tensor perturbations is obtained as where K is defined in Eq. (21) . These are the equations that will be solved numerically. The structure of the equations for the scalar field and the tensor perturbations are already in a suitable form for the leapfrog algorithm. However, the equations for the scale factors need a modification since the first time derivatives are one half step behind the corresponding variable at each step. In order to synchronize the variables and their first time derivatives we also keep their values from the previous step, meaning that we calculate the derivatives for this particular purpose as a ' 1 ( t ) ≈ [ a ' 1 ( t +∆ t/ 2) + a 1 ( t -∆ t/ 2)] / 2 where ∆ t is the time step. We use those values to calculate the expression given in Eq. (21) as well.",
"pages": [
6
]
},
{
"title": "3.2. Initial Conditions",
"content": "In order to start the simulations we use the thin-wall approximation [16] to determine the initial profile of the scalar field, that is, we implement where R c and l 0 are the critical radius and the bubble wall length, respectively, which can be found from the following expressions [23] Furthermore, the time derivative of the scalar field is taken initially to be zero, i.e. ψ ' ( t = 0) = 0. On the other hand, the bubble nucleation points in the lattice are randomly determined and the bubbles are nucleated simultaneously at the beginning of the simulations. As for the scale factors we choose and in order to determine the initial values for the derivatives of the scale factors we use Eq. (7) written with the new variables as with the help of Eq. (29) . Moreover, assuming that a ' 1 = a ' 2 = a ' 3 initially we get where Finally, we set v ij ( t = 0) = v ' ij ( t = 0) = 0 for the tensor perturbations.",
"pages": [
7
]
},
{
"title": "4. DENSITIES, POWER SPECTRA, AND SHEAR SCALAR",
"content": "Here we give the definitions for the densities, the power spectra for both the scalar field and the gravitational waves, and the shear scalar which will show the amount of anisotropy in simulations with different configurations. Starting with the densities for the scalar field we have which are the kinetic, the gradient, and the potential energies, respectively. On the other hand, for the energy density of the gravitational waves, the following expression is calculated Although we keep the gradient terms explicitly, we need to emphasize that they almost have no effect on the results. The power spectrum for the scalar field is expressed by and for the gravitational waves we use We also implement the following normalization where H ∗ is the average Hubble parameter value at the time of the transition, R ∗ is the mean bubble separation equals to ( V /N b ) 1 / 3 in which V and N b are the physical volume of the simulation box and the number of bubbles, respectively. We should emphasize that the above definitions for the power spectra are not entirely correct in the anisotropic case. However, we will represent them regardless and left those calculations to future studies. Because, in addition to investigating the mass scale for the scalar field, our main focus in this paper is to demonstrate the time evolution of the shear scalar defined as in order to quantify the anisotropy in the background. The Friedmann equation can then be written in the following form or in terms of energy density parameters 1 = Ω σ 2 + Ω ϕ where we have defined the energy density parameter for the shear scalar as follows and put all the terms of the scalar field into Ω ϕ since we will not use it further. ∗",
"pages": [
7,
8,
10
]
},
{
"title": "5. SIMULATION RESULTS",
"content": "We have simulated several configurations of the model with different parameter values as listed in Table (1) . In order to distinguish their effects on the results we have changed the number of nucleated bubbles (simulations 3-6) and the number of grid points (simulations 6, 7) as well as the mass scale (simulations 1, 2, 3) which is one of the main concern of this study in the context of free parameters. The values of the other constants that depend on the free parameters have also been calculated and given in the same table. The outcomes that will be reported and discussed in detail below are the Hubble parameters, densities, power spectra, and the shear scalar whose time evolution is another main interest of our work as it is one of the key indicators of anisotropy. First of all, we have run simulations to see the mass scale at which the phase transition can be completed successfully. We see from two-dimensional slices of the simulations represented in Fig. (1) that the phase transition does not complete for the configurations with M Pl /M = 1 , 10. This is due to the fact that the expansion rate is higher for relatively small values of M Pl /M as one can deduce this result from the right-hand side of Eqs. (25) . The high expansion rate causes that universe to outgrow with a speed much more than the enlargement of the bubbles, therefore, the bubble collision phase either can not be completed entirely or does not happen at all. For M Pl /M = 10 the bubbles expand for a while at the start of the simulation and the ones close to each other collide partially, but then, the expansion rate of the universe eventually dominates the dynamics, whereas the bubble collision does not even occur in the case of M Pl /M = 1. On the other hand, for M Pl /M = 100 we notice that this kind of effect does not take place and the bubble collision phase is completed successfully in a time less than H -1 ∗ . At this point it is necessary to emphasize that this mass value is beyond even the GUT scale let alone the EW phase transition epoch, which are around 10 15 GeV and 100 GeV, respectively. Therefore, the completion of a phase transition at either GUT or EW scale is not affected by the expansion of the Universe. On the other hand, this inference can also be supported by the time evolution of the average scale factors and the ∗ a × 10 - 0 1 2 3 4 5 6 7 8 9 10 tH ∗ × 10 - 0 1 2 3 4 5 6 7 8 9 10 × 10 - 0 1 2 3 4 5 6 7 8 9 10 tH ∗ average Hubble parameters represented in Fig. (2) . Higher mass ratios correspond to lower expansion rates in comparison at the same time scale. However, we should note that the scale factor of the case with M Pl /M = 10 is less than that of M Pl /M = 1 at the initial stages of their time evolution although it becomes slightly larger afterwards as can be seen in the figure as well. Nevertheless, the time evolution of those two curves are very similar and they differ from that of M Pl /M = 100 almost two orders of magnitude. However, the cases with M Pl /M = 1 , 10 are the scenarios we did not take into account further as they do not fulfill the requirements to complete the phase transition. Since we have confirmed that it is required to choose M Pl /M ≳ 100 roughly in order the transition to complete, now we will investigate the effect of the other parameters on the outcomes such as the number of initiated bubbles, N b , and the number of grid points, N . We examine the impact of N b through four different simulations with N b = 10 , 40 , 320 , 600 fixing M Pl /M = 100 and N = 640 for all runs. The results are shown in Fig. (3) for the differences in the directional Hubble parameters, in Fig. (4) for a sample of the gradient energy densities together with the differences in the directional components, and in Fig. (5) for the average Hubble parameters and the energy density parameter of the shear scalar. As seen from Fig. (3) although there is only one order of magnitude between them, the maximum value in the differences decreases with increasing number of bubbles except for the case of N b = 10 for which we have found that the transition does not complete and gives results similar to the case with M Pl /M = 10 given in Fig. (1) . Regarding the difference in the gradient energies, we have found that they are in the same order of magnitude for all four different simulations 6 N tH b ∗ = 600 i, j 1 , 2 1 , 3 2 , 3 5 M / | j H - i H | 0 1 N b ∗ = 320 i, j 1 , 2 1 , 3 2 , 3 M / | j H - 1 i H | M / | j H 2 - i H | 0 0 2 5 N b = 40 i, j 1 , 2 1 , 3 2 , 3 ∗ × 10 - 0 1 2 3 4 5 6 7 8 9 10 tH ∗ and we have represented one example of them in Fig. (4) . As seen from the figure, the directional quantities peak at early stages of the simulations corresponding to the bubble collision phase and then decreases smoothly throughout the run. Together with the corresponding Hubble parameters the results for the shear scalar defined in Eq. (38) are represented in Fig. (5) in terms of its energy density parameter given in Eq. (40) . The curves show that the value of the shear scalar increases to some extent with decreasing number of bubbles and then starts to get smaller after some value in accordance with discussion about the difference between the components of the directional Hubble parameters in the previous paragraph and we should recall that for N b = 10 the transition is not accomplished. The shear scalar has almost the same shape throughout its time evolution in different simulations as if it was shifted depending on the number of bubbles. Nevertheless, the maximum values occur around 10 -8 -10 -10 right after the completion of the bubble collision phase and then within our time scale for the simulations it reaches 10 -11 -10 -12 decreasing gradually. Moreover, we have also provided a result drawn with a dashed line on ∗ ∗ ac v /ρ | G,j ¯ ρ - G,i ¯ ρ | 0 2 6 N b = 320 i, j 1 , 2 1 , 3 2 , 3 ∗ the right panel of Fig. (5) in order to check the effect of resolution of the simulation box. We see that the shear scalar gets slightly smaller for higher resolution with the same number of bubbles. Nevertheless, for all cases the shear scalar increases during the bubble collision phase and then it decreases as the scalar field oscillates around its true vacuum. Since we do not expect an anisotropic structure to develop in this configuration at late times, we can conclude that the maximum value for the shear scalar energy density parameter that we have found is around 10 -8 . Additionally, we have also provided the result of a longer run for a simulation with N b = 320, N = 640, and M Pl /M = 100 in Fig. (6) . We see that the energy density parameter of the shear scalar reaches values around 10 -14 at the end of that simulation. We need to note that this value already matches one of the most stringent constraint on today's value for the energy density parameter of the shear scalar [32] , that is, in the order of 10 -15 , and, moreover, it continues to decrease. The results for the power spectra of the scalar field and the GW energy density, defined in Eqs. (35) and (36) respectively, are represented in Fig. (7) . We have shown only one example for the case of N b = 320 due to the fact that change in number of bubbles does not effect the shape of the spectrum neither for the scalar field nor for the GW energy density. We understand from the figures that the bubble collision phase is completed successfully before tH ∗ = 1 since the scalar field already oscillates around its true vacuum corresponding to a peak of its power spectrum near the mass value M t and the power spectrum of the GW energy density develops secondary peak there as well. The overall magnitude in both power spectra decreases due to the expansion while keeping the same shape. Therefore, the characteristic shapes for both power spectra are the same with the results of previous works [23, 26] . However, as we have mentioned before the tensor perturbations should be investigated in detail for Bianchi Type-I model and, in accordance with the spirit of the model, possible anisotropies in the power spectra with compatible definitions are needed to take into consideration which we left for future studies. ∗ ∗",
"pages": [
10,
11,
12,
13,
14
]
},
{
"title": "6. CONCLUSION",
"content": "In this paper, we have examined the cosmological first-order vacuum phase transitions in an anisotropic expanding universe modeled by Bianchi Type-I metric. To do this we have used a model with a scalar field that is minimally coupled to the gravity and has a typical potential for the first-order phase transitions. After representing the main equations in their analytical forms, we have put them into numerical set in accordance with the leapfrog algorithm. Then, we have integrated the equations of motion for the scalar field and for the directional scale factors as well as for the tensor perturbations, the results of which are valid up to a gauge transformation due to the fact that in Bianchi Type-I model the TT gauge should be modified [39, 40] . In addition to that it is also important to check the anisotropy in the GW power spectrum to either validate or eliminate a model or the source of the signal through possible upcoming observations even by taking the periodicity of the simulation box into account [44] . Nevertheless, main purpose of this work was to find out the mass scale at which the bubble collision phase is accomplished and, additionally, to track the anisotropy by determining the behavior of the shear scalar defined in Eq. (38) , in other words, to consider the anisotropy in the background evolution due to the scalar field responsible from the transition. We have run several simulations with different number of initiated bubbles which determines the initial conditions and correspondingly has the major impact for the time evolution of all variables. In addition to that due to the computational costs we have simulated only one configuration with higher resolution and the one with a longer run in comparison with the others. Before investigating the shear scalar, we have represented the results for three simulations with different mass scales, namely M Pl /M = 1 , 10 , 100, which have shown that the phase transition does not complete for the runs roughly M Pl /M ≲ 100. In those cases either the bubbles expand for a while and then the expansion of the universe prevents them to coalesce entirely or they do not find a chance to collide at all because of the expansion of the universe. We have given the results of examples for those two cases with the mass scales of M Pl /M = 10 and M Pl /M = 1, respectively, in Fig. (1) together with the case of M Pl /M = 100 that was adopted for the rest of the simulations. We did not use mass scales greater than that because of the computational costs and, moreover, this value is enough to examine the anisotropy in first place due to the fact that higher rates for M Pl /M suppress the expansion of the Universe more and more. After determining the order of minimum mass scale, that is M Pl /M ≈ 100, at which the phase transition can be completed successfully, we have run simulations to determine the time evolution of the energy density parameter for the shear scalar by examining the effect of different initial conditions created through different number of initiated bubbles. But before this we have shown in Fig. (3) that absolute differences in the directional Hubble parameters are in the order of 10 -5 for N b = 10 , 40 , 320 while it is around 10 -6 for N b = 600 at most. Additionally, we have also provided the directional gradient energies and their differences in Fig. (4) for the same configurations with the number of bubbles mentioned and have shown that their differences are in the order of 10 -6 . In Fig. (5) we have presented the results for the Hubble parameters and the shear scalars. Moreover, we have also given the outcomes for a longer run of a specific configuration in Fig. (6) . As indicated before from the results of the directional Hubble parameters it seems that the relatively small number of bubbles give rise to high values for the shear scalar except for the case of N b = 10 which seems to be a counter example for this conclusion at first glance, but the bubble collision phase is not completed for that simulation. Therefore, as one may guess before, in addition to the mass scale, the proportion of the number of initiated bubbles to whole simulation box is another quantity that also determines whether a phase transition can be completed or not. This can also be seen from Fig. (5) for the Hubble parameters where the case of N b = 10 is different from the others at the beginning of the simulation. Nevertheless, we have found that before decreasing smoothly, the energy density parameter for the shear scalar gains a peak between 10 -8 -10 -10 which occurs at bubble collision phase. With the aforementioned longer run we have shown that Ω σ 2 becomes close to one of the constraints obtained for its today's value [32] . Additionally, it seems that the expansion of the Universe does not effect the phase transition for a typical mass scales of M Pl /M ≳ 100 with a fairly distributed number of initiated bubbles, since hereby we have tested impact of the expansion itself as well besides the anisotropy.",
"pages": [
14,
15
]
},
{
"title": "Acknowledgements",
"content": "This work is supported by The Scientific and Technological Research Council of Turkiye (T UB ˙ ITAK) through grant number 121F066. Computing resources used in this work were provided by the National Center for High Performance Computing of Turkiye (UHeM) under grant number 5013072022 and the simulations were partially performed at TUBITAK ULAKBIM, High Performance and Grid Computing Center (TRUBA resources).",
"pages": [
15
]
}
]
2024PhLB..85538757A
https://arxiv.org/pdf/2308.03225.pdf
<document>
<section_header_level_1><location><page_1><loc_9><loc_86><loc_91><loc_89></location>Acceleration as a circular motion along an imaginary circle: Kubo-Martin-Schwinger condition for accelerating field theories in imaginary-time formalism</section_header_level_1>
<text><location><page_1><loc_35><loc_82><loc_65><loc_83></location>Victor E. Ambrus, a , Maxim N. Chernodub b,a</text>
<text><location><page_1><loc_32><loc_78><loc_68><loc_81></location>a Department of Physics, West University of Timis,oara, Bd. Vasile Pˆarvan 4, Timis,oara 300223, Romania b Institut Denis Poisson, Universit´e de Tours, Tours 37200, France</text>
<section_header_level_1><location><page_1><loc_6><loc_71><loc_13><loc_72></location>Abstract</section_header_level_1>
<text><location><page_1><loc_6><loc_57><loc_94><loc_70></location>We discuss the imaginary-time formalism for field theories in thermal equilibrium in uniformly accelerating frames. We show that under a Wick rotation of Minkowski spacetime, the Rindler event horizon shrinks to a point in a two-dimensional subspace tangential to the acceleration direction and the imaginary time. We demonstrate that the accelerated version of the Kubo-MartinSchwinger (KMS) condition implies an identification of all spacetime points related by integer-multiple rotations in the tangential subspace about this Euclidean Rindler event-horizon point, with the rotational quanta defined by the thermal acceleration, α = a / T . In the Wick-rotated Rindler hyperbolic coordinates, the KMS relations reduce to standard (anti-)periodic boundary conditions in terms of the imaginary proper time (rapidity) coordinate. Our findings pave the way to study, using first-principle lattice simulations, the Hawking-Unruh radiation in geometries with event horizons, phase transitions in accelerating Early Universe and early stages of quark-gluon plasma created in relativistic heavy-ion collisions.</text>
<text><location><page_1><loc_6><loc_55><loc_63><loc_56></location>Keywords: Acceleration, Unruh e ff ect, KMS relation, Finite temperature field theory</text>
<section_header_level_1><location><page_1><loc_6><loc_51><loc_17><loc_52></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_6><loc_29><loc_48><loc_50></location>In the past decades, there has been a renewed interest in studying systems with acceleration as toy models for understanding the dynamics of the quark-gluon plasma fireball created in ultrarelativistic (non-central) heavy-ion collisions [1]. Such systems exhibit large acceleration immediately after the collision [2] until the central rapidity plateau develops as in the Bjorken boost-invariant flow model [3], where the acceleration vanishes. A natural question that arises for such a system is to what extent these extreme kinematic regimes a ff ect the thermodynamics of the plasma fireball, which sets the stage for further evolution of the quark-gluon plasma. The environment of the 'Little Bangs' of high-energy heavy-ion collisions [4] sheds insights on the properties of a primordial quark-gluon matter that once emerged at the time of the Big Bang in the Early Universe [5].</text>
<text><location><page_1><loc_6><loc_14><loc_48><loc_28></location>Our knowledge of the non-perturbative properties of the quark-gluon plasma originates from first-principle numerical simulations of lattice QCD, which is formulated in Euclidean spacetime, by means of the imaginary-time formalism [6]. Acceleration is closely related to rotation due to the resemblance of the corresponding generators of Lorentz transformations of Minkowski spacetime. In the case of non-central collisions, the angular velocity of the quark-gluon fluid can reach values of the order of Ω ∼ 10 22 Hz [7] which translates to ℏ Ω ≃ 6 MeV ≪ Tc , where Tc is the transition temperature to the quark-gluon</text>
<text><location><page_1><loc_52><loc_31><loc_94><loc_52></location>plasma phase. The lattice studies have so far been limited to the case of uniformly rotating systems in Euclidean space-time, where the rotation parameter has to be analytically continued to imaginary values [8] in order to avoid the sign problem that also plagues lattice calculations at finite chemical potential [9]. Analytical analyses of the e ff ects of rotation on the phase diagram, performed in various e ff ective infrared models of QCD [10, 11, 12, 13, 14, 15, 16, 17], stay in persistent contradiction with the first-principle numerical results [18, 19, 20, 21, 22, 23], presumably due to numerically-observed rotational instability of quark-gluon plasma [21, 22, 23] (related to the thermal melting of the non-perturbative gluon condensate [21]), splitting of chiral and deconfining transitions [23, 24], or formation of a strongly inhomogeneous mixed hadronic-quark-gluon-plasma phase induced by rotation [17, 25].</text>
<text><location><page_1><loc_52><loc_20><loc_94><loc_31></location>Anearlier study of a Euclidean quantum field theory in an accelerating spacetime with the Friedmann-Lemaˆıtre-RobertsonWalker metric has also encountered the sign problem, which was avoided by considering a purely imaginary Hubble constant [26]. On the contrary, our formulation of acceleration in the imaginary-time formalism is free from the sign problem, and thus, it can be formulated for physical, real-valued acceleration. Throughout the paper, we use ℏ = c = kB = 1 units.</text>
<section_header_level_1><location><page_1><loc_52><loc_16><loc_84><loc_17></location>2. Global equilibrium in uniform acceleration</section_header_level_1>
<text><location><page_1><loc_52><loc_9><loc_94><loc_15></location>From a classical point of view, global equilibrium states in generic particle systems are characterized by the inverse temperature four-vector β µ ≡ u µ ( x ) / T ( x ), associated with the local fluid velocity u µ , with β µ satisfying the Killing equation,</text>
<text><location><page_2><loc_6><loc_80><loc_48><loc_90></location>∂µβν + ∂νβµ = 0 [27, 28]. For an accelerated system at equilibrium, one gets β µ ∂µ = β T [(1 + az ) ∂ t + at ∂ z ], with β T = 1 / T where 1 . T ≡ T ( 0 ) represents the temperature at the coordinate origin x ∥ ≡ ( t , z ) = 0 in the longitudinal plane spanned by the time coordinate t and the acceleration direction z . The local temperature T ( x ), the local fluid velocity u µ ( x ) and the local proper acceleration a µ ( x ) ≡ u ν ∂ν u µ , respectively,</text>
<formula><location><page_2><loc_15><loc_76><loc_48><loc_79></location>T ( x ) ≡ ( u µβ µ ) -1 = 1 β T p (1 + az ) 2 -( at ) 2 , (1)</formula>
<formula><location><page_2><loc_13><loc_74><loc_48><loc_76></location>u µ ( x ) ∂µ = T ( x ) β T GLYPH<2> (1 + az ) ∂ t + at ∂ z GLYPH<3> , (2)</formula>
<formula><location><page_2><loc_13><loc_72><loc_48><loc_74></location>a µ ( x ) ∂µ = aT 2 ( x ) β 2 T [ at ∂ t + (1 + az ) ∂ z ] , (3)</formula>
<text><location><page_2><loc_6><loc_70><loc_27><loc_71></location>diverge at the Rindler horizon:</text>
<formula><location><page_2><loc_16><loc_66><loc_48><loc_69></location>(1 + az ) 2 -( at ) 2 = 0 , z ⩾ -1 a . (4)</formula>
<text><location><page_2><loc_6><loc_61><loc_48><loc_65></location>It is convenient to define the dimensionless quantity called the proper thermal acceleration α = p -α µ αµ and the corresponding four-vector α µ = u ν ∂νβ µ = a µ / T ( x ), respectively:</text>
<formula><location><page_2><loc_9><loc_58><loc_48><loc_60></location>α = a β T , α µ ( x ) ∂µ = a β 2 T T ( x )[ at ∂ t + (1 + az ) ∂ z ] . (5)</formula>
<text><location><page_2><loc_6><loc_53><loc_48><loc_57></location>Note that, while the magnitude α of the thermal acceleration is a space-time constant, the local acceleration a ( x ) = p -a µ a µ = α T ( x ) depends on space and time coordinates.</text>
<text><location><page_2><loc_6><loc_50><loc_48><loc_53></location>In classical theory, the energy-momentum tensor for an accelerating fluid in thermal equilibrium reads</text>
<formula><location><page_2><loc_20><loc_48><loc_48><loc_49></location>T µν = E u µ u ν - P ∆ µν , (6)</formula>
<text><location><page_2><loc_6><loc_42><loc_48><loc_46></location>where ∆ µν = g µν -u µ u ν . The local energy density E and pressure P are characterized by the local temperature (1). For a conformal system,</text>
<formula><location><page_2><loc_20><loc_38><loc_48><loc_41></location>E = 3 P = ν e ff π 2 30 T 4 ( x ) , (7)</formula>
<text><location><page_2><loc_6><loc_29><loc_48><loc_37></location>where ν e ff is the e ff ective bosonic degrees of freedom. In the case of a massless, neutral scalar field, ν e ff = 1, while for Dirac fermions, ν e ff = 7 8 × 2 × 2 = 7 / 2, taking into account the di ff erence between Bose-Einstein and Fermi-Dirac statistics (7 / 8), spin degeneracy, as well as particle and anti-particle contributions.</text>
<section_header_level_1><location><page_2><loc_6><loc_26><loc_28><loc_27></location>3. Unruh and Hawking e ff ects</section_header_level_1>
<text><location><page_2><loc_6><loc_20><loc_48><loc_24></location>Unruh has found that in a frame subjected to a uniform acceleration a , an observer detects a thermal radiation with the temperature [29]:</text>
<formula><location><page_2><loc_22><loc_16><loc_48><loc_19></location>TU ≡ 1 β U = a 2 π , (8)</formula>
<text><location><page_2><loc_52><loc_87><loc_94><loc_90></location>where we also defined the Unruh length β U , which will be useful in our discussions below.</text>
<text><location><page_2><loc_52><loc_80><loc_94><loc_87></location>The Unruh e ff ect is closely related to the Hawking evaporation of black holes [30, 31], which proceeds via the quantum production of particle pairs near the event horizon of the black hole. The Hawking radiation has a thermal spectrum with an e ff ective temperature</text>
<formula><location><page_2><loc_69><loc_77><loc_94><loc_79></location>TH = κ 2 π , (9)</formula>
<text><location><page_2><loc_52><loc_62><loc_94><loc_76></location>where κ = 1 / (4 M ) is the acceleration due to gravity at the horizon of a black hole of mass M . The similarity of both e ff ects, suggested by the equivalence of formulas for the Unruh temperature (9) and the Hawking temperature (8), goes deeper as the thermal character of both phenomena apparently originates from the presence of appropriate event horizons [32, 33]. In an accelerating frame, the event horizon separates causally disconnected regions of spacetime, evident in the Rindler coordinates in which the metric of the accelerating frame is conformally flat [34].</text>
<text><location><page_2><loc_52><loc_42><loc_94><loc_62></location>Quantum e ff ects lead to acceleration-dependent corrections to Eq. (7) and may also produce extra (anisotropic) contributions to the energy-momentum tensor T µν of the system. Such corrections were already established using the Zubarev approach [35, 36] or Wigner function formalism [37, 38], and one remarkable conclusion is that the energy-momentum tensor Θ µν in an accelerating system exactly vanishes at the Unruh temperature (8), or, equivalently, when the thermal acceleration (3) reaches the critical value α = α c = 2 π : Θ µν ( T = TU ) = 0. A somewhat related property is satisfied by thermal correlation functions in the background of a Schwarzschild black hole, establishing the equivalence between Feynman and thermal Green's functions, with the latter one taken at the Hawking temperature (9), cf. Ref. [33, 32].</text>
<text><location><page_2><loc_52><loc_37><loc_94><loc_42></location>As noted earlier, the energy density receives quantum corrections. For the conformally-coupled massless real-valued Klein-Gordon scalar field and the Dirac field, we have, respectively [36, 37, 38, 39, 40]:</text>
<formula><location><page_2><loc_56><loc_33><loc_94><loc_36></location>E scalar = π 2 T 4 ( x ) 30 GLYPH<20> 1 -GLYPH<16> α 2 π GLYPH<17> 4 GLYPH<21> , (10a)</formula>
<formula><location><page_2><loc_56><loc_30><loc_94><loc_33></location>E Dirac = 7 π 2 T 4 ( x ) 60 GLYPH<20> 1 -GLYPH<16> α 2 π GLYPH<17> 2 GLYPH<21>GLYPH<20> 1 + 17 7 GLYPH<16> α 2 π GLYPH<17> 2 GLYPH<21> , (10b)</formula>
<text><location><page_2><loc_52><loc_25><loc_94><loc_29></location>where we specially rearranged terms to make it evident that at the Unruh temperature T = TU (or, equivalently, at α = 2 π ), the energy density vanishes.</text>
<text><location><page_2><loc_52><loc_9><loc_94><loc_25></location>The above discussion focused on the free-field theory. In the interacting case, a legitimate question is to what extent do the local kinematics influence the phase structure of phenomenologically relevant field theories, for example, to deconfinement and chiral thermal transitions of QCD. Central to lattice finitetemperature studies is how to set the Euclidean-space boundary conditions in the imaginary-time formalism. A static bosonic (fermionic) system at finite temperature can be implemented by imposing (anti-)periodicity in imaginary time τ = it with period given by the inverse temperature, τ → τ + β T . These boundary conditions are closely related to, and in fact, derived from</text>
<text><location><page_3><loc_6><loc_85><loc_48><loc_90></location>the usual Kubo-Martin-Schwinger (KMS) relation formulated for a finite-temperature state (at vanishing acceleration), which translates into a condition written for the scalar and fermionic thermal two-point functions [6, 41]:</text>
<formula><location><page_3><loc_12><loc_82><loc_48><loc_83></location>GF ( t ) = GF ( t + i β T ) , S F ( t ) = -S F ( t + i β T ) , (11)</formula>
<text><location><page_3><loc_6><loc_77><loc_48><loc_81></location>where we suppressed the dependence on the spatial coordinate x and the second four-point x ' . In the case of rotating states, the KMS relation (11) gets modified to [17, 40, 42]</text>
<formula><location><page_3><loc_13><loc_72><loc_48><loc_76></location>GF ( t , φ ) = GF ( t + i β T , φ + i β T Ω ) , S F ( t , φ ) = -e -β T Ω S z S F ( t + i β T , φ + i β T Ω ) , (12)</formula>
<text><location><page_3><loc_6><loc_64><loc_48><loc_71></location>where e -β T Ω S z is the spin part of the rotation with imaginary angle i β T Ω along the rotation ( z ) axis and S z = i 2 γ x γ y is the spin matrix. The purpose of the present paper is to uncover the KMS relation and subsequent conditions for fields and, consequently, for correlation functions in a uniformly accelerated state.</text>
<section_header_level_1><location><page_3><loc_6><loc_61><loc_41><loc_62></location>4. Quantum field theory at constant acceleration</section_header_level_1>
<text><location><page_3><loc_6><loc_57><loc_48><loc_60></location>In Minkowski space, the most general solution of the Killing equation reads</text>
<formula><location><page_3><loc_22><loc_56><loc_48><loc_57></location>β µ = b µ + ϖ µ ν x ν , (13)</formula>
<text><location><page_3><loc_6><loc_51><loc_48><loc_55></location>where b µ is a constant four-vector and ϖ µν is a constant, antisymmetric tensor. A quantum system in thermal equilibrium is characterized by the density operator</text>
<formula><location><page_3><loc_22><loc_48><loc_48><loc_50></location>ˆ ρ = e -b · ˆ P + ϖ : ˆ J / 2 , (14)</formula>
<text><location><page_3><loc_6><loc_39><loc_48><loc_47></location>where ˆ P µ and ˆ J µν are the conserved four-momentum and total angular momentum operator, representing the generators of translations and of Lorentz transformations. In order to derive the KMS relation, it is convenient to factorize ˆ ρ into a translation part and a Lorentz transformation part, as pointed out in Ref. [37]:</text>
<formula><location><page_3><loc_18><loc_37><loc_48><loc_39></location>e -b · ˆ P + ϖ : ˆ J / 2 = e -˜ b ( ϖ ) · ˆ P e ϖ : ˆ J / 2 , (15)</formula>
<text><location><page_3><loc_6><loc_35><loc_19><loc_37></location>where ˜ b is given by</text>
<formula><location><page_3><loc_10><loc_31><loc_48><loc_34></location>˜ b ( ϖ ) µ = ∞ X k = 0 i k ( k + 1)! ( ϖ µ ν 1 ϖ ν 1 ν 2 · · · ϖ ν k -1 ν k ) b ν k . (16)</formula>
<text><location><page_3><loc_6><loc_25><loc_48><loc_30></location>Focusing now on the accelerated system with reference inverse temperature β T = 1 / T , we have b µ = β T δ µ 0 and ϖ µ ν = α ( δ µ 3 g 0 ν -δ µ 0 g 3 ν ), such that ˜ b becomes</text>
<formula><location><page_3><loc_9><loc_22><loc_48><loc_24></location>˜ b µ = B δ µ 0 + A δ µ 3 , B = sin α a , A = i a (1 -cos α ) , (17)</formula>
<text><location><page_3><loc_6><loc_18><loc_48><loc_21></location>where α = a / T is the thermal acceleration (5). This observation allows ˆ ρ = e -β T ˆ H + α ˆ K z to be factorized as</text>
<formula><location><page_3><loc_21><loc_16><loc_48><loc_17></location>ˆ ρ = e -B ˆ H + A ˆ P z e α ˆ K z . (18)</formula>
<text><location><page_3><loc_6><loc_12><loc_48><loc_15></location>Arelativistic quantum field described by the field operator ˆ Φ transforms under Poincar'e transformations as</text>
<formula><location><page_3><loc_17><loc_9><loc_34><loc_11></location>e i ˜ b · ˆ P ˆ Φ ( x ) e -i ˜ b · ˆ P = ˆ Φ ( x + ˜ b ) ,</formula>
<formula><location><page_3><loc_64><loc_89><loc_94><loc_90></location>ˆ Λ ˆ Φ ( x ) ˆ Λ -1 = D [ Λ -1 ] ˆ Φ ( Λ x ) , (19)</formula>
<text><location><page_3><loc_52><loc_78><loc_94><loc_88></location>where Λ = e -i 2 ϖ : J is written in terms of the matrix generators ( J µν ) αβ = i ( δ µ α δ ν β -δ µ β δ ν α ), while D [ Λ ] -1 = e i 2 ϖ : S is the spin part of the inverse Lorentz transformation. Comparing Eq. (19) and (14), it can be seen that the density operator ˆ ρ acts like a Poincar'e transformation with imaginary parameters [37]. Using now the factorization (18), it can be seen that ˆ ρ acts on the field operator ˆ Φ as follows:</text>
<formula><location><page_3><loc_64><loc_75><loc_94><loc_77></location>ˆ ρ ˆ Φ ( t , z ) ˆ ρ -1 = e -α S 0 z ˆ Φ ( ˜ t , ˜ z ) , (20)</formula>
<text><location><page_3><loc_52><loc_73><loc_56><loc_74></location>where</text>
<formula><location><page_3><loc_59><loc_66><loc_94><loc_72></location>˜ t = cos( α ) t + i sin( α ) z + i a sin( α ) , ˜ z = i sin( α ) t + cos( α ) z -1 a [1 -cos( α )] . (21)</formula>
<text><location><page_3><loc_52><loc_62><loc_94><loc_65></location>The spin term evaluates to e -α S 0 z = 1 in the scalar case (since S 0 z = 0), while for the Dirac field, S 0 z = i 2 γ 0 γ 3 and</text>
<formula><location><page_3><loc_63><loc_59><loc_94><loc_61></location>e -α S 0 z = cos α 2 -i γ 0 γ 3 sin α 2 . (22)</formula>
<section_header_level_1><location><page_3><loc_52><loc_56><loc_87><loc_57></location>5. KMS relation at constant uniform acceleration</section_header_level_1>
<text><location><page_3><loc_52><loc_51><loc_94><loc_55></location>Consider now the Wightman functions G ± ( x , x ' ) and S ± ( x , x ' ) of the Klein-Gordon and Dirac theories, defined respectively as</text>
<formula><location><page_3><loc_53><loc_46><loc_94><loc_50></location>G + ( x , x ' ) = ⟨ ˆ Φ ( x ) ˆ Φ ( x ' ) ⟩ , S + ( x , x ' ) = ⟨ ˆ Ψ ( x ) ˆ Ψ ( x ' ) ⟩ , G -( x , x ' ) = ⟨ ˆ Φ ( x ' ) ˆ Φ ( x ) ⟩ , S -( x , x ' ) = -⟨ ˆ Ψ ( x ' ) ˆ Ψ ( x ) ⟩ . (23)</formula>
<text><location><page_3><loc_52><loc_42><loc_94><loc_44></location>When the expectation value ⟨·⟩ is taken at finite temperature and under acceleration, we derive the KMS relations:</text>
<formula><location><page_3><loc_62><loc_37><loc_94><loc_41></location>G + ( x , x ' ) = G -( ˜ t , ˜ z ; x ' ) , S + ( x , x ' ) = -e -α S 0 z S -( ˜ t , ˜ z ; x ' ) . (24)</formula>
<text><location><page_3><loc_52><loc_33><loc_94><loc_36></location>The KMS relations also imply natural boundary conditions for the thermal propagators:</text>
<formula><location><page_3><loc_62><loc_29><loc_94><loc_32></location>GF ( ˜ t , ˜ z ; x ' ) = GF ( t , z ; x ' ) , S F ( ˜ t , ˜ z ; x ' ) = -e α S 0 z S F ( t , z ; x ' ) , (25)</formula>
<text><location><page_3><loc_52><loc_26><loc_77><loc_27></location>which are solved formally by [34, 40]</text>
<formula><location><page_3><loc_55><loc_22><loc_94><loc_25></location>G ( α ) F ( t , z ; x ' ) = ∞ X j = -∞ G vac F ( t ( j ) , z ( j ); x ' ) , (26a)</formula>
<formula><location><page_3><loc_55><loc_18><loc_94><loc_21></location>S ( α ) F ( t , z ; x ' ) = ∞ X j = -∞ ( -1) j e -j α S 0 z S vac F ( t ( j ) , z ( j ); x ' ) , (26b)</formula>
<text><location><page_3><loc_52><loc_13><loc_94><loc_17></location>where G vac F ( x , x ' ) and S vac F ( x , x ' ) are the vacuum propagators, while t ( j ) and z ( j ) are obtained by applying the transformation in Eq. (21) j ∈ Z times:</text>
<formula><location><page_3><loc_59><loc_9><loc_83><loc_11></location>t ( j ) = t cos( j α ) + i a (1 + az ) sin( j α ) ,</formula>
<formula><location><page_4><loc_14><loc_88><loc_48><loc_90></location>z ( j ) = it sin( j α ) + 1 a (1 + az ) cos( j α ) -1 a . (27)</formula>
<text><location><page_4><loc_6><loc_82><loc_48><loc_87></location>In particular, ˜ t = t (1) and ˜ z = z (1). Due to the periodicity of the trigonometric functions appearing above, in the case when α/ 2 π = p / q is a rational number represented as an irreducible fraction, the sum over j in Eqs. (26) contains only q terms:</text>
<formula><location><page_4><loc_10><loc_77><loc_48><loc_81></location>G ( p , q ) F ( t , z ; x ' ) = q -1 X j = 0 G vac F ( t ( j ) , z ( j ); x ' ) , (28a)</formula>
<formula><location><page_4><loc_10><loc_73><loc_48><loc_76></location>S ( p , q ) F ( t , z ; x ' ) = q -1 X j = 0 ( -1) j e -j α S 0 z S vac F ( t ( j ) , z ( j ); x ' ) . (28b)</formula>
<text><location><page_4><loc_6><loc_64><loc_48><loc_72></location>In particular, the case α = 2 π corresponds to p = q = 1, while the thermal propagators reduce trivially to the vacuum ones: G (1 , 1) F = G vac F and S (1 , 1) F = S vac F . Since e -q α S 0 z = ( -1) p by virtue of Eq. (22), applying Eq. (25) q times shows that S ( p , q ) F ( t ( q ) , z ( q ); x ' ) = ( -1) p + q S ( p , q ) F ( t , z ; x ' ) and thus S ( p , q ) F cancels identically when p + q is an odd integer.</text>
<section_header_level_1><location><page_4><loc_6><loc_60><loc_40><loc_61></location>6. Imaginary-time formulation for acceleration</section_header_level_1>
<text><location><page_4><loc_6><loc_55><loc_48><loc_59></location>We now move to the Euclidean manifold by performing the Wick rotation to imaginary time, t → τ = it . Then, Eq. (25) becomes</text>
<formula><location><page_4><loc_15><loc_51><loc_48><loc_54></location>GE ( τ (1) , z (1); x ' ) = GE ( τ, z ; x ' ) , S E ( τ (1) , z (1); x ' ) = -e α S 0 z S E ( τ, z ; x ' ) , (29)</formula>
<text><location><page_4><loc_6><loc_47><loc_48><loc_50></location>and Eq. (26) reads, for the case when α/ 2 π is an irrational number,</text>
<formula><location><page_4><loc_9><loc_43><loc_48><loc_46></location>G ( α ) E ( τ, z ; x ' ) = ∞ X j = -∞ G vac E ( τ ( j ) , z ( j ); x ' ) , (30a)</formula>
<formula><location><page_4><loc_10><loc_39><loc_48><loc_42></location>S ( α ) E ( τ, z ; x ' ) = ∞ X j = -∞ ( -1) j e -j α S 0 z S vac E ( τ ( j ) , z ( j ); x ' ) . (30b)</formula>
<text><location><page_4><loc_6><loc_34><loc_48><loc_38></location>The case when α/ 2 π = p / q must be treated along the lines summarized in Eqs. (28) (see also discussion in Sec. 10). In the above, we considered j ∈ Z and</text>
<formula><location><page_4><loc_14><loc_30><loc_48><loc_33></location>τ ( j ) = τ cos( j α ) -1 a (1 + az ) sin( j α ) , (31a)</formula>
<formula><location><page_4><loc_14><loc_28><loc_48><loc_30></location>z ( j ) = τ sin( j α ) + 1 a (1 + az ) cos( j α ) -1 a . (31b)</formula>
<text><location><page_4><loc_6><loc_24><loc_48><loc_27></location>For the fields, the accelerated KMS conditions suggest the identification of the fields at the points:</text>
<formula><location><page_4><loc_15><loc_22><loc_48><loc_23></location>ϕ ( τ ( j ) , x ∥ , z ( j )) = ϕ ( τ, x ∥ , z ) , (32a)</formula>
<formula><location><page_4><loc_15><loc_20><loc_48><loc_22></location>ψ ( τ ( j ) , x ∥ , z ( j )) = ( -1) j e j α S 0 z ψ ( τ, x ∥ , z ) , (32b)</formula>
<text><location><page_4><loc_6><loc_9><loc_48><loc_19></location>where the identified coordinates ( τ ( j ) , z ( j )) in the longitudinal plane are given by Eq. (31) and x ∥ = ( x , y ) are the transverse coordinates which are unconstrained by acceleration. While the sums of the form (26) may formally be divergent, the modified conditions (31) and (32) give a finite solution to the accelerated KMSrelations. The points identified with the accelerated KMS condition (31) are illustrated in Fig. 1.</text>
<figure>
<location><page_4><loc_53><loc_63><loc_92><loc_90></location>
<caption>Figure 1: The cyclic paths determined by the accelerating KMS boundary condition (31) in the longitudinal plane spanned by the imaginary time τ and the acceleration direction z of Wick-rotated Minkowski spacetime. Each plot illustrates di ff erent accelerations a encoded in the ratio β U /β T ≡ 2 π T / a = 3 , 4 , 5 , 10 of the Unruh length β U , Eq. (8), to the thermal length β T = 1 / T . The starting point of each cyclic path, ( z , τ ) i = ( zi , 0), with zi /β U = -1 , -1 / 2 , . . . , 1, is denoted by a hollow circle. The position of the Rindler horizon, collapsed under the Wick rotation to a point (34), is denoted by the green star in each plot.</caption>
</figure>
<section_header_level_1><location><page_4><loc_52><loc_46><loc_94><loc_48></location>7. Geometrical meaning of the accelerated KMS relation in imaginary-time formalism</section_header_level_1>
<text><location><page_4><loc_52><loc_41><loc_94><loc_45></location>It is convenient, for a moment, to define a translationally shifted spatial coordinate, z = z + 1 / a , and rewrite Eq. (31) in the very simple and suggestive form:</text>
<formula><location><page_4><loc_63><loc_37><loc_94><loc_40></location>τ ( j ) = τ cos( j α ) -z sin( j α ) , z ( j ) = τ sin( j α ) + z cos( j α ) . (33)</formula>
<text><location><page_4><loc_52><loc_33><loc_94><loc_36></location>In the shifted coordinates, the condition (4) for the Rindler horizon becomes a 2 ( z 2 + τ 2 ) = 0, which is solved by</text>
<formula><location><page_4><loc_59><loc_30><loc_94><loc_32></location>τ = z = 0 ⇔ τ = 0 , z = -1 a . (34)</formula>
<text><location><page_4><loc_52><loc_18><loc_94><loc_29></location>Thus, we arrive at the following beautiful conclusion: in the Euclidean spacetime of the imaginary-time formalism, the Rindler horizon (4) shrinks to a single point (34). Thus, the accelerated KMS condition corresponds to the identification of all points obtained by the discrete rotation of the space around the Euclidean Rindler horizon point ( τ, z ) = (0 , -1 / a ) with the unit rotation angle defined by the reference thermal acceleration α = a / T .</text>
<text><location><page_4><loc_52><loc_9><loc_94><loc_18></location>Our accelerated KMS condition, given in Eqs. (31) and (32), recovers the usual finite-temperature KMS condition in the limit of vanishing acceleration. Figure 2 demonstrates that in this limit,with α = a / T → 0, the proposed KMS-type condition (27) for the acceleration is reduced to the standard finitetemperature KMS-boundary condition [6] for which imaginary</text>
<text><location><page_5><loc_6><loc_87><loc_48><loc_90></location>time τ is compactified to a circle of the length β T ≡ 1 / T with the points ( τ, x ) and ( τ + β T n , x ), n ∈ Z , identified.</text>
<figure>
<location><page_5><loc_10><loc_61><loc_45><loc_86></location>
<caption>Figure 2: The sets of points in the ( τ, z ) plane which are identified by our circular KMS condition (33) with the origin ( τ, z ) = (0 , 0) in a thermally equilibrated system which experiences a uniform acceleration a along the z axis. The color distinguishes di ff erent acceleration strength marked by di ff erent Unruh lengths β U = 2 π/ | a | . At vanishing acceleration ( β U /β T →±∞ ), condition (33) reduces to the standard thermodynamic requirement of compactification of imaginary time τ to a circle with the length β T = 1 / T , while the Euclidean Rindler horizon moves to (minus) spatial infinity. In the figure, each set of points, corresponding to various ratios β U /β T , is connected by a smooth line to guide the eye.</caption>
</figure>
<text><location><page_5><loc_6><loc_36><loc_48><loc_46></location>At the critical acceleration α = 2 π n (with n ∈ Z ), when the background temperature T equals to (an integer multiple of) the Unruh temperature (8), the accelerated KMS conditions (31) do not constrain the system anymore, τ ( j ) = τ and z ( j ) = z , so that the system becomes equivalent to a zero-temperature system in non-accelerated flat Minkowski spacetime. This property, for α = 2 π , has been observed in Refs. [35, 36, 37, 38].</text>
<text><location><page_5><loc_6><loc_23><loc_48><loc_36></location>In the situation where 2 π/α = β U /β T = n is an integer number, the accelerated state at finite temperature can be implemented in Euclidean space by imposing periodicity with respect to a specific set of points that form a regular polygon with n vertices located on the circle of radius τ 2 + z 2 . This is particularly convenient for lattice simulations since the Euclidean action remains the standard one, allowing accelerated systems to be modeled in the imaginary-time path integral formalism without encountering the infamous sign problem.</text>
<section_header_level_1><location><page_5><loc_6><loc_19><loc_35><loc_20></location>8. KMS relations in Rindler coordinates</section_header_level_1>
<text><location><page_5><loc_6><loc_9><loc_48><loc_18></location>In the Minkowski Lorentz frame that we considered so far, the accelerating KMS conditions (31) and (32) do not correspond to a boundary condition (as one would naively expect from the KMS condition in thermal field theory) but rather to a bulk condition: instead of relating the points at the boundary of the imaginary-time Euclidean system, the accelerated KMS</text>
<text><location><page_5><loc_52><loc_87><loc_94><loc_90></location>relations give us the identification of the spacetime points in its interior.</text>
<text><location><page_5><loc_52><loc_82><loc_94><loc_87></location>While seemingly non-trivial in the form written in Eq. (27), the displacements implied by the KMS relation correspond to the usual translation of the proper time (rapidity) coordinate η when employing the Rindler coordinates,</text>
<formula><location><page_5><loc_59><loc_79><loc_94><loc_80></location>at = e ζ sinh( a η ) , 1 + az = e ζ cosh( a η ) . (35)</formula>
<text><location><page_5><loc_52><loc_77><loc_65><loc_78></location>It is easy to see that</text>
<formula><location><page_5><loc_66><loc_74><loc_94><loc_75></location>at ( j ) = e ζ sinh( a η + i j α ) , (36a)</formula>
<formula><location><page_5><loc_63><loc_72><loc_94><loc_73></location>1 + az ( j ) = e ζ cosh( a η + i j α ) , (36b)</formula>
<text><location><page_5><loc_51><loc_70><loc_63><loc_71></location>which implies that</text>
<formula><location><page_5><loc_63><loc_67><loc_94><loc_68></location>η ( j ) = η + i j β T , ζ ( j ) = ζ, (37)</formula>
<text><location><page_5><loc_52><loc_60><loc_94><loc_66></location>in a seemingly perfect agreement with the usual KMS relation (11) for static systems in Minkowski. However, there is also an unusual particularity of the KMS conditions (37) in the Rindler coordinates (35).</text>
<text><location><page_5><loc_52><loc_52><loc_94><loc_60></location>The first relation in Eq. (37) suggests that the Wick rotation of the Minkowski time t = -i τ should be supplemented with the Wick rotation of the proper time in the accelerated frame η = -i θ/ a , where θ is the imaginary rapidity. 2 Then, the relation (35) in the imaginary (both Minkowski and Rindler) time becomes as follows:</text>
<formula><location><page_5><loc_62><loc_49><loc_94><loc_50></location>a τ = e ζ sin θ, 1 + az = e ζ cos θ, (38)</formula>
<text><location><page_5><loc_52><loc_45><loc_94><loc_48></location>which shows that the imaginary rapidity becomes an imaginary coordinate with the Euclidean Rindler KMS condition (37):</text>
<formula><location><page_5><loc_60><loc_43><loc_94><loc_44></location>θ ( j ) = θ + j α, ζ ( j ) = ζ, j ∈ Z . (39)</formula>
<text><location><page_5><loc_52><loc_33><loc_94><loc_41></location>Curiously, under the Wick transform, the rapidity becomes a cyclic compact variable, 0 ⩽ θ < 2 π , on which the imaginarytime condition (39) imposes the additional periodicity with the period equal to the thermal acceleration α . Expectedly, at α = 2 π (or, equivalently, at T = TU ), the boundary condition (39) becomes trivial.</text>
<text><location><page_5><loc_52><loc_22><loc_94><loc_33></location>The boundary conditions (39), characterized by the doublyperiodic imaginary rapidity coordinate θ , with periodicities θ → θ + 2 π and θ → θ + α (for 0 ⩽ α < 2 π ), can be easily implemented in lattice simulations. Notice that this double periodicity has a strong resemblance to the observation of Refs. [43, 44, 45] that the Euclidean Rindler space can be identified with the space of the cosmic string which possesses a conical singularity with the angular deficit ∆ φ = 2 π -α [46, 47].</text>
<text><location><page_5><loc_52><loc_13><loc_94><loc_21></location>The KMS periodicity (39) of the compact imaginary rapidity θ is formally sensitive to the rationality of the normalized thermal acceleration α/ (2 π ). Obviously, for α = 2 π p / q , where p < q are nonvanishing irreducible integer numbers, the interplay of the two periodicities will correspond to the single period θ → θ + 2 π/ q .</text>
<text><location><page_6><loc_6><loc_72><loc_48><loc_90></location>Interestingly, the sensitivity of an e ff ect to the denominator q (and not to the numerator p ) of a relevant parameter is a signature of the fractal nature of the e ff ect. Such fractality is noted, for example, in particle systems subjected to imaginary rotation implemented via rotwisted boundary conditions [17, 48, 49], which leads, in turn, to the appearance of 'ninionic' deformation of particle statistics [50]. The suggested fractality of acceleration in imaginary formalism is not surprising given the conceptual similarity of acceleration and rotation with imaginary angular frequency [37, 38]. Below, we will show that, despite the fractal property of the system, the KMS boundary condition (39) in Euclidean Rindler space correctly reproduces results for accelerated particle systems.</text>
<section_header_level_1><location><page_6><loc_6><loc_67><loc_48><loc_69></location>9. Energy-momentum tensor with the accelerated KMS conditions</section_header_level_1>
<text><location><page_6><loc_6><loc_57><loc_48><loc_65></location>Now let us come back to the Wick-rotated Minkowski spacetime and verify how the modified KMS conditions for the fields, Eqs. (31) and (32), and related solutions for their two-point functions (30), can recover the known results in field theories under acceleration. To this end, we start from a non-minimally coupled scalar field theory with the Lagrangian [51, 52, 53]</text>
<formula><location><page_6><loc_17><loc_53><loc_48><loc_56></location>L ξ = 1 2 ∂µϕ∂ µ ϕ -2 ξ∂µ ( ϕ∂ µ ϕ ) , (40)</formula>
<text><location><page_6><loc_6><loc_51><loc_41><loc_52></location>possessing the following energy-momentum tensor:</text>
<formula><location><page_6><loc_12><loc_45><loc_48><loc_49></location>Θ ξ µν = (1 -2 ξ ) ∂µϕ∂νϕ -2 ξϕ∂µ∂νϕ -1 2 (1 -4 ξ ) δµν∂λϕ∂λϕ, (41)</formula>
<text><location><page_6><loc_6><loc_38><loc_48><loc_44></location>where the values ξ = 0 and ξ = 1 / 6 of the coupling parameter correspond to the canonical and conformal energy-momentum tensors, respectively. In terms of the Euclidean Green's function, Θ ξ µν can be written as</text>
<formula><location><page_6><loc_8><loc_32><loc_48><loc_37></location>Θ ξ µν = lim x ' → x h (1 -2 ξ ) ∂ ( µ∂ν ' ) -1 2 (1 -4 ξ ) δµν∂λ∂λ ' -ξ ( ∂µ∂ν + ∂µ ' ∂ν ' ) i ∆ G ( α ) E ( x , x ' ) , (42)</formula>
<text><location><page_6><loc_6><loc_24><loc_48><loc_31></location>where ∆ G ( α ) E ( x , x ' ) = G ( α ) E ( x , x ' ) -G vac E ( x , x ' ) represents the thermal part of the Green's function. For the Dirac field, Θ µν = 1 2 ¯ ψγ E µ ←→ ∂ν ψ can be computed from the Euclidean two-point function S ( α ) E ( x , x ' ) via</text>
<formula><location><page_6><loc_15><loc_21><loc_48><loc_23></location>Θ µν = -1 2 lim x ' → x tr[ γ E µ ( ∂ν -∂ν ' ) ∆ S ( α ) E ] . (43)</formula>
<text><location><page_6><loc_6><loc_17><loc_48><loc_20></location>The vacuum propagators satisfying □ G vac E ( x , x ' ) = γ E µ ∂µ S vac E ( x , x ' ) = δ 4 ( x -x ' ) are given by</text>
<formula><location><page_6><loc_14><loc_13><loc_48><loc_15></location>G vac E ( ∆ x ) = 1 4 π 2 ∆ X 2 , (44)</formula>
<formula><location><page_6><loc_14><loc_9><loc_48><loc_12></location>S vac E ( ∆ x ) = γ E µ ∂µ G vac E ( ∆ x ) = -γ E µ ∂µ 2 π 2 ∆ X 4 , (45)</formula>
<text><location><page_6><loc_52><loc_86><loc_94><loc_90></location>with ∆ X 2 = ( ∆ τ ) 2 + ( ∆ x ) 2 . Using Eq. (30), the thermal expectation values of the normal-ordered energy-momentum operator can be obtained in the case of the Klein-Gordon field as:</text>
<formula><location><page_6><loc_53><loc_73><loc_94><loc_85></location>Θ µν ξ ( x ) = ∞ X j , 0 1 4 π 2 ∆ X 4 ( j ) h (1 -2 ξ )( R ( j ) µν + R ( j ) νµ ) -δµν (1 -4 ξ ) R ( j ) λλ + 2 ξ ( R ( j ) νλ R ( j ) µλ + δµν ) i -X j , 0 ∆ x ( j ) λ ∆ x ( j ) κ π 2 ∆ X 6 ( j ) h (1 -2 ξ )( δµλ R ( j ) νκ + δνλ R ( j ) µκ ) -δµν (1 -4 ξ ) R ( j ) λκ + 2 ξ ( R ( j ) µλ R ( j ) νκ + δµλδνκ ) i , (46)</formula>
<text><location><page_6><loc_52><loc_69><loc_94><loc_71></location>where ∆ X 2 ( j ) = 4 a 2 sin 2 j α 2 [( a τ ) 2 + (1 + az ) 2 ] and R ( j ) µν ≡ ∂µ ∆ x ( j ) ν is given by</text>
<formula><location><page_6><loc_60><loc_62><loc_94><loc_67></location>R ( j ) µν = cos( j α ) 0 0 sin( j α ) 0 1 0 0 0 0 1 0 -sin( j α ) 0 0 cos( j α ) , (47)</formula>
<text><location><page_6><loc_52><loc_59><loc_86><loc_61></location>such that R ( j ) µλ R ( j ) νλ = δµν . For the Dirac field, we find</text>
<formula><location><page_6><loc_53><loc_50><loc_94><loc_57></location>Θ µν = X j , 0 ( -1) j π 2 h δµλ cos j α 2 + GLYPH<16> δµ 0 δλ 3 -δµ 3 δλ 0 GLYPH<17> sin j α 2 i × R ( j ) νλ + δνλ ∆ X 4 ( j ) -4 ∆ X ( j ) λ ∆ X 6 ( j ) ( R ( j ) νκ + δνκ ) ∆ X ( j ) κ . (48)</formula>
<text><location><page_6><loc_52><loc_45><loc_94><loc_49></location>Taking advantage of the relation ( R ( j ) νκ + δνκ ) ∆ x ( j ) κ = -2 a sin( j α )[(1 + az ) δν 0 -a τδν 3] and after switching back to the real time t , we find</text>
<formula><location><page_6><loc_64><loc_42><loc_94><loc_43></location>Θ µν = E u µ u ν - P ∆ µν + π µν , (49)</formula>
<text><location><page_6><loc_52><loc_32><loc_94><loc_41></location>with E , P , and u µ being the energy density, isotropic pressure, and the fluid four-velocity (2), respectively. The shear-stress tensor π µν is by construction traceless, symmetric and orthogonal to u µ , discriminating between the energy-momentum tensors in classical (6) and quantum (49) fluids. Due to the symmetries of the problem, its tensor structure is fixed as</text>
<formula><location><page_6><loc_64><loc_29><loc_94><loc_31></location>π µν = π s 2 ∆ µν -3 α µ α ν α λ αλ ! , (50)</formula>
<text><location><page_6><loc_52><loc_23><loc_94><loc_27></location>with α µ ( x ) being the local thermal acceleration (3), such that the shear coe ffi cient π s is the only degree of freedom of π µν in Eq. (50). In the scalar case, we find for the components of (49):</text>
<formula><location><page_6><loc_58><loc_13><loc_94><loc_22></location>E ξ = 3[ α T ( x )] 4 16 π 2 GLYPH<2> G 4( α ) + 4 ξ G 2( α ) GLYPH<3> , P ξ = [ α T ( x ) 4 ] 16 π 2 " G 4( α ) + 4 3 (1 -3 ξ ) G 2( α ) # , π ξ s = -[ α T ( x )] 4 12 π 2 (1 -6 ξ ) G 2( α ) , (51)</formula>
<text><location><page_6><loc_52><loc_9><loc_94><loc_12></location>with Gn ( α ) = P ∞ j = 1 [sin( j α/ 2)] -n , in complete agreement with the results in Ref. [37]. Formally, Gn diverges, however its</text>
<text><location><page_7><loc_6><loc_86><loc_48><loc_90></location>value can be obtained from its analytical continuation to imaginary acceleration a = i ϕ , e Gn ( β T ϕ ) = i n Gn ( i β T ϕ ). The sum can be evaluated, in a certain domain around β T ϕ > 0 [37], to:</text>
<formula><location><page_7><loc_13><loc_78><loc_48><loc_85></location>e G 2( β T ϕ ) = 2 π 2 3 β 2 T ϕ 2 -2 β T ϕ + 1 6 , e G 4( β T ϕ ) = 8 π 4 45 β 4 T ϕ 4 -4 π 2 9 β 2 T ϕ 2 + 4 3 β T ϕ -11 90 . (52)</formula>
<text><location><page_7><loc_6><loc_72><loc_48><loc_77></location>Substituting now Gn ( α ) = Re[ i -n e Gn ( i β T ϕ ) ⌋ ϕ →-ia ] into Eq. (51) gives Eq. (10) for the conformal coupling ξ = 1 / 6. For minimal coupling ξ = 0 or a generic non-conformal coupling ξ , 1 / 6, we recover the results of Refs. [37, 54].</text>
<text><location><page_7><loc_6><loc_68><loc_48><loc_71></location>In the case of the Dirac field, one can easily check that E D = 3 P D and π s D = 0, while</text>
<formula><location><page_7><loc_20><loc_65><loc_48><loc_67></location>P D = [ α T ( x )] 4 4 π 2 S 4( α ) , (53)</formula>
<text><location><page_7><loc_6><loc_58><loc_48><loc_64></location>with Sn ( α ) = -P ∞ j = 1 ( -1) j cos( j α/ 2) / [sin( j α/ 2)] n → e Sn ( β T ϕ ) ≡ i n Sn ( i β T ϕ ) = -P ∞ j = 1 ( -1) j cosh( j β T ϕ/ 2) / [sinh( j β T ϕ/ 2)] n , which agrees with the results obtained in Ref. [38].</text>
<text><location><page_7><loc_6><loc_49><loc_48><loc_57></location>Finally, let us also illustrate the practical functionality of the accelerating KMS boundary conditions (39) formulated in the imaginary-rapidity Rindler space (38). For simplicity, we calculate the fluctuations of the scalar field ⟨ ϕ 2 ⟩ using pointsplitting and noticing that the same method can be used to calculate also other quantities.</text>
<text><location><page_7><loc_6><loc_45><loc_48><loc_49></location>When expressed with respect to Rindler coordinates X = ( θ/ a , x ⊥ , ζ ), the Euclidean vacuum two-point function G vac E , R ( X , X ' ) given in Eq. (44) reads as follows:</text>
<formula><location><page_7><loc_9><loc_41><loc_48><loc_44></location>G vac E , R = 1 4 π 2 " 2 a 2 e ζ + ζ ' (cosh ∆ ζ -cos ∆ θ ) + ∆ x 2 ⊥ # -1 . (54)</formula>
<text><location><page_7><loc_6><loc_31><loc_48><loc_39></location>The KMS condition (39) implies that the Euclidean two-point function under acceleration satisfies G ( α ) E , R = P j ∈ Z G vac E , R ( ∆ θ + j α ), where we consider vanishing spatial distance between the points: ζ ' → ζ and x ' ⊥ → x ⊥ . Subtracting the vacuum ( j = 0) term that diverges in the ∆ X → 0 limit, we get for the scalar fluctuations:</text>
<formula><location><page_7><loc_9><loc_25><loc_48><loc_30></location>⟨ ϕ 2 ⟩ = lim ∆ θ → 0 GLYPH<2> G ( α ) E , R ( ∆ θ ) -G vac E , R ( ∆ θ ) GLYPH<3> (55) = a 2 e -2 ζ 8 π 2 G 2( α ) = T 2 ( x ) 12 -a 2 ( x ) 48 π 2 , 0 ⩽ a ⩽ 2 π T ,</formula>
<text><location><page_7><loc_6><loc_23><loc_36><loc_24></location>which agrees with the known result [37, 55].</text>
<section_header_level_1><location><page_7><loc_6><loc_19><loc_34><loc_20></location>10. Fractalization of thermodynamics</section_header_level_1>
<text><location><page_7><loc_6><loc_13><loc_48><loc_18></location>Let us consider the case when α/ 2 π is a rational number, represented as the irreducible fraction p / q . Then, the functions Gn ( α ) → G ( p , q ) n ( α ) = 1 2 P q -1 j = 1 [sin( π jp / q )] -n are regular and evaluate in the relevant n = 2 and n = 4 cases to</text>
<formula><location><page_7><loc_13><loc_9><loc_48><loc_12></location>G ( p , q ) 2 = q 2 -1 6 , G ( p , q ) 4 = q 4 + 10 q 2 -11 90 . (56)</formula>
<text><location><page_7><loc_52><loc_86><loc_94><loc_90></location>The above results are independent of the numerator p of the irreducible fraction. The quadratic field fluctuations, shear stress coe ffi cient π s , energy density, and pressure reduce to</text>
<formula><location><page_7><loc_56><loc_82><loc_94><loc_85></location>⟨ ϕ 2 ⟩ ( p , q ) = [ α T ( x )] 2 96 π 2 ( q 2 -1) , (57a)</formula>
<formula><location><page_7><loc_58><loc_79><loc_94><loc_81></location>E ( p , q ) ξ = [ α T ( x )] 4 480 π 2 ( q 2 -1)( q 2 + 11 + 60 ξ ) , (57b)</formula>
<formula><location><page_7><loc_58><loc_75><loc_94><loc_78></location>P ( p , q ) ξ = [ α T ( x )] 4 1440 π 2 ( q 2 -1)( q 2 + 31 -60 ξ ) , (57c)</formula>
<formula><location><page_7><loc_58><loc_72><loc_94><loc_75></location>π ( p , q ) s ; ξ = -[ α T ( x )] 4 72 π 2 (1 -6 ξ )( q 2 -1) , (57d)</formula>
<text><location><page_7><loc_52><loc_70><loc_85><loc_71></location>manifestly vanishing when q 2 = 1, i.e. for α = 2 π .</text>
<text><location><page_7><loc_52><loc_56><loc_94><loc_70></location>In the case of the Dirac field, we have Sn ( α ) → S ( p , q ) n = -1 2 P q -1 j = 1 ( -1) j cos( π jp / q ) / [sin( π jp / q )] n . For the case n = 4, the relation ( -1) q -j cos[ π ( q -j ) p / q ] = ( -1) j + p + q cos( π jp / q ) implies that S ( p , q ) 4 vanishes when p + q is an odd number. This happens whenever q is an even number in order to maintain the fraction p / q irreducible. When q is odd, S ( p , q ) 4 vanishes for all even values of p . When both p and q are odd, S ( p , q ) 4 can be computed analytically and the final result can be summarized as</text>
<formula><location><page_7><loc_59><loc_53><loc_94><loc_56></location>S ( p , q ) 4 = 7 q 2 + 17 720 ( q 2 -1) × 1 + ( -1) p + q 2 . (58)</formula>
<text><location><page_7><loc_52><loc_51><loc_72><loc_52></location>The fermion pressure becomes</text>
<formula><location><page_7><loc_55><loc_47><loc_94><loc_50></location>P ( p , q ) D = [ α T ( x )] 4 2880 π 2 ( q 2 -1)(7 q 2 + 17) 1 + ( -1) p + q 2 . (59)</formula>
<section_header_level_1><location><page_7><loc_52><loc_43><loc_63><loc_44></location>11. Conclusions</section_header_level_1>
<text><location><page_7><loc_52><loc_21><loc_94><loc_40></location>In this paper, we derived the KMS relation for bosonic and fermionic quantum systems at finite temperature under uniform acceleration. In Wick-rotated Minkowski spacetime, the uniform acceleration requires the identification (31) of the points in the bulk of the system along the discrete points lying on circular orbits (32) about the Rindler horizon, which shrinks to a point (34) under the Wick rotation. In the Wickrotated Rindler coordinates, the KMS relations reduce to standard (anti-)periodic boundary conditions in terms of the imaginary rapidity coordinates. To illustrate the e ff ectiveness of the method, we considered the quantum thermal distributions of massless scalar and Dirac particles under acceleration and found perfect agreement with results previously derived in the literature.</text>
<text><location><page_7><loc_52><loc_9><loc_94><loc_20></location>Our work paves the way to systematic explorations of the influence of the kinematic state of a system on its global equilibrium thermodynamic properties. Our paper equips us with a rigorously formulated method in imaginary-time formalism which allows us to construct the ground state of a field theory in thermal equilibrium in a uniformly accelerating frame, opening, in particular, a way for first-principle lattice simulations of accelerated systems.</text>
<section_header_level_1><location><page_8><loc_6><loc_89><loc_20><loc_90></location>Acknowledgements</section_header_level_1>
<text><location><page_8><loc_6><loc_80><loc_48><loc_87></location>This work is supported by the European Union - NextGenerationEU through the grant No. 760079 / 23.05.2023, funded by the Romanian ministry of research, innovation and digitalization through Romania's National Recovery and Resilience Plan, call no. PNRR-III-C9-2022-I8.</text>
<section_header_level_1><location><page_8><loc_6><loc_76><loc_14><loc_77></location>References</section_header_level_1>
<unordered_list>
<list_item><location><page_8><loc_7><loc_71><loc_48><loc_74></location>[1] P. Castorina, D. Kharzeev, H. Satz, Thermal Hadronization and HawkingUnruh Radiation in QCD, Eur. Phys. J. C 52 (2007) 187-201. arXiv: 0704.1426 , doi:10.1140/epjc/s10052-007-0368-6 .</list_item>
<list_item><location><page_8><loc_7><loc_67><loc_48><loc_71></location>[2] D. Kharzeev, K. Tuchin, From color glass condensate to quark gluon plasma through the event horizon, Nucl. Phys. A 753 (2005) 316334. arXiv:hep-ph/0501234 , doi:10.1016/j.nuclphysa.2005. 03.001 .</list_item>
<list_item><location><page_8><loc_7><loc_63><loc_48><loc_66></location>[3] J. D. Bjorken, Highly Relativistic Nucleus-Nucleus Collisions: The Central Rapidity Region, Phys. Rev. D 27 (1983) 140-151. doi:10.1103/ PhysRevD.27.140 .</list_item>
<list_item><location><page_8><loc_7><loc_60><loc_48><loc_63></location>[4] F. Gelis, B. Schenke, Initial-state quantum fluctuations in the Little Bang, Annual Review of Nuclear and Particle Science 66 (1) (2016) 73-94. doi:10.1146/annurev-nucl-102115-044651 .</list_item>
<list_item><location><page_8><loc_9><loc_59><loc_48><loc_60></location>URL https://doi.org/10.1146/annurev-nucl-102115-044651</list_item>
<list_item><location><page_8><loc_7><loc_56><loc_48><loc_59></location>[5] K. Yagi, T. Hatsuda, Y. Miake, Quark-Gluon Plasma: From Big Bang to Little Bang, Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology, Cambridge University Press, 2008.</list_item>
<list_item><location><page_8><loc_9><loc_54><loc_44><loc_55></location>URL https://books.google.se/books?id=ZXIdOwAACAAJ</list_item>
<list_item><location><page_8><loc_7><loc_51><loc_48><loc_54></location>[6] J. I. Kapusta, C. Gale, Finite-temperature field theory: Principles and applications, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2011. doi:10.1017/CBO9780511535130 .</list_item>
<list_item><location><page_8><loc_7><loc_48><loc_48><loc_51></location>[7] L. Adamczyk, et al., Global Λ hyperon polarization in nuclear collisions: evidence for the most vortical fluid, Nature 548 (2017) 62-65. arXiv: 1701.06657 , doi:10.1038/nature23004 .</list_item>
<list_item><location><page_8><loc_7><loc_44><loc_48><loc_47></location>[8] A. Yamamoto, Y. Hirono, Lattice QCD in rotating frames, Phys. Rev. Lett. 111 (2013) 081601. arXiv:1303.6292 , doi:10.1103/PhysRevLett. 111.081601 .</list_item>
<list_item><location><page_8><loc_7><loc_40><loc_48><loc_44></location>[9] P. de Forcrand, O. Philipsen, The QCD phase diagram for small densities from imaginary chemical potential, Nucl. Phys. B 642 (2002) 290306. arXiv:hep-lat/0205016 , doi:10.1016/S0550-3213(02) 00626-0 .</list_item>
<list_item><location><page_8><loc_6><loc_35><loc_48><loc_40></location>[10] H.-L. Chen, K. Fukushima, X.-G. Huang, K. Mameda, Analogy between rotation and density for Dirac fermions in a magnetic field, Phys. Rev. D 93 (10) (2016) 104052. arXiv:1512.08974 , doi:10.1103/ PhysRevD.93.104052 .</list_item>
<list_item><location><page_8><loc_6><loc_32><loc_48><loc_35></location>[11] Y. Jiang, J. Liao, Pairing Phase Transitions of Matter under Rotation, Phys. Rev. Lett. 117 (19) (2016) 192302. arXiv:1606.03808 , doi: 10.1103/PhysRevLett.117.192302 .</list_item>
<list_item><location><page_8><loc_6><loc_29><loc_48><loc_32></location>[12] M. N. Chernodub, S. Gongyo, Interacting fermions in rotation: chiral symmetry restoration, moment of inertia and thermodynamics, JHEP 01 (2017) 136. arXiv:1611.02598 , doi:10.1007/JHEP01(2017)136 .</list_item>
<list_item><location><page_8><loc_6><loc_25><loc_48><loc_28></location>[13] X. Wang, M. Wei, Z. Li, M. Huang, Quark matter under rotation in the NJL model with vector interaction, Phys. Rev. D 99 (1) (2019) 016018. arXiv:1808.01931 , doi:10.1103/PhysRevD.99.016018 .</list_item>
<list_item><location><page_8><loc_6><loc_21><loc_48><loc_25></location>[14] N. Sadooghi, S. M. A. Tabatabaee Mehr, F. Taghinavaz, Inverse magnetorotational catalysis and the phase diagram of a rotating hot and magnetized quark matter, Phys. Rev. D 104 (11) (2021) 116022. arXiv: 2108.12760 , doi:10.1103/PhysRevD.104.116022 .</list_item>
<list_item><location><page_8><loc_6><loc_17><loc_48><loc_20></location>[15] X. Chen, L. Zhang, D. Li, D. Hou, M. Huang, Gluodynamics and deconfinement phase transition under rotation from holography, JHEP 07 (2021) 132. arXiv:2010.14478 , doi:10.1007/JHEP07(2021)132 .</list_item>
<list_item><location><page_8><loc_6><loc_13><loc_48><loc_17></location>[16] Y. Fujimoto, K. Fukushima, Y. Hidaka, Deconfining Phase Boundary of Rapidly Rotating Hot and Dense Matter and Analysis of Moment of Inertia, Phys. Lett. B 816 (2021) 136184. arXiv:2101.09173 , doi:10.1016/j.physletb.2021.136184 .</list_item>
<list_item><location><page_8><loc_6><loc_9><loc_48><loc_13></location>[17] M. N. Chernodub, Inhomogeneous confining-deconfining phases in rotating plasmas, Phys. Rev. D 103 (5) (2021) 054027. arXiv:2012.04924 , doi:10.1103/PhysRevD.103.054027 .</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_8><loc_52><loc_86><loc_94><loc_90></location>[18] V. V. Braguta, A. Y. Kotov, D. D. Kuznedelev, A. A. Roenko, Study of the Confinement / Deconfinement Phase Transition in Rotating Lattice SU(3) Gluodynamics, Pisma Zh. Eksp. Teor. Fiz. 112 (1) (2020) 9-16. doi: 10.31857/S1234567820130029 .</list_item>
<list_item><location><page_8><loc_52><loc_81><loc_94><loc_85></location>[19] V. V. Braguta, A. Y. Kotov, D. D. Kuznedelev, A. A. Roenko, Influence of relativistic rotation on the confinement-deconfinement transition in gluodynamics, Phys. Rev. D 103 (9) (2021) 094515. arXiv:2102.05084 , doi:10.1103/PhysRevD.103.094515 .</list_item>
<list_item><location><page_8><loc_52><loc_78><loc_94><loc_81></location>[20] V. V. Braguta, A. Kotov, A. Roenko, D. Sychev, Thermal phase transitions in rotating QCD with dynamical quarks, PoS LATTICE2022 (2023) 190. arXiv:2212.03224 , doi:10.22323/1.430.0190 .</list_item>
<list_item><location><page_8><loc_52><loc_74><loc_94><loc_77></location>[21] V. V. Braguta, M. N. Chernodub, A. A. Roenko, D. A. Sychev, Negative moment of inertia and rotational instability of gluon plasma (3 2023). arXiv:2303.03147 .</list_item>
<list_item><location><page_8><loc_52><loc_70><loc_94><loc_74></location>[22] V. V. Braguta, I. E. Kudrov, A. A. Roenko, D. A. Sychev, M. N. Chernodub, Lattice Study of the Equation of State of a Rotating Gluon Plasma, JETP Lett. 117 (9) (2023) 639-644. doi:10.1134/ S0021364023600830 .</list_item>
<list_item><location><page_8><loc_52><loc_68><loc_94><loc_70></location>[23] J.-C. Yang, X.-G. Huang, QCD on Rotating Lattice with Staggered Fermions (7 2023). arXiv:2307.05755 .</list_item>
<list_item><location><page_8><loc_52><loc_65><loc_94><loc_67></location>[24] F. Sun, K. Xu, M. Huang, Quarkyonic phase induced by Rotation (7 2023). arXiv:2307.14402 .</list_item>
<list_item><location><page_8><loc_52><loc_61><loc_94><loc_65></location>[25] M. N. Chernodub, V. A. Goy, A. V. Molochkov, Inhomogeneity of a rotating gluon plasma and the Tolman-Ehrenfest law in imaginary time: Lattice results for fast imaginary rotation, Phys. Rev. D 107 (11) (2023) 114502. arXiv:2209.15534 , doi:10.1103/PhysRevD.107.114502 .</list_item>
<list_item><location><page_8><loc_52><loc_57><loc_94><loc_61></location>[26] A. Yamamoto, Lattice QCD in curved spacetimes, Phys. Rev. D 90 (5) (2014) 054510. arXiv:1405.6665 , doi:10.1103/PhysRevD.90. 054510 .</list_item>
<list_item><location><page_8><loc_52><loc_55><loc_94><loc_57></location>[27] C. Cercignani, G. M. Kremer, The Relativistic Boltzmann Equation: Theory and Applications, Springer, 2002.</list_item>
<list_item><location><page_8><loc_52><loc_52><loc_94><loc_55></location>[28] F. Becattini, Covariant statistical mechanics and the stress-energy tensor, Phys. Rev. Lett. 108 (2012) 244502. arXiv:1201.5278 , doi: 10.1103/PhysRevLett.108.244502 .</list_item>
<list_item><location><page_8><loc_52><loc_50><loc_94><loc_52></location>[29] W. G. Unruh, Notes on black hole evaporation, Phys. Rev. D 14 (1976) 870. doi:10.1103/PhysRevD.14.870 .</list_item>
<list_item><location><page_8><loc_52><loc_47><loc_94><loc_49></location>[30] S. W. Hawking, Black hole explosions?, Nature 248 (5443) (1974) 30-31. doi:10.1038/248030a0 .</list_item>
<list_item><location><page_8><loc_55><loc_46><loc_80><loc_47></location>URL https://doi.org/10.1038/248030a0</list_item>
<list_item><location><page_8><loc_52><loc_43><loc_94><loc_46></location>[31] S. W. Hawking, Particle creation by black holes, Communications In Mathematical Physics 43 (3) (1975) 199-220. doi:10.1007/ bf02345020 .</list_item>
<list_item><location><page_8><loc_55><loc_42><loc_82><loc_43></location>URL https://doi.org/10.1007/bf02345020</list_item>
<list_item><location><page_8><loc_52><loc_38><loc_94><loc_41></location>[32] G. W. Gibbons, M. J. Perry, Black Holes and Thermal Green's Functions, Proc. Roy. Soc. Lond. A 358 (1978) 467-494. doi:10.1098/rspa. 1978.0022 .</list_item>
<list_item><location><page_8><loc_52><loc_36><loc_94><loc_38></location>[33] G. W. Gibbons, M. J. Perry, Black Holes in Thermal Equilibrium, Phys. Rev. Lett. 36 (1976) 985. doi:10.1103/PhysRevLett.36.985 .</list_item>
<list_item><location><page_8><loc_52><loc_33><loc_94><loc_36></location>[34] N. D. Birrell, P. C. W. Davies, Quantum Fields in Curved Space, Cambridge Monographs on Mathematical Physics, Cambridge Univ. Press, Cambridge, UK, 1984. doi:10.1017/CBO9780511622632 .</list_item>
<list_item><location><page_8><loc_52><loc_29><loc_94><loc_32></location>[35] G. Y. Prokhorov, O. V. Teryaev, V. I. Zakharov, Unruh e ff ect for fermions from the Zubarev density operator, Phys. Rev. D 99 (7) (2019) 071901. arXiv:1903.09697 , doi:10.1103/PhysRevD.99.071901 .</list_item>
<list_item><location><page_8><loc_52><loc_26><loc_94><loc_29></location>[36] G. Y. Prokhorov, O. V. Teryaev, V. I. Zakharov, Calculation of acceleration e ff ects using the Zubarev density operator, Particles 3 (1) (2020) 1-14. arXiv:1911.04563 , doi:10.3390/particles3010001 .</list_item>
<list_item><location><page_8><loc_52><loc_22><loc_94><loc_26></location>[37] F. Becattini, M. Buzzegoli, A. Palermo, Exact equilibrium distributions in statistical quantum field theory with rotation and acceleration: scalar field, JHEP 02 (2021) 101. arXiv:2007.08249 , doi:10.1007/ JHEP02(2021)101 .</list_item>
<list_item><location><page_8><loc_52><loc_17><loc_94><loc_21></location>[38] A. Palermo, M. Buzzegoli, F. Becattini, Exact equilibrium distributions in statistical quantum field theory with rotation and acceleration: Dirac field, JHEP 10 (2021) 077. arXiv:2106.08340 , doi:10.1007/ JHEP10(2021)077 .</list_item>
<list_item><location><page_8><loc_52><loc_15><loc_94><loc_17></location>[39] V. E. Ambrus, Dirac fermions on rotating space-times, Ph.D. thesis, She ffi eld U. (2014).</list_item>
<list_item><location><page_8><loc_52><loc_11><loc_94><loc_14></location>[40] V. E. Ambrus, E. Winstanley, Vortical E ff ects for Free Fermions on AntiDe Sitter Space-Time, Symmetry 13 (2021) 2019. arXiv:2107.06928 , doi:10.3390/sym13112019 .</list_item>
<list_item><location><page_8><loc_52><loc_10><loc_94><loc_11></location>[41] S. Mallik, S. Sarkar, Hadrons at Finite Temperature, Cambridge Univer-</list_item>
</unordered_list>
<text><location><page_9><loc_9><loc_89><loc_43><loc_90></location>sity Press, Cambridge, 2016. doi:10.1017/9781316535585 .</text>
<unordered_list>
<list_item><location><page_9><loc_6><loc_86><loc_48><loc_89></location>[42] V. E. Ambrus, Fermion condensation under rotation on anti-de Sitter space, Acta Phys. Polon. Supp. 13 (2020) 199. arXiv:1912.02014 , doi:10.5506/APhysPolBSupp.13.199 .</list_item>
<list_item><location><page_9><loc_6><loc_81><loc_48><loc_85></location>[43] G. Y. Prokhorov, O. V. Teryaev, V. I. Zakharov, Thermodynamics of accelerated fermion gases and their instability at the Unruh temperature, Phys. Rev. D 100 (12) (2019) 125009. arXiv:1906.03529 , doi: 10.1103/PhysRevD.100.125009 .</list_item>
<list_item><location><page_9><loc_6><loc_78><loc_48><loc_81></location>[44] G. Y. Prokhorov, O. V. Teryaev, V. I. Zakharov, Unruh e ff ect universality: emergent conical geometry from density operator, JHEP 03 (2020) 137. arXiv:1911.04545 , doi:10.1007/JHEP03(2020)137 .</list_item>
<list_item><location><page_9><loc_6><loc_74><loc_48><loc_77></location>[45] V. I. Zakharov, G. Y. Prokhorov, O. V. Teryaev, Acceleration and rotation in quantum statistical theory, Phys. Scripta 95 (8) (2020) 084001. doi: 10.1088/1402-4896/ab996b .</list_item>
<list_item><location><page_9><loc_6><loc_71><loc_48><loc_74></location>[46] J. S. Dowker, Vacuum Averages for Arbitrary Spin Around a Cosmic String, Phys. Rev. D 36 (1987) 3742. doi:10.1103/PhysRevD.36. 3742 .</list_item>
<list_item><location><page_9><loc_6><loc_68><loc_48><loc_71></location>[47] B. Linet, Euclidean spinor Green's functions in the space-time of a straight cosmic string, J. Math. Phys. 36 (1995) 3694-3703. arXiv: gr-qc/9412050 , doi:10.1063/1.530991 .</list_item>
<list_item><location><page_9><loc_6><loc_65><loc_48><loc_67></location>[48] S. Chen, K. Fukushima, Y. Shimada, Confinement in hot gluonic matter with imaginary and real rotation (7 2022). arXiv:2207.12665 .</list_item>
<list_item><location><page_9><loc_6><loc_62><loc_48><loc_65></location>[49] V. E. Ambrus¸, M. N. Chernodub, Rigidly rotating scalar fields: Between real divergence and imaginary fractalization, Phys. Rev. D 108 (8) (2023) 085016. arXiv:2304.05998 , doi:10.1103/PhysRevD.108.085016 .</list_item>
<list_item><location><page_9><loc_6><loc_59><loc_48><loc_62></location>[50] M. N. Chernodub, Fractal thermodynamics and ninionic statistics of coherent rotational states: realization via imaginary angular rotation in imaginary time formalism (10 2022). arXiv:2210.05651 .</list_item>
<list_item><location><page_9><loc_6><loc_55><loc_48><loc_58></location>[51] C. G. Callan, S. Coleman, R. Jackiw, A new improved energy-momentum tensor, Annals of Physics 59 (1) (1970) 42-73. doi:10.1016/ 0003-4916(70)90394-5 .</list_item>
</unordered_list>
<text><location><page_9><loc_9><loc_54><loc_44><loc_55></location>URL https://doi.org/10.1016/0003-4916(70)90394-5</text>
<unordered_list>
<list_item><location><page_9><loc_6><loc_51><loc_48><loc_54></location>[52] V. P. Frolov, E. M. Serebriany, Vacuum polarization in the gravitational field of a cosmic string, Physical Review D 35 (12) (1987) 3779-3782. doi:10.1103/physrevd.35.3779 .</list_item>
</unordered_list>
<text><location><page_9><loc_9><loc_50><loc_41><loc_50></location>URL https://doi.org/10.1103/physrevd.35.3779</text>
<unordered_list>
<list_item><location><page_9><loc_6><loc_46><loc_48><loc_49></location>[53] F. Becattini, E. Grossi, Quantum corrections to the stress-energy tensor in thermodynamic equilibrium with acceleration, Phys. Rev. D 92 (2015) 045037. arXiv:1505.07760 , doi:10.1103/PhysRevD.92.045037 .</list_item>
<list_item><location><page_9><loc_6><loc_43><loc_48><loc_46></location>[54] V. I. Zakharov, G. Y. Prokhorov, O. V. Teryaev, Acceleration and rotation in quantum statistical theory, Physica Scripta 95 (8) (2020) 084001. doi: 10.1088/1402-4896/ab996b .</list_item>
</unordered_list>
<text><location><page_9><loc_9><loc_42><loc_41><loc_43></location>URL https://doi.org/10.1088/1402-4896/ab996b</text>
<unordered_list>
<list_item><location><page_9><loc_6><loc_39><loc_48><loc_41></location>[55] D. V. Diakonov, K. V. Bazarov, Thermal loops in the accelerating frame (1 2023). arXiv:2301.07478 .</list_item>
</unordered_list>
</document>
[
{
"title": "Acceleration as a circular motion along an imaginary circle: Kubo-Martin-Schwinger condition for accelerating field theories in imaginary-time formalism",
"content": "Victor E. Ambrus, a , Maxim N. Chernodub b,a a Department of Physics, West University of Timis,oara, Bd. Vasile Pˆarvan 4, Timis,oara 300223, Romania b Institut Denis Poisson, Universit´e de Tours, Tours 37200, France",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "We discuss the imaginary-time formalism for field theories in thermal equilibrium in uniformly accelerating frames. We show that under a Wick rotation of Minkowski spacetime, the Rindler event horizon shrinks to a point in a two-dimensional subspace tangential to the acceleration direction and the imaginary time. We demonstrate that the accelerated version of the Kubo-MartinSchwinger (KMS) condition implies an identification of all spacetime points related by integer-multiple rotations in the tangential subspace about this Euclidean Rindler event-horizon point, with the rotational quanta defined by the thermal acceleration, α = a / T . In the Wick-rotated Rindler hyperbolic coordinates, the KMS relations reduce to standard (anti-)periodic boundary conditions in terms of the imaginary proper time (rapidity) coordinate. Our findings pave the way to study, using first-principle lattice simulations, the Hawking-Unruh radiation in geometries with event horizons, phase transitions in accelerating Early Universe and early stages of quark-gluon plasma created in relativistic heavy-ion collisions. Keywords: Acceleration, Unruh e ff ect, KMS relation, Finite temperature field theory",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "In the past decades, there has been a renewed interest in studying systems with acceleration as toy models for understanding the dynamics of the quark-gluon plasma fireball created in ultrarelativistic (non-central) heavy-ion collisions [1]. Such systems exhibit large acceleration immediately after the collision [2] until the central rapidity plateau develops as in the Bjorken boost-invariant flow model [3], where the acceleration vanishes. A natural question that arises for such a system is to what extent these extreme kinematic regimes a ff ect the thermodynamics of the plasma fireball, which sets the stage for further evolution of the quark-gluon plasma. The environment of the 'Little Bangs' of high-energy heavy-ion collisions [4] sheds insights on the properties of a primordial quark-gluon matter that once emerged at the time of the Big Bang in the Early Universe [5]. Our knowledge of the non-perturbative properties of the quark-gluon plasma originates from first-principle numerical simulations of lattice QCD, which is formulated in Euclidean spacetime, by means of the imaginary-time formalism [6]. Acceleration is closely related to rotation due to the resemblance of the corresponding generators of Lorentz transformations of Minkowski spacetime. In the case of non-central collisions, the angular velocity of the quark-gluon fluid can reach values of the order of Ω ∼ 10 22 Hz [7] which translates to ℏ Ω ≃ 6 MeV ≪ Tc , where Tc is the transition temperature to the quark-gluon plasma phase. The lattice studies have so far been limited to the case of uniformly rotating systems in Euclidean space-time, where the rotation parameter has to be analytically continued to imaginary values [8] in order to avoid the sign problem that also plagues lattice calculations at finite chemical potential [9]. Analytical analyses of the e ff ects of rotation on the phase diagram, performed in various e ff ective infrared models of QCD [10, 11, 12, 13, 14, 15, 16, 17], stay in persistent contradiction with the first-principle numerical results [18, 19, 20, 21, 22, 23], presumably due to numerically-observed rotational instability of quark-gluon plasma [21, 22, 23] (related to the thermal melting of the non-perturbative gluon condensate [21]), splitting of chiral and deconfining transitions [23, 24], or formation of a strongly inhomogeneous mixed hadronic-quark-gluon-plasma phase induced by rotation [17, 25]. Anearlier study of a Euclidean quantum field theory in an accelerating spacetime with the Friedmann-Lemaˆıtre-RobertsonWalker metric has also encountered the sign problem, which was avoided by considering a purely imaginary Hubble constant [26]. On the contrary, our formulation of acceleration in the imaginary-time formalism is free from the sign problem, and thus, it can be formulated for physical, real-valued acceleration. Throughout the paper, we use ℏ = c = kB = 1 units.",
"pages": [
1
]
},
{
"title": "2. Global equilibrium in uniform acceleration",
"content": "From a classical point of view, global equilibrium states in generic particle systems are characterized by the inverse temperature four-vector β µ ≡ u µ ( x ) / T ( x ), associated with the local fluid velocity u µ , with β µ satisfying the Killing equation, ∂µβν + ∂νβµ = 0 [27, 28]. For an accelerated system at equilibrium, one gets β µ ∂µ = β T [(1 + az ) ∂ t + at ∂ z ], with β T = 1 / T where 1 . T ≡ T ( 0 ) represents the temperature at the coordinate origin x ∥ ≡ ( t , z ) = 0 in the longitudinal plane spanned by the time coordinate t and the acceleration direction z . The local temperature T ( x ), the local fluid velocity u µ ( x ) and the local proper acceleration a µ ( x ) ≡ u ν ∂ν u µ , respectively, diverge at the Rindler horizon: It is convenient to define the dimensionless quantity called the proper thermal acceleration α = p -α µ αµ and the corresponding four-vector α µ = u ν ∂νβ µ = a µ / T ( x ), respectively: Note that, while the magnitude α of the thermal acceleration is a space-time constant, the local acceleration a ( x ) = p -a µ a µ = α T ( x ) depends on space and time coordinates. In classical theory, the energy-momentum tensor for an accelerating fluid in thermal equilibrium reads where ∆ µν = g µν -u µ u ν . The local energy density E and pressure P are characterized by the local temperature (1). For a conformal system, where ν e ff is the e ff ective bosonic degrees of freedom. In the case of a massless, neutral scalar field, ν e ff = 1, while for Dirac fermions, ν e ff = 7 8 × 2 × 2 = 7 / 2, taking into account the di ff erence between Bose-Einstein and Fermi-Dirac statistics (7 / 8), spin degeneracy, as well as particle and anti-particle contributions.",
"pages": [
1,
2
]
},
{
"title": "3. Unruh and Hawking e ff ects",
"content": "Unruh has found that in a frame subjected to a uniform acceleration a , an observer detects a thermal radiation with the temperature [29]: where we also defined the Unruh length β U , which will be useful in our discussions below. The Unruh e ff ect is closely related to the Hawking evaporation of black holes [30, 31], which proceeds via the quantum production of particle pairs near the event horizon of the black hole. The Hawking radiation has a thermal spectrum with an e ff ective temperature where κ = 1 / (4 M ) is the acceleration due to gravity at the horizon of a black hole of mass M . The similarity of both e ff ects, suggested by the equivalence of formulas for the Unruh temperature (9) and the Hawking temperature (8), goes deeper as the thermal character of both phenomena apparently originates from the presence of appropriate event horizons [32, 33]. In an accelerating frame, the event horizon separates causally disconnected regions of spacetime, evident in the Rindler coordinates in which the metric of the accelerating frame is conformally flat [34]. Quantum e ff ects lead to acceleration-dependent corrections to Eq. (7) and may also produce extra (anisotropic) contributions to the energy-momentum tensor T µν of the system. Such corrections were already established using the Zubarev approach [35, 36] or Wigner function formalism [37, 38], and one remarkable conclusion is that the energy-momentum tensor Θ µν in an accelerating system exactly vanishes at the Unruh temperature (8), or, equivalently, when the thermal acceleration (3) reaches the critical value α = α c = 2 π : Θ µν ( T = TU ) = 0. A somewhat related property is satisfied by thermal correlation functions in the background of a Schwarzschild black hole, establishing the equivalence between Feynman and thermal Green's functions, with the latter one taken at the Hawking temperature (9), cf. Ref. [33, 32]. As noted earlier, the energy density receives quantum corrections. For the conformally-coupled massless real-valued Klein-Gordon scalar field and the Dirac field, we have, respectively [36, 37, 38, 39, 40]: where we specially rearranged terms to make it evident that at the Unruh temperature T = TU (or, equivalently, at α = 2 π ), the energy density vanishes. The above discussion focused on the free-field theory. In the interacting case, a legitimate question is to what extent do the local kinematics influence the phase structure of phenomenologically relevant field theories, for example, to deconfinement and chiral thermal transitions of QCD. Central to lattice finitetemperature studies is how to set the Euclidean-space boundary conditions in the imaginary-time formalism. A static bosonic (fermionic) system at finite temperature can be implemented by imposing (anti-)periodicity in imaginary time τ = it with period given by the inverse temperature, τ → τ + β T . These boundary conditions are closely related to, and in fact, derived from the usual Kubo-Martin-Schwinger (KMS) relation formulated for a finite-temperature state (at vanishing acceleration), which translates into a condition written for the scalar and fermionic thermal two-point functions [6, 41]: where we suppressed the dependence on the spatial coordinate x and the second four-point x ' . In the case of rotating states, the KMS relation (11) gets modified to [17, 40, 42] where e -β T Ω S z is the spin part of the rotation with imaginary angle i β T Ω along the rotation ( z ) axis and S z = i 2 γ x γ y is the spin matrix. The purpose of the present paper is to uncover the KMS relation and subsequent conditions for fields and, consequently, for correlation functions in a uniformly accelerated state.",
"pages": [
2,
3
]
},
{
"title": "4. Quantum field theory at constant acceleration",
"content": "In Minkowski space, the most general solution of the Killing equation reads where b µ is a constant four-vector and ϖ µν is a constant, antisymmetric tensor. A quantum system in thermal equilibrium is characterized by the density operator where ˆ P µ and ˆ J µν are the conserved four-momentum and total angular momentum operator, representing the generators of translations and of Lorentz transformations. In order to derive the KMS relation, it is convenient to factorize ˆ ρ into a translation part and a Lorentz transformation part, as pointed out in Ref. [37]: where ˜ b is given by Focusing now on the accelerated system with reference inverse temperature β T = 1 / T , we have b µ = β T δ µ 0 and ϖ µ ν = α ( δ µ 3 g 0 ν -δ µ 0 g 3 ν ), such that ˜ b becomes where α = a / T is the thermal acceleration (5). This observation allows ˆ ρ = e -β T ˆ H + α ˆ K z to be factorized as Arelativistic quantum field described by the field operator ˆ Φ transforms under Poincar'e transformations as where Λ = e -i 2 ϖ : J is written in terms of the matrix generators ( J µν ) αβ = i ( δ µ α δ ν β -δ µ β δ ν α ), while D [ Λ ] -1 = e i 2 ϖ : S is the spin part of the inverse Lorentz transformation. Comparing Eq. (19) and (14), it can be seen that the density operator ˆ ρ acts like a Poincar'e transformation with imaginary parameters [37]. Using now the factorization (18), it can be seen that ˆ ρ acts on the field operator ˆ Φ as follows: where The spin term evaluates to e -α S 0 z = 1 in the scalar case (since S 0 z = 0), while for the Dirac field, S 0 z = i 2 γ 0 γ 3 and",
"pages": [
3
]
},
{
"title": "5. KMS relation at constant uniform acceleration",
"content": "Consider now the Wightman functions G ± ( x , x ' ) and S ± ( x , x ' ) of the Klein-Gordon and Dirac theories, defined respectively as When the expectation value ⟨·⟩ is taken at finite temperature and under acceleration, we derive the KMS relations: The KMS relations also imply natural boundary conditions for the thermal propagators: which are solved formally by [34, 40] where G vac F ( x , x ' ) and S vac F ( x , x ' ) are the vacuum propagators, while t ( j ) and z ( j ) are obtained by applying the transformation in Eq. (21) j ∈ Z times: In particular, ˜ t = t (1) and ˜ z = z (1). Due to the periodicity of the trigonometric functions appearing above, in the case when α/ 2 π = p / q is a rational number represented as an irreducible fraction, the sum over j in Eqs. (26) contains only q terms: In particular, the case α = 2 π corresponds to p = q = 1, while the thermal propagators reduce trivially to the vacuum ones: G (1 , 1) F = G vac F and S (1 , 1) F = S vac F . Since e -q α S 0 z = ( -1) p by virtue of Eq. (22), applying Eq. (25) q times shows that S ( p , q ) F ( t ( q ) , z ( q ); x ' ) = ( -1) p + q S ( p , q ) F ( t , z ; x ' ) and thus S ( p , q ) F cancels identically when p + q is an odd integer.",
"pages": [
3,
4
]
},
{
"title": "6. Imaginary-time formulation for acceleration",
"content": "We now move to the Euclidean manifold by performing the Wick rotation to imaginary time, t → τ = it . Then, Eq. (25) becomes and Eq. (26) reads, for the case when α/ 2 π is an irrational number, The case when α/ 2 π = p / q must be treated along the lines summarized in Eqs. (28) (see also discussion in Sec. 10). In the above, we considered j ∈ Z and For the fields, the accelerated KMS conditions suggest the identification of the fields at the points: where the identified coordinates ( τ ( j ) , z ( j )) in the longitudinal plane are given by Eq. (31) and x ∥ = ( x , y ) are the transverse coordinates which are unconstrained by acceleration. While the sums of the form (26) may formally be divergent, the modified conditions (31) and (32) give a finite solution to the accelerated KMSrelations. The points identified with the accelerated KMS condition (31) are illustrated in Fig. 1.",
"pages": [
4
]
},
{
"title": "7. Geometrical meaning of the accelerated KMS relation in imaginary-time formalism",
"content": "It is convenient, for a moment, to define a translationally shifted spatial coordinate, z = z + 1 / a , and rewrite Eq. (31) in the very simple and suggestive form: In the shifted coordinates, the condition (4) for the Rindler horizon becomes a 2 ( z 2 + τ 2 ) = 0, which is solved by Thus, we arrive at the following beautiful conclusion: in the Euclidean spacetime of the imaginary-time formalism, the Rindler horizon (4) shrinks to a single point (34). Thus, the accelerated KMS condition corresponds to the identification of all points obtained by the discrete rotation of the space around the Euclidean Rindler horizon point ( τ, z ) = (0 , -1 / a ) with the unit rotation angle defined by the reference thermal acceleration α = a / T . Our accelerated KMS condition, given in Eqs. (31) and (32), recovers the usual finite-temperature KMS condition in the limit of vanishing acceleration. Figure 2 demonstrates that in this limit,with α = a / T → 0, the proposed KMS-type condition (27) for the acceleration is reduced to the standard finitetemperature KMS-boundary condition [6] for which imaginary time τ is compactified to a circle of the length β T ≡ 1 / T with the points ( τ, x ) and ( τ + β T n , x ), n ∈ Z , identified. At the critical acceleration α = 2 π n (with n ∈ Z ), when the background temperature T equals to (an integer multiple of) the Unruh temperature (8), the accelerated KMS conditions (31) do not constrain the system anymore, τ ( j ) = τ and z ( j ) = z , so that the system becomes equivalent to a zero-temperature system in non-accelerated flat Minkowski spacetime. This property, for α = 2 π , has been observed in Refs. [35, 36, 37, 38]. In the situation where 2 π/α = β U /β T = n is an integer number, the accelerated state at finite temperature can be implemented in Euclidean space by imposing periodicity with respect to a specific set of points that form a regular polygon with n vertices located on the circle of radius τ 2 + z 2 . This is particularly convenient for lattice simulations since the Euclidean action remains the standard one, allowing accelerated systems to be modeled in the imaginary-time path integral formalism without encountering the infamous sign problem.",
"pages": [
4,
5
]
},
{
"title": "8. KMS relations in Rindler coordinates",
"content": "In the Minkowski Lorentz frame that we considered so far, the accelerating KMS conditions (31) and (32) do not correspond to a boundary condition (as one would naively expect from the KMS condition in thermal field theory) but rather to a bulk condition: instead of relating the points at the boundary of the imaginary-time Euclidean system, the accelerated KMS relations give us the identification of the spacetime points in its interior. While seemingly non-trivial in the form written in Eq. (27), the displacements implied by the KMS relation correspond to the usual translation of the proper time (rapidity) coordinate η when employing the Rindler coordinates, It is easy to see that which implies that in a seemingly perfect agreement with the usual KMS relation (11) for static systems in Minkowski. However, there is also an unusual particularity of the KMS conditions (37) in the Rindler coordinates (35). The first relation in Eq. (37) suggests that the Wick rotation of the Minkowski time t = -i τ should be supplemented with the Wick rotation of the proper time in the accelerated frame η = -i θ/ a , where θ is the imaginary rapidity. 2 Then, the relation (35) in the imaginary (both Minkowski and Rindler) time becomes as follows: which shows that the imaginary rapidity becomes an imaginary coordinate with the Euclidean Rindler KMS condition (37): Curiously, under the Wick transform, the rapidity becomes a cyclic compact variable, 0 ⩽ θ < 2 π , on which the imaginarytime condition (39) imposes the additional periodicity with the period equal to the thermal acceleration α . Expectedly, at α = 2 π (or, equivalently, at T = TU ), the boundary condition (39) becomes trivial. The boundary conditions (39), characterized by the doublyperiodic imaginary rapidity coordinate θ , with periodicities θ → θ + 2 π and θ → θ + α (for 0 ⩽ α < 2 π ), can be easily implemented in lattice simulations. Notice that this double periodicity has a strong resemblance to the observation of Refs. [43, 44, 45] that the Euclidean Rindler space can be identified with the space of the cosmic string which possesses a conical singularity with the angular deficit ∆ φ = 2 π -α [46, 47]. The KMS periodicity (39) of the compact imaginary rapidity θ is formally sensitive to the rationality of the normalized thermal acceleration α/ (2 π ). Obviously, for α = 2 π p / q , where p < q are nonvanishing irreducible integer numbers, the interplay of the two periodicities will correspond to the single period θ → θ + 2 π/ q . Interestingly, the sensitivity of an e ff ect to the denominator q (and not to the numerator p ) of a relevant parameter is a signature of the fractal nature of the e ff ect. Such fractality is noted, for example, in particle systems subjected to imaginary rotation implemented via rotwisted boundary conditions [17, 48, 49], which leads, in turn, to the appearance of 'ninionic' deformation of particle statistics [50]. The suggested fractality of acceleration in imaginary formalism is not surprising given the conceptual similarity of acceleration and rotation with imaginary angular frequency [37, 38]. Below, we will show that, despite the fractal property of the system, the KMS boundary condition (39) in Euclidean Rindler space correctly reproduces results for accelerated particle systems.",
"pages": [
5,
6
]
},
{
"title": "9. Energy-momentum tensor with the accelerated KMS conditions",
"content": "Now let us come back to the Wick-rotated Minkowski spacetime and verify how the modified KMS conditions for the fields, Eqs. (31) and (32), and related solutions for their two-point functions (30), can recover the known results in field theories under acceleration. To this end, we start from a non-minimally coupled scalar field theory with the Lagrangian [51, 52, 53] possessing the following energy-momentum tensor: where the values ξ = 0 and ξ = 1 / 6 of the coupling parameter correspond to the canonical and conformal energy-momentum tensors, respectively. In terms of the Euclidean Green's function, Θ ξ µν can be written as where ∆ G ( α ) E ( x , x ' ) = G ( α ) E ( x , x ' ) -G vac E ( x , x ' ) represents the thermal part of the Green's function. For the Dirac field, Θ µν = 1 2 ¯ ψγ E µ ←→ ∂ν ψ can be computed from the Euclidean two-point function S ( α ) E ( x , x ' ) via The vacuum propagators satisfying □ G vac E ( x , x ' ) = γ E µ ∂µ S vac E ( x , x ' ) = δ 4 ( x -x ' ) are given by with ∆ X 2 = ( ∆ τ ) 2 + ( ∆ x ) 2 . Using Eq. (30), the thermal expectation values of the normal-ordered energy-momentum operator can be obtained in the case of the Klein-Gordon field as: where ∆ X 2 ( j ) = 4 a 2 sin 2 j α 2 [( a τ ) 2 + (1 + az ) 2 ] and R ( j ) µν ≡ ∂µ ∆ x ( j ) ν is given by such that R ( j ) µλ R ( j ) νλ = δµν . For the Dirac field, we find Taking advantage of the relation ( R ( j ) νκ + δνκ ) ∆ x ( j ) κ = -2 a sin( j α )[(1 + az ) δν 0 -a τδν 3] and after switching back to the real time t , we find with E , P , and u µ being the energy density, isotropic pressure, and the fluid four-velocity (2), respectively. The shear-stress tensor π µν is by construction traceless, symmetric and orthogonal to u µ , discriminating between the energy-momentum tensors in classical (6) and quantum (49) fluids. Due to the symmetries of the problem, its tensor structure is fixed as with α µ ( x ) being the local thermal acceleration (3), such that the shear coe ffi cient π s is the only degree of freedom of π µν in Eq. (50). In the scalar case, we find for the components of (49): with Gn ( α ) = P ∞ j = 1 [sin( j α/ 2)] -n , in complete agreement with the results in Ref. [37]. Formally, Gn diverges, however its value can be obtained from its analytical continuation to imaginary acceleration a = i ϕ , e Gn ( β T ϕ ) = i n Gn ( i β T ϕ ). The sum can be evaluated, in a certain domain around β T ϕ > 0 [37], to: Substituting now Gn ( α ) = Re[ i -n e Gn ( i β T ϕ ) ⌋ ϕ →-ia ] into Eq. (51) gives Eq. (10) for the conformal coupling ξ = 1 / 6. For minimal coupling ξ = 0 or a generic non-conformal coupling ξ , 1 / 6, we recover the results of Refs. [37, 54]. In the case of the Dirac field, one can easily check that E D = 3 P D and π s D = 0, while with Sn ( α ) = -P ∞ j = 1 ( -1) j cos( j α/ 2) / [sin( j α/ 2)] n → e Sn ( β T ϕ ) ≡ i n Sn ( i β T ϕ ) = -P ∞ j = 1 ( -1) j cosh( j β T ϕ/ 2) / [sinh( j β T ϕ/ 2)] n , which agrees with the results obtained in Ref. [38]. Finally, let us also illustrate the practical functionality of the accelerating KMS boundary conditions (39) formulated in the imaginary-rapidity Rindler space (38). For simplicity, we calculate the fluctuations of the scalar field ⟨ ϕ 2 ⟩ using pointsplitting and noticing that the same method can be used to calculate also other quantities. When expressed with respect to Rindler coordinates X = ( θ/ a , x ⊥ , ζ ), the Euclidean vacuum two-point function G vac E , R ( X , X ' ) given in Eq. (44) reads as follows: The KMS condition (39) implies that the Euclidean two-point function under acceleration satisfies G ( α ) E , R = P j ∈ Z G vac E , R ( ∆ θ + j α ), where we consider vanishing spatial distance between the points: ζ ' → ζ and x ' ⊥ → x ⊥ . Subtracting the vacuum ( j = 0) term that diverges in the ∆ X → 0 limit, we get for the scalar fluctuations: which agrees with the known result [37, 55].",
"pages": [
6,
7
]
},
{
"title": "10. Fractalization of thermodynamics",
"content": "Let us consider the case when α/ 2 π is a rational number, represented as the irreducible fraction p / q . Then, the functions Gn ( α ) → G ( p , q ) n ( α ) = 1 2 P q -1 j = 1 [sin( π jp / q )] -n are regular and evaluate in the relevant n = 2 and n = 4 cases to The above results are independent of the numerator p of the irreducible fraction. The quadratic field fluctuations, shear stress coe ffi cient π s , energy density, and pressure reduce to manifestly vanishing when q 2 = 1, i.e. for α = 2 π . In the case of the Dirac field, we have Sn ( α ) → S ( p , q ) n = -1 2 P q -1 j = 1 ( -1) j cos( π jp / q ) / [sin( π jp / q )] n . For the case n = 4, the relation ( -1) q -j cos[ π ( q -j ) p / q ] = ( -1) j + p + q cos( π jp / q ) implies that S ( p , q ) 4 vanishes when p + q is an odd number. This happens whenever q is an even number in order to maintain the fraction p / q irreducible. When q is odd, S ( p , q ) 4 vanishes for all even values of p . When both p and q are odd, S ( p , q ) 4 can be computed analytically and the final result can be summarized as The fermion pressure becomes",
"pages": [
7
]
},
{
"title": "11. Conclusions",
"content": "In this paper, we derived the KMS relation for bosonic and fermionic quantum systems at finite temperature under uniform acceleration. In Wick-rotated Minkowski spacetime, the uniform acceleration requires the identification (31) of the points in the bulk of the system along the discrete points lying on circular orbits (32) about the Rindler horizon, which shrinks to a point (34) under the Wick rotation. In the Wickrotated Rindler coordinates, the KMS relations reduce to standard (anti-)periodic boundary conditions in terms of the imaginary rapidity coordinates. To illustrate the e ff ectiveness of the method, we considered the quantum thermal distributions of massless scalar and Dirac particles under acceleration and found perfect agreement with results previously derived in the literature. Our work paves the way to systematic explorations of the influence of the kinematic state of a system on its global equilibrium thermodynamic properties. Our paper equips us with a rigorously formulated method in imaginary-time formalism which allows us to construct the ground state of a field theory in thermal equilibrium in a uniformly accelerating frame, opening, in particular, a way for first-principle lattice simulations of accelerated systems.",
"pages": [
7
]
},
{
"title": "Acknowledgements",
"content": "This work is supported by the European Union - NextGenerationEU through the grant No. 760079 / 23.05.2023, funded by the Romanian ministry of research, innovation and digitalization through Romania's National Recovery and Resilience Plan, call no. PNRR-III-C9-2022-I8.",
"pages": [
8
]
},
{
"title": "References",
"content": "sity Press, Cambridge, 2016. doi:10.1017/9781316535585 . URL https://doi.org/10.1016/0003-4916(70)90394-5 URL https://doi.org/10.1103/physrevd.35.3779 URL https://doi.org/10.1088/1402-4896/ab996b",
"pages": [
9
]
}
]
2024PhLB..85538768L
https://arxiv.org/pdf/2211.05265.pdf
<document>
<section_header_level_1><location><page_1><loc_21><loc_89><loc_79><loc_91></location>On primordial universe in anti-de Sitter landscape</section_header_level_1>
<text><location><page_1><loc_18><loc_85><loc_82><loc_87></location>Pu-Xin Lin 1 , 2 ∗ , Hai-Long Huang 1 , 3 † , Jun Zhang 4 ‡ , and Yun-Song Piao 1 , 3 , 4 , 5 §</text>
<unordered_list>
<list_item><location><page_1><loc_24><loc_82><loc_76><loc_84></location>1 School of Fundamental Physics and Mathematical Sciences,</list_item>
</unordered_list>
<text><location><page_1><loc_19><loc_80><loc_81><loc_81></location>Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China</text>
<unordered_list>
<list_item><location><page_1><loc_24><loc_74><loc_75><loc_78></location>2 Department of Physics, University of Wisconsin-Madison, 1150 University Ave, Madison, WI 53706, U.S.A.</list_item>
<list_item><location><page_1><loc_27><loc_69><loc_72><loc_73></location>3 School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China</list_item>
<list_item><location><page_1><loc_12><loc_63><loc_88><loc_67></location>4 International Center for Theoretical Physics Asia-Pacific, Beijing/Hangzhou, China and 5 Institute of Theoretical Physics, Chinese Academy of Sciences,</list_item>
</unordered_list>
<text><location><page_1><loc_33><loc_60><loc_67><loc_62></location>P.O. Box 2735, Beijing 100190, China</text>
<section_header_level_1><location><page_1><loc_45><loc_57><loc_54><loc_58></location>Abstract</section_header_level_1>
<text><location><page_1><loc_12><loc_38><loc_88><loc_55></location>How the spacetime evolved non-perturbatively in a landscape with multiple anti-de Sitter (AdS) vacua, which is theoretically well-motivated, has always been a matter of concern. As a step towards this issue, we perform (3+1)D numerical relativity simulations for the inhomogeneous universe in an AdS landscape, and find that large inhomogeneity of scalar field can develop into not only sphere-like bubbles, but also novel tube-like structures. It is observed that the bubble or tube wall (across the potential barrier) likely inflates as a quasi-dS space, while the different regions separated by the walls are in different AdS vacua and will collapse towards singularity.</text>
<text><location><page_1><loc_12><loc_34><loc_23><loc_35></location>PACS numbers:</text>
<section_header_level_1><location><page_2><loc_12><loc_89><loc_32><loc_91></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_2><loc_12><loc_64><loc_88><loc_86></location>The success of inflation [1-5], consistent with current cosmological observations, suggests the existence of a de Sitter (dS) phase in the very early stage of our observable Universe. In string theory, although it is possible that dS vacua exist [6, 7], the construction of such vacua in string landscape is not straightforward. There are however valid concerns about the validity of such constructions in string theory, notably the swampland conjecture [8, 9], see also [10, 11] for recent reviews. In contrast, anti-de Sitter (AdS) vacua are easy to construct and are ubiquitous. Recently, it has been showed that AdS spacetime not only plays crucial roles as insights into our Universe, e.g.[12-17] but also have potential observable imprints [18-21].</text>
<text><location><page_2><loc_12><loc_43><loc_88><loc_62></location>The initial state of the universe might be highly inhomogeneous [22], i.e. the scalar field or spacetime metric can exhibit non-perturbative inhomogeneities before a region of space arrives at a certain vacuum, so that initially well-defined background in such highly inhomogeneous states does not exist. It is also often speculated that due to the past incompleteness of inflation [23], large fluctuations of spacetime will be inevitably excited near the initial cosmological singularity, in which case the initial conditions of scalar fields and spacetime are chaotic [5]. In such a non-perturbatively and highly inhomogeneous spacetime, perturbative calculation is not applicable any more.</text>
<text><location><page_2><loc_12><loc_24><loc_88><loc_41></location>In the past decades, numerical relativity (NR) has developed significantly and has been applied to studies on the merging of black hole binaries [24-26] and non-perturbative cosmologies, e.g.[27-31]. As pointed out, the study of AdS landscape is theoretically wellmotivated. It is well-known that the region in a single AdS vacuum is unstable and will eventually collapse into a singularity [32], see also e.g.[33]. Consequently, the non-perturbative evolution of the entire spacetime in such landscape has continued to be a matter of concern, e.g.[34].</text>
<text><location><page_2><loc_12><loc_8><loc_88><loc_23></location>As a step towards this issue, we perform (3+1)D NR simulations of the 'universe' in a simplified AdS landscape. Despite the application of NR simulations in cosmologies [27, 28, 35-37], few focused on the AdS landscape. Specially, in Ref. [37], we investigate the evolution of scalar field in an initial inhomogeneous expanding Universe and how a patch of spacetime evolves into the dS vacua that should happen at the early epoch of the Universe. In this letter, we further investigate the spacetime evolves in a landscape with multiple AdS</text>
<figure>
<location><page_3><loc_13><loc_57><loc_87><loc_91></location>
<caption>FIG. 1: Snapshots of spacetime. The upper and lower panel show how the sphere-like bubbles ( ϕ 0 = 0 . 89) and tube-like structures ( ϕ 0 = 0 . 92) form respectively, which are seen in light of the plane passing the center of the grid with normal ⃗n = (1 , -1 , 0). Initially, the whole space is expanding ( K < 0 at the red end of the colorbar). After a period of evolution, different regions of space will be separated by expanding walls (the bubble wall or tube-like 'ribbon' at which K < 0) into different AdS vacua, while the corresponding AdS regions will collapse ( K > 0 with a blue shifted color).</caption>
</figure>
<text><location><page_3><loc_12><loc_22><loc_88><loc_34></location>vacua, which are more generic in string theory. We explore potential new phenomenology and investigate whether a quasi-de Sitter space can emerge on the bubble wall separating different AdS regions. Our simulation results show novel phenomena not captured by conventional perturbative analyses, see Fig.1, which in certain sense highlights the important role of NR in comprehending the non-perturbative phenomena of spacetime.</text>
<section_header_level_1><location><page_3><loc_12><loc_17><loc_25><loc_18></location>II. METHOD</section_header_level_1>
<text><location><page_3><loc_12><loc_10><loc_88><loc_14></location>Throughout this work, we consider a 1D potential as a phenomenological simplification of the actual complex AdS landscape, setting a fourth-order polynomial barrier between the</text>
<figure>
<location><page_4><loc_27><loc_68><loc_73><loc_91></location>
<caption>FIG. 2: A simplified AdS landscape. The effective potential V ( ϕ ) has two AdS minima separated by a potential barrier with positive energy.</caption>
</figure>
<text><location><page_4><loc_12><loc_58><loc_63><loc_60></location>two AdS minima that have quadratic forms ∼ ϕ 2 (see Fig.2),</text>
<formula><location><page_4><loc_25><loc_46><loc_88><loc_57></location>V ( ϕ ) = 1 2 m 2 1 ( ϕ -ϕ 1 ) 2 + V 1 , ϕ < ϕ 1 , λ ( ϕ 2 + 1 λ ) 2 -2 ϕ 3 -1 λ + V 1 , ϕ 1 ≤ ϕ ≤ ϕ 2 , 1 2 m 2 2 ( ϕ -ϕ 2 ) 2 + V 2 , ϕ 2 < ϕ, (1)</formula>
<text><location><page_4><loc_12><loc_37><loc_88><loc_44></location>where V 1 , V 2 < 0. Here, we set m 2 1 = m 2 2 = 0 . 2, λ = 0 . 49 ( c = ℏ = 8 πG = 1). We thus have the AdS-like vacua located at ϕ 1 = 0 and ϕ 2 ≃ 2, where the potential is constructed to be differentiable, and the maximal value of potential barrier is at ϕ b ≃ 1.</text>
<text><location><page_4><loc_12><loc_29><loc_88><loc_36></location>The large non-perturbative inhomogeneity of fields will be excited inevitably near the initial singularity or the Planck scale, giving rise to the possibility of initially configuration ∂ i ϕ∂ i ϕ ≲ 1, or ∆ ϕ ≲ 1. As example, we set the initial inhomogeneity of scalar field as</text>
<formula><location><page_4><loc_35><loc_24><loc_88><loc_28></location>ϕ | t =0 = ϕ 0 +∆ ϕ ∑ ⃗x = x,y,z cos ( 2 π⃗x L ) , (2)</formula>
<text><location><page_4><loc_12><loc_13><loc_88><loc_22></location>where the amplitude of inhomogeneity is ∆ ϕ = O (0 . 1), the wavelength of inhomogeneity is taken to be equal to the length of the periodic cubic region (in our simulation L = 4). In our simulation, we set the periodic boundary condition 1 , and ∆ ϕ = 0 . 4 is the default value unless otherwise specified, with ϕ 0 varying from 0.8 to 0.98.</text>
<text><location><page_5><loc_12><loc_73><loc_88><loc_91></location>We modify the BSSN based [38, 39] NR package GRChombo 2 [40] to perform the simulation. The NR simulation is usually performed in the 3+1 decomposition context for spacetime, with the metric as g 00 = -α 2 + β i β i , g 0 i = β i and g ij = γ ij , where α is the lapse parameter, β i the shift vector and γ ij the spatial metric, while for the numerical stability, the spatial metric γ ij must be further factorized into its determinant χ (conformal factor) and a comformal metric ˜ γ ij = χγ ij . The evolutions of ˜ γ ij , the connections ˜ Γ i = ˜ γ jk ˜ Γ i jk and the extrinsic curvature K ij = 1 3 Kδ ij + A ij comply with the BSSN equations [38, 39].</text>
<text><location><page_5><loc_12><loc_52><loc_88><loc_72></location>Initially, the values of the BSSN variables are set as ˜ γ ij = δ ij , ˜ A ij = χA ij = 0, and χ = 1. In GRChombo code [40], to satisfy the Hamiltonian constraint involving the scalar field, we enforce the constraint by relaxing the conformal factor in the light of ∂ t χ = H 3 , e.g. see recent Refs.[36, 37]. In addition, we set the initial scalar field at rest ˙ ϕ = 0, satisfying the momentum constraint and the initial expansion rate to be uniform K init = -√ 3 ⟨ ρ ⟩ ≈ -1 . 08, where ⟨ ρ ⟩ indicates the average initial erergy density. In the special case of an isoptopic and homogeneous Universe, extrinsic curvature K is related to the Hubble constant as K = -3 H with geodesic observers.</text>
<section_header_level_1><location><page_5><loc_12><loc_47><loc_26><loc_48></location>III. RESULTS</section_header_level_1>
<text><location><page_5><loc_12><loc_29><loc_88><loc_44></location>The simulation results are presented in Fig.1 and Fig.3. The large inhomogeneity will spawn AdS bubbles in a background of a different AdS vacuum. As expected, if a bubble has subcritical initial radius, it will rapidly shrink and collapse into a black hole 4 , and eventually the whole space will evolve into a single AdS state and collapse integrally, see Fig.3-A. However, if the initial radii of the bubbles are supercritical, the regions with different AdS vacua will coexist, separated by not only near-spherical bubble walls (Fig.3-B) but also,</text>
<formula><location><page_5><loc_29><loc_15><loc_88><loc_18></location>H = ˜ D 2 χ -5 4 χ ˜ γ ij ˜ D i χ ˜ D j χ + χ 2 ˜ R + 1 3 K 2 -χ 3 2 ˜ A ij ˜ A ij -ρ. (3)</formula>
<text><location><page_6><loc_12><loc_87><loc_88><loc_91></location>interestingly, infinitely extending tube-like walls (Fig.3-C). Here, the critical radius of bubble refers to the minimal radius of bubble when the bubble phase can exist.</text>
<text><location><page_6><loc_12><loc_60><loc_88><loc_85></location>These tube-like walls can be understood more clearly in Fig.1. In our simulation, we replace an infinite space with the periodic boundary condition. This choice of boundary condition is motivated as follows: inhomogeneities occur at various scales, when considering evolution of those at a specific length scale, it is natural to expect the extent of the modes goes beyond just a single period. In light of this argument, the 'bubble' in the lower panel of Fig.1 actually correspond to the tube that extend along -→ x (= x, y, z ) directions through the region where the periodicity of the initial perturbation modes persists. Such tube-like structure looks 'as if' the adjacent spherical cells merged partly. It is intriguing to consider the more generic setup with perturbations at numerous wavelengths, a task beyond the scope of this letter, and to be pursed in future work.</text>
<text><location><page_6><loc_12><loc_39><loc_88><loc_59></location>It is significant to check the effect of the wavelength L of inhomogeneity (equivalently L/H -1 init ). A phase diagram is presented in Fig.4 5 . According to Fig.4, for ϕ 0 /ϕ b ≲ 0 . 9, if L/H -1 init ≲ 1 (i.e. the wavelength L is 'subhorizonal'), eventually the whole space will collapse integrally, while the bubble or tube phase comes into being only when the initial inhomogeneity is 'superhorizonal' (the radius of the bubble is supercritical and superhorizonal). It is also worth noting that if ϕ 0 /ϕ b ≳ 0 . 9, i.e. ϕ 0 is close to the maximal value of potential barrier, we will see the bubble or tube phase for L/H -1 init ≲ 1 (the radius of the bubble is supercritical but subhorizonal).</text>
<text><location><page_6><loc_12><loc_21><loc_88><loc_38></location>Physically, our results can be qualitatively understood as follows. Here, if a bubble has subcritical initial radius, it will rapidly shrink and disappear, and eventually the whole space will evolve into a single AdS state and collapse integrally, while if a bubble has supercritical and superhorizonal initial radius, the collapse of bubble wall will be prevented by the expansion of wall spacetime, as expected theoretically in Ref.[34], so a bubble phase will come into being. As the initial radii of bubbles gets bigger (than a certain critical value), the bubbles will get closer so that they can partly join together through the tube naturally.</text>
<text><location><page_6><loc_12><loc_15><loc_88><loc_19></location>Here, the initial 'universe' is set to be expanding homogeneously (all H local = const. > 0), but eventually different regions will tend to evolve into different AdS vacua and be separated</text>
<text><location><page_7><loc_12><loc_81><loc_88><loc_91></location>by bubble or tube walls. According to Fig.5, we see that though the AdS regions at both sides of the wall are collapsing, the existing walls will still expand and eventually arrive at a quasi-dS static profile, implying that the inflationary stage responsible for our observable Universe might happen at such walls, e.g.[41, 42].</text>
<text><location><page_7><loc_12><loc_76><loc_88><loc_80></location>It can be confirmed that in such AdS landscapes, inflation might occur. According to Refs.[41, 42], the width of the wall is δ 0 ∼ ( ϕ 2 -ϕ 1 ) V -1 / 2 b , and</text>
<formula><location><page_7><loc_41><loc_70><loc_88><loc_74></location>δ 0 > 1 H 0 ≃ ( 3 V b ) 1 / 2 (4)</formula>
<text><location><page_7><loc_12><loc_59><loc_88><loc_69></location>must be satisfied for the inflation occurring on the wall. Here, considering the longitudinal Hubble parameter H L = -u i u j K ij ( u i is the normal vector at the bubble wall), and ds 2 = γ ij e i e j dr 2 = γ 2 dr 2 ( r is the coordinate crossing the bubble wall and e i is the direction vector at the wall), we have</text>
<formula><location><page_7><loc_43><loc_56><loc_88><loc_59></location>δ 0 ≃ γ ∆ r ≳ 1 H L , (5)</formula>
<text><location><page_7><loc_12><loc_53><loc_48><loc_55></location>see Fig.6, which confirms the condition (4).</text>
<section_header_level_1><location><page_7><loc_12><loc_48><loc_32><loc_49></location>IV. CONCLUSIONS</section_header_level_1>
<text><location><page_7><loc_12><loc_30><loc_88><loc_45></location>We performed full relativistic simulations for non-perturbative evolution of spacetime in a simple AdS landscape, and found that large inhomogeneity of a scalar field inclines to develope into sphere-like bubbles, and more unexpectedly, into tube-like structures. It is observed that the spacetime evolution is completely different in the AdS landscape, the bubble or tube walls might inflate forever as a quasi-dS space, while the different regions separated by the walls are in different AdS vacua and will collapse towards singularity.</text>
<text><location><page_7><loc_12><loc_12><loc_88><loc_29></location>Though we take a simple potential with two AdS minima as example, actually for a spacetime in a landscape with multiple AdS vacua, its local region always can be described as that with only two neighbouring AdS minima. Thus our results might have effectively captured some of the essentials of non-perturbative phenomenology of spacetime in AdS landscape. In this sense, further research on the NR simulations in a more realistic landscape for the early universe is interesting and required, which will help to deepen insight into the origin of our observable Universe.</text>
<figure>
<location><page_8><loc_18><loc_59><loc_81><loc_91></location>
<caption>FIG. 3: The value of | ϕ -ϕ b | (1/8 of the whole computational grid). In subfig-A, B and C, the lower left and upper right regions correspond to the left and right side of the potential barrier in Fig.2, respectively, and the boundaries colored white mark the position of the barrier. In subfig-A, the snapshot is taken shortly before the upper right AdS region disappears. In subfig-B or C, the snapshot is taken when the bubble or tube wall eventually arrives at rest. The subfig-b and c show the interior of walls with 1/8 of the computational grid removed.</caption>
</figure>
<text><location><page_8><loc_12><loc_29><loc_88><loc_39></location>Acknowledgment This work is supported by the NSFC No.12075246 and by the Fundamental Research Funds for the Central Universities. We acknowledge the use of GRChombo code for our simulation and the Tianhe-2 supercomputer for providing computing resources. We would like to thank Tiago Fran¸ca, Hao-Hao Li and Hao-Yang Liu for useful discussions.</text>
<figure>
<location><page_9><loc_19><loc_54><loc_77><loc_87></location>
<caption>FIG. 4: The phase diagram of bubble-tube Phase Transition . 'Collapse' represents the phase in which the whole space will collapse integrally, 'Bubble' and 'Tube' represent phases with existence of bubbles and tube-like structures respectively. The boundaries between them (indicated by the orange and purple strips) are not sharp since we cannot know which will be the final state in our simulation precision. Here, we mainly focus on the parameter space in which the wavelengths of initial inhomogeneity are very close to the initial horizon, L ≈ H -1 init , in order to see the Phase Transition between different phases.</caption>
</figure>
<figure>
<location><page_10><loc_17><loc_71><loc_82><loc_91></location>
<caption>FIG. 5: The evolution of extrinsic curvature scalar K . The horizonal axis is the line from cubic center to the middle point on the edge of whole computational grids (upper right and left vertexes of the boxes in Fig.3). In subfig-b and c, the bubble and tube walls form, and eventually the walls will arrive a quasi-dS stationary profile ( K = const. < 0 on wall).</caption>
</figure>
<figure>
<location><page_10><loc_13><loc_37><loc_50><loc_56></location>
</figure>
<figure>
<location><page_10><loc_51><loc_37><loc_87><loc_56></location>
<caption>FIG. 6: The longitudinal Hubble rate H L and γ with respect to r across the bubble wall for different initial conditions. The solid line is the longitudinal Hubble rate (value shown on the left vertical axis) and the dotted line is γ (value shown on the right axis).</caption>
</figure>
<unordered_list>
<list_item><location><page_10><loc_13><loc_18><loc_48><loc_19></location>[1] A. H. Guth, Phys. Rev. D 23 , 347 (1981).</list_item>
<list_item><location><page_10><loc_13><loc_15><loc_50><loc_16></location>[2] A. D. Linde, Phys. Lett. B 108 , 389 (1982).</list_item>
<list_item><location><page_10><loc_13><loc_12><loc_68><loc_14></location>[3] A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. 48 , 1220 (1982).</list_item>
<list_item><location><page_10><loc_13><loc_10><loc_53><loc_11></location>[4] A. A. Starobinsky, Phys. Lett. B 91 , 99 (1980).</list_item>
<list_item><location><page_10><loc_13><loc_7><loc_50><loc_8></location>[5] A. D. Linde, Phys. Lett. B 129 , 177 (1983).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_11><loc_13><loc_87><loc_88><loc_91></location>[6] S. Kachru, R. Kallosh, A. D. Linde, and S. P. Trivedi, Phys. Rev. D 68 , 046005 (2003), hep-th/0301240.</list_item>
<list_item><location><page_11><loc_13><loc_84><loc_84><loc_85></location>[7] R. Kallosh, A. Linde, E. McDonough, and M. Scalisi, JHEP 03 , 134 (2019), 1901.02022.</list_item>
<list_item><location><page_11><loc_13><loc_81><loc_71><loc_82></location>[8] H. Ooguri and C. Vafa, Nucl. Phys. B 766 , 21 (2007), hep-th/0605264.</list_item>
<list_item><location><page_11><loc_13><loc_78><loc_70><loc_80></location>[9] G. Obied, H. Ooguri, L. Spodyneiko, and C. Vafa (2018), 1806.08362.</list_item>
<list_item><location><page_11><loc_12><loc_76><loc_78><loc_77></location>[10] T. D. Brennan, F. Carta, and C. Vafa, PoS TASI2017 , 015 (2017), 1711.00864.</list_item>
<list_item><location><page_11><loc_12><loc_73><loc_48><loc_74></location>[11] E. Palti, Contemp. Phys. 62 , 165 (2022).</list_item>
<list_item><location><page_11><loc_12><loc_70><loc_76><loc_71></location>[12] Y. Chen, V. Gorbenko, and J. Maldacena, JHEP 02 , 009 (2021), 2007.16091.</list_item>
<list_item><location><page_11><loc_12><loc_67><loc_77><loc_69></location>[13] T. Hartman, Y. Jiang, and E. Shaghoulian, JHEP 11 , 111 (2020), 2008.01022.</list_item>
<list_item><location><page_11><loc_12><loc_65><loc_55><loc_66></location>[14] A. Levine and E. Shaghoulian (2022), 2204.08503.</list_item>
<list_item><location><page_11><loc_12><loc_59><loc_88><loc_63></location>[15] S. Cooper, M. Rozali, B. Swingle, M. Van Raamsdonk, C. Waddell, and D. Wakeham, JHEP 07 , 065 (2019), 1810.10601.</list_item>
<list_item><location><page_11><loc_12><loc_57><loc_80><loc_58></location>[16] S. Antonini, P. Simidzija, B. Swingle, and M. Van Raamsdonk (2022), 2203.11220.</list_item>
<list_item><location><page_11><loc_12><loc_54><loc_80><loc_55></location>[17] S. Antonini, P. Simidzija, B. Swingle, and M. Van Raamsdonk (2022), 2206.14821.</list_item>
<list_item><location><page_11><loc_12><loc_51><loc_69><loc_52></location>[18] G. Ye and Y.-S. Piao, Phys. Rev. D 101 , 083507 (2020), 2001.02451.</list_item>
<list_item><location><page_11><loc_12><loc_48><loc_69><loc_50></location>[19] G. Ye and Y.-S. Piao, Phys. Rev. D 102 , 083523 (2020), 2008.10832.</list_item>
<list_item><location><page_11><loc_12><loc_46><loc_73><loc_47></location>[20] J.-Q. Jiang and Y.-S. Piao, Phys. Rev. D 104 , 103524 (2021), 2107.07128.</list_item>
<list_item><location><page_11><loc_12><loc_43><loc_56><loc_44></location>[21] G. Ye, J. Zhang, and Y.-S. Piao (2021), 2107.13391.</list_item>
<list_item><location><page_11><loc_12><loc_40><loc_88><loc_41></location>[22] J. Feldbrugge, J.-L. Lehners, and N. Turok, Phys. Rev. Lett. 119 , 171301 (2017), 1705.00192.</list_item>
<list_item><location><page_11><loc_12><loc_37><loc_87><loc_39></location>[23] A. Borde, A. H. Guth, and A. Vilenkin, Phys. Rev. Lett. 90 , 151301 (2003), gr-qc/0110012.</list_item>
<list_item><location><page_11><loc_12><loc_35><loc_66><loc_36></location>[24] F. Pretorius, Phys. Rev. Lett. 95 , 121101 (2005), gr-qc/0507014.</list_item>
<list_item><location><page_11><loc_12><loc_29><loc_88><loc_33></location>[25] M. Campanelli, C. O. Lousto, P. Marronetti, and Y. Zlochower, Phys. Rev. Lett. 96 , 111101 (2006), gr-qc/0511048.</list_item>
<list_item><location><page_11><loc_12><loc_24><loc_88><loc_28></location>[26] J. G. Baker, J. Centrella, D.-I. Choi, M. Koppitz, and J. van Meter, Phys. Rev. Lett. 96 , 111102 (2006), gr-qc/0511103.</list_item>
<list_item><location><page_11><loc_12><loc_18><loc_88><loc_22></location>[27] J. T. Giblin, J. B. Mertens, and G. D. Starkman, Phys. Rev. Lett. 116 , 251301 (2016), 1511.01105.</list_item>
<list_item><location><page_11><loc_12><loc_16><loc_77><loc_17></location>[28] E. Bentivegna and M. Bruni, Phys. Rev. Lett. 116 , 251302 (2016), 1511.05124.</list_item>
<list_item><location><page_11><loc_12><loc_13><loc_87><loc_14></location>[29] N. Musoke, S. Hotchkiss, and R. Easther, Phys. Rev. Lett. 124 , 061301 (2020), 1909.11678.</list_item>
<list_item><location><page_11><loc_12><loc_10><loc_88><loc_11></location>[30] H. J. Macpherson, P. D. Lasky, and D. J. Price, Phys. Rev. D 95 , 064028 (2017), 1611.05447.</list_item>
<list_item><location><page_11><loc_12><loc_7><loc_88><loc_9></location>[31] H. J. Macpherson, D. J. Price, and P. D. Lasky, Phys. Rev. D 99 , 063522 (2019), 1807.01711.</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_12><loc_12><loc_89><loc_66><loc_91></location>[32] L. F. Abbott and S. R. Coleman, Nucl. Phys. B 259 , 170 (1985).</list_item>
<list_item><location><page_12><loc_12><loc_87><loc_78><loc_88></location>[33] P. Bizon and A. Rostworowski, Phys. Rev. Lett. 107 , 031102 (2011), 1104.3702.</list_item>
<list_item><location><page_12><loc_12><loc_84><loc_80><loc_85></location>[34] J. J. Blanco-Pillado, H. Deng, and A. Vilenkin, JCAP 05 , 014 (2020), 1909.00068.</list_item>
<list_item><location><page_12><loc_12><loc_81><loc_83><loc_82></location>[35] W. E. East, M. Kleban, A. Linde, and L. Senatore, JCAP 09 , 010 (2016), 1511.05143.</list_item>
<list_item><location><page_12><loc_12><loc_76><loc_88><loc_80></location>[36] K. Clough, E. A. Lim, B. S. DiNunno, W. Fischler, R. Flauger, and S. Paban, JCAP 09 , 025 (2017), 1608.04408.</list_item>
<list_item><location><page_12><loc_12><loc_73><loc_72><loc_74></location>[37] P.-X. Lin and Y.-S. Piao, Phys. Rev. D 105 , 063534 (2022), 2111.09174.</list_item>
<list_item><location><page_12><loc_12><loc_70><loc_82><loc_71></location>[38] T. W. Baumgarte and S. L. Shapiro, Phys. Rev. D 59 , 024007 (1998), gr-qc/9810065.</list_item>
<list_item><location><page_12><loc_12><loc_67><loc_63><loc_69></location>[39] M. Shibata and T. Nakamura, Phys. Rev. D 52 , 5428 (1995).</list_item>
<list_item><location><page_12><loc_12><loc_62><loc_88><loc_66></location>[40] K. Clough, P. Figueras, H. Finkel, M. Kunesch, E. A. Lim, and S. Tunyasuvunakool, Class. Quant. Grav. 32 , 245011 (2015), 1503.03436.</list_item>
<list_item><location><page_12><loc_12><loc_59><loc_65><loc_61></location>[41] A. Vilenkin, Phys. Rev. Lett. 72 , 3137 (1994), hep-th/9402085.</list_item>
<list_item><location><page_12><loc_12><loc_57><loc_65><loc_58></location>[42] A. D. Linde, Phys. Lett. B 327 , 208 (1994), astro-ph/9402031.</list_item>
</unordered_list>
</document>
[
{
"title": "On primordial universe in anti-de Sitter landscape",
"content": "Pu-Xin Lin 1 , 2 ∗ , Hai-Long Huang 1 , 3 † , Jun Zhang 4 ‡ , and Yun-Song Piao 1 , 3 , 4 , 5 § Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China P.O. Box 2735, Beijing 100190, China",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "How the spacetime evolved non-perturbatively in a landscape with multiple anti-de Sitter (AdS) vacua, which is theoretically well-motivated, has always been a matter of concern. As a step towards this issue, we perform (3+1)D numerical relativity simulations for the inhomogeneous universe in an AdS landscape, and find that large inhomogeneity of scalar field can develop into not only sphere-like bubbles, but also novel tube-like structures. It is observed that the bubble or tube wall (across the potential barrier) likely inflates as a quasi-dS space, while the different regions separated by the walls are in different AdS vacua and will collapse towards singularity. PACS numbers:",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "The success of inflation [1-5], consistent with current cosmological observations, suggests the existence of a de Sitter (dS) phase in the very early stage of our observable Universe. In string theory, although it is possible that dS vacua exist [6, 7], the construction of such vacua in string landscape is not straightforward. There are however valid concerns about the validity of such constructions in string theory, notably the swampland conjecture [8, 9], see also [10, 11] for recent reviews. In contrast, anti-de Sitter (AdS) vacua are easy to construct and are ubiquitous. Recently, it has been showed that AdS spacetime not only plays crucial roles as insights into our Universe, e.g.[12-17] but also have potential observable imprints [18-21]. The initial state of the universe might be highly inhomogeneous [22], i.e. the scalar field or spacetime metric can exhibit non-perturbative inhomogeneities before a region of space arrives at a certain vacuum, so that initially well-defined background in such highly inhomogeneous states does not exist. It is also often speculated that due to the past incompleteness of inflation [23], large fluctuations of spacetime will be inevitably excited near the initial cosmological singularity, in which case the initial conditions of scalar fields and spacetime are chaotic [5]. In such a non-perturbatively and highly inhomogeneous spacetime, perturbative calculation is not applicable any more. In the past decades, numerical relativity (NR) has developed significantly and has been applied to studies on the merging of black hole binaries [24-26] and non-perturbative cosmologies, e.g.[27-31]. As pointed out, the study of AdS landscape is theoretically wellmotivated. It is well-known that the region in a single AdS vacuum is unstable and will eventually collapse into a singularity [32], see also e.g.[33]. Consequently, the non-perturbative evolution of the entire spacetime in such landscape has continued to be a matter of concern, e.g.[34]. As a step towards this issue, we perform (3+1)D NR simulations of the 'universe' in a simplified AdS landscape. Despite the application of NR simulations in cosmologies [27, 28, 35-37], few focused on the AdS landscape. Specially, in Ref. [37], we investigate the evolution of scalar field in an initial inhomogeneous expanding Universe and how a patch of spacetime evolves into the dS vacua that should happen at the early epoch of the Universe. In this letter, we further investigate the spacetime evolves in a landscape with multiple AdS vacua, which are more generic in string theory. We explore potential new phenomenology and investigate whether a quasi-de Sitter space can emerge on the bubble wall separating different AdS regions. Our simulation results show novel phenomena not captured by conventional perturbative analyses, see Fig.1, which in certain sense highlights the important role of NR in comprehending the non-perturbative phenomena of spacetime.",
"pages": [
2,
3
]
},
{
"title": "II. METHOD",
"content": "Throughout this work, we consider a 1D potential as a phenomenological simplification of the actual complex AdS landscape, setting a fourth-order polynomial barrier between the two AdS minima that have quadratic forms ∼ ϕ 2 (see Fig.2), where V 1 , V 2 < 0. Here, we set m 2 1 = m 2 2 = 0 . 2, λ = 0 . 49 ( c = ℏ = 8 πG = 1). We thus have the AdS-like vacua located at ϕ 1 = 0 and ϕ 2 ≃ 2, where the potential is constructed to be differentiable, and the maximal value of potential barrier is at ϕ b ≃ 1. The large non-perturbative inhomogeneity of fields will be excited inevitably near the initial singularity or the Planck scale, giving rise to the possibility of initially configuration ∂ i ϕ∂ i ϕ ≲ 1, or ∆ ϕ ≲ 1. As example, we set the initial inhomogeneity of scalar field as where the amplitude of inhomogeneity is ∆ ϕ = O (0 . 1), the wavelength of inhomogeneity is taken to be equal to the length of the periodic cubic region (in our simulation L = 4). In our simulation, we set the periodic boundary condition 1 , and ∆ ϕ = 0 . 4 is the default value unless otherwise specified, with ϕ 0 varying from 0.8 to 0.98. We modify the BSSN based [38, 39] NR package GRChombo 2 [40] to perform the simulation. The NR simulation is usually performed in the 3+1 decomposition context for spacetime, with the metric as g 00 = -α 2 + β i β i , g 0 i = β i and g ij = γ ij , where α is the lapse parameter, β i the shift vector and γ ij the spatial metric, while for the numerical stability, the spatial metric γ ij must be further factorized into its determinant χ (conformal factor) and a comformal metric ˜ γ ij = χγ ij . The evolutions of ˜ γ ij , the connections ˜ Γ i = ˜ γ jk ˜ Γ i jk and the extrinsic curvature K ij = 1 3 Kδ ij + A ij comply with the BSSN equations [38, 39]. Initially, the values of the BSSN variables are set as ˜ γ ij = δ ij , ˜ A ij = χA ij = 0, and χ = 1. In GRChombo code [40], to satisfy the Hamiltonian constraint involving the scalar field, we enforce the constraint by relaxing the conformal factor in the light of ∂ t χ = H 3 , e.g. see recent Refs.[36, 37]. In addition, we set the initial scalar field at rest ˙ ϕ = 0, satisfying the momentum constraint and the initial expansion rate to be uniform K init = -√ 3 ⟨ ρ ⟩ ≈ -1 . 08, where ⟨ ρ ⟩ indicates the average initial erergy density. In the special case of an isoptopic and homogeneous Universe, extrinsic curvature K is related to the Hubble constant as K = -3 H with geodesic observers.",
"pages": [
3,
4,
5
]
},
{
"title": "III. RESULTS",
"content": "The simulation results are presented in Fig.1 and Fig.3. The large inhomogeneity will spawn AdS bubbles in a background of a different AdS vacuum. As expected, if a bubble has subcritical initial radius, it will rapidly shrink and collapse into a black hole 4 , and eventually the whole space will evolve into a single AdS state and collapse integrally, see Fig.3-A. However, if the initial radii of the bubbles are supercritical, the regions with different AdS vacua will coexist, separated by not only near-spherical bubble walls (Fig.3-B) but also, interestingly, infinitely extending tube-like walls (Fig.3-C). Here, the critical radius of bubble refers to the minimal radius of bubble when the bubble phase can exist. These tube-like walls can be understood more clearly in Fig.1. In our simulation, we replace an infinite space with the periodic boundary condition. This choice of boundary condition is motivated as follows: inhomogeneities occur at various scales, when considering evolution of those at a specific length scale, it is natural to expect the extent of the modes goes beyond just a single period. In light of this argument, the 'bubble' in the lower panel of Fig.1 actually correspond to the tube that extend along -→ x (= x, y, z ) directions through the region where the periodicity of the initial perturbation modes persists. Such tube-like structure looks 'as if' the adjacent spherical cells merged partly. It is intriguing to consider the more generic setup with perturbations at numerous wavelengths, a task beyond the scope of this letter, and to be pursed in future work. It is significant to check the effect of the wavelength L of inhomogeneity (equivalently L/H -1 init ). A phase diagram is presented in Fig.4 5 . According to Fig.4, for ϕ 0 /ϕ b ≲ 0 . 9, if L/H -1 init ≲ 1 (i.e. the wavelength L is 'subhorizonal'), eventually the whole space will collapse integrally, while the bubble or tube phase comes into being only when the initial inhomogeneity is 'superhorizonal' (the radius of the bubble is supercritical and superhorizonal). It is also worth noting that if ϕ 0 /ϕ b ≳ 0 . 9, i.e. ϕ 0 is close to the maximal value of potential barrier, we will see the bubble or tube phase for L/H -1 init ≲ 1 (the radius of the bubble is supercritical but subhorizonal). Physically, our results can be qualitatively understood as follows. Here, if a bubble has subcritical initial radius, it will rapidly shrink and disappear, and eventually the whole space will evolve into a single AdS state and collapse integrally, while if a bubble has supercritical and superhorizonal initial radius, the collapse of bubble wall will be prevented by the expansion of wall spacetime, as expected theoretically in Ref.[34], so a bubble phase will come into being. As the initial radii of bubbles gets bigger (than a certain critical value), the bubbles will get closer so that they can partly join together through the tube naturally. Here, the initial 'universe' is set to be expanding homogeneously (all H local = const. > 0), but eventually different regions will tend to evolve into different AdS vacua and be separated by bubble or tube walls. According to Fig.5, we see that though the AdS regions at both sides of the wall are collapsing, the existing walls will still expand and eventually arrive at a quasi-dS static profile, implying that the inflationary stage responsible for our observable Universe might happen at such walls, e.g.[41, 42]. It can be confirmed that in such AdS landscapes, inflation might occur. According to Refs.[41, 42], the width of the wall is δ 0 ∼ ( ϕ 2 -ϕ 1 ) V -1 / 2 b , and must be satisfied for the inflation occurring on the wall. Here, considering the longitudinal Hubble parameter H L = -u i u j K ij ( u i is the normal vector at the bubble wall), and ds 2 = γ ij e i e j dr 2 = γ 2 dr 2 ( r is the coordinate crossing the bubble wall and e i is the direction vector at the wall), we have see Fig.6, which confirms the condition (4).",
"pages": [
5,
6,
7
]
},
{
"title": "IV. CONCLUSIONS",
"content": "We performed full relativistic simulations for non-perturbative evolution of spacetime in a simple AdS landscape, and found that large inhomogeneity of a scalar field inclines to develope into sphere-like bubbles, and more unexpectedly, into tube-like structures. It is observed that the spacetime evolution is completely different in the AdS landscape, the bubble or tube walls might inflate forever as a quasi-dS space, while the different regions separated by the walls are in different AdS vacua and will collapse towards singularity. Though we take a simple potential with two AdS minima as example, actually for a spacetime in a landscape with multiple AdS vacua, its local region always can be described as that with only two neighbouring AdS minima. Thus our results might have effectively captured some of the essentials of non-perturbative phenomenology of spacetime in AdS landscape. In this sense, further research on the NR simulations in a more realistic landscape for the early universe is interesting and required, which will help to deepen insight into the origin of our observable Universe. Acknowledgment This work is supported by the NSFC No.12075246 and by the Fundamental Research Funds for the Central Universities. We acknowledge the use of GRChombo code for our simulation and the Tianhe-2 supercomputer for providing computing resources. We would like to thank Tiago Fran¸ca, Hao-Hao Li and Hao-Yang Liu for useful discussions.",
"pages": [
7,
8
]
}
]
2024PhLB..85738993P
https://arxiv.org/pdf/2303.17621.pdf
<document>
<section_header_level_1><location><page_1><loc_18><loc_92><loc_82><loc_93></location>Diverging black hole entropy from quantum infrared non-localities</section_header_level_1>
<text><location><page_1><loc_31><loc_89><loc_70><loc_90></location>Alessia Platania 1, 2, ∗ and Jaime Redondo-Yuste 1, 3, †</text>
<text><location><page_1><loc_16><loc_86><loc_85><loc_88></location>1 Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON N2L 2Y5, Canada 2 Nordita, KTH Royal Institute of Technology and Stockholm University,</text>
<text><location><page_1><loc_32><loc_85><loc_69><loc_86></location>Hannes Alfvéns väg 12, SE-106 91 Stockholm, Sweden</text>
<text><location><page_1><loc_16><loc_83><loc_85><loc_84></location>3 Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark</text>
<text><location><page_1><loc_18><loc_74><loc_83><loc_82></location>Local higher-derivative corrections to the Einstein-Hilbert action yield sub-leading corrections to the Bekenstein-Hawking area law. Here we show that if the quantum effective action comprises a certain class of infrared non-localities, the entropy of large black holes generally diverges to either positive or negative infinity. In such theories, large spherically symmetric black holes would be either highly chaotic or thermodynamically impossible, respectively. In turn, this puts strong constraints on the Laurent expansion of the form factors in the effective action.</text>
<text><location><page_1><loc_9><loc_60><loc_49><loc_72></location>The theoretical prediction that black holes emit thermal radiation [1, 2] is considered one of Stephen Hawking's most important contributions to our understanding of black holes. Hawking radiation has two crucial consequences: (i) it entails that black holes evaporate, as the emission causes a reduction in their mass and rotational energy, and (ii) the entropy of a large black hole is, to leading order, proportional to its area [2, 3]</text>
<formula><location><page_1><loc_24><loc_56><loc_49><loc_59></location>S BH = A 4 l 2 P , (1)</formula>
<text><location><page_1><loc_9><loc_51><loc_49><loc_55></location>with l P being the Planck length. Corrections to this classical formula parametrize deviations from General Relativity (GR).</text>
<text><location><page_1><loc_9><loc_34><loc_49><loc_51></location>The first law of black hole mechanics connects the perturbations of the black hole mass, its angular momentum, and a combination of the surface gravity and a geometrical quantity that is later given the meaning of an entropy. The original derivation [4] used very explicit properties of the Einstein field equations, so a natural question was whether such a statement could be extended to any theory of gravity. Wald's seminal work [5] demonstrated that the first law holds for generic gravitational actions S , provided that some minimal assumptions on the (classical or effective) Lagrangian, such as diffeomorphism invariance, are fulfilled.</text>
<text><location><page_1><loc_9><loc_15><loc_49><loc_33></location>On the theoretical side, the form of the effective action depends on quantum gravitational effects, and potentially on the specific ultraviolet completion of gravity. In particular, owed to the quantum nature of the effective action, a common feature is the presence of nonlocalities. The latter could be exponential functions of the d'Alembert operator, as in non-local gravity [6], hyperbolic tangents, as in some asymptotically safe models [7], operators of the Polyakov type, as in string theory [8, 9] and in the high-energy limit of loop quantum gravity [10], or even higher inverse powers of the d'Alembert operator, as it could happen in causal dynamical triangulations [11]. Such non-localities also play</text>
<text><location><page_1><loc_52><loc_63><loc_92><loc_72></location>a role on the phenomenological side. Specifically, inverse powers of the d'Alembert operator seem to be important in cosmological settings [12-16], as an alternative explanation to dark energy, as well as in black hole physics [17], to accommodate for regular black holes stemming from a principle of least action in quantum gravity.</text>
<text><location><page_1><loc_52><loc_50><loc_92><loc_63></location>The entropy of black holes in the presence of local higher-derivative terms has been widely studied [1822]. As expected, adding such terms to the EinsteinHilbert action leads to sub-leading corrections to the area law (1). On dimensional grounds, one might expect that similar corrections stemming from inverse powers of the d'Alembert operator would dominate over the Bekenstein-Hawking scaling. The main motivation of our Letter is to test this expectation.</text>
<text><location><page_1><loc_52><loc_33><loc_92><loc_50></location>We provide analytical and numerical evidence that if the effective action contains infrared non-localities, not only the resulting corrections are dominant, but also divergent. The entropy of spherically symmetric configurations is thus generically divergent, barring unlikely cancellations of the infinities associated with each non-local term in the effective action. Depending on the sign of the divergence, black holes in such theories would be characterized either by infinitely many microstates or by none. This paradoxical behavior points to novel constraints on the Laurent expansion of the form factors in the effective action.</text>
<text><location><page_1><loc_52><loc_27><loc_92><loc_32></location>Setup - We shall focus on asymptotically flat spacetimes. The gravitational effective action of a generic diffeomorphism-invariant theory involving only the metric is</text>
<formula><location><page_1><loc_59><loc_23><loc_92><loc_26></location>Γ eff = ∫ d 4 x √ -g ( R 16 πG + L HD ) , (2)</formula>
<text><location><page_1><loc_52><loc_19><loc_92><loc_21></location>where the Einstein-Hilbert (EH) part is complemented by the higher-derivative (HD) Lagrangian</text>
<formula><location><page_1><loc_57><loc_13><loc_92><loc_17></location>L HD = 1 16 πG ( R F 1 ( glyph[square] ) R + R µν F 2 ( glyph[square] ) R µν (3) + R µνρσ F 3 ( glyph[square] ) R µνρσ + O ( R 3 ) ) .</formula>
<text><location><page_1><loc_52><loc_8><loc_92><loc_11></location>Above, G is the Newton constant and glyph[square] = -g µν D µ D ν is the d'Alembert operator constructed from the Levi-</text>
<text><location><page_2><loc_9><loc_83><loc_49><loc_93></location>Civita connection of the metric g . The form factors F i 1 result from integrating out quantum gravitational fluctuations in the functional integral, are typically non-local, and at one loop they ought to match the logarithmic behavior encountered in [22]. Their form and properties are strictly tied to the specific ultraviolet completion of gravity [24].</text>
<text><location><page_2><loc_9><loc_77><loc_49><loc_83></location>Computing the exact entropy corrections generated by these non-local form factors is generally involved, but within the annulus of convergence, where the form factors are holomorphic, one can exploit their Laurent expansion</text>
<formula><location><page_2><loc_21><loc_72><loc_49><loc_76></location>F i ( glyph[square] ) = ∞ ∑ n = -∞ c n glyph[square] n . (4)</formula>
<text><location><page_2><loc_9><loc_54><loc_49><loc_71></location>In practical applications only a few of the negative-degree terms play a role in quantum gravity and in cosmological models [8, 10-17]. In particular, such low-order terms appear to be compatible with cosmological observations [15, 16]. Entropy corrections coming from the first positive-degree terms ( n ≥ 0 ) in this Laurent series have been computed in [18-22]. Given the relevance of some of the negative-degree terms in (4)-the 'principal part' of the Laurent expansion-in quantum gravity [8, 10, 11], black hole physics [17], and cosmology [1216], investigating their impact on black hole thermodynamics is of great importance.</text>
<text><location><page_2><loc_9><loc_37><loc_49><loc_54></location>In order to compute the entropy corrections stemming from this type of infrared non-localities, while avoiding to compute the corresponding dressed field equations and solutions, we need to make some assumptions. First, we assume that the length scale at which quantum gravity effects become important is of the order of the Planck length, l QG glyph[similarequal] l P , such that no strong non-perturbative effects happen at the horizon scale for large black holes. Second, we limit ourselves to astrophysical black holes, for which M glyph[greatermuch] M P . This ensures that the event horizon is located within a few Planck lengths from the Schwarzschild radius, i.e., that r H glyph[similarequal] r s , with r s ≡ 2 MG .</text>
<text><location><page_2><loc_9><loc_32><loc_49><loc_36></location>The solutions to the quantum theory associated with the effective action (2) ought to be found by solving the quantum field equations</text>
<formula><location><page_2><loc_25><loc_28><loc_49><loc_31></location>δ Γ eff δg µν = 0 . (5)</formula>
<text><location><page_2><loc_9><loc_23><loc_49><loc_27></location>These solutions are generally different than those obtained from the vacuum Einstein equations (Ricci-flat metrics) 2 and are typically difficult to derive.</text>
<text><location><page_2><loc_52><loc_89><loc_92><loc_93></location>Even if the field equations (5) are different than in GR, they are expected to admit spherically symmetric solutions of the type 3</text>
<formula><location><page_2><loc_60><loc_85><loc_92><loc_88></location>ds 2 = -f ( r ) dt 2 + dr 2 g ( r ) + r 2 d Ω 2 , (6)</formula>
<text><location><page_2><loc_52><loc_79><loc_92><loc_83></location>where d Ω 2 is the line element on the 2 -sphere. Further, for large black holes the metric coefficients f ( r ) and g ( r ) can be written as</text>
<formula><location><page_2><loc_56><loc_75><loc_92><loc_78></location>f ( r ) glyph[similarequal] 1 -r s r + A r α , g ( r ) glyph[similarequal] 1 -r s r + B r β , (7)</formula>
<text><location><page_2><loc_52><loc_70><loc_92><loc_74></location>with α, β ≥ 2 , i.e., the metric only differs from the Schwarzschild one by sub-leading terms. In particular, g glyph[similarequal] f for large or massive black holes.</text>
<text><location><page_2><loc_52><loc_63><loc_92><loc_70></location>This geometry has a Killing horizon H , with bifurcation surface Σ , where the future and past horizons intersect. This is characterized by an antisymmetric bifurcation tensor glyph[epsilon1] µν , normalized such that glyph[epsilon1] µν glyph[epsilon1] µν = -2 , and given by</text>
<formula><location><page_2><loc_62><loc_58><loc_92><loc_61></location>glyph[epsilon1] µν = √ f ( r ) g ( r ) ( δ t µ δ r ν -δ r µ δ t ν ) . (8)</formula>
<text><location><page_2><loc_52><loc_48><loc_92><loc_56></location>A spacelike surface spanning from Σ to spatial infinity i 0 can be used to define a gravitational phase space consistent with general covariance 4 . Wald [5] showed that the first law of black hole thermodynamics follows, and the entropy acquires the interpretation of a Noether charge. For a static 5 , spherically-symmetric black hole it reads</text>
<formula><location><page_2><loc_59><loc_43><loc_92><loc_46></location>S W = -2 π ∫ Σ ( δ L δR µνρσ ) glyph[epsilon1] µν glyph[epsilon1] ρσ dV 2 2 , (9)</formula>
<text><location><page_2><loc_52><loc_30><loc_92><loc_42></location>where L is the (classical or effective) Lagrangian, dV 2 2 = r 2 sin θdθdφ is the line element on the bifurcation surface Σ , and the functional derivative is performed at a fixed metric. We shall employ this formula to compute the corrections to the entropy of the spherically symmetric spacetime (7) stemming from the quadratic higherderivative corrections (3) with form factors structurally given by the principle part of the expansion (4).</text>
<text><location><page_3><loc_9><loc_77><loc_49><loc_93></location>Localizing the effective action - In the derivation of the Wald entropy formula, there are several assumptions that deserve to be highlighted: (i) The theory is defined by a locally constructed [28], diffeomorphism invariant Lagrangian L built from dynamical fields and their derivatives on a Lorentzian manifold. (ii) There is some notion of asymptotic flatness, so that the black hole horizon can be defined as the (inner) boundary of the past region of the asymptotic region. (iii) The horizon is a Killing horizon, i.e., a null surface to which a Killing field is normal.</text>
<text><location><page_3><loc_9><loc_55><loc_49><loc_77></location>Out of these assumptions, the last one is always satisfied for static black holes, and in particular for all spherically symmetric solutions. The property of asymptotic flatness is necessary to precisely define the notion of an event horizon, but given that such a construction is possible, it does not enter later in the derivation 6 . The most critical assumption for our purposes is the first one, since the action used is not directly 'locally constructed'. This is a problem because the covariant phase space formalism relies on the properties of jet bundles on field space [28]. There is however a way to circumvent this issue: it consists in finding a 'localized' version of the action that on-shell yields the same configurations. Such a localized effective action is derived by translating the non-locality into a set of constrained auxiliary fields [30-33].</text>
<text><location><page_3><loc_9><loc_50><loc_49><loc_54></location>In order to proceed further, we now focus on the principal part of the expansion (4), and in particular on the form factors F i ( glyph[square] ) given by the following finite sums</text>
<formula><location><page_3><loc_15><loc_44><loc_49><loc_48></location>F i ( glyph[square] ) = N ∑ n =1 F i,n , F i,n ≡ c i,n glyph[square] -n , (10)</formula>
<text><location><page_3><loc_9><loc_31><loc_49><loc_42></location>where c i,n are real coefficients and the truncation order N can be systematically increased. Following standard procedures [30-33], it is straightforward to compute a localized version of the effective action, Γ local eff = Γ EH +Γ local eff,HD , made up of the Einstein-Hilbert part Γ EH and the localized higher-derivative action (see supplemental material). We will use such an action to compute the entropy corrections stemming from the form factors (10).</text>
<text><location><page_3><loc_9><loc_21><loc_49><loc_30></location>Entropy formula - The entropy of large black holes in a theory with infrared non-localities (10) is to be computed as a Noether charge, Eq. (9), with the action S ≡ Γ local eff = Γ EH + Γ local eff,HD being the localized version of the effective action. After some algebra (see supplemental material for details), the resulting dimensionless</text>
<text><location><page_3><loc_52><loc_92><loc_81><loc_93></location>entropy ˜ S W = (4 G/ A ) S W takes the form</text>
<formula><location><page_3><loc_53><loc_82><loc_92><loc_91></location>˜ S W = lim r → r H ( 1 + 2 F 1 ( glyph[square] ) R 1 + F 2 ( glyph[square] ) R 2 -4 g f F 3 ( glyph[square] ) R 3 -N ∑ n =1 ( ξ n, (1) R 1 -ξ n, (2) R 2 -4 g f ξ n, (3) R 3 )) . (11)</formula>
<text><location><page_3><loc_52><loc_80><loc_81><loc_82></location>where ξ n, ( i ) are Lagrange multipliers, and</text>
<formula><location><page_3><loc_53><loc_72><loc_91><loc_79></location>R 1 = gf ' 2 2 f 2 -2 -1 + g + rg ' r 2 -rf ' g ' +2 g (2 f ' + rf '' ) 2 rf , R 2 = rg ( f ' ) 2 -2 f 2 g ' -f [ rf ' g ' +2 g ( f ' + rf '' )] 2 rf 2 , ' ' '</formula>
<formula><location><page_3><loc_53><loc_68><loc_92><loc_72></location>R 3 = 1 4 ( -f 2 f + f g g +2 f '' ) . (12)</formula>
<text><location><page_3><loc_52><loc_66><loc_92><loc_68></location>are functionals of the metric components f ( r ) and g ( r ) .</text>
<text><location><page_3><loc_52><loc_53><loc_92><loc_66></location>Diverging black hole entropy - Exploiting the generalized formula (11), we now compute the corrections to the Bekenstein-Hawking area law stemming from the form factors (10). As the entropy is additive, corrections stemming from the individual glyph[square] -n -operators in the form factors (10) can be determined individually. We checked that the corrections in the second line of Eq. (11) are finite and unimportant to our conclusions. In the following we will thus focus on the dimensionless contributions</text>
<formula><location><page_3><loc_52><loc_48><loc_93><loc_53></location>˜ S ( n ) W ≡ lim r → r H ( 2 c 1 ,n 1 glyph[square] n R 1 + c 2 ,n 1 glyph[square] n R 2 -4 c 3 ,n g f 1 glyph[square] n R 3 ) . (13)</formula>
<text><location><page_3><loc_52><loc_42><loc_92><loc_48></location>To this end, we first need to determine how the glyph[square] -n terms act on the radial functions R i ( r ) . This requires solving glyph[square] n ψ ( r ) = R i ( r ) with respect to the function ψ . For a generic spherically symmetric metric (6) [17]</text>
<formula><location><page_3><loc_62><loc_40><loc_92><loc_41></location>glyph[square] -1 φ ( r ) = Φ[ φ, r, R x , R y ] , (14)</formula>
<text><location><page_3><loc_52><loc_36><loc_92><loc_39></location>where φ is a generic function of the radial coordinate and we have defined</text>
<formula><location><page_3><loc_53><loc_30><loc_92><loc_35></location>Φ[ φ, r, R x , R y ] ≡ ∫ R x r ∫ R y x dxdy -y 2 φ ( y ) x 2 √ g ( x ) f ( x ) √ f ( y ) g ( y ) . (15)</formula>
<text><location><page_3><loc_52><loc_17><loc_92><loc_30></location>This is the general solution to Eq. (14), and the 'cutoffs' R x and R y are related to its initial conditions. As in the case n = 1 we have φ ( r ) = R i ( r ) , the regularity properties of R i ( r ) and the absence of zero modes allow selecting special initial conditions, such that R x , R y → ∞ [17]. We will nevertheless show that our results are independent of the choice of initial conditions. Eq. (15) can be generalized to a recursion formula where φ n +1 ≡ glyph[square] -n -1 φ ( r ) is written in terms of φ n ≡ glyph[square] -n φ ( r ) ,</text>
<formula><location><page_3><loc_61><loc_14><loc_92><loc_16></location>φ n +1 ( r ) = Φ[ φ n , r, R ( n ) x , R ( n ) y ] , (16)</formula>
<text><location><page_3><loc_52><loc_9><loc_92><loc_13></location>where φ n ≡ glyph[square] -1 φ n -1 = glyph[square] -n φ ( r ) with n ≥ 1 , and { R ( n ) x , R ( n ) y } are initial conditions. With this, the starting point to compute the corrections to the area law is</text>
<text><location><page_4><loc_9><loc_85><loc_49><loc_93></location>to consider the lowest-order operators, F i, 1 = c i, 1 / glyph[square] , and then compute the effect of the others-the F i,n = c i,n glyph[square] -n in Eq. (10)-recursively. We anticipate that the divergence of the lowest-order operator implies, owed to the recursion formula (16), the divergence of all the higher-order terms.</text>
<text><location><page_4><loc_9><loc_82><loc_49><loc_84></location>The first three contributions to the entropy formula that we have to compute are thus</text>
<formula><location><page_4><loc_10><loc_78><loc_49><loc_81></location>A i, 1 ≡ F i, 1 R i = Φ[ c i, 1 R i , r H , R x , R y ] , i = 1 , 2 , 3 , (17)</formula>
<text><location><page_4><loc_9><loc_74><loc_49><loc_78></location>where we have considered the same ( R x , R y ) ∀ i = 1 , 2 , 3 for simplicity. This is justified since, as we shall see, our conclusions are independent of the initial conditions.</text>
<text><location><page_4><loc_9><loc_66><loc_49><loc_73></location>Based on our assumptions, the solution to the nonlocal effective field equation (5) is of the form (7). Since g glyph[similarequal] f for large and massive black holes, in what follows we shall set A = B and α = β . We checked that this simplification does not affect our conclusions.</text>
<text><location><page_4><loc_9><loc_56><loc_49><loc_66></location>It is useful to start from a simple example, where exact analytical formulas can be derived: the Schwarzschild black hole, corresponding to the case A = 0 , is an exact solution for F 3 = 0 . In this ideal case A 3 , 1 = 0 by construction, while the other two contributions in Eq. (17) can be computed with the aid of the formula (15). In particular, A 1 , 1 = 0 and</text>
<formula><location><page_4><loc_10><loc_46><loc_49><loc_55></location>Φ[ R 2 , r, R x , R y ] = 1 2 log ( R y R x ) log ( R y ( r s -R x ) 2 R x r 2 s ) -Li 2 ( R x r s ) -1 2 log ( R y r ) log ( R y ( r s -r ) 2 rr 2 s ) +Li 2 ( r r s ) , (18)</formula>
<text><location><page_4><loc_9><loc_36><loc_49><loc_45></location>where Li 2 is a polylogarithm of order two, which is real for r → r H ≤ r s . Analogous analytical formulas can also be derived in the case α = 2 (see supplemental material). Taking the limit r → r H = r s , one finds that A 2 , 1 diverges, with the sign of the divergence depending on the coefficient c 2 , 1 . In particular, divergences arise both at the horizon and at infinity (as R x , R y →∞ ).</text>
<text><location><page_4><loc_29><loc_31><loc_29><loc_32></location>glyph[negationslash]</text>
<text><location><page_4><loc_45><loc_31><loc_45><loc_32></location>glyph[negationslash]</text>
<text><location><page_4><loc_9><loc_28><loc_49><loc_35></location>Similar findings also hold when accounting for the asymptotic corrections to the Schwarzschild metric coefficients, in which case A,B = 0 in Eq. (7) and r H = r s . In these cases one has to account for all corrections (17). The event horizon, when it exists, is located at</text>
<formula><location><page_4><loc_18><loc_24><loc_49><loc_27></location>r H glyph[similarequal] ( 1 + a ( a -1) α -1 ) r s , (19)</formula>
<text><location><page_4><loc_9><loc_9><loc_49><loc_23></location>where a is the dimensionless constant a ≡ ( α +1) r -α s A , and a glyph[lessmuch] 1 for large black holes. Independent of the exponent α and of the value of a in the allowed range, whenever a horizon exists, the contributions (17) to the entropy ˜ S (1) W diverge as r → r H . This is shown in Fig. 1, where we account for the position of the horizon for different values of a . The presence of a horizon for large black holes requires a glyph[lessmuch] 1 . Indeed, when a is big enough, no horizon forms, and the integrals (17) are regular everywhere. Vice versa, when a horizon exists (dashed lines in</text>
<figure>
<location><page_4><loc_54><loc_75><loc_90><loc_93></location>
<caption>Figure 1. Dimensionless entropy correction ˜ S (1) W as a function of r/r s , for R x = R y = 10 3 , c 1 , 1 = c 2 , 1 = c 3 , 1 = 1 , α = 6 and different values of a . For each a , the location of the event horizon r H (when it exists) is displayed as a vertical dashed line. If an event horizon is present, the entropy diverges as r → r H .</caption>
</figure>
<figure>
<location><page_4><loc_54><loc_47><loc_90><loc_65></location>
<caption>Figure 2. Dimensionless entropy correction ˜ S (1) W , cf. Eq. (13), for a = 0 . 2 , α = 6 , c 1 , 1 = c 2 , 1 = c 3 , 1 = 1 and different values of R = R x = R y . The vertical dashed line denotes the position of the event horizon. The divergence of the entropy is independent of the initial conditions R .</caption>
</figure>
<text><location><page_4><loc_52><loc_26><loc_92><loc_35></location>Fig. 1), the entropy correction diverges (see supplemental material for further analytical and numerical evidence). In particular, as shown in Fig. 2, such a divergence is independent of the initial conditions ( R x , R y ) . Finally, higher-order terms in Eq. (10) lead to similar divergent contributions ˜ S ( n ) W owed to the recursion formula (16).</text>
<text><location><page_4><loc_52><loc_18><loc_92><loc_26></location>Whether the divergence is positive or negative infinity depends on the (relative) signs of the coefficients c i,n , with i = 1 , 2 , 3 and n = 1 , . . . , N . As discussed in the supplemental material, even in the simple case N = n = 1 and α = 2 , the condition on the coefficients c i, 1 to eliminate the divergence is non-trivial.</text>
<text><location><page_4><loc_52><loc_11><loc_92><loc_17></location>Plugging all corrections ˜ S ( n ) W into Eq. (11), and considering that the terms in the second line of Eq. (11) yield finite contributions, we finally obtain the surprising result that the total black hole entropy S W diverges,</text>
<formula><location><page_4><loc_68><loc_9><loc_92><loc_10></location>S W →±∞ , (20)</formula>
<text><location><page_5><loc_9><loc_86><loc_49><loc_93></location>unless the set of coefficients c i,n are fine-tuned such that all divergences generated by all glyph[square] -n operators in the form factors (10) cancel out. Such a special combination of coefficients is unlikely realized by the quantum gravitational dynamics.</text>
<text><location><page_5><loc_9><loc_83><loc_49><loc_86></location>Identifying the Wald entropy with the thermodynamical entropy given by the Boltzmann formula 7</text>
<formula><location><page_5><loc_21><loc_80><loc_49><loc_81></location>S W ≡ S B = k B log W, (21)</formula>
<text><location><page_5><loc_9><loc_66><loc_49><loc_78></location>our result entails that spherically symmetric configurations stemming from effective actions that display infrared non-localities are either highly chaotic or thermodinamically impossible. In turn, our result places strong constraints on the Laurent expansion of the gravitational effective action: finiteness of the black hole entropy requires the absence of quantum-induced infrared non-localities, or a very special set of coefficients resulting from quantum gravitational dynamics.</text>
<text><location><page_5><loc_9><loc_38><loc_49><loc_65></location>Conclusions - In our Letter, we tackled the problem of determining corrections to the black hole area law stemming from infrared non-localities in the gravitational effective action. Higher-derivative corrections to the effective action are generally expected from quantum gravity and are particularly important in the physics of large black holes, as a description in terms of an effective action has to be recovered irrespective of the specific ultraviolet completion of gravity. Local higher-derivative terms have been considered in the seminal work [18], and yield sub-leading contributions to the black hole area law. Our Letter complements the results in [18] by accounting for non-local higher derivative terms, the latter being of relevance in several quantum gravity models [8, 10, 11], black hole physics [17], and cosmology [12-16]. Despite some of these infrared non-localities seem compatible with observations [14-16], they appear to yield inconsistencies: we find that not only infrared non-localities lead to corrections that are dominant over the Bekenstein-Hawking</text>
<unordered_list>
<list_item><location><page_5><loc_10><loc_31><loc_38><loc_32></location>[1] S. W. Hawking, Nature 248 , 30 (1974).</list_item>
<list_item><location><page_5><loc_10><loc_29><loc_49><loc_31></location>[2] S. W. Hawking, Commun. Math. Phys. 43 , 199 (1975), [Erratum: Commun.Math.Phys. 46, 206 (1976)].</list_item>
<list_item><location><page_5><loc_10><loc_27><loc_45><loc_28></location>[3] J. D. Bekenstein, Lett. Nuovo Cim. 4 , 737 (1972).</list_item>
<list_item><location><page_5><loc_10><loc_25><loc_49><loc_27></location>[4] J. M. Bardeen, B. Carter, and S. W. Hawking, Commun. Math. Phys. 31 , 161 (1973).</list_item>
<list_item><location><page_5><loc_10><loc_22><loc_49><loc_24></location>[5] R. M. Wald, Phys. Rev. D 48 , R3427 (1993), arXiv:grqc/9307038.</list_item>
<list_item><location><page_5><loc_10><loc_20><loc_49><loc_22></location>[6] L. Modesto and L. Rachwał, Int. J. Mod. Phys. D 26 , 1730020 (2017).</list_item>
</unordered_list>
<text><location><page_5><loc_52><loc_79><loc_92><loc_93></location>term, as we expected, but they also yield diverging contributions. Exploiting the Boltzmann formula, one would then conclude that the corresponding black holes are made of either zero or infinitely many microstates, depending on the sign of the divergence. This paradoxical behavior points to novel constraints on the Laurent expansion of the effective action stemming from quantum gravity theories: its principle part has to vanish or be fine-tuned in order for the black hole entropy to remain finite.</text>
<text><location><page_5><loc_52><loc_69><loc_92><loc_79></location>It is still an open question whether similar cubic and higher-order terms in the curvature expansion (2) of the effective action could dramatically change this conclusion, and whether resumming all infinitely many corrections could provide a finite result. Yet, this is expected to come at the expense of fine-tuning infinitely many coefficients in the effective action.</text>
<section_header_level_1><location><page_5><loc_62><loc_65><loc_82><loc_66></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_5><loc_52><loc_38><loc_92><loc_62></location>The authors would like to thank I. Basile, B. Knorr, D. Pereñiguez, and A. Riello for interesting discussions, and I. Basile, B. Knorr, and V. Cardoso for feedback on our manuscript. The authors acknowledge support by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Colleges and Universities. JRY acknowledges support from the Villum Investigator program supported by the VILLUM Foundation (grant no. VIL37766) and the DNRF Chair program (grant no. DNRF162) by the Danish National Research Foundation. AP also acknowledges Nordita for support within the 'Nordita Distinguished Visitors' program and for hospitality during the last stages of development of this work. Nordita is supported in part by NordForsk.</text>
<unordered_list>
<list_item><location><page_5><loc_53><loc_30><loc_92><loc_32></location>[7] T. Draper, B. Knorr, C. Ripken, and F. Saueressig, Phys. Rev. Lett. 125 , 181301 (2020), arXiv:2007.00733.</list_item>
<list_item><location><page_5><loc_53><loc_29><loc_89><loc_30></location>[8] A. M. Polyakov, Mod. Phys. Lett. A 2 , 893 (1987).</list_item>
<list_item><location><page_5><loc_53><loc_26><loc_92><loc_28></location>[9] O. Aharony and M. Dodelson, JHEP 02 , 008, arXiv:1111.5758 [hep-th].</list_item>
<list_item><location><page_5><loc_52><loc_25><loc_92><loc_26></location>[10] J. Borissova and B. Dittrich, (2022), arXiv:2207.03307.</list_item>
<list_item><location><page_5><loc_52><loc_22><loc_92><loc_24></location>[11] B. Knorr and F. Saueressig, Phys. Rev. Lett. 121 , 161304 (2018), arXiv:1804.03846 [hep-th].</list_item>
<list_item><location><page_5><loc_52><loc_20><loc_92><loc_22></location>[12] C. Wetterich, Gen. Rel. Grav. 30 , 159 (1998), arXiv:grqc/9704052.</list_item>
<list_item><location><page_5><loc_52><loc_17><loc_92><loc_19></location>[13] S. Deser and R. P. Woodard, Phys. Rev. Lett. 99 , 111301 (2007), arXiv:0706.2151.</list_item>
<list_item><location><page_5><loc_52><loc_13><loc_92><loc_17></location>[14] H. Nersisyan, Y. Akrami, L. Amendola, T. S. Koivisto, J. Rubio, and A. R. Solomon, Phys. Rev. D 95 , 043539 (2017), arXiv:1610.01799 [gr-qc].</list_item>
<list_item><location><page_5><loc_52><loc_10><loc_92><loc_13></location>[15] V. Vardanyan, Y. Akrami, L. Amendola, and A. Silvestri, JCAP 03 , 048, arXiv:1702.08908 [gr-qc].</list_item>
<list_item><location><page_5><loc_52><loc_9><loc_92><loc_10></location>[16] E. Belgacem, Y. Dirian, A. Finke, S. Foffa, and M. Mag-</list_item>
<list_item><location><page_6><loc_12><loc_92><loc_38><loc_93></location>giore, JCAP 04 , 010, arXiv:2001.07619.</list_item>
<list_item><location><page_6><loc_9><loc_89><loc_49><loc_92></location>[17] B. Knorr and A. Platania, Phys. Rev. D 106 , L021901 (2022), arXiv:2202.01216 [hep-th].</list_item>
<list_item><location><page_6><loc_9><loc_85><loc_49><loc_89></location>[18] A. Conroy, A. Mazumdar, and A. Teimouri, Phys. Rev. Lett. 114 , 201101 (2015), [Erratum: Phys.Rev.Lett. 120, 039901 (2018)], arXiv:1503.05568 [hep-th].</list_item>
<list_item><location><page_6><loc_9><loc_83><loc_49><loc_85></location>[19] A. Conroy, A. Mazumdar, S. Talaganis, and A. Teimouri, Phys. Rev. D 92 , 124051 (2015), arXiv:1509.01247.</list_item>
<list_item><location><page_6><loc_9><loc_80><loc_49><loc_82></location>[20] S. Giaccari, L. Modesto, L. Rachwał, and Y. Zhu, Eur. Phys. J. C 78 , 459 (2018), arXiv:1512.06206 [hep-th].</list_item>
<list_item><location><page_6><loc_9><loc_77><loc_49><loc_80></location>[21] Y. S. Myung, Phys. Rev. D 95 , 106003 (2017), arXiv:1702.00915 [gr-qc].</list_item>
<list_item><location><page_6><loc_9><loc_75><loc_49><loc_77></location>[22] X. Calmet and F. Kuipers, Phys. Rev. D 104 , 066012 (2021), arXiv:2108.06824 [hep-th].</list_item>
<list_item><location><page_6><loc_9><loc_71><loc_49><loc_74></location>[23] T. Azeyanagi, G. Compere, N. Ogawa, Y. Tachikawa, and S. Terashima, Prog. Theor. Phys. 122 , 355 (2009), arXiv:0903.4176 [hep-th].</list_item>
<list_item><location><page_6><loc_9><loc_68><loc_49><loc_70></location>[24] B. Knorr, C. Ripken, and F. Saueressig, Nuovo Cim. C 45 , 28 (2022), arXiv:2111.12365 [hep-th].</list_item>
<list_item><location><page_6><loc_9><loc_67><loc_49><loc_68></location>[25] B. S. Kay and R. M. Wald, Phys. Rept. 207 , 49 (1991).</list_item>
<list_item><location><page_6><loc_9><loc_62><loc_49><loc_67></location>[26] T. Jacobson, G. Kang, and R. C. Myers, in 16th Annual MRST (Montreal-Rochester-Syracuse-Toronto) Meeting on High-energy Physics: What Next? Exploring the Future of High-energy Physics (1994) arXiv:gr-qc/9502009.</list_item>
<list_item><location><page_6><loc_9><loc_59><loc_49><loc_61></location>[27] V. Iyer and R. M. Wald, Phys. Rev. D 50 , 846 (1994), arXiv:gr-qc/9403028.</list_item>
<list_item><location><page_6><loc_9><loc_56><loc_49><loc_59></location>[28] R. M. Wald, Journal of mathematical physics 31 , 2378 (1990).</list_item>
<list_item><location><page_6><loc_9><loc_54><loc_49><loc_56></location>[29] R. Brustein and D. Gorbonos, Phys. Rev. D 79 , 126003 (2009), arXiv:0902.1553 [hep-th].</list_item>
<list_item><location><page_6><loc_9><loc_51><loc_49><loc_53></location>[30] A. O. Barvinsky, Phys. Rev. D 85 , 104018 (2012), arXiv:1112.4340 [hep-th].</list_item>
<list_item><location><page_6><loc_9><loc_48><loc_49><loc_51></location>[31] S. Deser and R. P. Woodard, JCAP 11 , 036, arXiv:1307.6639 [astro-ph.CO].</list_item>
<list_item><location><page_6><loc_9><loc_46><loc_49><loc_48></location>[32] V. Pestun and M. Zabzine, J. Phys. A 50 , 443001 (2017), arXiv:1608.02953 [hep-th].</list_item>
<list_item><location><page_6><loc_9><loc_44><loc_43><loc_45></location>[33] A. Teimouri, (2017), arXiv:1705.11164 [gr-qc].</list_item>
<list_item><location><page_6><loc_9><loc_43><loc_41><loc_44></location>[34] C. Rovelli, (2017), arXiv:1710.00218 [gr-qc].</list_item>
<list_item><location><page_6><loc_9><loc_42><loc_47><loc_43></location>[35] C. Rovelli, Entropy 21 , 839 (2019), arXiv:1902.03631.</list_item>
</unordered_list>
<section_header_level_1><location><page_7><loc_17><loc_92><loc_41><loc_93></location>SUPPLEMENTAL MATERIAL</section_header_level_1>
<text><location><page_7><loc_9><loc_84><loc_49><loc_90></location>In this appendix we provide technical details on the derivation of the entropy formula that have been omitted in the main text, as well as further analytical and numerical evidence supporting our results.</text>
<section_header_level_1><location><page_7><loc_14><loc_80><loc_44><loc_81></location>S.1. Localization of the effective action</section_header_level_1>
<text><location><page_7><loc_9><loc_67><loc_49><loc_78></location>We start our derivation by detailing the localization procedure of the effective action. The localization entails introducing a number of auxiliary fields satisfying specific constraints that are enforced by a set of Lagrange multipliers [30-33]. In our case, the localization requires introducing N scalars ψ n, (1) , N two-tensors ψ µν n, (2) , and N four-tensors ψ µνρσ n, (3) . The localized action reads</text>
<formula><location><page_7><loc_10><loc_53><loc_49><loc_66></location>Γ local eff,HD = α 16 πG ∫ d 4 x √ -g N ∑ n =0 ( c n, 1 Rψ n, (1) + Rξ n, (1) ( glyph[square] n ψ n, (1) -R ) + c n, 2 R µν ψ µν n, (2) + R µν ξ n, (2) ( glyph[square] n ψ µν n, (2) -R µν ) + c n, 3 R µνρσ ψ µνρσ n, (3) + R µνρσ ξ n, (3) ( glyph[square] n ψ µνρσ n, (3) -R µνρσ ) ) . (22)</formula>
<text><location><page_7><loc_9><loc_51><loc_49><loc_53></location>The 3 N scalars ξ n, ( i ) are Lagrange multipliers introduced in such a way that they enforce the on-shell constraints</text>
<formula><location><page_7><loc_21><loc_43><loc_49><loc_49></location>glyph[square] n ψ n, (1) -R = 0 , glyph[square] n ψ ab n, (2) -R ab = 0 , glyph[square] n ψ abcd n, (3) -R abcd = 0 . (23)</formula>
<text><location><page_7><loc_9><loc_38><loc_49><loc_42></location>One can check that when enforcing these constraints, as well as the field equations for the metric field, the localized action reproduces the original non-local one.</text>
<section_header_level_1><location><page_7><loc_16><loc_33><loc_41><loc_34></location>S.2. Derivation entropy formula</section_header_level_1>
<text><location><page_7><loc_9><loc_24><loc_49><loc_31></location>In this subsection we derive the expression (11) starting from the Wald entropy formula (9). For a general spherically-symmetric spacetime (6), the bifurcation tensor is given in terms of the metric functions as in Eq. (8). For f ( r ) = g ( r ) one can prove the exact relation</text>
<formula><location><page_7><loc_16><loc_21><loc_49><loc_23></location>( glyph[square] R µνσρ ) glyph[epsilon1] µν glyph[epsilon1] σρ ≡ glyph[square] ( R µνσρ glyph[epsilon1] µν glyph[epsilon1] σρ ) , (24)</formula>
<text><location><page_7><loc_9><loc_11><loc_49><loc_20></location>i.e., [ glyph[square] , glyph[epsilon1] µν ] = 0 . One can then use the same relation to show that higher powers of the d'Alembertian also commute with the bifurcation tensor. The argument also goes through for inverse powers of the d'Alembertian, as the commutator [ A,B ] of an operator A with B and its inverse are proportional,</text>
<formula><location><page_7><loc_19><loc_9><loc_49><loc_10></location>[ B,A -1 ] = -A -1 [ B,A ] A -1 . (25)</formula>
<text><location><page_7><loc_52><loc_90><loc_92><loc_93></location>Since asymptotically f ∼ g , Eq. (24) can be exploited to substantially simplify computations.</text>
<text><location><page_7><loc_52><loc_83><loc_92><loc_90></location>On this basis, the variation of the EH and HD Lagrangians in Eq. (2) with respect to the Riemann tensor at a fixed metric is obtained by following standard steps (see, e.g., [33]). We illustrate this procedure by explicitly showing the calculation of the Ricci term</text>
<formula><location><page_7><loc_52><loc_69><loc_92><loc_82></location>∂ L Local Ricci ∂R µνρσ glyph[epsilon1] µν glyph[epsilon1] ρσ = N ∑ n =1 ( c 2 ,n δR ab δR µνρσ glyph[epsilon1] µν glyph[epsilon1] ρσ ψ ab n, (2) + ξ n, (2) δR ab δR µνρσ glyph[epsilon1] µν glyph[epsilon1] ρσ ( glyph[square] n ψ ab n, (2) -2 R ab ) ) = -1 2 ( F 2 ( glyph[square] ) R 2 -N ∑ n =1 ξ n, (2) R 2 ) , (26)</formula>
<text><location><page_7><loc_52><loc_66><loc_92><loc_67></location>where the curvature invariant relevant for the entropy is</text>
<formula><location><page_7><loc_53><loc_61><loc_92><loc_65></location>R 2 = rg ( f ' ) 2 -2 f 2 g ' -f ( rf ' g ' +2 g ( f ' + rf '' ) ) 2 rf 2 . (27)</formula>
<text><location><page_7><loc_52><loc_41><loc_92><loc_59></location>The localization of the action modifies the entropy by a contribution that is proportional to the ξ n, ( i ) factors, but involving the same curvature invariant. We can anticipate from this expression that our result for the diverging black hole entropy does not depend on the localization procedure since, as we will show later, the first terms F i ( glyph[square] ) R i source the divergence whereas the term depending on the ξ n, ( i ) is finite. This contrasts with the procedure followed in [33], where the Lagrange multipliers are fixed to be ξ n, ( i ) = -c i,n ; despite allowing for some simplifications, this choice could obscure the dependence of the result on the localization procedure. We shall thereby leave them unspecified.</text>
<text><location><page_7><loc_52><loc_34><loc_92><loc_41></location>Repeating this procedure for all terms in the Lagrangian and integrating over the bifurcation surface we obtain that the entropy is given by S W = ( A / 4 G ) ˜ S W , where A = 4 πr H is the black hole area, r H stands for the event horizon, and the dimensionless entropy ˜ S W reads</text>
<formula><location><page_7><loc_53><loc_24><loc_92><loc_32></location>˜ S W = lim r → r H ( 1 + 2 F 1 ( glyph[square] ) R 1 + F 2 ( glyph[square] ) R 2 -4 g f F 3 ( glyph[square] ) R 3 -N ∑ n =1 ( ξ n, (1) R 1 -ξ n, (2) R 2 -4 g f ξ n, (3) R 3 )) . (28)</formula>
<text><location><page_7><loc_52><loc_21><loc_92><loc_23></location>The curvature invariants involved in the entropy formula can be computed directly for the metric (6) and read</text>
<formula><location><page_7><loc_53><loc_12><loc_92><loc_20></location>R 1 = gf ' 2 2 f 2 -2 -1 + g + rg ' r 2 -rf ' g ' +2 g (2 f ' + rf '' ) 2 rf , R 3 = 1 4 ( -f ' 2 f + f ' g ' g +2 f '' ) . (29)</formula>
<text><location><page_7><loc_52><loc_9><loc_92><loc_11></location>By requiring the metric to be Schwarzschild-like, i.e, setting f = g , the invariants of [21] are recovered.</text>
<section_header_level_1><location><page_8><loc_10><loc_92><loc_48><loc_93></location>S.3. Recursion formula for non-local corrections</section_header_level_1>
<text><location><page_8><loc_9><loc_84><loc_49><loc_90></location>In this subsection we detail the procedure to determine the action of a glyph[square] -n operator on a generic radial function. We first notice that the d'Alembertian operator acting on purely radial functions is</text>
<formula><location><page_8><loc_15><loc_79><loc_49><loc_83></location>glyph[square] = -1 r 2 √ g ( r ) f ( r ) ∂ r ( r 2 √ g ( r ) f ( r ) ∂ r ) . (30)</formula>
<text><location><page_8><loc_9><loc_74><loc_49><loc_78></location>Given a function φ ( r ) , computing φ 1 ( r ) = glyph[square] -1 φ ( r ) is equivalent to solving the differential equation glyph[square] ψ ( r ) = φ ( r ) . Its general solution reads [17]</text>
<formula><location><page_8><loc_10><loc_67><loc_49><loc_72></location>1 glyph[square] φ ( r ) = ∫ R x r dx -1 x 2 √ g ( x ) f ( x ) ∫ R y x dy √ f ( y ) g ( y ) y 2 φ ( y ) , (31)</formula>
<text><location><page_8><loc_9><loc_54><loc_49><loc_65></location>where ( R x , R y ) are related to the initial conditions of the above differential equation. If the function φ ( r ) is sufficiently regular at infinity, the absence of zero modes selects special initial conditions such that R x , R y → ∞ . This is the case when φ ( r ) = R i ( r ) , but not for higher orders, i.e., when φ = glyph[square] -n R i with n ≥ 1 . Setting φ n = glyph[square] -n φ , one can exploit the formula (31) to obtain a recursive relation to compute every φ n . It reads</text>
<formula><location><page_8><loc_9><loc_47><loc_49><loc_53></location>φ n +1 = ∫ R ( n ) y r dx -1 x 2 √ g ( x ) f ( x ) ∫ R ( n ) x x dy √ f ( y ) g ( y ) y 2 φ n ( y ) , (32)</formula>
<text><location><page_8><loc_9><loc_40><loc_49><loc_46></location>where ( R ( n ) x , R ( n ) y ) specify the initial conditions at the step n + 1 . Thus, in order to understand the divergent behavior of the glyph[square] -n correction, it is necessary to characterize the divergences of the glyph[square] -1 term.</text>
<section_header_level_1><location><page_8><loc_16><loc_36><loc_41><loc_37></location>S.4. Constraints on the horizon</section_header_level_1>
<text><location><page_8><loc_9><loc_25><loc_49><loc_34></location>We have introduced a general spherically symmetric metric (6), where the metric functions f ( r ) and g ( r ) are required to be approximately of the Schwarzschild form, according to the asymptotic expansion (7). Since the focus of our work is large black holes, we can place constraints on the parameters of such an expansion.</text>
<text><location><page_8><loc_9><loc_18><loc_49><loc_25></location>We can safely assume the outermost horizon r H to be located in the proximity of the Schwarzschild radius, so that r H = r s (1 + glyph[epsilon1] ) , with | glyph[epsilon1] | glyph[lessmuch] 1 . First, to identify glyph[epsilon1] , we can use the condition that for static black holes f ( r H ) = 0 , which to leading order in glyph[epsilon1] reads</text>
<formula><location><page_8><loc_17><loc_15><loc_49><loc_17></location>f ( r H ) glyph[similarequal] Ar -α s +(1 -Aαr -α s ) glyph[epsilon1] . (33)</formula>
<text><location><page_8><loc_9><loc_13><loc_44><loc_14></location>Thus, the spacetime can have an event horizon at</text>
<formula><location><page_8><loc_17><loc_8><loc_49><loc_11></location>r H = r s (1 + glyph[epsilon1] ) , glyph[epsilon1] glyph[similarequal] A Aα -r α s . (34)</formula>
<text><location><page_8><loc_52><loc_83><loc_92><loc_93></location>If it exists, the quantum-corrected horizon is located inside the Schwarzschild radius, so that glyph[epsilon1] ≤ 0 . This implies that A ∈ [0 , r α s /α ] . Further, the condition that the horizon is close to the classical Schwarzschild radius r s is | glyph[epsilon1] | glyph[lessmuch] 1 and entails | A | glyph[lessmuch] r α s / ( α +1) . In terms of the dimensionless constant a ≡ ( α + 1) r -α s A the two constraints read</text>
<formula><location><page_8><loc_62><loc_79><loc_92><loc_82></location>a glyph[lessmuch] 1 , a ∈ [ 0 , α +1 α ] . (35)</formula>
<text><location><page_8><loc_52><loc_69><loc_92><loc_78></location>While Eq. (34) and the conditions (35) allow to estimate the location of the event horizon of large black holes, its existence depends on the parameters ( a, α ) and has to be checked separately. The existence of a horizon is crucially related to the divergence of the entropy in the presence of infrared non-localities.</text>
<section_header_level_1><location><page_8><loc_59><loc_65><loc_84><loc_66></location>S.5. Analytic formulas for α = 2</section_header_level_1>
<text><location><page_8><loc_52><loc_55><loc_92><loc_63></location>In this subsection, we show analytically the divergent behavior of the corrections to the entropy for the simplest geometry that deviates from the Schwarzschild metric. We consider the spacetime described by the ansatz (7) with A = B and α = β = 2 . We obtain that the leadingorder contributions to the entropy are given by</text>
<formula><location><page_8><loc_52><loc_28><loc_92><loc_53></location>Φ[ R 1 , r, R x , R y ] = 0 , Φ[ R 2 , r, R x , R y ] = r + + r -r --r + ( log( R x ) log ( r -( r + -R x ) r + ( r --R x ) ) +log( r ) log ( ( r -r -) r + r -( r -r + ) ) -Li 2 ( R x r -) + Li 2 ( R x r + ) +log ( R 2 y ) tanh -1 ( r -+ r + -2 R x r --r + ) + Li 2 ( r r -) -log ( R 2 y ) tanh -1 ( r -+ r + -2 r r --r + ) -Li 2 ( r r + )) , Φ[ R 3 , r, R x , R y ] = 3 2 log ( r 2 ( A -( r -+ r + ) R x + R 2 x ) ( A + r 2 -( r -+ r + ) r ) R 2 x ) ))</formula>
<formula><location><page_8><loc_52><loc_15><loc_92><loc_33></location>+ 1 R y ( r + -r -) ( ( 6 A -( r -+ r + ) R y (log ( R 2 y ) -3) ) × ( tanh -1 ( r -+ r + -2 r r --r + ) -tanh -1 ( r -+ r + -2 R x r --r + +( r -+ r + ) R y ( + Li 2 ( R x r + ) -Li 2 ( R x r -) -log ( R x ) log ( A -r + R x A -r -R x ) + Li 2 ( r r -) +log( r ) log ( A -rr + A -rr -) -Li 2 ( r r + ))) . (36)</formula>
<text><location><page_8><loc_52><loc_9><loc_92><loc_14></location>where r ± = r s / 2 ± 1 / 2 √ r 2 s -4 A denotes the location of the outer and inner horizons, respectively, and Li 2 is a polylogarithm of order two. We have evaluated everything at a generic r , to show how the divergences occur</text>
<text><location><page_9><loc_9><loc_92><loc_43><loc_93></location>when evaluating r → r H = r + . Taking the limit</text>
<formula><location><page_9><loc_19><loc_86><loc_49><loc_90></location>lim r → r H Φ[ R 2 , r, R x , R y ] = ∞ , lim r → r H Φ[ R 3 , r, R x , R y ] = -∞ . (37)</formula>
<text><location><page_9><loc_9><loc_82><loc_49><loc_85></location>The divergent part of the entropy is therefore controlled by the contribution</text>
<formula><location><page_9><loc_12><loc_76><loc_49><loc_81></location>∆ S W ≡ ˜ S (1) W = lim r → r H ( c 2 , 1 Φ[ R 2 , r, R x , R y ] -4 c 3 , 1 Φ[ R 3 , r, R x , R y ]) . (38)</formula>
<text><location><page_9><loc_9><loc_68><loc_49><loc_75></location>The sign of the divergence thereby depends on the particular combination of the coefficients c i, 1 . In particular, the divergence may vanish for some special combinations of c i, 1 . We define the critical ratio c crit as the ratio c 2 , 1 /c 3 , 1 such that ∆ S W = 0 .</text>
<text><location><page_9><loc_9><loc_49><loc_49><loc_68></location>As an illustrative example, we study the critical ratio c crit in the case of a Reissner-Nordström black hole. The metric for the solution to the classical EinsteinMaxwell theory, with charge Q , is given by the ansatz (7) with A = B = Q and α = β = 2 . As for the Schwarzschild case, the corrections stemming from higher-order terms in the action could appear as subleading terms in the asymptotic expansion of the metric coefficients. As a first approximation, we shall neglect these sub-leading corrections. As shown by the above formulas, the entropy for Reissner-Nordström black holes in the presence of infrared non-localities diverges. In particular, we find that</text>
<formula><location><page_9><loc_10><loc_43><loc_49><loc_48></location>∆ S W = { sign( c 2 , 1 ) ×∞ , c 2 , 1 > c crit ( Q/ 2 M ) c 3 , 1 , -sign( c 2 , 1 ) ×∞ , c 2 , 1 < c crit ( Q/ 2 M ) c 3 , 1 , (39)</formula>
<text><location><page_9><loc_9><loc_38><loc_49><loc_42></location>where the critical threshold of the coefficients is a nontrivial function that is controlled by the charge of the black hole and its mass, as depicted in Fig. 3.</text>
<section_header_level_1><location><page_9><loc_9><loc_34><loc_48><loc_35></location>S.6. Parametric characterization of the divergence</section_header_level_1>
<text><location><page_9><loc_9><loc_25><loc_49><loc_32></location>We now extend the results from the previous subsection by numerically evaluating the dimensionless entropy correction ˜ S (1) W , cf. Eq. (13), in a large region of the parameter space spanned by ( a, α ) . Some of the results were already reported in the main text, in Fig. 1 and Fig. 5.</text>
<text><location><page_9><loc_9><loc_19><loc_49><loc_25></location>First, in Fig. 1 we fixed the value of the boundary condition R = R x = R y and showed that the corrected entropy diverges as r → r H , provided that a horizon exists. Moreover, these features are independent of the</text>
<text><location><page_9><loc_52><loc_89><loc_92><loc_93></location>exponent α and of the dimensionless coefficient a : the divergence as r → r H is present whenever a horizon exists (see Fig. 4).</text>
<text><location><page_9><loc_52><loc_83><loc_92><loc_88></location>In addition, such a divergence is independent of the initial conditions characterized by R = R x = R y , as is clearly visible in Fig. 2. We have also explicitly checked that the result also holds for α = β and A = B .</text>
<text><location><page_9><loc_74><loc_83><loc_74><loc_84></location>glyph[negationslash]</text>
<text><location><page_9><loc_82><loc_83><loc_82><loc_84></location>glyph[negationslash]</text>
<figure>
<location><page_9><loc_53><loc_62><loc_91><loc_81></location>
<caption>Figure 3. Critical value of the ratio between the coefficients c 2 , 1 and c 3 , 1 for which the entropy correction ∆ S W (c.f. Eq. (38)) vanishes, as a function of the dimensionless charge Q/ 2 M of a Reissner-Nordström black hole of mass M . In the plot we fixed R x = R y = 10 3 .</caption>
</figure>
<text><location><page_9><loc_52><loc_40><loc_92><loc_50></location>Secondly, in Fig. 5 we show that the corrected entropy with coefficients c 1 , 1 = c 2 , 1 = c 3 , 1 = 1 diverges to ˜ S (1) W → -∞ as R → ∞ . Specifically, the divergence is faster for smaller values of a . More importantly, the divergent behavior does not depend on a , as long as it satisfies the constraints described previously, and is also independent of α .</text>
<text><location><page_9><loc_52><loc_19><loc_92><loc_39></location>We conclude from our analytical and numerical results that the divergent behavior, both as r → r H and as R → ∞ is independent of the exact values of ( a, α ) , as long as the constraints on the horizon are satisfied, and modulo a special cancellation of divergences-as illustrated in the previous subsection. We emphasize that these results only account for the glyph[square] -1 correction in the form factors. If one were to include the whole series, there would be more divergent terms; hence a complicated relation between the c i,n coefficients would have to exist in order to potentially cancel all divergences. Such a cancellation is unlikely, as it would require a very special combination of Wilson coefficients in the gravitational effective action.</text>
<figure>
<location><page_10><loc_10><loc_52><loc_91><loc_93></location>
<caption>Figure 4. Dimensionless entropy correction ˜ S (1) W as a function of r/r s , for R x = R y = 10 3 , c 1 , 1 = c 2 , 1 = c 3 , 1 = 1 , and various combinations of ( a, α ) , as reported in the figures. The location of the corresponding event horizons is indicated by a vertical dashed line. Provided that a horizon exists, the entropy diverges as r → r H .</caption>
</figure>
<figure>
<location><page_10><loc_9><loc_20><loc_48><loc_39></location>
<caption>Figure 5. Dimensionless entropy correction ˜ S (1) W , cf. Eq. (13) as a function of the initial condition R = R x = R y . The correction is evaluated at r = 1 . 1 r s = r H and for c 1 , 1 = c 2 , 1 = c 3 , 1 = 1 . The left panel shows ˜ S (1) W for α = 2 and different values of a , whereas the right panel depicts the same entropy correction for a = 0 . 1 and different values of the exponent α . In all these cases, the entropy diverges as R →∞ .</caption>
</figure>
<figure>
<location><page_10><loc_52><loc_20><loc_92><loc_39></location>
</figure>
<text><location><page_10><loc_26><loc_15><loc_26><loc_16></location>glyph[negationslash]</text>
</document>
[
{
"title": "Diverging black hole entropy from quantum infrared non-localities",
"content": "Alessia Platania 1, 2, ∗ and Jaime Redondo-Yuste 1, 3, † 1 Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON N2L 2Y5, Canada 2 Nordita, KTH Royal Institute of Technology and Stockholm University, Hannes Alfvéns väg 12, SE-106 91 Stockholm, Sweden 3 Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark Local higher-derivative corrections to the Einstein-Hilbert action yield sub-leading corrections to the Bekenstein-Hawking area law. Here we show that if the quantum effective action comprises a certain class of infrared non-localities, the entropy of large black holes generally diverges to either positive or negative infinity. In such theories, large spherically symmetric black holes would be either highly chaotic or thermodynamically impossible, respectively. In turn, this puts strong constraints on the Laurent expansion of the form factors in the effective action. The theoretical prediction that black holes emit thermal radiation [1, 2] is considered one of Stephen Hawking's most important contributions to our understanding of black holes. Hawking radiation has two crucial consequences: (i) it entails that black holes evaporate, as the emission causes a reduction in their mass and rotational energy, and (ii) the entropy of a large black hole is, to leading order, proportional to its area [2, 3] with l P being the Planck length. Corrections to this classical formula parametrize deviations from General Relativity (GR). The first law of black hole mechanics connects the perturbations of the black hole mass, its angular momentum, and a combination of the surface gravity and a geometrical quantity that is later given the meaning of an entropy. The original derivation [4] used very explicit properties of the Einstein field equations, so a natural question was whether such a statement could be extended to any theory of gravity. Wald's seminal work [5] demonstrated that the first law holds for generic gravitational actions S , provided that some minimal assumptions on the (classical or effective) Lagrangian, such as diffeomorphism invariance, are fulfilled. On the theoretical side, the form of the effective action depends on quantum gravitational effects, and potentially on the specific ultraviolet completion of gravity. In particular, owed to the quantum nature of the effective action, a common feature is the presence of nonlocalities. The latter could be exponential functions of the d'Alembert operator, as in non-local gravity [6], hyperbolic tangents, as in some asymptotically safe models [7], operators of the Polyakov type, as in string theory [8, 9] and in the high-energy limit of loop quantum gravity [10], or even higher inverse powers of the d'Alembert operator, as it could happen in causal dynamical triangulations [11]. Such non-localities also play a role on the phenomenological side. Specifically, inverse powers of the d'Alembert operator seem to be important in cosmological settings [12-16], as an alternative explanation to dark energy, as well as in black hole physics [17], to accommodate for regular black holes stemming from a principle of least action in quantum gravity. The entropy of black holes in the presence of local higher-derivative terms has been widely studied [1822]. As expected, adding such terms to the EinsteinHilbert action leads to sub-leading corrections to the area law (1). On dimensional grounds, one might expect that similar corrections stemming from inverse powers of the d'Alembert operator would dominate over the Bekenstein-Hawking scaling. The main motivation of our Letter is to test this expectation. We provide analytical and numerical evidence that if the effective action contains infrared non-localities, not only the resulting corrections are dominant, but also divergent. The entropy of spherically symmetric configurations is thus generically divergent, barring unlikely cancellations of the infinities associated with each non-local term in the effective action. Depending on the sign of the divergence, black holes in such theories would be characterized either by infinitely many microstates or by none. This paradoxical behavior points to novel constraints on the Laurent expansion of the form factors in the effective action. Setup - We shall focus on asymptotically flat spacetimes. The gravitational effective action of a generic diffeomorphism-invariant theory involving only the metric is where the Einstein-Hilbert (EH) part is complemented by the higher-derivative (HD) Lagrangian Above, G is the Newton constant and glyph[square] = -g µν D µ D ν is the d'Alembert operator constructed from the Levi- Civita connection of the metric g . The form factors F i 1 result from integrating out quantum gravitational fluctuations in the functional integral, are typically non-local, and at one loop they ought to match the logarithmic behavior encountered in [22]. Their form and properties are strictly tied to the specific ultraviolet completion of gravity [24]. Computing the exact entropy corrections generated by these non-local form factors is generally involved, but within the annulus of convergence, where the form factors are holomorphic, one can exploit their Laurent expansion In practical applications only a few of the negative-degree terms play a role in quantum gravity and in cosmological models [8, 10-17]. In particular, such low-order terms appear to be compatible with cosmological observations [15, 16]. Entropy corrections coming from the first positive-degree terms ( n ≥ 0 ) in this Laurent series have been computed in [18-22]. Given the relevance of some of the negative-degree terms in (4)-the 'principal part' of the Laurent expansion-in quantum gravity [8, 10, 11], black hole physics [17], and cosmology [1216], investigating their impact on black hole thermodynamics is of great importance. In order to compute the entropy corrections stemming from this type of infrared non-localities, while avoiding to compute the corresponding dressed field equations and solutions, we need to make some assumptions. First, we assume that the length scale at which quantum gravity effects become important is of the order of the Planck length, l QG glyph[similarequal] l P , such that no strong non-perturbative effects happen at the horizon scale for large black holes. Second, we limit ourselves to astrophysical black holes, for which M glyph[greatermuch] M P . This ensures that the event horizon is located within a few Planck lengths from the Schwarzschild radius, i.e., that r H glyph[similarequal] r s , with r s ≡ 2 MG . The solutions to the quantum theory associated with the effective action (2) ought to be found by solving the quantum field equations These solutions are generally different than those obtained from the vacuum Einstein equations (Ricci-flat metrics) 2 and are typically difficult to derive. Even if the field equations (5) are different than in GR, they are expected to admit spherically symmetric solutions of the type 3 where d Ω 2 is the line element on the 2 -sphere. Further, for large black holes the metric coefficients f ( r ) and g ( r ) can be written as with α, β ≥ 2 , i.e., the metric only differs from the Schwarzschild one by sub-leading terms. In particular, g glyph[similarequal] f for large or massive black holes. This geometry has a Killing horizon H , with bifurcation surface Σ , where the future and past horizons intersect. This is characterized by an antisymmetric bifurcation tensor glyph[epsilon1] µν , normalized such that glyph[epsilon1] µν glyph[epsilon1] µν = -2 , and given by A spacelike surface spanning from Σ to spatial infinity i 0 can be used to define a gravitational phase space consistent with general covariance 4 . Wald [5] showed that the first law of black hole thermodynamics follows, and the entropy acquires the interpretation of a Noether charge. For a static 5 , spherically-symmetric black hole it reads where L is the (classical or effective) Lagrangian, dV 2 2 = r 2 sin θdθdφ is the line element on the bifurcation surface Σ , and the functional derivative is performed at a fixed metric. We shall employ this formula to compute the corrections to the entropy of the spherically symmetric spacetime (7) stemming from the quadratic higherderivative corrections (3) with form factors structurally given by the principle part of the expansion (4). Localizing the effective action - In the derivation of the Wald entropy formula, there are several assumptions that deserve to be highlighted: (i) The theory is defined by a locally constructed [28], diffeomorphism invariant Lagrangian L built from dynamical fields and their derivatives on a Lorentzian manifold. (ii) There is some notion of asymptotic flatness, so that the black hole horizon can be defined as the (inner) boundary of the past region of the asymptotic region. (iii) The horizon is a Killing horizon, i.e., a null surface to which a Killing field is normal. Out of these assumptions, the last one is always satisfied for static black holes, and in particular for all spherically symmetric solutions. The property of asymptotic flatness is necessary to precisely define the notion of an event horizon, but given that such a construction is possible, it does not enter later in the derivation 6 . The most critical assumption for our purposes is the first one, since the action used is not directly 'locally constructed'. This is a problem because the covariant phase space formalism relies on the properties of jet bundles on field space [28]. There is however a way to circumvent this issue: it consists in finding a 'localized' version of the action that on-shell yields the same configurations. Such a localized effective action is derived by translating the non-locality into a set of constrained auxiliary fields [30-33]. In order to proceed further, we now focus on the principal part of the expansion (4), and in particular on the form factors F i ( glyph[square] ) given by the following finite sums where c i,n are real coefficients and the truncation order N can be systematically increased. Following standard procedures [30-33], it is straightforward to compute a localized version of the effective action, Γ local eff = Γ EH +Γ local eff,HD , made up of the Einstein-Hilbert part Γ EH and the localized higher-derivative action (see supplemental material). We will use such an action to compute the entropy corrections stemming from the form factors (10). Entropy formula - The entropy of large black holes in a theory with infrared non-localities (10) is to be computed as a Noether charge, Eq. (9), with the action S ≡ Γ local eff = Γ EH + Γ local eff,HD being the localized version of the effective action. After some algebra (see supplemental material for details), the resulting dimensionless entropy ˜ S W = (4 G/ A ) S W takes the form where ξ n, ( i ) are Lagrange multipliers, and are functionals of the metric components f ( r ) and g ( r ) . Diverging black hole entropy - Exploiting the generalized formula (11), we now compute the corrections to the Bekenstein-Hawking area law stemming from the form factors (10). As the entropy is additive, corrections stemming from the individual glyph[square] -n -operators in the form factors (10) can be determined individually. We checked that the corrections in the second line of Eq. (11) are finite and unimportant to our conclusions. In the following we will thus focus on the dimensionless contributions To this end, we first need to determine how the glyph[square] -n terms act on the radial functions R i ( r ) . This requires solving glyph[square] n ψ ( r ) = R i ( r ) with respect to the function ψ . For a generic spherically symmetric metric (6) [17] where φ is a generic function of the radial coordinate and we have defined This is the general solution to Eq. (14), and the 'cutoffs' R x and R y are related to its initial conditions. As in the case n = 1 we have φ ( r ) = R i ( r ) , the regularity properties of R i ( r ) and the absence of zero modes allow selecting special initial conditions, such that R x , R y → ∞ [17]. We will nevertheless show that our results are independent of the choice of initial conditions. Eq. (15) can be generalized to a recursion formula where φ n +1 ≡ glyph[square] -n -1 φ ( r ) is written in terms of φ n ≡ glyph[square] -n φ ( r ) , where φ n ≡ glyph[square] -1 φ n -1 = glyph[square] -n φ ( r ) with n ≥ 1 , and { R ( n ) x , R ( n ) y } are initial conditions. With this, the starting point to compute the corrections to the area law is to consider the lowest-order operators, F i, 1 = c i, 1 / glyph[square] , and then compute the effect of the others-the F i,n = c i,n glyph[square] -n in Eq. (10)-recursively. We anticipate that the divergence of the lowest-order operator implies, owed to the recursion formula (16), the divergence of all the higher-order terms. The first three contributions to the entropy formula that we have to compute are thus where we have considered the same ( R x , R y ) ∀ i = 1 , 2 , 3 for simplicity. This is justified since, as we shall see, our conclusions are independent of the initial conditions. Based on our assumptions, the solution to the nonlocal effective field equation (5) is of the form (7). Since g glyph[similarequal] f for large and massive black holes, in what follows we shall set A = B and α = β . We checked that this simplification does not affect our conclusions. It is useful to start from a simple example, where exact analytical formulas can be derived: the Schwarzschild black hole, corresponding to the case A = 0 , is an exact solution for F 3 = 0 . In this ideal case A 3 , 1 = 0 by construction, while the other two contributions in Eq. (17) can be computed with the aid of the formula (15). In particular, A 1 , 1 = 0 and where Li 2 is a polylogarithm of order two, which is real for r → r H ≤ r s . Analogous analytical formulas can also be derived in the case α = 2 (see supplemental material). Taking the limit r → r H = r s , one finds that A 2 , 1 diverges, with the sign of the divergence depending on the coefficient c 2 , 1 . In particular, divergences arise both at the horizon and at infinity (as R x , R y →∞ ). glyph[negationslash] glyph[negationslash] Similar findings also hold when accounting for the asymptotic corrections to the Schwarzschild metric coefficients, in which case A,B = 0 in Eq. (7) and r H = r s . In these cases one has to account for all corrections (17). The event horizon, when it exists, is located at where a is the dimensionless constant a ≡ ( α +1) r -α s A , and a glyph[lessmuch] 1 for large black holes. Independent of the exponent α and of the value of a in the allowed range, whenever a horizon exists, the contributions (17) to the entropy ˜ S (1) W diverge as r → r H . This is shown in Fig. 1, where we account for the position of the horizon for different values of a . The presence of a horizon for large black holes requires a glyph[lessmuch] 1 . Indeed, when a is big enough, no horizon forms, and the integrals (17) are regular everywhere. Vice versa, when a horizon exists (dashed lines in Fig. 1), the entropy correction diverges (see supplemental material for further analytical and numerical evidence). In particular, as shown in Fig. 2, such a divergence is independent of the initial conditions ( R x , R y ) . Finally, higher-order terms in Eq. (10) lead to similar divergent contributions ˜ S ( n ) W owed to the recursion formula (16). Whether the divergence is positive or negative infinity depends on the (relative) signs of the coefficients c i,n , with i = 1 , 2 , 3 and n = 1 , . . . , N . As discussed in the supplemental material, even in the simple case N = n = 1 and α = 2 , the condition on the coefficients c i, 1 to eliminate the divergence is non-trivial. Plugging all corrections ˜ S ( n ) W into Eq. (11), and considering that the terms in the second line of Eq. (11) yield finite contributions, we finally obtain the surprising result that the total black hole entropy S W diverges, unless the set of coefficients c i,n are fine-tuned such that all divergences generated by all glyph[square] -n operators in the form factors (10) cancel out. Such a special combination of coefficients is unlikely realized by the quantum gravitational dynamics. Identifying the Wald entropy with the thermodynamical entropy given by the Boltzmann formula 7 our result entails that spherically symmetric configurations stemming from effective actions that display infrared non-localities are either highly chaotic or thermodinamically impossible. In turn, our result places strong constraints on the Laurent expansion of the gravitational effective action: finiteness of the black hole entropy requires the absence of quantum-induced infrared non-localities, or a very special set of coefficients resulting from quantum gravitational dynamics. Conclusions - In our Letter, we tackled the problem of determining corrections to the black hole area law stemming from infrared non-localities in the gravitational effective action. Higher-derivative corrections to the effective action are generally expected from quantum gravity and are particularly important in the physics of large black holes, as a description in terms of an effective action has to be recovered irrespective of the specific ultraviolet completion of gravity. Local higher-derivative terms have been considered in the seminal work [18], and yield sub-leading contributions to the black hole area law. Our Letter complements the results in [18] by accounting for non-local higher derivative terms, the latter being of relevance in several quantum gravity models [8, 10, 11], black hole physics [17], and cosmology [12-16]. Despite some of these infrared non-localities seem compatible with observations [14-16], they appear to yield inconsistencies: we find that not only infrared non-localities lead to corrections that are dominant over the Bekenstein-Hawking term, as we expected, but they also yield diverging contributions. Exploiting the Boltzmann formula, one would then conclude that the corresponding black holes are made of either zero or infinitely many microstates, depending on the sign of the divergence. This paradoxical behavior points to novel constraints on the Laurent expansion of the effective action stemming from quantum gravity theories: its principle part has to vanish or be fine-tuned in order for the black hole entropy to remain finite. It is still an open question whether similar cubic and higher-order terms in the curvature expansion (2) of the effective action could dramatically change this conclusion, and whether resumming all infinitely many corrections could provide a finite result. Yet, this is expected to come at the expense of fine-tuning infinitely many coefficients in the effective action.",
"pages": [
1,
2,
3,
4,
5
]
},
{
"title": "ACKNOWLEDGMENTS",
"content": "The authors would like to thank I. Basile, B. Knorr, D. Pereñiguez, and A. Riello for interesting discussions, and I. Basile, B. Knorr, and V. Cardoso for feedback on our manuscript. The authors acknowledge support by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Colleges and Universities. JRY acknowledges support from the Villum Investigator program supported by the VILLUM Foundation (grant no. VIL37766) and the DNRF Chair program (grant no. DNRF162) by the Danish National Research Foundation. AP also acknowledges Nordita for support within the 'Nordita Distinguished Visitors' program and for hospitality during the last stages of development of this work. Nordita is supported in part by NordForsk.",
"pages": [
5
]
},
{
"title": "SUPPLEMENTAL MATERIAL",
"content": "In this appendix we provide technical details on the derivation of the entropy formula that have been omitted in the main text, as well as further analytical and numerical evidence supporting our results.",
"pages": [
7
]
},
{
"title": "S.1. Localization of the effective action",
"content": "We start our derivation by detailing the localization procedure of the effective action. The localization entails introducing a number of auxiliary fields satisfying specific constraints that are enforced by a set of Lagrange multipliers [30-33]. In our case, the localization requires introducing N scalars ψ n, (1) , N two-tensors ψ µν n, (2) , and N four-tensors ψ µνρσ n, (3) . The localized action reads The 3 N scalars ξ n, ( i ) are Lagrange multipliers introduced in such a way that they enforce the on-shell constraints One can check that when enforcing these constraints, as well as the field equations for the metric field, the localized action reproduces the original non-local one.",
"pages": [
7
]
},
{
"title": "S.2. Derivation entropy formula",
"content": "In this subsection we derive the expression (11) starting from the Wald entropy formula (9). For a general spherically-symmetric spacetime (6), the bifurcation tensor is given in terms of the metric functions as in Eq. (8). For f ( r ) = g ( r ) one can prove the exact relation i.e., [ glyph[square] , glyph[epsilon1] µν ] = 0 . One can then use the same relation to show that higher powers of the d'Alembertian also commute with the bifurcation tensor. The argument also goes through for inverse powers of the d'Alembertian, as the commutator [ A,B ] of an operator A with B and its inverse are proportional, Since asymptotically f ∼ g , Eq. (24) can be exploited to substantially simplify computations. On this basis, the variation of the EH and HD Lagrangians in Eq. (2) with respect to the Riemann tensor at a fixed metric is obtained by following standard steps (see, e.g., [33]). We illustrate this procedure by explicitly showing the calculation of the Ricci term where the curvature invariant relevant for the entropy is The localization of the action modifies the entropy by a contribution that is proportional to the ξ n, ( i ) factors, but involving the same curvature invariant. We can anticipate from this expression that our result for the diverging black hole entropy does not depend on the localization procedure since, as we will show later, the first terms F i ( glyph[square] ) R i source the divergence whereas the term depending on the ξ n, ( i ) is finite. This contrasts with the procedure followed in [33], where the Lagrange multipliers are fixed to be ξ n, ( i ) = -c i,n ; despite allowing for some simplifications, this choice could obscure the dependence of the result on the localization procedure. We shall thereby leave them unspecified. Repeating this procedure for all terms in the Lagrangian and integrating over the bifurcation surface we obtain that the entropy is given by S W = ( A / 4 G ) ˜ S W , where A = 4 πr H is the black hole area, r H stands for the event horizon, and the dimensionless entropy ˜ S W reads The curvature invariants involved in the entropy formula can be computed directly for the metric (6) and read By requiring the metric to be Schwarzschild-like, i.e, setting f = g , the invariants of [21] are recovered.",
"pages": [
7
]
},
{
"title": "S.3. Recursion formula for non-local corrections",
"content": "In this subsection we detail the procedure to determine the action of a glyph[square] -n operator on a generic radial function. We first notice that the d'Alembertian operator acting on purely radial functions is Given a function φ ( r ) , computing φ 1 ( r ) = glyph[square] -1 φ ( r ) is equivalent to solving the differential equation glyph[square] ψ ( r ) = φ ( r ) . Its general solution reads [17] where ( R x , R y ) are related to the initial conditions of the above differential equation. If the function φ ( r ) is sufficiently regular at infinity, the absence of zero modes selects special initial conditions such that R x , R y → ∞ . This is the case when φ ( r ) = R i ( r ) , but not for higher orders, i.e., when φ = glyph[square] -n R i with n ≥ 1 . Setting φ n = glyph[square] -n φ , one can exploit the formula (31) to obtain a recursive relation to compute every φ n . It reads where ( R ( n ) x , R ( n ) y ) specify the initial conditions at the step n + 1 . Thus, in order to understand the divergent behavior of the glyph[square] -n correction, it is necessary to characterize the divergences of the glyph[square] -1 term.",
"pages": [
8
]
},
{
"title": "S.4. Constraints on the horizon",
"content": "We have introduced a general spherically symmetric metric (6), where the metric functions f ( r ) and g ( r ) are required to be approximately of the Schwarzschild form, according to the asymptotic expansion (7). Since the focus of our work is large black holes, we can place constraints on the parameters of such an expansion. We can safely assume the outermost horizon r H to be located in the proximity of the Schwarzschild radius, so that r H = r s (1 + glyph[epsilon1] ) , with | glyph[epsilon1] | glyph[lessmuch] 1 . First, to identify glyph[epsilon1] , we can use the condition that for static black holes f ( r H ) = 0 , which to leading order in glyph[epsilon1] reads Thus, the spacetime can have an event horizon at If it exists, the quantum-corrected horizon is located inside the Schwarzschild radius, so that glyph[epsilon1] ≤ 0 . This implies that A ∈ [0 , r α s /α ] . Further, the condition that the horizon is close to the classical Schwarzschild radius r s is | glyph[epsilon1] | glyph[lessmuch] 1 and entails | A | glyph[lessmuch] r α s / ( α +1) . In terms of the dimensionless constant a ≡ ( α + 1) r -α s A the two constraints read While Eq. (34) and the conditions (35) allow to estimate the location of the event horizon of large black holes, its existence depends on the parameters ( a, α ) and has to be checked separately. The existence of a horizon is crucially related to the divergence of the entropy in the presence of infrared non-localities.",
"pages": [
8
]
},
{
"title": "S.5. Analytic formulas for α = 2",
"content": "In this subsection, we show analytically the divergent behavior of the corrections to the entropy for the simplest geometry that deviates from the Schwarzschild metric. We consider the spacetime described by the ansatz (7) with A = B and α = β = 2 . We obtain that the leadingorder contributions to the entropy are given by where r ± = r s / 2 ± 1 / 2 √ r 2 s -4 A denotes the location of the outer and inner horizons, respectively, and Li 2 is a polylogarithm of order two. We have evaluated everything at a generic r , to show how the divergences occur when evaluating r → r H = r + . Taking the limit The divergent part of the entropy is therefore controlled by the contribution The sign of the divergence thereby depends on the particular combination of the coefficients c i, 1 . In particular, the divergence may vanish for some special combinations of c i, 1 . We define the critical ratio c crit as the ratio c 2 , 1 /c 3 , 1 such that ∆ S W = 0 . As an illustrative example, we study the critical ratio c crit in the case of a Reissner-Nordström black hole. The metric for the solution to the classical EinsteinMaxwell theory, with charge Q , is given by the ansatz (7) with A = B = Q and α = β = 2 . As for the Schwarzschild case, the corrections stemming from higher-order terms in the action could appear as subleading terms in the asymptotic expansion of the metric coefficients. As a first approximation, we shall neglect these sub-leading corrections. As shown by the above formulas, the entropy for Reissner-Nordström black holes in the presence of infrared non-localities diverges. In particular, we find that where the critical threshold of the coefficients is a nontrivial function that is controlled by the charge of the black hole and its mass, as depicted in Fig. 3.",
"pages": [
8,
9
]
},
{
"title": "S.6. Parametric characterization of the divergence",
"content": "We now extend the results from the previous subsection by numerically evaluating the dimensionless entropy correction ˜ S (1) W , cf. Eq. (13), in a large region of the parameter space spanned by ( a, α ) . Some of the results were already reported in the main text, in Fig. 1 and Fig. 5. First, in Fig. 1 we fixed the value of the boundary condition R = R x = R y and showed that the corrected entropy diverges as r → r H , provided that a horizon exists. Moreover, these features are independent of the exponent α and of the dimensionless coefficient a : the divergence as r → r H is present whenever a horizon exists (see Fig. 4). In addition, such a divergence is independent of the initial conditions characterized by R = R x = R y , as is clearly visible in Fig. 2. We have also explicitly checked that the result also holds for α = β and A = B . glyph[negationslash] glyph[negationslash] Secondly, in Fig. 5 we show that the corrected entropy with coefficients c 1 , 1 = c 2 , 1 = c 3 , 1 = 1 diverges to ˜ S (1) W → -∞ as R → ∞ . Specifically, the divergence is faster for smaller values of a . More importantly, the divergent behavior does not depend on a , as long as it satisfies the constraints described previously, and is also independent of α . We conclude from our analytical and numerical results that the divergent behavior, both as r → r H and as R → ∞ is independent of the exact values of ( a, α ) , as long as the constraints on the horizon are satisfied, and modulo a special cancellation of divergences-as illustrated in the previous subsection. We emphasize that these results only account for the glyph[square] -1 correction in the form factors. If one were to include the whole series, there would be more divergent terms; hence a complicated relation between the c i,n coefficients would have to exist in order to potentially cancel all divergences. Such a cancellation is unlikely, as it would require a very special combination of Wilson coefficients in the gravitational effective action. glyph[negationslash]",
"pages": [
9,
10
]
}
]
2024PhRvD.109b3512U
https://arxiv.org/pdf/2307.05600.pdf
<document>
<section_header_level_1><location><page_1><loc_13><loc_90><loc_88><loc_93></location>On the cosmological evolution of Scalar Field Dark Matter in the CLASS code: accuracy and precision of numerical solutions</section_header_level_1>
<text><location><page_1><loc_27><loc_87><loc_73><loc_89></location>L. Arturo Ure˜na-L'opez 1, ∗ and Francisco X. Linares Cede˜no 2, †</text>
<text><location><page_1><loc_14><loc_84><loc_87><loc_87></location>1 Departamento de F'ısica, DCI, Campus Le'on, Universidad de Guanajuato, 37150, Le'on, Guanajuato, M'exico 2 Instituto de F'ısica y Matem'aticas, Universidad Michoacana de San Nicol'as de Hidalgo,</text>
<text><location><page_1><loc_25><loc_83><loc_76><loc_84></location>Edificio C-3, Ciudad Universitaria, CP. 58040 Morelia, Michoac'an, M'exico.</text>
<text><location><page_1><loc_43><loc_81><loc_58><loc_82></location>(Dated: July 13, 2023)</text>
<text><location><page_1><loc_18><loc_70><loc_83><loc_80></location>We present a numerical analysis of the cosmological evolution of scalar field dark matter (SFDM) in the Boltzmann code CLASS , based on a dynamical system analysis of previous works. We show a detailed study of the evolution of the different dynamical variables, and in particular of the energy density and its corresponding linear perturbations. The numerical results are in good agreement with those of the original SFDM equations of motion, and have better accuracy than other approaches. In addition, we calculate the temperature and matter power spectra and discuss the reliability of their numerical results. We also give simple examples in which we can put constraints on the field mass using recent likelihoods incorporated in the Monte Carlo Markov Chain sampler MontePython .</text>
<section_header_level_1><location><page_1><loc_20><loc_66><loc_37><loc_67></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_47><loc_49><loc_64></location>In the modern era of cosmology [1, 2], it is mandatory to develop theoretical models able to describe the universe at large scales, with the precision that current data demand [2]. One key theoretical ingredient, particularly important for the formation of cosmic structure, is dark matter (DM), which must capture the physics of such a process in a mathematically consistent way [3]. This always starts, for the cosmological setting, with properly solving the Einstein-Boltzmann system describing the cosmological evolution for the initial perturbations of both the matter components and the metric tensor at the linear level [4].</text>
<text><location><page_1><loc_9><loc_31><loc_49><loc_47></location>Different Boltzmann solvers have been programmed for the linearized form of the Einstein equations in a Friedmann-Robertson-Walker-Lemaˆıtre (FRWL) universe, such as CMBFAST [5], CMBEASY [6], CAMB [7] and CLASS [8]. Only the latter two, CAMB and CLASS , have been kept up to date, and both are used by the community of cosmologists. The aforementioned codes consider the cold dark matter (CDM) model as the main matter component, and they are very well suited to explore most of its properties, although they have been amended to study alternative dark matter models in recent years.</text>
<text><location><page_1><loc_9><loc_15><loc_49><loc_31></location>One of these alternative models to CDM, which is currently one of the compelling proposals that has been explored for the last two decades, is that based on a scalar field. It is found in the literature under several names: Scalar Field Dark Matter (SFDM), Ultralight Axions, Fuzzy Dark Matter, Axion-like particle, Bose-Einstein Condansate, Wave Dark Matter (some initial works on this model are [9-15], whereas more recent work can be found at [16-24]). All these names reflect the particular properties of this DM particle. Its mathematical description is given by a scalar field ϕ (which can be real or com-</text>
<text><location><page_1><loc_52><loc_57><loc_92><loc_67></location>lex), with a fiducial ultralight mass of m ϕ c 2 ∼ 10 -22 eV that can be produced by the Peccei-Quinn mechanism for pseudo-Goldstone bosons (as is the case for QCD axions) and whose quantum nature manifests itself at cosmological scales through its imprint on the structure formation at small scales. Therefore, we shall refer to this model as SFDM hereafter.</text>
<text><location><page_1><loc_52><loc_37><loc_92><loc_57></location>In order to solve the cosmological dynamics of the linear perturbations of SFDM, CAMB have been modified to include a real scalar field as a DM component. This version has been called AxionCAMB [17], and it is written for a scalar field endowed with a quadratic potential V ( ϕ ) ∝ ϕ 2 . This is why this case is called the free case . On the other hand, CLASS has been modified by the authors of this work with the aim of including the SFDM model with quadratic potential: class.FreeSF [16]. Both codes deal with the background dynamics and linear perturbations of the SFDM, and it is possible to obtain the anisotropies of the Cosmic Microwave Background radiation (CMB), as well as the Matter Power Spectrum (MPS) for several mass values of the SFDM particle 1 .</text>
<text><location><page_1><loc_52><loc_21><loc_92><loc_37></location>Whereas the only SFDM model that has been treated with AxionCAMB is that of the noninteracting case, the authors made new amendments to CLASS to study potentials with self-interaction in the scalar field potential. For example, the full axion-like potential V ( ϕ ) ∝ cos( ϕ ) was fully implemented for the first time in CLASS in the work [25], where it was shown that this model presents an excess of power on small scales in the MPS with respect to CDM 2 , with a greater discussion of the cosmological signatures of this SFDM model in [27]. The other SFDM potential we have considered in CLASS is that of a</text>
<text><location><page_2><loc_9><loc_90><loc_49><loc_93></location>hyperbolic function V ( ϕ ) ∝ cosh( ϕ ) [28], where the selfinteracting term has the opposite sign to that of axions.</text>
<text><location><page_2><loc_9><loc_80><loc_49><loc_90></location>As we will show later, the mathematical treatment of the SFDM cosmological evolution used in the modified versions of CLASS makes use of new dynamical variables based on a dynamical system analysis. We have shown that this is useful for a unified description of different scalar potentials, showing our own method to deal with tree different cases in a one-parametric way [29].</text>
<text><location><page_2><loc_9><loc_69><loc_49><loc_80></location>The modified versions of CLASS mentioned above for SFDM deal with the linear evolution of density perturbations. This information (initial conditions at radiationdominated era, physical effects on observables such as CMB, MPS, Halo Mass Function and others) turns out to be crucial for the subsequent realization of realistic numerical simulations of structure formation in the nonlinear regime, see for example [30-34].</text>
<text><location><page_2><loc_9><loc_59><loc_49><loc_68></location>Our main goal in the present work is to give more details of the mathematical approach we have used in previously amended versions of CLASS , and in turn to show its robustness in describing the physical processes of SFDM up to the level of linear density perturbations. This is done here for SFDM endowed with a quadratic potential,</text>
<formula><location><page_2><loc_23><loc_54><loc_49><loc_57></location>V ( ϕ ) = 1 2 m 2 ϕ ϕ 2 , (1)</formula>
<text><location><page_2><loc_9><loc_43><loc_49><loc_53></location>where m ϕ is the mass of the scalar field particle and the only free parameter in the model. Note that we are using natural units with h = c = 1, and then m ϕ is given in units of eV. Whenever appropriate, we will compare our method to the common fluid approximation to the SFDM dynamics used in other works, following the description in Refs. [17, 35, 36]</text>
<text><location><page_2><loc_9><loc_32><loc_49><loc_43></location>This paper is organized as follows. In Sec. II, we develop the mathematical formulation for the evolution of the background and linear density perturbations, in terms of new dynamical variables that are appropriate to handle the particularities of SFDM. We also establish the appropriate initial conditions for the scalar field to behave as the DM component at late times, for both background and linear quantities.</text>
<text><location><page_2><loc_9><loc_10><loc_49><loc_32></location>Section III is dedicated to the description of the typical regime of rapid oscillations of SFDM at late times in its evolution and to the way in which we deal with them for their reliable numerical computation. In particular, we show a detailed study of the evolution of the (barotropic) equation of state, the energy density, and finally the linear density perturbations. As an application of our method in the Boltzmann code CLASS , we present the temperature and matter power spectra and discuss the reliability of our numerical results. Moreover, we also give simple examples in which we can put constraints on the field mass m ϕ using recent likelihoods incorporated in the Monte Carlo Markov Chain (MCMC) sampler MontePython [37, 38], taking advantage of its close interoperability with CLASS .</text>
<text><location><page_2><loc_10><loc_9><loc_49><loc_10></location>The comparison and equivalence between our approach</text>
<text><location><page_2><loc_52><loc_83><loc_92><loc_93></location>and the original formulation in terms of the scalar field itself is presented in Sec. IV. We explicitly show the transformation between our variables and the original field ones ( ϕ, ˙ ϕ ), and that the numerical results of our method are completely equivalent to the standard field approach. Finally, in Sec. V we summarize and discuss our results, highlighting the advantages of our method.</text>
<section_header_level_1><location><page_2><loc_56><loc_79><loc_88><loc_80></location>II. MATHEMATICAL BACKGROUND</section_header_level_1>
<text><location><page_2><loc_52><loc_68><loc_92><loc_77></location>In this section, we present the equations of motion for the SFDM model in the context of an expanding universe, and the subsequent transformations we use to make them more suitable for numerical computations. The original motivations and some extra details of the method described here can be found in [16, 25, 39].</text>
<section_header_level_1><location><page_2><loc_62><loc_64><loc_82><loc_65></location>A. Background evolution</section_header_level_1>
<text><location><page_2><loc_53><loc_60><loc_90><loc_62></location>Let us consider a spatially-flat FRWL line element,</text>
<formula><location><page_2><loc_54><loc_57><loc_92><loc_59></location>ds 2 = -dt 2 + a 2 ( t ) [ dr 2 + r 2 ( dθ 2 +sin 2 θdϕ 2 )] , (2)</formula>
<text><location><page_2><loc_52><loc_50><loc_92><loc_56></location>where a ( t ) is the scale factor. The background equations for ordinary matter, which is represented by perfect fluids with density ρ j and pressure p j , as well as for SFDM field ϕ endowed with the potential (1), are given by</text>
<formula><location><page_2><loc_57><loc_45><loc_92><loc_49></location>H 2 = κ 2 3 ∑ j ρ j + ρ ϕ , (3a)</formula>
<formula><location><page_2><loc_58><loc_40><loc_92><loc_44></location>˙ H = -κ 2 2 ∑ j ( ρ j + p j ) + ( ρ ϕ + p ϕ ) , (3b)</formula>
<formula><location><page_2><loc_58><loc_38><loc_92><loc_39></location>˙ ρ j = -3 H ( ρ j + p j ) , (3c)</formula>
<formula><location><page_2><loc_59><loc_35><loc_92><loc_37></location>¨ ϕ = -3 H ˙ ϕ -m 2 ϕ ϕ, (3d)</formula>
<text><location><page_2><loc_52><loc_28><loc_92><loc_34></location>where κ 2 = 8 πG . The dot denotes the derivative with respect to cosmic time t , and H = ˙ a/a is the Hubble parameter. In the equations above, the scalar field density ρ ϕ and pressure p ϕ are defined, respectively, as</text>
<formula><location><page_2><loc_56><loc_24><loc_92><loc_27></location>ρ ϕ = 1 2 ˙ ϕ 2 + 1 2 m 2 ϕ ϕ 2 , p ϕ = 1 2 ˙ ϕ 2 -1 2 m 2 ϕ ϕ 2 . (4)</formula>
<text><location><page_2><loc_52><loc_20><loc_92><loc_23></location>For the SFDM part, it is convenient to use the following change of variables [16, 25, 27, 39-41],</text>
<formula><location><page_2><loc_55><loc_12><loc_92><loc_19></location>κ ˙ ϕ √ 6 H = e β sin( θ/ 2) , -κm ϕ ϕ √ 6 H = e β cos( θ/ 2) , (5a) y 1 ≡ 2 m ϕ H . (5b)</formula>
<text><location><page_2><loc_52><loc_9><loc_92><loc_11></location>To understand the meaning of the new variables, we write here the SFDM density parameter Ω ϕ and equation</text>
<text><location><page_3><loc_9><loc_92><loc_22><loc_93></location>of state (EOS) w ϕ ,</text>
<formula><location><page_3><loc_19><loc_88><loc_49><loc_91></location>Ω ϕ = 8 πGρ ϕ 3 H 2 = e 2 β , (6a)</formula>
<formula><location><page_3><loc_19><loc_84><loc_49><loc_88></location>w ϕ = ˙ ϕ 2 -m 2 ϕ ϕ 2 ˙ ϕ 2 + m 2 ϕ ϕ 2 = -cos θ . (6b)</formula>
<text><location><page_3><loc_9><loc_74><loc_49><loc_82></location>Thus, β is the logarithm of the energy density parameter, and θ , being an internal polar angle, is directly related to w ϕ . Lastly, y 1 is simply the ratio of the field mass to the Hubble parameter (i.e. dimensionless by definition), which is an ubiquitous quantity in all methods for solutions of the SFDM equations of motion.</text>
<text><location><page_3><loc_9><loc_70><loc_49><loc_74></location>Using the new variables, the Klein-Gordon equation (3d) transforms into the following set of first order differential equations [16],</text>
<formula><location><page_3><loc_22><loc_67><loc_49><loc_68></location>θ ' = -3 sin θ + y 1 , (7a)</formula>
<formula><location><page_3><loc_22><loc_64><loc_36><loc_67></location>' 1 = 3 (1 + w tot ) y 1 ,</formula>
<formula><location><page_3><loc_21><loc_64><loc_49><loc_66></location>y 2 (7b)</formula>
<formula><location><page_3><loc_21><loc_61><loc_49><loc_64></location>β ' = 3 2 ( w tot +cos θ ) . (7c)</formula>
<text><location><page_3><loc_9><loc_50><loc_49><loc_60></location>Here, a prime denotes derivatives with respect to the number of e -folds of expansion N = ln a . The total EOS w tot , which is used in the foregoing equations, can be calculated from the ratio of the total pressure p tot to the total density ρ tot in the Universe, and its explicit expression is given in terms of the EOS of the different components of matter as</text>
<formula><location><page_3><loc_17><loc_45><loc_49><loc_48></location>w tot = p tot ρ tot = ∑ j w j ρ j + w ϕ ρ ϕ ∑ j ρ j + ρ ϕ . (8)</formula>
<text><location><page_3><loc_9><loc_35><loc_49><loc_44></location>Although the new dynamical variables are convenient for numerical purposes, we need to recover the standard quantities used in Boltzmann and cosmological codes in general, for instance, the density and pressure of the SFDM component. It can be shown that such quantities can be recovered in the form</text>
<formula><location><page_3><loc_15><loc_30><loc_49><loc_34></location>ρ ϕ = e 2 β 1 -e 2 β ∑ j ρ j , p ϕ = -cos θ ρ ϕ , (9)</formula>
<text><location><page_3><loc_9><loc_26><loc_49><loc_29></location>where the sum takes into account only the density components other than the SFDM one.</text>
<section_header_level_1><location><page_3><loc_16><loc_22><loc_41><loc_23></location>B. Linear density perturbations</section_header_level_1>
<text><location><page_3><loc_9><loc_16><loc_49><loc_20></location>When considering linear perturbations, the line element in the synchronous gauge, and the perturbed scalar field are given by</text>
<formula><location><page_3><loc_16><loc_13><loc_49><loc_14></location>ds 2 = -dt 2 + a 2 ( t )( δ ij + ¯ h ij ) dx i dx j , (10)</formula>
<formula><location><page_3><loc_20><loc_9><loc_49><loc_10></location>ϕ ( ⃗x, t ) = ϕ ( t ) + φ ( ⃗x, t ) . (11)</formula>
<text><location><page_3><loc_52><loc_88><loc_92><loc_93></location>where ¯ h ij and φ are the spatial part of the metric perturbation and scalar field fluctuation, respectively. The linearized Klein-Gordon equation for the scalar field perturbation is written (in Fourier space) as [42-45]</text>
<formula><location><page_3><loc_53><loc_83><loc_92><loc_86></location>¨ φ ( ⃗ k, t ) = -3 H ˙ φ ( ⃗ k, t ) -( m 2 ϕ + k 2 a 2 ) φ ( ⃗ k, t ) -1 2 ˙ ϕ ˙ ¯ h. (12)</formula>
<text><location><page_3><loc_52><loc_76><loc_92><loc_82></location>In a similar way to the procedure we have done for the Klein-Gordon equation (3b), we propose the following change of variables for the scalar field perturbation φ and its derivative ˙ φ [16, 25]</text>
<formula><location><page_3><loc_52><loc_70><loc_92><loc_75></location>√ 2 3 κ ˙ φ H = -e α + β cos( ϑ/ 2) , κy 1 φ √ 6 = -e α + β sin( ϑ/ 2) . (13)</formula>
<text><location><page_3><loc_52><loc_66><loc_92><loc_70></location>Field perturbations are then represented by the new variables α and ϑ . However, it is more convenient to further define two new variables in the form of</text>
<formula><location><page_3><loc_53><loc_61><loc_92><loc_64></location>δ 0 = -e α sin ( θ -ϑ 2 ) , δ 1 = -e α cos ( θ -ϑ 2 ) . (14)</formula>
<text><location><page_3><loc_52><loc_56><loc_92><loc_60></location>After some straightforward algebra, the linearized Klein-Gordon equation (12) is represented by the following set of first-order differential equations,</text>
<formula><location><page_3><loc_55><loc_48><loc_92><loc_55></location>δ ' 0 = [ -3 sin θ -k 2 k 2 J (1 -cos θ ) ] δ 1 + k 2 k 2 J sin θδ 0 -¯ h ' 2 (1 -cos θ ) , (15a)</formula>
<formula><location><page_3><loc_55><loc_41><loc_92><loc_48></location>δ ' 1 = [ -3 cos θ -k 2 k 2 J sin θ ] δ 1 + k 2 k 2 J (1 + cos θ ) δ 0 -¯ h ' 2 sin θ . (15b)</formula>
<text><location><page_3><loc_52><loc_36><loc_92><loc_40></location>In writing Eqs. (15), there appears a natural definition of a Jeans wavenumber given by k 2 J = a 2 H 2 y 1 , which .</text>
<text><location><page_3><loc_52><loc_26><loc_92><loc_37></location>also acts as a normalization factor of the wavenumber k Notice that for scales larger than the Jeans one, k 2 /k 2 J ≪ 1, the scale-dependent terms in Eqs. (15) can be neglected, whereas in the opposite case, k 2 /k 2 J ≫ 1, the evolution of the perturbations are different for each scale. Our definition of the Jeans wavenumber coincides with those given in the literature if we write it as k 2 J = 2 a 2 m ϕ H using y 1 = 2 m ϕ /H (see Eq. (5b)).</text>
<text><location><page_3><loc_52><loc_22><loc_92><loc_26></location>To understand the physical meaning of the new dynamical variables δ 0 , δ 1 , we first write their expressions in terms of the original field variables,</text>
<formula><location><page_3><loc_60><loc_17><loc_92><loc_20></location>δ 0 = m 2 ϕ κ 2 ρ ϕ [ κ ˙ φ m ϕ κ ˙ ϕ m ϕ +( κφ )( κϕ ) ] , (16a)</formula>
<formula><location><page_3><loc_60><loc_13><loc_92><loc_16></location>δ 1 = m 2 ϕ κ 2 ρ ϕ [ κ ˙ ϕ m ϕ ( κφ ) -κ ˙ φ m ϕ ( κϕ ) ] . (16b)</formula>
<text><location><page_3><loc_52><loc_9><loc_92><loc_11></location>Now, we recall the expressions for the scalar field density contrast δ ϕ and the velocity divergence Θ ϕ in the</text>
<text><location><page_4><loc_9><loc_92><loc_13><loc_93></location>form,</text>
<formula><location><page_4><loc_17><loc_88><loc_49><loc_91></location>δ ϕ = ˙ ϕ ˙ φ + m ϕ ϕφ ρ ϕ = δ 0 , (17a)</formula>
<formula><location><page_4><loc_9><loc_82><loc_49><loc_87></location>( ρ ϕ + p ϕ )Θ ϕ = k 2 a ˙ ϕφ = k 2 ρ ϕ aHy 1 [(1 -cos θ ) δ 1 -sin θδ 0 ] . (17b)</formula>
<text><location><page_4><loc_9><loc_74><loc_49><loc_81></location>Hence, the variable δ 0 is the scalar field density contrast, whereas the velocity divergence is based on a combination of the two variables δ 0 and δ 1 . See Appendix A for an extended discussion of the equivalence between our approach and the fluid approximation.</text>
<section_header_level_1><location><page_4><loc_21><loc_70><loc_37><loc_71></location>C. Initial conditions</section_header_level_1>
<text><location><page_4><loc_9><loc_61><loc_49><loc_68></location>Calculation of initial conditions implies an approximate solution of the equations of motion starting well within the epoch of radiation domination, with a corresponding initial value of the scale factor of the order of a i = 10 -14 [16].</text>
<text><location><page_4><loc_9><loc_58><loc_49><loc_61></location>We start with the initial condition of the auxiliary variable, which reads</text>
<formula><location><page_4><loc_10><loc_54><loc_49><loc_57></location>y 1 i = 2 m ϕ H 0 H 0 H i = 1 . 85 × 10 11 a 2 i √ Ω r 0 h 2 ( m ϕ 10 -22 eV ) , (18)</formula>
<text><location><page_4><loc_9><loc_40><loc_49><loc_52></location>where H 0 ( H i ) is the present (initial) value of the Hubble parameter, and h is its reduced value. Note that for the calculation of H i we assume radiation domination at early times. It is clear from Eq. (18) that the auxiliary variable is very small at early times, y 1 i ∼ 10 -14 , for the values of m ϕ that are of interest for SFDM models. On the contrary, its value at present is very large y 1 ∼ 10 10 , which means that it changes by almost 24 orders of magnitude during its evolution.</text>
<text><location><page_4><loc_9><loc_34><loc_49><loc_39></location>As for the polar angle, there is an atractor solution when the equations of motion are solved in the linear regime at early times, from which we obtain the following equations.</text>
<formula><location><page_4><loc_25><loc_30><loc_49><loc_33></location>θ i = 1 5 y 1 i . (19)</formula>
<text><location><page_4><loc_9><loc_23><loc_49><loc_29></location>The initial condition of the variable β is found by matching the early and late time solutions at the beginning of the rapid oscillations of the field. The resultant equation is</text>
<formula><location><page_4><loc_12><loc_18><loc_49><loc_22></location>e 2 β i = A × a i Ω ϕ 0 Ω r 0 [ 4 θ 2 i π 2 ( 1 + π 2 / 36 1 + θ 2 i / 9 )] 3 / 4 . (20)</formula>
<text><location><page_4><loc_9><loc_11><loc_49><loc_17></location>Here, A is a constant coefficient that is adjusted by the numerical code, typically with a shooting mechanism, to match the value of the desired density parameter Ω ϕ 0 at the present time.</text>
<text><location><page_4><loc_9><loc_9><loc_49><loc_11></location>On the other hand, the initial conditions of the density perturbations is a more involved procedure, but it reveals</text>
<text><location><page_4><loc_52><loc_90><loc_92><loc_93></location>the existence of an attractor solution for the dynamical variables in the form</text>
<formula><location><page_4><loc_61><loc_87><loc_92><loc_90></location>δ 0 i = 2 7 ¯ h sin( θ i / 2) sin( θ i / 12) , (21a)</formula>
<formula><location><page_4><loc_61><loc_84><loc_92><loc_87></location>δ 1 i = 2 7 ¯ h sin( θ i / 2) cos( θ i / 12) . (21b)</formula>
<text><location><page_4><loc_52><loc_76><loc_92><loc_83></location>The details of the numerical implementation of the polar method of the sections above in the amended version of the Boltzmann code CLASS are presented in Appendix C, from which we obtained the numerical solutions that are presented in the sections below.</text>
<section_header_level_1><location><page_4><loc_57><loc_70><loc_87><loc_73></location>III. THE STAGE OF RAPID FIELD OSCILLATIONS</section_header_level_1>
<text><location><page_4><loc_52><loc_61><loc_92><loc_68></location>For the field to behave as a CDM component, it should enter a phase of rapid oscillations around the minimum of the potential. Under our polar transformation (4), such fast oscillations are equivalent to the following averages during a Hubble time, ⟨ sin θ ⟩ = 0 and ⟨ cos θ ⟩ = 0.</text>
<text><location><page_4><loc_52><loc_54><loc_92><loc_61></location>However, we must be careful in the form the oscillations are dealt with, as the solutions at late times depend on the choices made for the averaged dynamical quantities. Here, we explain in detail our method for the cutoff of the rapid oscillations proposed in [16, 25, 27].</text>
<section_header_level_1><location><page_4><loc_59><loc_50><loc_85><loc_51></location>A. Outline of the general method</section_header_level_1>
<text><location><page_4><loc_52><loc_41><loc_92><loc_48></location>It is well known that the stage of rapid oscillations is difficult to solve numerically, and then we follow here the prescription in Ref. [16] in that the cosine and sine functions in the equations of motion are replaced by the cutoff trigonometric functions.</text>
<formula><location><page_4><loc_59><loc_38><loc_84><loc_40></location>cos ⋆ θ = 1 [1 -tanh( θ -θ ⋆ )] cos θ ,</formula>
<formula><location><page_4><loc_60><loc_35><loc_84><loc_37></location>sin ⋆ θ = 1 [1 -tanh( θ -θ ⋆ )] sin θ ,</formula>
<formula><location><page_4><loc_66><loc_34><loc_92><loc_39></location>2 (22a) 2 (22b)</formula>
<text><location><page_4><loc_52><loc_29><loc_92><loc_33></location>where θ ⋆ is a reference value. In this form, cos ⋆ θ = cos θ (sin ⋆ θ = sin θ ) if 0 ≤ θ < θ ⋆ , while cos ⋆ θ → 0 (sin ⋆ θ → 0) if θ ≫ θ ⋆ .</text>
<text><location><page_4><loc_52><loc_19><loc_92><loc_29></location>In the following, we will refer to t ⋆ as the time at which we apply the cutoff for the trigonometric functions and then to θ ( t ⋆ ) = θ ⋆ . We will also refer to t osc as the time for the beginning of the rapid oscillations. However, and in contrast to t ⋆ , the value of t osc cannot be precisely determined, and in our formalism it is just a reference value without a major effect on the numerical solutions.</text>
<text><location><page_4><loc_52><loc_9><loc_92><loc_19></location>The general method can be described as follows. We replace all the sine and cosine terms with the cutoff functions (22) in the equations of motion (7), and then solve them numerically. Note that the solutions are then continuous at t = t ⋆ by construction, and we only need to be sure that the cutoff is applied after the onset of the rapid oscillations so that t ⋆ > t osc .</text>
<text><location><page_5><loc_9><loc_79><loc_49><loc_93></location>However, it is difficult to determine the time at the start of the oscillations, and this also makes impractical the calculation of t ⋆ . The reason is that cosmic time is a dimensional quantity calculated from the integration of the Friedmann equation (3a), which depends on all dynamical variables in a cosmological model 3 . As we shall show in the following, it is better to calibrate the cutoff time using the polar variable θ , which is dimensionless and also a direct dynamical variable in our set of field equations.</text>
<section_header_level_1><location><page_5><loc_16><loc_75><loc_42><loc_76></location>B. The case of the polar angle θ</section_header_level_1>
<text><location><page_5><loc_9><loc_66><loc_49><loc_73></location>To understand the general behavior of the solutions after the cutoff of the trigonometric functions, we start with the equation of motion (7a) of the polar angle θ , which for convenience we write in terms of cosmic time t ,</text>
<formula><location><page_5><loc_21><loc_62><loc_49><loc_65></location>˙ θ = -3 2 t sin θ +2 m ϕ , (23)</formula>
<text><location><page_5><loc_9><loc_58><loc_49><loc_61></location>where we have considered H = 1 / (2 t ) for RD. Notice that we can write Eq. (23) in the form</text>
<formula><location><page_5><loc_19><loc_54><loc_49><loc_57></location>dθ d (2 m ϕ t ) = -3 2 sin θ (2 m ϕ t ) +1 , (24)</formula>
<text><location><page_5><loc_9><loc_46><loc_49><loc_53></location>which shows that the evolution of θ as a function of the dimensionless variable 2 m ϕ t is the same, regardless of the value of the field mass m ϕ . We will use this feature in the plots below, but we still refer to Eq. (23) to obtain semi-analytical expressions.</text>
<text><location><page_5><loc_9><loc_43><loc_49><loc_46></location>First, we assume at the beginning that 0 < θ ≪ 1, and then sin θ ≃ θ . As a consequence, Eq. (23) becomes</text>
<formula><location><page_5><loc_23><loc_39><loc_49><loc_42></location>˙ θ = -3 2 t θ +2 m ϕ , (25a)</formula>
<text><location><page_5><loc_9><loc_36><loc_49><loc_38></location>and the solution that satisfies the initial condition θ (0) = 0 is</text>
<formula><location><page_5><loc_21><loc_32><loc_49><loc_35></location>θ ( t ) = 4 5 m ϕ t . (Early) (25b)</formula>
<text><location><page_5><loc_9><loc_27><loc_49><loc_31></location>We now use Eq. (25b) on the right-hand side of Eq. (23), from which we obtain the new differential equation,</text>
<formula><location><page_5><loc_18><loc_23><loc_49><loc_26></location>˙ θ = -3 2 t sin ⋆ (4 m ϕ t/ 5) + 2 m ϕ , (26a)</formula>
<text><location><page_5><loc_9><loc_21><loc_21><loc_22></location>whose solution is</text>
<formula><location><page_5><loc_19><loc_17><loc_49><loc_20></location>θ ( t ) = 2 m ϕ t -3 2 Si(4 m ϕ t/ 5) , (26b)</formula>
<text><location><page_5><loc_52><loc_86><loc_92><loc_93></location>where Si( x ) is the sine integral. It can be shown that, at early times m ϕ t ≪ 1, we recover the solution (25b). Likewise, for the late-time evolution we can approximate the sine integral by its asymptotic behavior, Si( x ) ≃ π/ 2 -cos( x ) /x + O (1 /x 2 ) for x ≫ 1, to obtain</text>
<formula><location><page_5><loc_62><loc_82><loc_92><loc_85></location>θ ( t ) = 2 m ϕ t -3 π 4 . (Late) (26c)</formula>
<text><location><page_5><loc_52><loc_75><loc_92><loc_80></location>Any further iteration to integrate Eq. (23) cannot be expressed in closed form, but, as we present below, Eq. (26b) suffices to analyze the main properties in the time evolution of the polar angle θ .</text>
<text><location><page_5><loc_52><loc_60><loc_92><loc_75></location>The numerical solutions of Eq. (23) for different values of the field mass m ϕ , and as a function of the scale factor a , are shown in the top panel of Fig. 1. It can be seen that the polar variable shows two asymptotic behaviors that correspond to the semi-analytical solutions: at early times θ/ (2 m ϕ t ) → 5 / 4, while at late times θ/ (2 m ϕ t ) → 1, as indicated by Eqs. (25b) and (26c), respectively. These asymptotic limits are the same for any field mass m ϕ , and the latter only influences the time at which the transition occurs between the two values.</text>
<figure>
<location><page_5><loc_52><loc_40><loc_92><loc_58></location>
</figure>
<figure>
<location><page_5><loc_52><loc_19><loc_92><loc_38></location>
<caption>FIG. 1. Numerical solutions of the polar variable θ as a function of the dimensionless variable 2 m ϕ t . The cases correspond to different values of the field mass log( m ϕ / eV) = -24 , -23 , -22 , -21. Also shown are the semi-analytical solutions (26b) (Iterative, purple), (25b) (Early, dashed brown), and (26c) (late, dashed pink). See the text for more details.</caption>
</figure>
<text><location><page_6><loc_9><loc_85><loc_49><loc_93></location>In the bottom panel of Fig. 1 we have the evolution of the polar variable but now in terms of dimensionless cosmic time 2 m ϕ t . Although the numerical solutions are shown in different colors, it is clear that the corresponding curves are superimposed on each other because their behavior is the same.</text>
<text><location><page_6><loc_9><loc_67><loc_49><loc_84></location>It can be seen that the semi-analytical solutions agree well with the numerical ones. In particular, the iterative solution (26b) gives a reliable description of the early and late time trends of the solutions, and it even gives a good approximation to the oscillations of the numerical solutions at intermediate times 2 m ϕ t ≃ 4, which is also the time at which θ ≃ π/ 2. That is, it also corresponds to the time at which the scalar field EOS first crosses the zero value w ϕ ≃ 0. As this occurs within radiation domination, we also find 2 m ϕ t = m ϕ /H ≃ 4, which is the typical time for the start of the oscillations estimated for these field systems.</text>
<text><location><page_6><loc_9><loc_60><loc_49><loc_67></location>Surprisingly, the bottom panel of Fig. 1 also shows that the late-time expression (26c) also seems to work very well from the intermediate times onward, that is, almost from the start of the rapid oscillations. This means that we can safely write</text>
<formula><location><page_6><loc_20><loc_56><loc_49><loc_59></location>θ ( t > t osc ) = 2 m ϕ t -3 π 4 . (27a)</formula>
<text><location><page_6><loc_9><loc_44><loc_49><loc_54></location>That Eq. (27a) is also a very good approximation can be understood from the properties of the sine integral Si( x ), which rapidly converges to its asymptotic value of π/ 2, with small oscillations around it that rapidly decay away. In what follows, we will use Eq. (27a) to describe the behavior of the polar angle after the onset of rapid oscillations of the field ϕ .</text>
<text><location><page_6><loc_9><loc_37><loc_49><loc_44></location>We can also use Eq. (27a) also to convert the cutoff time t ⋆ into a cutoff polar angle θ ⋆ , which is both a dynamical variable and the argument in the modified trigonometric functions (22). Hence, the relation between the cutoff values t ⋆ and θ ⋆ is</text>
<formula><location><page_6><loc_22><loc_33><loc_49><loc_36></location>2 m ϕ t ⋆ = θ ⋆ + 3 π 4 . (27b)</formula>
<text><location><page_6><loc_9><loc_22><loc_49><loc_32></location>Equations (27) are a central result in the description of our method. First, Eq. (27a) shows that the polar angle evolves linearly with cosmic time t after the cutoff time. Second, Eq. (27b) allows us to determine the cutoff point of rapid oscillations via the polar angle θ ⋆ , which is more convenient from the numerical point of view and justifies the use of the cutoff expressions (22).</text>
<text><location><page_6><loc_9><loc_9><loc_49><loc_21></location>To finish this section, in Fig. 2 we show the numerical evolution of the scalar field EOS w ϕ as a function again of the dimensionless variable 2 m ϕ t , and we see that it first passes through zero (for θ = π/ 2) at around the time 2 m ϕ t osc ≃ 3 . 47, which we use to mark the time t osc for the start of the rapid oscillations. Notice that in terms of the usual mass-to-Hubble ratio, this is equivalent to m ϕ /H osc ≃ 3 . 47, a value used as a reference in other studies of field models.</text>
<text><location><page_6><loc_52><loc_84><loc_54><loc_85></location>w</text>
<figure>
<location><page_6><loc_52><loc_75><loc_92><loc_93></location>
<caption>FIG. 2. The numerical solution of the scalar field EOS w ϕ as a function of the variable 2 m ϕ t , under which all the different cases collapse into a single curve (blue curve). The dashed vertical lines mark the beginning of the rapid field oscillations at t osc . The curve of the semi-analytical approximation (28) (orange curve), which is well matched to the numerical solutions at t > t osc , is also plotted. See the text for more details.</caption>
</figure>
<text><location><page_6><loc_52><loc_52><loc_92><loc_60></location>For comparison, we also plot in Fig. 2 the result of the expression cos(2 m ϕ t -3 π/ 4). Notice that there is very good agreement of this curve with the original EOS w ϕ almost from the start of the rapid oscillations, which means that we can use the following expression for the field EOS,</text>
<formula><location><page_6><loc_59><loc_49><loc_92><loc_50></location>w ϕ ( t > t osc ) = -cos (2 m ϕ t -3 π/ 4) . (28)</formula>
<text><location><page_6><loc_52><loc_37><loc_92><loc_47></location>Equation (28) agrees with the common wisdom that the EOS oscillates with a frequency directly related to the field mass via 2 m ϕ . The phase of 3 π/ 4 becomes negligible at very late times, but as we shall see, it must be taken into account for a correct description of the dynamics at intermediate times of other variables after the onset of the rapid oscillations.</text>
<section_header_level_1><location><page_6><loc_57><loc_32><loc_86><loc_33></location>C. The case of the energy density ρ ϕ</section_header_level_1>
<text><location><page_6><loc_52><loc_19><loc_92><loc_30></location>We have found semi-analytical results to follow the evolution of the polar variable θ ( t ), which are in good agreement with the numerical results. However, one should worry about the numerical accuracy as the scalar field equations of motion must be solved together with other matter components in Boltzmann codes, covering an ample time interval for a complete description of diverse cosmological phenomena.</text>
<text><location><page_6><loc_52><loc_9><loc_92><loc_19></location>Here, we perform some accuracy tests using the amended version of the Boltzmann code CLASS , taking some guidelines from our semi-analytical results. Our main concern is the choice of the cutoff value θ ⋆ . As we shall see, the cutoff procedure leaves a residual difference with respect to the expected late-time evolution of a given variable that can be minimized if θ ⋆ ≫ 1.</text>
<text><location><page_7><loc_9><loc_89><loc_49><loc_93></location>An example of the effects of the cutoff on the evolution of different quantities is the energy density ρ ϕ , which obeys the equation</text>
<formula><location><page_7><loc_12><loc_86><loc_49><loc_88></location>˙ ρ ϕ = -3 H (1 + w ϕ ) ρ ϕ = -3 H (1 -cos θ ) ρ ϕ , (29)</formula>
<text><location><page_7><loc_9><loc_82><loc_49><loc_85></location>whose formal solution after the onset of the rapid oscillations can be written as</text>
<formula><location><page_7><loc_13><loc_79><loc_49><loc_81></location>ρ ϕ a 3 = ρ ϕ,osc a 3 osc exp[ F ( t ) -F ( t osc )] , (30a)</formula>
<text><location><page_7><loc_9><loc_77><loc_12><loc_78></location>with</text>
<formula><location><page_7><loc_12><loc_72><loc_49><loc_76></location>F ( t ) -F ( t osc ) = 3 ∫ t t osc H ( x ) cos( θ ( x )) dx. (30b)</formula>
<text><location><page_7><loc_9><loc_63><loc_49><loc_71></location>It suffices to understand the behavior of the density before the time of radiation-matter equality, and for that we proceed as follows. Equation (30b) can be written in a more convenient form if we use Eq. (27a) for the evolution of the polar angle, and then it can be shown that</text>
<formula><location><page_7><loc_14><loc_59><loc_49><loc_63></location>F ( t ) = 3 √ 2 4 [Si(2 m ϕ t ) -Ci(2 m ϕ t )] , (31a)</formula>
<text><location><page_7><loc_9><loc_55><loc_49><loc_58></location>where Si( x ) and Ci( x ) are the sine and cosine integrals, respectively.</text>
<text><location><page_7><loc_9><loc_48><loc_49><loc_55></location>We are interested in the evolution of the density at late times, that is, 2 m ϕ t ≫ 1. Given the asymptotic properties of the sine and cosine integrals for x ≫ 1, Si( x ) ≃ π/ 2 -cos( x ) /x and Ci( x ) ≃ sin( x ) /x , and substituting the polar angle θ using Eq. (27a), we find that</text>
<formula><location><page_7><loc_14><loc_43><loc_49><loc_47></location>F ( t ≫ t osc ) ≃ 3 √ 2 π 8 + 3 sin θ 2( θ +3 π/ 4) . (31b)</formula>
<text><location><page_7><loc_9><loc_38><loc_49><loc_42></location>The last term in Eq. (31b) will be responsible for a residual oscillatory term in the density, which will decay away. In fact, if we define</text>
<formula><location><page_7><loc_13><loc_33><loc_49><loc_37></location>ρ ϕ 0 ≡ ρ ϕ,osc a 3 osc exp [ 3 √ 2 π 8 -F ( t osc ) ] , (32)</formula>
<text><location><page_7><loc_9><loc_30><loc_45><loc_31></location>we can also write Eq. (30a) in a more neat form as</text>
<formula><location><page_7><loc_11><loc_26><loc_49><loc_29></location>ρ ϕ ( t ≫ t osc ) = ( ρ ϕ 0 /a 3 ) exp ( 3 sin θ 2( θ +3 π/ 4) ) , (33)</formula>
<text><location><page_7><loc_9><loc_14><loc_49><loc_24></location>which is correct for θ ≫ 1. There are two parts in the rhs of Eq. (33): one that evolves steadily at the same rate as a pressureless component ( ∼ a -3 ), and another one that contributes with a decaying oscillating term around unity provided by the exponential function. Moreover, ρ ϕ 0 represents the correct asymptotic value of the field density at late times.</text>
<text><location><page_7><loc_9><loc_9><loc_49><loc_14></location>The following question arises: can we be assured that our cutoff procedure of the rapid oscillations recovers the right evolution of the density at late times? First, notice that, in principle, ρ ϕ 0 in Eq. (32) is fixed at the onset</text>
<text><location><page_7><loc_52><loc_80><loc_92><loc_93></location>of the rapid oscillations, but we do not need to be very specific about the values of the parameters at this time. In our method, unlike others in the literature, we do not require knowing the precise value of t osc , and we can be completely oblivious to it as long as we ensure t ⋆ > t osc . The reason is simple: the cutoff of the rapid oscillations is made smoothly at the level of the equations of motion, and then there is no loss of continuity in the numerical variables.</text>
<text><location><page_7><loc_52><loc_76><loc_92><loc_80></location>To answer our question above, Eq. (33) should be compared with the truncated case. After the cutoff, the equation of motion for the density is</text>
<formula><location><page_7><loc_67><loc_74><loc_92><loc_75></location>˙ ρ ϕ = -3 Hρ ϕ , (34a)</formula>
<text><location><page_7><loc_52><loc_68><loc_92><loc_73></location>whose solution simply is ρ ( t > t ⋆ ) = ρ ⋆ /a 3 , where ρ ⋆ is the density value at t = t ⋆ . By the continuity of the solutions at t ⋆ for Eqs. (29) and (34a), we finally get</text>
<formula><location><page_7><loc_55><loc_64><loc_92><loc_67></location>ρ ( t > t ⋆ ) = ( ρ ϕ 0 /a 3 ) exp ( 3 sin θ ⋆ 2( θ ⋆ +3 π/ 4) ) . (34b)</formula>
<text><location><page_7><loc_52><loc_56><loc_92><loc_63></location>The result is quite direct: the cutoff introduces a small mismatch, and the correct asymptotic value of the density is not recovered from the solution (34b). But the discrepancy depends on the cutoff value θ ⋆ , and, in principle, it can be made as small as required if θ ⋆ ≫ 1.</text>
<text><location><page_7><loc_52><loc_47><loc_92><loc_56></location>However, Eq. (34b) itself suggests a faster route, which is to choose θ ⋆ = nπ , where n is an integer number, although a large enough one so that still θ ⋆ ≫ 1, as this also allows us to neglect other oscillatory terms in the sine and cosine integrals in Eq. (31a) that are of order O (1 /θ 2 ⋆ ) and smaller.</text>
<text><location><page_7><loc_52><loc_35><loc_92><loc_47></location>Numerical examples of the evolution of the field density, in the combination ( ρ ϕ /ρ ϕ 0 ) a 3 , are shown in Fig. 3, for different values of the field mass m ϕ but with a fixed value θ ⋆ = 30 π . Here, ρ 0 is the last value in each of the numerical solutions. Note that the asymptotic value is always unity. In the left panel, we see that the density makes a transition to a pressureless behavior once the rapid oscillations start, but as before, the transition time depends on the field mass m ϕ .</text>
<text><location><page_7><loc_52><loc_24><loc_92><loc_34></location>If the density is plotted as a function of the variable θ , as in the right panel of Fig. 3, we find that all curves collapse again into a single one, and there is a common evolution for all cases. Moreover, we also show the curve arising from Eq. (33) (denoted by exp[ B ( θ )]), and it can be seen that it quite well matches the numerical curves after the onset of the rapid oscillations.</text>
<text><location><page_7><loc_52><loc_11><loc_92><loc_24></location>Now, in Fig. 4 we show the effects arising from different choices of the cutoff value θ ⋆ and with a fixed mass m ϕ = 10 -24 eV. In the left panel, we take θ ⋆ = 10 π, 20 π, 30 π , and we see that the late-time evolution is the same for all cases (the curves are superimposed on each other) even though the cutoff of the oscillations appears at different times. Also, the correct asymptotic value of the density is recovered, and in all cases it corresponds to the expected average of the density oscillations.</text>
<text><location><page_7><loc_52><loc_9><loc_92><loc_11></location>For the right panel in Fig. 4 we take the cutoff values θ ⋆ = 10 π, 10 . 5 π . To have a good matching of the first</text>
<figure>
<location><page_8><loc_10><loc_74><loc_49><loc_92></location>
</figure>
<figure>
<location><page_8><loc_51><loc_74><loc_91><loc_93></location>
<caption>FIG. 3. Numerical solutions of the scalar field energy density ρ ϕ , in a combination that highlights the asymptotic, nonoscillatory value at late times; the curves represent the same cases as in Fig. 1. (Left) The energy density, as a function of the scale factor a , behaves as a pressureless component from the onset of rapid oscillations, which occur at times that depend on the field mass m ϕ , but the late-time behavior is the same: decaying oscillations around a fixed value. The cutoff value was set at θ ⋆ = 30 π for the four cases. (Right) If plotted as a function of the polar angle θ , all curves collapse into a single one. Also shown is the semi-analytical formula (33), with B ( θ ) ≡ 3 sin θ/ ( θ +3 π/ 4) (black curve), which is in good agreement with the numerical solutions from the start of the rapid oscillations at θ osc = π/ 2 (black dashed line). The cutoff value θ ⋆ = 30 π (red dashed line) is also shown as a reference. See the text for more details.</caption>
</figure>
<figure>
<location><page_8><loc_10><loc_39><loc_49><loc_59></location>
</figure>
<figure>
<location><page_8><loc_51><loc_39><loc_91><loc_59></location>
<caption>FIG. 4. The behavior of the normalized background density for different choices of the cutoff angle θ ⋆ . The density at late times matches the average value well if θ ⋆ is an integer multiple of π (left panel), while in any other cases there is a noticeable mismatch with respect to the average value (right panel). The dashed vertical lines mark the corresponding cutoff values θ ∗ of the numerical solutions. The field mass was fixed at m ϕ = 10 -24 eV in the numerical examples. See the text for more details.</caption>
</figure>
<text><location><page_8><loc_9><loc_20><loc_49><loc_29></location>density oscillations in the two cases and to facilitate the comparison of their asymptotic values, we applied the correction (exponential) factor that appears in Eq. (33) to the case θ ⋆ = 10 . 5 π . 4 It can be seen that the cutoff occurs at the maximum of the last oscillation, and hence the asymptotic value is larger than the correct one.</text>
<text><location><page_8><loc_52><loc_26><loc_92><loc_29></location>We can give an estimate of the error between the two asymptotic values, which according to Eq. (34b) is</text>
<formula><location><page_8><loc_60><loc_21><loc_92><loc_24></location>∆ ρ ϕ 0 ρ ϕ 0 = exp ( 3 sin θ ⋆ 2( θ ⋆ +3 π/ 4) ) -1 . (35a)</formula>
<text><location><page_8><loc_52><loc_19><loc_85><loc_20></location>For the particular case with θ ⋆ = 10 . 5 π we get</text>
<formula><location><page_8><loc_55><loc_14><loc_92><loc_17></location>100 × [ exp ( 3 sin(10 . 5 π ) 22 . 5 π ) -1 ] = 4 . 3% . (35b)</formula>
<text><location><page_8><loc_52><loc_9><loc_92><loc_13></location>This difference is not negligible if one desires high precision of the solution, and it clearly illustrates the necessity to choose the cutoff value θ ⋆ wisely.</text>
<section_header_level_1><location><page_9><loc_16><loc_92><loc_42><loc_93></location>D. Linear density perturbations</section_header_level_1>
<text><location><page_9><loc_9><loc_78><loc_49><loc_90></location>In contrast to the background quantities in the sections above, the analysis of Eqs. (15) is much more involved because the evolution of the quantities δ 0 and δ 1 is coupled to that of the so-called metric continuity ¯ h ' / 2 through the perturbed Einstein equations. It is not possible to make a clear separation of the oscillatory and non-oscillatory terms in the formal solution, and a wise decision on the cutoff value θ ⋆ cannot easily be decided.</text>
<text><location><page_9><loc_9><loc_62><loc_49><loc_78></location>In Fig. 5 we show the evolution of the density contrast δ 0 for a scale much larger than the Jeans wavenumber, so that k 2 ≪ k 2 J . In the upper panels, we see the cases with the same cutoff values θ ⋆ used previously in Fig. 4. We can see that the numerical solutions have a larger stage of rapid oscillations for larger values of θ ⋆ , as in the background case. Also, the choice θ ⋆ = nπ , see Eq. (34b) and the text below, does not make the numerical solution coincide with the nonoscillatory solutions at t ⋆ . There is a small, but visible, mismatch between the solutions for the different values of θ ⋆ considered in the graphs.</text>
<text><location><page_9><loc_9><loc_48><loc_49><loc_62></location>Nevertheless, the numerical solution is able to catch up with the non-oscillatory solution after the cutoff of the rapid oscillations, and this is because of the structure of the system (15): its solution is driven by the nonhomogeneous term involving the metric continuity ¯ h ' , which acts as an attractor solution even at early times. At late times, the value of | δ 0 | oscillates around ¯ h/ 2 with an amplitude that does not decay. It is only on average that the scalar field density contrast can be identified with the CDM one, ⟨ δ 0 ⟩ = δ CDM .</text>
<text><location><page_9><loc_9><loc_33><loc_49><loc_48></location>Other intrinsic oscillations, which we refer to as scale oscillations, are noticeable for scales smaller than the Jeans scale: k 2 /k 2 J ≳ 1, which appear even after the cutoff of the rapid oscillations. This is because the term k 2 /k 2 J plays the role of a frequency in terms of the number of e -folds N in Eqs. (15) and not in cosmic time t . For such small scales, from the very beginning there may be a combination of rapid oscillations with scale oscillations, and the choices of θ ⋆ at the transition time give different results for δ 0 ( t ⋆ ) and δ 1 ( t ⋆ ).</text>
<text><location><page_9><loc_9><loc_19><loc_49><loc_33></location>Our numerical results for the density contrast δ 0 for small scales are shown in the right panels of Fig. 5, where we see noticeable differences in the late-time behavior of the solutions. It is clear that it is necessary to follow the numerical solutions for longer before cutting off the rapid oscillations and to achieve some convergence of the solutions. However, the evolution of metric continuity ¯ h/ 2 is always smooth and the same regardless of the cutoff value θ ⋆ , although its amplitude is also highly suppressed with respect to the CDM case.</text>
<text><location><page_9><loc_9><loc_12><loc_49><loc_19></location>We also present the cases θ ⋆ = 10 π, 10 . 5 π in Fig. 6. Both cases show that at the cutoff time none of the numerical solutions agree with the CDM solution, but they join it quickly because of the driving term ¯ h ' / 2 in Eqs. (14).</text>
<text><location><page_9><loc_9><loc_9><loc_49><loc_11></location>In summary, the numerical solutions clearly show that the scalar field density contrast behaves on average like</text>
<text><location><page_9><loc_52><loc_85><loc_92><loc_93></location>the CDM one on large scales, and that the choice θ ⋆ = nπ only helps a little for the numerical solution to have a smooth transition at the cutoff time t ⋆ . There are no further consequences, because the attractor character of the equations of motion for the density perturbations eventually leads to the right numerical result 5 .</text>
<section_header_level_1><location><page_9><loc_55><loc_77><loc_88><loc_80></location>E. Mass power spectrum and temperature anisotropies</section_header_level_1>
<text><location><page_9><loc_52><loc_61><loc_92><loc_75></location>To study the issue of convergence in the solution of density perturbations for all scales, we plot the resultant mass power spectrum (MPS) in the upper panel of Fig. 7 the resultant mass power spectrum (MPS) for different choices of θ ⋆ , and also the relative differences in the numerical solutions. The field mass in these examples was fixed at m ϕ = 10 -24 eV. The first thing to notice is that there is complete agreement with the MPS of CDM for large scales, represented by the wavenumbers k < 0 . 5 h/ Mpc.</text>
<text><location><page_9><loc_52><loc_41><loc_92><loc_60></location>Regarding the convergence of numerical solutions for different choices of the cutoff value θ ⋆ , it can be seen that there is complete agreement for almost all scales, except for the interval k = 2 -20 h/ Mpc, where the difference can be as large as 100%. However, these discrepancies appear once the MPS is greatly suppressed compared to the CDM, although it should be noticed that the agreement is recovered once the MPS reaches a steady stage at small scales of the form P ( k ) ∼ k -3 . Although one could use the solution with the highest cutoff value θ ⋆ , the overall conclusion is that one can safely take θ ⋆ = 30 π for reliable results with the additional advantage of saving computational time.</text>
<text><location><page_9><loc_52><loc_21><loc_92><loc_41></location>We repeat the comparison between different cutoff values θ ⋆ for the case of the temperature spectrum in terms of the variable D ℓ = ℓ ( ℓ +1) C ℓ / 2 π in the lower panel of Fig. 7. This time the solutions are more similar, among themselves with different resolutions and also with the corresponding spectrum of CDM. This can be verified in the lower graph with the relative differences of the solutions with respect to choice θ ⋆ = 100 π , which are very small and below 0 . 1%. For reference, we also show the estimated rule of thumb for bias-free parameter inference, which is given by the curves ± 3 /ℓ at large ℓ . We can see that our numerical results are consistent with such a constraint for the cutoff values chosen for θ ⋆ (see also [5053]).</text>
<figure>
<location><page_10><loc_10><loc_73><loc_49><loc_93></location>
<caption>FIG. 5. Behavior of the field density contrast δ 0 (top panels) and of the metric continuity ¯ h/ 2 (bottom panels), for the different wavenumbers k in the plots and for a selection of values for θ ⋆ . It is clear that the behavior of both quantities is the same as that of their counterparts in the CDM for large scales (here represented by k = 0 . 01 / Mpc), whereas for small scales ( k = 10 / Mpc) they are considerably smaller than those of CDM at late times. The field mass was fixed at m ϕ = 10 -24 eV in the numerical examples. See the text for more details.</caption>
</figure>
<figure>
<location><page_10><loc_51><loc_73><loc_91><loc_93></location>
</figure>
<figure>
<location><page_10><loc_10><loc_52><loc_49><loc_72></location>
</figure>
<figure>
<location><page_10><loc_51><loc_52><loc_91><loc_71></location>
</figure>
<text><location><page_10><loc_69><loc_70><loc_70><loc_72></location>k</text>
<text><location><page_10><loc_70><loc_70><loc_75><loc_72></location>=10Mpc</text>
<text><location><page_10><loc_76><loc_71><loc_77><loc_72></location>1</text>
<section_header_level_1><location><page_10><loc_11><loc_37><loc_46><loc_40></location>F. Constraints on m ϕ from the matter power spectrum</section_header_level_1>
<text><location><page_10><loc_9><loc_24><loc_49><loc_35></location>Here, we describe possible constraints of SFDM models from the matter power spectrum, according to recent estimates of the UV galaxy luminous function [49, 54] with the package Gallumi 6 , and the effective field theory of the large scale structure (EFTofLSST) as in [55, 56] with the package PyBird 7 , respectively. These two are likelihoods of a recent addition to the MCMC software MontePython [37, 38], which are capable of exploring the</text>
<text><location><page_10><loc_52><loc_37><loc_92><loc_40></location>power spectrum at semilinear scales and can be as competitive as those of Lymanα observations.</text>
<text><location><page_10><loc_52><loc_25><loc_92><loc_37></location>We replicated the studies in [49, 55] with some shortcuts, as our aim was to focus our attention on the constraints on the field mass m ϕ . For the analysis, we used Gaussian priors on the following parameters: the angular scale for the sound horizon with 100 θ s = 1 . 0411 and σ θ s = 0 . 0003 8 , and the physical density of baryons ω b = 0 . 02233 with σ b = 0 . 00036 9 . Fixing the sound horizon θ s is known to also fix the combination Ω m h 3 . 4 ,</text>
<figure>
<location><page_11><loc_9><loc_73><loc_49><loc_93></location>
<caption>FIG. 6. Illustration of the behavior of the density contrast δ 0 , for one large scale and for different choices of the cutoff angle θ ⋆ . Although there is a mismatch between the solutions at the cutoff time, both follow exactly the CDM solution at late times. The dashed vertical line marks log( a eq ). The field mass was fixed to m ϕ = 10 -24 eV in the numerical examples. See the text for more details.</caption>
</figure>
<text><location><page_11><loc_9><loc_46><loc_49><loc_59></location>with Ω m the physical density of total matter and h the reduced Hubble constant [49, 62, 63]. This means that our Gaussian prior on 100 θ s , with h = 0 . 657, acts as an indirect prior on the combination Ω m = Ω b + Ω ϕ . For the field mass m ϕ , we considered a flat prior on the logarithmic scale in the range log( m ϕ / eV) = [ -26 , -18]. All other cosmological parameters in the models, such as the amplitude of the power spectrum, were fixed to their Planck 2018 CMB values.</text>
<text><location><page_11><loc_9><loc_30><loc_49><loc_46></location>The likelihood we chose for the UV luminosity function is that of the so-called Model I in [49], with its corresponding formalities and data. In the case of EFTofLSST, we selected the BOSS and eBOSS data sets as in[55]. It must be noted that some assumptions in the likelihoods have been made under the CDM paradigm only and would need to be amended for the case of SFDM. Taking into account these caveats, the results reported in the following may be stronger than in the case in which some of the assumptions are corrected for SFDM.</text>
<text><location><page_11><loc_9><loc_20><loc_49><loc_30></location>The resulting posterior distributions for the physical densities of baryons ω b and SFDM ω sfdm , and the field mass m ϕ , after marginalizing over the nuisance parameters of the likelihoods, are shown in Fig. 8. As expected, the separate constraints on the physical densities of baryons and SFDM are practically the same, as they are influenced mostly by the previously assumed priors.</text>
<text><location><page_11><loc_9><loc_10><loc_49><loc_20></location>Not surprisingly, the posterior distribution of the field mass shows that each data set only constrains m ϕ from below, which means that the likelihoods are insensitive to variations of m ϕ above a certain threshold value. The data set with the most constraining power is the UV luminosity function, whereas eBOSS is the less constraining one.</text>
<text><location><page_11><loc_10><loc_9><loc_49><loc_10></location>To properly calculate the lower bounds for m ϕ from the</text>
<text><location><page_11><loc_52><loc_83><loc_92><loc_93></location>confidence regions in Fig. 8, we use the method in [6466] to obtain prior independent constraints by means of the so-called shape distortion function R , appropriate for the so-called open likelihoods as in the present case. As explained in [66], the function R will allow us to use the data to define the region below which m ϕ is disfavored, regardless of the prior assumptions we have chosen.</text>
<text><location><page_11><loc_52><loc_61><loc_92><loc_83></location>An advantage of the function R is that for its calculation we only need to know the posterior distribution of the field mass m ϕ , which we obtained from the code MontePython . The resultant shape distortion function is shown in the lower panel of Fig. 8 on the logarithmic scale. Note that R → 1 for large values of the field mass, in this case log( m ϕ c 2 / eV) → -18 which is the upper value in our prior range. The sharp decay of R at lower values of m ϕ helps us to calculate an appropriate lower bound. Following the convention in [64-66], it can be seen that if ln R = -3 (moderate level according to Jeffrey's scale), we can say that the data strongly favor the regions log( m ϕ c 2 / eV) > -25 . 4 for eBOSS, log( m ϕ c 2 / eV) > -24 . 6 for BOSS, and log( m ϕ c 2 / eV) > -22 for UV LF 10 .</text>
<section_header_level_1><location><page_11><loc_54><loc_56><loc_90><loc_58></location>IV. COMPARISON WITH THE ORIGINAL FIELD VARIABLES ( ϕ, ˙ ϕ )</section_header_level_1>
<text><location><page_11><loc_52><loc_38><loc_92><loc_54></location>This section is dedicated to the comparison of the numerical solutions obtained from our polar method to those of the original scalar field equations of motion. To do this, we use the same Boltzmann code CLASS to provide the numerical results, so that they are subject to the same numerical methods and limitations of the code in the two cases. In the amended version, the polar variables are solved as a separate dark matter component, while the field variables are solved using the scalar field equations of the quintessence module already implemented in CLASS .</text>
<section_header_level_1><location><page_11><loc_62><loc_34><loc_82><loc_35></location>A. Background quantities</section_header_level_1>
<text><location><page_11><loc_52><loc_23><loc_92><loc_32></location>The first comparison of background quantities is for the variables ϕ and ˙ ϕ , which are the dynamical ones in the KG equation (3d) and the field potential. The relationship between the original and polar variables can be found from the transformation equations (5), in the form of</text>
<formula><location><page_11><loc_54><loc_18><loc_92><loc_23></location>κϕ = -2 √ 6 y 1 e β cos( θ/ 2) , κ ˙ ϕ m ϕ = 2 √ 6 y 1 e β sin( θ/ 2) . (36)</formula>
<text><location><page_12><loc_47><loc_92><loc_49><loc_93></location>log(</text>
<text><location><page_12><loc_49><loc_92><loc_50><loc_93></location>m</text>
<text><location><page_12><loc_51><loc_92><loc_55><loc_93></location>/eV) =</text>
<text><location><page_12><loc_56><loc_92><loc_58><loc_93></location>24</text>
<figure>
<location><page_12><loc_13><loc_21><loc_87><loc_92></location>
<caption>FIG. 7. MPS P ( k ) (top panel) and temperature spectrum D ℓ (bottom panel) calculated for the field mass of m ϕ = 10 -24 eV, together with their relative differences with respect to MPS P 100 π ( k ) and the CDM temperature spectrum D 100 π ℓ . The labels of the different curves refer to the cutoff variable θ ⋆ . Note that the difference for MPS given the choices of θ ⋆ can be as large as 100% in the range k = 2 -20 h/ Mpc (vertical gray-shaded region), while for the temperature spectrum it is less than 0 . 1% (horizontal gray-shaded region). For reference, we include in the temperature spectrum the (black) curves given by ± 3 /ℓ , which represents an estimated precision threshold beyond which the parameter biases may be significant. The field mass was fixed to m ϕ = 10 -24 eV in the numerical examples, but the black dashed line is for m ϕ = 10 -22 eV. The data points for the MPS are the measurements from the Planck 2018 CMB [1], DES cosmic shear [46], SDSS galaxy clustering [47], SDSS Lymanα [48], and UV LF [49] data sets. See the text for more details.</caption>
</figure>
<figure>
<location><page_13><loc_9><loc_45><loc_48><loc_93></location>
<caption>FIG. 8. (Top panel) Triangle plots obtained from the MCMC code MontePython using the likelihood of the UV galaxy luminous function, for parameters ω b , ω sfdm and log( m ϕ c 2 / eV). (Bottom panel) Plot of the shape distortion function R obtained from the posterior distribution obtained for the field mass m ϕ . See the text for more details.</caption>
</figure>
<text><location><page_13><loc_9><loc_16><loc_49><loc_32></location>We solved the KG equation (3d) separately, but with the same initial conditions as in the polar case through transformation (36). It is not possible to accurately follow the numerical evolution after the onset of the field oscillations, and then we only solved the KG equation (3d) up to the equivalent time to θ ⋆ = 100 π (2 m ϕ t ⋆ ≃ 100 π ). After this time, the equations of motion are set directly to ˙ ϕ = 0 and ¨ ϕ = 0, which means that the late-time solutions of the field variables are just ϕ ( t > t ⋆ ) = ϕ ( t ⋆ ) and ˙ ϕ ( t > t ⋆ ) = ˙ ϕ ( t ⋆ ) (and the density remains artificially constant afterwards).</text>
<text><location><page_13><loc_9><loc_9><loc_49><loc_16></location>The two sets of solutions, the original and the polar ones, are plotted in compact form in the phase space shown in Fig. 9. The thick curves correspond to the system ϕ -˙ ϕ , with the different colors representing the field mass m ϕ , while the solutions of the polar system,</text>
<text><location><page_13><loc_52><loc_89><loc_92><loc_93></location>all in black lines, are superimposed. We see that the agreement between the corresponding curves is exact up to t = t ⋆ .</text>
<figure>
<location><page_13><loc_52><loc_70><loc_91><loc_87></location>
<caption>FIG. 9. The phase space of the field variables ϕ and ˙ ϕ , written in their dimensionless form as in Eqs. (36). The colored curves are numerical solutions from the original field variables, whereas the black curves are those obtained from the polar method. The curve labels represent the value of the field mass m ϕ in units of eV. See the text for more details.</caption>
</figure>
<text><location><page_13><loc_52><loc_40><loc_92><loc_58></location>The same comparison exercise for field density ρ ϕ is shown in Fig. 10, using the same colors for the different curves as in Fig. 9. Furthermore, we normalize the density to the present value of the CDM density ρ CDM 0 , to highlight that the final value of the field densities coincides with the equivalent CDM case. The upper panel shows that the two sets of solutions coincide exactly, including the oscillatory phase, which is also confirmed by the comparison in the lower panel: the discrepancies appear at late times in the oscillatory phase and the cutoff in the solution of the field variables (see the explanation below Eq. (36)).</text>
<text><location><page_13><loc_52><loc_27><loc_92><loc_40></location>As a final example, we show in Fig. 11 the SFDM EOS calculated directly from the pressure-to-density ratio w ϕ = p ϕ /ρ ϕ , using the same unit system as in Fig. 2. The variable in the horizontal axis is the dimensionless quantity 2 m ϕ t , under which all curves corresponding to a given field mass m ϕ become the same curve. The EOS oscillates rapidly around the zero value, and we again see that there is excellent agreement between the numerical results of the two approaches.</text>
<section_header_level_1><location><page_13><loc_59><loc_23><loc_84><loc_24></location>B. Linear density perturbations</section_header_level_1>
<text><location><page_13><loc_52><loc_9><loc_92><loc_21></location>We repeated the comparison of the solutions with the case of linear density perturbations. This time we solved the linearly perturbed KG equation (12), again using the same initial conditions for the two sets of variables, the originals φ and ˙ φ and the polar ones δ 0 and δ 1 . Although the perturbed polar variables are α and ϑ , see Eqs. (13), recall that our final perturbed variables are those of Eqs. (14) and their corresponding equations of motion (15).</text>
<figure>
<location><page_14><loc_10><loc_55><loc_91><loc_93></location>
<caption>FIG. 10. The evolution of the field density ρ ϕ for the same cases as in Fig. 3. (Top panel) Colored curves represent the numerical solution of the original field variables ϕ and ˙ ϕ , while the corresponding solutions from the polar method are superimposed (black curves). The divergent behavior of the colored curves is due to the cutoff applied to the oscillations of the field solutions at θ ⋆ = 60 π . (Bottom panel) The relative difference between the curves in the upper panel for the same value of the field mass m ϕ . The difference is close to zero except for the last part, once the oscillations of the field solutions are cut off. See the text for more details.</caption>
</figure>
<figure>
<location><page_14><loc_9><loc_24><loc_48><loc_42></location>
<caption>FIG. 11. The numerical solution of the scalar field EOS w ϕ , calculated from the pressure to density ratio, as a function of the variable 2 m ϕ t , under which all cases collapse with different field masses m ϕ collapse into a single curve. The blue curve is the solution of the original variables ϕ -˙ ϕ (note that w ϕ = -1 after the oscillations are cut off is just an artifact), whereas the orange curve corresponds to the polar method. See the text for more details.</caption>
</figure>
<text><location><page_14><loc_52><loc_39><loc_92><loc_42></location>The transformation from the polar variables to the field ones is given by the expressions</text>
<formula><location><page_14><loc_58><loc_35><loc_92><loc_39></location>κφ = -√ 6 y 1 e β [ δ 0 cos( θ/ 2) -δ 1 sin( θ/ 2)] , (37a)</formula>
<formula><location><page_14><loc_57><loc_31><loc_92><loc_35></location>κ ˙ φ m ϕ = √ 6 y 1 e β [ δ 0 sin( θ/ 2) + δ 1 cos( θ/ 2)] . (37b)</formula>
<text><location><page_14><loc_52><loc_25><loc_92><loc_30></location>We only used Eqs. (37) to set the initial conditions of the field variables φ and ˙ φ in correspondence with those of the polar variables, and then the field equation (12) was solved separately.</text>
<text><location><page_14><loc_52><loc_17><loc_92><loc_24></location>We then show in Fig. 12 the evolution of the perturbation variables φ and ˙ φ as a function of the scale factor and for two values of the wavenumber k : 0 . 01 Mpc -1 for large scales (top panels) and 10 Mpc -1 for small scales (bottom panels).</text>
<text><location><page_14><loc_52><loc_9><loc_92><loc_17></location>In the two cases, the solutions from the polar variables via Eqs. (37) are superimposed on those of the field variables (obtained directly from Eq. (12)), which are identical up to the time the numerical solutions were followed ( θ ⋆ = 30 π for the polar variables and θ ⋆ = 100 π for the field ones). Note that the agreement goes beyond the</text>
<figure>
<location><page_15><loc_11><loc_73><loc_49><loc_93></location>
</figure>
<figure>
<location><page_15><loc_51><loc_73><loc_90><loc_93></location>
</figure>
<figure>
<location><page_15><loc_11><loc_52><loc_49><loc_71></location>
</figure>
<figure>
<location><page_15><loc_51><loc_52><loc_90><loc_71></location>
<caption>FIG. 12. The evolution of the perturbation variables φ (left column) and ˙ φ (right column), as obtained from both the original field variables via Eq. (12) and the polar method in Eqs. (15), see also Eqs. (37). The upper panels show the solutions for large scales, whereas the lower panels show them for small scales. It can be seen that the curves of the two methods agree completely even beyond the cutoff time for the polar variables at θ ⋆ = 30 π (indicated by the black-dashed vertical line). See the text for more details.</caption>
</figure>
<text><location><page_15><loc_9><loc_34><loc_49><loc_40></location>cutoff point of the polar variables, which means that the cutoff of the trigonometric functions (22) delivers from Eqs. (15) the expected results of the original field variables.</text>
<section_header_level_1><location><page_15><loc_22><loc_29><loc_36><loc_30></location>V. DISCUSSION</section_header_level_1>
<text><location><page_15><loc_9><loc_9><loc_49><loc_27></location>Among the diverse theoretical proposals to describe dark matter, SFDM constitutes a compelling candidate to play the role of the CDM component of the universe. The dynamics of such a particle is modeled through a scalar field endowed with a scalar field potential. In this work, we were particularly interested in the free case of a real scalar field. Given the oscillatory nature of the SFDM when it behaves as CDM, we must be careful to properly handle the differential equations for both background and linear perturbations, in such a way that numerically they were easily solved, allowing us to keep track of the evolution of the scalar field, and of all physical quantities built from it. By posing the Klein-Gordon</text>
<text><location><page_15><loc_52><loc_34><loc_92><loc_40></location>equations (background and linear perturbations) as a dynamical system once new variables are introduced, we were able to describe the evolution of the SFDM as a system of first-order differential equations.</text>
<text><location><page_15><loc_52><loc_9><loc_92><loc_34></location>With this prescription, instead of solving for the original scalar field variables ( ϕ, ˙ ϕ ; φ, ˙ φ ) , we solved a new set of dynamical variables: the polar variable θ , the scalar field energy density β , and the perturbations δ 0 , δ 1 . We have also added the variable y 1 which is proportional to the ratio of the mass of the scalar field and the Hubble parameter. It has been a standard procedure in the literature to deal with the rapid oscillations of the SFDM by averaging the oscillating functions in a Hubble time, and then writing down a new set of averaged equations of motion that resemble a standard cosmological fluid. Within our approach, this new set of equations is easier to solve numerically and to include in a standard Boltzmann code. Therefore, there was no need to invoke any approximation or average to cancel out the rapid oscillations of the scalar field. The only consideration of this kind was the introduction of the truncated trigonometric</text>
<text><location><page_16><loc_9><loc_86><loc_49><loc_93></location>functions that we used (see Eq. (22) in Section III A ). In addition, the average procedure introduces an undesirable mismatch between the early- and late-time solutions and leaves unanswered the question of the sound speed of the density perturbations of the averaged fluid.</text>
<text><location><page_16><loc_9><loc_64><loc_49><loc_86></location>Our method does not require a separate evolution of the SFDM equations of motion, but just a straightforward transformation of the original field equations. Our transformed system of equations remains the same throughout the full evolution, and this applies both for the background and for the linear density perturbations. Moreover, in the case of linear perturbations, we do not need to define an explicit expression for the sound speed, which is a slippery quantity in the fluid approximation. Nevertheless, in Appendices A and B we show the complete equivalence between our equations and the fluid ones, which only required the implicit definition of the sound speed. Our approach also deals successfully with the rapid oscillations in comparison with the original field variables.</text>
<text><location><page_16><loc_9><loc_40><loc_49><loc_64></location>The SFDM formulation we present in this work gives accurate predictions on observables such as the CMB anisotropies and the MPS. Moreover, it is possible to put constraints on the main parameter of the model, the SFDM mass m ϕ . This is of great relevance because, on the one hand, we need numerical solutions with a high precision level to put constraints on the SFDM model in light of new data from present and future surveys. On the other hand, the cosmological dynamics of the SFDM from the big bang until the linear perturbation regime needs to be properly solved in order to set the initial conditions for numerical simulations of the nonlinear process of structure formation. In this sense, our approach offers a mathematical and numerical treatment for the SFDM model, which is advantageous in that it suitably solves the cosmological dynamics of such scalar field coupled to the Einstein-Boltzmann system.</text>
<section_header_level_1><location><page_16><loc_19><loc_35><loc_39><loc_36></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_16><loc_9><loc_18><loc_49><loc_33></location>FXLC acknowledges Beca CONACYT and FORDECYT-PRONACES-CONACYT for support of the present research under Grant No. CF-MG-2558591. This work was partially supported by Programa para el Desarrollo Profesional Docente; Direcci'on de Apoyo a la Investigaci'on y al Posgrado, Universidad de Guanajuato; CONACyT M'exico under Grants No. A1-S-17899, No. 286897, No. 297771, No. 304001; and the Instituto Avanzado de Cosmolog'ıa Collaboration. We acknowledge the use of open-source software: Python [67, 68], numpy [69], scipy [70], astropy [71].</text>
<section_header_level_1><location><page_16><loc_14><loc_13><loc_43><loc_14></location>Appendix A: The fluid approximation</section_header_level_1>
<text><location><page_16><loc_9><loc_9><loc_49><loc_11></location>Here we discuss the possibility of rewriting the equations of motion (15) for the linear density perturbations</text>
<text><location><page_16><loc_52><loc_90><loc_92><loc_93></location>in their fluid counterparts. Using Eq. (17b), we find that Eq. (15a) becomes</text>
<formula><location><page_16><loc_53><loc_86><loc_92><loc_89></location>δ ' 0 = -1 aH (1 + w ϕ )Θ ϕ -3 sin θδ 1 -(1 + w ϕ ) ¯ h ' 2 , (A1)</formula>
<text><location><page_16><loc_52><loc_81><loc_92><loc_85></location>where we have also used the relation (6b). From the definition of the adiabatic sound speed c 2 s = δp ϕ /δρ ϕ we find that</text>
<formula><location><page_16><loc_63><loc_78><loc_92><loc_80></location>c 2 s δ 0 = sin θ δ 1 -cos θ δ 0 . (A2)</formula>
<text><location><page_16><loc_52><loc_73><loc_92><loc_77></location>The preceding equation can be used to substitute the second term on the right-hand side of Eq. (A1). After some manipulations, we get the following result,</text>
<formula><location><page_16><loc_53><loc_69><loc_92><loc_72></location>δ ' 0 = -(1+ w ϕ )Θ ϕ -3 H ( c 2 s -w ϕ ) δ 0 -(1+ w ϕ ) ¯ h ' 2 , (A3a)</formula>
<text><location><page_16><loc_52><loc_61><loc_92><loc_68></location>where now the primes denote the derivatives with respect to the conformal time and H is the corresponding Hubble parameter. These calculations show that Eq. (15a) is completely equivalent to the equation of motion for the density contrast in the fluid approximation.</text>
<text><location><page_16><loc_52><loc_55><loc_92><loc_61></location>Similarly, we can combine Eqs. (7), (15) and (A2) to write an equation for the momentum density. After some lengthy but otherwise straightforward manipulations, we find that</text>
<formula><location><page_16><loc_53><loc_52><loc_92><loc_54></location>[(1 + w ϕ )Θ ϕ ] ' = H (3 w ϕ -1)(1+ w ϕ )Θ ϕ + k 2 c 2 s δ 0 . (A3b)</formula>
<text><location><page_16><loc_52><loc_47><loc_92><loc_51></location>Equations (A3) are the standard fluid equations for linear density perturbations and then show that the polar equations (15) are their faithful representation.</text>
<section_header_level_1><location><page_16><loc_52><loc_42><loc_91><loc_44></location>Appendix B: Linear density perturbations after the cutoff point</section_header_level_1>
<text><location><page_16><loc_52><loc_37><loc_92><loc_40></location>It is illustrative to write Eqs. (15) after the cutoff on the rapid field oscillations. For times t > t ⋆ we find</text>
<formula><location><page_16><loc_59><loc_32><loc_92><loc_36></location>( δ 0 δ 1 ) ' ≃ k 2 k 2 J ( 0 -1 1 0 ) -( ¯ h ' / 2 0 ) . (B1)</formula>
<text><location><page_16><loc_52><loc_29><loc_92><loc_31></location>where again a prime denotes derivatives with respect to N = ln a .</text>
<text><location><page_16><loc_52><loc_17><loc_92><loc_28></location>The system (B1) has the same structure as a forced harmonic oscillator with frequency ω = k 2 /k 2 J . As explained in [16], if we assume that ω is a constant (as is the case during RD), for large scales, the solution is the same as that of CDM: δ 0 ≃ -¯ h/ 2 and δ 1 ≃ const . For small scales, the solutions are oscillatory and given by combinations of functions cos( ωN ) and sin( ωN ). These behaviors agree with the full numerical results in Fig. 5.</text>
<text><location><page_16><loc_52><loc_13><loc_92><loc_17></location>To obtain the corresponding fluid equations, we first notice that the cutoff method applied to Eqs. (12) and (A2) results in the following equations,</text>
<formula><location><page_16><loc_60><loc_8><loc_92><loc_11></location>Θ ϕ ≃ k 2 aHy 1 δ 1 , ( c 2 s -w ϕ ) δ 0 ≃ 0 . (B2)</formula>
<text><location><page_17><loc_9><loc_87><loc_49><loc_93></location>The first expression in Eqs. (B2) reveals a direct relation between the divergence of the fluid velocity and variable δ 1 , while the second suggests a non-varying value of the fluid's EOS (that is, dw ϕ /dρ ϕ = 0, see [4]).</text>
<text><location><page_17><loc_9><loc_83><loc_49><loc_87></location>Repeating the same procedure above for the calculation of Eqs. (A3), we find their counterpart for times t > t ⋆ ,</text>
<formula><location><page_17><loc_11><loc_79><loc_49><loc_83></location>δ ' 0 ≃ -Θ ϕ -¯ h ' 2 , Θ ' ϕ ≃ -H Θ ϕ + k 2 4 m 2 ϕ a 2 δ 0 , (B3)</formula>
<text><location><page_17><loc_9><loc_74><loc_49><loc_78></location>where a prime now denotes derivatives with respect to conformal time. Finally, the second of Eqs. (B3) suggests a scale-dependent sound speed given by</text>
<formula><location><page_17><loc_24><loc_70><loc_49><loc_73></location>c 2 s ≃ k 2 4 m 2 ϕ a 2 . (B4)</formula>
<text><location><page_17><loc_9><loc_58><loc_49><loc_69></location>One note on the sound speed of the perturbations is taken in turn. The discussion above gives a clear answer for the sound speed after the cutoff in the field oscillations, but there is none for the right expression of c 2 s before that. Our formalism for the perturbations does not require an explicit expression of c 2 s , but the latter is necessary for a correct fluid formulation of the SFDM linear perturbations.</text>
<text><location><page_17><loc_9><loc_46><loc_49><loc_58></location>A general expression of c 2 s was found in [72] in the comoving gauge of the scalar field, but the same authors argue in [73] that such expression cannot be transformed into the synchronous gauge, under which SFDM lacks a definite functional form of c 2 s . Despite this, the expression for c 2 s of [72] has been extensively used in the literature for the fluid approximation of SFDM in the synchronous gauge.</text>
<text><location><page_17><loc_9><loc_42><loc_49><loc_46></location>An example is the so-called Effective Fluid Approximation (EFA) discussed in [36]. The equations of motion of the EFA are</text>
<formula><location><page_17><loc_19><loc_38><loc_49><loc_42></location>δ ' 0 ≃ -Θ ϕ -3 H⟨ c 2 s ⟩ δ 0 -¯ h ' 2 , (B5a)</formula>
<formula><location><page_17><loc_18><loc_36><loc_49><loc_38></location>Θ ' ϕ ≃ -H Θ ϕ + k 2 ⟨ c 2 s ⟩ δ 0 , (B5b)</formula>
<text><location><page_17><loc_9><loc_35><loc_37><loc_36></location>where the cycle-averaged sound speed is</text>
<formula><location><page_17><loc_20><loc_30><loc_49><loc_34></location>⟨ c 2 s ⟩ = k 2 / (4 m 2 ϕ a 2 ) 1 + k 2 / (4 m 2 ϕ a 2 ) . (B5c)</formula>
<text><location><page_17><loc_9><loc_22><loc_49><loc_29></location>It can be seen that our Eqs. (B3) can be obtained from the EFA if the sound speed is given by the so-called nonrelativistic expression ⟨ c 2 s ⟩ ≃ k 2 / (4 m 2 ϕ a 2 ). However, the EFA considers an extra term in the equation of motion of δ 0 , see Eq. (B5a).</text>
<text><location><page_17><loc_9><loc_12><loc_49><loc_22></location>Noticing this discrepancy, a comparison was made between the fluid approximation and our method in [36], see their Appendix B, and no relevant differences were found in the numerical results. To understand this, we use Eqs. (B2) and write the EFA equations (B5a) and (B5b) in terms of our variables δ 0 , δ 1 and N = ln a . We find that</text>
<formula><location><page_17><loc_12><loc_8><loc_49><loc_12></location>( δ 0 δ 1 ) ' ≃ k 2 k 2 J ( -3 /y 1 -1 1 0 ) -( ¯ h ' / 2 0 ) . (B6)</formula>
<text><location><page_17><loc_52><loc_82><loc_92><loc_93></location>We now compare Eqs. (B6) with Eqs. (B1), and spot the extra term 3 /y 1 . The latter is very small, given that y 1 = 2 m ϕ /H and then y 1 ≫ 1 after the start of rapid field oscillations. This is the reason why the comparison in [36] did not find differences between our approach and the EFA. It can be concluded that our method correctly picks up the only terms in the equations of motion that are valid in the limit H/m ϕ ≪ 1. 11</text>
<section_header_level_1><location><page_17><loc_53><loc_78><loc_90><loc_79></location>Appendix C: Numerical implementation in CLASS</section_header_level_1>
<text><location><page_17><loc_52><loc_67><loc_92><loc_76></location>Following the design of CLASS , we introduced a new module for the SFDM equations of motion, replicating the same structure as for other dark matter components. In this form, the contribution of the SFDM component could be called with its own parameters in any given parameter file of CLASS (for example, explanatory.ini ).</text>
<text><location><page_17><loc_52><loc_53><loc_92><loc_67></location>The equations of motion for the background variables (12) were included in the file background.c , while those of the density perturbations (15) were included in the file perturbations.c . Similarly, a shooting routine was incorporated into the file input.ini to adjust the initial conditions of the dynamical variables. In each case, the SFDM quantities were added to the matter budget so that they contribute correctly to the right-hand side of the Einstein equations, for both the background and the linearly perturbed ones.</text>
<text><location><page_17><loc_52><loc_42><loc_92><loc_52></location>Given the oscillatory nature of all SFDM quantities, we applied the cutoff procedure to the sine and cosine functions that appear in their definition, see Eqs. (22). For example, the background pressure was written as p ϕ = -cos ⋆ θ · ρ ϕ , and then effectively p ϕ = 0 for t > t ⋆ . Another case would be the combination p ϕ + ρ ϕ = (1 -cos ⋆ θ ) ρ ϕ , so that p ϕ + ρ ϕ = ρ ϕ for t > t ⋆ .</text>
<text><location><page_17><loc_52><loc_34><loc_92><loc_42></location>The same was applied to the perturbed quantities, as in the density perturbation written in the form δρ ϕ = ρ ϕ · δ 0 , which is required for the right-hand side of the perturbed Einstein equations. Similarly, the momentum density perturbation ( p ϕ + ρ ϕ )Θ ϕ was written as in Eq. (17b), that is,</text>
<formula><location><page_17><loc_53><loc_29><loc_92><loc_33></location>( ρ ϕ + p ϕ )Θ ϕ = k 2 ρ ϕ aHy 1 [(1 -cos ⋆ θ ) δ 1 -sin ⋆ θ δ 0 ] . (C1a)</formula>
<text><location><page_17><loc_52><loc_26><loc_92><loc_28></location>The result for late times t > t ⋆ is well defined and is given by</text>
<formula><location><page_17><loc_63><loc_21><loc_92><loc_24></location>( ρ ϕ + p ϕ )Θ ϕ = k 2 ρ ϕ aHy 1 δ 1 . (C1b)</formula>
<text><location><page_18><loc_9><loc_76><loc_49><loc_93></location>It should be noted that in some parts of the file perturbations.c one requires to calculate the velocity divergence Θ, which is one of the fundamental variables in the fluid formalism for linear perturbations of density and momentum. However, the calculation of Θ ϕ itself is problematic for SFDM, as can be clearly seen in Eq. (C1a): One needs to divide by the quantity ( p ϕ + ρ ϕ ), which passes through zero during the phase of rapid field oscillations. As explained above, the calculation of Θ ϕ is not needed in the perturbed Einstein equations, and then we only used Eq. (C1b) as an additional source of the perturbed momentum density.</text>
<text><location><page_18><loc_9><loc_67><loc_49><loc_76></location>A technical aspect not always mentioned for SFDM models and its Boltzmann code implementation is that it is always required a small quantity of CDM component. This is because in order to be consistent with the synchronous gauge, the Einstein field equations have to be solved in the comoving frame of the CDM fluid, that</text>
<unordered_list>
<list_item><location><page_18><loc_10><loc_57><loc_49><loc_62></location>[1] N. Aghanim et al. (Planck), Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641 , A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO].</list_item>
<list_item><location><page_18><loc_10><loc_53><loc_49><loc_56></location>[2] P. J. E. Peebles, Anomalies in physical cosmology, Annals Phys. 447 , 169159 (2022), arXiv:2208.05018 [astroph.CO].</list_item>
<list_item><location><page_18><loc_10><loc_50><loc_49><loc_52></location>[3] G. Bertone and D. Hooper, History of dark matter, Rev. Mod. Phys. 90 , 045002 (2018).</list_item>
<list_item><location><page_18><loc_10><loc_45><loc_49><loc_50></location>[4] C.-P. Ma and E. Bertschinger, Cosmological perturbation theory in the synchronous and conformal Newtonian gauges, Astrophys. J. 455 , 7 (1995), arXiv:astroph/9506072.</list_item>
<list_item><location><page_18><loc_10><loc_41><loc_49><loc_44></location>[5] U. Seljak and M. Zaldarriaga, A Line of sight integration approach to cosmic microwave background anisotropies, Astrophys. J. 469 , 437 (1996), arXiv:astro-ph/9603033.</list_item>
<list_item><location><page_18><loc_10><loc_37><loc_49><loc_40></location>[6] M. Doran, CMBEASY: an object oriented code for the cosmic microwave background, JCAP 10 , 011, arXiv:astro-ph/0302138.</list_item>
<list_item><location><page_18><loc_10><loc_33><loc_49><loc_37></location>[7] A. Lewis, A. Challinor, and A. Lasenby, Efficient computation of CMB anisotropies in closed FRW models, Astrophys. J. 538 , 473 (2000), arXiv:astro-ph/9911177.</list_item>
<list_item><location><page_18><loc_10><loc_29><loc_49><loc_33></location>[8] J. Lesgourgues, The Cosmic Linear Anisotropy Solving System (CLASS) I: Overview, (2011), arXiv:1104.2932 [astro-ph.IM].</list_item>
<list_item><location><page_18><loc_10><loc_26><loc_49><loc_29></location>[9] L. F. Abbott and P. Sikivie, A Cosmological Bound on the Invisible Axion, Phys. Lett. B 120 , 133 (1983).</list_item>
<list_item><location><page_18><loc_9><loc_22><loc_49><loc_26></location>[10] T. Matos, From superstrings theory to the dark matter in galaxies, AIP Conf. Proc. 490 , 382 (1999), arXiv:astroph/9905024.</list_item>
<list_item><location><page_18><loc_9><loc_18><loc_49><loc_22></location>[11] T. Matos and F. S. Guzman, Scalar fields as dark matter in spiral galaxies, Class. Quant. Grav. 17 , L9 (2000), arXiv:gr-qc/9810028.</list_item>
<list_item><location><page_18><loc_9><loc_14><loc_49><loc_18></location>[12] T. Matos and L. A. Urena-Lopez, Quintessence and scalar dark matter in the universe, Class. Quant. Grav. 17 , L75 (2000), arXiv:astro-ph/0004332.</list_item>
<list_item><location><page_18><loc_9><loc_10><loc_49><loc_14></location>[13] W. Hu, R. Barkana, and A. Gruzinov, Fuzzy cold dark matter: the wave properties of ultralight particles, Physical Review Letters 85 , 1158 (2000).</list_item>
<list_item><location><page_18><loc_9><loc_9><loc_49><loc_10></location>[14] W. Hu, R. Barkana, and A. Gruzinov, Cold and</list_item>
</unordered_list>
<text><location><page_18><loc_52><loc_79><loc_92><loc_93></location>is, Θ cdm = 0. In fact, an automatic feature in the current versions of CLASS and its amended versions for SFDM, Ω cdm = 10 -10 once the amount of CDM is set to zero in the input file explanatory.ini . This ensures that there will be practically a null contribution of CDM, so that SFDM is the dominant non-relativistic matter component, and also helps us to avoid the known problem that the synchronous gauge is not completely fixed for density perturbations when the SFDM is the only DM component [73].</text>
<text><location><page_18><loc_52><loc_67><loc_92><loc_77></location>One last note is that the evolution of the perturbations should start well before the start of the rapid oscillations. We include an additional condition in the file perturbations.c to guarantee that the calculations begin only if m ϕ /H < 0 . 01. This gives the system (15) enough time to reach its attractor solution starting from the initial conditions (21).</text>
<unordered_list>
<list_item><location><page_18><loc_55><loc_59><loc_92><loc_62></location>fuzzy dark matter, Phys. Rev. Lett. 85 , 1158 (2000), arXiv:astro-ph/0003365.</list_item>
<list_item><location><page_18><loc_52><loc_57><loc_92><loc_59></location>[15] E. Masso, Axions and axion like particles, Nucl. Phys. B Proc. Suppl. 114 , 67 (2003), arXiv:hep-ph/0209132.</list_item>
<list_item><location><page_18><loc_52><loc_51><loc_92><loc_56></location>[16] L. A. Ure˜na L'opez and A. X. Gonzalez-Morales, Towards accurate cosmological predictions for rapidly oscillating scalar fields as dark matter, JCAP 07 , 048, arXiv:1511.08195 [astro-ph.CO].</list_item>
<list_item><location><page_18><loc_52><loc_46><loc_92><loc_51></location>[17] R. Hlozek, D. Grin, D. J. E. Marsh, and P. G. Ferreira, A search for ultralight axions using precision cosmological data, Phys. Rev. D 91 , 103512 (2015), arXiv:1410.2896 [astro-ph.CO].</list_item>
<list_item><location><page_18><loc_52><loc_42><loc_92><loc_46></location>[18] L. Hui, J. P. Ostriker, S. Tremaine, and E. Witten, Ultralight scalars as cosmological dark matter, Phys. Rev. D 95 , 043541 (2017), arXiv:1610.08297 [astro-ph.CO].</list_item>
<list_item><location><page_18><loc_52><loc_36><loc_92><loc_42></location>[19] V. Irˇsiˇc, M. Viel, M. G. Haehnelt, J. S. Bolton, and G. D. Becker, First constraints on fuzzy dark matter from Lymanα forest data and hydrodynamical simulations, Phys. Rev. Lett. 119 , 031302 (2017), arXiv:1703.04683 [astro-ph.CO].</list_item>
<list_item><location><page_18><loc_52><loc_33><loc_92><loc_35></location>[20] L. A. Ure˜na L'opez, Brief Review on Scalar Field Dark Matter Models, Front. Astron. Space Sci. 6 , 47 (2019).</list_item>
<list_item><location><page_18><loc_52><loc_28><loc_92><loc_33></location>[21] M. Kulkarni and J. P. Ostriker, What is the halo mass function in a fuzzy dark matter cosmology?, Mon. Not. Roy. Astron. Soc. 510 , 1425 (2021), arXiv:2011.02116 [astro-ph.CO].</list_item>
<list_item><location><page_18><loc_52><loc_25><loc_92><loc_27></location>[22] E. G. M. Ferreira, Ultra-light dark matter, Astron. Astrophys. Rev. 29 , 7 (2021), arXiv:2005.03254 [astro-ph.CO].</list_item>
<list_item><location><page_18><loc_52><loc_18><loc_92><loc_25></location>[23] H. Foidl and T. Rindler-Daller, Cosmological structure formation in complex scalar field dark matter versus real ultralight axions: A comparative study using class, Phys. Rev. D 105 , 123534 (2022), arXiv:2203.09396 [astroph.CO].</list_item>
<list_item><location><page_18><loc_52><loc_13><loc_92><loc_18></location>[24] S. T. H. Hartman, H. A. Winther, and D. F. Mota, Cosmological simulations of self-interacting Bose-Einstein condensate dark matter, Astron. Astrophys. 666 , A95 (2022), arXiv:2203.03946 [astro-ph.CO].</list_item>
<list_item><location><page_18><loc_52><loc_9><loc_92><loc_13></location>[25] F. X. L. Cede˜no, A. X. Gonz'alez-Morales, and L. A. Ure˜na L'opez, Cosmological signatures of ultralight dark matter with an axionlike potential, Phys. Rev. D 96 ,</list_item>
</unordered_list>
<text><location><page_19><loc_12><loc_92><loc_38><loc_93></location>061301 (2017), arXiv:1703.10180 [gr-qc].</text>
<unordered_list>
<list_item><location><page_19><loc_9><loc_87><loc_49><loc_92></location>[26] A. Chatrchyan, C. Eroncel, M. Koschnitzke, and G. Servant, ALP dark matter with non-periodic potentials: parametric resonance, halo formation and gravitational signatures, (2023), arXiv:2305.03756 [hep-ph].</list_item>
<list_item><location><page_19><loc_9><loc_81><loc_49><loc_86></location>[27] F. X. Linares Cede˜no, A. X. Gonz'alez-Morales, and L. A. Ure˜na L'opez, Ultralight DM bosons with an axion-like potential: scale-dependent constraints revisited, JCAP 01 , 051, arXiv:2006.05037 [astro-ph.CO].</list_item>
<list_item><location><page_19><loc_9><loc_77><loc_49><loc_81></location>[28] L. A. Ure˜na L'opez, Scalar field dark matter with a cosh potential, revisited, JCAP 06 , 009, arXiv:1904.03318 [astro-ph.CO].</list_item>
<list_item><location><page_19><loc_9><loc_72><loc_49><loc_77></location>[29] F. X. Linares Cede˜no and L. A. Ure˜na l'opez, Oneparametric description for scalar field dark matter potentials, Astron. Nachr. 342 , 404 (2021), arXiv:2102.05074 [astro-ph.CO].</list_item>
<list_item><location><page_19><loc_9><loc_66><loc_49><loc_72></location>[30] S. G. Medell'ın-Gonz'alez, L. Arturo Ure˜na L'opez, and A. X. Gonz'alez-Morales, Nonlinear cosmological structure with ultralight bosons via modified gravity, Phys. Rev. D 103 , 083509 (2021), arXiv:2010.13998 [astroph.CO].</list_item>
<list_item><location><page_19><loc_9><loc_60><loc_49><loc_65></location>[31] S. G. Medell'ın-Gonz'alez, A. X. Gonz'alez-Morales, and L. A. Ure˜na L'opez, Consistency tests of structure formation simulations of scalar field dark matter, Astron. Nachr. 344 , e230026 (2023).</list_item>
<list_item><location><page_19><loc_9><loc_55><loc_49><loc_60></location>[32] P. Mocz et al. , Cosmological structure formation and soliton phase transition in fuzzy dark matter with axion self-interactions, Mon. Not. Roy. Astron. Soc. 521 , 2608 (2023), arXiv:2301.10266 [astro-ph.CO].</list_item>
<list_item><location><page_19><loc_9><loc_50><loc_49><loc_55></location>[33] A. Ba˜nares Hern'andez, A. Castillo, J. Martin Camalich, and G. Iorio, Confronting fuzzy dark matter with the rotation curves of nearby dwarf irregular galaxies, (2023), arXiv:2304.05793 [astro-ph.GA].</list_item>
<list_item><location><page_19><loc_9><loc_44><loc_49><loc_49></location>[34] M. Sipp, P. LaChance, R. Croft, Y. Ni, and T. Di Matteo, Super-resolution simulation of the Fuzzy Dark Matter cosmological model, (2022), arXiv:2210.12907 [astroph.CO].</list_item>
<list_item><location><page_19><loc_9><loc_40><loc_49><loc_44></location>[35] S. Passaglia and W. Hu, Accurate effective fluid approximation for ultralight axions, Phys. Rev. D 105 , 123529 (2022), arXiv:2201.10238 [astro-ph.CO].</list_item>
<list_item><location><page_19><loc_9><loc_36><loc_49><loc_40></location>[36] J. Cookmeyer, D. Grin, and T. L. Smith, How sound are our ultralight axion approximations?, Phys. Rev. D 101 , 023501 (2020), arXiv:1909.11094 [astro-ph.CO].</list_item>
<list_item><location><page_19><loc_9><loc_30><loc_49><loc_36></location>[37] B. Audren, J. Lesgourgues, K. Benabed, and S. Prunet, Conservative Constraints on Early Cosmology: an illustration of the Monte Python cosmological parameter inference code, JCAP 1302 , 001, arXiv:1210.7183 [astroph.CO].</list_item>
<list_item><location><page_19><loc_9><loc_25><loc_49><loc_30></location>[38] T. Brinckmann and J. Lesgourgues, MontePython 3: boosted MCMC sampler and other features, Phys. Dark Univ. 24 , 100260 (2019), arXiv:1804.07261 [astroph.CO].</list_item>
<list_item><location><page_19><loc_9><loc_21><loc_49><loc_24></location>[39] L. A. Ure˜na L'opez, New perturbative method for analytical solutions in single-field models of inflation, Phys. Rev. D 94 , 063532 (2016), arXiv:1512.07142 [astro-ph.CO].</list_item>
<list_item><location><page_19><loc_9><loc_17><loc_49><loc_20></location>[40] E. J. Copeland, A. R. Liddle, and D. Wands, Exponential potentials and cosmological scaling solutions, Phys. Rev. D 57 , 4686 (1998), arXiv:gr-qc/9711068.</list_item>
<list_item><location><page_19><loc_9><loc_11><loc_49><loc_16></location>[41] N. Roy, A. X. Gonzalez-Morales, and L. A. Urena-Lopez, New general parametrization of quintessence fields and its observational constraints, Phys. Rev. D 98 , 063530 (2018), arXiv:1803.09204 [gr-qc].</list_item>
<list_item><location><page_19><loc_9><loc_9><loc_49><loc_11></location>[42] B. Ratra, Expressions for linearized perturbations in a massive scalar field dominated cosmological model, Phys.</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_19><loc_55><loc_92><loc_70><loc_93></location>Rev. D 44 , 352 (1991).</list_item>
<list_item><location><page_19><loc_52><loc_88><loc_92><loc_92></location>[43] P. G. Ferreira and M. Joyce, Structure formation with a selftuning scalar field, Phys. Rev. Lett. 79 , 4740 (1997), arXiv:astro-ph/9707286.</list_item>
<list_item><location><page_19><loc_52><loc_84><loc_92><loc_88></location>[44] P. G. Ferreira and M. Joyce, Cosmology with a primordial scaling field, Phys. Rev. D 58 , 023503 (1998), arXiv:astro-ph/9711102.</list_item>
<list_item><location><page_19><loc_52><loc_80><loc_92><loc_84></location>[45] F. Perrotta and C. Baccigalupi, Early time perturbations behavior in scalar field cosmologies, Phys. Rev. D 59 , 123508 (1999), arXiv:astro-ph/9811156.</list_item>
<list_item><location><page_19><loc_52><loc_75><loc_92><loc_80></location>[46] M. A. Troxel et al. (DES), Dark Energy Survey Year 1 results: Cosmological constraints from cosmic shear, Phys. Rev. D 98 , 043528 (2018), arXiv:1708.01538 [astroph.CO].</list_item>
<list_item><location><page_19><loc_52><loc_69><loc_92><loc_74></location>[47] B. A. Reid et al. , Cosmological constraints from the clustering of the sloan digital sky survey DR7 luminous red galaxies, Monthly Notices of the Royal Astronomical Society 10.1111/j.1365-2966.2010.16276.x (2010).</list_item>
<list_item><location><page_19><loc_52><loc_62><loc_92><loc_69></location>[48] B. Abolfathi et al. (SDSS), The Fourteenth Data Release of the Sloan Digital Sky Survey: First Spectroscopic Data from the Extended Baryon Oscillation Spectroscopic Survey and from the Second Phase of the Apache Point Observatory Galactic Evolution Experiment, Astrophys. J. Suppl. 235 , 42 (2018), arXiv:1707.09322 [astro-ph.GA].</list_item>
<list_item><location><page_19><loc_52><loc_58><loc_92><loc_61></location>[49] N. Sabti, J. B. Mu˜n oz, and D. Blas, Galaxy luminosity function pipeline for cosmology and astrophysics, Physical Review D 105 , 10.1103/physrevd.105.043518 (2022).</list_item>
<list_item><location><page_19><loc_52><loc_54><loc_92><loc_57></location>[50] T. Cookmeyer, D. Grin, and T. L. Smith, How sound are our ultralight axion approximations?, Phys. Rev. D 101 , 023501 (2020).</list_item>
<list_item><location><page_19><loc_52><loc_48><loc_92><loc_53></location>[51] U. c. v. Seljak, N. Sugiyama, M. White, and M. Zaldarriaga, Comparison of cosmological boltzmann codes: Are we ready for high precision cosmology?, Phys. Rev. D 68 , 083507 (2003).</list_item>
<list_item><location><page_19><loc_52><loc_44><loc_92><loc_48></location>[52] M. Shimon, N. Itzhaki, and Y. Rephaeli, Bias-limited extraction of cosmological parameters, Journal of Cosmology and Astroparticle Physics 2013 (03), 009.</list_item>
<list_item><location><page_19><loc_52><loc_39><loc_92><loc_44></location>[53] M. Shimon, S. Sadeh, and Y. Rephaeli, Cmb anisotropy due to filamentary gas: power spectrum and cosmological parameter bias, Journal of Cosmology and Astroparticle Physics 2012 (10), 038.</list_item>
<list_item><location><page_19><loc_52><loc_34><loc_92><loc_39></location>[54] N. Sabti, J. B. Mu˜n oz, and D. Blas, New roads to the small-scale universe: Measurements of the clustering of matter with the high-redshift UV galaxy luminosity function, The Astrophysical Journal Letters 928 , L20 (2022).</list_item>
<list_item><location><page_19><loc_52><loc_30><loc_92><loc_34></location>[55] T. Simon, P. Zhang, and V. Poulin, Cosmological inference from the EFTofLSS: the eBOSS QSO full-shape analysis, (2022), arXiv:2210.14931 [astro-ph.CO].</list_item>
<list_item><location><page_19><loc_52><loc_26><loc_92><loc_30></location>[56] G. D'Amico, L. Senatore, and P. Zhang, Limits on w CDM from the EFTofLSS with the PyBird code, JCAP 01 , 006, arXiv:2003.07956 [astro-ph.CO].</list_item>
<list_item><location><page_19><loc_52><loc_21><loc_92><loc_26></location>[57] A. Lague, J. R. Bond, R. Hloˇzek, K. K. Rogers, D. J. E. Marsh, and D. Grin, Constraining ultralight axions with galaxy surveys, JCAP 01 (01), 049, arXiv:2104.07802 [astro-ph.CO].</list_item>
<list_item><location><page_19><loc_52><loc_17><loc_92><loc_20></location>[58] X. Li, L. Hui, and G. L. Bryan, Numerical and perturbative computations of the fuzzy dark matter model, Phys. Rev. D 99 , 063509 (2019).</list_item>
<list_item><location><page_19><loc_52><loc_10><loc_92><loc_16></location>[59] M. Tsedrik, C. Moretti, P. Carrilho, F. Rizzo, and A. Pourtsidou, Interacting dark energy from the joint analysis of the power spectrum and bispectrum multipoles with the EFTofLSS 10.1093/mnras/stad260 (2022), arXiv:2207.13011 [astro-ph.CO].</list_item>
<list_item><location><page_19><loc_52><loc_9><loc_92><loc_10></location>[60] A. Semenaite, A. G. S'anchez, A. Pezzotta, J. Hou,</list_item>
<list_item><location><page_20><loc_12><loc_92><loc_49><loc_93></location>A. Eggemeier, M. Crocce, C. Zhao, J. R. Brownstein,</list_item>
<list_item><location><page_20><loc_12><loc_87><loc_49><loc_92></location>G. Rossi, and D. P. Schneider, Beyond - ΛCDM constraints from the full shape clustering measurements from BOSS and eBOSS, Mon. Not. Roy. Astron. Soc. 521 , 5013 (2023), arXiv:2210.07304 [astro-ph.CO].</list_item>
<list_item><location><page_20><loc_9><loc_81><loc_49><loc_86></location>[61] A. Glanville, C. Howlett, and T. M. Davis, Fullshape galaxy power spectra and the curvature tension, Mon. Not. Roy. Astron. Soc. 517 , 3087 (2022), arXiv:2205.05892 [astro-ph.CO].</list_item>
<list_item><location><page_20><loc_9><loc_76><loc_49><loc_81></location>[62] W. J. Percival et al. (2dFGRS Team), Parameter constraints for flat cosmologies from CMB and 2dFGRS power spectra, Mon. Not. Roy. Astron. Soc. 337 , 1068 (2002), arXiv:astro-ph/0206256.</list_item>
<list_item><location><page_20><loc_9><loc_71><loc_49><loc_76></location>[63] J. A. Kable, G. E. Addison, and C. L. Bennett, Quantifying the CMB degeneracy between the matter density and hubble constant in current experiments, The Astrophysical Journal 871 , 77 (2019).</list_item>
<list_item><location><page_20><loc_9><loc_66><loc_49><loc_70></location>[64] P. Astone and G. D'Agostini, Inferring the intensity of poisson processes at the limit of the detector sensitivity (with a case study on gravitational wave burst search) (1999), arXiv:hep-ex/9909047 [hep-ex].</list_item>
<list_item><location><page_20><loc_9><loc_62><loc_49><loc_65></location>[65] G. D'Agostini, Confidence limits: what is the problem? is there the solution? (2000), arXiv:hep-ex/0002055 [hepex].</list_item>
<list_item><location><page_20><loc_9><loc_58><loc_49><loc_61></location>[66] S. Gariazzo, Constraining power of open likelihoods, made prior-independent, Eur. Phys. J. C 80 , 552 (2020), arXiv:1910.06646 [astro-ph.CO].</list_item>
<list_item><location><page_20><loc_9><loc_54><loc_49><loc_57></location>[67] G. Van Rossum and F. L. Drake Jr, Python reference manual (Centrum voor Wiskunde en Informatica Amsterdam, 1995).</list_item>
<list_item><location><page_20><loc_9><loc_51><loc_49><loc_53></location>[68] J. D. Hunter, Matplotlib: A 2d graphics environment, Computing in Science & Engineering 9 , 90 (2007).</list_item>
<list_item><location><page_20><loc_9><loc_46><loc_49><loc_51></location>[69] S. van der Walt, S. C. Colbert, and G. Varoquaux, The NumPy array: A structure for efficient numerical computation, Computing in Science & Engineering 13 , 22 (2011).</list_item>
<list_item><location><page_20><loc_9><loc_42><loc_49><loc_45></location>[70] P. Virtanen et al. , SciPy 1.0-Fundamental Algorithms for Scientific Computing in Python, Nature Meth. 17 , 261 (2020), arXiv:1907.10121 [cs.MS].</list_item>
<list_item><location><page_20><loc_9><loc_36><loc_49><loc_41></location>[71] A. M. Price-Whelan et al. (Astropy), The Astropy Project: Building an Open-science Project and Status of the v2.0 Core Package, Astron. J. 156 , 123 (2018), arXiv:1801.02634.</list_item>
<list_item><location><page_20><loc_9><loc_33><loc_49><loc_36></location>[72] J.-c. Hwang and H. Noh, Axion as a Cold Dark Matter candidate, Phys. Lett. B 680 , 1 (2009), arXiv:0902.4738 [astro-ph.CO].</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_20><loc_52><loc_89><loc_92><loc_93></location>[73] J.-c. Hwang and H. Noh, Axion as a fuzzy dark matter candidate: proofs in different gauges, JCAP 03 (03), 001, arXiv:2109.05436 [astro-ph.CO].</list_item>
<list_item><location><page_20><loc_52><loc_85><loc_92><loc_89></location>[74] R. Trotta, Bayes in the sky: Bayesian inference and model selection in cosmology, Contemp. Phys. 49 , 71 (2008), arXiv:0803.4089 [astro-ph].</list_item>
<list_item><location><page_20><loc_52><loc_80><loc_92><loc_85></location>[75] M. Zaldarriaga, U. Seljak, and E. Bertschinger, Integral solution for the microwave background anisotropies in nonflat universes, Astrophys. J. 494 , 491 (1998), arXiv:astro-ph/9704265.</list_item>
<list_item><location><page_20><loc_52><loc_76><loc_92><loc_80></location>[76] M. Zaldarriaga and U. Seljak, Cmbfast for spatially closed universes, Astrophys. J. Suppl. 129 , 431 (2000), arXiv:astro-ph/9911219.</list_item>
<list_item><location><page_20><loc_52><loc_38><loc_92><loc_76></location>[77] P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, ˙ I. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H. Ribeiro, F. Pedregosa, P. van Mulbregt, A. Vijaykumar, A. P. Bardelli, A. Rothberg, A. Hilboll, A. Kloeckner, A. Scopatz, A. Lee, A. Rokem, C. N. Woods, C. Fulton, C. Masson, C. Haggstrom, C. Fitzgerald, D. A. Nicholson, D. R. Hagen, D. V. Pasechnik, E. Olivetti, E. Martin, E. Wieser, F. Silva, F. Lenders, F. Wilhelm, G. Young, G. A. Price, G.-L. Ingold, G. E. Allen, G. R. Lee, H. Audren, I. Probst, J. P. Dietrich, J. Silterra, J. T. Webber, J. Slaviˇc, J. Nothman, J. Buchner, J. Kulick, J. L. Schonberger, J. V. de Miranda Cardoso, J. Reimer, J. Harrington, J. L. C. Rodr'ıguez, J. NunezIglesias, J. Kuczynski, K. Tritz, M. Thoma, M. Newville, M. Kummerer, M. Bolingbroke, M. Tartre, M. Pak, N. J. Smith, N. Nowaczyk, N. Shebanov, O. Pavlyk, P. A. Brodtkorb, P. Lee, R. T. McGibbon, R. Feldbauer, S. Lewis, S. Tygier, S. Sievert, S. Vigna, S. Peterson, S. More, T. Pudlik, T. Oshima, T. J. Pingel, T. P. Robitaille, T. Spura, T. R. Jones, T. Cera, T. Leslie, T. Zito, T. Krauss, U. Upadhyay, Y. O. Halchenko, and Y. V.B. and, SciPy 1.0: fundamental algorithms for scientific computing in python, Nature Methods 17 , 261 (2020).</list_item>
<list_item><location><page_20><loc_52><loc_33><loc_92><loc_37></location>[78] R. Hloˇzek, D. J. E. Marsh, D. Grin, R. Allison, J. Dunkley, and E. Calabrese, Future CMB tests of dark matter: Ultralight axions and massive neutrinos, Phys. Rev. D 95 , 123511 (2017), arXiv:1607.08208 [astro-ph.CO].</list_item>
</document>
[
{
"title": "On the cosmological evolution of Scalar Field Dark Matter in the CLASS code: accuracy and precision of numerical solutions",
"content": "L. Arturo Ure˜na-L'opez 1, ∗ and Francisco X. Linares Cede˜no 2, † 1 Departamento de F'ısica, DCI, Campus Le'on, Universidad de Guanajuato, 37150, Le'on, Guanajuato, M'exico 2 Instituto de F'ısica y Matem'aticas, Universidad Michoacana de San Nicol'as de Hidalgo, Edificio C-3, Ciudad Universitaria, CP. 58040 Morelia, Michoac'an, M'exico. (Dated: July 13, 2023) We present a numerical analysis of the cosmological evolution of scalar field dark matter (SFDM) in the Boltzmann code CLASS , based on a dynamical system analysis of previous works. We show a detailed study of the evolution of the different dynamical variables, and in particular of the energy density and its corresponding linear perturbations. The numerical results are in good agreement with those of the original SFDM equations of motion, and have better accuracy than other approaches. In addition, we calculate the temperature and matter power spectra and discuss the reliability of their numerical results. We also give simple examples in which we can put constraints on the field mass using recent likelihoods incorporated in the Monte Carlo Markov Chain sampler MontePython .",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "In the modern era of cosmology [1, 2], it is mandatory to develop theoretical models able to describe the universe at large scales, with the precision that current data demand [2]. One key theoretical ingredient, particularly important for the formation of cosmic structure, is dark matter (DM), which must capture the physics of such a process in a mathematically consistent way [3]. This always starts, for the cosmological setting, with properly solving the Einstein-Boltzmann system describing the cosmological evolution for the initial perturbations of both the matter components and the metric tensor at the linear level [4]. Different Boltzmann solvers have been programmed for the linearized form of the Einstein equations in a Friedmann-Robertson-Walker-Lemaˆıtre (FRWL) universe, such as CMBFAST [5], CMBEASY [6], CAMB [7] and CLASS [8]. Only the latter two, CAMB and CLASS , have been kept up to date, and both are used by the community of cosmologists. The aforementioned codes consider the cold dark matter (CDM) model as the main matter component, and they are very well suited to explore most of its properties, although they have been amended to study alternative dark matter models in recent years. One of these alternative models to CDM, which is currently one of the compelling proposals that has been explored for the last two decades, is that based on a scalar field. It is found in the literature under several names: Scalar Field Dark Matter (SFDM), Ultralight Axions, Fuzzy Dark Matter, Axion-like particle, Bose-Einstein Condansate, Wave Dark Matter (some initial works on this model are [9-15], whereas more recent work can be found at [16-24]). All these names reflect the particular properties of this DM particle. Its mathematical description is given by a scalar field ϕ (which can be real or com- lex), with a fiducial ultralight mass of m ϕ c 2 ∼ 10 -22 eV that can be produced by the Peccei-Quinn mechanism for pseudo-Goldstone bosons (as is the case for QCD axions) and whose quantum nature manifests itself at cosmological scales through its imprint on the structure formation at small scales. Therefore, we shall refer to this model as SFDM hereafter. In order to solve the cosmological dynamics of the linear perturbations of SFDM, CAMB have been modified to include a real scalar field as a DM component. This version has been called AxionCAMB [17], and it is written for a scalar field endowed with a quadratic potential V ( ϕ ) ∝ ϕ 2 . This is why this case is called the free case . On the other hand, CLASS has been modified by the authors of this work with the aim of including the SFDM model with quadratic potential: class.FreeSF [16]. Both codes deal with the background dynamics and linear perturbations of the SFDM, and it is possible to obtain the anisotropies of the Cosmic Microwave Background radiation (CMB), as well as the Matter Power Spectrum (MPS) for several mass values of the SFDM particle 1 . Whereas the only SFDM model that has been treated with AxionCAMB is that of the noninteracting case, the authors made new amendments to CLASS to study potentials with self-interaction in the scalar field potential. For example, the full axion-like potential V ( ϕ ) ∝ cos( ϕ ) was fully implemented for the first time in CLASS in the work [25], where it was shown that this model presents an excess of power on small scales in the MPS with respect to CDM 2 , with a greater discussion of the cosmological signatures of this SFDM model in [27]. The other SFDM potential we have considered in CLASS is that of a hyperbolic function V ( ϕ ) ∝ cosh( ϕ ) [28], where the selfinteracting term has the opposite sign to that of axions. As we will show later, the mathematical treatment of the SFDM cosmological evolution used in the modified versions of CLASS makes use of new dynamical variables based on a dynamical system analysis. We have shown that this is useful for a unified description of different scalar potentials, showing our own method to deal with tree different cases in a one-parametric way [29]. The modified versions of CLASS mentioned above for SFDM deal with the linear evolution of density perturbations. This information (initial conditions at radiationdominated era, physical effects on observables such as CMB, MPS, Halo Mass Function and others) turns out to be crucial for the subsequent realization of realistic numerical simulations of structure formation in the nonlinear regime, see for example [30-34]. Our main goal in the present work is to give more details of the mathematical approach we have used in previously amended versions of CLASS , and in turn to show its robustness in describing the physical processes of SFDM up to the level of linear density perturbations. This is done here for SFDM endowed with a quadratic potential, where m ϕ is the mass of the scalar field particle and the only free parameter in the model. Note that we are using natural units with h = c = 1, and then m ϕ is given in units of eV. Whenever appropriate, we will compare our method to the common fluid approximation to the SFDM dynamics used in other works, following the description in Refs. [17, 35, 36] This paper is organized as follows. In Sec. II, we develop the mathematical formulation for the evolution of the background and linear density perturbations, in terms of new dynamical variables that are appropriate to handle the particularities of SFDM. We also establish the appropriate initial conditions for the scalar field to behave as the DM component at late times, for both background and linear quantities. Section III is dedicated to the description of the typical regime of rapid oscillations of SFDM at late times in its evolution and to the way in which we deal with them for their reliable numerical computation. In particular, we show a detailed study of the evolution of the (barotropic) equation of state, the energy density, and finally the linear density perturbations. As an application of our method in the Boltzmann code CLASS , we present the temperature and matter power spectra and discuss the reliability of our numerical results. Moreover, we also give simple examples in which we can put constraints on the field mass m ϕ using recent likelihoods incorporated in the Monte Carlo Markov Chain (MCMC) sampler MontePython [37, 38], taking advantage of its close interoperability with CLASS . The comparison and equivalence between our approach and the original formulation in terms of the scalar field itself is presented in Sec. IV. We explicitly show the transformation between our variables and the original field ones ( ϕ, ˙ ϕ ), and that the numerical results of our method are completely equivalent to the standard field approach. Finally, in Sec. V we summarize and discuss our results, highlighting the advantages of our method.",
"pages": [
1,
2
]
},
{
"title": "II. MATHEMATICAL BACKGROUND",
"content": "In this section, we present the equations of motion for the SFDM model in the context of an expanding universe, and the subsequent transformations we use to make them more suitable for numerical computations. The original motivations and some extra details of the method described here can be found in [16, 25, 39].",
"pages": [
2
]
},
{
"title": "A. Background evolution",
"content": "Let us consider a spatially-flat FRWL line element, where a ( t ) is the scale factor. The background equations for ordinary matter, which is represented by perfect fluids with density ρ j and pressure p j , as well as for SFDM field ϕ endowed with the potential (1), are given by where κ 2 = 8 πG . The dot denotes the derivative with respect to cosmic time t , and H = ˙ a/a is the Hubble parameter. In the equations above, the scalar field density ρ ϕ and pressure p ϕ are defined, respectively, as For the SFDM part, it is convenient to use the following change of variables [16, 25, 27, 39-41], To understand the meaning of the new variables, we write here the SFDM density parameter Ω ϕ and equation of state (EOS) w ϕ , Thus, β is the logarithm of the energy density parameter, and θ , being an internal polar angle, is directly related to w ϕ . Lastly, y 1 is simply the ratio of the field mass to the Hubble parameter (i.e. dimensionless by definition), which is an ubiquitous quantity in all methods for solutions of the SFDM equations of motion. Using the new variables, the Klein-Gordon equation (3d) transforms into the following set of first order differential equations [16], Here, a prime denotes derivatives with respect to the number of e -folds of expansion N = ln a . The total EOS w tot , which is used in the foregoing equations, can be calculated from the ratio of the total pressure p tot to the total density ρ tot in the Universe, and its explicit expression is given in terms of the EOS of the different components of matter as Although the new dynamical variables are convenient for numerical purposes, we need to recover the standard quantities used in Boltzmann and cosmological codes in general, for instance, the density and pressure of the SFDM component. It can be shown that such quantities can be recovered in the form where the sum takes into account only the density components other than the SFDM one.",
"pages": [
2,
3
]
},
{
"title": "B. Linear density perturbations",
"content": "We repeated the comparison of the solutions with the case of linear density perturbations. This time we solved the linearly perturbed KG equation (12), again using the same initial conditions for the two sets of variables, the originals φ and ˙ φ and the polar ones δ 0 and δ 1 . Although the perturbed polar variables are α and ϑ , see Eqs. (13), recall that our final perturbed variables are those of Eqs. (14) and their corresponding equations of motion (15). The transformation from the polar variables to the field ones is given by the expressions We only used Eqs. (37) to set the initial conditions of the field variables φ and ˙ φ in correspondence with those of the polar variables, and then the field equation (12) was solved separately. We then show in Fig. 12 the evolution of the perturbation variables φ and ˙ φ as a function of the scale factor and for two values of the wavenumber k : 0 . 01 Mpc -1 for large scales (top panels) and 10 Mpc -1 for small scales (bottom panels). In the two cases, the solutions from the polar variables via Eqs. (37) are superimposed on those of the field variables (obtained directly from Eq. (12)), which are identical up to the time the numerical solutions were followed ( θ ⋆ = 30 π for the polar variables and θ ⋆ = 100 π for the field ones). Note that the agreement goes beyond the cutoff point of the polar variables, which means that the cutoff of the trigonometric functions (22) delivers from Eqs. (15) the expected results of the original field variables.",
"pages": [
13,
14,
15
]
},
{
"title": "C. Initial conditions",
"content": "Calculation of initial conditions implies an approximate solution of the equations of motion starting well within the epoch of radiation domination, with a corresponding initial value of the scale factor of the order of a i = 10 -14 [16]. We start with the initial condition of the auxiliary variable, which reads where H 0 ( H i ) is the present (initial) value of the Hubble parameter, and h is its reduced value. Note that for the calculation of H i we assume radiation domination at early times. It is clear from Eq. (18) that the auxiliary variable is very small at early times, y 1 i ∼ 10 -14 , for the values of m ϕ that are of interest for SFDM models. On the contrary, its value at present is very large y 1 ∼ 10 10 , which means that it changes by almost 24 orders of magnitude during its evolution. As for the polar angle, there is an atractor solution when the equations of motion are solved in the linear regime at early times, from which we obtain the following equations. The initial condition of the variable β is found by matching the early and late time solutions at the beginning of the rapid oscillations of the field. The resultant equation is Here, A is a constant coefficient that is adjusted by the numerical code, typically with a shooting mechanism, to match the value of the desired density parameter Ω ϕ 0 at the present time. On the other hand, the initial conditions of the density perturbations is a more involved procedure, but it reveals the existence of an attractor solution for the dynamical variables in the form The details of the numerical implementation of the polar method of the sections above in the amended version of the Boltzmann code CLASS are presented in Appendix C, from which we obtained the numerical solutions that are presented in the sections below.",
"pages": [
4
]
},
{
"title": "III. THE STAGE OF RAPID FIELD OSCILLATIONS",
"content": "For the field to behave as a CDM component, it should enter a phase of rapid oscillations around the minimum of the potential. Under our polar transformation (4), such fast oscillations are equivalent to the following averages during a Hubble time, ⟨ sin θ ⟩ = 0 and ⟨ cos θ ⟩ = 0. However, we must be careful in the form the oscillations are dealt with, as the solutions at late times depend on the choices made for the averaged dynamical quantities. Here, we explain in detail our method for the cutoff of the rapid oscillations proposed in [16, 25, 27].",
"pages": [
4
]
},
{
"title": "A. Outline of the general method",
"content": "It is well known that the stage of rapid oscillations is difficult to solve numerically, and then we follow here the prescription in Ref. [16] in that the cosine and sine functions in the equations of motion are replaced by the cutoff trigonometric functions. where θ ⋆ is a reference value. In this form, cos ⋆ θ = cos θ (sin ⋆ θ = sin θ ) if 0 ≤ θ < θ ⋆ , while cos ⋆ θ → 0 (sin ⋆ θ → 0) if θ ≫ θ ⋆ . In the following, we will refer to t ⋆ as the time at which we apply the cutoff for the trigonometric functions and then to θ ( t ⋆ ) = θ ⋆ . We will also refer to t osc as the time for the beginning of the rapid oscillations. However, and in contrast to t ⋆ , the value of t osc cannot be precisely determined, and in our formalism it is just a reference value without a major effect on the numerical solutions. The general method can be described as follows. We replace all the sine and cosine terms with the cutoff functions (22) in the equations of motion (7), and then solve them numerically. Note that the solutions are then continuous at t = t ⋆ by construction, and we only need to be sure that the cutoff is applied after the onset of the rapid oscillations so that t ⋆ > t osc . However, it is difficult to determine the time at the start of the oscillations, and this also makes impractical the calculation of t ⋆ . The reason is that cosmic time is a dimensional quantity calculated from the integration of the Friedmann equation (3a), which depends on all dynamical variables in a cosmological model 3 . As we shall show in the following, it is better to calibrate the cutoff time using the polar variable θ , which is dimensionless and also a direct dynamical variable in our set of field equations.",
"pages": [
4,
5
]
},
{
"title": "B. The case of the polar angle θ",
"content": "To understand the general behavior of the solutions after the cutoff of the trigonometric functions, we start with the equation of motion (7a) of the polar angle θ , which for convenience we write in terms of cosmic time t , where we have considered H = 1 / (2 t ) for RD. Notice that we can write Eq. (23) in the form which shows that the evolution of θ as a function of the dimensionless variable 2 m ϕ t is the same, regardless of the value of the field mass m ϕ . We will use this feature in the plots below, but we still refer to Eq. (23) to obtain semi-analytical expressions. First, we assume at the beginning that 0 < θ ≪ 1, and then sin θ ≃ θ . As a consequence, Eq. (23) becomes and the solution that satisfies the initial condition θ (0) = 0 is We now use Eq. (25b) on the right-hand side of Eq. (23), from which we obtain the new differential equation, whose solution is where Si( x ) is the sine integral. It can be shown that, at early times m ϕ t ≪ 1, we recover the solution (25b). Likewise, for the late-time evolution we can approximate the sine integral by its asymptotic behavior, Si( x ) ≃ π/ 2 -cos( x ) /x + O (1 /x 2 ) for x ≫ 1, to obtain Any further iteration to integrate Eq. (23) cannot be expressed in closed form, but, as we present below, Eq. (26b) suffices to analyze the main properties in the time evolution of the polar angle θ . The numerical solutions of Eq. (23) for different values of the field mass m ϕ , and as a function of the scale factor a , are shown in the top panel of Fig. 1. It can be seen that the polar variable shows two asymptotic behaviors that correspond to the semi-analytical solutions: at early times θ/ (2 m ϕ t ) → 5 / 4, while at late times θ/ (2 m ϕ t ) → 1, as indicated by Eqs. (25b) and (26c), respectively. These asymptotic limits are the same for any field mass m ϕ , and the latter only influences the time at which the transition occurs between the two values. In the bottom panel of Fig. 1 we have the evolution of the polar variable but now in terms of dimensionless cosmic time 2 m ϕ t . Although the numerical solutions are shown in different colors, it is clear that the corresponding curves are superimposed on each other because their behavior is the same. It can be seen that the semi-analytical solutions agree well with the numerical ones. In particular, the iterative solution (26b) gives a reliable description of the early and late time trends of the solutions, and it even gives a good approximation to the oscillations of the numerical solutions at intermediate times 2 m ϕ t ≃ 4, which is also the time at which θ ≃ π/ 2. That is, it also corresponds to the time at which the scalar field EOS first crosses the zero value w ϕ ≃ 0. As this occurs within radiation domination, we also find 2 m ϕ t = m ϕ /H ≃ 4, which is the typical time for the start of the oscillations estimated for these field systems. Surprisingly, the bottom panel of Fig. 1 also shows that the late-time expression (26c) also seems to work very well from the intermediate times onward, that is, almost from the start of the rapid oscillations. This means that we can safely write That Eq. (27a) is also a very good approximation can be understood from the properties of the sine integral Si( x ), which rapidly converges to its asymptotic value of π/ 2, with small oscillations around it that rapidly decay away. In what follows, we will use Eq. (27a) to describe the behavior of the polar angle after the onset of rapid oscillations of the field ϕ . We can also use Eq. (27a) also to convert the cutoff time t ⋆ into a cutoff polar angle θ ⋆ , which is both a dynamical variable and the argument in the modified trigonometric functions (22). Hence, the relation between the cutoff values t ⋆ and θ ⋆ is Equations (27) are a central result in the description of our method. First, Eq. (27a) shows that the polar angle evolves linearly with cosmic time t after the cutoff time. Second, Eq. (27b) allows us to determine the cutoff point of rapid oscillations via the polar angle θ ⋆ , which is more convenient from the numerical point of view and justifies the use of the cutoff expressions (22). To finish this section, in Fig. 2 we show the numerical evolution of the scalar field EOS w ϕ as a function again of the dimensionless variable 2 m ϕ t , and we see that it first passes through zero (for θ = π/ 2) at around the time 2 m ϕ t osc ≃ 3 . 47, which we use to mark the time t osc for the start of the rapid oscillations. Notice that in terms of the usual mass-to-Hubble ratio, this is equivalent to m ϕ /H osc ≃ 3 . 47, a value used as a reference in other studies of field models. w For comparison, we also plot in Fig. 2 the result of the expression cos(2 m ϕ t -3 π/ 4). Notice that there is very good agreement of this curve with the original EOS w ϕ almost from the start of the rapid oscillations, which means that we can use the following expression for the field EOS, Equation (28) agrees with the common wisdom that the EOS oscillates with a frequency directly related to the field mass via 2 m ϕ . The phase of 3 π/ 4 becomes negligible at very late times, but as we shall see, it must be taken into account for a correct description of the dynamics at intermediate times of other variables after the onset of the rapid oscillations.",
"pages": [
5,
6
]
},
{
"title": "C. The case of the energy density ρ ϕ",
"content": "We have found semi-analytical results to follow the evolution of the polar variable θ ( t ), which are in good agreement with the numerical results. However, one should worry about the numerical accuracy as the scalar field equations of motion must be solved together with other matter components in Boltzmann codes, covering an ample time interval for a complete description of diverse cosmological phenomena. Here, we perform some accuracy tests using the amended version of the Boltzmann code CLASS , taking some guidelines from our semi-analytical results. Our main concern is the choice of the cutoff value θ ⋆ . As we shall see, the cutoff procedure leaves a residual difference with respect to the expected late-time evolution of a given variable that can be minimized if θ ⋆ ≫ 1. An example of the effects of the cutoff on the evolution of different quantities is the energy density ρ ϕ , which obeys the equation whose formal solution after the onset of the rapid oscillations can be written as with It suffices to understand the behavior of the density before the time of radiation-matter equality, and for that we proceed as follows. Equation (30b) can be written in a more convenient form if we use Eq. (27a) for the evolution of the polar angle, and then it can be shown that where Si( x ) and Ci( x ) are the sine and cosine integrals, respectively. We are interested in the evolution of the density at late times, that is, 2 m ϕ t ≫ 1. Given the asymptotic properties of the sine and cosine integrals for x ≫ 1, Si( x ) ≃ π/ 2 -cos( x ) /x and Ci( x ) ≃ sin( x ) /x , and substituting the polar angle θ using Eq. (27a), we find that The last term in Eq. (31b) will be responsible for a residual oscillatory term in the density, which will decay away. In fact, if we define we can also write Eq. (30a) in a more neat form as which is correct for θ ≫ 1. There are two parts in the rhs of Eq. (33): one that evolves steadily at the same rate as a pressureless component ( ∼ a -3 ), and another one that contributes with a decaying oscillating term around unity provided by the exponential function. Moreover, ρ ϕ 0 represents the correct asymptotic value of the field density at late times. The following question arises: can we be assured that our cutoff procedure of the rapid oscillations recovers the right evolution of the density at late times? First, notice that, in principle, ρ ϕ 0 in Eq. (32) is fixed at the onset of the rapid oscillations, but we do not need to be very specific about the values of the parameters at this time. In our method, unlike others in the literature, we do not require knowing the precise value of t osc , and we can be completely oblivious to it as long as we ensure t ⋆ > t osc . The reason is simple: the cutoff of the rapid oscillations is made smoothly at the level of the equations of motion, and then there is no loss of continuity in the numerical variables. To answer our question above, Eq. (33) should be compared with the truncated case. After the cutoff, the equation of motion for the density is whose solution simply is ρ ( t > t ⋆ ) = ρ ⋆ /a 3 , where ρ ⋆ is the density value at t = t ⋆ . By the continuity of the solutions at t ⋆ for Eqs. (29) and (34a), we finally get The result is quite direct: the cutoff introduces a small mismatch, and the correct asymptotic value of the density is not recovered from the solution (34b). But the discrepancy depends on the cutoff value θ ⋆ , and, in principle, it can be made as small as required if θ ⋆ ≫ 1. However, Eq. (34b) itself suggests a faster route, which is to choose θ ⋆ = nπ , where n is an integer number, although a large enough one so that still θ ⋆ ≫ 1, as this also allows us to neglect other oscillatory terms in the sine and cosine integrals in Eq. (31a) that are of order O (1 /θ 2 ⋆ ) and smaller. Numerical examples of the evolution of the field density, in the combination ( ρ ϕ /ρ ϕ 0 ) a 3 , are shown in Fig. 3, for different values of the field mass m ϕ but with a fixed value θ ⋆ = 30 π . Here, ρ 0 is the last value in each of the numerical solutions. Note that the asymptotic value is always unity. In the left panel, we see that the density makes a transition to a pressureless behavior once the rapid oscillations start, but as before, the transition time depends on the field mass m ϕ . If the density is plotted as a function of the variable θ , as in the right panel of Fig. 3, we find that all curves collapse again into a single one, and there is a common evolution for all cases. Moreover, we also show the curve arising from Eq. (33) (denoted by exp[ B ( θ )]), and it can be seen that it quite well matches the numerical curves after the onset of the rapid oscillations. Now, in Fig. 4 we show the effects arising from different choices of the cutoff value θ ⋆ and with a fixed mass m ϕ = 10 -24 eV. In the left panel, we take θ ⋆ = 10 π, 20 π, 30 π , and we see that the late-time evolution is the same for all cases (the curves are superimposed on each other) even though the cutoff of the oscillations appears at different times. Also, the correct asymptotic value of the density is recovered, and in all cases it corresponds to the expected average of the density oscillations. For the right panel in Fig. 4 we take the cutoff values θ ⋆ = 10 π, 10 . 5 π . To have a good matching of the first density oscillations in the two cases and to facilitate the comparison of their asymptotic values, we applied the correction (exponential) factor that appears in Eq. (33) to the case θ ⋆ = 10 . 5 π . 4 It can be seen that the cutoff occurs at the maximum of the last oscillation, and hence the asymptotic value is larger than the correct one. We can give an estimate of the error between the two asymptotic values, which according to Eq. (34b) is For the particular case with θ ⋆ = 10 . 5 π we get This difference is not negligible if one desires high precision of the solution, and it clearly illustrates the necessity to choose the cutoff value θ ⋆ wisely.",
"pages": [
6,
7,
8
]
},
{
"title": "D. Linear density perturbations",
"content": "In contrast to the background quantities in the sections above, the analysis of Eqs. (15) is much more involved because the evolution of the quantities δ 0 and δ 1 is coupled to that of the so-called metric continuity ¯ h ' / 2 through the perturbed Einstein equations. It is not possible to make a clear separation of the oscillatory and non-oscillatory terms in the formal solution, and a wise decision on the cutoff value θ ⋆ cannot easily be decided. In Fig. 5 we show the evolution of the density contrast δ 0 for a scale much larger than the Jeans wavenumber, so that k 2 ≪ k 2 J . In the upper panels, we see the cases with the same cutoff values θ ⋆ used previously in Fig. 4. We can see that the numerical solutions have a larger stage of rapid oscillations for larger values of θ ⋆ , as in the background case. Also, the choice θ ⋆ = nπ , see Eq. (34b) and the text below, does not make the numerical solution coincide with the nonoscillatory solutions at t ⋆ . There is a small, but visible, mismatch between the solutions for the different values of θ ⋆ considered in the graphs. Nevertheless, the numerical solution is able to catch up with the non-oscillatory solution after the cutoff of the rapid oscillations, and this is because of the structure of the system (15): its solution is driven by the nonhomogeneous term involving the metric continuity ¯ h ' , which acts as an attractor solution even at early times. At late times, the value of | δ 0 | oscillates around ¯ h/ 2 with an amplitude that does not decay. It is only on average that the scalar field density contrast can be identified with the CDM one, ⟨ δ 0 ⟩ = δ CDM . Other intrinsic oscillations, which we refer to as scale oscillations, are noticeable for scales smaller than the Jeans scale: k 2 /k 2 J ≳ 1, which appear even after the cutoff of the rapid oscillations. This is because the term k 2 /k 2 J plays the role of a frequency in terms of the number of e -folds N in Eqs. (15) and not in cosmic time t . For such small scales, from the very beginning there may be a combination of rapid oscillations with scale oscillations, and the choices of θ ⋆ at the transition time give different results for δ 0 ( t ⋆ ) and δ 1 ( t ⋆ ). Our numerical results for the density contrast δ 0 for small scales are shown in the right panels of Fig. 5, where we see noticeable differences in the late-time behavior of the solutions. It is clear that it is necessary to follow the numerical solutions for longer before cutting off the rapid oscillations and to achieve some convergence of the solutions. However, the evolution of metric continuity ¯ h/ 2 is always smooth and the same regardless of the cutoff value θ ⋆ , although its amplitude is also highly suppressed with respect to the CDM case. We also present the cases θ ⋆ = 10 π, 10 . 5 π in Fig. 6. Both cases show that at the cutoff time none of the numerical solutions agree with the CDM solution, but they join it quickly because of the driving term ¯ h ' / 2 in Eqs. (14). In summary, the numerical solutions clearly show that the scalar field density contrast behaves on average like the CDM one on large scales, and that the choice θ ⋆ = nπ only helps a little for the numerical solution to have a smooth transition at the cutoff time t ⋆ . There are no further consequences, because the attractor character of the equations of motion for the density perturbations eventually leads to the right numerical result 5 .",
"pages": [
9
]
},
{
"title": "E. Mass power spectrum and temperature anisotropies",
"content": "To study the issue of convergence in the solution of density perturbations for all scales, we plot the resultant mass power spectrum (MPS) in the upper panel of Fig. 7 the resultant mass power spectrum (MPS) for different choices of θ ⋆ , and also the relative differences in the numerical solutions. The field mass in these examples was fixed at m ϕ = 10 -24 eV. The first thing to notice is that there is complete agreement with the MPS of CDM for large scales, represented by the wavenumbers k < 0 . 5 h/ Mpc. Regarding the convergence of numerical solutions for different choices of the cutoff value θ ⋆ , it can be seen that there is complete agreement for almost all scales, except for the interval k = 2 -20 h/ Mpc, where the difference can be as large as 100%. However, these discrepancies appear once the MPS is greatly suppressed compared to the CDM, although it should be noticed that the agreement is recovered once the MPS reaches a steady stage at small scales of the form P ( k ) ∼ k -3 . Although one could use the solution with the highest cutoff value θ ⋆ , the overall conclusion is that one can safely take θ ⋆ = 30 π for reliable results with the additional advantage of saving computational time. We repeat the comparison between different cutoff values θ ⋆ for the case of the temperature spectrum in terms of the variable D ℓ = ℓ ( ℓ +1) C ℓ / 2 π in the lower panel of Fig. 7. This time the solutions are more similar, among themselves with different resolutions and also with the corresponding spectrum of CDM. This can be verified in the lower graph with the relative differences of the solutions with respect to choice θ ⋆ = 100 π , which are very small and below 0 . 1%. For reference, we also show the estimated rule of thumb for bias-free parameter inference, which is given by the curves ± 3 /ℓ at large ℓ . We can see that our numerical results are consistent with such a constraint for the cutoff values chosen for θ ⋆ (see also [5053]). k =10Mpc 1",
"pages": [
9,
10
]
},
{
"title": "F. Constraints on m ϕ from the matter power spectrum",
"content": "Here, we describe possible constraints of SFDM models from the matter power spectrum, according to recent estimates of the UV galaxy luminous function [49, 54] with the package Gallumi 6 , and the effective field theory of the large scale structure (EFTofLSST) as in [55, 56] with the package PyBird 7 , respectively. These two are likelihoods of a recent addition to the MCMC software MontePython [37, 38], which are capable of exploring the power spectrum at semilinear scales and can be as competitive as those of Lymanα observations. We replicated the studies in [49, 55] with some shortcuts, as our aim was to focus our attention on the constraints on the field mass m ϕ . For the analysis, we used Gaussian priors on the following parameters: the angular scale for the sound horizon with 100 θ s = 1 . 0411 and σ θ s = 0 . 0003 8 , and the physical density of baryons ω b = 0 . 02233 with σ b = 0 . 00036 9 . Fixing the sound horizon θ s is known to also fix the combination Ω m h 3 . 4 , with Ω m the physical density of total matter and h the reduced Hubble constant [49, 62, 63]. This means that our Gaussian prior on 100 θ s , with h = 0 . 657, acts as an indirect prior on the combination Ω m = Ω b + Ω ϕ . For the field mass m ϕ , we considered a flat prior on the logarithmic scale in the range log( m ϕ / eV) = [ -26 , -18]. All other cosmological parameters in the models, such as the amplitude of the power spectrum, were fixed to their Planck 2018 CMB values. The likelihood we chose for the UV luminosity function is that of the so-called Model I in [49], with its corresponding formalities and data. In the case of EFTofLSST, we selected the BOSS and eBOSS data sets as in[55]. It must be noted that some assumptions in the likelihoods have been made under the CDM paradigm only and would need to be amended for the case of SFDM. Taking into account these caveats, the results reported in the following may be stronger than in the case in which some of the assumptions are corrected for SFDM. The resulting posterior distributions for the physical densities of baryons ω b and SFDM ω sfdm , and the field mass m ϕ , after marginalizing over the nuisance parameters of the likelihoods, are shown in Fig. 8. As expected, the separate constraints on the physical densities of baryons and SFDM are practically the same, as they are influenced mostly by the previously assumed priors. Not surprisingly, the posterior distribution of the field mass shows that each data set only constrains m ϕ from below, which means that the likelihoods are insensitive to variations of m ϕ above a certain threshold value. The data set with the most constraining power is the UV luminosity function, whereas eBOSS is the less constraining one. To properly calculate the lower bounds for m ϕ from the confidence regions in Fig. 8, we use the method in [6466] to obtain prior independent constraints by means of the so-called shape distortion function R , appropriate for the so-called open likelihoods as in the present case. As explained in [66], the function R will allow us to use the data to define the region below which m ϕ is disfavored, regardless of the prior assumptions we have chosen. An advantage of the function R is that for its calculation we only need to know the posterior distribution of the field mass m ϕ , which we obtained from the code MontePython . The resultant shape distortion function is shown in the lower panel of Fig. 8 on the logarithmic scale. Note that R → 1 for large values of the field mass, in this case log( m ϕ c 2 / eV) → -18 which is the upper value in our prior range. The sharp decay of R at lower values of m ϕ helps us to calculate an appropriate lower bound. Following the convention in [64-66], it can be seen that if ln R = -3 (moderate level according to Jeffrey's scale), we can say that the data strongly favor the regions log( m ϕ c 2 / eV) > -25 . 4 for eBOSS, log( m ϕ c 2 / eV) > -24 . 6 for BOSS, and log( m ϕ c 2 / eV) > -22 for UV LF 10 .",
"pages": [
10,
11
]
},
{
"title": "IV. COMPARISON WITH THE ORIGINAL FIELD VARIABLES ( ϕ, ˙ ϕ )",
"content": "This section is dedicated to the comparison of the numerical solutions obtained from our polar method to those of the original scalar field equations of motion. To do this, we use the same Boltzmann code CLASS to provide the numerical results, so that they are subject to the same numerical methods and limitations of the code in the two cases. In the amended version, the polar variables are solved as a separate dark matter component, while the field variables are solved using the scalar field equations of the quintessence module already implemented in CLASS .",
"pages": [
11
]
},
{
"title": "A. Background quantities",
"content": "The first comparison of background quantities is for the variables ϕ and ˙ ϕ , which are the dynamical ones in the KG equation (3d) and the field potential. The relationship between the original and polar variables can be found from the transformation equations (5), in the form of log( m /eV) = 24 We solved the KG equation (3d) separately, but with the same initial conditions as in the polar case through transformation (36). It is not possible to accurately follow the numerical evolution after the onset of the field oscillations, and then we only solved the KG equation (3d) up to the equivalent time to θ ⋆ = 100 π (2 m ϕ t ⋆ ≃ 100 π ). After this time, the equations of motion are set directly to ˙ ϕ = 0 and ¨ ϕ = 0, which means that the late-time solutions of the field variables are just ϕ ( t > t ⋆ ) = ϕ ( t ⋆ ) and ˙ ϕ ( t > t ⋆ ) = ˙ ϕ ( t ⋆ ) (and the density remains artificially constant afterwards). The two sets of solutions, the original and the polar ones, are plotted in compact form in the phase space shown in Fig. 9. The thick curves correspond to the system ϕ -˙ ϕ , with the different colors representing the field mass m ϕ , while the solutions of the polar system, all in black lines, are superimposed. We see that the agreement between the corresponding curves is exact up to t = t ⋆ . The same comparison exercise for field density ρ ϕ is shown in Fig. 10, using the same colors for the different curves as in Fig. 9. Furthermore, we normalize the density to the present value of the CDM density ρ CDM 0 , to highlight that the final value of the field densities coincides with the equivalent CDM case. The upper panel shows that the two sets of solutions coincide exactly, including the oscillatory phase, which is also confirmed by the comparison in the lower panel: the discrepancies appear at late times in the oscillatory phase and the cutoff in the solution of the field variables (see the explanation below Eq. (36)). As a final example, we show in Fig. 11 the SFDM EOS calculated directly from the pressure-to-density ratio w ϕ = p ϕ /ρ ϕ , using the same unit system as in Fig. 2. The variable in the horizontal axis is the dimensionless quantity 2 m ϕ t , under which all curves corresponding to a given field mass m ϕ become the same curve. The EOS oscillates rapidly around the zero value, and we again see that there is excellent agreement between the numerical results of the two approaches.",
"pages": [
11,
12,
13
]
},
{
"title": "V. DISCUSSION",
"content": "Among the diverse theoretical proposals to describe dark matter, SFDM constitutes a compelling candidate to play the role of the CDM component of the universe. The dynamics of such a particle is modeled through a scalar field endowed with a scalar field potential. In this work, we were particularly interested in the free case of a real scalar field. Given the oscillatory nature of the SFDM when it behaves as CDM, we must be careful to properly handle the differential equations for both background and linear perturbations, in such a way that numerically they were easily solved, allowing us to keep track of the evolution of the scalar field, and of all physical quantities built from it. By posing the Klein-Gordon equations (background and linear perturbations) as a dynamical system once new variables are introduced, we were able to describe the evolution of the SFDM as a system of first-order differential equations. With this prescription, instead of solving for the original scalar field variables ( ϕ, ˙ ϕ ; φ, ˙ φ ) , we solved a new set of dynamical variables: the polar variable θ , the scalar field energy density β , and the perturbations δ 0 , δ 1 . We have also added the variable y 1 which is proportional to the ratio of the mass of the scalar field and the Hubble parameter. It has been a standard procedure in the literature to deal with the rapid oscillations of the SFDM by averaging the oscillating functions in a Hubble time, and then writing down a new set of averaged equations of motion that resemble a standard cosmological fluid. Within our approach, this new set of equations is easier to solve numerically and to include in a standard Boltzmann code. Therefore, there was no need to invoke any approximation or average to cancel out the rapid oscillations of the scalar field. The only consideration of this kind was the introduction of the truncated trigonometric functions that we used (see Eq. (22) in Section III A ). In addition, the average procedure introduces an undesirable mismatch between the early- and late-time solutions and leaves unanswered the question of the sound speed of the density perturbations of the averaged fluid. Our method does not require a separate evolution of the SFDM equations of motion, but just a straightforward transformation of the original field equations. Our transformed system of equations remains the same throughout the full evolution, and this applies both for the background and for the linear density perturbations. Moreover, in the case of linear perturbations, we do not need to define an explicit expression for the sound speed, which is a slippery quantity in the fluid approximation. Nevertheless, in Appendices A and B we show the complete equivalence between our equations and the fluid ones, which only required the implicit definition of the sound speed. Our approach also deals successfully with the rapid oscillations in comparison with the original field variables. The SFDM formulation we present in this work gives accurate predictions on observables such as the CMB anisotropies and the MPS. Moreover, it is possible to put constraints on the main parameter of the model, the SFDM mass m ϕ . This is of great relevance because, on the one hand, we need numerical solutions with a high precision level to put constraints on the SFDM model in light of new data from present and future surveys. On the other hand, the cosmological dynamics of the SFDM from the big bang until the linear perturbation regime needs to be properly solved in order to set the initial conditions for numerical simulations of the nonlinear process of structure formation. In this sense, our approach offers a mathematical and numerical treatment for the SFDM model, which is advantageous in that it suitably solves the cosmological dynamics of such scalar field coupled to the Einstein-Boltzmann system.",
"pages": [
15,
16
]
},
{
"title": "ACKNOWLEDGMENTS",
"content": "FXLC acknowledges Beca CONACYT and FORDECYT-PRONACES-CONACYT for support of the present research under Grant No. CF-MG-2558591. This work was partially supported by Programa para el Desarrollo Profesional Docente; Direcci'on de Apoyo a la Investigaci'on y al Posgrado, Universidad de Guanajuato; CONACyT M'exico under Grants No. A1-S-17899, No. 286897, No. 297771, No. 304001; and the Instituto Avanzado de Cosmolog'ıa Collaboration. We acknowledge the use of open-source software: Python [67, 68], numpy [69], scipy [70], astropy [71].",
"pages": [
16
]
},
{
"title": "Appendix A: The fluid approximation",
"content": "Here we discuss the possibility of rewriting the equations of motion (15) for the linear density perturbations in their fluid counterparts. Using Eq. (17b), we find that Eq. (15a) becomes where we have also used the relation (6b). From the definition of the adiabatic sound speed c 2 s = δp ϕ /δρ ϕ we find that The preceding equation can be used to substitute the second term on the right-hand side of Eq. (A1). After some manipulations, we get the following result, where now the primes denote the derivatives with respect to the conformal time and H is the corresponding Hubble parameter. These calculations show that Eq. (15a) is completely equivalent to the equation of motion for the density contrast in the fluid approximation. Similarly, we can combine Eqs. (7), (15) and (A2) to write an equation for the momentum density. After some lengthy but otherwise straightforward manipulations, we find that Equations (A3) are the standard fluid equations for linear density perturbations and then show that the polar equations (15) are their faithful representation.",
"pages": [
16
]
},
{
"title": "Appendix B: Linear density perturbations after the cutoff point",
"content": "It is illustrative to write Eqs. (15) after the cutoff on the rapid field oscillations. For times t > t ⋆ we find where again a prime denotes derivatives with respect to N = ln a . The system (B1) has the same structure as a forced harmonic oscillator with frequency ω = k 2 /k 2 J . As explained in [16], if we assume that ω is a constant (as is the case during RD), for large scales, the solution is the same as that of CDM: δ 0 ≃ -¯ h/ 2 and δ 1 ≃ const . For small scales, the solutions are oscillatory and given by combinations of functions cos( ωN ) and sin( ωN ). These behaviors agree with the full numerical results in Fig. 5. To obtain the corresponding fluid equations, we first notice that the cutoff method applied to Eqs. (12) and (A2) results in the following equations, The first expression in Eqs. (B2) reveals a direct relation between the divergence of the fluid velocity and variable δ 1 , while the second suggests a non-varying value of the fluid's EOS (that is, dw ϕ /dρ ϕ = 0, see [4]). Repeating the same procedure above for the calculation of Eqs. (A3), we find their counterpart for times t > t ⋆ , where a prime now denotes derivatives with respect to conformal time. Finally, the second of Eqs. (B3) suggests a scale-dependent sound speed given by One note on the sound speed of the perturbations is taken in turn. The discussion above gives a clear answer for the sound speed after the cutoff in the field oscillations, but there is none for the right expression of c 2 s before that. Our formalism for the perturbations does not require an explicit expression of c 2 s , but the latter is necessary for a correct fluid formulation of the SFDM linear perturbations. A general expression of c 2 s was found in [72] in the comoving gauge of the scalar field, but the same authors argue in [73] that such expression cannot be transformed into the synchronous gauge, under which SFDM lacks a definite functional form of c 2 s . Despite this, the expression for c 2 s of [72] has been extensively used in the literature for the fluid approximation of SFDM in the synchronous gauge. An example is the so-called Effective Fluid Approximation (EFA) discussed in [36]. The equations of motion of the EFA are where the cycle-averaged sound speed is It can be seen that our Eqs. (B3) can be obtained from the EFA if the sound speed is given by the so-called nonrelativistic expression ⟨ c 2 s ⟩ ≃ k 2 / (4 m 2 ϕ a 2 ). However, the EFA considers an extra term in the equation of motion of δ 0 , see Eq. (B5a). Noticing this discrepancy, a comparison was made between the fluid approximation and our method in [36], see their Appendix B, and no relevant differences were found in the numerical results. To understand this, we use Eqs. (B2) and write the EFA equations (B5a) and (B5b) in terms of our variables δ 0 , δ 1 and N = ln a . We find that We now compare Eqs. (B6) with Eqs. (B1), and spot the extra term 3 /y 1 . The latter is very small, given that y 1 = 2 m ϕ /H and then y 1 ≫ 1 after the start of rapid field oscillations. This is the reason why the comparison in [36] did not find differences between our approach and the EFA. It can be concluded that our method correctly picks up the only terms in the equations of motion that are valid in the limit H/m ϕ ≪ 1. 11",
"pages": [
16,
17
]
},
{
"title": "Appendix C: Numerical implementation in CLASS",
"content": "Following the design of CLASS , we introduced a new module for the SFDM equations of motion, replicating the same structure as for other dark matter components. In this form, the contribution of the SFDM component could be called with its own parameters in any given parameter file of CLASS (for example, explanatory.ini ). The equations of motion for the background variables (12) were included in the file background.c , while those of the density perturbations (15) were included in the file perturbations.c . Similarly, a shooting routine was incorporated into the file input.ini to adjust the initial conditions of the dynamical variables. In each case, the SFDM quantities were added to the matter budget so that they contribute correctly to the right-hand side of the Einstein equations, for both the background and the linearly perturbed ones. Given the oscillatory nature of all SFDM quantities, we applied the cutoff procedure to the sine and cosine functions that appear in their definition, see Eqs. (22). For example, the background pressure was written as p ϕ = -cos ⋆ θ · ρ ϕ , and then effectively p ϕ = 0 for t > t ⋆ . Another case would be the combination p ϕ + ρ ϕ = (1 -cos ⋆ θ ) ρ ϕ , so that p ϕ + ρ ϕ = ρ ϕ for t > t ⋆ . The same was applied to the perturbed quantities, as in the density perturbation written in the form δρ ϕ = ρ ϕ · δ 0 , which is required for the right-hand side of the perturbed Einstein equations. Similarly, the momentum density perturbation ( p ϕ + ρ ϕ )Θ ϕ was written as in Eq. (17b), that is, The result for late times t > t ⋆ is well defined and is given by It should be noted that in some parts of the file perturbations.c one requires to calculate the velocity divergence Θ, which is one of the fundamental variables in the fluid formalism for linear perturbations of density and momentum. However, the calculation of Θ ϕ itself is problematic for SFDM, as can be clearly seen in Eq. (C1a): One needs to divide by the quantity ( p ϕ + ρ ϕ ), which passes through zero during the phase of rapid field oscillations. As explained above, the calculation of Θ ϕ is not needed in the perturbed Einstein equations, and then we only used Eq. (C1b) as an additional source of the perturbed momentum density. A technical aspect not always mentioned for SFDM models and its Boltzmann code implementation is that it is always required a small quantity of CDM component. This is because in order to be consistent with the synchronous gauge, the Einstein field equations have to be solved in the comoving frame of the CDM fluid, that is, Θ cdm = 0. In fact, an automatic feature in the current versions of CLASS and its amended versions for SFDM, Ω cdm = 10 -10 once the amount of CDM is set to zero in the input file explanatory.ini . This ensures that there will be practically a null contribution of CDM, so that SFDM is the dominant non-relativistic matter component, and also helps us to avoid the known problem that the synchronous gauge is not completely fixed for density perturbations when the SFDM is the only DM component [73]. One last note is that the evolution of the perturbations should start well before the start of the rapid oscillations. We include an additional condition in the file perturbations.c to guarantee that the calculations begin only if m ϕ /H < 0 . 01. This gives the system (15) enough time to reach its attractor solution starting from the initial conditions (21). 061301 (2017), arXiv:1703.10180 [gr-qc].",
"pages": [
17,
18,
19
]
}
]
2024PhRvD.109b4050M
https://arxiv.org/pdf/2306.04703.pdf
<document>
<section_header_level_1><location><page_1><loc_33><loc_93><loc_68><loc_94></location>Octupolar test of general relativity</section_header_level_1>
<section_header_level_1><location><page_1><loc_40><loc_90><loc_61><loc_91></location>Parthapratim Mahapatra 1, ∗</section_header_level_1>
<text><location><page_1><loc_30><loc_87><loc_70><loc_89></location>1 Chennai Mathematical Institute, Siruseri, 603103, India (Dated: October 10, 2024)</text>
<text><location><page_1><loc_17><loc_61><loc_84><loc_85></location>Compact binaries with unequal masses and whose orbits are not aligned with the observer's line of sight are excellent probes of gravitational radiation beyond the quadrupole approximation. Among the compact binaries observed so far, strong evidence of octupolar modes is seen in GW190412 and GW190814, two binary black holes observed during the first half of the third observing run of LIGO/Virgo observatories. These two events, therefore, provide a unique opportunity to test the consistency of the octupolar modes with the predictions of general relativity (GR). In the postNewtonian (PN) approximation to GR, the gravitational-wave phasing has known dependencies on different radiative multipole moments, including the mass octupole. This permits the use of publicly released posteriors of the PN phase deformation parameters for placing constraints on the deformations to the different PN components of the radiative mass octupole denoted by δµ 3 n . Combining the posteriors on δµ 3 n from these two events, we deduce a joint bound (at 90% credibility) on the first three PN order terms in the radiative octupoles to be δµ 30 = -0 . 07 +0 . 11 -0 . 12 , δµ 32 = 0 . 48 +0 . 93 -1 . 15 , and δµ 33 = -0 . 32 +1 . 67 -0 . 62 , consistent with GR predictions. Among these, the measurement of δµ 33 for the first time confirms the well-known octupolar tail contribution, a novel nonlinear effect due to the scattering of the octupolar radiation by the background spacetime, is consistent with the predictions of GR. Detection of similar systems in the future observing runs should further tighten these constraints.</text>
<section_header_level_1><location><page_1><loc_20><loc_58><loc_37><loc_59></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_33><loc_49><loc_56></location>It is well known that the leading order gravitational wave (GW) emission is quadrupolar according to general relativity. However, subdominant higher multipoles get turned on if the binary has a mass asymmetry and when the line of sight of the observer is not aligned with the orbital angular momentum vector of the binary [1-9]. To date, the LIGO-Virgo-KAGRA collaboration has reported ∼ 90 confident detections of compact binary mergers [10-13]. Among these events, two compact binary mergers-GW190412 [14] and GW190814 [15]have shown clear evidence of the presence of octupolar ( ℓ = 3 , m = 3) mode, the first correction beyond the quadrupole. These two events, therefore, should facilitate a test of the gravitational octupolar structure of the compact binary dynamics.</text>
<text><location><page_1><loc_8><loc_11><loc_49><loc_32></location>The gravitational dynamics of a compact binary system is typically divided into three stages of evolution: inspiral, merger, and ringdown. While the post-Newtonian (PN) approximation to general relativity (GR) [16] is employed to model the adiabatic inspiral stage of a compact binary coalescence, one requires numerical solutions to the Einstein equations [17], and the black hole (BH) perturbation theory [18] to describe the highly nonlinear merger stage, and the ringdown phase, respectively. As numerical relativity simulations are computationally expensive, currently there are two main modeling approaches towards producing the complete gravitational waveform (i.e., a single waveform that captures all three stages of binary evolution) for parameter inference: ef-</text>
<text><location><page_1><loc_52><loc_53><loc_93><loc_59></location>fective one-body approach [19, 20] and phenomenological approach [21, 22]. Both these methods make the best use of the analytical and numerical understanding of compact binary dynamics.</text>
<text><location><page_1><loc_52><loc_48><loc_93><loc_53></location>The gravitational waveform from a coalescing compact binary within GR, in the frequency domain, has the following form:</text>
<formula><location><page_1><loc_54><loc_44><loc_93><loc_47></location>˜ h ( f ; ⃗ λ, ι, φ N ) = ∑ ℓ ≥ 2 ℓ ∑ m = -ℓ Y ℓm -2 ( ι, φ N ) ˜ h ℓm ( f ; ⃗ λ ) , (1)</formula>
<text><location><page_1><loc_52><loc_24><loc_93><loc_42></location>where Y ℓm -2 are spin-weighted spherical harmonics of spin weight -2, ( ι, φ N ) describes the location of the observer in the binary's sky and ⃗ λ denotes the intrinsic parameters (e.g., masses and spins) as well as other relevant extrinsic parameters (e.g., luminosity distance ( d L ), reference time and reference phase) of the binary. Each GW mode ( ˜ h ℓm ) has an amplitude, A ℓm ( f ; ⃗ λ ), and a phase, ψ ℓm ( f ; ⃗ λ ) (i.e., ˜ h ℓm = A ℓm ( f ; ⃗ λ ) e i ψ ℓm ( f ; ⃗ λ ) ). In alternative theories of gravity, the gravitational dynamics of a compact binary could differ from the prediction of GR during all the three stages and might modify the phase and amplitude in the waveform.</text>
<text><location><page_1><loc_52><loc_11><loc_93><loc_23></location>There exist proposals in the literature to probe the prediction of the harmonic structure of gravitational radiation from binary black hole coalescence in GR [2327]. Using GW190412 and GW190814, Ref. [28] tested the consistency between the dominant and subdominant modes and found the chirp mass estimated from the ℓ = 3 , m = 3 mode to be within ± 1% of the one estimated from the quadrupolar ℓ = 2 , m = 2 mode.</text>
<text><location><page_1><loc_52><loc_5><loc_93><loc_11></location>In a more recent work [29], the consistency of the amplitudes of the h 21 and h 33 modes of the GW spectrum with GR predictions was investigated using these two events and found no evidence for any violation of GR.</text>
<text><location><page_2><loc_8><loc_88><loc_49><loc_94></location>This test assumes the phases of subdominant harmonics ( ψ ℓm ; ℓ> 2 ) follow GR and investigates whether the amplitudes of subdominant harmonics ( A ℓm ; ℓ> 2 ) are consistent with the GR prediction.</text>
<text><location><page_2><loc_8><loc_62><loc_49><loc_88></location>In this paper, we argue that if a signal contains nonquadrupolar modes, apart from the amplitude, the phase evolution will also carry their unique imprints [1, 25, 26, 30]. As GW detectors are more sensitive to phase evolution , this could be used to test GR, complementing the approach of [29]. However, we will focus only on the inspiral phase in this work, which is well-modeled by PN approximation to GR, and discuss constraints on the PN structure of octupolar emission in GR. For this, we will make use of the unique map between the masstype octupole coefficients at different PN orders and the bounds on the 1PN, 2PN, and 2.5PN logarithmic phasing deformation parameters for these two events in the parametrized tests of GW phasing reported in [31, 32]. Further, we will consider only the leading order appearance of the octupole coefficients in the GW phase for this mapping.</text>
<text><location><page_2><loc_8><loc_47><loc_49><loc_61></location>The remainder of the paper is organized as follows. In Sec. II, we briefly review the parametrized tests of GW phasing. In Sec. III, we introduce the octupolar parametrization. We derive the relations between different PN pieces in the mass-type octupole moment and different PN phasing terms in Sec. IV. In Sec. V, we describe the Bayesian framework to infer the octupole parameters. Our results and conclusions are presented in Sec. VI.</text>
<section_header_level_1><location><page_2><loc_8><loc_43><loc_49><loc_44></location>II. PARAMETRIZED TESTS OF GW PHASING</section_header_level_1>
<text><location><page_2><loc_8><loc_35><loc_49><loc_41></location>The frequency domain GW phase from the inspiral part of the waveform (computed using the stationary phase approximation [33, 34]) for the leading quadrupolar harmonic [4, 35] takes the form</text>
<formula><location><page_2><loc_9><loc_27><loc_49><loc_34></location>Ψ( f ) = 2 πf t c -ϕ c + 3 128 ν v 5 [ i =7 ∑ i =0 ( ϕ i + ϕ il ln v ) v i + O ( v 8 ) ] , (2)</formula>
<text><location><page_2><loc_8><loc_18><loc_49><loc_26></location>where v = ( πGMf/c 3 ) 1 / 3 is the PN expansion parameter, M is the binary's redshifted total mass, ν is the symmetric mass ratio of the binary, ϕ i and ϕ il denote the nonlogarithmic and logarithmic PN phasing coefficients, respectively.</text>
<text><location><page_2><loc_8><loc_5><loc_49><loc_18></location>Due to the lack of accurate waveforms in alternative theories of gravity, 'theory-agnostic' approaches are often adopted to test GR with GW data. These 'null tests' of GR make use of our best knowledge of compact binary dynamics in GR and look for possible deviations from GR without reference to specific alternatives (see Refs. [31, 32, 36, 37] for more details.). One of the most generic tests of GR that has been routinely performed</text>
<text><location><page_2><loc_52><loc_91><loc_93><loc_94></location>with LIGO/Virgo data is the parametrized test of GW phasing [38-46].</text>
<text><location><page_2><loc_52><loc_80><loc_93><loc_91></location>The parametrized tests rely on measuring any deviations in the PN coefficients ϕ i and ϕ il in the GW phasing, which are uniquely predicted by GR, from compact binary mergers. A parametrized waveform model introduces additional degrees of freedom to capture signatures of possible GR violation by modifying the phasing coefficients as</text>
<formula><location><page_2><loc_65><loc_77><loc_93><loc_79></location>ϕ b = ϕ GR b (1 + δ ˆ ϕ b ) , (3)</formula>
<text><location><page_2><loc_52><loc_62><loc_93><loc_75></location>( b = i, il ) (see Sec. VA of Refs. [31, 32] for more details). In GR, these phenomenological dimensionless deviation parameters ( δ ˆ ϕ b ) are identically zero, whereas in alternative theories of gravity, one or more of these parameters could be different from zero. Combining data from different GW events detected during the first, second, and third observing runs of LIGO/Virgo, the current bound on the PN deviation parameters are found to be consistent with GR (see Figs. 6 and 7 of Ref. [32]).</text>
<text><location><page_2><loc_52><loc_37><loc_93><loc_61></location>For the two asymmetric binary events, GW190412 and GW190814, we will use the results of the parametrized tests, obtained by applying parametrized IMRPhenomPv3HM (denoted as ' Phenom ' in this paper) [8] and SEOBNRv4HM ROM (denoted as ' SEOB ')[9] waveform approximant to the data. Phenom waveform is a frequency-domain phenomenological waveform model that includes the effects of two-spin precession along with higher multipole moments [8], whereas SEOB is a frequency-domain nonprecessing reduced-order effective one-body model that incorporates the higher order modes [9]. In the current LIGO-Virgo-KAGRA analyses [31, 32, 37], the reported bounds on δ ˆ ϕ b come from the fractional deviations applied to the nonspinning portion of the phase (see Sec. VA of Refs. [31, 32] for detailed discussions).</text>
<section_header_level_1><location><page_2><loc_56><loc_31><loc_88><loc_33></location>III. PARAMETRIZED MULTIPOLAR GRAVITATIONAL WAVEFORMS</section_header_level_1>
<text><location><page_2><loc_52><loc_5><loc_93><loc_28></location>The radiative multipole moments of a compact binary system contain information about source physics (masses [16, 30, 47], spins [4, 16, 48-60], tidal deformability [6167], spin induced quadrupole moment [51, 68-71] etc.) and account for various nonlinear interactions and physical effects (such as 'tail' effects [30, 72, 73], tails of tails [74], tail square [75], memory effects [76-79], spinorbit effects [48, 80], spin-spin effects [48, 59] etc.) that occur at different PN orders in GR. In alternative theories of gravity, one or more radiative multipole moments of a compact binary could be different from those in GR (see for instance Refs. [81-83]). One can put constraints on such theories by studying the multipolar structure of asymmetric compact binary systems like GW190412 and GW190814.</text>
<text><location><page_3><loc_8><loc_62><loc_49><loc_94></location>References [25, 26] came up with a novel theoryagnostic method to test the multipolar structure of the gravitational field radiated from an inspiralling compact binary. Using the multipolar post-Minkowskian formalism [30, 47, 72, 84-92], Ref. [25] derived the parametrized multipolar gravitational wave phasing up to 3.5PN order for nonspinning binaries and Ref. [26] extended it for nonprecessing binaries. The multipolar postMinkowskian formalism relates the radiation content in the far zone, encoded in the mass- and current-type radiative multipole moments { U L , V L } , to the stress-energy tensor of the source. In order to model possible deviations in the multipole structure, Refs. [25, 26] adopted the following parametrization for radiative multipole moments: U L → µ l U L , V L → ϵ l V L . By construction, the phenomenological multipole parameters µ l , ϵ l are equal to unity in GR. With this parametrization, the contributions from various radiative multipole moments to the GWphasing can be tracked separately, thereby facilitating tests of the multipolar structure of the PN approximation to GR.</text>
<text><location><page_3><loc_8><loc_55><loc_49><loc_61></location>In this work, we go one step further and probe the different PN orders in the radiative mass octupole moment of a compact binary as it evolves through the adiabatic inspiral phase. We propose the parametrization</text>
<formula><location><page_3><loc_18><loc_51><loc_49><loc_54></location>U ijk -→ ∑ n 1 c n µ 3 n U ( n ) , GR ijk , (4)</formula>
<text><location><page_3><loc_8><loc_23><loc_49><loc_49></location>where U ijk is the mass-type radiative octupole moment, U ( n ) , GR ijk is the (n/2)th PN correction to U ijk in GR and µ 3 n is the corresponding octupole coefficient. Note that there is no 1/c (i.e., 0.5PN) contribution in the mass octupole moment in GR. The 1.5PN correction term in the octupole moment arises due to the tail effect, caused by the scattering of the outgoing octupolar wave off the background spacetime associated with the total [Arnowitt-Deser-Misner (ADM)] mass of the source [30, 72-75]. By definition µ 3 n is unity in GR and appears at different PN orders in the phasing formula. For instance, µ 30 first appears at 1PN, µ 32 at 2PN and µ 33 at 2.5PN (logarithmic) order. We next discuss how the existing bounds on the PN deformation coefficients δ ˆ ϕ b , based on the parametrization in Eq. (3), reported in [31, 32] can be mapped to the bounds on µ 3 n in the parametrization derived above.</text>
<section_header_level_1><location><page_3><loc_10><loc_17><loc_47><loc_19></location>IV. MAPPING THE PN BOUNDS TO THE OCTUPOLE PARAMETERS</section_header_level_1>
<text><location><page_3><loc_8><loc_5><loc_49><loc_14></location>Each of the parameters µ 3 n appears at multiple PN orders in the GW phasing. For example, µ 30 appears at 1PN, 2PN, 2.5PN (logarithmic), and 3.5PN orders. Therefore if there is a deviation from GR in one of the µ 3 n , it will result in a dephasing of each of the PN phasing coefficients at the order in which this octupole pa-</text>
<text><location><page_3><loc_52><loc_85><loc_93><loc_94></location>eter contributes. 1 Here we neglect the modification to all PN orders except the leading order at which they first appear. This is a reasonable assumption to make because if there is a deviation in any of the µ 3 n , the leading order at which they appear would be most sensitive to such a deviation.</text>
<text><location><page_3><loc_57><loc_71><loc_57><loc_72></location≯</text>
<text><location><page_3><loc_64><loc_71><loc_64><loc_72></location≯</text>
<text><location><page_3><loc_52><loc_68><loc_93><loc_85></location>Therefore, the goal now will be to obtain constraints on µ 30 , µ 32 , and µ 33 using the bounds on 1PN, 2PN, and 2.5PN logarithmic phase deformation parameters, respectively, along with other relevant intrinsic binary parameters for particular GW events. Further, while estimating bounds on one of the µ 3 n , we assume all other mass-type octupole parameters as well as rest of the multipole parameters to take their values in GR (i.e., µ 3 n ' ; n ' = n = µ l ; l =3 = ϵ l = 1) in the spirit of a singleparameter test, i.e., varying one deformation parameter at a time.</text>
<text><location><page_3><loc_52><loc_63><loc_93><loc_67></location>The expression for 1PN phasing coefficient in the parametrized multipolar GW phase [see Eq. (2.16) of Ref. [25]] is given by</text>
<formula><location><page_3><loc_53><loc_48><loc_93><loc_62></location>ϕ 2 = ( 1510 189 -130 21 ν ) + ( µ 30 µ 2 ) 2 ( -6835 2268 + 6835 567 ν ) + ( ϵ 2 µ 2 ) 2 ( -5 81 + 20 81 ν ) = ϕ GR 2 -K 1 ( ν ) [( µ 30 µ 2 ) 2 -1 ] -K 2 ( ν ) [( ϵ 2 µ 2 ) 2 -1 ] , (5)</formula>
<text><location><page_3><loc_52><loc_46><loc_56><loc_47></location>where</text>
<formula><location><page_3><loc_65><loc_42><loc_93><loc_44></location>ϕ GR 2 = ( 3715 756 + 55 9 ν ) , (6a)</formula>
<formula><location><page_3><loc_62><loc_38><loc_93><loc_41></location>K 1 ( ν ) = ( 6835 2268 -6835 567 ν ) , (6b)</formula>
<formula><location><page_3><loc_65><loc_35><loc_93><loc_38></location>K 2 ( ν ) = ( 5 81 -20 81 ν ) , (6c)</formula>
<text><location><page_3><loc_52><loc_31><loc_93><loc_34></location>and µ 2 and ϵ 2 are the mass and current quadrupole parameters, respectively.</text>
<text><location><page_3><loc_52><loc_26><loc_93><loc_30></location>Comparing the parametrized PN phasing of Eq. (3) to Eq. (5) we have the following relation between δ ˆ ϕ 2 and µ 30 :</text>
<formula><location><page_3><loc_53><loc_20><loc_93><loc_25></location>δ ˆ ϕ 2 = K 1 ( ν ) [ 1 -( µ 30 µ 2 ) 2 ] + [ 1 -( ϵ 2 µ 2 ) 2 ] K 2 ( ν ) ϕ GR 2 . (7)</formula>
<text><location><page_3><loc_52><loc_16><loc_93><loc_19></location>In the spirit of null tests, we find it more convenient to employ octupole deformation parameters δµ 3 n defined</text>
<text><location><page_4><loc_8><loc_88><loc_49><loc_94></location>for mass octupole by imposing µ 3 n = 1 + δµ 3 n for the different PN pieces. The aim now would be to derive bounds on δµ 30 , δµ 32 , and δµ 33 using GW observations assuming µ l ; l =3 = ϵ l = 1 and δµ 3 n ' ; n ' = n = 0. 2</text>
<text><location><page_4><loc_18><loc_88><loc_18><loc_89></location≯</text>
<text><location><page_4><loc_35><loc_88><loc_35><loc_89></location≯</text>
<text><location><page_4><loc_8><loc_85><loc_49><loc_88></location>With the above assumption we can express δµ 30 in terms of δ ˆ ϕ 2 as</text>
<formula><location><page_4><loc_17><loc_80><loc_49><loc_84></location>δµ 30 = -1 ± √ 1 -δ ˆ ϕ 2 ϕ GR 2 K 1 ( ν ) . (8)</formula>
<text><location><page_4><loc_8><loc_72><loc_49><loc_79></location>Among the two solutions, we will adopt the one which respects the GR limit (i.e., δµ 30 vanishes when δ ˆ ϕ 2 → 0). Similarly, we also obtain the expressions for δµ 32 and δµ 33 as (see Sec. 1 of the Supplemental Material for a derivation):</text>
<formula><location><page_4><loc_22><loc_68><loc_49><loc_71></location>δµ 32 = ϕ GR 4 δ ˆ ϕ 4 K 3 ( ν ) , (9)</formula>
<formula><location><page_4><loc_22><loc_64><loc_49><loc_67></location>δµ 33 = ϕ GR 5 l δ ˆ ϕ 5 l K 4 ( ν ) . (10)</formula>
<text><location><page_4><loc_8><loc_60><loc_49><loc_63></location>The expressions of ϕ GR 4 , K 3 ( ν ), ϕ GR 5 l , and K 4 ( ν ) are provided in Sec. 1 of the Supplemental Material.</text>
<text><location><page_4><loc_8><loc_52><loc_49><loc_60></location>Having obtained the mapping, now the problem essentially amounts to using the posterior samples of { δ ˆ ϕ b , ν } for any event and computing the corresponding posteriors on { δµ 3 n } using the above equations. The exact procedure followed is discussed next.</text>
<section_header_level_1><location><page_4><loc_10><loc_48><loc_48><loc_49></location>V. INFERRING OCTUPOLE PARAMETERS</section_header_level_1>
<text><location><page_4><loc_8><loc_25><loc_49><loc_46></location>Given the LIGO/Virgo data, d , we are interested in deriving ˜ P ( δµ 3 n | d, H ), the posterior probability distribution on δµ 3 n , for a flat prior on δµ 3 n (here H denotes the hypothesis, which is the parametric model we employ.). Towards this, we use Eqs. (8), (9), and (10), along with the two-dimensional posterior distribution P ( δ ˆ ϕ b , ν | d, H ), for different GW events. For example, to derive ˜ P ( δµ 30 | d, H ) we will use Eq. (8) and P ( δ ˆ ϕ 2 , ν | d, H ). The probability distribution P ( δ ˆ ϕ b , ν | d, H ) is computed for flat priors on δ ˆ ϕ b and mass ratio. Therefore in the Bayesian framework the samples of δµ 3 n , derived from P ( δ ˆ ϕ b , ν | d, H ), need to be reweighted to obtain posterior ˜ P ( δµ 3 n | d, H ) that assume flat priors on δµ 3 n as</text>
<formula><location><page_4><loc_9><loc_17><loc_49><loc_24></location>˜ P ( δµ 3 n | d, H ) ∝ [∫ dν d ( δ ˆ ϕ b ) P ( δµ 3 n | δ ˆ ϕ b , ν, H ) ˜ P ( δ ˆ ϕ b , ν | d, H ) ] × ˜ Π( δµ 3 n |H ) Π( δµ 3 n |H ) . (11)</formula>
<text><location><page_4><loc_52><loc_76><loc_93><loc_94></location>In the above equation, the tilde denotes flat priors or posteriors derived assuming flat prior on the corresponding parameters. Hence ˜ Π( δµ 3 n |H ) denotes flat prior on δµ 3 n and ˜ P ( δ ˆ ϕ b , ν | d, H ) denote posterior assuming flat priors on δ ˆ ϕ b and mass ratio. The prior distribution, ˜ Π( δµ 3 n |H ), is chosen to be uniform between [-30, 30]. While P ( δµ 3 n | δ ˆ ϕ b , ν, H ) takes care of the coordinate transformation between ( δ ˆ ϕ b , ν ) to δµ 3 n [using Eqs. (8)-(10)], Π( δµ 3 n |H ) in the above equation is simply P ( δµ 3 n | δ ˆ ϕ b , ν, H ) for the flat prior on δ ˆ ϕ b and mass ratio. A detailed derivation of the above equation is provided in Sec. 2 of the Supplemental Material.</text>
<text><location><page_4><loc_52><loc_63><loc_93><loc_75></location>As GW190814 and GW190412 are the only two events for which a confident detection of higher modes was possible, we will restrict to these two events for our purposes. We use the parameter estimation samples for δ ˆ ϕ b and symmetric mass ratio ( ν ) from the GWTC-3 Data Release [93] for these two events, analyzed with different waveform approximants, and estimate the bounds on the parameters δµ 3 n following this procedure.</text>
<text><location><page_4><loc_52><loc_57><loc_93><loc_63></location>While estimating the posterior probability distribution of δµ 30 , we need to ensure that the values of δµ 30 obtained from a pair of ( δ ˆ ϕ 2 , ν ) should be real by imposing the condition [see Eq. (8)],</text>
<formula><location><page_4><loc_67><loc_53><loc_93><loc_55></location>δ ˆ ϕ 2 ≤ K 1 ( ν ) ϕ GR 2 . (12)</formula>
<text><location><page_4><loc_52><loc_39><loc_93><loc_51></location>In order to realize this, we discard those samples of δµ 30 which do not meet the above condition. If we need to remove relatively large number of samples, that means the event is uninformative. We find that for events except GW190814 and GW190412, majority of the samples do not meet this condition. This simply is a reflection of the fact that we are trying to test deviation in octupole moment when its presence is barely there in the signal.</text>
<section_header_level_1><location><page_4><loc_57><loc_35><loc_88><loc_36></location>VI. RESULTS AND CONCLUSIONS</section_header_level_1>
<text><location><page_4><loc_52><loc_18><loc_93><loc_33></location>The posteriors of the leading and two subleading octupole deformation parameters δµ 30 , δµ 32 , and δµ 33 for GW190412 and GW190814 obtained by the abovementioned procedure are shown in Fig. 1. Among the detected events, GW190814 provides the tightest constraints on all the octupole parameters. This is expected as GW190814 is the most unequal mass binary (mass ratio, q = 0 . 112 +0 . 008 -0 . 009 ) among the GW events in the GWTC-3 and asymmetric systems get stronger contributions from non-quadrupolar moments.</text>
<text><location><page_4><loc_52><loc_5><loc_93><loc_18></location>In addition to the individual event analysis in Fig. 1, we have also obtained the combined bounds on δµ 3 n using data from multiple events under the assumption that the same value of δµ 3 n is shared across all the events. The joint constraints on these parameters are obtained by multiplying the individual likelihoods from the events, GW190412 and GW190814, analyzed with Phenom and SEOB waveforms. In the joint analysis the most tightly</text>
<figure>
<location><page_5><loc_10><loc_43><loc_91><loc_94></location>
<caption>FIG. 1. Bounds on δµ 3 n for GW190412 and GW190814 analyzed with Phenom (in red color) and SEOB (in blue color) waveform approximants are shown. Left-most panels show the bounds for GW190412, whereas the middle panels show the result for GW190814. The combined bounds are shown in the rightmost panels. The colored vertical dashed lines mark the 90% credible intervals and median values. The gray dashed vertical lines indicate the GR prediction ( δµ 3 n = 0). The posterior distributions of δµ 3 n show consistency with GR.</caption>
</figure>
<text><location><page_5><loc_8><loc_21><loc_49><loc_31></location>constrained parameter is δµ 30 and the most weakly constrained parameter is δµ 33 . The posterior on higherorder mass octupole deformation parameters, such as δµ 34 , are mostly uninformative and not shown here. Detections of unequal mass binaries in the future with a larger signal-to-noise ratio will enable us to probe higher PN pieces in the mass octupole moment.</text>
<text><location><page_5><loc_8><loc_5><loc_49><loc_19></location>Bounds from the two different kinds of waveform approximants show excellent agreement with each other. On all occasions the posterior distributions on δµ 3 n are statistically consistent with δµ 3 n = 0 within 90% credible interval. This is the first reported bound on the different PN pieces in mass-type octupole moment of compact binary complementing the previous consistency tests in Refs. [28, 29]. It is interesting that the bounds on δµ 33 also confirm the consistency of the octupolar tail</text>
<text><location><page_5><loc_52><loc_30><loc_79><loc_31></location>radiation with the predictions of GR.</text>
<text><location><page_5><loc_52><loc_18><loc_93><loc_29></location>The posterior distributions on current quadrupole deformation parameters are also largely uninformative and not reported here. In the future, the detections of high mass ratio and highly spinning binaries with larger signal-to-noise ratio will enhance the contribution of the current quadrupole to the flux making its measurement with good precision possible.</text>
<text><location><page_5><loc_52><loc_5><loc_93><loc_18></location>Last, it is instructive to ask if the GR violations in the δ ˆ ϕ b posteriors can be captured by the mapping proposed in this work. This is examined in Fig. 2 of the Supplemental Material (see the texts in Sec. 3 of the Supplemental Material for more details). We consider a GW190814-like system and simulate GR violations with different σ values in the δ ˆ ϕ b posteriors. We find that the derived posteriors on δµ 3 n , through this mapping, will</text>
<text><location><page_6><loc_8><loc_91><loc_49><loc_94></location>be able to detect deviations at different σ values in the δ ˆ ϕ b posteriors from GR.</text>
<text><location><page_6><loc_8><loc_64><loc_49><loc_91></location>To conclude, the parametrized multipolar waveforms could play a pivotal role in testing GR with current and next-generation GW detectors. The next-generation GW detectors will observe more diverse classes of compact binaries, thereby allowing us to probe even higher multipoles of compact binaries, and the parametrization introduced here will be crucial in such scenarios. Development of an infrastructure that can directly sample over the multipole parameters is planned for future work which should be able to probe the multipole structure without relying on the approximate mapping we have invoked here. The direct inference of multipole parameters from the GW data could provide even more stringent constraints as one will potentially gain information from multiple PN coefficients in the phase. Moreover, the inclusion of multipole parameters in the amplitude will likely play a critical role in this framework.</text>
<section_header_level_1><location><page_6><loc_19><loc_59><loc_39><loc_60></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_6><loc_8><loc_51><loc_49><loc_56></location>This material is based upon work supported by the NSF's LIGO Laboratory, which is a major facility fully funded by the National Science Foundation (NSF). The author is grateful for computational resources provided</text>
<unordered_list>
<list_item><location><page_6><loc_10><loc_43><loc_49><loc_45></location>[1] L. Blanchet, G. Faye, B. R. Iyer, and S. Sinha, Class. Quantum. Grav. 25 , 165003 (2008), 0802.1249.</list_item>
<list_item><location><page_6><loc_10><loc_40><loc_49><loc_42></location>[2] C. Van Den Broeck and A. S. Sengupta, Class. Quant. Grav. 24 , 155 (2007), gr-qc/0607092.</list_item>
<list_item><location><page_6><loc_10><loc_37><loc_49><loc_39></location>[3] C. Van Den Broeck and A. S. Sengupta, Class. Quant. Grav. 24 , 1089 (2007), gr-qc/0610126.</list_item>
<list_item><location><page_6><loc_10><loc_34><loc_49><loc_36></location>[4] K. G. Arun, A. Buonanno, G. Faye, and E. Ochsner, Phys. Rev. D 79 , 104023 (2009), 0810.5336.</list_item>
<list_item><location><page_6><loc_10><loc_31><loc_49><loc_34></location>[5] C. K. Mishra, A. Kela, K. G. Arun, and G. Faye, Phys. Rev. D93 , 084054 (2016), 1601.05588.</list_item>
<list_item><location><page_6><loc_10><loc_25><loc_49><loc_31></location>[6] L. London, S. Khan, E. Fauchon-Jones, C. Garc'ıa, M. Hannam, S. Husa, X. Jim'enez-Forteza, C. Kalaghatgi, F. Ohme, and F. Pannarale, Phys. Rev. Lett. 120 , 161102 (2018).</list_item>
<list_item><location><page_6><loc_10><loc_23><loc_49><loc_25></location>[7] S. Roy, A. S. Sengupta, and K. G. Arun, Phys. Rev. D 103 , 064012 (2021), 1910.04565.</list_item>
<list_item><location><page_6><loc_10><loc_20><loc_49><loc_22></location>[8] S. Khan, F. Ohme, K. Chatziioannou, and M. Hannam, Phys. Rev. D 101 , 024056 (2020), 1911.06050.</list_item>
<list_item><location><page_6><loc_10><loc_15><loc_49><loc_19></location>[9] R. Cotesta, A. Buonanno, A. Boh'e, A. Taracchini, I. Hinder, and S. Ossokine, Phys. Rev. D 98 , 084028 (2018), 1803.10701.</list_item>
<list_item><location><page_6><loc_9><loc_13><loc_49><loc_15></location>[10] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. Lett. 116 , 061102 (2016), 1602.03837.</list_item>
<list_item><location><page_6><loc_9><loc_10><loc_49><loc_12></location>[11] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. X 9 , 031040 (2019), 1811.12907.</list_item>
<list_item><location><page_6><loc_9><loc_7><loc_49><loc_9></location>[12] R. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. X 11 , 021053 (2021), 2010.14527.</list_item>
</unordered_list>
<text><location><page_6><loc_52><loc_51><loc_93><loc_94></location>by the LIGO Laboratory and supported by National Science Foundation Grants No. PHY-0757058 and No. PHY-0823459. It is a pleasure to thank K. G. Arun, who suggested this problem to me, encouraged me to pursue it, and for comments on the draft. The author is grateful to A. Gupta, S. Kastha, and B. S. Sathyaprakash for invaluable discussions and/or comments on the manuscript. We thank M. Saleem for critical reading of the manuscript and providing useful comments. We also thank P. Saini, S. A. Bhat, and P. D. Roy for comments on the manuscript. We are thankful to N. J. McDaniel, A. Laddha, and S. Datta for valuable discussions. P.M. acknowledge the support of the Core Research Grant No. CRG/2021/004565 of the Science and Engineering Research Board of India and a grant from the Infosys foundation. This research has made use of data obtained from the Gravitational Wave Open Science Center (www.gw-openscience.org), a service of LIGO Laboratory, the LIGO Scientific Collaboration and the Virgo Collaboration. Virgo is funded by the French Centre National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale della Fisica Nucleare (INFN), and the Dutch Nikhef, with contributions by Polish and Hungarian institutes. This manuscript has the LIGO preprint number P2300150 . Analyses in this paper made use of NumPy [94], SciPy [95], IPython [96], Matplotlib [97], Corner [98], Jupyter [99], Seaborn [100] software packages.</text>
<unordered_list>
<list_item><location><page_6><loc_53><loc_43><loc_93><loc_45></location>[13] R. Abbott et al. (LIGO Scientific, VIRGO, KAGRA) (2021), 2111.03606.</list_item>
<list_item><location><page_6><loc_53><loc_40><loc_93><loc_42></location>[14] R. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. D 102 , 043015 (2020), 2004.08342.</list_item>
<list_item><location><page_6><loc_53><loc_37><loc_93><loc_39></location>[15] R. Abbott et al. (LIGO Scientific, Virgo), Astrophys. J. Lett. 896 , L44 (2020), 2006.12611.</list_item>
<list_item><location><page_6><loc_53><loc_34><loc_93><loc_36></location>[16] L. Blanchet, Living Rev. Rel. 9 , 4 (2006), arXiv:1310.1528.</list_item>
<list_item><location><page_6><loc_53><loc_30><loc_93><loc_34></location>[17] F. Pretorius (2007), relativistic Objects in Compact Binaries: From Birth to Coalescence Editor: Colpi et al., arXiv:0710.1338.</list_item>
<list_item><location><page_6><loc_53><loc_27><loc_93><loc_29></location>[18] M. Sasaki and H. Tagoshi, Living Rev. Rel. 6 , 6 (2003), gr-qc/0306120.</list_item>
<list_item><location><page_6><loc_53><loc_24><loc_93><loc_26></location>[19] T. Damour and A. Nagar, Phys. Rev. D 79 , 081503 (2009).</list_item>
<list_item><location><page_6><loc_53><loc_20><loc_93><loc_24></location>[20] A. Buonanno, Y. Pan, J. G. Baker, J. Centrella, B. J. Kelly, S. T. McWilliams, and J. R. van Meter, Phys. Rev. D 76 , 104049 (2007).</list_item>
<list_item><location><page_6><loc_53><loc_15><loc_93><loc_19></location>[21] P. Ajith, S. Babak, Y. Chen, M. Hewitson, B. Krishnan, A. M. Sintes, J. T. Whelan, B. Brugmann, P. Diener, N. Dorband, et al., Phys. Rev. D 77 , 104017 (2008).</list_item>
<list_item><location><page_6><loc_53><loc_11><loc_93><loc_15></location>[22] P. Ajith, M. Hannam, S. Husa, Y. Chen, B. Bruegmann, et al., Phys.Rev.Lett. 106 , 241101 (2011), arXiv:0909.2867.</list_item>
<list_item><location><page_6><loc_53><loc_7><loc_93><loc_11></location>[23] S. Dhanpal, A. Ghosh, A. K. Mehta, P. Ajith, and B. S. Sathyaprakash, Phys. Rev. D 99 , 104056 (2019), 1804.03297.</list_item>
<list_item><location><page_7><loc_9><loc_90><loc_49><loc_94></location>[24] T. Islam, A. K. Mehta, A. Ghosh, V. Varma, P. Ajith, and B. S. Sathyaprakash, Phys. Rev. D 101 , 024032 (2020), 1910.14259.</list_item>
<list_item><location><page_7><loc_9><loc_86><loc_49><loc_90></location>[25] S. Kastha, A. Gupta, K. G. Arun, B. S. Sathyaprakash, and C. Van Den Broeck, Phys. Rev. D 98 , 124033 (2018), 1809.10465.</list_item>
<list_item><location><page_7><loc_9><loc_81><loc_49><loc_85></location>[26] S. Kastha, A. Gupta, K. G. Arun, B. S. Sathyaprakash, and C. Van Den Broeck, Phys. Rev. D 100 , 044007 (2019).</list_item>
<list_item><location><page_7><loc_9><loc_79><loc_49><loc_81></location>[27] S. Mezzasoma and N. Yunes, Phys. Rev. D 106 , 024026 (2022), 2203.15934.</list_item>
<list_item><location><page_7><loc_9><loc_76><loc_49><loc_78></location>[28] C. D. Capano and A. H. Nitz, Phys. Rev. D 102 , 124070 (2020), 2008.02248.</list_item>
<list_item><location><page_7><loc_9><loc_71><loc_49><loc_75></location>[29] A. Puecher, C. Kalaghatgi, S. Roy, Y. Setyawati, I. Gupta, B. S. Sathyaprakash, and C. Van Den Broeck, Phys. Rev. D 106 , 082003 (2022), 2205.09062.</list_item>
<list_item><location><page_7><loc_9><loc_69><loc_49><loc_71></location>[30] L. Blanchet, T. Damour, and B. R. Iyer, Phys. Rev. D 51 , 5360 (1995), gr-qc/9501029.</list_item>
<list_item><location><page_7><loc_9><loc_66><loc_49><loc_68></location>[31] R. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. D 103 , 122002 (2021), 2010.14529.</list_item>
<list_item><location><page_7><loc_9><loc_63><loc_49><loc_65></location>[32] R. Abbott et al. (LIGO Scientific, VIRGO, KAGRA) (2021), 2112.06861.</list_item>
<list_item><location><page_7><loc_9><loc_60><loc_49><loc_62></location>[33] C. Cutler and E. E. Flanagan, Phys. Rev. D 49 , 2658 (1994), gr-qc/9402014.</list_item>
<list_item><location><page_7><loc_9><loc_57><loc_49><loc_60></location>[34] T. Damour, B. R. Iyer, and B. S. Sathyaprakash, Phys. Rev. D 62 , 084036 (2000), gr-qc/0001023.</list_item>
<list_item><location><page_7><loc_9><loc_53><loc_49><loc_57></location>[35] A. Buonanno, B. Iyer, E. Ochsner, Y. Pan, and B. S. Sathyaprakash, Phys. Rev. D 80 , 084043 (2009), 0907.0700.</list_item>
<list_item><location><page_7><loc_9><loc_49><loc_49><loc_52></location>[36] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. Lett. 116 , 221101 (2016), [Erratum: Phys.Rev.Lett. 121, 129902 (2018)], 1602.03841.</list_item>
<list_item><location><page_7><loc_9><loc_46><loc_49><loc_48></location>[37] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. D 100 , 104036 (2019), 1903.04467.</list_item>
<list_item><location><page_7><loc_9><loc_43><loc_49><loc_45></location>[38] L. Blanchet and B. S. Sathyaprakash, Class. Quantum Grav. 11 , 2807 (1994).</list_item>
<list_item><location><page_7><loc_9><loc_40><loc_49><loc_42></location>[39] L. Blanchet and B. S. Sathyaprakash, Phys. Rev. Lett. 74 , 1067 (1995).</list_item>
<list_item><location><page_7><loc_9><loc_36><loc_49><loc_40></location>[40] K. G. Arun, B. R. Iyer, M. S. S. Qusailah, and B. S. Sathyaprakash, Class. Quantum Grav. 23 , L37 (2006), gr-qc/0604018.</list_item>
<list_item><location><page_7><loc_9><loc_31><loc_49><loc_35></location>[41] K. G. Arun, B. R. Iyer, M. S. S. Qusailah, and B. S. Sathyaprakash, Phys. Rev. D 74 , 024006 (2006), grqc/0604067.</list_item>
<list_item><location><page_7><loc_9><loc_29><loc_49><loc_31></location>[42] N. Yunes and F. Pretorius, Phys. Rev. D 80 , 122003 (2009), 0909.3328.</list_item>
<list_item><location><page_7><loc_9><loc_24><loc_49><loc_28></location>[43] C. K. Mishra, K. G. Arun, B. R. Iyer, and B. S. Sathyaprakash, Phys. Rev. D 82 , 064010 (2010), 1005.0304.</list_item>
<list_item><location><page_7><loc_9><loc_19><loc_49><loc_24></location>[44] T. G. F. Li, W. Del Pozzo, S. Vitale, C. Van Den Broeck, M. Agathos, J. Veitch, K. Grover, T. Sidery, R. Sturani, and A. Vecchio, Phys. Rev. D 85 , 082003 (2012), 1110.0530.</list_item>
<list_item><location><page_7><loc_9><loc_14><loc_49><loc_18></location>[45] M. Agathos, W. Del Pozzo, T. G. F. Li, C. V. D. Broeck, J. Veitch, et al., Phys.Rev. D89 , 082001 (2014), 1311.0420.</list_item>
<list_item><location><page_7><loc_9><loc_10><loc_49><loc_14></location>[46] A. K. Mehta, A. Buonanno, R. Cotesta, A. Ghosh, N. Sennett, and J. Steinhoff, Phys. Rev. D 107 , 044020 (2023), 2203.13937.</list_item>
<list_item><location><page_7><loc_9><loc_7><loc_49><loc_10></location>[47] L. Blanchet and T. Damour, Phys. Rev. D 46 , 4304 (1992).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_7><loc_53><loc_91><loc_93><loc_94></location>[48] L. Kidder, C. Will, and A. Wiseman, Phys. Rev. D 47 , R4183 (1993).</list_item>
<list_item><location><page_7><loc_53><loc_90><loc_89><loc_91></location>[49] T. A. Apostolatos, Phys. Rev. D 52 , 605 (1995).</list_item>
<list_item><location><page_7><loc_53><loc_89><loc_83><loc_90></location>[50] L. Kidder, Phys. Rev. D 52 , 821 (1995).</list_item>
<list_item><location><page_7><loc_53><loc_86><loc_93><loc_88></location>[51] E. Poisson, Phys. Rev. D 57 , 5287 (1998), grqc/9709032.</list_item>
<list_item><location><page_7><loc_53><loc_81><loc_93><loc_85></location>[52] L. Blanchet, A. Buonanno, and G. Faye, Phys. Rev. D 74 , 104034 (2006), erratum-ibid.D 75 , 049903 (E) (2007), gr-qc/0605140.</list_item>
<list_item><location><page_7><loc_53><loc_79><loc_93><loc_81></location>[53] L. Blanchet, A. Buonanno, and G. Faye, Phys. Rev. D 84 , 064041 (2011).</list_item>
<list_item><location><page_7><loc_53><loc_76><loc_93><loc_78></location>[54] A. Bohe, S. Marsat, G. Faye, and L. Blanchet, Class.Quant.Grav. 30 , 075017 (2013), arXiv:1212.5520.</list_item>
<list_item><location><page_7><loc_53><loc_73><loc_93><loc_76></location>[55] A. Boh ' E, S. Marsat, and L. Blanchet, Class.Quant.Grav. 30 , 135009 (2013), arXiv:1303.7412.</list_item>
<list_item><location><page_7><loc_53><loc_69><loc_93><loc_72></location>[56] S. Marsat, A. Bohe, G. Faye, and L. Blanchet, Class.Quantum Grav. 30 , 055007 (2013), arXiv:1210.4143.</list_item>
<list_item><location><page_7><loc_53><loc_66><loc_93><loc_68></location>[57] S. Marsat, A. Boh ' E, L. Blanchet, and A. Buonanno, Class.Quant.Grav. 31 , 025023 (2014), arXiv:1307.6793.</list_item>
<list_item><location><page_7><loc_53><loc_63><loc_93><loc_65></location>[58] S. Marsat, Class. Quant. Grav. 32 , 085008 (2015), 1411.4118.</list_item>
<list_item><location><page_7><loc_53><loc_60><loc_93><loc_62></location>[59] A. Buonanno, G. Faye, and T. Hinderer, Phys.Rev. D87 , 044009 (2013), 1209.6349.</list_item>
<list_item><location><page_7><loc_53><loc_59><loc_93><loc_60></location>[60] C. Will and A. Wiseman, Phys. Rev. D 54 , 4813 (1996).</list_item>
<list_item><location><page_7><loc_53><loc_56><loc_93><loc_58></location>[61] L. Bildsten and C. Cutler, Astrophys. J. 400 , 175 (1992).</list_item>
<list_item><location><page_7><loc_53><loc_54><loc_88><loc_55></location>[62] C. S. Kochanek, Astrophys. J. 398 , 234 (1992).</list_item>
<list_item><location><page_7><loc_53><loc_51><loc_93><loc_54></location>[63] D. Lai, Mon. Not. Roy. Astron. Soc. 270 , 611 (1994), astro-ph/9404062.</list_item>
<list_item><location><page_7><loc_53><loc_49><loc_93><loc_51></location>[64] K. D. Kokkotas and G. Schaefer, Mon. Not. Roy. Astron. Soc. 275 , 301 (1995), gr-qc/9502034.</list_item>
<list_item><location><page_7><loc_53><loc_46><loc_93><loc_48></location>[65] T. Mora and C. M. Will, Phys. Rev. D 69 , 104021 (2004).</list_item>
<list_item><location><page_7><loc_53><loc_43><loc_93><loc_45></location>[66] E. E. Flanagan and T. Hinderer, Phys. Rev. D77 , 021502 (2008), 0709.1915.</list_item>
<list_item><location><page_7><loc_53><loc_40><loc_93><loc_42></location>[67] Q. Henry, G. Faye, and L. Blanchet, Phys. Rev. D 102 , 044033 (2020).</list_item>
<list_item><location><page_7><loc_53><loc_39><loc_85><loc_40></location>[68] F. D. Ryan, Phys. Rev. D 55 , 6081 (1997).</list_item>
<list_item><location><page_7><loc_53><loc_36><loc_93><loc_38></location>[69] W. G. Laarakkers and E. Poisson, Astrophys. J. 512 , 282 (1999), gr-qc/9709033.</list_item>
<list_item><location><page_7><loc_53><loc_33><loc_93><loc_35></location>[70] G. Pappas and T. A. Apostolatos, Phys. Rev. Lett. 108 , 231104 (2012), 1201.6067.</list_item>
<list_item><location><page_7><loc_53><loc_30><loc_93><loc_32></location>[71] N. Uchikata and S. Yoshida, Class. Quant. Grav. 33 , 025005 (2016), 1506.06485.</list_item>
<list_item><location><page_7><loc_53><loc_27><loc_93><loc_30></location>[72] L. Blanchet and T. Damour, Phys. Rev. D 37 , 1410 (1988).</list_item>
<list_item><location><page_7><loc_53><loc_24><loc_93><loc_27></location>[73] L. Blanchet and G. Schafer, Class. Quantum Grav. 10 , 2699 (1993).</list_item>
<list_item><location><page_7><loc_53><loc_21><loc_93><loc_24></location>[74] L. Blanchet, Class. Quantum Grav. 15 , 113 (1998), grqc/9710038.</list_item>
<list_item><location><page_7><loc_53><loc_19><loc_93><loc_21></location>[75] L. Blanchet, Class. Quantum Grav. 15 , 89 (1998), grqc/9710037.</list_item>
<list_item><location><page_7><loc_53><loc_17><loc_91><loc_18></location>[76] D. Christodoulou, Phys. Rev. Lett. 67 , 1486 (1991).</list_item>
<list_item><location><page_7><loc_53><loc_16><loc_84><loc_17></location>[77] K. Thorne, Phys. Rev. D 45 , 520 (1992).</list_item>
<list_item><location><page_7><loc_53><loc_11><loc_93><loc_15></location>[78] K. G. Arun, L. Blanchet, B. R. Iyer, and M. S. S. Qusailah, Class. Quantum Grav. 21 , 3771 (2004), erratumibid. 22 , 3115 (2005), gr-qc/0404185.</list_item>
<list_item><location><page_7><loc_53><loc_10><loc_93><loc_11></location>[79] M. Favata, Phys. Rev. D 80 , 024002 (2009), 0812.0069.</list_item>
<list_item><location><page_7><loc_53><loc_6><loc_93><loc_10></location>[80] L. Blanchet, A. Buonanno, and G. Faye, Phys. Rev. D 74 , 104034 (2006), [Erratum: Phys.Rev.D 75, 049903 (2007), Erratum: Phys.Rev.D 81, 089901 (2010)], gr-</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_8><loc_9><loc_90><loc_49><loc_92></location>[81] S. Endlich, V. Gorbenko, J. Huang, and L. Senatore, JHEP 09 , 122 (2017), 1704.01590.</list_item>
<list_item><location><page_8><loc_9><loc_87><loc_49><loc_90></location>[82] E. Battista and V. De Falco, Phys. Rev. D 104 , 084067 (2021), 2109.01384.</list_item>
<list_item><location><page_8><loc_9><loc_84><loc_49><loc_87></location>[83] E. Battista and V. De Falco, Eur. Phys. J. C 82 , 628 (2022), 2206.12907.</list_item>
<list_item><location><page_8><loc_9><loc_83><loc_42><loc_84></location>[84] K. Thorne, Rev. Mod. Phys. 52 , 299 (1980).</list_item>
<list_item><location><page_8><loc_9><loc_80><loc_49><loc_82></location>[85] L. Blanchet and T. Damour, Phys. Lett. A 104 , 82 (1984).</list_item>
<list_item><location><page_8><loc_9><loc_77><loc_49><loc_80></location>[86] L. Blanchet and T. Damour, Phil. Trans. Roy. Soc. Lond. A 320 , 379 (1986).</list_item>
<list_item><location><page_8><loc_9><loc_76><loc_49><loc_77></location>[87] L. Blanchet, Proc. Roy. Soc. Lond. A 409 , 383 (1987).</list_item>
<list_item><location><page_8><loc_9><loc_73><loc_49><loc_75></location>[88] L. Blanchet and T. Damour, Annales Inst. H. Poincar'e Phys. Th'eor. 50 , 377 (1989).</list_item>
<list_item><location><page_8><loc_9><loc_70><loc_49><loc_72></location>[89] L. Blanchet, Phys. Rev. D 51 , 2559 (1995), grqc/9501030.</list_item>
<list_item><location><page_8><loc_9><loc_66><loc_49><loc_70></location>[90] L. Blanchet, B. R. Iyer, and B. Joguet, Phys. Rev. D 65 , 064005 (2002), Erratum-ibid 71 , 129903(E) (2005), gr-qc/0105098.</list_item>
<list_item><location><page_8><loc_9><loc_63><loc_49><loc_65></location>[91] T. Damour, P. Jaranowski, and G. Schafer, Phys. Lett. B 513 , 147 (2001).</list_item>
<list_item><location><page_8><loc_9><loc_59><loc_49><loc_62></location>[92] L. Blanchet, T. Damour, G. Esposito-Far'ese, and B. R. Iyer, Phys. Rev. Lett. 93 , 091101 (2004), grqc/0406012.</list_item>
<list_item><location><page_8><loc_9><loc_54><loc_49><loc_58></location>[93] LIGO Scientific Collaboration and Virgo Collaboration, GWTC-3 data release (2021), URL https://doi.org/ 10.7935/b024-1886 .</list_item>
<list_item><location><page_8><loc_9><loc_50><loc_49><loc_54></location>[94] S. van der Walt, S. C. Colbert, and G. Varoquaux, Computing in Science and Engineering 13 , 22 (2011), 1102.1523.</list_item>
<list_item><location><page_8><loc_9><loc_44><loc_49><loc_50></location>[95] P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, et al., Nature Methods 17 , 261 (2020), 1907.10121.</list_item>
<list_item><location><page_8><loc_9><loc_41><loc_49><loc_44></location>[96] F. Perez and B. E. Granger, Computing in Science & Engineering 9 , 21 (2007).</list_item>
<list_item><location><page_8><loc_9><loc_39><loc_49><loc_41></location>[97] J. D. Hunter, Computing in Science and Engineering 9 , 90 (2007).</list_item>
<list_item><location><page_8><loc_9><loc_36><loc_49><loc_38></location>[98] D. Foreman-Mackey, The Journal of Open Source Software 1 , 24 (2016).</list_item>
<list_item><location><page_8><loc_9><loc_27><loc_49><loc_35></location>[99] T. Kluyver, B. Ragan-Kelley, F. P'erez, B. Granger, M. Bussonnier, J. Frederic, K. Kelley, J. Hamrick, J. Grout, S. Corlay, et al., in Positioning and Power in Academic Publishing: Players, Agents and Agendas , edited by F. Loizides and B. Scmidt (IOS Press, 2016), pp. 87-90.</list_item>
<list_item><location><page_8><loc_8><loc_24><loc_49><loc_27></location>[100] M. L. Waskom, Journal of Open Source Software 6 , 3021 (2021).</list_item>
</unordered_list>
<section_header_level_1><location><page_9><loc_37><loc_93><loc_64><loc_94></location>Appendix: Supplemental Materials</section_header_level_1>
<section_header_level_1><location><page_9><loc_24><loc_90><loc_77><loc_91></location>1. Tracking different PN pieces of the mass type octupole moment</section_header_level_1>
<text><location><page_9><loc_8><loc_82><loc_93><loc_87></location>With the parametrization introduced in Eq. (4) of the main paper, the different PN contributions from the mass type radiative octupole moment to the GW flux, and hence to the GW phasing, can be separately kept track of. To derive such a parametrized octupolar GW phasing, we follow the MPM formalism [30] along the lines of Refs. [25, 26]. The parametrized multipolar flux schematically reads as</text>
<formula><location><page_9><loc_22><loc_76><loc_93><loc_79></location>F = 32 5 ν 2 x 5 µ 2 2 [ 1 + F 1PN x + F 1 . 5PN x 3 / 2 + F 2PN x 2 + F 2 . 5PN x 5 / 2 + O ( x 3 ) ] , (A.1)</formula>
<text><location><page_9><loc_8><loc_74><loc_75><loc_75></location>where x = ( GMω/c 3 ) 2 / 3 is a PN parameter, ω the orbital angular frequency of the binary.</text>
<text><location><page_9><loc_10><loc_72><loc_93><loc_73></location>The leading order contribution from the mass-type octupole moment ( U ijk ) to the GW flux first appears at 1PN:</text>
<formula><location><page_9><loc_38><loc_67><loc_93><loc_71></location>F Oct 1PN = ( µ 30 µ 2 ) 2 [ 1367 1008 -1367 252 ν ] . (A.2)</formula>
<text><location><page_9><loc_8><loc_63><loc_93><loc_66></location>Therefore the leading order contribution from U ijk to the GW phasing appears at 1PN and is given by Eq. (5) in the main paper.</text>
<text><location><page_9><loc_8><loc_60><loc_93><loc_63></location>The next PN contribution (i.e., 1PN correction to U ijk ; note that there is no 0.5PN correction to U ijk .) from U ijk to the GW flux makes appearance at 2PN which reads as</text>
<formula><location><page_9><loc_10><loc_56><loc_93><loc_59></location>F Oct 2PN = [ -1367 µ 2 30 168 µ 2 2 -8201 µ 30 µ 32 3024 µ 2 2 + ( 17771 µ 2 30 504 µ 2 2 + 1139 µ 30 µ 32 84 µ 2 2 ) ν -( 1367 µ 2 30 126 µ 2 2 + 2050 µ 30 µ 32 189 µ 2 2 ) ν 2 ] . (A.3)</formula>
<text><location><page_9><loc_8><loc_52><loc_93><loc_55></location>Thus the 1PN correction to the octupole moment first appears at 2PN in the phasing. The 2PN phasing coefficient reads</text>
<formula><location><page_9><loc_11><loc_36><loc_93><loc_51></location>ϕ 4 = ( 242245 5292 + 4525 5292 ν + 145445 5292 ν 2 + ( µ 30 µ 2 ) 2 [ -772355 21168 + 580975 3024 ν -977405 5292 ν 2 ] + ( µ 30 µ 32 µ 2 2 ) [ 41005 1512 -5695 42 ν + 20500 189 ν 2 ] + ( µ 30 µ 2 ) 2 ( ϵ 2 µ 2 ) 2 [ 6835 9072 -6835 1134 ν + 6835 ν 2 567 ] + ( µ 30 µ 2 ) 4 [ 9343445 508032 -9343445 63504 ν + 9343445 31752 ν 2 ] + ( µ 4 µ 2 ) 2 [ -89650 3969 + 179300 1323 ν -89650 441 ν 2 ] + ( ϵ 2 µ 2 ) 2 [ -785 378 + 7115 756 ν -835 189 ν 2 ] + ( ϵ 2 µ 2 ) 4 [ 5 648 -5 81 ν + 10 81 ν 2 ] + ( ϵ 3 µ 2 ) 2 [ -50 63 + 100 21 ν -50 7 ν 2 ]) . (A.4)</formula>
<text><location><page_9><loc_8><loc_33><loc_78><loc_34></location>With the assumptions µ l ; l =3 = ϵ l = 1 and µ 3 n ' ; n ' =2 = 1, the 2PN phasing coefficient reduces to</text>
<text><location><page_9><loc_27><loc_33><loc_27><loc_34></location≯</text>
<text><location><page_9><loc_44><loc_33><loc_44><loc_34></location≯</text>
<formula><location><page_9><loc_40><loc_30><loc_93><loc_31></location>ϕ 4 = ϕ GR 4 + K 3 ( ν ) ( µ 32 -1 ) , (A.5)</formula>
<text><location><page_9><loc_8><loc_27><loc_34><loc_28></location>where ϕ GR 4 and K 3 ( ν ) are given by</text>
<formula><location><page_9><loc_35><loc_23><loc_93><loc_26></location>ϕ GR 4 = ( 15293365 508032 + 27145 504 ν + 3085 72 ν 2 ) , (A.6)</formula>
<formula><location><page_9><loc_35><loc_20><loc_93><loc_22></location>K 3 ( ν ) = ( 41005 1512 -5695 42 ν + 20500 189 ν 2 ) . (A.7)</formula>
<text><location><page_9><loc_8><loc_17><loc_68><loc_18></location>Comparing the Eq. (A.5) with the parametrization, ϕ 4 → ϕ GR 4 (1 + δ ˆ ϕ 4 ), we have:</text>
<text><location><page_9><loc_42><loc_14><loc_42><loc_15></location>δ</text>
<text><location><page_9><loc_43><loc_14><loc_44><loc_15></location>ˆ</text>
<text><location><page_9><loc_43><loc_14><loc_44><loc_15></location>ϕ</text>
<text><location><page_9><loc_45><loc_14><loc_46><loc_15></location>=</text>
<text><location><page_9><loc_47><loc_14><loc_48><loc_15></location>K</text>
<text><location><page_9><loc_48><loc_14><loc_49><loc_15></location>3</text>
<text><location><page_9><loc_49><loc_14><loc_50><loc_15></location>(</text>
<text><location><page_9><loc_50><loc_14><loc_50><loc_15></location>ν</text>
<text><location><page_9><loc_50><loc_14><loc_52><loc_15></location>) (</text>
<text><location><page_9><loc_52><loc_14><loc_53><loc_15></location>µ</text>
<text><location><page_9><loc_53><loc_14><loc_55><loc_15></location>32</text>
<text><location><page_9><loc_52><loc_13><loc_54><loc_14></location>GR</text>
<text><location><page_9><loc_52><loc_13><loc_53><loc_13></location>4</text>
<text><location><page_9><loc_51><loc_13><loc_52><loc_14></location>ϕ</text>
<text><location><page_9><loc_8><loc_10><loc_84><loc_11></location>Further, substituting µ 32 → (1 + δµ 32 ) into the above equation we got the following expression for δµ 32 :</text>
<formula><location><page_9><loc_44><loc_6><loc_93><loc_9></location>δµ 32 = δ ˆ ϕ 4 ϕ GR 4 K 3 ( ν ) . (A.9)</formula>
<text><location><page_9><loc_55><loc_14><loc_56><loc_16></location>-</text>
<text><location><page_9><loc_57><loc_14><loc_58><loc_15></location>1)</text>
<text><location><page_9><loc_59><loc_14><loc_59><loc_15></location>.</text>
<text><location><page_9><loc_89><loc_14><loc_93><loc_15></location>(A.8)</text>
<text><location><page_9><loc_44><loc_13><loc_44><loc_14></location>4</text>
<text><location><page_10><loc_8><loc_93><loc_78><loc_94></location>The 1.5PN correction to U ijk , that is the octupolar tail, first appears at 2.5PN in the GW flux:</text>
<formula><location><page_10><loc_34><loc_88><loc_93><loc_92></location>F Oct 2 . 5PN = π ( µ 30 µ 33 µ 2 2 )( 16403 2016 -16403 504 ν ) . (A.10)</formula>
<text><location><page_10><loc_8><loc_85><loc_93><loc_87></location>Hence the tail contribution from U ijk to the GW phase shows up at the 2.5PN-logarithmic term. The 2.5PN logarithmic phasing coefficient reads,</text>
<formula><location><page_10><loc_17><loc_76><loc_93><loc_84></location>ϕ 5 l = π ( 12080 63 -3680 21 ν + ( µ 30 µ 2 ) 2 [ -27340 189 + 109360 189 ν ] + ( µ 30 µ 33 µ 2 2 ) [ 82015 756 -82015 189 ν ] + ( ϵ 2 µ 2 ) 2 [ -20 9 + 80 9 ν ]) . (A.11)</formula>
<text><location><page_10><loc_8><loc_74><loc_72><loc_75></location>Assuming µ l ; l =3 = ϵ l = 1 and µ 3 n ' ; n ' =3 = 1, the 2.5PN phasing coefficient simplifies to,</text>
<text><location><page_10><loc_18><loc_74><loc_18><loc_74></location≯</text>
<text><location><page_10><loc_35><loc_74><loc_35><loc_74></location≯</text>
<formula><location><page_10><loc_40><loc_71><loc_93><loc_72></location>ϕ 5 l = ϕ GR 5 l + K 4 ( ν ) ( µ 33 -1 ) , (A.12)</formula>
<text><location><page_10><loc_8><loc_68><loc_34><loc_70></location>where ϕ GR 5 l and K 4 ( ν ) are given by</text>
<formula><location><page_10><loc_39><loc_64><loc_93><loc_67></location>ϕ GR 5 l = π ( 38645 252 -65 3 ν ) , (A.13)</formula>
<formula><location><page_10><loc_39><loc_61><loc_93><loc_64></location>K 4 ( ν ) = π ( 82015 756 -82015 189 ν ) . (A.14)</formula>
<text><location><page_10><loc_8><loc_56><loc_93><loc_60></location>Again comparing the above equation (Eq. A.12) with the parametrization, ϕ 5 l → ϕ GR 5 l (1 + δ ˆ ϕ 5 l ) and replacing µ 33 with (1 + δµ 33 ), we obtained the following expression for δµ 33 :</text>
<formula><location><page_10><loc_44><loc_52><loc_93><loc_55></location>δµ 33 = δ ˆ ϕ 5 l ϕ GR 5 l K 4 ( ν ) . (A.15)</formula>
<section_header_level_1><location><page_10><loc_41><loc_48><loc_60><loc_49></location>2. Analysis framework</section_header_level_1>
<text><location><page_10><loc_8><loc_42><loc_93><loc_46></location>Within the framework of Bayesian inference, measuring the parameter δµ 3 n amounts to obtaining the posterior probability density function P ( δµ 3 n | d, H ), with d denotes the detector data and H denotes the model. Using Bayes' theorem,</text>
<formula><location><page_10><loc_35><loc_38><loc_93><loc_41></location>P ( δµ 3 n | d, H ) = P ( δµ 3 n |H ) P ( d | δµ 3 n , H ) P ( d |H ) , (A.16)</formula>
<text><location><page_10><loc_8><loc_32><loc_93><loc_37></location>where, P ( δµ 3 n |H ) is the prior probability density function, P ( d | δµ 3 n , H ) is the likelihood function, and P ( d |H ) [with P ( d |H ) = ∫ d ( δµ 3 n ) P ( δµ 3 n |H ) P ( d | δµ 3 n , H )] is the evidence. The above equation with uniform prior on δµ 3 n (i.e., P ( δµ 3 n |H ) = ˜ Π( δµ 3 n |H )) can be written as follows,</text>
<formula><location><page_10><loc_18><loc_17><loc_93><loc_31></location>˜ P ( δµ 3 n | d, H ) = ˜ Π( δµ 3 n |H ) P ( d | δµ 3 n , H ) ˜ P ( d |H ) = ˜ Π( δµ 3 n |H ) ∫ dν d ( δ ˆ ϕ b ) P ( d | δ ˆ ϕ b , ν, H ) P ( δ ˆ ϕ b , ν | δµ 3 n , H ) ˜ P ( d |H ) = ˜ Π( δµ 3 n |H ) ˜ P ( d |H ) × ∫ dν d ( δ ˆ ϕ b ) P ( d | δ ˆ ϕ b , ν, H ) ˜ Π( δ ˆ ϕ b , ν |H ) P ( δµ 3 n | δ ˆ ϕ b , ν, H ) Π( δµ 3 n |H ) ︸ ︷︷ ︸ P ( δ ˆ ϕ b ,ν | δµ 3 n , H ) , (A.17)</formula>
<text><location><page_10><loc_8><loc_9><loc_93><loc_15></location>where, ˜ P ( d |H ) is the evidence for the uniform prior on δµ 3 n [i.e., ˜ P ( d |H ) ≡ ∫ d ( δµ 3 n ) ˜ Π( δµ 3 n |H ) P ( d | δµ 3 n , H )], P ( d | δ ˆ ϕ b , ν, H ) is the likelihood function of the gravitational wave data given the parameters δ ˆ ϕ b and ν , P ( δµ 3 n | δ ˆ ϕ b , ν, H ) takes care of the coordinate transformation between { δ ˆ ϕ b , ν } and δµ 3 n , ˜ Π( δ ˆ ϕ b , ν |H ) is the flat prior on { δ ˆ ϕ b , ν } , and Π( δµ 3 n |H ) is given by,</text>
<formula><location><page_10><loc_30><loc_6><loc_93><loc_8></location>Π( δµ 3 n |H ) ≡ ∫ dν d ( δ ˆ ϕ b ) ˜ Π( δ ˆ ϕ b , ν |H ) P ( δµ 3 n | δ ˆ ϕ b , ν, H ) . (A.18)</formula>
<figure>
<location><page_11><loc_8><loc_72><loc_92><loc_94></location>
<caption>FIG. 2. The probability distributions of δµ 3 n for 'GR', 'nonGR 2 σ ' and 'nonGR 3 σ ' cases (see the texts in Sec. 3 of the Supplemental Material for more details). The widths of the probability distributions of { ν, δ ˆ ϕ b } are chosen similar to GW190814. In the 'nonGR 2 σ ' and 'nonGR 3 σ ' cases the means of the δ ˆ ϕ b -distributions are 2 σ and 3 σ away from zero respectively. The colored vertical dashed lines indicate the 90% confidence intervals and median values. The probability distributions of δµ 3 n are peaking at zero in the 'GR' cases while in the 'nonGR 2 σ ' and 'nonGR 3 σ ' cases the distributions of δµ 3 n are excluding zero at 90% credibility.</caption>
</figure>
<text><location><page_11><loc_8><loc_56><loc_93><loc_59></location>Therefore, Π( δµ 3 n |H ) is simply the distribution of δµ 3 n derived from the uniform prior on { δ ˆ ϕ b , ν } and the relation between δµ 3 n and { δ ˆ ϕ b , ν } . The Eq. A.17 can be further simplified as</text>
<formula><location><page_11><loc_12><loc_46><loc_93><loc_55></location>˜ P ( δµ 3 n | d, H ) = ˜ Π( δµ 3 n |H ) × ˜ P IT ( d |H ) ˜ P ( d |H ) × ∫ dν d ( δ ˆ ϕ b ) [ ˜ Π( δ ˆ ϕ b , ν |H ) P ( d | δ ˆ ϕ b , ν, H ) ˜ P IT ( d |H ) ︸ ︷︷ ︸ ˜ P ( δ ˆ ϕ b ,ν | d, H ) × P ( δµ 3 n | δ ˆ ϕ b , ν, H ) Π( δµ 3 n |H ) ] = [∫ dν d ( δ ˆ ϕ b ) P ( δµ 3 n | δ ˆ ϕ b , ν, H ) ˜ P ( δ ˆ ϕ b , ν | d, H ) ] × ˜ Π( δµ 3 n |H ) Π( δµ 3 n |H ) × ˜ P IT ( d |H ) ˜ P ( d |H ) , (A.19)</formula>
<text><location><page_11><loc_8><loc_37><loc_93><loc_44></location>where, ˜ P ( δ ˆ ϕ b , ν | d, H ) is the posterior probability density of { δ ˆ ϕ b , ν } and ˜ P IT ( d |H ) [with ˜ P IT ( d |H ) = ∫ dν d ( δ ˆ ϕ b ) ˜ Π( δ ˆ ϕ b , ν |H ) P ( d | δ ˆ ϕ b , ν, H )] is the corresponding evidence derived assuming a uniform prior on { δ ˆ ϕ b , ν } . The numerical factor ˜ P IT ( d |H ) ˜ P ( d |H ) is an overall normalization constant in the above equation and can be ignored if one is only interested in estimating ˜ P ( δµ 3 n | d, H ).</text>
<text><location><page_11><loc_8><loc_30><loc_93><loc_37></location>The probability density function P ( δµ 3 n | δ ˆ ϕ b , ν, H ) maps the posterior samples of { δ ˆ ϕ b , ν } to samples of δµ 3 n via the relations given by Eq. (8) in the main paper for { δ ˆ ϕ 2 , ν } → { δµ 30 } , Eq. (A.9) for { δ ˆ ϕ 4 , ν } → { δµ 32 } and Eq. (A.15) for { δ ˆ ϕ 5 l , ν } → { δµ 33 } . Moreover, δµ 3 n is a function f 3 n ( δ ˆ ϕ b , ν ) of δ ˆ ϕ b and ν ; hence given a value of the pair { δ ˆ ϕ b , ν } , δµ 3 n is uniquely determined using the relations mentioned above. Therefore, P ( δµ 3 n | δ ˆ ϕ b , ν, H ) becomes delta function:</text>
<formula><location><page_11><loc_35><loc_27><loc_93><loc_28></location>P ( δµ 3 n | δ ˆ ϕ b , ν, H ) = δ ( δµ 3 n -f 3 n ( δ ˆ ϕ b , ν )) . (A.20)</formula>
<text><location><page_11><loc_8><loc_22><loc_93><loc_25></location>In practice one needs to take the posterior samples of { δ ˆ ϕ b , ν } for different GW events and to estimate δµ 3 n for each sample through the relations mentioned above; and then reweight these samples by the probability ˜ Π( δµ 3 n |H ) Π( δµ 3 n |H ) .</text>
<section_header_level_1><location><page_11><loc_40><loc_18><loc_61><loc_19></location>3. Detecting GR violation</section_header_level_1>
<text><location><page_11><loc_8><loc_5><loc_93><loc_16></location>Here we investigate to what extent deviations in the phase deformation coefficients ( δ ˆ ϕ b ) will be propagated to the probability distributions of the octupole deformation parameters ( δµ 3 n ) through this mapping. For demonstration, we consider a GW190814-like system. In order to simulate the GR case, we have drawn samples for { ν, δ ˆ ϕ 2 } from a bivariate normal distribution with means, { 0 . 09 , 0 . 0 } , standard deviations, { 0 . 01 , 0 . 05 } , and correlation coefficient, -0 . 29. After generating the samples for { ν, δ ˆ ϕ 2 } , we estimate the probability distribution of δµ 30 (shown in the blue curve in the left-most panel of Fig. 2) following the above-mentioned procedure. As expected, we see that the probability distribution of δµ 30 is sharply peaking at zero (blue curve in the left-most panel of Fig. 2). In the second</text>
<text><location><page_12><loc_8><loc_85><loc_93><loc_94></location>scenario, which models a 2 σ violation of GR, we have chosen the mean associated with δ ˆ ϕ 2 to be 0.1 (2 σ deviation from zero i.e., GR value) and kept all other parameters of the bivariate normal distribution at the same values as before. Again we obtain the probability distribution of δµ 30 (shown in the green curve in the left-most panel of Fig. 2) and find that the distribution of δµ 30 excludes zero (GR value) at 90% credibility. To simulate another non-GR scenario, a GR violation at 3 σ , we consider the mean of δ ˆ ϕ 2 to be 0.15 (3 σ deviation from zero i.e., GR value) and obtain the probability distribution of δµ 30 (red curve in the left-most panel of Fig. 2).</text>
<text><location><page_12><loc_8><loc_76><loc_93><loc_85></location>We also performed the similar exercises and obtain the probability distributions on δµ 32 (middle panel of Fig. 2) and δµ 33 (rightmost panel of Fig. 2). We find that in the 'GR' cases the probability distributions of { δµ 3 n } are peaking at zero (or, very close to zero; the median values of the distributions are zero) while in the 'nonGR 2 σ ' and 'nonGR 3 σ ' cases the distributions of { δµ 3 n } are excluding zero at 90% credibility. This demonstrates that the proposed mapping is successful in mapping the posteriors of the PN deformations reliably to the posteriors of the corresponding octupole deformation parameters.</text>
</document>
[
{
"title": "Parthapratim Mahapatra 1, ∗",
"content": "1 Chennai Mathematical Institute, Siruseri, 603103, India (Dated: October 10, 2024) Compact binaries with unequal masses and whose orbits are not aligned with the observer's line of sight are excellent probes of gravitational radiation beyond the quadrupole approximation. Among the compact binaries observed so far, strong evidence of octupolar modes is seen in GW190412 and GW190814, two binary black holes observed during the first half of the third observing run of LIGO/Virgo observatories. These two events, therefore, provide a unique opportunity to test the consistency of the octupolar modes with the predictions of general relativity (GR). In the postNewtonian (PN) approximation to GR, the gravitational-wave phasing has known dependencies on different radiative multipole moments, including the mass octupole. This permits the use of publicly released posteriors of the PN phase deformation parameters for placing constraints on the deformations to the different PN components of the radiative mass octupole denoted by δµ 3 n . Combining the posteriors on δµ 3 n from these two events, we deduce a joint bound (at 90% credibility) on the first three PN order terms in the radiative octupoles to be δµ 30 = -0 . 07 +0 . 11 -0 . 12 , δµ 32 = 0 . 48 +0 . 93 -1 . 15 , and δµ 33 = -0 . 32 +1 . 67 -0 . 62 , consistent with GR predictions. Among these, the measurement of δµ 33 for the first time confirms the well-known octupolar tail contribution, a novel nonlinear effect due to the scattering of the octupolar radiation by the background spacetime, is consistent with the predictions of GR. Detection of similar systems in the future observing runs should further tighten these constraints.",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "It is well known that the leading order gravitational wave (GW) emission is quadrupolar according to general relativity. However, subdominant higher multipoles get turned on if the binary has a mass asymmetry and when the line of sight of the observer is not aligned with the orbital angular momentum vector of the binary [1-9]. To date, the LIGO-Virgo-KAGRA collaboration has reported ∼ 90 confident detections of compact binary mergers [10-13]. Among these events, two compact binary mergers-GW190412 [14] and GW190814 [15]have shown clear evidence of the presence of octupolar ( ℓ = 3 , m = 3) mode, the first correction beyond the quadrupole. These two events, therefore, should facilitate a test of the gravitational octupolar structure of the compact binary dynamics. The gravitational dynamics of a compact binary system is typically divided into three stages of evolution: inspiral, merger, and ringdown. While the post-Newtonian (PN) approximation to general relativity (GR) [16] is employed to model the adiabatic inspiral stage of a compact binary coalescence, one requires numerical solutions to the Einstein equations [17], and the black hole (BH) perturbation theory [18] to describe the highly nonlinear merger stage, and the ringdown phase, respectively. As numerical relativity simulations are computationally expensive, currently there are two main modeling approaches towards producing the complete gravitational waveform (i.e., a single waveform that captures all three stages of binary evolution) for parameter inference: ef- fective one-body approach [19, 20] and phenomenological approach [21, 22]. Both these methods make the best use of the analytical and numerical understanding of compact binary dynamics. The gravitational waveform from a coalescing compact binary within GR, in the frequency domain, has the following form: where Y ℓm -2 are spin-weighted spherical harmonics of spin weight -2, ( ι, φ N ) describes the location of the observer in the binary's sky and ⃗ λ denotes the intrinsic parameters (e.g., masses and spins) as well as other relevant extrinsic parameters (e.g., luminosity distance ( d L ), reference time and reference phase) of the binary. Each GW mode ( ˜ h ℓm ) has an amplitude, A ℓm ( f ; ⃗ λ ), and a phase, ψ ℓm ( f ; ⃗ λ ) (i.e., ˜ h ℓm = A ℓm ( f ; ⃗ λ ) e i ψ ℓm ( f ; ⃗ λ ) ). In alternative theories of gravity, the gravitational dynamics of a compact binary could differ from the prediction of GR during all the three stages and might modify the phase and amplitude in the waveform. There exist proposals in the literature to probe the prediction of the harmonic structure of gravitational radiation from binary black hole coalescence in GR [2327]. Using GW190412 and GW190814, Ref. [28] tested the consistency between the dominant and subdominant modes and found the chirp mass estimated from the ℓ = 3 , m = 3 mode to be within ± 1% of the one estimated from the quadrupolar ℓ = 2 , m = 2 mode. In a more recent work [29], the consistency of the amplitudes of the h 21 and h 33 modes of the GW spectrum with GR predictions was investigated using these two events and found no evidence for any violation of GR. This test assumes the phases of subdominant harmonics ( ψ ℓm ; ℓ> 2 ) follow GR and investigates whether the amplitudes of subdominant harmonics ( A ℓm ; ℓ> 2 ) are consistent with the GR prediction. In this paper, we argue that if a signal contains nonquadrupolar modes, apart from the amplitude, the phase evolution will also carry their unique imprints [1, 25, 26, 30]. As GW detectors are more sensitive to phase evolution , this could be used to test GR, complementing the approach of [29]. However, we will focus only on the inspiral phase in this work, which is well-modeled by PN approximation to GR, and discuss constraints on the PN structure of octupolar emission in GR. For this, we will make use of the unique map between the masstype octupole coefficients at different PN orders and the bounds on the 1PN, 2PN, and 2.5PN logarithmic phasing deformation parameters for these two events in the parametrized tests of GW phasing reported in [31, 32]. Further, we will consider only the leading order appearance of the octupole coefficients in the GW phase for this mapping. The remainder of the paper is organized as follows. In Sec. II, we briefly review the parametrized tests of GW phasing. In Sec. III, we introduce the octupolar parametrization. We derive the relations between different PN pieces in the mass-type octupole moment and different PN phasing terms in Sec. IV. In Sec. V, we describe the Bayesian framework to infer the octupole parameters. Our results and conclusions are presented in Sec. VI.",
"pages": [
1,
2
]
},
{
"title": "II. PARAMETRIZED TESTS OF GW PHASING",
"content": "The frequency domain GW phase from the inspiral part of the waveform (computed using the stationary phase approximation [33, 34]) for the leading quadrupolar harmonic [4, 35] takes the form where v = ( πGMf/c 3 ) 1 / 3 is the PN expansion parameter, M is the binary's redshifted total mass, ν is the symmetric mass ratio of the binary, ϕ i and ϕ il denote the nonlogarithmic and logarithmic PN phasing coefficients, respectively. Due to the lack of accurate waveforms in alternative theories of gravity, 'theory-agnostic' approaches are often adopted to test GR with GW data. These 'null tests' of GR make use of our best knowledge of compact binary dynamics in GR and look for possible deviations from GR without reference to specific alternatives (see Refs. [31, 32, 36, 37] for more details.). One of the most generic tests of GR that has been routinely performed with LIGO/Virgo data is the parametrized test of GW phasing [38-46]. The parametrized tests rely on measuring any deviations in the PN coefficients ϕ i and ϕ il in the GW phasing, which are uniquely predicted by GR, from compact binary mergers. A parametrized waveform model introduces additional degrees of freedom to capture signatures of possible GR violation by modifying the phasing coefficients as ( b = i, il ) (see Sec. VA of Refs. [31, 32] for more details). In GR, these phenomenological dimensionless deviation parameters ( δ ˆ ϕ b ) are identically zero, whereas in alternative theories of gravity, one or more of these parameters could be different from zero. Combining data from different GW events detected during the first, second, and third observing runs of LIGO/Virgo, the current bound on the PN deviation parameters are found to be consistent with GR (see Figs. 6 and 7 of Ref. [32]). For the two asymmetric binary events, GW190412 and GW190814, we will use the results of the parametrized tests, obtained by applying parametrized IMRPhenomPv3HM (denoted as ' Phenom ' in this paper) [8] and SEOBNRv4HM ROM (denoted as ' SEOB ')[9] waveform approximant to the data. Phenom waveform is a frequency-domain phenomenological waveform model that includes the effects of two-spin precession along with higher multipole moments [8], whereas SEOB is a frequency-domain nonprecessing reduced-order effective one-body model that incorporates the higher order modes [9]. In the current LIGO-Virgo-KAGRA analyses [31, 32, 37], the reported bounds on δ ˆ ϕ b come from the fractional deviations applied to the nonspinning portion of the phase (see Sec. VA of Refs. [31, 32] for detailed discussions).",
"pages": [
2
]
},
{
"title": "III. PARAMETRIZED MULTIPOLAR GRAVITATIONAL WAVEFORMS",
"content": "The radiative multipole moments of a compact binary system contain information about source physics (masses [16, 30, 47], spins [4, 16, 48-60], tidal deformability [6167], spin induced quadrupole moment [51, 68-71] etc.) and account for various nonlinear interactions and physical effects (such as 'tail' effects [30, 72, 73], tails of tails [74], tail square [75], memory effects [76-79], spinorbit effects [48, 80], spin-spin effects [48, 59] etc.) that occur at different PN orders in GR. In alternative theories of gravity, one or more radiative multipole moments of a compact binary could be different from those in GR (see for instance Refs. [81-83]). One can put constraints on such theories by studying the multipolar structure of asymmetric compact binary systems like GW190412 and GW190814. References [25, 26] came up with a novel theoryagnostic method to test the multipolar structure of the gravitational field radiated from an inspiralling compact binary. Using the multipolar post-Minkowskian formalism [30, 47, 72, 84-92], Ref. [25] derived the parametrized multipolar gravitational wave phasing up to 3.5PN order for nonspinning binaries and Ref. [26] extended it for nonprecessing binaries. The multipolar postMinkowskian formalism relates the radiation content in the far zone, encoded in the mass- and current-type radiative multipole moments { U L , V L } , to the stress-energy tensor of the source. In order to model possible deviations in the multipole structure, Refs. [25, 26] adopted the following parametrization for radiative multipole moments: U L → µ l U L , V L → ϵ l V L . By construction, the phenomenological multipole parameters µ l , ϵ l are equal to unity in GR. With this parametrization, the contributions from various radiative multipole moments to the GWphasing can be tracked separately, thereby facilitating tests of the multipolar structure of the PN approximation to GR. In this work, we go one step further and probe the different PN orders in the radiative mass octupole moment of a compact binary as it evolves through the adiabatic inspiral phase. We propose the parametrization where U ijk is the mass-type radiative octupole moment, U ( n ) , GR ijk is the (n/2)th PN correction to U ijk in GR and µ 3 n is the corresponding octupole coefficient. Note that there is no 1/c (i.e., 0.5PN) contribution in the mass octupole moment in GR. The 1.5PN correction term in the octupole moment arises due to the tail effect, caused by the scattering of the outgoing octupolar wave off the background spacetime associated with the total [Arnowitt-Deser-Misner (ADM)] mass of the source [30, 72-75]. By definition µ 3 n is unity in GR and appears at different PN orders in the phasing formula. For instance, µ 30 first appears at 1PN, µ 32 at 2PN and µ 33 at 2.5PN (logarithmic) order. We next discuss how the existing bounds on the PN deformation coefficients δ ˆ ϕ b , based on the parametrization in Eq. (3), reported in [31, 32] can be mapped to the bounds on µ 3 n in the parametrization derived above.",
"pages": [
2,
3
]
},
{
"title": "IV. MAPPING THE PN BOUNDS TO THE OCTUPOLE PARAMETERS",
"content": "Each of the parameters µ 3 n appears at multiple PN orders in the GW phasing. For example, µ 30 appears at 1PN, 2PN, 2.5PN (logarithmic), and 3.5PN orders. Therefore if there is a deviation from GR in one of the µ 3 n , it will result in a dephasing of each of the PN phasing coefficients at the order in which this octupole pa- eter contributes. 1 Here we neglect the modification to all PN orders except the leading order at which they first appear. This is a reasonable assumption to make because if there is a deviation in any of the µ 3 n , the leading order at which they appear would be most sensitive to such a deviation. ̸ ̸ Therefore, the goal now will be to obtain constraints on µ 30 , µ 32 , and µ 33 using the bounds on 1PN, 2PN, and 2.5PN logarithmic phase deformation parameters, respectively, along with other relevant intrinsic binary parameters for particular GW events. Further, while estimating bounds on one of the µ 3 n , we assume all other mass-type octupole parameters as well as rest of the multipole parameters to take their values in GR (i.e., µ 3 n ' ; n ' = n = µ l ; l =3 = ϵ l = 1) in the spirit of a singleparameter test, i.e., varying one deformation parameter at a time. The expression for 1PN phasing coefficient in the parametrized multipolar GW phase [see Eq. (2.16) of Ref. [25]] is given by where and µ 2 and ϵ 2 are the mass and current quadrupole parameters, respectively. Comparing the parametrized PN phasing of Eq. (3) to Eq. (5) we have the following relation between δ ˆ ϕ 2 and µ 30 : In the spirit of null tests, we find it more convenient to employ octupole deformation parameters δµ 3 n defined for mass octupole by imposing µ 3 n = 1 + δµ 3 n for the different PN pieces. The aim now would be to derive bounds on δµ 30 , δµ 32 , and δµ 33 using GW observations assuming µ l ; l =3 = ϵ l = 1 and δµ 3 n ' ; n ' = n = 0. 2 ̸ ̸ With the above assumption we can express δµ 30 in terms of δ ˆ ϕ 2 as Among the two solutions, we will adopt the one which respects the GR limit (i.e., δµ 30 vanishes when δ ˆ ϕ 2 → 0). Similarly, we also obtain the expressions for δµ 32 and δµ 33 as (see Sec. 1 of the Supplemental Material for a derivation): The expressions of ϕ GR 4 , K 3 ( ν ), ϕ GR 5 l , and K 4 ( ν ) are provided in Sec. 1 of the Supplemental Material. Having obtained the mapping, now the problem essentially amounts to using the posterior samples of { δ ˆ ϕ b , ν } for any event and computing the corresponding posteriors on { δµ 3 n } using the above equations. The exact procedure followed is discussed next.",
"pages": [
3,
4
]
},
{
"title": "V. INFERRING OCTUPOLE PARAMETERS",
"content": "Given the LIGO/Virgo data, d , we are interested in deriving ˜ P ( δµ 3 n | d, H ), the posterior probability distribution on δµ 3 n , for a flat prior on δµ 3 n (here H denotes the hypothesis, which is the parametric model we employ.). Towards this, we use Eqs. (8), (9), and (10), along with the two-dimensional posterior distribution P ( δ ˆ ϕ b , ν | d, H ), for different GW events. For example, to derive ˜ P ( δµ 30 | d, H ) we will use Eq. (8) and P ( δ ˆ ϕ 2 , ν | d, H ). The probability distribution P ( δ ˆ ϕ b , ν | d, H ) is computed for flat priors on δ ˆ ϕ b and mass ratio. Therefore in the Bayesian framework the samples of δµ 3 n , derived from P ( δ ˆ ϕ b , ν | d, H ), need to be reweighted to obtain posterior ˜ P ( δµ 3 n | d, H ) that assume flat priors on δµ 3 n as In the above equation, the tilde denotes flat priors or posteriors derived assuming flat prior on the corresponding parameters. Hence ˜ Π( δµ 3 n |H ) denotes flat prior on δµ 3 n and ˜ P ( δ ˆ ϕ b , ν | d, H ) denote posterior assuming flat priors on δ ˆ ϕ b and mass ratio. The prior distribution, ˜ Π( δµ 3 n |H ), is chosen to be uniform between [-30, 30]. While P ( δµ 3 n | δ ˆ ϕ b , ν, H ) takes care of the coordinate transformation between ( δ ˆ ϕ b , ν ) to δµ 3 n [using Eqs. (8)-(10)], Π( δµ 3 n |H ) in the above equation is simply P ( δµ 3 n | δ ˆ ϕ b , ν, H ) for the flat prior on δ ˆ ϕ b and mass ratio. A detailed derivation of the above equation is provided in Sec. 2 of the Supplemental Material. As GW190814 and GW190412 are the only two events for which a confident detection of higher modes was possible, we will restrict to these two events for our purposes. We use the parameter estimation samples for δ ˆ ϕ b and symmetric mass ratio ( ν ) from the GWTC-3 Data Release [93] for these two events, analyzed with different waveform approximants, and estimate the bounds on the parameters δµ 3 n following this procedure. While estimating the posterior probability distribution of δµ 30 , we need to ensure that the values of δµ 30 obtained from a pair of ( δ ˆ ϕ 2 , ν ) should be real by imposing the condition [see Eq. (8)], In order to realize this, we discard those samples of δµ 30 which do not meet the above condition. If we need to remove relatively large number of samples, that means the event is uninformative. We find that for events except GW190814 and GW190412, majority of the samples do not meet this condition. This simply is a reflection of the fact that we are trying to test deviation in octupole moment when its presence is barely there in the signal.",
"pages": [
4
]
},
{
"title": "VI. RESULTS AND CONCLUSIONS",
"content": "The posteriors of the leading and two subleading octupole deformation parameters δµ 30 , δµ 32 , and δµ 33 for GW190412 and GW190814 obtained by the abovementioned procedure are shown in Fig. 1. Among the detected events, GW190814 provides the tightest constraints on all the octupole parameters. This is expected as GW190814 is the most unequal mass binary (mass ratio, q = 0 . 112 +0 . 008 -0 . 009 ) among the GW events in the GWTC-3 and asymmetric systems get stronger contributions from non-quadrupolar moments. In addition to the individual event analysis in Fig. 1, we have also obtained the combined bounds on δµ 3 n using data from multiple events under the assumption that the same value of δµ 3 n is shared across all the events. The joint constraints on these parameters are obtained by multiplying the individual likelihoods from the events, GW190412 and GW190814, analyzed with Phenom and SEOB waveforms. In the joint analysis the most tightly constrained parameter is δµ 30 and the most weakly constrained parameter is δµ 33 . The posterior on higherorder mass octupole deformation parameters, such as δµ 34 , are mostly uninformative and not shown here. Detections of unequal mass binaries in the future with a larger signal-to-noise ratio will enable us to probe higher PN pieces in the mass octupole moment. Bounds from the two different kinds of waveform approximants show excellent agreement with each other. On all occasions the posterior distributions on δµ 3 n are statistically consistent with δµ 3 n = 0 within 90% credible interval. This is the first reported bound on the different PN pieces in mass-type octupole moment of compact binary complementing the previous consistency tests in Refs. [28, 29]. It is interesting that the bounds on δµ 33 also confirm the consistency of the octupolar tail radiation with the predictions of GR. The posterior distributions on current quadrupole deformation parameters are also largely uninformative and not reported here. In the future, the detections of high mass ratio and highly spinning binaries with larger signal-to-noise ratio will enhance the contribution of the current quadrupole to the flux making its measurement with good precision possible. Last, it is instructive to ask if the GR violations in the δ ˆ ϕ b posteriors can be captured by the mapping proposed in this work. This is examined in Fig. 2 of the Supplemental Material (see the texts in Sec. 3 of the Supplemental Material for more details). We consider a GW190814-like system and simulate GR violations with different σ values in the δ ˆ ϕ b posteriors. We find that the derived posteriors on δµ 3 n , through this mapping, will be able to detect deviations at different σ values in the δ ˆ ϕ b posteriors from GR. To conclude, the parametrized multipolar waveforms could play a pivotal role in testing GR with current and next-generation GW detectors. The next-generation GW detectors will observe more diverse classes of compact binaries, thereby allowing us to probe even higher multipoles of compact binaries, and the parametrization introduced here will be crucial in such scenarios. Development of an infrastructure that can directly sample over the multipole parameters is planned for future work which should be able to probe the multipole structure without relying on the approximate mapping we have invoked here. The direct inference of multipole parameters from the GW data could provide even more stringent constraints as one will potentially gain information from multiple PN coefficients in the phase. Moreover, the inclusion of multipole parameters in the amplitude will likely play a critical role in this framework.",
"pages": [
4,
5,
6
]
},
{
"title": "ACKNOWLEDGMENTS",
"content": "This material is based upon work supported by the NSF's LIGO Laboratory, which is a major facility fully funded by the National Science Foundation (NSF). The author is grateful for computational resources provided by the LIGO Laboratory and supported by National Science Foundation Grants No. PHY-0757058 and No. PHY-0823459. It is a pleasure to thank K. G. Arun, who suggested this problem to me, encouraged me to pursue it, and for comments on the draft. The author is grateful to A. Gupta, S. Kastha, and B. S. Sathyaprakash for invaluable discussions and/or comments on the manuscript. We thank M. Saleem for critical reading of the manuscript and providing useful comments. We also thank P. Saini, S. A. Bhat, and P. D. Roy for comments on the manuscript. We are thankful to N. J. McDaniel, A. Laddha, and S. Datta for valuable discussions. P.M. acknowledge the support of the Core Research Grant No. CRG/2021/004565 of the Science and Engineering Research Board of India and a grant from the Infosys foundation. This research has made use of data obtained from the Gravitational Wave Open Science Center (www.gw-openscience.org), a service of LIGO Laboratory, the LIGO Scientific Collaboration and the Virgo Collaboration. Virgo is funded by the French Centre National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale della Fisica Nucleare (INFN), and the Dutch Nikhef, with contributions by Polish and Hungarian institutes. This manuscript has the LIGO preprint number P2300150 . Analyses in this paper made use of NumPy [94], SciPy [95], IPython [96], Matplotlib [97], Corner [98], Jupyter [99], Seaborn [100] software packages.",
"pages": [
6
]
},
{
"title": "1. Tracking different PN pieces of the mass type octupole moment",
"content": "With the parametrization introduced in Eq. (4) of the main paper, the different PN contributions from the mass type radiative octupole moment to the GW flux, and hence to the GW phasing, can be separately kept track of. To derive such a parametrized octupolar GW phasing, we follow the MPM formalism [30] along the lines of Refs. [25, 26]. The parametrized multipolar flux schematically reads as where x = ( GMω/c 3 ) 2 / 3 is a PN parameter, ω the orbital angular frequency of the binary. The leading order contribution from the mass-type octupole moment ( U ijk ) to the GW flux first appears at 1PN: Therefore the leading order contribution from U ijk to the GW phasing appears at 1PN and is given by Eq. (5) in the main paper. The next PN contribution (i.e., 1PN correction to U ijk ; note that there is no 0.5PN correction to U ijk .) from U ijk to the GW flux makes appearance at 2PN which reads as Thus the 1PN correction to the octupole moment first appears at 2PN in the phasing. The 2PN phasing coefficient reads With the assumptions µ l ; l =3 = ϵ l = 1 and µ 3 n ' ; n ' =2 = 1, the 2PN phasing coefficient reduces to ̸ ̸ where ϕ GR 4 and K 3 ( ν ) are given by Comparing the Eq. (A.5) with the parametrization, ϕ 4 → ϕ GR 4 (1 + δ ˆ ϕ 4 ), we have: δ ˆ ϕ = K 3 ( ν ) ( µ 32 GR 4 ϕ Further, substituting µ 32 → (1 + δµ 32 ) into the above equation we got the following expression for δµ 32 : - 1) . (A.8) 4 The 1.5PN correction to U ijk , that is the octupolar tail, first appears at 2.5PN in the GW flux: Hence the tail contribution from U ijk to the GW phase shows up at the 2.5PN-logarithmic term. The 2.5PN logarithmic phasing coefficient reads, Assuming µ l ; l =3 = ϵ l = 1 and µ 3 n ' ; n ' =3 = 1, the 2.5PN phasing coefficient simplifies to, ̸ ̸ where ϕ GR 5 l and K 4 ( ν ) are given by Again comparing the above equation (Eq. A.12) with the parametrization, ϕ 5 l → ϕ GR 5 l (1 + δ ˆ ϕ 5 l ) and replacing µ 33 with (1 + δµ 33 ), we obtained the following expression for δµ 33 :",
"pages": [
9,
10
]
},
{
"title": "2. Analysis framework",
"content": "Within the framework of Bayesian inference, measuring the parameter δµ 3 n amounts to obtaining the posterior probability density function P ( δµ 3 n | d, H ), with d denotes the detector data and H denotes the model. Using Bayes' theorem, where, P ( δµ 3 n |H ) is the prior probability density function, P ( d | δµ 3 n , H ) is the likelihood function, and P ( d |H ) [with P ( d |H ) = ∫ d ( δµ 3 n ) P ( δµ 3 n |H ) P ( d | δµ 3 n , H )] is the evidence. The above equation with uniform prior on δµ 3 n (i.e., P ( δµ 3 n |H ) = ˜ Π( δµ 3 n |H )) can be written as follows, where, ˜ P ( d |H ) is the evidence for the uniform prior on δµ 3 n [i.e., ˜ P ( d |H ) ≡ ∫ d ( δµ 3 n ) ˜ Π( δµ 3 n |H ) P ( d | δµ 3 n , H )], P ( d | δ ˆ ϕ b , ν, H ) is the likelihood function of the gravitational wave data given the parameters δ ˆ ϕ b and ν , P ( δµ 3 n | δ ˆ ϕ b , ν, H ) takes care of the coordinate transformation between { δ ˆ ϕ b , ν } and δµ 3 n , ˜ Π( δ ˆ ϕ b , ν |H ) is the flat prior on { δ ˆ ϕ b , ν } , and Π( δµ 3 n |H ) is given by, Therefore, Π( δµ 3 n |H ) is simply the distribution of δµ 3 n derived from the uniform prior on { δ ˆ ϕ b , ν } and the relation between δµ 3 n and { δ ˆ ϕ b , ν } . The Eq. A.17 can be further simplified as where, ˜ P ( δ ˆ ϕ b , ν | d, H ) is the posterior probability density of { δ ˆ ϕ b , ν } and ˜ P IT ( d |H ) [with ˜ P IT ( d |H ) = ∫ dν d ( δ ˆ ϕ b ) ˜ Π( δ ˆ ϕ b , ν |H ) P ( d | δ ˆ ϕ b , ν, H )] is the corresponding evidence derived assuming a uniform prior on { δ ˆ ϕ b , ν } . The numerical factor ˜ P IT ( d |H ) ˜ P ( d |H ) is an overall normalization constant in the above equation and can be ignored if one is only interested in estimating ˜ P ( δµ 3 n | d, H ). The probability density function P ( δµ 3 n | δ ˆ ϕ b , ν, H ) maps the posterior samples of { δ ˆ ϕ b , ν } to samples of δµ 3 n via the relations given by Eq. (8) in the main paper for { δ ˆ ϕ 2 , ν } → { δµ 30 } , Eq. (A.9) for { δ ˆ ϕ 4 , ν } → { δµ 32 } and Eq. (A.15) for { δ ˆ ϕ 5 l , ν } → { δµ 33 } . Moreover, δµ 3 n is a function f 3 n ( δ ˆ ϕ b , ν ) of δ ˆ ϕ b and ν ; hence given a value of the pair { δ ˆ ϕ b , ν } , δµ 3 n is uniquely determined using the relations mentioned above. Therefore, P ( δµ 3 n | δ ˆ ϕ b , ν, H ) becomes delta function: In practice one needs to take the posterior samples of { δ ˆ ϕ b , ν } for different GW events and to estimate δµ 3 n for each sample through the relations mentioned above; and then reweight these samples by the probability ˜ Π( δµ 3 n |H ) Π( δµ 3 n |H ) .",
"pages": [
10,
11
]
},
{
"title": "3. Detecting GR violation",
"content": "Here we investigate to what extent deviations in the phase deformation coefficients ( δ ˆ ϕ b ) will be propagated to the probability distributions of the octupole deformation parameters ( δµ 3 n ) through this mapping. For demonstration, we consider a GW190814-like system. In order to simulate the GR case, we have drawn samples for { ν, δ ˆ ϕ 2 } from a bivariate normal distribution with means, { 0 . 09 , 0 . 0 } , standard deviations, { 0 . 01 , 0 . 05 } , and correlation coefficient, -0 . 29. After generating the samples for { ν, δ ˆ ϕ 2 } , we estimate the probability distribution of δµ 30 (shown in the blue curve in the left-most panel of Fig. 2) following the above-mentioned procedure. As expected, we see that the probability distribution of δµ 30 is sharply peaking at zero (blue curve in the left-most panel of Fig. 2). In the second scenario, which models a 2 σ violation of GR, we have chosen the mean associated with δ ˆ ϕ 2 to be 0.1 (2 σ deviation from zero i.e., GR value) and kept all other parameters of the bivariate normal distribution at the same values as before. Again we obtain the probability distribution of δµ 30 (shown in the green curve in the left-most panel of Fig. 2) and find that the distribution of δµ 30 excludes zero (GR value) at 90% credibility. To simulate another non-GR scenario, a GR violation at 3 σ , we consider the mean of δ ˆ ϕ 2 to be 0.15 (3 σ deviation from zero i.e., GR value) and obtain the probability distribution of δµ 30 (red curve in the left-most panel of Fig. 2). We also performed the similar exercises and obtain the probability distributions on δµ 32 (middle panel of Fig. 2) and δµ 33 (rightmost panel of Fig. 2). We find that in the 'GR' cases the probability distributions of { δµ 3 n } are peaking at zero (or, very close to zero; the median values of the distributions are zero) while in the 'nonGR 2 σ ' and 'nonGR 3 σ ' cases the distributions of { δµ 3 n } are excluding zero at 90% credibility. This demonstrates that the proposed mapping is successful in mapping the posteriors of the PN deformations reliably to the posteriors of the corresponding octupole deformation parameters.",
"pages": [
11,
12
]
}
]
2024PhRvD.109d3049T
https://arxiv.org/pdf/2311.04985.pdf
<document>
<section_header_level_1><location><page_1><loc_22><loc_90><loc_79><loc_93></location>Probing More Deeply in an All-Sky Search for Continuous Gravitational Waves in the LIGO O3 Data Set</section_header_level_1>
<text><location><page_1><loc_37><loc_87><loc_64><loc_88></location>Aashish Tripathee 1 and Keith Riles 1</text>
<text><location><page_1><loc_25><loc_84><loc_76><loc_87></location>1 University of Michigan Physics Department, Ann Arbor, MI 48109, USA (compiled February 27, 2024)</text>
<text><location><page_1><loc_18><loc_65><loc_83><loc_83></location>We report results from an all-sky search of the LIGO data from the third LIGO-Virgo-KAGRA run (O3) for continuous gravitational waves from isolated neutron stars in the frequency band [30, 150] Hz and spindown range of [ -1 × 10 -8 , +1 × 10 -9 ] Hz/s. This search builds upon a previous analysis of the first half of the O3 data using the same PowerFlux pipeline. We search more deeply here by using the full O3 data and by using loose coherence in the initial stage with fully coherent combination of LIGO Hanford (H1) and LIGO Livingston (L1) data, while limiting the frequency band searched and excluding narrow, highly disturbed spectral bands. We detect no signal and set strict frequentist upper limits on circularly polarized and on linearly polarized wave amplitudes, in addition to estimating population-averaged upper limits. The lowest upper limit obtained for circular polarization is ∼ 4 . 5 × 10 -26 , and the lowest linear polarization limit is ∼ 1 . 3 × 10 -25 (both near 144 Hz). The lowest estimated population-averaged upper limit is ∼ 1 . 0 × 10 -25 . In the frequency band and spindown range searched here, these limits improve upon the O3a PowerFlux search by a median factor of ∼ 1 . 4 and upon the best previous limits obtained for the full O3 data by a median factor of ∼ 1 . 1.</text>
<section_header_level_1><location><page_1><loc_20><loc_61><loc_37><loc_62></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_43><loc_49><loc_59></location>LIGO [1], VIRGO [2], and KAGRA [3-5] are interferometer-based experiments that search for gravitational waves. Nearly 100 transient gravitational waves have been detected from black hole (BH) mergers, neutron stars (NS) mergers, and BH-NS mergers since the first detection in 2015 [6-10]. An as yet undetected type of gravitational waves is continuous, long-lived, and nearly monochromatic. Such radiation is necessarily weaker than that detected already from transient mergers and requires long-period integration for detection in the presence of interferometer noise [11].</text>
<text><location><page_1><loc_9><loc_27><loc_49><loc_43></location>In this work we focus on one of the potential sources for continuous gravitational waves - rapidly spinning isolated neutron stars with non-axisymmetry. While previous all-sky searches in the O3 data [12, 13], including the O3a PowerFlux analysis, analyzed a broad band of approximately 20-2000 Hz, this work focuses on young rapidly spinning neutron stars in the Milky Way with gravitational wave frequencies of 30-150 Hz. The nonaxisymmetry in such stars could stem from different sources including crustal deformation, buried magnetic field energy, and excitation of r-modes [14-16].</text>
<text><location><page_1><loc_9><loc_16><loc_49><loc_27></location>This search uses the venerable PowerFlux pipeline [1724] with loose coherence [21, 25-27] to perform an all-sky search in the spindown range [ -1 × 10 -8 , +1 × 10 -9 ] Hz/s. Our search resulted in ≈ 3 . 4 × 10 6 outliers 1 in the first stage after excluding narrow, highly disturbed bands, a number that dropped to ≈ 1 . 2 × 10 4 by the final stage of the hierarchical search. These outliers were then followed up on with the Markov Chain Monte Carlo (MCMC)</text>
<text><location><page_1><loc_52><loc_49><loc_92><loc_62></location>based PyFstat framework [28, 29]. All but 10 of the surviving outliers can be attributed to hardware injections (simulated CW signals imposed upon the interferometer mirror motion) or to known spectral artifacts. Those 10 outliers, however, have Bayes factors obtained from PyFstat much less than the determined threshold expected for a plausible signal. As this search failed to detect a true signal, we set upper limits on strain amplitudes as a function of frequency, excluding excised bands.</text>
<text><location><page_1><loc_52><loc_38><loc_92><loc_49></location>This article is organized as follows: Section II describes the data set we used. Section III discusses the analysis method with PowerFlux and loose-coherence in addition to the data cleaning that was performed to suppress instrumental artifacts. Section IV shows the upper limits and outliers results obtained. Section V concludes the article with a discussion of the results and prospects for future searches including the recently started O4 search.</text>
<section_header_level_1><location><page_1><loc_63><loc_33><loc_81><loc_34></location>II. DATA SETS USED</section_header_level_1>
<text><location><page_1><loc_52><loc_17><loc_92><loc_31></location>This analysis uses data from the O3 run of Advanced LIGO. Advanced LIGO consists of two interferometers one at Hanford, Washington (H1) and the other at Livingston, Louisiana (L1), separated by a 3000 km baseline [1]. The O3 run took place between April 1, 2019 and March 27, 2020 [30]. The first part of the run (O3a) and the second part (O3b) were separated by a commissioning break in October 2019 [30]. This search builds upon the O3a PowerFlux analysis [12] but uses the ∼ 11 months of the full O3 dataset.</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_17></location>Before the analysis was performed, known spectral artifacts ('lines') [31] were excised and replaced with random Gaussian noise ('cleaned'). Additionally, loud single-detector artifacts of an unknown source were also cleaned [31]. Details of the cleaning are discussed below in Section III C.</text>
<text><location><page_2><loc_9><loc_77><loc_49><loc_93></location>Another artifact affecting the data were loud, relatively frequent glitches - short, high amplitude instrumental transients [32-34]. While also present in previous runs, they were much louder and more frequent in the O3 run [12, 34]. The effect of glitching was particularly troublesome for frequencies below 500 Hz. We started from 'self-gated' data in which a narrow inverse-Tukeywindow zeroing gate was applied to the time-domain data, details of which can be found in the technical document [32]. Gating applied here leads to livetime losses of about 3% and 12%</text>
<text><location><page_2><loc_9><loc_70><loc_49><loc_77></location>of the H1 and L1 observation times. The calibration uncertainties of this C01 gated data set are estimated to be < 7% in magnitude and < 4 deg in phase for O3a and < 11% in magnitude and < 9 deg in phase for O3b (68% confidence interval) [35, 36].</text>
<section_header_level_1><location><page_2><loc_18><loc_65><loc_40><loc_66></location>III. ANALYSIS METHOD</section_header_level_1>
<text><location><page_2><loc_9><loc_31><loc_49><loc_63></location>This analysis uses the PowerFlux pipeline to search for continuous gravitational waves. The strain data is divided into 7200 s segments for which discrete Fourier transforms are computed ('short' Fourier Transforms SFTs), using Hann windowing with 50% overlap. For a large number of templates that include corrections for spindown evolution and Doppler modulation, power is summed up over all the SFTs while using an inversenoise-weighting to disfavor SFTs with high noise levels [37]. For each 1/16 Hz search sub-band, the power is maximized across the sky and across spindown steps to produce 95% frequentist upper limits [23, 24, 37]. The first stage of the hierarchical search, called Stage 0, determines upper limits and yields a set of outliers defined by a signal-to-noise ratio (SNR) greater than 7. These outliers undergo two stages of loose coherence followup, applying tighter constraints and requiring increased SNR, after which persistent outliers are manually examined via strain histograms [12] to identify instrumental artifacts [38]. Surviving outliers are then followed up by the PyFstat pipeline which determines a Bayes factor of preference for signal over background for each outlier [28, 29].</text>
<section_header_level_1><location><page_2><loc_22><loc_27><loc_36><loc_28></location>A. Signal Model</section_header_level_1>
<text><location><page_2><loc_9><loc_9><loc_49><loc_24></location>Our analysis assumes a model in which a spinning neutron star with a time-varying quadrupole moment produces circularly polarized waves along the direction of the spin-axis and linearly polarized waves in the perpendicular directions [11, 23, 37] with a frequency twice that of the star's spin frequency. The polarization of the detected wave then depends on the inclination angle between the spin-axis and the line-of-sight to the Earth, ι [11]. Extreme cases are ι = 0 , π and ι = π/ 2. The former corresponds to circularly polarized waves which is the best-case scenario maximizing incident signal power.</text>
<text><location><page_2><loc_52><loc_90><loc_92><loc_93></location>The latter corresponds to linearly polarized waves which is the worst-case scenario.</text>
<text><location><page_2><loc_53><loc_89><loc_84><loc_90></location>Specifically, the signal model we assume is:</text>
<formula><location><page_2><loc_56><loc_82><loc_92><loc_87></location>h ( t ) = h 0 { F + ( t, α 0 , δ 0 , ψ ) 1 + cos ι 2 2 cos [Φ( t )] + F × ( t, α 0 , δ 0 , ψ ) cos ι sin [Φ( t )] } , (1)</formula>
<text><location><page_2><loc_52><loc_74><loc_92><loc_80></location>where h 0 is the intrinsic strain amplitude, Φ( t ) is the signal phase, the F + and F × are detector responses to signals with plus and cross polarizations, respectively [11, 23, 37].</text>
<text><location><page_2><loc_52><loc_69><loc_92><loc_74></location>In the loose coherence stages of the search, we assume a second-order Taylor series model 2 for the phase evolution in which we include only terms up to quadratic in offsets from a reference time τ :</text>
<formula><location><page_2><loc_54><loc_63><loc_92><loc_66></location>Φ( τ ) = Φ 0 +2 π { f ( τ -τ ref ) + 1 2 ˙ f ( τ -τ ref ) 2 } , (2)</formula>
<text><location><page_2><loc_52><loc_59><loc_92><loc_62></location>where Φ 0 is the initial phase and f, ˙ f are the frequency and spindowns at the chosen reference time τ ref :</text>
<formula><location><page_2><loc_59><loc_54><loc_92><loc_57></location>τ ref ( t ) = t + ⃗r ( t ) · ⃗n c +∆ E ⊙ -∆ S ⊙ , (3)</formula>
<text><location><page_2><loc_52><loc_46><loc_92><loc_53></location>where ⃗r ( t ) is the position vector of the detector in the Solar System Barycenter (SSB) frame, ∆ E ⊙ , and ∆ S ⊙ are the relativistic Einstein and Shapiro time delays respectively [13]. ⃗n is the vector pointing from the detector toward the source, ⃗n = (cos α cos δ, sin α cos δ, sin δ ).</text>
<text><location><page_2><loc_52><loc_43><loc_92><loc_46></location>In Equation 1, h 0 is the intrinsic strain amplitude and is a function of the ellipticity:</text>
<formula><location><page_2><loc_58><loc_31><loc_92><loc_41></location>h 0 = 4 π 2 G c 4 ϵI zz f 2 d ≈ 1 . 06 × 10 -26 ( ϵ 10 -6 ) ( I zz 10 38 kg m 2 ) ( f 100 Hz ) 2 ( 1 kpc d ) , (4)</formula>
<text><location><page_2><loc_52><loc_16><loc_92><loc_30></location>where ϵ is the equatorial ellipticity of the star, I xx -I yy I zz , which is a dimensionless measure of the non-axisymmetry of the star with I xx,yy,zz being the star's principal moments of inertia (spin axis along z ), d is the distance from the source to the detector, and f is the gravitational wave frequency [11, 37]. If we assume that the loss of rotational energy of the source is solely through gravitational wave emission, we can define the spindown limit on the maximum detectable strain, h sd [39]:</text>
<formula><location><page_3><loc_10><loc_83><loc_49><loc_91></location>h sd =(2 . 5 × 10 -25 ) ( 1 kpc d ) · √ √ √ √ ( 1 kHz f ) ( -˙ f 10 -10 Hz / s ) ( I zz 10 38 kg m 2 ) . (5)</formula>
<section_header_level_1><location><page_3><loc_17><loc_79><loc_41><loc_80></location>B. Parameter Space Analyzed</section_header_level_1>
<text><location><page_3><loc_9><loc_62><loc_49><loc_77></location>In this work, we search more deeply than in the 2021 O3a all sky LVK search [12], as discussed in Section III D, but limit the search band to lower frequencies (30-150 Hz) in order partly to make the total computational cost manageable and partly to focus on young neutron stars with lower spin frequencies (15-75 Hz). Excluding the 20-30 Hz band searched in the O3a analysis was a pragmatic choice, given the scarcity of clean spectral subbands in that range and a highly elevated noise floor in those subbands relative to that at higher frequencies.</text>
<text><location><page_3><loc_9><loc_36><loc_49><loc_62></location>We maintain the same spindown range of [ -1 × 10 -8 , +1 × 10 +9 ] Hz/s as in the O3a analysis [12]. Most isolated sources of continuous gravitational waves are expected to have a negative frequency derivative as they lose energy through the emission of gravitational waves [11, 12]. A small number of isolated pulsars in globular clusters are known to have slight apparent spin-ups however. These are believed to arise from their acceleration in the Earth's direction [11, 12]. Other reasons could include a strong slowly-varying Doppler shift from a long-period orbit with a binary companion [11, 12]. Exotic sources with spin-ups could include superradiance from a boson cloud in the vicinity of an isolated black hole [40]. Retaining a large search range in negative spin derivative allows probing sources at larger distances d as constrained by consistency among detectable h sd and d in Equation 5 and implied rotational energy loss.</text>
<section_header_level_1><location><page_3><loc_22><loc_32><loc_36><loc_33></location>C. Data Cleaning</section_header_level_1>
<text><location><page_3><loc_9><loc_13><loc_49><loc_30></location>The low-frequency spectrum, particularly that of Hanford H1 data, suffered from a large number of spectral artifacts. This led to an overwhelming number of outliers during the O3a analysis which made outlier followup difficult and computationally costly. Having performed an 'eyes wide open' search in the O3a data, we adopt a more pragmatic approach here, excising heavily polluted subbands, replacing the SFT coefficients by Gaussian random noise with target strain power based on neighboring frequency bands [41]. This approach avoids inundation by outliers in bands for which true signal detection likelihood is a priori low.</text>
<text><location><page_3><loc_9><loc_9><loc_49><loc_13></location>We perform line-cleaning in two steps. First, we clean out lines that have a known instrumental source using the list presented in [31]. One approach to determining</text>
<text><location><page_3><loc_52><loc_86><loc_92><loc_93></location>that the source of a line is unlikely to be astrophysical in origin involves looking for excess correlation between the strain channel and an environmental channel that is expected to be decoupled from the strain channel [42]. More details can be found in [42, 43].</text>
<text><location><page_3><loc_52><loc_56><loc_92><loc_86></location>A large number of single-detector lines of unknown source also affect the bands, however, some of which we clean in the second step. We remove only those lines clearly inconsistent with an astrophysical source, while accounting for the differing H1 and L1 noise floors. Specifically for a strain power spectral noise line peak in detector A of median-filter applied maximum value PSD A max and a visibly smooth (non-peaked) corresponding spectral range in detector B, we deem the peak in A to be non-astrophysical if median-filter applied PSD A max is greater than 2 . 5 times the counterpart noise level PSD B (also median-filter applied). We use this criterion to mark regions to clean, which we then refine slightly manually. Injection studies confirmed the safety of our vetoes, given that sky locations that may lie near an antenna pattern node for detector B at one moment, while retaining sensitivy in detector A will not sustain this imbalance when averaged over sidereal time. Figure 1 shows the bands we cleaned. Our data cleaning led to 6.3% of the highly contaminated H1 band and 1.2% of the L1 band being cleaned.</text>
<section_header_level_1><location><page_3><loc_65><loc_51><loc_79><loc_52></location>D. Methodology</section_header_level_1>
<text><location><page_3><loc_52><loc_12><loc_92><loc_49></location>We follow a hierarchical search strategy using the PowerFlux pipeline [17-24] with loose coherence [21, 25-27]. While the first stage (called Stage 0) establishes upper limits and produces first round of outliers, subsequent stages follow up on those outliers with increasing levels of constraints on signal coherence and parameter reconstruction. We steadily increase the effective coherence times by halving the phase mismatch parameter, δ between each successive stages, thereby increasing the sensitivity at each new stage [21, 25]. Additionally, we reduce the step-size for frequency, spindown, and sky location after each stage. As each successive stage improves sensitivity, we also require the SNR to increase while going from one stage to the next one. The exact factor of increase we require depends on the band and was empirically determined via software signal injections into the O3 data. Table I shows the values of phase-mismatch parameter δ , step-sizes, and SNR increase factors we require for each of the three stages. Outliers that survive all stages are then manually investigated through the use of strain histograms to look for contamination from known instrumental artifacts or single-interferometer lines with unknown origins. Outliers that survive all of these requirements are then followed up with the PyFstat pipeline [28], as in the O3a analysis [12].</text>
<text><location><page_3><loc_52><loc_9><loc_92><loc_11></location>PowerFlux iterates through all SFTs and computes the power for each template through a bilinear form of the</text>
<figure>
<location><page_4><loc_14><loc_21><loc_87><loc_90></location>
<caption>FIG. 1. Cleaned sub-bands for H1 (top) and L1 (bottom). The purple lines show the respective amplitude spectral density (ASDs), while the shaded vertical bars show regions where the data was replaced with randomly generated Gaussian noise. In most cases, the thickness of the vertical bar shown is larger than the width of the corresponding cleaned subband. The fractions of bandwidth cleaned are 6.3% and 1.2% for H1 and L1 data, respectively.</caption>
</figure>
<text><location><page_4><loc_49><loc_21><loc_59><loc_22></location>Frequency [Hz]</text>
<table>
<location><page_5><loc_10><loc_65><loc_91><loc_92></location>
<caption>TABLE I. Outlier followup parameters. All stages use loose coherence for demodulation and sum Hanford and Livingston data coherently.</caption>
</table>
<figure>
<location><page_5><loc_12><loc_38><loc_49><loc_55></location>
</figure>
<figure>
<location><page_5><loc_52><loc_38><loc_88><loc_55></location>
</figure>
<figure>
<location><page_5><loc_12><loc_19><loc_49><loc_36></location>
</figure>
<figure>
<location><page_5><loc_52><loc_19><loc_88><loc_36></location>
<caption>FIG. 2. Upper limits validation for the four major 30 Hz sub-bands - 30-60 Hz, 60-90 Hz, 90-120 Hz, and 120-150 Hz from top-left to bottom-right. Each point represents a unique injection with randomly generated parameter including polarization for which the produced upper limit is plotted against the injected strain. The red diagonal lines define the equality of upper limit with the injected strain.</caption>
</figure>
<figure>
<location><page_6><loc_12><loc_76><loc_49><loc_93></location>
</figure>
<figure>
<location><page_6><loc_52><loc_76><loc_88><loc_93></location>
</figure>
<figure>
<location><page_6><loc_12><loc_57><loc_49><loc_74></location>
</figure>
<figure>
<location><page_6><loc_52><loc_57><loc_88><loc_74></location>
<caption>FIG. 3. Injection (software simulations) recovery efficiencies in the 30-60 Hz, 60-90 Hz, 90-120 Hz, and 120-150 Hz frequency bands [top-left to bottom-right]. The horizontal axis shows the relative upper limit - the ratio of the injected strain to the 95% CL upper limit in the corresponding band without any injection. The vertical axis shows the fraction of surviving injections. The horizontal dashed lines correspond to a 95% recovery fraction while the vertical dashed lines represent a relative strain of 1 where the injected strain is equal to the 95% upper limit.</caption>
</figure>
<text><location><page_6><loc_9><loc_41><loc_49><loc_44></location>(SFT) input matrix { a t,f } , indexed by the SFT time t and template frequency f :</text>
<formula><location><page_6><loc_10><loc_37><loc_49><loc_40></location>P [ f ] = ∑ t 1 ,t 2 ,D i ,D j a ( D i ) t 1 ,f +∆ f ( t 1 ) a ( D j ) ∗ t 2 ,f +∆ f ( t 2 ) K D i D j t 1 ,t 2 ,f , (6)</formula>
<text><location><page_6><loc_9><loc_23><loc_49><loc_35></location>where ∆ f is the detector frequency shift arising from Doppler shift due to the Earth's motion and the neutron star's spindown [23, 24]. The factor K D i D j t 1 ,t 2 ,f includes a Lanczos kernel along with contributions from timedependent SFT weights, antenna patterns, signal polarization, and relative terms for the detectors D i,j [21, 2325]. The phase-mismatch parameter δ in the Lanczos kernel portion of K D i D j t 1 ,t 2 ,f :</text>
<formula><location><page_6><loc_11><loc_18><loc_49><loc_21></location>˜ K δ ( t 1 , t 2 ) = sinc [ δ ( t 1 -t 2 ) T coh ] sinc [ δ ( t 1 -t 2 ) 3 T coh ] , (7)</formula>
<text><location><page_6><loc_9><loc_9><loc_49><loc_17></location>where t 1 , t 2 are the mid-times of the two SFTs in the barycenter frame, and T coh = 7200 s , governs the degree of signal phase drift we tolerate [21, 25]. Loose coherence helps discard templates with non-physical phase evolution and hence, distinguish between signals that are possibly astrophysical in origin from instrumental arti-</text>
<text><location><page_6><loc_52><loc_37><loc_92><loc_44></location>facts [25]. Steadily decreasing the value of δ used reduces the amount of phase drift permitted in each successive stage, hence imposing tighter constraints on the templates that merit followup with another stage of still tighter constraints [21, 25].</text>
<text><location><page_6><loc_52><loc_22><loc_92><loc_35></location>The choice of the Lanczos kernel (or a sinc filter) for loose coherence essentially makes it equivalent to performing a coherent search with a low-pass filter that attenuates signals with rapidly varying phases [44, 45]. The cutoff frequency is determined by the tunable phase mismatch parameter δ , and hence lowering its value in follow-up stages of the search demand progressively higher coherence between SFTs, thereby disfavoring templates with non-physical phase evolutions [44, 46].</text>
<text><location><page_6><loc_52><loc_9><loc_92><loc_20></location>As our goal is to improve upon the earlier O3a PowerFlux search [12], we use the full O3 data set and exploit loose coherence in the first stage while combining SFT coefficients coherently from the two interferometers. The improved strain sensitivity comes at an increased computational cost, however, for a fixed frequency range. Restricting the search frequency band to 30-150 Hz makes the cost of this search acceptable.</text>
<section_header_level_1><location><page_7><loc_22><loc_92><loc_36><loc_93></location>E. Upper Limits</section_header_level_1>
<text><location><page_7><loc_9><loc_68><loc_49><loc_90></location>We use a universal statistic [47] to calculate upper limits from the first search stage, as in the O3a PowerFlux / search [12]. Owing to origin in an Markov inequality that is agnostic to the underlying probability distribution, our upper limits are valid even in the presence of non-Gaussian noise and give nearly optimum upper limits when the noise is indeed Gaussian [47]. The calculated upper limits are based on the overall noise level in each demodulated 1/16 Hz search band and the largest power value within that band [47]. We calculate 95% upper limits for circular polarization corresponding to the bestcase scenario, and linear polarization corresponding to worst-case scenario. Additionally, we provide estimated upper limits for a population averaged over random sky locations and polarizations.</text>
<text><location><page_7><loc_9><loc_52><loc_49><loc_68></location>The upper limits are naturally not valid for the cleaned regions shown in Figure 1. We use software injections to validate our upper limits in the retained subbands, as illustrated in Figure 2. We exclude vetoed frequency bins from all software injections. The plots show calculated linear-polarization upper limits as a function of the injected strains. Since our upper limits are 95% frequentist, we expect the points to be above the diagonal > 95% of the times. The flat regions on the left correspond to low strain values where we do not have enough sensitivity to produce useful upper limits.</text>
<section_header_level_1><location><page_7><loc_21><loc_48><loc_37><loc_49></location>F. Outlier Followup</section_header_level_1>
<text><location><page_7><loc_9><loc_30><loc_49><loc_46></location>Outliers produced from the first stage are analyzed by subsequent stages of loose coherence, with increasing constraints on the effective coherence time. Steadily decreasing values of the loose coherence phase-mismatch δ demands increasing temporal coherence with each iteration and hence, help distinguish between signals that could potentially be astrophysical in source from instrumental artifacts [21, 25]. Additionally, we step more finely in frequency, spindown, and sky points with each successive stage, which improves parameter reconstruction resolution.</text>
<text><location><page_7><loc_9><loc_9><loc_49><loc_30></location>Outliers obtained from stage 0 with SNR > 7 are clustered according to their frequency, spindown, and sky locations. The clustering algorithm we use works as follows: We start by sorting all the outliers in descending order of their SNRs. We then take the loudest outlier in the list and select all other outliers within a certain pre-defined distance. We assign all these outliers into a single cluster. We then remove all the outliers just assigned to a cluster from the list and select the next loudest outlier. We repeat this until all outliers have been assigned to their respective clusters. The distance measure is based on the frequency, spindown, and sky location and the thresholds are determined with injections. Once the clustering is complete, we take the loudest 10 outliers within each cluster and follow up on them with</text>
<text><location><page_7><loc_52><loc_85><loc_92><loc_93></location>stage 1. After each stage > 0, we require the SNR to exceed a threshold empirically determined from software injections and require the reconstructed parameters to be consistent with those of the previous stage. Table I shows the SNR increase constraints that were required, along with the search parameters for each stage.</text>
<text><location><page_7><loc_52><loc_64><loc_92><loc_84></location>We use software injections to determine suitable search parameters as well as to validate the followup process [21]. In addition to the above mentioned criteria between stages, an injection is considered recovered only if its inferred parameters lie within 2.5 mHz in frequency, 3 × 10 -10 Hz / s in spindown, and 28 . 5 rad Hz /f in sky location from the true injected values. Figure 3 shows injection recovery efficiencies for the major 30 Hz subbands (30-60 Hz, 60-90 Hz, 90-120 Hz, 120-150 Hz). Recovery for the three PowerFlux stages are shown separately. The vertical dashed lines represent a relative strain, the ratio of injected strain to upper limit on that band, equal to 1, and the horizontal dashed lines represent 95% recovery efficiency.</text>
<text><location><page_7><loc_52><loc_56><loc_92><loc_64></location>After three stages of PowerFlux , we manually inspect the outliers to look for contamination via strain histograms [38], as was done for the O3a search [12]. Ruling out clearly contaminated signals, we then use Markov Chain Monte Carlo (MCMC) with the PyFstat [28] pipeline to follow up on the final list of outliers.</text>
<text><location><page_7><loc_52><loc_18><loc_92><loc_55></location>The PyFstat pipeline uses semi-coherent summing of the F -statistic [48] with Markov Chain Monte Carlo sampling of the posterior to give a detection statistic as well as to produce a posterior for the reconstructed parameters [49]. We follow an implementation described in [29] where it was used to follow up on outliers from the O2 run. In particular, we create a ladder of steadily decreasing numbers of segments of increasing duration with which to subdivide the full O3 run and run multiple stages where the posterior from one stage is then used to shrink the prior for the next stage [29]. We use a random sample of 600 sky locations away with the same declination as the candidate source, but separated by more than 90 degrees in right ascension, in order to calculate a background distribution for each specific frequency band [29]. We then use this background distribution along with the change in the F -statistic value in the last stage of the search to compute a Bayes factor by comparing the probability distribution of having a signal in the given data with the probability distribution of its being due to a noise background [29]. We use software injections to calibrate the appropriate Bayes factor values to set a cutoff for what value would constitute large enough for an outlier to potentially be from a true signal. We obtain a threshold of 100, so any outlier with Bayes factor less than that can be safely discarded.</text>
<section_header_level_1><location><page_7><loc_66><loc_13><loc_78><loc_14></location>IV. RESULTS</section_header_level_1>
<text><location><page_7><loc_52><loc_9><loc_92><loc_11></location>The first stage of the PowerFlux search leads to upper limits and a list of outliers to follow up. The upper limit</text>
<figure>
<location><page_8><loc_13><loc_58><loc_87><loc_92></location>
<caption>FIG. 4. Upper limits on gravitational strain amplitude for this analysis compared to other O3 analysis including the PowerFlux O3a analysis. The green curve shows worst-case (linearly polarized) 95% CL upper limits in each 1/16 Hz sub-band. The lowest orange curve shows the best-case (circularly polarized) upper limits. The middle (purple) curve shows approximate populationaveraged all-sky upper limits inferred from the circularly polarized limits. As noted in Section IV A, these pipelines cover different parameter space volumes, and hence care must be used when comparing the results. Additionally, the Einstein@Home search sets 90% confidence upper limits unlike PowerFlux and SkyHough , which use 95% intervals.</caption>
</figure>
<table>
<location><page_8><loc_14><loc_34><loc_87><loc_45></location>
<caption>TABLE II. Parameters of hardware-injected simulated isolated-source continuous wave signals during the O3 run with frequencies falling within the analyzed 30-150 Hz range (epoch GPS 1253977218), along with upper limits in the nominal signal bins and averaged over the six nearest control bins. In the previous O3a PowerFlux search, only injections Inj5 and Inj6 were detected. In a previous full-O3 search using the FrequencyHough and SkyHough pipelines, only injections Inj3, Inj5, and Inj6 were detected. The successful recovery here of Inj12 confirms the improvement in sensitivity expected with the search method presented here.</caption>
</table>
<text><location><page_8><loc_9><loc_16><loc_49><loc_21></location>on a given band is valid regardless of whether or not it produces outliers [12, 22]. Additionally, the upper limits are robust with respect to non-Gaussian noise, owing to the use of the universal statistic [47].</text>
<text><location><page_8><loc_9><loc_9><loc_49><loc_13></location>Section IV A and Section IV C, respectively discuss the upper limits and outliers we obtain. In Section IV B, we present results from the recovery of hardware injections.</text>
<section_header_level_1><location><page_8><loc_65><loc_20><loc_79><loc_21></location>A. Upper Limits</section_header_level_1>
<text><location><page_8><loc_52><loc_9><loc_92><loc_17></location>Figure 4 shows a graph of upper limits as a function of the search frequency. The green curve shows the linearpolarization or worst-case upper limits, while the orange curve shows the circular-polarization or best-case upper limits. The dashed and dotted curves show O3 results from other search pipelines, including the O3a Power-</text>
<figure>
<location><page_9><loc_13><loc_58><loc_84><loc_92></location>
<caption>FIG. 5. Ranges (kpc) of this search as recast from the calculated upper limits on gravitational strain amplitudes. The green, orange, and purple curves show ranges for full-O3 circular-polarization, full-O3 population-averaged, and O3a populationaveraged results for assumed ellipticities of ϵ = 10 -4 , 10 -6 , and 10 -8 . The four gray curves show maximum ranges possible for maximum spindown magnitudes of 10 -8 , 2.6 × 10 -9 (Einstein@Home search maximum), 10 -10 , and 10 -12 Hz/s.</caption>
</figure>
<table>
<location><page_9><loc_29><loc_31><loc_72><loc_47></location>
<caption>TABLE III. Counts of outliers surviving different stages of the search. Survivors from stage-2 followup are first manually examined for contamination from instrumental artifacts through the use of strain histograms and then followed up with PyFstat as described in Section III F.</caption>
</table>
<text><location><page_9><loc_9><loc_12><loc_49><loc_22></location>Flux search [12], full-O3 FrequencyHough and SkyHough searches [13], and full-O3 Einstein@Home distributedcomputing search [50]. It is important, however, to use care in these comparisons as the searches cover different parameter spaces. For example, the Einstein@Home search covers only about 26% of the spindown range of this search. Results from similar but less sensitive</text>
<text><location><page_9><loc_52><loc_19><loc_92><loc_22></location>searches of the O1 and O2 data sets are not included. 3 To enable a straightforward comparison to the other</text>
<text><location><page_10><loc_9><loc_86><loc_49><loc_93></location>pipelines, we also present estimated population-averaged upper limits [12]. This estimate is produced by multiplying the circular upper limits by a factor of 2.3, determined empirically for control bands using an ensemble of random sky locations and random stellar orientations [12].</text>
<text><location><page_10><loc_9><loc_77><loc_49><loc_86></location>Each upper limit displayed is maximized over the entire sky except for a small region near the ecliptic poles [21, 23]. This region is excluded as it is prone to a large number of detector artifacts owing to stationary frequency evolution from the combination of frequency derivative and Doppler modulation [21, 23].</text>
<text><location><page_10><loc_9><loc_63><loc_49><loc_77></location>Our results improve upon the earlier PowerFlux O3a results by a median factor of ∼ 1 . 4. Improvements with respect to the FrequencyHough and SkyHough full-O3 searches are frequency-dependent, with median factors of ∼ 1 . 1 for both. In contrast, the full-O3 Einstein@Home distributed-computing search [50], which covers the 20800 Hz band improves upon these results by a median factor of ∼ 1 . 2 in the common range of frequency overlap and spindown overlap (26% of the spindown range probed here).</text>
<text><location><page_10><loc_9><loc_48><loc_49><loc_62></location>The improvements compared to the O3a PowerFlux analysis can be attributed to a number of reasons. First, this analysis uses the full O3 data set which is nearly twice as long as the O3a data set [30]. Second, unlike in the O3a search, we use loose coherence in the initial stage with a δ parameter value of π/ 2, which improves sensitivity via longer effective coherence time, at the expense of higher computational cost [21, 25]. Third, we combine the interferometer SFTs coherently in the initial stage [23].</text>
<text><location><page_10><loc_9><loc_38><loc_49><loc_48></location>As a large number of contaminating line artifacts were present, particularly in H1 data, we clean the SFTs by replacing polluted bands with randomly generated Gaussian noise [41]. This method naturally means that upper limits calculated for those bands would not be valid, so those bands are excluded from the results presented here. Figure 1 shows the bands that were cleaned out.</text>
<text><location><page_10><loc_9><loc_22><loc_49><loc_38></location>The upper limits on source strain amplitudes shown in Figure 4 can be recast into lower limits on distance ranges at which source neutron stars with certain assumed ellipticities can reside [11, 37]. Figure 5 shows these implied lower limits on ranges (kpc) for assumed ellipticity values of ϵ = 10 -4 , 10 -6 , and 10 -8 . The ranges are shown for the best-case circular popularization as well as population-averaged ranges. We also show our search's implied maximum ranges as a function of the frequency for a gravitar with maximum spindown magnitudes of 10 -8 , 10 -10 , and 10 -12 Hz/s.</text>
<section_header_level_1><location><page_10><loc_19><loc_18><loc_38><loc_19></location>B. Hardware Injections</section_header_level_1>
<text><location><page_10><loc_9><loc_9><loc_49><loc_16></location>A total of 18 hardware injections- 16 isolated and two binary sources- were added to the data in the O3 run [54]. Out of the 16 isolated-star injections, five injections fall within the frequency range we analyze [54]. We successfully recover four out of the five hardware injections. Ta-</text>
<table>
<location><page_10><loc_56><loc_36><loc_87><loc_93></location>
<caption>TABLE IV. Parameters of the outliers surviving PyFstat followup with Bayes Factor ≥ 15. Each outlier listed is the loudest after clustering in frequency, spin-down, and sky location. Outliers marked with an * are from hardware injections. A list of all hardware injections within the paramater space of this search can be found in Table II.</caption>
</table>
<text><location><page_10><loc_52><loc_9><loc_92><loc_23></location>ble II shows the list of relevant hardware injections, their parameters including injected strains, corresponding upper limits produced and the final Bayes factors we obtain. The control upper limits show 95% upper limits for the nearest six neighboring sub-bands to give a rough estimate of expected upper limit that would have been obtained in the absence of a signal. For reference, only two of the five injections were successfully recovered in the O3a PowerFlux search [12], and only three of the five were recovered in a previous full-O3 search [13] using</text>
<text><location><page_11><loc_9><loc_90><loc_49><loc_93></location>other pipelines, consistent with the improved sensitivity of the method presented here.</text>
<section_header_level_1><location><page_11><loc_24><loc_86><loc_34><loc_87></location>C. Outliers</section_header_level_1>
<text><location><page_11><loc_9><loc_70><loc_49><loc_84></location>In addition to producing upper limits, stage 0 produces an initial list of outliers, corresponding to templates with excess power. We follow up on these outliers via subsequent stages, in which we impose progressively tighter constraints. In the absence of a true signal or of a hardware injection, we would ideally expect the number of outliers to decrease in subsequent stages because of these tighter constraints. In practice, particularly severe detector artifacts can also persist. Table III shows the number of outliers we obtain for various stages of the search.</text>
<text><location><page_11><loc_9><loc_47><loc_49><loc_69></location>For the outliers that survive stage 2, we use strain histograms in which we plot the expected spectral shape overlaid upon the actual amplitude spectral density (ASD) to check if a particular band is clearly affected by an artifact. After the PowerFlux stages, we use Markov Chain Monte Carlo (MCMC) followup with the use of the PyFstat pipeline to obtain a Bayes factor for each of the surviving outliers. We use injections to determine appropriate Bayes factors as described in Section III F. Table IV shows the Bayes factors ≥ 15 for the final list of clusters of outliers. As can be seen, all outliers with appreciable Bayes factors have origins in hardware injections. The largest Bayes factor for an outlier not coming from a hardware injection and not having a visible artifact is 27, well below the threshold of 100 used to identify a plausible signal.</text>
<section_header_level_1><location><page_11><loc_21><loc_42><loc_37><loc_43></location>V. CONCLUSIONS</section_header_level_1>
<text><location><page_11><loc_9><loc_28><loc_49><loc_40></location>We have performed a deeper search for continuous gravitational waves from isolated neutrons stars, building upon a previous analysis [12]. Limiting the frequency band of the search allowed us to go deeper through the use of the full O3 data set, loose coherence, and coherent summing of SFTs across detectors in the initial stage. Additionally, line cleaning the SFTs before performing the analysis made the number of outliers to follow up on much more manageable.</text>
<text><location><page_11><loc_9><loc_22><loc_49><loc_27></location>While we failed to detect a credible signal as all surviving outliers have origins in either hardware injections or instrumental artifacts, we do set upper limits on the strain amplitudes as shown in Figure 4.</text>
<text><location><page_11><loc_9><loc_9><loc_49><loc_21></location>The best upper limit obtained for circular polarization is ∼ 4 . 5 × 10 -26 near 144 Hz, and for linearly polarization is ∼ 1 . 3 × 10 -25 . The best estimated population-averaged upper limit is ∼ 1 . 0 × 10 -25 , These limits improve on our O3a search by a median factor of ∼ 1 . 4. Improvements with respect to the FrequencyHough and SkyHough fullO3 searches [13] are frequency-dependent, with median factors of ∼ 1 . 1 for both. Our results are less constraining by a median factor of ∼ 1 . 2 than the full-O3 Ein-</text>
<text><location><page_11><loc_52><loc_88><loc_92><loc_93></location>ch, in the overlapping frequency band and spindown range, but cover a spindown range four times larger and hence probe to greater distances in the galaxy in the band searched.</text>
<text><location><page_11><loc_52><loc_79><loc_92><loc_87></location>The fourth LIGO-Virgo-KAGRA observation run O4 started May 24, 2023 with a planned duration of ∼ 20 months. The improved detector noise in the O4 data and longer observation period offer the prospect of improved strain sensitivity and, ideally, a continuous wave signal detection at last.</text>
<section_header_level_1><location><page_11><loc_59><loc_74><loc_84><loc_75></location>VI. ACKNOWLEDGEMENTS</section_header_level_1>
<text><location><page_11><loc_52><loc_55><loc_92><loc_72></location>We gratefully acknowledge useful discussions and long collaboration with current and former colleagues in the LIGO-Virgo-KAGRA continuous waves working group. For these results, in particular, we thank Evan Goetz, Ansel Neunzert, and Grant Weldon for spectral line investigations of the O3 data, Rodrigo Tenorio and David Keitel for creating the MCMC PyFstat pipeline, and Vladimir Dergachev for creating the PowerFlux pipeline. We also thank the two anonymous journal referees for constructive comments and suggestions. This work was supported in part by National Science Foundation Award No. PHY-2110181.</text>
<text><location><page_11><loc_52><loc_17><loc_92><loc_55></location>This research has made use of data or software obtained from the Gravitational Wave Open Science Center (gwosc.org), a service of the LIGO Scientific Collaboration, the Virgo Collaboration, and KAGRA. This material is based upon work supported by NSF's LIGO Laboratory which is a major facility fully funded by the National Science Foundation, as well as the Science and Technology Facilities Council (STFC) of the United Kingdom, the Max-Planck-Society (MPS), and the State of Niedersachsen/Germany for support of the construction of Advanced LIGO and construction and operation of the GEO600 detector. Additional support for Advanced LIGO was provided by the Australian Research Council. Virgo is funded, through the European Gravitational Observatory (EGO), by the French Centre National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale di Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions by institutions from Belgium, Germany, Greece, Hungary, Ireland, Japan, Monaco, Poland, Portugal, and Spain. KAGRA is supported by Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan Society for the Promotion of Science (JSPS) in Japan; National Research Foundation (NRF) and Ministry of Science and ICT (MSIT) in Korea; Academia Sinica (AS) and National Science and Technology Council (NSTC) in Taiwan.</text>
<text><location><page_11><loc_52><loc_12><loc_92><loc_17></location>The authors are grateful for computational resources provided by LIGO Laboratory and supported by National Science Foundation Grants PHY-0757058 and PHY-0823459.</text>
<text><location><page_11><loc_52><loc_9><loc_92><loc_11></location>This research was done in part using services provided by the OSG Consortium [55-58], which is supported by</text>
<text><location><page_12><loc_9><loc_90><loc_49><loc_93></location>the National Science Foundation Award Nos. 2030508 and 1836650.</text>
<text><location><page_12><loc_10><loc_89><loc_49><loc_90></location>This research was enabled in part by support pro-</text>
<unordered_list>
<list_item><location><page_12><loc_10><loc_81><loc_49><loc_83></location>[1] J. Aasi et al. (LIGO Scientific Collaboration), Classical Quantum Gravity 32 , 074001 (2015).</list_item>
<list_item><location><page_12><loc_10><loc_78><loc_49><loc_81></location>[2] F. Acernese et al. , Classical Quantum Gravity 32 , 024001 (2015).</list_item>
<list_item><location><page_12><loc_10><loc_77><loc_44><loc_78></location>[3] D. Castelvecchi, Nature (London) 565 , 9 (2019).</list_item>
<list_item><location><page_12><loc_10><loc_76><loc_34><loc_77></location>[4] H. Abe et al. , Galaxies 10 (2022).</list_item>
<list_item><location><page_12><loc_10><loc_73><loc_49><loc_75></location>[5] T. Akutsu et al. , Prog. Theor. Exp. Phys. 2021 , 05A101 (2020).</list_item>
<list_item><location><page_12><loc_10><loc_70><loc_49><loc_73></location>[6] B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), Phys. Rev. Lett. 116 , 061102 (2016).</list_item>
<list_item><location><page_12><loc_10><loc_68><loc_49><loc_70></location>[7] B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), Phys. Rev. X 9 , 031040 (2019).</list_item>
<list_item><location><page_12><loc_10><loc_65><loc_49><loc_67></location>[8] R. Abbott et al. (LIGO Scientific and Virgo Collaborations), Phys. Rev. X 11 , 021053 (2021).</list_item>
<list_item><location><page_12><loc_10><loc_60><loc_49><loc_65></location>[9] R. Abbott et al. (LIGO Scientific, Virgo, and KAGRA Collaborations), 'GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo During the Second Part of the Third Observing Run,' (2023).</list_item>
<list_item><location><page_12><loc_9><loc_57><loc_49><loc_60></location>[10] R. Abbott et al. (LIGO Scientific, Virgo, and KAGRA Collaborations), Astrophys. J. Lett. 915 , L5 (2021).</list_item>
<list_item><location><page_12><loc_9><loc_56><loc_46><loc_57></location>[11] K. Riles, Living Reviews in Relativity 26 , 3 (2023).</list_item>
<list_item><location><page_12><loc_9><loc_52><loc_49><loc_56></location>[12] R. Abbott et al. (LIGO Scientific, Virgo, and KAGRA Collaborations), Phys. Rev. D 104 (2021), 10.1103/physrevd.104.082004.</list_item>
<list_item><location><page_12><loc_9><loc_49><loc_49><loc_52></location>[13] R. Abbott et al. (LIGO Scientific, Virgo, and KAGRA Collaborations), Phys. Rev. D 106 , 102008 (2022).</list_item>
<list_item><location><page_12><loc_9><loc_47><loc_49><loc_49></location>[14] P. D. Lasky, Publ. Astron. Sco. Aust. 32 (2015), 10.1017/pasa.2015.35.</list_item>
<list_item><location><page_12><loc_9><loc_43><loc_49><loc_46></location>[15] K. Glampedakis and L. Gualtieri, in Astrophys. Space Sci. Libr. , Vol. 457, edited by L. Rezzolla, P. Pizzochero, D. I. Jones, N. Rea, and I. Vida˜na (2018) p. 673.</list_item>
<list_item><location><page_12><loc_9><loc_41><loc_47><loc_42></location>[16] M. Sieniawska and M. Bejger, Universe 5 , 217 (2019).</list_item>
<list_item><location><page_12><loc_9><loc_39><loc_49><loc_41></location>[17] V. Dergachev, 'Description of PowerFlux algorithms and implementation,' (2005), LIGO Report LIGO-T050186.</list_item>
<list_item><location><page_12><loc_9><loc_36><loc_49><loc_38></location>[18] V. Dergachev, 'Description of PowerFlux 2 algorithms and implementation,' LIGO Report T1000272 (2010).</list_item>
<list_item><location><page_12><loc_9><loc_33><loc_49><loc_36></location>[19] V. Dergachev and K. Riles, 'PowerFlux Polarization Analysis,' (2005), LIGO Report LIGO-T050187.</list_item>
<list_item><location><page_12><loc_9><loc_31><loc_49><loc_33></location>[20] B. P. Abbott et al. (LIGO Scientific), Phys. Rev. Lett. 102 , 111102 (2009).</list_item>
<list_item><location><page_12><loc_9><loc_28><loc_49><loc_30></location>[21] J. Abadie et al. (The LIGO Scientific Collaboration, The Virgo Collaboration), Phys. Rev. D 85 , 022001 (2012).</list_item>
<list_item><location><page_12><loc_9><loc_26><loc_49><loc_28></location>[22] B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), Phys. Rev. D 94 , 042002 (2016).</list_item>
<list_item><location><page_12><loc_9><loc_23><loc_49><loc_25></location>[23] B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. D 96 , 062002 (2017).</list_item>
<list_item><location><page_12><loc_9><loc_20><loc_49><loc_23></location>[24] B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), Phys. Rev. D 97 , 102003 (2018).</list_item>
<list_item><location><page_12><loc_9><loc_18><loc_49><loc_20></location>[25] V. Dergachev, Classical Quantum Gravity 27 , 205017 (2010).</list_item>
<list_item><location><page_12><loc_9><loc_15><loc_49><loc_17></location>[26] V. Dergachev, 'Loosely coherent searches for medium scale coherence lengths,' (2018).</list_item>
<list_item><location><page_12><loc_9><loc_14><loc_43><loc_15></location>[27] V. Dergachev, Phys. Rev. D 85 , 062003 (2012).</list_item>
<list_item><location><page_12><loc_9><loc_11><loc_49><loc_13></location>[28] D. Keitel, R. Tenorio, G. Ashton, and R. Prix, J. Open Source Software Source Software 6 , 3000 (2021).</list_item>
<list_item><location><page_12><loc_9><loc_10><loc_49><loc_11></location>[29] R. Tenorio, D. Keitel, and A. M. Sintes, Phys. Rev. D</list_item>
</unordered_list>
<text><location><page_12><loc_52><loc_89><loc_92><loc_93></location>vided by the Simon Fraser University Cedar Cluster (sfu.ca/research/supercomputer-cedar) and the Digital Research Alliance of Canada (alliancecan.ca).</text>
<unordered_list>
<list_item><location><page_12><loc_55><loc_82><loc_68><loc_83></location>104 , 084012 (2021).</list_item>
<list_item><location><page_12><loc_52><loc_78><loc_92><loc_82></location>[30] R. Abbott et al. (LIGO Scientific Collaboration, VIRGO Collaboration and KAGRA Collaboration), Astroph. J. Supp. 267 , 29 (2023).</list_item>
<list_item><location><page_12><loc_52><loc_76><loc_92><loc_78></location>[31] E. Goetz et al. , 'O3 lines and combs in found in self-gated C01 data,' LIGO Report T2100200 (2021).</list_item>
<list_item><location><page_12><loc_52><loc_72><loc_92><loc_75></location>[32] J. Zweizig and K. Riles, 'Information on self-gating of h ( t ) used in O3a continuous-wave searches,' LIGO Report T2000384 (2021).</list_item>
<list_item><location><page_12><loc_52><loc_69><loc_92><loc_71></location>[33] B. Steltner, M. A. Papa, and H.-B. Eggenstein, Phys. Rev. D 105 , 022005 (2022).</list_item>
<list_item><location><page_12><loc_52><loc_66><loc_92><loc_69></location>[34] D. Davis et al. , Classical Quantum Gravity 38 , 135014 (2021).</list_item>
<list_item><location><page_12><loc_52><loc_64><loc_92><loc_66></location>[35] L. Sun et al. , Classical Quantum Gravity 37 , 225008 (2020).</list_item>
<list_item><location><page_12><loc_52><loc_62><loc_91><loc_64></location>[36] L. Sun et al. , arXiv e-prints , arXiv:2107.00129 (2021).</list_item>
<list_item><location><page_12><loc_52><loc_59><loc_92><loc_62></location>[37] B. P. Abbott et al. (LIGO Scientific Collaboration), Phys. Rev. D 77 , 022001 (2008), [Erratum: Phys.Rev.D 80, 129904 (2009)].</list_item>
<list_item><location><page_12><loc_52><loc_55><loc_92><loc_58></location>[38] G. Weldon, K. Riles, and LIGO Team, in APS April Meeting Abstracts , APS Meeting Abstracts, Vol. 2019 (2019) p. Y10.007.</list_item>
<list_item><location><page_12><loc_52><loc_51><loc_92><loc_54></location>[39] R. Abbott and T. D. Abbott (LIGO Scientific, Virgo, and KAGRA Collaborations), Phys. Rev. D 76 , 082001 (2007), gr-qc/0605028.</list_item>
<list_item><location><page_12><loc_52><loc_48><loc_92><loc_50></location>[40] A. Arvanitaki and S. Dubovsky, Phys. Rev. D 83 , 044026 (2011).</list_item>
<list_item><location><page_12><loc_52><loc_44><loc_92><loc_48></location>[41] LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration, 'LVK Algorithm Library - LALSuite,' Free software (GPL) (2018).</list_item>
<list_item><location><page_12><loc_52><loc_41><loc_92><loc_44></location>[42] D. Davis et al. (LIGO), Classical Quantum Gravity 38 , 135014 (2021).</list_item>
<list_item><location><page_12><loc_52><loc_36><loc_92><loc_41></location>[43] P. Covas, A. Effler, E. Goetz, P. Meyers, A. Neunzert, M. Oliver, B. Pearlstone, V. Roma, R. Schofield, V. Adya, and et al., Phys. Rev. D 97 (2018), 10.1103/physrevd.97.082002.</list_item>
<list_item><location><page_12><loc_52><loc_33><loc_92><loc_36></location>[44] V. Dergachev, Classical Quantum Gravity 27 , 205017 (2010).</list_item>
<list_item><location><page_12><loc_52><loc_29><loc_92><loc_33></location>[45] A. Tripathee, An All-Sky Search for Continuous Gravitational Waves , Ph.D. thesis, University of Michigan (2022).</list_item>
<list_item><location><page_12><loc_52><loc_28><loc_86><loc_29></location>[46] V. Dergachev, Phys. Rev. D 85 , 062003 (2012).</list_item>
<list_item><location><page_12><loc_52><loc_27><loc_86><loc_28></location>[47] V. Dergachev, Phys. Rev. D 87 , 062001 (2013).</list_item>
<list_item><location><page_12><loc_52><loc_24><loc_92><loc_27></location>[48] P. Jaranowski, A. Kr'olak, and B. F. Schutz, Phys. Rev. D 58 , 063001 (1998).</list_item>
<list_item><location><page_12><loc_52><loc_23><loc_92><loc_24></location>[49] G. Ashton and R. Prix, Phys. Rev. D 97 , 103020 (2018).</list_item>
<list_item><location><page_12><loc_52><loc_19><loc_92><loc_23></location>[50] B. Steltner, M. A. Papa, H.-B. Eggenstein, R. Prix, M. Bensch, B. Allen, and B. Machenschalk, Astrophys. J. 952 , 55 (2023).</list_item>
<list_item><location><page_12><loc_52><loc_16><loc_92><loc_19></location>[51] V. Dergachev and M. A. Papa, Phys. Rev. Lett. 123 , 101101 (2019).</list_item>
<list_item><location><page_12><loc_52><loc_14><loc_92><loc_16></location>[52] V. Dergachev and M. A. Papa, Phys. Rev. D 104 , 043003 (2021).</list_item>
<list_item><location><page_12><loc_52><loc_11><loc_92><loc_13></location>[53] V. Dergachev and M. A. Papa, Phys. Rev. X 13 , 021020 (2023).</list_item>
<list_item><location><page_12><loc_52><loc_10><loc_92><loc_11></location>[54] C. Biwer, D. Barker, J. C. Batch, J. Betzwieser, R. P.</list_item>
</unordered_list>
<text><location><page_13><loc_12><loc_89><loc_49><loc_93></location>Fisher, E. Goetz, S. Kandhasamy, S. Karki, J. S. Kissel, A. P. Lundgren, and et al., Phys. Rev. D 95 , 062002 (2017).</text>
<unordered_list>
<list_item><location><page_13><loc_9><loc_84><loc_49><loc_89></location>[55] R. Pordes, D. Petravick, B. Kramer, D. Olson, M. Livny, A. Roy, P. Avery, K. Blackburn, T. Wenaus, F. Wurthwein, I. Foster, R. Gardner, M. Wilde, A. Blatecky, J. McGee, and R. Quick, in J. Phys. Conf. Ser. 78,</list_item>
</unordered_list>
<text><location><page_13><loc_55><loc_92><loc_66><loc_93></location>012057 ((2007)).</text>
<unordered_list>
<list_item><location><page_13><loc_52><loc_87><loc_92><loc_92></location>[56] I. Sfiligoi, D. C. Bradley, B. Holzman, P. Mhashilkar, S. Padhi, and F. Wurthwein, in 2009 WRI World Congress on Computer Science and Information Engineering , 2, Vol. 2 ((2009)) pp. 428-432.</list_item>
<list_item><location><page_13><loc_52><loc_85><loc_71><loc_86></location>[57] OSG, 'Ospool,' (2006).</list_item>
<list_item><location><page_13><loc_52><loc_84><loc_86><loc_85></location>[58] OSG, 'Open science data federation,' (2015).</list_item>
</document>
[
{
"title": "Probing More Deeply in an All-Sky Search for Continuous Gravitational Waves in the LIGO O3 Data Set",
"content": "Aashish Tripathee 1 and Keith Riles 1 1 University of Michigan Physics Department, Ann Arbor, MI 48109, USA (compiled February 27, 2024) We report results from an all-sky search of the LIGO data from the third LIGO-Virgo-KAGRA run (O3) for continuous gravitational waves from isolated neutron stars in the frequency band [30, 150] Hz and spindown range of [ -1 × 10 -8 , +1 × 10 -9 ] Hz/s. This search builds upon a previous analysis of the first half of the O3 data using the same PowerFlux pipeline. We search more deeply here by using the full O3 data and by using loose coherence in the initial stage with fully coherent combination of LIGO Hanford (H1) and LIGO Livingston (L1) data, while limiting the frequency band searched and excluding narrow, highly disturbed spectral bands. We detect no signal and set strict frequentist upper limits on circularly polarized and on linearly polarized wave amplitudes, in addition to estimating population-averaged upper limits. The lowest upper limit obtained for circular polarization is ∼ 4 . 5 × 10 -26 , and the lowest linear polarization limit is ∼ 1 . 3 × 10 -25 (both near 144 Hz). The lowest estimated population-averaged upper limit is ∼ 1 . 0 × 10 -25 . In the frequency band and spindown range searched here, these limits improve upon the O3a PowerFlux search by a median factor of ∼ 1 . 4 and upon the best previous limits obtained for the full O3 data by a median factor of ∼ 1 . 1.",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "LIGO [1], VIRGO [2], and KAGRA [3-5] are interferometer-based experiments that search for gravitational waves. Nearly 100 transient gravitational waves have been detected from black hole (BH) mergers, neutron stars (NS) mergers, and BH-NS mergers since the first detection in 2015 [6-10]. An as yet undetected type of gravitational waves is continuous, long-lived, and nearly monochromatic. Such radiation is necessarily weaker than that detected already from transient mergers and requires long-period integration for detection in the presence of interferometer noise [11]. In this work we focus on one of the potential sources for continuous gravitational waves - rapidly spinning isolated neutron stars with non-axisymmetry. While previous all-sky searches in the O3 data [12, 13], including the O3a PowerFlux analysis, analyzed a broad band of approximately 20-2000 Hz, this work focuses on young rapidly spinning neutron stars in the Milky Way with gravitational wave frequencies of 30-150 Hz. The nonaxisymmetry in such stars could stem from different sources including crustal deformation, buried magnetic field energy, and excitation of r-modes [14-16]. This search uses the venerable PowerFlux pipeline [1724] with loose coherence [21, 25-27] to perform an all-sky search in the spindown range [ -1 × 10 -8 , +1 × 10 -9 ] Hz/s. Our search resulted in ≈ 3 . 4 × 10 6 outliers 1 in the first stage after excluding narrow, highly disturbed bands, a number that dropped to ≈ 1 . 2 × 10 4 by the final stage of the hierarchical search. These outliers were then followed up on with the Markov Chain Monte Carlo (MCMC) based PyFstat framework [28, 29]. All but 10 of the surviving outliers can be attributed to hardware injections (simulated CW signals imposed upon the interferometer mirror motion) or to known spectral artifacts. Those 10 outliers, however, have Bayes factors obtained from PyFstat much less than the determined threshold expected for a plausible signal. As this search failed to detect a true signal, we set upper limits on strain amplitudes as a function of frequency, excluding excised bands. This article is organized as follows: Section II describes the data set we used. Section III discusses the analysis method with PowerFlux and loose-coherence in addition to the data cleaning that was performed to suppress instrumental artifacts. Section IV shows the upper limits and outliers results obtained. Section V concludes the article with a discussion of the results and prospects for future searches including the recently started O4 search.",
"pages": [
1
]
},
{
"title": "II. DATA SETS USED",
"content": "This analysis uses data from the O3 run of Advanced LIGO. Advanced LIGO consists of two interferometers one at Hanford, Washington (H1) and the other at Livingston, Louisiana (L1), separated by a 3000 km baseline [1]. The O3 run took place between April 1, 2019 and March 27, 2020 [30]. The first part of the run (O3a) and the second part (O3b) were separated by a commissioning break in October 2019 [30]. This search builds upon the O3a PowerFlux analysis [12] but uses the ∼ 11 months of the full O3 dataset. Before the analysis was performed, known spectral artifacts ('lines') [31] were excised and replaced with random Gaussian noise ('cleaned'). Additionally, loud single-detector artifacts of an unknown source were also cleaned [31]. Details of the cleaning are discussed below in Section III C. Another artifact affecting the data were loud, relatively frequent glitches - short, high amplitude instrumental transients [32-34]. While also present in previous runs, they were much louder and more frequent in the O3 run [12, 34]. The effect of glitching was particularly troublesome for frequencies below 500 Hz. We started from 'self-gated' data in which a narrow inverse-Tukeywindow zeroing gate was applied to the time-domain data, details of which can be found in the technical document [32]. Gating applied here leads to livetime losses of about 3% and 12% of the H1 and L1 observation times. The calibration uncertainties of this C01 gated data set are estimated to be < 7% in magnitude and < 4 deg in phase for O3a and < 11% in magnitude and < 9 deg in phase for O3b (68% confidence interval) [35, 36].",
"pages": [
1,
2
]
},
{
"title": "III. ANALYSIS METHOD",
"content": "This analysis uses the PowerFlux pipeline to search for continuous gravitational waves. The strain data is divided into 7200 s segments for which discrete Fourier transforms are computed ('short' Fourier Transforms SFTs), using Hann windowing with 50% overlap. For a large number of templates that include corrections for spindown evolution and Doppler modulation, power is summed up over all the SFTs while using an inversenoise-weighting to disfavor SFTs with high noise levels [37]. For each 1/16 Hz search sub-band, the power is maximized across the sky and across spindown steps to produce 95% frequentist upper limits [23, 24, 37]. The first stage of the hierarchical search, called Stage 0, determines upper limits and yields a set of outliers defined by a signal-to-noise ratio (SNR) greater than 7. These outliers undergo two stages of loose coherence followup, applying tighter constraints and requiring increased SNR, after which persistent outliers are manually examined via strain histograms [12] to identify instrumental artifacts [38]. Surviving outliers are then followed up by the PyFstat pipeline which determines a Bayes factor of preference for signal over background for each outlier [28, 29].",
"pages": [
2
]
},
{
"title": "A. Signal Model",
"content": "Our analysis assumes a model in which a spinning neutron star with a time-varying quadrupole moment produces circularly polarized waves along the direction of the spin-axis and linearly polarized waves in the perpendicular directions [11, 23, 37] with a frequency twice that of the star's spin frequency. The polarization of the detected wave then depends on the inclination angle between the spin-axis and the line-of-sight to the Earth, ι [11]. Extreme cases are ι = 0 , π and ι = π/ 2. The former corresponds to circularly polarized waves which is the best-case scenario maximizing incident signal power. The latter corresponds to linearly polarized waves which is the worst-case scenario. Specifically, the signal model we assume is: where h 0 is the intrinsic strain amplitude, Φ( t ) is the signal phase, the F + and F × are detector responses to signals with plus and cross polarizations, respectively [11, 23, 37]. In the loose coherence stages of the search, we assume a second-order Taylor series model 2 for the phase evolution in which we include only terms up to quadratic in offsets from a reference time τ : where Φ 0 is the initial phase and f, ˙ f are the frequency and spindowns at the chosen reference time τ ref : where ⃗r ( t ) is the position vector of the detector in the Solar System Barycenter (SSB) frame, ∆ E ⊙ , and ∆ S ⊙ are the relativistic Einstein and Shapiro time delays respectively [13]. ⃗n is the vector pointing from the detector toward the source, ⃗n = (cos α cos δ, sin α cos δ, sin δ ). In Equation 1, h 0 is the intrinsic strain amplitude and is a function of the ellipticity: where ϵ is the equatorial ellipticity of the star, I xx -I yy I zz , which is a dimensionless measure of the non-axisymmetry of the star with I xx,yy,zz being the star's principal moments of inertia (spin axis along z ), d is the distance from the source to the detector, and f is the gravitational wave frequency [11, 37]. If we assume that the loss of rotational energy of the source is solely through gravitational wave emission, we can define the spindown limit on the maximum detectable strain, h sd [39]:",
"pages": [
2
]
},
{
"title": "B. Parameter Space Analyzed",
"content": "In this work, we search more deeply than in the 2021 O3a all sky LVK search [12], as discussed in Section III D, but limit the search band to lower frequencies (30-150 Hz) in order partly to make the total computational cost manageable and partly to focus on young neutron stars with lower spin frequencies (15-75 Hz). Excluding the 20-30 Hz band searched in the O3a analysis was a pragmatic choice, given the scarcity of clean spectral subbands in that range and a highly elevated noise floor in those subbands relative to that at higher frequencies. We maintain the same spindown range of [ -1 × 10 -8 , +1 × 10 +9 ] Hz/s as in the O3a analysis [12]. Most isolated sources of continuous gravitational waves are expected to have a negative frequency derivative as they lose energy through the emission of gravitational waves [11, 12]. A small number of isolated pulsars in globular clusters are known to have slight apparent spin-ups however. These are believed to arise from their acceleration in the Earth's direction [11, 12]. Other reasons could include a strong slowly-varying Doppler shift from a long-period orbit with a binary companion [11, 12]. Exotic sources with spin-ups could include superradiance from a boson cloud in the vicinity of an isolated black hole [40]. Retaining a large search range in negative spin derivative allows probing sources at larger distances d as constrained by consistency among detectable h sd and d in Equation 5 and implied rotational energy loss.",
"pages": [
3
]
},
{
"title": "C. Data Cleaning",
"content": "The low-frequency spectrum, particularly that of Hanford H1 data, suffered from a large number of spectral artifacts. This led to an overwhelming number of outliers during the O3a analysis which made outlier followup difficult and computationally costly. Having performed an 'eyes wide open' search in the O3a data, we adopt a more pragmatic approach here, excising heavily polluted subbands, replacing the SFT coefficients by Gaussian random noise with target strain power based on neighboring frequency bands [41]. This approach avoids inundation by outliers in bands for which true signal detection likelihood is a priori low. We perform line-cleaning in two steps. First, we clean out lines that have a known instrumental source using the list presented in [31]. One approach to determining that the source of a line is unlikely to be astrophysical in origin involves looking for excess correlation between the strain channel and an environmental channel that is expected to be decoupled from the strain channel [42]. More details can be found in [42, 43]. A large number of single-detector lines of unknown source also affect the bands, however, some of which we clean in the second step. We remove only those lines clearly inconsistent with an astrophysical source, while accounting for the differing H1 and L1 noise floors. Specifically for a strain power spectral noise line peak in detector A of median-filter applied maximum value PSD A max and a visibly smooth (non-peaked) corresponding spectral range in detector B, we deem the peak in A to be non-astrophysical if median-filter applied PSD A max is greater than 2 . 5 times the counterpart noise level PSD B (also median-filter applied). We use this criterion to mark regions to clean, which we then refine slightly manually. Injection studies confirmed the safety of our vetoes, given that sky locations that may lie near an antenna pattern node for detector B at one moment, while retaining sensitivy in detector A will not sustain this imbalance when averaged over sidereal time. Figure 1 shows the bands we cleaned. Our data cleaning led to 6.3% of the highly contaminated H1 band and 1.2% of the L1 band being cleaned.",
"pages": [
3
]
},
{
"title": "D. Methodology",
"content": "We follow a hierarchical search strategy using the PowerFlux pipeline [17-24] with loose coherence [21, 25-27]. While the first stage (called Stage 0) establishes upper limits and produces first round of outliers, subsequent stages follow up on those outliers with increasing levels of constraints on signal coherence and parameter reconstruction. We steadily increase the effective coherence times by halving the phase mismatch parameter, δ between each successive stages, thereby increasing the sensitivity at each new stage [21, 25]. Additionally, we reduce the step-size for frequency, spindown, and sky location after each stage. As each successive stage improves sensitivity, we also require the SNR to increase while going from one stage to the next one. The exact factor of increase we require depends on the band and was empirically determined via software signal injections into the O3 data. Table I shows the values of phase-mismatch parameter δ , step-sizes, and SNR increase factors we require for each of the three stages. Outliers that survive all stages are then manually investigated through the use of strain histograms to look for contamination from known instrumental artifacts or single-interferometer lines with unknown origins. Outliers that survive all of these requirements are then followed up with the PyFstat pipeline [28], as in the O3a analysis [12]. PowerFlux iterates through all SFTs and computes the power for each template through a bilinear form of the Frequency [Hz] (SFT) input matrix { a t,f } , indexed by the SFT time t and template frequency f : where ∆ f is the detector frequency shift arising from Doppler shift due to the Earth's motion and the neutron star's spindown [23, 24]. The factor K D i D j t 1 ,t 2 ,f includes a Lanczos kernel along with contributions from timedependent SFT weights, antenna patterns, signal polarization, and relative terms for the detectors D i,j [21, 2325]. The phase-mismatch parameter δ in the Lanczos kernel portion of K D i D j t 1 ,t 2 ,f : where t 1 , t 2 are the mid-times of the two SFTs in the barycenter frame, and T coh = 7200 s , governs the degree of signal phase drift we tolerate [21, 25]. Loose coherence helps discard templates with non-physical phase evolution and hence, distinguish between signals that are possibly astrophysical in origin from instrumental arti- facts [25]. Steadily decreasing the value of δ used reduces the amount of phase drift permitted in each successive stage, hence imposing tighter constraints on the templates that merit followup with another stage of still tighter constraints [21, 25]. The choice of the Lanczos kernel (or a sinc filter) for loose coherence essentially makes it equivalent to performing a coherent search with a low-pass filter that attenuates signals with rapidly varying phases [44, 45]. The cutoff frequency is determined by the tunable phase mismatch parameter δ , and hence lowering its value in follow-up stages of the search demand progressively higher coherence between SFTs, thereby disfavoring templates with non-physical phase evolutions [44, 46]. As our goal is to improve upon the earlier O3a PowerFlux search [12], we use the full O3 data set and exploit loose coherence in the first stage while combining SFT coefficients coherently from the two interferometers. The improved strain sensitivity comes at an increased computational cost, however, for a fixed frequency range. Restricting the search frequency band to 30-150 Hz makes the cost of this search acceptable.",
"pages": [
3,
4,
6
]
},
{
"title": "E. Upper Limits",
"content": "We use a universal statistic [47] to calculate upper limits from the first search stage, as in the O3a PowerFlux / search [12]. Owing to origin in an Markov inequality that is agnostic to the underlying probability distribution, our upper limits are valid even in the presence of non-Gaussian noise and give nearly optimum upper limits when the noise is indeed Gaussian [47]. The calculated upper limits are based on the overall noise level in each demodulated 1/16 Hz search band and the largest power value within that band [47]. We calculate 95% upper limits for circular polarization corresponding to the bestcase scenario, and linear polarization corresponding to worst-case scenario. Additionally, we provide estimated upper limits for a population averaged over random sky locations and polarizations. The upper limits are naturally not valid for the cleaned regions shown in Figure 1. We use software injections to validate our upper limits in the retained subbands, as illustrated in Figure 2. We exclude vetoed frequency bins from all software injections. The plots show calculated linear-polarization upper limits as a function of the injected strains. Since our upper limits are 95% frequentist, we expect the points to be above the diagonal > 95% of the times. The flat regions on the left correspond to low strain values where we do not have enough sensitivity to produce useful upper limits.",
"pages": [
7
]
},
{
"title": "F. Outlier Followup",
"content": "Outliers produced from the first stage are analyzed by subsequent stages of loose coherence, with increasing constraints on the effective coherence time. Steadily decreasing values of the loose coherence phase-mismatch δ demands increasing temporal coherence with each iteration and hence, help distinguish between signals that could potentially be astrophysical in source from instrumental artifacts [21, 25]. Additionally, we step more finely in frequency, spindown, and sky points with each successive stage, which improves parameter reconstruction resolution. Outliers obtained from stage 0 with SNR > 7 are clustered according to their frequency, spindown, and sky locations. The clustering algorithm we use works as follows: We start by sorting all the outliers in descending order of their SNRs. We then take the loudest outlier in the list and select all other outliers within a certain pre-defined distance. We assign all these outliers into a single cluster. We then remove all the outliers just assigned to a cluster from the list and select the next loudest outlier. We repeat this until all outliers have been assigned to their respective clusters. The distance measure is based on the frequency, spindown, and sky location and the thresholds are determined with injections. Once the clustering is complete, we take the loudest 10 outliers within each cluster and follow up on them with stage 1. After each stage > 0, we require the SNR to exceed a threshold empirically determined from software injections and require the reconstructed parameters to be consistent with those of the previous stage. Table I shows the SNR increase constraints that were required, along with the search parameters for each stage. We use software injections to determine suitable search parameters as well as to validate the followup process [21]. In addition to the above mentioned criteria between stages, an injection is considered recovered only if its inferred parameters lie within 2.5 mHz in frequency, 3 × 10 -10 Hz / s in spindown, and 28 . 5 rad Hz /f in sky location from the true injected values. Figure 3 shows injection recovery efficiencies for the major 30 Hz subbands (30-60 Hz, 60-90 Hz, 90-120 Hz, 120-150 Hz). Recovery for the three PowerFlux stages are shown separately. The vertical dashed lines represent a relative strain, the ratio of injected strain to upper limit on that band, equal to 1, and the horizontal dashed lines represent 95% recovery efficiency. After three stages of PowerFlux , we manually inspect the outliers to look for contamination via strain histograms [38], as was done for the O3a search [12]. Ruling out clearly contaminated signals, we then use Markov Chain Monte Carlo (MCMC) with the PyFstat [28] pipeline to follow up on the final list of outliers. The PyFstat pipeline uses semi-coherent summing of the F -statistic [48] with Markov Chain Monte Carlo sampling of the posterior to give a detection statistic as well as to produce a posterior for the reconstructed parameters [49]. We follow an implementation described in [29] where it was used to follow up on outliers from the O2 run. In particular, we create a ladder of steadily decreasing numbers of segments of increasing duration with which to subdivide the full O3 run and run multiple stages where the posterior from one stage is then used to shrink the prior for the next stage [29]. We use a random sample of 600 sky locations away with the same declination as the candidate source, but separated by more than 90 degrees in right ascension, in order to calculate a background distribution for each specific frequency band [29]. We then use this background distribution along with the change in the F -statistic value in the last stage of the search to compute a Bayes factor by comparing the probability distribution of having a signal in the given data with the probability distribution of its being due to a noise background [29]. We use software injections to calibrate the appropriate Bayes factor values to set a cutoff for what value would constitute large enough for an outlier to potentially be from a true signal. We obtain a threshold of 100, so any outlier with Bayes factor less than that can be safely discarded.",
"pages": [
7
]
},
{
"title": "IV. RESULTS",
"content": "The first stage of the PowerFlux search leads to upper limits and a list of outliers to follow up. The upper limit on a given band is valid regardless of whether or not it produces outliers [12, 22]. Additionally, the upper limits are robust with respect to non-Gaussian noise, owing to the use of the universal statistic [47]. Section IV A and Section IV C, respectively discuss the upper limits and outliers we obtain. In Section IV B, we present results from the recovery of hardware injections.",
"pages": [
7,
8
]
},
{
"title": "A. Upper Limits",
"content": "Figure 4 shows a graph of upper limits as a function of the search frequency. The green curve shows the linearpolarization or worst-case upper limits, while the orange curve shows the circular-polarization or best-case upper limits. The dashed and dotted curves show O3 results from other search pipelines, including the O3a Power- Flux search [12], full-O3 FrequencyHough and SkyHough searches [13], and full-O3 Einstein@Home distributedcomputing search [50]. It is important, however, to use care in these comparisons as the searches cover different parameter spaces. For example, the Einstein@Home search covers only about 26% of the spindown range of this search. Results from similar but less sensitive searches of the O1 and O2 data sets are not included. 3 To enable a straightforward comparison to the other pipelines, we also present estimated population-averaged upper limits [12]. This estimate is produced by multiplying the circular upper limits by a factor of 2.3, determined empirically for control bands using an ensemble of random sky locations and random stellar orientations [12]. Each upper limit displayed is maximized over the entire sky except for a small region near the ecliptic poles [21, 23]. This region is excluded as it is prone to a large number of detector artifacts owing to stationary frequency evolution from the combination of frequency derivative and Doppler modulation [21, 23]. Our results improve upon the earlier PowerFlux O3a results by a median factor of ∼ 1 . 4. Improvements with respect to the FrequencyHough and SkyHough full-O3 searches are frequency-dependent, with median factors of ∼ 1 . 1 for both. In contrast, the full-O3 Einstein@Home distributed-computing search [50], which covers the 20800 Hz band improves upon these results by a median factor of ∼ 1 . 2 in the common range of frequency overlap and spindown overlap (26% of the spindown range probed here). The improvements compared to the O3a PowerFlux analysis can be attributed to a number of reasons. First, this analysis uses the full O3 data set which is nearly twice as long as the O3a data set [30]. Second, unlike in the O3a search, we use loose coherence in the initial stage with a δ parameter value of π/ 2, which improves sensitivity via longer effective coherence time, at the expense of higher computational cost [21, 25]. Third, we combine the interferometer SFTs coherently in the initial stage [23]. As a large number of contaminating line artifacts were present, particularly in H1 data, we clean the SFTs by replacing polluted bands with randomly generated Gaussian noise [41]. This method naturally means that upper limits calculated for those bands would not be valid, so those bands are excluded from the results presented here. Figure 1 shows the bands that were cleaned out. The upper limits on source strain amplitudes shown in Figure 4 can be recast into lower limits on distance ranges at which source neutron stars with certain assumed ellipticities can reside [11, 37]. Figure 5 shows these implied lower limits on ranges (kpc) for assumed ellipticity values of ϵ = 10 -4 , 10 -6 , and 10 -8 . The ranges are shown for the best-case circular popularization as well as population-averaged ranges. We also show our search's implied maximum ranges as a function of the frequency for a gravitar with maximum spindown magnitudes of 10 -8 , 10 -10 , and 10 -12 Hz/s.",
"pages": [
8,
9,
10
]
},
{
"title": "B. Hardware Injections",
"content": "A total of 18 hardware injections- 16 isolated and two binary sources- were added to the data in the O3 run [54]. Out of the 16 isolated-star injections, five injections fall within the frequency range we analyze [54]. We successfully recover four out of the five hardware injections. Ta- ble II shows the list of relevant hardware injections, their parameters including injected strains, corresponding upper limits produced and the final Bayes factors we obtain. The control upper limits show 95% upper limits for the nearest six neighboring sub-bands to give a rough estimate of expected upper limit that would have been obtained in the absence of a signal. For reference, only two of the five injections were successfully recovered in the O3a PowerFlux search [12], and only three of the five were recovered in a previous full-O3 search [13] using other pipelines, consistent with the improved sensitivity of the method presented here.",
"pages": [
10,
11
]
},
{
"title": "C. Outliers",
"content": "In addition to producing upper limits, stage 0 produces an initial list of outliers, corresponding to templates with excess power. We follow up on these outliers via subsequent stages, in which we impose progressively tighter constraints. In the absence of a true signal or of a hardware injection, we would ideally expect the number of outliers to decrease in subsequent stages because of these tighter constraints. In practice, particularly severe detector artifacts can also persist. Table III shows the number of outliers we obtain for various stages of the search. For the outliers that survive stage 2, we use strain histograms in which we plot the expected spectral shape overlaid upon the actual amplitude spectral density (ASD) to check if a particular band is clearly affected by an artifact. After the PowerFlux stages, we use Markov Chain Monte Carlo (MCMC) followup with the use of the PyFstat pipeline to obtain a Bayes factor for each of the surviving outliers. We use injections to determine appropriate Bayes factors as described in Section III F. Table IV shows the Bayes factors ≥ 15 for the final list of clusters of outliers. As can be seen, all outliers with appreciable Bayes factors have origins in hardware injections. The largest Bayes factor for an outlier not coming from a hardware injection and not having a visible artifact is 27, well below the threshold of 100 used to identify a plausible signal.",
"pages": [
11
]
},
{
"title": "V. CONCLUSIONS",
"content": "We have performed a deeper search for continuous gravitational waves from isolated neutrons stars, building upon a previous analysis [12]. Limiting the frequency band of the search allowed us to go deeper through the use of the full O3 data set, loose coherence, and coherent summing of SFTs across detectors in the initial stage. Additionally, line cleaning the SFTs before performing the analysis made the number of outliers to follow up on much more manageable. While we failed to detect a credible signal as all surviving outliers have origins in either hardware injections or instrumental artifacts, we do set upper limits on the strain amplitudes as shown in Figure 4. The best upper limit obtained for circular polarization is ∼ 4 . 5 × 10 -26 near 144 Hz, and for linearly polarization is ∼ 1 . 3 × 10 -25 . The best estimated population-averaged upper limit is ∼ 1 . 0 × 10 -25 , These limits improve on our O3a search by a median factor of ∼ 1 . 4. Improvements with respect to the FrequencyHough and SkyHough fullO3 searches [13] are frequency-dependent, with median factors of ∼ 1 . 1 for both. Our results are less constraining by a median factor of ∼ 1 . 2 than the full-O3 Ein- ch, in the overlapping frequency band and spindown range, but cover a spindown range four times larger and hence probe to greater distances in the galaxy in the band searched. The fourth LIGO-Virgo-KAGRA observation run O4 started May 24, 2023 with a planned duration of ∼ 20 months. The improved detector noise in the O4 data and longer observation period offer the prospect of improved strain sensitivity and, ideally, a continuous wave signal detection at last.",
"pages": [
11
]
},
{
"title": "VI. ACKNOWLEDGEMENTS",
"content": "We gratefully acknowledge useful discussions and long collaboration with current and former colleagues in the LIGO-Virgo-KAGRA continuous waves working group. For these results, in particular, we thank Evan Goetz, Ansel Neunzert, and Grant Weldon for spectral line investigations of the O3 data, Rodrigo Tenorio and David Keitel for creating the MCMC PyFstat pipeline, and Vladimir Dergachev for creating the PowerFlux pipeline. We also thank the two anonymous journal referees for constructive comments and suggestions. This work was supported in part by National Science Foundation Award No. PHY-2110181. This research has made use of data or software obtained from the Gravitational Wave Open Science Center (gwosc.org), a service of the LIGO Scientific Collaboration, the Virgo Collaboration, and KAGRA. This material is based upon work supported by NSF's LIGO Laboratory which is a major facility fully funded by the National Science Foundation, as well as the Science and Technology Facilities Council (STFC) of the United Kingdom, the Max-Planck-Society (MPS), and the State of Niedersachsen/Germany for support of the construction of Advanced LIGO and construction and operation of the GEO600 detector. Additional support for Advanced LIGO was provided by the Australian Research Council. Virgo is funded, through the European Gravitational Observatory (EGO), by the French Centre National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale di Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions by institutions from Belgium, Germany, Greece, Hungary, Ireland, Japan, Monaco, Poland, Portugal, and Spain. KAGRA is supported by Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan Society for the Promotion of Science (JSPS) in Japan; National Research Foundation (NRF) and Ministry of Science and ICT (MSIT) in Korea; Academia Sinica (AS) and National Science and Technology Council (NSTC) in Taiwan. The authors are grateful for computational resources provided by LIGO Laboratory and supported by National Science Foundation Grants PHY-0757058 and PHY-0823459. This research was done in part using services provided by the OSG Consortium [55-58], which is supported by the National Science Foundation Award Nos. 2030508 and 1836650. This research was enabled in part by support pro- vided by the Simon Fraser University Cedar Cluster (sfu.ca/research/supercomputer-cedar) and the Digital Research Alliance of Canada (alliancecan.ca). Fisher, E. Goetz, S. Kandhasamy, S. Karki, J. S. Kissel, A. P. Lundgren, and et al., Phys. Rev. D 95 , 062002 (2017). 012057 ((2007)).",
"pages": [
11,
12,
13
]
}
]
2024PhRvD.109d4050M
https://arxiv.org/pdf/2309.14874.pdf
<document>
<section_header_level_1><location><page_1><loc_16><loc_81><loc_84><loc_86></location>Abstract Formulation of the Spacetime Matching Problem and Null Thin Shells</section_header_level_1>
<text><location><page_1><loc_65><loc_78><loc_65><loc_79></location>†</text>
<text><location><page_1><loc_36><loc_77><loc_65><loc_79></location>Miguel Manzano ∗ and Marc Mars</text>
<text><location><page_1><loc_25><loc_68><loc_75><loc_75></location>Instituto de F'ısica Fundamental y Matem'aticas, IUFFyM Universidad de Salamanca Plaza de la Merced s/n 37008 Salamanca, Spain</text>
<text><location><page_1><loc_42><loc_65><loc_58><loc_66></location>September 27, 2023</text>
<section_header_level_1><location><page_1><loc_46><loc_58><loc_54><loc_59></location>Abstract</section_header_level_1>
<text><location><page_1><loc_13><loc_35><loc_87><loc_57></location>The formalism of hypersurface data is a framework to study hypersurfaces of any causal character abstractly (i.e. without the need of viewing them as embedded in an ambient space). In this paper we exploit this formalism to study the general problem of matching two spacetimes in a fully abstract manner, as this turns out to be advantageous over other approaches in several respects. We then concentrate on the case when the boundaries are null and prove that the whole matching is determined by a diffeomorphism ϕ on the abstract data set. By exploiting the gauge structure of the formalism we find explicit expressions for the gravitational/matter-energy content of any null thin shell. The results hold for arbitrary topology. A particular case of interest is when more than one matching is allowed. Assuming that one such matchings has already been solved, we provide explicit expressions for the gravitational/matter-energy content of any other shell in terms of the known one. This situation covers, in particular, all cut-and-paste constructions, where one can simply take as known matching the trivial re-attachment of the two regions. We include, as an example, the most general matching of two regions of the (anti-)de Sitter or Minkowski spacetime across a totally geodesic null hypersurface.</text>
<section_header_level_1><location><page_1><loc_9><loc_30><loc_28><loc_32></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_9><loc_16><loc_91><loc_28></location>The question of under which conditions two spacetimes can be matched across a hypersurface and give rise to a new spacetime is a fundamental problem in any metric theory of gravity. In particular, a matching theory is required in any physical situation where a substantial amount of gravitational/matterenergy content is located in a thin enough region of the spacetime (with respect to the dimensions of the problem). Then, the matter-content can be modelled as concentrated on a hypersurface. This thin shell of gravitational/matter-energy possesses its own gravity and hence affects the spacetime geometry, and it is worth finding the relationship between the shell's content and the properties of the spacetime.</text>
<text><location><page_1><loc_9><loc_12><loc_91><loc_15></location>Many authors have contributed to the matching problem in General Relativity, see e.g. the works [11], [34], [20], [18], [7], [10], [3], [31], [30], [26], [45]. The standard approach consists of considering two</text>
<text><location><page_2><loc_9><loc_74><loc_91><loc_91></location>spacetimes ( M ± , g ± ) with boundaries ˜ N ± . For the matching to be possible ˜ N ± must be diffeomorphic, i.e. there must exist a diffeomorphism Φ : ˜ N --/shortrightarrow ˜ N + , which we call matching map . One then defines the resulting spacetime M as the union of M + and M -with the corresponding identification of boundary points (ruled by Φ). The necessary and sufficient conditions for a metric g to exist on M are the so-called (preliminary) matching conditions (or junction conditions ) and require ( i ) that the first fundamental forms γ ± from both boundaries coincide, ( ii ) that there exists two riggings ζ ± (i.e. vector fields along ˜ N ± , everywhere transversal to them) with the same square norm and such that the one-forms g ± ( ζ ± , · ) coincide and ( iii ) that ζ ± are such that one points inwards and the other outwards. When these conditions are fulfilled, the matched spacetime exists. In general, this spacetime will contain a thin shell, which is ruled by the jump in the extrinsic geometry of the matching hypersurfaces.</text>
<text><location><page_2><loc_9><loc_61><loc_91><loc_73></location>In addition to this standard approach (also called ' a la Darmois ), one can also construct null thin shells with the so-called cut-and-paste method (see e.g. [35], [1], [36], [39], [38], [37], [17], [42], [43], [40], [41]), where the shell is described via a metric with a Dirac delta distribution with support on the matching hypersurface. The shell is built by taking a spacetime ( M , g ) with a null hypersurface ˜ N ⊂ M , then cutting M along ˜ N , which leaves two spacetimes ( M ± , g ± ), and finally reattaching (or pasting ) ( M ± , g ± ) by identifying the boundary points so that there exists a jump on the null direction on the matching hypersurface.</text>
<text><location><page_2><loc_9><loc_43><loc_91><loc_60></location>Be that as it may, null shells have been widely studied in the literature (for a sample, see [2], [8], [5], [33], [9], [6], [19], [13]), usually by imposing additional symmetries (such as spherical symmetry). In particular, the problem of matching two completely general spacetimes ( M ± , g ± ) with null boundaries ˜ N ± has been recently addressed in [23] under the only assumption that ˜ N ± admit a foliation by diffeomorphic spacelike cross-sections. One of the main results in [23] is that all the information about the matching is codified in a diffeomorphism Ψ between the set of null generators of ˜ N ± and function H , called step function , which corresponds to a shift along the null generators. Another result of interest is that, although generically two given spacetimes can be matched at most in one manner, sometimes multiple matchings are possible. A relevant case of the later, studied in detail in [24], is the matching across so-called Killing horizons of order zero.</text>
<text><location><page_2><loc_9><loc_26><loc_91><loc_42></location>The matching problem is studied in [23], [24] by means of the so-called formalism of hypersurface data [26], [27] (see also [31], [25], [29], [28], [21], [22]), which allows one to codify abstractly (i.e. in a detached way from an ambient manifold) the intrinsic and extrinsic geometric information of a hypersurface in terms of a data set D def = {N , γ, /lscript , /lscript (2) , Y } . The part {N , γ, /lscript , /lscript (2) } is called metric hypersurface data and codifies at the abstract level the would-be components of the full ambient metric g at the hypersurface. The tensor Y codifies extrisinc information. The formalism is equipped with a group of gauge transformations that accounts for the fact that, at the embedded level, the choice of a rigging is non-unique. Two data sets are equivalent if they are related by a gauge transformation. Each gauge group element G ( z,V ) ⊂ G is determined by a nowhere-zero function z and a vector field V in N .</text>
<text><location><page_2><loc_9><loc_10><loc_91><loc_25></location>In the language of the hypersurface data formalism, the matching can be performed if and only if [26] one can embed a single metric hypersurface data set in both spacetimes (and the corresponding matching riggings satisfy the orientation condition ( iii ) above). In that case, the gravitational/matter-energy content of the shell is fully codified by the jump of the tensors Y ± of each side, namely [ Y ] def = Y + -Y -[26]. The approach in [23], [24], whilst based on this formalism, still analyzes the matching in terms of the embeddings φ ± of the abstract manifold N in M ± and not directly at a detached level. Two questions arise naturally. The first one is whether there is a way of formulating the matching problem in a fully abstract manner (that is, exclusively in terms of objects defined in the abstract manifold N ) so that one does not need to make any reference to the actual spacetimes to be matched. The second</text>
<text><location><page_3><loc_9><loc_88><loc_91><loc_91></location>is whether one can generalize the results in [23], [24] to boundaries with arbitrary topology. The aim of this paper is to answer both questions.</text>
<text><location><page_3><loc_9><loc_66><loc_91><loc_87></location>The first question is solved in Theorem 3.1, where we provide a completely abstract version of the (spacetime) matching conditions. The theorem establishes that two given data sets D = {N , γ, /lscript , /lscript (2) , Y -} , ̂ D = {N , ̂ γ, ̂ /lscript , ̂ /lscript (2) , ̂ Y + } (each of them should be thought of as an abstraction of one of the boundaries) can be matched provided that there exists a diffeomorphism ϕ of N onto itself such that the metric hypersurface data sets {N , γ, /lscript , /lscript (2) } , {N , ϕ /star ̂ γ, ϕ /star ̂ /lscript , ϕ /star ̂ /lscript (2) } are related by a gauge transformation G ( z,V ) . This can be interpreted as follows. The map ϕ can be understood as an abstract version of the (spacetime) matching map Φ mentioned before. Since the matching requires that one single metric hypersurface data set is embedded in both spacetimes, D and ̂ D cannot be arbitrarily different. Instead, there must exist a gauge transformation that compensates for the change induced by ϕ so that, even after applying the pull-back ϕ /star , the metric part of the data are still equivalent. Theorem 3.1 also imposes a restriction on the sign of z . As we shall see, at the embedded level this restriction ensures that the orientation of the riggings to be identified in the matching process verifies condition ( iii ) above.</text>
<text><location><page_3><loc_9><loc_51><loc_91><loc_65></location>When the data sets D , ̂ D are embedded in two spacetimes, Theorem 3.1 is equivalent to the standard matching conditions ( i )-( iii ) above. This result is relevant for several reasons. First, it applies to (abstract) hypersurfaces of any causality and any topology. Secondly, the gauges of the data sets D , ̂ D are unfixed so that at the embedded level there is full freedom in the a priori choice of the riggings on each side. This gives a lot of flexibility to the framework. Finally, having formulated the matching problem abstractly allows one analyze in an independent manner thin shells with specific gravitational/matterenergy content and, on a second stage, study whether they can be embedded in a spacetime. This is useful e.g. for constructing examples of spacetimes containing certain types of shells.</text>
<text><location><page_3><loc_9><loc_24><loc_91><loc_51></location>We then concentrate on the null case. We intend to generalize the works [23], [24] so we impose no topological conditions on the boundaries We prove that a (null) metric hypersurface data set {N , γ, /lscript , /lscript (2) } is entirely codified by γ and that the remaining metric data is pure gauge. In these circumstances, the feasibility of the matching relies on the tensors { γ, ̂ γ } satisfying ϕ /star ̂ γ = γ (and z having suitable sign). One of our main results in the paper is that we find explicit expressions for the riggings to be identified in the matching process, as well as of the gravitational/matter-energy content of the resulting shell (Theorem 4.4). Specifically, we compute explicitly the jump [ Y ] and the energy-momentum tensor of the shell in terms of D , ̂ D and ϕ . In particular we provide fully geometric definitions of the energy density ρ , energy flux j and pressure p of the shell (Remark 2.16) and find explicit expression for them. We also codify the purely gravitational content of a null shell in a tensor Y G , which we also compute explicitly. We emphazise that all these result hold for any possible null thin shell. The pressure p of the null shell is worth studying in further detail. It turns out that it can be expressed as a difference of the surface gravities (i.e. the 'accelerations') of two null generators of N related by the push-forward map ϕ /star . This generalizes previous results in [23], [24], where in specific examples we noticed that p accounts for an effect of compression/stretching of points when crossing the matching hypersurface.</text>
<text><location><page_3><loc_9><loc_12><loc_91><loc_23></location>With the abstract matching formalism, one also recovers the property from [23], [24] that when the data sets D , ̂ D define abstract totally geodesic null hypersurfaces, then an infinite number of matchings are feasible. This situation is addressed in Section 5 where, assuming that all the information about one of the matchings is known, we prove that the gravitational/matter energy content of all the remaining matchings can be determined easily and explicity in terms of the known one and the map ϕ , with no need of performing additional calculations. The specific case when one of the matchings gives rise to no shell is of particular interest because it includes all cut-and-paste constructions 1 . In this context, we</text>
<text><location><page_4><loc_9><loc_86><loc_91><loc_91></location>find explicit expressions for the gravitational/matter-energy content of any null thin shell constructed with the cut-and-paste method. These results are applied in Section 7, where we study the matching of two regions of a constant-curvature spacetime across a totally geodesic null hypersurface.</text>
<text><location><page_4><loc_9><loc_80><loc_91><loc_85></location>For the sake of consistency, we devote Section 6 to showing how the results in [23] are recovered as a particular case of the general framework presented here in the specific case when N can be foliated by spacelike diffeomorphic cross-sections.</text>
<text><location><page_4><loc_9><loc_64><loc_91><loc_79></location>The structure of the paper is as follows. In Section 2 we review the results on the geometry of embedded null hypersurfaces, formalism of hypersurface data and matching of spacetimes that are needed later. In Section 3 we provide an abstract formulation of the matching problem. The rest of the paper concentrates on the null case. In particular, Section 4 is devoted to studying the properties of completely general null thin shells and finding explicit expressions for their gravitational/matter-energy content, while Section 5 addresses the case when multiple matchings are feasible. In Section 6, we establish the connection between the results in [23] and the abstract matching formalism developed here. The paper concludes with an example where we study all possible matchings involving two regions separated by a totally geodesic null hypersurface in the (anti-)de Sitter or Minkowski spacetimes (Section 7).</text>
<section_header_level_1><location><page_4><loc_9><loc_60><loc_39><loc_61></location>1.1 Notation and conventions</section_header_level_1>
<text><location><page_4><loc_9><loc_46><loc_91><loc_58></location>In this paper manifolds are smooth, connected and, unless otherwise indicated, without boundary. We use T M to denote the tangent bundle of a manifold M and Γ( T M ) for its sections (i.e. vector fields). We also let F ( M ) def = C ∞ ( M , R ) and F /star ( M ) ⊂ F ( M ) its subset of no-where zero functions. We use the symbols £ , d to denote Lie derivative and exterior derivative respectively. Both tensorial and abstract index notation will be employed depending on convenience. When index-free notation is used, we shall often use boldface for covariant tensors. In index notation we use standard font (not boldface) in all cases. We work in arbitrary dimension n , with the following values for different sets of indices:</text>
<formula><location><page_4><loc_23><loc_44><loc_91><loc_45></location>α, β, ... = 0 , 1 , 2 , ..., n ; a, b, ... = 1 , 2 , ..., n ; A,B,... = 2 , ..., n . (1.1)</formula>
<text><location><page_4><loc_9><loc_31><loc_91><loc_43></location>As usual, parenthesis (resp. brackets) will denote symmetrization (resp. antisymmetrization) of indices and we also use the notation A ⊗ s B ≡ 1 2 ( A ⊗ B + B ⊗ A ) for the symmetrized tensor product of two tensors A and B . When B is symmetric, 2-contravariant we write tr B A for the trace with respect to B of any 2-covariant tensor A . Given a semi-Riemannian manifold ( M , g ), the associated contravariant metric is called g /sharp and ∇ is the Levi-Civita derivative. Scalar products of two vectors are denoted indistinctly as g ( X,Y ) or 〈 X,Y 〉 g . Our convention for Lorentzian signature is ( -, + , ..., +). Finally, the abreviation 'w.r.t.' for 'with respect to' will be used sometimes.</text>
<section_header_level_1><location><page_4><loc_9><loc_26><loc_29><loc_28></location>2 Preliminaries</section_header_level_1>
<section_header_level_1><location><page_4><loc_9><loc_23><loc_56><loc_24></location>2.1 Geometry of embedded null hypersurfaces</section_header_level_1>
<text><location><page_4><loc_9><loc_6><loc_91><loc_21></location>In this subsection we review some facts about embedded null hypersurfaces, see e.g. [15], [16], [12]. This will serve to fix our notation. An embedded null hypersurface in a spacetime ( M , g ) of dimension n +1 is the image ˜ N = φ ( N ) of an embedding φ : N ↪ -/shortrightarrow M of an n -manifold N , such that the first fundamental form γ def = φ /star g of N is degenerate. Any choice of (nowhere zero) normal vector k to ˜ N defines a null direction tangent to ˜ N called null generator (and viceversa). The integral curves of k are geodesic and the surface gravity ˜ κ k ∈ F ( ˜ N ) of k is defined by ∇ k k = ˜ κ k k . The second fundamental form of ˜ N w.r.t k is the tensor ˜ K k ( X,Y ) def = g ( ∇ X k, Y ), ∀ X,Y ∈ Γ( T ˜ N ). Boundaries of manifolds are always</text>
<text><location><page_5><loc_9><loc_85><loc_91><loc_91></location>two-sided, so (cf. Lemma 1 in [26]) we shall always assume that ˜ N admits an everywhere transversal vector field L , i.e. verifying L / ∈ T p ˜ N ∀ p ∈ ˜ N . The vector L can always be taken null everywhere (see e.g. [23]).</text>
<text><location><page_5><loc_13><loc_73><loc_13><loc_75></location>/negationslash</text>
<formula><location><page_5><loc_9><loc_68><loc_91><loc_73></location>Given a transverse submanifold S ⊂ ˜ N , it is useful [23], [24] to define the following tensors on S Θ L ( X,Y ) | p def = 〈∇ X L, Y 〉 g | p , σ L ( X ) | p def = -1 g ( L, k ) 〈∇ X k, L 〉 g | p , ∀ X,Y ∈ T p S. (2.1)</formula>
<text><location><page_5><loc_9><loc_72><loc_91><loc_86></location>A transverse submanifold of ˜ N is any ( n -1)-dimensional submanifold S ⊂ ˜ N to which k is everywhere transverse. When, in addition, every integral curve of k crosses S exactly once S is called crosssection (or simply section ). The existence of a cross-section entails a strong topological restriction on ˜ N , as in such case there always exist functions v ∈ F ( ˜ N ), called foliation functions , whose level sets S v 0 def = { p ∈ ˜ N | v ( p ) = v 0 ∈ R } are cross-sections of ˜ N and { S v } define a foliation of ˜ N . Nevertheless existence of foliation functions is always granted in sufficiently local domains of ˜ N . Note that necessarily k ( v ) = 0 so we can always assume k ( v ) = 1 either by rescalling k or by changing v .</text>
<text><location><page_5><loc_9><loc_62><loc_91><loc_67></location>When L is chosen null and orthogonal to S then Θ L and σ L are the second fundamental form and torsion one-form of S w.r.t. L . For any choice of L , the tensors σ L , Θ L encode extrinsic information of S . However, Θ L is not symmetric in general.</text>
<text><location><page_5><loc_9><loc_56><loc_91><loc_61></location>Assuming that ˜ N admits a cross-section S , one can construct a foliation function v ∈ F ( ˜ N ) and (on local patches) a basis { L, k, v I } of Γ ( T M ) | ˜ N adapted to the foliation with the following properties:</text>
<unordered_list>
<list_item><location><page_5><loc_20><loc_49><loc_78><loc_50></location>(D) The basis vectors { k, v I } are such that [ k, v I ] = 0 and [ v I , v J ] = 0 .</list_item>
<list_item><location><page_5><loc_20><loc_50><loc_91><loc_57></location>(A) k is a future null generator with surface gravity ˜ κ k . (B) v ∈ F ( ˜ N ) is the only foliation function satisfying v | S = 0 , k ( v ) | ˜ N = 1 . (C) Each vector field v I is tangent to the foliation, i.e. v I ( v ) = 0 . (2.2)</list_item>
</unordered_list>
<text><location><page_5><loc_9><loc_44><loc_80><loc_49></location>(E) L is a past null vector field everywhere transversal to ˜ N . For any basis { L, k, v I } verifying (2.2), we also define n scalar functions { µ a } ⊂ F ( N ) as</text>
<formula><location><page_5><loc_29><loc_41><loc_91><loc_46></location>˜ µ 1 ( p ) def = g ( L, k ) | p , µ I ( p ) def = g ( L, v I ) | p ∀ p ∈ N . (2.3)</formula>
<text><location><page_5><loc_29><loc_40><loc_29><loc_41></location>/negationslash</text>
<text><location><page_5><loc_9><loc_33><loc_91><loc_44></location>˜ Note that necessarily µ 1 = 0 (this has already been used in (2.1)). The vectors { v A } are spacelike by construction and { k, v I } is a basis of Γ( T ˜ N ). Conditions (A) and (B) imply that v increases towards the future. We write h for the induced metric on the leaves { S v } and h IJ def = g ( v I , v J ) for its components in the basis { v I } . We use h IJ and its inverse h IJ to lower and raise Capital Latin indices irrespectively of whether they are tensorial or not (e.g. we let µ I def = h IJ µ J ). The property [ k, v I ] = 0 entails [23]</text>
<formula><location><page_5><loc_39><loc_27><loc_91><loc_33></location>k ( h ( v I , v J ) ) ˜ N =2 ˜ K k ( v I , v J ) . (2.4)</formula>
<section_header_level_1><location><page_5><loc_9><loc_26><loc_45><loc_27></location>2.2 Formalism of hypersurface data</section_header_level_1>
<text><location><page_5><loc_9><loc_21><loc_91><loc_24></location>The formalism of hypersurface data , which we introduce next, will allow us to analyze the matching of spacetimes at a fully abstract level. We refer to [26], [27], [29], [28], [25], [21], [22] for details.</text>
<section_header_level_1><location><page_5><loc_9><loc_17><loc_40><loc_18></location>2.2.1 General hypersurface data</section_header_level_1>
<text><location><page_5><loc_9><loc_10><loc_91><loc_16></location>The fundamental notion of the formalism is metric hypersurface data , defined to be a set {N , γ, /lscript , /lscript (2) } where N is an n -dimensional manifold, γ is a 2-covariant symmetric tensor, /lscript is a covector and /lscript (2) is a scalar function subject to the condition that the symmetric 2-covariant tensor A | p on T p N × R given by</text>
<formula><location><page_5><loc_11><loc_8><loc_91><loc_10></location>A | p (( W,a ) , ( Z, b )) def = γ | p ( W,Z ) + a /lscript | p ( Z ) + b /lscript | p ( W ) + ab/lscript (2) | p , W,Z ∈ T p N , a, b ∈ R (2.5)</formula>
<text><location><page_6><loc_9><loc_88><loc_91><loc_91></location>is non-degenerate at every p ∈ N . A priori any signature for A | p is allowed. Given metric hypersurface data, one can define unique tensor fields { P ab , n a , n (2) } , with P symmetric, by means of [26]</text>
<formula><location><page_6><loc_21><loc_86><loc_36><loc_87></location>γ ab n b + n (2) /lscript a = 0 ,</formula>
<formula><location><page_6><loc_45><loc_86><loc_91><loc_87></location>(2.6) P ab /lscript b + /lscript (2) n a = 0 , (2.8)</formula>
<formula><location><page_6><loc_21><loc_83><loc_91><loc_85></location>/lscript a n a + n (2) /lscript (2) = 1 , (2.7) P ab γ bc + n a /lscript c = δ a c . (2.9)</formula>
<text><location><page_6><loc_9><loc_74><loc_91><loc_82></location>No restriction is placed on γ , which in particular is allowed to be degenerate. However, A being nondegenerate forces γ to have at most one degeneration direction [27]. Specifically, the radical of γ at p ∈ N , defined by Rad γ | p def = { X ∈ T p N | γ ( X, · ) = 0 } , is either zero- or one-dimensional. The latter case occurs if and only if n (2) | p = 0, which by (2.6) means that Rad γ | p = 〈 n | p 〉 . A point p ∈ N is called null if dim(Rad γ | p ) = 1 and non-null otherwise.</text>
<text><location><page_6><loc_9><loc_68><loc_91><loc_73></location>The second basic notion of the formalism is hypersurface data which is just {N , γ, /lscript , /lscript (2) } equipped with an extra symmetric 2-covariant tensor Y , namely D def = {N , γ, /lscript , /lscript (2) , Y } . It is useful to define the following tensors (note that F , s and U only require metric hypersurface data)</text>
<formula><location><page_6><loc_20><loc_65><loc_27><loc_68></location>F def = 1 2 d /lscript ,</formula>
<formula><location><page_6><loc_44><loc_65><loc_91><loc_67></location>(2.10) 2 (2.12)</formula>
<formula><location><page_6><loc_20><loc_60><loc_49><loc_64></location>s def = F ( n, · ) , (2.11) r def = Y ( n, · )</formula>
<formula><location><page_6><loc_61><loc_65><loc_81><loc_68></location>U def = 1 £ n γ + /lscript ⊗ s dn (2) .</formula>
<formula><location><page_6><loc_61><loc_63><loc_75><loc_64></location>K def = n (2) Y + U .</formula>
<formula><location><page_6><loc_87><loc_63><loc_91><loc_64></location>(2.13)</formula>
<formula><location><page_6><loc_61><loc_60><loc_91><loc_62></location>κ n def = -Y ( n, n ) . (2.14)</formula>
<text><location><page_6><loc_9><loc_58><loc_88><loc_59></location>(Metric) hypersurface data has a built-in gauge group structure [27] with the following properties.</text>
<text><location><page_6><loc_9><loc_51><loc_91><loc_57></location>Definition 2.1. Let D = {N , γ, /lscript , /lscript (2) , Y } be hypersurface data, z ∈ F /star ( N ) and V ∈ Γ( T N ) . The gauge transformed data G ( z,V ) ( D ) def = { N , G ( z,V ) ( γ ) , G ( z,V ) ( /lscript ) , G ( z,V ) ( /lscript (2) ) , G ( z,V ) ( Y ) } is defined as</text>
<formula><location><page_6><loc_9><loc_47><loc_91><loc_50></location>G ( z,V ) ( Y ) def = z Y + /lscript ⊗ s dz + 1 2 £ zV γ = z Y +( /lscript + γ ( V, · )) ⊗ s dz + z 2 £ V γ. (2.16)</formula>
<formula><location><page_6><loc_9><loc_48><loc_91><loc_53></location>G ( z,V ) ( γ ) def = γ, G ( z,V ) ( /lscript ) def = z ( /lscript + γ ( V, · )) , G ( z,V ) ( /lscript (2) ) def = z 2 ( /lscript (2) +2 /lscript ( V ) + γ ( V, V ) ) , (2.15)</formula>
<text><location><page_6><loc_9><loc_42><loc_91><loc_46></location>The set of all possible gauge transformations forms a group G = F /star ( N ) × Γ( T N ) with composition law G ( z 2 ,V 2 ) · G ( z 1 ,V 1 ) = G ( z 1 z 2 ,V 2 + z -1 2 V 1 ) , identity G (1 , 0) and inverse G -1 ( z,V ) def = G ( z -1 , -zV ) .</text>
<text><location><page_6><loc_9><loc_34><loc_91><loc_42></location>All considerations so far make no reference to any ambient space where N is embedded. The abstract construction and the usual geometry of embedded hypersurfaces are connected through the notion of embeddedness of the data. Given a semi-Riemannian ( n + 1)-dimensional manifold ( M , g ) we say {N , γ, /lscript , /lscript (2) } is embedded with embedding φ and rigging ζ in ( M , g ) provided there exists an embedding φ : N ↪ -/shortrightarrow M and a rigging ζ (i.e. a vector field along φ ( N ), everywhere transversal to it) satisfying</text>
<formula><location><page_6><loc_28><loc_31><loc_91><loc_33></location>φ /star ( g ) = γ, φ /star ( g ( ζ, · )) = /lscript , φ /star ( g ( ζ, ζ )) = /lscript (2) . (2.17)</formula>
<text><location><page_6><loc_9><loc_29><loc_71><loc_30></location>The same notion for hypersurface data {N , γ, /lscript , /lscript (2) , Y } requires, in addition,</text>
<formula><location><page_6><loc_44><loc_25><loc_91><loc_29></location>1 2 φ /star ( £ ζ g ) = Y . (2.18)</formula>
<text><location><page_6><loc_9><loc_20><loc_91><loc_25></location>We often simplify the notation and say simply that the data is ' { φ, ζ } -embedded'. We also identify scalars and vectors in N with their corresponding images on φ ( N ) when there is no risk of confusion. The action of the gauge group in the data corresponds to a change of rigging according to [27]</text>
<formula><location><page_6><loc_41><loc_18><loc_91><loc_20></location>G ( z,V ) ( ζ ) def = z ( ζ + φ /star V ) . (2.19)</formula>
<text><location><page_6><loc_9><loc_14><loc_91><loc_17></location>More specifically, it holds that if {N , γ, /lscript , /lscript (2) } is { φ, ζ } -embedded in ( M , g ), then G ( z,V ) ( {N , γ, /lscript , /lscript (2) } ) is { φ, G ( z,V ) ( ζ ) } -embedded in the same space.</text>
<text><location><page_6><loc_9><loc_12><loc_91><loc_13></location>The hypersurface φ ( N ) admits a unique normal ν satisfying g ( ν, ζ ) = 1, which decomposes as [26], [27]</text>
<formula><location><page_6><loc_43><loc_9><loc_91><loc_11></location>ν = n (2) ζ + φ /star n. (2.20)</formula>
<text><location><page_7><loc_9><loc_90><loc_90><loc_91></location>It then turns out that K (defined in (2.13)) is the second fundamental form of φ ( N ) w.r.t. ν [27], i.e.</text>
<formula><location><page_7><loc_38><loc_88><loc_91><loc_89></location>K = φ /star ( ∇ ν ) , ν def = g ( ν, · ) . (2.21)</formula>
<text><location><page_7><loc_9><loc_80><loc_91><loc_87></location>Observe that K and U coincide at null points of N . Although generically N is not a semi-Riemannian manifold, it admits two useful covariant derivatives. The metric hypersurface connection · ∇ depends only on the metric part of the data and it is defined uniquely [26] by the properties of being torsion-free together with the expressions</text>
<formula><location><page_7><loc_19><loc_77><loc_91><loc_79></location>· ∇ a γ bc = -/lscript b U ac -/lscript c U ab , (2.22) · ∇ a /lscript b = F ab -/lscript (2) U ab . (2.23)</formula>
<text><location><page_7><loc_9><loc_71><loc_91><loc_77></location>The second connection is called hypersurface connection and denoted by ∇ . It is also torsion-free and relates to the former by ∇ X Z = · ∇ X Z -Y ( X,Z ) n for any X,Z ∈ Γ( T N ). When {N , γ, /lscript , /lscript (2) , Y } is { φ, ζ } -embedded in ( M , g ), the ambient Levi-Civita connection ∇ and the derivatives · ∇ , ∇ satisfy [26]</text>
<formula><location><page_7><loc_28><loc_69><loc_91><loc_71></location>∇ X Z = · ∇ X Z -Y ( X,Z ) ν -U ( X,Z ) ζ = ∇ X Z -K ( X,Z ) ζ, (2.24)</formula>
<formula><location><page_7><loc_24><loc_66><loc_91><loc_68></location>〈∇ X ζ, Z 〉 g = Y ( X,Z ) + F ( X,Z ) (2.25)</formula>
<text><location><page_7><loc_9><loc_62><loc_91><loc_66></location>for all X,Z ∈ Γ( T N ). Thus, ∇ is the connection induced from ∇ along the rigging [31]. Two consequences of the definition of · ∇ are</text>
<formula><location><page_7><loc_10><loc_55><loc_91><loc_60></location>n b ( · ∇ b θ d + · ∇ d θ b ) = £ n θ d + · ∇ d ( θ ( n )) -2 ( θ ( n ) ( s d -n (2) · ∇ d /lscript (2) ) + P ab θ b ( U da -n (2) F da )) . (2.27)</formula>
<formula><location><page_7><loc_20><loc_58><loc_91><loc_62></location>· ∇ b n c = n c ( s b -n (2) ( d/lscript (2) ) b ) + P ac ( U ba -n (2) F ba ) , (2.26)</formula>
<text><location><page_7><loc_9><loc_52><loc_91><loc_56></location>where θ a is an arbitrary one-form. Their explicit proof can be found in [27] and [25, Lem. 2.5] respectively. We shall also need the following lemma relating Lie and · ∇ derivatives.</text>
<text><location><page_7><loc_9><loc_48><loc_91><loc_51></location>Lemma 2.2. Let {N , γ, /lscript , /lscript (2) } be metric hypersurface data, V a any vector field and w a any covector field. Define V a def = γ ab V b and ˆ w a def = P ab w b . Then the following identities hold</text>
<formula><location><page_7><loc_14><loc_45><loc_91><loc_48></location>ˇ 1 2 £ V γ ab = /lscript ( V )U ab + · ∇ ( a ˇ V b ) , (2.28) 1 2 £ ˆ w γ ab = · ∇ ( a w b ) -/lscript ( a · ∇ b ) w ( n ) . (2.29)</formula>
<text><location><page_7><loc_9><loc_38><loc_91><loc_44></location>Proof. We first note that ∇ c γ ab -∇ a γ bc -∇ b γ ac = 2 /lscript c U ab as a direct consequence of (2.22). Moreover, since · ∇ has no torsion, the Lie derivative of any p -covariant tensor T along any direction V ∈ Γ( T N ) reads ( £ V T ) a 1 ··· a p = V b · ∇ b T a 1 ··· a p + p i =1 T a 1 ··· a i -1 ba i +1 ··· a p · ∇ a i V b . Particularizing this for T = γ we get</text>
<text><location><page_7><loc_21><loc_36><loc_85><loc_44></location>· · · ∑ · · · · · · ·</text>
<text><location><page_7><loc_9><loc_26><loc_91><loc_38></location>£ V γ ab = V c ∇ c γ ab +2 γ c ( a ∇ b ) V c = V c ( ∇ c γ ab -∇ a γ bc -∇ b γ ac ) +2 ∇ ( a ˇ V b ) = 2 /lscript ( V )U ab +2 ∇ ( a ˇ V b ) which is (2.28). To prove the second identity we apply (2.28) to V = ˆ w . Since by (2.9) we have γ ab P bc w c = w a -w ( n ) /lscript a , identity (2.28) gives 1 2 £ ˆ w γ ab = /lscript ( ˆ w )U ab + · ∇ ( a ( w b ) -w ( n ) /lscript b ) ) . From (2.8), we find /lscript ( ˆ w ) = -/lscript (2) w ( n ). Inserting above yields 1 2 £ ˆ w γ ab = -/lscript (2) w ( n )U ab + · ∇ ( a w b ) -/lscript ( a · ∇ b ) w ( n ) -w ( n ) · ∇ ( a /lscript b ) , which simplifies to (2.29) after taking into account (2.23).</text>
<text><location><page_7><loc_9><loc_23><loc_91><loc_25></location>From a covector and a function on N , one can build a unique vector field according to the next lemma.</text>
<text><location><page_7><loc_9><loc_17><loc_91><loc_22></location>Lemma 2.3. [26] Let {N , γ, /lscript , /lscript (2) } be metric hypersurface data. Given a covector field /rho1 ∈ Γ( T /star N ) and a scalar function u 0 ∈ F ( N ) , there exists a vector field W ∈ Γ( T N ) satisfying γ ( W, · ) = /rho1 , /lscript ( W ) = u 0 if and only if /rho1 ( n ) + n (2) u 0 = 0 . Such W is unique and reads W = P ( /rho1 , · ) + u 0 n .</text>
<section_header_level_1><location><page_7><loc_9><loc_13><loc_55><loc_14></location>Matter-hypersurface data and abstract thin shells</section_header_level_1>
<text><location><page_7><loc_9><loc_8><loc_91><loc_11></location>Hypersurface data encodes (abstractly) the intrinsic and extrinsic information of embedded hypersurfaces. In the context of gravity, knowning the matter contents of the spacetime determines part of</text>
<text><location><page_8><loc_9><loc_79><loc_91><loc_91></location>the curvature, typically by means of the Einstein tensor Ein g . Thus, to codify matter information abstractly we need to supplement the data with additional quantities. For general hypersurfaces, only the normal-transverse and the normal-tangential components of Ein g can be related exclusively to intrinsic and extrinsic information of the hypersurface [18], [3], [31], [26]. Hence, the additional (matter) data involves a scalar ρ and a covector J that, once the data is embedded, correspond to such components of Ein g . Their relation with the rest of the data needs to be imposed as constraint equations. They are well-known in the spacelike case (see e.g. [4]), and were generalized to arbitrary causal character in [26].</text>
<text><location><page_8><loc_9><loc_70><loc_91><loc_78></location>Note that although we refer to the variables ρ and J as matter variables, what we are actually prescribing are certain components of the Einstein tensor. The terminology is justified because in General Relativity (with vanishing cosmological constant) ρ and J indeed correspond to the matter four-momentum along the normal direction. However, we emphasize that we are not assuming any field equations and that the geometric approach that we take can be used in any theory of gravity.</text>
<text><location><page_8><loc_9><loc_68><loc_61><loc_69></location>The abstract definition of matter-hypersurface data is as follows.</text>
<text><location><page_8><loc_9><loc_61><loc_91><loc_66></location>Definition 2.4. [26] (Matter-Hypersurface data) A tuple {N , γ, /lscript , /lscript (2) , Y , ρ /lscript , J } formed by hypersurface data {N , γ, /lscript , /lscript (2) , Y } , a scalar ρ /lscript ∈ F ( N ) and a one-form J ∈ Γ( T /star N ) is matter-hypersurface data if G ( z,V ) ( ρ /lscript ) = ρ /lscript + J ( V ) , G ( z,V ) ( J ) = z -1 J and the following identities, called constraint equations, hold:</text>
<formula><location><page_8><loc_11><loc_48><loc_91><loc_61></location>ρ /lscript = 1 2 · R c bcd P bd + 1 2 /lscript a · R a bcd P bd n c + · ∇ d ( ( P bd n c -P bc n d )Y bc ) + n (2) P bd P ac Y b [ c Y d ] a + 1 2 ( P bd n c -P bc n d ) ( /lscript (2) · ∇ d U bc +(U bc + n (2) Y bc ) · ∇ d /lscript (2) +2Y bc (F df -Y df ) n f ) , (2.30) J c = /lscript a · R a bcd n b n d -2 · ∇ f ( ( n (2) P bd -n b n d )Y b [ c δ f d ] ) +2( P bd -/lscript (2) n b n d ) · ∇ [ c U d ] b -2 P bd n f U b [ c F d ] f -( n (2) P bd -n b n d ) ( (U b [ c + n (2) Y b [ c ) · ∇ d ] /lscript (2) +2Y b [ c F d ] f n f ) -( P bd n f -P bf n d )Y bd U cf . (2.31)</formula>
<text><location><page_8><loc_9><loc_46><loc_90><loc_48></location>The next theorem justifies both the gauge behaviour of { ρ /lscript , J } and the explicit form of (2.30)-(2.31).</text>
<text><location><page_8><loc_9><loc_42><loc_91><loc_45></location>Theorem 2.5. [26] Let {N , γ, /lscript , /lscript (2) , Y , ρ /lscript , J } be matter-hypersurface data and assume that the hypersurface data {N , γ, /lscript , /lscript (2) , Y } is { φ, ζ } -embedded in a semi-Riemannian manifold ( M , g ) . Then,</text>
<formula><location><page_8><loc_20><loc_39><loc_91><loc_41></location>-ρ /lscript = φ /star ( Ein g ( ζ, ν )) , (2.32) -J = φ /star ( Ein g ( · , ν )) , (2.33)</formula>
<text><location><page_8><loc_9><loc_35><loc_91><loc_38></location>where Ein g is the ( 2 -covariant) Einstein tensor of ( M , g ) and ν the (unique) normal vector field along φ ( N ) satisfying g ( ζ, ν ) = 1 .</text>
<text><location><page_8><loc_9><loc_30><loc_91><loc_33></location>As we shall see further on, the matching problem involves pairs of matter-hypersurface data. However, at this point we simply put forward various definitions and explore some of their consequences.</text>
<text><location><page_8><loc_9><loc_25><loc_91><loc_29></location>Definition 2.6. (Thin shell) A thin shell is a pair of matter-hypersurface data with same metric hypersurface data, i.e. of the form {N , γ, /lscript , /lscript (2) , Y ± , ρ ± /lscript , J ± , /epsilon1 } , where /epsilon1 is a sign with gauge behaviour:</text>
<formula><location><page_8><loc_44><loc_22><loc_91><loc_25></location>G ( z,V ) ( /epsilon1 ) = z | z | /epsilon1. (2.34)</formula>
<text><location><page_8><loc_9><loc_19><loc_90><loc_21></location>We write Q ± for quantities constructed from {N , γ, /lscript , /lscript (2) , Y ± } and let [ Q ] def = Q + -Q -be its jump.</text>
<text><location><page_8><loc_9><loc_15><loc_91><loc_18></location>One of the main properties of thin shells is that one can define an energy-momentum tensor encoding their matter-energy content. In a completely general case, this is done as follows.</text>
<text><location><page_8><loc_9><loc_11><loc_91><loc_14></location>Definition 2.7. (Energy-momentum tensor) For a thin shell {N , γ, /lscript , /lscript (2) , Y ± , ρ ± /lscript , J ± , /epsilon1 } , the energymomentum tensor is the symmetric 2 -covariant tensor τ defined by</text>
<formula><location><page_8><loc_13><loc_6><loc_91><loc_11></location>τ df def = /epsilon1 ( ( P af n d + P ad n f ) n b -( n (2) P af P bd + P df n a n b ) + P ab ( n (2) P df -n d n f ) ) [Y ab ] . (2.35)</formula>
<text><location><page_9><loc_9><loc_77><loc_91><loc_91></location>Remark 2.8. Definitions 2.6 and 2.7 are a modification of the previous ones introduced in [26] , which involved no /epsilon1 . The addition of the sign /epsilon1 is necessary in order for τ to retain its physical interpretation as energy-momentum tensor (density) in all gauges. Indeed, a change in the orientation of /lscript (or of rigging in the embedded picture) introduces a sign in [ Y ] (by (2.16) ). The value of τ cannot be sensitive to this, so one needs to introduce a sign /epsilon1 with gauge behaviour (2.34) to compensate the change of sign in [ Y ] (in fact, one checks easily that the gauge behaviour of τ is G ( z,V ) ( τ ) = | z | -1 τ ). To be more specific, when one deals with thin shell data {N , γ, /lscript , /lscript (2) , Y ± , ρ ± /lscript , J ± } { φ ± , ζ ± } -embedded in ( M ± , g ± ) , the sign /epsilon1 must be chosen positive if ζ -points outwards w.r.t. ( M -, g -) and negative otherwise.</text>
<text><location><page_9><loc_9><loc_60><loc_91><loc_75></location>The tensor field τ has the symmetries of an energy-momentum tensor and coincides with the Israel energy-momentum tensor of the shell [18] whenever it does not contain null points. Moreover, for null thin shells, the definition of energy-momentum tensor provided in [3, Eq. (31)] by Barrab'es and Israel yields precisely τ . In a spacetime ( M , g ) resulting from a matching, given a basis { e a } of Γ( T ˜ N ) where ˜ N is the matching hypersurface, one can also check that the quantity τ ab e µ a e ν b gives the singular part of the Einstein tensor of ( M , g ), as it is written in [31, Eq. (71)]. The gauge behaviour of τ is key in the embedded case, as it ensures that the singular part of the Einstein tensor of the matched spacetime remains invariant under rescaling the normal vector ν . All these reasons justify the Definition 2.7 for the energy-momentum tensor on a thin shell [26], irrespectively of whether the data is embedded.</text>
<text><location><page_9><loc_9><loc_47><loc_91><loc_59></location>At null points (and only there), τ = 0 is compatible with a non-trivial jump of the geometry. Indeed, in order to get τ = 0 when n (2) = 0, it suffices to require [ Y ]( n, · ) = 0 and tr P [ Y ] = 0, which does not mean that the whole tensor [ Y ] vanishes identically. Physically, this situation corresponds to an impulsive gravitational wave supported on the shell. This behaviour is possible only at null points. At non-null points τ = 0 implies, in addition, that P af P bd [Y] ab = 0 which entails 0 = γ fi γ dj P af P bd [Y] ab = ( δ a i -n a /lscript i )( δ b j -n b /lscript j )[Y] ab = [Y] ij , i.e. abscence of jumps in the geometry. In particular, this means that non-trivial thin shells with vanishing energy-momentum tensor can only exist on null points.</text>
<section_header_level_1><location><page_9><loc_9><loc_43><loc_36><loc_44></location>2.2.2 Null hypersurface data</section_header_level_1>
<text><location><page_9><loc_9><loc_38><loc_91><loc_41></location>A particular case of relevance for the matching problem is when the hypersurfaces are null everywhere. It is immediate to translate this notion to the abstract level.</text>
<text><location><page_9><loc_9><loc_33><loc_91><loc_37></location>Definition 2.9. (Null (metric) hypersurface data) A metric hypersurface data {N , γ, /lscript , /lscript (2) } or a hypersurface data {N , γ, /lscript , /lscript (2) , Y } is null if the scalar n (2) given by (2.6) -(2.9) is zero everywhere on N .</text>
<text><location><page_9><loc_9><loc_26><loc_91><loc_31></location>Let us describe the main properties of the formalism in the null case. We refer to [25] for proofs and additional results. We already know that n (2) = 0 implies Rad( γ ) = 〈 n 〉 and therefore γ ( n, · ) = 0. Moreover, the tensors s and U def = 1 2 £ n γ defined in (2.11) and (2.12) verify</text>
<formula><location><page_9><loc_40><loc_24><loc_91><loc_26></location>s ( n ) = 0 , U ( n, · ) = 0 . (2.36)</formula>
<text><location><page_9><loc_9><loc_20><loc_91><loc_23></location>When the data is { φ, ζ } -embedded, U becomes the second fundamental form of φ ( N ) w.r.t. the null normal ν = φ /star n (recall (2.20)). Inserting n (2) = 0 and (2.36) in the contraction of (2.26) with n b entails</text>
<formula><location><page_9><loc_46><loc_18><loc_91><loc_20></location>· ∇ n n = 0 , (2.37)</formula>
<text><location><page_9><loc_9><loc_15><loc_46><loc_17></location>which together with (2.14) and ν = φ /star n yields</text>
<formula><location><page_9><loc_30><loc_10><loc_91><loc_15></location>∇ ν ν (2.24) = φ /star ( · ∇ n n -Y ( n, n ) n ) (2.37) = κ n φ /star n = κ n ν. (2.38)</formula>
<text><location><page_9><loc_9><loc_9><loc_91><loc_12></location>Since ν is a null generator of Φ( N ), (2.38) means that κ n corresponds (at the abstract level) to the surface gravity of ν . Under the action of the gauge group, the surface gravity κ n transforms as follows.</text>
<text><location><page_10><loc_9><loc_88><loc_91><loc_91></location>Lemma 2.10. [25] Let {N , γ, /lscript , /lscript (2) , Y } be null hypersurface data and consider gauge parameters { z, V } . The gauge behaviour of the scalar function κ n defined in (2.14) is</text>
<formula><location><page_10><loc_38><loc_84><loc_91><loc_88></location>G ( z,V ) ( κ n ) = 1 z ( κ n -n ( z ) z ) . (2.39)</formula>
<text><location><page_10><loc_9><loc_78><loc_91><loc_83></location>We now state and prove a result that will be of particular relevance below, namely that by means of a gauge transformation one can always adapt the one-form /lscript and the scalar /lscript (2) to whatever pair { u ∈ F ( N ) , ϑ ∈ Γ( T /star N ) } one wishes, with the only restriction that ϑ ( n ) = 0 everywhere on N .</text>
<text><location><page_10><loc_30><loc_74><loc_30><loc_75></location>/negationslash</text>
<text><location><page_10><loc_68><loc_78><loc_68><loc_79></location>/negationslash</text>
<text><location><page_10><loc_9><loc_73><loc_91><loc_77></location>Lemma 2.11. Let {N , γ, /lscript , /lscript (2) } be null metric hypersurface data, u a function on N and ϑ ∈ Γ( T /star N ) a covector satisfying ϑ ( n ) = 0 everywhere. There exists a unique gauge transformation G ( z,V ) satisfying</text>
<formula><location><page_10><loc_36><loc_71><loc_91><loc_73></location>G ( z,V ) ( /lscript ) = ϑ , G ( z,V ) ( /lscript (2) ) = u. (2.40)</formula>
<text><location><page_10><loc_9><loc_69><loc_51><loc_70></location>Moreover, the gauge group element G ( z,V ) is given by</text>
<formula><location><page_10><loc_30><loc_65><loc_91><loc_68></location>z = ϑ ( n ) , V = 1 ϑ ( n ) P ( ϑ , · ) + u -P ( ϑ , ϑ ) 2( ϑ ( n )) 2 n. (2.41)</formula>
<text><location><page_10><loc_38><loc_63><loc_38><loc_64></location>/negationslash</text>
<text><location><page_10><loc_9><loc_61><loc_91><loc_64></location>Remark 2.12. The condition ϑ ( n ) = 0 is necessary because if ϑ ( n ) vanishes at any point then ϑ can never correspond to /lscript in any gauge, as</text>
<text><location><page_10><loc_45><loc_56><loc_45><loc_58></location>/negationslash</text>
<formula><location><page_10><loc_30><loc_56><loc_70><loc_61></location>1 = ( G ( z,V ) ( /lscript ) )( G ( z,V ) ( n ) ) | p = z -1 ( G ( z,V ) ( /lscript ))( n ) | p .</formula>
<text><location><page_10><loc_9><loc_56><loc_75><loc_58></location>which in particular states that ( G ( z,V ) ( /lscript ))( n ) = 0 for all possible gauge parameters.</text>
<text><location><page_10><loc_9><loc_48><loc_91><loc_54></location>Proof. We first assume that the gauge transformation exists and restrict its form up to a function yet to be determined. We then restrict to group elements of such a form and show that there exists one and only one of them that satisfies (2.40), namely (2.41). This will prove both the existence and uniqueness claims of the lemma. For the first part we impose (2.40):</text>
<text><location><page_10><loc_9><loc_36><loc_91><loc_44></location>Contracting the first with n gives z = ϑ ( n ), so w def = γ ( V, · ) = 1 ϑ ( n ) ϑ -/lscript . Observe that w ( n ) = 0. Moreover, the vector V -P ( w , · ) lies in the kernel of γ because γ ab ( V b -P bc w c ) = w a -( δ c a -n c /lscript a ) w c = 0. Therefore, there exists f ∈ F ( N ) such that V a = P ab w b + fn b = ( ϑ ( n )) -1 P ab ϑ b + ( /lscript (2) + f ) n a . Thus, it suffices to restrict oneself to gauge parameters in the class</text>
<formula><location><page_10><loc_28><loc_43><loc_91><loc_48></location>z ( /lscript + γ ( V, · )) = ϑ , z 2 ( /lscript (2) +2 /lscript ( V ) + γ ( V, V ) ) = u. (2.42)</formula>
<formula><location><page_10><loc_30><loc_32><loc_91><loc_37></location>{ ( z = ϑ ( n ) , V = 1 ϑ ( n ) P ( ϑ , · ) + qn ) , q ∈ F ( N ) } . (2.43)</formula>
<text><location><page_10><loc_9><loc_30><loc_91><loc_33></location>We now start anew and prove that there is precisely one function q such that the corresponding ( z, V ) in (2.43) fulfills conditions (2.40). For V as in (2.43) we get</text>
<formula><location><page_10><loc_22><loc_28><loc_23><loc_29></location>1</formula>
<formula><location><page_10><loc_13><loc_23><loc_86><loc_29></location>ϑ ( V ) = ϑ ( n ) P ( ϑ , ϑ ) + q ϑ ( n ) , /lscript ( V ) = -/lscript (2) + q, γ ( V, · ) = 1 ϑ ( n ) γ ( P ( ϑ , · ) , · ) = 1 ϑ ( n ) ϑ -/lscript , γ ( V, V ) = 1 ϑ ( n ) ϑ ( V ) -/lscript ( V ) = P ( ϑ , ϑ ) ϑ ( n ) 2 + /lscript (2)</formula>
<text><location><page_10><loc_9><loc_21><loc_77><loc_22></location>The first condition in (2.42) is satisfied for all q . The second is satisfied if and only if</text>
<formula><location><page_10><loc_26><loc_18><loc_28><loc_19></location>ϑ</formula>
<formula><location><page_10><loc_28><loc_16><loc_74><loc_21></location>( n ) 2 ( 2 q + P ( ϑ , ϑ ) ϑ ( n ) 2 ) = u ⇐⇒ q = u -P ( ϑ , ϑ ) 2 ϑ ( n ) 2 .</formula>
<text><location><page_10><loc_9><loc_15><loc_26><loc_16></location>which ends the proof.</text>
<text><location><page_10><loc_9><loc_8><loc_91><loc_13></location>In particular, Lemma 2.11 (together with (2.15)) means that two given null metric hypersurface data sets are related by a gauge transformation if and only if they both have the same data tensor γ . We prove this in the next corollary.</text>
<formula><location><page_10><loc_86><loc_23><loc_87><loc_25></location>.</formula>
<text><location><page_11><loc_9><loc_86><loc_91><loc_91></location>Corollary 2.13. Let D def = {N , γ, /lscript , /lscript (2) } , D def = {N , γ, /lscript , /lscript (2) } be two null metric hypersurface data. Then there is a gauge group element G ( z,V ) ∈ F /star ( N ) × Γ( T /star N ) such that G ( z,V ) ( D ) = D if and only if γ = γ . This gauge element is given by</text>
<formula><location><page_11><loc_31><loc_82><loc_91><loc_86></location>z = /lscript ( n ) , V = 1 /lscript ( n ) P ( /lscript , · ) + /lscript (2) -P ( /lscript , /lscript ) 2( /lscript ( n )) 2 n. (2.44)</formula>
<text><location><page_11><loc_9><loc_77><loc_91><loc_81></location>Proof. The necessity is obvious from the fact that γ remains unchanged by a gauge transformation. Sufficiency is a direct application of Lemma 2.11 to ϑ = /lscript and u = /lscript (2) .</text>
<text><location><page_11><loc_9><loc_71><loc_91><loc_75></location>Lemma 2.11 and Corollary 2.13 state that in the null case one can codify all the metric hypersurface data information exclusively in the tensor γ , and that /lscript and /lscript (2) are pure gauge. This fact will be key later in Section 3 when studying the matching of spacetimes with null boundaries.</text>
<text><location><page_11><loc_9><loc_68><loc_91><loc_70></location>We shall also need the decompositions of { γ, P } in a basis { n, e A } of Γ( T N ) and its corresponding dual.</text>
<text><location><page_11><loc_9><loc_62><loc_91><loc_67></location>Lemma 2.14. [25] Consider null metric hypersurface data {N , γ, /lscript , /lscript (2) } . Let { n, e A } be a basis of Γ( T N ) and { q , θ A } be its corresponding dual, i.e. q ( n ) = 1 , q ( e A ) = 0 , θ A ( n ) = 0 , θ A ( e B ) = δ A B . Define the functions ψ A def = /lscript ( e A ) ∈ F ( N ) . Then, the tensor fields γ and P decompose as</text>
<formula><location><page_11><loc_21><loc_60><loc_91><loc_61></location>γ = h AB θ A ⊗ θ B , (2.45)</formula>
<formula><location><page_11><loc_21><loc_57><loc_91><loc_59></location>P = P ( θ A , θ B ) e A ⊗ e B + P ( q , θ A )( n ⊗ e A + e A ⊗ n ) + P ( q , q ) n ⊗ n (2.46)</formula>
<text><location><page_11><loc_9><loc_52><loc_60><loc_54></location>where h AB def = γ ( e A , e B ) is a metric and h AB denotes its inverse.</text>
<formula><location><page_11><loc_23><loc_53><loc_91><loc_58></location>= h AB e A ⊗ e B -h AB ψ B ( n ⊗ e A + e A ⊗ n ) -( /lscript (2) -h AB ψ A ψ B ) n ⊗ n, (2.47)</formula>
<text><location><page_11><loc_9><loc_45><loc_91><loc_50></location>The concept of null thin shell arises naturally from Definitions 2.6 and 2.9. A thin shell is said to be null if its metric part {N , γ, /lscript , /lscript (2) } defines null metric hypersurface data . Moreover, as a corollary of Lemma 2.14, one can find a very simple form for the components of τ .</text>
<text><location><page_11><loc_9><loc_41><loc_91><loc_44></location>Corollary 2.15. In the setup of Lemma 2.14, let {N , γ, /lscript , /lscript (2) , Y ± , ρ ± /lscript , J ± , /epsilon1 } be a null thin shell. Then, the components of the energy-momentum tensor τ in the basis { q , θ A } read</text>
<formula><location><page_11><loc_10><loc_38><loc_91><loc_40></location>τ ( q , q ) = -/epsilon1 h AB [ Y ]( e A , e B ) , τ ( q , θ A ) = /epsilon1 h AB [ Y ]( n, e B ) , τ ( θ A , θ B ) = -/epsilon1 h AB [ Y ]( n, n ) . (2.48)</formula>
<text><location><page_11><loc_9><loc_35><loc_63><loc_36></location>Proof. Inserting the decomposition (2.47) into Definition 2.7 yields</text>
<text><location><page_11><loc_9><loc_29><loc_91><loc_32></location>after a simple but somewhat long computation in which several terms cancel out. Contracting with { q , θ A } it is immediate to get (2.48).</text>
<formula><location><page_11><loc_19><loc_31><loc_81><loc_35></location>τ df = -/epsilon1 h AB ( [ Y ]( e A , e B ) n d n f -[ Y ]( n, e A )( n d e f B + e d B n f ) + [ Y ]( n, n ) e d A e f B )</formula>
<text><location><page_11><loc_9><loc_19><loc_91><loc_27></location>Remark 2.16. In the literature, the different components of the energy-momentum tensor of a thin shell {N , γ, /lscript , /lscript (2) , Y ± , ρ ± /lscript , J ± , /epsilon1 } are interpreted physically as an energy density ρ , an energy-flux j and a pressure p (see e.g. [44] ). However, this is usually done in a context where {N , γ, /lscript , /lscript (2) , Y ± , ρ ± /lscript , J ± , /epsilon1 } are embedded with riggings ζ ± that are null and orthogonal to the basis vectors { e A } . In a completely general framework, we propose the following geometric definitions for the physical quantities { ρ, p, j } :</text>
<text><location><page_11><loc_9><loc_14><loc_88><loc_15></location>Definitions (2.49) are justified because in the null case (2.35) can be written in terms of { ρ, p, j } as</text>
<formula><location><page_11><loc_20><loc_14><loc_91><loc_19></location>ρ def = -/epsilon1 tr P [ Y ] , p def = -/epsilon1 [ Y ]( n, n ) , j def = /epsilon1 ( P ([ Y ]( n, · ) , · ) -/epsilon1/lscript (2) pn ) . (2.49)</formula>
<formula><location><page_11><loc_33><loc_9><loc_91><loc_14></location>τ = ρn ⊗ n + p ( P +2 /lscript (2) n ⊗ n ) +2 j ⊗ s n. (2.50)</formula>
<text><location><page_12><loc_9><loc_86><loc_91><loc_91></location>For null shells, the vector field j satisfies γ ( j, · ) = /epsilon1 [ Y ]( n, · )+ p /lscript and /lscript ( j ) = 0 , which makes the definitions (2.49) consistent since the one-form j def = γ ( j, · ) verifies j ( n ) = 0 . Moreover, a direct calculation based on (2.14) and (2.39) proves the following gauge behaviour for the pressure p :</text>
<formula><location><page_12><loc_44><loc_83><loc_91><loc_86></location>G ( z,V ) ( p ) = p | z | . (2.51)</formula>
<text><location><page_12><loc_9><loc_81><loc_74><loc_82></location>Whenever /lscript (2) = 0 and ψ A = 0 , it is straightforward to check that (2.49) becomes</text>
<formula><location><page_12><loc_23><loc_79><loc_91><loc_80></location>ρ = -/epsilon1 h AB [ Y ]( e A , e B ) , p = -/epsilon1 [ Y ]( n, n ) , j = /epsilon1 h AB [ Y ]( n, e B ) e A , (2.52)</formula>
<text><location><page_12><loc_9><loc_73><loc_91><loc_78></location>after using (2.45) and (2.47) . This allows one to recover the standard definitions for { ρ, p, j } introduced e.g. in [44, Eq. (3.99)] . Expressions (2.52) coincide with the definitions proposed in [44] whenever /epsilon1 = -1 which, as mentioned in Remark 2.8, corresponds to the rigging ζ -pointint inwards.</text>
<text><location><page_12><loc_9><loc_53><loc_91><loc_71></location>We conclude this subsection by recalling several aspects on the geometry of transverse submanifolds embedded in null metric hypersurface data sets. We again refer to [25] for proofs. Given null metric hypersurface data {N , γ, /lscript , /lscript (2) } , a transverse submanifold S is a codimension one embedded submanifold of N to which n is everywhere transverse . Letting ψ : S ↪ -/shortrightarrow N be the embedding of S in N we define h def = ψ /star γ . It is a fact [25] that h is a metric on S and we denote by ∇ h its Levi-Civita covariant derivative. When it is clear from the context we identify vectors and scalars on S with their counterpars on ψ ( S ). For any p -covariant tensor T along Ψ( S ) and given a basis { v A } of Γ( TS ), we define T ‖ def = ψ /star T and write T A 1 ...A p def = T ‖ ( v A 1 , . . . , v A p ) (without the parallel symbol). Capital Latin indices are raised with h IJ and its inverse h IJ . With the definition /lscript (2) ‖ := h IJ /lscript I /lscript J , the pull-back to S of the · ∇ derivative of any p -covariant tensor field T along ψ ( S ) takes the following explicit form [25, Lem. 3.15]:</text>
<formula><location><page_12><loc_22><loc_43><loc_91><loc_52></location>v a 1 A 1 . . . v a p A p v b B · ∇ b T a 1 ··· a p = ∇ h B T A 1 ··· A p -p ∑ i =1 /lscript J T A 1 ··· A i -1 JA i +1 ··· A p U A i B -p ∑ i =1 T a 1 ··· a p v a 1 A 1 . . . v a i -1 A i -1 n a i v a i +1 A i +1 . . . v a p A p ( ∇ h ( A i /lscript B ) +( /lscript (2) -/lscript (2) ‖ )U A i B ) . (2.53)</formula>
<section_header_level_1><location><page_12><loc_9><loc_40><loc_61><loc_41></location>2.3 Matching of spacetimes and junction conditions</section_header_level_1>
<text><location><page_12><loc_9><loc_35><loc_91><loc_38></location>From now on we focus on the problem of matching two spacetimes with boundary. In this section we recall known results, first for boundaries of any causality and secondly in the null case.</text>
<text><location><page_12><loc_9><loc_21><loc_91><loc_35></location>Consider two spacetimes ( M ± , g ± ) with boundaries ˜ N ± of any causal character. It is well-known (see e.g. [11], [34], [20], [18], [7], [10], [3], [31]) that the matching of ( M ± , g ± ) across ˜ N ± is possible if and only if the so-called junction conditions or matching conditions are satisfied. In the language of the formalism of hypersurface data, the matching requires [26] that there exist metric hypersurface data {N , γ, /lscript , /lscript (2) } that can be embedded in both spacetimes ( M ± , g ± ) with embeddings φ ± such that φ ± ( N ) = ˜ N ± and riggings ζ ± , i.e. there must exist two pairs { φ ± , ζ ± } satisfying γ = ( φ ± ) /star ( g ± ) , /lscript = ( φ ± ) /star ( g ± ( ζ ± , · )) , /lscript (2) = ( φ ± ) /star ( g ± ( ζ ± , ζ ± )) . (2.54)</text>
<text><location><page_12><loc_9><loc_12><loc_91><loc_21></location>In addition, the riggings ζ ± must fulfil an orientation condition (see item ( ii ) below). In these circumstances, it is always possible to select one of the embeddings freely by adapting N to one of the boundaries. In the following we shall make use of this freedom by fixing φ -at our convenience. This entails no loss of generality. Note that making the choice in the minus side is also of no relevance, as one can always switch the names of the spacetimes to be matched.</text>
<text><location><page_12><loc_9><loc_10><loc_91><loc_11></location>When the junction conditions are satisfied, the geometry of the shell [26] is determined by the jump of</text>
<text><location><page_13><loc_9><loc_90><loc_38><loc_91></location>the transverse tensors Y ± defined as</text>
<formula><location><page_13><loc_25><loc_85><loc_91><loc_90></location>Y ± def = (2.18) 1 2 ( φ ± ) /star ( £ ζ ± g ± ) , namely [ Y ] def = Y + -Y -. (2.55)</formula>
<text><location><page_13><loc_9><loc_83><loc_91><loc_86></location>In the literature, however, the matching conditions are not normally presented in terms of a hypersurface data set. Instead, they are usually formulated as follows (see e.g. [31]).</text>
<text><location><page_13><loc_9><loc_77><loc_85><loc_82></location>Junction Conditions. The matching of ( M ± , g ± ) across ˜ N ± can be performed if and only if</text>
<formula><location><page_13><loc_10><loc_74><loc_83><loc_78></location>( i ) There exist two riggings ζ ± along ˜ N ± and a diffeomorphism Φ : N --/shortrightarrow N + such that Φ /star ( g + ) = g -, Φ /star ( g + ( ζ + , · )) = g -( ζ -, · ) , Φ /star ( g + ( ζ + , ζ + )) = g -( ζ -, ζ -) .</formula>
<formula><location><page_13><loc_87><loc_74><loc_91><loc_76></location>(2.56)</formula>
<unordered_list>
<list_item><location><page_13><loc_10><loc_72><loc_74><loc_73></location>( ii ) One rigging must point inwards w.r.t. its boundary and the other outwards.</list_item>
</unordered_list>
<text><location><page_13><loc_9><loc_67><loc_91><loc_70></location>For the rest of the paper, two riggings ζ ± satisfying ( i )-( ii ) for a diffeomorphism Φ will be called matching riggings . The diffeomorphism Φ will be referred to as matching map .</text>
<text><location><page_13><loc_9><loc_61><loc_91><loc_66></location>If (2.56) holds for two riggings ζ ± then, for any other choice of rigging on one of the sides, (2.56) is fulfilled as well 2 (although different choices of rigging on one side will correspond to different riggings on the other side). We shall make use of this freedom to fix ζ -at will, again with no loss of generality.</text>
<text><location><page_13><loc_9><loc_44><loc_91><loc_60></location>As proven in Lemmas 2 and 3 of [30], given a rigging on one side (say ζ -) and a diffeomorphism Φ : ˜ N --/shortrightarrow ˜ N + satisfying Φ /star g + = g -, at non-null points the second and third equations of (2.56) yield either no solution for ζ + (hence the matching is not possible) or two solutions for ζ + with opposite orientation. At null points, on the other hand, if there exists a solution ζ + then it is unique. This means that at non-null points one can always make a suitable choice of rigging ζ + so that the junction condition ( ii ) is fulfilled, and hence one only needs to care about (2.56). In the null case, however, this is not so. It can happen that there exists a solution ζ + of (2.56) but with unsuitable orientation, and then the matching cannot be performed. Thus, at null points conditions (2.56) are necessary but not sufficient to guarantee that the matching is feasible [30].</text>
<text><location><page_13><loc_9><loc_35><loc_91><loc_43></location>When the matching is possible, the corresponding matching map Φ is the key object upon which the whole matching depends. This is so because once the point-to-point identification of the boundaries ˜ N ± (ruled by Φ) is known, one matching rigging can be selected at will (as we have seen) and the other is the unique solution that arises from enforcing both (2.56) and ( ii ). All the information about the matching is therefore codified by Φ, or equivalently by the embedding φ + (cf. (2.54)).</text>
<text><location><page_13><loc_9><loc_29><loc_91><loc_34></location>We now concentrate on the case when the boundaries ˜ N ± are null. From a spacetime viewpoint, this problem was addressed in [23] (see also [24], where the matching across Killing horizons of order zero was studied). In the remainder of the section we summarize the main results of [23].</text>
<text><location><page_13><loc_9><loc_13><loc_91><loc_28></location>Consider two ( n +1)-dimensional spacetimes ( M ± , g ± ) with null boundaries ˜ N ± that can be foliated by a family of diffeomorphic spacelike cross-sections. Assume further that one of the boundaries lies in the future of its corresponding spacetime while the other lies in its spacetime past. This entails no loss of generality, as explained in [23]. We construct foliation functions v ± ∈ F ( ˜ N ± ) and basis { L ± , k ± , v ± I } of Γ( T M ± ) | ˜ N ± according to (2.2). The surface gravities of k ± are ˜ κ ± k ± . As in Section 2.1, the leaves of the foliations are denoted by { S ± v ± } , while their corresponding induced metrics are h ± . We also let ˜ K k ± be the second fundamental forms of ˜ N ± w.r.t. k ± , and introduce the tensors Θ L ± , σ ± L on the leaves { S ± v ± } (cf. (2.1)). The scalar functions { µ ± a } ⊂ F ( N ± ) are defined by (2.3) w.r.t. the basis { L ± , k ± , v ± I } .</text>
<text><location><page_14><loc_9><loc_79><loc_91><loc_91></location>As we have seen, in order to perform a matching we need to embed a single metric hypersurface data set in both spacetimes. We codify the already described freedom in the choice of { φ -, ζ -} as follows. We first consider an abstract null hypersurface N and define coordinates { y 1 = λ, y A } therein. Then, we construct null embedded metric hypersurface data by enforcing that ( a ) the push-forwards { e -a def = φ -/star ( ∂ y a ) } coincide with the basis vectors { k -, v -I } (since { k -, v -I } are chosen at will, with this procedure we ensure that φ -is built at our convenience) and ( b ) that the rigging ζ -coincides with the basis vector L -. This amounts to impose</text>
<formula><location><page_14><loc_35><loc_77><loc_91><loc_79></location>e -1 = k -, e -I = v -I , ζ -= L -. (2.57)</formula>
<text><location><page_14><loc_9><loc_64><loc_91><loc_76></location>Thus, ζ -is a null past rigging (recall (2.2)) and λ is a coordinate along the degenerate direction of N . In fact, the subsets { λ = const. } ⊂ N are all diffeomorphic [23] and define a (spacelike) foliation of N . For the matching of ( M ± , g ± ) to be possible, there must exist another pair { φ + , ζ + } so that (2.54) hold (and the orientations of ζ ± are suitable). In that case, we can build another basis { e + a def = φ + /star ( ∂ y a ) } of Γ( T ˜ N + ) and then determining the matching requires that we find the explicit form of the vectors { e + a } (which fully codify φ + ). In the basis { k + , v + I } of Γ( T N + ), these vectors decompose as [23]</text>
<text><location><page_14><loc_9><loc_58><loc_38><loc_62></location>where f , a I , b J I ∈ F ( ˜ N + ) are given by</text>
<formula><location><page_14><loc_37><loc_62><loc_91><loc_67></location>˜ e + 1 = f k + , e + I = a I k + + b J I v + J , (2.58)</formula>
<formula><location><page_14><loc_27><loc_57><loc_91><loc_60></location>f = ∂H ( λ, y A ) ∂λ , a I = ∂H ( λ, y A ) ∂y I , b K I = ∂h K ( y A ) ∂y I (2.59)</formula>
<text><location><page_14><loc_9><loc_50><loc_91><loc_56></location>in terms of a set of functions { H ( λ, y B ) , h A ( y B ) } on N . The functions { H,h A } encode all the matching information and hence they determine φ + . In fact, given coordinates { v + , u I } on ˜ N + such that v + I = ∂ u I (i.e. { u I + } are constant along the null generators), the embedding φ + is such that [23]</text>
<text><location><page_14><loc_9><loc_43><loc_91><loc_49></location>The function H ( λ, y A ) is named step function because it measures a kind of jump along the null direction when crossing the matching hypersurface. It must satisfy the condition ∂ λ H > 0 [23]. The explicit form of the matching rigging ζ + was computed in [23, Cor. 1] and reads</text>
<formula><location><page_14><loc_33><loc_47><loc_91><loc_51></location>φ + ( λ, y I ) = ( v + = H ( λ, y I ) , u I = h I ( y J ) ) . (2.60)</formula>
<formula><location><page_14><loc_19><loc_39><loc_91><loc_43></location>ζ + = µ -1 ∂ λ H ( 1 µ + 1 L + -h AB + ( ( b -1 ) I A ( ∂ y I H -1 µ -1 ( ∂ λ H ) µ -I ) + 1 µ + 1 µ + A ) Z B ) , (2.61)</formula>
<text><location><page_14><loc_9><loc_33><loc_91><loc_36></location>The solvability of the first junction condition in (2.54) constitutes the core problem for the existence of a matching. In terms of the metrics h ± , it can be rewritten as</text>
<text><location><page_14><loc_9><loc_35><loc_78><loc_40></location>where ( b -1 ) J I def = ∂ h I y J and Z B def = 1 2 ( ( b -1 ) J B ( ∂ y J H -1 µ -1 ( ∂ λ H ) µ -J ) -1 µ + µ + B ) k + + v + B .</text>
<text><location><page_14><loc_9><loc_22><loc_91><loc_32></location>h -IJ | p = b L I b K J h + LK | Φ( p ) ∀ p ∈ ˜ N -. (2.62) Equation (2.62) is an isometry condition between each submanifold { v -= const. } ⊂ ˜ N -and its corresponding image on ˜ N + . On the other hand, the identification of { e ± 1 } requires the existence of a diffeomorphism Ψ (ruled by the coefficients b A B fulfilling (2.62)) between the set of null generators on both sides. Moreover, combining (2.4), (2.57)-(2.59), (2.62) and { e ± 1 def = φ ± /star ( ∂ y a ) } yields [23]</text>
<text><location><page_14><loc_9><loc_11><loc_91><loc_22></location>˜ K k -( v -I , v -J ) = ( ∂ λ H ) b A I b B J ˜ K k + ( v + A , v + B ) . (2.63) Thus, for each possible choice of Ψ (i.e. of { b A B } ), (2.63) determines a unique value for ∂ λ H unless the two second fundamental forms vanish simultaneously. In the latter case, the step function H cannot be restricted. Consequently, when ˜ N ± are totally geodesic, if a single matching of ( M ± , g ± ) can be performed then an infinite number of matchings (one for each possible step function H ) are feasible [23].</text>
<text><location><page_14><loc_9><loc_10><loc_91><loc_11></location>When the matching is possible, the matter-energy content of the shell is given by the next proposition.</text>
<text><location><page_15><loc_9><loc_84><loc_91><loc_91></location>Proposition 2.17. [23] Assume that the matching of ( M ± , g ± ) across ˜ N ± is possible and that it is determined by the functions { H ( λ, y A ) , h B ( y A ) } . Let h IJ be the induced metric on the leaves { λ = const. } ⊂ N , h IJ its inverse tensor and ∇ h its Levi-Civita covariant derivative. Define the vector fields { W A } , the scalars { µ + A } , the covector q ∈ Γ( T /star N ) and the vector field X = X a ∂ y a ∈ Γ( T N ) by</text>
<formula><location><page_15><loc_11><loc_82><loc_91><loc_84></location>µ + I def = b B I µ + B , W I def = b B I v + B , q I def = -µ + 1 ∇ h I H -µ + I , (2.64)</formula>
<formula><location><page_15><loc_11><loc_78><loc_91><loc_82></location>X 1 def = h IJ 2 µ + 1 ∂ λ H ( q I + µ + 1 µ -I µ -1 ∂ λ H )( q J -µ + 1 µ -J µ -1 ∂ λ H ) , X A def = h IA ( q I + µ + 1 µ -I µ -1 ∂ λ H ) . (2.65)</formula>
<text><location><page_15><loc_9><loc_76><loc_54><loc_77></location>Then, the components of the tensor [ Y ] def = Y + -Y -are</text>
<formula><location><page_15><loc_12><loc_73><loc_91><loc_75></location>Y ]( ∂ λ , ∂ λ ) = -µ -1 κ + k + ∂ λ H -κ -k -+ ∂ λ ∂ λ H , (2.66)</formula>
<formula><location><page_15><loc_11><loc_71><loc_49><loc_76></location>[ ( ∂ λ H )</formula>
<formula><location><page_15><loc_11><loc_62><loc_70><loc_68></location>[ Y ]( ∂ y I , ∂ y J ) = -µ -1 ( ˜ κ + k + ∇ h I H ∇ h J H ∂ λ H -∇ h ( I H ∂ λ µ + J ) µ + 1 ( ∂ λ H ) 2 -2 ∇ h ( I H σ + L ( W J ) ) ∂ λ H</formula>
<formula><location><page_15><loc_11><loc_65><loc_91><loc_74></location>˜ ˜ [ Y ]( ∂ λ , ∂ y J ) = -µ -1 ( ˜ κ + k + ∇ h J H -( σ + L ( W J ) -σ -L ( v -J ) ) + ∂ λ ∂ y J H ∂ λ H + X L ˜ K k -( v -J , v -L ) µ + 1 ∂ λ H ) , (2.67)</formula>
<formula><location><page_15><loc_12><loc_58><loc_91><loc_63></location>-( Θ L + ( W ( I , W J ) ) µ + 1 ∂ λ H -Θ L -( v -( I , v -J ) ) µ -1 ) -X 1 ˜ K k -( v -I , v -J ) µ + 1 ∂ λ H + ∇ h I ∇ h J H ∂ λ H + ∇ h ( I µ + J ) µ + 1 ∂ λ H -∇ h ( I µ -J ) µ -1 ) , (2.68)</formula>
<text><location><page_15><loc_9><loc_57><loc_86><loc_58></location>while the energy-momentum tensor of the shell is given by (the sign /epsilon1 is given by Definition 2.6)</text>
<formula><location><page_15><loc_10><loc_52><loc_90><loc_56></location>τ ( dλ, dλ ) = -/epsilon1 h IJ [ Y ]( ∂ y I , ∂ y J ) ( µ -1 ) 2 , τ ( dλ, dy I ) = /epsilon1 h IJ [ Y ]( ∂ λ , ∂ y J ) ( µ -1 ) 2 , τ ( dy I , dy J ) = -/epsilon1 h IJ [ Y ]( ∂ λ , ∂ λ ) ( µ -1 ) 2 .</formula>
<section_header_level_1><location><page_15><loc_9><loc_48><loc_68><loc_50></location>3 Abstract formulation of the matching problem</section_header_level_1>
<text><location><page_15><loc_9><loc_36><loc_91><loc_46></location>In the previous section, we have summarized the main aspects of the matching of two general spacetimes with null boundaries that admit a foliation by diffeomorphic spacelike sections. The matching conditions have been formulated from a spacetime viewpoint, and we have recalled the geometrical objects upon which the matching depends (namely the step function H and the diffeomorphism Ψ). We have also recollected the explicit expressions for the gravitational and matter-energy content of the resulting shells (Proposition 2.17).</text>
<text><location><page_15><loc_9><loc_27><loc_91><loc_35></location>The results we have just summarized leave (at least) two interesting problems unaddressed. The first one is whether one can obtain analogous results without the topological assumptions on the boundaries and the second is whether there is a way of formulating the matching problem in a fully abstract manner, namely without making any reference to the actual spacetimes to be matched. As already explained in the Introduction, addressing these problems is the key object of this paper.</text>
<text><location><page_15><loc_9><loc_9><loc_91><loc_26></location>Let us start with the abstract formulation of the junction conditions. For that purpose, we first consider that the boundaries ˜ N ± of the spacetimes ( M ± , g ± ) to be matched have any topology and any causal character . Since ˜ N -is embedded, there exists an abstract manifold N and an embedding ι -: N ↪ -/shortrightarrow M -such that ι -( N ) = ˜ N -. From the embedding ι -, one can construct an infinite number of embeddings simply by applying additional diffeomorphisms within N . To elude this unavoidable redundancy, we henceforth let ι -be one specific choice among all possible. As discussed before, two spacetimes ( M ± , g ± ) can be matched if there exists a pair of embeddings φ ± : N ↪ -/shortrightarrow M ± related to a matching map Φ by φ + = Φ · φ -. Moreover, the embedding and the rigging on one of the sides (say the minus side) can always be chosen freely. Suppose we enforce φ -= ι -and take a specific rigging ζ -. Then we can build embedded hypersurface data D def = {N , γ, /lscript , /lscript (2) , Y } by requiring (2.17)-(2.18),</text>
<text><location><page_16><loc_9><loc_90><loc_21><loc_91></location>i.e. by defining</text>
<formula><location><page_16><loc_12><loc_86><loc_91><loc_89></location>γ def = ( ι -) ∗ ( g -) , /lscript def = ( ι -) ∗ ( g -( ζ -, · )) , /lscript (2) def = ( ι -) ∗ ( g -( ζ -, ζ -)) , Y -def = 1 2 ( ι -) ∗ ( £ ζ -g -) . (3.1)</formula>
<text><location><page_16><loc_9><loc_76><loc_91><loc_86></location>Thus, all the information about the matching is encoded in φ + and the junction conditions are (2.54). These conditions, although of a more abstract nature than (2.56), still codify the matching information in the pair { φ + , ζ + } , which is not of abstract nature. In order to determine the matching in terms of objects defined at the abstract level, we must take one step further. The following theorem, based on the existence of a diffeomorphism ϕ of the abstract manifold N onto itself, sets up the corresponding construction.</text>
<text><location><page_16><loc_9><loc_57><loc_91><loc_75></location>Theorem 3.1. Consider two hypersurface data D def = {N , γ, /lscript , /lscript (2) , Y -} , ̂ D def = {N , ̂ γ, ̂ /lscript , ̂ /lscript (2) , ̂ Y + } embedded in two spacetimes ( M -, g -) , ( M + , g + ) with embeddings ι -, ι + and riggings L -, L + respectively. Assume that ι ± ( N ) def = ˜ N ± are boundaries of ( M ± , g ± ) and let /epsilon1 + = +1 (resp. /epsilon1 + = -1 ) if L + points outwards (resp. inwards) from M + . Define /epsilon1 -in the same way (i.e. /epsilon1 -= +1 if L -points outwards, /epsilon1 -= -1 if inwards). Then, the matching of ( M ± , g ± ) across ˜ N ± is possible if and only if (i) There exist a gauge group element G ( z,V ) and a diffeomorphism ϕ of N onto itself such that G ( z,V ) ( ϕ /star ̂ γ ) = γ, G ( z,V ) ( ϕ /star ̂ /lscript ) = /lscript , G ( z,V ) ( ϕ /star ̂ /lscript (2) ) = /lscript (2) ; (3.2) (ii) sign( z ) = -sign( /epsilon1 + )sign( /epsilon1 -) .</text>
<formula><location><page_16><loc_9><loc_50><loc_91><loc_56></location>Proof. The fact that D , ̂ D are embedded on ( M ± , g ± ) respectively means that γ def = ( ι -) /star ( g -) , /lscript def = ( ι -) /star ( g -( L -, · )) , /lscript (2) def = ( ι -) /star ( g -( L -, L -)) , Y -def = 1 2 ( ι -) /star ( £ L -g -) , (3.3)</formula>
<text><location><page_16><loc_9><loc_40><loc_91><loc_50></location>̂ γ def = ( ι + ) /star ( g + ) , ̂ /lscript def = ( ι + ) /star ( g + ( L + , · )) , ̂ /lscript (2) def = ( ι + ) /star ( g + ( L + , L + )) , ̂ Y + def = 1 2 ( ι + ) /star ( £ L + g + ) . (3.4) Since the spacetimes ( M ± , g ± ), the embeddings ι ± and the riggings L ± are all given, the tensor fields in (3.3)-(3.4) are known. To prove the first part of the theorem, we start by assuming ( i )-( ii ). Thus, there exist a pair { z ∈ F /star ( N ) , V ∈ Γ( T N ) } and a diffeomorphism ϕ : N -/shortrightarrow N so that (3.2) holds. These conditions can be rewritten as (recall (2.15), G -1 = G -1 )</text>
<text><location><page_16><loc_51><loc_40><loc_65><loc_41></location>( z,V ) ( z , -zV )</text>
<formula><location><page_16><loc_25><loc_34><loc_91><loc_39></location>ϕ /star ̂ γ = G -1 ( z,V ) ( γ ) = G ( z -1 , -zV ) ( γ ) = γ, (3.5) ϕ /star /lscript = G -1 ( z,V ) ( /lscript ) = G ( z -1 , -zV ) ( /lscript ) = /lscript z -γ ( V, · ) , (3.6)</formula>
<text><location><page_16><loc_9><loc_22><loc_91><loc_30></location>Let us define the map φ + def = ι + · ϕ , the vector field V ' def = ι + /star ( ϕ /star V ), the function z ' ∈ F /star ( ˜ N + ) given by ϕ /star (( ι + ) /star z ' ) def = z and the rigging ζ + def = z ' ( L + + V ' ) along ˜ N + . By definition of z ' , it holds that sign( z ) = sign( z ' ). On the other hand, combining (3.5)-(3.7) with the fact that ̂ D is embedded with embedding ι + and rigging L + , it follows</text>
<formula><location><page_16><loc_24><loc_28><loc_91><loc_36></location>̂ ϕ /star ̂ /lscript (2) = G -1 ( z,V ) ( /lscript (2) ) = G ( z -1 , -zV ) ( /lscript (2) ) = /lscript (2) z 2 -2 /lscript ( V ) z + γ ( V, V ) . (3.7)</formula>
<formula><location><page_16><loc_11><loc_16><loc_91><loc_23></location>γ = ϕ /star ̂ γ = ϕ /star (( ι + ) /star ( g + )) = ( φ + ) /star ( g + ) , (3.8) /lscript = z ( ϕ /star /lscript +( ϕ /star γ )( V, · ) ) = zϕ /star ( ( ι + ) /star ( g + ( L + , · ) + g + ( V ' , · ) )) = ( φ + ) /star ( g + ( ζ + , · )) , (3.9)</formula>
<text><location><page_16><loc_9><loc_9><loc_91><loc_12></location>The data D is therefore embedded in ( M + , g + ) with embedding φ + and rigging ζ + . Thus, conditions (2.54) are satisfied for φ -= ι -, φ + = ι + · ϕ and for the riggings ζ -= L -, ζ + . Moreover, combining</text>
<formula><location><page_16><loc_9><loc_12><loc_91><loc_20></location>̂ ̂ /lscript (2) = z 2 ( ϕ /star ̂ /lscript (2) +2( ϕ /star ̂ /lscript )( V ) + ( ϕ /star ̂ γ )( V, V ) ) = z 2 ϕ /star ( ( ι + ) /star ( g + ( L + , L + ) + 2 g + ( L + , V ' ) + g + ( V ' , V ' ) )) = ( φ + ) /star ( g + ( ζ + , ζ + )) . (3.10)</formula>
<text><location><page_17><loc_9><loc_90><loc_78><loc_91></location>( ii ) (which holds by assumption), the definition of ζ and sign( z ) = sign( z ), it follows</text>
<formula><location><page_17><loc_35><loc_88><loc_91><loc_89></location>ζ + = -sign( /epsilon1 + )sign( /epsilon1 ) | z | L + + V . (3.11)</formula>
<text><location><page_17><loc_9><loc_80><loc_91><loc_87></location>It is straightforward to check that (3.11) implies that whenever L -points inwards (resp. outwards) then ζ + points outwards (resp. inwards) irrespectively of the orientation of L + . Thus, D is embedded in ( M ± , g ± ) and L -, ζ + are such that one points inwards and the other outwards, which means that the matching of ( M ± , g ± ) is possible.</text>
<formula><location><page_17><loc_51><loc_85><loc_65><loc_91></location>+ ' -' ( ' )</formula>
<text><location><page_17><loc_9><loc_60><loc_91><loc_79></location>To prove the converse, we assume that the matching is possible for two pairs { φ ± , ζ ± } . We have already discussed the flexibility of selecting at will the embedding and the rigging on one side (say the minus side). Let us therefore set φ -= ι -, ζ -= L -. Since both L + and ζ + are riggings along ˜ N + , there exists a pair { z ' ∈ F /star ( ˜ N + ) , V ' ∈ Γ( T ˜ N + ) } such that ζ + = z ' ( L + + V ' ). Moreover, one can define a diffeomorphism ϕ : N -/shortrightarrow N by φ + def = ι + · ϕ . But then one can follow the arguments of (3.8)-(3.10) backwards and prove (3.2) for a function z ∈ F /star ( N ) defined by z def = ϕ /star (( ι + ) /star z ' ). As before, sign( z ) = sign( z ' ) so both ζ + = z ' ( L + + V ' ) and z ' L + = sign( z ) | z ' | L + have the same orientation (because V ' is tangent to ˜ N + ). By assumption the matching is possible, hence L -, ζ + are such that one points inwards and the other outwards. If L -points inwards (resp. outwards) then sign( z ) L + must point outwards (resp. inwards), so sign( z ) = sign( /epsilon1 + ) (sign( z ) = -sign( /epsilon1 + )) is forced. This means that ( i )-( ii ) are both fulfilled.</text>
<text><location><page_17><loc_9><loc_55><loc_91><loc_58></location>Remark 3.2. Theorem 3.1 does not impose any conditions on the topology of the abstract manifold N , except for the very mild one that hypersurface data sets can be defined on N .</text>
<text><location><page_17><loc_9><loc_46><loc_91><loc_54></location>Remark 3.3. In Theorem 3.1 we have not restricted the gauges of the data sets D , ̂ D (we let the two riggings L ± be given, but no conditions have been imposed on them). Each specific choice of L ± will fix a particular gauge on D , ̂ D . Moreover, Theorem 3.1 holds for data sets D , ̂ D of any causal nature. In particular, D , D are not required to contain non-null or null points exclusively.</text>
<text><location><page_17><loc_9><loc_32><loc_91><loc_49></location>̂ Remark 3.4. As proven in [27, Lem. 3.6] , given metric hypersurface data {N , γ, /lscript , /lscript (2) } and a point p ∈ N , the gauge group elements leaving {N , γ, /lscript , /lscript (2) } invariant at p are ( i ) G (1 , 0) | p if p is null and ( ii ) {G (1 , 0) | p , G ( -1 , -2 /lscript ) | p } if p is non-null, where the vector /lscript | p is obtained by raising index to /lscript | p with the inverse metric γ /sharp | p (which in that case exists). Since gauge parameters { z, V } are smooth by definition, it follows that when N contains a null point, only the identity element of G leaves the whole metric hypersurface data invariant. On the contrary, when N consists exclusively of non-null points there exist two gauge elements which do not transform the metric data. In this last case, the rigging G ( -1 , -2 /lscript ) ( ζ ) corresponds [27] to the reflection of ζ w.r.t. the tangent plane T q φ ( N ) at each point q ∈ φ ( N ) .</text>
<text><location><page_17><loc_9><loc_24><loc_91><loc_31></location>In view of the above, when there are no null points on N , condition ( ii ) can always be fulfilled once (i) is granted. Indeed, if there exists a gauge group element G ( z,V ) satisfying (i) then this also happens for G ( -1 , -2 /lscript ) · G ( z,V ) = G ( -z, -2 /lscript -V ) . Thus, there always exists a suitable choice of gauge parameter z for which ( i ) and ( ii ) hold.</text>
<text><location><page_17><loc_9><loc_17><loc_91><loc_23></location>On the contrary, when N contains null points only the gauge element G (1 , 0) leaves the hypersurface data invariant, which means that ( i ) can be fulfilled for a gauge group element G ( z,V ) but z may have the wrong sign. This is the underlying reason why the spacetime conditions (2.56) provide one unique solution for ζ + for given { ζ -, Φ } (see the corresponding discussion in Section 2.3).</text>
<text><location><page_17><loc_9><loc_9><loc_91><loc_15></location>Remark 3.5. In Theorem 3.1, we have expressed the junction conditions as a restriction over two data sets and a requirement on the sign of a gauge parameter. Theorem 3.1 therefore constitutes an abstract formulation of the standard matching conditions. In particular, a remarkable advantage of Theorem 3.1 is that it allows us to study different matchings in two different levels. At the first level one takes</text>
<text><location><page_18><loc_9><loc_79><loc_91><loc_91></location>whatever hypersurface data sets D , ̂ D satisfying ( i ) and studies its properties from a fully detached point of view. At this level, the spacetimes need not even exist. The problem can then move on and study whether or not one can construct spacetimes in which these data can be embedded so that condition ( ii ) holds. In other words, by Theorem 3.1 one can produce a thin shell of any causality with full freedom to prescribe the gravitational and matter-energy content, and then study the problem of constructing the resulting spacetime ( M , g ) which contains it. This is of great use, as it provides a framework to build examples of spacetimes with thin shells of any type.</text>
<text><location><page_18><loc_9><loc_65><loc_91><loc_77></location>In the setup of Theorem 3.1, the matching riggings are { L -, ζ + } , where ζ + is of the form (3.11). This means that the sign /epsilon1 -coincides with the sign /epsilon1 introduced in Definitions 2.6 and 2.7. It is convenient not to fix the signs /epsilon1 ± (or the riggings L ± ) a priori because it may well occur that transverse vectors L ± on each spacetime are already privileged or have been chosen for whatever other reason. The main point of the construction in Theorem 3.1 is firstly that it provides a fully abstract description of the matching and secondly that it keeps maximum flexibility so that one can adapt Theorem 3.1 to any particular scenario.</text>
<section_header_level_1><location><page_18><loc_9><loc_60><loc_88><loc_62></location>4 Abstract formulation of the matching problem: null boundaries</section_header_level_1>
<text><location><page_18><loc_9><loc_52><loc_91><loc_58></location>For the remainder of the paper, we focus on the case when both D and ̂ D are null hypersurface data. Under these circumstances, by Lemma 2.11 we know that there exists a pair { z, V } ensuring that the second and third equations in (3.2) are fulfilled. It follows that the only restrictions are therefore condition ( ii ) in Theorem 3.1 and the first equality in (3.2), namely</text>
<text><location><page_18><loc_9><loc_43><loc_91><loc_51></location>ϕ /star ̂ γ = γ. (4.1) Consequently, given two spacetimes ( M ± , g ± ) with null boundaries ˜ N ± , either there exists (at least) one diffeomorphism ϕ satisfying (4.1) or not. In the former case the matching is possible (provided ( ii ) holds) and, as we shall see next, all information about the matching is codified by ϕ .</text>
<text><location><page_18><loc_9><loc_36><loc_91><loc_42></location>From now on and without loss of generality, we again make the harmless assumption that one of the boundaries lies in the future of its corresponding spacetime while the other lies in its spacetime past (see the discussion in [23]). The following lemma provides the explicit form of the gauge parameters { z, V } and of the matching rigging ζ + in terms of the diffeomorphism ϕ .</text>
<text><location><page_18><loc_9><loc_28><loc_91><loc_34></location>Lemma 4.1. Assume that conditions ( i ) -( ii ) in Theorem 3.1 hold for a pair of embedded null hypersurface data D , ̂ D . Then, the gauge parameters { z, V } are given by</text>
<text><location><page_18><loc_9><loc_25><loc_82><loc_27></location>Moreover, the matching identifies the rigging L -with the following rigging in the plus side</text>
<formula><location><page_18><loc_26><loc_24><loc_91><loc_31></location>z = 1 ( ϕ /star ̂ /lscript )( n ) , V = -P ( ϕ /star ̂ /lscript , · ) + P ( ϕ /star ̂ /lscript , ϕ /star ̂ /lscript ) -ϕ /star ̂ /lscript (2) 2( ϕ /star ̂ /lscript )( n ) n. (4.2)</formula>
<text><location><page_18><loc_9><loc_17><loc_53><loc_22></location>where z ' ∈ F /star ( ˜ N + ) , µ ∈ F ( N + ) are given explicitly by</text>
<formula><location><page_18><loc_31><loc_20><loc_91><loc_26></location>ζ + = z ' ( L + -ι + /star ( ϕ /star ( P ( ϕ /star ̂ /lscript , · ) ) ) + µι + /star ( ϕ /star n ) ) , (4.3)</formula>
<text><location><page_18><loc_9><loc_11><loc_91><loc_14></location>Proof. The explicit form (4.2) for the function z follows from contracting (3.6) with n and using (2.7). The vector field V can be partially obtained also from (3.6) by particularizing Lemma 2.3 for W = V ,</text>
<formula><location><page_18><loc_23><loc_13><loc_91><loc_22></location>˜ ϕ /star (( ι + ) /star ( z ' )) = 1 ( ϕ /star ̂ /lscript )( n ) , ϕ /star (( ι + ) /star ( µ )) = P ( ϕ /star ̂ /lscript , ϕ /star ̂ /lscript ) -ϕ /star ̂ /lscript (2) 2( ϕ /star ̂ /lscript )( n ) . (4.4)</formula>
<text><location><page_19><loc_9><loc_87><loc_31><loc_91></location>/rho1 = z -1 /lscript -ϕ /star ̂ /lscript . This gives</text>
<text><location><page_19><loc_9><loc_81><loc_91><loc_85></location>where u 0 def = /lscript ( V ) is a function yet to be determined. This is done by substituting (4.5) into (3.7). First, γ ( V, V ) = /rho1 ( V ) = z -2 /lscript (2) + P ( ϕ /star /lscript , ϕ /star /lscript ) because of (2.7)-(2.8) and z -1 = ( ϕ /star /lscript )( n ). Thus,</text>
<formula><location><page_19><loc_26><loc_83><loc_91><loc_90></location>V = P ( /lscript z -ϕ /star ̂ /lscript , · ) + u 0 n (2.8) = -P ( ϕ /star ̂ /lscript , · ) + ( u 0 -/lscript (2) z ) n, (4.5)</formula>
<formula><location><page_19><loc_14><loc_75><loc_91><loc_83></location>̂ ̂ ̂ ϕ /star ̂ /lscript (2) = 2 z ( /lscript (2) z -u 0 ) + P ( ϕ /star ̂ /lscript , ϕ /star ̂ /lscript ) = ⇒ u 0 = /lscript (2) z + z 2 ( P ( ϕ /star ̂ /lscript , ϕ /star ̂ /lscript ) -ϕ /star ̂ /lscript (2) ) (4.6)</formula>
<text><location><page_19><loc_9><loc_73><loc_91><loc_76></location>so that substituting this into (4.5) proves (4.2). Equation (4.3) is a direct consequence of (4.2) and the fact that ζ + = z ' ( L + + ι + /star ( ϕ /star V )).</text>
<text><location><page_19><loc_9><loc_63><loc_91><loc_71></location>Whenever there exists a diffeomorphism ϕ solving (4.1) and given a basis { n, e A } of Γ( T N ), it is possible to obtain specific expressions for the push-forward vector fields { ϕ /star n, ϕ /star e A } . This is done in the next corollary. We use a hat for all objects defined in the data set ̂ D , in particular ̂ P and ̂ n are constructed in correspondence with (2.6)-(2.9).</text>
<formula><location><page_19><loc_19><loc_55><loc_91><loc_57></location>ϕ /star W A def = γ ( e A , · ) , ψ A def = /lscript ( e A ) , χ ( A ) def = ( ϕ -1 ) /star ( z -1 ψ A ) -W A ( ϕ /star V ) . (4.7)</formula>
<text><location><page_19><loc_9><loc_57><loc_91><loc_63></location>Corollary 4.2. Assume that conditions ( i ) -( ii ) in Theorem 3.1 hold for a pair of embedded null hypersurface data D , ̂ D . Let { n, e A } be a basis of Γ( T N ) and define the covectors { W A } and the functions { ψ A , χ ( A ) } along N by</text>
<text><location><page_19><loc_9><loc_53><loc_13><loc_55></location>Then,</text>
<text><location><page_19><loc_9><loc_46><loc_59><loc_51></location>Moreover, it holds that ̂ P ( W A , ̂ /lscript ) = 0 and ϕ /star χ ( A ) = ( ϕ /star ̂ /lscript )( e A ) .</text>
<formula><location><page_19><loc_21><loc_48><loc_82><loc_54></location>ϕ /star n = ( ( ϕ -1 ) /star z ) -1 ̂ n, (4.8) ϕ /star e A = ̂ P ( W A , · ) + χ ( A ) ̂ n,</formula>
<formula><location><page_19><loc_87><loc_52><loc_91><loc_53></location>(4.9)</formula>
<text><location><page_19><loc_9><loc_39><loc_91><loc_48></location>Proof. Consider any point p ∈ N . From (3.5) it follows that ̂ γ ( ϕ /star n, · ) | ϕ ( p ) = ( ϕ /star ̂ γ )( n, · ) | p = γ ( n, · ) | p = 0, so ϕ /star n = b ̂ n for some function b ∈ F ( N ). This, together with (4.2) and ̂ /lscript ( ̂ n ) = 1, entails that z -1 | p = ( ϕ /star ̂ /lscript )( n ) | p = ̂ /lscript ( ϕ /star n ) | ϕ ( p ) = b | ϕ ( p ) = ϕ /star b | p , which proves (4.8). On the other hand, any vector field X ∈ Γ( T N ) satisfies</text>
<text><location><page_19><loc_9><loc_27><loc_91><loc_34></location>which means that ̂ γ ( ϕ /star e A , · ) = W A , ̂ /lscript ( ϕ /star e A ) = ( ϕ -1 ) /star ( z -1 ψ A ) -W A ( ϕ /star V ). Particularizing Lemma 2.3 for the data ̂ D and for W = ϕ /star e A , /rho1 = W A and u 0 = ( ϕ -1 ) /star ( z -1 ψ A ) -W A ( ϕ /star V ) yields (4.9). Finally, P ( W A , /lscript ) = 0 because</text>
<formula><location><page_19><loc_20><loc_32><loc_80><loc_40></location>̂ γ ( ϕ /star e A , ϕ /star X ) | ϕ ( p ) = ( ϕ /star ̂ γ )( e A , X ) | p (3.5) = γ ( e A , X ) | p = ϕ /star W A ( X ) | p , ̂ /lscript ( ϕ /star e A ) | ϕ ( p ) = ( ϕ /star ̂ /lscript )( e A ) | p (3.6) = ψ A z -γ ( e A , V ) | p = ψ A z -ϕ /star W A ( V ) | p ,</formula>
<formula><location><page_19><loc_10><loc_23><loc_90><loc_31></location>̂ ̂ ̂ P ( W A , ̂ /lscript ) | ϕ ( p ) = -̂ /lscript (2) W A ( ̂ n ) | ϕ ( p ) = -̂ /lscript (2) (( ϕ -1 ) /star z ) W A ( ϕ /star n ) | ϕ ( p ) = -̂ /lscript (2) (( ϕ -1 ) /star z ) | ϕ ( p ) ( ϕ /star W A )( n ) | p = -/lscript (2) (( ϕ -1 ) /star z ) | ϕ ( p ) γ ( e A , n ) | p = 0 ,</formula>
<text><location><page_19><loc_9><loc_7><loc_91><loc_20></location>Remark 4.3. From (4.8) it follows that ϕ is a diffeormorphism which sends null generators into null generators. Moreover, since the vector fields { W A def = ̂ P ( W A , · ) } verify ̂ /lscript ( W A ) = 0 , it follows that W A / ∈ Rad ̂ γ . This, together with the fact that ϕ /star is necessarily of maximal rank, force the vector fields { W A } to be everywhere non-zero on N . In fact, { ̂ n, W A } constitutes a basis of Γ( T N ) , since { W A } are all linearly independent. We prove this by contradiction, i.e. we assume that one such vector field, e.g. W 2 , can be decomposed as W 2 = ∑ n r =3 c r W r . By (4.9) , this would mean that ϕ /star ( e 2 -∑ n r =3 c r e r ) =</text>
<text><location><page_19><loc_9><loc_18><loc_88><loc_26></location>̂ while χ ( A ) · ϕ = z -1 ψ A -( ϕ /star W A )( V ) = z -1 ψ A -γ ( e A , V ) (3.6) = ( ϕ /star ̂ /lscript )( e A ) yields ϕ /star χ ( A ) = ( ϕ /star ̂ /lscript )( e A ).</text>
<text><location><page_20><loc_9><loc_86><loc_91><loc_92></location>( χ (2) -∑ n r =3 c r χ ( r ) ) ̂ n , which we know it cannot occur, because only null generators can be mapped to null generators.</text>
<text><location><page_20><loc_9><loc_76><loc_91><loc_86></location>The point of introducing the objects { W A , χ ( A ) } will become clear later when we study the particular case when the boundaries have product topology S × R . For the moment, let us simply anticipate that in such case the property ̂ P ( W A , ̂ /lscript ) = 0 will allow us to conclude that the vector fields ̂ P ( W A , · ) are tangent to the leaves of a specific foliation of ˜ N + while from ϕ /star χ ( A ) = ( ϕ /star ̂ /lscript )( e A ) we will conclude that the functions { χ ( A ) } are actually spatial derivatives of the step function introduced in Section 2.3.</text>
<formula><location><page_20><loc_21><loc_64><loc_91><loc_70></location>ϕ /star ̂ U def = 1 2 ϕ /star ( £ ̂ n γ ) (4.8) = z 2 ϕ /star ( £ ϕ /star n γ ) (3.5) = z 2 £ n γ = z U = ⇒ U = ϕ /star ̂ U z , (4.10)</formula>
<text><location><page_20><loc_9><loc_68><loc_91><loc_77></location>One of the relevant results recalled in Section 2.3 is the relation (2.63) between the second fundamental forms of each side. It turns out that in this abstract framework with no topological assumptions one can also recover an equation of this form. To do that, we first note that £ f ̂ n ̂ γ = f £ ̂ n ̂ γ because ̂ n ∈ Rad ̂ γ . By direct computation one gets</text>
<text><location><page_20><loc_9><loc_59><loc_91><loc_68></location>̂ ̂ which connects the second fundamental forms U , ̂ U corresponding to the hypersurface data sets D , ̂ D . Equation (4.10) generalizes (2.63) to the case of boundaries with any topology, and has several implications that we discuss below.</text>
<text><location><page_20><loc_9><loc_53><loc_91><loc_60></location>In Theorem 3.1 we have seen that when the matching is possible there exists a diffeomorphism ϕ verifying (4.1). In such case, Lemma 4.1 and Corollary 4.2 provide explicit expressions for the gauge parameters { z, V } , the matching rigging ζ + and the push-forwards { ϕ /star n, ϕ /star e A } of any basis vector fields { n, e A } in terms of the map ϕ still to be determined.</text>
<text><location><page_20><loc_9><loc_37><loc_91><loc_52></location>However, as the reader may have noticed, condition (4.1) does not fix ϕ completely, firstly because there can be more than one diffeomorphism ϕ satisfying (4.1) and secondly because the tensor fields γ and ̂ γ are both degenerate. As happened in Section 2.3, where the step function could not be fixed directly by the isometry condition (2.62) but (2.63) was also required [23], here one also needs an extra condition in order to fix ϕ fully. This additional restriction is precisely (4.10). As in Section 2.3, this provides useful information only when U and ̂ U are non-zero. If both are zero then z (and hence part of ϕ , recall (4.2)) remains completely free. This means that one can find an infinite number of diffeomorphisms ϕ verifying (4.1), with which we recover (and extend to arbitrary topology) the property that whenever the boundaries are totally geodesic then the matching can be performed in an infinite number of ways.</text>
<text><location><page_20><loc_9><loc_33><loc_91><loc_36></location>One can obtain explicit expressions for the gravitational and matter-energy content of a general null shell in terms of the diffeomorphism ϕ . This is done in the following theorem.</text>
<text><location><page_20><loc_9><loc_28><loc_91><loc_31></location>Theorem 4.4. Assume that conditions ( i ) -( ii ) in Theorem 3.1 hold for a pair of embedded null hypersurface data D , D and let /epsilon1 = /epsilon1 -. Define</text>
<formula><location><page_20><loc_9><loc_17><loc_91><loc_23></location>[Y ab ] = z ( ( ϕ /star ̂ Y + ) ab + z 2 ( P ( ϕ /star ̂ /lscript , ϕ /star ̂ /lscript ) -ϕ /star ̂ /lscript (2) ) U ab -· ∇ ( a ( ϕ /star ̂ /lscript ) b ) ) -Y -ab , (4.11) where z ∈ F /star ( N ) is given by (4.2) . The components of [ Y ] in any basis { n, e A } of Γ( T N ) are</formula>
<formula><location><page_20><loc_9><loc_23><loc_84><loc_30></location>̂ Y -def = 1 2 ( ι -) /star ( £ L -g -) , ̂ Y + def = 1 2 ( ι + ) /star ( £ L + g + ) and Y + def = 1 2 ϕ /star ( ( ι + ) /star ( £ ζ + g + ) ) , where ζ + is given by (4.3) . Then, the tensor [ Y ] def = Y + -Y -reads</formula>
<formula><location><page_20><loc_9><loc_15><loc_10><loc_16></location>[</formula>
<formula><location><page_20><loc_10><loc_5><loc_89><loc_18></location>Y ]( e A , e B ) = ( z ( ϕ /star ̂ Y + ) -Y -) ( e A , e B ) + z 2 2 ( P ( ϕ /star ̂ /lscript , ϕ /star ̂ /lscript ) -ϕ /star ̂ /lscript (2) ) U ( e A , e B ) -ze a A e b B · ∇ ( a ( ϕ /star ̂ /lscript ) b ) , [ Y ]( n, e A ) = ( z ( ϕ /star ̂ Y + ) -Y -) ( n, e A ) -z 2 ( £ n ϕ /star ̂ /lscript )( e A ) + e A ( z ) 2 z + s ( e A ) + zP ( ϕ /star ̂ /lscript , U ( e A , · )) , [ Y ]( n, n ) = ( z ( ϕ /star ̂ Y + ) -Y -) ( n, n ) + n ( z ) z .</formula>
<formula><location><page_20><loc_87><loc_12><loc_91><loc_13></location>(4.12)</formula>
<text><location><page_21><loc_9><loc_88><loc_91><loc_91></location>The energy-momentum tensor τ is given by (2.48) in terms of the dual basis { q , θ A } of { n, e A } , while the purely gravitational content of the shell is ruled by the tensor</text>
<text><location><page_21><loc_9><loc_83><loc_12><loc_84></location>and</text>
<formula><location><page_21><loc_10><loc_83><loc_91><loc_88></location>Y G ( e A , e B ) def = [ Y ]( e A , e B ) + /epsilon1ρ n -1 γ ( e A , e B ) , where ρ def = ρ +2 P ( q , j ) + p ( 2 /lscript (2) + P ( q , q ) ) (4.13) { ρ, p, j } are defined as in Remark 2.16.</formula>
<formula><location><page_21><loc_9><loc_70><loc_91><loc_81></location>Proof. Applying Lemma 2.2 for ˇ V def = γ ( V, · ) = z -1 /lscript -ϕ /star ̂ /lscript (recall (3.6)) and u 0 def = /lscript ( V ) (cf. (4.6)) yields z 2 £ V γ ab = ( /lscript (2) + z 2 2 ( P ( ϕ /star ̂ /lscript , ϕ /star ̂ /lscript ) -ϕ /star ̂ /lscript (2) ) ) U ab + z · ∇ ( a ( /lscript b ) z -( ϕ /star ̂ /lscript ) b ) ) = z 2 2 ( P ( ϕ /star ̂ /lscript , ϕ /star ̂ /lscript ) -ϕ /star ̂ /lscript (2) ) U ab -1 z ( · ∇ ( a z ) /lscript b ) -z · ∇ ( a ( ϕ /star ̂ /lscript ) b ) , (4.14)</formula>
<text><location><page_21><loc_9><loc_61><loc_91><loc_72></location>where in the last step we used that · ∇ ( a /lscript b ) = -/lscript (2) U ab (cf. (2.23)). By hypothesis the matching of ( M ± , g ± ) is possible, so the data sets {N , ϕ /star ̂ γ, ϕ /star ̂ /lscript , ϕ /star ̂ /lscript (2) , ϕ /star ̂ Y + } , {N , γ, /lscript , /lscript (2) , Y + } are embedded in ( M + , g + ) with embedding ι + · ϕ and respective riggings L + , ζ + . This, together with (3.2), entails that the tensors ϕ /star ̂ Y + , Y + are related by Y + = G ( z,V ) ( ϕ /star ̂ Y + ), where { z, V } are given by (4.2). Thus (cf. (2.16), (4.1))</text>
<text><location><page_21><loc_9><loc_57><loc_55><loc_58></location>Inserting (4.14) into (4.15) yields the explicit form (4.11).</text>
<formula><location><page_21><loc_18><loc_56><loc_91><loc_62></location>Y + = zϕ /star ̂ Y + + dz ⊗ s ( ϕ /star ̂ /lscript + γ ( V, · )) + z 2 £ V γ (3.6) = zϕ /star ̂ Y + + dz z ⊗ s /lscript + z 2 £ V γ. (4.15)</formula>
<text><location><page_21><loc_9><loc_53><loc_91><loc_56></location>We now obtain the components of [ Y ] in the basis { n, e A } , for which we recall that U ( n, · ) = 0 and s ( n ) = 0. Particularizing (2.27) for θ = ϕ /star /lscript and using (4.2) gives</text>
<formula><location><page_21><loc_32><loc_43><loc_91><loc_50></location>= 1 2 £ n ( ϕ /star ̂ /lscript ) b -· ∇ b z 2 z 2 -s b z -P ac U bc ( ϕ /star ̂ /lscript ) a , (4.16)</formula>
<formula><location><page_21><loc_21><loc_47><loc_81><loc_55></location>̂ n a · ∇ ( a ( ϕ /star ̂ /lscript ) b ) = 1 2 £ n ( ϕ /star ̂ /lscript ) b + 1 2 · ∇ b (( ϕ /star ̂ /lscript )( n )) -( ϕ /star ̂ /lscript )( n ) s b -P ac U bc ( ϕ /star ̂ /lscript ) a ,</formula>
<formula><location><page_21><loc_19><loc_40><loc_91><loc_46></location>n a n b · ∇ ( a ( ϕ /star ̂ /lscript ) b ) = 1 2 £ n ( ( ϕ /star ̂ /lscript )( n ) ) -n ( z ) 2 z 2 = -n ( z ) z 2 . (4.17)</formula>
<text><location><page_21><loc_9><loc_37><loc_91><loc_42></location>Combining (4.16)-(4.17) with (4.11) yields (4.12). The components of the energy-momentum tensor being given by (2.48) is just the contents of Corollary 2.15. Finally, we prove (4.13) as follows. First, we note that the one-forms j (see Remark 2.16) and /lscript decompose in the basis { q , θ A } as</text>
<formula><location><page_21><loc_36><loc_35><loc_91><loc_37></location>j = j ( e A ) θ A , /lscript = q + /lscript ( e A ) θ A (4.18)</formula>
<text><location><page_21><loc_9><loc_29><loc_91><loc_34></location>because j ( n ) = 0 and /lscript ( n ) = 1. Also by Remark 2.16, we know that the one-form j verifies [ Y ]( n, e A ) = /epsilon1 ( j ( e A ) -p /lscript ( e A )). Thus, a direct computation based on the decomposition (2.46) of the tensor field P yields</text>
<formula><location><page_21><loc_19><loc_20><loc_81><loc_29></location>tr P [ Y ] = P ab [Y ab ] = h AB [ Y ]( e A , e B ) + 2 P ( q , θ A )[ Y ]( n, e A ) + P ( q , q )[ Y ]( n, n ) = h AB [ Y ]( e A , e B ) + 2 /epsilon1P ( q , j ( e A ) θ A -p /lscript ( e A ) θ A ) -/epsilon1pP ( q , q ) = h AB [ Y ]( e A , e B ) + 2 /epsilon1P ( q , j ) + /epsilon1p ( 2 /lscript (2) + P ( q , q ) )</formula>
<formula><location><page_21><loc_23><loc_14><loc_91><loc_19></location>h AB [ Y ]( e A , e B ) = -/epsilon1 ( ρ +2 P ( q , j ) + p ( 2 /lscript (2) + P ( q , q ) )) def = -/epsilon1ρ. (4.19)</formula>
<text><location><page_21><loc_9><loc_19><loc_91><loc_22></location>where we used that P ( θ A , θ B ) = h AB (by Lemma 2.14), P ( /lscript , · ) = -/lscript (2) n (cf. (2.8)) and (4.18) in this order. Taking into account the definition of the energy density ρ (see (2.49)), one finds</text>
<text><location><page_21><loc_9><loc_12><loc_91><loc_15></location>Now, from (2.48) it is clear that the only part of [ Y ] that does not contribute to the energy-momentum tensor is the h -traceless part of [ Y ]( e A , e B ). By Lemma 2.14, we know that h AB γ ( e A , e B ) = n -1.</text>
<text><location><page_22><loc_9><loc_90><loc_70><loc_91></location>Consequently, [ Y ]( e A , e B ) decomposes in a h -traceless and a h -trace part as</text>
<formula><location><page_22><loc_29><loc_86><loc_71><loc_89></location>[ Y ]( e A , e B ) = Y G ( e A , e B ) + h IJ [ Y ]( e I , e J ) n -1 γ ( e A , e B ) ,</formula>
<text><location><page_22><loc_9><loc_84><loc_53><loc_86></location>from where (4.13) follows at once after inserting (4.19).</text>
<text><location><page_22><loc_9><loc_76><loc_91><loc_83></location>Remark 4.5. We emphasize that we have not made any assumption on the topology of the boundaries ˜ N ± in Theorems 3.1 and 4.4 or in Lemma 4.1. The results above therefore describe the most general matching of two spacetimes across null hypersurfaces and generalize the results in [23] and [24] , where the existence of a foliation on the boundaries played an important role.</text>
<text><location><page_22><loc_9><loc_65><loc_91><loc_75></location>The gravitational/matter-energy content of the resulting null shell is given by Theorem 4.4, and the associated energy density ρ , energy flux j and pressure p are given by (2.49) . The reason why we refer to Y G ( e A , e B ) as the purely gravitational part of the shell is that only the components [ Y ]( n, n ) , [ Y ]( n, e A ) and the trace P ( θ A , θ B )[ Y ]( e A , e B ) contribute to the energy-momentum tensor τ (cf. (2.48) ). This means that even if τ vanishes identically Y G ( e A , e B ) does not need to be zero. Such a case corresponds to an impulsive gravitational wave propagating in the spacetime resulting from the matching.</text>
<text><location><page_22><loc_9><loc_55><loc_91><loc_64></location>Remark 4.6. By Lemma 2.14 we know that P ( q , · ) = 0 if and only if /lscript ( e A ) = 0 and /lscript (2) = 0 . In such case, the scalar ρ coincides with the energy density ρ of the shell. In the embedded picture, these restrictions amount to impose that the matching riggings ζ ± are null and orthogonal to the vector fields φ ± /star e A . In particular, in the setup of Section 2.3 this holds when the rigging ζ -(chosen according to (2.57) ) is null and orthogonal to the leaves of the foliation on the minus side (hence µ -A = 0 , cf. (2.3) ).</text>
<text><location><page_22><loc_9><loc_43><loc_91><loc_54></location>Remark 4.7. In Theorems 3.1 and 4.4 and Lemma 4.1, all expressions are fully explicit in terms of the diffeomorphism ϕ . The two data sets D , ̂ D are completely known (because the embeddings ι ± and the spacetimes ( M ± , g ± ) are given) and the rigging ζ + is determined by the pair { z, V } given by (4.2) in terms of ϕ . This is related to the results in [23] , [24] summarized in Section 2.3, where the whole matching depended upon the step function H and the coefficients b J I , which in turn determined the matching embedding φ + (recall (2.59) and (2.60) ) and the matching rigging ζ + (according to (2.61) ).</text>
<text><location><page_22><loc_9><loc_35><loc_91><loc_42></location>Expressions (4.12) involve the pull-back ϕ /star ̂ Y + , whose calculation can be cumbersome in general. It is more convenient to rewrite (4.12) in terms of pull-backs of scalar functions referred to the data ̂ D and objects defined with respect to D . We provide the corresponding expressions in the next lemma.</text>
<text><location><page_22><loc_9><loc_26><loc_91><loc_35></location>Lemma 4.8. Assume that conditions ( i ) -( ii ) in Theorem 3.1 hold for a pair of embedded null hypersurface data D , ̂ D and let /epsilon1 = /epsilon1 -. Define the tensors { Y -, ̂ Y + , Y + } as in Theorem 4.4, the covectors { W A } and the functions { χ ( A ) , ψ A } along N according to Corollary 4.2 and the vector field W A def = ̂ P ( W A , · ) . Let z be given by (4.2) and { n, e A } be a basis of Γ( T N ) with dual basis { q , θ A } . Then, equations (4.12) can be rewritten as</text>
<text><location><page_22><loc_9><loc_8><loc_91><loc_27></location>[ Y ]( n, n ) = 1 z ϕ /star ( ̂ Y + ( ̂ n, ̂ n ) ) -Y -( n, n ) + n ( z ) z , (4.20) [ Y ]( n, e A ) = ϕ /star ( ̂ Y + ( ̂ n, W A ) + χ ( A ) ̂ Y + ( ̂ n, ̂ n ) ) -Y -( n, e A ) -z 2 ( £ n ϕ /star ̂ /lscript ) ( e A ) + e A ( z ) 2 z + s ( e A ) + zP ( ϕ /star ̂ /lscript , U ( e A , · )) , (4.21) [ Y ]( e A , e B ) = zϕ /star ( ̂ Y + ( W A , W B ) + χ ( A ) ̂ Y + ( ̂ n, W B ) + χ ( B ) ̂ Y + ( ̂ n, W A ) + χ ( A ) χ ( B ) ̂ Y + ( ̂ n, ̂ n ) ) -Y -( e A , e B ) -ze a A e b B · ∇ ( a ( ϕ /star ̂ /lscript ) b ) + z 2 2 ( P ( ϕ /star ̂ /lscript , ϕ /star ̂ /lscript ) -ϕ /star ̂ /lscript (2) ) U ( e A , e B ) . (4.22) The energy-momentum tensor τ is given by (2.48) in terms of the dual basis { q , θ A } of { n, e A } .</text>
<text><location><page_23><loc_9><loc_86><loc_91><loc_92></location>Proof. Inserting ( ϕ /star ̂ Y + )( X,Y ) | p = ̂ Y + ( ϕ /star X,ϕ /star Y ) | ϕ ( p ) into (4.12) and using (4.8)-(4.9), equations (4.20)-(4.22) follow at once. We already know from Corollary 2.15 that τ is given by (2.48).</text>
<text><location><page_23><loc_9><loc_79><loc_91><loc_86></location>In Section 6, we shall recover the results of Proposition 2.17 by particularizing Lemma 4.8 to the case when the boundaries ˜ N ± have product topology. Lemma 4.8 therefore generalizes Proposition 2.17 to (null) boundaries of any topology, and determines the matter-energy content of any null thin shell arising from the matching of two spacetimes.</text>
<section_header_level_1><location><page_23><loc_9><loc_75><loc_34><loc_77></location>4.1 Pressure of the shell</section_header_level_1>
<text><location><page_23><loc_9><loc_61><loc_91><loc_73></location>In [23], [24] we discussed the effect and the importance of a non-zero pressure in a null thin shell. This, however, was done in very specific contexts (namely in the matching of two regions of Minkowski across a null hyperplane or for matchings across embedded AKH 0 s) and by following a non-fully geometric approach (i.e. by analyzing the effect of the pressure in some specific coordinates). Our aim in this section is to study the pressure of a completely general null shell at a fully abstract level, providing its explicit expression in terms of well-defined geometric quantities and reinforcing the geometric interpretation of [23] and [24].</text>
<text><location><page_23><loc_9><loc_57><loc_91><loc_60></location>In the following lemma we find explicit expressions for the pressure p in terms of the surface gravities of various null generators of N .</text>
<text><location><page_23><loc_9><loc_49><loc_91><loc_56></location>Lemma 4.9. Assume that conditions ( i ) -( ii ) in Theorem 3.1 hold for a pair of embedded null hypersurface data D = {N , γ, /lscript , /lscript (2) , Y -} , ̂ D = {N , ̂ γ, ̂ /lscript , ̂ /lscript (2) , ̂ Y + } and a diffeomorphism ϕ . Define z by (4.2) and introduce</text>
<text><location><page_23><loc_9><loc_46><loc_59><loc_48></location>Then, the pressure p of the corresponding null shell is given by</text>
<formula><location><page_23><loc_20><loc_45><loc_91><loc_51></location>κ n def = -Y -( n, n ) , ̂ κ n def = -̂ Y + ( ̂ n, ̂ n ) , ϕ /star κ ϕ /star n def = 1 z ( ϕ /star ̂ κ n -n ( z )) . (4.23)</formula>
<text><location><page_23><loc_9><loc_40><loc_91><loc_45></location>̂ where /epsilon1 = /epsilon1 -and V ∈ Γ( T N ) is a vector field that can be chosen at will. In particular, the pressure vanishes if and only if</text>
<formula><location><page_23><loc_20><loc_42><loc_91><loc_47></location>p = -/epsilon1 ( κ n -G ( z,V ) ( ϕ /star κ n ) ) or, equivalently p = -/epsilon1 ( κ n -ϕ /star κ ϕ /star n ) (4.24)</formula>
<formula><location><page_23><loc_28><loc_34><loc_91><loc_39></location>G ( z,V ) ( ϕ /star ̂ κ n ) = κ n or, equivalently ϕ /star κ ϕ /star n = κ n . (4.25) ) =</formula>
<formula><location><page_23><loc_10><loc_25><loc_91><loc_31></location>p = -/epsilon1 ( -1 z ϕ /star ̂ κ n + κ n + n ( z ) z ) = -/epsilon1 z ( G ( z -1 , -zV ) ( κ n ) -ϕ /star ̂ κ n ) = -/epsilon1 z ( G -1 ( z,V ) ( κ n ) -ϕ /star ̂ κ n ) . (4.26)</formula>
<text><location><page_23><loc_9><loc_29><loc_88><loc_35></location>Proof. Recall that G -1 ( z,V ) = G ( z -1 , -zV ) . We start by noticing that (2.39) implies that G -1 ( z,V ) ( κ n G ( z -1 , -zV ) ( κ n ) = z ( κ n + n ( z ) z ) . On the other hand, combining (2.49) and (4.20), it follows</text>
<text><location><page_23><loc_9><loc_20><loc_91><loc_27></location>Recalling the transformation law for /epsilon1 and p in (2.34) and (2.51) this expression can be written as -G -1 ( z,V ) ( /epsilon1p ) = G -1 ( z,V ) ( κ n ) -ϕ /star ̂ κ n . Applying G ( z,V ) on both sides one obtains the left part of (4.24). The right part of (4.24) is an immediate consequence of inserting the definition (4.23) of ϕ /star κ ϕ /star n into the first line of (4.26), while (4.25) is proven by setting p = 0 in (4.24).</text>
<text><location><page_23><loc_9><loc_10><loc_91><loc_18></location>Remark 4.10. The last expression in (4.23) defines a function κ ϕ /star n on N . However, we still need to justify this terminology. It turns out that κ ϕ /star n coincides with the surface gravity of the vector field ϕ /star n w.r.t. the hypersurface connection ̂ ∇ constructed from the data ̂ D . To prove this, we let z def = ( ϕ -1 ) /star z , so that (cf. (4.23) )</text>
<formula><location><page_23><loc_25><loc_5><loc_75><loc_11></location>κ ϕ /star n = 1 z ( ̂ κ n -( ϕ -1 ) /star ( n ( z )) ) and ( ϕ /star n )( z ) = ( ϕ -1 ) /star ( n ( z )) ,</formula>
<text><location><page_24><loc_9><loc_88><loc_91><loc_91></location>where the right part follows from ( ϕ /star n )( z ) | ϕ ( p ) = ( ϕ /star d z )( n ) | p = ( dϕ /star z )( n ) | p = n ( z ) | p = ( ϕ -1 ) /star ( n ( z )) | ϕ ( p ) . Then, the combination of (2.37) and (4.8) gives 3</text>
<text><location><page_24><loc_9><loc_75><loc_91><loc_80></location>Remark 4.11. The gauge parameter V is completely superfluous and plays no role in determining the pressure, which is only influenced by the function z given by (4.2) . We keep V in the expression to emphasize this fact.</text>
<formula><location><page_24><loc_19><loc_78><loc_81><loc_88></location>̂ ∇ ϕ /star n ϕ /star n = 1 z ̂ ∇ ̂ n ( ̂ n z ) = 1 z ( 1 z ̂ ∇ ̂ n ̂ n -̂ n ( z ) z 2 ̂ n ) = -1 z ( ̂ Y + ( ̂ n, ̂ n ) + ̂ n ( z ) z ) ϕ /star n = 1 z ( ̂ κ n -( ϕ /star n )( z ) ) ϕ /star n = 1 z ( ̂ κ n -( ϕ -1 ) /star ( n ( z )) ) ϕ /star n = κ ϕ /star n ϕ /star n.</formula>
<text><location><page_24><loc_9><loc_60><loc_91><loc_74></location>Remark 4.12. In [23] , [24] , we have introduced the notion of self-compression and self-stretching on the boundaries of the spacetimes to be matched. We have seen that this effect is completely ruled by the pressure, and that it has to do with the differences in the acceleration along the null generators of both sides. With (4.24) , we recover the same result but for the case of boundaries with any topology. Indeed, the surface gravities κ n and κ ϕ /star n verify ∇ n n = κ n n and ̂ ∇ ϕ /star n ϕ /star n = κ ϕ /star n ϕ /star n , so that the quantity -/epsilon1p is positive when κ n > ϕ /star κ ϕ /star n (namely when the 'acceleration' of n is greater than that of ϕ /star n ) and negative otherwise. The only scenario where there exists no pressure occurs when both surface gravities coincide, i.e. when the accelerations of n and ϕ /star n are the same.</text>
<section_header_level_1><location><page_24><loc_9><loc_55><loc_63><loc_57></location>5 Multiple matchings across null boundaries</section_header_level_1>
<text><location><page_24><loc_9><loc_41><loc_91><loc_53></location>We have already seen that generically there exists at most one way of matching two given spacetimes ( M ± , g ± ) (i.e. only one matching map Φ or one single diffeomorphism ϕ ). However, we have also mentioned that sometimes multiple (even infinite) matchings can be performed (e.g. when both second fundamental forms U , ̂ U vanish). In the language of (2.54), this means that given a choice of embedding φ -and matching rigging ζ -on the minus side, there exist several embeddings φ + for which the matching conditions hold, and each embedding gives rise to a unique solution for the rigging ζ + with suitable orientation.</text>
<text><location><page_24><loc_9><loc_26><loc_91><loc_40></location>In this section, our aim is to study the scenario of multiple matchings. The idea is to assume that all information about one of the matchings is known, in particular its corresponding diffeomorphism ϕ and hence the gravitational/matter-energy content. As we shall see, in these circumstances one only needs to consider a single hypersurface data set D (instead of two) and it is possible to provide explicit expressions for the jump [ Y ] and the energy-momentum tensor τ of any other shell in terms of their counterparts of the known matching. These results can be particularized to the case when the known matching gives rise to no-shell (i.e. when it is such that [ Y ] = 0). This precisely happens in all cut-and-paste constructions, where ( M ± , g ± ) are two regions of the same spacetime.</text>
<text><location><page_24><loc_9><loc_17><loc_91><loc_26></location>Our setup will be the following. We make a choice { φ -, ζ -} of embedding and rigging on the minus side and consider two matching embeddings φ + , ˜ φ + , each of them satisfying (2.54) for two riggings ζ + , ˜ ζ + respectively. We also assume that the information about one of the matchings is completely known, namely we let { φ + , ζ + } be given.</text>
<text><location><page_24><loc_9><loc_7><loc_91><loc_20></location>˜ ˜ From the spacetimes ( M ± , g ± ), we can construct two hypersurface data sets D = {N , γ, /lscript , /lscript (2) , Y -} , ̂ D = {N , ̂ γ, ̂ /lscript , ̂ /lscript (2) , ̂ Y + } and Theorem 3.1 ensures that we can find two diffeomorphisms ϕ , ˜ ϕ and two pairs { z, V } , { ˜ z, ˜ V } for which ( i )-( ii ) hold. Even more, since the pair { ˜ φ + , ˜ ζ + } is known, we can always make the choice { ι + = ˜ φ + , L + = ˜ ζ + } so that { ̂ γ, ̂ /lscript , ̂ /lscript (2) } = { γ, /lscript , /lscript (2) } and ˜ ϕ is the identity map, i.e. 3 Recall that the connections · ∇ , ∇ of a data set {N , γ, /lscript , /lscript (2) , Y } verify ∇ X Z = · ∇ X Z -Y ( X,Z ) n , ∀ X,Z ∈ Γ( T N ).</text>
<text><location><page_25><loc_9><loc_86><loc_91><loc_91></location>˜ ϕ = I N . In these circumstances, using (2.7)-(2.8) in (4.2) yields ˜ z = 1 and ˜ V = 0. Making the same choice of { ι + , L + } for the matching of ϕ transforms (3.2) into</text>
<formula><location><page_25><loc_25><loc_86><loc_91><loc_87></location>G ( z,V ) ( ϕ /star γ ) = γ, G ( z,V ) ( ϕ /star /lscript ) = /lscript , G ( z,V ) ( ϕ /star /lscript (2) ) = /lscript (2) , (5.1)</formula>
<formula><location><page_25><loc_9><loc_79><loc_91><loc_85></location>and forces the embedding φ + to be given by ˜ φ + · ϕ ≡ φ + . Equations (3.5)-(3.7) now read ϕ /star γ = γ, ϕ /star /lscript = /lscript z -γ ( V, · ) , ϕ /star /lscript (2) = /lscript (2) z 2 -2 /lscript ( V ) z + γ ( V, V ) , (5.2)</formula>
<text><location><page_25><loc_9><loc_78><loc_63><loc_79></location>while the expressions (4.2) for the gauge parameters { z, V } become</text>
<formula><location><page_25><loc_26><loc_74><loc_91><loc_77></location>z = 1 ( ϕ /star /lscript )( n ) , V = -P ( ϕ /star /lscript , · ) + P ( ϕ /star /lscript , ϕ /star /lscript ) -ϕ /star /lscript (2) 2( ϕ /star /lscript )( n ) n. (5.3)</formula>
<text><location><page_25><loc_9><loc_65><loc_91><loc_73></location>It is important to emphasize that whereas ˜ ϕ = I N forces the metric parts of D , ̂ D to be the same, the tensors Y -, ̂ Y + do not coincide in general. We let [ ˜ Y ] def = ̂ Y + -Y -, [ Y ] def = Y + -Y -be the jumps codifying the gravitational/matter-energy content of the null shells associated to ˜ ϕ and ϕ respectively. Then, by (4.11) we know that [ Y ] must be given by</text>
<text><location><page_25><loc_9><loc_58><loc_86><loc_63></location>The jumps [ Y ], [ ˜ Y ] can actually be related, as we shall see next. Indeed, by defining the tensor</text>
<formula><location><page_25><loc_20><loc_61><loc_91><loc_67></location>[Y ab ] = z ( ( ϕ /star ̂ Y + ) ab + z 2 ( P ( ϕ /star /lscript , ϕ /star /lscript ) -ϕ /star /lscript (2) ) U ab -· ∇ ( a ( ϕ /star /lscript ) b ) ) -Y -ab . (5.4)</formula>
<text><location><page_25><loc_9><loc_56><loc_37><loc_58></location>expression (5.4) can be rewritten as</text>
<formula><location><page_25><loc_23><loc_50><loc_91><loc_56></location>[Y ab ] = Y ab + z 2 2 ( P ( ϕ /star /lscript , ϕ /star /lscript ) -ϕ /star /lscript (2) ) U ab -z · ∇ ( a ( ϕ /star /lscript ) b ) +[ ˜ Y ab ] . (5.6)</formula>
<formula><location><page_25><loc_42><loc_56><loc_91><loc_61></location>Y def = zϕ /star ̂ Y + -̂ Y + , (5.5)</formula>
<text><location><page_25><loc_9><loc_49><loc_91><loc_52></location>Moreover, a direct calculation shows that the components (4.12) of [ Y ] in a basis { n, e A } of Γ( T N ) can be expressed in terms of Y as</text>
<text><location><page_25><loc_53><loc_44><loc_54><loc_46></location>e</text>
<text><location><page_25><loc_54><loc_44><loc_55><loc_45></location>A</text>
<text><location><page_25><loc_55><loc_44><loc_56><loc_46></location>(</text>
<text><location><page_25><loc_56><loc_44><loc_57><loc_46></location>z</text>
<text><location><page_25><loc_57><loc_44><loc_57><loc_46></location>)</text>
<text><location><page_25><loc_54><loc_42><loc_55><loc_44></location>2</text>
<text><location><page_25><loc_55><loc_42><loc_56><loc_44></location>z</text>
<text><location><page_25><loc_39><loc_42><loc_40><loc_44></location>2</text>
<text><location><page_25><loc_58><loc_43><loc_59><loc_45></location>+</text>
<text><location><page_25><loc_60><loc_43><loc_61><loc_45></location>s</text>
<text><location><page_25><loc_61><loc_43><loc_61><loc_45></location>(</text>
<text><location><page_25><loc_61><loc_43><loc_62><loc_45></location>e</text>
<text><location><page_25><loc_62><loc_43><loc_63><loc_44></location>A</text>
<text><location><page_25><loc_64><loc_43><loc_66><loc_45></location>) +</text>
<text><location><page_25><loc_66><loc_43><loc_69><loc_45></location>zP</text>
<text><location><page_25><loc_69><loc_43><loc_70><loc_45></location>(</text>
<text><location><page_25><loc_70><loc_43><loc_71><loc_45></location>ϕ</text>
<text><location><page_25><loc_72><loc_43><loc_72><loc_45></location>/lscript</text>
<text><location><page_25><loc_72><loc_43><loc_73><loc_45></location>,</text>
<text><location><page_25><loc_73><loc_43><loc_75><loc_45></location>U</text>
<text><location><page_25><loc_75><loc_43><loc_76><loc_45></location>(</text>
<text><location><page_25><loc_76><loc_43><loc_76><loc_45></location>e</text>
<text><location><page_25><loc_76><loc_43><loc_77><loc_44></location>A</text>
<text><location><page_25><loc_78><loc_43><loc_78><loc_45></location>,</text>
<text><location><page_25><loc_78><loc_43><loc_79><loc_45></location>·</text>
<text><location><page_25><loc_79><loc_43><loc_80><loc_45></location>))</text>
<text><location><page_25><loc_80><loc_43><loc_81><loc_45></location>,</text>
<text><location><page_25><loc_87><loc_43><loc_91><loc_45></location>(5.8)</text>
<text><location><page_25><loc_9><loc_34><loc_91><loc_39></location>Inserting (5.7)-(5.9) into (2.48) gives us the relation between the energy-momentum tensors τ , ˜ τ of the two shells. Specifically, for the dual basis { q , θ A } of { n, e A } one finds (recall that h AB def = γ ( e A , e B ))</text>
<formula><location><page_25><loc_8><loc_37><loc_91><loc_49></location>[ Y ]( n, n ) = Y ( n, n ) + [ ˜ Y ]( n, n ) + n ( z ) z , (5.7) [ Y ]( n, e A ) = Y ( n, e A ) + [ ˜ Y ]( n, e A ) -z ( £ n ϕ /star /lscript )( e A ) + [ Y ]( e A , e B ) = Y ( e A , e B ) + [ ˜ Y ]( e A , e B ) + z 2 2 ( P ( ϕ /star /lscript , ϕ /star /lscript ) -ϕ /star /lscript (2) ) U ( e A , e B ) -ze a A e b B · ∇ ( a ( ϕ /star /lscript ) b ) . (5.9)</formula>
<formula><location><page_25><loc_11><loc_29><loc_89><loc_36></location>τ ( q , q ) = ˜ τ ( q , q ) -/epsilon1 h AB ( Y ( e A , e B ) + z 2 2 ( P ( ϕ /star /lscript , ϕ /star /lscript ) -ϕ /star /lscript (2) ) U ( e A , e B ) -ze a A e b B · ∇ ( a ( ϕ /star /lscript ) b ) ) ,</formula>
<text><location><page_25><loc_9><loc_16><loc_91><loc_24></location>The results (5.7)-(5.10) turn out to be of particular interest when one of the matchings of ( M ± , g ± ) gives rise to no shell. In order to see this, let us assume that this is the case and take ˜ ϕ to be the diffeomorphism corresponding to the no-shell matching. Then, [ ˜ Y ] = 0 (i.e. ̂ Y + = Y -) holds, which means that the tensor Y is given by (cf. (5.5))</text>
<formula><location><page_25><loc_9><loc_22><loc_91><loc_32></location>τ ( q , θ A ) = ˜ τ ( q , θ A ) + /epsilon1 h AB ( Y ( n, e B ) -z 2 ( £ n ϕ /star /lscript )( e B ) + e B ( z ) 2 z + s ( e B ) + zP ( ϕ /star /lscript , U ( e B , · )) ) , (5.10) τ ( θ A , θ B ) = ˜ τ ( θ A , θ B ) -/epsilon1 h AB ( Y ( n, n ) + n ( z ) z ) .</formula>
<formula><location><page_25><loc_42><loc_15><loc_91><loc_17></location>Y = zϕ /star Y --Y -. (5.11)</formula>
<formula><location><page_25><loc_9><loc_9><loc_91><loc_15></location>Setting [ ˜ Y ] = 0 in equations (5.7)-(5.9) yields [ Y ]( n, n ) = Y ( n, n ) + n ( z ) z , (5.12)</formula>
<text><location><page_25><loc_71><loc_44><loc_71><loc_45></location>/star</text>
<formula><location><page_26><loc_15><loc_88><loc_91><loc_91></location>[ Y ]( n, e A ) = Y ( n, e A ) -z 2 ( £ n ϕ /star /lscript )( e A ) + e A ( z ) 2 z + s ( e A ) + zP ( ϕ /star /lscript , U ( e A , · )) , (5.13)</formula>
<text><location><page_26><loc_9><loc_75><loc_91><loc_85></location>Consequently, when a no-shell matching is possible, the jump [ Y ] corresponding to any other possible matching is given by (5.12)-(5.14) in terms of the data fields { γ, /lscript , /lscript (2) , Y -} and the diffeomorphism ϕ . In other words, knowing the information about the no-shell matching automatically allows one to obtain the gravitational/matter-energy content of the remaining matchings by simply determining ϕ . In particular, there is no need to compute the new matching rigging ζ + or the tensor Y + to determine the shell properties. One simple needs to compute the right-hand sides of (5.12)-(5.14) using (5.11).</text>
<formula><location><page_26><loc_14><loc_84><loc_91><loc_89></location>[ Y ]( e A , e B ) = Y ( e A , e B ) + z 2 2 ( P ( ϕ /star /lscript , ϕ /star /lscript ) -ϕ /star /lscript (2) ) U ( e A , e B ) -ze a A e b B · ∇ ( a ( ϕ /star /lscript ) b ) . (5.14)</formula>
<text><location><page_26><loc_9><loc_63><loc_91><loc_74></location>We emphasize that (5.12)-(5.14) apply, in particular, when ( M ± , g ± ) are two regions of the same spacetime ( M , g ) and more than one matching can be performed. Then, the existence of a no-shell matching is always guaranteed, as one can always recover the full spacetime ( M , g ) from the matching of ( M ± , g ± ). This in fact occurs in all cut-and-paste constructions, which means that (5.12) -(5.14) provide the matter content of a null shell generated by any cut-and-paste matching procedure, as long as the two regions ( M ± , g ± ) of ( M , g ) can be pasted in more than one way .</text>
<text><location><page_26><loc_9><loc_51><loc_91><loc_63></location>We conclude this section by discussing a particular situation of interest, namely the case when a null hypersurface data D = {N , γ, /lscript , /lscript (2) , Y -} can be embedded in two spacetimes ( M ± , g ± ) with embeddings ι ± (such that ι ± ( N ) are boundaries of M ± ) and riggings L ± with the appropriate orientation. This means that ( M ± , g ± ) can be matched so that the resulting spacetime contains no shell (because Y -is the same for both spacetimes). We assume, in addition, that D admits a vector field ξ ∈ Γ( T N ) with the property £ ξ γ = 0. The vector ξ defines a (local) one-parameter group of transformations { ϕ t } of N satisfying</text>
<formula><location><page_26><loc_47><loc_49><loc_91><loc_50></location>ϕ /star t γ = γ. (5.15)</formula>
<text><location><page_26><loc_9><loc_34><loc_91><loc_48></location>We now prove that, for each value of t , the diffeomorphism ϕ t gives rise to a matching. First, we define gauge parameters { z, V } according to (5.3) for ϕ = ϕ t . Then, it is immediate to check that (5.1) holds for ϕ = ϕ t and that z > 0 (because ϕ t depends continuously on t and ( ϕ /star t =0 /lscript )( n ) = /lscript ( n ) = 1). Therefore, conditions ( i ) and ( ii ) in Theorem 3.1 are both fulfilled (notice that, since L ± are matching riggings, one points inwards and the other outwards, so ( ii ) is just z > 0) and indeed each ϕ t corresponds to a different matching. The jump [ Y ] def = Y + -Y -where Y + def = 1 2 ϕ /star t ( ( ι + ) /star ( £ ζ + g + ) ) (and ζ + is given by (4.3)) rules the gravitational/matter-energy content of the resulting shell. The vector field ξ generates a multitude of new shells. The construction is further simplified when, in addtion to (5.15), it holds</text>
<formula><location><page_26><loc_45><loc_31><loc_91><loc_33></location>ϕ /star t Y -= Y -. (5.16)</formula>
<text><location><page_26><loc_9><loc_29><loc_25><loc_31></location>Then (5.11) implies</text>
<formula><location><page_26><loc_43><loc_28><loc_91><loc_29></location>Y = ( z -1) Y -, (5.17)</formula>
<text><location><page_26><loc_9><loc_21><loc_91><loc_26></location>which simplifies the expressions (5.12)-(5.14) considerably. One may wonder what is the final result when, in addition, ξ is the restriction to N of a Killing vector field ξ on M -(i.e. ι -/star ξ = ξ ) and £ ξ L -= 0 is fulfilled (so that (5.15) and (5.16) hold). It is straightforward to see that</text>
<formula><location><page_26><loc_40><loc_19><loc_91><loc_21></location>ϕ /star t /lscript = /lscript , ϕ /star t /lscript (2) = /lscript (2) , (5.18)</formula>
<text><location><page_26><loc_9><loc_10><loc_91><loc_18></location>which combined with (5.3) means that z = 1, and V = 0, so Y = 0 (cf. (5.17)). Moreover, one can easily check that the terms in the right-hand side of (5.12)-(5.14) cancel out. Thus, the procedure gives rise to another no-shell matching, as one would expect because the transformation induced by ξ does not affect in any geometric way the spacetime ( M -, g -). This constitutes a non-trivial consistency check of equations (5.12)-(5.14).</text>
<section_header_level_1><location><page_27><loc_9><loc_90><loc_67><loc_91></location>6 Null boundaries with product topology S × R</section_header_level_1>
<text><location><page_27><loc_9><loc_79><loc_91><loc_88></location>In order to connect the results in this paper with those from [23], [24] (see Section 2.3), we now consider the case when the boundaries of the spacetimes to be matched can be foliated by cross-sections. In particular, we shall construct a step function H and provide explicit expressions for the gauge parameters { z, V } (cf. (4.2)). The results for the jump [ Y ] will be then compared with their counterparts from Proposition 2.17.</text>
<text><location><page_27><loc_9><loc_67><loc_91><loc_78></location>Our setup for the present section is the following. We consider two spacetimes ( M ± , g ± ) with null boundaries ˜ N ± and assume that ˜ N ± have product topology S ± × R , where S ± are spacelike crosssections and the null generators are along R . We select two future null generators k ± ∈ Γ( T M ± ) | ˜ N ± of ˜ N ± and two cross-sections S ± 0 ⊂ ˜ N ± . We then construct foliation functions v ± ∈ F ( ˜ N ± ) by solving k ± ( v ± ) = 1 with initial values v ± | S ± 0 = 0. Finally, the riggings L ± are fixed by the conditions of being orthogonal to the respective leaves { v ± = const } , null and scaled to satisfy µ ± 1 def = g ± ( L ± , k ± ) = 1.</text>
<text><location><page_27><loc_9><loc_58><loc_91><loc_67></location>We assume that ( M ± , g ± ) can be matched, so that conditions ( i )-( ii ) in Theorem 3.1 are fulfilled for a diffeomorphism ϕ : N -/shortrightarrow N verifying (4.1). This allows us to take two embeddings ι ± : N ↪ -/shortrightarrow M ± and construct the hypersurface data sets D = {N , γ, /lscript , /lscript (2) , Y -} , ̂ D = {N , ̂ γ, ̂ /lscript , ̂ /lscript (2) , ̂ Y + } according to (3.3)-(3.4). We also introduce the functions</text>
<formula><location><page_27><loc_27><loc_57><loc_91><loc_59></location>λ def = ( ι -) /star ( v -) , v def = ( ι + ) /star ( v + ) , and H def = ϕ /star v (6.1)</formula>
<text><location><page_27><loc_9><loc_52><loc_91><loc_57></location>on N . Since by construction ι -/star ( n ) = k -and ι + /star ( ̂ n ) = k + (recall (2.20)), it is immediate to check that { λ, v } are foliation functions of N . Note that, also by construction, the data satisfies</text>
<text><location><page_27><loc_9><loc_49><loc_48><loc_50></location>which has the following immediate consequences</text>
<text><location><page_27><loc_9><loc_37><loc_91><loc_48></location>n ( λ ) = 1 , F = 0 , s = 0 , ̂ n ( v ) = 1 , ̂ F = 0 , ̂ s = 0 . (6.3) We now select vector fields { e A } tangent to the leaves { λ = const . } so that { n, e A } is a basis of Γ( T N ) satisfying [ n, e A ] = 0. As before, we let h be induced metric on { λ = const . } and ∇ h for its Levi-Civita derivative. In particular h AB def = γ ( e A , e B ) and we note that, for any f ∈ F ( N ), we can write e A ( f ) also as ∇ h A f . The pull-back of /lscript to the leaves of constant λ is zero, so /lscript A = ψ A = 0. This, together with /lscript (2) = 0 and (2.47), means that P = h AB e A ⊗ e B . Observe also that</text>
<formula><location><page_27><loc_29><loc_48><loc_91><loc_53></location>/lscript = dλ, /lscript (2) = 0 , ̂ /lscript = dv, ̂ /lscript (2) = 0 , (6.2)</formula>
<text><location><page_27><loc_9><loc_33><loc_34><loc_34></location>which in particular means that</text>
<formula><location><page_27><loc_33><loc_32><loc_91><loc_37></location>ϕ /star ̂ /lscript = ϕ /star dv = d ( ϕ /star v ) = dH, ϕ /star ̂ /lscript (2) = 0 , (6.4)</formula>
<text><location><page_27><loc_9><loc_28><loc_70><loc_30></location>Inserting these properties in (4.2) fixes the matching gauge parameters to be</text>
<formula><location><page_27><loc_35><loc_27><loc_91><loc_32></location>P ( ϕ /star ̂ /lscript , · ) = P ( dH, · ) = h AB ( ∇ h A H ) e B . (6.5)</formula>
<formula><location><page_27><loc_31><loc_24><loc_91><loc_28></location>z = 1 n ( H ) , V = h AB ∇ h A H ( ∇ h B H 2 n ( H ) n -e B ) . (6.6)</formula>
<text><location><page_27><loc_34><loc_18><loc_35><loc_20></location>n,</text>
<text><location><page_27><loc_9><loc_8><loc_91><loc_13></location>̂ ̂ Proof. Define the functions u def = ( ϕ -1 ) /star ( n ( H )) and χ ( A ) def = ( ϕ -1 ) /star ( e A ( H )), so that (6.7)-(6.8) can be</text>
<text><location><page_27><loc_9><loc_11><loc_91><loc_24></location>The push-forward vector fields { ϕ /star n, ϕ /star e A } can also be computed in terms of the function H and the vector fields W A def = ̂ P ( W A , · ). The result is an easy consequence of Corollary 4.2 and reads ϕ /star n = ( ϕ -1 ) /star ( n ( H )) ̂ (6.7) ϕ /star e A = W A +( ϕ -1 ) /star ( e A ( H )) ̂ n. (6.8) Observe that { W A } are tangent to the leaves { v = const. } (because by Corollary 4.2 we know that 0 = ̂ P ( W A , ̂ /lscript ) = ̂ /lscript ( W A ) = W A ( v )). Let us now prove that ̂ n and W A commute. Lemma 6.1. The vector fields n and W A satisfy [ n, W A ] = 0 .</text>
<text><location><page_28><loc_9><loc_90><loc_53><loc_91></location>written as n = u -1 ϕ /star n and W A = ϕ /star e A -χ ( A ) n . Thus,</text>
<text><location><page_28><loc_9><loc_80><loc_91><loc_85></location>where in the last equality we used [ n, e A ] = 0 and ̂ n = u -1 ϕ /star n . To prove the claim we just need to show that the last parenthesis is zero. Indeed,</text>
<formula><location><page_28><loc_16><loc_83><loc_91><loc_91></location>̂ ̂ [ ̂ n, W A ] = [ u -1 ϕ /star n, ϕ /star e A ] -[ ̂ n, χ ( A ) ̂ n ] = u -1 ϕ /star ([ n, e A ]) + u -2 ϕ /star e A ( u ) ϕ /star n -̂ n ( χ ( A ) ) ̂ n = u -2 ( ϕ /star e A ( u ) -ϕ /star n ( χ ( A ) ) ) ϕ /star n, (6.9)</formula>
<formula><location><page_28><loc_13><loc_77><loc_87><loc_81></location>ϕ /star e A ( u ) -ϕ /star n ( χ ( A ) ) = ( d u )( ϕ /star e A ) -( dχ ( A ) )( ϕ /star n ) = ϕ /star ( d u )( e A ) -ϕ /star ( dχ ( A ) )( n ) = ( dϕ /star u )( e A ) -( dϕ /star χ ( A ) )( n ) = e A ( n ( H )) -n ( e A ( H )) = [ e A , n ]( H ) = 0 .</formula>
<text><location><page_28><loc_9><loc_71><loc_88><loc_73></location>By Remark 4.3 we also know that { n, W A } constitute a basis of Γ( T N ) and hence the vector fields</text>
<text><location><page_28><loc_9><loc_63><loc_91><loc_68></location>form basis of Γ( T ˜ N ± ) respectively. Inserting (6.7)-(6.8) into (6.10) and using again that ι + /star ( ̂ n ) = k + , one obtains e + 1 = n ( H ) k + , e + A = e A ( H ) k + + ι + /star ( W A ) , (6.11)</text>
<formula><location><page_28><loc_24><loc_68><loc_91><loc_73></location>̂ { e -1 def = ι -/star n, e -A def = ι -/star e A } , { e + 1 def = ι + /star ( ϕ /star n ) , e + A def = ι + /star ( ϕ /star e A ) } (6.10)</formula>
<text><location><page_28><loc_9><loc_55><loc_91><loc_62></location>where for simplicity we have dropped pull-backs affecting functions. Given that { ι + /star W A } are linearly independent and tangent to the leaves { v + = const. } ⊂ ˜ N + , they can be decomposed in a basis { L + , k + , v + A } of Γ( T M + ) | ˜ N + satisfying (2.2) as ι + /star W A = b B A v + B , with { b B A } defining an invertible matrix. Moreover, b B A are constant along the null generators as a consequence of Lemma 6.1:</text>
<text><location><page_28><loc_40><loc_54><loc_42><loc_55></location>+</text>
<text><location><page_28><loc_43><loc_54><loc_44><loc_55></location>B</text>
<text><location><page_28><loc_45><loc_54><loc_46><loc_55></location>+</text>
<text><location><page_28><loc_51><loc_54><loc_52><loc_55></location>+</text>
<text><location><page_28><loc_52><loc_53><loc_52><loc_54></location>(</text>
<text><location><page_28><loc_52><loc_53><loc_53><loc_54></location>b</text>
<text><location><page_28><loc_53><loc_54><loc_54><loc_55></location>B</text>
<text><location><page_28><loc_53><loc_53><loc_54><loc_54></location>A</text>
<text><location><page_28><loc_54><loc_53><loc_55><loc_54></location>)</text>
<text><location><page_28><loc_55><loc_53><loc_56><loc_54></location>v</text>
<text><location><page_28><loc_56><loc_54><loc_57><loc_55></location>+</text>
<text><location><page_28><loc_56><loc_53><loc_57><loc_54></location>B</text>
<text><location><page_28><loc_61><loc_53><loc_65><loc_54></location>⇐⇒</text>
<text><location><page_28><loc_69><loc_53><loc_70><loc_54></location>k</text>
<text><location><page_28><loc_71><loc_53><loc_72><loc_54></location>(</text>
<text><location><page_28><loc_72><loc_53><loc_73><loc_54></location>b</text>
<text><location><page_28><loc_73><loc_54><loc_74><loc_55></location>B</text>
<text><location><page_28><loc_73><loc_53><loc_74><loc_54></location>A</text>
<text><location><page_28><loc_74><loc_53><loc_78><loc_54></location>) = 0</text>
<text><location><page_28><loc_78><loc_53><loc_79><loc_54></location>.</text>
<formula><location><page_28><loc_28><loc_46><loc_91><loc_51></location>ζ + = 1 n ( H ) ( L + -h AB ∇ h A H ( ι + /star ( W A ) + ∇ h B H 2 k + )) , (6.12)</formula>
<text><location><page_28><loc_9><loc_50><loc_91><loc_54></location>0 = [ ι /star ( ̂ n ) , ι /star ( W A )] = [ k , b A v A ] = k The matching rigging ζ + , obtained by inserting (6.6) into (4.3) and using (6.7)-(6.8), (6.10)-(6.11), reads</text>
<text><location><page_28><loc_9><loc_43><loc_91><loc_46></location>which one easily checks to be the same as (2.61) simply by noting that (in the notation of Section 2.3) our choice of L ± entails µ ± 1 = 1, µ ± A = 0 and that ι + /star W A = b B A v + B gives h AB = h IJ + ( b -1 ) A I ( b -1 ) B J .</text>
<text><location><page_28><loc_9><loc_41><loc_68><loc_42></location>The expressions for [ Y ] are obtained as a particular case of Theorem 4.4.</text>
<text><location><page_28><loc_9><loc_34><loc_91><loc_39></location>Theorem 6.2. In the setup and conditions of Theorem 4.4 suppose further that the boundaries ˜ N ± can be foliated by cross-sections and define λ, v, H ∈ F ( N ) as in (6.1) . Let h be the induced metric and ∇ h the corresponding Levi-Civita covariant derivative on the leaves { λ = const. } ⊂ N . Then,</text>
<text><location><page_28><loc_9><loc_27><loc_91><loc_30></location>Let { e A } be vector fields in N such that { n, e A } is a basis adapted to the foliation { λ = const. } and define W A by means of (6.8) . Then the components the jump [ Y ] can be written as</text>
<formula><location><page_28><loc_22><loc_28><loc_91><loc_34></location>[Y ab ] = 1 n ( H ) ( ( ϕ /star ̂ Y + ) ab + h AB ( ∇ h A H )( ∇ h B H ) 2 n ( H ) U ab -· ∇ a · ∇ b H ) -Y -ab . (6.13)</formula>
<formula><location><page_28><loc_11><loc_17><loc_91><loc_26></location>[ Y ]( n, n ) = n ( H ) ϕ /star ( ̂ Y + ( ̂ n, ̂ n ) ) -Y -( n, n ) -n ( n ( H )) n ( H ) , (6.14) [ Y ]( n, e A ) = ϕ /star ( ̂ Y + ( ̂ n, W A ) ) +( ∇ h A H ) ϕ /star ( ̂ Y + ( ̂ n, ̂ n ) ) -Y -( n, e A ) -∇ h A ( n ( H )) n ( H ) + h IJ ∇ h I H n ( H ) U ‖ ( e A , e J ) , (6.15)</formula>
<formula><location><page_28><loc_21><loc_8><loc_91><loc_13></location>-n ( H ) Y -( e A , e B ) + h IJ ( ∇ h I H )( ∇ h J H ) 2 n ( H ) U ‖ ( e A , e B ) -∇ h A ∇ h B H ) . (6.16)</formula>
<formula><location><page_28><loc_10><loc_10><loc_90><loc_17></location>[ Y ]( e A , e B ) = 1 n ( H ) ( ϕ /star ( ̂ Y + ( W A , W B ) ) +2( ∇ h ( A H ) ϕ /star ( ̂ Y + ( ̂ n, W B ) ) ) +( ∇ h A H )( ∇ h B H ) ϕ /star ( ̂ Y + ( ̂ n, ̂ n ) )</formula>
<text><location><page_28><loc_70><loc_54><loc_71><loc_55></location>+</text>
<text><location><page_29><loc_9><loc_83><loc_91><loc_91></location>Proof. Equation (6.13) follows at once after inserting (6.4)-(6.6) into (4.11). To obtain (6.14)-(6.16), it suffices to particularize (4.20)-(4.22) for z -1 = n ( H ), ϕ /star ̂ /lscript = dH , χ ( A ) = ( ϕ -1 ) /star ( e A ( H )), ϕ /star ̂ /lscript (2) = 0, s = 0 and P ( ϕ /star ̂ /lscript , · ) = h AB ( ∇ h A H ) e B and notice that £ n ( ϕ /star ̂ /lscript ) = £ n dH = d ( n ( H )), as well as e a A e b B · ∇ a · ∇ b H = ∇ h A ∇ h B H (see (2.53)).</text>
<text><location><page_29><loc_9><loc_79><loc_91><loc_82></location>Before establishing the connection between (6.14)-(6.16) and the corresponding expressions in Proposition 2.17 we need to relate hypersurface data quantities with the tensors defined in (2.1).</text>
<text><location><page_29><loc_9><loc_71><loc_91><loc_78></location>Lemma 6.3. Let {N , γ, /lscript , /lscript (2) , Y } be { φ, ζ } -embedded in ( M , g ) , k := φ /star n the corresponding null generator and ˜ κ k its surface gravity. Consider a transverse submanifold S ⊂ N and assume that the gauge is such that the rigging ζ is null and orthogonal to φ ( S ) . Then, for any basis { e A } of Γ( TS ) it holds (we identify scalars and vectors with their images on φ ( N ) )</text>
<formula><location><page_29><loc_10><loc_64><loc_76><loc_69></location>( a ) ˜ κ k = -Y ( n, n ) , ( b ) σ ζ ( e A ) = Y ( e A , n ) + F ( e A , n ) , ( c ) ˜ K k ( e A , e B ) = U ( e A , e B ) . ( d ) Θ ζ ( e ( A , e B ) ) = Y ( e A , e B ) ,</formula>
<text><location><page_29><loc_9><loc_60><loc_91><loc_63></location>Remark 6.4. This result is a particular case of a much more general analysis on the geometry of embedded submanifold in a hypersurface data set carried out in [28] . We include the proof for completeness.</text>
<text><location><page_29><loc_15><loc_56><loc_91><loc_58></location>Claim ( a ) follows at once from (2.14) and (2.38) (note that here ν = k ). To prove ( b ) we compute</text>
<formula><location><page_29><loc_9><loc_54><loc_75><loc_58></location>Proof. σ ζ ( e A ) (2.1) = -g ( ∇ e A k, ζ ) = g ( ∇ e A ζ, k ) (2.25) = Y ( e A , n ) + F ( e A , n ) .</formula>
<text><location><page_29><loc_9><loc_52><loc_70><loc_53></location>Item ( c ) has already been stated after definition (2.21) and ( d ) follows from</text>
<formula><location><page_29><loc_24><loc_46><loc_76><loc_51></location>Y ( e A , e B ) = 1 2 ( £ ζ g )( e A , e B ) = g ( ∇ e ( A ζ, e B ) ) (2.1) = = Θ ζ ( e ( A , e B ) ) .</formula>
<text><location><page_29><loc_9><loc_37><loc_91><loc_45></location>We are now in a position where the comparison can be made. We identify the vector fields { v -A } introduced in Section 2.3 with the push-forwards of { e A } , hence µ -1 = 1 and µ -A = 0. On the other hand, µ + 1 = 1 and µ + A = g + ( L + , v + A ) = ( b -1 ) B A ̂ /lscript ( W B ) = ( b -1 ) B A W B ( v ) = 0, so the covector q defined in Proposition 2.17 is simply q A = -∇ h A H . The vector X a in (2.65) is in turn given by</text>
<formula><location><page_29><loc_32><loc_34><loc_91><loc_37></location>X 1 = h AB ∇ h A H ∇ h B H 2 n ( H ) , X A = -h AB ∇ h B H. (6.17)</formula>
<text><location><page_29><loc_9><loc_32><loc_40><loc_33></location>Thus, expressions (2.66)-(2.68) become</text>
<formula><location><page_29><loc_12><loc_28><loc_91><loc_32></location>[ Y ]( n, n ) = -n ( H ) κ + k + + κ -k --n ( n ( H )) n ( H ) , (6.18)</formula>
<formula><location><page_29><loc_11><loc_15><loc_91><loc_27></location>˜ [ Y ]( e I , e J ) = 1 n ( H ) ( 2( ∇ h ( I H ) σ + L ( W J ) ) -˜ κ + k + ( ∇ h I H )( ∇ h J H ) + Θ L + ( W ( I , W J ) ) -n ( H ) Θ L -( v -( I , v -J ) ) + γ AB ∇ h A H ∇ h B H 2 n ( H ) ˜ K k -( v -I , v -J ) -∇ h I ∇ h J H ) . (6.20)</formula>
<formula><location><page_29><loc_11><loc_22><loc_91><loc_31></location>˜ ˜ [ Y ]( n, e J ) = σ + L ( W J ) -σ -L ( v -J ) -( ∇ h J H ) κ + k + -∇ h J ( n ( H )) n ( H ) + h LB ∇ h B H n ( H ) ˜ K k -( v -J , v -L ) , (6.19)</formula>
<text><location><page_29><loc_9><loc_11><loc_91><loc_17></location>Particularizing Lemma 6.3 to the sections { λ = const } of D (with basis e A ) and the sections { v = const } of ̂ D (with basis W A ), and recalling that F = ̂ F = 0 (see (6.3)), it is straightforward to check that (6.18)(6.20) coincide with (6.14)-(6.16).</text>
<section_header_level_1><location><page_30><loc_9><loc_90><loc_74><loc_91></location>7 Cut-and-paste matching: (anti-)de Sitter spacetime</section_header_level_1>
<text><location><page_30><loc_9><loc_77><loc_91><loc_88></location>We have already mentioned that (5.12)-(5.14) hold for the specific case when the two spacetimes to be matched are actually two regions of the same spacetime (and more than one matching is allowed). In this section, our aim is to provide an example of a cut-and-paste construction, namely the matching of two regions of a constant-curvature spacetime across a totally geodesic null hypersurface. For previous works on the cut-and-paste construction describing non-expanding impulsive gravitational waves in constant curvature backgrounds we refer e.g. to [39], [37], [17] [43], [40] and references therein.</text>
<text><location><page_30><loc_9><loc_68><loc_91><loc_76></location>In any constant curvature spacetime ( M , g ) there exists only one totally geodesic null hypersurface up to isometries (see e.g. [14], [32]). We denote one such hypersurface by ˜ N . Then, one can always construct coordinates {U , V , x A } adapted to ˜ N so that the metric is conformally flat and ˜ N def = {U = 0 } , namely</text>
<text><location><page_30><loc_9><loc_56><loc_91><loc_66></location>Here Λ stands for the cosmological constant, so Λ = 0, Λ > 0, Λ < 0 correspond to Minkowski, de Sitter and anti-de Sitter spacetimes respectively. When Λ ≤ 0, the coordinates {U , V , x A } cover a whole neighbourhood of ˜ N . However, for the de Sitter case one needs to remove one generator of ˜ N because the topology of ˜ N is S n × R while stereographic coordinates only cover the sphere minus one point. In this section, we will analyze the three cases Λ = 0, Λ < 0 and Λ > 0 at once with the matching formalism introduced before.</text>
<formula><location><page_30><loc_14><loc_65><loc_91><loc_70></location>g = g Mk µ 2 , where g Mk = -2 d U d V + δ AB dx A dx B , µ def = 1 + Λ 12 ( δ AB x A x B -2 UV ) . (7.1)</formula>
<text><location><page_30><loc_9><loc_48><loc_91><loc_56></location>The induced metric on ˜ N reads ds 2 ˜ N = ( 1 + Λ 12 δ AB x A x B ) -2 δ AB dx A dx B , and obviously the topology of ˜ N is S × R , S being a spacelike section and the null generators being along R . Therefore, all results from Section 6 can be applied.</text>
<text><location><page_30><loc_9><loc_42><loc_91><loc_49></location>Let us construct hypersurface data associated to ˜ N . Since ˜ N is embedded on ( M , g ), there exists an abstract manifold N and an embedding ι such that ι ( N ) = ˜ N . We can select ι to be as trivial as possible by constructing coordinates { λ, y A } on N so that</text>
<text><location><page_30><loc_39><loc_40><loc_39><loc_41></location>↦</text>
<text><location><page_30><loc_9><loc_34><loc_91><loc_39></location>We also need a choice of rigging vector field ζ along ˜ N . For convenience, we set ζ = -µ 2 ∂ U (observe that µ 2 | ˜ N = 0). The corresponding null metric hypersurface data (2.17) defined by {N , γ, /lscript , /lscript (2) } is</text>
<formula><location><page_30><loc_28><loc_39><loc_91><loc_43></location>ι : N ↪ -/shortrightarrow ˜ N ( λ, y A ) -/shortrightarrow ι ( λ, y A ) ≡ ( U = 0 , V = λ, x A = y A ) . (7.2)</formula>
<formula><location><page_30><loc_32><loc_32><loc_91><loc_35></location>γ = δ AB µ 2 N dy A ⊗ dy B , /lscript = dλ, /lscript (2) = 0 , (7.3)</formula>
<text><location><page_30><loc_9><loc_26><loc_91><loc_31></location>where µ N def = ι /star µ = 1 + Λ 12 δ AB y A y B . Observe that ∂ λ ∈ Rad γ and /lscript ( ∂ λ ) = 1 imply that n = ∂ λ . Moreover, F = 0 and s = 0 (cf. (2.10)-(2.11)) and U = 0 as a consequence of (2.12). The tensor Y is obtained from (2.18). A simple calculation gives</text>
<formula><location><page_30><loc_32><loc_21><loc_91><loc_26></location>Y = -Λ δ AB 6 µ N ( λdy A ⊗ dy B -2 y B dy A ⊗ s dλ ) . (7.4)</formula>
<text><location><page_30><loc_9><loc_19><loc_91><loc_22></location>Cutting the spacetime across the hypersurface {U = 0 } leaves two spacetimes ( M ± , g ± ) defined to be the regions U /greaterlessequal 0 endowed with the metrics</text>
<formula><location><page_30><loc_10><loc_13><loc_91><loc_18></location>g ± = g ± Mk µ 2 ± , where g ± Mk def = -2 d U ± d V ± + δ AB dx A ± dx B ± , µ ± def = 1 + Λ 12 ( δ AB x A ± x B ± -2 U ± V ± ) . (7.5)</formula>
<text><location><page_30><loc_9><loc_7><loc_91><loc_14></location>Obviously, the boundaries are ˜ N ± ≡ {U ± = 0 } . These two regions can clearly be matched so that the original spacetime (containing no shell) is obtained. Moreover, since ˜ N ± are totally geodesic we know that multiple matchings can be performed. We therefore proceed as in Section 5, i.e. we let the two</text>
<text><location><page_30><loc_17><loc_36><loc_17><loc_37></location>/negationslash</text>
<text><location><page_31><loc_9><loc_81><loc_91><loc_91></location>embeddings ι ± be given by ι ± = ι and take ζ -= -µ 2 -∂ U -, ˜ ζ + = -µ 2 + ∂ U + as the riggings defining the noshell matching, namely the matching for which [ ˜ Y ] = 0. Any other possible matching will be ruled by a diffeomorphism ϕ of N onto itself and it will correspond to a different rigging ζ + along ˜ N + . Specifically, the hypersurface data corresponding to the no-shell matching is D = {N , γ, /lscript , /lscript (2) , Y } , where { γ, /lscript , /lscript (2) } and Y are respectively given by (7.3) and (7.4), while the matter/gravitational content of the shell of any other possible matching (ruled by ϕ ) is given by the the jump [ Y ] def = Y + -Y with</text>
<formula><location><page_31><loc_40><loc_76><loc_91><loc_81></location>Y + def = 1 2 ϕ /star ( ι /star ( £ ζ + g + ) ) . (7.6)</formula>
<formula><location><page_31><loc_34><loc_67><loc_91><loc_71></location>( ∂ y a ϕ A )( ∂ y b ϕ B ) δ AB (1 + Λ 12 δ IJ ϕ I ϕ J ) 2 = δ A a δ B b δ AB (1 + Λ 12 δ IJ y I y J ) 2 . (7.7)</formula>
<text><location><page_31><loc_9><loc_70><loc_91><loc_77></location>From Section 5, we know that there is no need to compute the new rigging ζ + or its corresponding Y + to determine the jump [ Y ], which is explicitly given by (5.12)-(5.14). Consequently, we only need to worry about the diffeomorphism ϕ . The only restriction that ϕ must satisfy is ϕ /star γ = γ , which in coordinates reads</text>
<text><location><page_31><loc_9><loc_59><loc_91><loc_66></location>It follows that the components { ϕ A } cannot depend on the coordinate λ . In particular, if we let { h A ( y B ) } be a set of functions such that ( a ) the Jacobian matrix ∂ ( h 2 ,...,h n +1 ) ∂ ( y 2 ,...,y n +1 ) has non-zero determinant and ( b ) { h A ( y B ) } verify (1 + Λ 12 δ IJ y I y J ) -2 δ CD = (1 + Λ 12 δ IJ h I h J ) -2 ( ∂ y C h A )( ∂ y D h B ) δ AB , any diffeomorphism ϕ : N -/shortrightarrow N of the form</text>
<text><location><page_31><loc_41><loc_56><loc_41><loc_56></location>↦</text>
<formula><location><page_31><loc_30><loc_55><loc_91><loc_58></location>ϕ : N -/shortrightarrow N ( λ, y B ) -/shortrightarrow ϕ ( λ, y B ) ≡ ( H ( λ, y B ) , h A ( y B )) (7.8)</formula>
<text><location><page_31><loc_9><loc_42><loc_91><loc_55></location>with ∂ λ H = 0 fulfils ϕ /star γ = γ . A particular simple example is { h A = y A } , but many more exist. In fact since the metric on any section of ˜ N is of constant curvature, it is also maximally symmetric (and of dimension n -1) so h A ( y B ) can depend on n ( n -1) / 2 arbitrary parameters. For any possible choice of { h A ( y B ) } and an arbitrary step function H ( λ, y A ), the gauge parameters z and V are given by (6.6) for { n = ∂ λ , e A = ∂ y A } . In the present case the tensor Y is given by Y = 1 n ( H ) ϕ /star Y -Y (cf. (5.11)), so we need to compute the pull-back ϕ /star Y . Defining µ N def = 1 + Λ 12 δ AB h A h B , from (7.4) and (7.8) it is straightforward to get</text>
<formula><location><page_31><loc_26><loc_38><loc_91><loc_41></location>( ϕ /star Y ) λλ = 0 , ( ϕ /star Y ) λy B = Λ δ IJ h J 6 µ N ∂h I ∂y B ∂H ∂λ , (7.9)</formula>
<text><location><page_31><loc_9><loc_32><loc_70><loc_34></location>so that, multiplying (7.9)-(7.10) by 1 n ( H ) and subtracting Y (cf. (7.4)) yields</text>
<formula><location><page_31><loc_24><loc_34><loc_91><loc_39></location>( ϕ /star Y ) y A y B = Λ δ IJ 6 µ N ( h J ( ∂H ∂y A ∂h I ∂y B + ∂h I ∂y A ∂H ∂y B ) -H ∂h I ∂y A ∂h J ∂y B ) (7.10)</formula>
<formula><location><page_31><loc_18><loc_24><loc_91><loc_32></location>Y λλ = 0 , Y λy B = Λ δ IJ 6 ( h J µ N ∂h I ∂y B -δ I B y J µ N ) , (7.11) Y y A y B = Λ δ IJ 6 n ( H ) ( h J µ N ( ∂H ∂y A ∂h I ∂y B + ∂h I ∂y A ∂H ∂y B ) -H µ N ∂h I ∂y A ∂h J ∂y B + δ I A δ J B λ µ N n ( H ) ) .</formula>
<formula><location><page_31><loc_13><loc_17><loc_91><loc_22></location>̂ [Y λλ ] = -n ( n ( H )) n ( H ) , [Y λy A ] = Y λy A -∇ h A ( n ( H )) n ( H ) , [Y y A y B ] = Y y A y B -∇ h A ∇ h B H n ( H ) , (7.12)</formula>
<text><location><page_31><loc_9><loc_21><loc_91><loc_24></location>Inserting these expressions into (5.12)-(5.14) and using n = ∂ λ , s = 0, U = 0 together with the identity ( £ n ϕ /star /lscript ) ( e A ) = ( £ n dH ) ( e A ) = d ( n ( H ))( e A ) = e A ( n ( H )) (here ϕ /star /lscript = dH by (6.4) and /lscript = /lscript ), one finds</text>
<text><location><page_31><loc_9><loc_10><loc_91><loc_16></location>which can be interpreted as the sum of the jump corresponding to the matching of two regions of Minkowski across a null hyperplane (see [23, Eq. (6.6)]) plus the contribution of the tensor Y . Observe that Λ = 0 entails Y = 0, so in this way we recover expressions (6.6) in [23] for the most general planar shell in the spacetime of Minkowski</text>
<text><location><page_31><loc_17><loc_53><loc_17><loc_54></location>/negationslash</text>
<text><location><page_32><loc_9><loc_88><loc_91><loc_91></location>A direct computation that combines the definitions (2.49), (7.3) and (7.12) yields energy-density, energy flux and pressure (note that here we need to take /epsilon1 = -1)</text>
<text><location><page_32><loc_9><loc_75><loc_91><loc_84></location>Observe that only the pressure is independent of the value of the cosmological constant Λ ( ρ and j depend on the conformal factor µ N and on Y ). The pressure p takes the same value for the matchings of two regions of (anti-)de Sitter or Minkowski (in fact, p coincides with the pressure obtained in [23, Sect. 6]). In particular, in the case h A = y A (i.e. when the mapping between null generators of both sides is trivial), then Y λy B = 0 (cf. (7.11)) and (7.13) simplifies to</text>
<formula><location><page_32><loc_10><loc_83><loc_91><loc_88></location>ρ = µ 2 N δ AB ( Y y A y B -∇ h A ∇ h B H n ( H ) ) , j = µ 2 N δ AB ( ∇ h B ( n ( H )) n ( H ) -Y λy B ) ∂ y A , p = -n ( n ( H )) n ( H ) . (7.13)</formula>
<formula><location><page_32><loc_13><loc_70><loc_91><loc_75></location>ρ = µ 2 N δ AB ( Y y A y B -∇ h A ∇ h B H n ( H ) ) , j = µ 2 N δ AB ∇ h B ( n ( H )) n ( H ) ∂ y A , p = -n ( n ( H )) n ( H ) . (7.14)</formula>
<text><location><page_32><loc_9><loc_63><loc_91><loc_70></location>In the cut-and-paste constructions corresponding to constant-curvature spacetimes, the so-called Penrose's junction conditions (see e.g. [43], [40]) impose a jump in the coordinates across the shell. This jump is of the form V + | U + =0 = V -+ H ( x A -) | U -=0 . In the present case the matching embeddings φ -= ι and φ + = ι · ϕ are given by</text>
<text><location><page_32><loc_9><loc_50><loc_91><loc_60></location>so the step function corresponding to Penrose's jump is H ( λ, y A ) = λ + H ( y A ), H ∈ F ( N ). In order to recover such an H , one needs that there is no energy flux and no pressure on the shell. Indeed, imposing this in (7.14) and integrating for H yields H ( λ, y A ) = aλ + H ( y A ), where H ∈ F ( N ) and a is a positive 4 constant. Thus, in this more general context with arbitrary cosmological constant, the Penrose's jump still describes either purely gravitational waves (when ρ , j and p are all zero) or shells of null dust (when j and p vanish but ρ = 0), analogously to what happened in [23, Sect. 6] for the Minkowski spacetime.</text>
<formula><location><page_32><loc_10><loc_59><loc_90><loc_64></location>φ -( λ, y B ) = ( U -= 0 , V -= λ, x A -= y A ) , φ + ( λ, y B ) = ( U + = 0 , V + = H ( λ, y B ) , x A + = h A ( y B ) ) ,</formula>
<text><location><page_32><loc_26><loc_50><loc_26><loc_51></location>/negationslash</text>
<section_header_level_1><location><page_32><loc_9><loc_45><loc_31><loc_47></location>Acknowledgements</section_header_level_1>
<text><location><page_32><loc_9><loc_36><loc_91><loc_43></location>The authors acknowledge financial support under Grant PID2021-122938NB-I00 funded by MCIN/AEI /10.13039/501100011033 and by 'ERDF A way of making Europe' and SA096P20 (JCyL). M. Manzano also acknowledges the Ph.D. grant FPU17/03791 (Spanish Ministerio de Ciencia, Innovaci'on y Universidades).</text>
<section_header_level_1><location><page_32><loc_9><loc_31><loc_21><loc_33></location>References</section_header_level_1>
<unordered_list>
<list_item><location><page_32><loc_10><loc_28><loc_64><loc_29></location>[1] Battelle Rencontres, 1967 Lectures in mathematics and physics.</list_item>
<list_item><location><page_32><loc_10><loc_23><loc_91><loc_26></location>[2] Barrab'es, C., and Hogan, P. A. Singular null hypersurfaces in General Relativity: light-like signals from violent astrophysical events. World Scientific, 2003.</list_item>
<list_item><location><page_32><loc_10><loc_19><loc_91><loc_22></location>[3] Barrab'es, C., and Israel, W. Thin shells in general relativity and cosmology: the lightlike limit. Physical Review D 43 (1991), 1129-1142.</list_item>
</unordered_list>
<table>
<location><page_33><loc_9><loc_11><loc_91><loc_91></location>
</table>
<table>
<location><page_34><loc_9><loc_10><loc_91><loc_91></location>
</table>
<table>
<location><page_35><loc_9><loc_59><loc_91><loc_91></location>
</table>
</document>
[
{
"title": "Abstract Formulation of the Spacetime Matching Problem and Null Thin Shells",
"content": "† Miguel Manzano ∗ and Marc Mars Instituto de F'ısica Fundamental y Matem'aticas, IUFFyM Universidad de Salamanca Plaza de la Merced s/n 37008 Salamanca, Spain September 27, 2023",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "The formalism of hypersurface data is a framework to study hypersurfaces of any causal character abstractly (i.e. without the need of viewing them as embedded in an ambient space). In this paper we exploit this formalism to study the general problem of matching two spacetimes in a fully abstract manner, as this turns out to be advantageous over other approaches in several respects. We then concentrate on the case when the boundaries are null and prove that the whole matching is determined by a diffeomorphism ϕ on the abstract data set. By exploiting the gauge structure of the formalism we find explicit expressions for the gravitational/matter-energy content of any null thin shell. The results hold for arbitrary topology. A particular case of interest is when more than one matching is allowed. Assuming that one such matchings has already been solved, we provide explicit expressions for the gravitational/matter-energy content of any other shell in terms of the known one. This situation covers, in particular, all cut-and-paste constructions, where one can simply take as known matching the trivial re-attachment of the two regions. We include, as an example, the most general matching of two regions of the (anti-)de Sitter or Minkowski spacetime across a totally geodesic null hypersurface.",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "The question of under which conditions two spacetimes can be matched across a hypersurface and give rise to a new spacetime is a fundamental problem in any metric theory of gravity. In particular, a matching theory is required in any physical situation where a substantial amount of gravitational/matterenergy content is located in a thin enough region of the spacetime (with respect to the dimensions of the problem). Then, the matter-content can be modelled as concentrated on a hypersurface. This thin shell of gravitational/matter-energy possesses its own gravity and hence affects the spacetime geometry, and it is worth finding the relationship between the shell's content and the properties of the spacetime. Many authors have contributed to the matching problem in General Relativity, see e.g. the works [11], [34], [20], [18], [7], [10], [3], [31], [30], [26], [45]. The standard approach consists of considering two spacetimes ( M ± , g ± ) with boundaries ˜ N ± . For the matching to be possible ˜ N ± must be diffeomorphic, i.e. there must exist a diffeomorphism Φ : ˜ N --/shortrightarrow ˜ N + , which we call matching map . One then defines the resulting spacetime M as the union of M + and M -with the corresponding identification of boundary points (ruled by Φ). The necessary and sufficient conditions for a metric g to exist on M are the so-called (preliminary) matching conditions (or junction conditions ) and require ( i ) that the first fundamental forms γ ± from both boundaries coincide, ( ii ) that there exists two riggings ζ ± (i.e. vector fields along ˜ N ± , everywhere transversal to them) with the same square norm and such that the one-forms g ± ( ζ ± , · ) coincide and ( iii ) that ζ ± are such that one points inwards and the other outwards. When these conditions are fulfilled, the matched spacetime exists. In general, this spacetime will contain a thin shell, which is ruled by the jump in the extrinsic geometry of the matching hypersurfaces. In addition to this standard approach (also called ' a la Darmois ), one can also construct null thin shells with the so-called cut-and-paste method (see e.g. [35], [1], [36], [39], [38], [37], [17], [42], [43], [40], [41]), where the shell is described via a metric with a Dirac delta distribution with support on the matching hypersurface. The shell is built by taking a spacetime ( M , g ) with a null hypersurface ˜ N ⊂ M , then cutting M along ˜ N , which leaves two spacetimes ( M ± , g ± ), and finally reattaching (or pasting ) ( M ± , g ± ) by identifying the boundary points so that there exists a jump on the null direction on the matching hypersurface. Be that as it may, null shells have been widely studied in the literature (for a sample, see [2], [8], [5], [33], [9], [6], [19], [13]), usually by imposing additional symmetries (such as spherical symmetry). In particular, the problem of matching two completely general spacetimes ( M ± , g ± ) with null boundaries ˜ N ± has been recently addressed in [23] under the only assumption that ˜ N ± admit a foliation by diffeomorphic spacelike cross-sections. One of the main results in [23] is that all the information about the matching is codified in a diffeomorphism Ψ between the set of null generators of ˜ N ± and function H , called step function , which corresponds to a shift along the null generators. Another result of interest is that, although generically two given spacetimes can be matched at most in one manner, sometimes multiple matchings are possible. A relevant case of the later, studied in detail in [24], is the matching across so-called Killing horizons of order zero. The matching problem is studied in [23], [24] by means of the so-called formalism of hypersurface data [26], [27] (see also [31], [25], [29], [28], [21], [22]), which allows one to codify abstractly (i.e. in a detached way from an ambient manifold) the intrinsic and extrinsic geometric information of a hypersurface in terms of a data set D def = {N , γ, /lscript , /lscript (2) , Y } . The part {N , γ, /lscript , /lscript (2) } is called metric hypersurface data and codifies at the abstract level the would-be components of the full ambient metric g at the hypersurface. The tensor Y codifies extrisinc information. The formalism is equipped with a group of gauge transformations that accounts for the fact that, at the embedded level, the choice of a rigging is non-unique. Two data sets are equivalent if they are related by a gauge transformation. Each gauge group element G ( z,V ) ⊂ G is determined by a nowhere-zero function z and a vector field V in N . In the language of the hypersurface data formalism, the matching can be performed if and only if [26] one can embed a single metric hypersurface data set in both spacetimes (and the corresponding matching riggings satisfy the orientation condition ( iii ) above). In that case, the gravitational/matter-energy content of the shell is fully codified by the jump of the tensors Y ± of each side, namely [ Y ] def = Y + -Y -[26]. The approach in [23], [24], whilst based on this formalism, still analyzes the matching in terms of the embeddings φ ± of the abstract manifold N in M ± and not directly at a detached level. Two questions arise naturally. The first one is whether there is a way of formulating the matching problem in a fully abstract manner (that is, exclusively in terms of objects defined in the abstract manifold N ) so that one does not need to make any reference to the actual spacetimes to be matched. The second is whether one can generalize the results in [23], [24] to boundaries with arbitrary topology. The aim of this paper is to answer both questions. The first question is solved in Theorem 3.1, where we provide a completely abstract version of the (spacetime) matching conditions. The theorem establishes that two given data sets D = {N , γ, /lscript , /lscript (2) , Y -} , ̂ D = {N , ̂ γ, ̂ /lscript , ̂ /lscript (2) , ̂ Y + } (each of them should be thought of as an abstraction of one of the boundaries) can be matched provided that there exists a diffeomorphism ϕ of N onto itself such that the metric hypersurface data sets {N , γ, /lscript , /lscript (2) } , {N , ϕ /star ̂ γ, ϕ /star ̂ /lscript , ϕ /star ̂ /lscript (2) } are related by a gauge transformation G ( z,V ) . This can be interpreted as follows. The map ϕ can be understood as an abstract version of the (spacetime) matching map Φ mentioned before. Since the matching requires that one single metric hypersurface data set is embedded in both spacetimes, D and ̂ D cannot be arbitrarily different. Instead, there must exist a gauge transformation that compensates for the change induced by ϕ so that, even after applying the pull-back ϕ /star , the metric part of the data are still equivalent. Theorem 3.1 also imposes a restriction on the sign of z . As we shall see, at the embedded level this restriction ensures that the orientation of the riggings to be identified in the matching process verifies condition ( iii ) above. When the data sets D , ̂ D are embedded in two spacetimes, Theorem 3.1 is equivalent to the standard matching conditions ( i )-( iii ) above. This result is relevant for several reasons. First, it applies to (abstract) hypersurfaces of any causality and any topology. Secondly, the gauges of the data sets D , ̂ D are unfixed so that at the embedded level there is full freedom in the a priori choice of the riggings on each side. This gives a lot of flexibility to the framework. Finally, having formulated the matching problem abstractly allows one analyze in an independent manner thin shells with specific gravitational/matterenergy content and, on a second stage, study whether they can be embedded in a spacetime. This is useful e.g. for constructing examples of spacetimes containing certain types of shells. We then concentrate on the null case. We intend to generalize the works [23], [24] so we impose no topological conditions on the boundaries We prove that a (null) metric hypersurface data set {N , γ, /lscript , /lscript (2) } is entirely codified by γ and that the remaining metric data is pure gauge. In these circumstances, the feasibility of the matching relies on the tensors { γ, ̂ γ } satisfying ϕ /star ̂ γ = γ (and z having suitable sign). One of our main results in the paper is that we find explicit expressions for the riggings to be identified in the matching process, as well as of the gravitational/matter-energy content of the resulting shell (Theorem 4.4). Specifically, we compute explicitly the jump [ Y ] and the energy-momentum tensor of the shell in terms of D , ̂ D and ϕ . In particular we provide fully geometric definitions of the energy density ρ , energy flux j and pressure p of the shell (Remark 2.16) and find explicit expression for them. We also codify the purely gravitational content of a null shell in a tensor Y G , which we also compute explicitly. We emphazise that all these result hold for any possible null thin shell. The pressure p of the null shell is worth studying in further detail. It turns out that it can be expressed as a difference of the surface gravities (i.e. the 'accelerations') of two null generators of N related by the push-forward map ϕ /star . This generalizes previous results in [23], [24], where in specific examples we noticed that p accounts for an effect of compression/stretching of points when crossing the matching hypersurface. With the abstract matching formalism, one also recovers the property from [23], [24] that when the data sets D , ̂ D define abstract totally geodesic null hypersurfaces, then an infinite number of matchings are feasible. This situation is addressed in Section 5 where, assuming that all the information about one of the matchings is known, we prove that the gravitational/matter energy content of all the remaining matchings can be determined easily and explicity in terms of the known one and the map ϕ , with no need of performing additional calculations. The specific case when one of the matchings gives rise to no shell is of particular interest because it includes all cut-and-paste constructions 1 . In this context, we find explicit expressions for the gravitational/matter-energy content of any null thin shell constructed with the cut-and-paste method. These results are applied in Section 7, where we study the matching of two regions of a constant-curvature spacetime across a totally geodesic null hypersurface. For the sake of consistency, we devote Section 6 to showing how the results in [23] are recovered as a particular case of the general framework presented here in the specific case when N can be foliated by spacelike diffeomorphic cross-sections. The structure of the paper is as follows. In Section 2 we review the results on the geometry of embedded null hypersurfaces, formalism of hypersurface data and matching of spacetimes that are needed later. In Section 3 we provide an abstract formulation of the matching problem. The rest of the paper concentrates on the null case. In particular, Section 4 is devoted to studying the properties of completely general null thin shells and finding explicit expressions for their gravitational/matter-energy content, while Section 5 addresses the case when multiple matchings are feasible. In Section 6, we establish the connection between the results in [23] and the abstract matching formalism developed here. The paper concludes with an example where we study all possible matchings involving two regions separated by a totally geodesic null hypersurface in the (anti-)de Sitter or Minkowski spacetimes (Section 7).",
"pages": [
1,
2,
3,
4
]
},
{
"title": "1.1 Notation and conventions",
"content": "In this paper manifolds are smooth, connected and, unless otherwise indicated, without boundary. We use T M to denote the tangent bundle of a manifold M and Γ( T M ) for its sections (i.e. vector fields). We also let F ( M ) def = C ∞ ( M , R ) and F /star ( M ) ⊂ F ( M ) its subset of no-where zero functions. We use the symbols £ , d to denote Lie derivative and exterior derivative respectively. Both tensorial and abstract index notation will be employed depending on convenience. When index-free notation is used, we shall often use boldface for covariant tensors. In index notation we use standard font (not boldface) in all cases. We work in arbitrary dimension n , with the following values for different sets of indices: As usual, parenthesis (resp. brackets) will denote symmetrization (resp. antisymmetrization) of indices and we also use the notation A ⊗ s B ≡ 1 2 ( A ⊗ B + B ⊗ A ) for the symmetrized tensor product of two tensors A and B . When B is symmetric, 2-contravariant we write tr B A for the trace with respect to B of any 2-covariant tensor A . Given a semi-Riemannian manifold ( M , g ), the associated contravariant metric is called g /sharp and ∇ is the Levi-Civita derivative. Scalar products of two vectors are denoted indistinctly as g ( X,Y ) or 〈 X,Y 〉 g . Our convention for Lorentzian signature is ( -, + , ..., +). Finally, the abreviation 'w.r.t.' for 'with respect to' will be used sometimes.",
"pages": [
4
]
},
{
"title": "2.1 Geometry of embedded null hypersurfaces",
"content": "In this subsection we review some facts about embedded null hypersurfaces, see e.g. [15], [16], [12]. This will serve to fix our notation. An embedded null hypersurface in a spacetime ( M , g ) of dimension n +1 is the image ˜ N = φ ( N ) of an embedding φ : N ↪ -/shortrightarrow M of an n -manifold N , such that the first fundamental form γ def = φ /star g of N is degenerate. Any choice of (nowhere zero) normal vector k to ˜ N defines a null direction tangent to ˜ N called null generator (and viceversa). The integral curves of k are geodesic and the surface gravity ˜ κ k ∈ F ( ˜ N ) of k is defined by ∇ k k = ˜ κ k k . The second fundamental form of ˜ N w.r.t k is the tensor ˜ K k ( X,Y ) def = g ( ∇ X k, Y ), ∀ X,Y ∈ Γ( T ˜ N ). Boundaries of manifolds are always two-sided, so (cf. Lemma 1 in [26]) we shall always assume that ˜ N admits an everywhere transversal vector field L , i.e. verifying L / ∈ T p ˜ N ∀ p ∈ ˜ N . The vector L can always be taken null everywhere (see e.g. [23]). /negationslash A transverse submanifold of ˜ N is any ( n -1)-dimensional submanifold S ⊂ ˜ N to which k is everywhere transverse. When, in addition, every integral curve of k crosses S exactly once S is called crosssection (or simply section ). The existence of a cross-section entails a strong topological restriction on ˜ N , as in such case there always exist functions v ∈ F ( ˜ N ), called foliation functions , whose level sets S v 0 def = { p ∈ ˜ N | v ( p ) = v 0 ∈ R } are cross-sections of ˜ N and { S v } define a foliation of ˜ N . Nevertheless existence of foliation functions is always granted in sufficiently local domains of ˜ N . Note that necessarily k ( v ) = 0 so we can always assume k ( v ) = 1 either by rescalling k or by changing v . When L is chosen null and orthogonal to S then Θ L and σ L are the second fundamental form and torsion one-form of S w.r.t. L . For any choice of L , the tensors σ L , Θ L encode extrinsic information of S . However, Θ L is not symmetric in general. Assuming that ˜ N admits a cross-section S , one can construct a foliation function v ∈ F ( ˜ N ) and (on local patches) a basis { L, k, v I } of Γ ( T M ) | ˜ N adapted to the foliation with the following properties: (E) L is a past null vector field everywhere transversal to ˜ N . For any basis { L, k, v I } verifying (2.2), we also define n scalar functions { µ a } ⊂ F ( N ) as /negationslash ˜ Note that necessarily µ 1 = 0 (this has already been used in (2.1)). The vectors { v A } are spacelike by construction and { k, v I } is a basis of Γ( T ˜ N ). Conditions (A) and (B) imply that v increases towards the future. We write h for the induced metric on the leaves { S v } and h IJ def = g ( v I , v J ) for its components in the basis { v I } . We use h IJ and its inverse h IJ to lower and raise Capital Latin indices irrespectively of whether they are tensorial or not (e.g. we let µ I def = h IJ µ J ). The property [ k, v I ] = 0 entails [23]",
"pages": [
4,
5
]
},
{
"title": "2.2 Formalism of hypersurface data",
"content": "The formalism of hypersurface data , which we introduce next, will allow us to analyze the matching of spacetimes at a fully abstract level. We refer to [26], [27], [29], [28], [25], [21], [22] for details.",
"pages": [
5
]
},
{
"title": "2.2.1 General hypersurface data",
"content": "The fundamental notion of the formalism is metric hypersurface data , defined to be a set {N , γ, /lscript , /lscript (2) } where N is an n -dimensional manifold, γ is a 2-covariant symmetric tensor, /lscript is a covector and /lscript (2) is a scalar function subject to the condition that the symmetric 2-covariant tensor A | p on T p N × R given by is non-degenerate at every p ∈ N . A priori any signature for A | p is allowed. Given metric hypersurface data, one can define unique tensor fields { P ab , n a , n (2) } , with P symmetric, by means of [26] No restriction is placed on γ , which in particular is allowed to be degenerate. However, A being nondegenerate forces γ to have at most one degeneration direction [27]. Specifically, the radical of γ at p ∈ N , defined by Rad γ | p def = { X ∈ T p N | γ ( X, · ) = 0 } , is either zero- or one-dimensional. The latter case occurs if and only if n (2) | p = 0, which by (2.6) means that Rad γ | p = 〈 n | p 〉 . A point p ∈ N is called null if dim(Rad γ | p ) = 1 and non-null otherwise. The second basic notion of the formalism is hypersurface data which is just {N , γ, /lscript , /lscript (2) } equipped with an extra symmetric 2-covariant tensor Y , namely D def = {N , γ, /lscript , /lscript (2) , Y } . It is useful to define the following tensors (note that F , s and U only require metric hypersurface data) (Metric) hypersurface data has a built-in gauge group structure [27] with the following properties. Definition 2.1. Let D = {N , γ, /lscript , /lscript (2) , Y } be hypersurface data, z ∈ F /star ( N ) and V ∈ Γ( T N ) . The gauge transformed data G ( z,V ) ( D ) def = { N , G ( z,V ) ( γ ) , G ( z,V ) ( /lscript ) , G ( z,V ) ( /lscript (2) ) , G ( z,V ) ( Y ) } is defined as The set of all possible gauge transformations forms a group G = F /star ( N ) × Γ( T N ) with composition law G ( z 2 ,V 2 ) · G ( z 1 ,V 1 ) = G ( z 1 z 2 ,V 2 + z -1 2 V 1 ) , identity G (1 , 0) and inverse G -1 ( z,V ) def = G ( z -1 , -zV ) . All considerations so far make no reference to any ambient space where N is embedded. The abstract construction and the usual geometry of embedded hypersurfaces are connected through the notion of embeddedness of the data. Given a semi-Riemannian ( n + 1)-dimensional manifold ( M , g ) we say {N , γ, /lscript , /lscript (2) } is embedded with embedding φ and rigging ζ in ( M , g ) provided there exists an embedding φ : N ↪ -/shortrightarrow M and a rigging ζ (i.e. a vector field along φ ( N ), everywhere transversal to it) satisfying The same notion for hypersurface data {N , γ, /lscript , /lscript (2) , Y } requires, in addition, We often simplify the notation and say simply that the data is ' { φ, ζ } -embedded'. We also identify scalars and vectors in N with their corresponding images on φ ( N ) when there is no risk of confusion. The action of the gauge group in the data corresponds to a change of rigging according to [27] More specifically, it holds that if {N , γ, /lscript , /lscript (2) } is { φ, ζ } -embedded in ( M , g ), then G ( z,V ) ( {N , γ, /lscript , /lscript (2) } ) is { φ, G ( z,V ) ( ζ ) } -embedded in the same space. The hypersurface φ ( N ) admits a unique normal ν satisfying g ( ν, ζ ) = 1, which decomposes as [26], [27] It then turns out that K (defined in (2.13)) is the second fundamental form of φ ( N ) w.r.t. ν [27], i.e. Observe that K and U coincide at null points of N . Although generically N is not a semi-Riemannian manifold, it admits two useful covariant derivatives. The metric hypersurface connection · ∇ depends only on the metric part of the data and it is defined uniquely [26] by the properties of being torsion-free together with the expressions The second connection is called hypersurface connection and denoted by ∇ . It is also torsion-free and relates to the former by ∇ X Z = · ∇ X Z -Y ( X,Z ) n for any X,Z ∈ Γ( T N ). When {N , γ, /lscript , /lscript (2) , Y } is { φ, ζ } -embedded in ( M , g ), the ambient Levi-Civita connection ∇ and the derivatives · ∇ , ∇ satisfy [26] for all X,Z ∈ Γ( T N ). Thus, ∇ is the connection induced from ∇ along the rigging [31]. Two consequences of the definition of · ∇ are where θ a is an arbitrary one-form. Their explicit proof can be found in [27] and [25, Lem. 2.5] respectively. We shall also need the following lemma relating Lie and · ∇ derivatives. Lemma 2.2. Let {N , γ, /lscript , /lscript (2) } be metric hypersurface data, V a any vector field and w a any covector field. Define V a def = γ ab V b and ˆ w a def = P ab w b . Then the following identities hold Proof. We first note that ∇ c γ ab -∇ a γ bc -∇ b γ ac = 2 /lscript c U ab as a direct consequence of (2.22). Moreover, since · ∇ has no torsion, the Lie derivative of any p -covariant tensor T along any direction V ∈ Γ( T N ) reads ( £ V T ) a 1 ··· a p = V b · ∇ b T a 1 ··· a p + p i =1 T a 1 ··· a i -1 ba i +1 ··· a p · ∇ a i V b . Particularizing this for T = γ we get · · · ∑ · · · · · · · £ V γ ab = V c ∇ c γ ab +2 γ c ( a ∇ b ) V c = V c ( ∇ c γ ab -∇ a γ bc -∇ b γ ac ) +2 ∇ ( a ˇ V b ) = 2 /lscript ( V )U ab +2 ∇ ( a ˇ V b ) which is (2.28). To prove the second identity we apply (2.28) to V = ˆ w . Since by (2.9) we have γ ab P bc w c = w a -w ( n ) /lscript a , identity (2.28) gives 1 2 £ ˆ w γ ab = /lscript ( ˆ w )U ab + · ∇ ( a ( w b ) -w ( n ) /lscript b ) ) . From (2.8), we find /lscript ( ˆ w ) = -/lscript (2) w ( n ). Inserting above yields 1 2 £ ˆ w γ ab = -/lscript (2) w ( n )U ab + · ∇ ( a w b ) -/lscript ( a · ∇ b ) w ( n ) -w ( n ) · ∇ ( a /lscript b ) , which simplifies to (2.29) after taking into account (2.23). From a covector and a function on N , one can build a unique vector field according to the next lemma. Lemma 2.3. [26] Let {N , γ, /lscript , /lscript (2) } be metric hypersurface data. Given a covector field /rho1 ∈ Γ( T /star N ) and a scalar function u 0 ∈ F ( N ) , there exists a vector field W ∈ Γ( T N ) satisfying γ ( W, · ) = /rho1 , /lscript ( W ) = u 0 if and only if /rho1 ( n ) + n (2) u 0 = 0 . Such W is unique and reads W = P ( /rho1 , · ) + u 0 n .",
"pages": [
5,
6,
7
]
},
{
"title": "Matter-hypersurface data and abstract thin shells",
"content": "Hypersurface data encodes (abstractly) the intrinsic and extrinsic information of embedded hypersurfaces. In the context of gravity, knowning the matter contents of the spacetime determines part of the curvature, typically by means of the Einstein tensor Ein g . Thus, to codify matter information abstractly we need to supplement the data with additional quantities. For general hypersurfaces, only the normal-transverse and the normal-tangential components of Ein g can be related exclusively to intrinsic and extrinsic information of the hypersurface [18], [3], [31], [26]. Hence, the additional (matter) data involves a scalar ρ and a covector J that, once the data is embedded, correspond to such components of Ein g . Their relation with the rest of the data needs to be imposed as constraint equations. They are well-known in the spacelike case (see e.g. [4]), and were generalized to arbitrary causal character in [26]. Note that although we refer to the variables ρ and J as matter variables, what we are actually prescribing are certain components of the Einstein tensor. The terminology is justified because in General Relativity (with vanishing cosmological constant) ρ and J indeed correspond to the matter four-momentum along the normal direction. However, we emphasize that we are not assuming any field equations and that the geometric approach that we take can be used in any theory of gravity. The abstract definition of matter-hypersurface data is as follows. Definition 2.4. [26] (Matter-Hypersurface data) A tuple {N , γ, /lscript , /lscript (2) , Y , ρ /lscript , J } formed by hypersurface data {N , γ, /lscript , /lscript (2) , Y } , a scalar ρ /lscript ∈ F ( N ) and a one-form J ∈ Γ( T /star N ) is matter-hypersurface data if G ( z,V ) ( ρ /lscript ) = ρ /lscript + J ( V ) , G ( z,V ) ( J ) = z -1 J and the following identities, called constraint equations, hold: The next theorem justifies both the gauge behaviour of { ρ /lscript , J } and the explicit form of (2.30)-(2.31). Theorem 2.5. [26] Let {N , γ, /lscript , /lscript (2) , Y , ρ /lscript , J } be matter-hypersurface data and assume that the hypersurface data {N , γ, /lscript , /lscript (2) , Y } is { φ, ζ } -embedded in a semi-Riemannian manifold ( M , g ) . Then, where Ein g is the ( 2 -covariant) Einstein tensor of ( M , g ) and ν the (unique) normal vector field along φ ( N ) satisfying g ( ζ, ν ) = 1 . As we shall see further on, the matching problem involves pairs of matter-hypersurface data. However, at this point we simply put forward various definitions and explore some of their consequences. Definition 2.6. (Thin shell) A thin shell is a pair of matter-hypersurface data with same metric hypersurface data, i.e. of the form {N , γ, /lscript , /lscript (2) , Y ± , ρ ± /lscript , J ± , /epsilon1 } , where /epsilon1 is a sign with gauge behaviour: We write Q ± for quantities constructed from {N , γ, /lscript , /lscript (2) , Y ± } and let [ Q ] def = Q + -Q -be its jump. One of the main properties of thin shells is that one can define an energy-momentum tensor encoding their matter-energy content. In a completely general case, this is done as follows. Definition 2.7. (Energy-momentum tensor) For a thin shell {N , γ, /lscript , /lscript (2) , Y ± , ρ ± /lscript , J ± , /epsilon1 } , the energymomentum tensor is the symmetric 2 -covariant tensor τ defined by Remark 2.8. Definitions 2.6 and 2.7 are a modification of the previous ones introduced in [26] , which involved no /epsilon1 . The addition of the sign /epsilon1 is necessary in order for τ to retain its physical interpretation as energy-momentum tensor (density) in all gauges. Indeed, a change in the orientation of /lscript (or of rigging in the embedded picture) introduces a sign in [ Y ] (by (2.16) ). The value of τ cannot be sensitive to this, so one needs to introduce a sign /epsilon1 with gauge behaviour (2.34) to compensate the change of sign in [ Y ] (in fact, one checks easily that the gauge behaviour of τ is G ( z,V ) ( τ ) = | z | -1 τ ). To be more specific, when one deals with thin shell data {N , γ, /lscript , /lscript (2) , Y ± , ρ ± /lscript , J ± } { φ ± , ζ ± } -embedded in ( M ± , g ± ) , the sign /epsilon1 must be chosen positive if ζ -points outwards w.r.t. ( M -, g -) and negative otherwise. The tensor field τ has the symmetries of an energy-momentum tensor and coincides with the Israel energy-momentum tensor of the shell [18] whenever it does not contain null points. Moreover, for null thin shells, the definition of energy-momentum tensor provided in [3, Eq. (31)] by Barrab'es and Israel yields precisely τ . In a spacetime ( M , g ) resulting from a matching, given a basis { e a } of Γ( T ˜ N ) where ˜ N is the matching hypersurface, one can also check that the quantity τ ab e µ a e ν b gives the singular part of the Einstein tensor of ( M , g ), as it is written in [31, Eq. (71)]. The gauge behaviour of τ is key in the embedded case, as it ensures that the singular part of the Einstein tensor of the matched spacetime remains invariant under rescaling the normal vector ν . All these reasons justify the Definition 2.7 for the energy-momentum tensor on a thin shell [26], irrespectively of whether the data is embedded. At null points (and only there), τ = 0 is compatible with a non-trivial jump of the geometry. Indeed, in order to get τ = 0 when n (2) = 0, it suffices to require [ Y ]( n, · ) = 0 and tr P [ Y ] = 0, which does not mean that the whole tensor [ Y ] vanishes identically. Physically, this situation corresponds to an impulsive gravitational wave supported on the shell. This behaviour is possible only at null points. At non-null points τ = 0 implies, in addition, that P af P bd [Y] ab = 0 which entails 0 = γ fi γ dj P af P bd [Y] ab = ( δ a i -n a /lscript i )( δ b j -n b /lscript j )[Y] ab = [Y] ij , i.e. abscence of jumps in the geometry. In particular, this means that non-trivial thin shells with vanishing energy-momentum tensor can only exist on null points.",
"pages": [
7,
8,
9
]
},
{
"title": "2.2.2 Null hypersurface data",
"content": "A particular case of relevance for the matching problem is when the hypersurfaces are null everywhere. It is immediate to translate this notion to the abstract level. Definition 2.9. (Null (metric) hypersurface data) A metric hypersurface data {N , γ, /lscript , /lscript (2) } or a hypersurface data {N , γ, /lscript , /lscript (2) , Y } is null if the scalar n (2) given by (2.6) -(2.9) is zero everywhere on N . Let us describe the main properties of the formalism in the null case. We refer to [25] for proofs and additional results. We already know that n (2) = 0 implies Rad( γ ) = 〈 n 〉 and therefore γ ( n, · ) = 0. Moreover, the tensors s and U def = 1 2 £ n γ defined in (2.11) and (2.12) verify When the data is { φ, ζ } -embedded, U becomes the second fundamental form of φ ( N ) w.r.t. the null normal ν = φ /star n (recall (2.20)). Inserting n (2) = 0 and (2.36) in the contraction of (2.26) with n b entails which together with (2.14) and ν = φ /star n yields Since ν is a null generator of Φ( N ), (2.38) means that κ n corresponds (at the abstract level) to the surface gravity of ν . Under the action of the gauge group, the surface gravity κ n transforms as follows. Lemma 2.10. [25] Let {N , γ, /lscript , /lscript (2) , Y } be null hypersurface data and consider gauge parameters { z, V } . The gauge behaviour of the scalar function κ n defined in (2.14) is We now state and prove a result that will be of particular relevance below, namely that by means of a gauge transformation one can always adapt the one-form /lscript and the scalar /lscript (2) to whatever pair { u ∈ F ( N ) , ϑ ∈ Γ( T /star N ) } one wishes, with the only restriction that ϑ ( n ) = 0 everywhere on N . /negationslash /negationslash Lemma 2.11. Let {N , γ, /lscript , /lscript (2) } be null metric hypersurface data, u a function on N and ϑ ∈ Γ( T /star N ) a covector satisfying ϑ ( n ) = 0 everywhere. There exists a unique gauge transformation G ( z,V ) satisfying Moreover, the gauge group element G ( z,V ) is given by /negationslash Remark 2.12. The condition ϑ ( n ) = 0 is necessary because if ϑ ( n ) vanishes at any point then ϑ can never correspond to /lscript in any gauge, as /negationslash which in particular states that ( G ( z,V ) ( /lscript ))( n ) = 0 for all possible gauge parameters. Proof. We first assume that the gauge transformation exists and restrict its form up to a function yet to be determined. We then restrict to group elements of such a form and show that there exists one and only one of them that satisfies (2.40), namely (2.41). This will prove both the existence and uniqueness claims of the lemma. For the first part we impose (2.40): Contracting the first with n gives z = ϑ ( n ), so w def = γ ( V, · ) = 1 ϑ ( n ) ϑ -/lscript . Observe that w ( n ) = 0. Moreover, the vector V -P ( w , · ) lies in the kernel of γ because γ ab ( V b -P bc w c ) = w a -( δ c a -n c /lscript a ) w c = 0. Therefore, there exists f ∈ F ( N ) such that V a = P ab w b + fn b = ( ϑ ( n )) -1 P ab ϑ b + ( /lscript (2) + f ) n a . Thus, it suffices to restrict oneself to gauge parameters in the class We now start anew and prove that there is precisely one function q such that the corresponding ( z, V ) in (2.43) fulfills conditions (2.40). For V as in (2.43) we get The first condition in (2.42) is satisfied for all q . The second is satisfied if and only if which ends the proof. In particular, Lemma 2.11 (together with (2.15)) means that two given null metric hypersurface data sets are related by a gauge transformation if and only if they both have the same data tensor γ . We prove this in the next corollary. Corollary 2.13. Let D def = {N , γ, /lscript , /lscript (2) } , D def = {N , γ, /lscript , /lscript (2) } be two null metric hypersurface data. Then there is a gauge group element G ( z,V ) ∈ F /star ( N ) × Γ( T /star N ) such that G ( z,V ) ( D ) = D if and only if γ = γ . This gauge element is given by Proof. The necessity is obvious from the fact that γ remains unchanged by a gauge transformation. Sufficiency is a direct application of Lemma 2.11 to ϑ = /lscript and u = /lscript (2) . Lemma 2.11 and Corollary 2.13 state that in the null case one can codify all the metric hypersurface data information exclusively in the tensor γ , and that /lscript and /lscript (2) are pure gauge. This fact will be key later in Section 3 when studying the matching of spacetimes with null boundaries. We shall also need the decompositions of { γ, P } in a basis { n, e A } of Γ( T N ) and its corresponding dual. Lemma 2.14. [25] Consider null metric hypersurface data {N , γ, /lscript , /lscript (2) } . Let { n, e A } be a basis of Γ( T N ) and { q , θ A } be its corresponding dual, i.e. q ( n ) = 1 , q ( e A ) = 0 , θ A ( n ) = 0 , θ A ( e B ) = δ A B . Define the functions ψ A def = /lscript ( e A ) ∈ F ( N ) . Then, the tensor fields γ and P decompose as where h AB def = γ ( e A , e B ) is a metric and h AB denotes its inverse. The concept of null thin shell arises naturally from Definitions 2.6 and 2.9. A thin shell is said to be null if its metric part {N , γ, /lscript , /lscript (2) } defines null metric hypersurface data . Moreover, as a corollary of Lemma 2.14, one can find a very simple form for the components of τ . Corollary 2.15. In the setup of Lemma 2.14, let {N , γ, /lscript , /lscript (2) , Y ± , ρ ± /lscript , J ± , /epsilon1 } be a null thin shell. Then, the components of the energy-momentum tensor τ in the basis { q , θ A } read Proof. Inserting the decomposition (2.47) into Definition 2.7 yields after a simple but somewhat long computation in which several terms cancel out. Contracting with { q , θ A } it is immediate to get (2.48). Remark 2.16. In the literature, the different components of the energy-momentum tensor of a thin shell {N , γ, /lscript , /lscript (2) , Y ± , ρ ± /lscript , J ± , /epsilon1 } are interpreted physically as an energy density ρ , an energy-flux j and a pressure p (see e.g. [44] ). However, this is usually done in a context where {N , γ, /lscript , /lscript (2) , Y ± , ρ ± /lscript , J ± , /epsilon1 } are embedded with riggings ζ ± that are null and orthogonal to the basis vectors { e A } . In a completely general framework, we propose the following geometric definitions for the physical quantities { ρ, p, j } : Definitions (2.49) are justified because in the null case (2.35) can be written in terms of { ρ, p, j } as For null shells, the vector field j satisfies γ ( j, · ) = /epsilon1 [ Y ]( n, · )+ p /lscript and /lscript ( j ) = 0 , which makes the definitions (2.49) consistent since the one-form j def = γ ( j, · ) verifies j ( n ) = 0 . Moreover, a direct calculation based on (2.14) and (2.39) proves the following gauge behaviour for the pressure p : Whenever /lscript (2) = 0 and ψ A = 0 , it is straightforward to check that (2.49) becomes after using (2.45) and (2.47) . This allows one to recover the standard definitions for { ρ, p, j } introduced e.g. in [44, Eq. (3.99)] . Expressions (2.52) coincide with the definitions proposed in [44] whenever /epsilon1 = -1 which, as mentioned in Remark 2.8, corresponds to the rigging ζ -pointint inwards. We conclude this subsection by recalling several aspects on the geometry of transverse submanifolds embedded in null metric hypersurface data sets. We again refer to [25] for proofs. Given null metric hypersurface data {N , γ, /lscript , /lscript (2) } , a transverse submanifold S is a codimension one embedded submanifold of N to which n is everywhere transverse . Letting ψ : S ↪ -/shortrightarrow N be the embedding of S in N we define h def = ψ /star γ . It is a fact [25] that h is a metric on S and we denote by ∇ h its Levi-Civita covariant derivative. When it is clear from the context we identify vectors and scalars on S with their counterpars on ψ ( S ). For any p -covariant tensor T along Ψ( S ) and given a basis { v A } of Γ( TS ), we define T ‖ def = ψ /star T and write T A 1 ...A p def = T ‖ ( v A 1 , . . . , v A p ) (without the parallel symbol). Capital Latin indices are raised with h IJ and its inverse h IJ . With the definition /lscript (2) ‖ := h IJ /lscript I /lscript J , the pull-back to S of the · ∇ derivative of any p -covariant tensor field T along ψ ( S ) takes the following explicit form [25, Lem. 3.15]:",
"pages": [
9,
10,
11,
12
]
},
{
"title": "2.3 Matching of spacetimes and junction conditions",
"content": "From now on we focus on the problem of matching two spacetimes with boundary. In this section we recall known results, first for boundaries of any causality and secondly in the null case. Consider two spacetimes ( M ± , g ± ) with boundaries ˜ N ± of any causal character. It is well-known (see e.g. [11], [34], [20], [18], [7], [10], [3], [31]) that the matching of ( M ± , g ± ) across ˜ N ± is possible if and only if the so-called junction conditions or matching conditions are satisfied. In the language of the formalism of hypersurface data, the matching requires [26] that there exist metric hypersurface data {N , γ, /lscript , /lscript (2) } that can be embedded in both spacetimes ( M ± , g ± ) with embeddings φ ± such that φ ± ( N ) = ˜ N ± and riggings ζ ± , i.e. there must exist two pairs { φ ± , ζ ± } satisfying γ = ( φ ± ) /star ( g ± ) , /lscript = ( φ ± ) /star ( g ± ( ζ ± , · )) , /lscript (2) = ( φ ± ) /star ( g ± ( ζ ± , ζ ± )) . (2.54) In addition, the riggings ζ ± must fulfil an orientation condition (see item ( ii ) below). In these circumstances, it is always possible to select one of the embeddings freely by adapting N to one of the boundaries. In the following we shall make use of this freedom by fixing φ -at our convenience. This entails no loss of generality. Note that making the choice in the minus side is also of no relevance, as one can always switch the names of the spacetimes to be matched. When the junction conditions are satisfied, the geometry of the shell [26] is determined by the jump of the transverse tensors Y ± defined as In the literature, however, the matching conditions are not normally presented in terms of a hypersurface data set. Instead, they are usually formulated as follows (see e.g. [31]). Junction Conditions. The matching of ( M ± , g ± ) across ˜ N ± can be performed if and only if For the rest of the paper, two riggings ζ ± satisfying ( i )-( ii ) for a diffeomorphism Φ will be called matching riggings . The diffeomorphism Φ will be referred to as matching map . If (2.56) holds for two riggings ζ ± then, for any other choice of rigging on one of the sides, (2.56) is fulfilled as well 2 (although different choices of rigging on one side will correspond to different riggings on the other side). We shall make use of this freedom to fix ζ -at will, again with no loss of generality. As proven in Lemmas 2 and 3 of [30], given a rigging on one side (say ζ -) and a diffeomorphism Φ : ˜ N --/shortrightarrow ˜ N + satisfying Φ /star g + = g -, at non-null points the second and third equations of (2.56) yield either no solution for ζ + (hence the matching is not possible) or two solutions for ζ + with opposite orientation. At null points, on the other hand, if there exists a solution ζ + then it is unique. This means that at non-null points one can always make a suitable choice of rigging ζ + so that the junction condition ( ii ) is fulfilled, and hence one only needs to care about (2.56). In the null case, however, this is not so. It can happen that there exists a solution ζ + of (2.56) but with unsuitable orientation, and then the matching cannot be performed. Thus, at null points conditions (2.56) are necessary but not sufficient to guarantee that the matching is feasible [30]. When the matching is possible, the corresponding matching map Φ is the key object upon which the whole matching depends. This is so because once the point-to-point identification of the boundaries ˜ N ± (ruled by Φ) is known, one matching rigging can be selected at will (as we have seen) and the other is the unique solution that arises from enforcing both (2.56) and ( ii ). All the information about the matching is therefore codified by Φ, or equivalently by the embedding φ + (cf. (2.54)). We now concentrate on the case when the boundaries ˜ N ± are null. From a spacetime viewpoint, this problem was addressed in [23] (see also [24], where the matching across Killing horizons of order zero was studied). In the remainder of the section we summarize the main results of [23]. Consider two ( n +1)-dimensional spacetimes ( M ± , g ± ) with null boundaries ˜ N ± that can be foliated by a family of diffeomorphic spacelike cross-sections. Assume further that one of the boundaries lies in the future of its corresponding spacetime while the other lies in its spacetime past. This entails no loss of generality, as explained in [23]. We construct foliation functions v ± ∈ F ( ˜ N ± ) and basis { L ± , k ± , v ± I } of Γ( T M ± ) | ˜ N ± according to (2.2). The surface gravities of k ± are ˜ κ ± k ± . As in Section 2.1, the leaves of the foliations are denoted by { S ± v ± } , while their corresponding induced metrics are h ± . We also let ˜ K k ± be the second fundamental forms of ˜ N ± w.r.t. k ± , and introduce the tensors Θ L ± , σ ± L on the leaves { S ± v ± } (cf. (2.1)). The scalar functions { µ ± a } ⊂ F ( N ± ) are defined by (2.3) w.r.t. the basis { L ± , k ± , v ± I } . As we have seen, in order to perform a matching we need to embed a single metric hypersurface data set in both spacetimes. We codify the already described freedom in the choice of { φ -, ζ -} as follows. We first consider an abstract null hypersurface N and define coordinates { y 1 = λ, y A } therein. Then, we construct null embedded metric hypersurface data by enforcing that ( a ) the push-forwards { e -a def = φ -/star ( ∂ y a ) } coincide with the basis vectors { k -, v -I } (since { k -, v -I } are chosen at will, with this procedure we ensure that φ -is built at our convenience) and ( b ) that the rigging ζ -coincides with the basis vector L -. This amounts to impose Thus, ζ -is a null past rigging (recall (2.2)) and λ is a coordinate along the degenerate direction of N . In fact, the subsets { λ = const. } ⊂ N are all diffeomorphic [23] and define a (spacelike) foliation of N . For the matching of ( M ± , g ± ) to be possible, there must exist another pair { φ + , ζ + } so that (2.54) hold (and the orientations of ζ ± are suitable). In that case, we can build another basis { e + a def = φ + /star ( ∂ y a ) } of Γ( T ˜ N + ) and then determining the matching requires that we find the explicit form of the vectors { e + a } (which fully codify φ + ). In the basis { k + , v + I } of Γ( T N + ), these vectors decompose as [23] where f , a I , b J I ∈ F ( ˜ N + ) are given by in terms of a set of functions { H ( λ, y B ) , h A ( y B ) } on N . The functions { H,h A } encode all the matching information and hence they determine φ + . In fact, given coordinates { v + , u I } on ˜ N + such that v + I = ∂ u I (i.e. { u I + } are constant along the null generators), the embedding φ + is such that [23] The function H ( λ, y A ) is named step function because it measures a kind of jump along the null direction when crossing the matching hypersurface. It must satisfy the condition ∂ λ H > 0 [23]. The explicit form of the matching rigging ζ + was computed in [23, Cor. 1] and reads The solvability of the first junction condition in (2.54) constitutes the core problem for the existence of a matching. In terms of the metrics h ± , it can be rewritten as where ( b -1 ) J I def = ∂ h I y J and Z B def = 1 2 ( ( b -1 ) J B ( ∂ y J H -1 µ -1 ( ∂ λ H ) µ -J ) -1 µ + µ + B ) k + + v + B . h -IJ | p = b L I b K J h + LK | Φ( p ) ∀ p ∈ ˜ N -. (2.62) Equation (2.62) is an isometry condition between each submanifold { v -= const. } ⊂ ˜ N -and its corresponding image on ˜ N + . On the other hand, the identification of { e ± 1 } requires the existence of a diffeomorphism Ψ (ruled by the coefficients b A B fulfilling (2.62)) between the set of null generators on both sides. Moreover, combining (2.4), (2.57)-(2.59), (2.62) and { e ± 1 def = φ ± /star ( ∂ y a ) } yields [23] ˜ K k -( v -I , v -J ) = ( ∂ λ H ) b A I b B J ˜ K k + ( v + A , v + B ) . (2.63) Thus, for each possible choice of Ψ (i.e. of { b A B } ), (2.63) determines a unique value for ∂ λ H unless the two second fundamental forms vanish simultaneously. In the latter case, the step function H cannot be restricted. Consequently, when ˜ N ± are totally geodesic, if a single matching of ( M ± , g ± ) can be performed then an infinite number of matchings (one for each possible step function H ) are feasible [23]. When the matching is possible, the matter-energy content of the shell is given by the next proposition. Proposition 2.17. [23] Assume that the matching of ( M ± , g ± ) across ˜ N ± is possible and that it is determined by the functions { H ( λ, y A ) , h B ( y A ) } . Let h IJ be the induced metric on the leaves { λ = const. } ⊂ N , h IJ its inverse tensor and ∇ h its Levi-Civita covariant derivative. Define the vector fields { W A } , the scalars { µ + A } , the covector q ∈ Γ( T /star N ) and the vector field X = X a ∂ y a ∈ Γ( T N ) by Then, the components of the tensor [ Y ] def = Y + -Y -are while the energy-momentum tensor of the shell is given by (the sign /epsilon1 is given by Definition 2.6)",
"pages": [
12,
13,
14,
15
]
},
{
"title": "3 Abstract formulation of the matching problem",
"content": "In the previous section, we have summarized the main aspects of the matching of two general spacetimes with null boundaries that admit a foliation by diffeomorphic spacelike sections. The matching conditions have been formulated from a spacetime viewpoint, and we have recalled the geometrical objects upon which the matching depends (namely the step function H and the diffeomorphism Ψ). We have also recollected the explicit expressions for the gravitational and matter-energy content of the resulting shells (Proposition 2.17). The results we have just summarized leave (at least) two interesting problems unaddressed. The first one is whether one can obtain analogous results without the topological assumptions on the boundaries and the second is whether there is a way of formulating the matching problem in a fully abstract manner, namely without making any reference to the actual spacetimes to be matched. As already explained in the Introduction, addressing these problems is the key object of this paper. Let us start with the abstract formulation of the junction conditions. For that purpose, we first consider that the boundaries ˜ N ± of the spacetimes ( M ± , g ± ) to be matched have any topology and any causal character . Since ˜ N -is embedded, there exists an abstract manifold N and an embedding ι -: N ↪ -/shortrightarrow M -such that ι -( N ) = ˜ N -. From the embedding ι -, one can construct an infinite number of embeddings simply by applying additional diffeomorphisms within N . To elude this unavoidable redundancy, we henceforth let ι -be one specific choice among all possible. As discussed before, two spacetimes ( M ± , g ± ) can be matched if there exists a pair of embeddings φ ± : N ↪ -/shortrightarrow M ± related to a matching map Φ by φ + = Φ · φ -. Moreover, the embedding and the rigging on one of the sides (say the minus side) can always be chosen freely. Suppose we enforce φ -= ι -and take a specific rigging ζ -. Then we can build embedded hypersurface data D def = {N , γ, /lscript , /lscript (2) , Y } by requiring (2.17)-(2.18), i.e. by defining Thus, all the information about the matching is encoded in φ + and the junction conditions are (2.54). These conditions, although of a more abstract nature than (2.56), still codify the matching information in the pair { φ + , ζ + } , which is not of abstract nature. In order to determine the matching in terms of objects defined at the abstract level, we must take one step further. The following theorem, based on the existence of a diffeomorphism ϕ of the abstract manifold N onto itself, sets up the corresponding construction. Theorem 3.1. Consider two hypersurface data D def = {N , γ, /lscript , /lscript (2) , Y -} , ̂ D def = {N , ̂ γ, ̂ /lscript , ̂ /lscript (2) , ̂ Y + } embedded in two spacetimes ( M -, g -) , ( M + , g + ) with embeddings ι -, ι + and riggings L -, L + respectively. Assume that ι ± ( N ) def = ˜ N ± are boundaries of ( M ± , g ± ) and let /epsilon1 + = +1 (resp. /epsilon1 + = -1 ) if L + points outwards (resp. inwards) from M + . Define /epsilon1 -in the same way (i.e. /epsilon1 -= +1 if L -points outwards, /epsilon1 -= -1 if inwards). Then, the matching of ( M ± , g ± ) across ˜ N ± is possible if and only if (i) There exist a gauge group element G ( z,V ) and a diffeomorphism ϕ of N onto itself such that G ( z,V ) ( ϕ /star ̂ γ ) = γ, G ( z,V ) ( ϕ /star ̂ /lscript ) = /lscript , G ( z,V ) ( ϕ /star ̂ /lscript (2) ) = /lscript (2) ; (3.2) (ii) sign( z ) = -sign( /epsilon1 + )sign( /epsilon1 -) . ̂ γ def = ( ι + ) /star ( g + ) , ̂ /lscript def = ( ι + ) /star ( g + ( L + , · )) , ̂ /lscript (2) def = ( ι + ) /star ( g + ( L + , L + )) , ̂ Y + def = 1 2 ( ι + ) /star ( £ L + g + ) . (3.4) Since the spacetimes ( M ± , g ± ), the embeddings ι ± and the riggings L ± are all given, the tensor fields in (3.3)-(3.4) are known. To prove the first part of the theorem, we start by assuming ( i )-( ii ). Thus, there exist a pair { z ∈ F /star ( N ) , V ∈ Γ( T N ) } and a diffeomorphism ϕ : N -/shortrightarrow N so that (3.2) holds. These conditions can be rewritten as (recall (2.15), G -1 = G -1 ) ( z,V ) ( z , -zV ) Let us define the map φ + def = ι + · ϕ , the vector field V ' def = ι + /star ( ϕ /star V ), the function z ' ∈ F /star ( ˜ N + ) given by ϕ /star (( ι + ) /star z ' ) def = z and the rigging ζ + def = z ' ( L + + V ' ) along ˜ N + . By definition of z ' , it holds that sign( z ) = sign( z ' ). On the other hand, combining (3.5)-(3.7) with the fact that ̂ D is embedded with embedding ι + and rigging L + , it follows The data D is therefore embedded in ( M + , g + ) with embedding φ + and rigging ζ + . Thus, conditions (2.54) are satisfied for φ -= ι -, φ + = ι + · ϕ and for the riggings ζ -= L -, ζ + . Moreover, combining ( ii ) (which holds by assumption), the definition of ζ and sign( z ) = sign( z ), it follows It is straightforward to check that (3.11) implies that whenever L -points inwards (resp. outwards) then ζ + points outwards (resp. inwards) irrespectively of the orientation of L + . Thus, D is embedded in ( M ± , g ± ) and L -, ζ + are such that one points inwards and the other outwards, which means that the matching of ( M ± , g ± ) is possible. To prove the converse, we assume that the matching is possible for two pairs { φ ± , ζ ± } . We have already discussed the flexibility of selecting at will the embedding and the rigging on one side (say the minus side). Let us therefore set φ -= ι -, ζ -= L -. Since both L + and ζ + are riggings along ˜ N + , there exists a pair { z ' ∈ F /star ( ˜ N + ) , V ' ∈ Γ( T ˜ N + ) } such that ζ + = z ' ( L + + V ' ). Moreover, one can define a diffeomorphism ϕ : N -/shortrightarrow N by φ + def = ι + · ϕ . But then one can follow the arguments of (3.8)-(3.10) backwards and prove (3.2) for a function z ∈ F /star ( N ) defined by z def = ϕ /star (( ι + ) /star z ' ). As before, sign( z ) = sign( z ' ) so both ζ + = z ' ( L + + V ' ) and z ' L + = sign( z ) | z ' | L + have the same orientation (because V ' is tangent to ˜ N + ). By assumption the matching is possible, hence L -, ζ + are such that one points inwards and the other outwards. If L -points inwards (resp. outwards) then sign( z ) L + must point outwards (resp. inwards), so sign( z ) = sign( /epsilon1 + ) (sign( z ) = -sign( /epsilon1 + )) is forced. This means that ( i )-( ii ) are both fulfilled. Remark 3.2. Theorem 3.1 does not impose any conditions on the topology of the abstract manifold N , except for the very mild one that hypersurface data sets can be defined on N . Remark 3.3. In Theorem 3.1 we have not restricted the gauges of the data sets D , ̂ D (we let the two riggings L ± be given, but no conditions have been imposed on them). Each specific choice of L ± will fix a particular gauge on D , ̂ D . Moreover, Theorem 3.1 holds for data sets D , ̂ D of any causal nature. In particular, D , D are not required to contain non-null or null points exclusively. ̂ Remark 3.4. As proven in [27, Lem. 3.6] , given metric hypersurface data {N , γ, /lscript , /lscript (2) } and a point p ∈ N , the gauge group elements leaving {N , γ, /lscript , /lscript (2) } invariant at p are ( i ) G (1 , 0) | p if p is null and ( ii ) {G (1 , 0) | p , G ( -1 , -2 /lscript ) | p } if p is non-null, where the vector /lscript | p is obtained by raising index to /lscript | p with the inverse metric γ /sharp | p (which in that case exists). Since gauge parameters { z, V } are smooth by definition, it follows that when N contains a null point, only the identity element of G leaves the whole metric hypersurface data invariant. On the contrary, when N consists exclusively of non-null points there exist two gauge elements which do not transform the metric data. In this last case, the rigging G ( -1 , -2 /lscript ) ( ζ ) corresponds [27] to the reflection of ζ w.r.t. the tangent plane T q φ ( N ) at each point q ∈ φ ( N ) . In view of the above, when there are no null points on N , condition ( ii ) can always be fulfilled once (i) is granted. Indeed, if there exists a gauge group element G ( z,V ) satisfying (i) then this also happens for G ( -1 , -2 /lscript ) · G ( z,V ) = G ( -z, -2 /lscript -V ) . Thus, there always exists a suitable choice of gauge parameter z for which ( i ) and ( ii ) hold. On the contrary, when N contains null points only the gauge element G (1 , 0) leaves the hypersurface data invariant, which means that ( i ) can be fulfilled for a gauge group element G ( z,V ) but z may have the wrong sign. This is the underlying reason why the spacetime conditions (2.56) provide one unique solution for ζ + for given { ζ -, Φ } (see the corresponding discussion in Section 2.3). Remark 3.5. In Theorem 3.1, we have expressed the junction conditions as a restriction over two data sets and a requirement on the sign of a gauge parameter. Theorem 3.1 therefore constitutes an abstract formulation of the standard matching conditions. In particular, a remarkable advantage of Theorem 3.1 is that it allows us to study different matchings in two different levels. At the first level one takes whatever hypersurface data sets D , ̂ D satisfying ( i ) and studies its properties from a fully detached point of view. At this level, the spacetimes need not even exist. The problem can then move on and study whether or not one can construct spacetimes in which these data can be embedded so that condition ( ii ) holds. In other words, by Theorem 3.1 one can produce a thin shell of any causality with full freedom to prescribe the gravitational and matter-energy content, and then study the problem of constructing the resulting spacetime ( M , g ) which contains it. This is of great use, as it provides a framework to build examples of spacetimes with thin shells of any type. In the setup of Theorem 3.1, the matching riggings are { L -, ζ + } , where ζ + is of the form (3.11). This means that the sign /epsilon1 -coincides with the sign /epsilon1 introduced in Definitions 2.6 and 2.7. It is convenient not to fix the signs /epsilon1 ± (or the riggings L ± ) a priori because it may well occur that transverse vectors L ± on each spacetime are already privileged or have been chosen for whatever other reason. The main point of the construction in Theorem 3.1 is firstly that it provides a fully abstract description of the matching and secondly that it keeps maximum flexibility so that one can adapt Theorem 3.1 to any particular scenario.",
"pages": [
15,
16,
17,
18
]
},
{
"title": "4 Abstract formulation of the matching problem: null boundaries",
"content": "For the remainder of the paper, we focus on the case when both D and ̂ D are null hypersurface data. Under these circumstances, by Lemma 2.11 we know that there exists a pair { z, V } ensuring that the second and third equations in (3.2) are fulfilled. It follows that the only restrictions are therefore condition ( ii ) in Theorem 3.1 and the first equality in (3.2), namely ϕ /star ̂ γ = γ. (4.1) Consequently, given two spacetimes ( M ± , g ± ) with null boundaries ˜ N ± , either there exists (at least) one diffeomorphism ϕ satisfying (4.1) or not. In the former case the matching is possible (provided ( ii ) holds) and, as we shall see next, all information about the matching is codified by ϕ . From now on and without loss of generality, we again make the harmless assumption that one of the boundaries lies in the future of its corresponding spacetime while the other lies in its spacetime past (see the discussion in [23]). The following lemma provides the explicit form of the gauge parameters { z, V } and of the matching rigging ζ + in terms of the diffeomorphism ϕ . Lemma 4.1. Assume that conditions ( i ) -( ii ) in Theorem 3.1 hold for a pair of embedded null hypersurface data D , ̂ D . Then, the gauge parameters { z, V } are given by Moreover, the matching identifies the rigging L -with the following rigging in the plus side where z ' ∈ F /star ( ˜ N + ) , µ ∈ F ( N + ) are given explicitly by Proof. The explicit form (4.2) for the function z follows from contracting (3.6) with n and using (2.7). The vector field V can be partially obtained also from (3.6) by particularizing Lemma 2.3 for W = V , /rho1 = z -1 /lscript -ϕ /star ̂ /lscript . This gives where u 0 def = /lscript ( V ) is a function yet to be determined. This is done by substituting (4.5) into (3.7). First, γ ( V, V ) = /rho1 ( V ) = z -2 /lscript (2) + P ( ϕ /star /lscript , ϕ /star /lscript ) because of (2.7)-(2.8) and z -1 = ( ϕ /star /lscript )( n ). Thus, so that substituting this into (4.5) proves (4.2). Equation (4.3) is a direct consequence of (4.2) and the fact that ζ + = z ' ( L + + ι + /star ( ϕ /star V )). Whenever there exists a diffeomorphism ϕ solving (4.1) and given a basis { n, e A } of Γ( T N ), it is possible to obtain specific expressions for the push-forward vector fields { ϕ /star n, ϕ /star e A } . This is done in the next corollary. We use a hat for all objects defined in the data set ̂ D , in particular ̂ P and ̂ n are constructed in correspondence with (2.6)-(2.9). Corollary 4.2. Assume that conditions ( i ) -( ii ) in Theorem 3.1 hold for a pair of embedded null hypersurface data D , ̂ D . Let { n, e A } be a basis of Γ( T N ) and define the covectors { W A } and the functions { ψ A , χ ( A ) } along N by Then, Moreover, it holds that ̂ P ( W A , ̂ /lscript ) = 0 and ϕ /star χ ( A ) = ( ϕ /star ̂ /lscript )( e A ) . Proof. Consider any point p ∈ N . From (3.5) it follows that ̂ γ ( ϕ /star n, · ) | ϕ ( p ) = ( ϕ /star ̂ γ )( n, · ) | p = γ ( n, · ) | p = 0, so ϕ /star n = b ̂ n for some function b ∈ F ( N ). This, together with (4.2) and ̂ /lscript ( ̂ n ) = 1, entails that z -1 | p = ( ϕ /star ̂ /lscript )( n ) | p = ̂ /lscript ( ϕ /star n ) | ϕ ( p ) = b | ϕ ( p ) = ϕ /star b | p , which proves (4.8). On the other hand, any vector field X ∈ Γ( T N ) satisfies which means that ̂ γ ( ϕ /star e A , · ) = W A , ̂ /lscript ( ϕ /star e A ) = ( ϕ -1 ) /star ( z -1 ψ A ) -W A ( ϕ /star V ). Particularizing Lemma 2.3 for the data ̂ D and for W = ϕ /star e A , /rho1 = W A and u 0 = ( ϕ -1 ) /star ( z -1 ψ A ) -W A ( ϕ /star V ) yields (4.9). Finally, P ( W A , /lscript ) = 0 because Remark 4.3. From (4.8) it follows that ϕ is a diffeormorphism which sends null generators into null generators. Moreover, since the vector fields { W A def = ̂ P ( W A , · ) } verify ̂ /lscript ( W A ) = 0 , it follows that W A / ∈ Rad ̂ γ . This, together with the fact that ϕ /star is necessarily of maximal rank, force the vector fields { W A } to be everywhere non-zero on N . In fact, { ̂ n, W A } constitutes a basis of Γ( T N ) , since { W A } are all linearly independent. We prove this by contradiction, i.e. we assume that one such vector field, e.g. W 2 , can be decomposed as W 2 = ∑ n r =3 c r W r . By (4.9) , this would mean that ϕ /star ( e 2 -∑ n r =3 c r e r ) = ̂ while χ ( A ) · ϕ = z -1 ψ A -( ϕ /star W A )( V ) = z -1 ψ A -γ ( e A , V ) (3.6) = ( ϕ /star ̂ /lscript )( e A ) yields ϕ /star χ ( A ) = ( ϕ /star ̂ /lscript )( e A ). ( χ (2) -∑ n r =3 c r χ ( r ) ) ̂ n , which we know it cannot occur, because only null generators can be mapped to null generators. The point of introducing the objects { W A , χ ( A ) } will become clear later when we study the particular case when the boundaries have product topology S × R . For the moment, let us simply anticipate that in such case the property ̂ P ( W A , ̂ /lscript ) = 0 will allow us to conclude that the vector fields ̂ P ( W A , · ) are tangent to the leaves of a specific foliation of ˜ N + while from ϕ /star χ ( A ) = ( ϕ /star ̂ /lscript )( e A ) we will conclude that the functions { χ ( A ) } are actually spatial derivatives of the step function introduced in Section 2.3. One of the relevant results recalled in Section 2.3 is the relation (2.63) between the second fundamental forms of each side. It turns out that in this abstract framework with no topological assumptions one can also recover an equation of this form. To do that, we first note that £ f ̂ n ̂ γ = f £ ̂ n ̂ γ because ̂ n ∈ Rad ̂ γ . By direct computation one gets ̂ ̂ which connects the second fundamental forms U , ̂ U corresponding to the hypersurface data sets D , ̂ D . Equation (4.10) generalizes (2.63) to the case of boundaries with any topology, and has several implications that we discuss below. In Theorem 3.1 we have seen that when the matching is possible there exists a diffeomorphism ϕ verifying (4.1). In such case, Lemma 4.1 and Corollary 4.2 provide explicit expressions for the gauge parameters { z, V } , the matching rigging ζ + and the push-forwards { ϕ /star n, ϕ /star e A } of any basis vector fields { n, e A } in terms of the map ϕ still to be determined. However, as the reader may have noticed, condition (4.1) does not fix ϕ completely, firstly because there can be more than one diffeomorphism ϕ satisfying (4.1) and secondly because the tensor fields γ and ̂ γ are both degenerate. As happened in Section 2.3, where the step function could not be fixed directly by the isometry condition (2.62) but (2.63) was also required [23], here one also needs an extra condition in order to fix ϕ fully. This additional restriction is precisely (4.10). As in Section 2.3, this provides useful information only when U and ̂ U are non-zero. If both are zero then z (and hence part of ϕ , recall (4.2)) remains completely free. This means that one can find an infinite number of diffeomorphisms ϕ verifying (4.1), with which we recover (and extend to arbitrary topology) the property that whenever the boundaries are totally geodesic then the matching can be performed in an infinite number of ways. One can obtain explicit expressions for the gravitational and matter-energy content of a general null shell in terms of the diffeomorphism ϕ . This is done in the following theorem. Theorem 4.4. Assume that conditions ( i ) -( ii ) in Theorem 3.1 hold for a pair of embedded null hypersurface data D , D and let /epsilon1 = /epsilon1 -. Define The energy-momentum tensor τ is given by (2.48) in terms of the dual basis { q , θ A } of { n, e A } , while the purely gravitational content of the shell is ruled by the tensor and where in the last step we used that · ∇ ( a /lscript b ) = -/lscript (2) U ab (cf. (2.23)). By hypothesis the matching of ( M ± , g ± ) is possible, so the data sets {N , ϕ /star ̂ γ, ϕ /star ̂ /lscript , ϕ /star ̂ /lscript (2) , ϕ /star ̂ Y + } , {N , γ, /lscript , /lscript (2) , Y + } are embedded in ( M + , g + ) with embedding ι + · ϕ and respective riggings L + , ζ + . This, together with (3.2), entails that the tensors ϕ /star ̂ Y + , Y + are related by Y + = G ( z,V ) ( ϕ /star ̂ Y + ), where { z, V } are given by (4.2). Thus (cf. (2.16), (4.1)) Inserting (4.14) into (4.15) yields the explicit form (4.11). We now obtain the components of [ Y ] in the basis { n, e A } , for which we recall that U ( n, · ) = 0 and s ( n ) = 0. Particularizing (2.27) for θ = ϕ /star /lscript and using (4.2) gives Combining (4.16)-(4.17) with (4.11) yields (4.12). The components of the energy-momentum tensor being given by (2.48) is just the contents of Corollary 2.15. Finally, we prove (4.13) as follows. First, we note that the one-forms j (see Remark 2.16) and /lscript decompose in the basis { q , θ A } as because j ( n ) = 0 and /lscript ( n ) = 1. Also by Remark 2.16, we know that the one-form j verifies [ Y ]( n, e A ) = /epsilon1 ( j ( e A ) -p /lscript ( e A )). Thus, a direct computation based on the decomposition (2.46) of the tensor field P yields where we used that P ( θ A , θ B ) = h AB (by Lemma 2.14), P ( /lscript , · ) = -/lscript (2) n (cf. (2.8)) and (4.18) in this order. Taking into account the definition of the energy density ρ (see (2.49)), one finds Now, from (2.48) it is clear that the only part of [ Y ] that does not contribute to the energy-momentum tensor is the h -traceless part of [ Y ]( e A , e B ). By Lemma 2.14, we know that h AB γ ( e A , e B ) = n -1. Consequently, [ Y ]( e A , e B ) decomposes in a h -traceless and a h -trace part as from where (4.13) follows at once after inserting (4.19). Remark 4.5. We emphasize that we have not made any assumption on the topology of the boundaries ˜ N ± in Theorems 3.1 and 4.4 or in Lemma 4.1. The results above therefore describe the most general matching of two spacetimes across null hypersurfaces and generalize the results in [23] and [24] , where the existence of a foliation on the boundaries played an important role. The gravitational/matter-energy content of the resulting null shell is given by Theorem 4.4, and the associated energy density ρ , energy flux j and pressure p are given by (2.49) . The reason why we refer to Y G ( e A , e B ) as the purely gravitational part of the shell is that only the components [ Y ]( n, n ) , [ Y ]( n, e A ) and the trace P ( θ A , θ B )[ Y ]( e A , e B ) contribute to the energy-momentum tensor τ (cf. (2.48) ). This means that even if τ vanishes identically Y G ( e A , e B ) does not need to be zero. Such a case corresponds to an impulsive gravitational wave propagating in the spacetime resulting from the matching. Remark 4.6. By Lemma 2.14 we know that P ( q , · ) = 0 if and only if /lscript ( e A ) = 0 and /lscript (2) = 0 . In such case, the scalar ρ coincides with the energy density ρ of the shell. In the embedded picture, these restrictions amount to impose that the matching riggings ζ ± are null and orthogonal to the vector fields φ ± /star e A . In particular, in the setup of Section 2.3 this holds when the rigging ζ -(chosen according to (2.57) ) is null and orthogonal to the leaves of the foliation on the minus side (hence µ -A = 0 , cf. (2.3) ). Remark 4.7. In Theorems 3.1 and 4.4 and Lemma 4.1, all expressions are fully explicit in terms of the diffeomorphism ϕ . The two data sets D , ̂ D are completely known (because the embeddings ι ± and the spacetimes ( M ± , g ± ) are given) and the rigging ζ + is determined by the pair { z, V } given by (4.2) in terms of ϕ . This is related to the results in [23] , [24] summarized in Section 2.3, where the whole matching depended upon the step function H and the coefficients b J I , which in turn determined the matching embedding φ + (recall (2.59) and (2.60) ) and the matching rigging ζ + (according to (2.61) ). Expressions (4.12) involve the pull-back ϕ /star ̂ Y + , whose calculation can be cumbersome in general. It is more convenient to rewrite (4.12) in terms of pull-backs of scalar functions referred to the data ̂ D and objects defined with respect to D . We provide the corresponding expressions in the next lemma. Lemma 4.8. Assume that conditions ( i ) -( ii ) in Theorem 3.1 hold for a pair of embedded null hypersurface data D , ̂ D and let /epsilon1 = /epsilon1 -. Define the tensors { Y -, ̂ Y + , Y + } as in Theorem 4.4, the covectors { W A } and the functions { χ ( A ) , ψ A } along N according to Corollary 4.2 and the vector field W A def = ̂ P ( W A , · ) . Let z be given by (4.2) and { n, e A } be a basis of Γ( T N ) with dual basis { q , θ A } . Then, equations (4.12) can be rewritten as [ Y ]( n, n ) = 1 z ϕ /star ( ̂ Y + ( ̂ n, ̂ n ) ) -Y -( n, n ) + n ( z ) z , (4.20) [ Y ]( n, e A ) = ϕ /star ( ̂ Y + ( ̂ n, W A ) + χ ( A ) ̂ Y + ( ̂ n, ̂ n ) ) -Y -( n, e A ) -z 2 ( £ n ϕ /star ̂ /lscript ) ( e A ) + e A ( z ) 2 z + s ( e A ) + zP ( ϕ /star ̂ /lscript , U ( e A , · )) , (4.21) [ Y ]( e A , e B ) = zϕ /star ( ̂ Y + ( W A , W B ) + χ ( A ) ̂ Y + ( ̂ n, W B ) + χ ( B ) ̂ Y + ( ̂ n, W A ) + χ ( A ) χ ( B ) ̂ Y + ( ̂ n, ̂ n ) ) -Y -( e A , e B ) -ze a A e b B · ∇ ( a ( ϕ /star ̂ /lscript ) b ) + z 2 2 ( P ( ϕ /star ̂ /lscript , ϕ /star ̂ /lscript ) -ϕ /star ̂ /lscript (2) ) U ( e A , e B ) . (4.22) The energy-momentum tensor τ is given by (2.48) in terms of the dual basis { q , θ A } of { n, e A } . Proof. Inserting ( ϕ /star ̂ Y + )( X,Y ) | p = ̂ Y + ( ϕ /star X,ϕ /star Y ) | ϕ ( p ) into (4.12) and using (4.8)-(4.9), equations (4.20)-(4.22) follow at once. We already know from Corollary 2.15 that τ is given by (2.48). In Section 6, we shall recover the results of Proposition 2.17 by particularizing Lemma 4.8 to the case when the boundaries ˜ N ± have product topology. Lemma 4.8 therefore generalizes Proposition 2.17 to (null) boundaries of any topology, and determines the matter-energy content of any null thin shell arising from the matching of two spacetimes.",
"pages": [
18,
19,
20,
21,
22,
23
]
},
{
"title": "4.1 Pressure of the shell",
"content": "In [23], [24] we discussed the effect and the importance of a non-zero pressure in a null thin shell. This, however, was done in very specific contexts (namely in the matching of two regions of Minkowski across a null hyperplane or for matchings across embedded AKH 0 s) and by following a non-fully geometric approach (i.e. by analyzing the effect of the pressure in some specific coordinates). Our aim in this section is to study the pressure of a completely general null shell at a fully abstract level, providing its explicit expression in terms of well-defined geometric quantities and reinforcing the geometric interpretation of [23] and [24]. In the following lemma we find explicit expressions for the pressure p in terms of the surface gravities of various null generators of N . Lemma 4.9. Assume that conditions ( i ) -( ii ) in Theorem 3.1 hold for a pair of embedded null hypersurface data D = {N , γ, /lscript , /lscript (2) , Y -} , ̂ D = {N , ̂ γ, ̂ /lscript , ̂ /lscript (2) , ̂ Y + } and a diffeomorphism ϕ . Define z by (4.2) and introduce Then, the pressure p of the corresponding null shell is given by ̂ where /epsilon1 = /epsilon1 -and V ∈ Γ( T N ) is a vector field that can be chosen at will. In particular, the pressure vanishes if and only if Proof. Recall that G -1 ( z,V ) = G ( z -1 , -zV ) . We start by noticing that (2.39) implies that G -1 ( z,V ) ( κ n G ( z -1 , -zV ) ( κ n ) = z ( κ n + n ( z ) z ) . On the other hand, combining (2.49) and (4.20), it follows Recalling the transformation law for /epsilon1 and p in (2.34) and (2.51) this expression can be written as -G -1 ( z,V ) ( /epsilon1p ) = G -1 ( z,V ) ( κ n ) -ϕ /star ̂ κ n . Applying G ( z,V ) on both sides one obtains the left part of (4.24). The right part of (4.24) is an immediate consequence of inserting the definition (4.23) of ϕ /star κ ϕ /star n into the first line of (4.26), while (4.25) is proven by setting p = 0 in (4.24). Remark 4.10. The last expression in (4.23) defines a function κ ϕ /star n on N . However, we still need to justify this terminology. It turns out that κ ϕ /star n coincides with the surface gravity of the vector field ϕ /star n w.r.t. the hypersurface connection ̂ ∇ constructed from the data ̂ D . To prove this, we let z def = ( ϕ -1 ) /star z , so that (cf. (4.23) ) where the right part follows from ( ϕ /star n )( z ) | ϕ ( p ) = ( ϕ /star d z )( n ) | p = ( dϕ /star z )( n ) | p = n ( z ) | p = ( ϕ -1 ) /star ( n ( z )) | ϕ ( p ) . Then, the combination of (2.37) and (4.8) gives 3 Remark 4.11. The gauge parameter V is completely superfluous and plays no role in determining the pressure, which is only influenced by the function z given by (4.2) . We keep V in the expression to emphasize this fact. Remark 4.12. In [23] , [24] , we have introduced the notion of self-compression and self-stretching on the boundaries of the spacetimes to be matched. We have seen that this effect is completely ruled by the pressure, and that it has to do with the differences in the acceleration along the null generators of both sides. With (4.24) , we recover the same result but for the case of boundaries with any topology. Indeed, the surface gravities κ n and κ ϕ /star n verify ∇ n n = κ n n and ̂ ∇ ϕ /star n ϕ /star n = κ ϕ /star n ϕ /star n , so that the quantity -/epsilon1p is positive when κ n > ϕ /star κ ϕ /star n (namely when the 'acceleration' of n is greater than that of ϕ /star n ) and negative otherwise. The only scenario where there exists no pressure occurs when both surface gravities coincide, i.e. when the accelerations of n and ϕ /star n are the same.",
"pages": [
23,
24
]
},
{
"title": "5 Multiple matchings across null boundaries",
"content": "We have already seen that generically there exists at most one way of matching two given spacetimes ( M ± , g ± ) (i.e. only one matching map Φ or one single diffeomorphism ϕ ). However, we have also mentioned that sometimes multiple (even infinite) matchings can be performed (e.g. when both second fundamental forms U , ̂ U vanish). In the language of (2.54), this means that given a choice of embedding φ -and matching rigging ζ -on the minus side, there exist several embeddings φ + for which the matching conditions hold, and each embedding gives rise to a unique solution for the rigging ζ + with suitable orientation. In this section, our aim is to study the scenario of multiple matchings. The idea is to assume that all information about one of the matchings is known, in particular its corresponding diffeomorphism ϕ and hence the gravitational/matter-energy content. As we shall see, in these circumstances one only needs to consider a single hypersurface data set D (instead of two) and it is possible to provide explicit expressions for the jump [ Y ] and the energy-momentum tensor τ of any other shell in terms of their counterparts of the known matching. These results can be particularized to the case when the known matching gives rise to no-shell (i.e. when it is such that [ Y ] = 0). This precisely happens in all cut-and-paste constructions, where ( M ± , g ± ) are two regions of the same spacetime. Our setup will be the following. We make a choice { φ -, ζ -} of embedding and rigging on the minus side and consider two matching embeddings φ + , ˜ φ + , each of them satisfying (2.54) for two riggings ζ + , ˜ ζ + respectively. We also assume that the information about one of the matchings is completely known, namely we let { φ + , ζ + } be given. ˜ ˜ From the spacetimes ( M ± , g ± ), we can construct two hypersurface data sets D = {N , γ, /lscript , /lscript (2) , Y -} , ̂ D = {N , ̂ γ, ̂ /lscript , ̂ /lscript (2) , ̂ Y + } and Theorem 3.1 ensures that we can find two diffeomorphisms ϕ , ˜ ϕ and two pairs { z, V } , { ˜ z, ˜ V } for which ( i )-( ii ) hold. Even more, since the pair { ˜ φ + , ˜ ζ + } is known, we can always make the choice { ι + = ˜ φ + , L + = ˜ ζ + } so that { ̂ γ, ̂ /lscript , ̂ /lscript (2) } = { γ, /lscript , /lscript (2) } and ˜ ϕ is the identity map, i.e. 3 Recall that the connections · ∇ , ∇ of a data set {N , γ, /lscript , /lscript (2) , Y } verify ∇ X Z = · ∇ X Z -Y ( X,Z ) n , ∀ X,Z ∈ Γ( T N ). ˜ ϕ = I N . In these circumstances, using (2.7)-(2.8) in (4.2) yields ˜ z = 1 and ˜ V = 0. Making the same choice of { ι + , L + } for the matching of ϕ transforms (3.2) into while the expressions (4.2) for the gauge parameters { z, V } become It is important to emphasize that whereas ˜ ϕ = I N forces the metric parts of D , ̂ D to be the same, the tensors Y -, ̂ Y + do not coincide in general. We let [ ˜ Y ] def = ̂ Y + -Y -, [ Y ] def = Y + -Y -be the jumps codifying the gravitational/matter-energy content of the null shells associated to ˜ ϕ and ϕ respectively. Then, by (4.11) we know that [ Y ] must be given by The jumps [ Y ], [ ˜ Y ] can actually be related, as we shall see next. Indeed, by defining the tensor expression (5.4) can be rewritten as Moreover, a direct calculation shows that the components (4.12) of [ Y ] in a basis { n, e A } of Γ( T N ) can be expressed in terms of Y as e A ( z ) 2 z 2 + s ( e A ) + zP ( ϕ /lscript , U ( e A , · )) , (5.8) Inserting (5.7)-(5.9) into (2.48) gives us the relation between the energy-momentum tensors τ , ˜ τ of the two shells. Specifically, for the dual basis { q , θ A } of { n, e A } one finds (recall that h AB def = γ ( e A , e B )) The results (5.7)-(5.10) turn out to be of particular interest when one of the matchings of ( M ± , g ± ) gives rise to no shell. In order to see this, let us assume that this is the case and take ˜ ϕ to be the diffeomorphism corresponding to the no-shell matching. Then, [ ˜ Y ] = 0 (i.e. ̂ Y + = Y -) holds, which means that the tensor Y is given by (cf. (5.5)) /star Consequently, when a no-shell matching is possible, the jump [ Y ] corresponding to any other possible matching is given by (5.12)-(5.14) in terms of the data fields { γ, /lscript , /lscript (2) , Y -} and the diffeomorphism ϕ . In other words, knowing the information about the no-shell matching automatically allows one to obtain the gravitational/matter-energy content of the remaining matchings by simply determining ϕ . In particular, there is no need to compute the new matching rigging ζ + or the tensor Y + to determine the shell properties. One simple needs to compute the right-hand sides of (5.12)-(5.14) using (5.11). We emphasize that (5.12)-(5.14) apply, in particular, when ( M ± , g ± ) are two regions of the same spacetime ( M , g ) and more than one matching can be performed. Then, the existence of a no-shell matching is always guaranteed, as one can always recover the full spacetime ( M , g ) from the matching of ( M ± , g ± ). This in fact occurs in all cut-and-paste constructions, which means that (5.12) -(5.14) provide the matter content of a null shell generated by any cut-and-paste matching procedure, as long as the two regions ( M ± , g ± ) of ( M , g ) can be pasted in more than one way . We conclude this section by discussing a particular situation of interest, namely the case when a null hypersurface data D = {N , γ, /lscript , /lscript (2) , Y -} can be embedded in two spacetimes ( M ± , g ± ) with embeddings ι ± (such that ι ± ( N ) are boundaries of M ± ) and riggings L ± with the appropriate orientation. This means that ( M ± , g ± ) can be matched so that the resulting spacetime contains no shell (because Y -is the same for both spacetimes). We assume, in addition, that D admits a vector field ξ ∈ Γ( T N ) with the property £ ξ γ = 0. The vector ξ defines a (local) one-parameter group of transformations { ϕ t } of N satisfying We now prove that, for each value of t , the diffeomorphism ϕ t gives rise to a matching. First, we define gauge parameters { z, V } according to (5.3) for ϕ = ϕ t . Then, it is immediate to check that (5.1) holds for ϕ = ϕ t and that z > 0 (because ϕ t depends continuously on t and ( ϕ /star t =0 /lscript )( n ) = /lscript ( n ) = 1). Therefore, conditions ( i ) and ( ii ) in Theorem 3.1 are both fulfilled (notice that, since L ± are matching riggings, one points inwards and the other outwards, so ( ii ) is just z > 0) and indeed each ϕ t corresponds to a different matching. The jump [ Y ] def = Y + -Y -where Y + def = 1 2 ϕ /star t ( ( ι + ) /star ( £ ζ + g + ) ) (and ζ + is given by (4.3)) rules the gravitational/matter-energy content of the resulting shell. The vector field ξ generates a multitude of new shells. The construction is further simplified when, in addtion to (5.15), it holds Then (5.11) implies which simplifies the expressions (5.12)-(5.14) considerably. One may wonder what is the final result when, in addition, ξ is the restriction to N of a Killing vector field ξ on M -(i.e. ι -/star ξ = ξ ) and £ ξ L -= 0 is fulfilled (so that (5.15) and (5.16) hold). It is straightforward to see that which combined with (5.3) means that z = 1, and V = 0, so Y = 0 (cf. (5.17)). Moreover, one can easily check that the terms in the right-hand side of (5.12)-(5.14) cancel out. Thus, the procedure gives rise to another no-shell matching, as one would expect because the transformation induced by ξ does not affect in any geometric way the spacetime ( M -, g -). This constitutes a non-trivial consistency check of equations (5.12)-(5.14).",
"pages": [
24,
25,
26
]
},
{
"title": "6 Null boundaries with product topology S × R",
"content": "In order to connect the results in this paper with those from [23], [24] (see Section 2.3), we now consider the case when the boundaries of the spacetimes to be matched can be foliated by cross-sections. In particular, we shall construct a step function H and provide explicit expressions for the gauge parameters { z, V } (cf. (4.2)). The results for the jump [ Y ] will be then compared with their counterparts from Proposition 2.17. Our setup for the present section is the following. We consider two spacetimes ( M ± , g ± ) with null boundaries ˜ N ± and assume that ˜ N ± have product topology S ± × R , where S ± are spacelike crosssections and the null generators are along R . We select two future null generators k ± ∈ Γ( T M ± ) | ˜ N ± of ˜ N ± and two cross-sections S ± 0 ⊂ ˜ N ± . We then construct foliation functions v ± ∈ F ( ˜ N ± ) by solving k ± ( v ± ) = 1 with initial values v ± | S ± 0 = 0. Finally, the riggings L ± are fixed by the conditions of being orthogonal to the respective leaves { v ± = const } , null and scaled to satisfy µ ± 1 def = g ± ( L ± , k ± ) = 1. We assume that ( M ± , g ± ) can be matched, so that conditions ( i )-( ii ) in Theorem 3.1 are fulfilled for a diffeomorphism ϕ : N -/shortrightarrow N verifying (4.1). This allows us to take two embeddings ι ± : N ↪ -/shortrightarrow M ± and construct the hypersurface data sets D = {N , γ, /lscript , /lscript (2) , Y -} , ̂ D = {N , ̂ γ, ̂ /lscript , ̂ /lscript (2) , ̂ Y + } according to (3.3)-(3.4). We also introduce the functions on N . Since by construction ι -/star ( n ) = k -and ι + /star ( ̂ n ) = k + (recall (2.20)), it is immediate to check that { λ, v } are foliation functions of N . Note that, also by construction, the data satisfies which has the following immediate consequences n ( λ ) = 1 , F = 0 , s = 0 , ̂ n ( v ) = 1 , ̂ F = 0 , ̂ s = 0 . (6.3) We now select vector fields { e A } tangent to the leaves { λ = const . } so that { n, e A } is a basis of Γ( T N ) satisfying [ n, e A ] = 0. As before, we let h be induced metric on { λ = const . } and ∇ h for its Levi-Civita derivative. In particular h AB def = γ ( e A , e B ) and we note that, for any f ∈ F ( N ), we can write e A ( f ) also as ∇ h A f . The pull-back of /lscript to the leaves of constant λ is zero, so /lscript A = ψ A = 0. This, together with /lscript (2) = 0 and (2.47), means that P = h AB e A ⊗ e B . Observe also that which in particular means that Inserting these properties in (4.2) fixes the matching gauge parameters to be n, ̂ ̂ Proof. Define the functions u def = ( ϕ -1 ) /star ( n ( H )) and χ ( A ) def = ( ϕ -1 ) /star ( e A ( H )), so that (6.7)-(6.8) can be The push-forward vector fields { ϕ /star n, ϕ /star e A } can also be computed in terms of the function H and the vector fields W A def = ̂ P ( W A , · ). The result is an easy consequence of Corollary 4.2 and reads ϕ /star n = ( ϕ -1 ) /star ( n ( H )) ̂ (6.7) ϕ /star e A = W A +( ϕ -1 ) /star ( e A ( H )) ̂ n. (6.8) Observe that { W A } are tangent to the leaves { v = const. } (because by Corollary 4.2 we know that 0 = ̂ P ( W A , ̂ /lscript ) = ̂ /lscript ( W A ) = W A ( v )). Let us now prove that ̂ n and W A commute. Lemma 6.1. The vector fields n and W A satisfy [ n, W A ] = 0 . written as n = u -1 ϕ /star n and W A = ϕ /star e A -χ ( A ) n . Thus, where in the last equality we used [ n, e A ] = 0 and ̂ n = u -1 ϕ /star n . To prove the claim we just need to show that the last parenthesis is zero. Indeed, By Remark 4.3 we also know that { n, W A } constitute a basis of Γ( T N ) and hence the vector fields form basis of Γ( T ˜ N ± ) respectively. Inserting (6.7)-(6.8) into (6.10) and using again that ι + /star ( ̂ n ) = k + , one obtains e + 1 = n ( H ) k + , e + A = e A ( H ) k + + ι + /star ( W A ) , (6.11) where for simplicity we have dropped pull-backs affecting functions. Given that { ι + /star W A } are linearly independent and tangent to the leaves { v + = const. } ⊂ ˜ N + , they can be decomposed in a basis { L + , k + , v + A } of Γ( T M + ) | ˜ N + satisfying (2.2) as ι + /star W A = b B A v + B , with { b B A } defining an invertible matrix. Moreover, b B A are constant along the null generators as a consequence of Lemma 6.1: + B + + ( b B A ) v + B ⇐⇒ k ( b B A ) = 0 . 0 = [ ι /star ( ̂ n ) , ι /star ( W A )] = [ k , b A v A ] = k The matching rigging ζ + , obtained by inserting (6.6) into (4.3) and using (6.7)-(6.8), (6.10)-(6.11), reads which one easily checks to be the same as (2.61) simply by noting that (in the notation of Section 2.3) our choice of L ± entails µ ± 1 = 1, µ ± A = 0 and that ι + /star W A = b B A v + B gives h AB = h IJ + ( b -1 ) A I ( b -1 ) B J . The expressions for [ Y ] are obtained as a particular case of Theorem 4.4. Theorem 6.2. In the setup and conditions of Theorem 4.4 suppose further that the boundaries ˜ N ± can be foliated by cross-sections and define λ, v, H ∈ F ( N ) as in (6.1) . Let h be the induced metric and ∇ h the corresponding Levi-Civita covariant derivative on the leaves { λ = const. } ⊂ N . Then, Let { e A } be vector fields in N such that { n, e A } is a basis adapted to the foliation { λ = const. } and define W A by means of (6.8) . Then the components the jump [ Y ] can be written as + Proof. Equation (6.13) follows at once after inserting (6.4)-(6.6) into (4.11). To obtain (6.14)-(6.16), it suffices to particularize (4.20)-(4.22) for z -1 = n ( H ), ϕ /star ̂ /lscript = dH , χ ( A ) = ( ϕ -1 ) /star ( e A ( H )), ϕ /star ̂ /lscript (2) = 0, s = 0 and P ( ϕ /star ̂ /lscript , · ) = h AB ( ∇ h A H ) e B and notice that £ n ( ϕ /star ̂ /lscript ) = £ n dH = d ( n ( H )), as well as e a A e b B · ∇ a · ∇ b H = ∇ h A ∇ h B H (see (2.53)). Before establishing the connection between (6.14)-(6.16) and the corresponding expressions in Proposition 2.17 we need to relate hypersurface data quantities with the tensors defined in (2.1). Lemma 6.3. Let {N , γ, /lscript , /lscript (2) , Y } be { φ, ζ } -embedded in ( M , g ) , k := φ /star n the corresponding null generator and ˜ κ k its surface gravity. Consider a transverse submanifold S ⊂ N and assume that the gauge is such that the rigging ζ is null and orthogonal to φ ( S ) . Then, for any basis { e A } of Γ( TS ) it holds (we identify scalars and vectors with their images on φ ( N ) ) Remark 6.4. This result is a particular case of a much more general analysis on the geometry of embedded submanifold in a hypersurface data set carried out in [28] . We include the proof for completeness. Claim ( a ) follows at once from (2.14) and (2.38) (note that here ν = k ). To prove ( b ) we compute Item ( c ) has already been stated after definition (2.21) and ( d ) follows from We are now in a position where the comparison can be made. We identify the vector fields { v -A } introduced in Section 2.3 with the push-forwards of { e A } , hence µ -1 = 1 and µ -A = 0. On the other hand, µ + 1 = 1 and µ + A = g + ( L + , v + A ) = ( b -1 ) B A ̂ /lscript ( W B ) = ( b -1 ) B A W B ( v ) = 0, so the covector q defined in Proposition 2.17 is simply q A = -∇ h A H . The vector X a in (2.65) is in turn given by Thus, expressions (2.66)-(2.68) become Particularizing Lemma 6.3 to the sections { λ = const } of D (with basis e A ) and the sections { v = const } of ̂ D (with basis W A ), and recalling that F = ̂ F = 0 (see (6.3)), it is straightforward to check that (6.18)(6.20) coincide with (6.14)-(6.16).",
"pages": [
27,
28,
29
]
},
{
"title": "7 Cut-and-paste matching: (anti-)de Sitter spacetime",
"content": "We have already mentioned that (5.12)-(5.14) hold for the specific case when the two spacetimes to be matched are actually two regions of the same spacetime (and more than one matching is allowed). In this section, our aim is to provide an example of a cut-and-paste construction, namely the matching of two regions of a constant-curvature spacetime across a totally geodesic null hypersurface. For previous works on the cut-and-paste construction describing non-expanding impulsive gravitational waves in constant curvature backgrounds we refer e.g. to [39], [37], [17] [43], [40] and references therein. In any constant curvature spacetime ( M , g ) there exists only one totally geodesic null hypersurface up to isometries (see e.g. [14], [32]). We denote one such hypersurface by ˜ N . Then, one can always construct coordinates {U , V , x A } adapted to ˜ N so that the metric is conformally flat and ˜ N def = {U = 0 } , namely Here Λ stands for the cosmological constant, so Λ = 0, Λ > 0, Λ < 0 correspond to Minkowski, de Sitter and anti-de Sitter spacetimes respectively. When Λ ≤ 0, the coordinates {U , V , x A } cover a whole neighbourhood of ˜ N . However, for the de Sitter case one needs to remove one generator of ˜ N because the topology of ˜ N is S n × R while stereographic coordinates only cover the sphere minus one point. In this section, we will analyze the three cases Λ = 0, Λ < 0 and Λ > 0 at once with the matching formalism introduced before. The induced metric on ˜ N reads ds 2 ˜ N = ( 1 + Λ 12 δ AB x A x B ) -2 δ AB dx A dx B , and obviously the topology of ˜ N is S × R , S being a spacelike section and the null generators being along R . Therefore, all results from Section 6 can be applied. Let us construct hypersurface data associated to ˜ N . Since ˜ N is embedded on ( M , g ), there exists an abstract manifold N and an embedding ι such that ι ( N ) = ˜ N . We can select ι to be as trivial as possible by constructing coordinates { λ, y A } on N so that ↦ We also need a choice of rigging vector field ζ along ˜ N . For convenience, we set ζ = -µ 2 ∂ U (observe that µ 2 | ˜ N = 0). The corresponding null metric hypersurface data (2.17) defined by {N , γ, /lscript , /lscript (2) } is where µ N def = ι /star µ = 1 + Λ 12 δ AB y A y B . Observe that ∂ λ ∈ Rad γ and /lscript ( ∂ λ ) = 1 imply that n = ∂ λ . Moreover, F = 0 and s = 0 (cf. (2.10)-(2.11)) and U = 0 as a consequence of (2.12). The tensor Y is obtained from (2.18). A simple calculation gives Cutting the spacetime across the hypersurface {U = 0 } leaves two spacetimes ( M ± , g ± ) defined to be the regions U /greaterlessequal 0 endowed with the metrics Obviously, the boundaries are ˜ N ± ≡ {U ± = 0 } . These two regions can clearly be matched so that the original spacetime (containing no shell) is obtained. Moreover, since ˜ N ± are totally geodesic we know that multiple matchings can be performed. We therefore proceed as in Section 5, i.e. we let the two /negationslash embeddings ι ± be given by ι ± = ι and take ζ -= -µ 2 -∂ U -, ˜ ζ + = -µ 2 + ∂ U + as the riggings defining the noshell matching, namely the matching for which [ ˜ Y ] = 0. Any other possible matching will be ruled by a diffeomorphism ϕ of N onto itself and it will correspond to a different rigging ζ + along ˜ N + . Specifically, the hypersurface data corresponding to the no-shell matching is D = {N , γ, /lscript , /lscript (2) , Y } , where { γ, /lscript , /lscript (2) } and Y are respectively given by (7.3) and (7.4), while the matter/gravitational content of the shell of any other possible matching (ruled by ϕ ) is given by the the jump [ Y ] def = Y + -Y with From Section 5, we know that there is no need to compute the new rigging ζ + or its corresponding Y + to determine the jump [ Y ], which is explicitly given by (5.12)-(5.14). Consequently, we only need to worry about the diffeomorphism ϕ . The only restriction that ϕ must satisfy is ϕ /star γ = γ , which in coordinates reads It follows that the components { ϕ A } cannot depend on the coordinate λ . In particular, if we let { h A ( y B ) } be a set of functions such that ( a ) the Jacobian matrix ∂ ( h 2 ,...,h n +1 ) ∂ ( y 2 ,...,y n +1 ) has non-zero determinant and ( b ) { h A ( y B ) } verify (1 + Λ 12 δ IJ y I y J ) -2 δ CD = (1 + Λ 12 δ IJ h I h J ) -2 ( ∂ y C h A )( ∂ y D h B ) δ AB , any diffeomorphism ϕ : N -/shortrightarrow N of the form ↦ with ∂ λ H = 0 fulfils ϕ /star γ = γ . A particular simple example is { h A = y A } , but many more exist. In fact since the metric on any section of ˜ N is of constant curvature, it is also maximally symmetric (and of dimension n -1) so h A ( y B ) can depend on n ( n -1) / 2 arbitrary parameters. For any possible choice of { h A ( y B ) } and an arbitrary step function H ( λ, y A ), the gauge parameters z and V are given by (6.6) for { n = ∂ λ , e A = ∂ y A } . In the present case the tensor Y is given by Y = 1 n ( H ) ϕ /star Y -Y (cf. (5.11)), so we need to compute the pull-back ϕ /star Y . Defining µ N def = 1 + Λ 12 δ AB h A h B , from (7.4) and (7.8) it is straightforward to get so that, multiplying (7.9)-(7.10) by 1 n ( H ) and subtracting Y (cf. (7.4)) yields Inserting these expressions into (5.12)-(5.14) and using n = ∂ λ , s = 0, U = 0 together with the identity ( £ n ϕ /star /lscript ) ( e A ) = ( £ n dH ) ( e A ) = d ( n ( H ))( e A ) = e A ( n ( H )) (here ϕ /star /lscript = dH by (6.4) and /lscript = /lscript ), one finds which can be interpreted as the sum of the jump corresponding to the matching of two regions of Minkowski across a null hyperplane (see [23, Eq. (6.6)]) plus the contribution of the tensor Y . Observe that Λ = 0 entails Y = 0, so in this way we recover expressions (6.6) in [23] for the most general planar shell in the spacetime of Minkowski /negationslash A direct computation that combines the definitions (2.49), (7.3) and (7.12) yields energy-density, energy flux and pressure (note that here we need to take /epsilon1 = -1) Observe that only the pressure is independent of the value of the cosmological constant Λ ( ρ and j depend on the conformal factor µ N and on Y ). The pressure p takes the same value for the matchings of two regions of (anti-)de Sitter or Minkowski (in fact, p coincides with the pressure obtained in [23, Sect. 6]). In particular, in the case h A = y A (i.e. when the mapping between null generators of both sides is trivial), then Y λy B = 0 (cf. (7.11)) and (7.13) simplifies to In the cut-and-paste constructions corresponding to constant-curvature spacetimes, the so-called Penrose's junction conditions (see e.g. [43], [40]) impose a jump in the coordinates across the shell. This jump is of the form V + | U + =0 = V -+ H ( x A -) | U -=0 . In the present case the matching embeddings φ -= ι and φ + = ι · ϕ are given by so the step function corresponding to Penrose's jump is H ( λ, y A ) = λ + H ( y A ), H ∈ F ( N ). In order to recover such an H , one needs that there is no energy flux and no pressure on the shell. Indeed, imposing this in (7.14) and integrating for H yields H ( λ, y A ) = aλ + H ( y A ), where H ∈ F ( N ) and a is a positive 4 constant. Thus, in this more general context with arbitrary cosmological constant, the Penrose's jump still describes either purely gravitational waves (when ρ , j and p are all zero) or shells of null dust (when j and p vanish but ρ = 0), analogously to what happened in [23, Sect. 6] for the Minkowski spacetime. /negationslash",
"pages": [
30,
31,
32
]
},
{
"title": "Acknowledgements",
"content": "The authors acknowledge financial support under Grant PID2021-122938NB-I00 funded by MCIN/AEI /10.13039/501100011033 and by 'ERDF A way of making Europe' and SA096P20 (JCyL). M. Manzano also acknowledges the Ph.D. grant FPU17/03791 (Spanish Ministerio de Ciencia, Innovaci'on y Universidades).",
"pages": [
32
]
}
]
2024PhRvD.109f4036M
https://arxiv.org/pdf/2312.06444.pdf
<document>
<section_header_level_1><location><page_1><loc_13><loc_92><loc_87><loc_93></location>Multiparameter multipolar test of general relativity with gravitational waves</section_header_level_1>
<text><location><page_1><loc_11><loc_89><loc_90><loc_90></location>Parthapratim Mahapatra, 1, ∗ Shilpa Kastha, 2 Anuradha Gupta, 3 B. S. Sathyaprakash, 4, 5, 6 and K. G. Arun 1, 4</text>
<text><location><page_1><loc_31><loc_87><loc_70><loc_88></location>1 Chennai Mathematical Institute, Siruseri, 603103, India</text>
<unordered_list>
<list_item><location><page_1><loc_16><loc_86><loc_85><loc_87></location>2 Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark</list_item>
<list_item><location><page_1><loc_15><loc_85><loc_86><loc_86></location>3 Department of Physics and Astronomy, The University of Mississippi, University, Mississippi 38677, USA</list_item>
</unordered_list>
<text><location><page_1><loc_28><loc_83><loc_73><loc_84></location>4 Institute for Gravitation and the Cosmos, Department of Physics,</text>
<text><location><page_1><loc_28><loc_82><loc_73><loc_83></location>Penn State University, University Park, Pennsylvania 16802, USA</text>
<text><location><page_1><loc_13><loc_81><loc_88><loc_82></location>5 Department of Astronomy and Astrophysics, Penn State University, University Park, Pennsylvania 16802, USA</text>
<text><location><page_1><loc_19><loc_79><loc_81><loc_80></location>6 School of Physics and Astronomy, Cardiff University, Cardiff, CF24 3AA, United Kingdom</text>
<text><location><page_1><loc_42><loc_78><loc_59><loc_79></location>(Dated: October 14, 2024)</text>
<text><location><page_1><loc_18><loc_60><loc_83><loc_77></location>Amplitude and phase of the gravitational waveform from compact binary systems can be decomposed in terms of their mass- and current-type multipole moments. In a modified theory of gravity, one or more of these multipole moments could deviate from general theory of relativity. In this work, we show that a waveform model that parametrizes the amplitude and phase in terms of the multipole moments of the binary can facilitate a novel multiparameter test of general relativity with exquisite precision. Using a network of next-generation gravitational-wave observatories, simultaneous deviation in the leading seven multipoles of a GW190814-like binary can be bounded to within 6%-40% depending on the multipole order, while supermassive black hole mergers observed by the Laser Interferometer Space Antenna achieve a bound of 0.3%-2%. We further argue that bounds from multipoles can be uniquely mapped onto other parametrized tests of general relativity and have the potential to become a downstream analysis from which bounds of other parametric tests of general relativity can be derived. The set of multipole parameters, therefore, provides an excellent basis to carry out precision tests of general relativity.</text>
<section_header_level_1><location><page_1><loc_20><loc_56><loc_37><loc_57></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_31><loc_49><loc_54></location>Gravitational waveform from a compact binary coalescence is a nonlinear function of 'radiative mass-' and 'current-type' multipole moments [1] and their derivatives with respect to time. The 'adiabatic inspiral' of the binary is well described by the post-Newtonian (PN) approximation to the general theory of relativity (GR) where the mass ratio and the spins of the binary constituents determine which multipoles are excited and what their contributions are to the emitted flux and the phase evolution of the binary. After the leading quadrupole, the mass-octupole is the next dominant contribution to the phase. As the binary becomes more asymmetric, the contributions from higher-order multipole moments become significant. Spins of the binary constituents can further enhance the strengths of certain higher-order multipoles, especially the current-type ones.</text>
<text><location><page_1><loc_9><loc_14><loc_49><loc_31></location>In a modified theory of gravity, where the compact binary dynamics differs from GR, it is natural to expect that one or more of these radiative multipole moments will deviate from those of GR [2-10]. Therefore, asking whether the measured multipole moments of compact binaries are consistent with GR predictions is an excellent way to test GR. References [11, 12] first derived a multipolar parametrized gravitational-wave phase, which separately tracks the contribution from different radiative multipole moments within the PN approximation to GR. This is achieved by associating parameters µ l and ϵ l with the mass- and current-type radiative multipole moments,</text>
<text><location><page_1><loc_52><loc_47><loc_92><loc_57></location>respectively. Here l = 2 , 3 , . . . denote quadrupole, octupole, etc. The phenomenological multipole parameters are equal to unity in GR (i.e., µ GR l ≡ 1 and ϵ GR l ≡ 1), by definition. By introducing deviations to these multipole coefficients, denoted as δµ l and δϵ l (i.e., µ l ≡ 1+ δµ l and ϵ l ≡ 1 + δϵ l ), one can use the gravitational-wave data to obtain bounds on these two sets of parameters.</text>
<text><location><page_1><loc_52><loc_24><loc_92><loc_46></location>The radiative multipole moments of compact binaries are nonlinear functionals of the source multipole moments (i.e., moments of the stress-energy tensor of the material source and its gravitational fields) and contain time derivatives of the source moments [13]. These time derivatives of the source multipole moments are evaluated using the equation of motion of the compact binary. Therefore, in the gravitational-wave generation formalism the radiative multipole moments of compact binaries also carry information about the conservative dynamics of the binary. Hence, the parameters δµ l and δϵ l are sensitive to deviations from GR in both the dissipative and the conservative sectors of the compact binary dynamics. However, one can use the parametrization introduced in Eq. (3.2) of Ref. [12] to track explicitly different PN pieces in the conserved orbital energy.</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_23></location>The most general test of GR one can perform, within this framework, is the one where all the δµ l and δϵ l are simultaneously measured, which is often referred to as a 'multiparameter test' (multiparameter tests have been discussed in the context of PN phase expansion in Refs. [14-17]). We explore the possibility of simultaneously estimating the leading seven multipole parameters (i.e., the leading four mass-type and the leading three current-type moments) with the present and next-generation gravitational-wave detectors. This</text>
<text><location><page_2><loc_9><loc_83><loc_49><loc_93></location>generalizes the single-parameter projections reported in Refs. [11, 12] and complements the consistency tests proposed in Refs. [18, 19] and the results from GW190412 and GW190814 being reported in Refs. [20, 21]. This work also extends the single-parameter octupolar bounds from GW190412 and GW190814 reported recently in Ref. [22].</text>
<text><location><page_2><loc_9><loc_66><loc_49><loc_83></location>The crucial ingredient in this work is the introduction of new parametrized multipolar amplitudes up to 2PN order recently computed in a companion paper [23], which enables us to use the multipolar information in both the amplitude and the phase to derive the bounds on the multipole parameters. Unlike the parametrizations that look for deviations either in phase [14, 24-35] or in amplitude [19, 21] of gravitational waveform independently, the multipolar parametrization has the advantage that the number of independent parameters is smaller , the same as the number of multipole parameters that appear in the amplitude and phase .</text>
<text><location><page_2><loc_9><loc_44><loc_49><loc_65></location>What makes the multiparameter tests very difficult to perform is the strong degeneracies introduced by the simultaneous inclusion of more phenomenological deformation parameters. Multiband gravitational-wave observations [15, 16] and principal component analysis [17, 3638] have been argued to be two different approaches to carry out multiparameter tests of GR in terms of deformations introduced directly in the PN expansion coefficients of the signals's phase evolution. Here, we investigate the use of multipole parameters, as opposed to the usual deformation parameters in the signal's phase, to carry out multiparameter tests of GR. Apart from being a more downstream parameter set, orthogonality of the multipole parameters may help in lifting the abovementioned degeneracies.</text>
<text><location><page_2><loc_9><loc_28><loc_49><loc_44></location>In this work, we show that the multipolar framework is a viable route to carry out a very generic multiparameter test of GR. We further argue that the bounds on δµ l and δϵ l can be mapped to other parametrized tests of GR. Therefore, this new class of tests may be thought of as an 'all-in-one' test of GR, which may be mapped to any parametrized test of interest. We explicitly demonstrate this mapping in the context of parametrized tests of PN phasing, which is currently employed on the gravitational-wave data and used to obtain constraints on specific modified theories of gravity [39].</text>
<text><location><page_2><loc_9><loc_19><loc_49><loc_28></location>The remainder of the paper is organized as follows. In Sec. II, we briefly describe the parametrized multipolar waveform model. In Sec. III, we briefly explain the parameter estimation scheme used in our analysis. We discuss our results in Sec. IV. Our conclusions are presented in Sec. V.</text>
<section_header_level_1><location><page_2><loc_18><loc_15><loc_39><loc_16></location>II. WAVEFORM MODEL</section_header_level_1>
<text><location><page_2><loc_9><loc_9><loc_49><loc_13></location>We use the frequency-domain amplitude-corrected multipolar waveform for spinning, nonprecessing compact binaries recently reported in Ref. [23]. This wave-</text>
<text><location><page_2><loc_52><loc_86><loc_92><loc_93></location>form model is 3.5PN accurate in the phase and 2PN accurate in the amplitude (i.e., includes the contributions from the first six harmonics). The amplitude-corrected multipolar polarizations in the frequency domain up to 2PN schematically reads [40-42]</text>
<formula><location><page_2><loc_54><loc_79><loc_92><loc_85></location>˜ h + , × ( f ) = G 2 M 2 c 5 D L √ 5 π ν 48 4 ∑ n =0 6 ∑ k =1 V n -7 / 2 k H ( k,n ) + , × × e i ( k Ψ SPA ( f/k ) -π/ 4 ) . (1)</formula>
<text><location><page_2><loc_52><loc_39><loc_92><loc_77></location>Here M , ν (= q (1+ q ) 2 with q being the ratio between the primary and secondary mass), and D L denote the redshifted total mass, symmetric mass ratio, and the luminosity distance of the source, respectively. The indices n and k indicate the n 2 th PN order and harmonics of the orbital phase, respectively. The parameter V k = (2 π GM f/c 3 k ) 1 / 3 is the dimensionless gauge invariant PN parameter for the k th harmonic [40], G is the gravitational constant, c is the speed of light, and f is the gravitational-wave frequency. The coefficients H ( k,n ) + , × denote the amplitude corrections in the frequency-domain polarizations associated with the contribution from k th harmonic at n 2 th PN order. These amplitude coefficients are functions of the masses, spins, and orbital inclination angle ι and, in our parametrization, contain the multipole parameters µ l and ϵ l . The expressions for all the H ( k,n ) + , × can be found in Eqs. (10) and (11) of Ref. [23]. Lastly, Ψ SPA ( f ) represents the frequency-domain parametrized multipolar gravitational-wave phasing for the first harmonic. References [11, 12] obtained the 3.5PN accurate expression of Ψ SPA ( f ) for nonprecessing, spinning binaries using the stationary phase approximation. In the spirit of null tests, the multipolar polarizations in Eq. (1) are reexpressed in terms of { δµ l , δϵ l } with the goal of deducing projected bounds on them from gravitationalwave observations.</text>
<text><location><page_2><loc_52><loc_36><loc_92><loc_39></location>The gravitational-wave strain in the frequency domain measured by a detector D is given by</text>
<formula><location><page_2><loc_57><loc_30><loc_92><loc_35></location>˜ h D ( f ) = F lp ( f ; θ, ϕ ) [ ˜ h + ( f ) F + ( f ; θ, ϕ, ψ ) + ˜ h × ( f ) F × ( f ; θ, ϕ, ψ ) ] , (2)</formula>
<text><location><page_2><loc_52><loc_20><loc_92><loc_29></location>where F lp is the location phase factor of the detector, F + and F × are the antenna response functions that describe the detector's sensitivity to the two different polarizations, θ is the declination angle, ϕ is the right ascension, and ψ is the polarization angle (see Sec. III of Ref. [43] for more details).</text>
<text><location><page_2><loc_52><loc_9><loc_92><loc_20></location>Indeed, our inspiral-only waveform model ignores the contributions from the merger and ringdown phases of the compact binary dynamics, the inclusion of which can lead to a considerable increase in the signal-to-noise ratio (SNR). However, as we crucially make use of the multipole structure in PN theory, it is only natural to employ inspiral-only waveforms for a proof-of-concept study like this, provided we restrict ourselves to binaries</text>
<table>
<location><page_3><loc_9><loc_78><loc_49><loc_93></location>
<caption>TABLE I. A summary of the four networks of ground-based gravitational-wave detectors used in our analysis. The detector location determines the detector antenna patterns and location phase factors, whereas the PSD technology specifies the used power spectral density. Cosmic Explorer, CE; Einstein Telescope, ET. See Ref. [50] for more details.</caption>
</table>
<text><location><page_3><loc_9><loc_56><loc_49><loc_65></location>that are dominated by their inspiral. Finally, for simplicity, we only consider nonprecessing binary configurations in quasicircular orbits. It is likely that precessionand eccentricity-induced modulations may improve the bounds reported, though the magnitude of this needs to be quantified by a dedicated study.</text>
<section_header_level_1><location><page_3><loc_15><loc_50><loc_43><loc_51></location>III. PARAMETER ESTIMATION</section_header_level_1>
<text><location><page_3><loc_9><loc_36><loc_49><loc_47></location>To compute the statistical errors on various multipole deformation parameters and other relevant binary parameters, we use the semi-analytical Fisher information matrix formalism [51-54]. In the high SNR limit, the Fisher information matrix is a computationally inexpensive method to predict the statistical uncertainties (1 σ error bars) on the parameters of a signal model buried in stationary Gaussian noise.</text>
<text><location><page_3><loc_9><loc_31><loc_49><loc_35></location>For a frequency-domain gravitational-wave signal ˜ h D ( f ), described by a set of parameters ⃗ λ , the Fisher matrix is defined as</text>
<formula><location><page_3><loc_9><loc_24><loc_49><loc_29></location>Γ mn = 2 ∫ f max f min ˜ h D,m ( f ) ˜ h ∗ D,n ( f ) + ˜ h ∗ D,m ( f ) ˜ h D,n ( f ) S h ( f ) df , (3)</formula>
<text><location><page_3><loc_9><loc_12><loc_49><loc_23></location>where S h ( f ) is the one-sided noise power spectral density (PSD) of the detector, and f min and f max are the lower and upper limits of integration. In the above equation, ' ∗ ' denotes the operation of complex conjugation, and ',' denotes differentiation with respect to various elements in the parameter set ⃗ λ ≡ { λ m } . The 1 σ statistical error in λ m is σ m = √ Σ mm , where the covariance matrix Σ mn = (Γ mn ) -1 is the inverse of the Fisher matrix.</text>
<text><location><page_3><loc_9><loc_9><loc_49><loc_11></location>To estimate the errors on all multipole deformation parameters simultaneously, we have considered the fol-</text>
<text><location><page_3><loc_52><loc_92><loc_69><loc_93></location>lowing parameter space:</text>
<formula><location><page_3><loc_55><loc_86><loc_92><loc_91></location>⃗ λ = { t c , ϕ c , log M c , ν, χ 1 z , χ 2 z , log D L , cos ι, cos θ, ϕ, ψ, { δµ l , δϵ l } } , (4)</formula>
<text><location><page_3><loc_52><loc_79><loc_92><loc_84></location>where, t c is the time of coalescence, ϕ c is the phase at coalescence, M c = Mν 3 / 5 is the redshifted chirp mass, and χ 1 z and χ 2 z are the individual spin components along the orbital angular momentum. 1</text>
<text><location><page_3><loc_52><loc_43><loc_92><loc_79></location>For the computation of the statistical errors in the various parameters for different binary configurations and networks of ground-based gravitational-wave detectors, we use GWBENCH [43], a publicly available PYTHON -based package that computes the Fisher matrix and the corresponding covariance matrix for a given gravitationalwave network. The plus and cross polarizations in Eq. (1) are added into GWBENCH for this purpose. We have chosen f min to be 5 Hz and f max to be 6 F ISCO Hz for all the ground-based network configurations. Here F ISCO is the redshifted Kerr innermost stable circular orbit (ISCO) frequency [55-57] and its explicit expression for nonprecessing binaries can be found in Appendix C of Ref. [58]. For the sources observed by the space-based Laser Interferometer Space Antenna (LISA), we have used Eq. (2.15) of Ref. [59] and have taken f low = 10 -4 Hz and T obs = 4 yr to estimate f min . In the LISA band, f max is given by the smaller of 6 F ISCO and 0.1 Hz. We have summarized the different networks of ground-based detectors considered here in Table I. The noise PSDs of various ground-based detectors used here can be found in GWBENCH [43]. We have adopted the non-sky-averaged noise PSD of LISA reported in Ref. [60] [see Eqs. (1)-(5) of [60]] and ignored its orbital motion in our computation.</text>
<text><location><page_3><loc_52><loc_36><loc_92><loc_43></location>If we assume that all of the multipole deviation parameters take the same value for different events in a population, one can compute a joint bound on them by multiplying the corresponding 1D likelihoods. The width of the joint likelihood is given by</text>
<formula><location><page_3><loc_57><loc_30><loc_92><loc_34></location>σ a = [ N ∑ i =1 ( σ ( i ) a ) -2 ] -1 2 , a ∈ { δµ l , δϵ l } (5)</formula>
<text><location><page_3><loc_52><loc_27><loc_92><loc_29></location>where i = 1 , . . . , N denotes the events considered in the compact binary population.</text>
<section_header_level_1><location><page_3><loc_57><loc_23><loc_86><loc_24></location>IV. RESULTS AND DISCUSSIONS</section_header_level_1>
<text><location><page_3><loc_52><loc_18><loc_92><loc_20></location>We start by discussing the projected bounds on the multipole deformation parameters from GW190412- [61]</text>
<figure>
<location><page_4><loc_10><loc_71><loc_48><loc_93></location>
<caption>FIG. 1. Multiparameter bounds on different multipolar deformation parameters for GW190412- and GW190814-like systems in different networks of future gravitational-wave detectors. Median values from the synthesized population of 100 events is reported (see text for details). Different markers denote different networks considered here.</caption>
</figure>
<text><location><page_4><loc_9><loc_25><loc_49><loc_58></location>and GW190814-like systems [62], two asymmetric compact binary mergers detected in the third observing run by LIGO/Virgo observatories, in different networks of future ground-based gravitational-wave detectors. As these types of events have been confirmed to exist and extensively studied, they help us to understand the importance of the results. As the observed strengths of the higherorder multipoles depend crucially on the inclination angle ι and the SNR of the observed gravitational-wave signal depends on the location of the source, we synthesize a population for these two representative systems and use the median value of the resulting distribution to assess the measurement uncertainty in various multipole deformation parameters. Toward this, for each of the systems, we draw 100 samples distributed isotropically over the sphere for the orientation and location of the source. The component masses and spins and the luminosity distances are fixed at the median values reported by Refs. [61-63]. For each sample, we estimate the 1 σ statistical errors in the seven multipole deformation parameters simultaneously and then compute the median of these 1 σ errors from the 100 samples. The results for different detector networks are shown in Fig. 1.</text>
<text><location><page_4><loc_9><loc_9><loc_49><loc_24></location>We can measure all seven multipole deformation parameters simultaneously for a GW190814-like system to within ∼ 40% accuracy in 4020ET, whereas for GW190412-like binaries all multipole deformation parameters can be measured simultaneously to within ∼ 70% in 4020ET. Therefore, a single detection of a GW190412- or GW190814-like binary in the nextgeneration (XG) gravitational-wave detectors will allow us to measure all seven multipole deformation parameters simultaneously and hence to perform the most generic multiparameter test of gravitational-wave phase and am-</text>
<text><location><page_4><loc_52><loc_77><loc_92><loc_93></location>ude evolution in GR. It is seen that the mass-type multipole deformation parameters are always estimated better as compared to the current-type multipole deformation parameters. This should be due to the dominance of the mass-type moments over the current-type ones on the dynamics of the binary system. In terms of different detector networks, the 40LET bounds are comparable to those from 4020ET, which suggests that two thirdgeneration detectors already provide very precise bounds and the sensitivity of the third detector does not have a significant impact on the joint bounds.</text>
<text><location><page_4><loc_52><loc_63><loc_92><loc_77></location>Next, we consider three different classes of compact binary populations, neutron star-black holes (NSBHs), binary black holes (BBHs), and intermediate mass binary black holes (IMBBHs), reported in Ref. [50] (see Supplemental Material for details of the population). For each class of the compact binary population, we select 200 loudest events in the respective network of ground-based detectors and calculate the combined bounds on all seven multipole deformation parameters simultaneously using Eq. (5).</text>
<text><location><page_4><loc_52><loc_34><loc_92><loc_62></location>Figure 2 shows the combined bounds on multipole deformation parameters for these three types of compact binary populations in different networks. We can constrain all the multipole moments simultaneously within an accuracy of ∼ 20% in the XG era from the NSBH population. The BBH population considered here mostly contains equal-mass binaries, and therefore, they provide the best constraint on δµ 2 . Binaries in the IMBBH population are more massive than the other two populations and are also more asymmetric than the BBH population. As asymmetric massive binaries carry stronger signatures of higher-order multipoles, we obtain the best bounds on higher-order multipole deformation parameters from the IMBBH population-all multipole deformation parameters can be measured simultaneously to within ∼ 8% in the XG era. The NSBH population consists mainly of high mass ratio, but less massive, systems than the other two populations. As a result, they provide bounds similar to BBH population on higher-order multipole deformation parameters.</text>
<text><location><page_4><loc_52><loc_12><loc_92><loc_33></location>The merger rates of supermassive binary black holes (SMBBHs) and their detection rates in LISA are highly uncertain. Here we consider a few representative SMBBH systems and compute the projected error bars on various multipole deformation parameters. We consider merging SMBBHs at a luminosity distance of 3 Gpc with two different choices of spins ( χ 1 z = 0 . 2, χ 2 z = 0 . 1) and ( χ 1 z = 0 . 8, χ 2 z = 0 . 7). For each pair of spins, we choose two different mass ratios 2 and 5. All the angles (i.e., ι , θ , ϕ , ψ ) are set to be π/ 6. The 1 σ errors in all seven deformation parameters in the LISA band for various SMBBH configurations are shown in Fig. 3. We find that for most of the SMBBH systems considered here, LISA will be able to measure all seven multipole moments simultaneously to within ∼ 10%.</text>
<text><location><page_4><loc_52><loc_9><loc_92><loc_11></location>We next discuss how bounds on the PN deformations may be derived from the multipole bounds. In prin-</text>
<figure>
<location><page_5><loc_9><loc_74><loc_92><loc_93></location>
<caption>FIG. 2. Combined multiparameter bounds on different multipole deformation parameters for three distinct types of compact binary population in different networks of future ground-based gravitational-wave detectors. Population models described in Ref. [50] are employed, and the loudest 200 events in each category of the source population is combined to obtain the results shown.</caption>
</figure>
<text><location><page_5><loc_90><loc_56><loc_92><loc_57></location>/circledot</text>
<figure>
<location><page_5><loc_9><loc_39><loc_91><loc_65></location>
<caption>FIG. 3. Projected multiparameter constraints on various multipolar deformation parameters for SMBBHs in the LISA band. All the sources are considered to be at a fixed luminosity distance of 3 Gpc. All the angles specifying the binary's orientation and location in the sky are chosen to be π/ 6 as a representative angular configuration.</caption>
</figure>
<text><location><page_5><loc_9><loc_10><loc_49><loc_30></location>ciple, any PN parametrized test of gravitational-wave phase or amplitude can be effectively recast in terms of the multipole parameters. All we need for this is to derive a relation between those phenomenological parameters in the phase or amplitude and { δµ l , δϵ l } . If the parametric form of the phase or amplitude for any test and the contribution of different multipoles to the gravitational-wave phase [11, 12] and amplitude [23] are known, this derivation is straightforward. Here, as a proof-of-principle demonstration, we show how the constraints on { δµ l , δϵ l } can be mapped onto the different PN deformation parameters δ ˆ ϕ b in the phase evolution (where b ∈ 0 , 2 , 3 , 4 , 5 l, 6 , 6 l, 7 denotes different PN orders).</text>
<text><location><page_5><loc_52><loc_9><loc_92><loc_30></location>Given the gravitational-wave data d , we are interested in computing ˜ P ( δ ˆ ϕ b | d, H ), the posterior probability distribution of δ ˆ ϕ b , for a uniform prior on δ ˆ ϕ b ( H denotes the hypothesis, which is the parametric model we employ). Once we have the posterior samples for the joint probability distribution ˜ P ( ⃗ λ I , ⃗ λ T | d, H ) for uniform priors on ⃗ λ I ∈ { ν, χ 1 z , χ 2 z } and ⃗ λ T ∈ { δµ l , δϵ l } , we can compute the posteriors on δ ˆ ϕ b , P ( δ ˆ ϕ b | d, H ), using the relation between δ ˆ ϕ b and { ⃗ λ I , ⃗ λ T } . As δ ˆ ϕ b is a unique nonlinear function of { ⃗ λ I , ⃗ λ T } , a uniform prior on { ⃗ λ I , ⃗ λ T } does not translate into a uniform prior on δ ˆ ϕ b . Therefore, to obtain ˜ P ( δ ˆ ϕ b | d, H ) we need to reweight the samples of P ( δ ˆ ϕ b | d, H ) by the samples of δ ˆ ϕ b derived from the</text>
<figure>
<location><page_6><loc_10><loc_73><loc_48><loc_93></location>
<caption>FIG. 4. Violin plots for the posterior probability distributions of δ ˆ ϕ b obtained through the mapping of the multipole deformation bounds for a next-generation detector configuration consisting of two CE and one ET detector ( 4020ET ). The horizontal bars indicate the median values and 90% credible intervals.</caption>
</figure>
<text><location><page_6><loc_9><loc_55><loc_49><loc_61></location>uniform prior on { ⃗ λ I , ⃗ λ T } , using the relation between δ ˆ ϕ b and { ⃗ λ I , ⃗ λ T } . A more detailed discussion about the reweighting procedure is provided in the Supplemental Material.</text>
<text><location><page_6><loc_9><loc_28><loc_49><loc_54></location>We consider GW190412- and GW190814-like systems in the 4020ET network and compute the Fisher matrix Γ mn to construct the Gaussian probability distribution function p ( ⃗ λ ) ∝ e -1 2 Γ mn ( λ m -λ m inj )( λ n -λ n inj ) , where λ m inj are the injected parameter values. We marginalize the distribution p ( ⃗ λ ) over parameters other than { ⃗ λ I , ⃗ λ T } to get ˜ P ( ⃗ λ I , ⃗ λ T | d, H ). Next, we calculate P ( δ ˆ ϕ b | d, H ) using the samples of ˜ P ( ⃗ λ I , ⃗ λ T | d, H ). To obtain ˜ P ( δ ˆ ϕ b | d, H ) that assumes a uniform prior on δ ˆ ϕ b between [ -10 , 10], we reweight the distribution P ( δ ˆ ϕ b | d, H ) by the distribution of δ ˆ ϕ b derived from the following prior distributions: ν is uniform between [0 . 045 , 0 . 25], χ 1 z and χ 2 z are uniform between [ -0 . 99 , 0 . 99], and ⃗ λ T are uniform between [ -10 , 10]. The posterior distribution ˜ P ( δ ˆ ϕ b | d, H ) of different δ ˆ ϕ b are shown in Fig. 4. All the δ ˆ ϕ b probability distributions are constrained to better than 0.5 at 80% credibility.</text>
<text><location><page_6><loc_9><loc_9><loc_49><loc_27></location>Despite the reweighting employed, the mapped bounds derived here need not match with the regular multiparameter phasing tests using either ground-based or spacebased detector alone, where different phasing deformation parameters are treated as independent parameters. This should not be surprising, as the proposed mapping accounts only for the relation between the multipole and the phase deformation parameters and not the correlations these two sets of parameters would have with other binary parameters when the test is performed in the corresponding bases. We have checked that the bounds on the phase deformation parameters derived from the multipole bounds are overall much tighter than those that</text>
<text><location><page_6><loc_52><loc_90><loc_92><loc_93></location>follow from directly sampling over all of them simultaneously.</text>
<text><location><page_6><loc_52><loc_77><loc_92><loc_90></location>In the case of other parametrized tests of GR that rely on spin-induced multipole moments [64-67], modified dispersion relations [68-71], subdominant harmonics [19, 21], etc., the same method will work to derive the corresponding bounds from the multipole ones. In this case, one may visualize the test to be capturing a GR deviation via some effective multipolar deformation. A detailed study of these maps and their meanings will be taken up as a follow-up project.</text>
<section_header_level_1><location><page_6><loc_57><loc_72><loc_86><loc_74></location>V. CONCLUSIONS AND FUTURE DIRECTIONS</section_header_level_1>
<text><location><page_6><loc_52><loc_58><loc_92><loc_70></location>This work serves as a proof-of-concept for the ability of the multipolar framework to carry out a robust multiparameter test of GR with impressive precision, which is necessary to accomplish meaningful constraints on the parameter space of alternate theories of gravity. Moreover, as shown, the bounds from such tests can be uniquely mapped onto the other parametrized tests of GR that rely on amplitude or phase deformations.</text>
<text><location><page_6><loc_52><loc_41><loc_92><loc_58></location>In this work, we have employed the Fisher matrix formalism and a nonprecessing inspiral waveform for parameter estimation. While this paper is meant to illustrate the potential power of the multipolar approach, the results presented here should be revisited using the Bayesian framework with more realistic inspiral-mergerringdown waveforms. Moreover, the systematic biases induced due to the neglect of well-known effects such as spin precession and eccentricity need to be understood. Hence, the expected constraints that we report here are only indicative of the potential of the multipolar framework.</text>
<text><location><page_6><loc_52><loc_31><loc_92><loc_41></location>A natural next step is to construct a parametrized multipolar inspiral-merger-ringdown waveform that includes the effects of spin precession and eccentricity for gravitational-wave data analysis as well as employ stateof-the-art Bayesian parameter inference techniques to demonstrate the feasibility of the method and apply it on a selected subset of gravitational-wave events.</text>
<section_header_level_1><location><page_6><loc_62><loc_26><loc_82><loc_27></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_6><loc_52><loc_9><loc_92><loc_24></location>The authors thank Elisa Maggio for a critical reading of the manuscript and providing useful comments. We also thank Ish Gupta for a discussion of the compact binary populations used here. P.M. thanks Ssohrab Borhanian for a discussion of GWBENCH . P.M. also thanks Alok Laddha for valuable discussions. P.M. and K.G.A. acknowledge the support of the Core Research Grant No. CRG/2021/004565 of the Science and Engineering Research Board of India and a grant from the Infosys Foundation. K.G.A. acknowledges support from the Department of Science and Technology and Science</text>
<text><location><page_7><loc_9><loc_63><loc_49><loc_93></location>and Engineering Research Board (SERB) of India via the following grants: Swarnajayanti Fellowship Grant No. DST/SJF/PSA-01/2017-18 and MATRICS grant (Mathematical Research Impact Centric Support) No. MTR/2020/000177. S.K. acknowledges support from the Villum Investigator program supported by the VILLUM Foundation (Grant No. VIL37766) and the DNRF Chair program (Grant No. DNRF162) by the Danish National Research Foundation. This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie SklodowskaCurie Grant Agreement No. 101131233. K.G.A. and B.S.S. acknowledge the support of the Indo-U.S. Science and Technology Forum through the Indo-U.S. Centre for Gravitational-Physics and Astronomy, Grant No. IUSSTF/JC-142/2019. We also acknowledge NSF support via NSF Grants No. AST-2205920 and No. PHY2308887 to A.G. and No. AST-2307147, No. PHY2012083, No. PHY-2207638, No. PHY-2308886, and No. PHYS-2309064 to B.S.S. This manuscript has the LIGO preprint number P2300424.</text>
<text><location><page_7><loc_9><loc_57><loc_49><loc_63></location>The author is grateful for computational resources provided by the LIGO Laboratory and supported by National Science Foundation Grants No. PHY-0757058 and No. PHY-0823459.</text>
<text><location><page_7><loc_10><loc_56><loc_49><loc_57></location>This research has made use of data or software ob-</text>
<unordered_list>
<list_item><location><page_7><loc_10><loc_49><loc_43><loc_50></location>[1] K. S. Thorne, Rev. Mod. Phys. 52 , 299 (1980).</list_item>
<list_item><location><page_7><loc_10><loc_46><loc_49><loc_49></location>[2] S. Endlich, V. Gorbenko, J. Huang, and L. Senatore, JHEP 09 , 122 (2017), arXiv:1704.01590 [gr-qc].</list_item>
<list_item><location><page_7><loc_10><loc_44><loc_49><loc_46></location>[3] G. Comp'ere, R. Oliveri, and A. Seraj, JHEP 05 , 054 (2018), arXiv:1711.08806 [hep-th].</list_item>
<list_item><location><page_7><loc_10><loc_41><loc_49><loc_43></location>[4] L. Bernard, Phys. Rev. D 98 , 044004 (2018), arXiv:1802.10201 [gr-qc].</list_item>
<list_item><location><page_7><loc_10><loc_38><loc_49><loc_41></location>[5] F.-L. Juli'e and E. Berti, Phys. Rev. D 100 , 104061 (2019), arXiv:1909.05258 [gr-qc].</list_item>
<list_item><location><page_7><loc_10><loc_35><loc_49><loc_38></location>[6] B. Shiralilou, T. Hinderer, S. M. Nissanke, N. Ortiz, and H. Witek, Class. Quant. Grav. 39 , 035002 (2022), arXiv:2105.13972 [gr-qc].</list_item>
<list_item><location><page_7><loc_10><loc_32><loc_49><loc_34></location>[7] E. Battista and V. De Falco, Phys. Rev. D 104 , 084067 (2021), arXiv:2109.01384 [gr-qc].</list_item>
<list_item><location><page_7><loc_10><loc_29><loc_49><loc_32></location>[8] L. Bernard, L. Blanchet, and D. Trestini, JCAP 08 , 008 (2022), arXiv:2201.10924 [gr-qc].</list_item>
<list_item><location><page_7><loc_10><loc_25><loc_49><loc_29></location>[9] F.-L. Juli'e, V. Baibhav, E. Berti, and A. Buonanno, Phys. Rev. D 107 , 104044 (2023), arXiv:2212.13802 [grqc].</list_item>
<list_item><location><page_7><loc_9><loc_23><loc_49><loc_25></location>[10] R. F. Diedrichs, D. Schmitt, and L. Sagunski, (2023), arXiv:2311.04274 [gr-qc].</list_item>
<list_item><location><page_7><loc_9><loc_19><loc_49><loc_22></location>[11] S. Kastha, A. Gupta, K. G. Arun, B. S. Sathyaprakash, and C. Van Den Broeck, Phys. Rev. D 98 , 124033 (2018), arXiv:1809.10465 [gr-qc].</list_item>
<list_item><location><page_7><loc_9><loc_15><loc_49><loc_18></location>[12] S. Kastha, A. Gupta, K. G. Arun, B. S. Sathyaprakash, and C. Van Den Broeck, Phys. Rev. D 100 , 044007 (2019).</list_item>
<list_item><location><page_7><loc_9><loc_12><loc_49><loc_14></location>[13] L. Blanchet, T. Damour, and B. R. Iyer, Phys. Rev. D 51 , 5360 (1995), gr-qc/9501029.</list_item>
<list_item><location><page_7><loc_9><loc_9><loc_49><loc_12></location>[14] K. G. Arun, B. R. Iyer, M. S. S. Qusailah, and B. S. Sathyaprakash, Class. Quantum Grav. 23 , L37 (2006),</list_item>
</unordered_list>
<text><location><page_7><loc_52><loc_57><loc_92><loc_93></location>tained from the Gravitational Wave Open Science Center (gwosc.org), a service of the LIGO Scientific Collaboration, the Virgo Collaboration, and KAGRA. This material is based upon work supported by NSF's LIGO Laboratory which is a major facility fully funded by the National Science Foundation, as well as the Science and Technology Facilities Council (STFC) of the United Kingdom, the Max-Planck-Society (MPS), and the State of Niedersachsen/Germany for support of the construction of Advanced LIGO and construction and operation of the GEO600 detector. Additional support for Advanced LIGO was provided by the Australian Research Council. Virgo is funded, through the European Gravitational Observatory (EGO), by the French Centre National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale di Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions by institutions from Belgium, Germany, Greece, Hungary, Ireland, Japan, Monaco, Poland, Portugal, and Spain. KAGRA is supported by Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan Society for the Promotion of Science (JSPS) in Japan; National Research Foundation (NRF) and Ministry of Science and ICT (MSIT) in Korea; Academia Sinica (AS) and National Science and Technology Council (NSTC) in Taiwan.</text>
<text><location><page_7><loc_55><loc_49><loc_65><loc_50></location>gr-qc/0604018.</text>
<unordered_list>
<list_item><location><page_7><loc_52><loc_45><loc_92><loc_49></location>[15] A. Gupta, S. Datta, S. Kastha, S. Borhanian, K. G. Arun, and B. S. Sathyaprakash, Phys. Rev. Lett. 125 , 201101 (2020), arXiv:2005.09607 [gr-qc].</list_item>
<list_item><location><page_7><loc_52><loc_41><loc_92><loc_45></location>[16] S. Datta, A. Gupta, S. Kastha, K. G. Arun, and B. S. Sathyaprakash, Phys. Rev. D 103 , 024036 (2021), arXiv:2006.12137 [gr-qc].</list_item>
<list_item><location><page_7><loc_52><loc_37><loc_92><loc_41></location>[17] M. Saleem, S. Datta, K. G. Arun, and B. S. Sathyaprakash, Phys. Rev. D 105 , 084062 (2022), arXiv:2110.10147 [gr-qc].</list_item>
<list_item><location><page_7><loc_52><loc_33><loc_92><loc_37></location>[18] S. Dhanpal, A. Ghosh, A. K. Mehta, P. Ajith, and B. S. Sathyaprakash, Phys. Rev. D 99 , 104056 (2019), arXiv:1804.03297 [gr-qc].</list_item>
<list_item><location><page_7><loc_52><loc_29><loc_92><loc_33></location>[19] T. Islam, A. K. Mehta, A. Ghosh, V. Varma, P. Ajith, and B. S. Sathyaprakash, Phys. Rev. D 101 , 024032 (2020), arXiv:1910.14259 [gr-qc].</list_item>
<list_item><location><page_7><loc_52><loc_27><loc_92><loc_29></location>[20] C. D. Capano and A. H. Nitz, Phys. Rev. D 102 , 124070 (2020), arXiv:2008.02248 [gr-qc].</list_item>
<list_item><location><page_7><loc_52><loc_21><loc_92><loc_26></location>[21] A. Puecher, C. Kalaghatgi, S. Roy, Y. Setyawati, I. Gupta, B. S. Sathyaprakash, and C. Van Den Broeck, Phys. Rev. D 106 , 082003 (2022), arXiv:2205.09062 [grqc].</list_item>
<list_item><location><page_7><loc_52><loc_19><loc_92><loc_21></location>[22] P. Mahapatra, Phys. Rev. D 109 , 024050 (2024), arXiv:2306.04703 [gr-qc].</list_item>
<list_item><location><page_7><loc_52><loc_16><loc_92><loc_18></location>[23] P. Mahapatra and S. Kastha, Phys. Rev. D 109 , 084069 (2024), arXiv:2311.04672 [gr-qc].</list_item>
<list_item><location><page_7><loc_52><loc_13><loc_92><loc_16></location>[24] L. Blanchet and B. S. Sathyaprakash, Class. Quantum Grav. 11 , 2807 (1994).</list_item>
<list_item><location><page_7><loc_52><loc_11><loc_92><loc_13></location>[25] L. Blanchet and B. S. Sathyaprakash, Phys. Rev. Lett. 74 , 1067 (1995).</list_item>
<list_item><location><page_7><loc_52><loc_9><loc_92><loc_10></location>[26] K. G. Arun, B. R. Iyer, M. S. S. Qusailah, and B. S.</list_item>
<list_item><location><page_8><loc_12><loc_91><loc_49><loc_93></location>Sathyaprakash, Phys. Rev. D 74 , 024006 (2006), grqc/0604067.</list_item>
<list_item><location><page_8><loc_9><loc_88><loc_49><loc_90></location>[27] N. Yunes and F. Pretorius, Phys. Rev. D 80 , 122003 (2009), arXiv:0909.3328 [gr-qc].</list_item>
<list_item><location><page_8><loc_9><loc_84><loc_49><loc_88></location>[28] C. K. Mishra, K. G. Arun, B. R. Iyer, and B. S. Sathyaprakash, Phys. Rev. D 82 , 064010 (2010), arXiv:1005.0304 [gr-qc].</list_item>
<list_item><location><page_8><loc_9><loc_79><loc_49><loc_84></location>[29] T. G. F. Li, W. Del Pozzo, S. Vitale, C. Van Den Broeck, M. Agathos, J. Veitch, K. Grover, T. Sidery, R. Sturani, and A. Vecchio, Phys. Rev. D 85 , 082003 (2012), arXiv:1110.0530 [gr-qc].</list_item>
<list_item><location><page_8><loc_9><loc_75><loc_49><loc_78></location>[30] M. Agathos, W. Del Pozzo, T. G. F. Li, C. V. D. Broeck, J. Veitch, et al. , Phys.Rev. D89 , 082001 (2014), arXiv:1311.0420 [gr-qc].</list_item>
<list_item><location><page_8><loc_9><loc_71><loc_49><loc_74></location>[31] A. K. Mehta, A. Buonanno, R. Cotesta, A. Ghosh, N. Sennett, and J. Steinhoff, (2022), arXiv:2203.13937 [gr-qc].</list_item>
<list_item><location><page_8><loc_9><loc_67><loc_49><loc_70></location>[32] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. Lett. 116 , 221101 (2016), [Erratum: Phys.Rev.Lett. 121, 129902 (2018)], arXiv:1602.03841 [gr-qc].</list_item>
<list_item><location><page_8><loc_9><loc_64><loc_49><loc_67></location>[33] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. D 100 , 104036 (2019), arXiv:1903.04467 [gr-qc].</list_item>
<list_item><location><page_8><loc_9><loc_62><loc_49><loc_64></location>[34] R. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. D 103 , 122002 (2021), arXiv:2010.14529 [gr-qc].</list_item>
<list_item><location><page_8><loc_9><loc_59><loc_49><loc_61></location>[35] R. Abbott et al. (LIGO Scientific, VIRGO, KAGRA), (2021), arXiv:2112.06861 [gr-qc].</list_item>
<list_item><location><page_8><loc_9><loc_56><loc_49><loc_59></location>[36] A. Pai and K. Arun, Class.Quant.Grav. 30 , 025011 (2013), arXiv:1207.1943 [gr-qc].</list_item>
<list_item><location><page_8><loc_9><loc_54><loc_49><loc_56></location>[37] S. Datta, M. Saleem, K. G. Arun, and B. S. Sathyaprakash, (2022), arXiv:2208.07757 [gr-qc].</list_item>
<list_item><location><page_8><loc_9><loc_52><loc_40><loc_53></location>[38] S. Datta, (2023), arXiv:2303.04399 [gr-qc].</list_item>
<list_item><location><page_8><loc_9><loc_50><loc_49><loc_52></location>[39] N. Yunes, K. Yagi, and F. Pretorius, (2016), arXiv:1603.08955 [gr-qc].</list_item>
<list_item><location><page_8><loc_9><loc_47><loc_49><loc_49></location>[40] C. Van Den Broeck and A. S. Sengupta, Class. Quant. Grav. 24 , 1089 (2007), arXiv:gr-qc/0610126.</list_item>
<list_item><location><page_8><loc_9><loc_44><loc_49><loc_47></location>[41] K. G. Arun, B. R. Iyer, B. S. Sathyaprakash, and S. Sinha, Phys. Rev. D 75 , 124002 (2007), 0704.1086.</list_item>
<list_item><location><page_8><loc_9><loc_42><loc_49><loc_44></location>[42] K. G. Arun, A. Buonanno, G. Faye, and E. Ochsner, Phys. Rev. D 79 , 104023 (2009), arXiv:0810.5336 [gr-qc].</list_item>
<list_item><location><page_8><loc_9><loc_39><loc_49><loc_41></location>[43] S. Borhanian, Class. Quant. Grav. 38 , 175014 (2021), arXiv:2010.15202 [gr-qc].</list_item>
<list_item><location><page_8><loc_9><loc_33><loc_49><loc_39></location>[44] P. Fritschel, K. Kuns, J. Driggers, A. Effler, B. Lantz, D. Ottaway, S. Ballmer, K. Dooley, R. Adhikari, M. Evans, B. Farr, G. Gonzalez, P. Schmidt, and S. Raja, Report from the LSC Post-O5 Study Group , Tech. Rep. T2200287 (LIGO, 2022).</list_item>
<list_item><location><page_8><loc_9><loc_26><loc_49><loc_32></location>[45] B. Iyer, T. Souradeep, C. Unnikrishnan, S. Dhurandhar, S. Raja, and A. Sengupta, LIGO-India, Proposal of the Consortium for Indian Initiative in Gravitationalwave Observations (IndIGO) , Tech. Rep. M1100296-v2 (LIGO-India, 2011).</list_item>
<list_item><location><page_8><loc_9><loc_25><loc_49><loc_26></location>[46] M. Evans et al. , (2021), arXiv:2109.09882 [astro-ph.IM].</list_item>
<list_item><location><page_8><loc_9><loc_22><loc_49><loc_24></location>[47] S. Hild et al. , Class. Quant. Grav. 28 , 094013 (2011), arXiv:1012.0908 [gr-qc].</list_item>
<list_item><location><page_8><loc_9><loc_21><loc_49><loc_22></location>[48] M. Punturo et al. , Class. Quant. Grav. 27 , 194002 (2010).</list_item>
<list_item><location><page_8><loc_9><loc_18><loc_49><loc_20></location>[49] M. Branchesi et al. , JCAP 07 , 068 (2023), arXiv:2303.15923 [gr-qc].</list_item>
<list_item><location><page_8><loc_9><loc_15><loc_49><loc_18></location>[50] I. Gupta, C. Afle, K. G. Arun, et al. , arXiv e-prints , arXiv:2307.10421 (2023), arXiv:2307.10421 [gr-qc].</list_item>
<list_item><location><page_8><loc_9><loc_14><loc_45><loc_15></location>[51] C. Rao, Bullet. Calcutta Math. Soc 37 , 81 (1945).</list_item>
<list_item><location><page_8><loc_9><loc_10><loc_49><loc_14></location>[52] H. Cramer, Mathematical methods in statistics (Pergamon Press, Princeton University Press, NJ, U.S.A., 1946).</list_item>
<list_item><location><page_8><loc_9><loc_9><loc_49><loc_10></location>[53] C. Cutler and E. Flanagan, Phys. Rev. D 49 , 2658 (1994).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_8><loc_52><loc_92><loc_90><loc_93></location>[54] E. Poisson and C. Will, Phys. Rev. D 52 , 848 (1995).</list_item>
<list_item><location><page_8><loc_52><loc_89><loc_92><loc_92></location>[55] J. M. Bardeen, W. H. Press, and S. A. Teukolsky, Astrophys. J. 178 , 347 (1972).</list_item>
<list_item><location><page_8><loc_52><loc_85><loc_92><loc_89></location>[56] S. Husa, S. Khan, M. Hannam, M. Purrer, F. Ohme, X. Jim'enez Forteza, and A. Boh'e, Phys. Rev. D 93 , 044006 (2016), arXiv:1508.07250 [gr-qc].</list_item>
<list_item><location><page_8><loc_52><loc_83><loc_92><loc_85></location>[57] F. Hofmann, E. Barausse, and L. Rezzolla, Astrophys. J. Lett. 825 , L19 (2016), arXiv:1605.01938 [gr-qc].</list_item>
<list_item><location><page_8><loc_52><loc_79><loc_92><loc_82></location>[58] M. Favata, C. Kim, K. G. Arun, J. Kim, and H. W. Lee, Phys. Rev. D 105 , 023003 (2022), arXiv:2108.05861 [gr-qc].</list_item>
<list_item><location><page_8><loc_52><loc_76><loc_92><loc_78></location>[59] E. Berti, A. Buonanno, and C. M. Will, Phys. Rev. D 71 , 084025 (2005), gr-qc/0411129.</list_item>
<list_item><location><page_8><loc_52><loc_71><loc_92><loc_76></location>[60] A. Mangiagli, A. Klein, M. Bonetti, M. L. Katz, A. Sesana, M. Volonteri, M. Colpi, S. Marsat, and S. Babak, Phys. Rev. D 102 , 084056 (2020), arXiv:2006.12513 [astro-ph.HE].</list_item>
<list_item><location><page_8><loc_52><loc_68><loc_92><loc_70></location>[61] R. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. D 102 , 043015 (2020), arXiv:2004.08342 [astro-ph.HE].</list_item>
<list_item><location><page_8><loc_52><loc_66><loc_92><loc_68></location>[62] R. Abbott et al. (LIGO Scientific, Virgo), Astrophys. J. Lett. 896 , L44 (2020), arXiv:2006.12611 [astro-ph.HE].</list_item>
<list_item><location><page_8><loc_52><loc_62><loc_92><loc_65></location>[63] R. Abbott et al. (KAGRA, VIRGO, LIGO Scientific), Astrophys. J. Suppl. 267 , 29 (2023), arXiv:2302.03676 [gr-qc].</list_item>
<list_item><location><page_8><loc_52><loc_59><loc_92><loc_61></location>[64] N. V. Krishnendu, K. G. Arun, and C. K. Mishra, Phys. Rev. Lett. 119 , 091101 (2017), arXiv:1701.06318 [gr-qc].</list_item>
<list_item><location><page_8><loc_52><loc_56><loc_92><loc_59></location>[65] N. V. Krishnendu, C. K. Mishra, and K. G. Arun, Phys. Rev. D 99 , 064008 (2019), arXiv:1811.00317 [gr-qc].</list_item>
<list_item><location><page_8><loc_52><loc_52><loc_92><loc_56></location>[66] N. V. Krishnendu, M. Saleem, A. Samajdar, K. G. Arun, W. Del Pozzo, and C. K. Mishra, Phys. Rev. D 100 , 104019 (2019), arXiv:1908.02247 [gr-qc].</list_item>
<list_item><location><page_8><loc_52><loc_50><loc_92><loc_52></location>[67] P. Saini and N. V. Krishnendu, (2023), arXiv:2308.01309 [gr-qc].</list_item>
<list_item><location><page_8><loc_52><loc_47><loc_92><loc_49></location>[68] C. M. Will, Phys. Rev. D 57 , 2061 (1998), grqc/9709011.</list_item>
<list_item><location><page_8><loc_52><loc_44><loc_92><loc_47></location>[69] S. Mirshekari, N. Yunes, and C. M. Will, Phys. Rev. D 85 , 024041 (2012), arXiv:1110.2720 [gr-qc].</list_item>
<list_item><location><page_8><loc_52><loc_42><loc_92><loc_44></location>[70] V. A. Kosteleck'y and M. Mewes, Phys. Lett. B 757 , 510 (2016), arXiv:1602.04782 [gr-qc].</list_item>
<list_item><location><page_8><loc_52><loc_39><loc_92><loc_41></location>[71] A. Samajdar and K. G. Arun, Phys. Rev. D 96 , 104027 (2017), arXiv:1708.00671 [gr-qc].</list_item>
<list_item><location><page_8><loc_52><loc_36><loc_92><loc_39></location>[72] C. Talbot and E. Thrane, Astrophys. J. 856 , 173 (2018), arXiv:1801.02699 [astro-ph.HE].</list_item>
<list_item><location><page_8><loc_52><loc_34><loc_92><loc_36></location>[73] R. Abbott et al. (LIGO Scientific, Virgo), Astrophys. J. Lett. 913 , L7 (2021), arXiv:2010.14533 [astro-ph.HE].</list_item>
<list_item><location><page_8><loc_52><loc_30><loc_92><loc_34></location>[74] R. Abbott et al. (KAGRA, VIRGO, LIGO Scientific), Phys. Rev. X 13 , 011048 (2023), arXiv:2111.03634 [astroph.HE].</list_item>
<list_item><location><page_8><loc_52><loc_27><loc_92><loc_30></location>[75] M. Fishbach and D. E. Holz, Astrophys. J. Lett. 891 , L27 (2020), arXiv:1905.12669 [astro-ph.HE].</list_item>
<list_item><location><page_8><loc_52><loc_25><loc_92><loc_27></location>[76] D. Wysocki, J. Lange, and R. O'Shaughnessy, Phys. Rev. D 100 , 043012 (2019), arXiv:1805.06442 [gr-qc].</list_item>
<list_item><location><page_8><loc_52><loc_22><loc_92><loc_24></location>[77] P. Madau and M. Dickinson, Ann. Rev. Astron. Astrophys. 52 , 415 (2014), arXiv:1403.0007 [astro-ph.CO].</list_item>
<list_item><location><page_8><loc_52><loc_18><loc_92><loc_22></location>[78] R. Abbott et al. (LIGO Scientific, KAGRA, VIRGO), Astrophys. J. Lett. 915 , L5 (2021), arXiv:2106.15163 [astroph.HE].</list_item>
</unordered_list>
<section_header_level_1><location><page_9><loc_37><loc_92><loc_64><loc_93></location>Appendix: Supplemental Materials</section_header_level_1>
<text><location><page_9><loc_9><loc_85><loc_92><loc_90></location>In this Supplement, we briefly describe the three types of compact binary populations considered in the paper. The reader must refer to Sec. III of Ref. [50] for a more detailed description. We also provide the Bayesian framework for mapping the bounds on multipole deformation parameters { δµ l , δϵ l } to the PN phasing deformation parameters δ ˆ ϕ b .</text>
<section_header_level_1><location><page_9><loc_38><loc_81><loc_63><loc_82></location>1. Compact binary populations</section_header_level_1>
<section_header_level_1><location><page_9><loc_43><loc_78><loc_58><loc_79></location>a. Binary black holes</section_header_level_1>
<text><location><page_9><loc_9><loc_66><loc_92><loc_76></location>The primary black hole masses are drawn from the Power Law + Peak mass model [72, 73] with the following values of model parameters: α = -3 . 4, m min = 5 M ⊙ , m max = 87 M ⊙ , λ = 0 . 04, µ m = 34 M ⊙ , σ m = 3 . 6 M ⊙ , and δ m = 4 . 8 M ⊙ [74] [See Eq. (B3) in Appendix B of Ref. [74]]. The mass ratio follows a power-law distribution [75] with power law index of 1.1 [74], and respect the condition m min = 5 M ⊙ [See Eq. (B7) in Appendix B of Ref. [74]]. The aligned spins components of the binary ( χ 1 z , χ 2 z ) are drawn from a Beta distribution [76] with α χ = 2, β χ = 5 [74] [See Eq. (10) of Ref. [76]]. The merger rate of the binary black hole population is assumed to follow the Madau-Dickinson star formation rate [77] with a local merger rate density of 24 Gpc -3 yr -1 [73].</text>
<section_header_level_1><location><page_9><loc_37><loc_61><loc_64><loc_62></location>b. Intermediate mass binary black holes</section_header_level_1>
<text><location><page_9><loc_9><loc_52><loc_92><loc_59></location>The masses of the intermediate mass binary black hole population are drawn from a power law distribution with a power-law index of -2 . 5. The lightest and heaviest masses in the distribution are opted to be m min = 100 M ⊙ and m max = 1000 M ⊙ , respectively. The spin components along the orbital angular momentum follow a uniform distribution between [ -0 . 9 , 0 . 9]. The merger rate is chosen to follow the Madau-Dickinson star formation rate [77] with a local merger rate density of 1 Gpc -3 yr -1 .</text>
<section_header_level_1><location><page_9><loc_39><loc_47><loc_62><loc_48></location>c. Neutron star-black hole binaries</section_header_level_1>
<text><location><page_9><loc_9><loc_38><loc_92><loc_45></location>The black hole masses are drawn from the Power Law + Peak mass model [72, 73], same as the primary mass of the binary black hole population. The masses of the neutron star follow a uniform distribution between [1 , 2 . 2] M ⊙ . The aligned spin components of black hole are assumed to follow a normal distribution with mean 0 and standard deviation of 0.2. The aligned spin components of the neutron star are uniformly drawn between [ -0 . 1 , 0 . 1]. The merger rate follow the Madau-Dickinson star formation rate [77] with a local merger rate density of 45 Gpc -3 yr -1 [74, 78].</text>
<section_header_level_1><location><page_9><loc_16><loc_34><loc_84><loc_35></location>2. Mapping the multipole deformation bounds to the PN phase deformation parameters</section_header_level_1>
<text><location><page_9><loc_9><loc_27><loc_92><loc_32></location>In the Bayesian framework, measuring the PN phase deformation parameter δ ˆ ϕ b amounts to obtaining the posterior probability density function P ( δ ˆ ϕ b | d, H ), where d denotes the detector data and H denotes the model. Using Bayes' theorem</text>
<formula><location><page_9><loc_37><loc_23><loc_92><loc_26></location>P ( δ ˆ ϕ b | d, H ) = P ( δ ˆ ϕ b |H ) P ( d | δ ˆ ϕ b , H ) P ( d |H ) , (A.1)</formula>
<text><location><page_9><loc_9><loc_18><loc_92><loc_21></location>where, P ( δ ˆ ϕ b |H ) is the prior probability density function, P ( d | δ ˆ ϕ b , H ) is the likelihood function, and P ( d |H ) [with P ( d |H ) = ∫ d ( δ ˆ ϕ b ) P ( δ ˆ ϕ b |H ) P ( d | δ ˆ ϕ b , H )] is the evidence.</text>
<text><location><page_9><loc_9><loc_8><loc_92><loc_18></location>In the parametrized multipolar approach [11, 12] the different gravitational wave phasing coefficients ϕ b (also the amplitude [23]) are functions of µ l , ϵ l along with the other binary's intrinsic parameters. Therefore, different δ ˆ ϕ b are also function of { δµ l , δϵ l } and the intrinsic parameters of binary such as ν, χ 1 z , and χ 2 z . Here, we are interested in computing the posterior probability distribution function of δ ˆ ϕ b , ˜ P ( δ ˆ ϕ b | d, H ), assuming a uniform prior on δ ˆ ϕ b [i.e., P ( δ ˆ ϕ b |H ) = ˜ Π( δ ˆ ϕ b |H )] from the posterior distributions of { δµ l , δϵ l } and intrinsic binary parameters. The posterior probability function ˜ P ( δ ˆ ϕ b | d, H ) can be obtained by replacing P ( δ ˆ ϕ b |H ) = ˜ Π( δ ˆ ϕ b |H ) in Eq. (A.1) and can</text>
<text><location><page_10><loc_9><loc_92><loc_25><loc_93></location>be expressed as follows</text>
<formula><location><page_10><loc_20><loc_77><loc_92><loc_91></location>˜ P ( δ ˆ ϕ b | d, H ) = ˜ Π( δ ˆ ϕ b |H ) P ( d | δ ˆ ϕ b , H ) ˜ P ( d |H ) = ˜ Π( δ ˆ ϕ b |H ) ∫ d ⃗ λ I d ⃗ λ T P ( d | ⃗ λ I , ⃗ λ T , H ) P ( ⃗ λ I , ⃗ λ T | δ ˆ ϕ b , H ) ˜ P ( d |H ) = ˜ Π( δ ˆ ϕ b |H ) ˜ P ( d |H ) × ∫ d ⃗ λ I d ⃗ λ T P ( d | ⃗ λ I , ⃗ λ T , H ) ˜ Π( ⃗ λ I , ⃗ λ T |H ) P ( δ ˆ ϕ b | ⃗ λ I , ⃗ λ T , H ) Π( δ ˆ ϕ b |H ) ︸ ︷︷ ︸ P ( ⃗ λ I , ⃗ λ T | δ ˆ ϕ b , H ) , (A.2)</formula>
<text><location><page_10><loc_9><loc_69><loc_92><loc_75></location>where, the intrinsic binary parameters are represented by ⃗ λ I ∈ { ν, χ 1 z , χ 2 z } , the mulitpolar coefficients are denoted by ⃗ λ T ∈ { δµ l , δϵ l } . ˜ P ( d |H ) is the evidence for the uniform prior on δ ˆ ϕ b , P ( d | ⃗ λ I , ⃗ λ T , H ) is the likelihood function of the gravitational wave data given the parameters { ⃗ λ I , ⃗ λ T } , P ( δ ˆ ϕ b | ⃗ λ I , ⃗ λ T , H ) takes care of the coordinate transformation between { ⃗ λ I , ⃗ λ T } and δ ˆ ϕ b , and Π( δ ˆ ϕ b |H ) is given by</text>
<formula><location><page_10><loc_31><loc_65><loc_92><loc_68></location>Π( δ ˆ ϕ b |H ) ≡ ∫ d ⃗ λ I d ⃗ λ T ˜ Π( ⃗ λ I , ⃗ λ T |H ) P ( δ ˆ ϕ b | ⃗ λ I , ⃗ λ T , H ) . (A.3)</formula>
<text><location><page_10><loc_9><loc_60><loc_92><loc_63></location>Therefore, Π( δ ˆ ϕ b | H ) is simply the distribution of δ ˆ ϕ b derived from the uniform prior on ⃗ λ I and ⃗ λ T and the relation between δ ˆ ϕ b and { ⃗ λ I , ⃗ λ T } . In Eq. (A.2), we have used Bayes' theorem which can be further simplified as follows</text>
<formula><location><page_10><loc_14><loc_49><loc_92><loc_59></location>˜ P ( δ ˆ ϕ b | d, H ) = ˜ Π( δ ˆ ϕ b |H ) × ˜ P IT ( d |H ) ˜ P ( d |H ) × ∫ d ⃗ λ I d ⃗ λ T [ ˜ Π( ⃗ λ I , ⃗ λ T |H ) P ( d | ⃗ λ I , ⃗ λ T , H ) ˜ P IT ( d |H ) ︸ ︷︷ ︸ ˜ P ( ⃗ λ I , ⃗ λ T | d, H ) × P ( δ ˆ ϕ b | ⃗ λ I , ⃗ λ T , H ) Π( δ ˆ ϕ b |H ) ] = [∫ d ⃗ λ I d ⃗ λ T P ( δ ˆ ϕ b | ⃗ λ I , ⃗ λ T , H ) ˜ P ( ⃗ λ I , ⃗ λ T | d, H ) ] × ˜ Π( δ ˆ ϕ b |H ) Π( δ ˆ ϕ b |H ) × ˜ P IT ( d |H ) ˜ P ( d |H ) , (A.4)</formula>
<text><location><page_10><loc_9><loc_39><loc_92><loc_48></location>where, ˜ P ( ⃗ λ I , ⃗ λ T | d, H ) is the posterior probability density of { ⃗ λ I , ⃗ λ T } and ˜ P IT ( d |H ) [with ˜ P IT ( d |H ) = ∫ d ⃗ λ I d ⃗ λ T ˜ Π( ⃗ λ I , ⃗ λ T |H ) P ( d | ⃗ λ I , ⃗ λ T , H )] is the corresponding evidence derived assuming a uniform prior on { ⃗ λ I , ⃗ λ T } . The numerical factor ˜ P IT ( d |H ) ˜ P ( d |H ) is an overall normalization constant in the above equation and can be ignored if one is only interested in estimating ˜ P ( δ ˆ ϕ b | d, H ). The different steps for the derivation of the above equation are followed from Ref. [22] (See Sec. 2 of the Supplemental Material in Ref. [22]).</text>
<text><location><page_10><loc_9><loc_35><loc_92><loc_39></location>As mentioned earlier, δ ˆ ϕ b is a unique function of { ⃗ λ I , ⃗ λ T } , say f b ( ⃗ λ I , ⃗ λ T ) [11, 12]; hence given a value of { ⃗ λ I , ⃗ λ T } , δ ˆ ϕ b is particularly determined. Therefore, P ( δ ˆ ϕ b | ⃗ λ I , ⃗ λ T , H ) can simply be represented by a delta function,</text>
<formula><location><page_10><loc_35><loc_33><loc_92><loc_34></location>P ( δ ˆ ϕ b | ⃗ λ I , ⃗ λ T , H ) = δ ( δ ˆ ϕ b -f b ( ⃗ λ I , ⃗ λ T )) . (A.5)</formula>
<text><location><page_10><loc_9><loc_27><loc_92><loc_31></location>In practice, to compute ˜ P ( δ ˆ ϕ b | d, H ) one needs to take the posterior samples of { ⃗ λ I , ⃗ λ T } and estimate δ ˆ ϕ b for each sample through the functions f b ( ⃗ λ I , ⃗ λ T ); and then reweight these samples by the probability ˜ Π( δ ˆ ϕ b |H ) Π( δ ˆ ϕ b |H ) .</text>
</document>
[
{
"title": "Multiparameter multipolar test of general relativity with gravitational waves",
"content": "Parthapratim Mahapatra, 1, ∗ Shilpa Kastha, 2 Anuradha Gupta, 3 B. S. Sathyaprakash, 4, 5, 6 and K. G. Arun 1, 4 1 Chennai Mathematical Institute, Siruseri, 603103, India 4 Institute for Gravitation and the Cosmos, Department of Physics, Penn State University, University Park, Pennsylvania 16802, USA 5 Department of Astronomy and Astrophysics, Penn State University, University Park, Pennsylvania 16802, USA 6 School of Physics and Astronomy, Cardiff University, Cardiff, CF24 3AA, United Kingdom (Dated: October 14, 2024) Amplitude and phase of the gravitational waveform from compact binary systems can be decomposed in terms of their mass- and current-type multipole moments. In a modified theory of gravity, one or more of these multipole moments could deviate from general theory of relativity. In this work, we show that a waveform model that parametrizes the amplitude and phase in terms of the multipole moments of the binary can facilitate a novel multiparameter test of general relativity with exquisite precision. Using a network of next-generation gravitational-wave observatories, simultaneous deviation in the leading seven multipoles of a GW190814-like binary can be bounded to within 6%-40% depending on the multipole order, while supermassive black hole mergers observed by the Laser Interferometer Space Antenna achieve a bound of 0.3%-2%. We further argue that bounds from multipoles can be uniquely mapped onto other parametrized tests of general relativity and have the potential to become a downstream analysis from which bounds of other parametric tests of general relativity can be derived. The set of multipole parameters, therefore, provides an excellent basis to carry out precision tests of general relativity.",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "Gravitational waveform from a compact binary coalescence is a nonlinear function of 'radiative mass-' and 'current-type' multipole moments [1] and their derivatives with respect to time. The 'adiabatic inspiral' of the binary is well described by the post-Newtonian (PN) approximation to the general theory of relativity (GR) where the mass ratio and the spins of the binary constituents determine which multipoles are excited and what their contributions are to the emitted flux and the phase evolution of the binary. After the leading quadrupole, the mass-octupole is the next dominant contribution to the phase. As the binary becomes more asymmetric, the contributions from higher-order multipole moments become significant. Spins of the binary constituents can further enhance the strengths of certain higher-order multipoles, especially the current-type ones. In a modified theory of gravity, where the compact binary dynamics differs from GR, it is natural to expect that one or more of these radiative multipole moments will deviate from those of GR [2-10]. Therefore, asking whether the measured multipole moments of compact binaries are consistent with GR predictions is an excellent way to test GR. References [11, 12] first derived a multipolar parametrized gravitational-wave phase, which separately tracks the contribution from different radiative multipole moments within the PN approximation to GR. This is achieved by associating parameters µ l and ϵ l with the mass- and current-type radiative multipole moments, respectively. Here l = 2 , 3 , . . . denote quadrupole, octupole, etc. The phenomenological multipole parameters are equal to unity in GR (i.e., µ GR l ≡ 1 and ϵ GR l ≡ 1), by definition. By introducing deviations to these multipole coefficients, denoted as δµ l and δϵ l (i.e., µ l ≡ 1+ δµ l and ϵ l ≡ 1 + δϵ l ), one can use the gravitational-wave data to obtain bounds on these two sets of parameters. The radiative multipole moments of compact binaries are nonlinear functionals of the source multipole moments (i.e., moments of the stress-energy tensor of the material source and its gravitational fields) and contain time derivatives of the source moments [13]. These time derivatives of the source multipole moments are evaluated using the equation of motion of the compact binary. Therefore, in the gravitational-wave generation formalism the radiative multipole moments of compact binaries also carry information about the conservative dynamics of the binary. Hence, the parameters δµ l and δϵ l are sensitive to deviations from GR in both the dissipative and the conservative sectors of the compact binary dynamics. However, one can use the parametrization introduced in Eq. (3.2) of Ref. [12] to track explicitly different PN pieces in the conserved orbital energy. The most general test of GR one can perform, within this framework, is the one where all the δµ l and δϵ l are simultaneously measured, which is often referred to as a 'multiparameter test' (multiparameter tests have been discussed in the context of PN phase expansion in Refs. [14-17]). We explore the possibility of simultaneously estimating the leading seven multipole parameters (i.e., the leading four mass-type and the leading three current-type moments) with the present and next-generation gravitational-wave detectors. This generalizes the single-parameter projections reported in Refs. [11, 12] and complements the consistency tests proposed in Refs. [18, 19] and the results from GW190412 and GW190814 being reported in Refs. [20, 21]. This work also extends the single-parameter octupolar bounds from GW190412 and GW190814 reported recently in Ref. [22]. The crucial ingredient in this work is the introduction of new parametrized multipolar amplitudes up to 2PN order recently computed in a companion paper [23], which enables us to use the multipolar information in both the amplitude and the phase to derive the bounds on the multipole parameters. Unlike the parametrizations that look for deviations either in phase [14, 24-35] or in amplitude [19, 21] of gravitational waveform independently, the multipolar parametrization has the advantage that the number of independent parameters is smaller , the same as the number of multipole parameters that appear in the amplitude and phase . What makes the multiparameter tests very difficult to perform is the strong degeneracies introduced by the simultaneous inclusion of more phenomenological deformation parameters. Multiband gravitational-wave observations [15, 16] and principal component analysis [17, 3638] have been argued to be two different approaches to carry out multiparameter tests of GR in terms of deformations introduced directly in the PN expansion coefficients of the signals's phase evolution. Here, we investigate the use of multipole parameters, as opposed to the usual deformation parameters in the signal's phase, to carry out multiparameter tests of GR. Apart from being a more downstream parameter set, orthogonality of the multipole parameters may help in lifting the abovementioned degeneracies. In this work, we show that the multipolar framework is a viable route to carry out a very generic multiparameter test of GR. We further argue that the bounds on δµ l and δϵ l can be mapped to other parametrized tests of GR. Therefore, this new class of tests may be thought of as an 'all-in-one' test of GR, which may be mapped to any parametrized test of interest. We explicitly demonstrate this mapping in the context of parametrized tests of PN phasing, which is currently employed on the gravitational-wave data and used to obtain constraints on specific modified theories of gravity [39]. The remainder of the paper is organized as follows. In Sec. II, we briefly describe the parametrized multipolar waveform model. In Sec. III, we briefly explain the parameter estimation scheme used in our analysis. We discuss our results in Sec. IV. Our conclusions are presented in Sec. V.",
"pages": [
1,
2
]
},
{
"title": "II. WAVEFORM MODEL",
"content": "We use the frequency-domain amplitude-corrected multipolar waveform for spinning, nonprecessing compact binaries recently reported in Ref. [23]. This wave- form model is 3.5PN accurate in the phase and 2PN accurate in the amplitude (i.e., includes the contributions from the first six harmonics). The amplitude-corrected multipolar polarizations in the frequency domain up to 2PN schematically reads [40-42] Here M , ν (= q (1+ q ) 2 with q being the ratio between the primary and secondary mass), and D L denote the redshifted total mass, symmetric mass ratio, and the luminosity distance of the source, respectively. The indices n and k indicate the n 2 th PN order and harmonics of the orbital phase, respectively. The parameter V k = (2 π GM f/c 3 k ) 1 / 3 is the dimensionless gauge invariant PN parameter for the k th harmonic [40], G is the gravitational constant, c is the speed of light, and f is the gravitational-wave frequency. The coefficients H ( k,n ) + , × denote the amplitude corrections in the frequency-domain polarizations associated with the contribution from k th harmonic at n 2 th PN order. These amplitude coefficients are functions of the masses, spins, and orbital inclination angle ι and, in our parametrization, contain the multipole parameters µ l and ϵ l . The expressions for all the H ( k,n ) + , × can be found in Eqs. (10) and (11) of Ref. [23]. Lastly, Ψ SPA ( f ) represents the frequency-domain parametrized multipolar gravitational-wave phasing for the first harmonic. References [11, 12] obtained the 3.5PN accurate expression of Ψ SPA ( f ) for nonprecessing, spinning binaries using the stationary phase approximation. In the spirit of null tests, the multipolar polarizations in Eq. (1) are reexpressed in terms of { δµ l , δϵ l } with the goal of deducing projected bounds on them from gravitationalwave observations. The gravitational-wave strain in the frequency domain measured by a detector D is given by where F lp is the location phase factor of the detector, F + and F × are the antenna response functions that describe the detector's sensitivity to the two different polarizations, θ is the declination angle, ϕ is the right ascension, and ψ is the polarization angle (see Sec. III of Ref. [43] for more details). Indeed, our inspiral-only waveform model ignores the contributions from the merger and ringdown phases of the compact binary dynamics, the inclusion of which can lead to a considerable increase in the signal-to-noise ratio (SNR). However, as we crucially make use of the multipole structure in PN theory, it is only natural to employ inspiral-only waveforms for a proof-of-concept study like this, provided we restrict ourselves to binaries that are dominated by their inspiral. Finally, for simplicity, we only consider nonprecessing binary configurations in quasicircular orbits. It is likely that precessionand eccentricity-induced modulations may improve the bounds reported, though the magnitude of this needs to be quantified by a dedicated study.",
"pages": [
2,
3
]
},
{
"title": "III. PARAMETER ESTIMATION",
"content": "To compute the statistical errors on various multipole deformation parameters and other relevant binary parameters, we use the semi-analytical Fisher information matrix formalism [51-54]. In the high SNR limit, the Fisher information matrix is a computationally inexpensive method to predict the statistical uncertainties (1 σ error bars) on the parameters of a signal model buried in stationary Gaussian noise. For a frequency-domain gravitational-wave signal ˜ h D ( f ), described by a set of parameters ⃗ λ , the Fisher matrix is defined as where S h ( f ) is the one-sided noise power spectral density (PSD) of the detector, and f min and f max are the lower and upper limits of integration. In the above equation, ' ∗ ' denotes the operation of complex conjugation, and ',' denotes differentiation with respect to various elements in the parameter set ⃗ λ ≡ { λ m } . The 1 σ statistical error in λ m is σ m = √ Σ mm , where the covariance matrix Σ mn = (Γ mn ) -1 is the inverse of the Fisher matrix. To estimate the errors on all multipole deformation parameters simultaneously, we have considered the fol- lowing parameter space: where, t c is the time of coalescence, ϕ c is the phase at coalescence, M c = Mν 3 / 5 is the redshifted chirp mass, and χ 1 z and χ 2 z are the individual spin components along the orbital angular momentum. 1 For the computation of the statistical errors in the various parameters for different binary configurations and networks of ground-based gravitational-wave detectors, we use GWBENCH [43], a publicly available PYTHON -based package that computes the Fisher matrix and the corresponding covariance matrix for a given gravitationalwave network. The plus and cross polarizations in Eq. (1) are added into GWBENCH for this purpose. We have chosen f min to be 5 Hz and f max to be 6 F ISCO Hz for all the ground-based network configurations. Here F ISCO is the redshifted Kerr innermost stable circular orbit (ISCO) frequency [55-57] and its explicit expression for nonprecessing binaries can be found in Appendix C of Ref. [58]. For the sources observed by the space-based Laser Interferometer Space Antenna (LISA), we have used Eq. (2.15) of Ref. [59] and have taken f low = 10 -4 Hz and T obs = 4 yr to estimate f min . In the LISA band, f max is given by the smaller of 6 F ISCO and 0.1 Hz. We have summarized the different networks of ground-based detectors considered here in Table I. The noise PSDs of various ground-based detectors used here can be found in GWBENCH [43]. We have adopted the non-sky-averaged noise PSD of LISA reported in Ref. [60] [see Eqs. (1)-(5) of [60]] and ignored its orbital motion in our computation. If we assume that all of the multipole deviation parameters take the same value for different events in a population, one can compute a joint bound on them by multiplying the corresponding 1D likelihoods. The width of the joint likelihood is given by where i = 1 , . . . , N denotes the events considered in the compact binary population.",
"pages": [
3
]
},
{
"title": "IV. RESULTS AND DISCUSSIONS",
"content": "We start by discussing the projected bounds on the multipole deformation parameters from GW190412- [61] and GW190814-like systems [62], two asymmetric compact binary mergers detected in the third observing run by LIGO/Virgo observatories, in different networks of future ground-based gravitational-wave detectors. As these types of events have been confirmed to exist and extensively studied, they help us to understand the importance of the results. As the observed strengths of the higherorder multipoles depend crucially on the inclination angle ι and the SNR of the observed gravitational-wave signal depends on the location of the source, we synthesize a population for these two representative systems and use the median value of the resulting distribution to assess the measurement uncertainty in various multipole deformation parameters. Toward this, for each of the systems, we draw 100 samples distributed isotropically over the sphere for the orientation and location of the source. The component masses and spins and the luminosity distances are fixed at the median values reported by Refs. [61-63]. For each sample, we estimate the 1 σ statistical errors in the seven multipole deformation parameters simultaneously and then compute the median of these 1 σ errors from the 100 samples. The results for different detector networks are shown in Fig. 1. We can measure all seven multipole deformation parameters simultaneously for a GW190814-like system to within ∼ 40% accuracy in 4020ET, whereas for GW190412-like binaries all multipole deformation parameters can be measured simultaneously to within ∼ 70% in 4020ET. Therefore, a single detection of a GW190412- or GW190814-like binary in the nextgeneration (XG) gravitational-wave detectors will allow us to measure all seven multipole deformation parameters simultaneously and hence to perform the most generic multiparameter test of gravitational-wave phase and am- ude evolution in GR. It is seen that the mass-type multipole deformation parameters are always estimated better as compared to the current-type multipole deformation parameters. This should be due to the dominance of the mass-type moments over the current-type ones on the dynamics of the binary system. In terms of different detector networks, the 40LET bounds are comparable to those from 4020ET, which suggests that two thirdgeneration detectors already provide very precise bounds and the sensitivity of the third detector does not have a significant impact on the joint bounds. Next, we consider three different classes of compact binary populations, neutron star-black holes (NSBHs), binary black holes (BBHs), and intermediate mass binary black holes (IMBBHs), reported in Ref. [50] (see Supplemental Material for details of the population). For each class of the compact binary population, we select 200 loudest events in the respective network of ground-based detectors and calculate the combined bounds on all seven multipole deformation parameters simultaneously using Eq. (5). Figure 2 shows the combined bounds on multipole deformation parameters for these three types of compact binary populations in different networks. We can constrain all the multipole moments simultaneously within an accuracy of ∼ 20% in the XG era from the NSBH population. The BBH population considered here mostly contains equal-mass binaries, and therefore, they provide the best constraint on δµ 2 . Binaries in the IMBBH population are more massive than the other two populations and are also more asymmetric than the BBH population. As asymmetric massive binaries carry stronger signatures of higher-order multipoles, we obtain the best bounds on higher-order multipole deformation parameters from the IMBBH population-all multipole deformation parameters can be measured simultaneously to within ∼ 8% in the XG era. The NSBH population consists mainly of high mass ratio, but less massive, systems than the other two populations. As a result, they provide bounds similar to BBH population on higher-order multipole deformation parameters. The merger rates of supermassive binary black holes (SMBBHs) and their detection rates in LISA are highly uncertain. Here we consider a few representative SMBBH systems and compute the projected error bars on various multipole deformation parameters. We consider merging SMBBHs at a luminosity distance of 3 Gpc with two different choices of spins ( χ 1 z = 0 . 2, χ 2 z = 0 . 1) and ( χ 1 z = 0 . 8, χ 2 z = 0 . 7). For each pair of spins, we choose two different mass ratios 2 and 5. All the angles (i.e., ι , θ , ϕ , ψ ) are set to be π/ 6. The 1 σ errors in all seven deformation parameters in the LISA band for various SMBBH configurations are shown in Fig. 3. We find that for most of the SMBBH systems considered here, LISA will be able to measure all seven multipole moments simultaneously to within ∼ 10%. We next discuss how bounds on the PN deformations may be derived from the multipole bounds. In prin- /circledot ciple, any PN parametrized test of gravitational-wave phase or amplitude can be effectively recast in terms of the multipole parameters. All we need for this is to derive a relation between those phenomenological parameters in the phase or amplitude and { δµ l , δϵ l } . If the parametric form of the phase or amplitude for any test and the contribution of different multipoles to the gravitational-wave phase [11, 12] and amplitude [23] are known, this derivation is straightforward. Here, as a proof-of-principle demonstration, we show how the constraints on { δµ l , δϵ l } can be mapped onto the different PN deformation parameters δ ˆ ϕ b in the phase evolution (where b ∈ 0 , 2 , 3 , 4 , 5 l, 6 , 6 l, 7 denotes different PN orders). Given the gravitational-wave data d , we are interested in computing ˜ P ( δ ˆ ϕ b | d, H ), the posterior probability distribution of δ ˆ ϕ b , for a uniform prior on δ ˆ ϕ b ( H denotes the hypothesis, which is the parametric model we employ). Once we have the posterior samples for the joint probability distribution ˜ P ( ⃗ λ I , ⃗ λ T | d, H ) for uniform priors on ⃗ λ I ∈ { ν, χ 1 z , χ 2 z } and ⃗ λ T ∈ { δµ l , δϵ l } , we can compute the posteriors on δ ˆ ϕ b , P ( δ ˆ ϕ b | d, H ), using the relation between δ ˆ ϕ b and { ⃗ λ I , ⃗ λ T } . As δ ˆ ϕ b is a unique nonlinear function of { ⃗ λ I , ⃗ λ T } , a uniform prior on { ⃗ λ I , ⃗ λ T } does not translate into a uniform prior on δ ˆ ϕ b . Therefore, to obtain ˜ P ( δ ˆ ϕ b | d, H ) we need to reweight the samples of P ( δ ˆ ϕ b | d, H ) by the samples of δ ˆ ϕ b derived from the uniform prior on { ⃗ λ I , ⃗ λ T } , using the relation between δ ˆ ϕ b and { ⃗ λ I , ⃗ λ T } . A more detailed discussion about the reweighting procedure is provided in the Supplemental Material. We consider GW190412- and GW190814-like systems in the 4020ET network and compute the Fisher matrix Γ mn to construct the Gaussian probability distribution function p ( ⃗ λ ) ∝ e -1 2 Γ mn ( λ m -λ m inj )( λ n -λ n inj ) , where λ m inj are the injected parameter values. We marginalize the distribution p ( ⃗ λ ) over parameters other than { ⃗ λ I , ⃗ λ T } to get ˜ P ( ⃗ λ I , ⃗ λ T | d, H ). Next, we calculate P ( δ ˆ ϕ b | d, H ) using the samples of ˜ P ( ⃗ λ I , ⃗ λ T | d, H ). To obtain ˜ P ( δ ˆ ϕ b | d, H ) that assumes a uniform prior on δ ˆ ϕ b between [ -10 , 10], we reweight the distribution P ( δ ˆ ϕ b | d, H ) by the distribution of δ ˆ ϕ b derived from the following prior distributions: ν is uniform between [0 . 045 , 0 . 25], χ 1 z and χ 2 z are uniform between [ -0 . 99 , 0 . 99], and ⃗ λ T are uniform between [ -10 , 10]. The posterior distribution ˜ P ( δ ˆ ϕ b | d, H ) of different δ ˆ ϕ b are shown in Fig. 4. All the δ ˆ ϕ b probability distributions are constrained to better than 0.5 at 80% credibility. Despite the reweighting employed, the mapped bounds derived here need not match with the regular multiparameter phasing tests using either ground-based or spacebased detector alone, where different phasing deformation parameters are treated as independent parameters. This should not be surprising, as the proposed mapping accounts only for the relation between the multipole and the phase deformation parameters and not the correlations these two sets of parameters would have with other binary parameters when the test is performed in the corresponding bases. We have checked that the bounds on the phase deformation parameters derived from the multipole bounds are overall much tighter than those that follow from directly sampling over all of them simultaneously. In the case of other parametrized tests of GR that rely on spin-induced multipole moments [64-67], modified dispersion relations [68-71], subdominant harmonics [19, 21], etc., the same method will work to derive the corresponding bounds from the multipole ones. In this case, one may visualize the test to be capturing a GR deviation via some effective multipolar deformation. A detailed study of these maps and their meanings will be taken up as a follow-up project.",
"pages": [
3,
4,
5,
6
]
},
{
"title": "V. CONCLUSIONS AND FUTURE DIRECTIONS",
"content": "This work serves as a proof-of-concept for the ability of the multipolar framework to carry out a robust multiparameter test of GR with impressive precision, which is necessary to accomplish meaningful constraints on the parameter space of alternate theories of gravity. Moreover, as shown, the bounds from such tests can be uniquely mapped onto the other parametrized tests of GR that rely on amplitude or phase deformations. In this work, we have employed the Fisher matrix formalism and a nonprecessing inspiral waveform for parameter estimation. While this paper is meant to illustrate the potential power of the multipolar approach, the results presented here should be revisited using the Bayesian framework with more realistic inspiral-mergerringdown waveforms. Moreover, the systematic biases induced due to the neglect of well-known effects such as spin precession and eccentricity need to be understood. Hence, the expected constraints that we report here are only indicative of the potential of the multipolar framework. A natural next step is to construct a parametrized multipolar inspiral-merger-ringdown waveform that includes the effects of spin precession and eccentricity for gravitational-wave data analysis as well as employ stateof-the-art Bayesian parameter inference techniques to demonstrate the feasibility of the method and apply it on a selected subset of gravitational-wave events.",
"pages": [
6
]
},
{
"title": "ACKNOWLEDGMENTS",
"content": "The authors thank Elisa Maggio for a critical reading of the manuscript and providing useful comments. We also thank Ish Gupta for a discussion of the compact binary populations used here. P.M. thanks Ssohrab Borhanian for a discussion of GWBENCH . P.M. also thanks Alok Laddha for valuable discussions. P.M. and K.G.A. acknowledge the support of the Core Research Grant No. CRG/2021/004565 of the Science and Engineering Research Board of India and a grant from the Infosys Foundation. K.G.A. acknowledges support from the Department of Science and Technology and Science and Engineering Research Board (SERB) of India via the following grants: Swarnajayanti Fellowship Grant No. DST/SJF/PSA-01/2017-18 and MATRICS grant (Mathematical Research Impact Centric Support) No. MTR/2020/000177. S.K. acknowledges support from the Villum Investigator program supported by the VILLUM Foundation (Grant No. VIL37766) and the DNRF Chair program (Grant No. DNRF162) by the Danish National Research Foundation. This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie SklodowskaCurie Grant Agreement No. 101131233. K.G.A. and B.S.S. acknowledge the support of the Indo-U.S. Science and Technology Forum through the Indo-U.S. Centre for Gravitational-Physics and Astronomy, Grant No. IUSSTF/JC-142/2019. We also acknowledge NSF support via NSF Grants No. AST-2205920 and No. PHY2308887 to A.G. and No. AST-2307147, No. PHY2012083, No. PHY-2207638, No. PHY-2308886, and No. PHYS-2309064 to B.S.S. This manuscript has the LIGO preprint number P2300424. The author is grateful for computational resources provided by the LIGO Laboratory and supported by National Science Foundation Grants No. PHY-0757058 and No. PHY-0823459. This research has made use of data or software ob- tained from the Gravitational Wave Open Science Center (gwosc.org), a service of the LIGO Scientific Collaboration, the Virgo Collaboration, and KAGRA. This material is based upon work supported by NSF's LIGO Laboratory which is a major facility fully funded by the National Science Foundation, as well as the Science and Technology Facilities Council (STFC) of the United Kingdom, the Max-Planck-Society (MPS), and the State of Niedersachsen/Germany for support of the construction of Advanced LIGO and construction and operation of the GEO600 detector. Additional support for Advanced LIGO was provided by the Australian Research Council. Virgo is funded, through the European Gravitational Observatory (EGO), by the French Centre National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale di Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions by institutions from Belgium, Germany, Greece, Hungary, Ireland, Japan, Monaco, Poland, Portugal, and Spain. KAGRA is supported by Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan Society for the Promotion of Science (JSPS) in Japan; National Research Foundation (NRF) and Ministry of Science and ICT (MSIT) in Korea; Academia Sinica (AS) and National Science and Technology Council (NSTC) in Taiwan. gr-qc/0604018.",
"pages": [
6,
7
]
},
{
"title": "Appendix: Supplemental Materials",
"content": "In this Supplement, we briefly describe the three types of compact binary populations considered in the paper. The reader must refer to Sec. III of Ref. [50] for a more detailed description. We also provide the Bayesian framework for mapping the bounds on multipole deformation parameters { δµ l , δϵ l } to the PN phasing deformation parameters δ ˆ ϕ b .",
"pages": [
9
]
},
{
"title": "a. Binary black holes",
"content": "The primary black hole masses are drawn from the Power Law + Peak mass model [72, 73] with the following values of model parameters: α = -3 . 4, m min = 5 M ⊙ , m max = 87 M ⊙ , λ = 0 . 04, µ m = 34 M ⊙ , σ m = 3 . 6 M ⊙ , and δ m = 4 . 8 M ⊙ [74] [See Eq. (B3) in Appendix B of Ref. [74]]. The mass ratio follows a power-law distribution [75] with power law index of 1.1 [74], and respect the condition m min = 5 M ⊙ [See Eq. (B7) in Appendix B of Ref. [74]]. The aligned spins components of the binary ( χ 1 z , χ 2 z ) are drawn from a Beta distribution [76] with α χ = 2, β χ = 5 [74] [See Eq. (10) of Ref. [76]]. The merger rate of the binary black hole population is assumed to follow the Madau-Dickinson star formation rate [77] with a local merger rate density of 24 Gpc -3 yr -1 [73].",
"pages": [
9
]
},
{
"title": "b. Intermediate mass binary black holes",
"content": "The masses of the intermediate mass binary black hole population are drawn from a power law distribution with a power-law index of -2 . 5. The lightest and heaviest masses in the distribution are opted to be m min = 100 M ⊙ and m max = 1000 M ⊙ , respectively. The spin components along the orbital angular momentum follow a uniform distribution between [ -0 . 9 , 0 . 9]. The merger rate is chosen to follow the Madau-Dickinson star formation rate [77] with a local merger rate density of 1 Gpc -3 yr -1 .",
"pages": [
9
]
},
{
"title": "c. Neutron star-black hole binaries",
"content": "The black hole masses are drawn from the Power Law + Peak mass model [72, 73], same as the primary mass of the binary black hole population. The masses of the neutron star follow a uniform distribution between [1 , 2 . 2] M ⊙ . The aligned spin components of black hole are assumed to follow a normal distribution with mean 0 and standard deviation of 0.2. The aligned spin components of the neutron star are uniformly drawn between [ -0 . 1 , 0 . 1]. The merger rate follow the Madau-Dickinson star formation rate [77] with a local merger rate density of 45 Gpc -3 yr -1 [74, 78].",
"pages": [
9
]
},
{
"title": "2. Mapping the multipole deformation bounds to the PN phase deformation parameters",
"content": "In the Bayesian framework, measuring the PN phase deformation parameter δ ˆ ϕ b amounts to obtaining the posterior probability density function P ( δ ˆ ϕ b | d, H ), where d denotes the detector data and H denotes the model. Using Bayes' theorem where, P ( δ ˆ ϕ b |H ) is the prior probability density function, P ( d | δ ˆ ϕ b , H ) is the likelihood function, and P ( d |H ) [with P ( d |H ) = ∫ d ( δ ˆ ϕ b ) P ( δ ˆ ϕ b |H ) P ( d | δ ˆ ϕ b , H )] is the evidence. In the parametrized multipolar approach [11, 12] the different gravitational wave phasing coefficients ϕ b (also the amplitude [23]) are functions of µ l , ϵ l along with the other binary's intrinsic parameters. Therefore, different δ ˆ ϕ b are also function of { δµ l , δϵ l } and the intrinsic parameters of binary such as ν, χ 1 z , and χ 2 z . Here, we are interested in computing the posterior probability distribution function of δ ˆ ϕ b , ˜ P ( δ ˆ ϕ b | d, H ), assuming a uniform prior on δ ˆ ϕ b [i.e., P ( δ ˆ ϕ b |H ) = ˜ Π( δ ˆ ϕ b |H )] from the posterior distributions of { δµ l , δϵ l } and intrinsic binary parameters. The posterior probability function ˜ P ( δ ˆ ϕ b | d, H ) can be obtained by replacing P ( δ ˆ ϕ b |H ) = ˜ Π( δ ˆ ϕ b |H ) in Eq. (A.1) and can be expressed as follows where, the intrinsic binary parameters are represented by ⃗ λ I ∈ { ν, χ 1 z , χ 2 z } , the mulitpolar coefficients are denoted by ⃗ λ T ∈ { δµ l , δϵ l } . ˜ P ( d |H ) is the evidence for the uniform prior on δ ˆ ϕ b , P ( d | ⃗ λ I , ⃗ λ T , H ) is the likelihood function of the gravitational wave data given the parameters { ⃗ λ I , ⃗ λ T } , P ( δ ˆ ϕ b | ⃗ λ I , ⃗ λ T , H ) takes care of the coordinate transformation between { ⃗ λ I , ⃗ λ T } and δ ˆ ϕ b , and Π( δ ˆ ϕ b |H ) is given by Therefore, Π( δ ˆ ϕ b | H ) is simply the distribution of δ ˆ ϕ b derived from the uniform prior on ⃗ λ I and ⃗ λ T and the relation between δ ˆ ϕ b and { ⃗ λ I , ⃗ λ T } . In Eq. (A.2), we have used Bayes' theorem which can be further simplified as follows where, ˜ P ( ⃗ λ I , ⃗ λ T | d, H ) is the posterior probability density of { ⃗ λ I , ⃗ λ T } and ˜ P IT ( d |H ) [with ˜ P IT ( d |H ) = ∫ d ⃗ λ I d ⃗ λ T ˜ Π( ⃗ λ I , ⃗ λ T |H ) P ( d | ⃗ λ I , ⃗ λ T , H )] is the corresponding evidence derived assuming a uniform prior on { ⃗ λ I , ⃗ λ T } . The numerical factor ˜ P IT ( d |H ) ˜ P ( d |H ) is an overall normalization constant in the above equation and can be ignored if one is only interested in estimating ˜ P ( δ ˆ ϕ b | d, H ). The different steps for the derivation of the above equation are followed from Ref. [22] (See Sec. 2 of the Supplemental Material in Ref. [22]). As mentioned earlier, δ ˆ ϕ b is a unique function of { ⃗ λ I , ⃗ λ T } , say f b ( ⃗ λ I , ⃗ λ T ) [11, 12]; hence given a value of { ⃗ λ I , ⃗ λ T } , δ ˆ ϕ b is particularly determined. Therefore, P ( δ ˆ ϕ b | ⃗ λ I , ⃗ λ T , H ) can simply be represented by a delta function, In practice, to compute ˜ P ( δ ˆ ϕ b | d, H ) one needs to take the posterior samples of { ⃗ λ I , ⃗ λ T } and estimate δ ˆ ϕ b for each sample through the functions f b ( ⃗ λ I , ⃗ λ T ); and then reweight these samples by the probability ˜ Π( δ ˆ ϕ b |H ) Π( δ ˆ ϕ b |H ) .",
"pages": [
9,
10
]
}
]
2024PhRvD.109f4053D
https://arxiv.org/pdf/2401.01767.pdf
<document>
<section_header_level_1><location><page_1><loc_16><loc_92><loc_85><loc_93></location>Absorption and (unbounded) superradiance in a static regular black hole spacetime</section_header_level_1>
<text><location><page_1><loc_19><loc_89><loc_82><loc_90></location>Marco A. A. de Paula, 1, 2, ∗ Luiz C. S. Leite, 3, † Sam R. Dolan, 2, ‡ and Lu'ıs C. B. Crispino 1, 4, §</text>
<text><location><page_1><loc_19><loc_88><loc_19><loc_88></location>1</text>
<text><location><page_1><loc_19><loc_87><loc_82><loc_88></location>Programa de P'os-Graduac¸˜ao em F'ısica, Universidade Federal do Par'a, 66075-110, Bel'em, Par'a, Brazil.</text>
<text><location><page_1><loc_27><loc_86><loc_28><loc_87></location>2</text>
<text><location><page_1><loc_28><loc_86><loc_74><loc_87></location>Consortium for Fundamental Physics, School of Mathematics and Statistics,</text>
<text><location><page_1><loc_22><loc_85><loc_78><loc_86></location>University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, United Kingdom.</text>
<text><location><page_1><loc_26><loc_84><loc_26><loc_84></location>3</text>
<text><location><page_1><loc_26><loc_83><loc_75><loc_84></location>Campus Altamira, Instituto Federal do Par'a, 68377-630, Altamira, Par'a, Brazil.</text>
<text><location><page_1><loc_20><loc_81><loc_81><loc_83></location>4 Departamento de Matem'atica da Universidade de Aveiro and Centre for Research and Development in Mathematics and Applications (CIDMA), Campus de Santiago, 3810-183 Aveiro, Portugal.</text>
<text><location><page_1><loc_43><loc_79><loc_58><loc_80></location>(Dated: January 4, 2024)</text>
<text><location><page_1><loc_18><loc_60><loc_83><loc_78></location>Regular black holes (RBHs) - geometries free from curvature singularities - arise naturally in theories of non linear electrodynamics. Here we study the absorption, and superradiant amplification, of a monochromatic planar wave in a charged, massive scalar field impinging on the electrically-charged Ay'on-Beato-Garc'ıa (ABG) RBH. Comparisons are drawn with absorption and superradiance for the Reissner-Nordstrom (RN) black hole in linear electrodynamics. We find that, in a certain parameter regime, the ABG absorption cross section is negative, due to superradiance, and moreover it is unbounded from below as the momentum of the wave approaches zero; this phenomenon of 'unbounded superradiance' is absent in the RN case. We show how the parameter space can be divided into regions, using the bounded/unbounded and absorption/amplification boundaries. After introducing a high-frequency approximation based on particle trajectories, we calculate the absorption cross section numerically, via the partial-wave expansion, as function of wave frequency, and we present a gallery of results. The cross section of the ABG RBH is found to be larger (smaller) than in the RN case when the field charge has the same (opposite) sign as the black hole charge. We show that it is possible to find 'mimics': situations in which the cross sections of both black holes are very similar. We conclude with a discussion of unbounded superradiance, and superradiant instabilities.</text>
<section_header_level_1><location><page_1><loc_22><loc_56><loc_36><loc_57></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_37><loc_49><loc_54></location>General Relativity (GR) is a geometric theory in which gravity is associated with the spacetime curvature generated by the presence of energy and momentum. For more than a century, the physical predictions of this theory have been scrutinised and tested experimentally in various ways [1-3]. In the last decade, for instance, two important verifications of GR predictions in the strong-field regime were reported: The Laser Interferometer Gravitational-Wave Observatory (LIGO) Collaboration performed the first direct detection of gravitational waves [4], from black hole (BH) coalescences; and the Event Horizon Telescope (EHT) Consortium has obtained the first image of a supermassive BH shadow [5].</text>
<text><location><page_1><loc_9><loc_25><loc_49><loc_36></location>BHs are among the most fascinating predictions of GR. These objects are solutions of Einstein's field equations (EFEs) characterized by an event horizon (i.e., a non-return surface), which are typically formed by gravitational collapse [6]. Observational evidence indicates that BHs populate galaxies [7]; for example, the Milky Way galaxy harbors a supermassive BH (with 4 . 1 ± 0 . 4 × 10 6 M ⊙ [8, 9]) at its core, as well as myriad stellar-mass BHs.</text>
<text><location><page_1><loc_9><loc_18><loc_49><loc_25></location>In electrovacuum, the uniqueness theorems of GR [10] determine that stationary BH solutions are described by only three parameters: mass, charge, and angular momentum. Despite this apparent simplicity, the stationary BHs of GR are also paradoxical in nature: at their core is a curvature singu-</text>
<text><location><page_1><loc_52><loc_56><loc_85><loc_57></location>larity , where the classical field theory breaks down.</text>
<text><location><page_1><loc_52><loc_38><loc_92><loc_55></location>Key theorems that support the formation of curvature singularities in classical GR were established by Penrose [11] and Hawking [12]. These theorems show that spacetimes can become geodesically incomplete in rather general (i.e., nonsymmetric) collapse scenarios, within the classical field theory; and this, in turn, raises concerns about the global breakdown of causality in such spacetimes. As a remedy, the cosmic censorship hypothesis asserts that [11] all curvature singularities must be shrouded behind (apparent or event) horizons. Therefore, the spacetime outside this one-way membrane is not adversely affected by the presence of these hidden singularities.</text>
<text><location><page_1><loc_52><loc_26><loc_92><loc_38></location>It can be argued that the formation of singularities represents a flaw in classical field theory, and that the paradoxes associated with singularities will be fully resolved in a quantum theory of gravity. It is not necessary, however, to await a complete quantum theory before studying the properties of BH solutions that are free from singularities. Recently, there has been increasing interest in the properties of so-called regular black hole (RBH) solutions.</text>
<text><location><page_1><loc_52><loc_16><loc_92><loc_26></location>The first RBH model was proposed in 1968 by James Bardeen [13]. In this model, as well as others [14-17], the source term (i.e., the stress-energy tensor) in the EFEs did not have a clear physical motivation or origin. In 1998, Eloy Ay'on-Beato and Alberto Garc'ıa found that a RBH could arise in a physically well-motivated theory: nonlinear electrodynamics (NED) minimally coupled to GR [18].</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_15></location>NED models are, in essence, generalizations of Maxwell's linear theory to strong electromagnetic fields [19-22]. Two well-studied NED models are the Euler-Heisenberg model [23], which provides an effective description of Quantum Electrodynamics at the one-loop level; and the Born-</text>
<text><location><page_2><loc_9><loc_83><loc_49><loc_93></location>Infeld model [19, 20], introduced to remove the infinite selfenergy of the electron. Among the features and applications of NED models [19-32], there have recently been proposed several electrically [33-38] and magnetically [39-45] charged RBH solutions, in minimally-coupled GR, as well as in alternative theories of gravity [46-49]. For a review on NED and applications to BH physics see Refs. [50, 51].</text>
<text><location><page_2><loc_9><loc_67><loc_49><loc_83></location>Motivated in part by recent observational breakthroughs, there is growing interest in discerning how the key properties of regular holes will differ from those of irregular (i.e., singular) BHs, particularly in the observable region exterior to the horizon. A canonical example of the irregular class - and a key point of comparison for this study - is the ReissnerNordstrom (RN) spacetime: a spherically-symmetric solution to the EFEs for linear (i.e., Maxwell) electromagnetism minimally coupled with GR, describing a BH of mass M and charge Q with two horizons, at r ± = M ± √ M 2 -Q 2 , and a curvature singularity, at r = 0 .</text>
<text><location><page_2><loc_9><loc_44><loc_49><loc_66></location>It is well known that the RN BH exhibits the phenomenon of superradiance when interacting with a scalar field of charge q . Field modes with a frequency ω > 0 satisfying ω < qϕ + are amplified , rather than absorbed, by the BH, where ϕ + is the electric potential at the outer horizon. In the superradiant regime, the BH loses charge and mass (i.e., it flows out of the BH into the field), and yet the area of the BH ( A = 4 πr 2 + ) increases. In the thermodynamic interpretation, the horizon area is associated with the entropy of the BH, and superradiance is then a necessary consequence of the second law of thermodynamics. For studies about charged superradiance in static BHs, see, e.g., Refs. [52-57]. Superradiance can also occur with neutral fields if the BH is spinning. Superradiance has been studied in a range of BH scenarios over the past fifty years, leading to various interesting outcomes (see, e.g., Ref. [58] for a comprehensive review).</text>
<text><location><page_2><loc_9><loc_19><loc_49><loc_43></location>It is natural to ask whether superradiance persists for RBHs - and if so, whether it is enhanced or diminished. In the far-field region, r ≫ M , where the electromagnetic field is weak, NED models are expected to reduce to linear electromagnetism, and thus, NED RBHs to be locally equivalent to RN BHs. Conversely, in the near-horizon region, where the electromagnetic field is strong, NED models are likely to differ substantially from their linear counterparts, and ϕ + may differ substantially from ϕ RN + = Q/r + . Consequently, we would expect the condition for superradiance to depend on the precise form of the NED model in question and, potentially, for certain models, to display enhanced versions of superradiance and new phenomenology. Some recent works addressed superradiance in the background of rotating regular spacetimes, considering massive scalar fields [59, 60], but works considering charged scalar fields and superradiance in static RBH geometries are still lacking in the literature.</text>
<text><location><page_2><loc_9><loc_9><loc_49><loc_18></location>In this paper, we study the absorption of a charged massive test scalar field in the background of a RBH solution, namely, the first proposed exact charged RBH solution of Ay'on-Beato and Garc'ıa (ABG) [18]. Here we are particularly interested in characterizing the effect of superradiance on the absorption cross section (ACS). Over the last fifty years, much effort has been made to compute absorption and scattering in different</text>
<text><location><page_2><loc_52><loc_83><loc_92><loc_93></location>BH scenarios (cf., e.g., Refs. [61-69] and references therein). Although several works have been dedicated to chargeless test fields, few have dealt with the absorption of charged scalar fields [54, 55]. Recently, the absorption of chargeless test fields has been investigated for RBHs [70-73], but the absorption of charged massive scalar waves is still to be properly quantified.</text>
<text><location><page_2><loc_52><loc_64><loc_92><loc_83></location>The remainder of this paper is organized as follows. In Sec. II, we review the main aspects of the ABG RBH spacetime proposed in Ref. [18], and in Sec. III we investigate the dynamics of a massive and charged scalar field on this spacetime. In Sec. III B, we present an expression to compute the ACS via a sum over partial waves; in III C we partition the parameter space; and in III D we describe a high-frequency approximation. Our numerical results concerning the absorption and superradiance properties of the ABG RBH solution are presented in Sec. IV, and we also compare them with those obtained in the RN case. We conclude with our final remarks in Sec. V. Throughout the paper we use the natural units, for which G = c = ℏ = 1 , and metric signature -2 .</text>
<section_header_level_1><location><page_2><loc_65><loc_60><loc_78><loc_61></location>II. FRAMEWORK</section_header_level_1>
<text><location><page_2><loc_52><loc_55><loc_92><loc_58></location>The action associated with NED minimally coupled to GR can be written as [18]</text>
<formula><location><page_2><loc_59><loc_51><loc_92><loc_54></location>S = 1 4 π ∫ d 4 x ( 1 4 R -L ( F ) ) √ -g, (1)</formula>
<text><location><page_2><loc_52><loc_44><loc_92><loc_49></location>where R is the Ricci scalar, L ( F ) is a gauge-invariant electromagnetic Lagrangian density, and g is the determinant of the metric tensor g µν . The electromagnetic invariant F and the standard electromagnetic field tensor are given by</text>
<formula><location><page_2><loc_59><loc_39><loc_92><loc_42></location>F = 1 4 F µν F µν and F µν = 2 ∇ [ µ A ν ] , (2)</formula>
<text><location><page_2><loc_52><loc_37><loc_91><loc_38></location>respectively, where A ν is the electromagnetic four-potential.</text>
<text><location><page_2><loc_52><loc_34><loc_92><loc_37></location>It is possible to represent NED in a different framework by introducing an auxiliary anti-symmetric tensor</text>
<formula><location><page_2><loc_67><loc_31><loc_92><loc_32></location>P µν ≡ L F F µν , (3)</formula>
<text><location><page_2><loc_52><loc_27><loc_92><loc_29></location>where L F ≡ ∂ L /∂F ; and also a structural function H ( P ) through a Legendre transformation [74], namely</text>
<formula><location><page_2><loc_64><loc_24><loc_92><loc_25></location>H ( P ) ≡ 2 F L F -L ( F ) . (4)</formula>
<text><location><page_2><loc_52><loc_21><loc_82><loc_22></location>The invariant associated with P µν is defined as</text>
<formula><location><page_2><loc_67><loc_17><loc_92><loc_20></location>P ≡ 1 4 P µν P µν . (5)</formula>
<text><location><page_2><loc_52><loc_13><loc_92><loc_16></location>Among its applications [22, 74], this framework is useful to obtain electrically charged NED-based RBH solutions [37].</text>
<text><location><page_2><loc_53><loc_11><loc_84><loc_13></location>With the help of Eqs. (2)-(5), one can show that</text>
<formula><location><page_2><loc_54><loc_9><loc_92><loc_10></location>P = ( L F ) 2 F, H P L F = 1 and F µν = H P P µν , (6)</formula>
<text><location><page_3><loc_9><loc_89><loc_49><loc_93></location>where H P ≡ ∂ H /∂P . By varying the action (1) with respect to the metric tensor g µν and using Eqs. (6), it is possible to obtain</text>
<formula><location><page_3><loc_11><loc_87><loc_49><loc_88></location>G µ ν = -T µ ν = 2[ H P P να P µα -δ µ ν (2 P H P -H )] , (7)</formula>
<text><location><page_3><loc_9><loc_80><loc_49><loc_86></location>which are Einstein-NED field equations, and where G µ ν is the Einstein tensor and T µ ν is the energy-momentum tensor. The variation of the action (1) with respect to A µ leads to ∇ µ P µν = 0 (in the absence of electromagnetic sources).</text>
<text><location><page_3><loc_9><loc_76><loc_49><loc_80></location>For the solution proposed in Ref. [18], from now on simply referred to as ABG solution, the structural function (the NED source) is</text>
<formula><location><page_3><loc_9><loc_69><loc_50><loc_75></location>H ( P ) = P ( 1 -3 √ -2 Q 2 P ) (1 + √ -2 Q 2 P ) 3 -3 M | Q | Q 2 ( √ -2 Q 2 P 1 + √ -2 Q 2 P ) 5 2 , (8)</formula>
<text><location><page_3><loc_9><loc_62><loc_49><loc_69></location>with Q and M being the electric charge and mass of the central object, respectively. Considering a spherically symmetric and static line element as an ansatz for the spacetime, together with the Eqs. (7) and (8), one can obtain the ABG line element in spherical coordinates ( x µ = { t, r, θ, φ } ),</text>
<formula><location><page_3><loc_11><loc_58><loc_49><loc_61></location>ds 2 = f ( r ) dt 2 -1 f ( r ) dr 2 -r 2 ( dθ 2 +sin 2 θ dφ 2 ) , (9)</formula>
<text><location><page_3><loc_9><loc_56><loc_13><loc_57></location>where</text>
<formula><location><page_3><loc_10><loc_52><loc_49><loc_55></location>f ( r ) = f ABG ( r ) ≡ 1 -2 Mr 2 ( r 2 + Q 2 ) 3 / 2 + Q 2 r 2 ( r 2 + Q 2 ) 2 , (10)</formula>
<text><location><page_3><loc_9><loc_50><loc_38><loc_51></location>is the metric function of the ABG spacetime.</text>
<text><location><page_3><loc_9><loc_47><loc_49><loc_50></location>In the asymptotic limit r → ∞ , the metric function (10) has the following behavior</text>
<formula><location><page_3><loc_18><loc_45><loc_49><loc_46></location>f ABG ( r ) = f RN ( r ) + O ( r -3 ) , (11)</formula>
<text><location><page_3><loc_9><loc_43><loc_49><loc_44></location>where f RN ( r ) is the metric function of the RN spacetime [75],</text>
<formula><location><page_3><loc_20><loc_39><loc_49><loc_42></location>f RN ( r ) ≡ 1 -2 M r + Q 2 r 2 . (12)</formula>
<text><location><page_3><loc_9><loc_33><loc_49><loc_38></location>As argued earlier, this is expected because, in the far-field region, the electromagnetic field is weak and thus in the linear (i.e., Maxwell) regime. On the other hand, expanding the ABG metric function in powers of Q yields</text>
<formula><location><page_3><loc_15><loc_29><loc_49><loc_32></location>f ABG ( r ) = f RN ( r ) + 3 MQ 2 r 3 + O ( Q 4 ) . (13)</formula>
<text><location><page_3><loc_9><loc_16><loc_49><loc_28></location>When the condition | Q | ≤ Q ext ≈ 0 . 6341 M is fulfilled [18], the line element (9) describes an ABG RBH. For | Q | < Q ext , the ABG RBH possesses an inner (Cauchy) horizon at r -and an outer (event) horizon at r + , given by the real positive roots of f ABG ( r ) = 0 . For | Q | = Q ext , we have the so-called extreme ABGRBH, with r + = r -. For | Q | > Q ext , we have horizonless solutions. The ABG causal structure is similar to the RN one (for which Q RN ext = M ).</text>
<text><location><page_3><loc_9><loc_12><loc_49><loc_16></location>Throughout this work, we shall restrict our analysis to BH solutions ( | Q | ≤ Q ext ) , and exhibit our results in terms of the normalized electric charge</text>
<formula><location><page_3><loc_25><loc_8><loc_49><loc_11></location>α ≡ Q Q ext , (14)</formula>
<text><location><page_3><loc_52><loc_92><loc_83><loc_93></location>which satisfies 0 ≤ | α | ≤ 1 for BH geometries.</text>
<text><location><page_3><loc_52><loc_89><loc_92><loc_92></location>From F 01 = E ( r ) = H P Q/r 2 , one can show that the radial electrostatic field E ( r ) of the ABG solution is given by</text>
<formula><location><page_3><loc_53><loc_85><loc_92><loc_88></location>E ABG ( r ) = Qr 4 ( r 2 -5 Q 2 ( r 2 + Q 2 ) 4 + 15 M 2( r 2 + Q 2 ) 7 / 2 ) , (15)</formula>
<text><location><page_3><loc_52><loc_82><loc_92><loc_84></location>which is finite at the origin and asymptotically behaves as the electrostatic field in the RN case, given by</text>
<formula><location><page_3><loc_67><loc_78><loc_92><loc_81></location>E RN ( r ) = Q r 2 . (16)</formula>
<text><location><page_3><loc_52><loc_73><loc_92><loc_77></location>A detailed analysis of the metric function, electric field, and geodesics of massless particles of ABG and RN BHs is presented in Refs. [73, 76].</text>
<text><location><page_3><loc_52><loc_70><loc_92><loc_73></location>The covariant components of the electromagnetic fourpotential are given by</text>
<formula><location><page_3><loc_65><loc_68><loc_92><loc_69></location>A µ = ( ϕ ( r ) , 0 , 0 , 0) , (17)</formula>
<text><location><page_3><loc_52><loc_64><loc_92><loc_67></location>where ϕ is the electrostatic potential, which can be obtained using ϕ ( r ) = -∫ r ∞ E · d l and Eq. (15), to obtain 1</text>
<formula><location><page_3><loc_53><loc_59><loc_92><loc_64></location>ϕ ABG ( r ) = r 5 2 Q ( 3 M r 5 + 2 Q 2 ( Q 2 + r 2 ) 3 -3 M ( Q 2 + r 2 ) 5 / 2 ) . (18)</formula>
<text><location><page_3><loc_52><loc_56><loc_92><loc_59></location>In Fig. 1, we plot the ABG electrostatic potential, ϕ ABG ( r ) alongside the electrostatic potential of the RN BH,</text>
<formula><location><page_3><loc_67><loc_52><loc_92><loc_55></location>ϕ RN ( r ) = Q r . (19)</formula>
<text><location><page_3><loc_52><loc_46><loc_92><loc_52></location>Notably, ϕ ABG ( r ) is finite at r = 0 , whereas ϕ RN ( r ) diverges as r → 0 . In the far field ( r → ∞ ), ϕ ABG → ϕ RN . At the (outer) horizon, ϕ ( r + ) ABG > ϕ ( r + ) RN , and thus an enhanced superradiant regime may be anticipated.</text>
<figure>
<location><page_3><loc_52><loc_23><loc_92><loc_44></location>
<caption>FIG. 1. Comparison between the electrostatic potentials of ABG and RN BHs, considering different values of α .</caption>
</figure>
<section_header_level_1><location><page_4><loc_23><loc_92><loc_34><loc_93></location>III. ANALYSIS</section_header_level_1>
<section_header_level_1><location><page_4><loc_18><loc_89><loc_40><loc_90></location>A. Scalar fields and superradiance</section_header_level_1>
<text><location><page_4><loc_9><loc_83><loc_49><loc_87></location>We are interested in investigating a scalar field Φ with mass µ and charge q , propagating in a static (electric) RBH spacetime. Therefore, we shall consider the Klein-Gordon equation</text>
<formula><location><page_4><loc_15><loc_80><loc_49><loc_81></location>( ∇ ν + iqA ν ) ( ∇ ν + iqA ν ) Φ + µ 2 Φ = 0 . (20)</formula>
<text><location><page_4><loc_9><loc_76><loc_49><loc_78></location>Exploiting the separability of Eq. (20), we can write a particular mode of Φ as</text>
<formula><location><page_4><loc_20><loc_72><loc_49><loc_75></location>Φ ≡ Ψ ωl ( r ) r P l (cos θ ) e -iωt , (21)</formula>
<text><location><page_4><loc_9><loc_65><loc_49><loc_71></location>where Ψ ωl ( r ) are radial functions and P l (cos θ ) are the Legendre polynomials. The indexes ω and l denote the frequency and the angular momentum of the scalar wave, respectively. Inserting Eq. (21) into Eq. (20) leads to the radial equation</text>
<formula><location><page_4><loc_22><loc_61><loc_49><loc_64></location>d 2 dr 2 ⋆ Ψ ωl = V ( r )Ψ ωl , (22)</formula>
<text><location><page_4><loc_9><loc_57><loc_49><loc_59></location>where r ⋆ is the tortoise coordinate defined by dr ⋆ = dr/f ( r ) , and the potential function V ( r ) reads</text>
<formula><location><page_4><loc_9><loc_51><loc_49><loc_56></location>V ( r ) ≡ f ( r ) [ µ 2 + 1 r df ( r ) dr + l ( l +1) r 2 ] -( ω -qA 0 ( r )) 2 . (23)</formula>
<text><location><page_4><loc_9><loc_47><loc_49><loc_51></location>From the form of Eq. (22), it is clear that Φ is propagative (i.e., oscillatory) in regions where V ( r ) < 0 and evanescent (i.e., exponential) in regions where V ( r ) > 0 .</text>
<text><location><page_4><loc_9><loc_28><loc_49><loc_47></location>As the angular momentum l increases, the height of the potential barrier increases commensurately (as in the massless case [73]). Figure 2 shows V ( r ) as a function of the parameter qM for the particular case l = 0 , ω = µ and α = 0 . 5 (defined in Eq. (14)). The height of the local maximum value of V ( r ) increases with qM . As we increase α , the peak of V ( r ) increases (decreases) for qM > 0 ( qM < 0 ). This is anticipated from the Lorentz force: particles with the same charge sign of the BH are repelled, and consequently less absorbed, than particles with the opposite charge, which are attracted. Note also that for some values of qM , µM , and α , the peak of the radial function V ( r ) becomes negative (cf. curve qM = -0 . 3 in Fig. 2) [78].</text>
<text><location><page_4><loc_9><loc_25><loc_49><loc_28></location>In the (planar-wave) scattering problem, the wave satisfies the ingoing boundary conditions</text>
<formula><location><page_4><loc_12><loc_20><loc_49><loc_24></location>Ψ ωl ∼ { T ωl e -iζr ⋆ , r ⋆ →-∞ I ωl e -iκr ⋆ + R ωl e iκr ⋆ , r ⋆ → + ∞ , (24)</formula>
<text><location><page_4><loc_9><loc_16><loc_49><loc_19></location>where ζ ≡ ω -qϕ + (with ϕ + ≡ ϕ ( r + ) ), and κ ≡ √ ω 2 -µ 2 . The quantities T ωl , I ωl , and R ωl are complex coefficients.</text>
<text><location><page_4><loc_9><loc_9><loc_49><loc_16></location>Justification for the ingoing boundary condition in (24) runs as follows. In the near-horizon region, the general solution for the field Φ is a superposition of two terms, with behaviours e -i ( ωt ± ζr ⋆ ) . We seek fields Φ and A µ which are regular on the future horizon in a suitable gauge. Noting that, for A µ</text>
<figure>
<location><page_4><loc_52><loc_72><loc_92><loc_93></location>
<caption>FIG. 2. The function V ( r ) of charged massive scalar waves in the background of the ABG RBH, as a function of r/r + , considering different values of qM . In this panel, we have chosen l = 0 , ωM = µM = 0 . 1 , and α = 0 . 5 .</caption>
</figure>
<text><location><page_4><loc_52><loc_47><loc_92><loc_62></location>in Eq. (17), the Lorentz invariant A µ A µ is divergent, we can make a gauge transformation , A µ → A ' µ = A µ + q -1 ∇ µ χ and Φ → Φ ' = e iχ Φ , such that A ' µ = 0 on the horizon. This corresponds to the choice χ = qϕ + t . Hence the general solution for Φ ' is a superposition of two terms with behaviours e -iζ ( t ± r ∗ ) . The term with upper sign choice (+) is regular (irregular) on the future (past) horizon, and the term with the lower sign ( -) is regular (irregular) on the past (future) horizon. The boundary condition in Eq. (24) then follows from the requirement that Φ ' is regular on the future horizon.</text>
<text><location><page_4><loc_52><loc_43><loc_92><loc_47></location>For a propagating wave at infinity (unbounded modes), the condition κ > 0 holds, i.e., ω 2 > µ 2 . The transmission and reflection coefficients are defined, respectively, as</text>
<formula><location><page_4><loc_58><loc_39><loc_92><loc_42></location>|T ωl | 2 ≡ | T ωl | 2 | I ωl | 2 and |R ωl | 2 ≡ | R ωl | 2 | I ωl | 2 . (25)</formula>
<text><location><page_4><loc_52><loc_35><loc_92><loc_37></location>From the conservation of the flux, or using the Wronskian of Eq. (22), one can derive</text>
<formula><location><page_4><loc_64><loc_30><loc_92><loc_33></location>|R ωl | 2 + ζ κ |T ωl | 2 = 1 . (26)</formula>
<text><location><page_4><loc_52><loc_28><loc_72><loc_29></location>The amplification factor [58] is</text>
<formula><location><page_4><loc_61><loc_24><loc_92><loc_27></location>Z ωl ≡ |R ωl | 2 -1 = -ζ κ |T ωl | 2 . (27)</formula>
<text><location><page_4><loc_52><loc_16><loc_92><loc_23></location>This measures the fractional gain (or loss) of energy in a scattered wave, with positive values of Z ωl corresponding to superradiant amplification. Clearly, the sign of Z ωl is determined by the sign of ζ . Hence the critical frequency for superradiant scattering is</text>
<formula><location><page_4><loc_68><loc_13><loc_92><loc_14></location>ω c = qϕ + . (28)</formula>
<text><location><page_4><loc_52><loc_9><loc_92><loc_11></location>For frequencies ω > ω c , the wave is absorbed; conversely, for frequencies 0 < ω < ω c , the wave is amplified.</text>
<text><location><page_5><loc_9><loc_80><loc_49><loc_93></location>In Fig. 3, we show ϕ + , the electric potential at the horizon, for ABG and RN BHs. We note that (for Q > 0 ) ϕ + is always positive and increases monotonically with α . As a consequence, superradiance occurs whenever qϕ + > 0 . We also observe that ϕ ABG + is always greater than ϕ RN + . Therefore, for the same values of qM and α , the critical frequency of the ABG RBH is always larger than that of the RN BH. This implies a greater capacity for superradiant scattering in the ABG case.</text>
<figure>
<location><page_5><loc_9><loc_58><loc_48><loc_78></location>
<caption>FIG. 3. Electrostatic potential at the event horizon, ϕ + ≡ ϕ ( r + ) , as a function of the normalized charge α , for ABG and RN BHs.</caption>
</figure>
<section_header_level_1><location><page_5><loc_20><loc_47><loc_38><loc_48></location>B. Absorption cross section</section_header_level_1>
<text><location><page_5><loc_9><loc_42><loc_49><loc_45></location>The ACS σ , for a plane wave incident upon a sphericallysymmetric BH, can be expanded in partial waves as follows:</text>
<formula><location><page_5><loc_25><loc_37><loc_49><loc_41></location>σ = ∞ ∑ l =0 σ l , (29)</formula>
<text><location><page_5><loc_9><loc_35><loc_25><loc_36></location>where the partial ACS is</text>
<formula><location><page_5><loc_19><loc_31><loc_49><loc_34></location>σ l = π κ 2 (2 l +1)(1 -|R ωl | 2 ) . (30)</formula>
<text><location><page_5><loc_9><loc_29><loc_23><loc_30></location>Hence, using Eq. (26),</text>
<formula><location><page_5><loc_18><loc_25><loc_49><loc_28></location>σ l = π κ 3 (2 l +1)( ω -ω c ) |T ωl | 2 . (31)</formula>
<text><location><page_5><loc_9><loc_22><loc_49><loc_24></location>For superradiant modes (with 0 < ω < ω c = qϕ + ), σ l takes negative values, as the wave is amplified rather than absorbed.</text>
<text><location><page_5><loc_9><loc_12><loc_49><loc_21></location>In the limit ω → µ (from above), the momentum of the wave tends to zero, κ → 0 . Hence σ l in Eq. (31) will diverge in this limit, unless lim ω → µ |T ωl | 2 /κ 3 exists. In other words, σ l will diverge unless the transmission factor |T ωl | 2 approaches zero at least as rapidly as the cube of the momentum, κ 3 . We will call the divergent case unbounded and the finite case bounded .</text>
<text><location><page_5><loc_9><loc_9><loc_49><loc_11></location>It is clear that there are four possibilities to consider in the limit ω → µ , namely: (i) bounded absorption, (ii) bounded</text>
<text><location><page_5><loc_52><loc_83><loc_92><loc_93></location>superradiance, (iii) unbounded absorption and (iv) unbounded superradiance. Cases (i), (ii) and (iii) have been observed in absorption by a RN (irregular) BH [55]. The fourth possibility, unbounded superradiance, does not appear to occur for RN BHs; but it does arise for the regular ABG BH, as we demonstrate in Sec. IV. To understand why this arises, we now turn attention to the properties of the potential.</text>
<section_header_level_1><location><page_5><loc_64><loc_79><loc_80><loc_80></location>C. The parameter space</section_header_level_1>
<text><location><page_5><loc_52><loc_70><loc_92><loc_77></location>In this section, we argue that it is possible to divide the parameter space into regions where behaviours (i)-(iv) occur (see above) by examining the behaviour of an effective potential function. Considering Eq. (23) in the limit µ → ω , we define</text>
<formula><location><page_5><loc_53><loc_62><loc_92><loc_68></location>U ( r ) ≡ -V ( r ) | µ = ω , = ( µ -qϕ ( r )) 2 -f ( r ) [ µ 2 + 1 r df ( r ) dr + l ( l +1) r 2 ] . (32)</formula>
<text><location><page_5><loc_52><loc_53><loc_92><loc_60></location>In Fig. 4, we present the typical behavior of the function U ( r ) in the ABG RBH spacetime. In the region where U ( r ) is positive, the wave is propagative. The plot makes it clear that the existence of a propagative region extending to spatial infinity depends critically on the parameter values.</text>
<figure>
<location><page_5><loc_52><loc_31><loc_92><loc_52></location>
<caption>FIG. 4. The function U ( r ) of charged massive scalar waves in the background of the ABG RBH, as a function of r/r + , considering l = 0 , µM = 0 . 2 , and qM = 0 . 8 for distinct values of α . In this case, superradiance occurs whenever α > 0 . 2702 .</caption>
</figure>
<text><location><page_5><loc_52><loc_9><loc_92><loc_21></location>For the uncharged massive scalar field case, Jung and Park [78] defined the critical case as that in which the local maximum of U ( r ) is exactly zero. This idea extends naturally to absorption of a charged field: the critical case is shown as the blue dashed line in Fig. 4, and this case defines a critical charge α c (for fixed l , µM and qM ). For α < α c , a propagative region extends from a certain radius r c out to infinity (i.e., the region r ∈ ( r c , ∞ ) ), whereas for α > α c , the only propagative region is close to the horizon. It is natural to</text>
<text><location><page_6><loc_9><loc_89><loc_49><loc_93></location>anticipate qualitatively different absorption properties in the limit ω → µ , with the former (latter) case corresponding to unbounded (bounded) behaviour.</text>
<text><location><page_6><loc_9><loc_85><loc_49><loc_89></location>In fact, to determine the existence of a propagative region that extends to infinity, it is sufficient to examine the larger expansion of U ( r ) , given by</text>
<formula><location><page_6><loc_17><loc_80><loc_49><loc_83></location>U ( r ) = 2 µ ( µM -qQ ) r + O ( r -2 ) . (33)</formula>
<text><location><page_6><loc_9><loc_71><loc_49><loc_79></location>At leading order, the expansion is identical for the RN and ABG BHs (as one might expect in the weak-field/linear regime). For µM > qQ , Newtonian attraction dominates over the Coulomb repulsion and the propagative region exists; for µM < qQ , Coulomb repulsion is dominant and the propagative region does not exist. The critical case is at µM = qQ .</text>
<text><location><page_6><loc_9><loc_66><loc_49><loc_71></location>We can now divide the parameter space into regions using two separatrices: µM = qQ (the bounded/unbounded boundary) and µ = qϕ + (the absorption/amplification boundary).</text>
<text><location><page_6><loc_9><loc_43><loc_49><loc_66></location>Figure 5 shows the anticipated behaviour of the ACS in the limit ω → µ (i.e. κ → 0 ), in the parameter space. The horizontal blue line ( µM = qQ ) separates the bounded and unbounded regions. The solid red line demarcates the onset of superradiance. In the RN case, there is no overlap between the unbounded and superradiant regions (though the boundaries meet in the extremal case, Q = M ). This is consistent with an observed absence of unbounded superradiance. Conversely, for the ABG BH, the superradiant region is significantly larger than in the RN case (even for Q → 0 ), due to the increase in ϕ + (see Fig. 3). Consequently, the unbounded region overlaps with the superradiant region, and hence we should anticipate unbounded superradiance (that is, unbounded amplification) to occur in the limit ω → µ , in this region of the parameter space. These conclusions are supported by the numerical evidence presented in Sec. IV C.</text>
<section_header_level_1><location><page_6><loc_18><loc_39><loc_40><loc_40></location>D. High-frequency approximation</section_header_level_1>
<text><location><page_6><loc_9><loc_27><loc_49><loc_37></location>We now turn attention to absorption in the regime of high frequencies and short wavelengths. In this regime, the characteristics of the charged massive scalar wave can be associated with the trajectories of charged particles subject to the Lorentz force imparted by the electric background field. In order to obtain the equations of motion associated with our problem, we consider the following Lagrangian</text>
<formula><location><page_6><loc_19><loc_23><loc_49><loc_26></location>L cp = 1 2 g µν ˙ x µ ˙ x ν + q cp m A µ ˙ x µ , (34)</formula>
<text><location><page_6><loc_9><loc_19><loc_49><loc_22></location>where the overdot stands for the derivative with respect to the proper time, and q cp and m are the charged and mass of the</text>
<text><location><page_6><loc_52><loc_90><loc_92><loc_93></location>particle, respectively. From Eq. (34), one can introduce the conserved quantities</text>
<formula><location><page_6><loc_67><loc_86><loc_92><loc_89></location>E = m ∂ L cp ∂ ˙ t , (35)</formula>
<formula><location><page_6><loc_67><loc_83><loc_92><loc_86></location>L = -m ∂ L cp ∂ ˙ φ , (36)</formula>
<text><location><page_6><loc_52><loc_74><loc_92><loc_82></location>which are related to the energy and the angular momentum of the particle and, in the semiclassical limit, are associated with ω and l +1 / 2 , respectively. Using Eqs. (34)-(36) together with g µν ˙ x µ ˙ x ν = 1 (the condition that a massive particle follows a timelike path, parametrized by its proper time [79]), one can show that</text>
<formula><location><page_6><loc_54><loc_70><loc_92><loc_73></location>˙ r 2 ( m 2 L 2 ) = ( E -q cp A 0 ) 2 L 2 -f ( r ) ( m 2 L 2 + 1 r 2 ) , (37)</formula>
<text><location><page_6><loc_52><loc_65><loc_92><loc_68></location>in which, due to the spherical symmetry, we considered the motion in the equatorial plane ( θ = π/ 2) .</text>
<text><location><page_6><loc_52><loc_63><loc_92><loc_65></location>By defining the impact parameter b ≡ L/vE and K ( r ) ≡ ˙ r 2 ( m 2 /L 2 ) , we can rewrite Eq. (34) as</text>
<formula><location><page_6><loc_58><loc_55><loc_92><loc_61></location>K ( r ) = 1 b 2 v 2 ( 1 -√ 1 -v 2 q cp m A 0 ( r ) ) 2 -f ( r ) ( 1 -v 2 b 2 v 2 + 1 r 2 ) , (38)</formula>
<text><location><page_6><loc_52><loc_52><loc_83><loc_53></location>where v is a dimensionless parameter defined by</text>
<formula><location><page_6><loc_67><loc_47><loc_92><loc_51></location>v ≡ √ 1 -m 2 E 2 . (39)</formula>
<text><location><page_6><loc_52><loc_41><loc_92><loc_46></location>Since we are interested in the unbounded timelike paths, i.e., κ > 0 , this parameter is limited by 0 < v ≤ 1 . Considering that K ( r ) and its first derivative vanish at the critical radius r c , namely</text>
<formula><location><page_6><loc_68><loc_38><loc_92><loc_39></location>K ( r c ) = 0 , (40)</formula>
<formula><location><page_6><loc_67><loc_34><loc_92><loc_37></location>d K ( r c ) dr = 0 , (41)</formula>
<text><location><page_6><loc_52><loc_32><loc_81><loc_33></location>we may find the critical impact parameter b c ,</text>
<formula><location><page_6><loc_53><loc_22><loc_92><loc_30></location>b 2 c = r 2 c m 2 v 2 f ( r c ) [ m 2 ( 1 + ( v 2 -1) f ( r c ) ) + q cp A 0 ( r c ) ( q cp A 0 ( r c )(1 -v 2 ) -2 m √ 1 -v 2 ) ] , (42)</formula>
<text><location><page_6><loc_52><loc_19><loc_80><loc_21></location>and an equation that gives the values of r c ,</text>
<formula><location><page_6><loc_11><loc_9><loc_92><loc_15></location>2 f ( r c ) ( m 2 zf ( r c ) -( m -q cp √ zA 0 ( r c )) ( m -q cp √ z ( r c A ' 0 ( r c ) + A 0 ( r c ) ))) + r c f ' ( r c ) ( m -q cp √ zA 0 ( r c )) 2 m 2 zf ( r c ) -( m -q cp √ zA 0 ( r c )) 2 = 0 , (43)</formula>
<figure>
<location><page_7><loc_10><loc_72><loc_90><loc_93></location>
<caption>FIG. 5. Absorption parameter space for ABG (left panel) and RN (right panel) BHs, as functions of Q/M . The solid red curve corresponds to the superradiance threshold and the solid blue curve to the attractive/repulsive threshold in Eq. (33). In the left panel, the red line meets the vertical axis at µM/qQ = 23 / 16 . The points highlight situations in which we have unbounded superradiance. The total ACS corresponding to these situations is exhibited in Fig. 10.</caption>
</figure>
<text><location><page_7><loc_9><loc_55><loc_49><loc_61></location>where we defined z ≡ 1 -v 2 and the prime ( ' ) denotes derivative with respect to the radial coordinate r . For q cp = 0 , we recover the b c and r c of the massive chargeless case, which are given by [64]</text>
<formula><location><page_7><loc_20><loc_50><loc_49><loc_53></location>b 2 c = r 2 c ( 1 + ( v 2 -1 ) f ( r c ) ) v 2 f ( r c ) , (44)</formula>
<formula><location><page_7><loc_17><loc_46><loc_49><loc_49></location>2 ( 1 + ( v 2 -1 ) f ( r c ) ) = r c f ' ( r c ) f ( r c ) , (45)</formula>
<text><location><page_7><loc_9><loc_38><loc_49><loc_44></location>respectively. In the limit m → 0 , which implies in v → 1 , we obtain the results for the massless case [73]. The high-frequency absorption cross section, also called geometric cross section (GCS), σ gcs , is given by [79]</text>
<formula><location><page_7><loc_25><loc_34><loc_49><loc_36></location>σ gcs = πb 2 c . (46)</formula>
<section_header_level_1><location><page_7><loc_24><loc_28><loc_34><loc_29></location>IV. RESULTS</section_header_level_1>
<section_header_level_1><location><page_7><loc_21><loc_24><loc_36><loc_25></location>A. Numerical Analysis</section_header_level_1>
<text><location><page_7><loc_9><loc_9><loc_49><loc_21></location>We can obtain the reflection and transmission coefficients, given by Eqs. (25), by numerically integrating the Eq. (22) from very close to r + up to far from the BH, with the boundary conditions given by Eqs. (24) and their derivatives. Then we compute the total ACS using Eq. (29). The oscillatory character of the ACS is related to the partial waves contributions [see Eq. (30)]. We have chosen, in general, to perform the summation in Eq. (29) up to l = 20 . The GCS is obtained numerically through Eq. (46), using Eqs. (42) and (43).</text>
<section_header_level_1><location><page_7><loc_59><loc_60><loc_85><loc_61></location>B. ABG regular black hole cross sections</section_header_level_1>
<text><location><page_7><loc_52><loc_43><loc_92><loc_57></location>Figures 6 and 7 show the total ACS for different values of the charge ( qM ) and mass ( µM ) couplings, respectively. Generically, we can see that the total ACS oscillates around the GCS (black dotted lines), with good agreement in the high-frequency regime. Moreover, for a fixed value of µM and α , we observe that the absorption increases (diminishes) as we consider smaller (higher) values of qQ , as a consequence of the Lorentz force. As shown in Fig. 7, the total ACS increases as we increase µM , so that the increase of µM leads to a higher absorption of planar scalar waves.</text>
<figure>
<location><page_7><loc_52><loc_20><loc_92><loc_41></location>
<caption>FIG. 6. Total ACS of charged massive scalar waves in the background of the ABG RBH, as a function of ω/µ , for different choices of qM , considering α = 0 . 5 and µM = 0 . 2 . The ACS is compared with the geometric cross section (dashed black lines).</caption>
</figure>
<text><location><page_7><loc_53><loc_9><loc_92><loc_10></location>Figure 8 shows the total ACS together with the partial</text>
<figure>
<location><page_8><loc_9><loc_72><loc_49><loc_93></location>
<caption>FIG. 7. Total ACS of charged massive scalar waves in the background of the ABG RBH, as a function of ωM for α = 0 . 5 , qM = 0 . 1 , and different values of µM .</caption>
</figure>
<text><location><page_8><loc_9><loc_45><loc_49><loc_62></location>ACS for two choices of qM , with the normalized BH charge α = 0 . 5 and the field mass coupling µM = 0 . 2 . The plots show that the oscillatory pattern in the total ACS is related to the sequential contributions from partial ACSs l = 0 , 1 , 2 , . . . . The monopole ( l = 0) dominates the behavior of the total ACS in the low-frequency regime. Note that, although the values of α and µM are equal in both panels of Fig. 8, superradiance occurs only in the bottom panel case. We observe that the ACS is negative in the range µ < ω ≲ 2 µ . Physically, a negative cross section implies that the stimulation from the plane wave causes the BH to transmit mass-energy and charge into the field.</text>
<text><location><page_8><loc_9><loc_35><loc_49><loc_45></location>In Fig. 9, we present the amplification factor of massive charged scalar fields in the background of an ABG RBH. (We exhibit the amplification factor (27) in percentage, i.e., Z ωl [%] ≡ 100Z ωl , and restricted to the regions where Z ωl [%] ≥ 0 ). As we can see, the maximum superradiant amplification increases with the charge of the scalar field, for fixed values of the BH mass and (positive) charge.</text>
<section_header_level_1><location><page_8><loc_9><loc_31><loc_49><loc_32></location>C. Unbounded superradiance from a regular ABG black hole</section_header_level_1>
<text><location><page_8><loc_9><loc_17><loc_49><loc_29></location>In the previous section, we presented some typical absorption properties of charged massive scalar waves in the background of the ABG RBH. In the limit ω → µ we saw two types of behaviour: unbounded absorption (Figs. 6, 7 and 8, upper plot) and bounded superradiance (Fig. 8, lower plot). In this section, we show that the ABG RBH can also display unbounded superradiance for parameter choices informed by Fig. 5, and the discussion in Sec. III C.</text>
<figure>
<location><page_8><loc_52><loc_72><loc_92><loc_93></location>
<caption>Figure 10 shows the ACS of the ABG RBH for a selection of parameter choices (informed by Fig. 5) for which we would expect to see unbounded superradiance. As we can see, the results are consistent with the expectations of Fig. 5: in all cases, the cross section σ is negative (indicating superradiant amplification) and it grows without bound as ω → µ .</caption>
</figure>
<figure>
<location><page_8><loc_52><loc_51><loc_92><loc_71></location>
<caption>FIG. 8. Partial and total ACSs of charged massive scalar waves in the background of the ABG RBH, as functions of ω/µ , for α = 0 . 5 and µM = 0 . 2 , in two distinct scenarios: (i) qM = 0 . 2 (top panel) and (ii) qM = 0 . 8 (bottom panel) . The inset in the bottom panel emphasizes superradiance, that occurs for 1 < ω/µ ≲ 1 . 924 .</caption>
</figure>
<figure>
<location><page_8><loc_52><loc_17><loc_92><loc_38></location>
<caption>FIG. 9. Superradiant amplification of massive charged scalar fields by an ABG RBH with α = 0 . 5 , as a function of ω/µ . Here we consider the mode l = 0 , µM = 0 . 2 , and distinct values of qM .</caption>
</figure>
<text><location><page_9><loc_9><loc_82><loc_49><loc_93></location>The results in Fig. 10 reveal a remarkable implication of the electromagnetic fields associated with (electrically charged) NED-based RBH geometries: they generate a superradiant divergence in the ACS of a charged, massive scalar field. That is, an ABG BH stimulated by a massive plane wave of low momentum has an ACS which is unbounded from below. This is in stark contrast to the RN BH, where the cross section cannot obtain arbitrary negative values.</text>
<figure>
<location><page_9><loc_9><loc_60><loc_48><loc_79></location>
<caption>FIG. 10. Total ACS of charged massive scalar waves in the background of the ABG RBH, as a function of ω/µ , considering situations in which we have unbounded superradiance (the points in the left panel of Fig. 5). We focus on the region near the limit ωM → µM to emphasize the divergent behavior of the total ACS in the superradiant regime.</caption>
</figure>
<section_header_level_1><location><page_9><loc_13><loc_45><loc_45><loc_46></location>D. Comparison with the Reissner-Nordstrom BH</section_header_level_1>
<text><location><page_9><loc_9><loc_40><loc_49><loc_43></location>In this section, we compare the absorption properties of ABG RBHs with those of RN BHs [55].</text>
<text><location><page_9><loc_9><loc_19><loc_49><loc_40></location>Figure 11 shows a comparison between the total ACSs of ABG RBHs and RN BHs for α = 0 . 4 , qM = 1 . 6 , and two values of µM . We see that, for a fixed value of µM , the total ACS of the ABG RBH is smaller than the total ACS of the RN BH, across the frequency range. We also observe that for µM = 0 . 4 , the ABG RBH exhibits superradiant scattering (i.e. σ < 0 in some range of ω ), whereas the RN BH does not (for these parameters). This feature is due to the higher threshold frequency ( ω c = qϕ + ) for superradiance in the ABG case (see Sec. III, in particular, Eq. (28) and Fig. 3) and the condition for unbounded modes, namely ω 2 > µ 2 . For µM = 0 . 4 , both systems show unbounded absorption as ω → µ . Conversely, for µM = 0 . 8 , the RN shows bounded absorption, whereas the ABG RBH shows bounded superradiance, in this limit.</text>
<text><location><page_9><loc_9><loc_9><loc_49><loc_18></location>Referring again to the parameter space shown in Fig. 5, in the RN BH case the unbounded region and the superradiant region of the parameter space are disjoint. Therefore, we do not expected to observe unbounded superradiance (as ω → µ ) in the RN case. In this sense, absorption by the RN BH is qualitatively different to absorption by the ABG RBH, for which is possible to find a set of field and RBH parameters</text>
<figure>
<location><page_9><loc_52><loc_72><loc_92><loc_93></location>
<caption>FIG. 11. Comparison between the total ACSs of charged massive scalar waves in the background of ABG and RN BHs, as functions of ω/µ , considering α = 0 . 4 , qM = 1 . 6 , and different choices of µM . The inset highlights the range of frequency (1 < ω/µ ≲ 1 . 5 , for µM (ABG) = 0 . 4 ) for which the ACS becomes negative, denoting superradiance.</caption>
</figure>
<text><location><page_9><loc_52><loc_49><loc_92><loc_55></location>that leads to unbounded superradiance (see Fig. 10). A further notable feature of Fig. 5 is that bounded absorption does not occur in the ABG case (since the bounded case is necessarily superradiant) whereas it does in the RN case; see Fig. 11 for an example.</text>
<text><location><page_9><loc_52><loc_34><loc_92><loc_47></location>Figure 12 shows results for the ACSs in two distinct sets: (i) for α = 0 . 4 , µM = 0 . 4 , and different values of qM (top panel); and (ii) for µM = 0 . 2 , qM = 0 . 4 , and different values of α (bottom panel). We note that, similarly to the behavior presented in Fig. 11, for a given set of parameters, superradiance might occur only for the ABG RBH. We see again that the propagating waves are more absorbed in the RN case, when qQ > 0 ; however, for qQ < 0 , we observe the opposite behaviour.</text>
<text><location><page_9><loc_52><loc_24><loc_92><loc_33></location>Figure 13 compares the partial ACSs of ABG and RN BHs. In this case, for the chosen parameters we have superradiance for both BH types. The range of frequency in which σ < 0 is larger in the ABG case than in the RN case. The dominant contribution to superradiance comes from the monopole mode, l = 0 .</text>
<text><location><page_9><loc_52><loc_14><loc_92><loc_23></location>A comparison of the amplification factors in the ABG and RN geometries is exhibited in Fig. 14. The superradiant amplification in the background of the ABG RBH, for the same values of α , µM , and qM , is typically larger than that in the corresponding RN geometry, in agreement with the results presented in Figs. 11-13.</text>
<text><location><page_9><loc_52><loc_9><loc_92><loc_13></location>Weend this section by noting that all results presented here, as well as those not shown, are consistent with the parameter space for ABG and RN BHs introduced in Fig. 5.</text>
<figure>
<location><page_10><loc_9><loc_51><loc_49><loc_93></location>
<caption>FIG. 12. Comparison between the total ACSs of charged massive scalar waves in the background of ABG and RN BHs, as functions of ω/µ , in two different scenarios: (i) for α = 0 . 4 , µM = 0 . 4 , and different values of qM (top panel) ; and (ii) for µM = 0 . 2 , qM = 0 . 4 , and distinct values of α (bottom panel) .</caption>
</figure>
<figure>
<location><page_10><loc_9><loc_17><loc_49><loc_38></location>
<caption>FIG. 13. Comparison between the partial ACSs of charged massive scalar waves in the background of ABG and RN BHs, as functions of ω/µ , with α = 0 . 8 , µM = 0 . 4 , and qM = 0 . 8 .</caption>
</figure>
<figure>
<location><page_10><loc_52><loc_72><loc_92><loc_93></location>
<caption>FIG. 14. Superradiant amplification of charged massive scalar fields by ABG and RN BHs as a function of ω/µ . Here we consider α = 0 . 8 , l = 0 , µM = 0 . 3 , and qM = 0 . 8 in both geometries.</caption>
</figure>
<section_header_level_1><location><page_10><loc_64><loc_61><loc_80><loc_62></location>E. Mimic configurations</section_header_level_1>
<text><location><page_10><loc_52><loc_44><loc_92><loc_59></location>In this section, we show that it is possible to find combinations of the normalized charge of the BH solution, and the parameters (charge and mass) of the scalar field such that the absorption cross sections of ABG and RN BHs are very similar. We start by computing the values of α for which the geometric cross section (see Sec. III D) of the ABG RBH is equal to that of the RN BH, for fixed values of the charged massive particle parameters EM , q cp M , and mM . Next, we compute the absorption cross sections using the corresponding values of α , qM , and µM .</text>
<text><location><page_10><loc_52><loc_36><loc_92><loc_44></location>Figure 15 shows the total ACSs for specific pairs ( α ABG , α RN ) with µM = 0 . 6 and qM = 0 . 2 . As we can see, the total ACSs of the two types of BH (regular ABG and irregular RN) can be very similar in the middle-to-high frequency range, particularly for low-to-moderate values of α , but distinguishable in the low-frequency regime.</text>
<text><location><page_10><loc_52><loc_26><loc_92><loc_35></location>Figure 16 shows that, for a neutral field ( qM = 0 ), the ACSs of the two types of BH can be very similar across the whole frequency range, particularly for small-to-moderate values of α . Here we consider two pairs of choices of ( α ABG , α RN ) that lead to the same GCS. The field charge q increases the differences between the absorption pattern of ABG and RN BHs, particularly at lower frequencies.</text>
<text><location><page_10><loc_52><loc_21><loc_92><loc_25></location>It is possible to find configurations for which a massive and charged scalar field has the same value of the critical frequency ω c fo superradiance in the two spacetimes, that is,</text>
<formula><location><page_10><loc_67><loc_18><loc_92><loc_20></location>ω ABG c = ω RN c . (47)</formula>
<text><location><page_10><loc_52><loc_12><loc_92><loc_17></location>Figure 17 shows the values of the pair ( α ABG , α RN ) for which the critical frequency is the same in both backgrounds, considering a fixed value of qM . These configurations can be found up to α ABG ≲ 0 . 8716 .</text>
<text><location><page_10><loc_52><loc_9><loc_92><loc_11></location>Figure 18 shows the total ACS for pairs ( α ABG , α RN ) with the same critical frequency ω c . We see that the effect of su-</text>
<figure>
<location><page_11><loc_9><loc_72><loc_49><loc_93></location>
<caption>FIG. 15. Total ACSs of charged massive scalar waves for the pairs ( α ABG , α RN ) = (0 . 2 , 0 . 1794) and ( α ABG , α RN ) = (0 . 5 , 0 . 4498) as functions of ω/µ . In both cases, we set µM = 0 . 6 and qM = 0 . 2 .</caption>
</figure>
<figure>
<location><page_11><loc_9><loc_40><loc_49><loc_61></location>
<caption>FIG. 16. Total ACSs of charged massive scalar fields for the pairs ( α ABG , α RN ) = (0 . 2 , 0 . 1794) and ( α ABG , α RN ) = (0 . 5 , 0 . 4498) , as functions of ω/µ . In both cases, we set µM = 0 . 4 and qM = 0 .</caption>
</figure>
<text><location><page_11><loc_9><loc_25><loc_49><loc_28></location>radiance is enhanced in the ABG case, as one would expect given the stronger amplification shown in Fig. 14.</text>
<text><location><page_11><loc_9><loc_9><loc_49><loc_24></location>It is also possible to find situations in which scalar fields with different masses and charges have the same critical frequency ω c in the background of ABG and RN BH spacetimes. In Fig. 19, we consider a scalar field with qM = 1 and µM = 0 . 2 in the ABG spacetime and a scalar field with qM = 1 . 4 and µM = 0 . 4 in the RN geometry. As we can see, the distinct scalar fields are superradiantly scattered whenever ωM < 0 . 7 . Figure 20 shows the amplification factors for the same parameters, highlighting once again that superradiance amplification is enhanced for the ABG BH relative to the RN BH.</text>
<figure>
<location><page_11><loc_52><loc_62><loc_91><loc_92></location>
<caption>FIG. 17. Values of the pair ( α ABG , α RN ) for which a fixed choice of qM presents the same critical frequency in the background of ABG and RN BHs.</caption>
</figure>
<figure>
<location><page_11><loc_52><loc_32><loc_92><loc_53></location>
<caption>FIG. 18. Example of situations in which a massive and charged scalar field presents the same critical frequency in the background of ABG and RN BHs. The small disks denote the values of the critical frequency, namely ω c = 1 . 3333 (left) and ω c = 2 . 5071 (right).</caption>
</figure>
<section_header_level_1><location><page_11><loc_61><loc_19><loc_82><loc_20></location>V. CONCLUDING REMARKS</section_header_level_1>
<text><location><page_11><loc_52><loc_9><loc_92><loc_17></location>Within GR, the standard BH solutions (Schwarzschild, Reissner-Nordstrom, Kerr, and Kerr-Newman) possess a common feature in their core: a curvature singularity hidden by an event horizon. On the other hand, certain regular BH solutions, i.e., objects with an event horizon but with no curvature singularity, can be obtained by minimally coupling NED mod-e</text>
<figure>
<location><page_12><loc_9><loc_72><loc_49><loc_92></location>
<caption>FIG. 19. Total ACS for qM = 1 and µM = 0 . 2 in the ABG geometry and for qM = 1 . 4 and µM = 0 . 4 in the RN spacetime. Superradiance occurs when ωM < ω c M = 0 . 7 in both scenarios.</caption>
</figure>
<figure>
<location><page_12><loc_9><loc_42><loc_49><loc_62></location>
<caption>FIG. 20. Superradiant amplification of massive charged scalar fields, as a function of ωM , considering the same parameters used in Fig. 19, for which the critical frequency is ω c M = 0 . 7 .</caption>
</figure>
<text><location><page_12><loc_9><loc_25><loc_49><loc_32></location>s to GR. Much work is underway to determine the properties of RBHs. Contributing to this effort, we have scrutinised the absorption properties of charged massive scalar fields in the background of the electrically charged RBH solution proposed by Ay'on-Beato and Garc'ıa [18].</text>
<text><location><page_12><loc_9><loc_14><loc_49><loc_24></location>The most intriguing result of our study is that regular ABG BHs, unlike their RN counterparts, exhibit unbounded superradiance (as ω → µ ) in massive scalar fields, within a certain parameter range. More precisely, the cross section σ is unbounded from below as ω → µ ; this is shown clearly in Fig. 10. The region of parameter space in which unbounded superradiance occurs is clarified in Sec. III C and Fig. 5.</text>
<text><location><page_12><loc_9><loc_9><loc_49><loc_14></location>Some care is needed in interpreting the physical meaning of the divergence of the cross section σ as ω → µ . In the particle picture, a divergence arises naturally because particles of low momentum and large impact parameters are attracted by the</text>
<text><location><page_12><loc_52><loc_83><loc_92><loc_93></location>BH, with b c →∞ as κ → 0 . In the wave picture, a divergence in σ arises if the transmission factor does not go to zero as rapidly as the cube of the momentum κ in the denominator of Eq. (31); and if this occurs in the superradiant regime ω < ω c , then the unbounded superradiance phenomenon occurs. Notably, the amplification factor, Z ωl in Eq. (27), does not diverge in our numerical results.</text>
<text><location><page_12><loc_52><loc_68><loc_92><loc_82></location>In principle then, by stimulating the BH with a planar wave of low momentum ( κ → 0 ) in a massive charged field, one can extract (via superradiance) unbounded quantities of mass and charge from the ABG BH (within the limitations of the linearised regime of weak scalar fields). It is important to stress, however, that the divergence in σ is related to the fact that the BH is interacting with a planar wave of infinite lateral extent (and b c → ∞ as κ → 0 ). Therefore, one should not expect unbounded extraction of energy to be possible (even in principle) for a stimulating wave of finite width and duration.</text>
<text><location><page_12><loc_52><loc_65><loc_92><loc_67></location>Some further interesting aspects of the ACS are summarized below:</text>
<text><location><page_12><loc_52><loc_57><loc_92><loc_64></location>(i) Massive scalar waves are typically more absorbed than massless ones, and absorption increases with the value of µM . This result is expected since larger field masses lead to larger GCSs, and large field masses are associated with strongly absorbed modes [64, 78].</text>
<text><location><page_12><loc_52><loc_52><loc_92><loc_56></location>(ii) In the case of a charged scalar field, due to the Lorentz force, the absorption for qQ < 0 is typically larger than for qQ > 0 .</text>
<text><location><page_12><loc_52><loc_46><loc_92><loc_51></location>(iii) Low-frequency waves satisfying the condition ω < ω c [cf. Eq. (28)] can have a negative ACS. This occurs due to the superradiant amplification of low multipoles of the field (principally, in the l = 0 mode) [52].</text>
<text><location><page_12><loc_52><loc_39><loc_92><loc_45></location>(iv) The absorption of scalar waves by ABG RBHs is typically larger than for RN BHs (for equivalent q , M and α ), when the value of the charge coupling qQ is negative. Conversely, σ RN > σ ABG when qQ > 0 .</text>
<text><location><page_12><loc_52><loc_30><loc_92><loc_38></location>(v) The critical superradiant frequency of the ABG BH is always larger than that of the RN BH, for equivalent parameters. Moreover, superradiant amplification is stronger for the ABG BH. Both aspects are due to the enhanced electrostatic potential at the horizon, ϕ + , in the ABG case (i.e., ϕ ABG + > ϕ RN + ).</text>
<text><location><page_12><loc_52><loc_19><loc_92><loc_29></location>We showed in Sec. IV E that, for certain parameter choices, the ABG RBH solution can mimic the RN solution, from the point of view of absorption spectrum, reinforcing the results presented in Refs. [73, 76]. It is also possible to find configurations for which scalar fields with different masses and charges, in the background of ABG and RN BHs, have the same critical superradiant frequency.</text>
<text><location><page_12><loc_52><loc_9><loc_92><loc_18></location>Several avenues for further investigation are open. Firstly, we note that superradiant scattering is, in some sense, the wave analogue of the Penrose process. In light of the results here, it could be worth studying the Penrose process for the ABG BH in detail. That is, the scenario of a charged particle, incident from infinity, that splits into two parts in the vicinity of the BH, with one part ejected to infinity and the other ab-</text>
<text><location><page_13><loc_9><loc_86><loc_49><loc_93></location>sorbed 2 . In a Penrose process, the escaping particle has more mass-energy than the incident one. It would be interesting to compare the regions of parameter space in which energy extraction can occur, again drawing a comparison between the RN BH and the ABG BH.</text>
<text><location><page_13><loc_9><loc_61><loc_49><loc_86></location>Secondly, the existence of the 'unbounded superradiance' region in Fig. 5 strongly hints at the existence of superradiantly-unstable quasibound states in the spectrum of the massive charged scalar field on the ABG spacetime. Previous investigations on the RN spacetime have suggested that it is not possible to form quasibound states that are also superradiant in the RN case. Heuristically, the reason is clear: for bound states one needs an attractive potential in the far-field, which necessitates µM > qQ ; then modes with ω < µ do not lie in the superradiant regime ω < qϕ RN + = qQ/r + of the RN BH. Conversely, as shown in Fig. 5, modes satisfying µM/qQ > 1 can also be superradiant on the ABG spacetime, due to the increase in the electric potential at the horizon, ϕ + . This implies that certain quasibound modes of the massive charged scalar field will grow exponentially with time, and thus that the ABG BH suffers a superradiant instability. This is under active investigation.</text>
<text><location><page_13><loc_9><loc_56><loc_49><loc_61></location>Thirdly, the ABG BH is just one example in the regular class in NED. It would be interesting to clarify whether other solutions in this class also exhibit a stronger EM field at the horizon, and thus an enhanced region of superradiance with</text>
<unordered_list>
<list_item><location><page_13><loc_10><loc_47><loc_49><loc_50></location>[1] C. M. Will, The Confrontation between General Relativity and Experiment, Living Rev. Relativ. 17 , 4 (2014).</list_item>
<list_item><location><page_13><loc_10><loc_43><loc_49><loc_47></location>[2] L. C. B. Crispino and D. Kennefick, A hundred years of the first experimental test of general relativity, Nat. Phys. 15 , 416 (2019).</list_item>
<list_item><location><page_13><loc_10><loc_41><loc_49><loc_43></location>[3] L. C. B. Crispino and S. Paolantonio, The first attempts to measure light deflection by the Sun, Nat. Astron. 4 , 6 (2020).</list_item>
<list_item><location><page_13><loc_10><loc_37><loc_49><loc_41></location>[4] B. P. Abbott et al. [LIGO Scientific Collaboration and Virgo Collaboration], Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett. 116 , 061102 (2016).</list_item>
<list_item><location><page_13><loc_10><loc_33><loc_49><loc_37></location>[5] K. Akiyama et al. [Event Horizon Telescope Collaboration], First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole, Atrophys. J. 875 , L1 (2019).</list_item>
<list_item><location><page_13><loc_10><loc_30><loc_49><loc_33></location>[6] R. Narayan and J. E. McClintock, Observational Evidence for Black Holes, arXiv:1312.6698 [astro-ph.HE].</list_item>
<list_item><location><page_13><loc_10><loc_26><loc_49><loc_30></location>[7] J. Kormendy and L. C. Ho, Coevolution (Or Not) of Supermassive Black Holes and Host Galaxies, Annu. Rev. Astron. Astrophys. 51 , 511 (2013).</list_item>
<list_item><location><page_13><loc_10><loc_21><loc_49><loc_26></location>[8] A. M. Ghez, B. L. Klein, M. Morris, and E. E. Becklin, High Proper-Motion Stars in the Vicinity of Sagittarius A*: Evidence for a Supermassive Black Hole at the Center of our Galaxy, Astrophys. J. 509 , 678 (1998).</list_item>
<list_item><location><page_13><loc_10><loc_17><loc_49><loc_21></location>[9] A. M. Ghez et al. , Measuring Distance and Properties of the Milky Way's Central Supermassive Black Hole with Stellar Orbits, Astrophys. J. 689 , 1044 (2008).</list_item>
</unordered_list>
<text><location><page_13><loc_52><loc_90><loc_92><loc_93></location>associated phenomena; or whether the ABG BH stands alone in this respect.</text>
<text><location><page_13><loc_52><loc_85><loc_92><loc_90></location>Finally, real astrophysical BHs are known to be rotating. Future studies of absorption by spinning RBHs would clarify the interplay between charged superradiance (studied here) and rotational superradiance.</text>
<section_header_level_1><location><page_13><loc_64><loc_81><loc_80><loc_82></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_13><loc_52><loc_56><loc_92><loc_78></location>The authors thank Fundac¸˜ao Amazˆonia de Amparo a Estudos e Pesquisas (FAPESPA), Conselho Nacional de Desenvolvimento Cient'ıfico e Tecnol'ogico (CNPq) and Coordenac¸˜ao de Aperfeic¸oamento de Pessoal de N'ıvel Superior (Capes) - Finance Code 001, in Brazil, for partial financial support. M.P. and L.C. thank the University of Sheffield, in England, and University of Aveiro, in Portugal, respectively, for the kind hospitality during the completion of this work. This work has further been supported by the European Union's Horizon 2020 research and innovation (RISE) programme H2020-MSCA-RISE-2017 Grant No. FunFiCO-777740 and by the European Horizon Europe staff exchange (SE) programme HORIZON-MSCA-2021-SE01 Grant No. NewFunFiCO-101086251. S.D. acknowledges financial support from the Science and Technology Facilities Council (STFC) under Grant No. ST/T001038/1.</text>
<unordered_list>
<list_item><location><page_13><loc_52><loc_47><loc_92><loc_50></location>[10] M. Heusler, Black Hole Uniqueness Theorems (Cambridge University Press, Cambridge, England, 1996).</list_item>
<list_item><location><page_13><loc_52><loc_43><loc_92><loc_47></location>[11] R. Penrose, 'Golden Oldie': Gravitational Collapse: The Role of General Relativity, General Relativity and Gravitation 34 , 1141 (2002).</list_item>
<list_item><location><page_13><loc_52><loc_39><loc_92><loc_43></location>[12] S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time (Cambridge University Press, Cambridge, England, 1973).</list_item>
<list_item><location><page_13><loc_52><loc_36><loc_92><loc_39></location>[13] J. Bardeen, Non-singular General Relativistic Gravitational Collapse, Proceedings of the International Conference GR5, Tbilisi, U.S.S.R., 1968 (unpublished).</list_item>
<list_item><location><page_13><loc_52><loc_33><loc_92><loc_35></location>[14] A. Borde, Open and closed universes, initial singularities, and inflation, Phys. Rev. D 50 , 3692 (1994).</list_item>
<list_item><location><page_13><loc_52><loc_30><loc_92><loc_33></location>[15] C. Barrab'es and V. P. Frolov, How many new worlds are inside a black hole?, Phys. Rev. D 53 , 3215 (1996).</list_item>
<list_item><location><page_13><loc_52><loc_26><loc_92><loc_30></location>[16] M. Mars, M. M. Mart'ın-Prats, and J. M. M. Senovilla, Models of regular Schwarzschild black holes satisfying weak energy conditions, Class. Quantum Grav. 13 , L51 (1996).</list_item>
<list_item><location><page_13><loc_52><loc_24><loc_92><loc_26></location>[17] A. Cabo and E. Ay'on-Beato, About Black Holes with Nontrapping Interior, Int. J. Mod. Phys. A 14 , 2013 (1999).</list_item>
<list_item><location><page_13><loc_52><loc_20><loc_92><loc_23></location>[18] E. Ay'on-Beato and A. Garc'ıa, Regular Black Hole in General Relativity Coupled to Nonlinear Electrodynamics, Phys. Rev. Lett. 80 , 5056 (1998).</list_item>
<list_item><location><page_13><loc_52><loc_17><loc_92><loc_19></location>[19] M. Born, On the quantum theory of the electromagnetic field, Proc. R. Soc. A 143 , 410 (1934).</list_item>
<list_item><location><page_13><loc_52><loc_14><loc_92><loc_17></location>[20] M. Born and L. Infeld, Foundations of the new field theory, Proc. R. Soc. A 144 , 425 (1934).</list_item>
<list_item><location><page_13><loc_52><loc_12><loc_92><loc_14></location>[21] J. F. Pleba'nski, Lectures on Non-linear Electrodynamics (NORDITA, Copenhagen, Denmark, 1970).</list_item>
<list_item><location><page_13><loc_52><loc_9><loc_92><loc_11></location>[22] S. A. Guti'errez, A. L. Dudley, and J. F. Pleba'nski, Signals and discontinuities in general relativistic nonlinear electrodynam-</list_item>
<list_item><location><page_14><loc_12><loc_92><loc_33><loc_93></location>s, J. Math. Phys. 22 , 2835 (1981).</list_item>
<list_item><location><page_14><loc_9><loc_89><loc_49><loc_92></location>[23] W. Heisenberg and H. Euler, Folgerungen aus der Diracschen Theorie des Positrons, Z. Physik 98 , 714 (1936).</list_item>
<list_item><location><page_14><loc_9><loc_87><loc_49><loc_89></location>[24] E. S. Fradkin and A. A. Tseytlin, Non-linear electrodynamics from quantized strings, Phys. Lett. B 163 , 123 (1985).</list_item>
<list_item><location><page_14><loc_9><loc_84><loc_49><loc_86></location>[25] N. Seiberg and E. Witten, String theory and noncommutative geometry, J. High Energy Phys. 1999 , 032 (1999)</list_item>
<list_item><location><page_14><loc_9><loc_81><loc_49><loc_84></location>[26] A. A. Tseytlin, Born-Infeld action, supersymmetry and string theory, arXiv:hep-th/9908105 [hep-th].</list_item>
<list_item><location><page_14><loc_9><loc_77><loc_49><loc_81></location>[27] M. Novello, S. E. Perez Bergliaffa, and J. Salim, Nonlinear electrodynamics and the acceleration of the Universe, Phys. Rev. D 69 , 127301 (2004).</list_item>
<list_item><location><page_14><loc_9><loc_71><loc_49><loc_77></location>[28] R. P. Mignani, V. Testa, D. Gonz'alez Caniulef, R. Taverna, R. Turolla, S. Zane, and K. Wu, Evidence for vacuum birefringence from the first optical-polarimetry measurement of the isolated neutron star RX J1856.5-3754, Mon. Not. R. Astron. Soc. 465 , 492 (2017).</list_item>
<list_item><location><page_14><loc_9><loc_67><loc_49><loc_70></location>[29] M. Aaboud et al. [ATLAS Collaboration], Evidence for lightby-light scattering in heavy-ion collisions with the ATLAS detector at the LHC, Nature Phys. 13 , 852 (2017).</list_item>
<list_item><location><page_14><loc_9><loc_63><loc_49><loc_67></location>[30] G. Aad et al. [ATLAS Collaboration], Observation of Light-byLight Scattering in Ultraperipheral Pb + Pb Collisions with the ATLAS Detector, Phys. Rev. Lett. 123 , 052001 (2019).</list_item>
<list_item><location><page_14><loc_9><loc_58><loc_49><loc_63></location>[31] A. Ejlli, F. Della Valle, U. Gastaldi, G. Messineo, R. Pengo, G. Ruoso, and G. Zavattini, The PVLAS experiment: A 25 year effort to measure vacuum magnetic birefringence, Phys. Rep. 871 , 1 (2020).</list_item>
<list_item><location><page_14><loc_9><loc_55><loc_49><loc_57></location>[32] D. Delphenich, Nonlinear Electrodynamics and QED, arXiv:hep-th/0309108 [hep-th].</list_item>
<list_item><location><page_14><loc_9><loc_51><loc_49><loc_55></location>[33] E. Ay'on-Beato and A. Garc'ıa, Non-Singular Charged Black Hole Solution for Non-Linear Source, General Relativity and Gravitation 31 , 629 (1999).</list_item>
<list_item><location><page_14><loc_9><loc_48><loc_49><loc_51></location>[34] E. Ay'on-Beato and A. Garc'ıa, New regular black hole solution from nonlinear electrodynamics, Phys. Lett. B 464 , 25 (1999).</list_item>
<list_item><location><page_14><loc_9><loc_46><loc_49><loc_48></location>[35] E. Ay'on-Beato and A. Garc'ıa, The Bardeen model as a nonlinear magnetic monopole, Phys. Lett. B 493 , 149 (2000).</list_item>
<list_item><location><page_14><loc_9><loc_42><loc_49><loc_45></location>[36] I. Dymnikova, Regular electrically charged vacuum structures with de Sitter centre in nonlinear electrodynamics coupled to general relativity , Class. Quantum Grav. 21 , 4417 (2004).</list_item>
<list_item><location><page_14><loc_9><loc_39><loc_49><loc_41></location>[37] L. Balart and E. C. Vagenas, Regular black holes with a nonlinear electrodynamics source, Phys. Rev. D 90 , 124045 (2014).</list_item>
<list_item><location><page_14><loc_9><loc_36><loc_49><loc_39></location>[38] M. E. Rodrigues and M. V. de S. Silva, Bardeen regular black hole with an electric source, JCAP 06 , 025 (2018).</list_item>
<list_item><location><page_14><loc_9><loc_32><loc_49><loc_36></location>[39] K. A. Bronnikov, Regular magnetic black holes and monopoles from nonlinear electrodynamics, Phys. Rev. D 63 , 044005 (2001).</list_item>
<list_item><location><page_14><loc_9><loc_30><loc_49><loc_32></location>[40] J. Matyjasek, Extremal limit of the regular charged black holes in nonlinear electrodynamics, Phys. Rev. D 70 , 047504 (2004).</list_item>
<list_item><location><page_14><loc_9><loc_27><loc_49><loc_30></location>[41] M.-S. Ma, Magnetically charged regular black hole in a model of nonlinear electrodynamics, Ann. Phys. 362 , 529 (2015).</list_item>
<list_item><location><page_14><loc_9><loc_25><loc_49><loc_27></location>[42] Z.-Y. Fan and X. Wang, Construction of regular black holes in general relativity, Phys. Rev. D 94 , 124027 (2016).</list_item>
<list_item><location><page_14><loc_9><loc_22><loc_49><loc_24></location>[43] S. I. Kruglov, Black hole as a magnetic monopole within exponential nonlinear electrodynamics, Ann. Phys. 378 , 59 (2017).</list_item>
<list_item><location><page_14><loc_9><loc_19><loc_49><loc_22></location>[44] Z.-Y. Fan, Critical phenomena of regular black holes in anti-de Sitter space-time, Eur. Phys. J. C 77 , 266 (2017).</list_item>
<list_item><location><page_14><loc_9><loc_15><loc_49><loc_19></location>[45] B. Toshmatov, Z. Stuchl'ık, and B. Ahmedov, Comment on 'Construction of regular black holes in general relativity', Phys. Rev. D 94 , 028501 (2018).</list_item>
<list_item><location><page_14><loc_9><loc_13><loc_49><loc_15></location>[46] G. J. Olmo and D. Rubiera-Garcia, Palatini f ( R ) black holes in nonlinear electrodynamics, Phys. Rev. D 84 , 124059 (2011).</list_item>
<list_item><location><page_14><loc_9><loc_9><loc_49><loc_12></location>[47] E. L. B. Junior, M. E. Rodrigues, and M. J. S. Houndjo, Regular Black Holes in f ( T ) Gravity through a Nonlinear Electrodynamics Source, JCAP 10 , 060 (2015).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_14><loc_52><loc_88><loc_92><loc_93></location>[48] M. E. Rodrigues, E. L. B. Junior, G. T. Marques, and V. T. Zanchin, Regular black holes in f ( R ) gravity coupled to nonlinear electrodynamics, Phys. Rev. D 94 , 024062 (2016); Addendum, Phys. Rev. D 94 , 049904 (2016).</list_item>
<list_item><location><page_14><loc_52><loc_85><loc_92><loc_88></location>[49] M. V. de S. Silva and M. E. Rodrigues, Regular black holes in f ( G ) gravity, Eur. Phys. J. C 78 , 18 (2018).</list_item>
<list_item><location><page_14><loc_52><loc_83><loc_92><loc_85></location>[50] D. P. Sorokin, Introductory Notes on Non-linear Electrodynamics and its Applications, Fortsch. Phys. 70 , 2200092 (2022).</list_item>
<list_item><location><page_14><loc_52><loc_80><loc_92><loc_82></location>[51] K. A. Bronnikov, Regular black holes sourced by nonlinear electrodynamics, arXiv:2211.00743 [gr-qc] .</list_item>
<list_item><location><page_14><loc_52><loc_77><loc_92><loc_80></location>[52] J. Bekenstein, Extraction of Energy and Charge from a Black Hole, Phys. Rev. D 7 , 949 (1973).</list_item>
<list_item><location><page_14><loc_52><loc_75><loc_92><loc_77></location>[53] G. W. Gibbons, Vacuum polarization and the spontaneous loss of charge by black holes, Commun. Math. Phys. 44 , 245 (1975).</list_item>
<list_item><location><page_14><loc_52><loc_72><loc_92><loc_74></location>[54] T. Nakamura and H. Sato, Absorption of massive scalar field by a charged black hole, Phys. Lett. B 61 , 371 (1976).</list_item>
<list_item><location><page_14><loc_52><loc_69><loc_92><loc_72></location>[55] C. L. Benone and L. C. B. Crispino, Superradiance in static black hole spacetimes, Phys. Rev. D 93 , 024028 (2016).</list_item>
<list_item><location><page_14><loc_52><loc_66><loc_92><loc_69></location>[56] Olaf Baake and Oliver Rinne, Superradiance of a charged scalar field coupled to the Einstein-Maxwell equations, Phys. Rev. D 94 , 124016 (2016).</list_item>
<list_item><location><page_14><loc_52><loc_62><loc_92><loc_65></location>[57] V. Balakumar, E. Winstanley, R. P. Bernar, and L. C. B. Crispino, Quantum superradiance on static black hole spacetimes, Phys. Lett. B 811 , 135904 (2020).</list_item>
<list_item><location><page_14><loc_52><loc_59><loc_92><loc_61></location>[58] R. Brito, V. Cardoso, and P. Pani, Superradiance - the 2020 Edition, arXiv:1501.06570 [gr-qc].</list_item>
<list_item><location><page_14><loc_52><loc_54><loc_92><loc_59></location>[59] E. Franzin, S. Liberati, J. Mazza, R. Dey, and S. Chakraborty, Scalar perturbations around rotating regular black holes and wormholes: Quasinormal modes, ergoregion instability, and superradiance, Phys. Rev. D 105 , 124051 (2022).</list_item>
<list_item><location><page_14><loc_52><loc_50><loc_92><loc_53></location>[60] Z. Li, Scalar perturbation around rotating regular black hole: Superradiance instability and quasinormal modes, Phys. Rev. D 107 , 044013 (2023).</list_item>
<list_item><location><page_14><loc_52><loc_47><loc_92><loc_49></location>[61] W. Unruh, Absorption cross section of small black holes, Phys. Rev. D 14 , 3251 (1976).</list_item>
<list_item><location><page_14><loc_52><loc_43><loc_92><loc_47></location>[62] J. A. Futterman, F. A. Handler, and R. A. Matzner, Scattering from Black Holes (Cambridge University Press, Cambridge, England, 1988).</list_item>
<list_item><location><page_14><loc_52><loc_38><loc_92><loc_43></location>[63] E. S. Oliveira, L. C. B. Crispino, and A. Higuchi, Equality between gravitational and electromagnetic absorption cross sections of extreme Reissner-Nordstrom black holes, Phys. Rev. D 84 , 084048 (2011).</list_item>
<list_item><location><page_14><loc_52><loc_32><loc_92><loc_37></location>[64] C. L. Benone, E. S. de Oliveira, S. R. Dolan, and L. C. B. Crispino, Absorption of a massive scalar field by a charged black hole, Phys. Rev. D 89 , 104053 (2014); Addendum, Phys. Rev. D 95 , 044035 (2017).</list_item>
<list_item><location><page_14><loc_52><loc_29><loc_92><loc_32></location>[65] L. C. S. Leite, C. L. Benone, and L. C. B. Crispino, Scalar absorption by charged rotating black holes, Phys. Rev. D 96 , 044043 (2017).</list_item>
<list_item><location><page_14><loc_52><loc_25><loc_92><loc_28></location>[66] L. C. S. Leite, S. R. Dolan, and L. C. B. Crispino, Absorption of electromagnetic and gravitational waves by Kerr black holes, Phys. Lett. B 74 , 130 (2017).</list_item>
<list_item><location><page_14><loc_52><loc_21><loc_92><loc_24></location>[67] L. C. S. Leite, S. R. Dolan, and L. C. B. Crispino, Absorption of electromagnetic plane waves by rotating black holes, Phys. Rev. D 98 , 024046 (2018).</list_item>
<list_item><location><page_14><loc_52><loc_17><loc_92><loc_20></location>[68] C. L. Benone and L. C. B. Crispino, Massive and charged scalar field in Kerr-Newman spacetime: Absorption and superradiance, Phys. Rev. D 99 , 044009 (2019).</list_item>
<list_item><location><page_14><loc_52><loc_13><loc_92><loc_16></location>[69] S. V. M. C. B. Xavier, C. L. Benone, and L. C. B. Crispino, Absorption by stringy black holes, Eur. Phys. J. C 81 , 1127 (2021).</list_item>
<list_item><location><page_14><loc_52><loc_9><loc_92><loc_12></location>[70] C. F. B. Macedo and L. C. B. Crispino, Absorption of planar massless scalar waves by Bardeen regular black holes, Phys. Rev. D 90 , 064001 (2014).</list_item>
<list_item><location><page_15><loc_9><loc_89><loc_49><loc_93></location>[71] P. A. Sanchez, N. Bret'on, and S. E. P. Bergliaffa, Scattering and absorption of massless scalar waves by Born-Infeld black holes, Ann. Phys. 393 , 107 (2017).</list_item>
<list_item><location><page_15><loc_9><loc_87><loc_49><loc_89></location>[72] S. Fernando, Bardeen-de Sitter black holes, Int. J. Mod. Phys. D 26 , 1750071 (2017).</list_item>
<list_item><location><page_15><loc_9><loc_81><loc_49><loc_86></location>[73] M. A. A. Paula, L. C. S. Leite, and L. C. B. Crispino, Electrically charged black holes in linear and non-linear electrodynamics: Geodesic analysis and scalar absorption, Phys. Rev. D 102 , 104033 (2020).</list_item>
<list_item><location><page_15><loc_9><loc_77><loc_49><loc_81></location>[74] H. Salazar, A. Garc'ıa, and J. Plebanski, Duality rotations and type D solutions to Einstein equations with nonlinear electromagnetic sources, J. Math. Phys. 28 , 2171 (1987).</list_item>
<list_item><location><page_15><loc_9><loc_75><loc_49><loc_77></location>[75] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman, San Francisco, United States, 1973).</list_item>
<list_item><location><page_15><loc_9><loc_73><loc_49><loc_74></location>[76] M. A. A. de Paula, L. C. S. Leite, and L. C. B. Crispino,</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_15><loc_55><loc_91><loc_92><loc_93></location>Scattering properties of charged black holes in nonlinear and Maxwell's electrodynamics, Eur. Phys. J. Plus 137 , 785 (2022).</list_item>
<list_item><location><page_15><loc_52><loc_87><loc_92><loc_90></location>[77] A. Garc'ıa, E. Hackmann, J. Kunz, Claus Lammerzahl, and A. Mac'ıas, Motion of test particles in a regular black hole spacetime, J. Math. Phys. 56 , 032501 (2015).</list_item>
<list_item><location><page_15><loc_52><loc_83><loc_92><loc_86></location>[78] E. Jung and D. Park, Effect of scalar mass in the absorption and emission spectra of Schwarzschild black hole, Class. Quantum Grav. 21 , 3717 (2004)</list_item>
<list_item><location><page_15><loc_52><loc_80><loc_92><loc_82></location>[79] R. Wald, General Relativity (University of Chicago Press, Chicago, United States, 1984).</list_item>
<list_item><location><page_15><loc_52><loc_76><loc_92><loc_80></location>[80] B. Toshmatov, A. Abdujabbarov, B. Ahmedov, and Z. Stuchl'ık, Particle motion and Penrose processes around rotating regular black hole, Astrophys. Space Sci. 357 , 41 (2015).</list_item>
</document>
[
{
"title": "Absorption and (unbounded) superradiance in a static regular black hole spacetime",
"content": "Marco A. A. de Paula, 1, 2, ∗ Luiz C. S. Leite, 3, † Sam R. Dolan, 2, ‡ and Lu'ıs C. B. Crispino 1, 4, § 1 Programa de P'os-Graduac¸˜ao em F'ısica, Universidade Federal do Par'a, 66075-110, Bel'em, Par'a, Brazil. 2 Consortium for Fundamental Physics, School of Mathematics and Statistics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, United Kingdom. 3 Campus Altamira, Instituto Federal do Par'a, 68377-630, Altamira, Par'a, Brazil. 4 Departamento de Matem'atica da Universidade de Aveiro and Centre for Research and Development in Mathematics and Applications (CIDMA), Campus de Santiago, 3810-183 Aveiro, Portugal. (Dated: January 4, 2024) Regular black holes (RBHs) - geometries free from curvature singularities - arise naturally in theories of non linear electrodynamics. Here we study the absorption, and superradiant amplification, of a monochromatic planar wave in a charged, massive scalar field impinging on the electrically-charged Ay'on-Beato-Garc'ıa (ABG) RBH. Comparisons are drawn with absorption and superradiance for the Reissner-Nordstrom (RN) black hole in linear electrodynamics. We find that, in a certain parameter regime, the ABG absorption cross section is negative, due to superradiance, and moreover it is unbounded from below as the momentum of the wave approaches zero; this phenomenon of 'unbounded superradiance' is absent in the RN case. We show how the parameter space can be divided into regions, using the bounded/unbounded and absorption/amplification boundaries. After introducing a high-frequency approximation based on particle trajectories, we calculate the absorption cross section numerically, via the partial-wave expansion, as function of wave frequency, and we present a gallery of results. The cross section of the ABG RBH is found to be larger (smaller) than in the RN case when the field charge has the same (opposite) sign as the black hole charge. We show that it is possible to find 'mimics': situations in which the cross sections of both black holes are very similar. We conclude with a discussion of unbounded superradiance, and superradiant instabilities.",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "General Relativity (GR) is a geometric theory in which gravity is associated with the spacetime curvature generated by the presence of energy and momentum. For more than a century, the physical predictions of this theory have been scrutinised and tested experimentally in various ways [1-3]. In the last decade, for instance, two important verifications of GR predictions in the strong-field regime were reported: The Laser Interferometer Gravitational-Wave Observatory (LIGO) Collaboration performed the first direct detection of gravitational waves [4], from black hole (BH) coalescences; and the Event Horizon Telescope (EHT) Consortium has obtained the first image of a supermassive BH shadow [5]. BHs are among the most fascinating predictions of GR. These objects are solutions of Einstein's field equations (EFEs) characterized by an event horizon (i.e., a non-return surface), which are typically formed by gravitational collapse [6]. Observational evidence indicates that BHs populate galaxies [7]; for example, the Milky Way galaxy harbors a supermassive BH (with 4 . 1 ± 0 . 4 × 10 6 M ⊙ [8, 9]) at its core, as well as myriad stellar-mass BHs. In electrovacuum, the uniqueness theorems of GR [10] determine that stationary BH solutions are described by only three parameters: mass, charge, and angular momentum. Despite this apparent simplicity, the stationary BHs of GR are also paradoxical in nature: at their core is a curvature singu- larity , where the classical field theory breaks down. Key theorems that support the formation of curvature singularities in classical GR were established by Penrose [11] and Hawking [12]. These theorems show that spacetimes can become geodesically incomplete in rather general (i.e., nonsymmetric) collapse scenarios, within the classical field theory; and this, in turn, raises concerns about the global breakdown of causality in such spacetimes. As a remedy, the cosmic censorship hypothesis asserts that [11] all curvature singularities must be shrouded behind (apparent or event) horizons. Therefore, the spacetime outside this one-way membrane is not adversely affected by the presence of these hidden singularities. It can be argued that the formation of singularities represents a flaw in classical field theory, and that the paradoxes associated with singularities will be fully resolved in a quantum theory of gravity. It is not necessary, however, to await a complete quantum theory before studying the properties of BH solutions that are free from singularities. Recently, there has been increasing interest in the properties of so-called regular black hole (RBH) solutions. The first RBH model was proposed in 1968 by James Bardeen [13]. In this model, as well as others [14-17], the source term (i.e., the stress-energy tensor) in the EFEs did not have a clear physical motivation or origin. In 1998, Eloy Ay'on-Beato and Alberto Garc'ıa found that a RBH could arise in a physically well-motivated theory: nonlinear electrodynamics (NED) minimally coupled to GR [18]. NED models are, in essence, generalizations of Maxwell's linear theory to strong electromagnetic fields [19-22]. Two well-studied NED models are the Euler-Heisenberg model [23], which provides an effective description of Quantum Electrodynamics at the one-loop level; and the Born- Infeld model [19, 20], introduced to remove the infinite selfenergy of the electron. Among the features and applications of NED models [19-32], there have recently been proposed several electrically [33-38] and magnetically [39-45] charged RBH solutions, in minimally-coupled GR, as well as in alternative theories of gravity [46-49]. For a review on NED and applications to BH physics see Refs. [50, 51]. Motivated in part by recent observational breakthroughs, there is growing interest in discerning how the key properties of regular holes will differ from those of irregular (i.e., singular) BHs, particularly in the observable region exterior to the horizon. A canonical example of the irregular class - and a key point of comparison for this study - is the ReissnerNordstrom (RN) spacetime: a spherically-symmetric solution to the EFEs for linear (i.e., Maxwell) electromagnetism minimally coupled with GR, describing a BH of mass M and charge Q with two horizons, at r ± = M ± √ M 2 -Q 2 , and a curvature singularity, at r = 0 . It is well known that the RN BH exhibits the phenomenon of superradiance when interacting with a scalar field of charge q . Field modes with a frequency ω > 0 satisfying ω < qϕ + are amplified , rather than absorbed, by the BH, where ϕ + is the electric potential at the outer horizon. In the superradiant regime, the BH loses charge and mass (i.e., it flows out of the BH into the field), and yet the area of the BH ( A = 4 πr 2 + ) increases. In the thermodynamic interpretation, the horizon area is associated with the entropy of the BH, and superradiance is then a necessary consequence of the second law of thermodynamics. For studies about charged superradiance in static BHs, see, e.g., Refs. [52-57]. Superradiance can also occur with neutral fields if the BH is spinning. Superradiance has been studied in a range of BH scenarios over the past fifty years, leading to various interesting outcomes (see, e.g., Ref. [58] for a comprehensive review). It is natural to ask whether superradiance persists for RBHs - and if so, whether it is enhanced or diminished. In the far-field region, r ≫ M , where the electromagnetic field is weak, NED models are expected to reduce to linear electromagnetism, and thus, NED RBHs to be locally equivalent to RN BHs. Conversely, in the near-horizon region, where the electromagnetic field is strong, NED models are likely to differ substantially from their linear counterparts, and ϕ + may differ substantially from ϕ RN + = Q/r + . Consequently, we would expect the condition for superradiance to depend on the precise form of the NED model in question and, potentially, for certain models, to display enhanced versions of superradiance and new phenomenology. Some recent works addressed superradiance in the background of rotating regular spacetimes, considering massive scalar fields [59, 60], but works considering charged scalar fields and superradiance in static RBH geometries are still lacking in the literature. In this paper, we study the absorption of a charged massive test scalar field in the background of a RBH solution, namely, the first proposed exact charged RBH solution of Ay'on-Beato and Garc'ıa (ABG) [18]. Here we are particularly interested in characterizing the effect of superradiance on the absorption cross section (ACS). Over the last fifty years, much effort has been made to compute absorption and scattering in different BH scenarios (cf., e.g., Refs. [61-69] and references therein). Although several works have been dedicated to chargeless test fields, few have dealt with the absorption of charged scalar fields [54, 55]. Recently, the absorption of chargeless test fields has been investigated for RBHs [70-73], but the absorption of charged massive scalar waves is still to be properly quantified. The remainder of this paper is organized as follows. In Sec. II, we review the main aspects of the ABG RBH spacetime proposed in Ref. [18], and in Sec. III we investigate the dynamics of a massive and charged scalar field on this spacetime. In Sec. III B, we present an expression to compute the ACS via a sum over partial waves; in III C we partition the parameter space; and in III D we describe a high-frequency approximation. Our numerical results concerning the absorption and superradiance properties of the ABG RBH solution are presented in Sec. IV, and we also compare them with those obtained in the RN case. We conclude with our final remarks in Sec. V. Throughout the paper we use the natural units, for which G = c = ℏ = 1 , and metric signature -2 .",
"pages": [
1,
2
]
},
{
"title": "II. FRAMEWORK",
"content": "The action associated with NED minimally coupled to GR can be written as [18] where R is the Ricci scalar, L ( F ) is a gauge-invariant electromagnetic Lagrangian density, and g is the determinant of the metric tensor g µν . The electromagnetic invariant F and the standard electromagnetic field tensor are given by respectively, where A ν is the electromagnetic four-potential. It is possible to represent NED in a different framework by introducing an auxiliary anti-symmetric tensor where L F ≡ ∂ L /∂F ; and also a structural function H ( P ) through a Legendre transformation [74], namely The invariant associated with P µν is defined as Among its applications [22, 74], this framework is useful to obtain electrically charged NED-based RBH solutions [37]. With the help of Eqs. (2)-(5), one can show that where H P ≡ ∂ H /∂P . By varying the action (1) with respect to the metric tensor g µν and using Eqs. (6), it is possible to obtain which are Einstein-NED field equations, and where G µ ν is the Einstein tensor and T µ ν is the energy-momentum tensor. The variation of the action (1) with respect to A µ leads to ∇ µ P µν = 0 (in the absence of electromagnetic sources). For the solution proposed in Ref. [18], from now on simply referred to as ABG solution, the structural function (the NED source) is with Q and M being the electric charge and mass of the central object, respectively. Considering a spherically symmetric and static line element as an ansatz for the spacetime, together with the Eqs. (7) and (8), one can obtain the ABG line element in spherical coordinates ( x µ = { t, r, θ, φ } ), where is the metric function of the ABG spacetime. In the asymptotic limit r → ∞ , the metric function (10) has the following behavior where f RN ( r ) is the metric function of the RN spacetime [75], As argued earlier, this is expected because, in the far-field region, the electromagnetic field is weak and thus in the linear (i.e., Maxwell) regime. On the other hand, expanding the ABG metric function in powers of Q yields When the condition | Q | ≤ Q ext ≈ 0 . 6341 M is fulfilled [18], the line element (9) describes an ABG RBH. For | Q | < Q ext , the ABG RBH possesses an inner (Cauchy) horizon at r -and an outer (event) horizon at r + , given by the real positive roots of f ABG ( r ) = 0 . For | Q | = Q ext , we have the so-called extreme ABGRBH, with r + = r -. For | Q | > Q ext , we have horizonless solutions. The ABG causal structure is similar to the RN one (for which Q RN ext = M ). Throughout this work, we shall restrict our analysis to BH solutions ( | Q | ≤ Q ext ) , and exhibit our results in terms of the normalized electric charge which satisfies 0 ≤ | α | ≤ 1 for BH geometries. From F 01 = E ( r ) = H P Q/r 2 , one can show that the radial electrostatic field E ( r ) of the ABG solution is given by which is finite at the origin and asymptotically behaves as the electrostatic field in the RN case, given by A detailed analysis of the metric function, electric field, and geodesics of massless particles of ABG and RN BHs is presented in Refs. [73, 76]. The covariant components of the electromagnetic fourpotential are given by where ϕ is the electrostatic potential, which can be obtained using ϕ ( r ) = -∫ r ∞ E · d l and Eq. (15), to obtain 1 In Fig. 1, we plot the ABG electrostatic potential, ϕ ABG ( r ) alongside the electrostatic potential of the RN BH, Notably, ϕ ABG ( r ) is finite at r = 0 , whereas ϕ RN ( r ) diverges as r → 0 . In the far field ( r → ∞ ), ϕ ABG → ϕ RN . At the (outer) horizon, ϕ ( r + ) ABG > ϕ ( r + ) RN , and thus an enhanced superradiant regime may be anticipated.",
"pages": [
2,
3
]
},
{
"title": "A. Scalar fields and superradiance",
"content": "We are interested in investigating a scalar field Φ with mass µ and charge q , propagating in a static (electric) RBH spacetime. Therefore, we shall consider the Klein-Gordon equation Exploiting the separability of Eq. (20), we can write a particular mode of Φ as where Ψ ωl ( r ) are radial functions and P l (cos θ ) are the Legendre polynomials. The indexes ω and l denote the frequency and the angular momentum of the scalar wave, respectively. Inserting Eq. (21) into Eq. (20) leads to the radial equation where r ⋆ is the tortoise coordinate defined by dr ⋆ = dr/f ( r ) , and the potential function V ( r ) reads From the form of Eq. (22), it is clear that Φ is propagative (i.e., oscillatory) in regions where V ( r ) < 0 and evanescent (i.e., exponential) in regions where V ( r ) > 0 . As the angular momentum l increases, the height of the potential barrier increases commensurately (as in the massless case [73]). Figure 2 shows V ( r ) as a function of the parameter qM for the particular case l = 0 , ω = µ and α = 0 . 5 (defined in Eq. (14)). The height of the local maximum value of V ( r ) increases with qM . As we increase α , the peak of V ( r ) increases (decreases) for qM > 0 ( qM < 0 ). This is anticipated from the Lorentz force: particles with the same charge sign of the BH are repelled, and consequently less absorbed, than particles with the opposite charge, which are attracted. Note also that for some values of qM , µM , and α , the peak of the radial function V ( r ) becomes negative (cf. curve qM = -0 . 3 in Fig. 2) [78]. In the (planar-wave) scattering problem, the wave satisfies the ingoing boundary conditions where ζ ≡ ω -qϕ + (with ϕ + ≡ ϕ ( r + ) ), and κ ≡ √ ω 2 -µ 2 . The quantities T ωl , I ωl , and R ωl are complex coefficients. Justification for the ingoing boundary condition in (24) runs as follows. In the near-horizon region, the general solution for the field Φ is a superposition of two terms, with behaviours e -i ( ωt ± ζr ⋆ ) . We seek fields Φ and A µ which are regular on the future horizon in a suitable gauge. Noting that, for A µ in Eq. (17), the Lorentz invariant A µ A µ is divergent, we can make a gauge transformation , A µ → A ' µ = A µ + q -1 ∇ µ χ and Φ → Φ ' = e iχ Φ , such that A ' µ = 0 on the horizon. This corresponds to the choice χ = qϕ + t . Hence the general solution for Φ ' is a superposition of two terms with behaviours e -iζ ( t ± r ∗ ) . The term with upper sign choice (+) is regular (irregular) on the future (past) horizon, and the term with the lower sign ( -) is regular (irregular) on the past (future) horizon. The boundary condition in Eq. (24) then follows from the requirement that Φ ' is regular on the future horizon. For a propagating wave at infinity (unbounded modes), the condition κ > 0 holds, i.e., ω 2 > µ 2 . The transmission and reflection coefficients are defined, respectively, as From the conservation of the flux, or using the Wronskian of Eq. (22), one can derive The amplification factor [58] is This measures the fractional gain (or loss) of energy in a scattered wave, with positive values of Z ωl corresponding to superradiant amplification. Clearly, the sign of Z ωl is determined by the sign of ζ . Hence the critical frequency for superradiant scattering is For frequencies ω > ω c , the wave is absorbed; conversely, for frequencies 0 < ω < ω c , the wave is amplified. In Fig. 3, we show ϕ + , the electric potential at the horizon, for ABG and RN BHs. We note that (for Q > 0 ) ϕ + is always positive and increases monotonically with α . As a consequence, superradiance occurs whenever qϕ + > 0 . We also observe that ϕ ABG + is always greater than ϕ RN + . Therefore, for the same values of qM and α , the critical frequency of the ABG RBH is always larger than that of the RN BH. This implies a greater capacity for superradiant scattering in the ABG case.",
"pages": [
4,
5
]
},
{
"title": "B. Absorption cross section",
"content": "The ACS σ , for a plane wave incident upon a sphericallysymmetric BH, can be expanded in partial waves as follows: where the partial ACS is Hence, using Eq. (26), For superradiant modes (with 0 < ω < ω c = qϕ + ), σ l takes negative values, as the wave is amplified rather than absorbed. In the limit ω → µ (from above), the momentum of the wave tends to zero, κ → 0 . Hence σ l in Eq. (31) will diverge in this limit, unless lim ω → µ |T ωl | 2 /κ 3 exists. In other words, σ l will diverge unless the transmission factor |T ωl | 2 approaches zero at least as rapidly as the cube of the momentum, κ 3 . We will call the divergent case unbounded and the finite case bounded . It is clear that there are four possibilities to consider in the limit ω → µ , namely: (i) bounded absorption, (ii) bounded superradiance, (iii) unbounded absorption and (iv) unbounded superradiance. Cases (i), (ii) and (iii) have been observed in absorption by a RN (irregular) BH [55]. The fourth possibility, unbounded superradiance, does not appear to occur for RN BHs; but it does arise for the regular ABG BH, as we demonstrate in Sec. IV. To understand why this arises, we now turn attention to the properties of the potential.",
"pages": [
5
]
},
{
"title": "C. The parameter space",
"content": "In this section, we argue that it is possible to divide the parameter space into regions where behaviours (i)-(iv) occur (see above) by examining the behaviour of an effective potential function. Considering Eq. (23) in the limit µ → ω , we define In Fig. 4, we present the typical behavior of the function U ( r ) in the ABG RBH spacetime. In the region where U ( r ) is positive, the wave is propagative. The plot makes it clear that the existence of a propagative region extending to spatial infinity depends critically on the parameter values. For the uncharged massive scalar field case, Jung and Park [78] defined the critical case as that in which the local maximum of U ( r ) is exactly zero. This idea extends naturally to absorption of a charged field: the critical case is shown as the blue dashed line in Fig. 4, and this case defines a critical charge α c (for fixed l , µM and qM ). For α < α c , a propagative region extends from a certain radius r c out to infinity (i.e., the region r ∈ ( r c , ∞ ) ), whereas for α > α c , the only propagative region is close to the horizon. It is natural to anticipate qualitatively different absorption properties in the limit ω → µ , with the former (latter) case corresponding to unbounded (bounded) behaviour. In fact, to determine the existence of a propagative region that extends to infinity, it is sufficient to examine the larger expansion of U ( r ) , given by At leading order, the expansion is identical for the RN and ABG BHs (as one might expect in the weak-field/linear regime). For µM > qQ , Newtonian attraction dominates over the Coulomb repulsion and the propagative region exists; for µM < qQ , Coulomb repulsion is dominant and the propagative region does not exist. The critical case is at µM = qQ . We can now divide the parameter space into regions using two separatrices: µM = qQ (the bounded/unbounded boundary) and µ = qϕ + (the absorption/amplification boundary). Figure 5 shows the anticipated behaviour of the ACS in the limit ω → µ (i.e. κ → 0 ), in the parameter space. The horizontal blue line ( µM = qQ ) separates the bounded and unbounded regions. The solid red line demarcates the onset of superradiance. In the RN case, there is no overlap between the unbounded and superradiant regions (though the boundaries meet in the extremal case, Q = M ). This is consistent with an observed absence of unbounded superradiance. Conversely, for the ABG BH, the superradiant region is significantly larger than in the RN case (even for Q → 0 ), due to the increase in ϕ + (see Fig. 3). Consequently, the unbounded region overlaps with the superradiant region, and hence we should anticipate unbounded superradiance (that is, unbounded amplification) to occur in the limit ω → µ , in this region of the parameter space. These conclusions are supported by the numerical evidence presented in Sec. IV C.",
"pages": [
5,
6
]
},
{
"title": "D. High-frequency approximation",
"content": "We now turn attention to absorption in the regime of high frequencies and short wavelengths. In this regime, the characteristics of the charged massive scalar wave can be associated with the trajectories of charged particles subject to the Lorentz force imparted by the electric background field. In order to obtain the equations of motion associated with our problem, we consider the following Lagrangian where the overdot stands for the derivative with respect to the proper time, and q cp and m are the charged and mass of the particle, respectively. From Eq. (34), one can introduce the conserved quantities which are related to the energy and the angular momentum of the particle and, in the semiclassical limit, are associated with ω and l +1 / 2 , respectively. Using Eqs. (34)-(36) together with g µν ˙ x µ ˙ x ν = 1 (the condition that a massive particle follows a timelike path, parametrized by its proper time [79]), one can show that in which, due to the spherical symmetry, we considered the motion in the equatorial plane ( θ = π/ 2) . By defining the impact parameter b ≡ L/vE and K ( r ) ≡ ˙ r 2 ( m 2 /L 2 ) , we can rewrite Eq. (34) as where v is a dimensionless parameter defined by Since we are interested in the unbounded timelike paths, i.e., κ > 0 , this parameter is limited by 0 < v ≤ 1 . Considering that K ( r ) and its first derivative vanish at the critical radius r c , namely we may find the critical impact parameter b c , and an equation that gives the values of r c , where we defined z ≡ 1 -v 2 and the prime ( ' ) denotes derivative with respect to the radial coordinate r . For q cp = 0 , we recover the b c and r c of the massive chargeless case, which are given by [64] respectively. In the limit m → 0 , which implies in v → 1 , we obtain the results for the massless case [73]. The high-frequency absorption cross section, also called geometric cross section (GCS), σ gcs , is given by [79]",
"pages": [
6,
7
]
},
{
"title": "A. Numerical Analysis",
"content": "We can obtain the reflection and transmission coefficients, given by Eqs. (25), by numerically integrating the Eq. (22) from very close to r + up to far from the BH, with the boundary conditions given by Eqs. (24) and their derivatives. Then we compute the total ACS using Eq. (29). The oscillatory character of the ACS is related to the partial waves contributions [see Eq. (30)]. We have chosen, in general, to perform the summation in Eq. (29) up to l = 20 . The GCS is obtained numerically through Eq. (46), using Eqs. (42) and (43).",
"pages": [
7
]
},
{
"title": "B. ABG regular black hole cross sections",
"content": "Figures 6 and 7 show the total ACS for different values of the charge ( qM ) and mass ( µM ) couplings, respectively. Generically, we can see that the total ACS oscillates around the GCS (black dotted lines), with good agreement in the high-frequency regime. Moreover, for a fixed value of µM and α , we observe that the absorption increases (diminishes) as we consider smaller (higher) values of qQ , as a consequence of the Lorentz force. As shown in Fig. 7, the total ACS increases as we increase µM , so that the increase of µM leads to a higher absorption of planar scalar waves. Figure 8 shows the total ACS together with the partial ACS for two choices of qM , with the normalized BH charge α = 0 . 5 and the field mass coupling µM = 0 . 2 . The plots show that the oscillatory pattern in the total ACS is related to the sequential contributions from partial ACSs l = 0 , 1 , 2 , . . . . The monopole ( l = 0) dominates the behavior of the total ACS in the low-frequency regime. Note that, although the values of α and µM are equal in both panels of Fig. 8, superradiance occurs only in the bottom panel case. We observe that the ACS is negative in the range µ < ω ≲ 2 µ . Physically, a negative cross section implies that the stimulation from the plane wave causes the BH to transmit mass-energy and charge into the field. In Fig. 9, we present the amplification factor of massive charged scalar fields in the background of an ABG RBH. (We exhibit the amplification factor (27) in percentage, i.e., Z ωl [%] ≡ 100Z ωl , and restricted to the regions where Z ωl [%] ≥ 0 ). As we can see, the maximum superradiant amplification increases with the charge of the scalar field, for fixed values of the BH mass and (positive) charge.",
"pages": [
7,
8
]
},
{
"title": "C. Unbounded superradiance from a regular ABG black hole",
"content": "In the previous section, we presented some typical absorption properties of charged massive scalar waves in the background of the ABG RBH. In the limit ω → µ we saw two types of behaviour: unbounded absorption (Figs. 6, 7 and 8, upper plot) and bounded superradiance (Fig. 8, lower plot). In this section, we show that the ABG RBH can also display unbounded superradiance for parameter choices informed by Fig. 5, and the discussion in Sec. III C. The results in Fig. 10 reveal a remarkable implication of the electromagnetic fields associated with (electrically charged) NED-based RBH geometries: they generate a superradiant divergence in the ACS of a charged, massive scalar field. That is, an ABG BH stimulated by a massive plane wave of low momentum has an ACS which is unbounded from below. This is in stark contrast to the RN BH, where the cross section cannot obtain arbitrary negative values.",
"pages": [
8,
9
]
},
{
"title": "D. Comparison with the Reissner-Nordstrom BH",
"content": "In this section, we compare the absorption properties of ABG RBHs with those of RN BHs [55]. Figure 11 shows a comparison between the total ACSs of ABG RBHs and RN BHs for α = 0 . 4 , qM = 1 . 6 , and two values of µM . We see that, for a fixed value of µM , the total ACS of the ABG RBH is smaller than the total ACS of the RN BH, across the frequency range. We also observe that for µM = 0 . 4 , the ABG RBH exhibits superradiant scattering (i.e. σ < 0 in some range of ω ), whereas the RN BH does not (for these parameters). This feature is due to the higher threshold frequency ( ω c = qϕ + ) for superradiance in the ABG case (see Sec. III, in particular, Eq. (28) and Fig. 3) and the condition for unbounded modes, namely ω 2 > µ 2 . For µM = 0 . 4 , both systems show unbounded absorption as ω → µ . Conversely, for µM = 0 . 8 , the RN shows bounded absorption, whereas the ABG RBH shows bounded superradiance, in this limit. Referring again to the parameter space shown in Fig. 5, in the RN BH case the unbounded region and the superradiant region of the parameter space are disjoint. Therefore, we do not expected to observe unbounded superradiance (as ω → µ ) in the RN case. In this sense, absorption by the RN BH is qualitatively different to absorption by the ABG RBH, for which is possible to find a set of field and RBH parameters that leads to unbounded superradiance (see Fig. 10). A further notable feature of Fig. 5 is that bounded absorption does not occur in the ABG case (since the bounded case is necessarily superradiant) whereas it does in the RN case; see Fig. 11 for an example. Figure 12 shows results for the ACSs in two distinct sets: (i) for α = 0 . 4 , µM = 0 . 4 , and different values of qM (top panel); and (ii) for µM = 0 . 2 , qM = 0 . 4 , and different values of α (bottom panel). We note that, similarly to the behavior presented in Fig. 11, for a given set of parameters, superradiance might occur only for the ABG RBH. We see again that the propagating waves are more absorbed in the RN case, when qQ > 0 ; however, for qQ < 0 , we observe the opposite behaviour. Figure 13 compares the partial ACSs of ABG and RN BHs. In this case, for the chosen parameters we have superradiance for both BH types. The range of frequency in which σ < 0 is larger in the ABG case than in the RN case. The dominant contribution to superradiance comes from the monopole mode, l = 0 . A comparison of the amplification factors in the ABG and RN geometries is exhibited in Fig. 14. The superradiant amplification in the background of the ABG RBH, for the same values of α , µM , and qM , is typically larger than that in the corresponding RN geometry, in agreement with the results presented in Figs. 11-13. Weend this section by noting that all results presented here, as well as those not shown, are consistent with the parameter space for ABG and RN BHs introduced in Fig. 5.",
"pages": [
9
]
},
{
"title": "E. Mimic configurations",
"content": "In this section, we show that it is possible to find combinations of the normalized charge of the BH solution, and the parameters (charge and mass) of the scalar field such that the absorption cross sections of ABG and RN BHs are very similar. We start by computing the values of α for which the geometric cross section (see Sec. III D) of the ABG RBH is equal to that of the RN BH, for fixed values of the charged massive particle parameters EM , q cp M , and mM . Next, we compute the absorption cross sections using the corresponding values of α , qM , and µM . Figure 15 shows the total ACSs for specific pairs ( α ABG , α RN ) with µM = 0 . 6 and qM = 0 . 2 . As we can see, the total ACSs of the two types of BH (regular ABG and irregular RN) can be very similar in the middle-to-high frequency range, particularly for low-to-moderate values of α , but distinguishable in the low-frequency regime. Figure 16 shows that, for a neutral field ( qM = 0 ), the ACSs of the two types of BH can be very similar across the whole frequency range, particularly for small-to-moderate values of α . Here we consider two pairs of choices of ( α ABG , α RN ) that lead to the same GCS. The field charge q increases the differences between the absorption pattern of ABG and RN BHs, particularly at lower frequencies. It is possible to find configurations for which a massive and charged scalar field has the same value of the critical frequency ω c fo superradiance in the two spacetimes, that is, Figure 17 shows the values of the pair ( α ABG , α RN ) for which the critical frequency is the same in both backgrounds, considering a fixed value of qM . These configurations can be found up to α ABG ≲ 0 . 8716 . Figure 18 shows the total ACS for pairs ( α ABG , α RN ) with the same critical frequency ω c . We see that the effect of su- radiance is enhanced in the ABG case, as one would expect given the stronger amplification shown in Fig. 14. It is also possible to find situations in which scalar fields with different masses and charges have the same critical frequency ω c in the background of ABG and RN BH spacetimes. In Fig. 19, we consider a scalar field with qM = 1 and µM = 0 . 2 in the ABG spacetime and a scalar field with qM = 1 . 4 and µM = 0 . 4 in the RN geometry. As we can see, the distinct scalar fields are superradiantly scattered whenever ωM < 0 . 7 . Figure 20 shows the amplification factors for the same parameters, highlighting once again that superradiance amplification is enhanced for the ABG BH relative to the RN BH.",
"pages": [
10,
11
]
},
{
"title": "V. CONCLUDING REMARKS",
"content": "Within GR, the standard BH solutions (Schwarzschild, Reissner-Nordstrom, Kerr, and Kerr-Newman) possess a common feature in their core: a curvature singularity hidden by an event horizon. On the other hand, certain regular BH solutions, i.e., objects with an event horizon but with no curvature singularity, can be obtained by minimally coupling NED mod-e s to GR. Much work is underway to determine the properties of RBHs. Contributing to this effort, we have scrutinised the absorption properties of charged massive scalar fields in the background of the electrically charged RBH solution proposed by Ay'on-Beato and Garc'ıa [18]. The most intriguing result of our study is that regular ABG BHs, unlike their RN counterparts, exhibit unbounded superradiance (as ω → µ ) in massive scalar fields, within a certain parameter range. More precisely, the cross section σ is unbounded from below as ω → µ ; this is shown clearly in Fig. 10. The region of parameter space in which unbounded superradiance occurs is clarified in Sec. III C and Fig. 5. Some care is needed in interpreting the physical meaning of the divergence of the cross section σ as ω → µ . In the particle picture, a divergence arises naturally because particles of low momentum and large impact parameters are attracted by the BH, with b c →∞ as κ → 0 . In the wave picture, a divergence in σ arises if the transmission factor does not go to zero as rapidly as the cube of the momentum κ in the denominator of Eq. (31); and if this occurs in the superradiant regime ω < ω c , then the unbounded superradiance phenomenon occurs. Notably, the amplification factor, Z ωl in Eq. (27), does not diverge in our numerical results. In principle then, by stimulating the BH with a planar wave of low momentum ( κ → 0 ) in a massive charged field, one can extract (via superradiance) unbounded quantities of mass and charge from the ABG BH (within the limitations of the linearised regime of weak scalar fields). It is important to stress, however, that the divergence in σ is related to the fact that the BH is interacting with a planar wave of infinite lateral extent (and b c → ∞ as κ → 0 ). Therefore, one should not expect unbounded extraction of energy to be possible (even in principle) for a stimulating wave of finite width and duration. Some further interesting aspects of the ACS are summarized below: (i) Massive scalar waves are typically more absorbed than massless ones, and absorption increases with the value of µM . This result is expected since larger field masses lead to larger GCSs, and large field masses are associated with strongly absorbed modes [64, 78]. (ii) In the case of a charged scalar field, due to the Lorentz force, the absorption for qQ < 0 is typically larger than for qQ > 0 . (iii) Low-frequency waves satisfying the condition ω < ω c [cf. Eq. (28)] can have a negative ACS. This occurs due to the superradiant amplification of low multipoles of the field (principally, in the l = 0 mode) [52]. (iv) The absorption of scalar waves by ABG RBHs is typically larger than for RN BHs (for equivalent q , M and α ), when the value of the charge coupling qQ is negative. Conversely, σ RN > σ ABG when qQ > 0 . (v) The critical superradiant frequency of the ABG BH is always larger than that of the RN BH, for equivalent parameters. Moreover, superradiant amplification is stronger for the ABG BH. Both aspects are due to the enhanced electrostatic potential at the horizon, ϕ + , in the ABG case (i.e., ϕ ABG + > ϕ RN + ). We showed in Sec. IV E that, for certain parameter choices, the ABG RBH solution can mimic the RN solution, from the point of view of absorption spectrum, reinforcing the results presented in Refs. [73, 76]. It is also possible to find configurations for which scalar fields with different masses and charges, in the background of ABG and RN BHs, have the same critical superradiant frequency. Several avenues for further investigation are open. Firstly, we note that superradiant scattering is, in some sense, the wave analogue of the Penrose process. In light of the results here, it could be worth studying the Penrose process for the ABG BH in detail. That is, the scenario of a charged particle, incident from infinity, that splits into two parts in the vicinity of the BH, with one part ejected to infinity and the other ab- sorbed 2 . In a Penrose process, the escaping particle has more mass-energy than the incident one. It would be interesting to compare the regions of parameter space in which energy extraction can occur, again drawing a comparison between the RN BH and the ABG BH. Secondly, the existence of the 'unbounded superradiance' region in Fig. 5 strongly hints at the existence of superradiantly-unstable quasibound states in the spectrum of the massive charged scalar field on the ABG spacetime. Previous investigations on the RN spacetime have suggested that it is not possible to form quasibound states that are also superradiant in the RN case. Heuristically, the reason is clear: for bound states one needs an attractive potential in the far-field, which necessitates µM > qQ ; then modes with ω < µ do not lie in the superradiant regime ω < qϕ RN + = qQ/r + of the RN BH. Conversely, as shown in Fig. 5, modes satisfying µM/qQ > 1 can also be superradiant on the ABG spacetime, due to the increase in the electric potential at the horizon, ϕ + . This implies that certain quasibound modes of the massive charged scalar field will grow exponentially with time, and thus that the ABG BH suffers a superradiant instability. This is under active investigation. Thirdly, the ABG BH is just one example in the regular class in NED. It would be interesting to clarify whether other solutions in this class also exhibit a stronger EM field at the horizon, and thus an enhanced region of superradiance with associated phenomena; or whether the ABG BH stands alone in this respect. Finally, real astrophysical BHs are known to be rotating. Future studies of absorption by spinning RBHs would clarify the interplay between charged superradiance (studied here) and rotational superradiance.",
"pages": [
11,
12,
13
]
},
{
"title": "ACKNOWLEDGMENTS",
"content": "The authors thank Fundac¸˜ao Amazˆonia de Amparo a Estudos e Pesquisas (FAPESPA), Conselho Nacional de Desenvolvimento Cient'ıfico e Tecnol'ogico (CNPq) and Coordenac¸˜ao de Aperfeic¸oamento de Pessoal de N'ıvel Superior (Capes) - Finance Code 001, in Brazil, for partial financial support. M.P. and L.C. thank the University of Sheffield, in England, and University of Aveiro, in Portugal, respectively, for the kind hospitality during the completion of this work. This work has further been supported by the European Union's Horizon 2020 research and innovation (RISE) programme H2020-MSCA-RISE-2017 Grant No. FunFiCO-777740 and by the European Horizon Europe staff exchange (SE) programme HORIZON-MSCA-2021-SE01 Grant No. NewFunFiCO-101086251. S.D. acknowledges financial support from the Science and Technology Facilities Council (STFC) under Grant No. ST/T001038/1.",
"pages": [
13
]
}
]
2024PhRvD.109f5020B
https://arxiv.org/pdf/2309.14535.pdf
<document>
<section_header_level_1><location><page_1><loc_30><loc_92><loc_70><loc_93></location>A relativistic quantum broadcast channel</section_header_level_1>
<text><location><page_1><loc_29><loc_89><loc_71><loc_90></location>Ian Bernardes Barcellos 1, ∗ and Andr'e G. S. Landulfo 1, †</text>
<text><location><page_1><loc_23><loc_86><loc_77><loc_88></location>1 Centro de Ciˆencias Naturais e Humanas, Universidade Federal do ABC, Avenida dos Estados, 5001, 09210-580, Bangu, Santo Andr'e, S˜ao Paulo, Brazil</text>
<text><location><page_1><loc_43><loc_85><loc_58><loc_86></location>(Dated: March 1, 2024)</text>
<text><location><page_1><loc_18><loc_69><loc_83><loc_83></location>We investigate the transmission of classical and quantum information between three observers in a general globally hyperbolic spacetime using a quantum scalar field as a communication channel. We build a model for a quantum broadcast channel in which one observer (sender) wishes to transmit (classical and quantum) information to two other observers (receivers). They possess some localized two-level quantum system (a qubit) that can interact with the quantum field in order to prepare an input or receive the output of this channel. The field is supposed to be in an arbitrary quasifree state, the three observers may be in arbitrary states of motion, and no choice of representation of the field canonical commutation relations is made. The interaction of the field and qubits is such that it allows us to obtain the map that describes this channel in a non-perturbative manner. We conclude by analyzing the rates at which information can be transmitted through this channel and by investigating relativistic causality effects on such rates.</text>
<text><location><page_1><loc_18><loc_67><loc_45><loc_68></location>PACS numbers: 03.67.-a,03.67.Hk, 04.62.+v</text>
<section_header_level_1><location><page_1><loc_20><loc_63><loc_37><loc_64></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_45><loc_49><loc_61></location>Network information theory is the area of knowledge that studies classical communication problems involving multiple parts. Here, the word 'classical' stands not only for the fact that the information being transmitted is classic (bits) but also for the physical systems in which such information is encoded, i.e., systems that can be described by some area of classical physics (such as Electromagnetism). One particular case of interest is the broadcast channel, where typically one sender wishes to transmit information to multiple receivers (like radio and TV stations broadcasting their signals, for example).</text>
<text><location><page_1><loc_9><loc_25><loc_49><loc_45></location>Nowadays, one of the main goals of quantum information theory is to extend several results of information theory to the quantum world [1, 2], investigating any new features or advantages that can arise when one uses quantum systems to encode, process, and transmit information. The quantum network information theory comprises the studies of communication protocols using quantum systems to convey classical (bits) or quantum (qubits) information. In particular, the classical broadcast channels can be extended to the so-called quantum broadcast channels , where one sender transmits classical or quantum input information to many receivers using a quantum system as a communication channel with quantum outputs [3-6].</text>
<text><location><page_1><loc_9><loc_16><loc_49><loc_24></location>Such communication scenarios are very suitable for analyzing how relativistic effects can influence one's ability to communicate using quantum channels. This could be due to the existence of nontrivial spacetime structures such as black hole event horizons, Cauchy horizons, and causal horizons arising from the relativistic relative mo-</text>
<text><location><page_1><loc_52><loc_61><loc_92><loc_64></location>tion between senders and receivers or even due to the expansion of spacetime [7].</text>
<text><location><page_1><loc_52><loc_23><loc_92><loc_61></location>In order to consistently analyze quantum information theory in general spacetimes, one should use quantum field theory in curved spacetimes (QFTCS) [8]. This approach was used by several authors to analyze the communication process in relativistic settings, with particular attention being paid to Minkowski [9-21], Schwarzschild [22-26], or asymptotically flat cosmological spacetimes [27-29]. However, only recently [30] a communication model valid in general globally hyperbolic spacetimes and in which the parts that convey information can move in arbitrary worldlines and interact with the quantum field (used as communication channel) only in the vicinity of its worldlines was developed (and, since then, other works in this context have emerged as, for instance, Refs. [31, 32]). This is interesting for two reasons: firstly, it allows the analysis of information exchange between more general observers, not only observers following orbits of some Killing field (which does not even exist in spacetimes lacking timelike symmetries). Secondly, the model studied in [30] allows one to investigate the outputs of the quantum communication in a nonperturbative manner and thereby is suitable to investigate both the causality as well as the communication between parts lying in early and future asymptotic regions (limits that would invalidate results obtained by perturbative methods).</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_23></location>In the present paper, we generalize the analysis of [30]. This is done by constructing a model for classicalquantum, quantum-quantum, and entanglement-assisted quantum-quantum broadcast channels. We consider an arbitrary globally hyperbolic spacetime in which one observer (Alice) wants to convey classical (or quantum) information to two receivers (Bob and Charlie) using a quantum scalar field as a communication channel. The three observers will use two-level quantum systems (qubits) to locally interact with the quantum field in or-</text>
<text><location><page_2><loc_9><loc_76><loc_49><loc_93></location>der to send or receive information. The observers may be in arbitrary states of motion, the interaction between the detectors and the field is similar to the one given by the Unruh-DeWitt model [33], and the field may initially be in an arbitrary quasifree state [8]. We suppose, however, that the two levels of each qubit have the same energy. This model is interesting because the evolution of the system can be computed exactly, and therefore we will obtain nonperturbative results for the communication rates associated with such a broadcast channel. As we will see, causality in the information exchange is explicitly manifest in our results.</text>
<text><location><page_2><loc_9><loc_56><loc_49><loc_76></location>This work is organized as follows. In Sec. II we will present the quantization procedure of a free scalar field on a globally hyperbolic spacetime as well as the class of states we will be using. In Sec. III we describe the interaction between the qubits and the field and determine the quantum map that relates the information Alice wants to convey to the final joint state of Bob's and Charlie's qubits. In Sec. IV we investigate the rates at which information can be transmitted using this broadcast channel, as well as the influence of the spacetime curvature or relative motion of observers in the communication process. In Sec. V we give our final remarks. We assume metric signature (- + ++) and natural units in which c = /uni0335 h = G = k B = 1, unless stated otherwise.</text>
<section_header_level_1><location><page_2><loc_17><loc_52><loc_40><loc_53></location>II. FIELD QUANTIZATION</section_header_level_1>
<text><location><page_2><loc_9><loc_41><loc_49><loc_49></location>Let us consider a free, real scalar field ϕ propagating in an arbitrary four-dimensional globally hyperbolic spacetime (M , g ) , where M denotes the four-dimensional spacetime manifold and g its Lorentzian metric. Let the spacetime be foliated by Cauchy surfaces Σ t labeled by the real parameter t . The field is described by the action</text>
<formula><location><page_2><loc_14><loc_37><loc_49><loc_40></location>S ≡ -1 2 /integral.disp M ϵ M (∇ a ϕ ∇ a ϕ + m 2 ϕ 2 + ξRϕ 2 ) , (1)</formula>
<text><location><page_2><loc_9><loc_28><loc_49><loc_37></location>where ϵ M = √ -g dx 0 ∧ /uni22EF ∧ dx 3 is the spacetime volume 4-form, m is the field mass, ξ ∈ R , R is the scalar curvature, ∇ a is the torsion-free covariant derivative compatible with the metric g , and g ≡ det ( g µν ) in some arbitrary coordinate system { x µ } . The extremization of the action (1) gives rise to the Klein-Gordon equation</text>
<formula><location><page_2><loc_20><loc_25><loc_49><loc_27></location>(-∇ a ∇ a + m 2 + ξR ) ϕ = 0 . (2)</formula>
<text><location><page_2><loc_9><loc_20><loc_49><loc_24></location>In the canonical quantization procedure, we promote the real field ϕ to an operator 1 that satisfies the 'equaltime' canonical commutation relations (CCR)</text>
<formula><location><page_2><loc_13><loc_17><loc_49><loc_19></location>[ ϕ ( t, x ) , ϕ ( t, x ' )] Σ t = [ π ( t, x ) , π ( t, x ' )] Σ t = 0 , (3)</formula>
<formula><location><page_2><loc_18><loc_14><loc_49><loc_16></location>[ ϕ ( t, x ) , π ( t, x ' )] Σ t = iδ 3 ( x , x ' ) , (4)</formula>
<text><location><page_2><loc_52><loc_90><loc_92><loc_93></location>where x ≡ ( x 1 , x 2 , x 3 ) are spatial coordinates on Σ t and π ( x ) is the conjugate momentum defined as</text>
<formula><location><page_2><loc_69><loc_86><loc_92><loc_89></location>π ≡ δS δ ˙ ϕ , (5)</formula>
<text><location><page_2><loc_52><loc_82><loc_92><loc_85></location>with the notation ˙ ϕ ≡ ∂ t ϕ . In addition, we may formally write the canonical Hamiltonian of the field as</text>
<formula><location><page_2><loc_55><loc_78><loc_92><loc_80></location>H ϕ ( t ) ≡ /integral.disp Σ t d 3 x /parenleft.alt1 π ( t, x ) ˙ ϕ ( t, x ) - L[ ϕ, ∇ a ϕ ]/parenright.alt1 , (6)</formula>
<text><location><page_2><loc_52><loc_76><loc_55><loc_77></location>with</text>
<formula><location><page_2><loc_64><loc_73><loc_92><loc_74></location>d 3 x ≡ dx 1 ∧ dx 2 ∧ dx 3 (7)</formula>
<text><location><page_2><loc_52><loc_70><loc_54><loc_71></location>and</text>
<formula><location><page_2><loc_54><loc_66><loc_92><loc_69></location>L[ ϕ, ∇ a ϕ ] ≡ -1 2 √ -g (∇ a ϕ ∇ a ϕ + m 2 ϕ 2 + ξRϕ 2 ) (8)</formula>
<text><location><page_2><loc_52><loc_64><loc_73><loc_65></location>being the Lagrangian density.</text>
<text><location><page_2><loc_52><loc_60><loc_92><loc_64></location>To find a representation of the CCR, Eqs. (3) and (4), we define an antisymmetric bilinear map σ acting on the space S C of complex solutions of Eq. (2) as</text>
<formula><location><page_2><loc_57><loc_56><loc_92><loc_58></location>σ ( ψ 1 , ψ 2 ) ≡ /integral.disp Σ t ϵ Σ n a [ ψ 2 ∇ a ψ 1 -ψ 1 ∇ a ψ 2 ] , (9)</formula>
<text><location><page_2><loc_52><loc_49><loc_92><loc_55></location>where ϵ Σ represents the proper-volume 3-form on the Cauchy surface Σ t and n a its future-directed normal unit vector. It allows us to define the Klein-Gordon product as</text>
<formula><location><page_2><loc_64><loc_46><loc_92><loc_48></location>/uni27E8 ψ 1 , ψ 2 /uni27E9 ≡ -i σ ( ψ 1 , ψ 2 ) , (10)</formula>
<text><location><page_2><loc_52><loc_32><loc_92><loc_45></location>and, although this product is not positive-definite on S C , we may choose any subspace H ⊂ S C (the so-called one-particle Hilbert space) such that: (i) S C /uni2243 H/uni2295.big H ; 2 (ii) the KG product is positive definite on H , thus making (H , /uni27E8 , /uni27E9) a Hilbert space; 3 (iii) given any u ∈ H and v ∈ H , /uni27E8 u, v /uni27E9 = 0. (See [8] for details.) The Hilbert space that comprises the field states is defined as the symmetric Fock space F s (H) and the quantum field operator is formally defined as</text>
<formula><location><page_2><loc_55><loc_28><loc_92><loc_31></location>ϕ ( t, x ) ≡ /summation.disp j /bracketleft.alt1 u j ( t, x ) a ( u j ) + u j ( t, x ) a † ( u j )/bracketright.alt , (11)</formula>
<text><location><page_2><loc_52><loc_21><loc_92><loc_27></location>where { u j } comprise an orthonormal basis for H and a ( u ) / a † ( v ) are the usual annihilation/creation operators associated with the modes u / v , respectively. They satisfy the commutation relations</text>
<formula><location><page_2><loc_64><loc_18><loc_92><loc_20></location>/bracketleft.alt1 a ( u ) , a † ( v )/bracketright.alt = /uni27E8 u, v /uni27E9 I, (12)</formula>
<text><location><page_3><loc_9><loc_88><loc_49><loc_93></location>with I being the identity operator on F s (H) . The vacuum state associated with this representation of the CCR is the normalized vector /divides.alt0 0 /uni27E9 that satisfies a ( u )/divides.alt0 0 /uni27E9 = 0 for every mode u ∈ H .</text>
<text><location><page_3><loc_9><loc_73><loc_49><loc_87></location>In order to make it mathematically well-defined, the quantum field operator must be defined as an operatorvalued distribution. To this end, let S ⊂ S C be the space of real solutions of Eq. (2) whose restriction to Cauchy surfaces have compact support and K ∶ S → H be the projection operator that takes the positive-norm part of any ψ ∈ S . If C ∞ 0 (M) denote the set of all smooth compactly-supported real functions on M , we define the map E ∶ C ∞ 0 (M) → S acting on some test function f ∈ C ∞ 0 (M) as</text>
<formula><location><page_3><loc_20><loc_71><loc_49><loc_72></location>Ef ( x ) ≡ Af ( x ) -Rf ( x ) , (13)</formula>
<text><location><page_3><loc_9><loc_65><loc_49><loc_69></location>where Af and Rf are the advanced and retarded solutions of the Klein-Gordon equation with source f , respectively. Hence, they satisfy</text>
<formula><location><page_3><loc_22><loc_63><loc_49><loc_64></location>P ( Af ) = P ( Rf ) = f, (14)</formula>
<text><location><page_3><loc_9><loc_59><loc_49><loc_61></location>with P ≡ -∇ a ∇ a + m 2 + ξR representing the Klein-Gordon differential operator.</text>
<text><location><page_3><loc_9><loc_56><loc_49><loc_58></location>Now, for each test function f ∈ C ∞ 0 (M) , we define a smeared quantum field operator by</text>
<formula><location><page_3><loc_18><loc_53><loc_49><loc_55></location>ϕ ( f ) ≡ i /bracketleft.alt1 a ( KEf ) -a † ( KEf )/bracketright.alt , (15)</formula>
<text><location><page_3><loc_9><loc_51><loc_43><loc_52></location>which satisfies the covariant version of the CCR,</text>
<formula><location><page_3><loc_18><loc_48><loc_49><loc_49></location>[ ϕ ( f 1 ) , ϕ ( f 2 )] = -i ∆ ( f 1 , f 2 ) I, (16)</formula>
<text><location><page_3><loc_9><loc_46><loc_13><loc_47></location>where</text>
<formula><location><page_3><loc_18><loc_42><loc_49><loc_44></location>∆ ( f 1 , f 2 ) ≡ /integral.disp M ϵ M f 1 ( x ) Ef 2 ( x ) (17)</formula>
<text><location><page_3><loc_9><loc_37><loc_49><loc_41></location>for all f 1 , f 2 ∈ C ∞ 0 (M) . As shown in [8], Eq. (15) can be obtained by formally integrating Eq. (11) weighed by the test function f , i.e.,</text>
<formula><location><page_3><loc_20><loc_33><loc_49><loc_35></location>ϕ ( f ) = /integral.disp M ϵ M ϕ ( x ) f ( x ) . (18)</formula>
<text><location><page_3><loc_9><loc_22><loc_49><loc_32></location>The above construction has the downside that there are infinitely many choices of H satisfying properties (i) -(iii) listed below Eq. (10) and their respective Fock spaces are, in general, unitarily inequivalent. As discussed in [30], this issue can be avoided through the algebraic approach to quantum field theory (QFT). For more details, see Refs. [8, 34].</text>
<text><location><page_3><loc_9><loc_18><loc_49><loc_22></location>In this work, we will focus on a particular class of states: the quasifree states , defined as follows. Given a real inner product µ ∶ S × S → R satisfying</text>
<formula><location><page_3><loc_17><loc_15><loc_49><loc_17></location>/divides.alt0 σ ( φ 1 , φ 2 )/divides.alt0 2 ≤ 4 µ ( φ 1 , φ 1 ) µ ( φ 2 , φ 2 ) , (19)</formula>
<text><location><page_3><loc_9><loc_11><loc_49><loc_14></location>for all φ 1 , φ 2 ∈ S , we define a quasifree state ω µ associated with µ by the relation</text>
<formula><location><page_3><loc_19><loc_9><loc_49><loc_10></location>ω µ [ W ( Ef )] ≡ e -µ ( Ef,Ef )/slash.left 2 , (20)</formula>
<text><location><page_3><loc_52><loc_90><loc_92><loc_93></location>for all f ∈ C ∞ 0 (M) , where the so-called Weyl operators W ( Ef ) are defined by</text>
<formula><location><page_3><loc_61><loc_87><loc_92><loc_89></location>W ( Ef ) ≡ e iϕ ( f ) , f ∈ C ∞ 0 (M) . (21)</formula>
<text><location><page_3><loc_52><loc_83><loc_92><loc_86></location>The vacuum, n-particle, and thermal states are examples of quasifree states.</text>
<section_header_level_1><location><page_3><loc_52><loc_79><loc_91><loc_80></location>III. THE QUANTUM BROADCAST CHANNEL</section_header_level_1>
<text><location><page_3><loc_52><loc_57><loc_92><loc_77></location>A typical broadcast communication scenario involves the transmission of information between one station (sender) and several receivers who will decode the information independently. Let us consider a model in which one observer, Alice, wants to transmit separate information to two other observers, Bob and Charlie, using the quantum field ϕ as a broadcast channel. Suppose that the field is initially in some quasifree state ω µ 4 . Suppose also that the three observers follow arbitrary trajectories in the curved spacetime and that each one of them possesses a two-level quantum system that may interact with the quantum field at their will. The two-dimensional Hilbert spaces associated with Alice's, Bob's, and Charlie's qubits are denoted by H A , H B , and H C , respectively.</text>
<figure>
<location><page_3><loc_55><loc_36><loc_88><loc_55></location>
<caption>FIG. 1. The Figure depicts the quantum broadcast protocol being used. The dashed lines display the worldlines of the sender, Alice (A, Red), and receivers, Bob and Charlie (B and C, blue). The solid lines in each worldline depict the interaction interval of each observer's qubit with the quantum field. Here, Σ t 1 and Σ t 2 represent two Cauchy surfaces of the spacetime.</caption>
</figure>
<text><location><page_3><loc_52><loc_17><loc_92><loc_23></location>The communication setup, illustrated by Fig. 1, is as follows: In order to transmit information to Bob and Charlie, Alice prepares her qubit in some initial quantum state ρ A -∞ and switches on its interaction with the</text>
<text><location><page_4><loc_9><loc_83><loc_49><loc_93></location>field for a finite time interval ∆ t A (measured by the parameter t ). To measure the information imprinted by Alice on the field's state, Bob and Charlie initially prepare their qubits in suitable states ρ B -∞ and ρ C -∞ and then they switch on each of their qubit interaction with the field for finite time intervals ∆ t B and ∆ t C , respectively. For the sake of simplicity, we will consider here the case where</text>
<text><location><page_4><loc_7><loc_79><loc_49><loc_82></location>(QB1) Bob lets his qubit interact with the field only after Alice finishes her transmission;</text>
<text><location><page_4><loc_7><loc_75><loc_49><loc_78></location>(QB2) Charlie lets his qubit interact with the field only after Bob finishes his measurement process.</text>
<text><location><page_4><loc_9><loc_71><loc_49><loc_74></location>Such communication setup is implemented by means of the Hamiltonian</text>
<formula><location><page_4><loc_21><loc_69><loc_49><loc_70></location>H ( t ) ≡ H ϕ ( t ) + H int ( t ) , (22)</formula>
<text><location><page_4><loc_9><loc_62><loc_49><loc_67></location>where H ϕ is the field Hamiltonian in Eq. (6) and H int is the Hamiltonian that describes the interaction between each qubit and the field which, in the interaction picture, is given by</text>
<formula><location><page_4><loc_10><loc_57><loc_49><loc_60></location>H I int ( t ) ≡ /summation.disp j ϵ j ( t ) /integral.disp Σ t d 3 x √ -g ψ j ( t, x ) ϕ ( t, x ) ⊗ σ z j , (23)</formula>
<text><location><page_4><loc_9><loc_41><loc_49><loc_56></location>where j ∈ { A,B,C } , with A , B , and C labeling Alice's, Bob's, and Charlie's qubit, respectively. Here, σ z j is one of the Pauli matrices /braceleft.alt1 σ x j , σ y j , σ z j /braceright.alt1 associated with qubit j ; ψ j ( t, x ) is a smooth real function satisfying ψ j /divides.alt0 Σ t ∈ C ∞ 0 ( Σ t ) for all t , which models the finite range of interaction between qubit j and the field (i.e., the interaction occurs only at some vicinity of each qubit worldline); and ϵ j ( t ) is a smooth and compactly-supported real coupling function modeling the finite-time coupling of qubit j with the field. Each coupling function has support</text>
<formula><location><page_4><loc_22><loc_38><loc_49><loc_40></location>supp ϵ j = /bracketleft.alt2 T i j , T f j /bracketright.alt2 , (24)</formula>
<text><location><page_4><loc_9><loc_29><loc_49><loc_37></location>where T i j and T f j represent the time (with respect to the parameter t ) in which each qubit interaction with the field is switched-on and -off, respectively. Here, we denote ∆ t j ≡ T f j -T i j . Thus, the hypotheses (QB1) and (QB2) previously listed can be summarized as</text>
<formula><location><page_4><loc_22><loc_26><loc_49><loc_27></location>T i C ≥ T f B ≥ T i B ≥ T f A . (25)</formula>
<text><location><page_4><loc_9><loc_9><loc_49><loc_24></location>The interaction between each qubit and the field given by Eq. (23) is very similar to the Unruh-DeWitt model [33]. However, we assumed that the two levels of each qubit have the same (zero) energy. As we shall see, this assumption allows us to calculate the evolution operator of the system and trace out the field degrees of freedom in a nonperturbative manner, thus making this model interesting to investigate both the causality in the information exchange process as well as the communication between parts lying in early and future asymptotic spacetime regions. We note that one could also have</text>
<text><location><page_4><loc_52><loc_88><loc_92><loc_93></location>given an energy gap 2 δ j for each qubit j in z -direction by adding H j = δ j σ z j to the total Hamiltonian in Eq. (22) and still keep the model exactly solvable. This would change it to</text>
<formula><location><page_4><loc_60><loc_85><loc_92><loc_86></location>H = H ϕ + H A + H B + H C + H int , (26)</formula>
<text><location><page_4><loc_52><loc_79><loc_92><loc_83></location>but would keep the interaction Hamiltonian in the interaction picture, Eq. (23), unchanged. Hence, all the results we will describe below would remain the same.</text>
<text><location><page_4><loc_52><loc_75><loc_92><loc_79></location>The interaction-picture time-evolution operator at late times, associated with the foliation Σ t , can be written as the time-ordered expression</text>
<formula><location><page_4><loc_61><loc_71><loc_92><loc_74></location>U ≡ T exp /bracketleft.alt3i /integral.disp ∞ -∞ dt H I int ( t )/bracketright.alt3 . (27)</formula>
<text><location><page_4><loc_52><loc_67><loc_92><loc_70></location>It can be computed nonperturbatively by using the Magnus expansion [35]</text>
<formula><location><page_4><loc_66><loc_62><loc_92><loc_66></location>U = exp /bracketleft.alt4 ∞ /summation.disp n = 1 Ω n /bracketright.alt4 , (28)</formula>
<text><location><page_4><loc_52><loc_60><loc_56><loc_61></location>where</text>
<formula><location><page_4><loc_64><loc_56><loc_92><loc_59></location>Ω 1 = -i /integral.disp ∞ -∞ dt H I int ( t ) , (29)</formula>
<formula><location><page_4><loc_55><loc_51><loc_92><loc_54></location>Ω 2 = -1 2 /integral.disp ∞ -∞ dt /integral.disp t -∞ dt ' [ H I int ( t ) , H I int ( t ' )] , (30)</formula>
<formula><location><page_4><loc_51><loc_44><loc_92><loc_49></location>Ω 3 = i 6 /integral.disp ∞ -∞ dt /integral.disp t -∞ dt ' /integral.disp t ' -∞ dt '' {[ H I int ( t ) , [ H I int ( t ' ) , H I int ( t '' )]] +[ H I int ( t '' ) , [ H I int ( t ' ) , H I int ( t )]]} , (31)</formula>
<text><location><page_4><loc_52><loc_41><loc_90><loc_43></location>and so on. By using Eqs. (18), (23), and (29), we get</text>
<formula><location><page_4><loc_64><loc_38><loc_92><loc_40></location>Ω 1 = -i /summation.disp j ϕ ( f j ) ⊗ σ z j , (32)</formula>
<text><location><page_4><loc_52><loc_35><loc_68><loc_36></location>where we have defined</text>
<formula><location><page_4><loc_64><loc_32><loc_92><loc_34></location>f j ( t, x ) ≡ ϵ j ( t ) ψ j ( t, x ) . (33)</formula>
<text><location><page_4><loc_52><loc_28><loc_92><loc_31></location>Now, by making use of Eqs. (18) and (23) together with Eqs. (16), (25), and (30) we can cast Ω 2 as</text>
<formula><location><page_4><loc_53><loc_21><loc_92><loc_27></location>Ω 2 = i Ξ I -i 2 ∆ ( f A , f B ) σ z A ⊗ σ z B -i 2 ∆ ( f A , f C ) σ z A ⊗ σ z C -i 2 ∆ ( f B , f C ) σ z B ⊗ σ z C , (34)</formula>
<text><location><page_4><loc_52><loc_19><loc_69><loc_20></location>where Ξ is the c-number</text>
<formula><location><page_4><loc_56><loc_15><loc_87><loc_18></location>Ξ ≡ 1 2 /summation.disp j /integral.disp ∞ -∞ dt ϵ j ( t ) /integral.disp t -∞ dt ' ϵ j ( t ' ) ∆ j ( t, t ' ) ,</formula>
<text><location><page_4><loc_52><loc_12><loc_55><loc_13></location>with</text>
<formula><location><page_4><loc_52><loc_8><loc_92><loc_11></location>∆ j ( t, t ' )≡ /integral.disp Σ t d 3 x √ -g /integral.disp Σ t ' d 3 x ' /radical.alt1 -g ' ψ j ( t, x ) ∆ ( x, x ' ) ψ j ( t ' , x ' ) ,</formula>
<text><location><page_5><loc_9><loc_89><loc_49><loc_93></location>and we recall that [ ϕ ( x ) , ϕ ( x ' )] ≡ -i ∆ ( x, x ' ) I is the unsmeared version of Eq. (16). Finally, since [ H I int ( t ) , H I int ( t ' )] is proportional to the identity, we get</text>
<formula><location><page_5><loc_23><loc_86><loc_49><loc_88></location>Ω k = 0 for k ≥ 3 . (35)</formula>
<text><location><page_5><loc_9><loc_84><loc_30><loc_85></location>Using the Zassenhaus formula</text>
<formula><location><page_5><loc_21><loc_82><loc_49><loc_83></location>e A + B = e A e B e -1 2 [ A,B ] , (36)</formula>
<text><location><page_5><loc_9><loc_76><loc_49><loc_80></location>valid whenever [ A,B ] is a proportional to the identity, together with Eqs. (28), (32), (34), and (35) we obtain the following unitary evolution operator:</text>
<formula><location><page_5><loc_12><loc_74><loc_49><loc_75></location>U = e i Ξ e -iϕ ( f C )⊗ σ z C e -iϕ ( f B )⊗ σ z B e -iϕ ( f A )⊗ σ z A . (37)</formula>
<text><location><page_5><loc_9><loc_60><loc_49><loc_72></location>Now that we have the exact evolution operator U , we can use it to evolve the initial state of the 3 qubit + field system and then trace out the field and Alice's qubit degrees of freedom. This procedure allows us to obtain the final state of Bob's and Charlie's qubits after the communication protocol has ended. This is the state that they will measure to recover the information that Alice has sent. Explicitly, the final Bob+Charlie state is given by</text>
<formula><location><page_5><loc_13><loc_57><loc_49><loc_59></location>ρ BC ≡ tr ϕ,A /parenleft.alt1 Uρ A -∞ ⊗ ρ B -∞ ⊗ ρ C -∞ ⊗ ρ ω U † /parenright.alt1 , (38)</formula>
<text><location><page_5><loc_9><loc_53><loc_49><loc_56></location>where ρ j -∞ and ρ ω are the initial states of qubit j and the field, respectively.</text>
<text><location><page_5><loc_9><loc_50><loc_49><loc_53></location>To compute the trace in Eq. (38), let us cast the operators in Eq. (37) as</text>
<formula><location><page_5><loc_12><loc_47><loc_49><loc_49></location>e -iϕ ( f j )⊗ σ z j = cos [ ϕ ( f j )] -i sin [ ϕ ( f j )] ⊗ σ z j , (39)</formula>
<text><location><page_5><loc_9><loc_45><loc_13><loc_46></location>where</text>
<formula><location><page_5><loc_16><loc_41><loc_49><loc_44></location>cos [ ϕ ( f j )] ≡ 1 2 [ W ( Ef j ) + W (-Ef j )] (40)</formula>
<text><location><page_5><loc_9><loc_39><loc_11><loc_40></location>and</text>
<formula><location><page_5><loc_15><loc_36><loc_49><loc_39></location>sin [ ϕ ( f j )] ≡ 1 2 i [ W ( Ef j ) -W (-Ef j )] , (41)</formula>
<text><location><page_5><loc_9><loc_31><loc_49><loc_35></location>where W ( Ef ) is defined in Eq. (21). By plugging Eqs. (37) and (39) into Eq. (38) and then taking the partial traces on ϕ and A , a direct calculation yields</text>
<formula><location><page_5><loc_9><loc_9><loc_49><loc_30></location>ρ BC = ( Γ cccccc + Γ sccccs ) ρ BC -∞ +( Γ csccsc + Γ ssccss ) σ z B ρ BC -∞ σ z B +( Γ ccsscc + Γ scsscs ) σ z C ρ BC -∞ σ z C +( Γ cssssc + Γ ssssss ) σ z B ⊗ σ z C ρ BC -∞ σ z B ⊗ σ z C +[( Γ ccscsc + Γ scscss ) σ z B ρ BC -∞ σ z C + h . c . ] -[( Γ cssccc + Γ sssccs ) ρ BC -∞ σ z B ⊗ σ z C + h . c . ] (42) +[( Γ cscccs -Γ sscccc )/uni27E8 σ z A /uni27E9 ρ A -∞ ρ BC -∞ σ z B + h . c . ] +[( Γ ccsccs -Γ scsccc )/uni27E8 σ z A /uni27E9 ρ A -∞ ρ BC -∞ σ z C + h . c . ] +[( Γ csscss -Γ ssscsc )/uni27E8 σ z A /uni27E9 ρ A -∞ σ z B ρ BC -∞ σ z B ⊗ σ z C + h . c . ] +[( Γ cssscs -Γ sssscc )/uni27E8 σ z A /uni27E9 ρ A -∞ σ z C ρ BC -∞ σ z B ⊗ σ z C + h . c . ] ,</formula>
<text><location><page_5><loc_52><loc_90><loc_92><loc_93></location>where h . c . stands for Hermitian conjugation, and we have defined</text>
<formula><location><page_5><loc_66><loc_88><loc_92><loc_89></location>ρ BC -∞ ≡ ρ B -∞ ⊗ ρ C -∞ , (43)</formula>
<formula><location><page_5><loc_64><loc_83><loc_92><loc_85></location>/uni27E8 σ z A /uni27E9 ρ A -∞ ≡ tr /parenleft.alt1 σ z A ρ A -∞ /parenright.alt1 , (44)</formula>
<text><location><page_5><loc_52><loc_81><loc_54><loc_82></location>and</text>
<formula><location><page_5><loc_53><loc_76><loc_92><loc_80></location>Γ αβγδϵζ ≡ ω µ /parenleft.alt1F α [ ϕ ( f A )]F β [ ϕ ( f B )]F γ [ ϕ ( f C )] ×F δ [ ϕ ( f C )]F ϵ [ ϕ ( f B )]F ζ [ ϕ ( f A )]/parenright.alt1 , (45)</formula>
<text><location><page_5><loc_52><loc_65><loc_92><loc_75></location>with α,β,γ,δ,ϵ,ζ ∈ { c, s } , F c ( x ) ≡ cos x, and F s ( x ) ≡ sin x . We note that we have written the algebraic field state ω µ as a density matrix with tr [ ρ ω W ( Ef )] ≡ ω µ [ W ( Ef )] . Furthermore, we have used the fact that the expected value of odd functions of the field operator vanishes since we are assuming that ω µ is a quasifree state (a consequence of Wick's theorem).</text>
<text><location><page_5><loc_52><loc_61><loc_92><loc_65></location>Now, each Γ αβγδϵζ in Eq. (42) can be evaluated by substituting Eqs. (40) and (41) in Eq. (45) and then using the identity</text>
<formula><location><page_5><loc_55><loc_58><loc_92><loc_59></location>W ( Ef 1 ) W ( Ef 2 ) = e i 2 ∆ ( f 1 ,f 2 ) W [ E ( f 1 + f 2 )] , (46)</formula>
<text><location><page_5><loc_52><loc_46><loc_92><loc_56></location>for all f, f 1 , f 2 ∈ C ∞ 0 (M) , to simplify the product of the Weyl operators. By substituting these coefficients in Eq.(42) one finds the explicit form of the state ρ BC , which is given in Eq. (A1) of Appendix A. The expression in Eq. (A1) allows one to write the final joint state for Bob's and Charlie's qubits given any initial state configuration for the 3 qubits+field.</text>
<text><location><page_5><loc_52><loc_35><loc_92><loc_46></location>To define a quantum broadcast channel, we must choose suitable initial states for Bob and Charlie qubits in order to obtain a quantum map relating the initial state of Alice's qubit ρ A -∞ (which encodes the messages) to the final states that will be probed by them (to decode the messages). Since Bob only performs measurements in his own two-level system, we calculate the expression for the reduced state of his qubit, i.e.,</text>
<formula><location><page_5><loc_66><loc_32><loc_92><loc_33></location>ρ B ≡ tr C /parenleft.alt1 ρ BC /parenright.alt1 . (47)</formula>
<text><location><page_5><loc_52><loc_28><loc_92><loc_31></location>Taking the trace in Eq. (A1) relative to Charlie's degrees of freedom, we obtain</text>
<formula><location><page_5><loc_56><loc_18><loc_92><loc_27></location>ρ B = 1 2 ( 1 + ν B cos [ 2∆ ( f A , f B )]) ρ B -∞ + 1 2 ( 1 -ν B cos [ 2∆ ( f A , f B )]) σ z B ρ B -∞ σ z B (48) + i 2 ν B sin [ 2∆ ( f A , f B )]/uni27E8 σ z A /uni27E9 ρ A -∞ /bracketleft.alt1 ρ B -∞ , σ z B /bracketright.alt ,</formula>
<text><location><page_5><loc_52><loc_15><loc_56><loc_17></location>where</text>
<formula><location><page_5><loc_55><loc_13><loc_92><loc_14></location>ν B ≡ ω µ ( W [ E ( 2 f B )]) = e -2 µ ( KEf B ,KEf B ) , (49)</formula>
<text><location><page_5><loc_52><loc_9><loc_92><loc_11></location>with µ be the inner product associated with the field quasifree state ω µ as in Eq. (20). Note that it is the last</text>
<text><location><page_6><loc_9><loc_76><loc_49><loc_93></location>term in Eq. (48) that contains the information encoded by Alice, and thus it will be useless for Bob to choose the eigenstates /divides.alt0 0 /uni27E9 B and /divides.alt0 1 /uni27E9 B of σ z B as his initial state ρ B -∞ since this term would vanish. Furthermore, since σ z B commutes with the interaction Hamiltonian, he won't recover any information either if he performs projective measurements on this basis. To choose a suitable state ρ B -∞ that maximizes the chances of success in their communication, suppose for simplicity that Alice encodes a pair of messages in states ρ A -∞+ and ρ A -∞-which will be decoded by Bob using a set of projective measurements in the x -direction,</text>
<formula><location><page_6><loc_17><loc_73><loc_49><loc_75></location>{ F B + ≡ /divides.alt0+/uni27E9 BB /uni27E8+/divides.alt0 , F B -≡ /divides.alt0-/uni27E9 BB /uni27E8-/divides.alt0} , (50)</formula>
<text><location><page_6><loc_9><loc_68><loc_49><loc_72></location>where σ x B /divides.alt0±/uni27E9 B = ±/divides.alt0±/uni27E9 B . From Eq. (48), we conclude that the probability that Bob measures l = ± given that Alice has encoded the message k = ± in ρ A -∞ k is</text>
<formula><location><page_6><loc_16><loc_64><loc_49><loc_67></location>p ( l /divides.alt0 k ) ≡ tr /parenleft.alt1 F B l ρ B k /parenright.alt1 = 1 2 ( 1 + lν B Λ k ) , (51)</formula>
<text><location><page_6><loc_9><loc_62><loc_13><loc_63></location>where</text>
<formula><location><page_6><loc_8><loc_59><loc_49><loc_60></location>Λ k ≡ 2 R { β B ( cos [ 2∆ ( f A , f B )]-i /uni27E8 σ z A /uni27E9 ρ A -∞ k sin [ 2∆ ( f A , f B )])}</formula>
<text><location><page_6><loc_9><loc_47><loc_49><loc_57></location>and β B ≡ B /uni27E8 0 /divides.alt0 ρ B -∞ /divides.alt0 1 /uni27E9 B . From these two equations, we see that it is the second term Λ k that contains the information encoded by Alice on her qubit state, and thus we are motivated to choose a state ρ B -∞ that makes β B a pure imaginary number, which will make the first term of Λ k vanish while maximizing the amplitude of the second term. This motivates us to choose</text>
<formula><location><page_6><loc_23><loc_45><loc_49><loc_46></location>ρ B -∞ ≡ /divides.alt0 y + /uni27E9 BB /uni27E8 y + /divides.alt0 , (52)</formula>
<text><location><page_6><loc_9><loc_42><loc_13><loc_43></location>where</text>
<formula><location><page_6><loc_20><loc_38><loc_49><loc_41></location>/divides.alt0 y + /uni27E9 B ≡ 1 √ 2 (/divides.alt0 0 /uni27E9 B + i /divides.alt0 1 /uni27E9 B ) (53)</formula>
<text><location><page_6><loc_9><loc_34><loc_49><loc_37></location>is an eigenstate of σ y B (in this case, β B = -i /slash.left 2). With this choice, we can write Eq. (51) as</text>
<formula><location><page_6><loc_12><loc_30><loc_49><loc_33></location>p ( l /divides.alt0 k ) = 1 2 ( 1 -lν B /uni27E8 σ z A /uni27E9 ρ A -∞ k sin [ 2∆ ( f A , f B )]) . (54)</formula>
<text><location><page_6><loc_9><loc_27><loc_49><loc_29></location>Now we turn our attention to Charlie. The final reduced state for his qubit is</text>
<formula><location><page_6><loc_23><loc_24><loc_49><loc_25></location>ρ C ≡ tr B /parenleft.alt1 ρ BC /parenright.alt1 . (55)</formula>
<text><location><page_6><loc_9><loc_20><loc_49><loc_23></location>Taking the trace in Eq. (A1) relative to Bob's degrees of freedom and using Eq. (52) we obtain</text>
<formula><location><page_6><loc_9><loc_9><loc_49><loc_19></location>ρ C = 1 2 ( 1 + ν C cos [ 2∆ ( f A , f C )] cos [ 2∆ ( f B , f C )]) ρ C -∞ + 1 2 ( 1 -ν C cos [ 2∆ ( f A , f C )] cos [ 2∆ ( f B , f C )]) σ z C ρ C -∞ σ z C + i 2 ν C sin [ 2∆ ( f A , f C )] cos [ 2∆ ( f B , f C )]/uni27E8 σ z A /uni27E9 ρ A -∞ [ ρ C -∞ , σ z C ] , (56)</formula>
<text><location><page_6><loc_52><loc_92><loc_56><loc_93></location>where</text>
<formula><location><page_6><loc_55><loc_89><loc_92><loc_91></location>ν C ≡ ω µ ( W [ E ( 2 f C )]) = e -2 µ ( KEf C ,KEf C ) . (57)</formula>
<text><location><page_6><loc_52><loc_81><loc_92><loc_88></location>To obtain Eq. (56), we explicitly used the choice in Eq. (52), which implies that /uni27E8 σ z B /uni27E9 ρ B -∞ ≡ tr /parenleft.alt1 σ z B ρ B -∞ /parenright.alt1 = 0. By a completely similar reasoning as the one used to choose Bob's initial state, we are motivated to choose Charlie's initial qubit state as</text>
<formula><location><page_6><loc_66><loc_79><loc_92><loc_80></location>ρ C -∞ ≡ /divides.alt0 y + /uni27E9 CC /uni27E8 y + /divides.alt0 , (58)</formula>
<text><location><page_6><loc_52><loc_76><loc_68><loc_78></location>where σ y C /divides.alt0 y + /uni27E9 C = /divides.alt0 y + /uni27E9 C .</text>
<text><location><page_6><loc_52><loc_71><loc_92><loc_76></location>Now, the quantum broadcast channel is completely characterized by a linear, completely positive and tracepreserving (CPTP) quantum map E which takes ρ A -∞ into a final state ρ BC , i.e.,</text>
<formula><location><page_6><loc_67><loc_68><loc_92><loc_70></location>ρ BC = E( ρ A -∞ ) . (59)</formula>
<text><location><page_6><loc_52><loc_60><loc_92><loc_67></location>By substituting the initial states of Bob's and Charlie's qubits given in Eqs. (52) and (58) into Eq. (A1), we find the explicit expression for the quantum broadcast channel E . For the sake of clarity, due to its lengthy expression, we write its explicit form in Eq. (A7) of Appendix A.</text>
<text><location><page_6><loc_52><loc_57><loc_92><loc_60></location>For later use, we will denote the reduced channels E B ∶ A → B , E C ∶ A → C by</text>
<formula><location><page_6><loc_63><loc_55><loc_92><loc_56></location>E B ( ρ A -∞ ) ≡ tr C /bracketleft.alt1E ( ρ A -∞ )/bracketright.alt , (60)</formula>
<formula><location><page_6><loc_63><loc_53><loc_92><loc_54></location>E C ( ρ A -∞ ) ≡ tr B /bracketleft.alt1E ( ρ A -∞ )/bracketright.alt , (61)</formula>
<text><location><page_6><loc_52><loc_49><loc_92><loc_52></location>respectively. It then follows from Eqs. (A7), (60), and (61) that they can be explicitly written as</text>
<formula><location><page_6><loc_55><loc_42><loc_92><loc_48></location>E B ( ρ A -∞ ) = 1 2 I B + ν B 2 cos [ 2∆ ( f A , f B )] σ y B -ν B 2 sin [ 2∆ ( f A , f B )]/uni27E8 σ z A /uni27E9 ρ A -∞ σ x B (62)</formula>
<text><location><page_6><loc_52><loc_40><loc_54><loc_41></location>and</text>
<formula><location><page_6><loc_53><loc_31><loc_92><loc_40></location>E C ( ρ A -∞ ) = 1 2 I C + ν C 2 cos [ 2∆ ( f A , f C )] cos [ 2∆ ( f B , f C )] σ y C (63) -ν C 2 sin [ 2∆ ( f A , f C )] cos [ 2∆ ( f B , f C )]/uni27E8 σ z A /uni27E9 ρ A -∞ σ x C .</formula>
<text><location><page_6><loc_52><loc_26><loc_92><loc_30></location>Given an initial state ρ A -∞ prepared by Alice on her qubit, these expressions for E B and E C determine the final local states of Bob's and Charlie's qubit, respectively.</text>
<section_header_level_1><location><page_6><loc_52><loc_22><loc_91><loc_23></location>IV. ACHIEVABLE COMMUNICATION RATES</section_header_level_1>
<text><location><page_6><loc_52><loc_9><loc_92><loc_20></location>Now that we have constructed a model for a relativistic quantum broadcast channel, we can investigate at which rates classical and quantum information can be reliably transmitted by Alice to Bob and Charlie. We first review a few protocols for quantum broadcast communication published in the literature and then we investigate the achievable rates for our quantum broadcast channel E defined in Eq. (59).</text>
<section_header_level_1><location><page_7><loc_14><loc_92><loc_44><loc_93></location>A. Unassisted classical communication</section_header_level_1>
<text><location><page_7><loc_9><loc_83><loc_49><loc_90></location>Let us begin with the investigation of unassisted transmission of classical information. We follow the protocol presented in [3], where more details can be found. We evaluate achievable rates for our model and then we discuss how causality is explicitly manifest in our results.</text>
<text><location><page_7><loc_9><loc_76><loc_49><loc_83></location>Suppose Alice wishes to transmit a common message m ∈ M intended for both receivers while sending additional personal messages m B ∈ M B and m C ∈ M C intended for Bob and Charlie, respectively. Each message is chosen from one of the following sets,</text>
<formula><location><page_7><loc_16><loc_73><loc_49><loc_74></location>M = { 1 , /uni22EF , /divides.alt0 M /divides.alt0} , M j = { 1 , /uni22EF , /divides.alt0 M j /divides.alt0} , (64)</formula>
<text><location><page_7><loc_9><loc_55><loc_49><loc_71></location>with j ∈ { B,C } and /divides.alt0 M /divides.alt0 denoting the cardinality of M . Since the broadcast channel E is noisy, Alice needs to do a suitable block coding on the possible messages and then make n independent uses of the channel in order to be able to reliably convey the information. More precisely, Alice maps each message triple ( m B , m, m C ) to a codeword x n ( m B , m, m C ) which is then associated with a quantum state ρ A n x n ( m B ,m,m C ) defined in the space H ⊗ n A . Then, she transmits ρ A n x n ( m B ,m,m C ) by making n independent uses of the channel E . The output of the channel is the state</text>
<formula><location><page_7><loc_16><loc_52><loc_49><loc_54></location>ρ B n C n x n ( m B ,m,m C ) ≡ E ⊗ n /parenleft.alt2 ρ A n x n ( m B ,m,m C ) /parenright.alt2 (65)</formula>
<text><location><page_7><loc_9><loc_38><loc_49><loc_51></location>defined on H ⊗ n B ⊗ H ⊗ n C . To decode the message, Bob chooses a positive-operator valued measure (POVM) { F B n m B ,m /divides.alt0 ( m B , m ) ∈ M B × M } which acts on the system B n . Similarly, Charlie chooses a POVM { G C n m,m C /divides.alt0 ( m,m C ) ∈ M × M C } which acts on the system C n . We say that an error has occurred when at least one message is incorrectly decoded. Hence, the error probability associated with the transmission of the triple ( m B , m, m C ) is</text>
<formula><location><page_7><loc_9><loc_34><loc_49><loc_36></location>p e ( m B , m, m C ) ≡ 1 -tr /bracketleft.alt2( F B n m B ,m ⊗ G C n m,m C ) ρ B n C n x n ( m B ,m,m C ) /bracketright.alt2 .</formula>
<text><location><page_7><loc_9><loc_30><loc_49><loc_33></location>The transmission rates associated with each message are defined as</text>
<formula><location><page_7><loc_17><loc_27><loc_49><loc_29></location>R ≡ 1 n log 2 /divides.alt0 M /divides.alt0 , R j ≡ 1 n log 2 /divides.alt0 M j /divides.alt0 . (66)</formula>
<text><location><page_7><loc_9><loc_21><loc_49><loc_25></location>These rates essentially measure how many bits of classical information are sent per channel use. If, given an ϵ > 0, the average probability of error p e is bounded by ϵ , i.e.,</text>
<formula><location><page_7><loc_11><loc_17><loc_49><loc_20></location>p e ≡ 1 /divides.alt0 M B /divides.alt0/divides.alt0 M /divides.alt0/divides.alt0 M C /divides.alt0 /summation.disp m B ,m,m C p e ( m B , m, m C ) ≤ ϵ, (67)</formula>
<text><location><page_7><loc_9><loc_9><loc_49><loc_16></location>the classical-quantum broadcast channel coding protocol described above is said to be a ( n, R B , R, R C , ϵ ) code. We say that a rate triple ( R B , R, R C ) is achievable if given ϵ, δ > 0 there exists a ( n, R B -δ, R -δ, R C -δ, ϵ ) code for sufficiently large n . Hence, saying that a rate</text>
<text><location><page_7><loc_52><loc_90><loc_92><loc_93></location>triple is achievable means that classical information can be reliably transmitted at rates arbitrarily close to them.</text>
<text><location><page_7><loc_52><loc_82><loc_92><loc_90></location>The achievable rates depend highly on the coding and decoding techniques chosen by the sender and receivers. The best known achievable rate region for general broadcast channels is attained through the so-called Marton coding scheme . Following [3], we investigate here the quantum version of this protocol.</text>
<text><location><page_7><loc_52><loc_60><loc_92><loc_81></location>Suppose for simplicity that no common message is meant to be sent, i.e., let us consider a ( R B , 0 , R C ) quantum broadcast channel. In this scenario, one strategy they can use is the Marton coding scheme , where one chooses two correlated random variables U and V , with joint probability distribution denoted by p and reduced probability distributions denoted by p U and p V . Such a pair of random variables is usually referred to as binning variables . Then, for each m B ∈ M B and m C ∈ M C , one generates codewords u n ( m B ) and v n ( m C ) according to the reduced probability distributions p U ( u ) and p V ( v ) . Next, the codewords are mixed together into a single codeword x n ( m B , m C ) according to a deterministic function x = f ( u, v ) . With this approach, it follows that a rate pair ( R B , R C ) is achievable if it satisfies [3]</text>
<formula><location><page_7><loc_56><loc_57><loc_92><loc_58></location>0 ≤ R B ≤ I ( U ; B ) σ , (68)</formula>
<formula><location><page_7><loc_56><loc_55><loc_92><loc_56></location>0 ≤ R C ≤ I ( V ; C ) σ , (69)</formula>
<formula><location><page_7><loc_55><loc_53><loc_92><loc_55></location>R B + R C ≤ I ( U ; B ) σ + I ( V ; C ) σ -I ( U,V ) σ , (70)</formula>
<text><location><page_7><loc_52><loc_51><loc_56><loc_52></location>where</text>
<formula><location><page_7><loc_58><loc_48><loc_92><loc_49></location>I ( X ; Y ) ρ ≡ S ( X ) ρ + S ( Y ) ρ -S ( XY ) ρ (71)</formula>
<text><location><page_7><loc_52><loc_45><loc_85><loc_46></location>is the mutual information of a state ρ XY , with</text>
<formula><location><page_7><loc_63><loc_42><loc_80><loc_43></location>S ( α ) ρ = -tr ( ρ α log ρ α ) ,</formula>
<text><location><page_7><loc_52><loc_35><loc_92><loc_41></location>α = X,Y , being the von Neumann entropy of ρ α , α = X,Y . Here, ρ X = tr Y ρ XY and ρ Y = tr X ρ XY . The states σ in Eqs. (68)-(70) are obtained by suitably (partially) tracing out the degrees of freedom of the density matrix</text>
<formula><location><page_7><loc_53><loc_31><loc_92><loc_33></location>σ UVBC ≡ /summation.disp u,v p ( u, v )/divides.alt0 u /uni27E9/uni27E8 u /divides.alt0 U ⊗/divides.alt0 v /uni27E9/uni27E8 v /divides.alt0 V ⊗E /parenleft.alt1 ρ A f ( u,v ) /parenright.alt1 , (72)</formula>
<text><location><page_7><loc_52><loc_27><loc_92><loc_29></location>with p ( u, v ) being the joint probability distribution of the random variables U and V .</text>
<text><location><page_7><loc_52><loc_19><loc_92><loc_26></location>We begin our analysis by deriving bounds for the achievable rates through the Marton coding scheme applied to our relativistic quantum broadcast channel. To evaluate Eq. (68), we take partial traces relative to V and C in Eq. (72), obtaining</text>
<formula><location><page_7><loc_62><loc_15><loc_92><loc_18></location>σ UB ≡ /summation.disp u p U ( u )/divides.alt0 u /uni27E9/uni27E8 u /divides.alt0 U ⊗ ω B u , (73)</formula>
<text><location><page_7><loc_52><loc_12><loc_92><loc_14></location>where we have written p ( u, v ) = p V /divides.alt0 U ( v /divides.alt0 u ) p U ( u ) , whereas</text>
<formula><location><page_7><loc_61><loc_8><loc_92><loc_11></location>ω B u ≡ /summation.disp v p V /divides.alt0 U ( v /divides.alt0 u )E B /parenleft.alt1 ρ A f ( u,v ) /parenright.alt1 . (74)</formula>
<text><location><page_8><loc_9><loc_89><loc_49><loc_93></location>A state like σ UB in Eq. (73) is called a classical-quantum state . For this class of states, a straightforward calculation shows that [2]</text>
<formula><location><page_8><loc_11><loc_84><loc_49><loc_87></location>I ( U ; B ) σ = S /bracketleft.alt4/summation.disp u p U ( u ) ω B u /bracketright.alt4 -/summation.disp u p U ( u ) S /bracketleft.alt1 ω B u /bracketright.alt . (75)</formula>
<text><location><page_8><loc_9><loc_79><loc_49><loc_83></location>In order to compute ω B u and its von Neumann entropy, let us decompose the initial state of Alice's qubit in terms of Bloch vectors, i.e.,</text>
<formula><location><page_8><loc_18><loc_74><loc_49><loc_77></location>ρ A f ( u,v ) = 1 2 /parenleft.alt1 I A + r f ( u,v ) · σ A /parenright.alt1 , (76)</formula>
<text><location><page_8><loc_9><loc_67><loc_49><loc_73></location>where r f ( u,v ) ≡ ( x f ( u,v ) , y f ( u,v ) , z f ( u,v ) ) , I A is the identity in H A , σ A ≡ ( σ x A , σ y A , σ z A ) , and /parallel.alt1 r f ( u,v ) /parallel.alt1 2 = x 2 f ( u,v ) + y 2 f ( u,v ) + z 2 f ( u,v ) ≤ 1. From Eqs. (62), (74), and (76) we get</text>
<formula><location><page_8><loc_16><loc_60><loc_49><loc_66></location>ω B u = 1 2 I B + ν B 2 cos [ 2∆ ( f A , f B )] σ y B -z u ν B 2 sin [ 2∆ ( f A , f B )] σ x B , (77)</formula>
<text><location><page_8><loc_9><loc_56><loc_49><loc_59></location>where z u ≡ ∑ v p V /divides.alt0 U ( v /divides.alt0 u ) z f ( u,v ) , and thus we can further write</text>
<formula><location><page_8><loc_11><loc_49><loc_49><loc_55></location>ω B ≡ /summation.disp u p U ( u ) ω B u = 1 2 I B + ν B 2 cos [ 2∆ ( f A , f B )] σ y B -z ν B 2 sin [ 2∆ ( f A , f B )] σ x B , (78)</formula>
<text><location><page_8><loc_9><loc_46><loc_29><loc_48></location>where z ≡ ∑ u,v p ( u, v ) z f ( u,v ) .</text>
<text><location><page_8><loc_9><loc_43><loc_49><loc_46></location>Now, by using standard diagonalization, we find that ω B u has eigenvalues p B u and 1 -p B u , where</text>
<formula><location><page_8><loc_9><loc_38><loc_49><loc_42></location>p B u ≡ 1 2 + ν B 2 /radical.alt2 z 2 u sin 2 [ 2∆ ( f A , f B )] + cos 2 [ 2∆ ( f A , f B )] , (79)</formula>
<text><location><page_8><loc_9><loc_37><loc_43><loc_38></location>whereas ω B has eigenvalues p B and 1 -p B , with</text>
<formula><location><page_8><loc_9><loc_31><loc_49><loc_36></location>p B ≡ 1 2 + ν B 2 /radical.alt2 z 2 sin 2 [ 2∆ ( f A , f B )] + cos 2 [ 2∆ ( f A , f B )] . (80)</formula>
<text><location><page_8><loc_9><loc_30><loc_37><loc_31></location>Therefore, we can now write Eq. (75) as</text>
<formula><location><page_8><loc_15><loc_26><loc_49><loc_28></location>I ( U ; B ) σ = H /parenleft.alt1 p B /parenright.alt1 -/summation.disp u p U ( u ) H /parenleft.alt1 p B u /parenright.alt1 , (81)</formula>
<text><location><page_8><loc_9><loc_22><loc_49><loc_24></location>where H ( x ) ≡ -x log 2 x - ( 1 -x ) log 2 ( 1 -x ) , x ∈ [ 0 , 1 ] . Following similar steps, we can show that</text>
<formula><location><page_8><loc_15><loc_18><loc_49><loc_20></location>I ( V ; C ) σ = H /parenleft.alt1 p C /parenright.alt1 -/summation.disp v p V ( v ) H /parenleft.alt1 p C v /parenright.alt1 , (82)</formula>
<text><location><page_8><loc_9><loc_15><loc_13><loc_16></location>where</text>
<formula><location><page_8><loc_11><loc_9><loc_49><loc_14></location>p C v ≡ 1 2 + ν C 2 /divides.alt0 cos [ 2∆ ( f B , f C )]/divides.alt0 (83) × /radical.alt2 z 2 v sin 2 [ 2∆ ( f A , f C )] + cos 2 [ 2∆ ( f A , f C )]</formula>
<text><location><page_8><loc_52><loc_92><loc_76><loc_93></location>with z v ≡ ∑ u p U /divides.alt0 V ( u /divides.alt0 v ) z f ( u,v ) , and</text>
<formula><location><page_8><loc_53><loc_85><loc_92><loc_90></location>p C ≡ 1 2 + ν C 2 /divides.alt0 cos [ 2∆ ( f B , f C )]/divides.alt0 (84) × /radical.alt2 z 2 sin 2 [ 2∆ ( f A , f C )] + cos 2 [ 2∆ ( f A , f C )] .</formula>
<text><location><page_8><loc_52><loc_80><loc_92><loc_84></location>Now, let us note that H ( x ) is a monotonically decreasing function when x ≥ 1 /slash.left 2. From Eqs. (79) and (80), we have</text>
<formula><location><page_8><loc_67><loc_76><loc_92><loc_79></location>p B u ≤ 1 2 + ν B 2 (85)</formula>
<text><location><page_8><loc_52><loc_73><loc_54><loc_74></location>and</text>
<formula><location><page_8><loc_61><loc_69><loc_92><loc_72></location>p B ≥ 1 2 + ν B 2 /divides.alt0 cos [ 2∆ ( f A , f B )]/divides.alt0 , (86)</formula>
<text><location><page_8><loc_52><loc_67><loc_69><loc_68></location>and thus it follows that</text>
<formula><location><page_8><loc_64><loc_63><loc_92><loc_66></location>H ( p B u ) ≥ H /parenleft.alt3 1 2 + ν B 2 /parenright.alt3 (87)</formula>
<text><location><page_8><loc_52><loc_61><loc_54><loc_62></location>and</text>
<formula><location><page_8><loc_56><loc_57><loc_92><loc_60></location>H ( p B ) ≤ H /parenleft.alt3 1 2 + ν B 2 /divides.alt0 cos [ 2∆ ( f A , f B )]/divides.alt0/parenright.alt3 . (88)</formula>
<text><location><page_8><loc_53><loc_54><loc_85><loc_56></location>As a result, from Eq. (81), we conclude that</text>
<formula><location><page_8><loc_65><loc_52><loc_92><loc_53></location>I ( U ; B ) σ ≤ C(E B ) , (89)</formula>
<text><location><page_8><loc_52><loc_49><loc_56><loc_50></location>where</text>
<formula><location><page_8><loc_53><loc_44><loc_92><loc_48></location>C(E B ) ≡ H /parenleft.alt3 1 2 + ν B 2 /divides.alt0 cos [ 2∆ ( f A , f B )]/divides.alt0/parenright.alt3 -H /parenleft.alt3 1 2 + ν B 2 /parenright.alt3 (90)</formula>
<text><location><page_8><loc_52><loc_31><loc_92><loc_44></location>is the classical capacity of the reduced channel E B , given in Eq. (62), as shown in [30]. We note that the upper bound in Eq. (89) can be attained if we choose random variables U,V = { 0 , 1 } with p ( u, v ) = 1 /slash.left 4 for all u, v , associated with Bloch vectors r f ( 0 , 0 ) = r f ( 0 , 1 ) = ( 0 , 0 , + 1 ) and r f ( 1 , 0 ) = r f ( 1 , 1 ) = ( 0 , 0 , -1 ) . By using such choices together with Eq. (68), we conclude that Alice can reliably convey classical information to Bob at rates arbitrarily close to C(E B ) .</text>
<text><location><page_8><loc_53><loc_29><loc_87><loc_30></location>Similarly, we can show from Eqs. (82)-(84) that</text>
<formula><location><page_8><loc_65><loc_27><loc_92><loc_28></location>I ( V ; C ) σ ≤ C(E C ) , (91)</formula>
<text><location><page_8><loc_52><loc_24><loc_56><loc_25></location>where</text>
<formula><location><page_8><loc_52><loc_17><loc_92><loc_23></location>C(E C ) ≡ H /parenleft.alt3 1 2 + ν C 2 /divides.alt0 cos [ 2∆ ( f B , f C )] cos [ 2∆ ( f A , f C )]/divides.alt0/parenright.alt3 -H /parenleft.alt3 1 2 + ν C 2 /divides.alt0 cos [ 2∆ ( f B , f C )]/divides.alt0/parenright.alt3 , (92)</formula>
<text><location><page_8><loc_52><loc_8><loc_92><loc_16></location>is the classical capacity of the reduced channel E C given in Eq. (63). The upper bound can be attained, e.g., if we choose random variables U,V = { 0 , 1 } with p ( u, v ) = 1 /slash.left 4 for all u, v , associated with Bloch vectors r f ( 0 , 0 ) = r f ( 1 , 0 ) = ( 0 , 0 , + 1 ) and r f ( 0 , 1 ) = r f ( 1 , 1 ) = ( 0 , 0 , -1 ) .</text>
<text><location><page_9><loc_9><loc_89><loc_49><loc_93></location>Hence, from Eq. (69), we conclude that Alice can reliably convey classical information to Charlie as well at rates arbitrarily close to C(E C ) .</text>
<text><location><page_9><loc_9><loc_66><loc_49><loc_89></location>It is important to highlight that causality is explicitly manifest on the bounds of the achievable rates. First, we note that the achievable rates R B between Alice and Bob are bounded by C(E B ) , which does not depend on the interaction between Charlie's qubit and the quantum field. This should indeed be the case as, from hypothesis (QB2) in Sec. III, Charlie cannot influence the communication between Alice and Bob since he does not perform any actions before Bob finishes his measurement process. Furthermore, the presence of the commutator ∆ ( f B , f C ) in Eq. (92) indicates that when Bob and Charlie let their qubits interact with the quantum field in causally connected regions of the spacetime, noise from Bob's actions can influence on the rate R C of communication between Alice and Charlie. Additionally, we note that whenever ∆ ( f A , f j ) = 0, we have</text>
<formula><location><page_9><loc_26><loc_63><loc_49><loc_64></location>C(E j ) = 0 (93)</formula>
<text><location><page_9><loc_9><loc_56><loc_49><loc_62></location>for j = B,C . Hence, when Alice and Bob (or Charlie) interact with the field in causally disconnected regions of the spacetime, the achievable rate in Eq. (68) (or Eq. (69)) will reduce to R B = 0 (or R C = 0).</text>
<text><location><page_9><loc_9><loc_43><loc_49><loc_56></location>To this day, no one has been able to prove that the Marton rate region given by Eqs. (68)-(70) is optimal for general broadcast channels, not even in the classical case. However, it is generally conjectured that the Marton rate region indeed represents the full capacity region of general broadcast channels. If this is the case, then our analysis shows that causality will not be violated when transmitting classical information, no matter which communication protocol is chosen.</text>
<section_header_level_1><location><page_9><loc_9><loc_38><loc_49><loc_40></location>B. Unassisted and entanglement-assisted quantum communication</section_header_level_1>
<text><location><page_9><loc_9><loc_17><loc_49><loc_36></location>Let us now turn our attention to the communication of quantum information. Following [5], we present a father protocol for entanglement-assisted quantum communication through quantum broadcast channels that can be used to investigate at which rates Alice can send classical or quantum information to Bob and Charlie when they share an unlimited supply of entanglement. This protocol can also be adapted to investigate communication rates for quantum information transmission with no prior shared entanglement. After reviewing the father protocol, we investigate both quantum communication scenarios applied to our model of quantum broadcast channel constructed in Sec. III.</text>
<text><location><page_9><loc_9><loc_9><loc_49><loc_17></location>Let us suppose that Alice has access to two quantum systems T A and T A ' while Bob and Charlie possess similar quantum systems T B and T C , respectively. All systems possesses the same dimension d T A ≡ dim H T A . Suppose further that Alice shares maximally entangled states with both Bob and Charlie:</text>
<formula><location><page_9><loc_60><loc_87><loc_92><loc_91></location>/divides.alt1 Φ T A T k /uni27E9.alt1 = 1 /radical.alt1 d T A d T A -1 /summation.disp i = 0 /divides.alt0 i /uni27E9 T A ⊗/divides.alt0 i /uni27E9 T k , (94)</formula>
<text><location><page_9><loc_52><loc_82><loc_92><loc_86></location>where the above state is defined on H T A ⊗ H T k , with k = B,C and {/divides.alt0 i /uni27E9 T α } is an orthonormal set of vectors on H T α , α = A,B,C .</text>
<text><location><page_9><loc_52><loc_67><loc_92><loc_81></location>In order to study the transmission of quantum information, we first note that whenever Alice is able to transmit the entanglement she shares with some reference system to each receiver, she will be able to send arbitrary quantum states to each of them. Hence, suppose that Alice possesses two quantum systems A 1 and A 2 respectively entangled with reference systems R 1 and R 2 and that these systems are in states /divides.alt0 Φ A j R j /uni27E9 defined on H A j ⊗H R j for j = 1 , 2 5 . Her goal is to send her share of /divides.alt0 Φ A 1 R 1 /uni27E9 and /divides.alt0 Φ A 2 R 2 /uni27E9 to Bob and Charlie, respectively.</text>
<text><location><page_9><loc_53><loc_66><loc_81><loc_67></location>The initial global state of the system is</text>
<formula><location><page_9><loc_58><loc_63><loc_92><loc_64></location>/divides.alt0 φ /uni27E9 ≡ /divides.alt1 Φ A 1 R 1 /uni27E9.alt1 /divides.alt1 Φ A 2 R 2 /uni27E9.alt1 /divides.alt1 Φ T A T B /uni27E9.alt1 /divides.alt1 Φ T A ' T C /uni27E9.alt1 (95)</formula>
<text><location><page_9><loc_52><loc_52><loc_92><loc_61></location>and we will denote ρ φ ≡ /divides.alt0 φ /uni27E9 /uni27E8 φ /divides.alt0 . In order to use the quantum channel E to share her entanglement with R 1 and R 2 to Bob and Charlie (and hence, convey quantum information), Alice uses a CPTP map C ∶ H A 1 ⊗H A 2 ⊗H T A ⊗H T A ' →H ⊗ n A in order to encode her shares of the quantum systems ( T A , T A ' , A 1 , and A 2 ), into a state of n qubits. The global state will then reads</text>
<formula><location><page_9><loc_58><loc_49><loc_92><loc_50></location>˜ ρ A n R 1 R 2 T B T C ≡ /parenleft.alt1C ⊗ I R 1 R 2 T B T C /parenright.alt1 ( ρ φ ) , (96)</formula>
<text><location><page_9><loc_52><loc_42><loc_92><loc_48></location>where I R 1 R 2 T B T C is the identity operator of the joint system R 1 R 2 T B T C . Next, by making n independent uses of the channel E , Alice sends her total encoded state to Bob and Charlie, which results in the global state</text>
<formula><location><page_9><loc_53><loc_38><loc_92><loc_41></location>ω B n C n R 1 R 2 T B T C ≡ /parenleft.alt1E ⊗ n ⊗ I R 1 R 2 T B T C /parenright.alt1 /parenleft.alt1 ˜ ρ A n R 1 R 2 T B T C /parenright.alt1 . (97)</formula>
<text><location><page_9><loc_52><loc_32><loc_92><loc_37></location>Bob and Charlie decode their share of the global state by using the CPTP maps D B ∶ H ⊗ n B ⊗H T B → H B ' and D C ∶ H ⊗ n C ⊗H T C → H C ' , respectively. Hence, the final global state is</text>
<formula><location><page_9><loc_54><loc_28><loc_92><loc_31></location>ζ B ' C ' R 1 R 2 ≡ /parenleft.alt1D C ⊗D B ⊗ I R 1 R 2 /parenright.alt1 /parenleft.alt1 ω B n C n R 1 R 2 T B T C /parenright.alt1 . (98)</formula>
<text><location><page_9><loc_52><loc_25><loc_92><loc_27></location>We define the entanglement-assisted quantum communication rates as</text>
<formula><location><page_9><loc_59><loc_21><loc_92><loc_24></location>̃ Q B ≡ 1 n log 2 d A 1 , ̃ Q C ≡ 1 n log 2 d A 2 , (99)</formula>
<text><location><page_9><loc_52><loc_16><loc_92><loc_20></location>where d A j ≡ dim H A j and j = 1 , 2. These rates of quantum communication measure how many qubits are being sent per channel use.</text>
<text><location><page_10><loc_9><loc_90><loc_49><loc_93></location>The communication process will be good if given a small ϵ > 0 we have</text>
<formula><location><page_10><loc_18><loc_87><loc_49><loc_89></location>/parallel.alt3 ζ B ' C ' R 1 R 2 -ρ B ' C ' R 1 R 2 φ /parallel.alt3 1 ≤ ϵ, (100)</formula>
<text><location><page_10><loc_9><loc_85><loc_13><loc_86></location>where</text>
<formula><location><page_10><loc_22><loc_82><loc_49><loc_85></location>/parallel.alt1O /parallel.alt1 1 ≡ tr /parenleft.alt2 √ O † O/parenright.alt2 (101)</formula>
<text><location><page_10><loc_9><loc_75><loc_49><loc_81></location>is the trace norm of an operator O . Here, ρ B ' C ' R 1 R 2 φ is the analogous of the initial state in the composite system B ' C ' R 1 R 2 , i.e., given the initial state in Alice's laboratory</text>
<formula><location><page_10><loc_10><loc_73><loc_49><loc_74></location>ρ A 1 A 2 R 1 R 2 φ ≡ /divides.alt1 Φ A 1 R 1 /uni27E9.alt1 /uni27E8.alt Φ A 1 R 1 /divides.alt1 ⊗/divides.alt1 Φ A 2 R 2 /uni27E9.alt1 /uni27E8.alt Φ A 2 R 2 /divides.alt1 , (102)</formula>
<text><location><page_10><loc_9><loc_70><loc_15><loc_71></location>we define</text>
<formula><location><page_10><loc_11><loc_68><loc_49><loc_69></location>ρ B ' C ' R 1 R 2 φ ≡ /parenleft.alt2I A 1 → B ' ⊗I A 2 → C ' /parenright.alt2 /parenleft.alt1 ρ A 1 A 2 R 1 R 2 φ /parenright.alt1 , (103)</formula>
<text><location><page_10><loc_9><loc_63><loc_49><loc_66></location>where I A 1 → B ' (or I A 2 → C ' ) is the identity map between the quantum systems A 1 (or A 2 ) and B ' (or C ' ).</text>
<text><location><page_10><loc_9><loc_55><loc_49><loc_63></location>The communication protocol described here is named as a ( n, ̃ Q B , ̃ Q C , ϵ ) code if it satisfies Eq. (100) for every input state ρ A 1 A 2 R 1 R 2 φ . Again, we say that a rate pair ( ̃ Q B , ̃ Q C ) is achievable if given any ϵ, δ > 0 there exists a ( n, ̃ Q B -δ, ̃ Q C -δ, ϵ ) code for sufficiently large n .</text>
<text><location><page_10><loc_9><loc_50><loc_49><loc_55></location>Now, given a general broadcast channel E ∶ A → BC and an arbitrary mixed state ρ AA 1 A 2 defined on H A ⊗ H A 1 ⊗ H A 2 , it can be shown [5] that a entanglementassisted quantum rate pair ( ̃ Q B , ̃ Q C ) is achievable if</text>
<formula><location><page_10><loc_12><loc_46><loc_49><loc_48></location>0 ≤ ̃ Q B ≤ 1 2 I ( A 1 ; B ) σ , (104)</formula>
<formula><location><page_10><loc_12><loc_43><loc_28><loc_45></location>0 ≤ ̃ Q C ≤ 1 I ( A 2 ; C ) σ ,</formula>
<formula><location><page_10><loc_19><loc_43><loc_49><loc_45></location>2 (105)</formula>
<formula><location><page_10><loc_11><loc_38><loc_49><loc_42></location>̃ Q B + ̃ Q C ≤ 1 2 [ I ( A 1 ; B ) σ + I ( A 2 ; C ) σ -I ( A 1 ; A 2 ) σ ] , (106)</formula>
<text><location><page_10><loc_9><loc_34><loc_49><loc_37></location>where the mutual information quantities are evaluated relative to the state</text>
<formula><location><page_10><loc_17><loc_32><loc_49><loc_33></location>σ A 1 A 2 BC ≡ /parenleft.alt1E ⊗ I A 1 A 2 /parenright.alt1 /parenleft.alt1 ρ AA 1 A 2 /parenright.alt1 . (107)</formula>
<text><location><page_10><loc_9><loc_18><loc_49><loc_31></location>In addition to entanglement-assisted quantum communication, the father protocol presented here can be adapted to obtain achievable rates for unassisted quantum communication. To this end, we simply ignore the existence of the quantum systems T A , T A ' , T B , and T C and follow the exact same procedure. As shown in [5], given an arbitrary mixed state ρ AA 1 A 2 defined on H A ⊗H A 1 ⊗H A 2 , it follows that the following unassisted quantum rate region is achievable:</text>
<formula><location><page_10><loc_22><loc_15><loc_49><loc_17></location>0 ≤ Q B ≤ I ( A 1 /uni27E9 B ) σ , (108)</formula>
<formula><location><page_10><loc_22><loc_13><loc_49><loc_15></location>0 ≤ Q C ≤ I ( A 2 /uni27E9 C ) σ , (109)</formula>
<text><location><page_10><loc_9><loc_11><loc_33><loc_12></location>where σ is given by Eq. (107) and</text>
<formula><location><page_10><loc_19><loc_9><loc_39><loc_10></location>I ( A /uni27E9 B ) σ ≡ S ( B ) σ -S ( AB ) σ</formula>
<text><location><page_10><loc_52><loc_90><loc_92><loc_93></location>is the quantum coherent information between systems A and B .</text>
<text><location><page_10><loc_52><loc_82><loc_92><loc_90></location>Now, let us return to the relativistic quantum broadcast channel constructed in Sec. III. To analyze if Alice can send entanglement (and, as a result, an arbitrary state ρ A ) to Bob through the broadcast channel, let us note that we may purify the mixed state ρ AA 1 A 2 by adding an environment system E such that</text>
<formula><location><page_10><loc_57><loc_79><loc_92><loc_81></location>ρ AA 1 A 2 = tr E /parenleft.alt1/divides.alt0 ψ AA 1 A 2 E /uni27E9/uni27E8 ψ AA 1 A 2 E /divides.alt0/parenright.alt1 , (110)</formula>
<text><location><page_10><loc_52><loc_75><loc_92><loc_78></location>where /divides.alt0 ψ AA 1 A 2 E /uni27E9 ∈ H A ⊗H A 1 ⊗H A 2 ⊗H E is a pure state. Let us decompose it as</text>
<formula><location><page_10><loc_52><loc_69><loc_92><loc_74></location>/divides.alt0 ψ AA 1 A 2 E /uni27E9 = 1 /summation.disp a = 0 1 /summation.disp a 1 = 0 1 /summation.disp a 2 = 0 d -1 /summation.disp e = 0 c aa 1 a 2 e /divides.alt0 a /uni27E9 A /divides.alt0 a 1 /uni27E9 A 1 /divides.alt0 a 2 /uni27E9 A 2 /divides.alt0 e /uni27E9 E , (111)</formula>
<text><location><page_10><loc_52><loc_64><loc_92><loc_69></location>where /divides.alt0 a /uni27E9 A , /divides.alt0 a 1 /uni27E9 A 1 , and /divides.alt0 a 2 /uni27E9 A 2 are eigenstates of σ z A , σ z A 1 , and σ z A 2 , respectively. Furthermore, {/divides.alt0 e /uni27E9 E } is some orthonormal basis for H E , with d ≡ dim H E being as large as needed, and</text>
<formula><location><page_10><loc_62><loc_59><loc_92><loc_62></location>1 /summation.disp a = 0 1 /summation.disp a 1 = 0 1 /summation.disp a 2 = 0 d -1 /summation.disp e = 0 /divides.alt0 c aa 1 a 2 e /divides.alt0 2 = 1 . (112)</formula>
<text><location><page_10><loc_52><loc_57><loc_60><loc_58></location>By defining</text>
<formula><location><page_10><loc_54><loc_52><loc_92><loc_56></location>/divides.alt0 ζ a /uni27E9 A 1 A 2 E ≡ 1 /summation.disp a 1 = 0 1 /summation.disp a 2 = 0 d -1 /summation.disp e = 0 c aa 1 a 2 e /divides.alt0 a 1 /uni27E9 A 1 /divides.alt0 a 2 /uni27E9 A 2 /divides.alt0 e /uni27E9 E (113)</formula>
<text><location><page_10><loc_52><loc_50><loc_54><loc_51></location>and</text>
<formula><location><page_10><loc_57><loc_47><loc_92><loc_49></location>ζ A 1 A 2 aa ' ≡ tr E /parenleft.alt1/divides.alt0 ζ a /uni27E9 A 1 A 2 E A 1 A 2 E /uni27E8 ζ a ' /divides.alt0 /divides.alt0/parenright.alt1 , (114)</formula>
<text><location><page_10><loc_52><loc_45><loc_70><loc_46></location>we can write Eq. (110) as</text>
<formula><location><page_10><loc_60><loc_41><loc_92><loc_44></location>ρ AA 1 A 2 = 1 /summation.disp a = 0 1 /summation.disp a ' = 0 ζ A 1 A 2 aa ' ⊗/divides.alt0 a /uni27E9 AA /uni27E8 a ' /divides.alt0 . (115)</formula>
<text><location><page_10><loc_52><loc_37><loc_92><loc_40></location>By using Eq. (115) in Eq. (107) and taking the partial trace over C and A 2 , we obtain</text>
<formula><location><page_10><loc_61><loc_33><loc_92><loc_36></location>σ A 1 B = 1 /summation.disp a = 0 ζ A 1 aa ⊗E B (/divides.alt0 a /uni27E9 AA /uni27E8 a /divides.alt0) , (116)</formula>
<text><location><page_10><loc_52><loc_30><loc_92><loc_32></location>where ζ A 1 aa ≡ tr A 2 ( ζ A 1 A 2 aa ) and we have used the fact that</text>
<formula><location><page_10><loc_60><loc_28><loc_92><loc_29></location>E B (/divides.alt0 a /uni27E9 AA /uni27E8 a ' /divides.alt0) = δ aa ' E B (/divides.alt0 a /uni27E9 AA /uni27E8 a /divides.alt0) , (117)</formula>
<text><location><page_10><loc_52><loc_24><loc_92><loc_27></location>which can be proven by a direct calculation using Eq. (62). We define now the density matrices</text>
<formula><location><page_10><loc_65><loc_21><loc_92><loc_23></location>S B a ≡ E B (/divides.alt0 a /uni27E9 AA /uni27E8 a /divides.alt0) , (118)</formula>
<text><location><page_10><loc_52><loc_19><loc_54><loc_20></location>and</text>
<formula><location><page_10><loc_66><loc_16><loc_92><loc_18></location>τ A 1 a ≡ /parallel.alt1 ζ a /parallel.alt1 -2 ζ A 1 aa , (119)</formula>
<text><location><page_10><loc_52><loc_13><loc_92><loc_16></location>with /parallel.alt1 ζ a /parallel.alt1 2 ≡ tr ( ζ A 1 aa ) . This allows us to rewrite Eq. (116) as</text>
<formula><location><page_10><loc_62><loc_8><loc_92><loc_12></location>σ A 1 B = 1 /summation.disp a = 0 /parallel.alt1 ζ a /parallel.alt1 2 τ A 1 a ⊗ S B a , (120)</formula>
<text><location><page_11><loc_9><loc_92><loc_39><loc_93></location>and we note that tr ( S B a ) = tr ( τ A 1 a ) = 1 and</text>
<formula><location><page_11><loc_13><loc_88><loc_49><loc_91></location>1 /summation.disp a = 0 /parallel.alt1 ζ a /parallel.alt1 2 = 1 /summation.disp a = 0 1 /summation.disp a 1 = 0 1 /summation.disp a 2 = 0 d -1 /summation.disp e = 0 /divides.alt0 c aa 1 a 2 e /divides.alt0 2 = 1 . (121)</formula>
<text><location><page_11><loc_9><loc_80><loc_49><loc_87></location>Hence, we have shown that σ A 1 B is a separable state, which implies that the reduced channel from Alice to Bob lies in the class of the entanglement-breaking channels . As shown in [36], the coherent information is non-positive for separable states like σ A 1 B , i.e.,</text>
<formula><location><page_11><loc_24><loc_77><loc_49><loc_79></location>I ( A 1 /uni27E9 B ) σ ≤ 0 . (122)</formula>
<text><location><page_11><loc_9><loc_72><loc_49><loc_76></location>Following similar steps, one can also show that I ( A 2 /uni27E9 C ) σ ≤ 0. As a result, the unassisted achievable rate region given by Eqs. (108) and (109) reduces to</text>
<formula><location><page_11><loc_24><loc_70><loc_49><loc_71></location>Q B = Q C = 0 . (123)</formula>
<text><location><page_11><loc_9><loc_57><loc_49><loc_69></location>We do not know, to this day, if the region defined by Eqs. (108)-(109) characterizes the full capacity region for general quantum broadcast channels. If this is the case, our analysis implies that Alice cannot send qubits to the receivers without prior shared entanglement. Since the reduced channels E B and E C are entanglement-breaking, Alice cannot transmit the needed entanglement to establish quantum communication by using only the quantum broadcast channel E .</text>
<text><location><page_11><loc_9><loc_46><loc_49><loc_56></location>On the other hand, we can investigate if this limitation changes when the three observers perform an entanglement-assisted quantum communication protocol like the one in the beginning of this section. In this scenario, we recall that Eqs. (104)-(106) give an achievable (entanglement-assisted) quantum rate region that we shall analyze now.</text>
<text><location><page_11><loc_9><loc_42><loc_49><loc_46></location>First, we show that Alice can indeed send quantum information to Bob when they initially share entanglement as follows. We choose the initial input state to be</text>
<formula><location><page_11><loc_18><loc_40><loc_49><loc_41></location>ρ AA 1 A 2 ≡ /divides.alt0 Φ AA 1 /uni27E9/uni27E8 Φ AA 1 /divides.alt0 ⊗ ρ A 2 , (124)</formula>
<text><location><page_11><loc_9><loc_36><loc_49><loc_39></location>where ρ A 2 is arbitrary and /divides.alt0 Φ AA 1 /uni27E9 is the maximally entangled state</text>
<formula><location><page_11><loc_20><loc_32><loc_49><loc_36></location>/divides.alt0 Φ AA 1 /uni27E9 ≡ 1 √ 2 1 /summation.disp a = 0 /divides.alt0 a /uni27E9 A /divides.alt0 a /uni27E9 A 1 . (125)</formula>
<text><location><page_11><loc_9><loc_30><loc_46><loc_31></location>For this particular state, Eq. (120) can be written as</text>
<formula><location><page_11><loc_18><loc_26><loc_49><loc_29></location>σ A 1 B = 1 2 1 /summation.disp a = 0 /divides.alt0 a /uni27E9 A 1 A 1 /uni27E8 a /divides.alt0 ⊗ S B a , (126)</formula>
<text><location><page_11><loc_9><loc_23><loc_49><loc_25></location>which is a cq-state as the one in Eq. (73). Following the same steps that led to Eq. (81), one can show that</text>
<formula><location><page_11><loc_22><loc_20><loc_49><loc_22></location>I ( A 1 ; B ) σ = C(E B ) , (127)</formula>
<text><location><page_11><loc_9><loc_17><loc_49><loc_19></location>where C(E B ) is defined in Eq. (90). On the other hand, for this particular input state, we get</text>
<formula><location><page_11><loc_18><loc_13><loc_49><loc_16></location>σ A 2 C = /bracketleft.alt4 1 /summation.disp a = 0 1 2 S C a /bracketright.alt4 ⊗ ρ A 2 , (128)</formula>
<formula><location><page_11><loc_17><loc_9><loc_49><loc_12></location>σ A 1 A 2 = /bracketleft.alt4 1 /summation.disp a = 0 1 2 /divides.alt0 a /uni27E9 A 1 A 1 /uni27E8 a /divides.alt0/bracketright.alt4 ⊗ ρ A 2 , (129)</formula>
<text><location><page_11><loc_52><loc_90><loc_92><loc_93></location>where S C a ≡ E C (/divides.alt0 a /uni27E9 AA /uni27E8 a /divides.alt0) . Since we get these product states, it follows that</text>
<formula><location><page_11><loc_62><loc_88><loc_92><loc_89></location>I ( A 2 ; C ) σ = I ( A 1 ; A 2 ) σ = 0 . (130)</formula>
<text><location><page_11><loc_52><loc_82><loc_92><loc_86></location>In view of Eqs. (104)-(106), we conclude that Alice will be able to convey quantum information to Bob at a rate arbitrarily close to</text>
<formula><location><page_11><loc_67><loc_79><loc_92><loc_81></location>̃ Q B = C(E B )/slash.left 2 (131)</formula>
<text><location><page_11><loc_52><loc_68><loc_92><loc_78></location>when they initially share unlimited amounts of entanglement, provided that they let their qubits interact with the field in causally connected regions of the spacetime. Note that this is in contrast with the unassisted case previously discussed. On the other hand, this particular achievable rate region derived here implies that ̃ Q C = 0 in view of Eq. (130).</text>
<text><location><page_11><loc_52><loc_64><loc_92><loc_68></location>Similarly, by switching A 1 by A 2 in Eq. (124), one can show that Alice will be able to transmit quantum states to Charlie (but not to Bob) at a rate arbitrarily close to</text>
<formula><location><page_11><loc_67><loc_61><loc_92><loc_62></location>̃ Q C = C(E C )/slash.left 2 (132)</formula>
<text><location><page_11><loc_52><loc_57><loc_92><loc_59></location>when they communicate assisted by shared entanglement.</text>
<text><location><page_11><loc_52><loc_48><loc_92><loc_57></location>Furthermore, initial tripartite entangled states ρ AA 1 A 2 will, in general, lead to simultaneously nonvanishing rate pairs provided that sender and receivers interact with the field in causally connected regions of spacetime. For example, by choosing the initial input state to be a pure maximally entangled GHZ state</text>
<formula><location><page_11><loc_64><loc_45><loc_92><loc_47></location>ρ AA 1 A = /divides.alt0 GHZ /uni27E9/uni27E8 GHZ /divides.alt0 , (133)</formula>
<text><location><page_11><loc_52><loc_43><loc_56><loc_44></location>where</text>
<formula><location><page_11><loc_61><loc_38><loc_92><loc_41></location>/divides.alt0 GHZ /uni27E9 ≡ 1 √ 2 1 /summation.disp a = 0 /divides.alt0 a /uni27E9 A /divides.alt0 a /uni27E9 A 1 /divides.alt0 a /uni27E9 A 2 , (134)</formula>
<text><location><page_11><loc_52><loc_33><loc_92><loc_37></location>one can show by a similar direct calculation that the following entanglement-assisted quantum rate region is achievable:</text>
<formula><location><page_11><loc_64><loc_29><loc_92><loc_32></location>̃ Q B ≤ 1 2 C(E B ) , (135)</formula>
<formula><location><page_11><loc_64><loc_26><loc_92><loc_28></location>̃ Q C ≤ 1 2 C(E C ) , (136)</formula>
<formula><location><page_11><loc_60><loc_23><loc_92><loc_25></location>̃ Q B + ̃ Q C ≤ 1 2 [C(E B ) + C(E C ) -1 ] . (137)</formula>
<text><location><page_11><loc_52><loc_9><loc_92><loc_21></location>In [37], the authors investigate the classical channel capacity C(E B ) of the reduced channel E B from Alice to Bob for the case where both observers follow inertial or accelerated worldlines on Minkowski spacetime. In some scenarios, the authors show that one can tune the parameters of the channel to achievable capacities close to 1. Considering that C(E C ) has a similar expression, if we consider the case where ∆ ( f B , f C ) ≈ 0, we can argue in favor of also tune the channel parameters to make</text>
<text><location><page_12><loc_9><loc_83><loc_49><loc_93></location>C(E C ) close to 1. Under these assumptions, Eqs. (135)(137) imply that Alice would be able to simultaneously transmit quantum information to Bob and Charlie when assisted by prior entanglement. Hence, the relativistic quantum broadcast constructed in Sec. III seems to impose no limitation to simultaneous entanglement-assisted communication from Alice to both receivers.</text>
<section_header_level_1><location><page_12><loc_21><loc_79><loc_37><loc_80></location>V. CONCLUSIONS</section_header_level_1>
<text><location><page_12><loc_9><loc_66><loc_49><loc_77></location>In this paper, we have built a relativistic quantum broadcast channel by using a bosonic quantum field in a general globally hyperbolic spacetime. In this context, we have explored relativistic effects on the communication of classical and quantum information in a covariant manner, where the parts conveying the information are moving in arbitrary states of motion with the field being in an arbitrary (quasifree) state.</text>
<text><location><page_12><loc_9><loc_43><loc_49><loc_65></location>To construct the quantum broadcast channel, we have considered that Alice (the sender) prepares some input state ρ A -∞ for her qubit and switches on its interaction with the field for a finite time. After that, Bob (the first receiver) lets his qubit interact with the field for a finite time interval, thus obtaining a final state possibly containing information encoded by Alice. Similarly, after Bob finishes his measurement, Charlie performs an interaction between his qubit and the quantum field to try to recover information imprinted by Alice in the field state. We were able to trace the field degrees of freedom nonperturbatively and showed that suitable initial states for Bob's and Charlie's qubits can be chosen in order to maximize the signaling between Alice and the receivers. This procedure defines a fully relativistic quantum broadcast channel E .</text>
<text><location><page_12><loc_9><loc_34><loc_49><loc_42></location>With this channel, we were able to investigate at which rates Alice can reliably convey classical and quantum information to Bob and Charlie. By considering first a scenario where the three observers do not share prior entanglement, we found that Alice can reliably convey classical information to both Bob and Charlie and at which</text>
<text><location><page_12><loc_52><loc_79><loc_92><loc_93></location>rates she can perform this task. However, we have shown that the broadcast channel presented here breaks entanglement and thus, Alice cannot convey quantum information to Bob and Charlie following an unassisted strategy. Nevertheless, we have shown that this situation changes when they perform entanglement-assisted quantum communication. In this scenario, we were able to find achievable rates that Alice can achieve when sending qubits to the receivers provided that they initially share entangled states.</text>
<text><location><page_12><loc_52><loc_66><loc_92><loc_78></location>We were also able to show that all rates that were analyzed here vanish when the interactions between qubits and field occur in causally disconnected regions, an effect that is manifest in all expressions bounding the classical and quantum rates of communication even with the use of quantum resources like entanglement. Thus, our investigation provides good evidence that causality is not violated throughout the communication process, reinforcing the fundamental principles of relativistic physics.</text>
<text><location><page_12><loc_52><loc_45><loc_92><loc_65></location>Our study shows that quantum network information theory in general spacetimes is a rich and promising area of research, shedding light on several aspects of the interplay between quantum information theory and relativity. We believe that this work may provide tools to investigate open problems concerning quantum gravity, in particular, the fate of the information that has fallen in (evaporating) black holes. The preservation of causality observed in our analysis reaffirms the robustness of fundamental physical principles, even in the realm of quantum information theory in curved spacetimes. We hope that following the path we presented here could lead us to unveil fundamental aspects of physics that should be present in a full quantum theory of gravity.</text>
<section_header_level_1><location><page_12><loc_62><loc_40><loc_82><loc_41></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_12><loc_52><loc_34><loc_92><loc_38></location>I. B. and A. L. were fully and partially supported by S˜ao Paulo Research Foundation under Grants 2018/23355-2 and 2017/15084-6, respectively.</text>
<section_header_level_1><location><page_13><loc_24><loc_92><loc_77><loc_93></location>Appendix A: Full expression for the quantum broadcast channel map</section_header_level_1>
<text><location><page_13><loc_9><loc_87><loc_92><loc_90></location>As discussed in Sec. III, each Γ αβγδϵζ coefficient defined in Eq. (45) can be evaluated by using Eqs. (40) and (41) together with the product relation given by Eq. (46). Then, we substitute these coefficients in Eq. (42), obtaining</text>
<formula><location><page_13><loc_17><loc_52><loc_92><loc_86></location>ρ BC = 1 4 ( 1 + ν B cos [ 2∆ ( f A , f B )] + ν C cos [ 2∆ ( f A , f C )] cos [ 2∆ ( f B , f C )]) ρ BC -∞ + 1 4 ( 1 -ν B cos [ 2∆ ( f A , f B )] + ν C cos [ 2∆ ( f A , f C )] cos [ 2∆ ( f B , f C )]) σ z B ρ BC -∞ σ z B + 1 4 ( 1 + ν B cos [ 2∆ ( f A , f B )] -ν C cos [ 2∆ ( f A , f C )] cos [ 2∆ ( f B , f C )]) σ z C ρ BC -∞ σ z C + 1 4 ( 1 -ν B cos [ 2∆ ( f A , f B )] -ν C cos [ 2∆ ( f A , f C )] cos [ 2∆ ( f B , f C )]) σ z B ⊗ σ z C ρ BC -∞ σ z B ⊗ σ z C + iν B 4 sin [ 2∆ ( f A , f B )]/uni27E8 σ z A /uni27E9 ρ A -∞ [ ρ BC -∞ + σ z C ρ BC -∞ σ z C , σ z B ] + iν C 4 sin [ 2∆ ( f A , f C )] cos [ 2∆ ( f B , f C )]/uni27E8 σ z A /uni27E9 ρ A -∞ [ ρ BC -∞ + σ z B ρ BC -∞ σ z B , σ z C ] (A1) + Λ + c 8 /parenleft.alt1 ρ BC -∞ -σ z B ρ BC -∞ σ z B -σ z C ρ BC -∞ σ z C + σ z B ⊗ σ z C ρ BC -∞ σ z B ⊗ σ z C /parenright.alt1 + Λ -c 8 /parenleft.alt1{ ρ BC -∞ , σ z B ⊗ σ z C } -σ z B ρ BC -∞ σ z C -σ z C ρ BC -∞ σ z B /parenright.alt1 + i Λ + s 8 /uni27E8 σ z A /uni27E9 ρ A -∞ [ ρ BC -∞ -σ z C ρ BC -∞ σ z C , σ z B ] + i Λ -s 8 /uni27E8 σ z A /uni27E9 ρ A -∞ [ ρ BC -∞ -σ z B ρ BC -∞ σ z B , σ z C ] + iν C 4 cos [ 2∆ ( f A , f C )] sin [ 2∆ ( f B , f C )] /parenleft.alt1[ ρ BC -∞ , σ z B ⊗ σ z C ] + σ z B ρ BC -∞ σ z C -σ z C ρ BC -∞ σ z B /parenright.alt1 -ν C 4 sin [ 2∆ ( f A , f C )] sin [ 2∆ ( f B , f C )]/uni27E8 σ z A /uni27E9 ρ A -∞ { ρ BC -∞ -σ z C ρ BC -∞ σ z C , σ z B } ,</formula>
<text><location><page_13><loc_9><loc_50><loc_43><loc_52></location>where we have defined the following coefficients:</text>
<formula><location><page_13><loc_29><loc_48><loc_92><loc_49></location>Λ ± c ≡ ν + BC cos [ 2∆ ( f A , f B + f C )] ± ν -BC cos [ 2∆ ( f A , f B -f C )] , (A2)</formula>
<formula><location><page_13><loc_29><loc_46><loc_92><loc_47></location>Λ ± s ≡ ν + BC sin [ 2∆ ( f A , f B + f C )] ± ν -BC sin [ 2∆ ( f A , f B -f C )] , (A3)</formula>
<formula><location><page_13><loc_30><loc_44><loc_92><loc_45></location>ν j ≡ ω µ ( W [ E ( 2 f j )]) , (A4)</formula>
<formula><location><page_13><loc_29><loc_42><loc_92><loc_44></location>ν ± BC ≡ ω µ ( W [ E ( 2 f B ± 2 f C )]) . (A5)</formula>
<text><location><page_13><loc_9><loc_39><loc_92><loc_41></location>As discussed in Sec. III, we are motivated to fix the initial states for Bob's and Charlie's qubit as given in Eqs. (52) and (58). We write these states in terms of their Bloch vectors, i.e.,</text>
<formula><location><page_13><loc_45><loc_34><loc_92><loc_38></location>ρ j -∞ = I j + σ y j 2 , (A6)</formula>
<text><location><page_13><loc_9><loc_31><loc_92><loc_33></location>where j = B,C . By substituting Eq. (A6) in Eq. (A1), and by using the standard commutation relations of the Pauli matrices, we obtain the following expression describing the quantum broadcast channel map:</text>
<formula><location><page_13><loc_9><loc_9><loc_92><loc_29></location>E( ρ A -∞ ) = 1 16 ( 1 + ν B cos [ 2∆ ( f A , f B )] + ν C cos [ 2∆ ( f A , f C )] cos [ 2∆ ( f B , f C )]) /parenleft.alt1 I B + σ y B /parenright.alt1 ⊗/parenleft.alt1 I C + σ y C /parenright.alt1 + 1 16 ( 1 -ν B cos [ 2∆ ( f A , f B )] + ν C cos [ 2∆ ( f A , f C )] cos [ 2∆ ( f B , f C )]) /parenleft.alt1 I B -σ y B /parenright.alt1 ⊗/parenleft.alt1 I C + σ y C /parenright.alt1 + 1 16 ( 1 + ν B cos [ 2∆ ( f A , f B )] -ν C cos [ 2∆ ( f A , f C )] cos [ 2∆ ( f B , f C )]) /parenleft.alt1 I B + σ y B /parenright.alt1 ⊗/parenleft.alt1 I C -σ y C /parenright.alt1 + 1 16 ( 1 -ν B cos [ 2∆ ( f A , f B )] -ν C cos [ 2∆ ( f A , f C )] cos [ 2∆ ( f B , f C )]) /parenleft.alt1 I B -σ y B /parenright.alt1 ⊗/parenleft.alt1 I C -σ y C /parenright.alt1 (A7) -ν B 4 sin [ 2∆ ( f A , f B )]/uni27E8 σ z A /uni27E9 ρ A -∞ ( σ x B ⊗ I C ) -ν C 4 sin [ 2∆ ( f A , f C )] cos [ 2∆ ( f B , f C )]/uni27E8 σ z A /uni27E9 ρ A -∞ ( I B ⊗ σ x C ) + Λ + c 8 σ y B ⊗ σ y B -Λ -c 8 /parenleft.alt1 σ x B ⊗ σ y C /parenright.alt1 -Λ + s 8 /uni27E8 σ z A /uni27E9 ρ A -∞ /parenleft.alt1 σ x B ⊗ σ y C /parenright.alt1 -Λ -s 8 /uni27E8 σ z A /uni27E9 ρ A -∞ /parenleft.alt1 σ y B ⊗ σ x C /parenright.alt1 -ν C 4 cos [ 2∆ ( f A , f C )] sin [ 2∆ ( f B , f C )] ( σ z B ⊗ σ x C ) -ν C 4 sin [ 2∆ ( f A , f C )] sin [ 2∆ ( f B , f C )]/uni27E8 σ z A /uni27E9 ρ A -∞ /parenleft.alt1 σ z B ⊗ σ y C /parenright.alt1 .</formula>
<text><location><page_14><loc_9><loc_92><loc_65><loc_93></location>By taking partial traces relative to each qubit, one recovers Eqs. (62) and (63).</text>
<unordered_list>
<list_item><location><page_14><loc_10><loc_81><loc_49><loc_86></location>[1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, Cambridge, England, 2010).</list_item>
<list_item><location><page_14><loc_10><loc_79><loc_49><loc_81></location>[2] M. M. Wilde, Quantum Information Theory (Cambridge University Press, Cambridge, England, 2013).</list_item>
<list_item><location><page_14><loc_10><loc_76><loc_49><loc_78></location>[3] I. Savov and M. M. Wilde, IEEE Trans. Info. Theory 61 , 7017 (2015).</list_item>
<list_item><location><page_14><loc_10><loc_73><loc_49><loc_76></location>[4] J. Yard, P. Hayden, and I. Devetak, IEEE Trans. Info. Theory 57 , 7147 (2011).</list_item>
<list_item><location><page_14><loc_10><loc_71><loc_49><loc_73></location>[5] F. Dupuis, P. Hayden, and K. Li, IEEE Trans. Info. Theory 56 , 2946 (2010).</list_item>
</unordered_list>
<text><location><page_14><loc_12><loc_69><loc_12><loc_70></location>.</text>
<unordered_list>
<list_item><location><page_14><loc_10><loc_67><loc_49><loc_69></location>[6] W. Wang, S. Das, and M. M. Wilde, Quantum Information Processing 16 , 248 (2017).</list_item>
</unordered_list>
<text><location><page_14><loc_12><loc_65><loc_12><loc_66></location>.</text>
<unordered_list>
<list_item><location><page_14><loc_10><loc_63><loc_49><loc_65></location>[7] R. M. Wald, General Relativity (The University of Chicago Press, Chicago, USA, 1984).</list_item>
<list_item><location><page_14><loc_10><loc_59><loc_49><loc_62></location>[8] R. M. Wald, Quantum Field Theory in Curved SpaceTime and Black Hole Thermodynamics (The University of Chicago Press, Chicago, USA, 1994).</list_item>
<list_item><location><page_14><loc_10><loc_56><loc_49><loc_58></location>[9] P. M. Alsing and G. J. Milburn, Phys. Rev. Lett. 91 , 180404 (2003).</list_item>
<list_item><location><page_14><loc_9><loc_54><loc_49><loc_56></location>[10] K. Br'adler, P. Hayden, D. Touchette, and M. M. Wilde, Phys. Rev. A 81 , 062312 (2010).</list_item>
<list_item><location><page_14><loc_9><loc_51><loc_49><loc_53></location>[11] A. G. S. Landulfo and A. C. Torres, Phys. Rev. A 87 , 042339 (2013).</list_item>
<list_item><location><page_14><loc_9><loc_48><loc_49><loc_51></location>[12] E. Mart'ın-Mart'ınez, D. Hosler, and M. Montero, Phys. Rev. A 86 , 062307 (2012).</list_item>
<list_item><location><page_14><loc_9><loc_46><loc_49><loc_48></location>[13] K. Br'adler, P. Hayden, and P. Panangaden, Commun. Math. Phys. 312 , 361 (2012).</list_item>
<list_item><location><page_14><loc_9><loc_44><loc_49><loc_45></location>[14] M. Cliche and A. Kempf, Phys. Rev. A 81 , 012330 (2010).</list_item>
<list_item><location><page_14><loc_9><loc_42><loc_49><loc_44></location>[15] R. H. Jonsson, E. Mart'ın-Mart'ınez, and A. Kempf, Phys. Rev. A 89 , 022330 (2014).</list_item>
<list_item><location><page_14><loc_9><loc_40><loc_47><loc_41></location>[16] E. Mart'ın-Mart'ınez, Phys. Rev. D 92 , 104019 (2015).</list_item>
<list_item><location><page_14><loc_9><loc_38><loc_49><loc_40></location>[17] R. H. Jonsson, K. Ried, E. Mart'ın-Mart'ınez, and A. Kempf, J. Phys. A Math. Theor. 51 , 485301 (2018).</list_item>
<list_item><location><page_14><loc_9><loc_35><loc_49><loc_37></location>[18] R. H. Jonsson, J. Phys. A Math. Theor. 49 , 445402 (2016).</list_item>
<list_item><location><page_14><loc_9><loc_32><loc_49><loc_35></location>[19] B. L. Hu, S.-Y. Lin, and J. Louko, Class. Quantum Gravity 29 , 224005 (2012).</list_item>
<list_item><location><page_14><loc_9><loc_30><loc_49><loc_32></location>[20] P. Simidzija, A. Ahmadzadegan, A. Kempf, and E. Mart'ın-Mart'ınez, Phys. Rev. D 101 , 036014 (2020).</list_item>
<list_item><location><page_14><loc_52><loc_83><loc_92><loc_86></location>[21] K. Yamaguchi, A. Ahmadzadegan, P. Simidzija, A. Kempf, and E. Mart'ın-Mart'ınez, Phys. Rev. D 101 , 105009 (2020)</list_item>
</unordered_list>
<text><location><page_14><loc_55><loc_81><loc_55><loc_82></location>.</text>
<unordered_list>
<list_item><location><page_14><loc_52><loc_80><loc_92><loc_81></location>[22] H. Qi and M. M. Wilde, Phys. Rev. A 95 , 012339 (2017).</list_item>
<list_item><location><page_14><loc_52><loc_77><loc_92><loc_80></location>[23] D. Hosler, C. van de Bruck, and P. Kok, Phys. Rev. A 85 , 042312 (2012).</list_item>
<list_item><location><page_14><loc_52><loc_75><loc_92><loc_77></location>[24] K. Br'adler and C. Adami, J. High Energy Phys. 2014 (5), 95.</list_item>
<list_item><location><page_14><loc_52><loc_72><loc_92><loc_74></location>[25] K. Br'adler and C. Adami, Phys. Rev. D 92 , 025030 (2015).</list_item>
<list_item><location><page_14><loc_52><loc_69><loc_92><loc_72></location>[26] R. H. Jonsson, D. Q. Aruquipa, M. Casals, A. Kempf, and E. Mart'ın-Mart'ınez, Phys. Rev. D 101 , 125005 (2020).</list_item>
<list_item><location><page_14><loc_52><loc_67><loc_92><loc_69></location>[27] A. Blasco, L. J. Garay, M. Mart'ın-Benito, and E. Mart'ınMart'ınez, Phys. Rev. Lett. 114 , 141103 (2015).</list_item>
<list_item><location><page_14><loc_52><loc_64><loc_92><loc_66></location>[28] A. Blasco, L. J. Garay, M. Mart'ın-Benito, and E. Mart'ınMart'ınez, Phys. Rev. D 93 , 024055 (2016).</list_item>
<list_item><location><page_14><loc_52><loc_61><loc_92><loc_64></location>[29] P. Simidzija and E. Mart'ın-Mart'ınez, Phys. Rev. D 95 , 025002 (2017).</list_item>
<list_item><location><page_14><loc_52><loc_60><loc_89><loc_61></location>[30] A. G. S. Landulfo, Phys. Rev. D 93 , 104019 (2016).</list_item>
<list_item><location><page_14><loc_52><loc_57><loc_92><loc_60></location>[31] E. Tjoa and K. Gallock-Yoshimura, Phys. Rev. D 105 , 085011 (2022).</list_item>
<list_item><location><page_14><loc_52><loc_55><loc_92><loc_57></location>[32] L. Lapponi D. Moustos, D. E. Bruschi, and S. Mancini, Phys. Rev. D 107 , 125010 (2023).</list_item>
<list_item><location><page_14><loc_52><loc_50><loc_92><loc_55></location>[33] B. S. DeWitt, in General Relativity: An Einstein centenary survey , edited by S. W. Hawking and W. Israel (Cambridge University Press, Cambridge, England, 1979) pp. 680-745.</list_item>
<list_item><location><page_14><loc_52><loc_40><loc_92><loc_49></location>[34] I. Khavkine and V. Moretti, Algebraic QFT in Curved Spacetime and Quasifree Hadamard States: An Introduction, in Advances in Algebraic Quantum Field Theory , Mathematical Physics Studies, edited by R. Brunetti, C. Dappiaggi, K. Fredenhagen, and J. Yngvason (Springer International Publishing, Cham, Germany, 2015) pp. 191-251.</list_item>
<list_item><location><page_14><loc_52><loc_38><loc_92><loc_40></location>[35] S. Blanes, F. Casas, J. Oteo, and J. Ros, Physics Reports 470 , 151 (2009).</list_item>
<list_item><location><page_14><loc_52><loc_35><loc_92><loc_37></location>[36] A. S. Holevo, Problems of Information Transmission 44 , 171 (2008).</list_item>
<list_item><location><page_14><loc_52><loc_32><loc_92><loc_35></location>[37] I. B. Barcellos and A. G. S. Landulfo, Phys. Rev. D 104 , 105018 (2021).</list_item>
</unordered_list>
</document>
[
{
"title": "A relativistic quantum broadcast channel",
"content": "Ian Bernardes Barcellos 1, ∗ and Andr'e G. S. Landulfo 1, † 1 Centro de Ciˆencias Naturais e Humanas, Universidade Federal do ABC, Avenida dos Estados, 5001, 09210-580, Bangu, Santo Andr'e, S˜ao Paulo, Brazil (Dated: March 1, 2024) We investigate the transmission of classical and quantum information between three observers in a general globally hyperbolic spacetime using a quantum scalar field as a communication channel. We build a model for a quantum broadcast channel in which one observer (sender) wishes to transmit (classical and quantum) information to two other observers (receivers). They possess some localized two-level quantum system (a qubit) that can interact with the quantum field in order to prepare an input or receive the output of this channel. The field is supposed to be in an arbitrary quasifree state, the three observers may be in arbitrary states of motion, and no choice of representation of the field canonical commutation relations is made. The interaction of the field and qubits is such that it allows us to obtain the map that describes this channel in a non-perturbative manner. We conclude by analyzing the rates at which information can be transmitted through this channel and by investigating relativistic causality effects on such rates. PACS numbers: 03.67.-a,03.67.Hk, 04.62.+v",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "Network information theory is the area of knowledge that studies classical communication problems involving multiple parts. Here, the word 'classical' stands not only for the fact that the information being transmitted is classic (bits) but also for the physical systems in which such information is encoded, i.e., systems that can be described by some area of classical physics (such as Electromagnetism). One particular case of interest is the broadcast channel, where typically one sender wishes to transmit information to multiple receivers (like radio and TV stations broadcasting their signals, for example). Nowadays, one of the main goals of quantum information theory is to extend several results of information theory to the quantum world [1, 2], investigating any new features or advantages that can arise when one uses quantum systems to encode, process, and transmit information. The quantum network information theory comprises the studies of communication protocols using quantum systems to convey classical (bits) or quantum (qubits) information. In particular, the classical broadcast channels can be extended to the so-called quantum broadcast channels , where one sender transmits classical or quantum input information to many receivers using a quantum system as a communication channel with quantum outputs [3-6]. Such communication scenarios are very suitable for analyzing how relativistic effects can influence one's ability to communicate using quantum channels. This could be due to the existence of nontrivial spacetime structures such as black hole event horizons, Cauchy horizons, and causal horizons arising from the relativistic relative mo- tion between senders and receivers or even due to the expansion of spacetime [7]. In order to consistently analyze quantum information theory in general spacetimes, one should use quantum field theory in curved spacetimes (QFTCS) [8]. This approach was used by several authors to analyze the communication process in relativistic settings, with particular attention being paid to Minkowski [9-21], Schwarzschild [22-26], or asymptotically flat cosmological spacetimes [27-29]. However, only recently [30] a communication model valid in general globally hyperbolic spacetimes and in which the parts that convey information can move in arbitrary worldlines and interact with the quantum field (used as communication channel) only in the vicinity of its worldlines was developed (and, since then, other works in this context have emerged as, for instance, Refs. [31, 32]). This is interesting for two reasons: firstly, it allows the analysis of information exchange between more general observers, not only observers following orbits of some Killing field (which does not even exist in spacetimes lacking timelike symmetries). Secondly, the model studied in [30] allows one to investigate the outputs of the quantum communication in a nonperturbative manner and thereby is suitable to investigate both the causality as well as the communication between parts lying in early and future asymptotic regions (limits that would invalidate results obtained by perturbative methods). In the present paper, we generalize the analysis of [30]. This is done by constructing a model for classicalquantum, quantum-quantum, and entanglement-assisted quantum-quantum broadcast channels. We consider an arbitrary globally hyperbolic spacetime in which one observer (Alice) wants to convey classical (or quantum) information to two receivers (Bob and Charlie) using a quantum scalar field as a communication channel. The three observers will use two-level quantum systems (qubits) to locally interact with the quantum field in or- der to send or receive information. The observers may be in arbitrary states of motion, the interaction between the detectors and the field is similar to the one given by the Unruh-DeWitt model [33], and the field may initially be in an arbitrary quasifree state [8]. We suppose, however, that the two levels of each qubit have the same energy. This model is interesting because the evolution of the system can be computed exactly, and therefore we will obtain nonperturbative results for the communication rates associated with such a broadcast channel. As we will see, causality in the information exchange is explicitly manifest in our results. This work is organized as follows. In Sec. II we will present the quantization procedure of a free scalar field on a globally hyperbolic spacetime as well as the class of states we will be using. In Sec. III we describe the interaction between the qubits and the field and determine the quantum map that relates the information Alice wants to convey to the final joint state of Bob's and Charlie's qubits. In Sec. IV we investigate the rates at which information can be transmitted using this broadcast channel, as well as the influence of the spacetime curvature or relative motion of observers in the communication process. In Sec. V we give our final remarks. We assume metric signature (- + ++) and natural units in which c = /uni0335 h = G = k B = 1, unless stated otherwise.",
"pages": [
1,
2
]
},
{
"title": "II. FIELD QUANTIZATION",
"content": "Let us consider a free, real scalar field ϕ propagating in an arbitrary four-dimensional globally hyperbolic spacetime (M , g ) , where M denotes the four-dimensional spacetime manifold and g its Lorentzian metric. Let the spacetime be foliated by Cauchy surfaces Σ t labeled by the real parameter t . The field is described by the action where ϵ M = √ -g dx 0 ∧ /uni22EF ∧ dx 3 is the spacetime volume 4-form, m is the field mass, ξ ∈ R , R is the scalar curvature, ∇ a is the torsion-free covariant derivative compatible with the metric g , and g ≡ det ( g µν ) in some arbitrary coordinate system { x µ } . The extremization of the action (1) gives rise to the Klein-Gordon equation In the canonical quantization procedure, we promote the real field ϕ to an operator 1 that satisfies the 'equaltime' canonical commutation relations (CCR) where x ≡ ( x 1 , x 2 , x 3 ) are spatial coordinates on Σ t and π ( x ) is the conjugate momentum defined as with the notation ˙ ϕ ≡ ∂ t ϕ . In addition, we may formally write the canonical Hamiltonian of the field as with and being the Lagrangian density. To find a representation of the CCR, Eqs. (3) and (4), we define an antisymmetric bilinear map σ acting on the space S C of complex solutions of Eq. (2) as where ϵ Σ represents the proper-volume 3-form on the Cauchy surface Σ t and n a its future-directed normal unit vector. It allows us to define the Klein-Gordon product as and, although this product is not positive-definite on S C , we may choose any subspace H ⊂ S C (the so-called one-particle Hilbert space) such that: (i) S C /uni2243 H/uni2295.big H ; 2 (ii) the KG product is positive definite on H , thus making (H , /uni27E8 , /uni27E9) a Hilbert space; 3 (iii) given any u ∈ H and v ∈ H , /uni27E8 u, v /uni27E9 = 0. (See [8] for details.) The Hilbert space that comprises the field states is defined as the symmetric Fock space F s (H) and the quantum field operator is formally defined as where { u j } comprise an orthonormal basis for H and a ( u ) / a † ( v ) are the usual annihilation/creation operators associated with the modes u / v , respectively. They satisfy the commutation relations with I being the identity operator on F s (H) . The vacuum state associated with this representation of the CCR is the normalized vector /divides.alt0 0 /uni27E9 that satisfies a ( u )/divides.alt0 0 /uni27E9 = 0 for every mode u ∈ H . In order to make it mathematically well-defined, the quantum field operator must be defined as an operatorvalued distribution. To this end, let S ⊂ S C be the space of real solutions of Eq. (2) whose restriction to Cauchy surfaces have compact support and K ∶ S → H be the projection operator that takes the positive-norm part of any ψ ∈ S . If C ∞ 0 (M) denote the set of all smooth compactly-supported real functions on M , we define the map E ∶ C ∞ 0 (M) → S acting on some test function f ∈ C ∞ 0 (M) as where Af and Rf are the advanced and retarded solutions of the Klein-Gordon equation with source f , respectively. Hence, they satisfy with P ≡ -∇ a ∇ a + m 2 + ξR representing the Klein-Gordon differential operator. Now, for each test function f ∈ C ∞ 0 (M) , we define a smeared quantum field operator by which satisfies the covariant version of the CCR, where for all f 1 , f 2 ∈ C ∞ 0 (M) . As shown in [8], Eq. (15) can be obtained by formally integrating Eq. (11) weighed by the test function f , i.e., The above construction has the downside that there are infinitely many choices of H satisfying properties (i) -(iii) listed below Eq. (10) and their respective Fock spaces are, in general, unitarily inequivalent. As discussed in [30], this issue can be avoided through the algebraic approach to quantum field theory (QFT). For more details, see Refs. [8, 34]. In this work, we will focus on a particular class of states: the quasifree states , defined as follows. Given a real inner product µ ∶ S × S → R satisfying for all φ 1 , φ 2 ∈ S , we define a quasifree state ω µ associated with µ by the relation for all f ∈ C ∞ 0 (M) , where the so-called Weyl operators W ( Ef ) are defined by The vacuum, n-particle, and thermal states are examples of quasifree states.",
"pages": [
2,
3
]
},
{
"title": "III. THE QUANTUM BROADCAST CHANNEL",
"content": "A typical broadcast communication scenario involves the transmission of information between one station (sender) and several receivers who will decode the information independently. Let us consider a model in which one observer, Alice, wants to transmit separate information to two other observers, Bob and Charlie, using the quantum field ϕ as a broadcast channel. Suppose that the field is initially in some quasifree state ω µ 4 . Suppose also that the three observers follow arbitrary trajectories in the curved spacetime and that each one of them possesses a two-level quantum system that may interact with the quantum field at their will. The two-dimensional Hilbert spaces associated with Alice's, Bob's, and Charlie's qubits are denoted by H A , H B , and H C , respectively. The communication setup, illustrated by Fig. 1, is as follows: In order to transmit information to Bob and Charlie, Alice prepares her qubit in some initial quantum state ρ A -∞ and switches on its interaction with the field for a finite time interval ∆ t A (measured by the parameter t ). To measure the information imprinted by Alice on the field's state, Bob and Charlie initially prepare their qubits in suitable states ρ B -∞ and ρ C -∞ and then they switch on each of their qubit interaction with the field for finite time intervals ∆ t B and ∆ t C , respectively. For the sake of simplicity, we will consider here the case where (QB1) Bob lets his qubit interact with the field only after Alice finishes her transmission; (QB2) Charlie lets his qubit interact with the field only after Bob finishes his measurement process. Such communication setup is implemented by means of the Hamiltonian where H ϕ is the field Hamiltonian in Eq. (6) and H int is the Hamiltonian that describes the interaction between each qubit and the field which, in the interaction picture, is given by where j ∈ { A,B,C } , with A , B , and C labeling Alice's, Bob's, and Charlie's qubit, respectively. Here, σ z j is one of the Pauli matrices /braceleft.alt1 σ x j , σ y j , σ z j /braceright.alt1 associated with qubit j ; ψ j ( t, x ) is a smooth real function satisfying ψ j /divides.alt0 Σ t ∈ C ∞ 0 ( Σ t ) for all t , which models the finite range of interaction between qubit j and the field (i.e., the interaction occurs only at some vicinity of each qubit worldline); and ϵ j ( t ) is a smooth and compactly-supported real coupling function modeling the finite-time coupling of qubit j with the field. Each coupling function has support where T i j and T f j represent the time (with respect to the parameter t ) in which each qubit interaction with the field is switched-on and -off, respectively. Here, we denote ∆ t j ≡ T f j -T i j . Thus, the hypotheses (QB1) and (QB2) previously listed can be summarized as The interaction between each qubit and the field given by Eq. (23) is very similar to the Unruh-DeWitt model [33]. However, we assumed that the two levels of each qubit have the same (zero) energy. As we shall see, this assumption allows us to calculate the evolution operator of the system and trace out the field degrees of freedom in a nonperturbative manner, thus making this model interesting to investigate both the causality in the information exchange process as well as the communication between parts lying in early and future asymptotic spacetime regions. We note that one could also have given an energy gap 2 δ j for each qubit j in z -direction by adding H j = δ j σ z j to the total Hamiltonian in Eq. (22) and still keep the model exactly solvable. This would change it to but would keep the interaction Hamiltonian in the interaction picture, Eq. (23), unchanged. Hence, all the results we will describe below would remain the same. The interaction-picture time-evolution operator at late times, associated with the foliation Σ t , can be written as the time-ordered expression It can be computed nonperturbatively by using the Magnus expansion [35] where and so on. By using Eqs. (18), (23), and (29), we get where we have defined Now, by making use of Eqs. (18) and (23) together with Eqs. (16), (25), and (30) we can cast Ω 2 as where Ξ is the c-number with and we recall that [ ϕ ( x ) , ϕ ( x ' )] ≡ -i ∆ ( x, x ' ) I is the unsmeared version of Eq. (16). Finally, since [ H I int ( t ) , H I int ( t ' )] is proportional to the identity, we get Using the Zassenhaus formula valid whenever [ A,B ] is a proportional to the identity, together with Eqs. (28), (32), (34), and (35) we obtain the following unitary evolution operator: Now that we have the exact evolution operator U , we can use it to evolve the initial state of the 3 qubit + field system and then trace out the field and Alice's qubit degrees of freedom. This procedure allows us to obtain the final state of Bob's and Charlie's qubits after the communication protocol has ended. This is the state that they will measure to recover the information that Alice has sent. Explicitly, the final Bob+Charlie state is given by where ρ j -∞ and ρ ω are the initial states of qubit j and the field, respectively. To compute the trace in Eq. (38), let us cast the operators in Eq. (37) as where and where W ( Ef ) is defined in Eq. (21). By plugging Eqs. (37) and (39) into Eq. (38) and then taking the partial traces on ϕ and A , a direct calculation yields where h . c . stands for Hermitian conjugation, and we have defined and with α,β,γ,δ,ϵ,ζ ∈ { c, s } , F c ( x ) ≡ cos x, and F s ( x ) ≡ sin x . We note that we have written the algebraic field state ω µ as a density matrix with tr [ ρ ω W ( Ef )] ≡ ω µ [ W ( Ef )] . Furthermore, we have used the fact that the expected value of odd functions of the field operator vanishes since we are assuming that ω µ is a quasifree state (a consequence of Wick's theorem). Now, each Γ αβγδϵζ in Eq. (42) can be evaluated by substituting Eqs. (40) and (41) in Eq. (45) and then using the identity for all f, f 1 , f 2 ∈ C ∞ 0 (M) , to simplify the product of the Weyl operators. By substituting these coefficients in Eq.(42) one finds the explicit form of the state ρ BC , which is given in Eq. (A1) of Appendix A. The expression in Eq. (A1) allows one to write the final joint state for Bob's and Charlie's qubits given any initial state configuration for the 3 qubits+field. To define a quantum broadcast channel, we must choose suitable initial states for Bob and Charlie qubits in order to obtain a quantum map relating the initial state of Alice's qubit ρ A -∞ (which encodes the messages) to the final states that will be probed by them (to decode the messages). Since Bob only performs measurements in his own two-level system, we calculate the expression for the reduced state of his qubit, i.e., Taking the trace in Eq. (A1) relative to Charlie's degrees of freedom, we obtain where with µ be the inner product associated with the field quasifree state ω µ as in Eq. (20). Note that it is the last term in Eq. (48) that contains the information encoded by Alice, and thus it will be useless for Bob to choose the eigenstates /divides.alt0 0 /uni27E9 B and /divides.alt0 1 /uni27E9 B of σ z B as his initial state ρ B -∞ since this term would vanish. Furthermore, since σ z B commutes with the interaction Hamiltonian, he won't recover any information either if he performs projective measurements on this basis. To choose a suitable state ρ B -∞ that maximizes the chances of success in their communication, suppose for simplicity that Alice encodes a pair of messages in states ρ A -∞+ and ρ A -∞-which will be decoded by Bob using a set of projective measurements in the x -direction, where σ x B /divides.alt0±/uni27E9 B = ±/divides.alt0±/uni27E9 B . From Eq. (48), we conclude that the probability that Bob measures l = ± given that Alice has encoded the message k = ± in ρ A -∞ k is where and β B ≡ B /uni27E8 0 /divides.alt0 ρ B -∞ /divides.alt0 1 /uni27E9 B . From these two equations, we see that it is the second term Λ k that contains the information encoded by Alice on her qubit state, and thus we are motivated to choose a state ρ B -∞ that makes β B a pure imaginary number, which will make the first term of Λ k vanish while maximizing the amplitude of the second term. This motivates us to choose where is an eigenstate of σ y B (in this case, β B = -i /slash.left 2). With this choice, we can write Eq. (51) as Now we turn our attention to Charlie. The final reduced state for his qubit is Taking the trace in Eq. (A1) relative to Bob's degrees of freedom and using Eq. (52) we obtain where To obtain Eq. (56), we explicitly used the choice in Eq. (52), which implies that /uni27E8 σ z B /uni27E9 ρ B -∞ ≡ tr /parenleft.alt1 σ z B ρ B -∞ /parenright.alt1 = 0. By a completely similar reasoning as the one used to choose Bob's initial state, we are motivated to choose Charlie's initial qubit state as where σ y C /divides.alt0 y + /uni27E9 C = /divides.alt0 y + /uni27E9 C . Now, the quantum broadcast channel is completely characterized by a linear, completely positive and tracepreserving (CPTP) quantum map E which takes ρ A -∞ into a final state ρ BC , i.e., By substituting the initial states of Bob's and Charlie's qubits given in Eqs. (52) and (58) into Eq. (A1), we find the explicit expression for the quantum broadcast channel E . For the sake of clarity, due to its lengthy expression, we write its explicit form in Eq. (A7) of Appendix A. For later use, we will denote the reduced channels E B ∶ A → B , E C ∶ A → C by respectively. It then follows from Eqs. (A7), (60), and (61) that they can be explicitly written as and Given an initial state ρ A -∞ prepared by Alice on her qubit, these expressions for E B and E C determine the final local states of Bob's and Charlie's qubit, respectively.",
"pages": [
3,
4,
5,
6
]
},
{
"title": "IV. ACHIEVABLE COMMUNICATION RATES",
"content": "Now that we have constructed a model for a relativistic quantum broadcast channel, we can investigate at which rates classical and quantum information can be reliably transmitted by Alice to Bob and Charlie. We first review a few protocols for quantum broadcast communication published in the literature and then we investigate the achievable rates for our quantum broadcast channel E defined in Eq. (59).",
"pages": [
6
]
},
{
"title": "A. Unassisted classical communication",
"content": "Let us begin with the investigation of unassisted transmission of classical information. We follow the protocol presented in [3], where more details can be found. We evaluate achievable rates for our model and then we discuss how causality is explicitly manifest in our results. Suppose Alice wishes to transmit a common message m ∈ M intended for both receivers while sending additional personal messages m B ∈ M B and m C ∈ M C intended for Bob and Charlie, respectively. Each message is chosen from one of the following sets, with j ∈ { B,C } and /divides.alt0 M /divides.alt0 denoting the cardinality of M . Since the broadcast channel E is noisy, Alice needs to do a suitable block coding on the possible messages and then make n independent uses of the channel in order to be able to reliably convey the information. More precisely, Alice maps each message triple ( m B , m, m C ) to a codeword x n ( m B , m, m C ) which is then associated with a quantum state ρ A n x n ( m B ,m,m C ) defined in the space H ⊗ n A . Then, she transmits ρ A n x n ( m B ,m,m C ) by making n independent uses of the channel E . The output of the channel is the state defined on H ⊗ n B ⊗ H ⊗ n C . To decode the message, Bob chooses a positive-operator valued measure (POVM) { F B n m B ,m /divides.alt0 ( m B , m ) ∈ M B × M } which acts on the system B n . Similarly, Charlie chooses a POVM { G C n m,m C /divides.alt0 ( m,m C ) ∈ M × M C } which acts on the system C n . We say that an error has occurred when at least one message is incorrectly decoded. Hence, the error probability associated with the transmission of the triple ( m B , m, m C ) is The transmission rates associated with each message are defined as These rates essentially measure how many bits of classical information are sent per channel use. If, given an ϵ > 0, the average probability of error p e is bounded by ϵ , i.e., the classical-quantum broadcast channel coding protocol described above is said to be a ( n, R B , R, R C , ϵ ) code. We say that a rate triple ( R B , R, R C ) is achievable if given ϵ, δ > 0 there exists a ( n, R B -δ, R -δ, R C -δ, ϵ ) code for sufficiently large n . Hence, saying that a rate triple is achievable means that classical information can be reliably transmitted at rates arbitrarily close to them. The achievable rates depend highly on the coding and decoding techniques chosen by the sender and receivers. The best known achievable rate region for general broadcast channels is attained through the so-called Marton coding scheme . Following [3], we investigate here the quantum version of this protocol. Suppose for simplicity that no common message is meant to be sent, i.e., let us consider a ( R B , 0 , R C ) quantum broadcast channel. In this scenario, one strategy they can use is the Marton coding scheme , where one chooses two correlated random variables U and V , with joint probability distribution denoted by p and reduced probability distributions denoted by p U and p V . Such a pair of random variables is usually referred to as binning variables . Then, for each m B ∈ M B and m C ∈ M C , one generates codewords u n ( m B ) and v n ( m C ) according to the reduced probability distributions p U ( u ) and p V ( v ) . Next, the codewords are mixed together into a single codeword x n ( m B , m C ) according to a deterministic function x = f ( u, v ) . With this approach, it follows that a rate pair ( R B , R C ) is achievable if it satisfies [3] where is the mutual information of a state ρ XY , with α = X,Y , being the von Neumann entropy of ρ α , α = X,Y . Here, ρ X = tr Y ρ XY and ρ Y = tr X ρ XY . The states σ in Eqs. (68)-(70) are obtained by suitably (partially) tracing out the degrees of freedom of the density matrix with p ( u, v ) being the joint probability distribution of the random variables U and V . We begin our analysis by deriving bounds for the achievable rates through the Marton coding scheme applied to our relativistic quantum broadcast channel. To evaluate Eq. (68), we take partial traces relative to V and C in Eq. (72), obtaining where we have written p ( u, v ) = p V /divides.alt0 U ( v /divides.alt0 u ) p U ( u ) , whereas A state like σ UB in Eq. (73) is called a classical-quantum state . For this class of states, a straightforward calculation shows that [2] In order to compute ω B u and its von Neumann entropy, let us decompose the initial state of Alice's qubit in terms of Bloch vectors, i.e., where r f ( u,v ) ≡ ( x f ( u,v ) , y f ( u,v ) , z f ( u,v ) ) , I A is the identity in H A , σ A ≡ ( σ x A , σ y A , σ z A ) , and /parallel.alt1 r f ( u,v ) /parallel.alt1 2 = x 2 f ( u,v ) + y 2 f ( u,v ) + z 2 f ( u,v ) ≤ 1. From Eqs. (62), (74), and (76) we get where z u ≡ ∑ v p V /divides.alt0 U ( v /divides.alt0 u ) z f ( u,v ) , and thus we can further write where z ≡ ∑ u,v p ( u, v ) z f ( u,v ) . Now, by using standard diagonalization, we find that ω B u has eigenvalues p B u and 1 -p B u , where whereas ω B has eigenvalues p B and 1 -p B , with Therefore, we can now write Eq. (75) as where H ( x ) ≡ -x log 2 x - ( 1 -x ) log 2 ( 1 -x ) , x ∈ [ 0 , 1 ] . Following similar steps, we can show that where with z v ≡ ∑ u p U /divides.alt0 V ( u /divides.alt0 v ) z f ( u,v ) , and Now, let us note that H ( x ) is a monotonically decreasing function when x ≥ 1 /slash.left 2. From Eqs. (79) and (80), we have and and thus it follows that and As a result, from Eq. (81), we conclude that where is the classical capacity of the reduced channel E B , given in Eq. (62), as shown in [30]. We note that the upper bound in Eq. (89) can be attained if we choose random variables U,V = { 0 , 1 } with p ( u, v ) = 1 /slash.left 4 for all u, v , associated with Bloch vectors r f ( 0 , 0 ) = r f ( 0 , 1 ) = ( 0 , 0 , + 1 ) and r f ( 1 , 0 ) = r f ( 1 , 1 ) = ( 0 , 0 , -1 ) . By using such choices together with Eq. (68), we conclude that Alice can reliably convey classical information to Bob at rates arbitrarily close to C(E B ) . Similarly, we can show from Eqs. (82)-(84) that where is the classical capacity of the reduced channel E C given in Eq. (63). The upper bound can be attained, e.g., if we choose random variables U,V = { 0 , 1 } with p ( u, v ) = 1 /slash.left 4 for all u, v , associated with Bloch vectors r f ( 0 , 0 ) = r f ( 1 , 0 ) = ( 0 , 0 , + 1 ) and r f ( 0 , 1 ) = r f ( 1 , 1 ) = ( 0 , 0 , -1 ) . Hence, from Eq. (69), we conclude that Alice can reliably convey classical information to Charlie as well at rates arbitrarily close to C(E C ) . It is important to highlight that causality is explicitly manifest on the bounds of the achievable rates. First, we note that the achievable rates R B between Alice and Bob are bounded by C(E B ) , which does not depend on the interaction between Charlie's qubit and the quantum field. This should indeed be the case as, from hypothesis (QB2) in Sec. III, Charlie cannot influence the communication between Alice and Bob since he does not perform any actions before Bob finishes his measurement process. Furthermore, the presence of the commutator ∆ ( f B , f C ) in Eq. (92) indicates that when Bob and Charlie let their qubits interact with the quantum field in causally connected regions of the spacetime, noise from Bob's actions can influence on the rate R C of communication between Alice and Charlie. Additionally, we note that whenever ∆ ( f A , f j ) = 0, we have for j = B,C . Hence, when Alice and Bob (or Charlie) interact with the field in causally disconnected regions of the spacetime, the achievable rate in Eq. (68) (or Eq. (69)) will reduce to R B = 0 (or R C = 0). To this day, no one has been able to prove that the Marton rate region given by Eqs. (68)-(70) is optimal for general broadcast channels, not even in the classical case. However, it is generally conjectured that the Marton rate region indeed represents the full capacity region of general broadcast channels. If this is the case, then our analysis shows that causality will not be violated when transmitting classical information, no matter which communication protocol is chosen.",
"pages": [
7,
8,
9
]
},
{
"title": "B. Unassisted and entanglement-assisted quantum communication",
"content": "Let us now turn our attention to the communication of quantum information. Following [5], we present a father protocol for entanglement-assisted quantum communication through quantum broadcast channels that can be used to investigate at which rates Alice can send classical or quantum information to Bob and Charlie when they share an unlimited supply of entanglement. This protocol can also be adapted to investigate communication rates for quantum information transmission with no prior shared entanglement. After reviewing the father protocol, we investigate both quantum communication scenarios applied to our model of quantum broadcast channel constructed in Sec. III. Let us suppose that Alice has access to two quantum systems T A and T A ' while Bob and Charlie possess similar quantum systems T B and T C , respectively. All systems possesses the same dimension d T A ≡ dim H T A . Suppose further that Alice shares maximally entangled states with both Bob and Charlie: where the above state is defined on H T A ⊗ H T k , with k = B,C and {/divides.alt0 i /uni27E9 T α } is an orthonormal set of vectors on H T α , α = A,B,C . In order to study the transmission of quantum information, we first note that whenever Alice is able to transmit the entanglement she shares with some reference system to each receiver, she will be able to send arbitrary quantum states to each of them. Hence, suppose that Alice possesses two quantum systems A 1 and A 2 respectively entangled with reference systems R 1 and R 2 and that these systems are in states /divides.alt0 Φ A j R j /uni27E9 defined on H A j ⊗H R j for j = 1 , 2 5 . Her goal is to send her share of /divides.alt0 Φ A 1 R 1 /uni27E9 and /divides.alt0 Φ A 2 R 2 /uni27E9 to Bob and Charlie, respectively. The initial global state of the system is and we will denote ρ φ ≡ /divides.alt0 φ /uni27E9 /uni27E8 φ /divides.alt0 . In order to use the quantum channel E to share her entanglement with R 1 and R 2 to Bob and Charlie (and hence, convey quantum information), Alice uses a CPTP map C ∶ H A 1 ⊗H A 2 ⊗H T A ⊗H T A ' →H ⊗ n A in order to encode her shares of the quantum systems ( T A , T A ' , A 1 , and A 2 ), into a state of n qubits. The global state will then reads where I R 1 R 2 T B T C is the identity operator of the joint system R 1 R 2 T B T C . Next, by making n independent uses of the channel E , Alice sends her total encoded state to Bob and Charlie, which results in the global state Bob and Charlie decode their share of the global state by using the CPTP maps D B ∶ H ⊗ n B ⊗H T B → H B ' and D C ∶ H ⊗ n C ⊗H T C → H C ' , respectively. Hence, the final global state is We define the entanglement-assisted quantum communication rates as where d A j ≡ dim H A j and j = 1 , 2. These rates of quantum communication measure how many qubits are being sent per channel use. The communication process will be good if given a small ϵ > 0 we have where is the trace norm of an operator O . Here, ρ B ' C ' R 1 R 2 φ is the analogous of the initial state in the composite system B ' C ' R 1 R 2 , i.e., given the initial state in Alice's laboratory we define where I A 1 → B ' (or I A 2 → C ' ) is the identity map between the quantum systems A 1 (or A 2 ) and B ' (or C ' ). The communication protocol described here is named as a ( n, ̃ Q B , ̃ Q C , ϵ ) code if it satisfies Eq. (100) for every input state ρ A 1 A 2 R 1 R 2 φ . Again, we say that a rate pair ( ̃ Q B , ̃ Q C ) is achievable if given any ϵ, δ > 0 there exists a ( n, ̃ Q B -δ, ̃ Q C -δ, ϵ ) code for sufficiently large n . Now, given a general broadcast channel E ∶ A → BC and an arbitrary mixed state ρ AA 1 A 2 defined on H A ⊗ H A 1 ⊗ H A 2 , it can be shown [5] that a entanglementassisted quantum rate pair ( ̃ Q B , ̃ Q C ) is achievable if where the mutual information quantities are evaluated relative to the state In addition to entanglement-assisted quantum communication, the father protocol presented here can be adapted to obtain achievable rates for unassisted quantum communication. To this end, we simply ignore the existence of the quantum systems T A , T A ' , T B , and T C and follow the exact same procedure. As shown in [5], given an arbitrary mixed state ρ AA 1 A 2 defined on H A ⊗H A 1 ⊗H A 2 , it follows that the following unassisted quantum rate region is achievable: where σ is given by Eq. (107) and is the quantum coherent information between systems A and B . Now, let us return to the relativistic quantum broadcast channel constructed in Sec. III. To analyze if Alice can send entanglement (and, as a result, an arbitrary state ρ A ) to Bob through the broadcast channel, let us note that we may purify the mixed state ρ AA 1 A 2 by adding an environment system E such that where /divides.alt0 ψ AA 1 A 2 E /uni27E9 ∈ H A ⊗H A 1 ⊗H A 2 ⊗H E is a pure state. Let us decompose it as where /divides.alt0 a /uni27E9 A , /divides.alt0 a 1 /uni27E9 A 1 , and /divides.alt0 a 2 /uni27E9 A 2 are eigenstates of σ z A , σ z A 1 , and σ z A 2 , respectively. Furthermore, {/divides.alt0 e /uni27E9 E } is some orthonormal basis for H E , with d ≡ dim H E being as large as needed, and By defining and we can write Eq. (110) as By using Eq. (115) in Eq. (107) and taking the partial trace over C and A 2 , we obtain where ζ A 1 aa ≡ tr A 2 ( ζ A 1 A 2 aa ) and we have used the fact that which can be proven by a direct calculation using Eq. (62). We define now the density matrices and with /parallel.alt1 ζ a /parallel.alt1 2 ≡ tr ( ζ A 1 aa ) . This allows us to rewrite Eq. (116) as and we note that tr ( S B a ) = tr ( τ A 1 a ) = 1 and Hence, we have shown that σ A 1 B is a separable state, which implies that the reduced channel from Alice to Bob lies in the class of the entanglement-breaking channels . As shown in [36], the coherent information is non-positive for separable states like σ A 1 B , i.e., Following similar steps, one can also show that I ( A 2 /uni27E9 C ) σ ≤ 0. As a result, the unassisted achievable rate region given by Eqs. (108) and (109) reduces to We do not know, to this day, if the region defined by Eqs. (108)-(109) characterizes the full capacity region for general quantum broadcast channels. If this is the case, our analysis implies that Alice cannot send qubits to the receivers without prior shared entanglement. Since the reduced channels E B and E C are entanglement-breaking, Alice cannot transmit the needed entanglement to establish quantum communication by using only the quantum broadcast channel E . On the other hand, we can investigate if this limitation changes when the three observers perform an entanglement-assisted quantum communication protocol like the one in the beginning of this section. In this scenario, we recall that Eqs. (104)-(106) give an achievable (entanglement-assisted) quantum rate region that we shall analyze now. First, we show that Alice can indeed send quantum information to Bob when they initially share entanglement as follows. We choose the initial input state to be where ρ A 2 is arbitrary and /divides.alt0 Φ AA 1 /uni27E9 is the maximally entangled state For this particular state, Eq. (120) can be written as which is a cq-state as the one in Eq. (73). Following the same steps that led to Eq. (81), one can show that where C(E B ) is defined in Eq. (90). On the other hand, for this particular input state, we get where S C a ≡ E C (/divides.alt0 a /uni27E9 AA /uni27E8 a /divides.alt0) . Since we get these product states, it follows that In view of Eqs. (104)-(106), we conclude that Alice will be able to convey quantum information to Bob at a rate arbitrarily close to when they initially share unlimited amounts of entanglement, provided that they let their qubits interact with the field in causally connected regions of the spacetime. Note that this is in contrast with the unassisted case previously discussed. On the other hand, this particular achievable rate region derived here implies that ̃ Q C = 0 in view of Eq. (130). Similarly, by switching A 1 by A 2 in Eq. (124), one can show that Alice will be able to transmit quantum states to Charlie (but not to Bob) at a rate arbitrarily close to when they communicate assisted by shared entanglement. Furthermore, initial tripartite entangled states ρ AA 1 A 2 will, in general, lead to simultaneously nonvanishing rate pairs provided that sender and receivers interact with the field in causally connected regions of spacetime. For example, by choosing the initial input state to be a pure maximally entangled GHZ state where one can show by a similar direct calculation that the following entanglement-assisted quantum rate region is achievable: In [37], the authors investigate the classical channel capacity C(E B ) of the reduced channel E B from Alice to Bob for the case where both observers follow inertial or accelerated worldlines on Minkowski spacetime. In some scenarios, the authors show that one can tune the parameters of the channel to achievable capacities close to 1. Considering that C(E C ) has a similar expression, if we consider the case where ∆ ( f B , f C ) ≈ 0, we can argue in favor of also tune the channel parameters to make C(E C ) close to 1. Under these assumptions, Eqs. (135)(137) imply that Alice would be able to simultaneously transmit quantum information to Bob and Charlie when assisted by prior entanglement. Hence, the relativistic quantum broadcast constructed in Sec. III seems to impose no limitation to simultaneous entanglement-assisted communication from Alice to both receivers.",
"pages": [
9,
10,
11,
12
]
},
{
"title": "V. CONCLUSIONS",
"content": "In this paper, we have built a relativistic quantum broadcast channel by using a bosonic quantum field in a general globally hyperbolic spacetime. In this context, we have explored relativistic effects on the communication of classical and quantum information in a covariant manner, where the parts conveying the information are moving in arbitrary states of motion with the field being in an arbitrary (quasifree) state. To construct the quantum broadcast channel, we have considered that Alice (the sender) prepares some input state ρ A -∞ for her qubit and switches on its interaction with the field for a finite time. After that, Bob (the first receiver) lets his qubit interact with the field for a finite time interval, thus obtaining a final state possibly containing information encoded by Alice. Similarly, after Bob finishes his measurement, Charlie performs an interaction between his qubit and the quantum field to try to recover information imprinted by Alice in the field state. We were able to trace the field degrees of freedom nonperturbatively and showed that suitable initial states for Bob's and Charlie's qubits can be chosen in order to maximize the signaling between Alice and the receivers. This procedure defines a fully relativistic quantum broadcast channel E . With this channel, we were able to investigate at which rates Alice can reliably convey classical and quantum information to Bob and Charlie. By considering first a scenario where the three observers do not share prior entanglement, we found that Alice can reliably convey classical information to both Bob and Charlie and at which rates she can perform this task. However, we have shown that the broadcast channel presented here breaks entanglement and thus, Alice cannot convey quantum information to Bob and Charlie following an unassisted strategy. Nevertheless, we have shown that this situation changes when they perform entanglement-assisted quantum communication. In this scenario, we were able to find achievable rates that Alice can achieve when sending qubits to the receivers provided that they initially share entangled states. We were also able to show that all rates that were analyzed here vanish when the interactions between qubits and field occur in causally disconnected regions, an effect that is manifest in all expressions bounding the classical and quantum rates of communication even with the use of quantum resources like entanglement. Thus, our investigation provides good evidence that causality is not violated throughout the communication process, reinforcing the fundamental principles of relativistic physics. Our study shows that quantum network information theory in general spacetimes is a rich and promising area of research, shedding light on several aspects of the interplay between quantum information theory and relativity. We believe that this work may provide tools to investigate open problems concerning quantum gravity, in particular, the fate of the information that has fallen in (evaporating) black holes. The preservation of causality observed in our analysis reaffirms the robustness of fundamental physical principles, even in the realm of quantum information theory in curved spacetimes. We hope that following the path we presented here could lead us to unveil fundamental aspects of physics that should be present in a full quantum theory of gravity.",
"pages": [
12
]
},
{
"title": "ACKNOWLEDGMENTS",
"content": "I. B. and A. L. were fully and partially supported by S˜ao Paulo Research Foundation under Grants 2018/23355-2 and 2017/15084-6, respectively.",
"pages": [
12
]
},
{
"title": "Appendix A: Full expression for the quantum broadcast channel map",
"content": "As discussed in Sec. III, each Γ αβγδϵζ coefficient defined in Eq. (45) can be evaluated by using Eqs. (40) and (41) together with the product relation given by Eq. (46). Then, we substitute these coefficients in Eq. (42), obtaining where we have defined the following coefficients: As discussed in Sec. III, we are motivated to fix the initial states for Bob's and Charlie's qubit as given in Eqs. (52) and (58). We write these states in terms of their Bloch vectors, i.e., where j = B,C . By substituting Eq. (A6) in Eq. (A1), and by using the standard commutation relations of the Pauli matrices, we obtain the following expression describing the quantum broadcast channel map: By taking partial traces relative to each qubit, one recovers Eqs. (62) and (63). . . .",
"pages": [
13,
14
]
}
]
2024PhRvD.109h3009L
https://arxiv.org/pdf/2403.20038.pdf
<document>
<section_header_level_1><location><page_1><loc_14><loc_92><loc_87><loc_93></location>Probing solar modulation analytic models with cosmic ray periodic spectra</section_header_level_1>
<text><location><page_1><loc_38><loc_89><loc_62><loc_90></location>Wei-Cheng Long and Juan Wu ∗</text>
<text><location><page_1><loc_19><loc_88><loc_82><loc_89></location>School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China</text>
<text><location><page_1><loc_18><loc_58><loc_83><loc_86></location>The AMS02 experiment has published the periodic spectra of proton, helium and helium isotopes across the majority of the 24 solar cycle. These precise data exhibit temporal structures that correlate with solar modulation. In this study, we utilize these data to probe three analytic solar modulation models, including the force-field approximation, the convection-diffusion model and the extended force-field approximation with a drift effect. We adopt a method that eliminates the influence of interstellar cosmic ray spectra, and use the Earth-observed spectra at time t 1 to predict those at time t 2 . In order to explore the rigidity-dependence of solar modulation models, we substitute the conventional potential parameter φ with a modified parameter φ ' = R k 2 ( R ) φ for our analysis. Combining with the χ 2 minimization method, the best-fit modulation parameter φ ' can be evaluated. First, we test the validity of a rigidity-independent φ ' and find that both the force-field approximation (FFA) and the extended force-field approximation (EFFA) agree well with data near the solar minimum period. However, all models significantly deviate from the data during the solar maximum. Consequently, we assume a constant φ ' ( t 1 ) at solar minimum and calculate ∆ φ ' = φ ' ( t 2 ) -φ ' ( t 1 ) for each rigidity bin at time t 2 . It is found that ∆ φ ' generally adheres to a linear-logarithm relationship with rigidity at any given time. By adopting a linear-logarithm formula of ∆ φ ' , we further discover that both the modified FFA and EFFA can reconcile the observations during solar maxima. This suggests that at solar maximum, the parameter φ ' , which correlates with the diffusion pattern in the heliospheric magnetic fields, exhibits a rigidity dependence. Moreover, the modified EFFA enhances the concordance with data during periods of pronounced dips as observed by AMS02. This implies that the drift effect could significantly contribute to these solar transient phenomena.</text>
<text><location><page_1><loc_18><loc_56><loc_27><loc_57></location>PACS numbers:</text>
<text><location><page_1><loc_18><loc_53><loc_83><loc_55></location>Keywords: Particle astrophysics (96) - Cosmic rays (329) - Heliosphere (711) - Solar activity (1475) Solar cycle (1487) - Solar magnetic fields (1503)</text>
<section_header_level_1><location><page_1><loc_42><loc_47><loc_59><loc_48></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_36><loc_92><loc_45></location>Due to the interaction with the heliospheric magnetic fields (HMF) embedded in the solar wind [1], the cosmic ray (CR) energy spectra detected at the top of the atmosphere(TOA) of Earth differ from those at the local interstellar space (LIS). The entire process, which CRs undergo within the solar system, is referred to as solar modulation. During the passages of CR particles traversing through the interplanetary space, the modulation effects for them include convection driven by the solar wind, diffusion induced by the small-scale magnetic field irregularities, drift occurring in the large-scale magnetic structures and adiabatic losses due to the expansion of the solar wind [2].</text>
<text><location><page_1><loc_9><loc_25><loc_92><loc_36></location>Solar modulation effects are inherently influenced by the solar activities. The number and surface area of sunspots relate with the intensity of solar activity, and observations on them show a 11-year solar cycle. Meanwhile, the solar magnetic field undergoes a polarity reversal at the solar maximum, suggesting a 22-year cycle for solar activity [3, 4]. These periodic variations in solar activities causes both the solar modulation and the cosmic ray energy spectra, which are affected by solar modulation, to change periodically over time. Direct observation experiments such as PAMELA and AMS-02 have performed long-term measurements on CRs [5-11]. The unprecedented accurate periodic data reveal that the CR intensities display time structures that are anti-correlated with solar activities. These data offer substantial potential for us to reveal the features of solar modulation.</text>
<text><location><page_1><loc_9><loc_15><loc_92><loc_25></location>The CRs' transport processes in the heliosphere is usually described by Parker equation [12]. It can be solved either through numerical methods or analytical methods. While the numerical models provide more accurate and physically reasonable solutions [13-16], they necessitate a comprehensive understanding on the details of various physical quantities in the heliosphere, coupled with strong computational power. In contrast, analytical methods, which rely on a set of simple assumptions, may yield less precise results. However, these methods are computationally efficient and hence frequently employed by researchers. The most commonly used analytic models are the force-field approximation (FFA) and the convection-diffusion model (CD) [17]. These models describe solar modulation by using</text>
<text><location><page_2><loc_9><loc_89><loc_92><loc_93></location>a single parameter and greatly enhance the convenience of their applicability [18, 19]. A robust solar modulation model is crucial for comprehending CR acceleration and propagation [20-24] as well as for detecting dark matter signal [25-28].</text>
<text><location><page_2><loc_61><loc_79><loc_61><loc_80></location>glyph[negationslash]</text>
<text><location><page_2><loc_9><loc_72><loc_92><loc_89></location>As we known, the modulation effects might be very sensitive to the particle's rigidity[29]. For a CR particle, the diffusion coefficient is κ = vλ/ 3, in which v is the particle's velocity and λ is its mean free path. This coefficient is important for us to understand the scattering of particles on the random heliospheric magnetohydrodynamic waves and discontinuities. When the particle resonates with the HMF fluctuations with a spectrum w ( k ) ∼ k -η , where k is the resonant wave number, the diffusion mean free path can be expressed as λ ∼ R 2 -η [30-34]. A fundamental prerequisite for FFA is that the mean free path λ of a particle is proportional to its rigidity, i.e., η = 1. However, the real relationship between λ and the rigidity remains uncertain. If η = 1, the modulation parameter may have a different rigidity dependence. In recent years, there has been a lot of research focusing on the modification of FFA. Some studies have investigated the rigidity dependence of FFA and have updated FFA analytic formula by introducing additional parameters [35-37]. There are also some studies [38, 39] on FFA have considered a drift effect, which is prominent near the solar minimum and negligible during the HMF polarity reversal period [40]. The inclusion of this effect extend FFA into a charge-sign dependent model.</text>
<text><location><page_2><loc_9><loc_56><loc_92><loc_71></location>However, these studies usually firstly assumed CR LIS spectra, and then combined the experimental data to constrain the modulation parameters. It is known that CR LIS spectra have only been measured by Voyager-1 below a few hundred MeV [41]. Above this energy, no experimental data exist, leading to the adoption of different LIS models in the literature. Therefore, the analysis results might be biased due to inaccurate assumptions of LIS spectra. To get rid of the impact of LIS energy spectra, an alternative method (herein referred to as the Non-LIS method here) was proposed in [42]. In their work, they rewrote the FFA formula by removing the term of LIS spectra. Instead, they calculated the CR TOA intensity at time t 2 ( J ( t 2 )) based on the TOA intensity at t 1 ( J ( t 1 )). This method does not require an assumption of LIS spectra, but only needs to use the periodic CR data for the analysis. An important finding from their work was that the validity of FFA varies at different periods of solar activity. In this work, we will utilize the Non-LIS method to further investigate the rigidity dependence of the solar modulation effect for various analytic modulation models, including FFA, CD and an extended FFA (with a drift effect).</text>
<section_header_level_1><location><page_2><loc_22><loc_52><loc_78><loc_53></location>II. DESCRIPTION OF ANALYTIC SOLAR MODULATION MODELS</section_header_level_1>
<section_header_level_1><location><page_2><loc_39><loc_48><loc_62><loc_49></location>A. Force-field approximation</section_header_level_1>
<text><location><page_2><loc_10><loc_45><loc_88><loc_46></location>The basic transport equation(TPE) was first derived by Parker in the solar wind reference frame [17, 18, 43]:</text>
<formula><location><page_2><loc_32><loc_41><loc_92><loc_44></location>∂f ∂t + ∇· ( V f -K · ∇ f ) -1 3 p 2 ( ∇· V ) ∂f ∂p ( p 3 f ) = Q, (1)</formula>
<text><location><page_2><loc_9><loc_32><loc_92><loc_39></location>where f ( r , p, t ) is the phase space density or the omini-directional distribution function of CRs as a function of position r , momentum p and time t . It is linked to the differential intensity in terms of energy by: J T = p 2 f ( r , p, t ). In Eq. (1equation.2.1), V is the solar wind velocity, K is the diffusion tensor and Q is the local sources in the heliosphere. The diffusion tensor can be expressed as K = K a + K s , in which K s denotes the symmetrical component of the diffusion tensor and K a represents the asymmetrical component associated with the drift effect.</text>
<text><location><page_2><loc_10><loc_31><loc_71><loc_32></location>After that, an equivalent equation was derived in the observer's reference frame [44]:</text>
<formula><location><page_2><loc_33><loc_27><loc_92><loc_30></location>∂f ∂t + ∇· ( C V f -K · ∇ f ) + 1 p 2 ∂ ∂p ( p 2 〈 ˙ p 〉 f ) = Q. (2)</formula>
<text><location><page_2><loc_9><loc_23><loc_92><loc_25></location>Here C = -(1 / 3) ∂ ln f/∂ ln p is the Compton-getting coefficient, which corrects the anisotropy of transformation from the wind reference frame to the stationary reference frame [45].</text>
<text><location><page_2><loc_9><loc_16><loc_92><loc_23></location>FFA solves Eq. (2equation.2.2) under a series of assumptions: a) no local source of CRs, i.e. Q = 0; b) the existence of a steady state, i.e. ∂f/∂t = 0; c) spherically symmetric and ignoring the drift effect; d) an adiabatic rate 〈 ˙ p 〉 = ( p/ 3) V · ∇ f/f = 0. Additionally, the streaming term ( C V f -K · ∇ f ) is assumed to be divergence free and only the radial component of K , denoted as κ for convenience, is nonzero. Consequently, Eq. (2equation.2.2) can be written as the so-called force-field equation:</text>
<formula><location><page_2><loc_44><loc_11><loc_92><loc_14></location>∂f ∂r + V p 3 κ ∂f ∂p = 0 , (3)</formula>
<text><location><page_2><loc_9><loc_9><loc_92><loc_10></location>where κ is the one-dimension diffusion coefficient, r is the radial position and V is the radial component of solar wind</text>
<text><location><page_3><loc_9><loc_92><loc_15><loc_93></location>velocity.</text>
<text><location><page_3><loc_10><loc_90><loc_64><loc_92></location>It is a first-order partial differential equation with a characteristic equation</text>
<formula><location><page_3><loc_47><loc_86><loc_92><loc_89></location>d p d r = pV 3 κ . (4)</formula>
<text><location><page_3><loc_10><loc_84><loc_37><loc_85></location>The solution of Eq. (3equation.2.3) is</text>
<formula><location><page_3><loc_39><loc_81><loc_92><loc_82></location>f ( r TOA , p TOA ) = f ( r LIS , p LIS ) , (5)</formula>
<text><location><page_3><loc_9><loc_77><loc_92><loc_80></location>where r LIS , p LIS and r TOA , p TOA are the positions and momenta of CR particles in LIS and at TOA, respectively, and f ( r, p ) is the density of CRs along the characteristic curve described by Eq. (4equation.2.4).</text>
<text><location><page_3><loc_9><loc_73><loc_92><loc_77></location>Assuming the diffusion coefficient κ could be separated in radius distance r and rigidity R ≡ pc/Ze , i.e. κ ( r, R ) = βk 1 ( r ) k 2 ( R ), where Z is the charge number of the particle, β is the ratio of the particle velocity v and the speed of light c , we can define the modulation parameter φ as</text>
<formula><location><page_3><loc_35><loc_68><loc_92><loc_71></location>∫ R LIS R TOA k 2 ( R ) β R d R = ∫ r LIS r TOA V 3 k 1 ( r ) d r ≡ φ, (6)</formula>
<text><location><page_3><loc_9><loc_60><loc_92><loc_67></location>where R LIS and R TOA are the rigidities of CR particles in LIS and at TOA, respectively. Both φ and k 2 ( R ) are independent with the CR species but vary with time. Assuming that T LIS and T TOA represent the energies of a CR particle in LIS and at TOA, the energy loss experienced by this particle in the heliosphere can be denoted as Φ ≡ T LIS -T TOA . If Φ glyph[lessmuch] E 0 , where E 0 is the stationary energy of the particle, then the relationship between Φ and φ becomes [46]</text>
<formula><location><page_3><loc_43><loc_55><loc_92><loc_58></location>Φ( R ) = | Z | e R k 2 ( R ) φ. (7)</formula>
<text><location><page_3><loc_10><loc_53><loc_68><loc_54></location>From Eq. (7equation.2.7), we can define the modified modulation potential φ ' as</text>
<formula><location><page_3><loc_43><loc_48><loc_92><loc_51></location>φ ' ≡ Φ | Z | e = R k 2 ( R ) φ. (8)</formula>
<text><location><page_3><loc_9><loc_40><loc_92><loc_47></location>Taking λ ∝ R [47, 48], thus k 2 ( R ) = R . From the definition of Eq. (6equation.2.6), it can be found that φ is a rigidity-independent parameter. So Φ = Zeφ for any arbitrary rigidity. This leads to the conventional FFA, which condenses all the physical processes into a single parameter φ (or Φ) [46, 49]. It should be noticed that if k 2 ( R ) = R is not the condition, we may get more general results by substituting φ ' for φ . In this case, φ ' could become rigidity-dependent, and Φ can be expressed as Φ = Zeφ ' .</text>
<text><location><page_3><loc_10><loc_39><loc_90><loc_40></location>The modulated spectrum J ( r TOA , T TOA ) and the unmodulated LIS spectrum J ( r LIS , T TOA +Φ) are related as</text>
<formula><location><page_3><loc_31><loc_32><loc_92><loc_37></location>J ( r TOA , T TOA ) = T TOA ( T TOA +2 E 0 ) ( T TOA + Zeφ ' ) ( T TOA + Zeφ ' +2 E 0 ) × J ( r LIS , T TOA + Zeφ ' ) . (9)</formula>
<text><location><page_3><loc_9><loc_30><loc_92><loc_31></location>Therefore, by studying the modified potential parameter φ ' , we can investigate the FFA's rigidity dependence effect.</text>
<section_header_level_1><location><page_3><loc_37><loc_26><loc_64><loc_27></location>B. Convection-Diffusion equation</section_header_level_1>
<text><location><page_3><loc_9><loc_21><loc_92><loc_23></location>The other analytic modulation model CD can be directly derived from Eq. (1equation.2.1). The assumptions made in CD is similar as those given in FFA. Then Eq. (1equation.2.1) can be simplified into the CD equation</text>
<formula><location><page_3><loc_45><loc_17><loc_92><loc_19></location>V f -κ ∂f ∂r = 0 , (10)</formula>
<text><location><page_3><loc_9><loc_14><loc_26><loc_15></location>for which the solution is</text>
<formula><location><page_3><loc_37><loc_11><loc_92><loc_13></location>f ( r TOA , p TOA ) = f ( r LIS , p LIS ) e -M , (11)</formula>
<text><location><page_4><loc_9><loc_92><loc_13><loc_93></location>where</text>
<formula><location><page_4><loc_45><loc_88><loc_57><loc_91></location>M ≡ ∫ r LIS r TOA V κ d r.</formula>
<text><location><page_4><loc_9><loc_84><loc_92><loc_87></location>According to Eq. (6equation.2.6), Eq. (8equation.2.8) and Eq. (11equation.2.11), the relation between φ ( φ ' ) and M is [17, 50]</text>
<formula><location><page_4><loc_43><loc_79><loc_92><loc_82></location>M = 3 φ βk 2 ( R ) = 3 φ ' βR . (12)</formula>
<text><location><page_4><loc_9><loc_74><loc_92><loc_78></location>FFA and CD have been widely used due to their simplicity. Both of them can compress the modulation processes into one single parameter, which depends on the specific form of k 2 ( R ). Unlike FFA, which assumes k 2 ( R ) = R , CD does not place any constraints on k 2 ( R ). It may allow the modulation effect expected by CD to be rigidity-dependent.</text>
<section_header_level_1><location><page_4><loc_35><loc_70><loc_65><loc_71></location>C. Extended force-field approximation</section_header_level_1>
<text><location><page_4><loc_9><loc_65><loc_92><loc_68></location>Both FFA and CD neglect the drift effect. By incorporating the drift effect, Eq. (3equation.2.3) was led to Kuhlen's extended FFA (EFFA) equation [39]:</text>
<formula><location><page_4><loc_43><loc_61><loc_92><loc_64></location>∂f ∂r + pV 3 κ ∂f ∂p = v d,r κ f, (13)</formula>
<text><location><page_4><loc_9><loc_57><loc_89><loc_59></location>in which all the quantities are angular averaged, and v d,r is the radical component of the averaged drift velocity. The solution of Eq. (13equation.2.13) is</text>
<formula><location><page_4><loc_28><loc_52><loc_92><loc_56></location>f ( r TOA , p TOA ) = f ( r LIS , p LIS ) × exp [ -∫ r LIS r TOA dr v d,r ( r, p ) κ ( r, p ) ] . (14)</formula>
<text><location><page_4><loc_9><loc_47><loc_92><loc_51></location>whose characteristics curve is also Eq.(4equation.2.4). This solution's form seems to be a combination of FFA and CD, except for that the solar wind velocity in the integral term is replaced by the drift velocity. The rigidity dependence of v d,r correlates with the behavior of the anti-symmetrical diffusion coefficient κ a , and is given as [51]</text>
<formula><location><page_4><loc_43><loc_42><loc_92><loc_45></location>v d,r ∝ βR 3 B 10 R 2 1 + 10 R 2 , (15)</formula>
<text><location><page_4><loc_9><loc_40><loc_63><loc_41></location>where B is the magnitude of the heliospheric magnetic field on a large scale.</text>
<text><location><page_4><loc_9><loc_37><loc_92><loc_40></location>Given the similarity between the spatial integral part in Eq. (14equation.2.14) and the definition of M in Eq. (11equation.2.11), when Φ glyph[lessmuch] E 0 , this integral term can be associated with φ ' as</text>
<formula><location><page_4><loc_30><loc_32><loc_92><loc_36></location>∫ r LIS r TOA dr v d,r ( r, p ) κ ( r, p ) -→ g Rφ k 2 ( R ) 10 R 2 1 + 10 R 2 = gφ ' 10 R 2 1 + 10 R 2 , (16)</formula>
<text><location><page_4><loc_9><loc_28><loc_92><loc_31></location>where g is a scaling factor related to the magnitude of magnetic field B and the solar wind velocity V . A larger g indicates a stronger impact on the CR flux caused by the drift effect.</text>
<text><location><page_4><loc_9><loc_24><loc_92><loc_28></location>In summary, all three analytical models are derived from equivalent forms of TPE. Both FFA and CD are oneparameter models that disregard a term related to the adiabatic energy. But these terms differ as they are derived in different frames. Other than FFA and CD, Kuhlen's EFFA incorporates a drift effect and depends on two parameters.</text>
<section_header_level_1><location><page_4><loc_19><loc_20><loc_82><loc_21></location>III. MODULATION ANALYSIS BASED ON THE PERIODIC OBSERVATIONS</section_header_level_1>
<section_header_level_1><location><page_4><loc_40><loc_17><loc_60><loc_18></location>A. The Non-LIS method</section_header_level_1>
<text><location><page_4><loc_9><loc_9><loc_92><loc_15></location>In order to eliminate the influence of CR LIS energy spectra, we use the Non-LIS method to explore the general properties of above analytical models. Since the nature of the heliospheric diffusion effect remains incompletely understood, we refrain from specifying the formula for k 2 ( R ). Therefore, we use the modified modulation parameter φ ' to obtain the relationship between J ( t 1 ) and J ( t 2 ). For FFA, assuming ∆ φ ' = φ ' ( t 2 ) -φ ' ( t 1 ), Eq. (9equation.2.9)</text>
<text><location><page_5><loc_9><loc_92><loc_26><loc_93></location>can be transformed into</text>
<formula><location><page_5><loc_28><loc_86><loc_92><loc_91></location>J ( r TOA , T TOA , t 2 ) = T TOA ( T TOA +2 E 0 ) ( T TOA + Ze ∆ φ ' ) ( T TOA + Ze ∆ φ ' +2 E 0 ) × J ( r TOA , T TOA + Ze ∆ φ ' , t 1 ) . (17)</formula>
<text><location><page_5><loc_10><loc_83><loc_61><loc_84></location>Similar approaches can be applied to CD and EFFA. For CD, we yield</text>
<formula><location><page_5><loc_31><loc_79><loc_92><loc_82></location>J ( r TOA , T TOA , t 2 ) = J ( r TOA , T TOA , t 1 ) exp ( -3∆ φ ' βR ) . (18)</formula>
<text><location><page_5><loc_10><loc_76><loc_23><loc_78></location>For EFFA, we get</text>
<formula><location><page_5><loc_28><loc_69><loc_92><loc_75></location>J ( r TOA , T TOA , t 2 ) = T TOA ( T TOA +2 E 0 ) ( T TOA + Ze ∆ φ ' ) ( T TOA + Ze ∆ φ ' +2 E 0 ) × J ( r TOA , T TOA + Ze ∆ φ ' , t 1 ) exp ( -g 10 R 2 1 + 10 R 2 ∆ φ ' ) . (19)</formula>
<text><location><page_5><loc_9><loc_57><loc_92><loc_67></location>These three models are all written in terms of ∆ φ ' . As suggested in [42], traditional FFA may be reliable to describe the solar modulation effect around the solar minimum period. If we select t 1 near a period of minimal solar activity, it is reasonable to postulate that φ ' ( t 1 ) = φ ( t 1 ) is rigidity-independent. Therefore, the analysis on parameter ∆ φ ' can characterize the properties of parameter φ ' ( t 2 ). This approach will reveal the characteristics of solar modulation at any given time t 2 . For CD and EFFA, we also set t 1 near solar minimum and similarly assume φ ' ( t 1 ) is rigidityindependent. Subsequently, we use the free parameter ∆ φ ' to study modulation effect at other times. Note that in EFFA, except for ∆ φ ' , the factor g linked to the drift effect is also allowed to vary freely in the fittings.</text>
<section_header_level_1><location><page_5><loc_35><loc_53><loc_66><loc_54></location>B. Test of the rigidity-independent ∆ φ '</section_header_level_1>
<text><location><page_5><loc_9><loc_43><loc_92><loc_51></location>Synodic solar rotation causes the CR flux recurrent variations on the timescale of Bartels Rotations (BRs), which is 27 days for each BR. We use periodic data from AMS02 [8], which provides the measurements of proton (p) flux between 1 GV and 60 GV and helium (He) flux between 1.9 GV and 60 GV from May 2011 to May 2017. It also provides helium-3 ( 3 He) flux between 1.9 and 15 GV and helium-4 ( 4 He) flux between 2.1 GV and 21 GV from May 2011 to November 2017 [9]. The measurements covered most time of the 24 solar cycle, during which the solar maximum appeared in April 2014, and the HMF polarity reversed from A < 0 to A > 0 at that time.</text>
<text><location><page_5><loc_9><loc_29><loc_92><loc_42></location>For p and He, BR 2504 (February 18, 2017-March 16, 2017) is selected as t 1 in our work, since the measured flux at this time is higher than those at other phases. This indicates that the solar activity is weakest at this time. For 3 He and 4 He, since each of them has been measured in periods of 4 Bartels rotations (108 days), the period from BR 2502 to BR 2505 is selected as t 1 . It is assumed that the distribution of flux within rigidity bin ( R 1 , R 2 ) follows a power law. Consequently, the flux value at this bin is assigned to the interval center rigidity R = √ R 1 R 2 . We employed a cubic spline method to calculate the interpolated flux values for other rigidities within the rigidity range of observation, and utilize a power law distribution to extrapolate flux beyond this range. A least-square analysis using MINUIT package [52] is applied to obtain the ∆ φ ' values for each model, as well as the χ 2 /d.o.f values at time t 2 . For EFFA, the best-fit values of parameter g are also estimated.</text>
<text><location><page_5><loc_9><loc_15><loc_92><loc_30></location>First, we use AMS-02 p, He, 3 He and 4 He data to test the validity of a rigidity-independent ∆ φ ' for both FFA and CD. The results are shown in Fig. 1Top panel: the rigidity-independent parameter ∆ φ ' (=∆ φ ) estimated by using the p, He and He isotopes periodic data measured by AMS-02 for FFA and CD model. Bottom panel: the corresponding χ 2 /d.o.f at each BR. The shade area corresponds to the HMF polarity reversal period from A < 0 to A > 0 (2012 November-2014 March)figure.1. As we can see, for FFA, at any given time t 2 , He isotopes yield rather consistent values of ∆ φ ' with p and He. This is also the case for CD. For a given time t 2 , the best-fit ∆ φ ' parameter obtained in CD is higher than that given in FFA. The possible reason is that the integration of FFA from LIS to TOA is constrained by the characteristic curve, and the corresponding path of FFA is longer than that of CD for a same value of φ . In other words, the adiabatic energy term of TPE ignored in CD is larger than that ignored in FFA. For both models, the largest ∆ φ ' appears around 2014, which corresponds to a solar maximum.</text>
<text><location><page_5><loc_9><loc_9><loc_92><loc_15></location>During periods around 2011-2012 and 2016-2017, it is found that the χ 2 /d.o.f values are close to 1. This infers that FFA and CD can generally reproduce data during the low solar activity periods. But during the periods with high ∆ φ ' , the χ 2 /d.o.f values are much larger than 1. Especially, by using the p and He fluxes, the χ 2 /d.o.f values can achieve more than 20. It means that, for both FFA and CD, an rigidity-independent ∆ φ ' (or φ ' ) does not agree</text>
<text><location><page_6><loc_50><loc_93><loc_52><loc_93></location>Date</text>
<figure>
<location><page_6><loc_19><loc_63><loc_80><loc_93></location>
<caption>FIG. 1: Top panel: the rigidity-independent parameter ∆ φ ' (=∆ φ ) estimated by using the p, He and He isotopes periodic data measured by AMS-02 for FFA and CD model. Bottom panel: the corresponding χ 2 /d.o.f at each BR. The shade area corresponds to the HMF polarity reversal period from A < 0 to A > 0 (2012 November-2014 March).</caption>
</figure>
<text><location><page_6><loc_50><loc_55><loc_52><loc_55></location>Date</text>
<figure>
<location><page_6><loc_20><loc_23><loc_80><loc_55></location>
<caption>FIG. 2: Top and middle panels: the rigidity-independent parameter ∆ φ ' (=∆ φ ) and g estimated by using the p, He and He isotopes periodic data measured by AMS-02 for Kuhlen's EFFA. Bottom panel: the corresponding χ 2 /d.o.f at each BR. The shade area corresponds to the HMF polarity reversal period from A < 0 to A > 0 (2012 November-2014 March).</caption>
</figure>
<text><location><page_6><loc_9><loc_9><loc_92><loc_15></location>well with the p and He data during these periods with intense solar activity. This phenomenon is further confirmed in Fig. 3Top panels: The p and He fluxes at BR 2426 and BR 2504 expected from FFA, CD and Kuhlen's EFFA, in comparison with the AMS-02 measurements. Middle and bottom panels: the residuals of the model fittings to the p and He spectra for BR 2426 and BR 2463figure.3. At BR 2426 (May 5, 2011-June 10, 2011), a period after the</text>
<text><location><page_7><loc_9><loc_85><loc_92><loc_93></location>solar minimum in 2009, the solar activity is not strong. As we can see, at this BR, both FFA and CD generally give consistent results with the p data above 2 GV and the He data at the whole rigidity range , except for an obvious discrepancy exist between CD and the p data below 2 GV. But the p and He fluxes predicted by both models at BR 2463 (February 7, 2014-March 6, 2014), which is in a polarity reversal in solar cycle 24, show significant disagreements with the data. These disagreements indicate that both models with a rigidity-independent φ ' can not describe the solar modulation behavior well during the HMF polarity reversal period.</text>
<text><location><page_7><loc_9><loc_79><loc_92><loc_85></location>By using the 3 He or 4 He data, the calculated χ 2 /d.o.f values at solar maximum are not that high. This may be due to the large errors existing in the 3 He and 4 He data, which infers that using the He isotope data alone is not enough to test the validity of solar modulation models. Therefore, to further investigate EFFA, we use only the p and He data to run the analysis.</text>
<text><location><page_7><loc_9><loc_57><loc_92><loc_79></location>For EFFA, the time variations of ∆ φ and g are shown in Fig. 2Top and middle panels: the rigidity-independent parameter ∆ φ ' (=∆ φ ) and g estimated by using the p, He and He isotopes periodic data measured by AMS-02 for Kuhlen's EFFA. Bottom panel: the corresponding χ 2 /d.o.f at each BR. The shade area corresponds to the HMF polarity reversal period from A < 0 to A > 0 (2012 November-2014 March)figure.2. This model presents a similar tendency of ∆ φ ' ( φ ' ) in terms of time with FFA and CD. The scaling factor g does not show a clear variation with time. The minor fluctuations of g indicate the intensity of drift effect does not vary greatly in different periods. According to the χ 2 /d.o.f values given in Fig. 2Top and middle panels: the rigidity-independent parameter ∆ φ ' (=∆ φ ) and g estimated by using the p, He and He isotopes periodic data measured by AMS-02 for Kuhlen's EFFA. Bottom panel: the corresponding χ 2 /d.o.f at each BR. The shade area corresponds to the HMF polarity reversal period from A < 0 to A > 0 (2012 November-2014 March)figure.2, it can be found that EFFA can better fit the data than FFA and CD. At all the periods, EFFA reduces the χ 2 /d.o.f values by more than half. But EFFA still displays a poor performance during 2013-2014. This can also been seen in Fig. 3Top panels: The p and He fluxes at BR 2426 and BR 2504 expected from FFA, CD and Kuhlen's EFFA, in comparison with the AMS-02 measurements. Middle and bottom panels: the residuals of the model fittings to the p and He spectra for BR 2426 and BR 2463figure.3. It seems that including a drift effect still cannot explain the solar modulation effect during the polarity reversal period.</text>
<figure>
<location><page_7><loc_13><loc_21><loc_50><loc_56></location>
</figure>
<figure>
<location><page_7><loc_51><loc_21><loc_88><loc_56></location>
<caption>FIG. 3: Top panels: The p and He fluxes at BR 2426 and BR 2504 expected from FFA, CD and Kuhlen's EFFA, in comparison with the AMS-02 measurements. Middle and bottom panels: the residuals of the model fittings to the p and He spectra for BR 2426 and BR 2463.</caption>
</figure>
<section_header_level_1><location><page_8><loc_34><loc_92><loc_67><loc_93></location>C. Modulation with rigidity-dependent φ '</section_header_level_1>
<text><location><page_8><loc_9><loc_83><loc_92><loc_90></location>Since a rigidity-independent φ ' cannot accommodate the data well during the periods with intense solar activities, we only assume a constant φ ' at solar minimum period t 1 . But at other periods, we calculate ∆ φ ' at each rigidity bin to study the change of ∆ φ ' ( φ ' ( t 2 )) with rigidity. The analysis is performed both for AMS-02 p and He data and the results for FFA are presented in Fig. 4The variation of the best-fit parameter ∆ φ ' with rigidity and time for p (left panel) and He (right panel)figure.4. Noted that in this figure the x-axis is represented on a logarithmic scale.</text>
<figure>
<location><page_8><loc_13><loc_63><loc_47><loc_80></location>
</figure>
<figure>
<location><page_8><loc_51><loc_63><loc_84><loc_80></location>
<caption>FIG. 4: The variation of the best-fit parameter ∆ φ ' with rigidity and time for p (left panel) and He (right panel).</caption>
</figure>
<text><location><page_8><loc_9><loc_43><loc_92><loc_57></location>As the intensity of solar activity increases, the variation of ∆ φ ' with rigidity becomes more and more significant. This further confirms the necessary to introduce a rigidity-dependent φ ' during periods of high solar activity. We particularly show the relationships between ∆ φ ' and rigidity in Fig. 5The parameter ∆ φ ' as a function of rigidity in BR 2426, BR 2442, BR 2463, where BR 2426 is near the solar minimum, BR 2442 is one of the sharp dips observed in AMS02 [53], and BR 2463 is during the solar maximum periodfigure.5 for BR 2426, BR 2442 and BR 2463. It can be found that for all BRs, the curves are very close to straight lines in linear-logarithmic (lin-log) coordinates. Here we include BR 2442 in the plot because this BR is related to the location of sharp dips in the p and electron fluxes observed by AMS02 [53]. At this BR, ∆ φ ' has a slight downturn at very low rigidity. For CD and EFFA, ∆ φ ' have similar relations with rigidity. Therefore, we assume that ∆ φ ' has a lin-log relationship with rigidity, with the formula:</text>
<formula><location><page_8><loc_40><loc_40><loc_92><loc_42></location>∆ φ ' lin-log = φ 0 + φ 1 ln ( R/R 0 ) . (20)</formula>
<text><location><page_8><loc_35><loc_38><loc_65><loc_39></location>' '</text>
<text><location><page_8><loc_9><loc_19><loc_92><loc_39></location>where φ 0 is the normalization of ∆ φ at R 0 = 1 GV, and φ 1 is the slope of ∆ φ with ln R . They both vary with time. We adopt this lin-log formula of ∆ φ ' in FFA, CD and EFFA. In this case, the free parameters include φ 0 , φ 1 for FFA and CD, and φ 0 , φ 1 , g for EFFA. The predicted p and He flux are compared with the data measured at BR 2442 and BR 2463, as shown in Fig. 6Top panels: The p and He fluxes at BR 2426 and BR 2504 expected from the modified FFA, CD and Kuhlen's EFFA by introducing a linear-logarithm rigidity-dependent ∆ φ ' , in comparison with the AMS-02 measurements. Middle and bottom panels: the residuals of the model fittings to the p and He spectra for BR 2442 and BR 2463figure.6. At BR 2442 and 2463, the modified FFA can generally reproduce the p and He data at most rigidity range. It only gives slightly lower predictions than the p data below 2 GV. This might because ∆ φ lin-log does not give a perfect description of the p flux at very low rigidity, as exhibited in Fig. 5The parameter ∆ φ ' as a function of rigidity in BR 2426, BR 2442, BR 2463, where BR 2426 is near the solar minimum, BR 2442 is one of the sharp dips observed in AMS02 [53], and BR 2463 is during the solar maximum periodfigure.5. Nevertheless, the agreement between the modified FFA and the data is highly increased compared with the conventional FFA. This is also the case for modified EFFA. But the improvement of the modified CD is limited compared with the conventional CD. It give worse goodness of fit than the modified FFA and EFFA.</text>
<text><location><page_8><loc_9><loc_9><loc_92><loc_19></location>The χ 2 /d.o.f results of different models are summarized in Table IThe χ 2 /d.o.f results for different analytical models at BR 2426, BR 2442 and BR2463 based on the analysis of p or (and) He data. Here φ ' = φ assumes a rigidity-independent φ ' , and φ ' lin-log assumes a linear-logarithm rigidity-dependent φ ' table.1. As we can see, CD doesn't fit well with all the data. For BR 2426, both the FFA and EFFA with a rigidity-dependent or rigidityindependent φ ' agrees well with the p and He data. For BR 2442, the conventional FFA does not accommodate the data. The modified FFA improves the goodness-of-fit but still yield a χ 2 /d.o.f close to 2. By including a drift effect, both conventional EFFA and modified EFFA can reproduce the p and He data at BR 2442. It suggests that the flux</text>
<figure>
<location><page_9><loc_13><loc_73><loc_47><loc_92></location>
</figure>
<figure>
<location><page_9><loc_51><loc_73><loc_84><loc_91></location>
</figure>
<figure>
<location><page_9><loc_32><loc_53><loc_65><loc_71></location>
<caption>FIG. 5: The parameter ∆ φ ' as a function of rigidity in BR 2426, BR 2442, BR 2463, where BR 2426 is near the solar minimum, BR 2442 is one of the sharp dips observed in AMS02 [53], and BR 2463 is during the solar maximum period.</caption>
</figure>
<table>
<location><page_9><loc_31><loc_11><loc_69><loc_39></location>
<caption>TABLE I: The χ 2 /d.o.f results for different analytical models at BR 2426, BR 2442 and BR2463 based on the analysis of p or (and) He data. Here φ ' = φ assumes a rigidity-independent φ ' , and φ ' lin-log assumes a linear-logarithm rigidity-dependent φ ' .</caption>
</table>
<figure>
<location><page_10><loc_13><loc_58><loc_50><loc_94></location>
</figure>
<figure>
<location><page_10><loc_51><loc_58><loc_88><loc_94></location>
<caption>FIG. 6: Top panels: The p and He fluxes at BR 2426 and BR 2504 expected from the modified FFA, CD and Kuhlen's EFFA by introducing a linear-logarithm rigidity-dependent ∆ φ ' , in comparison with the AMS-02 measurements. Middle and bottom panels: the residuals of the model fittings to the p and He spectra for BR 2442 and BR 2463.</caption>
</figure>
<text><location><page_10><loc_9><loc_44><loc_92><loc_50></location>dips observed in AMS02 data may be associated with the drift effect. For BR 2463, the conventional FFA and EFFA have large disagreements with the data. But by adopting a rigidity-dependent φ ' , both models can explain the data well. For solar minimum and maximum phases, it is difficult do judge whether the drift effect needs to be introduced to interpret the data.</text>
<section_header_level_1><location><page_10><loc_32><loc_40><loc_69><loc_41></location>D. Compared with other modified FFA models</section_header_level_1>
<text><location><page_10><loc_9><loc_34><loc_92><loc_38></location>We compared our lin-log FFA and lin-log EFFA with other modified FFA models. One is Cholis' model [38, 54]. Instead of adding a drift term in the relationship between J ( r TOA , t 2 ) and J ( r TOA , t 1 ), they incorporated the drift term in φ . Based on their work, ∆ φ ' can be written as</text>
<formula><location><page_10><loc_37><loc_29><loc_92><loc_32></location>∆ φ ' Cholis = φ 0 + φ 1 ( 1 + ( R/R 0 ) 2 β ( R/R 0 ) 3 ) . (21)</formula>
<text><location><page_10><loc_9><loc_23><loc_92><loc_27></location>The other is Shen's model presented in [37]. In that paper, the authors attributed the variation of φ with energy to the behavior of diffusion coefficient. They used a double power-law empirical formula to describe φ . In Shen's model ∆ φ ' can be written as</text>
<formula><location><page_10><loc_34><loc_18><loc_92><loc_22></location>∆ φ ' Shen = φ 0 β -1 ( E E b ) φ 1 [ 1 + ( E E b 1 ) b 1 ] b 2 , (22)</formula>
<text><location><page_10><loc_9><loc_11><loc_92><loc_16></location>where E b = 1 GeV, φ 0 is a scaling factor in unit GV. The rest of the parameters are dimensionless. Both φ 0 and φ 1 vary with time, while E b 1 , b 1 and b 2 are time-independent parameters. It is worth to noticed that the estimations of all the parameters in Shen's model are adjustable to perform a good agreements with the data. This could result in overfitting and instability of the parameters [34, 55].</text>
<text><location><page_10><loc_10><loc_9><loc_92><loc_11></location>Both Cholis' and Shen's models include a β term from the diffusion coefficient into φ ' . The relationship between β</text>
<text><location><page_11><loc_9><loc_88><loc_92><loc_93></location>and rigidity shows that β is a function of A/Z . Thus for different particles, the same values of φ 0 and φ 1 may lead to different values of ∆ φ ' Shen (or ∆ φ ' Cholis ) for a given rigidity. This difference is slight in Cholis' model since β only exist in φ 1 term of ∆ φ ' Cholis . But from Eq. (8equation.2.8), we can see that a β term is unnecessary to be introduced in φ ' . The inclusion of β may be lack of rigorous theoretical basis.</text>
<text><location><page_11><loc_9><loc_82><loc_92><loc_87></location>Above models with a rigidity-dependent ∆ φ ' all contain two free parameters φ 0 and φ 1 . Other parameters are nuisances. The χ 2 minimization results of the lin-log FFA, the lin-log EFFA, Cholis' model and Shen's model are shown in Fig. 7The results of χ 2 /d.o.f over time obtained by fitting the p or (and) He data for the modified FFA, the modified EFFA, Cholis' model and Shen's modelfigure.7, respectively.</text>
<figure>
<location><page_11><loc_14><loc_41><loc_85><loc_80></location>
<caption>FIG. 7: The results of χ 2 /d.o.f over time obtained by fitting the p or (and) He data for the modified FFA, the modified EFFA, Cholis' model and Shen's model.</caption>
</figure>
<text><location><page_11><loc_9><loc_28><loc_92><loc_35></location>It can be found that the lin-log FFA give excellent goodness-of-fit for most periods. There are only a few BRs at which the values of χ 2 /d.o.f > 1. Peaks of χ 2 appears at BR 2437, BR 2442, BR 2453 and BR 2478. All these BRs happens only in A > 0 stage and corresponds to the sharp dips in AMS-02 p and electron fluxes. Notably, at these BRs, the lin-log EFFA agrees better with the AMS02 p and He data. It indicates that these solar transients on timescale of BRs maybe related with the drift effect.</text>
<text><location><page_11><loc_9><loc_21><loc_92><loc_28></location>For Cholis' model, there are much more BRs corresponding to χ 2 /d.o.f > 1, especially by fitting the p or p+He data. The largest χ 2 values exhibit during the solar reversal phase, which means Cholis' model is particularly poor to simulate the solar modulation during those stages. In this model, the χ 2 distribution over time is similar with that in conventional EFFA. This suggests that the introduction of a drift effect is not sufficient to explain the variation of φ ' with rigidity.</text>
<text><location><page_11><loc_9><loc_16><loc_92><loc_20></location>Shen's model could obtain good agreements with either p or He data. But when we combine p and He data to do the analysis, the resulted χ 2 /d.o.f values are large. The reason is that the estimated φ 0 and φ 1 deviate significantly between p and He. It reveals that Shen's model does not give consistent descriptions on p and He.</text>
<section_header_level_1><location><page_12><loc_34><loc_92><loc_66><loc_93></location>IV. CONCLUSION AND DISCUSSION</section_header_level_1>
<text><location><page_12><loc_9><loc_80><loc_92><loc_90></location>In this paper, we take into account three analytic solar modulation models: FFA, CD, and EFFA. To investigate these models, the Non-LIS method is employed to eliminate the impact of CR LIS spectra. The traditional potential parameter φ in FFA, CD and EFFA is a rigidity-independent parameter. However, since the radial diffusion coefficient may not be proportional to rigidity, the parameter Φ could be rigidity-dependent. Therefore, we introduce an alternative parameter φ ' = R k 2 ( R ) φ to revisit these models. By using ∆ φ ' = φ ' ( t 2 ) -φ ' ( t 1 ), we can determine the CR flux at time t 2 based on the observed CR flux at time t 1 . Then we use the χ 2 minimization analysis to estimate the best-fit ∆ φ ' .</text>
<text><location><page_12><loc_9><loc_68><loc_92><loc_80></location>First, it is found that the conventional FFA and EFFA with a rigidity-independent φ ' can describe the data well around solar minimum. But these models do not agree well with the data at HMF polarity reversal periods. Therefore, it is reasonable to assume a constant φ ' near solar minimum, but consider a rigidity-dependent φ ' (or ∆ φ ' ) for other periods. By calculating ∆ φ ' at different rigidity ranges, the results show that ∆ φ ' is not a constant but seems have a lin-log relationship with rigidity. By incorporating this lin-log formula of ∆ φ ' into FFA and EFFA models, we find that they can satisfactorily describe the data during the HMF polarity reversal stage. The CD models, no matter the conventional one or the modified one, cannot explain the data well. It infers that the ignored adiabatic term in CD plays a relatively important role in modulation, which could have a significant rigidity dependence.</text>
<text><location><page_12><loc_9><loc_54><loc_92><loc_68></location>The effect of drift may be important to explain the modulation during those solar transients detected by AMS02, since during those stages, the conventional and modified EFFA models can fit the date better than other models. The hysteresis-like loops (coinciding with the sharp dip times) between the proton and electron fluxes [53] or between the proton and antiproton fluxes [56] display a charge-sign-dependent solar modulation. This may be related to the fact that particles with opposite charge signs have different patterns of drift effects. Nevertheless, the variation of φ ' with rigidity is not mainly due to the drift effect, since Cholis' model have a worse performance than our lin-log FFA model. This suggests that the rigidity-dependence of the parameter φ ' mainly originates from the rigidity-dependence of diffusion coefficient. The lin-log FFA model is also better than Shen's model, in which they consider a double power-law φ ' . The specific form of φ ' is important for understanding of HMF fluctuations during the HMF polarity reversal periods.</text>
<text><location><page_12><loc_9><loc_48><loc_92><loc_53></location>A recent study by simultaneous scanning on the solar modulation parameter and other CR acceleration and propagation parameters, has suggested that the conventional FFA can describe well the CR spectra measured by AMS02 and Voyager-1 integrated over the entire detection period [57]. However, a rigidity-dependent φ ' may challenge our traditional understanding on CR acceleration and propagation mechanisms.</text>
<text><location><page_12><loc_9><loc_36><loc_92><loc_47></location>It should be noted that these analytic models are based on a series of assumptions. The dependence of modulation on A/Z is not studied in this work. But we find that during the sharp dips periods, the deviations of the He data from lin-log FFA are less significantly than those of the p data. This will be further studied in our future work. The solar modulation model proposed in our work enables us to place effective constraints on the CR source and propagation models. This allows for a reliable calculation on the CR LIS spectra. Some other studies have shown that it is also possible to derive the LIS spectra from synchrotron and gamma-ray observations without any assumption on solar modulation [58-61]. In our future work, we will further compare them for a better understanding on cosmic ray behaviors in Galaxy.</text>
<section_header_level_1><location><page_12><loc_44><loc_32><loc_57><loc_33></location>Acknowledgments</section_header_level_1>
<text><location><page_12><loc_9><loc_26><loc_92><loc_30></location>Thanks for Ilias Cholis, Claudio Corti and R.A. Caballero-Lopez for very helpful discussions. This work is supported by the Joint Funds of the National Natural Science Foundation of China (Grant No. U1738130). The use of the highperformance computing platform of China University of Geosciences is gratefully acknowledged.</text>
<unordered_list>
<list_item><location><page_12><loc_17><loc_18><loc_60><loc_19></location>[2] M. S. Potgieter, Living Reviews in Solar Physics 10 , 3 (2013).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_13><loc_17><loc_91><loc_92><loc_93></location>[6] M. Martucci, R. Munini, M. Boezio, V. D. Felice, O. Adriani, G. C. Barbarino, G. A. Bazilevskaya, R. Bellotti, M. Bongi, V. Bonvicini, et al., The Astrophysical Journal 854 , L2 (2018).</list_item>
</unordered_list>
<text><location><page_13><loc_17><loc_89><loc_92><loc_90></location>[7] N. Marcelli, M. Boezio, A. Lenni, W. Menn, R. Munini, O. P. M. Aslam, D. Bisschoff, M. D. Ngobeni, M. S.</text>
<text><location><page_13><loc_19><loc_88><loc_61><loc_89></location>Potgieter, O. Adriani, et al., The Astrophysical Journal Letters</text>
<text><location><page_13><loc_61><loc_88><loc_64><loc_89></location>925</text>
<text><location><page_13><loc_64><loc_88><loc_72><loc_89></location>, L24 (2022).</text>
<unordered_list>
<list_item><location><page_13><loc_17><loc_85><loc_92><loc_88></location>[8] AMS Collaboration, M. Aguilar, L. Ali Cavasonza, B. Alpat, G. Ambrosi, L. Arruda, N. Attig, S. Aupetit, P. Azzarello, A. Bachlechner, et al., Physical Review Letters 121 , 051101 (2018).</list_item>
</unordered_list>
<text><location><page_13><loc_17><loc_84><loc_92><loc_85></location>[9] AMS Collaboration, M. Aguilar, L. Ali Cavasonza, G. Ambrosi, L. Arruda, N. Attig, A. Bachlechner, F. Barao,</text>
<text><location><page_13><loc_19><loc_83><loc_54><loc_84></location>A. Barrau, L. Barrin, et al., Physical Review Letters</text>
<text><location><page_13><loc_54><loc_83><loc_57><loc_84></location>123</text>
<text><location><page_13><loc_57><loc_83><loc_67><loc_84></location>, 181102 (2019).</text>
<unordered_list>
<list_item><location><page_13><loc_16><loc_80><loc_92><loc_82></location>[10] AMS Collaboration, M. Aguilar, L. A. Cavasonza, G. Ambrosi, L. Arruda, N. Attig, F. Barao, L. Barrin, A. Bartoloni, S. Ba¸se˘gmez-du Pree, et al., Physical Review Letters 127 , 271102 (2021).</list_item>
<list_item><location><page_13><loc_16><loc_77><loc_92><loc_80></location>[11] AMS Collaboration, M. Aguilar, L. A. Cavasonza, G. Ambrosi, L. Arruda, N. Attig, F. Barao, L. Barrin, A. Bartoloni, S. Ba¸se˘gmez-du Pree, et al., Physical Review Letters 128 , 231102 (2022).</list_item>
<list_item><location><page_13><loc_16><loc_76><loc_56><loc_77></location>[12] E. N. Parker, Planetary and Space Science 13 , 9 (1965).</list_item>
<list_item><location><page_13><loc_16><loc_75><loc_64><loc_76></location>[13] R. Kappl, Journal of Physics: Conference Series 718 , 052020 (2016).</list_item>
<list_item><location><page_13><loc_16><loc_72><loc_92><loc_74></location>[14] A. Vittino, C. Evoli, and D. Gaggero, Proceedings of the 35th International Cosmic Ray Conference 301 , 024 (2017), arXiv:1707.09003 [astro-ph.HE].</list_item>
<list_item><location><page_13><loc_16><loc_69><loc_92><loc_72></location>[15] M. J. Boschini, S. Della Torre, M. Gervasi, G. La Vacca, and P. G. Rancoita, Advances in Space Research 62 , 2859 (2018).</list_item>
<list_item><location><page_13><loc_16><loc_67><loc_92><loc_69></location>[16] C. Corti, M. S. Potgieter, V. Bindi, C. Consolandi, C. Light, M. Palermo, and A. Popkow, The Astrophysical Journal 871 , 253 (2019).</list_item>
<list_item><location><page_13><loc_16><loc_66><loc_53><loc_67></location>[17] H. Moraal, Space Science Reviews 176 , 299 (2013).</list_item>
<list_item><location><page_13><loc_16><loc_64><loc_91><loc_65></location>[18] R. A. Caballero-Lopez and H. Moraal, Journal of Geophysical Research: Space Physics 109 , A01101 (2004).</list_item>
<list_item><location><page_13><loc_16><loc_63><loc_69><loc_64></location>[19] N. E. Engelbrecht and V. Di Felice, Physical Review D 102 , 103007 (2020).</list_item>
<list_item><location><page_13><loc_16><loc_60><loc_92><loc_63></location>[20] M. J. Boschini, S. Della Torre, M. Gervasi, D. Grandi, G. J'ohannesson, M. Kachelriess, G. La Vacca, N. Masi, I. V. Moskalenko, E. Orlando, et al., The Astrophysical Journal 840 , 115 (2017).</list_item>
<list_item><location><page_13><loc_16><loc_59><loc_54><loc_60></location>[21] N. Tomassetti, Physical Review D 96 , 103005 (2017).</list_item>
<list_item><location><page_13><loc_16><loc_58><loc_56><loc_59></location>[22] J. Wu and H. Chen, Physics Letters B 789 , 292 (2019).</list_item>
<list_item><location><page_13><loc_16><loc_56><loc_67><loc_57></location>[23] Y. Wang, J. Wu, and W.-C. Long, Chinese Physics C 46 , 095102 (2022).</list_item>
<list_item><location><page_13><loc_16><loc_55><loc_69><loc_56></location>[24] D. Maurin, F. B. E., and D. L., Astronomy and Astrophysics 667 , 1 (2022).</list_item>
<list_item><location><page_13><loc_16><loc_54><loc_70><loc_55></location>[25] J. Ellis, AIP Conference Proceedings 516 , 21 (2000), arXiv:astro-ph/9911440.</list_item>
<list_item><location><page_13><loc_16><loc_52><loc_78><loc_53></location>[26] Q. Yuan and X.-J. Bi, Journal of Cosmology and Astroparticle Physics 2015 , 033 (2015).</list_item>
<list_item><location><page_13><loc_16><loc_50><loc_92><loc_52></location>[27] H.-C. Cheng, W.-C. Huang, X. Huang, I. Low, Y.-L. Sming Tsai, and Q. Yuan, Journal of Cosmology and Astroparticle Physics 2017 , 041 (2017).</list_item>
<list_item><location><page_13><loc_16><loc_47><loc_92><loc_49></location>[28] Z. Cheng-Rui, C. Ming-Yang, X. Zi-Qing, Y. Zhao-Huan, H. Xiaoyuan, Y. Qiang, and F. Yi-Zhong, Physical Review Letters 129 , 231101 (2022).</list_item>
<list_item><location><page_13><loc_16><loc_46><loc_70><loc_47></location>[29] L. J. Gleeson and I. H. Urch, Astrophysics and Space Science 11 , 288 (1971).</list_item>
<list_item><location><page_13><loc_16><loc_44><loc_57><loc_45></location>[30] J. R. Jokipii, The Astrophysical Journal 146 , 480 (1966).</list_item>
<list_item><location><page_13><loc_16><loc_43><loc_57><loc_44></location>[31] J. R. Jokipii, The Astrophysical Journal 149 , 405 (1967).</list_item>
<list_item><location><page_13><loc_16><loc_42><loc_52><loc_43></location>[32] J. R. Jokipii, Reviews of Geophysics 9 , 27 (1971).</list_item>
<list_item><location><page_13><loc_16><loc_39><loc_92><loc_41></location>[33] N. E. Engelbrecht, F. Effenberger, V. Florinski, M. S. Potgieter, D. Ruffolo, R. Chhiber, A. V. Usmanov, J. S. Rankin, and P. L. Els, Space Science Reviews 218 , 33 (2022).</list_item>
<list_item><location><page_13><loc_16><loc_36><loc_92><loc_39></location>[34] N. Tomassetti, B. Bertucci, F. Donnini, M. Graziani, E. Fiandrini, B. Khiali, and A. Reina Conde, Rendiconti Lincei. Scienze Fisiche e Naturali 34 , 333 (2023).</list_item>
<list_item><location><page_13><loc_16><loc_35><loc_82><loc_36></location>[35] C. Corti, V. Bindi, C. Consolandi, and K. Whitman, The Astrophysical Journal 829 , 8 (2016).</list_item>
<list_item><location><page_13><loc_16><loc_34><loc_89><loc_35></location>[36] J. Gieseler, B. Heber, and K. Herbst, Journal of Geophysical Research: Space Physics 122 , 10,964 (2017).</list_item>
<list_item><location><page_13><loc_16><loc_33><loc_89><loc_34></location>[37] Z. Shen, H. Yang, P. Zuo, G. Qin, F. Wei, X. Xu, and Y. Xie, The Astrophysical Journal 921 , 109 (2021).</list_item>
<list_item><location><page_13><loc_16><loc_31><loc_69><loc_32></location>[38] I. Cholis, D. Hooper, and T. Linden, Physical Review D 93 , 043016 (2016).</list_item>
<list_item><location><page_13><loc_16><loc_30><loc_67><loc_31></location>[39] M. Kuhlen and P. Mertsch, Physical Review Letters 123 , 251104 (2019).</list_item>
<list_item><location><page_13><loc_16><loc_27><loc_92><loc_30></location>[40] O. P. M. Aslam, X. Luo, M. S. Potgieter, M. D. Ngobeni, and X. Song, The Astrophysical Journal 947 , 72 (2023).</list_item>
<list_item><location><page_13><loc_16><loc_25><loc_92><loc_27></location>[41] A. C. Cummings, E. C. Stone, B. C. Heikkila, N. Lal, W. R. Webber, G. J'ohannesson, I. V. Moskalenko, E. Orlando, and T. A. Porter, The Astrophysical Journal 831 , 18 (2016).</list_item>
<list_item><location><page_13><loc_16><loc_22><loc_92><loc_24></location>[42] C. Corti, V. Bindi, C. Consolandi, C. Freeman, A. Kuhlman, C. Light, M. Palermo, and S. Wang, Proceedings of the 36th International Cosmic Ray Conference 358 , 1070 (2019), arXiv:1910.00027 [astro-ph.HE].</list_item>
<list_item><location><page_13><loc_16><loc_21><loc_71><loc_22></location>[43] L. J. Gleeson and G. M. Webb, Astrophysics and Space Science 58 , 21 (1978).</list_item>
<list_item><location><page_13><loc_16><loc_19><loc_70><loc_20></location>[44] L. J. Gleeson and W. I. Axford, The Astrophysical Journal 149 , L115 (1967).</list_item>
<list_item><location><page_13><loc_16><loc_18><loc_71><loc_19></location>[45] L. J. Gleeson and W. I. Axford, Astrophysics and Space Science 2 , 431 (1968).</list_item>
<list_item><location><page_13><loc_16><loc_17><loc_70><loc_18></location>[46] L. J. Gleeson and W. I. Axford, The Astrophysical Journal 154 , 1011 (1968).</list_item>
<list_item><location><page_13><loc_16><loc_15><loc_55><loc_16></location>[47] G. Gloeckler, Physical Review Letters 17 , 203 (1966).</list_item>
<list_item><location><page_13><loc_16><loc_14><loc_53><loc_15></location>[48] N. Tomassetti, Physical Review Letters 121 (2018).</list_item>
<list_item><location><page_13><loc_16><loc_13><loc_70><loc_14></location>[49] L. J. Gleeson and I. H. Urch, Astrophysics and Space Science 25 , 387 (1973).</list_item>
<list_item><location><page_13><loc_16><loc_11><loc_54><loc_12></location>[50] J. J. Quenby, Space Science Reviews 37 , 201 (1984).</list_item>
<list_item><location><page_13><loc_16><loc_9><loc_92><loc_11></location>[51] R. A. Burger, M. S. Potgieter, and B. Heber, Journal of Geophysical Research: Space Physics 105 , 27447 (2000).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_14><loc_16><loc_92><loc_69><loc_93></location>[52] F. James and M. Roos, Computer Physics Communications 10 , 343 (1975).</list_item>
<list_item><location><page_14><loc_16><loc_89><loc_92><loc_92></location>[53] AMS Collaboration, M. Aguilar, L. A. Cavasonza, G. Ambrosi, L. Arruda, N. Attig, C. Bagwell, F. Barao, L. Barrin, A. Bartoloni, et al., Physical Review Letters 130 , 161001 (2023).</list_item>
<list_item><location><page_14><loc_16><loc_88><loc_88><loc_89></location>[54] I. Cholis, D. Hooper, and T. Linden, Journal of Cosmology and Astroparticle Physics 2022 , 051 (2022).</list_item>
<list_item><location><page_14><loc_16><loc_85><loc_92><loc_88></location>[55] X. Song, X. Luo, M. S. Potgieter, X. Liu, and Z. Geng, The Astrophysical Journal Supplement Series 257 , 48 (2021).</list_item>
<list_item><location><page_14><loc_16><loc_84><loc_89><loc_85></location>[56] O. P. M. Aslam, M. S. Potgieter, X. Luo, and M. D. Ngobeni, The Astrophysical Journal 953 , 101 (2023).</list_item>
<list_item><location><page_14><loc_16><loc_83><loc_62><loc_84></location>[57] E. Silver and E. Orlando (2024), arXiv:2401.06242 [astro-ph.HE].</list_item>
<list_item><location><page_14><loc_16><loc_81><loc_79><loc_82></location>[58] A. W. Strong, E. Orlando, and T. R. Jaffe, Astronomy and Astrophysics 534 , A54 (2011).</list_item>
<list_item><location><page_14><loc_16><loc_79><loc_92><loc_81></location>[59] M. Ackermann, M. Ajello, W. B. Atwood, L. Baldini, J. Ballet, G. Barbiellini, D. Bastieri, K. Bechtol, R. Bellazzini, B. Berenji, et al., The Astrophysical Journal 750 , 3 (2012).</list_item>
<list_item><location><page_14><loc_16><loc_77><loc_83><loc_78></location>[60] E. Orlando and A. Strong, Monthly Notices of the Royal Astronomical Society 436 , 2127 (2013).</list_item>
<list_item><location><page_14><loc_16><loc_76><loc_73><loc_77></location>[61] E. Orlando, Monthly Notices of the Royal Astronomical Society 475 , 2724 (2018).</list_item>
</document>
[
{
"title": "Probing solar modulation analytic models with cosmic ray periodic spectra",
"content": "Wei-Cheng Long and Juan Wu ∗ School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China The AMS02 experiment has published the periodic spectra of proton, helium and helium isotopes across the majority of the 24 solar cycle. These precise data exhibit temporal structures that correlate with solar modulation. In this study, we utilize these data to probe three analytic solar modulation models, including the force-field approximation, the convection-diffusion model and the extended force-field approximation with a drift effect. We adopt a method that eliminates the influence of interstellar cosmic ray spectra, and use the Earth-observed spectra at time t 1 to predict those at time t 2 . In order to explore the rigidity-dependence of solar modulation models, we substitute the conventional potential parameter φ with a modified parameter φ ' = R k 2 ( R ) φ for our analysis. Combining with the χ 2 minimization method, the best-fit modulation parameter φ ' can be evaluated. First, we test the validity of a rigidity-independent φ ' and find that both the force-field approximation (FFA) and the extended force-field approximation (EFFA) agree well with data near the solar minimum period. However, all models significantly deviate from the data during the solar maximum. Consequently, we assume a constant φ ' ( t 1 ) at solar minimum and calculate ∆ φ ' = φ ' ( t 2 ) -φ ' ( t 1 ) for each rigidity bin at time t 2 . It is found that ∆ φ ' generally adheres to a linear-logarithm relationship with rigidity at any given time. By adopting a linear-logarithm formula of ∆ φ ' , we further discover that both the modified FFA and EFFA can reconcile the observations during solar maxima. This suggests that at solar maximum, the parameter φ ' , which correlates with the diffusion pattern in the heliospheric magnetic fields, exhibits a rigidity dependence. Moreover, the modified EFFA enhances the concordance with data during periods of pronounced dips as observed by AMS02. This implies that the drift effect could significantly contribute to these solar transient phenomena. PACS numbers: Keywords: Particle astrophysics (96) - Cosmic rays (329) - Heliosphere (711) - Solar activity (1475) Solar cycle (1487) - Solar magnetic fields (1503)",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "Due to the interaction with the heliospheric magnetic fields (HMF) embedded in the solar wind [1], the cosmic ray (CR) energy spectra detected at the top of the atmosphere(TOA) of Earth differ from those at the local interstellar space (LIS). The entire process, which CRs undergo within the solar system, is referred to as solar modulation. During the passages of CR particles traversing through the interplanetary space, the modulation effects for them include convection driven by the solar wind, diffusion induced by the small-scale magnetic field irregularities, drift occurring in the large-scale magnetic structures and adiabatic losses due to the expansion of the solar wind [2]. Solar modulation effects are inherently influenced by the solar activities. The number and surface area of sunspots relate with the intensity of solar activity, and observations on them show a 11-year solar cycle. Meanwhile, the solar magnetic field undergoes a polarity reversal at the solar maximum, suggesting a 22-year cycle for solar activity [3, 4]. These periodic variations in solar activities causes both the solar modulation and the cosmic ray energy spectra, which are affected by solar modulation, to change periodically over time. Direct observation experiments such as PAMELA and AMS-02 have performed long-term measurements on CRs [5-11]. The unprecedented accurate periodic data reveal that the CR intensities display time structures that are anti-correlated with solar activities. These data offer substantial potential for us to reveal the features of solar modulation. The CRs' transport processes in the heliosphere is usually described by Parker equation [12]. It can be solved either through numerical methods or analytical methods. While the numerical models provide more accurate and physically reasonable solutions [13-16], they necessitate a comprehensive understanding on the details of various physical quantities in the heliosphere, coupled with strong computational power. In contrast, analytical methods, which rely on a set of simple assumptions, may yield less precise results. However, these methods are computationally efficient and hence frequently employed by researchers. The most commonly used analytic models are the force-field approximation (FFA) and the convection-diffusion model (CD) [17]. These models describe solar modulation by using a single parameter and greatly enhance the convenience of their applicability [18, 19]. A robust solar modulation model is crucial for comprehending CR acceleration and propagation [20-24] as well as for detecting dark matter signal [25-28]. glyph[negationslash] As we known, the modulation effects might be very sensitive to the particle's rigidity[29]. For a CR particle, the diffusion coefficient is κ = vλ/ 3, in which v is the particle's velocity and λ is its mean free path. This coefficient is important for us to understand the scattering of particles on the random heliospheric magnetohydrodynamic waves and discontinuities. When the particle resonates with the HMF fluctuations with a spectrum w ( k ) ∼ k -η , where k is the resonant wave number, the diffusion mean free path can be expressed as λ ∼ R 2 -η [30-34]. A fundamental prerequisite for FFA is that the mean free path λ of a particle is proportional to its rigidity, i.e., η = 1. However, the real relationship between λ and the rigidity remains uncertain. If η = 1, the modulation parameter may have a different rigidity dependence. In recent years, there has been a lot of research focusing on the modification of FFA. Some studies have investigated the rigidity dependence of FFA and have updated FFA analytic formula by introducing additional parameters [35-37]. There are also some studies [38, 39] on FFA have considered a drift effect, which is prominent near the solar minimum and negligible during the HMF polarity reversal period [40]. The inclusion of this effect extend FFA into a charge-sign dependent model. However, these studies usually firstly assumed CR LIS spectra, and then combined the experimental data to constrain the modulation parameters. It is known that CR LIS spectra have only been measured by Voyager-1 below a few hundred MeV [41]. Above this energy, no experimental data exist, leading to the adoption of different LIS models in the literature. Therefore, the analysis results might be biased due to inaccurate assumptions of LIS spectra. To get rid of the impact of LIS energy spectra, an alternative method (herein referred to as the Non-LIS method here) was proposed in [42]. In their work, they rewrote the FFA formula by removing the term of LIS spectra. Instead, they calculated the CR TOA intensity at time t 2 ( J ( t 2 )) based on the TOA intensity at t 1 ( J ( t 1 )). This method does not require an assumption of LIS spectra, but only needs to use the periodic CR data for the analysis. An important finding from their work was that the validity of FFA varies at different periods of solar activity. In this work, we will utilize the Non-LIS method to further investigate the rigidity dependence of the solar modulation effect for various analytic modulation models, including FFA, CD and an extended FFA (with a drift effect).",
"pages": [
1,
2
]
},
{
"title": "A. Force-field approximation",
"content": "The basic transport equation(TPE) was first derived by Parker in the solar wind reference frame [17, 18, 43]: where f ( r , p, t ) is the phase space density or the omini-directional distribution function of CRs as a function of position r , momentum p and time t . It is linked to the differential intensity in terms of energy by: J T = p 2 f ( r , p, t ). In Eq. (1equation.2.1), V is the solar wind velocity, K is the diffusion tensor and Q is the local sources in the heliosphere. The diffusion tensor can be expressed as K = K a + K s , in which K s denotes the symmetrical component of the diffusion tensor and K a represents the asymmetrical component associated with the drift effect. After that, an equivalent equation was derived in the observer's reference frame [44]: Here C = -(1 / 3) ∂ ln f/∂ ln p is the Compton-getting coefficient, which corrects the anisotropy of transformation from the wind reference frame to the stationary reference frame [45]. FFA solves Eq. (2equation.2.2) under a series of assumptions: a) no local source of CRs, i.e. Q = 0; b) the existence of a steady state, i.e. ∂f/∂t = 0; c) spherically symmetric and ignoring the drift effect; d) an adiabatic rate 〈 ˙ p 〉 = ( p/ 3) V · ∇ f/f = 0. Additionally, the streaming term ( C V f -K · ∇ f ) is assumed to be divergence free and only the radial component of K , denoted as κ for convenience, is nonzero. Consequently, Eq. (2equation.2.2) can be written as the so-called force-field equation: where κ is the one-dimension diffusion coefficient, r is the radial position and V is the radial component of solar wind velocity. It is a first-order partial differential equation with a characteristic equation The solution of Eq. (3equation.2.3) is where r LIS , p LIS and r TOA , p TOA are the positions and momenta of CR particles in LIS and at TOA, respectively, and f ( r, p ) is the density of CRs along the characteristic curve described by Eq. (4equation.2.4). Assuming the diffusion coefficient κ could be separated in radius distance r and rigidity R ≡ pc/Ze , i.e. κ ( r, R ) = βk 1 ( r ) k 2 ( R ), where Z is the charge number of the particle, β is the ratio of the particle velocity v and the speed of light c , we can define the modulation parameter φ as where R LIS and R TOA are the rigidities of CR particles in LIS and at TOA, respectively. Both φ and k 2 ( R ) are independent with the CR species but vary with time. Assuming that T LIS and T TOA represent the energies of a CR particle in LIS and at TOA, the energy loss experienced by this particle in the heliosphere can be denoted as Φ ≡ T LIS -T TOA . If Φ glyph[lessmuch] E 0 , where E 0 is the stationary energy of the particle, then the relationship between Φ and φ becomes [46] From Eq. (7equation.2.7), we can define the modified modulation potential φ ' as Taking λ ∝ R [47, 48], thus k 2 ( R ) = R . From the definition of Eq. (6equation.2.6), it can be found that φ is a rigidity-independent parameter. So Φ = Zeφ for any arbitrary rigidity. This leads to the conventional FFA, which condenses all the physical processes into a single parameter φ (or Φ) [46, 49]. It should be noticed that if k 2 ( R ) = R is not the condition, we may get more general results by substituting φ ' for φ . In this case, φ ' could become rigidity-dependent, and Φ can be expressed as Φ = Zeφ ' . The modulated spectrum J ( r TOA , T TOA ) and the unmodulated LIS spectrum J ( r LIS , T TOA +Φ) are related as Therefore, by studying the modified potential parameter φ ' , we can investigate the FFA's rigidity dependence effect.",
"pages": [
2,
3
]
},
{
"title": "B. Convection-Diffusion equation",
"content": "The other analytic modulation model CD can be directly derived from Eq. (1equation.2.1). The assumptions made in CD is similar as those given in FFA. Then Eq. (1equation.2.1) can be simplified into the CD equation for which the solution is where According to Eq. (6equation.2.6), Eq. (8equation.2.8) and Eq. (11equation.2.11), the relation between φ ( φ ' ) and M is [17, 50] FFA and CD have been widely used due to their simplicity. Both of them can compress the modulation processes into one single parameter, which depends on the specific form of k 2 ( R ). Unlike FFA, which assumes k 2 ( R ) = R , CD does not place any constraints on k 2 ( R ). It may allow the modulation effect expected by CD to be rigidity-dependent.",
"pages": [
3,
4
]
},
{
"title": "C. Extended force-field approximation",
"content": "Both FFA and CD neglect the drift effect. By incorporating the drift effect, Eq. (3equation.2.3) was led to Kuhlen's extended FFA (EFFA) equation [39]: in which all the quantities are angular averaged, and v d,r is the radical component of the averaged drift velocity. The solution of Eq. (13equation.2.13) is whose characteristics curve is also Eq.(4equation.2.4). This solution's form seems to be a combination of FFA and CD, except for that the solar wind velocity in the integral term is replaced by the drift velocity. The rigidity dependence of v d,r correlates with the behavior of the anti-symmetrical diffusion coefficient κ a , and is given as [51] where B is the magnitude of the heliospheric magnetic field on a large scale. Given the similarity between the spatial integral part in Eq. (14equation.2.14) and the definition of M in Eq. (11equation.2.11), when Φ glyph[lessmuch] E 0 , this integral term can be associated with φ ' as where g is a scaling factor related to the magnitude of magnetic field B and the solar wind velocity V . A larger g indicates a stronger impact on the CR flux caused by the drift effect. In summary, all three analytical models are derived from equivalent forms of TPE. Both FFA and CD are oneparameter models that disregard a term related to the adiabatic energy. But these terms differ as they are derived in different frames. Other than FFA and CD, Kuhlen's EFFA incorporates a drift effect and depends on two parameters.",
"pages": [
4
]
},
{
"title": "A. The Non-LIS method",
"content": "In order to eliminate the influence of CR LIS energy spectra, we use the Non-LIS method to explore the general properties of above analytical models. Since the nature of the heliospheric diffusion effect remains incompletely understood, we refrain from specifying the formula for k 2 ( R ). Therefore, we use the modified modulation parameter φ ' to obtain the relationship between J ( t 1 ) and J ( t 2 ). For FFA, assuming ∆ φ ' = φ ' ( t 2 ) -φ ' ( t 1 ), Eq. (9equation.2.9) can be transformed into Similar approaches can be applied to CD and EFFA. For CD, we yield For EFFA, we get These three models are all written in terms of ∆ φ ' . As suggested in [42], traditional FFA may be reliable to describe the solar modulation effect around the solar minimum period. If we select t 1 near a period of minimal solar activity, it is reasonable to postulate that φ ' ( t 1 ) = φ ( t 1 ) is rigidity-independent. Therefore, the analysis on parameter ∆ φ ' can characterize the properties of parameter φ ' ( t 2 ). This approach will reveal the characteristics of solar modulation at any given time t 2 . For CD and EFFA, we also set t 1 near solar minimum and similarly assume φ ' ( t 1 ) is rigidityindependent. Subsequently, we use the free parameter ∆ φ ' to study modulation effect at other times. Note that in EFFA, except for ∆ φ ' , the factor g linked to the drift effect is also allowed to vary freely in the fittings.",
"pages": [
4,
5
]
},
{
"title": "B. Test of the rigidity-independent ∆ φ '",
"content": "Synodic solar rotation causes the CR flux recurrent variations on the timescale of Bartels Rotations (BRs), which is 27 days for each BR. We use periodic data from AMS02 [8], which provides the measurements of proton (p) flux between 1 GV and 60 GV and helium (He) flux between 1.9 GV and 60 GV from May 2011 to May 2017. It also provides helium-3 ( 3 He) flux between 1.9 and 15 GV and helium-4 ( 4 He) flux between 2.1 GV and 21 GV from May 2011 to November 2017 [9]. The measurements covered most time of the 24 solar cycle, during which the solar maximum appeared in April 2014, and the HMF polarity reversed from A < 0 to A > 0 at that time. For p and He, BR 2504 (February 18, 2017-March 16, 2017) is selected as t 1 in our work, since the measured flux at this time is higher than those at other phases. This indicates that the solar activity is weakest at this time. For 3 He and 4 He, since each of them has been measured in periods of 4 Bartels rotations (108 days), the period from BR 2502 to BR 2505 is selected as t 1 . It is assumed that the distribution of flux within rigidity bin ( R 1 , R 2 ) follows a power law. Consequently, the flux value at this bin is assigned to the interval center rigidity R = √ R 1 R 2 . We employed a cubic spline method to calculate the interpolated flux values for other rigidities within the rigidity range of observation, and utilize a power law distribution to extrapolate flux beyond this range. A least-square analysis using MINUIT package [52] is applied to obtain the ∆ φ ' values for each model, as well as the χ 2 /d.o.f values at time t 2 . For EFFA, the best-fit values of parameter g are also estimated. First, we use AMS-02 p, He, 3 He and 4 He data to test the validity of a rigidity-independent ∆ φ ' for both FFA and CD. The results are shown in Fig. 1Top panel: the rigidity-independent parameter ∆ φ ' (=∆ φ ) estimated by using the p, He and He isotopes periodic data measured by AMS-02 for FFA and CD model. Bottom panel: the corresponding χ 2 /d.o.f at each BR. The shade area corresponds to the HMF polarity reversal period from A < 0 to A > 0 (2012 November-2014 March)figure.1. As we can see, for FFA, at any given time t 2 , He isotopes yield rather consistent values of ∆ φ ' with p and He. This is also the case for CD. For a given time t 2 , the best-fit ∆ φ ' parameter obtained in CD is higher than that given in FFA. The possible reason is that the integration of FFA from LIS to TOA is constrained by the characteristic curve, and the corresponding path of FFA is longer than that of CD for a same value of φ . In other words, the adiabatic energy term of TPE ignored in CD is larger than that ignored in FFA. For both models, the largest ∆ φ ' appears around 2014, which corresponds to a solar maximum. During periods around 2011-2012 and 2016-2017, it is found that the χ 2 /d.o.f values are close to 1. This infers that FFA and CD can generally reproduce data during the low solar activity periods. But during the periods with high ∆ φ ' , the χ 2 /d.o.f values are much larger than 1. Especially, by using the p and He fluxes, the χ 2 /d.o.f values can achieve more than 20. It means that, for both FFA and CD, an rigidity-independent ∆ φ ' (or φ ' ) does not agree Date Date well with the p and He data during these periods with intense solar activity. This phenomenon is further confirmed in Fig. 3Top panels: The p and He fluxes at BR 2426 and BR 2504 expected from FFA, CD and Kuhlen's EFFA, in comparison with the AMS-02 measurements. Middle and bottom panels: the residuals of the model fittings to the p and He spectra for BR 2426 and BR 2463figure.3. At BR 2426 (May 5, 2011-June 10, 2011), a period after the solar minimum in 2009, the solar activity is not strong. As we can see, at this BR, both FFA and CD generally give consistent results with the p data above 2 GV and the He data at the whole rigidity range , except for an obvious discrepancy exist between CD and the p data below 2 GV. But the p and He fluxes predicted by both models at BR 2463 (February 7, 2014-March 6, 2014), which is in a polarity reversal in solar cycle 24, show significant disagreements with the data. These disagreements indicate that both models with a rigidity-independent φ ' can not describe the solar modulation behavior well during the HMF polarity reversal period. By using the 3 He or 4 He data, the calculated χ 2 /d.o.f values at solar maximum are not that high. This may be due to the large errors existing in the 3 He and 4 He data, which infers that using the He isotope data alone is not enough to test the validity of solar modulation models. Therefore, to further investigate EFFA, we use only the p and He data to run the analysis. For EFFA, the time variations of ∆ φ and g are shown in Fig. 2Top and middle panels: the rigidity-independent parameter ∆ φ ' (=∆ φ ) and g estimated by using the p, He and He isotopes periodic data measured by AMS-02 for Kuhlen's EFFA. Bottom panel: the corresponding χ 2 /d.o.f at each BR. The shade area corresponds to the HMF polarity reversal period from A < 0 to A > 0 (2012 November-2014 March)figure.2. This model presents a similar tendency of ∆ φ ' ( φ ' ) in terms of time with FFA and CD. The scaling factor g does not show a clear variation with time. The minor fluctuations of g indicate the intensity of drift effect does not vary greatly in different periods. According to the χ 2 /d.o.f values given in Fig. 2Top and middle panels: the rigidity-independent parameter ∆ φ ' (=∆ φ ) and g estimated by using the p, He and He isotopes periodic data measured by AMS-02 for Kuhlen's EFFA. Bottom panel: the corresponding χ 2 /d.o.f at each BR. The shade area corresponds to the HMF polarity reversal period from A < 0 to A > 0 (2012 November-2014 March)figure.2, it can be found that EFFA can better fit the data than FFA and CD. At all the periods, EFFA reduces the χ 2 /d.o.f values by more than half. But EFFA still displays a poor performance during 2013-2014. This can also been seen in Fig. 3Top panels: The p and He fluxes at BR 2426 and BR 2504 expected from FFA, CD and Kuhlen's EFFA, in comparison with the AMS-02 measurements. Middle and bottom panels: the residuals of the model fittings to the p and He spectra for BR 2426 and BR 2463figure.3. It seems that including a drift effect still cannot explain the solar modulation effect during the polarity reversal period.",
"pages": [
5,
6,
7
]
},
{
"title": "C. Modulation with rigidity-dependent φ '",
"content": "Since a rigidity-independent φ ' cannot accommodate the data well during the periods with intense solar activities, we only assume a constant φ ' at solar minimum period t 1 . But at other periods, we calculate ∆ φ ' at each rigidity bin to study the change of ∆ φ ' ( φ ' ( t 2 )) with rigidity. The analysis is performed both for AMS-02 p and He data and the results for FFA are presented in Fig. 4The variation of the best-fit parameter ∆ φ ' with rigidity and time for p (left panel) and He (right panel)figure.4. Noted that in this figure the x-axis is represented on a logarithmic scale. As the intensity of solar activity increases, the variation of ∆ φ ' with rigidity becomes more and more significant. This further confirms the necessary to introduce a rigidity-dependent φ ' during periods of high solar activity. We particularly show the relationships between ∆ φ ' and rigidity in Fig. 5The parameter ∆ φ ' as a function of rigidity in BR 2426, BR 2442, BR 2463, where BR 2426 is near the solar minimum, BR 2442 is one of the sharp dips observed in AMS02 [53], and BR 2463 is during the solar maximum periodfigure.5 for BR 2426, BR 2442 and BR 2463. It can be found that for all BRs, the curves are very close to straight lines in linear-logarithmic (lin-log) coordinates. Here we include BR 2442 in the plot because this BR is related to the location of sharp dips in the p and electron fluxes observed by AMS02 [53]. At this BR, ∆ φ ' has a slight downturn at very low rigidity. For CD and EFFA, ∆ φ ' have similar relations with rigidity. Therefore, we assume that ∆ φ ' has a lin-log relationship with rigidity, with the formula: ' ' where φ 0 is the normalization of ∆ φ at R 0 = 1 GV, and φ 1 is the slope of ∆ φ with ln R . They both vary with time. We adopt this lin-log formula of ∆ φ ' in FFA, CD and EFFA. In this case, the free parameters include φ 0 , φ 1 for FFA and CD, and φ 0 , φ 1 , g for EFFA. The predicted p and He flux are compared with the data measured at BR 2442 and BR 2463, as shown in Fig. 6Top panels: The p and He fluxes at BR 2426 and BR 2504 expected from the modified FFA, CD and Kuhlen's EFFA by introducing a linear-logarithm rigidity-dependent ∆ φ ' , in comparison with the AMS-02 measurements. Middle and bottom panels: the residuals of the model fittings to the p and He spectra for BR 2442 and BR 2463figure.6. At BR 2442 and 2463, the modified FFA can generally reproduce the p and He data at most rigidity range. It only gives slightly lower predictions than the p data below 2 GV. This might because ∆ φ lin-log does not give a perfect description of the p flux at very low rigidity, as exhibited in Fig. 5The parameter ∆ φ ' as a function of rigidity in BR 2426, BR 2442, BR 2463, where BR 2426 is near the solar minimum, BR 2442 is one of the sharp dips observed in AMS02 [53], and BR 2463 is during the solar maximum periodfigure.5. Nevertheless, the agreement between the modified FFA and the data is highly increased compared with the conventional FFA. This is also the case for modified EFFA. But the improvement of the modified CD is limited compared with the conventional CD. It give worse goodness of fit than the modified FFA and EFFA. The χ 2 /d.o.f results of different models are summarized in Table IThe χ 2 /d.o.f results for different analytical models at BR 2426, BR 2442 and BR2463 based on the analysis of p or (and) He data. Here φ ' = φ assumes a rigidity-independent φ ' , and φ ' lin-log assumes a linear-logarithm rigidity-dependent φ ' table.1. As we can see, CD doesn't fit well with all the data. For BR 2426, both the FFA and EFFA with a rigidity-dependent or rigidityindependent φ ' agrees well with the p and He data. For BR 2442, the conventional FFA does not accommodate the data. The modified FFA improves the goodness-of-fit but still yield a χ 2 /d.o.f close to 2. By including a drift effect, both conventional EFFA and modified EFFA can reproduce the p and He data at BR 2442. It suggests that the flux dips observed in AMS02 data may be associated with the drift effect. For BR 2463, the conventional FFA and EFFA have large disagreements with the data. But by adopting a rigidity-dependent φ ' , both models can explain the data well. For solar minimum and maximum phases, it is difficult do judge whether the drift effect needs to be introduced to interpret the data.",
"pages": [
8,
10
]
},
{
"title": "D. Compared with other modified FFA models",
"content": "We compared our lin-log FFA and lin-log EFFA with other modified FFA models. One is Cholis' model [38, 54]. Instead of adding a drift term in the relationship between J ( r TOA , t 2 ) and J ( r TOA , t 1 ), they incorporated the drift term in φ . Based on their work, ∆ φ ' can be written as The other is Shen's model presented in [37]. In that paper, the authors attributed the variation of φ with energy to the behavior of diffusion coefficient. They used a double power-law empirical formula to describe φ . In Shen's model ∆ φ ' can be written as where E b = 1 GeV, φ 0 is a scaling factor in unit GV. The rest of the parameters are dimensionless. Both φ 0 and φ 1 vary with time, while E b 1 , b 1 and b 2 are time-independent parameters. It is worth to noticed that the estimations of all the parameters in Shen's model are adjustable to perform a good agreements with the data. This could result in overfitting and instability of the parameters [34, 55]. Both Cholis' and Shen's models include a β term from the diffusion coefficient into φ ' . The relationship between β and rigidity shows that β is a function of A/Z . Thus for different particles, the same values of φ 0 and φ 1 may lead to different values of ∆ φ ' Shen (or ∆ φ ' Cholis ) for a given rigidity. This difference is slight in Cholis' model since β only exist in φ 1 term of ∆ φ ' Cholis . But from Eq. (8equation.2.8), we can see that a β term is unnecessary to be introduced in φ ' . The inclusion of β may be lack of rigorous theoretical basis. Above models with a rigidity-dependent ∆ φ ' all contain two free parameters φ 0 and φ 1 . Other parameters are nuisances. The χ 2 minimization results of the lin-log FFA, the lin-log EFFA, Cholis' model and Shen's model are shown in Fig. 7The results of χ 2 /d.o.f over time obtained by fitting the p or (and) He data for the modified FFA, the modified EFFA, Cholis' model and Shen's modelfigure.7, respectively. It can be found that the lin-log FFA give excellent goodness-of-fit for most periods. There are only a few BRs at which the values of χ 2 /d.o.f > 1. Peaks of χ 2 appears at BR 2437, BR 2442, BR 2453 and BR 2478. All these BRs happens only in A > 0 stage and corresponds to the sharp dips in AMS-02 p and electron fluxes. Notably, at these BRs, the lin-log EFFA agrees better with the AMS02 p and He data. It indicates that these solar transients on timescale of BRs maybe related with the drift effect. For Cholis' model, there are much more BRs corresponding to χ 2 /d.o.f > 1, especially by fitting the p or p+He data. The largest χ 2 values exhibit during the solar reversal phase, which means Cholis' model is particularly poor to simulate the solar modulation during those stages. In this model, the χ 2 distribution over time is similar with that in conventional EFFA. This suggests that the introduction of a drift effect is not sufficient to explain the variation of φ ' with rigidity. Shen's model could obtain good agreements with either p or He data. But when we combine p and He data to do the analysis, the resulted χ 2 /d.o.f values are large. The reason is that the estimated φ 0 and φ 1 deviate significantly between p and He. It reveals that Shen's model does not give consistent descriptions on p and He.",
"pages": [
10,
11
]
},
{
"title": "IV. CONCLUSION AND DISCUSSION",
"content": "In this paper, we take into account three analytic solar modulation models: FFA, CD, and EFFA. To investigate these models, the Non-LIS method is employed to eliminate the impact of CR LIS spectra. The traditional potential parameter φ in FFA, CD and EFFA is a rigidity-independent parameter. However, since the radial diffusion coefficient may not be proportional to rigidity, the parameter Φ could be rigidity-dependent. Therefore, we introduce an alternative parameter φ ' = R k 2 ( R ) φ to revisit these models. By using ∆ φ ' = φ ' ( t 2 ) -φ ' ( t 1 ), we can determine the CR flux at time t 2 based on the observed CR flux at time t 1 . Then we use the χ 2 minimization analysis to estimate the best-fit ∆ φ ' . First, it is found that the conventional FFA and EFFA with a rigidity-independent φ ' can describe the data well around solar minimum. But these models do not agree well with the data at HMF polarity reversal periods. Therefore, it is reasonable to assume a constant φ ' near solar minimum, but consider a rigidity-dependent φ ' (or ∆ φ ' ) for other periods. By calculating ∆ φ ' at different rigidity ranges, the results show that ∆ φ ' is not a constant but seems have a lin-log relationship with rigidity. By incorporating this lin-log formula of ∆ φ ' into FFA and EFFA models, we find that they can satisfactorily describe the data during the HMF polarity reversal stage. The CD models, no matter the conventional one or the modified one, cannot explain the data well. It infers that the ignored adiabatic term in CD plays a relatively important role in modulation, which could have a significant rigidity dependence. The effect of drift may be important to explain the modulation during those solar transients detected by AMS02, since during those stages, the conventional and modified EFFA models can fit the date better than other models. The hysteresis-like loops (coinciding with the sharp dip times) between the proton and electron fluxes [53] or between the proton and antiproton fluxes [56] display a charge-sign-dependent solar modulation. This may be related to the fact that particles with opposite charge signs have different patterns of drift effects. Nevertheless, the variation of φ ' with rigidity is not mainly due to the drift effect, since Cholis' model have a worse performance than our lin-log FFA model. This suggests that the rigidity-dependence of the parameter φ ' mainly originates from the rigidity-dependence of diffusion coefficient. The lin-log FFA model is also better than Shen's model, in which they consider a double power-law φ ' . The specific form of φ ' is important for understanding of HMF fluctuations during the HMF polarity reversal periods. A recent study by simultaneous scanning on the solar modulation parameter and other CR acceleration and propagation parameters, has suggested that the conventional FFA can describe well the CR spectra measured by AMS02 and Voyager-1 integrated over the entire detection period [57]. However, a rigidity-dependent φ ' may challenge our traditional understanding on CR acceleration and propagation mechanisms. It should be noted that these analytic models are based on a series of assumptions. The dependence of modulation on A/Z is not studied in this work. But we find that during the sharp dips periods, the deviations of the He data from lin-log FFA are less significantly than those of the p data. This will be further studied in our future work. The solar modulation model proposed in our work enables us to place effective constraints on the CR source and propagation models. This allows for a reliable calculation on the CR LIS spectra. Some other studies have shown that it is also possible to derive the LIS spectra from synchrotron and gamma-ray observations without any assumption on solar modulation [58-61]. In our future work, we will further compare them for a better understanding on cosmic ray behaviors in Galaxy.",
"pages": [
12
]
},
{
"title": "Acknowledgments",
"content": "Thanks for Ilias Cholis, Claudio Corti and R.A. Caballero-Lopez for very helpful discussions. This work is supported by the Joint Funds of the National Natural Science Foundation of China (Grant No. U1738130). The use of the highperformance computing platform of China University of Geosciences is gratefully acknowledged. [7] N. Marcelli, M. Boezio, A. Lenni, W. Menn, R. Munini, O. P. M. Aslam, D. Bisschoff, M. D. Ngobeni, M. S. Potgieter, O. Adriani, et al., The Astrophysical Journal Letters 925 , L24 (2022). [9] AMS Collaboration, M. Aguilar, L. Ali Cavasonza, G. Ambrosi, L. Arruda, N. Attig, A. Bachlechner, F. Barao, A. Barrau, L. Barrin, et al., Physical Review Letters 123 , 181102 (2019).",
"pages": [
12,
13
]
}
]
2024PhRvD.109h3507P
https://arxiv.org/pdf/2311.07685.pdf
<document>
<section_header_level_1><location><page_1><loc_30><loc_92><loc_71><loc_93></location>Magnetogenesis from Anisotropic Universe</section_header_level_1>
<text><location><page_1><loc_29><loc_89><loc_72><loc_90></location>Sourav Pal, 1, ∗ Debaprasad Maity, 1, † and Tuan Q. Do 2, 3, ‡</text>
<text><location><page_1><loc_28><loc_87><loc_73><loc_88></location>1 Department of Physics, Indian Institute of Technology Guwahati,</text>
<text><location><page_1><loc_39><loc_86><loc_61><loc_87></location>Guwahati 781039, Assam, India</text>
<text><location><page_1><loc_22><loc_85><loc_79><loc_86></location>2 Phenikaa Institute for Advanced Study, Phenikaa University, Hanoi 12116, Vietnam</text>
<text><location><page_1><loc_26><loc_83><loc_75><loc_84></location>3 Faculty of Basic Sciences, Phenikaa University, Hanoi 12116, Vietnam</text>
<text><location><page_1><loc_41><loc_82><loc_60><loc_83></location>(Dated: November 15, 2023)</text>
<text><location><page_1><loc_18><loc_64><loc_83><loc_81></location>The existence of large-scale anisotropy can not be ruled out by the cosmic microwave background (CMB) radiation. Over the years, several models have been proposed in the context of anisotropic inflation to account for CMB's cold spot and hemispheric asymmetry. However, any small-scale anisotropy, if exists during inflation, is not constrained due to its nonlinear evolution in the subsequent phase. This small-scale anisotropy during inflation can play a non-trivial role in giving rise to the cosmic magnetic field, which is the subject of our present study. Assuming a particular phenomenological form of an anisotropic inflationary universe, we have shown that it can generate a large-scale magnetic field at 1-Mpc scale with a magnitude ∼ 4 × 10 -20 G , within the observed bound. Because of the anisotropy, the conformal flatness property is lost, and the Maxwell field is generated even without explicit coupling. This immediately resolves the strong coupling problem in the standard magnetogenesis scenario. In addition, assuming very low conductivity during the reheating era, we can further observe the evolution of the electromagnetic field with the equation of state (EoS) ω eff and its effects on the present-day magnetic field.</text>
<section_header_level_1><location><page_1><loc_20><loc_60><loc_37><loc_61></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_22><loc_49><loc_58></location>It is well known that our universe is magnetized on all observational scales, from planets and stars to large-scale galaxies and galaxy clusters. In particular, the magnetic field strength has been observed in the range from µG for galaxies and galaxy clusters to a few G for planets and 10 12 G for neutron stars. From Gamma-ray observations and Faraday rotation measurements, the magnetic field in the intergalactic medium (IGM) has also been shown to be bounded with the strength ranging from 10 -10 -10 -22 G [1-5]. It is possible that the primordial magnetic fields on a large scale ( ∼ 1 MPc ) were generated during the Big Bang or later and survived until today as a relic. The origin of the magnetic field in galaxies and galaxy clusters can be explained through classical magneto-hydrodynamic processes magnifying the tiny seed magnetic field. It is important to identify the origin of the primordial magnetic fields. There have been some proposed mechanisms for generating large-scale primordial magnetic fields, which can be found in the interesting review papers listed in Refs. [6-21]. Among them, the Ratra model [21] is the most accepted one, where the electromagnetic fields are generated during the inflationary era by breaking the conformal invariance of the Maxwell term, i.e., F µν F µν , through non-minimal coupling(s) with other fields such as scalar fields.</text>
<text><location><page_1><loc_9><loc_16><loc_49><loc_22></location>Inspired by the mechanism in [16, 18, 21, 40-42], we propose another mechanism for generating the primordial magnetic fields through anisotropic spacetime. In particular, the Maxwell field experiences the existence of</text>
<text><location><page_1><loc_52><loc_48><loc_92><loc_61></location>the anisotropy of spacetime during the inflationary era, leading to the genesis of primordial magnetic. In cosmology, there exists a nice classification of homogeneous but anisotropic spacetimes called the Bianchi universe [3133, 43, 44]. It turns out that the Bianchi type I metric is the simplest one and can be regarded as a straightforward extension of the Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime. Hence, we chose the Bianchi type I metric for our study.</text>
<text><location><page_1><loc_52><loc_12><loc_92><loc_48></location>Remarkably, it has long been argued that the very early universe, which is close to the initial singularity, should be strongly anisotropic [22-24]. During an inflationary phase of the early universe, all spatial anisotropies, which could happen in a pre-inflationary phase, should decrease very quickly such that the universe speedily approaches a locally isotropic state, as pointed out in Refs. [25, 26]. It should be noted that this scenario is consistent with the so-called cosmic no-hair conjecture, which claims that all initial anisotropies and inhomogeneities should disappear in a late-time universe [27, 28]. Very interestingly, some recent unavoidable anomalies in the cosmic microwave background (CMB) radiations confirmed by the Planck [29, 47] such as the cold spot and hemispheric asymmetry, have challenged the standard inflationary universe models, which are based on the cosmological principle stating, the universe should be homogeneous and isotropic on large scales. In addition, other interesting observational evidences, which have called the validity of cosmological principle into question, have been listed in a recent interesting review [30]. These remarkable points lead us to a possible scenario of an anisotropic inflationary universe in early times. Many papers have been working on anisotropic inflation, e.g., see Refs. [31-33] as well as Refs. [34-39].</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_11></location>This paper does not discuss the origin and evolution of anisotropy in spacetime. Instead, we treat it as a</text>
<text><location><page_2><loc_9><loc_72><loc_49><loc_93></location>perturbation over the FLRW background. As the spatial anisotropy breaks the conformal flatness of the background in the electromagnetic (EM) field, we do not need any explicit coupling with the scalar field as proposed in the literature [16-20] for gauge field production during inflation. In this context, it is important to mention the challenges in inflationary magnetogenesis, namely the strong-coupling problem and the backreaction problem [10, 16, 45, 46]. In the literature, several mechanisms and different types of coupling [45] have been introduced to overcome these problems. However, in our formalism, the strong coupling problem is readily solved in this paper since no explicit coupling is involved with the gauge field. However, there might still be a possibility of backreaction, which we will study in detail as we proceed.</text>
<text><location><page_2><loc_9><loc_54><loc_49><loc_71></location>The paper is organized as follows. (i) An introduction of the present paper has been written in Sec. I. (ii) In Sec. II, we describe the basic formalism and the quantization of the gauge field in an anisotropic background. (iii) In Sec. III, we show the evolution of the gauge field during the inflationary era and the strength of the present-day magnetic field under an instant reheating scenario. However, a scenario might occur when the universe undergoes a prolonged reheating era, affecting the magnetic field. (iv) We discuss the evolution in such a scenario in Sec. IV. (v) Finally, we discuss the findings and implications of this proposal in Sec. V.</text>
<section_header_level_1><location><page_2><loc_22><loc_50><loc_36><loc_51></location>II. THE SETUP</section_header_level_1>
<text><location><page_2><loc_9><loc_44><loc_49><loc_48></location>We introduce the spatial anisotropy in the background through the homogeneous but anisotropic Bianchi type I metric. In general, this type of metric can be written as</text>
<formula><location><page_2><loc_13><loc_41><loc_49><loc_43></location>ds 2 = a 2 ( η ) [ -dη 2 + b 2 ( η ) dx 2 + dy 2 + dz 2 ] , (1)</formula>
<text><location><page_2><loc_9><loc_26><loc_49><loc_40></location>where η is the conformal time, a ( η ) is the overall scale factor and b ( η ) is the anisotropic factor along the x -direction. In our phenomenological model, we impose the condition that the anisotropy in the spacetime exists only in the inflationary era, although the spacetime remains continuous. This ansatz guarantees that the conformal flatness is restored after inflation. These conditions can be satisfied by various models of the anisotropic factor b ( η ). However, we take a particular model that satisfies all the necessary conditions</text>
<formula><location><page_2><loc_21><loc_23><loc_49><loc_25></location>b ( η ) = 1 + α e -( η ηm ) 2 , (2)</formula>
<text><location><page_2><loc_9><loc_9><loc_49><loc_21></location>In the above Eq. (2), α is a dimensionless parameter that determines the strength of the anisotropy. Furthermore, η m is a parameter that dictates the overall behavior of the anisotropic background. An example of the anisotropic background is shown in Fig. 1. Here, α and η m are free parameters of the anisotropic model. In this paper, we do not discuss the origin of such anisotropy. However, the anisotropy, particularly near the end of inflation, maybe a combined effect of quantum field theory and the sudden</text>
<text><location><page_2><loc_52><loc_85><loc_53><loc_85></location></text>
<figure>
<location><page_2><loc_53><loc_74><loc_91><loc_93></location>
<caption>FIG. 1. Behaviour of the anisotropic factor b ( ˜ kη ) with ˜ kη for anisotropic free parameters α and ˜ kη m .</caption>
</figure>
<text><location><page_2><loc_52><loc_63><loc_92><loc_65></location>breakdown of slow-roll conditions. We will come back to this issue in the future.</text>
<text><location><page_2><loc_52><loc_60><loc_92><loc_62></location>The action of Einstein-scalar-vector theory can be given by</text>
<formula><location><page_2><loc_55><loc_52><loc_92><loc_59></location>S = ∫ d 4 x √ -g [ 1 16 πG R -1 2 ∂ µ ϕ∂ µ ϕ -V ( ϕ ) -1 4 F µν F µν ] , (3)</formula>
<text><location><page_2><loc_52><loc_35><loc_92><loc_51></location>where G is the gravitational constant, ϕ is a scalar field, and F µν ≡ ∂ µ A ν -∂ ν A µ is the field strength of the vector field A µ ( t, x ) describing the electromagnetic field. In this paper, the dynamics of the scalar field and the metric itself due to the anisotropy present are beyond our scope. Therefore, we will mainly discuss the dynamics of the EM field during the inflationary era due to the spatially anisotropic background. Therefore, the Lagrangian of interest here is the Lagrangian corresponding to the electromagnetic field, which, according to Eq. (3), is given by</text>
<formula><location><page_2><loc_58><loc_31><loc_92><loc_34></location>L em = 1 2 a 2 g jn A ' j A ' n -1 4 g im g jn F ij F mn , (4)</formula>
<text><location><page_2><loc_52><loc_20><loc_92><loc_30></location>where the prime denotes a derivative with respect to conformal time. In this paper, all the physical quantities are denoted with the lower index, e.g., the physical momentum is denoted by k i , and the vector potential is given by A i . We set A 0 = 0 as the choice of gauge, and unlike the case of conformally flat spacetime, the ν = 0 component satisfies the modified constraint equation,</text>
<formula><location><page_2><loc_66><loc_17><loc_92><loc_19></location>˜ g im ∂ i A ' m = 0 , (5)</formula>
<text><location><page_2><loc_52><loc_9><loc_92><loc_16></location>where we have defined ˜ g ij = a 2 g ij for simplicity of calculation. It can be further shown that the above equation boils down to the usual Coulomb condition for the conformally flat case. However, the raising (and lowering) of the indices are done through the metric component</text>
<text><location><page_3><loc_9><loc_89><loc_49><loc_93></location>g ij ( g ij ). Similarly, the dynamical equation of motion for the magnetic vector potential A i can be calculated from the ν = j component, which boils down to</text>
<formula><location><page_3><loc_14><loc_85><loc_49><loc_88></location>A '' n + b ' b A ' n +˜ g jn ˜ g ' jk A ' k -˜ g im ∂ i F mn = 0 . (6)</formula>
<text><location><page_3><loc_9><loc_73><loc_49><loc_84></location>It is important to note that the metric components play a crucial role in the dynamics of the field. In the case of a standard conformally flat background, the term with the metric component's derivative vanishes, giving us the regular plane wave solutions. Now, we will promote the fields and their conjugates as operators. The conjugate momentum operator corresponding to the field operator A i turns out as</text>
<formula><location><page_3><loc_19><loc_69><loc_49><loc_72></location>Π i = ∂ L em ∂A ' i = 1 a 2 g im A ' m . (7)</formula>
<text><location><page_3><loc_9><loc_65><loc_49><loc_68></location>To quantize the field, we decompose the magnetic vector potential A i as,</text>
<formula><location><page_3><loc_13><loc_58><loc_49><loc_65></location>A i ( η, x ) = ∑ p ∫ d 3 k (2 π ) 3 ( a ( p ) k u ( p ) i ( η ) e ik n x n + a † ( p ) k u ∗ ( p ) i ( η ) e -ik n x n ) . (8)</formula>
<text><location><page_3><loc_9><loc_50><loc_49><loc_56></location>In the above Eq. (8), ( p ) is the polarization index, a ( p ) k and a † ( p ) k are the annihilation and creation operators corresponding to the polarization mode ( p ). They follow the general commutation relation,</text>
<formula><location><page_3><loc_17><loc_47><loc_49><loc_49></location>[ a ( p ) k , a ( q ) k ' ] = δ pq (2 π ) 3 δ 3 ( k -k ' ) . (9)</formula>
<text><location><page_3><loc_9><loc_43><loc_49><loc_46></location>In this article, the boldface letters represent vector quantities.</text>
<text><location><page_3><loc_9><loc_36><loc_49><loc_43></location>In this context, it is important to discuss the commutation relation of the magnetic vector potential A i and its conjugate momentum Π i . We impose the commutation relation, such that they satisfy the constraint Eq. (5) on vector potential A i ,</text>
<formula><location><page_3><loc_11><loc_26><loc_49><loc_35></location>[ A i ( η, x ) , A j ( η, y )] = 0; [Π i ( η, x ) , Π j ( η, y )] = 0 , (10) [ A i ( η, x ) , Π j ( η, y ) ] = i √ -g ∫ d 3 k (2 π ) 3 e ik n ( x n -y n ) × ( δ j i -k i k j k n k n ) . (11)</formula>
<text><location><page_3><loc_9><loc_21><loc_49><loc_25></location>With the quantization of the field, we now get the mode function equations using Eq. (6). The mode functions satisfy the relation,</text>
<formula><location><page_3><loc_10><loc_17><loc_49><loc_20></location>u '' n + b ' b u ' n +˜ g ' jl ˜ g jn u ' l +˜ g im ( k m k i u n -k n k i u m ) = 0 . (12)</formula>
<text><location><page_3><loc_9><loc_11><loc_49><loc_16></location>The polarization index is omitted here, as all the polarization modes follow the same equation of motion. Similarly, the constraint Eq. (5) in terms of the mode function becomes</text>
<formula><location><page_3><loc_24><loc_8><loc_49><loc_10></location>˜ g in k i u ' n = 0 . (13)</formula>
<text><location><page_3><loc_52><loc_85><loc_92><loc_93></location>Interestingly, Eq. (12) contains the derivative of the metric coefficients, which works as the source of particle production during the inflationary era. As the mode function equation contains the derivative of the metric components, substituting the metric components, we get the modified mode function equations as,</text>
<formula><location><page_3><loc_53><loc_74><loc_92><loc_84></location>u '' 1 -b ' b u ' 1 + k 2 2 u 1 + k 2 3 u 1 -k 1 k 2 u 2 -k 1 k 3 u 3 = 0 , u '' 2 + b ' b u ' 2 + k 2 1 b 2 u 1 + k 2 3 u 2 -k 1 k 2 b 2 u 1 -k 2 k 3 u 3 = 0 , (14) u '' 3 + b ' b u ' 3 + k 2 1 b 2 u 3 + k 2 2 u 3 -k 1 k 3 b 2 u 1 -k 2 k 3 u 2 = 0 .</formula>
<text><location><page_3><loc_52><loc_71><loc_92><loc_74></location>Moreover, all the mode functions u 1 , u 2 , and u 3 satisfy the constraint in Eq. (13) which explicitly boils down to</text>
<formula><location><page_3><loc_63><loc_67><loc_92><loc_70></location>k 1 b 2 u ' 1 + k 2 u ' 2 + k 3 u ' 3 = 0 . (15)</formula>
<text><location><page_3><loc_52><loc_66><loc_91><loc_67></location>The mode functions follow the normalization condition,</text>
<formula><location><page_3><loc_53><loc_59><loc_92><loc_63></location>( u ( p ) i ˜ g jm u ∗ ' ( p ) m -u ∗ ( p ) i ˜ g jm u ' ( p ) m ) = i 2 b ( δ j i -k i k j k n k n ) . (16)</formula>
<text><location><page_3><loc_52><loc_55><loc_92><loc_59></location>Utilizing the above formalism of quantization along with the constraint relation, we evolve the mode function in different phases of the universe until the present epoch.</text>
<section_header_level_1><location><page_3><loc_54><loc_49><loc_90><loc_52></location>III. EVOLUTION OF ELECTROMAGNETIC FIELD DURING INFLATIONARY ERA</section_header_level_1>
<text><location><page_3><loc_52><loc_17><loc_92><loc_47></location>According to our model in Eq.(1), the anisotropy in the spacetime exists only towards the end of inflation. After the end of inflation, within a short period, the spacetime essentially becomes FLRW again, as seen from Fig. 1. However, it is important to mention that the largescale production of the EM field is not affected due to this short presence of anisotropy after inflation. It is also evident that b → 1 towards past infinity ensures that the Bunch-Davis vacuum condition is satisfied in the infinite past. Furthermore, we assume the background spacetime is de Sitter in nature, i.e., a = -1 / ( Hη ), where H is the Hubble parameter during inflation and remains constant throughout the entire inflation. Following these initial conditions, we numerically solve the mode function equations shown in Eq. (14). We consider k 1 = k 2 = k 3 = ˜ k ∼ 1 Mpc -1 for simplification and the Hubble parameter H = 10 -5 M pl remains constant throughout the inflationary era. Here M pl = √ 1 / (8 πG ) is the reduced Planck mass. We redefine the conformal time η as a dimensionless parameter x = ˜ kη . In terms of this new variable, Eq. (2) can be rewritten as,</text>
<formula><location><page_3><loc_64><loc_14><loc_92><loc_16></location>b ( x ) = 1 + αe -( x xm ) 2 , (17)</formula>
<text><location><page_3><loc_52><loc_9><loc_92><loc_13></location>where the parameters α and ˜ kη m are chosen accordingly to avoid the backreaction from anisotropy, which essentially means that the anisotropy acts as a perturbation</text>
<text><location><page_4><loc_9><loc_85><loc_11><loc_85></location></text>
<figure>
<location><page_4><loc_10><loc_74><loc_48><loc_93></location>
<caption>FIG. 2. Evolution of the mode functions ˜ u 1 , ˜ u 2 , and ˜ u 3 with x = ˜ kη for the value of anisotropic parameter α = 3 and x m = -2. As the anisotropy exists only along the x -direction, the mode function equation corresponding to the x -direction (˜ u 1 ) behaves differently than the other two (˜ u 2 , ˜ u 3 ).</caption>
</figure>
<text><location><page_4><loc_9><loc_54><loc_49><loc_61></location>over the FLRW universe. A detailed discussion of the anisotropic backreaction is done in the later section. In terms of the redefined variables x = ˜ kη, ˜ u i = √ ˜ ku i , and ˜ k = k 1 = k 2 = k 3 , the mode function equations can be written as,</text>
<formula><location><page_4><loc_15><loc_43><loc_49><loc_52></location>d 2 ˜ u 1 dx 2 -1 b db dx d ˜ u 1 dx +2˜ u 1 -˜ u 2 -˜ u 3 = 0 , d 2 ˜ u 2 dx 2 + 1 b db dx d ˜ u 2 dx + ˜ u 2 -˜ u 1 b 2 -˜ u 3 = 0 , (18) d 2 ˜ u 3 dx 2 + 1 b db dx d ˜ u 3 dx + ˜ u 3 -˜ u 1 b 2 -˜ u 2 = 0 ,</formula>
<text><location><page_4><loc_9><loc_40><loc_34><loc_42></location>along with the constraint equation,</text>
<formula><location><page_4><loc_20><loc_36><loc_49><loc_39></location>1 b 2 d ˜ u 1 dx + d ˜ u 2 dx + d ˜ u 3 dx = 0 . (19)</formula>
<text><location><page_4><loc_9><loc_24><loc_49><loc_35></location>By solving Eq. (18), we can obtain the mode function solution for different choices of the parameters α and x m as shown in Fig. 2. The above figure shows that the mode function grows in time due to anisotropy, particularly near the end of inflation. For values α < 0 . 03, field production stops altogether. Hence, we get a lower bound on the anisotropic parameter α ≥ 0 . 03. The upper bound on α is discussed in the later sections.</text>
<section_header_level_1><location><page_4><loc_10><loc_18><loc_47><loc_20></location>A. Power spectrum of the electromagnetic field during inflationary era</section_header_level_1>
<text><location><page_4><loc_9><loc_13><loc_49><loc_15></location>The stress energy-momentum tensor corresponding to the produced EM field is given by</text>
<formula><location><page_4><loc_20><loc_8><loc_49><loc_12></location>T µν = -2 √ -g δ [ √ -g L ] δg µν . (20)</formula>
<text><location><page_4><loc_52><loc_89><loc_92><loc_93></location>As a result, the energy-momentum tensor corresponding to the electromagnetic part of the Lagrangian boils down to</text>
<formula><location><page_4><loc_55><loc_85><loc_92><loc_88></location>T mn = -1 4 g mn g µα g νβ F µν F αβ + g µν F mµ F nν . (21)</formula>
<text><location><page_4><loc_52><loc_81><loc_92><loc_85></location>The total energy density of the system is given by the T tt component of the energy-momentum tensor. Therefore, the total electromagnetic energy density of the system is</text>
<formula><location><page_4><loc_54><loc_77><loc_92><loc_80></location>ρ = -⟨ T 0 0 ⟩ = 1 2 a 2 g ij ⟨ A ' i A ' j ⟩ + 1 4 g ij g ab ⟨ F ia F jb ⟩ . (22)</formula>
<text><location><page_4><loc_52><loc_74><loc_92><loc_76></location>Thus, we have the electric field and magnetic field energy densities as,</text>
<formula><location><page_4><loc_61><loc_67><loc_92><loc_73></location>ρ E ( x, η ) = 1 2 a 2 g ij ⟨ A ' i A ' j ⟩ , ρ B ( x, η ) = 1 4 g ij g mn ⟨ F ij F mn ⟩ , (23)</formula>
<text><location><page_4><loc_52><loc_61><loc_92><loc_66></location>respectively, where the expectation values are taken with respect to the initial Bunch-Davies (BD) vacuum. In the momentum space, these energy densities can be written as</text>
<formula><location><page_4><loc_55><loc_46><loc_92><loc_60></location>ρ E ( k, η ) = 1 2 a 4 ∑ p ∫ d 3 k (2 π ) 3 u ( p ) i ˜ g ij u ∗ ' ( p ) j , ρ B ( k, η ) = 1 4 a 4 ∑ p ∫ d 3 k (2 π ) 3 ˜ g ij ˜ g mn × [( k i k n u ( p ) m u ∗ ( p ) j -k i k j u ( p ) m u ∗ ( p ) n ) + ( k m k j u ( p ) i u ∗ ( p ) n -k m k n u i u ∗ ( p ) j ) ] . (24)</formula>
<text><location><page_4><loc_52><loc_40><loc_92><loc_44></location>In order to determine the strength of the magnetic field in the present era, we first define the power spectrum of the electromagnetic field as</text>
<formula><location><page_4><loc_64><loc_36><loc_92><loc_39></location>P E/B ( k, η ) = ∂ρ E/B ∂ ln k , (25)</formula>
<text><location><page_4><loc_52><loc_25><loc_92><loc_35></location>as already stated earlier, each polarization mode follows the same equation of motion. Therefore, all the polarization modes have equal contributions. Summing over all the polarization modes and using the assumption amplitude of all the momentum k 1 = k 2 = k 3 = ˜ k to be the same, we calculate the power spectrum of the electric and magnetic field as</text>
<formula><location><page_4><loc_53><loc_20><loc_92><loc_24></location>P E ( η, ˜ k ) = ˜ k 3 2 π 2 a 4 ( | u ' 1 ( η ) | 2 b 2 + | u ' 2 ( η ) | 2 + | u ' 3 ( η ) | 2 ) , (26)</formula>
<formula><location><page_4><loc_53><loc_10><loc_92><loc_19></location>P B ( η, ˜ k ) = ˜ k 5 2 π 2 a 4 [ 1 b 2 ( 2 | u 1 | 2 + | u 2 | 2 + | u 3 | 2 -2 ℜ ( u 1 u ∗ 2 ) -2 ℜ ( u 1 u ∗ 3 ) ) + ( | u 2 | 2 + | u 3 | 2 -2 ℜ ( u 2 u ∗ 3 ) )] . (27)</formula>
<text><location><page_5><loc_9><loc_86><loc_49><loc_93></location>With these forms of the power spectrum, our goal would be to calculate its strength at present. However, before that, we will calculate the condition for which the produced electromagnetic field should not back react to the background during inflation.</text>
<section_header_level_1><location><page_5><loc_10><loc_81><loc_47><loc_83></location>B. Backreaction of anisotropic background and generated EM field</section_header_level_1>
<text><location><page_5><loc_9><loc_37><loc_49><loc_79></location>In the previous section, we have briefly discussed the backreaction and strong coupling problem of inflationary magnetogenesis. In a general large-scale gauge field production scenario, a scalar field is coupled to the EM field to break the conformal invariance. Depending on the choice of the coupling function, it is possible to have a strong coupling problem, and different scenarios have been discussed in the literature [10, 16, 45, 46]. For the sake of completeness, we discuss it here briefly. In order to have a sustainable production of the electromagnetic field during inflation, the coupling function is often chosen to be an increasing function of time. However, it needs to revert to unity to restore the regular Maxwellian electromagnetism at the end of inflation. Hence, it needs to be very small at the start of the inflationary era, so the effective charge of electrons will be very large, and we cannot treat the gauge field as a free field during the inflationary era. In this proposal, there is no such direct coupling between the inflaton field and the EM field. Therefore, we do not need to worry about the strong coupling issue in this scenario. However, we must ensure that the anisotropy energy density or the generated EM field does not jeopardize the inflation. To this extent, we calculate the energy density produced by the anisotropic background and get a lower bound on the anisotropic parameter ˜ kη m and α introduced in Eq. (17). The energymomentum tensor of the background T µν is dictated by the Einstein equation in terms of the Einstein tensor G µν as</text>
<formula><location><page_5><loc_23><loc_34><loc_49><loc_35></location>G µν = 8 πGT µν . (28)</formula>
<text><location><page_5><loc_9><loc_29><loc_49><loc_33></location>In the case of Bianchi type I background as introduced in Eq. (1), the 00 component of the Einstein tensor can be calculated as</text>
<formula><location><page_5><loc_20><loc_25><loc_49><loc_28></location>G 00 = a ' (3 a ' b +2 ab ' ) a 2 b . (29)</formula>
<text><location><page_5><loc_9><loc_20><loc_49><loc_24></location>Thanks to this result, we can calculate the energy density corresponding to the anisotropic background. It turns out as</text>
<formula><location><page_5><loc_10><loc_12><loc_49><loc_19></location>ρ total = -T 0 0 = -1 8 πG G 0 0 = 1 8 πG ( 3 a ' 2 a 4 +2 a ' a 3 b ' b ) = 3 H 2 M 2 pl +2 HM 2 pl b ' ab , (30)</formula>
<text><location><page_5><loc_9><loc_9><loc_49><loc_11></location>where H ≡ a ' /a 2 is the Hubble parameter in conformal time during the inflation, a is the scale factor in de Sitter</text>
<text><location><page_5><loc_52><loc_83><loc_92><loc_93></location>spacetime ( a = -1 / ( Hη )). From the dynamics of the inflaton field during inflation, we already know that the total energy of the inflaton field is given by ρ inf = 3 H 2 M 2 pl . Therefore, the total background energy density in Eq. (30) consists of two parts. The first part we call inflationary energy density, and the second part is the energy density due to anisotropy in the background,</text>
<formula><location><page_5><loc_60><loc_79><loc_92><loc_82></location>ρ inf = 3 H 2 M 2 pl , ρ anis = 2 HM 2 pl b ' ab . (31)</formula>
<text><location><page_5><loc_52><loc_53><loc_92><loc_78></location>In our proposition, we have mentioned earlier that the anisotropy should act as a perturbation. Therefore, we must ensure that the anisotropic energy density must be much lower than the inflaton energy density. Furthermore, the electromagnetic energy density has to be lower than the anisotropic and inflaton energy densities. From the PLANCK data [47], we know that the temperature anisotropy in CMB is ∆ T T ∼ 10 -5 . If the anisotropic energy is closer to the perturbative limit towards the end of the inflationary era, it will not affect the CMB map, as observed by the PLANCK. We define e-folding number during the inflationary era as N = ln ( a a end ) , where a end is the scale factor at the end of inflation. By this definition, the e-folding number at the end of inflation N end = 0. Moreover, the total e-folding number during the inflation is N tot ≃ 60. The anisotropic factor b in terms of e-folding number can be written as</text>
<formula><location><page_5><loc_60><loc_50><loc_92><loc_52></location>b ( N ) = 1 + α exp [ -e 2( N m -N ) ] , (32)</formula>
<text><location><page_5><loc_52><loc_43><loc_92><loc_48></location>here N m is the e-folding number corresponding to the conformal time η m . We can calculate the ratio of the anisotropic energy density and the inflationary energy density in terms of the e-folding number N as follows</text>
<formula><location><page_5><loc_60><loc_39><loc_92><loc_42></location>∣ ∣ ∣ ∣ ρ anis ρ inf ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ 2 3 H b ' ab ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ 2 3 db dN 1 b ∣ ∣ ∣ ∣ . (33)</formula>
<text><location><page_5><loc_52><loc_32><loc_92><loc_37></location>In order to have sustainable inflation, such that the anisotropic energy density does not affect the inflation energy density, we need to have ∣ ∣ ∣ ∣ ρ anis ρ inf ∣ ∣ ∣ ∣ < 1 throughout</text>
<text><location><page_5><loc_52><loc_9><loc_92><loc_31></location>the entirety of the inflation. Thus, the ratio gives us an upper bound on α , which dictates the strength of the anisotropy. In Fig. 3, we can see that the ratio of the energy densities reaches its maximum towards the end of inflation. Thus, we can choose our parameters such that the ratio is up to the perturbative level ( ∼ 0 . 5). It gives us the upper bound α ≤ 1 . 48. Still, the CMB remains unaffected due to the presence of spatial anisotropy. However, it is worth mentioning here that we do not consider the dynamics of the anisotropy in the background. In order to make sure that the anisotropic background comes in towards the end of inflation, we take the upper limit on the parameter ˜ kη m ≥ -2. Furthermore, during inflation, the EM field also gets produced. It is also necessary to ensure that the generated gauge field energy density does not violate the inflationary energy density. We can see</text>
<figure>
<location><page_6><loc_10><loc_74><loc_48><loc_93></location>
<caption>FIG. 3. Evolution of the ratio of anisotropic energy density to inflationary energy density with the e-folding number N with different anisotropic parameters α and N m .</caption>
</figure>
<text><location><page_6><loc_9><loc_60><loc_49><loc_65></location>that the maximum production occurs towards the end of inflation from the nature of the coupling function introduced in Eq. (17). Thus, to avoid the backreaction problem, it is sufficient to satisfy</text>
<formula><location><page_6><loc_23><loc_57><loc_34><loc_58></location>ρ E + ρ B ⩽ ρ inf .</formula>
<text><location><page_6><loc_9><loc_50><loc_49><loc_56></location>We can obtain the values of the energy densities of the electric and magnetic fields from Eq. (23) and integrate over all the modes inside the horizon during the inflationary era. Which finally boils down to,</text>
<formula><location><page_6><loc_12><loc_32><loc_49><loc_49></location>ρ E = H 4 2 π 2 ∫ x f x i dxx 3 [ 1 b 2 ∣ ∣ ∣ ∣ d ˜ u 1 dx ∣ ∣ ∣ ∣ 2 + ∣ ∣ ∣ ∣ d ˜ u 2 dx ∣ ∣ ∣ ∣ 2 + ∣ ∣ ∣ ∣ d ˜ u 3 dx ∣ ∣ ∣ ∣ 2 ] , (34) ρ B = H 4 π 2 ∫ x f x i dxx 3 [ 1 b 2 ( 2 | ˜ u 1 | 2 + | ˜ u 2 | 2 + | ˜ u 3 | 2 -2 ℜ (˜ u 1 ˜ u ∗ 2 ) -2 ℜ (˜ u 1 ˜ u ∗ 3 ) ) + ( | ˜ u 2 | 2 + | ˜ u 3 | 2 -2 ℜ (˜ u 2 ˜ u ∗ 3 ) )] . (35)</formula>
<text><location><page_6><loc_9><loc_9><loc_49><loc_30></location>Where we recall the variables x = ˜ kη, ˜ u i = √ ˜ ku i , evaluating the integrations numerically, the ratio of the energy densities turns out to be ρ E + ρ B ρ inf ∼ 10 -9 , for the anisotropic parameter ˜ kη m = -1 and α = 1 . 45. As the generated electromagnetic energy density is very low compared to the background inflaton energy density, the backreaction problem is also avoided. Therefore, with this formalism, we can sustainably produce the EM field during inflation without worrying about the strong coupling or backreaction problem. On the other hand, ensuring that the generated EM field does not surpass the energy density of the inflationary background is also necessary. Eq. (17) shows that the maximum energy density occurs at η = 0. However, we have taken that the</text>
<text><location><page_6><loc_52><loc_72><loc_92><loc_93></location>inflation ends at η f , so there is no production of the large-scale magnetic field in the post-inflationary era. If the electromagnetic energy density is less than the anisotropic energy density at the end of inflation, all the sufficient conditions for no backreaction are satisfied. To this end, we reiterate that the anisotropic parameter α is so chosen that the anisotropic energy density remains subdominant compared to the inflaton energy density. We further show that the produced energy density of the electromagnetic field is less than the anisotropic energy density. In conclusion, the produced electromagnetic field affects neither the inflationary nor the anisotropic background. Therefore, this formalism effectively produces a magnetic field without special coupling to avoid the backreaction effect.</text>
<section_header_level_1><location><page_6><loc_55><loc_68><loc_89><loc_69></location>IV. POST INFLATIONARY EVOLUTION</section_header_level_1>
<text><location><page_6><loc_52><loc_44><loc_92><loc_65></location>The anisotropic factor b goes to unity after the end of inflation, and the spacetime becomes conformally flat. The EM field evolves as a usual Maxwellian field subsequently. However, depending on the evolution of the universe, we can have two different scenarios of field evolution: (i) In the first scenario, it is assumed that the universe instantly goes into radiation domination, i.e., the inflaton field instantly decays and produces radiation. (ii) In the second scenario, the inflaton decays within a finite time, and therefore, it goes through a brief period of reheating era having a non-zero e-folding number and very low conductivity. The dynamics of the subsequent evolution of the universe dictate the present strength of the observed magnetic field. We will discuss both scenarios in the next subsections.</text>
<section_header_level_1><location><page_6><loc_56><loc_40><loc_87><loc_41></location>A. The case of instantaneous reheating</section_header_level_1>
<text><location><page_6><loc_52><loc_15><loc_92><loc_38></location>Here in this section, we will find the strength of the magnetic field in the present time, considering an instantaneous reheating scenario. In this case, after the end of the inflationary era, the universe instantly thermalizes and goes to the radiation-dominated era. As the conductivity of the universe becomes very large, the electric field dies out instantly. However, the magnetic field produced during the inflationary era decays as a radiation density P B ∝ a -4 . Therefore, incorporating the conservation of entropy, we can compute the strength of the magnetic field, relating to the field strength at the end of inflation. We have already calculated the power spectra of the magnetic field during the inflationary era in Eq. (27). In terms of the mode functions, the explicit expression for the present-day magnetic field turns out as</text>
<formula><location><page_6><loc_52><loc_9><loc_92><loc_14></location>B 0 = ( -˜ kη f a f H π 2 a 0 ) 2 √ | ˜ u 1 ( x f ) | 2 + | ˜ u 2 ( x f ) | 2 + | ˜ u 3 ( x f ) | 2 . (36)</formula>
<text><location><page_7><loc_9><loc_70><loc_49><loc_93></location>The above expression is in the GeV 2 unit. Here, we recall the variable ˜ u i = √ ˜ ku i (with i = 1, 2, 3), H is the Hubble parameter during inflation, and ˜ k is the scale under consideration in which we will estimate the strength of the magnetic field. Furthermore, in order to calculate the value of B 0 from Eq. (36), we first need to evaluate the value of the ratio a f a 0 . We evaluate the value to be a 0 a f ≈ 10 30 ( H/ 10 -5 M pl ) 1 / 2 . Here, in particular, we have taken the value of the Hubble parameter to be H = 10 -5 M pl . With the numerical solution of the mode functions from Fig. 2 at the end of inflation ˜ kη f = -0 . 0001 and Eq. (36) we can evaluate the strength of the magnetic field at present-day using the conversion 1 G = 1 . 95 × 10 -20 GeV 2 for different values of the anisotropic parameters α and ˜ kη m .</text>
<text><location><page_7><loc_9><loc_49><loc_49><loc_70></location>In Fig. 4, we can see the variation of the present-day magnetic field B 0 with α for a fixed value of ˜ kη m as well as the variation of magnetic field strength with ˜ kη m . The maximum value of B 0 obtained in the instant reheating scenario for fixed value of ˜ kη m is B 0 = 2 . 86 × 10 -21 G varying α . Similarly for a fixed value of α , the maximum value of B 0 obtained is B 0 = 3 . 24 × 10 -21 G . Experiments like Faraday rotation and gamma-ray observation impose a bound on the present strength of primordial magnetic field 10 -10 G ≲ B 0 ≲ 10 -22 G [5]. Therefore, this proposal can generate a large-scale magnetic field within the experimental bound of the present-day intergalactic magnetic field. With the choice of the anisotropic parameter in the range 0 . 03 ≤ α ≤ 1 . 48, the backreaction or the strong coupling problem is also avoided.</text>
<section_header_level_1><location><page_7><loc_9><loc_43><loc_48><loc_46></location>B. The case of prolonged reheating with constant equation of state</section_header_level_1>
<text><location><page_7><loc_9><loc_11><loc_49><loc_41></location>In the last section, we saw that we can generate the required strength of the magnetic field in the instant reheating scenario. However, if we consider a reheating phase with a non-zero e-folding number, then the conductivity of the universe does not reach infinity instantly. Instead, during this period, the conductivity of the universe may remain to be very low. As a result, the electric field does not go to zero immediately and induces a magnetic field during this period. This conversion of the electric field into the magnetic field during the reheating phase occurs through Faraday induction [48]. This conversion of an electric field to a magnetic field makes it diluted slowly during the reheating era compared to the previous case of P B ∝ a -4 . Thus, a finite reheating era further strengthens the magnetic field on a large scale and gives us bounds on the EoS during the reheating era. After the inflation ends, the anisotropic factor b goes to unity after a very short period, and the EM field evolves in the usual manner. Following the regular Maxwellian evolution, the equation of motion of the mode functions u i ( ˜ k, η ) during the reheating becomes</text>
<formula><location><page_7><loc_17><loc_8><loc_49><loc_10></location>u '' ( re ) i ( ˜ k, η ) + 3 ˜ k 2 u ( re ) i ( ˜ k, η ) = 0 , (37)</formula>
<text><location><page_7><loc_52><loc_83><loc_92><loc_93></location>where i = 1 , 2 , 3 are the indices corresponding to three spatial components of the gauge field and u ( re ) i ( ˜ k, η ) are the mode functions during reheating. Furthermore, we consider the universe a poor conductor during this period. To be precise, we take the conductivity to be zero. The solution of the mode function from Eq. (37), along with the proper normalization condition, gives us</text>
<formula><location><page_7><loc_55><loc_63><loc_92><loc_77></location>u ( re ) 1 ( ˜ k, η ) = 1 √ 6 √ 3 ˜ k [ α 1 ( ˜ k ) e -i √ 3 ˜ k ( η -η f ) + β 1 ( ˜ k ) e i √ 3 ˜ k ( η -η f ) ] , u ( re ) 2 , 3 ( ˜ k, η ) = -1 2 √ 6 √ 3 ˜ k [ α 2 , 3 ( ˜ k ) e -i √ 3 ˜ k ( η -η f ) + β 2 , 3 ( ˜ k ) e i √ 3 ˜ k ( η -η f ) ] , (38)</formula>
<text><location><page_7><loc_52><loc_51><loc_92><loc_56></location>with α i and β i are the integration constants, and η f denotes the end of inflation. the integration constants are evaluated at the end of inflation η f by equating the junction conditions of inflationary and reheating era</text>
<formula><location><page_7><loc_53><loc_41><loc_92><loc_45></location>u ( re ) i ( ˜ k, η f ) = u i ( ˜ k, η f ) and u ' ( re ) i ( ˜ k, η f ) = u ' i ( ˜ k, η f ) . (39)</formula>
<text><location><page_7><loc_52><loc_34><loc_92><loc_41></location>In the above Eq. (39), u i ( ˜ k, η ) are mode functions during the inflationary era which follow Eq. (14) and u ( re ) i are mode functions during the reheating era following Eq. (37). This immediately leads to the integration constants,</text>
<formula><location><page_7><loc_63><loc_26><loc_79><loc_28></location>√ √</formula>
<formula><location><page_7><loc_54><loc_9><loc_92><loc_27></location>α 1 ( ˜ k ) = √ 3 3 ˜ k 2 u 1 ( ˜ k, η f ) + i √ 3 2 ˜ k u ' 1 ( ˜ k, η f ) , β 1 ( ˜ k ) = √ 3 √ 3 ˜ k 2 ˜ u 1 ( ˜ k, x f ) -i √ √ 3 2 ˜ k u ' 1 ( ˜ k, η f ) , α 2 , 3 ( ˜ k ) = -√ 6 √ 3 ˜ ku 2 , 3 ( ˜ k, η f ) -i √ 2 √ 3 ˜ k u ' 2 , 3 ( ˜ k, η f ) , β 2 , 3 ( ˜ k ) = -√ 6 √ 3 ˜ ku 2 , 3 ( ˜ k, η f ) + i √ 2 √ 3 ˜ k u ' 2 , 3 ( ˜ k, η f ) . (40)</formula>
<figure>
<location><page_8><loc_13><loc_71><loc_88><loc_93></location>
<caption>FIG. 4. (a) Variation of the present strength of the magnetic field B 0 with α for a fixed value of the anisotropic parameter ˜ kη m = -1. (b) Variation of the present strength of the magnetic field B 0 with ˜ kη m for a fixed value of the anisotropic parameter α = 1 . 45, the ratio of the anisotropic energy density in this case remains constant at ρ anis /ρ inf = 0 . 492.</caption>
</figure>
<text><location><page_8><loc_9><loc_58><loc_49><loc_61></location>With all these, we can now compute the time-evolving power spectrum during reheating as</text>
<formula><location><page_8><loc_11><loc_29><loc_49><loc_57></location>P B ( η, ˜ k ) = ˜ k 5 π 2 a 4 ( | u ( re ) 1 | 2 + | u ( re ) 2 | 2 + | u ( re ) 3 | 2 ) = ∑ i ˜ k 4 π 2 a 4 | ˜ u ( re ) i | 2 = ˜ k 4 π 2 a 4 [ 1 6 √ 3 ( | α 1 | 2 + | β 1 | 2 +2 | α 1 || β 1 | × cos[ Arg ( α 1 β ∗ 1 ) -2 √ 3 ˜ k ( η -η f )] ) + 1 24 √ 3 ( | α 2 | 2 + | β 2 | 2 +2 | α 2 || β 2 | × cos[ Arg ( α 2 β ∗ 2 ) -2 √ 3 ˜ k ( η -η f )] ) + 1 24 √ 3 ( | α 3 | 2 + | β 3 | 2 +2 | α 3 || β 3 | × cos[ Arg ( α 3 β ∗ 3 ) -2 √ 3 ˜ k ( η -η f ) )] . (41)</formula>
<text><location><page_8><loc_9><loc_22><loc_49><loc_28></location>In order to estimate the strength of the magnetic field during the reheating era, we first need to evaluate the term η -η f . Following Ref. [48] the term is calculated as</text>
<formula><location><page_8><loc_22><loc_18><loc_49><loc_22></location>η -η f = ∫ a a f da a 2 H . (42)</formula>
<text><location><page_8><loc_9><loc_9><loc_49><loc_17></location>As the Hubble constant H is present in the above equation, it is evident that the quantity η -η f depends on the background's evolution during the inflationary era. In particular, how the inflaton energy density is converted into radiation energy density. In general, there are two scenarios</text>
<unordered_list>
<list_item><location><page_8><loc_54><loc_58><loc_92><loc_61></location>· Evolution through time-independent, effective equation of state.</list_item>
<list_item><location><page_8><loc_54><loc_54><loc_92><loc_57></location>· Perturbative decay of inflaton into radiation (perturbative reheating scenario).</list_item>
</unordered_list>
<text><location><page_8><loc_52><loc_37><loc_92><loc_53></location>Here in this paper, we will only discuss evolution through an independent constant effective EoS. In this context, we follow the methodology proposed by Kamionkowski et al. in Ref. [49]. Here, the evolution of the background is parametrized by a constant effective EoS ω eff . Therefore, the Hubble parameter during the reheating evolves as H ∝ a -3 2 (1+ ω eff ) . The physical parameters of reheating, like the e-folding number of the reheating era N re and the reheating temperature T re , can be expressed in terms of the inflationary parameters and effective EoS ω eff as [51]</text>
<formula><location><page_8><loc_52><loc_30><loc_93><loc_36></location>N re = 1 3 ω eff -1 [ ln( ρ f ) -ln ( π 2 g re 30 ) -1 3 ln ( 43 11 g s,re ) -4 ln ( a 0 T 0 k ) +4ln( H k ) + 4 N k ] , (43)</formula>
<formula><location><page_8><loc_55><loc_24><loc_92><loc_27></location>T re = ( 43 11 g s,re ) 1 / 3 ( a 0 T 0 k H k e -N k e -N re ) , (44)</formula>
<text><location><page_8><loc_52><loc_9><loc_92><loc_23></location>where H k denotes the Hubble parameter at the time of horizon crossing, k/a 0 = 0 . 05 Mpc -1 is the pivot scale, g re is the degrees of freedom during reheating and N k is the total e-folding number from the end of inflation till horizon crossing. As we have not considered any particular inflation potential in this paper, we develop a modelindependent way to determine N k following Ref.[50]. In the calculation of N k (see Appendix C), we have taken the central values of scalar spectral index n s = 0 . 9649 and scalar perturbation amplitude ln[10 10 A s ] = 3 . 044,</text>
<text><location><page_9><loc_9><loc_74><loc_49><loc_93></location>considering the constraints provided by the PLANCK data [47] and as an input parameter we have chosen N k = 50. With this choice of n s , N k , we get an upper bound on the effective EoS ω eff < 0 . 164 from the BBN bound of reheating temperature T re ∼ 10 -2 GeV. Now, in order to connect the reheating parameters N re , T re to the strength of the primordial magnetic field, we need to evaluate the quantity η -η f in Eq. (42). It is evaluated following the evolution of the Hubble parameter during the reheating era. As the EoS is constant ω eff , the variation of the Hubble parameter during the reheating era ( H re ) is related to the Hubble parameter at the end of inflation ( H f ) as</text>
<formula><location><page_9><loc_19><loc_70><loc_49><loc_73></location>H re = H f ( a re a f ) -3 2 (1+ ω eff ) , (45)</formula>
<text><location><page_9><loc_9><loc_63><loc_49><loc_68></location>where the subscript ' re ' represents the end of reheating. Thus, a re and H re are the scale factor and Hubble parameter at the end of reheating, respectively. Following the above relation, the term in Eq. (42) boils down to</text>
<formula><location><page_9><loc_16><loc_58><loc_49><loc_61></location>η -η f = 2 1 + 3 ω eff ( 1 aH -1 a f H f ) . (46)</formula>
<text><location><page_9><loc_9><loc_42><loc_49><loc_57></location>Substituting the value of the extra reheating term η -η f , we can calculate the present strength of the magnetic field as a function of the effective EoS ω eff . After the end of reheating, the conductivity of the universe goes to infinity. Therefore, the electric field goes to zero, and the Faraday conversion of the electric field into the magnetic field stops at the end of reheating. And the magnetic field decays as radiation ( a -4 ) until now. From the conservation of magnetic energy density, the present strength of the magnetic field can be calculated from the relation as follows</text>
<formula><location><page_9><loc_19><loc_37><loc_49><loc_41></location>∂ρ B ∂ ln k ∣ ∣ ∣ ∣ 0 = ( a re a 0 ) 4 ∂ρ B ∂ ln k ∣ ∣ ∣ ∣ re . (47)</formula>
<text><location><page_9><loc_9><loc_33><loc_49><loc_36></location>Evolving through the reheating era, the strength of the magnetic field in the present era turns out as</text>
<formula><location><page_9><loc_13><loc_28><loc_49><loc_32></location>B 0 = √ 2 6 π √ 3 ( ˜ k a 0 ) 2 [ I 1 + 1 4 ( I 2 + I 3 ) ] 1 / 2 , (48)</formula>
<text><location><page_9><loc_9><loc_25><loc_13><loc_26></location>where,</text>
<formula><location><page_9><loc_11><loc_19><loc_49><loc_24></location>I i = | α i | 2 + | β i | 2 +2 | α i | | β i | cos(Arg( α i β ∗ i ) -Φ) , Φ = 4 ˜ k √ 3 (1 + 3 ω eff ) a f H f ( ( H f H re ) δ -1 ) . (49)</formula>
<text><location><page_9><loc_9><loc_9><loc_49><loc_17></location>In the above Eq. (48), δ = (3 ω eff + 1) / (3 ω eff + 3). Varying the EoS ω eff , we get the present-day strength of the magnetic field of B 0 ∼ 4 × 10 -20 G, which is one order higher than what was predicted for instantaneous reheating case, which is ∼ 3 × 10 -21 G. Furthermore, from the observed strength of the magnetic field, we also get</text>
<figure>
<location><page_9><loc_53><loc_76><loc_91><loc_94></location>
<caption>FIG. 5. Variation of present magnetic strength with effective equation of state ω eff for the choice of anisotropic parameters α = 1 . 45 and ˜ kη m = -1. With the model independent formalism input parameter N k = 50 and inflationary parameter n s = 0 . 9649.</caption>
</figure>
<text><location><page_9><loc_52><loc_53><loc_92><loc_63></location>a lower bound of EoS ω eff > 0 . 132. From Fig. 5, we see that the present strength of the magnetic field increases due to the Faraday conversion of the electric field into the magnetic field; such increment is quite insensitive to the reheating EoS. This increment is small since the strength of the electric field compared to the magnetic field at the end of inflation is not significantly higher.</text>
<section_header_level_1><location><page_9><loc_56><loc_48><loc_87><loc_49></location>V. SUMMARY AND CONCLUSIONS</section_header_level_1>
<text><location><page_9><loc_52><loc_9><loc_92><loc_46></location>This paper proposes a new formalism to generate largescale magnetic fields during the inflationary era. The novelty of the work lies in the generation of fields during inflation. Several works have been done in the context of inflationary magnetogenesis. However, all the previous works rely on the conformal breaking coupling of the EM field with some scalar field or gravity. In the present case, we have taken the underlying background to be an anisotropic one (Bianchi type I), keeping conformal property intact. Due to this, our model does not suffer from the usual strong coupling problem. In the process, we have introduced two parameters α and η m to characterize the behavior of the anisotropic scale factor b ( η ). By appropriately tuning those anisotropic parameters, we further addressed the backreaction problem. If we take the ratio of the anisotropic energy density ρ anis and the inflaton energy density ρ inf to be ρ anis /ρ inf ⩽ 0 . 5 we get an upper bound of α ≤ 1 . 48. Furthermore, to ensure that electromagnetic field gets produced during the inflationary era, we get a lower bound on the parameter α ≥ 0 . 03. The parameter η m is so chosen that the anisotropy appears towards the end of inflation. For this, we have taken ˜ kη m ≥ -2, ensuring that the anisotropy is localized and short-lived. With this choice of parameters, we find that the ratio of the energy density of the generated EM field to the total inflaton energy density is ∼ 10 -9 ,</text>
<text><location><page_10><loc_9><loc_69><loc_49><loc_93></location>which implies that the electromagnetic energy density is also lower than the anisotropic energy density. Therefore, the generated electromagnetic field neither back-reacts on the inflaton field nor the anisotropic background. Finally, this set of parameters gives us a present strength of magnetic field B 0 ∼ 3 × 10 -21 G , for α = 1 . 45 and ˜ kη m = -2, which is well in between the latest bound on present-day magnetic field strength. However, if we consider an elongated reheating period followed by inflation, the magnetic field strength further increases. This increase in strength occurs due to Faraday's conversion of the electric field to the magnetic field. By this prolonged reheating era, we get the present strength of magnetic field B 0 ∼ 4 × 10 -20 G . Through the introduction of the reheating era, we also get a tight constraint on the range of equation of state 0 . 132 < ω eff < 0 . 164 for the particular choice of inflationary parameter n s and N k . Due</text>
<text><location><page_10><loc_52><loc_85><loc_92><loc_93></location>to the presence of anisotropy, there might be interesting signatures of the anisotropy on gravitational waves at small scales. Further, the most interesting would be investigating the origin of such anisotropy, particularly near the end of inflation. All those questions we leave for our future study.</text>
<section_header_level_1><location><page_10><loc_62><loc_81><loc_82><loc_82></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_10><loc_52><loc_69><loc_92><loc_78></location>DM wishes to acknowledge support from the Science and Engineering Research Board (SERB), Department of Science, and Technology (DST), Government of India (GoI), through the Core Research Grant CRG/2020/003664. DM and SP also thanks the Gravity and High Energy Physics groups at IIT Guwahati for illuminating discussions.</text>
<section_header_level_1><location><page_10><loc_19><loc_63><loc_81><loc_64></location>Appendix A: Power spectrum of the electromagnetic field during inflationary era</section_header_level_1>
<text><location><page_10><loc_10><loc_59><loc_74><loc_61></location>We have the energy-momentum tensor corresponding to the free Maxwellian Lagrangian,</text>
<formula><location><page_10><loc_34><loc_56><loc_92><loc_59></location>T mn = -1 4 g mn g µα g νβ F µν F αβ + g µν F mµ F nν . (A1)</formula>
<text><location><page_10><loc_9><loc_53><loc_92><loc_55></location>The energy density of the electromagnetic field is obtained from the '00' component of the energy-momentum tensor, which boils down to</text>
<formula><location><page_10><loc_36><loc_49><loc_92><loc_52></location>T 00 = 1 2 g ij A ' i A ' j + a 2 4 g im g jn F ij F mn . (A2)</formula>
<text><location><page_10><loc_9><loc_46><loc_92><loc_49></location>Upon trading the EM field into the quantum operator and referring to Eq.(23), we have the expression for the electric and magnetic field energy densities as</text>
<formula><location><page_10><loc_39><loc_39><loc_92><loc_45></location>ρ E ( x, η ) = 1 2 a 2 g ij ⟨ A ' i A ' j ⟩ , ρ B ( x, η ) = 1 4 g ij g mn ⟨ F ij F mn ⟩ , (A3)</formula>
<text><location><page_10><loc_9><loc_36><loc_92><loc_39></location>where the expectation value is obtained over the BD vacuum. The expectation value of g im g jn F ij F mn in the BD vacuum in terms of mode functions boils down to</text>
<formula><location><page_10><loc_18><loc_20><loc_92><loc_36></location>⟨ g im g jn F ij F mn ⟩ = ∑ p ∫ d 3 ˜ k (2 π ) 3 [ 2 ˜ k 2 a 4 b 2 ( 2 | u 1 | 2 + | u 2 | 2 + | u 3 | 2 -u 1 u ∗ 2 -u 2 u ∗ 1 -u 1 u ∗ 3 -u 3 u ∗ 1 ) + 2 ˜ k 2 a 4 ( | u 2 | 2 + | u 3 | 2 -u 2 u ∗ 3 -u 3 u ∗ 2 ) ] = ∑ p ∫ d 3 ˜ k (2 π ) 3 [ 2 ˜ k 2 a 4 b 2 ( 2 | u 1 | 2 + | u 2 | 2 + | u 3 | 2 -2 ℜ ( u 1 u ∗ 2 ) -2 ℜ ( u 1 u ∗ 3 ) ) + 2 ˜ k 2 a 4 ( | u 2 | 2 + | u 3 | 2 -2 ℜ ( u 2 u ∗ 3 ) ) ] . (A4)</formula>
<text><location><page_10><loc_9><loc_16><loc_92><loc_19></location>Substituting the expectation value of the term g im g jn F ij F mn in Eq.(A3), we get the energy density of the magnetic field. With the polarization index, the expression for the magnetic field energy density turns out as</text>
<formula><location><page_10><loc_17><loc_9><loc_92><loc_16></location>ρ B ( ˜ k, η ) = ∑ p ∫ d 3 ˜ k (2 π ) 3 ˜ k 2 2 a 4 [ 1 b 2 ( 2 | u ( p ) 1 | 2 + | u ( p ) 2 | 2 + | u ( p ) 3 | 2 -2 ℜ ( u ( p ) 1 u ∗ ( p ) 2 ) -2 ℜ ( u ( p ) 1 u ∗ ( p ) 3 ) ) + ( | u ( p ) 2 | 2 + | u ( p ) 3 | 2 -2 ℜ ( u ( p ) 2 u ∗ ( p ) 3 ) ) ] . (A5)</formula>
<text><location><page_11><loc_9><loc_90><loc_92><loc_93></location>As all the polarization modes behave the same way, summing over all the polarization, we finally get the energy density of the magnetic field</text>
<formula><location><page_11><loc_24><loc_84><loc_92><loc_90></location>ρ B ( ˜ k, η ) = ∫ d 3 ˜ k (2 π ) 3 ˜ k 2 a 4 [ 1 b 2 ( 2 | u 1 | 2 + | u 2 | 2 + | u 3 | 2 -2 ℜ ( u 1 u ∗ 2 ) -2 ℜ ( u 1 u ∗ 3 ) ) + ( | u 2 | 2 + | u 3 | 2 -2 ℜ ( u 2 u ∗ 3 ) ) ] . (A6)</formula>
<text><location><page_11><loc_9><loc_79><loc_92><loc_82></location>Similarly, evaluating the expectation value of the term g ij A ' i A ' j and substituting it back in Eq.(A3), we get the energy density of the electric field in terms of mode functions as</text>
<formula><location><page_11><loc_34><loc_75><loc_92><loc_79></location>ρ E ( ˜ k, η ) = ∫ d 3 ˜ k (2 π ) 3 1 a 4 ( | u ' 1 | 2 b 2 + | u ' 2 | 2 + | u ' 3 | 2 ) . (A7)</formula>
<section_header_level_1><location><page_11><loc_16><loc_71><loc_84><loc_72></location>Appendix B: Backreaction of anisotropic background and generated electromagnetic field</section_header_level_1>
<text><location><page_11><loc_10><loc_68><loc_47><loc_69></location>The energy-momentum tensor of the background T</text>
<formula><location><page_11><loc_45><loc_66><loc_56><loc_67></location>G µν = 8 πGT µν ,</formula>
<formula><location><page_11><loc_47><loc_66><loc_92><loc_69></location>µν is dictated by the Einstein equation (B1)</formula>
<text><location><page_11><loc_9><loc_62><loc_92><loc_65></location>where G µν is the Einstein tensor and G is the gravitational constant. The Einstein tensor can be calculated in terms of the Riemann tensor ( R µν ) and Ricci scalar ( R ),</text>
<formula><location><page_11><loc_42><loc_59><loc_58><loc_61></location>G µν = R µν -1 2 Rg µν .</formula>
<text><location><page_11><loc_9><loc_55><loc_92><loc_58></location>In the case of Bianchi type I background as introduced in Eq. (1), the '00' component of the Einstein tensor turns out as</text>
<formula><location><page_11><loc_42><loc_52><loc_92><loc_55></location>G 00 = a ' (3 a ' b +2 ab ' ) a 2 b . (B2)</formula>
<text><location><page_11><loc_9><loc_50><loc_49><loc_51></location>This essentially gives us the background energy density,</text>
<formula><location><page_11><loc_27><loc_46><loc_92><loc_49></location>ρ total = -T 0 0 = 1 8 πG ( 3 a ' 2 a 4 +2 a ' a 3 b ' b ) = 3 H 2 M 2 pl +2 HM 2 pl b ' ab . (B3)</formula>
<text><location><page_11><loc_9><loc_44><loc_90><loc_45></location>The ratio of anisotropic energy density to the inflaton energy density is given in terms of the e-folding number N ,</text>
<formula><location><page_11><loc_39><loc_40><loc_62><loc_44></location>∣ ∣ ∣ ∣ ρ anis ρ inf ∣ ∣ ∣ ∣ = 2 HM 2 pl b ' ab 3 H 2 M 2 pl = 2 3 db dN 1 b ,</formula>
<text><location><page_11><loc_9><loc_36><loc_92><loc_39></location>with the e-folding number is defined as dN = d ln a , where a is the scale factor. From the above equation, we can get the ratio of the anisotropic and the inflationary energy densities.</text>
<text><location><page_11><loc_9><loc_33><loc_92><loc_36></location>The backreaction problem can be evaded if the total energy of the generated EM field is less than the energy density of the inflaton field, that is</text>
<formula><location><page_11><loc_45><loc_31><loc_92><loc_32></location>ρ E + ρ B ⩽ ρ inf . (B4)</formula>
<text><location><page_11><loc_9><loc_25><loc_92><loc_30></location>The total energy densities of the EM are given by Eq.(24). Integrating all the modes, we get the total energy density. The modes involved are given by ˜ k i = a i H , which crosses the horizon at the beginning of inflation, and k f = a f H are the modes that cross the horizon at the end of inflation. We have the total energy density of the electric field expressed as,</text>
<formula><location><page_11><loc_19><loc_8><loc_92><loc_24></location>ρ E = ∫ ˜ k f ˜ k i d ˜ k ˜ k ˜ k 3 2 π 2 a 4 ( | u ' 1 ( η ) | 2 b ( η ) 2 + | u ' 2 ( η ) | 2 + | u ' 3 ( η ) | 2 ) = 1 2 π 2 ∫ ˜ k f ˜ k i d ˜ k ˜ k 3 a 4 ( 1 b 2 ∣ ∣ ∣ ∣ d ˜ u 1 dx ∣ ∣ ∣ ∣ 2 + ∣ ∣ ∣ ∣ d ˜ u 2 dx ∣ ∣ ∣ ∣ 2 + ∣ ∣ ∣ ∣ d ˜ u 3 dx ∣ ∣ ∣ ∣ 2 ) = H 4 2 π 2 ∫ ˜ k f ˜ k i d ( ˜ kη )( ˜ k 3 η 3 ) ( 1 b 2 ∣ ∣ ∣ ∣ d ˜ u 1 dx ∣ ∣ ∣ ∣ 2 + ∣ ∣ ∣ ∣ d ˜ u 2 dx ∣ ∣ ∣ ∣ 2 + ∣ ∣ ∣ ∣ d ˜ u 3 dx ∣ ∣ ∣ ∣ 2 ) (Substituted a = -1 /Hη ) = H 4 2 π 2 ∫ x f x i dxx 3 ( 1 b 2 ∣ ∣ ∣ ∣ d ˜ u 1 dx ∣ ∣ ∣ ∣ 2 + ∣ ∣ ∣ ∣ d ˜ u 2 dx ∣ ∣ ∣ ∣ 2 + ∣ ∣ ∣ ∣ d ˜ u 3 dx ∣ ∣ ∣ ∣ 2 ) . (B5)</formula>
<text><location><page_12><loc_9><loc_92><loc_58><loc_93></location>Similarly, the total energy density of the magnetic field is calculated</text>
<formula><location><page_12><loc_12><loc_88><loc_92><loc_91></location>ρ B = H 4 2 π 2 ∫ x f x i dxx 3 [ 1 b 2 ( 2 | ˜ u 1 | 2 + | ˜ u 2 | 2 + | ˜ u 3 | 2 -2 ℜ (˜ u 1 ˜ u ∗ 2 ) -2 ℜ (˜ u 1 ˜ u ∗ 3 ) ) + ( | ˜ u 2 | 2 + | ˜ u 3 | 2 -2 ℜ (˜ u 2 ˜ u ∗ 3 ) )] , (B6)</formula>
<text><location><page_12><loc_9><loc_82><loc_92><loc_86></location>where x = ˜ kη and ˜ u i = √ ˜ ku i . Integrating over the limits numerically with the solutions of the mode functions, we get the total energy density of the generated EM field. Finally, comparing the energy density of the EM field to the inflaton field ( ρ inf = 3 H 2 M 2 pl ), we get</text>
<formula><location><page_12><loc_44><loc_78><loc_92><loc_81></location>ρ E + ρ B ρ inf ∼ 10 -9 . (B7)</formula>
<text><location><page_12><loc_9><loc_74><loc_92><loc_77></location>After the inflation, the production of electromagnetic field stops altogether. If we consider an instant reheating scenario, it essentially behaves as a radiation field. Thus, by conservation of entropy, we have</text>
<formula><location><page_12><loc_39><loc_65><loc_92><loc_73></location>a 4 0 ∂ρ B ∂ ln k ∣ ∣ ∣ ∣ 0 = a 4 f ∂ρ B ∂ ln k ∣ ∣ ∣ ∣ η f ⇒ ∂ρ B ∂ ln k ∣ ∣ ∣ ∣ 0 = ( a f a 0 ) 4 ∂ρ B ∂ ln k ∣ ∣ ∣ ∣ η f . (B8)</formula>
<text><location><page_12><loc_9><loc_60><loc_92><loc_64></location>Where '0' denotes the present epoch, a 0 represents the scale factor at present, η f is the conformal time at the end of inflation, and a f is the scale factor corresponding to η f . Implementing Eq. (B8), we can evaluate the present-day magnetic field strength.</text>
<section_header_level_1><location><page_12><loc_27><loc_56><loc_74><loc_57></location>Appendix C: Calculation of total e-folding number of inflation</section_header_level_1>
<text><location><page_12><loc_10><loc_53><loc_68><loc_54></location>We have the expression of the total e-folding number of inflation N k from [50] as</text>
<formula><location><page_12><loc_44><loc_48><loc_92><loc_52></location>N k = ∫ t f t k H ( t ) dt. (C1)</formula>
<text><location><page_12><loc_9><loc_43><loc_92><loc_47></location>The Hubble parameter explicitly depends on the background evolution. Therefore, to calculate the actual Hubble parameter, we will Taylor expand around the conformal time of horizon crossing t k to incorporate the background effects,</text>
<formula><location><page_12><loc_36><loc_40><loc_92><loc_42></location>H ( t ) = H k + ˙ H k ( t -t k ) + 1 2 H k ( t -t k ) 2 . (C2)</formula>
<text><location><page_12><loc_9><loc_36><loc_92><loc_39></location>In the above Eq. (C2), we consider only terms up to O ( H k ). The total duration of the inflation is represented as ∆ t = t -t k . Then, by Eq. (C2), the Hubble parameter at the end of inflation can be written as</text>
<formula><location><page_12><loc_40><loc_33><loc_92><loc_35></location>H f = H k + ˙ H k ∆ t + H k (∆ t ) 2 . (C3)</formula>
<text><location><page_12><loc_9><loc_29><loc_92><loc_32></location>Consequently, the duration of inflation (∆ t ) can be expressed in terms of the Hubble parameter and the derivative of it as</text>
<formula><location><page_12><loc_35><loc_25><loc_92><loc_28></location>∆ t = | ˙ H k | H k ( 1 -√ 1 -2 H k | ˙ H k | 2 ( H k -H f ) ) . (C4)</formula>
<text><location><page_12><loc_9><loc_22><loc_57><loc_23></location>Finally, the total e-folding number during inflation turns out to be,</text>
<formula><location><page_12><loc_19><loc_8><loc_92><loc_21></location>N k = ∫ t f t k H ( t ) dt = H k | ˙ H k | H k ( 1 -√ 1 -2 H k | ˙ H k | 2 ( H k -H f ) ) -| ˙ H k | 3 2 H 2 k ( 1 -√ 1 -2 H k | ˙ H k | 2 ( H k -H f ) ) 2 + | ˙ H k | 3 6 H 2 k ( 1 -√ 1 -2 H k | ˙ H k | 2 ( H k -H f ) ) 3 . (C5)</formula>
<text><location><page_13><loc_9><loc_90><loc_92><loc_93></location>The slow-roll parameters are also connected through the Hubble parameter and its derivatives. In terms of the inflationary Hubble parameter, the scalar perturbation amplitude A s and the scalar spectral index n s are related as</text>
<formula><location><page_13><loc_29><loc_86><loc_92><loc_89></location>| ˙ H k | = H 4 k 4 A s M 2 Pl , H k = H 5 k 4 A s M 2 Pl ( H 2 k A s M 2 Pl -(1 -n s ) ) . (C6)</formula>
<text><location><page_13><loc_9><loc_82><loc_92><loc_85></location>The Eq.(C5) can also be inverted to take N k as an input parameter, and correspondingly, we can calculate the quantity H f . For this study, we have taken N k = 50 and get H f ∼ 10 13 GeV.</text>
<section_header_level_1><location><page_13><loc_26><loc_78><loc_75><loc_79></location>Appendix D: Magnetic field power spectra during reheating era</section_header_level_1>
<text><location><page_13><loc_10><loc_75><loc_66><loc_76></location>The power spectrum of the magnetic field in the post-inflationary era becomes</text>
<formula><location><page_13><loc_19><loc_59><loc_92><loc_73></location>P B ( η, ˜ k ) = ˜ k 5 π 2 a 4 ( | u ( re ) 1 | 2 + | u ( re ) 2 | 2 + | u ( re ) 3 | 2 ) = ˜ k 4 π 2 a 4 ∑ i | ˜ u ( re ) i | 2 = ˜ k 4 π 2 a 4 [ 1 6 √ 3 ( 1 + 2 | β 1 | 2 +2 √ 1 + | β 1 | 2 | β 1 | cos[ Arg ( α 1 β ∗ 1 ) -2 √ 3 ˜ k ( η -η f )] ) + 1 24 √ 3 ( 1 + 2 | β 2 | 2 +2 √ 1 + | β 2 | 2 | β 2 | cos[ Arg ( α 2 β ∗ 2 ) -2 √ 3 ˜ k ( η -η f )] ) + 1 24 √ 3 ( 1 + 2 | β 3 | 2 +2 √ 1 + | β 3 | 2 | β 3 | cos[ Arg ( α 3 β ∗ 3 ) -2 √ 3 ˜ k ( η -η f ) )] , (D1)</formula>
<text><location><page_13><loc_9><loc_55><loc_92><loc_57></location>where we have substituted the mode function solutions during the reheating era in terms of the Bogoliubov coefficients. The term η -η f is calculated as</text>
<formula><location><page_13><loc_44><loc_51><loc_92><loc_53></location>η -η f = ∫ da a 2 H . (D2)</formula>
<text><location><page_13><loc_9><loc_47><loc_92><loc_49></location>Using the proper relations and substituting the value, we get the power spectrum of the magnetic field at the end of reheating as</text>
<formula><location><page_13><loc_10><loc_33><loc_92><loc_46></location>P B ( ˜ k, η ) ∣ ∣ ∣ ∣ re = ˜ k 4 π 2 a 4 re [ 1 6 √ 3 ( 1 + 2 | β 1 | 2 +2 √ 1 + | β 1 | 2 | β 1 | cos [ Arg ( α 1 β ∗ 1 ) -4 √ 3 ˜ k (1 + 3 ω eff ) a f H f ( ( H f H re ) δ -1 )] ) + 1 24 √ 3 ( 1 + 2 | β 2 | 2 +2 √ 1 + | β 2 | 2 | β 2 | cos [ Arg ( α 2 β ∗ 2 ) -4 √ 3 ˜ k (1 + 3 ω eff ) a f H f ( ( H f H re ) δ -1 )] ) + 1 24 √ 3 ( 1 + 2 | β 3 | 2 +2 √ 1 + | β 3 | 2 | β 3 | cos [ Arg ( α 3 β ∗ 3 ) -4 √ 3 ˜ k (1 + 3 ω eff ) a f H f ( ( H f H re ) δ -1 )] ) ] . (D3)</formula>
<unordered_list>
<list_item><location><page_13><loc_10><loc_21><loc_49><loc_24></location>[1] D. Grasso and H. R. Rubinstein, Magnetic fields in the early universe, Phys. Rept. 348 , 163 (2001) [astroph/0009061].</list_item>
<list_item><location><page_13><loc_10><loc_18><loc_49><loc_20></location>[2] R. Beck, Galactic and extragalactic magnetic fields, Space Sci. Rev. 99 , 243 (2001) [astro-ph/0012402].</list_item>
<list_item><location><page_13><loc_10><loc_14><loc_49><loc_18></location>[3] L. M. Widrow, Origin of galactic and extragalactic magnetic fields, Rev. Mod. Phys. 74 , 775 (2002) [astroph/0207240].</list_item>
<list_item><location><page_13><loc_10><loc_10><loc_49><loc_14></location>[4] A. Kandus, K. E. Kunze, and C. G. Tsagas, Primordial magnetogenesis, Phys. Rept. 505 , 1 (2011) [arXiv:1007.3891].</list_item>
<list_item><location><page_13><loc_10><loc_9><loc_49><loc_10></location>[5] R. Durrer and A. Neronov, Cosmological Magnetic</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_13><loc_53><loc_18><loc_92><loc_22></location>[6] K. Subramanian, The origin, evolution, and signatures of primordial magnetic fields, Rept. Prog. Phys. 79 , 076901 (2016) [arXiv:1504.02311].</list_item>
<list_item><location><page_13><loc_53><loc_14><loc_92><loc_18></location>[7] R. M. Kulsrud and E. G. Zweibel, The Origin of Astrophysical Magnetic Fields, Rept. Prog. Phys. 71 , 0046091 (2008) [arXiv:0707.2783].</list_item>
<list_item><location><page_13><loc_53><loc_10><loc_92><loc_14></location>[8] A. Brandenburg and K. Subramanian, Astrophysical magnetic fields and nonlinear dynamo theory, Phys. Rept. 417 , 1 (2005) [astro-ph/0405052].</list_item>
<list_item><location><page_13><loc_53><loc_9><loc_92><loc_10></location>[9] K. Subramanian, Magnetic fields in the early universe,</list_item>
</unordered_list>
<text><location><page_14><loc_12><loc_92><loc_45><loc_93></location>Astron. Nachr. 331 , 110 (2010) [arXiv:0911.4771].</text>
<unordered_list>
<list_item><location><page_14><loc_9><loc_85><loc_49><loc_92></location>[10] R. Sharma, S. Jagannathan, T. R. Seshadri, and K. Subramanian, Challenges in Inflationary Magnetogenesis: Constraints from Strong Coupling, Backreaction, the Schwinger Effect, Phys. Rev. D 96 , 083511 (2017) [arXiv:1708.08119].</list_item>
<list_item><location><page_14><loc_9><loc_80><loc_49><loc_85></location>[11] R. Sharma, K. Subramanian, and T. R. Seshadri, Generation of helical magnetic field in a viable scenario of inflationary magnetogenesis, Phys. Rev. D 97 , 083503 (2018) [arXiv:1802.04847].</list_item>
<list_item><location><page_14><loc_9><loc_76><loc_49><loc_80></location>[12] R. K. Jain and M. S. Sloth, Consistency relation for cosmic magnetic fields, Phys. Rev. D 86 , 123528 (2012) [arXiv:1207.4187].</list_item>
<list_item><location><page_14><loc_9><loc_72><loc_49><loc_76></location>[13] R. Durrer, L. Hollenstein, and R. K. Jain, Can slow roll inflation induce relevant helical magnetic fields?, JCAP 03 , 037 (2011) [arXiv:1005.5322].</list_item>
<list_item><location><page_14><loc_9><loc_68><loc_49><loc_72></location>[14] S. Kanno, J. Soda, and M. a. Watanabe, Cosmological Magnetic Fields from Inflation and Backreaction, JCAP 12 , 009 (2009) [arXiv:0908.3509].</list_item>
<list_item><location><page_14><loc_9><loc_66><loc_49><loc_68></location>[15] L. Campanelli, Helical Magnetic Fields from Inflation, Int. J. Mod. Phys. D 18 , 1395 (2009) [arXiv:0805.0575].</list_item>
<list_item><location><page_14><loc_9><loc_62><loc_49><loc_65></location>[16] V. Demozzi, V. Mukhanov, and H. Rubinstein, Magnetic fields from inflation?, JCAP 08 , 025 (2009) [arXiv:0907.1030].</list_item>
<list_item><location><page_14><loc_9><loc_58><loc_49><loc_61></location>[17] K. Bamba, C. Q. Geng, and L. W. Luo, Generation of large-scale magnetic fields from inflation in teleparallelism, JCAP 1210 , 058 (2012) [arXiv:1208.0665].</list_item>
<list_item><location><page_14><loc_9><loc_54><loc_49><loc_57></location>[18] K. Bamba and M. Sasaki, Large-scale magnetic fields in the inflationary universe, JCAP 02 , 030 (2007) [astroph/0611701].</list_item>
<list_item><location><page_14><loc_9><loc_50><loc_49><loc_53></location>[19] K. Bamba and J. Yokoyama, Large scale magnetic fields from inflation in dilaton electromagnetism, Phys. Rev. D 69 , 043507 (2004) [astro-ph/0310824].</list_item>
<list_item><location><page_14><loc_9><loc_46><loc_49><loc_49></location>[20] K. Bamba and J. Yokoyama, Large-scale magnetic fields from dilaton inflation in noncommutative spacetime, Phys. Rev. D 70 , 083508 (2004) [hep-ph/0409237].</list_item>
<list_item><location><page_14><loc_9><loc_43><loc_49><loc_45></location>[21] B. Ratra, Cosmological 'seed' magnetic field from inflation, Astrophys. J. 391 , L1 (1992).</list_item>
<list_item><location><page_14><loc_9><loc_39><loc_49><loc_43></location>[22] V. A. Belinsky, I. M. Khalatnikov, and E. M. Lifshitz, Oscillatory approach to a singular point in the relativistic cosmology, Adv. Phys. 19 , 525 (1970).</list_item>
<list_item><location><page_14><loc_9><loc_36><loc_49><loc_39></location>[23] C. B. Collins and S. W. Hawking, Why is the Universe isotropic?, Astrophys. J. 180 , 317 (1973).</list_item>
<list_item><location><page_14><loc_9><loc_33><loc_49><loc_36></location>[24] V. A. Belinsky, I. M. Khalatnikov, and E. M. Lifshitz, A general solution of the Einstein equations with a time singularity, Adv. Phys. 31 , 639 (1982).</list_item>
<list_item><location><page_14><loc_9><loc_29><loc_49><loc_32></location>[25] A. A. Starobinsky, Isotropization of arbitrary cosmological expansion given an effective cosmological constant, JETP Lett. 37 , 66 (1983).</list_item>
<list_item><location><page_14><loc_9><loc_25><loc_49><loc_28></location>[26] R. M. Wald, Asymptotic behavior of homogeneous cosmological models in the presence of a positive cosmological constant, Phys. Rev. D 28 , 2118 (1983).</list_item>
<list_item><location><page_14><loc_9><loc_21><loc_49><loc_24></location>[27] G. W. Gibbons and S. W. Hawking, Cosmological event horizons, thermodynamics, and particle creation, Phys. Rev. D 15 , 2738 (1977).</list_item>
<list_item><location><page_14><loc_9><loc_17><loc_49><loc_20></location>[28] S. W. Hawking and I. G. Moss, Supercooled phase transitions in the very early Universe, Phys. Lett. B 110 , 35 (1982).</list_item>
<list_item><location><page_14><loc_9><loc_13><loc_49><loc_16></location>[29] D. J. Schwarz, C. J. Copi, D. Huterer, and G. D. Starkman, CMB anomalies after Planck, Class. Quant. Grav. 33 , 184001 (2016) [arXiv:1510.07929].</list_item>
<list_item><location><page_14><loc_9><loc_9><loc_49><loc_12></location>[30] P. K. Aluri, P. Cea, P. Chingangbam, M. C. Chu, R. G. Clowes, D. Hutsem'ekers, J. P. Kochappan, A. M. Lopez, L. Liu, and N. C. M. Martens, et al. Is</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_14><loc_55><loc_89><loc_92><loc_93></location>the observable Universe consistent with the cosmological principle?, Class. Quant. Grav. 40 , 094001 (2023) [arXiv:2207.05765].</list_item>
<list_item><location><page_14><loc_52><loc_85><loc_92><loc_89></location>[31] M. a. Watanabe, S. Kanno, and J. Soda, Inflationary Universe with Anisotropic Hair, Phys. Rev. Lett. 102 , 191302 (2009) [arXiv:0902.2833].</list_item>
<list_item><location><page_14><loc_52><loc_81><loc_92><loc_85></location>[32] S. Kanno, J. Soda, and M. a. Watanabe, Anisotropic power-law inflation, J. Cosmol. Astropart. Phys. 12 (2010) 024 [arXiv:1010.5307].</list_item>
<list_item><location><page_14><loc_52><loc_77><loc_92><loc_81></location>[33] T. Q. Do and W. F. Kao, Anisotropic power-law inflation for the Dirac-Born-Infeld theory, Phys. Rev. D 84 , 123009 (2011).</list_item>
<list_item><location><page_14><loc_52><loc_75><loc_92><loc_77></location>[34] J. D. Barrow and S. Hervik, Anisotropically inflating universes, Phys. Rev. D 73 , 023007 (2006) [gr-qc/0511127].</list_item>
<list_item><location><page_14><loc_52><loc_69><loc_92><loc_74></location>[35] A. E. Gumrukcuoglu, C. R. Contaldi, and M. Peloso, Inflationary perturbations in anisotropic backgrounds and their imprint on the CMB, JCAP 11 , 005 (2007) [arXiv:0707.4179].</list_item>
<list_item><location><page_14><loc_52><loc_66><loc_92><loc_69></location>[36] C. Pitrou, T. S. Pereira, and J. P. Uzan, Predictions from an anisotropic inflationary era, JCAP 04 , 004 (2008) [arXiv:0801.3596].</list_item>
<list_item><location><page_14><loc_52><loc_62><loc_92><loc_65></location>[37] A. A. Starobinsky, S. V. Sushkov, and M. S. Volkov, Anisotropy screening in Horndeski cosmologies, Phys. Rev. D 101 , 064039 (2020) [arXiv:1912.12320].</list_item>
<list_item><location><page_14><loc_52><loc_56><loc_92><loc_61></location>[38] R. Galeev, R. Muharlyamov, A. A. Starobinsky, S. V. Sushkov, and M. S. Volkov, Anisotropic cosmological models in Horndeski gravity, Phys. Rev. D 103 , 104015 (2021) [arXiv:2102.10981].</list_item>
<list_item><location><page_14><loc_52><loc_51><loc_92><loc_56></location>[39] S. Nojiri, S. D. Odintsov, V. K. Oikonomou, and A. Constantini, Formalizing anisotropic inflation in modified gravity, Nucl. Phys. B 985 , 116011 (2022) [arXiv:2210.16383].</list_item>
<list_item><location><page_14><loc_52><loc_47><loc_92><loc_51></location>[40] M. S. Turner and L. M. Widrow, Inflation-produced, Large-scale Magnetic Fields, Phys. Rev. D 37 , 2743 (1988).</list_item>
<list_item><location><page_14><loc_52><loc_43><loc_92><loc_47></location>[41] A. Dolgov, Breaking of conformal invariance and electromagnetic field generation in the universe, Phys. Rev. D 48 , 2499 (1993) [hep-ph/9301280]</list_item>
<list_item><location><page_14><loc_52><loc_39><loc_92><loc_43></location>[42] J. Martin and J. Yokoyama, Generation of Large-Scale Magnetic Fields in Single-Field Inflation, JCAP 0801 , 025 (2008) [arXiv:0711.4307].</list_item>
<list_item><location><page_14><loc_52><loc_35><loc_92><loc_39></location>[43] G. F. R. Ellis and M. A. H. MacCallum, A Class of homogeneous cosmological models, Commun. Math. Phys. 12 , 108 (1969).</list_item>
<list_item><location><page_14><loc_52><loc_33><loc_92><loc_35></location>[44] G. F. R. Ellis, The Bianchi models: Then and now, Gen. Rel. Grav. 38 , 1003 (2006).</list_item>
<list_item><location><page_14><loc_52><loc_29><loc_92><loc_32></location>[45] R. J. Z. Ferreira, R. K. Jain, and M. S. Sloth, Inflationary magnetogenesis without the strong coupling problem, JCAP 10 (2013), 004 [arXiv:1305.7151].</list_item>
<list_item><location><page_14><loc_52><loc_25><loc_92><loc_28></location>[46] G. Tasinato, A scenario for inflationary magnetogenesis without strong coupling problem, JCAP 03 (2015), 040 [arXiv:1411.2803].</list_item>
<list_item><location><page_14><loc_52><loc_21><loc_92><loc_24></location>[47] Y. Akrami et al. [Planck], Planck 2018 results. X. Constraints on inflation, Astron. Astrophys. 641 , A10 (2020) [arXiv:1807.06211].</list_item>
<list_item><location><page_14><loc_52><loc_17><loc_92><loc_20></location>[48] T. Kobayashi and M. S. Sloth, Early Cosmological Evolution of Primordial Electromagnetic Fields, Phys. Rev. D 100 , 023524 (2019) [arXiv:1903.02561].</list_item>
<list_item><location><page_14><loc_52><loc_13><loc_92><loc_16></location>[49] L. Dai, M. Kamionkowski, and J. Wang, Reheating constraints to inflationary models, Phys. Rev. Lett. 113 , 041302 (2014) [arXiv:1404.6704].</list_item>
<list_item><location><page_14><loc_52><loc_9><loc_92><loc_12></location>[50] D. Maity, S. Pal, and T. Paul, Effective Theory of Inflationary Magnetogenesis and Constraints on Reheating, JCAP 05 , 045 (2021) [arXiv:2103.02411].</list_item>
</unordered_list>
<text><location><page_15><loc_9><loc_92><loc_49><loc_93></location>[51] J. L. Cook, E. Dimastrogiovanni, D. A. Easson, and</text>
<unordered_list>
<list_item><location><page_15><loc_55><loc_91><loc_92><loc_93></location>L. M. Krauss, Reheating predictions in single field inflation, JCAP 04 , 047 (2015) [arXiv:1502.04673].</list_item>
</document>
[
{
"title": "Magnetogenesis from Anisotropic Universe",
"content": "Sourav Pal, 1, ∗ Debaprasad Maity, 1, † and Tuan Q. Do 2, 3, ‡ 1 Department of Physics, Indian Institute of Technology Guwahati, Guwahati 781039, Assam, India 2 Phenikaa Institute for Advanced Study, Phenikaa University, Hanoi 12116, Vietnam 3 Faculty of Basic Sciences, Phenikaa University, Hanoi 12116, Vietnam (Dated: November 15, 2023) The existence of large-scale anisotropy can not be ruled out by the cosmic microwave background (CMB) radiation. Over the years, several models have been proposed in the context of anisotropic inflation to account for CMB's cold spot and hemispheric asymmetry. However, any small-scale anisotropy, if exists during inflation, is not constrained due to its nonlinear evolution in the subsequent phase. This small-scale anisotropy during inflation can play a non-trivial role in giving rise to the cosmic magnetic field, which is the subject of our present study. Assuming a particular phenomenological form of an anisotropic inflationary universe, we have shown that it can generate a large-scale magnetic field at 1-Mpc scale with a magnitude ∼ 4 × 10 -20 G , within the observed bound. Because of the anisotropy, the conformal flatness property is lost, and the Maxwell field is generated even without explicit coupling. This immediately resolves the strong coupling problem in the standard magnetogenesis scenario. In addition, assuming very low conductivity during the reheating era, we can further observe the evolution of the electromagnetic field with the equation of state (EoS) ω eff and its effects on the present-day magnetic field.",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "It is well known that our universe is magnetized on all observational scales, from planets and stars to large-scale galaxies and galaxy clusters. In particular, the magnetic field strength has been observed in the range from µG for galaxies and galaxy clusters to a few G for planets and 10 12 G for neutron stars. From Gamma-ray observations and Faraday rotation measurements, the magnetic field in the intergalactic medium (IGM) has also been shown to be bounded with the strength ranging from 10 -10 -10 -22 G [1-5]. It is possible that the primordial magnetic fields on a large scale ( ∼ 1 MPc ) were generated during the Big Bang or later and survived until today as a relic. The origin of the magnetic field in galaxies and galaxy clusters can be explained through classical magneto-hydrodynamic processes magnifying the tiny seed magnetic field. It is important to identify the origin of the primordial magnetic fields. There have been some proposed mechanisms for generating large-scale primordial magnetic fields, which can be found in the interesting review papers listed in Refs. [6-21]. Among them, the Ratra model [21] is the most accepted one, where the electromagnetic fields are generated during the inflationary era by breaking the conformal invariance of the Maxwell term, i.e., F µν F µν , through non-minimal coupling(s) with other fields such as scalar fields. Inspired by the mechanism in [16, 18, 21, 40-42], we propose another mechanism for generating the primordial magnetic fields through anisotropic spacetime. In particular, the Maxwell field experiences the existence of the anisotropy of spacetime during the inflationary era, leading to the genesis of primordial magnetic. In cosmology, there exists a nice classification of homogeneous but anisotropic spacetimes called the Bianchi universe [3133, 43, 44]. It turns out that the Bianchi type I metric is the simplest one and can be regarded as a straightforward extension of the Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime. Hence, we chose the Bianchi type I metric for our study. Remarkably, it has long been argued that the very early universe, which is close to the initial singularity, should be strongly anisotropic [22-24]. During an inflationary phase of the early universe, all spatial anisotropies, which could happen in a pre-inflationary phase, should decrease very quickly such that the universe speedily approaches a locally isotropic state, as pointed out in Refs. [25, 26]. It should be noted that this scenario is consistent with the so-called cosmic no-hair conjecture, which claims that all initial anisotropies and inhomogeneities should disappear in a late-time universe [27, 28]. Very interestingly, some recent unavoidable anomalies in the cosmic microwave background (CMB) radiations confirmed by the Planck [29, 47] such as the cold spot and hemispheric asymmetry, have challenged the standard inflationary universe models, which are based on the cosmological principle stating, the universe should be homogeneous and isotropic on large scales. In addition, other interesting observational evidences, which have called the validity of cosmological principle into question, have been listed in a recent interesting review [30]. These remarkable points lead us to a possible scenario of an anisotropic inflationary universe in early times. Many papers have been working on anisotropic inflation, e.g., see Refs. [31-33] as well as Refs. [34-39]. This paper does not discuss the origin and evolution of anisotropy in spacetime. Instead, we treat it as a perturbation over the FLRW background. As the spatial anisotropy breaks the conformal flatness of the background in the electromagnetic (EM) field, we do not need any explicit coupling with the scalar field as proposed in the literature [16-20] for gauge field production during inflation. In this context, it is important to mention the challenges in inflationary magnetogenesis, namely the strong-coupling problem and the backreaction problem [10, 16, 45, 46]. In the literature, several mechanisms and different types of coupling [45] have been introduced to overcome these problems. However, in our formalism, the strong coupling problem is readily solved in this paper since no explicit coupling is involved with the gauge field. However, there might still be a possibility of backreaction, which we will study in detail as we proceed. The paper is organized as follows. (i) An introduction of the present paper has been written in Sec. I. (ii) In Sec. II, we describe the basic formalism and the quantization of the gauge field in an anisotropic background. (iii) In Sec. III, we show the evolution of the gauge field during the inflationary era and the strength of the present-day magnetic field under an instant reheating scenario. However, a scenario might occur when the universe undergoes a prolonged reheating era, affecting the magnetic field. (iv) We discuss the evolution in such a scenario in Sec. IV. (v) Finally, we discuss the findings and implications of this proposal in Sec. V.",
"pages": [
1,
2
]
},
{
"title": "II. THE SETUP",
"content": "We introduce the spatial anisotropy in the background through the homogeneous but anisotropic Bianchi type I metric. In general, this type of metric can be written as where η is the conformal time, a ( η ) is the overall scale factor and b ( η ) is the anisotropic factor along the x -direction. In our phenomenological model, we impose the condition that the anisotropy in the spacetime exists only in the inflationary era, although the spacetime remains continuous. This ansatz guarantees that the conformal flatness is restored after inflation. These conditions can be satisfied by various models of the anisotropic factor b ( η ). However, we take a particular model that satisfies all the necessary conditions In the above Eq. (2), α is a dimensionless parameter that determines the strength of the anisotropy. Furthermore, η m is a parameter that dictates the overall behavior of the anisotropic background. An example of the anisotropic background is shown in Fig. 1. Here, α and η m are free parameters of the anisotropic model. In this paper, we do not discuss the origin of such anisotropy. However, the anisotropy, particularly near the end of inflation, maybe a combined effect of quantum field theory and the sudden breakdown of slow-roll conditions. We will come back to this issue in the future. The action of Einstein-scalar-vector theory can be given by where G is the gravitational constant, ϕ is a scalar field, and F µν ≡ ∂ µ A ν -∂ ν A µ is the field strength of the vector field A µ ( t, x ) describing the electromagnetic field. In this paper, the dynamics of the scalar field and the metric itself due to the anisotropy present are beyond our scope. Therefore, we will mainly discuss the dynamics of the EM field during the inflationary era due to the spatially anisotropic background. Therefore, the Lagrangian of interest here is the Lagrangian corresponding to the electromagnetic field, which, according to Eq. (3), is given by where the prime denotes a derivative with respect to conformal time. In this paper, all the physical quantities are denoted with the lower index, e.g., the physical momentum is denoted by k i , and the vector potential is given by A i . We set A 0 = 0 as the choice of gauge, and unlike the case of conformally flat spacetime, the ν = 0 component satisfies the modified constraint equation, where we have defined ˜ g ij = a 2 g ij for simplicity of calculation. It can be further shown that the above equation boils down to the usual Coulomb condition for the conformally flat case. However, the raising (and lowering) of the indices are done through the metric component g ij ( g ij ). Similarly, the dynamical equation of motion for the magnetic vector potential A i can be calculated from the ν = j component, which boils down to It is important to note that the metric components play a crucial role in the dynamics of the field. In the case of a standard conformally flat background, the term with the metric component's derivative vanishes, giving us the regular plane wave solutions. Now, we will promote the fields and their conjugates as operators. The conjugate momentum operator corresponding to the field operator A i turns out as To quantize the field, we decompose the magnetic vector potential A i as, In the above Eq. (8), ( p ) is the polarization index, a ( p ) k and a † ( p ) k are the annihilation and creation operators corresponding to the polarization mode ( p ). They follow the general commutation relation, In this article, the boldface letters represent vector quantities. In this context, it is important to discuss the commutation relation of the magnetic vector potential A i and its conjugate momentum Π i . We impose the commutation relation, such that they satisfy the constraint Eq. (5) on vector potential A i , With the quantization of the field, we now get the mode function equations using Eq. (6). The mode functions satisfy the relation, The polarization index is omitted here, as all the polarization modes follow the same equation of motion. Similarly, the constraint Eq. (5) in terms of the mode function becomes Interestingly, Eq. (12) contains the derivative of the metric coefficients, which works as the source of particle production during the inflationary era. As the mode function equation contains the derivative of the metric components, substituting the metric components, we get the modified mode function equations as, Moreover, all the mode functions u 1 , u 2 , and u 3 satisfy the constraint in Eq. (13) which explicitly boils down to The mode functions follow the normalization condition, Utilizing the above formalism of quantization along with the constraint relation, we evolve the mode function in different phases of the universe until the present epoch.",
"pages": [
2,
3
]
},
{
"title": "III. EVOLUTION OF ELECTROMAGNETIC FIELD DURING INFLATIONARY ERA",
"content": "According to our model in Eq.(1), the anisotropy in the spacetime exists only towards the end of inflation. After the end of inflation, within a short period, the spacetime essentially becomes FLRW again, as seen from Fig. 1. However, it is important to mention that the largescale production of the EM field is not affected due to this short presence of anisotropy after inflation. It is also evident that b → 1 towards past infinity ensures that the Bunch-Davis vacuum condition is satisfied in the infinite past. Furthermore, we assume the background spacetime is de Sitter in nature, i.e., a = -1 / ( Hη ), where H is the Hubble parameter during inflation and remains constant throughout the entire inflation. Following these initial conditions, we numerically solve the mode function equations shown in Eq. (14). We consider k 1 = k 2 = k 3 = ˜ k ∼ 1 Mpc -1 for simplification and the Hubble parameter H = 10 -5 M pl remains constant throughout the inflationary era. Here M pl = √ 1 / (8 πG ) is the reduced Planck mass. We redefine the conformal time η as a dimensionless parameter x = ˜ kη . In terms of this new variable, Eq. (2) can be rewritten as, where the parameters α and ˜ kη m are chosen accordingly to avoid the backreaction from anisotropy, which essentially means that the anisotropy acts as a perturbation over the FLRW universe. A detailed discussion of the anisotropic backreaction is done in the later section. In terms of the redefined variables x = ˜ kη, ˜ u i = √ ˜ ku i , and ˜ k = k 1 = k 2 = k 3 , the mode function equations can be written as, along with the constraint equation, By solving Eq. (18), we can obtain the mode function solution for different choices of the parameters α and x m as shown in Fig. 2. The above figure shows that the mode function grows in time due to anisotropy, particularly near the end of inflation. For values α < 0 . 03, field production stops altogether. Hence, we get a lower bound on the anisotropic parameter α ≥ 0 . 03. The upper bound on α is discussed in the later sections.",
"pages": [
3,
4
]
},
{
"title": "A. Power spectrum of the electromagnetic field during inflationary era",
"content": "The stress energy-momentum tensor corresponding to the produced EM field is given by As a result, the energy-momentum tensor corresponding to the electromagnetic part of the Lagrangian boils down to The total energy density of the system is given by the T tt component of the energy-momentum tensor. Therefore, the total electromagnetic energy density of the system is Thus, we have the electric field and magnetic field energy densities as, respectively, where the expectation values are taken with respect to the initial Bunch-Davies (BD) vacuum. In the momentum space, these energy densities can be written as In order to determine the strength of the magnetic field in the present era, we first define the power spectrum of the electromagnetic field as as already stated earlier, each polarization mode follows the same equation of motion. Therefore, all the polarization modes have equal contributions. Summing over all the polarization modes and using the assumption amplitude of all the momentum k 1 = k 2 = k 3 = ˜ k to be the same, we calculate the power spectrum of the electric and magnetic field as With these forms of the power spectrum, our goal would be to calculate its strength at present. However, before that, we will calculate the condition for which the produced electromagnetic field should not back react to the background during inflation.",
"pages": [
4,
5
]
},
{
"title": "B. Backreaction of anisotropic background and generated EM field",
"content": "In the previous section, we have briefly discussed the backreaction and strong coupling problem of inflationary magnetogenesis. In a general large-scale gauge field production scenario, a scalar field is coupled to the EM field to break the conformal invariance. Depending on the choice of the coupling function, it is possible to have a strong coupling problem, and different scenarios have been discussed in the literature [10, 16, 45, 46]. For the sake of completeness, we discuss it here briefly. In order to have a sustainable production of the electromagnetic field during inflation, the coupling function is often chosen to be an increasing function of time. However, it needs to revert to unity to restore the regular Maxwellian electromagnetism at the end of inflation. Hence, it needs to be very small at the start of the inflationary era, so the effective charge of electrons will be very large, and we cannot treat the gauge field as a free field during the inflationary era. In this proposal, there is no such direct coupling between the inflaton field and the EM field. Therefore, we do not need to worry about the strong coupling issue in this scenario. However, we must ensure that the anisotropy energy density or the generated EM field does not jeopardize the inflation. To this extent, we calculate the energy density produced by the anisotropic background and get a lower bound on the anisotropic parameter ˜ kη m and α introduced in Eq. (17). The energymomentum tensor of the background T µν is dictated by the Einstein equation in terms of the Einstein tensor G µν as In the case of Bianchi type I background as introduced in Eq. (1), the 00 component of the Einstein tensor can be calculated as Thanks to this result, we can calculate the energy density corresponding to the anisotropic background. It turns out as where H ≡ a ' /a 2 is the Hubble parameter in conformal time during the inflation, a is the scale factor in de Sitter spacetime ( a = -1 / ( Hη )). From the dynamics of the inflaton field during inflation, we already know that the total energy of the inflaton field is given by ρ inf = 3 H 2 M 2 pl . Therefore, the total background energy density in Eq. (30) consists of two parts. The first part we call inflationary energy density, and the second part is the energy density due to anisotropy in the background, In our proposition, we have mentioned earlier that the anisotropy should act as a perturbation. Therefore, we must ensure that the anisotropic energy density must be much lower than the inflaton energy density. Furthermore, the electromagnetic energy density has to be lower than the anisotropic and inflaton energy densities. From the PLANCK data [47], we know that the temperature anisotropy in CMB is ∆ T T ∼ 10 -5 . If the anisotropic energy is closer to the perturbative limit towards the end of the inflationary era, it will not affect the CMB map, as observed by the PLANCK. We define e-folding number during the inflationary era as N = ln ( a a end ) , where a end is the scale factor at the end of inflation. By this definition, the e-folding number at the end of inflation N end = 0. Moreover, the total e-folding number during the inflation is N tot ≃ 60. The anisotropic factor b in terms of e-folding number can be written as here N m is the e-folding number corresponding to the conformal time η m . We can calculate the ratio of the anisotropic energy density and the inflationary energy density in terms of the e-folding number N as follows In order to have sustainable inflation, such that the anisotropic energy density does not affect the inflation energy density, we need to have ∣ ∣ ∣ ∣ ρ anis ρ inf ∣ ∣ ∣ ∣ < 1 throughout the entirety of the inflation. Thus, the ratio gives us an upper bound on α , which dictates the strength of the anisotropy. In Fig. 3, we can see that the ratio of the energy densities reaches its maximum towards the end of inflation. Thus, we can choose our parameters such that the ratio is up to the perturbative level ( ∼ 0 . 5). It gives us the upper bound α ≤ 1 . 48. Still, the CMB remains unaffected due to the presence of spatial anisotropy. However, it is worth mentioning here that we do not consider the dynamics of the anisotropy in the background. In order to make sure that the anisotropic background comes in towards the end of inflation, we take the upper limit on the parameter ˜ kη m ≥ -2. Furthermore, during inflation, the EM field also gets produced. It is also necessary to ensure that the generated gauge field energy density does not violate the inflationary energy density. We can see that the maximum production occurs towards the end of inflation from the nature of the coupling function introduced in Eq. (17). Thus, to avoid the backreaction problem, it is sufficient to satisfy We can obtain the values of the energy densities of the electric and magnetic fields from Eq. (23) and integrate over all the modes inside the horizon during the inflationary era. Which finally boils down to, Where we recall the variables x = ˜ kη, ˜ u i = √ ˜ ku i , evaluating the integrations numerically, the ratio of the energy densities turns out to be ρ E + ρ B ρ inf ∼ 10 -9 , for the anisotropic parameter ˜ kη m = -1 and α = 1 . 45. As the generated electromagnetic energy density is very low compared to the background inflaton energy density, the backreaction problem is also avoided. Therefore, with this formalism, we can sustainably produce the EM field during inflation without worrying about the strong coupling or backreaction problem. On the other hand, ensuring that the generated EM field does not surpass the energy density of the inflationary background is also necessary. Eq. (17) shows that the maximum energy density occurs at η = 0. However, we have taken that the inflation ends at η f , so there is no production of the large-scale magnetic field in the post-inflationary era. If the electromagnetic energy density is less than the anisotropic energy density at the end of inflation, all the sufficient conditions for no backreaction are satisfied. To this end, we reiterate that the anisotropic parameter α is so chosen that the anisotropic energy density remains subdominant compared to the inflaton energy density. We further show that the produced energy density of the electromagnetic field is less than the anisotropic energy density. In conclusion, the produced electromagnetic field affects neither the inflationary nor the anisotropic background. Therefore, this formalism effectively produces a magnetic field without special coupling to avoid the backreaction effect.",
"pages": [
5,
6
]
},
{
"title": "IV. POST INFLATIONARY EVOLUTION",
"content": "The anisotropic factor b goes to unity after the end of inflation, and the spacetime becomes conformally flat. The EM field evolves as a usual Maxwellian field subsequently. However, depending on the evolution of the universe, we can have two different scenarios of field evolution: (i) In the first scenario, it is assumed that the universe instantly goes into radiation domination, i.e., the inflaton field instantly decays and produces radiation. (ii) In the second scenario, the inflaton decays within a finite time, and therefore, it goes through a brief period of reheating era having a non-zero e-folding number and very low conductivity. The dynamics of the subsequent evolution of the universe dictate the present strength of the observed magnetic field. We will discuss both scenarios in the next subsections.",
"pages": [
6
]
},
{
"title": "A. The case of instantaneous reheating",
"content": "Here in this section, we will find the strength of the magnetic field in the present time, considering an instantaneous reheating scenario. In this case, after the end of the inflationary era, the universe instantly thermalizes and goes to the radiation-dominated era. As the conductivity of the universe becomes very large, the electric field dies out instantly. However, the magnetic field produced during the inflationary era decays as a radiation density P B ∝ a -4 . Therefore, incorporating the conservation of entropy, we can compute the strength of the magnetic field, relating to the field strength at the end of inflation. We have already calculated the power spectra of the magnetic field during the inflationary era in Eq. (27). In terms of the mode functions, the explicit expression for the present-day magnetic field turns out as The above expression is in the GeV 2 unit. Here, we recall the variable ˜ u i = √ ˜ ku i (with i = 1, 2, 3), H is the Hubble parameter during inflation, and ˜ k is the scale under consideration in which we will estimate the strength of the magnetic field. Furthermore, in order to calculate the value of B 0 from Eq. (36), we first need to evaluate the value of the ratio a f a 0 . We evaluate the value to be a 0 a f ≈ 10 30 ( H/ 10 -5 M pl ) 1 / 2 . Here, in particular, we have taken the value of the Hubble parameter to be H = 10 -5 M pl . With the numerical solution of the mode functions from Fig. 2 at the end of inflation ˜ kη f = -0 . 0001 and Eq. (36) we can evaluate the strength of the magnetic field at present-day using the conversion 1 G = 1 . 95 × 10 -20 GeV 2 for different values of the anisotropic parameters α and ˜ kη m . In Fig. 4, we can see the variation of the present-day magnetic field B 0 with α for a fixed value of ˜ kη m as well as the variation of magnetic field strength with ˜ kη m . The maximum value of B 0 obtained in the instant reheating scenario for fixed value of ˜ kη m is B 0 = 2 . 86 × 10 -21 G varying α . Similarly for a fixed value of α , the maximum value of B 0 obtained is B 0 = 3 . 24 × 10 -21 G . Experiments like Faraday rotation and gamma-ray observation impose a bound on the present strength of primordial magnetic field 10 -10 G ≲ B 0 ≲ 10 -22 G [5]. Therefore, this proposal can generate a large-scale magnetic field within the experimental bound of the present-day intergalactic magnetic field. With the choice of the anisotropic parameter in the range 0 . 03 ≤ α ≤ 1 . 48, the backreaction or the strong coupling problem is also avoided.",
"pages": [
6,
7
]
},
{
"title": "B. The case of prolonged reheating with constant equation of state",
"content": "In the last section, we saw that we can generate the required strength of the magnetic field in the instant reheating scenario. However, if we consider a reheating phase with a non-zero e-folding number, then the conductivity of the universe does not reach infinity instantly. Instead, during this period, the conductivity of the universe may remain to be very low. As a result, the electric field does not go to zero immediately and induces a magnetic field during this period. This conversion of the electric field into the magnetic field during the reheating phase occurs through Faraday induction [48]. This conversion of an electric field to a magnetic field makes it diluted slowly during the reheating era compared to the previous case of P B ∝ a -4 . Thus, a finite reheating era further strengthens the magnetic field on a large scale and gives us bounds on the EoS during the reheating era. After the inflation ends, the anisotropic factor b goes to unity after a very short period, and the EM field evolves in the usual manner. Following the regular Maxwellian evolution, the equation of motion of the mode functions u i ( ˜ k, η ) during the reheating becomes where i = 1 , 2 , 3 are the indices corresponding to three spatial components of the gauge field and u ( re ) i ( ˜ k, η ) are the mode functions during reheating. Furthermore, we consider the universe a poor conductor during this period. To be precise, we take the conductivity to be zero. The solution of the mode function from Eq. (37), along with the proper normalization condition, gives us with α i and β i are the integration constants, and η f denotes the end of inflation. the integration constants are evaluated at the end of inflation η f by equating the junction conditions of inflationary and reheating era In the above Eq. (39), u i ( ˜ k, η ) are mode functions during the inflationary era which follow Eq. (14) and u ( re ) i are mode functions during the reheating era following Eq. (37). This immediately leads to the integration constants, With all these, we can now compute the time-evolving power spectrum during reheating as In order to estimate the strength of the magnetic field during the reheating era, we first need to evaluate the term η -η f . Following Ref. [48] the term is calculated as As the Hubble constant H is present in the above equation, it is evident that the quantity η -η f depends on the background's evolution during the inflationary era. In particular, how the inflaton energy density is converted into radiation energy density. In general, there are two scenarios Here in this paper, we will only discuss evolution through an independent constant effective EoS. In this context, we follow the methodology proposed by Kamionkowski et al. in Ref. [49]. Here, the evolution of the background is parametrized by a constant effective EoS ω eff . Therefore, the Hubble parameter during the reheating evolves as H ∝ a -3 2 (1+ ω eff ) . The physical parameters of reheating, like the e-folding number of the reheating era N re and the reheating temperature T re , can be expressed in terms of the inflationary parameters and effective EoS ω eff as [51] where H k denotes the Hubble parameter at the time of horizon crossing, k/a 0 = 0 . 05 Mpc -1 is the pivot scale, g re is the degrees of freedom during reheating and N k is the total e-folding number from the end of inflation till horizon crossing. As we have not considered any particular inflation potential in this paper, we develop a modelindependent way to determine N k following Ref.[50]. In the calculation of N k (see Appendix C), we have taken the central values of scalar spectral index n s = 0 . 9649 and scalar perturbation amplitude ln[10 10 A s ] = 3 . 044, considering the constraints provided by the PLANCK data [47] and as an input parameter we have chosen N k = 50. With this choice of n s , N k , we get an upper bound on the effective EoS ω eff < 0 . 164 from the BBN bound of reheating temperature T re ∼ 10 -2 GeV. Now, in order to connect the reheating parameters N re , T re to the strength of the primordial magnetic field, we need to evaluate the quantity η -η f in Eq. (42). It is evaluated following the evolution of the Hubble parameter during the reheating era. As the EoS is constant ω eff , the variation of the Hubble parameter during the reheating era ( H re ) is related to the Hubble parameter at the end of inflation ( H f ) as where the subscript ' re ' represents the end of reheating. Thus, a re and H re are the scale factor and Hubble parameter at the end of reheating, respectively. Following the above relation, the term in Eq. (42) boils down to Substituting the value of the extra reheating term η -η f , we can calculate the present strength of the magnetic field as a function of the effective EoS ω eff . After the end of reheating, the conductivity of the universe goes to infinity. Therefore, the electric field goes to zero, and the Faraday conversion of the electric field into the magnetic field stops at the end of reheating. And the magnetic field decays as radiation ( a -4 ) until now. From the conservation of magnetic energy density, the present strength of the magnetic field can be calculated from the relation as follows Evolving through the reheating era, the strength of the magnetic field in the present era turns out as where, In the above Eq. (48), δ = (3 ω eff + 1) / (3 ω eff + 3). Varying the EoS ω eff , we get the present-day strength of the magnetic field of B 0 ∼ 4 × 10 -20 G, which is one order higher than what was predicted for instantaneous reheating case, which is ∼ 3 × 10 -21 G. Furthermore, from the observed strength of the magnetic field, we also get a lower bound of EoS ω eff > 0 . 132. From Fig. 5, we see that the present strength of the magnetic field increases due to the Faraday conversion of the electric field into the magnetic field; such increment is quite insensitive to the reheating EoS. This increment is small since the strength of the electric field compared to the magnetic field at the end of inflation is not significantly higher.",
"pages": [
7,
8,
9
]
},
{
"title": "V. SUMMARY AND CONCLUSIONS",
"content": "This paper proposes a new formalism to generate largescale magnetic fields during the inflationary era. The novelty of the work lies in the generation of fields during inflation. Several works have been done in the context of inflationary magnetogenesis. However, all the previous works rely on the conformal breaking coupling of the EM field with some scalar field or gravity. In the present case, we have taken the underlying background to be an anisotropic one (Bianchi type I), keeping conformal property intact. Due to this, our model does not suffer from the usual strong coupling problem. In the process, we have introduced two parameters α and η m to characterize the behavior of the anisotropic scale factor b ( η ). By appropriately tuning those anisotropic parameters, we further addressed the backreaction problem. If we take the ratio of the anisotropic energy density ρ anis and the inflaton energy density ρ inf to be ρ anis /ρ inf ⩽ 0 . 5 we get an upper bound of α ≤ 1 . 48. Furthermore, to ensure that electromagnetic field gets produced during the inflationary era, we get a lower bound on the parameter α ≥ 0 . 03. The parameter η m is so chosen that the anisotropy appears towards the end of inflation. For this, we have taken ˜ kη m ≥ -2, ensuring that the anisotropy is localized and short-lived. With this choice of parameters, we find that the ratio of the energy density of the generated EM field to the total inflaton energy density is ∼ 10 -9 , which implies that the electromagnetic energy density is also lower than the anisotropic energy density. Therefore, the generated electromagnetic field neither back-reacts on the inflaton field nor the anisotropic background. Finally, this set of parameters gives us a present strength of magnetic field B 0 ∼ 3 × 10 -21 G , for α = 1 . 45 and ˜ kη m = -2, which is well in between the latest bound on present-day magnetic field strength. However, if we consider an elongated reheating period followed by inflation, the magnetic field strength further increases. This increase in strength occurs due to Faraday's conversion of the electric field to the magnetic field. By this prolonged reheating era, we get the present strength of magnetic field B 0 ∼ 4 × 10 -20 G . Through the introduction of the reheating era, we also get a tight constraint on the range of equation of state 0 . 132 < ω eff < 0 . 164 for the particular choice of inflationary parameter n s and N k . Due to the presence of anisotropy, there might be interesting signatures of the anisotropy on gravitational waves at small scales. Further, the most interesting would be investigating the origin of such anisotropy, particularly near the end of inflation. All those questions we leave for our future study.",
"pages": [
9,
10
]
},
{
"title": "ACKNOWLEDGMENTS",
"content": "DM wishes to acknowledge support from the Science and Engineering Research Board (SERB), Department of Science, and Technology (DST), Government of India (GoI), through the Core Research Grant CRG/2020/003664. DM and SP also thanks the Gravity and High Energy Physics groups at IIT Guwahati for illuminating discussions.",
"pages": [
10
]
},
{
"title": "Appendix A: Power spectrum of the electromagnetic field during inflationary era",
"content": "We have the energy-momentum tensor corresponding to the free Maxwellian Lagrangian, The energy density of the electromagnetic field is obtained from the '00' component of the energy-momentum tensor, which boils down to Upon trading the EM field into the quantum operator and referring to Eq.(23), we have the expression for the electric and magnetic field energy densities as where the expectation value is obtained over the BD vacuum. The expectation value of g im g jn F ij F mn in the BD vacuum in terms of mode functions boils down to Substituting the expectation value of the term g im g jn F ij F mn in Eq.(A3), we get the energy density of the magnetic field. With the polarization index, the expression for the magnetic field energy density turns out as As all the polarization modes behave the same way, summing over all the polarization, we finally get the energy density of the magnetic field Similarly, evaluating the expectation value of the term g ij A ' i A ' j and substituting it back in Eq.(A3), we get the energy density of the electric field in terms of mode functions as",
"pages": [
10,
11
]
},
{
"title": "Appendix B: Backreaction of anisotropic background and generated electromagnetic field",
"content": "The energy-momentum tensor of the background T where G µν is the Einstein tensor and G is the gravitational constant. The Einstein tensor can be calculated in terms of the Riemann tensor ( R µν ) and Ricci scalar ( R ), In the case of Bianchi type I background as introduced in Eq. (1), the '00' component of the Einstein tensor turns out as This essentially gives us the background energy density, The ratio of anisotropic energy density to the inflaton energy density is given in terms of the e-folding number N , with the e-folding number is defined as dN = d ln a , where a is the scale factor. From the above equation, we can get the ratio of the anisotropic and the inflationary energy densities. The backreaction problem can be evaded if the total energy of the generated EM field is less than the energy density of the inflaton field, that is The total energy densities of the EM are given by Eq.(24). Integrating all the modes, we get the total energy density. The modes involved are given by ˜ k i = a i H , which crosses the horizon at the beginning of inflation, and k f = a f H are the modes that cross the horizon at the end of inflation. We have the total energy density of the electric field expressed as, Similarly, the total energy density of the magnetic field is calculated where x = ˜ kη and ˜ u i = √ ˜ ku i . Integrating over the limits numerically with the solutions of the mode functions, we get the total energy density of the generated EM field. Finally, comparing the energy density of the EM field to the inflaton field ( ρ inf = 3 H 2 M 2 pl ), we get After the inflation, the production of electromagnetic field stops altogether. If we consider an instant reheating scenario, it essentially behaves as a radiation field. Thus, by conservation of entropy, we have Where '0' denotes the present epoch, a 0 represents the scale factor at present, η f is the conformal time at the end of inflation, and a f is the scale factor corresponding to η f . Implementing Eq. (B8), we can evaluate the present-day magnetic field strength.",
"pages": [
11,
12
]
},
{
"title": "Appendix C: Calculation of total e-folding number of inflation",
"content": "We have the expression of the total e-folding number of inflation N k from [50] as The Hubble parameter explicitly depends on the background evolution. Therefore, to calculate the actual Hubble parameter, we will Taylor expand around the conformal time of horizon crossing t k to incorporate the background effects, In the above Eq. (C2), we consider only terms up to O ( H k ). The total duration of the inflation is represented as ∆ t = t -t k . Then, by Eq. (C2), the Hubble parameter at the end of inflation can be written as Consequently, the duration of inflation (∆ t ) can be expressed in terms of the Hubble parameter and the derivative of it as Finally, the total e-folding number during inflation turns out to be, The slow-roll parameters are also connected through the Hubble parameter and its derivatives. In terms of the inflationary Hubble parameter, the scalar perturbation amplitude A s and the scalar spectral index n s are related as The Eq.(C5) can also be inverted to take N k as an input parameter, and correspondingly, we can calculate the quantity H f . For this study, we have taken N k = 50 and get H f ∼ 10 13 GeV.",
"pages": [
12,
13
]
},
{
"title": "Appendix D: Magnetic field power spectra during reheating era",
"content": "The power spectrum of the magnetic field in the post-inflationary era becomes where we have substituted the mode function solutions during the reheating era in terms of the Bogoliubov coefficients. The term η -η f is calculated as Using the proper relations and substituting the value, we get the power spectrum of the magnetic field at the end of reheating as Astron. Nachr. 331 , 110 (2010) [arXiv:0911.4771]. [51] J. L. Cook, E. Dimastrogiovanni, D. A. Easson, and",
"pages": [
13,
14,
15
]
}
]
2024PhRvD.109j4012K
https://arxiv.org/pdf/2308.05545.pdf
<document>
<section_header_level_1><location><page_1><loc_16><loc_90><loc_85><loc_93></location>Shadow and Weak Gravitational lensing of rotating traversable Wormhole in Non-homogeneous Plasma Space-time</section_header_level_1>
<text><location><page_1><loc_27><loc_87><loc_74><loc_89></location>Saurabh Kumar, 1, ∗ Akhil Uniyal, 1, 2, † and Sayan Chakrabarti 1, ‡</text>
<text><location><page_1><loc_25><loc_81><loc_75><loc_86></location>1 Department of Physics, Indian Institute of Technology, Guwahati 781039, India 2 Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shengrong Road 520, Shanghai, 201210, Peoples Republic of China (Dated: August 21, 2023)</text>
<text><location><page_1><loc_18><loc_63><loc_83><loc_80></location>In this work, we have studied the behavior of null geodesics within a rotating wormhole space-time in non-magnetized pressure-less plasma. By focusing on the dispersion relation of the plasma and disregarding its direct gravitational effects, we examine how light rays traverse in the mentioned spacetime. A key highlight of the work is the necessity of a specific plasma distribution profile to establish a generalized Carter's constant, shedding light on the importance of this parameter. Furthermore, we have derived analytical formulas to distinguish the shadow boundary across various plasma profiles, uncovering a fascinating trend of diminishing shadow size as plasma density increases. Intriguingly, certain limits of the plasma parameters result in the complete disappearance of the shadow. When calculating the deflection angle by a wormhole in plasma space-time, we observe a distinct pattern: the angle decreases as the plasma parameter rises in non-homogeneous plasma space-time, diverging from the behavior observed in homogeneous plasma space-time. Also, leveraging observational data from M 87 ∗ , we establish constraints on the throat radius. Furthermore, minimum shadow diameters provide valuable constraints for the radial and latitudinal plasma parameters.</text>
<section_header_level_1><location><page_1><loc_21><loc_59><loc_36><loc_60></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_24><loc_49><loc_57></location>The concept of a wormhole, a hypothetical structure in space-time that connects different regions in spacetime, has been extensively explored since the early work of Einstein and Rosen [1]. Subsequent developments by Wheeler [2] and the pioneering work of Morris and Thorne [3] on traversable static wormholes further fueled interest in these intriguing cosmic constructs. In a later work, Teo extended this concept to include rotation in the wormhole geometry [4]. The existence of wormholes challenges energy conditions and requires the presence of exotic matter within the throat [3]. Consequently, their plausibility has been a subject of debate. Various proposals, such as the existence of a thin layer of negative energy density inside the throat [5] or the incorporation of modified gravity theories [6, 7] have been put forth to address these challenges. Given the potential formation of wormholes in the early universe and their existence subject to specific conditions, it is essential to investigate them further and discern their unique characteristics from other compact objects. In the literature, there are different methods that exist in order to get the axisymmetric wormhole solution. One such example is using the Ehlers transformations [8].</text>
<text><location><page_1><loc_9><loc_17><loc_49><loc_24></location>It is believed that the center of the galaxies including ours contains supermassive black holes however the existence of the black hole can only be justified by the existence of the event horizon. Therefore a number of tests have been proposed in order to confirm the presence of</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_60></location>an event horizon in such compact objects [9-11]. In spite of all these proofs, a shred of conclusive evidence is still lacking [12]. It is worth mentioning that the existence of the event horizon along with the set of unstable light rings commonly known as the photon sphere in the exterior region of the compact object form the shadow of the object with the help of the radiation coming from the accretion disk around it [13]. Therefore, since the publication of the image of the M 87 ∗ supermassive black hole [14] and supermassive compact object at the center of our own galaxy known as Sagittarius A ∗ (Sgr A ∗ ) [15] by the Event Horizon Telescope (EHT), there has been extensive discussion among researchers regarding the nature of the object captured in the image including one of the papers by EHT [16]. However, first Synge [17] and Luminet [18] studied the Schwarzchild black hole shadow, and thereafter Bardeen [19] looked into the shadow of Kerr black hole. Consequently, the shadow in different geometrical backgrounds was studied in detail such as the Reissner-Nordstrom (RN) black hole [20], KerrNewman black hole [21], rotating regular black hole [22], Kerr black hole with scalar hair [23], regular black hole [24-26], Einstein-dilaton-Gauss-Bonnet black holes [27], Horava-Lifshitz black holes [28], non-Kerr black holes [29, 30], higher-dimensional black holes [31-33], black holes in theories of Non-linear electrodynamics [34-36], black holes in loop quantum gravity [37-39], black holes in the expanding Universe [40, 41], black holes in the presence of plasma [42, 43] etc. to name a few. It is important to understand that while the boundary of the shadow is only determined by the underlying space-time metric since it is formed only by the observed apparent shape of the photon sphere by the distant observer [44], the intensity map of the image is influenced by the accretion process around the compact object. Therefore, it is important to note that the presence of a shadow or a</text>
<text><location><page_2><loc_9><loc_70><loc_49><loc_93></location>photon ring does not provide conclusive evidence that the object is a black hole. This also has been shown in the recent simulations by using the general relativistic magnetohydrodynamical (GRMHD) and general relativistic radiative transfer (GRRT) calculations that distinguishing the shadow image of the Kerr black hole and non-rotating dilaton black hole is almost impossible within the present observations [45, 46]. In support of the argument, a number of other compact objects have been studied where it has been shown that the horizonless compact object such as naked singularities [47-49], a hard surface [50], and non-rotating wormholes [51-53] and rotating traversable wormhole [54] can also cast the similar shadows. Along with this wormhole shadow also has been discussed in arbitrary metric theory of gravity with parameterizing the wormhole space-time [55].</text>
<text><location><page_2><loc_9><loc_41><loc_49><loc_70></location>Previous studies have extensively investigated the shadows of wormholes, discussing their similarities with the shadow of Kerr black hole [54, 56, 57]. However, one crucial aspect that has been overlooked in these studies is the presence of plasma and its effects on wormhole shadows. The effects of plasma on the shadows of rotating wormholes space-time have been explored in [56], however, this study did not consider the contribution of the wormhole throat [57]. Therefore, in this work, we will be studying the Teo class of rotating wormholes [58] in the presence of the plasma and will be taking care of the contribution coming from the throat. Other than the shadow, a lot of studies were performed on gravitational lensing for the wormhole without the plasma medium [59-64] and with the plasma medium [65, 66]. Furthermore, investigations into weak lensing in plasma space-time have not been limited to compact objects alone, as some researchers have employed galaxy models to study the effects of non-uniform plasma, revealing an increasing impact on the deflection angle [67, 68].</text>
<text><location><page_2><loc_9><loc_15><loc_49><loc_40></location>Our goal is to derive analytical expressions for the shadow boundary of the rotating wormhole in plasmafilled space-time for the observer situated at infinity, similar to Bardeen's calculation of the Kerr black hole shadow [69]. It is known that including the plasma potential in the Hamiltonian can affect the existence of Carter's constant, so one crucial aspect of our work will be to find the necessary condition for the existence of Carter's constant [70]. Such a condition has also been pointed out for Kerr black hole and also for generalized axis-symmetric static-spacetime [71, 72]. Another aim is to derive the deflection angle by a wormhole in homogeneous and non-homogeneous plasma space-time and analyze their impacts on the deflection angle. At last, our final goal will be to constrain the wormhole and plasma parameters using the EHT results of black hole shadows at the center of M87*. A similar approach and calculations have been taken in [73].</text>
<text><location><page_2><loc_9><loc_9><loc_49><loc_14></location>The paper is structured as follows: In Section II, we provide an overview of the Hamiltonian formalism for null geodesics in plasma space-time and discuss the necessary conditions for the existence of light rays in the outer</text>
<text><location><page_2><loc_52><loc_61><loc_92><loc_93></location>communication of the Teo wormhole plasma space-time. Section III focuses on determining the specific forms of plasma profiles that satisfy the condition for the existence of Carter's constant. We derive the expressions for the null geodesic equations in this context. In Section IV, we delve into the role of the contribution of the wormhole throat to the wormhole shadow. We discuss the significance of the throat and derive the expressions for the celestial coordinates of the shadow boundary in generalized plasma space-time. Moving on to Section V, we explore specific plasma profiles that fulfill the separability condition. We also present a comparison of the shadows for various plasma densities of different plasma profiles. Section VI is dedicated to the calculation of the deflection angle of a rotating wormhole in weak field approximation for both homogeneous and non-homogeneous plasma space-time. Finally, in Section VII, we attempt to constrain the plasma parameters and throat size of the Teo wormhole using the observational data from M87*. Throughout the paper, we have considered units such that ℏ = G = c = M = 1 and our choice of signature is (-,+,+,+).</text>
<section_header_level_1><location><page_2><loc_53><loc_56><loc_91><loc_58></location>II. HAMILTON FORMALISM FOR LIGHT RAYS IN A PLASMA SPACE-TIME</section_header_level_1>
<text><location><page_2><loc_52><loc_51><loc_92><loc_54></location>The Hamiltonian describing the light ray traveling in non-magnetized pressureless plasma is given as [71]</text>
<formula><location><page_2><loc_59><loc_47><loc_92><loc_50></location>H ( x, p ) = 1 2 ( g αβ ( x ) p α p β + ω P ( x ) 2 ) , (1)</formula>
<text><location><page_2><loc_52><loc_42><loc_92><loc_46></location>here g µν are the contravariant components of the metric tensor and ω P represents the plasma electron frequency, which is defined as,</text>
<formula><location><page_2><loc_64><loc_38><loc_92><loc_41></location>ω P ( x ) 2 = 4 πe 2 m N ( x ) , (2)</formula>
<text><location><page_2><loc_52><loc_27><loc_92><loc_37></location>where m and e are the mass and charge of the electron respectively while N ( x ) defines the electron density distribution. Here x represents the space-time coordinates ( t, r, θ, ϕ ) while p represents the momentum coordinates ( p t , p r , p θ , p ϕ ) for the light ray. Please note that the plasma frequency ( ω P ) and the photon frequency ( ω ) is related by a general form,</text>
<formula><location><page_2><loc_62><loc_22><loc_92><loc_25></location>n ( x, ω ( x )) 2 = 1 -ω P ( x ) 2 ω ( x ) 2 , (3)</formula>
<text><location><page_2><loc_52><loc_12><loc_92><loc_21></location>where n is known as the refractive index, and it must be greater than 0 so that the light rays reach the observer [71]. Since the light rays reaching the observer are gravitationally redshifted, the observed redshifted frequency can be expressed in terms of the known constant of motion p t as</text>
<formula><location><page_2><loc_65><loc_8><loc_92><loc_12></location>ω ( x ) = p t √ -g tt ( x ) . (4)</formula>
<text><location><page_3><loc_9><loc_86><loc_49><loc_93></location>The necessary and sufficient condition for the existence of a light ray with a constant of motion p t is derived for the Kerr black hole by Volker et. al [71] and based on a similar approach, a similar condition for generalized rotating metric can be given by</text>
<formula><location><page_3><loc_22><loc_83><loc_49><loc_85></location>p 2 t > g tt ( x ) ω P ( x ) 2 . (5)</formula>
<text><location><page_3><loc_9><loc_79><loc_49><loc_82></location>Finally, the geodesics equations can be derived using Hamilton's equations, which are given as,</text>
<formula><location><page_3><loc_20><loc_75><loc_49><loc_78></location>˙ p α = -∂H ∂x α , ˙ x α = ∂H ∂p α . (6)</formula>
<text><location><page_3><loc_9><loc_70><loc_49><loc_74></location>For our analysis we have considered a stationary, axisymmetric rotating metric for Teo class traversable wormhole in the Boyer-Lindquist coordinates as [4],</text>
<formula><location><page_3><loc_11><loc_63><loc_49><loc_68></location>ds 2 = -N ( r ) 2 dt 2 + ( 1 -b 0 ( r ) r ) -1 dr 2 + r 2 K ( r ) 2 ( dθ 2 +sin 2 θ ( dϕ -ω T ( r ) dt ) 2 ) , (7)</formula>
<text><location><page_3><loc_9><loc_34><loc_49><loc_61></location>where r ≥ r 0 , r 0 is the throat radius of the wormhole. N , b 0 are known as the redshift factor and shape function respectively, K determines the proper radial distance which is given by R = rK and ω T is the measure for the angular velocity of the wormhole. N , b 0 , K and ω T are in general the functions of radial ( r ) and polar ( θ ) coordinates. For simplicity, in this work, we have only considered r dependency. Since the wormhole does not contain an event horizon, the metric component N ( r ) should be considered finite and non-zero throughout space-time. The shape function ( b 0 ) must satisfy the conditions ∂ θ b 0 | r = r 0 = 0 , ∂ r b 0 | r = r 0 < 1 and b 0 ≤ r [3] in order to have the geometry of a wormhole. The shape function also gives information about the mass of the wormhole [74] and the estimation of mass as M = r 0 / 2 is derived in great detail by Shaikh et. al [57]. In this work, we have considered the following form of the metric functions in order to get the traversable wormhole [4, 75].</text>
<formula><location><page_3><loc_16><loc_27><loc_49><loc_33></location>N = exp [ -r 0 r ] , b 0 ( r ) = r 0 = 2 M, K = 1 , ω T = 2 J r 3 , (8)</formula>
<text><location><page_3><loc_9><loc_20><loc_49><loc_26></location>where J is the angular momentum of the wormhole and M is the mass of the wormhole [57, 74]. We have used spin parameter of wormhole, a= J/M 2 which defines its rotation rate.</text>
<section_header_level_1><location><page_3><loc_9><loc_13><loc_49><loc_17></location>III. SEPARABILITY OF HAMILTON-JACOBI EQUATION FOR NULL GEODESICS IN PLASMA ON TEO WORMHOLESPACE-TIME</section_header_level_1>
<text><location><page_3><loc_9><loc_9><loc_49><loc_11></location>The geodesic motion in rotating space-time enables two constants of motion the angular momentum of the</text>
<text><location><page_3><loc_52><loc_77><loc_92><loc_93></location>particle about the axis of symmetry p ϕ and its energy p t due to the axisymmetric and stationary symmetries of the space-time. However, Carter et al. [70] showed that the geodesics in Kerr metric possess another constant of motion that governs the motion of geodesics in the latitudinal direction. Since the Kerr metric represents the rotating black hole space-time, Carter's constant should also exist in the rotating wormhole. This constant can be found using the method of separation of variables. Therefore, let's consider the Hamiltonian for the null geodesics as,</text>
<formula><location><page_3><loc_53><loc_72><loc_92><loc_75></location>H ( x, ∂S ∂x α ) = 1 2 g αβ ( x ) ∂S ∂x α ∂S ∂x β + 1 2 ω P ( x ) 2 = 0 , (9)</formula>
<text><location><page_3><loc_52><loc_69><loc_71><loc_70></location>with the separation ansatz</text>
<formula><location><page_3><loc_58><loc_66><loc_92><loc_67></location>S ( t, r, θ, ϕ ) = p t t + p ϕ ϕ + S r ( r ) + S θ ( θ ) , (10)</formula>
<text><location><page_3><loc_52><loc_60><loc_92><loc_64></location>where S r ( r ) and S θ ( θ ) are functions of r and θ coordinates respectively. Now substituting Eq. 10 into Eq. 9 will give,</text>
<formula><location><page_3><loc_55><loc_53><loc_92><loc_59></location>1 2 g tt ( ∂ t S ) 2 + g ϕt ( ∂ t S ) ( ∂ ϕ S ) + 1 2 g rr ( ∂ r S ) 2 + 1 2 g θθ ( ∂ θ S ) 2 + 1 2 g ϕϕ ( ∂ ϕ S ) 2 + 1 2 ω 2 P = 0 . (11)</formula>
<text><location><page_3><loc_52><loc_48><loc_92><loc_52></location>Nowconsidering p r = ∂ r S and p θ = ∂ θ S and solving the above equation for Teo rotating wormhole space-time (Eq. 7) will give,</text>
<formula><location><page_3><loc_54><loc_40><loc_92><loc_46></location>-1 N 2 p 2 t -2 ω T N 2 p t p ϕ + ( 1 -b 0 r ) p 2 r + 1 r 2 K 2 p 2 θ -( ω 2 T N 2 -1 r 2 K 2 sin 2 θ ) p 2 ϕ + ω 2 P = 0 . (12)</formula>
<text><location><page_3><loc_52><loc_33><loc_92><loc_38></location>Since we are considering plasma frequency which depends on both radial ( r ) and polar ( θ ) coordinates, the above equation is only separable if the general form of plasma frequency is considered as,</text>
<formula><location><page_3><loc_62><loc_29><loc_92><loc_31></location>ω P ( r, θ ) 2 = Ω r ( r ) + Ω θ ( θ ) r 2 K 2 , (13)</formula>
<text><location><page_3><loc_52><loc_25><loc_92><loc_27></location>where Ω r ( r ) and Ω θ ( θ ) are r and θ dependent functions respectively. Therefore, Eq. 12 can be rearranged as,</text>
<formula><location><page_3><loc_53><loc_13><loc_92><loc_23></location>-r 2 K 2 N 2 ( p t + ω T p ϕ ) 2 + r 2 K 2 ( 1 -b 0 r ) p 2 r +Ω r ( r ) ︸ ︷︷ ︸ f r ( r ) = -p 2 θ -p 2 ϕ sin 2 θ -Ω θ ( θ ) ︸ ︷︷ ︸ f θ ( θ ) , (14)</formula>
<text><location><page_3><loc_52><loc_9><loc_92><loc_11></location>here expressions f r ( r ) and f θ ( θ ) are only the function of r and θ respectively and therefore can be considered</text>
<text><location><page_4><loc_9><loc_89><loc_49><loc_93></location>as a constant since they are now separated by equality. This constant is known as Carter's constant and can be written as,</text>
<formula><location><page_4><loc_22><loc_87><loc_49><loc_88></location>f r ( r ) = f θ ( θ ) = -Q. (15)</formula>
<text><location><page_4><loc_9><loc_81><loc_49><loc_86></location>Therefore, by using these three constants of motion p t , p ϕ , and Q , one can write the impact parameters such as [75],</text>
<formula><location><page_4><loc_23><loc_77><loc_49><loc_81></location>η = L ω o , ξ = Q ω 2 o , (16)</formula>
<text><location><page_4><loc_9><loc_73><loc_49><loc_77></location>where we have considered p t = -ω o and p ϕ = L . Now solving for geodesics using Hamilton's Eqs. (6) for x µ = t, ϕ , we get:</text>
<formula><location><page_4><loc_22><loc_69><loc_49><loc_72></location>˙ t = 1 N 2 (1 -ηω T ) , (17)</formula>
<formula><location><page_4><loc_14><loc_63><loc_49><loc_67></location>˙ ϕ = 1 N 2 ( ω T (1 -ηω T ) + η N 2 r 2 K 2 sin 2 θ ) . (18)</formula>
<text><location><page_4><loc_9><loc_61><loc_49><loc_62></location>By calculating the expressions for p r and p θ using Eq. 14,</text>
<formula><location><page_4><loc_11><loc_54><loc_49><loc_60></location>p r = ± 1 N √ 1 -b 0 r √ (1 -ηω T ) 2 -N 2 r 2 K 2 ( ξ + Ω r ω 2 o ) , (19)</formula>
<formula><location><page_4><loc_20><loc_50><loc_49><loc_53></location>p θ = ± √ ξ -η 2 sin θ 2 -Ω θ ω 2 o , (20)</formula>
<text><location><page_4><loc_9><loc_46><loc_49><loc_49></location>we can calculate the remaining two geodesic Eqs. by solving Eqs 6 for x µ = r, θ and we get,</text>
<formula><location><page_4><loc_21><loc_41><loc_49><loc_45></location>˙ r = ± √ 1 -b 0 r N √ R ( r ) , (21)</formula>
<formula><location><page_4><loc_22><loc_37><loc_49><loc_39></location>˙ θ = ± 1 r 2 K 2 √ Θ( θ ) , (22)</formula>
<text><location><page_4><loc_9><loc_35><loc_36><loc_36></location>where R ( r ) and Θ( θ ) are expressed as,</text>
<formula><location><page_4><loc_15><loc_30><loc_49><loc_34></location>R ( r ) = (1 -ηω T ) 2 -N 2 r 2 K 2 ( ξ + Ω r ω 2 o ) , (23)</formula>
<formula><location><page_4><loc_20><loc_25><loc_49><loc_29></location>Θ( θ ) = ξ -Ω θ ω 2 o -η 2 sin θ 2 . (24)</formula>
<text><location><page_4><loc_9><loc_19><loc_49><loc_25></location>Since the shadow is formed due to last photon rings which are unstable in nature, so in order to have unstable spherical orbits, null rays must satisfy the following criteria [71],</text>
<formula><location><page_4><loc_21><loc_17><loc_49><loc_18></location>Θ( θ ) ≥ 0 , R '' ( r ) > 0 . (25)</formula>
<text><location><page_4><loc_9><loc_9><loc_49><loc_16></location>where the first condition ensures the existence of spherical orbits around the wormhole while the second condition is imposed to get unstable spherical orbits. A similar calculation for the general axially symmetric stationary space-time case can also be found in [72].</text>
<text><location><page_4><loc_52><loc_21><loc_57><loc_22></location>where</text>
<section_header_level_1><location><page_4><loc_54><loc_91><loc_89><loc_93></location>IV. SHADOWOFTEOWORMHOLEINPLASMA SPACE-TIME</section_header_level_1>
<text><location><page_4><loc_52><loc_74><loc_92><loc_88></location>Since photon orbits offer valuable insights into the optical appearance of wormholes, it would be insightful to study the boundary of the last photon ring in plasma space-time. In a non-rotating space-time, these orbits occur within the equatorial plane due to the spherical symmetry of the wormhole. However, in the case of rotating space-time, photon trajectories cross the equatorial plane repeatedly [76]. The Carter's constant which remains conserved in the latitudinal direction is crucial in order to determine the spherical orbits.</text>
<text><location><page_4><loc_52><loc_58><loc_92><loc_74></location>The primary objective is to identify the last photon orbits that distinguish between light rays moving outward and those moving inward. To accomplish this, we rely on the determination of critical orbits characterized by their impact parameters: η and ξ . These parameters play a pivotal role in delineating the boundary of the shadow cast by the wormhole. Remarkably, the last photon orbits correspond to the most unstable circular orbits, featuring the maximum value of the effective potential, V eff . Well-established criteria can be applied to identify this unstable circular photon orbits [77]:</text>
<formula><location><page_4><loc_61><loc_55><loc_92><loc_57></location>V eff ( r c ) = 0 , V ' eff ( r c ) = 0 , (26)</formula>
<text><location><page_4><loc_52><loc_50><loc_92><loc_54></location>where r c denotes the critical photon orbits and prime denotes the derivative with respect to r . The geodesic equation, Eq. 21 can be expressed as,</text>
<formula><location><page_4><loc_67><loc_47><loc_92><loc_48></location>˙ r 2 + V eff = 0 , (27)</formula>
<text><location><page_4><loc_52><loc_44><loc_56><loc_45></location>where</text>
<formula><location><page_4><loc_53><loc_40><loc_92><loc_43></location>V eff = -1 -b 0 /r N 2 [ (1 -ηω T ) 2 -N 2 r 2 K 2 ( ξ + Ω r ω 2 o ) ] , (28)</formula>
<text><location><page_4><loc_52><loc_35><loc_92><loc_38></location>Hence, calculating the impact parameter with the help of Eqs. 26, we get,</text>
<formula><location><page_4><loc_59><loc_30><loc_92><loc_34></location>ξ = [ r 2 K 2 N 2 (1 -ηω T ) 2 -Ω r ω 2 o ]∣ ∣ ∣ ∣ r = r c , (29)</formula>
<formula><location><page_4><loc_62><loc_24><loc_92><loc_28></location>η = B -√ B 2 -4 AC 2 A ∣ ∣ ∣ ∣ ∣ r = r c , (30)</formula>
<formula><location><page_4><loc_62><loc_18><loc_92><loc_20></location>A = ω T ω ' T -ω 2 T Σ , (31)</formula>
<formula><location><page_4><loc_62><loc_16><loc_92><loc_18></location>B = ω ' T -2 ω T Σ , (32)</formula>
<formula><location><page_4><loc_62><loc_12><loc_92><loc_16></location>C = ( Ω r ω 2 o N 2 r 2 K 2 -1 ) Σ+∆ , (33)</formula>
<formula><location><page_4><loc_62><loc_8><loc_92><loc_12></location>∆ = 1 2 d dr ( N 2 r 2 K 2 Ω r ω 2 o ) , (34)</formula>
<figure>
<location><page_5><loc_15><loc_70><loc_85><loc_93></location>
<caption>FIG. 1. (a) Effective potential of the Teo wormhole with β = 0 and α = -2 (red), 8 (blue). These plots reveal that for positive values of alpha, there is an extremum outside the throat, while for negative values of alpha, the extremum is located at the throat. This distinction helps us understand the respective contributions of the potential in the formation of the shadow. [See text for more details.] (b) Wormhole shadow (blue) due to maximum potential outside the throat and (red) due to unstable orbits at the throat. Here l is the proper radial distance given by l ( r ) = ± ∫ r r 0 dr √ 1 -r 0 r</caption>
</figure>
<formula><location><page_5><loc_19><loc_55><loc_49><loc_58></location>Σ = 1 2 d dr ( ln N 2 r 2 K 2 ) . (35)</formula>
<text><location><page_5><loc_9><loc_45><loc_49><loc_53></location>Since we have discussed that these critical orbits are crucial in determining the last photon rings, therefore, η and ξ can completely determine the boundary of the shadow however in order to look for the shadow in the observer's sky, we have used the following definitions of celestial coordinates [78]:</text>
<formula><location><page_5><loc_20><loc_41><loc_49><loc_44></location>α = lim r →∞ ( -r 2 sin θ dϕ dr ) , (36)</formula>
<formula><location><page_5><loc_20><loc_37><loc_49><loc_40></location>β = lim r →∞ r 2 dθ dr , (37)</formula>
<text><location><page_5><loc_9><loc_32><loc_49><loc_36></location>and these celestial coordinates can be calculated with the help of impact parameters, η , and ξ by following the geodesic Eqs. derived in section III and given as,</text>
<formula><location><page_5><loc_21><loc_28><loc_49><loc_31></location>α = -η sin θ , (38)</formula>
<formula><location><page_5><loc_21><loc_24><loc_49><loc_28></location>β = √ ξ -η 2 sin 2 θ -Ω θ ω 2 o . (39)</formula>
<text><location><page_5><loc_9><loc_19><loc_49><loc_23></location>These expressions are not valid for calculating the celestial coordinates in homogeneous plasma space-time which will be discussed in the next section.</text>
<text><location><page_5><loc_9><loc_9><loc_49><loc_18></location>Another crucial factor to consider in the case of the wormhole shadow is the existence of the extremum potential at the throat of the wormhole. It becomes apparent from Eq. 28 that the effective potential becomes zero at the throat when r = r 0 . This implies that stable or unstable spherical orbits may exist depending on the sign of the second derivative of the effective potential</text>
<text><location><page_5><loc_52><loc_11><loc_92><loc_58></location>( V '' eff ( r 0 ) ). To gain insights into the formation of the shadow, Fig. 1(a) showcases the effective potential for values of α equal to -2 (red), 8 (blue), and β equal to 0 . It has been observed that for positive values of α the potential exhibit two extrema however unstable orbits are located outside the throat. Consequently, the contribution to the shadow is solely derived from the outer region. On the other hand, for negative values of α , only one extremum is present at the throat. Please note that Fig. 1(a) is not solely responsible for the formation of complete shadow as shown in Fig. 1(b). The potential is for illustration purposes for showing the existence of extrema at the throat and outside the throat. It is noteworthy to point out that previous studies on the formation of wormhole shadows in plasma space-time failed to account for the contribution of the throat potential, despite the throat contributions being highlighted in rotating wormholes [57]. In our quest to understand the intricate interplay of factors contributing to shadow formation, we delve into the analysis of spherical photon orbits that satisfy constraints Eqs. 25. We solve the Eq. 39 for β = 0 along with satisfying the constraint equations to find out the minimum ( r min ) and maximum ( r max ) radius for the shadow formation. Therefore, the shadow will consist of the orbit formed by ( r min , r max ). However, it turns out that sometimes r min can be less than the throat radius r 0 . In such cases, the shadow will be formed by the orbits consisting of ( r 0 , r max ), highlighting the contribution of the throat. This nuanced differentiation ensures that we gain a comprehensive understanding of the shadow formation mechanism, considering the varying contributions from different regions of the wormhole's geometry.</text>
<text><location><page_5><loc_53><loc_9><loc_92><loc_10></location>Since at the throat, the potential vanishes which also</text>
<figure>
<location><page_6><loc_15><loc_69><loc_85><loc_93></location>
<caption>FIG. 2. Wormhole shadows in a vacuum with throat radius, r 0 = 2 (a) for various inclinations angles with spin parameter, a = 0 . 99 and (b) for various spin parameters with inclination angle, θ = 90 · .</caption>
</figure>
<text><location><page_6><loc_9><loc_58><loc_49><loc_61></location>corresponds to the extremum of the potential (see Fig. 1(a)) therefore using Eq. 28,</text>
<formula><location><page_6><loc_13><loc_54><loc_49><loc_57></location>[ (1 -ηω T ) 2 -N 2 r 2 K 2 ( ξ + Ω r ω 2 o )]∣ ∣ ∣ ∣ r = r 0 = 0 . (40)</formula>
<text><location><page_6><loc_9><loc_44><loc_49><loc_53></location>Celestial coordinates which are given by Eqs. 38 and 39 contribute to the incomplete shadow of the wormhole, as shown in the blue solid curve in Fig. 1(b). The remaining part of the shadow is contributed by the unstable orbits at the throat. Therefore from Eqs. 38 and 39, we can write,</text>
<formula><location><page_6><loc_22><loc_40><loc_49><loc_43></location>α 2 + β 2 + Ω θ ω 2 o = ξ (41)</formula>
<text><location><page_6><loc_9><loc_36><loc_49><loc_39></location>and using expressions of η from Eq. 38 and ξ from Eq. 41 into Eq. 40, we get,</text>
<formula><location><page_6><loc_10><loc_30><loc_49><loc_35></location>[ (1 + αω T sin θ ) 2 r 2 K 2 N 2 -α 2 -Ω r +Ω θ ω 2 o = β 2 ]∣ ∣ ∣ ∣ ∣ r = r 0 . (42)</formula>
<text><location><page_6><loc_9><loc_18><loc_49><loc_30></location>This contributes to the shadow which is shown in the red curve in Fig. 1(b), therefore the shadow will be the bounded region consisting of blue and red curves, indicated by the solid blue and solid red curves, respectively, while disregarding the dashed red portion. Here the extreme left point of the boundary of the shadow in the celestial plane is found by setting β = 0 in the expression 42 and using Ω r +Ω θ = r 2 K 2 ω 2 P from Eq. 13, we get,</text>
<formula><location><page_6><loc_9><loc_11><loc_49><loc_17></location>α L = -rK √ r 2 K 2 N 2 ω 2 T ω 2 P ω 2 o sin 2 θ + N 2 -N 4 ω 2 P ω 2 o -r 2 K 2 ω T sin θ r 2 K 2 ω 2 T sin 2 θ -N 2 . (43)</formula>
<text><location><page_6><loc_9><loc_9><loc_49><loc_11></location>However, this expression is not valid for the homogeneous plasma distribution which will be discussed in</text>
<text><location><page_6><loc_52><loc_54><loc_92><loc_61></location>the next section. The wormhole shadows in vacuum are shown in Fig. 2 for spin parameter a = 0 . 99 (left) and different inclination angles and similarly for fixed inclination angle θ = 90 · (right) and different spin for reference purposes.</text>
<section_header_level_1><location><page_6><loc_54><loc_50><loc_89><loc_51></location>V. SHADOWFORSPECIFIC PLASMA PROFILES</section_header_level_1>
<text><location><page_6><loc_52><loc_29><loc_92><loc_48></location>In this section, our focus shifts toward exploring the effects of commonly discussed plasma distribution profiles on the shadow of rotating wormhole space-time. A crucial criterion to consider is the satisfaction of the separability condition outlined in equation 13 while choosing the plasma distribution functions. Notably, Shapiro [79] made significant advancements in accretion studies involving black holes and determined that the plasma frequency is proportional to r -3 / 2 for pressureless plasma. It is imperative to acknowledge this radial decrease in plasma frequency when examining the dependence of plasma on θ , especially in the case of inhomogeneous plasma distributions.</text>
<text><location><page_6><loc_52><loc_17><loc_92><loc_29></location>Furthermore, we must emphasize the importance of investigating a generalized form of plasma distribution. By doing so, we can highlight the distinguishing characteristics and disparities it holds when compared to other plasma distribution profiles. This comprehensive analysis enables us to gain a deeper understanding of the intricate relationship between plasma and the unique properties of rotating wormhole space-time.</text>
<section_header_level_1><location><page_6><loc_59><loc_13><loc_85><loc_14></location>A. Homogeneous plasma distribution</section_header_level_1>
<text><location><page_6><loc_52><loc_9><loc_92><loc_11></location>Firstly, we have considered the homogeneous plasma distribution between the observer and the source which</text>
<text><location><page_7><loc_9><loc_90><loc_49><loc_93></location>is widely studied to understand the physical phenomena [66, 71],</text>
<formula><location><page_7><loc_26><loc_86><loc_49><loc_89></location>ω 2 P ω 2 o = k 0 , (44)</formula>
<text><location><page_7><loc_9><loc_79><loc_49><loc_84></location>where k 0 denotes the homogeneous plasma parameter and it varies from 0 to 1 in order to satisfy the constraint Eq. 5. By using Eqs. 13 and 44 we can write the following expressions,</text>
<formula><location><page_7><loc_19><loc_76><loc_49><loc_78></location>Ω r ( r ) = k 0 r 2 ω 2 o , Ω θ ( θ ) = 0 . (45)</formula>
<text><location><page_7><loc_9><loc_72><loc_49><loc_74></location>Hence, the celestial coordinates for homogeneous plasma are given by solving Eqs. 36 and 37 as,</text>
<formula><location><page_7><loc_15><loc_67><loc_49><loc_70></location>α = -η csc θ √ 1 -k 0 , β = √ ξ -η 2 csc 2 θ 1 -k 0 , (46)</formula>
<text><location><page_7><loc_9><loc_62><loc_49><loc_65></location>and the contribution from the wormhole throat for homogeneous plasma space-time is given by</text>
<formula><location><page_7><loc_12><loc_52><loc_49><loc_61></location>[ ( 1 + αω T sin θ √ 1 -k 0 ) 2 r 2 K 2 N 2 (1 -k 0 ) -α 2 -Ω r ω 2 o (1 -k 0 ) = β 2 ]∣ ∣ ∣ ∣ ∣ r = r 0 , (47)</formula>
<text><location><page_7><loc_9><loc_34><loc_49><loc_51></location>while α L as mentioned in Sec IV is found by solving β = 0 , Eq. 46. In Fig. 3(a), we have plotted this case for spin a = 0 . 99 and it provides valuable insights into the behavior of the last photon ring, revealing that its radius expands in conjunction with larger homogeneous plasma parameters. This observation leads us to the inference that the universe is not filled with homogeneous plasma. If that were the case, we would have been able to detect these compact objects using low-resolution radio telescopes, given that the radius of the photon ring increases as the plasma parameter rises. Similar behavior has been observed for the lower spin values as well.</text>
<section_header_level_1><location><page_7><loc_18><loc_29><loc_39><loc_31></location>B. Radial plasma distribution</section_header_level_1>
<text><location><page_7><loc_9><loc_22><loc_49><loc_27></location>For this analysis, we have specifically focused on the radial plasma profile, where Ω θ is set to zero. We adopted the plasma profile proposed by Shapiro et al. [79] to look at its effect on the rotating wormhole shadow as,</text>
<formula><location><page_7><loc_25><loc_17><loc_49><loc_20></location>ω 2 P ω 2 o = k r r 3 / 2 , (48)</formula>
<text><location><page_7><loc_9><loc_12><loc_49><loc_16></location>where k r denotes the radial plasma parameter and its value should be in accordance with Eq. 5. We can calculate Ω r and Ω θ by using Eqs. 13 and 48 as,</text>
<formula><location><page_7><loc_20><loc_8><loc_49><loc_10></location>Ω r = k r r 1 / 2 ω 2 o , Ω θ = 0 , (49)</formula>
<text><location><page_7><loc_52><loc_90><loc_92><loc_93></location>and the celestial coordinates for this plasma profile using Eqs. 38 and 39 are given by</text>
<formula><location><page_7><loc_60><loc_87><loc_92><loc_89></location>α = -η csc θ , β = √ ξ -η 2 csc 2 θ. (50)</formula>
<text><location><page_7><loc_52><loc_66><loc_92><loc_86></location>In this scenario, the contribution of the throat to the shadow can be determined using Eq. 42. We have demonstrated this case in Fig. 3(b) for spin parameter a = 0 . 99 which illustrates the shadow of the wormhole for various radial plasma parameters, with k r equal to 0 , 2 , 4 , 6 , and 8 . Notably, it becomes evident that the plasma density has a negative impact on the shadow, which contrasts with the behavior observed in the case of a homogeneous plasma distribution, as depicted in Fig. 3(a). As the plasma parameter increases, the shadow gradually becomes undetectable since the previously mentioned conditions (Eqs. 5 and 25) are no longer satisfied. These observations shed light on the intricate relationship between plasma density and the resulting shadow characteristics.</text>
<section_header_level_1><location><page_7><loc_60><loc_62><loc_84><loc_63></location>C. Latitudinal plasma distribution</section_header_level_1>
<text><location><page_7><loc_52><loc_50><loc_92><loc_59></location>Now, let's explore another example where the plasma distribution is dependent on the polar ( θ ) coordinate. In this scenario, we consider a distribution that exhibits a reduction in plasma density over increasing distances [79]. This choice is essential to distinguish it from a scenario involving homogeneous plasma. To illustrate this, let's denote the plasma distribution in this case as follows:</text>
<formula><location><page_7><loc_67><loc_45><loc_92><loc_48></location>ω 2 P ω 2 o = k θ sin 2 θ r 2 , (51)</formula>
<text><location><page_7><loc_52><loc_37><loc_92><loc_44></location>where k θ represents the latitudinal plasma parameter, just to differentiate it from the radial plasma parameter, k r and it is chosen such that it satisfies the constraint conditions, Eq. 5. With the help of Eqs. 13 and 51, we can write the following expressions,</text>
<formula><location><page_7><loc_62><loc_34><loc_92><loc_35></location>Ω r = 0 , Ω θ = k θ ω 2 o sin 2 θ, (52)</formula>
<text><location><page_7><loc_52><loc_29><loc_92><loc_32></location>and the celestial coordinates for this plasma profile using Eqs. 38 and 39 are given by</text>
<formula><location><page_7><loc_54><loc_26><loc_92><loc_28></location>α = -η csc θ , β = √ ξ -η 2 csc 2 θ -k θ sin 2 θ. (53)</formula>
<text><location><page_7><loc_52><loc_9><loc_92><loc_24></location>Fig. 3(c) provides a comparative visualization of the shadow cast by the wormhole for this particular plasma profile. Remarkably, it becomes evident that the dependence of plasma density on the θ coordinate exerts a significant influence on the size of the shadow, surpassing the impact of the radial profile. This finding is particularly noteworthy, as previous studies primarily concentrated on radial profiles [56] and omitted the analysis of such latitudinal profiles. It underscores the importance of considering a generalized plasma density distribution to gain a more profound understanding of the shadow</text>
<figure>
<location><page_8><loc_9><loc_53><loc_92><loc_93></location>
<caption>FIG. 3. Comparison of Wormhole shadows with throat radius r 0 = 2 , spin parameter a = 0 . 99 , and inclination angle θ = 90 · with various plasma parameters ( k 0 , k r , k θ ) for the plasma distributions : (a) ω 2 P = k 0 ω 2 o , (b) ω 2 P = k r r 3 / 2 ω 2 o , (c) ω 2 P = k θ sin 2 θ r 2 ω 2 o , and (d) ω 2 P = k r √ r + k θ sin 2 θ r 2 ω 2 o .</caption>
</figure>
<text><location><page_8><loc_9><loc_32><loc_49><loc_43></location>boundary in plasma space-time. Therefore, by incorporating the influence of plasma density variation with respect to θ , we can delve deeper into the intricacies of shadow formation and unravel more comprehensive insights into the behavior of wormholes in the presence of varying plasma distributions. Motivated by this, in the next subsection, we will be studying the more general case for plasma distribution.</text>
<section_header_level_1><location><page_8><loc_16><loc_28><loc_41><loc_29></location>D. Generalized plasma distribution</section_header_level_1>
<text><location><page_8><loc_9><loc_16><loc_49><loc_26></location>Now, let's consider the more comprehensive scenario where the plasma distribution depends on both the radial coordinate ( r ) and the angular coordinate ( θ ). This broader analysis allows us to gain further insights into the effects of plasma densities on wormhole shadows. For this case, we denote the plasma distribution as follows,</text>
<formula><location><page_8><loc_21><loc_12><loc_49><loc_16></location>ω 2 P ω 2 o = k r √ r + k θ sin 2 θ r 2 . (54)</formula>
<text><location><page_8><loc_9><loc_9><loc_49><loc_11></location>This particular profile is essentially a combination of the two previously discussed profiles. It incorporates the</text>
<text><location><page_8><loc_52><loc_35><loc_92><loc_43></location>additive contributions from each of them. We adopt a similar plasma distribution profile to the one proposed by Perlick et al. [71], which provides valuable insights into the behavior of the plasma distribution in relation to the formation of wormhole shadows. Now, using Eqs. 13 and 54,</text>
<formula><location><page_8><loc_61><loc_32><loc_92><loc_34></location>Ω r = k r rω 2 o , Ω θ = k θ ω 2 o sin 2 θ, (55)</formula>
<text><location><page_8><loc_52><loc_29><loc_92><loc_31></location>and the celestial coordinates for this generalized plasma profile using Eqs. 38 and 39 are given by</text>
<formula><location><page_8><loc_54><loc_25><loc_92><loc_28></location>α = -η csc θ , β = √ ξ -η 2 csc 2 θ -k θ sin 2 θ. (56)</formula>
<text><location><page_8><loc_52><loc_9><loc_92><loc_24></location>Fig. 3(d) showcases the shadow cast by the wormhole for different values of k r and k θ . Interestingly, the generalized plasma profile demonstrates superior performance compared to the other two profiles previously discussed. This emphasizes the significance of studying the more comprehensive generalized plasma profile rather than solely focusing on the radial profile. A similar kind of behavior has been observed for the low spin values. Hence, we have chosen to compare the different plasma density profiles with the higher spin case. Furthermore, it is noteworthy that the individual shadows resulting from the</text>
<figure>
<location><page_9><loc_10><loc_32><loc_91><loc_93></location>
<caption>FIG. 4. Wormhole shadows for generalized plasma distribution showing the disappearance of last photon ring (green curves) for various inclination angles and spin parameters with different radial plasma parameters, k r and longitudinal plasma parameter, k θ . The black curve represents the photon ring without plasma, blue curves represent shadow with low plasma parameters and are shown here for the illustration purpose of shrinking shadow size. Considering plasma profile as ω 2 P = k r √ r + k θ sin 2 θ r 2 ω 2 o .</caption>
</figure>
<text><location><page_9><loc_9><loc_9><loc_49><loc_21></location>specific plasma parameters are larger when compared to their combined effect. As we delve deeper into the analysis, we observe that the shadow progressively diminishes in size with increasing plasma parameters. At a certain critical value, the shadow may eventually vanish or become undetectable altogether. This phenomenon has been shown in Fig. 4 for different values of spin parameters as well as at different inclination angles. It can</text>
<text><location><page_9><loc_52><loc_12><loc_92><loc_21></location>be observed that as the plasma parameter increases, the wormhole shadow started shrinking and eventually disappears which has been shown by dashed green curves. Please note that the plasma parameter values corresponding to these green curves do not serve for maximum value after which the shadow gets disappear completely.</text>
<text><location><page_9><loc_52><loc_9><loc_92><loc_11></location>By considering these generalized plasma distributions and closely examining the changes in the resulting shad-</text>
<text><location><page_10><loc_9><loc_83><loc_49><loc_93></location>ows, we can potentially gain valuable insights into the plasma distribution along the observational path. This motivates us to further look into the photon trajectories in plasma space-time. Therefore, in the next section, we will be exploring the weak gravitational lensing within the effect of the plasma distribution around the rotating wormhole space-time.</text>
<section_header_level_1><location><page_10><loc_15><loc_78><loc_43><loc_79></location>VI. WEAKGRAVITATIONAL LENSING</section_header_level_1>
<text><location><page_10><loc_9><loc_43><loc_49><loc_76></location>In this section, we explore the influence of plasma distributions on the deflection angle within the framework of the weak field approximation. As we know, when light rays traverse the vicinity of massive objects, they experience deviations from their original paths. Here, we present the analytical expression for the deflection angle, focusing specifically on the case when the observer is situated in the equatorial plane ( θ = 90 · ) of the source. By examining the effects of plasma distributions on the deflection angle, we can gain a deeper understanding of how the presence of plasma affects the trajectory of light rays near massive objects. This analysis allows us to investigate the intricate interplay between plasma and gravity, shedding light on the nature of weak gravitational lensing in the presence of plasma. The derived analytic expression provides a valuable tool for predicting and analyzing the deflection of light in various astrophysical scenarios, contributing to our overall comprehension of the behavior of light in the presence of massive objects and plasma distributions. In order to analyze the deflection angle, we first calculate the geodesic equations with the help of Hamilton's equation 6 and the Hamiltonian for rotating plasma space-time given by Eq. 1 as,</text>
<formula><location><page_10><loc_11><loc_38><loc_49><loc_41></location>˙ ϕ = ∂H ∂p ϕ = g tϕ p t + g ϕϕ p ϕ , , ˙ r = ∂H ∂p r = g rr p r , (57)</formula>
<text><location><page_10><loc_9><loc_35><loc_39><loc_37></location>which can be further simplified as follows,</text>
<formula><location><page_10><loc_19><loc_30><loc_49><loc_34></location>( ˙ r ˙ ϕ ) 2 = ( g rr p r g ϕϕ p ϕ + g ϕt p t ) 2 . (58)</formula>
<text><location><page_10><loc_9><loc_26><loc_49><loc_29></location>Now, for the massless particles, the Hamiltonian should be zero ( H = 0 ). Therefore, Eq. 1 can be written by using</text>
<text><location><page_10><loc_52><loc_92><loc_88><loc_93></location>the definitions such that p t = -ω o , and p θ = L as,</text>
<formula><location><page_10><loc_55><loc_89><loc_92><loc_91></location>g rr p 2 r = -( g tt ω 2 o -2 g ϕt ω o L + g ϕϕ L 2 + ω 2 P ) . (59)</formula>
<text><location><page_10><loc_52><loc_86><loc_82><loc_88></location>Hence Eqs. 58 and 59 can be simplified to,</text>
<formula><location><page_10><loc_55><loc_79><loc_92><loc_85></location>( ˙ r ˙ ϕ ) 2 = -g rr ( g ϕϕ L -g ϕt ω o ) 2 ( g tt ω 2 o -2 g ϕt ω o L + g ϕϕ L 2 + ω 2 P ) , (60)</formula>
<text><location><page_10><loc_52><loc_75><loc_92><loc_78></location>and to simplify the above Eq. we have considered the following definitions,</text>
<formula><location><page_10><loc_65><loc_71><loc_92><loc_74></location>ω o L = λ , ω 2 P ω 2 o = X, (61)</formula>
<text><location><page_10><loc_52><loc_70><loc_78><loc_71></location>thus, equation 60 can be modified as,</text>
<formula><location><page_10><loc_53><loc_63><loc_92><loc_68></location>( ˙ r ˙ ϕ ) 2 = -g rr ( g ϕϕ -g ϕt λ ) 2 ( g tt λ 2 -2 g ϕt λ + g ϕϕ + Xλ 2 ) . (62)</formula>
<text><location><page_10><loc_52><loc_58><loc_92><loc_63></location>Since the deflection angle is calculated when the light deviates from its original path and consequently when it is at the closest approach ( r = R ) to the central object. Therefore, at the closet distance, we can define,</text>
<formula><location><page_10><loc_67><loc_53><loc_92><loc_56></location>( ˙ r ˙ ϕ )∣ ∣ ∣ ∣ r = R = 0 . (63)</formula>
<text><location><page_10><loc_52><loc_46><loc_92><loc_52></location>Now, the evaluation needs to be done at r = R as mention in Eq. 63, we have considered the following expressions for the metric and plasma functions, being evaluated at r = R :</text>
<formula><location><page_10><loc_58><loc_41><loc_92><loc_45></location>g tt ∣ ∣ R = G tt , g ϕt ∣ ∣ R = G ϕt , g ϕϕ ∣ ∣ R = G ϕϕ , g rr | R = G rr , X | R = Y. (64)</formula>
<text><location><page_10><loc_52><loc_36><loc_92><loc_40></location>Thus, the impact parameter, λ is calculated by using the expressions given in Eq. 64 with the help of Eqs. 62 and 63, and given as,</text>
<formula><location><page_10><loc_57><loc_31><loc_92><loc_35></location>λ = 2 G ϕt ± √ (2 G ϕt ) 2 -4 G ϕϕ ( G tt + Y ) 2 ( G tt + Y ) , (65)</formula>
<text><location><page_10><loc_52><loc_26><loc_92><loc_30></location>and finally, the integral form for the deflection angle of the light from its original trajectory can be given by solving further using Eqs. 62 and 65 as,</text>
<formula><location><page_10><loc_25><loc_18><loc_92><loc_22></location>∫ ¯ α 0 dϕ = ± 2 ∫ ∞ -∞ [ -g rr ( g ϕϕ -g ϕt λ ) 2 ( ( g tt + X ) λ 2 -2 g ϕt λ + g ϕϕ ) ] -1 / 2 dr. (66)</formula>
<text><location><page_10><loc_9><loc_10><loc_49><loc_14></location>It is important to note that the deflection angle for the light following its original trajectory will be π given that the center of coordinates corresponds to the compact ob-</text>
<text><location><page_10><loc_52><loc_11><loc_92><loc_14></location>ect. Therefore, the actual deflection angle is determined by α = ¯ α -π .</text>
<text><location><page_10><loc_53><loc_9><loc_92><loc_10></location>In the subsequent analysis, we proceed to calculate</text>
<figure>
<location><page_11><loc_14><loc_69><loc_52><loc_93></location>
<caption>FIG. 5. Weak deflection angle as a function of closest distance (R) with spin parameter, a = 0 . 5 in Teo wormhole space-time for plasma profiles: (a) ω 2 P = k 0 ω 2 o , and (b) ω 2 P = k r r 2 ω 2 o .</caption>
</figure>
<text><location><page_11><loc_80><loc_91><loc_84><loc_92></location>Kr=0</text>
<table>
<location><page_11><loc_11><loc_49><loc_89><loc_59></location>
<caption>TABLE I. Deflection angle in homogeneous and non-homogeneous plasma distributions for Teo wormhole space-time.</caption>
</table>
<text><location><page_11><loc_9><loc_32><loc_49><loc_46></location>the deflection angles for both homogeneous and nonhomogeneous plasma distributions. The homogeneous plasma distribution is characterized by uniform plasma density, while the radial plasma distribution exhibits a density variation in the non-homogeneous direction. By studying these specific cases, we can discern the effects of plasma distributions on the deflection of light and deepen our understanding of gravitational lensing phenomena in the presence of plasma. We have considered the following plasma distributions [66].</text>
<formula><location><page_11><loc_21><loc_28><loc_49><loc_31></location>ω 2 P ω 2 o = k 0 , ω 2 P ω 2 o = k r r 2 . (67)</formula>
<text><location><page_11><loc_9><loc_14><loc_49><loc_27></location>In the case of a homogeneous plasma distribution, the values of k 0 fall within the range of ( 0 , 1 ) as discussed in section V(A). Additionally, the choice of k r is determined to satisfy Eq. 5 which takes into account the gravitational redshift. It is worth noting that in previous studies [67, 68], the redshift condition has often been neglected. However, it is crucial to consider this condition as it significantly influences the trajectory of light and, consequently, the deflection angle.</text>
<text><location><page_11><loc_9><loc_9><loc_49><loc_14></location>We have considered the weak field limit and lower plasma densities for simplicity to calculate the weak deflection angle for the slow-rotating wormhole. The detailed derivation has been performed in Appendix A. The</text>
<text><location><page_11><loc_52><loc_31><loc_92><loc_46></location>resulting values for the deflection angle are presented in Table I, providing a comprehensive overview of the deflection angles for both the homogeneous and nonhomogeneous plasma profiles. By examining the values of the deflection angle, we can gain insights into the effects of plasma distributions on the path of light in the vicinity of massive objects. Therefore, we examine both the homogeneous and non-homogeneous plasma distributions around the rotating wormhole geometry and studied their effect on the deflection angle of the light rays.</text>
<text><location><page_11><loc_52><loc_17><loc_92><loc_30></location>The deflection angle exhibits a decrease with the closest distance to the wormhole (See Fig. 5(a) and 5(b)), indicating a reduced gravitational influence. It is noteworthy that at higher plasma densities, the deflection angle increases, as illustrated in Fig. 6(a) for all values of the spin parameter. This observation gives validation of the earlier observed phenomenon such that the shadow radius increases with the plasma density in uniform plasma distribution (see Fig. 3(a)).</text>
<text><location><page_11><loc_52><loc_9><loc_92><loc_17></location>In the case of a non-homogeneous plasma distribution, an intriguing observation is that the deflection angle decreases with increasing plasma densities, as shown in Figs. 5(b) and 6(b) for all values of the spin parameter. This stands in contrast to the homogeneous case and provides an explanation for the negative impact of plasma</text>
<figure>
<location><page_12><loc_14><loc_69><loc_52><loc_93></location>
</figure>
<figure>
<location><page_12><loc_53><loc_69><loc_86><loc_93></location>
<caption>FIG. 6. Weak deflection angle in Teo wormhole space-time for closest distance approach, R = 10 as a function of plasma parameters ( k 0 , k r ) for various spin parameters in case of plasma profiles: (a) ω 2 P = k 0 ω 2 o , and (b) ω 2 P = k r r 2 ω 2 o .</caption>
</figure>
<text><location><page_12><loc_9><loc_48><loc_49><loc_61></location>on the shadow which already has been observed in the case of the shadow (see Fig. 3(b)). Notably, the influence of non-homogeneous plasma distributions on the deflection angle has not been extensively explored in previous studies. Most investigations on the effects of plasma on the deflection angle by compact objects have focused on a single isothermal sphere model, commonly employed for galaxy modelling which yielded a positive impact of plasma on the deflection angle [68].</text>
<text><location><page_12><loc_9><loc_28><loc_49><loc_48></location>It is important to note that the choice of the plasma parameter value should ensure low plasma density and compliance with the condition given by Eq. 5. In previous studies, researchers have typically considered plasma parameter values ranging from 0 to 1 [68]. However, within the given impact parameter constraints along with the condition given by Eq. 5, a range of plasma parameter values can be chosen to study the deflection angle. Therefore, by analyzing the effects of plasma on the deflection angle, we can gain valuable insights into the distribution of plasma in the vicinity of compact objects. This investigation serves as a powerful tool for studying and understanding the properties of plasma surrounding these intriguing cosmic structures.</text>
<section_header_level_1><location><page_12><loc_10><loc_22><loc_48><loc_24></location>VII. CONSTRAINING THE WORMHOLE SHADOW ANDPLASMAPARAMETERS</section_header_level_1>
<text><location><page_12><loc_9><loc_9><loc_49><loc_20></location>To determine the plasma parameters and size of the wormhole, we employ observational data released by EHT [14] for the supermassive black hole located at the center of the elliptical galaxy Messier 87 (M87), also known as M 87 ∗ . By examining the average angular size of the shadow and its deviation from circularity, we can constrain shadow and plasma parameters. As the shadow possesses reflection symmetry around the α -axis</text>
<text><location><page_12><loc_52><loc_48><loc_92><loc_61></location>in the celestial plane, we calculate its geometric center ( α 0 , β 0 ) using the integrals α 0 = 1 /A ∫ αdA and β 0 = 0 . Here, dA represents an area element. Next, we introduced an angle ϕ defined as the angle between the α -axis and the vector connecting the geometric center ( α c , β c ) with a point ( α, β ) on the boundary of the shadow. This angle ϕ provides valuable information for our analysis as can be seen as follows. Therefore, the average radius ( R ) of the shadow is given by [73],</text>
<formula><location><page_12><loc_64><loc_44><loc_92><loc_47></location>R 2 = 1 2 π ∫ 2 π 0 l 2 ( ϕ ) dϕ, (68)</formula>
<text><location><page_12><loc_52><loc_38><loc_92><loc_43></location>where l ( ϕ ) = √ ( α ( ϕ ) -α 0 ) 2 + β ( ϕ ) 2 and ϕ = tan -1 ( β ( ϕ ) / ( α ( ϕ ) -α 0 )) . Following [14], we define the deviation ∆ C from circularity as [73],</text>
<formula><location><page_12><loc_60><loc_33><loc_92><loc_37></location>∆ C = 1 R √ 1 2 π ∫ 2 π 0 ( l ( ϕ ) -R ) 2 dϕ. (69)</formula>
<text><location><page_12><loc_52><loc_16><loc_92><loc_32></location>We should note that ∆ C represents the fractional root mean square distance from the average radius of the observed shadow. Based on the findings of the EHT collaboration [14], the angular size of the observed shadow is determined to be ∆ θ sh = 42 ± 3 µ as, with a deviation ∆ C of less than 10% . Additionally, following the same study [14], we adopt the distance to M87* as D = (16 . 8 ± 0 . 8) Mpc and the mass of the object as M = (6 . 5 ± 0 . 7) × 10 9 M ⊙ . With these values, we can estimate the average diameter of the shadow by considering the angular size,</text>
<formula><location><page_12><loc_53><loc_12><loc_92><loc_15></location>Diameter = d sh = 2 × D × tan ( ∆ θ sh 2 ) = 11 . 0 ± 1 . 5 . (70)</formula>
<text><location><page_12><loc_52><loc_9><loc_92><loc_11></location>This uncertainty in the shadow diameter can be determined by carefully propagating the uncertainties asso-</text>
<figure>
<location><page_13><loc_51><loc_68><loc_91><loc_93></location>
</figure>
<figure>
<location><page_13><loc_9><loc_68><loc_52><loc_93></location>
<caption>FIG. 7. Dependence of (a) the angular size and (b) the deviation of the shadow on radial plasma parameter ( k r ) and spin with θ = 17 · and r 0 = 2 . The two red dashed lines indicate the spin range 0 . 5 ≤ a ≤ 0 . 94 . Considering plasma profile as ω 2 P = k r r 3 / 2 ω 2 o .</caption>
</figure>
<figure>
<location><page_13><loc_52><loc_36><loc_91><loc_61></location>
</figure>
<figure>
<location><page_13><loc_9><loc_36><loc_53><loc_61></location>
<caption>FIG. 8. Dependence of (a) the angular size and (b) the deviation of the shadow on latitudinal plasma parameter ( k θ ) and spin with θ = 17 · and r 0 = 2 . The two red dashed lines indicate the spin range 0 . 5 ≤ a ≤ 0 . 94 . Considering plasma profile as ω 2 P = k θ sin 2 θ r 2 ω 2 o .</caption>
</figure>
<text><location><page_13><loc_9><loc_14><loc_49><loc_27></location>ciated with both the distance and angular size measurements. By accounting for these uncertainties, we can obtain more reliable estimates of the shadow's diameter, enabling us to extract valuable information about the physical characteristics of M 87 ∗ and delve into the intricacies of the shadow phenomenon. Therefore, these insights pave the way for further investigations and contribute to our ongoing exploration of the enigmatic nature of M 87 ∗ and its surrounding environment.</text>
<text><location><page_13><loc_9><loc_9><loc_49><loc_13></location>The determination of the average diameter and the deviation from the circularity of the observed shadow involves considering the errors in a combined manner,</text>
<text><location><page_13><loc_52><loc_10><loc_92><loc_27></location>with the uncertainties added in quadrature. It is important to ensure that this calculated quantity matches the expected value of the diameter (2R). Fig. 7 presents the results depicting the average diameter (left) and circularity deviation (right) of the shadow, considering various values for the spin parameter and the size of the wormhole throat concerning the radial plasma parameter ( k r ). In our analysis, we have taken an inclination angle of θ = 17 · , which represents the angle between the jet axis and the line of sight to M 87 ∗ . Additionally, based on the findings of the EHT collaboration, the spin parameter falls within the range of 0 . 5 ≤ a ≤ 0 . 94 which has been</text>
<figure>
<location><page_14><loc_9><loc_68><loc_52><loc_93></location>
</figure>
<figure>
<location><page_14><loc_53><loc_68><loc_91><loc_93></location>
<caption>FIG. 9. Dependence of (a) the angular size and (b) the deviation of the shadow on throat size and spin with θ = 17 · , k r = 3 . 5 and k θ = 20 . 4 . The two red dashed lines indicate the spin range 0 . 5 ≤ a ≤ 0 . 94 . Considering plasma profile as ω 2 P = k r √ r + k θ sin 2 θ r 2 ω 2 o .</caption>
</figure>
<text><location><page_14><loc_9><loc_59><loc_39><loc_60></location>shown in the Fig. 7 with vertical red lines.</text>
<text><location><page_14><loc_9><loc_9><loc_49><loc_58></location>Now, to determine the maximum plasma parameters that apply to M 87 ∗ , we analyzed the plasma density corresponding to the smallest observed shadow size. Initially, by looking only at the radial plasma profile, we observed that the maximum value for the radial plasma parameter is k rc = 4 . 75 when the shadow diameter matches with the observed M 87 ∗ diameter (Fig. 7). Similarly, the contour line corresponding to the lowest shadow diameter Fig. (8) indicates that the maximum longitudinal plasma parameter is k θc = 85 . 5 . furthermore, to cover the complete range of shadow diameters between 9 . 5 and 12 . 5 with the uncertainty in the M 87 ∗ shadow diameter, we considered k r = 3 . 5 and k θ = 20 . 4 in Fig. 9. These choices allow us to place constraints on the maximum possible value of the throat size ( r 0 c = 2 . 51 ) which corresponds to the maximum shadow diameter as shown in Fig. 9(a). It is worth noting that the deviation from circularity ( ∆ C ) in the wormhole shadows provides interesting features about the plasma parameters. From Figs. 7(b) and 8(b), we observed that ∆ C ≤ 10% required the spin range for M 87 ∗ to be less than 0 . 5 when considering either the radial or latitudinal plasma profiles. However, in the case of the generalized plasma distribution the observed circularity deviation from the shadow of M 87 ∗ ( ∆ C ≤ 10% ) can still be achieved for the known spin range while obtaining shadow diameters within the observed range for M 87 ∗ . It is important to note that this analysis restricts the throat size such that it should not exceed the value of r 0 c to maintain ∆ C < 10% and maximum shadow diameter of 12.5 within the allowed range of spin, the value of r 0 c should be less than 2.51 for the chosen value of k r and k θ in case of generalized plasma distributions (See Fig. 9(a) and 9(b)). Thus, the constraint on the maximum value of throat size may vary</text>
<text><location><page_14><loc_52><loc_52><loc_92><loc_60></location>depending on the choice of plasma parameters. These findings further contribute to our understanding of the possible spin range and plasma distributions associated with the observed shadow of M 87 ∗ . Please note that we have only considered those plasma profiles which have been discussed in Section V.</text>
<text><location><page_14><loc_52><loc_46><loc_92><loc_51></location>Therefore, this study ascertains the range of plasma parameters and throat size that are consistent with the observed shadows and provides further insights into the properties of the M 87 ∗ system.</text>
<section_header_level_1><location><page_14><loc_64><loc_41><loc_79><loc_42></location>VIII. CONCLUSION</section_header_level_1>
<text><location><page_14><loc_52><loc_28><loc_92><loc_39></location>In this study, our focus was on investigating the behavior of null geodesics in non-magnetized pressureless plasma within the context of a rotating wormhole spacetime. We specifically examined the gravitational influence of the wormhole while neglecting the direct gravitational effects of the plasma. Instead, we considered only the dispersive properties of the plasma, affecting the trajectory of light rays.</text>
<text><location><page_14><loc_52><loc_10><loc_92><loc_27></location>One key finding of our work, as discussed in Section III, is the requirement of a specific plasma distribution profile to establish a generalized Carter's constant. We also emphasized the importance of including potential contributions from both inside and outside the wormhole throat, as elaborated in Section IV. Furthermore, we derived analytical formulas for the boundary of the shadow for various plasma profiles in Section V. Notably, our results revealed that the shadow size decreases with increasing plasma density. Eventually, for certain upper limits of the plasma parameters, the shadow completely disappears.</text>
<text><location><page_14><loc_53><loc_9><loc_92><loc_10></location>Our primary objective throughout this study was to</text>
<text><location><page_15><loc_9><loc_69><loc_49><loc_93></location>obtain an analytical expression for the shadows observed in plasma space-time. By investigating the behavior of light rays in the presence of plasma, we aimed to enhance our understanding of the intricate interplay between gravitational and plasma effects in astrophysical phenomena. In Section VI, we conducted calculations to determine the deflection angle on a rotating wormhole in plasma space-time. Gravitational lensing phenomena have significant implications for astrophysical observations, and our study shed light on the impact of plasma on the deflection angle. Interestingly, we observed that as the plasma parameter increases, the deflection angle decreases in a non-homogeneous plasma space-time, contrary to the behavior observed in a homogeneous plasma profile. This intriguing result underscores the importance of further investigating the observational aspects and exploring the plasma distribution near compact objects.</text>
<text><location><page_15><loc_9><loc_53><loc_49><loc_68></location>Finally, we proceeded to constrain the size of the throat and plasma parameters mentioned in Section V by utilizing the observational data coming from EHT for M 87 ∗ . Our analysis revealed that the maximum allowed throat radius is determined to be r 0 c = 2 . 51 , which corresponds to the allowed range of shadow diameter, spin, and circularity deviation for M 87 ∗ as reported by EHT. On the other hand, by considering a minimum shadow diameter of 9 . 5 , we were able to place constraints on the radial and latitudinal plasma parameters, with maximum values k r = 4 . 75 and k θ = 85 . 5 , respectively. These constraints</text>
<unordered_list>
<list_item><location><page_15><loc_10><loc_46><loc_48><loc_47></location>[1] A. Einstein and N. Rosen, Physical Review 48 , 73 (1935).</list_item>
<list_item><location><page_15><loc_10><loc_45><loc_38><loc_46></location>[2] J. A. Wheeler, Phys. Rev. 128 , 919 (1962).</list_item>
<list_item><location><page_15><loc_10><loc_43><loc_49><loc_45></location>[3] M. S. Morris and K. S. Thorne, Am. J. Phys. 56 , 395 (1988).</list_item>
<list_item><location><page_15><loc_10><loc_42><loc_36><loc_43></location>[4] E. Teo, Phys. Rev. D 58 , 024014 (1998).</list_item>
<list_item><location><page_15><loc_10><loc_39><loc_49><loc_42></location>[5] M. Visser, S. Kar, and N. Dadhich, Physical Review Letters 90 (2003), 10.1103/physrevlett.90.201102.</list_item>
<list_item><location><page_15><loc_10><loc_38><loc_49><loc_39></location>[6] H. Maeda and M. Nozawa, Phys. Rev. D 78 , 024005 (2008).</list_item>
<list_item><location><page_15><loc_10><loc_36><loc_49><loc_38></location>[7] F. S. N. Lobo and M. A. Oliveira, Phys. Rev. D 80 , 104012 (2009).</list_item>
<list_item><location><page_15><loc_10><loc_33><loc_49><loc_35></location>[8] A. Cisterna, K. Muller, K. Pallikaris, and A. Vigan'o, Phys. Rev. D 108 , 024066 (2023), arXiv:2306.14541 [gr-qc].</list_item>
<list_item><location><page_15><loc_10><loc_29><loc_49><loc_33></location>[9] A. E. Broderick, R. Narayan, J. Kormendy, E. S. Perlman, M. J. Rieke, and S. S. Doeleman, Astrophys. J. 805 , 179 (2015), arXiv:1503.03873 [astro-ph.HE].</list_item>
<list_item><location><page_15><loc_9><loc_26><loc_49><loc_29></location>[10] A. E. Broderick, A. Loeb, and R. Narayan, Astrophys. J. 701 , 1357 (2009), arXiv:0903.1105 [astro-ph.HE].</list_item>
<list_item><location><page_15><loc_9><loc_24><loc_49><loc_26></location>[11] R. Narayan and J. E. McClintock, New Astron. Rev. 51 , 733 (2008), arXiv:0803.0322 [astro-ph].</list_item>
<list_item><location><page_15><loc_9><loc_21><loc_49><loc_23></location>[12] M. A. Abramowicz, W. Kluzniak, and J.-P. Lasota, Astron. Astrophys. 396 , L31 (2002), arXiv:astro-ph/0207270.</list_item>
<list_item><location><page_15><loc_9><loc_18><loc_49><loc_21></location>[13] R. Takahashi, J. Korean Phys. Soc. 45 , S1808 (2004), arXiv:astro-ph/0405099.</list_item>
<list_item><location><page_15><loc_9><loc_16><loc_49><loc_18></location>[14] The Event Horizon Telescope Collaboration, The Astrophysical Journal Letters 875 , L1 (2019).</list_item>
<list_item><location><page_15><loc_9><loc_13><loc_49><loc_16></location>[15] K. Akiyama et al. (Event Horizon Telescope), Astrophys. J. Lett. 930 , L12 (2022).</list_item>
<list_item><location><page_15><loc_9><loc_10><loc_49><loc_13></location>[16] K. Akiyama et al. (Event Horizon Telescope), Astrophys. J. Lett. 930 , L17 (2022).</list_item>
<list_item><location><page_15><loc_9><loc_9><loc_48><loc_10></location>[17] J. L. Synge, Mon. Not. Roy. Astron. Soc. 131 , 463 (1966).</list_item>
</unordered_list>
<text><location><page_15><loc_52><loc_83><loc_92><loc_93></location>provide valuable insights into the physical properties of the wormhole and the plasma surrounding it. By examining the maximum and minimum shadow diameters, we can better understand the range of possible sizes for the throat and the corresponding plasma parameters that are consistent with the observed shadows in the case of M 87 ∗ .</text>
<text><location><page_15><loc_52><loc_66><loc_92><loc_83></location>In our future research, we intend to investigate the impact of plasma on the shadow of a Kerr black hole. Additionally, we plan to compare the findings from the study of wormhole shadows to those of black hole shadows. This comparative analysis will provide further insights and potentially help discern whether M 87 ∗ is more likely to be a black hole or a wormhole. By delving into these investigations, we hope to contribute to the ongoing understanding of M 87 ∗ and its intriguing nature, paving the way for deeper insights into the astrophysical phenomena occurring in the vicinity of these enigmatic cosmic objects.</text>
<section_header_level_1><location><page_15><loc_63><loc_61><loc_80><loc_62></location>ACKNOWLEDGEMENT</section_header_level_1>
<text><location><page_15><loc_52><loc_53><loc_92><loc_58></location>The work of SC is supported by Mathematical Research Impact Centric Support (MATRICS) from the Science and Engineering Research Board (SERB) of India through grant MTR/2022/000318.</text>
<unordered_list>
<list_item><location><page_15><loc_52><loc_46><loc_86><loc_47></location>[18] J. P. Luminet, Astron. Astrophys. 75 , 228 (1979).</list_item>
<list_item><location><page_15><loc_52><loc_45><loc_78><loc_46></location>[19] Black Holes (Les Astres Occlus) (1973).</list_item>
<list_item><location><page_15><loc_52><loc_42><loc_92><loc_45></location>[20] A. F. Zakharov, Phys. Rev. D 90 , 062007 (2014), arXiv:1407.7457 [gr-qc].</list_item>
<list_item><location><page_15><loc_52><loc_39><loc_92><loc_42></location>[21] R. Takahashi, Publ. Astron. Soc. Jap. 57 , 273 (2005), arXiv:astro-ph/0505316.</list_item>
<list_item><location><page_15><loc_52><loc_38><loc_84><loc_39></location>[22] N. Tsukamoto, Phys. Rev. D 97 , 064021 (2018).</list_item>
<list_item><location><page_15><loc_52><loc_34><loc_92><loc_38></location>[23] P. V. P. Cunha, C. A. R. Herdeiro, E. Radu, and H. F. Runarsson, Phys. Rev. Lett. 115 , 211102 (2015), arXiv:1509.00021 [gr-qc].</list_item>
<list_item><location><page_15><loc_52><loc_32><loc_92><loc_34></location>[24] Z. Li and C. Bambi, JCAP 01 , 041 (2014), arXiv:1309.1606 [gr-qc].</list_item>
<list_item><location><page_15><loc_52><loc_29><loc_92><loc_31></location>[25] A. Abdujabbarov, M. Amir, B. Ahmedov, and S. G. Ghosh, Phys. Rev. D 93 , 104004 (2016), arXiv:1604.03809 [gr-qc].</list_item>
<list_item><location><page_15><loc_52><loc_26><loc_92><loc_29></location>[26] M. Amir and S. G. Ghosh, Phys. Rev. D 94 , 024054 (2016), arXiv:1603.06382 [gr-qc].</list_item>
<list_item><location><page_15><loc_52><loc_22><loc_92><loc_26></location>[27] P. V. P. Cunha, C. A. R. Herdeiro, B. Kleihaus, J. Kunz, and E. Radu, Phys. Lett. B 768 , 373 (2017), arXiv:1701.00079 [gr-qc].</list_item>
<list_item><location><page_15><loc_52><loc_20><loc_92><loc_22></location>[28] F. Atamurotov, A. Abdujabbarov, and B. Ahmedov, Astrophys. Space Sci. 348 , 179 (2013).</list_item>
<list_item><location><page_15><loc_52><loc_17><loc_92><loc_19></location>[29] F. Atamurotov, A. Abdujabbarov, and B. Ahmedov, Phys. Rev. D 88 , 064004 (2013).</list_item>
<list_item><location><page_15><loc_52><loc_14><loc_92><loc_17></location>[30] M. Wang, S. Chen, and J. Jing, JCAP 10 , 051 (2017), arXiv:1707.09451 [gr-qc].</list_item>
<list_item><location><page_15><loc_52><loc_12><loc_92><loc_14></location>[31] U. Papnoi, F. Atamurotov, S. G. Ghosh, and B. Ahmedov, Phys. Rev. D 90 , 024073 (2014), arXiv:1407.0834 [gr-qc].</list_item>
<list_item><location><page_15><loc_52><loc_9><loc_92><loc_12></location>[32] A. Uniyal, S. Kanzi, and I. Sakallı, Eur. Phys. J. C 83 , 668 (2023), arXiv:2207.10122 [hep-th].</list_item>
<list_item><location><page_16><loc_9><loc_89><loc_49><loc_93></location>[33] A. Abdujabbarov, F. Atamurotov, N. Dadhich, B. Ahmedov, and Z. Stuchl'ık, Eur. Phys. J. C 75 , 399 (2015), arXiv:1508.00331 [gr-qc].</list_item>
<list_item><location><page_16><loc_9><loc_87><loc_49><loc_89></location>[34] M. Okyay and A. Ovgun, JCAP 01 , 009 (2022), arXiv:2108.07766 [gr-qc].</list_item>
<list_item><location><page_16><loc_9><loc_84><loc_49><loc_87></location>[35] A. Uniyal, R. C. Pantig, and A. Ovgun, Phys. Dark Univ. 40 , 101178 (2023), arXiv:2205.11072 [gr-qc].</list_item>
<list_item><location><page_16><loc_9><loc_81><loc_49><loc_84></location>[36] A. Uniyal, S. Chakrabarti, R. C. Pantig, and A. Ovgun, (2023), arXiv:2303.07174 [gr-qc].</list_item>
<list_item><location><page_16><loc_9><loc_77><loc_49><loc_81></location>[37] C. Liu, T. Zhu, Q. Wu, K. Jusufi, M. Jamil, M. Azreg-A¨ınou, and A. Wang, Phys. Rev. D 101 , 084001 (2020), [Erratum: Phys.Rev.D 103, 089902 (2021)], arXiv:2003.00477 [gr-qc].</list_item>
<list_item><location><page_16><loc_9><loc_75><loc_49><loc_77></location>[38] S. Devi, A. N. S., S. Chakrabarti, and B. R. Majhi, Phys. Dark Univ. 39 , 101173 (2023), arXiv:2105.11847 [gr-qc].</list_item>
<list_item><location><page_16><loc_9><loc_72><loc_49><loc_74></location>[39] M. Afrin, S. Vagnozzi, and S. G. Ghosh, Astrophys. J. 944 , 149 (2023), arXiv:2209.12584 [gr-qc].</list_item>
<list_item><location><page_16><loc_9><loc_69><loc_49><loc_72></location>[40] V. Perlick, O. Y. Tsupko, and G. S. Bisnovatyi-Kogan, Phys. Rev. D 97 , 104062 (2018), arXiv:1804.04898 [gr-qc].</list_item>
<list_item><location><page_16><loc_9><loc_67><loc_49><loc_69></location>[41] R. Roy and S. Chakrabarti, Phys. Rev. D 102 , 024059 (2020), arXiv:2003.14107 [gr-qc].</list_item>
<list_item><location><page_16><loc_9><loc_64><loc_49><loc_67></location>[42] F. Atamurotov and B. Ahmedov, Phys. Rev. D 92 , 084005 (2015), arXiv:1507.08131 [gr-qc].</list_item>
<list_item><location><page_16><loc_9><loc_60><loc_49><loc_64></location>[43] A. Abdujabbarov, B. Toshmatov, Z. Stuchl'ık, and B. Ahmedov, Int. J. Mod. Phys. D 26 , 1750051 (2016), arXiv:1512.05206 [gr-qc].</list_item>
<list_item><location><page_16><loc_9><loc_58><loc_49><loc_60></location>[44] V. Perlick and O. Y. Tsupko, Phys. Rept. 947 , 1 (2022), arXiv:2105.07101 [gr-qc].</list_item>
<list_item><location><page_16><loc_9><loc_52><loc_49><loc_57></location>[45] Y. Mizuno, Z. Younsi, C. M. Fromm, O. Porth, M. De Laurentis, H. Olivares, H. Falcke, M. Kramer, and L. Rezzolla, Nature Astron. 2 , 585 (2018), arXiv:1804.05812 [astroph.GA].</list_item>
<list_item><location><page_16><loc_9><loc_48><loc_49><loc_52></location>[46] J. Roder, A. Cruz-Osorio, C. M. Fromm, Y. Mizuno, Z. Younsi, and L. Rezzolla, Astron. Astrophys. 671 , A143 (2023), arXiv:2301.09549 [astro-ph.HE].</list_item>
<list_item><location><page_16><loc_9><loc_46><loc_49><loc_48></location>[47] C. Bambi and K. Freese, Phys. Rev. D 79 , 043002 (2009), arXiv:0812.1328 [astro-ph].</list_item>
<list_item><location><page_16><loc_9><loc_43><loc_49><loc_45></location>[48] N. Ortiz, O. Sarbach, and T. Zannias, Phys. Rev. D 92 , 044035 (2015), arXiv:1505.07017 [gr-qc].</list_item>
<list_item><location><page_16><loc_9><loc_39><loc_49><loc_43></location>[49] R. Shaikh, P. Kocherlakota, R. Narayan, and P. S. Joshi, Mon. Not. Roy. Astron. Soc. 482 , 52 (2019), arXiv:1802.08060 [astro-ph.HE].</list_item>
<list_item><location><page_16><loc_9><loc_36><loc_49><loc_39></location>[50] A. E. Broderick and R. Narayan, Astrophys. J. Lett. 638 , L21 (2006), arXiv:astro-ph/0512211.</list_item>
<list_item><location><page_16><loc_9><loc_34><loc_49><loc_36></location>[51] C. Bambi, Phys. Rev. D 87 , 107501 (2013), arXiv:1304.5691 [gr-qc].</list_item>
<list_item><location><page_16><loc_9><loc_31><loc_49><loc_34></location>[52] T. Ohgami and N. Sakai, Phys. Rev. D 91 , 124020 (2015), arXiv:1704.07065 [gr-qc].</list_item>
<list_item><location><page_16><loc_9><loc_28><loc_49><loc_31></location>[53] M. Azreg-A¨ınou, JCAP 07 , 037 (2015), arXiv:1412.8282 [grqc].</list_item>
<list_item><location><page_16><loc_9><loc_26><loc_49><loc_28></location>[54] P. G. Nedkova, V. K. Tinchev, and S. S. Yazadjiev, Phys. Rev. D 88 , 124019 (2013), arXiv:1307.7647 [gr-qc].</list_item>
<list_item><location><page_16><loc_9><loc_23><loc_49><loc_26></location>[55] K. A. Bronnikov, R. A. Konoplya, and T. D. Pappas, Phys. Rev. D 103 , 124062 (2021), arXiv:2102.10679 [gr-qc].</list_item>
<list_item><location><page_16><loc_9><loc_21><loc_49><loc_23></location>[56] A. Abdujabbarov, M. Amir, B. Ahmedov, and S. G. Ghosh, Astrophysics and Space Science 361 , 226 (2016).</list_item>
<list_item><location><page_16><loc_9><loc_19><loc_39><loc_20></location>[57] R. Shaikh, Phys. Rev. D 98 , 024044 (2018).</list_item>
<list_item><location><page_16><loc_9><loc_18><loc_49><loc_19></location>[58] E. Teo, Phys. Rev. D 58 , 024014 (1998), arXiv:gr-qc/9803098.</list_item>
<list_item><location><page_16><loc_9><loc_14><loc_49><loc_18></location>[59] J. G. Cramer, R. L. Forward, M. S. Morris, M. Visser, G. Benford, and G. A. Landis, Phys. Rev. D 51 , 3117 (1995), arXiv:astro-ph/9409051.</list_item>
<list_item><location><page_16><loc_9><loc_11><loc_49><loc_14></location>[60] K. K. Nandi, Y.-Z. Zhang, and A. V. Zakharov, Phys. Rev. D 74 , 024020 (2006), arXiv:gr-qc/0602062.</list_item>
<list_item><location><page_16><loc_9><loc_9><loc_49><loc_11></location>[61] K. Nakajima and H. Asada, Phys. Rev. D 85 , 107501 (2012), arXiv:1204.3710 [gr-qc].</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_16><loc_52><loc_91><loc_92><loc_93></location>[62] N. Tsukamoto, T. Harada, and K. Yajima, Phys. Rev. D 86 , 104062 (2012), arXiv:1207.0047 [gr-qc].</list_item>
<list_item><location><page_16><loc_52><loc_88><loc_92><loc_90></location>[63] N. Tsukamoto and T. Harada, Phys. Rev. D 95 , 024030 (2017), arXiv:1607.01120 [gr-qc].</list_item>
<list_item><location><page_16><loc_52><loc_85><loc_92><loc_88></location>[64] K. Jusufi and A. Ovgun, Phys. Rev. D 97 , 024042 (2018), arXiv:1708.06725 [gr-qc].</list_item>
<list_item><location><page_16><loc_52><loc_80><loc_92><loc_85></location>[65] A. Rogers, Monthly Notices of the Royal Astronomical Society 451 , 17 (2015), https://academic.oup.com/mnras/articlepdf/451/1/17/4170188/stv903.pdf.</list_item>
<list_item><location><page_16><loc_52><loc_77><loc_92><loc_80></location>[66] G. S. Bisnovatyi-Kogan and O. Y. Tsupko, Physical Review D 82 (2010), 10.1103/physrevd.82.084024.</list_item>
<list_item><location><page_16><loc_52><loc_73><loc_92><loc_77></location>[67] G. S. Bisnovatyi-Kogan and O. Y. Tsupko, 'Gravitational lensing in plasmic medium,' (2015), arXiv:1507.08545 [grqc].</list_item>
<list_item><location><page_16><loc_52><loc_71><loc_92><loc_73></location>[68] F. Atamurotov, S. Shaymatov, and B. Ahmedov, Galaxies 9 , 54 (2021).</list_item>
<list_item><location><page_16><loc_52><loc_68><loc_92><loc_71></location>[69] J. M. Bardeen, in Black Holes (Les Astres Occlus) (1973) pp. 215-239.</list_item>
<list_item><location><page_16><loc_52><loc_67><loc_83><loc_68></location>[70] B. Carter, Physical Review 174 , 1559 (1968).</list_item>
<list_item><location><page_16><loc_52><loc_64><loc_92><loc_67></location>[71] V. Perlick and O. Y. Tsupko, Phys. Rev. D 95 , 104003 (2017), arXiv:1702.08768 [gr-qc].</list_item>
<list_item><location><page_16><loc_52><loc_61><loc_92><loc_64></location>[72] B. Bezdekova, V. Perlick, and J. Bicak, J. Math. Phys. 63 , 092501 (2022), arXiv:2204.05593 [gr-qc].</list_item>
<list_item><location><page_16><loc_52><loc_59><loc_92><loc_61></location>[73] F. Rahaman, K. N. Singh, R. Shaikh, T. Manna, and S. Aktar, Classical and Quantum Gravity 38 , 215007 (2021).</list_item>
<list_item><location><page_16><loc_52><loc_56><loc_92><loc_59></location>[74] M. Visser, Lorentzian Wormholes: From Einstein to Hawking (AIP Press, 1996).</list_item>
<list_item><location><page_16><loc_52><loc_54><loc_92><loc_56></location>[75] P. G. Nedkova, V. Tinchev, and S. S. Yazadjiev, Phys. Rev. D 88 , 124019 (2013).</list_item>
<list_item><location><page_16><loc_52><loc_52><loc_92><loc_53></location>[76] E. Teo, General Relativity and Gravitation 35 , 1909 (2003).</list_item>
<list_item><location><page_16><loc_52><loc_48><loc_92><loc_52></location>[77] V. P. Frolov and I. D. Novikov, Black Hole Physics: Basic concepts and new developments (Kluwer Academic Publishers, 1998).</list_item>
<list_item><location><page_16><loc_52><loc_46><loc_92><loc_48></location>[78] S. Vzquez and E. Esteban, Il Nuovo Cimento B 119 , 489519 (2004).</list_item>
<list_item><location><page_16><loc_52><loc_44><loc_86><loc_45></location>[79] S. Shapiro, Astrophysical Journal 189 , 343 (1974).</list_item>
</unordered_list>
<section_header_level_1><location><page_17><loc_12><loc_92><loc_45><loc_93></location>Appendix A: Calculations for the deflection angle</section_header_level_1>
<text><location><page_17><loc_9><loc_86><loc_49><loc_90></location>Let us expand equation 66 for the homogeneous plasma with low plasma density in the context of a slowrotating wormhole, assuming r 0 /R < 1 .</text>
<formula><location><page_17><loc_13><loc_81><loc_49><loc_85></location>¯ α = ∫ ∞ -∞ Φ 0 dr + ∫ ∞ -∞ Φ 1 dr + ∫ ∞ -∞ Φ 2 dr, (A1)</formula>
<text><location><page_17><loc_9><loc_79><loc_32><loc_80></location>where the integrand is given as,</text>
<formula><location><page_17><loc_10><loc_57><loc_49><loc_77></location>Φ 0 = 2 R r √ r 2 -R 2 , Φ 1 = r 0 ( 2 r 2 + rR + R 2 ) r 2 ( r + R ) √ r 2 -R 2 + a 2 r 0 ( 4 r 2 +7 rR +9 R 2 ) +4 rR ( r + R ) rR 2 ( r + R ) 2 √ r 2 -R 2 , Φ 2 = ( 2 r 0 ( r + R ) √ r 2 -R 2 + a r 0 ( 4 r 2 +3 rR +17 R 2 ) +2 rR ( r + R ) rR 2 ( r + R ) 2 √ r 2 -R 2 ) k 0 R 2 . (A2)</formula>
<text><location><page_17><loc_9><loc_51><loc_49><loc_55></location>and, upon solving while neglecting higher-order terms, we get the deflection angle by Teo wormhole in uniform plasma space-time as</text>
<formula><location><page_17><loc_12><loc_47><loc_49><loc_50></location>¯ α = π + ( 3 r 0 R + 4 a R 2 + ( 2 r 0 k 0 R + 2 ak 0 R 2 )) . (A3)</formula>
<text><location><page_17><loc_9><loc_43><loc_49><loc_45></location>Similarly, in the case of non-homogeneous plasma distribution let us again expand equation 66 in the context</text>
<text><location><page_17><loc_52><loc_90><loc_92><loc_93></location>of a slow-rotating wormhole by assuming r 0 /R < 1 and ω P ( R ) 2 /ω 2 o < 1 ,</text>
<formula><location><page_17><loc_56><loc_86><loc_92><loc_89></location>¯ α = ∫ ∞ -∞ ψ 0 dr + ∫ ∞ -∞ ψ 1 dr + ∫ ∞ -∞ ψ 2 dr, (A4)</formula>
<text><location><page_17><loc_52><loc_83><loc_75><loc_84></location>where the integrand is given as,</text>
<formula><location><page_17><loc_53><loc_61><loc_92><loc_81></location>ψ 0 = 2 R r √ r 2 -R 2 , ψ 1 = r 0 ( 2 r 2 + rR + R 2 ) r 2 ( r + R ) √ r 2 -R 2 + a 2 r 0 ( 4 r 2 +7 rR +9 R 2 ) +4 rR ( r + R ) rR 2 ( r + R ) 2 √ r 2 -R 2 , ψ 2 = ( r 0 ( 2 r 2 +3 rR -R 2 ) -2 rR ( r + R ) 2 r 2 ( r + R ) √ r 2 -R 2 -a 2 ( r 0 ( 4 r 2 +3 rR +5 R 2 ) +2 rR ( r + R ) ) r 3 ( r + R ) 2 √ r 2 -R 2 ) k 0 R 2 . (A5)</formula>
<text><location><page_17><loc_52><loc_55><loc_92><loc_59></location>and, upon solving while neglecting higher-order terms, we get the deflection angle by Teo wormhole in radial plasma space-time as</text>
<formula><location><page_17><loc_59><loc_45><loc_92><loc_52></location>¯ α = π + ( 3 r 0 R + 4 a R 2 + r 0 a (9 π -14) R 3 + k r 2 R 2 ( r 0 (2 π -3) R -π ) ) . (A6)</formula>
</document>
[
{
"title": "Shadow and Weak Gravitational lensing of rotating traversable Wormhole in Non-homogeneous Plasma Space-time",
"content": "Saurabh Kumar, 1, ∗ Akhil Uniyal, 1, 2, † and Sayan Chakrabarti 1, ‡ 1 Department of Physics, Indian Institute of Technology, Guwahati 781039, India 2 Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shengrong Road 520, Shanghai, 201210, Peoples Republic of China (Dated: August 21, 2023) In this work, we have studied the behavior of null geodesics within a rotating wormhole space-time in non-magnetized pressure-less plasma. By focusing on the dispersion relation of the plasma and disregarding its direct gravitational effects, we examine how light rays traverse in the mentioned spacetime. A key highlight of the work is the necessity of a specific plasma distribution profile to establish a generalized Carter's constant, shedding light on the importance of this parameter. Furthermore, we have derived analytical formulas to distinguish the shadow boundary across various plasma profiles, uncovering a fascinating trend of diminishing shadow size as plasma density increases. Intriguingly, certain limits of the plasma parameters result in the complete disappearance of the shadow. When calculating the deflection angle by a wormhole in plasma space-time, we observe a distinct pattern: the angle decreases as the plasma parameter rises in non-homogeneous plasma space-time, diverging from the behavior observed in homogeneous plasma space-time. Also, leveraging observational data from M 87 ∗ , we establish constraints on the throat radius. Furthermore, minimum shadow diameters provide valuable constraints for the radial and latitudinal plasma parameters.",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "The concept of a wormhole, a hypothetical structure in space-time that connects different regions in spacetime, has been extensively explored since the early work of Einstein and Rosen [1]. Subsequent developments by Wheeler [2] and the pioneering work of Morris and Thorne [3] on traversable static wormholes further fueled interest in these intriguing cosmic constructs. In a later work, Teo extended this concept to include rotation in the wormhole geometry [4]. The existence of wormholes challenges energy conditions and requires the presence of exotic matter within the throat [3]. Consequently, their plausibility has been a subject of debate. Various proposals, such as the existence of a thin layer of negative energy density inside the throat [5] or the incorporation of modified gravity theories [6, 7] have been put forth to address these challenges. Given the potential formation of wormholes in the early universe and their existence subject to specific conditions, it is essential to investigate them further and discern their unique characteristics from other compact objects. In the literature, there are different methods that exist in order to get the axisymmetric wormhole solution. One such example is using the Ehlers transformations [8]. It is believed that the center of the galaxies including ours contains supermassive black holes however the existence of the black hole can only be justified by the existence of the event horizon. Therefore a number of tests have been proposed in order to confirm the presence of an event horizon in such compact objects [9-11]. In spite of all these proofs, a shred of conclusive evidence is still lacking [12]. It is worth mentioning that the existence of the event horizon along with the set of unstable light rings commonly known as the photon sphere in the exterior region of the compact object form the shadow of the object with the help of the radiation coming from the accretion disk around it [13]. Therefore, since the publication of the image of the M 87 ∗ supermassive black hole [14] and supermassive compact object at the center of our own galaxy known as Sagittarius A ∗ (Sgr A ∗ ) [15] by the Event Horizon Telescope (EHT), there has been extensive discussion among researchers regarding the nature of the object captured in the image including one of the papers by EHT [16]. However, first Synge [17] and Luminet [18] studied the Schwarzchild black hole shadow, and thereafter Bardeen [19] looked into the shadow of Kerr black hole. Consequently, the shadow in different geometrical backgrounds was studied in detail such as the Reissner-Nordstrom (RN) black hole [20], KerrNewman black hole [21], rotating regular black hole [22], Kerr black hole with scalar hair [23], regular black hole [24-26], Einstein-dilaton-Gauss-Bonnet black holes [27], Horava-Lifshitz black holes [28], non-Kerr black holes [29, 30], higher-dimensional black holes [31-33], black holes in theories of Non-linear electrodynamics [34-36], black holes in loop quantum gravity [37-39], black holes in the expanding Universe [40, 41], black holes in the presence of plasma [42, 43] etc. to name a few. It is important to understand that while the boundary of the shadow is only determined by the underlying space-time metric since it is formed only by the observed apparent shape of the photon sphere by the distant observer [44], the intensity map of the image is influenced by the accretion process around the compact object. Therefore, it is important to note that the presence of a shadow or a photon ring does not provide conclusive evidence that the object is a black hole. This also has been shown in the recent simulations by using the general relativistic magnetohydrodynamical (GRMHD) and general relativistic radiative transfer (GRRT) calculations that distinguishing the shadow image of the Kerr black hole and non-rotating dilaton black hole is almost impossible within the present observations [45, 46]. In support of the argument, a number of other compact objects have been studied where it has been shown that the horizonless compact object such as naked singularities [47-49], a hard surface [50], and non-rotating wormholes [51-53] and rotating traversable wormhole [54] can also cast the similar shadows. Along with this wormhole shadow also has been discussed in arbitrary metric theory of gravity with parameterizing the wormhole space-time [55]. Previous studies have extensively investigated the shadows of wormholes, discussing their similarities with the shadow of Kerr black hole [54, 56, 57]. However, one crucial aspect that has been overlooked in these studies is the presence of plasma and its effects on wormhole shadows. The effects of plasma on the shadows of rotating wormholes space-time have been explored in [56], however, this study did not consider the contribution of the wormhole throat [57]. Therefore, in this work, we will be studying the Teo class of rotating wormholes [58] in the presence of the plasma and will be taking care of the contribution coming from the throat. Other than the shadow, a lot of studies were performed on gravitational lensing for the wormhole without the plasma medium [59-64] and with the plasma medium [65, 66]. Furthermore, investigations into weak lensing in plasma space-time have not been limited to compact objects alone, as some researchers have employed galaxy models to study the effects of non-uniform plasma, revealing an increasing impact on the deflection angle [67, 68]. Our goal is to derive analytical expressions for the shadow boundary of the rotating wormhole in plasmafilled space-time for the observer situated at infinity, similar to Bardeen's calculation of the Kerr black hole shadow [69]. It is known that including the plasma potential in the Hamiltonian can affect the existence of Carter's constant, so one crucial aspect of our work will be to find the necessary condition for the existence of Carter's constant [70]. Such a condition has also been pointed out for Kerr black hole and also for generalized axis-symmetric static-spacetime [71, 72]. Another aim is to derive the deflection angle by a wormhole in homogeneous and non-homogeneous plasma space-time and analyze their impacts on the deflection angle. At last, our final goal will be to constrain the wormhole and plasma parameters using the EHT results of black hole shadows at the center of M87*. A similar approach and calculations have been taken in [73]. The paper is structured as follows: In Section II, we provide an overview of the Hamiltonian formalism for null geodesics in plasma space-time and discuss the necessary conditions for the existence of light rays in the outer communication of the Teo wormhole plasma space-time. Section III focuses on determining the specific forms of plasma profiles that satisfy the condition for the existence of Carter's constant. We derive the expressions for the null geodesic equations in this context. In Section IV, we delve into the role of the contribution of the wormhole throat to the wormhole shadow. We discuss the significance of the throat and derive the expressions for the celestial coordinates of the shadow boundary in generalized plasma space-time. Moving on to Section V, we explore specific plasma profiles that fulfill the separability condition. We also present a comparison of the shadows for various plasma densities of different plasma profiles. Section VI is dedicated to the calculation of the deflection angle of a rotating wormhole in weak field approximation for both homogeneous and non-homogeneous plasma space-time. Finally, in Section VII, we attempt to constrain the plasma parameters and throat size of the Teo wormhole using the observational data from M87*. Throughout the paper, we have considered units such that ℏ = G = c = M = 1 and our choice of signature is (-,+,+,+).",
"pages": [
1,
2
]
},
{
"title": "II. HAMILTON FORMALISM FOR LIGHT RAYS IN A PLASMA SPACE-TIME",
"content": "The Hamiltonian describing the light ray traveling in non-magnetized pressureless plasma is given as [71] here g µν are the contravariant components of the metric tensor and ω P represents the plasma electron frequency, which is defined as, where m and e are the mass and charge of the electron respectively while N ( x ) defines the electron density distribution. Here x represents the space-time coordinates ( t, r, θ, ϕ ) while p represents the momentum coordinates ( p t , p r , p θ , p ϕ ) for the light ray. Please note that the plasma frequency ( ω P ) and the photon frequency ( ω ) is related by a general form, where n is known as the refractive index, and it must be greater than 0 so that the light rays reach the observer [71]. Since the light rays reaching the observer are gravitationally redshifted, the observed redshifted frequency can be expressed in terms of the known constant of motion p t as The necessary and sufficient condition for the existence of a light ray with a constant of motion p t is derived for the Kerr black hole by Volker et. al [71] and based on a similar approach, a similar condition for generalized rotating metric can be given by Finally, the geodesics equations can be derived using Hamilton's equations, which are given as, For our analysis we have considered a stationary, axisymmetric rotating metric for Teo class traversable wormhole in the Boyer-Lindquist coordinates as [4], where r ≥ r 0 , r 0 is the throat radius of the wormhole. N , b 0 are known as the redshift factor and shape function respectively, K determines the proper radial distance which is given by R = rK and ω T is the measure for the angular velocity of the wormhole. N , b 0 , K and ω T are in general the functions of radial ( r ) and polar ( θ ) coordinates. For simplicity, in this work, we have only considered r dependency. Since the wormhole does not contain an event horizon, the metric component N ( r ) should be considered finite and non-zero throughout space-time. The shape function ( b 0 ) must satisfy the conditions ∂ θ b 0 | r = r 0 = 0 , ∂ r b 0 | r = r 0 < 1 and b 0 ≤ r [3] in order to have the geometry of a wormhole. The shape function also gives information about the mass of the wormhole [74] and the estimation of mass as M = r 0 / 2 is derived in great detail by Shaikh et. al [57]. In this work, we have considered the following form of the metric functions in order to get the traversable wormhole [4, 75]. where J is the angular momentum of the wormhole and M is the mass of the wormhole [57, 74]. We have used spin parameter of wormhole, a= J/M 2 which defines its rotation rate.",
"pages": [
2,
3
]
},
{
"title": "III. SEPARABILITY OF HAMILTON-JACOBI EQUATION FOR NULL GEODESICS IN PLASMA ON TEO WORMHOLESPACE-TIME",
"content": "The geodesic motion in rotating space-time enables two constants of motion the angular momentum of the particle about the axis of symmetry p ϕ and its energy p t due to the axisymmetric and stationary symmetries of the space-time. However, Carter et al. [70] showed that the geodesics in Kerr metric possess another constant of motion that governs the motion of geodesics in the latitudinal direction. Since the Kerr metric represents the rotating black hole space-time, Carter's constant should also exist in the rotating wormhole. This constant can be found using the method of separation of variables. Therefore, let's consider the Hamiltonian for the null geodesics as, with the separation ansatz where S r ( r ) and S θ ( θ ) are functions of r and θ coordinates respectively. Now substituting Eq. 10 into Eq. 9 will give, Nowconsidering p r = ∂ r S and p θ = ∂ θ S and solving the above equation for Teo rotating wormhole space-time (Eq. 7) will give, Since we are considering plasma frequency which depends on both radial ( r ) and polar ( θ ) coordinates, the above equation is only separable if the general form of plasma frequency is considered as, where Ω r ( r ) and Ω θ ( θ ) are r and θ dependent functions respectively. Therefore, Eq. 12 can be rearranged as, here expressions f r ( r ) and f θ ( θ ) are only the function of r and θ respectively and therefore can be considered as a constant since they are now separated by equality. This constant is known as Carter's constant and can be written as, Therefore, by using these three constants of motion p t , p ϕ , and Q , one can write the impact parameters such as [75], where we have considered p t = -ω o and p ϕ = L . Now solving for geodesics using Hamilton's Eqs. (6) for x µ = t, ϕ , we get: By calculating the expressions for p r and p θ using Eq. 14, we can calculate the remaining two geodesic Eqs. by solving Eqs 6 for x µ = r, θ and we get, where R ( r ) and Θ( θ ) are expressed as, Since the shadow is formed due to last photon rings which are unstable in nature, so in order to have unstable spherical orbits, null rays must satisfy the following criteria [71], where the first condition ensures the existence of spherical orbits around the wormhole while the second condition is imposed to get unstable spherical orbits. A similar calculation for the general axially symmetric stationary space-time case can also be found in [72]. where",
"pages": [
3,
4
]
},
{
"title": "IV. SHADOWOFTEOWORMHOLEINPLASMA SPACE-TIME",
"content": "Since photon orbits offer valuable insights into the optical appearance of wormholes, it would be insightful to study the boundary of the last photon ring in plasma space-time. In a non-rotating space-time, these orbits occur within the equatorial plane due to the spherical symmetry of the wormhole. However, in the case of rotating space-time, photon trajectories cross the equatorial plane repeatedly [76]. The Carter's constant which remains conserved in the latitudinal direction is crucial in order to determine the spherical orbits. The primary objective is to identify the last photon orbits that distinguish between light rays moving outward and those moving inward. To accomplish this, we rely on the determination of critical orbits characterized by their impact parameters: η and ξ . These parameters play a pivotal role in delineating the boundary of the shadow cast by the wormhole. Remarkably, the last photon orbits correspond to the most unstable circular orbits, featuring the maximum value of the effective potential, V eff . Well-established criteria can be applied to identify this unstable circular photon orbits [77]: where r c denotes the critical photon orbits and prime denotes the derivative with respect to r . The geodesic equation, Eq. 21 can be expressed as, where Hence, calculating the impact parameter with the help of Eqs. 26, we get, Since we have discussed that these critical orbits are crucial in determining the last photon rings, therefore, η and ξ can completely determine the boundary of the shadow however in order to look for the shadow in the observer's sky, we have used the following definitions of celestial coordinates [78]: and these celestial coordinates can be calculated with the help of impact parameters, η , and ξ by following the geodesic Eqs. derived in section III and given as, These expressions are not valid for calculating the celestial coordinates in homogeneous plasma space-time which will be discussed in the next section. Another crucial factor to consider in the case of the wormhole shadow is the existence of the extremum potential at the throat of the wormhole. It becomes apparent from Eq. 28 that the effective potential becomes zero at the throat when r = r 0 . This implies that stable or unstable spherical orbits may exist depending on the sign of the second derivative of the effective potential ( V '' eff ( r 0 ) ). To gain insights into the formation of the shadow, Fig. 1(a) showcases the effective potential for values of α equal to -2 (red), 8 (blue), and β equal to 0 . It has been observed that for positive values of α the potential exhibit two extrema however unstable orbits are located outside the throat. Consequently, the contribution to the shadow is solely derived from the outer region. On the other hand, for negative values of α , only one extremum is present at the throat. Please note that Fig. 1(a) is not solely responsible for the formation of complete shadow as shown in Fig. 1(b). The potential is for illustration purposes for showing the existence of extrema at the throat and outside the throat. It is noteworthy to point out that previous studies on the formation of wormhole shadows in plasma space-time failed to account for the contribution of the throat potential, despite the throat contributions being highlighted in rotating wormholes [57]. In our quest to understand the intricate interplay of factors contributing to shadow formation, we delve into the analysis of spherical photon orbits that satisfy constraints Eqs. 25. We solve the Eq. 39 for β = 0 along with satisfying the constraint equations to find out the minimum ( r min ) and maximum ( r max ) radius for the shadow formation. Therefore, the shadow will consist of the orbit formed by ( r min , r max ). However, it turns out that sometimes r min can be less than the throat radius r 0 . In such cases, the shadow will be formed by the orbits consisting of ( r 0 , r max ), highlighting the contribution of the throat. This nuanced differentiation ensures that we gain a comprehensive understanding of the shadow formation mechanism, considering the varying contributions from different regions of the wormhole's geometry. Since at the throat, the potential vanishes which also corresponds to the extremum of the potential (see Fig. 1(a)) therefore using Eq. 28, Celestial coordinates which are given by Eqs. 38 and 39 contribute to the incomplete shadow of the wormhole, as shown in the blue solid curve in Fig. 1(b). The remaining part of the shadow is contributed by the unstable orbits at the throat. Therefore from Eqs. 38 and 39, we can write, and using expressions of η from Eq. 38 and ξ from Eq. 41 into Eq. 40, we get, This contributes to the shadow which is shown in the red curve in Fig. 1(b), therefore the shadow will be the bounded region consisting of blue and red curves, indicated by the solid blue and solid red curves, respectively, while disregarding the dashed red portion. Here the extreme left point of the boundary of the shadow in the celestial plane is found by setting β = 0 in the expression 42 and using Ω r +Ω θ = r 2 K 2 ω 2 P from Eq. 13, we get, However, this expression is not valid for the homogeneous plasma distribution which will be discussed in the next section. The wormhole shadows in vacuum are shown in Fig. 2 for spin parameter a = 0 . 99 (left) and different inclination angles and similarly for fixed inclination angle θ = 90 · (right) and different spin for reference purposes.",
"pages": [
4,
5,
6
]
},
{
"title": "V. SHADOWFORSPECIFIC PLASMA PROFILES",
"content": "In this section, our focus shifts toward exploring the effects of commonly discussed plasma distribution profiles on the shadow of rotating wormhole space-time. A crucial criterion to consider is the satisfaction of the separability condition outlined in equation 13 while choosing the plasma distribution functions. Notably, Shapiro [79] made significant advancements in accretion studies involving black holes and determined that the plasma frequency is proportional to r -3 / 2 for pressureless plasma. It is imperative to acknowledge this radial decrease in plasma frequency when examining the dependence of plasma on θ , especially in the case of inhomogeneous plasma distributions. Furthermore, we must emphasize the importance of investigating a generalized form of plasma distribution. By doing so, we can highlight the distinguishing characteristics and disparities it holds when compared to other plasma distribution profiles. This comprehensive analysis enables us to gain a deeper understanding of the intricate relationship between plasma and the unique properties of rotating wormhole space-time.",
"pages": [
6
]
},
{
"title": "A. Homogeneous plasma distribution",
"content": "Firstly, we have considered the homogeneous plasma distribution between the observer and the source which is widely studied to understand the physical phenomena [66, 71], where k 0 denotes the homogeneous plasma parameter and it varies from 0 to 1 in order to satisfy the constraint Eq. 5. By using Eqs. 13 and 44 we can write the following expressions, Hence, the celestial coordinates for homogeneous plasma are given by solving Eqs. 36 and 37 as, and the contribution from the wormhole throat for homogeneous plasma space-time is given by while α L as mentioned in Sec IV is found by solving β = 0 , Eq. 46. In Fig. 3(a), we have plotted this case for spin a = 0 . 99 and it provides valuable insights into the behavior of the last photon ring, revealing that its radius expands in conjunction with larger homogeneous plasma parameters. This observation leads us to the inference that the universe is not filled with homogeneous plasma. If that were the case, we would have been able to detect these compact objects using low-resolution radio telescopes, given that the radius of the photon ring increases as the plasma parameter rises. Similar behavior has been observed for the lower spin values as well.",
"pages": [
6,
7
]
},
{
"title": "B. Radial plasma distribution",
"content": "For this analysis, we have specifically focused on the radial plasma profile, where Ω θ is set to zero. We adopted the plasma profile proposed by Shapiro et al. [79] to look at its effect on the rotating wormhole shadow as, where k r denotes the radial plasma parameter and its value should be in accordance with Eq. 5. We can calculate Ω r and Ω θ by using Eqs. 13 and 48 as, and the celestial coordinates for this plasma profile using Eqs. 38 and 39 are given by In this scenario, the contribution of the throat to the shadow can be determined using Eq. 42. We have demonstrated this case in Fig. 3(b) for spin parameter a = 0 . 99 which illustrates the shadow of the wormhole for various radial plasma parameters, with k r equal to 0 , 2 , 4 , 6 , and 8 . Notably, it becomes evident that the plasma density has a negative impact on the shadow, which contrasts with the behavior observed in the case of a homogeneous plasma distribution, as depicted in Fig. 3(a). As the plasma parameter increases, the shadow gradually becomes undetectable since the previously mentioned conditions (Eqs. 5 and 25) are no longer satisfied. These observations shed light on the intricate relationship between plasma density and the resulting shadow characteristics.",
"pages": [
7
]
},
{
"title": "C. Latitudinal plasma distribution",
"content": "Now, let's explore another example where the plasma distribution is dependent on the polar ( θ ) coordinate. In this scenario, we consider a distribution that exhibits a reduction in plasma density over increasing distances [79]. This choice is essential to distinguish it from a scenario involving homogeneous plasma. To illustrate this, let's denote the plasma distribution in this case as follows: where k θ represents the latitudinal plasma parameter, just to differentiate it from the radial plasma parameter, k r and it is chosen such that it satisfies the constraint conditions, Eq. 5. With the help of Eqs. 13 and 51, we can write the following expressions, and the celestial coordinates for this plasma profile using Eqs. 38 and 39 are given by Fig. 3(c) provides a comparative visualization of the shadow cast by the wormhole for this particular plasma profile. Remarkably, it becomes evident that the dependence of plasma density on the θ coordinate exerts a significant influence on the size of the shadow, surpassing the impact of the radial profile. This finding is particularly noteworthy, as previous studies primarily concentrated on radial profiles [56] and omitted the analysis of such latitudinal profiles. It underscores the importance of considering a generalized plasma density distribution to gain a more profound understanding of the shadow boundary in plasma space-time. Therefore, by incorporating the influence of plasma density variation with respect to θ , we can delve deeper into the intricacies of shadow formation and unravel more comprehensive insights into the behavior of wormholes in the presence of varying plasma distributions. Motivated by this, in the next subsection, we will be studying the more general case for plasma distribution.",
"pages": [
7,
8
]
},
{
"title": "D. Generalized plasma distribution",
"content": "Now, let's consider the more comprehensive scenario where the plasma distribution depends on both the radial coordinate ( r ) and the angular coordinate ( θ ). This broader analysis allows us to gain further insights into the effects of plasma densities on wormhole shadows. For this case, we denote the plasma distribution as follows, This particular profile is essentially a combination of the two previously discussed profiles. It incorporates the additive contributions from each of them. We adopt a similar plasma distribution profile to the one proposed by Perlick et al. [71], which provides valuable insights into the behavior of the plasma distribution in relation to the formation of wormhole shadows. Now, using Eqs. 13 and 54, and the celestial coordinates for this generalized plasma profile using Eqs. 38 and 39 are given by Fig. 3(d) showcases the shadow cast by the wormhole for different values of k r and k θ . Interestingly, the generalized plasma profile demonstrates superior performance compared to the other two profiles previously discussed. This emphasizes the significance of studying the more comprehensive generalized plasma profile rather than solely focusing on the radial profile. A similar kind of behavior has been observed for the low spin values. Hence, we have chosen to compare the different plasma density profiles with the higher spin case. Furthermore, it is noteworthy that the individual shadows resulting from the specific plasma parameters are larger when compared to their combined effect. As we delve deeper into the analysis, we observe that the shadow progressively diminishes in size with increasing plasma parameters. At a certain critical value, the shadow may eventually vanish or become undetectable altogether. This phenomenon has been shown in Fig. 4 for different values of spin parameters as well as at different inclination angles. It can be observed that as the plasma parameter increases, the wormhole shadow started shrinking and eventually disappears which has been shown by dashed green curves. Please note that the plasma parameter values corresponding to these green curves do not serve for maximum value after which the shadow gets disappear completely. By considering these generalized plasma distributions and closely examining the changes in the resulting shad- ows, we can potentially gain valuable insights into the plasma distribution along the observational path. This motivates us to further look into the photon trajectories in plasma space-time. Therefore, in the next section, we will be exploring the weak gravitational lensing within the effect of the plasma distribution around the rotating wormhole space-time.",
"pages": [
8,
9,
10
]
},
{
"title": "VI. WEAKGRAVITATIONAL LENSING",
"content": "In this section, we explore the influence of plasma distributions on the deflection angle within the framework of the weak field approximation. As we know, when light rays traverse the vicinity of massive objects, they experience deviations from their original paths. Here, we present the analytical expression for the deflection angle, focusing specifically on the case when the observer is situated in the equatorial plane ( θ = 90 · ) of the source. By examining the effects of plasma distributions on the deflection angle, we can gain a deeper understanding of how the presence of plasma affects the trajectory of light rays near massive objects. This analysis allows us to investigate the intricate interplay between plasma and gravity, shedding light on the nature of weak gravitational lensing in the presence of plasma. The derived analytic expression provides a valuable tool for predicting and analyzing the deflection of light in various astrophysical scenarios, contributing to our overall comprehension of the behavior of light in the presence of massive objects and plasma distributions. In order to analyze the deflection angle, we first calculate the geodesic equations with the help of Hamilton's equation 6 and the Hamiltonian for rotating plasma space-time given by Eq. 1 as, which can be further simplified as follows, Now, for the massless particles, the Hamiltonian should be zero ( H = 0 ). Therefore, Eq. 1 can be written by using the definitions such that p t = -ω o , and p θ = L as, Hence Eqs. 58 and 59 can be simplified to, and to simplify the above Eq. we have considered the following definitions, thus, equation 60 can be modified as, Since the deflection angle is calculated when the light deviates from its original path and consequently when it is at the closest approach ( r = R ) to the central object. Therefore, at the closet distance, we can define, Now, the evaluation needs to be done at r = R as mention in Eq. 63, we have considered the following expressions for the metric and plasma functions, being evaluated at r = R : Thus, the impact parameter, λ is calculated by using the expressions given in Eq. 64 with the help of Eqs. 62 and 63, and given as, and finally, the integral form for the deflection angle of the light from its original trajectory can be given by solving further using Eqs. 62 and 65 as, It is important to note that the deflection angle for the light following its original trajectory will be π given that the center of coordinates corresponds to the compact ob- ect. Therefore, the actual deflection angle is determined by α = ¯ α -π . In the subsequent analysis, we proceed to calculate Kr=0 the deflection angles for both homogeneous and nonhomogeneous plasma distributions. The homogeneous plasma distribution is characterized by uniform plasma density, while the radial plasma distribution exhibits a density variation in the non-homogeneous direction. By studying these specific cases, we can discern the effects of plasma distributions on the deflection of light and deepen our understanding of gravitational lensing phenomena in the presence of plasma. We have considered the following plasma distributions [66]. In the case of a homogeneous plasma distribution, the values of k 0 fall within the range of ( 0 , 1 ) as discussed in section V(A). Additionally, the choice of k r is determined to satisfy Eq. 5 which takes into account the gravitational redshift. It is worth noting that in previous studies [67, 68], the redshift condition has often been neglected. However, it is crucial to consider this condition as it significantly influences the trajectory of light and, consequently, the deflection angle. We have considered the weak field limit and lower plasma densities for simplicity to calculate the weak deflection angle for the slow-rotating wormhole. The detailed derivation has been performed in Appendix A. The resulting values for the deflection angle are presented in Table I, providing a comprehensive overview of the deflection angles for both the homogeneous and nonhomogeneous plasma profiles. By examining the values of the deflection angle, we can gain insights into the effects of plasma distributions on the path of light in the vicinity of massive objects. Therefore, we examine both the homogeneous and non-homogeneous plasma distributions around the rotating wormhole geometry and studied their effect on the deflection angle of the light rays. The deflection angle exhibits a decrease with the closest distance to the wormhole (See Fig. 5(a) and 5(b)), indicating a reduced gravitational influence. It is noteworthy that at higher plasma densities, the deflection angle increases, as illustrated in Fig. 6(a) for all values of the spin parameter. This observation gives validation of the earlier observed phenomenon such that the shadow radius increases with the plasma density in uniform plasma distribution (see Fig. 3(a)). In the case of a non-homogeneous plasma distribution, an intriguing observation is that the deflection angle decreases with increasing plasma densities, as shown in Figs. 5(b) and 6(b) for all values of the spin parameter. This stands in contrast to the homogeneous case and provides an explanation for the negative impact of plasma on the shadow which already has been observed in the case of the shadow (see Fig. 3(b)). Notably, the influence of non-homogeneous plasma distributions on the deflection angle has not been extensively explored in previous studies. Most investigations on the effects of plasma on the deflection angle by compact objects have focused on a single isothermal sphere model, commonly employed for galaxy modelling which yielded a positive impact of plasma on the deflection angle [68]. It is important to note that the choice of the plasma parameter value should ensure low plasma density and compliance with the condition given by Eq. 5. In previous studies, researchers have typically considered plasma parameter values ranging from 0 to 1 [68]. However, within the given impact parameter constraints along with the condition given by Eq. 5, a range of plasma parameter values can be chosen to study the deflection angle. Therefore, by analyzing the effects of plasma on the deflection angle, we can gain valuable insights into the distribution of plasma in the vicinity of compact objects. This investigation serves as a powerful tool for studying and understanding the properties of plasma surrounding these intriguing cosmic structures.",
"pages": [
10,
11,
12
]
},
{
"title": "VII. CONSTRAINING THE WORMHOLE SHADOW ANDPLASMAPARAMETERS",
"content": "To determine the plasma parameters and size of the wormhole, we employ observational data released by EHT [14] for the supermassive black hole located at the center of the elliptical galaxy Messier 87 (M87), also known as M 87 ∗ . By examining the average angular size of the shadow and its deviation from circularity, we can constrain shadow and plasma parameters. As the shadow possesses reflection symmetry around the α -axis in the celestial plane, we calculate its geometric center ( α 0 , β 0 ) using the integrals α 0 = 1 /A ∫ αdA and β 0 = 0 . Here, dA represents an area element. Next, we introduced an angle ϕ defined as the angle between the α -axis and the vector connecting the geometric center ( α c , β c ) with a point ( α, β ) on the boundary of the shadow. This angle ϕ provides valuable information for our analysis as can be seen as follows. Therefore, the average radius ( R ) of the shadow is given by [73], where l ( ϕ ) = √ ( α ( ϕ ) -α 0 ) 2 + β ( ϕ ) 2 and ϕ = tan -1 ( β ( ϕ ) / ( α ( ϕ ) -α 0 )) . Following [14], we define the deviation ∆ C from circularity as [73], We should note that ∆ C represents the fractional root mean square distance from the average radius of the observed shadow. Based on the findings of the EHT collaboration [14], the angular size of the observed shadow is determined to be ∆ θ sh = 42 ± 3 µ as, with a deviation ∆ C of less than 10% . Additionally, following the same study [14], we adopt the distance to M87* as D = (16 . 8 ± 0 . 8) Mpc and the mass of the object as M = (6 . 5 ± 0 . 7) × 10 9 M ⊙ . With these values, we can estimate the average diameter of the shadow by considering the angular size, This uncertainty in the shadow diameter can be determined by carefully propagating the uncertainties asso- ciated with both the distance and angular size measurements. By accounting for these uncertainties, we can obtain more reliable estimates of the shadow's diameter, enabling us to extract valuable information about the physical characteristics of M 87 ∗ and delve into the intricacies of the shadow phenomenon. Therefore, these insights pave the way for further investigations and contribute to our ongoing exploration of the enigmatic nature of M 87 ∗ and its surrounding environment. The determination of the average diameter and the deviation from the circularity of the observed shadow involves considering the errors in a combined manner, with the uncertainties added in quadrature. It is important to ensure that this calculated quantity matches the expected value of the diameter (2R). Fig. 7 presents the results depicting the average diameter (left) and circularity deviation (right) of the shadow, considering various values for the spin parameter and the size of the wormhole throat concerning the radial plasma parameter ( k r ). In our analysis, we have taken an inclination angle of θ = 17 · , which represents the angle between the jet axis and the line of sight to M 87 ∗ . Additionally, based on the findings of the EHT collaboration, the spin parameter falls within the range of 0 . 5 ≤ a ≤ 0 . 94 which has been shown in the Fig. 7 with vertical red lines. Now, to determine the maximum plasma parameters that apply to M 87 ∗ , we analyzed the plasma density corresponding to the smallest observed shadow size. Initially, by looking only at the radial plasma profile, we observed that the maximum value for the radial plasma parameter is k rc = 4 . 75 when the shadow diameter matches with the observed M 87 ∗ diameter (Fig. 7). Similarly, the contour line corresponding to the lowest shadow diameter Fig. (8) indicates that the maximum longitudinal plasma parameter is k θc = 85 . 5 . furthermore, to cover the complete range of shadow diameters between 9 . 5 and 12 . 5 with the uncertainty in the M 87 ∗ shadow diameter, we considered k r = 3 . 5 and k θ = 20 . 4 in Fig. 9. These choices allow us to place constraints on the maximum possible value of the throat size ( r 0 c = 2 . 51 ) which corresponds to the maximum shadow diameter as shown in Fig. 9(a). It is worth noting that the deviation from circularity ( ∆ C ) in the wormhole shadows provides interesting features about the plasma parameters. From Figs. 7(b) and 8(b), we observed that ∆ C ≤ 10% required the spin range for M 87 ∗ to be less than 0 . 5 when considering either the radial or latitudinal plasma profiles. However, in the case of the generalized plasma distribution the observed circularity deviation from the shadow of M 87 ∗ ( ∆ C ≤ 10% ) can still be achieved for the known spin range while obtaining shadow diameters within the observed range for M 87 ∗ . It is important to note that this analysis restricts the throat size such that it should not exceed the value of r 0 c to maintain ∆ C < 10% and maximum shadow diameter of 12.5 within the allowed range of spin, the value of r 0 c should be less than 2.51 for the chosen value of k r and k θ in case of generalized plasma distributions (See Fig. 9(a) and 9(b)). Thus, the constraint on the maximum value of throat size may vary depending on the choice of plasma parameters. These findings further contribute to our understanding of the possible spin range and plasma distributions associated with the observed shadow of M 87 ∗ . Please note that we have only considered those plasma profiles which have been discussed in Section V. Therefore, this study ascertains the range of plasma parameters and throat size that are consistent with the observed shadows and provides further insights into the properties of the M 87 ∗ system.",
"pages": [
12,
13,
14
]
},
{
"title": "VIII. CONCLUSION",
"content": "In this study, our focus was on investigating the behavior of null geodesics in non-magnetized pressureless plasma within the context of a rotating wormhole spacetime. We specifically examined the gravitational influence of the wormhole while neglecting the direct gravitational effects of the plasma. Instead, we considered only the dispersive properties of the plasma, affecting the trajectory of light rays. One key finding of our work, as discussed in Section III, is the requirement of a specific plasma distribution profile to establish a generalized Carter's constant. We also emphasized the importance of including potential contributions from both inside and outside the wormhole throat, as elaborated in Section IV. Furthermore, we derived analytical formulas for the boundary of the shadow for various plasma profiles in Section V. Notably, our results revealed that the shadow size decreases with increasing plasma density. Eventually, for certain upper limits of the plasma parameters, the shadow completely disappears. Our primary objective throughout this study was to obtain an analytical expression for the shadows observed in plasma space-time. By investigating the behavior of light rays in the presence of plasma, we aimed to enhance our understanding of the intricate interplay between gravitational and plasma effects in astrophysical phenomena. In Section VI, we conducted calculations to determine the deflection angle on a rotating wormhole in plasma space-time. Gravitational lensing phenomena have significant implications for astrophysical observations, and our study shed light on the impact of plasma on the deflection angle. Interestingly, we observed that as the plasma parameter increases, the deflection angle decreases in a non-homogeneous plasma space-time, contrary to the behavior observed in a homogeneous plasma profile. This intriguing result underscores the importance of further investigating the observational aspects and exploring the plasma distribution near compact objects. Finally, we proceeded to constrain the size of the throat and plasma parameters mentioned in Section V by utilizing the observational data coming from EHT for M 87 ∗ . Our analysis revealed that the maximum allowed throat radius is determined to be r 0 c = 2 . 51 , which corresponds to the allowed range of shadow diameter, spin, and circularity deviation for M 87 ∗ as reported by EHT. On the other hand, by considering a minimum shadow diameter of 9 . 5 , we were able to place constraints on the radial and latitudinal plasma parameters, with maximum values k r = 4 . 75 and k θ = 85 . 5 , respectively. These constraints provide valuable insights into the physical properties of the wormhole and the plasma surrounding it. By examining the maximum and minimum shadow diameters, we can better understand the range of possible sizes for the throat and the corresponding plasma parameters that are consistent with the observed shadows in the case of M 87 ∗ . In our future research, we intend to investigate the impact of plasma on the shadow of a Kerr black hole. Additionally, we plan to compare the findings from the study of wormhole shadows to those of black hole shadows. This comparative analysis will provide further insights and potentially help discern whether M 87 ∗ is more likely to be a black hole or a wormhole. By delving into these investigations, we hope to contribute to the ongoing understanding of M 87 ∗ and its intriguing nature, paving the way for deeper insights into the astrophysical phenomena occurring in the vicinity of these enigmatic cosmic objects.",
"pages": [
14,
15
]
},
{
"title": "ACKNOWLEDGEMENT",
"content": "The work of SC is supported by Mathematical Research Impact Centric Support (MATRICS) from the Science and Engineering Research Board (SERB) of India through grant MTR/2022/000318.",
"pages": [
15
]
},
{
"title": "Appendix A: Calculations for the deflection angle",
"content": "Let us expand equation 66 for the homogeneous plasma with low plasma density in the context of a slowrotating wormhole, assuming r 0 /R < 1 . where the integrand is given as, and, upon solving while neglecting higher-order terms, we get the deflection angle by Teo wormhole in uniform plasma space-time as Similarly, in the case of non-homogeneous plasma distribution let us again expand equation 66 in the context of a slow-rotating wormhole by assuming r 0 /R < 1 and ω P ( R ) 2 /ω 2 o < 1 , where the integrand is given as, and, upon solving while neglecting higher-order terms, we get the deflection angle by Teo wormhole in radial plasma space-time as",
"pages": [
17
]
}
]
2024PhRvD.109k5026S
https://arxiv.org/pdf/2402.06416.pdf
<document>
<section_header_level_1><location><page_1><loc_20><loc_92><loc_80><loc_93></location>Reactor neutrino background in third-generation dark matter detectors</section_header_level_1>
<text><location><page_1><loc_24><loc_89><loc_76><loc_90></location>D. Aristizabal Sierra, 1, ∗ Valentina De Romeri, 2, † and Christoph A. Ternes 3, ‡</text>
<text><location><page_1><loc_29><loc_88><loc_29><loc_88></location>1</text>
<text><location><page_1><loc_29><loc_87><loc_72><loc_88></location>Universidad T'ecnica Federico Santa Mar'ıa - Departamento de F'ısica</text>
<text><location><page_1><loc_34><loc_86><loc_67><loc_87></location>Casilla 110-V, Avda. Espa˜na 1680, Valpara'ıso, Chile</text>
<text><location><page_1><loc_31><loc_84><loc_70><loc_86></location>2 Instituto de F'ısica Corpuscular (CSIC-Universitat de Val'encia),</text>
<text><location><page_1><loc_25><loc_83><loc_76><loc_84></location>Parc Cient'ıfic UV C/ Catedr'atico Jos'e Beltr'an, 2 E-46980 Paterna (Valencia) - Spain</text>
<text><location><page_1><loc_25><loc_82><loc_76><loc_83></location>3 Istituto Nazionale di Fisica Nucleare (INFN), Laboratori Nazionali del Gran Sasso,</text>
<text><location><page_1><loc_39><loc_81><loc_61><loc_81></location>67100 Assergi, L'Aquila (AQ), Italy</text>
<text><location><page_1><loc_18><loc_61><loc_83><loc_79></location>Third-generation dark matter detectors will be fully sensitive to the boron-8 solar neutrino flux. Because of this, the characterization of such a background has been the subject of extensive analyses over the last few years. In contrast, little is known about the impact of reactor neutrinos. In this letter we report on the implications of such a flux for dark matter direct detection searches. We consider five potential detector deployment sites envisioned by the recently established XLZD consortium: SURF, SNOLAB, Kamioka, LNGS and Boulby. By using public reactor data we construct five reactor clusters-involving about 100 currently operating commercial nuclear reactors each-and determine the net neutrino flux at each detector site. Assuming a xenon-based detector and a 50 tonne-year exposure, we show that in all cases the neutrino event rate may be sizable, depending on energy recoil thresholds. Of all possible detector sites, SURF and LNGS are those with the smallest reactor neutrino background. On the contrary, SNOLAB and Boulby are subject to the strongest reactor neutrino fluxes, with Kamioka being subject to a more moderate background. Our findings demonstrate that reactor neutrino fluxes should be taken into account in the next round of dark matter searches. We argue that this background may be particularly relevant for directional detectors, provided they meet the requirements we have employed in this analysis.</text>
<section_header_level_1><location><page_1><loc_23><loc_57><loc_35><loc_58></location>INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_31><loc_49><loc_55></location>A wealth of cosmological and astrophysical data supports the idea that the dominant form of matter in the Universe has feeble or none electromagnetic interactions. The conventional wisdom is that this new form of matter-dubbed dark matter (DM)-is of microscopical origin and its abundance is determined by fast-scattering processes with Standard Model (SM) particles at very early epochs, much before the onset of cosmic neutrino decoupling and primordial nucleosynthesis (for a review see e.g. Ref. [1]). Although at high temperatures DMis thermalized, as the temperature decreases-because of the expansion of the Universe -these scattering processes are unable to keep the species in thermodynamic equilibrium and so its abundance freezes out. This weakly interacting massive particle (WIMP) is a rather generic candidate appearing in a large class of particle physics models. It is a dominant paradigm that has driven DM searches.</text>
<text><location><page_1><loc_9><loc_8><loc_49><loc_31></location>DM direct detection is a subject that dates back to the mid 80's, when Goodman and Witten pointed out that WIMPs could be searched for by using the same detectors proposed by Drukier and Stodolsky for coherent elastic neutrino-nucleus scattering (CE ν NS) measurements [2, 3]. Since then, and because of the lack of a signal, detector technologies as well as fiducial volumes have dramatically evolved. At present, DMsearches in direct detection experiments are led by liquid xenon (LXe) dual-phase time projection chambers (secondgeneration DM detectors). Detectors at the INFN 'Laboratori Nazionali del Gran Sasso' (LNGS) in Italy (XENONnT), at the Sanford Underground Research Facility (SURF) in South Dakota in the US (LZ) and at the China Jinping Underground Laboratory in Sichuan, China (PandaX-4T) are using active volumes of the order of 5 tonne [4-6].</text>
<text><location><page_1><loc_52><loc_44><loc_92><loc_58></location>With their high capabilities for background rejection, along with low nuclear recoil energy thresholds, these secondgeneration DM detectors are sensitive to spin-independent WIMP-nucleon total cross sections of the order of 10 -48 cm 2 [7]. Indeed, XENONnT and LZ have recently published results where sensitivities of the order of σ WIMP-nuc ∼ 10 -47 cm 2 have been reported [4, 5]. PandaX-4T has set the most stringent upper limit in the low WIMP mass region ( ≲ 10 GeV), σ WIMP-nuc ∼ 10 -44 cm 2 [6].</text>
<text><location><page_1><loc_52><loc_30><loc_92><loc_45></location>A new generation of LXe detectors-third-generation DM detectors-is expected to pave the way for a discovery 1 . Recently the XENONnT, LZ and DARWIN collaborations have united forces and created the XLZD consortium [8]. Their goal is the construction of a 40-100 tonne detector with unprecedented sensitivities. With such active volume, a detector of this kind will be subject to an irreducible neutrino background dominated by 8 B solar neutrinos (for nuclearchannel signals) and by pp neutrinos (for electron-channel signals) [9].</text>
<text><location><page_1><loc_52><loc_16><loc_92><loc_30></location>The morphology and size of this background have been the subject of different analyses in recent years, first identified as the so-called 'neutrino floor' [9-13] and its more recent redefinition, the 'neutrino fog' [14], where a first estimation of the reactor neutrino background at LNGS was addressed. It is well known that the impact of the neutrino background on a WIMP discovery signal is mainly dominated by neutrino flux uncertainties, with uncertainties on the weak mixing angle and on the root-mean-square radii of the neutron distribu-</text>
<text><location><page_2><loc_9><loc_75><loc_49><loc_93></location>tions playing a rather subdominant role [15]. The presence of a neutrino background, however, does not mean that an identification of a WIMP signal is impossible. First of all, improvements in the determination of solar neutrino flux uncertainties are expected. Secondly, WIMP and neutrino spectra in general do not fully degenerate in most regions of parameter space. Even in regions where they strongly do, an identification is possible with sufficiently large data sets [12]. Furthermore, even if data is not abundant, directionality willpotentially-enable a distinction between WIMP and neutrino nuclear recoil spectra [16], if they turn out to be strongly degenerate.</text>
<text><location><page_2><loc_9><loc_56><loc_49><loc_74></location>Given this landscape, and the fact that DM direct detection will soon enter the third-generation detector phase, one should wonder whether other neutrino sources might contribute to the background and hence should be taken into account. This is a rather relevant question to raise, aiming to leverage the full discovery power of these types of detectors. Motivated by this question, in this Letter we assess the impact of nuclear reactor neutrinos. Since the reactor neutrino flux strongly depends on the geographical position of the detector-for definitiveness-we use LNGS, SURF, Boulby (UK), Kamioka (Japan) and SNOLAB (Canada) as possible deployment sites 2 .</text>
<section_header_level_1><location><page_2><loc_11><loc_44><loc_46><loc_47></location>NUCLEAR REACTOR SOURCES: LOCATIONS AND EVENT RATES</section_header_level_1>
<text><location><page_2><loc_9><loc_20><loc_49><loc_41></location>The data sets we employ follow from data provided on the Geoneutrinos.org website [17, 18]. We consider only commercial power plants (that involve the most powerful reactors) for which a non-zero operating power is reported. Reactors for which the thermal capacity is known but have zero operating power and those that have been permanently shut down are not included. Depending on the baseline, each detector site that we consider is 'surrounded' by a cluster of nuclear reactor power plants, at a certain distance Li . Table I shows the minimum and maximum baseline and power for each cluster, along with the number of reactors involved. For each detector site, we do not include reactors located at distances beyond L max, as their contribution to the event rates would be negligible.</text>
<table>
<location><page_2><loc_53><loc_83><loc_91><loc_94></location>
<caption>TABLE I. Minimum and maximum baselines ( L min and L min ) along with minimum and maximum reactor powers ( P min and P max) for the SURF, SNOLAB, Kamioka, LNGS and Boulby reactor clusters. The number of reactors in each cluster (NR) is also shown. Data has been extracted from the Geoneutrinos.org website.</caption>
</table>
<text><location><page_2><loc_52><loc_48><loc_92><loc_73></location>The largest clusters are those around the LNGS and Boulby detector sites (as expected, given that for these two cases the radius defining the cluster exceeds by about 1000 km the radius at the other sites). However, this does not necessarily mean that the largest flux is obtained for these two positions, as we now discuss. The reactor neutrino flux decreases rapidly with increasing baseline. So, a rather fair assumption is that the flux is dominated by the sub-cluster defined by all reactors included in a radius ≲ 1000 km. For the SURF and LNGS locations one finds that these sub-clusters involve only 5 reactors with a 2.1 GW and 1.8 GW average power, respectively. For the Kamioka, SNOLAB and Boulby locations, the subclusters are composed instead of 35, 59 and 49 reactors. The average power in each case (and in that order) is: 2.1 GW, 4.9 GW and 1.9 GW. Thus, already from these numbers one expects the SURF and LNGS location sites to involve a less intense reactor neutrino flux.</text>
<text><location><page_2><loc_52><loc_25><loc_92><loc_47></location>Fig. 1 shows the distribution of nuclear reactors in terms of baseline and power for the five different clusters we consider. The distributions involve the full data sets. From the graph, one can see that for the Boulby and SNOLAB clusters the reactor density for baselines below 1000 km is high, with a few of those reactors having powers above 3 GW. The distribution for the Kamioka cluster is somewhat different. Although below 1000 km there are a few reactors, their density is lower as well as their power. For the SURF and LNGS clusters, the reactor density for baselines below 1000 km is, instead, rather moderate. For these clusters, most reactors are at baselines above 1000 km. So, even without a dedicated calculation of the event rate, expectations are that in terms of increasing reactor neutrino fluxes the clusters can be sorted into three groups: SURF/LNGS, Kamioka, SNOLAB/Boulby.</text>
<text><location><page_2><loc_52><loc_19><loc_92><loc_25></location>The calculation of the differential nuclear recoil spectrum at each cluster (C) requires the convolution of the differential CE ν NS cross section [3, 19] with the reactor neutrino flux, namely</text>
<formula><location><page_2><loc_54><loc_14><loc_92><loc_18></location>dR C dEr = m det NA T η C m Xe mol ∫ E max ν E min ν d Φ ν e dE ν d σ dEr F 2 H ( Er ) dE ν . (1)</formula>
<text><location><page_2><loc_52><loc_9><loc_92><loc_13></location>Here, m det refers to the detector active volume mass, m Xe mol to the xenon molar mass, T to the exposure time, E min ν = √ mNEr / 2 ( Er and mN refer to nuclear recoil energy and</text>
<figure>
<location><page_3><loc_29><loc_69><loc_72><loc_93></location>
<caption>FIG. 1. Location of the different reactors within the SURF, SNOLAB, Kamioka, LNGS and Boulby clusters and their corresponding operating power.</caption>
</figure>
<text><location><page_3><loc_9><loc_48><loc_49><loc_61></location>mass), and E max ν to the neutrino spectrum kinematic 'highenergy' tail taken at 8MeV. The average nuclear mass is ⟨ m Xe ⟩ / GeV = 0 . 93 ×⟨ A ⟩ , ⟨ A ⟩ = ∑ i XiAi = 131 . 4 being the mass number averaged over the nine stable xenon isotopes. We include-for completeness-the weak-charge nuclear form factor, F H ( Er ) , parametrized 'a la Helm [20]. Note that if not included results would deviate from those presented here at most by ∼ 2%, because of the process occurring deep in the full coherent regime.</text>
<text><location><page_3><loc_9><loc_40><loc_49><loc_47></location>Regarding the electron antineutrino spectrum, we proceed as follows. For the 235 U and 238 U emission spectra we use results from Ref. [21]. For 239 Pu and 241 Pu we use instead results from Ref. [22]. The full electron antineutrino differential flux is then calculated according to</text>
<formula><location><page_3><loc_20><loc_35><loc_49><loc_39></location>d Φ ν e dE ν = ∑ i = Isotopes fi d Φ i ν e dE ν , (2)</formula>
<text><location><page_3><loc_9><loc_16><loc_49><loc_34></location>where fi = { f 235 U , f 238 U , f 239 Pu , f 241 Pu } = { 5 . 5 , 0 . 7 , 3 . 2 , 0 . 6 }× 10 -1 are the uranium and plutonium fission fractions [23]. Note that we do not include electron antineutrinos produced in neutron capture by 238 U. The reason is that the spectra for those neutrinos dominate at energies below ∼ 1 . 5 MeV, hence in a LXe detector would produce nuclear recoils below 0.04 keV (much below any realistic operation threshold). We assume the spectral function in Eq. (2) to be universal for all the reactors within the clusters 3 . Thus, the difference among clusters is determined only by the normalization factor, which we calculate assuming that in each fission process an energy of ε = 205 . 24 MeV is released and that neutrinos are emitted</text>
<text><location><page_3><loc_52><loc_60><loc_92><loc_61></location>isotropically. Explicitly, each normalization factor is given by</text>
<formula><location><page_3><loc_66><loc_56><loc_92><loc_59></location>η C = ∑ j Pj 4 π L 2 j ε , (3)</formula>
<text><location><page_3><loc_52><loc_48><loc_92><loc_55></location>where j runs over all reactors relevant for cluster C and Pj and Lj are the operating power and distance for reactor j . Their values are displayed in Tab. II, showing that SURF is subject to the least abundant neutrino flux, whereas Boulby to the most severe.</text>
<table>
<location><page_3><loc_53><loc_43><loc_90><loc_47></location>
<caption>TABLE II. Neutrino flux normalization factors for the five reactor clusters.</caption>
</table>
<text><location><page_3><loc_52><loc_15><loc_92><loc_37></location>With these results at hand, we are now in a position to calculate the differential event rate as well as the total event rate for each detector site. We assume a 50-tonne active volume LXe detector and 100% efficiency 4 . Since current realistic thresholds amount to 0.3 keV [25], we use E th,min r = 0 . 1keV as a value envisioned for future detector operations. Results are displayed in Fig. 2. The left (right) graph shows the differential event rate (total event rate) as a function of the recoil energy (recoil energy threshold) for the five different reactor clusters we have considered. The inset plot in the right panel is meant to zoom in on the bottom left corner. Inline with expectations, the differential and total event rates at the SURF (Boulby) detector site are the smallest (largest). The event rate at the LNGS detector location is slightly higher, followed by Kamioka and SNOLAB.</text>
<figure>
<location><page_4><loc_9><loc_70><loc_50><loc_93></location>
</figure>
<figure>
<location><page_4><loc_51><loc_70><loc_91><loc_93></location>
<caption>FIG. 2. Left graph : Reactor neutrino differential event rate for the five detector sites considered in this work: SURF, SNOLAB, Kamioka, LNGS, and Boulby as a function of nuclear recoil energy. Shown as well is the 8 B differential event rate. Right graph : Reactor neutrino total event rate for the same detector locations.</caption>
</figure>
<section_header_level_1><location><page_4><loc_24><loc_60><loc_33><loc_61></location>DISCUSSION</section_header_level_1>
<text><location><page_4><loc_9><loc_46><loc_49><loc_57></location>Naively one would expect the reactor neutrino flux to be suppressed and of little relevance. This expectation is mainly based on the fact that most reactors are far away from the detector sites. However, the fact that the clusters around each detector site involve a large number of active nuclear power plants (with in some cases powerful reactors), combined with a large active volume produces a non-zero event rate in all cases.</text>
<text><location><page_4><loc_9><loc_29><loc_49><loc_45></location>Ideally one would like a very low threshold to explore the small WIMP mass window and increase the WIMP-nucleus event rate. At 0.1 keV, we find that the total neutrino-nucleus event rate per year is: 16 (SURF), 44 (LNGS), 82 (Kamioka), 124 (SNOLAB) and 733 (Boulby). If that operation threshold is not achieved and instead the detector is operated at 0.3 keV, these numbers will be degraded by about a factor 7. In such an experimental scenario the reactor neutrino background becomes, of course, less severe. Thus, the question of whether the reactor neutrino background matters is-as anticipatedstrongly linked to operation thresholds.</text>
<text><location><page_4><loc_9><loc_18><loc_49><loc_28></location>It is worth emphasizing that variations of these estimated numbers are expected in the future, depending on the exact number of reactors that enter in either operation phase or are decommissioned. However, these results demonstrate that the reactor neutrino flux should be seriously taken into account in decision making as well as in data taken, contrary to expectations.</text>
<text><location><page_4><loc_9><loc_9><loc_49><loc_17></location>Finally, one might wonder how much this neutrino background matters compared to the boron-8 solar neutrino flux. For the detector configurations we have considered, with a 0.1 keV operation threshold, the number of boron-8 nuclear recoil induced events is overwhelming, 36500 events/year. So, of course, this will be the dominant background source. All</text>
<text><location><page_4><loc_52><loc_55><loc_92><loc_61></location>the efforts to understand the morphology of this background are indeed motivated by this fact. The question is then whether one should be concerned with the reactor neutrino background whatsoever.</text>
<text><location><page_4><loc_52><loc_36><loc_92><loc_55></location>It is well known that the boron-8 background can be to a certain degree circumvented. As we have already stressed, large data sets might enable differentiating neutrino from WIMP signals, if the WIMP parameters are such that the neutrino and WIMP event rates strongly degenerate. In general, however, directional detectors seem to be the most promising avenue [16, 26] 5 . For these detectors it seems that the reactor neutrino background might even become the most dominant background source. Therefore, if the boron-8 nuclear recoilinduced events can be efficiently discriminated, there will be yet another background source that will require careful identification and proper treatment, depending on statistics and operation capabilities.</text>
<section_header_level_1><location><page_4><loc_66><loc_32><loc_77><loc_33></location>CONCLUSIONS</section_header_level_1>
<text><location><page_4><loc_52><loc_18><loc_92><loc_30></location>With the advent of third-generation DM direct detection detectors, the quantification of reactor neutrino fluxes becomes of pivotal importance. In this work we have quantified the size of the neutrino flux produced by clusters of reactors surrounding five potential detector deployment sites. For definitiveness we have considered the locations envisioned by the recently established XLZD consortium: SURF, SNOLAB, Kamioka, LNGS, and Boulby.</text>
<text><location><page_4><loc_52><loc_15><loc_92><loc_17></location>Our findings show that detectors with active volumes of the order of 50 tonne and recoil energy thresholds of the or-</text>
<text><location><page_5><loc_9><loc_81><loc_49><loc_93></location>der of 0.1 keV, will be sensitive to a certain amount of reactor neutrino-induced events. The exact amount depends, to a large degree, on the energy threshold at which the detector is operated. However, even assuming a realistic threshold of 0.3 keV, the event rate turns out to be sizable in all cases. We find that the site with the smallest reactor neutrino background is SURF followed by LNGS, Kamioka, SNOLAB, and Boulby (in that order).</text>
<text><location><page_5><loc_9><loc_68><loc_49><loc_81></location>Although subdominant compared to the solar boron-8 neutrino background, we point out that the reactor neutrino background (and its corresponding events) should be-in principle-considered during data taken. Reactor neutrinoinduced events should be taken into account in background discrimination, regardless of the detector technique employed. This result will be particularly relevant for directional detection, if future detectors meet the requirements we have used here.</text>
<section_header_level_1><location><page_5><loc_21><loc_63><loc_37><loc_64></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_5><loc_9><loc_44><loc_48><loc_60></location>We thank P. Mart'ınez-Mirav'e for pointing out to us the Geoneutrinos.org website. The work of D.A.S. is funded by ANID under grant 'Fondecyt Regular' 1221445. He thanks 'Le Service de Physique Th'eorique (Universit'e Libre de Bruxelles)' and 'Instituto de F'ısica Corpuscular (CSIC y Universidad de Valencia)' for their kind hospitality and their stimulating research environment during the completion of this work. V.D.R. acknowledges financial support from the CIDEXG/2022/20 grant (project 'D'AMAGAT') funded by Generalitat Valenciana and by the Spanish grant PID2020-113775GB-I00</text>
<text><location><page_5><loc_9><loc_39><loc_48><loc_44></location>(MCIN/AEI/10.13039/501100011033). C.A.T. is very thankful for the hospitality at Universidad T'ecnica Federico Santa Mar'ıa, where this work was initiated.</text>
<unordered_list>
<list_item><location><page_5><loc_11><loc_31><loc_24><loc_32></location>† deromeri@ific.uv.es</list_item>
<list_item><location><page_5><loc_11><loc_30><loc_29><loc_31></location>‡ christoph.ternes@lngs.infn.it</list_item>
<list_item><location><page_5><loc_10><loc_23><loc_48><loc_29></location>[1] G. Arcadi, M. Dutra, P. Ghosh, M. Lindner, Y. Mambrini, M. Pierre, S. Profumo, and F. S. Queiroz, 'The waning of the WIMP? A review of models, searches, and constraints,' Eur. Phys. J. C 78 no. 3, (2018) 203, arXiv:1703.07364 [hep-ph] .</list_item>
<list_item><location><page_5><loc_10><loc_21><loc_49><loc_23></location>[2] M. W. Goodman and E. Witten, 'Detectability of Certain Dark Matter Candidates,' Phys. Rev. D 31 (1985) 3059.</list_item>
<list_item><location><page_5><loc_10><loc_17><loc_49><loc_20></location>[3] A. Drukier and L. Stodolsky, 'Principles and Applications of a Neutral Current Detector for Neutrino Physics and Astronomy,' Phys. Rev. D 30 (1984) 2295.</list_item>
<list_item><location><page_5><loc_10><loc_11><loc_47><loc_16></location>[4] XENON Collaboration, E. Aprile et al. , 'First Dark Matter Search with Nuclear Recoils from the XENONnT Experiment,' Phys. Rev. Lett. 131 no. 4, (2023) 041003, arXiv:2303.14729 [hep-ex] .</list_item>
<list_item><location><page_5><loc_10><loc_9><loc_48><loc_11></location>[5] LZ Collaboration, J. Aalbers et al. , 'First Dark Matter Search Results from the LUX-ZEPLIN (LZ) Experiment,' Phys. Rev.</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_5><loc_55><loc_91><loc_85><loc_93></location>Lett. 131 no. 4, (2023) 041002, arXiv:2207.03764 [hep-ex] .</list_item>
<list_item><location><page_5><loc_53><loc_85><loc_92><loc_90></location>[6] PandaX Collaboration, S. Li et al. , 'Search for Light Dark Matter with Ionization Signals in the PandaX-4T Experiment,' Phys. Rev. Lett. 130 no. 26, (2023) 261001, arXiv:2212.10067 [hep-ex] .</list_item>
<list_item><location><page_5><loc_53><loc_81><loc_89><loc_85></location>[7] J. Billard et al. , 'Direct detection of dark matter-APPEC committee report*,' Rept. Prog. Phys. 85 no. 5, (2022) 056201, arXiv:2104.07634 [hep-ex] .</list_item>
<list_item><location><page_5><loc_53><loc_80><loc_84><loc_81></location>[8] 'The XLZD Consortium.' https://xlzd.org/ .</list_item>
<list_item><location><page_5><loc_53><loc_76><loc_90><loc_80></location>[9] L. E. Strigari, 'Neutrino Coherent Scattering Rates at Direct Dark Matter Detectors,' New J. Phys. 11 (2009) 105011, arXiv:0903.3630 [astro-ph.CO] .</list_item>
<list_item><location><page_5><loc_52><loc_72><loc_88><loc_76></location>[10] J. Monroe and P. Fisher, 'Neutrino Backgrounds to Dark Matter Searches,' Phys. Rev. D 76 (2007) 033007, arXiv:0706.3019 [astro-ph] .</list_item>
<list_item><location><page_5><loc_52><loc_68><loc_92><loc_72></location>[11] J. D. Vergados and H. Ejiri, 'Can Solar Neutrinos be a Serious Background in Direct Dark Matter Searches?,' Nucl. Phys. B 804 (2008) 144-159, arXiv:0805.2583 [hep-ph] .</list_item>
<list_item><location><page_5><loc_52><loc_63><loc_91><loc_68></location>[12] J. Billard, L. Strigari, and E. Figueroa-Feliciano, 'Implication of neutrino backgrounds on the reach of next generation dark matter direct detection experiments,' Phys. Rev. D 89 no. 2, (2014) 023524, arXiv:1307.5458 [hep-ph] .</list_item>
<list_item><location><page_5><loc_52><loc_59><loc_91><loc_63></location>[13] C. A. J. O'Hare, 'Dark matter astrophysical uncertainties and the neutrino floor,' Phys. Rev. D 94 no. 6, (2016) 063527, arXiv:1604.03858 [astro-ph.CO] .</list_item>
<list_item><location><page_5><loc_52><loc_55><loc_89><loc_59></location>[14] C. A. J. O'Hare, 'New Definition of the Neutrino Floor for Direct Dark Matter Searches,' Phys. Rev. Lett. 127 no. 25, (2021) 251802, arXiv:2109.03116 [hep-ph] .</list_item>
<list_item><location><page_5><loc_52><loc_48><loc_90><loc_55></location>[15] D. Aristizabal Sierra, V. De Romeri, L. J. Flores, and D. K. Papoulias, 'Impact of COHERENT measurements, cross section uncertainties and new interactions on the neutrino floor,' JCAP 01 no. 01, (2022) 055, arXiv:2109.03247 [hep-ph] .</list_item>
<list_item><location><page_5><loc_52><loc_44><loc_90><loc_48></location>[16] S. E. Vahsen, C. A. J. O'Hare, and D. Loomba, 'Directional Recoil Detection,' Ann. Rev. Nucl. Part. Sci. 71 (2021) 189-224, arXiv:2102.04596 [physics.ins-det] .</list_item>
<list_item><location><page_5><loc_52><loc_42><loc_92><loc_44></location>[17] S. Dye and A. Barna, 'Global Antineutrino Modeling for a Web Application,' arXiv:1510.05633 [physics.ins-det] .</list_item>
<list_item><location><page_5><loc_52><loc_40><loc_69><loc_41></location>[18] 'Geoneutrinos website.'</list_item>
</unordered_list>
<text><location><page_5><loc_55><loc_39><loc_80><loc_40></location>https://reactors.geoneutrinos.org/</text>
<text><location><page_5><loc_80><loc_39><loc_81><loc_40></location>.</text>
<unordered_list>
<list_item><location><page_5><loc_52><loc_35><loc_91><loc_39></location>[19] D. Z. Freedman, 'Coherent Neutrino Nucleus Scattering as a Probe of the Weak Neutral Current,' Phys. Rev. D 9 (1974) 1389-1392.</list_item>
<list_item><location><page_5><loc_52><loc_31><loc_90><loc_35></location>[20] R. H. Helm, 'Inelastic and Elastic Scattering of 187-Mev Electrons from Selected Even-Even Nuclei,' Phys. Rev. 104 (1956) 1466-1475.</list_item>
<list_item><location><page_5><loc_52><loc_26><loc_90><loc_31></location>[21] V. Kopeikin, M. Skorokhvatov, and O. Titov, 'Reevaluating reactor antineutrino spectra with new measurements of the ratio between U235 and Pu239 β spectra,' Phys. Rev. D 104 no. 7, (2021) L071301, arXiv:2103.01684 [nucl-ex] .</list_item>
<list_item><location><page_5><loc_52><loc_21><loc_91><loc_26></location>[22] P. Huber, 'On the determination of anti-neutrino spectra from nuclear reactors,' Phys. Rev. C 84 (2011) 024617, arXiv:1106.0687 [hep-ph] . [Erratum: Phys.Rev.C 85, 029901 (2012)].</list_item>
<list_item><location><page_5><loc_52><loc_15><loc_91><loc_20></location>[23] TEXONO Collaboration, H. T. Wong et al. , 'A Search of Neutrino Magnetic Moments with a High-Purity Germanium Detector at the Kuo-Sheng Nuclear Power Station,' Phys. Rev. D 75 (2007) 012001, arXiv:hep-ex/0605006 .</list_item>
<list_item><location><page_5><loc_52><loc_10><loc_87><loc_15></location>[24] C. Giunti, Y. F. Li, C. A. Ternes, and Z. Xin, 'Reactor antineutrino anomaly in light of recent flux model refinements,' Phys. Lett. B 829 (2022) 137054, arXiv:2110.06820 [hep-ph] .</list_item>
<list_item><location><page_6><loc_9><loc_88><loc_48><loc_93></location>[25] B. Lenardo et al. , 'Measurement of the ionization yield from nuclear recoils in liquid xenon between 0.3 - 6 keV with single-ionization-electron sensitivity,' arXiv:1908.00518 [physics.ins-det] .</list_item>
<list_item><location><page_6><loc_9><loc_84><loc_48><loc_88></location>[26] S. E. Vahsen et al. , 'CYGNUS: Feasibility of a nuclear recoil observatory with directional sensitivity to dark matter and neutrinos,' arXiv:2008.12587 [physics.ins-det] .</list_item>
<list_item><location><page_6><loc_9><loc_79><loc_48><loc_84></location>[27] M. Abdullah, D. Aristizabal Sierra, B. Dutta, and L. E. Strigari, 'Coherent Elastic Neutrino-Nucleus Scattering with directional detectors,' Phys. Rev. D 102 no. 1, (2020) 015009, arXiv:2003.11510 [hep-ph] .</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_6><loc_52><loc_87><loc_91><loc_93></location>[28] D. Aristizabal Sierra, B. Dutta, D. Kim, D. Snowden-Ifft, and L. E. Strigari, 'Coherent elastic neutrino-nucleus scattering with the ν BDX-DRIFT directional detector at next generation neutrino facilities,' Phys. Rev. D 104 no. 3, (2021) 033004, arXiv:2103.10857 [hep-ph] .</list_item>
<list_item><location><page_6><loc_52><loc_80><loc_92><loc_86></location>[29] ν BDX-DRIFT Collaboration, D. Aristizabal Sierra, J. L. Barrow, B. Dutta, D. Kim, D. Snowden-Ifft, L. Strigari, and M. H. Wood, 'Rock neutron backgrounds from FNAL neutrino beamlines in the ν BDX-DRIFT detector,' Phys. Rev. D 107 no. 1, (2023) 013003, arXiv:2210.08612 [hep-ex] .</list_item>
</document>
[
{
"title": "Reactor neutrino background in third-generation dark matter detectors",
"content": "D. Aristizabal Sierra, 1, ∗ Valentina De Romeri, 2, † and Christoph A. Ternes 3, ‡ 1 Universidad T'ecnica Federico Santa Mar'ıa - Departamento de F'ısica Casilla 110-V, Avda. Espa˜na 1680, Valpara'ıso, Chile 2 Instituto de F'ısica Corpuscular (CSIC-Universitat de Val'encia), Parc Cient'ıfic UV C/ Catedr'atico Jos'e Beltr'an, 2 E-46980 Paterna (Valencia) - Spain 3 Istituto Nazionale di Fisica Nucleare (INFN), Laboratori Nazionali del Gran Sasso, 67100 Assergi, L'Aquila (AQ), Italy Third-generation dark matter detectors will be fully sensitive to the boron-8 solar neutrino flux. Because of this, the characterization of such a background has been the subject of extensive analyses over the last few years. In contrast, little is known about the impact of reactor neutrinos. In this letter we report on the implications of such a flux for dark matter direct detection searches. We consider five potential detector deployment sites envisioned by the recently established XLZD consortium: SURF, SNOLAB, Kamioka, LNGS and Boulby. By using public reactor data we construct five reactor clusters-involving about 100 currently operating commercial nuclear reactors each-and determine the net neutrino flux at each detector site. Assuming a xenon-based detector and a 50 tonne-year exposure, we show that in all cases the neutrino event rate may be sizable, depending on energy recoil thresholds. Of all possible detector sites, SURF and LNGS are those with the smallest reactor neutrino background. On the contrary, SNOLAB and Boulby are subject to the strongest reactor neutrino fluxes, with Kamioka being subject to a more moderate background. Our findings demonstrate that reactor neutrino fluxes should be taken into account in the next round of dark matter searches. We argue that this background may be particularly relevant for directional detectors, provided they meet the requirements we have employed in this analysis.",
"pages": [
1
]
},
{
"title": "INTRODUCTION",
"content": "A wealth of cosmological and astrophysical data supports the idea that the dominant form of matter in the Universe has feeble or none electromagnetic interactions. The conventional wisdom is that this new form of matter-dubbed dark matter (DM)-is of microscopical origin and its abundance is determined by fast-scattering processes with Standard Model (SM) particles at very early epochs, much before the onset of cosmic neutrino decoupling and primordial nucleosynthesis (for a review see e.g. Ref. [1]). Although at high temperatures DMis thermalized, as the temperature decreases-because of the expansion of the Universe -these scattering processes are unable to keep the species in thermodynamic equilibrium and so its abundance freezes out. This weakly interacting massive particle (WIMP) is a rather generic candidate appearing in a large class of particle physics models. It is a dominant paradigm that has driven DM searches. DM direct detection is a subject that dates back to the mid 80's, when Goodman and Witten pointed out that WIMPs could be searched for by using the same detectors proposed by Drukier and Stodolsky for coherent elastic neutrino-nucleus scattering (CE ν NS) measurements [2, 3]. Since then, and because of the lack of a signal, detector technologies as well as fiducial volumes have dramatically evolved. At present, DMsearches in direct detection experiments are led by liquid xenon (LXe) dual-phase time projection chambers (secondgeneration DM detectors). Detectors at the INFN 'Laboratori Nazionali del Gran Sasso' (LNGS) in Italy (XENONnT), at the Sanford Underground Research Facility (SURF) in South Dakota in the US (LZ) and at the China Jinping Underground Laboratory in Sichuan, China (PandaX-4T) are using active volumes of the order of 5 tonne [4-6]. With their high capabilities for background rejection, along with low nuclear recoil energy thresholds, these secondgeneration DM detectors are sensitive to spin-independent WIMP-nucleon total cross sections of the order of 10 -48 cm 2 [7]. Indeed, XENONnT and LZ have recently published results where sensitivities of the order of σ WIMP-nuc ∼ 10 -47 cm 2 have been reported [4, 5]. PandaX-4T has set the most stringent upper limit in the low WIMP mass region ( ≲ 10 GeV), σ WIMP-nuc ∼ 10 -44 cm 2 [6]. A new generation of LXe detectors-third-generation DM detectors-is expected to pave the way for a discovery 1 . Recently the XENONnT, LZ and DARWIN collaborations have united forces and created the XLZD consortium [8]. Their goal is the construction of a 40-100 tonne detector with unprecedented sensitivities. With such active volume, a detector of this kind will be subject to an irreducible neutrino background dominated by 8 B solar neutrinos (for nuclearchannel signals) and by pp neutrinos (for electron-channel signals) [9]. The morphology and size of this background have been the subject of different analyses in recent years, first identified as the so-called 'neutrino floor' [9-13] and its more recent redefinition, the 'neutrino fog' [14], where a first estimation of the reactor neutrino background at LNGS was addressed. It is well known that the impact of the neutrino background on a WIMP discovery signal is mainly dominated by neutrino flux uncertainties, with uncertainties on the weak mixing angle and on the root-mean-square radii of the neutron distribu- tions playing a rather subdominant role [15]. The presence of a neutrino background, however, does not mean that an identification of a WIMP signal is impossible. First of all, improvements in the determination of solar neutrino flux uncertainties are expected. Secondly, WIMP and neutrino spectra in general do not fully degenerate in most regions of parameter space. Even in regions where they strongly do, an identification is possible with sufficiently large data sets [12]. Furthermore, even if data is not abundant, directionality willpotentially-enable a distinction between WIMP and neutrino nuclear recoil spectra [16], if they turn out to be strongly degenerate. Given this landscape, and the fact that DM direct detection will soon enter the third-generation detector phase, one should wonder whether other neutrino sources might contribute to the background and hence should be taken into account. This is a rather relevant question to raise, aiming to leverage the full discovery power of these types of detectors. Motivated by this question, in this Letter we assess the impact of nuclear reactor neutrinos. Since the reactor neutrino flux strongly depends on the geographical position of the detector-for definitiveness-we use LNGS, SURF, Boulby (UK), Kamioka (Japan) and SNOLAB (Canada) as possible deployment sites 2 .",
"pages": [
1,
2
]
},
{
"title": "NUCLEAR REACTOR SOURCES: LOCATIONS AND EVENT RATES",
"content": "The data sets we employ follow from data provided on the Geoneutrinos.org website [17, 18]. We consider only commercial power plants (that involve the most powerful reactors) for which a non-zero operating power is reported. Reactors for which the thermal capacity is known but have zero operating power and those that have been permanently shut down are not included. Depending on the baseline, each detector site that we consider is 'surrounded' by a cluster of nuclear reactor power plants, at a certain distance Li . Table I shows the minimum and maximum baseline and power for each cluster, along with the number of reactors involved. For each detector site, we do not include reactors located at distances beyond L max, as their contribution to the event rates would be negligible. The largest clusters are those around the LNGS and Boulby detector sites (as expected, given that for these two cases the radius defining the cluster exceeds by about 1000 km the radius at the other sites). However, this does not necessarily mean that the largest flux is obtained for these two positions, as we now discuss. The reactor neutrino flux decreases rapidly with increasing baseline. So, a rather fair assumption is that the flux is dominated by the sub-cluster defined by all reactors included in a radius ≲ 1000 km. For the SURF and LNGS locations one finds that these sub-clusters involve only 5 reactors with a 2.1 GW and 1.8 GW average power, respectively. For the Kamioka, SNOLAB and Boulby locations, the subclusters are composed instead of 35, 59 and 49 reactors. The average power in each case (and in that order) is: 2.1 GW, 4.9 GW and 1.9 GW. Thus, already from these numbers one expects the SURF and LNGS location sites to involve a less intense reactor neutrino flux. Fig. 1 shows the distribution of nuclear reactors in terms of baseline and power for the five different clusters we consider. The distributions involve the full data sets. From the graph, one can see that for the Boulby and SNOLAB clusters the reactor density for baselines below 1000 km is high, with a few of those reactors having powers above 3 GW. The distribution for the Kamioka cluster is somewhat different. Although below 1000 km there are a few reactors, their density is lower as well as their power. For the SURF and LNGS clusters, the reactor density for baselines below 1000 km is, instead, rather moderate. For these clusters, most reactors are at baselines above 1000 km. So, even without a dedicated calculation of the event rate, expectations are that in terms of increasing reactor neutrino fluxes the clusters can be sorted into three groups: SURF/LNGS, Kamioka, SNOLAB/Boulby. The calculation of the differential nuclear recoil spectrum at each cluster (C) requires the convolution of the differential CE ν NS cross section [3, 19] with the reactor neutrino flux, namely Here, m det refers to the detector active volume mass, m Xe mol to the xenon molar mass, T to the exposure time, E min ν = √ mNEr / 2 ( Er and mN refer to nuclear recoil energy and mass), and E max ν to the neutrino spectrum kinematic 'highenergy' tail taken at 8MeV. The average nuclear mass is ⟨ m Xe ⟩ / GeV = 0 . 93 ×⟨ A ⟩ , ⟨ A ⟩ = ∑ i XiAi = 131 . 4 being the mass number averaged over the nine stable xenon isotopes. We include-for completeness-the weak-charge nuclear form factor, F H ( Er ) , parametrized 'a la Helm [20]. Note that if not included results would deviate from those presented here at most by ∼ 2%, because of the process occurring deep in the full coherent regime. Regarding the electron antineutrino spectrum, we proceed as follows. For the 235 U and 238 U emission spectra we use results from Ref. [21]. For 239 Pu and 241 Pu we use instead results from Ref. [22]. The full electron antineutrino differential flux is then calculated according to where fi = { f 235 U , f 238 U , f 239 Pu , f 241 Pu } = { 5 . 5 , 0 . 7 , 3 . 2 , 0 . 6 }× 10 -1 are the uranium and plutonium fission fractions [23]. Note that we do not include electron antineutrinos produced in neutron capture by 238 U. The reason is that the spectra for those neutrinos dominate at energies below ∼ 1 . 5 MeV, hence in a LXe detector would produce nuclear recoils below 0.04 keV (much below any realistic operation threshold). We assume the spectral function in Eq. (2) to be universal for all the reactors within the clusters 3 . Thus, the difference among clusters is determined only by the normalization factor, which we calculate assuming that in each fission process an energy of ε = 205 . 24 MeV is released and that neutrinos are emitted isotropically. Explicitly, each normalization factor is given by where j runs over all reactors relevant for cluster C and Pj and Lj are the operating power and distance for reactor j . Their values are displayed in Tab. II, showing that SURF is subject to the least abundant neutrino flux, whereas Boulby to the most severe. With these results at hand, we are now in a position to calculate the differential event rate as well as the total event rate for each detector site. We assume a 50-tonne active volume LXe detector and 100% efficiency 4 . Since current realistic thresholds amount to 0.3 keV [25], we use E th,min r = 0 . 1keV as a value envisioned for future detector operations. Results are displayed in Fig. 2. The left (right) graph shows the differential event rate (total event rate) as a function of the recoil energy (recoil energy threshold) for the five different reactor clusters we have considered. The inset plot in the right panel is meant to zoom in on the bottom left corner. Inline with expectations, the differential and total event rates at the SURF (Boulby) detector site are the smallest (largest). The event rate at the LNGS detector location is slightly higher, followed by Kamioka and SNOLAB.",
"pages": [
2,
3
]
},
{
"title": "DISCUSSION",
"content": "Naively one would expect the reactor neutrino flux to be suppressed and of little relevance. This expectation is mainly based on the fact that most reactors are far away from the detector sites. However, the fact that the clusters around each detector site involve a large number of active nuclear power plants (with in some cases powerful reactors), combined with a large active volume produces a non-zero event rate in all cases. Ideally one would like a very low threshold to explore the small WIMP mass window and increase the WIMP-nucleus event rate. At 0.1 keV, we find that the total neutrino-nucleus event rate per year is: 16 (SURF), 44 (LNGS), 82 (Kamioka), 124 (SNOLAB) and 733 (Boulby). If that operation threshold is not achieved and instead the detector is operated at 0.3 keV, these numbers will be degraded by about a factor 7. In such an experimental scenario the reactor neutrino background becomes, of course, less severe. Thus, the question of whether the reactor neutrino background matters is-as anticipatedstrongly linked to operation thresholds. It is worth emphasizing that variations of these estimated numbers are expected in the future, depending on the exact number of reactors that enter in either operation phase or are decommissioned. However, these results demonstrate that the reactor neutrino flux should be seriously taken into account in decision making as well as in data taken, contrary to expectations. Finally, one might wonder how much this neutrino background matters compared to the boron-8 solar neutrino flux. For the detector configurations we have considered, with a 0.1 keV operation threshold, the number of boron-8 nuclear recoil induced events is overwhelming, 36500 events/year. So, of course, this will be the dominant background source. All the efforts to understand the morphology of this background are indeed motivated by this fact. The question is then whether one should be concerned with the reactor neutrino background whatsoever. It is well known that the boron-8 background can be to a certain degree circumvented. As we have already stressed, large data sets might enable differentiating neutrino from WIMP signals, if the WIMP parameters are such that the neutrino and WIMP event rates strongly degenerate. In general, however, directional detectors seem to be the most promising avenue [16, 26] 5 . For these detectors it seems that the reactor neutrino background might even become the most dominant background source. Therefore, if the boron-8 nuclear recoilinduced events can be efficiently discriminated, there will be yet another background source that will require careful identification and proper treatment, depending on statistics and operation capabilities.",
"pages": [
4
]
},
{
"title": "CONCLUSIONS",
"content": "With the advent of third-generation DM direct detection detectors, the quantification of reactor neutrino fluxes becomes of pivotal importance. In this work we have quantified the size of the neutrino flux produced by clusters of reactors surrounding five potential detector deployment sites. For definitiveness we have considered the locations envisioned by the recently established XLZD consortium: SURF, SNOLAB, Kamioka, LNGS, and Boulby. Our findings show that detectors with active volumes of the order of 50 tonne and recoil energy thresholds of the or- der of 0.1 keV, will be sensitive to a certain amount of reactor neutrino-induced events. The exact amount depends, to a large degree, on the energy threshold at which the detector is operated. However, even assuming a realistic threshold of 0.3 keV, the event rate turns out to be sizable in all cases. We find that the site with the smallest reactor neutrino background is SURF followed by LNGS, Kamioka, SNOLAB, and Boulby (in that order). Although subdominant compared to the solar boron-8 neutrino background, we point out that the reactor neutrino background (and its corresponding events) should be-in principle-considered during data taken. Reactor neutrinoinduced events should be taken into account in background discrimination, regardless of the detector technique employed. This result will be particularly relevant for directional detection, if future detectors meet the requirements we have used here.",
"pages": [
4,
5
]
},
{
"title": "ACKNOWLEDGMENTS",
"content": "We thank P. Mart'ınez-Mirav'e for pointing out to us the Geoneutrinos.org website. The work of D.A.S. is funded by ANID under grant 'Fondecyt Regular' 1221445. He thanks 'Le Service de Physique Th'eorique (Universit'e Libre de Bruxelles)' and 'Instituto de F'ısica Corpuscular (CSIC y Universidad de Valencia)' for their kind hospitality and their stimulating research environment during the completion of this work. V.D.R. acknowledges financial support from the CIDEXG/2022/20 grant (project 'D'AMAGAT') funded by Generalitat Valenciana and by the Spanish grant PID2020-113775GB-I00 (MCIN/AEI/10.13039/501100011033). C.A.T. is very thankful for the hospitality at Universidad T'ecnica Federico Santa Mar'ıa, where this work was initiated. https://reactors.geoneutrinos.org/ .",
"pages": [
5
]
}
]
2024PhRvD.109l3009E
https://arxiv.org/pdf/2301.03619.pdf
<document>
<section_header_level_1><location><page_1><loc_23><loc_84><loc_77><loc_86></location>Revealing Phase Transition in Dense Matter with Gravitational Wave Spectroscopy of Binary Neutron Star Mergers</section_header_level_1>
<text><location><page_1><loc_16><loc_81><loc_83><loc_82></location>Pedro L. Espino, 1, 2 Aviral Prakash, 1, 3 David Radice, 1, 3, 4, ∗ and Domenico Logoteta 5, 6</text>
<text><location><page_1><loc_17><loc_75><loc_82><loc_80></location>1 Institute for Gravitation & the Cosmos, The Pennsylvania State University, University Park, PA 16802 2 Department of Physics, University of California, Berkeley, CA 94720, USA 3 Department of Physics, The Pennsylvania State University, University Park, PA 16802 4 Department of Astronomy & Astrophysics, The Pennsylvania State University,University Park, PA 16802</text>
<text><location><page_1><loc_23><loc_72><loc_77><loc_74></location>5 Dipartimento di Fisica, Università di Pisa, Largo B. Pontecorvo, 3 I-56127 Pisa, Italy 6 INFN, Sezione di Pisa, Largo B. Pontecorvo, 3 I-56127 Pisa, Italy</text>
<section_header_level_1><location><page_1><loc_45><loc_69><loc_55><loc_70></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_57><loc_86><loc_68></location>We use numerical relativity simulations of binary neutron star mergers to show that high density deconfinement phase transitions (PTs) to quark matter can be probed using multimodal postmerger gravitational wave (GW) spectroscopy. Hadron-quark PTs suppress the one-armed spiral instability in the remnant. This is manifested in an anti-correlation between the energy carried in the l = 2 , m = 1 GWmode and energy density gap which separates the two phases. Consequently, a single measurement of the signal-to-noise ratios of the l = 2 , m = 1 and l = 2 , m = 2 GWmodes could constrain the energy density gap of the PT.</text>
<text><location><page_1><loc_14><loc_53><loc_86><loc_54></location>Keywords: Neutron stars - Equation of state - Gravitational waves - Hydrodynamics - Instabilities</text>
<section_header_level_1><location><page_1><loc_20><loc_48><loc_36><loc_49></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_17><loc_48><loc_47></location>Binary neutron star (BNS) mergers produce some of the most extreme conditions in nature, compressing matter to several times the nuclear density and to temperatures of tens of MeV (Perego et al. 2019). More extreme conditions are only found in the early Universe and in the interior of black holes (BHs). Multimessenger observations of binary neutron star (BNS) mergers can be used to probe the properties of matter in these conditions, providing a unique avenue to study the nonperturbative regime of QCD (Shibata 2005; Hinderer et al. 2010; Damour et al. 2012; Sekiguchi et al. 2011; Hotokezaka et al. 2011; Bauswein et al. 2013; Radice et al. 2017; Abbott et al. 2017a; Margalit & Metzger 2017; Bauswein et al. 2017; Radice et al. 2018b; Most et al. 2019, 2020; Bauswein et al. 2019; Coughlin et al. 2019; De et al. 2018; Abbott et al. 2019, 2018; Radice & Dai 2019; Dietrich et al. 2020; Breschi et al. 2021, 2022; Kashyap et al. 2022; Perego et al. 2022; Fujimoto et al. 2022; Prakash et al. 2021).</text>
<text><location><page_1><loc_8><loc_14><loc_48><loc_17></location>Presently, there are large uncertainties in the fundamental physics of strongly-interacting matter at densi-</text>
<text><location><page_1><loc_9><loc_10><loc_23><loc_11></location>∗ Alfred P. Sloan fellow</text>
<text><location><page_1><loc_52><loc_21><loc_92><loc_49></location>ties of a few times nuclear saturation (Capano et al. 2020; Pang et al. 2021; Annala et al. 2022). It is not even clear what the relevant degrees of freedom are for the densities and temperatures reached in the core of remnant massive neutron stars (RMNS) of BNS mergers. It is possible that matter remains composed of nucleons, together with leptons (electrons, positrons, and muons) and photons (Perego et al. 2019; Loffredo et al. 2022). The appearance of more exotic baryons, such as hyperons, is not excluded (Sekiguchi et al. 2011; Radice et al. 2017; Logoteta 2021). It is also possible for a transition to the deconfined quark-gluon plasma phase to take place in BNS mergers (Most et al. 2019, 2020; Bauswein et al. 2019; Prakash et al. 2021). The determination of the state of matter formed in BNS mergers is one of the most pressing scientific objectives of multimessenger astronomy (Evans et al. 2021; Lovato et al. 2022).</text>
<text><location><page_1><loc_52><loc_10><loc_92><loc_21></location>Previous work has shown that the presence of phase transitions to deconfined quarks can be revealed by a shift of the postmerger gravitational wave (GW) peak frequency f 2 from the value expected for hadronic equations of state (EOSs) (Bauswein et al. 2019; Weih et al. 2020; Blacker et al. 2020; Kedia et al. 2022). However, such frequency shifts can be degenerate with deviations</text>
<text><location><page_2><loc_8><loc_67><loc_48><loc_91></location>from universal relations due to hadronic physics or other effects (Most et al. 2019; Weih et al. 2020; Liebling et al. 2021; Prakash et al. 2021; Fujimoto et al. 2022; Tootle et al. 2022). It has also been suggested that the presence of a phase transition could be inferred from a measurement of the threshold mass for prompt collapse of BNS systems (Bauswein et al. 2020, 2021; Perego et al. 2022; Kashyap et al. 2022). In this Letter , we use 8 stateof-the-art numerical relativity simulations to show, for the first time, that the presence and strength of a QCD phase transition could be unambiguously determined through multimodal GW spectroscopy of RMNS. Such measurements will be possible with the next-generation of GW experiments like Cosmic Explorer (Reitze et al. 2019), Einstein Telescope (Punturo et al. 2010), and NEMO (Ackley et al. 2020).</text>
<section_header_level_1><location><page_2><loc_23><loc_63><loc_33><loc_64></location>2. METHODS</section_header_level_1>
<text><location><page_2><loc_8><loc_28><loc_48><loc_62></location>We consider binaries in quasi-circular orbits and eccentric encounters on nearly parabolic orbits. Although BNS mergers with highly eccentric orbits are expected to be significantly more rare than those with quasi-circular inspirals, these events may still have appreciable rates of as high as 50 Gpc -3 yr -1 (Lee et al. 2010; Paschalidis et al. 2015); we include results from both types of mergers to consider as wide a variety of scenarios as possible. Initial data for the quasi-circular binaries is created using the conformal thin sandwich formalism (York 1999) and assuming a helical Killing vector and irrotational flows. The resulting elliptic equations are solved using the pseudo-spectral code LORENE (Gourgoulhon et al. 2001; Taniguchi et al. 2001; Taniguchi & Gourgoulhon 2002). Initial data for the eccentric encounters is constructed by superimposing two isolated, boosted, neutron stars, following Radice et al. (2016b). The initial separation of the stellar barycenters for parabolic encounters is set to 100 km , which is sufficiently large so that the level of constraint violation in the initial data is comparable to that of the quasi-circular binaries (Radice et al. 2016b).</text>
<text><location><page_2><loc_8><loc_9><loc_48><loc_28></location>We perform BNS merger simulations using the WhiskyTHC code (Radice & Rezzolla 2012; Radice et al. 2014a,b). WhiskyTHC makes use of the CTGamma spacetime solver (Pollney et al. 2011), which is a part of the Einstein Toolkit (Zlochower et al. 2022). The adaptive mesh refinement driver Carpet (Schnetter et al. 2004) is used to generate the dynamical grid structure employed in the simulations. All simulations considered in the present work have been performed using at least two grid resolutions. Although there are quantitative differences in the GW waveforms computed at different resolutions, the qualitative features discussed here are</text>
<text><location><page_2><loc_52><loc_81><loc_92><loc_91></location>robust across all simulations. Unless otherwise specified, we discuss results from simulations using the fiducial grid resolution (with grid spacing ∆ x glyph[similarequal] 184 . 6 m in the finest refinement level). The grid structure for the simulations is described in detail in Radice et al. (2018a) and Radice et al. (2016b) for the quasi-circular and eccentric simulations, respectively.</text>
<text><location><page_2><loc_52><loc_18><loc_92><loc_80></location>For a clear understanding of the role that high-density deconfinement phase transitions could play in the development of the one-armed spiral instability, we consider a total of 7 EOS models and run a total of 8 simulations with varying phase transition features. In particular, the size of the energy density gap which separates the hadronic and quark phases is a useful way to classify hybrid hadron-quark EOS models and provides a qualitative measure of the 'strength' of the phase transition (Alford & Han 2016). As such, we consider EOS models that cover several sizes of the energy density gap, ranging from non-existent (i.e., a purely hadronic EOS) to large, while maintaining consistency with current astrophysical constraints on the dense matter EOS. We consider both phenomenological EOS models (Paschalidis et al. 2018; Alvarez-Castillo & Blaschke 2017; Alford & Sedrakian 2017; Bozzola et al. 2019; Espino & Paschalidis 2022) (in the form of piecewise polytropic approximations using the prescription of Read et al. (2009)) and microphysical, finite temperature EOS models (Bastian 2021; Bombaci & Logoteta 2018; Logoteta et al. 2021; Prakash et al. 2021). We only consider equal-mass ratio binary configurations, with the total binary mass ranging from 2 . 6 M glyph[circledot] -2 . 7 M glyph[circledot] . The lack of π -rotational symmetry in BNS configurations with unequal-masses may be a suitable way of effectively seeding non-axisymmetric fluid instabilities that can take hold in the post-merger environment. Neutrino emission and reabsorption are not included for binaries in eccentric orbits, while all quasi-circular binaries include a neutrino treatment via the moment based M0 scheme (Radice et al. 2018a). However, neutrinos are not expected to influence the dynamics on the time scales considered in our study (Radice et al. 2020, 2022). Additionally, magnetic fields are not accounted for in any of our simulations, but these are also expected to be subdominant (Palenzuela et al. 2022). We find that, despite the diversity in binary properties and differences in the evolution, the effects presented in this work are robust.</text>
<section_header_level_1><location><page_2><loc_67><loc_14><loc_77><loc_16></location>3. RESULTS</section_header_level_1>
<text><location><page_2><loc_52><loc_9><loc_92><loc_13></location>The one-armed spiral instability is a nonaxisymmetric mode in a rapidly rotating fluid which, when saturated, leads to the dominance of a single</text>
<figure>
<location><page_3><loc_9><loc_68><loc_91><loc_91></location>
<caption>Figure 1. Left panel: Energy carried by GWs in the l = 2 , m = 1 mode as a function of time, scaled by the value at a fixed normalization time t norm = t mer +0 . 5 ms . The development of the one-armed spiral instability can be observed in the purely hadronic simulation as the energy in the l = 2 , m = 1 GW mode continues to grow, but is suppressed in the hadron-quark simulation. Right panel: The same quantity depicted in the left panel, but time-averaged over a fixed time window and shown as a function of the energy density gap ∆ e for each simulation. We normalize by the same quantity for the complementary hadronic EOS. We include approximate error bars, obtained using lower resolution simulations, to represent the uncertainty introduced by our numerical methods. The black solid line represents a linear fit to the data from our simulations. We find that the normalized energy emitted by the l = 2 , m = 1 GW mode decreases for simulations that employ EOS models with larger values of ∆ e .</caption>
</figure>
<text><location><page_3><loc_28><loc_67><loc_29><loc_69></location>-</text>
<text><location><page_3><loc_8><loc_11><loc_48><loc_53></location>high-density mode in the fluid density which is displaced from the fluid barycenter (Pickett et al. 1996; Centrella et al. 2001; Saijo et al. 2002; Ou & Tohline 2006). The one-armed spiral instability has been observed to develop commonly in BNS merger simulations that produce long-lived, massive post-merger remnants on timescales of O (10ms) (Paschalidis et al. 2015; East et al. 2016; East et al. 2016; Radice et al. 2016a; Lehner et al. 2016) and in simulations of many other astrophysical systems including supernovae (Ott et al. 2005; Kuroda et al. 2014), white dwarfs (Kashyap et al. 2015, 2017) and accretion disks (Kashyap et al. 2017; Wessel et al. 2021). Each fluid density mode that arises during the evolution of a massive NS remnant is associated with GW emission at characteristic frequencies stemming from its pattern speed. As such, the development of the one-armed spiral instability in astrophysical systems may be observed by considering multimodal GW spectroscopy (Radice et al. 2016a). For the simulations considered in this work we extract multimodal GW information within the Newman-Penrose formalism. We compute the coefficients of s = -2 spin-weighted spherical harmonic decompositions of the Newman-Penrose scalar Ψ 4 which we label as Ψ l,m 4 . The one-armed spiral instability can therefore be observed in the GW spectrum extracted from our simulations as a growth in the power and amplitude of the l = 2 , m = 1 GW mode</text>
<text><location><page_3><loc_52><loc_50><loc_92><loc_53></location>(i.e., Ψ 2 , 1 4 ) and simultaneous decay of the dominant l = 2 , m = 2 GW mode (i.e., Ψ 2 , 2 4 ).</text>
<text><location><page_3><loc_52><loc_17><loc_92><loc_50></location>Our simulations show that high-density deconfinement phase transitions act to suppress the one-armed spiral instability . Depending on the features of the phase transition, the one-armed spiral mode may either arise on a significantly longer timescale when compared to simulations which employ purely hadronic EOS models, or it may be suppressed altogether on the timescales probed by our simulations. There are several potential mechanisms via which the instability may be suppressed. For example, it has been shown that the physical extent of the remnant plays an important role in the development of the instability, with larger remnants being more conducive to the development of the instability on shorter timescales (Radice et al. 2016a; Lehner et al. 2016; Saijo & Yoshida 2016; Saijo 2018). The significant softening at high densities introduced by the phase transition results in more compact post-merger remnants (relative to scenarios that consider only hadronic degrees of freedom). As such, the more compact hybrid star remnants may see a weaker development of the one-armed spiral instability when compared to neutron star remnants.</text>
<text><location><page_3><loc_52><loc_10><loc_92><loc_17></location>In the left panel of Fig. 1 we show the energy carried by the l = 2 , m = 1 GW mode (scaled by the energy emitted at a time shortly after the merger t norm = t mer +0 . 5 ms ) as a function of time for simulations employing the DD2F (hadronic) and DD2F-SF5 (hybrid</text>
<text><location><page_4><loc_9><loc_78><loc_11><loc_79></location>√</text>
<text><location><page_4><loc_9><loc_73><loc_11><loc_74></location>√</text>
<figure>
<location><page_4><loc_9><loc_67><loc_49><loc_92></location>
<caption>Figure 2. Multimodal GW amplitude spectrum computed for symmetric binaries of total mass M=2 . 6 M glyph[circledot] in an edge-on configuration. Also shown are the noise sensitivity curves for advanced LIGO (aLIGO), Einstein Telescope (ET), the 20 km postmerger-optimized configuration for the Cosmic Explorer (CE20) and the 40 km configuration for Cosmic Explorer (CE40). A suppression in the amplitude spectral density (ASD) as a result of the deconfinement phase transition may be detectable with the third generation detectors and most cleanly with CE40 .</caption>
</figure>
<figure>
<location><page_4><loc_51><loc_67><loc_90><loc_92></location>
</figure>
<text><location><page_4><loc_50><loc_78><loc_53><loc_79></location>√</text>
<text><location><page_4><loc_50><loc_73><loc_53><loc_74></location>√</text>
<text><location><page_4><loc_8><loc_30><loc_48><loc_58></location>hadron-quark) EOSs. We find that the energy carried in the l = 2 , m = 1 mode of the GWs is significantly smaller in the simulation employing a hybrid hadronquark EOS, indicating that the one-armed spiral instability is suppressed in scenarios with deconfinement phase transitions at densities relevant for BNS mergers. We emphasize that in the left panel of Fig. 1 we showcase results for a set of EOS models which are identical below the threshold for a phase transition, and as such the simulations have identical initial conditions. In the right panel of Fig. 1, we show the time-averaged energy emitted by the l = 2 , m = 1 GW mode 〈 E 2 , 1 GW 〉 (again scaled by the energy emitted at a time shortly after the merger t norm = t mer +0 . 5 ms ) as a function of the energy density gap ∆ e , where we define the energy density gap as the difference between the energy density e at the end of the hadronic phase and beginning of the quark phase for cold matter in β -equilibrium (Alford & Han 2016),</text>
<formula><location><page_4><loc_17><loc_26><loc_48><loc_29></location>∆ e ≡ e quark , initial -e hadron , final , (1)</formula>
<text><location><page_4><loc_8><loc_8><loc_48><loc_26></location>where we assume units where the speed of light c = 1 . We identify the end and beginning of each phase by considering the change in the approximate adiabatic index Γ = dlog p/ dlog( ρ ) , where p is the fluid pressure of the cold, beta-equilibrium, barotropic EOS for each EOS model considered. The region corresponding to the phase transition is always unambiguously marked by discontinuities in, or sudden changes in the slope of, the adiabatic index for the EOS models we consider. For the results depicted in the right panel of Fig. 1, we time-average over a window of ∆ t ≈ 40 ms after the</text>
<text><location><page_4><loc_52><loc_25><loc_92><loc_58></location>merger except for cases that lead to a remnant collapse on shorter timescales (in such cases, we time-average until the collapse of the NS remnant). Additionally, for each simulation, we normalize by a complementary simulation that uses identical initial data but employs a hadronic EOS having the same low-density behavior below the phase transition threshold as the hybrid hadronquark EOS. As such, we depict the point corresponding to all hadronic EOS simulations with a black square at ∆ e = 0 . Each simulation is time-averaged to the same extent as its complementary hadronic simulation. We find an anti-correlation between the energy carried in the l = 2 , m = 1 GWmode and the size of the energy density gap . In other words, as the size of the energy density gap (and thereby the qualitative 'strength' of the phase transition) increases, GW emission in the l = 2 , m = 1 mode decreases, which signifies that the one-armed spiral instability is further suppressed for EOS models with 'stronger' deconfinement phase transitions. We elaborate on the choice of quantities depicted in Fig. 1 in App. A.</text>
<section_header_level_1><location><page_4><loc_66><loc_22><loc_78><loc_23></location>4. DISCUSSION</section_header_level_1>
<text><location><page_4><loc_52><loc_9><loc_92><loc_21></location>The characteristic frequency associated with peak emission in the l = 2 , m = 1 GW mode has half the value of that associated with the l = 2 , m = 2 mode (i.e., f 2 , 1 peak = f 2 , 2 peak / 2 ). Observationally, a GW signal would contain information at all contributing frequencies. However, the dominant GW emission associated with binary coalescence is always expected to be from the l = 2 , m = 2 contribution, such that f peak = f 2 , 2 peak .</text>
<text><location><page_5><loc_8><loc_79><loc_48><loc_91></location>Therefore, a potential observational signature of the one-armed spiral instability is the growth in power of an incoming GW signal at a frequency that is half of the dominant frequency; if it develops in the post-merger environment, the one-armed spiral instability will continuously power the emission of GWs at f peak / 2 , while emission in the dominant f peak decays in time (Bernuzzi et al. 2015).</text>
<text><location><page_5><loc_8><loc_42><loc_48><loc_79></location>In Fig. 2 we show the post-merger GW amplitude spectrum density (ASD) for a symmetric, edge-on binary situated at a distance of 40 Mpc , which is consistent with the luminosity distance observed for GW170817 (Abbott et al. 2017b). The edge-on configuration is the most optimal for the detection of an m = 1 mode. As expected, we see a relative suppression of power in the m = 1 mode (with respect to the complementing hadronic simulation) with the onset of a deconfinement phase transition. In this realistic configuration, coupled with the 40 km Cosmic Explorer detector (Reitze et al. 2019), the appearance of quarks in the post-merger remnant results in a suppression of the postmerger signal-to-noise ratio (SNR) of the ( l = 2 , m = 1) mode by a factor of 2, from 2.14 in the hadronic case to 1.08 in the hadron-quark case. The GW ASD peak of the l = 2 , m = 1 mode (between 1-2 kHz ) and the postmerger ASD peak of the l = 2 , m = 2 mode (between 2-4 kHz ), lie respectively in the most sensitive regions of the 40 km and the 20 km postmerger optimized Cosmic Explorer configurations. Our analysis recommends an increase in detector sensitivities in the high-frequency regimes (see also Zhang et al. (2022)) for best possible constraints on deconfinement phase transitions in BNS mergers.</text>
<text><location><page_5><loc_8><loc_12><loc_48><loc_41></location>In this letter we have highlighted, for the first time, that high-density deconfinement phase transitions act to suppress the one-armed spiral instability. We find an anti-correlation between the energy carried in the l = 2 , m = 1 GW mode and the size of the energy density gap which qualitatively separates the hadronic and quark phases. Our findings reveal a deep connection between observable multimodal GW emission and the microphysical description of matter in the post-merger environment. We expect the one-armed spiral instability to be detectable at distances of 40 Mpc using future generation detectors (Radice et al. 2016a). If evidence of a strong one-armed spiral mode can be inferred from GW observations of the post-BNS merger environment, our findings suggest that a strong high-density deconfinement phase transition at the densities relevant to BNS mergers would be disfavored. On the other hand, if evidence for the one-armed spiral instability is not found for close-by BNS mergers, this could also point to the</text>
<text><location><page_5><loc_52><loc_89><loc_92><loc_91></location>possibility of a deconfinement phase transition taking place at densities relevant to BNS mergers.</text>
<text><location><page_5><loc_52><loc_32><loc_92><loc_88></location>We point out that other effects relevant in the postmerger environment - such as the presence of strong magnetic fields (Franci et al. 2013) and additional degrees of freedom that can cause a sudden softening of the EOS - may affect the development of the one-armed spiral instability. However, the relevant timescales and extent to which the aforementioned phenomena can affect the development of non-axisymmetric instabilities or the GW spectrum remains uncertain (Radice et al. 2016a; Muhlberger et al. 2014), and may not impact our conclusions (Palenzuela et al. 2022; Zappa et al. 2022). The effects discussed in the present work arise on dynamical timescales ∼ O (10 ms) , and may be the dominant mechanism for suppression of the one-armed spiral instability. Additionally, although we find a trend in the decrease of energy carried by the l = 2 , m = 1 GW mode for larger values of ∆ e , additional studies will help establish a more robust trend and provide an understanding of the potential spread in the trend. In particular, future lines of investigation will include: (1) considering the combined effects of the mass ratio and high-density phase transitions on the development of the one-armed spiral instability; (2) considering the effects of accurate neutrino transport on high-density deconfinement phase transitions, as neutrinos may modify the composition of matter and thereby potentially affect the onset of the phase transition; (3) employing EOS models at systematically increasing values of ∆ e while holding the hadronic region of the EOS fixed, as a limitation of the present work is the assumption that the l = 2 , m = 1 GW mode is perfectly known in the case of hadronic EOSs; and (4) investigating the effects discussed in this work in scenarios with a crossover to quark matter, as our present work only considers EOS models with phase transitions. We leave such studies to future work.</text>
<text><location><page_5><loc_52><loc_10><loc_92><loc_30></location>PE acknowledges funding from the National Science Foundation under Grant No. PHY-2020275. DR acknowledges funding from the U.S. Department of Energy, Office of Science, Division of Nuclear Physics under Award Number(s) DE-SC0021177 and from the National Science Foundation under Grants No. PHY-2011725, PHY-2116686, and AST-2108467. Simulations were performed on Bridges2 and Expanse (NSF XSEDE allocation TG-PHY160025). This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.</text>
<section_header_level_1><location><page_6><loc_44><loc_90><loc_56><loc_91></location>REFERENCES</section_header_level_1>
<table>
<location><page_6><loc_8><loc_10><loc_48><loc_89></location>
</table>
<table>
<location><page_6><loc_52><loc_9><loc_92><loc_89></location>
</table>
<table>
<location><page_7><loc_8><loc_9><loc_48><loc_92></location>
</table>
<table>
<location><page_7><loc_52><loc_9><loc_92><loc_92></location>
</table>
<table>
<location><page_8><loc_8><loc_76><loc_48><loc_91></location>
</table>
<unordered_list>
<list_item><location><page_8><loc_52><loc_84><loc_90><loc_91></location>York, Jr., J. W. 1999, Phys. Rev. Lett., 82, 1350, doi: 10.1103/PhysRevLett.82.1350 Zappa, F., Bernuzzi, S., Radice, D., & Perego, A. 2022. https://arxiv.org/abs/2210.11491 Zhang, T., Yang, H., Martynov, D., Schmidt, P., & Miao,</list_item>
<list_item><location><page_8><loc_54><loc_82><loc_82><loc_83></location>H. 2022. https://arxiv.org/abs/2212.12144</list_item>
</unordered_list>
<text><location><page_8><loc_52><loc_77><loc_90><loc_81></location>Zlochower, Y., Brandt, S. R., Diener, P., et al. 2022, The Einstein Toolkit, The "Berhard Riemann" release, ET_2022_05, Zenodo, doi: 10.5281/zenodo.6588641</text>
<text><location><page_9><loc_28><loc_57><loc_30><loc_60></location>-</text>
<section_header_level_1><location><page_9><loc_26><loc_84><loc_74><loc_85></location>A. GW PROBE OF THE ONE-ARMED SPIRAL INSTABILITY</section_header_level_1>
<figure>
<location><page_9><loc_9><loc_58><loc_50><loc_82></location>
<caption>Figure 3. Left panel : Energy in the l = 2 , m = 1 GW mode as a function of time for simulations employing a hadronic (DD2F) and hadron-quark (BBKF1.5) EOS; the simulations use identical initial conditions and are run with a grid resolution of ∆ x = 369 . 2 m in the finest grid. These results showcase that the one-armed spiral instability may be seeded at different levels in the postmerger environment for different simulations. Right panel: Same quantity as the left panel, but normalized to the value at a time shortly after merger, t norm = t merger +0 . 5 ms . Normalizing at this time accounts for the one-armed spiral instability being seeded at disparate levels across simulations.</caption>
</figure>
<text><location><page_9><loc_69><loc_58><loc_69><loc_60></location>t</text>
<text><location><page_9><loc_70><loc_65><loc_82><loc_66></location>hadron (DD2F)</text>
<text><location><page_9><loc_70><loc_63><loc_89><loc_64></location>hadron-quark (BBKF1.5)</text>
<text><location><page_9><loc_70><loc_60><loc_71><loc_61></location>10</text>
<text><location><page_9><loc_70><loc_57><loc_71><loc_60></location>-</text>
<text><location><page_9><loc_71><loc_60><loc_72><loc_61></location>0</text>
<text><location><page_9><loc_72><loc_58><loc_72><loc_60></location>t</text>
<text><location><page_9><loc_72><loc_58><loc_76><loc_59></location>merger</text>
<text><location><page_9><loc_77><loc_58><loc_80><loc_60></location>(ms)</text>
<text><location><page_9><loc_8><loc_12><loc_92><loc_46></location>In Fig. 1 we depict an example of GW quantities that exhibit the suppression of the one-armed spiral instability for simulations that employ hadron-quark EOSs. In particular, we calculate the GW energy carried in the l = 2 , m = 1 mode. In Fig. 1 we show E 2 , 1 GW normalized to its value at a time shortly after the merger; we depict normalized quantities because of the variable nature in which the one-armed spiral instability is seeded in the immediate post-merger environment in the context of numerical studies. Unless it is explicitly excited as a non-axisymmetric perturbation of a known amplitude (e.g., as a fixed-amplitude perturbation in the rest mass density), the one-armed spiral instability arises numerically from error at the level of floating-point precision (Espino et al. 2019). As such, small differences in the early post-merger evolution of the fluid can result in the instability being seeded at different strengths. We do not explicitly seed the one-armed spiral instability using fluid perturbations in this work and, as a result, simulations that either run on different machines, use different grid resolutions, or use different numerical libraries result in different strengths for the initial instability seed. In Fig. 3 we show the energy in the l = 2 , m = 1 GW mode E 2 , 1 GW as a function of time for a set of low resolution simulations used to produce the error bars of Fig. 1. The left panel of Fig. 3 shows E 2 , 1 GW as extracted from our simulations and appears to show that the simulation employing a hadron-quark EOS produces a larger energy in the l = 2 , m = 1 GWmode. However, it is clear the energy at a time shortly after the merger E 2 , 1 GW ( t merger + glyph[epsilon1] ) (where glyph[epsilon1] is a small additive time) is larger for the hadron-quark simulation, suggesting that the one-armed spiral instability was seeded at a larger amplitude in that case. In order to account for the different levels at which the one-armed spiral instability is seeded in the immediate post-merger environment, we normalize the quantities depicted in Fig. 1 at a time shortly after the merger t norm = t mer + glyph[epsilon1] . We find that setting glyph[epsilon1] = 0 . 5 ms results in all simulations in our work having roughly equal values of E 2 , 1 GW in the few ms immediately following merger. Normalizing at a time shortly after merger ensures that all simulations have approximate parity in the level at which the one-armed spiral instability is seeded and leads to the robust trend established in the right panel of Fig. 1, regardless of grid resolution used.</text>
<text><location><page_9><loc_85><loc_60><loc_85><loc_61></location>1</text>
<text><location><page_9><loc_83><loc_60><loc_85><loc_61></location>10</text>
<text><location><page_9><loc_54><loc_81><loc_56><loc_82></location>10</text>
<text><location><page_9><loc_54><loc_78><loc_56><loc_79></location>10</text>
<text><location><page_9><loc_54><loc_75><loc_56><loc_76></location>10</text>
<text><location><page_9><loc_54><loc_72><loc_56><loc_74></location>10</text>
<text><location><page_9><loc_54><loc_70><loc_56><loc_71></location>10</text>
<text><location><page_9><loc_54><loc_67><loc_56><loc_68></location>10</text>
<text><location><page_9><loc_54><loc_64><loc_56><loc_65></location>10</text>
<text><location><page_9><loc_54><loc_61><loc_56><loc_63></location>10</text>
<text><location><page_9><loc_56><loc_81><loc_56><loc_82></location>7</text>
<text><location><page_9><loc_56><loc_79><loc_56><loc_79></location>6</text>
<text><location><page_9><loc_56><loc_76><loc_56><loc_77></location>5</text>
<text><location><page_9><loc_56><loc_73><loc_56><loc_74></location>4</text>
<text><location><page_9><loc_56><loc_70><loc_56><loc_71></location>3</text>
<text><location><page_9><loc_56><loc_67><loc_56><loc_68></location>2</text>
<text><location><page_9><loc_56><loc_65><loc_56><loc_66></location>1</text>
<text><location><page_9><loc_56><loc_62><loc_56><loc_63></location>0</text>
<text><location><page_9><loc_56><loc_60><loc_58><loc_61></location>10</text>
<text><location><page_9><loc_58><loc_60><loc_59><loc_61></location>-</text>
<text><location><page_9><loc_51><loc_77><loc_53><loc_77></location>)</text>
<text><location><page_9><loc_52><loc_74><loc_53><loc_76></location>norm</text>
<text><location><page_9><loc_51><loc_74><loc_53><loc_74></location>t</text>
<text><location><page_9><loc_51><loc_73><loc_53><loc_74></location>(</text>
<text><location><page_9><loc_51><loc_72><loc_52><loc_73></location>1</text>
<text><location><page_9><loc_51><loc_72><loc_52><loc_72></location>,</text>
<text><location><page_9><loc_51><loc_71><loc_52><loc_72></location>2</text>
<text><location><page_9><loc_52><loc_71><loc_53><loc_73></location>GW</text>
<text><location><page_9><loc_51><loc_70><loc_53><loc_71></location>/E</text>
<text><location><page_9><loc_51><loc_69><loc_52><loc_69></location>1</text>
<text><location><page_9><loc_51><loc_68><loc_52><loc_69></location>,</text>
<text><location><page_9><loc_51><loc_68><loc_52><loc_68></location>2</text>
<text><location><page_9><loc_52><loc_68><loc_53><loc_69></location>GW</text>
<text><location><page_9><loc_51><loc_67><loc_53><loc_68></location>E</text>
<text><location><page_9><loc_59><loc_60><loc_60><loc_61></location>1</text>
</document>
[
{
"title": "ABSTRACT",
"content": "We use numerical relativity simulations of binary neutron star mergers to show that high density deconfinement phase transitions (PTs) to quark matter can be probed using multimodal postmerger gravitational wave (GW) spectroscopy. Hadron-quark PTs suppress the one-armed spiral instability in the remnant. This is manifested in an anti-correlation between the energy carried in the l = 2 , m = 1 GWmode and energy density gap which separates the two phases. Consequently, a single measurement of the signal-to-noise ratios of the l = 2 , m = 1 and l = 2 , m = 2 GWmodes could constrain the energy density gap of the PT. Keywords: Neutron stars - Equation of state - Gravitational waves - Hydrodynamics - Instabilities",
"pages": [
1
]
},
{
"title": "Revealing Phase Transition in Dense Matter with Gravitational Wave Spectroscopy of Binary Neutron Star Mergers",
"content": "Pedro L. Espino, 1, 2 Aviral Prakash, 1, 3 David Radice, 1, 3, 4, ∗ and Domenico Logoteta 5, 6 1 Institute for Gravitation & the Cosmos, The Pennsylvania State University, University Park, PA 16802 2 Department of Physics, University of California, Berkeley, CA 94720, USA 3 Department of Physics, The Pennsylvania State University, University Park, PA 16802 4 Department of Astronomy & Astrophysics, The Pennsylvania State University,University Park, PA 16802 5 Dipartimento di Fisica, Università di Pisa, Largo B. Pontecorvo, 3 I-56127 Pisa, Italy 6 INFN, Sezione di Pisa, Largo B. Pontecorvo, 3 I-56127 Pisa, Italy",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "Binary neutron star (BNS) mergers produce some of the most extreme conditions in nature, compressing matter to several times the nuclear density and to temperatures of tens of MeV (Perego et al. 2019). More extreme conditions are only found in the early Universe and in the interior of black holes (BHs). Multimessenger observations of binary neutron star (BNS) mergers can be used to probe the properties of matter in these conditions, providing a unique avenue to study the nonperturbative regime of QCD (Shibata 2005; Hinderer et al. 2010; Damour et al. 2012; Sekiguchi et al. 2011; Hotokezaka et al. 2011; Bauswein et al. 2013; Radice et al. 2017; Abbott et al. 2017a; Margalit & Metzger 2017; Bauswein et al. 2017; Radice et al. 2018b; Most et al. 2019, 2020; Bauswein et al. 2019; Coughlin et al. 2019; De et al. 2018; Abbott et al. 2019, 2018; Radice & Dai 2019; Dietrich et al. 2020; Breschi et al. 2021, 2022; Kashyap et al. 2022; Perego et al. 2022; Fujimoto et al. 2022; Prakash et al. 2021). Presently, there are large uncertainties in the fundamental physics of strongly-interacting matter at densi- ∗ Alfred P. Sloan fellow ties of a few times nuclear saturation (Capano et al. 2020; Pang et al. 2021; Annala et al. 2022). It is not even clear what the relevant degrees of freedom are for the densities and temperatures reached in the core of remnant massive neutron stars (RMNS) of BNS mergers. It is possible that matter remains composed of nucleons, together with leptons (electrons, positrons, and muons) and photons (Perego et al. 2019; Loffredo et al. 2022). The appearance of more exotic baryons, such as hyperons, is not excluded (Sekiguchi et al. 2011; Radice et al. 2017; Logoteta 2021). It is also possible for a transition to the deconfined quark-gluon plasma phase to take place in BNS mergers (Most et al. 2019, 2020; Bauswein et al. 2019; Prakash et al. 2021). The determination of the state of matter formed in BNS mergers is one of the most pressing scientific objectives of multimessenger astronomy (Evans et al. 2021; Lovato et al. 2022). Previous work has shown that the presence of phase transitions to deconfined quarks can be revealed by a shift of the postmerger gravitational wave (GW) peak frequency f 2 from the value expected for hadronic equations of state (EOSs) (Bauswein et al. 2019; Weih et al. 2020; Blacker et al. 2020; Kedia et al. 2022). However, such frequency shifts can be degenerate with deviations from universal relations due to hadronic physics or other effects (Most et al. 2019; Weih et al. 2020; Liebling et al. 2021; Prakash et al. 2021; Fujimoto et al. 2022; Tootle et al. 2022). It has also been suggested that the presence of a phase transition could be inferred from a measurement of the threshold mass for prompt collapse of BNS systems (Bauswein et al. 2020, 2021; Perego et al. 2022; Kashyap et al. 2022). In this Letter , we use 8 stateof-the-art numerical relativity simulations to show, for the first time, that the presence and strength of a QCD phase transition could be unambiguously determined through multimodal GW spectroscopy of RMNS. Such measurements will be possible with the next-generation of GW experiments like Cosmic Explorer (Reitze et al. 2019), Einstein Telescope (Punturo et al. 2010), and NEMO (Ackley et al. 2020).",
"pages": [
1,
2
]
},
{
"title": "2. METHODS",
"content": "We consider binaries in quasi-circular orbits and eccentric encounters on nearly parabolic orbits. Although BNS mergers with highly eccentric orbits are expected to be significantly more rare than those with quasi-circular inspirals, these events may still have appreciable rates of as high as 50 Gpc -3 yr -1 (Lee et al. 2010; Paschalidis et al. 2015); we include results from both types of mergers to consider as wide a variety of scenarios as possible. Initial data for the quasi-circular binaries is created using the conformal thin sandwich formalism (York 1999) and assuming a helical Killing vector and irrotational flows. The resulting elliptic equations are solved using the pseudo-spectral code LORENE (Gourgoulhon et al. 2001; Taniguchi et al. 2001; Taniguchi & Gourgoulhon 2002). Initial data for the eccentric encounters is constructed by superimposing two isolated, boosted, neutron stars, following Radice et al. (2016b). The initial separation of the stellar barycenters for parabolic encounters is set to 100 km , which is sufficiently large so that the level of constraint violation in the initial data is comparable to that of the quasi-circular binaries (Radice et al. 2016b). We perform BNS merger simulations using the WhiskyTHC code (Radice & Rezzolla 2012; Radice et al. 2014a,b). WhiskyTHC makes use of the CTGamma spacetime solver (Pollney et al. 2011), which is a part of the Einstein Toolkit (Zlochower et al. 2022). The adaptive mesh refinement driver Carpet (Schnetter et al. 2004) is used to generate the dynamical grid structure employed in the simulations. All simulations considered in the present work have been performed using at least two grid resolutions. Although there are quantitative differences in the GW waveforms computed at different resolutions, the qualitative features discussed here are robust across all simulations. Unless otherwise specified, we discuss results from simulations using the fiducial grid resolution (with grid spacing ∆ x glyph[similarequal] 184 . 6 m in the finest refinement level). The grid structure for the simulations is described in detail in Radice et al. (2018a) and Radice et al. (2016b) for the quasi-circular and eccentric simulations, respectively. For a clear understanding of the role that high-density deconfinement phase transitions could play in the development of the one-armed spiral instability, we consider a total of 7 EOS models and run a total of 8 simulations with varying phase transition features. In particular, the size of the energy density gap which separates the hadronic and quark phases is a useful way to classify hybrid hadron-quark EOS models and provides a qualitative measure of the 'strength' of the phase transition (Alford & Han 2016). As such, we consider EOS models that cover several sizes of the energy density gap, ranging from non-existent (i.e., a purely hadronic EOS) to large, while maintaining consistency with current astrophysical constraints on the dense matter EOS. We consider both phenomenological EOS models (Paschalidis et al. 2018; Alvarez-Castillo & Blaschke 2017; Alford & Sedrakian 2017; Bozzola et al. 2019; Espino & Paschalidis 2022) (in the form of piecewise polytropic approximations using the prescription of Read et al. (2009)) and microphysical, finite temperature EOS models (Bastian 2021; Bombaci & Logoteta 2018; Logoteta et al. 2021; Prakash et al. 2021). We only consider equal-mass ratio binary configurations, with the total binary mass ranging from 2 . 6 M glyph[circledot] -2 . 7 M glyph[circledot] . The lack of π -rotational symmetry in BNS configurations with unequal-masses may be a suitable way of effectively seeding non-axisymmetric fluid instabilities that can take hold in the post-merger environment. Neutrino emission and reabsorption are not included for binaries in eccentric orbits, while all quasi-circular binaries include a neutrino treatment via the moment based M0 scheme (Radice et al. 2018a). However, neutrinos are not expected to influence the dynamics on the time scales considered in our study (Radice et al. 2020, 2022). Additionally, magnetic fields are not accounted for in any of our simulations, but these are also expected to be subdominant (Palenzuela et al. 2022). We find that, despite the diversity in binary properties and differences in the evolution, the effects presented in this work are robust.",
"pages": [
2
]
},
{
"title": "3. RESULTS",
"content": "The one-armed spiral instability is a nonaxisymmetric mode in a rapidly rotating fluid which, when saturated, leads to the dominance of a single - high-density mode in the fluid density which is displaced from the fluid barycenter (Pickett et al. 1996; Centrella et al. 2001; Saijo et al. 2002; Ou & Tohline 2006). The one-armed spiral instability has been observed to develop commonly in BNS merger simulations that produce long-lived, massive post-merger remnants on timescales of O (10ms) (Paschalidis et al. 2015; East et al. 2016; East et al. 2016; Radice et al. 2016a; Lehner et al. 2016) and in simulations of many other astrophysical systems including supernovae (Ott et al. 2005; Kuroda et al. 2014), white dwarfs (Kashyap et al. 2015, 2017) and accretion disks (Kashyap et al. 2017; Wessel et al. 2021). Each fluid density mode that arises during the evolution of a massive NS remnant is associated with GW emission at characteristic frequencies stemming from its pattern speed. As such, the development of the one-armed spiral instability in astrophysical systems may be observed by considering multimodal GW spectroscopy (Radice et al. 2016a). For the simulations considered in this work we extract multimodal GW information within the Newman-Penrose formalism. We compute the coefficients of s = -2 spin-weighted spherical harmonic decompositions of the Newman-Penrose scalar Ψ 4 which we label as Ψ l,m 4 . The one-armed spiral instability can therefore be observed in the GW spectrum extracted from our simulations as a growth in the power and amplitude of the l = 2 , m = 1 GW mode (i.e., Ψ 2 , 1 4 ) and simultaneous decay of the dominant l = 2 , m = 2 GW mode (i.e., Ψ 2 , 2 4 ). Our simulations show that high-density deconfinement phase transitions act to suppress the one-armed spiral instability . Depending on the features of the phase transition, the one-armed spiral mode may either arise on a significantly longer timescale when compared to simulations which employ purely hadronic EOS models, or it may be suppressed altogether on the timescales probed by our simulations. There are several potential mechanisms via which the instability may be suppressed. For example, it has been shown that the physical extent of the remnant plays an important role in the development of the instability, with larger remnants being more conducive to the development of the instability on shorter timescales (Radice et al. 2016a; Lehner et al. 2016; Saijo & Yoshida 2016; Saijo 2018). The significant softening at high densities introduced by the phase transition results in more compact post-merger remnants (relative to scenarios that consider only hadronic degrees of freedom). As such, the more compact hybrid star remnants may see a weaker development of the one-armed spiral instability when compared to neutron star remnants. In the left panel of Fig. 1 we show the energy carried by the l = 2 , m = 1 GW mode (scaled by the energy emitted at a time shortly after the merger t norm = t mer +0 . 5 ms ) as a function of time for simulations employing the DD2F (hadronic) and DD2F-SF5 (hybrid √ √ √ √ hadron-quark) EOSs. We find that the energy carried in the l = 2 , m = 1 mode of the GWs is significantly smaller in the simulation employing a hybrid hadronquark EOS, indicating that the one-armed spiral instability is suppressed in scenarios with deconfinement phase transitions at densities relevant for BNS mergers. We emphasize that in the left panel of Fig. 1 we showcase results for a set of EOS models which are identical below the threshold for a phase transition, and as such the simulations have identical initial conditions. In the right panel of Fig. 1, we show the time-averaged energy emitted by the l = 2 , m = 1 GW mode 〈 E 2 , 1 GW 〉 (again scaled by the energy emitted at a time shortly after the merger t norm = t mer +0 . 5 ms ) as a function of the energy density gap ∆ e , where we define the energy density gap as the difference between the energy density e at the end of the hadronic phase and beginning of the quark phase for cold matter in β -equilibrium (Alford & Han 2016), where we assume units where the speed of light c = 1 . We identify the end and beginning of each phase by considering the change in the approximate adiabatic index Γ = dlog p/ dlog( ρ ) , where p is the fluid pressure of the cold, beta-equilibrium, barotropic EOS for each EOS model considered. The region corresponding to the phase transition is always unambiguously marked by discontinuities in, or sudden changes in the slope of, the adiabatic index for the EOS models we consider. For the results depicted in the right panel of Fig. 1, we time-average over a window of ∆ t ≈ 40 ms after the merger except for cases that lead to a remnant collapse on shorter timescales (in such cases, we time-average until the collapse of the NS remnant). Additionally, for each simulation, we normalize by a complementary simulation that uses identical initial data but employs a hadronic EOS having the same low-density behavior below the phase transition threshold as the hybrid hadronquark EOS. As such, we depict the point corresponding to all hadronic EOS simulations with a black square at ∆ e = 0 . Each simulation is time-averaged to the same extent as its complementary hadronic simulation. We find an anti-correlation between the energy carried in the l = 2 , m = 1 GWmode and the size of the energy density gap . In other words, as the size of the energy density gap (and thereby the qualitative 'strength' of the phase transition) increases, GW emission in the l = 2 , m = 1 mode decreases, which signifies that the one-armed spiral instability is further suppressed for EOS models with 'stronger' deconfinement phase transitions. We elaborate on the choice of quantities depicted in Fig. 1 in App. A.",
"pages": [
2,
3,
4
]
},
{
"title": "4. DISCUSSION",
"content": "The characteristic frequency associated with peak emission in the l = 2 , m = 1 GW mode has half the value of that associated with the l = 2 , m = 2 mode (i.e., f 2 , 1 peak = f 2 , 2 peak / 2 ). Observationally, a GW signal would contain information at all contributing frequencies. However, the dominant GW emission associated with binary coalescence is always expected to be from the l = 2 , m = 2 contribution, such that f peak = f 2 , 2 peak . Therefore, a potential observational signature of the one-armed spiral instability is the growth in power of an incoming GW signal at a frequency that is half of the dominant frequency; if it develops in the post-merger environment, the one-armed spiral instability will continuously power the emission of GWs at f peak / 2 , while emission in the dominant f peak decays in time (Bernuzzi et al. 2015). In Fig. 2 we show the post-merger GW amplitude spectrum density (ASD) for a symmetric, edge-on binary situated at a distance of 40 Mpc , which is consistent with the luminosity distance observed for GW170817 (Abbott et al. 2017b). The edge-on configuration is the most optimal for the detection of an m = 1 mode. As expected, we see a relative suppression of power in the m = 1 mode (with respect to the complementing hadronic simulation) with the onset of a deconfinement phase transition. In this realistic configuration, coupled with the 40 km Cosmic Explorer detector (Reitze et al. 2019), the appearance of quarks in the post-merger remnant results in a suppression of the postmerger signal-to-noise ratio (SNR) of the ( l = 2 , m = 1) mode by a factor of 2, from 2.14 in the hadronic case to 1.08 in the hadron-quark case. The GW ASD peak of the l = 2 , m = 1 mode (between 1-2 kHz ) and the postmerger ASD peak of the l = 2 , m = 2 mode (between 2-4 kHz ), lie respectively in the most sensitive regions of the 40 km and the 20 km postmerger optimized Cosmic Explorer configurations. Our analysis recommends an increase in detector sensitivities in the high-frequency regimes (see also Zhang et al. (2022)) for best possible constraints on deconfinement phase transitions in BNS mergers. In this letter we have highlighted, for the first time, that high-density deconfinement phase transitions act to suppress the one-armed spiral instability. We find an anti-correlation between the energy carried in the l = 2 , m = 1 GW mode and the size of the energy density gap which qualitatively separates the hadronic and quark phases. Our findings reveal a deep connection between observable multimodal GW emission and the microphysical description of matter in the post-merger environment. We expect the one-armed spiral instability to be detectable at distances of 40 Mpc using future generation detectors (Radice et al. 2016a). If evidence of a strong one-armed spiral mode can be inferred from GW observations of the post-BNS merger environment, our findings suggest that a strong high-density deconfinement phase transition at the densities relevant to BNS mergers would be disfavored. On the other hand, if evidence for the one-armed spiral instability is not found for close-by BNS mergers, this could also point to the possibility of a deconfinement phase transition taking place at densities relevant to BNS mergers. We point out that other effects relevant in the postmerger environment - such as the presence of strong magnetic fields (Franci et al. 2013) and additional degrees of freedom that can cause a sudden softening of the EOS - may affect the development of the one-armed spiral instability. However, the relevant timescales and extent to which the aforementioned phenomena can affect the development of non-axisymmetric instabilities or the GW spectrum remains uncertain (Radice et al. 2016a; Muhlberger et al. 2014), and may not impact our conclusions (Palenzuela et al. 2022; Zappa et al. 2022). The effects discussed in the present work arise on dynamical timescales ∼ O (10 ms) , and may be the dominant mechanism for suppression of the one-armed spiral instability. Additionally, although we find a trend in the decrease of energy carried by the l = 2 , m = 1 GW mode for larger values of ∆ e , additional studies will help establish a more robust trend and provide an understanding of the potential spread in the trend. In particular, future lines of investigation will include: (1) considering the combined effects of the mass ratio and high-density phase transitions on the development of the one-armed spiral instability; (2) considering the effects of accurate neutrino transport on high-density deconfinement phase transitions, as neutrinos may modify the composition of matter and thereby potentially affect the onset of the phase transition; (3) employing EOS models at systematically increasing values of ∆ e while holding the hadronic region of the EOS fixed, as a limitation of the present work is the assumption that the l = 2 , m = 1 GW mode is perfectly known in the case of hadronic EOSs; and (4) investigating the effects discussed in this work in scenarios with a crossover to quark matter, as our present work only considers EOS models with phase transitions. We leave such studies to future work. PE acknowledges funding from the National Science Foundation under Grant No. PHY-2020275. DR acknowledges funding from the U.S. Department of Energy, Office of Science, Division of Nuclear Physics under Award Number(s) DE-SC0021177 and from the National Science Foundation under Grants No. PHY-2011725, PHY-2116686, and AST-2108467. Simulations were performed on Bridges2 and Expanse (NSF XSEDE allocation TG-PHY160025). This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.",
"pages": [
4,
5
]
},
{
"title": "REFERENCES",
"content": "Zlochower, Y., Brandt, S. R., Diener, P., et al. 2022, The Einstein Toolkit, The \"Berhard Riemann\" release, ET_2022_05, Zenodo, doi: 10.5281/zenodo.6588641 -",
"pages": [
8,
9
]
},
{
"title": "A. GW PROBE OF THE ONE-ARMED SPIRAL INSTABILITY",
"content": "t hadron (DD2F) hadron-quark (BBKF1.5) 10 - 0 t merger (ms) In Fig. 1 we depict an example of GW quantities that exhibit the suppression of the one-armed spiral instability for simulations that employ hadron-quark EOSs. In particular, we calculate the GW energy carried in the l = 2 , m = 1 mode. In Fig. 1 we show E 2 , 1 GW normalized to its value at a time shortly after the merger; we depict normalized quantities because of the variable nature in which the one-armed spiral instability is seeded in the immediate post-merger environment in the context of numerical studies. Unless it is explicitly excited as a non-axisymmetric perturbation of a known amplitude (e.g., as a fixed-amplitude perturbation in the rest mass density), the one-armed spiral instability arises numerically from error at the level of floating-point precision (Espino et al. 2019). As such, small differences in the early post-merger evolution of the fluid can result in the instability being seeded at different strengths. We do not explicitly seed the one-armed spiral instability using fluid perturbations in this work and, as a result, simulations that either run on different machines, use different grid resolutions, or use different numerical libraries result in different strengths for the initial instability seed. In Fig. 3 we show the energy in the l = 2 , m = 1 GW mode E 2 , 1 GW as a function of time for a set of low resolution simulations used to produce the error bars of Fig. 1. The left panel of Fig. 3 shows E 2 , 1 GW as extracted from our simulations and appears to show that the simulation employing a hadron-quark EOS produces a larger energy in the l = 2 , m = 1 GWmode. However, it is clear the energy at a time shortly after the merger E 2 , 1 GW ( t merger + glyph[epsilon1] ) (where glyph[epsilon1] is a small additive time) is larger for the hadron-quark simulation, suggesting that the one-armed spiral instability was seeded at a larger amplitude in that case. In order to account for the different levels at which the one-armed spiral instability is seeded in the immediate post-merger environment, we normalize the quantities depicted in Fig. 1 at a time shortly after the merger t norm = t mer + glyph[epsilon1] . We find that setting glyph[epsilon1] = 0 . 5 ms results in all simulations in our work having roughly equal values of E 2 , 1 GW in the few ms immediately following merger. Normalizing at a time shortly after merger ensures that all simulations have approximate parity in the level at which the one-armed spiral instability is seeded and leads to the robust trend established in the right panel of Fig. 1, regardless of grid resolution used. 1 10 10 10 10 10 10 10 10 10 7 6 5 4 3 2 1 0 10 - ) norm t ( 1 , 2 GW /E 1 , 2 GW E 1",
"pages": [
9
]
}
]
2024PhRvD.109l3513N
https://arxiv.org/pdf/2309.10554.pdf
<document>
<section_header_level_1><location><page_1><loc_9><loc_90><loc_92><loc_93></location>Signature of Non-Minimal Scalar-Gravity Coupling with an Early Matter Domination on the Power Spectrum of Gravitational Waves</section_header_level_1>
<text><location><page_1><loc_10><loc_87><loc_91><loc_89></location>Amirsalar Nikandish, 1, ∗ Shiva Rostam Zadeh, 2, † Reza Naderi, 3, ‡ Fatemeh Elahi, 4, § and Hadi Mehrabpour 4, 5, 6, ¶</text>
<text><location><page_1><loc_24><loc_86><loc_24><loc_87></location>1</text>
<text><location><page_1><loc_19><loc_75><loc_82><loc_86></location>Department of Physics, Shahid Beheshti University, 1983969411, Tehran, Iran 2 School of Particles and Accelerators, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran 3 Department of Physics, Faculty of Basic Sciences, Azarbaijan Shahid Madani University, 53714-161, Tabriz, Iran 4 PRISMA + Cluster of Excellence & Mainz Institute for Theoretical Physics, Johannes Gutenberg University, 55099 Mainz, Germany 5 Center for High Energy Physics, Peking University, Beijing 100871, China 6 Department of Physics and State Key Laboratory of Nuclear Physics and Technology,</text>
<text><location><page_1><loc_36><loc_74><loc_65><loc_75></location>Peking University, Beijing 100871, China</text>
<text><location><page_1><loc_18><loc_57><loc_83><loc_72></location>The signal strength of primordial gravitational waves experiencing an epoch of early scalar domination is reduced with respect to radiation domination. In this paper, we demonstrate that the specific pattern of this reduction is sensitive to the coupling between the dominant field and gravity. When this coupling is zero, the impact of early matter domination on gravitational waves is solely attributed to the alteration of the Hubble parameter and the scale factor. In the presence of non-zero couplings, on the other hand, the evolution of primordial gravitational waves is directly affected as well, resulting in a distinct step-like feature in the power spectrum of the gravitational wave as a function of frequency. This feature serves as a smoking gun signature of this model. In this paper, we provide an analytical expression of the power spectrum that illustrates the dependence of power spectrum on model parameters and initial conditions. Furthermore, we provide analytical relations that specify the frequency interval in which the step occurs. We compare the analytical estimates with numerical analysis and show they match well.</text>
<section_header_level_1><location><page_1><loc_42><loc_50><loc_59><loc_51></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_24><loc_92><loc_48></location>Since the first detection of gravitational waves (GWs) at the Laser Interferometer Gravitational-Wave Observatory (LIGO) and Virgo, a new window to unravel the mysteries of the cosmos has opened [1-4]. Even though the sources of the detected GWs have been astrophysical thus far [5], we hope to also detect the cosmological ones with the advance of the detectors [6]. Among the possible sources of GWs, the stochastic gravitational wave background (SGWB) originating from inflation is of particular interest, because detecting it may shed light on the history of the early universe[7-20]. The standard model of cosmology, under the assumption of radiation domination (RD) from inflation to matter-radiation equality, predicts an almost scale invariant power spectrum for the SGWB across all frequencies [21-24]. Yet, many well-motivated proposals predict a transient phase, dominated by a different energy component, intervening between inflation and the Big Bang Nucleosynthesis (BBN) [25, 26]. Depending on the temperature range and other specific features of this transient period, the SGWB profile is expected to vary. Inspired by numerous extensions to the standard model of particle physics, we focus our attention on early matter domination [27-39]. Specifically, we assume a scalar field, ϕ , which behaves like a pressureless fluid scaling as a -3 with a being the scale factor, causes a transient matter domination era. Once ϕ decays, the universe reverts to the RD era again. Assuming a minimal coupling between ϕ and gravity, SGWB is only affected indirectly and because of the alteration of the scale factor and the Hubble rate [25, 26]. We coin the term Indirect effect to denote this influence. In the context of early matter domination, the SGWB power spectrum is suppressed at high frequencies compared to standard cosmology, and the Equation of State (EoS) of the Universe governs the slope of the power spectrum [26].</text>
<text><location><page_1><loc_9><loc_20><loc_92><loc_24></location>In this paper, we study early matter domination where the dominating field has a non-minimal coupling with gravity; i.e., f ( R,ϕ ) = ( 1 8 πG -ξϕ 2 ) R . In the term ξRϕ 2 , ξ signifies the gravitational coupling constant and R</text>
<text><location><page_2><loc_9><loc_79><loc_92><loc_93></location>denotes the Ricci scalar. This coupling stands as the sole feasible local, scalar interaction of its kind with the appropriate dimensions [40]. A coupling between a scalar field and gravity is proposed in numerous models aiming to resolve some of the inherent problems with inflation, reheating, and baryogenesis [41-48]. For instance, in the context of non-oscillatory inflationary models[49-53], the term ξRϕ 2 has been recently used in an efficient reheating scenario denoted as Ricci reheating[54-57]. In the same context, a novel quintessential Affleck-Dine (AD) baryogenesis scenario has also been proposed[58] based on the term ξRϕ 2 , avoiding troublesome iso-curvature modes found in conventional AD scenarios. Moreover, the widely used self-interaction term for scalar fields, λϕ 4 , necessitates the inclusion of ξRϕ 2 in the Lagrangian for proper renormalization in curved space-time [40, 59]. Within the scalar sector of the Standard Model (SM) Lagrangian (Higgs) [54], ξRϕ 2 emerges as the missing term that upholds all the symmetries of both gravity and the SM.</text>
<text><location><page_2><loc_9><loc_67><loc_92><loc_79></location>As ξ is subject to running, it cannot be universally set to zero across all energy scales [60]. Two values of ξ = 0 for minimal coupling and ξ = 1 / 6 for conformal coupling are of particular interest. The latter, for the case m ϕ = 0, leads to conformal invariance of the action and hence the field equation of motion[40, 61]. Nonetheless, no compelling reason supports the presence of minimal or conformal coupling in the real world, as no symmetry is enhanced [46]. Furthermore, since ξ is dimensionless, there is no reason for it to be small. It could be non-minimal, i.e. of the order of unity or more. 1 [46]. Current experiments place a weak constraint on ξ ( ξ < 10 15 ) due to the feeble gravitational interaction with the SM fields [60, 62] However, GWs might offer insights into the existence and strength of such a coupling.</text>
<text><location><page_2><loc_9><loc_51><loc_92><loc_67></location>This paper delves into SGWB within a cosmological framework characterized by early matter domination, where the dominant field ϕ directly couples with gravity ( ξRϕ 2 ). D'Eramo et al. previously examined the case of ξ = 0 [26]. To highlight distinctions from [26], we specifically explore the non-minimal regime ( ξ ≥ 1). Studies indicate that such gravity couplings manifest as additional terms in the evolution equation of GWs [41, 63-66]. That is, alongside modifications to the scale factor and Hubble, i.e., indirect effect [67], an extra factor directly affects GW evolution. We term this extra impact the direct effect . Employing Wentzel-Kramers-Brillouin (WKB) analysis, we elucidate the power spectrum resulting from GW propagation in our cosmological setting. Our numerical findings demonstrate that the presence of the ξRϕ 2 term, with non-minimal coupling, deepens the kink shape in the power spectrum, compared with the case studied in [26]. Furthermore, owing to the direct effect , an additional step-like feature emerges, resulting in an enhanced reduction in the power spectrum. We highlight several benchmarks that could be probed by upcoming gravitational wave experiments.</text>
<text><location><page_2><loc_48><loc_37><loc_48><loc_38></location≯</text>
<text><location><page_2><loc_9><loc_36><loc_92><loc_51></location>To deepen our understanding of the ξRϕ 2 term, it is crucial to provide an analytical interpretation of the resulting power spectrum. Hence, we use reasonable approximations to get an analytical expression for the power spectrum compared with the case of the standard cosmology and explain how each part of the spectrum in a specific frequency interval is shaped due to a specific physical effect that was dominant in the corresponding temperature interval. To this end, the high-frequency modes that became sub at high temperatures and thus got affected by the changes in the cosmological evolution are of particular interest. In particular, we determine the fraction of the spectrum of high frequencies to that of low-frequency modes that did not experience any changes in the cosmological history. In this context, we utilize a physical quantity called the dilution factor , which was introduced in [26]. While this analytical interpretation was performed for the case of ξ = 0 in [26], exhibiting notable agreement with numerical results, our study expands this analysis to encompass the case of ξ = 0. The analytical formula we derive, expressed in terms of dilution and damping factors, closely aligns with numerical results, maintaining an acceptable level of accuracy.</text>
<text><location><page_2><loc_9><loc_25><loc_92><loc_35></location>Finally, we comment on the potential testability of this scenario, highlighting a selection of gravitational wave experiments that could probe these changes [68]. These include the Laser Interferometer Space Antenna (LISA), which operates within the frequency range of 10 -5 -1 Hz [69-71], the DECi-hertz Interferometer Gravitational wave Observatory (DECIGO) spanning 10 -3 -10 Hz [72-74], the Einstein Telescope (ET) operating within 1 -10 4 Hz [75-77], and the forthcoming Big Bang Observatory (BBO) encompassing 10 -3 -10 Hz[78-80]. Other experiments will explore intermediate frequencies like square kilometer array or SKA, probing 10 -9 -10 -6 Hz [81, 82] and NANOGrav collaboration[83, 84] that works based on pulsar timing arrays (PTA) measurements [64, 68]. 2</text>
<text><location><page_2><loc_9><loc_17><loc_92><loc_25></location>This paper is organized as follows: In Sec. II, we introduce the model and the cosmological framework in detail. In Sec. III, we study the evolution of GWs, highlighting the effect of ξRϕ 2 . We present the WKB solution for the modes that become sub, in Sec. III A. We follow in Sec. III B, by introducing the power spectrum as the observable of GWs, and in Sec. III C, we highlight two of the pivotal frequencies in the power spectrum. Sec. IV is dedicated to numerical results for the power spectrum and a comparison with the analytical estimates. Finally, the concluding remarks are presented in Sec. V.</text>
<section_header_level_1><location><page_3><loc_30><loc_92><loc_71><loc_93></location>II. NON-MINIMALLY COUPLED SCALAR FIELD</section_header_level_1>
<text><location><page_3><loc_9><loc_87><loc_92><loc_90></location>Our theory is defined by a real scalar field ϕ that has feeble interactions with other fields, but has a non-minimal coupling with gravity: f ( R,ϕ ) = ( 1 8 πG -ξϕ 2 ) R . The action of our theory is, thus, [56, 60]: 3</text>
<formula><location><page_3><loc_25><loc_82><loc_92><loc_85></location>S = ∫ d 4 x √ -g ( 1 16 πG R -1 2 g µν ∂ µ ϕ∂ ν ϕ -1 2 ξRϕ 2 -1 2 m 2 ϕ ϕ 2 + L M ) , (1)</formula>
<text><location><page_3><loc_9><loc_76><loc_92><loc_81></location>where g is the determinant of the metric g µν , G is the gravitational coupling constant, m ϕ is the mass of ϕ , and L M contains the kinetic component and the interactions of any other field in the cosmos, including its interaction with the scalar field, ϕ . Varying Eq. (1) with respect to g µν , one can obtain the gravitational equation as [60, 63]:</text>
<formula><location><page_3><loc_36><loc_73><loc_92><loc_75></location>G µν = 8 πGT (eff) µν = 8 πG ( T ( M ) µν + T ( ϕ ) µν ) . (2)</formula>
<text><location><page_3><loc_9><loc_70><loc_66><loc_71></location>In the definition of Eq. (2), the energy-momentum tensor of the scalar filed ϕ is:</text>
<formula><location><page_3><loc_24><loc_66><loc_92><loc_69></location>T ( ϕ ) µν = ∂ µ ϕ∂ ν ϕ -g µν ( 1 2 ∂ γ ϕ∂ γ ϕ + 1 2 m 2 ϕ ϕ 2 ) + ξ ( G µν + g µν □ -∇ µ ∇ ν ) ϕ 2 , (3)</formula>
<text><location><page_3><loc_9><loc_61><loc_92><loc_65></location>where we have □ ϕ 2 = 1 √ -g ∂ µ ( √ -g∂ µ ϕ 2 ). Varying Eq. (1) with respect to ϕ yields the scalar field equation of motion [63]:</text>
<formula><location><page_3><loc_40><loc_57><loc_92><loc_60></location>□ ϕ +( ξR + m 2 ϕ ) ϕ = -∂ L M ∂ϕ . (4)</formula>
<text><location><page_3><loc_9><loc_49><loc_92><loc_56></location>Using Eqs. (2, 4) and the Bianchi identity, ∇ µ G µ ν = ∇ µ T µ ( eff ) ν = 0, the continuity equation of the matter part can be obtained, ∇ µ T µ ( M ) ν = ∂ L M ∂ϕ ∂ ν ϕ [63]. In these equations, the field ϕ is only a function of time in order to respect the homogeneity and isotropy of the Universe, and L M describes the radiation. Therefore, the evolution equations for the scalar field ϕ and the energy density of radiation ρ R are:</text>
<formula><location><page_3><loc_38><loc_46><loc_92><loc_48></location>¨ ϕ +(3 H +Γ) ˙ ϕ +( ξR + m 2 ϕ ) ϕ = 0 , (5)</formula>
<formula><location><page_3><loc_47><loc_43><loc_92><loc_46></location>˙ ρ R +4 g s ⋆ g ρ ⋆ Hρ R = Γ ˙ ϕ 2 , (6)</formula>
<text><location><page_3><loc_9><loc_31><loc_92><loc_41></location>where to obtain these equations, we have employed the flat Friedmann-Lemaitre-Robertson-Walker (FLRW) metric, ds 2 = -dt 2 + a ( t ) 2 δ ij dx i dx j , 4 with i, j = 1 -3 specifying the spatial coordinates, explains the metric of a homogeneous and isotropic Universe, a is the cosmic scale factor, and accordingly the Ricci scalar is obtained as R = 6(˙ a 2 + a a ) /a 2 [63]. Furthermore, Γ represents the total decay rate of the scalar. In this paper, we are oblivious to the exact nature of the fields into which ϕ decays, and rather we assume it is part of radiation. The total number of relativistic degrees of freedom contributing to the energy (entropy) density of radiation is represented by g ρ ⋆ ( g s ⋆ ). The evolution of the Hubble rate, defined as H ≡ ˙ a a , is described by the first Friedmann equation, obtained from Eq. (2) as:</text>
<formula><location><page_3><loc_38><loc_27><loc_92><loc_30></location>H 2 = 1 3 M 2 Pl ρ tot = 1 3 M 2 Pl ( ρ ϕ + ρ R ) , (7)</formula>
<text><location><page_3><loc_9><loc_23><loc_92><loc_26></location>where M Pl is the reduced Planck mass, and the energy density of ϕ can be obtained from the T ( ϕ ) 00 component of Eq. (3) [56, 60]:</text>
<formula><location><page_3><loc_35><loc_19><loc_92><loc_21></location>ρ ϕ = 1 2 ˙ ϕ 2 + 1 2 m 2 ϕ ϕ 2 + ξ ( 3 H 2 ϕ 2 +6 Hϕ ˙ ϕ ) . (8)</formula>
<text><location><page_4><loc_9><loc_90><loc_92><loc_93></location>Using Eqs. (5), (6), and (7), the system of coupled differential equations for the background would be complete for three unknowns, ϕ, ρ R , and H . The free parameters in these equations are</text>
<formula><location><page_4><loc_30><loc_88><loc_70><loc_90></location>T in , ϕ in ≡ ϕ ( T in ) , ˙ ϕ in ≡ ˙ ϕ ( T in ) , ξ, m ϕ , Γ .</formula>
<text><location><page_4><loc_9><loc_79><loc_92><loc_87></location>In this project, we fix T in = 10 11 GeV, motivated by well-studied reheating scenarios which prefer T Reh ∈ (10 12 -10 15 ) GeV [54, 55, 89-92]. The initial value of ϕ in has an upper bound of M Pl to avoid undergoing the Universe to another inflationary period [26]. In addition, since the most noticeable deviations from standard cosmology occur when ϕ in ∼ M Pl , we choose ϕ in = 10 18 . The value of ξ , also, has an upper bound to make sure the Hubble rate stays real, i.e., ξ < ( M 2 pl /ϕ 2 in ), which in our case gives rise to ξ max ≃ 5 . 95. 5 Since our setup is not sensitive to ˙ ϕ in and any initial condition satisfying</text>
<formula><location><page_4><loc_36><loc_74><loc_92><loc_77></location>1 2 ˙ ϕ 2 in , 1 2 m 2 ϕ ϕ 2 in , 3 ξ 2 ˙ ϕ 2 in ϕ 2 in M 2 Pl -ξϕ 2 in ≪ ρ R ( T in ) , (9)</formula>
<text><location><page_4><loc_9><loc_67><loc_92><loc_73></location>leads to the total equation of state ω ≃ 1 / 3; and hence, a negligible ˙ ϕ (see Appendix A), we fix ˙ ϕ in = 0 for simplicity. Concentrating on one of the benchmarks of Ref. [26], i.e. m ϕ = 10GeV and Γ = 10 -8 GeV, we numerically solve the set of coupled equations and compare the results for the two cases of ξ = 0 (minimal coupling) [26] and ξ ≥ 1 (non-minimal coupling). 6</text>
<section_header_level_1><location><page_4><loc_34><loc_63><loc_66><loc_64></location>A. General Evolution of the Background</section_header_level_1>
<text><location><page_4><loc_9><loc_49><loc_92><loc_61></location>Given our initial conditions at high temperatures, the friction term in Eq. 5 ( more specifically the Hubble term, H ˙ ϕ ) forces ϕ to be stuck at its initial value, similar to the case of ξ = 0 [26]. As the Hubble rate decreases, eventually the friction term becomes less efficient and ϕ starts oscillating when H ≃ m eff ≡ √ ξR + m 2 ϕ . As demonstrated in Fig. 1, the effect of gravitational coupling causes ϕ to remain close to its initial value for a longer duration, because the Hubble rate is higher and m eff , because of negative R , is lower for larger ξ : H ≃ √ ρ R ( T )(3( M 2 Pl -ξϕ 2 in )) -1 (see Appendix A for this simplified expression of Hubble rate at high temperatures). Around the time when ϕ starts oscillating, the universe deviates from state of RD, or equivalently ω ≃ 1 / 3, or equivalently the state of Radiation Domination (RD). The temperatures at which this occurs is</text>
<formula><location><page_4><loc_27><loc_43><loc_92><loc_48></location>T osc ≃ ( 90 i osc π 2 g ρ ∗ osc ) 1 / 4 ( m ϕ M Pl ) 1 / 2 ( 1 -ξ 2 ( ϕ in M Pl ) 4 1 + ξ (6 ξ -1)( ϕ in M Pl ) 2 ) 1 / 4 , (10)</formula>
<text><location><page_4><loc_9><loc_30><loc_92><loc_43></location>when H and m eff lines meet ( H ≃ m eff ). The numerically determined factor i osc ≃ 6 has been implemented to obtain a more precise value for T osc , since ϕ oscillations start slightly after H ≃ m eff (e.g., for ξ = 0, oscillations start at T ∼ 10 9 and for ξ = 5 . 95, they start at T ∼ 10 8 ). After some oscillations, 7 ϕ behaves as sin( m ϕ t ) m ϕ t . Hence, the energy density of ϕ redshifts like cold matter, leading to an era of matter domination ( MD era ). This process continues until the decay of ϕ becomes more efficient than the Hubble rate, at which point ϕ depletes. We refer to this era, the decay era (DE). When ρ ϕ becomes negligible with respect to ρ R , we return to radiation domination, which we will refer to as the Late RD era . The temperature at which we return to late RD era can be found by setting ρ ϕ ∼ ρ R [26]:</text>
<formula><location><page_4><loc_35><loc_26><loc_92><loc_29></location>T RD ≃ ( α RD c RD ) 1 / 2 ( 90 2 π 2 g ρ ∗ RD ) 1 / 4 (Γ M Pl ) 1 / 2 , (12)</formula>
<formula><location><page_4><loc_44><loc_12><loc_92><loc_16></location>˙ ρ ϕ +3 Hρ ϕ = -Γ ρ ϕ , ˙ ρ R +4 g s ⋆ g ρ ⋆ Hρ R = Γ ρ ϕ . (11)</formula>
<text><location><page_4><loc_10><loc_9><loc_92><loc_11></location>Since g s ⋆ and g ρ ⋆ vary with temperature, especially at T EW ≃ 100 GeV ≤ T ≤ T BBN , we evaluate them as a function of ρ R using Ref. [93].</text>
<figure>
<location><page_5><loc_9><loc_69><loc_92><loc_91></location>
<caption>FIG. 1: (a) The behavior of the scalar field, ϕ , in terms of temperature, T . (b) The evolution of the Hubble parameter and the effective mass as a function of temperature. In both plots, the red line is for ξ = 0, and the blue line is for ξ = 5 . 95. As shown in these plots, the Hubble rate increases and m eff decreases with the value of ξ , and therefore the temperature at which H ≃ m eff is lower for larger values of ξ . Thereby, the oscillation of ϕ is delayed with non-minimal coupling of ϕ with gravity.</caption>
</figure>
<text><location><page_5><loc_9><loc_55><loc_92><loc_58></location>where α RD ≈ 0 . 64. 8 Assuming a constant ω throughout the MD era, c RD can be numerically determined as c RD ≃ 1 . 07. At this point, the evolution of Universe is followed by the standard model of cosmology.</text>
<text><location><page_5><loc_9><loc_49><loc_92><loc_55></location>To investigate the differences between minimal ( ξ = 0) and non-minimal ( ξ ≥ 1) ϕ -gravity coupling, GWs are the best candidates. The imprint of ξ = 0 case on GWs has been investigated in Ref. [26]. Due to the non-minimal coupling between ϕ and gravity, the evolution of ϕ significantly impacts in the evolution of GWs, which will be discussed in details in the following section.</text>
<section_header_level_1><location><page_5><loc_24><loc_45><loc_77><loc_46></location>III. THE EVOLUTION OF GWS AND THE POWER SPECTRUM</section_header_level_1>
<text><location><page_5><loc_10><loc_42><loc_80><loc_43></location>Gravitational waves, h ij , are perturbations in the space part of the FLRW metric [13, 15, 16, 94]:</text>
<formula><location><page_5><loc_35><loc_38><loc_92><loc_40></location>ds 2 = -dt 2 + a ( t ) 2 ( δ (3) ij + h ij ( t, x )) dx i dx j , (13)</formula>
<text><location><page_5><loc_9><loc_33><loc_92><loc_37></location>with | h ij | ≪ 1. Taking h ij to be transverse and traceless ( h TT ii = k i h TT ij = 0 ) , there are two degrees of freedom for h ij indicating two polarizations, ζ = + , × . By linearizing the gravitational equation, Eq. (2), the evolution equations of GWs is obtained:</text>
<formula><location><page_5><loc_35><loc_29><loc_92><loc_32></location>∂ 2 h TT ij ∂t 2 -1 a 2 ∇ 2 h TT ij +(3 H + A ) ∂h TT ij ∂t = 0 , (14)</formula>
<text><location><page_5><loc_9><loc_23><loc_92><loc_28></location>where A = -2 ξϕ ˙ ϕ ( 1 8 πG -ξϕ 2 ) -1 . It can be seen that an additional term, A , appears in the well-known evolution equations of the GWs [13-16], which comes from the ξRϕ 2 term. It is through this term that the evolution of ϕ directly impacts the evolution of the SGWB.</text>
<text><location><page_5><loc_9><loc_19><loc_92><loc_23></location>It is conventional to expand h TT ij as h TT ij ( t, x ) = ∑ ζ e ( ζ ) ij h ( ζ ) ( t, x ) where e ( ζ ) ij is the polarization tensor[13, 16, 94]. 9 Substituting the polarization expansion and h ( ζ ) ( t, x ) = ∫ d 3 k (2 π ) 3 e ik · x h ( ζ ) k ( t ) in Eq. (14), the evolution equation for the</text>
<figure>
<location><page_6><loc_26><loc_66><loc_77><loc_93></location>
<caption>FIG. 2: This plot illustrates the rates of 2 aH and aA , which represent the damping factors in the evolution of GWs, for the case of ξ = 5 . 95. At high temperatures, A ∝ ˙ ϕ is zero. However, as ϕ gains momentum, A rapidly grows and surpasses H within certain temperature intervals. In addition, the ripples of 2 aH and aA become comparable at the onset of the oscillation era, and thus cancel each other out.</caption>
</figure>
<text><location><page_6><loc_9><loc_54><loc_47><loc_56></location>amplitude of GWs, h k is obtained as following [63]: 10</text>
<formula><location><page_6><loc_39><loc_51><loc_92><loc_54></location>h k +(3 H + A ) ˙ h k + k 2 a 2 h k = 0 , (15)</formula>
<text><location><page_6><loc_9><loc_47><loc_92><loc_50></location>where h k ( t ) represents the Fourier transform of h ( x, t ). For a better understanding of the behavior of GWs, Eq. (15) is expressed in terms of conformal time, η , given by η = ∫ dt a , as in Refs. [64-66]:</text>
<formula><location><page_6><loc_36><loc_44><loc_92><loc_46></location>h '' k ( η ) + a (2 H + A ) h ' k ( η ) + k 2 h k ( η ) = 0 , (16)</formula>
<text><location><page_6><loc_9><loc_36><loc_92><loc_43></location>with the prime denoting differentiation with respect to conformal time. In what follows we do not mention the subscript k, since the term k 2 can highlight that the equation is written for a specific momentum. The above equation is a harmonic oscillator with the damping term a 2 (2 H + A ). Using the numerical results of the background as obtained in Sec. II, 11 we distinguish different regimes in the behavior of the damping term, by comparing the evolution of the terms aA and 2 aH as depicted in Fig. 2:</text>
<unordered_list>
<list_item><location><page_6><loc_11><loc_27><loc_92><loc_35></location>· A ≳ 2 H : Even though A is initially zero ( A ∝ ˙ ϕ and ˙ ϕ in = 0), it rapidly grows as ϕ gains momentum. Depending on the value of ξ and our initial conditions, A may eventually exceed the Hubble rate. We refer to this era as A Domination era (AD era) for short. 12 During AD era, the evolution of GWs is not only affected by the altered background but also directly by A . To highlight the effect of A , we refer to its contribution as the direct effect . Note that the direct effect goes to zero as ξ → 0. Conversely, for large values of ξ A remains comparable to H ( A ∼ H ) during the first few oscillations, and strongly damps the oscillations.</list_item>
</unordered_list>
<text><location><page_6><loc_20><loc_24><loc_20><loc_26></location≯</text>
<unordered_list>
<list_item><location><page_6><loc_11><loc_22><loc_92><loc_26></location>· 2 H ≫ A = 0 : The effect of A rapidly diminishes, and the Hubble rate dominates the the damping term. In this case, the evolution of GWs is mainly influenced by the modified background. Since it is the effect of ξRϕ 2 coupling that indirectly affects the evolution of GWs through a and H , we denote this effect as indirect effect .</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_7><loc_11><loc_90><loc_92><loc_93></location>· A → 0 : This is the case in the late RD era, when ϕ is almost gone. Therefore, not only A diminishes but also the evolution of background becomes that of the standard cosmology.</list_item>
</unordered_list>
<text><location><page_7><loc_9><loc_86><loc_92><loc_89></location>Different regimes leave different signatures on the GWs power spectrum, as we will show in Sec. III B. For this purpose, let us first describe the behaviour of GWs at different frequency regimes.</text>
<section_header_level_1><location><page_7><loc_42><loc_82><loc_59><loc_83></location>A. Evolution of GWs</section_header_level_1>
<text><location><page_7><loc_9><loc_76><loc_92><loc_80></location>Recalling the well-known theory of damping harmonic oscillators, the general behavior of a specific mode in Eq. (16) can be understood by comparing its momentum, k , with the damping term, a 2 (2 H + A ). In this regard, we categorize different modes as:</text>
<unordered_list>
<list_item><location><page_7><loc_11><loc_68><loc_92><loc_75></location>· Sup Modes: For a mode with k ≪ a 2 (2 H + A ), there exists a constant solution to Eq.(16), h Prim k , where the superscript 'Prim' stands for primordial. This solution describes the frozen behavior of the mode from the time it leaves the horizon during the inflationary epoch until it reaches the time η × when k ∼ a × 2 (2 H × + A × ). The solution h Prim k serves as an initial value for the subsequent evolution of the mode after k ≫ a 2 (2 H + A ).</list_item>
<list_item><location><page_7><loc_11><loc_62><loc_92><loc_68></location>· Sub Modes: For a mode with k ≫ a 2 (2 H + A ), it is conventional to parametrize the solution of Eq. (16) as h k ( η ) = h Prim k χ k ( η ), where χ k ( η ) represents the transfer function that describes the subsequent evolution of the mode from η × to the present time. By substituting this parametrization into Eq.(16), the evolution equation for the transfer function becomes:</list_item>
</unordered_list>
<formula><location><page_7><loc_42><loc_59><loc_92><loc_61></location>χ '' + a (2 H + A ) χ ' + k 2 χ = 0 , (17)</formula>
<text><location><page_7><loc_13><loc_55><loc_92><loc_58></location>which has a general solution presented in Appendix B. By imposing the initial conditions χ = 1 and χ ' = 0 for k ≪ a 2 (2 H + A ), the specific oscillatory solution for Eq. (17) can be expressed as:</text>
<formula><location><page_7><loc_44><loc_52><loc_92><loc_53></location>χ k ( η ) = e -d k ( η ) χ GR k ( η ) , (18)</formula>
<text><location><page_7><loc_13><loc_49><loc_17><loc_50></location>where</text>
<formula><location><page_7><loc_30><loc_45><loc_92><loc_48></location>χ GR k ( η ) = a × a ( η ) e ± ikη and d k ( η ) = 1 2 ∫ η η × aAdη ' = 1 2 ∫ t t × Adt ' , (19)</formula>
<text><location><page_7><loc_9><loc_40><loc_92><loc_43></location>with d k ( η ) denoting the damping factor for a specific mode k computed up to η . To calculate the observable, power spectrum of GWs, we set η = η 0 to obtain the WKB solution at present, where a 0 = 1.</text>
<text><location><page_7><loc_9><loc_34><loc_92><loc_40></location>The modes that become sub prior to the AD era, experience the whole AD era, and thus their amplitudes behave as | χ k | = e -d a × . Conversely, the k modes that become sub after the AD era when A is negligible, do not experience any suppression from the d factor and they go as | χ k | = a × . If a k mode becomes sub during the AD era, the suppression factor on its amplitude will depend on its k value: | χ k | = e -d k a × .</text>
<text><location><page_7><loc_9><loc_26><loc_92><loc_34></location>Finally, let us comment on the behavior of a × as a function of k to gain an intuition about | χ k | in different k regimes. Since for the cosmological eras with constant ω , a ∝ η 2 1+3 w , we can obtain a × given the functional form of η × in terms of k : k ∼ a × H × = 1 η × , leading to a × ∝ k -2 1+3 w . However, this argument is only valid when the effect of A is ignored (before and after AD era when H ≫ A ). For the modes that enter the horizon during AD , this simple analysis does not suffice and numerical analysis is required.</text>
<section_header_level_1><location><page_7><loc_41><loc_22><loc_60><loc_23></location>B. The Power Spectrum</section_header_level_1>
<text><location><page_7><loc_9><loc_18><loc_92><loc_20></location>To introduce the power spectrum of gravitational waves as an observable, we need their energy density given by [14, 93]</text>
<formula><location><page_7><loc_27><loc_12><loc_92><loc_16></location>ρ GW = 1 32 πG ⟨ ˙ h ij ( t, x ) ˙ h ij ( t, x ) ⟩ av = 1 32 πG ∑ ζ =+ , × ∫ d 3 k (2 π ) 3 2 | ˙ h ( ζ ) k | 2 , (20)</formula>
<figure>
<location><page_8><loc_24><loc_67><loc_77><loc_93></location>
<caption>FIG. 3: This plot demonstrates the evolution of a 2 (2 H + A ) with respect to temperature. The red line is for the case of ξ = 0, the blue line is for ξ = 5 . 95 while ignoring the A term, and the green line presents the case of ξ = 5 . 95 keeping the A term. Notice that since the A term is zero (ignored) in the red (blue) line, we have a 2 (2 H + A ) → aH . In this plot, T osc s are shown by stars, and T RD is represented by a triangle. As the definition of T osc does not depend on A , the temperature of oscillation for the blue and green lines coincide. Furthermore, T RD is independent of the value of ξ , and thus all of the lines in the plot share the same T RD .</caption>
</figure>
<text><location><page_8><loc_9><loc_49><loc_92><loc_53></location>where ⟨ ... ⟩ av denotes spatial averaging. The power spectrum, Ω GW ( t, k ), is defined as the energy density of GWs per logarithmic frequency interval, divided by the critical energy density, ρ crit = 3 H 2 (8 πG ) -1 , of the Universe [14, 26, 93]: Ω GW ( t, k ) = d ρ GW ( t, k ) / ( ρ crit dln k ). Using the parametrization h k ( η ) = h Prim k χ k ( η ), we obtain [14, 93]</text>
<formula><location><page_8><loc_34><loc_45><loc_92><loc_48></location>Ω GW ( η, k ) = 1 12 a 2 ( η ) H 2 ( η ) P T ( k ) ( χ ' ( η, k )) 2 . (21)</formula>
<text><location><page_8><loc_9><loc_37><loc_92><loc_44></location>The quantity P T ( k ) is the primordial tensor power spectrum which is a nearly scale-invariant power-law function of k around a pivot scale k ∗ as P T ( k ) = A T ( k/k ∗ ) n T . Here, A T is the amplitude and n T is the tilt of the spectrum [16, 26, 94]. Using the WKB solution, Eq. (18) for the sub modes k ≫ a 2 (2 H + A ), we get | χ ' ( η, k ) | ≃ k | χ ( η, k ) | (see Appendix B). By substituting this derivative in Eq. (21) and setting η = η 0 , the present power spectrum for GWs is obtained as</text>
<formula><location><page_8><loc_39><loc_32><loc_92><loc_36></location>Ω 0 GW ( f ) ≃ 1 12 k 2 a 2 0 H 2 0 P T ( k ) | χ k | 2 , (22)</formula>
<text><location><page_8><loc_9><loc_19><loc_92><loc_31></location>where f = k/ (2 πa 0 ), χ k , a 0 , and H 0 = 100 h km/s/Mpc are the present time quantities. In this work, we set A T ≃ 1 . 5 × 10 -10 , n T = 0 . 4 and k ⋆ = k CMB = 0 . 05 1 Mpc in P T ( k ) as is done in [26] to access the maximal reach of future detectors. These values are consistent with the upper bound on the tensor-to-scalar ratio r = A T A R ≤ 0 . 07, where A R ≃ 2 . 1 × 10 -9 is the amplitude of scalar perturbations obtained from observational data [95]. 13 Eq.(22) elegantly shows how the parametrization h k ( η ) = h Prim k χ k ( η ) shows up in the power spectrum. Therefore, Ω 0 GW is generally made by concerning two factors: 1) the initial condition h Prim k and 2) the value of | χ k | 2 (influenced by the indirect and direct effects). Note that GWs start from after becoming sub, coded in the primordial tensor power spectrum, P T ( k ). The form of power spectrum Ω 0 GW for the high-frequency modes (becoming sub before AD) and low-frequency</text>
<figure>
<location><page_9><loc_24><loc_67><loc_76><loc_93></location>
<caption>FIG. 4: The power spectrum of GWs for standard cosmology (dashed black line), ξ = 0 (red line), ξ = 5 . 95 ignoring the A term (blue line), and ξ = 5 . 95 preserving the A term (green line) with respect to frequency is presented. The frequencies corresponding to the beginning of ϕ oscillation (Eq. (26)), late RD era (Eq. (25)), and the step (Eq. (29)) are shown by star, triangle, and square, respectively. The ripples of the blue line are due to the intense oscillation of ϕ which is felt by the modes that become sub during the first RD era. In the case of the green line, the presence of A damps the ripples. Moreover, the A term leads to a noticeable drop in the power spectrum of the green line at high frequencies, manifesting a step-like feature.</caption>
</figure>
<text><location><page_9><loc_9><loc_51><loc_61><loc_53></location>ones (becoming sub after AD) using | χ k | obtained in Sec. III A would be:</text>
<formula><location><page_9><loc_42><loc_48><loc_92><loc_50></location>Ω 0( High ) GW ( k ) ∝ e -2 d k n , (23)</formula>
<formula><location><page_9><loc_43><loc_46><loc_92><loc_48></location>Ω 0( Low ) GW ( k ) ∝ k n , (24)</formula>
<text><location><page_9><loc_9><loc_41><loc_92><loc_45></location>where n = n T + 2(3 ω -1) 3 ω +1 . In the case of ξ = 0, or equivalently A = 0, the present power spectrum is roughly Ω 0 GW ( k ) ∝ k n [26].</text>
<section_header_level_1><location><page_9><loc_39><loc_37><loc_61><loc_38></location>C. Frequencies f osc and f RD</section_header_level_1>
<text><location><page_9><loc_9><loc_25><loc_92><loc_35></location>Before diving into the numerical results on power spectrum, let us discuss the relation between a k mode, or more precisely its corresponding frequency f = k/ (2 πa 0 ), and the temperature at which it becomes sub. Since this happens when k ≃ a 2 (2 H + A ), we can study Fig. 3 to find the transition frequencies. In this figure, the red line represents the ξ = 0 case, the green line is for ξ = 5 . 95, and we have included the blue line where the case of ξ = 5 . 95 is redrawn neglecting the effect of A . It is important to emphasize the blue line is not a physical case; however, it can illustrate the role of A more lucidly. Other than this numerical method using Fig. 3, we can use the following theoretical expression to find the frequency at T = T RD [26]:</text>
<formula><location><page_9><loc_33><loc_20><loc_92><loc_24></location>f RD ≃ bf 0 ( g s, 0 ∗ g s ∗ RD ) 1 3 ( g ρ ∗ RD g ρ, 0 ∗ ) 1 2 ( Ω 0 R 1 / 2 ) 1 2 T RD T 0 , (25)</formula>
<text><location><page_9><loc_9><loc_14><loc_92><loc_18></location>where T 0 ≃ 2 . 72548 K ≃ 2 . 3 × 10 -13 GeV is the CMB temperature [96], h 2 Ω 0 R ≃ 4 . 2 × 10 -5 is the present energy density parameter of radiation, and f 0 = H 0 2 π is the frequency of the GW mode, with comoving wavelength of one Hubble radius, f 0 /h ≃ 5 . 2 × 10 -19 Hz. The parameter b is obtained numerically as b ≃ 1 . 4, in our study. Furthermore,</text>
<figure>
<location><page_10><loc_23><loc_66><loc_77><loc_92></location>
<caption>FIG. 5: The power spectrum for different values of scalar-gravity coupling ξ is shown. Increasing ξ results in a decrease in the power spectrum at high frequencies. Since both the Hubble rate and A are proportional to ( 1 8 πG -ξϕ 2 ) -1 , increasing ξ such that ξϕ 2 ≃ 1 8 πG significantly increases the Hubble rate and A . Therefore, the drop at high frequencies for the case of ξ = 5 . 95 (green line) is considerably bigger than ξ = 5 (magenta line) or ξ = 4 (blue line).</caption>
</figure>
<text><location><page_10><loc_9><loc_53><loc_90><loc_54></location>the frequency f osc corresponding to the mode that becomes sub at T osc can be obtained using Eqs. (10) and (12):</text>
<formula><location><page_10><loc_28><loc_46><loc_92><loc_52></location>f osc ≃ f RD ( m ϕ c RD i osc α RD Γ ) 1 -α RD 1 + ξ ( ϕ in M Pl ) 2 1 + ξ (6 ξ -1) ( ϕ in M Pl ) 2 1 -α RD 2 . (26)</formula>
<text><location><page_10><loc_9><loc_43><loc_79><loc_44></location>Having these frequencies we can determine our scenario how narrates on power spectrum Ω 0 GW ( f ).</text>
<section_header_level_1><location><page_10><loc_44><loc_39><loc_56><loc_40></location>IV. RESULTS</section_header_level_1>
<text><location><page_10><loc_9><loc_25><loc_92><loc_37></location>To obtain the power spectrum Ω 0 GW , we need to evolve GWs till today. However, given that after ϕ decay, we return to the standard cosmology, we can use simple scaling arguments to determine Ω 0 GW from the power spectrum after ϕ 's effect has been diminished. To do this, we solve Eq. (17) from T in = 10 11 GeV to T f = 100 GeV, then scale | χ k ( T f ) | to its present value | χ k | using the conservation of entropy, S ( T ) ∝ g s ∗ ( T ) a 3 T 3 [93]. In this section, our main goal is to investigate how the non-minimal ( ξ ≥ 1) gravitational coupling of the scalar field, ξRϕ 2 , affects the power spectrum. In particular, we want to distinguish the direct and indirect effects on Ω 0 GW . This splitting can be done since they show up independently in the WKB solution, Eq. (18). To proceed, we study the evolution of χ k with and without the A term for ξ = 5 . 95, and contrast them with the case of ξ = 0. 14</text>
<text><location><page_10><loc_9><loc_16><loc_92><loc_25></location>Fig. 4 demonstrates the values of Ω 0 GW for the case of standard cosmology (dashed black line), the case of ξ = 0 (red line), and the non-minimal gravity coupling case ξ = 5 . 95, with (green line) and without (blue line) the A term. The frequencies that become sub during the RD era exhibit an almost scale-invariant spectrum. Introducing a transient MD era, causes high-frequency modes to be lower relative to the case of standard cosmology. That is because during the MD era, the damping term in Eq. (17) becomes a ' a = aH ∝ 2 η , which is greater than that of the RD era, a ' a = aH ∝ 1 η . Modes that become sub during the MD era, experience a weaker damping.</text>
<text><location><page_11><loc_9><loc_83><loc_92><loc_93></location>The aforementioned effects on Ω 0 GW due to the MD era are generally the same for both ξ = 0 and ξ > 0. However, the presence of the ξRϕ 2 term leaves a distinct imprint on the shape of the power spectrum. To better comprehend these effects, let us compare the blue line ( ξ = 5 . 95, while ignoring the A term) in Fig. 4 with the red line ( ξ = 0). The main difference between these two lines is the change in the Hubble rate due to the ξRϕ 2 term. The main points to highlight from the comparison of the blue and red lines are: 1) the oscillations of H appear in the power spectrum; 2) the MD era becomes longer 15 making a deeper kink in Ω 0 GW ; 3) the temperature at which the Late RD era begins, coincides with that of the case ξ = 0.</text>
<text><location><page_11><loc_9><loc_73><loc_92><loc_83></location>To grasp the effect of the A term on Ω 0 GW , we compare the green line ( ξ = 5 . 95, while preserving the A term) with the blue line ( ξ = 5 . 95, ignoring the A term) in Fig. 4. As can be seen from these two lines, the A term has a notable effect on the shape of the power spectrum. One effect is that the oscillations on Ω 0 GW vanish, since the oscillations of aA cancel those of 2 aH . Furthermore, an additional step-like feature appears in Ω 0 GW , which can be understood by the extra damping term in Eq. (17) during the AD era. We can find the temperature at which this step occurs, T step by setting A ≃ 2 H/β , where the parameter β depends on ξ . Using this along with the form of A in terms of ϕ and Eq. (A3), the exact expression for ˙ ϕ 2 at T step is obtained as:</text>
<formula><location><page_11><loc_42><loc_68><loc_92><loc_72></location>˙ ϕ 2 = m 2 ϕ ϕ 2 in +2 ρ R 6 β ( β +2) ξ 2 γ -1 , (27)</formula>
<text><location><page_11><loc_9><loc_65><loc_87><loc_67></location>where γ = ϕ 2 in M 2 Pl -ξϕ 2 in . Substituting A obtained from Eq. (27) and H ≃ √ ρ R 3( M 2 Pl -ξϕ 2 in ) in A ≃ 2 H/β , we obtain</text>
<formula><location><page_11><loc_26><loc_59><loc_92><loc_63></location>T step ≃ ( 90 β 2 ξ 2 γm 2 ϕ ϕ 2 in π 2 g ρ ∗ step (12 βξ 2 γ -1) ) 1 4 , (28)</formula>
<formula><location><page_11><loc_26><loc_52><loc_92><loc_59></location>f step ≃ f osc β 2 ξ 2 γ 2 i osc [ 1 + ξ (6 ξ -1) ( ϕ in M Pl ) 2 ] (12 βξ 2 γ -1) [ 1 + ξ ( ϕ in M Pl ) 2 ] 1 -α step 2 ( 1 + 1 β ) , (29)</formula>
<text><location><page_11><loc_9><loc_46><loc_92><loc_50></location>where α step ≃ 0 . 57. This frequency is shown by a square in Fig. 4. To complete this part of our study, we obtain Ω 0 GW for different values of ξ , and present the results in Fig. 5. As can be seen, the kink in Ω 0 GW deepens by increasing ξ , as expected.</text>
<text><location><page_11><loc_9><loc_42><loc_92><loc_46></location>Another way to interpret our results is by using dilution factor : D ≡ [ S ( T ≫ T osc ) /S ( T ≪ T RD )] 1 / 3 , where S = a 3 s with s = 2 π 2 45 g s ∗ T 3 being the comoving entropy density. Using a ∝ t α ≃ ( α H ) α , and H osc ≃ √ ρ osc R 3( M 2 Pl -ξϕ 2 in ) and</text>
<text><location><page_11><loc_9><loc_39><loc_79><loc_42></location>H RD = √ 2 ρ RD R 3 M 2 Pl , along with T osc and T RD given by Eqs. (10) and (12), the dilution factor becomes</text>
<formula><location><page_11><loc_16><loc_32><loc_92><loc_38></location>D ≃ ( m ϕ c RD i osc α RD Γ ) 1 -2 α D 2 ( g s ∗ osc g s ∗ RD ) 1 3 ( 2 g ρ ∗ RD g ρ ∗ osc ) 1 4 [ 1 -ξ ( ϕ in M Pl ) 2 ] 1 4 1 + ξ ( ϕ in M Pl ) 2 1 + ξ (6 ξ -1) ( ϕ in M Pl ) 2 1 -2 α D 4 , (30)</formula>
<text><location><page_11><loc_9><loc_24><loc_92><loc_30></location>with α D ≈ 0 . 68 obtained from numerical matching. To interpret the results of Fig. 5, we employ the WKB solution (see Appendix B). For the sub mode with the wave number k , we have | χ k | ≃ a k . The parameter a k is the scale factor at which the k mode becomes sub. Replacing | χ k | in Eq. (22) with a k = k/H ( a k ), for the modes with frequencies f osc and f RD , we obtain Ω 0 GW ( f osc ) / Ω 0 GW ( f RD ) (ignoring A term) as:</text>
<formula><location><page_11><loc_25><loc_20><loc_92><loc_23></location>Ω 0 GW ( f osc ) Ω 0 GW ( f RD ) ≃ ( f osc f RD ) n T D 4 ( g s ∗ RD g s ∗ osc ) 4 3 ( g ρ ∗ osc 2 g ρ ∗ RD ) [ 1 -ξ ( ϕ in M Pl ) 2 ] -1 , (31)</formula>
<text><location><page_11><loc_9><loc_17><loc_92><loc_18></location>which demonstrates the indirect effect of the term ξRϕ 2 . Second, if we set the wave number as k = k step , we have</text>
<table>
<location><page_12><loc_15><loc_54><loc_78><loc_84></location>
<caption>TABLE I: This table presents the values of (1) α RD,D,step and β obtained from numerical matching, (2) f RD,osc,step according to theoretical expectation and according to numerical analysis, (3) the dilution factor (4) the damping factor e -2 d , and (5) the ratio of Ω 0 GW for each of the mentioned frequencies according to both theory and data analysis, for different values of ξ . Albeit one needs some numerical input for the theoretical prediction, this table shows that Eqs. ( 31), (32), and (33) match the numerical results with satisfactory accuracy.</caption>
</table>
<text><location><page_12><loc_9><loc_46><loc_92><loc_50></location>| χ k | ≃ e -d a k . Replacing | χ k | in Eq. (22) with e -d k H ( a k )(1+ 1 β ) , for this mode, while using H osc H step ≃ [ f osc f step (1 + 1 β ) ] 1 1 -α step , we obtain</text>
<formula><location><page_12><loc_30><loc_41><loc_92><loc_45></location>Ω 0 GW ( f step ) Ω 0 GW ( f osc ) ≃ e -2 d ( f step f osc ) n T +4 -2 1 -α step ( 1 + 1 β ) 2 α step 1 -α step , (32)</formula>
<text><location><page_12><loc_9><loc_34><loc_92><loc_40></location>where d = 1 2 ∫ 10 5 m ϕ t step A d t . Eq. (32) exhibits the direct effect of the term ξRϕ 2 . Finally, implementing the correction parameter cr as the ratio of h 2 Ω 0 GW ( f osc ) in the power spectrum with A to that of the power spectrum without A , we can write</text>
<formula><location><page_12><loc_36><loc_30><loc_92><loc_33></location>Ω 0 GW ( f step ) Ω 0 GW ( f RD ) ≃ cr Ω 0 GW ( f step ) Ω 0 GW ( f osc ) Ω 0 GW ( f osc ) Ω 0 GW ( f RD ) , (33)</formula>
<text><location><page_12><loc_9><loc_17><loc_92><loc_28></location>where Ω 0 GW ( f step ) Ω 0 GW ( f osc ) and Ω 0 GW ( f osc ) Ω 0 GW ( f RD ) are given by Eqs. (32) and (31), respectively. To test our estimations, we compare some analytical results with numerical ones. Table I compares the analytical estimates and data driven pivotal frequencies f RD,osc,step for different values of ξ . Furthermore, the ratio of Ω 0 GW at these key frequencies according to the analytical expectations (Eqs. (31), (32), and (33)), and the data are provided in Table I as well. As it is illustrated, the predicted values are consistent with the numerical data. Albeit the analytical expressions depend on data driven parameters α RD,D,step and β , they prove to be valuable as they show the dependence of our observable Ω 0 GW ( f ) on model parameters and initial conditions.</text>
<text><location><page_12><loc_9><loc_9><loc_92><loc_17></location>For the sake of completeness, let us study the effect of m ϕ and Γ on the shape of Ω 0 GW , in the case of ξ = 0. Here we choose m ϕ = 10 GeV , 40 GeV and Γ = 10 -8 GeV , 4 × 10 -8 GeV. Since m ϕ determines the beginning of the MD era and Γ controls when it ends, only the position of the kinks change and the slopes in Ω 0 GW stay constant, as shown in Fig. 6. Finally, the effects of non-minimal gravitational coupling, ξRϕ 2 , of the scalar field on Ω 0 GW can be probed by future GWs experiments. Due to the shape of Ω 0 GW , we expect GWs to be detectable by some of the observatories focusing on low-frequencies but missed by high-frequency experiments. The shape of the GWs may also be probed by</text>
<figure>
<location><page_13><loc_23><loc_66><loc_77><loc_92></location>
<caption>FIG. 6: The power spectrum for four cases: (1) unchanged mass and decay rate with values as in Fig. 4 (red line), (2) increased mass (green line), (3) increased decay rate (magenta line), (4) increased mass and the decay rate but unchanged m/ Γ value (blue line).</caption>
</figure>
<text><location><page_13><loc_9><loc_55><loc_92><loc_57></location>some experiments that can detect relatively lower-intensity GWs. In Fig. 7, we have picked some benchmarks that may be probed by LISA [97], BBO [98], and DECIGO [99].</text>
<section_header_level_1><location><page_13><loc_43><loc_51><loc_58><loc_52></location>V. CONCLUSION</section_header_level_1>
<text><location><page_13><loc_9><loc_42><loc_92><loc_49></location>In this paper, we investigated the evolution of the SGWB originating from inflation within a cosmological framework. We considered a scenario where a scalar field ϕ dominates the energy density of the Universe at high temperatures and has a non-minimal coupling with gravity, represented as ξRϕ 2 . Previous studies have demonstrated that early matter domination leads to a reduced signal strength of SGWB in the present. Various experiments such as LISA, BBO, and DECIGO can probe SGWB, while others like HLVK and ET may miss it.</text>
<text><location><page_13><loc_75><loc_32><loc_75><loc_33></location≯</text>
<text><location><page_13><loc_9><loc_28><loc_92><loc_41></location>By introducing a non-minimal coupling between ϕ and gravity, not only does this decrease amplify, but non-trivial features also emerge at certain frequencies. When the coupling parameter ξ is large, the damping term in the evolution of SGWB is enhanced due to a larger Hubble rate and the presence of a ξ -dependent term denoted as A in this study. The distinctive step-like feature observed in Fig. 7 is caused by the dominance of A over the Hubble rate. Analytically, this feature can be understood by examining the ratio Ω 0 GW ( f step ) Ω 0 GW ( f RD ) , which includes not only the dilution factor but also an additional damping term e -∫ Adt compared to the case where ξ = 0. Additionally, when ξ = 0, the Hubble rate is relatively higher at the beginning of the MD era, resulting in a lower value of Ω GW h 2 , even if the A term is neglected, as the dilution factor discloses. We provided an analytical expression for Ω 0 GW at key frequencies and demonstrated that they align with the numerical results with acceptable precision.</text>
<text><location><page_13><loc_9><loc_18><loc_92><loc_27></location>In the scenario of early matter domination with ξ = 0, the power spectrum pattern exhibits deviations from a pure power law at frequencies determined by the mass and decay width of ϕ . Our study reveals that introducing the term ξRϕ 2 does not significantly alter the location of these deviations but introduces a distinct step feature that serves as a signature of this term in observational data. We identified a set of benchmarks that exhibit this feature within the parameter space where DECIGO and BBO experiments can probe. In general, the shape of the power spectrum of Primordial Gravitational Waves (PGWs) as a function of frequency provides valuable insights into the evolution of the early universe.</text>
<section_header_level_1><location><page_13><loc_44><loc_13><loc_57><loc_14></location>Acknowledgment</section_header_level_1>
<text><location><page_13><loc_9><loc_9><loc_92><loc_11></location>We thank Enrico Morgante, Nicklas Ramberg, Wolfram Ratzinger, Nematollah Riazi, Kai Schmitz, and Pedro Schwaller for useful discussions. F.E. and H.M. are grateful to CERN for their hospitality. The work in Mainz is sup-</text>
<figure>
<location><page_14><loc_17><loc_60><loc_84><loc_92></location>
<caption>FIG. 7: The sensitivity curves of existing and future GW experiments and the power spectrum of GWs in ξ = 0 and ξ = 0 scenarios. There is hope that the effect of direct and indirect effects of modified gravity ξRϕ 2 can be captured for specific regions in the parameter space of the model by future detectors. The curves are resulted using different benchmarks of ( ξ, m ϕ , Γ). Since we only can analyze the modes that become sub after T in = 10 11 GeV, the colored solid lines that are the results of the paper do not extend to arbitrarily large frequencies.</caption>
</figure>
<text><location><page_14><loc_14><loc_56><loc_14><loc_58></location≯</text>
<text><location><page_14><loc_9><loc_43><loc_92><loc_49></location>ported by the Cluster of Excellence Precision Physics, Fundamental Interactions and Structure of Matter (PRISMA + EXC 2118/1) within the German Excellence Strategy (project ID 39083149). Also, F.E. is supported by the grant 05H18UMCA1 of the German Federal Ministry for Education and Research (BMBF), and H.M. is supported by the National Natural Science Foundation of China under Grants No. 12247107 and 12075007.</text>
<section_header_level_1><location><page_14><loc_11><loc_38><loc_90><loc_40></location>Appendix A: Beginning of the evolution: Attractor Behavior, Hubble parameter, Ricci scalar, and EoS parameter</section_header_level_1>
<text><location><page_14><loc_15><loc_33><loc_15><loc_34></location≯</text>
<text><location><page_14><loc_9><loc_29><loc_92><loc_36></location>This appendix shows that the benchmarks satisfying Eq. (9) have an attractor behavior. That is, even if we start with ˙ ϕ in = 0, we will quickly merge to the path of ϕ = ϕ in and ˙ ϕ in = 0 in the phase space. Our other objective in this appendix is to show that even in the case of non-minimal coupling where ξ ≥ 1, the start of evolution is effectively RD defined as ω = 1 3 , even though the energy density of the scalar field dominates over that of radiation ρ ϕ > ρ R . For better understanding, we rewrite the EoM of ϕ , Eq.(5), with the assumption H ≫ Γ at the beginning as</text>
<formula><location><page_14><loc_42><loc_26><loc_92><loc_27></location>¨ ϕ +3 H ˙ ϕ + m 2 eff ϕ = 0 . (A1)</formula>
<text><location><page_14><loc_9><loc_19><loc_92><loc_24></location>This equation has the form of a harmonic oscillator equation, with m eff as the angular frequency and the friction term of 3 H 2 . In the limit where 3 H 2 ≫ m eff , and with the initial conditions ϕ ( T in ) = ϕ in and ˙ ϕ ( T in ) = 0 the solution for ϕ becomes ϕ = ϕ in = const , which means that the field is stuck at its initial value. If the initial velocity of the harmonic oscillator is non-zero, the solution to the equation in this regime becomes</text>
<formula><location><page_14><loc_40><loc_14><loc_92><loc_17></location>ϕ ( t ) = ϕ in + ˙ ϕ in 3 H -˙ ϕ 3 H e -3 Ht , (A2)</formula>
<text><location><page_14><loc_9><loc_8><loc_92><loc_13></location>which shows that in this case, the general solution does not have constant behavior and in fact is time-dependent. For any value of ˙ ϕ satisfying Eq. (9), the second term in Eq. (A2) becomes negligible compared with ϕ in and hence we can write the solution as ϕ ( t ) = ϕ in -˙ ϕ 3 H e -3 Ht . In addition, the third term rapidly approaches zero with time, and</text>
<text><location><page_15><loc_9><loc_90><loc_92><loc_93></location>hence ϕ approaches ϕ in again at high temperatures which obviously demonstrates the attractor behavior mentioned above.</text>
<text><location><page_15><loc_9><loc_85><loc_92><loc_90></location>To obtain a general understanding of the evolution of the background at the beginning, and besides that an estimation for the depth of the kink in the GWs power spectrum, it is essential to find an approximate relation for the Hubble parameter. Starting from the first Friedmann equation, Eq. (7) and using the energy density of scalar field with non-minimal gravitational coupling, Eq. (8), we have:</text>
<formula><location><page_15><loc_22><loc_78><loc_92><loc_83></location>H = √ 1 3 M 2 Pl ( ρ R + ρ ϕ ) = ξϕ ˙ ϕ + √ ξ 2 ϕ 2 ˙ ϕ 2 + 1 3 ( M 2 Pl -ξϕ 2 ) ( m 2 ϕ ϕ 2 2 + ˙ ϕ 2 2 + ρ R ) M 2 Pl -ξϕ 2 . (A3)</formula>
<text><location><page_15><loc_9><loc_74><loc_92><loc_77></location>We are considering a benchmark class that satisfies Eq. (9). Therefore, we can simplify the Hubble rate at T in to the following:</text>
<formula><location><page_15><loc_28><loc_65><loc_92><loc_73></location>H ( T ≃ T in ) ≃ ξϕ in ˙ ϕ in + √ ξ 2 ϕ 2 in ˙ ϕ 2 in + 1 3 ( M 2 Pl -ξϕ 2 in ) ρ R ( T in ) M 2 Pl -ξϕ 2 in ≃ √ ρ R ( T in ) 3( M 2 Pl -ξϕ 2 in ) , (A4)</formula>
<text><location><page_15><loc_9><loc_55><loc_92><loc_63></location>where in the first line m 2 ϕ ϕ 2 in / 2 , ˙ ϕ 2 in / 2 ≪ ρ R ( T in ), and in the second line 3 ξ 2 ˙ ϕ 2 in ϕ 2 in M 2 Pl -ξϕ 2 in ≪ ρ R ( T in ) is used. This equation demonstrates that the initial Hubble parameter increases by increasing ξ , which can be seen for the two values of ξ = 0 and ξ = 5 . 95 in Fig 1. As long as these conditions are satisfied, Eq. (A4) provides the approximate Hubble rate even if we move away from T in . With the Hubble parameter in hand, the EoS parameter at the beginning of the evolution can be found using the relation [55, 56]</text>
<formula><location><page_15><loc_44><loc_53><loc_92><loc_54></location>R = 3(1 -3 ω ) H 2 . (A5)</formula>
<text><location><page_15><loc_9><loc_49><loc_92><loc_51></location>Hence, to have an estimation of the ω at the beginning of the evolution, we need the behavior of R at high temperatures. To this end, we take the following steps. First, by contracting Eq. (2) with the metric, the trace of the equation yields</text>
<formula><location><page_15><loc_32><loc_44><loc_92><loc_47></location>R = -1 M 2 Pl g µν ( T ( M ) µν + T ( ϕ ) µν ) = -1 M 2 Pl ( T ( M ) + T ( ϕ ) ) . (A6)</formula>
<text><location><page_15><loc_9><loc_41><loc_75><loc_43></location>To use the above equation, we need to derive the trace of T ( ϕ ) µν by contacting with metric [60]</text>
<formula><location><page_15><loc_37><loc_39><loc_92><loc_40></location>T ( ϕ ) = (6 ξ -1)( ∂ µ ϕ∂ µ ϕ + ξRϕ 2 ) + 6 ξ, (A7)</formula>
<text><location><page_15><loc_9><loc_36><loc_59><loc_37></location>Now, substituting Eq. (A7) in Eq. (A6), gives the Ricci scalar as [60]:</text>
<formula><location><page_15><loc_31><loc_31><loc_92><loc_35></location>R = (1 -6 ξ ) ∂ µ ϕ∂ µ ϕ +4 ( 1 2 m 2 ϕ ϕ 2 ) -6 ξm 2 ϕ 2 ϕ 2 -T ( M ) M 2 Pl +(6 ξ -1) ξϕ 2 . (A8)</formula>
<text><location><page_15><loc_9><loc_25><loc_92><loc_29></location>In our study, the scalar field ϕ is assumed to be homogeneous and isotropic (and hence, just a function of time). On the other hand, T ( M ) = 0, since in our study, the matter part is only the radiation, and the trace of the energy-momentum tensor for relativistic fluid with ω = 1 3 is zero. As a result, the Ricci scalar at the beginning becomes</text>
<formula><location><page_15><loc_35><loc_20><loc_92><loc_24></location>R ( T ≃ T in ) = (1 -6 ξ ) ˙ ϕ 2 in +(2 -6 ξ ) m 2 ϕ ϕ 2 in M 2 Pl +(6 ξ -1) ξϕ 2 in . (A9)</formula>
<text><location><page_15><loc_9><loc_18><loc_79><loc_19></location>According to Eq. (A5), in order to evaluate ω , we need to find the dimensionless parameter R/H 2 :</text>
<formula><location><page_15><loc_26><loc_13><loc_92><loc_16></location>R H 2 ( T ≃ T in ) ≃ (1 -6 ξ ) ˙ ϕ 2 in +(2 -6 ξ ) m 2 ϕ ϕ 2 in M 2 Pl +(6 ξ -1) ξϕ 2 in × 3( M 2 Pl -ξϕ 2 in ) ρ R ( T in ) → 0 , (A10)</formula>
<text><location><page_15><loc_9><loc_9><loc_92><loc_12></location>because we are assuming ρ R ( T in ) ≫ ˙ ϕ in , m 2 ϕ ϕ 2 in . Now, given Eq. (A10), we see that ω ≃ 1 / 3 for the class of all benchmarks satisfying Eq. (9).</text>
<text><location><page_16><loc_9><loc_38><loc_13><loc_39></location>where</text>
<formula><location><page_16><loc_38><loc_34><loc_92><loc_37></location>C Mod ≡ e -d , d ≡ 1 2 ∫ M H d η, (B7)</formula>
<text><location><page_16><loc_9><loc_30><loc_92><loc_33></location>and χ GR = 1 a e ± ikη . The effect of the additional damping term in the equation of GWs appears in the additional exponential e -d . The parameter d can be written in cosmic time as:</text>
<formula><location><page_16><loc_35><loc_26><loc_92><loc_29></location>d = 1 2 ∫ M H d η = 1 2 ∫ MH d η = 1 2 ∫ A d t (B8)</formula>
<text><location><page_16><loc_9><loc_21><loc_92><loc_25></location>As before, all we need to know about the GWs are well coded in the observable of GWs, the power spectrum, introduced in Eq. (21). It will be more convenient if we try to write it in terms of the transfer function itself. It is in fact possible for sub-modes where we have k ≫ a 2 (2 H + A ), and A > 0 (in our specific case). Starting from</text>
<formula><location><page_16><loc_43><loc_18><loc_92><loc_20></location>χ = e -1 2 ∫ η η × aAdη χ GR , (B9)</formula>
<section_header_level_1><location><page_16><loc_38><loc_92><loc_63><loc_93></location>Appendix B: The WKB Analysis</section_header_level_1>
<text><location><page_16><loc_9><loc_83><loc_92><loc_90></location>In this appendix, we explain the WKB analysis for the evolution of the modes in sub-regime in the presence of the term, A[64-66]. Such an additional term appears in the various modified gravity theories and hence, our analysis in this appendix is quite general and we do not restrict ourselves to the case of ξRϕ 2 . This analysis provides us with physical intuition and parametrizes the solution in terms of the aforementioned direct and indirect effects on the evolution of GWs. Starting from Eq. (16), it is conventional to rewrite this equation as:</text>
<formula><location><page_16><loc_40><loc_80><loc_92><loc_82></location>χ '' + H (2 + M ) χ ' + k 2 χ = 0 , (B1)</formula>
<text><location><page_16><loc_9><loc_75><loc_92><loc_79></location>where we invoke the parametrization h ( η ) = h Prim χ ( η ), H = a ' a and M = A H . To obtain the WKB solution for high-frequency modes in this case, we consider an ansatz: χ = Ze iY where C = Zh prim . Substituting this ansatz for the transfer function in Eq. (B1), separates the equation into two equations for the imaginary and the real part:</text>
<formula><location><page_16><loc_37><loc_71><loc_92><loc_74></location>k 2 +(2 + M ) H Z ' Z -( Y ' ) 2 + Z '' Z = 0 , (B2)</formula>
<formula><location><page_16><loc_41><loc_67><loc_92><loc_71></location>(2 + M ) H +2 Z ' Z + Z '' Z ' = 0 . (B3)</formula>
<text><location><page_16><loc_9><loc_61><loc_92><loc_67></location>Since our aim here is to derive the WKB solution for high-frequency modes, we can neglect the second and fourth terms in Eq. (B2). That is because high-frequency modes correspond to the modes that enter the horizon at early times, and thus the terms involving Z are negligible compared to the terms that include the phase of χ . Hence, from Eq. (B2), we have:</text>
<formula><location><page_16><loc_47><loc_59><loc_92><loc_60></location>Y = ± kη. (B4)</formula>
<text><location><page_16><loc_9><loc_56><loc_66><loc_57></location>Substituting this result in Eq. (B3), we simply have the WKB solution [64-66]:</text>
<formula><location><page_16><loc_38><loc_51><loc_92><loc_55></location>χ ∝ e -∫ ( 1+ M 2 ) H d η e ± ikη = e -∫ H d η × e ± ikη × e -1 2 ∫ M H d η (B5)</formula>
<text><location><page_16><loc_9><loc_43><loc_92><loc_50></location>Concentrating on this solution, we see that the first two exponentials can be simplified to e -∫ H d η e ± ikη = e -∫ H d t e ± ikη = 1 a e ± ikη if we rewrite it in cosmic time. This part of the WKB solution is exactly the form of the WKB solution in the standard GR (cases of the standard cosmology and the case of ξ = 0) and hence, conventionally called χ GR . 16 As we can see χ GR demonstrates the oscillatory behavior of the high-frequency modes in the sub-regime. It is conventional to write the complete WKB solution for ξ = 0 as:</text>
<text><location><page_16><loc_61><loc_43><loc_61><loc_44></location≯</text>
<formula><location><page_16><loc_45><loc_40><loc_92><loc_41></location>χ = C Mod χ GR , (B6)</formula>
<text><location><page_17><loc_9><loc_92><loc_31><loc_93></location>and invoking the useful relation</text>
<text><location><page_17><loc_9><loc_85><loc_29><loc_86></location>we can simply conclude that</text>
<formula><location><page_17><loc_46><loc_83><loc_92><loc_84></location>| χ ' | ≃ k | χ | . (B11)</formula>
<text><location><page_17><loc_9><loc_80><loc_58><loc_81></location>By substituting the above equation in Eq. (21), Eq. (22) is obtained.</text>
<unordered_list>
<list_item><location><page_17><loc_10><loc_72><loc_89><loc_74></location>[1] B. P. Abbott, R. Abbott, T. Abbott, M. Abernathy, F. Acernese, K. Ackley et al., Observation of gravitational waves from a binary black hole merger , Physical review letters 116 (2016) 061102.</list_item>
<list_item><location><page_17><loc_10><loc_69><loc_87><loc_72></location>[2] G. M. Harry, forthe LIGO Scientific Collaboration et al., Advanced ligo: the next generation of gravitational wave detectors , Classical and Quantum Gravity 27 (2010) 084006.</list_item>
<list_item><location><page_17><loc_10><loc_67><loc_91><loc_69></location>[3] J. Aasi, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, K. Ackley et al., Advanced ligo , Classical and quantum gravity 32 (2015) 074001.</list_item>
<list_item><location><page_17><loc_10><loc_64><loc_86><loc_66></location>[4] F. a. Acernese, M. Agathos, K. Agatsuma, D. Aisa, N. Allemandou, A. Allocca et al., Advanced virgo: a second-generation interferometric gravitational wave detector , Classical and Quantum Gravity 32 (2014) 024001.</list_item>
<list_item><location><page_17><loc_10><loc_60><loc_91><loc_64></location>[5] B. Abbott, R. Abbott, T. Abbott, S. Abraham, F. Acernese, K. Ackley et al., Gwtc-1: a gravitational-wave transient catalog of compact binary mergers observed by ligo and virgo during the first and second observing runs , Physical Review X 9 (2019) 031040.</list_item>
</unordered_list>
<text><location><page_17><loc_10><loc_59><loc_21><loc_60></location>[6] M. Maggiore,</text>
<text><location><page_17><loc_21><loc_59><loc_61><loc_60></location>Gravitational wave experiments and early universe cosmology</text>
<text><location><page_17><loc_61><loc_59><loc_62><loc_60></location>,</text>
<text><location><page_17><loc_62><loc_59><loc_73><loc_60></location>Physics Reports</text>
<text><location><page_17><loc_73><loc_59><loc_76><loc_60></location>331</text>
<text><location><page_17><loc_76><loc_59><loc_84><loc_60></location>(2000) 283.</text>
<unordered_list>
<list_item><location><page_17><loc_10><loc_56><loc_85><loc_59></location>[7] L. Grishchuk, Amplification of gravitational waves in an isotropic universe , Soviet Journal of Experimental and Theoretical Physics 40 (1975) 409.</list_item>
<list_item><location><page_17><loc_10><loc_55><loc_92><loc_56></location>[8] A. Starobinskii, Spectrum of relict gravitational radiation and the early state of the universe , JETP Letters 30 (1979) 682.</list_item>
</unordered_list>
<text><location><page_17><loc_10><loc_54><loc_42><loc_55></location>[9] V. Rubakov, M. V. Sazhin and A. Veryaskin,</text>
<text><location><page_17><loc_42><loc_54><loc_89><loc_55></location>Graviton creation in the inflationary universe and the grand unification</text>
<text><location><page_17><loc_12><loc_52><loc_15><loc_53></location>scale</text>
<text><location><page_17><loc_15><loc_52><loc_15><loc_53></location>,</text>
<text><location><page_17><loc_16><loc_52><loc_27><loc_53></location>Physics Letters B</text>
<text><location><page_17><loc_28><loc_52><loc_31><loc_53></location>115</text>
<text><location><page_17><loc_31><loc_52><loc_38><loc_53></location>(1982) 189.</text>
<unordered_list>
<list_item><location><page_17><loc_9><loc_50><loc_90><loc_52></location>[10] S. T. McWilliams, R. Caldwell, K. Holley-Bockelmann, S. L. Larson and M. Vallisneri, Astro2020 decadal science white paper: The state of gravitational-wave astrophysics in 2020 , arXiv preprint arXiv:1903.04592 (2019) .</list_item>
<list_item><location><page_17><loc_9><loc_47><loc_88><loc_49></location>[11] M. C. Guzzetti, N. Bartolo, M. Liguori and S. Matarrese, Gravitational waves from inflation , La Rivista del Nuovo Cimento 39 (2016) 399.</list_item>
<list_item><location><page_17><loc_9><loc_44><loc_89><loc_47></location>[12] C. Caprini and D. G. Figueroa, Cosmological backgrounds of gravitational waves , Classical and Quantum Gravity 35 (2018) 163001.</list_item>
<list_item><location><page_17><loc_9><loc_43><loc_81><loc_44></location>[13] A. Riotto, Inflation and the theory of cosmological perturbations , arXiv preprint hep-ph/0210162 (2002) .</list_item>
<list_item><location><page_17><loc_9><loc_42><loc_90><loc_43></location>[14] R. R. Lino dos Santos and L. M. van Manen, Gravitational waves from the early universe , arXiv e-prints (2022) arXiv.</list_item>
<list_item><location><page_17><loc_9><loc_40><loc_65><loc_41></location>[15] D. Baumann, Tasi lectures on inflation , arXiv preprint arXiv:0907.5424 (2009) .</list_item>
<list_item><location><page_17><loc_9><loc_39><loc_83><loc_40></location>[16] M. Maggiore, Gravitational Waves: Volume 2: Astrophysics and Cosmology . Oxford University Press, 2018.</list_item>
<list_item><location><page_17><loc_9><loc_36><loc_87><loc_39></location>[17] R. Caldwell, M. Amin, C. Hogan, K. Holley-Bockelmann, D. Holz, P. Jetzer et al., Astro2020 science white paper: Cosmology with a space-based gravitational wave observatory , arXiv preprint arXiv:1903.04657 (2019) .</list_item>
<list_item><location><page_17><loc_9><loc_34><loc_90><loc_36></location>[18] V. Kalogera, C. P. Berry, M. Colpi, S. Fairhurst, S. Justham, I. Mandel et al., Deeper, wider, sharper: Next-generation ground-based gravitational-wave observations of binary black holes , arXiv preprint arXiv:1903.09220 (2019) .</list_item>
<list_item><location><page_17><loc_9><loc_31><loc_90><loc_33></location>[19] N. J. Cornish, E. Berti, K. Holley-Bockelmann, S. Larson, S. McWilliams, G. Mueller et al., The discovery potential of space-based gravitational wave astronomy , arXiv preprint arXiv:1904.01438 (2019) .</list_item>
</unordered_list>
<text><location><page_17><loc_9><loc_30><loc_39><loc_31></location>[20] D. Shoemaker, L. S. Collaboration et al.,</text>
<text><location><page_17><loc_39><loc_30><loc_91><loc_31></location>Gravitational wave astronomy with ligo and similar detectors in the next decade</text>
<text><location><page_17><loc_91><loc_30><loc_92><loc_31></location>,</text>
<text><location><page_17><loc_12><loc_28><loc_42><loc_29></location>Bulletin of the American Astronomical Society</text>
<text><location><page_17><loc_43><loc_28><loc_45><loc_29></location>51</text>
<text><location><page_17><loc_45><loc_28><loc_53><loc_29></location>(2019) 452.</text>
<unordered_list>
<list_item><location><page_17><loc_9><loc_26><loc_84><loc_28></location>[21] M. Kawasaki, K. Kohri and N. Sugiyama, Mev-scale reheating temperature and thermalization of the neutrino background , Physical Review D 62 (2000) 023506.</list_item>
<list_item><location><page_17><loc_9><loc_25><loc_80><loc_26></location>[22] S. Hannestad, What is the lowest possible reheating temperature? , Physical Review D 70 (2004) 043506.</list_item>
<list_item><location><page_17><loc_9><loc_22><loc_87><loc_24></location>[23] F. De Bernardis, L. Pagano and A. Melchiorri, New constraints on the reheating temperature of the universe after wmap-5 , Astroparticle Physics 30 (2008) 192.</list_item>
<list_item><location><page_17><loc_9><loc_19><loc_91><loc_22></location>[24] P. De Salas, M. Lattanzi, G. Mangano, G. Miele, S. Pastor and O. Pisanti, Bounds on very low reheating scenarios after planck , Physical Review D 92 (2015) 123534.</list_item>
<list_item><location><page_17><loc_9><loc_17><loc_84><loc_19></location>[25] F. Muia, F. Quevedo, A. Schachner and G. Villa, Testing bsm physics with gravitational waves , arXiv preprint arXiv:2303.01548 (2023) .</list_item>
<list_item><location><page_17><loc_9><loc_14><loc_91><loc_16></location>[26] F. D'Eramo and K. Schmitz, Imprint of a scalar era on the primordial spectrum of gravitational waves , Physical Review Research 1 (2019) 013010.</list_item>
<list_item><location><page_17><loc_9><loc_11><loc_90><loc_14></location>[27] R. D. Peccei, The strong cp problem and axions , in Axions: Theory, Cosmology, and Experimental Searches , pp. 3-17, Springer, (2008).</list_item>
<list_item><location><page_17><loc_9><loc_9><loc_89><loc_11></location>[28] T. Kugo, I. Ojima and T. Yanagida, Superpotential symmetries and pseudo nambu-goldstone supermultiplets , Physics Letters B 135 (1984) 402.</list_item>
</unordered_list>
<formula><location><page_17><loc_34><loc_87><loc_92><loc_91></location>d dx ∫ f ( x ) g ( x ) h ( t ) dt = h ( f ( x )) f ' ( x ) -h ( g ( x )) g ' ( x ) , (B10)</formula>
<unordered_list>
<list_item><location><page_18><loc_9><loc_92><loc_88><loc_93></location>[29] T. Banks, M. Dine and M. Graesser, Supersymmetry, axions, and cosmology , Physical Review D 68 (2003) 075011.</list_item>
<list_item><location><page_18><loc_9><loc_89><loc_87><loc_92></location>[30] M. Kawasaki and K. Nakayama, Solving cosmological problems of supersymmetric axion models in an inflationary universe , Physical Review D 77 (2008) 123524.</list_item>
<list_item><location><page_18><loc_9><loc_87><loc_87><loc_89></location>[31] K. Harigaya, M. Ibe, K. Schmitz and T. T. Yanagida, Peccei-quinn symmetry from a gauged discrete r symmetry , Physical Review D 88 (2013) 075022.</list_item>
<list_item><location><page_18><loc_9><loc_84><loc_90><loc_86></location>[32] K. Harigaya, M. Ibe, K. Schmitz and T. T. Yanagida, Peccei-quinn symmetry from dynamical supersymmetry breaking , Physical Review D 92 (2015) 075003.</list_item>
<list_item><location><page_18><loc_9><loc_81><loc_90><loc_84></location>[33] F. DEramo, L. J. Hall et al., Supersymmetric axion grand unified theories and their predictions , Physical Review D 94 (2016) 075001.</list_item>
<list_item><location><page_18><loc_9><loc_79><loc_90><loc_81></location>[34] R. T. Co, F. DEramo, L. J. Hall and K. Harigaya, Saxion cosmology for thermalized gravitino dark matter , Journal of High Energy Physics 2017 (2017) 1.</list_item>
<list_item><location><page_18><loc_9><loc_76><loc_92><loc_78></location>[35] G. Coughlan, W. Fischler, E. W. Kolb, S. Raby and G. G. Ross, Cosmological problems for the polonyi potential , Physics Letters B 131 (1983) 59.</list_item>
<list_item><location><page_18><loc_9><loc_73><loc_88><loc_76></location>[36] J. Ellis, D. V. Nanopoulos and M. Quir'os, On the axion, dilaton, polonyi, gravitino and shadow matter problems in supergravity and superstring models , Physics Letters B 174 (1986) 176.</list_item>
<list_item><location><page_18><loc_9><loc_71><loc_89><loc_73></location>[37] C. D. Froggatt and H. B. Nielsen, Hierarchy of quark masses, cabibbo angles and cp violation , Nuclear Physics B 147 (1979) 277.</list_item>
<list_item><location><page_18><loc_9><loc_69><loc_80><loc_70></location>[38] F. Elahi and S. R. Zadeh, Flavon magnetobaryogenesis , Phys. Rev. D 102 (2020) 096018 [ 2008.04434 ].</list_item>
<list_item><location><page_18><loc_9><loc_67><loc_91><loc_69></location>[39] F. Elahi and H. Mehrabpour, Magnetogenesis from baryon asymmetry during an early matter dominated era , Phys. Rev. D 104 (2021) 115030 [ 2104.04815 ].</list_item>
<list_item><location><page_18><loc_9><loc_66><loc_57><loc_67></location>[40] N. D. Birrell and P. C. W. Davies, Quantum fields in curved space , .</list_item>
<list_item><location><page_18><loc_9><loc_64><loc_69><loc_65></location>[41] A. De Felice and S. Tsujikawa, f (r) theories , Living Reviews in Relativity 13 (2010) 1.</list_item>
<list_item><location><page_18><loc_9><loc_63><loc_77><loc_64></location>[42] S. Capozziello and M. De Laurentis, Extended theories of gravity , Physics Reports 509 (2011) 167.</list_item>
<list_item><location><page_18><loc_9><loc_62><loc_88><loc_63></location>[43] T. Clifton, P. G. Ferreira, A. Padilla and C. Skordis, Modified gravity and cosmology , Physics reports 513 (2012) 1.</list_item>
<list_item><location><page_18><loc_9><loc_60><loc_76><loc_61></location>[44] T. P. Sotiriou and V. Faraoni, f (r) theories of gravity , Reviews of Modern Physics 82 (2010) 451.</list_item>
<list_item><location><page_18><loc_9><loc_59><loc_82><loc_60></location>[45] L. G. Jaime, L. Patino and M. Salgado, f (r) cosmology revisited , arXiv preprint arXiv:1206.1642 (2012) .</list_item>
<list_item><location><page_18><loc_9><loc_58><loc_61><loc_59></location>[46] S. M. Carroll, Spacetime and geometry . Cambridge University Press, 2019.</list_item>
<list_item><location><page_18><loc_9><loc_56><loc_81><loc_57></location>[47] V. Oikonomou and E. N. Saridakis, f (t) gravitational baryogenesis , Physical Review D 94 (2016) 124005.</list_item>
<list_item><location><page_18><loc_9><loc_55><loc_83><loc_56></location>[48] S. Odintsov and V. Oikonomou, Gauss-bonnet gravitational baryogenesis , Physics Letters B 760 (2016) 259.</list_item>
<list_item><location><page_18><loc_9><loc_52><loc_89><loc_55></location>[49] G. Felder, L. Kofman and A. Linde, Inflation and preheating in nonoscillatory models , Physical Review D 60 (1999) 103505.</list_item>
<list_item><location><page_18><loc_9><loc_51><loc_64><loc_52></location>[50] B. Spokoiny, Deflationary universe scenario , Physics Letters B 315 (1993) 40.</list_item>
<list_item><location><page_18><loc_9><loc_50><loc_71><loc_51></location>[51] P. Peebles and A. Vilenkin, Quintessential inflation , Physical Review D 59 (1999) 063505.</list_item>
<list_item><location><page_18><loc_9><loc_47><loc_88><loc_49></location>[52] J. Ellis, D. V. Nanopoulos, K. A. Olive and S. Verner, Non-oscillatory no-scale inflation , Journal of Cosmology and Astroparticle Physics 2021 (2021) 052.</list_item>
<list_item><location><page_18><loc_9><loc_46><loc_71><loc_47></location>[53] J. de Haro and L. Arest'e Sal'o, A review of quintessential inflation , Galaxies 9 (2021) 73.</list_item>
<list_item><location><page_18><loc_9><loc_43><loc_89><loc_45></location>[54] D. G. Figueroa and C. T. Byrnes, The standard model higgs as the origin of the hot big bang , Physics Letters B 767 (2017) 272.</list_item>
<list_item><location><page_18><loc_9><loc_40><loc_88><loc_43></location>[55] K. Dimopoulos and T. Markkanen, Non-minimal gravitational reheating during kination , Journal of Cosmology and Astroparticle Physics 2018 (2018) 021.</list_item>
<list_item><location><page_18><loc_9><loc_38><loc_90><loc_40></location>[56] T. Opferkuch, P. Schwaller and B. A. Stefanek, Ricci reheating , Journal of Cosmology and Astroparticle Physics 2019 (2019) 016.</list_item>
<list_item><location><page_18><loc_9><loc_36><loc_73><loc_37></location>[57] G. Laverda and J. Rubio, Ricci reheating reloaded , arXiv preprint arXiv:2307.03774 (2023) .</list_item>
<list_item><location><page_18><loc_9><loc_34><loc_89><loc_36></location>[58] D. Bettoni and J. Rubio, Quintessential affleck-dine baryogenesis with non-minimal couplings , Physics Letters B 784 (2018) 122.</list_item>
<list_item><location><page_18><loc_9><loc_31><loc_89><loc_34></location>[59] L. Parker and D. Toms, Quantum field theory in curved spacetime: quantized fields and gravity . Cambridge university press, 2009.</list_item>
<list_item><location><page_18><loc_9><loc_29><loc_89><loc_31></location>[60] D. G. Figueroa, A. Florio, T. Opferkuch and B. A. Stefanek, Dynamics of non-minimally coupled scalar fields in the jordan frame , arXiv preprint arXiv:2112.08388 (2021) .</list_item>
<list_item><location><page_18><loc_9><loc_27><loc_79><loc_28></location>[61] L. Ford, Cosmological particle production: a review , Reports on Progress in Physics 84 (2021) 116901.</list_item>
<list_item><location><page_18><loc_9><loc_25><loc_92><loc_27></location>[62] M. Atkins and X. Calmet, Bounds on the nonminimal coupling of the higgs boson to gravity , Physical Review Letters 110 (2013) 051301.</list_item>
<list_item><location><page_18><loc_9><loc_22><loc_88><loc_24></location>[63] J.-c. Hwang and H. Noh, Gauge-ready formulation of the cosmological kinetic theory in generalized gravity theories , Physical Review D 65 (2001) 023512.</list_item>
<list_item><location><page_18><loc_9><loc_19><loc_92><loc_22></location>[64] S. D. Odintsov, V. K. Oikonomou and R. Myrzakulov, Spectrum of primordial gravitational waves in modified gravities: a short overview , Symmetry 14 (2022) 729.</list_item>
<list_item><location><page_18><loc_9><loc_17><loc_89><loc_19></location>[65] A. Nishizawa, Generalized framework for testing gravity with gravitational-wave propagation. i. formulation , Physical Review D 97 (2018) 104037.</list_item>
<list_item><location><page_18><loc_9><loc_14><loc_90><loc_16></location>[66] S. Arai and A. Nishizawa, Generalized framework for testing gravity with gravitational-wave propagation. ii. constraints on horndeski theory , Physical Review D 97 (2018) 104038.</list_item>
<list_item><location><page_18><loc_9><loc_11><loc_89><loc_14></location>[67] N. Bernal, A. Ghoshal, F. Hajkarim and G. Lambiase, Primordial gravitational wave signals in modified cosmologies , Journal of Cosmology and Astroparticle Physics 2020 (2020) 051.</list_item>
<list_item><location><page_18><loc_9><loc_9><loc_87><loc_11></location>[68] M. R. Haque, D. Maity, T. Paul and L. Sriramkumar, Decoding the phases of early and late time reheating through imprints on primordial gravitational waves , Physical Review D 104 (2021) 063513.</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_19><loc_9><loc_91><loc_89><loc_93></location>[69] P. Amaro-Seoane, H. Audley, S. Babak, J. Baker, E. Barausse, P. Bender et al., Laser interferometer space antenna , arXiv preprint arXiv:1702.00786 (2017) .</list_item>
<list_item><location><page_19><loc_9><loc_88><loc_90><loc_90></location>[70] P. Amaro-Seoane, S. Aoudia, S. Babak, P. Binetruy, E. Berti, A. Boh'e et al., elisa: Astrophysics and cosmology in the millihertz regime , arXiv preprint arXiv:1201.3621 (2012) .</list_item>
<list_item><location><page_19><loc_9><loc_85><loc_88><loc_88></location>[71] R. R. Caldwell, T. L. Smith and D. G. Walker, Using a primordial gravitational wave background to illuminate new physics , Physical Review D 100 (2019) 043513.</list_item>
<list_item><location><page_19><loc_9><loc_83><loc_92><loc_85></location>[72] S. Sato, S. Kawamura, M. Ando, T. Nakamura, K. Tsubono, A. Araya et al., The status of decigo , in Journal of Physics: Conference Series , vol. 840, p. 012010, IOP Publishing, 2017.</list_item>
<list_item><location><page_19><loc_9><loc_80><loc_90><loc_82></location>[73] N. Seto, S. Kawamura and T. Nakamura, Possibility of direct measurement of the acceleration of the universe using 0.1 hz band laser interferometer gravitational wave antenna in space , Physical Review Letters 87 (2001) 221103.</list_item>
<list_item><location><page_19><loc_9><loc_77><loc_92><loc_80></location>[74] S. Kawamura, M. Ando, N. Seto, S. Sato, M. Musha, I. Kawano et al., Current status of space gravitational wave antenna decigo and b-decigo , Progress of Theoretical and Experimental Physics 2021 (2021) 05A105.</list_item>
<list_item><location><page_19><loc_9><loc_76><loc_82><loc_77></location>[75] M. Punturo, M. Abernathy, F. Acernese, B. Allen, N. Andersson, K. Arun et al., The einstein telescope: a</list_item>
<list_item><location><page_19><loc_12><loc_75><loc_77><loc_76></location>third-generation gravitational wave observatory , Classical and Quantum Gravity 27 (2010) 194002.</list_item>
<list_item><location><page_19><loc_9><loc_72><loc_92><loc_74></location>[76] B. Sathyaprakash, M. Abernathy, F. Acernese, P. Ajith, B. Allen, P. Amaro-Seoane et al., Scientific objectives of einstein telescope , Classical and Quantum Gravity 29 (2012) 124013.</list_item>
<list_item><location><page_19><loc_9><loc_69><loc_84><loc_72></location>[77] S. Hild, M. Abernathy, F. e. Acernese, P. Amaro-Seoane, N. Andersson, K. Arun et al., Sensitivity studies for third-generation gravitational wave observatories , Classical and Quantum gravity 28 (2011) 094013.</list_item>
<list_item><location><page_19><loc_9><loc_67><loc_90><loc_69></location>[78] J. Crowder and N. J. Cornish, Beyond lisa: Exploring future gravitational wave missions , Physical Review D 72 (2005) 083005.</list_item>
<list_item><location><page_19><loc_9><loc_64><loc_89><loc_67></location>[79] V. Corbin and N. J. Cornish, Detecting the cosmic gravitational wave background with the big bang observer , Classical and Quantum Gravity 23 (2006) 2435.</list_item>
<list_item><location><page_19><loc_9><loc_62><loc_86><loc_64></location>[80] T. L. Smith and R. Caldwell, Sensitivity to a frequency-dependent circular polarization in an isotropic stochastic gravitational wave background , Physical Review D 95 (2017) 044036.</list_item>
<list_item><location><page_19><loc_9><loc_59><loc_90><loc_61></location>[81] A. Weltman, P. Bull, S. Camera, K. Kelley, H. Padmanabhan, J. Pritchard et al., Fundamental physics with the square kilometre array , Publications of the Astronomical Society of Australia 37 (2020) e002.</list_item>
<list_item><location><page_19><loc_9><loc_56><loc_89><loc_59></location>[82] E. Barausse, E. Berti, T. Hertog, S. A. Hughes, P. Jetzer, P. Pani et al., Prospects for fundamental physics with lisa , General Relativity and Gravitation 52 (2020) 1.</list_item>
<list_item><location><page_19><loc_9><loc_54><loc_90><loc_56></location>[83] N. S. Pol, S. R. Taylor, L. Z. Kelley, S. J. Vigeland, J. Simon, S. Chen et al., Astrophysics milestones for pulsar timing array gravitational-wave detection , The Astrophysical Journal Letters 911 (2021) L34.</list_item>
<list_item><location><page_19><loc_9><loc_51><loc_92><loc_53></location>[84] Z. Arzoumanian, P. T. Baker, H. Blumer, B. B'ecsy, A. Brazier, P. R. Brook et al., The nanograv 12.5 yr data set: search for an isotropic stochastic gravitational-wave background , The Astrophysical Journal Letters 905 (2020) L34.</list_item>
<list_item><location><page_19><loc_9><loc_48><loc_90><loc_51></location>[85] K. Lee, Searching for the nano-hertz stochastic gravitational wave background with the chinese pulsar timing array data release i , Research in Astronomy and Astrophysics (2023) .</list_item>
<list_item><location><page_19><loc_9><loc_46><loc_91><loc_48></location>[86] J. Antoniadis, P. Arumugam, S. Arumugam, S. Babak, M. Bagchi, A.-S. B. Nielsen et al., The second data release from the european pulsar timing array iii. search for gravitational wave signals , arXiv preprint arXiv:2306.16214 (2023) .</list_item>
<list_item><location><page_19><loc_9><loc_43><loc_90><loc_45></location>[87] G. Agazie, A. Anumarlapudi, A. M. Archibald, Z. Arzoumanian, P. T. Baker, B. B'ecsy et al., The nanograv 15 yr data set: Evidence for a gravitational-wave background , The Astrophysical Journal Letters 951 (2023) L8.</list_item>
<list_item><location><page_19><loc_9><loc_40><loc_89><loc_43></location>[88] D. J. Reardon, A. Zic, R. M. Shannon, G. B. Hobbs, M. Bailes, V. Di Marco et al., Search for an isotropic gravitational-wave background with the parkes pulsar timing array , The Astrophysical Journal Letters 951 (2023) L6.</list_item>
<list_item><location><page_19><loc_9><loc_38><loc_92><loc_40></location>[89] L. Kofman, A. Linde and A. A. Starobinsky, Towards the theory of reheating after inflation , Physical Review D 56 (1997) 3258.</list_item>
<list_item><location><page_19><loc_9><loc_36><loc_73><loc_37></location>[90] G. Felder, L. Kofman and A. Linde, Instant preheating , Physical Review D 59 (1999) 123523.</list_item>
<list_item><location><page_19><loc_9><loc_34><loc_89><loc_36></location>[91] M. A. Amin, M. P. Hertzberg, D. I. Kaiser and J. Karouby, Nonperturbative dynamics of reheating after inflation: a review , International Journal of Modern Physics D 24 (2015) 1530003.</list_item>
<list_item><location><page_19><loc_9><loc_33><loc_74><loc_34></location>[92] K. D. Lozanov, Lectures on reheating after inflation , arXiv preprint arXiv:1907.04402 (2019) .</list_item>
<list_item><location><page_19><loc_9><loc_30><loc_90><loc_32></location>[93] K. Saikawa and S. Shirai, Primordial gravitational waves, precisely: The role of thermodynamics in the standard model , Journal of Cosmology and Astroparticle Physics 2018 (2018) 035.</list_item>
<list_item><location><page_19><loc_9><loc_27><loc_89><loc_30></location>[94] D. S. Gorbunov and V. A. Rubakov, Introduction to the theory of the early universe: Cosmological perturbations and inflationary theory . World Scientific, 2011.</list_item>
<list_item><location><page_19><loc_9><loc_25><loc_90><loc_27></location>[95] Y. Akrami, F. Arroja, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini et al., Planck 2018 results-x. constraints on inflation , Astronomy & Astrophysics 641 (2020) A10.</list_item>
<list_item><location><page_19><loc_9><loc_23><loc_76><loc_24></location>[96] D. J. Fixsen, The temperature of the cosmic microwave background , Astrophys. J. 707 (2009) 916.</list_item>
<list_item><location><page_19><loc_9><loc_22><loc_58><loc_23></location>[97] LISA collaboration, Laser Interferometer Space Antenna , 1702.00786 .</list_item>
<list_item><location><page_19><loc_9><loc_19><loc_92><loc_22></location>[98] G. M. Harry, P. Fritschel, D. A. Shaddock, W. Folkner and E. S. Phinney, Laser interferometry for the big bang observer , Class. Quant. Grav. 23 (2006) 4887.</list_item>
<list_item><location><page_19><loc_9><loc_18><loc_90><loc_19></location>[99] S. Kawamura et al., The Japanese space gravitational wave antenna: DECIGO , Class. Quant. Grav. 28 (2011) 094011.</list_item>
</unordered_list>
</document>
[
{
"title": "Signature of Non-Minimal Scalar-Gravity Coupling with an Early Matter Domination on the Power Spectrum of Gravitational Waves",
"content": "Amirsalar Nikandish, 1, ∗ Shiva Rostam Zadeh, 2, † Reza Naderi, 3, ‡ Fatemeh Elahi, 4, § and Hadi Mehrabpour 4, 5, 6, ¶ 1 Department of Physics, Shahid Beheshti University, 1983969411, Tehran, Iran 2 School of Particles and Accelerators, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran 3 Department of Physics, Faculty of Basic Sciences, Azarbaijan Shahid Madani University, 53714-161, Tabriz, Iran 4 PRISMA + Cluster of Excellence & Mainz Institute for Theoretical Physics, Johannes Gutenberg University, 55099 Mainz, Germany 5 Center for High Energy Physics, Peking University, Beijing 100871, China 6 Department of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China The signal strength of primordial gravitational waves experiencing an epoch of early scalar domination is reduced with respect to radiation domination. In this paper, we demonstrate that the specific pattern of this reduction is sensitive to the coupling between the dominant field and gravity. When this coupling is zero, the impact of early matter domination on gravitational waves is solely attributed to the alteration of the Hubble parameter and the scale factor. In the presence of non-zero couplings, on the other hand, the evolution of primordial gravitational waves is directly affected as well, resulting in a distinct step-like feature in the power spectrum of the gravitational wave as a function of frequency. This feature serves as a smoking gun signature of this model. In this paper, we provide an analytical expression of the power spectrum that illustrates the dependence of power spectrum on model parameters and initial conditions. Furthermore, we provide analytical relations that specify the frequency interval in which the step occurs. We compare the analytical estimates with numerical analysis and show they match well.",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "Since the first detection of gravitational waves (GWs) at the Laser Interferometer Gravitational-Wave Observatory (LIGO) and Virgo, a new window to unravel the mysteries of the cosmos has opened [1-4]. Even though the sources of the detected GWs have been astrophysical thus far [5], we hope to also detect the cosmological ones with the advance of the detectors [6]. Among the possible sources of GWs, the stochastic gravitational wave background (SGWB) originating from inflation is of particular interest, because detecting it may shed light on the history of the early universe[7-20]. The standard model of cosmology, under the assumption of radiation domination (RD) from inflation to matter-radiation equality, predicts an almost scale invariant power spectrum for the SGWB across all frequencies [21-24]. Yet, many well-motivated proposals predict a transient phase, dominated by a different energy component, intervening between inflation and the Big Bang Nucleosynthesis (BBN) [25, 26]. Depending on the temperature range and other specific features of this transient period, the SGWB profile is expected to vary. Inspired by numerous extensions to the standard model of particle physics, we focus our attention on early matter domination [27-39]. Specifically, we assume a scalar field, ϕ , which behaves like a pressureless fluid scaling as a -3 with a being the scale factor, causes a transient matter domination era. Once ϕ decays, the universe reverts to the RD era again. Assuming a minimal coupling between ϕ and gravity, SGWB is only affected indirectly and because of the alteration of the scale factor and the Hubble rate [25, 26]. We coin the term Indirect effect to denote this influence. In the context of early matter domination, the SGWB power spectrum is suppressed at high frequencies compared to standard cosmology, and the Equation of State (EoS) of the Universe governs the slope of the power spectrum [26]. In this paper, we study early matter domination where the dominating field has a non-minimal coupling with gravity; i.e., f ( R,ϕ ) = ( 1 8 πG -ξϕ 2 ) R . In the term ξRϕ 2 , ξ signifies the gravitational coupling constant and R denotes the Ricci scalar. This coupling stands as the sole feasible local, scalar interaction of its kind with the appropriate dimensions [40]. A coupling between a scalar field and gravity is proposed in numerous models aiming to resolve some of the inherent problems with inflation, reheating, and baryogenesis [41-48]. For instance, in the context of non-oscillatory inflationary models[49-53], the term ξRϕ 2 has been recently used in an efficient reheating scenario denoted as Ricci reheating[54-57]. In the same context, a novel quintessential Affleck-Dine (AD) baryogenesis scenario has also been proposed[58] based on the term ξRϕ 2 , avoiding troublesome iso-curvature modes found in conventional AD scenarios. Moreover, the widely used self-interaction term for scalar fields, λϕ 4 , necessitates the inclusion of ξRϕ 2 in the Lagrangian for proper renormalization in curved space-time [40, 59]. Within the scalar sector of the Standard Model (SM) Lagrangian (Higgs) [54], ξRϕ 2 emerges as the missing term that upholds all the symmetries of both gravity and the SM. As ξ is subject to running, it cannot be universally set to zero across all energy scales [60]. Two values of ξ = 0 for minimal coupling and ξ = 1 / 6 for conformal coupling are of particular interest. The latter, for the case m ϕ = 0, leads to conformal invariance of the action and hence the field equation of motion[40, 61]. Nonetheless, no compelling reason supports the presence of minimal or conformal coupling in the real world, as no symmetry is enhanced [46]. Furthermore, since ξ is dimensionless, there is no reason for it to be small. It could be non-minimal, i.e. of the order of unity or more. 1 [46]. Current experiments place a weak constraint on ξ ( ξ < 10 15 ) due to the feeble gravitational interaction with the SM fields [60, 62] However, GWs might offer insights into the existence and strength of such a coupling. This paper delves into SGWB within a cosmological framework characterized by early matter domination, where the dominant field ϕ directly couples with gravity ( ξRϕ 2 ). D'Eramo et al. previously examined the case of ξ = 0 [26]. To highlight distinctions from [26], we specifically explore the non-minimal regime ( ξ ≥ 1). Studies indicate that such gravity couplings manifest as additional terms in the evolution equation of GWs [41, 63-66]. That is, alongside modifications to the scale factor and Hubble, i.e., indirect effect [67], an extra factor directly affects GW evolution. We term this extra impact the direct effect . Employing Wentzel-Kramers-Brillouin (WKB) analysis, we elucidate the power spectrum resulting from GW propagation in our cosmological setting. Our numerical findings demonstrate that the presence of the ξRϕ 2 term, with non-minimal coupling, deepens the kink shape in the power spectrum, compared with the case studied in [26]. Furthermore, owing to the direct effect , an additional step-like feature emerges, resulting in an enhanced reduction in the power spectrum. We highlight several benchmarks that could be probed by upcoming gravitational wave experiments. ̸ To deepen our understanding of the ξRϕ 2 term, it is crucial to provide an analytical interpretation of the resulting power spectrum. Hence, we use reasonable approximations to get an analytical expression for the power spectrum compared with the case of the standard cosmology and explain how each part of the spectrum in a specific frequency interval is shaped due to a specific physical effect that was dominant in the corresponding temperature interval. To this end, the high-frequency modes that became sub at high temperatures and thus got affected by the changes in the cosmological evolution are of particular interest. In particular, we determine the fraction of the spectrum of high frequencies to that of low-frequency modes that did not experience any changes in the cosmological history. In this context, we utilize a physical quantity called the dilution factor , which was introduced in [26]. While this analytical interpretation was performed for the case of ξ = 0 in [26], exhibiting notable agreement with numerical results, our study expands this analysis to encompass the case of ξ = 0. The analytical formula we derive, expressed in terms of dilution and damping factors, closely aligns with numerical results, maintaining an acceptable level of accuracy. Finally, we comment on the potential testability of this scenario, highlighting a selection of gravitational wave experiments that could probe these changes [68]. These include the Laser Interferometer Space Antenna (LISA), which operates within the frequency range of 10 -5 -1 Hz [69-71], the DECi-hertz Interferometer Gravitational wave Observatory (DECIGO) spanning 10 -3 -10 Hz [72-74], the Einstein Telescope (ET) operating within 1 -10 4 Hz [75-77], and the forthcoming Big Bang Observatory (BBO) encompassing 10 -3 -10 Hz[78-80]. Other experiments will explore intermediate frequencies like square kilometer array or SKA, probing 10 -9 -10 -6 Hz [81, 82] and NANOGrav collaboration[83, 84] that works based on pulsar timing arrays (PTA) measurements [64, 68]. 2 This paper is organized as follows: In Sec. II, we introduce the model and the cosmological framework in detail. In Sec. III, we study the evolution of GWs, highlighting the effect of ξRϕ 2 . We present the WKB solution for the modes that become sub, in Sec. III A. We follow in Sec. III B, by introducing the power spectrum as the observable of GWs, and in Sec. III C, we highlight two of the pivotal frequencies in the power spectrum. Sec. IV is dedicated to numerical results for the power spectrum and a comparison with the analytical estimates. Finally, the concluding remarks are presented in Sec. V.",
"pages": [
1,
2
]
},
{
"title": "II. NON-MINIMALLY COUPLED SCALAR FIELD",
"content": "Our theory is defined by a real scalar field ϕ that has feeble interactions with other fields, but has a non-minimal coupling with gravity: f ( R,ϕ ) = ( 1 8 πG -ξϕ 2 ) R . The action of our theory is, thus, [56, 60]: 3 where g is the determinant of the metric g µν , G is the gravitational coupling constant, m ϕ is the mass of ϕ , and L M contains the kinetic component and the interactions of any other field in the cosmos, including its interaction with the scalar field, ϕ . Varying Eq. (1) with respect to g µν , one can obtain the gravitational equation as [60, 63]: In the definition of Eq. (2), the energy-momentum tensor of the scalar filed ϕ is: where we have □ ϕ 2 = 1 √ -g ∂ µ ( √ -g∂ µ ϕ 2 ). Varying Eq. (1) with respect to ϕ yields the scalar field equation of motion [63]: Using Eqs. (2, 4) and the Bianchi identity, ∇ µ G µ ν = ∇ µ T µ ( eff ) ν = 0, the continuity equation of the matter part can be obtained, ∇ µ T µ ( M ) ν = ∂ L M ∂ϕ ∂ ν ϕ [63]. In these equations, the field ϕ is only a function of time in order to respect the homogeneity and isotropy of the Universe, and L M describes the radiation. Therefore, the evolution equations for the scalar field ϕ and the energy density of radiation ρ R are: where to obtain these equations, we have employed the flat Friedmann-Lemaitre-Robertson-Walker (FLRW) metric, ds 2 = -dt 2 + a ( t ) 2 δ ij dx i dx j , 4 with i, j = 1 -3 specifying the spatial coordinates, explains the metric of a homogeneous and isotropic Universe, a is the cosmic scale factor, and accordingly the Ricci scalar is obtained as R = 6(˙ a 2 + a a ) /a 2 [63]. Furthermore, Γ represents the total decay rate of the scalar. In this paper, we are oblivious to the exact nature of the fields into which ϕ decays, and rather we assume it is part of radiation. The total number of relativistic degrees of freedom contributing to the energy (entropy) density of radiation is represented by g ρ ⋆ ( g s ⋆ ). The evolution of the Hubble rate, defined as H ≡ ˙ a a , is described by the first Friedmann equation, obtained from Eq. (2) as: where M Pl is the reduced Planck mass, and the energy density of ϕ can be obtained from the T ( ϕ ) 00 component of Eq. (3) [56, 60]: Using Eqs. (5), (6), and (7), the system of coupled differential equations for the background would be complete for three unknowns, ϕ, ρ R , and H . The free parameters in these equations are In this project, we fix T in = 10 11 GeV, motivated by well-studied reheating scenarios which prefer T Reh ∈ (10 12 -10 15 ) GeV [54, 55, 89-92]. The initial value of ϕ in has an upper bound of M Pl to avoid undergoing the Universe to another inflationary period [26]. In addition, since the most noticeable deviations from standard cosmology occur when ϕ in ∼ M Pl , we choose ϕ in = 10 18 . The value of ξ , also, has an upper bound to make sure the Hubble rate stays real, i.e., ξ < ( M 2 pl /ϕ 2 in ), which in our case gives rise to ξ max ≃ 5 . 95. 5 Since our setup is not sensitive to ˙ ϕ in and any initial condition satisfying leads to the total equation of state ω ≃ 1 / 3; and hence, a negligible ˙ ϕ (see Appendix A), we fix ˙ ϕ in = 0 for simplicity. Concentrating on one of the benchmarks of Ref. [26], i.e. m ϕ = 10GeV and Γ = 10 -8 GeV, we numerically solve the set of coupled equations and compare the results for the two cases of ξ = 0 (minimal coupling) [26] and ξ ≥ 1 (non-minimal coupling). 6",
"pages": [
3,
4
]
},
{
"title": "A. General Evolution of the Background",
"content": "Given our initial conditions at high temperatures, the friction term in Eq. 5 ( more specifically the Hubble term, H ˙ ϕ ) forces ϕ to be stuck at its initial value, similar to the case of ξ = 0 [26]. As the Hubble rate decreases, eventually the friction term becomes less efficient and ϕ starts oscillating when H ≃ m eff ≡ √ ξR + m 2 ϕ . As demonstrated in Fig. 1, the effect of gravitational coupling causes ϕ to remain close to its initial value for a longer duration, because the Hubble rate is higher and m eff , because of negative R , is lower for larger ξ : H ≃ √ ρ R ( T )(3( M 2 Pl -ξϕ 2 in )) -1 (see Appendix A for this simplified expression of Hubble rate at high temperatures). Around the time when ϕ starts oscillating, the universe deviates from state of RD, or equivalently ω ≃ 1 / 3, or equivalently the state of Radiation Domination (RD). The temperatures at which this occurs is when H and m eff lines meet ( H ≃ m eff ). The numerically determined factor i osc ≃ 6 has been implemented to obtain a more precise value for T osc , since ϕ oscillations start slightly after H ≃ m eff (e.g., for ξ = 0, oscillations start at T ∼ 10 9 and for ξ = 5 . 95, they start at T ∼ 10 8 ). After some oscillations, 7 ϕ behaves as sin( m ϕ t ) m ϕ t . Hence, the energy density of ϕ redshifts like cold matter, leading to an era of matter domination ( MD era ). This process continues until the decay of ϕ becomes more efficient than the Hubble rate, at which point ϕ depletes. We refer to this era, the decay era (DE). When ρ ϕ becomes negligible with respect to ρ R , we return to radiation domination, which we will refer to as the Late RD era . The temperature at which we return to late RD era can be found by setting ρ ϕ ∼ ρ R [26]: Since g s ⋆ and g ρ ⋆ vary with temperature, especially at T EW ≃ 100 GeV ≤ T ≤ T BBN , we evaluate them as a function of ρ R using Ref. [93]. where α RD ≈ 0 . 64. 8 Assuming a constant ω throughout the MD era, c RD can be numerically determined as c RD ≃ 1 . 07. At this point, the evolution of Universe is followed by the standard model of cosmology. To investigate the differences between minimal ( ξ = 0) and non-minimal ( ξ ≥ 1) ϕ -gravity coupling, GWs are the best candidates. The imprint of ξ = 0 case on GWs has been investigated in Ref. [26]. Due to the non-minimal coupling between ϕ and gravity, the evolution of ϕ significantly impacts in the evolution of GWs, which will be discussed in details in the following section.",
"pages": [
4,
5
]
},
{
"title": "III. THE EVOLUTION OF GWS AND THE POWER SPECTRUM",
"content": "Gravitational waves, h ij , are perturbations in the space part of the FLRW metric [13, 15, 16, 94]: with | h ij | ≪ 1. Taking h ij to be transverse and traceless ( h TT ii = k i h TT ij = 0 ) , there are two degrees of freedom for h ij indicating two polarizations, ζ = + , × . By linearizing the gravitational equation, Eq. (2), the evolution equations of GWs is obtained: where A = -2 ξϕ ˙ ϕ ( 1 8 πG -ξϕ 2 ) -1 . It can be seen that an additional term, A , appears in the well-known evolution equations of the GWs [13-16], which comes from the ξRϕ 2 term. It is through this term that the evolution of ϕ directly impacts the evolution of the SGWB. It is conventional to expand h TT ij as h TT ij ( t, x ) = ∑ ζ e ( ζ ) ij h ( ζ ) ( t, x ) where e ( ζ ) ij is the polarization tensor[13, 16, 94]. 9 Substituting the polarization expansion and h ( ζ ) ( t, x ) = ∫ d 3 k (2 π ) 3 e ik · x h ( ζ ) k ( t ) in Eq. (14), the evolution equation for the amplitude of GWs, h k is obtained as following [63]: 10 where h k ( t ) represents the Fourier transform of h ( x, t ). For a better understanding of the behavior of GWs, Eq. (15) is expressed in terms of conformal time, η , given by η = ∫ dt a , as in Refs. [64-66]: with the prime denoting differentiation with respect to conformal time. In what follows we do not mention the subscript k, since the term k 2 can highlight that the equation is written for a specific momentum. The above equation is a harmonic oscillator with the damping term a 2 (2 H + A ). Using the numerical results of the background as obtained in Sec. II, 11 we distinguish different regimes in the behavior of the damping term, by comparing the evolution of the terms aA and 2 aH as depicted in Fig. 2: ̸ Different regimes leave different signatures on the GWs power spectrum, as we will show in Sec. III B. For this purpose, let us first describe the behaviour of GWs at different frequency regimes.",
"pages": [
5,
6,
7
]
},
{
"title": "A. Evolution of GWs",
"content": "Recalling the well-known theory of damping harmonic oscillators, the general behavior of a specific mode in Eq. (16) can be understood by comparing its momentum, k , with the damping term, a 2 (2 H + A ). In this regard, we categorize different modes as: which has a general solution presented in Appendix B. By imposing the initial conditions χ = 1 and χ ' = 0 for k ≪ a 2 (2 H + A ), the specific oscillatory solution for Eq. (17) can be expressed as: where with d k ( η ) denoting the damping factor for a specific mode k computed up to η . To calculate the observable, power spectrum of GWs, we set η = η 0 to obtain the WKB solution at present, where a 0 = 1. The modes that become sub prior to the AD era, experience the whole AD era, and thus their amplitudes behave as | χ k | = e -d a × . Conversely, the k modes that become sub after the AD era when A is negligible, do not experience any suppression from the d factor and they go as | χ k | = a × . If a k mode becomes sub during the AD era, the suppression factor on its amplitude will depend on its k value: | χ k | = e -d k a × . Finally, let us comment on the behavior of a × as a function of k to gain an intuition about | χ k | in different k regimes. Since for the cosmological eras with constant ω , a ∝ η 2 1+3 w , we can obtain a × given the functional form of η × in terms of k : k ∼ a × H × = 1 η × , leading to a × ∝ k -2 1+3 w . However, this argument is only valid when the effect of A is ignored (before and after AD era when H ≫ A ). For the modes that enter the horizon during AD , this simple analysis does not suffice and numerical analysis is required.",
"pages": [
7
]
},
{
"title": "B. The Power Spectrum",
"content": "To introduce the power spectrum of gravitational waves as an observable, we need their energy density given by [14, 93] where ⟨ ... ⟩ av denotes spatial averaging. The power spectrum, Ω GW ( t, k ), is defined as the energy density of GWs per logarithmic frequency interval, divided by the critical energy density, ρ crit = 3 H 2 (8 πG ) -1 , of the Universe [14, 26, 93]: Ω GW ( t, k ) = d ρ GW ( t, k ) / ( ρ crit dln k ). Using the parametrization h k ( η ) = h Prim k χ k ( η ), we obtain [14, 93] The quantity P T ( k ) is the primordial tensor power spectrum which is a nearly scale-invariant power-law function of k around a pivot scale k ∗ as P T ( k ) = A T ( k/k ∗ ) n T . Here, A T is the amplitude and n T is the tilt of the spectrum [16, 26, 94]. Using the WKB solution, Eq. (18) for the sub modes k ≫ a 2 (2 H + A ), we get | χ ' ( η, k ) | ≃ k | χ ( η, k ) | (see Appendix B). By substituting this derivative in Eq. (21) and setting η = η 0 , the present power spectrum for GWs is obtained as where f = k/ (2 πa 0 ), χ k , a 0 , and H 0 = 100 h km/s/Mpc are the present time quantities. In this work, we set A T ≃ 1 . 5 × 10 -10 , n T = 0 . 4 and k ⋆ = k CMB = 0 . 05 1 Mpc in P T ( k ) as is done in [26] to access the maximal reach of future detectors. These values are consistent with the upper bound on the tensor-to-scalar ratio r = A T A R ≤ 0 . 07, where A R ≃ 2 . 1 × 10 -9 is the amplitude of scalar perturbations obtained from observational data [95]. 13 Eq.(22) elegantly shows how the parametrization h k ( η ) = h Prim k χ k ( η ) shows up in the power spectrum. Therefore, Ω 0 GW is generally made by concerning two factors: 1) the initial condition h Prim k and 2) the value of | χ k | 2 (influenced by the indirect and direct effects). Note that GWs start from after becoming sub, coded in the primordial tensor power spectrum, P T ( k ). The form of power spectrum Ω 0 GW for the high-frequency modes (becoming sub before AD) and low-frequency ones (becoming sub after AD) using | χ k | obtained in Sec. III A would be: where n = n T + 2(3 ω -1) 3 ω +1 . In the case of ξ = 0, or equivalently A = 0, the present power spectrum is roughly Ω 0 GW ( k ) ∝ k n [26].",
"pages": [
7,
8,
9
]
},
{
"title": "C. Frequencies f osc and f RD",
"content": "Before diving into the numerical results on power spectrum, let us discuss the relation between a k mode, or more precisely its corresponding frequency f = k/ (2 πa 0 ), and the temperature at which it becomes sub. Since this happens when k ≃ a 2 (2 H + A ), we can study Fig. 3 to find the transition frequencies. In this figure, the red line represents the ξ = 0 case, the green line is for ξ = 5 . 95, and we have included the blue line where the case of ξ = 5 . 95 is redrawn neglecting the effect of A . It is important to emphasize the blue line is not a physical case; however, it can illustrate the role of A more lucidly. Other than this numerical method using Fig. 3, we can use the following theoretical expression to find the frequency at T = T RD [26]: where T 0 ≃ 2 . 72548 K ≃ 2 . 3 × 10 -13 GeV is the CMB temperature [96], h 2 Ω 0 R ≃ 4 . 2 × 10 -5 is the present energy density parameter of radiation, and f 0 = H 0 2 π is the frequency of the GW mode, with comoving wavelength of one Hubble radius, f 0 /h ≃ 5 . 2 × 10 -19 Hz. The parameter b is obtained numerically as b ≃ 1 . 4, in our study. Furthermore, the frequency f osc corresponding to the mode that becomes sub at T osc can be obtained using Eqs. (10) and (12): Having these frequencies we can determine our scenario how narrates on power spectrum Ω 0 GW ( f ).",
"pages": [
9,
10
]
},
{
"title": "IV. RESULTS",
"content": "To obtain the power spectrum Ω 0 GW , we need to evolve GWs till today. However, given that after ϕ decay, we return to the standard cosmology, we can use simple scaling arguments to determine Ω 0 GW from the power spectrum after ϕ 's effect has been diminished. To do this, we solve Eq. (17) from T in = 10 11 GeV to T f = 100 GeV, then scale | χ k ( T f ) | to its present value | χ k | using the conservation of entropy, S ( T ) ∝ g s ∗ ( T ) a 3 T 3 [93]. In this section, our main goal is to investigate how the non-minimal ( ξ ≥ 1) gravitational coupling of the scalar field, ξRϕ 2 , affects the power spectrum. In particular, we want to distinguish the direct and indirect effects on Ω 0 GW . This splitting can be done since they show up independently in the WKB solution, Eq. (18). To proceed, we study the evolution of χ k with and without the A term for ξ = 5 . 95, and contrast them with the case of ξ = 0. 14 Fig. 4 demonstrates the values of Ω 0 GW for the case of standard cosmology (dashed black line), the case of ξ = 0 (red line), and the non-minimal gravity coupling case ξ = 5 . 95, with (green line) and without (blue line) the A term. The frequencies that become sub during the RD era exhibit an almost scale-invariant spectrum. Introducing a transient MD era, causes high-frequency modes to be lower relative to the case of standard cosmology. That is because during the MD era, the damping term in Eq. (17) becomes a ' a = aH ∝ 2 η , which is greater than that of the RD era, a ' a = aH ∝ 1 η . Modes that become sub during the MD era, experience a weaker damping. The aforementioned effects on Ω 0 GW due to the MD era are generally the same for both ξ = 0 and ξ > 0. However, the presence of the ξRϕ 2 term leaves a distinct imprint on the shape of the power spectrum. To better comprehend these effects, let us compare the blue line ( ξ = 5 . 95, while ignoring the A term) in Fig. 4 with the red line ( ξ = 0). The main difference between these two lines is the change in the Hubble rate due to the ξRϕ 2 term. The main points to highlight from the comparison of the blue and red lines are: 1) the oscillations of H appear in the power spectrum; 2) the MD era becomes longer 15 making a deeper kink in Ω 0 GW ; 3) the temperature at which the Late RD era begins, coincides with that of the case ξ = 0. To grasp the effect of the A term on Ω 0 GW , we compare the green line ( ξ = 5 . 95, while preserving the A term) with the blue line ( ξ = 5 . 95, ignoring the A term) in Fig. 4. As can be seen from these two lines, the A term has a notable effect on the shape of the power spectrum. One effect is that the oscillations on Ω 0 GW vanish, since the oscillations of aA cancel those of 2 aH . Furthermore, an additional step-like feature appears in Ω 0 GW , which can be understood by the extra damping term in Eq. (17) during the AD era. We can find the temperature at which this step occurs, T step by setting A ≃ 2 H/β , where the parameter β depends on ξ . Using this along with the form of A in terms of ϕ and Eq. (A3), the exact expression for ˙ ϕ 2 at T step is obtained as: where γ = ϕ 2 in M 2 Pl -ξϕ 2 in . Substituting A obtained from Eq. (27) and H ≃ √ ρ R 3( M 2 Pl -ξϕ 2 in ) in A ≃ 2 H/β , we obtain where α step ≃ 0 . 57. This frequency is shown by a square in Fig. 4. To complete this part of our study, we obtain Ω 0 GW for different values of ξ , and present the results in Fig. 5. As can be seen, the kink in Ω 0 GW deepens by increasing ξ , as expected. Another way to interpret our results is by using dilution factor : D ≡ [ S ( T ≫ T osc ) /S ( T ≪ T RD )] 1 / 3 , where S = a 3 s with s = 2 π 2 45 g s ∗ T 3 being the comoving entropy density. Using a ∝ t α ≃ ( α H ) α , and H osc ≃ √ ρ osc R 3( M 2 Pl -ξϕ 2 in ) and H RD = √ 2 ρ RD R 3 M 2 Pl , along with T osc and T RD given by Eqs. (10) and (12), the dilution factor becomes with α D ≈ 0 . 68 obtained from numerical matching. To interpret the results of Fig. 5, we employ the WKB solution (see Appendix B). For the sub mode with the wave number k , we have | χ k | ≃ a k . The parameter a k is the scale factor at which the k mode becomes sub. Replacing | χ k | in Eq. (22) with a k = k/H ( a k ), for the modes with frequencies f osc and f RD , we obtain Ω 0 GW ( f osc ) / Ω 0 GW ( f RD ) (ignoring A term) as: which demonstrates the indirect effect of the term ξRϕ 2 . Second, if we set the wave number as k = k step , we have | χ k | ≃ e -d a k . Replacing | χ k | in Eq. (22) with e -d k H ( a k )(1+ 1 β ) , for this mode, while using H osc H step ≃ [ f osc f step (1 + 1 β ) ] 1 1 -α step , we obtain where d = 1 2 ∫ 10 5 m ϕ t step A d t . Eq. (32) exhibits the direct effect of the term ξRϕ 2 . Finally, implementing the correction parameter cr as the ratio of h 2 Ω 0 GW ( f osc ) in the power spectrum with A to that of the power spectrum without A , we can write where Ω 0 GW ( f step ) Ω 0 GW ( f osc ) and Ω 0 GW ( f osc ) Ω 0 GW ( f RD ) are given by Eqs. (32) and (31), respectively. To test our estimations, we compare some analytical results with numerical ones. Table I compares the analytical estimates and data driven pivotal frequencies f RD,osc,step for different values of ξ . Furthermore, the ratio of Ω 0 GW at these key frequencies according to the analytical expectations (Eqs. (31), (32), and (33)), and the data are provided in Table I as well. As it is illustrated, the predicted values are consistent with the numerical data. Albeit the analytical expressions depend on data driven parameters α RD,D,step and β , they prove to be valuable as they show the dependence of our observable Ω 0 GW ( f ) on model parameters and initial conditions. For the sake of completeness, let us study the effect of m ϕ and Γ on the shape of Ω 0 GW , in the case of ξ = 0. Here we choose m ϕ = 10 GeV , 40 GeV and Γ = 10 -8 GeV , 4 × 10 -8 GeV. Since m ϕ determines the beginning of the MD era and Γ controls when it ends, only the position of the kinks change and the slopes in Ω 0 GW stay constant, as shown in Fig. 6. Finally, the effects of non-minimal gravitational coupling, ξRϕ 2 , of the scalar field on Ω 0 GW can be probed by future GWs experiments. Due to the shape of Ω 0 GW , we expect GWs to be detectable by some of the observatories focusing on low-frequencies but missed by high-frequency experiments. The shape of the GWs may also be probed by some experiments that can detect relatively lower-intensity GWs. In Fig. 7, we have picked some benchmarks that may be probed by LISA [97], BBO [98], and DECIGO [99].",
"pages": [
10,
11,
12,
13
]
},
{
"title": "V. CONCLUSION",
"content": "In this paper, we investigated the evolution of the SGWB originating from inflation within a cosmological framework. We considered a scenario where a scalar field ϕ dominates the energy density of the Universe at high temperatures and has a non-minimal coupling with gravity, represented as ξRϕ 2 . Previous studies have demonstrated that early matter domination leads to a reduced signal strength of SGWB in the present. Various experiments such as LISA, BBO, and DECIGO can probe SGWB, while others like HLVK and ET may miss it. ̸ By introducing a non-minimal coupling between ϕ and gravity, not only does this decrease amplify, but non-trivial features also emerge at certain frequencies. When the coupling parameter ξ is large, the damping term in the evolution of SGWB is enhanced due to a larger Hubble rate and the presence of a ξ -dependent term denoted as A in this study. The distinctive step-like feature observed in Fig. 7 is caused by the dominance of A over the Hubble rate. Analytically, this feature can be understood by examining the ratio Ω 0 GW ( f step ) Ω 0 GW ( f RD ) , which includes not only the dilution factor but also an additional damping term e -∫ Adt compared to the case where ξ = 0. Additionally, when ξ = 0, the Hubble rate is relatively higher at the beginning of the MD era, resulting in a lower value of Ω GW h 2 , even if the A term is neglected, as the dilution factor discloses. We provided an analytical expression for Ω 0 GW at key frequencies and demonstrated that they align with the numerical results with acceptable precision. In the scenario of early matter domination with ξ = 0, the power spectrum pattern exhibits deviations from a pure power law at frequencies determined by the mass and decay width of ϕ . Our study reveals that introducing the term ξRϕ 2 does not significantly alter the location of these deviations but introduces a distinct step feature that serves as a signature of this term in observational data. We identified a set of benchmarks that exhibit this feature within the parameter space where DECIGO and BBO experiments can probe. In general, the shape of the power spectrum of Primordial Gravitational Waves (PGWs) as a function of frequency provides valuable insights into the evolution of the early universe.",
"pages": [
13
]
},
{
"title": "Acknowledgment",
"content": "We thank Enrico Morgante, Nicklas Ramberg, Wolfram Ratzinger, Nematollah Riazi, Kai Schmitz, and Pedro Schwaller for useful discussions. F.E. and H.M. are grateful to CERN for their hospitality. The work in Mainz is sup- ̸ ported by the Cluster of Excellence Precision Physics, Fundamental Interactions and Structure of Matter (PRISMA + EXC 2118/1) within the German Excellence Strategy (project ID 39083149). Also, F.E. is supported by the grant 05H18UMCA1 of the German Federal Ministry for Education and Research (BMBF), and H.M. is supported by the National Natural Science Foundation of China under Grants No. 12247107 and 12075007.",
"pages": [
13,
14
]
},
{
"title": "Appendix A: Beginning of the evolution: Attractor Behavior, Hubble parameter, Ricci scalar, and EoS parameter",
"content": "̸ This appendix shows that the benchmarks satisfying Eq. (9) have an attractor behavior. That is, even if we start with ˙ ϕ in = 0, we will quickly merge to the path of ϕ = ϕ in and ˙ ϕ in = 0 in the phase space. Our other objective in this appendix is to show that even in the case of non-minimal coupling where ξ ≥ 1, the start of evolution is effectively RD defined as ω = 1 3 , even though the energy density of the scalar field dominates over that of radiation ρ ϕ > ρ R . For better understanding, we rewrite the EoM of ϕ , Eq.(5), with the assumption H ≫ Γ at the beginning as This equation has the form of a harmonic oscillator equation, with m eff as the angular frequency and the friction term of 3 H 2 . In the limit where 3 H 2 ≫ m eff , and with the initial conditions ϕ ( T in ) = ϕ in and ˙ ϕ ( T in ) = 0 the solution for ϕ becomes ϕ = ϕ in = const , which means that the field is stuck at its initial value. If the initial velocity of the harmonic oscillator is non-zero, the solution to the equation in this regime becomes which shows that in this case, the general solution does not have constant behavior and in fact is time-dependent. For any value of ˙ ϕ satisfying Eq. (9), the second term in Eq. (A2) becomes negligible compared with ϕ in and hence we can write the solution as ϕ ( t ) = ϕ in -˙ ϕ 3 H e -3 Ht . In addition, the third term rapidly approaches zero with time, and hence ϕ approaches ϕ in again at high temperatures which obviously demonstrates the attractor behavior mentioned above. To obtain a general understanding of the evolution of the background at the beginning, and besides that an estimation for the depth of the kink in the GWs power spectrum, it is essential to find an approximate relation for the Hubble parameter. Starting from the first Friedmann equation, Eq. (7) and using the energy density of scalar field with non-minimal gravitational coupling, Eq. (8), we have: We are considering a benchmark class that satisfies Eq. (9). Therefore, we can simplify the Hubble rate at T in to the following: where in the first line m 2 ϕ ϕ 2 in / 2 , ˙ ϕ 2 in / 2 ≪ ρ R ( T in ), and in the second line 3 ξ 2 ˙ ϕ 2 in ϕ 2 in M 2 Pl -ξϕ 2 in ≪ ρ R ( T in ) is used. This equation demonstrates that the initial Hubble parameter increases by increasing ξ , which can be seen for the two values of ξ = 0 and ξ = 5 . 95 in Fig 1. As long as these conditions are satisfied, Eq. (A4) provides the approximate Hubble rate even if we move away from T in . With the Hubble parameter in hand, the EoS parameter at the beginning of the evolution can be found using the relation [55, 56] Hence, to have an estimation of the ω at the beginning of the evolution, we need the behavior of R at high temperatures. To this end, we take the following steps. First, by contracting Eq. (2) with the metric, the trace of the equation yields To use the above equation, we need to derive the trace of T ( ϕ ) µν by contacting with metric [60] Now, substituting Eq. (A7) in Eq. (A6), gives the Ricci scalar as [60]: In our study, the scalar field ϕ is assumed to be homogeneous and isotropic (and hence, just a function of time). On the other hand, T ( M ) = 0, since in our study, the matter part is only the radiation, and the trace of the energy-momentum tensor for relativistic fluid with ω = 1 3 is zero. As a result, the Ricci scalar at the beginning becomes According to Eq. (A5), in order to evaluate ω , we need to find the dimensionless parameter R/H 2 : because we are assuming ρ R ( T in ) ≫ ˙ ϕ in , m 2 ϕ ϕ 2 in . Now, given Eq. (A10), we see that ω ≃ 1 / 3 for the class of all benchmarks satisfying Eq. (9). where and χ GR = 1 a e ± ikη . The effect of the additional damping term in the equation of GWs appears in the additional exponential e -d . The parameter d can be written in cosmic time as: As before, all we need to know about the GWs are well coded in the observable of GWs, the power spectrum, introduced in Eq. (21). It will be more convenient if we try to write it in terms of the transfer function itself. It is in fact possible for sub-modes where we have k ≫ a 2 (2 H + A ), and A > 0 (in our specific case). Starting from",
"pages": [
14,
15,
16
]
},
{
"title": "Appendix B: The WKB Analysis",
"content": "In this appendix, we explain the WKB analysis for the evolution of the modes in sub-regime in the presence of the term, A[64-66]. Such an additional term appears in the various modified gravity theories and hence, our analysis in this appendix is quite general and we do not restrict ourselves to the case of ξRϕ 2 . This analysis provides us with physical intuition and parametrizes the solution in terms of the aforementioned direct and indirect effects on the evolution of GWs. Starting from Eq. (16), it is conventional to rewrite this equation as: where we invoke the parametrization h ( η ) = h Prim χ ( η ), H = a ' a and M = A H . To obtain the WKB solution for high-frequency modes in this case, we consider an ansatz: χ = Ze iY where C = Zh prim . Substituting this ansatz for the transfer function in Eq. (B1), separates the equation into two equations for the imaginary and the real part: Since our aim here is to derive the WKB solution for high-frequency modes, we can neglect the second and fourth terms in Eq. (B2). That is because high-frequency modes correspond to the modes that enter the horizon at early times, and thus the terms involving Z are negligible compared to the terms that include the phase of χ . Hence, from Eq. (B2), we have: Substituting this result in Eq. (B3), we simply have the WKB solution [64-66]: Concentrating on this solution, we see that the first two exponentials can be simplified to e -∫ H d η e ± ikη = e -∫ H d t e ± ikη = 1 a e ± ikη if we rewrite it in cosmic time. This part of the WKB solution is exactly the form of the WKB solution in the standard GR (cases of the standard cosmology and the case of ξ = 0) and hence, conventionally called χ GR . 16 As we can see χ GR demonstrates the oscillatory behavior of the high-frequency modes in the sub-regime. It is conventional to write the complete WKB solution for ξ = 0 as: ̸ and invoking the useful relation we can simply conclude that By substituting the above equation in Eq. (21), Eq. (22) is obtained. [6] M. Maggiore, Gravitational wave experiments and early universe cosmology , Physics Reports 331 (2000) 283. [9] V. Rubakov, M. V. Sazhin and A. Veryaskin, Graviton creation in the inflationary universe and the grand unification scale , Physics Letters B 115 (1982) 189. [20] D. Shoemaker, L. S. Collaboration et al., Gravitational wave astronomy with ligo and similar detectors in the next decade , Bulletin of the American Astronomical Society 51 (2019) 452.",
"pages": [
16,
17
]
}
]
2024PhRvD.109l4047K
https://arxiv.org/pdf/2402.15336.pdf
<document>
<section_header_level_1><location><page_1><loc_20><loc_92><loc_81><loc_93></location>Probing black hole 'charge' from the binary black hole inspiral</section_header_level_1>
<text><location><page_1><loc_31><loc_89><loc_69><loc_90></location>N. V. Krishnendu 1, ∗ and Sumanta Chakraborty 2, †</text>
<text><location><page_1><loc_26><loc_87><loc_75><loc_88></location>1 International Centre for Theoretical Sciences (ICTS), Survey No. 151,</text>
<text><location><page_1><loc_29><loc_86><loc_72><loc_87></location>Shivakote, Hesaraghatta, Uttarahalli, Bengaluru, 560089, India</text>
<text><location><page_1><loc_16><loc_85><loc_84><loc_86></location>2 School of Physical Sciences, Indian Association for the Cultivation of Science, Kolkata 700032, India</text>
<text><location><page_1><loc_43><loc_83><loc_58><loc_84></location>(Dated: June 25, 2024)</text>
<text><location><page_1><loc_18><loc_57><loc_83><loc_82></location>Recent gravitational wave (GW) observations have enabled us to look beyond the standard paradigm of gravitational physics, namely general relativity (GR). Along with the mass and the angular momentum, which typical astrophysical black holes (BHs) are endowed with, theories beyond GR generically induce 'charge' to these BHs. Notably, for BHs carrying the extra 'charge' hair, we expect the BH absorption effects to modify accordingly and alter the tidal heating terms. Hence, the inclusion of the corrections in the GW waveform model, arising from the BH 'charge', allows us to test the consistency of the observed binaries with Kerr BHs in GR. We compute the explicit dependence of the binary inspiral phase on the 'charge' parameter arising from the tidal heating effect and study the measurability of the same from GW observations of binary mergers. Specifically, we employ the TaylorF2 waveform model, which accurately models the inspiral evolution of an aligned-spin binary merger, and Bayesian analysis-based GW data inference to measure the 'charge' parameter for a selected set of detected binaries. We also present a detailed simulation study to investigate the possibility of measuring the charge parameter from binaries with different masses, spins and source locations. The analysis of selected GW events from the third GW transient catalogue shows that the 'charge' parameter constraints are poor from the observed signals with the current sensitivity. In contrast, the simulation studies indicate that the spinning binaries with significant mass asymmetry provide the best constraints on the BH 'charge' parameter. Finally, we study the prospects of measuring the BH 'charge' parameter from a future GW detector with improved sensitivity.</text>
<section_header_level_1><location><page_1><loc_20><loc_53><loc_37><loc_54></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_25><loc_49><loc_51></location>After finishing the three successful observing runs, the ground-based gravitational wave (GW) detectors confirmed the detections of more than ninety compact binary coalescing events [1-9]. Then, the first part of the fourth observation run concluded by reporting around eighty GW events and details are available in the public alert system [10]. From the detection and further dedicated analysis, it is evident that the signals consist of three types of merger events - (a) binary black holes (BHs), (b) binary neutron stars (NSs), and (c) BH-NS systems [11-14]. Even though the observed signals are consistent with these compact objects being described by either BHs or NSs within the framework of general relativity (GR) and show no clear evidence of the presence of physics beyond the standard paradigm [15-18], however, given the statistical uncertainties in these measurements, there is still ample room for alternate theories of gravity and exotic compact objects [19, 20].</text>
<text><location><page_1><loc_9><loc_15><loc_49><loc_25></location>Despite the monumental success of GR in all the observational tests so far [18, 21], the prime reasons to look for alternatives of GR are threefold - (a) Einstein's equations predict singularities, where the theory itself breaks down, (b) solutions of Einstein's equations involve Cauchy horizon, leading to uncertain future for the classical theory, (c) consistency with observations require</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_55></location>postulating the existence of yet undetected dark matter and exotic dark energy. Besides, there are other important issues related to the energy scale at which gravity dominates, as this scale is much above the energy scales associated with the other fundamental interactions. This leads to the well-known hierarchy problem, where the stabilization of the Higgs mass requires an exorbitant fine-tuning of the order of one part in 10 15 . The models involving extra spatial dimensions are among several alternatives to get around this fine-tuning problem. In this work, we explore the implications of the existence of an extra spatial dimension on the inspiral regime of the binary BH merger events detected by the LIGO-VirgoKAGRAcollaboration. The presence of this extra spatial dimension modifies the gravitational field equations on the four-dimensional spacetime (known as the brane ), as the projection of the five-dimensional (known as the bulk ) Einstein's equations inherit additional contributions from the bulk. Consequently, the BHs on the brane are characterized by an additional hair, other than the mass and the spin. This hair is identical in appearance to that of the Maxwell charge but differs by an overall sign, which becomes a distinctive signature of the existence of extra spatial dimensions [22-25]. It is worth pointing out that a similar contribution, albeit with an opposite sign, appears in the context of Einstein-Maxwell theories, scalar coupled Maxwell theories [26-29] and also in f ( T ) theories of gravity [30, 31]. The presence of the extra 'charge' in the BH solution can induce differences in the evolution of a compact binary system, making their GW signatures utterly different from that of the Kerr BH. Specifically, by focusing on the inspiral dynamics of the binary, we</text>
<text><location><page_2><loc_9><loc_89><loc_49><loc_93></location>study the significance of the 'charge' hair on the tidal heating effects present in the GW waveform measured at infinity.</text>
<text><location><page_2><loc_9><loc_61><loc_49><loc_88></location>Tidal heating is an absorption effect, which arises due to the absorption of external GWs by compact objects in a binary, leading to time evolution of the mass and angular momentum of the compact objects, leaving observational imprints [32-45]. The existence of the 'charge' hair will modify the absorption of GWs by the BHs on the brane, which will reflect in the orbital evolution of the binary in the inspiral phase. Following this, we propose a novel test to measure the 'charge' parameter from the observed inspiral evolution of the binary coalescence events and demonstrate the measurability over a set of simulated binary signals. The inspiral measurements of the BH 'charge' provide a unique test of the nature of compact binary because for astrophysical BHs, the net electric charge is expected to be zero, or even if there is a small electric charge present, that gets shielded away quickly [46-48]. Therefore, the existence of a 'charge' hair, along with its overall sign, is a characteristic of the presence of an extra spatial dimension.</text>
<text><location><page_2><loc_9><loc_10><loc_49><loc_61></location>There have been several previous attempts in the literature to look for the existence of such a 'charge' in various observations involving BHs. These include - (a) the shadow of M87* and SgrA* as observed by the Event Horizon Telescope collaboration [49-51], (b) the X-ray luminosity from the accretion disc of quasars [52-55], and (c) the GW observations from binary coalescence [56-62]. Intriguingly, the observations involving BH shadow and accretion disc around quasars mildly favour the existence of a negative 'charge' compared to the Kerr scenario in GR. The ringdown signal, on the other hand, provides little information, and both GR and extra dimensions remain viable alternatives. In Ref. [63], the BH ringdown waveform has been obtained by numerically solving the perturbation equations for the braneworld BH and determining the associated quasi-normal modes [63] with the explicit BH 'charge' dependence. Subsequently, this analysis has been extended in [64], and constraints on the 'charge' parameter have been derived from the events observed through the first three observing runs of advanced LIGO and Virgo detectors. In a different perspective, [65] demonstrated yet another method to constrain the 'charge' parameter by looking into the GW micro-lensing signature of the charged lens. It is also possible through anti-de Sitter/conformal field theory correspondence (AdS/CFT) correspondence to argue that horizons for BHs on the brane must be reflective [66, 67], however, in this work, we will stick to the classical BH picture, returning to the reflective nature of horizon in future work. To summarise, we focus on the inspiral dynamics of the binary and study the distinctive features in the BH absorption spectra due to the presence of the 'charge' parameter arising from the existence of an extra spatial dimension considering stellar mass binary BH mergers.</text>
<text><location><page_2><loc_10><loc_9><loc_49><loc_10></location>The detection and parameter inference necessitate em-</text>
<text><location><page_2><loc_52><loc_57><loc_92><loc_93></location>ploying waveform models based on GR that accurately predict the binary evolution, including various physical effects of binary BHs in GR. One must validate these GW waveform models to regimes inaccessible to GR and its predictions to look for beyond-GR effects in the GW data. However, the theory-agnostic tests of GR are routinely employed to check the consistency of GR with the observed data where generic parametric deviations are introduced in the GW waveform model without assuming any specific modified gravity models [15-18]. Suppose GR is the correct theory of gravity; in this case, the measurement uncertainties on the parametric deviation coefficients will be consistent with GR prediction, and any departure will lead to further elaborate analyses. In the ideal scenario, one may start with an alternate gravity model, compute the GW waveform model for a merging binary in that particular theory, and use it for the analysis. Despite multiple efforts, such a complete model has yet to be available. In the middle ground, one can combine the orbital evolution information from GR, calculate additional contributions from beyond GR effects and estimate the GW waveform model. Analysis employing such a modified waveform model will provide a consistency check with the observed data and GR predictions. In this study, we focus on such a scenario.</text>
<text><location><page_2><loc_52><loc_34><loc_92><loc_57></location>To do so, we calculate the BH absorption effects of a braneworld BH and modify the post-Newtonian inspiral phase by appropriately adding the tidal charge contributions. We keep the 'charge' parameter as a free parameter and measure it from the data. Before discussing the real GW analysis, we demonstrate the method by simulating a set of binaries with various masses, spins and locations. The simulations indicate that it is possible to measure the 'charge' parameter for spinning binaries with mass asymmetry if we consider current-ground-based GW detectors with their plus-era (O5) sensitivity [68]. Further, we describe the method's applicability for inspiral-dominated GW events detected through the first three observing runs of LIGO-Virgo detectors. For completeness we will discuss both the scenarios involving positive as well as negative 'charged' hairs.</text>
<text><location><page_2><loc_52><loc_21><loc_92><loc_34></location>The paper is organized as follows: we give a brief outline of our geometrical setup in Sec. II, and then discuss the waveform model, a short description of the Bayesian analysis and a note on the details of binary simulations in Sec. III. The results from simulated binary signals are presented in Sec. IV, while our bounds from the GW signals of the detected binary merger events have been presented in Sec. V. Finally, we conclude by summarising our findings and listing future plans in Sec. VI.</text>
<section_header_level_1><location><page_2><loc_55><loc_13><loc_88><loc_17></location>II. BACKGROUND GEOMETRY WITH CHARGE: IMPLICATIONS FOR TIDAL HEATING</section_header_level_1>
<text><location><page_2><loc_52><loc_9><loc_92><loc_11></location>In this section, we present the background geometry of a braneworld BH, with the charge term, and also dis-</text>
<text><location><page_3><loc_9><loc_85><loc_49><loc_93></location>cuss the implications of this on the tidal heating phenomenon. In the braneworld scenario, the higher dimensional spacetime ( bulk ) satisfies Einstein's equations, while the gravitational field equations on the embedded four-dimensional hypersurface ( brane ), on which the standard model fields live become,</text>
<formula><location><page_3><loc_22><loc_82><loc_49><loc_83></location>(4) G µν + E µν = 0 . (1)</formula>
<text><location><page_3><loc_9><loc_54><loc_49><loc_80></location>Here, (4) G µν is the four-dimensional Einstein tensor on the brane hypersurface, E µν ≡ (5) W ABCD e A µ n B e C ν n D is the projection of the five-dimensional Weyl tensor (5) W ABCD on the brane hypersurface, where e A µ are the projectors and n B is the normal vector to the brane hypersurface. Owing to the symmetries of the Weyl tensor, it follows that E µ µ = 0, and Bianchi identity demands ∇ µ E µ ν = 0. Both of these properties are akin to the energy-momentum tensor of the electromagnetic field, except for an overall sign. This is because the energy-momentum tensor sits on the right-hand side of Einstein's equation, acting as the source of gravity, while the Weyl tensor E µν sits on the left-hand side, which mimics a source with energy-momentum tensor -E µν . Thus, braneworld BH depicts vacuum spacetime but resembles Kerr-Newman spacetime with an overall negative sign in front of the charge term. Therefore, the spacetime geometry of a rotating braneworld BH takes the form,</text>
<formula><location><page_3><loc_12><loc_46><loc_49><loc_53></location>ds 2 = -∆ Σ ( dt -a sin 2 θ dϕ ) 2 + Σ [ dr 2 ∆ + dθ 2 ] + sin 2 θ Σ [ adt -( r 2 + a 2 ) dϕ ] 2 . (2)</formula>
<text><location><page_3><loc_9><loc_25><loc_49><loc_45></location>The above metric depicts a rotating BH with mass M , angular momentum J = aM and braneworld 'charge' Q BH . The metric functions appearing in the above line element involve two unknown functions ∆ and Σ, defined as ∆ ≡ r 2 + a 2 -2 Mr - Q BH and Σ ≡ r 2 + a 2 cos 2 θ . Note that, for the case of Kerr-Newman BH, the parameter Q BH can be identified with the negative of the square of the electric charge Q of the BH, such that Q BH | KN = -Q 2 . The rotating braneworld BH inherits two horizons, located at r ± = M ± √ M 2 -a 2 + Q BH obtained by solving the equation ∆ = 0. Intriguingly, even if a > M , for non-zero values of Q BH , the outer horizon r + exists, in stark contrast to that of the Kerr-Newman BH.</text>
<text><location><page_3><loc_9><loc_9><loc_49><loc_24></location>Gravitational perturbation of the background geometry, depicting a rotating BH spacetime on the brane, can be described by Newman-Penrose scalars Ψ 0 and Ψ 4 . Since our interest is in the physics of the horizon, namely in determining the GW flux going down the horizon, we will work with the Newman-Penrose scalar Ψ 0 . In general, for the Kerr-Newman-like spacetimes, the angular and the radial parts of the Newman-Penrose scalars cannot be separated. However, in the present context, perturbation of the 'source' term E µν is directly proportional to the ratio of the bulk and the brane curvature length</text>
<text><location><page_3><loc_52><loc_88><loc_92><loc_93></location>scales, which can be ignored for all practical purposes. Therefore, the Weyl scalar Ψ 0 , describing gravitational perturbation of a rotating braneworld BH, can be expressed as,</text>
<formula><location><page_3><loc_54><loc_82><loc_92><loc_86></location>Ψ 0 = ∫ dω ∞ ∑ ℓ =0 ℓ ∑ m = -ℓ 2 S ℓm ( θ ) R ℓm ( r ) e -iωt e imϕ , (3)</formula>
<text><location><page_3><loc_52><loc_78><loc_92><loc_81></location>where the angular part 2 S ℓm ( θ ) satisfies the following differential equation,</text>
<formula><location><page_3><loc_54><loc_70><loc_92><loc_77></location>1 sin θ d dθ [ sin θ d 2 S ℓm dθ ] + [ ( aω cos θ ) 2 -4 aω cos θ +2 + 2 A ℓm -( m +2cos θ ) 2 1 -cos 2 θ ] 2 S ℓm = 0 , (4)</formula>
<text><location><page_3><loc_52><loc_62><loc_92><loc_69></location>which coincides with the equation satisfied by the spinweighted spherical harmonics. Here 2 A ℓm is the separation constant between the radial and the angular parts. The radial function R ℓm ( r ), on the other hand, satisfies the following differential equation [69].</text>
<formula><location><page_3><loc_55><loc_55><loc_92><loc_61></location>1 ∆ 2 d dr [ ∆ 3 dR ℓm dr ] + [ K 2 -4 i ( r -M ) K ∆ +4 i dK dr -λ ] R ℓm = 0 , (5)</formula>
<text><location><page_3><loc_52><loc_49><loc_92><loc_54></location>where, K ≡ ( r 2 + a 2 ) ω -am and λ = 2 A ℓm + ( aω ) 2 -2 amω , is related to the separation constant 2 A ℓm appearing in the angular part 2 S ℓm .</text>
<text><location><page_3><loc_52><loc_18><loc_92><loc_49></location>Consider now a binary system involving two braneworld BHs, characterized by masses M 1 and M 2 , angular momentum J 1 and J 2 , as well as 'charge' parameter Q BH . In the context of braneworld BH, the charge Q BH depends on the length of the extra dimension, and hence is an invariant quantity for all BHs. While for positive values of Q BH , we assume identical values of the charge for both the BHs in the binary, possibly due to some overall equilibrium. During the inspiral of this binary system around one another, each of these BHs will absorb a part of the emitted GW radiation in the centre of the mass frame, leading to a rate of change of mass M , angular momentum J , and area A . This is known as tidal heating. In order to determine the above rate of changes of BH parameters for the first BH, we must solve the radial equation near the horizon r +1 and impose purely ingoing boundary conditions at the horizon. This fixes one arbitrary constant appearing in the solution of the Teukolsky equation. In order to fix the other, we need to work in a regime where M 1 ≪ r ≪ b , with b being a typical distance between the binary BHs, and one imposes the following boundary condition,</text>
<formula><location><page_3><loc_54><loc_13><loc_92><loc_17></location>Ψ 0 = 8 π √ 6 M 2 5 b 3 2 ∑ m = -2 2 Y 2 m ( θ, ϕ ) 2 Y ∗ 2 m ( θ 0 , ϕ 0 ) . (6)</formula>
<text><location><page_3><loc_52><loc_8><loc_92><loc_11></location>To determine the tidal heating associated with the second BH, we simply have to interchange M 1 ↔ M 2 , and</text>
<text><location><page_4><loc_9><loc_82><loc_49><loc_93></location>J 1 ↔ J 2 , respectively. With the above boundary condition, one can uniquely solve for the Weyl scalar Ψ 0 and transform the same to the Hartle-Hawking frame, thereby determining the rate of change of area in terms of | Ψ 0 | 2 . The corresponding rate of change of mass can be derived using the laws of BH mechanics, which relates the rate of change of mass and angular momentum to the rate of change of area, yielding [70],</text>
<formula><location><page_4><loc_12><loc_69><loc_49><loc_81></location>dM 1 dt = ( dE dt ) N ( M 1 M ) 3 v 5 4 { -χ 1 ( ˆ L orb · ˆ J 1 ) +2 v 3 M ( r +1 + Q BH 2 M 1 ) }[ 1 + 3 χ 2 1 + Q BH M 2 1 ( 2 + 3 χ 2 1 + Q BH M 2 1 ) ] , (7)</formula>
<text><location><page_4><loc_9><loc_53><loc_49><loc_68></location>where ( dE/dt ) N = (32 / 5) η 2 v 10 is the energy loss due to GWs arising from the quadrupole approximation. Further, we have defined M ≡ M 1 + M 2 and η ≡ ( M 1 M 2 /M 2 ), while the relative velocity of the binary BH system is given by v = √ M/b . A similar expression can be derived for ( dM 2 /dt ), by simply the M 1 ↔ M 2 exchange. Thus, the rate of change of mass, in comparison to the quadrupolar rate of change of energy, depends on post-Newtonian (PN) terms of two distinct orders, at 2 . 5 PN (or, equivalently v 5 ),</text>
<formula><location><page_4><loc_11><loc_44><loc_49><loc_52></location>A (5) i ≡ ( M i M ) 3 χ i ( ˆ L orb · ˆ J i ) [ 1 + 3 χ 2 i + q BH ( M 2 M 2 i ){ 2 + 3 χ 2 i + q BH ( M 2 M 2 i )} ] , (8)</formula>
<text><location><page_4><loc_9><loc_42><loc_33><loc_43></location>and at 4 PN (or, equivalently v 8 ),</text>
<formula><location><page_4><loc_11><loc_29><loc_49><loc_41></location>A (8) i ≡ ( M i M ) 4 [ 1 + √ 1 -χ 2 i + q BH ( M 2 M 2 i ) + q BH 2 ( M 2 M 2 i ) ][ 1 + 3 χ 2 i + q BH ( M 2 M 2 i ) × { 2 + 3 χ 2 i + q BH ( M 2 M 2 i )} ] . (9)</formula>
<text><location><page_4><loc_9><loc_22><loc_49><loc_28></location>Here we have defined, q BH ≡ ( Q BH /M 2 ), χ i = a i /M i and i = { 1 , 2 } . Thus the rate of change of mass for the i th component of the binary BH system can be expressed as,</text>
<formula><location><page_4><loc_15><loc_18><loc_49><loc_21></location>dM i dt = ( dE dt ) N [ -A (5) i v 5 4 + A (8) i v 8 2 ] . (10)</formula>
<text><location><page_4><loc_9><loc_15><loc_49><loc_17></location>Therefore, the total flux going into the horizon of the braneworld BH becomes,</text>
<formula><location><page_4><loc_11><loc_9><loc_49><loc_14></location>F BH = ∑ i ( dM i dt ) = ( dE dt ) N [ -Ψ 5 v 5 4 +Ψ 8 v 8 2 ] . (11)</formula>
<text><location><page_4><loc_52><loc_87><loc_92><loc_93></location>Here, we have defined, Ψ 5 ≡ ∑ i A (5) i and Ψ 8 ≡ ∑ i A 8 i , for notational convenience. As we will demonstrate, these two quantities will be central to the phase evolution in the presence of tidal heating.</text>
<text><location><page_4><loc_52><loc_51><loc_92><loc_87></location>At this outset, let us discuss previous bounds on the tidal charge parameter q BH and the implications of such bounds on the size of the extra dimension [71]. In the context of GW observations, based on the ringdown part of the signal, [62-64] provides bounds on the tidal charge parameter. The 90% confidence contours, for the majority of GW observations, extended beyond q BH = ± 0 . 5. Besides, the lensing of GWs can also constrain the tidal charge q BH , but again for the braneworld models the constraints are weak q BH ≤ -0 . 9, while for electromagnetic theories the constraints are better q BH ≤ 0 . 5 [65]. From electromagnetic observations as well, e.g., the measurement of BH shadow constrains the tidal charge as q BH = -0 . 1 +0 . 6 -0 . 5 [72, 73]. These suggest that the constraints on the charge parameter, irrespective of its origin, are weak and are the prime motivation to choose the prior within the range q BH ∈ ( -1 , 1). Further note that, in the braneworld scenario, the brane is obtained by embedding the four-dimensional spacetime within a fivedimensional bulk. Therefore, the charge q BH gets naturally connected to χ , the size of the extra dimension. For example, the following bound: q BH ≤ -0 . 5, translates into ( χ/ℓ ) ≲ 0 . 63 [71], where ℓ depicts the ratio of the five-dimensional and the four-dimensional gravitational constants.</text>
<text><location><page_4><loc_52><loc_38><loc_92><loc_51></location>Having determined the effect of the charge term in the spacetime metric, arising from extra spatial dimension, on the perturbation equation governing the gravitational perturbation of the brane. The corresponding fluxes through the horizons of the braneworld BHs orbiting each other get corrected at 2.5 PN and 4 PN levels due to the charge term. Given the above modification to the horizon flux, we wish to determine the corresponding modifications to the GW phase in the next section.</text>
<section_header_level_1><location><page_4><loc_53><loc_32><loc_90><loc_34></location>III. DETAILS OF THE WAVEFORM MODEL AND PARAMETER ESTIMATION</section_header_level_1>
<text><location><page_4><loc_52><loc_23><loc_92><loc_30></location>In this section, we will present detailed discussion regarding the waveform modeling for braneworld BH, with special emphasis on the effect of charge inherited from extra spatial dimension. Then we proceed to the parameter estimation details.</text>
<section_header_level_1><location><page_4><loc_64><loc_18><loc_80><loc_19></location>A. Waveform model</section_header_level_1>
<text><location><page_4><loc_52><loc_12><loc_92><loc_16></location>The GW waveform model from a coalescing compact binary signal in the frequency domain can be schematically represented as,</text>
<formula><location><page_4><loc_61><loc_8><loc_92><loc_10></location>˜ h ( f ) = CA ( f ) e i { ψ test ( f )+ δψ ( f ) } , (12)</formula>
<text><location><page_5><loc_9><loc_57><loc_49><loc_93></location>where C is an overall constant, A ( f ) is the amplitude of the GW, ψ test ( f ) is the phase of the GW in the test particle approximation [74, 75], and δψ ( f ) is the contribution to the phase due to finite size effects of the binary BH system. In the inspiral regime, the amplitude of the GW follows from the relation A ( f ) ∼ D -1 L M 5 / 6 c f -7 / 6 at leading order, where D L is the luminosity distance between the observer and the source of the GW, with M c = ( M 1 M 2 ) 3 / 5 ( M 1 + M 2 ) -1 / 5 being the chirp mass of the binary. The frequency-independent factor C carries information about the source location and orientation of the source with respect to the detector, through the antenna pattern functions. For a compact binary signal, the GWphase plays a crucial role in detecting and analysing the signal. Hence, it is important to model them with the maximum available accuracy. In our case, the first term, ψ test ( f ), accounts for the 'point-particle' contributions, whereas δψ ( f ) represents the extra phase contributions that arise due to tidal heating, or, the BH absorption effect. Among these terms, ψ test ( f ) is taken to be accurate upto 3.5 PN [76], and δψ ( f ) has contributions at 2.5 PN, 3.5 PN and 4 PN orders. Altogether, these phase contributions accurately model the binary dynamics to the respective PN orders of aligned spin braneworld BHs in a binary system.</text>
<text><location><page_5><loc_9><loc_50><loc_49><loc_57></location>The explicit expression for the phase δψ ( f ), due to tidal heating, can be obtained using the phase formula of [37, 77], and then using the flux through the horizon due to tidal heating, derived in Eq. (11). The final expression reads (for a detailed derivation, see Appendix A),</text>
<formula><location><page_5><loc_10><loc_46><loc_49><loc_49></location>δψ = 3 128 η v -5 [ ψ 2 . 5PN v 5 + ψ 3 . 5PN v 7 + ψ 4PN v 8 ] , (13)</formula>
<text><location><page_5><loc_9><loc_35><loc_49><loc_45></location>where v is the relative velocity between the inspiralling braneworld BHs, acting as the PN parameter representing the PN order at which each coefficient would appear. So we have the tidal heating contributing at three postNewtonian orders, 2.5PN ( v 5 ), 3.5PN ( v 7 ) and 4PN ( v 8 ) with the explicit dependence to the binary parameters as (for a derivation, see Appendix A),</text>
<formula><location><page_5><loc_10><loc_31><loc_49><loc_34></location>ψ 2 . 5PN = -10 9 Ψ 5 (3 log v +1) , (14)</formula>
<formula><location><page_5><loc_10><loc_28><loc_49><loc_31></location>ψ 3 . 5PN = -5 168 Ψ 5 (952 η +995) , (15)</formula>
<formula><location><page_5><loc_10><loc_25><loc_49><loc_28></location>ψ 4PN = -20 9 (3 log v -1) [Ψ 5 ( F SO +4 π ) + Ψ 8 ] . (16)</formula>
<text><location><page_5><loc_9><loc_9><loc_49><loc_24></location>Here F SO is the spin-orbit coupling term (see Eq. (5) in [77]). The quantities Ψ 5 and Ψ 8 depend on the characteristic parameters associated with the BH spacetime through the relations defined below Eq. (11). Notice that the binary dynamics are also influenced by each other's gravitational field, and we neglect these tidalinduced deformation effects in the phase. This is justifiable in the current scenario as the tidal deformations is a higher post-Newtonian effect (it starts appearing at the five post-Newtonian order) with a lesser contribution compared to BH absorption effect [78].</text>
<figure>
<location><page_5><loc_53><loc_70><loc_92><loc_93></location>
<caption>FIG. 1. The mismatch between the GW waveform models corresponding to a charged BH and Kerr BH as a function of the charge parameter q BH for different binary configurations. The binary total mass is fixed to be 20 M ⊙ , with two different mass ratios q = ( M 1 /M 2 ) = 1 , 3 and dimensionless spins ( χ 1 , χ 2 ) taken to be (0 . 7 , 0 . 7) and (0 . 7 , 0 . 1), respectively.</caption>
</figure>
<text><location><page_5><loc_52><loc_32><loc_92><loc_56></location>In what follows, we have incorporated these modifications to the TaylorF2 waveform model in LALSuite [79]. TaylorF2 is an inspiral-only model for an aligned-spin binary, where the component spin angular momenta are either aligned or anti-aligned to the orbital angular momentum axis. Further, we analyse binaries of various kinds employing this waveform model after making necessary changes to the dynesty sampler implemented in Bilby [80]. We truncate the waveform model before the plunge to avoid any unmodelled effects appearing from the post-inspiral frequency region. The truncation frequency in each case is the corresponding inner-most stable circular frequency of a Kerr BH [81], and the estimate is based on calculating the final mass and spin of the remnant BH; hence, the angular frequency considering a circular equatorial orbit around the Kerr BH, given the component masses and spins.</text>
<section_header_level_1><location><page_5><loc_59><loc_27><loc_85><loc_28></location>B. Overview of Bayesian analysis</section_header_level_1>
<text><location><page_5><loc_52><loc_13><loc_92><loc_24></location>In the GW data analysis, we start with the data d, which contains both noise and signal. Here, noise is a random process while the signal is modelled following a particular hypothesis H and is a function of the complete set of binary parameters θ . The initial prior probability distribution, p( θ | H), restricts the range of θ . If we assume that the noise is Gaussian wide-sense stationary, the likelihood function takes the form,</text>
<formula><location><page_5><loc_60><loc_8><loc_92><loc_11></location>p(d | H , θ ) ∝ exp[ -(d -h | d -h) 2 ] , (17)</formula>
<text><location><page_6><loc_9><loc_86><loc_49><loc_93></location>where d and h are the frequency domain data and signal respectively. Once we estimate the likelihood function and the prior distribution on each parameter is known, Baye's theorem provides the posterior probability distribution on each parameter as follows:</text>
<formula><location><page_6><loc_18><loc_81><loc_49><loc_85></location>p( θ | H , d) = p( θ | H)p(d | H , θ ) p(d | H) . (18)</formula>
<text><location><page_6><loc_9><loc_75><loc_49><loc_81></location>In addition, the Bayesian evidence p(d | H) is a measure of how much the data supports the hypothesis, H, and is obtained by marginalizing the likelihood over the full prior volume,</text>
<formula><location><page_6><loc_16><loc_71><loc_49><loc_74></location>Z = p(d | H) = ∫ p( θ | H)p(d | θ, H)d θ . (19)</formula>
<text><location><page_6><loc_9><loc_56><loc_49><loc_71></location>For the usual analyses, we assume GR accurately models the signal, and we estimate the GR evidence Z GR . On the other hand, to check the consistency of the predictions from BHs beyond GR with the data, we include the waveform model described in Sec. III A introducing a free parameter, namely the charge q BH and obtain, Z nGR , the evidence for non-GR signature in the data. The ratio between Z nGR and Z GR provides the Bayes factor comparing the non-GR hypothesis over the GR hypothesis. That is,</text>
<formula><location><page_6><loc_23><loc_52><loc_49><loc_55></location>B nGR GR = Z nGR Z GR . (20)</formula>
<text><location><page_6><loc_9><loc_49><loc_49><loc_52></location>If the data is consistent with GR, the 1-dimensional marginalized posterior on q BH , which is defined as,</text>
<formula><location><page_6><loc_13><loc_45><loc_49><loc_48></location>p(q BH | H nGR , d) = ∫ p( θ, q BH | H nGR , d)d θ (21)</formula>
<text><location><page_6><loc_9><loc_38><loc_49><loc_44></location>will peak at zero (the GR value) and the Bayes factor B nGR GR will be less than zero. All of these computations require evaluating the noise-weighted inner product, which has first appeared in Eq. 17, and is of the form,</text>
<formula><location><page_6><loc_16><loc_34><loc_49><loc_37></location>( a | b ) = 4 Re ∫ f high f low a ∗ (f) × b(f) Sn(f) df . (22)</formula>
<text><location><page_6><loc_9><loc_22><loc_49><loc_33></location>As evident, the inner product depends on the detector characteristics through the noise power spectral density Sn(f) and the lower cut-off frequency f low . In our case, we fix f low to be 10Hz and the upper cut-off frequency f high will vary according to the masses and spins of the binary system. To perform the posterior evaluation, we will be choosing the power spectral density corresponding to the plus era sensitivity (O5) for the LIGO detector [68].</text>
<text><location><page_6><loc_13><loc_13><loc_13><loc_16></location≯</text>
<text><location><page_6><loc_9><loc_11><loc_49><loc_21></location>To visualise the effect of the extra phase term on the binary phasing and hence the GW waveform model, we show the mismatch between the two models in Fig. 1; one includes the effect of the charge parameter ˜ h nGR ( f ) ( q BH = 0), while the other depicts a waveform model representing a binary BH in GR ˜ h GR ( f ) ( q BH = 0), such that,</text>
<formula><location><page_6><loc_17><loc_8><loc_49><loc_10></location>1 -M = 1 -( ˜ h nGR ( f ) , ˜ h GR ( f )) , (23)</formula>
<text><location><page_6><loc_52><loc_80><loc_92><loc_93></location>where the noise weighted inner product is defined in Eq. 22. As Fig. 1 depicts, the mismatch between the two waveform models increases as we increase the q BH value for both the mass ratios, q = ( M 1 /M 2 ) = 1 , 3, and spins (0 . 7 , 0 . 7) , (0 . 7 , 0 . 1). The binary total mass is fixed to be 20 M ⊙ and q BH value is chosen from [ -1 , 1]. The mismatch for binaries with large spin and mass asymmetries is larger, indicating a better distinguishability from their Kerr BH counterparts.</text>
<section_header_level_1><location><page_6><loc_62><loc_76><loc_81><loc_77></location>C. Details of simulation</section_header_level_1>
<text><location><page_6><loc_52><loc_44><loc_92><loc_74></location>The simulations are motivated by the findings of mismatch studies, and we sub-divide these studies into different sets focusing on the masses, spins, signal-to-noise ratios (SNRs) and the ability to identify a non-GR signature if present. To show the effect of component spins, we choose ( χ 1 , χ 2 ) = (0 . 7 , 0 . 6) , (0 . 7 , 0 . 1) , (0 . 5 , 0 . 3) and (0 . 2 , 0 . 1), where χ 1 and χ 2 are the dimensionless spins of the BHs, assumed to be aligned to the orbital angular momentum axis of the binary. Further, we consider four mass ratios, q ≡ ( M 1 /M 2 ) = 1 , 2 , 3 , 4 to examine the effect of mass ratio on the q BH estimate. For all these cases, the total mass is fixed to 32 M ⊙ and the binary is fixed at a particular location to generate a signal-to-noise ratio of 120 in the detector. Moreover, we show posteriors on q BH by choosing signal-to-noises 40 and 80 along with 120 by varying the luminosity distance to quantify the measurability at different signal strengths. Moreover, the detectability of the tidal parameter is detailed by simulating injections with different values of q BH ranging from -0 . 7 to 0 . 7.</text>
<text><location><page_6><loc_52><loc_25><loc_92><loc_45></location>A uniform prior range between [-1, 1] is assumed for the charge parameter for the entire analysis. Prior for the component masses are also taken uniformly between [5 , 80] M ⊙ whereas for component spins uniform from [0, 0.99]. We fixed the luminosity distance, sky location (right ascension and declination), polarization angle, the inclination to the source fixed to the injected value while performing the parameter estimation analysis and verified that the q BH posteriors are unaffected by this choice. While creating simulated injections, the luminosity distance has been altered according to the signal-to-noise ratio requirement. However, the right ascension, declination, and polarization angle are chosen to be 0, and the inclination to the source is fixed at 0.5 rad.</text>
<text><location><page_6><loc_52><loc_9><loc_92><loc_24></location>It is worth mentioning that a full waveform model including inspiral-merger-ringdown regimes and spinprecession effects would be the best for our analysis. However, no such model is known in the context of braneworld and emplying the TaylorF2 waveform model is sufficient for our purpose. First of all, we have truncated the likelihood evaluation at the last stable circular frequency to ensure the validity of the PN approximation and to avoid any systematic that arises due to the presence of un-modelled physics contributing from the post-inspiral regime. Moreover, the absorption effects</text>
<figure>
<location><page_7><loc_10><loc_71><loc_48><loc_93></location>
<caption>FIG. 2. Posteriors on the charge parameter q BH from simulated binary signals of total mass 32M ⊙ and mass ratio q = ( M 1 /M 2 ) = 3 with spins (0 . 7 , 0 . 6), (0 . 7 , 0 . 1), (0 . 5 , 0 . 3) and (0 . 2 , 0 . 1). The luminosity distance to the source is chosen such that the binary generates a signal-to-noise ratio of 120 in the detector band.</caption>
</figure>
<text><location><page_7><loc_16><loc_41><loc_16><loc_44></location≯</text>
<text><location><page_7><loc_9><loc_35><loc_49><loc_58></location>start to appear at 2.5 PN for spinning binaries, which is relatively lower than other effects, such as tidal deformation (a 5PN effect). Therefore, the analysis demonstrated here is largely waveform-independent and can easily be extended to more generic waveform models by simply adding the phase corrections due to tidal heating appropriately. The only non-trivial part of the above analysis corresponds to the determination of the innermost stable circular orbit frequency, for which we have used the current fits available for Kerr BHs, even for simulations with q BH = 0. In the ideal case, one may use an upper cut-off frequency expression that includes the q BH effect, but that is currently unavailable. Also, as emphasized above, since the test applies to inspiral-dominated signals, a slight change in the upper cut-off frequency of the analysis will most likely leave the findings unaltered.</text>
<section_header_level_1><location><page_7><loc_12><loc_29><loc_46><loc_32></location>IV. CONSTRAINTS FROM SIMULATED BINARY SIGNALS</section_header_level_1>
<text><location><page_7><loc_9><loc_14><loc_49><loc_27></location>We generate a set of simulated binary signals (injections) in zero-noise to study the measurability of the charge parameter q BH , including binaries of different spins, mass ratios, and signal strengths or signal-tonoise ratios. Furthermore, a set of simulations is investigated, keeping non-zero values for q BH to quantify the detectability if present in the data. The analysis assumes that both the BHs in the binary spins are aligned/antialigned to the orbital angular momentum axis.</text>
<text><location><page_7><loc_9><loc_9><loc_49><loc_14></location>Figure 2, shows the posterior probability distribution on q BH for a binary of total mass 32 M ⊙ and mass ratio q = ( M 1 /M 2 ) = 3. Each curve corresponds to different spin configurations, namely (0 . 7 , 0 . 6), (0 . 7 , 0 . 1), (0 . 5 , 0 . 3)</text>
<figure>
<location><page_7><loc_53><loc_71><loc_91><loc_93></location>
<caption>FIG. 3. Posteriors on the charge parameter from simulated binary signals of total mass 32M ⊙ and spins (0 . 7 , 0 . 6) with various mass ratios q = ( M 1 /M 2 ) = 1 , 2 , 3 , 4. The signal-tonoise ratio of the signal is fixed to be 120.</caption>
</figure>
<text><location><page_7><loc_52><loc_46><loc_92><loc_60></location>and (0 . 2 , 0 . 1). Source locations of all these binaries are chosen such that the signal-to-noise ratio is 120. The black dotted line represents the GR value ( q BH = 0), and the dashed lines show the 90% bounds on q BH for each case. It is evident from the green and orange curves of Fig. 2 that the estimates are better when the spins are high, especially when the primary BH is largely spinning. The q BH estimate worsens as the spin magnitudes decrease. Especially, the negative prior side is not well constrained when the spin of the secondary BH is low 1 .</text>
<text><location><page_7><loc_52><loc_24><loc_92><loc_44></location>To study the effect of mass ratio on the estimates of the charge, we choose three mass ratios, q = ( M 1 /M 2 ) = 1 , 2 , 3 , 4 as shown in Fig. 3. As expected from the mismatch studies, the estimates are better as we move to larger mass ratios. Whereas equal mass binary provides the least interesting constrain on q BH . The total mass is fixed to 32 M ⊙ and dimensionless aligned-spin magnitudes to (0 . 7 , 0 . 6) and the luminosity distance and sky localization are chosen such that the signal-to-noise ratio is 120. As evident from Fig. 3, the 90% credible interval bound on the q BH parameter is estimated to be 0.59 for mass-ratio of q=3, and the bound on q BH is 0.4, when we consider mass-ratio to be q=4. Therefore, the bound on q BH is stronger for the q=4 case (blue curve) than</text>
<figure>
<location><page_8><loc_10><loc_48><loc_48><loc_93></location>
<caption>FIG. 4. Posteriors on the tidal charge parameter to demonstrate the detectability of the existence of the charge parameter q BH from the data. We choose a simulated signal from a binary BH of total mass 32M ⊙ and mass ratio q = ( M 1 /M 2 ) = 3 with spins (0 . 7 , 0 . 6). The positive values of the injected charge parameter is presented in the top figure, while the negative values of the charge parameter have been presented at the bottom. The injections are marked with dotted lines and assumes a signal-to-noise ratio of 120.</caption>
</figure>
<text><location><page_8><loc_9><loc_9><loc_49><loc_30></location>the q=3 case (green curve). This is because, the phase contributions for equal mass ratio are smaller than that of asymmetric mass ratio cases, leading to better estimates. This is consistent with earlier findings regarding BH absorption effects, See for example [45]. What is also interesting here is the slight shift in the peak of the posterior from the GR value, which is zero, with a median value of -0.13 for q=3 and 0.12 for q=4 case. This shift is arising because of the correlation between the charge parameter and the other intrinsic parameters of the binary (especially, we see that this effect is larger as the mass asymmetry of the binary increases). This implies that highly spinning binaries with large asymmetry in the mass ratio is the best candidate for detecting the existence of the charge parameter.</text>
<text><location><page_8><loc_52><loc_42><loc_92><loc_93></location>Finally, we have performed a detailed analyses on binary BH simulations by injecting different q BH values, namely q BH = ± 0 . 7 , ± 0 . 5 , ± 0 . 3 , ± 0 . 1. For demonstration, we fix the binary mass to be 32M ⊙ , the mass ratio to be 3, and source location and orientation in such a way that the signal produces a signal-to-noise ratio of 120 in the advanced LIGO detector. Figure 4 shows the probability distribution on the charge parameter q BH for various cases. The vertical dotted lines in Fig. 4 denote the initial injected values of q BH , while the central black line is the q BH estimate, assuming the true value to be zero, i.e., the GR value. From Fig. 4, it is clear that the q BH probability distribution function shows distinct features and can be distinguished from its GR value for | q BH | > 0 . 3. For smaller values of the charge, e.g., for | q BH | ∼ 0 . 1, the probability distributions are indistinguishable between positive and negative values of q BH and also with the GR value. Fig. 4 also indicates that the probability distribution functions are well separated for all the positive injections of q BH ≳ 0 . 3 and hence are distinguishable. While for the negative injections, though the distribution functions are different from GR value, they are indistinguishable among themselves. In other words, it is impossible to distinguish the probability distribution function for q BH = -0 . 5 and q BH = -0 . 7. Thus positive values of the charge parameter, namely those associated with extra dimensions are easier to distinguish, compared to the negative injections, associated with the electromagnetic origin. We would like to point out that there is an asymmetry between the probability distribution functions with positive and negative values of the injected tidal charge parameter. This happens because the phase change due to tidal heating depends on q BH as well as on ( q BH ) 2 (see Eqs. (8) and (9) for explicit expressions), and hence q BH →-q BH is not a symmetry.</text>
<text><location><page_8><loc_52><loc_9><loc_92><loc_42></location>For completeness, We have also studied the effect of the signal-to-noise ratio on the estimation of the charge parameter, and the resulting posteriors have been plotted in Fig. 5. To demonstrate the improvement in measuring the charge parameter with respect to the signal strength, we have considered four different signal-to-noise ratios, 260, 130, 80 and 40, adjusting the distance to the source accordingly. Following our earlier conclusion, we have assumed a binary BH with asymmetric mass ratio ( q = 3) and high spins ( χ 1 = 0 . 7 and χ 2 = 0 . 6), so that the charge parameter can be better estimated. As Fig. 5 demonstrates, unless we have a high signal-to-noise ratio ∼ 200, all possible values of q BH are allowed, while for high signal-to-noise ratio, the distribution functions start to rule out larger values of q BH . This gives the following three criteria for observing the existence of the charge parameter - (a) asymmetric mass ratio: more asymmetric the mass ratio, the better the chance of detection; (b) higher spins: higher is the spin of the binary BHs, better is the chance of observing them in future GW observations and (c) higher signal-to-noise ratio enhances the detection probability. With this input, we now discuss their implications for the third GW tran-</text>
<figure>
<location><page_9><loc_10><loc_71><loc_48><loc_93></location>
<caption>FIG. 5. Posteriors on the charge parameter have been presented for different signal-to-noise ratios, to demonstrate the effect of the same on the detectability of the charge parameter. We consider a binary system with masses (24 , 8) M ⊙ and spins (0 . 7 , 0 . 6). Four different signal-to-noise ratio values are considered, namely, SNR = 260 , 130 , 80 , 40.</caption>
</figure>
<text><location><page_9><loc_9><loc_57><loc_38><loc_58></location>t catalogue [82] and future detectors.</text>
<section_header_level_1><location><page_9><loc_11><loc_50><loc_46><loc_53></location>V. IMPLICATIONS FOR THE THIRD GW TRANSIENT CATALOG AND FUTURE DETECTORS</section_header_level_1>
<text><location><page_9><loc_9><loc_30><loc_49><loc_47></location>The GW detection from the advanced LIGO-Virgo detectors contains binary events of various types in terms of masses, spins, location, orientation and nature of the compact object. The test mentioned above for 'charged' BHs suits the inspiral-dominated binaries with non-zero spins and mass asymmetry - for instance, GW190412 [83]-like sources, which is one of the most asymmetric binary systems with ∼ 30 M ⊙ primary BH and ∼ 3 M ⊙ companion (see Tab.II of [83] for more details). Besides, the GW190412 is an inspiral-dominated event with evidence of non-zero spins, making it ideal to test the validity of the above claims against real data.</text>
<text><location><page_9><loc_9><loc_8><loc_49><loc_30></location>For this purpose, we employ the non-precessing GW waveform model TaylorF2 with the charp mass M c , effective spin parameter χ eff , tidal charge q BH and the luminosity distance as free parameters, and truncate the analysis at 210Hz. The result of such an analysis is an estimation of the q BH parameter, which reads 0 . 05 0 . 77 -0 . 86 within the 90% credible interval with respect to the mean value, 0 . 05. Even though a non-zero and positive mean value is a tantalizing indication of the existence of an extra spatial dimension, the errors are very large. In particular, the GR value is well within the 90% credible interval, and so are plenty of positive and negative values, reducing the robustness of the claim. We have also estimated the Bayes factor log B nGR GR supporting the non-GR hypothesis over the GR hypothesis as log e B nGR GR = -6 . 43.</text>
<text><location><page_9><loc_52><loc_90><loc_92><loc_93></location>Therefore, the results look promising, while the error bar needs to be further reduced.</text>
<text><location><page_9><loc_52><loc_64><loc_92><loc_90></location>Additionally, in the GW190412 analysis, we further notice that the inclusion of q BH introduces significant shifts in the intrinsic parameters of the binary, such as masses and spins 2 . In Fig. 6, we show the posteriors on the chirp mass (a combination of binary masses which is well estimated from the inspiral signal), the effective spin parameter (found to be the best representative of the alignedspin effects of the binary) and the luminosity distance to the source. As Fig. 6 demonstrates, the intrinsic binary parameters are significantly affected by the presence of q BH , while the extrinsic parameter remains unaltered. In particular, the percentage change in the chirp mass and the effective spin parameter becomes (∆ M c /M c ) ∼ 4 . 9% and ( δχ eff /χ eff ) ∼ 66 . 7%. Therefore, our results indicate that parameters beyond GR theories are highly entangled with other intrinsic parameters of the problem, particularly the spin. This makes the detection of any non-GR effect significantly challenging.</text>
<text><location><page_9><loc_52><loc_35><loc_92><loc_64></location>We finally discuss the constraints on the charge q BH from future GW detectors, such as Einstein Telescope and Cosmic Explorer. For this purpose we have performed the parameter analysis on simulated binaries with 200 and 400 as the signal-to-noise ratios, respectively and compare the constraints on q BH with lower signalto-noise ratio simulations. The corresponding posteriors on q BH , considering a simulated binary BH signal with total mass 32 M ⊙ , and spins (0 . 7 , 0 . 6) are shown in Fig. 7 for two possible injected values of the charge parameter, q BH inj = ± 0 . 3. As evident from Fig. 7, a signal-to-noise ratio of 200 and 400 significantly improves the detectability of the charge parameter. In particular, for signal-to-noise ratio of 400 and positive injected value of q BH (i.e., for q BH = 0 . 3), GR can be ruled out with more than 90% confidence. While for negative q BH , the GR remains within the 90% confidence interval. In summary, high signal-to-noise ratio is essential for detecting the charge parameter and ruling out GR with confidence, which requires next generation of GW detectors.</text>
<section_header_level_1><location><page_9><loc_65><loc_31><loc_79><loc_32></location>VI. SUMMARY</section_header_level_1>
<text><location><page_9><loc_52><loc_19><loc_92><loc_29></location>We have proposed the tidal heating phenomenon in the inspiral regime of a binary BH system to be a benchmark in distinguishing binaries composed of BHs with charge from Kerr BHs using their GW signatures. For this purpose, we have started with the GW waveform model for a binary BH, parameterized in terms of an extra parameter, namely the charge, q BH . The origin for</text>
<figure>
<location><page_10><loc_9><loc_73><loc_36><loc_93></location>
</figure>
<text><location><page_10><loc_25><loc_72><loc_26><loc_73></location>/circledot</text>
<figure>
<location><page_10><loc_38><loc_73><loc_63><loc_93></location>
</figure>
<figure>
<location><page_10><loc_64><loc_73><loc_91><loc_93></location>
<caption>FIG. 6. Posteriors on the chirp mass (a best measured component mass combination), effective spin parameter (a best measured aligned spin spin component) and luminosity distance of the GW190412 event [83] have been presented, with the GR analysis and also by including the charge parameter q BH . The upper cut-off frequency is calculated for providing the maximum probable values of mass-spin posteriors obtained from the GR analysis. As evident inclusion of charge affects the intrinsic parameters, the mass and the spin, significantly.</caption>
</figure>
<figure>
<location><page_10><loc_13><loc_41><loc_50><loc_62></location>
</figure>
<figure>
<location><page_10><loc_51><loc_41><loc_87><loc_62></location>
<caption>FIG. 7. The probability distribution functions of the charge parameter q BH have been presented for a binary of fixed total mass 32 M ⊙ , mass ratio q = 3 and spins (0 . 7 , 0 . 6) considering four different signal-to-noise ratios 400 , 200 , 100, and 50. The plot on the left shows the distribution function for the charge with a positive injected value of q BH ( q BH = 0 . 3), while the one on the right shows the distribution function for negative injected values of q BH ( q BH = -0 . 3) as indicated by the black dotted lines.</caption>
</figure>
<text><location><page_10><loc_9><loc_9><loc_49><loc_31></location>this charge can be from extra dimensions, in which case the charge is positive, or from scenarios involving simple electromagnetic interaction, in which case q BH takes negative values. To demonstrate the measurability and detectability of the existence of such a hair from the GW observations, we have considered simulated binary BH signals of varying masses, spins and signal strength. The findings of this simulation study suggest that if the BHs are highly spinning and mass asymmetric, we will be able to perform such a distinguishability test with the advanced LIGO sensitivity since the mismatch significantly increases. Moreover, if the charge parameter is present and its value is significant, e.g., greater than ± 0 . 3, we can detect its presence and distinguish it from GR with advanced sensitivity. Even though the current detectors</text>
<text><location><page_10><loc_52><loc_15><loc_92><loc_31></location>are not yet sensitive enough for this test, we show that a GW190412-like event would be an ideal candidate for testing the existence of 'charged' hair. Interestingly, the posteriors from the GW190412 event report a positive median value of the charge, consistent with the existence of an extra dimension, however, the error bars are huge, rendering any conclusive statement. Moreover, the Bayes factor supporting the GR hypothesis, on the other hand, was found to be log e 6 . 43, which is not a large number. This suggests that tidal heating provides an avenue to test theories beyond GR.</text>
<text><location><page_10><loc_52><loc_9><loc_92><loc_14></location>Besides providing simulated GW waveforms with the charge parameter and testing their distinguishability from pure GR waveforms, we have also provided a forecast analysis for the observability of the charge parame-</text>
<text><location><page_11><loc_9><loc_72><loc_49><loc_93></location>ter in future GW observations. We have shown that increasing the signal-to-noise ratio considerably improves the estimation of the charge parameter and enhances its detectability. In particular, the future detector sensitivity, with a signal-to-noise ratio of 400, can significantly constrain the charge parameter. Moreover, as we have demonstrated that the tests involving charged hair of BHs will better suit more asymmetric binaries such as those in intermediate and extreme mass ratio inspirals, the future space-based GW detectors, namely DECIGO and LISA can also play a vital role in looking for the existence of non-trivial charged hair. The adaptation of the analysis presented here, in the context of the GW detectors DECIGO and LISA, will be explored in the future.</text>
<section_header_level_1><location><page_11><loc_62><loc_92><loc_82><loc_93></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_11><loc_52><loc_57><loc_92><loc_90></location>S.C. and N.V.K. acknowledges discussions with Sayak Datta on several aspects of this manuscript. We also thank Khun Sang Phukon and Gregorio Carullo for reading the manuscript and providing useful comments. The research of S.C. is funded by the INSPIRE Faculty fellowship from DST, Government of India (Reg. No. DST/INSPIRE/04/2018/000893) and MATRICS grant from Science and Engineering Research Board (SERB), Government of India (Reg. No. MTR/2023/000049). N.V.K. acknowledges support from SERB for the National postdoctoral fellowship (Reg. No. PDF/2022/000379). N.V.K. acknowledges Max Planck Computing and Data Facility computing cluster Cobra and Raven for computations. We also thank the CIT cluster provided by the LIGO Laboratory. We acknowledge National Science Foundation Grants PHY-0757058 and PHY-0823459. This material is based upon work supported by NSF's LIGO Laboratory which is a major facility fully funded by the National Science Foundation. We used the following software packages: LALSuite [79], bilby [80], bilby pipe [84], NumPy [85], PESummary [86], Matplotlib [87], jupyter [88], dynesty [89]. This document has LIGO preprint number LIGO-P2400054 .</text>
<section_header_level_1><location><page_11><loc_33><loc_51><loc_67><loc_52></location>Appendix A: GW phase due to tidal heating</section_header_level_1>
<text><location><page_11><loc_10><loc_48><loc_59><loc_49></location>The phase of the GW, as a function of velocity can be given by [90],</text>
<formula><location><page_11><loc_35><loc_44><loc_92><loc_47></location>ψ ( v ) = -2 ∫ v d v ( v 3 -v 3 ) ( dE orb /d v ) F ∞ ( v ) + F H ( v ) , (A1)</formula>
<text><location><page_11><loc_9><loc_39><loc_92><loc_43></location>where, E orb is the orbital energy of the binary BH system, F ∞ is the GW flux at infinity and F H is the GW flux through the BH horizon. Since we are interested in the contribution from the tidal heating alone, the following PN expansion of the orbital energy suffices for our purpose (see Eq. (3.8) of [91])</text>
<formula><location><page_11><loc_39><loc_35><loc_92><loc_38></location>E orb ( v ) = -η 2 v 2 [ 1 -9 + η 12 v 2 ] . (A2)</formula>
<text><location><page_11><loc_9><loc_33><loc_16><loc_34></location>Such that,</text>
<formula><location><page_11><loc_40><loc_29><loc_92><loc_32></location>dE orb d v = -η v [ 1 -9 + η 6 v 2 ] . (A3)</formula>
<text><location><page_11><loc_9><loc_27><loc_80><loc_28></location>Similarly, for the GW flux through infinity, the following PN expansion suffices for our purpose [34],</text>
<formula><location><page_11><loc_29><loc_23><loc_92><loc_26></location>F ∞ ( v ) = 32 5 η 2 v 10 [ 1 -( 1247 336 + 35 12 η ) v 2 +(4 π + F SO ) v 3 ] , (A4)</formula>
<text><location><page_11><loc_9><loc_21><loc_42><loc_22></location>while the GW flux through the horizon yields,</text>
<formula><location><page_11><loc_37><loc_17><loc_92><loc_20></location>F H ( v ) = 32 5 η 2 v 10 [ -Ψ 5 4 v 5 + Ψ 8 2 v 8 ] . (A5)</formula>
<text><location><page_11><loc_9><loc_13><loc_92><loc_16></location>Therefore, the contribution of tidal heating from the (1 / Flux) term in the phase integral, presented in Eq. (A1), yields,</text>
<formula><location><page_11><loc_20><loc_9><loc_92><loc_13></location>1 F ∞ + F H = 5 32 η 2 1 v 10 [ Ψ 5 4 v 5 + ( 1247 336 + 35 12 η ) Ψ 5 2 v 7 -{ Ψ 8 2 + Ψ 5 2 (4 π + F SO ) } v 8 ] . (A6)</formula>
<text><location><page_12><loc_9><loc_92><loc_64><loc_93></location>Therefore the contribution of tidal heating to the phase of the GW becomes,</text>
<formula><location><page_12><loc_9><loc_74><loc_93><loc_91></location>ψ TH ( v ) = 10 32 η ∫ v d v ( v 3 -v 3 ) v 9 [ 1 -9 + η 6 v 2 ] [ Ψ 5 4 v 5 + ( 1247 336 + 35 12 η ) Ψ 5 2 v 7 -{ Ψ 8 2 + Ψ 5 2 (4 π + F SO ) } v 8 ] = 10 32 η ∫ v d v ( v 3 -v 3 ) v 9 [ Ψ 5 4 v 5 + { 2 ( 1247 336 + 35 12 η ) -9 + η 6 } Ψ 5 4 v 7 -{ Ψ 8 2 + Ψ 5 2 (4 π + F SO ) } v 8 ] = 10 32 η [ Ψ 5 4 ∫ v d v ( v 3 -v 3 ) v 4 + Ψ 5 4 ( 995 168 + 952 168 η )∫ v d v ( v 3 -v 3 ) v 2 -{ Ψ 8 2 + Ψ 5 2 (4 π + F SO ) }∫ v d v ( v 3 -v 3 ) v ] = 10 32 η [ -Ψ 5 12 (1 + 3 ln v ) -3Ψ 5 8 ( 995 168 + 952 168 η ) v 2 -{ Ψ 8 2 + Ψ 5 2 (4 π + F SO ) } v 3 3 (3 ln v -1) ] . (A7)</formula>
<text><location><page_12><loc_9><loc_70><loc_92><loc_73></location>This expression has been used in the main text for computing δψ and identifying the 2.5 PN, 3.5 PN and 4 PN terms in the phase factor due to tidal heating.</text>
<unordered_list>
<list_item><location><page_12><loc_10><loc_59><loc_49><loc_61></location>[1] J. Aasi et al. (LIGO Scientific), Class. Quant. Grav. 32 , 074001 (2015), 1411.4547.</list_item>
<list_item><location><page_12><loc_10><loc_56><loc_49><loc_59></location>[2] F. Acernese et al. (VIRGO), Class. Quant. Grav. 32 , 024001 (2015), 1408.3978.</list_item>
<list_item><location><page_12><loc_10><loc_55><loc_31><loc_56></location>[3] http://www.virgo.infn.it .</list_item>
<list_item><location><page_12><loc_10><loc_54><loc_33><loc_55></location>[4] http://www.ligo.caltech.edu .</list_item>
<list_item><location><page_12><loc_10><loc_51><loc_49><loc_54></location>[5] R. Abbott et al. (LIGO Scientific, VIRGO, KAGRA) (2021), 2111.03606.</list_item>
<list_item><location><page_12><loc_10><loc_48><loc_49><loc_51></location>[6] R. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. X 11 , 021053 (2021), 2010.14527.</list_item>
<list_item><location><page_12><loc_10><loc_46><loc_49><loc_48></location>[7] R. Abbott et al. (LIGO Scientific, VIRGO) (2021), 2108.01045.</list_item>
<list_item><location><page_12><loc_10><loc_42><loc_49><loc_46></location>[8] A. H. Nitz, S. Kumar, Y.-F. Wang, S. Kastha, S. Wu, M. Schafer, R. Dhurkunde, and C. D. Capano, Astrophys. J. 946 , 59 (2023), 2112.06878.</list_item>
<list_item><location><page_12><loc_10><loc_38><loc_49><loc_42></location>[9] A. K. Mehta, S. Olsen, D. Wadekar, J. Roulet, T. Venumadhav, J. Mushkin, B. Zackay, and M. Zaldarriaga (2023), 2311.06061.</list_item>
<list_item><location><page_12><loc_9><loc_33><loc_49><loc_38></location>[10] Rich Abbott, Thomas D. Abbott, et al., SoftwareX 13 , 100658 (2021), ISSN 2352-7110, URL https://www.sciencedirect.com/science/article/ pii/S2352711021000030 .</list_item>
<list_item><location><page_12><loc_9><loc_30><loc_49><loc_32></location>[11] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. Lett. 119 , 161101 (2017), 1710.05832.</list_item>
<list_item><location><page_12><loc_9><loc_27><loc_49><loc_30></location>[12] B. P. Abbott et al. (LIGO Scientific, Virgo), Astrophys. J. Lett. 892 , L3 (2020), 2001.01761.</list_item>
<list_item><location><page_12><loc_9><loc_25><loc_49><loc_27></location>[13] R. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. Lett. 125 , 101102 (2020), 2009.01075.</list_item>
<list_item><location><page_12><loc_9><loc_22><loc_49><loc_24></location>[14] R. Abbott et al. (LIGO Scientific, KAGRA, VIRGO), Astrophys. J. Lett. 915 , L5 (2021), 2106.15163.</list_item>
<list_item><location><page_12><loc_9><loc_19><loc_49><loc_22></location>[15] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. D 100 , 104036 (2019), 1903.04467.</list_item>
<list_item><location><page_12><loc_9><loc_17><loc_49><loc_19></location>[16] R. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. X 11 , 021053 (2021), 2010.14527.</list_item>
<list_item><location><page_12><loc_9><loc_14><loc_49><loc_17></location>[17] R. Abbott et al. (LIGO Scientific, VIRGO, KAGRA) (2021), 2111.03606.</list_item>
<list_item><location><page_12><loc_9><loc_11><loc_49><loc_14></location>[18] B. P. Abbott et al. (Virgo, LIGO Scientific), Phys. Rev. Lett. 116 , 221101 (2016), 1602.03841.</list_item>
<list_item><location><page_12><loc_9><loc_9><loc_49><loc_11></location>[19] E. Berti, E. Barausse, V. Cardoso, L. Gualtieri, P. Pani, U. Sperhake, L. C. Stein, N. Wex, K. Yagi, T. Baker,</list_item>
<list_item><location><page_12><loc_55><loc_58><loc_92><loc_61></location>et al., Classical and Quantum Gravity 32 , 243001 (2015), URL https://dx.doi.org/10.1088/0264-9381/32/24/ 243001 .</list_item>
<list_item><location><page_12><loc_52><loc_54><loc_92><loc_57></location>[20] V. Cardoso, E. Franzin, A. Maselli, P. Pani, and G. Raposo, Phys. Rev. D 95 , 084014 (2017), [Addendum: Phys.Rev.D 95, 089901 (2017)], 1701.01116.</list_item>
<list_item><location><page_12><loc_52><loc_51><loc_92><loc_54></location>[21] D. Psaltis et al. (Event Horizon Telescope), Phys. Rev. Lett. 125 , 141104 (2020), 2010.01055.</list_item>
<list_item><location><page_12><loc_52><loc_48><loc_92><loc_51></location>[22] A. N. Aliev and A. E. Gumrukcuoglu, Phys. Rev. D 71 , 104027 (2005), hep-th/0502223.</list_item>
<list_item><location><page_12><loc_52><loc_46><loc_92><loc_48></location>[23] T. Harko and M. K. Mak, Phys. Rev. D 69 , 064020 (2004), gr-qc/0401049.</list_item>
<list_item><location><page_12><loc_52><loc_43><loc_92><loc_46></location>[24] N. Dadhich, R. Maartens, P. Papadopoulos, and V. Rezania, Phys. Lett. B 487 , 1 (2000), hep-th/0003061.</list_item>
<list_item><location><page_12><loc_52><loc_41><loc_92><loc_43></location>[25] T. Shiromizu, K.-i. Maeda, and M. Sasaki, Phys. Rev. D 62 , 024012 (2000), gr-qc/9910076.</list_item>
<list_item><location><page_12><loc_52><loc_38><loc_92><loc_40></location>[26] E. Babichev and C. Charmousis, JHEP 08 , 106 (2014), 1312.3204.</list_item>
<list_item><location><page_12><loc_52><loc_34><loc_92><loc_38></location>[27] J. Barrientos, F. Cordonier-Tello, F. Izaurieta, P. Medina, D. Narbona, E. Rodr'ıguez, and O. Valdivia, Phys. Rev. D 96 , 084023 (2017), 1703.09686.</list_item>
<list_item><location><page_12><loc_52><loc_31><loc_92><loc_34></location>[28] E. Babichev, C. Charmousis, and M. Hassaine, JCAP 05 , 031 (2015), 1503.02545.</list_item>
<list_item><location><page_12><loc_52><loc_29><loc_92><loc_31></location>[29] H. Maeda and N. Dadhich, Phys. Rev. D 75 , 044007 (2007), hep-th/0611188.</list_item>
<list_item><location><page_12><loc_52><loc_23><loc_92><loc_28></location>[30] V. Faraoni and S. Capozziello, Beyond Einstein Gravity: A Survey of Gravitational Theories for Cosmology and Astrophysics (Springer, Dordrecht, 2011), ISBN 978-94007-0164-9, 978-94-007-0165-6.</list_item>
<list_item><location><page_12><loc_52><loc_21><loc_92><loc_23></location>[31] S. Capozziello, P. A. Gonzalez, E. N. Saridakis, and Y. Vasquez, JHEP 02 , 039 (2013), 1210.1098.</list_item>
<list_item><location><page_12><loc_52><loc_14><loc_92><loc_21></location>[32] S. A. Hughes, Phys. Rev. D 61 , 084004 (2000), [Erratum: Phys.Rev.D 63, 049902 (2001), Erratum: Phys.Rev.D 65, 069902 (2002), Erratum: Phys.Rev.D 67, 089901 (2003), Erratum: Phys.Rev.D 78, 109902 (2008), Erratum: Phys.Rev.D 90, 109904 (2014)], gr-qc/9910091.</list_item>
<list_item><location><page_12><loc_52><loc_11><loc_92><loc_14></location>[33] J. B. Hartle, Phys. Rev. D 8 , 1010 (1973), URL https: //link.aps.org/doi/10.1103/PhysRevD.8.1010 .</list_item>
<list_item><location><page_12><loc_52><loc_9><loc_92><loc_11></location>[34] S. Isoyama and H. Nakano, Class. Quant. Grav. 35 , 024001 (2018), 1705.03869.</list_item>
<list_item><location><page_13><loc_9><loc_91><loc_49><loc_93></location>[35] K. Chatziioannou, E. Poisson, and N. Yunes, Phys. Rev. D 87 , 044022 (2013), 1211.1686.</list_item>
<list_item><location><page_13><loc_9><loc_88><loc_49><loc_90></location>[36] S. Datta, R. Brito, S. Bose, P. Pani, and S. A. Hughes, Phys. Rev. D 101 , 044004 (2020), 1910.07841.</list_item>
<list_item><location><page_13><loc_9><loc_87><loc_49><loc_88></location>[37] S. Datta, Phys. Rev. D 102 , 064040 (2020), 2002.04480.</list_item>
<list_item><location><page_13><loc_9><loc_84><loc_49><loc_86></location>[38] S. Datta and S. Bose, Phys. Rev. D 99 , 084001 (2019), 1902.01723.</list_item>
<list_item><location><page_13><loc_9><loc_83><loc_49><loc_84></location>[39] K. Alvi, Phys. Rev. D 64 , 104020 (2001), gr-qc/0107080.</list_item>
<list_item><location><page_13><loc_9><loc_80><loc_49><loc_82></location>[40] E. Poisson, Phys. Rev. D 70 , 084044 (2004), grqc/0407050.</list_item>
<list_item><location><page_13><loc_9><loc_77><loc_49><loc_80></location>[41] M. V. S. Saketh, J. Steinhoff, J. Vines, and A. Buonanno, Phys. Rev. D 107 , 084006 (2023), 2212.13095.</list_item>
<list_item><location><page_13><loc_9><loc_76><loc_31><loc_77></location>[42] S. Datta (2023), 2305.03771.</list_item>
<list_item><location><page_13><loc_9><loc_73><loc_49><loc_76></location>[43] S. Mukherjee, S. Datta, K. S. Phukon, and S. Bose (2023), 2311.17554.</list_item>
<list_item><location><page_13><loc_9><loc_71><loc_49><loc_73></location>[44] S. Datta, K. S. Phukon, and S. Bose, Phys. Rev. D 104 , 084006 (2021), 2004.05974.</list_item>
<list_item><location><page_13><loc_9><loc_67><loc_49><loc_71></location>[45] A. Maselli, P. Pani, V. Cardoso, T. Abdelsalhin, L. Gualtieri, and V. Ferrari, Phys. Rev. Lett. 120 , 081101 (2018), 1703.10612.</list_item>
<list_item><location><page_13><loc_9><loc_64><loc_49><loc_67></location>[46] J. C. Feng, S. Chakraborty, and V. Cardoso, Phys. Rev. D 107 , 044050 (2023), 2211.05261.</list_item>
<list_item><location><page_13><loc_9><loc_62><loc_49><loc_64></location>[47] D. M. Pina, M. Orselli, and D. Pica, Phys. Rev. D 106 , 084012 (2022), 2204.08841.</list_item>
<list_item><location><page_13><loc_9><loc_59><loc_49><loc_61></location>[48] G. Bozzola and V. Paschalidis, Phys. Rev. Lett. 126 , 041103 (2021), 2006.15764.</list_item>
<list_item><location><page_13><loc_9><loc_56><loc_49><loc_59></location>[49] J. C. S. Neves, Eur. Phys. J. C 80 , 717 (2020), 2005.00483.</list_item>
<list_item><location><page_13><loc_9><loc_52><loc_49><loc_56></location>[50] M. Zajaˇcek, A. Tursunov, A. Eckart, S. Britzen, E. Hackmann, V. Karas, Z. Stuchl'ık, B. Czerny, and J. A. Zensus, J. Phys. Conf. Ser. 1258 , 012031 (2019), 1812.03574.</list_item>
<list_item><location><page_13><loc_9><loc_51><loc_46><loc_52></location>[51] A. F. Zakharov, Universe 8 , 141 (2022), 2108.01533.</list_item>
<list_item><location><page_13><loc_9><loc_48><loc_49><loc_51></location>[52] A. Maselli, L. Gualtieri, P. Pani, L. Stella, and V. Ferrari, Astrophys. J. 801 , 115 (2015), 1412.3473.</list_item>
<list_item><location><page_13><loc_9><loc_46><loc_49><loc_48></location>[53] Z. Stuchl'ık and A. Kotrlov'a, Gen. Rel. Grav. 41 , 1305 (2009), 0812.5066.</list_item>
<list_item><location><page_13><loc_9><loc_43><loc_49><loc_45></location>[54] I. Banerjee, S. Chakraborty, and S. SenGupta, Phys. Rev. D 96 , 084035 (2017), 1707.04494.</list_item>
<list_item><location><page_13><loc_9><loc_40><loc_49><loc_43></location>[55] I. Banerjee, S. Chakraborty, and S. SenGupta, JCAP 09 , 037 (2021), 2105.06636.</list_item>
<list_item><location><page_13><loc_9><loc_38><loc_49><loc_40></location>[56] E. Barausse and K. Yagi, Phys. Rev. Lett. 115 , 211105 (2015), 1509.04539.</list_item>
<list_item><location><page_13><loc_9><loc_35><loc_49><loc_38></location>[57] D. Andriot and G. Lucena G'omez, JCAP 06 , 048 (2017), [Erratum: JCAP 05, E01 (2019)], 1704.07392.</list_item>
<list_item><location><page_13><loc_9><loc_32><loc_49><loc_35></location>[58] S. Chakraborty, K. Chakravarti, S. Bose, and S. SenGupta, Phys. Rev. D 97 , 104053 (2018), 1710.05188.</list_item>
<list_item><location><page_13><loc_9><loc_28><loc_49><loc_32></location>[59] K. Chakravarti, S. Chakraborty, K. S. Phukon, S. Bose, and S. SenGupta, Class. Quant. Grav. 37 , 105004 (2020), 1903.10159.</list_item>
<list_item><location><page_13><loc_9><loc_26><loc_49><loc_28></location>[60] H.-P. Gu, H.-T. Wang, and L. Shao, Phys. Rev. D 109 , 024058 (2024), 2310.10447.</list_item>
<list_item><location><page_13><loc_9><loc_22><loc_49><loc_26></location>[61] P. K. Gupta, T. F. M. Spieksma, P. T. H. Pang, G. Koekoek, and C. V. D. Broeck, Phys. Rev. D 104 , 063041 (2021), 2107.12111.</list_item>
<list_item><location><page_13><loc_9><loc_18><loc_49><loc_22></location>[62] G. Carullo, D. Laghi, N. K. Johnson-McDaniel, W. Del Pozzo, O. J. C. Dias, M. Godazgar, and J. E. Santos, Phys. Rev. D 105 , 062009 (2022), 2109.13961.</list_item>
<list_item><location><page_13><loc_9><loc_15><loc_49><loc_18></location>[63] A. K. Mishra, A. Ghosh, and S. Chakraborty, Eur. Phys. J. C 82 , 820 (2022), 2106.05558.</list_item>
<list_item><location><page_13><loc_9><loc_13><loc_49><loc_15></location>[64] A. K. Mishra, G. Carullo, and S. Chakraborty, Phys. Rev. D 109 , 024025 (2024), 2311.03556.</list_item>
<list_item><location><page_13><loc_9><loc_10><loc_49><loc_12></location>[65] U. Deka, S. Chakraborty, S. J. Kapadia, M. A. Shaikh, and P. Ajith (2024), 2401.06553.</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_13><loc_52><loc_91><loc_92><loc_93></location>[66] R. Dey, S. Chakraborty, and N. Afshordi, Phys. Rev. D 101 , 104014 (2020), 2001.01301.</list_item>
<list_item><location><page_13><loc_52><loc_88><loc_92><loc_90></location>[67] R. Dey, S. Biswas, and S. Chakraborty, Phys. Rev. D 103 , 084019 (2021), 2010.07966.</list_item>
<list_item><location><page_13><loc_52><loc_85><loc_92><loc_88></location>[68] B. P. Abbott et al. (KAGRA, LIGO Scientific, Virgo, VIRGO), Living Rev. Rel. 21 , 3 (2018), 1304.0670.</list_item>
<list_item><location><page_13><loc_52><loc_81><loc_92><loc_85></location>[69] S. A. Teukolsky, Phys. Rev. Lett. 29 , 1114 (1972), URL https://link.aps.org/doi/10.1103/PhysRevLett.29. 1114 .</list_item>
<list_item><location><page_13><loc_52><loc_79><loc_92><loc_81></location>[70] S. Chakraborty, S. Datta, and S. Sau, Phys. Rev. D 104 , 104001 (2021), 2103.12430.</list_item>
<list_item><location><page_13><loc_52><loc_75><loc_92><loc_78></location>[71] A. Chamblin, H. S. Reall, H.-a. Shinkai, and T. Shiromizu, Phys. Rev. D 63 , 064015 (2001), hepth/0008177.</list_item>
<list_item><location><page_13><loc_52><loc_72><loc_92><loc_75></location>[72] I. Banerjee, S. Chakraborty, and S. SenGupta, Phys. Rev. D 106 , 084051 (2022), 2207.09003.</list_item>
<list_item><location><page_13><loc_52><loc_69><loc_92><loc_72></location>[73] I. Banerjee, S. Chakraborty, and S. SenGupta, Phys. Rev. D 101 , 041301 (2020), 1909.09385.</list_item>
<list_item><location><page_13><loc_52><loc_67><loc_92><loc_69></location>[74] K. G. Arun, A. Buonanno, G. Faye, and E. Ochsner, Phys. Rev. D 79 , 104023 (2009), 0810.5336.</list_item>
<list_item><location><page_13><loc_52><loc_63><loc_92><loc_67></location>[75] A. Buonanno, B. Iyer, E. Ochsner, Y. Pan, and B. S. Sathyaprakash, Phys. Rev. D 80 , 084043 (2009), 0907.0700.</list_item>
<list_item><location><page_13><loc_52><loc_60><loc_92><loc_63></location>[76] C. K. Mishra, A. Kela, K. G. Arun, and G. Faye, Phys. Rev. D 93 , 084054 (2016), 1601.05588.</list_item>
<list_item><location><page_13><loc_52><loc_58><loc_92><loc_60></location>[77] S. Mukherjee, S. Datta, S. Tiwari, K. S. Phukon, and S. Bose, Phys. Rev. D 106 , 104032 (2022), 2202.08661.</list_item>
<list_item><location><page_13><loc_52><loc_55><loc_92><loc_57></location>[78] K. Chakravarti, S. Chakraborty, S. Bose, and S. SenGupta, Phys. Rev. D 99 , 024036 (2019), 1811.11364.</list_item>
<list_item><location><page_13><loc_52><loc_52><loc_92><loc_55></location>[79] LIGO Scientific Collaboration, LIGO Algorithm Library - LALSuite , free software (GPL) (2018).</list_item>
<list_item><location><page_13><loc_52><loc_50><loc_92><loc_52></location>[80] G. Ashton et al., Astrophys. J. Suppl. 241 , 27 (2019), 1811.02042.</list_item>
<list_item><location><page_13><loc_52><loc_47><loc_92><loc_49></location>[81] M. Favata, C. Kim, K. G. Arun, J. Kim, and H. W. Lee, Phys. Rev. D 105 , 023003 (2022), 2108.05861.</list_item>
<list_item><location><page_13><loc_52><loc_44><loc_92><loc_47></location>[82] R. Abbott et al. (KAGRA, VIRGO, LIGO Scientific), Astrophys. J. Suppl. 267 , 29 (2023), 2302.03676.</list_item>
<list_item><location><page_13><loc_52><loc_42><loc_92><loc_44></location>[83] R. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. D 102 , 043015 (2020), 2004.08342.</list_item>
<list_item><location><page_13><loc_52><loc_39><loc_92><loc_42></location>[84] I. M. Romero-Shaw et al., Mon. Not. Roy. Astron. Soc. 499 , 3295 (2020), arXiv:2006.00714 [astro-ph.IM].</list_item>
<list_item><location><page_13><loc_52><loc_34><loc_92><loc_39></location>[85] C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gommers, P. Virtanen, D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N. J. Smith, et al., Nature (London) 585 , 357 (2020), arXiv:2006.10256 [cs.MS].</list_item>
<list_item><location><page_13><loc_52><loc_31><loc_92><loc_34></location>[86] C. Hoy and V. Raymond, SoftwareX 15 , 100765 (2021), arXiv:2006.06639 [astro-ph.IM].</list_item>
<list_item><location><page_13><loc_52><loc_28><loc_92><loc_31></location>[87] J. D. Hunter, Computing in Science and Engineering 9 , 90 (2007).</list_item>
<list_item><location><page_13><loc_52><loc_21><loc_92><loc_28></location>[88] T. Kluyver, B. Ragan-Kelley, F. P'erez, B. Granger, M. Bussonnier, J. Frederic, K. Kelley, J. Hamrick, J. Grout, S. Corlay, et al., in Positioning and Power in Academic Publishing: Players, Agents and Agendas , edited by F. Loizides and B. Scmidt (IOS Press, 2016), pp. 87-90.</list_item>
<list_item><location><page_13><loc_52><loc_18><loc_92><loc_20></location>[89] J. S. Speagle, Monthly Notices of the Royal Astronomical Society 493 , 3132 (2020).</list_item>
<list_item><location><page_13><loc_52><loc_15><loc_92><loc_18></location>[90] W. Tichy, E. E. Flanagan, and E. Poisson, Phys. Rev. D 61 , 104015 (2000), gr-qc/9912075.</list_item>
<list_item><location><page_13><loc_52><loc_13><loc_92><loc_15></location>[91] T. Damour, B. R. Iyer, and B. S. Sathyaprakash, Phys. Rev. D 57 , 885 (1998), gr-qc/9708034.</list_item>
</document>
[
{
"title": "Probing black hole 'charge' from the binary black hole inspiral",
"content": "N. V. Krishnendu 1, ∗ and Sumanta Chakraborty 2, † 1 International Centre for Theoretical Sciences (ICTS), Survey No. 151, Shivakote, Hesaraghatta, Uttarahalli, Bengaluru, 560089, India 2 School of Physical Sciences, Indian Association for the Cultivation of Science, Kolkata 700032, India (Dated: June 25, 2024) Recent gravitational wave (GW) observations have enabled us to look beyond the standard paradigm of gravitational physics, namely general relativity (GR). Along with the mass and the angular momentum, which typical astrophysical black holes (BHs) are endowed with, theories beyond GR generically induce 'charge' to these BHs. Notably, for BHs carrying the extra 'charge' hair, we expect the BH absorption effects to modify accordingly and alter the tidal heating terms. Hence, the inclusion of the corrections in the GW waveform model, arising from the BH 'charge', allows us to test the consistency of the observed binaries with Kerr BHs in GR. We compute the explicit dependence of the binary inspiral phase on the 'charge' parameter arising from the tidal heating effect and study the measurability of the same from GW observations of binary mergers. Specifically, we employ the TaylorF2 waveform model, which accurately models the inspiral evolution of an aligned-spin binary merger, and Bayesian analysis-based GW data inference to measure the 'charge' parameter for a selected set of detected binaries. We also present a detailed simulation study to investigate the possibility of measuring the charge parameter from binaries with different masses, spins and source locations. The analysis of selected GW events from the third GW transient catalogue shows that the 'charge' parameter constraints are poor from the observed signals with the current sensitivity. In contrast, the simulation studies indicate that the spinning binaries with significant mass asymmetry provide the best constraints on the BH 'charge' parameter. Finally, we study the prospects of measuring the BH 'charge' parameter from a future GW detector with improved sensitivity.",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "After finishing the three successful observing runs, the ground-based gravitational wave (GW) detectors confirmed the detections of more than ninety compact binary coalescing events [1-9]. Then, the first part of the fourth observation run concluded by reporting around eighty GW events and details are available in the public alert system [10]. From the detection and further dedicated analysis, it is evident that the signals consist of three types of merger events - (a) binary black holes (BHs), (b) binary neutron stars (NSs), and (c) BH-NS systems [11-14]. Even though the observed signals are consistent with these compact objects being described by either BHs or NSs within the framework of general relativity (GR) and show no clear evidence of the presence of physics beyond the standard paradigm [15-18], however, given the statistical uncertainties in these measurements, there is still ample room for alternate theories of gravity and exotic compact objects [19, 20]. Despite the monumental success of GR in all the observational tests so far [18, 21], the prime reasons to look for alternatives of GR are threefold - (a) Einstein's equations predict singularities, where the theory itself breaks down, (b) solutions of Einstein's equations involve Cauchy horizon, leading to uncertain future for the classical theory, (c) consistency with observations require postulating the existence of yet undetected dark matter and exotic dark energy. Besides, there are other important issues related to the energy scale at which gravity dominates, as this scale is much above the energy scales associated with the other fundamental interactions. This leads to the well-known hierarchy problem, where the stabilization of the Higgs mass requires an exorbitant fine-tuning of the order of one part in 10 15 . The models involving extra spatial dimensions are among several alternatives to get around this fine-tuning problem. In this work, we explore the implications of the existence of an extra spatial dimension on the inspiral regime of the binary BH merger events detected by the LIGO-VirgoKAGRAcollaboration. The presence of this extra spatial dimension modifies the gravitational field equations on the four-dimensional spacetime (known as the brane ), as the projection of the five-dimensional (known as the bulk ) Einstein's equations inherit additional contributions from the bulk. Consequently, the BHs on the brane are characterized by an additional hair, other than the mass and the spin. This hair is identical in appearance to that of the Maxwell charge but differs by an overall sign, which becomes a distinctive signature of the existence of extra spatial dimensions [22-25]. It is worth pointing out that a similar contribution, albeit with an opposite sign, appears in the context of Einstein-Maxwell theories, scalar coupled Maxwell theories [26-29] and also in f ( T ) theories of gravity [30, 31]. The presence of the extra 'charge' in the BH solution can induce differences in the evolution of a compact binary system, making their GW signatures utterly different from that of the Kerr BH. Specifically, by focusing on the inspiral dynamics of the binary, we study the significance of the 'charge' hair on the tidal heating effects present in the GW waveform measured at infinity. Tidal heating is an absorption effect, which arises due to the absorption of external GWs by compact objects in a binary, leading to time evolution of the mass and angular momentum of the compact objects, leaving observational imprints [32-45]. The existence of the 'charge' hair will modify the absorption of GWs by the BHs on the brane, which will reflect in the orbital evolution of the binary in the inspiral phase. Following this, we propose a novel test to measure the 'charge' parameter from the observed inspiral evolution of the binary coalescence events and demonstrate the measurability over a set of simulated binary signals. The inspiral measurements of the BH 'charge' provide a unique test of the nature of compact binary because for astrophysical BHs, the net electric charge is expected to be zero, or even if there is a small electric charge present, that gets shielded away quickly [46-48]. Therefore, the existence of a 'charge' hair, along with its overall sign, is a characteristic of the presence of an extra spatial dimension. There have been several previous attempts in the literature to look for the existence of such a 'charge' in various observations involving BHs. These include - (a) the shadow of M87* and SgrA* as observed by the Event Horizon Telescope collaboration [49-51], (b) the X-ray luminosity from the accretion disc of quasars [52-55], and (c) the GW observations from binary coalescence [56-62]. Intriguingly, the observations involving BH shadow and accretion disc around quasars mildly favour the existence of a negative 'charge' compared to the Kerr scenario in GR. The ringdown signal, on the other hand, provides little information, and both GR and extra dimensions remain viable alternatives. In Ref. [63], the BH ringdown waveform has been obtained by numerically solving the perturbation equations for the braneworld BH and determining the associated quasi-normal modes [63] with the explicit BH 'charge' dependence. Subsequently, this analysis has been extended in [64], and constraints on the 'charge' parameter have been derived from the events observed through the first three observing runs of advanced LIGO and Virgo detectors. In a different perspective, [65] demonstrated yet another method to constrain the 'charge' parameter by looking into the GW micro-lensing signature of the charged lens. It is also possible through anti-de Sitter/conformal field theory correspondence (AdS/CFT) correspondence to argue that horizons for BHs on the brane must be reflective [66, 67], however, in this work, we will stick to the classical BH picture, returning to the reflective nature of horizon in future work. To summarise, we focus on the inspiral dynamics of the binary and study the distinctive features in the BH absorption spectra due to the presence of the 'charge' parameter arising from the existence of an extra spatial dimension considering stellar mass binary BH mergers. The detection and parameter inference necessitate em- ploying waveform models based on GR that accurately predict the binary evolution, including various physical effects of binary BHs in GR. One must validate these GW waveform models to regimes inaccessible to GR and its predictions to look for beyond-GR effects in the GW data. However, the theory-agnostic tests of GR are routinely employed to check the consistency of GR with the observed data where generic parametric deviations are introduced in the GW waveform model without assuming any specific modified gravity models [15-18]. Suppose GR is the correct theory of gravity; in this case, the measurement uncertainties on the parametric deviation coefficients will be consistent with GR prediction, and any departure will lead to further elaborate analyses. In the ideal scenario, one may start with an alternate gravity model, compute the GW waveform model for a merging binary in that particular theory, and use it for the analysis. Despite multiple efforts, such a complete model has yet to be available. In the middle ground, one can combine the orbital evolution information from GR, calculate additional contributions from beyond GR effects and estimate the GW waveform model. Analysis employing such a modified waveform model will provide a consistency check with the observed data and GR predictions. In this study, we focus on such a scenario. To do so, we calculate the BH absorption effects of a braneworld BH and modify the post-Newtonian inspiral phase by appropriately adding the tidal charge contributions. We keep the 'charge' parameter as a free parameter and measure it from the data. Before discussing the real GW analysis, we demonstrate the method by simulating a set of binaries with various masses, spins and locations. The simulations indicate that it is possible to measure the 'charge' parameter for spinning binaries with mass asymmetry if we consider current-ground-based GW detectors with their plus-era (O5) sensitivity [68]. Further, we describe the method's applicability for inspiral-dominated GW events detected through the first three observing runs of LIGO-Virgo detectors. For completeness we will discuss both the scenarios involving positive as well as negative 'charged' hairs. The paper is organized as follows: we give a brief outline of our geometrical setup in Sec. II, and then discuss the waveform model, a short description of the Bayesian analysis and a note on the details of binary simulations in Sec. III. The results from simulated binary signals are presented in Sec. IV, while our bounds from the GW signals of the detected binary merger events have been presented in Sec. V. Finally, we conclude by summarising our findings and listing future plans in Sec. VI.",
"pages": [
1,
2
]
},
{
"title": "II. BACKGROUND GEOMETRY WITH CHARGE: IMPLICATIONS FOR TIDAL HEATING",
"content": "In this section, we present the background geometry of a braneworld BH, with the charge term, and also dis- cuss the implications of this on the tidal heating phenomenon. In the braneworld scenario, the higher dimensional spacetime ( bulk ) satisfies Einstein's equations, while the gravitational field equations on the embedded four-dimensional hypersurface ( brane ), on which the standard model fields live become, Here, (4) G µν is the four-dimensional Einstein tensor on the brane hypersurface, E µν ≡ (5) W ABCD e A µ n B e C ν n D is the projection of the five-dimensional Weyl tensor (5) W ABCD on the brane hypersurface, where e A µ are the projectors and n B is the normal vector to the brane hypersurface. Owing to the symmetries of the Weyl tensor, it follows that E µ µ = 0, and Bianchi identity demands ∇ µ E µ ν = 0. Both of these properties are akin to the energy-momentum tensor of the electromagnetic field, except for an overall sign. This is because the energy-momentum tensor sits on the right-hand side of Einstein's equation, acting as the source of gravity, while the Weyl tensor E µν sits on the left-hand side, which mimics a source with energy-momentum tensor -E µν . Thus, braneworld BH depicts vacuum spacetime but resembles Kerr-Newman spacetime with an overall negative sign in front of the charge term. Therefore, the spacetime geometry of a rotating braneworld BH takes the form, The above metric depicts a rotating BH with mass M , angular momentum J = aM and braneworld 'charge' Q BH . The metric functions appearing in the above line element involve two unknown functions ∆ and Σ, defined as ∆ ≡ r 2 + a 2 -2 Mr - Q BH and Σ ≡ r 2 + a 2 cos 2 θ . Note that, for the case of Kerr-Newman BH, the parameter Q BH can be identified with the negative of the square of the electric charge Q of the BH, such that Q BH | KN = -Q 2 . The rotating braneworld BH inherits two horizons, located at r ± = M ± √ M 2 -a 2 + Q BH obtained by solving the equation ∆ = 0. Intriguingly, even if a > M , for non-zero values of Q BH , the outer horizon r + exists, in stark contrast to that of the Kerr-Newman BH. Gravitational perturbation of the background geometry, depicting a rotating BH spacetime on the brane, can be described by Newman-Penrose scalars Ψ 0 and Ψ 4 . Since our interest is in the physics of the horizon, namely in determining the GW flux going down the horizon, we will work with the Newman-Penrose scalar Ψ 0 . In general, for the Kerr-Newman-like spacetimes, the angular and the radial parts of the Newman-Penrose scalars cannot be separated. However, in the present context, perturbation of the 'source' term E µν is directly proportional to the ratio of the bulk and the brane curvature length scales, which can be ignored for all practical purposes. Therefore, the Weyl scalar Ψ 0 , describing gravitational perturbation of a rotating braneworld BH, can be expressed as, where the angular part 2 S ℓm ( θ ) satisfies the following differential equation, which coincides with the equation satisfied by the spinweighted spherical harmonics. Here 2 A ℓm is the separation constant between the radial and the angular parts. The radial function R ℓm ( r ), on the other hand, satisfies the following differential equation [69]. where, K ≡ ( r 2 + a 2 ) ω -am and λ = 2 A ℓm + ( aω ) 2 -2 amω , is related to the separation constant 2 A ℓm appearing in the angular part 2 S ℓm . Consider now a binary system involving two braneworld BHs, characterized by masses M 1 and M 2 , angular momentum J 1 and J 2 , as well as 'charge' parameter Q BH . In the context of braneworld BH, the charge Q BH depends on the length of the extra dimension, and hence is an invariant quantity for all BHs. While for positive values of Q BH , we assume identical values of the charge for both the BHs in the binary, possibly due to some overall equilibrium. During the inspiral of this binary system around one another, each of these BHs will absorb a part of the emitted GW radiation in the centre of the mass frame, leading to a rate of change of mass M , angular momentum J , and area A . This is known as tidal heating. In order to determine the above rate of changes of BH parameters for the first BH, we must solve the radial equation near the horizon r +1 and impose purely ingoing boundary conditions at the horizon. This fixes one arbitrary constant appearing in the solution of the Teukolsky equation. In order to fix the other, we need to work in a regime where M 1 ≪ r ≪ b , with b being a typical distance between the binary BHs, and one imposes the following boundary condition, To determine the tidal heating associated with the second BH, we simply have to interchange M 1 ↔ M 2 , and J 1 ↔ J 2 , respectively. With the above boundary condition, one can uniquely solve for the Weyl scalar Ψ 0 and transform the same to the Hartle-Hawking frame, thereby determining the rate of change of area in terms of | Ψ 0 | 2 . The corresponding rate of change of mass can be derived using the laws of BH mechanics, which relates the rate of change of mass and angular momentum to the rate of change of area, yielding [70], where ( dE/dt ) N = (32 / 5) η 2 v 10 is the energy loss due to GWs arising from the quadrupole approximation. Further, we have defined M ≡ M 1 + M 2 and η ≡ ( M 1 M 2 /M 2 ), while the relative velocity of the binary BH system is given by v = √ M/b . A similar expression can be derived for ( dM 2 /dt ), by simply the M 1 ↔ M 2 exchange. Thus, the rate of change of mass, in comparison to the quadrupolar rate of change of energy, depends on post-Newtonian (PN) terms of two distinct orders, at 2 . 5 PN (or, equivalently v 5 ), and at 4 PN (or, equivalently v 8 ), Here we have defined, q BH ≡ ( Q BH /M 2 ), χ i = a i /M i and i = { 1 , 2 } . Thus the rate of change of mass for the i th component of the binary BH system can be expressed as, Therefore, the total flux going into the horizon of the braneworld BH becomes, Here, we have defined, Ψ 5 ≡ ∑ i A (5) i and Ψ 8 ≡ ∑ i A 8 i , for notational convenience. As we will demonstrate, these two quantities will be central to the phase evolution in the presence of tidal heating. At this outset, let us discuss previous bounds on the tidal charge parameter q BH and the implications of such bounds on the size of the extra dimension [71]. In the context of GW observations, based on the ringdown part of the signal, [62-64] provides bounds on the tidal charge parameter. The 90% confidence contours, for the majority of GW observations, extended beyond q BH = ± 0 . 5. Besides, the lensing of GWs can also constrain the tidal charge q BH , but again for the braneworld models the constraints are weak q BH ≤ -0 . 9, while for electromagnetic theories the constraints are better q BH ≤ 0 . 5 [65]. From electromagnetic observations as well, e.g., the measurement of BH shadow constrains the tidal charge as q BH = -0 . 1 +0 . 6 -0 . 5 [72, 73]. These suggest that the constraints on the charge parameter, irrespective of its origin, are weak and are the prime motivation to choose the prior within the range q BH ∈ ( -1 , 1). Further note that, in the braneworld scenario, the brane is obtained by embedding the four-dimensional spacetime within a fivedimensional bulk. Therefore, the charge q BH gets naturally connected to χ , the size of the extra dimension. For example, the following bound: q BH ≤ -0 . 5, translates into ( χ/ℓ ) ≲ 0 . 63 [71], where ℓ depicts the ratio of the five-dimensional and the four-dimensional gravitational constants. Having determined the effect of the charge term in the spacetime metric, arising from extra spatial dimension, on the perturbation equation governing the gravitational perturbation of the brane. The corresponding fluxes through the horizons of the braneworld BHs orbiting each other get corrected at 2.5 PN and 4 PN levels due to the charge term. Given the above modification to the horizon flux, we wish to determine the corresponding modifications to the GW phase in the next section.",
"pages": [
2,
3,
4
]
},
{
"title": "III. DETAILS OF THE WAVEFORM MODEL AND PARAMETER ESTIMATION",
"content": "In this section, we will present detailed discussion regarding the waveform modeling for braneworld BH, with special emphasis on the effect of charge inherited from extra spatial dimension. Then we proceed to the parameter estimation details.",
"pages": [
4
]
},
{
"title": "A. Waveform model",
"content": "The GW waveform model from a coalescing compact binary signal in the frequency domain can be schematically represented as, where C is an overall constant, A ( f ) is the amplitude of the GW, ψ test ( f ) is the phase of the GW in the test particle approximation [74, 75], and δψ ( f ) is the contribution to the phase due to finite size effects of the binary BH system. In the inspiral regime, the amplitude of the GW follows from the relation A ( f ) ∼ D -1 L M 5 / 6 c f -7 / 6 at leading order, where D L is the luminosity distance between the observer and the source of the GW, with M c = ( M 1 M 2 ) 3 / 5 ( M 1 + M 2 ) -1 / 5 being the chirp mass of the binary. The frequency-independent factor C carries information about the source location and orientation of the source with respect to the detector, through the antenna pattern functions. For a compact binary signal, the GWphase plays a crucial role in detecting and analysing the signal. Hence, it is important to model them with the maximum available accuracy. In our case, the first term, ψ test ( f ), accounts for the 'point-particle' contributions, whereas δψ ( f ) represents the extra phase contributions that arise due to tidal heating, or, the BH absorption effect. Among these terms, ψ test ( f ) is taken to be accurate upto 3.5 PN [76], and δψ ( f ) has contributions at 2.5 PN, 3.5 PN and 4 PN orders. Altogether, these phase contributions accurately model the binary dynamics to the respective PN orders of aligned spin braneworld BHs in a binary system. The explicit expression for the phase δψ ( f ), due to tidal heating, can be obtained using the phase formula of [37, 77], and then using the flux through the horizon due to tidal heating, derived in Eq. (11). The final expression reads (for a detailed derivation, see Appendix A), where v is the relative velocity between the inspiralling braneworld BHs, acting as the PN parameter representing the PN order at which each coefficient would appear. So we have the tidal heating contributing at three postNewtonian orders, 2.5PN ( v 5 ), 3.5PN ( v 7 ) and 4PN ( v 8 ) with the explicit dependence to the binary parameters as (for a derivation, see Appendix A), Here F SO is the spin-orbit coupling term (see Eq. (5) in [77]). The quantities Ψ 5 and Ψ 8 depend on the characteristic parameters associated with the BH spacetime through the relations defined below Eq. (11). Notice that the binary dynamics are also influenced by each other's gravitational field, and we neglect these tidalinduced deformation effects in the phase. This is justifiable in the current scenario as the tidal deformations is a higher post-Newtonian effect (it starts appearing at the five post-Newtonian order) with a lesser contribution compared to BH absorption effect [78]. In what follows, we have incorporated these modifications to the TaylorF2 waveform model in LALSuite [79]. TaylorF2 is an inspiral-only model for an aligned-spin binary, where the component spin angular momenta are either aligned or anti-aligned to the orbital angular momentum axis. Further, we analyse binaries of various kinds employing this waveform model after making necessary changes to the dynesty sampler implemented in Bilby [80]. We truncate the waveform model before the plunge to avoid any unmodelled effects appearing from the post-inspiral frequency region. The truncation frequency in each case is the corresponding inner-most stable circular frequency of a Kerr BH [81], and the estimate is based on calculating the final mass and spin of the remnant BH; hence, the angular frequency considering a circular equatorial orbit around the Kerr BH, given the component masses and spins.",
"pages": [
4,
5
]
},
{
"title": "B. Overview of Bayesian analysis",
"content": "In the GW data analysis, we start with the data d, which contains both noise and signal. Here, noise is a random process while the signal is modelled following a particular hypothesis H and is a function of the complete set of binary parameters θ . The initial prior probability distribution, p( θ | H), restricts the range of θ . If we assume that the noise is Gaussian wide-sense stationary, the likelihood function takes the form, where d and h are the frequency domain data and signal respectively. Once we estimate the likelihood function and the prior distribution on each parameter is known, Baye's theorem provides the posterior probability distribution on each parameter as follows: In addition, the Bayesian evidence p(d | H) is a measure of how much the data supports the hypothesis, H, and is obtained by marginalizing the likelihood over the full prior volume, For the usual analyses, we assume GR accurately models the signal, and we estimate the GR evidence Z GR . On the other hand, to check the consistency of the predictions from BHs beyond GR with the data, we include the waveform model described in Sec. III A introducing a free parameter, namely the charge q BH and obtain, Z nGR , the evidence for non-GR signature in the data. The ratio between Z nGR and Z GR provides the Bayes factor comparing the non-GR hypothesis over the GR hypothesis. That is, If the data is consistent with GR, the 1-dimensional marginalized posterior on q BH , which is defined as, will peak at zero (the GR value) and the Bayes factor B nGR GR will be less than zero. All of these computations require evaluating the noise-weighted inner product, which has first appeared in Eq. 17, and is of the form, As evident, the inner product depends on the detector characteristics through the noise power spectral density Sn(f) and the lower cut-off frequency f low . In our case, we fix f low to be 10Hz and the upper cut-off frequency f high will vary according to the masses and spins of the binary system. To perform the posterior evaluation, we will be choosing the power spectral density corresponding to the plus era sensitivity (O5) for the LIGO detector [68]. ̸ To visualise the effect of the extra phase term on the binary phasing and hence the GW waveform model, we show the mismatch between the two models in Fig. 1; one includes the effect of the charge parameter ˜ h nGR ( f ) ( q BH = 0), while the other depicts a waveform model representing a binary BH in GR ˜ h GR ( f ) ( q BH = 0), such that, where the noise weighted inner product is defined in Eq. 22. As Fig. 1 depicts, the mismatch between the two waveform models increases as we increase the q BH value for both the mass ratios, q = ( M 1 /M 2 ) = 1 , 3, and spins (0 . 7 , 0 . 7) , (0 . 7 , 0 . 1). The binary total mass is fixed to be 20 M ⊙ and q BH value is chosen from [ -1 , 1]. The mismatch for binaries with large spin and mass asymmetries is larger, indicating a better distinguishability from their Kerr BH counterparts.",
"pages": [
5,
6
]
},
{
"title": "C. Details of simulation",
"content": "The simulations are motivated by the findings of mismatch studies, and we sub-divide these studies into different sets focusing on the masses, spins, signal-to-noise ratios (SNRs) and the ability to identify a non-GR signature if present. To show the effect of component spins, we choose ( χ 1 , χ 2 ) = (0 . 7 , 0 . 6) , (0 . 7 , 0 . 1) , (0 . 5 , 0 . 3) and (0 . 2 , 0 . 1), where χ 1 and χ 2 are the dimensionless spins of the BHs, assumed to be aligned to the orbital angular momentum axis of the binary. Further, we consider four mass ratios, q ≡ ( M 1 /M 2 ) = 1 , 2 , 3 , 4 to examine the effect of mass ratio on the q BH estimate. For all these cases, the total mass is fixed to 32 M ⊙ and the binary is fixed at a particular location to generate a signal-to-noise ratio of 120 in the detector. Moreover, we show posteriors on q BH by choosing signal-to-noises 40 and 80 along with 120 by varying the luminosity distance to quantify the measurability at different signal strengths. Moreover, the detectability of the tidal parameter is detailed by simulating injections with different values of q BH ranging from -0 . 7 to 0 . 7. A uniform prior range between [-1, 1] is assumed for the charge parameter for the entire analysis. Prior for the component masses are also taken uniformly between [5 , 80] M ⊙ whereas for component spins uniform from [0, 0.99]. We fixed the luminosity distance, sky location (right ascension and declination), polarization angle, the inclination to the source fixed to the injected value while performing the parameter estimation analysis and verified that the q BH posteriors are unaffected by this choice. While creating simulated injections, the luminosity distance has been altered according to the signal-to-noise ratio requirement. However, the right ascension, declination, and polarization angle are chosen to be 0, and the inclination to the source is fixed at 0.5 rad. It is worth mentioning that a full waveform model including inspiral-merger-ringdown regimes and spinprecession effects would be the best for our analysis. However, no such model is known in the context of braneworld and emplying the TaylorF2 waveform model is sufficient for our purpose. First of all, we have truncated the likelihood evaluation at the last stable circular frequency to ensure the validity of the PN approximation and to avoid any systematic that arises due to the presence of un-modelled physics contributing from the post-inspiral regime. Moreover, the absorption effects ̸ start to appear at 2.5 PN for spinning binaries, which is relatively lower than other effects, such as tidal deformation (a 5PN effect). Therefore, the analysis demonstrated here is largely waveform-independent and can easily be extended to more generic waveform models by simply adding the phase corrections due to tidal heating appropriately. The only non-trivial part of the above analysis corresponds to the determination of the innermost stable circular orbit frequency, for which we have used the current fits available for Kerr BHs, even for simulations with q BH = 0. In the ideal case, one may use an upper cut-off frequency expression that includes the q BH effect, but that is currently unavailable. Also, as emphasized above, since the test applies to inspiral-dominated signals, a slight change in the upper cut-off frequency of the analysis will most likely leave the findings unaltered.",
"pages": [
6,
7
]
},
{
"title": "IV. CONSTRAINTS FROM SIMULATED BINARY SIGNALS",
"content": "We generate a set of simulated binary signals (injections) in zero-noise to study the measurability of the charge parameter q BH , including binaries of different spins, mass ratios, and signal strengths or signal-tonoise ratios. Furthermore, a set of simulations is investigated, keeping non-zero values for q BH to quantify the detectability if present in the data. The analysis assumes that both the BHs in the binary spins are aligned/antialigned to the orbital angular momentum axis. Figure 2, shows the posterior probability distribution on q BH for a binary of total mass 32 M ⊙ and mass ratio q = ( M 1 /M 2 ) = 3. Each curve corresponds to different spin configurations, namely (0 . 7 , 0 . 6), (0 . 7 , 0 . 1), (0 . 5 , 0 . 3) and (0 . 2 , 0 . 1). Source locations of all these binaries are chosen such that the signal-to-noise ratio is 120. The black dotted line represents the GR value ( q BH = 0), and the dashed lines show the 90% bounds on q BH for each case. It is evident from the green and orange curves of Fig. 2 that the estimates are better when the spins are high, especially when the primary BH is largely spinning. The q BH estimate worsens as the spin magnitudes decrease. Especially, the negative prior side is not well constrained when the spin of the secondary BH is low 1 . To study the effect of mass ratio on the estimates of the charge, we choose three mass ratios, q = ( M 1 /M 2 ) = 1 , 2 , 3 , 4 as shown in Fig. 3. As expected from the mismatch studies, the estimates are better as we move to larger mass ratios. Whereas equal mass binary provides the least interesting constrain on q BH . The total mass is fixed to 32 M ⊙ and dimensionless aligned-spin magnitudes to (0 . 7 , 0 . 6) and the luminosity distance and sky localization are chosen such that the signal-to-noise ratio is 120. As evident from Fig. 3, the 90% credible interval bound on the q BH parameter is estimated to be 0.59 for mass-ratio of q=3, and the bound on q BH is 0.4, when we consider mass-ratio to be q=4. Therefore, the bound on q BH is stronger for the q=4 case (blue curve) than the q=3 case (green curve). This is because, the phase contributions for equal mass ratio are smaller than that of asymmetric mass ratio cases, leading to better estimates. This is consistent with earlier findings regarding BH absorption effects, See for example [45]. What is also interesting here is the slight shift in the peak of the posterior from the GR value, which is zero, with a median value of -0.13 for q=3 and 0.12 for q=4 case. This shift is arising because of the correlation between the charge parameter and the other intrinsic parameters of the binary (especially, we see that this effect is larger as the mass asymmetry of the binary increases). This implies that highly spinning binaries with large asymmetry in the mass ratio is the best candidate for detecting the existence of the charge parameter. Finally, we have performed a detailed analyses on binary BH simulations by injecting different q BH values, namely q BH = ± 0 . 7 , ± 0 . 5 , ± 0 . 3 , ± 0 . 1. For demonstration, we fix the binary mass to be 32M ⊙ , the mass ratio to be 3, and source location and orientation in such a way that the signal produces a signal-to-noise ratio of 120 in the advanced LIGO detector. Figure 4 shows the probability distribution on the charge parameter q BH for various cases. The vertical dotted lines in Fig. 4 denote the initial injected values of q BH , while the central black line is the q BH estimate, assuming the true value to be zero, i.e., the GR value. From Fig. 4, it is clear that the q BH probability distribution function shows distinct features and can be distinguished from its GR value for | q BH | > 0 . 3. For smaller values of the charge, e.g., for | q BH | ∼ 0 . 1, the probability distributions are indistinguishable between positive and negative values of q BH and also with the GR value. Fig. 4 also indicates that the probability distribution functions are well separated for all the positive injections of q BH ≳ 0 . 3 and hence are distinguishable. While for the negative injections, though the distribution functions are different from GR value, they are indistinguishable among themselves. In other words, it is impossible to distinguish the probability distribution function for q BH = -0 . 5 and q BH = -0 . 7. Thus positive values of the charge parameter, namely those associated with extra dimensions are easier to distinguish, compared to the negative injections, associated with the electromagnetic origin. We would like to point out that there is an asymmetry between the probability distribution functions with positive and negative values of the injected tidal charge parameter. This happens because the phase change due to tidal heating depends on q BH as well as on ( q BH ) 2 (see Eqs. (8) and (9) for explicit expressions), and hence q BH →-q BH is not a symmetry. For completeness, We have also studied the effect of the signal-to-noise ratio on the estimation of the charge parameter, and the resulting posteriors have been plotted in Fig. 5. To demonstrate the improvement in measuring the charge parameter with respect to the signal strength, we have considered four different signal-to-noise ratios, 260, 130, 80 and 40, adjusting the distance to the source accordingly. Following our earlier conclusion, we have assumed a binary BH with asymmetric mass ratio ( q = 3) and high spins ( χ 1 = 0 . 7 and χ 2 = 0 . 6), so that the charge parameter can be better estimated. As Fig. 5 demonstrates, unless we have a high signal-to-noise ratio ∼ 200, all possible values of q BH are allowed, while for high signal-to-noise ratio, the distribution functions start to rule out larger values of q BH . This gives the following three criteria for observing the existence of the charge parameter - (a) asymmetric mass ratio: more asymmetric the mass ratio, the better the chance of detection; (b) higher spins: higher is the spin of the binary BHs, better is the chance of observing them in future GW observations and (c) higher signal-to-noise ratio enhances the detection probability. With this input, we now discuss their implications for the third GW tran- t catalogue [82] and future detectors.",
"pages": [
7,
8,
9
]
},
{
"title": "V. IMPLICATIONS FOR THE THIRD GW TRANSIENT CATALOG AND FUTURE DETECTORS",
"content": "The GW detection from the advanced LIGO-Virgo detectors contains binary events of various types in terms of masses, spins, location, orientation and nature of the compact object. The test mentioned above for 'charged' BHs suits the inspiral-dominated binaries with non-zero spins and mass asymmetry - for instance, GW190412 [83]-like sources, which is one of the most asymmetric binary systems with ∼ 30 M ⊙ primary BH and ∼ 3 M ⊙ companion (see Tab.II of [83] for more details). Besides, the GW190412 is an inspiral-dominated event with evidence of non-zero spins, making it ideal to test the validity of the above claims against real data. For this purpose, we employ the non-precessing GW waveform model TaylorF2 with the charp mass M c , effective spin parameter χ eff , tidal charge q BH and the luminosity distance as free parameters, and truncate the analysis at 210Hz. The result of such an analysis is an estimation of the q BH parameter, which reads 0 . 05 0 . 77 -0 . 86 within the 90% credible interval with respect to the mean value, 0 . 05. Even though a non-zero and positive mean value is a tantalizing indication of the existence of an extra spatial dimension, the errors are very large. In particular, the GR value is well within the 90% credible interval, and so are plenty of positive and negative values, reducing the robustness of the claim. We have also estimated the Bayes factor log B nGR GR supporting the non-GR hypothesis over the GR hypothesis as log e B nGR GR = -6 . 43. Therefore, the results look promising, while the error bar needs to be further reduced. Additionally, in the GW190412 analysis, we further notice that the inclusion of q BH introduces significant shifts in the intrinsic parameters of the binary, such as masses and spins 2 . In Fig. 6, we show the posteriors on the chirp mass (a combination of binary masses which is well estimated from the inspiral signal), the effective spin parameter (found to be the best representative of the alignedspin effects of the binary) and the luminosity distance to the source. As Fig. 6 demonstrates, the intrinsic binary parameters are significantly affected by the presence of q BH , while the extrinsic parameter remains unaltered. In particular, the percentage change in the chirp mass and the effective spin parameter becomes (∆ M c /M c ) ∼ 4 . 9% and ( δχ eff /χ eff ) ∼ 66 . 7%. Therefore, our results indicate that parameters beyond GR theories are highly entangled with other intrinsic parameters of the problem, particularly the spin. This makes the detection of any non-GR effect significantly challenging. We finally discuss the constraints on the charge q BH from future GW detectors, such as Einstein Telescope and Cosmic Explorer. For this purpose we have performed the parameter analysis on simulated binaries with 200 and 400 as the signal-to-noise ratios, respectively and compare the constraints on q BH with lower signalto-noise ratio simulations. The corresponding posteriors on q BH , considering a simulated binary BH signal with total mass 32 M ⊙ , and spins (0 . 7 , 0 . 6) are shown in Fig. 7 for two possible injected values of the charge parameter, q BH inj = ± 0 . 3. As evident from Fig. 7, a signal-to-noise ratio of 200 and 400 significantly improves the detectability of the charge parameter. In particular, for signal-to-noise ratio of 400 and positive injected value of q BH (i.e., for q BH = 0 . 3), GR can be ruled out with more than 90% confidence. While for negative q BH , the GR remains within the 90% confidence interval. In summary, high signal-to-noise ratio is essential for detecting the charge parameter and ruling out GR with confidence, which requires next generation of GW detectors.",
"pages": [
9
]
},
{
"title": "VI. SUMMARY",
"content": "We have proposed the tidal heating phenomenon in the inspiral regime of a binary BH system to be a benchmark in distinguishing binaries composed of BHs with charge from Kerr BHs using their GW signatures. For this purpose, we have started with the GW waveform model for a binary BH, parameterized in terms of an extra parameter, namely the charge, q BH . The origin for /circledot this charge can be from extra dimensions, in which case the charge is positive, or from scenarios involving simple electromagnetic interaction, in which case q BH takes negative values. To demonstrate the measurability and detectability of the existence of such a hair from the GW observations, we have considered simulated binary BH signals of varying masses, spins and signal strength. The findings of this simulation study suggest that if the BHs are highly spinning and mass asymmetric, we will be able to perform such a distinguishability test with the advanced LIGO sensitivity since the mismatch significantly increases. Moreover, if the charge parameter is present and its value is significant, e.g., greater than ± 0 . 3, we can detect its presence and distinguish it from GR with advanced sensitivity. Even though the current detectors are not yet sensitive enough for this test, we show that a GW190412-like event would be an ideal candidate for testing the existence of 'charged' hair. Interestingly, the posteriors from the GW190412 event report a positive median value of the charge, consistent with the existence of an extra dimension, however, the error bars are huge, rendering any conclusive statement. Moreover, the Bayes factor supporting the GR hypothesis, on the other hand, was found to be log e 6 . 43, which is not a large number. This suggests that tidal heating provides an avenue to test theories beyond GR. Besides providing simulated GW waveforms with the charge parameter and testing their distinguishability from pure GR waveforms, we have also provided a forecast analysis for the observability of the charge parame- ter in future GW observations. We have shown that increasing the signal-to-noise ratio considerably improves the estimation of the charge parameter and enhances its detectability. In particular, the future detector sensitivity, with a signal-to-noise ratio of 400, can significantly constrain the charge parameter. Moreover, as we have demonstrated that the tests involving charged hair of BHs will better suit more asymmetric binaries such as those in intermediate and extreme mass ratio inspirals, the future space-based GW detectors, namely DECIGO and LISA can also play a vital role in looking for the existence of non-trivial charged hair. The adaptation of the analysis presented here, in the context of the GW detectors DECIGO and LISA, will be explored in the future.",
"pages": [
9,
10,
11
]
},
{
"title": "ACKNOWLEDGMENTS",
"content": "S.C. and N.V.K. acknowledges discussions with Sayak Datta on several aspects of this manuscript. We also thank Khun Sang Phukon and Gregorio Carullo for reading the manuscript and providing useful comments. The research of S.C. is funded by the INSPIRE Faculty fellowship from DST, Government of India (Reg. No. DST/INSPIRE/04/2018/000893) and MATRICS grant from Science and Engineering Research Board (SERB), Government of India (Reg. No. MTR/2023/000049). N.V.K. acknowledges support from SERB for the National postdoctoral fellowship (Reg. No. PDF/2022/000379). N.V.K. acknowledges Max Planck Computing and Data Facility computing cluster Cobra and Raven for computations. We also thank the CIT cluster provided by the LIGO Laboratory. We acknowledge National Science Foundation Grants PHY-0757058 and PHY-0823459. This material is based upon work supported by NSF's LIGO Laboratory which is a major facility fully funded by the National Science Foundation. We used the following software packages: LALSuite [79], bilby [80], bilby pipe [84], NumPy [85], PESummary [86], Matplotlib [87], jupyter [88], dynesty [89]. This document has LIGO preprint number LIGO-P2400054 .",
"pages": [
11
]
},
{
"title": "Appendix A: GW phase due to tidal heating",
"content": "The phase of the GW, as a function of velocity can be given by [90], where, E orb is the orbital energy of the binary BH system, F ∞ is the GW flux at infinity and F H is the GW flux through the BH horizon. Since we are interested in the contribution from the tidal heating alone, the following PN expansion of the orbital energy suffices for our purpose (see Eq. (3.8) of [91]) Such that, Similarly, for the GW flux through infinity, the following PN expansion suffices for our purpose [34], while the GW flux through the horizon yields, Therefore, the contribution of tidal heating from the (1 / Flux) term in the phase integral, presented in Eq. (A1), yields, Therefore the contribution of tidal heating to the phase of the GW becomes, This expression has been used in the main text for computing δψ and identifying the 2.5 PN, 3.5 PN and 4 PN terms in the phase factor due to tidal heating.",
"pages": [
11,
12
]
}
]
2024PhRvD.109l4059F
https://arxiv.org/pdf/2203.01267.pdf
<document>
<section_header_level_1><location><page_1><loc_14><loc_73><loc_85><loc_83></location>Generating gravitational waveform libraries of exotic compact binaries with deep learning</section_header_level_1>
<text><location><page_1><loc_14><loc_65><loc_87><loc_70></location>Felipe F. Freitas a,b Carlos A. R. Herdeiro c,b António P. Morais a,b António Onofre d Roman Pasechnik e Eugen Radu c,b Nicolas Sanchis-Gual c,b,f Rui Santos g,h</text>
<unordered_list>
<list_item><location><page_1><loc_15><loc_62><loc_53><loc_64></location>a Departamento de Física da Universidade de Aveiro,</list_item>
<list_item><location><page_1><loc_16><loc_61><loc_51><loc_62></location>Campus de Santiago, 3810-183 Aveiro, Portugal.</list_item>
<list_item><location><page_1><loc_15><loc_58><loc_77><loc_61></location>b Centre for Research and Development in Mathematics and Applications (CIDMA), Campus de Santiago, 3810-183 Aveiro, Portugal.</list_item>
<list_item><location><page_1><loc_15><loc_56><loc_58><loc_58></location>c Departamento de Matemática da Universidade de Aveiro,</list_item>
</unordered_list>
<text><location><page_1><loc_16><loc_55><loc_51><loc_56></location>Campus de Santiago, 3810-183 Aveiro, Portugal.</text>
<unordered_list>
<list_item><location><page_1><loc_15><loc_54><loc_68><loc_55></location>d Centro de Física das Universidades do Minho e do Porto (CF-UM-UP),</list_item>
</unordered_list>
<text><location><page_1><loc_16><loc_52><loc_53><loc_53></location>Universidade do Minho, 4710-057 Braga, Portugal.</text>
<unordered_list>
<list_item><location><page_1><loc_16><loc_51><loc_66><loc_52></location>Department of Astronomy and Theoretical Physics, Lund University,</list_item>
<list_item><location><page_1><loc_15><loc_49><loc_32><loc_52></location>e 221 00 Lund, Sweden.</list_item>
<list_item><location><page_1><loc_16><loc_48><loc_66><loc_49></location>Departamento de Astronomía y Astrofísica, Universitat de València,</list_item>
<list_item><location><page_1><loc_15><loc_46><loc_53><loc_49></location>f Dr. Moliner 50, 46100, Burjassot (València), Spain</list_item>
<list_item><location><page_1><loc_15><loc_45><loc_53><loc_46></location>g ISEL - Instituto Superior de Engenharia de Lisboa,</list_item>
</unordered_list>
<text><location><page_1><loc_16><loc_43><loc_58><loc_44></location>Instituto Politécnico de Lisboa 1959-007 Lisboa, Portugal.</text>
<unordered_list>
<list_item><location><page_1><loc_15><loc_42><loc_16><loc_43></location>h</list_item>
<list_item><location><page_1><loc_16><loc_40><loc_82><loc_43></location>Centro de Física Teórica e Computacional, Faculdade de Ciências,Universidade de Lisboa, Campo Grande, Edifício C8 1749-016 Lisboa, Portugal.</list_item>
</unordered_list>
<text><location><page_1><loc_16><loc_37><loc_80><loc_39></location>E-mail: felipefreitas@ua.pt, herdeiro@ua.pt, aapmorais@ua.pt, Antonio.Onofre@cern.ch, roman.pasechnik@thep.lu.se, eugen.radu@ua.pt, nicolas.sanchis@uv.es, rasantos@fc.ul.pt</text>
<text><location><page_1><loc_14><loc_16><loc_88><loc_35></location>Abstract. Current gravitational wave (GW) detections rely on the existence of libraries of theoretical waveforms. Consequently, finding new physics with GWs requires libraries of non-standard models, which are computationally demanding. We discuss how deep learning frameworks can be used to generate new waveforms "learned" from a simulation dataset obtained, say, from numerical relativity simulations. Concretely, we use the WaveGAN architecture of a generative adversarial network (GAN). As a proof of concept we provide this neural network (NN) with a sample of ( > 500 ) waveforms from the collisions of exotic compact objects (Proca stars), obtained from numerical relativity simulations. Dividing the sample into a training and a validation set, we show that after a sufficiently large number of training epochs the NN can produce from 12% to 25% of the synthetic waveforms with an overlapping match of at least 95% with the ones from the validation set. We also demonstrate that a NN can be used to predict the overlapping match score, with 90% of accuracy, of new synthetic samples. These are encouraging results for using GANs for data augmentation and interpolation in the context of GWs, to cover the full parameter space of, say, exotic compact binaries, without the need of intensive numerical relativity simulations.</text>
<section_header_level_1><location><page_2><loc_14><loc_86><loc_23><loc_87></location>Contents</section_header_level_1>
<table>
<location><page_2><loc_14><loc_69><loc_88><loc_84></location>
</table>
<section_header_level_1><location><page_2><loc_14><loc_66><loc_30><loc_67></location>1 Introduction</section_header_level_1>
<text><location><page_2><loc_14><loc_55><loc_88><loc_65></location>The advent of the gravitational wave (GW) era [1-3] opens new possibilities not only for relativistic astrophysics, cosmology and strong gravity, but also for fundamental physics. Alongside with the unveiling of the population of black holes and neutron stars in the Universe, see e.g. , [4], it is possible that smoking guns about the nature of dark energy, dark matter and even quantum gravity will emerge from this new channel. In fact, particular events have already established concrete illustrations of how dark energy models can be constrained, see e.g. [5-7] and dark matter could be identified, see e.g. [8].</text>
<text><location><page_2><loc_14><loc_41><loc_88><loc_55></location>The interpretation of GW signals relies on matched filtering [9]. Therefore, libraries of theoretical templates are mandatory. The construction of such libraries is a non-trivial, time-consuming process. For the vanilla black hole binary problem in (vacuum) general relativity, the construction of full waveforms, including inspiral, merger and ringdown, became under control after the numerical relativity breakthroughs of 2005 - see the review in [10]. However, a dense scanning of the full parameter space (of the black hole binary problem) only with numerical relativity simulations is computationally impossible. Thus, a community effort drove the scanning of the parameter space using state of the art numerical relativity simulations, patched together with approximation methods for both the inspiral and the ringdown - see e.g. [11-14], building approximants for the theoretical waveforms with generic parameters.</text>
<text><location><page_2><loc_14><loc_29><loc_88><loc_40></location>An under-emphasised caveat of current GW interpretations is the degeneracy problem: can non-standard waveforms fit the data better? Non-standard means waveforms from exotic compact objects, which could either be non-Kerr black holes or horizonless compact objects. Moreover, such exotic compact objects could originate either from general relativity with matter sources or from modified gravity. The difficulty in tackling the above question is, however, the almost complete lack of alternative waveform libraries that can be compared with real events to determine whether the vanilla (Kerr black holes or neutron star binaries) waveforms are indeed the ones selected within a larger library, when employing matched filtering and Bayesian analysis.</text>
<text><location><page_2><loc_14><loc_14><loc_88><loc_29></location>At the time of writing, the one non-standard model of compact binaries for which there has been a more consistent and successful effort to produce waveforms is the case of bosonic ( i.e. scalar [15-17] or vector [18-20]) stars. The dynamical evolution of these models is theoretically and technically under control [21] and presents a variety of motivations: bosonic stars emerge in sound physical models, can be dynamically robust [21, 22] and have been put forward as "fuzzy" dark matter [23] lumps and black hole imitators, e.g. [24-26]. In the context of GWs, several studies of waveforms from collisions and binaries of bosonic stars have been reported, e.g. [27-30]. As an application to the ongoing detections, the massive GW event GW190521 [31] was shown to fit well a collision of two vector bosonic ( a.k.a. as Proca) stars [8]. This effort relied on scanning a library of 89 Proca star collision waveforms (in the meantime enlarged to nearly 800 waveform) [32]. Still, this only scratches the surface of the full parameter space of the model. As such, looking for efficient computational</text>
<text><location><page_3><loc_14><loc_87><loc_88><loc_90></location>methodologies that can transform a coarse sampling of the parameter space into a dense coverage is of paramount importance.</text>
<text><location><page_3><loc_14><loc_79><loc_88><loc_87></location>The goal of this paper is to start an exploration of such a methodology using deep learning techniques. Moreover, the method can, in principle, be used for waveforms produced from generic non-standard compact binaries. Thus, the Proca model explicitly discussed herein can be taken both as interesting in its own right, but simultaneously as a proof of concept of the application of the method, illustrating it but not-exhausting it. To be concrete, we shall be making use of Generative Adversarial Networks (GANs) [33], a particular class of deep learning frameworks.</text>
<text><location><page_3><loc_14><loc_62><loc_88><loc_78></location>GANs can be described as unsupervised methods for mapping low-dimensional latent vectors to high-dimensional data. In our case, this means mapping known waveforms, corresponding to a prior distribution, p model , to a larger space of waveforms, the generated data distribution, p data . In a nutshell, GANs are based on a game-theoretic scenario where we have two networks competing against each other. On the one hand, we have the generator , responsible for mapping the low dimensional vector z into the high dimensional samples we want to reproduce x = g ( z , θ g ) ( i.e. the waveforms in our case). Here, θ g are the parameters from the generator network to be adjusted during the training phase. Competing against the generator we have, on the other hand, the discriminator network, whose sole purpose is to distinguish between samples drawn from the original dataset and samples drawn from the generator. The discriminator provides a probability, d ( x ; θ d ) ∈ [0 ., 1 . ] , of a given sample x being real, as opposed to a fake one drawn from the generator model. Here, θ d are the parameters from the discriminator network to be adjusted during the training phase.</text>
<text><location><page_3><loc_14><loc_54><loc_88><loc_61></location>The simplest way to describe the learning process of a GAN is a zero-sum game, in which a function L ( θ g , θ d ) determines the payoff of the discriminator. The generator receives -L ( θ g , θ d ) as its own payoff. During the training phase, each player attempts to maximize its own payoff, so that the generator is trained to maximize L ( θ g , θ d ) , whereas the discriminator is trained to minimize it. The original proposal [33] for the function L ( θ g , θ d ) is :</text>
<formula><location><page_3><loc_25><loc_51><loc_88><loc_53></location>arg min g max d L ( θ g , θ d ) = E x ∼ p data [log d ( x )] + E z ∼ p model [log(1 -d ( g ( z ))] , (1.1)</formula>
<text><location><page_3><loc_14><loc_48><loc_88><loc_50></location>where E x ∼ p data and E z ∼ p model are the expected values for a sample to be drawn from the data and the generator, respectively.</text>
<text><location><page_3><loc_14><loc_41><loc_88><loc_47></location>This drives the discriminator to learn to correctly classify samples as real or fake. Meanwhile, the generator attempts to fool the discriminator by producing fake samples with features as close as possible to the features from real samples. At convergence, the generator's samples are indistinguishable from the real ones, and the discriminator outputs a probability of 50 % for every sample. The discriminator may be discarded or its parameters can be reused for other purposes later on.</text>
<text><location><page_3><loc_14><loc_32><loc_88><loc_40></location>GANs have shown great success in generating high-quality synthetic images [34-36] indistinguishable from real images. This has encouraged the use of GANs for synthetic data generation in broader contexts, in particular in high-energy physics, where in some instances the data generation can be a computational intensive task [37-40]. In this regard, while GANs were developed for image generation [33], there have been attempts to adapt this approach for other formats, such as tabular data [34], time series [41], video content augmentation [42] and audio synthesis [43-45].</text>
<text><location><page_3><loc_14><loc_21><loc_88><loc_32></location>In this article, we shall examine the potential of GANs to produce a larger waveform catalogue from a limited dataset of the corresponding waveforms. We shall focus on the case of waveforms produced by Proca star binaries. For this purpose, we shall modify WaveGAN [45], a GAN initially designed to provide an unsupervised synthesis of raw-waveform audio, such that it could learn and produce Proca waveforms from an initial dataset obtained from numerical relativity simulations. Dividing the sample into a training and a validation set, we show that after a sufficiently large number of training epochs the neural network (NN) can generate synthetic data with at least 95% overlapping match with reference samples from the validation set.</text>
<text><location><page_3><loc_14><loc_14><loc_88><loc_20></location>This article is organized as follows. In Sec. 2, we briefly review the Proca model of bosonic stars and describe the dataset and methodology explored in this study. We also discuss the issue of waveform normalization. In Sec. 3, we describe the WaveGAN architecture and the training methodology. Then, in Sec. 4 we discuss the evaluation methodology, i.e. how to assess the quality of the generated waveforms and use the trained discriminator architectures to predict the match score for new synthetic</text>
<text><location><page_4><loc_14><loc_87><loc_88><loc_90></location>samples. Sec. 5 presents our results after applying the chosen architecture, training and evaluation to the initial dataset. Finally, Sec. 6 provides a final discussion on the approach proposed herein.</text>
<section_header_level_1><location><page_4><loc_14><loc_83><loc_51><loc_84></location>2 The Proca model and the dataset</section_header_level_1>
<text><location><page_4><loc_14><loc_77><loc_88><loc_81></location>The Proca stars, their dynamics and the corresponding GWs will be considered in the simplest model: a complex, massive Proca field minimally coupled to Einstein's gravity. The action reads (with c = 1 = G )</text>
<formula><location><page_4><loc_32><loc_73><loc_88><loc_76></location>S = ∫ d 4 x √ -g [ R 16 π -1 4 F αβ ¯ F αβ -µ 2 2 A α ¯ A α ] , (2.1)</formula>
<text><location><page_4><loc_14><loc_68><loc_88><loc_72></location>where R is the Ricci scalar of the spacetime metric g , A is a complex 4-potential, with the field strength F αβ = ∂ α A β -∂ β A α , µ > 0 corresponds to the mass of the Proca field, and the overbar denotes complex conjugation.</text>
<text><location><page_4><loc_14><loc_62><loc_88><loc_67></location>Spinning Proca stars (the fundamental solutions, in the stable branch [22]) can be labelled by their ADM mass, Mµ or, alternatively, by their oscillating frequency ω/µ , both in units of the Proca field mass. In the following, for simplicity, we shall set µ = 1 and label the solutions via M . The fundamental solutions in the stable branch have M and ω in the interval(s) [20]:</text>
<formula><location><page_4><loc_40><loc_59><loc_88><loc_60></location>( M,ω ) ∈ ([0 , 1 . 125] , [0 . 469 , 1]) . (2.2)</formula>
<text><location><page_4><loc_14><loc_53><loc_88><loc_57></location>Note that the upper (lower) limit in the M interval corresponds to the lower (upper) limit in the ω interval. The angular momentum of the solutions is determined by M . For the considered solutions the total angular momentum is in the range [20] J ∈ [0 , 1 . 259]</text>
<text><location><page_4><loc_14><loc_43><loc_88><loc_53></location>The collision of two Proca stars generates GWs. These are extracted via the Newman-Penrose (complex) scalar Ψ 4 . Both the real ( R (Ψ 4 ) ) and imaginary parts of this scalar (corresponding to the two GW polarizations) can be decomposed into harmonics. The dominant GW modes, i.e. with higher amplitude, have harmonic indices ( l, m ) = (2 , 2) and ( l, m ) = (2 , 0) . For simplicity, we shall consider only the ( l, m ) = (2 , 2) waveforms for each collision, focusing on the real part of the scalar (the "+" polarization). Each waveform is a time series for r R (Ψ 4 ) , since Ψ 4 falls as 1 /r , with r being the distance to the source.</text>
<text><location><page_4><loc_14><loc_35><loc_88><loc_43></location>Our dataset consists of waveforms generated from the merger of two spinning Proca stars with aligned spin axes. This sort of collisions were recently studied in [8, 46]. Although the stars start from rest, due to frame dragging the binary describes an eccentric (rather than precisely head-on) trajectory. The end point depends on the progenitor Proca stars. In the region of the parameter space explored here, the Proca star progenitors are sufficiently massive to trigger black hole formation after the merger.</text>
<text><location><page_4><loc_14><loc_23><loc_88><loc_34></location>The waveforms are generated from numerical evolutions using the Einstein toolkit infrastructure [47-49], together with the carpet package [50, 51] for mesh-refinement. The Proca evolution equations are solved via a modified Proca thorn [22, 30, 52, 53] to include a complex field. We have performed numerical simulations of equal and unequal mass Proca stars. The initial data consists in the superposition of two equilibrium solutions separated by D = 40 /µ [8], in geometrized units. This guarantees an admissible initial constraint violation. The equilibrium spinning Proca stars are numerically constructed using the solver fidisol/cadsol for non-linear Partial Differential Equations of elliptic type, via a Newton Raphson method - see [18-20] for more details.</text>
<text><location><page_4><loc_18><loc_22><loc_42><loc_23></location>We divide our data into two sets:</text>
<unordered_list>
<list_item><location><page_4><loc_14><loc_18><loc_88><loc_20></location>i ) one set contains 98 waveforms generated from the merger of two equal mass Proca stars ( M 1 = M 2 ). For each collision, we consider waveforms of the ( l, m ) = (2 , 2) mode.</list_item>
</unordered_list>
<text><location><page_4><loc_22><loc_14><loc_22><loc_15></location>glyph[negationslash]</text>
<unordered_list>
<list_item><location><page_4><loc_14><loc_14><loc_88><loc_16></location>ii ) The other dataset consists of 457 waveforms from the merger of two unequal mass Proca stars ( M 1 = M 2 ), with the same ( l, m ) = (2 , 2) mode.</list_item>
</unordered_list>
<text><location><page_5><loc_14><loc_87><loc_88><loc_90></location>In figure 1 and 2 some samples for both data sets are illustrated, for the dominant quadrupolar mode ( l, m ) = (2 , 2) .</text>
<figure>
<location><page_5><loc_15><loc_69><loc_34><loc_85></location>
<caption>FIG. 2 . Samples of r R (Ψ l =2 ,m =2 4 ) for the unequal mass data set ( M 1 = M 2 ). The amplitude of r R (Ψ l =2 ,m =2 4 ) is normalized to be within range of [ -1 ., 1 . ] .</caption>
</figure>
<figure>
<location><page_5><loc_36><loc_69><loc_52><loc_85></location>
</figure>
<figure>
<location><page_5><loc_54><loc_69><loc_70><loc_85></location>
</figure>
<figure>
<location><page_5><loc_72><loc_69><loc_88><loc_85></location>
<caption>FIG. 1 . Samples of r R (Ψ l =2 ,m =2 4 ) for the equal mass data set ( M = M 1 = M 2 ). The amplitude of r R (Ψ l =2 ,m =2 4 ) is normalized to be within range of [ -1 ., 1 . ] .</caption>
</figure>
<figure>
<location><page_5><loc_15><loc_45><loc_34><loc_61></location>
</figure>
<figure>
<location><page_5><loc_36><loc_45><loc_52><loc_61></location>
</figure>
<figure>
<location><page_5><loc_72><loc_45><loc_88><loc_61></location>
</figure>
<figure>
<location><page_5><loc_54><loc_45><loc_70><loc_61></location>
</figure>
<text><location><page_5><loc_68><loc_42><loc_68><loc_44></location>glyph[negationslash]</text>
<text><location><page_5><loc_14><loc_23><loc_88><loc_39></location>In figure 3 we display the mass distribution for both datasets (equal mass case in figure 3a and different mass in figure 3b). Both datasets are pre-processed to be sampled at 2048 Hz. Due to the feature that the dataset have different y -ranges we need to normalize them. Having the samples scaled to a similar range helps to prevent or at least mitigate bias and to speed up the optimization process by preventing the model parameter weights to either vanish or explode [54]. We have tested different methods of scaling, including standard scaling 1 , Robust Scaler 2 , the min-max scaler 3 and max-absolute scaler 4 which scale the features into the [ -1 ., 1 . ] range without breaking the sparsity of the dataset. We have chosen to scale the datasets according to the max-absolute, since we want to preserve the sparsity of our dataset. It is important to mention that all features are scaled only after the train/validation split happens, to avoid any bias in the training procedure, and their true amplitude range are stored for a later use to transform back the normalized samples into their original</text>
<text><location><page_6><loc_14><loc_86><loc_88><loc_90></location>values. Then each dataset is shuffled and split into training (80 % of the total dataset) and validation (20 % of the total dataset) datasets; these are then fed into the NN model, as further explained in the next Section.</text>
<text><location><page_6><loc_14><loc_83><loc_88><loc_85></location>Each sample consist of a time series representing the real part of the Newman-Penrose scalar Ψ l =2 ,m =2 4 , together with the value of the mass - as show in figure 1 - and the feature scale.</text>
<figure>
<location><page_6><loc_14><loc_48><loc_88><loc_81></location>
<caption>FIG. 3 . Mass distribution for the equal (a) and unequal (b) mass dataset with one (red) and two (blue) sigmas. (a) The green and red crosses at the bottom show the sample values for the training and validation datasets, respectively. (b) The green and black dots display the sample values for the training and validation datasets, respectively, while the red and blue ellipsis display the one and two sigmas regions for M 1 and M 2 distributions.</caption>
</figure>
<text><location><page_6><loc_56><loc_46><loc_56><loc_47></location>glyph[negationslash]</text>
<paragraph><location><page_6><loc_51><loc_44><loc_88><loc_47></location>(b) M 1 = M 2 dataset. The green (black) points represent the samples for the training (validation) set.</paragraph>
<section_header_level_1><location><page_6><loc_14><loc_32><loc_63><loc_34></location>3 Model architecture and training methodology</section_header_level_1>
<text><location><page_6><loc_14><loc_25><loc_88><loc_31></location>Inspired by the use GANs for audio generation, we employ WaveGAN [45] to generate new waveforms "learned" from our simulations dataset. Our purpose is to test whether this method is useful for data augmentation and data interpolation to cover the full parameter space without the need of intensive computational simulations.</text>
<text><location><page_6><loc_14><loc_17><loc_88><loc_25></location>The WaveGAN architecture is based on deep convolutional GAN (DCGAN) [55] which popularized the usage of GANs for image synthesis. The DCGAN generator uses the transposed convolution operation to iteratively upsample low-resolution feature maps into high-resolution images. The WaveGAN uses a modified transposed convolution operation to widen its receptive fields 5 . We keep the longer one-dimensional filters of 25, as proposed in [45], however we set the number of layers to 4 and channels to 1 at first, and upsample by a factor of 4 at each layer. The discriminator network is</text>
<table>
<location><page_7><loc_31><loc_70><loc_70><loc_87></location>
<caption>TABLE 1 . WaveGAN generator architecture.</caption>
</table>
<text><location><page_7><loc_14><loc_65><loc_88><loc_68></location>modified in a similar fashion, using length-25 filters. The output length from the generator, as well the input length to the discriminator, is set to 2048, to be the same length as the waveforms samples.</text>
<text><location><page_7><loc_14><loc_37><loc_88><loc_65></location>The usual GAN generate samples similar to the ones learned in the training. However, this approach is not the most practical if one wants to produce synthetic samples from particular classes present in the dataset, i.e. augment classes or generate samples to interpolate missing regions from the dataset. One way to overcome such problem is to condition our generator and discriminator models. To promote a generator and discriminator to its conditional model forms, one must provide additional information about the training samples, which can be any kind of auxiliary information, such as class labels or, in our case, the mass value M for each sample waveform. We conditioned our WaveGAN using the values of the mass M as labels y and the feature scale max | x i | used to normalize the sample with the intent to restrict the model to generate samples within these parameter constraints. We can perform the conditioning by feeding y and max | x i | into both the discriminator and the generator. We use a similar approach as in [56]. To include the label y and max | x i | , we scale the feature maps output from each hidden layers based on the conditioning representation; in our case we scale the feature maps by the mass and feature scale values provided by the sample labels. It is important to mention that this approach is applicable in our case due to the low variance of our labels. In order to deal with high variance labels the best approach is to either normalize them or encode it using a linear layer. Meanwhile, the scale factors max | x i | help us to constrain the amplitude scale of the synthetic samples, in a sense that when we produce the new samples their amplitude values will be within a region allowed by their physical parameters in the dataset. This is required in order to avoid producing samples which are not permitted by physics or artefacts that can be produced by such methods [57].</text>
<text><location><page_7><loc_14><loc_27><loc_88><loc_36></location>Our WaveGAN is implemented in PyTorch[58], and we train our model for 1050 epochs using WGAN-GP [59] strategy, with ADAM [60] as an optimizer, for both generator and discriminator, with learning rate of 10 -4 for the generator and 3 × 10 -4 for the discriminator. We train our networks using batches of size 32, while the validation set has batches of size 16, on a single GPU NVIDIA Tesla V100. As a first task, we set the generators and discriminators to one channel in order to generate the synthetic r Ψ l =2 ,m =2 modes for the equal and different masses datasets. The results for equal and unequal mass datasets are presented in Section 5.</text>
<section_header_level_1><location><page_7><loc_14><loc_23><loc_42><loc_25></location>4 Evaluation methodology</section_header_level_1>
<text><location><page_7><loc_14><loc_14><loc_88><loc_22></location>The evaluation of generative models is an ongoing topic in the community [61]. Just as important as choosing the right strategy to train a generative model, is selecting the right metric to evaluate the quality of the generated samples. A direct comparison between the synthetic samples and the real ones can be a useful diagnostic, often allowing us to build intuition of how the generative model is working, how it is failing and how it can be improved. However, qualitative as well quantitative analysis based on this approach can be misleading about the performance of the generator. In order</text>
<table>
<location><page_8><loc_33><loc_67><loc_69><loc_87></location>
<caption>TABLE 2 . WaveGAN discriminator architecture.</caption>
</table>
<text><location><page_8><loc_14><loc_45><loc_88><loc_63></location>to evaluate the quality, and therefore how trustworthy is our generator, we shall employ the following strategy. Using the PyCBC [62] matched filtering module, we estimate the overlap over time and phase between the synthetic and real samples for a given value of the parameters. We generate a set of 1000 synthetic samples for a given set of parameters, ( M ) for equal mass dataset or ( M 1 , M 2 ) for unequal mass dataset, and compute the overlapping match for each sample to the real equivalent samples. The overlapping match is computed with the real and synthetic normalized samples, so we can ensure that features generated for the synthetic samples are as close as possible to the expected real features. With these values, we estimate the probability of a generated sample to be above a certain threshold of match. In figures 4-9 we show the evolution of the match between synthetic and original samples throughout the training epochs of our NN model. To visualize the overlapping between the real and synthetic samples, we plot various samples from the equal and different mass datasets and select synthetic samples according to their overlapping matches to check against the real ones; these plots are shown in figure 10 and figure 11.</text>
<text><location><page_8><loc_14><loc_34><loc_88><loc_44></location>Using the overlapping match we build a separate dataset with synthetic samples and their respective match score. This dataset is further used to train another NN with the intent of predict the match score for a given waveform sample. This new dataset consists of 85000 synthetic samples, and their matched scores are evaluated using the validation dataset for the equal and unequal mass dataset, each sample consisting of the normalized r Ψ l =2 ,m =2 4 time-series, the value of the match score for the synthetic sample when compared to the real one and the mass M parameter value. This dataset is again randomly shuffled and divided by 80% for training and 20% for validation dataset.</text>
<text><location><page_8><loc_14><loc_14><loc_88><loc_33></location>We use the discriminator architecture as in Table 2, with the inclusion of dropout layers, with 0.2 probability of an element to be zeroed, in between each convolution layer, the dropout layers are included so we can further use the Monte-Carlo dropout method [63] to estimate the predictions' uncertainties. We train this modified discriminator using the mean squared loss since the new task now - to predict the match score for a given sample - is similar to a regression problem. We also take advantage of the transfer learning method and use the weights from the trained discriminators to speed up the training of the new NN. We use the Adam optimizer and this time we train the model using the OneCycle [63] learning rate policy; the model is trained for 11 epochs with an initial learning rate 2 e -3 and uses the root mean squared error as main metric. The final model can predict the match values for a sample with an accuracy around 90%. Using the Monte-Carlo dropout scheme we can further estimate the uncertainty of the model and determine the minimum and maximum predicted values of matches. Figures 12 and 13 display three synthetic samples generated from our model using M values which are not present in the original dataset, but within the one sigma range from the mean M value, and their minimum and maximum predicted match values.</text>
<section_header_level_1><location><page_9><loc_14><loc_88><loc_41><loc_90></location>5 Results and discussions</section_header_level_1>
<text><location><page_9><loc_14><loc_62><loc_88><loc_87></location>The results of our evaluation are shown in Tables 3 and 4. In figures 10 and 11 we sample waveforms and rank them according to their respective matches to their waveform reference, from the validation dataset. We are able to generate waveforms with a 95% to 99% match with the reference waveforms, despite the model never having had access to the samples from the validation dataset. From Tables 3 and 4, we see an interesting and understandable pattern: our model has a higher chance of producing samples which are closer to the expected ones whenever the sample parameters desired are close to the mean of the mass M parameters of the dataset, i.e. the region where we have more samples and, by and large, the model is "learning" the underlying features which appear more often in the data. Nevertheless, in the parameter region away from the expected mean we still have a relatively high chance of producing samples with, at least, 90% match with the expected real samples. Such results show the potential of the method presented in this work. Moreover, one key aspect of such methodology that should be emphasised is the speed of generation of the samples. By using this technique we can generate 1000 synthetic samples in 188 ms 6 , assuming that we can have from 16% to 25% (depending on the mass parameter values) samples which are 95% similar to the expected real waveforms. In other words, we can quickly build a catalogue of waveforms, to bridge the gaps within the parameter space. Such numbers show a clear advantage of the method to help the task of exploring the parameter space for the case of Proca stars waveforms, or more generically, gravitational waveforms generation in any model.</text>
<text><location><page_9><loc_14><loc_50><loc_88><loc_61></location>Although the methodology put forward here shows promising results, one should consider this proposal as a proof of concept, with necessary improvements yet to be done. The evaluation methodology using a NN can be improved by experimenting with different architectures for this task, and use Bayesian methods to estimate the error on the predicted match score. Additionally, the number of samples is a factor that heavily influences the quality of the synthetic sample; a rich dataset with not only more samples but samples with different sample rates can greatly improve the quality of the synthetic samples. New architecture models are already being tested and the Transformer [64] based architectures are showing promising results that will be explored in a future project.</text>
<figure>
<location><page_9><loc_14><loc_23><loc_86><loc_48></location>
<caption>(b) Probabilities of generating samples for a given match accuracy for different mass parameters.</caption>
</figure>
<paragraph><location><page_9><loc_14><loc_16><loc_88><loc_19></location>FIG. 4 . Equal mass dataset, epoch 0. Evaluation of the quality of the generated samples compared to the real samples of r R (Ψ l =2 ,m =2 4 ) and probabilities to generate good quality samples.</paragraph>
<table>
<location><page_10><loc_33><loc_63><loc_69><loc_90></location>
<caption>TABLE 3 . Probabilities for a generator to produce r R (Ψ l =2 ,m =2 4 ) with an overlapping match to the original one above 0.8 (second column), 0.9 (third column) and 0.95 (last column) for each mass ( M = M 1 = M 2 ) and ω values of the validation equal mass dataset. The colors show where the values fall into the 1 and 2 σ mass distribution in figure 3a.</caption>
</table>
<table>
<location><page_10><loc_35><loc_23><loc_67><loc_54></location>
<caption>TABLE 4 . Same as Table 3 but for the unequal mass case providing each mass ( M 1 = M 2 ) and values of the validation dataset. The colors show where the values fall into the 1 and 2 σ mass distribution in figure 3b.</caption>
</table>
<text><location><page_10><loc_75><loc_21><loc_75><loc_22></location>glyph[negationslash]</text>
<figure>
<location><page_11><loc_15><loc_72><loc_51><loc_82></location>
<caption>(a) Real (yellow) and generated (blue) NewmanPenrose scalar Ψ l =2 ,m =2 4 .</caption>
</figure>
<figure>
<location><page_11><loc_55><loc_65><loc_86><loc_88></location>
<caption>(b) Probabilities of generating samples for a given match accuracy, for different mass parameters.</caption>
</figure>
<figure>
<location><page_11><loc_14><loc_21><loc_86><loc_47></location>
<caption>FIG. 5 . Evaluation of the quality of the generated samples compared to the real samples of the Newman-Penrose scalar Ψ l =2 ,m =2 4 and probabilities to generate good quality samples at epoch 0 with a different mass dataset.(b) Probabilities of generating samples for a given match accuracy for different mass parameters.</caption>
</figure>
<paragraph><location><page_11><loc_14><loc_14><loc_88><loc_17></location>FIG. 6 . Evaluation of the quality of the generated samples compared to the real samples of the Newman-Penrose scalar Ψ l =2 ,m =2 4 and probabilities to generate good quality samples at epoch 500.</paragraph>
<figure>
<location><page_12><loc_14><loc_72><loc_51><loc_82></location>
<caption>(a) Real (yellow) and generated (blue) NewmanPenrose scalar Ψ l =2 ,m =2 4 .</caption>
</figure>
<figure>
<location><page_12><loc_55><loc_65><loc_86><loc_88></location>
<caption>(b) Probabilities of generating samples for a given match accuracy for different mass parameters.</caption>
</figure>
<figure>
<location><page_12><loc_14><loc_21><loc_86><loc_47></location>
<caption>FIG. 7 . Evaluation of the quality of the generated samples compared to the real samples of the Newman-Penrose scalar Ψ l =2 ,m =2 4 and probabilities to generate good quality samples at epoch 500 with the different mass dataset.(b) Probabilities of generating samples for a given match accuracy for different mass parameters.</caption>
</figure>
<paragraph><location><page_12><loc_14><loc_14><loc_88><loc_17></location>FIG. 8 . Evaluation of the quality of the generated samples compared to the real samples of the Newman-Penrose scalar Ψ l =2 ,m =2 4 and probabilities to generate good quality samples at epoch 1050.</paragraph>
<figure>
<location><page_13><loc_14><loc_72><loc_51><loc_82></location>
<caption>(a) Real (yellow) and generated (blue) NewmanPenrose scalar Ψ l =2 ,m =2 4 .</caption>
</figure>
<figure>
<location><page_13><loc_55><loc_65><loc_86><loc_88></location>
<caption>(b) Probabilities of generating samples for a given match accuracy for different mass parameters.</caption>
</figure>
<paragraph><location><page_13><loc_14><loc_56><loc_88><loc_60></location>FIG. 9 . Evaluation of the quality of the generated samples compared to the real samples of the Newman-Penrose scalar Ψ l =2 ,m =2 4 and probabilities to generate good quality samples at epoch 1050 with the different mass dataset.</paragraph>
<figure>
<location><page_14><loc_18><loc_74><loc_40><loc_89></location>
</figure>
<figure>
<location><page_14><loc_66><loc_74><loc_84><loc_88></location>
</figure>
<figure>
<location><page_14><loc_18><loc_58><loc_40><loc_72></location>
</figure>
<figure>
<location><page_14><loc_66><loc_58><loc_84><loc_72></location>
</figure>
<figure>
<location><page_14><loc_18><loc_42><loc_40><loc_56></location>
</figure>
<figure>
<location><page_14><loc_18><loc_26><loc_40><loc_40></location>
<caption>M1 = Mz = M: 0.67 match:0.91</caption>
</figure>
<figure>
<location><page_14><loc_44><loc_74><loc_62><loc_89></location>
<caption>M1 match:0.91</caption>
</figure>
<figure>
<location><page_14><loc_44><loc_58><loc_62><loc_72></location>
<caption>M1 = Mz = M: 0.81</caption>
</figure>
<figure>
<location><page_14><loc_44><loc_42><loc_62><loc_56></location>
<caption>match:0.93</caption>
</figure>
<figure>
<location><page_14><loc_66><loc_42><loc_84><loc_56></location>
<caption>match:0.94</caption>
</figure>
<text><location><page_14><loc_72><loc_40><loc_77><loc_40></location>match:0.99</text>
<figure>
<location><page_14><loc_44><loc_26><loc_62><loc_39></location>
</figure>
<figure>
<location><page_14><loc_66><loc_26><loc_84><loc_40></location>
<caption>FIG. 10 . Real (dotted blue) and generated (continuous yellow) r R (Ψ l =2 ,m =2 4 ) for overlapping matches in the regions 0 . 8 ≤ match ≤ 0 . 85 (left column), 0 . 9 ≤ match ≤ 0 . 95 (middle column) and match > 0 . 95 (right column) of the equal mass dataset.</caption>
</figure>
<text><location><page_14><loc_72><loc_56><loc_77><loc_56></location>match:0.97</text>
<figure>
<location><page_15><loc_18><loc_74><loc_40><loc_89></location>
</figure>
<text><location><page_15><loc_28><loc_72><loc_33><loc_73></location>match:0.81</text>
<text><location><page_15><loc_72><loc_72><loc_77><loc_73></location>match:0.98</text>
<figure>
<location><page_15><loc_18><loc_58><loc_40><loc_72></location>
</figure>
<figure>
<location><page_15><loc_44><loc_58><loc_62><loc_72></location>
</figure>
<figure>
<location><page_15><loc_66><loc_58><loc_84><loc_72></location>
</figure>
<text><location><page_15><loc_28><loc_56><loc_34><loc_56></location>match:0.84</text>
<paragraph><location><page_15><loc_50><loc_56><loc_56><loc_56></location>match:0.91</paragraph>
<text><location><page_15><loc_72><loc_56><loc_77><loc_56></location>match:0.99</text>
<figure>
<location><page_15><loc_18><loc_42><loc_40><loc_56></location>
</figure>
<figure>
<location><page_15><loc_44><loc_42><loc_62><loc_56></location>
</figure>
<figure>
<location><page_15><loc_66><loc_42><loc_84><loc_56></location>
</figure>
<figure>
<location><page_15><loc_18><loc_26><loc_40><loc_40></location>
<caption>match:0.92</caption>
</figure>
<text><location><page_15><loc_72><loc_40><loc_77><loc_40></location>match:0.96</text>
<figure>
<location><page_15><loc_44><loc_26><loc_62><loc_40></location>
</figure>
<figure>
<location><page_15><loc_66><loc_26><loc_84><loc_40></location>
<caption>FIG. 11 . Same as figure 11 but for the unequal mass dataset.</caption>
</figure>
<section_header_level_1><location><page_15><loc_14><loc_18><loc_29><loc_19></location>6 Conclusions</section_header_level_1>
<text><location><page_15><loc_14><loc_14><loc_88><loc_16></location>In this work, we have presented a unique application of a particular GAN architecture - WaveGan - in the context of unsupervised gravitational waveforms generation. The model presented here can</text>
<figure>
<location><page_15><loc_44><loc_74><loc_62><loc_88></location>
<caption>M1:0.62, M2:0.65 match:0.90</caption>
</figure>
<figure>
<location><page_15><loc_66><loc_74><loc_84><loc_88></location>
<caption>M1:0.62, M2:0.81 match:0.95</caption>
</figure>
<figure>
<location><page_16><loc_14><loc_73><loc_38><loc_90></location>
</figure>
<text><location><page_16><loc_50><loc_89><loc_51><loc_90></location>M</text>
<text><location><page_16><loc_51><loc_89><loc_55><loc_90></location>:0.7224</text>
<figure>
<location><page_16><loc_42><loc_73><loc_63><loc_89></location>
</figure>
<text><location><page_16><loc_75><loc_89><loc_76><loc_90></location>M</text>
<text><location><page_16><loc_76><loc_89><loc_80><loc_90></location>:0.7796</text>
<figure>
<location><page_16><loc_67><loc_73><loc_88><loc_90></location>
<caption>FIG. 12 . Synthetic samples and their predicted matched samples for parameter values of M not present, but within the one sigma range from the mean, in the current equal mass dataset.</caption>
</figure>
<figure>
<location><page_16><loc_14><loc_51><loc_38><loc_67></location>
</figure>
<text><location><page_16><loc_47><loc_66><loc_48><loc_67></location>M</text>
<text><location><page_16><loc_48><loc_66><loc_53><loc_67></location>1:0.7258,</text>
<text><location><page_16><loc_53><loc_66><loc_54><loc_67></location>M</text>
<text><location><page_16><loc_54><loc_66><loc_58><loc_67></location>2:0.8646</text>
<figure>
<location><page_16><loc_42><loc_51><loc_63><loc_67></location>
</figure>
<text><location><page_16><loc_72><loc_66><loc_72><loc_67></location>M</text>
<text><location><page_16><loc_72><loc_66><loc_77><loc_67></location>1:0.7262,</text>
<text><location><page_16><loc_77><loc_66><loc_78><loc_67></location>M</text>
<text><location><page_16><loc_78><loc_66><loc_83><loc_67></location>2:0.8680</text>
<figure>
<location><page_16><loc_67><loc_51><loc_88><loc_66></location>
<caption>FIG. 13 . Synthetic samples and their predicted matched samples for parameter values of M 1 and M 2 not present, but within the one sigma range from the mean, in the current unequal mass dataset.</caption>
</figure>
<text><location><page_16><loc_14><loc_40><loc_88><loc_44></location>generates hundreds of thousands of waveforms within very short time intervals, labelled by the physical parameters of the Proca stars that sourced them - with equal ( M ) or different ( M 1 , M 2 ) masses with a high probability that such "fake" samples are 95%, or higher, similar to the expected real ones.</text>
<text><location><page_16><loc_14><loc_33><loc_88><loc_39></location>We have also explored the use of the trained discriminator architectures to assist in the task of estimating the overlap score of synthetic samples, which can be used to select the synthetic waveforms which show closer features to the expected real ones. The methods presented here show that it is possible to use such techniques to accelerate the generation of the waveforms, in particular for the case of binaries of exotic compact objects.</text>
<text><location><page_16><loc_14><loc_28><loc_88><loc_32></location>In a future work we plan to extend and refine the method to produce samples with higher quality and automatically assign the overlapping match factor. We are also working on applying such methods in the generation of waveforms from core-collapse supernovae.</text>
<section_header_level_1><location><page_16><loc_14><loc_23><loc_32><loc_25></location>Acknowledgments</section_header_level_1>
<text><location><page_16><loc_14><loc_14><loc_88><loc_22></location>This work is supported by the Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT - Fundação para a Ciência e a Tecnologia), references UIDB/04106/2020 and, UIDP/04106/2020, and by national funds (OE), through FCT, I.P., in the scope of the framework contract foreseen in the numbers 4, 5 and 6 of the article 23, of the Decree-Law 57/2016, of August 29, changed by Law 57/2017, of July 19. This work is also supported by CFTC-UL through FCT, references UIDB/00618/2020</text>
<text><location><page_17><loc_14><loc_73><loc_88><loc_90></location>and UIDP/00618/2020. The author(s) gratefully acknowledges the computer resources at Artemisa, funded by the European Union ERDF and Comunitat Valenciana as well as the technical support provided by the Instituto de Física Corpuscular, IFIC (CSIC-UV). We acknowledge support from the projects PTDC/FIS-OUT/28407/2017, PTDC/FIS-PAR/31000/2017, CERN/FIS-PAR/0027/2019, CERN/FIS-PAR/0002/2019 and PTDC/FIS-AST/3041/2020. This work has further been supported by the European Union's Horizon 2020 research and innovation (RISE) programme H2020-MSCARISE-2017 Grant No. FunFiCO-777740. NSG was also supported by the Spanish Ministerio de Universidades, reference UP2021-044, within the European Union-Next Generation EU. This work is also supported by FCT under contracts UIDB/00618/2020, UIDP/00618/2020, CERN/FISPAR/0002/2017 and CERN/FIS-PAR/0014/2019. This work has also been supported in part by the Swedish Research Council grant, contract number 2016-05996 and by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 668679).</text>
<section_header_level_1><location><page_17><loc_14><loc_69><loc_25><loc_70></location>References</section_header_level_1>
<unordered_list>
<list_item><location><page_17><loc_15><loc_64><loc_84><loc_68></location>[1] LIGO Scientific, Virgo Collaboration, B. P. Abbott et al., GWTC-1: A Gravitational-Wave Transient Catalog of Compact Binary Mergers Observed by LIGO and Virgo during the First and Second Observing Runs , Phys. Rev. X 9 (2019), no. 3 031040, [ 1811.12907 ].</list_item>
<list_item><location><page_17><loc_15><loc_59><loc_85><loc_63></location>[2] LIGO Scientific, Virgo Collaboration, R. Abbott et al., GWTC-2: Compact Binary Coalescences Observed by LIGO and Virgo During the First Half of the Third Observing Run , Phys. Rev. X 11 (2021) 021053, [ 2010.14527 ].</list_item>
<list_item><location><page_17><loc_15><loc_55><loc_86><loc_58></location>[3] LIGO Scientific, VIRGO, KAGRA Collaboration, R. Abbott et al., GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo During the Second Part of the Third Observing Run , 2111.03606 .</list_item>
<list_item><location><page_17><loc_15><loc_50><loc_88><loc_54></location>[4] LIGO Scientific, Virgo Collaboration, B. P. Abbott et al., Binary Black Hole Population Properties Inferred from the First and Second Observing Runs of Advanced LIGO and Advanced Virgo , Astrophys. J. Lett. 882 (2019), no. 2 L24, [ 1811.12940 ].</list_item>
<list_item><location><page_17><loc_15><loc_47><loc_87><loc_49></location>[5] P. Creminelli and F. Vernizzi, Dark Energy after GW170817 and GRB170817A , Phys. Rev. Lett. 119 (2017), no. 25 251302, [ 1710.05877 ].</list_item>
<list_item><location><page_17><loc_15><loc_42><loc_87><loc_46></location>[6] T. Baker, E. Bellini, P. G. Ferreira, M. Lagos, J. Noller, and I. Sawicki, Strong constraints on cosmological gravity from GW170817 and GRB 170817A , Phys. Rev. Lett. 119 (2017), no. 25 251301, [ 1710.06394 ].</list_item>
<list_item><location><page_17><loc_15><loc_39><loc_84><loc_42></location>[7] J. M. Ezquiaga and M. Zumalacárregui, Dark Energy After GW170817: Dead Ends and the Road Ahead , Phys. Rev. Lett. 119 (2017), no. 25 251304, [ 1710.05901 ].</list_item>
<list_item><location><page_17><loc_15><loc_34><loc_87><loc_38></location>[8] J. C. Bustillo, N. Sanchis-Gual, A. Torres-Forné, J. A. Font, A. Vajpeyi, R. Smith, C. Herdeiro, E. Radu, and S. H. W. Leong, GW190521 as a Merger of Proca Stars: A Potential New Vector Boson of 8 . 7 × 10 -13 eV , Phys. Rev. Lett. 126 (2021), no. 8 081101, [ 2009.05376 ].</list_item>
<list_item><location><page_17><loc_15><loc_31><loc_88><loc_34></location>[9] B. J. Owen and B. S. Sathyaprakash, Matched filtering of gravitational waves from inspiraling compact binaries: Computational cost and template placement , Phys. Rev. D 60 (1999) 022002, [ gr-qc/9808076 ].</list_item>
<list_item><location><page_17><loc_14><loc_29><loc_56><loc_31></location>[10] F. Pretorius, Binary Black Hole Coalescence , 0710.1338 .</list_item>
<list_item><location><page_17><loc_14><loc_25><loc_88><loc_29></location>[11] P. Ajith et al., A Template bank for gravitational waveforms from coalescing binary black holes. I. Non-spinning binaries , Phys. Rev. D 77 (2008) 104017, [ 0710.2335 ]. [Erratum: Phys.Rev.D 79, 129901 (2009)].</list_item>
<list_item><location><page_17><loc_14><loc_21><loc_87><loc_24></location>[12] S. A. Usman et al., The PyCBC search for gravitational waves from compact binary coalescence , Class. Quant. Grav. 33 (2016), no. 21 215004, [ 1508.02357 ].</list_item>
<list_item><location><page_17><loc_14><loc_18><loc_88><loc_21></location>[13] M. Boyle et al., The SXS Collaboration catalog of binary black hole simulations , Class. Quant. Grav. 36 (2019), no. 19 195006, [ 1904.04831 ].</list_item>
<list_item><location><page_17><loc_14><loc_14><loc_87><loc_18></location>[14] V. Varma, S. E. Field, M. A. Scheel, J. Blackman, D. Gerosa, L. C. Stein, L. E. Kidder, and H. P. Pfeiffer, Surrogate models for precessing binary black hole simulations with unequal masses , Phys. Rev. Research. 1 (2019) 033015, [ 1905.09300 ].</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_18><loc_14><loc_88><loc_59><loc_90></location>[15] D. J. Kaup, Klein-gordon geon , Phys. Rev. 172 (1968) 1331.</list_item>
<list_item><location><page_18><loc_14><loc_85><loc_87><loc_88></location>[16] R. Ruffini and S. Bonazzola, System of self-gravitating particles in general relativity and the concept of an equation of state , Phys. Rev. 187 (1969) 1767-1783.</list_item>
<list_item><location><page_18><loc_14><loc_82><loc_85><loc_84></location>[17] F. E. Schunck and E. W. Mielke, General relativistic boson stars , Class. Quantum Grav. 20 (2003) R301-R356, [ 0801.0307 ].</list_item>
<list_item><location><page_18><loc_14><loc_78><loc_87><loc_81></location>[18] R. Brito, V. Cardoso, C. A. Herdeiro, and E. Radu, Proca stars: gravitating bose-einstein condensates of massive spin 1 particles , Physics Letters B 752 (2016) 291-295.</list_item>
<list_item><location><page_18><loc_14><loc_75><loc_84><loc_78></location>[19] C. A. R. Herdeiro, A. M. Pombo, and E. Radu, Asymptotically flat scalar, Dirac and Proca stars: discrete vs. continuous families of solutions , Phys. Lett. B 773 (2017) 654-662, [ 1708.05674 ].</list_item>
<list_item><location><page_18><loc_14><loc_72><loc_84><loc_74></location>[20] C. Herdeiro, I. Perapechka, E. Radu, and Y. Shnir, Asymptotically flat spinning scalar, Dirac and Proca stars , Phys. Lett. B 797 (2019) 134845, [ 1906.05386 ].</list_item>
<list_item><location><page_18><loc_14><loc_69><loc_86><loc_71></location>[21] S. L. Liebling and C. Palenzuela, Dynamical Boson Stars , Living Rev.Rel. 15 (2012) 6, [ 1202.5809 ].</list_item>
<list_item><location><page_18><loc_14><loc_65><loc_86><loc_69></location>[22] N. Sanchis-Gual, F. Di Giovanni, M. Zilhão, C. Herdeiro, P. Cerdá-Durán, J. Font, and E. Radu, Nonlinear dynamics of spinning bosonic stars: Formation and stability , Physical Review Letters 123 (2019), no. 22 221101.</list_item>
<list_item><location><page_18><loc_14><loc_61><loc_88><loc_64></location>[23] L. Hui, J. P. Ostriker, S. Tremaine, and E. Witten, Ultralight scalars as cosmological dark matter , Phys. Rev. D 95 (2017), no. 4 043541, [ 1610.08297 ].</list_item>
<list_item><location><page_18><loc_14><loc_58><loc_87><loc_61></location>[24] F. H. Vincent, Z. Meliani, P. Grandclement, E. Gourgoulhon, and O. Straub, Imaging a boson star at the Galactic center , Class. Quant. Grav. 33 (2016), no. 10 105015, [ 1510.04170 ].</list_item>
<list_item><location><page_18><loc_14><loc_53><loc_88><loc_57></location>[25] H. Olivares, Z. Younsi, C. M. Fromm, M. De Laurentis, O. Porth, Y. Mizuno, H. Falcke, M. Kramer, and L. Rezzolla, How to tell an accreting boson star from a black hole , Mon. Not. Roy. Astron. Soc. 497 (2020), no. 1 521-535, [ 1809.08682 ].</list_item>
<list_item><location><page_18><loc_14><loc_50><loc_87><loc_53></location>[26] C. A. R. Herdeiro, A. M. Pombo, E. Radu, P. V. P. Cunha, and N. Sanchis-Gual, The imitation game: Proca stars that can mimic the Schwarzschild shadow , JCAP 04 (2021) 051, [ 2102.01703 ].</list_item>
<list_item><location><page_18><loc_14><loc_47><loc_87><loc_49></location>[27] M. Bezares, C. Palenzuela, and C. Bona, Final fate of compact boson star mergers , Physical Review D 95 (2017), no. 12 124005.</list_item>
<list_item><location><page_18><loc_14><loc_43><loc_87><loc_46></location>[28] M. Bezares and C. Palenzuela, Gravitational waves from dark boson star binary mergers , Classical and Quantum Gravity 35 (2018), no. 23 234002.</list_item>
<list_item><location><page_18><loc_14><loc_39><loc_88><loc_42></location>[29] M. Bezares, M. Bošković, S. Liebling, C. Palenzuela, P. Pani, and E. Barausse, Gravitational waves and kicks from the merger of unequal mass, highly compact boson stars , arXiv preprint arXiv:2201.06113 (2022).</list_item>
<list_item><location><page_18><loc_14><loc_35><loc_87><loc_38></location>[30] N. Sanchis-Gual, C. Herdeiro, J. A. Font, E. Radu, and F. Di Giovanni, Head-on collisions and orbital mergers of proca stars , Physical Review D 99 (2019), no. 2 024017.</list_item>
<list_item><location><page_18><loc_14><loc_32><loc_86><loc_34></location>[31] LIGO Scientific, Virgo Collaboration, R. Abbott et al., GW190521: A Binary Black Hole Merger with a Total Mass of 150 M glyph[circledot] , Phys. Rev. Lett. 125 (2020), no. 10 101102, [ 2009.01075 ].</list_item>
<list_item><location><page_18><loc_14><loc_28><loc_87><loc_31></location>[32] N. Sanchis-Gual and et al., A numerical-relativity gravitational-wave catalogue of proca-star collisions , To appear (2022).</list_item>
<list_item><location><page_18><loc_14><loc_25><loc_88><loc_28></location>[33] I. J. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio, Generative Adversarial Networks , arXiv e-prints (June, 2014) arXiv:1406.2661, [ 1406.2661 ].</list_item>
<list_item><location><page_18><loc_14><loc_22><loc_81><loc_24></location>[34] A. Hu, R. Xie, Z. Lu, A. Hu, and M. Xue, Tablegan-mca: Evaluating membership collisions of gan-synthesized tabular data releasing , CoRR abs/2107.13190 (2021) [ 2107.13190 ].</list_item>
<list_item><location><page_18><loc_14><loc_17><loc_85><loc_21></location>[35] C. Ledig, L. Theis, F. Huszar, J. Caballero, A. P. Aitken, A. Tejani, J. Totz, Z. Wang, and W. Shi, Photo-realistic single image super-resolution using a generative adversarial network , CoRR abs/1609.04802 (2016) [ 1609.04802 ].</list_item>
<list_item><location><page_18><loc_14><loc_14><loc_85><loc_16></location>[36] S. E. Reed, Z. Akata, X. Yan, L. Logeswaran, B. Schiele, and H. Lee, Generative adversarial text to image synthesis , CoRR abs/1605.05396 (2016) [ 1605.05396 ].</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_19><loc_14><loc_86><loc_87><loc_90></location>[37] M. Paganini, L. de Oliveira, and B. Nachman, CaloGAN : Simulating 3D high energy particle showers in multilayer electromagnetic calorimeters with generative adversarial networks , Phys. Rev. D 97 (2018), no. 1 014021, [ 1712.10321 ].</list_item>
<list_item><location><page_19><loc_14><loc_81><loc_87><loc_85></location>[38] L. de Oliveira, M. Paganini, and B. Nachman, Learning Particle Physics by Example: Location-Aware Generative Adversarial Networks for Physics Synthesis , Comput. Softw. Big Sci. 1 (2017), no. 1 4, [ 1701.05927 ].</list_item>
<list_item><location><page_19><loc_14><loc_77><loc_84><loc_81></location>[39] L. de Oliveira, M. Paganini, and B. Nachman, Controlling Physical Attributes in GAN-Accelerated Simulation of Electromagnetic Calorimeters , J. Phys. Conf. Ser. 1085 (2018), no. 4 042017, [ 1711.08813 ].</list_item>
<list_item><location><page_19><loc_14><loc_72><loc_85><loc_76></location>[40] M. Paganini, L. de Oliveira, and B. Nachman, Accelerating Science with Generative Adversarial Networks: An Application to 3D Particle Showers in Multilayer Calorimeters , Phys. Rev. Lett. 120 (2018), no. 4 042003, [ 1705.02355 ].</list_item>
<list_item><location><page_19><loc_14><loc_69><loc_85><loc_72></location>[41] M. Wiese, R. Knobloch, R. Korn, and P. Kretschmer, Quant GANs: Deep Generation of Financial Time Series , arXiv e-prints (July, 2019) arXiv:1907.06673, [ 1907.06673 ].</list_item>
<list_item><location><page_19><loc_14><loc_66><loc_82><loc_69></location>[42] T. Lau, S. Xu, and X. Wang, Video Content Swapping Using GAN , arXiv e-prints (Nov., 2021) arXiv:2111.10916, [ 2111.10916 ].</list_item>
<list_item><location><page_19><loc_14><loc_61><loc_86><loc_65></location>[43] A. van den Oord, S. Dieleman, H. Zen, K. Simonyan, O. Vinyals, A. Graves, N. Kalchbrenner, A. Senior, and K. Kavukcuoglu, WaveNet: A Generative Model for Raw Audio , arXiv e-prints (Sept., 2016) arXiv:1609.03499, [ 1609.03499 ].</list_item>
<list_item><location><page_19><loc_14><loc_57><loc_84><loc_61></location>[44] S. Mehri, K. Kumar, I. Gulrajani, R. Kumar, S. Jain, J. Sotelo, A. C. Courville, and Y. Bengio, Samplernn: An unconditional end-to-end neural audio generation model , CoRR abs/1612.07837 (2016) [ 1612.07837 ].</list_item>
<list_item><location><page_19><loc_14><loc_54><loc_82><loc_56></location>[45] C. Donahue, J. J. McAuley, and M. S. Puckette, Synthesizing audio with generative adversarial networks , CoRR abs/1802.04208 (2018) [ 1802.04208 ].</list_item>
<list_item><location><page_19><loc_14><loc_49><loc_83><loc_53></location>[46] N. Sanchis-Gual, M. Zilhão, C. Herdeiro, F. Di Giovanni, J. A. Font, and E. Radu, Synchronized gravitational atoms from mergers of bosonic stars , Phys. Rev. D 102 (2020), no. 10 101504, [ 2007.11584 ].</list_item>
<list_item><location><page_19><loc_14><loc_47><loc_52><loc_49></location>[47] 'Einstein toolkit: http://www.einsteintoolkit.org.'</list_item>
<list_item><location><page_19><loc_14><loc_43><loc_85><loc_47></location>[48] F. Loffler, J. Faber, E. Bentivegna, T. Bode, P. Diener, et al., The Einstein Toolkit: A Community Computational Infrastructure for Relativistic Astrophysics , Class.Quant.Grav. 29 (2012) 115001, [ 1111.3344 ].</list_item>
<list_item><location><page_19><loc_14><loc_40><loc_83><loc_42></location>[49] M. Zilhão and F. Löffler, An Introduction to the Einstein Toolkit , Int. J. Mod. Phys. A28 (2013) 1340014, [ 1305.5299 ].</list_item>
<list_item><location><page_19><loc_14><loc_37><loc_84><loc_39></location>[50] E. Schnetter, S. H. Hawley, and I. Hawke, Evolutions in 3-D numerical relativity using fixed mesh refinement , Class. Quant. Grav. 21 (2004) 1465-1488, [ gr-qc/0310042 ].</list_item>
<list_item><location><page_19><loc_14><loc_35><loc_44><loc_36></location>[51] 'Cactus: http://www.cactuscode.org..'</list_item>
<list_item><location><page_19><loc_14><loc_31><loc_86><loc_34></location>[52] H. Witek, M. Zilhao, G. Ficarra, and M. Elley, Canuda: a public numerical relativity library to probe fundamental physics , May, 2020.</list_item>
<list_item><location><page_19><loc_14><loc_28><loc_84><loc_31></location>[53] M. Zilhao, H. Witek, and V. Cardoso, Nonlinear interactions between black holes and Proca fields , Class. Quant. Grav. 32 (2015) 234003, [ 1505.00797 ].</list_item>
<list_item><location><page_19><loc_14><loc_25><loc_83><loc_28></location>[54] Y. Bengio, P. Simard, and P. Frasconi, Learning long-term dependencies with gradient descent is difficult , IEEE Transactions on Neural Networks 5 (1994), no. 2 157-166.</list_item>
<list_item><location><page_19><loc_14><loc_22><loc_87><loc_25></location>[55] A. Radford, L. Metz, and S. Chintala, Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks , arXiv e-prints (Nov., 2015) arXiv:1511.06434, [ 1511.06434 ].</list_item>
<list_item><location><page_19><loc_14><loc_19><loc_85><loc_21></location>[56] C. Y. Lee, A. Toffy, G. J. Jung, and W. Han, Conditional wavegan , CoRR abs/1809.10636 (2018) [ 1809.10636 ].</list_item>
<list_item><location><page_19><loc_14><loc_17><loc_82><loc_18></location>[57] A. Odena, V. Dumoulin, and C. Olah, Deconvolution and checkerboard artifacts , Distill (2016).</list_item>
<list_item><location><page_19><loc_14><loc_14><loc_86><loc_16></location>[58] A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. Kopf, E. Yang, Z. DeVito, M. Raison, A. Tejani, S. Chilamkurthy,</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_20><loc_18><loc_86><loc_88><loc_90></location>B. Steiner, L. Fang, J. Bai, and S. Chintala, Pytorch: An imperative style, high-performance deep learning library , in Advances in Neural Information Processing Systems 32 (H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, and R. Garnett, eds.), pp. 8024-8035. Curran Associates, Inc.,</list_item>
<list_item><location><page_20><loc_18><loc_84><loc_21><loc_86></location>2019.</list_item>
<list_item><location><page_20><loc_14><loc_81><loc_81><loc_84></location>[59] I. Gulrajani, F. Ahmed, M. Arjovsky, V. Dumoulin, and A. C. Courville, Improved training of wasserstein gans , CoRR abs/1704.00028 (2017) [ 1704.00028 ].</list_item>
<list_item><location><page_20><loc_14><loc_78><loc_85><loc_81></location>[60] D. P. Kingma and J. Ba, Adam: A Method for Stochastic Optimization , arXiv e-prints (Dec., 2014) arXiv:1412.6980, [ 1412.6980 ].</list_item>
<list_item><location><page_20><loc_14><loc_75><loc_85><loc_78></location>[61] A. Borji, Pros and cons of GAN evaluation measures: New developments , CoRR abs/2103.09396 (2021) [ 2103.09396 ].</list_item>
<list_item><location><page_20><loc_14><loc_68><loc_88><loc_74></location>[62] A. Nitz, I. Harry, D. Brown, C. M. Biwer, J. Willis, T. D. Canton, C. Capano, L. Pekowsky, T. Dent, A. R. Williamson, G. S. Davies, S. De, M. Cabero, B. Machenschalk, P. Kumar, S. Reyes, D. Macleod, dfinstad, F. Pannarale, T. Massinger, M. Tápai, L. Singer, S. Kumar, S. Khan, S. Fairhurst, A. Nielsen, SSingh087, shasvath, B. U. V. Gadre, and I. Dorrington, gwastro/pycbc: Pycbc release v1.16.11 , Oct., 2020.</list_item>
<list_item><location><page_20><loc_14><loc_65><loc_85><loc_67></location>[63] L. N. Smith and N. Topin, Super-Convergence: Very Fast Training of Neural Networks Using Large Learning Rates , arXiv e-prints (Aug., 2017) arXiv:1708.07120, [ 1708.07120 ].</list_item>
<list_item><location><page_20><loc_14><loc_61><loc_88><loc_64></location>[64] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin, Attention Is All You Need , arXiv e-prints (June, 2017) arXiv:1706.03762, [ 1706.03762 ].</list_item>
</unordered_list>
</document>
[
{
"title": "Generating gravitational waveform libraries of exotic compact binaries with deep learning",
"content": "Felipe F. Freitas a,b Carlos A. R. Herdeiro c,b António P. Morais a,b António Onofre d Roman Pasechnik e Eugen Radu c,b Nicolas Sanchis-Gual c,b,f Rui Santos g,h Campus de Santiago, 3810-183 Aveiro, Portugal. Universidade do Minho, 4710-057 Braga, Portugal. Instituto Politécnico de Lisboa 1959-007 Lisboa, Portugal. E-mail: felipefreitas@ua.pt, herdeiro@ua.pt, aapmorais@ua.pt, Antonio.Onofre@cern.ch, roman.pasechnik@thep.lu.se, eugen.radu@ua.pt, nicolas.sanchis@uv.es, rasantos@fc.ul.pt Abstract. Current gravitational wave (GW) detections rely on the existence of libraries of theoretical waveforms. Consequently, finding new physics with GWs requires libraries of non-standard models, which are computationally demanding. We discuss how deep learning frameworks can be used to generate new waveforms \"learned\" from a simulation dataset obtained, say, from numerical relativity simulations. Concretely, we use the WaveGAN architecture of a generative adversarial network (GAN). As a proof of concept we provide this neural network (NN) with a sample of ( > 500 ) waveforms from the collisions of exotic compact objects (Proca stars), obtained from numerical relativity simulations. Dividing the sample into a training and a validation set, we show that after a sufficiently large number of training epochs the NN can produce from 12% to 25% of the synthetic waveforms with an overlapping match of at least 95% with the ones from the validation set. We also demonstrate that a NN can be used to predict the overlapping match score, with 90% of accuracy, of new synthetic samples. These are encouraging results for using GANs for data augmentation and interpolation in the context of GWs, to cover the full parameter space of, say, exotic compact binaries, without the need of intensive numerical relativity simulations.",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "The advent of the gravitational wave (GW) era [1-3] opens new possibilities not only for relativistic astrophysics, cosmology and strong gravity, but also for fundamental physics. Alongside with the unveiling of the population of black holes and neutron stars in the Universe, see e.g. , [4], it is possible that smoking guns about the nature of dark energy, dark matter and even quantum gravity will emerge from this new channel. In fact, particular events have already established concrete illustrations of how dark energy models can be constrained, see e.g. [5-7] and dark matter could be identified, see e.g. [8]. The interpretation of GW signals relies on matched filtering [9]. Therefore, libraries of theoretical templates are mandatory. The construction of such libraries is a non-trivial, time-consuming process. For the vanilla black hole binary problem in (vacuum) general relativity, the construction of full waveforms, including inspiral, merger and ringdown, became under control after the numerical relativity breakthroughs of 2005 - see the review in [10]. However, a dense scanning of the full parameter space (of the black hole binary problem) only with numerical relativity simulations is computationally impossible. Thus, a community effort drove the scanning of the parameter space using state of the art numerical relativity simulations, patched together with approximation methods for both the inspiral and the ringdown - see e.g. [11-14], building approximants for the theoretical waveforms with generic parameters. An under-emphasised caveat of current GW interpretations is the degeneracy problem: can non-standard waveforms fit the data better? Non-standard means waveforms from exotic compact objects, which could either be non-Kerr black holes or horizonless compact objects. Moreover, such exotic compact objects could originate either from general relativity with matter sources or from modified gravity. The difficulty in tackling the above question is, however, the almost complete lack of alternative waveform libraries that can be compared with real events to determine whether the vanilla (Kerr black holes or neutron star binaries) waveforms are indeed the ones selected within a larger library, when employing matched filtering and Bayesian analysis. At the time of writing, the one non-standard model of compact binaries for which there has been a more consistent and successful effort to produce waveforms is the case of bosonic ( i.e. scalar [15-17] or vector [18-20]) stars. The dynamical evolution of these models is theoretically and technically under control [21] and presents a variety of motivations: bosonic stars emerge in sound physical models, can be dynamically robust [21, 22] and have been put forward as \"fuzzy\" dark matter [23] lumps and black hole imitators, e.g. [24-26]. In the context of GWs, several studies of waveforms from collisions and binaries of bosonic stars have been reported, e.g. [27-30]. As an application to the ongoing detections, the massive GW event GW190521 [31] was shown to fit well a collision of two vector bosonic ( a.k.a. as Proca) stars [8]. This effort relied on scanning a library of 89 Proca star collision waveforms (in the meantime enlarged to nearly 800 waveform) [32]. Still, this only scratches the surface of the full parameter space of the model. As such, looking for efficient computational methodologies that can transform a coarse sampling of the parameter space into a dense coverage is of paramount importance. The goal of this paper is to start an exploration of such a methodology using deep learning techniques. Moreover, the method can, in principle, be used for waveforms produced from generic non-standard compact binaries. Thus, the Proca model explicitly discussed herein can be taken both as interesting in its own right, but simultaneously as a proof of concept of the application of the method, illustrating it but not-exhausting it. To be concrete, we shall be making use of Generative Adversarial Networks (GANs) [33], a particular class of deep learning frameworks. GANs can be described as unsupervised methods for mapping low-dimensional latent vectors to high-dimensional data. In our case, this means mapping known waveforms, corresponding to a prior distribution, p model , to a larger space of waveforms, the generated data distribution, p data . In a nutshell, GANs are based on a game-theoretic scenario where we have two networks competing against each other. On the one hand, we have the generator , responsible for mapping the low dimensional vector z into the high dimensional samples we want to reproduce x = g ( z , θ g ) ( i.e. the waveforms in our case). Here, θ g are the parameters from the generator network to be adjusted during the training phase. Competing against the generator we have, on the other hand, the discriminator network, whose sole purpose is to distinguish between samples drawn from the original dataset and samples drawn from the generator. The discriminator provides a probability, d ( x ; θ d ) ∈ [0 ., 1 . ] , of a given sample x being real, as opposed to a fake one drawn from the generator model. Here, θ d are the parameters from the discriminator network to be adjusted during the training phase. The simplest way to describe the learning process of a GAN is a zero-sum game, in which a function L ( θ g , θ d ) determines the payoff of the discriminator. The generator receives -L ( θ g , θ d ) as its own payoff. During the training phase, each player attempts to maximize its own payoff, so that the generator is trained to maximize L ( θ g , θ d ) , whereas the discriminator is trained to minimize it. The original proposal [33] for the function L ( θ g , θ d ) is : where E x ∼ p data and E z ∼ p model are the expected values for a sample to be drawn from the data and the generator, respectively. This drives the discriminator to learn to correctly classify samples as real or fake. Meanwhile, the generator attempts to fool the discriminator by producing fake samples with features as close as possible to the features from real samples. At convergence, the generator's samples are indistinguishable from the real ones, and the discriminator outputs a probability of 50 % for every sample. The discriminator may be discarded or its parameters can be reused for other purposes later on. GANs have shown great success in generating high-quality synthetic images [34-36] indistinguishable from real images. This has encouraged the use of GANs for synthetic data generation in broader contexts, in particular in high-energy physics, where in some instances the data generation can be a computational intensive task [37-40]. In this regard, while GANs were developed for image generation [33], there have been attempts to adapt this approach for other formats, such as tabular data [34], time series [41], video content augmentation [42] and audio synthesis [43-45]. In this article, we shall examine the potential of GANs to produce a larger waveform catalogue from a limited dataset of the corresponding waveforms. We shall focus on the case of waveforms produced by Proca star binaries. For this purpose, we shall modify WaveGAN [45], a GAN initially designed to provide an unsupervised synthesis of raw-waveform audio, such that it could learn and produce Proca waveforms from an initial dataset obtained from numerical relativity simulations. Dividing the sample into a training and a validation set, we show that after a sufficiently large number of training epochs the neural network (NN) can generate synthetic data with at least 95% overlapping match with reference samples from the validation set. This article is organized as follows. In Sec. 2, we briefly review the Proca model of bosonic stars and describe the dataset and methodology explored in this study. We also discuss the issue of waveform normalization. In Sec. 3, we describe the WaveGAN architecture and the training methodology. Then, in Sec. 4 we discuss the evaluation methodology, i.e. how to assess the quality of the generated waveforms and use the trained discriminator architectures to predict the match score for new synthetic samples. Sec. 5 presents our results after applying the chosen architecture, training and evaluation to the initial dataset. Finally, Sec. 6 provides a final discussion on the approach proposed herein.",
"pages": [
2,
3,
4
]
},
{
"title": "2 The Proca model and the dataset",
"content": "The Proca stars, their dynamics and the corresponding GWs will be considered in the simplest model: a complex, massive Proca field minimally coupled to Einstein's gravity. The action reads (with c = 1 = G ) where R is the Ricci scalar of the spacetime metric g , A is a complex 4-potential, with the field strength F αβ = ∂ α A β -∂ β A α , µ > 0 corresponds to the mass of the Proca field, and the overbar denotes complex conjugation. Spinning Proca stars (the fundamental solutions, in the stable branch [22]) can be labelled by their ADM mass, Mµ or, alternatively, by their oscillating frequency ω/µ , both in units of the Proca field mass. In the following, for simplicity, we shall set µ = 1 and label the solutions via M . The fundamental solutions in the stable branch have M and ω in the interval(s) [20]: Note that the upper (lower) limit in the M interval corresponds to the lower (upper) limit in the ω interval. The angular momentum of the solutions is determined by M . For the considered solutions the total angular momentum is in the range [20] J ∈ [0 , 1 . 259] The collision of two Proca stars generates GWs. These are extracted via the Newman-Penrose (complex) scalar Ψ 4 . Both the real ( R (Ψ 4 ) ) and imaginary parts of this scalar (corresponding to the two GW polarizations) can be decomposed into harmonics. The dominant GW modes, i.e. with higher amplitude, have harmonic indices ( l, m ) = (2 , 2) and ( l, m ) = (2 , 0) . For simplicity, we shall consider only the ( l, m ) = (2 , 2) waveforms for each collision, focusing on the real part of the scalar (the \"+\" polarization). Each waveform is a time series for r R (Ψ 4 ) , since Ψ 4 falls as 1 /r , with r being the distance to the source. Our dataset consists of waveforms generated from the merger of two spinning Proca stars with aligned spin axes. This sort of collisions were recently studied in [8, 46]. Although the stars start from rest, due to frame dragging the binary describes an eccentric (rather than precisely head-on) trajectory. The end point depends on the progenitor Proca stars. In the region of the parameter space explored here, the Proca star progenitors are sufficiently massive to trigger black hole formation after the merger. The waveforms are generated from numerical evolutions using the Einstein toolkit infrastructure [47-49], together with the carpet package [50, 51] for mesh-refinement. The Proca evolution equations are solved via a modified Proca thorn [22, 30, 52, 53] to include a complex field. We have performed numerical simulations of equal and unequal mass Proca stars. The initial data consists in the superposition of two equilibrium solutions separated by D = 40 /µ [8], in geometrized units. This guarantees an admissible initial constraint violation. The equilibrium spinning Proca stars are numerically constructed using the solver fidisol/cadsol for non-linear Partial Differential Equations of elliptic type, via a Newton Raphson method - see [18-20] for more details. We divide our data into two sets: glyph[negationslash] In figure 1 and 2 some samples for both data sets are illustrated, for the dominant quadrupolar mode ( l, m ) = (2 , 2) . glyph[negationslash] In figure 3 we display the mass distribution for both datasets (equal mass case in figure 3a and different mass in figure 3b). Both datasets are pre-processed to be sampled at 2048 Hz. Due to the feature that the dataset have different y -ranges we need to normalize them. Having the samples scaled to a similar range helps to prevent or at least mitigate bias and to speed up the optimization process by preventing the model parameter weights to either vanish or explode [54]. We have tested different methods of scaling, including standard scaling 1 , Robust Scaler 2 , the min-max scaler 3 and max-absolute scaler 4 which scale the features into the [ -1 ., 1 . ] range without breaking the sparsity of the dataset. We have chosen to scale the datasets according to the max-absolute, since we want to preserve the sparsity of our dataset. It is important to mention that all features are scaled only after the train/validation split happens, to avoid any bias in the training procedure, and their true amplitude range are stored for a later use to transform back the normalized samples into their original values. Then each dataset is shuffled and split into training (80 % of the total dataset) and validation (20 % of the total dataset) datasets; these are then fed into the NN model, as further explained in the next Section. Each sample consist of a time series representing the real part of the Newman-Penrose scalar Ψ l =2 ,m =2 4 , together with the value of the mass - as show in figure 1 - and the feature scale. glyph[negationslash]",
"pages": [
4,
5,
6
]
},
{
"title": "3 Model architecture and training methodology",
"content": "Inspired by the use GANs for audio generation, we employ WaveGAN [45] to generate new waveforms \"learned\" from our simulations dataset. Our purpose is to test whether this method is useful for data augmentation and data interpolation to cover the full parameter space without the need of intensive computational simulations. The WaveGAN architecture is based on deep convolutional GAN (DCGAN) [55] which popularized the usage of GANs for image synthesis. The DCGAN generator uses the transposed convolution operation to iteratively upsample low-resolution feature maps into high-resolution images. The WaveGAN uses a modified transposed convolution operation to widen its receptive fields 5 . We keep the longer one-dimensional filters of 25, as proposed in [45], however we set the number of layers to 4 and channels to 1 at first, and upsample by a factor of 4 at each layer. The discriminator network is modified in a similar fashion, using length-25 filters. The output length from the generator, as well the input length to the discriminator, is set to 2048, to be the same length as the waveforms samples. The usual GAN generate samples similar to the ones learned in the training. However, this approach is not the most practical if one wants to produce synthetic samples from particular classes present in the dataset, i.e. augment classes or generate samples to interpolate missing regions from the dataset. One way to overcome such problem is to condition our generator and discriminator models. To promote a generator and discriminator to its conditional model forms, one must provide additional information about the training samples, which can be any kind of auxiliary information, such as class labels or, in our case, the mass value M for each sample waveform. We conditioned our WaveGAN using the values of the mass M as labels y and the feature scale max | x i | used to normalize the sample with the intent to restrict the model to generate samples within these parameter constraints. We can perform the conditioning by feeding y and max | x i | into both the discriminator and the generator. We use a similar approach as in [56]. To include the label y and max | x i | , we scale the feature maps output from each hidden layers based on the conditioning representation; in our case we scale the feature maps by the mass and feature scale values provided by the sample labels. It is important to mention that this approach is applicable in our case due to the low variance of our labels. In order to deal with high variance labels the best approach is to either normalize them or encode it using a linear layer. Meanwhile, the scale factors max | x i | help us to constrain the amplitude scale of the synthetic samples, in a sense that when we produce the new samples their amplitude values will be within a region allowed by their physical parameters in the dataset. This is required in order to avoid producing samples which are not permitted by physics or artefacts that can be produced by such methods [57]. Our WaveGAN is implemented in PyTorch[58], and we train our model for 1050 epochs using WGAN-GP [59] strategy, with ADAM [60] as an optimizer, for both generator and discriminator, with learning rate of 10 -4 for the generator and 3 × 10 -4 for the discriminator. We train our networks using batches of size 32, while the validation set has batches of size 16, on a single GPU NVIDIA Tesla V100. As a first task, we set the generators and discriminators to one channel in order to generate the synthetic r Ψ l =2 ,m =2 modes for the equal and different masses datasets. The results for equal and unequal mass datasets are presented in Section 5.",
"pages": [
6,
7
]
},
{
"title": "4 Evaluation methodology",
"content": "The evaluation of generative models is an ongoing topic in the community [61]. Just as important as choosing the right strategy to train a generative model, is selecting the right metric to evaluate the quality of the generated samples. A direct comparison between the synthetic samples and the real ones can be a useful diagnostic, often allowing us to build intuition of how the generative model is working, how it is failing and how it can be improved. However, qualitative as well quantitative analysis based on this approach can be misleading about the performance of the generator. In order to evaluate the quality, and therefore how trustworthy is our generator, we shall employ the following strategy. Using the PyCBC [62] matched filtering module, we estimate the overlap over time and phase between the synthetic and real samples for a given value of the parameters. We generate a set of 1000 synthetic samples for a given set of parameters, ( M ) for equal mass dataset or ( M 1 , M 2 ) for unequal mass dataset, and compute the overlapping match for each sample to the real equivalent samples. The overlapping match is computed with the real and synthetic normalized samples, so we can ensure that features generated for the synthetic samples are as close as possible to the expected real features. With these values, we estimate the probability of a generated sample to be above a certain threshold of match. In figures 4-9 we show the evolution of the match between synthetic and original samples throughout the training epochs of our NN model. To visualize the overlapping between the real and synthetic samples, we plot various samples from the equal and different mass datasets and select synthetic samples according to their overlapping matches to check against the real ones; these plots are shown in figure 10 and figure 11. Using the overlapping match we build a separate dataset with synthetic samples and their respective match score. This dataset is further used to train another NN with the intent of predict the match score for a given waveform sample. This new dataset consists of 85000 synthetic samples, and their matched scores are evaluated using the validation dataset for the equal and unequal mass dataset, each sample consisting of the normalized r Ψ l =2 ,m =2 4 time-series, the value of the match score for the synthetic sample when compared to the real one and the mass M parameter value. This dataset is again randomly shuffled and divided by 80% for training and 20% for validation dataset. We use the discriminator architecture as in Table 2, with the inclusion of dropout layers, with 0.2 probability of an element to be zeroed, in between each convolution layer, the dropout layers are included so we can further use the Monte-Carlo dropout method [63] to estimate the predictions' uncertainties. We train this modified discriminator using the mean squared loss since the new task now - to predict the match score for a given sample - is similar to a regression problem. We also take advantage of the transfer learning method and use the weights from the trained discriminators to speed up the training of the new NN. We use the Adam optimizer and this time we train the model using the OneCycle [63] learning rate policy; the model is trained for 11 epochs with an initial learning rate 2 e -3 and uses the root mean squared error as main metric. The final model can predict the match values for a sample with an accuracy around 90%. Using the Monte-Carlo dropout scheme we can further estimate the uncertainty of the model and determine the minimum and maximum predicted values of matches. Figures 12 and 13 display three synthetic samples generated from our model using M values which are not present in the original dataset, but within the one sigma range from the mean M value, and their minimum and maximum predicted match values.",
"pages": [
7,
8
]
},
{
"title": "5 Results and discussions",
"content": "The results of our evaluation are shown in Tables 3 and 4. In figures 10 and 11 we sample waveforms and rank them according to their respective matches to their waveform reference, from the validation dataset. We are able to generate waveforms with a 95% to 99% match with the reference waveforms, despite the model never having had access to the samples from the validation dataset. From Tables 3 and 4, we see an interesting and understandable pattern: our model has a higher chance of producing samples which are closer to the expected ones whenever the sample parameters desired are close to the mean of the mass M parameters of the dataset, i.e. the region where we have more samples and, by and large, the model is \"learning\" the underlying features which appear more often in the data. Nevertheless, in the parameter region away from the expected mean we still have a relatively high chance of producing samples with, at least, 90% match with the expected real samples. Such results show the potential of the method presented in this work. Moreover, one key aspect of such methodology that should be emphasised is the speed of generation of the samples. By using this technique we can generate 1000 synthetic samples in 188 ms 6 , assuming that we can have from 16% to 25% (depending on the mass parameter values) samples which are 95% similar to the expected real waveforms. In other words, we can quickly build a catalogue of waveforms, to bridge the gaps within the parameter space. Such numbers show a clear advantage of the method to help the task of exploring the parameter space for the case of Proca stars waveforms, or more generically, gravitational waveforms generation in any model. Although the methodology put forward here shows promising results, one should consider this proposal as a proof of concept, with necessary improvements yet to be done. The evaluation methodology using a NN can be improved by experimenting with different architectures for this task, and use Bayesian methods to estimate the error on the predicted match score. Additionally, the number of samples is a factor that heavily influences the quality of the synthetic sample; a rich dataset with not only more samples but samples with different sample rates can greatly improve the quality of the synthetic samples. New architecture models are already being tested and the Transformer [64] based architectures are showing promising results that will be explored in a future project. glyph[negationslash] match:0.99 match:0.97 match:0.81 match:0.98 match:0.84 match:0.99 match:0.96",
"pages": [
9,
10,
14,
15
]
},
{
"title": "6 Conclusions",
"content": "In this work, we have presented a unique application of a particular GAN architecture - WaveGan - in the context of unsupervised gravitational waveforms generation. The model presented here can M :0.7224 M :0.7796 M 1:0.7258, M 2:0.8646 M 1:0.7262, M 2:0.8680 generates hundreds of thousands of waveforms within very short time intervals, labelled by the physical parameters of the Proca stars that sourced them - with equal ( M ) or different ( M 1 , M 2 ) masses with a high probability that such \"fake\" samples are 95%, or higher, similar to the expected real ones. We have also explored the use of the trained discriminator architectures to assist in the task of estimating the overlap score of synthetic samples, which can be used to select the synthetic waveforms which show closer features to the expected real ones. The methods presented here show that it is possible to use such techniques to accelerate the generation of the waveforms, in particular for the case of binaries of exotic compact objects. In a future work we plan to extend and refine the method to produce samples with higher quality and automatically assign the overlapping match factor. We are also working on applying such methods in the generation of waveforms from core-collapse supernovae.",
"pages": [
15,
16
]
},
{
"title": "Acknowledgments",
"content": "This work is supported by the Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT - Fundação para a Ciência e a Tecnologia), references UIDB/04106/2020 and, UIDP/04106/2020, and by national funds (OE), through FCT, I.P., in the scope of the framework contract foreseen in the numbers 4, 5 and 6 of the article 23, of the Decree-Law 57/2016, of August 29, changed by Law 57/2017, of July 19. This work is also supported by CFTC-UL through FCT, references UIDB/00618/2020 and UIDP/00618/2020. The author(s) gratefully acknowledges the computer resources at Artemisa, funded by the European Union ERDF and Comunitat Valenciana as well as the technical support provided by the Instituto de Física Corpuscular, IFIC (CSIC-UV). We acknowledge support from the projects PTDC/FIS-OUT/28407/2017, PTDC/FIS-PAR/31000/2017, CERN/FIS-PAR/0027/2019, CERN/FIS-PAR/0002/2019 and PTDC/FIS-AST/3041/2020. This work has further been supported by the European Union's Horizon 2020 research and innovation (RISE) programme H2020-MSCARISE-2017 Grant No. FunFiCO-777740. NSG was also supported by the Spanish Ministerio de Universidades, reference UP2021-044, within the European Union-Next Generation EU. This work is also supported by FCT under contracts UIDB/00618/2020, UIDP/00618/2020, CERN/FISPAR/0002/2017 and CERN/FIS-PAR/0014/2019. This work has also been supported in part by the Swedish Research Council grant, contract number 2016-05996 and by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 668679).",
"pages": [
16,
17
]
}
]
2024PhRvD.110b3541B
https://arxiv.org/pdf/2401.16910.pdf
<document>
<section_header_level_1><location><page_1><loc_17><loc_92><loc_84><loc_93></location>Production of ultralight dark matter from inflationary spectator fields</section_header_level_1>
<text><location><page_1><loc_31><loc_89><loc_69><loc_90></location>Alessio Belfiglio 1, 2, ∗ and Orlando Luongo 1, 2, 3, 4, 5, †</text>
<text><location><page_1><loc_30><loc_87><loc_71><loc_88></location>1 School of Science and Technology, University of Camerino,</text>
<text><location><page_1><loc_32><loc_86><loc_68><loc_87></location>Via Madonna delle Carceri, Camerino, 62032, Italy.</text>
<text><location><page_1><loc_20><loc_85><loc_80><loc_86></location>2 Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Perugia, Perugia, 06123, Italy.</text>
<text><location><page_1><loc_30><loc_83><loc_71><loc_84></location>3 SUNY Polytechnic Institute, 13502 Utica, New York, USA.</text>
<text><location><page_1><loc_30><loc_82><loc_71><loc_83></location>4 INAF - Osservatorio Astronomico di Brera, Milano, Italy.</text>
<text><location><page_1><loc_21><loc_81><loc_80><loc_82></location>5 Al-Farabi Kazakh National University, Al-Farabi av. 71, 050040 Almaty, Kazakhstan.</text>
<text><location><page_1><loc_18><loc_64><loc_83><loc_79></location>We investigate inflationary particle production associated with a spectator ultralight scalar field, which has been recently proposed as a plausible dark matter candidate. In this framework, we select the Starobinsky potential to drive the inflationary epoch, also discussing the case of a nonminimally coupled inflaton field fueled by a quartic symmetry-breaking potential. We focus on particle production arising from spacetime perturbations, which are induced by inflaton fluctuations during the quasi-de Sitter stage of inflation. In particular, we construct the first order Lagrangian describing interaction between inhomogeneities and the spectator field, quantifying superhorizon particle production during slow-roll. We then compare this mechanism with gravitational particle production associated with an instantaneous transition from inflation to the radiation dominated era. We show that the amount of particles obtained from perturbations is typically non-negligible and it is significantly enhanced on super-Hubble scales by the nonadiabatic inflationary expansion. Possible implications for primordial entanglement generation are also debated.</text>
<text><location><page_1><loc_18><loc_61><loc_51><loc_62></location>PACS numbers: 04.62.+v, 98.80.-k, 98.80.Cq, 03.67.bg</text>
<section_header_level_1><location><page_1><loc_20><loc_57><loc_37><loc_58></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_47><loc_49><loc_55></location>Dark matter (DM) is undoubtedly a key ingredient to explain the cosmological large-scale dynamics and clustering [1]. Its nature, however, remains mysterious: we are still lacking any conclusive experiment able to unambiguously identify its properties, with slight agreement among the plethora of theoretical proposals [2].</text>
<text><location><page_1><loc_9><loc_37><loc_49><loc_46></location>Within this scenario, recent efforts in the search for weakly interacting massive particles from a few up to 100 GeV have unfortunately proved unsuccessful [3-6]. Thus, this lack of evidence has also revived the interest in ultralight DM candidates such as axions [7-12], axionlike particles [13-15], 'fuzzy' DM models [16-19] and so on.</text>
<text><location><page_1><loc_9><loc_23><loc_49><loc_36></location>In this respect, several treatments have been proposed to explain the origin of DM particles. Among them, gravitational particle production (GPP) represents a plausible and widely-investigated approach [20-23] as it creates particles directly from vacuum fluctuations and does not require any coupling between DM and generic quantum fields 1 . For this reason, GPP of DM has been studied in various cosmological contexts, with particular interest on inflationary [24-28] and reheating [29-31] phases.</text>
<text><location><page_1><loc_9><loc_19><loc_49><loc_23></location>Gravitational production of ultralight particles has been recently discussed in Ref. [32], focusing on the dynamics of a spectator scalar field in the transition be-</text>
<text><location><page_1><loc_52><loc_44><loc_92><loc_59></location>tween inflation and the radiation dominated era. There, assuming an instantaneous transition, and thus neglecting the details of reheating, a significant particle production may take place for super-Hubble wavelength modes after inflation, if the field starts from the Bunch-Davies vacuum [33-35] and it is minimally coupled to spacetime curvature. Specifically, the matching conditions between the two epochs provide the Bogoliubov coefficients from which one obtains the number density of produced particles, compatible with a cold ultralight DM candidate 2 .</text>
<text><location><page_1><loc_52><loc_25><loc_92><loc_44></location>This approach, however, neglects the slow-roll of the inflaton field and the dynamics of its quantum fluctuations, which represent the fundamental seeds for structure formation in our universe [36-39]. In fact, inflationary fluctuations induce perturbations on the de Sitter dynamics of background, and the presence of inhomogeneities, by virtue of expansion, may result in the production of additional 'geometric' particles, due to purely gravitational effects [40, 41]. In Ref. [42], it was argued that DM could be reinterpreted as geometric quasiparticles, in the attempt of solving the cosmological constant problem [43], extending a mechanism of direct cancellation between fields [44, 45].</text>
<text><location><page_1><loc_52><loc_19><loc_92><loc_24></location>Thus, at least two more points require additional investigations. First, how the nonminimal coupling acts on the inflationary dynamics and particle production, see e.g. Refs. [46, 47], and, second, whether gravitational</text>
<text><location><page_2><loc_9><loc_86><loc_49><loc_93></location>production furnishes stable particles, or more broadly, stable quasi-particles, since their dynamics at the end of the slow-roll regime can be altered by possible couplings of the inflaton to other quantum fields, as expected in the standard picture of reheating [48-50].</text>
<text><location><page_2><loc_9><loc_65><loc_49><loc_85></location>For the above reasons, we here aim to generalize inflationary geometric production to the case of a spectator scalar field. Assuming a quasi-de Sitter background evolution to properly account for the slow-roll of the inflaton, we show how spacetime perturbations generated by inflaton fluctuations couple to the energy-momentum tensor of a given spectator field, leading to particle creation during the slow-roll regime. We focus in particular on the Starobinsky model of inflation [51-53], also discussing the case of a nonminimally coupled inflaton field driven by a quartic potential [54, 55]. We single out these paradigms since both the above models are among the best options to describe inflation, as certified by the Planck satellite measurements [56].</text>
<text><location><page_2><loc_9><loc_26><loc_49><loc_65></location>We observe that the amount of created particles depends on the mass of the spectator field and on the details of its coupling to the background. In particular, the presence of a small but still non-negligible coupling to the scalar curvature of spacetime is able to produce a significant amount of particles for super-Hubble modes 3 . Low-momentum enhancement of particle production is a peculiar trait of bosonic fields, and it is similarly found in unperturbed GPP scenarios. However, the presence of inhomogeneities allows for mode-mixing in particle production 4 , with the Hubble radius emerging as the natural separation scale for modes during inflation. Consequently, motivated by these facts, we investigate particle production across the Hubble horizon, showing that a perturbative treatment is possible for super-Hubble modes that crossed the horizon well before the end of inflation. Since this approach was recently employed to study the entropy of cosmological perturbations [59-62], we accordingly work the inclusion of geometric and perturbative effects in primordial particle creation mechanisms out, showing that this is not only required for computing the correct abundance of DM candidates, but it may also shed further light on the quantum properties and entropy associated with the created particles. Physical consequences of our approach are promising, confirming that the nature of DM may arise from a spectator field, subdominant throughout the inflationary evolution.</text>
<text><location><page_2><loc_9><loc_23><loc_49><loc_25></location>The work is organized as follows. In Sec. II, we discuss the features of the spectator DM field, computing the</text>
<text><location><page_2><loc_52><loc_82><loc_92><loc_93></location>Bogoliubov coefficients related to GPP. In Sec. III, we focus on the geometric contribution to GPP. In Sec. IV, we analyze the main consequences of our results throughout inflationary stages, whereas in Sec. V we emphasize the main results inferred from our findings, highlighting possible quantum signatures, detectable at late-times. Conclusions and perspectives are reported in Sec. VI.</text>
<section_header_level_1><location><page_2><loc_57><loc_76><loc_87><loc_77></location>II. SPECTATOR FIELD DYNAMICS</section_header_level_1>
<text><location><page_2><loc_52><loc_66><loc_92><loc_74></location>We assume that, besides the inflaton, a subdominant field is present throughout the inflationary phase, with no interaction with the inflaton itself. Consequently, we exclusively consider gravitational interactions, disregarding potential couplings to other quantum fields both during inflation and the subsequent radiation era.</text>
<text><location><page_2><loc_52><loc_63><loc_92><loc_65></location>In this respect, we consider the Lagrangian density for the spectator, φ , with mass m ,</text>
<formula><location><page_2><loc_58><loc_59><loc_92><loc_62></location>L S = 1 2 [ g µν φ ,µ φ ,ν -( m 2 + ξ φ R ) φ 2 ] , (1)</formula>
<text><location><page_2><loc_52><loc_52><loc_92><loc_58></location>where, as above stated, the only interaction is with the background, i.e., we include a nonminimal coupling between the curvature and φ , while the subscript 'S' indicates the nature of φ .</text>
<text><location><page_2><loc_52><loc_48><loc_92><loc_52></location>Further, in Eq. (1), g is the spacetime metric determinant, while ξ φ describes the coupling strength between the spectator field and the Ricci background scalar, R .</text>
<text><location><page_2><loc_52><loc_35><loc_92><loc_48></location>To claim that φ represents a spectator field, we require that its energy density is sufficiently small not to affect the inflationary dynamics. Accordingly, its evolution during the slow-roll phase is solely determined by the background potential driving inflation. To this end, we now describe the inflaton dynamics, then selecting the corresponding inflationary potentials in fulfillment of the most recent developments provided by the Planck satellite measurements [56].</text>
<section_header_level_1><location><page_2><loc_57><loc_31><loc_87><loc_32></location>A. Setting up the inflationary scenario</section_header_level_1>
<text><location><page_2><loc_52><loc_25><loc_92><loc_29></location>During inflation, the background evolution is governed by the inflaton field, ϕ , which we assume of scalar nature. The corresponding Lagrangian density reads</text>
<formula><location><page_2><loc_63><loc_21><loc_92><loc_23></location>L I = 1 2 g µν ϕ ,µ ϕ ,ν -V ( ϕ ) , (2)</formula>
<text><location><page_2><loc_52><loc_16><loc_92><loc_20></location>where the potential V ( ϕ ) dominates over the other species and the subscript 'I' denotes the inflationary epoch.</text>
<text><location><page_2><loc_52><loc_11><loc_92><loc_15></location>The inflationary standard paradigm predicts the seeds for gravitational small perturbations, induced by perturbing the inflaton through the standard ansatz [36, 37]</text>
<formula><location><page_2><loc_62><loc_9><loc_92><loc_10></location>ϕ ( x , τ ) = ϕ 0 ( τ ) + δϕ ( x , τ ) , (3)</formula>
<text><location><page_3><loc_9><loc_88><loc_49><loc_93></location>where the homogeneous background term, ϕ 0 , is distinct from its corresponding quantum fluctuations, denoted by δϕ , depending on the position and conformal time, τ = ∫ dt/a ( t ), with t the measurable cosmic time.</text>
<text><location><page_3><loc_9><loc_82><loc_49><loc_87></location>The inflaton background field, hereafter denoted by ϕ instead of ϕ 0 , for simplicity, speeds the universe up by virtue of a quasi-de Sitter phase , yielding the unperturbed metric tensor,</text>
<formula><location><page_3><loc_23><loc_79><loc_49><loc_81></location>g µν = a 2 ( τ ) η µν , (4)</formula>
<text><location><page_3><loc_9><loc_75><loc_49><loc_78></location>where η µν is the Minkowski metric tensor and we assume [32]</text>
<formula><location><page_3><loc_20><loc_71><loc_49><loc_74></location>a ( τ ) = -1 H I ( τ -2 τ R ) 1+ ϵ . (5)</formula>
<text><location><page_3><loc_9><loc_60><loc_49><loc_69></location>At this stage, Eq. (5) deserves some additional comments. Particularly, the time τ R describes transition to the radiation dominated era, while H I provides the Hubble parameter during inflation, up to corrections of first order. Moreover, ϵ represents slight deviations from a purely de Sitter phase. By calculating the slow-roll parameter, we can precisely identify it with the latter.</text>
<text><location><page_3><loc_9><loc_55><loc_49><loc_59></location>As stated above, the presence of inflaton fluctuations induces perturbations on the background spacetime, leading to the perturbed metric tensor</text>
<formula><location><page_3><loc_15><loc_52><loc_49><loc_54></location>g µν = a 2 ( τ ) ( η µν + h µν ) , | h µν | ≪ 1 . (6)</formula>
<text><location><page_3><loc_9><loc_47><loc_49><loc_51></location>Selecting now the longitudinal, or conformal Newtonian gauge [38], it can be shown that scalar perturbations associated with ϕ become particularly simple,</text>
<formula><location><page_3><loc_19><loc_44><loc_49><loc_46></location>h µν = diag (2Ψ , 2Ψ , 2Ψ , 2Ψ) , (7)</formula>
<text><location><page_3><loc_9><loc_42><loc_43><loc_43></location>where the perturbation potential Ψ satisfies [39]</text>
<formula><location><page_3><loc_22><loc_38><loc_49><loc_40></location>Ψ ' + H Ψ = ϵ H 2 δϕ ϕ ' , (8)</formula>
<text><location><page_3><loc_9><loc_34><loc_49><loc_36></location>with H = a ' /a and the prime denoting derivative with respect to conformal time.</text>
<text><location><page_3><loc_9><loc_28><loc_49><loc_33></location>Before studying spacetime perturbations for some specific models of inflation and computing the corresponding particle production, we briefly discuss the evolution of the spectator field, focusing on</text>
<unordered_list>
<list_item><location><page_3><loc_12><loc_25><loc_42><loc_26></location>- its dynamics during the slow-roll regime,</list_item>
<list_item><location><page_3><loc_12><loc_21><loc_49><loc_24></location>- the subsequent transition of this field to the radiation era.</list_item>
</unordered_list>
<text><location><page_3><loc_9><loc_9><loc_49><loc_20></location>We underline that, by assuming an instantaneous transition to the radiation era, we are neglecting the effects of reheating on GPP. Nevertheless, we will see that superHubble modes feature very slow dynamics at the end of inflation, so they are typically unaffected by the microphysical processes of thermalization that should take place after the slow-roll regime [32]. It is clear, however, that this picture induces an approximation that</text>
<text><location><page_3><loc_52><loc_83><loc_92><loc_93></location>may overestimate the production itself. A more comprehensive analysis needs to take into account more refined matching conditions and the effects of backreaction, which will be both subject of future investigations. For the moment, we focus on the slow-roll dynamics of the spectator field and the transition occurring between inflation and radiation epochs.</text>
<section_header_level_1><location><page_3><loc_55><loc_79><loc_89><loc_80></location>B. Spectator field dynamics during inflation</section_header_level_1>
<text><location><page_3><loc_52><loc_74><loc_92><loc_77></location>Following standard approaches [21, 32], we consider the conformally rescaled spectator field,</text>
<formula><location><page_3><loc_64><loc_72><loc_92><loc_73></location>χ ( x , τ ) = a ( τ ) φ ( x , τ ) , (9)</formula>
<text><location><page_3><loc_52><loc_69><loc_61><loc_70></location>quantized by</text>
<formula><location><page_3><loc_52><loc_63><loc_92><loc_68></location>ˆ χ ( x , τ ) = 1 (2 π ) 3 / 2 ∫ d 3 k [ ˆ a k g k ( τ ) e -i k · x +ˆ a † k g ∗ k ( τ ) e i k · x ] , (10)</formula>
<text><location><page_3><loc_52><loc_60><loc_92><loc_63></location>where we introduce the comoving momentum, k , and the field modes, g k ( τ ), satisfying the differential equation,</text>
<formula><location><page_3><loc_54><loc_56><loc_92><loc_59></location>g '' k ( τ ) + [ k 2 + m 2 a 2 -a '' a (1 -6 ξ ) ] g k ( τ ) = 0 . (11)</formula>
<text><location><page_3><loc_52><loc_54><loc_61><loc_55></location>Defining now</text>
<formula><location><page_3><loc_60><loc_49><loc_92><loc_52></location>g k ( τ ) = { g < k ( τ ) for τ < τ R , g > k ( τ ) for τ > τ R , (12)</formula>
<text><location><page_3><loc_52><loc_45><loc_92><loc_47></location>we recall the ansatz of Eq. (5) to obtain the mode evolution during inflation, namely</text>
<formula><location><page_3><loc_53><loc_39><loc_92><loc_43></location>d 2 dη 2 g < k + [ k 2 -1 η 2 ( (1 -6 ξ )(2 + 3 ϵ ) -m 2 H 2 I )] g < k = 0 , (13)</formula>
<text><location><page_3><loc_52><loc_34><loc_92><loc_39></location>where η = τ -2 τ R . Notice that we exploit the fact that a '' /a ≃ (2 + 3 ϵ ) /η 2 , since ϵ ≪ 1 throughout the slow-roll phase.</text>
<text><location><page_3><loc_53><loc_33><loc_92><loc_34></location>The solutions of Eq. (13) can be expressed in the form</text>
<formula><location><page_3><loc_54><loc_28><loc_92><loc_32></location>g < k ( η ) = √ -η [ c 1 ( k ) H (1) ν ( -kη ) + c 2 ( k ) H (2) ν ( -kη ) ] , (14)</formula>
<text><location><page_3><loc_52><loc_26><loc_85><loc_28></location>where H (1) ν and H (2) ν are Hankel functions and</text>
<formula><location><page_3><loc_60><loc_21><loc_92><loc_25></location>ν = √ 1 4 +(1 -6 ξ )(2 + 3 ϵ ) -m 2 H 2 I . (15)</formula>
<text><location><page_3><loc_52><loc_13><loc_92><loc_20></location>The integration constants, c 1 ( k ) and c 2 ( k ), are determined by choosing the in vacuum state for the field. A common ansatz consists in employing the Bunch-Davies vacuum state [33-35], that requires the asymptotic condition,</text>
<formula><location><page_3><loc_65><loc_8><loc_92><loc_12></location>g k ( η ) -----→ η →-∞ e -ikη √ 2 k . (16)</formula>
<text><location><page_4><loc_9><loc_90><loc_49><loc_94></location>This choice implies c 1 ( k ) = √ πe i ( ν + 1 2 ) π 2 / 2 and c 2 ( k ) = 0, so the rescaled field modes take the form</text>
<formula><location><page_4><loc_16><loc_87><loc_49><loc_90></location>g < k ( η ) = √ -πη 2 e i ( ν + 1 2 ) π 2 H (1) ν ( -kη ) . (17)</formula>
<text><location><page_4><loc_9><loc_80><loc_49><loc_86></location>Exploiting the asymptotic behaviour of Hankel functions, one can show that on super-Hubble scales, k ≪ aH I , the original field modes are nearly frozen, while they oscillate on sub-Hubble scales, k ≫ aH I [39, 58].</text>
<section_header_level_1><location><page_4><loc_10><loc_76><loc_48><loc_77></location>C. The spectator field transition to radiation era</section_header_level_1>
<text><location><page_4><loc_9><loc_68><loc_49><loc_74></location>At τ = τ R , we assume an instantaneous transition from inflation to the radiation dominated phase, whose dynamics is still described by the metric tensor of Eq. (4), with a ( τ ) = H R τ .</text>
<text><location><page_4><loc_10><loc_67><loc_49><loc_68></location>From the continuity of the scale factor at τ R , we have 5</text>
<formula><location><page_4><loc_22><loc_63><loc_49><loc_66></location>1 H I ( τ R ) 1+ ϵ = H R τ R , (18)</formula>
<text><location><page_4><loc_9><loc_60><loc_37><loc_61></location>that, under the assumption ϵ ≪ 1, gives</text>
<formula><location><page_4><loc_24><loc_56><loc_49><loc_59></location>τ R ≃ 1 √ H I H R , (19)</formula>
<text><location><page_4><loc_9><loc_52><loc_49><loc_55></location>where H R ≃ 10 -35 eV [32] and H I can be derived by fixing the energy scales of inflation.</text>
<text><location><page_4><loc_10><loc_51><loc_48><loc_52></location>During the radiation era, from Eq. (11) we can write</text>
<formula><location><page_4><loc_17><loc_46><loc_49><loc_49></location>d 2 dτ 2 g > k + [ k 2 + m 2 H 2 R τ 2 ] g > k = 0 , (20)</formula>
<text><location><page_4><loc_9><loc_41><loc_49><loc_45></location>which is solved in terms of parabolic cylinder functions. The general solution of Eq. (20) can be expressed in the form</text>
<formula><location><page_4><loc_19><loc_38><loc_49><loc_40></location>g > k ( τ ) = α k f k ( τ ) + β k f ∗ k ( τ ) , (21)</formula>
<text><location><page_4><loc_9><loc_33><loc_49><loc_37></location>where α k , β k are known as Bogoliubov coefficients. The modes f k ( τ ) satisfy Eq. (20) with asymptotic boundary condition</text>
<formula><location><page_4><loc_18><loc_29><loc_49><loc_32></location>f k ( τ ) -----→ τ →-∞ e -i ∫ τ ω k ( τ ' ) dτ ' √ 2 ω k ( τ ) , (22)</formula>
<text><location><page_4><loc_9><loc_23><loc_49><loc_27></location>where ω k ( τ ) = √ k 2 + m 2 H 2 R τ 2 . In order to properly define the notion of particle and vacuum state during the radiation phase, the adiabatic condition</text>
<formula><location><page_4><loc_25><loc_19><loc_49><loc_22></location>ω ' k ( τ ) ω 2 k ( τ ) ≪ 1 . (23)</formula>
<text><location><page_4><loc_52><loc_89><loc_92><loc_93></location>should be satisfied [21]. An upper bound to this ratio is given by modes with negligible momentum, for which the adiabatic approximation gives</text>
<formula><location><page_4><loc_66><loc_86><loc_92><loc_88></location>a ( τ ) ≫ √ H R /m. (24)</formula>
<text><location><page_4><loc_52><loc_81><loc_92><loc_85></location>It can be shown that this condition is verified well before matter-radiation equality, even in case of ultralight DM candidates with m ≪ 1 eV [64].</text>
<text><location><page_4><loc_52><loc_75><loc_92><loc_79></location>Accordingly, we can properly associate out particle states to the modes f k ( τ ), which are normalized via the Wronskian condition</text>
<formula><location><page_4><loc_61><loc_72><loc_92><loc_73></location>f ' k ( τ ) f ∗ k ( τ ) -f k ( τ ) f '∗ k ( τ ) = -i. (25)</formula>
<text><location><page_4><loc_52><loc_66><loc_92><loc_70></location>Spectator field modes also require proper matching conditions at τ = τ R , to ensure continuity of the field energy density at the transition [32]. Thus, imposing</text>
<formula><location><page_4><loc_62><loc_59><loc_92><loc_65></location>g < k ( τ R ) = g > k ( τ R ) , d dτ g < k ( τ ) ∣ ∣ ∣ ∣ τ R = d dτ g > k ( τ ) ∣ ∣ ∣ ∣ τ R , (26)</formula>
<text><location><page_4><loc_52><loc_56><loc_92><loc_58></location>and exploiting Eq. (25), one obtains the Bogoliubov coefficients associated with this transition, namely</text>
<formula><location><page_4><loc_56><loc_53><loc_92><loc_54></location>α k = i [ g ' < k ( τ R ) f ∗ k ( τ R ) -g < k ( τ R ) f '∗ k ( τ R ) ] , (27)</formula>
<formula><location><page_4><loc_56><loc_51><loc_92><loc_52></location>β k = -i [ g ' < k ( τ R ) f k ( τ R ) -g < k ( τ R ) f ' k ( τ R ) ] . (28)</formula>
<text><location><page_4><loc_52><loc_47><loc_92><loc_49></location>This implies that the field expansion at τ > τ R can be written as</text>
<formula><location><page_4><loc_52><loc_38><loc_93><loc_46></location>ˆ χ ( x , τ ) = 1 (2 π ) 3 / 2 ∫ d 3 k [ ˆ a k g > k ( τ ) e -i k · x +ˆ a † k g ∗ > k ( τ ) e i k · x ] = 1 (2 π ) 3 / 2 ∫ d 3 k [ ˆ b k f k ( τ ) e -i k · x + ˆ b † k f ∗ k ( τ ) e i k · x ] , (29)</formula>
<text><location><page_4><loc_52><loc_34><loc_81><loc_36></location>where we introduced ˆ b k = α k ˆ a k + β ∗ k ˆ a † -k .</text>
<section_header_level_1><location><page_4><loc_53><loc_31><loc_90><loc_32></location>D. Producing particles from the spectator field</section_header_level_1>
<text><location><page_4><loc_52><loc_20><loc_92><loc_28></location>By virtue of the above results, we can now identify ˆ b k , ˆ b † k as the ladder operators corresponding to out particle states, obeying canonical quantization conditions. Since in and out vacua are different in general, due to the background expansion, a certain amount of particles is produced via the GPP mechanism.</text>
<text><location><page_4><loc_52><loc_17><loc_92><loc_20></location>In the Heisenberg picture, the final comoving number density of spectator field particles reads</text>
<formula><location><page_4><loc_59><loc_13><loc_92><loc_16></location>N (0) k ≡ 1 a 2 ( τ R ) ⟨ 0 | ˆ b † k ˆ b k | 0 ⟩ = | β k | 2 a 2 ( τ R ) , (30)</formula>
<text><location><page_4><loc_52><loc_9><loc_92><loc_11></location>where | 0 ⟩ is the initial Bunch-Davies vacuum state, satisfying the condition, ˆ a k | 0 ⟩ = 0, ∀ k .</text>
<figure>
<location><page_5><loc_9><loc_75><loc_49><loc_94></location>
<caption>FIG. 1: Rescaled number density ¯ N (0) k as function of the momentum k ∈ [ 10 -5 /τ R , 10 -4 /τ R ] , for typical values of the Hubble parameter during inflation. We set τ R ≃ 2 . 05 × 10 15 GeV -1 , m = 10 -14 GeV and ξ = 0. In case of bosonic fields, GPP is generally more efficient for modes which are superHubble at the end of inflation.</caption>
</figure>
<text><location><page_5><loc_30><loc_75><loc_30><loc_76></location>k</text>
<text><location><page_5><loc_31><loc_75><loc_31><loc_76></location>(</text>
<text><location><page_5><loc_31><loc_75><loc_32><loc_76></location>GeV</text>
<text><location><page_5><loc_32><loc_75><loc_33><loc_76></location>)</text>
<text><location><page_5><loc_9><loc_54><loc_49><loc_63></location>Thus, Eq. (30) implies that the initial vacuum state of the field is no longer seen as a vacuum in the out region. Hence, we can interpret N (0) k as the number density of particles asymptotically produced from cosmic expansion, i.e., in terms of a gravitational production obtained from vacuum.</text>
<text><location><page_5><loc_10><loc_53><loc_31><loc_54></location>Defining now the quantities,</text>
<formula><location><page_5><loc_18><loc_46><loc_49><loc_51></location>R k = 2 3 / 4 | α | 1 / 4 ∣ ∣ ∣ ∣ ∣ ∣ Γ ( 3 4 -i | α | 2 ) Γ ( 1 4 -i | α | 2 ) ∣ ∣ ∣ ∣ ∣ ∣ 1 / 2 , (31)</formula>
<formula><location><page_5><loc_18><loc_44><loc_49><loc_45></location>κ = √ 1 + e -2 π | α | -e -π | α | , (32)</formula>
<formula><location><page_5><loc_18><loc_40><loc_49><loc_43></location>α = -k 2 2 mH R , , (33)</formula>
<text><location><page_5><loc_9><loc_35><loc_49><loc_39></location>it can be shown that, in the limit of minimal coupling ξ = 0, the number density of gravitationally produced particles for super-Hubble modes kτ R ≪ 1 reads [32, 64]</text>
<formula><location><page_5><loc_9><loc_28><loc_49><loc_33></location>N (0) k = 1 4 R 2 k δ 4 a 2 ( τ R ) [ κ ( R 2 k δ 2 -1 ) 2 + 1 κ ( R 2 k δ 2 +1 ) 2 ] (34)</formula>
<text><location><page_5><loc_9><loc_27><loc_49><loc_28></location>where we introduced the additional parameter, δ ≡ kτ R .</text>
<text><location><page_5><loc_9><loc_24><loc_49><loc_26></location>It is quite convenient to quantify the rescaled number density,</text>
<formula><location><page_5><loc_22><loc_21><loc_49><loc_22></location>¯ N (0) k ≡ N (0) k a 2 ( τ R ) , (35)</formula>
<text><location><page_5><loc_9><loc_15><loc_49><loc_19></location>drawn in Fig. 1, where we explore super-Hubble momenta within the range k ∈ [ 10 -5 /τ R , 10 -4 /τ R ] , by assuming different values for H I .</text>
<text><location><page_5><loc_9><loc_11><loc_49><loc_15></location>We remark that the number density is strongly peaked at low momentum, due to the bosonic nature of the spectator field considered. At the same time, GPP is more</text>
<text><location><page_5><loc_50><loc_31><loc_50><loc_32></location>,</text>
<text><location><page_5><loc_52><loc_90><loc_92><loc_93></location>efficient at larger H I , since in this case there is more energy to be converted into particles.</text>
<text><location><page_5><loc_52><loc_83><loc_92><loc_90></location>In the next section, we will include inflationary perturbations in this framework, showing how the presence of spacetime inhomogeneities is able to enhance the total number of particles produced, also allowing for modemixing in particle creation.</text>
<section_header_level_1><location><page_5><loc_55><loc_76><loc_88><loc_78></location>III. GEOMETRIC CONTRIBUTION TO PARTICLE CREATION</section_header_level_1>
<text><location><page_5><loc_52><loc_64><loc_92><loc_74></location>Particle production from spacetime perturbations represents an alternative mechanism to the widely-studied GPP approach [40, 41]. In particular, during inflation the presence of inhomogeneities can be traced back to the quantum fluctuations of the inflaton field, which are the fundamental seeds for structure formation in our universe.</text>
<text><location><page_5><loc_52><loc_59><loc_92><loc_63></location>From Eq. (6), the first-order interaction Lagrangian density describing the coupling between perturbations and a given quantum field can be written in the form</text>
<formula><location><page_5><loc_63><loc_54><loc_92><loc_57></location>L I = -1 2 √ -g (0) H µν T (0) µν , (36)</formula>
<text><location><page_5><loc_52><loc_45><loc_92><loc_53></location>where T (0) µν is the zero-order energy-momentum tensor for the field, g (0) the determinant of the background unperturbed metric tensor and H µν = a 2 ( τ ) h µν . When dealing with the spectator scalar field φ introduced in Sec. II, we have [40]</text>
<formula><location><page_5><loc_53><loc_36><loc_92><loc_43></location>T (0) µν = ∂ µ φ∂ ν φ -1 2 g (0) µν [ g ρσ (0) ∂ ρ φ ∂ σ φ -m 2 φ 2 ] -ξ [ ∇ µ ∂ ν -g (0) µν ∇ ρ ∇ ρ + R (0) µν -1 2 R (0) g (0) µν ] φ 2 . (37)</formula>
<text><location><page_5><loc_52><loc_29><loc_92><loc_33></location>Moving now to the interaction picture, it can be shown that the first-order S matrix in Dyson's expansion associated with L I reads ˆ S ≃ 1 + i ˆ T ∫ d 4 x L I .</text>
<text><location><page_5><loc_52><loc_23><loc_92><loc_29></location>Since both the field potential and the field-curvature coupling term are quadratic in φ , particles are produced in pairs at first perturbative order. We can write the corresponding probability amplitude as [58]</text>
<formula><location><page_5><loc_53><loc_14><loc_92><loc_21></location>C k 1 , k 2 ≡ ⟨ k 1 , k 2 | ˆ S | 0 ⟩ = -i 2 (2 π ) 3 ∫ d 4 x 2 a 2 ( A 0 ( x , τ ) + A 1 ( x , τ ) + A 2 ( x , τ ) + A 3 ( x , τ ) ) , (38)</formula>
<text><location><page_5><loc_52><loc_11><loc_56><loc_12></location>where</text>
<text><location><page_6><loc_9><loc_79><loc_19><loc_80></location>and, similarly,</text>
<formula><location><page_6><loc_23><loc_84><loc_78><loc_91></location>A 0 ( x , τ ) = 2Ψ [ ∂ 0 φ ∗ k 1 ∂ 0 φ ∗ k 2 -1 2 ( η ρσ ∂ ρ φ ∗ k 1 ∂ σ φ ∗ k 2 -m 2 a 2 φ k 1 φ k 2 ) -ξ ( ∂ 0 ∂ 0 -a ' a ∂ 0 -η ρσ ∂ ρ ∂ σ -3 ( a ' a ) 2 ) φ ∗ k 1 φ ∗ k 2 ] e -i ( k 1 + k 2 ) · x</formula>
<text><location><page_6><loc_89><loc_82><loc_92><loc_83></location>(39)</text>
<formula><location><page_6><loc_21><loc_69><loc_92><loc_78></location>A i ( x , τ ) = 2Ψ [ ∂ i φ ∗ k 1 ∂ i φ ∗ k 2 + 1 2 ( η ρσ ∂ ρ φ ∗ k 1 ∂ σ φ ∗ k 2 -m 2 a 2 φ k 1 φ k 2 ) -ξ ( ∂ i ∂ i + 3 a ' a ∂ 0 + 2 a '' a + η ρσ ∂ ρ ∂ σ -( a ' a ) 2 ) φ ∗ k 1 φ ∗ k 2 ] e -i ( k 1 + k 2 ) · x , (40)</formula>
<text><location><page_6><loc_9><loc_62><loc_49><loc_65></location>for i = 1 , 2 , 3. In Eqs. (39)-(40), we reintroduced the original field modes during inflation,</text>
<formula><location><page_6><loc_23><loc_58><loc_49><loc_61></location>φ k ( τ ) = g < k ( τ ) a ( τ ) , (41)</formula>
<text><location><page_6><loc_9><loc_52><loc_49><loc_56></location>in order to properly compute the amount of spectator particles produced. For each particle pair, the final state can be written in the form,</text>
<formula><location><page_6><loc_9><loc_46><loc_49><loc_50></location>| Ψ ⟩ = ˆ S | 0 k 1 ; 0 k 2 ⟩ = N ( | 0 k 1 ; 0 k 2 ⟩ + 1 2 C k 1 , k 2 | 1 k 1 ; 1 k 2 ⟩ ) , (42)</formula>
<text><location><page_6><loc_9><loc_43><loc_49><loc_46></location>where the normalization factor N is derived as usual from the condition ⟨ Ψ | Ψ ⟩ = 1.</text>
<text><location><page_6><loc_9><loc_39><loc_49><loc_43></location>The comoving number density associated with a geometric production of particles can be then computed at first and second perturbative order, giving respectively</text>
<formula><location><page_6><loc_9><loc_34><loc_49><loc_37></location>N (1) k = |N| 2 δ 3 ( k 1 + k 2 ) Re [ C k 1 , k 2 ( α k 1 β k 1 + α k 2 β k 2 )] , (43)</formula>
<formula><location><page_6><loc_9><loc_31><loc_49><loc_33></location>N (2) k 1 ,k 2 = |N| 2 |C k 1 , k 2 | 2 ( 1 + | β k 1 | 2 + | β k 2 | 2 ) . (44)</formula>
<section_header_level_1><location><page_6><loc_10><loc_26><loc_48><loc_27></location>A. Amplitudes and orders of particle production</section_header_level_1>
<text><location><page_6><loc_9><loc_9><loc_49><loc_24></location>It is relevant to stress that probability amplitudes for pair production are typically small in our perturbative approach. We thus have |N| 2 ≃ 1 and we can neglect the normalization constant in the further computations. Analogously, we underline that, when computing number densities in the interaction picture, the zero-order term of Eq. (30) also acquires a normalization factor. Specifically, the final state of the system is modified by the interaction itself, implying that the contribution of Eqs. (43)-(44) with respect to the background GPP term is always independent from the normalization procedure.</text>
<text><location><page_6><loc_52><loc_52><loc_92><loc_65></location>In particular, we notice that the first order term in Eq. (43) involves creation of particles with opposite momenta, thus only increasing the total number of particleantiparticle pairs. For this reason, we will focus on the second order term, which instead introduces modemixing in particle production. In particular, we are interested in superhorizon pair production, so we pick one mode on super-Hubble scales and the other on subHubble ones, namely</text>
<formula><location><page_6><loc_63><loc_47><loc_92><loc_50></location>a ( τ i ) H I < | k 1 | < a ( τ ) H I , a ( τ ) H I < | k 2 | < a ( τ ) M pl , (45)</formula>
<text><location><page_6><loc_52><loc_40><loc_92><loc_45></location>where τ i is the initial time for inflation and the ultraviolet cutoff is given by the Planck mass M pl . Moreover, we assume that there are no super-Hubble modes at the beginning of inflation 6 .</text>
<text><location><page_6><loc_52><loc_37><loc_92><loc_39></location>Exploiting the properties of Hankel functions, from Eqs. (17) and (41), we can write [58]</text>
<formula><location><page_6><loc_54><loc_30><loc_92><loc_35></location>φ super k ≃ e i ( ν -1 2 ) π 2 2 ( ν -3 2 ) Γ( ν ) Γ ( 3 2 ) H I √ 2 k 3 ( k aH I ) 3 2 -ν , (46)</formula>
<formula><location><page_6><loc_54><loc_24><loc_92><loc_27></location>φ sub k ≃ 1 √ 2 k e i ( ν + 1 2 ) π 2 e i ( -kτ -π 2 ν -π 4 ) a . (47)</formula>
<text><location><page_6><loc_52><loc_17><loc_92><loc_22></location>In the following, we specify our calculations to some relevant inflationary potentials, in order to compute the amount of geometric particle produced during the slowroll phase.</text>
<section_header_level_1><location><page_7><loc_11><loc_91><loc_47><loc_93></location>IV. THEORETICAL CONSEQUENCES OF INFLATIONARY PARTICLE PRODUCTION</section_header_level_1>
<text><location><page_7><loc_9><loc_77><loc_49><loc_88></location>As above stated, our particle computation depends on the underlying inflationary potential. Thus, to accurately compute the production of geometric particles during the slow-roll phase, it is mandatory to meticulously identify the most promising approaches that agree with current observations. The Planck satellite's numerical findings suggest that two categories of potentials remain viable, namely large and small field potentials [56].</text>
<figure>
<location><page_7><loc_9><loc_57><loc_49><loc_76></location>
<caption>FIG. 2: Ratio between the number density for 'geometric' particles N (2) k 1 ,k 2 and the unperturbed density N (0) k , assuming the Starobinsky potential to drive inflation. The ratio is plotted as function of the super-Hubble mode | k 1 | ∈ [ 10 -5 /τ R , 10 -4 /τ R ] . We set ξ φ = 10 -4 , ϵ = 10 -3 , ϕ ( τ i ) = 5 M pl , Λ 4 = 10 64 GeV 4 and m = 10 -14 GeV.</caption>
</figure>
<text><location><page_7><loc_30><loc_57><loc_30><loc_57></location>1</text>
<text><location><page_7><loc_9><loc_38><loc_49><loc_45></location>Even though appealing, the class of small field potentials is expected to provide a very small geometric particle production across the Hubble horizon, being incompatible with the possibility that DM can arise from perturbative approaches, see e.g. [42, 58].</text>
<text><location><page_7><loc_9><loc_31><loc_49><loc_37></location>In other words, inflationary particle production from inhomogeneities is typically inefficient if the energy in the inflaton field is not large enough. This implies that the substantial energy released during inflation has the potential to be physically converted into particles.</text>
<text><location><page_7><loc_9><loc_28><loc_49><loc_30></location>In this respect, we focus on two main large-field inflationary potentials:</text>
<unordered_list>
<list_item><location><page_7><loc_12><loc_21><loc_49><loc_26></location>- The Starobinsky potential [51], that is characterized by the inclusion of the quadratic term R 2 in the Hilbert-Einstein action, currently representing the leading candidate to describe inflation.</list_item>
<list_item><location><page_7><loc_12><loc_13><loc_49><loc_19></location>- The nonminimally coupled fourth order chaotic potential. Here, the fourth order potential alone is unsuitable to describe inflation [56], albeit its coupling with R quite evidently candidates it as a still viable inflationary framework.</list_item>
</unordered_list>
<text><location><page_7><loc_9><loc_9><loc_49><loc_11></location>Below, we discuss the production of particles in both the aforementioned schemes.</text>
<section_header_level_1><location><page_7><loc_52><loc_92><loc_91><loc_93></location>A. Particles produced from Starobinsky potential</section_header_level_1>
<text><location><page_7><loc_52><loc_86><loc_92><loc_90></location>Here, the metric tensor can be conformally rescaled into the Einstein frame, where the action takes the form of Eq. (2), with corresponding potential [52, 53]</text>
<formula><location><page_7><loc_60><loc_82><loc_92><loc_85></location>V ( ϕ ) = Λ 4 ( 1 -e -√ 2 / 3 ϕ/M pl ) 2 , (48)</formula>
<text><location><page_7><loc_52><loc_76><loc_92><loc_80></location>and Λ 4 describes the energy scales of inflation. Recalling Eq. (3), the background dynamics of this effective field during slow-roll is given by</text>
<formula><location><page_7><loc_66><loc_73><loc_92><loc_75></location>3 H ϕ ' ≃ -V ,ϕ a 2 , (49)</formula>
<text><location><page_7><loc_52><loc_66><loc_92><loc_72></location>where we have introduced the compact notation V ,ϕ ≡ ∂V/∂ϕ and the scale factor during inflation has been defined in Eq. (5). The corresponding fluctuation modes are described by [39]</text>
<formula><location><page_7><loc_55><loc_62><loc_92><loc_65></location>δχ '' k + [ k 2 -1 η 2 ( 2 + 9 ϵ -V ϕϕ H 2 I )] δχ k = 0 , (50)</formula>
<text><location><page_7><loc_52><loc_55><loc_92><loc_60></location>where the fluctuation field has been rescaled as usual by δχ k = δϕ k a . This equation admits solutions in terms of Hankel functions, provided the potential term is substituted by its mean value during slow-roll [58].</text>
<figure>
<location><page_7><loc_52><loc_34><loc_92><loc_53></location>
<caption>FIG. 3: Ratio between the number density for 'geometric' particles N (2) k 1 ,k 2 and the unperturbed density N (0) k , assuming a quartic symmetry-breaking potential. The ratio is plotted as function of the super-Hubble mode | k 1 | ∈ [ 10 -5 /τ R , 10 -4 /τ R ] . We set ξ φ = ξ ϕ = 10 -4 , ϵ = 10 -3 , ϕ ( τ i ) = 5 M pl , λ = 2 . 9 × 10 -15 and m = 10 -14 GeV.</caption>
</figure>
<text><location><page_7><loc_52><loc_17><loc_92><loc_22></location>Once obtained the background and fluctuation dynamics, the perturbation potential Ψ can be derived from Eq. (8) and inserted in Eq. (36) after proper normalization 7 , to compute the amount of spectator field particles</text>
<text><location><page_8><loc_9><loc_79><loc_49><loc_93></location>arising from perturbations. In Fig. 2, we show the ratio N (2) /N (0) as function of the super-Hubble mode k 1 . In particular, the probability amplitude for perturbative production is evaluated in the range τ ∈ [0 , τ R ], in order to exploit the simplified expression of Eq. (46) for the modes under investigation 8 . At the same time, we neglect perturbative production during the radiation era, where the contribution of inhomogeneities is expected to be much smaller due to the presence of other quantum fields and possible backreaction mechanisms.</text>
<section_header_level_1><location><page_8><loc_10><loc_73><loc_48><loc_75></location>B. Particles produced from a nonminimal fourth order chaotic potential</section_header_level_1>
<text><location><page_8><loc_9><loc_66><loc_49><loc_70></location>As an alternative scenario, we discuss a nonminimally coupled inflaton field driven by a quartic symmetrybreaking potential. This gives</text>
<formula><location><page_8><loc_17><loc_62><loc_49><loc_65></location>V ( ϕ ) = λ 4 ( ϕ 2 -v 2 ) 2 + 1 2 ξ ϕ Rϕ 2 , (51)</formula>
<text><location><page_8><loc_9><loc_48><loc_49><loc_61></location>where v is the vacuum expectation value of the inflaton field and λ a self-coupling constant. In case of positive coupling constant, quartic chaotic inflation is not expected to work, unless the inflaton coupling to curvature is sufficiently small [54, 55] (see also [66]). This model has been recently considered for perturbative particle production in inflationary scenarios [47, 58] and it may also allow to identify the Standard Model Higgs field as the inflaton [67-69].</text>
<text><location><page_8><loc_9><loc_38><loc_49><loc_47></location>Following the same steps of Sec. IV A, we can derive the dynamics of the perturbation potential in this model and then compute the corresponding number densities of particles arising from inhomogeneities. In Fig. 3 we show again the number density of geometric particles produced at second perturbative order, normalized with respect to the unperturbed GPP contribution.</text>
<section_header_level_1><location><page_8><loc_10><loc_31><loc_48><loc_34></location>V. CONSEQUENCES AND PREDICTIONS OF OUR SCENARIOS</section_header_level_1>
<text><location><page_8><loc_9><loc_25><loc_49><loc_29></location>We here discuss the main implications of our findings in inflationary stages and possible signatures of our scenarios.</text>
<text><location><page_8><loc_9><loc_19><loc_49><loc_25></location>Particularly, let us first observe, from Figs. 2-3, that perturbative particle production is non-negligible for modes that crossed the Hubble horizon mainly before the cutoff time, τ = 0. Second, we also notice that, when</text>
<text><location><page_8><loc_52><loc_73><loc_92><loc_93></location>approaching the infrared cutoff | k 1 | ≃ a ( τ i ) H I , the contribution of inhomogeneities is typically larger. In particular, it can be shown that a perturbative treatment is no longer possible at sufficiently small | k 1 | , thus requiring a different technique to evaluate the effects of inflaton fluctuations. The above issue is also related to the normalization procedure for the perturbation potential Ψ, whose amplitude is typically fixed at horizon crossing [70]. Specifically, a correct normalization procedure might depend upon modes in order to guarantee that all the perturbation magnitudes appear the same at the horizon crossing. Likely, this would imply to reformulate the correct vacuum, modifying the Bunch-Davies choice with a more refined approach.</text>
<text><location><page_8><loc_52><loc_59><loc_92><loc_73></location>Further, we notice that the contribution of inhomogeneities is typically enhanced in case of larger fieldcurvature coupling constants. In the limit of conformal coupling, ξ = 1 / 6, GPP results in negligible densities [32], implying that geometric production would become the dominant mechanism for primordial particle creation. A similar result was obtained for the gravitational production of massless fermions during preheating [71], showing that metric perturbations may have also played an important role at the end of inflation.</text>
<text><location><page_8><loc_52><loc_48><loc_92><loc_58></location>To summarize, particle production arising from inhomogeneities can significantly affect the total number density of spectator field particles created up to the radiation era. For this reason, if DM has been produced via purely gravitational mechanisms, the presence of inhomogeneities should be taken into account when computing the corresponding particle abundance.</text>
<text><location><page_8><loc_52><loc_34><loc_92><loc_48></location>Last but not least, we also remarked how spacetime inhomogeneities are responsible for mode-mixing in particle production, that is not conversely found in unperturbed GPP scenarios, where only particle-antiparticle pairs can be generated, i.e., with opposite momenta. More specifically, during inflation the Hubble horizon emerges as a natural separation scale for modes and superhorizon particle production has been recently investigated for inflaton fluctuations, showing that quantum entanglement can be generated in this process [58].</text>
<text><location><page_8><loc_52><loc_19><loc_92><loc_33></location>Remarkably, since we focused on particle production across the Hubble horizon, we conclude that plausible detectable quantum 'signatures' at late times can occur. Indeed, DM is expected to weakly interact with Standard Model fields, so that some entanglement entropy associated with particle production may have survived after the inflationary epoch. Hence, we underline that the role of entanglement could help to understand the quantum properties of produced particles, thus opening new avenues in the search for DM.</text>
<section_header_level_1><location><page_8><loc_53><loc_15><loc_91><loc_16></location>VI. FINAL OUTLOOKS AND PERSPECTIVES</section_header_level_1>
<text><location><page_8><loc_52><loc_9><loc_92><loc_13></location>In this work, we investigated the particle production associated with a spectator scalar field, i.e., subdominant with respect to the inflaton, during and after the slow-roll</text>
<text><location><page_9><loc_9><loc_88><loc_49><loc_93></location>regime. To show how particle production is influenced by the universe expansion, we pictured an instantaneous transition from inflation to the radiation dominated era, neglecting the effects due to reheating in GPP regimes.</text>
<text><location><page_9><loc_9><loc_76><loc_49><loc_87></location>In particular, we focused on the contribution associated with inhomogeneous particle production across the Hubble horizon, that can be traced back to the fluctuations of the inflaton field during slow-roll. We thus showed that the number density of particles arising from perturbations is typically non-negligible with respect to the widely-studied quantum GPP contribution, obtained from the unperturbed universe expansion.</text>
<text><location><page_9><loc_9><loc_62><loc_49><loc_76></location>We reobtained this outcome in the realms of large-field inflation and, particularly, we focused on two among the most consolidate paradigms describing the inflationary speed up. Specifically, we worked out the Starobinsky and the fourth order nonminimally coupled potentials. The latter represents the most viable large-field model of inflation, conformally equivalent to an extended theory of gravity, whereas the second is a suitable example of chaotic inflation, overcoming the Planck satellite observational constraints.</text>
<text><location><page_9><loc_9><loc_51><loc_49><loc_61></location>We discussed the physical results obtained and, particularly, we showed that the amount of particles obtained is similar in both the aforementioned scenarios. We also argued that geometric particle production across the horizon is expected to be negligible in small-field approaches, since in that case the energy in the inflaton field is significantly smaller throughout the slow-roll regime.</text>
<text><location><page_9><loc_9><loc_43><loc_49><loc_51></location>In addition, we observed that the presence of inhomogeneities allows for mode-mixing in particle production, that instead is not found in unperturbed GPP processes, where the total momentum of created particles is necessarily conserved. The presence of mode-mixing may lead to entanglement generation across the Hubble horizon,</text>
<unordered_list>
<list_item><location><page_9><loc_10><loc_33><loc_49><loc_37></location>[1] A. Arbey, F. Mahmoudi, Dark matter and the early Universe: a review , Progr. Part. Nucl. Phys. 19 , 103865 (2021).</list_item>
<list_item><location><page_9><loc_10><loc_30><loc_49><loc_33></location>[2] J. M. Gaskins, A review of indirect searches for particle dark matter , Contemp. Phys. 57 (2106) 496.</list_item>
<list_item><location><page_9><loc_10><loc_28><loc_49><loc_30></location>[3] G. Bertone and T. M. P. Tait, A new era in the search for dark matter , Nature 562 , 51 (2018).</list_item>
<list_item><location><page_9><loc_10><loc_25><loc_49><loc_27></location>[4] F. Kahlhoefer, Review of LHC dark matter searches , Int. J. Mod. Phys. A 32 , 1730006 (2017).</list_item>
<list_item><location><page_9><loc_10><loc_21><loc_49><loc_25></location>[5] D. S. Akerib et al. (LUX collaboration), Results from a search for dark matter in the complete LUX exposure , Phys. Rev. Lett. 118 , 021303 (2017).</list_item>
<list_item><location><page_9><loc_10><loc_17><loc_49><loc_21></location>[6] E. Caprile et al., Dark matter search results from a one ton-year exposure of XENON1T , Phys. Rev. Lett. 121 , 111302 (2018).</list_item>
<list_item><location><page_9><loc_10><loc_13><loc_49><loc_17></location>[7] M. P. Hertzberg, M. Tegmark, and F. Wilczek, Axion cosmology and the enrgy scale of inflation , Phys. Rev. D 78 , 083507 (2008).</list_item>
<list_item><location><page_9><loc_10><loc_11><loc_49><loc_13></location>[8] P. Sikivie, Dark matter axions , Int. J. Mod. Phys. A 25 , 554 (2010).</list_item>
<list_item><location><page_9><loc_10><loc_9><loc_49><loc_10></location>[9] A. Arvanitaki, S. Dimopoulos, S. Dubovsky, N. Kaloper,</list_item>
</unordered_list>
<text><location><page_9><loc_52><loc_88><loc_92><loc_93></location>and we argued that such quantum correlations could have survived after the inflationary epoch due to the weakly interacting nature of DM. Possibility of detecting such particles through entanglement are also discussed above.</text>
<text><location><page_9><loc_52><loc_76><loc_92><loc_87></location>At the same time, we noticed that a perturbative approach to inhomogeneous particle production is not always possible, since the magnitude of inflaton fluctuations becomes typically large on super-Hubble scales. We also pointed out that a more refined approach is needed for the normalization of the perturbation potential, in order to obtain correct amplitudes at horizon crossing for all the modes involved.</text>
<text><location><page_9><loc_52><loc_67><loc_92><loc_75></location>As perspectives, further steps would include the effects of reheating in geometric production. Although such effects may be negligible at a first sight, they may affect the total amount of produced particles via the dynamics of preheating metric perturbations, especially for subHubble modes.</text>
<text><location><page_9><loc_52><loc_59><loc_92><loc_66></location>In addition, we intend to study the possible backreaction effects associated with the dynamics of perturbations at the end of inflation and shed further light on a more general non-perturbative approach to inhomogeneous particle production.</text>
<text><location><page_9><loc_52><loc_55><loc_92><loc_59></location>Finally, we plan to extend our treatment to higher spin spectator fields, starting from fermionic ones, with the aim of evaluating other possible DM candidates.</text>
<section_header_level_1><location><page_9><loc_65><loc_49><loc_79><loc_50></location>Acknowledgements</section_header_level_1>
<text><location><page_9><loc_52><loc_43><loc_92><loc_47></location>The work of OL is partially financed by the Ministry of Education and Science of the Republic of Kazakhstan, Grant: IRN AP19680128.</text>
<unordered_list>
<list_item><location><page_9><loc_55><loc_34><loc_92><loc_37></location>and J. March-Russell, String axiverse , Phys. Rev. D 81 , 123530 (2010).</list_item>
<list_item><location><page_9><loc_52><loc_32><loc_92><loc_34></location>[10] O. Wantz and E. P. S. Shellard, Axion cosmology revisited , Phys. Rev. D 82 , 123508 (2010).</list_item>
<list_item><location><page_9><loc_52><loc_28><loc_92><loc_31></location>[11] J. E. Kim and G. Carosi, Axions and the strong CP problem , Rev. Mod. Phys. 82 , 557 (2010); Erratum: Rev. Mod. Phys. 91 , 049902 (2019).</list_item>
<list_item><location><page_9><loc_52><loc_25><loc_92><loc_27></location>[12] D. J. E. Marsh, Axion cosmology , Phys. Rept. 643 , 1 (2016).</list_item>
<list_item><location><page_9><loc_52><loc_20><loc_92><loc_25></location>[13] P. W. Graham, I. G. Irastorza, S. T. Lamoreaux, A. Lindner, and K. A. van Bibber, Experimental searches for the axion and axion-like particles , Annu. Rev. Nucl. Part. Sci. 65 (2015) 485.</list_item>
<list_item><location><page_9><loc_52><loc_16><loc_92><loc_20></location>[14] J. C. Niemeyer, Small-scale structure of fuzzy and axionlike dark matter , Prog. Part. Nucl. Phys. 113 , 103787 (2020).</list_item>
<list_item><location><page_9><loc_52><loc_12><loc_92><loc_16></location>[15] K. Choi, S. Hui Im, and C. Sub Shin, Recent progress in the physics of axions and axion-like particles , Ann. Rev. Nucl. Part. Sci. 71 (2021) 225.</list_item>
<list_item><location><page_9><loc_52><loc_9><loc_92><loc_12></location>[16] A. Su'arez, V. H. Robles, and T. Matos, A review on the scalar field/Bose-Einstein condensate dark matter model ,</list_item>
<list_item><location><page_10><loc_12><loc_92><loc_40><loc_93></location>Astrophys. Space Sci. Proc. 38 (2014) 107.</list_item>
<list_item><location><page_10><loc_9><loc_88><loc_49><loc_92></location>[17] L. Hui, J. P. Ostriker, S. Tremaine, and E. Witten, Ulralight scalars as cosmological dark matter , Phys. Rev. D 95 , 043541 (2017).</list_item>
<list_item><location><page_10><loc_9><loc_85><loc_49><loc_88></location>[18] L. A. Ure˜na-L'opez, Brief review on scalar field dark matter models , Front. Astron. Space Sci. 6 , (2019).</list_item>
<list_item><location><page_10><loc_9><loc_83><loc_49><loc_85></location>[19] E. G. M. Ferreira, Ultra-light dark matter , Astron. Astrophys. Rev. 29 (2021) 1, 7.</list_item>
<list_item><location><page_10><loc_9><loc_76><loc_49><loc_82></location>[20] L. Parker, Particle creation in expanding universes , Phys. Rev. Lett. 21 , 562 (1968); Quantized fields and particle creation in expanding universes. I , Phys. Rev 183 , 1057 (1969); Quantized fields and particle creation in expanding universes. II , Phys. Rev. D 3 , 346 (1971).</list_item>
<list_item><location><page_10><loc_9><loc_73><loc_49><loc_76></location>[21] N. Birrell and P. Davies, Quantum Fields in Curved Space , Cambridge Univ. Press, Cambridge, UK (1982).</list_item>
<list_item><location><page_10><loc_9><loc_69><loc_49><loc_73></location>[22] V. Mukhanov and S. Winitzki, Introduction to Quantum Effects in Gravity , Cambridge University Press, Cambridge (2012).</list_item>
<list_item><location><page_10><loc_9><loc_67><loc_49><loc_69></location>[23] L. H. Ford, Cosmological particle production: a review , Rep. Prog. Phys. 84 , 116901 (2021).</list_item>
<list_item><location><page_10><loc_9><loc_64><loc_49><loc_67></location>[24] L. H. Ford, Gravitational particle creation and inflation , Phys. Rev. D 35 , 2955 (1987).</list_item>
<list_item><location><page_10><loc_9><loc_62><loc_49><loc_64></location>[25] D. J. H. Chung, E. W. Kolb, and A. Riotto, Superheavy dark matter , Phys. Rev. D 59 , 023501 (1998).</list_item>
<list_item><location><page_10><loc_9><loc_58><loc_49><loc_61></location>[26] D. J. H. Chung, P. Crotty, E. W. Kolb, and A. Riotto, Gravitational prodution of superheavy dark matter , Phys. Rev. D 64 , 043503 (2001).</list_item>
<list_item><location><page_10><loc_9><loc_52><loc_49><loc_57></location>[27] Y. Ema, R. Jinno, K. Mukaida, and K. Nakayama, Gravitational particle production in oscillating backgrounds and its cosmological implications , Phys. Rev. D 94 , 063517 (2016).</list_item>
<list_item><location><page_10><loc_9><loc_50><loc_49><loc_52></location>[28] Y. Ema, K. Nakayama, and Y. Tong, Production of purely gravitational dark matter , JHEP 09 (2018) 135.</list_item>
<list_item><location><page_10><loc_9><loc_46><loc_49><loc_49></location>[29] S. Hashiba and J. Yokoyama, Gravitational particle creation for dark matter and reheating , Phys. Rev. D 99 , 043008 (2019).</list_item>
<list_item><location><page_10><loc_9><loc_42><loc_49><loc_45></location>[30] J. A. R. Cembranos, L. J. Garay, and J. M. S'anchez Vel'asquez, Gravitational production of scalar dark matter , JHEP 06 (2020) 084.</list_item>
<list_item><location><page_10><loc_9><loc_38><loc_49><loc_41></location>[31] J. Lankinen, O. Kerrpo, and I. Vilja, Reheating via gravitational particle production in the kination epoch , Phys. Rev. D 101 , 063529 (2020).</list_item>
<list_item><location><page_10><loc_9><loc_34><loc_49><loc_37></location>[32] N. Herring, D. Boyanovsky, and A. R. Zentner, Nonadiabatic cosmological production of ultralight dark matter , Phys. Rev. D 101 , 083516 (2020).</list_item>
<list_item><location><page_10><loc_9><loc_30><loc_49><loc_34></location>[33] T. S. Bunch and P. Davies, Quantum field theory in de Sitter space: Renormalization by point splitting , Proc. R. Soc. A 360 , 117 (1978).</list_item>
<list_item><location><page_10><loc_9><loc_27><loc_49><loc_30></location>[34] U. H. Danielsson and M. E. Olsson, On the thermalization in de Sitter space , JHEP 03 (2004) 036.</list_item>
<list_item><location><page_10><loc_9><loc_23><loc_49><loc_27></location>[35] B. R. Greene, M. K. Parikh, and J. P. van der Schaar, Universal correction to the inflationary vacuum , JHEP 04 (2006) 057.</list_item>
<list_item><location><page_10><loc_9><loc_19><loc_49><loc_23></location>[36] V. F. Mukhanov, H. A. Feldman, and R. H. Brandenberger, Theory of cosmological perturbations , Phys. Rep. 215, 203 (1992).</list_item>
<list_item><location><page_10><loc_9><loc_17><loc_49><loc_19></location>[37] R. H. Brandenberger, Lectures on the theory of cosmological perturbations , Lect. Notes Phys. 646 , 127 (2004).</list_item>
<list_item><location><page_10><loc_9><loc_14><loc_49><loc_16></location>[38] D. Baumann, TASI lectures on inflation , arXiv:hepth/0907.5424 (2012).</list_item>
<list_item><location><page_10><loc_9><loc_11><loc_49><loc_14></location>[39] A. Riotto, Inflation and the theory of cosmological perturbations , arXiv[hep-ph]:0210162 (2017).</list_item>
<list_item><location><page_10><loc_9><loc_9><loc_49><loc_11></location>[40] J. A. Frieman, Particle creation in inhomogeneous spacetimes , Phys. Rev. D 39 , 389 (1989).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_10><loc_52><loc_91><loc_92><loc_93></location>[41] J. C'espedes and E. Verdaguer, Particle production in inhomogeneous cosmologies , Phys. Rev. D 41 , 1022 (1990).</list_item>
<list_item><location><page_10><loc_52><loc_87><loc_92><loc_90></location>[42] A. Belfiglio, R. Giamb'o, and O. Luongo, Alleviating the cosmological constant problem from particle production , Class. Quantum Grav. 40 (2023) 105004.</list_item>
<list_item><location><page_10><loc_52><loc_83><loc_92><loc_86></location>[43] J. Martin, Everything You Always Wanted To Know About The Cosmological Constant Problem (But Were Afraid To Ask) , Compt. Rend. Phys., 13 , 566-665 (2012).</list_item>
<list_item><location><page_10><loc_52><loc_75><loc_92><loc_82></location>[44] O. Luongo, M. Muccino, Speeding up the universe using dust with pressure , Phys. Rev. D 98 10, 103520, (2018); O. Luongo, H. Quevedo, A Unified Dark Energy Model from a Vanishing Speed of Sound with Emergent Cosmological Constant , Int. J. Mod. Phys. D 23 , 1450012 (2014).</list_item>
<list_item><location><page_10><loc_52><loc_66><loc_92><loc_74></location>[45] O. Luongo, Revising the cosmological constant problem through a fluid different from the quintessence , Phys. Scien. and Tech., 10 , 17, (2023); R. D'Agostino, O. Luongo, M. Muccino, Healing the cosmological constant problem during inflation through a unified quasiquintessence matter field , Class. Quant. Grav., 39 , 19, 195014 (2022).</list_item>
<list_item><location><page_10><loc_52><loc_62><loc_92><loc_65></location>[46] O. Luongo, T. Mengoni, Quasi-quintessence inflation with non-minimal coupling to curvature in the Jordan and Einstein frames , arXiv:2309.03065 [gr-qc]</list_item>
<list_item><location><page_10><loc_52><loc_56><loc_92><loc_61></location>[47] A. Belfiglio, Y. Carloni, and O. Luongo, Particle production from non-minimal coupling in a symmetry-breaking potential transporting vacuum energy , arXiv:2307.04739 [gr-qc]</list_item>
<list_item><location><page_10><loc_52><loc_54><loc_92><loc_56></location>[48] L. Kofman, A. Linde, and A. A. Starobinsky, Reheating after inflation , Phys. Rev. Lett. 73 , 3195 (1994).</list_item>
<list_item><location><page_10><loc_52><loc_50><loc_92><loc_53></location>[49] L. Kofman, A. Linde, and A. A. Starobinsky, Towards the theory of reheating after inflation , Phys. Rev. D 56 , 3258 (1997).</list_item>
<list_item><location><page_10><loc_52><loc_44><loc_92><loc_49></location>[50] R. Allahverdi, R. Brandenberger, F. Cyr-Racine, A. Mazumdar, Reheating in inflationary cosmology: theory and applications , Ann. Rev. Nucl. Part. Sci. 60 (2010) 27.</list_item>
<list_item><location><page_10><loc_52><loc_42><loc_92><loc_44></location>[51] A. Starobinsky, A new type of isotropic cosmological models without singularity , Phys. Lett. B 91 , 99 (1980).</list_item>
<list_item><location><page_10><loc_52><loc_38><loc_92><loc_41></location>[52] S. S. Mishra, V. Sahni, and A. V. Toporensky, Initial conditions for inflation in an FRW universe , Phys. Rev. D 98 , 083538 (2018).</list_item>
<list_item><location><page_10><loc_52><loc_34><loc_92><loc_37></location>[53] S. S. Mishra, D. Muller, and A. V. Toporensky, Generality of Starobinsky and Higgs inflation in the Jordan frame , Phys. Rev. D 102 , 063523 (2020).</list_item>
<list_item><location><page_10><loc_52><loc_30><loc_92><loc_34></location>[54] T. Futamase and K. Maeda, Chaotic inflationary scenario of the Universe with a nonminimally coupled 'inflaton' field , Phys. Rev. D 39 , 399 (1989).</list_item>
<list_item><location><page_10><loc_52><loc_26><loc_92><loc_30></location>[55] R. Fakir and W. G. Unruh, Improvement on cosmological chaotic inflation through nonminimal coupling , Phys. Rev. D 41 , 1783 (1990).</list_item>
<list_item><location><page_10><loc_52><loc_23><loc_92><loc_26></location>[56] Planck Collaboration, Y. Akrami, et al., Constraints on inflation , Astron. Astrophys., 641 , A10 (2020).</list_item>
<list_item><location><page_10><loc_52><loc_21><loc_92><loc_23></location>[57] L. Boubekeur and D. H. Lyth, Hilltop inflation , JCAP 07 (2005) 010.</list_item>
<list_item><location><page_10><loc_52><loc_17><loc_92><loc_20></location>[58] A. Belfiglio, O. Luongo, and S. Mancini, Superhorizon entanglement from inflationary particle production , arXiv: 2312.11419 [gr-qc]</list_item>
<list_item><location><page_10><loc_52><loc_11><loc_92><loc_16></location>[59] V. Balasubramanian, M. B. McDermott, and M. Van Raamsdonk, Momentum-space entanglement and renormalization in quantum field theory , Phys. Rev. D 86 , 045014 (2012).</list_item>
<list_item><location><page_10><loc_52><loc_9><loc_92><loc_11></location>[60] S. S. Kumar and S. Shankaranarayanan, Role of spatial higher order derivatives in momentum space entangle-</list_item>
</unordered_list>
<text><location><page_11><loc_12><loc_92><loc_38><loc_93></location>ment , Phys. Rev. D 95 , 065023 (2017).</text>
<unordered_list>
<list_item><location><page_11><loc_9><loc_88><loc_49><loc_92></location>[61] S. Brahma, O. Alaryani, and R. Brandenberger, Entanglement entropy of cosmological perturbations , Phys. Rev. D 102 , 043529 (2020).</list_item>
<list_item><location><page_11><loc_9><loc_84><loc_49><loc_88></location>[62] S. Brahma, J. Calder'on-Figueroa, M. Hassan, and X. Mi, Momentum-space entanglement entropy in de Sitter spacetime , Phys. Rev. D 108 , 043522 (2023).</list_item>
<list_item><location><page_11><loc_9><loc_81><loc_49><loc_84></location>[63] S. Schander and T. Thiemann, Backreaction in cosmology , Front. Astron. Space Sci. 8 , 692198 (2021).</list_item>
<list_item><location><page_11><loc_9><loc_77><loc_49><loc_81></location>[64] M. Rai and D. Boyanovsky, Origin of entropy of gravitationally produced dark matter: the entanglement entropy , Phys. Rev. D 102 , 063532 (2020).</list_item>
<list_item><location><page_11><loc_9><loc_75><loc_49><loc_77></location>[65] A. Belfiglio, O. Luongo, and S. Mancini, Inflationary entanglement , Phys. Rev. D 107 , 103512 (2023).</list_item>
<list_item><location><page_11><loc_9><loc_72><loc_49><loc_74></location>[66] S. Tsujikawa and B. Gumjudpai, Density perturbations in generalized Einstein scenarios and constraints on</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_11><loc_55><loc_91><loc_92><loc_93></location>nonminimal couplings from the cosmic microwave background , Phys. Rev. D 69 , 123523 (2004).</list_item>
<list_item><location><page_11><loc_52><loc_87><loc_92><loc_90></location>[67] F. L. Bezrukov and M. Shaposhnikov, The Standard Model Higgs boson as the inflaton , Phys. Lett. B 659 , 703 (2008).</list_item>
<list_item><location><page_11><loc_52><loc_83><loc_92><loc_86></location>[68] F. L. Bezrukov, J. Rubio, and M. Shaposhnikov, Living beyond the edge: Higgs inflation and vacuum metastability , Phys. Rev. D 92 , 083512 (2015).</list_item>
<list_item><location><page_11><loc_52><loc_80><loc_92><loc_82></location>[69] J. Rubio, Higgs inflation , Front. Astron. Space Sci. 5 (2019) 50.</list_item>
<list_item><location><page_11><loc_52><loc_77><loc_92><loc_80></location>[70] N. Kaloper and J. Scargill, Quantum cosmic no-hair theorem and inflation , Phys. Rev. D 99 , 103514 (2019).</list_item>
<list_item><location><page_11><loc_52><loc_73><loc_92><loc_77></location>[71] B. A. Bassett, M. Peloso, L. Sorbo and S. Tsujikawa, Fermion production from preheating-amplified metric perturbations , Nucl. Phys. B 622 (2002) 393.</list_item>
</document>
[
{
"title": "Production of ultralight dark matter from inflationary spectator fields",
"content": "Alessio Belfiglio 1, 2, ∗ and Orlando Luongo 1, 2, 3, 4, 5, † 1 School of Science and Technology, University of Camerino, Via Madonna delle Carceri, Camerino, 62032, Italy. 2 Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Perugia, Perugia, 06123, Italy. 3 SUNY Polytechnic Institute, 13502 Utica, New York, USA. 4 INAF - Osservatorio Astronomico di Brera, Milano, Italy. 5 Al-Farabi Kazakh National University, Al-Farabi av. 71, 050040 Almaty, Kazakhstan. We investigate inflationary particle production associated with a spectator ultralight scalar field, which has been recently proposed as a plausible dark matter candidate. In this framework, we select the Starobinsky potential to drive the inflationary epoch, also discussing the case of a nonminimally coupled inflaton field fueled by a quartic symmetry-breaking potential. We focus on particle production arising from spacetime perturbations, which are induced by inflaton fluctuations during the quasi-de Sitter stage of inflation. In particular, we construct the first order Lagrangian describing interaction between inhomogeneities and the spectator field, quantifying superhorizon particle production during slow-roll. We then compare this mechanism with gravitational particle production associated with an instantaneous transition from inflation to the radiation dominated era. We show that the amount of particles obtained from perturbations is typically non-negligible and it is significantly enhanced on super-Hubble scales by the nonadiabatic inflationary expansion. Possible implications for primordial entanglement generation are also debated. PACS numbers: 04.62.+v, 98.80.-k, 98.80.Cq, 03.67.bg",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "Dark matter (DM) is undoubtedly a key ingredient to explain the cosmological large-scale dynamics and clustering [1]. Its nature, however, remains mysterious: we are still lacking any conclusive experiment able to unambiguously identify its properties, with slight agreement among the plethora of theoretical proposals [2]. Within this scenario, recent efforts in the search for weakly interacting massive particles from a few up to 100 GeV have unfortunately proved unsuccessful [3-6]. Thus, this lack of evidence has also revived the interest in ultralight DM candidates such as axions [7-12], axionlike particles [13-15], 'fuzzy' DM models [16-19] and so on. In this respect, several treatments have been proposed to explain the origin of DM particles. Among them, gravitational particle production (GPP) represents a plausible and widely-investigated approach [20-23] as it creates particles directly from vacuum fluctuations and does not require any coupling between DM and generic quantum fields 1 . For this reason, GPP of DM has been studied in various cosmological contexts, with particular interest on inflationary [24-28] and reheating [29-31] phases. Gravitational production of ultralight particles has been recently discussed in Ref. [32], focusing on the dynamics of a spectator scalar field in the transition be- tween inflation and the radiation dominated era. There, assuming an instantaneous transition, and thus neglecting the details of reheating, a significant particle production may take place for super-Hubble wavelength modes after inflation, if the field starts from the Bunch-Davies vacuum [33-35] and it is minimally coupled to spacetime curvature. Specifically, the matching conditions between the two epochs provide the Bogoliubov coefficients from which one obtains the number density of produced particles, compatible with a cold ultralight DM candidate 2 . This approach, however, neglects the slow-roll of the inflaton field and the dynamics of its quantum fluctuations, which represent the fundamental seeds for structure formation in our universe [36-39]. In fact, inflationary fluctuations induce perturbations on the de Sitter dynamics of background, and the presence of inhomogeneities, by virtue of expansion, may result in the production of additional 'geometric' particles, due to purely gravitational effects [40, 41]. In Ref. [42], it was argued that DM could be reinterpreted as geometric quasiparticles, in the attempt of solving the cosmological constant problem [43], extending a mechanism of direct cancellation between fields [44, 45]. Thus, at least two more points require additional investigations. First, how the nonminimal coupling acts on the inflationary dynamics and particle production, see e.g. Refs. [46, 47], and, second, whether gravitational production furnishes stable particles, or more broadly, stable quasi-particles, since their dynamics at the end of the slow-roll regime can be altered by possible couplings of the inflaton to other quantum fields, as expected in the standard picture of reheating [48-50]. For the above reasons, we here aim to generalize inflationary geometric production to the case of a spectator scalar field. Assuming a quasi-de Sitter background evolution to properly account for the slow-roll of the inflaton, we show how spacetime perturbations generated by inflaton fluctuations couple to the energy-momentum tensor of a given spectator field, leading to particle creation during the slow-roll regime. We focus in particular on the Starobinsky model of inflation [51-53], also discussing the case of a nonminimally coupled inflaton field driven by a quartic potential [54, 55]. We single out these paradigms since both the above models are among the best options to describe inflation, as certified by the Planck satellite measurements [56]. We observe that the amount of created particles depends on the mass of the spectator field and on the details of its coupling to the background. In particular, the presence of a small but still non-negligible coupling to the scalar curvature of spacetime is able to produce a significant amount of particles for super-Hubble modes 3 . Low-momentum enhancement of particle production is a peculiar trait of bosonic fields, and it is similarly found in unperturbed GPP scenarios. However, the presence of inhomogeneities allows for mode-mixing in particle production 4 , with the Hubble radius emerging as the natural separation scale for modes during inflation. Consequently, motivated by these facts, we investigate particle production across the Hubble horizon, showing that a perturbative treatment is possible for super-Hubble modes that crossed the horizon well before the end of inflation. Since this approach was recently employed to study the entropy of cosmological perturbations [59-62], we accordingly work the inclusion of geometric and perturbative effects in primordial particle creation mechanisms out, showing that this is not only required for computing the correct abundance of DM candidates, but it may also shed further light on the quantum properties and entropy associated with the created particles. Physical consequences of our approach are promising, confirming that the nature of DM may arise from a spectator field, subdominant throughout the inflationary evolution. The work is organized as follows. In Sec. II, we discuss the features of the spectator DM field, computing the Bogoliubov coefficients related to GPP. In Sec. III, we focus on the geometric contribution to GPP. In Sec. IV, we analyze the main consequences of our results throughout inflationary stages, whereas in Sec. V we emphasize the main results inferred from our findings, highlighting possible quantum signatures, detectable at late-times. Conclusions and perspectives are reported in Sec. VI.",
"pages": [
1,
2
]
},
{
"title": "II. SPECTATOR FIELD DYNAMICS",
"content": "We assume that, besides the inflaton, a subdominant field is present throughout the inflationary phase, with no interaction with the inflaton itself. Consequently, we exclusively consider gravitational interactions, disregarding potential couplings to other quantum fields both during inflation and the subsequent radiation era. In this respect, we consider the Lagrangian density for the spectator, φ , with mass m , where, as above stated, the only interaction is with the background, i.e., we include a nonminimal coupling between the curvature and φ , while the subscript 'S' indicates the nature of φ . Further, in Eq. (1), g is the spacetime metric determinant, while ξ φ describes the coupling strength between the spectator field and the Ricci background scalar, R . To claim that φ represents a spectator field, we require that its energy density is sufficiently small not to affect the inflationary dynamics. Accordingly, its evolution during the slow-roll phase is solely determined by the background potential driving inflation. To this end, we now describe the inflaton dynamics, then selecting the corresponding inflationary potentials in fulfillment of the most recent developments provided by the Planck satellite measurements [56].",
"pages": [
2
]
},
{
"title": "A. Setting up the inflationary scenario",
"content": "During inflation, the background evolution is governed by the inflaton field, ϕ , which we assume of scalar nature. The corresponding Lagrangian density reads where the potential V ( ϕ ) dominates over the other species and the subscript 'I' denotes the inflationary epoch. The inflationary standard paradigm predicts the seeds for gravitational small perturbations, induced by perturbing the inflaton through the standard ansatz [36, 37] where the homogeneous background term, ϕ 0 , is distinct from its corresponding quantum fluctuations, denoted by δϕ , depending on the position and conformal time, τ = ∫ dt/a ( t ), with t the measurable cosmic time. The inflaton background field, hereafter denoted by ϕ instead of ϕ 0 , for simplicity, speeds the universe up by virtue of a quasi-de Sitter phase , yielding the unperturbed metric tensor, where η µν is the Minkowski metric tensor and we assume [32] At this stage, Eq. (5) deserves some additional comments. Particularly, the time τ R describes transition to the radiation dominated era, while H I provides the Hubble parameter during inflation, up to corrections of first order. Moreover, ϵ represents slight deviations from a purely de Sitter phase. By calculating the slow-roll parameter, we can precisely identify it with the latter. As stated above, the presence of inflaton fluctuations induces perturbations on the background spacetime, leading to the perturbed metric tensor Selecting now the longitudinal, or conformal Newtonian gauge [38], it can be shown that scalar perturbations associated with ϕ become particularly simple, where the perturbation potential Ψ satisfies [39] with H = a ' /a and the prime denoting derivative with respect to conformal time. Before studying spacetime perturbations for some specific models of inflation and computing the corresponding particle production, we briefly discuss the evolution of the spectator field, focusing on We underline that, by assuming an instantaneous transition to the radiation era, we are neglecting the effects of reheating on GPP. Nevertheless, we will see that superHubble modes feature very slow dynamics at the end of inflation, so they are typically unaffected by the microphysical processes of thermalization that should take place after the slow-roll regime [32]. It is clear, however, that this picture induces an approximation that may overestimate the production itself. A more comprehensive analysis needs to take into account more refined matching conditions and the effects of backreaction, which will be both subject of future investigations. For the moment, we focus on the slow-roll dynamics of the spectator field and the transition occurring between inflation and radiation epochs.",
"pages": [
2,
3
]
},
{
"title": "B. Spectator field dynamics during inflation",
"content": "Following standard approaches [21, 32], we consider the conformally rescaled spectator field, quantized by where we introduce the comoving momentum, k , and the field modes, g k ( τ ), satisfying the differential equation, Defining now we recall the ansatz of Eq. (5) to obtain the mode evolution during inflation, namely where η = τ -2 τ R . Notice that we exploit the fact that a '' /a ≃ (2 + 3 ϵ ) /η 2 , since ϵ ≪ 1 throughout the slow-roll phase. The solutions of Eq. (13) can be expressed in the form where H (1) ν and H (2) ν are Hankel functions and The integration constants, c 1 ( k ) and c 2 ( k ), are determined by choosing the in vacuum state for the field. A common ansatz consists in employing the Bunch-Davies vacuum state [33-35], that requires the asymptotic condition, This choice implies c 1 ( k ) = √ πe i ( ν + 1 2 ) π 2 / 2 and c 2 ( k ) = 0, so the rescaled field modes take the form Exploiting the asymptotic behaviour of Hankel functions, one can show that on super-Hubble scales, k ≪ aH I , the original field modes are nearly frozen, while they oscillate on sub-Hubble scales, k ≫ aH I [39, 58].",
"pages": [
3,
4
]
},
{
"title": "C. The spectator field transition to radiation era",
"content": "At τ = τ R , we assume an instantaneous transition from inflation to the radiation dominated phase, whose dynamics is still described by the metric tensor of Eq. (4), with a ( τ ) = H R τ . From the continuity of the scale factor at τ R , we have 5 that, under the assumption ϵ ≪ 1, gives where H R ≃ 10 -35 eV [32] and H I can be derived by fixing the energy scales of inflation. During the radiation era, from Eq. (11) we can write which is solved in terms of parabolic cylinder functions. The general solution of Eq. (20) can be expressed in the form where α k , β k are known as Bogoliubov coefficients. The modes f k ( τ ) satisfy Eq. (20) with asymptotic boundary condition where ω k ( τ ) = √ k 2 + m 2 H 2 R τ 2 . In order to properly define the notion of particle and vacuum state during the radiation phase, the adiabatic condition should be satisfied [21]. An upper bound to this ratio is given by modes with negligible momentum, for which the adiabatic approximation gives It can be shown that this condition is verified well before matter-radiation equality, even in case of ultralight DM candidates with m ≪ 1 eV [64]. Accordingly, we can properly associate out particle states to the modes f k ( τ ), which are normalized via the Wronskian condition Spectator field modes also require proper matching conditions at τ = τ R , to ensure continuity of the field energy density at the transition [32]. Thus, imposing and exploiting Eq. (25), one obtains the Bogoliubov coefficients associated with this transition, namely This implies that the field expansion at τ > τ R can be written as where we introduced ˆ b k = α k ˆ a k + β ∗ k ˆ a † -k .",
"pages": [
4
]
},
{
"title": "D. Producing particles from the spectator field",
"content": "By virtue of the above results, we can now identify ˆ b k , ˆ b † k as the ladder operators corresponding to out particle states, obeying canonical quantization conditions. Since in and out vacua are different in general, due to the background expansion, a certain amount of particles is produced via the GPP mechanism. In the Heisenberg picture, the final comoving number density of spectator field particles reads where | 0 ⟩ is the initial Bunch-Davies vacuum state, satisfying the condition, ˆ a k | 0 ⟩ = 0, ∀ k . k ( GeV ) Thus, Eq. (30) implies that the initial vacuum state of the field is no longer seen as a vacuum in the out region. Hence, we can interpret N (0) k as the number density of particles asymptotically produced from cosmic expansion, i.e., in terms of a gravitational production obtained from vacuum. Defining now the quantities, it can be shown that, in the limit of minimal coupling ξ = 0, the number density of gravitationally produced particles for super-Hubble modes kτ R ≪ 1 reads [32, 64] where we introduced the additional parameter, δ ≡ kτ R . It is quite convenient to quantify the rescaled number density, drawn in Fig. 1, where we explore super-Hubble momenta within the range k ∈ [ 10 -5 /τ R , 10 -4 /τ R ] , by assuming different values for H I . We remark that the number density is strongly peaked at low momentum, due to the bosonic nature of the spectator field considered. At the same time, GPP is more , efficient at larger H I , since in this case there is more energy to be converted into particles. In the next section, we will include inflationary perturbations in this framework, showing how the presence of spacetime inhomogeneities is able to enhance the total number of particles produced, also allowing for modemixing in particle creation.",
"pages": [
4,
5
]
},
{
"title": "III. GEOMETRIC CONTRIBUTION TO PARTICLE CREATION",
"content": "Particle production from spacetime perturbations represents an alternative mechanism to the widely-studied GPP approach [40, 41]. In particular, during inflation the presence of inhomogeneities can be traced back to the quantum fluctuations of the inflaton field, which are the fundamental seeds for structure formation in our universe. From Eq. (6), the first-order interaction Lagrangian density describing the coupling between perturbations and a given quantum field can be written in the form where T (0) µν is the zero-order energy-momentum tensor for the field, g (0) the determinant of the background unperturbed metric tensor and H µν = a 2 ( τ ) h µν . When dealing with the spectator scalar field φ introduced in Sec. II, we have [40] Moving now to the interaction picture, it can be shown that the first-order S matrix in Dyson's expansion associated with L I reads ˆ S ≃ 1 + i ˆ T ∫ d 4 x L I . Since both the field potential and the field-curvature coupling term are quadratic in φ , particles are produced in pairs at first perturbative order. We can write the corresponding probability amplitude as [58] where and, similarly, (39) for i = 1 , 2 , 3. In Eqs. (39)-(40), we reintroduced the original field modes during inflation, in order to properly compute the amount of spectator particles produced. For each particle pair, the final state can be written in the form, where the normalization factor N is derived as usual from the condition ⟨ Ψ | Ψ ⟩ = 1. The comoving number density associated with a geometric production of particles can be then computed at first and second perturbative order, giving respectively",
"pages": [
5,
6
]
},
{
"title": "A. Amplitudes and orders of particle production",
"content": "It is relevant to stress that probability amplitudes for pair production are typically small in our perturbative approach. We thus have |N| 2 ≃ 1 and we can neglect the normalization constant in the further computations. Analogously, we underline that, when computing number densities in the interaction picture, the zero-order term of Eq. (30) also acquires a normalization factor. Specifically, the final state of the system is modified by the interaction itself, implying that the contribution of Eqs. (43)-(44) with respect to the background GPP term is always independent from the normalization procedure. In particular, we notice that the first order term in Eq. (43) involves creation of particles with opposite momenta, thus only increasing the total number of particleantiparticle pairs. For this reason, we will focus on the second order term, which instead introduces modemixing in particle production. In particular, we are interested in superhorizon pair production, so we pick one mode on super-Hubble scales and the other on subHubble ones, namely where τ i is the initial time for inflation and the ultraviolet cutoff is given by the Planck mass M pl . Moreover, we assume that there are no super-Hubble modes at the beginning of inflation 6 . Exploiting the properties of Hankel functions, from Eqs. (17) and (41), we can write [58] In the following, we specify our calculations to some relevant inflationary potentials, in order to compute the amount of geometric particle produced during the slowroll phase.",
"pages": [
6
]
},
{
"title": "IV. THEORETICAL CONSEQUENCES OF INFLATIONARY PARTICLE PRODUCTION",
"content": "As above stated, our particle computation depends on the underlying inflationary potential. Thus, to accurately compute the production of geometric particles during the slow-roll phase, it is mandatory to meticulously identify the most promising approaches that agree with current observations. The Planck satellite's numerical findings suggest that two categories of potentials remain viable, namely large and small field potentials [56]. 1 Even though appealing, the class of small field potentials is expected to provide a very small geometric particle production across the Hubble horizon, being incompatible with the possibility that DM can arise from perturbative approaches, see e.g. [42, 58]. In other words, inflationary particle production from inhomogeneities is typically inefficient if the energy in the inflaton field is not large enough. This implies that the substantial energy released during inflation has the potential to be physically converted into particles. In this respect, we focus on two main large-field inflationary potentials: Below, we discuss the production of particles in both the aforementioned schemes.",
"pages": [
7
]
},
{
"title": "A. Particles produced from Starobinsky potential",
"content": "Here, the metric tensor can be conformally rescaled into the Einstein frame, where the action takes the form of Eq. (2), with corresponding potential [52, 53] and Λ 4 describes the energy scales of inflation. Recalling Eq. (3), the background dynamics of this effective field during slow-roll is given by where we have introduced the compact notation V ,ϕ ≡ ∂V/∂ϕ and the scale factor during inflation has been defined in Eq. (5). The corresponding fluctuation modes are described by [39] where the fluctuation field has been rescaled as usual by δχ k = δϕ k a . This equation admits solutions in terms of Hankel functions, provided the potential term is substituted by its mean value during slow-roll [58]. Once obtained the background and fluctuation dynamics, the perturbation potential Ψ can be derived from Eq. (8) and inserted in Eq. (36) after proper normalization 7 , to compute the amount of spectator field particles arising from perturbations. In Fig. 2, we show the ratio N (2) /N (0) as function of the super-Hubble mode k 1 . In particular, the probability amplitude for perturbative production is evaluated in the range τ ∈ [0 , τ R ], in order to exploit the simplified expression of Eq. (46) for the modes under investigation 8 . At the same time, we neglect perturbative production during the radiation era, where the contribution of inhomogeneities is expected to be much smaller due to the presence of other quantum fields and possible backreaction mechanisms.",
"pages": [
7,
8
]
},
{
"title": "B. Particles produced from a nonminimal fourth order chaotic potential",
"content": "As an alternative scenario, we discuss a nonminimally coupled inflaton field driven by a quartic symmetrybreaking potential. This gives where v is the vacuum expectation value of the inflaton field and λ a self-coupling constant. In case of positive coupling constant, quartic chaotic inflation is not expected to work, unless the inflaton coupling to curvature is sufficiently small [54, 55] (see also [66]). This model has been recently considered for perturbative particle production in inflationary scenarios [47, 58] and it may also allow to identify the Standard Model Higgs field as the inflaton [67-69]. Following the same steps of Sec. IV A, we can derive the dynamics of the perturbation potential in this model and then compute the corresponding number densities of particles arising from inhomogeneities. In Fig. 3 we show again the number density of geometric particles produced at second perturbative order, normalized with respect to the unperturbed GPP contribution.",
"pages": [
8
]
},
{
"title": "V. CONSEQUENCES AND PREDICTIONS OF OUR SCENARIOS",
"content": "We here discuss the main implications of our findings in inflationary stages and possible signatures of our scenarios. Particularly, let us first observe, from Figs. 2-3, that perturbative particle production is non-negligible for modes that crossed the Hubble horizon mainly before the cutoff time, τ = 0. Second, we also notice that, when approaching the infrared cutoff | k 1 | ≃ a ( τ i ) H I , the contribution of inhomogeneities is typically larger. In particular, it can be shown that a perturbative treatment is no longer possible at sufficiently small | k 1 | , thus requiring a different technique to evaluate the effects of inflaton fluctuations. The above issue is also related to the normalization procedure for the perturbation potential Ψ, whose amplitude is typically fixed at horizon crossing [70]. Specifically, a correct normalization procedure might depend upon modes in order to guarantee that all the perturbation magnitudes appear the same at the horizon crossing. Likely, this would imply to reformulate the correct vacuum, modifying the Bunch-Davies choice with a more refined approach. Further, we notice that the contribution of inhomogeneities is typically enhanced in case of larger fieldcurvature coupling constants. In the limit of conformal coupling, ξ = 1 / 6, GPP results in negligible densities [32], implying that geometric production would become the dominant mechanism for primordial particle creation. A similar result was obtained for the gravitational production of massless fermions during preheating [71], showing that metric perturbations may have also played an important role at the end of inflation. To summarize, particle production arising from inhomogeneities can significantly affect the total number density of spectator field particles created up to the radiation era. For this reason, if DM has been produced via purely gravitational mechanisms, the presence of inhomogeneities should be taken into account when computing the corresponding particle abundance. Last but not least, we also remarked how spacetime inhomogeneities are responsible for mode-mixing in particle production, that is not conversely found in unperturbed GPP scenarios, where only particle-antiparticle pairs can be generated, i.e., with opposite momenta. More specifically, during inflation the Hubble horizon emerges as a natural separation scale for modes and superhorizon particle production has been recently investigated for inflaton fluctuations, showing that quantum entanglement can be generated in this process [58]. Remarkably, since we focused on particle production across the Hubble horizon, we conclude that plausible detectable quantum 'signatures' at late times can occur. Indeed, DM is expected to weakly interact with Standard Model fields, so that some entanglement entropy associated with particle production may have survived after the inflationary epoch. Hence, we underline that the role of entanglement could help to understand the quantum properties of produced particles, thus opening new avenues in the search for DM.",
"pages": [
8
]
},
{
"title": "VI. FINAL OUTLOOKS AND PERSPECTIVES",
"content": "In this work, we investigated the particle production associated with a spectator scalar field, i.e., subdominant with respect to the inflaton, during and after the slow-roll regime. To show how particle production is influenced by the universe expansion, we pictured an instantaneous transition from inflation to the radiation dominated era, neglecting the effects due to reheating in GPP regimes. In particular, we focused on the contribution associated with inhomogeneous particle production across the Hubble horizon, that can be traced back to the fluctuations of the inflaton field during slow-roll. We thus showed that the number density of particles arising from perturbations is typically non-negligible with respect to the widely-studied quantum GPP contribution, obtained from the unperturbed universe expansion. We reobtained this outcome in the realms of large-field inflation and, particularly, we focused on two among the most consolidate paradigms describing the inflationary speed up. Specifically, we worked out the Starobinsky and the fourth order nonminimally coupled potentials. The latter represents the most viable large-field model of inflation, conformally equivalent to an extended theory of gravity, whereas the second is a suitable example of chaotic inflation, overcoming the Planck satellite observational constraints. We discussed the physical results obtained and, particularly, we showed that the amount of particles obtained is similar in both the aforementioned scenarios. We also argued that geometric particle production across the horizon is expected to be negligible in small-field approaches, since in that case the energy in the inflaton field is significantly smaller throughout the slow-roll regime. In addition, we observed that the presence of inhomogeneities allows for mode-mixing in particle production, that instead is not found in unperturbed GPP processes, where the total momentum of created particles is necessarily conserved. The presence of mode-mixing may lead to entanglement generation across the Hubble horizon, and we argued that such quantum correlations could have survived after the inflationary epoch due to the weakly interacting nature of DM. Possibility of detecting such particles through entanglement are also discussed above. At the same time, we noticed that a perturbative approach to inhomogeneous particle production is not always possible, since the magnitude of inflaton fluctuations becomes typically large on super-Hubble scales. We also pointed out that a more refined approach is needed for the normalization of the perturbation potential, in order to obtain correct amplitudes at horizon crossing for all the modes involved. As perspectives, further steps would include the effects of reheating in geometric production. Although such effects may be negligible at a first sight, they may affect the total amount of produced particles via the dynamics of preheating metric perturbations, especially for subHubble modes. In addition, we intend to study the possible backreaction effects associated with the dynamics of perturbations at the end of inflation and shed further light on a more general non-perturbative approach to inhomogeneous particle production. Finally, we plan to extend our treatment to higher spin spectator fields, starting from fermionic ones, with the aim of evaluating other possible DM candidates.",
"pages": [
8,
9
]
},
{
"title": "Acknowledgements",
"content": "The work of OL is partially financed by the Ministry of Education and Science of the Republic of Kazakhstan, Grant: IRN AP19680128. ment , Phys. Rev. D 95 , 065023 (2017).",
"pages": [
9,
11
]
}
]
2024PhRvD.110b4058T
https://arxiv.org/pdf/2311.17767.pdf
<document>
<section_header_level_1><location><page_1><loc_9><loc_90><loc_92><loc_93></location>Forecasting constraints on the no-hair theorem from the stochastic gravitational wave background</section_header_level_1>
<text><location><page_1><loc_38><loc_87><loc_63><loc_89></location>Chen Tan 1 , 2 , 3 and Ke Wang 1 , 2 , 3 ∗</text>
<text><location><page_1><loc_14><loc_85><loc_88><loc_87></location>Institute of Theoretical Physics & Research Center of Gravitation, Lanzhou University, Lanzhou 730000, China 2 Key Laboratory of Quantum Theory and Applications of MoE,</text>
<text><location><page_1><loc_34><loc_83><loc_67><loc_84></location>Lanzhou University, Lanzhou 730000, China and</text>
<text><location><page_1><loc_26><loc_82><loc_75><loc_83></location>Lanzhou Center for Theoretical Physics & Key Laboratory of Theoretical</text>
<text><location><page_1><loc_13><loc_82><loc_26><loc_87></location>1 3</text>
<text><location><page_1><loc_26><loc_81><loc_75><loc_82></location>Physics of Gansu Province, Lanzhou University, Lanzhou 730000, China</text>
<text><location><page_1><loc_43><loc_79><loc_57><loc_80></location>(Dated: July 1, 2024)</text>
<text><location><page_1><loc_18><loc_61><loc_83><loc_78></location>Although the constraints on general relativity (GR) from each individual gravitational-wave (GW) event can be combined to form a cumulative estimate of the deviations from GR, the ever-increasing number of GW events used also leads to the ever-increasing computational cost during the parameter estimation. Therefore, in this paper, we will introduce the deviations from GR into GWs from all events in advance and then create a modified stochastic gravitational-wave background (SGWB) to perform tests of GR. More precisely, we use the pSEOBNRv4HM PA model to include the modelindependent hairs and calculate the corresponding SGWB with a given merger rate. Then we turn to the Fisher information matrix to forecast the constraints on the no-hair theorem from SGWB at frequency 10[Hz] ≲ f ≲ 10 3 [Hz] detected by the third-generation ground-based GW detectors, such as the Cosmic Explorer. We find that the forecasting constraints on hairs at 68% confidence range are δω 220 = 0 ± 0 . 1296 and δτ 220 = 0 ± 0 . 0678 when the flat priors about the merger rate are added but δω 220 = 0 ± 0 . 0903 and δτ 220 = 0 ± 0 . 0608 when the non-flat priors about the merger rate are added.</text>
<section_header_level_1><location><page_1><loc_20><loc_58><loc_37><loc_59></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_22><loc_49><loc_55></location>There are about 90 compact binary coalescences observed by the LIGO Scientific Collaboration [1], the Virgo Collaboration [2] and the KAGRA Collaboration [3] during the first three observing runs [4-6]. Through these observed gravitational-wave (GW) transient events, the full population of merging compact binaries surrounding us can be further inferred [7, 8]. Consequently, a superposition of GWs from the surrounding population of these astrophysical sources will create a stochastic gravitational-wave background (SGWB) with energy density Ω GW ( f ) ∼ 10 -9 at frequency 10[Hz] ≲ f ≲ 10 3 [Hz] [7]. Due to the limited sensitivity of the current ground-based GW detectors, however, no evidence for a SGWB at this frequency span was found [9-11]. Encouragingly, there are multiple lines of evidence for an excess SGWB signal with amplitude A SGWB ∼ 10 -14 at frequency f ∼ 10 -8 [Hz] recently in the NANOGrav 15-year data [12] and the second data release from EPTA [13]. Therefore, there is an excellent probability that a SGWB at frequency 10[Hz] ≲ f ≲ 10 3 [Hz] can be detected by the third-generation ground-based GW detectors, such as the Cosmic Explorer (CE) [14] and the Einstein Telescope (ET) [15].</text>
<text><location><page_1><loc_9><loc_14><loc_49><loc_21></location>The ever-increasing number of detections of GWs from compact binaries by the current ground-based GW detectors allows the ever-more sensitive tests of general relativity (GR) with GW generation, propagation or polarization [16-18]. More precisely, one can per-</text>
<text><location><page_1><loc_52><loc_33><loc_92><loc_59></location>form residuals test [19, 20], inspiral-merger-ringdown (IMR) consistency test [21, 22], extra polarization contents searches [23, 24], echoes searches [25, 26] as well as parametrized tests of GW generation and propagation [27-29], spin-induced multipole moment effects [30, 31], modified GW dispersion relation [32-37] and the no-hair theorem [38-40]. All of there tests will introduce some ad hoc parameters to account for the deviations from GR. Since these new-introduced parameters are independent of the individual sources by construction and cleanly encode information about the underlying theory of gravity, the constraints on them from each individual GW event can be combined to form a cumulative estimate of the deviations from GR. The more GW events are taken into consideration, the tighter cumulative constraints are obtained. However, the ever-increasing number of GW events used also leads to the ever-increasing computational cost during the parameter estimation.</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_31></location>To avoid the time-consuming parameter estimation procedure, one can combine GWs from all events in advance and then perform tests of GR. That is to say, we can use SGWB at frequency 10[Hz] ≲ f ≲ 10 3 [Hz] to test GR. While the information of GW propagation may wash out in SGWB, the information of GW generation and polarization must survive SGWB and some deviations from GR's GW generation and polarization will lead to some suppressions or enhancements in SGWB. If such suppressions or enhancements have no particular distinguishing features, testing GR with SGWB must deals with the degeneracy between models. However, if one would not confine oneself to specific modified gravities, one can search for extra polarization contents in SGWB model-independently [41, 42]. Similarly, in this paper, we will probe the deviations from GR's GW gen-</text>
<text><location><page_2><loc_9><loc_89><loc_49><loc_93></location>ion with SGWB directly. More precisely, we will take the violation of the no-hair theorem as a heuristic example.</text>
<text><location><page_2><loc_9><loc_51><loc_49><loc_89></location>Under the no-hair theorem, the ringdown frequencies and damping times of a perturbed Kerr black hole (BH) in GR can be predicted by its mass and dimensionless spin [43, 48]. Without the no-hair conjecture, a perturbed Kerr BH's ringdown frequencies and damping times are also dependent on the extra hairs/parameters [38-40]. In fact, violating the no-hair theorem may also modify the whole IMR waveform for a BH binary. Such overall modifications will degenerate with other overall effects, for example changing the total mass of the BH binary, hence the final mass of its remnant Kerr BH. That is to say, when the extra hairs dominate the inspiral and merger regions of a BH binary, the mass and dimensionless spin of its remnant Kerr BH can't be predicted by its intrinsic parameters only, hence an unknown ringdown region. Therefore, for simplicity, we assume that the effects of the extra hairs on the inspiral and merger regions can be summarized by a set of effective intrinsic parameters under the no-hair theorem and then the residual effects of these extra hairs will just appear during an effective ringdown region. It is generally the case when different modified gravity theories are considered. Here we validate our above assumption with a specific modified theory of gravity, as shown in Fig. 1. Finally the effective complete IMR time-domain waveform can be described by pSEOBNRv4HM PA model [46, 47] explicitly.</text>
<text><location><page_2><loc_9><loc_42><loc_49><loc_51></location>This paper is organized as follows. In section II, we calculate the stochastic gravitational wave background at frequency 10[Hz] ≲ f ≲ 10 3 [Hz] without the no-hair conjecture. In section III, we forecast the constraints on the no-hair theorem from SGWB. Finally, a brief summary and discussions are included in section IV.</text>
<section_header_level_1><location><page_2><loc_11><loc_35><loc_47><loc_39></location>II. STOCHASTIC GRAVITATIONAL WAVE BACKGROUND WITHOUT THE NO-HAIR CONJECTURE</section_header_level_1>
<section_header_level_1><location><page_2><loc_19><loc_32><loc_39><loc_33></location>A. Analytical Calculation</section_header_level_1>
<text><location><page_2><loc_9><loc_25><loc_49><loc_29></location>The radiative degrees of freedom in transversetraceless (TT) gauge can be written in terms of two polarizations h + and h × :</text>
<formula><location><page_2><loc_21><loc_19><loc_49><loc_24></location>h ij = h + h × 0 h × -h + 0 0 0 0 . (1)</formula>
<text><location><page_2><loc_9><loc_15><loc_49><loc_17></location>After Fourier transform, their representation in the frequency domain is ˜ h ij = F ( h ij )</text>
<formula><location><page_2><loc_21><loc_9><loc_49><loc_13></location>˜ h ij = ˜ h + ˜ h × 0 ˜ h × -˜ h + 0 0 0 0 . (2)</formula>
<text><location><page_2><loc_52><loc_92><loc_85><loc_93></location>Then the energy-momentum tensor of GWs is</text>
<formula><location><page_2><loc_62><loc_88><loc_92><loc_91></location>T µν = c 4 32 πG 〈 ∂ µ h ij ∂ ν h ij 〉 , (3)</formula>
<text><location><page_2><loc_52><loc_82><loc_92><loc_87></location>where ⟨ ... ⟩ indicates averaging over several wavelengths or periods. The energy density of GWs is just the 00 component. So the energy flux through a sphere of radius R is</text>
<formula><location><page_2><loc_62><loc_78><loc_92><loc_81></location>dE dt = c 3 R 2 32 πG ∫ d Ω 〈 ˙ h ij ˙ h ij 〉 , (4)</formula>
<text><location><page_2><loc_52><loc_76><loc_74><loc_77></location>and the total energy emitted is</text>
<formula><location><page_2><loc_61><loc_71><loc_92><loc_75></location>E = c 3 R 2 32 πG ∫ d Ω ∫ ∞ -∞ dt ˙ h ij ˙ h ij , (5)</formula>
<text><location><page_2><loc_52><loc_64><loc_92><loc_71></location>where dots denote derivative with respect to physical time t and d Ω is a solid angle element. Here we are considering all time. So ⟨ ... ⟩ disappears in the integral with respect to t . We can rewrite the total energy in the frequency domain as</text>
<formula><location><page_2><loc_58><loc_59><loc_92><loc_63></location>E = πc 3 R 2 4 G ∫ d Ω ∫ ∞ 0 dff 2 ˜ h ij ( f ) ˜ h ∗ ij ( f ) . (6)</formula>
<text><location><page_2><loc_52><loc_57><loc_77><loc_59></location>So the energy spectrum is given by</text>
<formula><location><page_2><loc_55><loc_50><loc_92><loc_56></location>dE df = πc 3 R 2 f 2 4 G ∫ d Ω ˜ h ij ( f ) ˜ h ∗ ij ( f ) = πc 3 R 2 f 2 2 G ∫ d Ω ( ∣ ∣ ∣ ˜ h + ( f ) ∣ ∣ ∣ 2 + ∣ ∣ ∣ ˜ h × ( f ) ∣ ∣ ∣ 2 ) . (7)</formula>
<text><location><page_2><loc_52><loc_43><loc_92><loc_49></location>Next we will do the integral with respect to d Ω. The general GW signal can be described as a linear combination of the two polarization states or expressed in terms of series of spin-weighted spherical harmonics</text>
<formula><location><page_2><loc_63><loc_37><loc_92><loc_42></location>h ≡ h + -ih × = ∑ lm h lm Y -2 lm ( φ, ι ) , (8)</formula>
<text><location><page_2><loc_52><loc_22><loc_92><loc_36></location>where φ is the azimuthal direction to the observer and ι is the inclination angle. Here we use the spin-weighted spherical harmonics to describe the whole IMR waveform including the ringdown waveform even thought the spin-weighted spheroidal harmonics are more accurate for the remnant Kerr BHs. In fact, the spherical version is a sufficiently good approximation of its spheroidal counterpart [48, 49]. For the two dominant modes with l = 2 and m = ± 2, the spin-weighted spherical harmonics are [50, 51]</text>
<formula><location><page_2><loc_58><loc_16><loc_92><loc_21></location>Y -2 2 -2 ( φ, ι ) ≡ √ 5 64 π (1 -cos ι ) 2 e -2 iφ , Y -2 22 ( φ, ι ) ≡ √ 5 64 π (1 + cos ι ) 2 e 2 iφ . (9)</formula>
<text><location><page_2><loc_52><loc_12><loc_92><loc_15></location>When φ = 0 and ι = 0, for example, we have the GW signal as</text>
<formula><location><page_2><loc_62><loc_8><loc_92><loc_11></location>h ( φ = ι = 0) ≈ 4 √ 5 64 π h 22 . (10)</formula>
<figure>
<location><page_3><loc_18><loc_79><loc_80><loc_93></location>
<caption>FIG. 1: The IMR time-domain waveforms predicted by GR and four-derivative scalar-tensor theories (4 ∂ ST) [45]. The black dashed curve is plotted with { m 1 = m 2 = 50 M ⊙ , s 1 , z = s 2 , z = 0 . 40 } in GR and its 4 ∂ ST counterpart (the blue dotted curve) is also given. The effects of the extra scalar hairs from 4 ∂ ST on the inspiral and merger regions can be summarized by a set of effective intrinsic parameters { m 1 = 53 . 5 M ⊙ , m 2 = 50 . 6 M ⊙ , s 1 , z = 0 . 32 , s 2 , z = 0 . 45 } in GR and the residual effects of these extra scalar hairs just appear in ringdown region (the red solid curve). More precisely, there are obvious differences between the red solid curve and the blue dotted curve in ringdown region while there is an almost complete overlap between these two curves before merger.</caption>
</figure>
<text><location><page_3><loc_9><loc_61><loc_32><loc_62></location>According to Eq. (8), we rewrite</text>
<formula><location><page_3><loc_9><loc_55><loc_52><loc_60></location>dE df ≈ πc 3 R 2 f 2 2 G ∫ d Ω ( ∣ ∣ ∣ ˜ h 22 Y -2 22 ( φ, ι ) ∣ ∣ ∣ 2 + ∣ ∣ ∣ ˜ h 2 -2 Y -2 2 -2 ( φ, ι ) ∣ ∣ ∣ 2 ) (11)</formula>
<text><location><page_3><loc_52><loc_57><loc_92><loc_62></location>. fixed angles, for example φ = 0 and ι = π/ 2. Unless otherwise stated, in our following calculations, we used the (2 , 2) mode only.</text>
<text><location><page_3><loc_52><loc_55><loc_92><loc_58></location>The dimensionless GW energy density per logarithmic frequency interval is</text>
<text><location><page_3><loc_9><loc_52><loc_49><loc_55></location>Due to the orthonormality of spin-weighted spherical harmonics, the integral can be done easily</text>
<formula><location><page_3><loc_16><loc_44><loc_49><loc_51></location>dE df ≈ πc 3 R 2 f 2 2 G ( ∣ ∣ ∣ ˜ h 22 ∣ ∣ ∣ 2 + ∣ ∣ ∣ ˜ h 2 -2 ∣ ∣ ∣ 2 ) ≈ πc 3 R 2 f 2 G ∣ ∣ ∣ ˜ h 22 ∣ ∣ ∣ 2 , (12)</formula>
<text><location><page_3><loc_9><loc_39><loc_49><loc_43></location>where we have used the equatorial symmetry h 22 = ( h 2 -2 ) ∗ . According to Eq. (8) and Eq. (10), we can express the energy spectrum approximately as</text>
<formula><location><page_3><loc_66><loc_51><loc_92><loc_54></location>Ω GW = f ρ c dρ GW df , (15)</formula>
<text><location><page_3><loc_52><loc_46><loc_92><loc_50></location>where ρ c = 3 H 2 0 c 2 / 8 πG is the critical energy density and H 0 = 67 . 36[km s -1 Mpc -1 ] [52]. The contribution of BH binary mergers can be estimated as [9]</text>
<formula><location><page_3><loc_54><loc_41><loc_92><loc_44></location>Ω GW , BH = f ρ c ∫ z max 0 dz R BH ( z ) ⟨ dE s /df s ⟩ BH (1 + z ) H ( z ) , (16)</formula>
<formula><location><page_3><loc_9><loc_31><loc_52><loc_38></location>dE df ≈ 4 π 2 c 3 R 2 f 2 5 G ∣ ∣ ∣ ˜ h ( φ = ι = 0) ∣ ∣ ∣ 2 (13) ≈ 4 π 2 c 3 R 2 f 2 5 G ( ∣ ∣ ∣ ˜ h + ( φ = ι = 0) ∣ ∣ ∣ 2 + ∣ ∣ ∣ ˜ h × ( φ = ι = 0) ∣ ∣ ∣ 2 )</formula>
<text><location><page_3><loc_9><loc_23><loc_49><loc_30></location>Although here we have fixed the two angles φ = ι = 0, the values of them don't affect the total energy emitted. When the higher-order modes are taken into consideration, we can express the energy spectrum more accurately as</text>
<formula><location><page_3><loc_14><loc_14><loc_49><loc_22></location>dE df = πc 3 R 2 f 2 2 G ∑ lm ∫ d Ω ∣ ∣ ∣ ˜ h lm Y -2 lm ( φ, ι ) ∣ ∣ ∣ 2 = πc 3 R 2 f 2 G ∑ l | m | ∣ ∣ ∣ ˜ h lm ∣ ∣ ∣ 2 , (14)</formula>
<text><location><page_3><loc_9><loc_8><loc_49><loc_13></location>where we have used the equatorial symmetry of all modes ( l, | m | ) and ˜ h lm can be numerically obtained because ˜ h lm Y -2 lm ( φ, ι ) as a whole can be numerically obtained at</text>
<text><location><page_3><loc_51><loc_28><loc_92><loc_40></location>. where f s = f (1 + z ) is frequency in the source frame. The Hubble parameter is H ( z ) = H 0 √ Ω m (1 + z ) 3 +Ω Λ , where Ω m = 0 . 3153 is the matter density parameter and Ω Λ = 0 . 6847 is the dark energy density parameter [52]. The quantity ⟨ dE s /df s ⟩ GW , BBHs is the source-frame energy radiated by a single source, which should be averaged over the ensemble properties of the full BH binary population</text>
<formula><location><page_3><loc_59><loc_24><loc_92><loc_27></location>⟨ dE s /df s ⟩ BH = ∫ dα p BH ( α ) dE s df s ( α ) , (17)</formula>
<text><location><page_3><loc_52><loc_9><loc_92><loc_23></location>where p BH ( α ) is the probability distribution of intrinsic parameters α (e.g. masses, spins, etc.) for the full BH binary population. The merger rate R BH ( z ) can be obtained by a convolution of the BH binary formation rate R f BH ( z f ) with the distribution of the time delays P ( t d ) between BH binary formation and merger [53, 54]. Also R BH ( z ) can be obtained from a merger rate R BH ( z quoted ) at a quoted redshift z quoted = 0 [55] or z quoted = 0 . 2 [7]. Here we turn to the fiducial Power Law + Peak (PP) model [7, 8], as shown in Fig. 10 of [7], where</text>
<text><location><page_4><loc_9><loc_85><loc_49><loc_93></location>R BH ( z quoted = 0 . 2) = 28 . 3[Gpc -3 yr -1 ] is obtained by integrating the primary mass distribution d R BH d m 1 or the mass ratio distribution d R BH d q . After integrating over the contribution of spins, p BH ( α ) ≈ 1 R 2 BH d R BH d m 1 d R BH d q serves as a good approximation.</text>
<section_header_level_1><location><page_4><loc_19><loc_80><loc_39><loc_81></location>B. Numerical Calculation</section_header_level_1>
<text><location><page_4><loc_9><loc_63><loc_49><loc_78></location>Given the waveform of every GW event, we can calculate the total SGWB energy-density spectrum numerically according to Eq. (16). Here we will consider the deviations from GR's GW generation, namely violating the no-hair theorem. First we turn to the PyCBC package[56] and the pSEOBNRv4HM PA model [46, 47] to generate the IMR time-domain waveforms for every possible BH binary with different primary mass and mass ratio allowed by a given fiducial PP model [7, 8] individually. In the pSEOBNRv4HM PA model, the ad hoc hairs δω lmn and δτ lmn are introduced as</text>
<formula><location><page_4><loc_20><loc_58><loc_49><loc_61></location>ω lmn = ω GR lmn (1 + δω lmn ) , τ lmn = τ GR lmn (1 + δτ lmn ) , (18)</formula>
<text><location><page_4><loc_9><loc_54><loc_49><loc_56></location>where ω lmn and τ lmn are the ringdown frequencies and damping times respectively.</text>
<text><location><page_4><loc_9><loc_20><loc_49><loc_53></location>Why we can fix the fiducial PP model for different values of { δω lmn , δτ lmn } ? The fiducial PP model, in our paper, just provides a effective mass distribution when we summarize the effects of { δω lmn , δτ lmn } on the inspiral and merger regions with a set of effective intrinsic parameters. Given a modified theory of gravity, the effective mass distribution will vary with the values of { δω lmn , δτ lmn } to compensate for the changes of the inspiral and merger regions. Similarly, given the values of { δω lmn , δτ lmn } , the effective mass distribution will also vary with the different modified theory of gravity to compensate for the corresponding changes of the inspiral and merger regions. Under different situations of the extra hairs, however, the effective mass distribution provided by the fiducial PP model can be unchanged in certain cases. For example, we can fix the effective mass distribution as the realistic one picked out by the observations of GWtransients during our calculations for different values of { δω lmn , δτ lmn } through adjusting the modified theory of gravity simultaneously. That is to say, our modelindependent parameterization of Eq. (18) is not confined to a specific modified theory of gravity, but aims to identify any deviation from GR.</text>
<text><location><page_4><loc_9><loc_9><loc_49><loc_20></location>We plot the IMR time-domain waveforms from a GW150914-like event for '+'-polarization in Fig. 2, where the luminosity distance is 1[Mpc]. By definition, δω lmn affects the ringdown frequencies (green curves) and δτ lmn affects the ringdown damping times (cyan curves). Then we make the Fourier transform of the IMR time-domain waveforms to get the IMR frequencydomain waveforms. We plot the IMR frequency-domain</text>
<text><location><page_4><loc_52><loc_57><loc_92><loc_93></location>waveforms from a GW150914-like event for '+'-polarization in Fig. 3, where the black dashed curve is its GR counterpart and plotted directly by the IMRPhenomD model [57]. We find that the effects of δω lmn on the IMR frequency-domain waveform is lager than that of δτ lmn . Next, according to Eq. (13), we can get the energy spectrum for every BH binary. We plot the energy spectrum for a GW150914-like event in Fig. 4, where the black dashed curve is the analytical energy spectrum [50, 55]. Again we find that the effects of δω lmn on the energy spectrum is lager than that of δτ lmn . Finally, we plot the total SGWB energy-density spectrua at frequency 10[Hz] ≲ f ≲ 10 3 [Hz] in Fig. 5, where the ratio of the contribution of BH binary mergers to the contribution of neutron star (NS) binary and neutron star-black hole (NS-BH) mergers is supposed to be 1 : 0 . 3 which is consistent with the forecast by [7]. Compared with Fig. 3 and Fig. 4, we confirm that the effects of δω lmn on the total SGWB energy-density spectrum is lager than that of δτ lmn . However, the frequency where their effects on the total SGWB energy-density spectrum become obvious is dependent on the values of them because SGWB results from the superposition of many GWs. For example, the blue dashed curve just deviates the black solid curve obviously at f > 10 3 [Hz].</text>
<text><location><page_4><loc_52><loc_41><loc_92><loc_57></location>It is worth noting that the pSEOBNRv4HM PA model is for spin-aligned compact binaries moving in quasicircular orbits. For a given fiducial PP model, this approximation of the inspiral region will only affects the total SGWB energy-density spectra at the lower frequency span 10[Hz] ≲ f ≲ 10 2 [Hz] while the effects of { δω lmn , δτ lmn } dominate at the higher frequency span 10 2 [Hz] ≲ f ≲ 10 3 [Hz]. As shown in Fig. 5, these two parts don't correlate with each other directly and our forecasting constraints on { δω lmn , δτ lmn } will not be influenced by this approximation.</text>
<section_header_level_1><location><page_4><loc_53><loc_36><loc_91><loc_38></location>III. FORECASTING CONSTRAINTS ON THE NO-HAIR THEOREM</section_header_level_1>
<text><location><page_4><loc_52><loc_28><loc_92><loc_34></location>A stationary, Gaussian, unpolarized and isotropic SGWB can be detected by a cross-correlation statistic C ( f ) between two GW detectors [9-11]. The Gaussian likelihood for the measured cross-correlations C ( f ) is</text>
<formula><location><page_4><loc_55><loc_23><loc_92><loc_27></location>L ∝ exp ( -1 2 ∑ k [ C ( f k ) -Ω GW ( f k ; θ )] 2 σ 2 ( f k ) ) , (19)</formula>
<text><location><page_4><loc_52><loc_21><loc_73><loc_22></location>where the variance of C ( f ) is</text>
<formula><location><page_4><loc_58><loc_16><loc_92><loc_20></location>σ 2 ( f ) ≈ 1 2 Tdf P 1 ( f ) P 2 ( f ) γ 2 T ( f ) ( 10 π 2 f 3 3 H 2 0 ) 2 . (20)</formula>
<text><location><page_4><loc_52><loc_9><loc_92><loc_16></location>Here we assume one year's worth of data T = 1year and a frequency bin width of df = 0 . 25[Hz]. P ( f ) 1 , 2 are the one-sided noise power spectral densities of the two GW detectors, which are choosen as CE's [58]. Since the actual locations and arm orientations for CE are yet to be</text>
<figure>
<location><page_5><loc_22><loc_66><loc_78><loc_91></location>
<caption>FIG. 2: The IMR time-domain waveforms from a GW150914-like event for '+'-polarization given by the pSEOBNRv4HM PA model [46, 47], where the luminosity distance to the binary is set as 1Mpc. The GR version with { δτ 220 = 0 , δω 220 = 0 } is the black solid curve; the non-GR version with { δτ 220 = 0 . 5 , δω 220 = 0 . 5 } , { δτ 220 = -0 . 5 , δω 220 = -0 . 5 } , { δτ 220 = -0 . 5 , δω 220 = 0 . 5 } , { δτ 220 = 0 . 5 , δω 220 = -0 . 5 } , { δτ 220 = 0 , δω 220 = 0 . 5 } , { δτ 220 = 0 , δω 220 = -0 . 5 } , { δτ 220 = 0 . 5 , δω 220 = 0 } or { δτ 220 = -0 . 5 , δω 220 = 0 } is plotted by the red dashed, red dotted, blue dashed, blue dotted, green dashed, green dotted, cyan dashed or cyan dotted curve respectively.</caption>
</figure>
<figure>
<location><page_5><loc_21><loc_24><loc_77><loc_49></location>
<caption>FIG. 3: The IMR frequency-domain waveforms from a GW150914-like event for '+'-polarization calculated by the Fourier transform of Fig. 2. The GR version with { δτ 220 = 0 , δω 220 = 0 } is the black solid curve; the non-GR version with { δτ 220 = 0 . 5 , δω 220 = 0 . 5 } , { δτ 220 = -0 . 5 , δω 220 = -0 . 5 } , { δτ 220 = -0 . 5 , δω 220 = 0 . 5 } , { δτ 220 = 0 . 5 , δω 220 = -0 . 5 } , { δτ 220 = 0 , δω 220 = 0 . 5 } , { δτ 220 = 0 , δω 220 = -0 . 5 } , { δτ 220 = 0 . 5 , δω 220 = 0 } or { δτ 220 = -0 . 5 , δω 220 = 0 } is plotted by the red dashed, red dotted, blue dashed, blue dotted, green dashed, green dotted, cyan dashed or cyan dotted curve respectively; the black dashed curve is the GR IMR frequency-domain waveform from a GW150914-like event for '+'-polarization and plotted directly by the IMRPhenomD model [57].</caption>
</figure>
<figure>
<location><page_6><loc_21><loc_67><loc_77><loc_92></location>
<caption>FIG. 4: The energy spectrum for a GW150914-like event calculated numerically with the pSEOBNRv4HM PA model's [46, 47] waveforms. The GR version with { δτ 220 = 0 , δω 220 = 0 } is the black solid curve; the non-GR version with { δτ 220 = 0 . 5 , δω 220 = 0 . 5 } , { δτ 220 = -0 . 5 , δω 220 = -0 . 5 } , { δτ 220 = -0 . 5 , δω 220 = 0 . 5 } , { δτ 220 = 0 . 5 , δω 220 = -0 . 5 } , { δτ 220 = 0 , δω 220 = 0 . 5 } , { δτ 220 = 0 , δω 220 = -0 . 5 } , { δτ 220 = 0 . 5 , δω 220 = 0 } or { δτ 220 = -0 . 5 , δω 220 = 0 } is plotted by the red dashed, red dotted, blue dashed, blue dotted, green dashed, green dotted, cyan dashed or cyan dotted curve respectively; the black dashed curve is the analytical energy spectrum [50, 55].</caption>
</figure>
<figure>
<location><page_6><loc_21><loc_26><loc_78><loc_50></location>
<caption>FIG. 5: The total SGWB energy-density spectra at frequency 10[Hz] ≲ f ≲ 10 3 [Hz], where the ratio of the contribution of BH binary mergers to the contribution of NS binary and NS-BH mergers is about 1 : 0 . 3 [7]. The GR version with { δτ 220 = 0 , δω 220 = 0 } is the black solid curve; the non-GR version with { δτ 220 = 0 . 5 , δω 220 = 0 . 5 } , { δτ 220 = -0 . 5 , δω 220 = -0 . 5 } , { δτ 220 = -0 . 5 , δω 220 = 0 . 5 } , { δτ 220 = 0 . 5 , δω 220 = -0 . 5 } , { δτ 220 = 0 , δω 220 = 0 . 5 } , { δτ 220 = 0 , δω 220 = -0 . 5 } , { δτ 220 = 0 . 5 , δω 220 = 0 } or { δτ 220 = -0 . 5 , δω 220 = 0 } is plotted by the red dashed, red dotted, blue dashed, blue dotted, green dashed, green dotted, cyan dashed or cyan dotted curve respectively; the black dashed curve is the expected sensitivity of LIGO's anticipated 'A+' configuration [59-61]; the black dotted curve is the sensitivity of CE [58] by assuming one year's worth of data, a frequency bin width of 0 . 25[Hz] and the same overlap reduction function as LIGO's.</caption>
</figure>
<text><location><page_7><loc_9><loc_80><loc_49><loc_93></location>determined, we assume that CE shares the same overlap reduction function γ T ( f ) with LIGO. As shown in Fig. 5, the expected sensitivity of LIGO's anticipated 'A+' configuration [59-61] is not much lower than the total SGWB energy-density spectrum. Although the sensitivity of CE is much lower than the total SGWB energy-density spectrum, there is no measured C ( f ) for CE now. Therefore, we just turn to the Fisher information matrix to obtain the forecasting constraints</text>
<formula><location><page_7><loc_12><loc_75><loc_49><loc_79></location>F ij = ∑ k 1 σ 2 ( f k ) ∂ Ω GW ( f k ; θ ) ∂ θ i ∂ Ω GW ( f k ; θ ) ∂ θ j , (21)</formula>
<text><location><page_7><loc_9><loc_73><loc_45><loc_74></location>where θ is a vector consisting of 5 free parameters</text>
<formula><location><page_7><loc_18><loc_70><loc_40><loc_71></location>θ = { δω 220 , δτ 220 , z R , z m 1 , z q } ,</formula>
<text><location><page_7><loc_9><loc_66><loc_49><loc_69></location>and the derivative of the total SGWB energy-density spectrum with respect to each parameter is defined as</text>
<formula><location><page_7><loc_12><loc_62><loc_49><loc_65></location>∂ Ω GW ∂ θ i = Ω GW ( θ i + d θ i ) -Ω GW ( θ i -d θ i ) 2 d θ i , (22)</formula>
<text><location><page_7><loc_9><loc_59><loc_46><loc_61></location>and is calculated numerically by choosing d θ i = 0 . 1.</text>
<text><location><page_7><loc_9><loc_55><loc_49><loc_59></location>Here we introduce three new parameters { z R , z m 1 , z q } to summarize the original eight parameters of the fiducial PP model as</text>
<formula><location><page_7><loc_10><loc_35><loc_49><loc_54></location>d R BH d m 1 → A d((1 + z R ) R BH ) d ( m 1 ( m 1 27 . 9[M ⊙ ] ) z m 1 ) , d R BH d q → B d((1 + z R ) R BH ) d ( q ( q 1 ) z q ) , A = ∫ d R BH d m 1 dm 1 ∫ d R BH d ( m 1 ( m 1 27 . 9[M ⊙ ] ) z m 1 ) dm 1 -1 , B = ∫ d R BH d q dq [ ∫ d R BH d ( q ( q 1 ) z q ) dq ] -1 , (23)</formula>
<text><location><page_7><loc_9><loc_16><loc_49><loc_35></location>where z R scales the merger rate R BH ( z quoted = 0 . 2) directly, z m 1 scales the primary mass distribution around the pivot mass m 1 = 27 . 9[M ⊙ ], z q scales the mass ratio distribution around the pivot ratio q = 1 and A (or B ) is introduced to guarantee that z m 1 (or z q ) doesn't modify R BH ( z quoted = 0 . 2). That is to say, these three new parameters are independent of each other by construction. Therefore, we can obtain the non-flat priors of them from the constraints on the fiducial PP model as { z R = 0 ± 0 . 6672 , z m 1 = 0 ± 0 . 1062 , z q = 0 ± 0 . 1965 } (90% C.L.), where we have ignored the correlations between them. In Fig. 6, we show that the effects of { z R , z m 1 , z q } are similar to ones of the original fiducial PP model.</text>
<text><location><page_7><loc_9><loc_9><loc_49><loc_16></location>Weplot the absolute value of these derivatives in Fig. 7. While both | ∂ Ω GW ∂δτ 220 | and | ∂ Ω GW ∂δω 220 | are suppressed with f becoming lower which is consistent with the colored curves in Fig. 5, | ∂ Ω GW ∂z R | , | ∂ Ω GW ∂z m 1 | and | ∂ Ω GW ∂z q | are not very sensitive to f . The root mean square errors of these parame-</text>
<text><location><page_7><loc_52><loc_92><loc_64><loc_93></location></text>
<formula><location><page_7><loc_66><loc_89><loc_92><loc_90></location>σ i = √ ( F -1 ) ii . (24)</formula>
<text><location><page_7><loc_52><loc_79><loc_92><loc_87></location>Since the observations of GW transients and the direct observation of SGWB are independent of each other, the constraints on R BH from them should be independent of each other too. Here we rewrite the former constraints on R BH [7] as a set of non-flat priors which can be added to F ij in the form of</text>
<formula><location><page_7><loc_52><loc_68><loc_97><loc_78></location> σ -2 δω 220 = 0 0 0 0 0 0 σ -2 δτ 220 = 0 0 0 0 0 0 σ -2 z R = 6 . 1 0 0 0 0 0 σ -2 z m 1 = 239 . 9 0 0 0 0 0 σ -2 z q = 70 . 1 (25)</formula>
<text><location><page_7><loc_52><loc_37><loc_92><loc_68></location>In Fig. 8, we show the forecasting constraints on { δω 220 , δτ 220 , z R , z m 1 , z q } with the Fisher . py [62] package, where δω 220 = 0 ± 0 . 1296, δτ 220 = 0 ± 0 . 0678, z R = 0 ± 0 . 1219, z m 1 = 0 ± 0 . 0597 and z q = 0 ± 0 . 5155 at 68% confidence range (inner black contour) when only (2 , 2) mode and its hairs are considered and the flat priors of { z R , z m 1 , z q } are added, δω 220 = 0 ± 0 . 0903, δτ 220 = 0 ± 0 . 0608, z R = 0 ± 0 . 0283, z m 1 = 0 ± 0 . 0434 and z q = 0 ± 0 . 1161 at 68% confidence range (inner blue contour) when only (2 , 2) mode and its hairs are considered and the non-flat priors of { z R , z m 1 , z q } are added, δω 220 = 0 ± 0 . 0907, δτ 220 = 0 ± 0 . 0610, z R = 0 ± 0 . 0284, z m 1 = 0 ± 0 . 0435 and z q = 0 ± 0 . 1167 at 68% confidence range (inner green contour) when the higher modes are also considered according to Eq. (14) and the non-flat priors of { z R , z m 1 , z q } are added and δω 220 = 0 ± 0 . 0908, δτ 220 = 0 ± 0 . 0611, z R = 0 ± 0 . 0284, z m 1 = 0 ± 0 . 0435 and z q = 0 ± 0 . 1167 at 68% confidence range (inner red contour) when the higher modes and their corresponding hairs are considered simultaneously and the non-flat priors of { z R , z m 1 , z q } are added.</text>
<text><location><page_7><loc_52><loc_9><loc_92><loc_37></location>We can find that the green contours almost completely overlap the blue contours because the higher modes without hairs just shift the values of Ω GW ( f ) but don't affect their derivative with respect to each hair ∂ Ω GW ∂ θ i ( f ). Since the contributions of the higher modes' hairs should be smaller than the contributions of the higher modes themselves, the red contours also almost completely overlap the blue contours. Interestingly, there is a slight correlation between (2 , 2) mode's two hairs. It means that certain deviations from GR probably affect the ringdown frequencies and damping times simultaneously. There is an obvious positive correlation between z R and z q . Although (1 + z R ) serves as a factor of R BH (or Ω GW , BH ) according to Eq. (23) (or Eq. (16)), z q doesn't affect R BH directly according to the definition of B in Eq. (23). Therefore, z q must affects Ω GW , BH via ⟨ dE s /df s ⟩ BH according to Eq. (17). For example, z q = 0 . 1965 widens the mass ratio distribution to 0 . 1 ≲ q ≤ 1 as shown in the right subplot of Fig. 6 then suppresses the probability of 0 . 3 ≲ q ≤ 1 overall due to B , hence a smaller</text>
<text><location><page_7><loc_97><loc_73><loc_97><loc_74></location>.</text>
<figure>
<location><page_8><loc_9><loc_75><loc_48><loc_92></location>
</figure>
<figure>
<location><page_8><loc_51><loc_75><loc_88><loc_92></location>
<caption>FIG. 6: The primary mass (left) and mass ratio (right) distributions of the astrophysical BH binaries for the fiducial PP model at z quoted = 0 . 2. The original constraints from the fiducial PP model [7] are the black curves. They are well mimicked by the blue ones which are obtained from the approximate median curves (red) by zooming in or out with parameters { z R = 0 ± 0 . 6672 , z m 1 = 0 ± 0 . 1062 , z q = 0 ± 0 . 1965 } (90% C.L.).</caption>
</figure>
<figure>
<location><page_8><loc_29><loc_49><loc_68><loc_64></location>
<caption>FIG. 7: The absolute value of the derivatives of the total SGWB energy-density spectrum with respect to δτ 220 (black curve), δω 220 (blue curve), z R (red curve), z m 1 (green curve) and z q (cyan curve) respectively.</caption>
</figure>
<text><location><page_8><loc_9><loc_10><loc_49><loc_40></location>⟨ dE s /df s ⟩ BH due to the suppression of dE s /df s by the smaller mass ratio 0 . 1 ≲ q ≲ 0 . 3. For z q = -0 . 1965, the reverse applies. There are correlations between { z R , z q } and { δω 220 , δτ 220 } in the black contours when the flat priors of { z R , z q } are added but these correlations almost disappear in the blue contours when the non-flat priors of { z R , z q } are added. The reason for this disappearance is that both z R and z q can serve as a factor of Ω GW , BH and make an almost frequency-independent contribution to Ω GW , BH . That is to say, the absolute contribution of { z R = 0 ± 0 . 0283 , z q = 0 ± 0 . 1161 } is much larger at lower frequency but much smaller at higher frequency than that of { δω 220 = 0 ± 0 . 0903 , δτ 220 = 0 ± 0 . 0608 } , as hinted in Fig. 7. These totally different behaviors of them lead to the negligible correlations between them. There are obvious correlations between z m 1 and { δω 220 , δτ 220 } in the black and blue contours. It means that Ω GW , BH at higher frequency is sensitive to z m 1 which results from the dramatic influence of z m 1 on the primary mass distribution around m 1 ≲ 10[M ⊙ ], hence indirect correlations between z m 1 and { δω 220 , δτ 220 } at higher frequency. Fi-</text>
<text><location><page_8><loc_52><loc_33><loc_92><loc_40></location>m the present observations of GW transients is similar to the its forecasting constraint from the future observation of SGWB only, which leads to two similar constraints in the z m 1 -δω 220 plane and z m 1 -δτ 220 plane respectively.</text>
<section_header_level_1><location><page_8><loc_57><loc_28><loc_87><loc_29></location>IV. SUMMARY AND DISCUSSION</section_header_level_1>
<text><location><page_8><loc_52><loc_9><loc_92><loc_26></location>In this paper, we assume that the effects of the extra hairs on the inspiral and merger regions can be summarized by a set of effective intrinsic parameters under the no-hair theorem and then the residual effects of these extra hairs will just appear during an effective ringdown region. Then we turn to the PyCBC package[56] and the pSEOBNRv4HM PA model [46, 47] to obtain the effective complete IMR time-domain waveform with hairs, as shown in Fig. 2. After the Fourier transform (as shown in Fig. 3), we calculate the energy spectrum for every possible BH binary with different primary mass and mass ratio individually, as shown in Fig. 4. Combining these</text>
<figure>
<location><page_9><loc_23><loc_50><loc_78><loc_90></location>
<caption>FIG. 8: Error ellipses for { δω 220 , δτ 220 , z R , z m 1 , z q } , where δω 220 = 0 ± 0 . 1296, δτ 220 = 0 ± 0 . 0678, z R = 0 ± 0 . 1219, z m 1 = 0 ± 0 . 0597 and z q = 0 ± 0 . 5155 at 68% confidence range (inner black contour) when only (2 , 2) mode and its hairs are considered and the flat priors of { z R , z m 1 , z q } are added, δω 220 = 0 ± 0 . 0903, δτ 220 = 0 ± 0 . 0608, z R = 0 ± 0 . 0283, z m 1 = 0 ± 0 . 0434 and z q = 0 ± 0 . 1161 at 68% confidence range (inner blue contour) when only (2 , 2) mode and its hairs are considered and the non-flat priors of { z R , z m 1 , z q } are added, δω 220 = 0 ± 0 . 0907, δτ 220 = 0 ± 0 . 0610, z R = 0 ± 0 . 0284, z m 1 = 0 ± 0 . 0435 and z q = 0 ± 0 . 1167 at 68% confidence range (inner green contour) when the higher modes are also considered and the non-flat priors of { z R , z m 1 , z q } are added, δω 220 = 0 ± 0 . 0908, δτ 220 = 0 ± 0 . 0611, z R = 0 ± 0 . 0284, z m 1 = 0 ± 0 . 0435 and z q = 0 ± 0 . 1167 at 68% confidence range (inner red contour) when the higher modes and their corresponding hairs are considered simultaneously and the non-flat priors of { z R , z m 1 , z q } are added and these outer lines are their corresponding 95% confidence range. It is worth noting that the red, green and blue contours almost completely overlap each other.</caption>
</figure>
<text><location><page_9><loc_9><loc_9><loc_49><loc_26></location>energy spectra under a fixed fiducial PP model [7, 8] , we obtain the modified total SGWB energy-density spectrum at frequency 10[Hz] ≲ f ≲ 10 3 [Hz] for given hairs, as shown in Fig. 5. Here we suppose that the ratio of the contribution of BH binary mergers to the contribution of NS binary and NS-BH mergers is about 1 : 0 . 3 [7]. To further take the uncertainties of the fiducial PP model [7, 8] into consideration, the Fisher information matrix should also include the parameters of the fiducial PP model. For simplicity, we reduce the original eight parameters of the fiducial PP model to { z R , z m 1 , z q } and change the uncertainties of the original eight ones to</text>
<text><location><page_9><loc_52><loc_9><loc_92><loc_26></location>the non-flat priors of the latter three ones. By choosing the all free parameters as ∼ 0 . 1, we calculate numerically the derivative of the total SGWB energy-density spectrum with respect to each one respectively, as shown in Fig. 7. To obtain the variance of SGWB, we assume that CE shares the same overlap reduction function with LIGO. Finally, the forecasting constraints on hairs at 68% confidence range are δω 220 = 0 ± 0 . 1296 and δτ 220 = 0 ± 0 . 0678 when the flat priors of { z R , z m 1 , z q } are added but δω 220 = 0 ± 0 . 0903 and δτ 220 = 0 ± 0 . 0608 when the non-flat priors of { z R , z m 1 , z q } are added, as shown in Fig. 8. As for the higher modes, we find that they hardly</text>
<text><location><page_10><loc_9><loc_89><loc_49><loc_93></location>affect the forecasting constraints on { δω 220 , δτ 220 } while they do contribute to and shift the total SGWB energydensity spectrum. And so do their corresponding hairs.</text>
<text><location><page_10><loc_9><loc_72><loc_49><loc_89></location>There are three caveats. The first one is that we turn to the pSEOBNRv4HM PA model [46, 47] where the extra hairs appear only during the ringdown region. In fact, we have mimicked the effects of the extra hairs on the inspiral and merger regions with an effective GR's IMR waveform and then leaved the residual effects on the ringdown region alone. Therefore, the chosen fiducial PP model [7, 8] for the mass distribution is also an effective one which has included the effects of the extra hairs on the inspiral and merger regions. The second one is that we assume the ratio of the contribution of BH binary mergers to the contribution of NS binary and NS-BH mergers is</text>
<unordered_list>
<list_item><location><page_10><loc_10><loc_65><loc_47><loc_66></location>[1] LIGO scientific collaboration , https://www.ligo.org/.</list_item>
<list_item><location><page_10><loc_10><loc_64><loc_43><loc_65></location>[2] Virgo collaboration , https://www.virgo-gw.eu/.</list_item>
<list_item><location><page_10><loc_10><loc_61><loc_49><loc_63></location>[3] KAGRA collaboration , https://gwcenter.icrr.utokyo.ac.jp/.</list_item>
<list_item><location><page_10><loc_10><loc_54><loc_49><loc_61></location>[4] B. P. Abbott et al. [LIGO Scientific and Virgo], 'GWTC1: A Gravitational-Wave Transient Catalog of Compact Binary Mergers Observed by LIGO and Virgo during the First and Second Observing Runs,' Phys. Rev. X 9 , no.3, 031040 (2019) [arXiv:1811.12907 [astro-ph.HE]].</list_item>
<list_item><location><page_10><loc_10><loc_48><loc_49><loc_54></location>[5] R. Abbott et al. [LIGO Scientific and Virgo], 'GWTC2: Compact Binary Coalescences Observed by LIGO and Virgo During the First Half of the Third Observing Run,' Phys. Rev. X 11 , 021053 (2021) [arXiv:2010.14527 [grqc]].</list_item>
<list_item><location><page_10><loc_10><loc_42><loc_49><loc_47></location>[6] R. Abbott et al. [LIGO Scientific, VIRGO and KAGRA], 'GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo During the Second Part of the Third Observing Run,' [arXiv:2111.03606 [gr-qc]].</list_item>
<list_item><location><page_10><loc_10><loc_36><loc_49><loc_42></location>[7] R. Abbott et al. [KAGRA, VIRGO and LIGO Scientific], 'Population of Merging Compact Binaries Inferred Using Gravitational Waves through GWTC-3,' Phys. Rev. X 13 , no.1, 011048 (2023) [arXiv:2111.03634 [astroph.HE]].</list_item>
<list_item><location><page_10><loc_10><loc_29><loc_49><loc_36></location>[8] R. Abbott et al. [LIGO Scientific and Virgo], 'Population Properties of Compact Objects from the Second LIGO-Virgo Gravitational-Wave Transient Catalog,' Astrophys. J. Lett. 913 , no.1, L7 (2021) [arXiv:2010.14533 [astro-ph.HE]].</list_item>
<list_item><location><page_10><loc_10><loc_23><loc_49><loc_29></location>[9] R. Abbott et al. [KAGRA, Virgo and LIGO Scientific], 'Upper limits on the isotropic gravitational-wave background from Advanced LIGO and Advanced Virgo's third observing run,' Phys. Rev. D 104 , no.2, 022004 (2021) [arXiv:2101.12130 [gr-qc]].</list_item>
<list_item><location><page_10><loc_9><loc_17><loc_49><loc_22></location>[10] B. P. Abbott et al. [LIGO Scientific and Virgo], 'Search for the isotropic stochastic background using data from Advanced LIGO's second observing run,' Phys. Rev. D 100 , no.6, 061101 (2019) [arXiv:1903.02886 [gr-qc]].</list_item>
<list_item><location><page_10><loc_9><loc_9><loc_49><loc_17></location>[11] B. P. Abbott et al. [LIGO Scientific and Virgo], 'Upper Limits on the Stochastic Gravitational-Wave Background from Advanced LIGO's First Observing Run,' Phys. Rev. Lett. 118 , no.12, 121101 (2017) [erratum: Phys. Rev. Lett. 119 , no.2, 029901 (2017)] [arXiv:1612.02029 [gr-qc]].</list_item>
</unordered_list>
<text><location><page_10><loc_52><loc_82><loc_92><loc_93></location>about 1 : 0 . 3 according to Fig. 23 of [7]. It is just a temporary assumption and will be improved according to the newest GW observations. The third one is that CE shares the same overlap reduction function γ T ( f ) with LIGO. It is also a temporary assumption. Because γ T ( f ) is determined by the relative positions and orientations of a pair of detectors and the actual locations and arm orientations for CE are yet to be determined.</text>
<section_header_level_1><location><page_10><loc_65><loc_77><loc_79><loc_78></location>Acknowledgments</section_header_level_1>
<text><location><page_10><loc_52><loc_72><loc_92><loc_74></location>Ke Wang is supported by grants from NSFC (grant No. 12005084 and grant No.12247101).</text>
<unordered_list>
<list_item><location><page_10><loc_52><loc_61><loc_92><loc_66></location>[12] G. Agazie et al. [NANOGrav], 'The NANOGrav 15 yr Data Set: Evidence for a Gravitational-wave Background,' Astrophys. J. Lett. 951 , no.1, L8 (2023) [arXiv:2306.16213 [astro-ph.HE]].</list_item>
<list_item><location><page_10><loc_52><loc_56><loc_92><loc_61></location>[13] J. Antoniadis et al. [EPTA and InPTA:], 'The second data release from the European Pulsar Timing Array III. Search for gravitational wave signals,' Astron. Astrophys. 678 , A50 (2023) [arXiv:2306.16214 [astro-ph.HE]].</list_item>
<list_item><location><page_10><loc_52><loc_48><loc_92><loc_55></location>[14] D. Reitze, R. X. Adhikari, S. Ballmer, B. Barish, L. Barsotti, G. Billingsley, D. A. Brown, Y. Chen, D. Coyne and R. Eisenstein, et al. 'Cosmic Explorer: The U.S. Contribution to Gravitational-Wave Astronomy beyond LIGO,' Bull. Am. Astron. Soc. 51 , no.7, 035 (2019) [arXiv:1907.04833 [astro-ph.IM]].</list_item>
<list_item><location><page_10><loc_52><loc_41><loc_92><loc_47></location>[15] M. Punturo, M. Abernathy, F. Acernese, B. Allen, N. Andersson, K. Arun, F. Barone, B. Barr, M. Barsuglia and M. Beker, et al. 'The Einstein Telescope: A third-generation gravitational wave observatory,' Class. Quant. Grav. 27 , 194002 (2010)</list_item>
<list_item><location><page_10><loc_52><loc_37><loc_92><loc_41></location>[16] R. Abbott et al. [LIGO Scientific, VIRGO and KAGRA], 'Tests of General Relativity with GWTC-3,' [arXiv:2112.06861 [gr-qc]].</list_item>
<list_item><location><page_10><loc_52><loc_31><loc_92><loc_37></location>[17] R. Abbott et al. [LIGO Scientific and Virgo], 'Tests of general relativity with binary black holes from the second LIGO-Virgo gravitational-wave transient catalog,' Phys. Rev. D 103 , no.12, 122002 (2021) [arXiv:2010.14529 [grqc]].</list_item>
<list_item><location><page_10><loc_52><loc_25><loc_92><loc_30></location>[18] B. P. Abbott et al. [LIGO Scientific and Virgo], 'Tests of General Relativity with the Binary Black Hole Signals from the LIGO-Virgo Catalog GWTC-1,' Phys. Rev. D 100 , no.10, 104036 (2019) [arXiv:1903.04467 [gr-qc]].</list_item>
<list_item><location><page_10><loc_52><loc_20><loc_92><loc_25></location>[19] B. P. Abbott et al. [LIGO Scientific and Virgo], 'Tests of general relativity with GW150914,' Phys. Rev. Lett. 116 , no.22, 221101 (2016) [erratum: Phys. Rev. Lett. 121 , no.12, 129902 (2018)] [arXiv:1602.03841 [gr-qc]].</list_item>
<list_item><location><page_10><loc_52><loc_15><loc_92><loc_20></location>[20] R. Abbott et al. [LIGO Scientific and Virgo], 'Properties and Astrophysical Implications of the 150 M ⊙ Binary Black Hole Merger GW190521,' Astrophys. J. Lett. 900 , no.1, L13 (2020) [arXiv:2009.01190 [astro-ph.HE]].</list_item>
<list_item><location><page_10><loc_52><loc_9><loc_92><loc_14></location>[21] A. Ghosh, A. Ghosh, N. K. Johnson-McDaniel, C. K. Mishra, P. Ajith, W. Del Pozzo, D. A. Nichols, Y. Chen, A. B. Nielsen and C. P. L. Berry, et al. 'Testing general relativity using golden black-hole binaries,'</list_item>
</unordered_list>
<text><location><page_11><loc_12><loc_91><loc_49><loc_93></location>Phys. Rev. D 94 , no.2, 021101 (2016) [arXiv:1602.02453 [gr-qc]].</text>
<unordered_list>
<list_item><location><page_11><loc_9><loc_83><loc_49><loc_90></location>[22] A. Ghosh, N. K. Johnson-Mcdaniel, A. Ghosh, C. K. Mishra, P. Ajith, W. Del Pozzo, C. P. L. Berry, A. B. Nielsen and L. London, 'Testing general relativity using gravitational wave signals from the inspiral, merger and ringdown of binary black holes,' Class. Quant. Grav. 35 , no.1, 014002 (2018) [arXiv:1704.06784 [gr-qc]].</list_item>
<list_item><location><page_11><loc_9><loc_76><loc_49><loc_82></location>[23] P. T. H. Pang, R. K. L. Lo, I. C. F. Wong, T. G. F. Li and C. Van Den Broeck, 'Generic searches for alternative gravitational wave polarizations with networks of interferometric detectors,' Phys. Rev. D 101 , no.10, 104055 (2020) [arXiv:2003.07375 [gr-qc]].</list_item>
<list_item><location><page_11><loc_9><loc_71><loc_49><loc_76></location>[24] H. Takeda, S. Morisaki and A. Nishizawa, 'Pure polarization test of GW170814 and GW170817 using waveforms consistent with modified theories of gravity,' Phys. Rev. D 103 , no.6, 064037 (2021) [arXiv:2010.14538 [gr-qc]].</list_item>
<list_item><location><page_11><loc_9><loc_67><loc_49><loc_70></location>[25] R. S. Conklin, B. Holdom and J. Ren, 'Gravitational wave echoes through new windows,' Phys. Rev. D 98 , no.4, 044021 (2018) [arXiv:1712.06517 [gr-qc]].</list_item>
<list_item><location><page_11><loc_9><loc_59><loc_49><loc_67></location>[26] N. Uchikata, T. Narikawa, H. Nakano, N. Sago, H. Tagoshi and T. Tanaka, 'Searching for gravitational wave echoes from black hole binary events in the third observing run of LIGO, Virgo, and KAGRA collaborations,' Phys. Rev. D 108 , no.10, 104040 (2023) [arXiv:2309.01894 [gr-qc]].</list_item>
<list_item><location><page_11><loc_9><loc_54><loc_49><loc_59></location>[27] N. Yunes and F. Pretorius, 'Fundamental Theoretical Bias in Gravitational Wave Astrophysics and the Parameterized Post-Einsteinian Framework,' Phys. Rev. D 80 , 122003 (2009) [arXiv:0909.3328 [gr-qc]].</list_item>
<list_item><location><page_11><loc_9><loc_48><loc_49><loc_53></location>[28] A. Nishizawa, 'Generalized framework for testing gravity with gravitational-wave propagation. I. Formulation,' Phys. Rev. D 97 , no.10, 104037 (2018) [arXiv:1710.04825 [gr-qc]].</list_item>
<list_item><location><page_11><loc_9><loc_38><loc_49><loc_48></location>[29] J. Meidam, K. W. Tsang, J. Goldstein, M. Agathos, A. Ghosh, C. J. Haster, V. Raymond, A. Samajdar, P. Schmidt and R. Smith, et al. 'Parametrized tests of the strong-field dynamics of general relativity using gravitational wave signals from coalescing binary black holes: Fast likelihood calculations and sensitivity of the method,' Phys. Rev. D 97 , no.4, 044033 (2018) [arXiv:1712.08772 [gr-qc]].</list_item>
<list_item><location><page_11><loc_9><loc_35><loc_49><loc_37></location>[30] B. Carter, 'Axisymmetric Black Hole Has Only Two Degrees of Freedom,' Phys. Rev. Lett. 26 , 331-333 (1971)</list_item>
<list_item><location><page_11><loc_9><loc_31><loc_49><loc_35></location>[31] N. Gurlebeck, 'No-hair theorem for Black Holes in Astrophysical Environments,' Phys. Rev. Lett. 114 , no.15, 151102 (2015) [arXiv:1503.03240 [gr-qc]].</list_item>
<list_item><location><page_11><loc_9><loc_26><loc_49><loc_31></location>[32] C. M. Will, 'Bounding the mass of the graviton using gravitational wave observations of inspiralling compact binaries,' Phys. Rev. D 57 , 2061-2068 (1998) [arXiv:grqc/9709011 [gr-qc]].</list_item>
<list_item><location><page_11><loc_9><loc_21><loc_49><loc_26></location>[33] S. Mirshekari, N. Yunes and C. M. Will, 'Constraining Generic Lorentz Violation and the Speed of the Graviton with Gravitational Waves,' Phys. Rev. D 85 , 024041 (2012) [arXiv:1110.2720 [gr-qc]].</list_item>
<list_item><location><page_11><loc_9><loc_15><loc_49><loc_20></location>[34] S. Wang and Z. C. Zhao, 'Tests of CPT invariance in gravitational waves with LIGO-Virgo catalog GWTC-1,' Eur. Phys. J. C 80 , no.11, 1032 (2020) [arXiv:2002.00396 [gr-qc]].</list_item>
<list_item><location><page_11><loc_9><loc_10><loc_49><loc_15></location>[35] Z. C. Zhao, Z. Cao and S. Wang, 'Search for the Birefringence of Gravitational Waves with the Third Observing Run of Advanced LIGO-Virgo,' Astrophys. J. 930 , no.2, 139 (2022) [arXiv:2201.02813 [gr-qc]].</list_item>
<list_item><location><page_11><loc_9><loc_9><loc_49><loc_10></location>[36] R. Niu, T. Zhu and W. Zhao, 'Testing Lorentz invariance</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_11><loc_55><loc_91><loc_92><loc_93></location>of gravity in the Standard-Model Extension with GWTC3,' JCAP 12 , 011 (2022) [arXiv:2202.05092 [gr-qc]].</list_item>
<list_item><location><page_11><loc_52><loc_85><loc_92><loc_90></location>[37] Z. Wang, L. Shao and C. Liu, 'New Limits on the Lorentz/CPT Symmetry Through 50 Gravitationalwave Events,' Astrophys. J. 921 , no.2, 158 (2021) [arXiv:2108.02974 [gr-qc]].</list_item>
<list_item><location><page_11><loc_52><loc_80><loc_92><loc_85></location>[38] M. Isi, M. Giesler, W. M. Farr, M. A. Scheel and S. A. Teukolsky, 'Testing the no-hair theorem with GW150914,' Phys. Rev. Lett. 123 , no.11, 111102 (2019) [arXiv:1905.00869 [gr-qc]].</list_item>
<list_item><location><page_11><loc_52><loc_73><loc_92><loc_80></location>[39] G. Carullo, W. Del Pozzo and J. Veitch, 'Observational Black Hole Spectroscopy: A time-domain multimode analysis of GW150914,' Phys. Rev. D 99 , no.12, 123029 (2019) [erratum: Phys. Rev. D 100 , no.8, 089903 (2019)] [arXiv:1902.07527 [gr-qc]].</list_item>
<list_item><location><page_11><loc_52><loc_69><loc_92><loc_73></location>[40] K. Wang, 'Retesting the no-hair theorem with GW150914,' Eur. Phys. J. C 82 , no.2, 125 (2022) [arXiv:2111.00953 [gr-qc]].</list_item>
<list_item><location><page_11><loc_52><loc_63><loc_92><loc_69></location>[41] B. P. Abbott et al. [LIGO Scientific and Virgo], 'Search for Tensor, Vector, and Scalar Polarizations in the Stochastic Gravitational-Wave Background,' Phys. Rev. Lett. 120 , no.20, 201102 (2018) [arXiv:1802.10194 [grqc]].</list_item>
<list_item><location><page_11><loc_52><loc_58><loc_92><loc_63></location>[42] Y. Jiang and Q. G. Huang, 'Upper limits on the polarized isotropic stochastic gravitational-wave background from advanced LIGO-Virgo's first three observing runs,' JCAP 02 , 026 (2023) [arXiv:2210.09952 [astro-ph.CO]].</list_item>
<list_item><location><page_11><loc_52><loc_52><loc_92><loc_57></location>[43] E. Berti, V. Cardoso and C. M. Will, 'On gravitationalwave spectroscopy of massive black holes with the space interferometer LISA,' Phys. Rev. D 73 , 064030 (2006) [arXiv:gr-qc/0512160 [gr-qc]].</list_item>
<list_item><location><page_11><loc_52><loc_50><loc_92><loc_52></location>[44] M. Isi and W. M. Farr, 'Analyzing black-hole ringdowns,' [arXiv:2107.05609 [gr-qc]].</list_item>
<list_item><location><page_11><loc_52><loc_43><loc_92><loc_49></location>[45] L. Arest'e Sal'o, K. Clough and P. Figueras, 'Puncture gauge formulation for Einstein-Gauss-Bonnet gravity and four-derivative scalar-tensor theories in d+1 spacetime dimensions,' Phys. Rev. D 108 , no.8, 084018 (2023) [arXiv:2306.14966 [gr-qc]].</list_item>
<list_item><location><page_11><loc_52><loc_38><loc_92><loc_43></location>[46] A. Ghosh, R. Brito and A. Buonanno, 'Constraints on quasinormal-mode frequencies with LIGO-Virgo binary-black-hole observations,' Phys. Rev. D 103 , no.12, 124041 (2021) [arXiv:2104.01906 [gr-qc]].</list_item>
<list_item><location><page_11><loc_52><loc_31><loc_92><loc_37></location>[47] D. P. Mihaylov, S. Ossokine, A. Buonanno and A. Ghosh, 'Fast post-adiabatic waveforms in the time domain: Applications to compact binary coalescences in LIGO and Virgo,' Phys. Rev. D 104 , no.12, 124087 (2021) [arXiv:2105.06983 [gr-qc]].</list_item>
<list_item><location><page_11><loc_52><loc_29><loc_92><loc_31></location>[48] M. Isi and W. M. Farr, 'Analyzing black-hole ringdowns,' [arXiv:2107.05609 [gr-qc]].</list_item>
<list_item><location><page_11><loc_52><loc_22><loc_92><loc_28></location>[49] E. Berti, V. Cardoso and M. Casals, 'Eigenvalues and eigenfunctions of spin-weighted spheroidal harmonics in four and higher dimensions,' Phys. Rev. D 73 , 024013 (2006) [erratum: Phys. Rev. D 73 , 109902 (2006)] [arXiv:gr-qc/0511111 [gr-qc]].</list_item>
<list_item><location><page_11><loc_52><loc_13><loc_92><loc_22></location>[50] P. Ajith, S. Babak, Y. Chen, M. Hewitson, B. Krishnan, A. M. Sintes, J. T. Whelan, B. Bruegmann, P. Diener and N. Dorband, et al. 'A Template bank for gravitational waveforms from coalescing binary black holes. I. Nonspinning binaries,' Phys. Rev. D 77 , 104017 (2008) [erratum: Phys. Rev. D 79 , 129901 (2009)] [arXiv:0710.2335 [gr-qc]].</list_item>
<list_item><location><page_11><loc_52><loc_9><loc_92><loc_12></location>[51] Y. Wiaux, L. Jacques and P. Vandergheynst, 'Fast spin +-2 spherical harmonics transforms,' J. Comput. Phys. 226 , 2359-2371 (2007) [arXiv:astro-ph/0508514 [astro-</list_item>
<list_item><location><page_12><loc_12><loc_92><loc_15><loc_93></location>ph]].</list_item>
<list_item><location><page_12><loc_9><loc_87><loc_49><loc_92></location>[52] N. Aghanim et al. [Planck], 'Planck 2018 results. VI. Cosmological parameters,' Astron. Astrophys. 641 , A6 (2020) [erratum: Astron. Astrophys. 652 , C4 (2021)] [arXiv:1807.06209 [astro-ph.CO]].</list_item>
<list_item><location><page_12><loc_9><loc_80><loc_49><loc_86></location>[53] B. P. Abbott et al. [LIGO Scientific and Virgo], 'GW150914: Implications for the stochastic gravitational wave background from binary black holes,' Phys. Rev. Lett. 116 , no.13, 131102 (2016) [arXiv:1602.03847 [grqc]].</list_item>
<list_item><location><page_12><loc_9><loc_76><loc_49><loc_80></location>[54] I. Cholis, 'On the Gravitational Wave Background from Black Hole Binaries after the First LIGO Detections,' JCAP 06 , 037 (2017) [arXiv:1609.03565 [astro-ph.HE]].</list_item>
<list_item><location><page_12><loc_9><loc_71><loc_49><loc_76></location>[55] T. Callister, L. Sammut, S. Qiu, I. Mandel and E. Thrane, 'The limits of astrophysics with gravitationalwave backgrounds,' Phys. Rev. X 6 , no.3, 031018 (2016) [arXiv:1604.02513 [gr-qc]].</list_item>
<list_item><location><page_12><loc_9><loc_63><loc_49><loc_70></location>[56] C. M. Biwer, C. D. Capano, S. De, M. Cabero, D. A. Brown, A. H. Nitz and V. Raymond, 'PyCBC Inference: A Python-based parameter estimation toolkit for compact binary coalescence signals,' Publ. Astron. Soc. Pac. 131 , no.996, 024503 (2019) [arXiv:1807.10312 [astro-ph.IM]].</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_12><loc_52><loc_85><loc_92><loc_93></location>[57] S. Khan, S. Husa, M. Hannam, F. Ohme, M. Purrer, X. Jim'enez Forteza and A. Boh'e, 'Frequency-domain gravitational waves from nonprecessing black-hole binaries. II. A phenomenological model for the advanced detector era,' Phys. Rev. D 93 , no.4, 044007 (2016) [arXiv:1508.07253 [gr-qc]].</list_item>
<list_item><location><page_12><loc_52><loc_81><loc_92><loc_85></location>[58] Exploring the Sensitivity of Next Generation Gravitational Wave Detectors , https://dcc.ligo.org/LIGOP1600143/public.</list_item>
<list_item><location><page_12><loc_52><loc_76><loc_92><loc_81></location>[59] E. Thrane and J. D. Romano, 'Sensitivity curves for searches for gravitational-wave backgrounds,' Phys. Rev. D 88 , no.12, 124032 (2013) [arXiv:1310.5300 [astroph.IM]].</list_item>
<list_item><location><page_12><loc_52><loc_72><loc_92><loc_76></location>[60] Sensitivity curves for searches for gravitationalwave backgrounds , https://dcc.ligo.org/LIGOP1300115/public.</list_item>
<list_item><location><page_12><loc_52><loc_69><loc_92><loc_72></location>[61] The A+ design curve , https://dcc.ligo.org/LIGOT1800042/public.</list_item>
<list_item><location><page_12><loc_52><loc_66><loc_92><loc_69></location>[62] D. Coe, 'Fisher Matrices and Confidence Ellipses: A Quick-Start Guide and Software,' [arXiv:0906.4123 [astro-ph.IM]].</list_item>
</document>
[
{
"title": "Forecasting constraints on the no-hair theorem from the stochastic gravitational wave background",
"content": "Chen Tan 1 , 2 , 3 and Ke Wang 1 , 2 , 3 ∗ Institute of Theoretical Physics & Research Center of Gravitation, Lanzhou University, Lanzhou 730000, China 2 Key Laboratory of Quantum Theory and Applications of MoE, Lanzhou University, Lanzhou 730000, China and Lanzhou Center for Theoretical Physics & Key Laboratory of Theoretical 1 3 Physics of Gansu Province, Lanzhou University, Lanzhou 730000, China (Dated: July 1, 2024) Although the constraints on general relativity (GR) from each individual gravitational-wave (GW) event can be combined to form a cumulative estimate of the deviations from GR, the ever-increasing number of GW events used also leads to the ever-increasing computational cost during the parameter estimation. Therefore, in this paper, we will introduce the deviations from GR into GWs from all events in advance and then create a modified stochastic gravitational-wave background (SGWB) to perform tests of GR. More precisely, we use the pSEOBNRv4HM PA model to include the modelindependent hairs and calculate the corresponding SGWB with a given merger rate. Then we turn to the Fisher information matrix to forecast the constraints on the no-hair theorem from SGWB at frequency 10[Hz] ≲ f ≲ 10 3 [Hz] detected by the third-generation ground-based GW detectors, such as the Cosmic Explorer. We find that the forecasting constraints on hairs at 68% confidence range are δω 220 = 0 ± 0 . 1296 and δτ 220 = 0 ± 0 . 0678 when the flat priors about the merger rate are added but δω 220 = 0 ± 0 . 0903 and δτ 220 = 0 ± 0 . 0608 when the non-flat priors about the merger rate are added.",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "There are about 90 compact binary coalescences observed by the LIGO Scientific Collaboration [1], the Virgo Collaboration [2] and the KAGRA Collaboration [3] during the first three observing runs [4-6]. Through these observed gravitational-wave (GW) transient events, the full population of merging compact binaries surrounding us can be further inferred [7, 8]. Consequently, a superposition of GWs from the surrounding population of these astrophysical sources will create a stochastic gravitational-wave background (SGWB) with energy density Ω GW ( f ) ∼ 10 -9 at frequency 10[Hz] ≲ f ≲ 10 3 [Hz] [7]. Due to the limited sensitivity of the current ground-based GW detectors, however, no evidence for a SGWB at this frequency span was found [9-11]. Encouragingly, there are multiple lines of evidence for an excess SGWB signal with amplitude A SGWB ∼ 10 -14 at frequency f ∼ 10 -8 [Hz] recently in the NANOGrav 15-year data [12] and the second data release from EPTA [13]. Therefore, there is an excellent probability that a SGWB at frequency 10[Hz] ≲ f ≲ 10 3 [Hz] can be detected by the third-generation ground-based GW detectors, such as the Cosmic Explorer (CE) [14] and the Einstein Telescope (ET) [15]. The ever-increasing number of detections of GWs from compact binaries by the current ground-based GW detectors allows the ever-more sensitive tests of general relativity (GR) with GW generation, propagation or polarization [16-18]. More precisely, one can per- form residuals test [19, 20], inspiral-merger-ringdown (IMR) consistency test [21, 22], extra polarization contents searches [23, 24], echoes searches [25, 26] as well as parametrized tests of GW generation and propagation [27-29], spin-induced multipole moment effects [30, 31], modified GW dispersion relation [32-37] and the no-hair theorem [38-40]. All of there tests will introduce some ad hoc parameters to account for the deviations from GR. Since these new-introduced parameters are independent of the individual sources by construction and cleanly encode information about the underlying theory of gravity, the constraints on them from each individual GW event can be combined to form a cumulative estimate of the deviations from GR. The more GW events are taken into consideration, the tighter cumulative constraints are obtained. However, the ever-increasing number of GW events used also leads to the ever-increasing computational cost during the parameter estimation. To avoid the time-consuming parameter estimation procedure, one can combine GWs from all events in advance and then perform tests of GR. That is to say, we can use SGWB at frequency 10[Hz] ≲ f ≲ 10 3 [Hz] to test GR. While the information of GW propagation may wash out in SGWB, the information of GW generation and polarization must survive SGWB and some deviations from GR's GW generation and polarization will lead to some suppressions or enhancements in SGWB. If such suppressions or enhancements have no particular distinguishing features, testing GR with SGWB must deals with the degeneracy between models. However, if one would not confine oneself to specific modified gravities, one can search for extra polarization contents in SGWB model-independently [41, 42]. Similarly, in this paper, we will probe the deviations from GR's GW gen- ion with SGWB directly. More precisely, we will take the violation of the no-hair theorem as a heuristic example. Under the no-hair theorem, the ringdown frequencies and damping times of a perturbed Kerr black hole (BH) in GR can be predicted by its mass and dimensionless spin [43, 48]. Without the no-hair conjecture, a perturbed Kerr BH's ringdown frequencies and damping times are also dependent on the extra hairs/parameters [38-40]. In fact, violating the no-hair theorem may also modify the whole IMR waveform for a BH binary. Such overall modifications will degenerate with other overall effects, for example changing the total mass of the BH binary, hence the final mass of its remnant Kerr BH. That is to say, when the extra hairs dominate the inspiral and merger regions of a BH binary, the mass and dimensionless spin of its remnant Kerr BH can't be predicted by its intrinsic parameters only, hence an unknown ringdown region. Therefore, for simplicity, we assume that the effects of the extra hairs on the inspiral and merger regions can be summarized by a set of effective intrinsic parameters under the no-hair theorem and then the residual effects of these extra hairs will just appear during an effective ringdown region. It is generally the case when different modified gravity theories are considered. Here we validate our above assumption with a specific modified theory of gravity, as shown in Fig. 1. Finally the effective complete IMR time-domain waveform can be described by pSEOBNRv4HM PA model [46, 47] explicitly. This paper is organized as follows. In section II, we calculate the stochastic gravitational wave background at frequency 10[Hz] ≲ f ≲ 10 3 [Hz] without the no-hair conjecture. In section III, we forecast the constraints on the no-hair theorem from SGWB. Finally, a brief summary and discussions are included in section IV.",
"pages": [
1,
2
]
},
{
"title": "A. Analytical Calculation",
"content": "The radiative degrees of freedom in transversetraceless (TT) gauge can be written in terms of two polarizations h + and h × : After Fourier transform, their representation in the frequency domain is ˜ h ij = F ( h ij ) Then the energy-momentum tensor of GWs is where ⟨ ... ⟩ indicates averaging over several wavelengths or periods. The energy density of GWs is just the 00 component. So the energy flux through a sphere of radius R is and the total energy emitted is where dots denote derivative with respect to physical time t and d Ω is a solid angle element. Here we are considering all time. So ⟨ ... ⟩ disappears in the integral with respect to t . We can rewrite the total energy in the frequency domain as So the energy spectrum is given by Next we will do the integral with respect to d Ω. The general GW signal can be described as a linear combination of the two polarization states or expressed in terms of series of spin-weighted spherical harmonics where φ is the azimuthal direction to the observer and ι is the inclination angle. Here we use the spin-weighted spherical harmonics to describe the whole IMR waveform including the ringdown waveform even thought the spin-weighted spheroidal harmonics are more accurate for the remnant Kerr BHs. In fact, the spherical version is a sufficiently good approximation of its spheroidal counterpart [48, 49]. For the two dominant modes with l = 2 and m = ± 2, the spin-weighted spherical harmonics are [50, 51] When φ = 0 and ι = 0, for example, we have the GW signal as According to Eq. (8), we rewrite . fixed angles, for example φ = 0 and ι = π/ 2. Unless otherwise stated, in our following calculations, we used the (2 , 2) mode only. The dimensionless GW energy density per logarithmic frequency interval is Due to the orthonormality of spin-weighted spherical harmonics, the integral can be done easily where we have used the equatorial symmetry h 22 = ( h 2 -2 ) ∗ . According to Eq. (8) and Eq. (10), we can express the energy spectrum approximately as where ρ c = 3 H 2 0 c 2 / 8 πG is the critical energy density and H 0 = 67 . 36[km s -1 Mpc -1 ] [52]. The contribution of BH binary mergers can be estimated as [9] Although here we have fixed the two angles φ = ι = 0, the values of them don't affect the total energy emitted. When the higher-order modes are taken into consideration, we can express the energy spectrum more accurately as where we have used the equatorial symmetry of all modes ( l, | m | ) and ˜ h lm can be numerically obtained because ˜ h lm Y -2 lm ( φ, ι ) as a whole can be numerically obtained at . where f s = f (1 + z ) is frequency in the source frame. The Hubble parameter is H ( z ) = H 0 √ Ω m (1 + z ) 3 +Ω Λ , where Ω m = 0 . 3153 is the matter density parameter and Ω Λ = 0 . 6847 is the dark energy density parameter [52]. The quantity ⟨ dE s /df s ⟩ GW , BBHs is the source-frame energy radiated by a single source, which should be averaged over the ensemble properties of the full BH binary population where p BH ( α ) is the probability distribution of intrinsic parameters α (e.g. masses, spins, etc.) for the full BH binary population. The merger rate R BH ( z ) can be obtained by a convolution of the BH binary formation rate R f BH ( z f ) with the distribution of the time delays P ( t d ) between BH binary formation and merger [53, 54]. Also R BH ( z ) can be obtained from a merger rate R BH ( z quoted ) at a quoted redshift z quoted = 0 [55] or z quoted = 0 . 2 [7]. Here we turn to the fiducial Power Law + Peak (PP) model [7, 8], as shown in Fig. 10 of [7], where R BH ( z quoted = 0 . 2) = 28 . 3[Gpc -3 yr -1 ] is obtained by integrating the primary mass distribution d R BH d m 1 or the mass ratio distribution d R BH d q . After integrating over the contribution of spins, p BH ( α ) ≈ 1 R 2 BH d R BH d m 1 d R BH d q serves as a good approximation.",
"pages": [
2,
3,
4
]
},
{
"title": "B. Numerical Calculation",
"content": "Given the waveform of every GW event, we can calculate the total SGWB energy-density spectrum numerically according to Eq. (16). Here we will consider the deviations from GR's GW generation, namely violating the no-hair theorem. First we turn to the PyCBC package[56] and the pSEOBNRv4HM PA model [46, 47] to generate the IMR time-domain waveforms for every possible BH binary with different primary mass and mass ratio allowed by a given fiducial PP model [7, 8] individually. In the pSEOBNRv4HM PA model, the ad hoc hairs δω lmn and δτ lmn are introduced as where ω lmn and τ lmn are the ringdown frequencies and damping times respectively. Why we can fix the fiducial PP model for different values of { δω lmn , δτ lmn } ? The fiducial PP model, in our paper, just provides a effective mass distribution when we summarize the effects of { δω lmn , δτ lmn } on the inspiral and merger regions with a set of effective intrinsic parameters. Given a modified theory of gravity, the effective mass distribution will vary with the values of { δω lmn , δτ lmn } to compensate for the changes of the inspiral and merger regions. Similarly, given the values of { δω lmn , δτ lmn } , the effective mass distribution will also vary with the different modified theory of gravity to compensate for the corresponding changes of the inspiral and merger regions. Under different situations of the extra hairs, however, the effective mass distribution provided by the fiducial PP model can be unchanged in certain cases. For example, we can fix the effective mass distribution as the realistic one picked out by the observations of GWtransients during our calculations for different values of { δω lmn , δτ lmn } through adjusting the modified theory of gravity simultaneously. That is to say, our modelindependent parameterization of Eq. (18) is not confined to a specific modified theory of gravity, but aims to identify any deviation from GR. We plot the IMR time-domain waveforms from a GW150914-like event for '+'-polarization in Fig. 2, where the luminosity distance is 1[Mpc]. By definition, δω lmn affects the ringdown frequencies (green curves) and δτ lmn affects the ringdown damping times (cyan curves). Then we make the Fourier transform of the IMR time-domain waveforms to get the IMR frequencydomain waveforms. We plot the IMR frequency-domain waveforms from a GW150914-like event for '+'-polarization in Fig. 3, where the black dashed curve is its GR counterpart and plotted directly by the IMRPhenomD model [57]. We find that the effects of δω lmn on the IMR frequency-domain waveform is lager than that of δτ lmn . Next, according to Eq. (13), we can get the energy spectrum for every BH binary. We plot the energy spectrum for a GW150914-like event in Fig. 4, where the black dashed curve is the analytical energy spectrum [50, 55]. Again we find that the effects of δω lmn on the energy spectrum is lager than that of δτ lmn . Finally, we plot the total SGWB energy-density spectrua at frequency 10[Hz] ≲ f ≲ 10 3 [Hz] in Fig. 5, where the ratio of the contribution of BH binary mergers to the contribution of neutron star (NS) binary and neutron star-black hole (NS-BH) mergers is supposed to be 1 : 0 . 3 which is consistent with the forecast by [7]. Compared with Fig. 3 and Fig. 4, we confirm that the effects of δω lmn on the total SGWB energy-density spectrum is lager than that of δτ lmn . However, the frequency where their effects on the total SGWB energy-density spectrum become obvious is dependent on the values of them because SGWB results from the superposition of many GWs. For example, the blue dashed curve just deviates the black solid curve obviously at f > 10 3 [Hz]. It is worth noting that the pSEOBNRv4HM PA model is for spin-aligned compact binaries moving in quasicircular orbits. For a given fiducial PP model, this approximation of the inspiral region will only affects the total SGWB energy-density spectra at the lower frequency span 10[Hz] ≲ f ≲ 10 2 [Hz] while the effects of { δω lmn , δτ lmn } dominate at the higher frequency span 10 2 [Hz] ≲ f ≲ 10 3 [Hz]. As shown in Fig. 5, these two parts don't correlate with each other directly and our forecasting constraints on { δω lmn , δτ lmn } will not be influenced by this approximation.",
"pages": [
4
]
},
{
"title": "III. FORECASTING CONSTRAINTS ON THE NO-HAIR THEOREM",
"content": "A stationary, Gaussian, unpolarized and isotropic SGWB can be detected by a cross-correlation statistic C ( f ) between two GW detectors [9-11]. The Gaussian likelihood for the measured cross-correlations C ( f ) is where the variance of C ( f ) is Here we assume one year's worth of data T = 1year and a frequency bin width of df = 0 . 25[Hz]. P ( f ) 1 , 2 are the one-sided noise power spectral densities of the two GW detectors, which are choosen as CE's [58]. Since the actual locations and arm orientations for CE are yet to be determined, we assume that CE shares the same overlap reduction function γ T ( f ) with LIGO. As shown in Fig. 5, the expected sensitivity of LIGO's anticipated 'A+' configuration [59-61] is not much lower than the total SGWB energy-density spectrum. Although the sensitivity of CE is much lower than the total SGWB energy-density spectrum, there is no measured C ( f ) for CE now. Therefore, we just turn to the Fisher information matrix to obtain the forecasting constraints where θ is a vector consisting of 5 free parameters and the derivative of the total SGWB energy-density spectrum with respect to each parameter is defined as and is calculated numerically by choosing d θ i = 0 . 1. Here we introduce three new parameters { z R , z m 1 , z q } to summarize the original eight parameters of the fiducial PP model as where z R scales the merger rate R BH ( z quoted = 0 . 2) directly, z m 1 scales the primary mass distribution around the pivot mass m 1 = 27 . 9[M ⊙ ], z q scales the mass ratio distribution around the pivot ratio q = 1 and A (or B ) is introduced to guarantee that z m 1 (or z q ) doesn't modify R BH ( z quoted = 0 . 2). That is to say, these three new parameters are independent of each other by construction. Therefore, we can obtain the non-flat priors of them from the constraints on the fiducial PP model as { z R = 0 ± 0 . 6672 , z m 1 = 0 ± 0 . 1062 , z q = 0 ± 0 . 1965 } (90% C.L.), where we have ignored the correlations between them. In Fig. 6, we show that the effects of { z R , z m 1 , z q } are similar to ones of the original fiducial PP model. Weplot the absolute value of these derivatives in Fig. 7. While both | ∂ Ω GW ∂δτ 220 | and | ∂ Ω GW ∂δω 220 | are suppressed with f becoming lower which is consistent with the colored curves in Fig. 5, | ∂ Ω GW ∂z R | , | ∂ Ω GW ∂z m 1 | and | ∂ Ω GW ∂z q | are not very sensitive to f . The root mean square errors of these parame- Since the observations of GW transients and the direct observation of SGWB are independent of each other, the constraints on R BH from them should be independent of each other too. Here we rewrite the former constraints on R BH [7] as a set of non-flat priors which can be added to F ij in the form of In Fig. 8, we show the forecasting constraints on { δω 220 , δτ 220 , z R , z m 1 , z q } with the Fisher . py [62] package, where δω 220 = 0 ± 0 . 1296, δτ 220 = 0 ± 0 . 0678, z R = 0 ± 0 . 1219, z m 1 = 0 ± 0 . 0597 and z q = 0 ± 0 . 5155 at 68% confidence range (inner black contour) when only (2 , 2) mode and its hairs are considered and the flat priors of { z R , z m 1 , z q } are added, δω 220 = 0 ± 0 . 0903, δτ 220 = 0 ± 0 . 0608, z R = 0 ± 0 . 0283, z m 1 = 0 ± 0 . 0434 and z q = 0 ± 0 . 1161 at 68% confidence range (inner blue contour) when only (2 , 2) mode and its hairs are considered and the non-flat priors of { z R , z m 1 , z q } are added, δω 220 = 0 ± 0 . 0907, δτ 220 = 0 ± 0 . 0610, z R = 0 ± 0 . 0284, z m 1 = 0 ± 0 . 0435 and z q = 0 ± 0 . 1167 at 68% confidence range (inner green contour) when the higher modes are also considered according to Eq. (14) and the non-flat priors of { z R , z m 1 , z q } are added and δω 220 = 0 ± 0 . 0908, δτ 220 = 0 ± 0 . 0611, z R = 0 ± 0 . 0284, z m 1 = 0 ± 0 . 0435 and z q = 0 ± 0 . 1167 at 68% confidence range (inner red contour) when the higher modes and their corresponding hairs are considered simultaneously and the non-flat priors of { z R , z m 1 , z q } are added. We can find that the green contours almost completely overlap the blue contours because the higher modes without hairs just shift the values of Ω GW ( f ) but don't affect their derivative with respect to each hair ∂ Ω GW ∂ θ i ( f ). Since the contributions of the higher modes' hairs should be smaller than the contributions of the higher modes themselves, the red contours also almost completely overlap the blue contours. Interestingly, there is a slight correlation between (2 , 2) mode's two hairs. It means that certain deviations from GR probably affect the ringdown frequencies and damping times simultaneously. There is an obvious positive correlation between z R and z q . Although (1 + z R ) serves as a factor of R BH (or Ω GW , BH ) according to Eq. (23) (or Eq. (16)), z q doesn't affect R BH directly according to the definition of B in Eq. (23). Therefore, z q must affects Ω GW , BH via ⟨ dE s /df s ⟩ BH according to Eq. (17). For example, z q = 0 . 1965 widens the mass ratio distribution to 0 . 1 ≲ q ≤ 1 as shown in the right subplot of Fig. 6 then suppresses the probability of 0 . 3 ≲ q ≤ 1 overall due to B , hence a smaller . ⟨ dE s /df s ⟩ BH due to the suppression of dE s /df s by the smaller mass ratio 0 . 1 ≲ q ≲ 0 . 3. For z q = -0 . 1965, the reverse applies. There are correlations between { z R , z q } and { δω 220 , δτ 220 } in the black contours when the flat priors of { z R , z q } are added but these correlations almost disappear in the blue contours when the non-flat priors of { z R , z q } are added. The reason for this disappearance is that both z R and z q can serve as a factor of Ω GW , BH and make an almost frequency-independent contribution to Ω GW , BH . That is to say, the absolute contribution of { z R = 0 ± 0 . 0283 , z q = 0 ± 0 . 1161 } is much larger at lower frequency but much smaller at higher frequency than that of { δω 220 = 0 ± 0 . 0903 , δτ 220 = 0 ± 0 . 0608 } , as hinted in Fig. 7. These totally different behaviors of them lead to the negligible correlations between them. There are obvious correlations between z m 1 and { δω 220 , δτ 220 } in the black and blue contours. It means that Ω GW , BH at higher frequency is sensitive to z m 1 which results from the dramatic influence of z m 1 on the primary mass distribution around m 1 ≲ 10[M ⊙ ], hence indirect correlations between z m 1 and { δω 220 , δτ 220 } at higher frequency. Fi- m the present observations of GW transients is similar to the its forecasting constraint from the future observation of SGWB only, which leads to two similar constraints in the z m 1 -δω 220 plane and z m 1 -δτ 220 plane respectively.",
"pages": [
4,
7,
8
]
},
{
"title": "IV. SUMMARY AND DISCUSSION",
"content": "In this paper, we assume that the effects of the extra hairs on the inspiral and merger regions can be summarized by a set of effective intrinsic parameters under the no-hair theorem and then the residual effects of these extra hairs will just appear during an effective ringdown region. Then we turn to the PyCBC package[56] and the pSEOBNRv4HM PA model [46, 47] to obtain the effective complete IMR time-domain waveform with hairs, as shown in Fig. 2. After the Fourier transform (as shown in Fig. 3), we calculate the energy spectrum for every possible BH binary with different primary mass and mass ratio individually, as shown in Fig. 4. Combining these energy spectra under a fixed fiducial PP model [7, 8] , we obtain the modified total SGWB energy-density spectrum at frequency 10[Hz] ≲ f ≲ 10 3 [Hz] for given hairs, as shown in Fig. 5. Here we suppose that the ratio of the contribution of BH binary mergers to the contribution of NS binary and NS-BH mergers is about 1 : 0 . 3 [7]. To further take the uncertainties of the fiducial PP model [7, 8] into consideration, the Fisher information matrix should also include the parameters of the fiducial PP model. For simplicity, we reduce the original eight parameters of the fiducial PP model to { z R , z m 1 , z q } and change the uncertainties of the original eight ones to the non-flat priors of the latter three ones. By choosing the all free parameters as ∼ 0 . 1, we calculate numerically the derivative of the total SGWB energy-density spectrum with respect to each one respectively, as shown in Fig. 7. To obtain the variance of SGWB, we assume that CE shares the same overlap reduction function with LIGO. Finally, the forecasting constraints on hairs at 68% confidence range are δω 220 = 0 ± 0 . 1296 and δτ 220 = 0 ± 0 . 0678 when the flat priors of { z R , z m 1 , z q } are added but δω 220 = 0 ± 0 . 0903 and δτ 220 = 0 ± 0 . 0608 when the non-flat priors of { z R , z m 1 , z q } are added, as shown in Fig. 8. As for the higher modes, we find that they hardly affect the forecasting constraints on { δω 220 , δτ 220 } while they do contribute to and shift the total SGWB energydensity spectrum. And so do their corresponding hairs. There are three caveats. The first one is that we turn to the pSEOBNRv4HM PA model [46, 47] where the extra hairs appear only during the ringdown region. In fact, we have mimicked the effects of the extra hairs on the inspiral and merger regions with an effective GR's IMR waveform and then leaved the residual effects on the ringdown region alone. Therefore, the chosen fiducial PP model [7, 8] for the mass distribution is also an effective one which has included the effects of the extra hairs on the inspiral and merger regions. The second one is that we assume the ratio of the contribution of BH binary mergers to the contribution of NS binary and NS-BH mergers is about 1 : 0 . 3 according to Fig. 23 of [7]. It is just a temporary assumption and will be improved according to the newest GW observations. The third one is that CE shares the same overlap reduction function γ T ( f ) with LIGO. It is also a temporary assumption. Because γ T ( f ) is determined by the relative positions and orientations of a pair of detectors and the actual locations and arm orientations for CE are yet to be determined.",
"pages": [
8,
9,
10
]
},
{
"title": "Acknowledgments",
"content": "Ke Wang is supported by grants from NSFC (grant No. 12005084 and grant No.12247101). Phys. Rev. D 94 , no.2, 021101 (2016) [arXiv:1602.02453 [gr-qc]].",
"pages": [
10,
11
]
}
]
2024PhRvD.110b4067H
https://arxiv.org/pdf/2404.19470.pdf
<document>
<section_header_level_1><location><page_1><loc_10><loc_90><loc_91><loc_93></location>New phenomenology in the first-order thermodynamics of scalar-tensor gravity for Bianchi universes</section_header_level_1>
<text><location><page_1><loc_36><loc_87><loc_65><loc_89></location>Julien Houle 1, ∗ and Valerio Faraoni 1, †</text>
<text><location><page_1><loc_30><loc_84><loc_71><loc_87></location>1 Department of Physics & Astronomy, Bishop's University, 2600 College Street, Sherbrooke, Qu'ebec, Canada J1M 1Z7</text>
<text><location><page_1><loc_18><loc_78><loc_83><loc_83></location>The phase space of Bianchi I universes in vacuum Brans-Dicke gravity is analyzed in terms of physical variables. The behaviour of the solutions of the field equations near the fixed points (which are solutions of Einstein gravity) is compared with basic ideas of the recent first-order thermodynamics of scalar-tensor gravity, elucidating new phenomenology.</text>
<section_header_level_1><location><page_1><loc_20><loc_74><loc_37><loc_75></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_39><loc_49><loc_72></location>Shortly after the introduction of general relativity (GR), researchers began looking for alternative theories of gravity, moved by pure curiosity about how things could be different in nature [1, 2]. More concrete motivation emerged with the birth of quantum field theory, when the question arose of how to reconcile the two biggest physics discoveries of the twentieth century, quantum mechanics and GR. The reason is that, as soon as one introduces the lowest-order quantum corrections to GR, one simultaneously causes deviations from it in the form of higher derivative terms in the field equations or extra degrees of freedom [3, 4]. This situation does not change in string theory: the simplest string theory, the bosonic string, has a low-energy limit that reproduces not GR, but an ω = -1 Brans-Dicke gravity [5, 6]. The prototype of alternative gravity is scalar-tensor gravity, which contains only a scalar degree of freedom φ in addition to the two massless spin two modes familiar from GR. The original Jordan-Brans-Dicke theory [7] was later generalized to wider scalar-tensor theories [8-11] in which the 'Brans-Dicke coupling' parameter became a function ω ( φ ) of the scalar field φ , which was also endowed with a potential V ( φ ).</text>
<text><location><page_1><loc_9><loc_16><loc_49><loc_39></location>There is independent motivation for the study of alternative theories of gravity coming from cosmological observations. The 1998 discovery, made with type Ia supernovae, that the present expansion of the universe is accelerated led to introducing overnight a completely ad hoc dark energy with very exotic properties (equation of state parameter close to w /similarequal -1), of completely unkown nature and comprising approximately 70% of the energy content of the universe (see [12] for a review). A wide range of dark energy models have been proposed in the literature, but none is compelling and this state of affairs is deeply unsatisfactory from the theoretical point of view. For this reason, many cosmologists have turned to questioning whether, instead, we do not understand gravity on the largest (cosmological) scales and dark energy simply does not exist [13, 14]. This idea had led to</text>
<text><location><page_1><loc_52><loc_59><loc_92><loc_75></location>formulating and testing modified gravity models. Among a spectrum of possibilities, the most popular models belong to the so-called metric f ( R ) gravity class (see [1517] for reviews). Metric f ( R ) gravity contains only one extra massive, propagating, scalar degree of freedom and, therefore, falls into the wider category of scalar-tensor gravity [15-17]. Even Starobinski inflation [18], the first scenario of inflation and the one currently favored by observations [19], is based on quadratic corrections to the Einstein-Hilbert action and is ultimately a scalar-tensor theory.</text>
<text><location><page_1><loc_52><loc_45><loc_92><loc_59></location>In the past decade, the problem of finding the most general scalar-tensor theory expressed by field equations that are at most of second order led to the rediscovery and intense study of the older Horndeski gravity [20]. This sought-for property was found to belong not to Horndeski gravity but to the more general Degenerate Higher Order Scalar-Tensor (DHOST) theories, a subclass of higher order gravities in which a degeneracy condition brings the order of the field equations back to two ( e.g. , [21-33], see [34-36] for reviews).</text>
<text><location><page_1><loc_52><loc_11><loc_92><loc_45></location>Given the wide spectrum of scalar-tensor gravities (not to mention other alternatives richer in propagating degrees of freedom that are difficult to identify and count [37]), what is the role of GR in this landscape? A proposal is well-known in the context of emergent gravity, in which the field equations can be deduced as an emergent or collective property of underlying degrees of freedom and are not fundamental. The seminal paper by Jacobson [38] derived the Einstein equations of GR from purely thermodynamical considerations, an idea referred to as 'thermodynamics of spacetime'. This feat was repeated with quadratic f ( R ) gravity producing a new picture: GR is somehow a state of 'thermal equilibrium' of gravity, while alternative theories correspond to entropy generation and to excited thermal states [39]. This view has been very influential and has generated a large literature but, unfortunately, no substantial progress has been made since its early days. Specifically, the 'temperature of gravity' (or other order parameter) has not been identified and no equation describing the relaxation of alternative gravity to GR has been proposed, although there are reasons to believe that such phenomena could have occurred in the early universe [40, 41].</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_11></location>Recently a new proposal has been advanced, known as the first-order thermodynamics of scalar-tensor gravity</text>
<text><location><page_2><loc_9><loc_70><loc_49><loc_93></location>[42-46]: it has nothing to do with Jacobson's thermodynamics of spacetime, although some of its ideas share the same spirit. It is a far-reaching analogy (but, still, only an analogy) between the description of the effective stress-energy tensor of the scalar degree of freedom φ of scalar-tensor gravity and the stress-energy tensor of a dissipative fluid. By writing the field equations of scalar-tensor gravity as effective Einstein equations, the contributions of φ and of its first and second derivatives form an effective stress-energy tensor T ( φ ) ab which has the form of a dissipative fluid stress-energy tensor [42] (this fact has been known for a long time for special theories or special geometries [47, 48] and has been recently recognized also for 'viable' Horndeski gravity [49, 50]). Specifically, if the scalar field gradient ∇ a φ is timelike and future-oriented [45], its normalized version</text>
<formula><location><page_2><loc_22><loc_65><loc_49><loc_69></location>u a ≡ ∇ a φ √ -∇ c φ ∇ c φ (1.1)</formula>
<text><location><page_2><loc_9><loc_61><loc_49><loc_64></location>can be seen as the four-velocity of an effective fluid with stress-energy tensor of the form</text>
<formula><location><page_2><loc_12><loc_58><loc_49><loc_60></location>T ( φ ) ab = ρu a u b + Ph ab + π ab + q a u b + q b u a , (1.2)</formula>
<text><location><page_2><loc_9><loc_47><loc_49><loc_57></location>where ρ is an effective energy density, P is an effective isotropic pressure, π ab is an effective anisotropic, tracefree, stress tensor, and q a is an effective heat flux density. Here h ab ≡ g ab + u a u b is the Riemannian three-space metric seen by observers comoving with this fluid (with h a b the projector onto this 3-space), while π ab and q a are purely spatial:</text>
<formula><location><page_2><loc_10><loc_42><loc_49><loc_45></location>h ab u a = h ab u b = 0 , π ab u a = π ab u b = 0 , q c u c = 0 . (1.3)</formula>
<text><location><page_2><loc_9><loc_32><loc_49><loc_42></location>The fact that this T ( φ ) ab has the dissipative fluid form contains no physics: the decomposition (1.2) holds true for any symmetric two-index tensor [46]. However, when one takes seriously this dissipative fluid structure and tries to apply to it Eckart's [51] theory of dissipative fluids [4244] one discovers that, miraculously, Eckart's constitutive relation</text>
<formula><location><page_2><loc_20><loc_27><loc_49><loc_30></location>q a = -K h ab ( ∇ b T + T ˙ u b ) (1.4)</formula>
<text><location><page_2><loc_9><loc_12><loc_49><loc_27></location>holds. Here T is the temperature of the dissipative fluid, K is the thermal conductivity, and ˙ u a is the fluid fouracceleration. Equation (1.4) is nothing but the relativistic generalization of Fourier's law with the addition of an inertial term proportional to the four-acceleration, which takes into account the fact that heat is a form of energy and its transport contributes to the energy flux in a way that was absent in pre-relativistic physics [51]. The unexpected fact that this relation holds for the effective φ -fluid makes it possible to define the product KT for scalar-tensor gravity.</text>
<text><location><page_2><loc_9><loc_9><loc_49><loc_11></location>Let us refer, for simplicity, to 'first-generation' ( i.e. , pre-Horndeski) scalar-tensor gravity: in the Jordan</text>
<text><location><page_2><loc_52><loc_90><loc_92><loc_93></location>frame, the gravitational sector of the theory is described by the action 1</text>
<formula><location><page_2><loc_53><loc_85><loc_92><loc_89></location>S ST = 1 16 π ∫ d 4 x √ -g [ φR -ω ( φ ) φ ∇ c φ ∇ c φ -V ( φ ) ] (1.7)</formula>
<text><location><page_2><loc_52><loc_72><loc_92><loc_84></location>where φ > 0 is the Brans-Dicke scalar (approximately equivalent to the inverse of the effective gravitational coupling strength G -1 ), the function ω ( φ ) (which was a strictly constant parameter in the original Brans-Dicke theory) is the 'Brans-Dicke coupling', and V ( φ ) is a scalar field potential (absent in the original Brans-Dicke theory). When ∇ a φ is timelike and future-oriented, the product of effective thermal conductivity and effective temperature is found to be [43, 44]</text>
<formula><location><page_2><loc_65><loc_67><loc_92><loc_71></location>KT = √ -∇ c φ ∇ c φ 8 πφ . (1.8)</formula>
<text><location><page_2><loc_52><loc_56><loc_92><loc_66></location>It is apparent that GR, reproduced for φ = const. > 0, corresponds to KT = 0. It is rather intuitive that, when extra degrees of freedom with respect to GR are excited, gravity is in some sense excited and 'hotter' than the GR state in which the extra degrees of freedom are absent. The 'temperature of gravity' T is, in this sense, a temperature relative to the GR state of equilibrium .</text>
<text><location><page_2><loc_52><loc_24><loc_92><loc_56></location>The first-order thermodynamics of scalar-tensor gravity has been extended [50] to 'viable' Horndeski gravity, i.e. , to the subclass in which gravitational waves propagate at the speed of light, and applied to various situations in cosmology and other contexts [46, 5360] (see [61] for a recent review). Two basic qualitative ideas emerge from these studies: near spacetime singularities gravity is 'hot', i.e. , it diverges from GR; the expansion of the three-space perceived by comoving observers (with 3-metric h ab ) 'cools' gravity, bringing it closer to GR. These ideas have been tested against various situations of physical interest or mathematical convenience (for which it is possible to draw analytically definite conclusions). Here we continue this program. While it was natural to apply the first-order thermodynamics of scalar-tensor gravity to Friedmann-LemaˆıtreRobertson-Walker (FLRW) cosmology [54, 62], here we extend the description to anisotropic Bianchi universes. For simplicity, we confine ourselves to Brans-Dicke gravity and to the simplest anisotropic cosmologies described by spatially flat Bianchi I geometries. To describe the dynamics we resort to a phase space view and, contrary</text>
<formula><location><page_2><loc_55><loc_13><loc_92><loc_14></location>R ab ≡ R c acb = ∂ c Γ c ba -∂ b Γ c ca +Γ c cd Γ d ba -Γ c bd Γ d ca , (1.5)</formula>
<formula><location><page_2><loc_58><loc_11><loc_92><loc_12></location>R ≡ R a a = ∂ c Γ c ba -∂ b Γ c ca +Γ c cd Γ d ba -Γ c bd Γ d ca . (1.6)</formula>
<text><location><page_3><loc_9><loc_83><loc_49><loc_93></location>to the literature that we are aware of, we choose as phase space variables the average Hubble function H , the scalar field φ , and its time derivative ˙ φ that have a direct physical meaning instead of other variables obtained by various non-linear combinations of physical ones (for this reason we cannot avail ourselves of existing phase space analyses).</text>
<section_header_level_1><location><page_3><loc_12><loc_76><loc_46><loc_78></location>II. FIELD EQUATIONS AND BIANCHI I GEOMETRY</section_header_level_1>
<text><location><page_3><loc_9><loc_69><loc_49><loc_73></location>The (Jordan frame) vacuum field equations obtained by varying the action (1.7) with respect to the inverse metric g ab and the scalar φ are</text>
<formula><location><page_3><loc_11><loc_59><loc_49><loc_67></location>R ab -1 2 g ab R = ω φ 2 ( ∇ a φ ∇ b φ -1 2 g ab ∇ c φ ∇ c φ ) + 1 φ ( ∇ a ∇ b φ -g ab /square φ ) -V 2 φ g ab , (2.1)</formula>
<formula><location><page_3><loc_11><loc_51><loc_49><loc_55></location>/square φ = 1 2 ω +3 ( φ dV dφ -2 V -dω dφ ∇ c φ ∇ c φ ) . (2.2)</formula>
<text><location><page_3><loc_9><loc_45><loc_49><loc_50></location>In the following we assume V ( φ ) ≥ 0, constant BransDicke coupling ω , and 2 ω +3 > 0 to avoid phantom scalar fields φ .</text>
<text><location><page_3><loc_10><loc_44><loc_41><loc_45></location>The line element of a Bianchi I universe is</text>
<formula><location><page_3><loc_11><loc_39><loc_49><loc_42></location>ds 2 = -dt 2 + A 2 ( t ) dx 2 + B 2 ( t ) dy 2 + C 2 ( t ) dz 2 (2.3)</formula>
<text><location><page_3><loc_9><loc_31><loc_49><loc_38></location>in Cartesian comoving coordinates ( t, x, y, z ), where A ( t ) , B ( t ), and C ( t ) are the scale factors associated with the three orthogonal spatial directions. This anisotropic universe has average scale factor a ( t ) ≡ ( ABC ) 1 / 3 and average Hubble parameter</text>
<formula><location><page_3><loc_10><loc_24><loc_49><loc_29></location>H ≡ ˙ a a = 1 3 ( ˙ A A + ˙ B B + ˙ C C ) ≡ 1 3 ( H A + H B + H C ) , (2.4)</formula>
<text><location><page_3><loc_9><loc_19><loc_49><loc_24></location>where an overdot denotes differentiation with respect to the cosmic time t , H i ≡ ˙ A i /A (where A i = A,B , or C ), and</text>
<formula><location><page_3><loc_11><loc_12><loc_49><loc_18></location>H 2 = 1 9 ( H 2 A + H 2 B + H 2 C +2 H A H B +2 H A H C +2 H B H C ) . (2.5)</formula>
<text><location><page_3><loc_9><loc_9><loc_49><loc_10></location>The shear scalar Σ for this scalar-tensor gravity, com-</text>
<text><location><page_3><loc_52><loc_92><loc_91><loc_93></location>uted in terms of φ and its derivatives, is given by [42]</text>
<formula><location><page_3><loc_52><loc_71><loc_96><loc_89></location>σ ≡ √ 1 2 σ ab σ ab = ( -∇ e φ ∇ e φ ) -3 / 2 { 1 3 ( ∇ a ∇ b φ ∇ a φ ∇ b φ ) 2 + 1 2 ( ∇ e φ ∇ e φ ) 2 [ ∇ a ∇ b φ ∇ a ∇ b φ -1 3 ( /square φ ) 2 ] -( ∇ e φ ∇ e φ ) × ( ∇ a ∇ b φ ∇ b ∇ c φ -1 3 /square φ ∇ a ∇ c φ ) ∇ a φ ∇ c φ } 1 / 2 , (2.6)</formula>
<text><location><page_3><loc_52><loc_64><loc_92><loc_68></location>where σ ab is the shear tensor [52]. For clarity, we use the shear variable Σ ≡ 1 2 σ 2 which, in the Bianchi I geometry (2.3), assumes the form</text>
<formula><location><page_3><loc_52><loc_54><loc_87><loc_62></location>Σ = 1 6 A 2 B 2 C 2 ( B 2 C 2 ˙ A 2 + A 2 C 2 ˙ B 2 + A 2 B 2 ˙ C 2 -ABC 2 ˙ A ˙ B -AB 2 C ˙ A ˙ C -A 2 BC ˙ B ˙ C )</formula>
<formula><location><page_3><loc_54><loc_50><loc_79><loc_54></location>= B 2 C 2 ˙ A 2 + A 2 C 2 ˙ B 2 + A 2 B 2 ˙ C 2 6 A 2 B 2 C 2</formula>
<formula><location><page_3><loc_58><loc_48><loc_73><loc_49></location>C ˙ A ˙ B + B ˙ A ˙ C + A ˙ B ˙ C</formula>
<formula><location><page_3><loc_54><loc_34><loc_92><loc_48></location>-6 ABC = 1 6 ( H 2 A + H 2 B + H 2 C -H A H B -H A H C -H B H C ) = 1 4 ( H 2 A + H 2 B + H 2 C ) -3 4 H 2 = 3 H 2 2 -1 2 ( H A H B + H A H C + H B H C ) . (2.7)</formula>
<text><location><page_3><loc_52><loc_27><loc_92><loc_32></location>Σ vanishes if and only if all the components of σ ab vanish [63], in which case the Bianchi geometry reduces to a FLRW one.</text>
<text><location><page_3><loc_52><loc_24><loc_92><loc_26></location>The only non-vanishing Christoffel symbols of the geometry (2.3) are</text>
<formula><location><page_3><loc_57><loc_12><loc_92><loc_21></location>Γ t xx = A ˙ A, Γ t yy = B ˙ B, Γ t zz = C ˙ C , Γ x tx = Γ x xt = ˙ A A , Γ y ty = Γ y yt = ˙ B B , Γ z tz = Γ z zt = ˙ C C , (2.8)</formula>
<text><location><page_3><loc_52><loc_9><loc_92><loc_10></location>while the non-vanishing components of the Ricci tensor</text>
<formula><location><page_4><loc_16><loc_87><loc_49><loc_91></location>R tt = -( A A + B B + C C ) , (2.9)</formula>
<formula><location><page_4><loc_15><loc_82><loc_49><loc_86></location>R xx = ABC A + ( AC ˙ B + AB ˙ C ) ˙ A BC , (2.10)</formula>
<formula><location><page_4><loc_15><loc_77><loc_49><loc_81></location>R yy = ABC B + ( BC ˙ A + AB ˙ C ) ˙ B AC , (2.11)</formula>
<formula><location><page_4><loc_15><loc_72><loc_49><loc_76></location>R zz = ABC C + ( BC ˙ A + AC ˙ B ) ˙ C AB , (2.12)</formula>
<text><location><page_4><loc_9><loc_70><loc_27><loc_71></location>and the Ricci scalar reads</text>
<formula><location><page_4><loc_10><loc_63><loc_49><loc_68></location>R = 2 ( A A + B B + C C + H A H B + H B H C + H A H C ) (2.13)</formula>
<formula><location><page_4><loc_12><loc_59><loc_49><loc_62></location>= 6 ( ˙ H +2 H 2 + 2Σ 3 ) . (2.14)</formula>
<text><location><page_4><loc_9><loc_55><loc_49><loc_58></location>The time-time component of the Brans-Dicke field equations (2.1) is</text>
<formula><location><page_4><loc_14><loc_50><loc_49><loc_54></location>H 2 = ω 6 ( ˙ φ φ ) 2 + V 6 φ + 2Σ 3 -H ˙ φ φ , (2.15)</formula>
<text><location><page_4><loc_9><loc_47><loc_33><loc_48></location>while the spatial components read</text>
<formula><location><page_4><loc_10><loc_39><loc_49><loc_46></location>-ωABC ˙ φ 2 + ABCVφ -2 φ 2 ( A BC + A ˙ B ˙ C + AB C ) -2 ABCφ ¨ φ -2 φ ˙ φ ( A ˙ BC + AB ˙ C ) = 0 , (2.16)</formula>
<formula><location><page_4><loc_10><loc_31><loc_49><loc_37></location>-ωABC ˙ φ 2 + ABCVφ -2 φ 2 ( ABC + ˙ AB ˙ C + AB C ) -2 ABCφ ¨ φ -2 φ ˙ φ ( ˙ ABC + AB ˙ C ) = 0 , (2.17)</formula>
<formula><location><page_4><loc_10><loc_22><loc_49><loc_29></location>-ωABC ˙ φ 2 + ABCVφ -2 φ 2 ( ABC + ˙ A ˙ BC + A BC ) -2 ABCφ ¨ φ -2 φ ˙ φ ( ˙ ABC + A ˙ BC ) = 0 . (2.18)</formula>
<text><location><page_4><loc_9><loc_21><loc_39><loc_22></location>and the trace of the field equation (2.1) is</text>
<formula><location><page_4><loc_10><loc_15><loc_49><loc_20></location>˙ H = -ω 6 ( ˙ φ φ ) 2 + V 3 φ -2Σ 3 -2 H 2 -( ¨ φ +3 H ˙ φ ) 2 φ . (2.19)</formula>
<text><location><page_4><loc_10><loc_13><loc_36><loc_14></location>Equation (2.2) for the scalar field is</text>
<formula><location><page_4><loc_19><loc_8><loc_49><loc_11></location>¨ φ +3 H ˙ φ + φV ' -2 V 2 ω +3 = 0 , (2.20)</formula>
<text><location><page_4><loc_52><loc_90><loc_92><loc_93></location>where a prime denotes differentiation with respect to φ . Combining these three equations yields</text>
<formula><location><page_4><loc_54><loc_85><loc_92><loc_89></location>˙ H = -ω 2 ( ˙ φ φ ) 2 -2Σ + 2 H ˙ φ φ + ( φV ' -2 V ) 2 φ (2 ω +3) . (2.21)</formula>
<text><location><page_4><loc_52><loc_81><loc_92><loc_83></location>By inserting Eq. (2.20) for the scalar field φ into (2.19) and combining the result with Eq. (2.15), one obtains</text>
<formula><location><page_4><loc_54><loc_76><loc_92><loc_79></location>˙ H = -3 H 2 -H ˙ φ φ + φV ' ( φ ) + (2 ω +1) V ( φ ) 2 φ (2 ω +3) . (2.22)</formula>
<text><location><page_4><loc_52><loc_69><loc_92><loc_74></location>To summarize, the field equations to be solved for the scalar field φ ( t ) and the Hubble function H ( t ) are Eq. (2.20) and Eq. (2.22), respectively. Once these quantities are known, Eq. (2.15) gives the shear Σ.</text>
<section_header_level_1><location><page_4><loc_64><loc_64><loc_80><loc_65></location>III. PHASE SPACE</section_header_level_1>
<text><location><page_4><loc_52><loc_31><loc_92><loc_62></location>Let us discuss the phase space of Bianchi I cosmologies in vacuum Brans-Dicke gravity, where the dynamics is due entirely to the Brans-Dicke scalar φ . We use the variables ( H,φ, ˙ φ ) which are physical: the Hubble parameter H is a cosmological observable (although its actual value is subject to a very significant tension [64, 65]), while φ is the extra scalar degree of freedom of scalartensor gravity in addition to the two spin zero massless modes of GR contained in the metric g ab . The strength of the gravitational coupling G /similarequal φ -1 is measured directly by Cavendish experiments and its time variation ( i.e. , ˙ G and, consequently, ˙ φ ) is subject to observational constraints [66, 67]. By contrast, much of the existing literature on Bianchi cosmologies uses variables which are complicated functions of H,φ , and ˙ φ and do not have direct physical interpretation. Although they may make the study of the phase space dynamics more convenient from the formal point of view, they have no direct physical meaning. Here we want to interpret the dynamics and compare it with the first-order thermodynamics of spacetime, therefore we must use physical variables.</text>
<text><location><page_4><loc_53><loc_30><loc_76><loc_31></location>Equation (2.15) yields the shear</text>
<formula><location><page_4><loc_53><loc_22><loc_92><loc_28></location>Σ ( H,φ, ˙ φ ) = 3 2 H 2 + H ˙ φ φ -ω 6 ( ˙ φ φ ) 2 -V 6 φ (3.1)</formula>
<text><location><page_4><loc_52><loc_18><loc_92><loc_22></location>as a function of the three phase space variables, therefore Σ is not an independent variable, although it will be relevant in our analysis.</text>
<text><location><page_4><loc_52><loc_9><loc_92><loc_18></location>With our choice of variables the fixed points in the phase space, if they exist, have necessarily the form ( H,φ, ˙ φ ) = ( H 0 , φ 0 , 0), with H 0 and φ 0 > 0 constants. They are solutions of the Einstein equations located on the 'GR plane' ˙ φ = 0 of the phase space identified by constant scalar field, the condition that reproduces GR.</text>
<text><location><page_5><loc_9><loc_90><loc_49><loc_93></location>Using the notation V ( φ 0 ) ≡ V 0 and V ' ( φ 0 ) ≡ V ' 0 , Eq. (3.1) that must be satisfied by the fixed points gives</text>
<formula><location><page_5><loc_21><loc_85><loc_49><loc_89></location>Σ 0 = 3 2 ( H 2 0 -V 0 6 φ 0 ) , (3.2)</formula>
<text><location><page_5><loc_9><loc_82><loc_49><loc_85></location>while the mix of Eqs. (2.19) and (2.20) gives, using ˙ H = ω ' = 0,</text>
<formula><location><page_5><loc_17><loc_77><loc_49><loc_80></location>H 2 0 = V 0 6 φ 0 -Σ 0 3 + φ 0 V ' 0 -2 V 0 4 φ 0 (2 ω +3) (3.3)</formula>
<text><location><page_5><loc_9><loc_74><loc_41><loc_76></location>and the scalar field equation degenerates into</text>
<formula><location><page_5><loc_25><loc_70><loc_49><loc_73></location>V ' 0 = 2 V 0 φ 0 (3.4)</formula>
<text><location><page_5><loc_9><loc_66><loc_49><loc_69></location>(which is non-negative since V ≥ 0 and φ > 0) reducing Eq. (3.3) to</text>
<formula><location><page_5><loc_23><loc_62><loc_49><loc_65></location>H 2 0 = V 0 6 φ 0 -Σ 0 3 . (3.5)</formula>
<text><location><page_5><loc_9><loc_53><loc_49><loc_60></location>Comparing Eqs. (3.2) and (3.5), one obtains Σ 0 = 0: as expected, the shear vanishes at the fixed points, which have A = B = C ≡ a ( t ) and the average Hubble function H ( t ) coincides with the FLRW one. Equation (3.3) then becomes H 2 0 = V 0 / (6 φ 0 ), or</text>
<formula><location><page_5><loc_20><loc_48><loc_49><loc_52></location>H 0 = ± √ V 0 6 φ 0 = ± √ V ' 0 12 . (3.6)</formula>
<text><location><page_5><loc_9><loc_37><loc_49><loc_47></location>The degenerate fixed points corresponding to V 0 = V ' 0 = H 0 = 0 are Minkowski spaces, while those corresponding to V 0 > 0, V ' 0 > 0, and H 0 = ± √ V 0 6 φ 0 are de Sitter spaces. 2 When φ becomes a constant φ 0 and V ( φ 0 ) ≡ V 0 is positive, the theory of gravity reduces to GR with a positive cosmological constant Λ = V 0 .</text>
<text><location><page_5><loc_9><loc_17><loc_49><loc_37></location>In the study of exact solutions of the field equations, one sometimes find solutions ( H ( t ) , φ ( t ) , ˙ φ ( t ) ) with φ ( t ) → 0 + at late times t → + ∞ : these are pathological as the effective gravitational coupling G eff → + ∞ . The line φ = 0 in the 'GR plane' ˙ φ = 0 corresponds to singularities at which G eff changes sign, but it does so by going through G eff = ∞ (a similar situation occurs with conformally coupled scalar fields [68, 69]). Exact solutions with these properties are unphysical and cannot be regarded as GR solutions, even though the firstorder thermodynamics of scalar-tensor gravity does not, strictly speaking, indicate a pathology or a gross deviation from GR in this situation (see Appendix A).</text>
<section_header_level_1><location><page_5><loc_57><loc_92><loc_87><loc_93></location>A. Stability of the equilibrium points</section_header_level_1>
<text><location><page_5><loc_52><loc_87><loc_92><loc_90></location>Let us examine the stability of the fixed points with respect to homogeneous perturbations described by</text>
<formula><location><page_5><loc_65><loc_85><loc_92><loc_86></location>φ ( t ) = φ 0 + δφ ( t ) , (3.7)</formula>
<formula><location><page_5><loc_64><loc_81><loc_92><loc_82></location>H ( t ) = H 0 + δH ( t ) . (3.8)</formula>
<text><location><page_5><loc_52><loc_76><loc_92><loc_80></location>Evolution equations for the perturbations δφ ( t ) , δH ( t ) are obtained from Eq. (2.20) for the scalar field φ and Eq. (2.22) for H .</text>
<text><location><page_5><loc_52><loc_70><loc_92><loc_76></location>Let us begin with the stability of the scalar field. By expanding the scalar field potential, V ( φ ) /similarequal V 0 + V ' 0 δφ , and using the zero-order field equation (2.20), one obtains the linearized equation for δφ</text>
<formula><location><page_5><loc_63><loc_67><loc_92><loc_69></location>δ ¨ φ +3 H 0 δ ˙ φ + ω 2 0 δφ = 0 , (3.9)</formula>
<text><location><page_5><loc_52><loc_65><loc_56><loc_66></location>where</text>
<formula><location><page_5><loc_53><loc_60><loc_92><loc_64></location>ω 2 0 ≡ φ 0 V '' 0 -V ' 0 2 ω +3 = φ 0 V '' 0 -2 V 0 /φ 0 2 ω +3 = φ 0 V '' 0 -12 H 2 0 2 ω +3 . (3.10)</formula>
<text><location><page_5><loc_52><loc_59><loc_60><loc_60></location>The ansatz</text>
<formula><location><page_5><loc_67><loc_56><loc_92><loc_58></location>δφ ( t ) = δ 0 e αt (3.11)</formula>
<text><location><page_5><loc_52><loc_54><loc_90><loc_55></location>with δ 0 and α constants yields the algebraic equation</text>
<formula><location><page_5><loc_65><loc_51><loc_92><loc_53></location>α 2 +3 H 0 α + ω 2 0 = 0 (3.12)</formula>
<text><location><page_5><loc_52><loc_49><loc_59><loc_50></location>with roots</text>
<formula><location><page_5><loc_54><loc_44><loc_92><loc_48></location>α ( ± ) = -3 H 0 ± √ 9 H 2 0 -4 ω 2 0 2 ≡ 1 2 ( -3 H 0 ± √ ∆ ) (3.13)</formula>
<text><location><page_5><loc_91><loc_40><loc_91><loc_42></location>/negationslash</text>
<text><location><page_5><loc_52><loc_40><loc_92><loc_43></location>and two modes δφ ( ± ) ( t ) = δ 0 e α ( ± ) t (this applies if α ( ± ) = 0; the case α ( ± ) = 0 is discussed separately).</text>
<unordered_list>
<list_item><location><page_5><loc_54><loc_36><loc_92><loc_38></location>· If ∆ < 0, corresponding to ω 2 0 > 9 H 2 0 / 4 and ω 0 real, then</list_item>
</unordered_list>
<formula><location><page_5><loc_60><loc_33><loc_92><loc_35></location>δφ ( ± ) ( t ) = δ 0 e -3 H 0 t 2 e ± i 2 √ | ∆ | t : (3.14)</formula>
<text><location><page_5><loc_56><loc_24><loc_92><loc_32></location>the second exponential in the right-hand side oscillates while, as t → + ∞ , the first exponential diverges if H 0 < 0 and decays if H 0 > 0 (it remains constant if H 0 = 0). In this case the fixed point is stable if H 0 ≥ 0 and unstable if H 0 < 0.</text>
<unordered_list>
<list_item><location><page_5><loc_54><loc_18><loc_92><loc_24></location>· If ∆ = 0, corresponding to ω 2 0 = 9 H 2 0 / 4 (and ω 0 real), the scalar field perturbation is simply δφ ( t ) = δ 0 e -3 H 0 t 2 and is stable if H 0 ≥ 0, unstable if H 0 < 0.</list_item>
<list_item><location><page_5><loc_54><loc_14><loc_92><loc_17></location>· If ∆ > 0, corresponding to ω 2 0 < 9 H 2 0 / 4 then it is convenient to write</list_item>
</unordered_list>
<formula><location><page_5><loc_56><loc_7><loc_92><loc_13></location>α ( ± ) = 3 H 0 2 -1 ± √ 1 -( 2 ω 0 3 H 0 ) 2 . (3.15)</formula>
<text><location><page_6><loc_13><loc_83><loc_49><loc_93></location>If ω 0 is real, corresponding to φ 0 V '' 0 ≥ 12 H 2 0 , then √ 1 -( 2 ω 0 3 H 0 ) 2 < 1 and -1 ± √ 1 -( 2 ω 0 3 H 0 ) 2 < 0, hence α ( ± ) < 0 if H 0 > 0, or α ( ± ) = 0 if H 0 = 0 (corresponding to a Minkowski fixed point), or α ( ± ) > 0 if H 0 < 0. Fixed points with H 0 ≥ 0 are stable while those with H 0 < 0 are unstable.</text>
<text><location><page_6><loc_13><loc_75><loc_49><loc_82></location>If instead ω 0 is imaginary, ω 0 = i | ω 0 | , corresponding to φ 0 V '' 0 < 12 H 2 0 , then we have √ -1 + ( 2 ω 0 3 H 0 ) 2 > 0 and -1 -√ 1 -( 2 ω 0 3 H 0 ) 2 < 0.</text>
<text><location><page_6><loc_27><loc_70><loc_27><loc_72></location>/negationslash</text>
<text><location><page_6><loc_13><loc_69><loc_49><loc_76></location>The mode δφ (+) is unstable ( i.e. , α (+) > 0) if H 0 > 0, while the other mode δφ ( -) is unstable ( i.e. , α ( -) > 0) if H 0 < 0. In short, when ω 0 is imaginary and H 0 = 0 there is always a unstable mode and the fixed point is unstable.</text>
<text><location><page_6><loc_13><loc_65><loc_49><loc_69></location>If instead ω 0 is imaginary and H 0 = 0 (Minkowski fixed point), the equation for the scalar field perturbations reduces to</text>
<formula><location><page_6><loc_23><loc_61><loc_49><loc_64></location>δ ¨ φ -| ω 2 0 | δφ = 0 , (3.16)</formula>
<text><location><page_6><loc_13><loc_58><loc_49><loc_61></location>which describes an unstable inverted harmonic oscillator.</text>
<text><location><page_6><loc_9><loc_50><loc_49><loc_57></location>An exception not included in the previous discussion is the situation in which V = 0 and H 0 = 0, ω 0 = 0 corresponding to a Minkowski space. In this case, Eq. (3.9) reduces to ¨ φ = 0, which has a linear solution and this Minkowski space is unstable.</text>
<text><location><page_6><loc_9><loc_46><loc_49><loc_50></location>Let us consider now the perturbation δH ( t ). Using the zero-order equations, Eq. (2.22) gives the linearized equation of motion for δH</text>
<formula><location><page_6><loc_11><loc_40><loc_49><loc_45></location>δ ˙ H +6 H 0 δH = -H 0 φ 0 δ ˙ φ + [ V '' 0 φ 0 +(2 ω +1) V 0 φ 0 ] 2 φ 0 (2 ω +3) δφ (3.17)</formula>
<text><location><page_6><loc_9><loc_37><loc_24><loc_38></location>and Eq. (3.10) yields</text>
<formula><location><page_6><loc_11><loc_32><loc_49><loc_36></location>δ ˙ H +6 H 0 δH = -H 0 φ 0 δ ˙ φ + ( ω 2 0 +6 H 2 0 ) 2 φ 0 δφ. (3.18)</formula>
<text><location><page_6><loc_9><loc_29><loc_49><loc_32></location>Using the explicit form δφ ( ± ) ( t ) of the scalar field perturbation gives</text>
<formula><location><page_6><loc_20><loc_27><loc_49><loc_28></location>δ ˙ H +6 H 0 δH = β ( ± ) δφ, (3.19)</formula>
<text><location><page_6><loc_9><loc_24><loc_13><loc_26></location>where</text>
<formula><location><page_6><loc_18><loc_20><loc_49><loc_24></location>β ( ± ) = ω 2 0 +6 H 2 0 -2 α ( ± ) H 0 2 φ 0 . (3.20)</formula>
<text><location><page_6><loc_9><loc_17><loc_49><loc_19></location>The solution of this inhomogeneous ordinary differential equation is</text>
<formula><location><page_6><loc_14><loc_8><loc_49><loc_16></location>δH ( ± ) ( t ) = β ( ± ) α ( ± ) +6 H 0 δφ + C e -6 H 0 t = ( H 0 -α ( ± ) ) 2 φ 0 δφ + C e -6 H 0 t , (3.21)</formula>
<text><location><page_6><loc_52><loc_89><loc_92><loc_93></location>where C is an integration constant. The perturbation δH ( t ) diverges for H 0 < 0 regardless of the behaviour of the scalar field perturbation δφ .</text>
<text><location><page_6><loc_53><loc_88><loc_64><loc_89></location>To summarize:</text>
<unordered_list>
<list_item><location><page_6><loc_54><loc_81><loc_92><loc_86></location>· Contracting de Sitter spaces are always unstable fixed points, which can be understood as the effect of anti-friction in the (anti-)damped harmonic oscillator equation (3.9).</list_item>
<list_item><location><page_6><loc_54><loc_77><loc_92><loc_80></location>· Expanding de Sitter fixed points are stable if φ 0 V '' 0 > 12 H 2 0 and unstable if φ 0 V '' 0 < 12 H 2 0 .</list_item>
<list_item><location><page_6><loc_54><loc_69><loc_92><loc_76></location>· Minkowski fixed points are (marginally) stable if ω 0 is real (corresponding to φ 0 V '' 0 ≥ 12 H 2 0 ) and unstable otherwise. The exception is the Minkowski space obtained for V ≡ 0 , H 0 = 0, which is unstable.</list_item>
</unordered_list>
<text><location><page_6><loc_52><loc_57><loc_92><loc_67></location>Let us consider now the ratio of the shear variable to the expansion variable Σ /H 2 0 , which quantifies the amount of anisotropy and the departure from GR (remember that the fixed points, when they exist, all lie in the GR plane ˙ φ = 0 and Σ = 0). Since the equilibrium points are isotropic de Sitter spaces, the shear (3.1) is purely perturbative and given by</text>
<formula><location><page_6><loc_54><loc_45><loc_92><loc_56></location>Σ = δ Σ = 3 2 [ ( H 0 + δH ) 2 + δ ˙ φ φ 0 + δφ ( H 0 + δH ) -ω 6 ( δ ˙ φ φ 0 + δφ ) 2 -( V 0 + V ' 0 δφ ) 6 ( φ 0 + δφ ) (3.22)</formula>
<text><location><page_6><loc_52><loc_44><loc_74><loc_45></location>which, to first order, reduces to</text>
<formula><location><page_6><loc_57><loc_39><loc_92><loc_43></location>δ Σ H 2 0 = 3 2 H 0 [ 2 δH + δ ˙ φ φ 0 -H 0 φ 0 δφ ] . (3.23)</formula>
<text><location><page_6><loc_52><loc_35><loc_92><loc_38></location>Inserting the solution for the perturbations δφ, δH into this equation and using Eq. (3.12) to express ω 2 0 yields</text>
<formula><location><page_6><loc_66><loc_31><loc_92><loc_34></location>δ Σ H 2 0 = 3 C H 0 e -6 H 0 t , (3.24)</formula>
<text><location><page_6><loc_52><loc_27><loc_92><loc_29></location>(Since Σ /similarequal δ Σ ≥ 0, one deduces that sign( C ) = sign ( H 0 ).)</text>
<text><location><page_6><loc_52><loc_21><loc_92><loc_27></location>The ratio Σ /H 2 0 vanishes as t → + ∞ for all solutions that are perturbations of expanding de Sitter fixed points, and diverges in the same late-time limit near contracting de Sitter fixed points.</text>
<text><location><page_6><loc_52><loc_11><loc_92><loc_21></location>Let us now examine the significance of these results with respect to the first-order thermodynamics of spacetime of Refs. [42-46]. We emphasize that the following discussion is meaningful only because physical phase space variables ( H,φ, ˙ φ ) have been chosen at the outset.</text>
<text><location><page_6><loc_52><loc_9><loc_92><loc_13></location>de Sitter solutions with non-constant scalar field, which are possible in scalar-tensor gravity but not in GR, have been discussed in Ref. [57].</text>
<section_header_level_1><location><page_7><loc_13><loc_91><loc_44><loc_93></location>B. Comparison with the first-order thermodynamics of scalar-tensor gravity</section_header_level_1>
<text><location><page_7><loc_9><loc_83><loc_49><loc_88></location>In Brans-Dicke theory, the temperature of gravity relative to the GR state of equilibrium [42-46] is given by Eq. (1.8). For linear homogeneous perturbations of the fixed points, it reads</text>
<formula><location><page_7><loc_20><loc_78><loc_49><loc_81></location>KT = | α ( ± ) || δφ ( ± ) | 8 πφ 0 ≥ 0 . (3.25)</formula>
<text><location><page_7><loc_9><loc_71><loc_49><loc_77></location>The fixed points, which lie in the GR plane with ˙ φ = 0 clearly correspond to KT = 0. KT assumes the same form as in a FLRW universe, but the solution φ ( t ) is, in general, different in Bianchi I and in FLRW universes.</text>
<text><location><page_7><loc_9><loc_47><loc_49><loc_71></location>If the orbit of a solution in the ( H,φ, ˙ φ ) phase space lies near an expanding de Sitter fixed point and is attracted to it, the anisotropic three-space expands, and the solution converges to the zero-temperature state of equilibrium, while Σ /H 2 0 → 0 and this three-space isotropizes. The cooling of gravity ( KT → 0) is indeed another way of saying that GR is a late-time attractor of the dynamics. The situation is not so trivial, however, because there are exceptions for φ 0 V '' 0 < 12 H 2 0 . In this case three-space still expands exponentially but the de Sitter fixed point nearby is a repellor. It is still the case that H 0 > 0 and Σ /H 2 0 → 0 as t → + ∞ . How do we understand this situation in the light of scalar-tensor thermodynamics? The answer comes from examining the equation ruling the approach to/departure from the GR equilibrium state derived in Refs. [42-46]</text>
<formula><location><page_7><loc_14><loc_42><loc_49><loc_45></location>d ( KT ) dτ = 8 π ( KT ) 2 -Θ KT + /square φ 8 πφ , (3.26)</formula>
<text><location><page_7><loc_9><loc_30><loc_49><loc_40></location>where τ is the comoving time of the effective φ -fluid ( i.e. , the proper time of observers comoving with this fluid and with four-velocity (1.1)) and Θ is the expansion scalar of the fluid [52]. In a Bianchi I universe Θ = 3 H and τ = t . Near a fixed point (which lies in the GR plane) KT is a first-order quantity and Eq. (3.26) reduces, to linear order, to</text>
<formula><location><page_7><loc_13><loc_25><loc_49><loc_29></location>d ( KT ) dt = -3 H 0 KT -( δ ¨ φ +3 H 0 δ ˙ φ ) 8 πφ 0 (3.27)</formula>
<text><location><page_7><loc_9><loc_22><loc_38><loc_23></location>or, in the light of the previous discussion,</text>
<formula><location><page_7><loc_18><loc_17><loc_49><loc_21></location>d ( KT ) dt = -3 H 0 | δ ˙ φ | 8 πφ 0 + ω 2 0 δφ 8 πφ 0 . (3.28)</formula>
<text><location><page_7><loc_9><loc_9><loc_49><loc_16></location>For expanding de Sitter spaces with purely imaginary ω 0 it is ω 2 0 < 0. In order for the scalar field gradient ∇ a φ = -˙ φδ a 0 to be future-oriented it must be ˙ φ < 0, which implies that δφ = δ 0 e αt < 0, or δ 0 < 0. Then, for the exceptional expanding de Sitter fixed points with</text>
<text><location><page_7><loc_52><loc_92><loc_68><loc_93></location>imaginary ω 0 , we have</text>
<formula><location><page_7><loc_60><loc_79><loc_92><loc_91></location>d ( KT ) dt = -3 H 0 α | δφ | 8 πφ 0 + ω 2 0 δφ 8 πφ 0 = | δφ | 8 πφ 0 ( -3 H 0 α -ω 2 0 ) = α 2 8 πφ 0 | δφ | > 0 (3.29)</formula>
<text><location><page_7><loc_52><loc_73><loc_92><loc_78></location>using Eq. (3.12): KT always grows near these repellors, describing the departure from the GR equilibrium state. The reason for this behaviour is clearly due to the third term /square φ/ (8 πφ ) in the right-hand side of Eq. (3.26).</text>
<section_header_level_1><location><page_7><loc_63><loc_69><loc_80><loc_70></location>IV. CONCLUSIONS</section_header_level_1>
<text><location><page_7><loc_52><loc_57><loc_92><loc_67></location>Two basic insights have been obtained thus far in the first-order thermodynamics of scalar-tensor (including viable Horndeski) gravity [42-46]. The first one is that the expansion of the three-space seen by observers comoving with the effective φ -fluid 'cools' gravity. The second is that gravity is 'hot' ( i.e. , KT → + ∞ ) near spacetime singularities.</text>
<text><location><page_7><loc_84><loc_50><loc_84><loc_52></location>/negationslash</text>
<text><location><page_7><loc_52><loc_41><loc_92><loc_57></location>The idea that 'expansion cools gravity' ( i.e. , KT → 0) was deduced in Refs. [42-46] using situations in which /square φ = 0. The lesson from the present study is that this statement is not always true when /square φ = 0. The term /square φ/ (8 πφ ) in Eq. (3.26) cannot be expressed unambigously in terms of KT or its powers or derivatives. This third term in the right-hand side of (3.26) is reminiscent of entropy generation terms in non-equilibrium thermodynamics and it is fair to say that it is the dynamics of the scalar field itself, embodied in /square φ/φ , that drives gravity away from the GR equilibrium state.</text>
<text><location><page_7><loc_52><loc_34><loc_92><loc_41></location>When ω = const., V ( φ ) ≡ 0, and in the presence of conformally invariant matter (for example, in the radiation era), it is /square φ = 0 because Eq. (2.2) in the presence of matter and with a quadratic potential V ( φ ) = m 2 φ 2 / 2 becomes</text>
<formula><location><page_7><loc_67><loc_30><loc_92><loc_33></location>/square φ = 8 πT (m) 2 ω +3 (4.1)</formula>
<text><location><page_7><loc_52><loc_13><loc_92><loc_29></location>where T (m) is the trace of the matter energy-momentum tensor. /square φ vanishes in the presence of conformally invariant matter with zero trace, such as a radiation fluid in the radiation era, during which the expansion of space causes gravity to approach GR. This phenomenon was indeed reported in FLRW scalar-tensor cosmology [40, 41]. However, was not expected in other cosmological eras. The convergence of scalar-tensor to GR cosmology has been debated at length and we hope that our approach can shed some light on this issue, which we will discuss in a future publication.</text>
<text><location><page_7><loc_52><loc_9><loc_92><loc_13></location>Another lesson garnered from the discussion of the previous section is that the degree of anisotropy Σ /H 2 0 commonly used in the literature on Bianchi universes does</text>
<text><location><page_8><loc_9><loc_87><loc_49><loc_93></location>not tell the full story about the approach to, or departure from the GR state because it tends to zero for the exceptional expanding de Sitter fixed points with imaginary ω 0 that are phase space repellors.</text>
<text><location><page_8><loc_9><loc_74><loc_49><loc_87></location>To conclude, more research is needed to understand scalar-tensor gravity (and even more for Horndeski gravity) from the point of view of first-order thermodynamics. We remind the reader that this formalism is, ultimately, only an analogy; nevertheless, it is proving useful from the theoretical point of view and it is building up to a consistent framework to understand at least scalartensor gravity in the increasingly wider spectrum of alternatives to GR.</text>
<section_header_level_1><location><page_8><loc_19><loc_70><loc_39><loc_71></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_8><loc_9><loc_60><loc_49><loc_68></location>V. F. is grateful to Peter Dunsby, Andrea Giusti, Orlando Luongo, and Lavinia Heisenberg for discussions. This work is supported, in part, by the Natural Sciences & Engineering Research Council of Canada (grant 202303234 to V. F.) and by a Bishop's University Graduate Entrance Scholarship (J. H.).</text>
<section_header_level_1><location><page_8><loc_10><loc_54><loc_48><loc_57></location>Appendix A THE PATHOLOGICAL LINE φ = 0 IN THE ˙ φ = 0 PLANE OF THE PHASE SPACE</section_header_level_1>
<text><location><page_8><loc_9><loc_47><loc_49><loc_52></location>Let us consider an exact Bianchi I solution of BransDicke gravity that asymptotes to a φ = 0 solution. Assuming V ( φ ) ≡ 0, which yields /square φ = 0, consider the power-law ansatz for the scalar field</text>
<formula><location><page_8><loc_25><loc_44><loc_49><loc_46></location>φ ( t ) = φ 0 t α (A.1)</formula>
<unordered_list>
<list_item><location><page_8><loc_10><loc_35><loc_49><loc_38></location>[1] H. Weyl,'A New Extension of Relativity Theory,'Annalen Phys. 59 , 101-133 (1919)doi:10.1002/andp.19193641002</list_item>
<list_item><location><page_8><loc_10><loc_32><loc_49><loc_34></location>[2] A. S. Eddington, Mathematical Theory of Relativity (Cambridge University Press, Cambridge, 1923).</list_item>
<list_item><location><page_8><loc_10><loc_28><loc_49><loc_32></location>[3] K. S. Stelle,'Renormalization of Higher Derivative Quantum Gravity,'Phys. Rev. D 16 , 953-969 (1977)doi:10.1103/PhysRevD.16.953</list_item>
<list_item><location><page_8><loc_10><loc_24><loc_49><loc_28></location>[4] K. S. Stelle,'Classical Gravity with Higher Derivatives,'Gen. Rel. Grav. 9 , 353-371 (1978)doi:10.1007/BF00760427</list_item>
<list_item><location><page_8><loc_10><loc_20><loc_49><loc_24></location>[5] C. G. Callan, Jr., . J. Martinec, M. J. Perry and D. Friedan, 'Strings in Background Fields,' Nucl. Phys. B 262 , 593-609 (1985) doi:10.1016/0550-3213(85)90506-1</list_item>
<list_item><location><page_8><loc_10><loc_15><loc_49><loc_20></location>[6] E. S. Fradkin and A. A. Tseytlin, 'Quantum String Theory Effective Action,' Nucl. Phys. B 261 , 1-27 (1985) [erratum: Nucl. Phys. B 269 , 745-745 (1986)] doi:10.1016/0550-3213(85)90559-0</list_item>
<list_item><location><page_8><loc_10><loc_11><loc_49><loc_15></location>[7] C. Brans and R. H. Dicke, 'Mach's principle and a relativistic theory of gravitation', Phys. Rev. 124 , 925-935 (1961) doi:10.1103/PhysRev.124.925.</list_item>
<list_item><location><page_8><loc_10><loc_10><loc_49><loc_11></location>[8] P. G. Bergmann, 'Comments on the scalar ten-</list_item>
</unordered_list>
<text><location><page_8><loc_52><loc_89><loc_92><loc_93></location>where φ 0 is a positive constant, t > 0, and α is assumed to be negative to guarantee that the gradient ∇ a φ is futureoriented. The corresponding Hubble function</text>
<formula><location><page_8><loc_67><loc_85><loc_92><loc_88></location>H ( t ) = 1 -α 3 t (A.2)</formula>
<text><location><page_8><loc_52><loc_80><loc_92><loc_84></location>is always positive, describing an expanding universe, and H ( t ) → 0 + as t → + ∞ . The shear</text>
<formula><location><page_8><loc_60><loc_76><loc_92><loc_80></location>Σ( t ) = -( 3 α 2 ω +4 α 2 -2 α -2 ) 12 t 2 (A.3)</formula>
<text><location><page_8><loc_52><loc_74><loc_60><loc_75></location>is positive if</text>
<formula><location><page_8><loc_63><loc_70><loc_92><loc_73></location>ω < -2 (2 α +1)( α -1) 3 α 2 . (A.4)</formula>
<text><location><page_8><loc_52><loc_61><loc_92><loc_69></location>It is interesting that the quantity Σ /H 2 , which measures the ratio of anisotropy to expansion, remains exactly constant during the evolution of this universe, signalling that GR (which corresponds to exactly vanishing Σ) is not approached. Formally, for this solution it is</text>
<formula><location><page_8><loc_56><loc_57><loc_92><loc_60></location>KT = | ˙ φ | 8 πφ = | α | 8 πt → 0 + as t → + ∞ . (A.5)</formula>
<text><location><page_8><loc_52><loc_43><loc_92><loc_57></location>Although the expansion of 3-space 'cools' this BransDicke gravity, the zero temperature limit is not GR and is indeed a physical pathology corresponding to infinite G eff , which should be excluded from the range of physical possibilities. This means that a grain of salt is needed in the physical interpretation of the first-order thermodynamics of scalar-tensor gravity (which is not defined for φ = 0). In any case, the Minkowski space obtained for V ≡ 0, H 0 = 0, ω 0 = 0 is unstable, as seen in Sec. III A.</text>
<text><location><page_8><loc_55><loc_36><loc_92><loc_38></location>sor theory', Int. J. Theor. Phys. 1 , 25-36 (1968) doi:10.1007/BF00668828.</text>
<unordered_list>
<list_item><location><page_8><loc_53><loc_32><loc_92><loc_36></location>[9] K. Nordtvedt, 'Equivalence Principle for Massive Bodies. 2. Theory', Phys. Rev. 169 , 1017-1025 (1968) doi:10.1103/PhysRev.169.1017.</list_item>
<list_item><location><page_8><loc_52><loc_28><loc_92><loc_32></location>[10] R. V. Wagoner, 'Scalar tensor theory and gravitational waves', Phys. Rev. D 1 , 3209-3216 (1970) doi:10.1103/PhysRevD.1.3209.</list_item>
<list_item><location><page_8><loc_52><loc_23><loc_92><loc_28></location>[11] K. Nordtvedt, Jr., 'PostNewtonian metric for a general class of scalar tensor gravitational theories and observational consequences', Astrophys. J. 161 , 1059-1067 (1970) doi:10.1086/150607.</list_item>
<list_item><location><page_8><loc_52><loc_19><loc_92><loc_23></location>[12] L. Amendola and S. Tsujikawa, Dark Energy, Theory and Observations (Cambridge University Press, Cambridge, 2010).</list_item>
<list_item><location><page_8><loc_52><loc_15><loc_92><loc_19></location>[13] S. Capozziello, S. Carloni and A. Troisi, 'Quintessence without scalar fields,' Recent Res. Dev. Astron. Astrophys. 1 , 625 (2003) [arXiv:astro-ph/0303041 [astro-ph]].</list_item>
<list_item><location><page_8><loc_52><loc_10><loc_92><loc_15></location>[14] S. M. Carroll, V. Duvvuri, M. Trodden and M. S. Turner, 'Is cosmic speed -up due to new gravitational physics?,' Phys. Rev. D 70 , 043528 (2004) doi:10.1103/PhysRevD.70.043528</list_item>
</unordered_list>
<text><location><page_9><loc_12><loc_92><loc_35><loc_93></location>[arXiv:astro-ph/0306438 [astro-ph]].</text>
<unordered_list>
<list_item><location><page_9><loc_9><loc_87><loc_49><loc_92></location>[15] T. P. Sotiriou and V. Faraoni, ' f ( R ) Theories of Gravity,' Rev. Mod. Phys. 82 , 451-497 (2010) doi:10.1103/RevModPhys.82.451 [arXiv:0805.1726 [grqc]].</list_item>
<list_item><location><page_9><loc_9><loc_83><loc_49><loc_86></location>[16] A. De Felice and S. Tsujikawa,' f ( R ) theories,'Living Rev. Rel. 13 , 3 (2010)doi:10.12942/lrr-20103[arXiv:1002.4928 [gr-qc]].</list_item>
<list_item><location><page_9><loc_9><loc_76><loc_49><loc_82></location>[17] S. Nojiri and S. D. Odintsov,'Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models,'Phys. Rept. 505 , 59-144 (2011)doi:10.1016/j.physrep.2011.04.001[arXiv:1011.0544 [gr-qc]].</list_item>
<list_item><location><page_9><loc_9><loc_72><loc_49><loc_76></location>[18] A. A. Starobinsky, 'A New Type of Isotropic Cosmological Models Without Singularity,' Phys. Lett. B 91 , 99-102 (1980) doi:10.1016/0370-2693(80)90670-X</list_item>
<list_item><location><page_9><loc_9><loc_67><loc_49><loc_72></location>[19] C. L. Bennett et al. (WMAP Collaboration), 'Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Final Maps and Results', Astrophys. J. Suppl. 208 , 20 (2013).</list_item>
<list_item><location><page_9><loc_9><loc_63><loc_49><loc_67></location>[20] G. W. Horndeski, 'Second-order scalar-tensor field equations in a four-dimensional space', Int. J. Theor. Phys. 10 , 363 (1974), doi:10.1007/BF01807638.</list_item>
<list_item><location><page_9><loc_9><loc_59><loc_49><loc_63></location>[21] C. Deffayet, G. Esposito-Far'ese and A. Vikman, 'Covariant Galileon', Phys. Rev. D 79 , 084003 (2009) arXiv:0901.1314.</list_item>
<list_item><location><page_9><loc_9><loc_52><loc_49><loc_59></location>[22] C. Deffayet, S. Deser and G. Esposito-Far'ese, 'Generalized Galileons: All scalar models whose curved background extensions maintain second-order field equations and stress-tensors', Phys. Rev. D 80 , 064015 (2009), arXiv:0906.1967.</list_item>
<list_item><location><page_9><loc_9><loc_48><loc_49><loc_52></location>[23] C. Deffayet, X. Gao, D. A. Steer and G. Zahariade, 'From k-essence to generalised Galileons', Phys. Rev. D 84 , 064039 (2011), arXiv:1103.3260.</list_item>
<list_item><location><page_9><loc_9><loc_44><loc_49><loc_48></location>[24] P. Creminelli, M. Lewandowski, G. Tambalo and F. Vernizzi, 'Gravitational Wave Decay into Dark Energy', JCAP 1812 , no. 12, 025 (2018) arXiv:1809.03484.</list_item>
<list_item><location><page_9><loc_9><loc_40><loc_49><loc_44></location>[25] J. Gleyzes, D. Langlois, F. Piazza and F. Vernizzi, 'Healthy theories beyond Horndeski', Phys. Rev. Lett. 114 , no. 21, 211101 (2015) arXiv:1404.6495.</list_item>
<list_item><location><page_9><loc_9><loc_36><loc_49><loc_40></location>[26] J. Gleyzes, D. Langlois, F. Piazza and F. Vernizzi, 'Exploring gravitational theories beyond Horndeski', JCAP 1502 , 018 (2015) arXiv:1408.1952.</list_item>
<list_item><location><page_9><loc_9><loc_31><loc_49><loc_36></location>[27] D. Langlois and K. Noui, 'Degenerate higher derivative theories beyond Horndeski: evading the Ostrogradski instability', JCAP 1602 , no. 02, 034 (2016) arXiv:1510.06930.</list_item>
<list_item><location><page_9><loc_9><loc_27><loc_49><loc_31></location>[28] D. Langlois and K. Noui, 'Hamiltonian analysis of higher derivative scalar-tensor theories', JCAP 1607 , no. 07, 016 (2016) arXiv:1512.06820.</list_item>
<list_item><location><page_9><loc_9><loc_22><loc_49><loc_27></location>[29] J. Ben Achour, D. Langlois and K. Noui, 'Degenerate higher order scalar-tensor theories beyond Horndeski and disformal transformations', Phys. Rev. D 93 , no. 12, 124005 (2016) arXiv:1602.08398.</list_item>
<list_item><location><page_9><loc_9><loc_18><loc_49><loc_22></location>[30] M. Crisostomi, K. Koyama and G. Tasinato, 'Extended Scalar-Tensor Theories of Gravity', JCAP 1604 , no. 04, 044 (2016) arXiv:1602.03119.</list_item>
<list_item><location><page_9><loc_9><loc_13><loc_49><loc_18></location>[31] H. Motohashi, K. Noui, T. Suyama, M. Yamaguchi and D. Langlois, 'Healthy degenerate theories with higher derivatives', JCAP 1607 , no. 07, 033 (2016) arXiv:1603.09355.</list_item>
<list_item><location><page_9><loc_9><loc_9><loc_49><loc_12></location>[32] J. Ben Achour, M. Crisostomi, K. Koyama, D. Langlois, K. Noui and G. Tasinato, 'Degenerate higher order scalar-tensor theories beyond Horndeski up to cubic or-</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_9><loc_55><loc_92><loc_87><loc_93></location>der', JHEP 1612 , 100 (2016) arXiv:1608.08135.</list_item>
<list_item><location><page_9><loc_52><loc_88><loc_92><loc_92></location>[33] M. Crisostomi, R. Klein and D. Roest, 'Higher Derivative Field Theories: Degeneracy Conditions and Classes', JHEP 1706 , 124 (2017) arXiv:1703.01623.</list_item>
<list_item><location><page_9><loc_52><loc_83><loc_92><loc_88></location>[34] D. Langlois, R. Saito, D. Yamauchi and K. Noui, 'Scalartensor theories and modified gravity in the wake of GW170817', Phys. Rev. D 97 , no. 6, 061501 (2018) arXiv:1711.07403.</list_item>
<list_item><location><page_9><loc_52><loc_80><loc_92><loc_82></location>[35] D. Langlois, 'Degenerate Higher-Order Scalar-Tensor (DHOST) theories', arXiv:1707.03625.</list_item>
<list_item><location><page_9><loc_52><loc_75><loc_92><loc_80></location>[36] D. Langlois, 'Dark energy and modified gravity in degenerate higher-order scalar-tensor (DHOST) theories: A review', Int. J. Mod. Phys. D 28 , no. 05, 1942006 (2019) arXiv:1811.06271.</list_item>
<list_item><location><page_9><loc_52><loc_69><loc_92><loc_75></location>[37] C. de Rham, S. Garcia-Saenz, L. Heisenberg, V. Pozsgay and X. Wang,'To Half-Be or Not To Be?,'JHEP 06 , 088 (2023)doi:10.1007/JHEP06(2023)088[arXiv:2303.05354 [hep-th]].</list_item>
<list_item><location><page_9><loc_52><loc_64><loc_92><loc_69></location>[38] T. Jacobson, 'Thermodynamics of space-time: The Einstein equation of state,' Phys. Rev. Lett. 75 (1995) 1260, doi:10.1103/PhysRevLett.75.1260 [arXiv:gr-qc/9504004 [gr-qc]].</list_item>
<list_item><location><page_9><loc_52><loc_58><loc_92><loc_64></location>[39] C. Eling, R. Guedens, and T. Jacobson, 'Non-equilibrium thermodynamics of spacetime,' Phys. Rev. Lett. 96 (2006) 121301, doi:10.1103/PhysRevLett.96.121301 [arXiv:gr-qc/0602001 [gr-qc]].</list_item>
<list_item><location><page_9><loc_52><loc_52><loc_92><loc_57></location>[40] T. Damour and K. Nordtvedt,'Tensor -scalar cosmological models and their relaxation toward general relativity,'Phys. Rev. D 48 , 3436-3450 (1993)doi:10.1103/PhysRevD.48.3436</list_item>
<list_item><location><page_9><loc_52><loc_47><loc_92><loc_52></location>[41] T. Damour and K. Nordtvedt,'General relativity as a cosmological attractor of tensor scalar theories,'Phys. Rev. Lett. 70 , 2217-2219 (1993)doi:10.1103/PhysRevLett.70.2217</list_item>
<list_item><location><page_9><loc_52><loc_42><loc_92><loc_47></location>[42] V. Faraoni and J. Cˆot'e, 'Imperfect fluid description of modified gravities,' Phys. Rev. D 98 no. 8, 084019 (2018) doi:10.1103/PhysRevD.98.084019 [arXiv:1808.02427 [grqc]].</list_item>
<list_item><location><page_9><loc_52><loc_36><loc_92><loc_42></location>[43] V. Faraoni and A. Giusti, 'Thermodynamics of scalartensor gravity,' Phys. Rev. D 103 , no.12, L121501 (2021) doi:10.1103/PhysRevD.103.L121501 [arXiv:2103.05389 [gr-qc]].</list_item>
<list_item><location><page_9><loc_52><loc_30><loc_92><loc_36></location>[44] V. Faraoni, A. Giusti and A. Mentrelli, 'New approach to the thermodynamics of scalar-tensor gravity,' Phys. Rev. D 104 , no.12, 124031 (2021) doi:10.1103/PhysRevD.104.124031 [arXiv:2110.02368 [gr-qc]].</list_item>
<list_item><location><page_9><loc_52><loc_23><loc_92><loc_30></location>[45] A. Giusti, S. Giardino and V. Faraoni,'Pastdirected scalar field gradients and scalar-tensor thermodynamics,'Gen. Rel. Grav. 55 , no.3, 47 (2023)doi:10.1007/s10714-023-03095-7[arXiv:2210.15348 [gr-qc]].</list_item>
<list_item><location><page_9><loc_52><loc_18><loc_92><loc_23></location>[46] V. Faraoni and J. Houle,'More on the first-order thermodynamics of scalar-tensor and Horndeski gravity,'Eur. Phys. J. C 83 , no.6, 521 (2023)doi:10.1140/epjc/s10052023-11712-7[arXiv:2302.01442 [gr-qc]].</list_item>
<list_item><location><page_9><loc_52><loc_14><loc_92><loc_18></location>[47] M. S. Madsen, 'Scalar Fields in Curved Space-times,' Class. Quant. Grav. 5 , 627-639 (1988) doi:10.1088/02649381/5/4/010</list_item>
<list_item><location><page_9><loc_52><loc_10><loc_92><loc_14></location>[48] L. O. Pimentel, 'Energy Momentum Tensor in the General Scalar-Tensor Theory,' Class. Quant. Grav. 6 , L263L265 (1989) doi:10.1088/0264-9381/6/12/005</list_item>
<list_item><location><page_9><loc_52><loc_9><loc_92><loc_10></location>[49] U. Nucamendi, R. De Arcia, T. Gonzalez, F. A. Horta-</list_item>
<list_item><location><page_10><loc_12><loc_87><loc_49><loc_93></location>Rangel and I. Quiros, 'Equivalence between Horndeski and beyond Horndeski theories and imperfect fluids,' Phys. Rev. D 102 (2020) no.8, 084054, doi:10.1103/PhysRevD.102.084054 [arXiv:1910.13026 [gr-qc]].</list_item>
<list_item><location><page_10><loc_9><loc_80><loc_50><loc_86></location>[50] A. Giusti, S. Zentarra, L. Heisenberg and V. Faraoni,'First-order thermodynamics of Horndeski gravity,'Phys. Rev. D 105 , no.12, 124011 (2022)doi:10.1103/PhysRevD.105.124011[arXiv:2108.10706 [gr-qc]].</list_item>
<list_item><location><page_10><loc_9><loc_76><loc_49><loc_80></location>[51] C. Eckart, 'The thermodynamics of irreversible processes. 3. Relativistic theory of the simple fluid,' Phys. Rev. 58 (1940), 919-924 doi:10.1103/PhysRev.58.919</list_item>
<list_item><location><page_10><loc_9><loc_73><loc_49><loc_76></location>[52] R. M. Wald, General Relativity (Chicago University Press, Chicago, 1984).</list_item>
<list_item><location><page_10><loc_9><loc_67><loc_49><loc_73></location>[53] M. Miranda, D. Vernieri, S. Capozziello and V. Faraoni,'Fluid nature constrains Horndeski gravity,'Gen. Rel. Grav. 55 , no.7, 84 (2023)doi:10.1007/s10714-023-03128-1[arXiv:2209.02727 [gr-qc]].</list_item>
<list_item><location><page_10><loc_9><loc_62><loc_49><loc_67></location>[54] S. Giardino, V. Faraoni and A. Giusti,'Firstorder thermodynamics of scalar-tensor cosmology,'JCAP 04 , no.04, 053 (2022)doi:10.1088/14757516/2022/04/053[arXiv:2202.07393 [gr-qc]].</list_item>
<list_item><location><page_10><loc_9><loc_54><loc_49><loc_61></location>[55] V. Faraoni, S. Giardino, A. Giusti and R. Vanderwee,'Scalar field as a perfect fluid: thermodynamics of minimally coupled scalars and Einstein frame scalar-tensor gravity,'Eur. Phys. J. C 83 , no.1, 24 (2023)doi:10.1140/epjc/s10052-023-111867[arXiv:2208.04051 [gr-qc]].</list_item>
<list_item><location><page_10><loc_9><loc_48><loc_49><loc_53></location>[56] M. Miranda, P. A. Graham and V. Faraoni,'Effective fluid mixture of tensor-multi-scalar gravity,'Eur. Phys. J. Plus 138 , no.5, 387 (2023)doi:10.1140/epjp/s13360023-03984-5[arXiv:2211.03958 [gr-qc]].</list_item>
<list_item><location><page_10><loc_9><loc_42><loc_50><loc_48></location>[57] V. Faraoni, A. Giusti, S. Jose and S. Giardino,'Peculiar thermal states in the first-order thermodynamics of gravity,'Phys. Rev. D 106 , no.2, 024049 (2022)doi:10.1103/PhysRevD.106.024049[arXiv:2206.02046 [gr-qc]].</list_item>
<list_item><location><page_10><loc_9><loc_35><loc_50><loc_42></location>[58] V. Faraoni and T. B. Fran¸connet,'Stealth metastable state of scalar-tensor thermodynamics,'Phys. Rev. D 105 , no.10, 104006 (2022)doi:10.1103/PhysRevD.105.104006[arXiv:2203.14934 [gr-qc]].</list_item>
<list_item><location><page_10><loc_9><loc_34><loc_49><loc_35></location>[59] S. Giardino, A. Giusti and V. Faraoni,'Thermal</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_10><loc_55><loc_88><loc_92><loc_93></location>stability of stealth and de Sitter spacetimes in scalar-tensor gravity,'Eur. Phys. J. C 83 , no.7, 621 (2023)doi:10.1140/epjc/s10052-023-116973[arXiv:2302.08550 [gr-qc]].</list_item>
<list_item><location><page_10><loc_52><loc_80><loc_93><loc_88></location>[60] V. Faraoni, P. A. Graham and A. Leblanc,'Critical solutions of nonminimally coupled scalar field theory and first-order thermodynamics of gravity,'Phys. Rev. D 106 , no.8, 084008 (2022)doi:10.1103/PhysRevD.106.084008[arXiv:2207.03841 [gr-qc]].</list_item>
<list_item><location><page_10><loc_52><loc_75><loc_92><loc_80></location>[61] S. Giardino and A. Giusti, 'First-order thermodynamics of scalar-tensor gravity,' Ricerche di Matematica (2023) doi:10.1007/s11587-023-00801-0 [arXiv:2306.01580 [grqc]].</list_item>
<list_item><location><page_10><loc_52><loc_71><loc_92><loc_75></location>[62] M. Miranda, S. Giardino, A. Giusti and L. Heisenberg,'First-order thermodynamics of Horndeski cosmology,'[arXiv:2401.10351 [gr-qc]].</list_item>
<list_item><location><page_10><loc_52><loc_67><loc_92><loc_71></location>[63] G. F. R. Ellis, 'Relativistic cosmology,' Proc. Int. Sch. Phys. Fermi 47 (1971), 104-182 doi:10.1007/s10714-0090760-7</list_item>
<list_item><location><page_10><loc_52><loc_62><loc_92><loc_67></location>[64] L. Verde, T. Treu and A. G. Riess,'Tensions between the Early and the Late Universe,'Nature Astron. 3 , 891doi:10.1038/s41550-019-0902-0[arXiv:1907.10625 [astro-ph.CO]].</list_item>
<list_item><location><page_10><loc_52><loc_54><loc_92><loc_61></location>[65] E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri, D. F. Mota, A. G. Riess and J. Silk,'In the realm of the Hubble tension-a review of solutions,'Class. Quant. Grav. 38 , no.15, 153001 (2021)doi:10.1088/1361-6382/ac086d[arXiv:2103.01183 [astro-ph.CO]].</list_item>
<list_item><location><page_10><loc_52><loc_50><loc_92><loc_53></location>[66] C. M. Will, Theory and Experiment In Gravitational Physics , second edition (Cambridge University Press, Cambridge, 2018).</list_item>
<list_item><location><page_10><loc_52><loc_46><loc_92><loc_49></location>[67] C. M. Will,'The Confrontation between General Relativity and Experiment,'Living Rev. Rel. 17 , 4 (2014)doi:10.12942/lrr-2014-4[arXiv:1403.7377 [gr-qc]].</list_item>
<list_item><location><page_10><loc_52><loc_42><loc_92><loc_45></location>[68] A. A. Starobinsky, 'Can the effective gravitational constant become negative?', Sov. Astron. (Lett.) 7 , 36 (1981).</list_item>
<list_item><location><page_10><loc_52><loc_36><loc_92><loc_42></location>[69] O. Hrycyna and M. Szydlowski,'Dynamics of the Bianchi I model with non-minimally coupled scalar field near the singularity,'AIP Conf. Proc. 1514 , no.1, 191-194 (2013)doi:10.1063/1.4791754[arXiv:1212.6408 [gr-qc]].</list_item>
</document>
[
{
"title": "New phenomenology in the first-order thermodynamics of scalar-tensor gravity for Bianchi universes",
"content": "Julien Houle 1, ∗ and Valerio Faraoni 1, † 1 Department of Physics & Astronomy, Bishop's University, 2600 College Street, Sherbrooke, Qu'ebec, Canada J1M 1Z7 The phase space of Bianchi I universes in vacuum Brans-Dicke gravity is analyzed in terms of physical variables. The behaviour of the solutions of the field equations near the fixed points (which are solutions of Einstein gravity) is compared with basic ideas of the recent first-order thermodynamics of scalar-tensor gravity, elucidating new phenomenology.",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "Shortly after the introduction of general relativity (GR), researchers began looking for alternative theories of gravity, moved by pure curiosity about how things could be different in nature [1, 2]. More concrete motivation emerged with the birth of quantum field theory, when the question arose of how to reconcile the two biggest physics discoveries of the twentieth century, quantum mechanics and GR. The reason is that, as soon as one introduces the lowest-order quantum corrections to GR, one simultaneously causes deviations from it in the form of higher derivative terms in the field equations or extra degrees of freedom [3, 4]. This situation does not change in string theory: the simplest string theory, the bosonic string, has a low-energy limit that reproduces not GR, but an ω = -1 Brans-Dicke gravity [5, 6]. The prototype of alternative gravity is scalar-tensor gravity, which contains only a scalar degree of freedom φ in addition to the two massless spin two modes familiar from GR. The original Jordan-Brans-Dicke theory [7] was later generalized to wider scalar-tensor theories [8-11] in which the 'Brans-Dicke coupling' parameter became a function ω ( φ ) of the scalar field φ , which was also endowed with a potential V ( φ ). There is independent motivation for the study of alternative theories of gravity coming from cosmological observations. The 1998 discovery, made with type Ia supernovae, that the present expansion of the universe is accelerated led to introducing overnight a completely ad hoc dark energy with very exotic properties (equation of state parameter close to w /similarequal -1), of completely unkown nature and comprising approximately 70% of the energy content of the universe (see [12] for a review). A wide range of dark energy models have been proposed in the literature, but none is compelling and this state of affairs is deeply unsatisfactory from the theoretical point of view. For this reason, many cosmologists have turned to questioning whether, instead, we do not understand gravity on the largest (cosmological) scales and dark energy simply does not exist [13, 14]. This idea had led to formulating and testing modified gravity models. Among a spectrum of possibilities, the most popular models belong to the so-called metric f ( R ) gravity class (see [1517] for reviews). Metric f ( R ) gravity contains only one extra massive, propagating, scalar degree of freedom and, therefore, falls into the wider category of scalar-tensor gravity [15-17]. Even Starobinski inflation [18], the first scenario of inflation and the one currently favored by observations [19], is based on quadratic corrections to the Einstein-Hilbert action and is ultimately a scalar-tensor theory. In the past decade, the problem of finding the most general scalar-tensor theory expressed by field equations that are at most of second order led to the rediscovery and intense study of the older Horndeski gravity [20]. This sought-for property was found to belong not to Horndeski gravity but to the more general Degenerate Higher Order Scalar-Tensor (DHOST) theories, a subclass of higher order gravities in which a degeneracy condition brings the order of the field equations back to two ( e.g. , [21-33], see [34-36] for reviews). Given the wide spectrum of scalar-tensor gravities (not to mention other alternatives richer in propagating degrees of freedom that are difficult to identify and count [37]), what is the role of GR in this landscape? A proposal is well-known in the context of emergent gravity, in which the field equations can be deduced as an emergent or collective property of underlying degrees of freedom and are not fundamental. The seminal paper by Jacobson [38] derived the Einstein equations of GR from purely thermodynamical considerations, an idea referred to as 'thermodynamics of spacetime'. This feat was repeated with quadratic f ( R ) gravity producing a new picture: GR is somehow a state of 'thermal equilibrium' of gravity, while alternative theories correspond to entropy generation and to excited thermal states [39]. This view has been very influential and has generated a large literature but, unfortunately, no substantial progress has been made since its early days. Specifically, the 'temperature of gravity' (or other order parameter) has not been identified and no equation describing the relaxation of alternative gravity to GR has been proposed, although there are reasons to believe that such phenomena could have occurred in the early universe [40, 41]. Recently a new proposal has been advanced, known as the first-order thermodynamics of scalar-tensor gravity [42-46]: it has nothing to do with Jacobson's thermodynamics of spacetime, although some of its ideas share the same spirit. It is a far-reaching analogy (but, still, only an analogy) between the description of the effective stress-energy tensor of the scalar degree of freedom φ of scalar-tensor gravity and the stress-energy tensor of a dissipative fluid. By writing the field equations of scalar-tensor gravity as effective Einstein equations, the contributions of φ and of its first and second derivatives form an effective stress-energy tensor T ( φ ) ab which has the form of a dissipative fluid stress-energy tensor [42] (this fact has been known for a long time for special theories or special geometries [47, 48] and has been recently recognized also for 'viable' Horndeski gravity [49, 50]). Specifically, if the scalar field gradient ∇ a φ is timelike and future-oriented [45], its normalized version can be seen as the four-velocity of an effective fluid with stress-energy tensor of the form where ρ is an effective energy density, P is an effective isotropic pressure, π ab is an effective anisotropic, tracefree, stress tensor, and q a is an effective heat flux density. Here h ab ≡ g ab + u a u b is the Riemannian three-space metric seen by observers comoving with this fluid (with h a b the projector onto this 3-space), while π ab and q a are purely spatial: The fact that this T ( φ ) ab has the dissipative fluid form contains no physics: the decomposition (1.2) holds true for any symmetric two-index tensor [46]. However, when one takes seriously this dissipative fluid structure and tries to apply to it Eckart's [51] theory of dissipative fluids [4244] one discovers that, miraculously, Eckart's constitutive relation holds. Here T is the temperature of the dissipative fluid, K is the thermal conductivity, and ˙ u a is the fluid fouracceleration. Equation (1.4) is nothing but the relativistic generalization of Fourier's law with the addition of an inertial term proportional to the four-acceleration, which takes into account the fact that heat is a form of energy and its transport contributes to the energy flux in a way that was absent in pre-relativistic physics [51]. The unexpected fact that this relation holds for the effective φ -fluid makes it possible to define the product KT for scalar-tensor gravity. Let us refer, for simplicity, to 'first-generation' ( i.e. , pre-Horndeski) scalar-tensor gravity: in the Jordan frame, the gravitational sector of the theory is described by the action 1 where φ > 0 is the Brans-Dicke scalar (approximately equivalent to the inverse of the effective gravitational coupling strength G -1 ), the function ω ( φ ) (which was a strictly constant parameter in the original Brans-Dicke theory) is the 'Brans-Dicke coupling', and V ( φ ) is a scalar field potential (absent in the original Brans-Dicke theory). When ∇ a φ is timelike and future-oriented, the product of effective thermal conductivity and effective temperature is found to be [43, 44] It is apparent that GR, reproduced for φ = const. > 0, corresponds to KT = 0. It is rather intuitive that, when extra degrees of freedom with respect to GR are excited, gravity is in some sense excited and 'hotter' than the GR state in which the extra degrees of freedom are absent. The 'temperature of gravity' T is, in this sense, a temperature relative to the GR state of equilibrium . The first-order thermodynamics of scalar-tensor gravity has been extended [50] to 'viable' Horndeski gravity, i.e. , to the subclass in which gravitational waves propagate at the speed of light, and applied to various situations in cosmology and other contexts [46, 5360] (see [61] for a recent review). Two basic qualitative ideas emerge from these studies: near spacetime singularities gravity is 'hot', i.e. , it diverges from GR; the expansion of the three-space perceived by comoving observers (with 3-metric h ab ) 'cools' gravity, bringing it closer to GR. These ideas have been tested against various situations of physical interest or mathematical convenience (for which it is possible to draw analytically definite conclusions). Here we continue this program. While it was natural to apply the first-order thermodynamics of scalar-tensor gravity to Friedmann-LemaˆıtreRobertson-Walker (FLRW) cosmology [54, 62], here we extend the description to anisotropic Bianchi universes. For simplicity, we confine ourselves to Brans-Dicke gravity and to the simplest anisotropic cosmologies described by spatially flat Bianchi I geometries. To describe the dynamics we resort to a phase space view and, contrary to the literature that we are aware of, we choose as phase space variables the average Hubble function H , the scalar field φ , and its time derivative ˙ φ that have a direct physical meaning instead of other variables obtained by various non-linear combinations of physical ones (for this reason we cannot avail ourselves of existing phase space analyses).",
"pages": [
1,
2,
3
]
},
{
"title": "II. FIELD EQUATIONS AND BIANCHI I GEOMETRY",
"content": "The (Jordan frame) vacuum field equations obtained by varying the action (1.7) with respect to the inverse metric g ab and the scalar φ are In the following we assume V ( φ ) ≥ 0, constant BransDicke coupling ω , and 2 ω +3 > 0 to avoid phantom scalar fields φ . The line element of a Bianchi I universe is in Cartesian comoving coordinates ( t, x, y, z ), where A ( t ) , B ( t ), and C ( t ) are the scale factors associated with the three orthogonal spatial directions. This anisotropic universe has average scale factor a ( t ) ≡ ( ABC ) 1 / 3 and average Hubble parameter where an overdot denotes differentiation with respect to the cosmic time t , H i ≡ ˙ A i /A (where A i = A,B , or C ), and The shear scalar Σ for this scalar-tensor gravity, com- uted in terms of φ and its derivatives, is given by [42] where σ ab is the shear tensor [52]. For clarity, we use the shear variable Σ ≡ 1 2 σ 2 which, in the Bianchi I geometry (2.3), assumes the form Σ vanishes if and only if all the components of σ ab vanish [63], in which case the Bianchi geometry reduces to a FLRW one. The only non-vanishing Christoffel symbols of the geometry (2.3) are while the non-vanishing components of the Ricci tensor and the Ricci scalar reads The time-time component of the Brans-Dicke field equations (2.1) is while the spatial components read and the trace of the field equation (2.1) is Equation (2.2) for the scalar field is where a prime denotes differentiation with respect to φ . Combining these three equations yields By inserting Eq. (2.20) for the scalar field φ into (2.19) and combining the result with Eq. (2.15), one obtains To summarize, the field equations to be solved for the scalar field φ ( t ) and the Hubble function H ( t ) are Eq. (2.20) and Eq. (2.22), respectively. Once these quantities are known, Eq. (2.15) gives the shear Σ.",
"pages": [
3,
4
]
},
{
"title": "III. PHASE SPACE",
"content": "Let us discuss the phase space of Bianchi I cosmologies in vacuum Brans-Dicke gravity, where the dynamics is due entirely to the Brans-Dicke scalar φ . We use the variables ( H,φ, ˙ φ ) which are physical: the Hubble parameter H is a cosmological observable (although its actual value is subject to a very significant tension [64, 65]), while φ is the extra scalar degree of freedom of scalartensor gravity in addition to the two spin zero massless modes of GR contained in the metric g ab . The strength of the gravitational coupling G /similarequal φ -1 is measured directly by Cavendish experiments and its time variation ( i.e. , ˙ G and, consequently, ˙ φ ) is subject to observational constraints [66, 67]. By contrast, much of the existing literature on Bianchi cosmologies uses variables which are complicated functions of H,φ , and ˙ φ and do not have direct physical interpretation. Although they may make the study of the phase space dynamics more convenient from the formal point of view, they have no direct physical meaning. Here we want to interpret the dynamics and compare it with the first-order thermodynamics of spacetime, therefore we must use physical variables. Equation (2.15) yields the shear as a function of the three phase space variables, therefore Σ is not an independent variable, although it will be relevant in our analysis. With our choice of variables the fixed points in the phase space, if they exist, have necessarily the form ( H,φ, ˙ φ ) = ( H 0 , φ 0 , 0), with H 0 and φ 0 > 0 constants. They are solutions of the Einstein equations located on the 'GR plane' ˙ φ = 0 of the phase space identified by constant scalar field, the condition that reproduces GR. Using the notation V ( φ 0 ) ≡ V 0 and V ' ( φ 0 ) ≡ V ' 0 , Eq. (3.1) that must be satisfied by the fixed points gives while the mix of Eqs. (2.19) and (2.20) gives, using ˙ H = ω ' = 0, and the scalar field equation degenerates into (which is non-negative since V ≥ 0 and φ > 0) reducing Eq. (3.3) to Comparing Eqs. (3.2) and (3.5), one obtains Σ 0 = 0: as expected, the shear vanishes at the fixed points, which have A = B = C ≡ a ( t ) and the average Hubble function H ( t ) coincides with the FLRW one. Equation (3.3) then becomes H 2 0 = V 0 / (6 φ 0 ), or The degenerate fixed points corresponding to V 0 = V ' 0 = H 0 = 0 are Minkowski spaces, while those corresponding to V 0 > 0, V ' 0 > 0, and H 0 = ± √ V 0 6 φ 0 are de Sitter spaces. 2 When φ becomes a constant φ 0 and V ( φ 0 ) ≡ V 0 is positive, the theory of gravity reduces to GR with a positive cosmological constant Λ = V 0 . In the study of exact solutions of the field equations, one sometimes find solutions ( H ( t ) , φ ( t ) , ˙ φ ( t ) ) with φ ( t ) → 0 + at late times t → + ∞ : these are pathological as the effective gravitational coupling G eff → + ∞ . The line φ = 0 in the 'GR plane' ˙ φ = 0 corresponds to singularities at which G eff changes sign, but it does so by going through G eff = ∞ (a similar situation occurs with conformally coupled scalar fields [68, 69]). Exact solutions with these properties are unphysical and cannot be regarded as GR solutions, even though the firstorder thermodynamics of scalar-tensor gravity does not, strictly speaking, indicate a pathology or a gross deviation from GR in this situation (see Appendix A).",
"pages": [
4,
5
]
},
{
"title": "A. Stability of the equilibrium points",
"content": "Let us examine the stability of the fixed points with respect to homogeneous perturbations described by Evolution equations for the perturbations δφ ( t ) , δH ( t ) are obtained from Eq. (2.20) for the scalar field φ and Eq. (2.22) for H . Let us begin with the stability of the scalar field. By expanding the scalar field potential, V ( φ ) /similarequal V 0 + V ' 0 δφ , and using the zero-order field equation (2.20), one obtains the linearized equation for δφ where The ansatz with δ 0 and α constants yields the algebraic equation with roots /negationslash and two modes δφ ( ± ) ( t ) = δ 0 e α ( ± ) t (this applies if α ( ± ) = 0; the case α ( ± ) = 0 is discussed separately). the second exponential in the right-hand side oscillates while, as t → + ∞ , the first exponential diverges if H 0 < 0 and decays if H 0 > 0 (it remains constant if H 0 = 0). In this case the fixed point is stable if H 0 ≥ 0 and unstable if H 0 < 0. If ω 0 is real, corresponding to φ 0 V '' 0 ≥ 12 H 2 0 , then √ 1 -( 2 ω 0 3 H 0 ) 2 < 1 and -1 ± √ 1 -( 2 ω 0 3 H 0 ) 2 < 0, hence α ( ± ) < 0 if H 0 > 0, or α ( ± ) = 0 if H 0 = 0 (corresponding to a Minkowski fixed point), or α ( ± ) > 0 if H 0 < 0. Fixed points with H 0 ≥ 0 are stable while those with H 0 < 0 are unstable. If instead ω 0 is imaginary, ω 0 = i | ω 0 | , corresponding to φ 0 V '' 0 < 12 H 2 0 , then we have √ -1 + ( 2 ω 0 3 H 0 ) 2 > 0 and -1 -√ 1 -( 2 ω 0 3 H 0 ) 2 < 0. /negationslash The mode δφ (+) is unstable ( i.e. , α (+) > 0) if H 0 > 0, while the other mode δφ ( -) is unstable ( i.e. , α ( -) > 0) if H 0 < 0. In short, when ω 0 is imaginary and H 0 = 0 there is always a unstable mode and the fixed point is unstable. If instead ω 0 is imaginary and H 0 = 0 (Minkowski fixed point), the equation for the scalar field perturbations reduces to which describes an unstable inverted harmonic oscillator. An exception not included in the previous discussion is the situation in which V = 0 and H 0 = 0, ω 0 = 0 corresponding to a Minkowski space. In this case, Eq. (3.9) reduces to ¨ φ = 0, which has a linear solution and this Minkowski space is unstable. Let us consider now the perturbation δH ( t ). Using the zero-order equations, Eq. (2.22) gives the linearized equation of motion for δH and Eq. (3.10) yields Using the explicit form δφ ( ± ) ( t ) of the scalar field perturbation gives where The solution of this inhomogeneous ordinary differential equation is where C is an integration constant. The perturbation δH ( t ) diverges for H 0 < 0 regardless of the behaviour of the scalar field perturbation δφ . To summarize: Let us consider now the ratio of the shear variable to the expansion variable Σ /H 2 0 , which quantifies the amount of anisotropy and the departure from GR (remember that the fixed points, when they exist, all lie in the GR plane ˙ φ = 0 and Σ = 0). Since the equilibrium points are isotropic de Sitter spaces, the shear (3.1) is purely perturbative and given by which, to first order, reduces to Inserting the solution for the perturbations δφ, δH into this equation and using Eq. (3.12) to express ω 2 0 yields (Since Σ /similarequal δ Σ ≥ 0, one deduces that sign( C ) = sign ( H 0 ).) The ratio Σ /H 2 0 vanishes as t → + ∞ for all solutions that are perturbations of expanding de Sitter fixed points, and diverges in the same late-time limit near contracting de Sitter fixed points. Let us now examine the significance of these results with respect to the first-order thermodynamics of spacetime of Refs. [42-46]. We emphasize that the following discussion is meaningful only because physical phase space variables ( H,φ, ˙ φ ) have been chosen at the outset. de Sitter solutions with non-constant scalar field, which are possible in scalar-tensor gravity but not in GR, have been discussed in Ref. [57].",
"pages": [
5,
6
]
},
{
"title": "B. Comparison with the first-order thermodynamics of scalar-tensor gravity",
"content": "In Brans-Dicke theory, the temperature of gravity relative to the GR state of equilibrium [42-46] is given by Eq. (1.8). For linear homogeneous perturbations of the fixed points, it reads The fixed points, which lie in the GR plane with ˙ φ = 0 clearly correspond to KT = 0. KT assumes the same form as in a FLRW universe, but the solution φ ( t ) is, in general, different in Bianchi I and in FLRW universes. If the orbit of a solution in the ( H,φ, ˙ φ ) phase space lies near an expanding de Sitter fixed point and is attracted to it, the anisotropic three-space expands, and the solution converges to the zero-temperature state of equilibrium, while Σ /H 2 0 → 0 and this three-space isotropizes. The cooling of gravity ( KT → 0) is indeed another way of saying that GR is a late-time attractor of the dynamics. The situation is not so trivial, however, because there are exceptions for φ 0 V '' 0 < 12 H 2 0 . In this case three-space still expands exponentially but the de Sitter fixed point nearby is a repellor. It is still the case that H 0 > 0 and Σ /H 2 0 → 0 as t → + ∞ . How do we understand this situation in the light of scalar-tensor thermodynamics? The answer comes from examining the equation ruling the approach to/departure from the GR equilibrium state derived in Refs. [42-46] where τ is the comoving time of the effective φ -fluid ( i.e. , the proper time of observers comoving with this fluid and with four-velocity (1.1)) and Θ is the expansion scalar of the fluid [52]. In a Bianchi I universe Θ = 3 H and τ = t . Near a fixed point (which lies in the GR plane) KT is a first-order quantity and Eq. (3.26) reduces, to linear order, to or, in the light of the previous discussion, For expanding de Sitter spaces with purely imaginary ω 0 it is ω 2 0 < 0. In order for the scalar field gradient ∇ a φ = -˙ φδ a 0 to be future-oriented it must be ˙ φ < 0, which implies that δφ = δ 0 e αt < 0, or δ 0 < 0. Then, for the exceptional expanding de Sitter fixed points with imaginary ω 0 , we have using Eq. (3.12): KT always grows near these repellors, describing the departure from the GR equilibrium state. The reason for this behaviour is clearly due to the third term /square φ/ (8 πφ ) in the right-hand side of Eq. (3.26).",
"pages": [
7
]
},
{
"title": "IV. CONCLUSIONS",
"content": "Two basic insights have been obtained thus far in the first-order thermodynamics of scalar-tensor (including viable Horndeski) gravity [42-46]. The first one is that the expansion of the three-space seen by observers comoving with the effective φ -fluid 'cools' gravity. The second is that gravity is 'hot' ( i.e. , KT → + ∞ ) near spacetime singularities. /negationslash The idea that 'expansion cools gravity' ( i.e. , KT → 0) was deduced in Refs. [42-46] using situations in which /square φ = 0. The lesson from the present study is that this statement is not always true when /square φ = 0. The term /square φ/ (8 πφ ) in Eq. (3.26) cannot be expressed unambigously in terms of KT or its powers or derivatives. This third term in the right-hand side of (3.26) is reminiscent of entropy generation terms in non-equilibrium thermodynamics and it is fair to say that it is the dynamics of the scalar field itself, embodied in /square φ/φ , that drives gravity away from the GR equilibrium state. When ω = const., V ( φ ) ≡ 0, and in the presence of conformally invariant matter (for example, in the radiation era), it is /square φ = 0 because Eq. (2.2) in the presence of matter and with a quadratic potential V ( φ ) = m 2 φ 2 / 2 becomes where T (m) is the trace of the matter energy-momentum tensor. /square φ vanishes in the presence of conformally invariant matter with zero trace, such as a radiation fluid in the radiation era, during which the expansion of space causes gravity to approach GR. This phenomenon was indeed reported in FLRW scalar-tensor cosmology [40, 41]. However, was not expected in other cosmological eras. The convergence of scalar-tensor to GR cosmology has been debated at length and we hope that our approach can shed some light on this issue, which we will discuss in a future publication. Another lesson garnered from the discussion of the previous section is that the degree of anisotropy Σ /H 2 0 commonly used in the literature on Bianchi universes does not tell the full story about the approach to, or departure from the GR state because it tends to zero for the exceptional expanding de Sitter fixed points with imaginary ω 0 that are phase space repellors. To conclude, more research is needed to understand scalar-tensor gravity (and even more for Horndeski gravity) from the point of view of first-order thermodynamics. We remind the reader that this formalism is, ultimately, only an analogy; nevertheless, it is proving useful from the theoretical point of view and it is building up to a consistent framework to understand at least scalartensor gravity in the increasingly wider spectrum of alternatives to GR.",
"pages": [
7,
8
]
},
{
"title": "ACKNOWLEDGMENTS",
"content": "V. F. is grateful to Peter Dunsby, Andrea Giusti, Orlando Luongo, and Lavinia Heisenberg for discussions. This work is supported, in part, by the Natural Sciences & Engineering Research Council of Canada (grant 202303234 to V. F.) and by a Bishop's University Graduate Entrance Scholarship (J. H.).",
"pages": [
8
]
},
{
"title": "Appendix A THE PATHOLOGICAL LINE φ = 0 IN THE ˙ φ = 0 PLANE OF THE PHASE SPACE",
"content": "Let us consider an exact Bianchi I solution of BransDicke gravity that asymptotes to a φ = 0 solution. Assuming V ( φ ) ≡ 0, which yields /square φ = 0, consider the power-law ansatz for the scalar field where φ 0 is a positive constant, t > 0, and α is assumed to be negative to guarantee that the gradient ∇ a φ is futureoriented. The corresponding Hubble function is always positive, describing an expanding universe, and H ( t ) → 0 + as t → + ∞ . The shear is positive if It is interesting that the quantity Σ /H 2 , which measures the ratio of anisotropy to expansion, remains exactly constant during the evolution of this universe, signalling that GR (which corresponds to exactly vanishing Σ) is not approached. Formally, for this solution it is Although the expansion of 3-space 'cools' this BransDicke gravity, the zero temperature limit is not GR and is indeed a physical pathology corresponding to infinite G eff , which should be excluded from the range of physical possibilities. This means that a grain of salt is needed in the physical interpretation of the first-order thermodynamics of scalar-tensor gravity (which is not defined for φ = 0). In any case, the Minkowski space obtained for V ≡ 0, H 0 = 0, ω 0 = 0 is unstable, as seen in Sec. III A. sor theory', Int. J. Theor. Phys. 1 , 25-36 (1968) doi:10.1007/BF00668828. [arXiv:astro-ph/0306438 [astro-ph]].",
"pages": [
8,
9
]
}
]
2024PhRvD.110d3510N
https://arxiv.org/pdf/2407.03766.pdf
<document>
<section_header_level_1><location><page_1><loc_13><loc_90><loc_88><loc_93></location>Growth of matter perturbations in the Interacting Dark Energy/Dark Matter Scenarios</section_header_level_1>
<text><location><page_1><loc_43><loc_87><loc_57><loc_89></location>N.Nazari Pooya 1, ∗</text>
<text><location><page_1><loc_22><loc_85><loc_78><loc_87></location>1 Department of Physics, Faculty of Science, Bu-Ali Sina University, Hamedan Iran</text>
<text><location><page_1><loc_18><loc_61><loc_83><loc_84></location>In this study, we investigate two widely recognized Interacting Dark Energy(IDE) models and assess their compatibility with observational data, focusing on the growth rate of matter perturbations. We explore IDE models with different equations of state (EoS) parameters for Dark Energy (DE), including the CPL parameterization and a constant value for w de . To constrain the parameters of the IDE models using background data, we employ a Markov Chain Monte Carlo (MCMC) analysis. Our results show that both IDE-I and IDE-II models are Compatible with observational data, although with slight variations influenced by the homogeneity or clustering of DE. Following that, we investigate the growth of matter perturbations and perform a comprehensive statistical analysis utilizing both the background and growth rate data. The growth rate in IDE models exhibits deviations compared to the ΛCDM model due to the impact of homogeneity or clustering of DE, as well as the selection of the EoS parameter. However, we find that the IDE models show good compatibility with the growth rate data. Furthermore, we explore how the clustering or homogeneity of DE and the selection of the EoS parameter affect the evolution of the relative difference in the growth rate of IDE models, ∆ f , in comparison to the ΛCDM model. Lastly, we employ the AIC and BIC criteria to evaluate and identify the best model that is compatible with the observational data. The selection of the model depends on the homogeneity or clustering of DE, the EoS parameter, and the dataset used. Overall, the IDE-I and IDE-II models exhibit agreement with the data, with slight deviations depending on specific scenarios and parameters.</text>
<section_header_level_1><location><page_1><loc_20><loc_57><loc_37><loc_58></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_26><loc_49><loc_55></location>Observational evidence from various sources strongly supports the notion that the expansion of the Universe is accelerating. This evidence encompasses diverse measurements, including Supernovae type Ia (SnIa) [1-3], Cosmic Microwave Background (CMB) [4-7], Baryon Acoustic Oscillations (BAO) [8-12], high redshift galaxy clusters [13, 14], weak lensing surveys [15-17], and other sources. These diverse observations consistently support the idea of accelerated expansion, shedding light on the evolution of the Universe and the role of Dark Energy (DE). However, despite the strong support for the ΛCDM model provided by these observations, several challenges persist. The nature of DE itself remains a mystery, as its origin and properties are not yet fully understood. The cosmological constant's fine-tuning and the cosmic coincidence problem raise questions about why DE dominates the Universe's energy density at the present epoch. Moreover, tensions related to S 8 [18-20] and H 0 [21-26] further complicate matters, posing both theoretical and observational challenges.</text>
<text><location><page_1><loc_9><loc_15><loc_49><loc_26></location>Observationally, there are discrepancies in the formation of cosmic structures at smaller scales compared to the predictions of the ΛCDM model. These inconsistencies and tensions necessitate exploring alternative theories and modifications to address these issues and refine our understanding expansion of the Universe. Scientists are actively investigating various approaches, including modified gravity theories [27-33], Unified DE models</text>
<text><location><page_1><loc_52><loc_54><loc_92><loc_58></location>[34-38], and Interacting DE (IDE) models [39-46], and many other alternative cosmological scenarios, to overcome these theoretical and observational challenges.</text>
<text><location><page_1><loc_52><loc_31><loc_92><loc_53></location>IDE models, which involve the interaction between DE and Dark Matter(DM), have significant implications for the evolution of the Universe and the behavior of these enigmatic components. Extensive research has been conducted on these models to comprehend their impact on the expansion history of the Universe, the growth of large-scale structures, and observational constraints. Since the exact form of the interaction cannot be deduced from fundamental principles due to the unknown nature of DE and DM, a phenomenological approach is often employed to determine the nature of their interaction. Recent observational data indicates that a direct interaction between DE and DM cannot be ruled out. These models introduce additional parameters to describe the strength of the interaction and its effects on observables related to large-scale structures.</text>
<text><location><page_1><loc_52><loc_13><loc_92><loc_30></location>The growth of matter perturbations in the Universe can be influenced by DE through various mechanisms, even without considering the interaction between two dark sectors. One such mechanism is the deceleration of the growth rate due to the accelerating expansion of the Universe, resulting in a slower evolution of matter perturbations. Additionally, DE can exhibit perturbations that grow in a similar manner to DM, leading to changes in the distribution and clustering of DM throughout the Universe. These mechanisms demonstrate how DE can impact the growth of matter perturbations in the Universe[11, 54-63, 103].</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_12></location>The growth of DM perturbations in IDE models can be influenced by perturbations in DE, where an exchange of energy between the two dark components impacts their</text>
<text><location><page_2><loc_9><loc_85><loc_49><loc_93></location>evolutions. To comprehend the growth of structures caused by this interaction, researchers utilize theoretical modeling, simulations, and analysis of observational data. These investigations significantly advance our understanding of the Universe's evolution by uncovering the interaction between DE and DM.</text>
<text><location><page_2><loc_9><loc_61><loc_49><loc_84></location>This study aims to examine the effects of small DE perturbations on the growth of DM perturbations in IDE models. By studying the growth of DM perturbations in the presence of DE perturbations, valuable insights are gained into the complex interaction between these dark components and their influence on the evolution of cosmic structures. To explore the impact of DE perturbations on the growth of DM perturbations, we consider different interaction terms and Equation of State (EoS) parameters for DE in the conservation equations related to DE and DM. Subsequently, we solve the coupled equations governing the evolution of DE and DM, and finally, we compare the resulting outcomes with observational data. This analysis enables us to measure and evaluate any deviations between the predictions of the standard ΛCDM model and the observational data.</text>
<text><location><page_2><loc_9><loc_31><loc_49><loc_61></location>The article's structure is outlined as follows: In Sec. II, we derive the necessary equations for the background evolution of the Universe and introduce the IDE models investigated in this study. In Sec. III, we give a brief overview of the current observational datasets at the background level. We then utilize numerical Markov Chain Monte Carlo (MCMC) analysis to constrain the free parameters of the IDE models examined in this research. In Sec. IV, we establish the fundamental equations governing the evolution of DE and DM in the linear regime within the IDE model scenarios. We also investigate the growth rate of matter perturbations in this section. Additionally, we incorporate growth rate data along with background data to obtain more comprehensive constraints on the free parameters of models. In Sec. V, we perform a comparison between the IDE models and the ΛCDM model, using significant cosmological quantities. This comparison is performed using the bestfit values obtained from the likelihood analysis for the free parameters. Finally, in Sec. VI, we present the conclusions derived from our study.</text>
<section_header_level_1><location><page_2><loc_11><loc_24><loc_46><loc_26></location>II. BASIC EQUATIONS IN IDE MODELS: BACKGROUND LEVEL</section_header_level_1>
<text><location><page_2><loc_9><loc_9><loc_49><loc_22></location>In this study, the background of the Universe is described using the Friedmann-Robertson-Walker (FRW) metric. The FRW metric, which is defined in terms of conformal time, can be expressed as ds 2 = a 2 ( η ) ( -d η 2 + δ ij dx i dx j ) . where a ( η ) is the scale factor. Also, the energy-momentum tensor of DE and DM, denoted by ¯ T µν = ¯ p ¯ g µν (¯ p + ¯ ρ )¯ u µ ¯ v ν . where the bars denote that the quantities are unperturbed. In the context of IDE models, the conservation equations for the energy-momentum</text>
<text><location><page_2><loc_52><loc_92><loc_90><loc_93></location>of these components can be expressed as follows [64]:</text>
<formula><location><page_2><loc_67><loc_89><loc_92><loc_91></location>∇ µ ¯ T µ,i ν = ¯ Q i ν (1)</formula>
<text><location><page_2><loc_52><loc_71><loc_92><loc_88></location>where i = de , dm, and ¯ Q i ν is the phenomenological interaction term among the DM and DE. Due to the conservation of total energy-momentum, we can conclude ¯ Q de ν = -¯ Q dm ν . Also, in the case of non-interaction between DE and DM, we have ¯ Q de ν = -¯ Q dm ν = 0. This means that the energy transfer rate from DE to DM or vice versa is zero. Furthermore, because of the homogeneous and isotropic of background, the spatial components of ¯ Q i ν are zero. Consequently, the evolution of the energy density of dark energy ( ρ de ), dark matter ( ρ dm ), baryons ( ρ b ), and radiation ( ρ r ) over time can be determined using the following conservation equations.</text>
<formula><location><page_2><loc_61><loc_69><loc_92><loc_70></location>˙ ρ dm +3 H ρ dm = ¯ Q 0 (2)</formula>
<formula><location><page_2><loc_61><loc_67><loc_92><loc_68></location>˙ ρ de +3 H (1 + w de ) ρ de = -¯ Q 0 (3)</formula>
<formula><location><page_2><loc_61><loc_65><loc_92><loc_66></location>˙ ρ b +3 H ρ b = 0 (4)</formula>
<formula><location><page_2><loc_61><loc_63><loc_92><loc_64></location>˙ ρ r +4 H ρ r = 0 (5)</formula>
<text><location><page_2><loc_52><loc_53><loc_92><loc_62></location>where dots denote derivative with respect to the conformal time, H is the conformal Hubble parameter( H = aH ), and w de = p de /ρ de is the EoS parameter of DE. Moreover, the evolution of a spatially flat FRW Universe with a homogeneous and isotropic background is governed by the following equation:</text>
<formula><location><page_2><loc_60><loc_49><loc_92><loc_52></location>H 2 = 8 π G 3 a 2 ( ρ dm + ρ de + ρ b + ρ r ) (6)</formula>
<text><location><page_2><loc_52><loc_34><loc_92><loc_49></location>The solutions of Eqs. (2 & 3) depend on the particular forms of ¯ Q 0 and w de . In this study, two phenomenological interaction terms for ¯ Q 0 are considered: ¯ Q 01 = ξ 1 H ρ de and ¯ Q 02 = ξ 2 H ρ dm . Where ξ 1 and ξ 2 are dimensionless coupling parameter describing the strength of interaction between DE and DM. In recent years, there has been a significant amount of research devoted to models resembling these, in which the interaction term exhibits proportionality to the energy densities ( ρ de , ρ dm ), or a combination of both[65-68].</text>
<text><location><page_2><loc_52><loc_23><loc_92><loc_34></location>Moreover, we investigate two distinct cases for the EoS parameter associated with each of the interaction terms. In the first case, we use a well-known parameterization called the Chevallier-Polarski-Linder (CPL) parameterization, defined as w de = w 0 + w 1 (1 -a )[69, 70]. In the second case, we assume that the parameter w de is constant. In the following, we examine these contents in more detail.</text>
<text><location><page_2><loc_52><loc_17><loc_92><loc_23></location>Interaction Term I: ¯ Q 01 = ξ 1 H ρ de : By employing this interaction term and assuming the CPL parameterization for DE in equations (2 & 3), we can obtain the solutions for these equations as follows:</text>
<formula><location><page_2><loc_55><loc_11><loc_92><loc_16></location>ρ dm = ρ 0 de ξ 1 a -3 ∫ a 1 e 3 w 1 ( a -1) a -3( w 0 + w 1 ) -ξ 1 -1 da + ρ 0 dm a -3 ; (7)</formula>
<formula><location><page_2><loc_55><loc_9><loc_92><loc_11></location>ρ de = ρ 0 de a -3(1+ w 0 + w 1 ) -ξ 1 e 3( a -1) w 1 (8)</formula>
<table>
<location><page_3><loc_10><loc_83><loc_48><loc_88></location>
<caption>TABLE I. Phenomenological interaction models and their related equations of state that have been investigated in this research.</caption>
</table>
<text><location><page_3><loc_9><loc_76><loc_49><loc_78></location>Furthermore, if we assume that w de is a constant, we can determine the solutions for equations (2 & 3) as below:</text>
<formula><location><page_3><loc_14><loc_66><loc_49><loc_74></location>ρ dm = a -3((1+ w de ) -ξ 1 3 w de + ξ 1 [ ξ 1 ρ 0 de (a 3 w de + ξ 1 -1) +(3 w de + ξ 1 ) ρ 0 dm a 3 w de + ξ 1 ] (9) ρ de = ρ 0 de a -3((1+ w de ) -ξ 1 (10)</formula>
<text><location><page_3><loc_9><loc_59><loc_49><loc_65></location>Interaction Term II: ¯ Q 02 = ξ 2 H ρ dm : By assuming this form of the interaction term and utilizing the CPL parameterization for DE in equations (2 & 3), we can derive the solutions for these equations as follows:</text>
<formula><location><page_3><loc_13><loc_51><loc_49><loc_58></location>ρ dm = ρ 0 dm a -3+ ξ 2 ; (11) ρ de = a -3(1+ w 0 + w 1 ) [ ρ 0 de e 3(a -1) w 1 -ρ 0 dm e 3 aw 1 × ξ 2 ∫ a e -3 aw 1 a 3( w 0 + w 1 )+ ξ 2 -1 da ] (12)</formula>
<formula><location><page_3><loc_20><loc_50><loc_21><loc_51></location>1</formula>
<text><location><page_3><loc_9><loc_46><loc_49><loc_48></location>when w de is constant, the solutions to Eqs.(2 & 3) can be derived as follows.</text>
<formula><location><page_3><loc_9><loc_38><loc_49><loc_44></location>ρ dm = ρ 0 dm a -3+ ξ 2 ρ de = a -3(1+ w de ) 3 w de + ξ 2 [ ρ 0 de (3 w de + ξ 2 ) + ξ 2 ρ 0 dm (1 -a 3 w de + ξ 2 ) ] (13)</formula>
<text><location><page_3><loc_9><loc_32><loc_49><loc_36></location>In the following section, we analyze the observational constraints that are imposed on these IDE models at the background level.</text>
<section_header_level_1><location><page_3><loc_9><loc_28><loc_48><loc_29></location>III. DATA ANALYSIS: BACKGROUND LEVEL</section_header_level_1>
<text><location><page_3><loc_9><loc_23><loc_49><loc_26></location>In this section, we provide an overview of the steps in analyzing observational data. The steps are as follows:</text>
<unordered_list>
<list_item><location><page_3><loc_9><loc_17><loc_49><loc_23></location>I . Background Analysis: The total likelihood at the background level is calculated using the χ 2 bac equation, which combines the contributions from different observational data sets, which is given by:</list_item>
</unordered_list>
<formula><location><page_3><loc_10><loc_14><loc_49><loc_16></location>χ 2 bac ( p ) = χ 2 H ( p ) + χ 2 BAO ( p ) + χ 2 SN ( p ) + χ 2 CMB ( p ) (14)</formula>
<text><location><page_3><loc_9><loc_9><loc_49><loc_13></location>where, p = { Ω b h 2 , Ω c h 2 , H 0 , w 0 , w 1 , ξ 1 , ξ 2 } represents the free parameters of the models, and the subscripts H, BAO, SN, and CMB denote the contributions from the</text>
<text><location><page_3><loc_52><loc_83><loc_92><loc_93></location>Hubble parameter, Baryon Acoustic Oscillations, Supernova type Ia, and Cosmic Microwave Background, respectively. In this analysis, we utilize 1098 observational data points related to the background. These data points consist of 1048 data points from the Pantheon catalog for supernova type Ia (SnIa), 3 data points for the CMB, 11 data points for the BAO, and 36 data points for the H(z).</text>
<text><location><page_3><loc_52><loc_77><loc_92><loc_83></location>II . Growth Rate Analysis: In the second analysis, the growth rate data is incorporated. The total χ 2 value ( χ 2 tot ) combining the background and growth rate components is given by:</text>
<formula><location><page_3><loc_62><loc_75><loc_92><loc_76></location>χ 2 tot ( q ) = χ 2 bac ( p ) + χ 2 growth (15)</formula>
<text><location><page_3><loc_52><loc_65><loc_92><loc_74></location>where, q = { p , σ 8 , 0 } represents the free parameters of the models at both the background and perturbation levels. The χ 2 growth term represents the contribution from the growth rate data. In this step, 44 growth rate data points are added to the background data points. Additionally, the details of χ 2 growth are explained in Subsec. IV A.</text>
<text><location><page_3><loc_52><loc_51><loc_92><loc_65></location>III . Statistical Tools: The χ 2 statistic is commonly used to assess the level of agreement between theoretical models and observational data. Therefore, we utilize MCMC analysis, which explores the parameter space to determine uncertainties and correlations among the parameters. These statistical techniques are employed to analyze observational data and constrain model parameters based on their compatibility with the observational data. In the subsequent sections, we present a concise description of the datasets employed in this study.</text>
<section_header_level_1><location><page_3><loc_58><loc_46><loc_86><loc_47></location>A. Type Ia Supernovae(SnIa) data</section_header_level_1>
<text><location><page_3><loc_52><loc_35><loc_92><loc_44></location>The dataset of Type Ia Supernovae (SnIa) plays a crucial role in studying the dynamic background of the Universe and continues to provide valuable constraints for DE models. The SnIa dataset involves comparing the apparent magnitude with the absolute magnitude of observed SnIa, which is known as the distance modulus and theoretically is given by:</text>
<formula><location><page_3><loc_56><loc_32><loc_92><loc_33></location>µ th ( z ) = 5 log 10 d L ( z ) + 42 . 384 -5 log 10 h (16)</formula>
<text><location><page_3><loc_52><loc_28><loc_92><loc_31></location>Where, d L ( z ) represents the luminosity distance, which is defined in the following manner:</text>
<formula><location><page_3><loc_62><loc_24><loc_92><loc_27></location>d L ( z ) = c H 0 (1 + z ) ∫ z 0 dz ' E ( z ' ) (17)</formula>
<text><location><page_3><loc_52><loc_17><loc_92><loc_23></location>In this analysis, we employ the Pantheon SnIa dataset, containing 1048 data points sourced from the Pantheon sample [71]. Additionally, we obtain the respective χ 2 SN using the following relation:</text>
<formula><location><page_3><loc_59><loc_12><loc_92><loc_16></location>χ 2 sn ( p ) = 1048 ∑ i =1 [ µ th ( p , z i ) -µ obs ( z i )] 2 σ 2 µ,i (18)</formula>
<text><location><page_3><loc_52><loc_9><loc_92><loc_11></location>Where, the µ th ( p , z i ) refers to the theoretical prediction of the distance modulus at a specific redshift z i . On</text>
<text><location><page_4><loc_9><loc_89><loc_49><loc_93></location>the other hand, µ obs ( z i ) represents the distance modulus determined through observations and, σ µ,i indicates the uncertainty related to the observational data.</text>
<section_header_level_1><location><page_4><loc_12><loc_85><loc_46><loc_86></location>B. Baryon Acoustic Oscillations(BAO) data</section_header_level_1>
<text><location><page_4><loc_9><loc_57><loc_49><loc_83></location>Recent investigations have highlighted the significance of BAO as a valuable geometric probe for examining DE. The precise position of the BAO peak in the CMB power spectrum is indeed dependent on the ratio of D V ( z ) to the comoving sound horizon size r s ( z ) at the drag epoch, denoted as z d which represents the epoch when baryons decoupled from photons. In their study, Komatsu et al.[72] noted that the drag epoch, characterized by z d , occurs slightly later than the epoch of photon decoupling, represented by z ∗ . During this epoch, the gravitational potential well affects the behavior of baryons. As a result, the sound horizon size during the drag era is slightly larger compared to the photon decoupling era. Various researchers have reported their measurements of the BAO feature using different observable quantities. Some measurements included constraints on the ratio r s ( z d ) /D V ( z ) or its inverse. The comoving sound horizon r s ( z d ) is given by[72]</text>
<formula><location><page_4><loc_19><loc_52><loc_49><loc_56></location>r s ( z d ) = c H 0 ∫ ∞ z d c s ( z ' ) dz ' E ( z ' ) (19)</formula>
<text><location><page_4><loc_9><loc_43><loc_49><loc_51></location>where c s ( z ) = 1 / [3(1 + 3Ω b 0 4(1+ z )Ω γ 0 )] 1 2 and E ( z ) is given by Eq. (6). We adopt the approximate function for z d as described in [73]. Furthermore, we set Ω γ 0 = 2 . 469 × 10 -5 h -2 (for T cmb =2.725 K) according to [72, 79]. Also, the expression for D V ( z ) is provided in[72] as follows:</text>
<formula><location><page_4><loc_16><loc_39><loc_49><loc_42></location>D V ( z ) = c H 0 [ (1 + z ) 2 D 2 A ( z ) z E ( z ) ] 1 3 (20)</formula>
<text><location><page_4><loc_9><loc_34><loc_49><loc_38></location>where D A ( z ) is the angular diameter distance. When the curvature density, Ω K , is zero, we can calculate D A ( z ) by using the following formula [72]:</text>
<formula><location><page_4><loc_18><loc_29><loc_49><loc_32></location>D A ( z ) = c H 0 (1 + z ) ∫ z 0 dz E ( z ) (21)</formula>
<text><location><page_4><loc_9><loc_21><loc_49><loc_28></location>We utilize two datasets, one in the old format presented in Table II and the other in the new format shown in Table III. Since the data points listed in Tables II and III are uncorrelated, we calculate χ 2 bao , 1 for the first case as follows:</text>
<formula><location><page_4><loc_16><loc_16><loc_49><loc_20></location>χ 2 bao , 1 = 4 ∑ i =1 [ d z ( z i ) | th -( d z,i ) | obs ] 2 σ 2 i (22)</formula>
<text><location><page_4><loc_9><loc_9><loc_49><loc_15></location>In this case, the theoretical prediction is expressed as d z ( z ) = r s ( z d ) D V ( z eff ) , where r s ( z d ) represents the comoving sound horizon size at the drag epoch, and D V ( z eff ) denotes the effective volume distance. In the second case,</text>
<table>
<location><page_4><loc_58><loc_82><loc_85><loc_90></location>
<caption>TABLE II. The old format of the BAO data points, along with their Survey details and References.TABLE III. The new format of the BAO data points, along with their Survey details and References.</caption>
</table>
<table>
<location><page_4><loc_58><loc_67><loc_86><loc_75></location>
</table>
<text><location><page_4><loc_52><loc_61><loc_84><loc_63></location>χ 2 bao , 2 is obtained from the following relation.</text>
<formula><location><page_4><loc_60><loc_56><loc_92><loc_60></location>χ 2 bao , 2 = 4 ∑ i =1 [ β ∗ ( z i ) | th -β ∗ z,i | obs ] 2 σ 2 i (23)</formula>
<text><location><page_4><loc_52><loc_50><loc_92><loc_55></location>In this case, the theoretical prediction is represented by β ∗ ( z ) = D V ( z eff ) r s ( z d ) r fid s . Consequently, the total χ 2 bao is given by χ 2 bao = χ 2 bao , 1 + χ 2 bao , 2 .</text>
<section_header_level_1><location><page_4><loc_53><loc_46><loc_90><loc_47></location>C. Cosmic Microwave Background(CMB) data</section_header_level_1>
<text><location><page_4><loc_52><loc_33><loc_92><loc_44></location>The location of the CMB acoustic peak is valuable tool for constraining models of DE as it depends on the angular diameter distance in dynamical DE models. The specific position of this peak in the power spectrum of temperature anisotropy in the CMB is determined by three parameters: l a , R , and Ω b h 2 . Where, l a represents the angular scale of the sound horizon at the decoupling era, which can be calculated using the equation:</text>
<formula><location><page_4><loc_64><loc_29><loc_92><loc_32></location>l a = (1 + z ∗ ) πD A ( z ∗ ) r s ( z ∗ ) (24)</formula>
<text><location><page_4><loc_52><loc_16><loc_92><loc_27></location>In this equation, z ∗ refers to the redshift at the decoupling time, and a fitting formula from Hu[82] is used to determine it. The coefficient of (1 + z ∗ ) is included because D A ( z ∗ ) represents the physical angular diameter distance (see Eq. 21), while r s ( z ∗ ) represents the comoving sound horizon at z ∗ (see Eq. 19). The scale distance or shift parameter at the decoupling epoch, denoted as R , is defined as follows[81]:</text>
<formula><location><page_4><loc_59><loc_12><loc_92><loc_15></location>R ( z ∗ ) = 1 c √ Ω m 0 H 0 (1 + z ∗ ) D A ( z ∗ ) (25)</formula>
<text><location><page_4><loc_52><loc_9><loc_92><loc_11></location>Chen et al. [83] conducted a comparison between the full CMB power spectrum analysis and the distance prior</text>
<table>
<location><page_5><loc_12><loc_62><loc_46><loc_88></location>
<caption>TABLE IV. The H(z) data used in the current analysis (in units of km s -1 Mpc -1 ). This compilation is partly based on those of Ref [91].</caption>
</table>
<text><location><page_5><loc_9><loc_49><loc_49><loc_59></location>method to constrain different DE models. The results of both methods were found to be completely consistent. Therefore, in this study, we utilize the combined CMB likelihood (Planck 2018 TT, TE, EE + lowE) based on the observed values X obs i = { R,l a , Ω b h 2 } = { 1 . 7493 , 301 . 462 , 0 . 02239 } , as obtained by Chen et al. [83]. The χ 2 cmb is expressed as follows:</text>
<formula><location><page_5><loc_21><loc_46><loc_49><loc_48></location>χ 2 cmb = ∆ X i Σ -1 ij ∆ X T i (26)</formula>
<text><location><page_5><loc_9><loc_39><loc_49><loc_45></location>Where, ∆ X i = { X th i -X obs i } represents the difference between the theoretical value X th i and the observed value X obs i . The inverse of the covariance matrix Σ -1 ij associated with ∆ X i is given by:</text>
<formula><location><page_5><loc_10><loc_33><loc_48><loc_38></location>Σ -1 ij = 94392 . 3971 -1360 . 4913 1664517 . 2916 -1360 . 4913 161 . 4349 3671 . 6180 1664517 . 2916 3671 . 6180 79719182 . 5162 </formula>
<section_header_level_1><location><page_5><loc_22><loc_29><loc_35><loc_30></location>D. Hubble data</section_header_level_1>
<text><location><page_5><loc_9><loc_21><loc_49><loc_27></location>In our analysis, we utilize 36 data points of H ( z ) from Table (IV), spanning the redshift range 0 . 07 ⩽ z ⩽ 2 . 34. Since the measurements of H ( z ) are uncorrelated, we can express the χ 2 H statistic as follows:</text>
<formula><location><page_5><loc_16><loc_16><loc_49><loc_19></location>χ 2 H ( p ) = 36 ∑ i =1 [ H th ( p , z i ) -H obs ( z i )] 2 σ 2 i (27)</formula>
<text><location><page_5><loc_9><loc_9><loc_49><loc_14></location>Here, H th ( p , z i ) represents the model predictions at the redshift z i , while H obs ( z i ) and σ i denote the measured values and Gaussian errors, respectively, corresponding to the data points listed in Table (IV).</text>
<section_header_level_1><location><page_5><loc_54><loc_91><loc_89><loc_93></location>IV. BASIC EQUATIONS IN IDE MODELS: PERTURBATIONS LEVEL</section_header_level_1>
<text><location><page_5><loc_52><loc_80><loc_92><loc_88></location>The perturbed Friedmann-Robertson-Walker (FRW) metric is used to describe the spacetime geometry in cosmology, taking into account small perturbations from the homogeneous and isotropic background Universe. In the conformal Newtonian gauge, the metric can be expressed as follows:</text>
<formula><location><page_5><loc_54><loc_78><loc_92><loc_79></location>ds 2 = a( η ) 2 [ -(1 + 2 ψ ) dη 2 +(1 -2 ϕ ) δ ij dx i dx j ] , (28)</formula>
<text><location><page_5><loc_52><loc_55><loc_92><loc_77></location>where a ( η ) is the scale factor depending on conformal time η , and ψ and ϕ are scalar potentials representing gravitational potential and spatial curvature, respectively. The (1 + 2 ψ ) and (1 -2 ϕ ) terms modify the temporal and spatial components of the metric, accounting for small perturbations in the spacetime geometry. This gauge simplifies calculations and is commonly used to study linear perturbations. Furthermore, in the absence of anisotropic stresses, the equations of Einstein's gravity theory require that the metric potentials ϕ and ψ are equal. However, this equality does not generally hold in models of modified gravity. The perturbed conservation equations, taking into account perturbed metrics and perturbed energy-momentum tensors, yield the following evolution equations for the perturbations[64, 96, 97]:</text>
<formula><location><page_5><loc_54><loc_48><loc_92><loc_54></location>˙ δ = -[ 3 H ( δp δρ -w de ) -¯ Q 0 ¯ ρ ] δ -(1 + w de ) ( θ -3 ˙ ϕ ) -δ ¯ Q 0 ¯ ρ , (29)</formula>
<formula><location><page_5><loc_54><loc_42><loc_92><loc_47></location>˙ θ = -[ H ( 1 -3 c 2 a ) -¯ Q 0 ¯ ρ ] θ + δp δρ k 2 δ (1 + w de ) + k 2 ϕ ¯ (30)</formula>
<formula><location><page_5><loc_57><loc_41><loc_67><loc_44></location>+ ik i δ Q i ¯ ρ (1 + w de ) .</formula>
<text><location><page_5><loc_52><loc_21><loc_92><loc_40></location>where dot denotes the derivative with respect to the conformal time, η , which is related to the physical time, t , through the scale factor, a ( adη = dt ). The variables k i , δ ≡ δρ/ρ , and θ represent the components of the wavevector in Fourier space, density contrast, and divergence of the peculiar velocity, respectively. The parameter w de corresponds to the EoS of DE, taking different values depending on whether the perturbations are associated with dust( w de =0) or DE. And, δQ µ are the perturbations to the exchange of energy -momentum in the perturbed conservation equations. Lastly, the parameter c 2 a represents the squared adiabatic sound speed of the DE perturbations, and its definition is as follows:</text>
<formula><location><page_5><loc_63><loc_17><loc_92><loc_20></location>c 2 a = w de -˙ w de 3 H (1 + w de ) (31)</formula>
<text><location><page_5><loc_52><loc_12><loc_92><loc_16></location>To investigate perturbations of DE, it is useful to introduce an effective sound speed, c eff , specifically for DE perturbations. This quantity is defined as follows[98]:</text>
<formula><location><page_5><loc_58><loc_8><loc_92><loc_11></location>δp δρ = c 2 eff +3 H (1 + w d ) ( c 2 eff -c 2 a ) θ δ 1 k 2 (32)</formula>
<text><location><page_6><loc_9><loc_90><loc_49><loc_93></location>Additionally, in this context, the Poisson equation can be expressed as follows[99]:</text>
<formula><location><page_6><loc_20><loc_88><loc_49><loc_89></location>k 2 ϕ = -4 πGa 2 ( δρ +3 δp ) (33)</formula>
<text><location><page_6><loc_9><loc_81><loc_49><loc_87></location>where δρ = δρ dm + δρ de and δp = δp dm + δp de . After that, using quantities δp dm = 0, δp de = c 2 eff δρ de , δρ de = ρ de δ de , δρ dm = ρ dm δ dm in Eq. (33), the Poisson equation can be written as:</text>
<formula><location><page_6><loc_12><loc_77><loc_49><loc_80></location>-k 2 ϕ = 3 2 H 2 [ Ω dm δ dm +(1 + 3 c 2 eff )Ω de δ de ] , (34)</formula>
<text><location><page_6><loc_9><loc_69><loc_49><loc_77></location>where, Ω dm and Ω de represent the fractional densities of DM and DE respectively. These fractional densities are defined as the ratio of the densities ρ dm and ρ de to the critical density of the Universe. Also, the ρ 0 crit , is defined as ρ 0 crit = 3 H 2 0 / 8 πG .</text>
<text><location><page_6><loc_9><loc_54><loc_49><loc_69></location>In a matter-dominated Universe with a small DE component, the gravitational potential ϕ can be approximated as a constant in the linear perturbation regime on sub-horizon scales( k 2 ≫ H 2 ). This assumption is confirmed by the fact that most observed structures, which formed during the matter-dominated era, align with this assumption. This simplification allows for easier analysis of the evolution of perturbations and the growth of structures. However, this assumption is only valid under specific conditions and may not hold in other regimes or on larger scales[100].</text>
<text><location><page_6><loc_9><loc_48><loc_49><loc_53></location>To obtain second-order coupled differential equations describing the evolution of DE and (29 & 30) as follows: Firstly, by manipulating Eq. (32), we can obtain the following relation:</text>
<formula><location><page_6><loc_12><loc_42><loc_49><loc_47></location>-3 H δp δρ δ = -3 H c 2 eff δ -9 H 2 k 2 (1 + w de )( c 2 eff -c 2 a ) θ ≃ -3 H c 2 eff δ (35)</formula>
<text><location><page_6><loc_9><loc_35><loc_49><loc_41></location>In the regime of sub-horizon scales ( k 2 ≫ H 2 ), we can neglect the second term on the right-hand side of Eq. (35). Secondly, according to Eq.(32), we can express this relation as:</text>
<formula><location><page_6><loc_13><loc_31><loc_49><loc_34></location>k 2 δp δρ δ = k 2 c 2 eff δ +3 H (1 + w de )( c 2 eff -c 2 a ) θ (36)</formula>
<text><location><page_6><loc_9><loc_27><loc_49><loc_30></location>Now, by substituting Eqs.(35 & 36) into Eqs.(29 & 30), we can express them in the following form:</text>
<formula><location><page_6><loc_10><loc_23><loc_49><loc_26></location>˙ δ +3 H c 2 eff δ -3 H w de δ -¯ Q 0 ¯ ρ δ +(1 + w de ) θ = 0 , (37)</formula>
<formula><location><page_6><loc_10><loc_20><loc_49><loc_23></location>˙ θ + [ H ( 1 -3 c 2 eff ) -¯ Q 0 ¯ ρ ] θ -k 2 ϕ -k 2 c 2 eff 1 + w de δ = 0 . (38)</formula>
<text><location><page_6><loc_9><loc_9><loc_49><loc_18></location>Morermore, in order to derive Eqs. (37 & 38) (see also [64]), we ignore δ ¯ Q µ . We remind that Eqs.(29 & 30) or their equivalent Eqs.(37 & 38) can be used separately for the components of DE and DM. Based on this, we initially utilize Eqs.(37 & 38) to obtain a second-order equation that describes the evolution of DE perturbations. By eliminating θ from the system of Eqs.(37 &</text>
<text><location><page_6><loc_52><loc_90><loc_92><loc_93></location>38), we can derive following equation for δ de in terms of conformal time.</text>
<formula><location><page_6><loc_62><loc_88><loc_92><loc_89></location>¨ δ de + ˜ A de ˙ δ de + ˜ B de δ de = ˜ S de (39)</formula>
<text><location><page_6><loc_52><loc_84><loc_92><loc_87></location>where the coefficients ˜ A de , ˜ B de , and ˜ S de are defined as follows:</text>
<formula><location><page_6><loc_53><loc_67><loc_92><loc_83></location>˜ A de = H (1 -3 w de ) + 3 H ( c 2 a -w de ) -2 ¯ Q 0 ρ de ˜ B de = 3 H 2 ( c 2 eff -w de ) [ 1 + ˙ H H 2 -3( w de + c 2 eff -c 2 a ) ] + k 2 c 2 eff -3 H ˙ w de + H [ 3 ( 2 w de -c 2 a ) -1 ] ¯ Q 0 ρ de + ( ¯ Q 0 ρ de ) 2 -d dη ( ¯ Q 0 ρ de ) ˜ S de = -(1 + w de ) k 2 ϕ (40)</formula>
<text><location><page_6><loc_52><loc_59><loc_92><loc_66></location>where -k 2 ϕ is expressed by Poisson Eq. (34). Likewise, by utilizing Eqs. (37 & 38), we can derive a second-order equation that describes the evolution of DM perturbations. In this case, we set w d = c 2 eff = c 2 a = 0. The resulting equation is obtained as follows:</text>
<formula><location><page_6><loc_60><loc_57><loc_92><loc_58></location>¨ δ dm + ˜ A dm ˙ δ dm + ˜ B dm δ dm = ˜ S dm (41)</formula>
<text><location><page_6><loc_52><loc_52><loc_92><loc_55></location>where, in this case, the coefficients ˜ A dm , ˜ B dm , and ˜ S dm are defined as follows:</text>
<formula><location><page_6><loc_58><loc_43><loc_92><loc_52></location>˜ A dm = H2 ¯ Q 0 ρ dm ˜ B dm = -H ¯ Q 0 ρ dm + ( ¯ Q 0 ρ dm ) 2 -d dη ( ¯ Q 0 ρ dm ) ˜ S dm = -k 2 ϕ (42)</formula>
<text><location><page_6><loc_52><loc_35><loc_92><loc_42></location>Additionally, by utilizing the expressions d dη = a H d da and d 2 dη 2 = ( a H 2 + a ˙ H ) d da + a 2 H 2 d 2 da 2 , along with Eq. (34), one can represent Eqs. (39 & 41) in terms of the scale factor. Thus, we obtain the following equations:</text>
<formula><location><page_6><loc_60><loc_33><loc_92><loc_34></location>δ '' de + A de δ ' de + B de δ de = S de (43)</formula>
<formula><location><page_6><loc_60><loc_31><loc_92><loc_32></location>δ '' dm + A dm δ ' dm + B dm δ dm = S dm (44)</formula>
<text><location><page_6><loc_52><loc_26><loc_92><loc_30></location>where, the prime denotes the derivative with respect to the scale factor. The coefficients A de , B de , and S de are defined as follows:</text>
<formula><location><page_6><loc_53><loc_8><loc_92><loc_25></location>A de = 3 a + H ' H + 3 a ( c 2 a -2 w de ) -2 a 2 H ¯ Q 0 ρ de B de = 3 a ( c 2 eff -w de ) [ 2 a + H ' H -3 a ( w de + c 2 eff -c 2 a ) ] + k 2 c 2 eff a 4 H 2 -3 a w ' de + 1 a 3 H 2 [ 3 ( 2 w de -c 2 a ) -1 ] ¯ Q 0 ρ de + 1 a 4 H 2 ( ¯ Q 0 ρ de ) 2 -1 a 2 H d da ( ¯ Q 0 ρ de ) (45) S de = 3 2 a 2 (1 + w de ) [ Ω dm δ dm +Ω de δ de ( 1 + 3 c 2 eff ) ]</formula>
<text><location><page_7><loc_9><loc_90><loc_49><loc_93></location>In addition, the coefficients A dm , B dm , and S dm can be defined as follows:</text>
<formula><location><page_7><loc_10><loc_79><loc_49><loc_89></location>A dm = 3 a + H ' H -2 a 2 H ¯ Q 0 ρ dm B dm = -1 a 3 H 2 ¯ Q 0 ρ dm + 1 a 4 H 2 ( ¯ Q 0 ρ dm ) 2 -1 a 2 H d da ( ¯ Q 0 ρ dm ) S dm = 3 2 a 2 [ Ω dm δ dm +Ω de δ de ] (46)</formula>
<text><location><page_7><loc_9><loc_71><loc_49><loc_78></location>Where ¯ Q 0 , ¯ Q 0 ρ dm , and ¯ Q 0 ρ de for both models IDE I and IDE II are summarized in Table I. By solving the the coupled Eqs. (43 and 44) numerically from an initial scale factor of a i = 10 -3 to the current time ( a = 1), we can obtain the density contrasts of the δ dm and δ de .</text>
<text><location><page_7><loc_9><loc_58><loc_49><loc_70></location>The effect of clustered and non-clustered DE on DM perturbations can be explored by considering the effective sound speed parameter c eff , where c eff ≃ 0 for clustered DE and c eff ≃ 1 for non-clustered or homogeneous DE. Also, in the case of non-clustered DE, we can simplify the equations by setting δ de = 0. This allows us to determine the evolution of the density contrasts δ dm and δ de as a functions of the scale factor via numerical integration with following appropriate initial conditions [61, 100].</text>
<formula><location><page_7><loc_11><loc_53><loc_43><loc_56></location>δ dm , i = -2 ϕ i ( 1 + k 2 3 H i 2 ) ; δ ' dm , i = -2 3 k 2 H 2 ϕ i</formula>
<formula><location><page_7><loc_11><loc_49><loc_49><loc_55></location>i (47) δ de , i = (1 + w di ) δ dm , i δ ' de , i = (1 + w di ) δ ' dm , i + w ' di δ dm , i (48)</formula>
<text><location><page_7><loc_9><loc_22><loc_49><loc_48></location>Where w di means the value of w de at a i . The choice of k = 0 . 1hMpc -1 ensures that the analysis remains in the linear regime because it falls within the range of scales where the linear approximation is valid. This choice is supported by the assumption that the shape of the power spectrum recovered from galaxy surveys matches the linear matter power spectrum shape for scales k ≤ 0 . 15hMpc -1 . Additionally, it is consistent with the power-spectrum normalization σ 8 , which corresponds to k = 0 . 125hMpc -1 . The specific value chosen for ϕ i , such as ϕ i = -2 × 10 -6 , corresponds to δ dm = 0 . 08 at the present time for k = 0 . 11hMpc -1 . Therefore, the choice of k = 0 . 1hMpc -1 allows for a reliable examination of the growth rate of clustering in the linear regime[11, 101, 102]. In the following section, we will utilize the numerical results derived from solving Eqs. (43 & 44) to examine the growth rate related to DM.</text>
<section_header_level_1><location><page_7><loc_15><loc_18><loc_43><loc_19></location>A. Growth of matter perturbations</section_header_level_1>
<text><location><page_7><loc_9><loc_9><loc_49><loc_16></location>By numerically solving Eqs.(43 & 44), we can determine the theoretical prediction for the quantity fσ 8 . The quantity f represents the linear growth rate of matter perturbations as a function of redshift (z). It quantifies how structures form and evolve, and is defined as</text>
<table>
<location><page_7><loc_53><loc_59><loc_91><loc_91></location>
<caption>TABLE V. The fσ 8 ( z ) data points and their References.</caption>
</table>
<text><location><page_7><loc_52><loc_54><loc_60><loc_55></location>follows[125]:</text>
<formula><location><page_7><loc_61><loc_50><loc_92><loc_52></location>f = d ln δ dm d ln a = -1 + z z d ln δ m d ln z (49)</formula>
<text><location><page_7><loc_52><loc_18><loc_92><loc_49></location>On the other hand, σ 8 ( z ) quantifies the growth of rootmean-square mass fluctuations in spheres with radius 8Mpch -1 [124], and can be calculated in the linear regime as σ 8 ( z ) = σ 8 , 0 δ m ( z ) δ m ( z =0) . Also, σ 8 ( z ) characterizes the level of clustering or fluctuations in the distribution of matter on large scales. Furthermore, we can rescale the parameter σ 8 , 0 as σ 8 , 0 = δ m ( z =0) δ m, Λ ( z =0) σ 8 , Λ to obtain appropriate parameters for evaluating different cosmological models, particularly in the context of IDE models. The f ( z ) σ 8 ( z ) measurement provides insights into the perturbations of the galaxy density, represented as δ g , which is related to the perturbations in DM through the bias factor b , defined as b = δ g /δ m [108]. The independence of f ( z ) σ 8 ( z ) from the bias factor, as shown by Song and Percival[115], is significant because it allows for more reliable and robust discrimination between different IDE models based on this quantity. In conclusion, the validity of various IDE models can be assessed by comparing the theoretical predictions of fσ 8 ( z ) with observational data. This is accomplished by calculating the χ 2 growth statistic, which can be expressed as follows:</text>
<formula><location><page_7><loc_55><loc_12><loc_92><loc_16></location>χ 2 growth = 44 ∑ i =1 [( fσ 8 ) th ( z i ) -( fσ 8 ) obs ( z i )] 2 σ 2 obs ( z i ) (50)</formula>
<text><location><page_7><loc_52><loc_9><loc_92><loc_11></location>where, ( fσ 8 ) th ( z i ) represents the theoretical prediction at the redshift z i , while ( fσ 8 ) obs ( z i ) and σ obs ( z i ) denote</text>
<figure>
<location><page_8><loc_9><loc_35><loc_49><loc_93></location>
<caption>FIG. 1. Upper panel: The ∆ E (%) of the IDE models compared to the ΛCDM model (see Eq. 51). Middle panel: The evolution of the deceleration parameter for various models. Lower panel: The ∆ T (%) of the IDE models compared to its value in the standard ΛCDM model(see Eq. 54), using the best-fit values listed in Tab.VIII for the IDE models. The various IDE models have been specified by different colors and line styles in the inner panels of the figure. The dashed (solid) line represents the homogeneous (clustered) case of DE.</caption>
</figure>
<text><location><page_8><loc_30><loc_35><loc_31><loc_36></location>z</text>
<text><location><page_8><loc_9><loc_9><loc_49><loc_13></location>the measured values and uncertainties, respectively. The dataset used in this study, consisting of 44 measurements of fσ 8 ( z ), is displayed in Table V.</text>
<section_header_level_1><location><page_8><loc_53><loc_92><loc_90><loc_93></location>V. IDE MODELS VERSUS DATA ANALYSIS</section_header_level_1>
<text><location><page_8><loc_52><loc_87><loc_92><loc_90></location>In this section, we will examine the IDE models considered in this study by following a two-step approach.</text>
<text><location><page_8><loc_52><loc_71><loc_92><loc_87></location>Initially, we perform an MCMC analysis to constrain the free parameters of the models based on the latest available background data (see Sec. III and Eq. (14)). Subsequently, we provide a concise overview of our data analysis results pertaining to the IDE models, which can be found in Table VIII. Furthermore, left panels of Fig. 5 illustrates the confidence levels for 1 σ and 2 σ constraints on the IDE models based on the background datasets. These triangular plots are particularly valuable as they visually indicate the correlations between each pair of free parameters in the models.</text>
<text><location><page_8><loc_52><loc_54><loc_92><loc_71></location>The Hubble parameter plays an important role in characterizing the background evolution of the Universe. Moreover, how the Hubble parameter evolves can influence the growth of matter perturbations. Therefore, it is very important to investigate the behavior of the Hubble parameter in the context of IDE models. In light of this, the upper panel of Fig. 1 illustrates the evolution of the percentage deviation of the normalized Hubble parameter E(z) of the models in comparison to the standard ΛCDM model. In other words, it shows the relative deviation of the normalized Hubble parameter of the models from the concordance ΛCDM model, i.e.</text>
<formula><location><page_8><loc_60><loc_50><loc_92><loc_53></location>∆ E (%) = 100 × [ E ( z ) model E ( z ) ΛCDM -1 ] (51)</formula>
<text><location><page_8><loc_52><loc_24><loc_92><loc_48></location>In the top panel of Fig.1, it is obvious that the value of quantity ∆ E (%) associated with IDE-I model, considering the CPL parameterization and a constant value for w de , exhibits negative values in comparison to ΛCDM model for all z. This finding holds for both the scenarios of homogeneous and clustered DE. Moreover, in the case of the IDE-II model assuming the CPL parameterization and a constant value for w de of DE, the ∆ E (%) is positive at z ≲ 1 . 14 and z ≲ 2 . 93, respectively. This is true for both homogeneous and clustered DE. Being positive (negative) value of quantity ∆E(%) relative to the ΛCDM model means that the cosmic expansion in the corresponding IDE model is larger (smaller) compared to the ΛCDM model. Moreover, in the right panel of Fig. 3, we present a comparison between the theoretical evolution of the Hubble parameter, H(z), and a set of 36 cosmic chronometer data points listed in Table IV.</text>
<text><location><page_8><loc_52><loc_20><loc_92><loc_24></location>Here, we explore the deceleration parameter, which can be utilized for evaluating IDE models. This parameter is defined as follows:</text>
<formula><location><page_8><loc_58><loc_16><loc_92><loc_18></location>q ( z ) = -a aH 2 = 1 H ( z ) dH ( z ) dz (1 + z ) -1 (52)</formula>
<text><location><page_8><loc_52><loc_9><loc_92><loc_14></location>By utilizing Equation (52), we can calculate the transition time, denoted as z t , when the Universe undergoes a shift from a decelerated expansion phase ( q > 0) to an accelerated expansion phase ( q < 0). This transition</text>
<figure>
<location><page_9><loc_9><loc_54><loc_49><loc_94></location>
<caption>FIG. 2. Upper panel: Evolution of the linear growth rate of matter perturbations (see Eq. 49) for the various IDE models in terms of the redshift z. Lower panel: The ∆ f (%) of models compared to standard ΛCDM model as a function of redshift z (see Eq. 55). The color lines and styles utilized in the inner panels of the figure resemble those depicted in Fig.1</caption>
</figure>
<text><location><page_9><loc_9><loc_31><loc_49><loc_41></location>time is determined by setting either q = 0 or a = 0. The middle panel of Fig. 1, illustrates the evolution of the deceleration parameter for the IDE models as a function of the redshift z . The values of the transition redshift, z t , pertaining to the IDE-I and IDE-II models, considering the CPL parameterization and a constant value for w de , are as follows:</text>
<table>
<location><page_9><loc_9><loc_22><loc_47><loc_30></location>
</table>
<text><location><page_9><loc_9><loc_9><loc_49><loc_20></location>Moreover, the transition redshift, z t , for ΛCDM model, is z t = 0 . 692. It is evident that during the early times, when matter was the dominant component in the Universe, the quantity q approaches a value of 1 2 . Hence, during the early matter-dominated era, the value of q indicates a decelerating but slowing expansion. These outcomes are consistent with the findings reported in the study by Farooq et al. in [126].</text>
<text><location><page_9><loc_52><loc_86><loc_92><loc_93></location>The age of the Universe can be used as another parameter for assessing and comparing different models of IDE. By utilizing the best-fitting values provided in Table VIII and applying the following equation, we can compute the age of the Universe.</text>
<formula><location><page_9><loc_63><loc_82><loc_92><loc_85></location>t U = 1 H 0 ∫ ∞ 0 dz (1 + z ) E ( z ) (53)</formula>
<text><location><page_9><loc_52><loc_62><loc_92><loc_80></location>which E ( z ) is given by Eq. (6). The age of the Universe, determined by the Eq. (53), yields the following results for the IDE models analyzed in this study. The t U is computed for both homogeneous and clustered DE scenarios. In the case of the homogeneous DE, t U for IDE-I(CPL, w de ) = (13 . 33 , 13 . 43)Gyr and for IDE-II (CPL, w de ) =(13 . 41 , 13 . 53)Gyr. Similarly, in the case of clustered DE, the t U for IDE-I (CPL, w de ) =(13 . 28 , 13 . 48)Gyr and for IDE-II (CPL, w de ) = (13 . 35 , 13 . 57)Gyr. In addition, we indicated that the value of t U for the ΛCDM model is 13 . 642 Gyr. It is worth noting that the age of the Universe, as determined by the Planck (2018) results, is 13 . 78 Gyr [7].</text>
<text><location><page_9><loc_52><loc_56><loc_92><loc_61></location>Additionally, the lower panel in Fig. 1 illustrates the percentage of the relative deviation in the age of the Universe for the IDE models compared to the standard ΛCDM model. This quantity is defined as follows:</text>
<formula><location><page_9><loc_60><loc_51><loc_92><loc_54></location>∆ T (%) = 100 × [ ( t U ) model ( t U ) ΛCDM -1 ] (54)</formula>
<text><location><page_9><loc_52><loc_46><loc_92><loc_50></location>In the lower panel of Fig. 1, we see that the results of our analysis for the IDE models investigated in this study, the values of ∆ T (%), are as follows:</text>
<table>
<location><page_9><loc_52><loc_36><loc_90><loc_44></location>
</table>
<text><location><page_9><loc_52><loc_20><loc_92><loc_34></location>In the subsequent step of our investigation, we focus on the growth of matter perturbations. This involves numerically solving Eqs.(43 & 44) for both homogeneous and clustered cases of DE in the context of IDE models. To constrain the values of σ 8 and other free parameters associated with the IDE models, we perform a combined statistical analysis that incorporates background and growth rate data obtained from RSD (refer to Eq. (15) and Subsec. IVA). The outcomes of this data analysis are presented in Table IX.</text>
<text><location><page_9><loc_52><loc_9><loc_92><loc_20></location>With the obtained best-fit values listed in Table (IX), we examine the evolution of the growth rate of matter perturbations, f ( z ), and the percentage deviation ∆ f (%) compared to the ΛCDM model. The evolution of the linear growth rate of matter perturbations for different models as a function of redshift z is displayed in the upper panel of Fig. 2. The line f = 1 corresponds to the Einstein-de Sitter (EdS) Universe, characterized by</text>
<figure>
<location><page_10><loc_10><loc_73><loc_91><loc_93></location>
<caption>FIG. 3. Left panel: comparison of the observational growth rate data points (see Table V) and theoretical prediction of the growth rate fσ 8 ( z )(see Eq.49) and explanations after it) as a function of redshift z . Right panel: comparison of the 36 cosmic chronometer data points given in Table IV and the theoretical evolution of the Hubble parameter of the IDE models in terms of the redshift z . The color lines and styles utilized in the inner panels of the figure are similar to those depicted in Fig.1.</caption>
</figure>
<text><location><page_10><loc_9><loc_51><loc_49><loc_64></location>Ω dm = 1 and Ω de = 0. It is evident that as redshift increases, the linear growth rates of matter perturbations associated with all models approach and converge towards the constant EdS line. We observe that the IDEII model, incorporating both homogeneous and clustered DE, with the CPL parameterization and a constant value of w de , demonstrates a relatively smaller deviation from the evolution of the linear growth rate of matter perturbations observed in the ΛCDM model.</text>
<text><location><page_10><loc_9><loc_45><loc_49><loc_51></location>Furthermore, we can quantify the difference in the growth rate of the IDE models compared to its value in the ΛCDM model by calculating the percentage relative difference as follows:</text>
<formula><location><page_10><loc_18><loc_41><loc_49><loc_44></location>∆ f (%) = 100 × [ f model f ΛCDM -1 ] (55)</formula>
<text><location><page_10><loc_9><loc_27><loc_49><loc_40></location>The ∆ f (%), as a function of redshift z is illustrated in the lower panel of Fig. 2. These values are calculated using the best-fit parameters provided in Table IX. A positive (negative) ∆ f indicates that the corresponding IDE models shows a higher (lower) linear growth rate of matter perturbations compared to the ΛCDM model. Listed below are the obtained results for ∆ f (%) at the present time for both homogeneous and clustered IDE models:</text>
<table>
<location><page_10><loc_10><loc_17><loc_46><loc_26></location>
</table>
<text><location><page_10><loc_9><loc_9><loc_49><loc_16></location>As illustrated in the lower panel of Fig. 2, the evolution of the ∆ f (%) value is influenced by the clustering or homogeneity of DE, as well as the choice of the parameter for the EoS of DE. In the IDE-I model, when the parameter considered is CPL, the ∆ f (%) value associated with</text>
<text><location><page_10><loc_52><loc_48><loc_92><loc_64></location>the homogeneous DE surpasses the ∆ f (%) value related to the clustered DE at z ≳ 0 . 53. However, if a constant w de is assumed for the EoS, the ∆ f (%)value for the homogeneous DE is smaller compared to the clustered DE. Furthermore, in the IDE-II model, when we consider the CPL parameterization, ∆ f (%) associated with clustered DE exceeds the value associated with homogeneous DE at z ≲ 1 . 40, and the opposite behavior is observed at z ≳ 1 . 40. On the other hand, if we assume a constant value for w de , the ∆ f (%) value for homogeneous DE is larger than that for clustered DE at z ≳ 0 . 3.</text>
<text><location><page_10><loc_52><loc_38><loc_92><loc_48></location>By examining the lower panel of Fig. 2 and the upper panel of Fig. 1, It is observed that when the ∆E of IDE models is positive, there is a corresponding negative ∆ f . This indicates an inverse relation between the evolution of ∆E and ∆ f . In other words, an increase in ∆E leads to a decrease in ∆ f , and vice versa. Also, It is observed that when ∆E reaches its maximum value, ∆ f is minimized.</text>
<text><location><page_10><loc_52><loc_28><loc_92><loc_38></location>Moreover, in the left panel of Fig. 3, a comparison is presented between the observed data points (listed in Table V) and the theoretical prediction of the growth rate of matter perturbations, fσ 8 ( z ), (refer to Eq.(49) and the explanation after it). This analysis includes both homogeneous and clustered DE scenarios within the IDE models.</text>
<section_header_level_1><location><page_10><loc_61><loc_24><loc_82><loc_25></location>A. AIC and BIC Criteria</section_header_level_1>
<text><location><page_10><loc_52><loc_8><loc_92><loc_22></location>When comparing models, the χ 2 min can be used if the models have the same degrees of freedom. In this case, a smaller χ 2 min indicates a better fit to the observational data. If the degrees of freedom are not equal, the reduced chi-square statistic χ 2 red = χ 2 min / ( N -k ) can be used, where k is the number of free parameters in the model and N is the total number of data points. When χ 2 red is around 1, it suggests a good fit to the data. Values significantly smaller or larger than 1 ( χ 2 red ≪ 1 or χ 2 red ≫</text>
<table>
<location><page_11><loc_29><loc_75><loc_72><loc_89></location>
<caption>TABLE VI. The numerical results of model selection were conducted using background data for both Homogeneous (H) and Clustered (C) Dark Energy within the context of IDE models studied in this work.(see Eq. 14).TABLE VII. The numerical outcomes of model selection based on background and growth rate data jointly for both Homogeneous (H) and Clustered (C) Dark Energy in the context of IDE models studied in this work(see Eq. 15).</caption>
</table>
<table>
<location><page_11><loc_28><loc_55><loc_72><loc_69></location>
</table>
<text><location><page_11><loc_9><loc_49><loc_49><loc_52></location>1) indicate that the model is not desirable and should be discarded.</text>
<text><location><page_11><loc_9><loc_23><loc_49><loc_49></location>Additionally, the Akaike Information Criterion(AIC) [127] and Bayesian Information Criterion(BIC)[128] can be used to select the most appropriate model based on its compatibility with the observational data. The AIC and BIC are defined as: AIC = -2 ln L max + 2 k and BIC = -2 ln L max + k ln N , where L max represents the maximum value of the likelihood, which is related to χ 2 min as χ 2 min = -2 ln L max . Both the AIC and BIC consider the number of free parameters (k) and the total number of data points (N). In this case, N specifically refers to 1098 data points for the background data and expands to 1142 when accounting for both the background and growth data jointly (see Sec. III). By calculating the differences between the AIC and BIC of models and a reference model (often chosen as the bestfitting model), we can assess the relative support for each model. The differences ∆AIC and ∆BIC are calculated as follows[129, 130]:</text>
<formula><location><page_11><loc_10><loc_18><loc_49><loc_22></location>∆AIC = AIC model -AIC ΛCDM = ∆ χ 2 min +2∆ k (56) ∆BIC = BIC model -BIC ΛCDM = ∆ χ 2 min +∆ k (ln N)</formula>
<text><location><page_11><loc_9><loc_9><loc_49><loc_17></location>The interpretations of these criteria suggest different levels of support or evidence against a model based on the magnitudes of ∆AIC and ∆BIC. For example, small values of ∆AIC or ∆BIC indicate substantial support or weak evidence against the model, respectively. Larger values of ∆BIC indicate better agreement with the ob-</text>
<text><location><page_11><loc_52><loc_35><loc_92><loc_52></location>l data. In summary, the χ 2 min , AIC, and BIC are used to compare models. The choice of which criterion to use depends on the degrees of freedom and the emphasis placed on goodness of fit versus model complexity. In [129], guidelines are presented to assess model support using ∆AIC and ∆BIC. Substantial support is given when | ∆AIC | is in (0, 2], while considerably less support is in [4, 7]. Models with | ∆AIC | exceeding 10 are considered inappropriate. Similar criteria apply to | ∆BIC | , indicating weak, positive, strong, or very strong evidence against the model. Larger ∆BIC values suggest better consistency with the observational data.</text>
<text><location><page_11><loc_52><loc_26><loc_92><loc_34></location>We present the computed results in Tables (VI & VII). These tables are obtained using the numerical values from Tables (VIII & IX), taking into account the CPL parametrization and a constant w de for the investigated IDE models. Also, the analysis assumes both the homogeneity and clustering of DE.</text>
<text><location><page_11><loc_52><loc_14><loc_92><loc_26></location>According to the analysis of AIC and BIC, it can be inferred that the selection of a model that is more consistent with the observational data (including background and growth rate data) depends on two factors: the homogeneity or clustering of DE and the EoS parameter of DE ( Specifically, in this study, the CPL parametrization and a constant w de ). Moreover, the model selection also depends on the dataset utilized.</text>
<text><location><page_11><loc_52><loc_9><loc_92><loc_14></location>For instance, if we perform AIC analysis and focus solely on the background data, we can deduce that IDEI( w de ) and IDE-II( w de ) models demonstrate a higher level of compatibility with the observational data com-</text>
<table>
<location><page_12><loc_13><loc_68><loc_88><loc_88></location>
<caption>TABLE VIII. The numerical outcomes of parameter fitting with a 1 σ confidence level, are obtained for the IDE models examined in this study for both Homogeneous(H) and Clustered(C) DE. These results are obtained from the combination of a dataset, as defined in Eq. (14), which includes background dataset, BAO+ SnIa+CMB +H.TABLE IX. Numerical results of parameter fitting with 1 σ confidence level in the IDE models investigated in this work assuming Homogeneous (H) and Clustered (C) DE from combining the data set using Eq.(15) based on background and growth rate data, BAO+SnIa+CMB +H+RSD.</caption>
</table>
<table>
<location><page_12><loc_9><loc_42><loc_91><loc_62></location>
</table>
<text><location><page_12><loc_9><loc_34><loc_49><loc_39></location>pared to the other models. This holds for both homogeneous and clustered DE. However, when considering homogeneous DE, there is a slightly better agreement with the observational data(see Table VI).</text>
<text><location><page_12><loc_9><loc_19><loc_49><loc_33></location>Moreover, when conducting AIC analysis and considering both the background and growth rate data simultaneously, it can be inferred that models IDE-I( w de ) and IDEII( w de ) show greater compatibility with the observational data compared to the other models. This finding holds true for both homogeneous and clustered DE. Notably, when specifically examining clustered DE, model IDEI( w de ) demonstrates a slightly better agreement with the observational data in comparison to homogeneous DE (see Table VII).</text>
<text><location><page_12><loc_9><loc_9><loc_49><loc_18></location>In summary, when utilizing the background data, the AIC analysis shows that for homogeneous DE, the models IDE-I(wd), IDE-II(wd), IDE-I(CPL), and IDE-II(CPL) exhibit a better fit with the observational data, respectively. Conversely, when considering clustered DE, the models IDE-II(wd), IDE-I(wd), IDE-I(CPL), and IDEII(CPL) demonstrate better compatibility with the data</text>
<text><location><page_12><loc_52><loc_32><loc_92><loc_39></location>(see Table VI). Also, when using the background and growth data jointly, the assumption of either homogeneous or clustered DE not only affects the fitting of the models to the observational data but also modifies the order in which the models fit the observational data.</text>
<section_header_level_1><location><page_12><loc_63><loc_28><loc_80><loc_29></location>VI. CONCLUSIONS</section_header_level_1>
<text><location><page_12><loc_52><loc_20><loc_92><loc_26></location>In this study, we utilized a two-step approach to investigate two well-known IDE models, considering two distinct cases for the EoS parameter of DE (CPL parameterization and a constant value for w de ).</text>
<text><location><page_12><loc_52><loc_9><loc_92><loc_20></location>Firstly, we performed an MCMC analysis to constrain the free parameters of the models based on the latest available background data (see Sec. III and Eq. (14)). The results of our data analysis pertaining to the IDE models were summarized in Table VIII. Additionally, the left panels of Fig. 5 illustrated the confidence levels for 1 σ and 2 σ constraints on the IDE models based on the background datasets. These triangular plots visually depicted</text>
<figure>
<location><page_13><loc_10><loc_54><loc_91><loc_93></location>
<caption>FIG. 4. The confidence levels of the 1 σ and 2 σ limits for the IDE-I and IDE-II models. The upper panels depict the EoS parameter of DE using the CPL parameterization for both IDE-I (upper left panel) and IDE-II (upper right panel) models. The lower panels focus on the fixed EoS parameter, w d , for both IDE-I (lower left panel) and IDE-II (lower right panel) models. These confidence levels are determined using the background dataset alone for both homogeneous (red) and clustered DE (blue) scenarios. Additionally, the combined background and growth rate dataset is utilized for both homogeneous (cyan) and clustered DE (green) scenarios. For more details, refer to Eqs.(14 & 15), as well as Table IX for numerical values.</caption>
</figure>
<text><location><page_13><loc_9><loc_39><loc_49><loc_42></location>the correlations between each pair of free parameters in the models.</text>
<text><location><page_13><loc_9><loc_31><loc_49><loc_38></location>Following that, we utilized the best-fit values obtained from the data analysis to investigate significant background parameters such as ∆ E , q , and ∆ T . These parameters were examined to compare the models with each other as well as with the ΛCDM model.</text>
<text><location><page_13><loc_9><loc_11><loc_49><loc_30></location>Concerning the Hubble parameter, we concluded that the IDE-I model, with the CPL parameterization and a constant value for w de , exhibited negative values of ∆ E (%) compared to the ΛCDM model for all redshifts. This result is true for both homogeneous and clustered DE scenarios (see top panel of Fig. 1). For the IDE-II model with the same parameterization, positive values of ∆ E (%) were obtained at specific redshift ranges. These findings indicate that the cosmic expansion in the IDE models can be either larger or smaller than the ΛCDM model, depending on the specific model and parameters. Additionally, a comparison between the theoretical evolution of the Hubble parameter and cosmic chronometer data was presented in the right panel of Fig. 3.</text>
<text><location><page_13><loc_10><loc_9><loc_49><loc_10></location>Moreover, the calculated transition time from the de-</text>
<text><location><page_13><loc_52><loc_35><loc_92><loc_42></location>celerated expansion phase ( q > 0) to the accelerated expansion phase ( q < 0) in the studied IDE models was found to be comparable to the transition time obtained in the ΛCDM model. This comparison is presented in the middle panel of Fig. 1.</text>
<text><location><page_13><loc_52><loc_23><loc_92><loc_34></location>In addition to the previously mentioned quantities, we also calculated the age of the Universe within each of the IDE models. Interestingly, we observed that the age of the Universe in the IDE-II ( w de ) models exhibited better comparability to the age of the Universe in the standard ΛCDM model, for both the homogeneous and clustered DE scenarios. The evolution of ∆ T (%) as a function of redshift ( z ) is presented in the lower panel of Fig. 1.</text>
<text><location><page_13><loc_52><loc_11><loc_92><loc_22></location>In the subsequent step of our investigation, we focused on matter perturbation growth in the context of IDE models. We solved the relevant equations numerically for both homogeneous and clustered cases of DE. To constrain the parameters of the IDE models, including σ 8 , we performed a combined statistical analysis using background and growth rate data obtained from RSD. The results are summarized in Table IX.</text>
<text><location><page_13><loc_53><loc_9><loc_92><loc_10></location>Using the best-fit values from Table IX, we analyzed</text>
<figure>
<location><page_14><loc_10><loc_54><loc_90><loc_93></location>
<caption>FIG. 5. The confidence levels for 1 σ and 2 σ constraints on the IDE-I and IDE-2 models using different datasets. The upper left panel shows confidence levels obtained solely from background data, while the upper right panel displays constraints obtained by combining background and growth rate data. These upper panel constraints are presented for both homogeneous and clustered cases of DE, assuming the equation of state (EoS) parameter of DE follows the CPL parameterization. Similarly, the lower panels (left and right) depict constraints when the EoS parameter of DE, w d , is considered constant.</caption>
</figure>
<text><location><page_14><loc_9><loc_32><loc_49><loc_43></location>the evolution of the growth rate of matter perturbations, f ( z ) and its deviation, ∆ f (%) from the ΛCDM model. The lower panel of Fig. 2 displays ∆ f (%) as a function of redshift. Positive (negative) values indicate higher (lower) growth rates compared to the ΛCDM model. The clustering or homogeneity of DE, as well as the choice of the parameter for the EoS of DE, influence the evolution of ∆ f (%).</text>
<text><location><page_14><loc_9><loc_15><loc_49><loc_31></location>In the IDE-I model with the CPL parameterization, we concluded that the ∆ f (%) value for homogeneous DE surpasses the value for clustered DE at z ≳ 0 . 53. However, assuming a constant w de for the EoS, the ∆ f (%) value for homogeneous DE is smaller than that for clustered DE. Also, In the IDE-II model with the CPL parameterization, the ∆ f (%) value related to the clustered DE exceeds the value for homogeneous DE at z ≲ 1 . 40, while the opposite behavior is observed at z ≳ 1 . 40. Assuming a constant w de , the ∆ f (%) value for homogeneous DE is larger than that for clustered DE at z ≳ 0 . 3.</text>
<text><location><page_14><loc_9><loc_9><loc_49><loc_14></location>Following that, a comparison was conducted between the growth rate data (Table V) and the theoretical prediction of the growth rate, fσ 8 ( z ). we observed that the IDE models demonstrated good compatibility with the</text>
<text><location><page_14><loc_52><loc_35><loc_92><loc_43></location>growth rate data (refer to the left panel of Fig. 3). Additionally, in panels of Figs. 4 and 5 , we illustrated the confidence levels representing the 1 σ and 2 σ constraints on the IDE models for both homogeneous and clustered DE. These constraints were determined through an analysis of background and growth rate data.</text>
<text><location><page_14><loc_52><loc_25><loc_92><loc_34></location>Eventually, the analysis of AIC and BIC revealed that the selection of a model consistent with the observational data depended on the homogeneity or clustering of DE and the EoS parameter of DE (specifically, CPL parametrization and a constant w de ). The choice of dataset also influenced the model selection.</text>
<text><location><page_14><loc_52><loc_18><loc_92><loc_25></location>For background data analysis alone, IDE-I( w de ) and IDE-II( w de ) models demonstrated higher compatibility with the observational data, regardless of homogeneity or clustering of DE. However, homogeneous DE showed slightly better agreement with the data.</text>
<text><location><page_14><loc_52><loc_9><loc_92><loc_17></location>Considering both background and growth rate data, IDE-I( w de ) and IDE-II( w de ) models exhibited greater compatibility with the observational data for both homogeneous and clustered DE. However, IDE-I( w de ) showed slightly better agreement with the data in the case of clustered DE.</text>
<text><location><page_15><loc_9><loc_85><loc_49><loc_93></location>In summary, when utilizing the background data, the AIC analysis indicated that for homogeneous DE, the models IDE-I(wd), IDE-II(wd), IDE-I(CPL), and IDEII(CPL) provided a better fit with the observational data, respectively. In adition, for the clustered DE, the models IDE-II(wd), IDE-I(wd), IDE-I(CPL), and IDE-II(CPL)</text>
<text><location><page_15><loc_52><loc_85><loc_92><loc_93></location>demonstrated better compatibility with the data (see Table VI). Also, when using the background and growth data jointly, the assumptions of homogeneity or clustering of DE not only affected the model fitting to the data but also modified the order in which the models fit the observational data (see Table VII).</text>
<unordered_list>
<list_item><location><page_15><loc_10><loc_77><loc_49><loc_79></location>[1] A.G. Riess, A.V. Filippenko, P. Challis, et al. AJ, 116 ,1009 (1998).</list_item>
<list_item><location><page_15><loc_10><loc_74><loc_48><loc_76></location>[2] S. Perlmutter, G. Aldering, G. Goldhaber, et al., ApJ, 517 , 565 (1999).</list_item>
<list_item><location><page_15><loc_10><loc_71><loc_49><loc_74></location>[3] M. Kowalski, D. Rubin, G. Aldering, et al., APJ., 686 , 749 (2008) .</list_item>
<list_item><location><page_15><loc_10><loc_69><loc_49><loc_71></location>[4] N. Jarosik, C.L. Bennett, J. Dunkley, B. Gold, M.R., et al., ApJS, 192 , 14 (2011).</list_item>
<list_item><location><page_15><loc_10><loc_66><loc_49><loc_68></location>[5] E. Komatsu, K.M. Smith, J. Dunkley, et al., ApJS, 192 , 18 (2011).</list_item>
<list_item><location><page_15><loc_10><loc_65><loc_48><loc_66></location>[6] Planck Collaboration XIV., A& A, 594 , A14 (2016) .</list_item>
<list_item><location><page_15><loc_10><loc_62><loc_49><loc_64></location>[7] N. Aghanim, et al., (Planck Collaboration), Astron. Astrophys. 641 , A6 (2020) .</list_item>
<list_item><location><page_15><loc_10><loc_59><loc_49><loc_62></location>[8] B.A. Reid, L. Samushia, M. White, et al., MNRAS, 426 , 2719 (2012).</list_item>
<list_item><location><page_15><loc_10><loc_57><loc_49><loc_59></location>[9] C. Blake, S. Brough, M. Colless, et al., MNRAS, 415 , 2876 (2011b).</list_item>
<list_item><location><page_15><loc_10><loc_54><loc_49><loc_57></location>[10] W.J. Percival, B.A. Reid, D.J. Eisenstein, et al., MNRAS, 401 , 2148 (2010) .</list_item>
<list_item><location><page_15><loc_10><loc_52><loc_49><loc_54></location>[11] M. Tegmark, M. Strauss, M. Blanton, et al., PhRvD, 69 , 103501(2004) .</list_item>
<list_item><location><page_15><loc_10><loc_49><loc_49><loc_51></location>[12] S. Cole, W.J. Percival, J.A. Peacock, et al., MNRAS, 362 , 505 (2005).</list_item>
<list_item><location><page_15><loc_10><loc_48><loc_45><loc_49></location>[13] L. Wang, P.J. Steinhardt, ApJ, 508 , 483 (1998a).</list_item>
<list_item><location><page_15><loc_10><loc_45><loc_49><loc_47></location>[14] S.W. Allen, R.W. Schmidt., H. Ebeling, A.C. Fabian., Mon. Not. Roy. Astron. Soc., 353 , 457(2004).</list_item>
<list_item><location><page_15><loc_10><loc_44><loc_44><loc_45></location>[15] L. Fu, et al., Astron. Astrophys., 479 , 9(2008).</list_item>
<list_item><location><page_15><loc_10><loc_41><loc_49><loc_43></location>[16] L. Amendola, M. Kunz, D. Sapone, JCAP., 0804 , 013 (2008) .</list_item>
<list_item><location><page_15><loc_10><loc_38><loc_49><loc_41></location>[17] J. Benjamin, C. Heymans, E. Semboloni, et al., MNRAS, 381 , 702 (2007).</list_item>
<list_item><location><page_15><loc_10><loc_36><loc_49><loc_38></location>[18] S. Joudaki et al., Mon. Not. Roy. Astron. Soc. 474 , 4894(2018).</list_item>
<list_item><location><page_15><loc_10><loc_33><loc_49><loc_35></location>[19] T. M. C. Abbott et al. (DES), Phys. Rev. D 98 , 043526(2018).</list_item>
<list_item><location><page_15><loc_10><loc_30><loc_49><loc_33></location>[20] S. Basilakos and S. Nesseris, Phys. Rev. D 96 , 063517(2017).</list_item>
<list_item><location><page_15><loc_10><loc_29><loc_39><loc_30></location>[21] E. Abdalla et al., JHEAp 34 , 49 (2022).</list_item>
<list_item><location><page_15><loc_10><loc_26><loc_49><loc_29></location>[22] L. Kazantzidis and L. Perivolaropoulos 10.1007/978-3030 -83715-0 33 (2019).</list_item>
<list_item><location><page_15><loc_10><loc_22><loc_49><loc_26></location>[23] E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri, D. F. Mota, A. G. Riess, and J. Silk, Class. Quant. Grav. 38 , 153001 (2021).</list_item>
<list_item><location><page_15><loc_10><loc_19><loc_49><loc_22></location>[24] N. Schoneberg, G. Franco Abell'an, A. P'erez S'anchez, S. J. Witte, V. Poulin, and J. Lesgourgues, Phys. Rept. 984 , 1 (2022).</list_item>
<list_item><location><page_15><loc_10><loc_16><loc_49><loc_18></location>[25] G. Alestas, L. Kazantzidis, and L. Perivolaropoulos, Phys. Rev. D 101 , 123516 (2020).</list_item>
<list_item><location><page_15><loc_10><loc_13><loc_49><loc_16></location>[26] P. Shah, P. Lemos, and O. Lahav, Astron. Astrophys. Rev. 29 , 9 (2021).</list_item>
<list_item><location><page_15><loc_10><loc_12><loc_38><loc_13></location>[27] S. Capozziello, IJMP D 11 , 483 (2002).</list_item>
<list_item><location><page_15><loc_10><loc_9><loc_49><loc_12></location>[28] S.M. Carroll, V. Duvvuri, M. Trodden, M.S.Turner, PRD 70 , 043528 (2004).</list_item>
<list_item><location><page_15><loc_53><loc_77><loc_92><loc_79></location>[29] A. Nicolis, R. Rattazzi, E. Trincherini, PRD 79 , 064036 (2009).</list_item>
<list_item><location><page_15><loc_53><loc_74><loc_92><loc_76></location>[30] C. Deffayet, G. Esposito-Farese, A. Vikman, PRD 79 , 084003 (2009).</list_item>
<list_item><location><page_15><loc_53><loc_71><loc_92><loc_74></location>[31] G.R. Dvali, G. Gabadadze, M. Porrati, PLB 485 , 208 (2000).</list_item>
<list_item><location><page_15><loc_53><loc_69><loc_92><loc_71></location>[32] S. Nojiri, S.D. Odintsov, M. Sasaki, PRD 71 , 123509 (2005).</list_item>
<list_item><location><page_15><loc_53><loc_67><loc_87><loc_68></location>[33] T. Koivisto, D.F. Mota, PLB, 644 , 104 (2007).</list_item>
<list_item><location><page_15><loc_53><loc_66><loc_80><loc_67></location>[34] J. S. Farnes, A& A 620 , A92 (2018).</list_item>
<list_item><location><page_15><loc_53><loc_63><loc_92><loc_66></location>[35] V.F. Cardone, A. Troisi, S. Capozziello, Phys.Rev. D 69 083517(2004).</list_item>
<list_item><location><page_15><loc_53><loc_62><loc_88><loc_63></location>[36] A. Paliathanasis, Eur. Phys. J. C 83 :756 (2023).</list_item>
<list_item><location><page_15><loc_53><loc_59><loc_92><loc_62></location>[37] S. Ansoldi, and E. I. Guendelman, JCAP, 05 , 036(2013).</list_item>
<list_item><location><page_15><loc_53><loc_56><loc_92><loc_59></location>[38] M. Bousder, Z. Sakhi, M. Bennai, International Journal of Geometric Methods in Modern Physics, 17(13) , 2050183 (2020).</list_item>
<list_item><location><page_15><loc_53><loc_53><loc_92><loc_55></location>[39] Y. Liu, S. Liao, X. Liu, J. Zhang, Z. Fan, 511 , 3076 (2022).</list_item>
<list_item><location><page_15><loc_53><loc_50><loc_92><loc_53></location>[40] Z.-K. Guo and Y.-Z. Zhang, Phys. Rev. D. 71 , 023501(2005).</list_item>
<list_item><location><page_15><loc_53><loc_49><loc_86><loc_50></location>[41] R.-G. Cai and A. Wang, JCAP 0503 , (2002).</list_item>
<list_item><location><page_15><loc_53><loc_46><loc_92><loc_49></location>[42] X.-J. Bi, B. Feng, H. Li, and X. Zhang, Phys. Rev. D. 72 , 123523 (2005).</list_item>
<list_item><location><page_15><loc_53><loc_44><loc_92><loc_46></location>[43] E. G. M. Ferreira, J. Quintin, A. A. Costa, E. Abdalla, and B. Wang, Phys. Rev. D, 95 , 043520 (2017).</list_item>
<list_item><location><page_15><loc_53><loc_42><loc_86><loc_43></location>[44] R. G. Landim, Eur. Phys. J. C 79 , 889 (2019).</list_item>
<list_item><location><page_15><loc_53><loc_40><loc_92><loc_42></location>[45] S. Vagnozzi, L. Visinelli, O. Mena, and D. F. Mota, Mon. Not. Roy. Astron. Soc. 493 , 1139 (2020).</list_item>
<list_item><location><page_15><loc_53><loc_37><loc_92><loc_39></location>[46] A. A. Costa, X.-D. Xu, B. Wang, E. G. M. Ferreira, and E. Abdalla, Phys. Rev. D, 89 , 103531 (2014).</list_item>
<list_item><location><page_15><loc_53><loc_34><loc_92><loc_37></location>[47] E.V. Linder, D. Huterer., Phys. Rev. D 72 , 043509 (2005).</list_item>
<list_item><location><page_15><loc_53><loc_32><loc_92><loc_34></location>[48] S. Nesseris, L. Perivolaropoulos., JCAP , 0701 , 018 (2007).</list_item>
<list_item><location><page_15><loc_53><loc_29><loc_92><loc_31></location>[49] J. Frieman, M. Turner, D. Huterer., Ann. Rev. Astron. Astrophys. 46 , 385, (2088).</list_item>
<list_item><location><page_15><loc_53><loc_28><loc_89><loc_29></location>[50] Tsujikawa S. 2010, book series, ASSL,volume 370.</list_item>
<list_item><location><page_15><loc_53><loc_26><loc_92><loc_27></location>[51] P.H. Frampton, K.J. Ludwick., EPJC 71 , 1735 (2011).</list_item>
<list_item><location><page_15><loc_53><loc_25><loc_89><loc_26></location>[52] D.H. Weinberg., et al., Phys. Rept. 530 , 87 (2013).</list_item>
<list_item><location><page_15><loc_53><loc_22><loc_92><loc_25></location>[53] Q.J. Zhang, Y.L. Wu, Mod. Phys. Lett. A 27 , 1250030 (2012).</list_item>
<list_item><location><page_15><loc_53><loc_20><loc_92><loc_22></location>[54] F. Pace, J.C. Waizmann, M. Bartelmann., MNRAS, 406 , 1865 (2010).</list_item>
<list_item><location><page_15><loc_53><loc_19><loc_91><loc_20></location>[55] J. Garriga, V.F. Mukhanov., PhLB, 458 , 219 (1999).</list_item>
<list_item><location><page_15><loc_53><loc_16><loc_92><loc_18></location>[56] D. Sapone, E. Majerotto., Phys. Rev. D, 85 , 123529 (2012).</list_item>
<list_item><location><page_15><loc_53><loc_13><loc_92><loc_16></location>[57] T. Basse, O.E. Bjaelde, J. Hamann, S. Hannestad., JCAP, 1405 , 021(2014).</list_item>
<list_item><location><page_15><loc_53><loc_12><loc_90><loc_13></location>[58] S. Nesseris, D. Sapone., IJMPD, 24 , 1550045 (2015).</list_item>
<list_item><location><page_15><loc_53><loc_9><loc_92><loc_12></location>[59] D.F. Mota, J.R. Kristiansen, T. Koivisto, N.E. Groeneboom., MNRAS, 382 , 793 (2007).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_16><loc_10><loc_92><loc_46><loc_93></location>[60] J. Dossett, M. Ishak., PhRvD, 88 , 103008 (2013).</list_item>
<list_item><location><page_16><loc_10><loc_91><loc_43><loc_92></location>[61] R. Batista, F. Pace., JCAP, 1306 , 044 (2013).</list_item>
<list_item><location><page_16><loc_10><loc_89><loc_40><loc_90></location>[62] R.C. Batista., PhRvD, 89 , 123508 (2014).</list_item>
<list_item><location><page_16><loc_10><loc_87><loc_49><loc_89></location>[63] G. Ballesteros, A. Riotto., Phys. Lett. B, 668 , 171 (2008).</list_item>
<list_item><location><page_16><loc_10><loc_84><loc_49><loc_86></location>[64] R. J. F. Marcondes, R. C. G. Landim, A. A. Costa, B. Wang, and E. Abdalla, JCAP 1612 , 009 (2016).</list_item>
<list_item><location><page_16><loc_10><loc_81><loc_49><loc_84></location>[65] E. Abdalla, L. R. Abramo, L. Sodre Jr., and B. Wang, Phys. Lett. B, 673 ,107, (2009).</list_item>
<list_item><location><page_16><loc_10><loc_79><loc_49><loc_81></location>[66] E. Abdalla, L. R. Abramo, and J. C. de Souza, Phys. Rev. D, 82 , 023508, (2010).</list_item>
<list_item><location><page_16><loc_10><loc_76><loc_49><loc_78></location>[67] S. Cao, N. Liang, and Z.-H. Zhu, Mon. Not. Roy. Astron. Soc., 416 , 1099, (2011).</list_item>
<list_item><location><page_16><loc_10><loc_73><loc_49><loc_76></location>[68] B. Wang , E. Abdalla, F. Atrio-Barandela, and D. Pavon, Rep. Prog. Phys., 79 , no. 9 096901, (2016).</list_item>
<list_item><location><page_16><loc_10><loc_71><loc_49><loc_73></location>[69] D. Polarski, M.Chevallier., Int. J. Mod. Phys. D 10 , 213 (2001).</list_item>
<list_item><location><page_16><loc_10><loc_69><loc_45><loc_70></location>[70] E.V. Linder., Phys. Rev. Lett. 90 , 91301 (2003).</list_item>
<list_item><location><page_16><loc_10><loc_68><loc_45><loc_69></location>[71] D. Scolnic, et al., Astrophys. J., 859 , 101 (2018).</list_item>
<list_item><location><page_16><loc_10><loc_66><loc_49><loc_68></location>[72] E. Komatsu, J. Dunkley, et al., ApJS, 180 :330-376 (2008).</list_item>
<list_item><location><page_16><loc_10><loc_64><loc_43><loc_65></location>[73] D.J. Eisenstein, W.Hu., ApJ, 496 , 605 (1998).</list_item>
<list_item><location><page_16><loc_10><loc_62><loc_49><loc_64></location>[74] F. Beutler, C. Blake, M. Colless, et al., MNRAS, 416 , 3017 (2011).</list_item>
<list_item><location><page_16><loc_10><loc_59><loc_49><loc_61></location>[75] C. Blake, E. Kazin, F. Beutler, T. Davis, et al., MNRAS, 415 , 2892 (2011).</list_item>
<list_item><location><page_16><loc_10><loc_56><loc_49><loc_59></location>[76] L. Anderson, E. Aubourg, S. Bailey, et al., MNRAS, 427 , 3435 (2012).</list_item>
<list_item><location><page_16><loc_10><loc_55><loc_47><loc_56></location>[77] S. Alam, M. Ata, et al., MNRAS., 470 , 2617 (2017).</list_item>
<list_item><location><page_16><loc_10><loc_52><loc_49><loc_55></location>[78] A.J. Ross, L. Samushia, C.Howlett., MNRAS. 449 , 835 (2015).</list_item>
<list_item><location><page_16><loc_10><loc_51><loc_40><loc_52></location>[79] G. Hinshaw, et al., ApJS, 208 , 19(2013) .</list_item>
<list_item><location><page_16><loc_10><loc_50><loc_42><loc_51></location>[80] D.J. Eisenstein, et al., ApJ, 633 , 560 (2005).</list_item>
<list_item><location><page_16><loc_10><loc_47><loc_49><loc_49></location>[81] J.R. Bond, G. Efstathiou, M. Tegmark., MNRAS, 291 , L33 (1997).</list_item>
<list_item><location><page_16><loc_10><loc_46><loc_42><loc_47></location>[82] W. Hu, N. Sugiyama., ApJ, 471 , 542 (1996).</list_item>
<list_item><location><page_16><loc_10><loc_44><loc_49><loc_45></location>[83] L. Chen, Q.G. Huang, K.Wang., JCAP., 02 , 028 (2019).</list_item>
<list_item><location><page_16><loc_10><loc_43><loc_43><loc_44></location>[84] S. Cao, B. Ratra., MNRAS, 513 , 5686 (2022).</list_item>
<list_item><location><page_16><loc_10><loc_40><loc_49><loc_43></location>[85] R.C. Nunes, A. Bernui., Eur. Phys. J. C, 80 , 1025 (2020).</list_item>
<list_item><location><page_16><loc_10><loc_39><loc_49><loc_40></location>[86] E.G. Adelberger, et al., Rev. Mod. Phys. 83 , 195 (2011).</list_item>
<list_item><location><page_16><loc_10><loc_36><loc_49><loc_39></location>[87] C. Zhang, H. Zhang, S. Yuan, T.J. Zhang, Y.C. Sun, Res. Astron. Astrophys. 14 (10), 1221 (2014).</list_item>
<list_item><location><page_16><loc_10><loc_34><loc_49><loc_36></location>[88] D. Stern, R. Jimenez, L. Verde, M. Kamionkowski, S.A. Stanford, JCAP 02, 008 (2010).</list_item>
<list_item><location><page_16><loc_10><loc_29><loc_49><loc_34></location>[89] M. Moresco, A. Cimatti, R. Jimenez, L. Pozzetti, G. Zamorani, M. Bolzonella, J. Dunlop, F. Lamareille, M. Mignoli, H. Pearce, et al., Journal of Cosmology and Astroparticle Physics 12(08) (2012).</list_item>
<list_item><location><page_16><loc_10><loc_26><loc_49><loc_28></location>[90] C.H. Chuang, Y. Wang, Mon. Not. Roy. Astron. Soc.435, 255 (2013).</list_item>
<list_item><location><page_16><loc_10><loc_22><loc_49><loc_26></location>[91] M. Moresco, L. Pozzetti, A. Cimatti, R. Jimenez, C. Maraston, L. Verde, D. Thomas, A. Citro, R. Tojeiro, D. Wilkinson, JCAP, 1605 , 014 (2016).</list_item>
<list_item><location><page_16><loc_10><loc_19><loc_49><loc_22></location>[92] C. Blake, et al., Mon. Not. Roy. Astron. Soc.425, 405 (2012).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_16><loc_53><loc_91><loc_92><loc_93></location>[93] L. Anderson, et al., Mon. Not. Roy. Astron. Soc.441(1), 24 (2014).</list_item>
<list_item><location><page_16><loc_53><loc_88><loc_92><loc_90></location>[94] M. Moresco, Mon. Not. Roy. Astron. Soc. 450(1), L16 (2015).</list_item>
<list_item><location><page_16><loc_53><loc_87><loc_84><loc_88></location>[95] A. Font-Ribera, et al., JCAP05, 027 (2014).</list_item>
<list_item><location><page_16><loc_53><loc_85><loc_92><loc_86></location>[96] C.P. Ma, E. Bertschinger., Astrophys. J. 455 , 7 (1995).</list_item>
<list_item><location><page_16><loc_53><loc_83><loc_92><loc_85></location>[97] R.de Putter, D. Huterer, and E. Linder, Phys. Rev. D. 81 , 103513 (2010).</list_item>
<list_item><location><page_16><loc_53><loc_81><loc_89><loc_82></location>[98] R. Bean, O. Dore, Phys. Rev., D 69 , 083503 (2004).</list_item>
<list_item><location><page_16><loc_53><loc_79><loc_92><loc_81></location>[99] J. Lima, V.Zanchin, R.H.Brandenberger, MNRAS, 291 , L1 (1997).</list_item>
<list_item><location><page_16><loc_52><loc_76><loc_92><loc_78></location>[100] L. R. Abramo, R. C. Batista, L. Liberato, R. Rosenfeld., PRD, 79 , 023516 (2009).</list_item>
<list_item><location><page_16><loc_52><loc_75><loc_86><loc_76></location>[101] Smith R. E. et al., MNRAS, 341 , 1311( 2003).</list_item>
<list_item><location><page_16><loc_52><loc_73><loc_84><loc_74></location>[102] Percival W. J. et al., ApJ, 657 , 645(2007).</list_item>
<list_item><location><page_16><loc_52><loc_71><loc_92><loc_73></location>[103] L.R. Abramo, R. C. Batista, L. Liberato, R. Rosenfeld, PRevD, 77 , 067301 (2008).</list_item>
<list_item><location><page_16><loc_52><loc_68><loc_92><loc_70></location>[104] D. Huterer, D. Shafer, D. Scolnic, F. Schmidt, JCAP, 1705 , 015 (2017).</list_item>
<list_item><location><page_16><loc_52><loc_67><loc_90><loc_68></location>[105] R. Tojeiro, et al., MNRAS, 424 , 2339, 2344 (2012).</list_item>
<list_item><location><page_16><loc_52><loc_66><loc_89><loc_67></location>[106] M.J. Hudson, S.J. Turnbull, ApJ, 751 , L30 (2013).</list_item>
<list_item><location><page_16><loc_52><loc_63><loc_92><loc_65></location>[107] Florian Beutler, et al. (BOSS), Mon. Not. Roy. Astron. Soc. 466 , 2242, 2260 (2017).</list_item>
<list_item><location><page_16><loc_52><loc_62><loc_91><loc_63></location>[108] Michael J. Wilson, Ph.D. thesis, Edinburgh U. (2016).</list_item>
<list_item><location><page_16><loc_52><loc_59><loc_92><loc_61></location>[109] Hector Gil-Marin, Will J. Percival, et.al. Mon. Not. Roy. Astron. Soc. 465 , 1757, 1788 (2017).</list_item>
<list_item><location><page_16><loc_52><loc_56><loc_92><loc_59></location>[110] Shadab Alam, et al. (BOSS), Mon. Not. Roy. Astron. Soc. 470 , 26172652 (2017).</list_item>
<list_item><location><page_16><loc_52><loc_55><loc_85><loc_56></location>[111] Yuting Wang, Gong-Bo Zhao, et al., (2017).</list_item>
<list_item><location><page_16><loc_52><loc_54><loc_76><loc_55></location>[112] Hector Gil-Marin et al., (2018).</list_item>
<list_item><location><page_16><loc_52><loc_51><loc_92><loc_53></location>[113] M. Feix, A. Nusser, E. Branchini, Phys. Rev. Lett., 115, 011301 (2015)</list_item>
<list_item><location><page_16><loc_52><loc_48><loc_92><loc_51></location>[114] C. Howlett, A. Ross, L.Samushia, W. Percival, M. Manera, Mon. Not. Roy. Astron. Soc., 449 , 848 (2015).</list_item>
<list_item><location><page_16><loc_52><loc_47><loc_89><loc_48></location>[115] Y.S. Song, W.J. Percival, JCAP, 0910 , 004 (2009).</list_item>
<list_item><location><page_16><loc_52><loc_46><loc_85><loc_47></location>[116] C. Blake, et al., MNRAS., 436 , 3089 (2013).</list_item>
<list_item><location><page_16><loc_52><loc_43><loc_92><loc_45></location>[117] L. Samushia, W.J. Percival, A. Raccanelli, MNRAS, 420 , 2102 (2012).</list_item>
<list_item><location><page_16><loc_52><loc_42><loc_88><loc_43></location>[118] A.G. Sanchez, et al., MNRAS., 440 , 2692 (2014).</list_item>
<list_item><location><page_16><loc_52><loc_40><loc_88><loc_41></location>[119] C.H. Chuang, et al., MNRAS., 461 , 3781 (2016).</list_item>
<list_item><location><page_16><loc_52><loc_39><loc_84><loc_40></location>[120] C. Blake, et al., MNRAS., 425 , 405 (2012).</list_item>
<list_item><location><page_16><loc_52><loc_36><loc_92><loc_39></location>[121] A. Pezzotta, et al., Astron. Astrophys., 604 , A33 (2017).</list_item>
<list_item><location><page_16><loc_52><loc_34><loc_92><loc_36></location>[122] T. Okumura, et al., Publ. Astron. Soc. Jap., 68 , 38 (2016).</list_item>
<list_item><location><page_16><loc_52><loc_33><loc_89><loc_34></location>[123] G.B. Zhao, et al., MNRAS 482 , 3497-3513 (2019).</list_item>
<list_item><location><page_16><loc_52><loc_30><loc_92><loc_32></location>[124] S. Nesseris, L. Perivolaropoulos, Phys. Rev. D, 77 , 023504 (2008).</list_item>
<list_item><location><page_16><loc_52><loc_27><loc_92><loc_30></location>[125] S. Nesseris, G. Pantazis, Phys. Rev., D, 96 , 023542 (2017).</list_item>
<list_item><location><page_16><loc_52><loc_25><loc_92><loc_27></location>[126] O. Farooq, F.R.Madiyar, S. Crandall, B. Ratra, ApJ, 835 , 26 (2017).</list_item>
<list_item><location><page_16><loc_52><loc_23><loc_78><loc_24></location>[127] H. Akaike, ITAC, 19 , 716 (1974).</list_item>
<list_item><location><page_16><loc_52><loc_22><loc_78><loc_23></location>[128] G. Schwarz, AnSta, 6 , 461(1978).</list_item>
<list_item><location><page_16><loc_52><loc_21><loc_91><loc_22></location>[129] A.B. Rivera, J.G. Farieta, IJMPD, 28 , 1950118(2019).</list_item>
<list_item><location><page_16><loc_52><loc_19><loc_81><loc_20></location>[130] A.R. Liddle, MNRAS., 377 , L74(2007).</list_item>
</document>
[
{
"title": "Growth of matter perturbations in the Interacting Dark Energy/Dark Matter Scenarios",
"content": "N.Nazari Pooya 1, ∗ 1 Department of Physics, Faculty of Science, Bu-Ali Sina University, Hamedan Iran In this study, we investigate two widely recognized Interacting Dark Energy(IDE) models and assess their compatibility with observational data, focusing on the growth rate of matter perturbations. We explore IDE models with different equations of state (EoS) parameters for Dark Energy (DE), including the CPL parameterization and a constant value for w de . To constrain the parameters of the IDE models using background data, we employ a Markov Chain Monte Carlo (MCMC) analysis. Our results show that both IDE-I and IDE-II models are Compatible with observational data, although with slight variations influenced by the homogeneity or clustering of DE. Following that, we investigate the growth of matter perturbations and perform a comprehensive statistical analysis utilizing both the background and growth rate data. The growth rate in IDE models exhibits deviations compared to the ΛCDM model due to the impact of homogeneity or clustering of DE, as well as the selection of the EoS parameter. However, we find that the IDE models show good compatibility with the growth rate data. Furthermore, we explore how the clustering or homogeneity of DE and the selection of the EoS parameter affect the evolution of the relative difference in the growth rate of IDE models, ∆ f , in comparison to the ΛCDM model. Lastly, we employ the AIC and BIC criteria to evaluate and identify the best model that is compatible with the observational data. The selection of the model depends on the homogeneity or clustering of DE, the EoS parameter, and the dataset used. Overall, the IDE-I and IDE-II models exhibit agreement with the data, with slight deviations depending on specific scenarios and parameters.",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "Observational evidence from various sources strongly supports the notion that the expansion of the Universe is accelerating. This evidence encompasses diverse measurements, including Supernovae type Ia (SnIa) [1-3], Cosmic Microwave Background (CMB) [4-7], Baryon Acoustic Oscillations (BAO) [8-12], high redshift galaxy clusters [13, 14], weak lensing surveys [15-17], and other sources. These diverse observations consistently support the idea of accelerated expansion, shedding light on the evolution of the Universe and the role of Dark Energy (DE). However, despite the strong support for the ΛCDM model provided by these observations, several challenges persist. The nature of DE itself remains a mystery, as its origin and properties are not yet fully understood. The cosmological constant's fine-tuning and the cosmic coincidence problem raise questions about why DE dominates the Universe's energy density at the present epoch. Moreover, tensions related to S 8 [18-20] and H 0 [21-26] further complicate matters, posing both theoretical and observational challenges. Observationally, there are discrepancies in the formation of cosmic structures at smaller scales compared to the predictions of the ΛCDM model. These inconsistencies and tensions necessitate exploring alternative theories and modifications to address these issues and refine our understanding expansion of the Universe. Scientists are actively investigating various approaches, including modified gravity theories [27-33], Unified DE models [34-38], and Interacting DE (IDE) models [39-46], and many other alternative cosmological scenarios, to overcome these theoretical and observational challenges. IDE models, which involve the interaction between DE and Dark Matter(DM), have significant implications for the evolution of the Universe and the behavior of these enigmatic components. Extensive research has been conducted on these models to comprehend their impact on the expansion history of the Universe, the growth of large-scale structures, and observational constraints. Since the exact form of the interaction cannot be deduced from fundamental principles due to the unknown nature of DE and DM, a phenomenological approach is often employed to determine the nature of their interaction. Recent observational data indicates that a direct interaction between DE and DM cannot be ruled out. These models introduce additional parameters to describe the strength of the interaction and its effects on observables related to large-scale structures. The growth of matter perturbations in the Universe can be influenced by DE through various mechanisms, even without considering the interaction between two dark sectors. One such mechanism is the deceleration of the growth rate due to the accelerating expansion of the Universe, resulting in a slower evolution of matter perturbations. Additionally, DE can exhibit perturbations that grow in a similar manner to DM, leading to changes in the distribution and clustering of DM throughout the Universe. These mechanisms demonstrate how DE can impact the growth of matter perturbations in the Universe[11, 54-63, 103]. The growth of DM perturbations in IDE models can be influenced by perturbations in DE, where an exchange of energy between the two dark components impacts their evolutions. To comprehend the growth of structures caused by this interaction, researchers utilize theoretical modeling, simulations, and analysis of observational data. These investigations significantly advance our understanding of the Universe's evolution by uncovering the interaction between DE and DM. This study aims to examine the effects of small DE perturbations on the growth of DM perturbations in IDE models. By studying the growth of DM perturbations in the presence of DE perturbations, valuable insights are gained into the complex interaction between these dark components and their influence on the evolution of cosmic structures. To explore the impact of DE perturbations on the growth of DM perturbations, we consider different interaction terms and Equation of State (EoS) parameters for DE in the conservation equations related to DE and DM. Subsequently, we solve the coupled equations governing the evolution of DE and DM, and finally, we compare the resulting outcomes with observational data. This analysis enables us to measure and evaluate any deviations between the predictions of the standard ΛCDM model and the observational data. The article's structure is outlined as follows: In Sec. II, we derive the necessary equations for the background evolution of the Universe and introduce the IDE models investigated in this study. In Sec. III, we give a brief overview of the current observational datasets at the background level. We then utilize numerical Markov Chain Monte Carlo (MCMC) analysis to constrain the free parameters of the IDE models examined in this research. In Sec. IV, we establish the fundamental equations governing the evolution of DE and DM in the linear regime within the IDE model scenarios. We also investigate the growth rate of matter perturbations in this section. Additionally, we incorporate growth rate data along with background data to obtain more comprehensive constraints on the free parameters of models. In Sec. V, we perform a comparison between the IDE models and the ΛCDM model, using significant cosmological quantities. This comparison is performed using the bestfit values obtained from the likelihood analysis for the free parameters. Finally, in Sec. VI, we present the conclusions derived from our study.",
"pages": [
1,
2
]
},
{
"title": "II. BASIC EQUATIONS IN IDE MODELS: BACKGROUND LEVEL",
"content": "In this study, the background of the Universe is described using the Friedmann-Robertson-Walker (FRW) metric. The FRW metric, which is defined in terms of conformal time, can be expressed as ds 2 = a 2 ( η ) ( -d η 2 + δ ij dx i dx j ) . where a ( η ) is the scale factor. Also, the energy-momentum tensor of DE and DM, denoted by ¯ T µν = ¯ p ¯ g µν (¯ p + ¯ ρ )¯ u µ ¯ v ν . where the bars denote that the quantities are unperturbed. In the context of IDE models, the conservation equations for the energy-momentum of these components can be expressed as follows [64]: where i = de , dm, and ¯ Q i ν is the phenomenological interaction term among the DM and DE. Due to the conservation of total energy-momentum, we can conclude ¯ Q de ν = -¯ Q dm ν . Also, in the case of non-interaction between DE and DM, we have ¯ Q de ν = -¯ Q dm ν = 0. This means that the energy transfer rate from DE to DM or vice versa is zero. Furthermore, because of the homogeneous and isotropic of background, the spatial components of ¯ Q i ν are zero. Consequently, the evolution of the energy density of dark energy ( ρ de ), dark matter ( ρ dm ), baryons ( ρ b ), and radiation ( ρ r ) over time can be determined using the following conservation equations. where dots denote derivative with respect to the conformal time, H is the conformal Hubble parameter( H = aH ), and w de = p de /ρ de is the EoS parameter of DE. Moreover, the evolution of a spatially flat FRW Universe with a homogeneous and isotropic background is governed by the following equation: The solutions of Eqs. (2 & 3) depend on the particular forms of ¯ Q 0 and w de . In this study, two phenomenological interaction terms for ¯ Q 0 are considered: ¯ Q 01 = ξ 1 H ρ de and ¯ Q 02 = ξ 2 H ρ dm . Where ξ 1 and ξ 2 are dimensionless coupling parameter describing the strength of interaction between DE and DM. In recent years, there has been a significant amount of research devoted to models resembling these, in which the interaction term exhibits proportionality to the energy densities ( ρ de , ρ dm ), or a combination of both[65-68]. Moreover, we investigate two distinct cases for the EoS parameter associated with each of the interaction terms. In the first case, we use a well-known parameterization called the Chevallier-Polarski-Linder (CPL) parameterization, defined as w de = w 0 + w 1 (1 -a )[69, 70]. In the second case, we assume that the parameter w de is constant. In the following, we examine these contents in more detail. Interaction Term I: ¯ Q 01 = ξ 1 H ρ de : By employing this interaction term and assuming the CPL parameterization for DE in equations (2 & 3), we can obtain the solutions for these equations as follows: Furthermore, if we assume that w de is a constant, we can determine the solutions for equations (2 & 3) as below: Interaction Term II: ¯ Q 02 = ξ 2 H ρ dm : By assuming this form of the interaction term and utilizing the CPL parameterization for DE in equations (2 & 3), we can derive the solutions for these equations as follows: when w de is constant, the solutions to Eqs.(2 & 3) can be derived as follows. In the following section, we analyze the observational constraints that are imposed on these IDE models at the background level.",
"pages": [
2,
3
]
},
{
"title": "III. DATA ANALYSIS: BACKGROUND LEVEL",
"content": "In this section, we provide an overview of the steps in analyzing observational data. The steps are as follows: where, p = { Ω b h 2 , Ω c h 2 , H 0 , w 0 , w 1 , ξ 1 , ξ 2 } represents the free parameters of the models, and the subscripts H, BAO, SN, and CMB denote the contributions from the Hubble parameter, Baryon Acoustic Oscillations, Supernova type Ia, and Cosmic Microwave Background, respectively. In this analysis, we utilize 1098 observational data points related to the background. These data points consist of 1048 data points from the Pantheon catalog for supernova type Ia (SnIa), 3 data points for the CMB, 11 data points for the BAO, and 36 data points for the H(z). II . Growth Rate Analysis: In the second analysis, the growth rate data is incorporated. The total χ 2 value ( χ 2 tot ) combining the background and growth rate components is given by: where, q = { p , σ 8 , 0 } represents the free parameters of the models at both the background and perturbation levels. The χ 2 growth term represents the contribution from the growth rate data. In this step, 44 growth rate data points are added to the background data points. Additionally, the details of χ 2 growth are explained in Subsec. IV A. III . Statistical Tools: The χ 2 statistic is commonly used to assess the level of agreement between theoretical models and observational data. Therefore, we utilize MCMC analysis, which explores the parameter space to determine uncertainties and correlations among the parameters. These statistical techniques are employed to analyze observational data and constrain model parameters based on their compatibility with the observational data. In the subsequent sections, we present a concise description of the datasets employed in this study.",
"pages": [
3
]
},
{
"title": "A. Type Ia Supernovae(SnIa) data",
"content": "The dataset of Type Ia Supernovae (SnIa) plays a crucial role in studying the dynamic background of the Universe and continues to provide valuable constraints for DE models. The SnIa dataset involves comparing the apparent magnitude with the absolute magnitude of observed SnIa, which is known as the distance modulus and theoretically is given by: Where, d L ( z ) represents the luminosity distance, which is defined in the following manner: In this analysis, we employ the Pantheon SnIa dataset, containing 1048 data points sourced from the Pantheon sample [71]. Additionally, we obtain the respective χ 2 SN using the following relation: Where, the µ th ( p , z i ) refers to the theoretical prediction of the distance modulus at a specific redshift z i . On the other hand, µ obs ( z i ) represents the distance modulus determined through observations and, σ µ,i indicates the uncertainty related to the observational data.",
"pages": [
3,
4
]
},
{
"title": "B. Baryon Acoustic Oscillations(BAO) data",
"content": "Recent investigations have highlighted the significance of BAO as a valuable geometric probe for examining DE. The precise position of the BAO peak in the CMB power spectrum is indeed dependent on the ratio of D V ( z ) to the comoving sound horizon size r s ( z ) at the drag epoch, denoted as z d which represents the epoch when baryons decoupled from photons. In their study, Komatsu et al.[72] noted that the drag epoch, characterized by z d , occurs slightly later than the epoch of photon decoupling, represented by z ∗ . During this epoch, the gravitational potential well affects the behavior of baryons. As a result, the sound horizon size during the drag era is slightly larger compared to the photon decoupling era. Various researchers have reported their measurements of the BAO feature using different observable quantities. Some measurements included constraints on the ratio r s ( z d ) /D V ( z ) or its inverse. The comoving sound horizon r s ( z d ) is given by[72] where c s ( z ) = 1 / [3(1 + 3Ω b 0 4(1+ z )Ω γ 0 )] 1 2 and E ( z ) is given by Eq. (6). We adopt the approximate function for z d as described in [73]. Furthermore, we set Ω γ 0 = 2 . 469 × 10 -5 h -2 (for T cmb =2.725 K) according to [72, 79]. Also, the expression for D V ( z ) is provided in[72] as follows: where D A ( z ) is the angular diameter distance. When the curvature density, Ω K , is zero, we can calculate D A ( z ) by using the following formula [72]: We utilize two datasets, one in the old format presented in Table II and the other in the new format shown in Table III. Since the data points listed in Tables II and III are uncorrelated, we calculate χ 2 bao , 1 for the first case as follows: In this case, the theoretical prediction is expressed as d z ( z ) = r s ( z d ) D V ( z eff ) , where r s ( z d ) represents the comoving sound horizon size at the drag epoch, and D V ( z eff ) denotes the effective volume distance. In the second case, χ 2 bao , 2 is obtained from the following relation. In this case, the theoretical prediction is represented by β ∗ ( z ) = D V ( z eff ) r s ( z d ) r fid s . Consequently, the total χ 2 bao is given by χ 2 bao = χ 2 bao , 1 + χ 2 bao , 2 .",
"pages": [
4
]
},
{
"title": "C. Cosmic Microwave Background(CMB) data",
"content": "The location of the CMB acoustic peak is valuable tool for constraining models of DE as it depends on the angular diameter distance in dynamical DE models. The specific position of this peak in the power spectrum of temperature anisotropy in the CMB is determined by three parameters: l a , R , and Ω b h 2 . Where, l a represents the angular scale of the sound horizon at the decoupling era, which can be calculated using the equation: In this equation, z ∗ refers to the redshift at the decoupling time, and a fitting formula from Hu[82] is used to determine it. The coefficient of (1 + z ∗ ) is included because D A ( z ∗ ) represents the physical angular diameter distance (see Eq. 21), while r s ( z ∗ ) represents the comoving sound horizon at z ∗ (see Eq. 19). The scale distance or shift parameter at the decoupling epoch, denoted as R , is defined as follows[81]: Chen et al. [83] conducted a comparison between the full CMB power spectrum analysis and the distance prior method to constrain different DE models. The results of both methods were found to be completely consistent. Therefore, in this study, we utilize the combined CMB likelihood (Planck 2018 TT, TE, EE + lowE) based on the observed values X obs i = { R,l a , Ω b h 2 } = { 1 . 7493 , 301 . 462 , 0 . 02239 } , as obtained by Chen et al. [83]. The χ 2 cmb is expressed as follows: Where, ∆ X i = { X th i -X obs i } represents the difference between the theoretical value X th i and the observed value X obs i . The inverse of the covariance matrix Σ -1 ij associated with ∆ X i is given by:",
"pages": [
4,
5
]
},
{
"title": "D. Hubble data",
"content": "In our analysis, we utilize 36 data points of H ( z ) from Table (IV), spanning the redshift range 0 . 07 ⩽ z ⩽ 2 . 34. Since the measurements of H ( z ) are uncorrelated, we can express the χ 2 H statistic as follows: Here, H th ( p , z i ) represents the model predictions at the redshift z i , while H obs ( z i ) and σ i denote the measured values and Gaussian errors, respectively, corresponding to the data points listed in Table (IV).",
"pages": [
5
]
},
{
"title": "IV. BASIC EQUATIONS IN IDE MODELS: PERTURBATIONS LEVEL",
"content": "The perturbed Friedmann-Robertson-Walker (FRW) metric is used to describe the spacetime geometry in cosmology, taking into account small perturbations from the homogeneous and isotropic background Universe. In the conformal Newtonian gauge, the metric can be expressed as follows: where a ( η ) is the scale factor depending on conformal time η , and ψ and ϕ are scalar potentials representing gravitational potential and spatial curvature, respectively. The (1 + 2 ψ ) and (1 -2 ϕ ) terms modify the temporal and spatial components of the metric, accounting for small perturbations in the spacetime geometry. This gauge simplifies calculations and is commonly used to study linear perturbations. Furthermore, in the absence of anisotropic stresses, the equations of Einstein's gravity theory require that the metric potentials ϕ and ψ are equal. However, this equality does not generally hold in models of modified gravity. The perturbed conservation equations, taking into account perturbed metrics and perturbed energy-momentum tensors, yield the following evolution equations for the perturbations[64, 96, 97]: where dot denotes the derivative with respect to the conformal time, η , which is related to the physical time, t , through the scale factor, a ( adη = dt ). The variables k i , δ ≡ δρ/ρ , and θ represent the components of the wavevector in Fourier space, density contrast, and divergence of the peculiar velocity, respectively. The parameter w de corresponds to the EoS of DE, taking different values depending on whether the perturbations are associated with dust( w de =0) or DE. And, δQ µ are the perturbations to the exchange of energy -momentum in the perturbed conservation equations. Lastly, the parameter c 2 a represents the squared adiabatic sound speed of the DE perturbations, and its definition is as follows: To investigate perturbations of DE, it is useful to introduce an effective sound speed, c eff , specifically for DE perturbations. This quantity is defined as follows[98]: Additionally, in this context, the Poisson equation can be expressed as follows[99]: where δρ = δρ dm + δρ de and δp = δp dm + δp de . After that, using quantities δp dm = 0, δp de = c 2 eff δρ de , δρ de = ρ de δ de , δρ dm = ρ dm δ dm in Eq. (33), the Poisson equation can be written as: where, Ω dm and Ω de represent the fractional densities of DM and DE respectively. These fractional densities are defined as the ratio of the densities ρ dm and ρ de to the critical density of the Universe. Also, the ρ 0 crit , is defined as ρ 0 crit = 3 H 2 0 / 8 πG . In a matter-dominated Universe with a small DE component, the gravitational potential ϕ can be approximated as a constant in the linear perturbation regime on sub-horizon scales( k 2 ≫ H 2 ). This assumption is confirmed by the fact that most observed structures, which formed during the matter-dominated era, align with this assumption. This simplification allows for easier analysis of the evolution of perturbations and the growth of structures. However, this assumption is only valid under specific conditions and may not hold in other regimes or on larger scales[100]. To obtain second-order coupled differential equations describing the evolution of DE and (29 & 30) as follows: Firstly, by manipulating Eq. (32), we can obtain the following relation: In the regime of sub-horizon scales ( k 2 ≫ H 2 ), we can neglect the second term on the right-hand side of Eq. (35). Secondly, according to Eq.(32), we can express this relation as: Now, by substituting Eqs.(35 & 36) into Eqs.(29 & 30), we can express them in the following form: Morermore, in order to derive Eqs. (37 & 38) (see also [64]), we ignore δ ¯ Q µ . We remind that Eqs.(29 & 30) or their equivalent Eqs.(37 & 38) can be used separately for the components of DE and DM. Based on this, we initially utilize Eqs.(37 & 38) to obtain a second-order equation that describes the evolution of DE perturbations. By eliminating θ from the system of Eqs.(37 & 38), we can derive following equation for δ de in terms of conformal time. where the coefficients ˜ A de , ˜ B de , and ˜ S de are defined as follows: where -k 2 ϕ is expressed by Poisson Eq. (34). Likewise, by utilizing Eqs. (37 & 38), we can derive a second-order equation that describes the evolution of DM perturbations. In this case, we set w d = c 2 eff = c 2 a = 0. The resulting equation is obtained as follows: where, in this case, the coefficients ˜ A dm , ˜ B dm , and ˜ S dm are defined as follows: Additionally, by utilizing the expressions d dη = a H d da and d 2 dη 2 = ( a H 2 + a ˙ H ) d da + a 2 H 2 d 2 da 2 , along with Eq. (34), one can represent Eqs. (39 & 41) in terms of the scale factor. Thus, we obtain the following equations: where, the prime denotes the derivative with respect to the scale factor. The coefficients A de , B de , and S de are defined as follows: In addition, the coefficients A dm , B dm , and S dm can be defined as follows: Where ¯ Q 0 , ¯ Q 0 ρ dm , and ¯ Q 0 ρ de for both models IDE I and IDE II are summarized in Table I. By solving the the coupled Eqs. (43 and 44) numerically from an initial scale factor of a i = 10 -3 to the current time ( a = 1), we can obtain the density contrasts of the δ dm and δ de . The effect of clustered and non-clustered DE on DM perturbations can be explored by considering the effective sound speed parameter c eff , where c eff ≃ 0 for clustered DE and c eff ≃ 1 for non-clustered or homogeneous DE. Also, in the case of non-clustered DE, we can simplify the equations by setting δ de = 0. This allows us to determine the evolution of the density contrasts δ dm and δ de as a functions of the scale factor via numerical integration with following appropriate initial conditions [61, 100]. Where w di means the value of w de at a i . The choice of k = 0 . 1hMpc -1 ensures that the analysis remains in the linear regime because it falls within the range of scales where the linear approximation is valid. This choice is supported by the assumption that the shape of the power spectrum recovered from galaxy surveys matches the linear matter power spectrum shape for scales k ≤ 0 . 15hMpc -1 . Additionally, it is consistent with the power-spectrum normalization σ 8 , which corresponds to k = 0 . 125hMpc -1 . The specific value chosen for ϕ i , such as ϕ i = -2 × 10 -6 , corresponds to δ dm = 0 . 08 at the present time for k = 0 . 11hMpc -1 . Therefore, the choice of k = 0 . 1hMpc -1 allows for a reliable examination of the growth rate of clustering in the linear regime[11, 101, 102]. In the following section, we will utilize the numerical results derived from solving Eqs. (43 & 44) to examine the growth rate related to DM.",
"pages": [
5,
6,
7
]
},
{
"title": "A. Growth of matter perturbations",
"content": "By numerically solving Eqs.(43 & 44), we can determine the theoretical prediction for the quantity fσ 8 . The quantity f represents the linear growth rate of matter perturbations as a function of redshift (z). It quantifies how structures form and evolve, and is defined as follows[125]: On the other hand, σ 8 ( z ) quantifies the growth of rootmean-square mass fluctuations in spheres with radius 8Mpch -1 [124], and can be calculated in the linear regime as σ 8 ( z ) = σ 8 , 0 δ m ( z ) δ m ( z =0) . Also, σ 8 ( z ) characterizes the level of clustering or fluctuations in the distribution of matter on large scales. Furthermore, we can rescale the parameter σ 8 , 0 as σ 8 , 0 = δ m ( z =0) δ m, Λ ( z =0) σ 8 , Λ to obtain appropriate parameters for evaluating different cosmological models, particularly in the context of IDE models. The f ( z ) σ 8 ( z ) measurement provides insights into the perturbations of the galaxy density, represented as δ g , which is related to the perturbations in DM through the bias factor b , defined as b = δ g /δ m [108]. The independence of f ( z ) σ 8 ( z ) from the bias factor, as shown by Song and Percival[115], is significant because it allows for more reliable and robust discrimination between different IDE models based on this quantity. In conclusion, the validity of various IDE models can be assessed by comparing the theoretical predictions of fσ 8 ( z ) with observational data. This is accomplished by calculating the χ 2 growth statistic, which can be expressed as follows: where, ( fσ 8 ) th ( z i ) represents the theoretical prediction at the redshift z i , while ( fσ 8 ) obs ( z i ) and σ obs ( z i ) denote z the measured values and uncertainties, respectively. The dataset used in this study, consisting of 44 measurements of fσ 8 ( z ), is displayed in Table V.",
"pages": [
7,
8
]
},
{
"title": "V. IDE MODELS VERSUS DATA ANALYSIS",
"content": "In this section, we will examine the IDE models considered in this study by following a two-step approach. Initially, we perform an MCMC analysis to constrain the free parameters of the models based on the latest available background data (see Sec. III and Eq. (14)). Subsequently, we provide a concise overview of our data analysis results pertaining to the IDE models, which can be found in Table VIII. Furthermore, left panels of Fig. 5 illustrates the confidence levels for 1 σ and 2 σ constraints on the IDE models based on the background datasets. These triangular plots are particularly valuable as they visually indicate the correlations between each pair of free parameters in the models. The Hubble parameter plays an important role in characterizing the background evolution of the Universe. Moreover, how the Hubble parameter evolves can influence the growth of matter perturbations. Therefore, it is very important to investigate the behavior of the Hubble parameter in the context of IDE models. In light of this, the upper panel of Fig. 1 illustrates the evolution of the percentage deviation of the normalized Hubble parameter E(z) of the models in comparison to the standard ΛCDM model. In other words, it shows the relative deviation of the normalized Hubble parameter of the models from the concordance ΛCDM model, i.e. In the top panel of Fig.1, it is obvious that the value of quantity ∆ E (%) associated with IDE-I model, considering the CPL parameterization and a constant value for w de , exhibits negative values in comparison to ΛCDM model for all z. This finding holds for both the scenarios of homogeneous and clustered DE. Moreover, in the case of the IDE-II model assuming the CPL parameterization and a constant value for w de of DE, the ∆ E (%) is positive at z ≲ 1 . 14 and z ≲ 2 . 93, respectively. This is true for both homogeneous and clustered DE. Being positive (negative) value of quantity ∆E(%) relative to the ΛCDM model means that the cosmic expansion in the corresponding IDE model is larger (smaller) compared to the ΛCDM model. Moreover, in the right panel of Fig. 3, we present a comparison between the theoretical evolution of the Hubble parameter, H(z), and a set of 36 cosmic chronometer data points listed in Table IV. Here, we explore the deceleration parameter, which can be utilized for evaluating IDE models. This parameter is defined as follows: By utilizing Equation (52), we can calculate the transition time, denoted as z t , when the Universe undergoes a shift from a decelerated expansion phase ( q > 0) to an accelerated expansion phase ( q < 0). This transition time is determined by setting either q = 0 or a = 0. The middle panel of Fig. 1, illustrates the evolution of the deceleration parameter for the IDE models as a function of the redshift z . The values of the transition redshift, z t , pertaining to the IDE-I and IDE-II models, considering the CPL parameterization and a constant value for w de , are as follows: Moreover, the transition redshift, z t , for ΛCDM model, is z t = 0 . 692. It is evident that during the early times, when matter was the dominant component in the Universe, the quantity q approaches a value of 1 2 . Hence, during the early matter-dominated era, the value of q indicates a decelerating but slowing expansion. These outcomes are consistent with the findings reported in the study by Farooq et al. in [126]. The age of the Universe can be used as another parameter for assessing and comparing different models of IDE. By utilizing the best-fitting values provided in Table VIII and applying the following equation, we can compute the age of the Universe. which E ( z ) is given by Eq. (6). The age of the Universe, determined by the Eq. (53), yields the following results for the IDE models analyzed in this study. The t U is computed for both homogeneous and clustered DE scenarios. In the case of the homogeneous DE, t U for IDE-I(CPL, w de ) = (13 . 33 , 13 . 43)Gyr and for IDE-II (CPL, w de ) =(13 . 41 , 13 . 53)Gyr. Similarly, in the case of clustered DE, the t U for IDE-I (CPL, w de ) =(13 . 28 , 13 . 48)Gyr and for IDE-II (CPL, w de ) = (13 . 35 , 13 . 57)Gyr. In addition, we indicated that the value of t U for the ΛCDM model is 13 . 642 Gyr. It is worth noting that the age of the Universe, as determined by the Planck (2018) results, is 13 . 78 Gyr [7]. Additionally, the lower panel in Fig. 1 illustrates the percentage of the relative deviation in the age of the Universe for the IDE models compared to the standard ΛCDM model. This quantity is defined as follows: In the lower panel of Fig. 1, we see that the results of our analysis for the IDE models investigated in this study, the values of ∆ T (%), are as follows: In the subsequent step of our investigation, we focus on the growth of matter perturbations. This involves numerically solving Eqs.(43 & 44) for both homogeneous and clustered cases of DE in the context of IDE models. To constrain the values of σ 8 and other free parameters associated with the IDE models, we perform a combined statistical analysis that incorporates background and growth rate data obtained from RSD (refer to Eq. (15) and Subsec. IVA). The outcomes of this data analysis are presented in Table IX. With the obtained best-fit values listed in Table (IX), we examine the evolution of the growth rate of matter perturbations, f ( z ), and the percentage deviation ∆ f (%) compared to the ΛCDM model. The evolution of the linear growth rate of matter perturbations for different models as a function of redshift z is displayed in the upper panel of Fig. 2. The line f = 1 corresponds to the Einstein-de Sitter (EdS) Universe, characterized by Ω dm = 1 and Ω de = 0. It is evident that as redshift increases, the linear growth rates of matter perturbations associated with all models approach and converge towards the constant EdS line. We observe that the IDEII model, incorporating both homogeneous and clustered DE, with the CPL parameterization and a constant value of w de , demonstrates a relatively smaller deviation from the evolution of the linear growth rate of matter perturbations observed in the ΛCDM model. Furthermore, we can quantify the difference in the growth rate of the IDE models compared to its value in the ΛCDM model by calculating the percentage relative difference as follows: The ∆ f (%), as a function of redshift z is illustrated in the lower panel of Fig. 2. These values are calculated using the best-fit parameters provided in Table IX. A positive (negative) ∆ f indicates that the corresponding IDE models shows a higher (lower) linear growth rate of matter perturbations compared to the ΛCDM model. Listed below are the obtained results for ∆ f (%) at the present time for both homogeneous and clustered IDE models: As illustrated in the lower panel of Fig. 2, the evolution of the ∆ f (%) value is influenced by the clustering or homogeneity of DE, as well as the choice of the parameter for the EoS of DE. In the IDE-I model, when the parameter considered is CPL, the ∆ f (%) value associated with the homogeneous DE surpasses the ∆ f (%) value related to the clustered DE at z ≳ 0 . 53. However, if a constant w de is assumed for the EoS, the ∆ f (%)value for the homogeneous DE is smaller compared to the clustered DE. Furthermore, in the IDE-II model, when we consider the CPL parameterization, ∆ f (%) associated with clustered DE exceeds the value associated with homogeneous DE at z ≲ 1 . 40, and the opposite behavior is observed at z ≳ 1 . 40. On the other hand, if we assume a constant value for w de , the ∆ f (%) value for homogeneous DE is larger than that for clustered DE at z ≳ 0 . 3. By examining the lower panel of Fig. 2 and the upper panel of Fig. 1, It is observed that when the ∆E of IDE models is positive, there is a corresponding negative ∆ f . This indicates an inverse relation between the evolution of ∆E and ∆ f . In other words, an increase in ∆E leads to a decrease in ∆ f , and vice versa. Also, It is observed that when ∆E reaches its maximum value, ∆ f is minimized. Moreover, in the left panel of Fig. 3, a comparison is presented between the observed data points (listed in Table V) and the theoretical prediction of the growth rate of matter perturbations, fσ 8 ( z ), (refer to Eq.(49) and the explanation after it). This analysis includes both homogeneous and clustered DE scenarios within the IDE models.",
"pages": [
8,
9,
10
]
},
{
"title": "A. AIC and BIC Criteria",
"content": "When comparing models, the χ 2 min can be used if the models have the same degrees of freedom. In this case, a smaller χ 2 min indicates a better fit to the observational data. If the degrees of freedom are not equal, the reduced chi-square statistic χ 2 red = χ 2 min / ( N -k ) can be used, where k is the number of free parameters in the model and N is the total number of data points. When χ 2 red is around 1, it suggests a good fit to the data. Values significantly smaller or larger than 1 ( χ 2 red ≪ 1 or χ 2 red ≫ 1) indicate that the model is not desirable and should be discarded. Additionally, the Akaike Information Criterion(AIC) [127] and Bayesian Information Criterion(BIC)[128] can be used to select the most appropriate model based on its compatibility with the observational data. The AIC and BIC are defined as: AIC = -2 ln L max + 2 k and BIC = -2 ln L max + k ln N , where L max represents the maximum value of the likelihood, which is related to χ 2 min as χ 2 min = -2 ln L max . Both the AIC and BIC consider the number of free parameters (k) and the total number of data points (N). In this case, N specifically refers to 1098 data points for the background data and expands to 1142 when accounting for both the background and growth data jointly (see Sec. III). By calculating the differences between the AIC and BIC of models and a reference model (often chosen as the bestfitting model), we can assess the relative support for each model. The differences ∆AIC and ∆BIC are calculated as follows[129, 130]: The interpretations of these criteria suggest different levels of support or evidence against a model based on the magnitudes of ∆AIC and ∆BIC. For example, small values of ∆AIC or ∆BIC indicate substantial support or weak evidence against the model, respectively. Larger values of ∆BIC indicate better agreement with the ob- l data. In summary, the χ 2 min , AIC, and BIC are used to compare models. The choice of which criterion to use depends on the degrees of freedom and the emphasis placed on goodness of fit versus model complexity. In [129], guidelines are presented to assess model support using ∆AIC and ∆BIC. Substantial support is given when | ∆AIC | is in (0, 2], while considerably less support is in [4, 7]. Models with | ∆AIC | exceeding 10 are considered inappropriate. Similar criteria apply to | ∆BIC | , indicating weak, positive, strong, or very strong evidence against the model. Larger ∆BIC values suggest better consistency with the observational data. We present the computed results in Tables (VI & VII). These tables are obtained using the numerical values from Tables (VIII & IX), taking into account the CPL parametrization and a constant w de for the investigated IDE models. Also, the analysis assumes both the homogeneity and clustering of DE. According to the analysis of AIC and BIC, it can be inferred that the selection of a model that is more consistent with the observational data (including background and growth rate data) depends on two factors: the homogeneity or clustering of DE and the EoS parameter of DE ( Specifically, in this study, the CPL parametrization and a constant w de ). Moreover, the model selection also depends on the dataset utilized. For instance, if we perform AIC analysis and focus solely on the background data, we can deduce that IDEI( w de ) and IDE-II( w de ) models demonstrate a higher level of compatibility with the observational data com- pared to the other models. This holds for both homogeneous and clustered DE. However, when considering homogeneous DE, there is a slightly better agreement with the observational data(see Table VI). Moreover, when conducting AIC analysis and considering both the background and growth rate data simultaneously, it can be inferred that models IDE-I( w de ) and IDEII( w de ) show greater compatibility with the observational data compared to the other models. This finding holds true for both homogeneous and clustered DE. Notably, when specifically examining clustered DE, model IDEI( w de ) demonstrates a slightly better agreement with the observational data in comparison to homogeneous DE (see Table VII). In summary, when utilizing the background data, the AIC analysis shows that for homogeneous DE, the models IDE-I(wd), IDE-II(wd), IDE-I(CPL), and IDE-II(CPL) exhibit a better fit with the observational data, respectively. Conversely, when considering clustered DE, the models IDE-II(wd), IDE-I(wd), IDE-I(CPL), and IDEII(CPL) demonstrate better compatibility with the data (see Table VI). Also, when using the background and growth data jointly, the assumption of either homogeneous or clustered DE not only affects the fitting of the models to the observational data but also modifies the order in which the models fit the observational data.",
"pages": [
10,
11,
12
]
},
{
"title": "VI. CONCLUSIONS",
"content": "In this study, we utilized a two-step approach to investigate two well-known IDE models, considering two distinct cases for the EoS parameter of DE (CPL parameterization and a constant value for w de ). Firstly, we performed an MCMC analysis to constrain the free parameters of the models based on the latest available background data (see Sec. III and Eq. (14)). The results of our data analysis pertaining to the IDE models were summarized in Table VIII. Additionally, the left panels of Fig. 5 illustrated the confidence levels for 1 σ and 2 σ constraints on the IDE models based on the background datasets. These triangular plots visually depicted the correlations between each pair of free parameters in the models. Following that, we utilized the best-fit values obtained from the data analysis to investigate significant background parameters such as ∆ E , q , and ∆ T . These parameters were examined to compare the models with each other as well as with the ΛCDM model. Concerning the Hubble parameter, we concluded that the IDE-I model, with the CPL parameterization and a constant value for w de , exhibited negative values of ∆ E (%) compared to the ΛCDM model for all redshifts. This result is true for both homogeneous and clustered DE scenarios (see top panel of Fig. 1). For the IDE-II model with the same parameterization, positive values of ∆ E (%) were obtained at specific redshift ranges. These findings indicate that the cosmic expansion in the IDE models can be either larger or smaller than the ΛCDM model, depending on the specific model and parameters. Additionally, a comparison between the theoretical evolution of the Hubble parameter and cosmic chronometer data was presented in the right panel of Fig. 3. Moreover, the calculated transition time from the de- celerated expansion phase ( q > 0) to the accelerated expansion phase ( q < 0) in the studied IDE models was found to be comparable to the transition time obtained in the ΛCDM model. This comparison is presented in the middle panel of Fig. 1. In addition to the previously mentioned quantities, we also calculated the age of the Universe within each of the IDE models. Interestingly, we observed that the age of the Universe in the IDE-II ( w de ) models exhibited better comparability to the age of the Universe in the standard ΛCDM model, for both the homogeneous and clustered DE scenarios. The evolution of ∆ T (%) as a function of redshift ( z ) is presented in the lower panel of Fig. 1. In the subsequent step of our investigation, we focused on matter perturbation growth in the context of IDE models. We solved the relevant equations numerically for both homogeneous and clustered cases of DE. To constrain the parameters of the IDE models, including σ 8 , we performed a combined statistical analysis using background and growth rate data obtained from RSD. The results are summarized in Table IX. Using the best-fit values from Table IX, we analyzed the evolution of the growth rate of matter perturbations, f ( z ) and its deviation, ∆ f (%) from the ΛCDM model. The lower panel of Fig. 2 displays ∆ f (%) as a function of redshift. Positive (negative) values indicate higher (lower) growth rates compared to the ΛCDM model. The clustering or homogeneity of DE, as well as the choice of the parameter for the EoS of DE, influence the evolution of ∆ f (%). In the IDE-I model with the CPL parameterization, we concluded that the ∆ f (%) value for homogeneous DE surpasses the value for clustered DE at z ≳ 0 . 53. However, assuming a constant w de for the EoS, the ∆ f (%) value for homogeneous DE is smaller than that for clustered DE. Also, In the IDE-II model with the CPL parameterization, the ∆ f (%) value related to the clustered DE exceeds the value for homogeneous DE at z ≲ 1 . 40, while the opposite behavior is observed at z ≳ 1 . 40. Assuming a constant w de , the ∆ f (%) value for homogeneous DE is larger than that for clustered DE at z ≳ 0 . 3. Following that, a comparison was conducted between the growth rate data (Table V) and the theoretical prediction of the growth rate, fσ 8 ( z ). we observed that the IDE models demonstrated good compatibility with the growth rate data (refer to the left panel of Fig. 3). Additionally, in panels of Figs. 4 and 5 , we illustrated the confidence levels representing the 1 σ and 2 σ constraints on the IDE models for both homogeneous and clustered DE. These constraints were determined through an analysis of background and growth rate data. Eventually, the analysis of AIC and BIC revealed that the selection of a model consistent with the observational data depended on the homogeneity or clustering of DE and the EoS parameter of DE (specifically, CPL parametrization and a constant w de ). The choice of dataset also influenced the model selection. For background data analysis alone, IDE-I( w de ) and IDE-II( w de ) models demonstrated higher compatibility with the observational data, regardless of homogeneity or clustering of DE. However, homogeneous DE showed slightly better agreement with the data. Considering both background and growth rate data, IDE-I( w de ) and IDE-II( w de ) models exhibited greater compatibility with the observational data for both homogeneous and clustered DE. However, IDE-I( w de ) showed slightly better agreement with the data in the case of clustered DE. In summary, when utilizing the background data, the AIC analysis indicated that for homogeneous DE, the models IDE-I(wd), IDE-II(wd), IDE-I(CPL), and IDEII(CPL) provided a better fit with the observational data, respectively. In adition, for the clustered DE, the models IDE-II(wd), IDE-I(wd), IDE-I(CPL), and IDE-II(CPL) demonstrated better compatibility with the data (see Table VI). Also, when using the background and growth data jointly, the assumptions of homogeneity or clustering of DE not only affected the model fitting to the data but also modified the order in which the models fit the observational data (see Table VII).",
"pages": [
12,
13,
14,
15
]
}
]
2024PhRvD.110d4059K
https://arxiv.org/pdf/2406.15127.pdf
<document>
<section_header_level_1><location><page_1><loc_12><loc_88><loc_90><loc_93></location>Uniqueness of the static vacuum asymptotically flat spacetimes with massive particle spheres</section_header_level_1>
<text><location><page_1><loc_20><loc_82><loc_75><loc_86></location>Kirill Kobialko 1 ∗ , Igor Bogush 2 † , and Dmitri Gal'tsov 1 ‡ 1</text>
<unordered_list>
<list_item><location><page_1><loc_22><loc_81><loc_82><loc_83></location>Faculty of Physics, Moscow State University, 119899, Moscow, Russia</list_item>
</unordered_list>
<text><location><page_1><loc_18><loc_79><loc_85><loc_80></location>Moldova State University, strada Alexei Mateevici 60, 2009, Chi¸sin˘au, Moldova</text>
<text><location><page_1><loc_16><loc_79><loc_17><loc_80></location>2</text>
<section_header_level_1><location><page_1><loc_47><loc_75><loc_56><loc_77></location>Abstract</section_header_level_1>
<text><location><page_1><loc_12><loc_56><loc_91><loc_74></location>In this paper, we establish that a four-dimensional static vacuum asymptotically flat spacetime containing a massive particle sphere is isometric to the Schwarzschild spacetime. Our results expand upon existing uniqueness theorems for static vacuum asymptotically flat spacetimes, which focus on scenarios featuring event horizons or photon spheres. Similarly to the uniqueness theorems concerning photon spheres or event horizons, only a single massive particle sphere is sufficient to obtain a unique solution. However, in contrast to previous theorems, our result leads to the existence of an entire spacetime foliation sliced by a set of massive particle spheres spanning various energies.</text>
<section_header_level_1><location><page_2><loc_12><loc_92><loc_32><loc_93></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_2><loc_12><loc_82><loc_90><loc_89></location>Black hole shadows offer a direct means to observe the optical characteristics of immensely strong gravitational fields. The theoretical comprehension of these shadows is closely intertwined with the photon and massive particle surfaces [1-7].</text>
<text><location><page_2><loc_12><loc_50><loc_91><loc_81></location>Since the inception of general relativity, it has been widely recognized that the spherically symmetric Schwarzschild solution contains a series of circular null orbits, collectively constituting a complete photon sphere due to its inherent symmetry. The profound implications of these surfaces began to crystallize in the late 1990s. The seminal work by Virbhadra and Ellis [8] delineated the correlation between the properties of photon spheres and the intricacies of strong gravitational lensing, leading to the formal characterization of the photon sphere as a timelike hypersurface in spacetime, where the light beam's deflection angle at the closest approach distance tends towards infinity. Subsequently, Claudel, Virbhadra, and Ellis [9] introduced a comprehensive definition of the general photon surface as a timelike surface wherein any null geodesic, touching it tangentially, remains entirely within it, and established a theorem linking this definition to the hypersurface's geometry. The equivalence of these definitions was demonstrated in Ref. [10] for general static spherically symmetric metrics.</text>
<text><location><page_2><loc_12><loc_35><loc_91><loc_49></location>Notably, it was revealed that the intimate connection between photon spheres and strong lensing persists even in the context of naked singularities, thereby suggesting their categorization into weak and strong variants [10, 11]. Recently, in close correlation with photon surfaces, several significant relationships have been unveiled regarding the geometric attributes of relativistic images [12, 13], the compactness of supermassive dark objects at galactic cores [14], and the impact of the cosmological constant on the photon sphere [15].</text>
<text><location><page_2><loc_12><loc_21><loc_91><loc_33></location>An important property of the photon surfaces is established by the theorem asserting that these are timelike totally umbilic hypersurfaces in spacetime [16] exhibiting proportionality of their first and second fundamental forms. This purely geometric approach serves as a constructive definition for analyzing photon surfaces instead of solving geodesic equations and plays a decisive role in the analysis of the black hole uniqueness.</text>
<text><location><page_2><loc_12><loc_8><loc_91><loc_20></location>The first black hole uniqueness theorem was established by Israel in Ref. [17]. It states that the Schwarzschild solution is the only static asymptotically flat vacuum spacetime which has a nonsingular closed simply connected event horizon. Then, similar uniqueness theorem were suggested to generalize the result, e.g., the Kerr solution was shown to be the only rotating vacuum black hole by Robinson in Ref. [18]. The focus on the event horizon in the uniqueness</text>
<text><location><page_3><loc_12><loc_57><loc_91><loc_93></location>proofs was then moved to photon surfaces in work by Cederbaum [19], where the Schwarzschild spacetime was shown to be the only static vacuum asymptotically flat spacetime that possesses a suitably defined photon sphere. Later, the uniqueness theoremes for photon spheres were established for Einstein-scalar [20], Einstein-Maxwell [21-23], Einstein-Maxwell-dilaton [24, 25], Einstein-multiple-scalar [26] models, for wormholes [27, 28], and in higher dimensions [29, 30]. The condition of constancy of the lapse function that was used in first papers was weakened to constancy of lapse function in each spatial slice in Ref. [31]. An alternative perturbative approach to considering the uniqueness of photon spheres was suggested in Ref. [32, 33]. Gibbons and Warnick [34] discovered that photon surfaces may exist in less symmetric spacetimes. This was extended in Ref. [35], where the photon surfaces were demonstrated to be possible if the spacetime admits a nontrivial Killing tensor. This might suggest the potential challenge of formulating a uniqueness theorem solely relying on the umbilicity without constancy of the lapse function. Similar connections in stationary spacetimes was established in the Refs. [36, 37].</text>
<text><location><page_3><loc_12><loc_31><loc_91><loc_56></location>The framework of photon spheres has been extended to encompass massive particle surfaces , which share analogous properties for timelike geodesics associated with massive particles interacting with black holes or other ultra-compact gravitating objects [38-40]. Although the flow of massive particles is not directly observable from distant points (except perhaps for neutrinos, whose detection remains a considerable challenge), these surfaces can be indirectly observed through their proper radiation, which may become visible under certain conditions. Moreover, the significance of massive particle surfaces lies in their relevance to photons traversing through plasma environments that may surround black holes [6]. In environments with inhomogeneous plasma, besides the gravitational deflection of light, electromagnetic refraction also plays a role [41-44], which can be integrated into a unified lensing theory.</text>
<text><location><page_3><loc_12><loc_15><loc_91><loc_30></location>In this paper, we adapt proofs from Refs. [20, 21, 24, 26] suggested by Yazadjiev et. al. Within the framework of the same assumptions (there is only one connected photon sphere and the lapse function regularly foliates spacetime), we prove the uniqueness theorem for massive particle spheres in static vacuum asymptotically flat spacetimes. In contrast to previous uniqueness theorems, our result leads to the existence of an entire spacetime foliation sliced by a set of massive particle spheres spanning various energies.</text>
<section_header_level_1><location><page_4><loc_12><loc_92><loc_46><loc_93></location>II. MASSIVE PARTICLE SPHERE</section_header_level_1>
<text><location><page_4><loc_12><loc_71><loc_91><loc_89></location>We start with considering a four-dimensional static vacuum asymptotically flat spacetime M with given ADM mass M > 0 and Levi-Civita connection ∇ α . In a static spacetime, there is a timelike Killing vector field k α = αm α , where α > 0 is a lapse function and m α is a future-directed timelike unit vector ( k α k α = -α 2 ). One can define spatial slices Σ orthogonal to k α . The induced metric on Σ is ¯ g αβ = g αβ + m α m β which defines the corresponding LeviCivita connection ¯ ∇ α . Here and further bars will denote quantities associated with the slice Σ. Vacuum Einstein equations R αβ = 0 after dimensional reduction along k α read [19, 20]</text>
<formula><location><page_4><loc_37><loc_66><loc_91><loc_69></location>¯ R αβ = α -1 ¯ ∇ α ¯ ∇ β α, ¯ ∇ α ¯ ∇ α α = 0 . (1)</formula>
<text><location><page_4><loc_12><loc_59><loc_90><loc_65></location>Since we are interested in asymptotically flat spacetimes, the lapse function and the metric have the following asymptotics for r →∞ [19, 20]:</text>
<formula><location><page_4><loc_32><loc_54><loc_91><loc_59></location>α = 1 -M r + O ( r -2 ) , g αβ = η αβ + O ( r -1 ) , (2)</formula>
<text><location><page_4><loc_12><loc_47><loc_91><loc_54></location>where r is a suitable radial coordinate and η αβ is a flat Minkowski four-dimensional metric. For vacuum spacetime with a time-like Killing vector k α , there is an alternative in determination of the ADM mass of the solution through the Komar integral [45]</text>
<formula><location><page_4><loc_32><loc_41><loc_91><loc_45></location>M = -1 8 π ∫ ¯ S ∇ α k β dS αβ , dS αβ = n [ α m β ] d ¯ S, (3)</formula>
<text><location><page_4><loc_12><loc_33><loc_90><loc_41></location>where ¯ S is an arbitrary closed two-dimensional surface in Σ with an outer normal vector n α (vector n α lies in the tangent space of slice Σ) and d ¯ S is a volume form associated with the induced metric on ¯ S .</text>
<text><location><page_4><loc_12><loc_28><loc_90><loc_32></location>The definition of a massive particle surface for static vacuum spacetimes can be formulated as follows [38]:</text>
<text><location><page_4><loc_12><loc_16><loc_91><loc_26></location>Definition 2.1 : A massive particle surface is a timelike hypersurface S of M such that, for every point p ∈ S and every vector v α | p ∈ T p S such that v α k α | p = -E and v α v α | p = -m 2 , there exists a geodesic γ of M for a particle with mass m and energy E such that ˙ γ α (0) = v α | p and γ ⊂ S .</text>
<text><location><page_4><loc_12><loc_8><loc_91><loc_15></location>We omit the charge from the definition in Ref. [38] since the solution is vacuum and there is no electromagnetic force acting on particles. Here, we focus on static massive particle surface , which are additionally tangent to the timelike Killing vector k α .</text>
<text><location><page_5><loc_12><loc_84><loc_91><loc_93></location>In other words, the definition states that any geodesic of a particle with mass m and energy E initially tangent to the corresponding massive particle surface S will remain tangent to S . For static massive particle surface S with normal n α the first and second fundamental forms read as [38]</text>
<formula><location><page_5><loc_31><loc_78><loc_91><loc_83></location>h αβ = g αβ -n α n β , χ αβ = H ( h αβ + m 2 E 2 k α k β ) , (4)</formula>
<text><location><page_5><loc_12><loc_71><loc_90><loc_77></location>where H is some scalar function on S and D α is a Levi-Civita connection in S . Since the Killing vector field k α is tangent to the hypersurface S everywhere ( k α n α = 0), the following Lie derivatives are equal to zero (see App. A):</text>
<formula><location><page_5><loc_30><loc_65><loc_91><loc_68></location>L k n α = 0 , L k h αβ = 0 , L k χ αβ = 0 , L k H = 0 . (5)</formula>
<text><location><page_5><loc_12><loc_60><loc_90><loc_64></location>If ¯ S is a spatial section of a surface S sliced by Σ, from general geometric considerations we have [46]:</text>
<formula><location><page_5><loc_29><loc_55><loc_91><loc_58></location>χ αβ = ¯ χ αβ -m α m β · n α ∇ α ln α, h αβ = ¯ h αβ -m α m β . (6)</formula>
<text><location><page_5><loc_12><loc_52><loc_38><loc_53></location>Comparing (4) and (6), we find</text>
<formula><location><page_5><loc_27><loc_46><loc_91><loc_50></location>¯ χ αβ = H ¯ h αβ + α -2 k α k β ( n α ∇ α ln α -H ( 1 -α 2 m 2 E 2 )) , (7)</formula>
<text><location><page_5><loc_12><loc_41><loc_90><loc_45></location>and since ¯ χ αβ is tangent to the spatial section and k α is orthogonal to it, we find the following expressions:</text>
<formula><location><page_5><loc_32><loc_35><loc_91><loc_40></location>¯ χ αβ = H ¯ h αβ , n α ∇ α ln α = H ( 1 -α 2 m 2 E 2 ) . (8)</formula>
<text><location><page_5><loc_12><loc_25><loc_91><loc_35></location>The spatial section of static massive particle hypersurface is a totally umbilical surface with a spatial mean curvature H = 1 2 ¯ χ α α . However, unlike the photon sphere [9], the principal curvature in the time direction is different from the spatial ones. In what follows we also assume that the spatial section is connected, compact and closed.</text>
<text><location><page_5><loc_12><loc_20><loc_91><loc_24></location>Equations (5), (8) and Refs. [19-22, 24, 26, 28] inspire us to introduce two important definitions - a massive particle sphere and a non-extremal massive particle surface.</text>
<text><location><page_5><loc_12><loc_13><loc_91><loc_18></location>Definition 2.2 : The massive particle surface S is a massive particle sphere if and only if D α α = 0 on S .</text>
<text><location><page_5><loc_12><loc_6><loc_91><loc_11></location>Definition 2.3 : The massive particle surface S is a non-extremal massive particle surface if and only if m 2 α 2 / E 2 < 1 on S .</text>
<text><location><page_6><loc_12><loc_85><loc_91><loc_93></location>Further we will be interested in non-extreme massive particle spheres only. The latter has a number of important geometric properties. First, it has a constant spatial mean curvature, i.e. D α H = 0. Indeed, consider the Codazzi equation [16] in spacetime</text>
<formula><location><page_6><loc_25><loc_73><loc_91><loc_84></location>0 = n ρ h σ α R ρσ = D β χ α β -D α χ β β = (9) = m 2 E 2 k α L k H + Hm 2 E 2 D β ( k α k β ) + D α H -( 3 -m 2 α 2 E 2 ) D α H = -( 2 -m 2 α 2 E 2 ) D α H.</formula>
<text><location><page_6><loc_12><loc_70><loc_91><loc_72></location>Here, we used Eq. (4) to rewrite the second fundamental form χ αβ . Then, we used Eq. (5) and</text>
<formula><location><page_6><loc_36><loc_64><loc_91><loc_69></location>D β ( k α k β ) = 1 2 D α α 2 + k α · D β k β = 0 , (10)</formula>
<text><location><page_6><loc_12><loc_56><loc_91><loc_64></location>to get rid of the first two terms in the second line. Also, we used the condition D α α = 0 and Killing equations D ( α k β ) = 0. Since we consider a non-extremal sphere m 2 α 2 / E 2 < 1, the term in brackets (2 -m 2 α 2 / E 2 ) is non-zero, and we get D α H = 0.</text>
<text><location><page_6><loc_12><loc_49><loc_91><loc_56></location>Since we have proven that H is constant at the sphere, this allows us to obtain useful geometric identities. First, consider the Komar integral (3) over the spatial sections ¯ S of a massive particle sphere:</text>
<formula><location><page_6><loc_29><loc_39><loc_91><loc_47></location>M = -1 8 π ∫ ¯ S ∇ α k β dS αβ = -1 8 π ∫ ¯ S ∇ α k β n [ α m β ] d ¯ S = 1 4 π ∫ ¯ S n α ∇ α αd ¯ S = 1 4 π ∫ ¯ S αH ( 1 -α 2 m 2 E 2 ) d ¯ S. (11)</formula>
<text><location><page_6><loc_12><loc_34><loc_90><loc_38></location>Since the integrand expression is constant, there is an algebraic relation between the mass M and the spatial section area of the massive particle sphere A S :</text>
<formula><location><page_6><loc_32><loc_28><loc_91><loc_33></location>4 πM = αH ( 1 -α 2 m 2 E 2 ) A S , A S = ∫ ¯ S d ¯ S. (12)</formula>
<text><location><page_6><loc_12><loc_19><loc_90><loc_28></location>Particularly, sphere ¯ S has a positive constant mean curvature H > 0 if a physical assumption of positive mass M is taken into consideration. By virtue of Eq. (8), this means that on the sphere n α ∇ α α > 0, i.e., the norm of spatial gradient ¯ ∇ α α does not vanish anywhere on ¯ S .</text>
<text><location><page_6><loc_12><loc_7><loc_91><loc_20></location>Consider an outer space region Σ ext outside the massive particle sphere ¯ S or equivalently a spacetime region M ext outside S . In this case, the massive particle sphere is an inner boundary ∂ M ext . Similarly to Ref. [20], we introduce an additional assumption that α = const regularly foliate the manifold M ext . It is worth noting that the condition for the existence of a regular foliation is technical and, in principle, open to relaxation, as discussed in Ref. [22].</text>
<text><location><page_7><loc_12><loc_76><loc_90><loc_93></location>By definition, the function α is constant at the massive particle sphere. As we will show, the massive particle sphere ¯ S has a topology of a sphere. Given the regularity of the foliation, any slice in the outer region Σ ext is a topological sphere as well. Equations of motion (1) necessitate α to be a harmonic function, while the boundary conditions at asymptotics dictate that α must approach 1 as it tends to infinity. Following the maximum principle for the harmonic functions, α monotonically increases to 1 moving from the sphere ¯ S to infinity along the flow of slices, i.e., 0 < α < 1.</text>
<text><location><page_7><loc_12><loc_67><loc_90><loc_75></location>The second key identity can be obtained from the Gauss-Bonnet theorem. The trace of the Gauss equations gives an expression for the scalar curvature ¯ R of the spatial section ¯ S the following (see Eq. (C13) in Ref. [46], keeping in mind that R αβ = 0, D α α = 0 and (8))</text>
<text><location><page_7><loc_12><loc_53><loc_90><loc_63></location>Therefore, the non-extremal ( m 2 α 2 / E 2 < 1) sphere ¯ S has a constant and positive scalar curvature ¯ R > 0, representing a round sphere [20]. Then, integrating (13) over ¯ S and applying Gauss-Bonnet theorem ∫ ¯ S ¯ R d ¯ S = 8 π , we find the second useful identity:</text>
<formula><location><page_7><loc_27><loc_62><loc_91><loc_67></location>¯ R = ¯ χ 2 -¯ χ αβ ¯ χ αβ +2¯ χ β β n α ∇ α ln α = 4 ( 3 2 -α 2 m 2 E 2 ) H 2 . (13)</formula>
<formula><location><page_7><loc_40><loc_50><loc_91><loc_54></location>2 π = ( 3 2 -α 2 m 2 E 2 ) H 2 A S . (14)</formula>
<text><location><page_7><loc_12><loc_46><loc_90><loc_50></location>Dividing the equation (14) by (12), the following algebraic connection between the mean curvature H and the lapse function α on S can be found:</text>
<formula><location><page_7><loc_41><loc_40><loc_91><loc_45></location>H = α M · 1 -α 2 m 2 / E 2 3 -2 α 2 m 2 / E 2 . (15)</formula>
<section_header_level_1><location><page_7><loc_12><loc_37><loc_42><loc_38></location>III. UNIQUENESS THEOREM</section_header_level_1>
<text><location><page_7><loc_12><loc_29><loc_90><loc_34></location>Having completed all the preparations, we are ready to formulate and prove the main result of this article.</text>
<text><location><page_7><loc_12><loc_12><loc_91><loc_28></location>Theorem 3.1 : Let M ext be a four-dimensional static and asymptotically flat spacetime with given ADM mass M > 0 , satisfying the vacuum Einstein equations R αβ = 0 and possessing a non-extremal massive particle sphere as an inner boundary of M ext . Assume that the lapse function α regularly foliates M ext . Then, M ext is an isometric to the Schwarzschild spacetime with mass M , and the area radius r S = √ A S / 4 π of the massive particle sphere satisfies the equation E 2 /m 2 = ( r S -2 M ) 2 / ( r 2 S -3 Mr S ) .</text>
<text><location><page_7><loc_12><loc_7><loc_91><loc_11></location>Proof: The proof is based on a modification of the proof presented in Refs. [20, 21, 24, 26] for the case of photon spheres. The main problem is to prove the spherical symmetry of the</text>
<text><location><page_8><loc_12><loc_89><loc_90><loc_93></location>spacetime M ext . First, let us perform a Weyl transformation ˜ g αβ = α 2 ¯ g αβ . In this case, Eq. (1) turns into</text>
<formula><location><page_8><loc_33><loc_85><loc_91><loc_87></location>˜ R αβ = 2 ˜ ∇ α ln α · ˜ ∇ β ln α, ˜ ∇ α ˜ ∇ α ln α = 0 , (16)</formula>
<text><location><page_8><loc_12><loc_77><loc_91><loc_84></location>where ˜ ∇ and ˜ R αβ are the Levi-Civita connection and the Ricci tensor for ˜ g αβ . Our goal is to show that metric ˜ g αβ is conformally flat. For this purpose, one can use the Cotton tensor [47] over a 3-dimensional Riemannian manifold which is defined by</text>
<formula><location><page_8><loc_39><loc_72><loc_91><loc_76></location>˜ R αβγ = ˜ ∇ [ α ˜ R β ] γ -1 4 ˜ ∇ [ α ˜ R ˜ g β ] γ . (17)</formula>
<text><location><page_8><loc_12><loc_70><loc_64><loc_71></location>Using Eq. (16), the following divergences can be obtained [20]:</text>
<formula><location><page_8><loc_41><loc_64><loc_91><loc_69></location>˜ ∇ α ( Ω -1 ˜ ∇ α ω ) = 1 16 ω -7 Ω 3 ˜ R αβγ ˜ R αβγ , (18a)</formula>
<formula><location><page_8><loc_29><loc_60><loc_91><loc_65></location>˜ ∇ α ( Ω -1 ( U ˜ ∇ α ω -ω ˜ ∇ α U )) = 1 16 Hω -7 Ω 3 ˜ R αβγ ˜ R αβγ , (18b)</formula>
<formula><location><page_8><loc_29><loc_53><loc_91><loc_58></location>ω = ( ˜ ∇ α U ˜ ∇ α U ) 1 / 4 , U = 1 -α 2 1 + α 2 , Ω = 4 α 2 (1 + α 2 ) 2 . (19)</formula>
<text><location><page_8><loc_12><loc_59><loc_17><loc_60></location>where</text>
<text><location><page_8><loc_12><loc_49><loc_90><loc_53></location>Since we have shown that 0 < α < 1, we also have 0 < U < 1. Which after integration over the entire spatial slice Σ lead to the following two inequalities (equality if and only if ˜ R αβγ = 0)</text>
<formula><location><page_8><loc_24><loc_43><loc_91><loc_48></location>∫ Σ ˜ ∇ α ( Ω -1 ˜ ∇ α ω ) d ˜ Σ ≥ ∫ Σ ˜ ∇ α ( Ω -1 ( U ˜ ∇ α ω -ω ˜ ∇ α U )) d ˜ Σ ≥ 0 , (20)</formula>
<text><location><page_8><loc_12><loc_39><loc_90><loc_44></location>where d ˜ Σ volume form associated with metric ˜ g αβ . Let us now apply Stokes' theorem to them using massive particle sphere ¯ S and asymptotic sphere ¯ S ∞ as boundary surfaces:</text>
<formula><location><page_8><loc_23><loc_33><loc_91><loc_38></location>∫ ¯ S ∞ -¯ S α Ω -1 n α ¯ ∇ α ωd ¯ S ≥ ∫ ¯ S ∞ -¯ S α Ω -1 ( Un α ¯ ∇ α ω -ωn α ¯ ∇ α U ) d ¯ S ≥ 0 , (21)</formula>
<text><location><page_8><loc_12><loc_26><loc_91><loc_34></location>where we used d ˜ S = α 2 d ¯ S and ˜ n α = α -1 n α , and -¯ S means that the orientation of the normal to the inner boundary is opposite to the foliation. Given asymptotics (2), each surface term reads (see App. B for some details)</text>
<formula><location><page_8><loc_13><loc_21><loc_91><loc_26></location>∫ ¯ S ∞ α Ω -1 n α ¯ ∇ α ωd ¯ S = -4 π √ M, (22a)</formula>
<formula><location><page_8><loc_13><loc_11><loc_91><loc_17></location>∫ -¯ S α Ω -1 n α ¯ ∇ α ωd ¯ S = A S 2 √ H 3 α ( 1 -α 2 m 2 E 2 )( ( 1 + 3 α 2 ) -2 α 4 m 2 E 2 ) , (22c)</formula>
<formula><location><page_8><loc_13><loc_16><loc_91><loc_21></location>∫ ¯ S ∞ α Ω -1 ( Un α ¯ ∇ α ω -ωn α ¯ ∇ α U ) d ¯ S = 0 , (22b)</formula>
<formula><location><page_8><loc_13><loc_6><loc_91><loc_12></location>∫ -¯ S α Ω -1 ( Un α ¯ ∇ α ω -ωn α ¯ ∇ α U ) d ¯ S = A S 2 √ H 3 α ( 1 -α 2 m 2 E 2 )( ( 1 -3 α 2 ) + 2 α 4 m 2 E 2 ) . (22d)</formula>
<text><location><page_9><loc_12><loc_92><loc_59><loc_93></location>Then, the right inequality in (21) immediately results in</text>
<formula><location><page_9><loc_42><loc_86><loc_91><loc_91></location>3 α 2 -1 -2 α 4 m 2 E 2 ≤ 0 . (23)</formula>
<text><location><page_9><loc_12><loc_84><loc_38><loc_86></location>The left inequality in (21) gives</text>
<formula><location><page_9><loc_29><loc_77><loc_91><loc_83></location>4 π √ M -A S √ α 3 H 3 ( 1 -α 2 m 2 E 2 )( 3 -2 α 2 m 2 E 2 ) ≤ 0 , (24)</formula>
<text><location><page_9><loc_12><loc_75><loc_79><loc_77></location>which can be transformed using Eqs. (14) and (15) into the following expression</text>
<formula><location><page_9><loc_42><loc_69><loc_91><loc_74></location>3 α 2 -1 -2 α 4 m 2 E 2 ≥ 0 , (25)</formula>
<text><location><page_9><loc_12><loc_65><loc_91><loc_69></location>where we took into account non-extrimality m 2 α 2 / E 2 < 1. The inequalities (23) and (25) are compatible if and only if they degenerate into equalities</text>
<formula><location><page_9><loc_42><loc_59><loc_91><loc_64></location>3 α 2 -1 -2 α 4 m 2 E 2 = 0 . (26)</formula>
<text><location><page_9><loc_12><loc_49><loc_91><loc_59></location>On the other hand, inequalities can degenerate into equalities if and only if the Cotton tensor vanishes ˜ R αβγ = 0. For a three-dimensional Riemannian manifold, this is a necessary and sufficient condition for the metric ˜ g αβ to be conformally flat [47]. Hence, the metric ¯ g αβ is also conformally flat and ¯ R αβγ = 0. In particular, we have the identity [20]</text>
<formula><location><page_9><loc_13><loc_43><loc_91><loc_48></location>0 = ¯ R αβγ ¯ R αβγ = 8 α 4 ϕ 4 (( α ¯ χ αβ -α ¯ χ 2 · α ¯ h αβ )( α ¯ χ αβ -α ¯ χ 2 · α ¯ h αβ ) + 1 2 ϕ 2 α ¯ h αβ ∂ α ϕ∂ β ϕ ) , (27)</formula>
<text><location><page_9><loc_12><loc_33><loc_91><loc_43></location>where α ¯ χ αβ , α ¯ h αβ and ϕ -1 = n α ∇ α α are the induced metric, the second fundamental forms and the lapse function of slices α = const respectively, and the trace is denoted as α ¯ χ = α ¯ χ α α . Since the induced metric possesses the Euclidean signature, each square in the brackets is equal to zero, yielding the following expressions:</text>
<formula><location><page_9><loc_39><loc_28><loc_91><loc_32></location>α ¯ χ αβ = α ¯ χ 2 · α ¯ h αβ , ¯ D α ϕ = 0 . (28)</formula>
<text><location><page_9><loc_12><loc_12><loc_91><loc_27></location>Thus, all slices are totally umbilic and the lapse function is constant on them. As in the case of photon spheres [20], this implies that all slices of the foliation α = const have constant mean and scalar curvatures, i.e. slices are round spheres. As a result, the entire spacetime M ext is spherically symmetric and therefore isometric to the Schwarzschild vacuum asymptotically flat spacetime (due to Birkhoff's theorem). In particular, resolving Eqs. (12), (14), (26) and introducing the area radius</text>
<formula><location><page_9><loc_46><loc_7><loc_91><loc_11></location>r S = √ A S 4 π , (29)</formula>
<text><location><page_10><loc_12><loc_89><loc_90><loc_93></location>we get standard expressions for the radius of the massive particles sphere [38] and lapse functions in the Schwarzschild spacetime ( m = 0)</text>
<text><location><page_10><loc_41><loc_88><loc_41><loc_91></location>/negationslash</text>
<formula><location><page_10><loc_26><loc_83><loc_91><loc_88></location>E 2 m 2 = ( r S -2 M ) 2 r S ( r S -3 M ) , α 2 = 1 -2 M r S , H 2 = 1 r 2 S ( 1 -2 M r S ) . (30)</formula>
<text><location><page_10><loc_12><loc_81><loc_46><loc_82></location>This completes the proof of the theorem.</text>
<text><location><page_10><loc_12><loc_67><loc_91><loc_79></location>On the one hand, substitution of Eq. (30) into the condition of nonextremality m 2 α 2 / E 2 < 1 results in M/ ( r S -2 M ) > 0, i.e. holds outside the horizon. On the other hand, massive particle spheres exists for r S > 3 M , otherwise E 2 /m 2 is negative. There is a photon sphere at r S = 3 M , so massive particle spheres are located outside the photon sphere, which is a physically reasonable.</text>
<text><location><page_10><loc_12><loc_40><loc_91><loc_66></location>We also emphasize the need to have only one massive particle sphere to prove the theorem. However, the result of the theorem suggests that the entire spacetime M ext is sliced by the massive particle spheres, each with distinct energy. Indeed, by virtue of (28) all spatial slices α = const are totally umbilic and have a constant mean curvature and lapse function ϕ -1 = n α ∇ α α . These slices represent a massive particle sphere when we additionally demand only that Eq. (8) admits a real solution for E , as it will automatically remain constant on the slice. From our previous discussion, it is clear that such a solution will exist for all slices at r S > 3 M . Future inquiries may find it intriguing to explore the flows of massive particle surfaces, parameterized by particle energy, rather than adhering to a regular foliation α = const. Such a shift could potentially weaken several technical assumptions of the theorem.</text>
<section_header_level_1><location><page_10><loc_12><loc_35><loc_32><loc_36></location>IV. CONCLUSIONS</section_header_level_1>
<text><location><page_10><loc_12><loc_15><loc_91><loc_32></location>In this paper, we have established (under some technical assumptions) that a fourdimensional static vacuum asymptotically flat spacetime admitting a massive particle sphere is isometric to the Schwarzschild spacetime. This broadens the scope of uniqueness theorems applicable to static vacuum asymptotically flat spacetimes containing regular event horizons or photon spheres, now encompassing a more general case of massive particles. Moreover, this result allows further generalization to other theories like Einstein-scalar, Einstein-Maxwell, and Einstein-Maxwell-dilaton-axion theories and others.</text>
<text><location><page_10><loc_12><loc_9><loc_91><loc_13></location>It is worth noting that the new theorem offers the possibility of extending the uniqueness theorem to encompass strongly naked singularities [10], wherein neither a photon sphere nor a</text>
<text><location><page_11><loc_12><loc_84><loc_90><loc_93></location>horizon exists, but a massive particle sphere is present. For instance, in a superextreme electrovacuum spacetime, the massive particle sphere exists in a broader range of parameters compared to the photon sphere and, particularly, can be detected in close proximity to a strongly naked singularity [38].</text>
<text><location><page_11><loc_12><loc_63><loc_91><loc_83></location>While the assumption of a regular foliation is just technical [20, 22], the constancy of the lapse function and the static nature of the sphere plays a key role in the proof of the uniqueness theorem. Recent work has explored the notion of equipotential surfaces as a potential dynamic alternative to static sphere [31]. However, whether solely relying on the concept of a massive particle surface is sufficient for the uniqueness theorems, remains uncertain. Unlike unique photon surfaces, massive particle surfaces form entire flows (for varying energies) that extend to infinity, passing in asymptotic spheres. Analysis of such surface flows can provide additional information and advances in this area of research.</text>
<text><location><page_11><loc_12><loc_52><loc_91><loc_62></location>In addition, there is considerable interest in the prospect of extending the result to stationary spacetime, where there is a suitable geometric definition of the surfaces of massive particles [39]. Such generalizations could expand understanding of the role of massive particle surfaces and hidden symmetries in the discussion of uniqueness.</text>
<section_header_level_1><location><page_11><loc_14><loc_47><loc_30><loc_48></location>Acknowledgments</section_header_level_1>
<text><location><page_11><loc_12><loc_39><loc_90><loc_44></location>KK and DG acknowledge the support of the Russian Science Foundation under Contract No. 23-22-00424.</text>
<section_header_level_1><location><page_11><loc_14><loc_34><loc_37><loc_35></location>Appendix A: Proposition</section_header_level_1>
<text><location><page_11><loc_12><loc_26><loc_91><loc_30></location>Proposition A.1 : Let the Killing vector field k α be everywhere tangent ( k α n α = 0 ) to the hypersurface S , then</text>
<formula><location><page_11><loc_35><loc_21><loc_91><loc_23></location>L k n α = 0 , L k h αβ = 0 , L k χ αβ = 0 . (A1)</formula>
<text><location><page_11><loc_12><loc_14><loc_61><loc_16></location>Proof: Calculate the Lie derivative of the normal covector</text>
<formula><location><page_11><loc_40><loc_9><loc_91><loc_12></location>L k n α = k β ∇ β n α + n β ∇ α k β . (A2)</formula>
<text><location><page_12><loc_12><loc_92><loc_60><loc_93></location>The projection of this equation onto the normal n α reads</text>
<formula><location><page_12><loc_22><loc_87><loc_91><loc_90></location>n α L k n α = n α k β ∇ β n α + n α n β ∇ α k β = 1 2 k β ∇ β ( n α n α ) + n α n β ∇ α k β = 0 , (A3)</formula>
<text><location><page_12><loc_12><loc_81><loc_90><loc_85></location>by virtue of the Killing equations ∇ ( α k β ) = 0 and normalization of the normal vector on the surface n α n α = 1. The tangent projection reads</text>
<formula><location><page_12><loc_14><loc_76><loc_91><loc_79></location>h α γ L k n α = k β h α γ ∇ β n α + n β h α γ ∇ α k β = k β h α γ ∇ β n α -k β h α γ ∇ α n β = -k σ h α γ h β σ ∇ [ α n β ] = 0 , (A4)</formula>
<text><location><page_12><loc_12><loc_74><loc_34><loc_75></location>where we used the relation</text>
<formula><location><page_12><loc_33><loc_69><loc_91><loc_71></location>0 = h α γ ∇ α ( k β n β ) = n β h α γ ∇ α k β + k β h α γ ∇ α n β , (A5)</formula>
<text><location><page_12><loc_12><loc_63><loc_90><loc_67></location>and the involutive property h α γ h β σ ∇ [ α n β ] = 0. Thus, the expression L k n α = 0 is proved. We also derive the following straightforward yet valuable corollaries</text>
<formula><location><page_12><loc_31><loc_58><loc_91><loc_61></location>L k n α = g αβ L k n β = 0 , (A6a)</formula>
<formula><location><page_12><loc_30><loc_55><loc_91><loc_58></location>L k h αβ = L k ( g αβ -n α n β ) = -n ( α L k n β ) = 0 , (A6b)</formula>
<formula><location><page_12><loc_30><loc_52><loc_91><loc_55></location>L k χ αβ = 1 2 L k L n h αβ = 1 2 L n L k h αβ + 1 2 L L k n h αβ = 0 . (A6c)</formula>
<text><location><page_12><loc_12><loc_49><loc_86><loc_50></location>Calculating the Lie derivative of equations (4) we immediately find that H is also static:</text>
<formula><location><page_12><loc_47><loc_44><loc_91><loc_47></location>L k H = 0 . (A7)</formula>
<section_header_level_1><location><page_12><loc_14><loc_35><loc_37><loc_36></location>Appendix B: Calculations</section_header_level_1>
<text><location><page_12><loc_12><loc_28><loc_91><loc_32></location>Here, we give explicit expressions for ω and its derivative along n α . First, from D α α = 0 follows the expression</text>
<formula><location><page_12><loc_31><loc_22><loc_91><loc_27></location>ω = ( -α -1 n α ¯ ∇ α U ) 1 / 2 = ( 4 (1 + α ) 2 n α ∇ α α ) 1 / 2 , (B1)</formula>
<text><location><page_12><loc_12><loc_17><loc_90><loc_21></location>where relations ˜ ∇ α U = ¯ ∇ α U = n α n β ¯ ∇ β U and ˜ g αβ = α -2 ¯ g αβ are used. Then, the remaining derivatives read</text>
<formula><location><page_12><loc_22><loc_12><loc_91><loc_15></location>n β ¯ ∇ β U = -4 α (1 + α 2 ) 2 n β ¯ ∇ β α, (B2a)</formula>
<formula><location><page_12><loc_22><loc_7><loc_91><loc_11></location>n β ¯ ∇ β ω = 1 2 ( -α -1 n α ¯ ∇ α U ) -1 / 2 ( -16 α ( n α ∇ α α ) 2 ( α 2 +1) 3 -8 H ( α 2 +1) 2 n α ∇ α α ) , (B2b)</formula>
<text><location><page_13><loc_12><loc_89><loc_91><loc_94></location>where we used identity n α ¯ ∇ α ( n β ¯ ∇ β α ) = -2 Hn γ ∇ γ α . Using equations (B1), (B2a), (B2b) and identity (8), some algebraic calculations lead us to the result (22).</text>
<unordered_list>
<list_item><location><page_13><loc_13><loc_81><loc_69><loc_82></location>[1] V. Perlick and O. Y. Tsupko, Phys. Rept. 947 , 1 (2022), 2105.07101.</list_item>
<list_item><location><page_13><loc_13><loc_78><loc_88><loc_79></location>[2] A. Grenzebach, V. Perlick, and C. Lammerzahl, Phys. Rev. D 89 , 124004 (2014), 1403.5234.</list_item>
<list_item><location><page_13><loc_13><loc_75><loc_62><loc_77></location>[3] A. A. Shoom, Phys. Rev. D 96 , 084056 (2017), 1708.00019.</list_item>
<list_item><location><page_13><loc_13><loc_73><loc_80><loc_74></location>[4] P. V. P. Cunha and C. A. R. Herdeiro, Gen. Rel. Grav. 50 , 42 (2018), 1801.00860.</list_item>
<list_item><location><page_13><loc_13><loc_70><loc_67><loc_71></location>[5] Y. Song and C. Zhang, Eur. Phys. J. C 83 , 50 (2023), 2208.03661.</list_item>
<list_item><location><page_13><loc_13><loc_67><loc_84><loc_68></location>[6] K. Kobialko, I. Bogush, and D. Gal'tsov, Phys. Rev. D 109 , 024060 (2024), 2312.07498.</list_item>
<list_item><location><page_13><loc_13><loc_62><loc_90><loc_66></location>[7] S. Chen, J. Jing, W.-L. Qian, and B. Wang, Sci. China Phys. Mech. Astron. 66 , 260401 (2023), 2301.00113.</list_item>
<list_item><location><page_13><loc_13><loc_59><loc_84><loc_60></location>[8] K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D 62 , 084003 (2000), astro-ph/9904193.</list_item>
<list_item><location><page_13><loc_13><loc_56><loc_90><loc_58></location>[9] C.-M. Claudel, K. S. Virbhadra, and G. F. R. Ellis, J. Math. Phys. 42 , 818 (2001), gr-qc/0005050.</list_item>
<list_item><location><page_13><loc_12><loc_53><loc_69><loc_55></location>[10] K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D 65 , 103004 (2002).</list_item>
<list_item><location><page_13><loc_12><loc_51><loc_78><loc_52></location>[11] K. S. Virbhadra and C. R. Keeton, Phys. Rev. D 77 , 124014 (2008), 0710.2333.</list_item>
<list_item><location><page_13><loc_12><loc_48><loc_65><loc_49></location>[12] K. S. Virbhadra, Phys. Rev. D 106 , 064038 (2022), 2204.01879.</list_item>
<list_item><location><page_13><loc_12><loc_45><loc_44><loc_47></location>[13] K. S. Virbhadra (2024), 2402.17190.</list_item>
<list_item><location><page_13><loc_12><loc_42><loc_44><loc_44></location>[14] K. S. Virbhadra (2022), 2204.01792.</list_item>
<list_item><location><page_13><loc_12><loc_40><loc_75><loc_41></location>[15] S. L. Adler and K. S. Virbhadra, Gen. Rel. Grav. 54 , 93 (2022), 2205.04628.</list_item>
<list_item><location><page_13><loc_12><loc_34><loc_91><loc_38></location>[16] B.-Y. Chen, Pseudo-Riemannian Geometry, δ -Invariants and Applications (World Scientific Publishing, Hackensack, NJ, 2011), ISBN 978-981-4329-63-7, 978-981-4462-48-8.</list_item>
<list_item><location><page_13><loc_12><loc_32><loc_46><loc_33></location>[17] W. Israel, Phys. Rev. 164 , 1776 (1967).</list_item>
<list_item><location><page_13><loc_12><loc_29><loc_54><loc_30></location>[18] D. C. Robinson, Phys. Rev. Lett. 34 , 905 (1975).</list_item>
<list_item><location><page_13><loc_12><loc_26><loc_41><loc_27></location>[19] C. Cederbaum (2014), 1406.5475.</list_item>
<list_item><location><page_13><loc_12><loc_23><loc_61><loc_25></location>[20] S. Yazadjiev, Phys. Rev. D 91 , 123013 (2015), 1501.06837.</list_item>
<list_item><location><page_13><loc_12><loc_21><loc_77><loc_22></location>[21] S. Yazadjiev and B. Lazov, Class. Quant. Grav. 32 , 165021 (2015), 1503.06828.</list_item>
<list_item><location><page_13><loc_12><loc_18><loc_84><loc_19></location>[22] C. Cederbaum and G. J. Galloway, Class. Quant. Grav. 33 , 075006 (2016), 1508.00355.</list_item>
<list_item><location><page_13><loc_12><loc_15><loc_62><loc_16></location>[23] M. Rogatko, Phys. Rev. D 109 , 024056 (2024), 2401.14116.</list_item>
<list_item><location><page_13><loc_12><loc_12><loc_72><loc_14></location>[24] S. Yazadjiev and B. Lazov, Phys. Rev. D 93 , 083002 (2016), 1510.04022.</list_item>
<list_item><location><page_13><loc_12><loc_10><loc_61><loc_11></location>[25] M. Rogatko, Phys. Rev. D 93 , 064003 (2016), 1602.03270.</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_14><loc_12><loc_92><loc_62><loc_93></location>[26] S. Yazadjiev, Phys. Rev. D 104 , 124070 (2021), 2109.02945.</list_item>
<list_item><location><page_14><loc_12><loc_89><loc_59><loc_90></location>[27] Y. Koga, Phys. Rev. D 101 , 104022 (2020), 2003.10859.</list_item>
<list_item><location><page_14><loc_12><loc_86><loc_61><loc_88></location>[28] S. Yazadjiev, Phys. Rev. D 96 , 044045 (2017), 1707.03654.</list_item>
<list_item><location><page_14><loc_12><loc_84><loc_63><loc_85></location>[29] S. Jahns, Class. Quant. Grav. 36 , 235019 (2019), 1910.10691.</list_item>
<list_item><location><page_14><loc_12><loc_81><loc_78><loc_82></location>[30] C. Cederbaum, A. Cogo, B. Leandro, and J. a. P. d. Santos (2024), 2403.14422.</list_item>
<list_item><location><page_14><loc_12><loc_78><loc_80><loc_80></location>[31] C. Cederbaum and G. J. Galloway, J. Math. Phys. 62 , 032504 (2021), 1910.04220.</list_item>
<list_item><location><page_14><loc_12><loc_75><loc_60><loc_77></location>[32] H. Yoshino, Phys. Rev. D 95 , 044047 (2017), 1607.07133.</list_item>
<list_item><location><page_14><loc_12><loc_73><loc_40><loc_74></location>[33] H. Yoshino (2023), 2309.14318.</list_item>
<list_item><location><page_14><loc_12><loc_70><loc_78><loc_71></location>[34] G. W. Gibbons and C. M. Warnick, Phys. Lett. B 763 , 169 (2016), 1609.01673.</list_item>
<list_item><location><page_14><loc_12><loc_67><loc_80><loc_69></location>[35] Y. Koga, T. Igata, and K. Nakashi, Phys. Rev. D 103 , 044003 (2021), 2011.10234.</list_item>
<list_item><location><page_14><loc_12><loc_64><loc_84><loc_66></location>[36] K. Kobialko, I. Bogush, and D. Gal'tsov, Phys. Rev. D 104 , 044009 (2021), 2104.02167.</list_item>
<list_item><location><page_14><loc_12><loc_62><loc_84><loc_63></location>[37] K. Kobialko, I. Bogush, and D. Gal'tsov, Phys. Rev. D 106 , 024006 (2022), 2202.09126.</list_item>
<list_item><location><page_14><loc_12><loc_59><loc_84><loc_60></location>[38] K. Kobialko, I. Bogush, and D. Gal'tsov, Phys. Rev. D 106 , 084032 (2022), 2208.02690.</list_item>
<list_item><location><page_14><loc_12><loc_56><loc_84><loc_58></location>[39] I. Bogush, K. Kobialko, and D. Gal'tsov, Phys. Rev. D 108 , 044070 (2023), 2306.12888.</list_item>
<list_item><location><page_14><loc_12><loc_54><loc_82><loc_55></location>[40] I. Bogush, K. Kobialko, and D. Gal'tsov, Eur. Phys. J. C 84 , 387 (2024), 2402.03266.</list_item>
<list_item><location><page_14><loc_12><loc_48><loc_90><loc_52></location>[41] G. S. Bisnovatyi-Kogan and O. Y. Tsupko, Mon. Not. Roy. Astron. Soc. 404 , 1790 (2010), 1006.2321.</list_item>
<list_item><location><page_14><loc_12><loc_45><loc_47><loc_47></location>[42] J. Wagner and X. Er (2020), 2006.16263.</list_item>
<list_item><location><page_14><loc_12><loc_43><loc_85><loc_44></location>[43] M. Fathi, M. Olivares, and J. R. Villanueva, Eur. Phys. J. C 81 , 987 (2021), 2104.07721.</list_item>
<list_item><location><page_14><loc_12><loc_40><loc_63><loc_41></location>[44] S. Kumar, A. Uniyal, and S. Chakrabarti (2023), 2308.05545.</list_item>
<list_item><location><page_14><loc_12><loc_37><loc_47><loc_38></location>[45] A. Komar, Phys. Rev. 129 , 1873 (1963).</list_item>
<list_item><location><page_14><loc_12><loc_34><loc_90><loc_36></location>[46] H. Yoshino, K. Izumi, T. Shiromizu, and Y. Tomikawa, PTEP 2017 , 063E01 (2017), 1704.04637.</list_item>
<list_item><location><page_14><loc_12><loc_29><loc_90><loc_33></location>[47] A. Garcia, F. W. Hehl, C. Heinicke, and A. Macias, Class. Quant. Grav. 21 , 1099 (2004), grqc/0309008.</list_item>
</unordered_list>
</document>
[
{
"title": "Uniqueness of the static vacuum asymptotically flat spacetimes with massive particle spheres",
"content": "Kirill Kobialko 1 ∗ , Igor Bogush 2 † , and Dmitri Gal'tsov 1 ‡ 1 Moldova State University, strada Alexei Mateevici 60, 2009, Chi¸sin˘au, Moldova 2",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "In this paper, we establish that a four-dimensional static vacuum asymptotically flat spacetime containing a massive particle sphere is isometric to the Schwarzschild spacetime. Our results expand upon existing uniqueness theorems for static vacuum asymptotically flat spacetimes, which focus on scenarios featuring event horizons or photon spheres. Similarly to the uniqueness theorems concerning photon spheres or event horizons, only a single massive particle sphere is sufficient to obtain a unique solution. However, in contrast to previous theorems, our result leads to the existence of an entire spacetime foliation sliced by a set of massive particle spheres spanning various energies.",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "Black hole shadows offer a direct means to observe the optical characteristics of immensely strong gravitational fields. The theoretical comprehension of these shadows is closely intertwined with the photon and massive particle surfaces [1-7]. Since the inception of general relativity, it has been widely recognized that the spherically symmetric Schwarzschild solution contains a series of circular null orbits, collectively constituting a complete photon sphere due to its inherent symmetry. The profound implications of these surfaces began to crystallize in the late 1990s. The seminal work by Virbhadra and Ellis [8] delineated the correlation between the properties of photon spheres and the intricacies of strong gravitational lensing, leading to the formal characterization of the photon sphere as a timelike hypersurface in spacetime, where the light beam's deflection angle at the closest approach distance tends towards infinity. Subsequently, Claudel, Virbhadra, and Ellis [9] introduced a comprehensive definition of the general photon surface as a timelike surface wherein any null geodesic, touching it tangentially, remains entirely within it, and established a theorem linking this definition to the hypersurface's geometry. The equivalence of these definitions was demonstrated in Ref. [10] for general static spherically symmetric metrics. Notably, it was revealed that the intimate connection between photon spheres and strong lensing persists even in the context of naked singularities, thereby suggesting their categorization into weak and strong variants [10, 11]. Recently, in close correlation with photon surfaces, several significant relationships have been unveiled regarding the geometric attributes of relativistic images [12, 13], the compactness of supermassive dark objects at galactic cores [14], and the impact of the cosmological constant on the photon sphere [15]. An important property of the photon surfaces is established by the theorem asserting that these are timelike totally umbilic hypersurfaces in spacetime [16] exhibiting proportionality of their first and second fundamental forms. This purely geometric approach serves as a constructive definition for analyzing photon surfaces instead of solving geodesic equations and plays a decisive role in the analysis of the black hole uniqueness. The first black hole uniqueness theorem was established by Israel in Ref. [17]. It states that the Schwarzschild solution is the only static asymptotically flat vacuum spacetime which has a nonsingular closed simply connected event horizon. Then, similar uniqueness theorem were suggested to generalize the result, e.g., the Kerr solution was shown to be the only rotating vacuum black hole by Robinson in Ref. [18]. The focus on the event horizon in the uniqueness proofs was then moved to photon surfaces in work by Cederbaum [19], where the Schwarzschild spacetime was shown to be the only static vacuum asymptotically flat spacetime that possesses a suitably defined photon sphere. Later, the uniqueness theoremes for photon spheres were established for Einstein-scalar [20], Einstein-Maxwell [21-23], Einstein-Maxwell-dilaton [24, 25], Einstein-multiple-scalar [26] models, for wormholes [27, 28], and in higher dimensions [29, 30]. The condition of constancy of the lapse function that was used in first papers was weakened to constancy of lapse function in each spatial slice in Ref. [31]. An alternative perturbative approach to considering the uniqueness of photon spheres was suggested in Ref. [32, 33]. Gibbons and Warnick [34] discovered that photon surfaces may exist in less symmetric spacetimes. This was extended in Ref. [35], where the photon surfaces were demonstrated to be possible if the spacetime admits a nontrivial Killing tensor. This might suggest the potential challenge of formulating a uniqueness theorem solely relying on the umbilicity without constancy of the lapse function. Similar connections in stationary spacetimes was established in the Refs. [36, 37]. The framework of photon spheres has been extended to encompass massive particle surfaces , which share analogous properties for timelike geodesics associated with massive particles interacting with black holes or other ultra-compact gravitating objects [38-40]. Although the flow of massive particles is not directly observable from distant points (except perhaps for neutrinos, whose detection remains a considerable challenge), these surfaces can be indirectly observed through their proper radiation, which may become visible under certain conditions. Moreover, the significance of massive particle surfaces lies in their relevance to photons traversing through plasma environments that may surround black holes [6]. In environments with inhomogeneous plasma, besides the gravitational deflection of light, electromagnetic refraction also plays a role [41-44], which can be integrated into a unified lensing theory. In this paper, we adapt proofs from Refs. [20, 21, 24, 26] suggested by Yazadjiev et. al. Within the framework of the same assumptions (there is only one connected photon sphere and the lapse function regularly foliates spacetime), we prove the uniqueness theorem for massive particle spheres in static vacuum asymptotically flat spacetimes. In contrast to previous uniqueness theorems, our result leads to the existence of an entire spacetime foliation sliced by a set of massive particle spheres spanning various energies.",
"pages": [
2,
3
]
},
{
"title": "II. MASSIVE PARTICLE SPHERE",
"content": "We start with considering a four-dimensional static vacuum asymptotically flat spacetime M with given ADM mass M > 0 and Levi-Civita connection ∇ α . In a static spacetime, there is a timelike Killing vector field k α = αm α , where α > 0 is a lapse function and m α is a future-directed timelike unit vector ( k α k α = -α 2 ). One can define spatial slices Σ orthogonal to k α . The induced metric on Σ is ¯ g αβ = g αβ + m α m β which defines the corresponding LeviCivita connection ¯ ∇ α . Here and further bars will denote quantities associated with the slice Σ. Vacuum Einstein equations R αβ = 0 after dimensional reduction along k α read [19, 20] Since we are interested in asymptotically flat spacetimes, the lapse function and the metric have the following asymptotics for r →∞ [19, 20]: where r is a suitable radial coordinate and η αβ is a flat Minkowski four-dimensional metric. For vacuum spacetime with a time-like Killing vector k α , there is an alternative in determination of the ADM mass of the solution through the Komar integral [45] where ¯ S is an arbitrary closed two-dimensional surface in Σ with an outer normal vector n α (vector n α lies in the tangent space of slice Σ) and d ¯ S is a volume form associated with the induced metric on ¯ S . The definition of a massive particle surface for static vacuum spacetimes can be formulated as follows [38]: Definition 2.1 : A massive particle surface is a timelike hypersurface S of M such that, for every point p ∈ S and every vector v α | p ∈ T p S such that v α k α | p = -E and v α v α | p = -m 2 , there exists a geodesic γ of M for a particle with mass m and energy E such that ˙ γ α (0) = v α | p and γ ⊂ S . We omit the charge from the definition in Ref. [38] since the solution is vacuum and there is no electromagnetic force acting on particles. Here, we focus on static massive particle surface , which are additionally tangent to the timelike Killing vector k α . In other words, the definition states that any geodesic of a particle with mass m and energy E initially tangent to the corresponding massive particle surface S will remain tangent to S . For static massive particle surface S with normal n α the first and second fundamental forms read as [38] where H is some scalar function on S and D α is a Levi-Civita connection in S . Since the Killing vector field k α is tangent to the hypersurface S everywhere ( k α n α = 0), the following Lie derivatives are equal to zero (see App. A): If ¯ S is a spatial section of a surface S sliced by Σ, from general geometric considerations we have [46]: Comparing (4) and (6), we find and since ¯ χ αβ is tangent to the spatial section and k α is orthogonal to it, we find the following expressions: The spatial section of static massive particle hypersurface is a totally umbilical surface with a spatial mean curvature H = 1 2 ¯ χ α α . However, unlike the photon sphere [9], the principal curvature in the time direction is different from the spatial ones. In what follows we also assume that the spatial section is connected, compact and closed. Equations (5), (8) and Refs. [19-22, 24, 26, 28] inspire us to introduce two important definitions - a massive particle sphere and a non-extremal massive particle surface. Definition 2.2 : The massive particle surface S is a massive particle sphere if and only if D α α = 0 on S . Definition 2.3 : The massive particle surface S is a non-extremal massive particle surface if and only if m 2 α 2 / E 2 < 1 on S . Further we will be interested in non-extreme massive particle spheres only. The latter has a number of important geometric properties. First, it has a constant spatial mean curvature, i.e. D α H = 0. Indeed, consider the Codazzi equation [16] in spacetime Here, we used Eq. (4) to rewrite the second fundamental form χ αβ . Then, we used Eq. (5) and to get rid of the first two terms in the second line. Also, we used the condition D α α = 0 and Killing equations D ( α k β ) = 0. Since we consider a non-extremal sphere m 2 α 2 / E 2 < 1, the term in brackets (2 -m 2 α 2 / E 2 ) is non-zero, and we get D α H = 0. Since we have proven that H is constant at the sphere, this allows us to obtain useful geometric identities. First, consider the Komar integral (3) over the spatial sections ¯ S of a massive particle sphere: Since the integrand expression is constant, there is an algebraic relation between the mass M and the spatial section area of the massive particle sphere A S : Particularly, sphere ¯ S has a positive constant mean curvature H > 0 if a physical assumption of positive mass M is taken into consideration. By virtue of Eq. (8), this means that on the sphere n α ∇ α α > 0, i.e., the norm of spatial gradient ¯ ∇ α α does not vanish anywhere on ¯ S . Consider an outer space region Σ ext outside the massive particle sphere ¯ S or equivalently a spacetime region M ext outside S . In this case, the massive particle sphere is an inner boundary ∂ M ext . Similarly to Ref. [20], we introduce an additional assumption that α = const regularly foliate the manifold M ext . It is worth noting that the condition for the existence of a regular foliation is technical and, in principle, open to relaxation, as discussed in Ref. [22]. By definition, the function α is constant at the massive particle sphere. As we will show, the massive particle sphere ¯ S has a topology of a sphere. Given the regularity of the foliation, any slice in the outer region Σ ext is a topological sphere as well. Equations of motion (1) necessitate α to be a harmonic function, while the boundary conditions at asymptotics dictate that α must approach 1 as it tends to infinity. Following the maximum principle for the harmonic functions, α monotonically increases to 1 moving from the sphere ¯ S to infinity along the flow of slices, i.e., 0 < α < 1. The second key identity can be obtained from the Gauss-Bonnet theorem. The trace of the Gauss equations gives an expression for the scalar curvature ¯ R of the spatial section ¯ S the following (see Eq. (C13) in Ref. [46], keeping in mind that R αβ = 0, D α α = 0 and (8)) Therefore, the non-extremal ( m 2 α 2 / E 2 < 1) sphere ¯ S has a constant and positive scalar curvature ¯ R > 0, representing a round sphere [20]. Then, integrating (13) over ¯ S and applying Gauss-Bonnet theorem ∫ ¯ S ¯ R d ¯ S = 8 π , we find the second useful identity: Dividing the equation (14) by (12), the following algebraic connection between the mean curvature H and the lapse function α on S can be found:",
"pages": [
4,
5,
6,
7
]
},
{
"title": "III. UNIQUENESS THEOREM",
"content": "Having completed all the preparations, we are ready to formulate and prove the main result of this article. Theorem 3.1 : Let M ext be a four-dimensional static and asymptotically flat spacetime with given ADM mass M > 0 , satisfying the vacuum Einstein equations R αβ = 0 and possessing a non-extremal massive particle sphere as an inner boundary of M ext . Assume that the lapse function α regularly foliates M ext . Then, M ext is an isometric to the Schwarzschild spacetime with mass M , and the area radius r S = √ A S / 4 π of the massive particle sphere satisfies the equation E 2 /m 2 = ( r S -2 M ) 2 / ( r 2 S -3 Mr S ) . Proof: The proof is based on a modification of the proof presented in Refs. [20, 21, 24, 26] for the case of photon spheres. The main problem is to prove the spherical symmetry of the spacetime M ext . First, let us perform a Weyl transformation ˜ g αβ = α 2 ¯ g αβ . In this case, Eq. (1) turns into where ˜ ∇ and ˜ R αβ are the Levi-Civita connection and the Ricci tensor for ˜ g αβ . Our goal is to show that metric ˜ g αβ is conformally flat. For this purpose, one can use the Cotton tensor [47] over a 3-dimensional Riemannian manifold which is defined by Using Eq. (16), the following divergences can be obtained [20]: where Since we have shown that 0 < α < 1, we also have 0 < U < 1. Which after integration over the entire spatial slice Σ lead to the following two inequalities (equality if and only if ˜ R αβγ = 0) where d ˜ Σ volume form associated with metric ˜ g αβ . Let us now apply Stokes' theorem to them using massive particle sphere ¯ S and asymptotic sphere ¯ S ∞ as boundary surfaces: where we used d ˜ S = α 2 d ¯ S and ˜ n α = α -1 n α , and -¯ S means that the orientation of the normal to the inner boundary is opposite to the foliation. Given asymptotics (2), each surface term reads (see App. B for some details) Then, the right inequality in (21) immediately results in The left inequality in (21) gives which can be transformed using Eqs. (14) and (15) into the following expression where we took into account non-extrimality m 2 α 2 / E 2 < 1. The inequalities (23) and (25) are compatible if and only if they degenerate into equalities On the other hand, inequalities can degenerate into equalities if and only if the Cotton tensor vanishes ˜ R αβγ = 0. For a three-dimensional Riemannian manifold, this is a necessary and sufficient condition for the metric ˜ g αβ to be conformally flat [47]. Hence, the metric ¯ g αβ is also conformally flat and ¯ R αβγ = 0. In particular, we have the identity [20] where α ¯ χ αβ , α ¯ h αβ and ϕ -1 = n α ∇ α α are the induced metric, the second fundamental forms and the lapse function of slices α = const respectively, and the trace is denoted as α ¯ χ = α ¯ χ α α . Since the induced metric possesses the Euclidean signature, each square in the brackets is equal to zero, yielding the following expressions: Thus, all slices are totally umbilic and the lapse function is constant on them. As in the case of photon spheres [20], this implies that all slices of the foliation α = const have constant mean and scalar curvatures, i.e. slices are round spheres. As a result, the entire spacetime M ext is spherically symmetric and therefore isometric to the Schwarzschild vacuum asymptotically flat spacetime (due to Birkhoff's theorem). In particular, resolving Eqs. (12), (14), (26) and introducing the area radius we get standard expressions for the radius of the massive particles sphere [38] and lapse functions in the Schwarzschild spacetime ( m = 0) /negationslash This completes the proof of the theorem. On the one hand, substitution of Eq. (30) into the condition of nonextremality m 2 α 2 / E 2 < 1 results in M/ ( r S -2 M ) > 0, i.e. holds outside the horizon. On the other hand, massive particle spheres exists for r S > 3 M , otherwise E 2 /m 2 is negative. There is a photon sphere at r S = 3 M , so massive particle spheres are located outside the photon sphere, which is a physically reasonable. We also emphasize the need to have only one massive particle sphere to prove the theorem. However, the result of the theorem suggests that the entire spacetime M ext is sliced by the massive particle spheres, each with distinct energy. Indeed, by virtue of (28) all spatial slices α = const are totally umbilic and have a constant mean curvature and lapse function ϕ -1 = n α ∇ α α . These slices represent a massive particle sphere when we additionally demand only that Eq. (8) admits a real solution for E , as it will automatically remain constant on the slice. From our previous discussion, it is clear that such a solution will exist for all slices at r S > 3 M . Future inquiries may find it intriguing to explore the flows of massive particle surfaces, parameterized by particle energy, rather than adhering to a regular foliation α = const. Such a shift could potentially weaken several technical assumptions of the theorem.",
"pages": [
7,
8,
9,
10
]
},
{
"title": "IV. CONCLUSIONS",
"content": "In this paper, we have established (under some technical assumptions) that a fourdimensional static vacuum asymptotically flat spacetime admitting a massive particle sphere is isometric to the Schwarzschild spacetime. This broadens the scope of uniqueness theorems applicable to static vacuum asymptotically flat spacetimes containing regular event horizons or photon spheres, now encompassing a more general case of massive particles. Moreover, this result allows further generalization to other theories like Einstein-scalar, Einstein-Maxwell, and Einstein-Maxwell-dilaton-axion theories and others. It is worth noting that the new theorem offers the possibility of extending the uniqueness theorem to encompass strongly naked singularities [10], wherein neither a photon sphere nor a horizon exists, but a massive particle sphere is present. For instance, in a superextreme electrovacuum spacetime, the massive particle sphere exists in a broader range of parameters compared to the photon sphere and, particularly, can be detected in close proximity to a strongly naked singularity [38]. While the assumption of a regular foliation is just technical [20, 22], the constancy of the lapse function and the static nature of the sphere plays a key role in the proof of the uniqueness theorem. Recent work has explored the notion of equipotential surfaces as a potential dynamic alternative to static sphere [31]. However, whether solely relying on the concept of a massive particle surface is sufficient for the uniqueness theorems, remains uncertain. Unlike unique photon surfaces, massive particle surfaces form entire flows (for varying energies) that extend to infinity, passing in asymptotic spheres. Analysis of such surface flows can provide additional information and advances in this area of research. In addition, there is considerable interest in the prospect of extending the result to stationary spacetime, where there is a suitable geometric definition of the surfaces of massive particles [39]. Such generalizations could expand understanding of the role of massive particle surfaces and hidden symmetries in the discussion of uniqueness.",
"pages": [
10,
11
]
},
{
"title": "Acknowledgments",
"content": "KK and DG acknowledge the support of the Russian Science Foundation under Contract No. 23-22-00424.",
"pages": [
11
]
},
{
"title": "Appendix A: Proposition",
"content": "Proposition A.1 : Let the Killing vector field k α be everywhere tangent ( k α n α = 0 ) to the hypersurface S , then Proof: Calculate the Lie derivative of the normal covector The projection of this equation onto the normal n α reads by virtue of the Killing equations ∇ ( α k β ) = 0 and normalization of the normal vector on the surface n α n α = 1. The tangent projection reads where we used the relation and the involutive property h α γ h β σ ∇ [ α n β ] = 0. Thus, the expression L k n α = 0 is proved. We also derive the following straightforward yet valuable corollaries Calculating the Lie derivative of equations (4) we immediately find that H is also static:",
"pages": [
11,
12
]
},
{
"title": "Appendix B: Calculations",
"content": "Here, we give explicit expressions for ω and its derivative along n α . First, from D α α = 0 follows the expression where relations ˜ ∇ α U = ¯ ∇ α U = n α n β ¯ ∇ β U and ˜ g αβ = α -2 ¯ g αβ are used. Then, the remaining derivatives read where we used identity n α ¯ ∇ α ( n β ¯ ∇ β α ) = -2 Hn γ ∇ γ α . Using equations (B1), (B2a), (B2b) and identity (8), some algebraic calculations lead us to the result (22).",
"pages": [
12,
13
]
}
]
2024PhRvD.110d6023M
https://arxiv.org/pdf/2307.11348.pdf
<document>
<section_header_level_1><location><page_1><loc_31><loc_92><loc_70><loc_93></location>All holographic systems have scar states</section_header_level_1>
<section_header_level_1><location><page_1><loc_44><loc_89><loc_56><loc_90></location>Alexey Milekhin</section_header_level_1>
<text><location><page_1><loc_23><loc_88><loc_78><loc_89></location>UC Santa Barbara, Physics Department, Broida Hall, Santa Barbara, CA 93106 ∗</text>
<section_header_level_1><location><page_1><loc_45><loc_85><loc_56><loc_86></location>Nikolay Sukhov</section_header_level_1>
<text><location><page_1><loc_24><loc_82><loc_77><loc_84></location>Princeton University, Physics Department, Jadwin Hall, Princeton, NJ 08540 (Dated: June 19, 2024)</text>
<text><location><page_1><loc_18><loc_73><loc_83><loc_81></location>Scar states are special finite-energy density, but non-thermal states of chaotic Hamiltonians. We argue that all holographic quantum field theories, including N = 4 super Yang-Mills, have scar states. Their presence is tied to the existence of non-topological, horizonless soliton solutions in gravity: oscillons and a novel family of excited boson stars. We demonstrate that these solutions have periodic oscillations in the correlation functions and posses low-entanglement entropy as expected for scar states. Also we find that they can be very easily prepared with Euclidean path integral.</text>
<section_header_level_1><location><page_1><loc_24><loc_69><loc_34><loc_70></location>CONTENTS</section_header_level_1>
<list_item><location><page_1><loc_10><loc_66><loc_21><loc_67></location>I. Introduction</list_item>
<text><location><page_1><loc_48><loc_66><loc_49><loc_67></location>1</text>
<list_item><location><page_1><loc_9><loc_63><loc_18><loc_64></location>II. Oscillons</list_item>
<text><location><page_1><loc_48><loc_63><loc_49><loc_64></location>3</text>
<list_item><location><page_1><loc_12><loc_62><loc_30><loc_63></location>A. Entanglement entropy</list_item>
<text><location><page_1><loc_48><loc_62><loc_49><loc_63></location>5</text>
<list_item><location><page_1><loc_12><loc_60><loc_43><loc_62></location>B. A comment on N = 4 super Yang-Mills</list_item>
<text><location><page_1><loc_48><loc_60><loc_49><loc_62></location>5</text>
<list_item><location><page_1><loc_9><loc_58><loc_20><loc_59></location>III. Boson stars</list_item>
<text><location><page_1><loc_48><loc_58><loc_49><loc_59></location>6</text>
<list_item><location><page_1><loc_9><loc_55><loc_24><loc_56></location>IV. State preparation</list_item>
<text><location><page_1><loc_48><loc_55><loc_49><loc_56></location>9</text>
<list_item><location><page_1><loc_9><loc_51><loc_43><loc_53></location>V. Volume-law entanglement and weak energy conditions</list_item>
<text><location><page_1><loc_47><loc_51><loc_49><loc_52></location>10</text>
<list_item><location><page_1><loc_9><loc_48><loc_20><loc_49></location>VI. Conclusion</list_item>
<text><location><page_1><loc_47><loc_48><loc_49><loc_49></location>11</text>
<text><location><page_1><loc_12><loc_45><loc_24><loc_47></location>Acknowledgment</text>
<text><location><page_1><loc_47><loc_45><loc_49><loc_47></location>11</text>
<text><location><page_1><loc_12><loc_43><loc_19><loc_44></location>References</text>
<text><location><page_1><loc_47><loc_43><loc_49><loc_44></location>12</text>
<section_header_level_1><location><page_1><loc_20><loc_39><loc_37><loc_40></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_14><loc_49><loc_36></location>Everyday experience tells us that most systems thermalize over time, but there are exceptions. For instance, consider a single classical particle moving inside a reflecting cavity - Figure 1. For a cavity of random shape, all trajectories will look essentially random and they will explore of all the phase space - Figure 1 (a). This is a one-particle counterpart of thermalization: time average along such trajectory will be equal to the phase space average. We may call such cavity ergodic. Of course, there are special (integrable) shapes for which all trajectories have a short period of oscillations - Figure 1 (b). However, there are intriguing cases like the Bunimovich stadium[1]: a generic trajectory is thermal - Figure 1 (c), but there is a set of short periodic trajectories which do not explore all of the phase space - Figure 1 (d). We will refer to them as classical scars .</text>
<figure>
<location><page_1><loc_58><loc_49><loc_85><loc_71></location>
<caption>Figure 1. Illustration of possible one-particle motions in a cavity. (a) is a fully chaotic cavity, (b) is integrable circular cavity. The cavity in (c) and (d) is the Bunimovich stadium. Case (d) is the scar trajectory.</caption>
</figure>
<text><location><page_1><loc_52><loc_28><loc_92><loc_39></location>There is an exponentially small number of such trajectories and they are unstable, so one might expect that on a quantum level they are not important for anything. It turns out to be incorrect [2]. A number of energy eigenstate wavefunctions are concentrated around the classical scar trajectories. In other words, short classical unstable orbits permanently 'scar' the wavefunctions. This is the phenomena of single-particle quantum scars .</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_27></location>Recent interest towards scars started from discovering a similar phenomena experimentally in a many-body quantum system of cold Rydberg atoms [3, 4](see also [5]). In the many-body setup, there is no classical analogue and the scarring occurs in the Hilbert space: evolution of certain states | Ψ(0) ⟩ shows short-period revivals when the fidelity |⟨ Ψ( t ) | Ψ(0) ⟩| ≈ 1. It is important to emphasize that scars were observed in ergodic Hamiltonians, for which most of the states are thermalizing: the fidelity exponentially decays to zero and it stays zero till the Poincare recurrence time. A hallmark of many-body quantum scars are scarred energy eigenstates with abnormally low entanglement compared to states of the same</text>
<text><location><page_2><loc_9><loc_89><loc_49><loc_93></location>energy density. Typically they represent an exponentially small fraction of the Hilbert space. We refer to [6-9] for an overview.</text>
<text><location><page_2><loc_9><loc_59><loc_49><loc_89></location>The main objective of this paper is to explore the scarring phenomena in gravity and quantum field theory. AdS/CFT correspondence [10-13] says that certain strongly-coupled large N conformal field theories (CFTs) are dual to classical gravitational theories in the asymptotically anti de-Sitter (AdS) spacetime. The gravitational dual of thermalization process is the formation of a black hole. It is known [14-18] that AdS spacetime is unstable: a small perturbation (either matter or metric) inside AdS quickly collapses into a black hole. However, there are special initial conditions which do not lead to a collapse. Instead the perturbation oscillates inside AdS forever [19, 20]. This is a clear analogue of a classical scar trajectory. It is important to emphasize that these oscillating gravity solutions and 'small' perturbations are large from the boundary CFT point of view, they correspond to the energy density of the order of the central charge (in the units of boundary volume). The simplest example, which we study in the present paper, involves Einstein gravity minimally coupled to a matter scalar field.</text>
<text><location><page_2><loc_9><loc_56><loc_49><loc_58></location>Given that there are classical scar solutions in classical gravity, one [21] can naturally ask two question:</text>
<unordered_list>
<list_item><location><page_2><loc_11><loc_52><loc_49><loc_54></location>· Is there associated quantum scarring phenomena in the wavefunctions of quantum gravity?</list_item>
<list_item><location><page_2><loc_11><loc_48><loc_49><loc_50></location>· What is the CFT state dual to classical gravity scar?</list_item>
</unordered_list>
<text><location><page_2><loc_9><loc_35><loc_49><loc_46></location>In this paper we would like to answer the second question: Such periodic classical gravity solutions, known as boson stars or oscillons, are holographically dual to manybody scar states at the boundary. Moreover, we would like to argue that it is a feature of all holographic systems, whenever the bulk has 3 or mode spacetime dimensions and has a scalar field. One such example is N = 4 super Yang-Mills which we will discuss in detail.</text>
<text><location><page_2><loc_9><loc_9><loc_49><loc_34></location>Why is this interesting? Holographic systems are supposed to be not just chaotic [22, 23], but maximally chaotic [24-26]. Having scar states is not a generic feature of chaotic Hamiltonians. However, our result indicates that scars, which break erodicity, are generic for holographic systems. Also, it is expected that the presence of scars is associated with hidden symmetries (or more generally, spectrum-generating algebras) [27-35]. The question of a possible hidden symmetry behind oscillons has been extensively discussed in the literature before and we give a small overview in the Conclusion. Also there is an interesting difference with classical scars. It is known that boson stars and oscillons are linearly stable and exhibit slow thermalization: adding an extra perturbation on top does not immediately lead to black hole formation [36]. In contrast, classical scars are associated with unstable periodic orbits. Finally, the presence of boson stars and oscillons gives predictions for</text>
<text><location><page_2><loc_52><loc_89><loc_92><loc_93></location>certain boundary CFT correlation functions involving a non-primary operator e ε O , O being the single-trace CFT scalar, dual to the scalar field in the bulk.</text>
<text><location><page_2><loc_52><loc_80><loc_92><loc_89></location>Oscillons and boson stars require scalar matter fields. Even more generally, there are geon solutions [20] which are made purely from the metric, that is, from the CFT stress-energy tensor. However, they are more complicated and they break translational symmetry at the boundary so we do not consider them here.</text>
<text><location><page_2><loc_52><loc_50><loc_92><loc_80></location>How hard is it to prepare these states? The main difference between oscillons and boson stars is whether the scalar field is real or complex: for oscillons the field is real and the metric is time-dependent. For boson stars the field is complex and has a harmonic time-dependence e -i Ω t , so the stress-energy tensor and the metric are timeindependent. From the gravity point of view, oscillons and boson stars are very different solutions which are usually discussed separately. Interestingly, we find that from the boundary CFT viewpoint they are very similar and both of them can be very easily prepared using the Euclidean path integral on a hemisphere with a single operator insertion e ˜ ε O at the pole - Figure 2. In case of oscillons the single-trace operator O is hermitian, whereas for boson stars it is complex. In our regime of interest, ˜ ε can be large, of order the square-root of central charge if the two-point function of O is normalized to 1. The resulting CFT state lives on sphere S d -1 . We find evidence that for boson stars it is possible to take the infinite volume limit to get a homogeneous state on R d -1 . For oscillons we were not able to find such limit.</text>
<text><location><page_2><loc_52><loc_34><loc_92><loc_50></location>It is important to emphasize that oscillons and boson stars are not energy eigenstates for the boundary CFT. However, they are very close to being energy eigenstates: their energy density scales with the (large) central charge, whereas the variance is expected to scale at most as the square-root of the central charge, essentially because the classical gravitational solution provides a dominant saddle-point for the path-integral. Nonetheless, they are non-thermalizing, uniform energy density, pure states which support eternal oscillations in the 1-point function of a scalar single-trace operator O :</text>
<formula><location><page_2><loc_57><loc_28><loc_92><loc_33></location>oscillon: ⟨ Ψ |O| Ψ ⟩ = ∑ i =1 ε i cos((2 i -1)Ω t ) , (1) boson star: ⟨ Ψ |O| Ψ ⟩ = εe i Ω t .</formula>
<text><location><page_2><loc_52><loc_9><loc_92><loc_27></location>Such behavior, by definition, implies the violation of the eigenstate thermalization hypothesis (ETH) [37, 38], which only allows such oscillating terms to be exponentially small in the thermal entropy. More interestingly, it was recently argued [39] that under certain mild assumptions in discrete local systems, the presence of such revivals in pure states possessing area-law entanglement imply the presence of scarred energy eigenstates in the spectrum. In this paper we indeed find evidence that boson stars possess area law entanglement at the boundary: entanglement of a CFT subregion scales as the area of the boundary of that region. In contrast, for a thermal (black hole) state the entropy scales as the volume</text>
<figure>
<location><page_3><loc_22><loc_85><loc_35><loc_94></location>
<caption>Figure 2. Euclidean path-integral state preparation of oscillons and boson stars: we put CFT on a Euclidean hemisphere and insert e ˜ ε O at the south pole.</caption>
</figure>
<text><location><page_3><loc_9><loc_69><loc_49><loc_76></location>of a CFT subregion. Hence we can expect the presence of exact scars in the spectrum. The simplest oscillons we study here only exist in a finite volume and up to some critical energy density, so it is not clear how to separate volume and area law entanglement for them.</text>
<text><location><page_3><loc_10><loc_67><loc_35><loc_68></location>Let us summarize out key findings:</text>
<unordered_list>
<list_item><location><page_3><loc_11><loc_63><loc_49><loc_66></location>· Boson stars and oscillons can be easily prepared with Euclidean path integral.</list_item>
<list_item><location><page_3><loc_11><loc_55><loc_49><loc_61></location>· They have lower boundary entanglement entropy compared to a black hole of the same mass. In case of boson star this separation is parametric: area law instead of volume law.</list_item>
<list_item><location><page_3><loc_11><loc_47><loc_49><loc_53></location>· We find that oscillons have bounded mass. However, we uncover a novel family of linearly-stable excited boson stars, which we call C-stars , for which the mass is unbounded.</list_item>
</unordered_list>
<text><location><page_3><loc_9><loc_41><loc_49><loc_45></location>In Section III we will explain why one needs to study excited boson stars, rather than fundamental (non-excited ones) to find scars.</text>
<text><location><page_3><loc_9><loc_28><loc_49><loc_40></location>Recently the phenomena of scar states in quantum field theories (QFT) has been addressed in a number of papers. Previous discussions of oscillons and boson stars within the AdS/CFT include [17, 40-44]. Scars based on Virasoro symmetry and their relation to AdS 3 /CFT 2 are discussed in [45, 46]. For a general discussion of scar states within the QFT framework we refer to [47-49]. Another recent discussion of scar states [50] is based on stable orbits around black holes.</text>
<text><location><page_3><loc_9><loc_9><loc_49><loc_27></location>The rest of the paper is organized as follows. Section II is dedicated to oscillon solutions. We discuss their generic properties and then switch to the entanglement entropy in Section II A. In Section II B we argue that oscillons exist in the supergravity dual to N = 4 super Yang-Mills. Section III is devoted to boson stars. We briefly describe their properties and demonstrate that they have area-law entanglement. The CFT state preparation of oscillons and boson stars is discussed in Section IV. In Section V we turn away from discussing specific solutions and argue generally that black holes maximize entanglement entropy due to weak energy condition. In Conclusion we summarize our findings and outline open question.</text>
<section_header_level_1><location><page_3><loc_65><loc_92><loc_79><loc_93></location>II. OSCILLONS</section_header_level_1>
<text><location><page_3><loc_52><loc_86><loc_92><loc_90></location>In this Section we study the oscillons states first found in [19]. We will review their perturbative construction and then compute subsystem entanglement entropy.</text>
<text><location><page_3><loc_52><loc_79><loc_92><loc_85></location>Consider Einstein gravity with negative cosmological constant plus a minimally-coupled scalar field ϕ , which is spherically symmetric. The main statement of [19] is that such soliton can exist forever, without collapsing into a black hole. The Lagrangian is</text>
<formula><location><page_3><loc_58><loc_74><loc_92><loc_77></location>L = 1 16 πG N ( R +2Λ) -( ∂ µ ϕ ) 2 -m 2 ϕ 2 . (2)</formula>
<text><location><page_3><loc_52><loc_70><loc_92><loc_73></location>A general ansatz for the metric, preserving the spherical symmetry is</text>
<formula><location><page_3><loc_53><loc_65><loc_92><loc_69></location>ds 2 = l 2 cos 2 x ( -dt 2 Ae -2 δ + A -1 dx 2 +sin 2 xd Ω 2 d -1 ) . (3)</formula>
<text><location><page_3><loc_52><loc_53><loc_92><loc_65></location>Usual (undeformed) AdS d +1 is A ( x, t ) = 1 , δ ( x, t ) = 0. AdS radius l is determined by l 2 = d ( d +1) 2Λ . The boundary is at x = π/ 2 and the center is at x = 0. We impose a gauge constraint that at the boundary δ is zero: δ ( t, π/ 2) = 0, so the dimensionless coordinate t is the boundary time. This setup corresponds to ( d -1)+1dimensional CFT located at the asymptotic boundary S d -1 × R t .</text>
<text><location><page_3><loc_52><loc_40><loc_92><loc_53></location>It is very important to discuss units in this paper because all results we present will be in dimensionless units. The conformal metric at the boundary has unit radius. Correspondingly, t and the frequencies are measured in the units of the boundary radius. AdS radius l drops out from the equations and we can reabsorb 8 πG N = l d -1 p into ϕ . So the scalar field is measured in the units of l -( d -1) / 2 p . With Dirichlet boundary conditions for the scalar field, function A has the following expansion:</text>
<formula><location><page_3><loc_62><loc_37><loc_92><loc_38></location>A = 1 -2 M ( π/ 2 -x ) d + . . . (4)</formula>
<text><location><page_3><loc_52><loc_27><loc_92><loc_35></location>CFT energy density T tt is proportional to M times ( l/l p ) d -1 . The ratio ( l/l p ) d -1 is proportional to the CFT central charge, which is large. Similarly, using RT/HRT prescription [51, 52], entanglement entropy of boundary subregions is given by the area of extremal co-dimension two surfaces in the bulk with minimal area:</text>
<formula><location><page_3><loc_67><loc_23><loc_92><loc_26></location>S vN = Area 4 G N . (5)</formula>
<text><location><page_3><loc_52><loc_9><loc_92><loc_22></location>Technically it is always infinite, because AdS boundary is infinitely far, so we will always compute the difference with the vacuum (empty AdS answer). So in this paper we compute it in the units of ( l/l p ) d -1 . The upshot is that we are interested in large CFT perturbations, when the energy density and entropy are proportional to the central charge. The solutions are parametrized only by the (dimensionless) value of the scalar field (in the l -( d -1) / 2 p units).</text>
<text><location><page_4><loc_9><loc_89><loc_49><loc_93></location>Without gravitational backreaction, a minimally coupled scalar of mass m 2 = ∆(∆ -d ) has a set of (spherically symmetric) normal modes:</text>
<formula><location><page_4><loc_14><loc_80><loc_49><loc_88></location>e j ( x ) = n j cos( x ) ∆ P d/ 2 -1 , ∆ -d/ 2 j (cos(2 x )) , (6) j = 0 , 1 , . . . , n 2 j = 2(2 j +∆)Γ( j +∆) j ! Γ( j + d/ 2)Γ( j +∆ -d/ 2 + 1) ,</formula>
<text><location><page_4><loc_9><loc_65><loc_49><loc_79></location>with frequencies ω j = ∆ + 2 j and P being the Jacobi polynomials. With this choice of normalization, they are orthonormal with respect to the tan( x ) d -1 . The fundamental mode j = 0 has no zeroes, and the excited ones have j zeroes. The question is what happens with this solution once we include backreaction into account. The key statement is that solutions with a single dominant frequency do not collapse into a black hole. Solutions which have several frequencies collapse very quickly, that is, they thermalize. Let us list a few other facts:</text>
<unordered_list>
<list_item><location><page_4><loc_11><loc_59><loc_49><loc_64></location>· Non-spherically symmetric configurations are more prone to instability: only very special modes can be dressed to yield a periodic solution [53].</list_item>
<list_item><location><page_4><loc_11><loc_52><loc_49><loc_58></location>· For generic perturbations (not oscillons), secularly growing corrections arise in the 3rd order of perturbation theory, ε 3 t . So thermalization time is of order 1 /ε 2 .</list_item>
<list_item><location><page_4><loc_11><loc_42><loc_49><loc_51></location>· Even if the scalar field has zero self-interaction, tree-level graviton exchange induces it. Hence one might expect that explicit self-interaction should not change the picture much. This logic was verified numerically in [54-56] by considering λϕ 4 interaction in the bulk.</list_item>
<list_item><location><page_4><loc_11><loc_33><loc_49><loc_41></location>· Finally, one can ask about the influence of higher-derivative terms in the gravity action. It was argued based on numerical analysis that in Einstein-Gauss-Bonnet [57] gravity [41, 58, 59] and Starobinsky R 2 gravity [60], oscillons continue to exist.</list_item>
</unordered_list>
<text><location><page_4><loc_9><loc_27><loc_49><loc_31></location>All this suggests that oscillons have lifetime nonperturbative in 1 /G N . Meaning that their lifetime is e # /G N , non-perturbatively large in 1 /G N .</text>
<text><location><page_4><loc_9><loc_23><loc_49><loc_27></location>For example, one can start from the lowest mode solution in asymptotically AdS 5 spacetime with j = 0 , Ω = 4:</text>
<formula><location><page_4><loc_22><loc_20><loc_49><loc_21></location>ϕ = εe 0 ( x ) cos(4 t ) , (7)</formula>
<text><location><page_4><loc_9><loc_17><loc_45><loc_18></location>and then try to find the fully backreacted solution:</text>
<formula><location><page_4><loc_17><loc_13><loc_49><loc_16></location>ϕ = ∑ i,j f i,j e j ( x ) cos((2 i +1)Ω t ) . (8)</formula>
<text><location><page_4><loc_9><loc_9><loc_49><loc_11></location>Functions e j ( x ) form a basis, so the only special property of this ansatz is the periodic time-dependence. By</text>
<figure>
<location><page_4><loc_53><loc_74><loc_91><loc_93></location>
</figure>
<figure>
<location><page_4><loc_53><loc_55><loc_91><loc_72></location>
<caption>Figure 3. Boundary energy density (in units of ( l/l p ) d -1 ) and frequency (in units of boundary radius) of oscillon as the function of the amplitude (in units of l -( d -1) / 2 p ). At finite value of the amplitude the frequency blows up, but the mass stays constant. These plots are made for a massless scalar in asymptotically AdS 5 but similar behavior occurs for massive fields and in other dimensions.</caption>
</figure>
<text><location><page_4><loc_52><loc_15><loc_92><loc_38></location>AdS/CFT dictionary, such field profile leads to the oscillating expectation value of the dual operator O in the form (1). One can add backreaction either perturbatively or perform numerics. Following the approach of [19], we constructed such solutions numerically. In short, one truncates the expansion (8) at some big values of i, j and then requires the Einstein equations to be satisfied on a set of collocation points in space and time. We fix the amplitude ε by requiring f 0 , 0 = ε . The resulting mass M and frequency Ω for asymptotically AdS 5 space and massless scalar field (which is the case relevant for N = 4 super Yang-Mills) are shown on Figure 3. One distinct feature we find for various spacetime dimensions and various masses is that the frequency Ω blows up at a finite value of the scalar field amplitude, whereas the mass stays finite.</text>
<text><location><page_4><loc_52><loc_9><loc_92><loc_14></location>One interesting question is whether for AdS 3 these solutions can have mass above the BTZ black hole threshold. For a massless field with Dirichlet boundary conditions the maximal mass appears to be much below.</text>
<section_header_level_1><location><page_5><loc_19><loc_92><loc_39><loc_93></location>A. Entanglement entropy</section_header_level_1>
<text><location><page_5><loc_9><loc_79><loc_49><loc_90></location>Let us discuss the entanglement entropy. Since the metric is time-dependent the entanglement entropy is expected to be time-dependent too. However, we do not expect it to change a lot during the period of one oscillation, therefore we will concentrate on the time-symmetric t = 0 slice for which we can use a simple RT prescription. For small subsystems of linear size s the entanglement entropy grows very slowly:</text>
<formula><location><page_5><loc_21><loc_76><loc_49><loc_77></location>S vN ∼ ε 2 s d , ε 2 s d ≪ 1 . (9)</formula>
<text><location><page_5><loc_9><loc_61><loc_49><loc_75></location>(as usual, in the units of ( l/l p ) d -1 ). The origin of this equation is the following. Oscillon has a finite energy density M ∼ ⟨ T tt ⟩ at the boundary, proportional to ε 2 . The behavior ⟨ T tt ⟩ s d for small subsystems was previously proved in [61, 62]: for small subsystems the RT surface lies near the boundary and the only important parameter in the metric is M ∼ ⟨ T tt ⟩ , s d comes from dimension analysis. Of course, this does not imply volume law entanglement or area law, for that we need to look at large subsystems.</text>
<text><location><page_5><loc_9><loc_35><loc_49><loc_60></location>The problem is that we are dealing with global AdS , so the boundary CFT lives on a sphere. But what is a large subsystem of a sphere? Similar problem arises in condensed matter setups because they study systems of finite number of spins. In principle, we can do a Weyl transformation to map the state of the CFT from a sphere to a plane. On the gravity side it corresponds to appropriately selecting a Poincare path inside global AdS . However, the resulting state will be time-dependent and inhomogeneous [63, 64], so it is not very useful. One natural thing to do is to compute the entropy for half of the system. Then we have only two parameters because we have a CFT: radius of the boundary sphere r (which we set to unity) and energy density M ∼ ⟨ T tt ⟩ . The entanglement entropy depends only on the effective dimensionless length r ⟨ T tt ⟩ 1 /d ∼ rM 1 /d . Then the volume-law entanglement in d -1 spacial dimension can be associated with</text>
<formula><location><page_5><loc_12><loc_32><loc_49><loc_33></location>volume law: S vN ∼ ( rM 1 /d ) d -1 ∼ M ( d -1) /d (10)</formula>
<text><location><page_5><loc_9><loc_30><loc_40><loc_31></location>growth for large M , whereas the area-law is</text>
<formula><location><page_5><loc_13><loc_27><loc_49><loc_28></location>area law: S vN ∼ ( rM 1 /d ) d -2 ∼ M ( d -2) /d . (11)</formula>
<text><location><page_5><loc_9><loc_17><loc_49><loc_26></location>This is the same as conventional area and volume law entanglement: we can fix a smaller subsystem of size s and send M to infinity, s to zero, keeping s d M fixed, but big. In this limit CFT is effectively decompactified and large s d M governs the entanglement behavior of large subsystems.</text>
<text><location><page_5><loc_9><loc_9><loc_49><loc_17></location>Black holes yield volume-law M ( d -1) / 2 . This can be understood without any computations: in this regime the horizon lies very close to x = π/ 2, namely x h ∼ π/ 2 -1 /M 1 /d . The RT surface will wrap around the horizon and most of its length will come from a disk x = x h , which area is proportional to 1 / cos d -1 ( x h ) ∼ M ( d -1) /d .</text>
<figure>
<location><page_5><loc_52><loc_75><loc_93><loc_93></location>
<caption>Figure 4. Entanglement entropy (again, in units of ( l/l p ) d -1 ) of half the system for AdS 5 black hole and oscillon.</caption>
</figure>
<text><location><page_5><loc_52><loc_56><loc_92><loc_67></location>Unfortunately, for oscillons M has a maximum value, so we cannot distinguish the area law and the volume law. The only thing we can verify is that oscillons have lower entanglement entropy, compared to black holes. This is indeed the case as illustrated by Figure 4. In the next Section we will study boson stars for which the mass M can be unbounded. We will see that they indeed exhibit area law.</text>
<section_header_level_1><location><page_5><loc_55><loc_52><loc_88><loc_53></location>B. A comment on N = 4 super Yang-Mills</section_header_level_1>
<text><location><page_5><loc_52><loc_38><loc_92><loc_50></location>N = 4 super Yang-Mills is dual to AdS 5 × S 5 solution in IIB ten-dimensional supergravity. This background is sourced by (self-dual) 4-form field H . We would like to claim that there exist oscillons in this background which only propagate along AdS 5 part. Meaning that this solution keeps the radius of S 5 constant. The relevant oscillating scalar field is the dilaton ϕ or the axion χ . In the Einstein frame the Lagrangian looks like</text>
<formula><location><page_5><loc_53><loc_33><loc_92><loc_37></location>1 16 πG (10) N ( R -1 2 ( ∂ϕ ) 2 -1 2 e 2 ϕ ( ∂χ ) 2 -1 4 ( dH ) 2 ) , (12)</formula>
<text><location><page_5><loc_52><loc_30><loc_87><loc_32></location>where G (10) N is ten-dimensional Newton constant.</text>
<text><location><page_5><loc_52><loc_25><loc_92><loc_30></location>There is non-trivial flux of H through S 5 , but the dilaton and axion are constant. Since H has traceless stressenergy tensor, Einstein equations with non-constant ϕ and χ can be written as</text>
<formula><location><page_5><loc_59><loc_22><loc_92><loc_23></location>R µν = ∂ µ ϕ∂ ν ϕ + e 2 ϕ ∂ µ χ∂ ν χ + T H µν . (13)</formula>
<text><location><page_5><loc_52><loc_16><loc_92><loc_20></location>Hence, if ϕ and χ do not vary over the S 5 , one can make an ansatz for AdS 5 deformation like eq. (3), but with S 5 having a constant radius.</text>
<text><location><page_5><loc_52><loc_9><loc_92><loc_16></location>In these units (which are different from the rest of the paper) dilaton ϕ is dimensionless, the corresponding dual operator is Tr F 2 /g 2 Y M , its two-point function is proportional to N 2 ( F is the gauge field strength). In Section IV we will discuss the state preparation. In order</text>
<text><location><page_6><loc_9><loc_87><loc_49><loc_93></location>to produce order 1 correction to the metric, the operator insertion at the Euclidean disk should have the form exp ( ˜ ε 1 g 2 Y M Tr F 2 ) , where ˜ ε is order 1 small number (say, 0.01).</text>
<text><location><page_6><loc_9><loc_72><loc_49><loc_87></location>Strictly speaking[65], the stability of oscillons has been shown only for the AdS metric and the scalar field perturbations. In the linear regime the S 5 part will add extra scalar fields charged under SO (6) symmetry. We do not expect that these extra fields will destabilize the oscillon. For empty AdS the extra fields have normal modes with frequencies away from zero and for small oscillon amplitudes the shift of the normal mode frequencies will be small. But it would be instructive to study this question more carefully.</text>
<section_header_level_1><location><page_6><loc_21><loc_68><loc_37><loc_69></location>III. BOSON STARS</section_header_level_1>
<text><location><page_6><loc_9><loc_43><loc_49><loc_66></location>In this Section we make a minimal modification and study a 'phenomenological' holographic model: Einstein-Maxwell theory minimally coupled to a complex scalar field. We refer to [43, 66-69] for more 'realistic' Einstein-Maxwell-scalar theories arising from higherdimensional supergravities. For simplicity, we consider massless scalar in 1 + 3 dimensions. This theory has a plethora of different phases and solutions, including hairy black holes and boson stars [70-72], depending on the value of the charge. In this paper we would like to point out the existence of an extra family of heavy boson stars, which we call C-stars . It was first discovered in [70] but then overlooked in the subsequent works. Here we study their properties in more detail, demonstrate their linear stability and argue that they represent scar states in the dual CFT.</text>
<text><location><page_6><loc_9><loc_33><loc_49><loc_42></location>It is expected that there are no global symmetries is quantum gravity, this is why we do not study a complex field with global U (1). By AdS/CFT correspondence gauged U (1) symmetry in the bulk corresponds to a global symmetry in the CFT. Gauge coupling constant is related to the coefficient in the operator product expansion (OPE) of two current operators in the CFT.</text>
<text><location><page_6><loc_10><loc_31><loc_23><loc_32></location>The Lagrangian is</text>
<formula><location><page_6><loc_12><loc_27><loc_49><loc_30></location>L = 1 16 πG N ( R +2Λ) -| D µ ϕ | 2 -1 4 F µν F µν , (14)</formula>
<formula><location><page_6><loc_21><loc_23><loc_36><loc_24></location>D µ ϕ = ∂ µ ϕ -ieA µ ϕ.</formula>
<text><location><page_6><loc_9><loc_9><loc_49><loc_21></location>We again put l d -1 p = 8 πG N = 1 and measure the scalar field in unites of l -( d -1) / 2 p and gauge field A µ in units of ll -( d -1) / 2 p . We can do rescaling of the boson star equations, which reveals that the only important parameter (in addition to the value of the fields) is e eff = el/l ( d -1) / 2 p . In holography we expect it to be of order 1, because the interaction strength is of order the gravitational one (suppressed by the CFT central charge).</text>
<text><location><page_6><loc_52><loc_89><loc_92><loc_93></location>We again study spherically-symmetric solutions in the form (3), but the scalar field has a simple one-harmonic time-behavior:</text>
<formula><location><page_6><loc_65><loc_86><loc_92><loc_88></location>ϕ ( t, x ) = e -i Ω t ϕ ( x ) . (15)</formula>
<text><location><page_6><loc_52><loc_55><loc_92><loc_85></location>In the limit of vanishing backreaction (very small amplitude), ϕ ( x ) are the normal modes (6) inside empty AdS d +1 . Because the stress-energy tensor is proportional to | ϕ | 2 , the actual metric is time-independent. This is why to find the solutions we can use a simple shooting method. Since the equations of motion for the scalar field are singular both at the origin x = 0 and at the AdS boundary x = π/ 2, we step away from the origin using power series expansion at the origin, numerically integrate the equations up to a point x 1 close to the boundary and then use the scalar field, the gauge field and the metric functions values at x 1 to fit an asymptotic power series expansion near the boundary. We then shoot for the scalar field frequency Ω to match the scalar field derivative ϕ ' l ( x 1 ) found by numeric integration to the scalar field derivative found from the asymptotic ϕ ' r ( x 1 ). We again impose Dirichlet boundary conditions for the scalar, such that near the boundary ˜ ϕ ∼ ( π/ 2 -x ) d . The only non-zero component of the vector potential is A t . This component and the metric has the following expansion near the boundary:</text>
<formula><location><page_6><loc_60><loc_51><loc_92><loc_53></location>A t ( x ) ≈ Ql d -2 ( π/ 2 -x ) d -2 + . . . , (16)</formula>
<formula><location><page_6><loc_61><loc_47><loc_92><loc_48></location>A ( x ) ≈ 1 -2 M ( π/ 2 -x ) d + . . . (17)</formula>
<text><location><page_6><loc_52><loc_40><loc_92><loc_45></location>Before diving inside the details, let us discuss the expectations from the CFT side. From the CFT perspective such state has non-zero global U (1) charge density ∝ Q and 1-point expectation value of a charged operator O :</text>
<formula><location><page_6><loc_66><loc_37><loc_92><loc_39></location>⟨ Ψ |O| Ψ ⟩ ∼ e -i Ω t . (18)</formula>
<text><location><page_6><loc_52><loc_25><loc_92><loc_36></location>As in the case of oscillons, we are interested in the 'large' masses and electric charges, of the order of the CFT central charge. In this case in a given charge sector, the lowest energy state has non-zero energy density, proportional to the mass M . In the limit of large charge Q there are many field-theory results relating Q to M [73-80]. Specifically for 2 + 1 CFT it is expected that M ∼ Q 3 / 2 and in 3 + 1 dimensions M ∼ Q 4 / 3 .</text>
<text><location><page_6><loc_52><loc_17><loc_92><loc_24></location>It would be convenient to explore the space of gravity solutions by fixing the amplitude | ϕ (0) | ≡ ε of the scalar field at the center x = 0 and the asking what discrete set of frequencies Ω are allowed. In short, there are three phases, depending on the effective charge [71, 72, 81] e eff .</text>
<text><location><page_6><loc_52><loc_9><loc_92><loc_17></location>'Boring' weak coupling phase: e eff < e eff crit , 1 : In this case we can start from a normal mode solution e j ( x ) and increase the amplitude. It turns out that all solutions have a bounded mass: at first the mass grows with the amplitude | ϕ (0) | , but then reaches the maximum and decreases. For amplitudes above the maximum of the</text>
<figure>
<location><page_7><loc_14><loc_72><loc_49><loc_92></location>
</figure>
<figure>
<location><page_7><loc_14><loc_51><loc_50><loc_71></location>
</figure>
<figure>
<location><page_7><loc_51><loc_51><loc_87><loc_71></location>
<caption>Figure 5. Asymptotically AdS 4 , intermediate coupling e eff crit , 1 < e eff = 2 . 2 < e eff crit , 2 : frequency Ω and mass M of fundamental boson stars (blue) and C-stars (orange and green, almost coincide) as a function of the scalar field value at the center | ϕ (0) | = ε . Both the fundamental (dashed blue), the second branch fundamental (solid blue) and C-stars (orange, green) has a local maximum in the mass, after which the solutions with bigger amplitudes are linearly unstable.</caption>
</figure>
<text><location><page_7><loc_9><loc_27><loc_49><loc_40></location>mass the solution is linearly unstable. Technically the Figure 5 illustrates the intermediate coupling phase, but the qualitative behavior of the normal modes is the same. Dashed blue (representing fundamental mode e 0 ( x )) and orange (first excited mode e 1 ( x )) shows the behavior of the frequency and ADM mass. Analytical arguments suggest [82] that e eff crit , 1 = √ 3 / 2. Our numerical results are consistent with this prediction, although the shooting becomes increasingly hard near the critical point.</text>
<text><location><page_7><loc_9><loc_9><loc_49><loc_26></location>Intermediate coupling: e eff crit , 1 < e eff < e eff crit , 2 , illustrated by Figure 5. In this regime the solutions connected to the perturbative normal modes e j ( x ) behave qualitatively similar (dashed lines). Interestingly, above certain amplitude additional solutions appear (solid lines). These solutions can have different number of zeros. The one with no zeros (solid blue) is usually called 'the second branch of the fundamental mode' in the literature [71, 72, 81]. Why it is called the second branch will become apparent from its behavior in the strong coupling phase. Take the fundamental second branch with no zeros (solid blue). This branch has unbounded mass, but</text>
<text><location><page_7><loc_52><loc_10><loc_92><loc_40></location>is linearly stable. Does it signal the presence of a scar? Our answer for this question is: probably not. This state has a lower mass compared to the extremal ReissnerNordstrom (RN) black hole of the same mass [72], it satisfies [83] the relation M ∼ Q 3 / 2 expected for the ground state of a CFT. Moreover, it is horizonless, hence it is dual to a pure state of the boundary theory. Hence, we can expect that it is actually dual to (or at least very close to) a ground state of the CFT, as was proposed in [83]. Below we will also show that it has area law entanglement. Interestingly, we find solutions with unbounded mass which has more than one zero in the scalar profile: 1-node (solid orange) and 2-node (green) in the Figure 5, although they almost coincide. However, we will abstain from calling them 'the second branch of excited modes'. Instead, we call them 'C-stars'. Again, the reason for this will become clear from the strong coupling behavior. They are significantly heavier than the fundamental (0-node) solution, so they are not close to the ground state at a fixed charge. We claim that these C-stars are approximate scar states. Approximate means</text>
<figure>
<location><page_7><loc_52><loc_72><loc_87><loc_92></location>
</figure>
<figure>
<location><page_8><loc_13><loc_72><loc_49><loc_92></location>
<caption>Figure 6. Asymptotically AdS 4 , strong coupling e eff = 3 > e eff crit , 2 . Upper panel: frequency Ω, mass M . Lower panel: charge Q and the lowest frequency of linearized perturbations. Orange and green solid curve is the C-star: it is a solution where the scalar field has one (orange) or two (green) zeros, but it is not continuously connected to the normal modes e 1 , 2 ( x ) of the scalar, which is shown in dashed. C-star is linearly stable: its perturbations frequencies are close to zero, but they are real.</caption>
</figure>
<figure>
<location><page_8><loc_10><loc_21><loc_47><loc_40></location>
<caption>Figure 7. Asymptotically AdS 4 : boundary EE of half the system for extremal RN black hole, C-star and the (second branch) fundamental boson star in the intermediate regime when e eff crit , 1 < e eff = 2 . 2 < e eff crit , 2 . Dots are numerical data, solid lines represent a fit. The corresponding phase diagram is shown in Figure 5.</caption>
</figure>
<text><location><page_8><loc_12><loc_21><loc_25><loc_22></location>Fund. second branch</text>
<text><location><page_8><loc_28><loc_21><loc_35><loc_22></location>Black hole</text>
<text><location><page_8><loc_38><loc_21><loc_39><loc_22></location>1</text>
<text><location><page_8><loc_39><loc_21><loc_39><loc_22></location>-</text>
<text><location><page_8><loc_39><loc_21><loc_44><loc_22></location>node C</text>
<text><location><page_8><loc_44><loc_21><loc_45><loc_22></location>-</text>
<text><location><page_8><loc_45><loc_21><loc_47><loc_22></location>star</text>
<text><location><page_8><loc_52><loc_13><loc_92><loc_40></location>that they are not exact energy eigenstates, as discussed in the Introduction. To backup this statement, we evaluated the entanglement entropy of half the system for different values of M - Figure 7. As we explained in Section II A, this probes the entanglement structure in the infinite-volume limit. We indeed find area-law entanglement M ( d -2) /d = M 1 / 3 , in contrast to the volume law M ( d -1) /d = M 2 / 3 of the extremal RN black hole. Actually, depending on the mass, the relevant RT surface can have different configurations - Figure 9. There is a simple RT surface (pink line) slicing through the equatorial plane, which yields area-law entanglement M 1 / 3 . We found that it always dominates for large enough M , for both C-stars and the second branch of the fundamental mode. However, there is another RT surface which avoids the strong gravity region by curving around it (red line). For some C-stars, it dominates if M is not too large, resulting in a entanglement shadow [84-86]: an area in the bulk which cannot be probed with extremal surfaces.</text>
<text><location><page_8><loc_52><loc_9><loc_92><loc_11></location>The planar limit of a heavy fundamental boson star should coincide with zero temperature limit of a holo-</text>
<figure>
<location><page_9><loc_10><loc_71><loc_48><loc_93></location>
<caption>Figure 8. Asymptotically AdS 4 : boundary EE of half the system for extremal RN black hole, C-star and fundamental boson star in the strong coupling regime when e eff = 3 > e eff crit , 2 . Dots are numerical data, solid lines represent a fit. The corresponding phase diagram is shown in Figure 6.</caption>
</figure>
<text><location><page_9><loc_9><loc_49><loc_49><loc_58></location>graphic superconductor [87]. One can verify independently that those have sub-volume law entanglement. However, they seem to posses an interesting phenomena that at intermediate distances there is volume law scaling (c.f. Figures 7,8). It would be interesting to understand this entanglement pattern from the boundary perspective.</text>
<text><location><page_9><loc_10><loc_47><loc_36><loc_48></location>Our numerics suggests e eff crit , 2 ≈ 2 . 3.</text>
<text><location><page_9><loc_9><loc_28><loc_49><loc_46></location>Strong coupling: e eff > e eff crit , 2 . In this case the two branches of the fundamental mode merge: one can start from small-amplitude normal mode solution and monotonically increase mass to infinity - Figure 6 (blue). This is the origin of the name 'second branch' in the intermediate coupling phase. In contrast, this does not happen (at least for e eff ≤ 20, because we are limited by numerics) to the excited modes and C-stars - Figure 6 (orange, green). This is why we do not call C-stars 'the second branch of the excited modes'. Both C-stars and the fundamental boson star are superextremal - Figure 10 and have area-law entanglement - Figure 8. So they continue being scar states at strong coupling.</text>
<text><location><page_9><loc_9><loc_9><loc_49><loc_27></location>One important question is the stability of the solutions we found. This question is important because unstable solutions can be highly sensitive to the parameters of the theory. For example, in this paper we neglected a possible explicit self-interaction of the scalar field (apart from the one mediated by gauge field and gravity). In holographic theories we expect the bulk fields interactions to be small, but non-zero. The first step to understand boson star stability is to consider normal modes of the linearized boson star perturbations. We follow the method outlined by [72], which reduces the system of equations for spherically symmetric linearized perturbations of a boson star to a system of three linear equations, two for</text>
<figure>
<location><page_9><loc_55><loc_66><loc_89><loc_93></location>
<caption>Figure 9. Time-slice ( x, θ ) of a C-star solution, ds 2 = 1 cos( x ) 2 ( A ( x ) -1 dx 2 +sin( x ) 2 ( dθ 2 +sin( θ ) 2 dφ 2 ) ) which is asymptotically global AdS 4 (each point contains a circle φ which we suppressed). The gray circle indicates the conformal boundary at x = π/ 2.</caption>
</figure>
<text><location><page_9><loc_59><loc_65><loc_61><loc_66></location>area</text>
<text><location><page_9><loc_61><loc_65><loc_62><loc_66></location>-</text>
<text><location><page_9><loc_62><loc_65><loc_70><loc_66></location>law RT surface</text>
<text><location><page_9><loc_74><loc_65><loc_78><loc_66></location>volume</text>
<text><location><page_9><loc_78><loc_65><loc_79><loc_66></location>-</text>
<text><location><page_9><loc_79><loc_65><loc_87><loc_66></location>law RT surface</text>
<text><location><page_9><loc_52><loc_23><loc_92><loc_53></location>real and imaginary scalar field perturbations and one for gauge potential perturbations, and then uses Chebyshev pseudospectral collocation method to find normal modes of these equations. The star is stable when all the modes have real frequencies and becomes unstable when at least one of the frequencies becomes imaginary. The point of transition between the stable and the unstable parts of the branches of boson stars happens when the mass curve encounters an extremum M ' ( ε ) = 0, there is a good argument to that [72] that follows a well-known argument for fluid stars [88] and it stays valid for all the AdS boson stars we have encountered. Furthermore, linearly stable boson stars in AdS are known to be non-linearly stable as well [19, 40, 72]. Figures 5 and 6 show the square of the frequency of the lowest normal mode, all the parts of the branches with unbounded mass (separated by a mass extremum) are stable. Because they are linearly stable, we do not expect that self-interaction or higher-derivative curvature terms will affect the solutions. Direct studies of fundamental boson stars in various theories [40, 89-97] support this intuition.</text>
<section_header_level_1><location><page_9><loc_60><loc_18><loc_83><loc_19></location>IV. STATE PREPARATION</section_header_level_1>
<text><location><page_9><loc_52><loc_9><loc_92><loc_16></location>We have identified a bulk field configuration which has scar properties. How do we prepare this state using CFT path integral? The calculations we presented above concerned purely classical system of Einstein gravity with a scalar field. It means that in the QFT language we are</text>
<figure>
<location><page_10><loc_9><loc_72><loc_48><loc_93></location>
<caption>Figure 10. Mass M vs charge Q in the strong coupling regime e eff > e eff , 2 . 1-node C-star solution lies slightly below the extremal RN curve.</caption>
</figure>
<text><location><page_10><loc_9><loc_59><loc_49><loc_63></location>dealing with a coherent state of that scalar field. The question of preparing such coherent states was addressed in a series of papers [98-103].</text>
<text><location><page_10><loc_9><loc_35><loc_49><loc_59></location>Since we want a single-mode configuration at t = 0, which is spherically symmetric, we prepare CFT state on a sphere using disk (hemisphere) path integral, where we insert a single e ˜ ε O operator at the center of the disk - Figure 2. O is CFT single-trace operator dual to the field ϕ in the bulk. For N = 4 super Yang-Mills such operator could be Tr F 2 or Tr ˜ FF ( F being the gauge field strength) which is marginal and has no R -charge in order to leave the S 5 part of the bulk geometry intact. Constant ˜ ε is proportional (up to an order-one number) to the bulk scalar field amplitude ε . If the twopoint function of O is normalized to 1, then ˜ ε can be large, of order the square-root central charge, such that gravity backreaction becomes important. This state is fine-tuned: moving the operator away from the pole will create a spherically non-symmetric configuration which will collapse to a black hole at times of order 1 /ε 2 .</text>
<text><location><page_10><loc_9><loc_32><loc_49><loc_34></location>In order to read off the bulk configuration we need bulk-to-boundary propagator which for global AdS is</text>
<formula><location><page_10><loc_10><loc_27><loc_49><loc_31></location>K ( x, ˆ e, t ; ˆ e ' , t ' ) = ( cos( x ) cos( t -t ' ) + sin( x )(ˆ e, ˆ e ' ) ) ∆ , (19)</formula>
<text><location><page_10><loc_9><loc_9><loc_49><loc_26></location>where unit vector ˆ e parameterize S d -1 of AdS d +1 . Also it is important to keep in mind that we are interested in expectation values in the form ⟨ Ψ | · | Ψ ⟩ , so the path integral involves both south and north hemispheres. Putting the operators at the poles basically [104] sets (ˆ e, ˆ e ' ) = 0. This yields ϕ ∝ ˜ ε cos( x ) ∆ , ∂ t ϕ = 0 profile at t = 0, which is what we want for an oscillon. In case of a complex field, we get ϕ ∝ ˜ ε cos( x ) ∆ , ∂ t ϕ ∝ i ∆ ˜ ε cos( x ) ∆ , which is the relevant configuration for a boson star. Excitations with higher radial numbers can be obtained by acting with derivatives. For example, for the first excited mode we need to insert ∂ 2 O (0), ∂ 2 being the Laplacian on the</text>
<text><location><page_10><loc_52><loc_84><loc_92><loc_93></location>sphere. In a generic quantum field theory an insertion in the form e ˜ ε O (0) is not well-defined. Thanks to the stability of boson stars/oscillons one can introduce a small smearing e ˜ ε ∫ d d z O . Such operator is well-defined and it would be interesting to investigate whether such operators lead to scars beyond holographic CFTs.</text>
<text><location><page_10><loc_52><loc_77><loc_92><loc_84></location>Of course, this is just a leading order in ˜ ε (that is, in ε ) answer. One can solve bulk equations of motions perturbatively and then prepare the configuration at t = 0 exactly by placing the appropriate Euclidean sources. We refer to [103] for a discussion.</text>
<section_header_level_1><location><page_10><loc_54><loc_71><loc_90><loc_74></location>V. VOLUME-LAW ENTANGLEMENT AND WEAK ENERGY CONDITIONS</section_header_level_1>
<text><location><page_10><loc_52><loc_59><loc_92><loc_69></location>In the previous sections we demonstrated that oscillons have smaller entanglement entropy compared to black holes and boson stars have parametrically smaller entanglement entropy compared to a black hole of the same mass. Namely, for half-system the black hole answer is 'volume-law' M ( d -1) /d , whereas for C-stars it is 'arealaw' M ( d -2) /d .</text>
<text><location><page_10><loc_52><loc_46><loc_92><loc_59></location>In this Section we show that imposing the weak-energy condition in the bulk guarantees that the CFT entanglement entropy is smaller compared to the one of a thermal state of the same mass, which is a well-known statement from the statistical mechanics. Unfortunately, our arguments are not sensitive to the presence/absence of the horizon. It would be interesting to understand how the requirement of horizon absence further bounds the entanglement.</text>
<text><location><page_10><loc_52><loc_42><loc_92><loc_46></location>Consider a static space-time (3) with some matter fields, with or without a horizon. Weak energy condition T tt ≥ 0 for the matter stress-energy tensor yields</text>
<formula><location><page_10><loc_60><loc_37><loc_92><loc_40></location>∂ x A ≤ d -2 + 2 sin 2 ( x ) sin( x ) cos( x ) (1 -A ) . (20)</formula>
<text><location><page_10><loc_52><loc_33><loc_92><loc_36></location>Gronwall theorem implies[105] that A can be bounded by the solution of the corresponding differential equation:</text>
<formula><location><page_10><loc_64><loc_29><loc_92><loc_32></location>A ≥ 1 -2 M cos( x ) d sin( x ) d -2 . (21)</formula>
<text><location><page_10><loc_52><loc_9><loc_92><loc_27></location>The right hand side is the value of A for AdS black hole of mass M . In particular, it implies that if there is a horizon, it lies inside a would-be black hole of the same mass[106]. Imagine now, that we fix a boundary subregion and the energy density M . Then the RT prescription in the corresponding black hole background with this mass will yield a certain co-dimension 2 surface. Now, if we compute the area of the same surface in the geometry of interest of the same energy density, the area will be lower because of the inequality (21). But the RT prescription instructs us to find the minimum over all possible surfaces, so the actual RT answer will be even lower.</text>
<section_header_level_1><location><page_11><loc_21><loc_92><loc_37><loc_93></location>VI. CONCLUSION</section_header_level_1>
<text><location><page_11><loc_9><loc_68><loc_49><loc_90></location>In this paper we studied the properties of oscillons and boson stars in asymptotically AdS spacetime. These are time-periodic, horizonless, solitonic solutions and we argued that they are linearly stable, have low-entanglement and are easily prepared with the Euclidean path integral. In the dual CFT they signal the presence of scars states. We demonstrated that excited boson stars have area law entanglement. In contrast, in low dimensional spin-chains scar states often have logarithmic scaling of entanglement. It is an interesting question if it is possible to obtain something like this in holographic theories in higher dimensions. Also it would be interesting to go beyond entanglement and understand other properties of scar states, both holographic and not. For example, entanglement can be used as a probe for confinement [107].</text>
<text><location><page_11><loc_9><loc_58><loc_49><loc_68></location>Oscillons and boson stars only require the presence of a scalar field in a theory, hence they represent a generic phenomenon for holographic theories. As we mentioned in the Introduction, one can even build solitonic objects (geons) purely from the gravitational degrees of freedom [20]. It would be interesting to perform the analysis of this paper for geons.</text>
<text><location><page_11><loc_9><loc_55><loc_49><loc_58></location>There are two important take-aways, one for the holographic CFTs and one for the gravity.</text>
<text><location><page_11><loc_9><loc_48><loc_49><loc_55></location>The gravity predicts that for holographic CFTs the states corresponding to e ˜ ε O (prepared by Figure 2) are non-thermalizing finite energy-density states. It means that in any holographic CFT, 4-point correlation function of the form</text>
<formula><location><page_11><loc_24><loc_45><loc_49><loc_47></location>⟨ e ˜ ε O LLe ˜ ε O ⟩ , (22)</formula>
<text><location><page_11><loc_9><loc_28><loc_49><loc_44></location>where L are light fields evolved in Lorentzian signature, and e ˜ ε O are inserted at the poles of the Euclidean sphere, will not look thermal in any number of dimensions. It is important to emphasize that ˜ ε can be large, of the order of square-root central charge (if ⟨OO⟩ is normalized to 1), so this operator causes huge backreaction. In contrast, states prepared with the insertion finitely away from the pole will look thermal. The question of thermality of CFT correlators has been extensively studied before. For the case of heavy conformal primaries H in 2d CFT, it was argued in [108, 109] that 4-point function of the form</text>
<formula><location><page_11><loc_25><loc_26><loc_49><loc_27></location>⟨ HLLH ⟩ , (23)</formula>
<text><location><page_11><loc_9><loc_11><loc_49><loc_24></location>does look thermal. Also [110, 111] studied more complicated CFT states which effectively prepare collapsing dust shells in the dual gravity. In both cases, the large central charge limit and the vacuum dominance in the 4point function was enough to link the CFT results to the gravity black hole computation. It would be interesting to perform a similar CFT calculation for the (22) correlator and map the results to the oscillon (boson star) background.</text>
<text><location><page_11><loc_52><loc_85><loc_92><loc_93></location>On the other hand, all current examples of scar states in the condensed matter literature involve the presence of hidden symmetries [29-35, 112]. A possible hidden symmetry in behind oscillons was discussed in [113116]. Without a backreaction, a scalar field of mass m 2 = ∆(∆ -d ) in AdS d +1 posses a set of normal modes:</text>
<formula><location><page_11><loc_60><loc_81><loc_92><loc_82></location>ω jl = ∆ + l +2 j, j, l = 0 , 1 , . . . , (24)</formula>
<text><location><page_11><loc_52><loc_60><loc_92><loc_78></location>where l is the (integer) angular momentum and j is the (integer) radial quantum number. Notice that they enter only in l + 2 j combination. This highly resonant spectrum is a direct consequence of the conformal SO (2 , d ) symmetry of AdS . Such resonant spectrum is the reason why AdS is unstable, because non-linearities coming from gravity may produce secular corrections growing linearly in time. Surprisingly, it was argued in [117] that the same SO (2 , d ) symmetry forces the secular terms to vanish. This symmetry constrains a lot the leading nonlinear correction to the oscillon solution, but it would be interesting to understand what role this possibly weaklybroken symmetry plays for the full non-linear solution.</text>
<text><location><page_11><loc_52><loc_38><loc_92><loc_59></location>As we mentioned many times, oscillons and boson stars are only approximate energy eigenstates which only signal the presence of scar energy eigenstates in the spectrum. Is it is possible to construct a geometry dual to the actual scar eigenstate? One possible set of candidates are Lin-Lunin-Maldacena (LLM) [118] geometries. They are dual to half-BPS boundary operators [119, 120]. These geometries are complicated, but it would be nice to understand what entanglement law they have, some preliminary steps in this direction were made in [121]. However, LLM solutions are very special: they preserve supersymmetry and explicitly use the compact S 5 part of the geometry. Unlike oscillons and boson stars, we do not expect to find something like this in more generic holographic theories.</text>
<section_header_level_1><location><page_11><loc_63><loc_31><loc_81><loc_32></location>ACKNOWLEDGMENT</section_header_level_1>
<text><location><page_11><loc_52><loc_11><loc_92><loc_28></location>We would like to thank F. Popov and K. Pakrouski for for collaboration at the early states. We are grateful to D. Berenstein, A. Buchel, E. Colafranceschi, S. ColinEllerin, A. Dymarsky, A. Gorsky, A. Holguin, L. Lehner, D. Marolf, F. Pretorius, M. Srednicki, A. Zhiboedov and specially to S. Hellerman, G. Horowitz, I. Klebanov for comments and discussions. This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-19-1-0360. It was also supported in part by funds from the University of California. AM would like to thank C. King for moral support.</text>
<unordered_list>
<list_item><location><page_12><loc_10><loc_87><loc_49><loc_89></location>[1] L. A. Bunimovich, Communications in Mathematical Physics 65 , 295 (1979).</list_item>
<list_item><location><page_12><loc_10><loc_86><loc_43><loc_87></location>[2] E. J. Heller, Phys. Rev. Lett. 53 , 1515 (1984).</list_item>
<list_item><location><page_12><loc_10><loc_80><loc_49><loc_85></location>[3] H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletic, and M. D. Lukin, Nature 551 , 579 (2017).</list_item>
<list_item><location><page_12><loc_10><loc_78><loc_49><loc_80></location>[4] C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papi'c, Nature Physics 14 , 745 (2018).</list_item>
<list_item><location><page_12><loc_10><loc_72><loc_49><loc_77></location>[5] G.-X. Su, H. Sun, A. Hudomal, J.-Y. Desaules, Z.-Y. Zhou, B. Yang, J. C. Halimeh, Z.-S. Yuan, Z. Papi'c, and J.-W. Pan, Physical Review Research 5 (2023), 10.1103/physrevresearch.5.023010.</list_item>
<list_item><location><page_12><loc_10><loc_70><loc_49><loc_72></location>[6] M. Serbyn, D. A. Abanin, and Z. Papi'c, Nature Physics 17 , 675 (2021).</list_item>
<list_item><location><page_12><loc_10><loc_66><loc_49><loc_69></location>[7] S. Moudgalya, B. A. Bernevig, and N. Regnault, Rept. Prog. Phys. 85 , 086501 (2022), arXiv:2109.00548 [condmat.str-el].</list_item>
<list_item><location><page_12><loc_10><loc_58><loc_49><loc_65></location>[8] Z. Papic, 'Weak ergodicity breaking through the lens of quantum entanglement,' in Entanglement in Spin Chains: From Theory to Quantum Technology Applications , edited by A. Bayat, S. Bose, and H. Johannesson (Springer International Publishing, Cham, 2022) pp. 341-395.</list_item>
<list_item><location><page_12><loc_10><loc_54><loc_49><loc_57></location>[9] A. Chandran, T. Iadecola, V. Khemani, and R. Moessner, (2022), 10.1146/annurev-conmatphys031620-101617, arXiv:2206.11528 [cond-mat.str-el].</list_item>
<list_item><location><page_12><loc_10><loc_51><loc_49><loc_54></location>[10] L. Susskind, J. Math. Phys. 36 , 6377 (1995), arXiv:hepth/9409089 [hep-th].</list_item>
<list_item><location><page_12><loc_10><loc_47><loc_49><loc_51></location>[11] J. M. Maldacena, Int. J. Theor. Phys. 38 , 1113 (1999), [Adv. Theor. Math. Phys.2,231(1998)], arXiv:hepth/9711200 [hep-th].</list_item>
<list_item><location><page_12><loc_10><loc_45><loc_49><loc_47></location>[12] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Phys. Lett. B428 , 105 (1998), arXiv:hep-th/9802109 [hep-th].</list_item>
<list_item><location><page_12><loc_10><loc_42><loc_49><loc_44></location>[13] E. Witten, Adv. Theor. Math. Phys. 2 , 253 (1998), arXiv:hep-th/9802150 [hep-th].</list_item>
<list_item><location><page_12><loc_10><loc_39><loc_49><loc_42></location>[14] P. Bizon and A. Rostworowski, Phys. Rev. Lett. 107 , 031102 (2011), arXiv:1104.3702 [gr-qc].</list_item>
<list_item><location><page_12><loc_10><loc_37><loc_49><loc_39></location>[15] D. Garfinkle and L. A. Pando Zayas, Phys. Rev. D 84 , 066006 (2011), arXiv:1106.2339 [hep-th].</list_item>
<list_item><location><page_12><loc_10><loc_34><loc_49><loc_36></location>[16] J. Jalmuzna, A. Rostworowski, and P. Bizon, Phys. Rev. D 84 , 085021 (2011), arXiv:1108.4539 [gr-qc].</list_item>
<list_item><location><page_12><loc_10><loc_31><loc_49><loc_34></location>[17] A. Buchel, L. Lehner, and S. L. Liebling, Phys. Rev. D 86 , 123011 (2012), arXiv:1210.0890 [gr-qc].</list_item>
<list_item><location><page_12><loc_10><loc_27><loc_49><loc_31></location>[18] O. J. C. Dias, G. T. Horowitz, and J. E. Santos, Class. Quant. Grav. 29 , 194002 (2012), arXiv:1109.1825 [hepth].</list_item>
<list_item><location><page_12><loc_10><loc_25><loc_49><loc_27></location>[19] M. Maliborski and A. Rostworowski, Phys. Rev. Lett. 111 , 051102 (2013), arXiv:1303.3186 [gr-qc].</list_item>
<list_item><location><page_12><loc_10><loc_22><loc_49><loc_24></location>[20] G. T. Horowitz and J. E. Santos, Surveys Diff. Geom. 20 , 321 (2015), arXiv:1408.5906 [gr-qc].</list_item>
<list_item><location><page_12><loc_10><loc_18><loc_49><loc_22></location>[21] We are indebted to F. Popov for asking the first question and explaining to us the difference between the various scar phenomena.</list_item>
<list_item><location><page_12><loc_10><loc_16><loc_49><loc_18></location>[22] P. Hayden and J. Preskill, JHEP 09 , 120 (2007), arXiv:0708.4025 [hep-th].</list_item>
<list_item><location><page_12><loc_10><loc_13><loc_49><loc_15></location>[23] Y. Sekino and L. Susskind, JHEP 10 , 065 (2008), arXiv:0808.2096 [hep-th].</list_item>
<list_item><location><page_12><loc_10><loc_10><loc_49><loc_13></location>[24] S. H. Shenker and D. Stanford, JHEP 03 , 067 (2014), arXiv:1306.0622 [hep-th].</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_12><loc_53><loc_87><loc_92><loc_89></location>[25] S. H. Shenker and D. Stanford, JHEP 12 , 046 (2014), arXiv:1312.3296 [hep-th].</list_item>
<list_item><location><page_12><loc_53><loc_84><loc_92><loc_87></location>[26] J. Maldacena, S. H. Shenker, and D. Stanford, (2015), arXiv:1503.01409 [hep-th].</list_item>
<list_item><location><page_12><loc_53><loc_82><loc_92><loc_84></location>[27] B. Buˇca, J. Tindall, and D. Jaksch, Nature Communications 10 (2019), 10.1038/s41467-019-09757-y.</list_item>
<list_item><location><page_12><loc_53><loc_79><loc_92><loc_81></location>[28] M. Medenjak, B. Buˇca, and D. Jaksch, Physical Review B 102 (2020), 10.1103/physrevb.102.041117.</list_item>
<list_item><location><page_12><loc_53><loc_76><loc_92><loc_79></location>[29] C.-J. Lin and O. I. Motrunich, Physical Review Letters 122 (2019), 10.1103/physrevlett.122.173401.</list_item>
<list_item><location><page_12><loc_53><loc_74><loc_92><loc_76></location>[30] D. K. Mark, C.-J. Lin, and O. I. Motrunich, Physical Review B 101 (2020), 10.1103/physrevb.101.195131.</list_item>
<list_item><location><page_12><loc_53><loc_71><loc_92><loc_73></location>[31] J. Ren, C. Liang, and C. Fang, Phys. Rev. Lett. 126 , 120604 (2021), arXiv:2007.10380 [cond-mat.str-el].</list_item>
<list_item><location><page_12><loc_53><loc_67><loc_92><loc_71></location>[32] K. Pakrouski, P. N. Pallegar, F. K. Popov, and I. R. Klebanov, Phys. Rev. Res. 3 , 043156 (2021), arXiv:2106.10300 [cond-mat.str-el].</list_item>
<list_item><location><page_12><loc_53><loc_64><loc_92><loc_67></location>[33] J. Ren, C. Liang, and C. Fang, Phys. Rev. Res. 4 , 013155 (2022), arXiv:2108.07817 [cond-mat.str-el].</list_item>
<list_item><location><page_12><loc_53><loc_62><loc_92><loc_64></location>[34] S. Moudgalya and O. I. Motrunich, (2022), arXiv:2209.03377 [cond-mat.str-el].</list_item>
<list_item><location><page_12><loc_53><loc_59><loc_92><loc_61></location>[35] Z. Sun, F. K. Popov, I. R. Klebanov, and K. Pakrouski, (2022), arXiv:2212.11914 [cond-mat.str-el].</list_item>
<list_item><location><page_12><loc_53><loc_55><loc_92><loc_59></location>[36] O. J. C. Dias, G. T. Horowitz, D. Marolf, and J. E. Santos, Class. Quant. Grav. 29 , 235019 (2012), arXiv:1208.5772 [gr-qc].</list_item>
<list_item><location><page_12><loc_53><loc_52><loc_92><loc_55></location>[37] M. Srednicki, (1994), 10.1103/PhysRevE.50.888, arXiv:cond-mat/9403051.</list_item>
<list_item><location><page_12><loc_53><loc_51><loc_86><loc_52></location>[38] J. M. Deutsch, Phys. Rev. A 43 , 2046 (1991).</list_item>
<list_item><location><page_12><loc_53><loc_47><loc_92><loc_51></location>[39] A. M. Alhambra, A. Anshu, and H. Wilming, Phys. Rev. B 101 , 205107 (2020), arXiv:1911.05637 [quantph].</list_item>
<list_item><location><page_12><loc_53><loc_45><loc_92><loc_47></location>[40] A. Buchel, S. L. Liebling, and L. Lehner, Phys. Rev. D 87 , 123006 (2013), arXiv:1304.4166 [gr-qc].</list_item>
<list_item><location><page_12><loc_53><loc_42><loc_92><loc_44></location>[41] A. Buchel, S. R. Green, L. Lehner, and S. L. Liebling, (2014), arXiv:1410.5381 [hep-th].</list_item>
<list_item><location><page_12><loc_53><loc_38><loc_92><loc_42></location>[42] V. Balasubramanian, A. Buchel, S. R. Green, L. Lehner, and S. L. Liebling, Phys. Rev. Lett. 113 , 071601 (2014), arXiv:1403.6471 [hep-th].</list_item>
<list_item><location><page_12><loc_53><loc_37><loc_86><loc_38></location>[43] A. Buchel, (2015), arXiv:1510.08415 [hep-th].</list_item>
<list_item><location><page_12><loc_53><loc_34><loc_92><loc_36></location>[44] B. Craps, M. De Clerck, and O. Evnin, JHEP 09 , 030 (2021), arXiv:2103.12798 [hep-th].</list_item>
<list_item><location><page_12><loc_53><loc_31><loc_92><loc_34></location>[45] P. Caputa and D. Ge, (2022), arXiv:2211.03630 [hepth].</list_item>
<list_item><location><page_12><loc_53><loc_29><loc_92><loc_31></location>[46] D. Liska, V. Gritsev, W. Vleeshouwers, and J. Min'aˇr, (2022), arXiv:2212.05962 [hep-th].</list_item>
<list_item><location><page_12><loc_53><loc_26><loc_92><loc_28></location>[47] J. Cotler and A. Y. Wei, (2022), arXiv:2212.01637 [hepth].</list_item>
<list_item><location><page_12><loc_53><loc_22><loc_92><loc_26></location>[48] J.-Y. Desaules, D. Banerjee, A. Hudomal, Z. Papi'c, A. Sen, and J. C. Halimeh, Phys. Rev. B 107 , L201105 (2023), arXiv:2203.08830 [cond-mat.str-el].</list_item>
<list_item><location><page_12><loc_53><loc_18><loc_92><loc_22></location>[49] L. V. Delacretaz, A. L. Fitzpatrick, E. Katz, and M. T. Walters, JHEP 02 , 045 (2023), arXiv:2207.11261 [hepth].</list_item>
<list_item><location><page_12><loc_53><loc_16><loc_92><loc_18></location>[50] M. Dodelson and A. Zhiboedov, JHEP 12 , 163 (2022), arXiv:2204.09749 [hep-th].</list_item>
<list_item><location><page_12><loc_53><loc_13><loc_92><loc_15></location>[51] S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96 , 181602 (2006), arXiv:hep-th/0603001 [hep-th].</list_item>
<list_item><location><page_12><loc_53><loc_10><loc_92><loc_13></location>[52] V. E. Hubeny, M. Rangamani, and T. Takayanagi, JHEP 07 , 062 (2007), arXiv:0705.0016 [hep-th].</list_item>
<list_item><location><page_13><loc_10><loc_91><loc_49><loc_93></location>[53] O. J. C. Dias and J. E. Santos, Class. Quant. Grav. 35 , 185006 (2018), arXiv:1705.03065 [hep-th].</list_item>
<list_item><location><page_13><loc_10><loc_88><loc_49><loc_90></location>[54] N. Kim, Phys. Lett. B 742 , 274 (2015), arXiv:1411.1633 [hep-th].</list_item>
<list_item><location><page_13><loc_10><loc_85><loc_49><loc_88></location>[55] N. Kim and J. Hun Lee, J. Korean Phys. Soc. 69 , 623 (2016), arXiv:1512.02816 [hep-th].</list_item>
<list_item><location><page_13><loc_10><loc_83><loc_49><loc_85></location>[56] R.-G. Cai, L.-W. Ji, and R.-Q. Yang, Commun. Theor. Phys. 65 , 329 (2016), arXiv:1511.00868 [gr-qc].</list_item>
<list_item><location><page_13><loc_10><loc_77><loc_49><loc_82></location>[57] Einstein-Gauss-Bonnet gravity has a mass gap for black holes, making AdS stable again. The claim is that there are initial conditions, above this gap, which do not lead to black hole formation.</list_item>
<list_item><location><page_13><loc_10><loc_73><loc_49><loc_77></location>[58] N. Deppe, A. Kolly, A. Frey, and G. Kunstatter, Phys. Rev. Lett. 114 , 071102 (2015), arXiv:1410.1869 [hepth].</list_item>
<list_item><location><page_13><loc_10><loc_71><loc_49><loc_73></location>[59] N. Deppe, A. Kolly, A. R. Frey, and G. Kunstatter, JHEP 10 , 087 (2016), arXiv:1608.05402 [hep-th].</list_item>
<list_item><location><page_13><loc_10><loc_68><loc_49><loc_70></location>[60] C.-Y. Zhang, Z.-Y. Tang, and B. Wang, Phys. Rev. D 94 , 104013 (2016), arXiv:1608.04836 [gr-qc].</list_item>
<list_item><location><page_13><loc_10><loc_64><loc_49><loc_68></location>[61] J. Bhattacharya, M. Nozaki, T. Takayanagi, and T. Ugajin, Phys. Rev. Lett. 110 , 091602 (2013), arXiv:1212.1164 [hep-th].</list_item>
<list_item><location><page_13><loc_10><loc_60><loc_49><loc_64></location>[62] T. Faulkner, M. Guica, T. Hartman, R. C. Myers, and M. Van Raamsdonk, JHEP 03 , 051 (2014), arXiv:1312.7856 [hep-th].</list_item>
<list_item><location><page_13><loc_10><loc_58><loc_49><loc_60></location>[63] G. T. Horowitz and N. Itzhaki, JHEP 02 , 010 (1999), arXiv:hep-th/9901012 [hep-th].</list_item>
<list_item><location><page_13><loc_10><loc_55><loc_49><loc_57></location>[64] M. Nozaki, T. Numasawa, and T. Takayanagi, JHEP 05 , 080 (2013), arXiv:1302.5703 [hep-th].</list_item>
<list_item><location><page_13><loc_10><loc_52><loc_49><loc_55></location>[65] We are grateful to A. Buchel for a discussion on this point.</list_item>
<list_item><location><page_13><loc_10><loc_50><loc_49><loc_52></location>[66] M. Cvetic, H. Lu, and C. N. Pope, Phys. Rev. Lett. 83 , 5226 (1999), arXiv:hep-th/9906221.</list_item>
<list_item><location><page_13><loc_10><loc_44><loc_49><loc_49></location>[67] M. Cvetic, M. J. Duff, P. Hoxha, J. T. Liu, H. Lu, J. X. Lu, R. Martinez-Acosta, C. N. Pope, H. Sati, and T. A. Tran, Nucl. Phys. B 558 , 96 (1999), arXiv:hepth/9903214.</list_item>
<list_item><location><page_13><loc_10><loc_42><loc_49><loc_44></location>[68] M. Cvetic, H. Lu, and C. N. Pope, Phys. Rev. D 62 , 064028 (2000), arXiv:hep-th/0003286.</list_item>
<list_item><location><page_13><loc_10><loc_38><loc_49><loc_41></location>[69] M. Cvetic, H. Lu, C. N. Pope, A. Sadrzadeh, and T. A. Tran, Nucl. Phys. B 586 , 275 (2000), arXiv:hepth/0003103.</list_item>
<list_item><location><page_13><loc_10><loc_35><loc_49><loc_37></location>[70] S. Hu, J. T. Liu, and L. A. Pando Zayas, (2012), arXiv:1209.2378 [hep-th].</list_item>
<list_item><location><page_13><loc_10><loc_31><loc_49><loc_35></location>[71] O. J. C. Dias, P. Figueras, S. Minwalla, P. Mitra, R. Monteiro, and J. E. Santos, JHEP 08 , 117 (2012), arXiv:1112.4447 [hep-th].</list_item>
<list_item><location><page_13><loc_10><loc_29><loc_49><loc_31></location>[72] R. Arias, J. Mas, and A. Serantes, JHEP 09 , 024 (2016), arXiv:1606.00830 [hep-th].</list_item>
<list_item><location><page_13><loc_10><loc_26><loc_49><loc_28></location>[73] S. Hellerman, D. Orlando, S. Reffert, and M. Watanabe, JHEP 12 , 071 (2015), arXiv:1505.01537 [hep-th].</list_item>
<list_item><location><page_13><loc_10><loc_23><loc_49><loc_26></location>[74] S. Hellerman and S. Maeda, JHEP 12 , 135 (2017), arXiv:1710.07336 [hep-th].</list_item>
<list_item><location><page_13><loc_10><loc_21><loc_49><loc_23></location>[75] D. Jafferis, B. Mukhametzhanov, and A. Zhiboedov, JHEP 05 , 043 (2018), arXiv:1710.11161 [hep-th].</list_item>
<list_item><location><page_13><loc_10><loc_18><loc_49><loc_20></location>[76] G. Badel, G. Cuomo, A. Monin, and R. Rattazzi, Phys. Lett. B 802 , 135202 (2020), arXiv:1911.08505 [hep-th].</list_item>
<list_item><location><page_13><loc_10><loc_14><loc_49><loc_18></location>[77] S. Hellerman, S. Maeda, D. Orlando, S. Reffert, and M. Watanabe, JHEP 04 , 287 (2021), arXiv:2005.03021 [hep-th].</list_item>
<list_item><location><page_13><loc_10><loc_11><loc_49><loc_14></location>[78] S. Hellerman and D. Orlando, (2021), arXiv:2103.05642 [hep-th].</list_item>
<list_item><location><page_13><loc_10><loc_9><loc_49><loc_11></location>[79] L. Alvarez-Gaume, D. Orlando, and S. Reffert, JHEP 12 , 142 (2019), arXiv:1909.02571 [hep-th].</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_13><loc_53><loc_91><loc_92><loc_93></location>[80] G. Cuomo, Phys. Lett. B 812 , 136014 (2021), arXiv:2010.00407 [hep-th].</list_item>
<list_item><location><page_13><loc_53><loc_88><loc_92><loc_90></location>[81] S. A. Gentle, M. Rangamani, and B. Withers, JHEP 05 , 106 (2012), arXiv:1112.3979 [hep-th].</list_item>
<list_item><location><page_13><loc_53><loc_85><loc_92><loc_88></location>[82] O. J. C. Dias and R. Masachs, JHEP 02 , 128 (2017), arXiv:1610.03496 [hep-th].</list_item>
<list_item><location><page_13><loc_53><loc_83><loc_92><loc_85></location>[83] A. de la Fuente and J. Zosso, JHEP 06 , 178 (2020), arXiv:2005.06169 [hep-th].</list_item>
<list_item><location><page_13><loc_53><loc_79><loc_92><loc_82></location>[84] B. Czech, J. L. Karczmarek, F. Nogueira, and M. Van Raamsdonk, Class. Quant. Grav. 29 , 155009 (2012), arXiv:1204.1330 [hep-th].</list_item>
<list_item><location><page_13><loc_53><loc_75><loc_92><loc_78></location>[85] V. Balasubramanian, B. D. Chowdhury, B. Czech, and J. de Boer, JHEP 01 , 048 (2015), arXiv:1406.5859 [hepth].</list_item>
<list_item><location><page_13><loc_53><loc_71><loc_92><loc_74></location>[86] B. Freivogel, R. Jefferson, L. Kabir, B. Mosk, and I.-S. Yang, Phys. Rev. D 91 , 086013 (2015), arXiv:1412.5175 [hep-th].</list_item>
<list_item><location><page_13><loc_53><loc_68><loc_92><loc_70></location>[87] G. T. Horowitz and M. M. Roberts, JHEP 11 , 015 (2009), arXiv:0908.3677 [hep-th].</list_item>
<list_item><location><page_13><loc_53><loc_64><loc_92><loc_68></location>[88] S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (John Wiley and Sons, New York, 1972).</list_item>
<list_item><location><page_13><loc_53><loc_62><loc_92><loc_64></location>[89] M. Colpi, S. L. Shapiro, and I. Wasserman, Phys. Rev. Lett. 57 , 2485 (1986).</list_item>
<list_item><location><page_13><loc_53><loc_60><loc_83><loc_61></location>[90] M. Gleiser, Phys. Rev. D 38 , 2376 (1988).</list_item>
<list_item><location><page_13><loc_53><loc_58><loc_92><loc_60></location>[91] J. Balakrishna, E. Seidel, and W.-M. Suen, Phys. Rev. D 58 , 104004 (1998), arXiv:gr-qc/9712064.</list_item>
<list_item><location><page_13><loc_53><loc_56><loc_92><loc_57></location>[92] E. Seidel and W.-M. Suen, Phys. Rev. D 42 , 384 (1990).</list_item>
<list_item><location><page_13><loc_53><loc_54><loc_92><loc_56></location>[93] F. E. Schunck and D. F. Torres, Int. J. Mod. Phys. D 9 , 601 (2000), arXiv:gr-qc/9911038.</list_item>
<list_item><location><page_13><loc_53><loc_50><loc_92><loc_53></location>[94] N. Sanchis-Gual, C. Herdeiro, and E. Radu, Class. Quant. Grav. 39 , 064001 (2022), arXiv:2110.03000 [grqc].</list_item>
<list_item><location><page_13><loc_53><loc_47><loc_92><loc_49></location>[95] Y. Brihaye, B. Hartmann, and J. Riedel, Phys. Rev. D 92 , 044049 (2015), arXiv:1404.1874 [gr-qc].</list_item>
<list_item><location><page_13><loc_53><loc_44><loc_92><loc_47></location>[96] B. Hartmann, J. Riedel, and R. Suciu, Phys. Lett. B 726 , 906 (2013), arXiv:1308.3391 [gr-qc].</list_item>
<list_item><location><page_13><loc_53><loc_42><loc_92><loc_44></location>[97] L. J. Henderson, R. B. Mann, and S. Stotyn, Phys. Rev. D 91 , 024009 (2015), arXiv:1403.1865 [gr-qc].</list_item>
<list_item><location><page_13><loc_53><loc_39><loc_92><loc_41></location>[98] K. Skenderis and B. C. van Rees, JHEP 05 , 085 (2009), arXiv:0812.2909 [hep-th].</list_item>
<list_item><location><page_13><loc_53><loc_36><loc_92><loc_39></location>[99] K. Skenderis and B. C. van Rees, Phys. Rev. Lett. 101 , 081601 (2008), arXiv:0805.0150 [hep-th].</list_item>
<list_item><location><page_13><loc_52><loc_34><loc_92><loc_36></location>[100] M. Botta-Cantcheff, P. Mart'ınez, and G. A. Silva, JHEP 02 , 171 (2016), arXiv:1512.07850 [hep-th].</list_item>
<list_item><location><page_13><loc_52><loc_31><loc_92><loc_34></location>[101] A. Christodoulou and K. Skenderis, JHEP 04 , 096 (2016), arXiv:1602.02039 [hep-th].</list_item>
<list_item><location><page_13><loc_52><loc_27><loc_92><loc_31></location>[102] T. Faulkner, F. M. Haehl, E. Hijano, O. Parrikar, C. Rabideau, and M. Van Raamsdonk, JHEP 08 , 057 (2017), arXiv:1705.03026 [hep-th].</list_item>
<list_item><location><page_13><loc_52><loc_23><loc_92><loc_27></location>[103] D. Marolf, O. Parrikar, C. Rabideau, A. Izadi Rad, and M. Van Raamsdonk, JHEP 06 , 077 (2018), arXiv:1709.10101 [hep-th].</list_item>
<list_item><location><page_13><loc_52><loc_18><loc_92><loc_23></location>[104] This involves an auxiliary conformal transformation. For example, for AdS 3 , one introduces z = e it ' + il ' , z = e it ' -il ' , where l ' is angle in the equatorial plane. Center of the disk is z = z = 0. The propagator looks like</list_item>
</unordered_list>
<formula><location><page_13><loc_56><loc_14><loc_92><loc_17></location>K = ( 2 cos( x ) √ zz e it + e -it zz +sin( x )[ e il z + e -il z ] ) ∆ . (25)</formula>
<text><location><page_13><loc_56><loc_13><loc_56><loc_14></location>.</text>
<text><location><page_13><loc_52><loc_9><loc_92><loc_12></location>[105] Specifically we need to apply it for the interval [0 , π/ 2) in the vicinity of x = π/ 2 to connect constant M with spacetime mass.</text>
<unordered_list>
<list_item><location><page_14><loc_9><loc_91><loc_49><loc_93></location>[106] We can transfer surfaces from one geometry to another because we gauge-fixed the metric to be in the form (3).</list_item>
<list_item><location><page_14><loc_9><loc_88><loc_49><loc_90></location>[107] I. R. Klebanov, D. Kutasov, and A. Murugan, Nucl. Phys. B 796 , 274 (2008), arXiv:0709.2140 [hep-th].</list_item>
<list_item><location><page_14><loc_9><loc_85><loc_49><loc_88></location>[108] A. L. Fitzpatrick, J. Kaplan, and M. T. Walters, JHEP 11 , 200 (2015), arXiv:1501.05315 [hep-th].</list_item>
<list_item><location><page_14><loc_9><loc_83><loc_49><loc_85></location>[109] A. L. Fitzpatrick, J. Kaplan, D. Li, and J. Wang, JHEP 05 , 109 (2016), arXiv:1603.08925 [hep-th].</list_item>
<list_item><location><page_14><loc_9><loc_80><loc_49><loc_82></location>[110] T. Anous, T. Hartman, A. Rovai, and J. Sonner, JHEP 07 , 123 (2016), arXiv:1603.04856 [hep-th].</list_item>
<list_item><location><page_14><loc_9><loc_77><loc_49><loc_80></location>[111] T. Anous, T. Hartman, A. Rovai, and J. Sonner, JHEP 09 , 009 (2017), arXiv:1706.02668 [hep-th].</list_item>
<list_item><location><page_14><loc_9><loc_73><loc_49><loc_77></location>[112] K. Pakrouski, P. N. Pallegar, F. K. Popov, and I. R. Klebanov, Phys. Rev. Lett. 125 , 230602 (2020), arXiv:2007.00845 [cond-mat.str-el].</list_item>
<list_item><location><page_14><loc_9><loc_71><loc_49><loc_73></location>[113] A. Buchel, S. R. Green, L. Lehner, and S. L. Liebling, Phys. Rev. D 91 , 064026 (2015), arXiv:1412.4761 [gr-</list_item>
</unordered_list>
<text><location><page_14><loc_56><loc_92><loc_58><loc_93></location>qc].</text>
<unordered_list>
<list_item><location><page_14><loc_52><loc_89><loc_92><loc_92></location>[114] O. Evnin and C. Krishnan, Phys. Rev. D 91 , 126010 (2015), arXiv:1502.03749 [hep-th].</list_item>
<list_item><location><page_14><loc_52><loc_87><loc_92><loc_89></location>[115] V. Cardoso, T. Houri, and M. Kimura, Phys. Rev. D 96 , 024044 (2017), arXiv:1706.07339 [hep-th].</list_item>
<list_item><location><page_14><loc_52><loc_84><loc_92><loc_86></location>[116] B. Craps, O. Evnin, and J. Vanhoof, JHEP 10 , 048 (2014), arXiv:1407.6273 [gr-qc].</list_item>
<list_item><location><page_14><loc_52><loc_81><loc_92><loc_84></location>[117] O. Evnin and R. Nivesvivat, JHEP 01 , 151 (2016), arXiv:1512.00349 [hep-th].</list_item>
<list_item><location><page_14><loc_52><loc_79><loc_92><loc_81></location>[118] H. Lin, O. Lunin, and J. M. Maldacena, JHEP 10 , 025 (2004), arXiv:hep-th/0409174.</list_item>
<list_item><location><page_14><loc_52><loc_76><loc_92><loc_78></location>[119] S. Corley, A. Jevicki, and S. Ramgoolam, Adv. Theor. Math. Phys. 5 , 809 (2002), arXiv:hep-th/0111222.</list_item>
<list_item><location><page_14><loc_52><loc_73><loc_92><loc_76></location>[120] D. Berenstein, JHEP 07 , 018 (2004), arXiv:hepth/0403110.</list_item>
<list_item><location><page_14><loc_52><loc_69><loc_92><loc_73></location>[121] V. Balasubramanian, A. Lawrence, A. Rolph, and S. Ross, JHEP 11 , 159 (2017), arXiv:1704.03448 [hepth].</list_item>
</document>
[
{
"title": "Alexey Milekhin",
"content": "UC Santa Barbara, Physics Department, Broida Hall, Santa Barbara, CA 93106 ∗",
"pages": [
1
]
},
{
"title": "Nikolay Sukhov",
"content": "Princeton University, Physics Department, Jadwin Hall, Princeton, NJ 08540 (Dated: June 19, 2024) Scar states are special finite-energy density, but non-thermal states of chaotic Hamiltonians. We argue that all holographic quantum field theories, including N = 4 super Yang-Mills, have scar states. Their presence is tied to the existence of non-topological, horizonless soliton solutions in gravity: oscillons and a novel family of excited boson stars. We demonstrate that these solutions have periodic oscillations in the correlation functions and posses low-entanglement entropy as expected for scar states. Also we find that they can be very easily prepared with Euclidean path integral.",
"pages": [
1
]
},
{
"title": "CONTENTS",
"content": "1 3 5 5 6 9 10 11 Acknowledgment 11 References 12",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "Everyday experience tells us that most systems thermalize over time, but there are exceptions. For instance, consider a single classical particle moving inside a reflecting cavity - Figure 1. For a cavity of random shape, all trajectories will look essentially random and they will explore of all the phase space - Figure 1 (a). This is a one-particle counterpart of thermalization: time average along such trajectory will be equal to the phase space average. We may call such cavity ergodic. Of course, there are special (integrable) shapes for which all trajectories have a short period of oscillations - Figure 1 (b). However, there are intriguing cases like the Bunimovich stadium[1]: a generic trajectory is thermal - Figure 1 (c), but there is a set of short periodic trajectories which do not explore all of the phase space - Figure 1 (d). We will refer to them as classical scars . There is an exponentially small number of such trajectories and they are unstable, so one might expect that on a quantum level they are not important for anything. It turns out to be incorrect [2]. A number of energy eigenstate wavefunctions are concentrated around the classical scar trajectories. In other words, short classical unstable orbits permanently 'scar' the wavefunctions. This is the phenomena of single-particle quantum scars . Recent interest towards scars started from discovering a similar phenomena experimentally in a many-body quantum system of cold Rydberg atoms [3, 4](see also [5]). In the many-body setup, there is no classical analogue and the scarring occurs in the Hilbert space: evolution of certain states | Ψ(0) ⟩ shows short-period revivals when the fidelity |⟨ Ψ( t ) | Ψ(0) ⟩| ≈ 1. It is important to emphasize that scars were observed in ergodic Hamiltonians, for which most of the states are thermalizing: the fidelity exponentially decays to zero and it stays zero till the Poincare recurrence time. A hallmark of many-body quantum scars are scarred energy eigenstates with abnormally low entanglement compared to states of the same energy density. Typically they represent an exponentially small fraction of the Hilbert space. We refer to [6-9] for an overview. The main objective of this paper is to explore the scarring phenomena in gravity and quantum field theory. AdS/CFT correspondence [10-13] says that certain strongly-coupled large N conformal field theories (CFTs) are dual to classical gravitational theories in the asymptotically anti de-Sitter (AdS) spacetime. The gravitational dual of thermalization process is the formation of a black hole. It is known [14-18] that AdS spacetime is unstable: a small perturbation (either matter or metric) inside AdS quickly collapses into a black hole. However, there are special initial conditions which do not lead to a collapse. Instead the perturbation oscillates inside AdS forever [19, 20]. This is a clear analogue of a classical scar trajectory. It is important to emphasize that these oscillating gravity solutions and 'small' perturbations are large from the boundary CFT point of view, they correspond to the energy density of the order of the central charge (in the units of boundary volume). The simplest example, which we study in the present paper, involves Einstein gravity minimally coupled to a matter scalar field. Given that there are classical scar solutions in classical gravity, one [21] can naturally ask two question: In this paper we would like to answer the second question: Such periodic classical gravity solutions, known as boson stars or oscillons, are holographically dual to manybody scar states at the boundary. Moreover, we would like to argue that it is a feature of all holographic systems, whenever the bulk has 3 or mode spacetime dimensions and has a scalar field. One such example is N = 4 super Yang-Mills which we will discuss in detail. Why is this interesting? Holographic systems are supposed to be not just chaotic [22, 23], but maximally chaotic [24-26]. Having scar states is not a generic feature of chaotic Hamiltonians. However, our result indicates that scars, which break erodicity, are generic for holographic systems. Also, it is expected that the presence of scars is associated with hidden symmetries (or more generally, spectrum-generating algebras) [27-35]. The question of a possible hidden symmetry behind oscillons has been extensively discussed in the literature before and we give a small overview in the Conclusion. Also there is an interesting difference with classical scars. It is known that boson stars and oscillons are linearly stable and exhibit slow thermalization: adding an extra perturbation on top does not immediately lead to black hole formation [36]. In contrast, classical scars are associated with unstable periodic orbits. Finally, the presence of boson stars and oscillons gives predictions for certain boundary CFT correlation functions involving a non-primary operator e ε O , O being the single-trace CFT scalar, dual to the scalar field in the bulk. Oscillons and boson stars require scalar matter fields. Even more generally, there are geon solutions [20] which are made purely from the metric, that is, from the CFT stress-energy tensor. However, they are more complicated and they break translational symmetry at the boundary so we do not consider them here. How hard is it to prepare these states? The main difference between oscillons and boson stars is whether the scalar field is real or complex: for oscillons the field is real and the metric is time-dependent. For boson stars the field is complex and has a harmonic time-dependence e -i Ω t , so the stress-energy tensor and the metric are timeindependent. From the gravity point of view, oscillons and boson stars are very different solutions which are usually discussed separately. Interestingly, we find that from the boundary CFT viewpoint they are very similar and both of them can be very easily prepared using the Euclidean path integral on a hemisphere with a single operator insertion e ˜ ε O at the pole - Figure 2. In case of oscillons the single-trace operator O is hermitian, whereas for boson stars it is complex. In our regime of interest, ˜ ε can be large, of order the square-root of central charge if the two-point function of O is normalized to 1. The resulting CFT state lives on sphere S d -1 . We find evidence that for boson stars it is possible to take the infinite volume limit to get a homogeneous state on R d -1 . For oscillons we were not able to find such limit. It is important to emphasize that oscillons and boson stars are not energy eigenstates for the boundary CFT. However, they are very close to being energy eigenstates: their energy density scales with the (large) central charge, whereas the variance is expected to scale at most as the square-root of the central charge, essentially because the classical gravitational solution provides a dominant saddle-point for the path-integral. Nonetheless, they are non-thermalizing, uniform energy density, pure states which support eternal oscillations in the 1-point function of a scalar single-trace operator O : Such behavior, by definition, implies the violation of the eigenstate thermalization hypothesis (ETH) [37, 38], which only allows such oscillating terms to be exponentially small in the thermal entropy. More interestingly, it was recently argued [39] that under certain mild assumptions in discrete local systems, the presence of such revivals in pure states possessing area-law entanglement imply the presence of scarred energy eigenstates in the spectrum. In this paper we indeed find evidence that boson stars possess area law entanglement at the boundary: entanglement of a CFT subregion scales as the area of the boundary of that region. In contrast, for a thermal (black hole) state the entropy scales as the volume of a CFT subregion. Hence we can expect the presence of exact scars in the spectrum. The simplest oscillons we study here only exist in a finite volume and up to some critical energy density, so it is not clear how to separate volume and area law entanglement for them. Let us summarize out key findings: In Section III we will explain why one needs to study excited boson stars, rather than fundamental (non-excited ones) to find scars. Recently the phenomena of scar states in quantum field theories (QFT) has been addressed in a number of papers. Previous discussions of oscillons and boson stars within the AdS/CFT include [17, 40-44]. Scars based on Virasoro symmetry and their relation to AdS 3 /CFT 2 are discussed in [45, 46]. For a general discussion of scar states within the QFT framework we refer to [47-49]. Another recent discussion of scar states [50] is based on stable orbits around black holes. The rest of the paper is organized as follows. Section II is dedicated to oscillon solutions. We discuss their generic properties and then switch to the entanglement entropy in Section II A. In Section II B we argue that oscillons exist in the supergravity dual to N = 4 super Yang-Mills. Section III is devoted to boson stars. We briefly describe their properties and demonstrate that they have area-law entanglement. The CFT state preparation of oscillons and boson stars is discussed in Section IV. In Section V we turn away from discussing specific solutions and argue generally that black holes maximize entanglement entropy due to weak energy condition. In Conclusion we summarize our findings and outline open question.",
"pages": [
1,
2,
3
]
},
{
"title": "II. OSCILLONS",
"content": "In this Section we study the oscillons states first found in [19]. We will review their perturbative construction and then compute subsystem entanglement entropy. Consider Einstein gravity with negative cosmological constant plus a minimally-coupled scalar field ϕ , which is spherically symmetric. The main statement of [19] is that such soliton can exist forever, without collapsing into a black hole. The Lagrangian is A general ansatz for the metric, preserving the spherical symmetry is Usual (undeformed) AdS d +1 is A ( x, t ) = 1 , δ ( x, t ) = 0. AdS radius l is determined by l 2 = d ( d +1) 2Λ . The boundary is at x = π/ 2 and the center is at x = 0. We impose a gauge constraint that at the boundary δ is zero: δ ( t, π/ 2) = 0, so the dimensionless coordinate t is the boundary time. This setup corresponds to ( d -1)+1dimensional CFT located at the asymptotic boundary S d -1 × R t . It is very important to discuss units in this paper because all results we present will be in dimensionless units. The conformal metric at the boundary has unit radius. Correspondingly, t and the frequencies are measured in the units of the boundary radius. AdS radius l drops out from the equations and we can reabsorb 8 πG N = l d -1 p into ϕ . So the scalar field is measured in the units of l -( d -1) / 2 p . With Dirichlet boundary conditions for the scalar field, function A has the following expansion: CFT energy density T tt is proportional to M times ( l/l p ) d -1 . The ratio ( l/l p ) d -1 is proportional to the CFT central charge, which is large. Similarly, using RT/HRT prescription [51, 52], entanglement entropy of boundary subregions is given by the area of extremal co-dimension two surfaces in the bulk with minimal area: Technically it is always infinite, because AdS boundary is infinitely far, so we will always compute the difference with the vacuum (empty AdS answer). So in this paper we compute it in the units of ( l/l p ) d -1 . The upshot is that we are interested in large CFT perturbations, when the energy density and entropy are proportional to the central charge. The solutions are parametrized only by the (dimensionless) value of the scalar field (in the l -( d -1) / 2 p units). Without gravitational backreaction, a minimally coupled scalar of mass m 2 = ∆(∆ -d ) has a set of (spherically symmetric) normal modes: with frequencies ω j = ∆ + 2 j and P being the Jacobi polynomials. With this choice of normalization, they are orthonormal with respect to the tan( x ) d -1 . The fundamental mode j = 0 has no zeroes, and the excited ones have j zeroes. The question is what happens with this solution once we include backreaction into account. The key statement is that solutions with a single dominant frequency do not collapse into a black hole. Solutions which have several frequencies collapse very quickly, that is, they thermalize. Let us list a few other facts: All this suggests that oscillons have lifetime nonperturbative in 1 /G N . Meaning that their lifetime is e # /G N , non-perturbatively large in 1 /G N . For example, one can start from the lowest mode solution in asymptotically AdS 5 spacetime with j = 0 , Ω = 4: and then try to find the fully backreacted solution: Functions e j ( x ) form a basis, so the only special property of this ansatz is the periodic time-dependence. By AdS/CFT dictionary, such field profile leads to the oscillating expectation value of the dual operator O in the form (1). One can add backreaction either perturbatively or perform numerics. Following the approach of [19], we constructed such solutions numerically. In short, one truncates the expansion (8) at some big values of i, j and then requires the Einstein equations to be satisfied on a set of collocation points in space and time. We fix the amplitude ε by requiring f 0 , 0 = ε . The resulting mass M and frequency Ω for asymptotically AdS 5 space and massless scalar field (which is the case relevant for N = 4 super Yang-Mills) are shown on Figure 3. One distinct feature we find for various spacetime dimensions and various masses is that the frequency Ω blows up at a finite value of the scalar field amplitude, whereas the mass stays finite. One interesting question is whether for AdS 3 these solutions can have mass above the BTZ black hole threshold. For a massless field with Dirichlet boundary conditions the maximal mass appears to be much below.",
"pages": [
3,
4
]
},
{
"title": "A. Entanglement entropy",
"content": "Let us discuss the entanglement entropy. Since the metric is time-dependent the entanglement entropy is expected to be time-dependent too. However, we do not expect it to change a lot during the period of one oscillation, therefore we will concentrate on the time-symmetric t = 0 slice for which we can use a simple RT prescription. For small subsystems of linear size s the entanglement entropy grows very slowly: (as usual, in the units of ( l/l p ) d -1 ). The origin of this equation is the following. Oscillon has a finite energy density M ∼ ⟨ T tt ⟩ at the boundary, proportional to ε 2 . The behavior ⟨ T tt ⟩ s d for small subsystems was previously proved in [61, 62]: for small subsystems the RT surface lies near the boundary and the only important parameter in the metric is M ∼ ⟨ T tt ⟩ , s d comes from dimension analysis. Of course, this does not imply volume law entanglement or area law, for that we need to look at large subsystems. The problem is that we are dealing with global AdS , so the boundary CFT lives on a sphere. But what is a large subsystem of a sphere? Similar problem arises in condensed matter setups because they study systems of finite number of spins. In principle, we can do a Weyl transformation to map the state of the CFT from a sphere to a plane. On the gravity side it corresponds to appropriately selecting a Poincare path inside global AdS . However, the resulting state will be time-dependent and inhomogeneous [63, 64], so it is not very useful. One natural thing to do is to compute the entropy for half of the system. Then we have only two parameters because we have a CFT: radius of the boundary sphere r (which we set to unity) and energy density M ∼ ⟨ T tt ⟩ . The entanglement entropy depends only on the effective dimensionless length r ⟨ T tt ⟩ 1 /d ∼ rM 1 /d . Then the volume-law entanglement in d -1 spacial dimension can be associated with growth for large M , whereas the area-law is This is the same as conventional area and volume law entanglement: we can fix a smaller subsystem of size s and send M to infinity, s to zero, keeping s d M fixed, but big. In this limit CFT is effectively decompactified and large s d M governs the entanglement behavior of large subsystems. Black holes yield volume-law M ( d -1) / 2 . This can be understood without any computations: in this regime the horizon lies very close to x = π/ 2, namely x h ∼ π/ 2 -1 /M 1 /d . The RT surface will wrap around the horizon and most of its length will come from a disk x = x h , which area is proportional to 1 / cos d -1 ( x h ) ∼ M ( d -1) /d . Unfortunately, for oscillons M has a maximum value, so we cannot distinguish the area law and the volume law. The only thing we can verify is that oscillons have lower entanglement entropy, compared to black holes. This is indeed the case as illustrated by Figure 4. In the next Section we will study boson stars for which the mass M can be unbounded. We will see that they indeed exhibit area law.",
"pages": [
5
]
},
{
"title": "B. A comment on N = 4 super Yang-Mills",
"content": "N = 4 super Yang-Mills is dual to AdS 5 × S 5 solution in IIB ten-dimensional supergravity. This background is sourced by (self-dual) 4-form field H . We would like to claim that there exist oscillons in this background which only propagate along AdS 5 part. Meaning that this solution keeps the radius of S 5 constant. The relevant oscillating scalar field is the dilaton ϕ or the axion χ . In the Einstein frame the Lagrangian looks like where G (10) N is ten-dimensional Newton constant. There is non-trivial flux of H through S 5 , but the dilaton and axion are constant. Since H has traceless stressenergy tensor, Einstein equations with non-constant ϕ and χ can be written as Hence, if ϕ and χ do not vary over the S 5 , one can make an ansatz for AdS 5 deformation like eq. (3), but with S 5 having a constant radius. In these units (which are different from the rest of the paper) dilaton ϕ is dimensionless, the corresponding dual operator is Tr F 2 /g 2 Y M , its two-point function is proportional to N 2 ( F is the gauge field strength). In Section IV we will discuss the state preparation. In order to produce order 1 correction to the metric, the operator insertion at the Euclidean disk should have the form exp ( ˜ ε 1 g 2 Y M Tr F 2 ) , where ˜ ε is order 1 small number (say, 0.01). Strictly speaking[65], the stability of oscillons has been shown only for the AdS metric and the scalar field perturbations. In the linear regime the S 5 part will add extra scalar fields charged under SO (6) symmetry. We do not expect that these extra fields will destabilize the oscillon. For empty AdS the extra fields have normal modes with frequencies away from zero and for small oscillon amplitudes the shift of the normal mode frequencies will be small. But it would be instructive to study this question more carefully.",
"pages": [
5,
6
]
},
{
"title": "III. BOSON STARS",
"content": "In this Section we make a minimal modification and study a 'phenomenological' holographic model: Einstein-Maxwell theory minimally coupled to a complex scalar field. We refer to [43, 66-69] for more 'realistic' Einstein-Maxwell-scalar theories arising from higherdimensional supergravities. For simplicity, we consider massless scalar in 1 + 3 dimensions. This theory has a plethora of different phases and solutions, including hairy black holes and boson stars [70-72], depending on the value of the charge. In this paper we would like to point out the existence of an extra family of heavy boson stars, which we call C-stars . It was first discovered in [70] but then overlooked in the subsequent works. Here we study their properties in more detail, demonstrate their linear stability and argue that they represent scar states in the dual CFT. It is expected that there are no global symmetries is quantum gravity, this is why we do not study a complex field with global U (1). By AdS/CFT correspondence gauged U (1) symmetry in the bulk corresponds to a global symmetry in the CFT. Gauge coupling constant is related to the coefficient in the operator product expansion (OPE) of two current operators in the CFT. The Lagrangian is We again put l d -1 p = 8 πG N = 1 and measure the scalar field in unites of l -( d -1) / 2 p and gauge field A µ in units of ll -( d -1) / 2 p . We can do rescaling of the boson star equations, which reveals that the only important parameter (in addition to the value of the fields) is e eff = el/l ( d -1) / 2 p . In holography we expect it to be of order 1, because the interaction strength is of order the gravitational one (suppressed by the CFT central charge). We again study spherically-symmetric solutions in the form (3), but the scalar field has a simple one-harmonic time-behavior: In the limit of vanishing backreaction (very small amplitude), ϕ ( x ) are the normal modes (6) inside empty AdS d +1 . Because the stress-energy tensor is proportional to | ϕ | 2 , the actual metric is time-independent. This is why to find the solutions we can use a simple shooting method. Since the equations of motion for the scalar field are singular both at the origin x = 0 and at the AdS boundary x = π/ 2, we step away from the origin using power series expansion at the origin, numerically integrate the equations up to a point x 1 close to the boundary and then use the scalar field, the gauge field and the metric functions values at x 1 to fit an asymptotic power series expansion near the boundary. We then shoot for the scalar field frequency Ω to match the scalar field derivative ϕ ' l ( x 1 ) found by numeric integration to the scalar field derivative found from the asymptotic ϕ ' r ( x 1 ). We again impose Dirichlet boundary conditions for the scalar, such that near the boundary ˜ ϕ ∼ ( π/ 2 -x ) d . The only non-zero component of the vector potential is A t . This component and the metric has the following expansion near the boundary: Before diving inside the details, let us discuss the expectations from the CFT side. From the CFT perspective such state has non-zero global U (1) charge density ∝ Q and 1-point expectation value of a charged operator O : As in the case of oscillons, we are interested in the 'large' masses and electric charges, of the order of the CFT central charge. In this case in a given charge sector, the lowest energy state has non-zero energy density, proportional to the mass M . In the limit of large charge Q there are many field-theory results relating Q to M [73-80]. Specifically for 2 + 1 CFT it is expected that M ∼ Q 3 / 2 and in 3 + 1 dimensions M ∼ Q 4 / 3 . It would be convenient to explore the space of gravity solutions by fixing the amplitude | ϕ (0) | ≡ ε of the scalar field at the center x = 0 and the asking what discrete set of frequencies Ω are allowed. In short, there are three phases, depending on the effective charge [71, 72, 81] e eff . 'Boring' weak coupling phase: e eff < e eff crit , 1 : In this case we can start from a normal mode solution e j ( x ) and increase the amplitude. It turns out that all solutions have a bounded mass: at first the mass grows with the amplitude | ϕ (0) | , but then reaches the maximum and decreases. For amplitudes above the maximum of the mass the solution is linearly unstable. Technically the Figure 5 illustrates the intermediate coupling phase, but the qualitative behavior of the normal modes is the same. Dashed blue (representing fundamental mode e 0 ( x )) and orange (first excited mode e 1 ( x )) shows the behavior of the frequency and ADM mass. Analytical arguments suggest [82] that e eff crit , 1 = √ 3 / 2. Our numerical results are consistent with this prediction, although the shooting becomes increasingly hard near the critical point. Intermediate coupling: e eff crit , 1 < e eff < e eff crit , 2 , illustrated by Figure 5. In this regime the solutions connected to the perturbative normal modes e j ( x ) behave qualitatively similar (dashed lines). Interestingly, above certain amplitude additional solutions appear (solid lines). These solutions can have different number of zeros. The one with no zeros (solid blue) is usually called 'the second branch of the fundamental mode' in the literature [71, 72, 81]. Why it is called the second branch will become apparent from its behavior in the strong coupling phase. Take the fundamental second branch with no zeros (solid blue). This branch has unbounded mass, but is linearly stable. Does it signal the presence of a scar? Our answer for this question is: probably not. This state has a lower mass compared to the extremal ReissnerNordstrom (RN) black hole of the same mass [72], it satisfies [83] the relation M ∼ Q 3 / 2 expected for the ground state of a CFT. Moreover, it is horizonless, hence it is dual to a pure state of the boundary theory. Hence, we can expect that it is actually dual to (or at least very close to) a ground state of the CFT, as was proposed in [83]. Below we will also show that it has area law entanglement. Interestingly, we find solutions with unbounded mass which has more than one zero in the scalar profile: 1-node (solid orange) and 2-node (green) in the Figure 5, although they almost coincide. However, we will abstain from calling them 'the second branch of excited modes'. Instead, we call them 'C-stars'. Again, the reason for this will become clear from the strong coupling behavior. They are significantly heavier than the fundamental (0-node) solution, so they are not close to the ground state at a fixed charge. We claim that these C-stars are approximate scar states. Approximate means Fund. second branch Black hole 1 - node C - star that they are not exact energy eigenstates, as discussed in the Introduction. To backup this statement, we evaluated the entanglement entropy of half the system for different values of M - Figure 7. As we explained in Section II A, this probes the entanglement structure in the infinite-volume limit. We indeed find area-law entanglement M ( d -2) /d = M 1 / 3 , in contrast to the volume law M ( d -1) /d = M 2 / 3 of the extremal RN black hole. Actually, depending on the mass, the relevant RT surface can have different configurations - Figure 9. There is a simple RT surface (pink line) slicing through the equatorial plane, which yields area-law entanglement M 1 / 3 . We found that it always dominates for large enough M , for both C-stars and the second branch of the fundamental mode. However, there is another RT surface which avoids the strong gravity region by curving around it (red line). For some C-stars, it dominates if M is not too large, resulting in a entanglement shadow [84-86]: an area in the bulk which cannot be probed with extremal surfaces. The planar limit of a heavy fundamental boson star should coincide with zero temperature limit of a holo- graphic superconductor [87]. One can verify independently that those have sub-volume law entanglement. However, they seem to posses an interesting phenomena that at intermediate distances there is volume law scaling (c.f. Figures 7,8). It would be interesting to understand this entanglement pattern from the boundary perspective. Our numerics suggests e eff crit , 2 ≈ 2 . 3. Strong coupling: e eff > e eff crit , 2 . In this case the two branches of the fundamental mode merge: one can start from small-amplitude normal mode solution and monotonically increase mass to infinity - Figure 6 (blue). This is the origin of the name 'second branch' in the intermediate coupling phase. In contrast, this does not happen (at least for e eff ≤ 20, because we are limited by numerics) to the excited modes and C-stars - Figure 6 (orange, green). This is why we do not call C-stars 'the second branch of the excited modes'. Both C-stars and the fundamental boson star are superextremal - Figure 10 and have area-law entanglement - Figure 8. So they continue being scar states at strong coupling. One important question is the stability of the solutions we found. This question is important because unstable solutions can be highly sensitive to the parameters of the theory. For example, in this paper we neglected a possible explicit self-interaction of the scalar field (apart from the one mediated by gauge field and gravity). In holographic theories we expect the bulk fields interactions to be small, but non-zero. The first step to understand boson star stability is to consider normal modes of the linearized boson star perturbations. We follow the method outlined by [72], which reduces the system of equations for spherically symmetric linearized perturbations of a boson star to a system of three linear equations, two for area - law RT surface volume - law RT surface real and imaginary scalar field perturbations and one for gauge potential perturbations, and then uses Chebyshev pseudospectral collocation method to find normal modes of these equations. The star is stable when all the modes have real frequencies and becomes unstable when at least one of the frequencies becomes imaginary. The point of transition between the stable and the unstable parts of the branches of boson stars happens when the mass curve encounters an extremum M ' ( ε ) = 0, there is a good argument to that [72] that follows a well-known argument for fluid stars [88] and it stays valid for all the AdS boson stars we have encountered. Furthermore, linearly stable boson stars in AdS are known to be non-linearly stable as well [19, 40, 72]. Figures 5 and 6 show the square of the frequency of the lowest normal mode, all the parts of the branches with unbounded mass (separated by a mass extremum) are stable. Because they are linearly stable, we do not expect that self-interaction or higher-derivative curvature terms will affect the solutions. Direct studies of fundamental boson stars in various theories [40, 89-97] support this intuition.",
"pages": [
6,
7,
8,
9
]
},
{
"title": "IV. STATE PREPARATION",
"content": "We have identified a bulk field configuration which has scar properties. How do we prepare this state using CFT path integral? The calculations we presented above concerned purely classical system of Einstein gravity with a scalar field. It means that in the QFT language we are dealing with a coherent state of that scalar field. The question of preparing such coherent states was addressed in a series of papers [98-103]. Since we want a single-mode configuration at t = 0, which is spherically symmetric, we prepare CFT state on a sphere using disk (hemisphere) path integral, where we insert a single e ˜ ε O operator at the center of the disk - Figure 2. O is CFT single-trace operator dual to the field ϕ in the bulk. For N = 4 super Yang-Mills such operator could be Tr F 2 or Tr ˜ FF ( F being the gauge field strength) which is marginal and has no R -charge in order to leave the S 5 part of the bulk geometry intact. Constant ˜ ε is proportional (up to an order-one number) to the bulk scalar field amplitude ε . If the twopoint function of O is normalized to 1, then ˜ ε can be large, of order the square-root central charge, such that gravity backreaction becomes important. This state is fine-tuned: moving the operator away from the pole will create a spherically non-symmetric configuration which will collapse to a black hole at times of order 1 /ε 2 . In order to read off the bulk configuration we need bulk-to-boundary propagator which for global AdS is where unit vector ˆ e parameterize S d -1 of AdS d +1 . Also it is important to keep in mind that we are interested in expectation values in the form ⟨ Ψ | · | Ψ ⟩ , so the path integral involves both south and north hemispheres. Putting the operators at the poles basically [104] sets (ˆ e, ˆ e ' ) = 0. This yields ϕ ∝ ˜ ε cos( x ) ∆ , ∂ t ϕ = 0 profile at t = 0, which is what we want for an oscillon. In case of a complex field, we get ϕ ∝ ˜ ε cos( x ) ∆ , ∂ t ϕ ∝ i ∆ ˜ ε cos( x ) ∆ , which is the relevant configuration for a boson star. Excitations with higher radial numbers can be obtained by acting with derivatives. For example, for the first excited mode we need to insert ∂ 2 O (0), ∂ 2 being the Laplacian on the sphere. In a generic quantum field theory an insertion in the form e ˜ ε O (0) is not well-defined. Thanks to the stability of boson stars/oscillons one can introduce a small smearing e ˜ ε ∫ d d z O . Such operator is well-defined and it would be interesting to investigate whether such operators lead to scars beyond holographic CFTs. Of course, this is just a leading order in ˜ ε (that is, in ε ) answer. One can solve bulk equations of motions perturbatively and then prepare the configuration at t = 0 exactly by placing the appropriate Euclidean sources. We refer to [103] for a discussion.",
"pages": [
9,
10
]
},
{
"title": "V. VOLUME-LAW ENTANGLEMENT AND WEAK ENERGY CONDITIONS",
"content": "In the previous sections we demonstrated that oscillons have smaller entanglement entropy compared to black holes and boson stars have parametrically smaller entanglement entropy compared to a black hole of the same mass. Namely, for half-system the black hole answer is 'volume-law' M ( d -1) /d , whereas for C-stars it is 'arealaw' M ( d -2) /d . In this Section we show that imposing the weak-energy condition in the bulk guarantees that the CFT entanglement entropy is smaller compared to the one of a thermal state of the same mass, which is a well-known statement from the statistical mechanics. Unfortunately, our arguments are not sensitive to the presence/absence of the horizon. It would be interesting to understand how the requirement of horizon absence further bounds the entanglement. Consider a static space-time (3) with some matter fields, with or without a horizon. Weak energy condition T tt ≥ 0 for the matter stress-energy tensor yields Gronwall theorem implies[105] that A can be bounded by the solution of the corresponding differential equation: The right hand side is the value of A for AdS black hole of mass M . In particular, it implies that if there is a horizon, it lies inside a would-be black hole of the same mass[106]. Imagine now, that we fix a boundary subregion and the energy density M . Then the RT prescription in the corresponding black hole background with this mass will yield a certain co-dimension 2 surface. Now, if we compute the area of the same surface in the geometry of interest of the same energy density, the area will be lower because of the inequality (21). But the RT prescription instructs us to find the minimum over all possible surfaces, so the actual RT answer will be even lower.",
"pages": [
10
]
},
{
"title": "VI. CONCLUSION",
"content": "In this paper we studied the properties of oscillons and boson stars in asymptotically AdS spacetime. These are time-periodic, horizonless, solitonic solutions and we argued that they are linearly stable, have low-entanglement and are easily prepared with the Euclidean path integral. In the dual CFT they signal the presence of scars states. We demonstrated that excited boson stars have area law entanglement. In contrast, in low dimensional spin-chains scar states often have logarithmic scaling of entanglement. It is an interesting question if it is possible to obtain something like this in holographic theories in higher dimensions. Also it would be interesting to go beyond entanglement and understand other properties of scar states, both holographic and not. For example, entanglement can be used as a probe for confinement [107]. Oscillons and boson stars only require the presence of a scalar field in a theory, hence they represent a generic phenomenon for holographic theories. As we mentioned in the Introduction, one can even build solitonic objects (geons) purely from the gravitational degrees of freedom [20]. It would be interesting to perform the analysis of this paper for geons. There are two important take-aways, one for the holographic CFTs and one for the gravity. The gravity predicts that for holographic CFTs the states corresponding to e ˜ ε O (prepared by Figure 2) are non-thermalizing finite energy-density states. It means that in any holographic CFT, 4-point correlation function of the form where L are light fields evolved in Lorentzian signature, and e ˜ ε O are inserted at the poles of the Euclidean sphere, will not look thermal in any number of dimensions. It is important to emphasize that ˜ ε can be large, of the order of square-root central charge (if ⟨OO⟩ is normalized to 1), so this operator causes huge backreaction. In contrast, states prepared with the insertion finitely away from the pole will look thermal. The question of thermality of CFT correlators has been extensively studied before. For the case of heavy conformal primaries H in 2d CFT, it was argued in [108, 109] that 4-point function of the form does look thermal. Also [110, 111] studied more complicated CFT states which effectively prepare collapsing dust shells in the dual gravity. In both cases, the large central charge limit and the vacuum dominance in the 4point function was enough to link the CFT results to the gravity black hole computation. It would be interesting to perform a similar CFT calculation for the (22) correlator and map the results to the oscillon (boson star) background. On the other hand, all current examples of scar states in the condensed matter literature involve the presence of hidden symmetries [29-35, 112]. A possible hidden symmetry in behind oscillons was discussed in [113116]. Without a backreaction, a scalar field of mass m 2 = ∆(∆ -d ) in AdS d +1 posses a set of normal modes: where l is the (integer) angular momentum and j is the (integer) radial quantum number. Notice that they enter only in l + 2 j combination. This highly resonant spectrum is a direct consequence of the conformal SO (2 , d ) symmetry of AdS . Such resonant spectrum is the reason why AdS is unstable, because non-linearities coming from gravity may produce secular corrections growing linearly in time. Surprisingly, it was argued in [117] that the same SO (2 , d ) symmetry forces the secular terms to vanish. This symmetry constrains a lot the leading nonlinear correction to the oscillon solution, but it would be interesting to understand what role this possibly weaklybroken symmetry plays for the full non-linear solution. As we mentioned many times, oscillons and boson stars are only approximate energy eigenstates which only signal the presence of scar energy eigenstates in the spectrum. Is it is possible to construct a geometry dual to the actual scar eigenstate? One possible set of candidates are Lin-Lunin-Maldacena (LLM) [118] geometries. They are dual to half-BPS boundary operators [119, 120]. These geometries are complicated, but it would be nice to understand what entanglement law they have, some preliminary steps in this direction were made in [121]. However, LLM solutions are very special: they preserve supersymmetry and explicitly use the compact S 5 part of the geometry. Unlike oscillons and boson stars, we do not expect to find something like this in more generic holographic theories.",
"pages": [
11
]
},
{
"title": "ACKNOWLEDGMENT",
"content": "We would like to thank F. Popov and K. Pakrouski for for collaboration at the early states. We are grateful to D. Berenstein, A. Buchel, E. Colafranceschi, S. ColinEllerin, A. Dymarsky, A. Gorsky, A. Holguin, L. Lehner, D. Marolf, F. Pretorius, M. Srednicki, A. Zhiboedov and specially to S. Hellerman, G. Horowitz, I. Klebanov for comments and discussions. This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-19-1-0360. It was also supported in part by funds from the University of California. AM would like to thank C. King for moral support. . [105] Specifically we need to apply it for the interval [0 , π/ 2) in the vicinity of x = π/ 2 to connect constant M with spacetime mass. qc].",
"pages": [
11,
13,
14
]
}
]
2024PhRvD.110j3537I
https://arxiv.org/pdf/2406.14982.pdf
<document>
<section_header_level_1><location><page_1><loc_9><loc_90><loc_92><loc_93></location>Non-thermal particle production in Einstein-Cartan gravity with modified Holst term and non-minimal couplings</section_header_level_1>
<text><location><page_1><loc_43><loc_87><loc_57><loc_89></location>Tomohiro Inagaki ∗</text>
<text><location><page_1><loc_31><loc_85><loc_70><loc_87></location>Graduate School of Advanced Science and Engineering, Hiroshima University, Higashi-Hiroshima 739-8526, Japan</text>
<text><location><page_1><loc_20><loc_83><loc_81><loc_84></location>Information Media Center, Hiroshima University, Higashi-Hiroshima 739-8521, Japan and</text>
<text><location><page_1><loc_16><loc_82><loc_85><loc_83></location>Core of Research for the Energetic Universe, Hiroshima University, Higashi-Hiroshima 739-8526, Japan</text>
<section_header_level_1><location><page_1><loc_44><loc_79><loc_56><loc_80></location>Naoki Yoshioka †</section_header_level_1>
<text><location><page_1><loc_31><loc_76><loc_70><loc_79></location>Graduate School of Advanced Science and Engineering, Hiroshima University, Higashi-Hiroshima 739-8526, Japan</text>
<text><location><page_1><loc_42><loc_75><loc_59><loc_76></location>(Dated: October 18, 2024)</text>
<text><location><page_1><loc_18><loc_61><loc_83><loc_74></location>Non-thermal fermionic particle production is investigated in Einstein-Cartan modified gravity with a modified Holst term and non-minimal couplings between the spin connection and a fermion. By using the auxiliary field method, the theory is rewritten into a pseudoscalar-tensor theory with Einstein-Hilbert action and canonical kinetic and potential terms for a pseudoscalar field. The introduced field is called Einstein-Cartan pseudoscalaron. If the potential energy of the EinsteinCartan pseudoscalaron dominates the energy density of the early universe, it causes inflationary expansion. After the end of inflation, the pseudoscalaron develops a large value and the non-minimal couplings destabilize the vacuum. Evaluating the non-thermal fermionic particle production process, we obtain the mass and the helicity dependences of the produced particle number density. We show the model parameters to enhance the preheating and reheating processes.</text>
<section_header_level_1><location><page_1><loc_20><loc_57><loc_37><loc_58></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_30><loc_49><loc_55></location>The extension of general relativity (GR) from a geometrical perspective is one of the candidates for solving cosmological problems. In F ( R ) theories, the gravitational action is replaced by a function of the curvature, R → F ( R ), and it has been shown to describe well various cosmological phenomena [1-4]. For example, the idea explains the inflationary expansion of the universe, and the gravitational interaction produces particles necessary to reheat the universe [5, 6]. Through the conformal transformation, the additional degree of freedom in the modified action can be represented as a dynamical scalar field that plays a role in the inflaton. After the end of the inflation, the interaction between the inflaton and the matter converts the inflaton energy into the matter and reheats the universe [7]. Whether the dominant reheating process is perturbative or non-perturbative particle production depends on the structure of the interaction.</text>
<text><location><page_1><loc_9><loc_16><loc_49><loc_30></location>Recently [8, 9], in the metric-affine gravity [10] where the metric g µν and the affine connection Γ µ νρ are independent variables, it is shown that a dynamical pseudoscalaron from the modification of geometrical quantity are obtained due to the existence of the Holst term ϵR = ϵ µνρσ R µνρσ / √ -g [11-13] which consists of the antisymmetric part of the affine connection(torsion) Γ µ νρ -Γ µ ρν . Consequently, that pseudoscalaron can be used as the inflaton [14, 15]. Thus, even in Einstein-Cartan (EC) gravity (metric-compatible metric-affine gravity) [16-20], the</text>
<text><location><page_1><loc_52><loc_45><loc_92><loc_58></location>inflation can be realized through the pseudoscalar field [21, 22] since there exists the torsion. The interaction between torsion and matter fields can lead to non-thermal particle production. Several relevant studies exist in this area. For instance, the preheating process has been investigated within Einstein-Cartan gravity incorporating the Nieh-Yan topological invariant [23]. It has also been noted that the Holst term can either suppress or enhance the rate of vacuum decay [24].</text>
<text><location><page_1><loc_52><loc_29><loc_92><loc_45></location>By introducing an auxiliary pseudoscalar field, the Euler-Lagrange equation about affine connection gives the algebra equation regarding the torsion. Since, in this case, the torsion is represented by the metric, the auxiliary pseudoscalar field, and the matter coupling to the affine connection, integrating out the torsion yields the effective metric theory as the Palatini f ( R ) gravity [2]. In this effective metric theory, one can get the pseudoscalaron and the interaction between the pseudoscalaron and the matter therefore can realize the inflation and the reheating of the universe.</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_28></location>In this paper, we discuss the non-thermal particle production in EC gravity with the modified Holst term. Since fermions naturally couple to the torsion in EC gravity, they are considered as matter fields. Additionally, the natural extension of the kinetic term of the fermion [25] is considered. To discuss the non-thermal fermionic particle production, we follow a way of previous work about fermionic preheating [26-30]. In the discussion, it is assumed that the fermionic field operator is composed of (anti) particles. By this assumption, one of the new parameters α must vanish. Numerical calculations eventually reveal that particle production occurs when the value of β is large, and its behavior depends on the mass and helicity of the particle.</text>
<text><location><page_2><loc_9><loc_67><loc_49><loc_93></location>The overview of this paper is as follows. The section 2 reviews the Einstein-Cartan pseudoscalaron with matter fields and introduces the ( ϵR ) 2 model [14] effective for inflation. In Sec. 3, we introduce the extension of the kinetic term of the fermion. From this extension in Einstein-Cartan pseudoscalaron theory, the equation of motion(EoM) of fermions is non-trivial and thus nonthermal particle production occurs even in the FRLW universe. In Sec. 4, several results of numerical calculations about non-thermal particle production are exhibited. By these results, it can be concluded that more larger value of β contributes to more particle production. Also, it is observed that the behavior of the number density is very different depending on whether the mass of the fermion is lighter, heavier or intermediate compared to the inflaton mass. In Sec. 5, we apply the particle production to reheat the universe. Finally, a discussion and summary of this paper are presented in Sec. 6.</text>
<text><location><page_2><loc_9><loc_57><loc_49><loc_67></location>Calculations in this paper are based on the following notations. m p = √ 1 / 8 πG N is the planck mass and m ϕ is the inflaton mass. G N means the gravitational Newton constant. The gamma matrix is defiend by { γ i , γ j } = 2 η ij where η ij is the minkowski metric η = diag( -, + , + , +). The definition of gamma matrice is</text>
<formula><location><page_2><loc_22><loc_53><loc_36><loc_56></location>γ µ = -i ( 0 σ µ σ µ 0 ) ,</formula>
<formula><location><page_2><loc_26><loc_50><loc_26><loc_51></location>µ</formula>
<formula><location><page_2><loc_24><loc_47><loc_34><loc_50></location>σ = (1 , σ ) , σ µ = (1 , -σ ) ,</formula>
<formula><location><page_2><loc_23><loc_43><loc_35><loc_45></location>γ 5 = iγ 0 γ 1 γ 2 γ 3 .</formula>
<text><location><page_2><loc_9><loc_41><loc_30><loc_42></location>Charge conjugate matrix C is</text>
<formula><location><page_2><loc_25><loc_38><loc_33><loc_40></location>C = iγ 2 γ 0 .</formula>
<text><location><page_2><loc_9><loc_28><loc_49><loc_37></location>ϵ µνρσ is Levi-Civita antisymmetric symbol ϵ 0123 ≡ 1. We use the natural units( ℏ = c = 1). The symmetrization A { i,j } and the anti-symmetrization A [ i,j ] are respectively 1 2 ( A ij + A ji ) and 1 2 ( A ij -A ji ). Greek indices mean the coordinate of spacetime while Roman indices mean that of local Minkowski spacetime. ˆ O means a q-number.</text>
<section_header_level_1><location><page_2><loc_10><loc_23><loc_48><loc_25></location>II. EINSTEIN-CARTAN PSEUDOSCALARON INFLATION FROM MODIFIED HOLST TERM</section_header_level_1>
<text><location><page_2><loc_9><loc_14><loc_49><loc_20></location>EC gravity is a metric-compatible metric-affine theory [10, 31], where the metric and the connection are independent variables. Fundamental conditions in this theory are the tetrad hypothesis and the metric compatible condition,</text>
<formula><location><page_2><loc_19><loc_11><loc_49><loc_12></location>∂ µ e i ν + ω i jµ e j ν -Γ α νµ e i α = 0 , (1)</formula>
<formula><location><page_2><loc_31><loc_9><loc_49><loc_10></location>∇ µ g νρ = 0 , (2)</formula>
<text><location><page_2><loc_52><loc_54><loc_55><loc_55></location>with</text>
<text><location><page_2><loc_52><loc_48><loc_54><loc_49></location>and</text>
<formula><location><page_2><loc_58><loc_44><loc_92><loc_47></location>T = 1 4 T ρµν T ρµν -1 2 T ρµν T µνρ -T µ T µ , (8)</formula>
<text><location><page_2><loc_52><loc_35><loc_92><loc_43></location>where the script ⋄ indicates the torsionless parts which consist of metric and tetrad, Γ ⋄ µ νρ is Levi-Civita symbol and ω ⋄ ij µ ≡ -e jν ∇ ⋄ µ e i ν . The torsionful parts consist of the contorsion, K µ νρ , defined by K µ νρ ≡ 1 2 ( T µ νρ + T νρ µ + T ρν µ ) and K ij µ ≡ e i ρ e jν K ρ νµ .</text>
<text><location><page_2><loc_52><loc_33><loc_92><loc_35></location>In our research, we start from the action with a function of the Holst term, ϵR ≡ ϵ µνρσ R µνρσ /e ,</text>
<formula><location><page_2><loc_56><loc_28><loc_92><loc_31></location>S = ∫ ed 4 x { m 2 p 2 R + m 2 p 4 H ( ϵR ) + L matter } , (9)</formula>
<text><location><page_2><loc_52><loc_21><loc_92><loc_27></location>where e = det( e i µ ). Since the modification of the curvature scalar R can not be represented by a dynamical scalar field [8, 14, 22], F ( R ) regime is not considered. Introducing an auxiliary field χ , we obtain the action</text>
<formula><location><page_2><loc_53><loc_15><loc_92><loc_20></location>S = ∫ ed 4 x { m 2 p 2 R + m 2 p 4 H ' ( χ ) ϵR -V ( χ ) + L matter } , (10)</formula>
<text><location><page_2><loc_52><loc_9><loc_92><loc_14></location>where V ( χ ) = m 2 p 4 ( H ' ( χ ) χ -H ( χ )) and H ' ( χ ) = dH ( χ ) dχ . χ is a pseudoscalar field. The action (9) is reproduced by substituting the Euler-Lagrange equation with respect to</text>
<text><location><page_2><loc_52><loc_77><loc_92><loc_93></location>where, e i µ represents the tetrad which satisfies g µν = e i µ e j ν η ij . The inverse of e i µ serves as a basis component of local Minkowski spacetime, e µ i e ν j g µν = η ij . Γ µ νρ is the affine connection and ω ij µ is the gauge field of local Lorentz transformation that is called spin connection. Thus, the covariant derivative ∇ of spacetime is defined by ∇ µ A ν ≡ ∂ µ A ν + Γ ν αµ A α , and the covariant derivative D of local Minkowski spacetime is defined by D µ B i = ∂ µ B i + ω i j µ B j . By Eq. (1), the curvature tensor R µ νρσ and the strength of local Lorentz transformation R ij µν are connected by</text>
<formula><location><page_2><loc_52><loc_70><loc_92><loc_76></location>R µ νρσ = ∂ ρ Γ µ νσ -∂ σ Γ µ νρ +Γ µ αρ Γ α νσ -Γ µ ασ Γ α νρ = e µ i e νj ( ∂ ρ ω ij σ -∂ σ ω ij ρ + ω i kρ ω kj σ -ω i kσ ω kj ρ ) = e µ i e νj R ij ρσ . (3)</formula>
<text><location><page_2><loc_52><loc_63><loc_92><loc_68></location>In this theory, the existence of the antisymmetric part of the affine connection(torsion) is not prohibited. Thus, the connections Γ, ω and the curvature scalar R can be generally separated into torsionless and torsionful parts,</text>
<formula><location><page_2><loc_64><loc_60><loc_92><loc_62></location>Γ µ νρ = Γ ⋄ µ νρ + K µ νρ , (4)</formula>
<formula><location><page_2><loc_65><loc_58><loc_92><loc_60></location>ω ij µ = ω ⋄ ij µ + K ij µ , (5)</formula>
<formula><location><page_2><loc_64><loc_56><loc_92><loc_58></location>R = R ⋄ + T -2 ∇ ⋄ µ T µ , (6)</formula>
<formula><location><page_2><loc_64><loc_51><loc_92><loc_52></location>T ρ µν ≡ Γ ρ µν -Γ ρ νµ , (7)</formula>
<figure>
<location><page_3><loc_11><loc_76><loc_46><loc_93></location>
<caption>FIG. 1: The potential of ϕ in the ( ϵR ) 2 model with parameters</caption>
</figure>
<formula><location><page_3><loc_9><loc_70><loc_49><loc_72></location>b = -320 , m ϕ = √ d 2 V dϕ 2 | ϕ =0 = √ (1+ b 2 ) 12 c ∼ 2 . 76 ∗ 10 13 GeV</formula>
<text><location><page_3><loc_9><loc_64><loc_49><loc_67></location>χ into Eq. (10). The Cartan equation is obtained as the Euler-Lagrange equation with respect to ω ij µ ,</text>
<formula><location><page_3><loc_11><loc_60><loc_49><loc_63></location>T µ ij -T i e µ j + T j e µ i + H ' ( χ ) ϵ µαβγ e δ [ i e j ] α T δ βγ /e = S µ ij + ϵ µαβγ e αi e βj ( ∂ γ H ' ( χ )) /e, (11)</formula>
<text><location><page_3><loc_9><loc_49><loc_49><loc_59></location>where T µ ≡ T ν µν is the torsion vector and S µ ij ≡ 2 m 2 p ∂ L matter ∂ω ij µ is the spin density. Thus, the torsion is rewritten in terms of tetrad e i µ , matter field and auxiliary field χ . The effective metric theory is obtained by inserting the solution Eq. (11) into Eq. (10). For the vacuum ( S µ mn = 0), it becomes</text>
<formula><location><page_3><loc_10><loc_45><loc_49><loc_48></location>S = ∫ ed 4 x { m 2 p 2 R ⋄ -3 m 2 p 4 ∂ µ H ' ∂ µ H ' (1 + H ' 2 ) -V ( χ ) } . (12)</formula>
<text><location><page_3><loc_9><loc_41><loc_49><loc_44></location>We introduce the pseudoscalaron ϕ by the redefinition, H ' ( χ ) = sinh ( √ 2( ϕ + δ ) √ 3 m p ) , and obtain</text>
<formula><location><page_3><loc_11><loc_36><loc_49><loc_40></location>S = ∫ ed 4 x { m 2 p 2 R ⋄ -1 2 ∂ µ ϕ∂ µ ϕ -V ( χ ( ϕ )) } , (13)</formula>
<text><location><page_3><loc_9><loc_33><loc_49><loc_35></location>where δ is a constant to impose V ( ϕ = 0) = 0. For example, in the ( ϵR ) 2 model [14]</text>
<formula><location><page_3><loc_21><loc_30><loc_49><loc_32></location>H ( ϵR ) = bϵR + c ( ϵR ) 2 , (14)</formula>
<text><location><page_3><loc_9><loc_20><loc_49><loc_29></location>we obtain the pseudoscalaron with a potential, V ( ϕ ) = m 2 p 16 c ( sinh ( √ 2( ϕ + δ ) √ 3 m p ) -b ) 2 . The constant δ is fixed to satisfy sinh ( √ 2 δ √ 3 m p ) = b . As is shown in Fig.1, a certain value of parameters b, c can realize the potential with a plateau.</text>
<section_header_level_1><location><page_3><loc_11><loc_16><loc_47><loc_17></location>A. Pseudoscalaron inflation in FLRW universe</section_header_level_1>
<text><location><page_3><loc_9><loc_11><loc_49><loc_14></location>We consider the homogeneous and isotropic universe described by the FLRW metric,</text>
<formula><location><page_3><loc_16><loc_9><loc_49><loc_10></location>ds 2 = -dt 2 + a 2 ( t )( dx 2 + dy 2 + dz 2 ) . (15)</formula>
<text><location><page_3><loc_52><loc_88><loc_92><loc_93></location>where x µ = ( t, x, y, z ) represents the cosmological time and the conformal space, and a ( t ) is the scale factor. By introducing the conformal time, dη = a -1 dt , the spacetime is represented by</text>
<formula><location><page_3><loc_58><loc_85><loc_92><loc_86></location>ds 2 = a 2 ( η )( -dη 2 + dx 2 + dy 2 + dz 2 ) , (16)</formula>
<text><location><page_3><loc_52><loc_75><loc_92><loc_83></location>The dot and the prime represent the derivative with respect to the cosmological time t , ˙ () = d dt (), and the conformal time η , () ' = d dη (), respectively. The metric (16) is used for the analysis in Sec. 4. In the metric (15), the scale factor is developed through the FriedmannRobertson equations,</text>
<formula><location><page_3><loc_70><loc_70><loc_92><loc_74></location>H 2 = ρ 3 m 2 p , (17)</formula>
<formula><location><page_3><loc_65><loc_67><loc_92><loc_70></location>˙ H = -1 2 m 2 p ( ρ + p ) , (18)</formula>
<text><location><page_3><loc_52><loc_60><loc_92><loc_66></location>where H is the Hubble parameter defiend by H = ˙ a a , and ρ and p respectively denote the energy density and the pressure of the matter. From Eqs. (17) and (18), we derive</text>
<formula><location><page_3><loc_65><loc_56><loc_92><loc_59></location>a a = -1 6 m 2 p ( ρ +3 p ) . (19)</formula>
<text><location><page_3><loc_52><loc_49><loc_92><loc_55></location>When the scalar field φ distributes homogeneously, the energy density and the pressure are described as ρ = 1 2 ˙ φ 2 + V ( φ ) and p = 1 2 ˙ φ 2 -V ( φ ). The accelerated expansion takes place for V ( φ ) > ˙ φ 2 .</text>
<text><location><page_3><loc_52><loc_31><loc_92><loc_49></location>If the potential V ( φ ) has a plateau, the scalr field starting from the plateau induces an inflationary expansion. To solve the horizon and flatness problems encountered in the expanding universe, the total e-folding number N e , defined as N e = log( a f /a i ), should exceed 50 ∼ 60. Here, a f represents the value of the scale factor at the end of inflation, while a i represents the value of the scale factor at the start of inflation. To obtain the number, we often employ the slow-roll inflation scenario. In the slow-roll approximation, the end of inflation is fixed by the slow-roll parameters ε and η , defined by ε = m 2 p 2 ( 1 V dV dφ ) 2 , η = m 2 p 1 V d 2 V dφ 2 .</text>
<text><location><page_3><loc_52><loc_17><loc_92><loc_30></location>The quantum fluctuations of φ induce the curvature perturbation, P , during inflation. In terms of the conformal momentum space, a component of the fourieor decomposition of the curvature perturbation is represented as P ( k ) = P r k n s . Applying the slow-roll approximation, the amplitude P r of the curvature perturbation can be derived by P r = V ( φ ) 24 π 2 m 4 p ε ( φ ) . The spectral index n s is derived by n s = 1 -6 ε ( φ ) + 2 η ( φ ). The scalar-tensor ratio r is calculated by r = 16 ε ( φ ).</text>
<text><location><page_3><loc_52><loc_9><loc_92><loc_17></location>The potential has a plateau in the ( ϵR ) 2 model with b = -320 , m ϕ = √ d 2 V dϕ 2 | ϕ =0 = √ (1+ b 2 ) 12 c ∼ 2 . 76 ∗ 10 13 GeV (Fig.1), and the slow-roll inflation scenario can be adopted . We assume that the pseudoscalar field regarded as an inflaton dominates the energy density of the</text>
<text><location><page_4><loc_9><loc_82><loc_49><loc_93></location>early universe. Consequently, we obtain values such as n s ∼ 0 . 969 , r ∼ 0 . 003 , P r ∼ 2 . 1 ∗ 10 -9 , N e ∼ 60 that agree with the observation regarding the Cosmic Microwave Background (CMB). Below, we adapt this model to the non-thermal particle production after the end of inflation. In our analysis, we consider that the particle production starts at ϕ ∼ 0 . 94 m p , dϕ dt ∼ -0 . 293 m p m ϕ where the slow-roll parameter ε ( ϕ ) becomes unity.</text>
<section_header_level_1><location><page_4><loc_9><loc_76><loc_49><loc_79></location>B. The dynamics of background field after the end of inflation</section_header_level_1>
<text><location><page_4><loc_9><loc_61><loc_49><loc_74></location>After the end of inflation, the oscillating inflaton ϕ dominates the energy density of the universe. The potential is approximated to be V ∼ m 2 ϕ 2 ϕ 2 during the particle production. The energy density of the inflaton and the scale factor are fixed by the Friedmann equations (17) and (18). Since the contribution to the pressure is cancelled between the kinetic and the potential energy, ρ ϕ follows the ˙ ρ ϕ +3 Hρ ϕ = 0. The solution of this equation with (17) is given by</text>
<formula><location><page_4><loc_18><loc_56><loc_49><loc_60></location>ρ ϕ = m 2 p m 2 ϕ ( √ 3 2 m ϕ t + A ) -2 . (20)</formula>
<text><location><page_4><loc_9><loc_53><loc_49><loc_55></location>From the Eqs. (17) and (20), the scale factor is derived as</text>
<formula><location><page_4><loc_19><loc_48><loc_49><loc_52></location>a ( t ) = A -2 3 ( √ 3 2 m ϕ t + A ) 2 3 . (21)</formula>
<text><location><page_4><loc_9><loc_45><loc_49><loc_47></location>The relation between cosmic time and conformal time is determined by dt = adη and a ( t = 0) = a ( η = 0) = 1,</text>
<formula><location><page_4><loc_15><loc_40><loc_49><loc_44></location>( √ 3 2 m ϕ η +3 A ) = 3 A 2 3 ( √ 3 2 m ϕ t + A ) 1 3 . (22)</formula>
<text><location><page_4><loc_52><loc_90><loc_92><loc_93></location>We assume that the oscillating part of the inflaton can be factored out,</text>
<formula><location><page_4><loc_55><loc_83><loc_92><loc_88></location>ϕ ( t ) /m p = CH ( t ) /m ϕ sin( m ϕ t + B ) = C √ 3 ( √ 3 2 m ϕ t + A ) sin( m ϕ t + B ) . (23)</formula>
<text><location><page_4><loc_52><loc_75><loc_92><loc_81></location>It should be noted that Eq. (23) satisfies the EoM of the inflaton ¨ ϕ +3 H ˙ ϕ + dV dϕ ≃ ¨ ϕ +3 H ˙ ϕ + m 2 ϕ ϕ = 0. Arbitrary constants A , B and C are fixed by the initial values of ϕ and ˙ ϕ ,</text>
<formula><location><page_4><loc_54><loc_68><loc_92><loc_73></location>ρ ϕ | t =0 /m 2 p m 2 ϕ = A -2 = ( 1 2 ˙ ϕ 2 + 1 2 m 2 ϕ ϕ 2 ) | t =0 /m 2 p m 2 ϕ , (24)</formula>
<formula><location><page_4><loc_63><loc_62><loc_92><loc_66></location>cot( B ) = ˙ ϕ/ϕ | t =0 + √ 3 2 A , (25)</formula>
<formula><location><page_4><loc_65><loc_56><loc_92><loc_59></location>ϕ | t =0 = C √ 3 A sin B. (26)</formula>
<text><location><page_4><loc_52><loc_52><loc_92><loc_54></location>We employ these formula as a simple background for the universe after the end of inflation.</text>
<section_header_level_1><location><page_4><loc_55><loc_49><loc_88><loc_51></location>III. A MODEL OF THE NON-MINIMAL COUPLINGS TO FERMION</section_header_level_1>
<text><location><page_4><loc_52><loc_41><loc_92><loc_47></location>In Einstein-Cartan gravity, the matter action is generalized with non-minimal gravitational interactions. We consider a general fermion Lagrangian constructed with operators up to four dimensions [25],</text>
<formula><location><page_4><loc_26><loc_33><loc_92><loc_36></location>L matter = -1 2 ( ψ (1 -iα -iβγ 5 ) γ µ ( ∂ µ + 1 4 ω ij µ γ ij ) ψ +h.c.) -m ψ ψψ, (27)</formula>
<text><location><page_4><loc_9><loc_22><loc_49><loc_29></location>where the Dirac conjugate ψ is defined by ψ = iγ 0 ψ † and h.c. means the hermitian conjugate. This Lagrangian reduces to the ordinary one in the absence of gravity. The extension causes the parity violation observed in the various astrophysical and elementary particle phenomena.</text>
<text><location><page_4><loc_10><loc_21><loc_30><loc_22></location>The spin density is given by</text>
<formula><location><page_4><loc_11><loc_13><loc_49><loc_19></location>S µ ij = 2 m 2 p ∂ L matter ∂ω ij µ = -1 m 2 p ( ϵ ijkl A l e µk -2 e µ [ i ( αV j ] + βA j ] )) . (28)</formula>
<text><location><page_4><loc_9><loc_9><loc_49><loc_12></location>A i ≡ ¯ ψiγ 5 γ i ψ and V i ≡ ¯ ψiγ i ψ denote the axial vector and the vector current, respectivly. Performing the</text>
<text><location><page_4><loc_52><loc_27><loc_92><loc_29></location>partial integration, the Lagrangian density (27) can be decomposed into</text>
<formula><location><page_4><loc_53><loc_23><loc_92><loc_26></location>L matter = L ⋄ matter + 1 8 ˆ T µ A µ -1 2 T µ ( αV µ + βA µ ) , (29)</formula>
<text><location><page_4><loc_52><loc_21><loc_72><loc_22></location>where the torsionless part is</text>
<formula><location><page_4><loc_57><loc_15><loc_92><loc_21></location>L ⋄ matter = -ψγ µ ∂ µ ψ -1 2 ψω ⋄ ijk η ik γ j ψ -i 4 ϵ ijkl ω ⋄ ijk ψγ 5 γ l ψ -m ψ ψψ. (30)</formula>
<text><location><page_4><loc_52><loc_9><loc_92><loc_14></location>Since non-minimal coupling parameters don't appear in the torsionless part (30), these parameters only contribute to the interaction between the torsion and the fermion.</text>
<text><location><page_5><loc_9><loc_90><loc_49><loc_93></location>By solving Eq. (11), the torsion is represented by the inflaton and the fermion,</text>
<formula><location><page_5><loc_18><loc_86><loc_49><loc_89></location>T ijk = -2 3 η i [ j η k ] l T l + 1 6 ϵ ijkl ˆ T l , (31)</formula>
<text><location><page_5><loc_9><loc_84><loc_12><loc_85></location>with</text>
<formula><location><page_5><loc_10><loc_77><loc_49><loc_83></location>T l = sech 2 ( X ( ϕ )) 2 ( 3 2 m 2 p ( αV l +( β +sinh X ( ϕ )) A l ) + 3sinh X ( ϕ ) ∂ l (sinh X ( ϕ )) ) , (32)</formula>
<formula><location><page_5><loc_11><loc_69><loc_49><loc_74></location>ˆ T l = -3 sech 2 ( X ( ϕ )) ( 1 m 2 p (( -1 + sinh X ( ϕ ) β ) A l +sinh X ( ϕ ) αV l ) -2 ∂ l (sinh X ( ϕ )) ) , (33)</formula>
<text><location><page_5><loc_9><loc_65><loc_49><loc_68></location>where we write X ( ϕ ) = √ 2( ϕ + δ ) √ 3 m p . Inserting the solution</text>
<text><location><page_5><loc_52><loc_92><loc_89><loc_93></location>(31) into (10), we obtain the effective metric action,</text>
<formula><location><page_5><loc_55><loc_84><loc_92><loc_90></location>S = ∫ ed 4 x { m 2 p 2 R ⋄ -1 2 ∂ µ ϕ∂ µ ϕ -V ( ϕ ) + L ⋄ ψ + f µ ( ϕ ) A µ + g µ ( ϕ ) V µ + L 4-fermi } , (34)</formula>
<text><location><page_5><loc_52><loc_81><loc_55><loc_82></location>with</text>
<formula><location><page_5><loc_53><loc_74><loc_92><loc_80></location>f µ = √ 3 2 √ 2 ∂ µ ( ϕ m p ){ -sech( X ( ϕ )) + β tanh( X ( ϕ )) } , (35)</formula>
<formula><location><page_5><loc_68><loc_69><loc_92><loc_74></location>g µ = √ 3 α 2 √ 2 ∂ µ ( ϕ m p ) tanh( X ( ϕ )) , (36)</formula>
<text><location><page_5><loc_52><loc_65><loc_73><loc_66></location>and four-fermion interactions,</text>
<formula><location><page_5><loc_18><loc_57><loc_92><loc_60></location>L 4-fermi = 3 sech 2 ( X ( ϕ )) 16 m 2 p ( α 2 V 2 +2( αβ + α sinh( X ( ϕ ))) V A +( β 2 +2 β sinh( X ( ϕ )) -1) A 2 ) , (37)</formula>
<text><location><page_5><loc_9><loc_50><loc_49><loc_53></location>where A 2 , V A and V 2 are defined by A 2 = A µ A µ , V A = V µ A µ and V 2 = V µ V µ .</text>
<text><location><page_5><loc_9><loc_46><loc_49><loc_50></location>The four-fermion interactions, (37), are suppressed by the factor m -2 p . On the other hand, the five-dimensional interactions,</text>
<formula><location><page_5><loc_19><loc_43><loc_49><loc_44></location>L ϕψψ = f µ ( ϕ ) A µ + g µ ( ϕ ) V µ , (38)</formula>
<text><location><page_5><loc_9><loc_36><loc_49><loc_42></location>are suppressed by m -1 p . Therefore, L ϕψψ becomes the leading order term in the reheatig era. We neglect the higher order terms and apply the formalism developed in the previous works of the fermionic preheating [26-30].</text>
<text><location><page_5><loc_10><loc_34><loc_45><loc_35></location>In the previous works, Yukawa type interaction,</text>
<formula><location><page_5><loc_27><loc_32><loc_49><loc_33></location>ϕψψ, (39)</formula>
<text><location><page_5><loc_9><loc_29><loc_49><loc_30></location>is considered. In our model, the form of the interactions,</text>
<formula><location><page_5><loc_20><loc_26><loc_49><loc_28></location>∂ µ ϕψiγ µ ψ, ∂ µ ϕψiγ 5 γ µ ψ, (40)</formula>
<text><location><page_5><loc_9><loc_22><loc_49><loc_25></location>are different from the Yukawa type interaction and violate the parity.</text>
<section_header_level_1><location><page_5><loc_16><loc_18><loc_41><loc_19></location>A. The EoM in FLRW universe</section_header_level_1>
<text><location><page_5><loc_9><loc_8><loc_49><loc_16></location>For discussing the fermionic non-thermal particle production after the end of inflation, we derive the EoM of the classical fermionic field ψ and the Heisenberg operator ˆ ψ . In the spatially flat and homogeneous FLRW metric (16), the tetrad is given by e i µ = aδ i µ from its definition</text>
<text><location><page_5><loc_52><loc_48><loc_92><loc_53></location>g µν = η ij e i µ e j ν , and components regarding the torsionless spin connection are η ik ω ⋄ ijk = 3 H a δ j 0 and ϵ ijkl ω ⋄ ijk = 0. Thus, Eq. (30) becomes</text>
<formula><location><page_5><loc_56><loc_44><loc_92><loc_47></location>L ⋄ matter = -ψγ µ ∂ µ ψ -3 H 2 a ψγ 0 ψ -m ψ ψψ, (41)</formula>
<text><location><page_5><loc_52><loc_41><loc_92><loc_44></location>where H is defined by H = a ' a . Therefore, the Lagrangian density of ψ is denoted as</text>
<formula><location><page_5><loc_53><loc_34><loc_92><loc_40></location>e L matter = a 4 { -1 a ψδ µ i γ i ∂ µ ψ -3 H 2 a ψγ 0 ψ -m ψ ψψ + 1 a δ µ i ( f µ A i + g µ V i ) } , (42)</formula>
<text><location><page_5><loc_52><loc_29><loc_92><loc_33></location>where f µ ( ϕ ) and g µ ( ϕ ) are functions of the inflaton defined by Eqs. (35) and (36). Rescaling the fermion as a 3 2 ψ → ψ , one can finally obtain the Lagrangian density,</text>
<formula><location><page_5><loc_53><loc_25><loc_92><loc_28></location>e L matter = -ψδ µ i γ i ∂ µ ψ -m ψ aψψ + δ µ i ( f µ A i + g µ V i ) . (43)</formula>
<text><location><page_5><loc_52><loc_19><loc_92><loc_23></location>The inflaton ϕ after the end of inflation is assumed to be a homogeneous field, ∂ x ϕ = ∂ y ϕ = ∂ z ϕ = 0. The EoM of ψ is</text>
<formula><location><page_5><loc_55><loc_17><loc_92><loc_18></location>-γ µ ∂ µ ψ -m ψ aψ + i ( g 0 γ 0 + f 0 γ 5 γ 0 ) ψ = 0 . (44)</formula>
<text><location><page_5><loc_52><loc_12><loc_92><loc_16></location>We note that γ µ in Eq. (44) is the gamma matrices in the local Lorentz frame. Since the Heisenberg operator ˆ ψ also satisfies the identical equation,</text>
<formula><location><page_5><loc_55><loc_9><loc_92><loc_11></location>-γ µ ∂ µ ˆ ψ -m ψ a ˆ ψ + i ( g 0 γ 0 + f 0 γ 5 γ 0 ) ˆ ψ = 0 , (45)</formula>
<text><location><page_6><loc_9><loc_90><loc_49><loc_93></location>we can obtain the EoM of the spinors u s ( k , η ) and v s ( k , η ) by the decomposition of the operator ˆ ψ ,</text>
<formula><location><page_6><loc_12><loc_82><loc_49><loc_88></location>ˆ ψ = ∫ d 3 k √ 2(2 π ) 3 ∑ s [ u s ( k , η )ˆ a s ( k ) e i kx + v s ( k , η ) ˆ b † s ( k ) e -i kx ] , (46)</formula>
<text><location><page_6><loc_9><loc_77><loc_49><loc_80></location>where ˆ a s ( k ) , ˆ b s ( k ) indicate (anti) particle annihilation operator. They satisfy the anti-commutation relations,</text>
<formula><location><page_6><loc_17><loc_74><loc_49><loc_75></location>{ ˆ a s ( k ) , ˆ a † r ( l ) } = (2 π ) 3 δ 3 ( k -l ) δ rs , (47)</formula>
<formula><location><page_6><loc_17><loc_72><loc_49><loc_73></location>{ ˆ b s ( k ) , ˆ b † r ( l ) } = (2 π ) 3 δ 3 ( k -l ) δ rs , (48)</formula>
<formula><location><page_6><loc_22><loc_70><loc_49><loc_71></location>{ ˆ a s ( k ) , ˆ a r ( l ) } = 0 , (49)</formula>
<formula><location><page_6><loc_22><loc_68><loc_49><loc_69></location>{ ˆ b s ( k ) , ˆ b r ( l ) } = 0 . (50)</formula>
<text><location><page_6><loc_9><loc_63><loc_49><loc_66></location>We consider the case where the charge conjugate of fermion,</text>
<formula><location><page_6><loc_23><loc_59><loc_49><loc_61></location>C ˆ ψ T = iγ 2 γ 0 ˆ ψ T , (51)</formula>
<text><location><page_6><loc_9><loc_53><loc_49><loc_57></location>is well-defined and interchanges the particle and the antiparticle. Then, we obtain the relation between u s ( k , η ) and v s ( k , η ),</text>
<formula><location><page_6><loc_21><loc_50><loc_49><loc_51></location>u s ( k , η ) = C v s ( k , η ) T . (52)</formula>
<text><location><page_6><loc_9><loc_45><loc_49><loc_48></location>From Eq. (45) and its charge conjugate with (46) and (52), v s ( -k , η ) should satisfy</text>
<formula><location><page_6><loc_9><loc_40><loc_49><loc_43></location>( -γ 0 ∂ 0 -iγ k -am ψ + i ( g 0 γ 0 + f 0 γ 5 γ 0 )) v s ( -k , η ) = 0 , (53)</formula>
<formula><location><page_6><loc_9><loc_37><loc_50><loc_40></location>( -γ 0 ∂ 0 -iγ k -am ψ + i ( -g 0 γ 0 + f 0 γ 5 γ 0 )) v s ( -k , η ) = 0 . (54)</formula>
<text><location><page_6><loc_9><loc_32><loc_49><loc_35></location>Since g 0 is proportional to α , these equations are satisfied for α = 0.</text>
<text><location><page_6><loc_53><loc_92><loc_76><loc_93></location>We define the spinor u s ( k , η ) as</text>
<formula><location><page_6><loc_63><loc_87><loc_92><loc_91></location>u s ( k , η ) = ( u + s,k ( η ) ξ s, k u -s,k ( η ) ξ s, k ) , (55)</formula>
<text><location><page_6><loc_52><loc_83><loc_92><loc_86></location>where s indicates the spin direction and ξ s, k describes the eigen-spinor of helicity,</text>
<formula><location><page_6><loc_66><loc_80><loc_92><loc_82></location>σk ξ s, k = skξ s, k , (56)</formula>
<formula><location><page_6><loc_66><loc_76><loc_92><loc_78></location>ξ s, -k = -iσ 2 ξ ∗ s, k . (57)</formula>
<text><location><page_6><loc_52><loc_74><loc_75><loc_75></location>The equation (45) is rewritten as</text>
<formula><location><page_6><loc_54><loc_71><loc_92><loc_73></location>u ± s,k ( η ) ' = ± i ( ks + f 0 ) u ± s,k ( η ) -iam ψ u ∓ s,k ( η ) . (58)</formula>
<text><location><page_6><loc_52><loc_67><loc_92><loc_70></location>Performing the time derivative, we obtain the EoMs of the amplitude of the spinor, u + , -s,k ( η ),</text>
<formula><location><page_6><loc_55><loc_62><loc_92><loc_65></location>u ± s,k ( η ) '' = -(( ks + f 0 ) 2 + a 2 m 2 ψ ∓ if ' 0 ) u ± s,k ( η ) -ia ' m ψ u ∓ ( η ) . (59)</formula>
<formula><location><page_6><loc_80><loc_62><loc_82><loc_62></location>s,k</formula>
<text><location><page_6><loc_52><loc_54><loc_92><loc_60></location>The EoM (45) guarantees the relation | u + s,k | 2 + | u -s,k | 2 = 2 with the anti-commutation relations (47)-(50), the assumption (55) and the canonical anti-commutation relation { ψ ( η, x ) , ψ † ( η, y ) } = δ 3 ( x -y ).</text>
<text><location><page_6><loc_53><loc_53><loc_91><loc_54></location>Below, we evaluate the particle production for α = 0.</text>
<section_header_level_1><location><page_6><loc_58><loc_51><loc_85><loc_52></location>B. The number density of fermion</section_header_level_1>
<text><location><page_6><loc_52><loc_46><loc_92><loc_49></location>From the Lagrangian density (43), we can define the Hamiltonian operator ˆ H ψ ,</text>
<formula><location><page_6><loc_53><loc_41><loc_92><loc_45></location>ˆ H ψ = 1 a ∫ d 3 x ( ˆ ψγ I ∂ I ˆ ψ + m ψ a ˆ ψ ˆ ψ -f 0 ( ϕ ) ˆ ψiγ 5 γ 0 ˆ ψ ) , (60)</formula>
<text><location><page_6><loc_52><loc_32><loc_92><loc_39></location>where the script I denotes the space components, I ∈ { x, y, z } , and the factor a -1 comes from the fact that the Hamiltonian is the generator of the translation regarding the cosmic time t , and the relation dt = a ( η ) dη . By inserting (46) into (60), the Hamiltonian becomes</text>
<formula><location><page_6><loc_19><loc_24><loc_92><loc_28></location>ˆ H ψ = 1 a ∫ d 3 k (2 π ) 3 ∑ s [ E s k (ˆ a † s ( k )ˆ a s ( k ) -ˆ b s ( k ) ˆ b † s ( k )) + F s k ˆ b s ( -k )ˆ a s ( k ) + F s ∗ k ˆ b † s ( -k )ˆ a † s ( k )] , (61)</formula>
<text><location><page_6><loc_9><loc_18><loc_35><loc_20></location>where the coefficients E s k and F s k are</text>
<formula><location><page_6><loc_10><loc_13><loc_49><loc_18></location>E s k = ( ks + f 0 )(1 -| u + s,k | 2 ) + am ψ Re( u + s,k u -∗ s,k ) , (62) F s k = -( ks + f 0 ) u + s,k u -s,k + am ψ ( -u + s,k u + s,k + u -s,k u -s,k ) . (63)</formula>
<text><location><page_6><loc_9><loc_9><loc_49><loc_12></location>Since they satisfy | E s k | 2 + | F s k | 2 = ω s 2 k = a 2 m 2 ψ + ( ks + f 0 ) 2 , the Hamiltonian operator (61) is diagonalyzed by</text>
<text><location><page_6><loc_52><loc_17><loc_92><loc_20></location>introducing the time-dependent annihilation operators, ˆ a s ( k , η ) and ˆ b s ( k , η ),</text>
<formula><location><page_6><loc_52><loc_9><loc_92><loc_13></location>ˆ H ψ = ∫ d 3 k (2 π ) 3 ∑ s ω s k a (ˆ a † s ( k , η )ˆ a s ( k , η ) -ˆ b s ( k , η ) ˆ b † s ( k , η )) . (64)</formula>
<text><location><page_7><loc_9><loc_90><loc_49><loc_93></location>The Bogoliubov transformation connects the operators ˆ a s ( k ) , ˆ b † s ( -k ) and ˆ a s ( k , η ) , ˆ b † s ( k , η ),</text>
<formula><location><page_7><loc_10><loc_86><loc_49><loc_89></location>( ˆ a s ( k , η ) ˆ b † s ( k , η ) ) = ( α s ( k, η ) β s ( k, η ) -β ∗ s ( k, η ) α ∗ s ( k, η ) )( ˆ a s ( k ) ˆ b † s ( -k ) ) , (65)</formula>
<text><location><page_7><loc_9><loc_82><loc_49><loc_84></location>where α s ( k, η ) and β s ( k, η ) are the Bogoliubov coefficients defined by</text>
<formula><location><page_7><loc_21><loc_77><loc_49><loc_81></location>| α s ( k, η ) | 2 = ω s k + E s k 2 ω s k , (66)</formula>
<formula><location><page_7><loc_21><loc_74><loc_49><loc_77></location>| β s ( k, η ) | 2 = ω s k -E s k 2 ω s k . (67)</formula>
<text><location><page_7><loc_9><loc_67><loc_49><loc_73></location>From the definition of the Bogoliubov coefficients and the anti-commutation relations (47)-(50), the anti-commutation relations for the operators, ˆ a s ( k , η ) , ˆ b † s ( -k , η ), are derived to be</text>
<formula><location><page_7><loc_15><loc_64><loc_49><loc_66></location>{ ˆ a s ( k , η ) , ˆ a † r ( l , η ) } = (2 π ) 3 δ 3 ( k -l ) δ rs , (68)</formula>
<formula><location><page_7><loc_15><loc_62><loc_49><loc_64></location>{ ˆ b s ( k , η ) , ˆ b † r ( l , η ) } = (2 π ) 3 δ 3 ( k -l ) δ rs , (69)</formula>
<formula><location><page_7><loc_21><loc_60><loc_49><loc_62></location>{ ˆ a s ( k , η ) , ˆ a r ( l , η ) } = 0 , (70)</formula>
<formula><location><page_7><loc_21><loc_58><loc_49><loc_60></location>{ ˆ b s ( k , η ) , ˆ b r ( l , η ) } = 0 . (71)</formula>
<text><location><page_7><loc_9><loc_54><loc_49><loc_57></location>We redefine the Hamiltonian operator so that a minimum of its expectation value is zero,</text>
<formula><location><page_7><loc_9><loc_49><loc_49><loc_53></location>ˆ H ψ = ∫ d 3 k (2 π ) 3 ∑ s ω s k a (ˆ a † s ( k , η )ˆ a s ( k , η )+ ˆ b † s ( k , η ) ˆ b s ( k , η )) .</formula>
<text><location><page_7><loc_46><loc_48><loc_49><loc_49></location>(72)</text>
<text><location><page_7><loc_52><loc_87><loc_92><loc_93></location>We consider the vacuum state defined by ˆ a s ( k )( , ˆ b s ( k )) | 0 ⟩ η 0 = 0 at η = η 0 . Under the state, the expectation value of the number operators are developed</text>
<formula><location><page_7><loc_53><loc_82><loc_92><loc_86></location>〈 0 ∣ ∣ ∣ η 0 ˆ a † s ( k , η )ˆ a s ( k , η ) ∣ ∣ ∣ 0 〉 η 0 = 〈 0 ∣ ∣ ∣ η 0 ˆ b † s ( k , η ) ˆ b s ( k , η ) ∣ ∣ ∣ 0 〉 η 0 = | β s ( k, η ) | 2 . (73)</formula>
<text><location><page_7><loc_52><loc_72><loc_92><loc_80></location>The behavior of the expectation value of the number operator for the particle and the antiparticle is equivalent. Even if the expectation value of Hamiltonian ⟨ ˆ H ψ ( η ) ⟩ η 0 is zero at η 0 , it is not necessary to be so at η > η 0 . This means a non-thermal particle production. From the conditions of the amplitude of the spinor,</text>
<formula><location><page_7><loc_65><loc_69><loc_92><loc_70></location>| u + s,k | 2 + | u -s,k | 2 = 2 , (74)</formula>
<formula><location><page_7><loc_66><loc_66><loc_92><loc_68></location>⟨ ˆ H ψ ( η 0 ) ⟩ η 0 = 0 , (75)</formula>
<text><location><page_7><loc_52><loc_62><loc_92><loc_65></location>the initial values of the amplitude of the spinor are derived,</text>
<formula><location><page_7><loc_61><loc_57><loc_92><loc_61></location>u + s,k (0) = √ 1 -( ks + f 0 ( ϕ (0))) ω s k (0) , (76)</formula>
<formula><location><page_7><loc_55><loc_53><loc_92><loc_57></location>u -s,k (0) = m ψ a (0) ω s k (0) ( 1 -( ks + f 0 ( ϕ (0))) ω s k (0) ) -1 2 . (77)</formula>
<text><location><page_7><loc_52><loc_48><loc_92><loc_52></location>We define the quantities to observe whether the particle production occurs. The expectation value of total number density is given by</text>
<formula><location><page_7><loc_18><loc_39><loc_92><loc_43></location>〈 0 ∣ ∣ ∣ η 0 ˆ N s particle a 3 V ∣ ∣ ∣ 0 〉 η 0 ≡ ∫ d 3 k (2 π ) 3 a 3 V 〈 0 ∣ ∣ ∣ η 0 ˆ a † s ( k , η )ˆ a s ( k , η ) ∣ ∣ ∣ 0 〉 η 0 = 1 2 π 2 a 3 ∫ dkk 2 ( 1 2 -E s k 2 ω s k ) , (78)</formula>
<text><location><page_7><loc_9><loc_33><loc_49><loc_35></location>where V is the volume of conformal space. The number density regarding the conformal momentum space is</text>
<formula><location><page_7><loc_22><loc_28><loc_49><loc_32></location>n s m ψ ,k ≡ 1 2 -E s k 2 ω s k . (79)</formula>
<text><location><page_7><loc_9><loc_23><loc_49><loc_27></location>It vanishes at η = η 0 and must not exceed unity at anytime by Pauli blocking. The total number of (anti) particles is defined by</text>
<formula><location><page_7><loc_21><loc_19><loc_49><loc_22></location>N s ψ ≡ ∫ dk k 2 n s m ψ ,k 2 π 2 . (80)</formula>
<text><location><page_7><loc_9><loc_16><loc_31><loc_18></location>We evaluate the energy density,</text>
<formula><location><page_7><loc_21><loc_13><loc_49><loc_15></location>ρ m ψ = 2Σ s ∫ dkρ s m ψ ,k , (81)</formula>
<formula><location><page_7><loc_21><loc_9><loc_49><loc_12></location>ρ s m ψ ,k = k 2 ω s k n s m ψ ,k 2 π 2 a 4 , (82)</formula>
<text><location><page_7><loc_52><loc_33><loc_92><loc_35></location>where the factor 2 comes from the degree of freedom of the particle and the antiparticle.</text>
<section_header_level_1><location><page_7><loc_56><loc_27><loc_88><loc_29></location>IV. ANALYTICAL AND NUMERICAL RESULTS</section_header_level_1>
<text><location><page_7><loc_52><loc_18><loc_92><loc_25></location>In this section, an analytical implication regarding the behavior of the number density and some numerical results are exhibited. The former suggests the behavior for particles lighter than the inflaton. We will attempt a specific application of the latter results in Sec. 5.</text>
<section_header_level_1><location><page_7><loc_65><loc_13><loc_79><loc_14></location>A. Massless limit</section_header_level_1>
<text><location><page_7><loc_52><loc_9><loc_92><loc_11></location>We can analytically evaluate the particle production for a simple case. Here, we consider the massless limit,</text>
<text><location><page_8><loc_9><loc_90><loc_49><loc_93></location>m ψ = 0, which has several differences from the massive particle.</text>
<text><location><page_8><loc_9><loc_85><loc_49><loc_90></location>First, the initial conditions (76) and (77) are not appropriate for the massless limit. The initial conditions of u + s,k and u -s,k are divided into four cases. For f 0 (0) > 0, the initial conditions become</text>
<formula><location><page_8><loc_25><loc_82><loc_49><loc_84></location>u + ↑ ,k (0) = 0 , (83)</formula>
<formula><location><page_8><loc_23><loc_80><loc_49><loc_82></location>u -↑ ,k (0) = √ 2 e iθ , (84)</formula>
<formula><location><page_8><loc_18><loc_75><loc_49><loc_79></location>u + ↓ ,k (0) = { √ 2 e iψ , k > | f 0 (0) | 0 , k < | f 0 (0) | (85)</formula>
<formula><location><page_8><loc_18><loc_69><loc_49><loc_72></location>u -↓ ,k (0) = { 0 , k > | f 0 (0) | √ 2 e iψ , k < | f 0 (0) | (86)</formula>
<text><location><page_8><loc_9><loc_65><loc_49><loc_68></location>where θ and ψ are arbitrary phases. For f 0 (0) < 0, these are given by</text>
<formula><location><page_8><loc_23><loc_62><loc_49><loc_65></location>u + ↓ ,k (0) = √ 2 e iψ , (87)</formula>
<formula><location><page_8><loc_25><loc_60><loc_49><loc_62></location>u -↓ ,k (0) = 0 , (88)</formula>
<formula><location><page_8><loc_18><loc_55><loc_49><loc_58></location>u + ↑ ,k (0) = { 0 , k > | f 0 (0) | √ 2 e iθ , k < | f 0 (0) | (89)</formula>
<formula><location><page_8><loc_18><loc_49><loc_49><loc_53></location>u -↑ ,k (0) = { √ 2 e iθ , k > | f 0 (0) | 0 . k < | f 0 (0) | (90)</formula>
<text><location><page_8><loc_9><loc_47><loc_43><loc_48></location>Second, the number densities are represented by</text>
<formula><location><page_8><loc_11><loc_43><loc_49><loc_46></location>n ↑ ,k ( η ) = 1 2 (1 -sign( k + f 0 )(1 -| u + ↑ ,k ( η ) | 2 )) , (91)</formula>
<formula><location><page_8><loc_11><loc_39><loc_49><loc_42></location>n ↓ ,k ( η ) = 1 2 (1 -sign( k -f 0 )(1 -| u -↓ ,k ( η ) | 2 )) . (92)</formula>
<text><location><page_8><loc_9><loc_37><loc_49><loc_38></location>Third, the 1st order differential equations (58) reduce to</text>
<formula><location><page_8><loc_17><loc_34><loc_49><loc_36></location>u ± s,k ( η ) ' = ± i ( ks + f 0 ( η )) u ± s,k ( η ) . (93)</formula>
<text><location><page_8><loc_9><loc_31><loc_49><loc_34></location>From Eq. (93), the time derivative of | u ± s,k ( η ) | 2 vanishes,</text>
<formula><location><page_8><loc_24><loc_28><loc_49><loc_30></location>| u ± s,k ( η ) | 2 ' = 0 . (94)</formula>
<text><location><page_8><loc_9><loc_22><loc_49><loc_27></location>Because of a damped oscillation of the inflaton ϕ , | f 0 | becomes smaller than k after a sufficient amount of time. The signature of k + sf 0 is positive and Eqs. (91) and (92) are simplified to</text>
<formula><location><page_8><loc_19><loc_15><loc_49><loc_21></location>n ↑ ,k = 1 2 (1 -(1 -| u + ↑ ,k | 2 )) = 1 2 | u + ↑ ,k | 2 η =0 , (95)</formula>
<formula><location><page_8><loc_19><loc_9><loc_49><loc_15></location>n ↓ ,k = 1 2 (1 -(1 -| u -↓ ,k | 2 )) = 1 2 | u -↓ ,k | 2 η =0 . (96)</formula>
<table>
<location><page_8><loc_58><loc_89><loc_86><loc_94></location>
<caption>TABLE I: particle production in massless limit</caption>
</table>
<text><location><page_8><loc_52><loc_70><loc_92><loc_83></location>Thus, the number densities are fixed by the initial values of | u ± s,k | 2 η =0 . The non-thermal excitations strongly depend on the initial conditions at the massless limit. The hericity of the produced particle is determined by the initial condition f 0 (0) (Table.I). From the numerical simulation, it is observed that lighter particles have a similiar property. If the reheating starts at ϕ ' (0) = 0, the production of lighter particles hardly occur due to the property in Tab. I with f 0 (0) ∝ ϕ ' (0) = 0.</text>
<section_header_level_1><location><page_8><loc_63><loc_66><loc_80><loc_67></location>B. Numerical results</section_header_level_1>
<text><location><page_8><loc_52><loc_46><loc_92><loc_64></location>Since it is difficult to solve Eq. (59) analytically, we evaluate the behavior of the number density through numerical calculations. Here, we set ρ ψ (0) = 0 and employ the ( ϵR ) 2 model for the dynamics of the inflaton governing the evolution of the universe. As is shown in Sec. 2, initial values are fixed at ϕ (0) ∼ 0 . 940 m p and dϕ dη (0) ∼ -0 . 293 m p m ϕ , and the inflaton dynamics is determined by the values ϕ (0), ϕ ' (0). We solve the Eq. (59) to obtain n s m ψ ,k , N s ψ and ρ m ψ . Numerical calculations in our research are performed by Julia and an algorithm specified by Verner9 solver in the package DifferentialEq.jl.</text>
<section_header_level_1><location><page_8><loc_63><loc_41><loc_81><loc_42></location>1. Heavy and light fermion</section_header_level_1>
<text><location><page_8><loc_52><loc_16><loc_92><loc_39></location>We evaluate the time evolution of the number density n s m ψ ,k for the helicity up and down particles with a certain conformal momentum k . Fig.2 clearly shows that the non-minimal coupling β contributes to the nonthermal particle production. Because of the interaction between the pseudoscalaron and the pseudovector current ψiγ 5 γ µ ψ in Eq. (34), the amount of the produced particles depend on the helicity. In Fig.3, we draw the distribution of n s m ψ ,k for m ψ = 4 m ϕ as a function of the conformal momentum k/m ϕ for different β at η = 20 m -1 ϕ . The upper limit of the conformal momentum for the non-thermal particle production is observed to be higher as β increases. Thus, a larger | β | can lead to a higher conformal momentum excitation. The extreme oscillations in Fig.3 are confirmed to be unaltered by the initial values of the inflaton and the solvers.</text>
<text><location><page_8><loc_52><loc_9><loc_92><loc_16></location>After the end of of the particle production, the total number of the produced particles remains at a certain value (Fig.4). In Figs. 3 and 4, different behavior is observed between up and down particles. Both up and down particles are produced for m ψ = 4 m ϕ . However,</text>
<figure>
<location><page_9><loc_11><loc_78><loc_46><loc_93></location>
<caption>(a) m ψ = 4 m ϕ , β =100 , k = 10 m ϕ</caption>
</figure>
<figure>
<location><page_9><loc_11><loc_59><loc_46><loc_75></location>
<caption>Fig.5 shows that extremely heavy particles ( m ψ = 10 m ϕ ) are not produced.</caption>
</figure>
<text><location><page_9><loc_18><loc_57><loc_20><loc_58></location>(b)</text>
<text><location><page_9><loc_20><loc_57><loc_22><loc_58></location>m</text>
<text><location><page_9><loc_22><loc_57><loc_22><loc_57></location>ψ</text>
<text><location><page_9><loc_23><loc_57><loc_25><loc_58></location>= 4</text>
<text><location><page_9><loc_25><loc_57><loc_27><loc_58></location>m</text>
<text><location><page_9><loc_27><loc_57><loc_27><loc_57></location>ϕ</text>
<text><location><page_9><loc_27><loc_57><loc_28><loc_58></location>,</text>
<text><location><page_9><loc_28><loc_57><loc_29><loc_58></location>β</text>
<text><location><page_9><loc_30><loc_57><loc_32><loc_58></location>=0</text>
<text><location><page_9><loc_32><loc_57><loc_33><loc_58></location>,</text>
<text><location><page_9><loc_33><loc_57><loc_34><loc_58></location>k</text>
<text><location><page_9><loc_35><loc_57><loc_38><loc_58></location>= 10</text>
<text><location><page_9><loc_38><loc_57><loc_39><loc_58></location>m</text>
<text><location><page_9><loc_39><loc_57><loc_40><loc_57></location>ϕ</text>
<paragraph><location><page_9><loc_10><loc_52><loc_47><loc_55></location>FIG. 2: The time evolution of n s m ψ ,k with k = 10 m ϕ and m ψ = 4 m ϕ .</paragraph>
<text><location><page_9><loc_9><loc_36><loc_49><loc_45></location>We also examine the number density distribution for lighter particles production. Fig.6 shows the number density distribution for m ψ = 0 . 01 m ϕ at η = 100 m -1 ϕ . It is observed that the higher conformal momentum particles can be produced with increasing β , as is mentioned in Sec. 4-1. Compared with Fig.3, the property in Tab. I is almost confirmed for the lighter particles.</text>
<text><location><page_9><loc_9><loc_27><loc_49><loc_35></location>From Eq. (58), the helicity is inverted when the sign of f 0 is reversed. A dominant contribution to f 0 comes from the second term in Eq. (35). Thus, the up and down particles are exchanged and the aforementioned numerical results are nearly inverted for up and down with the sign of β flipped.</text>
<section_header_level_1><location><page_9><loc_19><loc_22><loc_39><loc_23></location>2. Intermediate-mass particle</section_header_level_1>
<text><location><page_9><loc_9><loc_9><loc_49><loc_20></location>For intermediate-mass particles, m ψ = 0 . 1 m ϕ ∼ 0 . 5 m ϕ , we observe an alternative property that does not appear in light and heavy particles. A fermionic field with a higher conformal momentum is gradually generated and the Boltzmann-type distribution is established. According to the distribution of the number density (Fig.7) for m ψ = 0 . 3 m ϕ at η = 110 m ϕ -1 , particles with greater conformal momentum are excited than that</text>
<text><location><page_9><loc_52><loc_92><loc_79><loc_93></location>in the case of light and heavy particle.</text>
<text><location><page_9><loc_52><loc_83><loc_92><loc_92></location>We also evaluate the number density distribution in the non-expanding universe with a constant scale factor. The higher conformal momentum excitation is not observed in Fig. 8 for the non-expanding case. It means that the higher conformal momentum excitation is due to the expansion of the universe.</text>
<section_header_level_1><location><page_9><loc_55><loc_79><loc_88><loc_80></location>V. COSMOLOGICAL CONSEQUENCES</section_header_level_1>
<text><location><page_9><loc_52><loc_67><loc_92><loc_77></location>In this section, we apply the particle production to the reheating phenomena in the early universe. In the standard history of the universe, the energy density of the matter field must exceed the energy density of the inflaton after the end of inflation. Therefore, we examine if the condition ρ ψ ≫ ρ ϕ can be achieved with the nonthermal and thermal particle production.</text>
<section_header_level_1><location><page_9><loc_66><loc_63><loc_78><loc_64></location>A. Preheating</section_header_level_1>
<text><location><page_9><loc_52><loc_53><loc_92><loc_61></location>The preheating is the thermal process of the universe due to the non-thermal particle production before the reheating. As we show in Chapter 4, a larger | β | makes higher conformal momentum particles excited. From Fig.9, ρ ψ /ρ ϕ grows as β increases. The condition ρ ψ /ρ ϕ ≫ 1 can be achieved for sufficiently large β .</text>
<text><location><page_9><loc_52><loc_43><loc_92><loc_52></location>However, it is necessary to discuss thermalization due to the decay of produced particles into relativistic particles to define the reheating temperature. Thus, the results presented in this section only indicate that the energy can be sufficiently transferred from the inflaton to the matter. Further analyses are required to estimate the reheating temperature.</text>
<section_header_level_1><location><page_9><loc_66><loc_38><loc_77><loc_39></location>B. Reheating</section_header_level_1>
<text><location><page_9><loc_52><loc_28><loc_92><loc_36></location>We adapt the perturvative calculations of the zero temperature quantum field theory to the fluctuation of ϕ after a long time from the end of inflation. If the energy density of the inflaton ρ ψ transfers to that of relativistic matter ρ r through the decay with the decay rate Γ and ρ r | η =0 = 0, the energy density, ρ r , is estimated to be</text>
<formula><location><page_9><loc_65><loc_24><loc_92><loc_27></location>ρ r ( t r ) = 27 25 Γ 2 m 2 p , (97)</formula>
<text><location><page_9><loc_52><loc_19><loc_92><loc_23></location>at the moment for ρ ϕ = ρ r . From the Stefan-Boltzman law ρ r ∝ T 4 , the reheating temparature T R is proportional to √ Γ.</text>
<text><location><page_9><loc_52><loc_15><loc_92><loc_19></location>In Einstein-Cartan pseudoscalaron model with nonminimal couplings to fermion, the decay of the ϕ into the fermions due to the interaction term</text>
<formula><location><page_9><loc_58><loc_9><loc_92><loc_15></location>f µ ( ϕ ) A µ ∼ √ 3 2 √ 2 m p 1 + βb √ 1 + b 2 ψiγ 5 γ µ ψ∂ µ ϕ ≡ c ϕψψ ψiγ 5 γ µ ψ∂ µ ϕ, (98)</formula>
<figure>
<location><page_10><loc_13><loc_74><loc_88><loc_93></location>
<caption>FIG. 3: The distribution of n s m ψ ,k as a function of the conformal momentum k/m ϕ at m ϕ η = 20 ( m ψ = 4 m ϕ ).</caption>
</figure>
<figure>
<location><page_10><loc_13><loc_50><loc_87><loc_70></location>
<caption>FIG. 4: The time evolution of N s ψ ( m ψ = 4 m ϕ ).</caption>
</figure>
<text><location><page_10><loc_9><loc_42><loc_49><loc_44></location>dominates the particle production. The decay rate Γ ϕψψ from this interaction is found to be</text>
<formula><location><page_10><loc_13><loc_33><loc_49><loc_41></location>Γ ϕψψ = | c ϕψψ | 2 m ϕ m 2 ψ 2 π √ 1 -( 2 m ψ m ϕ ) 2 = 3 m ϕ m 2 ψ (1 + bβ ) 2 16 πm 2 p (1 + b 2 ) √ 1 -( 2 m ψ m ϕ ) 2 . (99)</formula>
<text><location><page_10><loc_9><loc_23><loc_49><loc_31></location>At β = 0 the result (99) reproduces the one derived in the previous work [14]. Thus, even in thermal particle production, the effect of non-minimal coupling to fermion is significant for | βb | ≫ 1. A reheating temperature is tuned by non-minimal coupling, β . For m ϕ ≫ m ψ and | bβ | ≫ 1, reheating temperature is estimated as</text>
<formula><location><page_10><loc_13><loc_19><loc_49><loc_22></location>T R = ( 30 ∗ 27 25 π 2 g ∗ ) 1 4 √ 3 16 π β ( m ψ m ϕ ) 1 / 2 m ϕ , (100)</formula>
<text><location><page_10><loc_9><loc_13><loc_49><loc_18></location>where k B is the Boltzmann constant and g ∗ shows the physical degree of freedom. The interaction of inflaton and the fermion vector current,</text>
<formula><location><page_10><loc_13><loc_9><loc_49><loc_13></location>g µ ( ϕ ) V µ ∼ √ 3 α 2 √ 2 m p b √ 1 + b 2 ψiγ µ ψ∂ µ ϕ, (101)</formula>
<text><location><page_10><loc_52><loc_39><loc_92><loc_44></location>has no contribution to the decay rate for ϕ → ψψ , at the tree level. Thus, α does not appear in Eq. (99). The contribution from α dependent terms appear from the next to leading order.</text>
<section_header_level_1><location><page_10><loc_57><loc_34><loc_87><loc_35></location>VI. SUMMARY AND DISCUSSION</section_header_level_1>
<text><location><page_10><loc_52><loc_26><loc_92><loc_32></location>In this paper, we have investigated the non-thermal fermionic particle production in Einstein-Cartan gravity with modified Holst term and non-minimal couplings to fermion.</text>
<text><location><page_10><loc_52><loc_9><loc_92><loc_26></location>In EC gravity, the only imposed condition is metric compatiblity, ∇ µ g νρ = 0. Thus, the antisymmetric part of the affine connection T µ νρ ≡ Γ µ νρ -Γ µ ρν (i.e., torsion) can have a non-vanishing value. The existence of the torsion introduces invariant scalar components which do not exist in GR. One of these quantities is the Holst term ϵR = ϵ µνρσ R µνρσ / √ -g . The derivative of the affine connection in this term introduces a dynamical degree of freedom through the EoM. Following the auxiliary field method, the EoM for the affine connection becomes an algebraic equation for the torsion. After solving the EoM of the torsion, one can get the effective metric action with</text>
<figure>
<location><page_11><loc_13><loc_74><loc_88><loc_93></location>
<caption>FIG. 5: The distribution of n s m ψ ,k as a function of the conformal momentum k/m ϕ at m ϕ η = 20( m ψ = 10 m ϕ ).</caption>
</figure>
<figure>
<location><page_11><loc_13><loc_50><loc_87><loc_69></location>
<caption>FIG. 6: The distribution of n s m ψ ,k as a function of the conformal momentum k/m ϕ at m ϕ η = 100( m ψ = 0 . 01 m ϕ ).</caption>
</figure>
<text><location><page_11><loc_9><loc_39><loc_49><loc_44></location>the dynamical pseudoscalaron ϕ [8, 9, 14, 21, 22]. The potential energy of the pseudoscalaron can induce inflation in EC gravity. In this paper, we utilize the ( ϵR ) 2 model, which is consistent with the CMB observations.</text>
<text><location><page_11><loc_9><loc_11><loc_49><loc_38></location>The coupling between the matter fields and the affine connection in the original action yields an interaction between the matter fields and the inflaton in the effective metric action. In this study, we have employed a theory with non-minimal couplings to fermion where two parameters denoted as α and β are introduced [25]. This extension introduces interactions between the inflaton and the fermionic field ψ described in Eq. (34). Since the inflaton ϕ has a large value after the end of inflation, the interactions between the ϕ and the ψ destabilize the vacuum. Through the instability, the non-thermal particle production occurs. Since the parameter α must be zero to satisfy the condition that C ψ T represents a field where particles and antiparticles are exchanged, we have shown that the non-minimal coupling β in Eq. (34) contributes to the non-thermal particle production after the end of inflation. In this paper, only ϕψψ terms in Eq. (46) that introduce linear term with respect to ψ to the EoM of ψ is focused.</text>
<text><location><page_11><loc_10><loc_9><loc_49><loc_10></location>We have examined how many particles are produced</text>
<text><location><page_11><loc_52><loc_13><loc_92><loc_44></location>due to the existence of β through numerical calculations and observed several properties. First, a larger value of β leads to the excitation of particles with higher conformal momentum, and the excitation persists over time. It is also observed that fermions much heavier than the inflaton are hardly excited. Second, there is generally a difference in the amount of produced particles between helicity up and down particles. It should be noted that the produced numbers of particles and antiparticles are identical, and the total spin of the universe is conserved. Third, for the lighter mass fermion, m ψ ⪅ 0 . 1 m ϕ , the difference in the number of created particles between helicities becomes more pronounced. This property is also suggested analytically at the massless limit. If the initial value of dϕ dη is small enough, light particles are not excited non-thermally. Eventually, for the intermediatemass fermion 0 . 1 m ϕ ⪅ m ψ ⪅ 0 . 5 m ϕ , the number density is exponentially supressed for a higher conformal momentum like a Boltzmann distribution. Therefore, particles with higher conformal momentum can be excited than for heavier and lighter fermions. This property is not observed in a non-expanding universe.</text>
<text><location><page_11><loc_52><loc_9><loc_92><loc_11></location>We have applied the particle production to the reheating and the preheating of the universe. From the</text>
<figure>
<location><page_12><loc_13><loc_74><loc_88><loc_93></location>
<caption>FIG. 7: The distribution of n s m ψ ,k as a function of the conformal momentum k/m ϕ at m ϕ η = 110( m ψ = 0 . 3 m ϕ ).</caption>
</figure>
<figure>
<location><page_12><loc_13><loc_51><loc_50><loc_70></location>
<caption>FIG. 8: The distribution of n s m ψ ,k as a function of the conformal momentum k/m ϕ at m ϕ η = 110( m ψ = 0 . 3 m ϕ ) in non-expanding case.</caption>
</figure>
<figure>
<location><page_12><loc_10><loc_27><loc_48><loc_43></location>
<caption>FIG. 9: The time evolution of the ratio of energy density ( ρ up ψ + ρ down ψ ) /ρ ϕ ( m ψ = 1 m ϕ ).</caption>
</figure>
<text><location><page_12><loc_9><loc_9><loc_49><loc_17></location>consistency with the CMB observations, we set a model paremeter b ∼ -320 in our analysis. In the preheating era, sufficiently large values of β have the potential to make the energy density of ψ dominant over that of the inflaton. In the reheating era, we derive the formula of the reheating temperature T R,β with non-minimal cou-</text>
<text><location><page_12><loc_52><loc_39><loc_92><loc_43></location>. About | βb | times larger reheating temperature is predicted than that for the minimal coupling [14]. Thus, the contribution of β is important in both eras.</text>
<text><location><page_12><loc_52><loc_9><loc_92><loc_37></location>Future research directions include further development of analytical and numerical discussions and applications to phenomenology. Due to the complexity of the function of the inflaton coupling to fermion as given in Eq. (35), the dynamics of fermions during inflation and the backreaction to the evolution of the universe during particle production are not considered. To discuss a realistic universe, consideration of these two aspects can not be avoidable. Moreover, analytical solutions in some limits are necessary to verify the numerical results. Potential applications of our research include the production of the dark matter and the matter-antimatter asymmetry. The particle production investigated in our research is induced by gravitational effect. Heavy fermions that could be dark matter may be produced after the end of inflation. Our asymmetric helicity production has a possibility to describe the matter-antimatter asymmetry through particle production of fermions with lepton numbers [32, 33]. Our analysis can be applied to Majorana fermion which directly induces lepton number asymme-</text>
<section_header_level_1><location><page_13><loc_19><loc_88><loc_39><loc_89></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_13><loc_9><loc_83><loc_49><loc_86></location>The authors would like to thank H. Sakamoto, M. Alwan, M. Taniguchi and S. Takahashi for valuable discus-</text>
<unordered_list>
<list_item><location><page_13><loc_10><loc_76><loc_43><loc_77></location>[1] A. A. Starobinsky, Phys. Lett. B 91 , 99 (1980).</list_item>
<list_item><location><page_13><loc_10><loc_73><loc_49><loc_76></location>[2] T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys. 82 , 451 (2010), arXiv:0805.1726 [gr-qc].</list_item>
<list_item><location><page_13><loc_10><loc_71><loc_49><loc_73></location>[3] S. Nojiri and S. D. Odintsov, Phys. Rept. 505 , 59 (2011), arXiv:1011.0544 [gr-qc].</list_item>
<list_item><location><page_13><loc_10><loc_68><loc_49><loc_70></location>[4] S. Nojiri, S. D. Odintsov, and V. K. Oikonomou, Phys. Rept. 692 , 1 (2017), arXiv:1705.11098 [gr-qc].</list_item>
<list_item><location><page_13><loc_10><loc_65><loc_49><loc_68></location>[5] L. Kofman, A. D. Linde, and A. A. Starobinsky, Phys. Rev. Lett. 73 , 3195 (1994), arXiv:hep-th/9405187.</list_item>
<list_item><location><page_13><loc_10><loc_63><loc_49><loc_65></location>[6] L. Kofman, A. D. Linde, and A. A. Starobinsky, Phys. Rev. D 56 , 3258 (1997), arXiv:hep-ph/9704452.</list_item>
<list_item><location><page_13><loc_10><loc_60><loc_49><loc_62></location>[7] H. Motohashi and A. Nishizawa, Phys. Rev. D 86 , 083514 (2012), arXiv:1204.1472 [astro-ph.CO].</list_item>
<list_item><location><page_13><loc_10><loc_58><loc_49><loc_60></location>[8] G. Pradisi and A. Salvio, Eur. Phys. J. C 82 , 840 (2022), arXiv:2206.15041 [hep-th].</list_item>
<list_item><location><page_13><loc_10><loc_55><loc_49><loc_57></location>[9] W. Barker and S. Zell, (2024), arXiv:2402.14917 [hepth].</list_item>
<list_item><location><page_13><loc_9><loc_51><loc_49><loc_55></location>[10] F. W. Hehl, J. D. McCrea, E. W. Mielke, and Y. Ne'eman, Phys. Rept. 258 , 1 (1995), arXiv:grqc/9402012.</list_item>
<list_item><location><page_13><loc_9><loc_50><loc_40><loc_51></location>[11] P. C. Nelson, Phys. Lett. A 79 , 285 (1980).</list_item>
<list_item><location><page_13><loc_9><loc_47><loc_49><loc_49></location>[12] R. Hojman, C. Mukku, and W. A. Sayed, Phys. Rev. D 22 , 1915 (1980).</list_item>
<list_item><location><page_13><loc_9><loc_44><loc_49><loc_47></location>[13] S. Holst, Phys. Rev. D 53 , 5966 (1996), arXiv:grqc/9511026.</list_item>
<list_item><location><page_13><loc_9><loc_42><loc_49><loc_44></location>[14] A. Salvio, Phys. Rev. D 106 , 103510 (2022), arXiv:2207.08830 [hep-ph].</list_item>
<list_item><location><page_13><loc_9><loc_39><loc_49><loc_41></location>[15] I. D. Gialamas and K. Tamvakis, JCAP 03 , 042 (2023), arXiv:2212.09896 [gr-qc].</list_item>
<list_item><location><page_13><loc_9><loc_38><loc_48><loc_39></location>[16] E. Cartan, Annales Sci. Ecole Norm. Sup. 41 , 1 (1924).</list_item>
<list_item><location><page_13><loc_9><loc_36><loc_43><loc_37></location>[17] T. W. B. Kibble, J. Math. Phys. 2 , 212 (1961).</list_item>
<list_item><location><page_13><loc_9><loc_35><loc_49><loc_36></location>[18] D. W. Sciama, Rev. Mod. Phys. 36 , 463 (1964), [Erra-</list_item>
</unordered_list>
<text><location><page_13><loc_55><loc_76><loc_84><loc_77></location>tum: Rev.Mod.Phys. 36, 1103-1103 (1964)].</text>
<unordered_list>
<list_item><location><page_13><loc_52><loc_73><loc_92><loc_76></location>[19] F. W. Hehl and B. K. Datta, J. Math. Phys. 12 , 1334 (1971).</list_item>
<list_item><location><page_13><loc_52><loc_71><loc_92><loc_73></location>[20] F. W. Hehl, P. Von Der Heyde, G. D. Kerlick, and J. M. Nester, Rev. Mod. Phys. 48 , 393 (1976).</list_item>
<list_item><location><page_13><loc_52><loc_68><loc_92><loc_70></location>[21] A. Di Marco, E. Orazi, and G. Pradisi, Eur. Phys. J. C 84 , 146 (2024), arXiv:2309.11345 [hep-th].</list_item>
<list_item><location><page_13><loc_52><loc_65><loc_92><loc_68></location>[22] M. He, M. Hong, and K. Mukaida, JCAP 05 , 107 (2024), arXiv:2402.05358 [gr-qc].</list_item>
<list_item><location><page_13><loc_52><loc_63><loc_92><loc_65></location>[23] M. Piani and J. Rubio, JCAP 12 , 002 (2023), arXiv:2304.13056 [hep-ph].</list_item>
<list_item><location><page_13><loc_52><loc_60><loc_92><loc_62></location>[24] I. D. Gialamas and H. Veermae, Phys. Lett. B 844 , 138109 (2023), arXiv:2305.07693 [hep-th].</list_item>
<list_item><location><page_13><loc_52><loc_58><loc_92><loc_60></location>[25] L. Freidel, D. Minic, and T. Takeuchi, Phys. Rev. D 72 , 104002 (2005), arXiv:hep-th/0507253.</list_item>
<list_item><location><page_13><loc_52><loc_55><loc_92><loc_57></location>[26] P. B. Greene and L. Kofman, Phys. Lett. B 448 , 6 (1999), arXiv:hep-ph/9807339.</list_item>
<list_item><location><page_13><loc_52><loc_52><loc_92><loc_55></location>[27] P. B. Greene and L. Kofman, Phys. Rev. D 62 , 123516 (2000), arXiv:hep-ph/0003018.</list_item>
<list_item><location><page_13><loc_52><loc_50><loc_92><loc_52></location>[28] G. F. Giudice, M. Peloso, A. Riotto, and I. Tkachev, JHEP 08 , 014 (1999), arXiv:hep-ph/9905242.</list_item>
<list_item><location><page_13><loc_52><loc_47><loc_92><loc_49></location>[29] M. Peloso and L. Sorbo, JHEP 05 , 016 (2000), arXiv:hepph/0003045.</list_item>
<list_item><location><page_13><loc_52><loc_44><loc_92><loc_47></location>[30] N. Herring and D. Boyanovsky, Phys. Rev. D 101 , 123522 (2020), arXiv:2005.00391 [astro-ph.CO].</list_item>
<list_item><location><page_13><loc_52><loc_42><loc_92><loc_44></location>[31] F. Gronwald, Int. J. Mod. Phys. D 6 , 263 (1997), arXiv:gr-qc/9702034.</list_item>
<list_item><location><page_13><loc_52><loc_39><loc_92><loc_41></location>[32] P. Adshead and E. I. Sfakianakis, JCAP 11 , 021 (2015), arXiv:1508.00891 [hep-ph].</list_item>
<list_item><location><page_13><loc_52><loc_36><loc_92><loc_39></location>[33] P. Adshead and E. I. Sfakianakis, Phys. Rev. Lett. 116 , 091301 (2016), arXiv:1508.00881 [hep-ph].</list_item>
<list_item><location><page_13><loc_52><loc_34><loc_92><loc_36></location>[34] M. Fukugita and T. Yanagida, Phys. Lett. B 174 , 45 (1986).</list_item>
</unordered_list>
</document>
[
{
"title": "Non-thermal particle production in Einstein-Cartan gravity with modified Holst term and non-minimal couplings",
"content": "Tomohiro Inagaki ∗ Graduate School of Advanced Science and Engineering, Hiroshima University, Higashi-Hiroshima 739-8526, Japan Information Media Center, Hiroshima University, Higashi-Hiroshima 739-8521, Japan and Core of Research for the Energetic Universe, Hiroshima University, Higashi-Hiroshima 739-8526, Japan",
"pages": [
1
]
},
{
"title": "Naoki Yoshioka †",
"content": "Graduate School of Advanced Science and Engineering, Hiroshima University, Higashi-Hiroshima 739-8526, Japan (Dated: October 18, 2024) Non-thermal fermionic particle production is investigated in Einstein-Cartan modified gravity with a modified Holst term and non-minimal couplings between the spin connection and a fermion. By using the auxiliary field method, the theory is rewritten into a pseudoscalar-tensor theory with Einstein-Hilbert action and canonical kinetic and potential terms for a pseudoscalar field. The introduced field is called Einstein-Cartan pseudoscalaron. If the potential energy of the EinsteinCartan pseudoscalaron dominates the energy density of the early universe, it causes inflationary expansion. After the end of inflation, the pseudoscalaron develops a large value and the non-minimal couplings destabilize the vacuum. Evaluating the non-thermal fermionic particle production process, we obtain the mass and the helicity dependences of the produced particle number density. We show the model parameters to enhance the preheating and reheating processes.",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "The extension of general relativity (GR) from a geometrical perspective is one of the candidates for solving cosmological problems. In F ( R ) theories, the gravitational action is replaced by a function of the curvature, R → F ( R ), and it has been shown to describe well various cosmological phenomena [1-4]. For example, the idea explains the inflationary expansion of the universe, and the gravitational interaction produces particles necessary to reheat the universe [5, 6]. Through the conformal transformation, the additional degree of freedom in the modified action can be represented as a dynamical scalar field that plays a role in the inflaton. After the end of the inflation, the interaction between the inflaton and the matter converts the inflaton energy into the matter and reheats the universe [7]. Whether the dominant reheating process is perturbative or non-perturbative particle production depends on the structure of the interaction. Recently [8, 9], in the metric-affine gravity [10] where the metric g µν and the affine connection Γ µ νρ are independent variables, it is shown that a dynamical pseudoscalaron from the modification of geometrical quantity are obtained due to the existence of the Holst term ϵR = ϵ µνρσ R µνρσ / √ -g [11-13] which consists of the antisymmetric part of the affine connection(torsion) Γ µ νρ -Γ µ ρν . Consequently, that pseudoscalaron can be used as the inflaton [14, 15]. Thus, even in Einstein-Cartan (EC) gravity (metric-compatible metric-affine gravity) [16-20], the inflation can be realized through the pseudoscalar field [21, 22] since there exists the torsion. The interaction between torsion and matter fields can lead to non-thermal particle production. Several relevant studies exist in this area. For instance, the preheating process has been investigated within Einstein-Cartan gravity incorporating the Nieh-Yan topological invariant [23]. It has also been noted that the Holst term can either suppress or enhance the rate of vacuum decay [24]. By introducing an auxiliary pseudoscalar field, the Euler-Lagrange equation about affine connection gives the algebra equation regarding the torsion. Since, in this case, the torsion is represented by the metric, the auxiliary pseudoscalar field, and the matter coupling to the affine connection, integrating out the torsion yields the effective metric theory as the Palatini f ( R ) gravity [2]. In this effective metric theory, one can get the pseudoscalaron and the interaction between the pseudoscalaron and the matter therefore can realize the inflation and the reheating of the universe. In this paper, we discuss the non-thermal particle production in EC gravity with the modified Holst term. Since fermions naturally couple to the torsion in EC gravity, they are considered as matter fields. Additionally, the natural extension of the kinetic term of the fermion [25] is considered. To discuss the non-thermal fermionic particle production, we follow a way of previous work about fermionic preheating [26-30]. In the discussion, it is assumed that the fermionic field operator is composed of (anti) particles. By this assumption, one of the new parameters α must vanish. Numerical calculations eventually reveal that particle production occurs when the value of β is large, and its behavior depends on the mass and helicity of the particle. The overview of this paper is as follows. The section 2 reviews the Einstein-Cartan pseudoscalaron with matter fields and introduces the ( ϵR ) 2 model [14] effective for inflation. In Sec. 3, we introduce the extension of the kinetic term of the fermion. From this extension in Einstein-Cartan pseudoscalaron theory, the equation of motion(EoM) of fermions is non-trivial and thus nonthermal particle production occurs even in the FRLW universe. In Sec. 4, several results of numerical calculations about non-thermal particle production are exhibited. By these results, it can be concluded that more larger value of β contributes to more particle production. Also, it is observed that the behavior of the number density is very different depending on whether the mass of the fermion is lighter, heavier or intermediate compared to the inflaton mass. In Sec. 5, we apply the particle production to reheat the universe. Finally, a discussion and summary of this paper are presented in Sec. 6. Calculations in this paper are based on the following notations. m p = √ 1 / 8 πG N is the planck mass and m ϕ is the inflaton mass. G N means the gravitational Newton constant. The gamma matrix is defiend by { γ i , γ j } = 2 η ij where η ij is the minkowski metric η = diag( -, + , + , +). The definition of gamma matrice is Charge conjugate matrix C is ϵ µνρσ is Levi-Civita antisymmetric symbol ϵ 0123 ≡ 1. We use the natural units( ℏ = c = 1). The symmetrization A { i,j } and the anti-symmetrization A [ i,j ] are respectively 1 2 ( A ij + A ji ) and 1 2 ( A ij -A ji ). Greek indices mean the coordinate of spacetime while Roman indices mean that of local Minkowski spacetime. ˆ O means a q-number.",
"pages": [
1,
2
]
},
{
"title": "II. EINSTEIN-CARTAN PSEUDOSCALARON INFLATION FROM MODIFIED HOLST TERM",
"content": "EC gravity is a metric-compatible metric-affine theory [10, 31], where the metric and the connection are independent variables. Fundamental conditions in this theory are the tetrad hypothesis and the metric compatible condition, with and where the script ⋄ indicates the torsionless parts which consist of metric and tetrad, Γ ⋄ µ νρ is Levi-Civita symbol and ω ⋄ ij µ ≡ -e jν ∇ ⋄ µ e i ν . The torsionful parts consist of the contorsion, K µ νρ , defined by K µ νρ ≡ 1 2 ( T µ νρ + T νρ µ + T ρν µ ) and K ij µ ≡ e i ρ e jν K ρ νµ . In our research, we start from the action with a function of the Holst term, ϵR ≡ ϵ µνρσ R µνρσ /e , where e = det( e i µ ). Since the modification of the curvature scalar R can not be represented by a dynamical scalar field [8, 14, 22], F ( R ) regime is not considered. Introducing an auxiliary field χ , we obtain the action where V ( χ ) = m 2 p 4 ( H ' ( χ ) χ -H ( χ )) and H ' ( χ ) = dH ( χ ) dχ . χ is a pseudoscalar field. The action (9) is reproduced by substituting the Euler-Lagrange equation with respect to where, e i µ represents the tetrad which satisfies g µν = e i µ e j ν η ij . The inverse of e i µ serves as a basis component of local Minkowski spacetime, e µ i e ν j g µν = η ij . Γ µ νρ is the affine connection and ω ij µ is the gauge field of local Lorentz transformation that is called spin connection. Thus, the covariant derivative ∇ of spacetime is defined by ∇ µ A ν ≡ ∂ µ A ν + Γ ν αµ A α , and the covariant derivative D of local Minkowski spacetime is defined by D µ B i = ∂ µ B i + ω i j µ B j . By Eq. (1), the curvature tensor R µ νρσ and the strength of local Lorentz transformation R ij µν are connected by In this theory, the existence of the antisymmetric part of the affine connection(torsion) is not prohibited. Thus, the connections Γ, ω and the curvature scalar R can be generally separated into torsionless and torsionful parts, χ into Eq. (10). The Cartan equation is obtained as the Euler-Lagrange equation with respect to ω ij µ , where T µ ≡ T ν µν is the torsion vector and S µ ij ≡ 2 m 2 p ∂ L matter ∂ω ij µ is the spin density. Thus, the torsion is rewritten in terms of tetrad e i µ , matter field and auxiliary field χ . The effective metric theory is obtained by inserting the solution Eq. (11) into Eq. (10). For the vacuum ( S µ mn = 0), it becomes We introduce the pseudoscalaron ϕ by the redefinition, H ' ( χ ) = sinh ( √ 2( ϕ + δ ) √ 3 m p ) , and obtain where δ is a constant to impose V ( ϕ = 0) = 0. For example, in the ( ϵR ) 2 model [14] we obtain the pseudoscalaron with a potential, V ( ϕ ) = m 2 p 16 c ( sinh ( √ 2( ϕ + δ ) √ 3 m p ) -b ) 2 . The constant δ is fixed to satisfy sinh ( √ 2 δ √ 3 m p ) = b . As is shown in Fig.1, a certain value of parameters b, c can realize the potential with a plateau.",
"pages": [
2,
3
]
},
{
"title": "A. Pseudoscalaron inflation in FLRW universe",
"content": "We consider the homogeneous and isotropic universe described by the FLRW metric, where x µ = ( t, x, y, z ) represents the cosmological time and the conformal space, and a ( t ) is the scale factor. By introducing the conformal time, dη = a -1 dt , the spacetime is represented by The dot and the prime represent the derivative with respect to the cosmological time t , ˙ () = d dt (), and the conformal time η , () ' = d dη (), respectively. The metric (16) is used for the analysis in Sec. 4. In the metric (15), the scale factor is developed through the FriedmannRobertson equations, where H is the Hubble parameter defiend by H = ˙ a a , and ρ and p respectively denote the energy density and the pressure of the matter. From Eqs. (17) and (18), we derive When the scalar field φ distributes homogeneously, the energy density and the pressure are described as ρ = 1 2 ˙ φ 2 + V ( φ ) and p = 1 2 ˙ φ 2 -V ( φ ). The accelerated expansion takes place for V ( φ ) > ˙ φ 2 . If the potential V ( φ ) has a plateau, the scalr field starting from the plateau induces an inflationary expansion. To solve the horizon and flatness problems encountered in the expanding universe, the total e-folding number N e , defined as N e = log( a f /a i ), should exceed 50 ∼ 60. Here, a f represents the value of the scale factor at the end of inflation, while a i represents the value of the scale factor at the start of inflation. To obtain the number, we often employ the slow-roll inflation scenario. In the slow-roll approximation, the end of inflation is fixed by the slow-roll parameters ε and η , defined by ε = m 2 p 2 ( 1 V dV dφ ) 2 , η = m 2 p 1 V d 2 V dφ 2 . The quantum fluctuations of φ induce the curvature perturbation, P , during inflation. In terms of the conformal momentum space, a component of the fourieor decomposition of the curvature perturbation is represented as P ( k ) = P r k n s . Applying the slow-roll approximation, the amplitude P r of the curvature perturbation can be derived by P r = V ( φ ) 24 π 2 m 4 p ε ( φ ) . The spectral index n s is derived by n s = 1 -6 ε ( φ ) + 2 η ( φ ). The scalar-tensor ratio r is calculated by r = 16 ε ( φ ). The potential has a plateau in the ( ϵR ) 2 model with b = -320 , m ϕ = √ d 2 V dϕ 2 | ϕ =0 = √ (1+ b 2 ) 12 c ∼ 2 . 76 ∗ 10 13 GeV (Fig.1), and the slow-roll inflation scenario can be adopted . We assume that the pseudoscalar field regarded as an inflaton dominates the energy density of the early universe. Consequently, we obtain values such as n s ∼ 0 . 969 , r ∼ 0 . 003 , P r ∼ 2 . 1 ∗ 10 -9 , N e ∼ 60 that agree with the observation regarding the Cosmic Microwave Background (CMB). Below, we adapt this model to the non-thermal particle production after the end of inflation. In our analysis, we consider that the particle production starts at ϕ ∼ 0 . 94 m p , dϕ dt ∼ -0 . 293 m p m ϕ where the slow-roll parameter ε ( ϕ ) becomes unity.",
"pages": [
3,
4
]
},
{
"title": "B. The dynamics of background field after the end of inflation",
"content": "After the end of inflation, the oscillating inflaton ϕ dominates the energy density of the universe. The potential is approximated to be V ∼ m 2 ϕ 2 ϕ 2 during the particle production. The energy density of the inflaton and the scale factor are fixed by the Friedmann equations (17) and (18). Since the contribution to the pressure is cancelled between the kinetic and the potential energy, ρ ϕ follows the ˙ ρ ϕ +3 Hρ ϕ = 0. The solution of this equation with (17) is given by From the Eqs. (17) and (20), the scale factor is derived as The relation between cosmic time and conformal time is determined by dt = adη and a ( t = 0) = a ( η = 0) = 1, We assume that the oscillating part of the inflaton can be factored out, It should be noted that Eq. (23) satisfies the EoM of the inflaton ¨ ϕ +3 H ˙ ϕ + dV dϕ ≃ ¨ ϕ +3 H ˙ ϕ + m 2 ϕ ϕ = 0. Arbitrary constants A , B and C are fixed by the initial values of ϕ and ˙ ϕ , We employ these formula as a simple background for the universe after the end of inflation.",
"pages": [
4
]
},
{
"title": "III. A MODEL OF THE NON-MINIMAL COUPLINGS TO FERMION",
"content": "In Einstein-Cartan gravity, the matter action is generalized with non-minimal gravitational interactions. We consider a general fermion Lagrangian constructed with operators up to four dimensions [25], where the Dirac conjugate ψ is defined by ψ = iγ 0 ψ † and h.c. means the hermitian conjugate. This Lagrangian reduces to the ordinary one in the absence of gravity. The extension causes the parity violation observed in the various astrophysical and elementary particle phenomena. The spin density is given by A i ≡ ¯ ψiγ 5 γ i ψ and V i ≡ ¯ ψiγ i ψ denote the axial vector and the vector current, respectivly. Performing the partial integration, the Lagrangian density (27) can be decomposed into where the torsionless part is Since non-minimal coupling parameters don't appear in the torsionless part (30), these parameters only contribute to the interaction between the torsion and the fermion. By solving Eq. (11), the torsion is represented by the inflaton and the fermion, with where we write X ( ϕ ) = √ 2( ϕ + δ ) √ 3 m p . Inserting the solution (31) into (10), we obtain the effective metric action, with and four-fermion interactions, where A 2 , V A and V 2 are defined by A 2 = A µ A µ , V A = V µ A µ and V 2 = V µ V µ . The four-fermion interactions, (37), are suppressed by the factor m -2 p . On the other hand, the five-dimensional interactions, are suppressed by m -1 p . Therefore, L ϕψψ becomes the leading order term in the reheatig era. We neglect the higher order terms and apply the formalism developed in the previous works of the fermionic preheating [26-30]. In the previous works, Yukawa type interaction, is considered. In our model, the form of the interactions, are different from the Yukawa type interaction and violate the parity.",
"pages": [
4,
5
]
},
{
"title": "A. The EoM in FLRW universe",
"content": "For discussing the fermionic non-thermal particle production after the end of inflation, we derive the EoM of the classical fermionic field ψ and the Heisenberg operator ˆ ψ . In the spatially flat and homogeneous FLRW metric (16), the tetrad is given by e i µ = aδ i µ from its definition g µν = η ij e i µ e j ν , and components regarding the torsionless spin connection are η ik ω ⋄ ijk = 3 H a δ j 0 and ϵ ijkl ω ⋄ ijk = 0. Thus, Eq. (30) becomes where H is defined by H = a ' a . Therefore, the Lagrangian density of ψ is denoted as where f µ ( ϕ ) and g µ ( ϕ ) are functions of the inflaton defined by Eqs. (35) and (36). Rescaling the fermion as a 3 2 ψ → ψ , one can finally obtain the Lagrangian density, The inflaton ϕ after the end of inflation is assumed to be a homogeneous field, ∂ x ϕ = ∂ y ϕ = ∂ z ϕ = 0. The EoM of ψ is We note that γ µ in Eq. (44) is the gamma matrices in the local Lorentz frame. Since the Heisenberg operator ˆ ψ also satisfies the identical equation, we can obtain the EoM of the spinors u s ( k , η ) and v s ( k , η ) by the decomposition of the operator ˆ ψ , where ˆ a s ( k ) , ˆ b s ( k ) indicate (anti) particle annihilation operator. They satisfy the anti-commutation relations, We consider the case where the charge conjugate of fermion, is well-defined and interchanges the particle and the antiparticle. Then, we obtain the relation between u s ( k , η ) and v s ( k , η ), From Eq. (45) and its charge conjugate with (46) and (52), v s ( -k , η ) should satisfy Since g 0 is proportional to α , these equations are satisfied for α = 0. We define the spinor u s ( k , η ) as where s indicates the spin direction and ξ s, k describes the eigen-spinor of helicity, The equation (45) is rewritten as Performing the time derivative, we obtain the EoMs of the amplitude of the spinor, u + , -s,k ( η ), The EoM (45) guarantees the relation | u + s,k | 2 + | u -s,k | 2 = 2 with the anti-commutation relations (47)-(50), the assumption (55) and the canonical anti-commutation relation { ψ ( η, x ) , ψ † ( η, y ) } = δ 3 ( x -y ). Below, we evaluate the particle production for α = 0.",
"pages": [
5,
6
]
},
{
"title": "B. The number density of fermion",
"content": "From the Lagrangian density (43), we can define the Hamiltonian operator ˆ H ψ , where the script I denotes the space components, I ∈ { x, y, z } , and the factor a -1 comes from the fact that the Hamiltonian is the generator of the translation regarding the cosmic time t , and the relation dt = a ( η ) dη . By inserting (46) into (60), the Hamiltonian becomes where the coefficients E s k and F s k are Since they satisfy | E s k | 2 + | F s k | 2 = ω s 2 k = a 2 m 2 ψ + ( ks + f 0 ) 2 , the Hamiltonian operator (61) is diagonalyzed by introducing the time-dependent annihilation operators, ˆ a s ( k , η ) and ˆ b s ( k , η ), The Bogoliubov transformation connects the operators ˆ a s ( k ) , ˆ b † s ( -k ) and ˆ a s ( k , η ) , ˆ b † s ( k , η ), where α s ( k, η ) and β s ( k, η ) are the Bogoliubov coefficients defined by From the definition of the Bogoliubov coefficients and the anti-commutation relations (47)-(50), the anti-commutation relations for the operators, ˆ a s ( k , η ) , ˆ b † s ( -k , η ), are derived to be We redefine the Hamiltonian operator so that a minimum of its expectation value is zero, (72) We consider the vacuum state defined by ˆ a s ( k )( , ˆ b s ( k )) | 0 ⟩ η 0 = 0 at η = η 0 . Under the state, the expectation value of the number operators are developed The behavior of the expectation value of the number operator for the particle and the antiparticle is equivalent. Even if the expectation value of Hamiltonian ⟨ ˆ H ψ ( η ) ⟩ η 0 is zero at η 0 , it is not necessary to be so at η > η 0 . This means a non-thermal particle production. From the conditions of the amplitude of the spinor, the initial values of the amplitude of the spinor are derived, We define the quantities to observe whether the particle production occurs. The expectation value of total number density is given by where V is the volume of conformal space. The number density regarding the conformal momentum space is It vanishes at η = η 0 and must not exceed unity at anytime by Pauli blocking. The total number of (anti) particles is defined by We evaluate the energy density, where the factor 2 comes from the degree of freedom of the particle and the antiparticle.",
"pages": [
6,
7
]
},
{
"title": "IV. ANALYTICAL AND NUMERICAL RESULTS",
"content": "In this section, an analytical implication regarding the behavior of the number density and some numerical results are exhibited. The former suggests the behavior for particles lighter than the inflaton. We will attempt a specific application of the latter results in Sec. 5.",
"pages": [
7
]
},
{
"title": "A. Massless limit",
"content": "We can analytically evaluate the particle production for a simple case. Here, we consider the massless limit, m ψ = 0, which has several differences from the massive particle. First, the initial conditions (76) and (77) are not appropriate for the massless limit. The initial conditions of u + s,k and u -s,k are divided into four cases. For f 0 (0) > 0, the initial conditions become where θ and ψ are arbitrary phases. For f 0 (0) < 0, these are given by Second, the number densities are represented by Third, the 1st order differential equations (58) reduce to From Eq. (93), the time derivative of | u ± s,k ( η ) | 2 vanishes, Because of a damped oscillation of the inflaton ϕ , | f 0 | becomes smaller than k after a sufficient amount of time. The signature of k + sf 0 is positive and Eqs. (91) and (92) are simplified to Thus, the number densities are fixed by the initial values of | u ± s,k | 2 η =0 . The non-thermal excitations strongly depend on the initial conditions at the massless limit. The hericity of the produced particle is determined by the initial condition f 0 (0) (Table.I). From the numerical simulation, it is observed that lighter particles have a similiar property. If the reheating starts at ϕ ' (0) = 0, the production of lighter particles hardly occur due to the property in Tab. I with f 0 (0) ∝ ϕ ' (0) = 0.",
"pages": [
7,
8
]
},
{
"title": "B. Numerical results",
"content": "Since it is difficult to solve Eq. (59) analytically, we evaluate the behavior of the number density through numerical calculations. Here, we set ρ ψ (0) = 0 and employ the ( ϵR ) 2 model for the dynamics of the inflaton governing the evolution of the universe. As is shown in Sec. 2, initial values are fixed at ϕ (0) ∼ 0 . 940 m p and dϕ dη (0) ∼ -0 . 293 m p m ϕ , and the inflaton dynamics is determined by the values ϕ (0), ϕ ' (0). We solve the Eq. (59) to obtain n s m ψ ,k , N s ψ and ρ m ψ . Numerical calculations in our research are performed by Julia and an algorithm specified by Verner9 solver in the package DifferentialEq.jl.",
"pages": [
8
]
},
{
"title": "1. Heavy and light fermion",
"content": "We evaluate the time evolution of the number density n s m ψ ,k for the helicity up and down particles with a certain conformal momentum k . Fig.2 clearly shows that the non-minimal coupling β contributes to the nonthermal particle production. Because of the interaction between the pseudoscalaron and the pseudovector current ψiγ 5 γ µ ψ in Eq. (34), the amount of the produced particles depend on the helicity. In Fig.3, we draw the distribution of n s m ψ ,k for m ψ = 4 m ϕ as a function of the conformal momentum k/m ϕ for different β at η = 20 m -1 ϕ . The upper limit of the conformal momentum for the non-thermal particle production is observed to be higher as β increases. Thus, a larger | β | can lead to a higher conformal momentum excitation. The extreme oscillations in Fig.3 are confirmed to be unaltered by the initial values of the inflaton and the solvers. After the end of of the particle production, the total number of the produced particles remains at a certain value (Fig.4). In Figs. 3 and 4, different behavior is observed between up and down particles. Both up and down particles are produced for m ψ = 4 m ϕ . However, (b) m ψ = 4 m ϕ , β =0 , k = 10 m ϕ We also examine the number density distribution for lighter particles production. Fig.6 shows the number density distribution for m ψ = 0 . 01 m ϕ at η = 100 m -1 ϕ . It is observed that the higher conformal momentum particles can be produced with increasing β , as is mentioned in Sec. 4-1. Compared with Fig.3, the property in Tab. I is almost confirmed for the lighter particles. From Eq. (58), the helicity is inverted when the sign of f 0 is reversed. A dominant contribution to f 0 comes from the second term in Eq. (35). Thus, the up and down particles are exchanged and the aforementioned numerical results are nearly inverted for up and down with the sign of β flipped.",
"pages": [
8,
9
]
},
{
"title": "2. Intermediate-mass particle",
"content": "For intermediate-mass particles, m ψ = 0 . 1 m ϕ ∼ 0 . 5 m ϕ , we observe an alternative property that does not appear in light and heavy particles. A fermionic field with a higher conformal momentum is gradually generated and the Boltzmann-type distribution is established. According to the distribution of the number density (Fig.7) for m ψ = 0 . 3 m ϕ at η = 110 m ϕ -1 , particles with greater conformal momentum are excited than that in the case of light and heavy particle. We also evaluate the number density distribution in the non-expanding universe with a constant scale factor. The higher conformal momentum excitation is not observed in Fig. 8 for the non-expanding case. It means that the higher conformal momentum excitation is due to the expansion of the universe.",
"pages": [
9
]
},
{
"title": "V. COSMOLOGICAL CONSEQUENCES",
"content": "In this section, we apply the particle production to the reheating phenomena in the early universe. In the standard history of the universe, the energy density of the matter field must exceed the energy density of the inflaton after the end of inflation. Therefore, we examine if the condition ρ ψ ≫ ρ ϕ can be achieved with the nonthermal and thermal particle production.",
"pages": [
9
]
},
{
"title": "A. Preheating",
"content": "The preheating is the thermal process of the universe due to the non-thermal particle production before the reheating. As we show in Chapter 4, a larger | β | makes higher conformal momentum particles excited. From Fig.9, ρ ψ /ρ ϕ grows as β increases. The condition ρ ψ /ρ ϕ ≫ 1 can be achieved for sufficiently large β . However, it is necessary to discuss thermalization due to the decay of produced particles into relativistic particles to define the reheating temperature. Thus, the results presented in this section only indicate that the energy can be sufficiently transferred from the inflaton to the matter. Further analyses are required to estimate the reheating temperature.",
"pages": [
9
]
},
{
"title": "B. Reheating",
"content": "We adapt the perturvative calculations of the zero temperature quantum field theory to the fluctuation of ϕ after a long time from the end of inflation. If the energy density of the inflaton ρ ψ transfers to that of relativistic matter ρ r through the decay with the decay rate Γ and ρ r | η =0 = 0, the energy density, ρ r , is estimated to be at the moment for ρ ϕ = ρ r . From the Stefan-Boltzman law ρ r ∝ T 4 , the reheating temparature T R is proportional to √ Γ. In Einstein-Cartan pseudoscalaron model with nonminimal couplings to fermion, the decay of the ϕ into the fermions due to the interaction term dominates the particle production. The decay rate Γ ϕψψ from this interaction is found to be At β = 0 the result (99) reproduces the one derived in the previous work [14]. Thus, even in thermal particle production, the effect of non-minimal coupling to fermion is significant for | βb | ≫ 1. A reheating temperature is tuned by non-minimal coupling, β . For m ϕ ≫ m ψ and | bβ | ≫ 1, reheating temperature is estimated as where k B is the Boltzmann constant and g ∗ shows the physical degree of freedom. The interaction of inflaton and the fermion vector current, has no contribution to the decay rate for ϕ → ψψ , at the tree level. Thus, α does not appear in Eq. (99). The contribution from α dependent terms appear from the next to leading order.",
"pages": [
9,
10
]
},
{
"title": "VI. SUMMARY AND DISCUSSION",
"content": "In this paper, we have investigated the non-thermal fermionic particle production in Einstein-Cartan gravity with modified Holst term and non-minimal couplings to fermion. In EC gravity, the only imposed condition is metric compatiblity, ∇ µ g νρ = 0. Thus, the antisymmetric part of the affine connection T µ νρ ≡ Γ µ νρ -Γ µ ρν (i.e., torsion) can have a non-vanishing value. The existence of the torsion introduces invariant scalar components which do not exist in GR. One of these quantities is the Holst term ϵR = ϵ µνρσ R µνρσ / √ -g . The derivative of the affine connection in this term introduces a dynamical degree of freedom through the EoM. Following the auxiliary field method, the EoM for the affine connection becomes an algebraic equation for the torsion. After solving the EoM of the torsion, one can get the effective metric action with the dynamical pseudoscalaron ϕ [8, 9, 14, 21, 22]. The potential energy of the pseudoscalaron can induce inflation in EC gravity. In this paper, we utilize the ( ϵR ) 2 model, which is consistent with the CMB observations. The coupling between the matter fields and the affine connection in the original action yields an interaction between the matter fields and the inflaton in the effective metric action. In this study, we have employed a theory with non-minimal couplings to fermion where two parameters denoted as α and β are introduced [25]. This extension introduces interactions between the inflaton and the fermionic field ψ described in Eq. (34). Since the inflaton ϕ has a large value after the end of inflation, the interactions between the ϕ and the ψ destabilize the vacuum. Through the instability, the non-thermal particle production occurs. Since the parameter α must be zero to satisfy the condition that C ψ T represents a field where particles and antiparticles are exchanged, we have shown that the non-minimal coupling β in Eq. (34) contributes to the non-thermal particle production after the end of inflation. In this paper, only ϕψψ terms in Eq. (46) that introduce linear term with respect to ψ to the EoM of ψ is focused. We have examined how many particles are produced due to the existence of β through numerical calculations and observed several properties. First, a larger value of β leads to the excitation of particles with higher conformal momentum, and the excitation persists over time. It is also observed that fermions much heavier than the inflaton are hardly excited. Second, there is generally a difference in the amount of produced particles between helicity up and down particles. It should be noted that the produced numbers of particles and antiparticles are identical, and the total spin of the universe is conserved. Third, for the lighter mass fermion, m ψ ⪅ 0 . 1 m ϕ , the difference in the number of created particles between helicities becomes more pronounced. This property is also suggested analytically at the massless limit. If the initial value of dϕ dη is small enough, light particles are not excited non-thermally. Eventually, for the intermediatemass fermion 0 . 1 m ϕ ⪅ m ψ ⪅ 0 . 5 m ϕ , the number density is exponentially supressed for a higher conformal momentum like a Boltzmann distribution. Therefore, particles with higher conformal momentum can be excited than for heavier and lighter fermions. This property is not observed in a non-expanding universe. We have applied the particle production to the reheating and the preheating of the universe. From the consistency with the CMB observations, we set a model paremeter b ∼ -320 in our analysis. In the preheating era, sufficiently large values of β have the potential to make the energy density of ψ dominant over that of the inflaton. In the reheating era, we derive the formula of the reheating temperature T R,β with non-minimal cou- . About | βb | times larger reheating temperature is predicted than that for the minimal coupling [14]. Thus, the contribution of β is important in both eras. Future research directions include further development of analytical and numerical discussions and applications to phenomenology. Due to the complexity of the function of the inflaton coupling to fermion as given in Eq. (35), the dynamics of fermions during inflation and the backreaction to the evolution of the universe during particle production are not considered. To discuss a realistic universe, consideration of these two aspects can not be avoidable. Moreover, analytical solutions in some limits are necessary to verify the numerical results. Potential applications of our research include the production of the dark matter and the matter-antimatter asymmetry. The particle production investigated in our research is induced by gravitational effect. Heavy fermions that could be dark matter may be produced after the end of inflation. Our asymmetric helicity production has a possibility to describe the matter-antimatter asymmetry through particle production of fermions with lepton numbers [32, 33]. Our analysis can be applied to Majorana fermion which directly induces lepton number asymme-",
"pages": [
10,
11,
12
]
},
{
"title": "ACKNOWLEDGMENTS",
"content": "The authors would like to thank H. Sakamoto, M. Alwan, M. Taniguchi and S. Takahashi for valuable discus- tum: Rev.Mod.Phys. 36, 1103-1103 (1964)].",
"pages": [
13
]
}
]
2024PhRvD.110l2001R
https://arxiv.org/pdf/2307.13099.pdf
<document>
<section_header_level_1><location><page_1><loc_11><loc_90><loc_90><loc_93></location>Measuring Gravitational Wave Speed and Lorentz Violation with the First Three Gravitational-Wave Catalogs</section_header_level_1>
<text><location><page_1><loc_16><loc_87><loc_84><loc_89></location>Anarya Ray , 1, ∗ Pinchen Fan , 2 Vincent F. He , 2 Malachy Bloom , 2 Suyu Michael Yang,</text>
<text><location><page_1><loc_32><loc_86><loc_84><loc_88></location>2 Jay D. Tasson , 2, † and Jolien D. E. Creighton 1</text>
<text><location><page_1><loc_20><loc_84><loc_80><loc_85></location>1 Department of Physics, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA</text>
<text><location><page_1><loc_21><loc_83><loc_80><loc_84></location>2 Department of Physics and Astronomy, Carleton College, Northfield, MN 55057, USA</text>
<text><location><page_1><loc_18><loc_61><loc_83><loc_81></location>The speed of gravitational waves v g can be measured with the time delay between gravitationalwave detectors. Our study provides a more precise measurement of v g using gravitational-wave signals only, compared with previous studies. We select 52 gravitational-wave events that were detected with high confidence by at least two detectors in the first three observing runs (O1, O2, and O3) of Advanced LIGO and Advanced Virgo. We use Markov chain Monte Carlo and nested sampling to estimate the v g posterior distribution for each of those events. We then combine their posterior distributions to find the 90% credible interval of the combined v g distribution for which we obtain 0 . 99 +0 . 02 -0 . 02 c without the use of more accurate sky localization from the electromagnetic signal associated with GW170817. Restricting attention to the 50 binary black hole events generates the same result, while the use of the electromagnetic sky localization for GW170817 gives a tighter constraint of 0 . 99 +0 . 01 -0 . 02 c . The abundance of gravitational wave events allows us to apply hierarchical Bayesian inference on the posterior samples to simultaneously constrain all nine coefficients for Lorentz violation in the nondispersive, nonbirefringent limit of the gravitational sector of the Standard-Model Extension test framework. We compare the hierarchical Bayesian inference method with other methods of combining limits on Lorentz violation in the gravity sector that are found in the literature.</text>
<section_header_level_1><location><page_1><loc_20><loc_57><loc_37><loc_58></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_39><loc_49><loc_55></location>The third observing run (O3) of Advanced LIGO [1] and Advanced Virgo [2] was the first complete run in which all three detectors were used [3, 4]. In total, O3 adds 79 gravitational-wave (GW) candidates, more than seven times the 11 GW candidates from the first (O1) and second (O2) observing runs combined [5]. With the availability of many more GW events, it becomes possible to measure the speed of gravitational waves v g more precisely than previous works that used similar methods [6, 7]. Furthermore, it allows a direct and comprehensive exploration of the isotropy of v g for the first time.</text>
<text><location><page_1><loc_9><loc_15><loc_49><loc_39></location>General relativity predicts that the speed of GWs is the same as the vacuum speed of light c . The GWs detected by Advanced LIGO and Advanced Virgo can be used to make statistical inferences about v g , thereby testing the theory of general relativity. The first measurement of v g using the time delay between the GW detectors was performed by Ref. [6]. By applying Bayesian inference, the 90% credible interval of v g distribution was constrained to be (0 . 55 c, 1 . 42 c ) [6]. Reference [7] further constrained the 90% credible interval to (0 . 97 c, 1 . 05 c ), by applying similar methods to 11 events from O1 and O2. With a total of 52 high-confidence multi-detector GW events accrued through the end of O3, we are able to perform a similar analysis using more events, more robustly testing the theory of general relativity. While the method used here remains considerably less sensitive than a multimessenger astronomy approach in Ref. [8], the latter remains</text>
<text><location><page_1><loc_52><loc_54><loc_92><loc_58></location>but one measurement, and the approach used here offers the robustness of multiple measurements attained in the context of an independent method.</text>
<text><location><page_1><loc_52><loc_24><loc_92><loc_54></location>The large body of events now available, which arrive from a multitude of sky directions, allows for a complete exploration of the isotropy of v g in the context of the Lorentz invariance test framework provided by the gravitational Standard-Model Extension (SME). 1 Reference [7] simultaneously constrained the first four of nine coefficients for Lorentz violation in the nondispersive, nonbirefringent limit of the gravity sector using four GW events from O1 and O2. Other recent works [13-15] have sought the effects of birefringence and dispersion using the SME. In this paper, we use 24 of 52 high-significance multi-detector GW events to simultaneously constrain all nine coefficients in the nondispersive, nonbirefringent limit. While our constraints are much weaker than previous works such as Ref. [8], which have constrained the coefficients for Lorentz violation in the gravity sector down to the order of 10 -15 to 10 -14 via multimessenger astronomy, these constraints were obtained using models with only one parameter each. Therefore, our work is the first to provide direct limits from GW observations on all nine coefficients simultaneously.</text>
<text><location><page_1><loc_52><loc_16><loc_92><loc_23></location>The remainder of this paper is organized as follows. In Sec. II, we discuss the methods used to extract v g estimates for each event and present the results. Section III presents and compares a number of methods for extracting simultaneous limits on the nine coefficients for</text>
<text><location><page_2><loc_9><loc_90><loc_49><loc_93></location>Lorentz violation before presenting our final estimate of these coefficients from the O1-O3 data.</text>
<section_header_level_1><location><page_2><loc_12><loc_86><loc_46><loc_87></location>II. SPEED OF GRAVITATIONAL WAVES</section_header_level_1>
<section_header_level_1><location><page_2><loc_16><loc_83><loc_41><loc_84></location>A. Bayesian Inference Methods</section_header_level_1>
<text><location><page_2><loc_9><loc_77><loc_49><loc_81></location>Here, we briefly describe our method for obtaining the speed of GWs. Interested readers are invited to refer to Ref. [7] for full details.</text>
<text><location><page_2><loc_9><loc_65><loc_49><loc_76></location>When a GW passes through Earth, if two or more detectors detect the signal, we can use the relative locations of the detectors and the difference in detection times from those detectors to simultaneously estimate the sky location of the GW event and v g . With only one detector, we cannot find any v g information, as there is no difference in detection times in this case. Therefore, we select those events that are detected by at least two GW detectors.</text>
<text><location><page_2><loc_9><loc_48><loc_49><loc_65></location>Furthermore, we only consider those events whose median signal-to-noise ratios (SNR) are no smaller than 10.0, as reported in the GWTC-2 and GWTC-3 catalog papers [3, 4]. In total, 41 O3 events meet our selection criteria and are listed in Tables I and II. All O1 and O2 events meet these two selection criteria, so we include their posterior distributions used in Ref. [7] in our analysis. Note that the SNR values used to select the O1 and O2 events (which is the same as what was used in Ref. [7]) correspond to the network SNR with which the events were found by the GstLAL search pipeline as reported in Ref. [5].</text>
<text><location><page_2><loc_9><loc_36><loc_49><loc_47></location>The standard parameter estimation using GW data from multiple detectors imposes the constraint that GWs travel at the speed of light [16]. In this work, we remove this constraint such that v g becomes a parameter to be estimated with all other signal parameters. This causes wider distributions for certain parameter estimations. For example, the calculated sky area is often larger because a defined v g aids sky localization.</text>
<text><location><page_2><loc_9><loc_33><loc_49><loc_36></location>Gravitational wave data d , can be decomposed into a pure GW signal h ( t ) plus random noise n ( t ),</text>
<formula><location><page_2><loc_22><loc_30><loc_49><loc_31></location>d ( t ) = h ( t ) + n ( t ) . (1)</formula>
<text><location><page_2><loc_9><loc_13><loc_49><loc_29></location>Within the framework of Bayesian inference, the posterior distribution of the parameters ⃗ θ characterizing a GW signal is computed from the likelihood of obtaining GW data given particular values of said parameters and the a priori knowledge of what we expect those values to be. The likelihood function is constructed by assuming the noise n ( t ) to be stationary and Gaussian distributed. For details regarding the exact forms of the likelihood see Ref. [7]. Once obtained, the joint posterior distribution of the signal parameters can be used to compute the marginalized posterior distribution of v g as in:</text>
<formula><location><page_2><loc_21><loc_9><loc_49><loc_11></location>p ( v g | d ) = ∫ p ( ⃗ θ | d ) d ⃗ θ ' , (2)</formula>
<text><location><page_2><loc_52><loc_92><loc_91><loc_93></location>where ⃗ θ ' is the set of parameters in ⃗ θ except for v g [7].</text>
<text><location><page_2><loc_52><loc_31><loc_92><loc_92></location>To carry out PE for each event that passes our selection criteria, we use public data [17, 18] from GWTC-1 through GWTC-3. We use lalinference mcmc [16, 1921] which implements MCMC with Metropolis-Hastings algorithm and lalinference nest which implements nested sampling to run the Bayesian parameter estimation [16, 22, 23]. For our purposes of extracting v g distributions, these two algorithms generate comparable results. We use the publicly available power spectral densities and calibration envelopes from the LIGO Scientific, Virgo and KAGRA (LVK) collaborations in our analysis. In this paper, we use a uniform prior in v g between 0 . 1 c and 10 c . When the v g posterior rails against the prior, we increase the upper limit of the prior by another 10 c . The broadest prior we use is from 0 . 1 c to 30 c , which we only use for one event, GW190929 012149. For parameters such as binary masses and spins, we use the same uniform and isotropic priors as those used by the LVK [3-5]. We choose a distance prior that is proportional to luminosity distance squared, similar to Ref. [5]. We do not use the more complicated cosmological priors used by Refs. [3, 4]. For O1 and O2 events, we use the posterior samples from Ref. [7], which used the IMRPhenomPv2 [24-26] waveform for all events except for the binary neutron star (BNS) event GW170817 which was analyzed with the TaylorF2 waveform [27-32]. For most O3 events, we use the IMRPhenomD waveform [24, 25], which is an aligned spin waveform model for black-hole binaries. We do not use the more sophisticated IMRPhenomPv2 model for these events since in the context of our study, we do not expect any siginificant change in v g measurements to result from the additional intricacies of the more sophisticated model. We have verified this lack of change for a subset of these events and hence chosen to stick to the IMRPhenomD model consistently for all O3 events except for GW190521. For the extremely high-mass BBH event GW190521, we use the NRSur7dq4 waveform [33] which is one of the waveform models used by Ref. [34] for inferring this event's source properties. We note that IMRPhenomPv2, IMRPhenomD, and NRSur7dq4 are all waveform models with inspiral, merger as well as ringdown.</text>
<text><location><page_2><loc_52><loc_20><loc_92><loc_31></location>We can achieve a more precise measurement of v g by combing data from multiple GW events. By interpreting each observation as an independent experiment, we can multiply the marginalized likelihood as a function of v g corresponding to each event and obtain the joint posterior distribution of v g given data from multiple events. For a uniform prior on v g , the joint posterior can be expressed as a product of individual event posteriors.</text>
<text><location><page_2><loc_52><loc_14><loc_92><loc_19></location>Suppose the GW detectors observe n independent GW events with data d 1 , d 2 , ..., d n . For a uniform prior distribution of v g , the combined posterior distribution of v g is</text>
<formula><location><page_2><loc_53><loc_11><loc_92><loc_12></location>p ( v g | d 1 , d 2 , ..., d n ) ∝ p ( v g | d 1 ) p ( v g | d 2 ) · · · p ( v g | d n ) . (3)</formula>
<text><location><page_2><loc_53><loc_9><loc_92><loc_10></location>The single event posterior distributions p ( v g | d i ) are ob-</text>
<text><location><page_3><loc_9><loc_86><loc_49><loc_93></location>tained as a numerical function of v g from its PE samples by means of Gaussian Kernel Density estimation (KDE) [35, 36]. We use the package Scipy 's implementation of Gaussian KDE to obtained the posteriors [37]. The joint posterior distribution is then obtained through Eq. (3).</text>
<text><location><page_3><loc_9><loc_82><loc_49><loc_86></location>Then, for individual and combined posteriors, we calculate Bayes factors K , via the Savage-Dickey density ratio</text>
<formula><location><page_3><loc_20><loc_78><loc_49><loc_81></location>K = p ( v g = c | d 1 , d 2 , ... ) p ( v g = c ) , (4)</formula>
<text><location><page_3><loc_9><loc_71><loc_49><loc_77></location>where p ( v g = c | d 1 , d 2 , ... ) is the posterior probability of v g = c , and p ( v g = c ) is the prior probability of v g = c [38]. Higher Bayes factors suggest stronger evidence for v g = c .</text>
<section_header_level_1><location><page_3><loc_24><loc_67><loc_33><loc_68></location>B. Results</section_header_level_1>
<text><location><page_3><loc_9><loc_48><loc_49><loc_65></location>In Tables I and II, we show the v g estimates with 90% credible intervals, network SNRs, sky areas at 90% credible level, and Bayes factors for these selected 41 O3 events. Also shown are the analogous quantities obtained from their combined posteriors. Out of the 41 selected O3 events, 40 events are binary black hole (BBH) candidate events. GW200115 042309 is a neutron starblack hole (NSBH) event, with masses of 5 . 9 +2 . 0 -2 . 5 M ⊙ and 1 . 44 +0 . 85 -0 . 29 M ⊙ at 90% credible interval [4]. Here, by combining the 41 selected O3 events, we constrain the 90% credible interval of v g to be 0 . 99 +0 . 02 -0 . 03 c , with a Bayes factor of 205 . 9.</text>
<text><location><page_3><loc_9><loc_12><loc_49><loc_47></location>We combine the O3 results with the O1 and O2 results discussed in Ref. [7]. The eleven O1 and O2 events are run with lalinference mcmc , which shows results that are consistent with lalinference nest used for O3a runs [3, 5, 7]. In Table III, we show the v g estimates with 90% credible intervals, network SNRs, sky areas at 90% credible level, and Bayes factors for the 11 O1 and O2 events and their combined posteriors. We use the same posterior samples as used by Ref. [7], but Table III shows slightly different 90% credible intervals from those in Ref. [7], because we use Gaussian KDE smoothing in this study to extract the credible intervals while Ref. [7] directly used the posterior samples without KDE smoothing [7]. These 11 events were detected by at least two detectors and had median GstLAL network SNR values greater than 10.0 [5]. GWTC-2.1 [39] shows network SNR values for O1 and O2 events based on lalinference parameter estimations, but we choose GstLAL SNR values to be consistent with Ref. [7] from which we obtain the v g posterior samples. GW170817 is a BNS event that was also detected in the electromagnetic spectrum [8, 40]. The 'fixed' label means that the result uses the sky localization from the electromagnetic detections, which is much more precise than the localization generated by GW detection pipelines.</text>
<text><location><page_3><loc_9><loc_9><loc_49><loc_11></location>Combining the 41 O3 events and 11 O1 and O2 events without fixing GW170817's sky localization at the de-</text>
<text><location><page_3><loc_52><loc_83><loc_92><loc_93></location>tected EM signal, we obtain the 90% credible interval of v g to be 0 . 99 +0 . 02 -0 . 02 c , with a Bayes factor of 291 . 9. With GW170817 sky localization fixed, v g is 0 . 99 +0 . 01 -0 . 02 c , with a Bayes factor of 249 . 0. For a total of 49 BBH events, i.e. excluding GW170817, GW190924, and GW200115, v g is 0 . 99 +0 . 02 -0 . 02 c , with a Bayes factor of 221 . 2. FIG. 1 shows the combined posterior of v g .</text>
<table>
<location><page_3><loc_52><loc_27><loc_92><loc_77></location>
<caption>TABLE I. The 90% credible intervals of v g from individual O3a events posteriors. Median network SNR values are reported from GWTC-2 [3]. The 90% credible regions of the sky localization (Ω) without fixing v g at c are calculated from the individual posteriors [3]. The Bayes factor K indicates how strong the posterior distributions support v g = c . The asterisks ( ∗ ) in front of GW event names represent the GW events chosen for obtaining constraints on all nine coefficients for Lorentz violation in Sec. III.</caption>
</table>
<table>
<location><page_4><loc_9><loc_52><loc_49><loc_93></location>
<caption>TABLE II. The 90% credible intervals of v g from individual O3b events posteriors and combined posteriors using all O3 events. Median network SNR values are reported from GWTC-3 [3]. The 90% credible regions of the sky localization (Ω) without fixing v g at c are calculated from the individual posteriors [4]. The Bayes factor K indicates how strong the posterior distributions support v g = c . The asterisks ( ∗ ) in front of GW event names represent the GW events chosen for obtaining constraints on all nine coefficients for Lorentz violation in Sec. III.</caption>
</table>
<section_header_level_1><location><page_4><loc_23><loc_32><loc_35><loc_33></location>C. Discussion</section_header_level_1>
<text><location><page_4><loc_9><loc_10><loc_49><loc_29></location>In Ref. [7], with 11 O1 and O2 events and GW170817's sky localization unfixed, the combined posterior distribution of v g was measured to be 1 . 01 +0 . 04 -0 . 05 c , while here we measure v g to be 0 . 99 +0 . 02 -0 . 02 c with the 52 selected events. With GW170817's localization fixed, in Ref. [7], the combined posterior distribution of v g was measured to be 0 . 99 +0 . 02 -0 . 02 c for 11 events, while here we find 0 . 99 +0 . 02 -0 . 01 c for the 52 events. Given that 1 c is the relativistic prediction of v g , the combined posterior distributions of v g measured using 52 selected events show no evidence for a violation of general relativity. All of these combined results have Bayes factors on the order of 10 2 , providing strong evidence for v g = c .</text>
<text><location><page_4><loc_10><loc_9><loc_49><loc_10></location>Here, our measured distribution of v g is much narrower</text>
<table>
<location><page_4><loc_52><loc_53><loc_92><loc_93></location>
<caption>TABLE III. The 90% credible intervals of v g from individual O1 and O2 events posteriors. Combined posteriors are for all selected O2 and O3 events. Network SNR values are reported from the GstLAL search pipeline in GWTC-1 [5, 7]. The 90% credible regions of the sky localization (Ω) without fixing v g at c are calculated from the individual posteriors [5, 7]. The Bayes factor indicates how strong the posterior distributions support v g = c . The asterisks ( ∗ ) in front of GW event names represent the GW events chosen for obtaining constraints on all nine coefficients for Lorentz violation in Sec. III.</caption>
</table>
<text><location><page_4><loc_52><loc_9><loc_92><loc_36></location>than that measured with 11 O1 and O2 events in Ref. [7] using GW signals alone. This is reasonable, given the larger sample size of events included in this study. When we assume that the v g distributions of individual events are independent and identically distributed, we expect that the measurement errors would decrease by 1 / √ n . In our calculations, we find that the combined v g distribution roughly follows such a pattern as more events are added. For example, with 11 O1 and O2 events, the combined v g posterior had an error bar of 0 . 09 c . With 52 events in total, the combined posterior had an error bar of 0 . 04 c , which follows 0 . 09 c/ √ 52 / 11 ≈ 0 . 04 c . With GW170817's sky localization unfixed, we find that Bayes factor more than doubles from the value of 149 . 0 obtained from 11 O1 and O2 events to the value of 291 . 9 obtained with all 52 events. This, in conjunction with the error bar being reduced by half, implies that our measurement with 52 GW events in total has provided approximately twice stronger evidence for v g = c .</text>
<figure>
<location><page_5><loc_9><loc_70><loc_50><loc_93></location>
<caption>FIG. 1. Posterior distributions of v g inferred jointly from all 41 events. The blue dashed lines show the 90% credible interval, which includes 1 c as predicted by general relativity.</caption>
</figure>
<text><location><page_5><loc_9><loc_42><loc_49><loc_61></location>Interestingly, we find that the combined 90% v g credible interval using the 41 O3 events is approximately the same as the 90% v g credible interval obtained by only considering GW170817 with the fixed sky localization. GW170817 had an SNR of 33 . 0, while only four of the 41 O3 events had SNRs above 20 . 0, with the highest being 26 . 8 for GW200129 065458. The similarity between the v g posterior of GW170817 alone and the 41 O3 events combined suggests that some combination of higher SNRs and better sky localization do help put tighter constraints on v g . This shows that our decision to exclude events with SNRs lower than 10 . 0 should not have a high impact on the v g estimates.</text>
<text><location><page_5><loc_9><loc_26><loc_49><loc_42></location>Looking to the future, additional two- and threedetector BBH events with SNRs typical of those above will lead to a slow improvement in v g measurements as improvements proceed as 1 / √ n . However, as GW detectors become more sensitive and the network of detectors expands, we expect more high-SNR, multi-detector GW events that would likely lead to a more rapid pace of progress in v g estimations via the methods used here. Meanwhile, future multimessenger detections can provide more precise sky localizations, which will likely improve the error bars on the 90% v g credible interval further.</text>
<section_header_level_1><location><page_5><loc_14><loc_21><loc_43><loc_22></location>III. SIMULTANEOUS SME LIMITS</section_header_level_1>
<section_header_level_1><location><page_5><loc_25><loc_18><loc_33><loc_19></location>A. Basics</section_header_level_1>
<text><location><page_5><loc_9><loc_9><loc_49><loc_16></location>In the non-birefringent, non-dispersive limit of the SME (mass dimension d = 4), using natural units and assuming that the nongravitational sectors, including the photon sector, are Lorentz invariant, the difference between the group velocities of gravity and light takes the</text>
<text><location><page_5><loc_52><loc_92><loc_59><loc_93></location>form [41]:</text>
<formula><location><page_5><loc_60><loc_87><loc_92><loc_90></location>∆ v = -∑ lm Y lm (ˆ n ) 1 2 ( -1) 1+ l ¯ s lm , (5)</formula>
<text><location><page_5><loc_52><loc_78><loc_92><loc_86></location>where the Y lm 's are the spherical harmonics with l ≤ 2. Here the nine Lorentz-violating degrees of freedom are characterized by the spherical coefficients for Lorentz violation ¯ s lm , and ˆ n is the sky location of the source of the GWs. We can expand Eq.(5) over positive m to get its equivalent expression:</text>
<formula><location><page_5><loc_52><loc_71><loc_92><loc_76></location>∆ v = ∑ l ( -1) l ( 1 2 s l 0 Y l 0 + ∑ m> 0 [Re s lm Re Y lm -Im s lm Im Y lm ] ) . (6)</formula>
<text><location><page_5><loc_52><loc_45><loc_92><loc_69></location>The SME is a broad and general test framework for testing Lorentz invariance. Unlike models that attempt to describe specific effects with a small number of parameters, test frameworks, because of their generality, have a large number of undetermined coefficients to be explored in experimental data. While a number of studies have proceeded under a simplified approach, sometimes referred to as a maximum reach analysis [42], in which only one coefficient at a time is considered, it is also common to study a family of coefficients together in what is sometimes referred to as a coefficient separation approach [42]. In the context of the maximum reach approach, many coefficients can sometimes be constrained one at a time using a single measurement, while a number of measurements that is greater than or equal to the number of coefficients considered is typically required to simultaneously measure the entire family.</text>
<text><location><page_5><loc_52><loc_36><loc_92><loc_45></location>A number of approaches to simultaneously estimating multiple coefficients exist in the literature. One approach involves directly fitting a single data stream to a model involving all of the coefficients in the family. 2 This approach is well-suited to experiments that take data as the lab is boosted and rotated.</text>
<text><location><page_5><loc_52><loc_16><loc_92><loc_36></location>In the context of astrophysical observations, each individual event provides a measurement of a linear combination of coefficients for Lorentz violation. A system of these inequalities must then be solved, or otherwise disentangled, for estimates of the coefficients for Lorentz violation. Several methods of addressing this issue exist in the literature. In this section, we will compare the implications of several of these approaches in the context of the speed of gravitational wave data, as well as introduce new methods based on hierarchical Bayesian inference. Our goal is to consolidate information about these methods and help illuminate their relative merits. We achieve that goal by performing a Mock Data Challenge (MDC), where-in we generate synthetic data corresponding to a</text>
<text><location><page_6><loc_9><loc_89><loc_49><loc_93></location>chosen set of 'true' values of the SME coefficients and test the efficacy of each method in recovering the true values from the synthetic data.</text>
<text><location><page_6><loc_9><loc_73><loc_49><loc_89></location>Given their performance in the MDC, along with other considerations, we choose one of these methods whose merits outweigh that of the others and use it to analyze the real speed of gravity data from a subset of the events analyzed in Sec. II to generate the final results of our SME analysis. Because well-localized events are most informative for the SME analysis, we choose the 24-event subset of those considered in Sec. II with 90% credible posterior sky areas under 2000 square degrees as obtained from our parameter estimation with v g as a free parameter.</text>
<section_header_level_1><location><page_6><loc_16><loc_69><loc_42><loc_70></location>B. Linear Programming Method</section_header_level_1>
<text><location><page_6><loc_9><loc_58><loc_49><loc_66></location>A number of past studies that have performed maximum reach analysis using limits from astrophysical events have taken a linear programming approach. See, for example, Refs. [44-46]. The basic idea translated to the speed of gravitational waves problem proceeds as follows.</text>
<text><location><page_6><loc_9><loc_38><loc_49><loc_58></location>From a given event we have an upper and lower bound on ∆ v . If we suppose that we know an exact sky location, as is effectively the case for GW170817 when the electromagnetic signal's localization is used, then Eq. (6) can be understood as generating a pair of hyperplanes in s lm space that are the boundaries of the parameter space excluded by the event. A subsequent event at a different sky location will generate a distinct pair of hyperplanes. Once a set of n events are collected at distinct sky locations, where n is greater than or equal to the dimensionality of the coefficient space, then a finite maximum and minimum allowed value for each coefficient can be identified via a linear programming scheme such as the simplex method.</text>
<text><location><page_6><loc_9><loc_28><loc_49><loc_37></location>In the applications of Refs. [44-46], the sky localizations were sufficiently well known that analysis could proceed directly via the above prescription. In the current problem, for all events except GW170817, the sky localization is comparatively poorly known. This makes the slopes of the hyperplanes bounding the allowed region poorly known.</text>
<text><location><page_6><loc_9><loc_9><loc_49><loc_27></location>To address our uncertainty in sky positions, the linear programming scheme can be adapted as follows. The linear programming process can be applied with all possible hyperplanes generated by samples from our inference that fall within the 68% credible sky localization bands. The worst-case limits generated by the set of linear programming analysis can then be taken as bounds. As might be expected, this method generates very conservative bounds relative to the methods to follow. Testing this approach using four test events and a skymap resolution of N side = 64, which corresponds to 12 × 64 2 = 49152 pixels on the celestial sphere [47, 48], we generate bounds that are about an order of magnitude greater than the</text>
<text><location><page_6><loc_52><loc_86><loc_92><loc_93></location>1 σ credible intervals found via the application of the random draw method that we present in the next subsection. Hence we do not consider this approach further as a method of extracting SME limits from the speed of gravitational waves data at this time.</text>
<section_header_level_1><location><page_6><loc_61><loc_82><loc_82><loc_83></location>C. Random Draw Method</section_header_level_1>
<text><location><page_6><loc_52><loc_67><loc_92><loc_79></location>In Ref. [7] the random draw method for extracting simultaneous limits on coefficients for Lorentz violation was first used. In that work, simultaneous limits were achieved for the set of four l = 0 and l = 1 s lm coefficients using the 4 high-confidence, well-localized events available at the time. In this section, we review this method and discuss ways of extending it to cases in which the number of events exceeds the number of coefficients to be estimated.</text>
<text><location><page_6><loc_52><loc_49><loc_92><loc_66></location>The result of the inference discussed in Sec. II A is a set of samples with each sample consisting of values for each of the parameters including the speed of GWs and the sky localization. Hence distributions for each of the sampled parameters are generated. If one randomly draws one sample associated with each event, one can then solve for the coefficients for Lorentz violation that are consistent with that set of samples using Eq. (6). The process of randomly drawing one sample from each event and solving for the coefficients can be iterated to build up a set of samples for the s lm coefficients. In other words, a set of points in s lm space is built up.</text>
<text><location><page_6><loc_52><loc_15><loc_92><loc_49></location>The process described above is straightforward when the number of events observed is equal to the number of coefficients for Lorentz violation to be estimated. Furthermore, in such a scenario, quantile ranges of s lm computed from the set of samples of s lm , accurately represent the uncertainty in our measurement of the SME coefficients. This is because using one posterior sample of (∆ v, θ, ϕ ) from each event and exactly calculating s lm from them by solving a set of non-degenerate linear equations, is equivalent to computing and multiplying the posterior distributions of s lm for each event and then drawing one sample from that joint posterior. However, in the case where the number of GW observations exceeds the number of SME coefficients, the linear equations become degenerate and hence no longer exactly solvable. While one can be tempted to cherry-pick the top 9 events with the highest SNRs and lowest sky areas from the set of observations and perform random draw on those, such an analysis will not be maximally informative given the data we have. We can do better using Bayesian hierarchical inference techniques which can combine information from a large number of events, producing much more informative bounds on the SME coefficients with accurate estimation of measurement uncertainties.</text>
<text><location><page_6><loc_52><loc_9><loc_92><loc_14></location>Before discussing our robust Bayesian methods we show how the random draw method can be extended to the case in which the number of observations exceeds the number of coefficients for Lorentz violation by means</text>
<text><location><page_7><loc_9><loc_77><loc_49><loc_93></location>of Singular Value Decomposition (SVD). However, this extension of the random draw method is susceptible to the limitations of the approximation used to perform the SVD and hence cannot produce reliable uncertainty estimates for the measured Lorentz violation parameters. We elaborate more on this near the end of this section while informing the reader beforehand that this SVD-assisted random draw generalization is useful in the present context only as a consistency check and an optimization tool for the hierarchical Bayesian methods on which we rely for our final results.</text>
<text><location><page_7><loc_9><loc_72><loc_49><loc_77></location>For n SME number of Lorentz Violation coefficients and N E number of events with n SME < N E , for each random draw, we need to solve the degenerate system of linear equations:</text>
<formula><location><page_7><loc_23><loc_69><loc_49><loc_70></location>A [ ¯ s lm ] = [ ∆ v ] . (7)</formula>
<text><location><page_7><loc_9><loc_45><loc_49><loc_68></location>Here A is an N E × n SME matrix in which each row corresponds to one of the N E events under consideration. The entries in each of the n SME columns moving across a given row consist of the coefficients of ¯ s lm in Eq. (6), computed for a random sample of θ, ϕ drawn from the event corresponding to that row. The n SME SME coefficients to be computed are organized into a column vector denoted [ ¯ s lm ], while [ ∆ v ] denotes a column vector of the randomly drawn ∆ v corresponding to the samples used in constructing the rows of A . Before factorizing the nonsquare matrix, we scale both sides of each line of Eq. (7) by the standard deviation of the ∆ v samples corresponding to that event. We define [ σ ∆ v ] to be a column vector in which each element corresponds to the standard deviation of the ∆ v samples from that particular event, then we can write the scaled version of Eq. (7) as:</text>
<formula><location><page_7><loc_22><loc_42><loc_49><loc_44></location>A ' [ ¯ s lm ] = [ ∆ v ' ] , (8)</formula>
<text><location><page_7><loc_9><loc_39><loc_13><loc_41></location>where</text>
<formula><location><page_7><loc_25><loc_36><loc_49><loc_38></location>A ' ij = A ij [ σ ∆ v ] i (9)</formula>
<formula><location><page_7><loc_23><loc_34><loc_49><loc_36></location>[ ∆ v ' ] i = [ ∆ v ] i [ σ ∆ v ] i . (10)</formula>
<text><location><page_7><loc_9><loc_27><loc_49><loc_33></location>The SVD factorizes the non-square matrix A ' into two orthogonal square matrices U and V , that are N E × N E and n SME × n SME respectively, and a diagonal N E × n SME matrix [Σ] with non-negative entries:</text>
<formula><location><page_7><loc_23><loc_25><loc_49><loc_26></location>A ' = U Σ V T , (11)</formula>
<text><location><page_7><loc_9><loc_22><loc_25><loc_23></location>where Σ has the form:</text>
<formula><location><page_7><loc_24><loc_18><loc_49><loc_21></location>Σ = ( S 0 0 0 ) (12)</formula>
<text><location><page_7><loc_9><loc_15><loc_12><loc_16></location>with</text>
<formula><location><page_7><loc_19><loc_13><loc_49><loc_14></location>S = diagonal { σ 1 , ..., σ n SME } . (13)</formula>
<text><location><page_7><loc_9><loc_9><loc_49><loc_11></location>The non-negative values σ 1 > σ 2 > ... > σ n SME are known as singular values and are estimated along with U</text>
<text><location><page_7><loc_52><loc_82><loc_92><loc_93></location>and V by a linear least squares algorithm[49]. The scaling with the standard deviation of ∆ v essentially transforms a least square minimized SVD on A ' into a Chisquare minimized SVD on A . This allows us to properly account for the fact that some events in our list are less significant than others. Proceeding without this scaling biases the SVD. Once computed, the singular values can be used to solve for ¯ s lm in Eq. (7) :</text>
<formula><location><page_7><loc_61><loc_76><loc_92><loc_80></location>[ ¯ s lm ] i = 1 σ i n SME ∑ k =1 V ik [ U T ∆ v ' ] k (14)</formula>
<text><location><page_7><loc_52><loc_71><loc_92><loc_75></location>for each draw. We can then estimate the densities of the SME parameters from all draws and produce constraints on them.</text>
<text><location><page_7><loc_52><loc_36><loc_92><loc_71></location>We note that despite being a computationally cheap method for computing constraints on the SME coefficients from multiple GW events, the SVD-assisted random draw method has certain inadequacies. There is ambiguity in the exact meaning and interpretation of the uncertainty estimates produced by this method. In the case where the number of events is larger than the number of SME coefficients, this implementation of the random draw method boils down to randomly choosing a posterior sample of (∆ v, θ, ϕ ) from each event and doing a least chi-square fit for the SME parameters. This procedure is then repeated a large number of times, producing a least chi-square fit of the SME coefficients for each draw. However this is not equivalent to the multiplication of posterior probabilities of the SME coefficients, over all events, and drawing samples from that joint posterior. Thus the quantile ranges of the set of chi-square fitted SME coefficients do not hold the same meaning as Bayesian credible intervals. While the Bayesian intervals represent regions of the SME parameter space wherein their true values lie with a particular posterior probability given the data, the SVD-based random draw constraints can be expected to have a different meaning, the exact nature of which remains ambiguous.</text>
<text><location><page_7><loc_52><loc_9><loc_92><loc_36></location>Due to these considerations, we conclude that the weighted SVD-assisted random draw method produces constraints that are unreliable and are likely to be underestimates of the true uncertainties in the measurement of SME coefficients. We verify this claim by testing this method against its Bayesian counterparts in an MDC that we describe later in this work. The results of the MDC show that the samples of SME parameters produced by this method are concentrated in a narrow region around the true values of the parameters, which also coincide with the peaks of the posterior distributions inferred by the Bayesian methods. Therein lies the merit of this method in the present context and its potential to serve as a rapid consistency check for the Bayesian methods. Furthermore, this method is extremely fast and computationally cheap and hence can be used to quickly find the narrow region in the parameter space inside which the peak of the posteriors lies. The stochastic MCMC sampling employed by our Bayesian methods</text>
<text><location><page_8><loc_9><loc_72><loc_49><loc_93></location>is expected to converge much faster if the MCMC chains are initialized near the maxima of the posterior being sampled. Thus the SVD-assisted random draw method can be used to optimize the MCMC sampling used in our Bayesian methods with significant speed-up gains for narrowly peaked SME posteriors. Given the large number of events expected to be observed in O4 and the width of the Bayesian intervals we compute using our current set of events, the posterior distributions of the SME coefficients can be expected to be very narrow post O4, and hence lead to a drastic increase in the computational cost and latency of the Bayesian methods being applied to such a data set. This will likely make the optimization of the Bayesian methods as offered by the SVD-assisted random draw method a necessary tool in the near future.</text>
<section_header_level_1><location><page_8><loc_15><loc_67><loc_43><loc_68></location>D. Hierarchical Bayesian Inference</section_header_level_1>
<text><location><page_8><loc_9><loc_57><loc_49><loc_65></location>Since the SME coefficients are properties that are expected to be the same for all events, one can perform Bayesian Hierarchical Inference on them from the GW data of multiple events. To do so, we can construct the marginalized likelihood of GW data given a particular value of the SME coefficients, jointly from multiple events</text>
<formula><location><page_8><loc_9><loc_49><loc_49><loc_55></location>L (¯ s lm ) = ∏ i ∈{ events } ∫ L ( d i | ∆ v ' , θ, ϕ )Π(∆ v ' , θ, ϕ | ¯ s lm ) d ∆ v ' dθdϕ, (15)</formula>
<text><location><page_8><loc_9><loc_43><loc_49><loc_47></location>where the SME sensitive part of the prior imposes the relationship (6) on ∆ v, θ, ϕ for a given value of the SME coefficients :</text>
<formula><location><page_8><loc_11><loc_39><loc_49><loc_42></location>Π(∆ v ' , θ, ϕ | ¯ s lm ) = δ (∆ v ' -∆ v (¯ s lm , θ, ϕ )) π ( θ ) π ( ϕ ) . (16)</formula>
<text><location><page_8><loc_9><loc_25><loc_49><loc_37></location>Here ∆ v (¯ s lm , θ, ϕ ) is the right hand side of Eq. (6). Note that we have chosen to represent the deviation of the speed of gravity from the speed of light by the dummy variable ∆ v ' whenever a probabilistic quantity (such as likelihood, posterior, prior, or detection fraction) is expressed as a function of it, so as to distinguish it from the quantity ∆ v (¯ s lm , θ, ϕ ). The presence of the delta function in Eq. (16) is due to the deterministic nature of the Eq. (6).</text>
<text><location><page_8><loc_9><loc_13><loc_49><loc_24></location>By Bayes' theorem, for a uniform prior on ¯ s lm , the likelihood L (¯ s lm ) is proportional to the posterior of these parameters given GW data. We can now sample this posterior using MCMC to produce joint SME constraints from multiple GW observations. However, this procedure involves a very large number of evaluations of the likelihoods L ( d i | ∆ v ' , θ, ϕ ) which is so computationally expensive that it's practically infeasible.</text>
<text><location><page_8><loc_9><loc_9><loc_49><loc_13></location>To get around this problem, one can again use Bayes' theorem to write the likelihood L ( d i | ∆ v ' , θ, ϕ ) as proportional to the ratio of the posterior p (∆ v ' , θ, ϕ | d i ) to the</text>
<text><location><page_8><loc_52><loc_92><loc_56><loc_93></location>prior:</text>
<formula><location><page_8><loc_59><loc_88><loc_92><loc_91></location>L ( d i | ∆ v ' , θ, ϕ ) ∝ p (∆ v ' , θ, ϕ | d i ) π (∆ v ' ) π ( θ ) π ( ϕ ) (17)</formula>
<text><location><page_8><loc_52><loc_86><loc_80><loc_87></location>Substituting this into Eq. (15) gives us:</text>
<formula><location><page_8><loc_53><loc_77><loc_92><loc_85></location>L (¯ s lm ) ∝ ∏ i ∈{ events } ∫ p (∆ v ' , θ, ϕ | d i ) δ (∆ v ' -∆ v (¯ s lm , θ, ϕ )) d ∆ v ' dθdϕ. (18)</formula>
<text><location><page_8><loc_52><loc_53><loc_92><loc_76></location>We can now use the samples drawn from the posterior p (∆ v ' , θ, ϕ | d i ) obtained using the parameter estimation run described above to evaluate the integral in Eq. (18). Note that we have ignored a factor of 1 /π (∆ v ' ) in Eq. (18) which is constant since we choose π (∆ v ' ) to be uniform in our parameter estimation runs. However, the presence of the Dirac delta makes it slightly complicated to evaluate this integral directly as a sum over posterior samples. We describe shortly two approximation schemes that can be used to smooth out the discrete sum of Dirac deltas over posterior samples that would entail the evaluation of the integral in Eq. (18) and hence constrain the SME coefficients jointly from multiple GW observations. Before that, we first describe why Bayesian Inference of this form is subject to selection biases and how we account for them.</text>
<text><location><page_8><loc_52><loc_12><loc_92><loc_53></location>Bayesian Hierarchical Inference from a set of GW events selected based on a particular criterion introduces selection biases into the inferred posterior distribution of hyper-parameters [50, 51]. Since we are selecting events based on whether they were found with a Signal to Noise Ratio (SNR) greater than some threshold in at least 3 detectors, and since each detector has an antenna pattern that makes it more sensitive to certain sky directions than others at the time of detection[52], our analysis might be biased towards some values s lm against others. Particularly, the fact that GW search pipelines such as GsTLAL only report multi-detector coincidences based on whether or not the time-delays between the detectors being triggered are smaller than the light travel time between detectors plus a 5 millisecond window, has the potential to bias our results greatly [53]. Furthermore, non-coincident events are down-ranked in significance by means of single's penalties [53], making events even less likely to be detectable for certain cases. Other pipelines such as PyCBC use similar methods for identifying multi-detector coincidences albeit with a different value for the timing error window (which is 2 milliseconds for PyCBC [54]). The existence of this restriction for coincidence formation in search pipelines implies that we are more likely to discover a multi-detector event if the speed of gravitational waves is greater than or equal to c , as compared to if it were lower than c . Thus, our speed of gravitational wave measurements may be biased towards measuring ∆ v ≥ 0 against ∆ v < 0 along any particular sky position.</text>
<text><location><page_8><loc_52><loc_9><loc_92><loc_11></location>To account for this bias, we must normalize our hierarchical likelihood over the true rate of events as opposed</text>
<text><location><page_9><loc_9><loc_86><loc_49><loc_93></location>to the detected rate, with the latter being different from the former, due to selection biases. The constant of normalization is the fraction of events that are detectable given a particular value of the hyper-parameters and the detection criteria:</text>
<formula><location><page_9><loc_10><loc_79><loc_49><loc_86></location>L (¯ s lm ) ∝ 1 β N det (¯ s lm ) × N ∏ i ∈{ events } ∫ p (∆ v ' , θ, ϕ | d i ) δ (∆ v ' -∆ v (¯ s lm , θ, ϕ )) (19)</formula>
<text><location><page_9><loc_18><loc_77><loc_19><loc_78></location>d</text>
<text><location><page_9><loc_19><loc_77><loc_20><loc_78></location>∆</text>
<text><location><page_9><loc_20><loc_77><loc_21><loc_78></location>v</text>
<text><location><page_9><loc_21><loc_78><loc_21><loc_79></location>'</text>
<text><location><page_9><loc_21><loc_77><loc_25><loc_78></location>dθdϕ,</text>
<text><location><page_9><loc_9><loc_47><loc_49><loc_76></location>where β det (¯ s lm ) = R det (¯ s lm ) R true , the fraction of detectable events is the ratio of the detectable rate of events to the true Rate of events [55]. To calculate the fraction accurately we must simulate a large number of events whose parameters are drawn from broad enough distributions, inject them into the detector noise realizations, and see what fraction of them are recovered given our selection criteria. To do that we must first quantify our selection criteria in terms of the parameters that characterize the GW signal. Accurate modeling would require us to recalculate the search pipeline's ranking statistic of a simulated event while allowing for non-zero ∆ v and to find the corresponding False Alarm Rate(FAR) of that trigger from said ranking statistics. One can then apply a threshold on the combined FAR of the event to classify them as detectable or non-detectable. However, such a calculation would require a pipeline-specific analysis which is beyond the scope of this work. Instead, we use an approximated selection criteria: for the i -th event to be detectable, its recovered parameters must satisfy:</text>
<formula><location><page_9><loc_9><loc_41><loc_49><loc_46></location>det = ⇒ { ρ H ≥ ρ th , ρ L ≥ ρ th , ρ V ≥ ρ th , ρ net ≥ ρ net , th , ∆ t HL (∆ v ) ≤ ∆ t HL (0) + 5 ms,t HV (∆ v ) ≤ ∆ t HV (0) + 5 ms,t V L (∆ v ) ≤ ∆ t V L (0) + 5 ms } , (20)</formula>
<text><location><page_9><loc_9><loc_12><loc_49><loc_40></location>where ρ A is the SNR in detector A , ρ net is the network SNR, ∆ t AB (∆ v ) is the time-delay of signal arrival between detectors A and B as a function of ∆ v and ρ th is the SNR threshold used for selecting events. Even though we do not select events depending on which search pipeline found them, we use GstLAL 's timing error window to quantify our selection criteria, instead of say PyCBC's, due to the following reason. Among the events that survive our three detector SNR thresholds, most are found by both GstLAL and PyCBC except for GW170818, GW190701, and GW190814 which are found only by GstLAL . Hence, it is sufficient to model the selection biases that might have appeared in this particular study based on GstLAL's value of the timing error window. This would not have been possible if there were events found by PyCBC and not GstLAL with SNR greater than 10 in three detectors during O3. In such a scenario, a more generalized treatment of selection biases would have been necessary, one that accounts for the difference in timing errors allowed by GstLAL and PyCBC.</text>
<text><location><page_9><loc_9><loc_9><loc_49><loc_11></location>Now that we have a quantifiable detection criterion, we can carry out our simulations. Once the simulated events</text>
<text><location><page_9><loc_52><loc_88><loc_92><loc_93></location>are injected into detector noise realizations and classified as detectable or non-detectable depending on their recovered parameters, it is possible to compute the fraction of events detectable given a choice of CBC parameters:</text>
<formula><location><page_9><loc_58><loc_83><loc_92><loc_86></location>f det (∆ v ' , θ, ϕ, ⃗γ ) ∝ p (∆ v ' , θ, ϕ, ⃗γ | det ) Π sim (∆ v ' , θ, ϕ, ⃗γ ) . (21)</formula>
<text><location><page_9><loc_52><loc_69><loc_92><loc_82></location>Here, ⃗γ are additional CBC parameters such as masses, spins, etc. that characterize the waveform, p (∆ v ' , θ, ϕ, ⃗γ | det ) is the probability of detection, which can be calculated from the set of simulated events that are detectable, and Π sim is the prior from which the simulations are drawn, which has to be broad enough so that we have enough events in both the detectable and non-detectable parts of the parameter space. We can marginalize Eq. (21) over suitable priors to get:</text>
<formula><location><page_9><loc_54><loc_61><loc_92><loc_66></location>β det (¯ s lm ) = ∫ f det (∆ v ' , θ, ϕ, ⃗γ )Π(∆ v ' , θ, ϕ | ¯ s lm ) × Π( ⃗γ ) d ∆ v ' dθdϕd⃗γ. (22)</formula>
<text><location><page_9><loc_52><loc_51><loc_92><loc_60></location>If we choose Π sim (∆ v ' , θ, ϕ, ⃗γ ) = π (∆ v ' ) π ( θ ) π ( ϕ )Π( ⃗γ ), where π (∆ v ' ) , π, ( θ ) , π ( ϕ ) are the same as the ones defined in Eqs. (16) and (17), then priors in the denominator and numerator of the integrand in (21) cancel out and we can define the marginalized fraction of detectable events (up to the factors that cancel out later):</text>
<formula><location><page_9><loc_56><loc_47><loc_92><loc_50></location>f marg det (∆ v ' , θ, ϕ ) ∝ ∫ p (∆ v ' , θ, ϕ, ⃗γ | det ) d⃗γ. (23)</formula>
<text><location><page_9><loc_52><loc_41><loc_92><loc_45></location>As in the case of Eq. (18), we have ignored a factor of 1 /π (∆ v ' ) in Eq. (23) for the same reason mentioned before. In terms of this marginalized fraction, β det becomes:</text>
<formula><location><page_9><loc_52><loc_33><loc_92><loc_40></location>β det (¯ s lm ) ∝ ∫ f marg det (∆ v ' , θ, ϕ ) δ (∆ v ' -∆ v ' ( θ, ϕ, ¯ s lm )) d ∆ v ' dθdϕ. (24)</formula>
<text><location><page_9><loc_52><loc_15><loc_92><loc_32></location>To estimate p (∆ v ' , θ, ϕ, ⃗γ | det ) and hence f marg det (∆ v ' , θ, ϕ ) we simulate a large number of events whose parameters are drawn from a broad distribution. We then inject the corresponding signals into detector noise realizations and record their SNRs and arrival times. We then apply our selection criteria to find which of these simulated events are detectable given our criteria and estimate p (∆ v ' , θ, ϕ, ⃗γ | det ). The estimation schemes will depend on which of the two approximations referred to before are used to smooth out the delta function integral and are hence described in more detail in the corresponding subsections below.</text>
<text><location><page_9><loc_52><loc_9><loc_92><loc_14></location>The priors we use to draw the simulated events are truncated power-law in the primary mass and mass ratio, uniform in spin, sky position, orientation, co-moving volume, geocentric time, and speed of gravitational waves.</text>
<text><location><page_10><loc_9><loc_73><loc_49><loc_93></location>Particularly, for each observing run, the mass distributions are chosen to be consistent with corresponding population analyses performed by the LVC such that the distributions used have are have support in regions of the mass space where the events being analyzed are found. For O2, we choose p ( m 1 ) ∝ m -1 . 6 1 , m 1 ∈ (7 . 9 M ⊙ , 42 M ⊙ ) and p ( q ) ∝ q 6 . 7 where q = m 2 m 1 which is consistent with Ref. [56], and is identical to the mass distributions used for similar selection function computations [52]. For O3 we choose p ( m 1 ) ∝ m -1 . 6 1 , m 1 ∈ (7 M ⊙ , 80 M ⊙ ) and p ( q ) ∝ q 6 . 7 , which is broad enough for the O3 events as evident from Ref. [57]. In the next two subsections, we describe the details of our smoothing approximations and the computation β det in each approximation scheme.</text>
<section_header_level_1><location><page_10><loc_19><loc_69><loc_39><loc_70></location>1. Narrow Gaussian Method</section_header_level_1>
<text><location><page_10><loc_9><loc_57><loc_49><loc_67></location>The approach introduced here involves estimating the delta function in Eq. (18) as a narrow Gaussian distribution. For each sample with measured speed difference ∆ v ' and sky location θ and ϕ . We construct a Gaussian distribution for the random variable ∆ v ' -∆ v (¯ s lm , θ, ϕ ) with mean zero and standard deviation σ . Thus, Eq. (19) becomes:</text>
<formula><location><page_10><loc_9><loc_48><loc_49><loc_56></location>L (¯ s lm ) = 1 β N det (¯ s lm ) N ∏ i ∈{ events } ∫ p (∆ v ' , θ, ϕ | d i ) ×N (∆ v ' -∆ v (¯ s lm , θ, ϕ )) d ∆ v ' dθdϕ, (25)</formula>
<text><location><page_10><loc_9><loc_43><loc_49><loc_47></location>where N represents Gaussian distributions. Similarly, we can also apply the Narrow Gaussian approximation to the computation of β det in Eq. (24):</text>
<formula><location><page_10><loc_9><loc_37><loc_49><loc_42></location>β det (¯ s lm ) = ∫ f marg det (∆ v ' , θ, ϕ ) N (∆ v ' -∆ v (¯ s lm , θ, ϕ )) × d ∆ v ' dθdϕ. (26)</formula>
<text><location><page_10><loc_9><loc_23><loc_49><loc_36></location>Since N is a smooth function of its arguments we can evaluate the two integrals in Eqns. (25) and (26) as a Monte Carlo sum over samples drawn from p (∆ v ' , θ, ϕ | d i ) and f marg det (∆ v ' , θ, ϕ ) respectively. Since we already have posterior samples drawn from p (∆ v ' , θ, ϕ | d i ) for each event during the v g inference described in Sec. II, and since the samples drawn from f marg det (∆ v ' , θ, ϕ ) are the parameters of simulated events that survive our selection criteria, we can compute the log-likelihood of ¯ s lm :</text>
<formula><location><page_10><loc_17><loc_15><loc_49><loc_22></location>ln L (¯ s lm ) = ∑ i ∈{ events } (27) ln ∑ { j } N (∆ v ' j -∆ v (¯ s lm , θ j , ϕ j )) ∑ k N (∆ v ' k -∆ v (¯ s lm , θ k , ϕ k ) ,</formula>
<text><location><page_10><loc_9><loc_9><loc_49><loc_14></location>where the sum in the numerator is over posterior samples corresponding to the i th event while the one in the denominator is over detectable samples. After choosing a width σ for our Gaussian N appropriately, we can</text>
<text><location><page_10><loc_52><loc_85><loc_92><loc_93></location>thus use Eq. (27) for fast evaluation of the log-likelihood ln L (¯ s lm ) as a numerical function of the SME coefficients. Hence we can use MCMC algorithms to draw samples from ln L (¯ s lm ) and interpret the quantile ranges of said samples as Bayesian credible intervals of the SME coefficients given GW data.</text>
<text><location><page_10><loc_52><loc_70><loc_92><loc_84></location>To determine the appropriate width of our Gaussian distribution σ , we consider the effect of varying its size. Because the Gaussian distribution is an estimation of the delta distribution, theoretically, as the size of σ decreases, the approximation should be more accurate. However, because we sample the log-likelihood with an MCMC algorithm, we encounter numerical difficulties when the σ is too small. Thus our choice of σ has to be tuned in accordance with how the MCMC is implemented numerically.</text>
<text><location><page_10><loc_52><loc_47><loc_92><loc_69></location>In the MCMC process, the walkers only make use of local information at each step. Thus, it is possible for walkers to be trapped inside of islands of high likelihood. This is what happens when σ is set too small. Since most samples have high likelihood around zero, walkers can explore freely the region near zero. However, at more peripheral locations in the parameter space, the peaks are usually scattered. Thus, when σ is too small, these peripheral samples form isolated islands of high likelihood. In this case, the walkers will not be able to explore these isolated islands, resulting in false small constraints. On the other hand, as σ gets larger, our approximation becomes less accurate and distributions are artificially broadened. Therefore, we aim to find a σ such that it is big enough for the walkers to explore the sample space fully and small enough such that it gives us useful results.</text>
<text><location><page_10><loc_52><loc_9><loc_92><loc_46></location>One way to determine the appropriate value of σ is by applying both the random draw method and the Narrow Gaussian method on the same set of data and comparing the results. We divide our list of events into subsets of nine events and apply both methods to each subset using various sizes of σ . The solid lines in Fig. 2 represent the average uncertainty of the resultant ¯ s lm 's against σ for the nine O3 events from this paper with the smallest sky areas. The average uncertainty is calculated by taking the average of the absolute value of the upper and lower one-standard-deviation value for each ¯ s lm . For the same sets of events, the random draw method produces uncertainties on the order of 10 0 -10 1 , which corresponds to dashed lines in Fig. 2. To select a suitable σ , we use superimposed plots such as Fig. 2 to select a σ such that each uncertainty produced by the Narrow Gaussian method is marginally larger than its counterpart from the random draw method. In the example shown, σ = 0 . 0005 is a good choice because every solid line lies marginally above the corresponding dashed line in the same color, which means that the choice σ does not artificially tighten the constraints. We perform such analysis for every subset of events and produce a 'good σ '. We pick the largest of such 'good σ 's' as the final choice. Even though this final σ produces wider constraints for each subset of events as compared to the random draw method, because the</text>
<text><location><page_11><loc_9><loc_89><loc_49><loc_93></location>Narrow Gaussian method incorporates information from more than 9 events, ideally we could still produce tighter constraints than the random draw method.</text>
<text><location><page_11><loc_9><loc_66><loc_49><loc_89></location>Another procedure for determining σ is to use information directly from the distribution of ∆ v ' samples for each event. One specific procedure is to sort the list of ∆ v ' samples and compute the average difference between adjacent values. Choosing σ as half this value produces results that align well with the prior method for the specific distributions tested. That is, the value of σ for a given event is half of the average of the differences between adjacent ∆ v ' samples for that event. This procedure is used in the MDC shown in Fig. 3, and has the advantage of not needing to construct plots to determine σ . However, there still is subjectivity in choosing what fraction of the average to use. Moreover, the distribution and quantity of samples will impact the value of σ . Both considerations will affect the final credible intervals for ¯ s lm .</text>
<text><location><page_11><loc_9><loc_48><loc_49><loc_65></location>This method performs much better than the SVDassisted random draw in the MDC performed in Sec. III F. However, we note that both processes for choosing σ involve a significant amount of user-controlled fine-tuning and can potentially lead to an over/underestimate of measurement uncertainties of the SME coefficients. Due to these reasons, we do not choose this method for our final results. We instead choose a different smoothing approximation to the Bayesian method by means of Kernel Density Estimation (KDE) which can be shown to produce either equally or more accurate results, while requiring almost no user-controlled fine-tuning.</text>
<section_header_level_1><location><page_11><loc_23><loc_44><loc_35><loc_45></location>2. KDE Methods</section_header_level_1>
<text><location><page_11><loc_9><loc_29><loc_49><loc_42></location>In this section we outline a different approach from the one in the previous section, to perform Bayesian Hierarchical Inference of the SME coefficients from GW data. In this method, instead of smearing out the delta function in Π(∆ v ' , θ, ϕ | ¯ s lm ) with a Gaussian, we approximate the marginalized posterior of ∆ v, θ, ϕ given GW data, for each event, as a fast evaluating function of these quantities, from their single event PE samples via Gaussian KDE.</text>
<text><location><page_11><loc_9><loc_9><loc_49><loc_29></location>The KDE approximation of the posterior is constructed by fitting a multi-variate Gaussian around each posterior sample and then writing the estimate of the posterior as a sum of these individual Gaussians. The covariance matrix of each of the Gaussians is approximated from the sample covariance matrix of the posterior samples themselves up to a constant of proportionality. The constant of proportionality is known as the bandwidth of the estimator and is computed, under reasonable assumptions regarding the true distribution being estimated(see [36]). We use SciPy's Gaussian KDE algorithm to obtain our estimate of the marginalized posterior as a fast evaluating function p KDE , i (∆ v ' = ∆ v ( θ, ϕ, ¯ s lm ) , θ, ϕ ) of the relevant parameters [37]. We then perform the ∆ v ' inte-</text>
<text><location><page_11><loc_52><loc_85><loc_92><loc_93></location>) analytically using the delta function and compute the remaining two integrals (over θ, ϕ ) numerically using the trapezoidal rule. We loop over multiple events by multiplying the value of the integral obtained using the KDE corresponding to each event, to evaluate L (¯ s lm ) :</text>
<formula><location><page_11><loc_52><loc_75><loc_93><loc_82></location>L (¯ s lm ) ≈ ∏ iϵ { events } ∫ p KDE,i (∆ v ' = ∆ v ( θ, ϕ, ¯ s lm ) , θ, ϕ ) dθdϕ. (28)</formula>
<text><location><page_11><loc_52><loc_60><loc_92><loc_74></location>We then sample from it using the same MCMC method described in the previous section to constrain the SME coefficients. To incorporate selection effects in the KDE method, we estimate f marg det (∆ v ' , θ, ϕ | det ) by performing a KDE on the samples of (∆ v ' , θ, ϕ ) for which the simulated events are detectable given our detection criteria. By restricting our KDE to only these parameters and ignoring other parameters that characterize a simulated event, we effectively marginalize over those other parameters thus implicitly performing the integral in Eq. (23):</text>
<formula><location><page_11><loc_61><loc_57><loc_92><loc_59></location>f marg det ≈ p KDE , det (∆ v ' , θ, ϕ ) , (29)</formula>
<text><location><page_11><loc_52><loc_50><loc_92><loc_56></location>where the subscript det in p KDE,det represents the fact that this KDE was performed on only those samples for which the simulated events are detectable given our detection criteria. Substituting into Eq. (24) we get:</text>
<formula><location><page_11><loc_52><loc_44><loc_92><loc_49></location>β det (¯ s lm ) ≈ ∫ p KDE , det (∆ v ' = ∆ v ( θ, ϕ, ¯ s l m ) , θ, ϕ ) dθdϕ. (30)</formula>
<text><location><page_11><loc_52><loc_9><loc_92><loc_43></location>We note that the KDE's bandwidth acts like a control parameter with potential room for user-controlled finetuning in the computation of its value, somewhat analogous to the σ of the narrow Gaussian method. However, unlike the narrow Gaussian method where σ can in principle be chosen to be anything, the bandwidth of the KDE is computed directly from the properties of the samples (such as the number of samples and dimensionality of the parameter space) under reasonable assumptions regarding the true density. Thus the user's choice is restricted to a number of discrete such assumptions (for example Scott's rule [36], Silverman's rule [35] etc.). Furthermore, the effects of choosing a bandwidth on the estimated density (and hence the remainder of the inference) is limited in the sense that the covariance matrix of the Gaussians is determined from the samples themselves with the bandwidth only acting as a scaling parameter that is usually of order unity. This additionally restricts the effects of user-controlled fine tuning on the inference as compared to the narrow Gaussian method wherein the width of the Gaussian that approximates the delta function is completely determined by the user's choice. A more detailed discussion of this comparison between the two methods in the context of the MDC can be found</text>
<figure>
<location><page_12><loc_9><loc_58><loc_92><loc_93></location>
<caption>FIG. 2. A portion of the superimposed plot of average uncertainties produced by the random draw method (dashed lines) and the Bayesian method (solid lines). From this plot, we can see σ = 0 . 0005 is a potential choice for σ because every solid line is marginally above the corresponding dashed line of the same color.</caption>
</figure>
<text><location><page_12><loc_9><loc_42><loc_49><loc_49></location>in Sec. III F . For our chosen bandwidth approximation scheme (Scott's rule) the KDE method can be seen to perform extremely well in the MDC. For these reasons, we choose the KDE method for our final results on the SME constraints.</text>
<text><location><page_12><loc_9><loc_34><loc_49><loc_42></location>We present the results of the KDE method upon its use in the analysis of the events marked * listed on tables I through III (except for GW170817 for which we do not use the fixed posteriors due to the inability of the KDE to estimate very narrow densities) in Fig. 4 as our final result.</text>
<section_header_level_1><location><page_12><loc_23><loc_29><loc_34><loc_30></location>E. Discussion</section_header_level_1>
<text><location><page_12><loc_9><loc_9><loc_49><loc_27></location>The random draw method, which was originally presented in [7], is very time efficient. However, the number of events that can be used in the analysis is limited by the number of ¯ s lm to be estimated. Hence, it is only useful in the scenario where in we have exactly the same number of events available to be used in the analysis as the number of ¯ s lm coefficients being simultaneously measured. On the other hand, Unlike any of the following methods, this method does not involve any approximation of the delta distribution. Thus, this method can be used as a quick feasibility test. For example, we used this method to estimate the size of σ for the narrow Gaussian approach.</text>
<text><location><page_12><loc_52><loc_37><loc_92><loc_49></location>With the use of SVD, we were able to take data from a larger set of events. However, the presence of one 'bad' event, i.e. a low-significance event with biased posteriors, can disturb the entire analysis since all events are given equal weights. With the SVD chi-square method, this problem is solved by weighting each event with its uncertainty in the v g measurement. However, this leads to artificially narrower bounds with ambiguity in the meaning of those bounds.</text>
<text><location><page_12><loc_52><loc_25><loc_92><loc_36></location>The Bayesian methods are free of all aforementioned pathologies that plague the other methods. They can efficiently handle a large number of events and is unaffected by a small number of bad events if any. Furthermore, the Bayesian credible intervals have the clear and unambiguous meaning of being regions of the parameter space that contain the true value of said parameters with a certain posterior probability given data.</text>
<text><location><page_12><loc_52><loc_9><loc_92><loc_24></location>Among the two approximate Bayesian techniques described in this paper, the Narrow Gaussian method has the following issue. The process to find σ is cumbersome and somewhat subjective as it involves partitioning the set of events into subsets and estimating σ from plots, or developing an algorithm that needs to be compared to an independent method. On the other hand, the KDE method has less user-controlled finetuning than the narrow Gaussian method as it estimates its control parameter, i.e. the bandwidth, quantitatively from properties of the posterior samples themselves, with its variation</text>
<figure>
<location><page_13><loc_9><loc_48><loc_91><loc_93></location>
<caption>FIG. 3. Distribution of all 9 ¯ s lm for the Mock data. The Exact Bayesian, KDE, Narrow Gaussian and SVD chi-square methods are color-coded blue, orgnge, green, and magenta respectively. The mock true values of ¯ s lm are marked by orange lines. Numbers above the plots show median values with 90% equal tail credible intervals. The displayed constraints were obtained from the inferred ¯ s lm samples using the python package corner [58].</caption>
</figure>
<text><location><page_13><loc_9><loc_33><loc_49><loc_38></location>having a much more restrictive effect on the estimated density. Hence we claim that the Bayesian analysis implemented by the KDE method produces the most trustworthy measurements of the SME coefficients.</text>
<section_header_level_1><location><page_13><loc_11><loc_27><loc_46><loc_29></location>F. Mock Data Challenge and Comparison of Methods</section_header_level_1>
<text><location><page_13><loc_9><loc_12><loc_49><loc_24></location>In this section, we describe the MDC that was set up to compare the different methods of SME measurements from GW data in order to verify our claims regarding them that were made in the previous section. To construct the MDC, we choose a fiducial value of the SME coefficients as their true values, say ¯ s lm,tr and generate data for 15 mock events. The true sky positions of the mock events are chosen to be the mean values of the ( θ, ϕ ) samples of the real events.</text>
<text><location><page_13><loc_9><loc_9><loc_49><loc_11></location>We choose 15 of the 'best' real events, i.e. the ones with the most precise sky localizations and v g measure-</text>
<text><location><page_13><loc_52><loc_26><loc_92><loc_38></location>ments, to be represented by our mock events in the MDC. We then calculate the true value of ∆ v for each mock event from the true values of their sky positions and those of the SME coefficients. We then generate mock posterior samples of ( θ, ϕ, ∆ v ) by adding un-correlated Gaussian fluctuations to the true sky positions and true ∆ v 's. The width of the fluctuations, for each mock event is chosen to be the standard deviations of the posterior samples of the corresponding real event.</text>
<text><location><page_13><loc_52><loc_18><loc_92><loc_25></location>This allows us to create a controlled numerical experiment wherein we know the true answer. For Gaussian distributions of θ, ϕ, ∆ v about known true values, one can write down the exact functional form of the likelihood of these parameters given mock data:</text>
<formula><location><page_13><loc_52><loc_12><loc_92><loc_17></location>L mdc ( d i | ∆ v ' , θ, ϕ ) = 1 (2 π ) 3 / 2 σ θ,i σ ϕ,i σ ∆ v,i exp -1 2 { ( θ -θ tr,i ) 2 σ 2 θ,i + ( ϕ -ϕ tr,i ) 2 σ 2 ϕ,i + (∆ v ' -∆ v tr,i ) 2 σ 2 ∆ v,i } , (31)</formula>
<text><location><page_13><loc_52><loc_9><loc_92><loc_11></location>where ( θ tr,i , ϕ tr,i ) are the true values of the sky positions of the i th mock event, σ θ,i , σ ϕ,i , σ ∆ v,i are the</text>
<figure>
<location><page_14><loc_9><loc_49><loc_91><loc_93></location>
<caption>FIG. 4. Distribution of all 9 ¯ s lm using the KDE method from the 24 chosen GW events in Tables I, II, and III. The numbers above the plots show median values with 90% equal tail credible intervals. The displayed constraints were obtained from the inferred ¯ s lm samples using the python package corner [58].</caption>
</figure>
<text><location><page_14><loc_9><loc_31><loc_49><loc_40></location>widths of the Gaussian fluctuations used to generate the mock posterior samples of the ith mock event and ∆ v tr,i = ∆ v (¯ s lm,tr , θ tr,i , ϕ tr,i ). Knowledge of these quantities allows us to exactly write down and evaluate Eq. (31) as a function of θ, ϕ without any smoothing approximations.</text>
<text><location><page_14><loc_9><loc_26><loc_49><loc_31></location>We can then substitute L mdc ( d i | ∆ v ' , θ, ϕ ) in place of L ( d i | ∆ v ' , θ, ϕ ) in Eq. (15) and carry out the integral numerically to obtain the 'exact Bayesian' likelihood of our mock data given the SME coefficients:</text>
<formula><location><page_14><loc_9><loc_15><loc_49><loc_24></location>L mdc (¯ s lm ) = ∏ i 1 (2 π ) 3 / 2 σ θ,i σ ϕ,i σ ∆ v,i ∫ exp -1 2 { ( θ -θ tr,i ) 2 σ 2 θ,i + ( ϕ -ϕ tr,i ) 2 σ 2 ϕ,i (32) + (∆ v (¯ s lm ,θ,ϕ ) -∆ v (¯ s lm,tr ,θ tr,i ,ϕ tr,i )) 2 σ 2 ∆ v,i } dθdϕ.</formula>
<text><location><page_14><loc_9><loc_9><loc_49><loc_14></location>We can then sample the likelihood in Eq. (32), after applying suitable priors on ¯ s lm , using the MCMC techniques described above and obtain what can be thought of as the 'true' posterior distribution of the SME coef-</text>
<text><location><page_14><loc_52><loc_34><loc_92><loc_40></location>icients given the mock data. We can then compute the constraints obtained from the approximate methods being applied to the mock posterior samples and compare those results with the true posterior.</text>
<text><location><page_14><loc_52><loc_9><loc_92><loc_27></location>The results of this comparison is displayed in Fig. 3. We can see that the KDE method agrees remarkably well with the exact Bayesian method, while the narrow Gaussian method deviates from it slightly for some coefficients. We note that a different choice of σ for the narrow Gaussian method leading to better agreement with the exact Bayesian result is possible. However, we conclude that the KDE method's agreement with the exact Bayesian method, independent of any external finetuning, justifies its use on the real data for producing our final result. We also note that our claim regarding the SVD chisquare method producing artificially narrower bounds is also verified by this comparison.</text>
<section_header_level_1><location><page_15><loc_22><loc_92><loc_36><loc_93></location>G. Final Results</section_header_level_1>
<text><location><page_15><loc_9><loc_86><loc_49><loc_90></location>With the 24 chosen GW events in Tables I, II, and III, we are able to constrain all nine ¯ s lm coefficients. We obtain the results shown in Fig. 4.</text>
<text><location><page_15><loc_9><loc_79><loc_49><loc_85></location>Note that the measurements of ¯ s lm shown in Fig.4 are consistent with zero. Given that zero lies within the 90% credible intervals for all coefficients, we consider these results to be consistent with existing constraints on ¯ s lm [9].</text>
<text><location><page_15><loc_9><loc_66><loc_49><loc_78></location>These limits are considerably weaker than some found in the literature. However, they are valuable as independent tests. Moreover, this is also the first attempt to simultaneously constrain all s lm using gravitational-wave measurements, thus putting direct limits on the full potential anisotropy of the speed of gravitational waves. Additionally, this method can theoretically incorporate as many events as available and thus improve in precision as additional high-quality events become available.</text>
<section_header_level_1><location><page_15><loc_21><loc_61><loc_37><loc_62></location>IV. CONCLUSION</section_header_level_1>
<text><location><page_15><loc_9><loc_32><loc_49><loc_59></location>In our study, we select 52 high-SNR gravitationalwave events that were detected by at least two detectors from the first three observing runs of Advanced LIGO and Advanced Virgo. We use lalinference nest and lalinference mcmc to construct posterior distributions of the speeds of GWs for each event. We find the 90% credible interval of the combined v g posterior distribution to be 0 . 99 +0 . 01 -0 . 02 c . This interval is narrower than the similar one constructed with O1 and O2 events in previous studies, suggesting a more precise measurement of v g [7]. However, even with the inclusion of a high-SNR BNS event GW170817 with its pinpoint sky localization, we were only able to narrow the 90% credible interval to 0 . 99 +0 . 01 -0 . 02 c . We then explore multiple methods of extracting SME constraints from v g -like data. Based on the conclusions of that investigation, we use hierarchical Bayesian inference implemented with KDE methods to simultaneously constrain all nine coefficients for Lorentz violation in the SME framework. The resultant con-</text>
<unordered_list>
<list_item><location><page_15><loc_10><loc_23><loc_49><loc_26></location>[1] J. Aasi et al. (LIGO Scientific Collaboration), Advanced LIGO, Class. Quantum Grav. 32 , 074001 (2015), arXiv:1411.4547 [gr-qc].</list_item>
<list_item><location><page_15><loc_10><loc_19><loc_49><loc_22></location>[2] F. Acernese et al. , Advanced Virgo: a second-generation interferometric gravitational wave detector, Class. Quantum Grav. 32 , 024001 (2014), arXiv:1408.3978 [gr-qc].</list_item>
<list_item><location><page_15><loc_10><loc_12><loc_49><loc_18></location>[3] R. Abbott et al. (LIGO-Virgo Collaboration), GWTC2: Compact binary coalescences observed by LIGO and Virgo during the first half of the third observing run, Phys. Rev. X 11 , 021053 (2021), arXiv:2010.14527 [grqc].</list_item>
<list_item><location><page_15><loc_10><loc_9><loc_49><loc_12></location>[4] R. Abbott et al. (LIGO-Virgo-KAGRA Collaboration), GWTC-3: Compact binary coalescences observed by</list_item>
</unordered_list>
<text><location><page_15><loc_52><loc_92><loc_90><loc_93></location>straints do not exhibit evidence for Lorentz violation.</text>
<section_header_level_1><location><page_15><loc_62><loc_87><loc_82><loc_88></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_15><loc_52><loc_70><loc_92><loc_85></location>This work was supported by NSF awards PHY1806990, PHY-1912649, and PHY-2207728. This material is based upon work supported by NSF's LIGO Laboratory which is a major facility fully funded by the National Science Foundation. The analysis in this work used the LALSuite software library [16]. We thank LIGO and Virgo Collaboration for providing the data from the first, second, and third observing runs. The authors are grateful for computational resources provided by the LIGO Laboratory and supported by National Science Foundation Grants PHY-0757058 and PHY-0823459.</text>
<text><location><page_15><loc_52><loc_32><loc_92><loc_69></location>This research has made use of data or software obtained from the Gravitational Wave Open Science Center (gwosc.org), a service of LIGO Laboratory, the LIGO Scientific Collaboration, the Virgo Collaboration, and KAGRA. LIGO Laboratory and Advanced LIGO are funded by the United States National Science Foundation (NSF) as well as the Science and Technology Facilities Council (STFC) of the United Kingdom, the Max-Planck-Society (MPS), and the State of Niedersachsen/Germany for support of the construction of Advanced LIGO and construction and operation of the GEO600 detector. Additional support for Advanced LIGO was provided by the Australian Research Council. Virgo is funded, through the European Gravitational Observatory (EGO), by the French Centre National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale di Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions by institutions from Belgium, Germany, Greece, Hungary, Ireland, Japan, Monaco, Poland, Portugal, Spain. KAGRA is supported by Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan Society for the Promotion of Science (JSPS) in Japan; National Research Foundation (NRF) and Ministry of Science and ICT (MSIT) in Korea; Academia Sinica (AS) and National Science and Technology Council (NSTC) in Taiwan.</text>
<text><location><page_15><loc_55><loc_24><loc_92><loc_26></location>LIGO and Virgo during the second part of the third observing run (2021), arXiv:2111.03606 [gr-qc].</text>
<unordered_list>
<list_item><location><page_15><loc_53><loc_17><loc_92><loc_24></location>[5] B. P. Abbott et al. (LIGO-Virgo Collaboration), GWTC1: A gravitational-wave transient catalog of compact binary mergers observed by LIGO and Virgo during the first and second observing runs , Phys. Rev. X 9 , 031040 (2019), arXiv:1811.12907 [astro-ph.HE].</list_item>
<list_item><location><page_15><loc_53><loc_13><loc_92><loc_17></location>[6] N. Cornish, D. Blas, and G. Nardini, Bounding the speed of gravity with gravitational wave observations, Phys. Rev. Lett. 119 , 161102 (2017), arXiv:1707.06101 [gr-qc].</list_item>
<list_item><location><page_15><loc_53><loc_9><loc_92><loc_13></location>[7] X. Liu, V. F. He, T. M. Mikulski, D. Palenova, C. E. Williams, J. Creighton, and J. D. Tasson, Measuring the speed of gravitational waves from the first and second</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_16><loc_12><loc_89><loc_49><loc_93></location>observing run of Advanced LIGO and Advanced Virgo, Phys. Rev. D 102 , 024028 (2020), arXiv:2005.03121 [grqc].</list_item>
<list_item><location><page_16><loc_10><loc_84><loc_49><loc_89></location>[8] B. P. Abbott et al. (LIGO-Virgo Collaboration), Multimessenger observations of a binary neutron star merger, Astrophys. J 848 , 2 (2017), arXiv:1710.05833 [astroph.HE].</list_item>
<list_item><location><page_16><loc_10><loc_81><loc_49><loc_84></location>[9] V. A. Kosteleck'y and N. Russell, Data tables for Lorentz and CPT violation, arXiv:0801.0287 [hep-ph].</list_item>
<list_item><location><page_16><loc_9><loc_77><loc_49><loc_81></location>[10] D. Colladay and V. A. Kosteleck'y, Lorentz-violating extension of the standard model, Phys. Rev. D 58 , 116002 (1998), arXiv:hep-ph/9809521 [hep-ph].</list_item>
<list_item><location><page_16><loc_9><loc_73><loc_49><loc_77></location>[11] V. A. Kosteleck'y, Gravity, Lorentz violation, and the standard model, Phys. Rev. D 69 , 105009 (2004), arXiv:hep-ph/0312310 [hep-ph].</list_item>
<list_item><location><page_16><loc_9><loc_69><loc_49><loc_73></location>[12] Q. G. Bailey and V. A. Kosteleck'y, Signals for Lorentz violation in post-Newtonian gravity, Phys. Rev. D 74 , 045001 (2006), arXiv:1905.00409 [gr-qc].</list_item>
<list_item><location><page_16><loc_9><loc_63><loc_49><loc_69></location>[13] L. Haegel, K. O'Neal-Ault, Q. G. Bailey, J. D. Tasson, M. Bloom, and L. Shao, Search for anisotropic, birefringent spacetime-symmetry breaking in gravitational wave propagation from GWTC-3, Phys. Rev. D 107 , 064031 (2023), arXiv:2210.04481 [gr-qc].</list_item>
<list_item><location><page_16><loc_9><loc_58><loc_49><loc_63></location>[14] R. Niu, T. Zhu, and W. Zhao, Testing Lorentz invariance of gravity in the Standard-Model Extension with GWTC-3, J. Cosmol. Astropart. Phys. 2022 (12), 011, arXiv:2202.05092 [gr-qc].</list_item>
<list_item><location><page_16><loc_9><loc_51><loc_49><loc_57></location>[15] C. Gong, T. Zhu, R. Niu, Q. Wu, J.-L. Cui, X. Zhang, W. Zhao, and A. Wang, Gravitational wave constraints on nonbirefringent dispersions of gravitational waves due to Lorentz violations with GWTC-3 events, Phys. Rev. D 107 , 124015 (2023), arXiv:2302.05077 [gr-qc].</list_item>
<list_item><location><page_16><loc_9><loc_48><loc_49><loc_51></location>[16] LIGO Scientific Collaboration, LIGO Algorithm Library (2018).</list_item>
<list_item><location><page_16><loc_9><loc_43><loc_49><loc_48></location>[17] R. Abbott et al. (LIGO-Virgo-KAGRA Collaboration), Open data from the first and second observing runs of Advanced LIGO and Advanced Virgo, SoftwareX 13 , 100658 (2021), arXiv:1912.11716 [gr-qc].</list_item>
<list_item><location><page_16><loc_9><loc_39><loc_49><loc_43></location>[18] R. Abbott et al. (LIGO-Virgo-KAGRA Collaboration), Open data from the third observing run of LIGO, Virgo, KAGRA and GEO (2023), arXiv:2302.03676 [gr-qc].</list_item>
<list_item><location><page_16><loc_9><loc_34><loc_49><loc_39></location>[19] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, Equation of state calculations by fast computing machines, J. Chem. Phys. 21 , 1087 (1953).</list_item>
<list_item><location><page_16><loc_9><loc_30><loc_49><loc_34></location>[20] W. K. Hastings, Monte Carlo sampling methods using Markov chains and their applications, Biometrika 57 , 97 (1970).</list_item>
<list_item><location><page_16><loc_9><loc_25><loc_49><loc_30></location>[21] J. Veitch et al. , Parameter estimation for compact binaries with ground-based gravitational-wave observations using the LALInference software library, Phys. Rev. D 91 , 042003 (2015), arXiv:1409.7215 [gr-qc].</list_item>
<list_item><location><page_16><loc_9><loc_19><loc_49><loc_24></location>[22] J. Veitch and A. Vecchio, Bayesian coherent analysis of in-spiral gravitational wave signals with a detector network, Phys. Rev. D 81 , 062003 (2010), arXiv:0911.3820 [astro-ph.CO].</list_item>
<list_item><location><page_16><loc_9><loc_17><loc_49><loc_19></location>[23] J. Skilling, Nested sampling for general Bayesian computation, Bayesian Anal. 1 , 833 (2006).</list_item>
<list_item><location><page_16><loc_9><loc_9><loc_49><loc_16></location>[24] S. Khan, S. Husa, M. Hannam, F. Ohme, M. Purrer, X. J. Forteza, and A. Boh'e, Frequency-domain gravitational waves from nonprecessing black-hole binaries. II. A phenomenological model for the advanced detector era, Phys. Rev. D 93 , 044007 (2016), arXiv:1508.07253 [grqc].</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_16><loc_52><loc_87><loc_92><loc_93></location>[25] S. Husa, S. Khan, M. Hannam, M. Purrer, F. Ohme, X. J. Forteza, and A. Boh'e, Frequency-domain gravitational waves from nonprecessing black-hole binaries. I. New numerical waveforms and anatomy of the signal, Phys. Rev. D 93 , 044006 (2016), arXiv:1508.07250 [gr-qc].</list_item>
<list_item><location><page_16><loc_52><loc_80><loc_92><loc_86></location>[26] M. Hannam, P. Schmidt, A. Boh'e, L. Haegel, S. Husa, F. Ohme, G. Pratten, and M. Purrer, Simple model of complete precessing black-hole-binary gravitational waveforms, Phys. Rev. Lett. 113 , 151101 (2014), arXiv:1308.3271 [gr-qc].</list_item>
<list_item><location><page_16><loc_52><loc_73><loc_92><loc_80></location>[27] K. G. Arun, A. Buonanno, G. Faye, and E. Ochsner, Higher-order spin effects in the amplitude and phase of gravitational waveforms emitted by inspiraling compact binaries: Ready-to-use gravitational waveforms, Phys. Rev. D 79 , 104023 (2009), arXiv:0810.5336 [gr-qc].</list_item>
<list_item><location><page_16><loc_52><loc_67><loc_92><loc_73></location>[28] A. Buonanno, B. R. Iyer, E. Ochsner, Y. Pan, and B. S. Sathyaprakash, Comparison of post-newtonian templates for compact binary inspiral signals in gravitationalwave detectors, Phys. Rev. D 80 , 084043 (2009), arXiv:0907.0700 [gr-qc].</list_item>
<list_item><location><page_16><loc_52><loc_62><loc_92><loc_67></location>[29] B. Mik'oczi, M. Vas'uth, and L. A. Gergely, Selfinteraction spin effects in inspiralling compact binaries, Phys. Rev. D 71 , 124043 (2005), arXiv:astro-ph/0504538 [astro-ph].</list_item>
<list_item><location><page_16><loc_52><loc_56><loc_92><loc_61></location>[30] J. Vines, E. E. Flanagan, and T. Hinderer, Post-1newtonian tidal effects in the gravitational waveform from binary inspirals, Phys. Rev. D 83 , 084051 (2011), arXiv:1101.1673 [gr-qc].</list_item>
<list_item><location><page_16><loc_52><loc_50><loc_92><loc_56></location>[31] A. Boh'e, S. Marsat, and L. Blanchet, Next-to-nextto-leading order spin-orbit effects in the gravitational wave flux and orbital phasing of compact binaries, Class. Quantum Grav. 30 , 135009 (2013), arXiv:1303.7412 [grqc].</list_item>
<list_item><location><page_16><loc_52><loc_43><loc_92><loc_49></location>[32] A. Boh'e, G. Faye, S. Marsat, and E. K. Porter, Quadratic-in-spin effects in the orbital dynamics and gravitational-wave energy flux of compact binaries at the 3pn order, Class. Quantum Grav. 32 , 195010 (2015), arXiv:1501.01529 [gr-qc].</list_item>
<list_item><location><page_16><loc_52><loc_36><loc_92><loc_43></location>[33] V. Varma, S. E. Field, M. A. Scheel, J. Blackman, D. Gerosa, L. C. Stein, L. E. Kidder, and H. P. Pfeiffer, Surrogate models for precessing binary black hole simulations with unequal masses, Phys. Rev. Res. 1 , 033015 (2019), arXiv:1905.09300 [gr-qc].</list_item>
<list_item><location><page_16><loc_52><loc_31><loc_92><loc_36></location>[34] R. Abbott et al. (LIGO-Virgo Collaboration), GW190521: A binary black hole merger with a total mass of 150 M ⊙ , Phys. Rev. Lett. 125 , 101102 (2020).</list_item>
<list_item><location><page_16><loc_52><loc_29><loc_92><loc_31></location>[35] B. W. Silverman, Density Estimation for Statistics and Data Analysis (Chapman & Hall/CRC, 1986).</list_item>
<list_item><location><page_16><loc_52><loc_26><loc_92><loc_28></location>[36] D. W. Scott, Multivariate density estimation (Wiley, 1992).</list_item>
<list_item><location><page_16><loc_52><loc_22><loc_92><loc_26></location>[37] P. Virtanen et al. , SciPy 1.0: Fundamental algorithms for scientific computing in python, Nature Methods 17 , 261 (2020), arXiv:1907.10121 [cs.MS].</list_item>
<list_item><location><page_16><loc_52><loc_17><loc_92><loc_22></location>[38] E.-J. Wagenmakers, T. Lodewyckx, H. Kuriyal, and R. Grasman, Bayesian hypothesis testing for psychologists: A tutorial on the Savage-Dickey method, Cogn. Psychol. 60 , 158 (2010).</list_item>
<list_item><location><page_16><loc_52><loc_10><loc_92><loc_16></location>[39] R. Abbott et al. (LIGO-Virgo Collaboration), GWTC2.1: Deep extended catalog of compact binary coalescences observed by LIGO and Virgo during the first half of the third observing run (2021), arXiv:2108.01045 [grqc].</list_item>
<list_item><location><page_16><loc_52><loc_9><loc_92><loc_10></location>[40] B. P. Abbott et al. (LIGO-Virgo Collaboration),</list_item>
<list_item><location><page_17><loc_12><loc_89><loc_49><loc_93></location>GW170817: Observation of gravitational waves from a binary neutron star inspiral, Phys. Rev. Lett. 119 , 161101 (2017), arXiv:1710.05832 [gr-qc].</list_item>
<list_item><location><page_17><loc_9><loc_85><loc_49><loc_89></location>[41] V. A. Kosteleck'y and M. Mewes, Testing local Lorentz invariance with gravitational waves, Phys. Lett. B 757 , 510 (2016), arXiv:1602.04782 [gr-qc].</list_item>
<list_item><location><page_17><loc_9><loc_80><loc_49><loc_85></location>[42] N. A. Flowers, C. Goodge, and J. D. Tasson, Superconducting-gravimeter tests of local Lorentz invariance, Phys. Rev. Lett. 119 , 201101 (2017), arXiv:1612.08495 [gr-qc].</list_item>
<list_item><location><page_17><loc_9><loc_76><loc_49><loc_80></location>[43] H. Pihan-le Bars et al. , New Test of Lorentz invariance using the MICROSCOPE space mission, Phys. Rev. Lett. 123 , 231102 (2019), arXiv:1912.03030 [physics.space-ph].</list_item>
<list_item><location><page_17><loc_9><loc_71><loc_49><loc_76></location>[44] J. S. D'ıaz, V. A. Kosteleck'y, and M. Mewes, Testing relativity with high-energy astrophysical neutrinos, Phys. Rev. D 89 , 043005 (2014), arXiv:1308.6344 [astroph.HE].</list_item>
<list_item><location><page_17><loc_9><loc_67><loc_49><loc_70></location>[45] K. N. Lau and M. D. Seifert, Direct-coupling lensing by antisymmetric tensor monopoles, Phys. Rev. D 95 , 025023 (2017), arXiv:1309.2241 [hep-th].</list_item>
<list_item><location><page_17><loc_9><loc_63><loc_49><loc_67></location>[46] V. A. Kosteleck'y and J. D. Tasson, Constraints on Lorentz violation from gravitational ˇ Cerenkov radiation, Phys. Lett. B 749 , 551 (2015), arXiv:1508.07007 [gr-qc].</list_item>
<list_item><location><page_17><loc_9><loc_58><loc_49><loc_63></location>[47] A. Zonca, L. Singer, D. Lenz, M. Reinecke, C. Rosset, E. Hivon, and K. G'orski, Healpy: equal area pixelization and spherical harmonics transforms for data on the sphere in Python, J. Open Source Softw. 4 , 1298 (2019).</list_item>
<list_item><location><page_17><loc_9><loc_50><loc_49><loc_57></location>[48] K. M. G'orski, E. Hivon, A. J. Banday, B. D. Wandelt, F. K. Hansen, M. Reinecke, and M. Bartelmann, HEALPix: A framework for high-resolution discretization and fast analysis of data distributed on the sphere, Astrophys. J. 622 , 759 (2005), arXiv:astro-ph/0409513 [astro-ph].</list_item>
<list_item><location><page_17><loc_9><loc_46><loc_49><loc_49></location>[49] G. H. Golub and C. Reinsch, Singular value decomposition and least squares solutions, Numer. Math. 14 , 403 (1970).</list_item>
<list_item><location><page_17><loc_9><loc_44><loc_49><loc_45></location>[50] I. Mandel, W. M. Farr, and J. R. Gair, Extracting distri-</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_17><loc_55><loc_89><loc_92><loc_93></location>bution parameters from multiple uncertain observations with selection biases, Mon. Not. R. Astron. Soc. 486 , 1086 (2019), arXiv:1809.02063 [physics.data-an].</list_item>
<list_item><location><page_17><loc_52><loc_83><loc_92><loc_89></location>[51] S. Vitale, D. Gerosa, W. M. Farr, and S. R. Taylor, Inferring the properties of a population of compact binaries in presence of selection effects, in Handbook of Gravitational Wave Astronomy (Springer Singapore, Singapore, 2020) pp. 1-60, arXiv:2007.05579 [astro-ph.IM].</list_item>
<list_item><location><page_17><loc_52><loc_77><loc_92><loc_82></location>[52] E. Payne, S. Banagiri, P. D. Lasky, and E. Thrane, Searching for anisotropy in the distribution of binary black hole mergers, Phys. Rev. D 102 , 102004 (2020), arXiv:2006.11957 [astro-ph.CO].</list_item>
<list_item><location><page_17><loc_52><loc_72><loc_92><loc_77></location>[53] C. Messick et al. , Analysis framework for the prompt discovery of compact binary mergers in gravitational-wave data, Phys. Rev. D 95 , 042001 (2017), arXiv:1604.04324 [astro-ph.IM].</list_item>
<list_item><location><page_17><loc_52><loc_66><loc_92><loc_72></location>[54] G. S. Davies, T. Dent, M. T'apai, I. Harry, C. McIsaac, and A. H. Nitz, Extending the PyCBC search for gravitational waves from compact binary mergers to a global network, Phys. Rev. D 102 , 022004 (2020), arXiv:2002.08291 [astro-ph.HE].</list_item>
<list_item><location><page_17><loc_52><loc_62><loc_92><loc_65></location>[55] W. M. Farr, Accuracy requirements for empirically measured selection functions, Res. Notes Am. Astron. Soc. 3 , 66 (2019), arXiv:1904.10879 [astro-ph.IM].</list_item>
<list_item><location><page_17><loc_52><loc_55><loc_92><loc_61></location>[56] B. P. Abbott et al. (LIGO-Virgo Collaboration), Binary black hole population properties inferred from the first and second observing runs of Advanced LIGO and Advanced Virgo, Astrophys. J. 882 , L24 (2019), arXiv:1811.12940 [astro-ph.HE].</list_item>
<list_item><location><page_17><loc_52><loc_50><loc_92><loc_55></location>[57] R. Abbott et al. (LIGO-Virgo-KAGRA Collaboration), Population of merging compact binaries inferred using gravitational waves through GWTC-3, Phys. Rev. X 13 , 011048 (2023), arXiv:2111.03634 [astro-ph.HE].</list_item>
<list_item><location><page_17><loc_52><loc_47><loc_92><loc_49></location>[58] D. Foreman-Mackey, corner.py: Scatterplot matrices in python, J. Open Source Softw. 1 , 24 (2016).</list_item>
</document>
[
{
"title": "Measuring Gravitational Wave Speed and Lorentz Violation with the First Three Gravitational-Wave Catalogs",
"content": "Anarya Ray , 1, ∗ Pinchen Fan , 2 Vincent F. He , 2 Malachy Bloom , 2 Suyu Michael Yang, 2 Jay D. Tasson , 2, † and Jolien D. E. Creighton 1 1 Department of Physics, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA 2 Department of Physics and Astronomy, Carleton College, Northfield, MN 55057, USA The speed of gravitational waves v g can be measured with the time delay between gravitationalwave detectors. Our study provides a more precise measurement of v g using gravitational-wave signals only, compared with previous studies. We select 52 gravitational-wave events that were detected with high confidence by at least two detectors in the first three observing runs (O1, O2, and O3) of Advanced LIGO and Advanced Virgo. We use Markov chain Monte Carlo and nested sampling to estimate the v g posterior distribution for each of those events. We then combine their posterior distributions to find the 90% credible interval of the combined v g distribution for which we obtain 0 . 99 +0 . 02 -0 . 02 c without the use of more accurate sky localization from the electromagnetic signal associated with GW170817. Restricting attention to the 50 binary black hole events generates the same result, while the use of the electromagnetic sky localization for GW170817 gives a tighter constraint of 0 . 99 +0 . 01 -0 . 02 c . The abundance of gravitational wave events allows us to apply hierarchical Bayesian inference on the posterior samples to simultaneously constrain all nine coefficients for Lorentz violation in the nondispersive, nonbirefringent limit of the gravitational sector of the Standard-Model Extension test framework. We compare the hierarchical Bayesian inference method with other methods of combining limits on Lorentz violation in the gravity sector that are found in the literature.",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "The third observing run (O3) of Advanced LIGO [1] and Advanced Virgo [2] was the first complete run in which all three detectors were used [3, 4]. In total, O3 adds 79 gravitational-wave (GW) candidates, more than seven times the 11 GW candidates from the first (O1) and second (O2) observing runs combined [5]. With the availability of many more GW events, it becomes possible to measure the speed of gravitational waves v g more precisely than previous works that used similar methods [6, 7]. Furthermore, it allows a direct and comprehensive exploration of the isotropy of v g for the first time. General relativity predicts that the speed of GWs is the same as the vacuum speed of light c . The GWs detected by Advanced LIGO and Advanced Virgo can be used to make statistical inferences about v g , thereby testing the theory of general relativity. The first measurement of v g using the time delay between the GW detectors was performed by Ref. [6]. By applying Bayesian inference, the 90% credible interval of v g distribution was constrained to be (0 . 55 c, 1 . 42 c ) [6]. Reference [7] further constrained the 90% credible interval to (0 . 97 c, 1 . 05 c ), by applying similar methods to 11 events from O1 and O2. With a total of 52 high-confidence multi-detector GW events accrued through the end of O3, we are able to perform a similar analysis using more events, more robustly testing the theory of general relativity. While the method used here remains considerably less sensitive than a multimessenger astronomy approach in Ref. [8], the latter remains but one measurement, and the approach used here offers the robustness of multiple measurements attained in the context of an independent method. The large body of events now available, which arrive from a multitude of sky directions, allows for a complete exploration of the isotropy of v g in the context of the Lorentz invariance test framework provided by the gravitational Standard-Model Extension (SME). 1 Reference [7] simultaneously constrained the first four of nine coefficients for Lorentz violation in the nondispersive, nonbirefringent limit of the gravity sector using four GW events from O1 and O2. Other recent works [13-15] have sought the effects of birefringence and dispersion using the SME. In this paper, we use 24 of 52 high-significance multi-detector GW events to simultaneously constrain all nine coefficients in the nondispersive, nonbirefringent limit. While our constraints are much weaker than previous works such as Ref. [8], which have constrained the coefficients for Lorentz violation in the gravity sector down to the order of 10 -15 to 10 -14 via multimessenger astronomy, these constraints were obtained using models with only one parameter each. Therefore, our work is the first to provide direct limits from GW observations on all nine coefficients simultaneously. The remainder of this paper is organized as follows. In Sec. II, we discuss the methods used to extract v g estimates for each event and present the results. Section III presents and compares a number of methods for extracting simultaneous limits on the nine coefficients for Lorentz violation before presenting our final estimate of these coefficients from the O1-O3 data.",
"pages": [
1,
2
]
},
{
"title": "A. Bayesian Inference Methods",
"content": "Here, we briefly describe our method for obtaining the speed of GWs. Interested readers are invited to refer to Ref. [7] for full details. When a GW passes through Earth, if two or more detectors detect the signal, we can use the relative locations of the detectors and the difference in detection times from those detectors to simultaneously estimate the sky location of the GW event and v g . With only one detector, we cannot find any v g information, as there is no difference in detection times in this case. Therefore, we select those events that are detected by at least two GW detectors. Furthermore, we only consider those events whose median signal-to-noise ratios (SNR) are no smaller than 10.0, as reported in the GWTC-2 and GWTC-3 catalog papers [3, 4]. In total, 41 O3 events meet our selection criteria and are listed in Tables I and II. All O1 and O2 events meet these two selection criteria, so we include their posterior distributions used in Ref. [7] in our analysis. Note that the SNR values used to select the O1 and O2 events (which is the same as what was used in Ref. [7]) correspond to the network SNR with which the events were found by the GstLAL search pipeline as reported in Ref. [5]. The standard parameter estimation using GW data from multiple detectors imposes the constraint that GWs travel at the speed of light [16]. In this work, we remove this constraint such that v g becomes a parameter to be estimated with all other signal parameters. This causes wider distributions for certain parameter estimations. For example, the calculated sky area is often larger because a defined v g aids sky localization. Gravitational wave data d , can be decomposed into a pure GW signal h ( t ) plus random noise n ( t ), Within the framework of Bayesian inference, the posterior distribution of the parameters ⃗ θ characterizing a GW signal is computed from the likelihood of obtaining GW data given particular values of said parameters and the a priori knowledge of what we expect those values to be. The likelihood function is constructed by assuming the noise n ( t ) to be stationary and Gaussian distributed. For details regarding the exact forms of the likelihood see Ref. [7]. Once obtained, the joint posterior distribution of the signal parameters can be used to compute the marginalized posterior distribution of v g as in: where ⃗ θ ' is the set of parameters in ⃗ θ except for v g [7]. To carry out PE for each event that passes our selection criteria, we use public data [17, 18] from GWTC-1 through GWTC-3. We use lalinference mcmc [16, 1921] which implements MCMC with Metropolis-Hastings algorithm and lalinference nest which implements nested sampling to run the Bayesian parameter estimation [16, 22, 23]. For our purposes of extracting v g distributions, these two algorithms generate comparable results. We use the publicly available power spectral densities and calibration envelopes from the LIGO Scientific, Virgo and KAGRA (LVK) collaborations in our analysis. In this paper, we use a uniform prior in v g between 0 . 1 c and 10 c . When the v g posterior rails against the prior, we increase the upper limit of the prior by another 10 c . The broadest prior we use is from 0 . 1 c to 30 c , which we only use for one event, GW190929 012149. For parameters such as binary masses and spins, we use the same uniform and isotropic priors as those used by the LVK [3-5]. We choose a distance prior that is proportional to luminosity distance squared, similar to Ref. [5]. We do not use the more complicated cosmological priors used by Refs. [3, 4]. For O1 and O2 events, we use the posterior samples from Ref. [7], which used the IMRPhenomPv2 [24-26] waveform for all events except for the binary neutron star (BNS) event GW170817 which was analyzed with the TaylorF2 waveform [27-32]. For most O3 events, we use the IMRPhenomD waveform [24, 25], which is an aligned spin waveform model for black-hole binaries. We do not use the more sophisticated IMRPhenomPv2 model for these events since in the context of our study, we do not expect any siginificant change in v g measurements to result from the additional intricacies of the more sophisticated model. We have verified this lack of change for a subset of these events and hence chosen to stick to the IMRPhenomD model consistently for all O3 events except for GW190521. For the extremely high-mass BBH event GW190521, we use the NRSur7dq4 waveform [33] which is one of the waveform models used by Ref. [34] for inferring this event's source properties. We note that IMRPhenomPv2, IMRPhenomD, and NRSur7dq4 are all waveform models with inspiral, merger as well as ringdown. We can achieve a more precise measurement of v g by combing data from multiple GW events. By interpreting each observation as an independent experiment, we can multiply the marginalized likelihood as a function of v g corresponding to each event and obtain the joint posterior distribution of v g given data from multiple events. For a uniform prior on v g , the joint posterior can be expressed as a product of individual event posteriors. Suppose the GW detectors observe n independent GW events with data d 1 , d 2 , ..., d n . For a uniform prior distribution of v g , the combined posterior distribution of v g is The single event posterior distributions p ( v g | d i ) are ob- tained as a numerical function of v g from its PE samples by means of Gaussian Kernel Density estimation (KDE) [35, 36]. We use the package Scipy 's implementation of Gaussian KDE to obtained the posteriors [37]. The joint posterior distribution is then obtained through Eq. (3). Then, for individual and combined posteriors, we calculate Bayes factors K , via the Savage-Dickey density ratio where p ( v g = c | d 1 , d 2 , ... ) is the posterior probability of v g = c , and p ( v g = c ) is the prior probability of v g = c [38]. Higher Bayes factors suggest stronger evidence for v g = c .",
"pages": [
2,
3
]
},
{
"title": "B. Results",
"content": "In Tables I and II, we show the v g estimates with 90% credible intervals, network SNRs, sky areas at 90% credible level, and Bayes factors for these selected 41 O3 events. Also shown are the analogous quantities obtained from their combined posteriors. Out of the 41 selected O3 events, 40 events are binary black hole (BBH) candidate events. GW200115 042309 is a neutron starblack hole (NSBH) event, with masses of 5 . 9 +2 . 0 -2 . 5 M ⊙ and 1 . 44 +0 . 85 -0 . 29 M ⊙ at 90% credible interval [4]. Here, by combining the 41 selected O3 events, we constrain the 90% credible interval of v g to be 0 . 99 +0 . 02 -0 . 03 c , with a Bayes factor of 205 . 9. We combine the O3 results with the O1 and O2 results discussed in Ref. [7]. The eleven O1 and O2 events are run with lalinference mcmc , which shows results that are consistent with lalinference nest used for O3a runs [3, 5, 7]. In Table III, we show the v g estimates with 90% credible intervals, network SNRs, sky areas at 90% credible level, and Bayes factors for the 11 O1 and O2 events and their combined posteriors. We use the same posterior samples as used by Ref. [7], but Table III shows slightly different 90% credible intervals from those in Ref. [7], because we use Gaussian KDE smoothing in this study to extract the credible intervals while Ref. [7] directly used the posterior samples without KDE smoothing [7]. These 11 events were detected by at least two detectors and had median GstLAL network SNR values greater than 10.0 [5]. GWTC-2.1 [39] shows network SNR values for O1 and O2 events based on lalinference parameter estimations, but we choose GstLAL SNR values to be consistent with Ref. [7] from which we obtain the v g posterior samples. GW170817 is a BNS event that was also detected in the electromagnetic spectrum [8, 40]. The 'fixed' label means that the result uses the sky localization from the electromagnetic detections, which is much more precise than the localization generated by GW detection pipelines. Combining the 41 O3 events and 11 O1 and O2 events without fixing GW170817's sky localization at the de- tected EM signal, we obtain the 90% credible interval of v g to be 0 . 99 +0 . 02 -0 . 02 c , with a Bayes factor of 291 . 9. With GW170817 sky localization fixed, v g is 0 . 99 +0 . 01 -0 . 02 c , with a Bayes factor of 249 . 0. For a total of 49 BBH events, i.e. excluding GW170817, GW190924, and GW200115, v g is 0 . 99 +0 . 02 -0 . 02 c , with a Bayes factor of 221 . 2. FIG. 1 shows the combined posterior of v g .",
"pages": [
3
]
},
{
"title": "C. Discussion",
"content": "In Ref. [7], with 11 O1 and O2 events and GW170817's sky localization unfixed, the combined posterior distribution of v g was measured to be 1 . 01 +0 . 04 -0 . 05 c , while here we measure v g to be 0 . 99 +0 . 02 -0 . 02 c with the 52 selected events. With GW170817's localization fixed, in Ref. [7], the combined posterior distribution of v g was measured to be 0 . 99 +0 . 02 -0 . 02 c for 11 events, while here we find 0 . 99 +0 . 02 -0 . 01 c for the 52 events. Given that 1 c is the relativistic prediction of v g , the combined posterior distributions of v g measured using 52 selected events show no evidence for a violation of general relativity. All of these combined results have Bayes factors on the order of 10 2 , providing strong evidence for v g = c . Here, our measured distribution of v g is much narrower than that measured with 11 O1 and O2 events in Ref. [7] using GW signals alone. This is reasonable, given the larger sample size of events included in this study. When we assume that the v g distributions of individual events are independent and identically distributed, we expect that the measurement errors would decrease by 1 / √ n . In our calculations, we find that the combined v g distribution roughly follows such a pattern as more events are added. For example, with 11 O1 and O2 events, the combined v g posterior had an error bar of 0 . 09 c . With 52 events in total, the combined posterior had an error bar of 0 . 04 c , which follows 0 . 09 c/ √ 52 / 11 ≈ 0 . 04 c . With GW170817's sky localization unfixed, we find that Bayes factor more than doubles from the value of 149 . 0 obtained from 11 O1 and O2 events to the value of 291 . 9 obtained with all 52 events. This, in conjunction with the error bar being reduced by half, implies that our measurement with 52 GW events in total has provided approximately twice stronger evidence for v g = c . Interestingly, we find that the combined 90% v g credible interval using the 41 O3 events is approximately the same as the 90% v g credible interval obtained by only considering GW170817 with the fixed sky localization. GW170817 had an SNR of 33 . 0, while only four of the 41 O3 events had SNRs above 20 . 0, with the highest being 26 . 8 for GW200129 065458. The similarity between the v g posterior of GW170817 alone and the 41 O3 events combined suggests that some combination of higher SNRs and better sky localization do help put tighter constraints on v g . This shows that our decision to exclude events with SNRs lower than 10 . 0 should not have a high impact on the v g estimates. Looking to the future, additional two- and threedetector BBH events with SNRs typical of those above will lead to a slow improvement in v g measurements as improvements proceed as 1 / √ n . However, as GW detectors become more sensitive and the network of detectors expands, we expect more high-SNR, multi-detector GW events that would likely lead to a more rapid pace of progress in v g estimations via the methods used here. Meanwhile, future multimessenger detections can provide more precise sky localizations, which will likely improve the error bars on the 90% v g credible interval further.",
"pages": [
4,
5
]
},
{
"title": "A. Basics",
"content": "In the non-birefringent, non-dispersive limit of the SME (mass dimension d = 4), using natural units and assuming that the nongravitational sectors, including the photon sector, are Lorentz invariant, the difference between the group velocities of gravity and light takes the form [41]: where the Y lm 's are the spherical harmonics with l ≤ 2. Here the nine Lorentz-violating degrees of freedom are characterized by the spherical coefficients for Lorentz violation ¯ s lm , and ˆ n is the sky location of the source of the GWs. We can expand Eq.(5) over positive m to get its equivalent expression: The SME is a broad and general test framework for testing Lorentz invariance. Unlike models that attempt to describe specific effects with a small number of parameters, test frameworks, because of their generality, have a large number of undetermined coefficients to be explored in experimental data. While a number of studies have proceeded under a simplified approach, sometimes referred to as a maximum reach analysis [42], in which only one coefficient at a time is considered, it is also common to study a family of coefficients together in what is sometimes referred to as a coefficient separation approach [42]. In the context of the maximum reach approach, many coefficients can sometimes be constrained one at a time using a single measurement, while a number of measurements that is greater than or equal to the number of coefficients considered is typically required to simultaneously measure the entire family. A number of approaches to simultaneously estimating multiple coefficients exist in the literature. One approach involves directly fitting a single data stream to a model involving all of the coefficients in the family. 2 This approach is well-suited to experiments that take data as the lab is boosted and rotated. In the context of astrophysical observations, each individual event provides a measurement of a linear combination of coefficients for Lorentz violation. A system of these inequalities must then be solved, or otherwise disentangled, for estimates of the coefficients for Lorentz violation. Several methods of addressing this issue exist in the literature. In this section, we will compare the implications of several of these approaches in the context of the speed of gravitational wave data, as well as introduce new methods based on hierarchical Bayesian inference. Our goal is to consolidate information about these methods and help illuminate their relative merits. We achieve that goal by performing a Mock Data Challenge (MDC), where-in we generate synthetic data corresponding to a chosen set of 'true' values of the SME coefficients and test the efficacy of each method in recovering the true values from the synthetic data. Given their performance in the MDC, along with other considerations, we choose one of these methods whose merits outweigh that of the others and use it to analyze the real speed of gravity data from a subset of the events analyzed in Sec. II to generate the final results of our SME analysis. Because well-localized events are most informative for the SME analysis, we choose the 24-event subset of those considered in Sec. II with 90% credible posterior sky areas under 2000 square degrees as obtained from our parameter estimation with v g as a free parameter.",
"pages": [
5,
6
]
},
{
"title": "B. Linear Programming Method",
"content": "A number of past studies that have performed maximum reach analysis using limits from astrophysical events have taken a linear programming approach. See, for example, Refs. [44-46]. The basic idea translated to the speed of gravitational waves problem proceeds as follows. From a given event we have an upper and lower bound on ∆ v . If we suppose that we know an exact sky location, as is effectively the case for GW170817 when the electromagnetic signal's localization is used, then Eq. (6) can be understood as generating a pair of hyperplanes in s lm space that are the boundaries of the parameter space excluded by the event. A subsequent event at a different sky location will generate a distinct pair of hyperplanes. Once a set of n events are collected at distinct sky locations, where n is greater than or equal to the dimensionality of the coefficient space, then a finite maximum and minimum allowed value for each coefficient can be identified via a linear programming scheme such as the simplex method. In the applications of Refs. [44-46], the sky localizations were sufficiently well known that analysis could proceed directly via the above prescription. In the current problem, for all events except GW170817, the sky localization is comparatively poorly known. This makes the slopes of the hyperplanes bounding the allowed region poorly known. To address our uncertainty in sky positions, the linear programming scheme can be adapted as follows. The linear programming process can be applied with all possible hyperplanes generated by samples from our inference that fall within the 68% credible sky localization bands. The worst-case limits generated by the set of linear programming analysis can then be taken as bounds. As might be expected, this method generates very conservative bounds relative to the methods to follow. Testing this approach using four test events and a skymap resolution of N side = 64, which corresponds to 12 × 64 2 = 49152 pixels on the celestial sphere [47, 48], we generate bounds that are about an order of magnitude greater than the 1 σ credible intervals found via the application of the random draw method that we present in the next subsection. Hence we do not consider this approach further as a method of extracting SME limits from the speed of gravitational waves data at this time.",
"pages": [
6
]
},
{
"title": "C. Random Draw Method",
"content": "In Ref. [7] the random draw method for extracting simultaneous limits on coefficients for Lorentz violation was first used. In that work, simultaneous limits were achieved for the set of four l = 0 and l = 1 s lm coefficients using the 4 high-confidence, well-localized events available at the time. In this section, we review this method and discuss ways of extending it to cases in which the number of events exceeds the number of coefficients to be estimated. The result of the inference discussed in Sec. II A is a set of samples with each sample consisting of values for each of the parameters including the speed of GWs and the sky localization. Hence distributions for each of the sampled parameters are generated. If one randomly draws one sample associated with each event, one can then solve for the coefficients for Lorentz violation that are consistent with that set of samples using Eq. (6). The process of randomly drawing one sample from each event and solving for the coefficients can be iterated to build up a set of samples for the s lm coefficients. In other words, a set of points in s lm space is built up. The process described above is straightforward when the number of events observed is equal to the number of coefficients for Lorentz violation to be estimated. Furthermore, in such a scenario, quantile ranges of s lm computed from the set of samples of s lm , accurately represent the uncertainty in our measurement of the SME coefficients. This is because using one posterior sample of (∆ v, θ, ϕ ) from each event and exactly calculating s lm from them by solving a set of non-degenerate linear equations, is equivalent to computing and multiplying the posterior distributions of s lm for each event and then drawing one sample from that joint posterior. However, in the case where the number of GW observations exceeds the number of SME coefficients, the linear equations become degenerate and hence no longer exactly solvable. While one can be tempted to cherry-pick the top 9 events with the highest SNRs and lowest sky areas from the set of observations and perform random draw on those, such an analysis will not be maximally informative given the data we have. We can do better using Bayesian hierarchical inference techniques which can combine information from a large number of events, producing much more informative bounds on the SME coefficients with accurate estimation of measurement uncertainties. Before discussing our robust Bayesian methods we show how the random draw method can be extended to the case in which the number of observations exceeds the number of coefficients for Lorentz violation by means of Singular Value Decomposition (SVD). However, this extension of the random draw method is susceptible to the limitations of the approximation used to perform the SVD and hence cannot produce reliable uncertainty estimates for the measured Lorentz violation parameters. We elaborate more on this near the end of this section while informing the reader beforehand that this SVD-assisted random draw generalization is useful in the present context only as a consistency check and an optimization tool for the hierarchical Bayesian methods on which we rely for our final results. For n SME number of Lorentz Violation coefficients and N E number of events with n SME < N E , for each random draw, we need to solve the degenerate system of linear equations: Here A is an N E × n SME matrix in which each row corresponds to one of the N E events under consideration. The entries in each of the n SME columns moving across a given row consist of the coefficients of ¯ s lm in Eq. (6), computed for a random sample of θ, ϕ drawn from the event corresponding to that row. The n SME SME coefficients to be computed are organized into a column vector denoted [ ¯ s lm ], while [ ∆ v ] denotes a column vector of the randomly drawn ∆ v corresponding to the samples used in constructing the rows of A . Before factorizing the nonsquare matrix, we scale both sides of each line of Eq. (7) by the standard deviation of the ∆ v samples corresponding to that event. We define [ σ ∆ v ] to be a column vector in which each element corresponds to the standard deviation of the ∆ v samples from that particular event, then we can write the scaled version of Eq. (7) as: where The SVD factorizes the non-square matrix A ' into two orthogonal square matrices U and V , that are N E × N E and n SME × n SME respectively, and a diagonal N E × n SME matrix [Σ] with non-negative entries: where Σ has the form: with The non-negative values σ 1 > σ 2 > ... > σ n SME are known as singular values and are estimated along with U and V by a linear least squares algorithm[49]. The scaling with the standard deviation of ∆ v essentially transforms a least square minimized SVD on A ' into a Chisquare minimized SVD on A . This allows us to properly account for the fact that some events in our list are less significant than others. Proceeding without this scaling biases the SVD. Once computed, the singular values can be used to solve for ¯ s lm in Eq. (7) : for each draw. We can then estimate the densities of the SME parameters from all draws and produce constraints on them. We note that despite being a computationally cheap method for computing constraints on the SME coefficients from multiple GW events, the SVD-assisted random draw method has certain inadequacies. There is ambiguity in the exact meaning and interpretation of the uncertainty estimates produced by this method. In the case where the number of events is larger than the number of SME coefficients, this implementation of the random draw method boils down to randomly choosing a posterior sample of (∆ v, θ, ϕ ) from each event and doing a least chi-square fit for the SME parameters. This procedure is then repeated a large number of times, producing a least chi-square fit of the SME coefficients for each draw. However this is not equivalent to the multiplication of posterior probabilities of the SME coefficients, over all events, and drawing samples from that joint posterior. Thus the quantile ranges of the set of chi-square fitted SME coefficients do not hold the same meaning as Bayesian credible intervals. While the Bayesian intervals represent regions of the SME parameter space wherein their true values lie with a particular posterior probability given the data, the SVD-based random draw constraints can be expected to have a different meaning, the exact nature of which remains ambiguous. Due to these considerations, we conclude that the weighted SVD-assisted random draw method produces constraints that are unreliable and are likely to be underestimates of the true uncertainties in the measurement of SME coefficients. We verify this claim by testing this method against its Bayesian counterparts in an MDC that we describe later in this work. The results of the MDC show that the samples of SME parameters produced by this method are concentrated in a narrow region around the true values of the parameters, which also coincide with the peaks of the posterior distributions inferred by the Bayesian methods. Therein lies the merit of this method in the present context and its potential to serve as a rapid consistency check for the Bayesian methods. Furthermore, this method is extremely fast and computationally cheap and hence can be used to quickly find the narrow region in the parameter space inside which the peak of the posteriors lies. The stochastic MCMC sampling employed by our Bayesian methods is expected to converge much faster if the MCMC chains are initialized near the maxima of the posterior being sampled. Thus the SVD-assisted random draw method can be used to optimize the MCMC sampling used in our Bayesian methods with significant speed-up gains for narrowly peaked SME posteriors. Given the large number of events expected to be observed in O4 and the width of the Bayesian intervals we compute using our current set of events, the posterior distributions of the SME coefficients can be expected to be very narrow post O4, and hence lead to a drastic increase in the computational cost and latency of the Bayesian methods being applied to such a data set. This will likely make the optimization of the Bayesian methods as offered by the SVD-assisted random draw method a necessary tool in the near future.",
"pages": [
6,
7,
8
]
},
{
"title": "D. Hierarchical Bayesian Inference",
"content": "Since the SME coefficients are properties that are expected to be the same for all events, one can perform Bayesian Hierarchical Inference on them from the GW data of multiple events. To do so, we can construct the marginalized likelihood of GW data given a particular value of the SME coefficients, jointly from multiple events where the SME sensitive part of the prior imposes the relationship (6) on ∆ v, θ, ϕ for a given value of the SME coefficients : Here ∆ v (¯ s lm , θ, ϕ ) is the right hand side of Eq. (6). Note that we have chosen to represent the deviation of the speed of gravity from the speed of light by the dummy variable ∆ v ' whenever a probabilistic quantity (such as likelihood, posterior, prior, or detection fraction) is expressed as a function of it, so as to distinguish it from the quantity ∆ v (¯ s lm , θ, ϕ ). The presence of the delta function in Eq. (16) is due to the deterministic nature of the Eq. (6). By Bayes' theorem, for a uniform prior on ¯ s lm , the likelihood L (¯ s lm ) is proportional to the posterior of these parameters given GW data. We can now sample this posterior using MCMC to produce joint SME constraints from multiple GW observations. However, this procedure involves a very large number of evaluations of the likelihoods L ( d i | ∆ v ' , θ, ϕ ) which is so computationally expensive that it's practically infeasible. To get around this problem, one can again use Bayes' theorem to write the likelihood L ( d i | ∆ v ' , θ, ϕ ) as proportional to the ratio of the posterior p (∆ v ' , θ, ϕ | d i ) to the prior: Substituting this into Eq. (15) gives us: We can now use the samples drawn from the posterior p (∆ v ' , θ, ϕ | d i ) obtained using the parameter estimation run described above to evaluate the integral in Eq. (18). Note that we have ignored a factor of 1 /π (∆ v ' ) in Eq. (18) which is constant since we choose π (∆ v ' ) to be uniform in our parameter estimation runs. However, the presence of the Dirac delta makes it slightly complicated to evaluate this integral directly as a sum over posterior samples. We describe shortly two approximation schemes that can be used to smooth out the discrete sum of Dirac deltas over posterior samples that would entail the evaluation of the integral in Eq. (18) and hence constrain the SME coefficients jointly from multiple GW observations. Before that, we first describe why Bayesian Inference of this form is subject to selection biases and how we account for them. Bayesian Hierarchical Inference from a set of GW events selected based on a particular criterion introduces selection biases into the inferred posterior distribution of hyper-parameters [50, 51]. Since we are selecting events based on whether they were found with a Signal to Noise Ratio (SNR) greater than some threshold in at least 3 detectors, and since each detector has an antenna pattern that makes it more sensitive to certain sky directions than others at the time of detection[52], our analysis might be biased towards some values s lm against others. Particularly, the fact that GW search pipelines such as GsTLAL only report multi-detector coincidences based on whether or not the time-delays between the detectors being triggered are smaller than the light travel time between detectors plus a 5 millisecond window, has the potential to bias our results greatly [53]. Furthermore, non-coincident events are down-ranked in significance by means of single's penalties [53], making events even less likely to be detectable for certain cases. Other pipelines such as PyCBC use similar methods for identifying multi-detector coincidences albeit with a different value for the timing error window (which is 2 milliseconds for PyCBC [54]). The existence of this restriction for coincidence formation in search pipelines implies that we are more likely to discover a multi-detector event if the speed of gravitational waves is greater than or equal to c , as compared to if it were lower than c . Thus, our speed of gravitational wave measurements may be biased towards measuring ∆ v ≥ 0 against ∆ v < 0 along any particular sky position. To account for this bias, we must normalize our hierarchical likelihood over the true rate of events as opposed to the detected rate, with the latter being different from the former, due to selection biases. The constant of normalization is the fraction of events that are detectable given a particular value of the hyper-parameters and the detection criteria: d ∆ v ' dθdϕ, where β det (¯ s lm ) = R det (¯ s lm ) R true , the fraction of detectable events is the ratio of the detectable rate of events to the true Rate of events [55]. To calculate the fraction accurately we must simulate a large number of events whose parameters are drawn from broad enough distributions, inject them into the detector noise realizations, and see what fraction of them are recovered given our selection criteria. To do that we must first quantify our selection criteria in terms of the parameters that characterize the GW signal. Accurate modeling would require us to recalculate the search pipeline's ranking statistic of a simulated event while allowing for non-zero ∆ v and to find the corresponding False Alarm Rate(FAR) of that trigger from said ranking statistics. One can then apply a threshold on the combined FAR of the event to classify them as detectable or non-detectable. However, such a calculation would require a pipeline-specific analysis which is beyond the scope of this work. Instead, we use an approximated selection criteria: for the i -th event to be detectable, its recovered parameters must satisfy: where ρ A is the SNR in detector A , ρ net is the network SNR, ∆ t AB (∆ v ) is the time-delay of signal arrival between detectors A and B as a function of ∆ v and ρ th is the SNR threshold used for selecting events. Even though we do not select events depending on which search pipeline found them, we use GstLAL 's timing error window to quantify our selection criteria, instead of say PyCBC's, due to the following reason. Among the events that survive our three detector SNR thresholds, most are found by both GstLAL and PyCBC except for GW170818, GW190701, and GW190814 which are found only by GstLAL . Hence, it is sufficient to model the selection biases that might have appeared in this particular study based on GstLAL's value of the timing error window. This would not have been possible if there were events found by PyCBC and not GstLAL with SNR greater than 10 in three detectors during O3. In such a scenario, a more generalized treatment of selection biases would have been necessary, one that accounts for the difference in timing errors allowed by GstLAL and PyCBC. Now that we have a quantifiable detection criterion, we can carry out our simulations. Once the simulated events are injected into detector noise realizations and classified as detectable or non-detectable depending on their recovered parameters, it is possible to compute the fraction of events detectable given a choice of CBC parameters: Here, ⃗γ are additional CBC parameters such as masses, spins, etc. that characterize the waveform, p (∆ v ' , θ, ϕ, ⃗γ | det ) is the probability of detection, which can be calculated from the set of simulated events that are detectable, and Π sim is the prior from which the simulations are drawn, which has to be broad enough so that we have enough events in both the detectable and non-detectable parts of the parameter space. We can marginalize Eq. (21) over suitable priors to get: If we choose Π sim (∆ v ' , θ, ϕ, ⃗γ ) = π (∆ v ' ) π ( θ ) π ( ϕ )Π( ⃗γ ), where π (∆ v ' ) , π, ( θ ) , π ( ϕ ) are the same as the ones defined in Eqs. (16) and (17), then priors in the denominator and numerator of the integrand in (21) cancel out and we can define the marginalized fraction of detectable events (up to the factors that cancel out later): As in the case of Eq. (18), we have ignored a factor of 1 /π (∆ v ' ) in Eq. (23) for the same reason mentioned before. In terms of this marginalized fraction, β det becomes: To estimate p (∆ v ' , θ, ϕ, ⃗γ | det ) and hence f marg det (∆ v ' , θ, ϕ ) we simulate a large number of events whose parameters are drawn from a broad distribution. We then inject the corresponding signals into detector noise realizations and record their SNRs and arrival times. We then apply our selection criteria to find which of these simulated events are detectable given our criteria and estimate p (∆ v ' , θ, ϕ, ⃗γ | det ). The estimation schemes will depend on which of the two approximations referred to before are used to smooth out the delta function integral and are hence described in more detail in the corresponding subsections below. The priors we use to draw the simulated events are truncated power-law in the primary mass and mass ratio, uniform in spin, sky position, orientation, co-moving volume, geocentric time, and speed of gravitational waves. Particularly, for each observing run, the mass distributions are chosen to be consistent with corresponding population analyses performed by the LVC such that the distributions used have are have support in regions of the mass space where the events being analyzed are found. For O2, we choose p ( m 1 ) ∝ m -1 . 6 1 , m 1 ∈ (7 . 9 M ⊙ , 42 M ⊙ ) and p ( q ) ∝ q 6 . 7 where q = m 2 m 1 which is consistent with Ref. [56], and is identical to the mass distributions used for similar selection function computations [52]. For O3 we choose p ( m 1 ) ∝ m -1 . 6 1 , m 1 ∈ (7 M ⊙ , 80 M ⊙ ) and p ( q ) ∝ q 6 . 7 , which is broad enough for the O3 events as evident from Ref. [57]. In the next two subsections, we describe the details of our smoothing approximations and the computation β det in each approximation scheme.",
"pages": [
8,
9,
10
]
},
{
"title": "1. Narrow Gaussian Method",
"content": "The approach introduced here involves estimating the delta function in Eq. (18) as a narrow Gaussian distribution. For each sample with measured speed difference ∆ v ' and sky location θ and ϕ . We construct a Gaussian distribution for the random variable ∆ v ' -∆ v (¯ s lm , θ, ϕ ) with mean zero and standard deviation σ . Thus, Eq. (19) becomes: where N represents Gaussian distributions. Similarly, we can also apply the Narrow Gaussian approximation to the computation of β det in Eq. (24): Since N is a smooth function of its arguments we can evaluate the two integrals in Eqns. (25) and (26) as a Monte Carlo sum over samples drawn from p (∆ v ' , θ, ϕ | d i ) and f marg det (∆ v ' , θ, ϕ ) respectively. Since we already have posterior samples drawn from p (∆ v ' , θ, ϕ | d i ) for each event during the v g inference described in Sec. II, and since the samples drawn from f marg det (∆ v ' , θ, ϕ ) are the parameters of simulated events that survive our selection criteria, we can compute the log-likelihood of ¯ s lm : where the sum in the numerator is over posterior samples corresponding to the i th event while the one in the denominator is over detectable samples. After choosing a width σ for our Gaussian N appropriately, we can thus use Eq. (27) for fast evaluation of the log-likelihood ln L (¯ s lm ) as a numerical function of the SME coefficients. Hence we can use MCMC algorithms to draw samples from ln L (¯ s lm ) and interpret the quantile ranges of said samples as Bayesian credible intervals of the SME coefficients given GW data. To determine the appropriate width of our Gaussian distribution σ , we consider the effect of varying its size. Because the Gaussian distribution is an estimation of the delta distribution, theoretically, as the size of σ decreases, the approximation should be more accurate. However, because we sample the log-likelihood with an MCMC algorithm, we encounter numerical difficulties when the σ is too small. Thus our choice of σ has to be tuned in accordance with how the MCMC is implemented numerically. In the MCMC process, the walkers only make use of local information at each step. Thus, it is possible for walkers to be trapped inside of islands of high likelihood. This is what happens when σ is set too small. Since most samples have high likelihood around zero, walkers can explore freely the region near zero. However, at more peripheral locations in the parameter space, the peaks are usually scattered. Thus, when σ is too small, these peripheral samples form isolated islands of high likelihood. In this case, the walkers will not be able to explore these isolated islands, resulting in false small constraints. On the other hand, as σ gets larger, our approximation becomes less accurate and distributions are artificially broadened. Therefore, we aim to find a σ such that it is big enough for the walkers to explore the sample space fully and small enough such that it gives us useful results. One way to determine the appropriate value of σ is by applying both the random draw method and the Narrow Gaussian method on the same set of data and comparing the results. We divide our list of events into subsets of nine events and apply both methods to each subset using various sizes of σ . The solid lines in Fig. 2 represent the average uncertainty of the resultant ¯ s lm 's against σ for the nine O3 events from this paper with the smallest sky areas. The average uncertainty is calculated by taking the average of the absolute value of the upper and lower one-standard-deviation value for each ¯ s lm . For the same sets of events, the random draw method produces uncertainties on the order of 10 0 -10 1 , which corresponds to dashed lines in Fig. 2. To select a suitable σ , we use superimposed plots such as Fig. 2 to select a σ such that each uncertainty produced by the Narrow Gaussian method is marginally larger than its counterpart from the random draw method. In the example shown, σ = 0 . 0005 is a good choice because every solid line lies marginally above the corresponding dashed line in the same color, which means that the choice σ does not artificially tighten the constraints. We perform such analysis for every subset of events and produce a 'good σ '. We pick the largest of such 'good σ 's' as the final choice. Even though this final σ produces wider constraints for each subset of events as compared to the random draw method, because the Narrow Gaussian method incorporates information from more than 9 events, ideally we could still produce tighter constraints than the random draw method. Another procedure for determining σ is to use information directly from the distribution of ∆ v ' samples for each event. One specific procedure is to sort the list of ∆ v ' samples and compute the average difference between adjacent values. Choosing σ as half this value produces results that align well with the prior method for the specific distributions tested. That is, the value of σ for a given event is half of the average of the differences between adjacent ∆ v ' samples for that event. This procedure is used in the MDC shown in Fig. 3, and has the advantage of not needing to construct plots to determine σ . However, there still is subjectivity in choosing what fraction of the average to use. Moreover, the distribution and quantity of samples will impact the value of σ . Both considerations will affect the final credible intervals for ¯ s lm . This method performs much better than the SVDassisted random draw in the MDC performed in Sec. III F. However, we note that both processes for choosing σ involve a significant amount of user-controlled fine-tuning and can potentially lead to an over/underestimate of measurement uncertainties of the SME coefficients. Due to these reasons, we do not choose this method for our final results. We instead choose a different smoothing approximation to the Bayesian method by means of Kernel Density Estimation (KDE) which can be shown to produce either equally or more accurate results, while requiring almost no user-controlled fine-tuning.",
"pages": [
10,
11
]
},
{
"title": "2. KDE Methods",
"content": "In this section we outline a different approach from the one in the previous section, to perform Bayesian Hierarchical Inference of the SME coefficients from GW data. In this method, instead of smearing out the delta function in Π(∆ v ' , θ, ϕ | ¯ s lm ) with a Gaussian, we approximate the marginalized posterior of ∆ v, θ, ϕ given GW data, for each event, as a fast evaluating function of these quantities, from their single event PE samples via Gaussian KDE. The KDE approximation of the posterior is constructed by fitting a multi-variate Gaussian around each posterior sample and then writing the estimate of the posterior as a sum of these individual Gaussians. The covariance matrix of each of the Gaussians is approximated from the sample covariance matrix of the posterior samples themselves up to a constant of proportionality. The constant of proportionality is known as the bandwidth of the estimator and is computed, under reasonable assumptions regarding the true distribution being estimated(see [36]). We use SciPy's Gaussian KDE algorithm to obtain our estimate of the marginalized posterior as a fast evaluating function p KDE , i (∆ v ' = ∆ v ( θ, ϕ, ¯ s lm ) , θ, ϕ ) of the relevant parameters [37]. We then perform the ∆ v ' inte- ) analytically using the delta function and compute the remaining two integrals (over θ, ϕ ) numerically using the trapezoidal rule. We loop over multiple events by multiplying the value of the integral obtained using the KDE corresponding to each event, to evaluate L (¯ s lm ) : We then sample from it using the same MCMC method described in the previous section to constrain the SME coefficients. To incorporate selection effects in the KDE method, we estimate f marg det (∆ v ' , θ, ϕ | det ) by performing a KDE on the samples of (∆ v ' , θ, ϕ ) for which the simulated events are detectable given our detection criteria. By restricting our KDE to only these parameters and ignoring other parameters that characterize a simulated event, we effectively marginalize over those other parameters thus implicitly performing the integral in Eq. (23): where the subscript det in p KDE,det represents the fact that this KDE was performed on only those samples for which the simulated events are detectable given our detection criteria. Substituting into Eq. (24) we get: We note that the KDE's bandwidth acts like a control parameter with potential room for user-controlled finetuning in the computation of its value, somewhat analogous to the σ of the narrow Gaussian method. However, unlike the narrow Gaussian method where σ can in principle be chosen to be anything, the bandwidth of the KDE is computed directly from the properties of the samples (such as the number of samples and dimensionality of the parameter space) under reasonable assumptions regarding the true density. Thus the user's choice is restricted to a number of discrete such assumptions (for example Scott's rule [36], Silverman's rule [35] etc.). Furthermore, the effects of choosing a bandwidth on the estimated density (and hence the remainder of the inference) is limited in the sense that the covariance matrix of the Gaussians is determined from the samples themselves with the bandwidth only acting as a scaling parameter that is usually of order unity. This additionally restricts the effects of user-controlled fine tuning on the inference as compared to the narrow Gaussian method wherein the width of the Gaussian that approximates the delta function is completely determined by the user's choice. A more detailed discussion of this comparison between the two methods in the context of the MDC can be found in Sec. III F . For our chosen bandwidth approximation scheme (Scott's rule) the KDE method can be seen to perform extremely well in the MDC. For these reasons, we choose the KDE method for our final results on the SME constraints. We present the results of the KDE method upon its use in the analysis of the events marked * listed on tables I through III (except for GW170817 for which we do not use the fixed posteriors due to the inability of the KDE to estimate very narrow densities) in Fig. 4 as our final result.",
"pages": [
11,
12
]
},
{
"title": "E. Discussion",
"content": "The random draw method, which was originally presented in [7], is very time efficient. However, the number of events that can be used in the analysis is limited by the number of ¯ s lm to be estimated. Hence, it is only useful in the scenario where in we have exactly the same number of events available to be used in the analysis as the number of ¯ s lm coefficients being simultaneously measured. On the other hand, Unlike any of the following methods, this method does not involve any approximation of the delta distribution. Thus, this method can be used as a quick feasibility test. For example, we used this method to estimate the size of σ for the narrow Gaussian approach. With the use of SVD, we were able to take data from a larger set of events. However, the presence of one 'bad' event, i.e. a low-significance event with biased posteriors, can disturb the entire analysis since all events are given equal weights. With the SVD chi-square method, this problem is solved by weighting each event with its uncertainty in the v g measurement. However, this leads to artificially narrower bounds with ambiguity in the meaning of those bounds. The Bayesian methods are free of all aforementioned pathologies that plague the other methods. They can efficiently handle a large number of events and is unaffected by a small number of bad events if any. Furthermore, the Bayesian credible intervals have the clear and unambiguous meaning of being regions of the parameter space that contain the true value of said parameters with a certain posterior probability given data. Among the two approximate Bayesian techniques described in this paper, the Narrow Gaussian method has the following issue. The process to find σ is cumbersome and somewhat subjective as it involves partitioning the set of events into subsets and estimating σ from plots, or developing an algorithm that needs to be compared to an independent method. On the other hand, the KDE method has less user-controlled finetuning than the narrow Gaussian method as it estimates its control parameter, i.e. the bandwidth, quantitatively from properties of the posterior samples themselves, with its variation having a much more restrictive effect on the estimated density. Hence we claim that the Bayesian analysis implemented by the KDE method produces the most trustworthy measurements of the SME coefficients.",
"pages": [
12,
13
]
},
{
"title": "F. Mock Data Challenge and Comparison of Methods",
"content": "In this section, we describe the MDC that was set up to compare the different methods of SME measurements from GW data in order to verify our claims regarding them that were made in the previous section. To construct the MDC, we choose a fiducial value of the SME coefficients as their true values, say ¯ s lm,tr and generate data for 15 mock events. The true sky positions of the mock events are chosen to be the mean values of the ( θ, ϕ ) samples of the real events. We choose 15 of the 'best' real events, i.e. the ones with the most precise sky localizations and v g measure- ments, to be represented by our mock events in the MDC. We then calculate the true value of ∆ v for each mock event from the true values of their sky positions and those of the SME coefficients. We then generate mock posterior samples of ( θ, ϕ, ∆ v ) by adding un-correlated Gaussian fluctuations to the true sky positions and true ∆ v 's. The width of the fluctuations, for each mock event is chosen to be the standard deviations of the posterior samples of the corresponding real event. This allows us to create a controlled numerical experiment wherein we know the true answer. For Gaussian distributions of θ, ϕ, ∆ v about known true values, one can write down the exact functional form of the likelihood of these parameters given mock data: where ( θ tr,i , ϕ tr,i ) are the true values of the sky positions of the i th mock event, σ θ,i , σ ϕ,i , σ ∆ v,i are the widths of the Gaussian fluctuations used to generate the mock posterior samples of the ith mock event and ∆ v tr,i = ∆ v (¯ s lm,tr , θ tr,i , ϕ tr,i ). Knowledge of these quantities allows us to exactly write down and evaluate Eq. (31) as a function of θ, ϕ without any smoothing approximations. We can then substitute L mdc ( d i | ∆ v ' , θ, ϕ ) in place of L ( d i | ∆ v ' , θ, ϕ ) in Eq. (15) and carry out the integral numerically to obtain the 'exact Bayesian' likelihood of our mock data given the SME coefficients: We can then sample the likelihood in Eq. (32), after applying suitable priors on ¯ s lm , using the MCMC techniques described above and obtain what can be thought of as the 'true' posterior distribution of the SME coef- icients given the mock data. We can then compute the constraints obtained from the approximate methods being applied to the mock posterior samples and compare those results with the true posterior. The results of this comparison is displayed in Fig. 3. We can see that the KDE method agrees remarkably well with the exact Bayesian method, while the narrow Gaussian method deviates from it slightly for some coefficients. We note that a different choice of σ for the narrow Gaussian method leading to better agreement with the exact Bayesian result is possible. However, we conclude that the KDE method's agreement with the exact Bayesian method, independent of any external finetuning, justifies its use on the real data for producing our final result. We also note that our claim regarding the SVD chisquare method producing artificially narrower bounds is also verified by this comparison.",
"pages": [
13,
14
]
},
{
"title": "G. Final Results",
"content": "With the 24 chosen GW events in Tables I, II, and III, we are able to constrain all nine ¯ s lm coefficients. We obtain the results shown in Fig. 4. Note that the measurements of ¯ s lm shown in Fig.4 are consistent with zero. Given that zero lies within the 90% credible intervals for all coefficients, we consider these results to be consistent with existing constraints on ¯ s lm [9]. These limits are considerably weaker than some found in the literature. However, they are valuable as independent tests. Moreover, this is also the first attempt to simultaneously constrain all s lm using gravitational-wave measurements, thus putting direct limits on the full potential anisotropy of the speed of gravitational waves. Additionally, this method can theoretically incorporate as many events as available and thus improve in precision as additional high-quality events become available.",
"pages": [
15
]
},
{
"title": "IV. CONCLUSION",
"content": "In our study, we select 52 high-SNR gravitationalwave events that were detected by at least two detectors from the first three observing runs of Advanced LIGO and Advanced Virgo. We use lalinference nest and lalinference mcmc to construct posterior distributions of the speeds of GWs for each event. We find the 90% credible interval of the combined v g posterior distribution to be 0 . 99 +0 . 01 -0 . 02 c . This interval is narrower than the similar one constructed with O1 and O2 events in previous studies, suggesting a more precise measurement of v g [7]. However, even with the inclusion of a high-SNR BNS event GW170817 with its pinpoint sky localization, we were only able to narrow the 90% credible interval to 0 . 99 +0 . 01 -0 . 02 c . We then explore multiple methods of extracting SME constraints from v g -like data. Based on the conclusions of that investigation, we use hierarchical Bayesian inference implemented with KDE methods to simultaneously constrain all nine coefficients for Lorentz violation in the SME framework. The resultant con- straints do not exhibit evidence for Lorentz violation.",
"pages": [
15
]
},
{
"title": "ACKNOWLEDGMENTS",
"content": "This work was supported by NSF awards PHY1806990, PHY-1912649, and PHY-2207728. This material is based upon work supported by NSF's LIGO Laboratory which is a major facility fully funded by the National Science Foundation. The analysis in this work used the LALSuite software library [16]. We thank LIGO and Virgo Collaboration for providing the data from the first, second, and third observing runs. The authors are grateful for computational resources provided by the LIGO Laboratory and supported by National Science Foundation Grants PHY-0757058 and PHY-0823459. This research has made use of data or software obtained from the Gravitational Wave Open Science Center (gwosc.org), a service of LIGO Laboratory, the LIGO Scientific Collaboration, the Virgo Collaboration, and KAGRA. LIGO Laboratory and Advanced LIGO are funded by the United States National Science Foundation (NSF) as well as the Science and Technology Facilities Council (STFC) of the United Kingdom, the Max-Planck-Society (MPS), and the State of Niedersachsen/Germany for support of the construction of Advanced LIGO and construction and operation of the GEO600 detector. Additional support for Advanced LIGO was provided by the Australian Research Council. Virgo is funded, through the European Gravitational Observatory (EGO), by the French Centre National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale di Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions by institutions from Belgium, Germany, Greece, Hungary, Ireland, Japan, Monaco, Poland, Portugal, Spain. KAGRA is supported by Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan Society for the Promotion of Science (JSPS) in Japan; National Research Foundation (NRF) and Ministry of Science and ICT (MSIT) in Korea; Academia Sinica (AS) and National Science and Technology Council (NSTC) in Taiwan. LIGO and Virgo during the second part of the third observing run (2021), arXiv:2111.03606 [gr-qc].",
"pages": [
15
]
}
]
2024PhRvL.132r1801B
https://arxiv.org/pdf/2305.05681.pdf
<document>
<section_header_level_1><location><page_1><loc_21><loc_92><loc_79><loc_93></location>Absorption of Axion Dark Matter in a Magnetized Medium</section_header_level_1>
<text><location><page_1><loc_19><loc_88><loc_81><loc_90></location>Asher Berlin ∗ and Tanner Trickle † Theoretical Physics Division, Fermi National Accelerator Laboratory, Batavia, Illinois 60510</text>
<text><location><page_1><loc_43><loc_86><loc_58><loc_87></location>(Dated: May 11, 2023)</text>
<text><location><page_1><loc_18><loc_72><loc_83><loc_85></location>Detection of axion dark matter heavier than a meV is hindered by its small wavelength, which limits the useful volume of traditional experiments. This problem can be avoided by directly detecting in-medium excitations, whose ∼ meV -eV energies are decoupled from the detector size. We show that for any target inside a magnetic field, the absorption rate of electromagnetically-coupled axions into in-medium excitations is determined by the dielectric function. As a result, the plethora of candidate targets previously identified for sub-GeV dark matter searches can be repurposed as broadband axion detectors. We find that a kg · yr exposure with noise levels comparable to recent measurements is sufficient to probe parameter space currently unexplored by laboratory tests. Noise reduction by only a few orders of magnitude can enable sensitivity to the QCD axion in the ∼ 10 meV -10 eV mass range.</text>
<text><location><page_1><loc_9><loc_58><loc_49><loc_69></location>Introduction.Despite constituting roughly 27% of the energy density of the universe [1], the fundamental nature of dark matter (DM) remains elusive. Of the theoretically motivated DM candidates, the QCD axion is particularly remarkable since its existence would also solve the longstanding strong CP problem [2-5]. A generic feature of QCD axion DM models is a coupling between the axion field a and electromagnetism,</text>
<formula><location><page_1><loc_15><loc_54><loc_49><loc_57></location>L ⊃ 1 4 g aγγ aF µν ˜ F µν = g aγγ a E · B . (1)</formula>
<text><location><page_1><loc_9><loc_33><loc_49><loc_53></location>In the presence of a static magnetic field B 0 , the interaction in Eq. (1) converts an axion to an oscillating electromagnetic field [6]. Directly detecting this field is the underlying principle of many ongoing and planned experiments [7]. Traditional detection schemes utilize cavities with electromagnetic modes resonantly matched to axion masses of m a ∼ (10 -6 -10 -5 ) eV, as motivated by postinflationary misalignment production and a standard cosmological history [8-12]. However, searches across a larger parameter space are motivated by alternative production mechanisms [13-33] and axions that couple to the Standard Model similarly to the QCD axion but without the strict connection between coupling strength and mass [34-44].</text>
<text><location><page_1><loc_9><loc_16><loc_49><loc_33></location>Cavities are an exceptional tool to search for axion DM. However, they are fundamentally limited in the axion mass that they can probe. This is because the axion mass must be matched to a resonant frequency of the cavity, which are inversely related to its size. Therefore, to resonantly search for higher axion masses, the cavity must be prohibitively small, limiting the total exposure. Recent strategies to boost exposure to high-mass axions include non-resonant detection of single-photons in a large volume dish antenna [45], and modifications to the photon's dispersion relation in dielectric [46-48] or plasma [49-51] structures tuned to a specific mass.</text>
<text><location><page_1><loc_52><loc_36><loc_92><loc_69></location>While these searches are focused on photon detection, another possibility is to directly detect the in-medium excitations in crystal targets involving, e.g., electrons [5280], phonons [76, 77, 81-87], and magnons [82, 88-92]. Since the energy of these modes ( ∼ meV -eV) is not set by the target size, but rather by the physics of the local environment, they are ideal for high-mass axion searches. Furthermore, the manufacturing of low-noise targets and the technology required to detect single quanta of such excitations is at the forefront of the DM direct detection community and is thus an active area of development [93]. In particular, current experiments, such as CDEX [94], DAMIC [95-100], EDELWEISS [101-103], SENSEI [104-106], and SuperCDMS [107-109], utilize eV-scale electronic excitations in Si and Ge targets. More novel targets with sub-eV electronic excitations have also been proposed, such as narrow gap semiconductors [110], Dirac materials [61, 63, 65, 69-72], spin-orbit coupled materials [57, 79], and doped semiconductors [73]. Additionally, phonon excitations have been studied in a wide variety of target materials [76, 80, 82-85, 111], including GaAs and Al 2 O 3 as planned for the TESSARACT experiment [112].</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_36></location>In this Letter , we demonstrate that the entirety of these ideas can be used to search for the axion-photon coupling in Eq. (1), provided that the target can be placed inside a magnetic field, thus creating a 'magnetized medium.' In particular, we show that in a magnetized medium the inclusive axion absorption rate into in-medium excitations is directly related to the dielectric function. While certain signals of axion DM have previously been found to be related to the dielectric function on a case-by-case basis, we show here that this is universal. This is important both experimentally, since the dielectric can be measured, and theoretically, because it broadly captures the absorption rate into any in-medium excitation, abstracting away from calculations specific to any single excitation. This allows us to easily evaluate the sensitivity of various materials, as well as identify a larger scope of relevant signals that have previously been overlooked, such as low-energy electronic excitations. Below, we begin by deriving the absorption rate with two meth-</text>
<figure>
<location><page_2><loc_20><loc_66><loc_81><loc_93></location>
<caption>FIG. 1. Projected sensitivity to electromagnetically-coupled axion dark matter, for a kg · yr exposure of various targets inside a 10 T magnetic field. Solid colored lines assume negligible backgrounds, and dotted colored lines assume a dark count (DC) rate of R DC = 10 10 / kg · yr. Targets utilizing single-phonon excitations include GaAs (light blue) and Al 2 O 3 (purple) (proposed for the TESSARACT experiment [112]), as well as SiO 2 (turquoise). The Si (dark green) and Ge (light green) targets correspond to single-electron excitations currently searched for by the CDEX [94], DAMIC [95-100], EDELWEISS [101-103], SENSEI [104106], and SuperCDMS [107-109] experiments. Doped Si (Si ∗ , pink) [73] and ZrTe 5 (red) [57] correspond to novel targets utilizing low-energy electronic excitations. Shaded gray regions are existing limits derived from horizontal branch (HB) star cooling [113], the CAST helioscope [114, 115], and searches for a → 2 γ decays by the MUSE [116], HST [117], and VIMOS [118] telescopes. Also shown as gray lines are projections from the IAXO (solid) [119], BREAD (dashed) (assuming DCs of ∼ 10 4 or < ∼ 1 for masses below or above ∼ 100 meV, respectively, over 10 3 days) [45], and LAMPOST (dotted) (assuming ∼ 10 DCs per 10 6 s run and ∼ 10 4 runs) [48] experiments. The orange band denotes the range of couplings and masses as motivated by the QCD axion.</caption>
</figure>
<text><location><page_2><loc_9><loc_34><loc_49><loc_46></location>ods. The first derivation involves self-energies, analogous to calculations performed in the context of direct detection experiments; the second is provided within the language of classical axion electrodynamics. These derivations provide complementary ways to understand the underlying physics. We then discuss the projections shown in Fig. 1, which illustrate the promising ability to explore new, high-mass, QCD axion parameter space.</text>
<text><location><page_2><loc_9><loc_27><loc_49><loc_32></location>Absorption Rate.Before deriving the rate for axion absorption in a magnetized medium, we begin with a synopsis of the final result for isotropic targets. The total axion absorption rate, per unit exposure, is given by</text>
<formula><location><page_2><loc_16><loc_22><loc_49><loc_26></location>R glyph[similarequal] ( g aγγ B 0 m a ) 2 ρ DM ρ T Im [ -1 ε ( m a ) ] , (2)</formula>
<text><location><page_2><loc_9><loc_9><loc_49><loc_22></location>where ρ DM glyph[similarequal] 0 . 4 GeV / cm 3 is the local axion DM energy density, ρ T is the mass density of the target, and ε ( m a ) is the dielectric function evaluated at energy ω = m a and momentum deposition q = 0, appropriate for absorption kinematics ( q glyph[lessmuch] ω ) which are assumed throughout. The simplicity of this expression derives from the separability of the axion absorption process as a + B 0 → E followed by absorption of the corresponding electric field. 1 The former process is governed by the strength of the external</text>
<text><location><page_2><loc_52><loc_26><loc_92><loc_46></location>magnetic field and g aγγ , while the latter is determined by the dielectric function, independent of both the axion physics and, in the q glyph[lessmuch] ω limit, the magnetic permeability. This separability is advantageous since, in principle, the dielectric function of the target can be measured directly. In the absence of measurement, this parameterization is useful as a bridge between particle physics and first principles condensed matter calculations. First principles calculations are a useful tool to understand the contributions of individual excitations, which cannot be understood from a measurement of the inclusive dielectric function. However, this generally ceases to be a problem when the various excitations are sufficiently separated in energy.</text>
<text><location><page_2><loc_52><loc_16><loc_92><loc_25></location>The idea of relating the dielectric function to the DM absorption rate into in-medium excitations has been used for other DM models [52-54, 56, 79, 80], as well as in calculations of the DM absorption rate into in-medium photon states [50, 51, 120]. For example, for kineticallymixed dark photon DM, A ' , the absorption rate into inmedium excitations is [52, 56, 80]</text>
<formula><location><page_2><loc_62><loc_11><loc_92><loc_14></location>R glyph[similarequal] κ 2 ρ DM ρ T Im [ -1 ε ( m A ' ) ] , (3)</formula>
<text><location><page_2><loc_52><loc_9><loc_92><loc_10></location>where κ is the kinetic-mixing parameter and m A ' is the A '</text>
<figure>
<location><page_3><loc_10><loc_84><loc_46><loc_93></location>
<caption>FIG. 2. An illustrative Feynman diagram for the axion a self-energy in a magnetized medium. The optical theorem relates the imaginary part of this diagram to the axion absorption rate. The external magnetic field B 0 is represented as a background source, and the shaded 'blob' represents any in-medium modifications to the photon propagator (e.g., from electron and phonon excitations).</caption>
</figure>
<text><location><page_3><loc_9><loc_63><loc_49><loc_72></location>mass. The similarity between the dark photon absorption rate in Eq. (3) and the axion absorption rate in Eq. (2) is immediately clear. As a result, for DM particles of the same mass, the sensitivity to electromagnetically-coupled axions can be simply rescaled via the mapping g aγγ B 0 ↔ κm a , corresponding to</text>
<formula><location><page_3><loc_10><loc_59><loc_49><loc_62></location>g aγγ ∼ 10 -10 GeV -1 × ( κ 10 -14 ) ( m a meV )( T B 0 ) . (4)</formula>
<text><location><page_3><loc_9><loc_53><loc_49><loc_58></location>We now turn to the details of the calculation, beginning with a self-energy treatment similar to Refs. [53, 56, 121], then turning to an alternative derivation employing classical equations of motion, similar to Refs. [87, 122].</text>
<text><location><page_3><loc_9><loc_36><loc_49><loc_51></location>Self-Energy Calculation.The starting point of the selfenergy derivation is the optical theorem, which relates the absorption rate of a particle to the imaginary part of its self-energy. The self-energy can then be computed by summing over all the relevant Feynman diagrams. The Feynman diagram for axion absorption in a magnetized medium is shown in Fig. 2; the vertex Feynman rule is derived from the Lagrangian in Eq. (1), and the photon propagator is modified due to the presence of the medium.</text>
<text><location><page_3><loc_9><loc_23><loc_49><loc_36></location>Following Refs. [53, 56, 121], instead of directly computing the diagram in Fig. 2, we identify the diagrams which mix the axion and photon, and those which do not. If the axion and photon mix, the in-medium fields are those that diagonalize the 2 × 2 self-energy matrix of a and A . The diagonalizing fields ˆ a and ˆ A (the axion-like and photon-like fields, respectively) are a linear combination of a and A . The probability of axion absorption per unit time is then related to the self-energy of ˆ a ,</text>
<formula><location><page_3><loc_15><loc_18><loc_49><loc_22></location>Γ glyph[similarequal] -1 m a Im [ Π aa + ∑ λ Π λ aA Π λ Aa m 2 a -Π λ AA ] , (5)</formula>
<text><location><page_3><loc_52><loc_82><loc_92><loc_93></location>written in terms of the unmixed self-energies of the a, A fields, Π aa and Π AA , respectively, and the mixing terms, Π aA , Π Aa . Π λ AA is the self-energy of A projected onto the λ th polarization e λ µ , defined to diagonalize Π µν AA : Π λ AA = -e λ µ Π µν AA e λ ν . Π λ aA is the self-energy mixing a and A , projected onto the same photon polarization vector, i.e., Π λ aA = -e λ µ Π µ aA .</text>
<text><location><page_3><loc_52><loc_58><loc_92><loc_83></location>In the absence of direct axion couplings to electrons, the only non-zero self-energies (ignoring vacuum processes) are Π µν AA and Π µ aA . Furthermore, since the Ward identities ( Q µ Π µν AA = Q µ Π µ aA = 0, where Q µ = ( ω, q )) relate the temporal and spatial components, only Π ij AA and Π i aA need to be computed. The photon self-energy is determined by the dielectric tensor ε ij through Π ij AA = -ω 2 (1 -ε ij ). Note that the spatial component of the photon polarization vectors diagonalize the dielectric tensor, ε ij = ∑ λ ε λ e i λ e j λ , where ε λ ≡ e λ i ε ij e λ j , such that Π λ AA glyph[similarequal] ω 2 (1 -ε λ ). The axion-photon self-energies are determined by the term in Eq. (1) involving the vector potential which, after integrating by parts, is g aγγ ˙ a A · B 0 . The mixed self-energies are then given by Π i aA = ig aγγ m a B i 0 = -Π i Aa . Substituting Π AA and Π aA into Eq. (5), and taking the q → 0 limit, results in</text>
<formula><location><page_3><loc_58><loc_54><loc_92><loc_58></location>Γ glyph[similarequal] g 2 aγγ m a ∑ λ ( e λ · B 0 ) 2 Im [ -1 ε λ ( m a ) ] . (6)</formula>
<text><location><page_3><loc_52><loc_41><loc_92><loc_52></location>This expression can be further simplified in the limit of an isotropic target. The dielectric of an isotropic target is independent of polarization, ε λ = ε , and the photon polarization vectors are the standard transverse e µ ± = (0 , ˆ q ± ) and longitudinal e µ L = ( q, ω ˆ q ) / √ Q 2 ones, where ˆ q ± are two vectors mutually orthonormal to ˆ q . The sum over polarizations can be performed using ∑ λ e i λ e j λ = δ ij . Applying these approximations, Eq. (6) simplifies to</text>
<formula><location><page_3><loc_61><loc_36><loc_92><loc_39></location>Γ glyph[similarequal] ( g aγγ B 0 ) 2 m a Im [ -1 ε ( m a ) ] . (7)</formula>
<text><location><page_3><loc_52><loc_30><loc_92><loc_34></location>The rate per unit exposure R in Eq. (2) is then obtained by multiplying Γ by the number of axions in the target and dividing by the target mass.</text>
<text><location><page_3><loc_52><loc_14><loc_92><loc_28></location>Classical Equations of Motion.Alternatively, the absorption rate can be derived using classical axion electrodynamics. Throughout, we will implicitly work in Lorenz gauge. Our starting point is the wave equation for the vector potential, which is approximately ε ∂ 2 t A glyph[similarequal] j a , where j a = g aγγ ˙ a B 0 is the axion effective current and ε is the dielectric tensor. This equation is trivially solved for A by switching to momentum space and projecting onto the photon polarization vectors e λ , which determines the corresponding electric field to be</text>
<formula><location><page_3><loc_61><loc_9><loc_92><loc_12></location>E glyph[similarequal] -g aγγ a ∑ λ ( e λ · B 0 ) ε λ e λ . (8)</formula>
<text><location><page_4><loc_9><loc_90><loc_49><loc_93></location>The rate for axion absorption is governed by the axion equation of motion, which is approximately</text>
<formula><location><page_4><loc_19><loc_87><loc_49><loc_89></location>( ∂ 2 t + m 2 a ) a glyph[similarequal] g aγγ E · B 0 , (9)</formula>
<text><location><page_4><loc_9><loc_80><loc_49><loc_87></location>with E given by Eq. (8). The probability per unit time for axion absorption is determined by solving for the imaginary component of the axion frequency in Eq. (9), Γ glyph[similarequal] Im( -ω 2 ) /m a . This leads to a result in agreement with Eq. (6). 2</text>
<text><location><page_4><loc_9><loc_49><loc_49><loc_78></location>Projected Sensitivity.The sensitivity of a variety of targets is shown in Fig. 1, assuming a kg · yr exposure and B 0 = 10 T. In our estimates, we demand N > 3 √ 1 + N DC + δ 2 N 2 DC , where N = RM T t and N DC = R DC M T t are the the number of signal events and dark counts (DCs), respectively, R DC is the DC rate, M T is the target mass, t is the exposure time, and δ ≤ 1 is the systematic uncertainty in the DC. In the absence of background, N DC glyph[lessmuch] 1, the sensitivity to the axionphoton coupling scales as g aγγ ∝ ( M T t ) -1 / 2 . If backgrounds are instead significant, N DC glyph[greatermuch] 1, the reliance of the signal on B 0 allows these DCs to be directly measured by removing the magnetic field, thereby suppressing systematic uncertainties, δ glyph[lessmuch] 1. In the statisticallylimited regime, δ glyph[lessmuch] 1 / √ N DC , an observable signal only needs to overcome Poisson fluctuations in noise, such that g aγγ ∝ ( R DC /M T t ) 1 / 4 . Below, we begin by discussing the sensitivity of various targets assuming negligible backgrounds, and then proceed to examine the impact of currently measured noise sources.</text>
<text><location><page_4><loc_9><loc_24><loc_49><loc_49></location>For m a > ∼ eV, enough energy is deposited to excite an electron across the ∼ 1 eV band gap in standard semiconductors, such as Si and Ge, which is then read out by drifting the charge to a sensing output. This is the operating principle of many ongoing experiments, such as CDEX [94], DAMIC [95-100], EDELWEISS [101-103], SENSEI [104-106], and SuperCDMS [107-109]. In our estimate of the signal rate in Eq. (2), we use the measured dielectric functions of Si and Ge from Ref. [127], and do not incorporate multi-phonon responses at lower energies [80] since these are subdominant to the singlephonon responses of the polar materials discussed below. As shown in Fig. 1, background-free Ge targets have the potential to be the best laboratory-based search for the QCD axion for masses greater than that probed by the CAST helioscope [114, 115] and smaller than that probed by astrophysical searches for a → 2 γ decays [116-118].</text>
<text><location><page_4><loc_52><loc_69><loc_92><loc_93></location>While the energy of electronic excitations is limited to ∼ eV scales in standard semiconductors, novel targets have lower ∼ meV electronic excitations. For example, Dirac [61, 63, 65, 69-72] and spin-orbit coupled materials [57, 79] have small bulk band gaps. One such target that falls under both categories is ZrTe 5 ; while its Dirac character is somewhat debated [57], the presence of its small band gap has been firmly established [128]. However, since its dielectric response has not been accurately measured, we adopt the first-principles calculation performed in Ref. [57]. In addition to pure targets, doping is another method to create electronic states below the band gap. Recently, Ref. [73] studied Si doped with phosphorus as a candidate target, using an analytic model for the dielectric response consistent with measurements [129, 130]. In Fig. 1, we rescale their quoted dark photon sensitivity, using the mapping described above.</text>
<text><location><page_4><loc_52><loc_51><loc_92><loc_68></location>Phonon excitations [76, 80, 82-85, 111] in the ∼ (1 -100) meV energy range have also been studied as an avenue to detect axions [82, 87]. The results shown in Fig. 1 are consistent with the first principles calculation done in Ref. [82]. We have chosen to focus on GaAs, Al 2 O 3 , and SiO 2 targets, as they are of active investigation in the sub-GeV DM community; the first two are currently planned for the TESSARACT experiment [112], and SiO 2 has also been identified as an optimal target for light DM scattering [76]. The measured dielectric data for these targets in the phonon energy regime is taken from Ref. [80].</text>
<text><location><page_4><loc_52><loc_25><loc_92><loc_49></location>Backgrounds.The sensitivity of detectors focused on readout of single-electron excitations are currently limited by DCs [93]. The SENSEI experiment [105, 131], utilizing a Si Skipper CCD detector, is the current stateof-the-art for measuring charge from single-electron excitations. Recent measurements indicate a DC rate of 10 8 / kg · yr and 10 6 / kg · yr for energy ranges of < ∼ 4 . 7 eV and ∼ (4 . 7 -8 . 3) eV, respectively, and rates consistent with zero for larger energies. Assuming that similar background levels in the two-electron bin can be achieved in a Ge based detector, noise reduction by three orders of magnitude ( R DC ∼ 10 3 / kg · yr) is necessary to attain sensitivity to the QCD axion at eV-scale masses. Similar noise levels, but at much lower energies, are also necessary for doped Si and ZrTe 5 targets to attain sensitivity to the QCD axion at ∼ 100 meV masses.</text>
<text><location><page_4><loc_52><loc_9><loc_92><loc_26></location>Calorimetric detectors (e.g., SuperCDMS CPD [132], which measures phonons produced from single-electron excitations) have registered significantly higher noise levels, R DC ∼ 10 10 / kg · yr. As an estimate of the backgrounds that will contaminate sensors based on detection of single-phonon excitations, we assume R DC = 10 10 / kg · yr and δ < ∼ 1 / √ N DC for the dotted colored lines shown in Fig. 1. If the DCs are reduced to just ∼ 10 8 / kg · yr at ∼ 100 meV energies, an Al 2 O 3 or SiO 2 target can be sensitive to the QCD axion at couplings smaller than that bounded by considerations of stellar energy loss.</text>
<text><location><page_5><loc_9><loc_77><loc_49><loc_93></location>While their origin is unknown, noise levels are generally peaked towards smaller energies [133]. Recent work has focused on progressing the understanding of such backgrounds. For instance, it has been calculated that a subdominant fraction of DCs arises from secondary radiation generated by high-energy tracks [134] and photons [135] in low-threshold charge and phonon detectors, respectively. Additionally, a bulk of eV-scale phonon backgrounds in superconducting calorimetric detectors has recently been shown to emerge from cooling-induced micro-fractures in auxiliary detector components [136].</text>
<text><location><page_5><loc_9><loc_43><loc_49><loc_77></location>The setup investigated in our work differs from previously proposed applications of these targets in the use of a large external magnetic field, whose dominant effect will be disrupting detection technology based on superconducting devices (measurements of typical semiconducting sensors have found minor changes to their operation when exposed to ∼ 1 T magnetic fields [137]). For example, transition edge sensors (TES) (the main technology for reading out single-phonon excitations in the TESSARACT experiment) are not operative in > ∼ µ T magnetic fields [138]. However, there is a strong dependence on the direction of the magnetic field relative to the face of the TES. Additionally, if the region of bulk target within the external magnetic field can be physically separated from the superconducting detector, these problems can be avoided. Other complications arising from, e.g., additional radioactive components or Lorentz-force induced mechanical stress may also be introduced. A detailed investigation of these effects is beyond the scope of this work and these will also need to be confronted in other axion experiments operating in the meV -eV energy range [45, 48]. However, we do not expect fundamental roadblocks in this approach since ∼ 10 T magnetic fields only change electronic energies at the level of</text>
<unordered_list>
<list_item><location><page_5><loc_10><loc_33><loc_49><loc_37></location>[1] Particle Data Group Collaboration, P. A. Zyla et al. , 'Review of Particle Physics,' PTEP 2020 no. 8, (2020) 083C01.</list_item>
<list_item><location><page_5><loc_10><loc_29><loc_48><loc_33></location>[2] R. D. Peccei and H. R. Quinn, 'Constraints imposed by cp conservation in the presence of instantons,' Phys. Rev. D 16 (1977) 1791-1797.</list_item>
<list_item><location><page_5><loc_10><loc_25><loc_49><loc_29></location>[3] R. D. Peccei and H. R. Quinn, 'Cp conservation in the presence of instantons,' Phys. Rev. Lett. 38 (1977) 1440-1443.</list_item>
<list_item><location><page_5><loc_10><loc_21><loc_49><loc_25></location>[4] F. Wilczek, 'Problem of strong p and t invariance in the presence of instantons,' Phys. Rev. Lett. 40 (1978) 279-282.</list_item>
<list_item><location><page_5><loc_10><loc_19><loc_46><loc_21></location>[5] S. Weinberg, 'The u(1) problem,' Phys. Rev. D 11 (1975) 3583-3593.</list_item>
<list_item><location><page_5><loc_10><loc_15><loc_48><loc_18></location>[6] P. Sikivie, 'Experimental tests of the invisible axion,' Phys. Rev. Lett. 51 (1983) 1415-1417. [Erratum: Phys.Rev.Lett. 52, 695 (1984)].</list_item>
<list_item><location><page_5><loc_10><loc_12><loc_49><loc_14></location>[7] C. B. Adams et al. , 'Axion dark matter,' in Snowmass 2021 . 3, 2022. arXiv:2203.14923 [hep-ex] .</list_item>
<list_item><location><page_5><loc_10><loc_10><loc_49><loc_12></location>[8] L. F. Abbott and P. Sikivie, 'A cosmological bound on the invisible axion,' Phys. Lett. B 120 (1983) 133-136.</list_item>
</unordered_list>
<text><location><page_5><loc_52><loc_91><loc_87><loc_93></location>∼ meV, well below the energies investigated here.</text>
<text><location><page_5><loc_52><loc_64><loc_92><loc_89></location>Discussion.The past decade has seen a meteoric rise in target proposals to hunt for sub-GeV DM. In an external magnetic field these targets are also powerful probes of axions in a mass range that is notoriously difficult to explore, corresponding to m a ∼ 10 meV -10 eV. Searching for axions with these targets has the intrinsic advantage of directly utilizing all future experimental improvements in background reduction, an effort which has assiduously driven direct detection experiments to incredible precision in recent history. The axion absorption rate in a magnetized medium can be simply written in terms of the measurable dielectric function, encoding all in-medium responses. This synergizes with future developments towards optimizing the energy loss function of the material, Im( -1 /ε ), as well as further study into other novel low-energy excitations, such as axion quasiparticles [139, 140] or chiral phonons [141].</text>
<text><location><page_5><loc_52><loc_43><loc_92><loc_61></location>Acknowledgements. We would like to thank Aaron Chou, Juan Estrada, Roni Harnik, Yoni Kahn, Alex Millar, Tongyan Lin, Andrea Mitridate, Kris Pardo, and Kathryn Zurek for helpful conversations, and Ciaran O'Hare for the compilation of axion limits in Ref. [142]. This material is based upon work supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Superconducting Quantum Materials and Systems Center (SQMS) under contract number DE-AC02-07CH11359. Fermilab is operated by the Fermi Research Alliance, LLC under Contract DE-AC02-07CH11359 with the U.S. Department of Energy.</text>
<unordered_list>
<list_item><location><page_5><loc_53><loc_35><loc_91><loc_37></location>[9] M. S. Turner, 'Coherent scalar field oscillations in an expanding universe,' Phys. Rev. D 28 (1983) 1243.</list_item>
<list_item><location><page_5><loc_53><loc_32><loc_89><loc_34></location>[10] M. S. Turner, 'Cosmic and local mass density of invisible axions,' Phys. Rev. D 33 (1986) 889-896.</list_item>
<list_item><location><page_5><loc_53><loc_29><loc_92><loc_32></location>[11] J. Preskill, M. B. Wise, and F. Wilczek, 'Cosmology of the invisible axion,' Phys. Lett. B 120 (1983) 127-132.</list_item>
<list_item><location><page_5><loc_53><loc_27><loc_87><loc_29></location>[12] M. Dine and W. Fischler, 'The not so harmless axion,' Phys. Lett. B 120 (1983) 137-141.</list_item>
<list_item><location><page_5><loc_53><loc_23><loc_91><loc_26></location>[13] R. T. Co and K. Harigaya, 'Axiogenesis,' Phys. Rev. Lett. 124 no. 11, (2020) 111602, arXiv:1910.02080 [hep-ph] .</list_item>
<list_item><location><page_5><loc_53><loc_17><loc_92><loc_22></location>[14] R. T. Co, L. J. Hall, K. Harigaya, K. A. Olive, and S. Verner, 'Axion kinetic misalignment and parametric resonance from inflation,' JCAP 08 (2020) 036, arXiv:2004.00629 [hep-ph] .</list_item>
<list_item><location><page_5><loc_53><loc_13><loc_91><loc_17></location>[15] R. T. Co, L. J. Hall, and K. Harigaya, 'Axion kinetic misalignment mechanism,' Phys. Rev. Lett. 124 no. 25, (2020) 251802, arXiv:1910.14152 [hep-ph] .</list_item>
<list_item><location><page_5><loc_53><loc_9><loc_91><loc_13></location>[16] C. Hagmann, S. Chang, and P. Sikivie, 'Axions from string decay,' Nucl. Phys. B Proc. Suppl. 72 (1999) 81-86, arXiv:hep-ph/9807428 .</list_item>
<list_item><location><page_6><loc_10><loc_89><loc_49><loc_93></location>[17] M. Gorghetto, E. Hardy, and G. Villadoro, 'Axions from strings: the attractive solution,' JHEP 07 (2018) 151, arXiv:1806.04677 [hep-ph] .</list_item>
<list_item><location><page_6><loc_10><loc_84><loc_47><loc_89></location>[18] R. A. Battye and E. P. S. Shellard, 'Axion string constraints,' Phys. Rev. Lett. 73 (1994) 2954-2957, arXiv:astro-ph/9403018 . [Erratum: Phys.Rev.Lett. 76, 2203-2204 (1996)].</list_item>
<list_item><location><page_6><loc_10><loc_80><loc_47><loc_84></location>[19] M. Hindmarsh, J. Lizarraga, A. Lopez-Eiguren, and J. Urrestilla, 'Comment on 'more axions from strings',' arXiv:2109.09679 [astro-ph.CO] .</list_item>
<list_item><location><page_6><loc_10><loc_75><loc_48><loc_80></location>[20] M. Dine, N. Fernandez, A. Ghalsasi, and H. H. Patel, 'Comments on axions, domain walls, and cosmic strings,' JCAP 11 (2021) 041, arXiv:2012.13065 [hep-ph] .</list_item>
<list_item><location><page_6><loc_10><loc_68><loc_48><loc_74></location>[21] M. Buschmann, J. W. Foster, A. Hook, A. Peterson, D. E. Willcox, W. Zhang, and B. R. Safdi, 'Dark matter from axion strings with adaptive mesh refinement,' Nature Commun. 13 no. 1, (2022) 1049, arXiv:2108.05368 [hep-ph] .</list_item>
<list_item><location><page_6><loc_10><loc_64><loc_48><loc_68></location>[22] A. Vaquero, J. Redondo, and J. Stadler, 'Early seeds of axion miniclusters,' JCAP 04 (2019) 012, arXiv:1809.09241 [astro-ph.CO] .</list_item>
<list_item><location><page_6><loc_10><loc_59><loc_49><loc_64></location>[23] M. Buschmann, J. W. Foster, and B. R. Safdi, 'Early-universe simulations of the cosmological axion,' Phys. Rev. Lett. 124 no. 16, (2020) 161103, arXiv:1906.00967 [astro-ph.CO] .</list_item>
<list_item><location><page_6><loc_10><loc_55><loc_49><loc_59></location>[24] V. B. Klaer and G. D. Moore, 'How to simulate global cosmic strings with large string tension,' JCAP 10 (2017) 043, arXiv:1707.05566 [hep-ph] .</list_item>
<list_item><location><page_6><loc_10><loc_48><loc_45><loc_55></location>[25] T. Hiramatsu, M. Kawasaki, T. Sekiguchi, M. Yamaguchi, and J. Yokoyama, 'Improved estimation of radiated axions from cosmological axionic strings,' Phys. Rev. D 83 (2011) 123531, arXiv:1012.5502 [hep-ph] .</list_item>
<list_item><location><page_6><loc_10><loc_44><loc_47><loc_48></location>[26] M. Gorghetto, E. Hardy, and G. Villadoro, 'More axions from strings,' SciPost Phys. 10 no. 2, (2021) 050, arXiv:2007.04990 [hep-ph] .</list_item>
<list_item><location><page_6><loc_10><loc_40><loc_48><loc_44></location>[27] C.-F. Chang and Y. Cui, 'New perspectives on axion misalignment mechanism,' Phys. Rev. D 102 no. 1, (2020) 015003, arXiv:1911.11885 [hep-ph] .</list_item>
<list_item><location><page_6><loc_10><loc_36><loc_47><loc_40></location>[28] K. Harigaya and J. M. Leedom, 'Qcd axion dark matter from a late time phase transition,' JHEP 06 (2020) 034, arXiv:1910.04163 [hep-ph] .</list_item>
<list_item><location><page_6><loc_10><loc_31><loc_48><loc_36></location>[29] R. T. Co, L. J. Hall, and K. Harigaya, 'Qcd axion dark matter with a small decay constant,' Phys. Rev. Lett. 120 no. 21, (2018) 211602, arXiv:1711.10486 [hep-ph] .</list_item>
<list_item><location><page_6><loc_10><loc_26><loc_48><loc_31></location>[30] M. Hindmarsh, J. Lizarraga, A. Lopez-Eiguren, and J. Urrestilla, 'Scaling density of axion strings,' Phys. Rev. Lett. 124 no. 2, (2020) 021301, arXiv:1908.03522 [astro-ph.CO] .</list_item>
<list_item><location><page_6><loc_10><loc_22><loc_48><loc_26></location>[31] R. Daido, F. Takahashi, and W. Yin, 'The alp miracle: unified inflaton and dark matter,' JCAP 05 (2017) 044, arXiv:1702.03284 [hep-ph] .</list_item>
<list_item><location><page_6><loc_10><loc_18><loc_49><loc_22></location>[32] R. Daido, F. Takahashi, and W. Yin, 'The alp miracle revisited,' JHEP 02 (2018) 104, arXiv:1710.11107 [hep-ph] .</list_item>
<list_item><location><page_6><loc_10><loc_14><loc_48><loc_18></location>[33] V. B. . Klaer and G. D. Moore, 'The dark-matter axion mass,' JCAP 11 (2017) 049, arXiv:1708.07521 [hep-ph] .</list_item>
<list_item><location><page_6><loc_10><loc_10><loc_47><loc_14></location>[34] K.-S. Choi, H. P. Nilles, S. Ramos-Sanchez, and P. K. S. Vaudrevange, 'Accions,' Phys. Lett. B 675 (2009) 381-386, arXiv:0902.3070 [hep-th] .</list_item>
<list_item><location><page_6><loc_10><loc_9><loc_45><loc_10></location>[35] B. S. Acharya, K. Bobkov, and P. Kumar, 'An m</list_item>
<list_item><location><page_6><loc_56><loc_89><loc_89><loc_93></location>theory solution to the strong cp problem and constraints on the axiverse,' JHEP 11 (2010) 105, arXiv:1004.5138 [hep-th] .</list_item>
<list_item><location><page_6><loc_53><loc_84><loc_89><loc_89></location>[36] Y. Chikashige, R. N. Mohapatra, and R. D. Peccei, 'Are there real goldstone bosons associated with broken lepton number?' Phys. Lett. B 98 (1981) 265-268.</list_item>
<list_item><location><page_6><loc_53><loc_81><loc_90><loc_84></location>[37] P. Svrcek and E. Witten, 'Axions in string theory,' JHEP 06 (2006) 051, arXiv:hep-th/0605206 .</list_item>
<list_item><location><page_6><loc_53><loc_77><loc_91><loc_81></location>[38] C. D. Froggatt and H. B. Nielsen, 'Hierarchy of quark masses, cabibbo angles and cp violation,' Nucl. Phys. B 147 (1979) 277-298.</list_item>
<list_item><location><page_6><loc_53><loc_73><loc_92><loc_77></location>[39] R. T. Co, L. J. Hall, and K. Harigaya, 'Predictions for axion couplings from alp cogenesis,' JHEP 01 (2021) 172, arXiv:2006.04809 [hep-ph] .</list_item>
<list_item><location><page_6><loc_53><loc_71><loc_90><loc_73></location>[40] E. Witten, 'Some properties of o(32) superstrings,' Phys. Lett. B 149 (1984) 351-356.</list_item>
<list_item><location><page_6><loc_53><loc_65><loc_90><loc_70></location>[41] A. Arvanitaki, S. Dimopoulos, S. Dubovsky, N. Kaloper, and J. March-Russell, 'String axiverse,' Phys. Rev. D 81 (2010) 123530, arXiv:0905.4720 [hep-th] .</list_item>
<list_item><location><page_6><loc_53><loc_62><loc_83><loc_65></location>[42] J. P. Conlon, 'The qcd axion and moduli stabilisation,' JHEP 05 (2006) 078, arXiv:hep-th/0602233 .</list_item>
<list_item><location><page_6><loc_53><loc_58><loc_92><loc_61></location>[43] M. Cicoli, M. Goodsell, and A. Ringwald, 'The type iib string axiverse and its low-energy phenomenology,' JHEP 10 (2012) 146, arXiv:1206.0819 [hep-th] .</list_item>
<list_item><location><page_6><loc_53><loc_52><loc_91><loc_57></location>[44] J. Halverson, C. Long, and P. Nath, 'Ultralight axion in supersymmetry and strings and cosmology at small scales,' Phys. Rev. D 96 no. 5, (2017) 056025, arXiv:1703.07779 [hep-ph] .</list_item>
<list_item><location><page_6><loc_53><loc_47><loc_89><loc_52></location>[45] BREAD Collaboration, J. Liu et al. , 'Broadband solenoidal haloscope for terahertz axion detection,' Phys. Rev. Lett. 128 no. 13, (2022) 131801, arXiv:2111.12103 [physics.ins-det] .</list_item>
<list_item><location><page_6><loc_53><loc_38><loc_90><loc_47></location>[46] MADMAX Working Group Collaboration, A. Caldwell, G. Dvali, B. Majorovits, A. Millar, G. Raffelt, J. Redondo, O. Reimann, F. Simon, and F. Steffen, 'Dielectric Haloscopes: A New Way to Detect Axion Dark Matter,' Phys. Rev. Lett. 118 no. 9, (2017) 091801, arXiv:1611.05865 [physics.ins-det] .</list_item>
<list_item><location><page_6><loc_53><loc_32><loc_89><loc_37></location>[47] A. J. Millar, G. G. Raffelt, J. Redondo, and F. D. Steffen, 'Dielectric Haloscopes to Search for Axion Dark Matter: Theoretical Foundations,' JCAP 01 (2017) 061, arXiv:1612.07057 [hep-ph] .</list_item>
<list_item><location><page_6><loc_53><loc_27><loc_92><loc_32></location>[48] M. Baryakhtar, J. Huang, and R. Lasenby, 'Axion and hidden photon dark matter detection with multilayer optical haloscopes,' Phys. Rev. D 98 no. 3, (2018) 035006, arXiv:1803.11455 [hep-ph] .</list_item>
<list_item><location><page_6><loc_53><loc_22><loc_90><loc_27></location>[49] M. Lawson, A. J. Millar, M. Pancaldi, E. Vitagliano, and F. Wilczek, 'Tunable axion plasma haloscopes,' Phys. Rev. Lett. 123 no. 14, (2019) 141802, arXiv:1904.11872 [hep-ph] .</list_item>
<list_item><location><page_6><loc_53><loc_17><loc_92><loc_22></location>[50] A. Caputo, A. J. Millar, and E. Vitagliano, 'Revisiting longitudinal plasmon-axion conversion in external magnetic fields,' Phys. Rev. D 101 no. 12, (2020) 123004, arXiv:2005.00078 [hep-ph] .</list_item>
<list_item><location><page_6><loc_53><loc_11><loc_92><loc_16></location>[51] ALPHA Collaboration, A. J. Millar et al. , 'Searching for dark matter with plasma haloscopes,' Phys. Rev. D 107 no. 5, (2023) 055013, arXiv:2210.00017 [hep-ph] .</list_item>
<list_item><location><page_6><loc_53><loc_9><loc_92><loc_11></location>[52] Y. Hochberg, T. Lin, and K. M. Zurek, 'Absorption of light dark matter in semiconductors,' Phys. Rev. D 95</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_7><loc_13><loc_92><loc_47><loc_93></location>no. 2, (2017) 023013, arXiv:1608.01994 [hep-ph] .</list_item>
<list_item><location><page_7><loc_10><loc_88><loc_48><loc_92></location>[53] G. Krnjaic and T. Trickle, 'Absorption of vector dark matter beyond kinetic mixing,' arXiv:2303.11344 [hep-ph] .</list_item>
<list_item><location><page_7><loc_10><loc_84><loc_47><loc_88></location>[54] A. Derevianko, V. A. Dzuba, V. V. Flambaum, and M. Pospelov, 'Axio-electric effect,' Phys. Rev. D 82 (2010) 065006, arXiv:1007.1833 [hep-ph] .</list_item>
<list_item><location><page_7><loc_10><loc_79><loc_48><loc_84></location>[55] R. Catena, T. Emken, M. Matas, N. A. Spaldin, and E. Urdshals, 'Crystal responses to general dark matter-electron interactions,' Phys. Rev. Res. 3 no. 3, (2021) 033149, arXiv:2105.02233 [hep-ph] .</list_item>
<list_item><location><page_7><loc_10><loc_75><loc_48><loc_78></location>[56] A. Mitridate, T. Trickle, Z. Zhang, and K. M. Zurek, 'Dark matter absorption via electronic excitations,' JHEP 09 (2021) 123, arXiv:2106.12586 [hep-ph] .</list_item>
<list_item><location><page_7><loc_10><loc_68><loc_48><loc_74></location>[57] H.-Y. Chen, A. Mitridate, T. Trickle, Z. Zhang, M. Bernardi, and K. M. Zurek, 'Dark matter direct detection in materials with spin-orbit coupling,' Phys. Rev. D 106 no. 1, (2022) 015024, arXiv:2202.11716 [hep-ph] .</list_item>
<list_item><location><page_7><loc_10><loc_62><loc_49><loc_68></location>[58] R. Catena, D. Cole, T. Emken, M. Matas, N. Spaldin, W. Tarantino, and E. Urdshals, 'Dark matter-electron interactions in materials beyond the dark photon model,' JCAP 03 (2023) 052, arXiv:2210.07305 [hep-ph] .</list_item>
<list_item><location><page_7><loc_10><loc_56><loc_49><loc_61></location>[59] C. Blanco, J. I. Collar, Y. Kahn, and B. Lillard, 'Dark matter-electron scattering from aromatic organic targets,' Phys. Rev. D 101 no. 5, (2020) 056001, arXiv:1912.02822 [hep-ph] .</list_item>
<list_item><location><page_7><loc_10><loc_51><loc_49><loc_56></location>[60] S. Knapen, J. Kozaczuk, and T. Lin, 'Dark matter-electron scattering in dielectrics,' Phys. Rev. D 104 no. 1, (2021) 015031, arXiv:2101.08275 [hep-ph] .</list_item>
<list_item><location><page_7><loc_10><loc_46><loc_46><loc_51></location>[61] Y. Hochberg, M. Pyle, Y. Zhao, and K. M. Zurek, 'Detecting superlight dark matter with fermi-degenerate materials,' JHEP 08 (2016) 057, arXiv:1512.04533 [hep-ph] .</list_item>
<list_item><location><page_7><loc_10><loc_40><loc_46><loc_45></location>[62] Y. Hochberg, T. Lin, and K. M. Zurek, 'Detecting ultralight bosonic dark matter via absorption in superconductors,' Phys. Rev. D 94 no. 1, (2016) 015019, arXiv:1604.06800 [hep-ph] .</list_item>
<list_item><location><page_7><loc_10><loc_34><loc_49><loc_40></location>[63] Y. Hochberg, Y. Kahn, M. Lisanti, K. M. Zurek, A. G. Grushin, R. Ilan, S. M. Griffin, Z.-F. Liu, S. F. Weber, and J. B. Neaton, 'Detection of sub-mev dark matter with three-dimensional dirac materials,' Phys. Rev. D 97 no. 1, (2018) 015004, arXiv:1708.08929 [hep-ph] .</list_item>
<list_item><location><page_7><loc_10><loc_27><loc_48><loc_34></location>[64] Y. Hochberg, Y. Kahn, N. Kurinsky, B. V. Lehmann, T. C. Yu, and K. K. Berggren, 'Determining dark-matter-electron scattering rates from the dielectric function,' Phys. Rev. Lett. 127 no. 15, (2021) 151802, arXiv:2101.08263 [hep-ph] .</list_item>
<list_item><location><page_7><loc_10><loc_21><loc_47><loc_27></location>[65] R. M. Geilhufe, F. Kahlhoefer, and M. W. Winkler, 'Dirac materials for sub-mev dark matter detection: New targets and improved formalism,' Phys. Rev. D 101 no. 5, (2020) 055005, arXiv:1910.02091 [hep-ph] .</list_item>
<list_item><location><page_7><loc_10><loc_17><loc_47><loc_20></location>[66] R. Essig, J. Mardon, and T. Volansky, 'Direct detection of sub-gev dark matter,' Phys. Rev. D 85 (2012) 076007, arXiv:1108.5383 [hep-ph] .</list_item>
<list_item><location><page_7><loc_10><loc_11><loc_47><loc_16></location>[67] S. Derenzo, R. Essig, A. Massari, A. Soto, and T.-T. Yu, 'Direct detection of sub-gev dark matter with scintillating targets,' Phys. Rev. D 96 no. 1, (2017) 016026, arXiv:1607.01009 [hep-ph] .</list_item>
<list_item><location><page_7><loc_10><loc_9><loc_46><loc_11></location>[68] R. Essig, M. Fernandez-Serra, J. Mardon, A. Soto, T. Volansky, and T.-T. Yu, 'Direct detection of</list_item>
<list_item><location><page_7><loc_56><loc_91><loc_90><loc_93></location>sub-gev dark matter with semiconductor targets,' JHEP 05 (2016) 046, arXiv:1509.01598 [hep-ph] .</list_item>
<list_item><location><page_7><loc_53><loc_85><loc_91><loc_90></location>[69] A. Coskuner, A. Mitridate, A. Olivares, and K. M. Zurek, 'Directional dark matter detection in anisotropic dirac materials,' Phys. Rev. D 103 no. 1, (2021) 016006, arXiv:1909.09170 [hep-ph] .</list_item>
<list_item><location><page_7><loc_53><loc_80><loc_90><loc_85></location>[70] Y. Hochberg, Y. Kahn, M. Lisanti, C. G. Tully, and K. M. Zurek, 'Directional detection of dark matter with two-dimensional targets,' Phys. Lett. B 772 (2017) 239-246, arXiv:1606.08849 [hep-ph] .</list_item>
<list_item><location><page_7><loc_53><loc_73><loc_91><loc_80></location>[71] R. Catena, T. Emken, M. Matas, N. A. Spaldin, and E. Urdshals, 'Direct searches for general dark matter-electron interactions with graphene detectors: Part i. electronic structure calculations,' arXiv:2303.15497 [hep-ph] .</list_item>
<list_item><location><page_7><loc_53><loc_67><loc_91><loc_73></location>[72] R. Catena, T. Emken, M. Matas, N. A. Spaldin, and E. Urdshals, 'Direct searches for general dark matter-electron interactions with graphene detectors: Part ii. sensitivity studies,' arXiv:2303.15509 [hep-ph] .</list_item>
<list_item><location><page_7><loc_53><loc_62><loc_90><loc_67></location>[73] P. Du, D. Ega˜na Ugrinovic, R. Essig, and M. Sholapurkar, 'Doped semiconductor devices for sub-mev dark matter detection,' arXiv:2212.04504 [hep-ph] .</list_item>
<list_item><location><page_7><loc_53><loc_55><loc_90><loc_61></location>[74] S. M. Griffin, K. Inzani, T. Trickle, Z. Zhang, and K. M. Zurek, 'Extended calculation of dark matter-electron scattering in crystal targets,' Phys. Rev. D 104 no. 9, (2021) 095015, arXiv:2105.05253 [hep-ph] .</list_item>
<list_item><location><page_7><loc_53><loc_50><loc_91><loc_55></location>[75] T. Trickle, 'Extended calculation of electronic excitations for direct detection of dark matter,' Phys. Rev. D 107 no. 3, (2023) 035035, arXiv:2210.14917 [hep-ph] .</list_item>
<list_item><location><page_7><loc_53><loc_44><loc_90><loc_49></location>[76] S. M. Griffin, K. Inzani, T. Trickle, Z. Zhang, and K. M. Zurek, 'Multichannel direct detection of light dark matter: Target comparison,' Phys. Rev. D 101 no. 5, (2020) 055004, arXiv:1910.10716 [hep-ph] .</list_item>
<list_item><location><page_7><loc_53><loc_39><loc_92><loc_44></location>[77] T. Trickle, Z. Zhang, K. M. Zurek, K. Inzani, and S. M. Griffin, 'Multi-channel direct detection of light dark matter: Theoretical framework,' JHEP 03 (2020) 036, arXiv:1910.08092 [hep-ph] .</list_item>
<list_item><location><page_7><loc_53><loc_34><loc_90><loc_39></location>[78] P. W. Graham, D. E. Kaplan, S. Rajendran, and M. T. Walters, 'Semiconductor probes of light dark matter,' Phys. Dark Univ. 1 (2012) 32-49, arXiv:1203.2531 [hep-ph] .</list_item>
<list_item><location><page_7><loc_53><loc_29><loc_92><loc_34></location>[79] K. Inzani, A. Faghaninia, and S. M. Griffin, 'Prediction of tunable spin-orbit gapped materials for dark matter detection,' Phys. Rev. Res. 3 no. 1, (2021) 013069, arXiv:2008.05062 [cond-mat.mtrl-sci] .</list_item>
<list_item><location><page_7><loc_53><loc_23><loc_91><loc_28></location>[80] S. Knapen, J. Kozaczuk, and T. Lin, 'python package for dark matter scattering in dielectric targets,' Phys. Rev. D 105 no. 1, (2022) 015014, arXiv:2104.12786 [hep-ph] .</list_item>
<list_item><location><page_7><loc_53><loc_19><loc_89><loc_23></location>[81] P. Cox, T. Melia, and S. Rajendran, 'Dark matter phonon coupling,' Phys. Rev. D 100 no. 5, (2019) 055011, arXiv:1905.05575 [hep-ph] .</list_item>
<list_item><location><page_7><loc_53><loc_14><loc_91><loc_19></location>[82] A. Mitridate, T. Trickle, Z. Zhang, and K. M. Zurek, 'Detectability of axion dark matter with phonon polaritons and magnons,' Phys. Rev. D 102 no. 9, (2020) 095005, arXiv:2005.10256 [hep-ph] .</list_item>
<list_item><location><page_7><loc_53><loc_9><loc_92><loc_14></location>[83] S. Knapen, T. Lin, M. Pyle, and K. M. Zurek, 'Detection of light dark matter with optical phonons in polar materials,' Phys. Lett. B 785 (2018) 386-390, arXiv:1712.06598 [hep-ph] .</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_8><loc_10><loc_87><loc_49><loc_93></location>[84] A. Coskuner, T. Trickle, Z. Zhang, and K. M. Zurek, 'Directional detectability of dark matter with single phonon excitations: Target comparison,' Phys. Rev. D 105 no. 1, (2022) 015010, arXiv:2102.09567 [hep-ph] .</list_item>
<list_item><location><page_8><loc_10><loc_81><loc_48><loc_86></location>[85] S. Griffin, S. Knapen, T. Lin, and K. M. Zurek, 'Directional detection of light dark matter with polar materials,' Phys. Rev. D 98 no. 11, (2018) 115034, arXiv:1807.10291 [hep-ph] .</list_item>
<list_item><location><page_8><loc_10><loc_76><loc_49><loc_81></location>[86] T. Trickle, Z. Zhang, and K. M. Zurek, 'Effective field theory of dark matter direct detection with collective excitations,' Phys. Rev. D 105 no. 1, (2022) 015001, arXiv:2009.13534 [hep-ph] .</list_item>
<list_item><location><page_8><loc_10><loc_71><loc_48><loc_76></location>[87] D. J. E. Marsh, J. I. McDonald, A. J. Millar, and J. Schutte-Engel, 'Axion detection with phonon-polaritons revisited,' Phys. Rev. D 107 no. 3, (2023) 035036, arXiv:2209.12909 [hep-ph] .</list_item>
<list_item><location><page_8><loc_10><loc_66><loc_49><loc_70></location>[88] R. Barbieri, M. Cerdonio, G. Fiorentini, and S. Vitale, 'Axion to magnon conversion: A scheme for the detection of galactic axions,' Phys. Lett. B 226 (1989) 357-360.</list_item>
<list_item><location><page_8><loc_10><loc_59><loc_49><loc_65></location>[89] G. Flower, J. Bourhill, M. Goryachev, and M. E. Tobar, 'Broadening frequency range of a ferromagnetic axion haloscope with strongly coupled cavity-magnon polaritons,' Phys. Dark Univ. 25 (2019) 100306, arXiv:1811.09348 [physics.ins-det] .</list_item>
<list_item><location><page_8><loc_10><loc_54><loc_47><loc_59></location>[90] S. Chigusa, T. Moroi, and K. Nakayama, 'Detecting light boson dark matter through conversion into a magnon,' Phys. Rev. D 101 no. 9, (2020) 096013, arXiv:2001.10666 [hep-ph] .</list_item>
<list_item><location><page_8><loc_10><loc_50><loc_49><loc_53></location>[91] T. Trickle, Z. Zhang, and K. M. Zurek, 'Detecting light dark matter with magnons,' Phys. Rev. Lett. 124 no. 20, (2020) 201801, arXiv:1905.13744 [hep-ph] .</list_item>
<list_item><location><page_8><loc_10><loc_46><loc_46><loc_49></location>[92] A. Esposito and S. Pavaskar, 'Optimal anti-ferromagnets for light dark matter detection,' arXiv:2210.13516 [hep-ph] .</list_item>
<list_item><location><page_8><loc_10><loc_38><loc_49><loc_45></location>[93] R. Essig, G. K. Giovanetti, N. Kurinsky, D. McKinsey, K. Ramanathan, K. Stifter, and T.-T. Yu, 'Snowmass2021 Cosmic Frontier: The landscape of low-threshold dark matter direct detection in the next decade,' in Snowmass 2021 . 3, 2022. arXiv:2203.08297 [hep-ph] .</list_item>
<list_item><location><page_8><loc_10><loc_32><loc_49><loc_37></location>[94] CDEX Collaboration, Z. Y. Zhang et al. , 'Constraints on sub-gev dark matter-electron scattering from the cdex-10 experiment,' Phys. Rev. Lett. 129 no. 22, (2022) 221301, arXiv:2206.04128 [hep-ex] .</list_item>
<list_item><location><page_8><loc_10><loc_26><loc_49><loc_32></location>[95] DAMIC Collaboration, A. Aguilar-Arevalo et al. , 'Constraints on light dark matter particles interacting with electrons from damic at snolab,' Phys. Rev. Lett. 123 no. 18, (2019) 181802, arXiv:1907.12628 [astro-ph.CO] .</list_item>
<list_item><location><page_8><loc_10><loc_21><loc_47><loc_26></location>[96] DAMIC-M Collaboration, I. Arnquist et al. , 'First constraints from damic-m on sub-gev dark-matter particles interacting with electrons,' arXiv:2302.02372 [hep-ex] .</list_item>
<list_item><location><page_8><loc_10><loc_14><loc_46><loc_20></location>[97] DAMIC Collaboration, A. Aguilar-Arevalo et al. , 'First direct-detection constraints on ev-scale hidden-photon dark matter with damic at snolab,' Phys. Rev. Lett. 118 no. 14, (2017) 141803, arXiv:1611.03066 [astro-ph.CO] .</list_item>
<list_item><location><page_8><loc_10><loc_9><loc_49><loc_14></location>[98] DAMIC Collaboration, A. Aguilar-Arevalo et al. , 'Results on low-mass weakly interacting massive particles from a 11 kg-day target exposure of damic at snolab,' Phys. Rev. Lett. 125 (2020) 241803,</list_item>
<list_item><location><page_8><loc_56><loc_92><loc_79><loc_93></location>arXiv:2007.15622 [astro-ph.CO] .</list_item>
<list_item><location><page_8><loc_53><loc_85><loc_91><loc_92></location>[99] DAMIC, DAMIC-M Collaboration, M. Settimo, 'Search for low-mass dark matter with the damic experiment,' in 16th Rencontres du Vietnam: Theory meeting experiment: Particle Astrophysics and Cosmology . 4, 2020. arXiv:2003.09497 [hep-ex] .</list_item>
<list_item><location><page_8><loc_52><loc_80><loc_91><loc_85></location>[100] DAMIC Collaboration, J. R. T. de Mello Neto et al. , 'The damic dark matter experiment,' PoS ICRC2015 (2016) 1221, arXiv:1510.02126 [physics.ins-det] .</list_item>
<list_item><location><page_8><loc_52><loc_73><loc_92><loc_80></location>[101] EDELWEISS Collaboration, Q. Arnaud et al. , 'First germanium-based constraints on sub-mev dark matter with the edelweiss experiment,' Phys. Rev. Lett. 125 no. 14, (2020) 141301, arXiv:2003.01046 [astro-ph.GA] .</list_item>
<list_item><location><page_8><loc_52><loc_67><loc_90><loc_73></location>[102] EDELWEISS Collaboration, E. Armengaud et al. , 'Searches for electron interactions induced by new physics in the edelweiss-iii germanium bolometers,' Phys. Rev. D 98 no. 8, (2018) 082004, arXiv:1808.02340 [hep-ex] .</list_item>
<list_item><location><page_8><loc_52><loc_60><loc_90><loc_67></location>[103] EDELWEISS Collaboration, E. Armengaud et al. , 'Searching for low-mass dark matter particles with a massive ge bolometer operated above-ground,' Phys. Rev. D 99 no. 8, (2019) 082003, arXiv:1901.03588 [astro-ph.GA] .</list_item>
<list_item><location><page_8><loc_52><loc_54><loc_90><loc_60></location>[104] SENSEI Collaboration, O. Abramoff et al. , 'Sensei: Direct-detection constraints on sub-gev dark matter from a shallow underground run using a prototype skipper-ccd,' Phys. Rev. Lett. 122 no. 16, (2019) 161801, arXiv:1901.10478 [hep-ex] .</list_item>
<list_item><location><page_8><loc_52><loc_47><loc_89><loc_53></location>[105] SENSEI Collaboration, L. Barak et al. , 'SENSEI: Direct-Detection Results on sub-GeV Dark Matter from a New Skipper-CCD,' Phys. Rev. Lett. 125 no. 17, (2020) 171802, arXiv:2004.11378 [astro-ph.CO] .</list_item>
<list_item><location><page_8><loc_52><loc_39><loc_92><loc_47></location>[106] SENSEI Collaboration, M. Crisler, R. Essig, J. Estrada, G. Fernandez, J. Tiffenberg, M. Sofo haro, T. Volansky, and T.-T. Yu, 'Sensei: First direct-detection constraints on sub-gev dark matter from a surface run,' Phys. Rev. Lett. 121 no. 6, (2018) 061803, arXiv:1804.00088 [hep-ex] .</list_item>
<list_item><location><page_8><loc_52><loc_32><loc_92><loc_39></location>[107] SuperCDMS Collaboration, D. W. Amaral et al. , 'Constraints on low-mass, relic dark matter candidates from a surface-operated SuperCDMS single-charge sensitive detector,' Phys. Rev. D 102 no. 9, (2020) 091101, arXiv:2005.14067 [hep-ex] .</list_item>
<list_item><location><page_8><loc_52><loc_26><loc_91><loc_32></location>[108] SuperCDMS Collaboration, R. Agnese et al. , 'First dark matter constraints from a supercdms single-charge sensitive detector,' Phys. Rev. Lett. 121 no. 5, (2018) 051301, arXiv:1804.10697 [hep-ex] . [Erratum: Phys.Rev.Lett. 122, 069901 (2019)].</list_item>
<list_item><location><page_8><loc_52><loc_22><loc_90><loc_26></location>[109] CDMS Collaboration, Z. Ahmed et al. , 'Search for axions with the cdms experiment,' Phys. Rev. Lett. 103 (2009) 141802, arXiv:0902.4693 [hep-ex] .</list_item>
<list_item><location><page_8><loc_52><loc_15><loc_92><loc_22></location>[110] SuperCDMS Collaboration, M. F. Albakry et al. , 'A strategy for low-mass dark matter searches with cryogenic detectors in the supercdms snolab facility,' in Snowmass 2021 . 3, 2022. arXiv:2203.08463 [physics.ins-det] .</list_item>
<list_item><location><page_8><loc_52><loc_10><loc_91><loc_15></location>[111] S. M. Griffin, Y. Hochberg, K. Inzani, N. Kurinsky, T. Lin, and T. Chin, 'Silicon carbide detectors for sub-gev dark matter,' Phys. Rev. D 103 no. 7, (2021) 075002, arXiv:2008.08560 [hep-ph] .</list_item>
<list_item><location><page_8><loc_52><loc_9><loc_89><loc_10></location>[112] C. Chang et al. , 'Snowmass 2021 letter of interest:</list_item>
</unordered_list>
<text><location><page_9><loc_13><loc_92><loc_40><loc_93></location>The tessaract dark matter project,' 2020.</text>
<unordered_list>
<list_item><location><page_9><loc_9><loc_87><loc_43><loc_92></location>[113] M. J. Dolan, F. J. Hiskens, and R. R. Volkas, 'Advancing globular cluster constraints on the axion-photon coupling,' JCAP 10 (2022) 096, arXiv:2207.03102 [hep-ph] .</list_item>
<list_item><location><page_9><loc_9><loc_81><loc_48><loc_86></location>[114] CAST Collaboration, S. Andriamonje et al. , 'An improved limit on the axion-photon coupling from the cast experiment,' JCAP 04 (2007) 010, arXiv:hep-ex/0702006 .</list_item>
<list_item><location><page_9><loc_9><loc_76><loc_49><loc_81></location>[115] CAST Collaboration, V. Anastassopoulos et al. , 'New cast limit on the axion-photon interaction,' Nature Phys. 13 (2017) 584-590, arXiv:1705.02290 [hep-ex] .</list_item>
<list_item><location><page_9><loc_9><loc_68><loc_49><loc_76></location>[116] M. Regis, M. Taoso, D. Vaz, J. Brinchmann, S. L. Zoutendijk, N. F. Bouch'e, and M. Steinmetz, 'Searching for light in the darkness: Bounds on alp dark matter with the optical muse-faint survey,' Phys. Lett. B 814 (2021) 136075, arXiv:2009.01310 [astro-ph.CO] .</list_item>
<list_item><location><page_9><loc_9><loc_64><loc_47><loc_68></location>[117] P. Carenza, G. Lucente, and E. Vitagliano, 'Probing the blue axion with cosmic optical background anisotropies,' arXiv:2301.06560 [hep-ph] .</list_item>
<list_item><location><page_9><loc_9><loc_59><loc_49><loc_64></location>[118] D. Grin, G. Covone, J.-P. Kneib, M. Kamionkowski, A. Blain, and E. Jullo, 'A telescope search for decaying relic axions,' Phys. Rev. D 75 (2007) 105018, arXiv:astro-ph/0611502 .</list_item>
<list_item><location><page_9><loc_9><loc_51><loc_49><loc_59></location>[119] I. Shilon, A. Dudarev, H. Silva, and H. H. J. ten Kate, 'Conceptual design of a new large superconducting toroid for iaxo, the new international axion observatory,' IEEE Trans. Appl. Supercond. 23 no. 3, (2013) 4500604, arXiv:1212.4633 [physics.ins-det] .</list_item>
<list_item><location><page_9><loc_9><loc_46><loc_47><loc_51></location>[120] G. B. Gelmini, A. J. Millar, V. Takhistov, and E. Vitagliano, 'Probing dark photons with plasma haloscopes,' Phys. Rev. D 102 no. 4, (2020) 043003, arXiv:2006.06836 [hep-ph] .</list_item>
<list_item><location><page_9><loc_9><loc_42><loc_48><loc_45></location>[121] E. Hardy and R. Lasenby, 'Stellar cooling bounds on new light particles: plasma mixing effects,' JHEP 02 (2017) 033, arXiv:1611.05852 [hep-ph] .</list_item>
<list_item><location><page_9><loc_9><loc_38><loc_48><loc_41></location>[122] S. Dubovsky and G. Hern'andez-Chifflet, 'Heating up the Galaxy with Hidden Photons,' JCAP 12 (2015) 054, arXiv:1509.00039 [hep-ph] .</list_item>
<list_item><location><page_9><loc_9><loc_32><loc_48><loc_37></location>[123] S. Chaudhuri, P. W. Graham, K. Irwin, J. Mardon, S. Rajendran, and Y. Zhao, 'Radio for hidden-photon dark matter detection,' Phys. Rev. D 92 no. 7, (2015) 075012, arXiv:1411.7382 [hep-ph] .</list_item>
<list_item><location><page_9><loc_9><loc_27><loc_47><loc_32></location>[124] R. Balafendiev, C. Simovski, A. J. Millar, and P. Belov, 'Wire metamaterial filled metallic resonators,' Phys. Rev. B 106 no. 7, (2022) 075106, arXiv:2203.10083 [hep-ph] .</list_item>
<list_item><location><page_9><loc_9><loc_22><loc_46><loc_27></location>[125] P. W. Graham, J. Mardon, S. Rajendran, and Y. Zhao, 'Parametrically enhanced hidden photon search,' Phys. Rev. D 90 no. 7, (2014) 075017, arXiv:1407.4806 [hep-ph] .</list_item>
<list_item><location><page_9><loc_9><loc_18><loc_47><loc_22></location>[126] A. Berlin, R. Harnik, and R. Janish, 'Light Shining Through a Thin Wall: Evanescent Hidden Photon Detection,' arXiv:2303.00014 [hep-ph] .</list_item>
<list_item><location><page_9><loc_9><loc_14><loc_49><loc_18></location>[127] D. W. Lynch and W. R. Hunter, 'Handbook of optical constants of solids,' in Handbook of Optical Constants of Solids . 1985.</list_item>
<list_item><location><page_9><loc_9><loc_9><loc_49><loc_14></location>[128] L. Moreschini, J. C. Johannsen, H. Berger, J. Denlinger, C. Jozwiak, E. Rotenberg, K. S. Kim, A. Bostwick, and M. Grioni, 'Nature and topology of the low-energy states in zrte 5 ,' Phys. Rev. B 94 (Aug,</list_item>
</unordered_list>
<text><location><page_9><loc_56><loc_92><loc_70><loc_93></location>2016) 081101. https:</text>
<text><location><page_9><loc_56><loc_91><loc_91><loc_92></location>//link.aps.org/doi/10.1103/PhysRevB.94.081101 .</text>
<unordered_list>
<list_item><location><page_9><loc_52><loc_84><loc_92><loc_90></location>[129] A. Gaymann, H. P. Geserich, and H. v. Lohneysen, 'Temperature dependence of the far-infrared reflectance spectra of si:p near the metal-insulator transition,' Phys. Rev. B 52 (Dec, 1995) 16486-16493. https:</list_item>
</unordered_list>
<text><location><page_9><loc_56><loc_83><loc_90><loc_84></location>//link.aps.org/doi/10.1103/PhysRevB.52.16486 .</text>
<unordered_list>
<list_item><location><page_9><loc_52><loc_77><loc_91><loc_82></location>[130] A. Gaymann, H. P. Geserich, and H. v. Lohneysen, 'Far-infrared reflectance spectra of si:p near the metal-insulator transition,' Phys. Rev. Lett. 71 (Nov, 1993) 3681-3684. https:</list_item>
</unordered_list>
<text><location><page_9><loc_56><loc_76><loc_91><loc_77></location>//link.aps.org/doi/10.1103/PhysRevLett.71.3681 .</text>
<unordered_list>
<list_item><location><page_9><loc_52><loc_69><loc_91><loc_76></location>[131] SENSEI Collaboration, L. Barak et al. , 'SENSEI: Characterization of Single-Electron Events Using a Skipper Charge-Coupled Device,' Phys. Rev. Applied 17 no. 1, (2022) 014022, arXiv:2106.08347 [physics.ins-det] .</list_item>
<list_item><location><page_9><loc_52><loc_63><loc_92><loc_69></location>[132] SuperCDMS Collaboration, I. Alkhatib et al. , 'Light Dark Matter Search with a High-Resolution Athermal Phonon Detector Operated Above Ground,' Phys. Rev. Lett. 127 (2021) 061801, arXiv:2007.14289 [hep-ex] .</list_item>
<list_item><location><page_9><loc_52><loc_59><loc_90><loc_63></location>[133] P. Adari et al. , 'EXCESS workshop: Descriptions of rising low-energy spectra,' SciPost Phys. Proc. 9 (2022) 001, arXiv:2202.05097 [astro-ph.IM] .</list_item>
<list_item><location><page_9><loc_52><loc_52><loc_91><loc_59></location>[134] P. Du, D. Egana-Ugrinovic, R. Essig, and M. Sholapurkar, 'Sources of Low-Energy Events in Low-Threshold Dark-Matter and Neutrino Detectors,' Phys. Rev. X 12 no. 1, (2022) 011009, arXiv:2011.13939 [hep-ph] .</list_item>
<list_item><location><page_9><loc_52><loc_46><loc_91><loc_52></location>[135] K. V. Berghaus, R. Essig, Y. Hochberg, Y. Shoji, and M. Sholapurkar, 'Phonon background from gamma rays in sub-GeV dark matter detectors,' Phys. Rev. D 106 no. 2, (2022) 023026, arXiv:2112.09702 [hep-ph] .</list_item>
<list_item><location><page_9><loc_52><loc_42><loc_91><loc_45></location>[136] R. Anthony-Petersen et al. , 'A Stress Induced Source of Phonon Bursts and Quasiparticle Poisoning,' arXiv:2208.02790 [physics.ins-det] .</list_item>
<list_item><location><page_9><loc_52><loc_36><loc_92><loc_41></location>[137] K. Kondo, T. Dotani, M. Ozaki, and M. Iwai, 'Performance improvement of x-ray ccds by applying a magnetic field,' vol. 9144, p. 91443W. SPIE, 2014. https://doi.org/10.1117/12.2055705 .</list_item>
<list_item><location><page_9><loc_52><loc_27><loc_92><loc_36></location>[138] M. de Wit, L. Gottardi, M. L. Ridder, K. Nagayoshi, E. Taralli, H. Akamatsu, D. Vaccaro, J.-W. A. d. Herder, M. P. Bruijn, and J.-R. Gao, 'Mitigation of the Magnetic Field Susceptibility of Transition-Edge Sensors Using a Superconducting Groundplane,' Phys. Rev. Applied 18 no. 2, (2022) 024066, arXiv:2208.10775 [astro-ph.IM] .</list_item>
<list_item><location><page_9><loc_52><loc_21><loc_92><loc_27></location>[139] D. J. E. Marsh, K.-C. Fong, E. W. Lentz, L. Smejkal, and M. N. Ali, 'Proposal to Detect Dark Matter using Axionic Topological Antiferromagnets,' Phys. Rev. Lett. 123 no. 12, (2019) 121601, arXiv:1807.08810 [hep-ph] .</list_item>
<list_item><location><page_9><loc_52><loc_14><loc_91><loc_20></location>[140] J. Schutte-Engel, D. J. E. Marsh, A. J. Millar, A. Sekine, F. Chadha-Day, S. Hoof, M. N. Ali, K.-C. Fong, E. Hardy, and L. ˇ Smejkal, 'Axion quasiparticles for axion dark matter detection,' JCAP 08 (2021) 066, arXiv:2102.05366 [hep-ph] .</list_item>
<list_item><location><page_9><loc_52><loc_10><loc_92><loc_14></location>[141] C. P. Romao, R. Catena, N. A. Spaldin, and M. Matas, 'Chiral phonons as dark matter detectors,' arXiv:2301.07617 [hep-ph] .</list_item>
<list_item><location><page_9><loc_52><loc_9><loc_87><loc_10></location>[142] C. O'Hare, 'cajohare/axionlimits: Axionlimits,'</list_item>
</unordered_list>
<text><location><page_10><loc_56><loc_92><loc_59><loc_93></location>2020.</text>
</document>
[
{
"title": "Absorption of Axion Dark Matter in a Magnetized Medium",
"content": "Asher Berlin ∗ and Tanner Trickle † Theoretical Physics Division, Fermi National Accelerator Laboratory, Batavia, Illinois 60510 (Dated: May 11, 2023) Detection of axion dark matter heavier than a meV is hindered by its small wavelength, which limits the useful volume of traditional experiments. This problem can be avoided by directly detecting in-medium excitations, whose ∼ meV -eV energies are decoupled from the detector size. We show that for any target inside a magnetic field, the absorption rate of electromagnetically-coupled axions into in-medium excitations is determined by the dielectric function. As a result, the plethora of candidate targets previously identified for sub-GeV dark matter searches can be repurposed as broadband axion detectors. We find that a kg · yr exposure with noise levels comparable to recent measurements is sufficient to probe parameter space currently unexplored by laboratory tests. Noise reduction by only a few orders of magnitude can enable sensitivity to the QCD axion in the ∼ 10 meV -10 eV mass range. Introduction.Despite constituting roughly 27% of the energy density of the universe [1], the fundamental nature of dark matter (DM) remains elusive. Of the theoretically motivated DM candidates, the QCD axion is particularly remarkable since its existence would also solve the longstanding strong CP problem [2-5]. A generic feature of QCD axion DM models is a coupling between the axion field a and electromagnetism, In the presence of a static magnetic field B 0 , the interaction in Eq. (1) converts an axion to an oscillating electromagnetic field [6]. Directly detecting this field is the underlying principle of many ongoing and planned experiments [7]. Traditional detection schemes utilize cavities with electromagnetic modes resonantly matched to axion masses of m a ∼ (10 -6 -10 -5 ) eV, as motivated by postinflationary misalignment production and a standard cosmological history [8-12]. However, searches across a larger parameter space are motivated by alternative production mechanisms [13-33] and axions that couple to the Standard Model similarly to the QCD axion but without the strict connection between coupling strength and mass [34-44]. Cavities are an exceptional tool to search for axion DM. However, they are fundamentally limited in the axion mass that they can probe. This is because the axion mass must be matched to a resonant frequency of the cavity, which are inversely related to its size. Therefore, to resonantly search for higher axion masses, the cavity must be prohibitively small, limiting the total exposure. Recent strategies to boost exposure to high-mass axions include non-resonant detection of single-photons in a large volume dish antenna [45], and modifications to the photon's dispersion relation in dielectric [46-48] or plasma [49-51] structures tuned to a specific mass. While these searches are focused on photon detection, another possibility is to directly detect the in-medium excitations in crystal targets involving, e.g., electrons [5280], phonons [76, 77, 81-87], and magnons [82, 88-92]. Since the energy of these modes ( ∼ meV -eV) is not set by the target size, but rather by the physics of the local environment, they are ideal for high-mass axion searches. Furthermore, the manufacturing of low-noise targets and the technology required to detect single quanta of such excitations is at the forefront of the DM direct detection community and is thus an active area of development [93]. In particular, current experiments, such as CDEX [94], DAMIC [95-100], EDELWEISS [101-103], SENSEI [104-106], and SuperCDMS [107-109], utilize eV-scale electronic excitations in Si and Ge targets. More novel targets with sub-eV electronic excitations have also been proposed, such as narrow gap semiconductors [110], Dirac materials [61, 63, 65, 69-72], spin-orbit coupled materials [57, 79], and doped semiconductors [73]. Additionally, phonon excitations have been studied in a wide variety of target materials [76, 80, 82-85, 111], including GaAs and Al 2 O 3 as planned for the TESSARACT experiment [112]. In this Letter , we demonstrate that the entirety of these ideas can be used to search for the axion-photon coupling in Eq. (1), provided that the target can be placed inside a magnetic field, thus creating a 'magnetized medium.' In particular, we show that in a magnetized medium the inclusive axion absorption rate into in-medium excitations is directly related to the dielectric function. While certain signals of axion DM have previously been found to be related to the dielectric function on a case-by-case basis, we show here that this is universal. This is important both experimentally, since the dielectric can be measured, and theoretically, because it broadly captures the absorption rate into any in-medium excitation, abstracting away from calculations specific to any single excitation. This allows us to easily evaluate the sensitivity of various materials, as well as identify a larger scope of relevant signals that have previously been overlooked, such as low-energy electronic excitations. Below, we begin by deriving the absorption rate with two meth- ods. The first derivation involves self-energies, analogous to calculations performed in the context of direct detection experiments; the second is provided within the language of classical axion electrodynamics. These derivations provide complementary ways to understand the underlying physics. We then discuss the projections shown in Fig. 1, which illustrate the promising ability to explore new, high-mass, QCD axion parameter space. Absorption Rate.Before deriving the rate for axion absorption in a magnetized medium, we begin with a synopsis of the final result for isotropic targets. The total axion absorption rate, per unit exposure, is given by where ρ DM glyph[similarequal] 0 . 4 GeV / cm 3 is the local axion DM energy density, ρ T is the mass density of the target, and ε ( m a ) is the dielectric function evaluated at energy ω = m a and momentum deposition q = 0, appropriate for absorption kinematics ( q glyph[lessmuch] ω ) which are assumed throughout. The simplicity of this expression derives from the separability of the axion absorption process as a + B 0 → E followed by absorption of the corresponding electric field. 1 The former process is governed by the strength of the external magnetic field and g aγγ , while the latter is determined by the dielectric function, independent of both the axion physics and, in the q glyph[lessmuch] ω limit, the magnetic permeability. This separability is advantageous since, in principle, the dielectric function of the target can be measured directly. In the absence of measurement, this parameterization is useful as a bridge between particle physics and first principles condensed matter calculations. First principles calculations are a useful tool to understand the contributions of individual excitations, which cannot be understood from a measurement of the inclusive dielectric function. However, this generally ceases to be a problem when the various excitations are sufficiently separated in energy. The idea of relating the dielectric function to the DM absorption rate into in-medium excitations has been used for other DM models [52-54, 56, 79, 80], as well as in calculations of the DM absorption rate into in-medium photon states [50, 51, 120]. For example, for kineticallymixed dark photon DM, A ' , the absorption rate into inmedium excitations is [52, 56, 80] where κ is the kinetic-mixing parameter and m A ' is the A ' mass. The similarity between the dark photon absorption rate in Eq. (3) and the axion absorption rate in Eq. (2) is immediately clear. As a result, for DM particles of the same mass, the sensitivity to electromagnetically-coupled axions can be simply rescaled via the mapping g aγγ B 0 ↔ κm a , corresponding to We now turn to the details of the calculation, beginning with a self-energy treatment similar to Refs. [53, 56, 121], then turning to an alternative derivation employing classical equations of motion, similar to Refs. [87, 122]. Self-Energy Calculation.The starting point of the selfenergy derivation is the optical theorem, which relates the absorption rate of a particle to the imaginary part of its self-energy. The self-energy can then be computed by summing over all the relevant Feynman diagrams. The Feynman diagram for axion absorption in a magnetized medium is shown in Fig. 2; the vertex Feynman rule is derived from the Lagrangian in Eq. (1), and the photon propagator is modified due to the presence of the medium. Following Refs. [53, 56, 121], instead of directly computing the diagram in Fig. 2, we identify the diagrams which mix the axion and photon, and those which do not. If the axion and photon mix, the in-medium fields are those that diagonalize the 2 × 2 self-energy matrix of a and A . The diagonalizing fields ˆ a and ˆ A (the axion-like and photon-like fields, respectively) are a linear combination of a and A . The probability of axion absorption per unit time is then related to the self-energy of ˆ a , written in terms of the unmixed self-energies of the a, A fields, Π aa and Π AA , respectively, and the mixing terms, Π aA , Π Aa . Π λ AA is the self-energy of A projected onto the λ th polarization e λ µ , defined to diagonalize Π µν AA : Π λ AA = -e λ µ Π µν AA e λ ν . Π λ aA is the self-energy mixing a and A , projected onto the same photon polarization vector, i.e., Π λ aA = -e λ µ Π µ aA . In the absence of direct axion couplings to electrons, the only non-zero self-energies (ignoring vacuum processes) are Π µν AA and Π µ aA . Furthermore, since the Ward identities ( Q µ Π µν AA = Q µ Π µ aA = 0, where Q µ = ( ω, q )) relate the temporal and spatial components, only Π ij AA and Π i aA need to be computed. The photon self-energy is determined by the dielectric tensor ε ij through Π ij AA = -ω 2 (1 -ε ij ). Note that the spatial component of the photon polarization vectors diagonalize the dielectric tensor, ε ij = ∑ λ ε λ e i λ e j λ , where ε λ ≡ e λ i ε ij e λ j , such that Π λ AA glyph[similarequal] ω 2 (1 -ε λ ). The axion-photon self-energies are determined by the term in Eq. (1) involving the vector potential which, after integrating by parts, is g aγγ ˙ a A · B 0 . The mixed self-energies are then given by Π i aA = ig aγγ m a B i 0 = -Π i Aa . Substituting Π AA and Π aA into Eq. (5), and taking the q → 0 limit, results in This expression can be further simplified in the limit of an isotropic target. The dielectric of an isotropic target is independent of polarization, ε λ = ε , and the photon polarization vectors are the standard transverse e µ ± = (0 , ˆ q ± ) and longitudinal e µ L = ( q, ω ˆ q ) / √ Q 2 ones, where ˆ q ± are two vectors mutually orthonormal to ˆ q . The sum over polarizations can be performed using ∑ λ e i λ e j λ = δ ij . Applying these approximations, Eq. (6) simplifies to The rate per unit exposure R in Eq. (2) is then obtained by multiplying Γ by the number of axions in the target and dividing by the target mass. Classical Equations of Motion.Alternatively, the absorption rate can be derived using classical axion electrodynamics. Throughout, we will implicitly work in Lorenz gauge. Our starting point is the wave equation for the vector potential, which is approximately ε ∂ 2 t A glyph[similarequal] j a , where j a = g aγγ ˙ a B 0 is the axion effective current and ε is the dielectric tensor. This equation is trivially solved for A by switching to momentum space and projecting onto the photon polarization vectors e λ , which determines the corresponding electric field to be The rate for axion absorption is governed by the axion equation of motion, which is approximately with E given by Eq. (8). The probability per unit time for axion absorption is determined by solving for the imaginary component of the axion frequency in Eq. (9), Γ glyph[similarequal] Im( -ω 2 ) /m a . This leads to a result in agreement with Eq. (6). 2 Projected Sensitivity.The sensitivity of a variety of targets is shown in Fig. 1, assuming a kg · yr exposure and B 0 = 10 T. In our estimates, we demand N > 3 √ 1 + N DC + δ 2 N 2 DC , where N = RM T t and N DC = R DC M T t are the the number of signal events and dark counts (DCs), respectively, R DC is the DC rate, M T is the target mass, t is the exposure time, and δ ≤ 1 is the systematic uncertainty in the DC. In the absence of background, N DC glyph[lessmuch] 1, the sensitivity to the axionphoton coupling scales as g aγγ ∝ ( M T t ) -1 / 2 . If backgrounds are instead significant, N DC glyph[greatermuch] 1, the reliance of the signal on B 0 allows these DCs to be directly measured by removing the magnetic field, thereby suppressing systematic uncertainties, δ glyph[lessmuch] 1. In the statisticallylimited regime, δ glyph[lessmuch] 1 / √ N DC , an observable signal only needs to overcome Poisson fluctuations in noise, such that g aγγ ∝ ( R DC /M T t ) 1 / 4 . Below, we begin by discussing the sensitivity of various targets assuming negligible backgrounds, and then proceed to examine the impact of currently measured noise sources. For m a > ∼ eV, enough energy is deposited to excite an electron across the ∼ 1 eV band gap in standard semiconductors, such as Si and Ge, which is then read out by drifting the charge to a sensing output. This is the operating principle of many ongoing experiments, such as CDEX [94], DAMIC [95-100], EDELWEISS [101-103], SENSEI [104-106], and SuperCDMS [107-109]. In our estimate of the signal rate in Eq. (2), we use the measured dielectric functions of Si and Ge from Ref. [127], and do not incorporate multi-phonon responses at lower energies [80] since these are subdominant to the singlephonon responses of the polar materials discussed below. As shown in Fig. 1, background-free Ge targets have the potential to be the best laboratory-based search for the QCD axion for masses greater than that probed by the CAST helioscope [114, 115] and smaller than that probed by astrophysical searches for a → 2 γ decays [116-118]. While the energy of electronic excitations is limited to ∼ eV scales in standard semiconductors, novel targets have lower ∼ meV electronic excitations. For example, Dirac [61, 63, 65, 69-72] and spin-orbit coupled materials [57, 79] have small bulk band gaps. One such target that falls under both categories is ZrTe 5 ; while its Dirac character is somewhat debated [57], the presence of its small band gap has been firmly established [128]. However, since its dielectric response has not been accurately measured, we adopt the first-principles calculation performed in Ref. [57]. In addition to pure targets, doping is another method to create electronic states below the band gap. Recently, Ref. [73] studied Si doped with phosphorus as a candidate target, using an analytic model for the dielectric response consistent with measurements [129, 130]. In Fig. 1, we rescale their quoted dark photon sensitivity, using the mapping described above. Phonon excitations [76, 80, 82-85, 111] in the ∼ (1 -100) meV energy range have also been studied as an avenue to detect axions [82, 87]. The results shown in Fig. 1 are consistent with the first principles calculation done in Ref. [82]. We have chosen to focus on GaAs, Al 2 O 3 , and SiO 2 targets, as they are of active investigation in the sub-GeV DM community; the first two are currently planned for the TESSARACT experiment [112], and SiO 2 has also been identified as an optimal target for light DM scattering [76]. The measured dielectric data for these targets in the phonon energy regime is taken from Ref. [80]. Backgrounds.The sensitivity of detectors focused on readout of single-electron excitations are currently limited by DCs [93]. The SENSEI experiment [105, 131], utilizing a Si Skipper CCD detector, is the current stateof-the-art for measuring charge from single-electron excitations. Recent measurements indicate a DC rate of 10 8 / kg · yr and 10 6 / kg · yr for energy ranges of < ∼ 4 . 7 eV and ∼ (4 . 7 -8 . 3) eV, respectively, and rates consistent with zero for larger energies. Assuming that similar background levels in the two-electron bin can be achieved in a Ge based detector, noise reduction by three orders of magnitude ( R DC ∼ 10 3 / kg · yr) is necessary to attain sensitivity to the QCD axion at eV-scale masses. Similar noise levels, but at much lower energies, are also necessary for doped Si and ZrTe 5 targets to attain sensitivity to the QCD axion at ∼ 100 meV masses. Calorimetric detectors (e.g., SuperCDMS CPD [132], which measures phonons produced from single-electron excitations) have registered significantly higher noise levels, R DC ∼ 10 10 / kg · yr. As an estimate of the backgrounds that will contaminate sensors based on detection of single-phonon excitations, we assume R DC = 10 10 / kg · yr and δ < ∼ 1 / √ N DC for the dotted colored lines shown in Fig. 1. If the DCs are reduced to just ∼ 10 8 / kg · yr at ∼ 100 meV energies, an Al 2 O 3 or SiO 2 target can be sensitive to the QCD axion at couplings smaller than that bounded by considerations of stellar energy loss. While their origin is unknown, noise levels are generally peaked towards smaller energies [133]. Recent work has focused on progressing the understanding of such backgrounds. For instance, it has been calculated that a subdominant fraction of DCs arises from secondary radiation generated by high-energy tracks [134] and photons [135] in low-threshold charge and phonon detectors, respectively. Additionally, a bulk of eV-scale phonon backgrounds in superconducting calorimetric detectors has recently been shown to emerge from cooling-induced micro-fractures in auxiliary detector components [136]. The setup investigated in our work differs from previously proposed applications of these targets in the use of a large external magnetic field, whose dominant effect will be disrupting detection technology based on superconducting devices (measurements of typical semiconducting sensors have found minor changes to their operation when exposed to ∼ 1 T magnetic fields [137]). For example, transition edge sensors (TES) (the main technology for reading out single-phonon excitations in the TESSARACT experiment) are not operative in > ∼ µ T magnetic fields [138]. However, there is a strong dependence on the direction of the magnetic field relative to the face of the TES. Additionally, if the region of bulk target within the external magnetic field can be physically separated from the superconducting detector, these problems can be avoided. Other complications arising from, e.g., additional radioactive components or Lorentz-force induced mechanical stress may also be introduced. A detailed investigation of these effects is beyond the scope of this work and these will also need to be confronted in other axion experiments operating in the meV -eV energy range [45, 48]. However, we do not expect fundamental roadblocks in this approach since ∼ 10 T magnetic fields only change electronic energies at the level of ∼ meV, well below the energies investigated here. Discussion.The past decade has seen a meteoric rise in target proposals to hunt for sub-GeV DM. In an external magnetic field these targets are also powerful probes of axions in a mass range that is notoriously difficult to explore, corresponding to m a ∼ 10 meV -10 eV. Searching for axions with these targets has the intrinsic advantage of directly utilizing all future experimental improvements in background reduction, an effort which has assiduously driven direct detection experiments to incredible precision in recent history. The axion absorption rate in a magnetized medium can be simply written in terms of the measurable dielectric function, encoding all in-medium responses. This synergizes with future developments towards optimizing the energy loss function of the material, Im( -1 /ε ), as well as further study into other novel low-energy excitations, such as axion quasiparticles [139, 140] or chiral phonons [141]. Acknowledgements. We would like to thank Aaron Chou, Juan Estrada, Roni Harnik, Yoni Kahn, Alex Millar, Tongyan Lin, Andrea Mitridate, Kris Pardo, and Kathryn Zurek for helpful conversations, and Ciaran O'Hare for the compilation of axion limits in Ref. [142]. This material is based upon work supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Superconducting Quantum Materials and Systems Center (SQMS) under contract number DE-AC02-07CH11359. Fermilab is operated by the Fermi Research Alliance, LLC under Contract DE-AC02-07CH11359 with the U.S. Department of Energy. The tessaract dark matter project,' 2020. 2016) 081101. https: //link.aps.org/doi/10.1103/PhysRevB.94.081101 . //link.aps.org/doi/10.1103/PhysRevB.52.16486 . //link.aps.org/doi/10.1103/PhysRevLett.71.3681 . 2020.",
"pages": [
1,
2,
3,
4,
5,
9,
10
]
}
]
2024RAA....24a5001S
https://arxiv.org/pdf/2310.20318.pdf
<document>
<text><location><page_1><loc_12><loc_89><loc_47><loc_90></location>Research in Astron. Astrophys. Vol.0 (20xx) No.0, 000-000</text>
<text><location><page_1><loc_12><loc_87><loc_28><loc_88></location>http://www.raa-journal.org</text>
<text><location><page_1><loc_32><loc_87><loc_51><loc_88></location>http://www.iop.org/journals/raa</text>
<text><location><page_1><loc_12><loc_86><loc_13><loc_87></location>(L</text>
<text><location><page_1><loc_13><loc_86><loc_13><loc_87></location>A</text>
<text><location><page_1><loc_13><loc_86><loc_14><loc_87></location>T</text>
<text><location><page_1><loc_14><loc_86><loc_49><loc_87></location>X: ms2023-0221.tex; printed on November 1, 2023; 1:19)</text>
<text><location><page_1><loc_14><loc_86><loc_15><loc_87></location>E</text>
<text><location><page_1><loc_68><loc_86><loc_78><loc_89></location>R esearchin A stronomyand A strophysics</text>
<section_header_level_1><location><page_1><loc_12><loc_73><loc_78><loc_77></location>Formation of a rapidly rotating classical Be-star in a massive close binary system</section_header_level_1>
<text><location><page_1><loc_12><loc_69><loc_24><loc_70></location>Evgeny Staritsin</text>
<text><location><page_1><loc_12><loc_61><loc_74><loc_63></location>K.A. Barkhatova Kourovka Astronomical Observatory, B.N. Yeltsin Ural Federal University, pr. Lenina 51, Ekaterinburg 620000, Russia; Evgeny.Staritsin@urfu.ru</text>
<text><location><page_1><loc_16><loc_40><loc_74><loc_55></location>Abstract This paper investigates the spin-up of a mass-accreting star in a close binary system passing through the first stage of mass exchange in the Hertzsprung gap. Inside an accreting star, angular momentum is carried by meridional circulation and shear turbulence. The circulation carries part of the angular momentum entering the accretor to its surface. The greater the rate of arrival of angular momentum in the accretor is, the greater this part. It is assumed that this part of the angular momentum can be removed by the disk further from the accretor. If the angular momentum in the matter entering the accretor is more than half the Keplerian value, then the angular momentum obtained by the accretor during mass exchange stage does not depend on the rate of arrival of angular momentum. The accretor may have the characteristics of a Be-star immediately after the end of mass exchange.</text>
<text><location><page_1><loc_16><loc_35><loc_74><loc_37></location>Key words: stars: binaries: close - stars: rotation - stars: early-type - stars: emission line, Be</text>
<section_header_level_1><location><page_1><loc_12><loc_30><loc_27><loc_31></location>1 INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_12><loc_20><loc_79><loc_28></location>Classical Be-stars include OBA stars with observed or previously observed emissions in the Balmer lines of hydrogen (Porter & Rivinius 2003). These stars are not supergiants and have large rotational velocities. Among Be-stars, there is the group of early spectral subclasses (B3-O9). The surface rotational velocities of these stars range widely. The lower range limit is 40%-60% of the Keplerian value, while the upper limit is 90%-100% (Cranmer 2005). The origin of the large rotational velocities of Be-stars is not clear.</text>
<text><location><page_1><loc_12><loc_12><loc_79><loc_19></location>Young B-stars in the early spectral subclasses and O-stars are characterized by lower rotational velocities (Huang et al. 2010). 70% of these stars are observed in binary and multiple systems (Chini et al. 2012; Sana et al. 2012). All these stars are expected to form binary and multiple systems, considering selection effects. Mass exchange in a binary system may be the reason for the rapid rotation of the star receiving mass. The synthesis of the Be-stars population in binary systems makes it possible to</text>
<text><location><page_2><loc_12><loc_84><loc_79><loc_87></location>reproduce the observed number of these stars in the Galaxy (Pols et al. 1991; Portegies Zwart 1995; Van Bever & Vanbeveren 1997; Shao & Li 2014; Hastings et al. 2021).</text>
<text><location><page_2><loc_12><loc_51><loc_79><loc_82></location>A simple estimation made assuming the instantaneous redistribution of angular momentum in the stellar interior to solid-state rotation shows that a 5% - 10% increase to the star's mass due to accretion of mass with Keplerian velocity leads to a critical rotation state (Packet 1981). The question of what happens when there is continued accretion into a star close to a state of critical rotation has been discussed in Paczy ' nski (1991), Popham & Narayan (1991), Colpi et al. (1991), and Bisnovatyi-Kogan (1993). Paczy ' nski (1991), Popham and Narayan (1991), and Colpi et al. (1991) used various approaches. All authors agree that accretion does not stop when the star's speed of rotation reaches a critical value. Paczy ' nski (1991) studied the whole star-boundary layer-accretion disk system for various rotations of the central star. For models rotating slightly above critical, mass accretion is accompanied by the loss of angular momentum from the star to the disk, mediated by viscous stresses. However, the solutions obtained in Paczy ' nski (1991), Popham and Narayan (1991), and Colpi et al. (1991) are not self-consistent. The condition for a self-consistent solution for a system consisting of a star in a state of critical rotation and an accretion disk is that 'the star absorbs accreted matter with a certain angular momentum, such that the star remains in a state of critical rotation' (Bisnovaty-Kogan 1993). Let J ( M ) be the angular momentum of a star with mass M in a state of critical rotation and let j Kep e be the specific Keplerian angular momentum at the equator of the star. Then j a = dJ/dM < j Kep e . A mass-accreting star can move along the sequence of stars in a state of critical rotation J ( M ) if the excess angular momentum of /triangle j = j Kep e -j a is eliminated. Bisnovaty-Kogan (1993) constructed models of accretion disks that remove excess angular momentum from the surface of a star. At the same time, the speed of rotation at the star's surface remains critical. So an increase in the mass and angular momentum of a star in a critical rotation state may occur due to the removal of excess angular momentum from the star by the accretion disk (Paczy ' nski 1991; Bisnovatyi-Kogan 1993).</text>
<text><location><page_2><loc_12><loc_33><loc_79><loc_48></location>Physical processes, such as meridional circulation and turbulence, require finite amounts of time to transfer angular momentum (Staritsin 2019, 2021). At the very beginning of accretion, only the outer layers of the star, including the accreted mass, have a fast rotation. The star surface gains critical rotation shortly after the start of accretion. Later, at the accretion stage in a state of accretor critical rotation, the circulation carries part of the angular momentum brought along with the accreted mass from subsurface layers to the star's surface (Staritsin 2022). Thus, accreted layers can shrink, as usually happens during accretion. The angular momentum transferred by circulation to the star's surface can be removed through an accretion disk (Paczy ' nski 1991; Bisnovatyi-Kogan 1993). Thus, the mass and angular momentum of an accretor in a state of critical rotation increase due to the removal of excess angular momentum from the accreted layers to the accretor surface and the further removal of this angular momentum from the star.</text>
<text><location><page_2><loc_12><loc_22><loc_79><loc_31></location>In Staritsin (2022), the transfer of angular momentum in the accretor interior was carried out only by meridional circulation. The turbulence was artificially suppressed. This made it possible to elucidate the transport properties of circulation in an accreting star's interior. The role of turbulence in angular momentum transport within the accretor remained unclear. As to the angular momentum input, only one option has been considered, the effective transport of angular momentum from the disk's boundary layer to the accretor's upper layer.</text>
<text><location><page_2><loc_12><loc_12><loc_79><loc_20></location>In this paper, we consider two mechanisms of angular momentum transfer in an accreting star interior, namely circulation and turbulence. This allows us to find the role of turbulence in the spinning up of a star. We also took into account the possible reduction of input angular momentum. This decrease can be attributed both to the transfer of angular momentum from the boundary layer to the outer parts of the disk, and to sub-Keplerian rotation in the disk. The accretor's rotation, obtained as a result of mass exchange, has been studied depending on the angular momentum introduced during mass exchange.</text>
<section_header_level_1><location><page_3><loc_12><loc_86><loc_47><loc_87></location>2 BASIC EQUATIONS AND SIMPLIFICATIONS</section_header_level_1>
<section_header_level_1><location><page_3><loc_12><loc_83><loc_36><loc_84></location>2.1 The angular momentum input</section_header_level_1>
<text><location><page_3><loc_12><loc_73><loc_79><loc_82></location>The matter lost by the donor due to the filling of the Roche lobe falls into the accretor's gravitational field and swirls around it. The formation of gas structures around the accretor, in particular the formation of a disk and a velocity field in it, depends on the ratio between three factors : the size of the accretor R , the minimum distance ω min from the center of the accretor to the central line of the stream of matter falling from donor point L 1 , and the distance from the accretor's center to the edge of the inviscid disk ω d (Lubow & Shu 1975).</text>
<text><location><page_3><loc_12><loc_66><loc_79><loc_73></location>Transient disks with sub-Keplerian rotation have been found (for example, RW Tau (Kaitchuck & Honeycutt 1982) and β Per (Cugier & Molaro 1984, Richards 1992)) in direct-impact systems ( ω d < R ). Three-dimensional hydrodynamic calculations show disk formation in such systems; the rotation velocity is 80% and 60% of the Keplerian value at the inner and outer edges of the disk, respectively (Raymer 2012).</text>
<text><location><page_3><loc_12><loc_56><loc_79><loc_66></location>Both transient disks (SW Cyg) and permanent, but variable, accretion disks (RY Gem, TT Hya, AU Mon) in grazing-impact systems ( ω min < R < ω d ) have been discovered. The velocity fields in the transient disk of the SW Cyg system and in the permanent disk of the RY Gem system are subKeplerian (Kaitchuck 1988, 1989). Asymmetric parts were found in the disks of the TT Hya and AU Mon systems; the gas in the disk's asymmetric part in the AU Mon system moves at a sub-Keplerian velocity (Richards et al. 2014). Hydrodynamic calculations also show the possibility of disk formation at sub-Keplerian velocities in these systems (Richards & Ratliff 1998).</text>
<text><location><page_3><loc_12><loc_52><loc_79><loc_56></location>Permanent disks are found in systems with R < ω min . The radial component of the matter velocity in the disk is directed towards the accretor and is 10-30 km/s . The change in the tangential component with distance from the accretor may differ from the Keplerian one (Etzel et al. 1995).</text>
<text><location><page_3><loc_12><loc_33><loc_79><loc_51></location>The aforementioned observational data and the results of hydrodynamic calculations relate to systems with the low mass of accreting components ( M ≤ 6 M /circledot ) and with a ratio of donor mass to accretor mass within the range of 0.2 to 0.3. The formation of Be-stars in the early spectral subclass occurs in systems with large component masses. The ratio of donor mass to accretor mass varies widely. Mass transfer in such systems is non-conservative (Van Rensbergen et al. 2011; Deschamps et al. 2015). The star receiving mass increases in volume (Benson 1970; Kippenhahn & Meyer-Hofmeister 1977). The distance between two stars depends on how the system loses mass and angular momentum. So, the formation of gas structures in the Roche lobe of an accretor depends on the loss of mass and angular momentum from the system. A quantitative theory of mass and angular momentum losses from a close binary system has not yet been developed. The formation of conditions for sub-Kelerian rotation in an accretion disk due to the loss of mass and angular momentum from the binary system cannot be ruled out. Thus, the possibility of mass accretion with sub-Keplerian velocities of rotation should be considered.</text>
<text><location><page_3><loc_12><loc_27><loc_79><loc_33></location>At the very beginning of accretion, when accretor rotation velocity is low, the rotation velocity of disk matter decreases in the narrow boundary layer from the maximum value in the disk Ω max to the value on the star's surface Ω s (Paczy ' nski 1991). Turbulence can remove angular momentum from the boundary layer to an accretor's upper layers at a rate of:</text>
<formula><location><page_3><loc_36><loc_23><loc_79><loc_26></location>dJ dt = 2 3 R 2 (Ω max -Ω s ) ˙ M, (1)</formula>
<text><location><page_3><loc_12><loc_20><loc_66><loc_22></location>where J - angular momentum of the accretor, t - time, and ˙ M - mass accretion rate.</text>
<text><location><page_3><loc_12><loc_15><loc_79><loc_20></location>Supersonic shear flow in the boundary layer is a source of acoustic waves. The waves can carry the angular momentum out of the boundary layer both into the accretor's outer part and the disk's outer part (Dittmann 2021, Coleman et al. 2022). In this case, the amount of angular momentum coming from the boundary layer into the accretor is less than the Keplerian one.</text>
<text><location><page_3><loc_12><loc_12><loc_79><loc_14></location>In an earlier study (Staritsin 2022), we considered as follows: when at the stage of subcritical rotation, angular momentum enters the accretor through two channels, namely together with matter having</text>
<text><location><page_4><loc_12><loc_81><loc_79><loc_87></location>the same rotation velocity as the accretor's surface and due to turbulence within the rate (1). This is a case of high efficiency of angular momentum transfer from the boundary layer to the accretor's upper part. The transfer of angular momentum in the accretor's interior was carried out by meridional circulation; turbulence was artificially suppressed.</text>
<text><location><page_4><loc_12><loc_77><loc_79><loc_81></location>In the current calculations, angular momentum transfer in the accretor's interior can be carried out both by meridional circulation and turbulence. We have studied two variants for the arrival of angular momentum into the accretor.</text>
<text><location><page_4><loc_12><loc_65><loc_79><loc_77></location>In the first variant, to clarify the influence of angular momentum transport by turbulence in the accretor's interior on the spinning up of the accretor, we calculated accretion with the same rate of the arrival of angular momentum into the accretor as in Staritsin (2022). At the stage of subcritical rotation, the parameter Ω max in the angular momentum source (1) is equal to α Ω Kep , where α = 0 . 8 ; here, Ω Kep is the Keplerian velocity of the star's surface at the equator. After the angular velocity of the accretor's surface increases to α Ω Kep value, the arrival of angular momentum from the boundary layer (1) stops. The angular velocity of the adding matter is set equal to α Ω Kep for the remainder of the mass exchange.</text>
<text><location><page_4><loc_12><loc_54><loc_79><loc_65></location>In the second variant, the case of extremely low efficiency of angular momentum transfer from the boundary layer to the accretor's upper part is considered. The angular momentum's source (1) in this case is assumed to be zero. As long as the angular velocity of the star's surface is less than α Ω Kep , the star accretes matter with the same angular velocity as that of the star's surface. After the surface angular velocity increases to the value of α Ω Kep , the angular velocity of the adding matter remains equal to α Ω Kep . In order to determine the dependence of the accretor's rotation state after the end of mass exchange on the content of angular momentum in adding mass, calculations were carried out at four values of α : 0 . 8 , 0 . 6 , 0 . 4 , and 0 . 2 .</text>
<section_header_level_1><location><page_4><loc_12><loc_51><loc_52><loc_52></location>2.2 Angular momentum transfer in the accretor's interior</section_header_level_1>
<text><location><page_4><loc_12><loc_44><loc_79><loc_49></location>Angular momentum transfer in the radiative layers of a star is taken into account in the framework of the shellular rotation model (Zahn 1992). In terms of this model, two mechanisms of angular momentum transfer are considered: meridional circulation and shear turbulence. The angular momentum transfer is described by the law of conservation of angular momentum (Tassoul 1978):</text>
<formula><location><page_4><loc_29><loc_40><loc_60><loc_43></location>∂ ( ρ/pi1 2 Ω) ∂t + div ( ρ/pi1 2 Ω u ) = div ( ρν v /pi1 2 grad Ω) .</formula>
<text><location><page_4><loc_12><loc_36><loc_79><loc_39></location>The meridional circulation velocity u is determined from the law of conservation of energy in a stationary form (Maeder & Zahn 1998):</text>
<formula><location><page_4><loc_30><loc_34><loc_59><loc_35></location>ρT u grad s = ρε n + div ( χ grad T ) -div F h .</formula>
<text><location><page_4><loc_12><loc_21><loc_79><loc_33></location>In these equations, ρ - density, /pi1 - distance to the axis of rotation, Ω - angular velocity ν v - turbulent viscosity in the vertical direction, T - temperature, s - specific entropy, ε n - nuclear energy release rate, χ - thermal conductivity, F h - turbulent enthalpy flow in the horizontal direction: F h = -ν h ρT∂s/∂ i θ and ν h - turbulent viscosity in the horizontal direction. The coefficients of turbulent viscosity were determined by Talon and Zahn (1997), Maeder (2003), and Mathis et al. (2004). The convective core rotates solid-state. These equations are solved together with equations related to the structure and evolution of stars. We used a set of programs from Paczy ' nski (1970) modified to calculate the evolution of rotating stars (Staritsin 1999, 2005, 2007, 2009, 2014).</text>
<section_header_level_1><location><page_4><loc_12><loc_18><loc_33><loc_19></location>3 CALCULATION RESULTS</section_header_level_1>
<section_header_level_1><location><page_4><loc_12><loc_16><loc_33><loc_17></location>3.1 Binary system parameters</section_header_level_1>
<text><location><page_4><loc_12><loc_12><loc_79><loc_14></location>We consider mass exchange in a binary system with the component masses of 13.4 M /circledot and 10.7 M /circledot and the period P = 35 d as in Staritsin (2022). By the beginning of mass exchange, star rotation with</text>
<figure>
<location><page_5><loc_12><loc_53><loc_78><loc_86></location>
<caption>Fig. 1: Angular velocity at the bottom of the outer cell of the meridional circulation at the beginning of mass exchange.</caption>
</figure>
<text><location><page_5><loc_12><loc_42><loc_79><loc_46></location>a mass of 10.7 M /circledot is synchronized with orbital motion. The star angular momentum is equal to 1 . 3 × 10 51 g · cm 2 s -1 . A star with a mass of 13.4 M /circledot loses 10.5 M /circledot for 12,000 years. After that, the star decouples its Roche lobe and the mass exchange stage ceases.</text>
<text><location><page_5><loc_12><loc_28><loc_79><loc_42></location>The second star accretes 5.3 M /circledot . The final mass of the accretor is 16.0 M /circledot . The accretion rate was set constant, equal to the average value of ∼ 4 . 4 × 10 -4 M /circledot /year . We consider a case when the entropy of the added matter is the same as the surface layers of the second star. The thermal timescale of the second star is longer than mass exchange duration. The star's reaction to the increase in mass in this case is well understood (Benson 1970; Flannery and Ulrich 1977; Neo et al. 1977). The second star is driven out of thermal equilibrium by mass accretion. Nuclear power output in the center of the second star increases, and some of the nuclear energy release is spent on an increase in entropy in the second star's central parts. Gravitational energy release in the surface layers is added to nuclear energy release in the center. The typical luminosity distribution in the second star's interior is shown in Staritsin (2022) (see Fig. 4).</text>
<text><location><page_5><loc_12><loc_22><loc_79><loc_27></location>The remaining part of the mass lost by the first star leaves the system. The tidal interaction between the two stars is unable to synchronize the accreting star with the orbit due to the long period of the system and the short accretion timescale. The accretion of matter and angular momentum, as well as transport processes inside the accretor and in the disk, determine the accretor's angular momentum.</text>
<section_header_level_1><location><page_5><loc_12><loc_17><loc_77><loc_20></location>3.2 The case of the high efficiency of angular momentum transfer from the boundary layer to the accretor's upper part</section_header_level_1>
<text><location><page_5><loc_12><loc_12><loc_79><loc_16></location>With the beginning of mass exchange, a circulation cell is formed in the subsurface layer of the accretor, in which the circulation carries the incoming angular momentum downwards. The cell consists of accreted layers and the swirled layers of the accretor located below. In the cell's upper part, angular ve-</text>
<figure>
<location><page_6><loc_12><loc_53><loc_78><loc_86></location>
<caption>Fig. 2: Turbulent (dashed-line), advective (dot-and-dashed line), and total (solid line) angular momentum flux in the accretor's interior at the stage of subcritical rotation.</caption>
</figure>
<text><location><page_6><loc_12><loc_43><loc_79><loc_47></location>locity has an almost constant value, but near the bottom of the cell, it sharply reduces to the initial value (Fig. 1). Therefore, in the lower part of the cell, the contribution of turbulence to angular momentum transfer is greater and exceeds the contribution of meridional circulation (Fig. 2). The bottom of the cell</text>
<table>
<location><page_6><loc_26><loc_30><loc_65><loc_38></location>
<caption>Table 1: Angular momentum balance.</caption>
</table>
<unordered_list>
<list_item><location><page_6><loc_12><loc_26><loc_56><loc_27></location>( J 1 ) the angular momentum that entered the accretor during mass exchange;</list_item>
<list_item><location><page_6><loc_12><loc_25><loc_58><loc_26></location>( J 2 ) the angular momentum removed from the accretor during mass exchange;</list_item>
<list_item><location><page_6><loc_12><loc_24><loc_47><loc_25></location>( J 3 ) the angular momentum remaining in the accreted mass;</list_item>
<list_item><location><page_6><loc_12><loc_23><loc_68><loc_24></location>( J 4 ) the angular momentum transferred to the part of the accretor that made up the star initially;</list_item>
<list_item><location><page_6><loc_12><loc_22><loc_50><loc_23></location>( J 5 ) the angular momentum of the accretor after mass exchange,</list_item>
<list_item><location><page_6><loc_12><loc_20><loc_47><loc_21></location>the angular momentum is given in units of 10 52 g · cm 2 s -1 .</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_6><loc_12><loc_17><loc_79><loc_19></location>(Case 1) angular momentum transfer from the boundary layer is considered, turbulence in the accretor's interior is artificially suppressed (Staritsin 2022);</list_item>
<list_item><location><page_6><loc_12><loc_14><loc_79><loc_17></location>(Case 2) angular momentum transfer from the boundary layer is considered, turbulence in the accretor interior is considered;</list_item>
</unordered_list>
<text><location><page_6><loc_12><loc_12><loc_79><loc_14></location>(Case 3), (Case 4), (Case 5), and (Case 6) angular momentum transfer from the boundary layer is not considered, turbulence in the accretor interior is present, and α is equal to 0.8, 0.6, 0.4, and 0.2 respectively.</text>
<figure>
<location><page_7><loc_12><loc_53><loc_78><loc_86></location>
<caption>Fig. 3: Angular velocity when the rotation of the accretor's surface becomes critical and when active turbulence (solid line) and artificially suppressed turbulence are present (Staritsin 2022) (dashed line).</caption>
</figure>
<text><location><page_7><loc_12><loc_39><loc_79><loc_46></location>goes down into the star faster than when turbulence is artificially suppressed. The angular momentum entering the accretor is distributed over a larger mass of matter than in the case of suppressed turbulence. The rotation of the accretor's surface becomes critical when its mass increases to 11.3 M /circledot (in the case of suppressed turbulence - up to 11.0 M /circledot (Staritsin 2022)). The distribution of angular velocity in the accretor's interior at this moment is shown in Fig. 3.</text>
<text><location><page_7><loc_12><loc_31><loc_79><loc_39></location>At the stage of critical rotation, the mass of the accretor increases by another 4.7 M /circledot . Another circulation cell is formed in the accreted matter. In this cell, the circulation transfers the some part of the angular momentum that came along with the accreted mass to the surface of the accretor (Fig. 4). It is assumed that this part of the angular momentum is removed from the accretor by the accretion disk (Paczy ' nski 1991; Bisnovatyi-Kogan 1993). As a result of a decrease in angular momentum, the accreted layers are contracted. The velocity of their rotation is permanently lower than the Keplerian velocity.</text>
<text><location><page_7><loc_12><loc_21><loc_79><loc_30></location>In the circulation cell formed at the beginning of mass exchange, the transfer of angular momentum inside the accretor continues. The mass of the matter in this cell increases as the upper boundary of the cell moves up along the accretor mass, and the bottom of the cell moves down. The bottom of the cell goes down to the convective core when the accretor mass increases to 11.9 M /circledot (in the case of suppressed turbulence - up to 15 M /circledot (Staritsin 2022)). The role of turbulence lies in the rapid lowering of the bottom of the circulation cell, in which the circulation carries angular momentum into the star's interior.</text>
<text><location><page_7><loc_12><loc_12><loc_79><loc_20></location>The amount of angular momentum removed from the accretor during mass exchange depends slightly on processes of angular momentum transfer within the accretor (Fig. 5). When turbulence is present, the amount of angular momentum transferred to the accretor's inner layers increases, and the amount of angular momentum carried to the accretor's surface decreases compared to the case of suppressed turbulence (Table 1). The angular momentum brought into the accretor during mass exchange is 1 . 72 × 10 53 g · cm 2 s -1 . 12% of this value enters the inner layers that made up the accretor initially,</text>
<figure>
<location><page_8><loc_12><loc_53><loc_76><loc_86></location>
<caption>Fig. 4: Angular momentum flux in the accretor's interior when its mass is equal to 12 M /circledot , 14 M /circledot , and 16 M /circledot .</caption>
</figure>
<text><location><page_8><loc_12><loc_41><loc_79><loc_46></location>31% remains in the accreted mass, and 57% is carried to the accretor's surface and is removed by the disk. In the case of suppressed turbulence, the corresponding values are 5%, 30%, and 65% (Staritsin 2022). After the end of mass exchange, the accretor's angular momentum is greater when turbulence is present (Table 1).</text>
<section_header_level_1><location><page_8><loc_12><loc_36><loc_79><loc_39></location>3.3 The case of extremely low efficiency of angular momentum transfer from the boundary layer to the accretor's upper part</section_header_level_1>
<text><location><page_8><loc_12><loc_28><loc_79><loc_35></location>At the beginning of mass exchange, the rotation velocity of the incoming mass and the accretor's surface coincide. The rate of angular momentum arrival into the accretor is significantly less than when turbulence and/or waves transfer angular momentum from the boundary layer to the accretor's outer part. Due to the low rate at which angular momentum enters the accretor, the accretor's angular momentum increases slowly at the beginning of accretion (Fig. 5).</text>
<text><location><page_8><loc_12><loc_12><loc_79><loc_28></location>The general picture of angular momentum transfer in the accretor's interior at α equal to 0.8 and 0.6 is the same as when angular momentum goes from the boundary layer to the accretor's upper part.The difference is that the total amount of angular momentum that has entered the accretor during mass exchange decreases (Table 1). Reasons for the decrease are associated with the absence of a source (1) and a decrease in parameter α . However, with α equal to 0.8 and 0.6, the accretor surface's rotation velocity increases to a critical value. This occurs when accretor mass increases to 12.9 M /circledot and 13.3 M /circledot , respectively (Fig. 6). In these cases, a circulation cell is formed in the accretor's outer layer, in which the circulation transfers part of the angular momentum of the accreted layers to the accretor's surface. The amount of angular momentum removed from the accreted layers and lost by the accretor in these cases is less than in calculations with a source (1) (Table 1). The state of accretor rotation once mass exchange finishes is approximately the same as when the angular momentum's arrival from the boundary layer to</text>
<figure>
<location><page_9><loc_12><loc_53><loc_76><loc_86></location>
<caption>Fig. 5: The angular momentum amount entering the accretor (dashed line), the angular momentum of the accretor (solid line) depending on its mass at α = 0 . 8 when the source of the angular momentum (1) is considered (blue color) and is not taken into account (black color).The case of artificially suppressed turbulence (Staritsin 2022) is also shown (dot-and-dashed line).</caption>
</figure>
<text><location><page_9><loc_12><loc_39><loc_79><loc_43></location>the accretor's upper part was considered. A decrease in angular momentum entering the accretor only results in decreases in the angular momentum taken out of the accretor at the stage of accretion during accretor critical rotation.</text>
<text><location><page_9><loc_12><loc_23><loc_79><loc_39></location>At α equal to 0.4 and 0.2, a smaller amount of angular momentum enters the accretor (Table 1). The rotation velocity of the accretor's surface remains subcritical throughout the entire mass exchange stage; at α equal to 0.4, it approaches the critical value by the end of this stage. In both cases, at the beginning of mass exchange, a circulation cell is formed in the accretor's subsurface layer, in which the angular momentum of the accreted matter is transferred inside the accretor. The bottom of the cell goes down to the convective core when the mass of the accretor increases to 13.1 M /circledot at α equal to 0.4 and up to 13.9 M /circledot at α equal to 0.2. In both cases, the angular momentum of the accreted mass is transferred inside the star throughout the mass exchange stage. The accretor retains all the angular momentum received with the accreted mass (Fig. 6). Once mass exchange finishes, the angular momentum of the accretor at α equal to 0.4 is little less than at α equal to 0.6 and 0.8, and at α equal to 0.2 is significantly less (Table 1).</text>
<section_header_level_1><location><page_9><loc_12><loc_20><loc_45><loc_21></location>3.4 Accretor rotation state after mass exchange</section_header_level_1>
<text><location><page_9><loc_12><loc_12><loc_79><loc_19></location>The distribution of the angular velocity of rotation in the accretor's interior immediately after the end of mass exchange is shown in Fig. 7. In all cases, angular velocity decreases rapidly in a layer of variable chemical composition located between the chemically homogeneous part of the radiative envelope and the convective core. A similar jump is formed in cases where angular momentum enters the accretor in a short time - in the donor's thermal timescale or faster (Staritsin 2021). The thermal timescale of</text>
<figure>
<location><page_10><loc_12><loc_53><loc_76><loc_86></location>
<caption>Fig. 6: The angular momentum amount entering the accretor (dashed line), the angular momentum of the accretor (solid line) depending on its mass at α = 0 . 8 (black), α = 0 . 6 (red), α = 0 . 4 (green), and α = 0 . 2 (orange) when the source of angular momentum (1) is not considered. At α = 0 . 4 and α = 0 . 2 , the accretor retains the entire angular momentum obtained with the accreted mass, so the dashed lines coincide with the solid ones.</caption>
</figure>
<text><location><page_10><loc_12><loc_38><loc_79><loc_42></location>the accretor is longer than that of the donor in the cases considered in Staritsin (2021, 2022). After the end of mass exchange, the jump gradually decreases and disappears during the thermal timescale of the accretor (see, for example, Fig. 3 in Staritsin (2021)).</text>
<text><location><page_10><loc_12><loc_28><loc_79><loc_38></location>The angular velocity in the accretor's interior after mass exchange when α is equal to 0.8 and α is equal to 0.6 almost does not depend on what the content of the angular momentum was in the adding mass and on whether angular momentum is transferred from the boundary layer to the accretor's upper layer (Fig. 7). In these cases, the accretor's angular momentum is almost equal to ∼ 7 . 5 × 10 52 g · cm 2 s -1 (Table 1) after the mass exchange. An isolated star with a mass of 16 M /circledot has a critical rotation throughout the stage of hydrogen burning in the core with this angular momentum value (Staritsin 2007). Consequently, due to the exchange of mass, the accretor receives a rotation typical for Be-stars.</text>
<text><location><page_10><loc_12><loc_23><loc_79><loc_28></location>At α equal to 0.4, the accretor receives almost the same angular momentum with accreted mass as the angular momentum that remains in the accretor when α is 0.8 and α is 0.6 (Table 1). Therefore, at α equal to 0.4, the accretor also has a rotation typical for Be-stars.</text>
<text><location><page_10><loc_12><loc_12><loc_79><loc_23></location>At α equal to 0.2, the accreted mass brings a much lower angular momentum (Table 1). The angular velocity in the accretor's interior is lower than in other cases (Fig. 7). The rotation of the accretor's surface immediately after mass exchange in this case is lower than that of Be-type stars. In an isolated star with the same mass and angular momentum as the accretor, the removal of angular momentum from the inner layers to the outside occurs intensively at the stage of hydrogen burning in the core (Staritsin 2007, 2009). The angular velocity of the star's surface, expressed in Keplerian angular velocity, increases; at the last steps of this stage, the star acquires a rotation typical for Be-stars of the early spectral subclass. If tidal interaction is low, then even in this case the accretor can obtain the characteristics of a Be-star</text>
<figure>
<location><page_11><loc_12><loc_53><loc_77><loc_87></location>
<caption>Fig. 7: The distribution of angular velocity in the accretor's interior after the end of mass exchange in cases when the source of angular momentum (1) is considered (blue) and not considered at α = 0 . 8 (black), α = 0 . 6 (red), α = 0 . 4 (green), and α = 0 . 2 (orange).</caption>
</figure>
<text><location><page_11><loc_12><loc_42><loc_79><loc_45></location>after the end of mass exchange, but only after a long period of time on the order of part of the hydrogen burning stage in the core.</text>
<section_header_level_1><location><page_11><loc_12><loc_39><loc_25><loc_40></location>4 CONCLUSIONS</section_header_level_1>
<text><location><page_11><loc_12><loc_22><loc_79><loc_38></location>Meridional circulation is a flexible mechanism for the transfer of angular momentum in the stellar interior of a rotating star. The direction and rate of angular momentum transfer by circulation may vary widely at the stage of mass accretion, depending on the star rotation state and the rate of angular momentum arrival along with the accreted mass and waves and/or due to turbulence. The two main circulation cells are formed due to the accretion of mass and angular momentum. In the cell, which is formed at the stage of subcritical rotation of the accretor, circulation transfers the angular momentum inside the accretor. Only in the lower part of this cell does turbulence make the main contribution to the transfer of angular momentum. Due to turbulence, the cell bottom quickly goes downwards into the accretor's interior. In a cell formed at the stage of critical rotation, the circulation transfers part of the angular momentum of the accreted mass to the surface of the star; the more the content of the angular momentum in the entering matter is, the greater this part.</text>
<text><location><page_11><loc_12><loc_12><loc_79><loc_22></location>We have considered the case of mass exchange in a binary system, when half of the mass lost by the donor falls into the accretor. If the angular momentum of the mass falling into the accretor exceeds half the Keplerian value at the boundary of the accretor, the state of rotation of the accretor after the end of mass exchange does not depend on the angular momentum entering the accretor. In other words, processes that could reduce the angular momentum of the matter located around the accretor to no more than half the Keplerian value do not affect the angular momentum and the state of rotation that the accretor receives by the end of the mass exchange stage. These processes impact on the amount of</text>
<text><location><page_12><loc_12><loc_84><loc_79><loc_87></location>angular momentum removed by circulation from the accreted mass to the accretor's surface and removed further from the accretor by a disk or other processes.</text>
<text><location><page_12><loc_12><loc_80><loc_79><loc_84></location>In the considered system with the initial component masses of 13.4 M /circledot and 10.7 M /circledot , the accretor has a rotation typical for Be-stars immediately after the end of mass exchange, if during mass exchange the angular momentum of the mass added to the accretor exceeded 40% of the Keplerian value.</text>
<text><location><page_12><loc_12><loc_76><loc_79><loc_78></location>Acknowledgements This work was supported by the Ministry of Science and Education, FEUZ-20230019</text>
<section_header_level_1><location><page_12><loc_12><loc_73><loc_19><loc_74></location>References</section_header_level_1>
<code><location><page_12><loc_12><loc_12><loc_79><loc_71></location>Benson, R.S. 1970, BAAS, 2, 295 Bisnovatyi-Kogan, G.S. 1993, A&A, 274, 796 Chini, R., Hoffmeister, V.H., Nasseri, A., Stahl, O., & Zinnecker, H. 2012, MNRAS, 424, 1925 Coleman, M.S.B., Rafikov, R.R., & Philippov, A.A. 2022, MNRAS, 512, 2945 Colpi, M., Nannurelli, M., & Calvani, M. 1991, MNRAS, 253, 55 Cranmer, S.R. 2005, ApJ, 634, 585 Cugier, H., & Molaro, P. 1984, A&A, 140, 105 Deschamps, R., Braun, K., Jorissen, A., et al. 2015, A&A, 528, A16 Dittmann, A.J. 2021, MNRAS, 508, 1842 Etzel, P.B., Olson, E.C., & Senay, M.C. 1995 AJ, 109, 1269 Flannery, B.P., & Ulrich, R.K. 1977, ApJ, 212, 533 Hastings, B., Langer, N., Wang, C., Schootemeijer, A., & Milone, A.P. 2021, A&A, 653, A144 Huang, W., Gies, D.R., & McSwain, M.V. 2010, ApJ, 722, 605 Kaitchuck, R.H., & Honeycutt, R.K. 1982, ApJ, 258, 224 Kaitchuck, R.H. 1988, PASP, 100, 594 Kaitchuck, R.H. 1989, Space Sci. Rev., 50, 51 Kippenhahn, R., & Meyer-Hofmeister, E. 1977, A&A, 54, 539 Lubow, S.H., & Shu, F.H. 1975, ApJ, 198, 383 Maeder, A. 2003, A&A, 399, 263 Maeder, A., & Zahn, J.-P. 1998, A&A, 334, 1000 Mathis, S., Palacios, A., & Zahn, J.-P. 2004, A&A, 425, 243 Neo, S., Miyaji, S., Nomoto, K., & Sugimoto, D. 1977, PASJ, 29, 249 Packet, W. 1981, A&A, 102, 17 Paczy ´ nski, B. 1970, Acta Astronomica, 20, 47 Paczy ´ nski, B. 1991, ApJ, 370, 597 Pols, O.R., Cote, J., Waters, L.B.F.M., & Heise, J. 1991, A&A, 241, 419 Popham, R., & Narayan, R. 1991, ApJ, 370, 604 Portegies Zwart, S.F. 1995, A&A, 241, 419 Porter, J.M., & Rivinius, T. 2003, PASP, 115, 1153 Raymer, E. 2012, MNRAS, 427, 702 Richards, M.T. 1992, ApJ, 387, 329 Richards, M.T., & Ratliff, M.A. 1998, ApJ, 493, 326 Richards, M.T., Cocking, A.S., Fisher, J.G., & Conover, M.J. 2014, ApJ, 795, 160 Sana, H., de Mink, S.E., de Koter, A., Langer, N., Evans, C.J., Gieles, M., Gosset, E., Izzard, R.G., Le Bouquin, J.-B., & Schneider, F.R.N. 2012, Science, 337, 444 Shao, Y., & Li, X.-D. 2014, ApJ, 796, 37</code>
<text><location><page_13><loc_12><loc_86><loc_40><loc_87></location>Staritsin, E.I. 1999, Astronomy Reports, 43, 592</text>
<text><location><page_13><loc_12><loc_67><loc_72><loc_68></location>Van Rensbergen, W., De Greve, J.P., Mennekens, N., Jansen, K., & De Loore, C. 2011, A&A, 528, A16</text>
<text><location><page_13><loc_12><loc_66><loc_52><loc_85></location>Staritsin, E.I. 2005, Astronomy Reports, 49, 634 Staritsin, E.I. 2007, Astronomy Letters, 33, 93 Staritsin, E.I. 2009, Astronomy Letters, 35, 4133 Staritsin, E.I. 2014, Astronomy Reports, 58, 808 Staritsin, E.I. 2019, Ap&SS, 364, 110 Staritsin, E. 2021, A&A, 646, A90 Staritsin, E. 2022, RAA, 22, 105015 Talon, S., & Zahn, J.-P. 1997, A&A, 317, 749 Tassoul, J.-L. 1978 Theory of Rotating Stars (Princeton Univ. Press) Van Bever, J., & Vanbeveren, D. 1997, A&A, 322, 116 Zahn, J.-P. 1992, A&A, 265, 115</text>
</document>
[
{
"title": "ABSTRACT",
"content": "Research in Astron. Astrophys. Vol.0 (20xx) No.0, 000-000 http://www.raa-journal.org http://www.iop.org/journals/raa (L A T X: ms2023-0221.tex; printed on November 1, 2023; 1:19) E R esearchin A stronomyand A strophysics",
"pages": [
1
]
},
{
"title": "Formation of a rapidly rotating classical Be-star in a massive close binary system",
"content": "Evgeny Staritsin K.A. Barkhatova Kourovka Astronomical Observatory, B.N. Yeltsin Ural Federal University, pr. Lenina 51, Ekaterinburg 620000, Russia; Evgeny.Staritsin@urfu.ru Abstract This paper investigates the spin-up of a mass-accreting star in a close binary system passing through the first stage of mass exchange in the Hertzsprung gap. Inside an accreting star, angular momentum is carried by meridional circulation and shear turbulence. The circulation carries part of the angular momentum entering the accretor to its surface. The greater the rate of arrival of angular momentum in the accretor is, the greater this part. It is assumed that this part of the angular momentum can be removed by the disk further from the accretor. If the angular momentum in the matter entering the accretor is more than half the Keplerian value, then the angular momentum obtained by the accretor during mass exchange stage does not depend on the rate of arrival of angular momentum. The accretor may have the characteristics of a Be-star immediately after the end of mass exchange. Key words: stars: binaries: close - stars: rotation - stars: early-type - stars: emission line, Be",
"pages": [
1
]
},
{
"title": "1 INTRODUCTION",
"content": "Classical Be-stars include OBA stars with observed or previously observed emissions in the Balmer lines of hydrogen (Porter & Rivinius 2003). These stars are not supergiants and have large rotational velocities. Among Be-stars, there is the group of early spectral subclasses (B3-O9). The surface rotational velocities of these stars range widely. The lower range limit is 40%-60% of the Keplerian value, while the upper limit is 90%-100% (Cranmer 2005). The origin of the large rotational velocities of Be-stars is not clear. Young B-stars in the early spectral subclasses and O-stars are characterized by lower rotational velocities (Huang et al. 2010). 70% of these stars are observed in binary and multiple systems (Chini et al. 2012; Sana et al. 2012). All these stars are expected to form binary and multiple systems, considering selection effects. Mass exchange in a binary system may be the reason for the rapid rotation of the star receiving mass. The synthesis of the Be-stars population in binary systems makes it possible to reproduce the observed number of these stars in the Galaxy (Pols et al. 1991; Portegies Zwart 1995; Van Bever & Vanbeveren 1997; Shao & Li 2014; Hastings et al. 2021). A simple estimation made assuming the instantaneous redistribution of angular momentum in the stellar interior to solid-state rotation shows that a 5% - 10% increase to the star's mass due to accretion of mass with Keplerian velocity leads to a critical rotation state (Packet 1981). The question of what happens when there is continued accretion into a star close to a state of critical rotation has been discussed in Paczy ' nski (1991), Popham & Narayan (1991), Colpi et al. (1991), and Bisnovatyi-Kogan (1993). Paczy ' nski (1991), Popham and Narayan (1991), and Colpi et al. (1991) used various approaches. All authors agree that accretion does not stop when the star's speed of rotation reaches a critical value. Paczy ' nski (1991) studied the whole star-boundary layer-accretion disk system for various rotations of the central star. For models rotating slightly above critical, mass accretion is accompanied by the loss of angular momentum from the star to the disk, mediated by viscous stresses. However, the solutions obtained in Paczy ' nski (1991), Popham and Narayan (1991), and Colpi et al. (1991) are not self-consistent. The condition for a self-consistent solution for a system consisting of a star in a state of critical rotation and an accretion disk is that 'the star absorbs accreted matter with a certain angular momentum, such that the star remains in a state of critical rotation' (Bisnovaty-Kogan 1993). Let J ( M ) be the angular momentum of a star with mass M in a state of critical rotation and let j Kep e be the specific Keplerian angular momentum at the equator of the star. Then j a = dJ/dM < j Kep e . A mass-accreting star can move along the sequence of stars in a state of critical rotation J ( M ) if the excess angular momentum of /triangle j = j Kep e -j a is eliminated. Bisnovaty-Kogan (1993) constructed models of accretion disks that remove excess angular momentum from the surface of a star. At the same time, the speed of rotation at the star's surface remains critical. So an increase in the mass and angular momentum of a star in a critical rotation state may occur due to the removal of excess angular momentum from the star by the accretion disk (Paczy ' nski 1991; Bisnovatyi-Kogan 1993). Physical processes, such as meridional circulation and turbulence, require finite amounts of time to transfer angular momentum (Staritsin 2019, 2021). At the very beginning of accretion, only the outer layers of the star, including the accreted mass, have a fast rotation. The star surface gains critical rotation shortly after the start of accretion. Later, at the accretion stage in a state of accretor critical rotation, the circulation carries part of the angular momentum brought along with the accreted mass from subsurface layers to the star's surface (Staritsin 2022). Thus, accreted layers can shrink, as usually happens during accretion. The angular momentum transferred by circulation to the star's surface can be removed through an accretion disk (Paczy ' nski 1991; Bisnovatyi-Kogan 1993). Thus, the mass and angular momentum of an accretor in a state of critical rotation increase due to the removal of excess angular momentum from the accreted layers to the accretor surface and the further removal of this angular momentum from the star. In Staritsin (2022), the transfer of angular momentum in the accretor interior was carried out only by meridional circulation. The turbulence was artificially suppressed. This made it possible to elucidate the transport properties of circulation in an accreting star's interior. The role of turbulence in angular momentum transport within the accretor remained unclear. As to the angular momentum input, only one option has been considered, the effective transport of angular momentum from the disk's boundary layer to the accretor's upper layer. In this paper, we consider two mechanisms of angular momentum transfer in an accreting star interior, namely circulation and turbulence. This allows us to find the role of turbulence in the spinning up of a star. We also took into account the possible reduction of input angular momentum. This decrease can be attributed both to the transfer of angular momentum from the boundary layer to the outer parts of the disk, and to sub-Keplerian rotation in the disk. The accretor's rotation, obtained as a result of mass exchange, has been studied depending on the angular momentum introduced during mass exchange.",
"pages": [
1,
2
]
},
{
"title": "2.1 The angular momentum input",
"content": "The matter lost by the donor due to the filling of the Roche lobe falls into the accretor's gravitational field and swirls around it. The formation of gas structures around the accretor, in particular the formation of a disk and a velocity field in it, depends on the ratio between three factors : the size of the accretor R , the minimum distance ω min from the center of the accretor to the central line of the stream of matter falling from donor point L 1 , and the distance from the accretor's center to the edge of the inviscid disk ω d (Lubow & Shu 1975). Transient disks with sub-Keplerian rotation have been found (for example, RW Tau (Kaitchuck & Honeycutt 1982) and β Per (Cugier & Molaro 1984, Richards 1992)) in direct-impact systems ( ω d < R ). Three-dimensional hydrodynamic calculations show disk formation in such systems; the rotation velocity is 80% and 60% of the Keplerian value at the inner and outer edges of the disk, respectively (Raymer 2012). Both transient disks (SW Cyg) and permanent, but variable, accretion disks (RY Gem, TT Hya, AU Mon) in grazing-impact systems ( ω min < R < ω d ) have been discovered. The velocity fields in the transient disk of the SW Cyg system and in the permanent disk of the RY Gem system are subKeplerian (Kaitchuck 1988, 1989). Asymmetric parts were found in the disks of the TT Hya and AU Mon systems; the gas in the disk's asymmetric part in the AU Mon system moves at a sub-Keplerian velocity (Richards et al. 2014). Hydrodynamic calculations also show the possibility of disk formation at sub-Keplerian velocities in these systems (Richards & Ratliff 1998). Permanent disks are found in systems with R < ω min . The radial component of the matter velocity in the disk is directed towards the accretor and is 10-30 km/s . The change in the tangential component with distance from the accretor may differ from the Keplerian one (Etzel et al. 1995). The aforementioned observational data and the results of hydrodynamic calculations relate to systems with the low mass of accreting components ( M ≤ 6 M /circledot ) and with a ratio of donor mass to accretor mass within the range of 0.2 to 0.3. The formation of Be-stars in the early spectral subclass occurs in systems with large component masses. The ratio of donor mass to accretor mass varies widely. Mass transfer in such systems is non-conservative (Van Rensbergen et al. 2011; Deschamps et al. 2015). The star receiving mass increases in volume (Benson 1970; Kippenhahn & Meyer-Hofmeister 1977). The distance between two stars depends on how the system loses mass and angular momentum. So, the formation of gas structures in the Roche lobe of an accretor depends on the loss of mass and angular momentum from the system. A quantitative theory of mass and angular momentum losses from a close binary system has not yet been developed. The formation of conditions for sub-Kelerian rotation in an accretion disk due to the loss of mass and angular momentum from the binary system cannot be ruled out. Thus, the possibility of mass accretion with sub-Keplerian velocities of rotation should be considered. At the very beginning of accretion, when accretor rotation velocity is low, the rotation velocity of disk matter decreases in the narrow boundary layer from the maximum value in the disk Ω max to the value on the star's surface Ω s (Paczy ' nski 1991). Turbulence can remove angular momentum from the boundary layer to an accretor's upper layers at a rate of: where J - angular momentum of the accretor, t - time, and ˙ M - mass accretion rate. Supersonic shear flow in the boundary layer is a source of acoustic waves. The waves can carry the angular momentum out of the boundary layer both into the accretor's outer part and the disk's outer part (Dittmann 2021, Coleman et al. 2022). In this case, the amount of angular momentum coming from the boundary layer into the accretor is less than the Keplerian one. In an earlier study (Staritsin 2022), we considered as follows: when at the stage of subcritical rotation, angular momentum enters the accretor through two channels, namely together with matter having the same rotation velocity as the accretor's surface and due to turbulence within the rate (1). This is a case of high efficiency of angular momentum transfer from the boundary layer to the accretor's upper part. The transfer of angular momentum in the accretor's interior was carried out by meridional circulation; turbulence was artificially suppressed. In the current calculations, angular momentum transfer in the accretor's interior can be carried out both by meridional circulation and turbulence. We have studied two variants for the arrival of angular momentum into the accretor. In the first variant, to clarify the influence of angular momentum transport by turbulence in the accretor's interior on the spinning up of the accretor, we calculated accretion with the same rate of the arrival of angular momentum into the accretor as in Staritsin (2022). At the stage of subcritical rotation, the parameter Ω max in the angular momentum source (1) is equal to α Ω Kep , where α = 0 . 8 ; here, Ω Kep is the Keplerian velocity of the star's surface at the equator. After the angular velocity of the accretor's surface increases to α Ω Kep value, the arrival of angular momentum from the boundary layer (1) stops. The angular velocity of the adding matter is set equal to α Ω Kep for the remainder of the mass exchange. In the second variant, the case of extremely low efficiency of angular momentum transfer from the boundary layer to the accretor's upper part is considered. The angular momentum's source (1) in this case is assumed to be zero. As long as the angular velocity of the star's surface is less than α Ω Kep , the star accretes matter with the same angular velocity as that of the star's surface. After the surface angular velocity increases to the value of α Ω Kep , the angular velocity of the adding matter remains equal to α Ω Kep . In order to determine the dependence of the accretor's rotation state after the end of mass exchange on the content of angular momentum in adding mass, calculations were carried out at four values of α : 0 . 8 , 0 . 6 , 0 . 4 , and 0 . 2 .",
"pages": [
3,
4
]
},
{
"title": "2.2 Angular momentum transfer in the accretor's interior",
"content": "Angular momentum transfer in the radiative layers of a star is taken into account in the framework of the shellular rotation model (Zahn 1992). In terms of this model, two mechanisms of angular momentum transfer are considered: meridional circulation and shear turbulence. The angular momentum transfer is described by the law of conservation of angular momentum (Tassoul 1978): The meridional circulation velocity u is determined from the law of conservation of energy in a stationary form (Maeder & Zahn 1998): In these equations, ρ - density, /pi1 - distance to the axis of rotation, Ω - angular velocity ν v - turbulent viscosity in the vertical direction, T - temperature, s - specific entropy, ε n - nuclear energy release rate, χ - thermal conductivity, F h - turbulent enthalpy flow in the horizontal direction: F h = -ν h ρT∂s/∂ i θ and ν h - turbulent viscosity in the horizontal direction. The coefficients of turbulent viscosity were determined by Talon and Zahn (1997), Maeder (2003), and Mathis et al. (2004). The convective core rotates solid-state. These equations are solved together with equations related to the structure and evolution of stars. We used a set of programs from Paczy ' nski (1970) modified to calculate the evolution of rotating stars (Staritsin 1999, 2005, 2007, 2009, 2014).",
"pages": [
4
]
},
{
"title": "3.1 Binary system parameters",
"content": "We consider mass exchange in a binary system with the component masses of 13.4 M /circledot and 10.7 M /circledot and the period P = 35 d as in Staritsin (2022). By the beginning of mass exchange, star rotation with a mass of 10.7 M /circledot is synchronized with orbital motion. The star angular momentum is equal to 1 . 3 × 10 51 g · cm 2 s -1 . A star with a mass of 13.4 M /circledot loses 10.5 M /circledot for 12,000 years. After that, the star decouples its Roche lobe and the mass exchange stage ceases. The second star accretes 5.3 M /circledot . The final mass of the accretor is 16.0 M /circledot . The accretion rate was set constant, equal to the average value of ∼ 4 . 4 × 10 -4 M /circledot /year . We consider a case when the entropy of the added matter is the same as the surface layers of the second star. The thermal timescale of the second star is longer than mass exchange duration. The star's reaction to the increase in mass in this case is well understood (Benson 1970; Flannery and Ulrich 1977; Neo et al. 1977). The second star is driven out of thermal equilibrium by mass accretion. Nuclear power output in the center of the second star increases, and some of the nuclear energy release is spent on an increase in entropy in the second star's central parts. Gravitational energy release in the surface layers is added to nuclear energy release in the center. The typical luminosity distribution in the second star's interior is shown in Staritsin (2022) (see Fig. 4). The remaining part of the mass lost by the first star leaves the system. The tidal interaction between the two stars is unable to synchronize the accreting star with the orbit due to the long period of the system and the short accretion timescale. The accretion of matter and angular momentum, as well as transport processes inside the accretor and in the disk, determine the accretor's angular momentum.",
"pages": [
4,
5
]
},
{
"title": "3.2 The case of the high efficiency of angular momentum transfer from the boundary layer to the accretor's upper part",
"content": "With the beginning of mass exchange, a circulation cell is formed in the subsurface layer of the accretor, in which the circulation carries the incoming angular momentum downwards. The cell consists of accreted layers and the swirled layers of the accretor located below. In the cell's upper part, angular ve- locity has an almost constant value, but near the bottom of the cell, it sharply reduces to the initial value (Fig. 1). Therefore, in the lower part of the cell, the contribution of turbulence to angular momentum transfer is greater and exceeds the contribution of meridional circulation (Fig. 2). The bottom of the cell (Case 3), (Case 4), (Case 5), and (Case 6) angular momentum transfer from the boundary layer is not considered, turbulence in the accretor interior is present, and α is equal to 0.8, 0.6, 0.4, and 0.2 respectively. goes down into the star faster than when turbulence is artificially suppressed. The angular momentum entering the accretor is distributed over a larger mass of matter than in the case of suppressed turbulence. The rotation of the accretor's surface becomes critical when its mass increases to 11.3 M /circledot (in the case of suppressed turbulence - up to 11.0 M /circledot (Staritsin 2022)). The distribution of angular velocity in the accretor's interior at this moment is shown in Fig. 3. At the stage of critical rotation, the mass of the accretor increases by another 4.7 M /circledot . Another circulation cell is formed in the accreted matter. In this cell, the circulation transfers the some part of the angular momentum that came along with the accreted mass to the surface of the accretor (Fig. 4). It is assumed that this part of the angular momentum is removed from the accretor by the accretion disk (Paczy ' nski 1991; Bisnovatyi-Kogan 1993). As a result of a decrease in angular momentum, the accreted layers are contracted. The velocity of their rotation is permanently lower than the Keplerian velocity. In the circulation cell formed at the beginning of mass exchange, the transfer of angular momentum inside the accretor continues. The mass of the matter in this cell increases as the upper boundary of the cell moves up along the accretor mass, and the bottom of the cell moves down. The bottom of the cell goes down to the convective core when the accretor mass increases to 11.9 M /circledot (in the case of suppressed turbulence - up to 15 M /circledot (Staritsin 2022)). The role of turbulence lies in the rapid lowering of the bottom of the circulation cell, in which the circulation carries angular momentum into the star's interior. The amount of angular momentum removed from the accretor during mass exchange depends slightly on processes of angular momentum transfer within the accretor (Fig. 5). When turbulence is present, the amount of angular momentum transferred to the accretor's inner layers increases, and the amount of angular momentum carried to the accretor's surface decreases compared to the case of suppressed turbulence (Table 1). The angular momentum brought into the accretor during mass exchange is 1 . 72 × 10 53 g · cm 2 s -1 . 12% of this value enters the inner layers that made up the accretor initially, 31% remains in the accreted mass, and 57% is carried to the accretor's surface and is removed by the disk. In the case of suppressed turbulence, the corresponding values are 5%, 30%, and 65% (Staritsin 2022). After the end of mass exchange, the accretor's angular momentum is greater when turbulence is present (Table 1).",
"pages": [
5,
6,
7,
8
]
},
{
"title": "3.3 The case of extremely low efficiency of angular momentum transfer from the boundary layer to the accretor's upper part",
"content": "At the beginning of mass exchange, the rotation velocity of the incoming mass and the accretor's surface coincide. The rate of angular momentum arrival into the accretor is significantly less than when turbulence and/or waves transfer angular momentum from the boundary layer to the accretor's outer part. Due to the low rate at which angular momentum enters the accretor, the accretor's angular momentum increases slowly at the beginning of accretion (Fig. 5). The general picture of angular momentum transfer in the accretor's interior at α equal to 0.8 and 0.6 is the same as when angular momentum goes from the boundary layer to the accretor's upper part.The difference is that the total amount of angular momentum that has entered the accretor during mass exchange decreases (Table 1). Reasons for the decrease are associated with the absence of a source (1) and a decrease in parameter α . However, with α equal to 0.8 and 0.6, the accretor surface's rotation velocity increases to a critical value. This occurs when accretor mass increases to 12.9 M /circledot and 13.3 M /circledot , respectively (Fig. 6). In these cases, a circulation cell is formed in the accretor's outer layer, in which the circulation transfers part of the angular momentum of the accreted layers to the accretor's surface. The amount of angular momentum removed from the accreted layers and lost by the accretor in these cases is less than in calculations with a source (1) (Table 1). The state of accretor rotation once mass exchange finishes is approximately the same as when the angular momentum's arrival from the boundary layer to the accretor's upper part was considered. A decrease in angular momentum entering the accretor only results in decreases in the angular momentum taken out of the accretor at the stage of accretion during accretor critical rotation. At α equal to 0.4 and 0.2, a smaller amount of angular momentum enters the accretor (Table 1). The rotation velocity of the accretor's surface remains subcritical throughout the entire mass exchange stage; at α equal to 0.4, it approaches the critical value by the end of this stage. In both cases, at the beginning of mass exchange, a circulation cell is formed in the accretor's subsurface layer, in which the angular momentum of the accreted matter is transferred inside the accretor. The bottom of the cell goes down to the convective core when the mass of the accretor increases to 13.1 M /circledot at α equal to 0.4 and up to 13.9 M /circledot at α equal to 0.2. In both cases, the angular momentum of the accreted mass is transferred inside the star throughout the mass exchange stage. The accretor retains all the angular momentum received with the accreted mass (Fig. 6). Once mass exchange finishes, the angular momentum of the accretor at α equal to 0.4 is little less than at α equal to 0.6 and 0.8, and at α equal to 0.2 is significantly less (Table 1).",
"pages": [
8,
9
]
},
{
"title": "3.4 Accretor rotation state after mass exchange",
"content": "The distribution of the angular velocity of rotation in the accretor's interior immediately after the end of mass exchange is shown in Fig. 7. In all cases, angular velocity decreases rapidly in a layer of variable chemical composition located between the chemically homogeneous part of the radiative envelope and the convective core. A similar jump is formed in cases where angular momentum enters the accretor in a short time - in the donor's thermal timescale or faster (Staritsin 2021). The thermal timescale of the accretor is longer than that of the donor in the cases considered in Staritsin (2021, 2022). After the end of mass exchange, the jump gradually decreases and disappears during the thermal timescale of the accretor (see, for example, Fig. 3 in Staritsin (2021)). The angular velocity in the accretor's interior after mass exchange when α is equal to 0.8 and α is equal to 0.6 almost does not depend on what the content of the angular momentum was in the adding mass and on whether angular momentum is transferred from the boundary layer to the accretor's upper layer (Fig. 7). In these cases, the accretor's angular momentum is almost equal to ∼ 7 . 5 × 10 52 g · cm 2 s -1 (Table 1) after the mass exchange. An isolated star with a mass of 16 M /circledot has a critical rotation throughout the stage of hydrogen burning in the core with this angular momentum value (Staritsin 2007). Consequently, due to the exchange of mass, the accretor receives a rotation typical for Be-stars. At α equal to 0.4, the accretor receives almost the same angular momentum with accreted mass as the angular momentum that remains in the accretor when α is 0.8 and α is 0.6 (Table 1). Therefore, at α equal to 0.4, the accretor also has a rotation typical for Be-stars. At α equal to 0.2, the accreted mass brings a much lower angular momentum (Table 1). The angular velocity in the accretor's interior is lower than in other cases (Fig. 7). The rotation of the accretor's surface immediately after mass exchange in this case is lower than that of Be-type stars. In an isolated star with the same mass and angular momentum as the accretor, the removal of angular momentum from the inner layers to the outside occurs intensively at the stage of hydrogen burning in the core (Staritsin 2007, 2009). The angular velocity of the star's surface, expressed in Keplerian angular velocity, increases; at the last steps of this stage, the star acquires a rotation typical for Be-stars of the early spectral subclass. If tidal interaction is low, then even in this case the accretor can obtain the characteristics of a Be-star after the end of mass exchange, but only after a long period of time on the order of part of the hydrogen burning stage in the core.",
"pages": [
9,
10,
11
]
},
{
"title": "4 CONCLUSIONS",
"content": "Meridional circulation is a flexible mechanism for the transfer of angular momentum in the stellar interior of a rotating star. The direction and rate of angular momentum transfer by circulation may vary widely at the stage of mass accretion, depending on the star rotation state and the rate of angular momentum arrival along with the accreted mass and waves and/or due to turbulence. The two main circulation cells are formed due to the accretion of mass and angular momentum. In the cell, which is formed at the stage of subcritical rotation of the accretor, circulation transfers the angular momentum inside the accretor. Only in the lower part of this cell does turbulence make the main contribution to the transfer of angular momentum. Due to turbulence, the cell bottom quickly goes downwards into the accretor's interior. In a cell formed at the stage of critical rotation, the circulation transfers part of the angular momentum of the accreted mass to the surface of the star; the more the content of the angular momentum in the entering matter is, the greater this part. We have considered the case of mass exchange in a binary system, when half of the mass lost by the donor falls into the accretor. If the angular momentum of the mass falling into the accretor exceeds half the Keplerian value at the boundary of the accretor, the state of rotation of the accretor after the end of mass exchange does not depend on the angular momentum entering the accretor. In other words, processes that could reduce the angular momentum of the matter located around the accretor to no more than half the Keplerian value do not affect the angular momentum and the state of rotation that the accretor receives by the end of the mass exchange stage. These processes impact on the amount of angular momentum removed by circulation from the accreted mass to the accretor's surface and removed further from the accretor by a disk or other processes. In the considered system with the initial component masses of 13.4 M /circledot and 10.7 M /circledot , the accretor has a rotation typical for Be-stars immediately after the end of mass exchange, if during mass exchange the angular momentum of the mass added to the accretor exceeded 40% of the Keplerian value. Acknowledgements This work was supported by the Ministry of Science and Education, FEUZ-20230019",
"pages": [
11,
12
]
},
{
"title": "References",
"content": "Staritsin, E.I. 1999, Astronomy Reports, 43, 592 Van Rensbergen, W., De Greve, J.P., Mennekens, N., Jansen, K., & De Loore, C. 2011, A&A, 528, A16 Staritsin, E.I. 2005, Astronomy Reports, 49, 634 Staritsin, E.I. 2007, Astronomy Letters, 33, 93 Staritsin, E.I. 2009, Astronomy Letters, 35, 4133 Staritsin, E.I. 2014, Astronomy Reports, 58, 808 Staritsin, E.I. 2019, Ap&SS, 364, 110 Staritsin, E. 2021, A&A, 646, A90 Staritsin, E. 2022, RAA, 22, 105015 Talon, S., & Zahn, J.-P. 1997, A&A, 317, 749 Tassoul, J.-L. 1978 Theory of Rotating Stars (Princeton Univ. Press) Van Bever, J., & Vanbeveren, D. 1997, A&A, 322, 116 Zahn, J.-P. 1992, A&A, 265, 115",
"pages": [
13
]
}
]
2024RAA....24h5012A
https://arxiv.org/pdf/2407.12314.pdf
<document>
<text><location><page_1><loc_12><loc_88><loc_51><loc_89></location>Research in Astronomy and Astrophysics manuscript no.</text>
<text><location><page_1><loc_12><loc_86><loc_16><loc_88></location>(L A T E</text>
<text><location><page_1><loc_15><loc_87><loc_49><loc_88></location>X: Manuscript.tex; printed on July 18, 2024; 0:23)</text>
<section_header_level_1><location><page_1><loc_12><loc_73><loc_75><loc_78></location>Charged Particles Capture Cross-Section by a Weakly Charged Schwarzschild Black Hole</section_header_level_1>
<text><location><page_1><loc_12><loc_69><loc_38><loc_71></location>A. M. Al Zahrani, 1 and A. Al-Jama 2</text>
<text><location><page_1><loc_12><loc_64><loc_78><loc_67></location>Physics Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia; amz@kfupm.edu.sa 1 ; ahmad.aljama1905@gmail.com 2</text>
<text><location><page_1><loc_12><loc_61><loc_43><loc_62></location>Received 20xx month day; accepted 20xx month day</text>
<text><location><page_1><loc_16><loc_42><loc_77><loc_57></location>Abstract We study the capture cross-section of charged particles by a weakly charged Schwarzschild black hole. The dependence of the maximum impact parameter for capture on the particle's energy is investigated numerically for different values of the electromagnetic coupling strength between the particle and the black hole. The capture cross-section is then calculated. We show that the capture cross-section is independent of the electromagnetic coupling for ultra-relativistic particles. The astrophysical implications of our results are discussed.</text>
<text><location><page_1><loc_16><loc_36><loc_77><loc_39></location>Key words: stars: black holes - Stars, accretion, accretion disks - Physical Data and Processes, black hole physics - Physical Data and Processes</text>
<section_header_level_1><location><page_1><loc_12><loc_32><loc_27><loc_33></location>1 INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_12><loc_19><loc_81><loc_29></location>Studying the capture cross-section of black holes is central to understand how matter interacts with them. It helps us understand the process of matter accretion by a black hole which in turn determines how its mass, angular momentum and charge evolve. It can also help us understand the environment near black holes. Moreover, scrutinizing capture cross-section can be used to test theories of gravity in strong gravitational fields.</text>
<text><location><page_1><loc_12><loc_1><loc_81><loc_18></location>Astrophysicists generally assume black holes are electrically neutral. This is because they would quickly attract oppositely charged matter to balance out any access charge. However, there are compelling reasons why weakly charged black holes might exist as discussed in Al Zahrani (2021, 2022); Zajaˇcek & Tursunov (2019); Zajaˇcek et al. (2019, 2018); Carter (1973) and the references therein. The differences in how a black hole accretes electrons and protons within its plasma environment, influenced by radiation, could render it charged. Also, the spin of a black hole in the presence of a magnetic field can induce the accretion of charged particles. In fact, using the EHT observations, it was inferred that Sgr A* and M87 can be charged Ghosh & Afrin (2023); Kocherlakota et al. (2021). The black hole's charge is weak in the sense</text>
<text><location><page_2><loc_12><loc_76><loc_81><loc_88></location>There are numerous astrophysical scenarios wherein charged particles are drawn into black holes. Stars within the Roche limits near black holes often contribute matter through tidal interactions. Additionally, stars emit streams of charged particles as stellar winds. Highly energetic charged particles, resulting from supernovae, gamma-ray bursts, and bipolar jets from compact objects, frequently find their way into the vicinity of black holes. These processes collectively enrich the environment around black holes with a significant population of charged particles.</text>
<text><location><page_2><loc_12><loc_48><loc_81><loc_75></location>The concept of capture cross-sections has been explored extensively for various black hole types. Foundational treatment which examine photon and neutral particle capture by Schwarzschild black holes was given in several monographs, such as Frolov & Zelnikov (2011). Further work addressed capture cross-sections of charged and neutral particles by Kerr-Newman black holes, including the implications for black hole spin and charge evolution Young (1976). Capture by Reissner-Nordstrom black holes was also investigated Zakharov (1994). In the context of higher-dimensional black holes, studies have focused on calculating photon critical impact parameters for Schwarzschild-Tangherlini black holes Connell & Frolov (2008); Tsukamoto (2014); Singh & Ghosh (2018); Bugden (2020). The capture cross-section for massive particles was determined in Ahmedov et al. (2021). Additionally, research extends to particle capture in Myers-Perry rotating spacetime which describes rotating black holes in five-dimensions Gooding & Frolov (2008). Moreover, wave capture cross-sections have been studied for various black hole configurations (see Anacleto, et al. (2023) and the references within).</text>
<text><location><page_2><loc_12><loc_30><loc_81><loc_47></location>In this research, we examine the capture cross-section of charged particles by a weakly charged Schwarzschild black hole and discuss the astrophysical consequences of our findings. The paper is organized as follows: In Sec. 2, we review the dynamics of charged particles in the background of a weakly charged black hole. We then review the capture cross-section of neutral particles in Sec. 3. The capture cross-section of charged particles is calculated for different coupling strengths and particle energies in Secs. 4. Finally, we summarize our main findings and discuss their astrophysical consequences in Sec 5. We use the sign conventions adopted in Misner et al. (1973) and geometrized units where c = G = k = 1 , where k is the electrostatic constant.</text>
<section_header_level_1><location><page_2><loc_12><loc_26><loc_80><loc_28></location>2 CHARGED PARTICLES NEAR A WEAKLY CHARGED SCHWARZSCHILD BLACK HOLE</section_header_level_1>
<text><location><page_2><loc_12><loc_19><loc_81><loc_24></location>Here, we review the dynamics of charged particles near a weakly charged black hole. The spacetime geometry around a black hole of mass M and charge Q is described by the Schwarzschild Reissner-Nordstron metric which reads Misner et al. (1973)</text>
<formula><location><page_2><loc_30><loc_15><loc_81><loc_17></location>ds 2 = -hdt 2 + h -1 dr 2 + r 2 dθ 2 + r 2 sin 2 θdϕ 2 , (1)</formula>
<text><location><page_2><loc_12><loc_10><loc_81><loc_13></location>where h = 1 -r S /r + Q 2 /r 2 and r S = 2 M is the Schwarzschild radius. The electromagnetic 4-potential is</text>
<formula><location><page_2><loc_42><loc_7><loc_81><loc_9></location>A µ = -Q r δ 0 µ . (2)</formula>
<text><location><page_2><loc_12><loc_2><loc_81><loc_5></location>However, when the charge is weak we can ignore the curvature due to it and use the Schwarzschild metric, which reads Misner et al. (1973)</text>
<text><location><page_3><loc_12><loc_85><loc_81><loc_88></location>where f = 1 -r S /r and r S = 2 M is the Schwarzschild radius. This weak charge approximation is valid unless the charge creates curvature comparable to that due to the black hole's mass. This happens when</text>
<formula><location><page_3><loc_43><loc_81><loc_81><loc_83></location>Q 2 ∼ M 2 . (4)</formula>
<text><location><page_3><loc_12><loc_78><loc_54><loc_79></location>In conventional units, the weak charge approximation fails when</text>
<formula><location><page_3><loc_34><loc_74><loc_81><loc_77></location>Q ∼ G 1 / 2 M k 1 / 2 ∼ 10 20 M M ⊙ coloumbs . (5)</formula>
<text><location><page_3><loc_12><loc_67><loc_81><loc_72></location>This charge is way greater than the greatest estimated change on any black hole. Although the black hole charge is tiny, its effect on charged particles dynamics is profound because it is multiplied by the charge-tomass ratio of these particles ( ∼ 10 21 m -1 for electrons and ∼ 10 18 m -1 for protons).</text>
<text><location><page_3><loc_12><loc_62><loc_81><loc_66></location>The Lagrangian describing a charged particle of charge q and mass m in a spacetime described by a metric g µν and an electromagnetic field produced by a 4-potential A µ reads Chandrasekhar (1983)</text>
<formula><location><page_3><loc_37><loc_58><loc_81><loc_61></location>L = 1 2 mg µν u µ u ν + qu µ A µ , (6)</formula>
<text><location><page_3><loc_12><loc_56><loc_81><loc_57></location>where u µ ≡ dx µ /dτ is the particle's 4-velocity and τ is its proper time. In our case, the Lagrangian becomes</text>
<formula><location><page_3><loc_22><loc_47><loc_81><loc_54></location>L = 1 2 m [ -f ( dt dτ ) 2 + f -1 ( dr dτ ) 2 + r 2 ( dθ dτ ) 2 + r 2 sin 2 θ ( dϕ dτ ) 2 ] -qQ dt dτ . (7)</formula>
<text><location><page_3><loc_12><loc_40><loc_81><loc_46></location>This Lagrangian is cyclic in t and ϕ , which means that the particle's energy and azimuthal angular momentum are constants of motion. The specific energy and azimuthal angular momentum are, respectively, given by</text>
<formula><location><page_3><loc_35><loc_36><loc_81><loc_39></location>E = -1 m ∂L ∂ ( dt dτ ) = f dt dτ + qQ mr , (8)</formula>
<formula><location><page_3><loc_36><loc_32><loc_81><loc_35></location>ℓ = 1 m ∂L ∂ ( dϕ dτ ) = r 2 sin 2 θ dϕ dτ (9)</formula>
<text><location><page_3><loc_12><loc_29><loc_80><loc_30></location>Combining these equations with the normalization condition g µν u µ u ν = -1 and solving for dr/dτ give</text>
<formula><location><page_3><loc_27><loc_24><loc_81><loc_28></location>( dr dτ ) 2 = ( E qQ mr ) 2 -f [ r 2 ( dθ dτ ) 2 + ℓ 2 r 2 sin 2 θ +1 ] . (10)</formula>
<text><location><page_3><loc_12><loc_21><loc_52><loc_23></location>In the equatorial plane where θ = π/ 2 , the equation becomes</text>
<formula><location><page_3><loc_33><loc_17><loc_81><loc_20></location>( dr dτ ) 2 = ( E qQ mr ) 2 -f ( ℓ 2 r 2 +1 ) . (11)</formula>
<text><location><page_3><loc_12><loc_12><loc_81><loc_16></location>Let us rewrite the last equation in a dimensionless form. We first introduce the following dimensionless quantities:</text>
<formula><location><page_3><loc_33><loc_9><loc_81><loc_12></location>T = τ r S , ρ = r r S , L = ℓ r S . (12)</formula>
<text><location><page_3><loc_12><loc_7><loc_29><loc_8></location>Equation 11 then becomes</text>
<text><location><page_3><loc_12><loc_1><loc_16><loc_3></location>where</text>
<formula><location><page_3><loc_33><loc_4><loc_81><loc_7></location>( dρ d T ) 2 = ( E α ρ ) 2 -f ( L 2 ρ 2 +1 ) , (13)</formula>
<text><location><page_3><loc_47><loc_0><loc_49><loc_1></location>qQ</text>
<text><location><page_4><loc_12><loc_85><loc_81><loc_88></location>The parameter α represent the relative strength of the electromagnetic force to the Newtonian gravitational force. We can rewrite Eq. 13 as</text>
<formula><location><page_4><loc_36><loc_81><loc_81><loc_85></location>( dρ d T ) 2 = ( E V + )( E V -) , (15)</formula>
<text><location><page_4><loc_12><loc_79><loc_16><loc_80></location>where</text>
<formula><location><page_4><loc_37><loc_76><loc_81><loc_79></location>V ± = α ρ ± √ f ( L 2 ρ 2 +1 ) , (16)</formula>
<text><location><page_4><loc_12><loc_71><loc_81><loc_74></location>is an effective potential. It is V + that corresponds to physical, future-directed motion and hence will be used in all of the analyses below. Without loss of generality, we will consider L > 0 only.</text>
<text><location><page_4><loc_12><loc_64><loc_81><loc_70></location>It was estimated in Ref. Zajaˇcek et al. (2019) that the charge of Sgr A* is 10 8 -10 15 coulomb. Using the lower limit of charge, the coupling constant for electrons α e and protons α p near Sgr A*, which has a mass of M = 4 . 3 × 10 6 M ⊙ according to Ref. GRAVITY Collaboration (2023), are</text>
<formula><location><page_4><loc_42><loc_60><loc_81><loc_62></location>α e ∼ 10 9 , (17)</formula>
<formula><location><page_4><loc_42><loc_58><loc_81><loc_59></location>α p ∼ 10 6 . (18)</formula>
<section_header_level_1><location><page_4><loc_12><loc_54><loc_56><loc_55></location>3 CAPTURE CROSS-SECTION OF NEUTRAL PARTICLES</section_header_level_1>
<text><location><page_4><loc_12><loc_48><loc_81><loc_52></location>Before we tackle the main problem, let us find the capture cross-cross section for neutral particles first. Setting α = 0 , the effective potential V + reduces to</text>
<formula><location><page_4><loc_39><loc_43><loc_81><loc_47></location>V + = √ f ( L 2 ρ 2 +1 ) . (19)</formula>
<text><location><page_4><loc_12><loc_38><loc_81><loc_42></location>Capture occurs whenever the particle's energy is greater than the maximum of V + . The function V + is at an extremum when dV + /dρ = 0 or</text>
<formula><location><page_4><loc_39><loc_36><loc_81><loc_37></location>ρ 2 +(3 -2 ρ ) L 2 = 0 , (20)</formula>
<text><location><page_4><loc_12><loc_33><loc_48><loc_34></location>which gives the position of the extrema in terms of L as</text>
<formula><location><page_4><loc_39><loc_29><loc_81><loc_31></location>ρ ± = L 2 ±L √ L 2 -3 , (21)</formula>
<text><location><page_4><loc_12><loc_21><loc_81><loc_28></location>where L ∈ [ √ 3 , ∞ ) . When L = √ 3 ( ≡ L min), ρ + and ρ -meet at a saddle point. Inspecting d 2 V + /dρ 2 reveals that ρ -corresponds to the position of the local maximum of V + . In terms of L , the escape condition E = V + | ρ = ρ -becomes</text>
<formula><location><page_4><loc_28><loc_16><loc_81><loc_20></location>E = √ 2 27 [ L ( √ L 2 -3 + L ) -3 √ L 2 -3 L +9 ] 1 / 2 , (22)</formula>
<text><location><page_4><loc_12><loc_13><loc_30><loc_14></location>Inverting this equation gives</text>
<formula><location><page_4><loc_30><loc_8><loc_81><loc_12></location>L = [ 27 E 4 -36 E 2 + ( 9 E 2 -8 ) 3 / 2 E +8 8 ( E 2 -1) ] 1 / 2 (23)</formula>
<text><location><page_4><loc_12><loc_3><loc_81><loc_6></location>The impact parameter b is defined as the perpendicular distance between the center of force and the incident velocity Goldstein et al. (2001). It can be written as</text>
<text><location><page_4><loc_43><loc_0><loc_44><loc_1></location>L</text>
<text><location><page_4><loc_49><loc_0><loc_50><loc_1></location>L</text>
<figure>
<location><page_5><loc_22><loc_63><loc_71><loc_88></location>
<caption>Fig. 1: The maximum impact parameter for capture b max vs. E for a neutral particle.</caption>
</figure>
<figure>
<location><page_5><loc_22><loc_34><loc_71><loc_59></location>
<caption>Fig. 2: The capture cross-section σ cap vs. E for a neutral particle.</caption>
</figure>
<text><location><page_5><loc_12><loc_27><loc_79><loc_28></location>where P is the specific linear momentum. The maximum impact parameter for capture b max is given by</text>
<formula><location><page_5><loc_30><loc_21><loc_81><loc_26></location>b max = [ 27 E 4 -36 E 2 + ( 9 E 2 -8 ) 3 / 2 E +8 ] 2 √ 2 ( E 2 -1) 1 / 2 . (25)</formula>
<text><location><page_5><loc_12><loc_19><loc_39><loc_20></location>The capture cross-section σ cap is given by</text>
<formula><location><page_5><loc_28><loc_13><loc_81><loc_17></location>σ cap = πb 2 max = π 8 27 E 4 -36 E 2 + ( 9 E 2 -8 ) 3 / 2 E +8 ( E 2 -1) 2 . (26)</formula>
<text><location><page_5><loc_12><loc_8><loc_81><loc_12></location>Figures 1 and 2 are plots of b max and the capture cross-section σ cap vs. E , respectively. For ultra-relativistic particles ( E ≫ 1 ), √ √</text>
<formula><location><page_5><loc_35><loc_5><loc_81><loc_8></location>b max = 3 3 2 + 3 2 E 2 + O ( 1 E 3 ) . (27)</formula>
<text><location><page_5><loc_12><loc_3><loc_49><loc_4></location>The corresponding capture cross-section σ cap is therefore</text>
<formula><location><page_5><loc_41><loc_0><loc_57><loc_1></location>27 π 9 π ( 1 )</formula>
<text><location><page_6><loc_12><loc_87><loc_43><loc_88></location>For a slowly moving particle with speed v ≪ 1 ,</text>
<formula><location><page_6><loc_42><loc_83><loc_81><loc_86></location>E ≈ 1 + v 2 2 , (29)</formula>
<text><location><page_6><loc_12><loc_81><loc_17><loc_82></location>and thus</text>
<formula><location><page_6><loc_32><loc_77><loc_81><loc_81></location>b max = √ 2 √ E 1 + O ( √ E 1) = 2 v + O ( v ) , (30)</formula>
<text><location><page_6><loc_12><loc_75><loc_37><loc_77></location>and the capture cross-section becomes</text>
<formula><location><page_6><loc_40><loc_71><loc_81><loc_74></location>σ cap = 4 π v 2 + O ( v 0 ) . (31)</formula>
<section_header_level_1><location><page_6><loc_12><loc_69><loc_56><loc_70></location>4 CAPTURE CROSS-SECTION OF CHARGED PARTICLES</section_header_level_1>
<text><location><page_6><loc_12><loc_58><loc_81><loc_66></location>We will now follow the same procedure we used for the neutral particle. However, analytic expressions are not viable in this case and we will resort to numerical solutions, except in the ultra-relativistic particle case. The structure of the effective potential V + is generically similar to the neutral particle's. The effect of α is to raise (lower) the peak of V + for positive (negative) α . The effective potential V + is at an extremum when</text>
<formula><location><page_6><loc_28><loc_55><loc_81><loc_57></location>2 α √ ( ρ ± -1) ( L 2 + ρ 2 ± ) ρ ± -L 2 (3 -2 ρ ± ) -ρ 2 ± = 0 . (32)</formula>
<text><location><page_6><loc_12><loc_52><loc_35><loc_53></location>The extremum is a maximum when</text>
<formula><location><page_6><loc_29><loc_47><loc_81><loc_51></location>α [ L 2 (1 -2 ρ ± ) + (3 -4 ρ ± ) ρ 2 ± ] √ ( ρ ± -1) ( L 2 + ρ 2 ± ) ρ ± +2 ρ ± -2 L 2 < 0 . (33)</formula>
<text><location><page_6><loc_12><loc_40><loc_81><loc_46></location>To be consistent with the notation of the previous section, we let ρ + correspond to the minimum of V + and ρ -correspond to the maximum. Here, L min (the value at which ρ -and ρ + meet) depends on the value of α . The two parameters are related by the relation</text>
<formula><location><page_6><loc_21><loc_37><loc_81><loc_39></location>-α 8 +6 α 4 L 2 min ( L 2 min -3 ) -8 α 2 L 4 min ( L 2 min +9 ) +3 L 4 min ( L 2 min -3 ) 2 = 0 . (34)</formula>
<text><location><page_6><loc_12><loc_29><loc_81><loc_35></location>Figure 3 is a plot of L min vs α . When α = 1 / 2 , L min approaches zero. This is because V + ceases to have a local minimum for α ≥ 1 / 2 . Physically, this limit corresponds to the case when the Coulomb repulsion becomes too strong for stable orbits to exist as discussed in Ref. Al Zahrani (2021).</text>
<text><location><page_6><loc_12><loc_18><loc_81><loc_28></location>Figure 4 shows how b max depends on E for several negative values of the coupling parameter α . The effect of increasing | α | is to increase the values of b max for all energies. This is expected because the Coulombs attraction makes it easier for a charges particle to get captured. In all cases, b max is a monotonic function of E . In the ultra-relativistic limit, b max approaches 3 √ 3 / 2 , the limit in the neutral particle case, for any finite value of α , provided that α is not too large compared to E .</text>
<text><location><page_6><loc_12><loc_2><loc_81><loc_17></location>Figure 5 shows how b max depends on E for several values of α between 0 and 0 . 5 . In this range, there is competition between that gravitational 'attraction' and the Coulomb repulsion. The curves have richer structure. They falls quickly as E goes beyond 1 and reach a minimum. After that, the curves rise and reach 3 √ 3 / 2 asymptotically. Figure 6 shows how b max depends on E for several positive values of α greater than 0 . 5 . Generally, b max becomes smaller as α increases. This is expected because the greater the Coulomb repulsion the more difficult it is for a charged particle to be captured. In fact, there is a threshold energy E thr below which capture cannot occur. It is given by</text>
<text><location><page_7><loc_30><loc_90><loc_63><loc_91></location>Capture Cross-Section by Weakly Charged Black Hole</text>
<figure>
<location><page_7><loc_22><loc_64><loc_71><loc_89></location>
<caption>Fig. 3: The value of L at which ρ + and ρ -meet ( L min) vs. the electromagnetic coupling parameter α .</caption>
</figure>
<figure>
<location><page_7><loc_22><loc_34><loc_71><loc_59></location>
<caption>Fig. 4: The maximum impact parameter for capture b max vs. E for a charged particle with α = 0 (black), α = -1 (blue), α = -2 (green), α = -3 (red).</caption>
</figure>
<text><location><page_7><loc_12><loc_24><loc_60><loc_25></location>This equation is valid for α ≥ 0 . 5 only. Fig. 7 shows how E thr vary with α .</text>
<text><location><page_7><loc_12><loc_17><loc_81><loc_21></location>The capture cross-section σ cap corresponding to Figs. 4, 5 and 6 is shown in Figs. 8, 10 and 9, respectively. In all cases, σ cap vs. E curves inherent the features of the b min vs. E curves.</text>
<text><location><page_7><loc_12><loc_15><loc_45><loc_16></location>For ultra-relativistic particles, we can write b max as</text>
<formula><location><page_7><loc_31><loc_10><loc_81><loc_14></location>b max = 3 √ 3 2 -√ 3 α E + 9 -2 α 2 6 √ 3 E 2 + O ( 1 E 3 ) . (36)</formula>
<text><location><page_7><loc_12><loc_7><loc_43><loc_8></location>The corresponding capture cross-section is then</text>
<formula><location><page_7><loc_30><loc_2><loc_81><loc_5></location>σ cap = 27 π 4 -9 πα E + ( 4 α 2 +9 ) π 2 E 2 + O ( 1 E 3 ) . (37)</formula>
<section_header_level_1><location><page_8><loc_37><loc_90><loc_56><loc_91></location>A. M. Al Zaharni & A. Al-Jamaa</section_header_level_1>
<figure>
<location><page_8><loc_22><loc_63><loc_71><loc_88></location>
<caption>Fig. 5: The maximum impact parameter for capture b max vs. E for a charged particle with α = 0 (black), α = 0 . 1 (blue), α = 0 . 3 (green), α = 0 . 5 (red).</caption>
</figure>
<figure>
<location><page_8><loc_22><loc_30><loc_71><loc_55></location>
<caption>Fig. 6: The max impact parameter for capture b max vs. E for a charged particle with with α = 0 (black), α = 1 (blue), α = 2 (green), α = 3 (red).</caption>
</figure>
<section_header_level_1><location><page_8><loc_12><loc_19><loc_25><loc_20></location>5 CONCLUSION</section_header_level_1>
<text><location><page_8><loc_12><loc_13><loc_81><loc_16></location>We have studied the capture cross-section of charged particles by a weakly charged Schwarzschild black hole. We have shown that a trace charge on the black hole can have prominent effects.</text>
<text><location><page_8><loc_12><loc_1><loc_81><loc_11></location>When the Coulomb force between a charged particle and the black hole is attractive, it enlarges the capture cross-section significantly. This is expected since the Coulomb attraction enhances the capture of charged particles. However, when the Coulomb force between a charged particle and the black hole is repulsive, it shrinks the capture cross-section significantly. When the electromagnetic coupling strength is below a critical value, capture is possible for all values of the particle's energy. When the electromagnetic</text>
<figure>
<location><page_9><loc_22><loc_65><loc_71><loc_89></location>
<caption>Fig. 7: The energy threshold for escape E thr vs. electromagnetic coupling parameter α .</caption>
</figure>
<figure>
<location><page_9><loc_22><loc_34><loc_71><loc_59></location>
<caption>Fig. 8: The capture cross-section σ cap vs. E for a charged particle with α = 0 (black), α = -1 (blue), α = -2 (green), α = -3 (red).</caption>
</figure>
<text><location><page_9><loc_12><loc_20><loc_81><loc_24></location>capture is impossible. This is because the Coulomb repulsion surpasses the gravitational attraction unless the particle's radial momentum is large enough.</text>
<text><location><page_9><loc_12><loc_6><loc_81><loc_19></location>Our results emphasizes the assertion that charged black holes will favorably accretes charges of the opposite sign. However, it is still possible for the black hole charge to grow if the plunging charged particles are energetic enough to the limit that the capture cross-section becomes independent of the sign of the charges. Moreover, the fact that the electromagnetic coupling constant is three orders of magnitudes greater for electrons than protons suggests that it is relatively easier for a black hole to accumulate positive charge than negative charge.</text>
<text><location><page_9><loc_12><loc_1><loc_81><loc_4></location>It will be an astrophysically interesting to study the energies of charged particles near an astrophysical black hole to understand better how the black hole's charge evolves. The problem can be astrophyically</text>
<figure>
<location><page_10><loc_22><loc_64><loc_71><loc_89></location>
<caption>A. M. Al Zaharni & A. Al-JamaaFig. 9: The capture cross-section σ cap vs. E for a charged particle with α = 0 (black), α = 0 . 1 (blue), α = 0 . 3 (green), α = 0 . 5 (red).</caption>
</figure>
<figure>
<location><page_10><loc_22><loc_32><loc_71><loc_57></location>
<caption>Fig. 10: The capture cross-section σ cap vs. E for a charged particle with α = 0 (black), α = 1 (blue), α = 2 (green), α = 3 (red).</caption>
</figure>
<section_header_level_1><location><page_10><loc_12><loc_22><loc_19><loc_24></location>References</section_header_level_1>
<text><location><page_10><loc_12><loc_19><loc_58><loc_20></location>Ahmedov B., Rahimov O., Toshmatov B., 2021, Universe, 7(8) , 307. 2</text>
<text><location><page_10><loc_12><loc_17><loc_41><loc_18></location>Al Zahrani A., 2021, Phys. Rev. D, 103. 1, 6</text>
<text><location><page_10><loc_12><loc_15><loc_33><loc_16></location>Al Zahrani A., 2022, ApJ, 937. 1</text>
<text><location><page_10><loc_12><loc_12><loc_49><loc_13></location>Anacleto M., et al., 2023, arXiv:2307.09536v1 [gr-qc]. 2</text>
<text><location><page_10><loc_12><loc_10><loc_50><loc_11></location>Bugden M., 2020, Class. Quantum Gravity, 37, 015001. 2</text>
<text><location><page_10><loc_12><loc_8><loc_81><loc_9></location>Carter B., 1973, Black Hole Equilibrium States, Black Holes, eds. C. DeWitt and B. S. DeWitt (Gordon and</text>
<text><location><page_10><loc_13><loc_6><loc_47><loc_7></location>Breach Science Publishers, Inc. New York, p. 57. 1</text>
<text><location><page_10><loc_12><loc_3><loc_74><loc_4></location>Chandrasekhar S., 1983, The Mathematical Theory of Black Holes, Oxford University Press. 3</text>
<text><location><page_10><loc_12><loc_1><loc_49><loc_2></location>Connell P., Frolov V., 2008, Phys. Rev. D, 78, 024032. 2</text>
<text><location><page_11><loc_12><loc_62><loc_74><loc_88></location>Ghosh S. and Afrin M., 2023, ApJ, 944, 174. 1 Goldstein H., Poole C. and Safko J., 2001, Classical Mechanics, third eddition, Pearson. 4 Gooding C., Frolov A., 2008, Phys. Rev. D, 77, 104026. 2 GRAVITY Collaboration, 2023, A & A, 677, L10. 4 Kocherlakota P. et al., 2021, (EHT Collaboration), Phys. Rev. D, 103, 104047. 1 Misner C., Thorne K., Wheeler J., 1973, Gravitation, W. H. Freeman and Co., San Francisco. 2 Singh B. , Ghosh S., 2018, Annals of Physics 395, 127. 2 Tsukamoto N., et al., 2014, Phys. Rev. D, 90, 064043. 2 Young P., 1976, Phys. Rev. D, 14, 3281. 2 Zajaˇcek M. et al., 2018, Monthly Notices of the Royal Astronomical Society, 480 4, 4408. 1 Zajaˇcek M. et al., 2019, J. Phys.: Conf. Ser., 1258, 012031. 1, 4 Zajaˇcek M., Tursunov A., arXiv:1904.04654. 1</text>
<text><location><page_11><loc_12><loc_60><loc_47><loc_61></location>Zakharov A., 1994, Class. Quantum Grav., 11, 1027. 2</text>
</document>
[
{
"title": "ABSTRACT",
"content": "Research in Astronomy and Astrophysics manuscript no. (L A T E X: Manuscript.tex; printed on July 18, 2024; 0:23)",
"pages": [
1
]
},
{
"title": "Charged Particles Capture Cross-Section by a Weakly Charged Schwarzschild Black Hole",
"content": "A. M. Al Zahrani, 1 and A. Al-Jama 2 Physics Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia; amz@kfupm.edu.sa 1 ; ahmad.aljama1905@gmail.com 2 Received 20xx month day; accepted 20xx month day Abstract We study the capture cross-section of charged particles by a weakly charged Schwarzschild black hole. The dependence of the maximum impact parameter for capture on the particle's energy is investigated numerically for different values of the electromagnetic coupling strength between the particle and the black hole. The capture cross-section is then calculated. We show that the capture cross-section is independent of the electromagnetic coupling for ultra-relativistic particles. The astrophysical implications of our results are discussed. Key words: stars: black holes - Stars, accretion, accretion disks - Physical Data and Processes, black hole physics - Physical Data and Processes",
"pages": [
1
]
},
{
"title": "1 INTRODUCTION",
"content": "Studying the capture cross-section of black holes is central to understand how matter interacts with them. It helps us understand the process of matter accretion by a black hole which in turn determines how its mass, angular momentum and charge evolve. It can also help us understand the environment near black holes. Moreover, scrutinizing capture cross-section can be used to test theories of gravity in strong gravitational fields. Astrophysicists generally assume black holes are electrically neutral. This is because they would quickly attract oppositely charged matter to balance out any access charge. However, there are compelling reasons why weakly charged black holes might exist as discussed in Al Zahrani (2021, 2022); Zajaˇcek & Tursunov (2019); Zajaˇcek et al. (2019, 2018); Carter (1973) and the references therein. The differences in how a black hole accretes electrons and protons within its plasma environment, influenced by radiation, could render it charged. Also, the spin of a black hole in the presence of a magnetic field can induce the accretion of charged particles. In fact, using the EHT observations, it was inferred that Sgr A* and M87 can be charged Ghosh & Afrin (2023); Kocherlakota et al. (2021). The black hole's charge is weak in the sense There are numerous astrophysical scenarios wherein charged particles are drawn into black holes. Stars within the Roche limits near black holes often contribute matter through tidal interactions. Additionally, stars emit streams of charged particles as stellar winds. Highly energetic charged particles, resulting from supernovae, gamma-ray bursts, and bipolar jets from compact objects, frequently find their way into the vicinity of black holes. These processes collectively enrich the environment around black holes with a significant population of charged particles. The concept of capture cross-sections has been explored extensively for various black hole types. Foundational treatment which examine photon and neutral particle capture by Schwarzschild black holes was given in several monographs, such as Frolov & Zelnikov (2011). Further work addressed capture cross-sections of charged and neutral particles by Kerr-Newman black holes, including the implications for black hole spin and charge evolution Young (1976). Capture by Reissner-Nordstrom black holes was also investigated Zakharov (1994). In the context of higher-dimensional black holes, studies have focused on calculating photon critical impact parameters for Schwarzschild-Tangherlini black holes Connell & Frolov (2008); Tsukamoto (2014); Singh & Ghosh (2018); Bugden (2020). The capture cross-section for massive particles was determined in Ahmedov et al. (2021). Additionally, research extends to particle capture in Myers-Perry rotating spacetime which describes rotating black holes in five-dimensions Gooding & Frolov (2008). Moreover, wave capture cross-sections have been studied for various black hole configurations (see Anacleto, et al. (2023) and the references within). In this research, we examine the capture cross-section of charged particles by a weakly charged Schwarzschild black hole and discuss the astrophysical consequences of our findings. The paper is organized as follows: In Sec. 2, we review the dynamics of charged particles in the background of a weakly charged black hole. We then review the capture cross-section of neutral particles in Sec. 3. The capture cross-section of charged particles is calculated for different coupling strengths and particle energies in Secs. 4. Finally, we summarize our main findings and discuss their astrophysical consequences in Sec 5. We use the sign conventions adopted in Misner et al. (1973) and geometrized units where c = G = k = 1 , where k is the electrostatic constant.",
"pages": [
1,
2
]
},
{
"title": "2 CHARGED PARTICLES NEAR A WEAKLY CHARGED SCHWARZSCHILD BLACK HOLE",
"content": "Here, we review the dynamics of charged particles near a weakly charged black hole. The spacetime geometry around a black hole of mass M and charge Q is described by the Schwarzschild Reissner-Nordstron metric which reads Misner et al. (1973) where h = 1 -r S /r + Q 2 /r 2 and r S = 2 M is the Schwarzschild radius. The electromagnetic 4-potential is However, when the charge is weak we can ignore the curvature due to it and use the Schwarzschild metric, which reads Misner et al. (1973) where f = 1 -r S /r and r S = 2 M is the Schwarzschild radius. This weak charge approximation is valid unless the charge creates curvature comparable to that due to the black hole's mass. This happens when In conventional units, the weak charge approximation fails when This charge is way greater than the greatest estimated change on any black hole. Although the black hole charge is tiny, its effect on charged particles dynamics is profound because it is multiplied by the charge-tomass ratio of these particles ( ∼ 10 21 m -1 for electrons and ∼ 10 18 m -1 for protons). The Lagrangian describing a charged particle of charge q and mass m in a spacetime described by a metric g µν and an electromagnetic field produced by a 4-potential A µ reads Chandrasekhar (1983) where u µ ≡ dx µ /dτ is the particle's 4-velocity and τ is its proper time. In our case, the Lagrangian becomes This Lagrangian is cyclic in t and ϕ , which means that the particle's energy and azimuthal angular momentum are constants of motion. The specific energy and azimuthal angular momentum are, respectively, given by Combining these equations with the normalization condition g µν u µ u ν = -1 and solving for dr/dτ give In the equatorial plane where θ = π/ 2 , the equation becomes Let us rewrite the last equation in a dimensionless form. We first introduce the following dimensionless quantities: Equation 11 then becomes where qQ The parameter α represent the relative strength of the electromagnetic force to the Newtonian gravitational force. We can rewrite Eq. 13 as where is an effective potential. It is V + that corresponds to physical, future-directed motion and hence will be used in all of the analyses below. Without loss of generality, we will consider L > 0 only. It was estimated in Ref. Zajaˇcek et al. (2019) that the charge of Sgr A* is 10 8 -10 15 coulomb. Using the lower limit of charge, the coupling constant for electrons α e and protons α p near Sgr A*, which has a mass of M = 4 . 3 × 10 6 M ⊙ according to Ref. GRAVITY Collaboration (2023), are",
"pages": [
2,
3,
4
]
},
{
"title": "3 CAPTURE CROSS-SECTION OF NEUTRAL PARTICLES",
"content": "Before we tackle the main problem, let us find the capture cross-cross section for neutral particles first. Setting α = 0 , the effective potential V + reduces to Capture occurs whenever the particle's energy is greater than the maximum of V + . The function V + is at an extremum when dV + /dρ = 0 or which gives the position of the extrema in terms of L as where L ∈ [ √ 3 , ∞ ) . When L = √ 3 ( ≡ L min), ρ + and ρ -meet at a saddle point. Inspecting d 2 V + /dρ 2 reveals that ρ -corresponds to the position of the local maximum of V + . In terms of L , the escape condition E = V + | ρ = ρ -becomes Inverting this equation gives The impact parameter b is defined as the perpendicular distance between the center of force and the incident velocity Goldstein et al. (2001). It can be written as L L where P is the specific linear momentum. The maximum impact parameter for capture b max is given by The capture cross-section σ cap is given by Figures 1 and 2 are plots of b max and the capture cross-section σ cap vs. E , respectively. For ultra-relativistic particles ( E ≫ 1 ), √ √ The corresponding capture cross-section σ cap is therefore For a slowly moving particle with speed v ≪ 1 , and thus and the capture cross-section becomes",
"pages": [
4,
5,
6
]
},
{
"title": "4 CAPTURE CROSS-SECTION OF CHARGED PARTICLES",
"content": "We will now follow the same procedure we used for the neutral particle. However, analytic expressions are not viable in this case and we will resort to numerical solutions, except in the ultra-relativistic particle case. The structure of the effective potential V + is generically similar to the neutral particle's. The effect of α is to raise (lower) the peak of V + for positive (negative) α . The effective potential V + is at an extremum when The extremum is a maximum when To be consistent with the notation of the previous section, we let ρ + correspond to the minimum of V + and ρ -correspond to the maximum. Here, L min (the value at which ρ -and ρ + meet) depends on the value of α . The two parameters are related by the relation Figure 3 is a plot of L min vs α . When α = 1 / 2 , L min approaches zero. This is because V + ceases to have a local minimum for α ≥ 1 / 2 . Physically, this limit corresponds to the case when the Coulomb repulsion becomes too strong for stable orbits to exist as discussed in Ref. Al Zahrani (2021). Figure 4 shows how b max depends on E for several negative values of the coupling parameter α . The effect of increasing | α | is to increase the values of b max for all energies. This is expected because the Coulombs attraction makes it easier for a charges particle to get captured. In all cases, b max is a monotonic function of E . In the ultra-relativistic limit, b max approaches 3 √ 3 / 2 , the limit in the neutral particle case, for any finite value of α , provided that α is not too large compared to E . Figure 5 shows how b max depends on E for several values of α between 0 and 0 . 5 . In this range, there is competition between that gravitational 'attraction' and the Coulomb repulsion. The curves have richer structure. They falls quickly as E goes beyond 1 and reach a minimum. After that, the curves rise and reach 3 √ 3 / 2 asymptotically. Figure 6 shows how b max depends on E for several positive values of α greater than 0 . 5 . Generally, b max becomes smaller as α increases. This is expected because the greater the Coulomb repulsion the more difficult it is for a charged particle to be captured. In fact, there is a threshold energy E thr below which capture cannot occur. It is given by Capture Cross-Section by Weakly Charged Black Hole This equation is valid for α ≥ 0 . 5 only. Fig. 7 shows how E thr vary with α . The capture cross-section σ cap corresponding to Figs. 4, 5 and 6 is shown in Figs. 8, 10 and 9, respectively. In all cases, σ cap vs. E curves inherent the features of the b min vs. E curves. For ultra-relativistic particles, we can write b max as The corresponding capture cross-section is then",
"pages": [
6,
7
]
},
{
"title": "5 CONCLUSION",
"content": "We have studied the capture cross-section of charged particles by a weakly charged Schwarzschild black hole. We have shown that a trace charge on the black hole can have prominent effects. When the Coulomb force between a charged particle and the black hole is attractive, it enlarges the capture cross-section significantly. This is expected since the Coulomb attraction enhances the capture of charged particles. However, when the Coulomb force between a charged particle and the black hole is repulsive, it shrinks the capture cross-section significantly. When the electromagnetic coupling strength is below a critical value, capture is possible for all values of the particle's energy. When the electromagnetic capture is impossible. This is because the Coulomb repulsion surpasses the gravitational attraction unless the particle's radial momentum is large enough. Our results emphasizes the assertion that charged black holes will favorably accretes charges of the opposite sign. However, it is still possible for the black hole charge to grow if the plunging charged particles are energetic enough to the limit that the capture cross-section becomes independent of the sign of the charges. Moreover, the fact that the electromagnetic coupling constant is three orders of magnitudes greater for electrons than protons suggests that it is relatively easier for a black hole to accumulate positive charge than negative charge. It will be an astrophysically interesting to study the energies of charged particles near an astrophysical black hole to understand better how the black hole's charge evolves. The problem can be astrophyically",
"pages": [
8,
9
]
},
{
"title": "References",
"content": "Ahmedov B., Rahimov O., Toshmatov B., 2021, Universe, 7(8) , 307. 2 Al Zahrani A., 2021, Phys. Rev. D, 103. 1, 6 Al Zahrani A., 2022, ApJ, 937. 1 Anacleto M., et al., 2023, arXiv:2307.09536v1 [gr-qc]. 2 Bugden M., 2020, Class. Quantum Gravity, 37, 015001. 2 Carter B., 1973, Black Hole Equilibrium States, Black Holes, eds. C. DeWitt and B. S. DeWitt (Gordon and Breach Science Publishers, Inc. New York, p. 57. 1 Chandrasekhar S., 1983, The Mathematical Theory of Black Holes, Oxford University Press. 3 Connell P., Frolov V., 2008, Phys. Rev. D, 78, 024032. 2 Ghosh S. and Afrin M., 2023, ApJ, 944, 174. 1 Goldstein H., Poole C. and Safko J., 2001, Classical Mechanics, third eddition, Pearson. 4 Gooding C., Frolov A., 2008, Phys. Rev. D, 77, 104026. 2 GRAVITY Collaboration, 2023, A & A, 677, L10. 4 Kocherlakota P. et al., 2021, (EHT Collaboration), Phys. Rev. D, 103, 104047. 1 Misner C., Thorne K., Wheeler J., 1973, Gravitation, W. H. Freeman and Co., San Francisco. 2 Singh B. , Ghosh S., 2018, Annals of Physics 395, 127. 2 Tsukamoto N., et al., 2014, Phys. Rev. D, 90, 064043. 2 Young P., 1976, Phys. Rev. D, 14, 3281. 2 Zajaˇcek M. et al., 2018, Monthly Notices of the Royal Astronomical Society, 480 4, 4408. 1 Zajaˇcek M. et al., 2019, J. Phys.: Conf. Ser., 1258, 012031. 1, 4 Zajaˇcek M., Tursunov A., arXiv:1904.04654. 1 Zakharov A., 1994, Class. Quantum Grav., 11, 1027. 2",
"pages": [
10,
11
]
}
]
2024RMxAA..60..403A
https://arxiv.org/pdf/2409.19649.pdf
<document>
<section_header_level_1><location><page_1><loc_24><loc_77><loc_76><loc_84></location>X-RAY OBSERVATIONS OF THE VERY-FAINT X-RAY TRANSIENT XMMSL1 J171900.4-353217: A NEW CANDIDATE NEUTRON STAR LOW-MASS X-RAY BINARY</section_header_level_1>
<text><location><page_1><loc_25><loc_74><loc_74><loc_77></location>O. Ahmed, 1,2 N. Degenaar, 3 R. Wijnands, 3 and M. Armas Padilla 4,5 Received: November 14 2023; Accepted: August 15 2024</text>
<section_header_level_1><location><page_1><loc_46><loc_70><loc_54><loc_72></location>RESUMEN</section_header_level_1>
<text><location><page_1><loc_23><loc_49><loc_77><loc_70></location>XMMSL1 J171900.4-353217 es una binaria de rayos-X transitoria poco luminosa descubierta en marzo de 2010 durante una erupci'on. Presentamos 7 observaciones obtenidas entre mayo y octubre de 2010 con el Telescopio de Rayos-X (XRT) a bordo del Observatorio Neil Gehrels Swift. Mediante el ajuste de los espectros del XRT con un modelo de ley de potencias absorbido, obtenemos un 'ındice fot'onico de Γ = 1 . 8 -2 . 7 y una densidad de la columna de hidr'ogeno de N H = (4 . 6 -7 . 9) × 10 22 cm -2 . La luminosidad, en el rango 0 . 5 -10 keV, fluctu'o irregularmente, con picos de L X ≈ 10 35 -10 36 erg s -1 para una distancia de 4 -12 kpc. Bas'andonos en la evoluci'on del 'ındice fot'onico con la luminosidad, proponemos que la fuente es probablemente una binaria de rayos-X poco masiva con estrella de neutrones situada a varios kpc. De ser cierto, esta fuente ser'ıa una buena candidata para buscar pulsaciones coherentes de milisegundos cuando entre de nuevo en erupci'on.</text>
<section_header_level_1><location><page_1><loc_45><loc_46><loc_55><loc_47></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_23><loc_26><loc_77><loc_46></location>XMMSL1 J171900.4-353217 is a very-faint X-ray transient that was discovered in 2010 March when it exhibited an outburst. We report on 7 observations, obtained with the X-Ray Telescope (XRT) aboard the Neil Gehrels Swift Observatory between 2010 May to October. By fitting a single absorbed power-law model to the XRT spectra, we infer power-law indices of Γ = 1 . 8 -2 . 7 and an absorption column density of N H = (4 . 6 -7 . 9) × 10 22 cm -2 . The inferred 0 . 5 -10 keV luminosity fluctuated irregularly and peaked at L X /similarequal 10 35 -10 36 erg s -1 for a distance of 4 -12 kpc. Based on the evolution of the power-law index with varying luminosity, we propose that the source most likely is a transient neutron star low-mass X-ray binary located at several kpc. If true, it would be a good candidate to search for coherent millisecond pulsations when it enters a new accretion outburst.</text>
<text><location><page_2><loc_23><loc_85><loc_77><loc_88></location>Key Words: ISM: abundances - (stars:) binaries: general - stars: individual (XMMSL1 J171900.4-353217 ) - stars: neutron - X-rays</text>
<section_header_level_1><location><page_2><loc_42><loc_81><loc_58><loc_82></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_2><loc_23><loc_70><loc_77><loc_79></location>X-ray binaries are binary systems in which a compact object, either a black hole (BH) or neutron star (NS), accretes matter from a companion star. When the companion is a low-mass star (M ∼ < 1M /circledot ), the system is known as a low-mass X-ray binary (LMXB). Many LMXBs are transient: they become bright only during outbursts of active accretion, but are more often found in a dim quiescent state.</text>
<text><location><page_2><loc_23><loc_51><loc_77><loc_69></location>In quiescence, LMXBs have a low X-ray luminosity of L X ∼ < 33 erg s -1 (e.g., Wijnands et al. 2017). The maximum luminosity that is reached in outburst can vary a lot from source to source, and even from outburst to outburst for a single object. While many LMXBs are bright, with 2-10 keV peak luminosities of L X /similarequal 10 37 -10 39 erg s -1 , some also exhibit 'mini outbursts' that reach much lower peak luminosities of L X /similarequal 10 34 -10 36 erg s -1 (e.g., Degenaar & Wijnands 2009; Wijnands & Degenaar 2013; Coti Zelati et al. 2014; Zhang et al. 2019). These are often shorter than regular bright outbursts, although there are also LMXBs that accrete at such a low-luminosity level for extended periods of time ( ˇ Simon 2004; Degenaar et al. 2014; Allen et al. 2015; Parikh et al. 2018).</text>
<text><location><page_2><loc_23><loc_35><loc_77><loc_51></location>Interestingly, a growing number of systems has been discovered that exhibit maximum outburst luminosities of L X /similarequal 10 34 -10 36 erg s -1 and seemingly never exhibit brighter outbursts (Sakano et al. 2005; Muno et al. 2005b; Degenaar & Wijnands 2009; Bozzo et al. 2015; Bahramian et al. 2021). These LMXBs belong to the class of very faint X-ray transients (VFXTs; Wijnands et al. 2006). Many of these VFXTs are found near the Galactic center, but this is very likely a selection bias since this region has been regularly surveyed by many X-ray missions hence the brief and dim outbursts of VFXTs are more easily discovered than in other regions of our Galaxy (e.g., Muno et al. 2005b; Sakano et al. 2005; Wijnands et al. 2006; Degenaar et al. 2015).</text>
<text><location><page_2><loc_23><loc_23><loc_77><loc_34></location>While VFXTs could be intrinsically brighter sources located at large distances (tens of kpc) within Milky Way, estimates from thermonuclear bursts 6 . In addition, while inclination effects could possibly make these systems appear fainter than they intrinsically are (e.g., Muno et al. 2005a), this can likely only account for a small fraction of the VFXTs (see King & Wijnands 2006). Many VFXTs are thus expected to be intrinsically faint, i.e. accrete at low rates. This makes them interesting for a number of scientific reasons. For</text>
<text><location><page_3><loc_23><loc_77><loc_77><loc_88></location>instance, they probe a little explored mass-accretion regime, hence are valuable for studying accretion physics (e.g., Armas Padilla et al. 2013a; Weng & Zhang 2015; Degenaar et al. 2017). In addition, VFXTs are interesting for testing and improving binary evolution models (e.g. King & Wijnands 2006; Degenaar & Wijnands 2010; Maccarone et al. 2015), and for increasing our understanding of nuclear burning on the surface of accreting NS (e.g., Peng et al. 2007; Degenaar et al. 2010b).</text>
<text><location><page_3><loc_23><loc_55><loc_77><loc_76></location>Despite that the number of VFXTs has now grown to a few tens of systems (e.g., Bahramian & Degenaar 2023) and detailed studies of several systems have been performed over the past decade, still much remains to be learned about this source class. For instance, there is no clear picture yet about the distribution of system properties such as the nature of the compact accretor, type of companion star, and the size of the orbit. Determining whether an LMXB harbors a NS or a BH requires direct measurements of the physical properties of the compact object, such as its mass, or to detect the presence of a solid surface (e.g., through X-ray pulsations or thermonuclear bursts). However, such measurements are often challenging for VFXTs due to their faintness (e.g., making pulsation searches challenging; van den Eijnden et al. 2018) and low accretion rates (e.g., rendering thermonuclear bursts rare; Degenaar et al. 2011).</text>
<text><location><page_3><loc_23><loc_35><loc_77><loc_54></location>For some VFXTs, indirect approaches of using the ratio between the Xray and radio or optical/infrared luminosity have been employed to assess the nature of the compact accretor (e.g., Armas Padilla et al. 2011a; Paizis et al. 2011). However, their short outbursts often make it difficult to identify a counterpart for VFXTs at other wavelengths (e.g., Shaw et al. 2020). Furthermore, due to their low accretion rates, not many VXFTs have been detected in the radio band (van den Eijnden et al. 2021) and for finding optical/infrared counterparts added complications arise from their biased locations in the direction of the Galactic center (i.e., high extinction and crowding; e.g., Bandyopadhyay et al. 2005). Fortunately, an indication of the nature of the accretor can also be obtained by studying the X-ray spectral evolution of VFXTs (Armas Padilla et al. 2011b; Beri et al. 2019; Stoop et al. 2021).</text>
<text><location><page_3><loc_23><loc_21><loc_77><loc_34></location>The X-ray properties of LMXBs harboring a NS can be very similar to those containing a BH. However, when comparing their X-ray spectra at low luminosities of L X /similarequal 10 34 -10 36 erg s -1 , it turns out that confirmed or candidate BH systems have significantly harder spectra than confirmed NSs. In addition, the BH spectra show a strong softening when the X-ray lumninosity evolves below /similarequal 10 34 erg s -1 , while NSs start to show clear softening already at higher X-ray luminosities of L X /similarequal 10 36 erg s -1 (e.g., Wijnands et al. 2015; Parikh et al. 2017).</text>
<text><location><page_3><loc_23><loc_13><loc_77><loc_21></location>Over the last few years, detailed studies have been performed for a growing number of VFXTs and the general conclusion is that due to low statistics on their X-ray spectra, such systems can be satisfactorily described with a simple power-law model, with a soft (black body) component only being distinguishable when high quality (i.e. many counts) data are available (e.g., Armas</text>
<text><location><page_4><loc_23><loc_82><loc_77><loc_88></location>Padilla et al. 2011b). However, irrespectively of what model is fitted to the spectra, VFXTs also become softer with decreasing X-ray luminosity. Their X-ray spectral evolution during an outburst can thus be used as a diagnostic for the nature of the compact accretor.</text>
<section_header_level_1><location><page_4><loc_33><loc_78><loc_66><loc_79></location>1.1 . Discovery of XMMSL1 J171900.4-353217</section_header_level_1>
<text><location><page_4><loc_23><loc_65><loc_77><loc_76></location>XMMSL1 J171900.4-353217was discovered as an X-ray transient in XMMNewton slew data obtained on 2010 March 10 (Read et al. 2010a). The source location was in FOV of INTEGRAL observations performed around the same time, but it was not detected (20-40 keV; Bozzo et al. 2010). Markwardt et al. (2010) pointed out that XMMSL1 J171900.4-353217 was likely associated to a faint transient source, XTE J1719-356, detected in RXTE /PCA scans of the Galactic bulge since 2010 March.</text>
<text><location><page_4><loc_23><loc_45><loc_77><loc_64></location>Observations performed with the X-Ray Telescope (XRT; Burrows et al. 2005) onboard the Neil Gehrels Swift Observatory ( Swift ; Gehrels et al. 2004) in 2010 May, showed that the source was still active, i.e. two months after the initial discovery (Read et al. 2010b). Armas Padilla et al. (2010b,a) reported on further Swift /XRT observations, performed in 2010 May and June, which showed that the source remained active in soft X-rays albeit with varying flux. While Swift /XRT did no longer detect the source in 2010 July, suggesting it had returned to quiescence (Armas Padilla et al. 2010c), INTEGRAL serendipitously detected the source in hard X-rays in August 2010 (20-40 keV; Ishibashi et al. 2010). It was also detected again in soft X-rays with Swift /XRT around that time (Pavan et al. 2010). Nothing more was reported on the source after this.</text>
<text><location><page_4><loc_23><loc_40><loc_77><loc_45></location>In this work we investigate the nature of the compact accretor in the VFXT XMMSL1 J171900.4-353217 by studying its X-ray spectral evolution as seen with Swift /XRT.</text>
<section_header_level_1><location><page_4><loc_34><loc_36><loc_66><loc_37></location>2. OBSERVATIONS AND DATA ANALYSIS</section_header_level_1>
<text><location><page_4><loc_23><loc_28><loc_77><loc_34></location>XMMSL1 J171900.4-353217 was observed over a 157 days time-span with Swift , between 2010 May 11 and October 15 (see Table 1). Seven pointed observations were carried out during this time and we investigate the data collected using the XRT.</text>
<section_header_level_1><location><page_4><loc_37><loc_24><loc_63><loc_25></location>2.1 . Description of the data reduction</section_header_level_1>
<text><location><page_4><loc_23><loc_13><loc_77><loc_22></location>All XRT data were collected in photon counting (PC) mode. We reduced the data and obtained science products using the heasoft software package (v. 6.26). We cleaned the data by running the xrtpipeline task in which standard event grades of 0-12 were selected. For every observation, images, count rates and spectra were obtained with the xselect (v.2.4) package. We extracted the source events using a circular region with a radius of 52</text>
<table>
<location><page_5><loc_22><loc_65><loc_82><loc_84></location>
<caption>TABLE 1 LOG OF SWIFT /XRT OBSERVATIONS.TABLE 2</caption>
</table>
<section_header_level_1><location><page_5><loc_27><loc_60><loc_73><loc_62></location>RESULTS FROM ANALYSING THE SWIFT /XRT SPECTRA.</section_header_level_1>
<table>
<location><page_5><loc_13><loc_42><loc_78><loc_59></location>
</table>
<text><location><page_5><loc_23><loc_37><loc_77><loc_40></location>arcseconds. The background emission was averaged over three circular regions of similar size that were placed on nearby, source-free parts of the image.</text>
<text><location><page_5><loc_23><loc_28><loc_77><loc_36></location>The source was detected in 5 of the 7 observations (see Table 1) and for these we extracted spectra. Using grppha , three spectra were grouped to have 10 counts per energy bin, one was grouped to 20 counts per bin (observation ID 00031719004, when the source was brightest) and one to 5 counts per bin (observation ID 00031719002 when the source was faintest).</text>
<text><location><page_5><loc_23><loc_20><loc_77><loc_28></location>The spectra were corrected for the fractional exposure loss due to bad columns on the CCD. For this, we created exposure maps with the xrtexpomap task, which were then used as an input to generate the ancillary response files (arf) with the xrtmkarf task. We acquired the response matrix file (rmf) from the heasarc calibration database (v.12).</text>
<section_header_level_1><location><page_5><loc_46><loc_17><loc_54><loc_18></location>2.2 . Pile-Up</section_header_level_1>
<text><location><page_5><loc_23><loc_13><loc_77><loc_16></location>Observation 00031719004 has the highest count rate (0.7 ct s -1 ) and is affected by pile-up. We tested this following the steps outlined in the dedicated</text>
<table>
<location><page_6><loc_13><loc_72><loc_93><loc_84></location>
<caption>TABLE 3 OTHER REPORTED X-RAY FLUX MEASUREMENTS.</caption>
</table>
<text><location><page_6><loc_13><loc_66><loc_83><loc_72></location>a The quoted count rates were converted to 0.5-10 keV fluxes using webpimms and assuming a power-law spectral model. For the first two table entries we used N H = 5 . 45 × 10 22 cm -2 and Γ = 1 . 90, for the last two N H = 6 . 67 × 10 22 cm -2 and Γ = 1 . 80. These parameter values match those found from our spectral fitting of Swift /XRT data obtained around that time (see Table 2).</text>
<text><location><page_6><loc_13><loc_63><loc_83><loc_66></location>b References: 1=Read et al. (2010a), 2=Bozzo et al. (2010), 3=Ishibashi et al. (2010), 4=Pavan et al. (2010).</text>
<text><location><page_6><loc_23><loc_53><loc_77><loc_59></location>XRT analysis thread 7 . Following these guidelines, we found that five pixels had to be excluded in the bright core to mitigate the effect of pile-up. The remaining six observations have < 0 . 5 ct s -1 and are not affected by pile-up (see Table 1).</text>
<section_header_level_1><location><page_6><loc_45><loc_49><loc_55><loc_50></location>3. RESULTS</section_header_level_1>
<section_header_level_1><location><page_6><loc_41><loc_46><loc_59><loc_47></location>3.1 . X-ray spectral fitting</section_header_level_1>
<text><location><page_6><loc_23><loc_25><loc_77><loc_44></location>To fit the X-ray spectra, we used xspec 8 (v.12.10.1; Arnaud 1996). Given the low count rates (Table 1), we used simple power law ( pegpwrlw ) and black body ( bbodyrad ) models to describe the data. For both models, we took into account interstellar extinction by including the tbabs model and used C-statistics due to low data counts. For this absorption model we used abundances set to those of Wilms et al. (2000) and the cross-sections from Verner et al. (1996). Both models yielded the same quality of fit, so that we cannot statistically prefer one model over the other. However, in order to use the Wijnands et al. (2015) method to probe the nature of the compact accretor, we need to use the power-law model. Therefore, we here report on the results from fitting the absorbed power-law model, but we include the results for the absorbed black-body fits in the Appendix for completeness.</text>
<text><location><page_6><loc_23><loc_17><loc_77><loc_24></location>We also briefly explored whether the spectrum could be composed of two emission components, such as has been seen for VFXTs that have highquality data available (e.g., Armas Padilla et al. 2011b, 2013a; Degenaar et al. 2017). For this we used the observation with the highest flux (observation ID 00031719004). We first fitted this to an absorbed powerlaw, then added a</text>
<text><location><page_7><loc_23><loc_75><loc_77><loc_88></location>black body component and re-fitted. This resulted in similar fit parameters as for the single absorbed power-law model. The two-component model adequately fits the spectra by eye, but the extra thermal component is not statistically required (F-test probability > 0.99). It is likely that the low number of counts in the spectrum does not allow us to detect a second component, even if it is present. We therefore did not test this for the other observations, since these have even lower count rates. We conclude that a single-component model can adequately fit the Swift spectra.</text>
<text><location><page_7><loc_23><loc_63><loc_77><loc_75></location>Using the convolution model cflux within XSpec , setting the energy range to 0.5 to 10 keV, we determined both the absorbed (F X , abs ) and unabsorbed fluxes (F X , unabs ). The results are listed in Table 2. In Figure 1 we show the light curve constructed from the unabsorbed fluxes. From the seven XRT observations, the highest unabsorbed flux we measure is F X , unabs = 13 . 8 × 10 -11 erg cm -2 s -1 in observation 00031719004 (see Figure 1 and Table 2). In Figure 2 we show the Swift /XRT spectrum of this observation.</text>
<section_header_level_1><location><page_7><loc_34><loc_60><loc_66><loc_61></location>3.2 . Flux upper limits for XRT non-detections</section_header_level_1>
<text><location><page_7><loc_23><loc_38><loc_77><loc_58></location>In observations 00031719005 and 00031719007 the source was not detected with the XRT. For these observations, we determined the net counts detected at the source position with xselect (using similar source and background extraction regions as for the other observations; see Section 2.1). For observation 00031719005 (1 ks) we detect 4 counts at the source position and 1 count averaged over the background regions. For observation 00031719007 (1.3 ks), we measure 6 counts for the source and none for the background. Accounting for small number statistics using the tables of Gehrels (1986), we determine 95% upper limit on the detected net source counts of 7.75 and 11.84 for observations 00031719005 and 00031719007, respectively. Dividing by the exposure times then gives 95% confidence count rate upper limits of < 7 . 9 × 10 -3 ct s -1 (00031719005) and < 9 . 1 × 10 -3 ct s -1 (00031719007).</text>
<text><location><page_7><loc_23><loc_30><loc_77><loc_38></location>To estimate flux upper limits for the non-detections, we used webpimms 9 to convert the count rate upper limits, assuming an absorbed power-law model with a photon index of Γ = 2 . 49 and a hydrogen column density of N H = 5 . 81 × 10 22 cm -2 . We choose those values because these are the ones we obtained for the observation with the lowest flux (observation 00031719002).</text>
<section_header_level_1><location><page_7><loc_40><loc_26><loc_60><loc_28></location>3.3 . X-ray spectral evolution</section_header_level_1>
<text><location><page_7><loc_23><loc_15><loc_77><loc_25></location>The distance of XMMSL1 J171900.4-353217 is unknown. We therefore took three different values of 4, 8 and 12 kpc, to calculate the 0.5-10 keV luminosity from the unabsorbed flux. These results are included in Table 2. In Figure 3 we plot the evolution of the power-law index versus luminosity of XMMSL1 J171900.4-353217 along with the sample of NS (red filled circles) and BH (black crosses) LMXBs of Wijnands et al. (2015). We then overplot</text>
<figure>
<location><page_8><loc_23><loc_40><loc_72><loc_67></location>
<caption>Fig. 1. Evolution of 0.5 - 10 keV unabsorbed flux of XMMSL1 J171900.4-353217, inferred from our spectral analysis of the Swift /XRT data (blue filled circles). To show the full outburst, we include points reported in the literature from INTEGRAL (red open diamonds) and XMM-Newton (green open square), which were converted to 0.5-10 keV for this purpose (see Section 3.3 and Table 3).</caption>
</figure>
<text><location><page_9><loc_23><loc_85><loc_77><loc_88></location>XMMSL1 J171900.4-353217 as blue open diamonds for different distances of 4, 8, and 12 kpc in sub-panels a, b, and c, respectively.</text>
<text><location><page_9><loc_23><loc_72><loc_77><loc_85></location>We find that for all distances, our data points fall among the NS sample, but above the BH track. This would suggest that the source is either a proximate ( ∼ < 4 kpc) BH, or a NS located around or beyond 4 kpc. Considering the high N H towards the source, both as inferred from our X-ray spectral fitting (N H /similarequal 5 × 10 22 cm -2 ) and from Galactic extinction maps (N H /similarequal 1 × 10 22 cm -2 ;Bekhti et al. (2016)), we consider a larger distance more likely, and hence tentatively favor a NS nature. However, the reader should bear in mind that a BH nature cannot be excluded.</text>
<section_header_level_1><location><page_9><loc_38><loc_66><loc_62><loc_67></location>3.4 . Time-averaged accretion rate</section_header_level_1>
<text><location><page_9><loc_23><loc_53><loc_77><loc_64></location>We continue to calculate the time-averaged accretion rate for XMMSL1 J171900.4-353217, since this is an interesting parameter to understand the possible evolution paths of VFXTs (King & Wijnands 2006). We initially assume that the source contains a NS primary and then calculate the mean outburst accretion rate, 〈 ˙ M ob 〉 , from the mean unabsorbed flux measured during the outburst. For this purpose, we add the INTEGRAL and XMMNewton fluxes reported in the literature to our results obtained with Swift .</text>
<text><location><page_9><loc_23><loc_36><loc_77><loc_52></location>We used webpimms to convert reported instrument count rates or 20-40 keV fluxes to unabsorbed 0.5-10 keV fluxes. All information used for these conversions are listed in Table 3. We assumed an absorbed power-law spectral shape. For the XMM-Newton and first INTEGRAL observations, both performed in March 2010, we used N H = 5 . 45 × 10 22 cm -2 and Γ = 1 . 90, which are the values we obtained from spectral fitting for the Swift /XRT observations performed closest in time (observation 00000031719001; see Tables 1-3). For the other two INTEGRAL observations, both performed in 2010 August, we assumed N H = 6 . 67 × 10 22 cm -2 and Γ = 1 . 80 as found from fitting the Swift /XRT spectrum obtained closest in time (observation 00000031719006).</text>
<text><location><page_9><loc_23><loc_21><loc_77><loc_36></location>The resulting 0.5-10 keV flux light curve is shown in Figure 1. From all these data points we determine a mean 0.5-10 keV outburst flux of 8 . 9 × 10 -11 erg cm -2 s -1 . Based on this, we estimate the 0.1-100 keV accretion luminosity by assuming a bolometric correction factor of 3 (following in't Zand et al. 2007). The mass transfer rate of the outburst was then computed using the equation 〈 ˙ M ob 〉 = R Ns L acc / GM Ns , where G = 6 . 67 × 10 -8 cm 3 g -1 s -2 is the gravitational constant. Assuming R Ns = 1 . 1 × 10 6 cm = 11 km and M Ns = 1 . 5 M /circledot , the outburst mass accretion rate we obtain is 〈 ˙ M ob 〉 /similarequal 1 . 7 × 10 -10 M /circledot yr -1 .</text>
<text><location><page_9><loc_23><loc_13><loc_77><loc_21></location>We can next estimate the mean long-term averaged accretion rate using 〈 ˙ M long 〉 = 〈 ˙ M ob 〉 × t ob / t rec , where t ob is the outburst duration, t rec is the system's recurrence time, and the ratio of the two represents its duty cycle. Neither the onset nor the fading of the outburst into quiescence have been observed for XMMSL1 J171900.4-353217, so the total outburst duration is</text>
<text><location><page_10><loc_23><loc_72><loc_77><loc_88></location>unconstrained. 10 If we assume that the source was continuously active (i.e., only occasionally dropping to non-detectable flux levels) between its first and last detection on 2010 March 9 and 2010 August 20, the minimum outburst duration is 164 days. Since this was the first and only outburst ever observed for the source, its recurrence time is also unconstrained. For the present purpose we assume a duty cycle of 1-10% based on long-term X-ray monitoring of VFXTs in the Galactic center (Degenaar & Wijnands 2009, 2010). This would imply an outburst recurrence time of 4.5-45 yr for XMMSL1 J171900.4353217, and yields a mean long-term accretion rate of 〈 ˙ M long 〉 /similarequal 0 . 17 -1 . 7 × 10 -11 M /circledot yr -1 .</text>
<text><location><page_10><loc_23><loc_67><loc_77><loc_71></location>We note that if the source harbors a black hole accretor, the above estimates for the (long-term) mass accretion rate would be a factor /similarequal 10 lower due to the mass difference between neutron stars and black holes.</text>
<section_header_level_1><location><page_10><loc_44><loc_63><loc_56><loc_65></location>4. DISCUSSION</section_header_level_1>
<text><location><page_10><loc_23><loc_51><loc_77><loc_62></location>We report on the properties of the discovery outburst of the X-ray transient XMMSL1 J171900.4-353217, which lasted more than 164 days in 2010. We studied the X-ray spectral evolution of the source using the Swift/XRT data and used the method of Wijnands et al. (2015) to investigate the nature of the accreting object. Based on the evolution of its power-law index with 0.5-10 keV luminosity, we conclude that XMMSL1 J171900.4-353217 is most likely a NS LMXB located at several kpc.</text>
<text><location><page_10><loc_23><loc_39><loc_77><loc_50></location>Adding to our Swift /XRT results flux measurements reported in the literature (from XMM-Newton and INTEGRAL observations), we constructed the light curve of the 2010 outburst (see Figure 1). Over the 5.5 months that the source was observed to be active, the maximum 0.5-10 keV unabsorbed flux detected with Swift /XRT was F peak X , unabs = 13 . 8 × 10 -11 erg cm -2 s -1 . For a distance of 8 kpc, this peak flux translates into a luminosity of L peak X /similarequal 1 . 1 × 10 36 erg s -1 . This classifies XMMSL1 J171900.4-353217 as a VFXT. 11</text>
<text><location><page_10><loc_23><loc_26><loc_77><loc_38></location>We furthermore estimated a mean unabsorbed flux along the observations of F avg X /similarequal 8 . 9 × 10 -11 erg cm -2 s -1 . For a distance of 8 kpc, this translates into a luminosity of L avg X /similarequal 6 . 8 × 10 35 erg s -1 . We used this information to estimate the average accretion rate along the outburst as < ˙ M ob > /similarequal 1 . 7 × 10 -10 M /circledot yr -1 . If the source has a duty cycle of 110%, which is not uncommon for LMXBs and VFXTs (Degenaar & Wijnands 2010), we can then estimate a long-term average accretion rate of < ˙ M long > /similarequal 0 . 17 -1 . 7 × 10 -11 M /circledot yr -1 . This is in the same range as</text>
<figure>
<location><page_11><loc_22><loc_39><loc_72><loc_68></location>
<caption>Fig. 2. An X-ray spectrum of XMMSL1 J171900.4-353217 detected with Swift /XRT. Upper panel: shown is the brightest observation, 00031719004, fitted with an absorbed power law model. Bottom panel: the corresponding fit residuals in units of σ .</caption>
</figure>
<text><location><page_12><loc_24><loc_73><loc_24><loc_73></location>3.25</text>
<text><location><page_12><loc_24><loc_71><loc_24><loc_71></location>3.00</text>
<text><location><page_12><loc_24><loc_70><loc_24><loc_70></location>2.75</text>
<text><location><page_12><loc_24><loc_68><loc_24><loc_68></location>2.50</text>
<text><location><page_12><loc_24><loc_66><loc_24><loc_67></location>2.25</text>
<text><location><page_12><loc_24><loc_65><loc_24><loc_65></location>2.00</text>
<text><location><page_12><loc_24><loc_63><loc_24><loc_64></location>1.75</text>
<text><location><page_12><loc_24><loc_62><loc_24><loc_62></location>1.50</text>
<text><location><page_12><loc_24><loc_60><loc_24><loc_60></location>1.25</text>
<text><location><page_12><loc_33><loc_73><loc_34><loc_73></location>Black H(l s )()ulati(n</text>
<text><location><page_12><loc_33><loc_72><loc_35><loc_72></location>N utr(n Stars )()ulati(n</text>
<text><location><page_12><loc_33><loc_71><loc_34><loc_72></location>XMMSL1 J171900.4</text>
<text><location><page_12><loc_39><loc_71><loc_40><loc_72></location>-353217 at 4 kpc</text>
<text><location><page_12><loc_52><loc_73><loc_52><loc_73></location>3.25</text>
<text><location><page_12><loc_52><loc_71><loc_52><loc_71></location>3.00</text>
<text><location><page_12><loc_52><loc_70><loc_52><loc_70></location>2.75</text>
<text><location><page_12><loc_52><loc_68><loc_52><loc_68></location>2.50</text>
<text><location><page_12><loc_52><loc_66><loc_52><loc_67></location>2.25</text>
<text><location><page_12><loc_52><loc_65><loc_52><loc_65></location>2.00</text>
<text><location><page_12><loc_52><loc_63><loc_52><loc_64></location>1.75</text>
<text><location><page_12><loc_52><loc_62><loc_52><loc_62></location>1.50</text>
<text><location><page_12><loc_52><loc_60><loc_52><loc_60></location>1.25</text>
<text><location><page_12><loc_61><loc_73><loc_62><loc_73></location>Black H(l s )()ulati(n</text>
<text><location><page_12><loc_61><loc_72><loc_62><loc_72></location>N utr(n Stars )()ulati(n</text>
<text><location><page_12><loc_61><loc_71><loc_62><loc_72></location>XMMSL1 J171900.4</text>
<text><location><page_12><loc_67><loc_71><loc_68><loc_72></location>-353217 at 8 kpc</text>
<text><location><page_12><loc_23><loc_68><loc_24><loc_68></location>(Γ)</text>
<text><location><page_12><loc_23><loc_65><loc_24><loc_65></location>x</text>
<text><location><page_12><loc_23><loc_65><loc_24><loc_65></location>Photon Inde</text>
<text><location><page_12><loc_27><loc_59><loc_27><loc_59></location>31</text>
<text><location><page_12><loc_28><loc_59><loc_30><loc_59></location>X-ray Luminosity</text>
<text><location><page_12><loc_34><loc_59><loc_34><loc_59></location>(LogLx; 0</text>
<text><location><page_12><loc_37><loc_59><loc_38><loc_59></location>.5-10 keV; ergs/s)</text>
<text><location><page_12><loc_48><loc_58><loc_50><loc_59></location>(a)</text>
<text><location><page_12><loc_56><loc_59><loc_57><loc_59></location>X-ray Luminosity</text>
<text><location><page_12><loc_61><loc_59><loc_62><loc_59></location>(LogLx; 0</text>
<text><location><page_12><loc_64><loc_59><loc_65><loc_59></location>.5-10 keV; ergs/s)</text>
<text><location><page_12><loc_75><loc_58><loc_77><loc_59></location>(b)</text>
<figure>
<location><page_12><loc_31><loc_36><loc_69><loc_55></location>
<caption>Fig. 3. Power-law index versus X-ray luminosity in the 0.5-10 keV range for XMMSL1 J171900.4-353217 as well as a sample of NS (red circles) and BH (grey crosses) LMXBs (from Wijnands et al. 2015). For XMMSL1 J171900.4-353217 we used three different distances of 4 kpc (panel a), 8 kpc (panel b) and 12 kpc (panel c).</caption>
</figure>
<text><location><page_12><loc_30><loc_59><loc_30><loc_59></location>32</text>
<text><location><page_12><loc_32><loc_59><loc_32><loc_59></location>33</text>
<text><location><page_12><loc_34><loc_59><loc_35><loc_59></location>34</text>
<text><location><page_12><loc_37><loc_59><loc_37><loc_59></location>35</text>
<text><location><page_12><loc_39><loc_59><loc_40><loc_59></location>36</text>
<text><location><page_12><loc_42><loc_59><loc_42><loc_59></location>37</text>
<text><location><page_12><loc_44><loc_59><loc_44><loc_59></location>38</text>
<text><location><page_12><loc_51><loc_68><loc_51><loc_68></location>(Γ)</text>
<text><location><page_12><loc_51><loc_65><loc_51><loc_65></location>x</text>
<text><location><page_12><loc_51><loc_65><loc_51><loc_65></location>Photon Inde</text>
<text><location><page_12><loc_55><loc_59><loc_55><loc_59></location>31</text>
<text><location><page_12><loc_57><loc_59><loc_57><loc_59></location>32</text>
<text><location><page_12><loc_60><loc_59><loc_60><loc_59></location>33</text>
<text><location><page_12><loc_62><loc_59><loc_62><loc_59></location>34</text>
<text><location><page_12><loc_65><loc_59><loc_65><loc_59></location>35</text>
<text><location><page_12><loc_67><loc_59><loc_67><loc_59></location>36</text>
<text><location><page_12><loc_69><loc_59><loc_70><loc_59></location>37</text>
<text><location><page_12><loc_72><loc_59><loc_72><loc_59></location>38</text>
<text><location><page_13><loc_23><loc_78><loc_77><loc_88></location>inferred for the VFXTs in the Galactic Center (Degenaar & Wijnands 2009, 2010). Very low long-term average accretion rates can only be explained if these systems have hydrogen poor companions or are born with low companion masses (King & Wijnands 2006). However, the current (faint) accretion activity may not necessarily be representative for the long-term behavior of these systems (Wijnands et al. 2013).</text>
<text><location><page_13><loc_23><loc_47><loc_77><loc_78></location>In the past years, several NS LMXBs with similarly low outburst luminosities (hence accretion rates) as XMMSL1 J171900.4-353217 were uncovered to harbor accreting millisecond X-ray pulsars (AMXPs). Examples are IGR J17062-6143 (Strohmayer & Keek 2017), IGR J17591-2342 (Sanna et al. 2018b), IGR J17379-3747 (Sanna et al. 2018a) and IGR J17494-3030 (Ng et al. 2020). All were previously known VXFTs that were observed during (new) outbursts with NICER , which detected the X-ray pulsations. Given the similar X-ray spectral properties of XMMSL1 J171900.4-353217 with those sources, we hypothesize that it may also harbor a millisecond X-ray pulsar. Indeed, one of the sources mentioned above, was proposed to be a NS based on the same method as we employ here (Armas Padilla et al. 2013b) and later found to be an AMXP (Ng et al. 2020). Therefore, should XMMSL1 J171900.4-353217 enter a new accretion outburst in the future, we encourage X-ray observations (in particular with NICER ) to search for pulsations that would confirm the NS nature of this source and allow for a measurement of its orbital period. In case a new outburst occurs, we also encourage dense monitoring of the outburst decay (in particular with Swift ), since this can also provide an indication of the orbital period and nature of the compact accretor (e.g., Armas Padilla et al. 2011b; Heinke et al. 2015; Stoop et al. 2021).</text>
<section_header_level_1><location><page_13><loc_41><loc_43><loc_59><loc_45></location>ACKNOWLEGEMENTS</section_header_level_1>
<text><location><page_13><loc_23><loc_26><loc_77><loc_42></location>OA is grateful to Sera Markoff and the Anton Pannekoek Institute for organizing and hosting the Advanced Theoretical Astrophysics summer school in 2019, which fostered the collaboration that led to this work. ND was partly supported by a Vidi grant awarded by the Netherlands organization for scientific research (NWO). This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester. M. A. P. acknowledges support from the Spanish ministry of science under grant PID2020120323GB-I00. M. A. P. acknowledges support from the Consejeria de Economia, Conocimiento y Empleo del Gobierno de Canarias and the European Regional Development Fund under grant ProID2021010132.</text>
<section_header_level_1><location><page_13><loc_45><loc_22><loc_55><loc_23></location>APPENDICES</section_header_level_1>
<section_header_level_1><location><page_13><loc_31><loc_19><loc_69><loc_20></location>A. BLACK-BODY SPECTRAL FITTING RESULTS</section_header_level_1>
<text><location><page_13><loc_23><loc_13><loc_77><loc_17></location>For completeness we here report on the results of fitting the Swift /XRT spectra of XMMSL1 J171900.4-353217 with an absorbed black body model. For the upper limit calculation of the two XRT non-detections, we now used</text>
<paragraph><location><page_14><loc_16><loc_85><loc_84><loc_88></location>TABLE 4 RESULTS FROM XRT SPECTRAL ANALYSIS USING A BLACK-BODY MODEL. X-RAY</paragraph>
<table>
<location><page_14><loc_13><loc_65><loc_83><loc_82></location>
<caption>FLUXES AND LUMINOSITIES ARE GIVEN IN THE 0.5-10 KEV ENERGY BAND.</caption>
</table>
<text><location><page_14><loc_13><loc_64><loc_45><loc_65></location>Quoted errors reflect 1 - σ confidence intervals.</text>
<text><location><page_14><loc_23><loc_56><loc_77><loc_61></location>kT = 1 . 06 keV and N H = 2 . 29 × 10 22 cm -2 . These are the values we obtained for the observation with the lowest flux (observation ID 00031719002). All results are listed in Table 4.</text>
<section_header_level_1><location><page_14><loc_44><loc_52><loc_56><loc_53></location>REFERENCES</section_header_level_1>
<text><location><page_14><loc_23><loc_48><loc_77><loc_51></location>Allen, J. L., Linares, M., Homan, J., & Chakrabarty, D. 2015, The Astrophysical Journal</text>
<unordered_list>
<list_item><location><page_14><loc_23><loc_45><loc_77><loc_48></location>Armas Padilla, M., Degenaar, N., Kaur, R., Wijnands, R., & Yang, Y. 2010a, The Astronomer's Telegram, 2738, 1</list_item>
<list_item><location><page_14><loc_23><loc_40><loc_77><loc_44></location>Armas Padilla, M., Degenaar, N., Patruno, A., Russell, D. M., Linares, M., Maccarone, T. J., Homan, J., & Wijnands, R. 2011a, MNRAS, 417, 659 -. 2011b, MNRAS, 417, 659</list_item>
</unordered_list>
<text><location><page_14><loc_23><loc_39><loc_73><loc_40></location>Armas Padilla, M., Degenaar, N., & Wijnands, R. 2013a, MNRAS, 434, 1586</text>
<unordered_list>
<list_item><location><page_14><loc_23><loc_36><loc_77><loc_38></location>Armas Padilla, M., Degenaar, N., Yang, Y., Patruno, A., & Wijnands, R. 2010b, The Astronomer's Telegram, 2656, 1</list_item>
</unordered_list>
<text><location><page_14><loc_23><loc_33><loc_77><loc_35></location>Armas Padilla, M., Kaur, R., Degenaar, N., Wijnands, R., Lewis, F., & Russell, D. M. 2010c, The Astronomer's Telegram, 2722, 1</text>
<text><location><page_14><loc_23><loc_31><loc_73><loc_32></location>Armas Padilla, M., Wijnands, R., & Degenaar, N. 2013b, MNRAS, 436, L89</text>
<unordered_list>
<list_item><location><page_14><loc_23><loc_27><loc_77><loc_31></location>Arnaud, K. A. in , Astronomical Society of the Pacific Conference Series, Vol. 101, Astronomical Data Analysis Software and Systems V, ed. G. H. JacobyJ. Barnes, 17</list_item>
</unordered_list>
<text><location><page_14><loc_23><loc_24><loc_77><loc_26></location>Bahramian, A. & Degenaar, N. 2023, in Handbook of X-ray and Gamma-ray Astrophysics. Edited by Cosimo Bambi and Andrea Santangelo, 120</text>
<text><location><page_14><loc_23><loc_17><loc_77><loc_23></location>Bahramian, A., Heinke, C. O., Kennea, J. A., Maccarone, T. J., Evans, P. A., Wijnands, R., Degenaar, N., in't Zand, J. J. M., Shaw, A. W., Rivera Sandoval, L. E., McClure, S., Tetarenko, A. J., Strader, J., Kuulkers, E., & Sivakoff, G. R. 2021, MNRAS, 501, 2790</text>
<text><location><page_14><loc_23><loc_13><loc_77><loc_17></location>Bandyopadhyay, R. M., Miller-Jones, J. C. A., Blundell, K. M., Bauer, F. E., Podsiadlowski, P., Gosling, A. J., Wang, Q. D., Pfahl, E., & Rappaport, S. 2005, MNRAS, 364, 1195</text>
<text><location><page_15><loc_23><loc_84><loc_77><loc_88></location>Bekhti, N. B., Floer, L., Keller, R., Kerp, J., Lenz, D., Winkel, B., Bailin, J., Calabretta, M., Dedes, L., Ford, H., et al. 2016, Astronomy & Astrophysics, 594, A116</text>
<text><location><page_15><loc_23><loc_81><loc_77><loc_83></location>Beri, A., Altamirano, D., Wijnands, R., Degenaar, N., Parikh, A. S., & Yamaoka, K. 2019, MNRAS, 486, 1620</text>
<text><location><page_15><loc_23><loc_78><loc_77><loc_80></location>Bozzo, E., Romano, P., Falanga, M., Ferrigno, C., Papitto, A., & Krimm, H. A. 2015, A&A, 579, A56</text>
<text><location><page_15><loc_23><loc_75><loc_77><loc_77></location>Bozzo, E., Weidenspointner, G., Kuulkers, E., Terrier, R., & Carmona, P. K. A. 2010, The Astronomer's Telegram, 2616, 1</text>
<text><location><page_15><loc_23><loc_67><loc_77><loc_74></location>Burrows, D. N., Hill, J. E., Nousek, J. A., Kennea, J. A., Wells, A., Osborne, J. P., Abbey, A. F., Beardmore, A., Mukerjee, K., Short, A. D. T., Chincarini, G., Campana, S., Citterio, O., Moretti, A., Pagani, C., Tagliaferri, G., Giommi, P., Capalbi, M., Tamburelli, F., Angelini, L., Cusumano, G., Brauninger, H. W., Burkert, W., & Hartner, G. D. 2005, Space Sci. Rev., 120, 165</text>
<text><location><page_15><loc_23><loc_61><loc_77><loc_67></location>Cornelisse, R., Verbunt, F., in 't Zand, J., Kuulkers, E., Heise, J., Remillard, R., Cocchi, M., Natalucci, L., Bazzano, A., & Ubertini, P. 2002, A&A, 392, 885 Coti Zelati, F., Campana, S., D'Avanzo, P., & Melandri, A. 2014, MNRAS, 438, 2634</text>
<text><location><page_15><loc_23><loc_56><loc_77><loc_60></location>Degenaar, N., Jonker, P., Torres, M., Kaur, R., Rea, N., Israel, G., Patruno, A., Trap, G., Cackett, E., D'Avanzo, P., Lo Curto, G., Novara, G., Krimm, H., Holland, S., de Luca, A., Esposito, P., & Wijnands, R. 2010a, MNRAS, 404, 1591</text>
<text><location><page_15><loc_23><loc_50><loc_77><loc_56></location>Degenaar, N., Jonker, P. G., Torres, M. A. P., Kaur, R., Rea, N., Israel, G. L., Patruno, A., Trap, G., Cackett, E. M., D'Avanzo, P., Lo Curto, G., Novara, G., Krimm, H., Holland, S. T., de Luca, A., Esposito, P., & Wijnands, R. 2010b, MNRAS, 404, 1591</text>
<text><location><page_15><loc_23><loc_47><loc_77><loc_50></location>Degenaar, N., Pinto, C., Miller, J. M., Wijnands, R., Altamirano, D., Paerels, F., Fabian, A. C., & Chakrabarty, D. 2017, MNRAS, 464, 398</text>
<text><location><page_15><loc_23><loc_46><loc_71><loc_47></location>Degenaar, N. & Wijnands, R. 2009, Astronomy & Astrophysics, 495, 547</text>
<unordered_list>
<list_item><location><page_15><loc_23><loc_44><loc_53><loc_45></location>-. 2010, Astronomy & Astrophysics, 524, A69</list_item>
<list_item><location><page_15><loc_23><loc_43><loc_67><loc_44></location>Degenaar, N., Wijnands, R., & Kaur, R. 2011, MNRAS, 414, L104</list_item>
<list_item><location><page_15><loc_23><loc_40><loc_77><loc_42></location>Degenaar, N., Wijnands, R., Miller, J. M., Reynolds, M. T., Kennea, J., & Gehrels, N. 2015, Journal of High Energy Astrophysics, 7, 137</list_item>
<list_item><location><page_15><loc_23><loc_35><loc_77><loc_39></location>Degenaar, N., Wijnands, R., Reynolds, M. T., Miller, J. M., Altamirano, D., Kennea, J., Gehrels, N., Haggard, D., & Ponti, G. 2014, The Astrophysical Journal Gehrels, N. 1986, ApJ, 303, 336</list_item>
</unordered_list>
<text><location><page_15><loc_23><loc_14><loc_77><loc_35></location>Gehrels, N., Chincarini, G., Giommi, P., Mason, K. O., Nousek, J. A., Wells, A. A., White, N. E., Barthelmy, S. D., Burrows, D. N., Cominsky, L. R., Hurley, K. C., Marshall, F. E., M'esz'aros, P., Roming, P. W. A., Angelini, L., Barbier, L. M., Belloni, T., Campana, S., Caraveo, P. A., Chester, M. M., Citterio, O., Cline, T. L., Cropper, M. S., Cummings, J. R., Dean, A. J., Feigelson, E. D., Fenimore, E. E., Frail, D. A., Fruchter, A. S., Garmire, G. P., Gendreau, K., Ghisellini, G., Greiner, J., Hill, J. E., Hunsberger, S. D., Krimm, H. A., Kulkarni, S. R., Kumar, P., Lebrun, F., Lloyd-Ronning, N. M., Markwardt, C. B., Mattson, B. J., Mushotzky, R. F., Norris, J. P., Osborne, J., Paczynski, B., Palmer, D. M., Park, H. S., Parsons, A. M., Paul, J., Rees, M. J., Reynolds, C. S., Rhoads, J. E., Sasseen, T. P., Schaefer, B. E., Short, A. T., Smale, A. P., Smith, I. A., Stella, L., Tagliaferri, G., Takahashi, T., Tashiro, M., Townsley, L. K., Tueller, J., Turner, M. J. L., Vietri, M., Voges, W., Ward, M. J., Willingale, R., Zerbi, F. M., & Zhang, W. W. 2004, ApJ, 611, 1005</text>
<text><location><page_16><loc_23><loc_85><loc_77><loc_88></location>Heinke, C. O., Bahramian, A., Degenaar, N., & Wijnands, R. 2015, MNRAS, 447, 3034</text>
<text><location><page_16><loc_23><loc_84><loc_73><loc_85></location>in't Zand, J. J. M., Jonker, P. G., & Markwardt, C. B. 2007, A&A, 465, 953</text>
<text><location><page_16><loc_23><loc_75><loc_77><loc_83></location>Ishibashi, W., Bozzo, E., Terrier, R., Mereghetti, S., Paizis, A., Ducci, L., Gotz, D., Bazzano, A., Fiocchi, M., de Rosa, A., Tarana, A., Del Santo, M., Natalucci, L., Panessa, F., Capitanio, F., Sguera, V., Bianchin, V., Watanabe, K., Kuiper, L., Barragan, L., Chenevez, J., Caballero, I., Shrader, C., Bird, A., Puehlhofer, G., Sanchez-Fernandez, C., Skinner, G., Hartog, P. R. D., Pottschmidt, K., Negueruela, I., & Prat, L. 2010, The Astronomer's Telegram, 2803, 1</text>
<text><location><page_16><loc_23><loc_71><loc_77><loc_74></location>Keek, L., Iwakiri, W., Serino, M., Ballantyne, D. R., in't Zand, J. J. M., & Strohmayer, T. E. 2017, ApJ, 836, 111</text>
<text><location><page_16><loc_23><loc_68><loc_77><loc_71></location>King, A. R. & Wijnands, R. 2006, Monthly Notices of the Royal Astronomical Society: Letters, 366, L31</text>
<text><location><page_16><loc_23><loc_65><loc_77><loc_68></location>Kuulkers, E., den Hartog, P. R., in't Zand, J. J. M., Verbunt, F. W. M., Harris, W. E., & Cocchi, M. 2003, A&A, 399, 663</text>
<text><location><page_16><loc_23><loc_59><loc_77><loc_65></location>Lutovinov, A., Revnivtsev, M., Molkov, S., & Sunyaev, R. 2005, A&A, 430, 997 Maccarone, T. J., Wijnands, R. A. M., Degenaar, N., Archibald, A., Watts, A., Vaughan, S., Wynn, G., Knevitt, G., Farr, W., Andersson, N., van der Klis, M., Patruno, A., & Tauris, T. M. 2015, arXiv e-prints, arXiv:1501.02769</text>
<text><location><page_16><loc_23><loc_56><loc_77><loc_59></location>Markwardt, C. B., Strohmayer, T. E., & Swank, J. H. 2010, The Astronomer's Telegram, 2615, 1</text>
<text><location><page_16><loc_23><loc_52><loc_77><loc_56></location>Muno, M. P., Lu, J. R., Baganoff, F. K., Brandt, W. N., Garmire, G. P., Ghez, A. M., Hornstein, S. D., & Morris, M. R. 2005a, The Astrophysical Journal, 633, 228</text>
<text><location><page_16><loc_23><loc_49><loc_77><loc_51></location>Muno, M. P., Pfahl, E., Baganoff, F. K., Brandt, W. N., Ghez, A., Lu, J., & Morris, M. R. 2005b, ApJ, 622, L113</text>
<text><location><page_16><loc_23><loc_44><loc_77><loc_48></location>Ng, M., Ray, P. S., Strohmayer, T. E., Bult, P. M., Chakrabarty, D., Altamirano, D., Jaisawal, G. K., Malacaria, C., Bogdanov, S., Gendreau, K. C., & Arzoumanian, Z. 2020, The Astronomer's Telegram, 14124, 1</text>
<text><location><page_16><loc_23><loc_41><loc_77><loc_44></location>Paizis, A., Nowak, M. A., Wilms, J., Chaty, S., Corbel, S., Rodriguez, J., Del Santo, M., Ubertini, P., & Chini, R. 2011, ApJ, 738, 183</text>
<text><location><page_16><loc_23><loc_38><loc_77><loc_41></location>Parikh, A. S., Wijnands, R., Degenaar, N., & Altamirano, D. 2018, The Astronomer's Telegram, 11869, 1</text>
<text><location><page_16><loc_23><loc_35><loc_77><loc_38></location>Parikh, A. S., Wijnands, R., Degenaar, N., Altamirano, D., Patruno, A., Gusinskaia, N. V., & Hessels, J. W. T. 2017, MNRAS, 468, 3979</text>
<text><location><page_16><loc_23><loc_24><loc_77><loc_35></location>Pavan, L., Terrier, R., Bozzo, E., Ferrigno, C., Mereghetti, S., Paizis, A., Ducci, L., Gotz, D., Bazzano, A., Fiocchi, M., De Rosa, A., Tarana, A., Del Santo, M., Natalucci, L., Panessa, F., Capitanio, F., Sguera, V., Bianchin, V., Watanabe, K., Kuiper, L., Barragan, L., Chenevez, J., Caballero, I., Shrader, C., Bird, A., Puehlhofer, G., Sanchez-Fernandez, C., Skinner, G., den Hartog, P. R., Pottschmidt, K., Negueruela, I., & Prat, L. 2010, The Astronomer's Telegram, 2807, 1</text>
<text><location><page_16><loc_23><loc_23><loc_64><loc_24></location>Peng, F., Brown, E. F., & Truran, J. W. 2007, ApJ, 654, 1022</text>
<unordered_list>
<list_item><location><page_16><loc_23><loc_20><loc_77><loc_22></location>Read, A. M., Saxton, R. D., & Esquej, P. 2010a, The Astronomer's Telegram, 2607, 1</list_item>
<list_item><location><page_16><loc_23><loc_17><loc_77><loc_19></location>Read, A. M., Saxton, R. D., Esquej, P., & Evans, P. A. 2010b, The Astronomer's Telegram, 2627, 1</list_item>
</unordered_list>
<text><location><page_16><loc_23><loc_14><loc_77><loc_16></location>Sakano, M., Warwick, R. S., Decourchelle, A., & Wang, Q. D. 2005, MNRAS, 357, 1211</text>
<text><location><page_17><loc_23><loc_85><loc_77><loc_88></location>Sanna, A., Bozzo, E., Papitto, A., Riggio, A., Ferrigno, C., Di Salvo, T., Iaria, R., Mazzola, S. M., D'Amico, N., & Burderi, L. 2018a, A&A, 616, L17</text>
<text><location><page_17><loc_23><loc_78><loc_77><loc_85></location>Sanna, A., Ferrigno, C., Ray, P. S., Ducci, L., Jaisawal, G. K., Enoto, T., Bozzo, E., Altamirano, D., Di Salvo, T., Strohmayer, T. E., Papitto, A., Riggio, A., Burderi, L., Bult, P. M., Bogdanov, S., Gambino, A. F., Marino, A., Iaria, R., Arzoumanian, Z., Chakrabarty, D., Gendreau, K. C., Guillot, S., Markwardt, C., & Wolff, M. T. 2018b, A&A, 617, L8</text>
<text><location><page_17><loc_23><loc_71><loc_77><loc_77></location>Shaw, A. W., Heinke, C. O., Maccarone, T. J., Sivakoff, G. R., Strader, J., Bahramian, A., Degenaar, N., Kennea, J. A., Kuulkers, E., Rau, A., Rivera Sandoval, L. E., Shishkovsky, L., Swihart, S. J., Tetarenko, A. J., Wijnands, R., & in't Zand, J. J. M. 2020, MNRAS, 492, 4344</text>
<text><location><page_17><loc_23><loc_64><loc_77><loc_71></location>Stoop, M., van den Eijnden, J., Degenaar, N., Bahramian, A., Swihart, S. J., Strader, J., Jim'enez-Ibarra, F., Mu˜noz-Darias, T., Armas Padilla, M., Shaw, A. W., Maccarone, T. J., Wijnands, R., Russell, T. D., Hern'andez Santisteban, J. V., MillerJones, J. C. A., Russell, D. M., Maitra, D., Heinke, C. O., Sivakoff, G. R., Lewis, F., & Bramich, D. M. 2021, MNRAS, 507, 330</text>
<text><location><page_17><loc_23><loc_62><loc_54><loc_63></location>Strohmayer, T. & Keek, L. 2017, ApJ, 836, L23</text>
<text><location><page_17><loc_23><loc_58><loc_77><loc_62></location>van den Eijnden, J., Degenaar, N., Pinto, C., Patruno, A., Wette, K., Messenger, C., Hern'andez Santisteban, J. V., Wijnands, R., Miller, J. M., Altamirano, D., Paerels, F., Chakrabarty, D., & Fabian, A. C. 2018, MNRAS, 475, 2027</text>
<text><location><page_17><loc_23><loc_50><loc_77><loc_57></location>van den Eijnden, J., Degenaar, N., Russell, T. D., Wijnands, R., Bahramian, A., Miller-Jones, J. C. A., Hern'andez Santisteban, J. V., Gallo, E., Atri, P., Plotkin, R. M., Maccarone, T. J., Sivakoff, G., Miller, J. M., Reynolds, M., Russell, D. M., Maitra, D., Heinke, C. O., Armas Padilla, M., & Shaw, A. W. 2021, MNRAS, 507, 3899</text>
<text><location><page_17><loc_23><loc_47><loc_77><loc_50></location>Verner, D. A., Ferland, G. J., Korista, K. T., & Yakovlev, D. G. 1996, The Astrophysical Journal, 465, 487</text>
<text><location><page_17><loc_23><loc_46><loc_43><loc_47></location>ˇ Simon, V. 2004, A&A, 418, 617</text>
<text><location><page_17><loc_23><loc_44><loc_57><loc_45></location>Weng, S.-S. & Zhang, S.-N. 2015, MNRAS, 447, 486</text>
<text><location><page_17><loc_23><loc_43><loc_59><loc_44></location>Wijnands, R. & Degenaar, N. 2013, MNRAS, 434, 1599</text>
<text><location><page_17><loc_23><loc_38><loc_77><loc_42></location>Wijnands, R., Degenaar, N., Padilla, M. A., Altamirano, D., Cavecchi, Y., Linares, M., Bahramian, A., & Heinke, C. O. 2015, Monthly Notices of the Royal Astronomical Society, 454, 1371</text>
<text><location><page_17><loc_23><loc_35><loc_66><loc_38></location>Wijnands, R., Degenaar, N., & Page, D. 2013, MNRAS, 432, 2366 -. 2017, Journal of Astrophysics and Astronomy, 38, 49</text>
<text><location><page_17><loc_23><loc_29><loc_77><loc_35></location>Wijnands, R., in ' t Zand, J. J. M., Rupen, M., Maccarone, T., Homan, J., Cornelisse, R., Fender, R., Grindlay, J., van der Klis, M., Kuulkers, E., Markwardt, C. B., Miller-Jones, J. C. A., & Wang, Q. D. 2006, Astronomy & Astrophysics, 449, 1117</text>
<text><location><page_17><loc_23><loc_23><loc_77><loc_28></location>Wilms, J., Allen, A., & McCray, R. 2000, The Astrophysical Journal, 542, 914 Zhang, G. B., Bernardini, F., Russell, D. M., Gelfand , J. D., Lasota, J. P., Qasim, A. A., AlMannaei, A., Koljonen, K. I. I., Shaw, A. W., Lewis, F., Tomsick, J. A., Plotkin, R. M., Miller-Jones, J. C. A., Maitra, D., Homan, J., Charles, P. A.,</text>
<text><location><page_17><loc_25><loc_21><loc_59><loc_22></location>Kobel, P., Perez, D., & Doran, R. 2019, ApJ, 876, 5</text>
<text><location><page_18><loc_23><loc_83><loc_77><loc_88></location>Nathalie Degenaar and Rudy Wijnands: Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH, Amsterdam, The Netherlands.</text>
<text><location><page_18><loc_23><loc_79><loc_77><loc_83></location>Osman Ahmed: Astronomy and Space Science Department, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Kingdom of Saudi Arabia</text>
<text><location><page_18><loc_23><loc_74><loc_77><loc_78></location>Department of Physics, Faculty of Natural and computational Sciences, Debre Tabor University, P.O. Box 272, South Gondar, Ethiopia Montserrat Armas Padilla:</text>
<text><location><page_18><loc_23><loc_72><loc_77><loc_73></location>Instituto de Astrof ˜ Asica de Canarias, 38205, San Cristobal de La Laguna,</text>
<text><location><page_18><loc_25><loc_70><loc_29><loc_71></location>Spain</text>
<text><location><page_18><loc_25><loc_67><loc_77><loc_70></location>Departamento de Astrof'ısica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain</text>
</document>
[
{
"title": "ABSTRACT",
"content": "XMMSL1 J171900.4-353217 is a very-faint X-ray transient that was discovered in 2010 March when it exhibited an outburst. We report on 7 observations, obtained with the X-Ray Telescope (XRT) aboard the Neil Gehrels Swift Observatory between 2010 May to October. By fitting a single absorbed power-law model to the XRT spectra, we infer power-law indices of Γ = 1 . 8 -2 . 7 and an absorption column density of N H = (4 . 6 -7 . 9) × 10 22 cm -2 . The inferred 0 . 5 -10 keV luminosity fluctuated irregularly and peaked at L X /similarequal 10 35 -10 36 erg s -1 for a distance of 4 -12 kpc. Based on the evolution of the power-law index with varying luminosity, we propose that the source most likely is a transient neutron star low-mass X-ray binary located at several kpc. If true, it would be a good candidate to search for coherent millisecond pulsations when it enters a new accretion outburst. Key Words: ISM: abundances - (stars:) binaries: general - stars: individual (XMMSL1 J171900.4-353217 ) - stars: neutron - X-rays",
"pages": [
1,
2
]
},
{
"title": "X-RAY OBSERVATIONS OF THE VERY-FAINT X-RAY TRANSIENT XMMSL1 J171900.4-353217: A NEW CANDIDATE NEUTRON STAR LOW-MASS X-RAY BINARY",
"content": "O. Ahmed, 1,2 N. Degenaar, 3 R. Wijnands, 3 and M. Armas Padilla 4,5 Received: November 14 2023; Accepted: August 15 2024",
"pages": [
1
]
},
{
"title": "RESUMEN",
"content": "XMMSL1 J171900.4-353217 es una binaria de rayos-X transitoria poco luminosa descubierta en marzo de 2010 durante una erupci'on. Presentamos 7 observaciones obtenidas entre mayo y octubre de 2010 con el Telescopio de Rayos-X (XRT) a bordo del Observatorio Neil Gehrels Swift. Mediante el ajuste de los espectros del XRT con un modelo de ley de potencias absorbido, obtenemos un 'ındice fot'onico de Γ = 1 . 8 -2 . 7 y una densidad de la columna de hidr'ogeno de N H = (4 . 6 -7 . 9) × 10 22 cm -2 . La luminosidad, en el rango 0 . 5 -10 keV, fluctu'o irregularmente, con picos de L X ≈ 10 35 -10 36 erg s -1 para una distancia de 4 -12 kpc. Bas'andonos en la evoluci'on del 'ındice fot'onico con la luminosidad, proponemos que la fuente es probablemente una binaria de rayos-X poco masiva con estrella de neutrones situada a varios kpc. De ser cierto, esta fuente ser'ıa una buena candidata para buscar pulsaciones coherentes de milisegundos cuando entre de nuevo en erupci'on.",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "X-ray binaries are binary systems in which a compact object, either a black hole (BH) or neutron star (NS), accretes matter from a companion star. When the companion is a low-mass star (M ∼ < 1M /circledot ), the system is known as a low-mass X-ray binary (LMXB). Many LMXBs are transient: they become bright only during outbursts of active accretion, but are more often found in a dim quiescent state. In quiescence, LMXBs have a low X-ray luminosity of L X ∼ < 33 erg s -1 (e.g., Wijnands et al. 2017). The maximum luminosity that is reached in outburst can vary a lot from source to source, and even from outburst to outburst for a single object. While many LMXBs are bright, with 2-10 keV peak luminosities of L X /similarequal 10 37 -10 39 erg s -1 , some also exhibit 'mini outbursts' that reach much lower peak luminosities of L X /similarequal 10 34 -10 36 erg s -1 (e.g., Degenaar & Wijnands 2009; Wijnands & Degenaar 2013; Coti Zelati et al. 2014; Zhang et al. 2019). These are often shorter than regular bright outbursts, although there are also LMXBs that accrete at such a low-luminosity level for extended periods of time ( ˇ Simon 2004; Degenaar et al. 2014; Allen et al. 2015; Parikh et al. 2018). Interestingly, a growing number of systems has been discovered that exhibit maximum outburst luminosities of L X /similarequal 10 34 -10 36 erg s -1 and seemingly never exhibit brighter outbursts (Sakano et al. 2005; Muno et al. 2005b; Degenaar & Wijnands 2009; Bozzo et al. 2015; Bahramian et al. 2021). These LMXBs belong to the class of very faint X-ray transients (VFXTs; Wijnands et al. 2006). Many of these VFXTs are found near the Galactic center, but this is very likely a selection bias since this region has been regularly surveyed by many X-ray missions hence the brief and dim outbursts of VFXTs are more easily discovered than in other regions of our Galaxy (e.g., Muno et al. 2005b; Sakano et al. 2005; Wijnands et al. 2006; Degenaar et al. 2015). While VFXTs could be intrinsically brighter sources located at large distances (tens of kpc) within Milky Way, estimates from thermonuclear bursts 6 . In addition, while inclination effects could possibly make these systems appear fainter than they intrinsically are (e.g., Muno et al. 2005a), this can likely only account for a small fraction of the VFXTs (see King & Wijnands 2006). Many VFXTs are thus expected to be intrinsically faint, i.e. accrete at low rates. This makes them interesting for a number of scientific reasons. For instance, they probe a little explored mass-accretion regime, hence are valuable for studying accretion physics (e.g., Armas Padilla et al. 2013a; Weng & Zhang 2015; Degenaar et al. 2017). In addition, VFXTs are interesting for testing and improving binary evolution models (e.g. King & Wijnands 2006; Degenaar & Wijnands 2010; Maccarone et al. 2015), and for increasing our understanding of nuclear burning on the surface of accreting NS (e.g., Peng et al. 2007; Degenaar et al. 2010b). Despite that the number of VFXTs has now grown to a few tens of systems (e.g., Bahramian & Degenaar 2023) and detailed studies of several systems have been performed over the past decade, still much remains to be learned about this source class. For instance, there is no clear picture yet about the distribution of system properties such as the nature of the compact accretor, type of companion star, and the size of the orbit. Determining whether an LMXB harbors a NS or a BH requires direct measurements of the physical properties of the compact object, such as its mass, or to detect the presence of a solid surface (e.g., through X-ray pulsations or thermonuclear bursts). However, such measurements are often challenging for VFXTs due to their faintness (e.g., making pulsation searches challenging; van den Eijnden et al. 2018) and low accretion rates (e.g., rendering thermonuclear bursts rare; Degenaar et al. 2011). For some VFXTs, indirect approaches of using the ratio between the Xray and radio or optical/infrared luminosity have been employed to assess the nature of the compact accretor (e.g., Armas Padilla et al. 2011a; Paizis et al. 2011). However, their short outbursts often make it difficult to identify a counterpart for VFXTs at other wavelengths (e.g., Shaw et al. 2020). Furthermore, due to their low accretion rates, not many VXFTs have been detected in the radio band (van den Eijnden et al. 2021) and for finding optical/infrared counterparts added complications arise from their biased locations in the direction of the Galactic center (i.e., high extinction and crowding; e.g., Bandyopadhyay et al. 2005). Fortunately, an indication of the nature of the accretor can also be obtained by studying the X-ray spectral evolution of VFXTs (Armas Padilla et al. 2011b; Beri et al. 2019; Stoop et al. 2021). The X-ray properties of LMXBs harboring a NS can be very similar to those containing a BH. However, when comparing their X-ray spectra at low luminosities of L X /similarequal 10 34 -10 36 erg s -1 , it turns out that confirmed or candidate BH systems have significantly harder spectra than confirmed NSs. In addition, the BH spectra show a strong softening when the X-ray lumninosity evolves below /similarequal 10 34 erg s -1 , while NSs start to show clear softening already at higher X-ray luminosities of L X /similarequal 10 36 erg s -1 (e.g., Wijnands et al. 2015; Parikh et al. 2017). Over the last few years, detailed studies have been performed for a growing number of VFXTs and the general conclusion is that due to low statistics on their X-ray spectra, such systems can be satisfactorily described with a simple power-law model, with a soft (black body) component only being distinguishable when high quality (i.e. many counts) data are available (e.g., Armas Padilla et al. 2011b). However, irrespectively of what model is fitted to the spectra, VFXTs also become softer with decreasing X-ray luminosity. Their X-ray spectral evolution during an outburst can thus be used as a diagnostic for the nature of the compact accretor.",
"pages": [
2,
3,
4
]
},
{
"title": "1.1 . Discovery of XMMSL1 J171900.4-353217",
"content": "XMMSL1 J171900.4-353217was discovered as an X-ray transient in XMMNewton slew data obtained on 2010 March 10 (Read et al. 2010a). The source location was in FOV of INTEGRAL observations performed around the same time, but it was not detected (20-40 keV; Bozzo et al. 2010). Markwardt et al. (2010) pointed out that XMMSL1 J171900.4-353217 was likely associated to a faint transient source, XTE J1719-356, detected in RXTE /PCA scans of the Galactic bulge since 2010 March. Observations performed with the X-Ray Telescope (XRT; Burrows et al. 2005) onboard the Neil Gehrels Swift Observatory ( Swift ; Gehrels et al. 2004) in 2010 May, showed that the source was still active, i.e. two months after the initial discovery (Read et al. 2010b). Armas Padilla et al. (2010b,a) reported on further Swift /XRT observations, performed in 2010 May and June, which showed that the source remained active in soft X-rays albeit with varying flux. While Swift /XRT did no longer detect the source in 2010 July, suggesting it had returned to quiescence (Armas Padilla et al. 2010c), INTEGRAL serendipitously detected the source in hard X-rays in August 2010 (20-40 keV; Ishibashi et al. 2010). It was also detected again in soft X-rays with Swift /XRT around that time (Pavan et al. 2010). Nothing more was reported on the source after this. In this work we investigate the nature of the compact accretor in the VFXT XMMSL1 J171900.4-353217 by studying its X-ray spectral evolution as seen with Swift /XRT.",
"pages": [
4
]
},
{
"title": "2. OBSERVATIONS AND DATA ANALYSIS",
"content": "XMMSL1 J171900.4-353217 was observed over a 157 days time-span with Swift , between 2010 May 11 and October 15 (see Table 1). Seven pointed observations were carried out during this time and we investigate the data collected using the XRT.",
"pages": [
4
]
},
{
"title": "2.1 . Description of the data reduction",
"content": "All XRT data were collected in photon counting (PC) mode. We reduced the data and obtained science products using the heasoft software package (v. 6.26). We cleaned the data by running the xrtpipeline task in which standard event grades of 0-12 were selected. For every observation, images, count rates and spectra were obtained with the xselect (v.2.4) package. We extracted the source events using a circular region with a radius of 52",
"pages": [
4
]
},
{
"title": "RESULTS FROM ANALYSING THE SWIFT /XRT SPECTRA.",
"content": "arcseconds. The background emission was averaged over three circular regions of similar size that were placed on nearby, source-free parts of the image. The source was detected in 5 of the 7 observations (see Table 1) and for these we extracted spectra. Using grppha , three spectra were grouped to have 10 counts per energy bin, one was grouped to 20 counts per bin (observation ID 00031719004, when the source was brightest) and one to 5 counts per bin (observation ID 00031719002 when the source was faintest). The spectra were corrected for the fractional exposure loss due to bad columns on the CCD. For this, we created exposure maps with the xrtexpomap task, which were then used as an input to generate the ancillary response files (arf) with the xrtmkarf task. We acquired the response matrix file (rmf) from the heasarc calibration database (v.12).",
"pages": [
5
]
},
{
"title": "2.2 . Pile-Up",
"content": "Observation 00031719004 has the highest count rate (0.7 ct s -1 ) and is affected by pile-up. We tested this following the steps outlined in the dedicated a The quoted count rates were converted to 0.5-10 keV fluxes using webpimms and assuming a power-law spectral model. For the first two table entries we used N H = 5 . 45 × 10 22 cm -2 and Γ = 1 . 90, for the last two N H = 6 . 67 × 10 22 cm -2 and Γ = 1 . 80. These parameter values match those found from our spectral fitting of Swift /XRT data obtained around that time (see Table 2). b References: 1=Read et al. (2010a), 2=Bozzo et al. (2010), 3=Ishibashi et al. (2010), 4=Pavan et al. (2010). XRT analysis thread 7 . Following these guidelines, we found that five pixels had to be excluded in the bright core to mitigate the effect of pile-up. The remaining six observations have < 0 . 5 ct s -1 and are not affected by pile-up (see Table 1).",
"pages": [
5,
6
]
},
{
"title": "3.1 . X-ray spectral fitting",
"content": "To fit the X-ray spectra, we used xspec 8 (v.12.10.1; Arnaud 1996). Given the low count rates (Table 1), we used simple power law ( pegpwrlw ) and black body ( bbodyrad ) models to describe the data. For both models, we took into account interstellar extinction by including the tbabs model and used C-statistics due to low data counts. For this absorption model we used abundances set to those of Wilms et al. (2000) and the cross-sections from Verner et al. (1996). Both models yielded the same quality of fit, so that we cannot statistically prefer one model over the other. However, in order to use the Wijnands et al. (2015) method to probe the nature of the compact accretor, we need to use the power-law model. Therefore, we here report on the results from fitting the absorbed power-law model, but we include the results for the absorbed black-body fits in the Appendix for completeness. We also briefly explored whether the spectrum could be composed of two emission components, such as has been seen for VFXTs that have highquality data available (e.g., Armas Padilla et al. 2011b, 2013a; Degenaar et al. 2017). For this we used the observation with the highest flux (observation ID 00031719004). We first fitted this to an absorbed powerlaw, then added a black body component and re-fitted. This resulted in similar fit parameters as for the single absorbed power-law model. The two-component model adequately fits the spectra by eye, but the extra thermal component is not statistically required (F-test probability > 0.99). It is likely that the low number of counts in the spectrum does not allow us to detect a second component, even if it is present. We therefore did not test this for the other observations, since these have even lower count rates. We conclude that a single-component model can adequately fit the Swift spectra. Using the convolution model cflux within XSpec , setting the energy range to 0.5 to 10 keV, we determined both the absorbed (F X , abs ) and unabsorbed fluxes (F X , unabs ). The results are listed in Table 2. In Figure 1 we show the light curve constructed from the unabsorbed fluxes. From the seven XRT observations, the highest unabsorbed flux we measure is F X , unabs = 13 . 8 × 10 -11 erg cm -2 s -1 in observation 00031719004 (see Figure 1 and Table 2). In Figure 2 we show the Swift /XRT spectrum of this observation.",
"pages": [
6,
7
]
},
{
"title": "3.2 . Flux upper limits for XRT non-detections",
"content": "In observations 00031719005 and 00031719007 the source was not detected with the XRT. For these observations, we determined the net counts detected at the source position with xselect (using similar source and background extraction regions as for the other observations; see Section 2.1). For observation 00031719005 (1 ks) we detect 4 counts at the source position and 1 count averaged over the background regions. For observation 00031719007 (1.3 ks), we measure 6 counts for the source and none for the background. Accounting for small number statistics using the tables of Gehrels (1986), we determine 95% upper limit on the detected net source counts of 7.75 and 11.84 for observations 00031719005 and 00031719007, respectively. Dividing by the exposure times then gives 95% confidence count rate upper limits of < 7 . 9 × 10 -3 ct s -1 (00031719005) and < 9 . 1 × 10 -3 ct s -1 (00031719007). To estimate flux upper limits for the non-detections, we used webpimms 9 to convert the count rate upper limits, assuming an absorbed power-law model with a photon index of Γ = 2 . 49 and a hydrogen column density of N H = 5 . 81 × 10 22 cm -2 . We choose those values because these are the ones we obtained for the observation with the lowest flux (observation 00031719002).",
"pages": [
7
]
},
{
"title": "3.3 . X-ray spectral evolution",
"content": "The distance of XMMSL1 J171900.4-353217 is unknown. We therefore took three different values of 4, 8 and 12 kpc, to calculate the 0.5-10 keV luminosity from the unabsorbed flux. These results are included in Table 2. In Figure 3 we plot the evolution of the power-law index versus luminosity of XMMSL1 J171900.4-353217 along with the sample of NS (red filled circles) and BH (black crosses) LMXBs of Wijnands et al. (2015). We then overplot XMMSL1 J171900.4-353217 as blue open diamonds for different distances of 4, 8, and 12 kpc in sub-panels a, b, and c, respectively. We find that for all distances, our data points fall among the NS sample, but above the BH track. This would suggest that the source is either a proximate ( ∼ < 4 kpc) BH, or a NS located around or beyond 4 kpc. Considering the high N H towards the source, both as inferred from our X-ray spectral fitting (N H /similarequal 5 × 10 22 cm -2 ) and from Galactic extinction maps (N H /similarequal 1 × 10 22 cm -2 ;Bekhti et al. (2016)), we consider a larger distance more likely, and hence tentatively favor a NS nature. However, the reader should bear in mind that a BH nature cannot be excluded.",
"pages": [
7,
9
]
},
{
"title": "3.4 . Time-averaged accretion rate",
"content": "We continue to calculate the time-averaged accretion rate for XMMSL1 J171900.4-353217, since this is an interesting parameter to understand the possible evolution paths of VFXTs (King & Wijnands 2006). We initially assume that the source contains a NS primary and then calculate the mean outburst accretion rate, 〈 ˙ M ob 〉 , from the mean unabsorbed flux measured during the outburst. For this purpose, we add the INTEGRAL and XMMNewton fluxes reported in the literature to our results obtained with Swift . We used webpimms to convert reported instrument count rates or 20-40 keV fluxes to unabsorbed 0.5-10 keV fluxes. All information used for these conversions are listed in Table 3. We assumed an absorbed power-law spectral shape. For the XMM-Newton and first INTEGRAL observations, both performed in March 2010, we used N H = 5 . 45 × 10 22 cm -2 and Γ = 1 . 90, which are the values we obtained from spectral fitting for the Swift /XRT observations performed closest in time (observation 00000031719001; see Tables 1-3). For the other two INTEGRAL observations, both performed in 2010 August, we assumed N H = 6 . 67 × 10 22 cm -2 and Γ = 1 . 80 as found from fitting the Swift /XRT spectrum obtained closest in time (observation 00000031719006). The resulting 0.5-10 keV flux light curve is shown in Figure 1. From all these data points we determine a mean 0.5-10 keV outburst flux of 8 . 9 × 10 -11 erg cm -2 s -1 . Based on this, we estimate the 0.1-100 keV accretion luminosity by assuming a bolometric correction factor of 3 (following in't Zand et al. 2007). The mass transfer rate of the outburst was then computed using the equation 〈 ˙ M ob 〉 = R Ns L acc / GM Ns , where G = 6 . 67 × 10 -8 cm 3 g -1 s -2 is the gravitational constant. Assuming R Ns = 1 . 1 × 10 6 cm = 11 km and M Ns = 1 . 5 M /circledot , the outburst mass accretion rate we obtain is 〈 ˙ M ob 〉 /similarequal 1 . 7 × 10 -10 M /circledot yr -1 . We can next estimate the mean long-term averaged accretion rate using 〈 ˙ M long 〉 = 〈 ˙ M ob 〉 × t ob / t rec , where t ob is the outburst duration, t rec is the system's recurrence time, and the ratio of the two represents its duty cycle. Neither the onset nor the fading of the outburst into quiescence have been observed for XMMSL1 J171900.4-353217, so the total outburst duration is unconstrained. 10 If we assume that the source was continuously active (i.e., only occasionally dropping to non-detectable flux levels) between its first and last detection on 2010 March 9 and 2010 August 20, the minimum outburst duration is 164 days. Since this was the first and only outburst ever observed for the source, its recurrence time is also unconstrained. For the present purpose we assume a duty cycle of 1-10% based on long-term X-ray monitoring of VFXTs in the Galactic center (Degenaar & Wijnands 2009, 2010). This would imply an outburst recurrence time of 4.5-45 yr for XMMSL1 J171900.4353217, and yields a mean long-term accretion rate of 〈 ˙ M long 〉 /similarequal 0 . 17 -1 . 7 × 10 -11 M /circledot yr -1 . We note that if the source harbors a black hole accretor, the above estimates for the (long-term) mass accretion rate would be a factor /similarequal 10 lower due to the mass difference between neutron stars and black holes.",
"pages": [
9,
10
]
},
{
"title": "4. DISCUSSION",
"content": "We report on the properties of the discovery outburst of the X-ray transient XMMSL1 J171900.4-353217, which lasted more than 164 days in 2010. We studied the X-ray spectral evolution of the source using the Swift/XRT data and used the method of Wijnands et al. (2015) to investigate the nature of the accreting object. Based on the evolution of its power-law index with 0.5-10 keV luminosity, we conclude that XMMSL1 J171900.4-353217 is most likely a NS LMXB located at several kpc. Adding to our Swift /XRT results flux measurements reported in the literature (from XMM-Newton and INTEGRAL observations), we constructed the light curve of the 2010 outburst (see Figure 1). Over the 5.5 months that the source was observed to be active, the maximum 0.5-10 keV unabsorbed flux detected with Swift /XRT was F peak X , unabs = 13 . 8 × 10 -11 erg cm -2 s -1 . For a distance of 8 kpc, this peak flux translates into a luminosity of L peak X /similarequal 1 . 1 × 10 36 erg s -1 . This classifies XMMSL1 J171900.4-353217 as a VFXT. 11 We furthermore estimated a mean unabsorbed flux along the observations of F avg X /similarequal 8 . 9 × 10 -11 erg cm -2 s -1 . For a distance of 8 kpc, this translates into a luminosity of L avg X /similarequal 6 . 8 × 10 35 erg s -1 . We used this information to estimate the average accretion rate along the outburst as < ˙ M ob > /similarequal 1 . 7 × 10 -10 M /circledot yr -1 . If the source has a duty cycle of 110%, which is not uncommon for LMXBs and VFXTs (Degenaar & Wijnands 2010), we can then estimate a long-term average accretion rate of < ˙ M long > /similarequal 0 . 17 -1 . 7 × 10 -11 M /circledot yr -1 . This is in the same range as 3.25 3.00 2.75 2.50 2.25 2.00 1.75 1.50 1.25 Black H(l s )()ulati(n N utr(n Stars )()ulati(n XMMSL1 J171900.4 -353217 at 4 kpc 3.25 3.00 2.75 2.50 2.25 2.00 1.75 1.50 1.25 Black H(l s )()ulati(n N utr(n Stars )()ulati(n XMMSL1 J171900.4 -353217 at 8 kpc (Γ) x Photon Inde 31 X-ray Luminosity (LogLx; 0 .5-10 keV; ergs/s) (a) X-ray Luminosity (LogLx; 0 .5-10 keV; ergs/s) (b) 32 33 34 35 36 37 38 (Γ) x Photon Inde 31 32 33 34 35 36 37 38 inferred for the VFXTs in the Galactic Center (Degenaar & Wijnands 2009, 2010). Very low long-term average accretion rates can only be explained if these systems have hydrogen poor companions or are born with low companion masses (King & Wijnands 2006). However, the current (faint) accretion activity may not necessarily be representative for the long-term behavior of these systems (Wijnands et al. 2013). In the past years, several NS LMXBs with similarly low outburst luminosities (hence accretion rates) as XMMSL1 J171900.4-353217 were uncovered to harbor accreting millisecond X-ray pulsars (AMXPs). Examples are IGR J17062-6143 (Strohmayer & Keek 2017), IGR J17591-2342 (Sanna et al. 2018b), IGR J17379-3747 (Sanna et al. 2018a) and IGR J17494-3030 (Ng et al. 2020). All were previously known VXFTs that were observed during (new) outbursts with NICER , which detected the X-ray pulsations. Given the similar X-ray spectral properties of XMMSL1 J171900.4-353217 with those sources, we hypothesize that it may also harbor a millisecond X-ray pulsar. Indeed, one of the sources mentioned above, was proposed to be a NS based on the same method as we employ here (Armas Padilla et al. 2013b) and later found to be an AMXP (Ng et al. 2020). Therefore, should XMMSL1 J171900.4-353217 enter a new accretion outburst in the future, we encourage X-ray observations (in particular with NICER ) to search for pulsations that would confirm the NS nature of this source and allow for a measurement of its orbital period. In case a new outburst occurs, we also encourage dense monitoring of the outburst decay (in particular with Swift ), since this can also provide an indication of the orbital period and nature of the compact accretor (e.g., Armas Padilla et al. 2011b; Heinke et al. 2015; Stoop et al. 2021).",
"pages": [
10,
12,
13
]
},
{
"title": "ACKNOWLEGEMENTS",
"content": "OA is grateful to Sera Markoff and the Anton Pannekoek Institute for organizing and hosting the Advanced Theoretical Astrophysics summer school in 2019, which fostered the collaboration that led to this work. ND was partly supported by a Vidi grant awarded by the Netherlands organization for scientific research (NWO). This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester. M. A. P. acknowledges support from the Spanish ministry of science under grant PID2020120323GB-I00. M. A. P. acknowledges support from the Consejeria de Economia, Conocimiento y Empleo del Gobierno de Canarias and the European Regional Development Fund under grant ProID2021010132.",
"pages": [
13
]
},
{
"title": "A. BLACK-BODY SPECTRAL FITTING RESULTS",
"content": "For completeness we here report on the results of fitting the Swift /XRT spectra of XMMSL1 J171900.4-353217 with an absorbed black body model. For the upper limit calculation of the two XRT non-detections, we now used Quoted errors reflect 1 - σ confidence intervals. kT = 1 . 06 keV and N H = 2 . 29 × 10 22 cm -2 . These are the values we obtained for the observation with the lowest flux (observation ID 00031719002). All results are listed in Table 4.",
"pages": [
13,
14
]
},
{
"title": "REFERENCES",
"content": "Allen, J. L., Linares, M., Homan, J., & Chakrabarty, D. 2015, The Astrophysical Journal Armas Padilla, M., Degenaar, N., & Wijnands, R. 2013a, MNRAS, 434, 1586 Armas Padilla, M., Kaur, R., Degenaar, N., Wijnands, R., Lewis, F., & Russell, D. M. 2010c, The Astronomer's Telegram, 2722, 1 Armas Padilla, M., Wijnands, R., & Degenaar, N. 2013b, MNRAS, 436, L89 Bahramian, A. & Degenaar, N. 2023, in Handbook of X-ray and Gamma-ray Astrophysics. Edited by Cosimo Bambi and Andrea Santangelo, 120 Bahramian, A., Heinke, C. O., Kennea, J. A., Maccarone, T. J., Evans, P. A., Wijnands, R., Degenaar, N., in't Zand, J. J. M., Shaw, A. W., Rivera Sandoval, L. E., McClure, S., Tetarenko, A. J., Strader, J., Kuulkers, E., & Sivakoff, G. R. 2021, MNRAS, 501, 2790 Bandyopadhyay, R. M., Miller-Jones, J. C. A., Blundell, K. M., Bauer, F. E., Podsiadlowski, P., Gosling, A. J., Wang, Q. D., Pfahl, E., & Rappaport, S. 2005, MNRAS, 364, 1195 Bekhti, N. B., Floer, L., Keller, R., Kerp, J., Lenz, D., Winkel, B., Bailin, J., Calabretta, M., Dedes, L., Ford, H., et al. 2016, Astronomy & Astrophysics, 594, A116 Beri, A., Altamirano, D., Wijnands, R., Degenaar, N., Parikh, A. S., & Yamaoka, K. 2019, MNRAS, 486, 1620 Bozzo, E., Romano, P., Falanga, M., Ferrigno, C., Papitto, A., & Krimm, H. A. 2015, A&A, 579, A56 Bozzo, E., Weidenspointner, G., Kuulkers, E., Terrier, R., & Carmona, P. K. A. 2010, The Astronomer's Telegram, 2616, 1 Burrows, D. N., Hill, J. E., Nousek, J. A., Kennea, J. A., Wells, A., Osborne, J. P., Abbey, A. F., Beardmore, A., Mukerjee, K., Short, A. D. T., Chincarini, G., Campana, S., Citterio, O., Moretti, A., Pagani, C., Tagliaferri, G., Giommi, P., Capalbi, M., Tamburelli, F., Angelini, L., Cusumano, G., Brauninger, H. W., Burkert, W., & Hartner, G. D. 2005, Space Sci. Rev., 120, 165 Cornelisse, R., Verbunt, F., in 't Zand, J., Kuulkers, E., Heise, J., Remillard, R., Cocchi, M., Natalucci, L., Bazzano, A., & Ubertini, P. 2002, A&A, 392, 885 Coti Zelati, F., Campana, S., D'Avanzo, P., & Melandri, A. 2014, MNRAS, 438, 2634 Degenaar, N., Jonker, P., Torres, M., Kaur, R., Rea, N., Israel, G., Patruno, A., Trap, G., Cackett, E., D'Avanzo, P., Lo Curto, G., Novara, G., Krimm, H., Holland, S., de Luca, A., Esposito, P., & Wijnands, R. 2010a, MNRAS, 404, 1591 Degenaar, N., Jonker, P. G., Torres, M. A. P., Kaur, R., Rea, N., Israel, G. L., Patruno, A., Trap, G., Cackett, E. M., D'Avanzo, P., Lo Curto, G., Novara, G., Krimm, H., Holland, S. T., de Luca, A., Esposito, P., & Wijnands, R. 2010b, MNRAS, 404, 1591 Degenaar, N., Pinto, C., Miller, J. M., Wijnands, R., Altamirano, D., Paerels, F., Fabian, A. C., & Chakrabarty, D. 2017, MNRAS, 464, 398 Degenaar, N. & Wijnands, R. 2009, Astronomy & Astrophysics, 495, 547 Gehrels, N., Chincarini, G., Giommi, P., Mason, K. O., Nousek, J. A., Wells, A. A., White, N. E., Barthelmy, S. D., Burrows, D. N., Cominsky, L. R., Hurley, K. C., Marshall, F. E., M'esz'aros, P., Roming, P. W. A., Angelini, L., Barbier, L. M., Belloni, T., Campana, S., Caraveo, P. A., Chester, M. M., Citterio, O., Cline, T. L., Cropper, M. S., Cummings, J. R., Dean, A. J., Feigelson, E. D., Fenimore, E. E., Frail, D. A., Fruchter, A. S., Garmire, G. P., Gendreau, K., Ghisellini, G., Greiner, J., Hill, J. E., Hunsberger, S. D., Krimm, H. A., Kulkarni, S. R., Kumar, P., Lebrun, F., Lloyd-Ronning, N. M., Markwardt, C. B., Mattson, B. J., Mushotzky, R. F., Norris, J. P., Osborne, J., Paczynski, B., Palmer, D. M., Park, H. S., Parsons, A. M., Paul, J., Rees, M. J., Reynolds, C. S., Rhoads, J. E., Sasseen, T. P., Schaefer, B. E., Short, A. T., Smale, A. P., Smith, I. A., Stella, L., Tagliaferri, G., Takahashi, T., Tashiro, M., Townsley, L. K., Tueller, J., Turner, M. J. L., Vietri, M., Voges, W., Ward, M. J., Willingale, R., Zerbi, F. M., & Zhang, W. W. 2004, ApJ, 611, 1005 Heinke, C. O., Bahramian, A., Degenaar, N., & Wijnands, R. 2015, MNRAS, 447, 3034 in't Zand, J. J. M., Jonker, P. G., & Markwardt, C. B. 2007, A&A, 465, 953 Ishibashi, W., Bozzo, E., Terrier, R., Mereghetti, S., Paizis, A., Ducci, L., Gotz, D., Bazzano, A., Fiocchi, M., de Rosa, A., Tarana, A., Del Santo, M., Natalucci, L., Panessa, F., Capitanio, F., Sguera, V., Bianchin, V., Watanabe, K., Kuiper, L., Barragan, L., Chenevez, J., Caballero, I., Shrader, C., Bird, A., Puehlhofer, G., Sanchez-Fernandez, C., Skinner, G., Hartog, P. R. D., Pottschmidt, K., Negueruela, I., & Prat, L. 2010, The Astronomer's Telegram, 2803, 1 Keek, L., Iwakiri, W., Serino, M., Ballantyne, D. R., in't Zand, J. J. M., & Strohmayer, T. E. 2017, ApJ, 836, 111 King, A. R. & Wijnands, R. 2006, Monthly Notices of the Royal Astronomical Society: Letters, 366, L31 Kuulkers, E., den Hartog, P. R., in't Zand, J. J. M., Verbunt, F. W. M., Harris, W. E., & Cocchi, M. 2003, A&A, 399, 663 Lutovinov, A., Revnivtsev, M., Molkov, S., & Sunyaev, R. 2005, A&A, 430, 997 Maccarone, T. J., Wijnands, R. A. M., Degenaar, N., Archibald, A., Watts, A., Vaughan, S., Wynn, G., Knevitt, G., Farr, W., Andersson, N., van der Klis, M., Patruno, A., & Tauris, T. M. 2015, arXiv e-prints, arXiv:1501.02769 Markwardt, C. B., Strohmayer, T. E., & Swank, J. H. 2010, The Astronomer's Telegram, 2615, 1 Muno, M. P., Lu, J. R., Baganoff, F. K., Brandt, W. N., Garmire, G. P., Ghez, A. M., Hornstein, S. D., & Morris, M. R. 2005a, The Astrophysical Journal, 633, 228 Muno, M. P., Pfahl, E., Baganoff, F. K., Brandt, W. N., Ghez, A., Lu, J., & Morris, M. R. 2005b, ApJ, 622, L113 Ng, M., Ray, P. S., Strohmayer, T. E., Bult, P. M., Chakrabarty, D., Altamirano, D., Jaisawal, G. K., Malacaria, C., Bogdanov, S., Gendreau, K. C., & Arzoumanian, Z. 2020, The Astronomer's Telegram, 14124, 1 Paizis, A., Nowak, M. A., Wilms, J., Chaty, S., Corbel, S., Rodriguez, J., Del Santo, M., Ubertini, P., & Chini, R. 2011, ApJ, 738, 183 Parikh, A. S., Wijnands, R., Degenaar, N., & Altamirano, D. 2018, The Astronomer's Telegram, 11869, 1 Parikh, A. S., Wijnands, R., Degenaar, N., Altamirano, D., Patruno, A., Gusinskaia, N. V., & Hessels, J. W. T. 2017, MNRAS, 468, 3979 Pavan, L., Terrier, R., Bozzo, E., Ferrigno, C., Mereghetti, S., Paizis, A., Ducci, L., Gotz, D., Bazzano, A., Fiocchi, M., De Rosa, A., Tarana, A., Del Santo, M., Natalucci, L., Panessa, F., Capitanio, F., Sguera, V., Bianchin, V., Watanabe, K., Kuiper, L., Barragan, L., Chenevez, J., Caballero, I., Shrader, C., Bird, A., Puehlhofer, G., Sanchez-Fernandez, C., Skinner, G., den Hartog, P. R., Pottschmidt, K., Negueruela, I., & Prat, L. 2010, The Astronomer's Telegram, 2807, 1 Peng, F., Brown, E. F., & Truran, J. W. 2007, ApJ, 654, 1022 Sakano, M., Warwick, R. S., Decourchelle, A., & Wang, Q. D. 2005, MNRAS, 357, 1211 Sanna, A., Bozzo, E., Papitto, A., Riggio, A., Ferrigno, C., Di Salvo, T., Iaria, R., Mazzola, S. M., D'Amico, N., & Burderi, L. 2018a, A&A, 616, L17 Sanna, A., Ferrigno, C., Ray, P. S., Ducci, L., Jaisawal, G. K., Enoto, T., Bozzo, E., Altamirano, D., Di Salvo, T., Strohmayer, T. E., Papitto, A., Riggio, A., Burderi, L., Bult, P. M., Bogdanov, S., Gambino, A. F., Marino, A., Iaria, R., Arzoumanian, Z., Chakrabarty, D., Gendreau, K. C., Guillot, S., Markwardt, C., & Wolff, M. T. 2018b, A&A, 617, L8 Shaw, A. W., Heinke, C. O., Maccarone, T. J., Sivakoff, G. R., Strader, J., Bahramian, A., Degenaar, N., Kennea, J. A., Kuulkers, E., Rau, A., Rivera Sandoval, L. E., Shishkovsky, L., Swihart, S. J., Tetarenko, A. J., Wijnands, R., & in't Zand, J. J. M. 2020, MNRAS, 492, 4344 Stoop, M., van den Eijnden, J., Degenaar, N., Bahramian, A., Swihart, S. J., Strader, J., Jim'enez-Ibarra, F., Mu˜noz-Darias, T., Armas Padilla, M., Shaw, A. W., Maccarone, T. J., Wijnands, R., Russell, T. D., Hern'andez Santisteban, J. V., MillerJones, J. C. A., Russell, D. M., Maitra, D., Heinke, C. O., Sivakoff, G. R., Lewis, F., & Bramich, D. M. 2021, MNRAS, 507, 330 Strohmayer, T. & Keek, L. 2017, ApJ, 836, L23 van den Eijnden, J., Degenaar, N., Pinto, C., Patruno, A., Wette, K., Messenger, C., Hern'andez Santisteban, J. V., Wijnands, R., Miller, J. M., Altamirano, D., Paerels, F., Chakrabarty, D., & Fabian, A. C. 2018, MNRAS, 475, 2027 van den Eijnden, J., Degenaar, N., Russell, T. D., Wijnands, R., Bahramian, A., Miller-Jones, J. C. A., Hern'andez Santisteban, J. V., Gallo, E., Atri, P., Plotkin, R. M., Maccarone, T. J., Sivakoff, G., Miller, J. M., Reynolds, M., Russell, D. M., Maitra, D., Heinke, C. O., Armas Padilla, M., & Shaw, A. W. 2021, MNRAS, 507, 3899 Verner, D. A., Ferland, G. J., Korista, K. T., & Yakovlev, D. G. 1996, The Astrophysical Journal, 465, 487 ˇ Simon, V. 2004, A&A, 418, 617 Weng, S.-S. & Zhang, S.-N. 2015, MNRAS, 447, 486 Wijnands, R. & Degenaar, N. 2013, MNRAS, 434, 1599 Wijnands, R., Degenaar, N., Padilla, M. A., Altamirano, D., Cavecchi, Y., Linares, M., Bahramian, A., & Heinke, C. O. 2015, Monthly Notices of the Royal Astronomical Society, 454, 1371 Wijnands, R., Degenaar, N., & Page, D. 2013, MNRAS, 432, 2366 -. 2017, Journal of Astrophysics and Astronomy, 38, 49 Wijnands, R., in ' t Zand, J. J. M., Rupen, M., Maccarone, T., Homan, J., Cornelisse, R., Fender, R., Grindlay, J., van der Klis, M., Kuulkers, E., Markwardt, C. B., Miller-Jones, J. C. A., & Wang, Q. D. 2006, Astronomy & Astrophysics, 449, 1117 Wilms, J., Allen, A., & McCray, R. 2000, The Astrophysical Journal, 542, 914 Zhang, G. B., Bernardini, F., Russell, D. M., Gelfand , J. D., Lasota, J. P., Qasim, A. A., AlMannaei, A., Koljonen, K. I. I., Shaw, A. W., Lewis, F., Tomsick, J. A., Plotkin, R. M., Miller-Jones, J. C. A., Maitra, D., Homan, J., Charles, P. A., Kobel, P., Perez, D., & Doran, R. 2019, ApJ, 876, 5 Nathalie Degenaar and Rudy Wijnands: Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH, Amsterdam, The Netherlands. Osman Ahmed: Astronomy and Space Science Department, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Kingdom of Saudi Arabia Department of Physics, Faculty of Natural and computational Sciences, Debre Tabor University, P.O. Box 272, South Gondar, Ethiopia Montserrat Armas Padilla: Instituto de Astrof ˜ Asica de Canarias, 38205, San Cristobal de La Laguna, Spain Departamento de Astrof'ısica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain",
"pages": [
14,
15,
16,
17,
18
]
}
]
2024SCPMA..6750412W
https://arxiv.org/pdf/2308.11886.pdf
<document>
<section_header_level_1><location><page_1><loc_20><loc_92><loc_80><loc_93></location>Thermodynamic Nature of Black Holes in Coexistence Region</section_header_level_1>
<text><location><page_1><loc_38><loc_89><loc_62><loc_90></location>Shao-Wen Wei ∗ , Yu-Xiao Liu †</text>
<text><location><page_1><loc_17><loc_82><loc_84><loc_89></location>1 Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province, and Key Laboratory of Quantum Theory and Applications of MoE, Lanzhou University, Lanzhou, Gansu 730000, China, 2 Institute of Theoretical Physics & Research Center of Gravitation, Lanzhou University, Lanzhou 730000, People's Republic of China</text>
<text><location><page_1><loc_41><loc_81><loc_60><loc_82></location>(Dated: December 10, 2024)</text>
<text><location><page_1><loc_18><loc_66><loc_83><loc_80></location>Studying the system state of coexistence regions will peek into to reveal microscopic interactions between different phases of a thermodynamic system. However, there is no effective method to study thermodynamic nature of the coexistence black hole regions for the failure of the equation of state. Aiming at these coexistence states, in this work, we develop a general approach by introducing two new ratio parameters. The first one is the ratio of the horizon radii of the saturated coexistence small and large black holes, and the second one measures that of the small black hole molecule number to the total molecule number. We demonstrate that the first parameter can serve as an order parameter to characterize the first-order phase transition. The study also shows that the black hole state in the coexistence region is uniquely determined by these two introduced parameters bounded between 0 and 1. These results are quite significant in the analytical study of phase transition and the microscopic nature of black hole in the coexistence regions.</text>
<text><location><page_1><loc_36><loc_63><loc_64><loc_64></location>PACS numbers: 04.70.Dy, 04.70.Bw, 05.70.Ce</text>
<section_header_level_1><location><page_1><loc_20><loc_59><loc_37><loc_60></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_34><loc_49><loc_57></location>Thermodynamics has been one of the active areas in modern black hole physics. The study of black hole thermodynamics shall provide insight into the quantum gravity and nature of black hole. Recently, it was extensively observed that distinct phase transitions exist in anti-de Sitter (AdS) black hole systems, where the cosmological constant was treated as pressure [1]. The small-large black hole phase transition [2-4], analogous to the gasliquid phase transition of the van der Waals (VdW) fluid, was found to be universal in the charged AdS black hole systems. Further combining with the thermodynamical geometry, it was shown that both the repulsive and attractive interactions can dominate the two neighboring black hole molecules [5-7]. Other interesting issues including the Euler relation and dual conformal field theory have also been examined in Refs. [8-16].</text>
<text><location><page_1><loc_9><loc_28><loc_49><loc_34></location>The starting point of black hole phase transition comes from the analogy of the equation of state (EoS) of the VdW fluid. For a general AdS black hole system, the EoS can be expressed as</text>
<formula><location><page_1><loc_16><loc_24><loc_49><loc_27></location>P = T v + f 2 ( α ; T ) v 2 + ∑ i ≥ 3 f i ( α ; T ) v i , (1)</formula>
<text><location><page_1><loc_9><loc_15><loc_49><loc_22></location>where v is the specific volume of the system, and f i denote the functions that depend on the black hole parameters α and temperature. If taking f 2 = -a , f i ≥ 3 = 0 and replacing v with v -b in the first term on the right side of the equation, it shall recover the EoS of the VdW</text>
<text><location><page_1><loc_52><loc_20><loc_92><loc_60></location>fluid model with a and b measuring the attraction and nonzero size of the molecules of the VdW fluid. Although the VdW fluid is developed to describe the phase transition, it has limitation in exploring the critical phenomena of an actual system. Significantly, since black holes have the similar EoS of the VdW fluid, one can naturally demonstrate that black hole phase transitions are of the VdW-like type. In fact, many works has confirmed this point. Besides, more interesting black hole phase diagrams are exposed. Nevertheless, black hole systems also hold their unique features. For examples, the gravitational constant G appears in the functions f i , which indicates that the EoS describes a gravity system. Adopting the Planck constant l P = 1, it is easy to find that the specific volume shown in (1) is v ∼ r h with r h denoting the length scale of the black hole. This is quite different from the ordinary system, where the specific volume is proportional to the cubic power of the system characteristic length. Such difference leads to the inconsistence in the Maxwell equal area law in determining the phase transition point. In particular, the higher orders f i ≥ 3 in the EoS for the black hole systems is quite universal. For the static and spherically symmetric black holes, entropy is proportional to the power of specific volume leading to the disappearance of the specific heat at constant volume. All these features reveal the quantum and gravity nature of black hole thermodynamics included in the EoS (1).</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_20></location>Despite the success, there is a huge challenge that the thermodynamical nature of the coexistence black hole region remains unknown, mainly caused by the failure of the EoS. Since in the coexistence region, the phenomena of the system are entirely determined by the coexistence and competition of the microscopic molecules. Therefore, studying it can help us insight into the unique microstructure of black holes. With this in mind, our goal</text>
<figure>
<location><page_2><loc_15><loc_80><loc_43><loc_94></location>
<caption>FIG. 1: Sketch of the phase diagram of the first-order phase transition near the critical point. The red and blue curves represent the statured coexistence small and large black holes, respectively. Black dot denotes the critical point. The small and large black holes locate at the left and right sides. Below the solid curve is the coexistence region where the small and large black holes coexist. The small circles represent different phase molecules.</caption>
</figure>
<text><location><page_2><loc_9><loc_59><loc_49><loc_64></location>is to investigate the universal properties of the small and large black holes in the coexistence region, which will lead to a comprehensive understanding of black hole thermodynamics within the phase space.</text>
<text><location><page_2><loc_9><loc_34><loc_49><loc_58></location>As well known, the small and large black holes are characterized by the values of their horizon radii r h . For clarity, we sketch the features of the small-large black hole phase diagram in the T -r h plane in Fig. 1. The small and large black hole phases are, respectively, located at the left and right sides. Comparing with the large black hole, the small black hole admits small size molecules while high density [5]. Below the coexistence curve, constituted by the left red curve (coexistence saturated small black hole) and right blue curve (coexistence saturated large black hole), is the coexistence black hole region. As the EoS remains unknown in this particular region, previous studies have neglected the corresponding thermodynamic properties. However, we will demonstrate that these properties can be effectively explored by utilizing the properties of the coexistence saturated small and large black holes.</text>
<text><location><page_2><loc_10><loc_33><loc_42><loc_34></location>First, let us introduce a key ratio parameter</text>
<formula><location><page_2><loc_24><loc_28><loc_49><loc_31></location>/epsilon1 = 1 -r hs r hl , (2)</formula>
<text><location><page_2><loc_9><loc_9><loc_49><loc_27></location>where r hs and r hl denote the horizon radii of the coexistence saturated small and large black holes at the same temperature and pressure, respectively. They coincide and give /epsilon1 = 0 at the critical point. Since r hs ≤ r hl , one always has /epsilon1 ∈ [0, 1]. More importantly, there are extremal black holes with vanished temperature. For this case, r hl tends to infinity, while r hs remains a finite value. As a result, we have /epsilon1 = 0 for the extremal black holes. Further adjusting the parameters such that the black hole horizon disappears and a naked singularity is exposed, the thermodynamics will fail, so no any phase transition exists. We shall show that this ratio provides us with a favorable parameter for our following study. Here, we</text>
<text><location><page_2><loc_52><loc_74><loc_92><loc_93></location>emphasize its advantages: i) It acts as an order parameter for characterizing the small-large black hole transitions. ii) The coexistence curves of the black hole and the VdW fluid can be parameterized analytically by /epsilon1 . iii) By further combining with the ratio of the molecule numbers of the small and large black holes, denoted by x , the state of the black hole system within the coexistence region can be uniquely determined. The effective EoS will also be given. On the other hand, the difference of the radii ∆ = r hl -r hs is also an order parameter and can be used to characterize the small-large black hole phase transition. However, its value is unbounded and thus it is very difficult to parameterize coexistence curves via it.</text>
<text><location><page_2><loc_52><loc_60><loc_92><loc_74></location>The present work is organized as follows. In Sec. II, we examine the ratio /epsilon1 near the critical point and confirm that it indeed can act the order parameter to describe the small-large black hole phase transition. The analytical parameterized coexistence curve of the small and large black hole phase transitions, as well as the liquid and gas phase transition shall be given in Sec. III. Then the equation of state in the coexistence region will be effectively investigated in Sec. IV. Finally, we summarize and discuss our results in Sec. V.</text>
<section_header_level_1><location><page_2><loc_61><loc_56><loc_83><loc_56></location>II. ORDER PARAMETER</section_header_level_1>
<text><location><page_2><loc_52><loc_46><loc_92><loc_53></location>It is extensively known that the difference of the thermodynamical volumes of the coexistence saturated small and large black holes is an order parameter for this VdW like phase transition. Here, we demonstrate that the ratio /epsilon1 also acts as a significant order parameter.</text>
<text><location><page_2><loc_52><loc_43><loc_92><loc_46></location>Near the critical point, the reduced pressure can be expanded as [3]</text>
<formula><location><page_2><loc_56><loc_40><loc_92><loc_42></location>p = 1 + a 10 t + a 11 tω + a 03 ω 3 + O ( tω 2 , ω 4 ) , (3)</formula>
<text><location><page_2><loc_52><loc_30><loc_92><loc_39></location>where the coefficients a 10 , a 11 , and a 03 depend on the parameters of the specific black holes. The reduced quantities are defined as p = P/P c , t = T/T c -1, and ω = V/V c -1. Solving the Maxwell equal area law ∮ ωdp = 0, or equivalently ∫ ω l ω s ( ω∂ ω p ) dω = 0, we have</text>
<formula><location><page_2><loc_58><loc_27><loc_92><loc_30></location>a 11 tω 2 l + 3 2 a 03 ω 4 l = a 11 tω 2 s + 3 2 a 03 ω 4 s . (4)</formula>
<text><location><page_2><loc_52><loc_23><loc_92><loc_26></location>Along the coexistence curve, the EoS holds for the saturated small and large black holes</text>
<formula><location><page_2><loc_60><loc_20><loc_92><loc_22></location>p = 1 + a 10 t + a 11 tω s + a 03 ω 3 s , (5)</formula>
<formula><location><page_2><loc_60><loc_18><loc_92><loc_20></location>p = 1 + a 10 t + a 11 tω l + a 03 ω 3 l . (6)</formula>
<text><location><page_2><loc_52><loc_16><loc_89><loc_17></location>Combining with these equations, it is easy to obtain</text>
<formula><location><page_2><loc_64><loc_11><loc_92><loc_14></location>ω s,l = ∓ √ a 11 a 03 √ -t, (7)</formula>
<text><location><page_2><loc_52><loc_9><loc_92><loc_10></location>for the coexistence saturated small and large black holes.</text>
<text><location><page_3><loc_10><loc_92><loc_26><loc_93></location>From (7), one arrives</text>
<formula><location><page_3><loc_14><loc_83><loc_49><loc_91></location>/epsilon1 = 1 -( ω s +1 ω l +1 ) 1 d -3 = 1 -( a 03 -a 11 t -2 √ -a 03 a 11 t a 03 + a 11 t ) 1 d -1 , (8)</formula>
<text><location><page_3><loc_9><loc_80><loc_43><loc_82></location>for d -dimensional case. Note that we have used</text>
<formula><location><page_3><loc_22><loc_76><loc_49><loc_79></location>r hs r hl = ( V s V l ) 1 d -1 . (9)</formula>
<text><location><page_3><loc_9><loc_73><loc_46><loc_75></location>Expanding it near the critical point t = 0, we obtain</text>
<formula><location><page_3><loc_18><loc_68><loc_49><loc_72></location>/epsilon1 = 2 d -1 √ a 11 a 03 √ -t + O ( t ) . (10)</formula>
<text><location><page_3><loc_9><loc_59><loc_49><loc_68></location>Obviously, /epsilon1 exhibits a critical exponent 1/2, strongly indicating that it can serve an order parameter, and can be utilized to characterize the small-large black hole phase transition. This shall be further confirmed by the results of the VdW fluid and charged AdS black holes in what follows.</text>
<text><location><page_3><loc_9><loc_40><loc_49><loc_59></location>Before ending this section, we would like to give a note on the Maxwell equal area law. Actually, it mainly lies in the equality of the free energy of the coexistence saturated small and large black holes. Meanwhile, one should note that ω is related with the thermodynamical volume rather than the specific volume. The details can be found in Ref. [17]. For the VdW fluid system, one can obtain its free energy by combining with the micro-model. However, for the black hole systems, since the micro-model is unclear, one could not derive its free energy via the EoS or the Hawking temperature. For a specific black hole system, its free energy can be calculated through the action.</text>
<section_header_level_1><location><page_3><loc_13><loc_35><loc_45><loc_37></location>III. ANALYTICAL PARAMETERIZED COEXISTENCE CURVE</section_header_level_1>
<text><location><page_3><loc_9><loc_19><loc_49><loc_33></location>Now, we turn to analytically study the coexistence curve by using the parameter /epsilon1 . Let us focus on the charged AdS black hole. In Ref. [2], it was found that the free energy demonstrates the swallow tail behaviors indicating the existence of the phase transition. Treating the cosmological constant as the pressure, the small-large black hole phase transition of the VdW type was observed in the extended phase transition [3]. Starting with the d -dimensional case, the EoS in the reduced parameter space was given by [4]</text>
<formula><location><page_3><loc_9><loc_13><loc_49><loc_17></location>p = 4( d -2) τ (2 d -5)˜ r h -d -2 ( d -3)˜ r 2 h + 1 ( d -3)(2 d -5)˜ r 2 d -4 h . (11)</formula>
<text><location><page_3><loc_9><loc_9><loc_49><loc_13></location>Similar to the VdW fluid, the black hole charge has been scaled out in the reduced parameter space. It is worthwhile noting that, the Maxwell equal area law is in the</text>
<text><location><page_3><loc_52><loc_50><loc_58><loc_51></location>and d =5</text>
<figure>
<location><page_3><loc_57><loc_77><loc_87><loc_93></location>
<caption>FIG. 2: The ratio /epsilon1 as a function of the shifted temperature 1τ . The critical behavior can be observed near the coordinate origin.</caption>
</figure>
<text><location><page_3><loc_52><loc_61><loc_92><loc_68></location>form of ∮ ˜ r ( d -1) h dp = 0 for the d -dimensional charged AdS black hole [17]. By making use of it, the analytical parameterized curve can be obtained for any d . However, its expression is long and so we will not show it here [18]. For clarity, we list the result for d = 4</text>
<formula><location><page_3><loc_63><loc_55><loc_92><loc_59></location>p = 36(1 -/epsilon1 ) 2 ( /epsilon1 2 -6 /epsilon1 +6) 2 , (12)</formula>
<formula><location><page_3><loc_63><loc_52><loc_92><loc_57></location>τ = 3 √ 6(2 -/epsilon1 )(1 -/epsilon1 ) ( /epsilon1 2 -6 /epsilon1 +6) 3 / 2 . (13)</formula>
<formula><location><page_3><loc_58><loc_45><loc_92><loc_50></location>p = 3 √ 15(1 -/epsilon1 ) 2 ( /epsilon1 2 -5 /epsilon1 +5) ( /epsilon1 4 -7 /epsilon1 3 +22 /epsilon1 2 -30 /epsilon1 +15) 3 / 2 , (14)</formula>
<formula><location><page_3><loc_58><loc_41><loc_92><loc_46></location>τ = 15 4 √ 15(2 -/epsilon1 ) 3 (1 -/epsilon1 ) 8( /epsilon1 4 -7 /epsilon1 3 +22 /epsilon1 2 -30 /epsilon1 +15) 5 / 4 . (15)</formula>
<text><location><page_3><loc_52><loc_36><loc_92><loc_40></location>Fortunately, for the four-dimensional case, we can solve the ratio /epsilon1 from the pressure (12), and then express the temperature as</text>
<formula><location><page_3><loc_64><loc_32><loc_92><loc_35></location>τ = √ 1 2 p (3 -√ p ) , (16)</formula>
<text><location><page_3><loc_52><loc_22><loc_92><loc_30></location>which is exactly the result given in Ref. [20]. Adopting the analytical parameterized formulas, one can easily obtain the corresponding phase diagram for the black holes. Moreover, we find that when the dimension number d is larger than 100, the phase diagrams almost hold unchanged in the reduced parameter space.</text>
<text><location><page_3><loc_52><loc_19><loc_92><loc_22></location>Employing with the parameterized formula, we can expand the ratio near the critical temperature as</text>
<formula><location><page_3><loc_52><loc_13><loc_92><loc_18></location>/epsilon1 = 2 √ 6 √ 2 d -5 (1 -τ ) 1 2 -12 2 d -5 (1 -τ ) + O ((1 -τ ) 3 2 ) , (17)</formula>
<text><location><page_3><loc_52><loc_9><loc_92><loc_13></location>which exactly confirms the result (10), and indicates that the ratio /epsilon1 acts as an order parameter for any dimension d .</text>
<text><location><page_4><loc_9><loc_89><loc_49><loc_93></location>More interestingly, applying the study to the VdW fluid, we obtain the analytical parameterized coexistence curve for the VdW fluid</text>
<formula><location><page_4><loc_12><loc_84><loc_49><loc_88></location>p = 27(1 -/epsilon1 ) [2 /epsilon1 +(2 -/epsilon1 ) /lscript /epsilon1 ] 2 [ /epsilon1 2 -(1 -/epsilon1 ) /lscript 2 /epsilon1 ] /epsilon1 2 [ /epsilon1 + /lscript /epsilon1 ] 2 [ /epsilon1 +(1 + /epsilon1 ) /lscript /epsilon1 ] 2 , (18)</formula>
<formula><location><page_4><loc_12><loc_80><loc_49><loc_84></location>τ = 27 [( /epsilon1 -2) /lscript /epsilon1 -2 /epsilon1 ] [(2 -/epsilon1 ) /epsilon1 +2(1 -/epsilon1 ) /lscript /epsilon1 ] 2 8 /epsilon1 [ /epsilon1 + /lscript /epsilon1 ] 2 [ /epsilon1 -( /epsilon1 -1) /lscript /epsilon1 ] 2 , (19)</formula>
<text><location><page_4><loc_9><loc_66><loc_49><loc_79></location>where /lscript /epsilon1 = ln(1 -/epsilon1 ). When /epsilon1 = 1 -ν l -1 3 ν g -1 3 → 1, it gives p = τ = 0, and when /epsilon1 → 0, the critical point at p = τ = 1 is given. Employing this analytical formula, it is possible to precisely study various properties of the VdW fluid in all parameter spaces without using any approximation. This approach shall represent a significant improvement over earlier approximate studies [21, 22], which are limited to the τ → 0 and 1 regimes.</text>
<text><location><page_4><loc_9><loc_64><loc_49><loc_67></location>Interestingly, the ratio parameter /epsilon1 can also be obtained by expanding near the critical temperature</text>
<formula><location><page_4><loc_14><loc_60><loc_49><loc_63></location>/epsilon1 = 6 √ 1 -τ -18(1 -τ ) + O ((1 -τ ) 3 2 ) . (20)</formula>
<text><location><page_4><loc_9><loc_53><loc_49><loc_59></location>Moreover, we also show the ratio /epsilon1 as a function of 1 -τ in Fig. 2. It is evident that, for the VdW fluid and charged AdS black holes with dimensions d = 4 and 5, /epsilon1 approaches zero as (1 -τ ) 1 2 .</text>
<text><location><page_4><loc_9><loc_41><loc_49><loc_54></location>To summarize, the ratio /epsilon1 that we have introduced can serve as an order parameter to characterize the smalllarge black hole phase transition. Significantly, the coexistence curves for both the VdW fluid and d -dimensional charged AdS black holes can be analytically obtained in parameterized forms. In the following sections, we shall demonstrate that this ratio can also serve as a parameter for characterizing the black hole state within the coexistence region.</text>
<section_header_level_1><location><page_4><loc_10><loc_35><loc_48><loc_37></location>IV. STATE PARAMETERS IN COEXISTENCE REGION</section_header_level_1>
<text><location><page_4><loc_9><loc_26><loc_49><loc_33></location>The exploration of the coexistence region in black hole systems has been neglected in previous studies due to the violation of the EoS. Now, we aim to explore this issue by utilizing the ratio /epsilon1 and taking the charged AdS black hole as an illustrative example.</text>
<text><location><page_4><loc_9><loc_9><loc_49><loc_26></location>Let us consider one isobaric curve with p = 0 . 9 shown for d =4 dimensional charged AdS black hole in Fig. 3. The gray curve denotes the coexistence curve, and below which is the coexistence region. The oscillatory part of the isobaric curve in green color should be replaced by a horizontal line, and which describes a phase transition between the small and large black holes. Points A and B are the saturated small and large black hole states. Meanwhile, each point on the horizontal line represents a system state with the same temperature and pressure. For example, the state denoted by point C shares the same temperature and pressure of the states A and B,</text>
<figure>
<location><page_4><loc_57><loc_79><loc_87><loc_94></location>
<caption>FIG. 3: The isobaric curve with p =0.9 for the fourdimensional charged AdS black hole. The gray solid curve is the coexistence curve and the black dot is the critical point. Points A and B denote two saturated coexistence small and large black holes. Point C in the coexistence region represents a state of coexistence small and large black holes sharing with the same temperature and pressure as points A and B. The phase transition from saturated small black hole to large black hole is described by the horizontal line.</caption>
</figure>
<text><location><page_4><loc_52><loc_54><loc_92><loc_62></location>which are not independent and given in (12) and (13), respectively. The ratio /epsilon1 serves as one characteristic parameter of state C, but only using it cannot uniquely determine the state. Since state C actually contains both the coexistence small and large black holes. Hence, a second parameter is needed to characterize this property.</text>
<text><location><page_4><loc_52><loc_44><loc_92><loc_54></location>During the transition from state A to B, the molecules of the saturated small black hole gradually transform into those of the saturated large black hole. To characterize this property, we introduce another parameter x = N s / ( N s + N l ), which measures the number of the saturated small black hole molecules to the total number of molecules. Further combining with the lever rule</text>
<formula><location><page_4><loc_65><loc_38><loc_92><loc_43></location>N s N l = ˜ V hl -˜ V hi ˜ V hi -˜ V hs , (21)</formula>
<text><location><page_4><loc_52><loc_36><loc_92><loc_38></location>we can express the parameter ˜ r hi of an intermediate state, such as state C, as</text>
<formula><location><page_4><loc_56><loc_30><loc_92><loc_35></location>˜ r hi = √ /epsilon1 2 -6 /epsilon1 +6 √ 6(1 -/epsilon1 ) 3 √ 1 -x (1 -(1 -/epsilon1 ) 3 ) . (22)</formula>
<text><location><page_4><loc_52><loc_13><loc_92><loc_30></location>Here ˜ r hi is bounded by the radii of the saturated small and large black holes. We need to point out that it just denotes the location of the system in the phase diagram instead of the actual radius of the coexistence black holes. It is easy to find that x =1 and 0 correspond to states A and B, respectively. So, analogous to the fluid systems, x gives the ratio of the saturated small and large black hole molecules located at this system state. On the other hand, understanding these coexistence black holes from a gravitational perspective is worth future investigation via the potential relationship between the black hole thermodynamics and gravity.</text>
<text><location><page_4><loc_52><loc_9><loc_92><loc_13></location>Finally, we find that each coexistence state can be parameterized by two quantities /epsilon1 and x , both of which are bounded between 0 and 1. In Fig. 4, we show these</text>
<figure>
<location><page_5><loc_14><loc_76><loc_44><loc_93></location>
<caption>FIG. 4: Coexistence states characterized by the ratios /epsilon1 and x shown in τ -˜ r hi . These curves are plotted by taking x =0.3, 0.5, 0.7, 0.9, 0.99, 0.999, 0.9999 from right to left. They start at the critical point with /epsilon1 =0 and extend to the right with the increase of /epsilon1 . The coexistence region is marked in light green color.</caption>
</figure>
<text><location><page_5><loc_9><loc_55><loc_49><loc_64></location>coexistence states by taking x =0.3, 0.5, 0.7, 0.9, 0.99, 0.999, 0.9999 from right to left. The coexistence region is marked in light green color. Note that the parameter x heavily relies on the saturated small and large black holes and is not directly related to the extremal black holes or naked singularities.</text>
<text><location><page_5><loc_9><loc_44><loc_49><loc_55></location>All of these curves coincide at the critical temperature with /epsilon1 =0, and extend to the right. In particular, with the increase of x , the curve shifts towards the saturated small black hole curve. The monotonic behavior is also broken. Nevertheless, each state within the coexistence region is characterized by a pair of parameters, namely ( /epsilon1 , x ). This study can also be easily generalized to higher dimensional cases.</text>
<section_header_level_1><location><page_5><loc_21><loc_40><loc_37><loc_41></location>V. CONCLUSIONS</section_header_level_1>
<text><location><page_5><loc_9><loc_28><loc_49><loc_38></location>In this study, we have investigated the thermodynamic nature of the coexisting black hole phases of the VdW type by introducing two new ratio parameters, /epsilon1 and x . Each system state within the coexistence region can be uniquely determined by these parameters, offering an opportunity to explore the coexisting system states that have been previously overlooked.</text>
<text><location><page_5><loc_9><loc_25><loc_49><loc_27></location>Our findings suggest that the ratio /epsilon1 plays a crucial role in the study of coexisting physics. By utilizing this</text>
<text><location><page_5><loc_52><loc_73><loc_92><loc_93></location>parameter, we obtained, for the first time, an analytical parameterized form for the coexisting saturated liquid and gas phases. This shall greatly improve the fitting formula given by Ref. [23]. Consequently, all other properties of the VdW fluid can be analyzed analytically without any approximation. For the d -dimensional charged AdS black holes, the analytical parameterized form of the coexistence curve was also obtained. By performing the general calculations, we demonstrated that the ratio /epsilon1 can serve as an order parameter for characterizing the liquid-gas phase transition or the small-large black hole phase transition. This conclusion is further supported by the results of the VdW fluid and charged AdS black holes.</text>
<text><location><page_5><loc_52><loc_61><loc_92><loc_73></location>Employing with the lever rule, we introduced another parameter, x , which measures the ratio of the small black hole molecule number to the total molecule number. Our result indicates that a black hole state within the coexistence region can be uniquely determined by the ratios /epsilon1 and x , thereby providing an effective EoS. This approach offers insight into the coexistent physics that has been neglected in previous studies.</text>
<text><location><page_5><loc_52><loc_52><loc_92><loc_61></location>While our study is limited to the VdW type phase transition, the approach we have developed can be generalized to the local analysis of other phase structures near the critical point, representing a second-order phase transition. Actually, this feature is quite prevalent in black hole chemistry.</text>
<text><location><page_5><loc_52><loc_44><loc_92><loc_52></location>On the other hand, although we gave an effective EoS in the coexistence region, several issues are still not resolved. For examples, the first law still remains to be established and the free energy corresponding to the effective EoS is still unclear. We expect to address these issues in future study.</text>
<text><location><page_5><loc_52><loc_38><loc_92><loc_43></location>In conclusion, we have developed a general analytical approach to gain insight into the coexistent physics. The effective EoS represented by the two ratio parameters can be used to test other underlying properties.</text>
<section_header_level_1><location><page_5><loc_59><loc_33><loc_84><loc_34></location>VI. ACKNOWLEDGEMENTS</section_header_level_1>
<text><location><page_5><loc_52><loc_25><loc_92><loc_30></location>This work was supported by the National Natural Science Foundation of China (Grants No. 12075103, No. 11875151, and No. 12247101) and the Major Science and Technology Projects of Gansu Province.</text>
<unordered_list>
<list_item><location><page_6><loc_12><loc_89><loc_49><loc_93></location>structure of an AdS black hole from thermodynamical phase transition , Phys. Rev. Lett. 115 , 111302 (2015), [arXiv:1502.00386 [gr-qc]].</list_item>
<list_item><location><page_6><loc_10><loc_84><loc_49><loc_89></location>[6] S.-W. Wei, Y.-X. Liu, and R. B. Mann, Repulsive Interactions and Universal Properties of Charged AdS Black Hole Microstructures , Phys. Rev. Lett. 123 , 071103 (2019), [arXiv:1906.10840 [gr-qc]].</list_item>
<list_item><location><page_6><loc_10><loc_79><loc_49><loc_84></location>[7] S.-W. Wei, Y.-X. Liu, and R. B. Mann, Ruppeiner Geometry, Phase Transitions, and the Microstructure of Charged AdS Black Holes , Phys. Rev. D 100 , 124033 (2019) [arXiv:1909.03887[gr-qc]].</list_item>
<list_item><location><page_6><loc_10><loc_73><loc_49><loc_78></location>[8] A. M. Frassino, J. F. Pedraza, A. Svesko, and M. R. Visser, Higher-dimensional origin of extended black hole thermodynamics , Phys. Rev. Lett. 130 , 161501 (2023), [arXiv:2212.14055 [hep-th]].</list_item>
<list_item><location><page_6><loc_10><loc_68><loc_49><loc_73></location>[9] M. B. Ahmed, W. Cong, D. Kubiznak, R. B. Mann, and M. R. Visser, Holographic dual of extended black hole thermodynamics , Phys. Rev. Lett. 130 , 181401 (2023), [arXiv: 2302.08163 [hep-th]].</list_item>
<list_item><location><page_6><loc_9><loc_64><loc_49><loc_68></location>[10] T.-F. Gong, J. Jiang, and M. Zhang, Holographic thermodynamics of rotating black holes , J. High Energy Phys. 06 , 105 (2023), [arXiv:2305.00267 [hep-th]].</list_item>
<list_item><location><page_6><loc_9><loc_59><loc_49><loc_64></location>[11] M. B. Ahmed, W. Cong, D. Kubiznak, R. B. Mann, and M. R. Visser, Holographic CFT Phase Transitions and Criticality for Rotating AdS Black Holes , JHEP 08 , 142 (2023), [arXiv:2305.03161 [hep-th]].</list_item>
<list_item><location><page_6><loc_9><loc_55><loc_49><loc_59></location>[12] Z. Gao and L. Zhao, Restricted phase space thermodynamics for AdS black holes via holography , Class. Quany. Grav. 39 , 075019 (2022), [arXiv:2112.02386 [gr-qc]].</list_item>
<list_item><location><page_6><loc_9><loc_51><loc_49><loc_55></location>[13] Z.-M. Xu, B. Wu, and W.-L. Yang, Rate of the phase transition for a charged anti-de Sitter black hole , Sci. China Phys. Mech. Astron. 66 , 240411 (2023),</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_6><loc_55><loc_92><loc_72><loc_93></location>[arXiv:2211.03512 [gr-qc]].</list_item>
<list_item><location><page_6><loc_52><loc_88><loc_92><loc_92></location>[14] Y.-X. Liu, Characteristic process of the black hole phase transition , Sci. China Phys. Mech. Astron. 66 , 240431 (2023).</list_item>
<list_item><location><page_6><loc_52><loc_84><loc_92><loc_88></location>[15] N.-C. Bai, L. Li, and J. Tao, Superfluid λ transition in charged AdS black holes , Sci. China Phys. Mech. Astron. 66 , 120411 (2023), [arXiv:2305.15258 [hep-th]].</list_item>
<list_item><location><page_6><loc_52><loc_80><loc_92><loc_84></location>[16] L. Ma, Y. Pang, and H. Lu, Negative corrections to black hole entropy from string theory , Sci. China Phys. Mech. Astron. 66 , 121011 (2023), [arXiv:2212.03262 [hep-th]].</list_item>
<list_item><location><page_6><loc_52><loc_75><loc_92><loc_80></location>[17] S.-W. Wei and Y.-X. Liu, Clapeyron equations and fitting formula of the coexistence curve in the extended phase space of charged AdS black holes , Phys. Rev. D 91 , 044018 (2015), [arXiv:1411.5749[hep-th]].</list_item>
<list_item><location><page_6><loc_52><loc_71><loc_92><loc_75></location>[18] By treating the ratio of the radii of the coexisting small and large black holes as an intermediate variable, an equivalent coexistence curve was presented in Ref. [19].</list_item>
<list_item><location><page_6><loc_52><loc_65><loc_92><loc_71></location>[19] L.-C. Zhang, H.-H. Zhao, R. Zhao, and M.-S. Ma, Equal Area Laws and Latent Heat for d -Dimensional RNAdS Black Hole , Adv. High Energy Phys. 2014 , 816728 (2014).</list_item>
<list_item><location><page_6><loc_52><loc_62><loc_92><loc_65></location>[20] E. Spallucci and A. Smailagic, Maxwell's equal area law for charged Anti-deSitter black holes , Phys. Lett. B 723 , 436 (2013), [arXiv:1305.3379[hep-th]].</list_item>
<list_item><location><page_6><loc_52><loc_58><loc_92><loc_61></location>[21] J. Lekner, Parametric solution of the van der Waals liquid-vapor coexistence curve Am. J. Phys. 50 , 161 (1982).</list_item>
<list_item><location><page_6><loc_52><loc_54><loc_92><loc_57></location>[22] M. N. Berberan-Santos, E. N. Bodunov, and L. Pogliani, The van der Waals equation: analytical and approximate solutions , J. Math. Chem. 43 , 1437 (2008).</list_item>
<list_item><location><page_6><loc_52><loc_51><loc_92><loc_53></location>[23] D. C. Johnston, Thermodynamic properties of the van der Waals fluid , [arXiv:1402.1205[cond-mat.soft]].</list_item>
</document>
[
{
"title": "Thermodynamic Nature of Black Holes in Coexistence Region",
"content": "Shao-Wen Wei ∗ , Yu-Xiao Liu † 1 Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province, and Key Laboratory of Quantum Theory and Applications of MoE, Lanzhou University, Lanzhou, Gansu 730000, China, 2 Institute of Theoretical Physics & Research Center of Gravitation, Lanzhou University, Lanzhou 730000, People's Republic of China (Dated: December 10, 2024) Studying the system state of coexistence regions will peek into to reveal microscopic interactions between different phases of a thermodynamic system. However, there is no effective method to study thermodynamic nature of the coexistence black hole regions for the failure of the equation of state. Aiming at these coexistence states, in this work, we develop a general approach by introducing two new ratio parameters. The first one is the ratio of the horizon radii of the saturated coexistence small and large black holes, and the second one measures that of the small black hole molecule number to the total molecule number. We demonstrate that the first parameter can serve as an order parameter to characterize the first-order phase transition. The study also shows that the black hole state in the coexistence region is uniquely determined by these two introduced parameters bounded between 0 and 1. These results are quite significant in the analytical study of phase transition and the microscopic nature of black hole in the coexistence regions. PACS numbers: 04.70.Dy, 04.70.Bw, 05.70.Ce",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "Thermodynamics has been one of the active areas in modern black hole physics. The study of black hole thermodynamics shall provide insight into the quantum gravity and nature of black hole. Recently, it was extensively observed that distinct phase transitions exist in anti-de Sitter (AdS) black hole systems, where the cosmological constant was treated as pressure [1]. The small-large black hole phase transition [2-4], analogous to the gasliquid phase transition of the van der Waals (VdW) fluid, was found to be universal in the charged AdS black hole systems. Further combining with the thermodynamical geometry, it was shown that both the repulsive and attractive interactions can dominate the two neighboring black hole molecules [5-7]. Other interesting issues including the Euler relation and dual conformal field theory have also been examined in Refs. [8-16]. The starting point of black hole phase transition comes from the analogy of the equation of state (EoS) of the VdW fluid. For a general AdS black hole system, the EoS can be expressed as where v is the specific volume of the system, and f i denote the functions that depend on the black hole parameters α and temperature. If taking f 2 = -a , f i ≥ 3 = 0 and replacing v with v -b in the first term on the right side of the equation, it shall recover the EoS of the VdW fluid model with a and b measuring the attraction and nonzero size of the molecules of the VdW fluid. Although the VdW fluid is developed to describe the phase transition, it has limitation in exploring the critical phenomena of an actual system. Significantly, since black holes have the similar EoS of the VdW fluid, one can naturally demonstrate that black hole phase transitions are of the VdW-like type. In fact, many works has confirmed this point. Besides, more interesting black hole phase diagrams are exposed. Nevertheless, black hole systems also hold their unique features. For examples, the gravitational constant G appears in the functions f i , which indicates that the EoS describes a gravity system. Adopting the Planck constant l P = 1, it is easy to find that the specific volume shown in (1) is v ∼ r h with r h denoting the length scale of the black hole. This is quite different from the ordinary system, where the specific volume is proportional to the cubic power of the system characteristic length. Such difference leads to the inconsistence in the Maxwell equal area law in determining the phase transition point. In particular, the higher orders f i ≥ 3 in the EoS for the black hole systems is quite universal. For the static and spherically symmetric black holes, entropy is proportional to the power of specific volume leading to the disappearance of the specific heat at constant volume. All these features reveal the quantum and gravity nature of black hole thermodynamics included in the EoS (1). Despite the success, there is a huge challenge that the thermodynamical nature of the coexistence black hole region remains unknown, mainly caused by the failure of the EoS. Since in the coexistence region, the phenomena of the system are entirely determined by the coexistence and competition of the microscopic molecules. Therefore, studying it can help us insight into the unique microstructure of black holes. With this in mind, our goal is to investigate the universal properties of the small and large black holes in the coexistence region, which will lead to a comprehensive understanding of black hole thermodynamics within the phase space. As well known, the small and large black holes are characterized by the values of their horizon radii r h . For clarity, we sketch the features of the small-large black hole phase diagram in the T -r h plane in Fig. 1. The small and large black hole phases are, respectively, located at the left and right sides. Comparing with the large black hole, the small black hole admits small size molecules while high density [5]. Below the coexistence curve, constituted by the left red curve (coexistence saturated small black hole) and right blue curve (coexistence saturated large black hole), is the coexistence black hole region. As the EoS remains unknown in this particular region, previous studies have neglected the corresponding thermodynamic properties. However, we will demonstrate that these properties can be effectively explored by utilizing the properties of the coexistence saturated small and large black holes. First, let us introduce a key ratio parameter where r hs and r hl denote the horizon radii of the coexistence saturated small and large black holes at the same temperature and pressure, respectively. They coincide and give /epsilon1 = 0 at the critical point. Since r hs ≤ r hl , one always has /epsilon1 ∈ [0, 1]. More importantly, there are extremal black holes with vanished temperature. For this case, r hl tends to infinity, while r hs remains a finite value. As a result, we have /epsilon1 = 0 for the extremal black holes. Further adjusting the parameters such that the black hole horizon disappears and a naked singularity is exposed, the thermodynamics will fail, so no any phase transition exists. We shall show that this ratio provides us with a favorable parameter for our following study. Here, we emphasize its advantages: i) It acts as an order parameter for characterizing the small-large black hole transitions. ii) The coexistence curves of the black hole and the VdW fluid can be parameterized analytically by /epsilon1 . iii) By further combining with the ratio of the molecule numbers of the small and large black holes, denoted by x , the state of the black hole system within the coexistence region can be uniquely determined. The effective EoS will also be given. On the other hand, the difference of the radii ∆ = r hl -r hs is also an order parameter and can be used to characterize the small-large black hole phase transition. However, its value is unbounded and thus it is very difficult to parameterize coexistence curves via it. The present work is organized as follows. In Sec. II, we examine the ratio /epsilon1 near the critical point and confirm that it indeed can act the order parameter to describe the small-large black hole phase transition. The analytical parameterized coexistence curve of the small and large black hole phase transitions, as well as the liquid and gas phase transition shall be given in Sec. III. Then the equation of state in the coexistence region will be effectively investigated in Sec. IV. Finally, we summarize and discuss our results in Sec. V.",
"pages": [
1,
2
]
},
{
"title": "II. ORDER PARAMETER",
"content": "It is extensively known that the difference of the thermodynamical volumes of the coexistence saturated small and large black holes is an order parameter for this VdW like phase transition. Here, we demonstrate that the ratio /epsilon1 also acts as a significant order parameter. Near the critical point, the reduced pressure can be expanded as [3] where the coefficients a 10 , a 11 , and a 03 depend on the parameters of the specific black holes. The reduced quantities are defined as p = P/P c , t = T/T c -1, and ω = V/V c -1. Solving the Maxwell equal area law ∮ ωdp = 0, or equivalently ∫ ω l ω s ( ω∂ ω p ) dω = 0, we have Along the coexistence curve, the EoS holds for the saturated small and large black holes Combining with these equations, it is easy to obtain for the coexistence saturated small and large black holes. From (7), one arrives for d -dimensional case. Note that we have used Expanding it near the critical point t = 0, we obtain Obviously, /epsilon1 exhibits a critical exponent 1/2, strongly indicating that it can serve an order parameter, and can be utilized to characterize the small-large black hole phase transition. This shall be further confirmed by the results of the VdW fluid and charged AdS black holes in what follows. Before ending this section, we would like to give a note on the Maxwell equal area law. Actually, it mainly lies in the equality of the free energy of the coexistence saturated small and large black holes. Meanwhile, one should note that ω is related with the thermodynamical volume rather than the specific volume. The details can be found in Ref. [17]. For the VdW fluid system, one can obtain its free energy by combining with the micro-model. However, for the black hole systems, since the micro-model is unclear, one could not derive its free energy via the EoS or the Hawking temperature. For a specific black hole system, its free energy can be calculated through the action.",
"pages": [
2,
3
]
},
{
"title": "III. ANALYTICAL PARAMETERIZED COEXISTENCE CURVE",
"content": "Now, we turn to analytically study the coexistence curve by using the parameter /epsilon1 . Let us focus on the charged AdS black hole. In Ref. [2], it was found that the free energy demonstrates the swallow tail behaviors indicating the existence of the phase transition. Treating the cosmological constant as the pressure, the small-large black hole phase transition of the VdW type was observed in the extended phase transition [3]. Starting with the d -dimensional case, the EoS in the reduced parameter space was given by [4] Similar to the VdW fluid, the black hole charge has been scaled out in the reduced parameter space. It is worthwhile noting that, the Maxwell equal area law is in the and d =5 form of ∮ ˜ r ( d -1) h dp = 0 for the d -dimensional charged AdS black hole [17]. By making use of it, the analytical parameterized curve can be obtained for any d . However, its expression is long and so we will not show it here [18]. For clarity, we list the result for d = 4 Fortunately, for the four-dimensional case, we can solve the ratio /epsilon1 from the pressure (12), and then express the temperature as which is exactly the result given in Ref. [20]. Adopting the analytical parameterized formulas, one can easily obtain the corresponding phase diagram for the black holes. Moreover, we find that when the dimension number d is larger than 100, the phase diagrams almost hold unchanged in the reduced parameter space. Employing with the parameterized formula, we can expand the ratio near the critical temperature as which exactly confirms the result (10), and indicates that the ratio /epsilon1 acts as an order parameter for any dimension d . More interestingly, applying the study to the VdW fluid, we obtain the analytical parameterized coexistence curve for the VdW fluid where /lscript /epsilon1 = ln(1 -/epsilon1 ). When /epsilon1 = 1 -ν l -1 3 ν g -1 3 → 1, it gives p = τ = 0, and when /epsilon1 → 0, the critical point at p = τ = 1 is given. Employing this analytical formula, it is possible to precisely study various properties of the VdW fluid in all parameter spaces without using any approximation. This approach shall represent a significant improvement over earlier approximate studies [21, 22], which are limited to the τ → 0 and 1 regimes. Interestingly, the ratio parameter /epsilon1 can also be obtained by expanding near the critical temperature Moreover, we also show the ratio /epsilon1 as a function of 1 -τ in Fig. 2. It is evident that, for the VdW fluid and charged AdS black holes with dimensions d = 4 and 5, /epsilon1 approaches zero as (1 -τ ) 1 2 . To summarize, the ratio /epsilon1 that we have introduced can serve as an order parameter to characterize the smalllarge black hole phase transition. Significantly, the coexistence curves for both the VdW fluid and d -dimensional charged AdS black holes can be analytically obtained in parameterized forms. In the following sections, we shall demonstrate that this ratio can also serve as a parameter for characterizing the black hole state within the coexistence region.",
"pages": [
3,
4
]
},
{
"title": "IV. STATE PARAMETERS IN COEXISTENCE REGION",
"content": "The exploration of the coexistence region in black hole systems has been neglected in previous studies due to the violation of the EoS. Now, we aim to explore this issue by utilizing the ratio /epsilon1 and taking the charged AdS black hole as an illustrative example. Let us consider one isobaric curve with p = 0 . 9 shown for d =4 dimensional charged AdS black hole in Fig. 3. The gray curve denotes the coexistence curve, and below which is the coexistence region. The oscillatory part of the isobaric curve in green color should be replaced by a horizontal line, and which describes a phase transition between the small and large black holes. Points A and B are the saturated small and large black hole states. Meanwhile, each point on the horizontal line represents a system state with the same temperature and pressure. For example, the state denoted by point C shares the same temperature and pressure of the states A and B, which are not independent and given in (12) and (13), respectively. The ratio /epsilon1 serves as one characteristic parameter of state C, but only using it cannot uniquely determine the state. Since state C actually contains both the coexistence small and large black holes. Hence, a second parameter is needed to characterize this property. During the transition from state A to B, the molecules of the saturated small black hole gradually transform into those of the saturated large black hole. To characterize this property, we introduce another parameter x = N s / ( N s + N l ), which measures the number of the saturated small black hole molecules to the total number of molecules. Further combining with the lever rule we can express the parameter ˜ r hi of an intermediate state, such as state C, as Here ˜ r hi is bounded by the radii of the saturated small and large black holes. We need to point out that it just denotes the location of the system in the phase diagram instead of the actual radius of the coexistence black holes. It is easy to find that x =1 and 0 correspond to states A and B, respectively. So, analogous to the fluid systems, x gives the ratio of the saturated small and large black hole molecules located at this system state. On the other hand, understanding these coexistence black holes from a gravitational perspective is worth future investigation via the potential relationship between the black hole thermodynamics and gravity. Finally, we find that each coexistence state can be parameterized by two quantities /epsilon1 and x , both of which are bounded between 0 and 1. In Fig. 4, we show these coexistence states by taking x =0.3, 0.5, 0.7, 0.9, 0.99, 0.999, 0.9999 from right to left. The coexistence region is marked in light green color. Note that the parameter x heavily relies on the saturated small and large black holes and is not directly related to the extremal black holes or naked singularities. All of these curves coincide at the critical temperature with /epsilon1 =0, and extend to the right. In particular, with the increase of x , the curve shifts towards the saturated small black hole curve. The monotonic behavior is also broken. Nevertheless, each state within the coexistence region is characterized by a pair of parameters, namely ( /epsilon1 , x ). This study can also be easily generalized to higher dimensional cases.",
"pages": [
4,
5
]
},
{
"title": "V. CONCLUSIONS",
"content": "In this study, we have investigated the thermodynamic nature of the coexisting black hole phases of the VdW type by introducing two new ratio parameters, /epsilon1 and x . Each system state within the coexistence region can be uniquely determined by these parameters, offering an opportunity to explore the coexisting system states that have been previously overlooked. Our findings suggest that the ratio /epsilon1 plays a crucial role in the study of coexisting physics. By utilizing this parameter, we obtained, for the first time, an analytical parameterized form for the coexisting saturated liquid and gas phases. This shall greatly improve the fitting formula given by Ref. [23]. Consequently, all other properties of the VdW fluid can be analyzed analytically without any approximation. For the d -dimensional charged AdS black holes, the analytical parameterized form of the coexistence curve was also obtained. By performing the general calculations, we demonstrated that the ratio /epsilon1 can serve as an order parameter for characterizing the liquid-gas phase transition or the small-large black hole phase transition. This conclusion is further supported by the results of the VdW fluid and charged AdS black holes. Employing with the lever rule, we introduced another parameter, x , which measures the ratio of the small black hole molecule number to the total molecule number. Our result indicates that a black hole state within the coexistence region can be uniquely determined by the ratios /epsilon1 and x , thereby providing an effective EoS. This approach offers insight into the coexistent physics that has been neglected in previous studies. While our study is limited to the VdW type phase transition, the approach we have developed can be generalized to the local analysis of other phase structures near the critical point, representing a second-order phase transition. Actually, this feature is quite prevalent in black hole chemistry. On the other hand, although we gave an effective EoS in the coexistence region, several issues are still not resolved. For examples, the first law still remains to be established and the free energy corresponding to the effective EoS is still unclear. We expect to address these issues in future study. In conclusion, we have developed a general analytical approach to gain insight into the coexistent physics. The effective EoS represented by the two ratio parameters can be used to test other underlying properties.",
"pages": [
5
]
},
{
"title": "VI. ACKNOWLEDGEMENTS",
"content": "This work was supported by the National Natural Science Foundation of China (Grants No. 12075103, No. 11875151, and No. 12247101) and the Major Science and Technology Projects of Gansu Province.",
"pages": [
5
]
}
]
2024SCPMA..6760412L
https://arxiv.org/pdf/2403.09211.pdf
<document>
<section_header_level_1><location><page_1><loc_13><loc_86><loc_87><loc_91></location>Energy flux and waveforms by coalescing spinless binary system in effective one-body theory</section_header_level_1>
<text><location><page_1><loc_27><loc_82><loc_73><loc_83></location>Sheng Long , 1 Weike Deng a , 1 and Jiliang Jing , 1, 2, †</text>
<text><location><page_1><loc_24><loc_79><loc_76><loc_80></location>1 Department of Physics, Key Laboratory of Low Dimensional</text>
<text><location><page_1><loc_19><loc_63><loc_81><loc_77></location>Quantum Structures and Quantum Control of Ministry of Education, and Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha, Hunan 410081, P. R. China 2 Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University, Yangzhou 225009, P. R. China</text>
<section_header_level_1><location><page_1><loc_45><loc_59><loc_54><loc_61></location>Abstract</section_header_level_1>
<text><location><page_1><loc_12><loc_35><loc_88><loc_58></location>We present a study on the energy radiation rate and waveforms of the gravitational wave generated by coalescing spinless binary systems up to the third post-Minkowskian approximation in the effective one-body theory. To derive an analytical expansion of the null tetrad components of the gravitational perturbed Weyl tensor Ψ 4 in the effective spacetime, we utilize the method proposed by Sasaki et al. During this investigation, we discover more general integral formulas that provide a theoretical framework for computing the results in any order. Subsequently, we successfully compute the energy radiation rate and waveforms of the gravitational wave, which include the results of the Schwarzschild case and the correction terms resulting from the dimensionless parameters a 2 and a 3 in the effective metric.</text>
<text><location><page_1><loc_12><loc_31><loc_44><loc_32></location>PACS numbers: 04.25.Nx, 04.30.Db, 04.20.Cv</text>
<text><location><page_1><loc_12><loc_28><loc_87><loc_29></location>Keywords: post-Minkowskian approximation, effective one-body theory, gravitational waveform template</text>
<section_header_level_1><location><page_2><loc_12><loc_89><loc_32><loc_91></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_2><loc_12><loc_77><loc_88><loc_86></location>Gravitational waveform templates play an important role in the detection of gravitational wave events generated by coalescing binary systems [1-10]. The foundation of gravitational waveform templates is the theoretical model of gravitational radiation, in which the key point is studying the late-stage dynamical evolution of a coalescing binary system.</text>
<text><location><page_2><loc_12><loc_58><loc_88><loc_75></location>Damour and Buonanno [11, 12] proposed an effective one-body (EOB) theory that maps the real two-body problem with masses m 1 and m 2 to a test particle of mass µ = m 1 m 2 m 1 + m 2 moving around an effective spacetime of mass M = m 1 + m 2 (and we denote the symmetric mass ratio as ν = µ/M ). This theory enables the study of gravitational radiation produced by merging binary systems. Based on the EOB theory with the post-Newtonian (PN) approximation, Damour et al. provided an estimate of the gravitational waveforms emitted throughout the inspiral, plunge, and coalescence phases [13, 14].</text>
<text><location><page_2><loc_12><loc_34><loc_88><loc_56></location>To release the assumption that v/c is a small quantity, in 2016, Damour introduced another theoretical model by combining the EOB theory with the post-Minkowskian (PM) approximation [15, 16]. Damour and Rettegno [17] compared numerical relativistic (NR) data for equal-mass binary black hole scattering with analytical predictions based on the fourth PM (4PM) dynamics [18-25] and pointed out that the reconstruction of PM information in terms of EOB radial potentials leads to remarkable agreement with NR data, especially when using radiation-reacted 4PM information. Therefore, this new model may lead to a theoretically improved version of the EOB conservative dynamics and may be useful in the upcoming era of high-signal-to-noise-ratio gravitational wave observations.</text>
<text><location><page_2><loc_12><loc_10><loc_88><loc_32></location>The dynamical evolution of a coalescing binary system for a spinless EOB theory can be described by the Hamilton equation [26], and the Hamiltonian H [ g eff µν ] is dependent on the effective metric. The radiation reaction forces F R [ g eff µν ] and F φ [ g eff µν ] in the Hamilton equation can be described by the energy radiation rate as follows: d E d t = 1 4 πG 2 ω 2 ∫ | Ψ B 4 | 2 r 2 dΩ [13, 27]. Furthermore, the 'plus' and 'cross' modes of gravitational waves are related to the null tetrad components of the gravitational perturbed Weyl tensor Ψ B 4 in the NewmanPenrose formalism as follows: Ψ B 4 = 1 2 ( h + -i h × ). Thus, as long as we obtain the effective metric and the solution of Ψ B 4 in the Newman-Penrose formalism, we can calculate the energy radiation rate and construct gravitational waveforms.</text>
<text><location><page_2><loc_14><loc_7><loc_88><loc_8></location>In previous work, we attempted to develop a self-consistent EOB theory for spinless</text>
<text><location><page_3><loc_12><loc_68><loc_88><loc_91></location>and spinning binaries based on the PM approximation [28-31]. Furthermore, in a recent paper [32], we obtained the effective metric up to the 4PM order. We adopted the black hole perturbation method used by Teukolsky [33, 34] and decomposed all quantities into background and perturbation (denoted with a superscript B ) parts in the Newman-Penrose formalism. After choosing a shadow gauge [29, 35, 36] with Ψ 1 and Ψ 3 set to 0, we can decouple the equations for the null tetrad components of the gravitational perturbed Weyl tensor Ψ B 4 . Subsequently, upon separating the variables in the equations, we obtained a radial equation, which is the so-called Teukolsky-like equation, and an angular equation that features spin-weighted spherical harmonics.</text>
<text><location><page_3><loc_12><loc_50><loc_88><loc_67></location>This Teukolsky-like equation is more complex compared to the Teukolsky equation in Kerr and Schwarzschild spacetimes. We were unable to find a similar transformation to convert the homogeneous Teukolsky-like equation into hypergeometric or Heun equations; thus, we did not choose to adopt the so-called MST [37, 38] method or the Heun function [39-44]. Instead, we follow the approach used by Sasaki. Several researchers [45-57] have employed numerical methods to solve the Teukolsky equation and achieved significant success by combining the EOB theory with numerical relativity.</text>
<text><location><page_3><loc_12><loc_20><loc_88><loc_48></location>We initially applied the Sasaki-Nakamura-Chandrasekhar-like (S-N-C-like) transformation [35, 58] to convert the homogeneous Teukolsky-like equation into a homogeneous SasakiNakamura-like (S-N-like) equation [38, 58-60]. In an asymptotically flat spacetime, this homogeneous S-N-like equation can be simplified to the Klein-Gordon equation. Subsequently, we performed a Taylor expansion with respect to η = 2 GMω . The equation of the zeroth order is the spherical Bessel equation, and its solutions are linear combinations of the first and second kind spherical Bessel functions, denoted as j ℓ and n ℓ , respectively, allowing us to construct higher-order solutions based on the zeroth-order solution. By performing an inverse transformation, we can deduce the solutions of the homogeneous Teukolsky-like equation. This framework enables us to construct the solutions of the inhomogeneous Teukolsky equation, which includes a source term, using Green's function method.</text>
<text><location><page_3><loc_12><loc_10><loc_88><loc_19></location>However, due to the complexity of Green's function method and the fact that the integral formulas provided in the previous work of Sasaki [60] were insufficient for our needs in computing higher-order solutions, we found some new integral formulas presented in Appendix B of this article, which are crucial for our journey toward calculating higher-order solutions.</text>
<text><location><page_3><loc_14><loc_7><loc_88><loc_8></location>Section II introduces the effective metric of 3PM, while Sec. III discusses the solutions</text>
<text><location><page_4><loc_12><loc_70><loc_88><loc_91></location>of the equation for Ψ B 4 in the effective spacetime. Specifically, Section IIIA summarizes the general structure of the solutions of the radial equation (Teukolsky-like equation) of Ψ B 4 . Section IIIB provides a comprehensive explanation of the calculation of the homogeneous S-N-like equation. Subsequently, we employ boundary conditions to determine the amplitudes. Section IIIC presents the source terms for quasi-circular orbits; by combining the homogeneous solutions provided in Section IIIB and utilizing Eq. (3.8), we can obtain the solution of Ψ B 4 under quasi-circular orbits. In Section IV, we present the energy radiation rate d E d t and the gravitational waveforms h ℓm .</text>
<section_header_level_1><location><page_4><loc_12><loc_65><loc_62><loc_67></location>II. EFFECTIVE METRIC FOR THE EOB THEORY</section_header_level_1>
<text><location><page_4><loc_12><loc_50><loc_88><loc_62></location>In the EOB theory, the main idea is to map the two-body problem onto an EOB problem, that is, a test particle orbits around a massive black hole described by an effective metric. With the help of the scattering angles, we found that the effective metric for spinless binaries with radiation reaction effects in the EOB theory, up to the 3PM approximation, can be expressed as follows [32]:</text>
<formula><location><page_4><loc_25><loc_46><loc_88><loc_49></location>ds 2 eff = g eff µν dx µ dx ν = ∆ r r 2 dt 2 -r 2 ∆ r dr 2 -r 2 ( dθ 2 +sin 2 θdφ 2 ) , (2.1)</formula>
<text><location><page_4><loc_12><loc_43><loc_15><loc_45></location>with</text>
<formula><location><page_4><loc_30><loc_36><loc_64><loc_43></location>∆ r = r 2 -2 GMr + a 2 ( GM ) 2 + a 3 ( GM ) 3 r =</formula>
<formula><location><page_4><loc_35><loc_35><loc_88><loc_39></location>1 r ( r -2 c h GM )( r -2 c 1 GM )( r -2 c 2 GM ) , (2.2)</formula>
<text><location><page_4><loc_12><loc_33><loc_49><loc_34></location>the definitions of c 1 , c 2 , and c h are as follows:</text>
<formula><location><page_4><loc_20><loc_29><loc_88><loc_32></location>c 1 = 1 3 -1 2 [ (1 -i √ 3)( Q + √ P 3 + Q 2 ) 1 3 +(1 + i √ 3)( Q -√ P 3 + Q 2 ) 1 3 ] , (2.3)</formula>
<formula><location><page_4><loc_20><loc_25><loc_88><loc_29></location>c 2 = 1 3 -1 2 [ (1 + i √ 3)( Q + √ P 3 + Q 2 ) 1 3 +(1 -i √ 3)( Q -√ P 3 + Q 2 ) 1 3 ] , (2.4)</formula>
<formula><location><page_4><loc_20><loc_21><loc_88><loc_25></location>c h = 1 3 +( Q + √ P 3 + Q 2 ) 1 3 +( Q -√ P 3 + Q 2 ) 1 3 , (2.5)</formula>
<text><location><page_4><loc_12><loc_16><loc_88><loc_21></location>where Q = 1 27 -a 2 24 -a 3 16 , P = 1 3 ( a 2 4 -1 3 ) and a 2 and a 3 are dimensionless parameters expressed as follows:</text>
<formula><location><page_4><loc_15><loc_12><loc_88><loc_15></location>a 2 = 3(1 -Γ)(1 -5 γ 2 ) Γ(3 γ 2 -1) , (2.6)</formula>
<formula><location><page_4><loc_15><loc_7><loc_88><loc_11></location>a 3 = 3 2(4 γ 2 -1) [ 3 -2 Γ -3(15 -8 Γ) γ 2 +6(25 -16 Γ) γ 4 Γ(3 γ 2 -1) -2 P 30 -2 χ rr 3 √ γ 2 -1 ] , (2.7)</formula>
<text><location><page_5><loc_12><loc_86><loc_88><loc_91></location>in which γ = E µ = 1 2 E 2 -m 2 1 -m 2 2 m 1 m 2 is the Lorentz factor variable, E is the real two-body energy [32, 61], E is the effective energy, Γ = E/M = √ 1 + 2 ν ( γ -1) is the rescaled energy, and</text>
<formula><location><page_5><loc_14><loc_77><loc_88><loc_85></location>P 30 = 18 γ 2 -1 2 Γ 2 + 8 ν (3 + 12 γ 2 -4 γ 4 ) Γ 2 √ γ 2 -1 arcsinh √ γ -1 2 + ν Γ 2 ( 1 -103 3 γ -48 γ 2 -2 3 γ 3 + 3 Γ (1 -2 γ 2 )(1 -5 γ 2 ) (1 + Γ)(1 + γ ) ) , (2.8)</formula>
<formula><location><page_5><loc_15><loc_72><loc_88><loc_76></location>χ rr 3 = -2 ν Γ 2 γ (2 γ 2 -1) 2 3( γ 2 -1) 3 / 2 { (5 γ 2 -8) γ √ γ 2 -1 + 2(9 -6 γ 2 )arcsinh √ γ -1 2 } . (2.9)</formula>
<text><location><page_5><loc_12><loc_64><loc_88><loc_71></location>In Eq. (2.7), the term χ rr 3 , described by Eq. (2.9), represents the 3PM radiation reaction effects, which shows that the structure of the effective spacetime is affected by the radiation reaction effect.</text>
<section_header_level_1><location><page_5><loc_12><loc_58><loc_81><loc_60></location>III. SOLUTIONS OF EQUATION FOR Ψ B 4 IN EFFECTIVE SPACETIME</section_header_level_1>
<text><location><page_5><loc_12><loc_44><loc_88><loc_56></location>In this section, we first present the formal solution of the radial equation for Ψ B 4 . Then, we transform the radial equation without source to the corresponding S-N-like equation, and we look for its solution. At last, we work out the solution of the radial equation of Ψ B 4 with the source, which describes the gravitational radiation induced by the motion of an effective particle in an effective background.</text>
<section_header_level_1><location><page_5><loc_14><loc_38><loc_78><loc_40></location>A. Formal solution of the radial equation of Ψ B 4 in effective spacetime</section_header_level_1>
<text><location><page_5><loc_12><loc_23><loc_88><loc_35></location>In the EOB theory for the spinless real two-body system, we have found a decoupled equation of Ψ B 4 for the gravitational perturbation in the effective spacetime (2.1) using the gauge transform property of the tetrad components of the perturbed Weyl tensors and separated the decoupled equation in the radial and angular parts, in which the radial part of Ψ B 4 is given by [28, 29]</text>
<formula><location><page_5><loc_30><loc_18><loc_88><loc_22></location>[ ∆ 2 r f d d r ( f ∆ r d d r ) -V ( r ) ] R lmω ( r ) = T ℓmω ( r ) , (3.1)</formula>
<text><location><page_5><loc_12><loc_15><loc_15><loc_17></location>with</text>
<formula><location><page_5><loc_13><loc_6><loc_81><loc_14></location>f = -3 GM r 3 F 4 , V ( r ) = -r 2 ω ( r 2 ω +2 i ∆ ' r ) ∆ r + irω ( 5 + 2 r 3 F 1 ∆ r ) -3∆ r r 2 -3∆ ' r r +6 rF 1 -2 r 2 F 4 + λF 2</formula>
<formula><location><page_5><loc_81><loc_8><loc_81><loc_9></location>,</formula>
<formula><location><page_6><loc_13><loc_85><loc_88><loc_91></location>T ℓmω ( r ) = -µ ∫ ∞ -∞ dte iωt -imφ ( t ) ∆ 2 r { A 0 δ ( r -r ( t )) + [ A 1 δ ( r -r ( t )) ] ' + [ A 2 δ ( r -r ( t )) ] '' } , (3.2)</formula>
<text><location><page_6><loc_12><loc_79><loc_88><loc_83></location>where the prime ' denotes derivation with respect to r , and A 0 = A nn 0 + A mn 0 + A mm 0 , A 1 = A mn 1 + A mm 1 , and A 2 = A mm 2</text>
<formula><location><page_6><loc_29><loc_77><loc_29><loc_78></location>4</formula>
<formula><location><page_6><loc_27><loc_53><loc_29><loc_55></location>2 π</formula>
<formula><location><page_6><loc_16><loc_54><loc_88><loc_78></location>A nn 0 = -2 r √ 2 π ∆ 2 r C nn F 2 L + 1 [ L + 2 ( -2 Y ℓm ( θ ) )] , A mn 0 = r 3 √ π ∆ r C mn [ ( 1 + F 2 ) ir 2 ω ∆ r + F 2 r -F ' 2 -F ' 4 F 4 ] L † 2 ( -2 Y ℓm ( θ ) ) , A mm 0 = r 2 √ 2 π C mm -2 Y ℓm ( θ ) [ i ( r 2 ω ∆ r ) ' + r 4 ω 2 ∆ 2 r + ir 2 ω ∆ r ( 1 r + F ' 4 F 4 ) + F ' 4 rF 4 + ( F ' 4 F 4 ) ' ] , A mn 1 = r 3 √ π ∆ r C mn ( 1 + F 2 ) L † 2 ( -2 Y ℓm ( θ ) ) , A mm 1 = r 2 √ 2 π C mm -2 Y ℓm ( θ ) ( -2 i r 2 ω ∆ r + 1 r + F ' 4 F 4 ) , A mm 2 = -r 2 √ C mm -2 Y ℓm ( θ ) , (3.3)</formula>
<text><location><page_6><loc_12><loc_48><loc_88><loc_52></location>the explicitly definitions of F a ( a = 1 , 2 , 3 , 4), L n (or L † n ), and C b ( b is nn or mn or mm ) can be found in Ref. [29], and -2 Y ℓm ( θ ) is the spin-weighted spherical harmonics [38, 62].</text>
<text><location><page_6><loc_12><loc_43><loc_88><loc_47></location>The radial equation, i.e., Eq. (3.1), can be solved using Green's function method. That is, based on the homogeneous solutions of Eq. (3.1)</text>
<formula><location><page_6><loc_23><loc_36><loc_88><loc_41></location>R in ℓmω → B trans ℓmω ∆ 2 r e -iωr ∗ , for r → r + , r 3 B ref ℓmω e iωr ∗ + r -1 B in ℓmω e -iωr ∗ , for r → + ∞ , (3.4)</formula>
<formula><location><page_6><loc_25><loc_28><loc_88><loc_34></location>R up ℓmω → C up ℓmω e iωr ∗ +∆ 2 r C ref ℓmω e -iωr ∗ , for r → r + , C trans ℓmω r 3 e iωr ∗ , for r → + ∞ , (3.5)</formula>
<text><location><page_6><loc_12><loc_22><loc_88><loc_27></location>where r ∗ denotes the tortoise coordinate defined by r ∗ = ∫ r 2 ∆ r dr . The inhomogeneous solution of the radial Eq. (3.1) can be expressed as follows:</text>
<formula><location><page_6><loc_14><loc_17><loc_86><loc_21></location>R ℓmω = 1 2 iωC trans ℓmω B inc ℓmω { R up ℓmω ∫ r r + d ˜ r f R in ℓmω (˜ r ) T ℓmω (˜ r ) ∆ 2 r (˜ r ) + R in ℓmω ∫ ∞ r d ˜ r f R up ℓmω (˜ r ) T ℓmω (˜ r ) ∆ 2 r (˜ r ) } ,</formula>
<formula><location><page_6><loc_84><loc_15><loc_88><loc_16></location>(3.6)</formula>
<text><location><page_6><loc_12><loc_12><loc_65><loc_13></location>whereas the counterpart at infinity can be expressed as follows:</text>
<formula><location><page_6><loc_23><loc_7><loc_88><loc_11></location>R ℓmω ( r →∞ ) → r 3 e i ωr ∗ 2i ωB inc ℓmω ∫ ∞ r + d˜ r T ℓmω (˜ r ) R in ℓmω (˜ r ) ˜ r 3 F 4 (˜ r )∆ 2 r (˜ r ) ≡ ˆ Z ℓmω r 3 e i ωr ∗ . (3.7)</formula>
<text><location><page_7><loc_12><loc_84><loc_88><loc_91></location>As discussed in Ref. [29], for the point source case, after a lengthy calculation, we can obtain the expression for ˆ Z ℓmω . If we focus our attention just on the quasi-circular orbit, we have ˆ Z ℓmω n = Z ℓmω δ ( ω -ω n ), in which</text>
<formula><location><page_7><loc_23><loc_79><loc_88><loc_83></location>Z ℓmω = πνGM iωB inc ℓmω [ A 0 f R in ℓmω -A 1 d dr ( f R in ℓmω ) + A 2 d 2 dr 2 ( f R in ℓmω )] . (3.8)</formula>
<text><location><page_7><loc_12><loc_76><loc_46><loc_78></location>Then, the solution of Ψ B 4 is described by</text>
<formula><location><page_7><loc_34><loc_71><loc_88><loc_75></location>Ψ B 4 = 1 R ∑ ℓmn ˆ Z ℓmω n -2 Y ℓm √ 2 π e iω n ( r ∗ -t )+ imφ . (3.9)</formula>
<text><location><page_7><loc_12><loc_66><loc_88><loc_70></location>Equations (3.9) and (3.8) show that to get the explicit expression for Ψ B 4 , we should work out A i ( i = 1 , 2 , 3), B inc ℓmω , and R in ℓmω .</text>
<section_header_level_1><location><page_7><loc_14><loc_60><loc_77><loc_62></location>B. S-N-like equation and its solution of the third PM approximation</section_header_level_1>
<text><location><page_7><loc_12><loc_42><loc_88><loc_58></location>In Eq. (3.8), R in ℓmω is the solution of the homogeneous equation without the source term. To get the solution, we do not treat the Teukolsky-like equation directly because the potential function in the equation is a long-range potential. Instead, we transform the radial equation, i.e., Eq. (refEqTS), without source into the S-N-like equation with a new function X ℓmω , which has a short-range potential. Then, using the solution of X in ℓmω , we find out B in ℓmω and R in ℓmω .</text>
<unordered_list>
<list_item><location><page_7><loc_14><loc_37><loc_60><loc_39></location>1. S-N-like equation and relation between R in ℓmω and X in ℓmω</list_item>
</unordered_list>
<text><location><page_7><loc_14><loc_33><loc_48><loc_34></location>Taking an S-N-C-like transformation as 1</text>
<formula><location><page_7><loc_32><loc_28><loc_88><loc_32></location>X ( r ) = rf 1 / 2 ∆ r ( α ( r ) R ( r ) + β ( r ) ∆ r R ' ( r ) ) , (3.10)</formula>
<text><location><page_7><loc_12><loc_25><loc_17><loc_27></location>where</text>
<formula><location><page_7><loc_31><loc_17><loc_88><loc_24></location>α ( r ) =6 ∆ r r 2 + V ( r ) + 2 irω + ir 2 ω ∆ r ' ∆ r -r 4 ω 2 ∆ r , β ( r ) =∆ r ( -2 ir 2 ω -4 ∆ r r -∆ r f ' f +∆ r ) . (3.11)</formula>
<text><location><page_8><loc_12><loc_84><loc_88><loc_91></location>and considering the coordinate transformation r → r ∗ , the radial equation (Eq. (3.1)) without the source term ( T ℓmω = 0) can be rewritten as the so-called S-N-like equation, as follows:</text>
<formula><location><page_8><loc_38><loc_81><loc_88><loc_83></location>X ,r ∗ r ∗ -F X ,r ∗ -U X = 0 , (3.12)</formula>
<text><location><page_8><loc_12><loc_78><loc_15><loc_79></location>with</text>
<text><location><page_8><loc_12><loc_66><loc_17><loc_67></location>where</text>
<formula><location><page_8><loc_25><loc_53><loc_88><loc_65></location>γ = α ( α + β ' ∆ r -β ∆ r f ' f ) -β ∆ r ( α ' + β ∆ 2 r V ( r ) ) , U = ∆ 2 r β ( (2 α + β ' ∆ r -β ∆ r f ' f ) ' -γ ' γ ( α + β ' ∆ r -β ∆ r f ' f ) ) + V ( r ) , G = ∆ r r 3 + ∆ r f ' 2 r 2 f -∆ r ' r 2 . (3.14)</formula>
<text><location><page_8><loc_12><loc_49><loc_62><loc_51></location>The asymptotic solution of X in ℓ can be expressed as follows:</text>
<formula><location><page_8><loc_31><loc_43><loc_88><loc_48></location>X in ℓ ( r ) = A trans ℓ e -iωr ∗ r ∗ →-∞ , A out ℓ e iωr ∗ + A in ℓ e -iωr ∗ r ∗ → + ∞ . (3.15)</formula>
<text><location><page_8><loc_12><loc_40><loc_78><loc_41></location>Meanwhile, the inverse transformation is described by the following expression:</text>
<formula><location><page_8><loc_25><loc_35><loc_88><loc_38></location>R ( r ) = 1 γ [ ( α + β ' ∆ r -β ∆ r f ' f ) ∆ r r f 1 / 2 X ( r ) -β ∆ r ( ∆ r r f 1 / 2 X ( r )) ' ] . (3.16)</formula>
<text><location><page_8><loc_12><loc_29><loc_88><loc_33></location>Using a method similar to that used in Refs. [38, 63], the coefficient A in ℓ in (3.15) is related to B in ℓ in Eq. (3.4) as follows:</text>
<formula><location><page_8><loc_42><loc_24><loc_88><loc_27></location>B in ℓ = -1 4 ω 2 A in ℓ . (3.17)</formula>
<section_header_level_1><location><page_8><loc_14><loc_19><loc_30><loc_21></location>2. Solution of X in ℓ</section_header_level_1>
<text><location><page_8><loc_12><loc_10><loc_88><loc_16></location>We now look for the solution of X in ℓ and amplitude A in ℓ of the S-N-like equation. The method employed in this subsection is based on the work of Sasaki [38] and Mino [60]. We first take the following ansatz:</text>
<formula><location><page_8><loc_42><loc_7><loc_88><loc_9></location>X in ℓ = e -iϕ ( z ) zξ ℓ ( z ) , (3.18)</formula>
<formula><location><page_8><loc_36><loc_69><loc_88><loc_77></location>F = ∆ r r 2 γ ' γ , U = ∆ r r 4 U + G 2 + G ,r ∗ -∆ r r 2 γ ' γ G, (3.13)</formula>
<text><location><page_9><loc_12><loc_88><loc_84><loc_91></location>where z = ωr , η = 2 GMω , b 1 = c 3 1 ( c 1 -c 2 )( c 1 -c h ) , b 2 = c 3 2 ( c 2 -c 1 )( c 2 -c h ) , b h = c 3 h ( c 1 -c h )( c h -c 2 ) , and</text>
<formula><location><page_9><loc_26><loc_81><loc_88><loc_87></location>ϕ ( z ) = ∫ ( r 2 ω ∆ r -ω )d r = η ( b 1 ln( z -c 1 η ) + b 2 ln( z -c 2 η ) -b h ln( z -c h η )) . (3.19)</formula>
<text><location><page_9><loc_12><loc_74><loc_88><loc_78></location>With this choice of the phase function, ξ ℓm is regular and finite at z = ηc h . Then, we determine that Eq. (3.12) can be expressed as follows:</text>
<formula><location><page_9><loc_28><loc_70><loc_88><loc_71></location>L (0) [ ξ ℓ ] = ηL (1) [ ξ ℓ ] + η 2 L (2) [ ξ ℓ ] + η 3 L (3) [ ξ ℓ ] + O ( η 4 ) , (3.20)</formula>
<text><location><page_9><loc_12><loc_66><loc_15><loc_67></location>with</text>
<formula><location><page_9><loc_15><loc_48><loc_88><loc_64></location>L (0) = d 2 d z 2 + 2 z d d z + [ 1 -ℓ ( ℓ +1) z 2 ] , L (1) = 1 z d 2 d z 2 + 1 + 2 3 a 2 +2 iz z 2 d d z -4 + z 2 + a 2 ( ℓ 2 + ℓ +2) 3 -iz z 3 , L (2) = -a 2 4 z 2 d 2 d z 2 + ( i z 2 a (2) ℓ + b (2) ℓ z 3 ) d d z + ( c (2) ℓ z 2 + i z 3 d (2) ℓ + e (2) ℓ z 4 ) , L (3) = -a 3 8 z 3 d 2 d z 2 + ( a (3) ℓ z 2 + i z 3 b (3) ℓ + c (3) ℓ z 4 ) d d z + ( i z 2 d (3) ℓ + e (3) ℓ z 3 + i z 4 f (3) ℓ + g (3) ℓ z 5 ) , (3.21)</formula>
<text><location><page_9><loc_12><loc_44><loc_72><loc_46></location>where the definitions of a ( n ) ℓ and other terms are shown in Appendix A.</text>
<text><location><page_9><loc_12><loc_39><loc_88><loc_43></location>In the low-frequency limit and noting that η = 2 GMω only appears on the right-hand side of Eq. (3.20), we may look for the solution of the ξ ℓ ( z ) perturbative in terms of η , i.e.,</text>
<formula><location><page_9><loc_40><loc_33><loc_88><loc_38></location>ξ ℓ ( z ) = ∞ ∑ n =0 η n ξ ( n ) ℓ ( z ) , (3.22)</formula>
<text><location><page_9><loc_12><loc_30><loc_65><loc_32></location>and we obtain the recursive equation from Eq. (3.20) as follows:</text>
<formula><location><page_9><loc_42><loc_26><loc_88><loc_28></location>L (0) [ ξ ( n ) ℓ ] = W ( n ) ℓ , (3.23)</formula>
<text><location><page_9><loc_12><loc_22><loc_17><loc_23></location>where</text>
<formula><location><page_9><loc_35><loc_18><loc_88><loc_20></location>W (0) ℓ = 0 , (3.24)</formula>
<formula><location><page_9><loc_34><loc_13><loc_88><loc_17></location>W ( n ) ℓ = n ∑ i =1 L ( i ) [ ξ ( n -i ) ℓ ] , n = 1 , 2 , 3 . (3.25)</formula>
<text><location><page_9><loc_12><loc_7><loc_88><loc_11></location>The solution of ξ (0) ℓ can be expressed as a linear combination of the spherical Bessel functions j ℓ and n ℓ , i.e., ξ (0) ℓ = α (0) j ℓ + β (0) n ℓ . Because n ℓ does not match with the horizon</text>
<text><location><page_10><loc_12><loc_78><loc_88><loc_91></location>solution at the leading order of η , we should take β (0) = 0. Furthermore, because the constant α (0) represents the overall normalization of the solution, which can be chosen arbitrarily, we set α (0) = 1. That is, for the zeroth-order solution, we have f (0) ℓ = j ℓ and g (0) ℓ = 0. Then, one can immediately write the integral expression for ξ ( n ) ℓ (where n > 0). Noting that j ℓ n ℓ ' -n ℓ j ℓ ' = 1 /z 2 , we derive the expression of ξ ( β ) ℓ for β ≥ 1 as follows:</text>
<formula><location><page_10><loc_29><loc_74><loc_88><loc_77></location>ξ ( β ) ℓ = n ℓ ∫ z d z ( z 2 j ℓ W ( β ) ℓ ) -j ℓ ∫ z d z ( z 2 n ℓ W ( β ) ℓ ) . (3.26)</formula>
<text><location><page_10><loc_12><loc_66><loc_88><loc_70></location>In general, ξ ( β ) ℓ can be decomposed into the real and imaginary parts ξ ( β ) ℓ = f ( β ) ℓ + ig ( β ) ℓ , in which</text>
<formula><location><page_10><loc_25><loc_61><loc_88><loc_64></location>f ( β ) ℓ = n ℓ ∫ z d z ( z 2 j ℓ Re [ W ( β ) ℓ ] ) -j ℓ ∫ z d z ( z 2 n ℓ Re [ W ( β ) ℓ ] ) , (3.27)</formula>
<formula><location><page_10><loc_25><loc_57><loc_88><loc_60></location>g ( β ) ℓ = n ℓ ∫ z d z ( z 2 j ℓ Im [ W ( β ) ℓ ] ) -j ℓ ∫ z d z ( z 2 n ℓ Im [ W ( β ) ℓ ] ) , (3.28)</formula>
<formula><location><page_10><loc_28><loc_46><loc_88><loc_50></location>Re [ W ( β ) ℓ ] = β ∑ i =1 ( Re [ L ( i ) ][ f ( β -i ) ℓ ] -Im [ L ( i ) ][ g ( β -i ) ℓ ] ) , (3.29)</formula>
<formula><location><page_10><loc_28><loc_40><loc_88><loc_45></location>Im [ W ( β ) ℓ ] = β ∑ i =1 ( Im [ L ( i ) ][ f ( β -i ) ℓ ] + Re [ L ( i ) ][ g ( β -i ) ℓ ] ) , (3.30)</formula>
<text><location><page_10><loc_12><loc_18><loc_88><loc_38></location>where Re [ x ] and Im [ x ] are the real and imaginary parts of x , respectively. Using the previously presented formula and the method discussed in Appendix B, after some tedious calculations, we derive closed analytical formulas of the ingoing-wave S-N-like function for arbitrary ℓ to the first order of η . At the second order of η 2 , we can calculate results for any order utilizing equation (3.27). However, generalizing these results to encompass all values of ℓ is unattainable. Therefore, for the higher-order results, we only provide results for specific values of ℓ , for ℓ = 2 , 3 to η 2 order, and for ℓ = 2 to η 3 order. We express the real parts as follows:</text>
<formula><location><page_10><loc_14><loc_7><loc_88><loc_16></location>f (1) ℓ = ( ℓ -1)( ℓ +3) 2( ℓ +1)(2 ℓ +1) j ℓ +1 -( ℓ 2 -4 2 ℓ (2 ℓ +1) + 2 ℓ -1 ℓ ( ℓ -1) ) j ℓ -1 + R ℓ, 0 j 0 -2 D nj ℓ + ℓ -2 ∑ m =1 ( 1 m + 1 m +1 ) R ℓ,m j m -a 2 3 ( ℓ 2 +3 ℓ +4 2( ℓ +1)(2 ℓ +1) j ℓ +1 -ℓ 2 -ℓ +2 2 ℓ (2 ℓ +1) j ℓ -1 ) , (3.31)</formula>
<text><location><page_10><loc_12><loc_52><loc_15><loc_54></location>with</text>
<text><location><page_11><loc_12><loc_89><loc_85><loc_91></location>where R m,k is the Lommel polynomial ( R m,k = -R k,m for m<k ), expressed as follows:</text>
<formula><location><page_11><loc_18><loc_79><loc_88><loc_87></location>R m,k = z 2 ( n m j k -j m n k ) ( m>k ) = -[ ( m -k -1) 2 ] ∑ r =0 ( -1) r ( m -k -1 -r )!Γ ( m + 1 2 -r ) r !( m -k -1 -2 r )!Γ ( k + 3 2 + r ) ( 2 z ) m -k -1 -2 r , (3.32)</formula>
<text><location><page_11><loc_12><loc_73><loc_88><loc_77></location>B J is the generalized integral sinusoidal function, and D J ℓ is the generalized spherical Bessel function in Appendix B.</text>
<formula><location><page_11><loc_12><loc_55><loc_82><loc_71></location>f (2) 2 = ( -193 a 2 2 1890 z + 45 a 2 14 z -257 a 3 1008 z -113 420 z ) j 1 + ( -17 a 2 2 1890 z -10 a 2 63 z -59 a 3 336 z + 1 7 z ) j 3 + ( -16 a 2 2 945 + a 2 21 -5 a 3 126 -107 210 ) j 2 ln z + ( 32 a 2 2 315 z -55 a 2 21 z + 5 a 3 21 z ) n 0 + ( 32 a 2 2 945 -2 a 2 21 + 5 a 3 63 + 107 105 ) D nj -3 + ( 10 3 -2 a 2 15 ) D nj 1 + 14 a 2 45 D nj 3 -389 j 0 70 z 2 -1 j 2 (ln z ) 2 + 6 D nj 0 -5 D nj 2 +4 D nnj 2 ,</formula>
<formula><location><page_11><loc_12><loc_38><loc_90><loc_57></location>2 z 3 z (3.33) f (2) 3 = ( 20 a 2 2 81 z -635 a 2 72 z + 5 a 3 9 z -445 14 z 3 -1031 588 z ) j 0 + ( -197 a 2 2 1134 z + 1093 a 2 168 z -415 a 3 1008 z + 323 49 z ) j 2 + ( 2 a 2 2 189 z -199 a 2 840 z -65 a 3 504 z + 1 4 z ) j 4 + j 3 ln z ( -5 a 2 2 378 + a 2 42 -5 a 3 168 -13 42 ) +4 D nnj 3 + ( 25 a 2 2 63 z -405 a 2 28 z + 25 a 3 28 z -65 6 z ) n 1 + ( -5 a 2 2 189 + a 2 21 -5 a 3 84 -13 21 ) D nj -4 + ( 13 3 -8 a 2 63 ) D nj 2 + 11 a 2 42 D nj 4 -5065 j 1 294 z 2 -1 2 j 3 (ln z ) 2 + 65 n 0 6 z 2 + 30 D nj 0 z 2 + 9 D nj 1 z -3 D nj 3 z . (3.34)</formula>
<formula><location><page_11><loc_13><loc_7><loc_88><loc_35></location>f (3) 2 = ( 9 4 -a 2 30 ) j 1 (ln z ) 2 + ( 7 a 2 90 -1 12 ) j 3 (ln z ) 2 + D nj 2 (ln z ) 2 + ( -16 a 3 2 14175 + 5 a 2 2 63 -a 3 a 2 378 -887 a 2 3150 + 5 a 3 28 + 349 140 ) j 1 ln z + c 2 1 +( c 2 -1)( c 1 + c 2 ) z j 2 ln z + ( 2 3 -28 a 2 45 ) D nnj 3 + ( 16 a 3 2 6075 -11 a 2 2 315 + a 3 a 2 162 + 1543 a 2 18900 -10 a 3 189 + 29 252 ) j 3 ln z + ( -16 a 2 2 315 + a 2 7 -5 a 3 42 -107 70 ) n 0 ln z + ( 32 a 2 2 945 -2 a 2 21 + 5 a 3 63 + 107 105 ) D nj 2 ln z + ( -2381 a 3 2 66150 + 8207 a 2 2 132300 -185 a 3 a 2 2352 -83821 a 2 44100 + 21 100 -187 a 3 168 ) j 1 + ( 97 a 3 2 36450 + 5339 a 2 2 170100 + 35 a 3 a 2 3888 + 9053 a 2 25200 + 4609 a 3 18144 -457 1050 ) j 3 + ( -11 a 3 2 142884 + 139 a 2 2 476280 -43 a 3 a 2 95256 -277 a 2 105840 + 1 504 -109 a 3 45360 ) j 5 + ( 193 a 3 2 10206 -110 a 2 2 1701 + 1033 a 3 a 2 27216 + 6911 a 2 5670 + 48353 a 3 90720 -3(ln z ) 2 2 -2539 3780 ) n 0 +</formula>
<text><location><page_12><loc_12><loc_51><loc_15><loc_53></location>with</text>
<formula><location><page_12><loc_27><loc_44><loc_88><loc_48></location>T 1 =1 + c 2 1 -( c 1 + c 2 )(1 -c 2 ) , T 2 = c 1 c 2 ( -c 1 -c 2 +3) + 2( c 1 -1) c 1 +2( c 2 -1) c 2 +1 , (3.39)</formula>
<text><location><page_12><loc_12><loc_39><loc_52><loc_41></location>where ς ( n ) ℓ for ℓ = 2 can be expressed as follows:</text>
<formula><location><page_12><loc_12><loc_19><loc_88><loc_37></location>ς (2) 2 = ( 2 a 2 2 81 + 5 a 3 108 + a 2 180 ) j 3 + a 2 30 j 1 , (3.40) ς (3) 2 = ( -4 a 2 2 81 -a 2 90 -5 a 3 54 ) D nj 3 + ( 176 a 3 2 2835 z -1469 a 2 2 3780 z + 40 a 2 a 3 189 z -5 a 2 24 z -181 a 3 252 z ) j 1 -a 2 15 D nj 1 + ( 5 a 3 2 1701 z + 83 a 2 2 3780 z + 20 a 2 a 3 567 z + a 2 144 z + 13 a 3 252 z ) j 3 + ( 22 a 3 2 2835 + a 2 2 105 + 5 a 2 a 3 189 + a 3 56 ) j 2 ln z + ( -44 a 3 2 945 z + 296 a 2 2 945 z -10 a 2 a 3 63 z + a 2 12 z + 37 a 3 63 z ) n 0 + ( -44 a 3 2 2835 -2 a 2 2 105 -10 a 2 a 3 189 -a 3 28 ) D nj -3 . (3.41)</formula>
<text><location><page_12><loc_12><loc_12><loc_88><loc_16></location>Inserting these expressions into Eq. (3.18) and expanding the result with respect to η , we find that</text>
<formula><location><page_12><loc_33><loc_7><loc_88><loc_9></location>X in ℓmω = X (0) ℓ + ηX (1) ℓ + η 2 X (2) ℓ + η 3 X (3) ℓ , (3.42)</formula>
<formula><location><page_12><loc_18><loc_71><loc_92><loc_91></location>( -32 a 3 2 1215 + 65 a 2 2 1134 -5 a 3 a 2 81 -1574 a 2 945 + 197 126 -2455 a 3 3024 ) n 2 + ( 32 a 3 2 6075 -58 a 2 2 2835 + a 3 a 2 81 + 824 a 2 4725 -5 a 3 378 -107 630 ) D nj -4 + ( -32 a 3 2 14175 -2 a 2 2 45 -a 3 a 2 189 + 7468 a 2 1575 -5 a 3 42 -457 70 ) D nj -2 + ( -19 a 2 2 567 + 59 a 2 21 -2629 630 -103 a 3 1512 ) D nj 0 + ( 1402 a 2 2 19845 -925 a 2 441 + 1165 a 3 5292 + 16949 4410 ) D nj 2 + ( 17 a 2 2 6615 + 20 a 2 441 + 59 a 3 1176 -2 49 ) D nj 4 + ( -64 a 2 2 945 + 4 a 2 21 -10 a 3 63 -214 105 ) D njj 2 + ( -64 a 2 2 945 + 4 a 2 21 -10 a 3 63 -214 105 ) D nnj -3 -12 D nnj -1 + ( 4 a 2 15 -18 ) D nnj 1 -8 D nnnj 2 . (3.35)</formula>
<text><location><page_12><loc_12><loc_68><loc_62><loc_69></location>The corresponding imaginary parts are expressed as follows:</text>
<formula><location><page_12><loc_29><loc_63><loc_88><loc_65></location>g (1) ℓ = j ℓ ln z, (3.36)</formula>
<formula><location><page_12><loc_29><loc_59><loc_88><loc_62></location>g (2) ℓ = f (1) ℓ ln z -T 1 z j ℓ + ς (2) ℓ , (3.37)</formula>
<formula><location><page_12><loc_29><loc_55><loc_88><loc_59></location>g (3) ℓ = f (2) ℓ ln z -T 1 f (1) ℓ z -T 2 j ℓ 2 z 2 + 1 3 j ℓ (ln z ) 3 + ς (3) ℓ , (3.38)</formula>
<text><location><page_13><loc_12><loc_89><loc_17><loc_91></location>where</text>
<formula><location><page_13><loc_26><loc_74><loc_88><loc_87></location>X (0) ℓ = zj ℓ , X (1) ℓ = zf (1) ℓ , X (2) ℓ = z [ f (2) ℓ + 1 2 j ℓ (ln z ) 2 + iς (2) ℓ ] , X (3) ℓ = z [ f (3) ℓ + 1 2 f (1) ℓ (ln z ) 2 -T 1 z j ℓ ln z + ς (2) ℓ ln z + iς (3) ℓ ] . (3.43)</formula>
<section_header_level_1><location><page_13><loc_14><loc_68><loc_39><loc_70></location>3. Coefficient of amplitude A in ℓ</section_header_level_1>
<text><location><page_13><loc_12><loc_61><loc_88><loc_66></location>Noting e -iη ( -b 1 ln( z -c 1 η )+ b 2 ln( z -c 2 η )+ b h ln( z -c h η )) = e -iz ∗ e iz z →∞ -→ 1, taking the expressions of the spherical Hankel functions of the first and second kinds h (1) ℓ and h (2) ℓ as</text>
<formula><location><page_13><loc_37><loc_56><loc_88><loc_60></location>h (1) ℓ = j ℓ + in ℓ z →∞ ---→ ( -i ) ℓ +1 e iz z , (3.44)</formula>
<formula><location><page_13><loc_37><loc_52><loc_88><loc_56></location>h (2) ℓ = j ℓ -in ℓ z →∞ ---→ i ℓ +1 e -iz z , (3.45)</formula>
<text><location><page_13><loc_12><loc_46><loc_88><loc_50></location>and using the asymptotic behavior of B J and D J ℓ in Ref. [60], we obtain the following expression:</text>
<formula><location><page_13><loc_14><loc_37><loc_88><loc_44></location>A in ℓ = 1 2 i ℓ +1 e -iη (ln 2 η + elg ) e i [ ηp (0) ℓ -πη 2 p (1) ℓ + η 3 ( p (2) ℓ -π 2 p (3) ℓ + p (4) ℓ RiZ (3) )] { 1 -π 2 η + η 2 [ 2( elg +ln2) p (1) ℓ + q (1) ℓ + 5 π 2 24 ] + η 3 [ πq (2) ℓ + π 3 q (4) ℓ + π ( elg +ln2) q (3) ℓ ] } . (3.46)</formula>
<text><location><page_13><loc_12><loc_30><loc_88><loc_35></location>where elg is the EulerGamma constant ( elg = 0 . 57721 · · · ), RiZ ( n ) is the Riemann zeta function ( RiZ (3) = 1 . 202 · · · ), and the coefficients of A in 2 are</text>
<formula><location><page_13><loc_18><loc_6><loc_88><loc_29></location>p (0) 2 = 15 -2 a 2 9 , p (1) 2 = 32 a 2 2 -90 a 2 +75 a 3 +963 3780 , p (2) 2 = -292 a 3 2 +1962 a 2 2 -765 a 2 a 3 -1782 a 2 +3564 a 3 +1566 34992 , p (3) 2 = 32 a 2 2 -90 a 2 +75 a 3 +963 11340 , p (4) 2 = 1 3 , q (1) 2 = -37 a 2 -5 a 3 +150 108 , q (4) 2 = -1 16 , q (2) 2 = -176 a 3 2 -216 a 2 2 -600 a 2 a 3 +7770 a 2 +645 a 3 -31500 45360 , and q (3) 2 = -32 a 2 2 +90 a 2 -75 a 3 -963 3780 -i 176 a 3 2 +216 a 2 2 +600 a 2 a 3 +405 a 3 22680 π . (3.47)</formula>
<section_header_level_1><location><page_14><loc_14><loc_89><loc_72><loc_91></location>C. Quasi-circular orbit on the equatorial plane around an EOB</section_header_level_1>
<text><location><page_14><loc_12><loc_79><loc_88><loc_86></location>In this section, we consider a quasi-circular orbit. In this case, we assume that the orbit lies on the equatorial plane ( θ = π/ 2) without loss of generality. By setting V r ( r 0 ) = ∂V r /∂r ( r 0 ) = 0, the effective energy E and effective angular momentum L are given by</text>
<formula><location><page_14><loc_24><loc_73><loc_76><loc_79></location>E/µ = √ 2 [ a 3 ( GM ) 3 + r 0 ( a 2 ( GM ) 2 + r 0 ( r 0 -2 GM ) )] r 0 √ r 0 √ 5 a 3 ( GM ) 3 +2 r 0 ( 2 a 2 ( GM ) 2 + r 0 ( r 0 -3 GM ) ) ,</formula>
<formula><location><page_14><loc_24><loc_68><loc_71><loc_72></location>L/µ = r 0 √ -3 a 3 ( GM ) 3 +2 GMr 0 ( r 0 -a 2 GM ) √ 5 a 3 ( GM ) 3 +2 r 0 ( 2 a 2 ( GM ) 2 + r 0 ( r 0 -3 GM ) ) .</formula>
<formula><location><page_14><loc_83><loc_69><loc_88><loc_76></location>(3.48) (3.49)</formula>
<text><location><page_14><loc_12><loc_64><loc_47><loc_65></location>where r 0 is the orbital radius. By defining</text>
<formula><location><page_14><loc_12><loc_46><loc_87><loc_62></location>0 b ℓm = √ ( ℓ -1) ℓ ( ℓ +1)( ℓ +2) 2 √ 2 π ∆ r 15 a 3 ( GM ) 2 +8 a 2 GMr 0 -6 r 2 0 5 a 3 ( GM ) 2 +4 a 2 GMr 0 -6 r 2 0 0 Y ℓm ( π 2 , 0) × r 2 0 E, -1 b ℓm = 2 √ ( ℓ -1)( ℓ +2) √ 2 πr 0 5 a 3 ( GM ) 2 +3 a 2 GMr 0 -3 r 2 0 ( 5 a 3 ( GM ) 2 +4 a 2 GMr 0 -6 r 2 0 ) 3 -1 Y ℓm ( π 2 , 0) · P r · L, -2 b ℓm = ∆ r √ 2 πr 4 E -1 Y ℓm ( π 2 , 0) L 2 , P r = 75 a 2 3 ( GM ) 4 +10 a 3 ( GM ) 2 r 0 ( 11 a 2 GM -18 r 0 ) +4 r 2 0 ( 8 a 2 2 ( GM ) 2 -21 a 2 GMr 0 +9 r 2 0 ) ,</formula>
<formula><location><page_14><loc_12><loc_43><loc_88><loc_45></location>B r = r 0 ∆ r ( 8 a 2 r 0 +30 a 3 GM ) +6 a 3 r 2 0 ( GM ) 2 -8 a 2 a 3 r 0 ( GM ) 3 -15 a 2 3 ( GM ) 4 , (3.50)</formula>
<text><location><page_14><loc_12><loc_39><loc_20><loc_40></location>we obtain</text>
<formula><location><page_14><loc_23><loc_6><loc_88><loc_37></location>A 0 = 1 2 r 2 0 · { 2 0 b ℓm +4 i -1 b ℓm [ 1 + i 2 ω r 3 0 ∆ r ( 1 + 2 GMr 0 P r × ( 6 a 2 r 2 0 +30 a 3 GMr 0 -5 a 2 a 3 ( GM ) 2 ) )] -2 i -2 b ℓm ωr 3 0 ∆ 2 r [ r 2 0 -GMr 0 + GM ×B r 15 a 3 ( GM ) 2 +8 a 2 GMr 0 -6 r 2 0 + 1 2 iωr 3 0 + 6 i GM ∆ 2 r · ( 2 a 2 r 2 0 +15 a 3 GMr 0 -5 a 2 a 3 ( GM ) 2 ) r 2 0 ω ( 15 a 3 ( GM ) 2 +8 a 2 GMr 0 -6 r 2 0 ) 2 ]} , (3.51) A 1 = i -1 b ℓm r 0 (1 + 2 GMr 0 ( 6 a 2 r 2 0 +30 a 3 GMr 0 -5 a 2 a 3 ( GM ) 2 ) P r ) --2 b ℓm r 0 (1 + GM · (15 a 3 GM +4 a 2 r 0 ) 15 a 3 ( GM ) 2 +8 a 2 GMr 0 -6 r 2 0 + i r 3 0 ω ∆ r ) , (3.52) A 2 = --2 b ℓm 2 . (3.53)</formula>
<section_header_level_1><location><page_15><loc_12><loc_89><loc_83><loc_91></location>IV. ENERGY RADIATION RATE AND GRAVITATIONAL WAVEFORMS</section_header_level_1>
<text><location><page_15><loc_12><loc_82><loc_88><loc_86></location>Inserting the aforementioned result of A i ( i = 1 , 2 , 3), B in ℓ , and R in ℓ into Eq. (3.8), we can obtain the following expression:</text>
<formula><location><page_15><loc_28><loc_78><loc_88><loc_80></location>Z ℓmω 0 = Z ( N,ϵ ) ℓmω 0 ˜ Z ℓmω 0 , (4.1)</formula>
<formula><location><page_15><loc_28><loc_74><loc_88><loc_78></location>Z ( N,ϵ ) ℓmω 0 =( -1) ℓ + ϵ +1 m 2 ν 2 M n ( ϵ ) ℓm v ℓ + ϵ +6 0 Y ℓ -ϵ, -m ( π/ 2 , φ ) , (4.2)</formula>
<formula><location><page_15><loc_30><loc_68><loc_88><loc_74></location>n ( ϵ ) ℓm = ( im ) ℓ 8 π (2 ℓ +1)!! √ ( ℓ +1)( ℓ +2) ℓ ( ℓ -1) , ϵ = 0 , -( im ) ℓ 16 π (2 ℓ +1)!! √ (2 ℓ +1)( ℓ +2)( ℓ 2 -m 2 ) (2 ℓ -1)( ℓ +1) ℓ ( ℓ -1) , ϵ = 1 . (4.3)</formula>
<text><location><page_15><loc_12><loc_59><loc_88><loc_67></location>where we define v = ( GM Ω) 1 / 3 , ω 0 = m Ω, ϵ = 1 when ℓ + m = 1, and ϵ = 0 when ℓ + m = 0. We can divide the higher-order term ˜ Z ℓmω 0 into two parts: the ˜ Z ( S ) ℓmω 0 is computed in the Schwarzschild case [64] and the ˜ Z ( R ) ℓmω 0 is the 2PM and 3PM perturbation terms:</text>
<formula><location><page_15><loc_40><loc_56><loc_88><loc_58></location>˜ Z ℓmω 0 = ˜ Z ( S ) ℓmω 0 + ˜ Z ( R ) ℓmω 0 . (4.4)</formula>
<text><location><page_15><loc_12><loc_47><loc_88><loc_54></location>The explicit expression of ˜ Z ( R ) ℓmω 0 is presented in Appendix C. In the test particle limit, i.e., ν → 0, we note that ˜ Z ( R ) ℓmω 0 vanishes completely because a 2 and a 3 approach 0. That is, our results revert to the Schwarzschild case in the test particle limit.</text>
<text><location><page_15><loc_12><loc_36><loc_90><loc_46></location>In Eq. (3.9), utilizing the symmetry of the spin-weighted spherical harmonics, s Y ℓ, -m ( π 2 , 0) = ( -1) s + ℓ s Y ℓm ( π 2 , 0), we know that Z ℓ ( -m ) ω = ( -1) ℓ Z ∗ ℓmω , where Z ∗ ℓmω is the complex conjugate of Z ℓmω . In terms of the amplitude Z ℓmω , we find from Eq. (3.9) that the gravitational waveform [13, 27, 65] at infinity is given by</text>
<formula><location><page_15><loc_34><loc_31><loc_88><loc_35></location>h + -ih × = ∑ ℓm h ℓm -2 Y ℓm √ 2 π e iω 0 ( r ∗ -t )+ imφ , (4.5)</formula>
<text><location><page_15><loc_12><loc_28><loc_15><loc_30></location>with</text>
<text><location><page_15><loc_12><loc_21><loc_17><loc_22></location>where</text>
<formula><location><page_15><loc_36><loc_23><loc_88><loc_27></location>h ℓm = -2 R ω 2 0 Z ℓmω 0 = h ( S ) ℓm + h ( R ) ℓm , (4.6)</formula>
<formula><location><page_15><loc_31><loc_17><loc_88><loc_19></location>h ( S ) ℓm = h ( N,ϵ ) ℓm ˜ Z ( S ) ℓmω 0 , h ( R ) ℓm = h ( N,ϵ ) ℓm ˜ Z ( R ) ℓmω 0 , (4.7)</formula>
<formula><location><page_15><loc_31><loc_13><loc_88><loc_16></location>h ( N,ϵ ) ℓm = GMν R n ( ϵ ) ℓm c ℓ + ϵ ( ν ) v ℓ + ϵ 0 Y ℓ -ϵ, -m ( π/ 2 , φ ) . (4.8)</formula>
<text><location><page_15><loc_12><loc_7><loc_88><loc_12></location>The energy loss rate along any orbit, in polar coordinates, can be expressed as dE [ g eff µν ] dt = ˙ R F R [ g eff µν ] + ˙ φ F φ [ g eff µν ]. By simply replacing the radial component with zero, an excellent</text>
<text><location><page_16><loc_12><loc_84><loc_88><loc_91></location>approximation of the radiation reaction forces can be obtained [66]. Thus, from Eq. (4.5), we know that, for given energy ω n , the energy loss rate [13, 27] for the 'plus' and 'cross' modes of the gravitational wave is described by the following expression:</text>
<formula><location><page_16><loc_36><loc_79><loc_88><loc_83></location>dE [ g eff µν ] dt = 1 2 ( d E d t ) N ∞ ∑ ℓ =2 ℓ ∑ m =1 Π ℓm , (4.9)</formula>
<text><location><page_16><loc_12><loc_76><loc_15><loc_78></location>with</text>
<formula><location><page_16><loc_23><loc_73><loc_88><loc_75></location>Π ℓm = E ( S ) ℓm + E ( R ) ℓm , (4.10)</formula>
<formula><location><page_16><loc_23><loc_68><loc_88><loc_72></location>E ( S ) ℓm = | Z ( N,ϵ ) ℓmω 0 ˜ Z ( S ) ℓmω 0 | 2 2 πG 2 ω 2 0 ( d E d t ) N , (4.11)</formula>
<formula><location><page_16><loc_23><loc_63><loc_88><loc_67></location>E ( R ) ℓm = | Z ( N,ϵ ) ℓmω 0 | 2 2 πG 2 ω 2 0 ( d E d t ) N { ˜ Z ( S ) ℓmω 0 ˜ Z ( R ) ∗ ℓmω 0 + ˜ Z ( R ) ℓmω 0 ˜ Z ( S ) ∗ ℓmω 0 + ˜ Z ( R ) ℓmω 0 ˜ Z ( R ) ∗ ℓmω 0 } . (4.12)</formula>
<text><location><page_16><loc_12><loc_55><loc_88><loc_62></location>where (d E/ d t ) N = 32 ν 2 v 10 / 5 is the Newtonian quadrupole luminosity and the superscript ∗ denotes the complex conjugation of the corresponding expression. In Figure 1, we present the curves of Π 22 and Π 33 as the symmetric mass ratio ν takes different values.</text>
<figure>
<location><page_16><loc_11><loc_31><loc_89><loc_54></location>
<caption>Figure 1: Π ℓm = Π ℓm ( v 2 , ν ) with respect to v 2 and ν , where the curve of ν = 0 corresponds to the 4.5PN result obtained for the Schwarzschild case. The figure on the left is for Π 22 = Π 22 ( v 2 , ν ), and the right one is for Π 33 = Π 33 ( v 2 , ν )</caption>
</figure>
<text><location><page_16><loc_12><loc_12><loc_88><loc_16></location>Then, we determine the radiation reaction forces for the 'plus' and 'cross' modes of the gravitational wave as follows:</text>
<formula><location><page_16><loc_24><loc_7><loc_88><loc_11></location>F circ φ [ g eff µν ] ≃ 1 ˙ φ dE [ g eff µν ] dt = 1 2 ( d E d t ) N ∞ ∑ ℓ =2 ℓ ∑ m =1 ( F ( S ) ℓm + F ( R ) ℓm ) , (4.13)</formula>
<formula><location><page_17><loc_12><loc_89><loc_46><loc_91></location>where F ( S ) ℓm = E ( S ) ℓm / 1 ˙ φ and F ( R ) ℓm = E ( R ) ℓm / 1 ˙ φ .</formula>
<text><location><page_17><loc_12><loc_84><loc_88><loc_88></location>Eqs.(4.5), (4.9), and (4.13) indicate that all of the gravitational waveforms, the energy radiation rate, and the radiation reaction forces are based on the effective spacetime.</text>
<section_header_level_1><location><page_17><loc_12><loc_78><loc_31><loc_79></location>V. CONCLUSIONS</section_header_level_1>
<text><location><page_17><loc_12><loc_49><loc_88><loc_74></location>In this paper, we investigate the waveforms and energy radiation rate of gravitational waves generated by coalescing spinless binary systems up to 3PM approximation in the EOB theory. We focus on the radiation reaction forces in the Hamilton equation, which can be described by the energy radiation rate d E d t = 1 4 πGω 2 ∫ | Ψ B 4 | 2 r 2 dΩ and the 'plus' and 'cross' modes of gravitational waves, which are related to the null tetrad components of the gravitational perturbed Weyl tensor by Ψ B 4 = 1 2 ( h + -i h × ). Clearly, to find the energy radiation rate and construct gravitational waveforms, the key step is to seek the solution of Ψ B 4 . Therefore, the main task of this paper is to solve the decoupled and separated equations of the null tetrad components of gravitational perturbed Weyl tensor Ψ B 4 in the effective spacetime by employing the Green's function method.</text>
<text><location><page_17><loc_12><loc_15><loc_88><loc_48></location>To achieve this goal, noting that the potential function in the radial Teukolsky-like equation is a long-range potential, we first transform it into an S-N-like equation, which has a short-range potential. Then, by expanding the homogeneous S-N-like equation with η = 2 GMω , where ω denotes the angular frequency of the wave, we derive closed analytical expressions for the solutions of each order. These solutions are essential for constructing the Green function and asymptotic amplitude. The lowest-order solution is expressed as a linear combination of spherical Bessel functions, which allows us to perform iterative calculations to obtain higher-order solutions. This approach simplifies the problem and enables us to efficiently study the radial Teukolsky-like equation. In the calculation process, we use a low-frequency approximation and considered the conditions of quasi-circular orbits. These conditions are represented by the relationships z ∝ v , η ∝ v 3 and ω = m Ω. As a result, the obtained results are accurate to O ( v 9 -2( ℓ -2) -ϵ ), which means that the accuracy of the results of this paper reaches the 4.5PN order[67, 68].</text>
<text><location><page_17><loc_12><loc_7><loc_88><loc_14></location>This work also presents a more general integral formula than that given by Sasaki [60], which can be extended to higher orders or even arbitrary orders without additional treatments. In Appendix B, the general integral formulas, which can theoretically derive the</text>
<text><location><page_18><loc_12><loc_71><loc_88><loc_91></location>series solution of the homogeneous S-N-like equation to any order, are presented. However, when constructing the general solution of the nonhomogeneous equation using Green's function method, it is necessary to specify the amplitude at infinity, which requires finding the asymptotic behavior of B J as z → ∞ . Although we know what needs to be done at each step, we have not yet been able to implement our ideas using a computer, but we can obtain specific results through complex calculations. Therefore, combining the outstanding works of Sasaki et al. with the useful formulas presented in Appendix B, we have confidence that this method will yield good results in the future.</text>
<text><location><page_18><loc_12><loc_50><loc_88><loc_70></location>From the analysis provided, it is evident that the effective metric degenerates into the Schwarzschild case in the test particle limit ( ν → 0). This limit is characterized by vanishing of the coefficients a 2 and a 3 . Therefore, the gravitational waveforms and energy radiation rate calculated in this study were divided into two parts: the Schwarzschild part and the correction part related to PM parameters a 2 and a 3 . Handling spinning binary systems will involve additional complexities. However, the results of this paper and the listed mathematical techniques will be valuable for understanding the energy flux and waveforms in spinning binary systems.</text>
<section_header_level_1><location><page_18><loc_14><loc_44><loc_37><loc_45></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_18><loc_12><loc_32><loc_88><loc_41></location>We would like to thank professors S. Chen and Q. Pan for useful discussions on the manuscript. This work was supported by the Grant of Natural Science Foundation of China No. 12035005, and the National Key Research and Development Program of China No. 2020YFC2201400.</text>
<section_header_level_1><location><page_19><loc_12><loc_89><loc_28><loc_91></location>VI. APPENDIX</section_header_level_1>
<section_header_level_1><location><page_19><loc_14><loc_85><loc_49><loc_86></location>A. The definition of coefficient of L ( n )</section_header_level_1>
<text><location><page_19><loc_14><loc_80><loc_52><loc_82></location>Associated with L (2) are expressed as follows</text>
<formula><location><page_19><loc_14><loc_75><loc_88><loc_79></location>a (2) ℓ = -36 a 2 +8 λa 2 2 +3 λ [ 5 a 3 -6( λ +2) ( c 2 1 +( c 1 + c 2 )( c 2 -1) ) ] 9 λ ( λ +2) , (A.1)</formula>
<formula><location><page_19><loc_14><loc_71><loc_88><loc_74></location>b (2) ℓ = 4 a 2 2 λ ( λ +4) + 5 a 3 λ (2 λ +7) -4 a 2 ( λ 2 +2 λ +6) 6 λ ( λ +2) , (A.2)</formula>
<formula><location><page_19><loc_15><loc_67><loc_88><loc_70></location>c (2) ℓ = 3 λa 2 2 +60 λa 3 +9 a 2 ( λ 2 +2 λ +16) 36 λ ( λ +2) , (A.3)</formula>
<formula><location><page_19><loc_14><loc_62><loc_88><loc_66></location>d (2) ℓ = 8 a 2 2 λ ( λ +4) + 15 a 3 λ ( λ +4) -24 a 2 ( λ 2 +2 λ +12) 36 λ ( λ +2) , (A.4)</formula>
<formula><location><page_19><loc_15><loc_58><loc_88><loc_62></location>e (2) ℓ = -8 a 2 2 λ ( λ 2 +19 λ +22) -15 a 3 λ (2 λ 2 +27 λ +34) + 12 a 2 (23 λ 2 +34 λ -48) 72 λ ( λ +2) . (A.5)</formula>
<text><location><page_19><loc_14><loc_56><loc_32><loc_57></location>Related with L (3) are</text>
<formula><location><page_19><loc_13><loc_51><loc_88><loc_55></location>a (3) ℓ = -2( λa 2 -9)(36 a 2 +8 a 2 2 λ +15 a 3 λ ) 27 λ 2 (2 + λ ) 2 (A.6)</formula>
<formula><location><page_19><loc_13><loc_47><loc_73><loc_50></location>b (3) ℓ = -1 18 λ 2 ( λ +2) 2 { 16 a 3 2 λ (2 λ +5) -8 a 2 2 λ (2 λ 2 -5 λ +6) -3 a 2 [ 3 λ 4 +</formula>
<formula><location><page_19><loc_18><loc_41><loc_88><loc_46></location>(12 -35 a 3 ) λ 3 +(36 -80 a 3 ) λ 2 +48 λ -144 ] +3 λ [ -a 3 ( 10 λ 2 -55 λ -60 ) + 12( c 1 -1)( c 2 -1)( c 1 + c 2 ) λ ( λ +2) 2 ]} , (A.7)</formula>
<formula><location><page_19><loc_13><loc_37><loc_80><loc_40></location>c (3) ℓ = 1 216 λ ( λ +2) { 16 a 3 2 λ (7 λ +62) -9 a 3 λ (37 λ +224) -36 a 2 2 ( 3 λ 2 +26 λ -24 ) +</formula>
<formula><location><page_19><loc_18><loc_34><loc_88><loc_36></location>90 a 2 ( 24 + a 3 λ (5 λ +38) ) } , (A.8)</formula>
<formula><location><page_19><loc_13><loc_29><loc_88><loc_33></location>d (3) ℓ = -2( λa 2 -9)(36 a 2 +8 a 2 2 λ +15 a 3 λ ) 27 λ 2 ( λ +2) 2 , (A.9)</formula>
<formula><location><page_19><loc_83><loc_19><loc_88><loc_20></location>(A.10)</formula>
<formula><location><page_19><loc_13><loc_19><loc_84><loc_29></location>e (3) ℓ = 1 216 λ 2 ( λ +2) 2 { -96 a 2 2 λ 2 (2 λ -5) + 64 a 3 2 λ 2 (6 λ +13) + 12 a 2 [ 9 λ 4 -144 λ +864+ 3(35 a 3 +12) λ 3 +4(55 a 3 -9) λ 2 ] +9 λ [ a 3 (3 λ 3 -28 λ 2 +232 λ +480)+ 48 ( c 2 1 +( c 2 -1)( c 1 + c 2 ) ) λ ( λ +2) 2 ]} ,</formula>
<formula><location><page_19><loc_13><loc_7><loc_88><loc_18></location>f (3) ℓ = 1 108 λ 2 ( λ +2) 2 { 8 a 3 2 λ 2 ( 3 λ 2 +44 λ +64 ) -24 a 2 2 λ ( 3 λ 3 +23 λ 2 +34 λ -48 ) + 3 a 2 [ 3 λ 4 ( 10 a 3 -8( c 2 1 +( c 2 -1)( c 1 + c 2 )) -3 ) +576 λ -1728 + λ 3 ( 370 a 3 -12 ( 8( c 2 1 + ( c 2 -1)( c 1 + c 2 )) + 3 ) ) + λ 2 ( 560 a 3 -12 ( 8( c 2 1 +( c 2 -1)( c 1 + c 2 )) -39 ) )]</formula>
<formula><location><page_20><loc_13><loc_80><loc_88><loc_91></location>-9 λ [ 5 a 3 ( 4 λ 3 +27 λ 2 +80 λ +48 ) -12( c 1 -1)( c 2 -1)( c 1 + c 2 ) λ ( λ +2) 2 ] } , (A.11) g (3) ℓ = 1 216 λ ( λ +2) { -8 a 3 2 λ ( λ 2 +62 λ +48 ) +12 a 2 2 ( 31 λ 2 +2 λ +192 ) -3 a 2 ( 15 a 3 λ 3 +650 a 3 λ 2 +8(65 a 3 -18) λ -1440 ) +72 a 3 λ (23 λ +16) } . (A.12)</formula>
<text><location><page_20><loc_12><loc_76><loc_33><loc_77></location>where λ = ( ℓ -1)( ℓ +2).</text>
<section_header_level_1><location><page_20><loc_14><loc_70><loc_40><loc_72></location>B. How to calculate the ξ ( n ) ℓ</section_header_level_1>
<figure>
<location><page_20><loc_20><loc_41><loc_80><loc_66></location>
<caption>Figure 2: The figure shows how to calculate the integral in Eq. (3.26). ζ i and ζ ∗ i stands for j i or n i , i is an arbitrary integer. 1 involves transforming the integrand using the properties of spherical Bessel functions, while 2 yields different integral expressions depending on the type of Q ( z ).</caption>
</figure>
<text><location><page_20><loc_12><loc_22><loc_88><loc_26></location>The integral appearing in Eq. (3.26) involves spherical Bessel functions with different quantum numbers. By utilizing the following formula,</text>
<formula><location><page_20><loc_35><loc_17><loc_88><loc_20></location>2 ℓ +1 z ζ ℓ = ζ ℓ -1 + ζ ℓ +1 , ζ is j or n, (B.1)</formula>
<formula><location><page_20><loc_35><loc_14><loc_88><loc_16></location>n ℓ = ( -1) ℓ +1 j -ℓ -1 , (B.2)</formula>
<formula><location><page_20><loc_35><loc_11><loc_88><loc_13></location>j ℓ = ( -1) ℓ n -ℓ -1 , (B.3)</formula>
<formula><location><page_20><loc_35><loc_7><loc_88><loc_10></location>j 0 = sin z z , n 0 = -cos z z . (B.4)</formula>
<text><location><page_21><loc_12><loc_76><loc_88><loc_91></location>we can express any quantum number ℓ of j ℓ and n ℓ in the form of poly1 ( z ) j 0 + poly2 ( z ) n 0 . poly1 ( z ) and poly2 ( z ) are both polynomials in 1 /z , while j 0 and n 0 have elementary function representations. This transformation allows for easier computation of the integral expression. Based on our research in mathematical manuals and insights from Sasaki et al. [60], we have classified these integrals and derived formal integral formulas, as depicted in Fig. 2, outlining the specific implementation route.</text>
<section_header_level_1><location><page_21><loc_14><loc_70><loc_82><loc_72></location>1. Generalized sine integral function B J and generalized spherical Bessel function D J ℓ</section_header_level_1>
<text><location><page_21><loc_12><loc_63><loc_88><loc_67></location>We first introduce some special integral formulas from Ref. [60], the generalized sine integral function B J</text>
<formula><location><page_21><loc_42><loc_58><loc_88><loc_62></location>B jJ = ∫ z z ∗ zj 0 D J 0 d z, (B.5)</formula>
<formula><location><page_21><loc_41><loc_54><loc_88><loc_58></location>B nJ = ∫ z z ∗ zn 0 D J 0 d z. (B.6)</formula>
<text><location><page_21><loc_12><loc_51><loc_54><loc_52></location>And the generalized spherical Bessel functions D J ℓ :</text>
<formula><location><page_21><loc_42><loc_46><loc_88><loc_48></location>D j ℓ = j ℓ , D n ℓ = n ℓ , (B.7)</formula>
<formula><location><page_21><loc_41><loc_43><loc_88><loc_45></location>D nJ ℓ = n ℓ B jJ -j ℓ B nJ , (B.8)</formula>
<formula><location><page_21><loc_41><loc_40><loc_88><loc_42></location>D jJ ℓ = j ℓ B jJ + n ℓ B nJ . (B.9)</formula>
<text><location><page_21><loc_12><loc_37><loc_81><loc_38></location>From the above definitions, we can derive the following expressions when J = j, n :</text>
<formula><location><page_21><loc_36><loc_31><loc_88><loc_35></location>B jj = ∫ z 0 zj 0 j 0 d z = -1 2 C ( z ) , (B.10)</formula>
<formula><location><page_21><loc_36><loc_27><loc_88><loc_31></location>B nj = ∫ z 0 zn 0 j 0 d z = -1 2 S ( z ) , (B.11)</formula>
<formula><location><page_21><loc_36><loc_23><loc_88><loc_27></location>B jn = ∫ z 0 zj 0 n 0 d z = -1 2 S ( z ) , (B.12)</formula>
<formula><location><page_21><loc_36><loc_19><loc_88><loc_23></location>B nn = ∫ z z ∗ zn 0 n 0 d z = -B jj +ln z, (B.13)</formula>
<formula><location><page_21><loc_30><loc_11><loc_88><loc_15></location>S ( z ) = Si (2 z ) = ∫ 2 z 0 sin t t d t ( | z | < ∞ ) , (B.14)</formula>
<formula><location><page_21><loc_30><loc_7><loc_88><loc_11></location>C ( z ) = -∫ ∞ 2 z cos t t d t -γ -ln 2 z | arg z | < π. (B.15)</formula>
<text><location><page_21><loc_12><loc_16><loc_17><loc_18></location>where</text>
<text><location><page_22><loc_12><loc_81><loc_88><loc_91></location>B J is essential to note that when J ends with j , there may be a logarithmic divergence issue when the integrand is combined with n 0 . This occurs because we use n 2 0 = 1 z 2 -j 2 0 , which results in 1 z 2 diverging when z ∗ = 0. Therefore, the terms related to j 2 0 have their lower limit set at z ∗ = 0, while those related to 1 z 2 are set at z ∗ = 1.</text>
<text><location><page_22><loc_12><loc_76><loc_88><loc_80></location>It is worth noting that all strings J ending with the character n in B J can be expressed in terms of those whose J end with j , for example,</text>
<formula><location><page_22><loc_38><loc_72><loc_88><loc_73></location>B jn = B nj , (B.16)</formula>
<formula><location><page_22><loc_37><loc_69><loc_88><loc_70></location>B jjn = -B nnj + B nj ln z, (B.17)</formula>
<formula><location><page_22><loc_37><loc_65><loc_88><loc_67></location>B jnn =2 B jjj + B nnj -B jj ln z, (B.18)</formula>
<formula><location><page_22><loc_37><loc_62><loc_88><loc_64></location>B nnn =2 B njj -B jnj -B nj ln z. (B.19)</formula>
<text><location><page_22><loc_12><loc_56><loc_88><loc_60></location>Using these relations, we can express all the D J ℓ whose J end with n in terms of those whose J end with j .</text>
<figure>
<location><page_22><loc_12><loc_35><loc_88><loc_54></location>
<caption>Figure 3: The image of B J for J of length 2 is given on the left, and the image of B J for J of length 3 is given on the right.</caption>
</figure>
<section_header_level_1><location><page_22><loc_14><loc_20><loc_43><loc_21></location>2. the general integral form of Q ( z )</section_header_level_1>
<text><location><page_22><loc_12><loc_13><loc_88><loc_17></location>For k > -1 and k ∈ N , with Q ( z ) being differentiable functions, we can establish the following integral properties:</text>
<formula><location><page_22><loc_14><loc_7><loc_88><loc_11></location>∫ Q ( z ) z k j 2 0 d z = 2 1 + k { -∫ Q ( z ) z k -1 n 0 j 0 d z -1 2 Q ( z ) z k -1 j 2 0 + 1 2 ∫ Q ( z ) ' z k -1 j 2 0 d z } , (B.20)</formula>
<formula><location><page_23><loc_14><loc_87><loc_74><loc_91></location>∫ Q ( z ) z k n 2 0 d z = 2 1 + k { ∫ Q ( z ) z k -1 n 0 j 0 d z -1 2 Q ( z ) z k -1 n 2 0 + 1 2 ∫ Q ( z ) ' z k -1 n 2 0 d z } ,</formula>
<formula><location><page_23><loc_14><loc_75><loc_88><loc_89></location>(B.21) ∫ Q ( z ) z k n 0 j 0 d z = 2 1+ k { ∫ Q ( z ) z k -1 j 2 0 d z -1 2 ∫ Q ( z ) z k +1 d z -1 2 Q ( z ) z k -1 n 0 j 0 + 1 2 ∫ Q ( z ) ' z k -1 n 0 j 0 d z } , 2 1+ k { 1 2 ∫ Q ( z ) z k +1 d z -∫ Q ( z ) z k -1 n 2 0 d z -1 2 Q ( z ) z k -1 n 0 j 0 + 1 2 ∫ Q ( z ) ' z k -1 n 0 j 0 d z } . (B.22)</formula>
<text><location><page_23><loc_12><loc_66><loc_88><loc_73></location>and for k = -1, we define ∫ zQ ( z ) ζ 0 ζ ∗ 0 d z as a new function. This enables us to establish a complete recursive relationship, allowing us to obtain the desired expressions by letting the computer perform a finite number of iterations for a specific level.</text>
<formula><location><page_23><loc_14><loc_60><loc_51><loc_62></location>3. the case of Q ( z ) = (ln z ) m , ( m ⩾ 0 , m ∈ N )</formula>
<text><location><page_23><loc_12><loc_51><loc_89><loc_58></location>Defining E m k = ∫ j 2 0 (ln z ) m z k d z and F m k = ∫ j 0 n 0 (ln z ) m z k d z , m = 0 , 1 , 2 , . . . and k = -1 , 0 , 1 , 2 , . . . , we can derive the following expressions utilizing the integral formulas (B.20)-(B.22) mentioned above:</text>
<formula><location><page_23><loc_31><loc_42><loc_88><loc_49></location>-1 m k ' = ⇒ E m -1 , E m k ' k ' = 0 , 1 , 2 , . . . (B.23)</formula>
<formula><location><page_23><loc_31><loc_30><loc_88><loc_41></location>-1 m 0 k '' = ⇒ F m -1 , F m 0 , F m k '' k '' = 1 , 2 , . . . (B.24)</formula>
<formula><location><page_23><loc_18><loc_22><loc_88><loc_24></location>E m -1 = GenlogS j m , F m -1 = GenlogU j m , (B.25)</formula>
<formula><location><page_23><loc_18><loc_18><loc_88><loc_21></location>F m 0 = 2 { E m -1 -1 2( m +1) (ln z ) m +1 -1 2 zn 0 j 0 (ln z ) m + m 2 F m -1 0 } , (B.26)</formula>
<formula><location><page_23><loc_18><loc_14><loc_88><loc_18></location>E m k ' = 2 1 + k ' { -F m k ' -1 -1 2 j 2 0 (ln z ) m z k ' -1 + m 2 E m -1 k ' } , (B.27)</formula>
<formula><location><page_23><loc_18><loc_10><loc_88><loc_14></location>F m k '' = 2 1 + k '' { E m k '' -1 + 1 2 ei k '' ln z -m (ln z ) m +1 -1 2 n 0 j 0 (ln z ) m z k '' -1 + m 2 F m -1 k '' } , (B.28)</formula>
<text><location><page_23><loc_12><loc_6><loc_60><loc_9></location>ei z n is an exponential integral of order n , ei z n = ∫ ∞ 1 e -zt t n d t .</text>
<text><location><page_23><loc_12><loc_27><loc_17><loc_28></location>where</text>
<formula><location><page_24><loc_14><loc_89><loc_65><loc_91></location>4. the case of Q ( z ) = (ln z ) m B jJ or (ln z ) m B nJ , ( m ⩾ 0 , m ∈ N )</formula>
<text><location><page_24><loc_12><loc_82><loc_88><loc_87></location>Defining S m,J k = -∫ j 0 D nJ 0 (ln z ) m z k d z and U m,J k = -∫ n 0 D nJ 0 (ln z ) m z k d z , we can derive the following expressions utilizing the integral formulas mentioned above:</text>
<formula><location><page_24><loc_18><loc_69><loc_88><loc_80></location>-1 m j 0 J ' k '' = ⇒ S m,J -1 , S m,J 0 , S m,J k '' U m,J -1 , U m,J 0 , U m,J k '' = ⇒ Π m,jJ 1 , Π m,jJ ˜ k Π m,nJ 1 , Π m,nJ ˜ k (B.29)</formula>
<text><location><page_24><loc_12><loc_58><loc_88><loc_67></location>where J ' is a string of length greater than two, and ˜ k = 2 , 3 , 4 , . . . , the terms S m,J 0 , S m,J k '' can actually be merged into a single term S m,J k ' . The same holds true for U m,J 0 , U m,J k '' , as the expressions are entirely determined by Π m,J k '' due to the parameters k ' and J . In the following, we provide the corresponding expressions.</text>
<formula><location><page_24><loc_23><loc_53><loc_88><loc_55></location>S m,J -1 = -GenlogS nJ m , U m,J -1 = -GenlogU nJ m , (B.30)</formula>
<formula><location><page_24><loc_23><loc_49><loc_88><loc_53></location>S m,J k ' = 2 1 + k ' { 1 2 j 0 D nJ 0 (ln z ) m z k ' -1 -1 2 Π m,jJ k ' +1 -U m,J k ' -1 + m 2 S m -1 ,J k ' } , (B.31)</formula>
<formula><location><page_24><loc_23><loc_45><loc_88><loc_49></location>U m,J k ' = 2 1 + k ' { S m,J k ' -1 -1 2 Π m,nJ k ' +1 + 1 2 n 0 D nJ 0 (ln z ) m z k ' -1 + m 2 U m -1 ,J k ' } , (B.32)</formula>
<text><location><page_24><loc_12><loc_42><loc_45><loc_44></location>The definitions of Π m,jJ k ' and Π m,nJ k ' are,</text>
<formula><location><page_24><loc_13><loc_34><loc_84><loc_40></location>Π m,jJ k ' = ∫ B jJ (ln z ) m z k ' d z = 1 1 -k ' ( B jJ (ln z ) m z k ' -1 -m Π m -1 ,jJ k ' -∫ j 0 D J 0 z k ' -2 (ln z ) m d z ) , k ' > 1 1 m +1 ( B jJ (ln z ) m +1 -GenlogS J m +1 ) , k ' = 1</formula>
<text><location><page_24><loc_83><loc_32><loc_88><loc_33></location>(B.33)</text>
<formula><location><page_24><loc_13><loc_22><loc_88><loc_31></location>Π m,nJ k ' = ∫ B nJ (ln z ) m z k ' d z = 1 1 -k ' ( B nJ (ln z ) m z k ' -1 -m Π m -1 ,nJ k ' -∫ n 0 D J 0 z k ' -2 (ln z ) m d z ) , k ' > 1 1 m +1 ( B nJ (ln z ) m +1 -GenlogU J m +1 ) , k ' = 1 (B.34)</formula>
<text><location><page_24><loc_12><loc_16><loc_88><loc_20></location>We will then provide the specific expressions of Π m,jJ k ' and Π m,nJ k ' for different values of k ' and J :</text>
<formula><location><page_24><loc_13><loc_11><loc_88><loc_14></location>Π m,jj 1 = B jj (ln z ) m +1 -GenlogS j m +1 m +1 , Π m,nj 1 = B nj (ln z ) m +1 -GenlogU j m +1 m +1 , (B.35)</formula>
<formula><location><page_24><loc_13><loc_7><loc_88><loc_10></location>Π m,jjJ 1 = B jjJ (ln z ) m +1 -GenlogS jJ m +1 m +1 , Π m,njJ 1 = B njJ (ln z ) m +1 -GenlogU jJ m +1 m +1 , (B.36)</formula>
<formula><location><page_25><loc_13><loc_87><loc_88><loc_91></location>Π m,jnJ 1 = B jnJ (ln z ) m +1 -GenlogS nJ m +1 m +1 , Π m,nnJ 1 = B nnJ (ln z ) m +1 -GenlogU nJ m +1 m +1 , (B.37)</formula>
<formula><location><page_25><loc_13><loc_83><loc_88><loc_87></location>Π m,jj ˜ k = 1 1 -˜ k ( B jj (ln z ) m z ˜ k -1 -m Π m -1 ,jj ˜ k -E m ˜ k -2 ) , (B.38)</formula>
<formula><location><page_25><loc_13><loc_79><loc_88><loc_83></location>Π m,nj ˜ k = 1 1 -˜ k ( B nj (ln z ) m z ˜ k -1 -m Π m -1 ,nj ˜ k -F m ˜ k -2 ) , (B.39)</formula>
<formula><location><page_25><loc_13><loc_75><loc_88><loc_79></location>Π m,jjJ ˜ k = 1 1 -˜ k ( B jjJ (ln z ) m z ˜ k -1 -m Π m -1 ,jjJ ˜ k -Π m,jJ ˜ k -U m,J ˜ k -2 ) , (B.40)</formula>
<formula><location><page_25><loc_13><loc_71><loc_88><loc_75></location>Π m,njJ ˜ k = 1 1 -˜ k ( B njJ (ln z ) m z ˜ k -1 -m Π m -1 ,njJ ˜ k -Π m,nJ ˜ k + S m,J ˜ k -2 ) , (B.41)</formula>
<formula><location><page_25><loc_13><loc_67><loc_88><loc_71></location>Π m,jnJ ˜ k = 1 1 -˜ k ( B jnJ (ln z ) m z ˜ k -1 -m Π m -1 ,jnJ ˜ k + S m,J ˜ k -2 ) , (B.42)</formula>
<formula><location><page_25><loc_13><loc_63><loc_88><loc_67></location>Π m,nnJ ˜ k = 1 1 -˜ k ( B nnJ (ln z ) m z ˜ k -1 -m Π m -1 ,nnJ ˜ k + U m,J ˜ k -2 ) , (B.43)</formula>
<text><location><page_25><loc_12><loc_60><loc_54><loc_61></location>The definitions of GenlogS J m and GenlogU J m are,</text>
<formula><location><page_25><loc_18><loc_54><loc_88><loc_58></location>GenlogS J m = ∫ zj 0 D J 0 (ln z ) m d z = m ∑ i =0 ( -1) i m ! ( m -i )! (ln z ) m -i theB jJ i +1 , (B.44)</formula>
<formula><location><page_25><loc_17><loc_49><loc_88><loc_53></location>GenlogU J m = ∫ zn 0 D J 0 (ln z ) m d z = m ∑ i =0 ( -1) i m ! ( m -i )! (ln z ) m -i theB nJ i +1 , (B.45)</formula>
<text><location><page_25><loc_12><loc_46><loc_17><loc_47></location>where</text>
<formula><location><page_25><loc_14><loc_41><loc_88><loc_43></location>ϑ 1 [ A ] = { (1 , A ) } , (B.46)</formula>
<formula><location><page_25><loc_14><loc_28><loc_83><loc_40></location>ϑ n [ A ] 2 µ -1 , ϑ n [ A ] 2 µ = 1 . ( ϑ n -1 [ A ] µ 1 , jjϑ n -1 [ A ] µ 2 | -1 2 ) , ( ϑ n -1 [ A ] µ 1 , nnϑ n -1 [ A ] µ 2 | -1 2 ) , when ϑ n -1 [ A ] µ 2 | 1 1 = j, 2 . ( ϑ n -1 [ A ] µ 1 , njϑ n -1 [ A ] µ 2 | -1 2 ) , ( -ϑ n -1 [ A ] µ 1 , jnϑ n -1 [ A ] µ 2 | -1 2 ) , when ϑ n -1 [ A ] µ 2 | 1 1 = n.</formula>
<text><location><page_25><loc_83><loc_25><loc_88><loc_27></location>(B.47)</text>
<formula><location><page_25><loc_14><loc_22><loc_88><loc_24></location>thB jJ i = ϑ i [ jJ ] ν 1 B ϑ i [ jJ ] ν 2 , (B.48)</formula>
<formula><location><page_25><loc_14><loc_19><loc_88><loc_21></location>thB nJ i = ϑ i [ nJ ] ν 1 B ϑ i [ nJ ] ν 2 , (B.49)</formula>
<text><location><page_25><loc_12><loc_7><loc_88><loc_17></location>in this context, µ = 1 , 2 , . . . , dim[ ϑ n -1 [ A ]], ϑ n [ A ] ij represents the ( i, j )-th element of ϑ n [ A ], dim[ ϑ n ] denotes the dimension of ϑ n [ A ], ϑ n -1 [ A ] µ 2 | b a refers to the characters within the range from the a -th to the b -th in ϑ n -1 [ A ] µ 2 , and ϑ i [ A ] ν 1 B ϑ i [ A ] ν 2 implies summation over ν . We shall provide expressions for the first five terms of thB jJ i and thB nJ i , even though even these</text>
<text><location><page_26><loc_12><loc_89><loc_38><loc_91></location>initial terms are quite intricate:</text>
<formula><location><page_26><loc_13><loc_85><loc_88><loc_87></location>thB jJ 1 = B jJ , thB nJ 1 = B nJ , (B.50)</formula>
<formula><location><page_26><loc_13><loc_82><loc_88><loc_84></location>thB jJ 2 = B jjJ + B nnJ , thB nJ 2 = B njJ -B jnJ , (B.51)</formula>
<formula><location><page_26><loc_13><loc_79><loc_83><loc_81></location>thB jJ 3 = B jjjJ + B nnjJ + B njnJ -B jnnJ , thB nJ 3 = B njjJ -B jnjJ -B jjnJ -B nnnJ ,</formula>
<text><location><page_26><loc_83><loc_76><loc_88><loc_78></location>(B.52)</text>
<formula><location><page_26><loc_13><loc_73><loc_88><loc_75></location>thB jJ 4 = B jjjjJ + B nnjjJ + B njnjJ -B jnnjJ + B njjnJ -B jnjnJ -B jjnnJ -B nnnnJ , (B.53)</formula>
<formula><location><page_26><loc_13><loc_70><loc_88><loc_72></location>thB nJ 4 = B njjjJ -B jnjjJ -B jjnjJ -B nnnjJ -B jjjnJ -B nnjnJ -B njnnJ + B jnnnJ , (B.54)</formula>
<formula><location><page_26><loc_13><loc_67><loc_86><loc_69></location>thB jJ 5 = B jjjjjJ + B nnjjjJ + B njnjjJ -B jnnjjJ + B njjnjJ -B jnjnjJ -B jjnnjJ -B nnnnjJ +</formula>
<text><location><page_26><loc_20><loc_64><loc_22><loc_65></location>B</text>
<text><location><page_26><loc_22><loc_64><loc_26><loc_65></location>njjjnJ</text>
<text><location><page_26><loc_27><loc_64><loc_28><loc_65></location>-</text>
<text><location><page_26><loc_29><loc_64><loc_30><loc_65></location>B</text>
<text><location><page_26><loc_30><loc_64><loc_34><loc_65></location>jnjjnJ</text>
<text><location><page_26><loc_35><loc_64><loc_37><loc_65></location>-</text>
<text><location><page_26><loc_37><loc_64><loc_38><loc_65></location>B</text>
<text><location><page_26><loc_38><loc_64><loc_43><loc_65></location>jjnjnJ</text>
<text><location><page_26><loc_43><loc_64><loc_45><loc_65></location>-</text>
<text><location><page_26><loc_45><loc_64><loc_47><loc_65></location>B</text>
<text><location><page_26><loc_47><loc_64><loc_52><loc_65></location>nnnjnJ</text>
<text><location><page_26><loc_52><loc_64><loc_54><loc_65></location>-</text>
<text><location><page_26><loc_54><loc_64><loc_56><loc_65></location>B</text>
<text><location><page_26><loc_56><loc_64><loc_60><loc_65></location>jjjnnJ</text>
<text><location><page_26><loc_61><loc_64><loc_62><loc_65></location>-</text>
<text><location><page_26><loc_63><loc_64><loc_64><loc_65></location>B</text>
<text><location><page_26><loc_64><loc_64><loc_69><loc_65></location>nnjnnJ</text>
<text><location><page_26><loc_69><loc_64><loc_71><loc_65></location>-</text>
<text><location><page_26><loc_71><loc_64><loc_73><loc_65></location>B</text>
<text><location><page_26><loc_73><loc_64><loc_78><loc_65></location>njnnnJ</text>
<text><location><page_26><loc_78><loc_64><loc_80><loc_65></location>+</text>
<text><location><page_26><loc_80><loc_64><loc_82><loc_65></location>B</text>
<text><location><page_26><loc_82><loc_64><loc_86><loc_65></location>jnnnnJ</text>
<text><location><page_26><loc_86><loc_64><loc_87><loc_65></location>,</text>
<text><location><page_26><loc_83><loc_61><loc_88><loc_63></location>(B.55)</text>
<formula><location><page_26><loc_13><loc_53><loc_88><loc_60></location>thB nJ 5 = B njjjjJ -B jnjjjJ -B jjnjjJ -B nnnjjJ -B jjjnjJ -B nnjnjJ -B njnnjJ + B jnnnjJ -B jjjjnJ -B nnjjnJ -B njnjnJ + B jnnjnJ -B njjnnJ + B jnjnnJ + B jjnnnJ + B nnnnnJ . (B.56)</formula>
<section_header_level_1><location><page_26><loc_14><loc_46><loc_39><loc_48></location>C. the expressions of ˜ Z ( R ) ℓmω 0</section_header_level_1>
<text><location><page_26><loc_12><loc_39><loc_88><loc_44></location>We present the expression of ˜ Z ( R ) ℓmω 0 for mode ( ℓ, m ) = (2 , 2) , (2 , 1) , (3 , 3) in which each mode contains terms up to O ( v 9 -2( ℓ -2) -ϵ ).</text>
<formula><location><page_26><loc_12><loc_6><loc_83><loc_37></location>˜ Z ( R ) 22 ω 0 = a 2 v 2 +2 ia 2 v 3 + 308 a 2 2 -919 a 2 +420 a 3 378 v 4 + ( 41 ia 2 2 18 + a 2 ( 12 i ln v -238 i 27 + 4 i elg +2 π +12 i ln 2 ) + 11 ia 3 6 ) v 5 + ( 44 a 3 2 81 -( elg +ln2 v ) ( 256 a 2 2 945 + 152 a 2 21 + 40 a 3 63 ) + π ( 128 ia 2 2 945 + 76 ia 2 21 + 20 ia 3 63 ) +ln2 ( -256 a 2 2 945 -320 a 2 21 -40 a 3 63 ) -173707 a 2 2 66150 + 85 a 2 a 3 54 -16 a 2 ln v + 891691 a 2 158760 -4553 a 3 5292 ) v 6 + ( 505 ia 3 2 243 -54365 ia 2 2 6804 +( elg +ln2 v ) ( 88 ia 2 2 27 -1838 ia 2 189 + 40 ia 3 9 ) + ( 176 ia 2 2 27 -3676 ia 2 189 + 80 ia 3 9 ) ln 2 v + π ( 44 a 2 2 27 -919 a 2 189 + 20 a 3 9 ) + 403 ia 2 a 3 81 + 67805 ia 2 6804 -17005 ia 3 2268 ) v 7 + ( 2 a 4 2 27 -291169 a 3 2 132300 + 55 a 2 2 a 3 162 + 16150588 a 2 2 2083725 + ( -256 a 3 2 945 -513614 a 2 2 19845 -40 a 2 a 3 63 + 70328 a 2 735 -26966 a 3 1323 ) ln v + π ( 128 ia 3 2 945 + 75997 ia 2 2 19845 + 20 ia 2 a 3 63 -83236 ia 2 6615 + 3781 ia 3 1323 ) +</formula>
<text><location><page_27><loc_16><loc_41><loc_88><loc_91></location>elg ( -256 a 3 2 945 -151994 a 2 2 19845 -40 a 2 a 3 63 + 166472 a 2 6615 -7562 a 3 1323 ) +ln2 ( -512 a 3 2 945 -484798 a 2 2 19845 -80 a 2 a 3 63 + 188728 a 2 2205 -24826 a 3 1323 ) -1885 a 2 a 3 441 -72 a 2 (ln 2 v ) 2 + 24 iπa 2 ln 2 v -48 elg a 2 ln 2 v -8 elg 2 a 2 + 2 π 2 a 2 3 + 2319196003 a 2 366735600 +8 elg iπa 2 + 125 a 2 3 864 + 1009699 a 3 222264 ) v 8 + ( 352 ia 4 2 243 -8946247 ia 3 2 1786050 -( 1 315 1024 ia 2 2 + 944 ia 2 7 + 160 ia 3 21 ) (ln v ) 2 + ln 2 ( -1 105 1024 ia 2 2 -1824 ia 2 7 -160 ia 3 7 ) ln v + π ln v ( -2048 a 2 2 945 -880 a 2 21 -320 a 3 63 ) + elg ln v ( -1 945 4096 ia 2 2 -1760 ia 2 21 -640 ia 3 63 ) + 440 81 ia 2 2 a 3 -elg 2 ( 1024 ia 2 2 945 + 272 ia 2 21 + 160 ia 3 63 ) + π 2 ( 256 ia 2 2 567 + 4 ia 2 63 + 200 ia 3 189 ) -elg π ( 1024 a 2 2 945 + 272 a 2 21 + 160 a 3 63 ) -(ln 2) 2 ( 2048 ia 2 2 315 + 880 ia 2 7 + 320 ia 3 21 ) -π ln 2 ( 512 a 2 2 189 + 848 a 2 21 + 400 a 3 63 ) -260495 ia 2 a 3 23814 elg ln 2 ( 1024 ia 2 2 189 + 1696 ia 2 21 + 800 ia 3 63 ) + 1183663 ia 2 2 138915 + ( 17456 ia 3 2 2835 -2942726 ia 2 2 99225 + 3410 ia 2 a 3 189 + 716683 ia 2 13230 -11059 ia 3 1323 -5136 i 245 ) ln v + π ( 3016 a 3 2 2835 -339121 a 2 2 99225 + 635 a 2 a 3 189 + 14639 a 2 15876 + 191 a 3 294 ) + elg ( 6032 ia 3 2 2835 -678242 ia 2 2 99225 + 1270 ia 2 a 3 189 + 14639 ia 2 7938 + 191 ia 3 147 ) +ln2 ( 5776 ia 3 2 945 -2578726 ia 2 2 99225 + 3490 ia 2 a 3 189 + 443851 ia 2 13230 -4787 ia 3 1323 -5136 i 245 ) + 2503003411 ia 2 91683900 + 175 ia 2 3 108 -15511 ia 3 16464 ) v 9 , (C.1)</text>
<formula><location><page_27><loc_12><loc_6><loc_90><loc_38></location>˜ Z ( R ) 21 ω 0 = a 2 3 v 2 + 23 ia 2 36 v 3 + ( 5 a 2 2 72 -20 a 2 63 + a 3 12 ) v 4 + ( 101 ia 2 2 432 +2 ia 2 ln v + 2 ia 2 ( elg +ln4) 3 -37 ia 2 56 + πa 2 3 + ia 3 6 ) v 5 + ( 23 a 3 2 324 +( elg +ln2 v ) ( -64 a 2 2 945 -137 a 2 126 -10 a 3 63 ) + π ( 32 ia 2 2 945 + 137 ia 2 252 + 5 ia 3 63 ) + 139673 a 2 2 1587600 + 17 a 2 a 3 108 -23 9 a 2 ln v + 3709 a 2 158760 -23 18 a 2 ln 2 + 2143 a 3 5292 ) v 6 + ( 199 ia 3 2 2592 + ( 5 ia 2 2 36 -40 ia 2 63 + ia 3 6 )( ln 4 v 3 + elg ) + π ( 5 a 2 2 72 -20 a 2 63 + a 3 12 ) -3581 ia 2 2 18144 + 55 ia 2 a 3 216 + 5291 ia 2 4536 + 2533 ia 3 3024 ) v 7 + ( -( 64 a 3 2 2835 + 10 a 2 a 3 189 ) (ln 2 v + elg ) + 452071 a 3 2 9525600 -( 101 a 2 2 216 -37 a 2 28 + a 3 3 ) ln(2 v 2 ) + ( -6403 a 2 2 17640 + 4643 a 2 8820 -209 a 3 882 ) (ln 2 v + elg ) + 15354247 a 2 2 133358400 + π ( 32 ia 3 2 2835 + 6403 ia 2 2 35280 + 5 ia 2 a 3 189 -4643 ia 2 17640 + 209 ia 3 1764 ) + 7037 a 2 a 3 31752 -8 3 a 2 ln 2 ( ln 2 v 3 + elg -iπ 2 ) -6 a 2 (ln v ) 2 -4 ( elg -iπ 2 ) a 2 ln v +</formula>
<formula><location><page_28><loc_16><loc_87><loc_88><loc_91></location>π 2 a 2 18 -2 elg 2 a 2 3 + 159578843 a 2 122245200 + 2 i elg πa 2 3 + 3985 a 3 296352 ) v 8 . (C.2)</formula>
<formula><location><page_28><loc_12><loc_66><loc_90><loc_81></location>˜ Z ( R ) 33 ω 0 = 4 a 2 3 v 2 + 29 ia 2 12 v 3 + ( 4 a 2 2 3 -5 a 2 + 3 a 3 2 ) v 4 + ( 511 ia 2 2 144 +8 ia 2 ( ln(12 v 3 ) + elg -iπ 2 ) -2951 ia 2 120 + 47 ia 3 24 ) v 5 + ( 92 a 3 2 81 -( 10 a 2 2 21 + 191 a 2 14 + 15 a 3 14 )( ln 6 v + elg -iπ 2 ) -122987 a 2 2 21168 + 77 a 2 a 3 27 -29 2 a 2 ln 2 v 2 + 571751 a 2 21560 -16229 a 3 7056 ) v 6 + ( 1117 ia 3 2 288 -745 ia 3 32 + 10256 ia 2 165 + ( 8 ia 2 2 -30 ia 2 +9 ia 3 )( ln(12 v 3 ) + elg -iπ 2 ) -39121 ia 2 2 1440 + 353 ia 2 a 3 48 ) v 7 . (C.3)</formula>
<unordered_list>
<list_item><location><page_28><loc_13><loc_51><loc_88><loc_57></location>[1] B. P. Abbott et al. (LIGO Scientific, Virgo), GW151226: Observation of Gravitational Waves from a 22-Solar-Mass Binary Black Hole Coalescence, Phys. Rev. Lett. 116 , 241103 (2016), arXiv:1606.04855 [gr-qc].</list_item>
<list_item><location><page_28><loc_13><loc_42><loc_88><loc_49></location>[2] B. P. Abbott et al. (LIGO Scientific and Virgo Collaboration), Gw170104: Observation of a 50-solar-mass binary black hole coalescence at redshift 0.2, Phys. Rev. Lett. 118 , 221101 (2017).</list_item>
<list_item><location><page_28><loc_13><loc_34><loc_88><loc_41></location>[3] B. P. Abbott et al. (LIGO Scientific, Virgo), GW170814: A Three-Detector Observation of Gravitational Waves from a Binary Black Hole Coalescence, Phys. Rev. Lett. 119 , 141101 (2017), arXiv:1709.09660 [gr-qc].</list_item>
<list_item><location><page_28><loc_13><loc_26><loc_88><loc_33></location>[4] B. P. Abbott et al. (LIGO Scientific, Virgo), GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral, Phys. Rev. Lett. 119 , 161101 (2017), arXiv:1710.05832 [gr-qc].</list_item>
<list_item><location><page_28><loc_13><loc_18><loc_88><loc_25></location>[5] R. Abbott et al. (LIGO Scientific, Virgo), GW190412: Observation of a Binary-Black-Hole Coalescence with Asymmetric Masses, Phys. Rev. D 102 , 043015 (2020), arXiv:2004.08342 [astro-ph.HE].</list_item>
<list_item><location><page_28><loc_13><loc_10><loc_88><loc_16></location>[6] R. Abbott et al. (LIGO Scientific, Virgo), GWTC-2: Compact Binary Coalescences Observed by LIGO and Virgo During the First Half of the Third Observing Run, Phys. Rev. X 11 , 021053 (2021), arXiv:2010.14527 [gr-qc].</list_item>
<list_item><location><page_28><loc_13><loc_7><loc_88><loc_8></location>[7] R. Abbott et al. (KAGRA, Virgo, LIGO Scientific), All-sky, all-frequency directional search</list_item>
</unordered_list>
<text><location><page_29><loc_16><loc_87><loc_88><loc_91></location>for persistent gravitational waves from Advanced LIGO's and Advanced Virgo's first three observing runs, Phys. Rev. D 105 , 122001 (2022), arXiv:2110.09834 [gr-qc].</text>
<unordered_list>
<list_item><location><page_29><loc_13><loc_78><loc_88><loc_85></location>[8] A. H. Nitz, C. D. Capano, S. Kumar, Y.-F. Wang, S. Kastha, M. Schafer, R. Dhurkunde, and M. Cabero, 3-OGC: Catalog of gravitational waves from compact-binary mergers, The Astrophysical Journal 922 , 76 (2021).</list_item>
<list_item><location><page_29><loc_13><loc_70><loc_88><loc_77></location>[9] R. Abbott et al. (LIGO Scientific, VIRGO, KAGRA), GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo During the Second Part of the Third Observing Run, (2021), arXiv:2111.03606 [gr-qc].</list_item>
<list_item><location><page_29><loc_12><loc_62><loc_88><loc_69></location>[10] R. Abbott et al. (LIGO Scientific, VIRGO), GWTC-2.1: Deep Extended Catalog of Compact Binary Coalescences Observed by LIGO and Virgo During the First Half of the Third Observing Run, (2021), arXiv:2108.01045 [gr-qc].</list_item>
<list_item><location><page_29><loc_12><loc_57><loc_88><loc_61></location>[11] T. Damour and G. Schaeer, Higher-order relativistic periastron advances and binary pulsars, Il Nuovo Cimento B 101 , 127 (1988).</list_item>
<list_item><location><page_29><loc_12><loc_51><loc_88><loc_55></location>[12] A. Buonanno and T. Damour, Effective one-body approach to general relativistic two-body dynamics, Physical Review D 59 , 10.1103/physrevd.59.084006 (1999).</list_item>
<list_item><location><page_29><loc_12><loc_46><loc_88><loc_50></location>[13] A. Buonanno, G. B. Cook, and F. Pretorius, Inspiral, merger, and ring-down of equal-mass black-hole binaries, Physical Review D 75 , 10.1103/physrevd.75.124018 (2007).</list_item>
<list_item><location><page_29><loc_12><loc_37><loc_88><loc_44></location>[14] T. Damour, P. Jaranowski, and G. Schafer, Determination of the last stable orbit for circular general relativistic binaries at the third post-newtonian approximation, Physical Review D 62 , 10.1103/physrevd.62.084011 (2000).</list_item>
<list_item><location><page_29><loc_12><loc_32><loc_88><loc_36></location>[15] T. Damour, Gravitational scattering, post-minkowskian approximation, and effective-onebody theory, Physical Review D 94 , 10.1103/physrevd.94.104015 (2016).</list_item>
<list_item><location><page_29><loc_12><loc_26><loc_88><loc_30></location>[16] D. Bini and T. Damour, Gravitational spin-orbit coupling in binary systems at the second post-minkowskian approximation, Physical Review D 98 , 10.1103/physrevd.98.044036 (2018).</list_item>
<list_item><location><page_29><loc_12><loc_18><loc_88><loc_25></location>[17] T. Damour and P. Rettegno, Strong-field scattering of two black holes: Numerical relativity meets post-minkowskian gravity, Physical Review D 107 , 10.1103/physrevd.107.064051 (2023).</list_item>
<list_item><location><page_29><loc_12><loc_10><loc_88><loc_17></location>[18] C. Dlapa, G. Kalin, Z. Liu, and R. A. Porto, Conservative dynamics of binary systems at fourth post-minkowskian order in the large-eccentricity expansion, Physical Review Letters 128 , 10.1103/physrevlett.128.161104 (2022).</list_item>
<list_item><location><page_29><loc_12><loc_7><loc_88><loc_9></location>[19] C. Dlapa, G. Kalin, Z. Liu, J. Neef, and R. A. Porto, Radiation reaction and gravita-t</list_item>
</unordered_list>
<text><location><page_30><loc_16><loc_87><loc_88><loc_91></location>onal waves at fourth post-minkowskian order, Physical Review Letters 130 , 10.1103/physrevlett.130.101401 (2023).</text>
<unordered_list>
<list_item><location><page_30><loc_12><loc_78><loc_88><loc_85></location>[20] G. Kalin, Z. Liu, and R. A. Porto, Conservative dynamics of binary systems to third postminkowskian order from the effective field theory approach, Physical Review Letters 125 , 10.1103/physrevlett.125.261103 (2020).</list_item>
<list_item><location><page_30><loc_12><loc_73><loc_88><loc_77></location>[21] T. Damour, Introductory lectures on the effective one body formalism, International Journal of Modern Physics A 23 , 1130 (2008).</list_item>
<list_item><location><page_30><loc_12><loc_67><loc_88><loc_71></location>[22] T. Damour and A. Nagar, Comparing effective-one-body gravitational waveforms to accurate numerical data, Physical Review D 77 , 10.1103/physrevd.77.024043 (2008).</list_item>
<list_item><location><page_30><loc_12><loc_62><loc_88><loc_66></location>[23] T. Damour and A. Nagar, The Effective One Body description of the Two-Body problem, Fundam. Theor. Phys. 162 , 211 (2011), arXiv:0906.1769 [gr-qc].</list_item>
<list_item><location><page_30><loc_12><loc_57><loc_88><loc_61></location>[24] T. Damour and A. Nagar, Faithful effective-one-body waveforms of small-mass-ratio coalescing black hole binaries, Physical Review D 76 , 10.1103/physrevd.76.064028 (2007).</list_item>
<list_item><location><page_30><loc_12><loc_51><loc_88><loc_55></location>[25] T. Damour, Classical and quantum scattering in post-Minkowskian gravity, Phys. Rev. D 102 , 024060 (2020).</list_item>
<list_item><location><page_30><loc_12><loc_46><loc_88><loc_50></location>[26] A. Buonanno and T. Damour, Transition from inspiral to plunge in binary black hole coalescences, Physical Review D 62 , 10.1103/physrevd.62.064015 (2000).</list_item>
<list_item><location><page_30><loc_12><loc_40><loc_88><loc_44></location>[27] E. Poisson, Gravitational radiation from a particle in circular orbit around a black hole. i. analytical results for the nonrotating case, Physical Review D 47 , 1497 (1993).</list_item>
<list_item><location><page_30><loc_12><loc_32><loc_88><loc_39></location>[28] S. Long, Y. Zou, and J. Jing, Reconstruction of gravitational waveforms of coalescing spinless binaries in EOB theory based on PM approximation, Classical and Quantum Gravity 10.1088/1361-6382/acfdee (2023).</list_item>
<list_item><location><page_30><loc_12><loc_24><loc_88><loc_30></location>[29] J. Jing, S. Long, W. Deng, M. Wang, and J. Wang, New self-consistent effective one-body theory for spinless binaries based on the post-Minkowskian approximation, Sci. China Phys. Mech. Astron. 65 , 100411 (2022), arXiv:2208.02420 [gr-qc].</list_item>
<list_item><location><page_30><loc_12><loc_16><loc_88><loc_22></location>[30] J. Jing, W. Deng, S. Long, and J. Wang, Self-consistent effective-one-body theory for spinning binaries based on post-minkowskian approximation, Science China Physics, Mechanics, and Astronomy 66 , 270411 (2023).</list_item>
<list_item><location><page_30><loc_12><loc_7><loc_88><loc_14></location>[31] X. He, M. Sun, J. Jing, and Z. Cao, Energy map and effective metric in an effective-one-body theory based on the second-post-Minkowskian approximation, European Physical Journal C 81 , 97 (2021).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_31><loc_12><loc_84><loc_88><loc_91></location>[32] J. Jing, W. Deng, S. Long, and J. Wang, Effective metric of spinless binaries with radiationreaction effect up to fourth post-minkowskian order in effective-one-body theory, The European Physical Journal C 83 , 10.1140/epjc/s10052-023-11705-6 (2023).</list_item>
<list_item><location><page_31><loc_12><loc_78><loc_88><loc_82></location>[33] S. A. Teukolsky, Perturbations of a Rotating Black Hole. I. Fundamental Equations for Gravitational, Electromagnetic, and Neutrino-Field Perturbations, Astrophys. J. 185 , 635 (1973).</list_item>
<list_item><location><page_31><loc_12><loc_73><loc_88><loc_77></location>[34] J. M. Bardeen and W. H. Press, Radiation fields in the Schwarzschild background, Journal of Mathematical Physics 14 , 7 (1973).</list_item>
<list_item><location><page_31><loc_12><loc_65><loc_88><loc_71></location>[35] S. Chandrasekhar, On the equations governing the perturbations of the schwarzschild black hole, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 343 , 289 (1975).</list_item>
<list_item><location><page_31><loc_12><loc_62><loc_67><loc_63></location>[36] S. Chandrasekhar, The mathematical theory of black holes (1983).</list_item>
<list_item><location><page_31><loc_12><loc_57><loc_88><loc_61></location>[37] S. Mano, H. Suzuki, and E. Takasugi, Analytic solutions of the teukolsky equation and their low frequency expansions, Progress of Theoretical Physics 95 , 1079 (1996).</list_item>
<list_item><location><page_31><loc_12><loc_51><loc_88><loc_55></location>[38] M. Sasaki and H. Tagoshi, Analytic black hole perturbation approach to gravitational radiation, Living Reviews in Relativity 6 , 10.12942/lrr-2003-6 (2003).</list_item>
<list_item><location><page_31><loc_12><loc_46><loc_88><loc_50></location>[39] G. B. Cook and M. Zalutskiy, Gravitational perturbations of the kerr geometry: High-accuracy study, Physical Review D 90 , 10.1103/physrevd.90.124021 (2014).</list_item>
<list_item><location><page_31><loc_12><loc_37><loc_88><loc_44></location>[40] P. Fiziev and D. Staicova, Application of the confluent heun functions for finding the quasinormal modes of nonrotating black holes, Physical Review D 84 , 10.1103/physrevd.84.127502 (2011).</list_item>
<list_item><location><page_31><loc_12><loc_29><loc_88><loc_36></location>[41] E. W. Leaver, Solutions to a generalized spheroidal wave equation: Teukolsky's equations in general relativity, and the two-center problem in molecular quantum mechanics, Journal of Mathematical Physics 27 , 1238 (1986).</list_item>
<list_item><location><page_31><loc_12><loc_24><loc_88><loc_28></location>[42] An analytic representation for the quasi-normal modes of kerr black holes, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 402 , 285 (1985).</list_item>
<list_item><location><page_31><loc_12><loc_18><loc_88><loc_22></location>[43] E. W. Leaver, Spectral decomposition of the perturbation response of the schwarzschild geometry, Physical Review D 34 , 384 (1986).</list_item>
<list_item><location><page_31><loc_12><loc_10><loc_88><loc_17></location>[44] C. Chen and J. Jing, Radiation fluxes of gravitational, electromagnetic, and scalar perturbations in type-d black holes: an exact approach, J. Cosmol. Astropart. Phys. 11 , 070, arXiv:arXiv: 2307.14616 [gr-qc].</list_item>
<list_item><location><page_31><loc_12><loc_7><loc_88><loc_9></location>[45] W.-B. Han and Z. Cao, Constructing effective one-body dynamics with numerical energy</list_item>
</unordered_list>
<text><location><page_32><loc_16><loc_87><loc_88><loc_91></location>flux for intermediate-mass-ratio inspirals, Physical Review D 84 , 10.1103/physrevd.84.044014 (2011).</text>
<unordered_list>
<list_item><location><page_32><loc_12><loc_78><loc_88><loc_85></location>[46] A. Nagar, G. Riemenschneider, and G. Pratten, Impact of numerical relativity information on effective-one-body waveform models, Physical Review D 96 , 10.1103/physrevd.96.084045 (2017).</list_item>
<list_item><location><page_32><loc_12><loc_70><loc_88><loc_77></location>[47] A. Nagar, J. Healy, C. O. Lousto, S. Bernuzzi, and A. Albertini, Numerical-relativity validation of effective-one-body waveforms in the intermediate-mass-ratio regime, Physical Review D 105 , 10.1103/physrevd.105.124061 (2022).</list_item>
<list_item><location><page_32><loc_12><loc_62><loc_88><loc_69></location>[48] A. Boh'e et al., Improved effective-one-body model of spinning, nonprecessing binary black holes for the era of gravitational-wave astrophysics with advanced detectors, Phys. Rev. D 95 , 044028 (2017), arXiv:1611.03703 [gr-qc].</list_item>
<list_item><location><page_32><loc_12><loc_51><loc_88><loc_61></location>[49] S. Xin, B. Chen, R. K. L. Lo, L. Sun, W.-B. Han, X. Zhong, M. Srivastava, S. Ma, Q. Wang, and Y. Chen, Gravitational-wave echoes from spinning exotic compact objects: Numerical waveforms from the Teukolsky equation, Phys. Rev. D 104 , 104005 (2021), arXiv:2105.12313 [gr-qc].</list_item>
<list_item><location><page_32><loc_12><loc_43><loc_88><loc_50></location>[50] C. Zhang, W.-B. Han, X.-Y. Zhong, and G. Wang, Geometrized effective-one-body formalism for extreme-mass-ratio limits: Generic orbits, Phys. Rev. D 104 , 024050 (2021), arXiv:2102.05391 [gr-qc].</list_item>
<list_item><location><page_32><loc_12><loc_35><loc_88><loc_41></location>[51] C. Zhang, W.-B. Han, and S.-C. Yang, Analytical effective one-body formalism for extrememass-ratio inspirals with eccentric orbits, Commun. Theor. Phys. 73 , 085401 (2021), arXiv:2001.06763 [gr-qc].</list_item>
<list_item><location><page_32><loc_12><loc_29><loc_88><loc_33></location>[52] R. Cheng and W.-B. Han, Accurate recalibated waveforms for extreme-mass-ratio inspirals in effective-one-body frame, (2017), arXiv:1706.03884 [gr-qc].</list_item>
<list_item><location><page_32><loc_12><loc_21><loc_88><loc_28></location>[53] Z. Cao and W.-B. Han, Waveform model for an eccentric binary black hole based on the effective-one-body-numerical-relativity formalism, Phys. Rev. D 96 , 044028 (2017), arXiv:1708.00166 [gr-qc].</list_item>
<list_item><location><page_32><loc_12><loc_13><loc_88><loc_19></location>[54] R. Gamba, S. Ak¸cay, S. Bernuzzi, and J. Williams, Effective-one-body waveforms for precessing coalescing compact binaries with post-Newtonian twist, Phys. Rev. D 106 , 024020 (2022), arXiv:2111.03675 [gr-qc].</list_item>
<list_item><location><page_32><loc_12><loc_7><loc_88><loc_11></location>[55] S. Ossokine et al., Multipolar Effective-One-Body Waveforms for Precessing Binary Black Holes: Construction and Validation, Phys. Rev. D 102 , 044055 (2020), arXiv:2004.09442 [gr-</list_item>
</unordered_list>
<section_header_level_1><location><page_33><loc_16><loc_89><loc_18><loc_91></location>qc].</section_header_level_1>
<unordered_list>
<list_item><location><page_33><loc_12><loc_84><loc_88><loc_88></location>[56] T. Damour and A. Nagar, New effective-one-body description of coalescing nonprecessing spinning black-hole binaries, Phys. Rev. D 90 , 044018 (2014), arXiv:1406.6913 [gr-qc].</list_item>
<list_item><location><page_33><loc_12><loc_78><loc_88><loc_82></location>[57] N. Afshordi et al. (LISA Consortium Waveform Working Group), Waveform Modelling for the Laser Interferometer Space Antenna, (2023), arXiv:2311.01300 [gr-qc].</list_item>
<list_item><location><page_33><loc_12><loc_73><loc_88><loc_77></location>[58] M. Sasaki, Post-newtonian expansion of the ingoing-wave regge-wheeler function, Progress of Theoretical Physics 92 , 17 (1994).</list_item>
<list_item><location><page_33><loc_12><loc_67><loc_88><loc_71></location>[59] E. Poisson, Gravitational radiation from a particle in circular orbit around a black hole. i. analytical results for the nonrotating case, Physical Review D 47 , 1497 (1993).</list_item>
<list_item><location><page_33><loc_12><loc_62><loc_88><loc_66></location>[60] Y. Mino, M. Sasaki, M. Shibata, H. Tagoshi, and T. Tanaka, Chapter 1. black hole perturbation, Progress of Theoretical Physics Supplement 128 , 1 (1997).</list_item>
<list_item><location><page_33><loc_12><loc_54><loc_88><loc_61></location>[61] T. Damour and P. Rettegno, Strong-field scattering of two black holes: Numerical relativity meets post-minkowskian gravity, Physical Review D 107 , 10.1103/physrevd.107.064051 (2023).</list_item>
<list_item><location><page_33><loc_12><loc_48><loc_88><loc_52></location>[62] T. Dray, The relationship between monopole harmonics and spin-weighted spherical harmonics, Journal of Mathematical Physics 26 , 1030 (1985).</list_item>
<list_item><location><page_33><loc_12><loc_43><loc_88><loc_47></location>[63] E. Poisson, Gravitational radiation from infall into a black hole: Regularization of the teukolsky equation, Physical Review D 55 , 639 (1997).</list_item>
<list_item><location><page_33><loc_12><loc_35><loc_88><loc_41></location>[64] Y. Pan, A. Buonanno, R. Fujita, E. Racine, and H. Tagoshi, Post-newtonian factorized multipolar waveforms for spinning, nonprecessing black-hole binaries, Phys. Rev. D 83 , 064003 (2011).</list_item>
<list_item><location><page_33><loc_12><loc_29><loc_88><loc_33></location>[65] M. Sasaki, Post-newtonian expansion of the ingoing-wave regge-wheeler function, Progress of Theoretical Physics 92 , 17 (1994).</list_item>
<list_item><location><page_33><loc_12><loc_24><loc_88><loc_28></location>[66] A. Buonanno and T. Damour, Transition from inspiral to plunge in binary black hole coalescences, Physical Review D 62 , 10.1103/physrevd.62.064015 (2000).</list_item>
<list_item><location><page_33><loc_12><loc_16><loc_88><loc_22></location>[67] Y. Pan, A. Buonanno, R. Fujita, E. Racine, and H. Tagoshi, Post-newtonian factorized multipolar waveforms for spinning, nonprecessing black-hole binaries, Phys. Rev. D 83 , 064003 (2011).</list_item>
<list_item><location><page_33><loc_12><loc_7><loc_88><loc_14></location>[68] T. Damour, B. R. Iyer, and A. Nagar, Improved resummation of post-newtonian multipolar waveforms from circularized compact binaries, Physical Review D 79 , 10.1103/physrevd.79.064004 (2009).</list_item>
</unordered_list>
</document>
[
{
"title": "Energy flux and waveforms by coalescing spinless binary system in effective one-body theory",
"content": "Sheng Long , 1 Weike Deng a , 1 and Jiliang Jing , 1, 2, † 1 Department of Physics, Key Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education, and Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha, Hunan 410081, P. R. China 2 Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University, Yangzhou 225009, P. R. China",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "We present a study on the energy radiation rate and waveforms of the gravitational wave generated by coalescing spinless binary systems up to the third post-Minkowskian approximation in the effective one-body theory. To derive an analytical expansion of the null tetrad components of the gravitational perturbed Weyl tensor Ψ 4 in the effective spacetime, we utilize the method proposed by Sasaki et al. During this investigation, we discover more general integral formulas that provide a theoretical framework for computing the results in any order. Subsequently, we successfully compute the energy radiation rate and waveforms of the gravitational wave, which include the results of the Schwarzschild case and the correction terms resulting from the dimensionless parameters a 2 and a 3 in the effective metric. PACS numbers: 04.25.Nx, 04.30.Db, 04.20.Cv Keywords: post-Minkowskian approximation, effective one-body theory, gravitational waveform template",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "Gravitational waveform templates play an important role in the detection of gravitational wave events generated by coalescing binary systems [1-10]. The foundation of gravitational waveform templates is the theoretical model of gravitational radiation, in which the key point is studying the late-stage dynamical evolution of a coalescing binary system. Damour and Buonanno [11, 12] proposed an effective one-body (EOB) theory that maps the real two-body problem with masses m 1 and m 2 to a test particle of mass µ = m 1 m 2 m 1 + m 2 moving around an effective spacetime of mass M = m 1 + m 2 (and we denote the symmetric mass ratio as ν = µ/M ). This theory enables the study of gravitational radiation produced by merging binary systems. Based on the EOB theory with the post-Newtonian (PN) approximation, Damour et al. provided an estimate of the gravitational waveforms emitted throughout the inspiral, plunge, and coalescence phases [13, 14]. To release the assumption that v/c is a small quantity, in 2016, Damour introduced another theoretical model by combining the EOB theory with the post-Minkowskian (PM) approximation [15, 16]. Damour and Rettegno [17] compared numerical relativistic (NR) data for equal-mass binary black hole scattering with analytical predictions based on the fourth PM (4PM) dynamics [18-25] and pointed out that the reconstruction of PM information in terms of EOB radial potentials leads to remarkable agreement with NR data, especially when using radiation-reacted 4PM information. Therefore, this new model may lead to a theoretically improved version of the EOB conservative dynamics and may be useful in the upcoming era of high-signal-to-noise-ratio gravitational wave observations. The dynamical evolution of a coalescing binary system for a spinless EOB theory can be described by the Hamilton equation [26], and the Hamiltonian H [ g eff µν ] is dependent on the effective metric. The radiation reaction forces F R [ g eff µν ] and F φ [ g eff µν ] in the Hamilton equation can be described by the energy radiation rate as follows: d E d t = 1 4 πG 2 ω 2 ∫ | Ψ B 4 | 2 r 2 dΩ [13, 27]. Furthermore, the 'plus' and 'cross' modes of gravitational waves are related to the null tetrad components of the gravitational perturbed Weyl tensor Ψ B 4 in the NewmanPenrose formalism as follows: Ψ B 4 = 1 2 ( h + -i h × ). Thus, as long as we obtain the effective metric and the solution of Ψ B 4 in the Newman-Penrose formalism, we can calculate the energy radiation rate and construct gravitational waveforms. In previous work, we attempted to develop a self-consistent EOB theory for spinless and spinning binaries based on the PM approximation [28-31]. Furthermore, in a recent paper [32], we obtained the effective metric up to the 4PM order. We adopted the black hole perturbation method used by Teukolsky [33, 34] and decomposed all quantities into background and perturbation (denoted with a superscript B ) parts in the Newman-Penrose formalism. After choosing a shadow gauge [29, 35, 36] with Ψ 1 and Ψ 3 set to 0, we can decouple the equations for the null tetrad components of the gravitational perturbed Weyl tensor Ψ B 4 . Subsequently, upon separating the variables in the equations, we obtained a radial equation, which is the so-called Teukolsky-like equation, and an angular equation that features spin-weighted spherical harmonics. This Teukolsky-like equation is more complex compared to the Teukolsky equation in Kerr and Schwarzschild spacetimes. We were unable to find a similar transformation to convert the homogeneous Teukolsky-like equation into hypergeometric or Heun equations; thus, we did not choose to adopt the so-called MST [37, 38] method or the Heun function [39-44]. Instead, we follow the approach used by Sasaki. Several researchers [45-57] have employed numerical methods to solve the Teukolsky equation and achieved significant success by combining the EOB theory with numerical relativity. We initially applied the Sasaki-Nakamura-Chandrasekhar-like (S-N-C-like) transformation [35, 58] to convert the homogeneous Teukolsky-like equation into a homogeneous SasakiNakamura-like (S-N-like) equation [38, 58-60]. In an asymptotically flat spacetime, this homogeneous S-N-like equation can be simplified to the Klein-Gordon equation. Subsequently, we performed a Taylor expansion with respect to η = 2 GMω . The equation of the zeroth order is the spherical Bessel equation, and its solutions are linear combinations of the first and second kind spherical Bessel functions, denoted as j ℓ and n ℓ , respectively, allowing us to construct higher-order solutions based on the zeroth-order solution. By performing an inverse transformation, we can deduce the solutions of the homogeneous Teukolsky-like equation. This framework enables us to construct the solutions of the inhomogeneous Teukolsky equation, which includes a source term, using Green's function method. However, due to the complexity of Green's function method and the fact that the integral formulas provided in the previous work of Sasaki [60] were insufficient for our needs in computing higher-order solutions, we found some new integral formulas presented in Appendix B of this article, which are crucial for our journey toward calculating higher-order solutions. Section II introduces the effective metric of 3PM, while Sec. III discusses the solutions of the equation for Ψ B 4 in the effective spacetime. Specifically, Section IIIA summarizes the general structure of the solutions of the radial equation (Teukolsky-like equation) of Ψ B 4 . Section IIIB provides a comprehensive explanation of the calculation of the homogeneous S-N-like equation. Subsequently, we employ boundary conditions to determine the amplitudes. Section IIIC presents the source terms for quasi-circular orbits; by combining the homogeneous solutions provided in Section IIIB and utilizing Eq. (3.8), we can obtain the solution of Ψ B 4 under quasi-circular orbits. In Section IV, we present the energy radiation rate d E d t and the gravitational waveforms h ℓm .",
"pages": [
2,
3,
4
]
},
{
"title": "II. EFFECTIVE METRIC FOR THE EOB THEORY",
"content": "In the EOB theory, the main idea is to map the two-body problem onto an EOB problem, that is, a test particle orbits around a massive black hole described by an effective metric. With the help of the scattering angles, we found that the effective metric for spinless binaries with radiation reaction effects in the EOB theory, up to the 3PM approximation, can be expressed as follows [32]: with the definitions of c 1 , c 2 , and c h are as follows: where Q = 1 27 -a 2 24 -a 3 16 , P = 1 3 ( a 2 4 -1 3 ) and a 2 and a 3 are dimensionless parameters expressed as follows: in which γ = E µ = 1 2 E 2 -m 2 1 -m 2 2 m 1 m 2 is the Lorentz factor variable, E is the real two-body energy [32, 61], E is the effective energy, Γ = E/M = √ 1 + 2 ν ( γ -1) is the rescaled energy, and In Eq. (2.7), the term χ rr 3 , described by Eq. (2.9), represents the 3PM radiation reaction effects, which shows that the structure of the effective spacetime is affected by the radiation reaction effect.",
"pages": [
4,
5
]
},
{
"title": "III. SOLUTIONS OF EQUATION FOR Ψ B 4 IN EFFECTIVE SPACETIME",
"content": "In this section, we first present the formal solution of the radial equation for Ψ B 4 . Then, we transform the radial equation without source to the corresponding S-N-like equation, and we look for its solution. At last, we work out the solution of the radial equation of Ψ B 4 with the source, which describes the gravitational radiation induced by the motion of an effective particle in an effective background.",
"pages": [
5
]
},
{
"title": "A. Formal solution of the radial equation of Ψ B 4 in effective spacetime",
"content": "In the EOB theory for the spinless real two-body system, we have found a decoupled equation of Ψ B 4 for the gravitational perturbation in the effective spacetime (2.1) using the gauge transform property of the tetrad components of the perturbed Weyl tensors and separated the decoupled equation in the radial and angular parts, in which the radial part of Ψ B 4 is given by [28, 29] with where the prime ' denotes derivation with respect to r , and A 0 = A nn 0 + A mn 0 + A mm 0 , A 1 = A mn 1 + A mm 1 , and A 2 = A mm 2 the explicitly definitions of F a ( a = 1 , 2 , 3 , 4), L n (or L † n ), and C b ( b is nn or mn or mm ) can be found in Ref. [29], and -2 Y ℓm ( θ ) is the spin-weighted spherical harmonics [38, 62]. The radial equation, i.e., Eq. (3.1), can be solved using Green's function method. That is, based on the homogeneous solutions of Eq. (3.1) where r ∗ denotes the tortoise coordinate defined by r ∗ = ∫ r 2 ∆ r dr . The inhomogeneous solution of the radial Eq. (3.1) can be expressed as follows: whereas the counterpart at infinity can be expressed as follows: As discussed in Ref. [29], for the point source case, after a lengthy calculation, we can obtain the expression for ˆ Z ℓmω . If we focus our attention just on the quasi-circular orbit, we have ˆ Z ℓmω n = Z ℓmω δ ( ω -ω n ), in which Then, the solution of Ψ B 4 is described by Equations (3.9) and (3.8) show that to get the explicit expression for Ψ B 4 , we should work out A i ( i = 1 , 2 , 3), B inc ℓmω , and R in ℓmω .",
"pages": [
5,
6,
7
]
},
{
"title": "B. S-N-like equation and its solution of the third PM approximation",
"content": "In Eq. (3.8), R in ℓmω is the solution of the homogeneous equation without the source term. To get the solution, we do not treat the Teukolsky-like equation directly because the potential function in the equation is a long-range potential. Instead, we transform the radial equation, i.e., Eq. (refEqTS), without source into the S-N-like equation with a new function X ℓmω , which has a short-range potential. Then, using the solution of X in ℓmω , we find out B in ℓmω and R in ℓmω . Taking an S-N-C-like transformation as 1 where and considering the coordinate transformation r → r ∗ , the radial equation (Eq. (3.1)) without the source term ( T ℓmω = 0) can be rewritten as the so-called S-N-like equation, as follows: with where The asymptotic solution of X in ℓ can be expressed as follows: Meanwhile, the inverse transformation is described by the following expression: Using a method similar to that used in Refs. [38, 63], the coefficient A in ℓ in (3.15) is related to B in ℓ in Eq. (3.4) as follows:",
"pages": [
7,
8
]
},
{
"title": "2. Solution of X in ℓ",
"content": "We now look for the solution of X in ℓ and amplitude A in ℓ of the S-N-like equation. The method employed in this subsection is based on the work of Sasaki [38] and Mino [60]. We first take the following ansatz: where z = ωr , η = 2 GMω , b 1 = c 3 1 ( c 1 -c 2 )( c 1 -c h ) , b 2 = c 3 2 ( c 2 -c 1 )( c 2 -c h ) , b h = c 3 h ( c 1 -c h )( c h -c 2 ) , and With this choice of the phase function, ξ ℓm is regular and finite at z = ηc h . Then, we determine that Eq. (3.12) can be expressed as follows: with where the definitions of a ( n ) ℓ and other terms are shown in Appendix A. In the low-frequency limit and noting that η = 2 GMω only appears on the right-hand side of Eq. (3.20), we may look for the solution of the ξ ℓ ( z ) perturbative in terms of η , i.e., and we obtain the recursive equation from Eq. (3.20) as follows: where The solution of ξ (0) ℓ can be expressed as a linear combination of the spherical Bessel functions j ℓ and n ℓ , i.e., ξ (0) ℓ = α (0) j ℓ + β (0) n ℓ . Because n ℓ does not match with the horizon solution at the leading order of η , we should take β (0) = 0. Furthermore, because the constant α (0) represents the overall normalization of the solution, which can be chosen arbitrarily, we set α (0) = 1. That is, for the zeroth-order solution, we have f (0) ℓ = j ℓ and g (0) ℓ = 0. Then, one can immediately write the integral expression for ξ ( n ) ℓ (where n > 0). Noting that j ℓ n ℓ ' -n ℓ j ℓ ' = 1 /z 2 , we derive the expression of ξ ( β ) ℓ for β ≥ 1 as follows: In general, ξ ( β ) ℓ can be decomposed into the real and imaginary parts ξ ( β ) ℓ = f ( β ) ℓ + ig ( β ) ℓ , in which where Re [ x ] and Im [ x ] are the real and imaginary parts of x , respectively. Using the previously presented formula and the method discussed in Appendix B, after some tedious calculations, we derive closed analytical formulas of the ingoing-wave S-N-like function for arbitrary ℓ to the first order of η . At the second order of η 2 , we can calculate results for any order utilizing equation (3.27). However, generalizing these results to encompass all values of ℓ is unattainable. Therefore, for the higher-order results, we only provide results for specific values of ℓ , for ℓ = 2 , 3 to η 2 order, and for ℓ = 2 to η 3 order. We express the real parts as follows: with where R m,k is the Lommel polynomial ( R m,k = -R k,m for m R m,k = z 2 ( n m j k -j m n k ) ( m>k ) = -[ ( m -k -1) 2 ] ∑ r =0 ( -1) r ( m -k -1 -r )!Γ ( m + 1 2 -r ) r !( m -k -1 -2 r )!Γ ( k + 3 2 + r ) ( 2 z ) m -k -1 -2 r , (3.32) B J is the generalized integral sinusoidal function, and D J ℓ is the generalized spherical Bessel function in Appendix B. f (2) 2 = ( -193 a 2 2 1890 z + 45 a 2 14 z -257 a 3 1008 z -113 420 z ) j 1 + ( -17 a 2 2 1890 z -10 a 2 63 z -59 a 3 336 z + 1 7 z ) j 3 + ( -16 a 2 2 945 + a 2 21 -5 a 3 126 -107 210 ) j 2 ln z + ( 32 a 2 2 315 z -55 a 2 21 z + 5 a 3 21 z ) n 0 + ( 32 a 2 2 945 -2 a 2 21 + 5 a 3 63 + 107 105 ) D nj -3 + ( 10 3 -2 a 2 15 ) D nj 1 + 14 a 2 45 D nj 3 -389 j 0 70 z 2 -1 j 2 (ln z ) 2 + 6 D nj 0 -5 D nj 2 +4 D nnj 2 , 2 z 3 z (3.33) f (2) 3 = ( 20 a 2 2 81 z -635 a 2 72 z + 5 a 3 9 z -445 14 z 3 -1031 588 z ) j 0 + ( -197 a 2 2 1134 z + 1093 a 2 168 z -415 a 3 1008 z + 323 49 z ) j 2 + ( 2 a 2 2 189 z -199 a 2 840 z -65 a 3 504 z + 1 4 z ) j 4 + j 3 ln z ( -5 a 2 2 378 + a 2 42 -5 a 3 168 -13 42 ) +4 D nnj 3 + ( 25 a 2 2 63 z -405 a 2 28 z + 25 a 3 28 z -65 6 z ) n 1 + ( -5 a 2 2 189 + a 2 21 -5 a 3 84 -13 21 ) D nj -4 + ( 13 3 -8 a 2 63 ) D nj 2 + 11 a 2 42 D nj 4 -5065 j 1 294 z 2 -1 2 j 3 (ln z ) 2 + 65 n 0 6 z 2 + 30 D nj 0 z 2 + 9 D nj 1 z -3 D nj 3 z . (3.34) f (3) 2 = ( 9 4 -a 2 30 ) j 1 (ln z ) 2 + ( 7 a 2 90 -1 12 ) j 3 (ln z ) 2 + D nj 2 (ln z ) 2 + ( -16 a 3 2 14175 + 5 a 2 2 63 -a 3 a 2 378 -887 a 2 3150 + 5 a 3 28 + 349 140 ) j 1 ln z + c 2 1 +( c 2 -1)( c 1 + c 2 ) z j 2 ln z + ( 2 3 -28 a 2 45 ) D nnj 3 + ( 16 a 3 2 6075 -11 a 2 2 315 + a 3 a 2 162 + 1543 a 2 18900 -10 a 3 189 + 29 252 ) j 3 ln z + ( -16 a 2 2 315 + a 2 7 -5 a 3 42 -107 70 ) n 0 ln z + ( 32 a 2 2 945 -2 a 2 21 + 5 a 3 63 + 107 105 ) D nj 2 ln z + ( -2381 a 3 2 66150 + 8207 a 2 2 132300 -185 a 3 a 2 2352 -83821 a 2 44100 + 21 100 -187 a 3 168 ) j 1 + ( 97 a 3 2 36450 + 5339 a 2 2 170100 + 35 a 3 a 2 3888 + 9053 a 2 25200 + 4609 a 3 18144 -457 1050 ) j 3 + ( -11 a 3 2 142884 + 139 a 2 2 476280 -43 a 3 a 2 95256 -277 a 2 105840 + 1 504 -109 a 3 45360 ) j 5 + ( 193 a 3 2 10206 -110 a 2 2 1701 + 1033 a 3 a 2 27216 + 6911 a 2 5670 + 48353 a 3 90720 -3(ln z ) 2 2 -2539 3780 ) n 0 + with T 1 =1 + c 2 1 -( c 1 + c 2 )(1 -c 2 ) , T 2 = c 1 c 2 ( -c 1 -c 2 +3) + 2( c 1 -1) c 1 +2( c 2 -1) c 2 +1 , (3.39) where ς ( n ) ℓ for ℓ = 2 can be expressed as follows: ς (2) 2 = ( 2 a 2 2 81 + 5 a 3 108 + a 2 180 ) j 3 + a 2 30 j 1 , (3.40) ς (3) 2 = ( -4 a 2 2 81 -a 2 90 -5 a 3 54 ) D nj 3 + ( 176 a 3 2 2835 z -1469 a 2 2 3780 z + 40 a 2 a 3 189 z -5 a 2 24 z -181 a 3 252 z ) j 1 -a 2 15 D nj 1 + ( 5 a 3 2 1701 z + 83 a 2 2 3780 z + 20 a 2 a 3 567 z + a 2 144 z + 13 a 3 252 z ) j 3 + ( 22 a 3 2 2835 + a 2 2 105 + 5 a 2 a 3 189 + a 3 56 ) j 2 ln z + ( -44 a 3 2 945 z + 296 a 2 2 945 z -10 a 2 a 3 63 z + a 2 12 z + 37 a 3 63 z ) n 0 + ( -44 a 3 2 2835 -2 a 2 2 105 -10 a 2 a 3 189 -a 3 28 ) D nj -3 . (3.41) Inserting these expressions into Eq. (3.18) and expanding the result with respect to η , we find that X in ℓmω = X (0) ℓ + ηX (1) ℓ + η 2 X (2) ℓ + η 3 X (3) ℓ , (3.42) ( -32 a 3 2 1215 + 65 a 2 2 1134 -5 a 3 a 2 81 -1574 a 2 945 + 197 126 -2455 a 3 3024 ) n 2 + ( 32 a 3 2 6075 -58 a 2 2 2835 + a 3 a 2 81 + 824 a 2 4725 -5 a 3 378 -107 630 ) D nj -4 + ( -32 a 3 2 14175 -2 a 2 2 45 -a 3 a 2 189 + 7468 a 2 1575 -5 a 3 42 -457 70 ) D nj -2 + ( -19 a 2 2 567 + 59 a 2 21 -2629 630 -103 a 3 1512 ) D nj 0 + ( 1402 a 2 2 19845 -925 a 2 441 + 1165 a 3 5292 + 16949 4410 ) D nj 2 + ( 17 a 2 2 6615 + 20 a 2 441 + 59 a 3 1176 -2 49 ) D nj 4 + ( -64 a 2 2 945 + 4 a 2 21 -10 a 3 63 -214 105 ) D njj 2 + ( -64 a 2 2 945 + 4 a 2 21 -10 a 3 63 -214 105 ) D nnj -3 -12 D nnj -1 + ( 4 a 2 15 -18 ) D nnj 1 -8 D nnnj 2 . (3.35) The corresponding imaginary parts are expressed as follows: g (1) ℓ = j ℓ ln z, (3.36) g (2) ℓ = f (1) ℓ ln z -T 1 z j ℓ + ς (2) ℓ , (3.37) g (3) ℓ = f (2) ℓ ln z -T 1 f (1) ℓ z -T 2 j ℓ 2 z 2 + 1 3 j ℓ (ln z ) 3 + ς (3) ℓ , (3.38) where X (0) ℓ = zj ℓ , X (1) ℓ = zf (1) ℓ , X (2) ℓ = z [ f (2) ℓ + 1 2 j ℓ (ln z ) 2 + iς (2) ℓ ] , X (3) ℓ = z [ f (3) ℓ + 1 2 f (1) ℓ (ln z ) 2 -T 1 z j ℓ ln z + ς (2) ℓ ln z + iς (3) ℓ ] . (3.43) 3. Coefficient of amplitude A in ℓ Noting e -iη ( -b 1 ln( z -c 1 η )+ b 2 ln( z -c 2 η )+ b h ln( z -c h η )) = e -iz ∗ e iz z →∞ -→ 1, taking the expressions of the spherical Hankel functions of the first and second kinds h (1) ℓ and h (2) ℓ as h (1) ℓ = j ℓ + in ℓ z →∞ ---→ ( -i ) ℓ +1 e iz z , (3.44) h (2) ℓ = j ℓ -in ℓ z →∞ ---→ i ℓ +1 e -iz z , (3.45) and using the asymptotic behavior of B J and D J ℓ in Ref. [60], we obtain the following expression: A in ℓ = 1 2 i ℓ +1 e -iη (ln 2 η + elg ) e i [ ηp (0) ℓ -πη 2 p (1) ℓ + η 3 ( p (2) ℓ -π 2 p (3) ℓ + p (4) ℓ RiZ (3) )] { 1 -π 2 η + η 2 [ 2( elg +ln2) p (1) ℓ + q (1) ℓ + 5 π 2 24 ] + η 3 [ πq (2) ℓ + π 3 q (4) ℓ + π ( elg +ln2) q (3) ℓ ] } . (3.46) where elg is the EulerGamma constant ( elg = 0 . 57721 · · · ), RiZ ( n ) is the Riemann zeta function ( RiZ (3) = 1 . 202 · · · ), and the coefficients of A in 2 are p (0) 2 = 15 -2 a 2 9 , p (1) 2 = 32 a 2 2 -90 a 2 +75 a 3 +963 3780 , p (2) 2 = -292 a 3 2 +1962 a 2 2 -765 a 2 a 3 -1782 a 2 +3564 a 3 +1566 34992 , p (3) 2 = 32 a 2 2 -90 a 2 +75 a 3 +963 11340 , p (4) 2 = 1 3 , q (1) 2 = -37 a 2 -5 a 3 +150 108 , q (4) 2 = -1 16 , q (2) 2 = -176 a 3 2 -216 a 2 2 -600 a 2 a 3 +7770 a 2 +645 a 3 -31500 45360 , and q (3) 2 = -32 a 2 2 +90 a 2 -75 a 3 -963 3780 -i 176 a 3 2 +216 a 2 2 +600 a 2 a 3 +405 a 3 22680 π . (3.47) C. Quasi-circular orbit on the equatorial plane around an EOB In this section, we consider a quasi-circular orbit. In this case, we assume that the orbit lies on the equatorial plane ( θ = π/ 2) without loss of generality. By setting V r ( r 0 ) = ∂V r /∂r ( r 0 ) = 0, the effective energy E and effective angular momentum L are given by E/µ = √ 2 [ a 3 ( GM ) 3 + r 0 ( a 2 ( GM ) 2 + r 0 ( r 0 -2 GM ) )] r 0 √ r 0 √ 5 a 3 ( GM ) 3 +2 r 0 ( 2 a 2 ( GM ) 2 + r 0 ( r 0 -3 GM ) ) , L/µ = r 0 √ -3 a 3 ( GM ) 3 +2 GMr 0 ( r 0 -a 2 GM ) √ 5 a 3 ( GM ) 3 +2 r 0 ( 2 a 2 ( GM ) 2 + r 0 ( r 0 -3 GM ) ) . (3.48) (3.49) where r 0 is the orbital radius. By defining 0 b ℓm = √ ( ℓ -1) ℓ ( ℓ +1)( ℓ +2) 2 √ 2 π ∆ r 15 a 3 ( GM ) 2 +8 a 2 GMr 0 -6 r 2 0 5 a 3 ( GM ) 2 +4 a 2 GMr 0 -6 r 2 0 0 Y ℓm ( π 2 , 0) × r 2 0 E, -1 b ℓm = 2 √ ( ℓ -1)( ℓ +2) √ 2 πr 0 5 a 3 ( GM ) 2 +3 a 2 GMr 0 -3 r 2 0 ( 5 a 3 ( GM ) 2 +4 a 2 GMr 0 -6 r 2 0 ) 3 -1 Y ℓm ( π 2 , 0) · P r · L, -2 b ℓm = ∆ r √ 2 πr 4 E -1 Y ℓm ( π 2 , 0) L 2 , P r = 75 a 2 3 ( GM ) 4 +10 a 3 ( GM ) 2 r 0 ( 11 a 2 GM -18 r 0 ) +4 r 2 0 ( 8 a 2 2 ( GM ) 2 -21 a 2 GMr 0 +9 r 2 0 ) , B r = r 0 ∆ r ( 8 a 2 r 0 +30 a 3 GM ) +6 a 3 r 2 0 ( GM ) 2 -8 a 2 a 3 r 0 ( GM ) 3 -15 a 2 3 ( GM ) 4 , (3.50) we obtain A 0 = 1 2 r 2 0 · { 2 0 b ℓm +4 i -1 b ℓm [ 1 + i 2 ω r 3 0 ∆ r ( 1 + 2 GMr 0 P r × ( 6 a 2 r 2 0 +30 a 3 GMr 0 -5 a 2 a 3 ( GM ) 2 ) )] -2 i -2 b ℓm ωr 3 0 ∆ 2 r [ r 2 0 -GMr 0 + GM ×B r 15 a 3 ( GM ) 2 +8 a 2 GMr 0 -6 r 2 0 + 1 2 iωr 3 0 + 6 i GM ∆ 2 r · ( 2 a 2 r 2 0 +15 a 3 GMr 0 -5 a 2 a 3 ( GM ) 2 ) r 2 0 ω ( 15 a 3 ( GM ) 2 +8 a 2 GMr 0 -6 r 2 0 ) 2 ]} , (3.51) A 1 = i -1 b ℓm r 0 (1 + 2 GMr 0 ( 6 a 2 r 2 0 +30 a 3 GMr 0 -5 a 2 a 3 ( GM ) 2 ) P r ) --2 b ℓm r 0 (1 + GM · (15 a 3 GM +4 a 2 r 0 ) 15 a 3 ( GM ) 2 +8 a 2 GMr 0 -6 r 2 0 + i r 3 0 ω ∆ r ) , (3.52) A 2 = --2 b ℓm 2 . (3.53) IV. ENERGY RADIATION RATE AND GRAVITATIONAL WAVEFORMS Inserting the aforementioned result of A i ( i = 1 , 2 , 3), B in ℓ , and R in ℓ into Eq. (3.8), we can obtain the following expression: Z ℓmω 0 = Z ( N,ϵ ) ℓmω 0 ˜ Z ℓmω 0 , (4.1) Z ( N,ϵ ) ℓmω 0 =( -1) ℓ + ϵ +1 m 2 ν 2 M n ( ϵ ) ℓm v ℓ + ϵ +6 0 Y ℓ -ϵ, -m ( π/ 2 , φ ) , (4.2) n ( ϵ ) ℓm = ( im ) ℓ 8 π (2 ℓ +1)!! √ ( ℓ +1)( ℓ +2) ℓ ( ℓ -1) , ϵ = 0 , -( im ) ℓ 16 π (2 ℓ +1)!! √ (2 ℓ +1)( ℓ +2)( ℓ 2 -m 2 ) (2 ℓ -1)( ℓ +1) ℓ ( ℓ -1) , ϵ = 1 . (4.3) where we define v = ( GM Ω) 1 / 3 , ω 0 = m Ω, ϵ = 1 when ℓ + m = 1, and ϵ = 0 when ℓ + m = 0. We can divide the higher-order term ˜ Z ℓmω 0 into two parts: the ˜ Z ( S ) ℓmω 0 is computed in the Schwarzschild case [64] and the ˜ Z ( R ) ℓmω 0 is the 2PM and 3PM perturbation terms: ˜ Z ℓmω 0 = ˜ Z ( S ) ℓmω 0 + ˜ Z ( R ) ℓmω 0 . (4.4) The explicit expression of ˜ Z ( R ) ℓmω 0 is presented in Appendix C. In the test particle limit, i.e., ν → 0, we note that ˜ Z ( R ) ℓmω 0 vanishes completely because a 2 and a 3 approach 0. That is, our results revert to the Schwarzschild case in the test particle limit. In Eq. (3.9), utilizing the symmetry of the spin-weighted spherical harmonics, s Y ℓ, -m ( π 2 , 0) = ( -1) s + ℓ s Y ℓm ( π 2 , 0), we know that Z ℓ ( -m ) ω = ( -1) ℓ Z ∗ ℓmω , where Z ∗ ℓmω is the complex conjugate of Z ℓmω . In terms of the amplitude Z ℓmω , we find from Eq. (3.9) that the gravitational waveform [13, 27, 65] at infinity is given by h + -ih × = ∑ ℓm h ℓm -2 Y ℓm √ 2 π e iω 0 ( r ∗ -t )+ imφ , (4.5) with where h ℓm = -2 R ω 2 0 Z ℓmω 0 = h ( S ) ℓm + h ( R ) ℓm , (4.6) h ( S ) ℓm = h ( N,ϵ ) ℓm ˜ Z ( S ) ℓmω 0 , h ( R ) ℓm = h ( N,ϵ ) ℓm ˜ Z ( R ) ℓmω 0 , (4.7) h ( N,ϵ ) ℓm = GMν R n ( ϵ ) ℓm c ℓ + ϵ ( ν ) v ℓ + ϵ 0 Y ℓ -ϵ, -m ( π/ 2 , φ ) . (4.8) The energy loss rate along any orbit, in polar coordinates, can be expressed as dE [ g eff µν ] dt = ˙ R F R [ g eff µν ] + ˙ φ F φ [ g eff µν ]. By simply replacing the radial component with zero, an excellent approximation of the radiation reaction forces can be obtained [66]. Thus, from Eq. (4.5), we know that, for given energy ω n , the energy loss rate [13, 27] for the 'plus' and 'cross' modes of the gravitational wave is described by the following expression: dE [ g eff µν ] dt = 1 2 ( d E d t ) N ∞ ∑ ℓ =2 ℓ ∑ m =1 Π ℓm , (4.9) with Π ℓm = E ( S ) ℓm + E ( R ) ℓm , (4.10) E ( S ) ℓm = | Z ( N,ϵ ) ℓmω 0 ˜ Z ( S ) ℓmω 0 | 2 2 πG 2 ω 2 0 ( d E d t ) N , (4.11) E ( R ) ℓm = | Z ( N,ϵ ) ℓmω 0 | 2 2 πG 2 ω 2 0 ( d E d t ) N { ˜ Z ( S ) ℓmω 0 ˜ Z ( R ) ∗ ℓmω 0 + ˜ Z ( R ) ℓmω 0 ˜ Z ( S ) ∗ ℓmω 0 + ˜ Z ( R ) ℓmω 0 ˜ Z ( R ) ∗ ℓmω 0 } . (4.12) where (d E/ d t ) N = 32 ν 2 v 10 / 5 is the Newtonian quadrupole luminosity and the superscript ∗ denotes the complex conjugation of the corresponding expression. In Figure 1, we present the curves of Π 22 and Π 33 as the symmetric mass ratio ν takes different values. Figure 1: Π ℓm = Π ℓm ( v 2 , ν ) with respect to v 2 and ν , where the curve of ν = 0 corresponds to the 4.5PN result obtained for the Schwarzschild case. The figure on the left is for Π 22 = Π 22 ( v 2 , ν ), and the right one is for Π 33 = Π 33 ( v 2 , ν ) Figure 1: Π ℓm = Π ℓm ( v 2 , ν ) with respect to v 2 and ν , where the curve of ν = 0 corresponds to the 4.5PN result obtained for the Schwarzschild case. The figure on the left is for Π 22 = Π 22 ( v 2 , ν ), and the right one is for Π 33 = Π 33 ( v 2 , ν ) Then, we determine the radiation reaction forces for the 'plus' and 'cross' modes of the gravitational wave as follows: F circ φ [ g eff µν ] ≃ 1 ˙ φ dE [ g eff µν ] dt = 1 2 ( d E d t ) N ∞ ∑ ℓ =2 ℓ ∑ m =1 ( F ( S ) ℓm + F ( R ) ℓm ) , (4.13) where F ( S ) ℓm = E ( S ) ℓm / 1 ˙ φ and F ( R ) ℓm = E ( R ) ℓm / 1 ˙ φ . Eqs.(4.5), (4.9), and (4.13) indicate that all of the gravitational waveforms, the energy radiation rate, and the radiation reaction forces are based on the effective spacetime. V. CONCLUSIONS In this paper, we investigate the waveforms and energy radiation rate of gravitational waves generated by coalescing spinless binary systems up to 3PM approximation in the EOB theory. We focus on the radiation reaction forces in the Hamilton equation, which can be described by the energy radiation rate d E d t = 1 4 πGω 2 ∫ | Ψ B 4 | 2 r 2 dΩ and the 'plus' and 'cross' modes of gravitational waves, which are related to the null tetrad components of the gravitational perturbed Weyl tensor by Ψ B 4 = 1 2 ( h + -i h × ). Clearly, to find the energy radiation rate and construct gravitational waveforms, the key step is to seek the solution of Ψ B 4 . Therefore, the main task of this paper is to solve the decoupled and separated equations of the null tetrad components of gravitational perturbed Weyl tensor Ψ B 4 in the effective spacetime by employing the Green's function method. To achieve this goal, noting that the potential function in the radial Teukolsky-like equation is a long-range potential, we first transform it into an S-N-like equation, which has a short-range potential. Then, by expanding the homogeneous S-N-like equation with η = 2 GMω , where ω denotes the angular frequency of the wave, we derive closed analytical expressions for the solutions of each order. These solutions are essential for constructing the Green function and asymptotic amplitude. The lowest-order solution is expressed as a linear combination of spherical Bessel functions, which allows us to perform iterative calculations to obtain higher-order solutions. This approach simplifies the problem and enables us to efficiently study the radial Teukolsky-like equation. In the calculation process, we use a low-frequency approximation and considered the conditions of quasi-circular orbits. These conditions are represented by the relationships z ∝ v , η ∝ v 3 and ω = m Ω. As a result, the obtained results are accurate to O ( v 9 -2( ℓ -2) -ϵ ), which means that the accuracy of the results of this paper reaches the 4.5PN order[67, 68]. This work also presents a more general integral formula than that given by Sasaki [60], which can be extended to higher orders or even arbitrary orders without additional treatments. In Appendix B, the general integral formulas, which can theoretically derive the series solution of the homogeneous S-N-like equation to any order, are presented. However, when constructing the general solution of the nonhomogeneous equation using Green's function method, it is necessary to specify the amplitude at infinity, which requires finding the asymptotic behavior of B J as z → ∞ . Although we know what needs to be done at each step, we have not yet been able to implement our ideas using a computer, but we can obtain specific results through complex calculations. Therefore, combining the outstanding works of Sasaki et al. with the useful formulas presented in Appendix B, we have confidence that this method will yield good results in the future. From the analysis provided, it is evident that the effective metric degenerates into the Schwarzschild case in the test particle limit ( ν → 0). This limit is characterized by vanishing of the coefficients a 2 and a 3 . Therefore, the gravitational waveforms and energy radiation rate calculated in this study were divided into two parts: the Schwarzschild part and the correction part related to PM parameters a 2 and a 3 . Handling spinning binary systems will involve additional complexities. However, the results of this paper and the listed mathematical techniques will be valuable for understanding the energy flux and waveforms in spinning binary systems. ACKNOWLEDGMENTS We would like to thank professors S. Chen and Q. Pan for useful discussions on the manuscript. This work was supported by the Grant of Natural Science Foundation of China No. 12035005, and the National Key Research and Development Program of China No. 2020YFC2201400. VI. APPENDIX A. The definition of coefficient of L ( n ) Associated with L (2) are expressed as follows a (2) ℓ = -36 a 2 +8 λa 2 2 +3 λ [ 5 a 3 -6( λ +2) ( c 2 1 +( c 1 + c 2 )( c 2 -1) ) ] 9 λ ( λ +2) , (A.1) b (2) ℓ = 4 a 2 2 λ ( λ +4) + 5 a 3 λ (2 λ +7) -4 a 2 ( λ 2 +2 λ +6) 6 λ ( λ +2) , (A.2) c (2) ℓ = 3 λa 2 2 +60 λa 3 +9 a 2 ( λ 2 +2 λ +16) 36 λ ( λ +2) , (A.3) d (2) ℓ = 8 a 2 2 λ ( λ +4) + 15 a 3 λ ( λ +4) -24 a 2 ( λ 2 +2 λ +12) 36 λ ( λ +2) , (A.4) e (2) ℓ = -8 a 2 2 λ ( λ 2 +19 λ +22) -15 a 3 λ (2 λ 2 +27 λ +34) + 12 a 2 (23 λ 2 +34 λ -48) 72 λ ( λ +2) . (A.5) Related with L (3) are a (3) ℓ = -2( λa 2 -9)(36 a 2 +8 a 2 2 λ +15 a 3 λ ) 27 λ 2 (2 + λ ) 2 (A.6) b (3) ℓ = -1 18 λ 2 ( λ +2) 2 { 16 a 3 2 λ (2 λ +5) -8 a 2 2 λ (2 λ 2 -5 λ +6) -3 a 2 [ 3 λ 4 + (12 -35 a 3 ) λ 3 +(36 -80 a 3 ) λ 2 +48 λ -144 ] +3 λ [ -a 3 ( 10 λ 2 -55 λ -60 ) + 12( c 1 -1)( c 2 -1)( c 1 + c 2 ) λ ( λ +2) 2 ]} , (A.7) c (3) ℓ = 1 216 λ ( λ +2) { 16 a 3 2 λ (7 λ +62) -9 a 3 λ (37 λ +224) -36 a 2 2 ( 3 λ 2 +26 λ -24 ) + 90 a 2 ( 24 + a 3 λ (5 λ +38) ) } , (A.8) d (3) ℓ = -2( λa 2 -9)(36 a 2 +8 a 2 2 λ +15 a 3 λ ) 27 λ 2 ( λ +2) 2 , (A.9) (A.10) e (3) ℓ = 1 216 λ 2 ( λ +2) 2 { -96 a 2 2 λ 2 (2 λ -5) + 64 a 3 2 λ 2 (6 λ +13) + 12 a 2 [ 9 λ 4 -144 λ +864+ 3(35 a 3 +12) λ 3 +4(55 a 3 -9) λ 2 ] +9 λ [ a 3 (3 λ 3 -28 λ 2 +232 λ +480)+ 48 ( c 2 1 +( c 2 -1)( c 1 + c 2 ) ) λ ( λ +2) 2 ]} , f (3) ℓ = 1 108 λ 2 ( λ +2) 2 { 8 a 3 2 λ 2 ( 3 λ 2 +44 λ +64 ) -24 a 2 2 λ ( 3 λ 3 +23 λ 2 +34 λ -48 ) + 3 a 2 [ 3 λ 4 ( 10 a 3 -8( c 2 1 +( c 2 -1)( c 1 + c 2 )) -3 ) +576 λ -1728 + λ 3 ( 370 a 3 -12 ( 8( c 2 1 + ( c 2 -1)( c 1 + c 2 )) + 3 ) ) + λ 2 ( 560 a 3 -12 ( 8( c 2 1 +( c 2 -1)( c 1 + c 2 )) -39 ) )] -9 λ [ 5 a 3 ( 4 λ 3 +27 λ 2 +80 λ +48 ) -12( c 1 -1)( c 2 -1)( c 1 + c 2 ) λ ( λ +2) 2 ] } , (A.11) g (3) ℓ = 1 216 λ ( λ +2) { -8 a 3 2 λ ( λ 2 +62 λ +48 ) +12 a 2 2 ( 31 λ 2 +2 λ +192 ) -3 a 2 ( 15 a 3 λ 3 +650 a 3 λ 2 +8(65 a 3 -18) λ -1440 ) +72 a 3 λ (23 λ +16) } . (A.12) where λ = ( ℓ -1)( ℓ +2). B. How to calculate the ξ ( n ) ℓ Figure 2: The figure shows how to calculate the integral in Eq. (3.26). ζ i and ζ ∗ i stands for j i or n i , i is an arbitrary integer. 1 involves transforming the integrand using the properties of spherical Bessel functions, while 2 yields different integral expressions depending on the type of Q ( z ). Figure 2: The figure shows how to calculate the integral in Eq. (3.26). ζ i and ζ ∗ i stands for j i or n i , i is an arbitrary integer. 1 involves transforming the integrand using the properties of spherical Bessel functions, while 2 yields different integral expressions depending on the type of Q ( z ). The integral appearing in Eq. (3.26) involves spherical Bessel functions with different quantum numbers. By utilizing the following formula, 2 ℓ +1 z ζ ℓ = ζ ℓ -1 + ζ ℓ +1 , ζ is j or n, (B.1) n ℓ = ( -1) ℓ +1 j -ℓ -1 , (B.2) j ℓ = ( -1) ℓ n -ℓ -1 , (B.3) j 0 = sin z z , n 0 = -cos z z . (B.4) we can express any quantum number ℓ of j ℓ and n ℓ in the form of poly1 ( z ) j 0 + poly2 ( z ) n 0 . poly1 ( z ) and poly2 ( z ) are both polynomials in 1 /z , while j 0 and n 0 have elementary function representations. This transformation allows for easier computation of the integral expression. Based on our research in mathematical manuals and insights from Sasaki et al. [60], we have classified these integrals and derived formal integral formulas, as depicted in Fig. 2, outlining the specific implementation route. 1. Generalized sine integral function B J and generalized spherical Bessel function D J ℓ We first introduce some special integral formulas from Ref. [60], the generalized sine integral function B J B jJ = ∫ z z ∗ zj 0 D J 0 d z, (B.5) B nJ = ∫ z z ∗ zn 0 D J 0 d z. (B.6) And the generalized spherical Bessel functions D J ℓ : D j ℓ = j ℓ , D n ℓ = n ℓ , (B.7) D nJ ℓ = n ℓ B jJ -j ℓ B nJ , (B.8) D jJ ℓ = j ℓ B jJ + n ℓ B nJ . (B.9) From the above definitions, we can derive the following expressions when J = j, n : B jj = ∫ z 0 zj 0 j 0 d z = -1 2 C ( z ) , (B.10) B nj = ∫ z 0 zn 0 j 0 d z = -1 2 S ( z ) , (B.11) B jn = ∫ z 0 zj 0 n 0 d z = -1 2 S ( z ) , (B.12) B nn = ∫ z z ∗ zn 0 n 0 d z = -B jj +ln z, (B.13) S ( z ) = Si (2 z ) = ∫ 2 z 0 sin t t d t ( | z | < ∞ ) , (B.14) C ( z ) = -∫ ∞ 2 z cos t t d t -γ -ln 2 z | arg z | < π. (B.15) where B J is essential to note that when J ends with j , there may be a logarithmic divergence issue when the integrand is combined with n 0 . This occurs because we use n 2 0 = 1 z 2 -j 2 0 , which results in 1 z 2 diverging when z ∗ = 0. Therefore, the terms related to j 2 0 have their lower limit set at z ∗ = 0, while those related to 1 z 2 are set at z ∗ = 1. It is worth noting that all strings J ending with the character n in B J can be expressed in terms of those whose J end with j , for example, B jn = B nj , (B.16) B jjn = -B nnj + B nj ln z, (B.17) B jnn =2 B jjj + B nnj -B jj ln z, (B.18) B nnn =2 B njj -B jnj -B nj ln z. (B.19) Using these relations, we can express all the D J ℓ whose J end with n in terms of those whose J end with j . Figure 3: The image of B J for J of length 2 is given on the left, and the image of B J for J of length 3 is given on the right. Figure 3: The image of B J for J of length 2 is given on the left, and the image of B J for J of length 3 is given on the right. 2. the general integral form of Q ( z ) For k > -1 and k ∈ N , with Q ( z ) being differentiable functions, we can establish the following integral properties: ∫ Q ( z ) z k j 2 0 d z = 2 1 + k { -∫ Q ( z ) z k -1 n 0 j 0 d z -1 2 Q ( z ) z k -1 j 2 0 + 1 2 ∫ Q ( z ) ' z k -1 j 2 0 d z } , (B.20) ∫ Q ( z ) z k n 2 0 d z = 2 1 + k { ∫ Q ( z ) z k -1 n 0 j 0 d z -1 2 Q ( z ) z k -1 n 2 0 + 1 2 ∫ Q ( z ) ' z k -1 n 2 0 d z } , (B.21) ∫ Q ( z ) z k n 0 j 0 d z = 2 1+ k { ∫ Q ( z ) z k -1 j 2 0 d z -1 2 ∫ Q ( z ) z k +1 d z -1 2 Q ( z ) z k -1 n 0 j 0 + 1 2 ∫ Q ( z ) ' z k -1 n 0 j 0 d z } , 2 1+ k { 1 2 ∫ Q ( z ) z k +1 d z -∫ Q ( z ) z k -1 n 2 0 d z -1 2 Q ( z ) z k -1 n 0 j 0 + 1 2 ∫ Q ( z ) ' z k -1 n 0 j 0 d z } . (B.22) and for k = -1, we define ∫ zQ ( z ) ζ 0 ζ ∗ 0 d z as a new function. This enables us to establish a complete recursive relationship, allowing us to obtain the desired expressions by letting the computer perform a finite number of iterations for a specific level. 3. the case of Q ( z ) = (ln z ) m , ( m ⩾ 0 , m ∈ N ) Defining E m k = ∫ j 2 0 (ln z ) m z k d z and F m k = ∫ j 0 n 0 (ln z ) m z k d z , m = 0 , 1 , 2 , . . . and k = -1 , 0 , 1 , 2 , . . . , we can derive the following expressions utilizing the integral formulas (B.20)-(B.22) mentioned above: -1 m k ' = ⇒ E m -1 , E m k ' k ' = 0 , 1 , 2 , . . . (B.23) -1 m 0 k '' = ⇒ F m -1 , F m 0 , F m k '' k '' = 1 , 2 , . . . (B.24) E m -1 = GenlogS j m , F m -1 = GenlogU j m , (B.25) F m 0 = 2 { E m -1 -1 2( m +1) (ln z ) m +1 -1 2 zn 0 j 0 (ln z ) m + m 2 F m -1 0 } , (B.26) E m k ' = 2 1 + k ' { -F m k ' -1 -1 2 j 2 0 (ln z ) m z k ' -1 + m 2 E m -1 k ' } , (B.27) F m k '' = 2 1 + k '' { E m k '' -1 + 1 2 ei k '' ln z -m (ln z ) m +1 -1 2 n 0 j 0 (ln z ) m z k '' -1 + m 2 F m -1 k '' } , (B.28) ei z n is an exponential integral of order n , ei z n = ∫ ∞ 1 e -zt t n d t . where 4. the case of Q ( z ) = (ln z ) m B jJ or (ln z ) m B nJ , ( m ⩾ 0 , m ∈ N ) Defining S m,J k = -∫ j 0 D nJ 0 (ln z ) m z k d z and U m,J k = -∫ n 0 D nJ 0 (ln z ) m z k d z , we can derive the following expressions utilizing the integral formulas mentioned above: -1 m j 0 J ' k '' = ⇒ S m,J -1 , S m,J 0 , S m,J k '' U m,J -1 , U m,J 0 , U m,J k '' = ⇒ Π m,jJ 1 , Π m,jJ ˜ k Π m,nJ 1 , Π m,nJ ˜ k (B.29) where J ' is a string of length greater than two, and ˜ k = 2 , 3 , 4 , . . . , the terms S m,J 0 , S m,J k '' can actually be merged into a single term S m,J k ' . The same holds true for U m,J 0 , U m,J k '' , as the expressions are entirely determined by Π m,J k '' due to the parameters k ' and J . In the following, we provide the corresponding expressions. S m,J -1 = -GenlogS nJ m , U m,J -1 = -GenlogU nJ m , (B.30) S m,J k ' = 2 1 + k ' { 1 2 j 0 D nJ 0 (ln z ) m z k ' -1 -1 2 Π m,jJ k ' +1 -U m,J k ' -1 + m 2 S m -1 ,J k ' } , (B.31) U m,J k ' = 2 1 + k ' { S m,J k ' -1 -1 2 Π m,nJ k ' +1 + 1 2 n 0 D nJ 0 (ln z ) m z k ' -1 + m 2 U m -1 ,J k ' } , (B.32) The definitions of Π m,jJ k ' and Π m,nJ k ' are, Π m,jJ k ' = ∫ B jJ (ln z ) m z k ' d z = 1 1 -k ' ( B jJ (ln z ) m z k ' -1 -m Π m -1 ,jJ k ' -∫ j 0 D J 0 z k ' -2 (ln z ) m d z ) , k ' > 1 1 m +1 ( B jJ (ln z ) m +1 -GenlogS J m +1 ) , k ' = 1 (B.33) Π m,nJ k ' = ∫ B nJ (ln z ) m z k ' d z = 1 1 -k ' ( B nJ (ln z ) m z k ' -1 -m Π m -1 ,nJ k ' -∫ n 0 D J 0 z k ' -2 (ln z ) m d z ) , k ' > 1 1 m +1 ( B nJ (ln z ) m +1 -GenlogU J m +1 ) , k ' = 1 (B.34) We will then provide the specific expressions of Π m,jJ k ' and Π m,nJ k ' for different values of k ' and J : Π m,jj 1 = B jj (ln z ) m +1 -GenlogS j m +1 m +1 , Π m,nj 1 = B nj (ln z ) m +1 -GenlogU j m +1 m +1 , (B.35) Π m,jjJ 1 = B jjJ (ln z ) m +1 -GenlogS jJ m +1 m +1 , Π m,njJ 1 = B njJ (ln z ) m +1 -GenlogU jJ m +1 m +1 , (B.36) Π m,jnJ 1 = B jnJ (ln z ) m +1 -GenlogS nJ m +1 m +1 , Π m,nnJ 1 = B nnJ (ln z ) m +1 -GenlogU nJ m +1 m +1 , (B.37) Π m,jj ˜ k = 1 1 -˜ k ( B jj (ln z ) m z ˜ k -1 -m Π m -1 ,jj ˜ k -E m ˜ k -2 ) , (B.38) Π m,nj ˜ k = 1 1 -˜ k ( B nj (ln z ) m z ˜ k -1 -m Π m -1 ,nj ˜ k -F m ˜ k -2 ) , (B.39) Π m,jjJ ˜ k = 1 1 -˜ k ( B jjJ (ln z ) m z ˜ k -1 -m Π m -1 ,jjJ ˜ k -Π m,jJ ˜ k -U m,J ˜ k -2 ) , (B.40) Π m,njJ ˜ k = 1 1 -˜ k ( B njJ (ln z ) m z ˜ k -1 -m Π m -1 ,njJ ˜ k -Π m,nJ ˜ k + S m,J ˜ k -2 ) , (B.41) Π m,jnJ ˜ k = 1 1 -˜ k ( B jnJ (ln z ) m z ˜ k -1 -m Π m -1 ,jnJ ˜ k + S m,J ˜ k -2 ) , (B.42) Π m,nnJ ˜ k = 1 1 -˜ k ( B nnJ (ln z ) m z ˜ k -1 -m Π m -1 ,nnJ ˜ k + U m,J ˜ k -2 ) , (B.43) The definitions of GenlogS J m and GenlogU J m are, GenlogS J m = ∫ zj 0 D J 0 (ln z ) m d z = m ∑ i =0 ( -1) i m ! ( m -i )! (ln z ) m -i theB jJ i +1 , (B.44) GenlogU J m = ∫ zn 0 D J 0 (ln z ) m d z = m ∑ i =0 ( -1) i m ! ( m -i )! (ln z ) m -i theB nJ i +1 , (B.45) where ϑ 1 [ A ] = { (1 , A ) } , (B.46) ϑ n [ A ] 2 µ -1 , ϑ n [ A ] 2 µ = 1 . ( ϑ n -1 [ A ] µ 1 , jjϑ n -1 [ A ] µ 2 | -1 2 ) , ( ϑ n -1 [ A ] µ 1 , nnϑ n -1 [ A ] µ 2 | -1 2 ) , when ϑ n -1 [ A ] µ 2 | 1 1 = j, 2 . ( ϑ n -1 [ A ] µ 1 , njϑ n -1 [ A ] µ 2 | -1 2 ) , ( -ϑ n -1 [ A ] µ 1 , jnϑ n -1 [ A ] µ 2 | -1 2 ) , when ϑ n -1 [ A ] µ 2 | 1 1 = n. (B.47) thB jJ i = ϑ i [ jJ ] ν 1 B ϑ i [ jJ ] ν 2 , (B.48) thB nJ i = ϑ i [ nJ ] ν 1 B ϑ i [ nJ ] ν 2 , (B.49) in this context, µ = 1 , 2 , . . . , dim[ ϑ n -1 [ A ]], ϑ n [ A ] ij represents the ( i, j )-th element of ϑ n [ A ], dim[ ϑ n ] denotes the dimension of ϑ n [ A ], ϑ n -1 [ A ] µ 2 | b a refers to the characters within the range from the a -th to the b -th in ϑ n -1 [ A ] µ 2 , and ϑ i [ A ] ν 1 B ϑ i [ A ] ν 2 implies summation over ν . We shall provide expressions for the first five terms of thB jJ i and thB nJ i , even though even these initial terms are quite intricate: thB jJ 1 = B jJ , thB nJ 1 = B nJ , (B.50) thB jJ 2 = B jjJ + B nnJ , thB nJ 2 = B njJ -B jnJ , (B.51) thB jJ 3 = B jjjJ + B nnjJ + B njnJ -B jnnJ , thB nJ 3 = B njjJ -B jnjJ -B jjnJ -B nnnJ , (B.52) thB jJ 4 = B jjjjJ + B nnjjJ + B njnjJ -B jnnjJ + B njjnJ -B jnjnJ -B jjnnJ -B nnnnJ , (B.53) thB nJ 4 = B njjjJ -B jnjjJ -B jjnjJ -B nnnjJ -B jjjnJ -B nnjnJ -B njnnJ + B jnnnJ , (B.54) thB jJ 5 = B jjjjjJ + B nnjjjJ + B njnjjJ -B jnnjjJ + B njjnjJ -B jnjnjJ -B jjnnjJ -B nnnnjJ + B njjjnJ - B jnjjnJ - B jjnjnJ - B nnnjnJ - B jjjnnJ - B nnjnnJ - B njnnnJ + B jnnnnJ , (B.55) thB nJ 5 = B njjjjJ -B jnjjjJ -B jjnjjJ -B nnnjjJ -B jjjnjJ -B nnjnjJ -B njnnjJ + B jnnnjJ -B jjjjnJ -B nnjjnJ -B njnjnJ + B jnnjnJ -B njjnnJ + B jnjnnJ + B jjnnnJ + B nnnnnJ . (B.56) C. the expressions of ˜ Z ( R ) ℓmω 0 We present the expression of ˜ Z ( R ) ℓmω 0 for mode ( ℓ, m ) = (2 , 2) , (2 , 1) , (3 , 3) in which each mode contains terms up to O ( v 9 -2( ℓ -2) -ϵ ). ˜ Z ( R ) 22 ω 0 = a 2 v 2 +2 ia 2 v 3 + 308 a 2 2 -919 a 2 +420 a 3 378 v 4 + ( 41 ia 2 2 18 + a 2 ( 12 i ln v -238 i 27 + 4 i elg +2 π +12 i ln 2 ) + 11 ia 3 6 ) v 5 + ( 44 a 3 2 81 -( elg +ln2 v ) ( 256 a 2 2 945 + 152 a 2 21 + 40 a 3 63 ) + π ( 128 ia 2 2 945 + 76 ia 2 21 + 20 ia 3 63 ) +ln2 ( -256 a 2 2 945 -320 a 2 21 -40 a 3 63 ) -173707 a 2 2 66150 + 85 a 2 a 3 54 -16 a 2 ln v + 891691 a 2 158760 -4553 a 3 5292 ) v 6 + ( 505 ia 3 2 243 -54365 ia 2 2 6804 +( elg +ln2 v ) ( 88 ia 2 2 27 -1838 ia 2 189 + 40 ia 3 9 ) + ( 176 ia 2 2 27 -3676 ia 2 189 + 80 ia 3 9 ) ln 2 v + π ( 44 a 2 2 27 -919 a 2 189 + 20 a 3 9 ) + 403 ia 2 a 3 81 + 67805 ia 2 6804 -17005 ia 3 2268 ) v 7 + ( 2 a 4 2 27 -291169 a 3 2 132300 + 55 a 2 2 a 3 162 + 16150588 a 2 2 2083725 + ( -256 a 3 2 945 -513614 a 2 2 19845 -40 a 2 a 3 63 + 70328 a 2 735 -26966 a 3 1323 ) ln v + π ( 128 ia 3 2 945 + 75997 ia 2 2 19845 + 20 ia 2 a 3 63 -83236 ia 2 6615 + 3781 ia 3 1323 ) + elg ( -256 a 3 2 945 -151994 a 2 2 19845 -40 a 2 a 3 63 + 166472 a 2 6615 -7562 a 3 1323 ) +ln2 ( -512 a 3 2 945 -484798 a 2 2 19845 -80 a 2 a 3 63 + 188728 a 2 2205 -24826 a 3 1323 ) -1885 a 2 a 3 441 -72 a 2 (ln 2 v ) 2 + 24 iπa 2 ln 2 v -48 elg a 2 ln 2 v -8 elg 2 a 2 + 2 π 2 a 2 3 + 2319196003 a 2 366735600 +8 elg iπa 2 + 125 a 2 3 864 + 1009699 a 3 222264 ) v 8 + ( 352 ia 4 2 243 -8946247 ia 3 2 1786050 -( 1 315 1024 ia 2 2 + 944 ia 2 7 + 160 ia 3 21 ) (ln v ) 2 + ln 2 ( -1 105 1024 ia 2 2 -1824 ia 2 7 -160 ia 3 7 ) ln v + π ln v ( -2048 a 2 2 945 -880 a 2 21 -320 a 3 63 ) + elg ln v ( -1 945 4096 ia 2 2 -1760 ia 2 21 -640 ia 3 63 ) + 440 81 ia 2 2 a 3 -elg 2 ( 1024 ia 2 2 945 + 272 ia 2 21 + 160 ia 3 63 ) + π 2 ( 256 ia 2 2 567 + 4 ia 2 63 + 200 ia 3 189 ) -elg π ( 1024 a 2 2 945 + 272 a 2 21 + 160 a 3 63 ) -(ln 2) 2 ( 2048 ia 2 2 315 + 880 ia 2 7 + 320 ia 3 21 ) -π ln 2 ( 512 a 2 2 189 + 848 a 2 21 + 400 a 3 63 ) -260495 ia 2 a 3 23814 elg ln 2 ( 1024 ia 2 2 189 + 1696 ia 2 21 + 800 ia 3 63 ) + 1183663 ia 2 2 138915 + ( 17456 ia 3 2 2835 -2942726 ia 2 2 99225 + 3410 ia 2 a 3 189 + 716683 ia 2 13230 -11059 ia 3 1323 -5136 i 245 ) ln v + π ( 3016 a 3 2 2835 -339121 a 2 2 99225 + 635 a 2 a 3 189 + 14639 a 2 15876 + 191 a 3 294 ) + elg ( 6032 ia 3 2 2835 -678242 ia 2 2 99225 + 1270 ia 2 a 3 189 + 14639 ia 2 7938 + 191 ia 3 147 ) +ln2 ( 5776 ia 3 2 945 -2578726 ia 2 2 99225 + 3490 ia 2 a 3 189 + 443851 ia 2 13230 -4787 ia 3 1323 -5136 i 245 ) + 2503003411 ia 2 91683900 + 175 ia 2 3 108 -15511 ia 3 16464 ) v 9 , (C.1) ˜ Z ( R ) 21 ω 0 = a 2 3 v 2 + 23 ia 2 36 v 3 + ( 5 a 2 2 72 -20 a 2 63 + a 3 12 ) v 4 + ( 101 ia 2 2 432 +2 ia 2 ln v + 2 ia 2 ( elg +ln4) 3 -37 ia 2 56 + πa 2 3 + ia 3 6 ) v 5 + ( 23 a 3 2 324 +( elg +ln2 v ) ( -64 a 2 2 945 -137 a 2 126 -10 a 3 63 ) + π ( 32 ia 2 2 945 + 137 ia 2 252 + 5 ia 3 63 ) + 139673 a 2 2 1587600 + 17 a 2 a 3 108 -23 9 a 2 ln v + 3709 a 2 158760 -23 18 a 2 ln 2 + 2143 a 3 5292 ) v 6 + ( 199 ia 3 2 2592 + ( 5 ia 2 2 36 -40 ia 2 63 + ia 3 6 )( ln 4 v 3 + elg ) + π ( 5 a 2 2 72 -20 a 2 63 + a 3 12 ) -3581 ia 2 2 18144 + 55 ia 2 a 3 216 + 5291 ia 2 4536 + 2533 ia 3 3024 ) v 7 + ( -( 64 a 3 2 2835 + 10 a 2 a 3 189 ) (ln 2 v + elg ) + 452071 a 3 2 9525600 -( 101 a 2 2 216 -37 a 2 28 + a 3 3 ) ln(2 v 2 ) + ( -6403 a 2 2 17640 + 4643 a 2 8820 -209 a 3 882 ) (ln 2 v + elg ) + 15354247 a 2 2 133358400 + π ( 32 ia 3 2 2835 + 6403 ia 2 2 35280 + 5 ia 2 a 3 189 -4643 ia 2 17640 + 209 ia 3 1764 ) + 7037 a 2 a 3 31752 -8 3 a 2 ln 2 ( ln 2 v 3 + elg -iπ 2 ) -6 a 2 (ln v ) 2 -4 ( elg -iπ 2 ) a 2 ln v + π 2 a 2 18 -2 elg 2 a 2 3 + 159578843 a 2 122245200 + 2 i elg πa 2 3 + 3985 a 3 296352 ) v 8 . (C.2) ˜ Z ( R ) 33 ω 0 = 4 a 2 3 v 2 + 29 ia 2 12 v 3 + ( 4 a 2 2 3 -5 a 2 + 3 a 3 2 ) v 4 + ( 511 ia 2 2 144 +8 ia 2 ( ln(12 v 3 ) + elg -iπ 2 ) -2951 ia 2 120 + 47 ia 3 24 ) v 5 + ( 92 a 3 2 81 -( 10 a 2 2 21 + 191 a 2 14 + 15 a 3 14 )( ln 6 v + elg -iπ 2 ) -122987 a 2 2 21168 + 77 a 2 a 3 27 -29 2 a 2 ln 2 v 2 + 571751 a 2 21560 -16229 a 3 7056 ) v 6 + ( 1117 ia 3 2 288 -745 ia 3 32 + 10256 ia 2 165 + ( 8 ia 2 2 -30 ia 2 +9 ia 3 )( ln(12 v 3 ) + elg -iπ 2 ) -39121 ia 2 2 1440 + 353 ia 2 a 3 48 ) v 7 . (C.3) [1] B. P. Abbott et al. (LIGO Scientific, Virgo), GW151226: Observation of Gravitational Waves from a 22-Solar-Mass Binary Black Hole Coalescence, Phys. Rev. Lett. 116 , 241103 (2016), arXiv:1606.04855 [gr-qc]. [2] B. P. Abbott et al. (LIGO Scientific and Virgo Collaboration), Gw170104: Observation of a 50-solar-mass binary black hole coalescence at redshift 0.2, Phys. Rev. Lett. 118 , 221101 (2017). [3] B. P. Abbott et al. (LIGO Scientific, Virgo), GW170814: A Three-Detector Observation of Gravitational Waves from a Binary Black Hole Coalescence, Phys. Rev. Lett. 119 , 141101 (2017), arXiv:1709.09660 [gr-qc]. [4] B. P. Abbott et al. (LIGO Scientific, Virgo), GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral, Phys. Rev. Lett. 119 , 161101 (2017), arXiv:1710.05832 [gr-qc]. [5] R. Abbott et al. (LIGO Scientific, Virgo), GW190412: Observation of a Binary-Black-Hole Coalescence with Asymmetric Masses, Phys. Rev. D 102 , 043015 (2020), arXiv:2004.08342 [astro-ph.HE]. [6] R. Abbott et al. (LIGO Scientific, Virgo), GWTC-2: Compact Binary Coalescences Observed by LIGO and Virgo During the First Half of the Third Observing Run, Phys. Rev. X 11 , 021053 (2021), arXiv:2010.14527 [gr-qc]. [7] R. Abbott et al. (KAGRA, Virgo, LIGO Scientific), All-sky, all-frequency directional search for persistent gravitational waves from Advanced LIGO's and Advanced Virgo's first three observing runs, Phys. Rev. D 105 , 122001 (2022), arXiv:2110.09834 [gr-qc]. [8] A. H. Nitz, C. D. Capano, S. Kumar, Y.-F. Wang, S. Kastha, M. Schafer, R. Dhurkunde, and M. Cabero, 3-OGC: Catalog of gravitational waves from compact-binary mergers, The Astrophysical Journal 922 , 76 (2021). [9] R. Abbott et al. (LIGO Scientific, VIRGO, KAGRA), GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo During the Second Part of the Third Observing Run, (2021), arXiv:2111.03606 [gr-qc]. [10] R. Abbott et al. (LIGO Scientific, VIRGO), GWTC-2.1: Deep Extended Catalog of Compact Binary Coalescences Observed by LIGO and Virgo During the First Half of the Third Observing Run, (2021), arXiv:2108.01045 [gr-qc]. [11] T. Damour and G. Schaeer, Higher-order relativistic periastron advances and binary pulsars, Il Nuovo Cimento B 101 , 127 (1988). [12] A. Buonanno and T. Damour, Effective one-body approach to general relativistic two-body dynamics, Physical Review D 59 , 10.1103/physrevd.59.084006 (1999). [13] A. Buonanno, G. B. Cook, and F. Pretorius, Inspiral, merger, and ring-down of equal-mass black-hole binaries, Physical Review D 75 , 10.1103/physrevd.75.124018 (2007). [14] T. Damour, P. Jaranowski, and G. Schafer, Determination of the last stable orbit for circular general relativistic binaries at the third post-newtonian approximation, Physical Review D 62 , 10.1103/physrevd.62.084011 (2000). [15] T. Damour, Gravitational scattering, post-minkowskian approximation, and effective-onebody theory, Physical Review D 94 , 10.1103/physrevd.94.104015 (2016). [16] D. Bini and T. Damour, Gravitational spin-orbit coupling in binary systems at the second post-minkowskian approximation, Physical Review D 98 , 10.1103/physrevd.98.044036 (2018). [17] T. Damour and P. Rettegno, Strong-field scattering of two black holes: Numerical relativity meets post-minkowskian gravity, Physical Review D 107 , 10.1103/physrevd.107.064051 (2023). [18] C. Dlapa, G. Kalin, Z. Liu, and R. A. Porto, Conservative dynamics of binary systems at fourth post-minkowskian order in the large-eccentricity expansion, Physical Review Letters 128 , 10.1103/physrevlett.128.161104 (2022). [19] C. Dlapa, G. Kalin, Z. Liu, J. Neef, and R. A. Porto, Radiation reaction and gravita-t onal waves at fourth post-minkowskian order, Physical Review Letters 130 , 10.1103/physrevlett.130.101401 (2023). [20] G. Kalin, Z. Liu, and R. A. Porto, Conservative dynamics of binary systems to third postminkowskian order from the effective field theory approach, Physical Review Letters 125 , 10.1103/physrevlett.125.261103 (2020). [21] T. Damour, Introductory lectures on the effective one body formalism, International Journal of Modern Physics A 23 , 1130 (2008). [22] T. Damour and A. Nagar, Comparing effective-one-body gravitational waveforms to accurate numerical data, Physical Review D 77 , 10.1103/physrevd.77.024043 (2008). [23] T. Damour and A. Nagar, The Effective One Body description of the Two-Body problem, Fundam. Theor. Phys. 162 , 211 (2011), arXiv:0906.1769 [gr-qc]. [24] T. Damour and A. Nagar, Faithful effective-one-body waveforms of small-mass-ratio coalescing black hole binaries, Physical Review D 76 , 10.1103/physrevd.76.064028 (2007). [25] T. Damour, Classical and quantum scattering in post-Minkowskian gravity, Phys. Rev. D 102 , 024060 (2020). [26] A. Buonanno and T. Damour, Transition from inspiral to plunge in binary black hole coalescences, Physical Review D 62 , 10.1103/physrevd.62.064015 (2000). [27] E. Poisson, Gravitational radiation from a particle in circular orbit around a black hole. i. analytical results for the nonrotating case, Physical Review D 47 , 1497 (1993). [28] S. Long, Y. Zou, and J. Jing, Reconstruction of gravitational waveforms of coalescing spinless binaries in EOB theory based on PM approximation, Classical and Quantum Gravity 10.1088/1361-6382/acfdee (2023). [29] J. Jing, S. Long, W. Deng, M. Wang, and J. Wang, New self-consistent effective one-body theory for spinless binaries based on the post-Minkowskian approximation, Sci. China Phys. Mech. Astron. 65 , 100411 (2022), arXiv:2208.02420 [gr-qc]. [30] J. Jing, W. Deng, S. Long, and J. Wang, Self-consistent effective-one-body theory for spinning binaries based on post-minkowskian approximation, Science China Physics, Mechanics, and Astronomy 66 , 270411 (2023). [31] X. He, M. Sun, J. Jing, and Z. Cao, Energy map and effective metric in an effective-one-body theory based on the second-post-Minkowskian approximation, European Physical Journal C 81 , 97 (2021). [32] J. Jing, W. Deng, S. Long, and J. Wang, Effective metric of spinless binaries with radiationreaction effect up to fourth post-minkowskian order in effective-one-body theory, The European Physical Journal C 83 , 10.1140/epjc/s10052-023-11705-6 (2023). [33] S. A. Teukolsky, Perturbations of a Rotating Black Hole. I. Fundamental Equations for Gravitational, Electromagnetic, and Neutrino-Field Perturbations, Astrophys. J. 185 , 635 (1973). [34] J. M. Bardeen and W. H. Press, Radiation fields in the Schwarzschild background, Journal of Mathematical Physics 14 , 7 (1973). [35] S. Chandrasekhar, On the equations governing the perturbations of the schwarzschild black hole, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 343 , 289 (1975). [36] S. Chandrasekhar, The mathematical theory of black holes (1983). [37] S. Mano, H. Suzuki, and E. Takasugi, Analytic solutions of the teukolsky equation and their low frequency expansions, Progress of Theoretical Physics 95 , 1079 (1996). [38] M. Sasaki and H. Tagoshi, Analytic black hole perturbation approach to gravitational radiation, Living Reviews in Relativity 6 , 10.12942/lrr-2003-6 (2003). [39] G. B. Cook and M. Zalutskiy, Gravitational perturbations of the kerr geometry: High-accuracy study, Physical Review D 90 , 10.1103/physrevd.90.124021 (2014). [40] P. Fiziev and D. Staicova, Application of the confluent heun functions for finding the quasinormal modes of nonrotating black holes, Physical Review D 84 , 10.1103/physrevd.84.127502 (2011). [41] E. W. Leaver, Solutions to a generalized spheroidal wave equation: Teukolsky's equations in general relativity, and the two-center problem in molecular quantum mechanics, Journal of Mathematical Physics 27 , 1238 (1986). [42] An analytic representation for the quasi-normal modes of kerr black holes, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 402 , 285 (1985). [43] E. W. Leaver, Spectral decomposition of the perturbation response of the schwarzschild geometry, Physical Review D 34 , 384 (1986). [44] C. Chen and J. Jing, Radiation fluxes of gravitational, electromagnetic, and scalar perturbations in type-d black holes: an exact approach, J. Cosmol. Astropart. Phys. 11 , 070, arXiv:arXiv: 2307.14616 [gr-qc]. [45] W.-B. Han and Z. Cao, Constructing effective one-body dynamics with numerical energy flux for intermediate-mass-ratio inspirals, Physical Review D 84 , 10.1103/physrevd.84.044014 (2011). [46] A. Nagar, G. Riemenschneider, and G. Pratten, Impact of numerical relativity information on effective-one-body waveform models, Physical Review D 96 , 10.1103/physrevd.96.084045 (2017). [47] A. Nagar, J. Healy, C. O. Lousto, S. Bernuzzi, and A. Albertini, Numerical-relativity validation of effective-one-body waveforms in the intermediate-mass-ratio regime, Physical Review D 105 , 10.1103/physrevd.105.124061 (2022). [48] A. Boh'e et al., Improved effective-one-body model of spinning, nonprecessing binary black holes for the era of gravitational-wave astrophysics with advanced detectors, Phys. Rev. D 95 , 044028 (2017), arXiv:1611.03703 [gr-qc]. [49] S. Xin, B. Chen, R. K. L. Lo, L. Sun, W.-B. Han, X. Zhong, M. Srivastava, S. Ma, Q. Wang, and Y. Chen, Gravitational-wave echoes from spinning exotic compact objects: Numerical waveforms from the Teukolsky equation, Phys. Rev. D 104 , 104005 (2021), arXiv:2105.12313 [gr-qc]. [50] C. Zhang, W.-B. Han, X.-Y. Zhong, and G. Wang, Geometrized effective-one-body formalism for extreme-mass-ratio limits: Generic orbits, Phys. Rev. D 104 , 024050 (2021), arXiv:2102.05391 [gr-qc]. [51] C. Zhang, W.-B. Han, and S.-C. Yang, Analytical effective one-body formalism for extrememass-ratio inspirals with eccentric orbits, Commun. Theor. Phys. 73 , 085401 (2021), arXiv:2001.06763 [gr-qc]. [52] R. Cheng and W.-B. Han, Accurate recalibated waveforms for extreme-mass-ratio inspirals in effective-one-body frame, (2017), arXiv:1706.03884 [gr-qc]. [53] Z. Cao and W.-B. Han, Waveform model for an eccentric binary black hole based on the effective-one-body-numerical-relativity formalism, Phys. Rev. D 96 , 044028 (2017), arXiv:1708.00166 [gr-qc]. [54] R. Gamba, S. Ak¸cay, S. Bernuzzi, and J. Williams, Effective-one-body waveforms for precessing coalescing compact binaries with post-Newtonian twist, Phys. Rev. D 106 , 024020 (2022), arXiv:2111.03675 [gr-qc]. [55] S. Ossokine et al., Multipolar Effective-One-Body Waveforms for Precessing Binary Black Holes: Construction and Validation, Phys. Rev. D 102 , 044055 (2020), arXiv:2004.09442 [gr- qc]. [56] T. Damour and A. Nagar, New effective-one-body description of coalescing nonprecessing spinning black-hole binaries, Phys. Rev. D 90 , 044018 (2014), arXiv:1406.6913 [gr-qc]. [57] N. Afshordi et al. (LISA Consortium Waveform Working Group), Waveform Modelling for the Laser Interferometer Space Antenna, (2023), arXiv:2311.01300 [gr-qc]. [58] M. Sasaki, Post-newtonian expansion of the ingoing-wave regge-wheeler function, Progress of Theoretical Physics 92 , 17 (1994). [59] E. Poisson, Gravitational radiation from a particle in circular orbit around a black hole. i. analytical results for the nonrotating case, Physical Review D 47 , 1497 (1993). [60] Y. Mino, M. Sasaki, M. Shibata, H. Tagoshi, and T. Tanaka, Chapter 1. black hole perturbation, Progress of Theoretical Physics Supplement 128 , 1 (1997). [61] T. Damour and P. Rettegno, Strong-field scattering of two black holes: Numerical relativity meets post-minkowskian gravity, Physical Review D 107 , 10.1103/physrevd.107.064051 (2023). [62] T. Dray, The relationship between monopole harmonics and spin-weighted spherical harmonics, Journal of Mathematical Physics 26 , 1030 (1985). [63] E. Poisson, Gravitational radiation from infall into a black hole: Regularization of the teukolsky equation, Physical Review D 55 , 639 (1997). [64] Y. Pan, A. Buonanno, R. Fujita, E. Racine, and H. Tagoshi, Post-newtonian factorized multipolar waveforms for spinning, nonprecessing black-hole binaries, Phys. Rev. D 83 , 064003 (2011). [65] M. Sasaki, Post-newtonian expansion of the ingoing-wave regge-wheeler function, Progress of Theoretical Physics 92 , 17 (1994). [66] A. Buonanno and T. Damour, Transition from inspiral to plunge in binary black hole coalescences, Physical Review D 62 , 10.1103/physrevd.62.064015 (2000). [67] Y. Pan, A. Buonanno, R. Fujita, E. Racine, and H. Tagoshi, Post-newtonian factorized multipolar waveforms for spinning, nonprecessing black-hole binaries, Phys. Rev. D 83 , 064003 (2011). [68] T. Damour, B. R. Iyer, and A. Nagar, Improved resummation of post-newtonian multipolar waveforms from circularized compact binaries, Physical Review D 79 , 10.1103/physrevd.79.064004 (2009). B J is the generalized integral sinusoidal function, and D J ℓ is the generalized spherical Bessel function in Appendix B. with where ς ( n ) ℓ for ℓ = 2 can be expressed as follows: Inserting these expressions into Eq. (3.18) and expanding the result with respect to η , we find that The corresponding imaginary parts are expressed as follows: where",
"pages": [
8,
9,
10,
11,
12,
13
]
},
{
"title": "3. Coefficient of amplitude A in ℓ",
"content": "Noting e -iη ( -b 1 ln( z -c 1 η )+ b 2 ln( z -c 2 η )+ b h ln( z -c h η )) = e -iz ∗ e iz z →∞ -→ 1, taking the expressions of the spherical Hankel functions of the first and second kinds h (1) ℓ and h (2) ℓ as and using the asymptotic behavior of B J and D J ℓ in Ref. [60], we obtain the following expression: where elg is the EulerGamma constant ( elg = 0 . 57721 · · · ), RiZ ( n ) is the Riemann zeta function ( RiZ (3) = 1 . 202 · · · ), and the coefficients of A in 2 are",
"pages": [
13
]
},
{
"title": "C. Quasi-circular orbit on the equatorial plane around an EOB",
"content": "In this section, we consider a quasi-circular orbit. In this case, we assume that the orbit lies on the equatorial plane ( θ = π/ 2) without loss of generality. By setting V r ( r 0 ) = ∂V r /∂r ( r 0 ) = 0, the effective energy E and effective angular momentum L are given by where r 0 is the orbital radius. By defining we obtain",
"pages": [
14
]
},
{
"title": "IV. ENERGY RADIATION RATE AND GRAVITATIONAL WAVEFORMS",
"content": "Inserting the aforementioned result of A i ( i = 1 , 2 , 3), B in ℓ , and R in ℓ into Eq. (3.8), we can obtain the following expression: where we define v = ( GM Ω) 1 / 3 , ω 0 = m Ω, ϵ = 1 when ℓ + m = 1, and ϵ = 0 when ℓ + m = 0. We can divide the higher-order term ˜ Z ℓmω 0 into two parts: the ˜ Z ( S ) ℓmω 0 is computed in the Schwarzschild case [64] and the ˜ Z ( R ) ℓmω 0 is the 2PM and 3PM perturbation terms: The explicit expression of ˜ Z ( R ) ℓmω 0 is presented in Appendix C. In the test particle limit, i.e., ν → 0, we note that ˜ Z ( R ) ℓmω 0 vanishes completely because a 2 and a 3 approach 0. That is, our results revert to the Schwarzschild case in the test particle limit. In Eq. (3.9), utilizing the symmetry of the spin-weighted spherical harmonics, s Y ℓ, -m ( π 2 , 0) = ( -1) s + ℓ s Y ℓm ( π 2 , 0), we know that Z ℓ ( -m ) ω = ( -1) ℓ Z ∗ ℓmω , where Z ∗ ℓmω is the complex conjugate of Z ℓmω . In terms of the amplitude Z ℓmω , we find from Eq. (3.9) that the gravitational waveform [13, 27, 65] at infinity is given by with where The energy loss rate along any orbit, in polar coordinates, can be expressed as dE [ g eff µν ] dt = ˙ R F R [ g eff µν ] + ˙ φ F φ [ g eff µν ]. By simply replacing the radial component with zero, an excellent approximation of the radiation reaction forces can be obtained [66]. Thus, from Eq. (4.5), we know that, for given energy ω n , the energy loss rate [13, 27] for the 'plus' and 'cross' modes of the gravitational wave is described by the following expression: with where (d E/ d t ) N = 32 ν 2 v 10 / 5 is the Newtonian quadrupole luminosity and the superscript ∗ denotes the complex conjugation of the corresponding expression. In Figure 1, we present the curves of Π 22 and Π 33 as the symmetric mass ratio ν takes different values. Then, we determine the radiation reaction forces for the 'plus' and 'cross' modes of the gravitational wave as follows: Eqs.(4.5), (4.9), and (4.13) indicate that all of the gravitational waveforms, the energy radiation rate, and the radiation reaction forces are based on the effective spacetime.",
"pages": [
15,
16,
17
]
},
{
"title": "V. CONCLUSIONS",
"content": "In this paper, we investigate the waveforms and energy radiation rate of gravitational waves generated by coalescing spinless binary systems up to 3PM approximation in the EOB theory. We focus on the radiation reaction forces in the Hamilton equation, which can be described by the energy radiation rate d E d t = 1 4 πGω 2 ∫ | Ψ B 4 | 2 r 2 dΩ and the 'plus' and 'cross' modes of gravitational waves, which are related to the null tetrad components of the gravitational perturbed Weyl tensor by Ψ B 4 = 1 2 ( h + -i h × ). Clearly, to find the energy radiation rate and construct gravitational waveforms, the key step is to seek the solution of Ψ B 4 . Therefore, the main task of this paper is to solve the decoupled and separated equations of the null tetrad components of gravitational perturbed Weyl tensor Ψ B 4 in the effective spacetime by employing the Green's function method. To achieve this goal, noting that the potential function in the radial Teukolsky-like equation is a long-range potential, we first transform it into an S-N-like equation, which has a short-range potential. Then, by expanding the homogeneous S-N-like equation with η = 2 GMω , where ω denotes the angular frequency of the wave, we derive closed analytical expressions for the solutions of each order. These solutions are essential for constructing the Green function and asymptotic amplitude. The lowest-order solution is expressed as a linear combination of spherical Bessel functions, which allows us to perform iterative calculations to obtain higher-order solutions. This approach simplifies the problem and enables us to efficiently study the radial Teukolsky-like equation. In the calculation process, we use a low-frequency approximation and considered the conditions of quasi-circular orbits. These conditions are represented by the relationships z ∝ v , η ∝ v 3 and ω = m Ω. As a result, the obtained results are accurate to O ( v 9 -2( ℓ -2) -ϵ ), which means that the accuracy of the results of this paper reaches the 4.5PN order[67, 68]. This work also presents a more general integral formula than that given by Sasaki [60], which can be extended to higher orders or even arbitrary orders without additional treatments. In Appendix B, the general integral formulas, which can theoretically derive the series solution of the homogeneous S-N-like equation to any order, are presented. However, when constructing the general solution of the nonhomogeneous equation using Green's function method, it is necessary to specify the amplitude at infinity, which requires finding the asymptotic behavior of B J as z → ∞ . Although we know what needs to be done at each step, we have not yet been able to implement our ideas using a computer, but we can obtain specific results through complex calculations. Therefore, combining the outstanding works of Sasaki et al. with the useful formulas presented in Appendix B, we have confidence that this method will yield good results in the future. From the analysis provided, it is evident that the effective metric degenerates into the Schwarzschild case in the test particle limit ( ν → 0). This limit is characterized by vanishing of the coefficients a 2 and a 3 . Therefore, the gravitational waveforms and energy radiation rate calculated in this study were divided into two parts: the Schwarzschild part and the correction part related to PM parameters a 2 and a 3 . Handling spinning binary systems will involve additional complexities. However, the results of this paper and the listed mathematical techniques will be valuable for understanding the energy flux and waveforms in spinning binary systems.",
"pages": [
17,
18
]
},
{
"title": "ACKNOWLEDGMENTS",
"content": "We would like to thank professors S. Chen and Q. Pan for useful discussions on the manuscript. This work was supported by the Grant of Natural Science Foundation of China No. 12035005, and the National Key Research and Development Program of China No. 2020YFC2201400.",
"pages": [
18
]
},
{
"title": "A. The definition of coefficient of L ( n )",
"content": "Associated with L (2) are expressed as follows Related with L (3) are where λ = ( ℓ -1)( ℓ +2).",
"pages": [
19,
20
]
},
{
"title": "B. How to calculate the ξ ( n ) ℓ",
"content": "The integral appearing in Eq. (3.26) involves spherical Bessel functions with different quantum numbers. By utilizing the following formula, we can express any quantum number ℓ of j ℓ and n ℓ in the form of poly1 ( z ) j 0 + poly2 ( z ) n 0 . poly1 ( z ) and poly2 ( z ) are both polynomials in 1 /z , while j 0 and n 0 have elementary function representations. This transformation allows for easier computation of the integral expression. Based on our research in mathematical manuals and insights from Sasaki et al. [60], we have classified these integrals and derived formal integral formulas, as depicted in Fig. 2, outlining the specific implementation route.",
"pages": [
20,
21
]
},
{
"title": "1. Generalized sine integral function B J and generalized spherical Bessel function D J ℓ",
"content": "We first introduce some special integral formulas from Ref. [60], the generalized sine integral function B J And the generalized spherical Bessel functions D J ℓ : From the above definitions, we can derive the following expressions when J = j, n : where B J is essential to note that when J ends with j , there may be a logarithmic divergence issue when the integrand is combined with n 0 . This occurs because we use n 2 0 = 1 z 2 -j 2 0 , which results in 1 z 2 diverging when z ∗ = 0. Therefore, the terms related to j 2 0 have their lower limit set at z ∗ = 0, while those related to 1 z 2 are set at z ∗ = 1. It is worth noting that all strings J ending with the character n in B J can be expressed in terms of those whose J end with j , for example, Using these relations, we can express all the D J ℓ whose J end with n in terms of those whose J end with j .",
"pages": [
21,
22
]
},
{
"title": "2. the general integral form of Q ( z )",
"content": "For k > -1 and k ∈ N , with Q ( z ) being differentiable functions, we can establish the following integral properties: and for k = -1, we define ∫ zQ ( z ) ζ 0 ζ ∗ 0 d z as a new function. This enables us to establish a complete recursive relationship, allowing us to obtain the desired expressions by letting the computer perform a finite number of iterations for a specific level. Defining E m k = ∫ j 2 0 (ln z ) m z k d z and F m k = ∫ j 0 n 0 (ln z ) m z k d z , m = 0 , 1 , 2 , . . . and k = -1 , 0 , 1 , 2 , . . . , we can derive the following expressions utilizing the integral formulas (B.20)-(B.22) mentioned above: ei z n is an exponential integral of order n , ei z n = ∫ ∞ 1 e -zt t n d t . where Defining S m,J k = -∫ j 0 D nJ 0 (ln z ) m z k d z and U m,J k = -∫ n 0 D nJ 0 (ln z ) m z k d z , we can derive the following expressions utilizing the integral formulas mentioned above: where J ' is a string of length greater than two, and ˜ k = 2 , 3 , 4 , . . . , the terms S m,J 0 , S m,J k '' can actually be merged into a single term S m,J k ' . The same holds true for U m,J 0 , U m,J k '' , as the expressions are entirely determined by Π m,J k '' due to the parameters k ' and J . In the following, we provide the corresponding expressions. The definitions of Π m,jJ k ' and Π m,nJ k ' are, (B.33) We will then provide the specific expressions of Π m,jJ k ' and Π m,nJ k ' for different values of k ' and J : The definitions of GenlogS J m and GenlogU J m are, where (B.47) in this context, µ = 1 , 2 , . . . , dim[ ϑ n -1 [ A ]], ϑ n [ A ] ij represents the ( i, j )-th element of ϑ n [ A ], dim[ ϑ n ] denotes the dimension of ϑ n [ A ], ϑ n -1 [ A ] µ 2 | b a refers to the characters within the range from the a -th to the b -th in ϑ n -1 [ A ] µ 2 , and ϑ i [ A ] ν 1 B ϑ i [ A ] ν 2 implies summation over ν . We shall provide expressions for the first five terms of thB jJ i and thB nJ i , even though even these initial terms are quite intricate: (B.52) B njjjnJ - B jnjjnJ - B jjnjnJ - B nnnjnJ - B jjjnnJ - B nnjnnJ - B njnnnJ + B jnnnnJ , (B.55)",
"pages": [
22,
23,
24,
25,
26
]
},
{
"title": "C. the expressions of ˜ Z ( R ) ℓmω 0",
"content": "We present the expression of ˜ Z ( R ) ℓmω 0 for mode ( ℓ, m ) = (2 , 2) , (2 , 1) , (3 , 3) in which each mode contains terms up to O ( v 9 -2( ℓ -2) -ϵ ). elg ( -256 a 3 2 945 -151994 a 2 2 19845 -40 a 2 a 3 63 + 166472 a 2 6615 -7562 a 3 1323 ) +ln2 ( -512 a 3 2 945 -484798 a 2 2 19845 -80 a 2 a 3 63 + 188728 a 2 2205 -24826 a 3 1323 ) -1885 a 2 a 3 441 -72 a 2 (ln 2 v ) 2 + 24 iπa 2 ln 2 v -48 elg a 2 ln 2 v -8 elg 2 a 2 + 2 π 2 a 2 3 + 2319196003 a 2 366735600 +8 elg iπa 2 + 125 a 2 3 864 + 1009699 a 3 222264 ) v 8 + ( 352 ia 4 2 243 -8946247 ia 3 2 1786050 -( 1 315 1024 ia 2 2 + 944 ia 2 7 + 160 ia 3 21 ) (ln v ) 2 + ln 2 ( -1 105 1024 ia 2 2 -1824 ia 2 7 -160 ia 3 7 ) ln v + π ln v ( -2048 a 2 2 945 -880 a 2 21 -320 a 3 63 ) + elg ln v ( -1 945 4096 ia 2 2 -1760 ia 2 21 -640 ia 3 63 ) + 440 81 ia 2 2 a 3 -elg 2 ( 1024 ia 2 2 945 + 272 ia 2 21 + 160 ia 3 63 ) + π 2 ( 256 ia 2 2 567 + 4 ia 2 63 + 200 ia 3 189 ) -elg π ( 1024 a 2 2 945 + 272 a 2 21 + 160 a 3 63 ) -(ln 2) 2 ( 2048 ia 2 2 315 + 880 ia 2 7 + 320 ia 3 21 ) -π ln 2 ( 512 a 2 2 189 + 848 a 2 21 + 400 a 3 63 ) -260495 ia 2 a 3 23814 elg ln 2 ( 1024 ia 2 2 189 + 1696 ia 2 21 + 800 ia 3 63 ) + 1183663 ia 2 2 138915 + ( 17456 ia 3 2 2835 -2942726 ia 2 2 99225 + 3410 ia 2 a 3 189 + 716683 ia 2 13230 -11059 ia 3 1323 -5136 i 245 ) ln v + π ( 3016 a 3 2 2835 -339121 a 2 2 99225 + 635 a 2 a 3 189 + 14639 a 2 15876 + 191 a 3 294 ) + elg ( 6032 ia 3 2 2835 -678242 ia 2 2 99225 + 1270 ia 2 a 3 189 + 14639 ia 2 7938 + 191 ia 3 147 ) +ln2 ( 5776 ia 3 2 945 -2578726 ia 2 2 99225 + 3490 ia 2 a 3 189 + 443851 ia 2 13230 -4787 ia 3 1323 -5136 i 245 ) + 2503003411 ia 2 91683900 + 175 ia 2 3 108 -15511 ia 3 16464 ) v 9 , (C.1) for persistent gravitational waves from Advanced LIGO's and Advanced Virgo's first three observing runs, Phys. Rev. D 105 , 122001 (2022), arXiv:2110.09834 [gr-qc]. onal waves at fourth post-minkowskian order, Physical Review Letters 130 , 10.1103/physrevlett.130.101401 (2023). flux for intermediate-mass-ratio inspirals, Physical Review D 84 , 10.1103/physrevd.84.044014 (2011).",
"pages": [
26,
27,
29,
30,
32
]
}
]
[
{
"title": "Optical scattering imaging with sub-nanometer precision based on position-ultra-sensitive giant Lamb shift",
"content": "ZEYANG LIAO, 1,# YUWEI LU, 1,# AND XUE-HUA WANG 1,* 1 State Key Laboratory of Optoelectronic Materials and Technologies, School of Physics, Sun Yat-sen University, Guangzhou, 510275, Peoples Republic of China Abstract: The Lamb shift of a quantum emitter very close to a plasmonic nanostructure, mainly induced by the higher-order plasmonic dark modes, can be three or more orders of magnitude larger than that in the free space and it is ultra-sensitive to the emitter position and polarization. We show that this giant Lamb shift can be sensitively observed from the scattering spectrum dip shift of coupled system when the plasmonic nanoparticle or tip scans through the emitter. Based on these observations, we propose an optical localization and polarization microscopy scheme with sub-nanometer precision for a quantum emitter via detecting the scattering spectrum instead of fluorescence. Our method is free of fluorescence quenching problem and it is relatively easier to be implemented in the plasmon-emitter coupling system. Moreover, the sample in our method does not need to be placed inside a plasmonic picocavity to enhance the radiative fluorescence rate and it also works even if the quantum emitter is slightly below a dielectric surface which can bring about broader applications in various fields, such as physics, chemistry, medicine, life science and materials science.",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "The optical microscope is widely used to localize and image objects which cannot be directly observed by the human eyes. However, the resolution of the conventional optical microscope is subjected to the well-known Abbe's diffraction limit, i.e., the resolvable smallest distance is about half wavelength of the detection light [1]. For visible light, the spatial resolution is about 250-300 nm. In the past few decades, several superresolution methods have been proposed to surpass the diffraction limit, such as the stimulated emission depletion microscopy (STED) [2,3], structured illumination microscopy (SIM) [4-7], the single-molecule localization microscopy (SMLM) [8,9], and stochastic optical fluctuation imaging (SOFI) [10]. These methods have been widely used for biological imaging with typical resolution being about 20-50 nm. The near-field scanning optical microscope (NSOM) can collect the evanescent field and is in principle not subjected to the diffraction limit, but the achievable minimum resolution to date is about 10 nm in practice [11-13]. With the help of quantum effects, such as using quantum entanglement [14-16], quantum Rabi oscillations [17-19], quantum statistical imaging [20-22], and spatial mode demultiplexing techniques [23-25], the optical diffraction limit can also be overcome, but the resolutions demonstrated by these methods are still far from a few nanometers. To achieve atomic-level resolution, the scanning tunneling microscopy (STM) and atomic force microscopy (AFM) are commonly used [26-28]. The resolution gains of STM and AFM are, however, at the expense of losing the spectroscopic properties of the sample and the requirement of extreme environment conditions. Using the highly localized tunneling electrons as a source of excitation, a technique called STM-induced luminescence (STML) can achieve spectral characterization with sub-nanometer spatial resolution [29-32]. However, since the electronic tunneling excitation has some limitations, it is a long-standing pursuing goal to achieve sub-nanometer resolution using all-optical methods. It is well-known that the plasmonic system can confine the light field in an extremely tiny volume [33-35], which is widely used to enhance the interaction between light and quantum emitters (atom,molecule,quantum dot and so on) [36-45,47]. The typical STM junction is actually a plasmonic picocavity which can confine the light field down to sub-nanometer scale [48,49] and it can also enhance the field strength orders of magnitude which has been used for demonstrating the tip-enhanced Raman spectroscopy (TERS) with atomic resolution [50-53]. In addition, the plasmonic picocavity can act as an antenna which can enhance the fluorescence emission rate of a quantum emitter inside the cavity [54-57] and it was also applied to demonstrate the tip-enhanced photoluminescence (TEPL) with sub-nanometer resolution [58]. In both the TERS and TEPL, the target emitters need to be placed inside the plasmonic picocavity which may limit its applications. Considering that the scattering spectrum is much easier to be observed in the plasmonic-emitter coupling system [59, 60], it is thus very interesting to devise a plasmonic scanning localization microscopy with ultra-high resolution via its scattering spectrum. In this article, we propose an optical scattering imaging (OSI) method with sub-nanometer resolution based on the scattering spectrum shift induced by the plasmon-enhanced Lamb shift when a plasmonic nanoparticle or tip scans through a quantum emitter. As is known, quantum theory predicts that vacuum is not truly empty, but full of fluctuations where virtual particles are constantly created and annihilated [61] which can lead to many interesting quantum effects. Among the observable effects of the electromagnetic vacuum, the Lamb shift (LS) is one of the most important effects which directly stimulated the emerging of modern quantum electrodynamics theory, and its precious measurement becomes an important testbed for quantum field theory [61, 63]. Through vacuum engineering, the LS of a quantum emitter can be significantly modified [64-70], e.g., a QE near band edge of photonic crystal can have energy shift one to two orders of magnitudes larger than that in the normal vacuum [71-73]. Due to the much stronger field confinement, the Lamb shift of an emitter very close to a plasmonic nanostructure can be strongly amplified [74-76] and has also been experimentally observed [32,58]. Here, we show that the giant Lamb shift in the plasmon-emitter coupling system, mainly induced by the higher-order plasmonic dark modes when the emitter is very close to the plasmonic nanostructure, is ultra-sensitive to the emitter position and it can be observed from the scattering spectrum dip shift of the coupled system with variation of several meV/nm when the metal nanoparticle or tip scans through the quantum emitter. Due to these observations, we propose that this quantum effect can be exploited to localize the position of an emitter precisely and construct the Lamb shift imaging by scanning the plasmonic nanoparticle or tip through the emitter. The obtained imaging spot is of angstrom size and its shape evidently depends on the dipole orientation of the emitter. Since our method is based on scattering spectrum instead of fluorescence, it is free of fluorescence quenching problem and compared with the usual STML and TEPL methods, our scheme does not require that the sample is placed inside a plasmonic picocavity. The findings here can in principle be developed as a Lamb-shift-based superresolution scanning optical microscope with atomic-level resolution and it also works even when the emitter is embedded slightly below a dielectric substrate, which can bring about broader applications in various fields. The article is organized as follows. In Sec. 2, we first describe the model under study and illustrate the theory used to calculate the emission and scattering spectra of this system including the effect of Lamb shift. Particularly, we study the scattering spectrum shift as a function of emitter-nanoparticle distance and propose an experimentally feasible method to observe the giant Lamb shift in this system. In Sec. 3, we propose a possible experimental scheme based on tip-scattering method to detect the giant Lamb shift which can be used to localize the position of an emitter and its polarization even if it is embedded inside a substrate with ultra-high sensitivity. Finally, we summarize our results.",
"pages": [
1,
2
]
},
{
"title": "2. Model and theory",
"content": "The schematic system considered here is shown in Fig. 1(a) where a QE interacts with a MNP with radius 𝑅 and r 𝑎 is the position of the QE with distance ℎ away from the surface of the MNP. In this paper, we mainly consider that ℎ is larger than 1 𝑛𝑚 where the electron tunnelling effect and nonlocal optical response can be neglected [77,78]. The MNP can support many localized surface plasmon modes with the dipole mode being the bright mode and the higher-order modes (HOMs) being the dark modes [46, 47]. Considering that the dipole mode can be effectively excited by the incident light and be scattered into the far field while the HOMs do not [79], it is convenient to treat the dipole mode as a quantized pseudo-mode separately [80-82] and leave the HOMs as background reservoir fields which can be well described by the macroscopic quantization method based on dyadic Green's function [83,84]. The effective Hamiltonian is then given by where the first term is the effective emitter energy, the second term is the effective plasmon dipole energy, the third term is the interaction between the emitter and the dipole mode, and the four term desribes the effect of the HOMs with The first term in Eq. (2) is the Hamiltonian of the HOMs and the second term is the interaction between the HOMs and QE where the non-rotating interaction terms are retained for the correct calculation of Lamb shift. ˆ 𝜎 + ( ˆ 𝜎 -) is the Pauli lowering (raising) operator of the QE with transition frequency 𝜔 𝑒 , and 𝛾 𝑒 = 𝛾 0 𝑒 + 𝛾 𝑛𝑟 𝑒 is the emitter decay rate including the radiative part 𝛾 0 𝑒 and nonradiative part 𝛾 𝑛𝑟 𝑒 ; ˆ 𝑑 + ( ˆ 𝑑 ) is the creation (annihilation) operator of the plasmon dipole mode with frequency 𝜔 𝑑 and linewidth 𝛾 𝑑 which is the combination of radiative decay 𝛾 0 𝑑 and nonradiative decay 𝛾 𝑛𝑟 𝑑 ; 𝑔 𝑑𝑒 = 𝜔 2 d 𝜇 0 𝝁 𝑒 · 𝑮 0 ( 𝒓 𝑒 , 𝒓 𝑑 ; 𝜔 𝑑 ) · 𝝁 𝑑 / ℏ is the coupling strength between the emitter and the plasmon dipole mode, here 𝜇 e and 𝜇 d are the dipole moments of the QE and the nanoparticle, respectively. Here, we assume that | 𝜇 e | = 24 𝐷 [85] and the magnitude of the effective plasmon dipole moment GLYPH<12> GLYPH<12> 𝝁 𝑑 GLYPH<12> GLYPH<12> = 𝜖 𝑏 √︁ 12 𝜋𝜖 0 ℏ 𝜂 1 𝑅 3 where 𝜂 1 = n 𝑑 𝑑𝜔 𝑅𝑒 [ 𝜖 𝑚 ( 𝜔 )] GLYPH<12> GLYPH<12> 𝜔 = 𝜔 𝑑 o -1 and 𝜖 𝑚 ( 𝜔 ) = 𝜖 ∞ -𝜔 2 𝑝 / GLYPH<0> 𝜔 2 + 𝑖𝜔𝛾 𝑝 GLYPH<1> is the effective permittivity of the nanoparticle in the Drude model with 𝜖 𝑏 being the relative permittivity of the surrounding environment and 𝜖 0 is the vacuum permittivity [86]. In all the numerical calculations throughout this paper, we choose typical parameters for silver with 𝜖 ∞ = 6 . 0, 𝜔 𝑝 = 7 . 9eV, 𝛾 𝑝 = 51meV and 𝛾 𝑒 = 15meV [75]. Since the free vacuum and the dipolar field have been accounted separately, ˆ 𝑓 '+ ( 𝑟, 𝜔 𝜆 ) GLYPH<16> ˆ 𝑓 ' ( 𝑟, 𝜔 𝜆 ) GLYPH<17> denotes the continuum bosonic-field creation (annihilation) operator of the HOMs and b 𝑬 ' ( 𝒓 ) = b 𝑬 '+ ( 𝒓 ) + b 𝑬 '-( 𝒓 ) is the electric field operator of the HOMs with being the positive part of the field where 𝜖 𝐼 ( 𝒓 ' , 𝜔 𝜆 ) is the imaginary part of the material permittivity and G S 𝑛 ≥ 2 ( 𝒓 , 𝒓 ' ; 𝜔 𝜆 ) is the higher-order scattering Green function with n ≥ 2. The dyadic Green function 𝑮 ( 𝒓 , 𝒓 ' ; 𝜔 𝜆 ) satisfies the equation Without plasmonic structure, 𝑮 ( 𝒓 , 𝒓 ' ; 𝜔 ) = 𝑮 0 ( 𝒓 , 𝒓 ' ; 𝜔 ) . In the presence of plasmonic structure, 𝑮 ( 𝒓 , 𝒓 ' ; 𝜔 ) = 𝑮 0 ( 𝒓 , 𝒓 ' ; 𝜔 ) + 𝑮 𝑆 ( 𝒓 , 𝒓 ' ; 𝜔 ) where 𝑮 0 ( 𝒓 , 𝒓 ' ; 𝜔 ) and 𝑮 𝑆 ( 𝒓 , 𝒓 ' ; 𝜔 ) are the free and scattering parts of the dyadic Green function, respectively and their analytical expressions are given in the Supplementary Information (SI) Sec. I. The normalized spectral density of the nth-order plasmonic mode at position r is given by Jn ( 𝒓 , 𝜔 ) = Im GLYPH<2> ˆ 𝒆 𝒊 · 𝑮 𝑆 𝑛 ( 𝒓 , 𝒓 ; 𝜔 ) · ˆ 𝒆 𝒊 GLYPH<3> / 𝐺 0 where 𝑮 𝑆 𝑛 ( 𝒓 , 𝒓 ; 𝜔 ) is the nth-order scattering Green function of the nanoparticle, 𝐺 0 = 𝑘 / 6 𝜋 and ˆ 𝒆 𝒊 is a unit vector [75]. When the distance between the emitter and the nanoparticle is of the order of or larger than the radius of the nanoparticle, the dipole mode dominates (see upper panel of Fig. 1(b)). However, when the emitter is very close to the nanoparticle, the HOMs can have larger spectral density than that of the dipole mode (see lower panel of Fig. 1(b)) and therefore they should be taken into accounts. For spontaneous decay process where only the emitter is initially excited, we can obtain the emission spectrum of the emitter given by (see SI Sec. II) Here, we define the effective transition frequency of the quantum emitter 𝜔 ' 𝑒 = 𝜔 𝑒 + Δ ' 𝑒 ( 𝜔 ) - and are the Lamb shift and decay rate induced by the HOMs vacuum fields, respectively. The term 𝑔 2 𝑑𝑒 / GLYPH<0> 𝜔 -𝜔 ' 𝑑 GLYPH<1> with 𝜔 ' 𝑑 = 𝜔 𝑑 -𝑖𝛾 𝑑 / 2 is the contribution from the plasmon dipole mode. The total LS of the emitter is given by Δ ' 𝑒 ( 𝜔 ) + Re GLYPH<2> 𝑔 2 𝑑𝑒 / GLYPH<0> 𝜔 -𝜔 ' 𝑑 GLYPH<1> GLYPH<3> and the total decay rate is given by 𝛾 𝑒 + 𝛾 ' e ( 𝜔 ) -2 Im GLYPH<2> 𝑔 2 𝑑𝑒 / GLYPH<0> 𝜔 -𝜔 ' 𝑑 GLYPH<1> GLYPH<3> . The emission spectrum shown in Equation (5) is actually equivalent to that obtained from the original Hamiltonian of the system in the rotating wave approximation, i.e. 𝑆 𝑒𝑚𝑖 ( 𝜔 ) = GLYPH<8> [ 𝜔 -𝜔 𝑒 -Δ 𝑒 ( 𝜔 )] 2 + Γ 2 𝑒 ( 𝜔 )/ 4 GLYPH<9> -1 [87] where Δ 𝑒 ( 𝜔 ) = -𝜔 2 ℏ 𝜖 0 𝑐 2 𝝁 𝑒 · Re GLYPH<2> 𝑮 𝑆 ( 𝒓 𝑒 , 𝒓 𝑒 ; 𝜔 ) GLYPH<3> · 𝝁 𝑒 and Γ 𝑒 ( 𝜔 ) = 2 𝜔 2 ℏ 𝜖 0 𝑐 2 𝝁 𝑒 · Im GLYPH<2> 𝑮 𝑆 ( 𝒓 𝑒 , 𝒓 𝑒 ; 𝜔 ) GLYPH<3> · 𝝁 𝑒 are the LS and the effective decay rate calculated by the full scattering Green function. The equivalence can be seen from Fig. 1(c) where the well agreement of both the real and imaginary parts of 𝑔 2 𝑑𝑒 / GLYPH<0> 𝜔 -𝜔 ' 𝑑 GLYPH<1> with -GLYPH<0> 𝜔 2 / ℏ 𝜖 0 𝑐 2 GLYPH<1> 𝝁 𝑒 · 𝑮 𝑆 𝑛 = 1 ( 𝒓 𝑒 , 𝒓 𝑒 ; 𝜔 ) · 𝝁 𝑒 are shown (also see SI Sec.IV). Around the dipolar mode frequency 𝜔 𝑑 , the LS induced by the dipole mode vanishes but the decay rate is maximum. On the contrary, the HOMs have little effect on the decay rate but have significant effect on the LS even though the bare emitter transition frequency is far-off resonant from the HOMs' frequencies, as shown in Fig. 1(d). The LS induced by the HOMs can be high up to 34.1 meV in current example which is more than three orders of magnitude larger than that in the free vacuum. The decay rate and Lamb shift of the quantum emitter as a function of ℎ / 𝑅 for two different nanoparticle radii (i.e., 𝑅 = 10 nm and 𝑅 = 15 nm) are shown in Figs. 1(e) and (f), respectively. For comparisons, the results in the dipolar approximations are also shown as the dashed curves. We can see that both the total decay rate and the Lamb shift increase rapidly when ℎ / 𝑅 decreases and their values especially the Lamb shift when ℎ / 𝑅 is small can be very different from those in the dipole approximation. We also find that both the decay rate and the Lamb shift increase when 𝑅 decreases for the same ℎ / 𝑅 . By extracting the peak of the density of state of the dipolar component, we find that the plasmon dipolar frequency does not depend on ℎ but is red-shifted when 𝑅 increases (see the inset of Fig. 1(e) and Fig. 1(f)). In the small radius limit, the dipolar frequency approaches the value of 𝜔 𝑝 / √ 𝜖 ∞ + 2 𝜖 𝑏 in the quasistatic-limit which is 2 . 793 eV in our example. Since 𝑮 𝑆 ( 𝒓 𝑒 , 𝒓 𝑒 ; 𝜔 ) strongly depends on the position of the emitter, the LS of the emitter varies with emitter's position. From Fig. 2(a), we can see that the emission spectrum peak is red-shifted and the linewidth is also broaden when the nanoparticle moves towards the emitter which provides a method to observe the LS in this system and similar phenomena have also been experimentally observed in the plasmonic nanocavity system [58]. Although the coupling strength increases when the emitter is closer to the nanoparticle, we do not observe fluorescence splitting because the damping rate also increases rapidly and thus no spectrum splitting is observed in Fig. 2(a). We should also mention that in Fig. 2(a) we show the emitter fluorescence spectrum shape but its real intensity in the far field also depends on the quantum yield. In this coupling system, since the emitter radiative decay 𝛾 0 𝑒 is usually much less than its non-radiative decay rate 𝛾 𝑛𝑟 𝑒 especially when the quenching effect is considered in the case that the emitter is very close to the metal nanoparticle [88], the quantum yield of the fluorescence is actually very small and most of the emitted energy are absorbed by the metal nanoparticle. In addition, the spectrum shown in Equation (5) is the fluorescence spectrum when only the emitter is initially excited which is not easy to be realized in this system because the cross section of the nanoparticle is usually much larger than that of the quantum emitter. To enhance the quantum yield of the fluorescence, a plasmonic cavity together with a dielectric spacer is usually required [58] which may limit its application scenarios. Next, we show that the LS can also be observed from the scattering spectrum of the system under weak driving field, which is much easier to be experimentally implemented. According to the input-output theory [89], the output field operator ˆ 𝑎 𝑜𝑢𝑡 ( 𝑡 ) = √︃ 𝛾 0 𝑑 ˆ 𝑑 ( 𝑡 ) + √︁ 𝛾 0 𝑒 𝜎 -( 𝑡 ) and the scattering light spectrum 𝑆 ( 𝜔 ) ∝ ⟨ ˆ 𝑎 + 𝑜𝑢𝑡 ( 𝜔 ) ˆ 𝑎 𝑜𝑢𝑡 ( 𝜔 )⟩ where ˆ 𝑎 𝑜𝑢𝑡 ( 𝜔 ) is the Fourier transformation of ˆ 𝑎 𝑜𝑢𝑡 ( 𝑡 ) . In the typical nanoparticle-emitter coupling system, 𝛾 0 𝑑 ≫ 𝛾 0 𝑒 and therefore the output field is mainly due to the emission of the palsmon dipole, i.e., ˆ 𝑎 𝑜𝑢𝑡 ( 𝑡 ) ≈ √︃ 𝛾 0 𝑑 ˆ 𝑑 ( 𝑡 ) [88]. Thus, the scattering spectrum of the system in the stationary limit Ssca ( 𝜔 ) ∝ lim 𝑡 →∞ Re h ∫ ∞ 0 ⟨ 𝑑 + ( 𝑡 ) d ( 𝑡 + 𝜏 )⟩ 𝑒 𝑖 𝜔𝜏 𝑑𝑡 i and in the weak excitation limit is given by (see Sec. III in SM) If the coupling strength vanishes (i.e., 𝑔 𝑑𝑒 = 0), the scattering spectrum 𝑆 𝑠𝑐𝑎 ( 𝜔 ) ∝ -Im GLYPH<0> 𝜔 -𝜔 ' 𝑑 GLYPH<1> -1 which has Lorentzian lineshape with frequency 𝜔 𝑑 and linewidth 𝛾 𝑑 . Without coupling, the scattering spectrum of the nanoparticle does not contain any information of the emitter. However, if 𝑔 𝑑𝑒 ≠ 0, the effective transition frequency of the plasmon dipole is modified in the presence of the QE and hence 𝑆 𝑠𝑐𝑎 ( 𝜔 ) includes the emitter's spectral information. The scattering spectrum shown in Equation (8) can be rewritten as where 𝜔 ± = 1 2 GLYPH<0> 𝜔 ' 𝑒 + 𝜔 ' 𝑑 ± Δ 𝑙𝑠 GLYPH<1> are two eigenfrequencies of the coupled system, with Δ 𝑙𝑠 = √︂ GLYPH<16> 𝜔 ' 𝑑 -𝜔 ' 𝑒 GLYPH<17> 2 + 4 𝑔 2 𝑑𝑒 and f ± = 1 2 ± 𝜔 ' 𝑑 -𝜔 ' 𝑒 Δ 𝑙𝑠 are two constant coefficients [60]. It is clearly seen that the scattering spectrum is the superposition of two eigen-channels and quantum interference between these two channels can induce a spectrum dip even if the system is in the pseudo-strong coupling regime. If 𝑓 + = 𝑓 -and they are real, the spectrum dip occurs at exactly the center of the two eigen-frequencies, i.e., 1 2 GLYPH<0> 𝜔 𝑒 + Δ ' 𝑒 + 𝜔 𝑑 GLYPH<1> which is a linear function of the LS. However, in the usual case, 𝑓 + ≠ 𝑓 -and both of them may be complex number, the spectrum dip usually deviates from 1 2 GLYPH<0> 𝜔 𝑒 + Δ ' 𝑒 + 𝜔 𝑑 GLYPH<1> but it is still a monotonic function of the emitter LS (see SI Sec. V). The scattering spectrum as a function of the emitter distance is shown in Fig. 2(b). Different from the spontaneous emission spectrum, there is usually a spectrum dip in the scattering spectrum due to the Fano-like interfernce even if the strong coupling condition is not met [32,60]. From Fig. 2(b) we can see that the LS can be observed from the shift of the spectrum dip when the nanoparticle moves towards the emitter (the curves with filling area). In contrast, without considering the LS, the position of spectrum dip does not change with the emitter position (dotted curves) although the peak separation increases with decreasing distance. The LS, emission peak shift and scattering spectrum dip shift as a function of emitter position are shown in Fig. 2(c). It is clearly seen that both the emission peak and the scattering spectrum dip are rapidly red-shifted together with the LS when ℎ decreases and the gradients are about 50 ∼ 60 meV/nm when ℎ ≈ 2 nm. Therefore, either the emission peak shift or the scattering dip shift can reveal the LS of the emitter. The method discussed above also works when the emitter transition frequency is slightly different from the plasmon dipole frequency (see SI Sec. VI). Since the LS highly depends on the relative position between the emitter and the nanoparticle, it is possible to determine an emitter's position from the scattering spectrum. From Fig. 2(c) we can see that when the distance decreases from 20 nm to 1.5 nm, the magnitude of LS increases from almost 0 to 77.4 meV which can be observed from the scattering spectrum dip shift of about 53 meV. In particular, when the emitter-nanoparticle distance changes from 2 nm to 1.5 nm, the magnitude of LS increases by about 48 meV and the spectrum dip is red-shifted by about 27 meV. In a typical high-resolution spectrometer, 0.02 nm wavelength difference (about 0.1 meV energy shift) can be resolved. Thus, from the scattering spectrum dip shift we can in principle measure emitter position change with angstrom or even sub-angstrom precision which is only limited by the precision position control of the nanoparticle. The precision of the scheme may be further improved if the emitter-nanoparticle distance is less than 1 nm, while in this regime the quantum effects such as quantum tunneling and nonlocal effects may play an important role which needs to be considered [33,77,78,90-92]. In addition to the distance, the polarization direction of the quantum emitter can also affect the LS and therefore the scattering spectrum is changed as the emitter polarization angle. In the near-field regime, G 0 ( 𝒓 𝑒 , 𝒓 𝑑 ; 𝜔 ) ≈ 𝑒 𝑖𝑘𝑟 (-𝑰 + 3 e r e r ) / 4 𝜋𝑘 2 𝑟 3 where r = | 𝒓 𝑒 -𝒓 𝑑 | . The coupling strength 𝑔 de ≈ 𝜇 𝑑 𝜇 𝑒 √ 1 + 3 cos 2 𝜃 / 4 𝜋𝜖 0 ℏ ( 𝑅 + ℎ ) 3 which decreases when 𝜃 increases from 0 to 𝜋 /2. In addition to the coupling strength, the magnitude of LS also decreases when 𝜃 increases from 0 to 𝜋 /2 (see SI Sec. VII). The scattering spectra for different emitter polarizations are shown in Fig. 2(d) where it is shown that the spectrum dip is shifted when 𝜃 varies. The dip shift is maximum when the polarization of the emitter is along the z-direction while it is minimum a e when it is along the x-direction (Fig. 2(d)). The spectrum dip shift with respect to 𝜃 is also shown in the inset of Fig. 2(d). The average gradient of the shift is about 0.17 meV per degree. Hence it is possible to detect the polarization of the emitter with about 1 o resolution from the scattering spectrum.",
"pages": [
3,
4,
5,
6,
7,
8
]
},
{
"title": "3. Experiment proposal",
"content": "Finally, we propose a possible realization based on the theory described above to detect the position of a quantum emitter sitting on or inside a substrate (such as SiN) as shown in Fig. 3(a). Here, as an example the emitter is assumed to be x-polarized and 0.5 nm below the surface. The field intensity distribution when the tip is 0.5 nm above the surface is also shown in right panel of Fig. 3(a), where we can see that the field is strongly localized around the tip. A light beam is applied to the tip and the scattering spectra for different tip lateral positions are shown in Fig. 3(b) when the z distance between the emitter and tip (z-offset) is fixed to be 1 nm. When the tip is closer to the emitter, the spectrum dip is shifted to the lower frequency due to the LS discussed above. The dip shift and the LS as a function of lateral offset are shown in Fig. 3(c) when the z-offset is fixed to be 1 nm. When the tip is right on top of the emitter, the spectrum dip has a maximum shift. The gradient around the maximum shift is about 1 meV/nm near the center. When the tip is right on top of the emitter, the spectrum dip shift as a function of z distance is shown in Fig. 3(d). When the tip moves towards the emitter, the spectrum dip is red-shifted with gradient about 15.4 meV/nm, which indicates a much higher sensitivity for detecting the longitudinal distance. For a spectrometer with 0.02 nm resolution, the lateral and longitudinal resolutions in this setup can in principle be about 1 ˚ 𝐴 and 0.1 ˚ 𝐴 , respectively. If the substrate is replaced by a material with lower reflective index or the emitter is placed above the substrate surface, the LS can be even larger and the resolution can be further enhanced (see SI Sec. VIII). We can also extract a two-dimensional image of the emitter from the dip shift of the scattering spectra at different positions when the emitter is x-polarized (Fig. 3(e)) or z-polarized (Fig. 3(f)). When the emitter is z-polarized, the image is symmetric. However, when the emitter is x-polarized, the image has an elliptical shape with major axis along the polarization direction. Thus, in our scheme, we can not only detect the position of the emitter but also its polarization direction. When the z-offset increases, the major axis of the imaging spot also increases which indicates the reduction of resolution (see Fig. S6 in SM). Different from the typical STML and TEPL methods [29-31,58], here our scheme does not require the formation of plasmonic picocavity to enhance the fluorescence rate and it also works even when the emitter is embedded slightly below a dielectric substrate which may find more application scenarios.",
"pages": [
8,
9
]
},
{
"title": "4. Summary and discussion",
"content": "To conclude, we show that a quantum emitter very close to a metal nanoparticle or tip can have huge Lamb shift mainly induced by the higher-order plasmonic dark modes and this energy shift is ultra-sensitive to the emitter position. We also show that this giant Lamb shift can be sensitively observed from the scattering spectrum dip shift when the metal nanoparticle or tip scans through the emitter. Moreover, we propose that this quantum effect can be exploited for all-optical detection of an emitter position with angstrom precision which is comparable to the typical STM, STML and TEPL methods and we can also determine the polarization direction of the quantum emitter. In contrast to the typical STML and TEPL methods which are usually surface bound and only detect the sample in a plasmonic picocavity, we can detect the position of the emitter precisely even if it is slightly below a dielectric substrate. Due to the fact that the experimental measurement of the scattering spectrum is much easier than the fluorescence in the plasmon-emitter coupling system, our work here can find important applications in many areas and can stimulate extensive theoretical and experimental researches in the future. Finally, we should mention that the emitter is treated as a point dipole in this work to demonstrate the basic principle, the full scattering spectrum considering the non-dipolar effect of the quantum emitter and other quantum effects should be further studied in the future [93]. Funding. The authors thank R. Liu and X. Zeng for helpful discussions. This work was supported by the National Key R&D Program of China (Grant No. 2021YFA1400800), the Key-Area Research and Development Program of Guangdong Province (Grant No.2018B030329001), the Guangdong Special Support Program (Grant No.2019JC05X397), the Natural Science Foundations of Guangdong (Grant Nos. 2021A1515010039 and 2018A030313722). Disclosures. The authors declare no conflicts of interest. Data availability. Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request. Supplemental document. See Supplement 1 for supporting content.",
"pages": [
9
]
},
{
"title": "References",
"content": "fluctuations in an electrical circuit by measuring the lamb shift, Science 322 (2008), 1357. 85. S. Savasta, R. Saija, A. Ridolfo, O. Di Stefano, P. Denti, and F. Borghese, Nanopolaritons: Vacuum rabi splitting with a single quantum dot in the center of a dimer nanoantenna, ACS Nano 4 (2010), 6369.",
"pages": [
12
]
}
]
2024SCPMA..6790312Y
https://arxiv.org/pdf/2205.02515.pdf
<document>
<section_header_level_1><location><page_1><loc_17><loc_92><loc_84><loc_93></location>Quantum Control for Time-dependent Noise by Inverse Geometric Optimization</section_header_level_1>
<text><location><page_1><loc_21><loc_89><loc_80><loc_90></location>Xiaodong Yang, 1, 2, 3 Xinfang Nie, 4, 1, 2, 3 Tao Xin, 1, 2, 3 Dawei Lu, 4, 1, 2, 3, ∗ and Jun Li 1, 2, 3, †</text>
<text><location><page_1><loc_32><loc_87><loc_68><loc_89></location>1 Shenzhen Institute for Quantum Science and Engineering,</text>
<text><location><page_1><loc_28><loc_86><loc_73><loc_87></location>Southern University of Science and Technology, Shenzhen, 518055, China</text>
<text><location><page_1><loc_32><loc_84><loc_69><loc_86></location>2 International Quantum Academy, Shenzhen, 518055, China</text>
<text><location><page_1><loc_27><loc_83><loc_74><loc_85></location>3 Guangdong Provincial Key Laboratory of Quantum Science and Engineering,</text>
<text><location><page_1><loc_28><loc_82><loc_73><loc_83></location>Southern University of Science and Technology, Shenzhen, 518055, China</text>
<text><location><page_1><loc_21><loc_81><loc_80><loc_82></location>4 Department of Physics, Southern University of Science and Technology, Shenzhen, 518055, China</text>
<text><location><page_1><loc_18><loc_69><loc_83><loc_80></location>Quantum systems are exceedingly difficult to engineer because they are sensitive to various types of noises. In particular, time-dependent noises are frequently encountered in experiments but how to overcome them remains a challenging problem. In this work, we extend and apply the recently proposed robust control technique of inverse geometric optimization to time-dependent noises by working it in the filter-function formalism. The basic idea is to parameterize the control filter function geometrically and minimize its overlap with the noise spectral density. This then effectively reduces the noise susceptibility of the controlled system evolution. We show that the proposed method can produce high-quality robust pulses for realizing desired quantum evolutions under realistic noise models, and thus will find practical applications for current physical platforms.</text>
<text><location><page_1><loc_9><loc_41><loc_49><loc_66></location>Introduction.The ability to precisely manipulate quantum systems against noise is central to practical quantum information processing [1]. There have been developed a variety of robust quantum control methods, such as composite pulses [2-4], dynamical decoupling [5-7], sampling-based learning control [8, 9], geometric-formalism-based pulse control [1015]. Many of these methods assume the considered noise to be quasi-static, i.e., slow enough compared to the operation time, which is however not always a valid noise model in reality. Actually, time-dependent noises are routinely encountered in experiments. For example, 1 /f type noise, which contains wide distribution of correlation times [16], is present in many solid-state qubit platforms such as superconducting qubits [17, 18] and semiconductor quantum dots [19, 20]. Therefore, in order to further enhance experimental control fidelities, it is of vital importance to develop robust quantum control for general time-dependent noises.</text>
<text><location><page_1><loc_9><loc_10><loc_49><loc_40></location>Attempts to address errors induced by time-dependent noises in quantum system engineering are challenging. Results to date suggest that conventional methods usually have their limitations. For example, composite pulses, originally designed to tackle static, systematic errors, were found to be robust to fluctuating noises up to as fast as around 10% of the Rabi frequency [21]. Dynamical decoupling can protect quantum coherence in a fluctuating environment, but it requires rapid and strong control modulation, which might be problematic to realize experimentally. Moreover, how to incorporate dynamical decoupling into the task of realizing arbitrary quantum operations is still not fully clear [22]. Optimal control provides a flexible and generically applicable approach, in which the requirements of pulse smoothness and robustness can be added as optimization constraints [23]. Usually, the control variables to be optimized are temporal pulse parameters such as amplitudes and phases. Alternatively, optimization can be done in the dynamical variable space with a geometric flavor, as proposed and developed in Refs. [10-15], yet only static errors have been considered therein.</text>
<text><location><page_1><loc_10><loc_8><loc_49><loc_10></location>In this work, we consider combing the geometric-based op-</text>
<text><location><page_1><loc_52><loc_45><loc_92><loc_66></location>timal control method with the filter function (FF) formalism [24-26] to overcome these limitations for the purpose of resisting time-dependent noises. FFs were originally introduced to evaluate operational infidelities under stationary stochastic noises, and have proven very useful in quantum control, especially for designing dynamical decoupling sequences [27-32]. Recently, there have been studies on incorporating FF into gradient-based optimal control [33]. Here, we take the geometric approach, that is, we first parameterize the controlled system evolution trajectory with dynamical variables, which corresponds to a parameterized filter function in the frequency domain, and then minimize the overlap of the filter function and the noise spectral density; see Fig. 1 for an illustration of the basic idea.</text>
<text><location><page_1><loc_52><loc_29><loc_92><loc_45></location>We give test examples of finding robust optimal control (ROC) pulses for producing target quantum gate and state transfer under realistic, experimentally relevant noise environments. It is found that our robust pulses outperform typical composite pulses in that their resultant FFs are suppressed at the characteristic frequencies of the considered noises, thus having much improved control fidelities. A separate section is devoted to treat the case of Markovian noise based directly on the Bloch equation, and the optimization results show that the T 1 and T 2 limit can be surpassed in the quantum state transfer task. Finally, discussions and implications are presented.</text>
<text><location><page_1><loc_52><loc_16><loc_92><loc_29></location>Inverse geometric engineering.We consider a prototypical robust quantum control model, i.e., a resonantly controlled two-level system under time-dependent detuning noise and control amplitude noise. By convention, we parameterize the control field as Ω( t )[cos φ ( t ) , sin φ ( t )] ( t ∈ [0 , T ] ), with Ω( t ) ( | Ω( t ) | ≤ Ω max ) being the pulsed Rabi frequency and φ ( t ) ∈ [ -π, π ] the phase. Taking into account of noises, we have the following resonant frame Hamiltonian</text>
<formula><location><page_1><loc_53><loc_12><loc_92><loc_15></location>H ( t ) = Ω(1 + glyph[epsilon1] a ( t )) [ cos φ σ x 2 +sin φ σ y 2 ] + glyph[epsilon1] d ( t ) σ z 2 , (1)</formula>
<text><location><page_1><loc_52><loc_7><loc_92><loc_11></location>where glyph[epsilon1] a ( t ) , glyph[epsilon1] d ( t ) represent fluctuating noises on control amplitude and detuning, respectively, and we introduce E a ( t ) ≡</text>
<figure>
<location><page_2><loc_9><loc_76><loc_49><loc_94></location>
<caption>FIG. 1. Schematic diagram of the method of geometric- and FFbased pulse optimization for resisting time-dependent, stochastic noise. Since there exist many evolution trajectories realizing the same target but with different extent of noise filtering capabilities, the goal is hence to find a noise-resilient trajectory. For example, in the single-qubit case, the geometric trajectories generated by rectangular wave (blue line), composite pulse (green line), and a robust shaped pulse (red line) are plotted on the Bloch sphere for comparison, all implementing the same state transfer | 0 〉 → | 1 〉 . Robust trajectory is found by minimizing the overlap of its associated control filter function with the noise spectral density. The pulse shape that generates this trajectory is then obtained through inverse engineering.</caption>
</figure>
<text><location><page_2><loc_9><loc_37><loc_49><loc_57></location>Ω[cos φσ x / 2 + sin φσ y / 2] and E d ≡ σ z / 2 as their corresponding noise operators. Physically, control amplitude noise is usually due to imperfect fabricated components, noisy electronics or varied fields [34], while detuning may originate from, e.g., random shifts in control driving frequency, or Overhauser effects on an electron spin by its surrounding nuclear spins [20]. In the following, we shall assume that glyph[epsilon1] a ( t ) , glyph[epsilon1] d ( t ) are mutually independent stationary Gaussian processes with zero means. Under this assumption, each noise is fully characterized in terms of its own power spectral density S µ ( ω ) = ∫ ∞ -∞ dte -iωt 〈 glyph[epsilon1] µ (0) glyph[epsilon1] µ ( t ) 〉 , µ ∈ { a, d } . For practical applications, S µ ( ω ) will be determined from noise spectroscopy measurements in real experiments [35, 36].</text>
<text><location><page_2><loc_9><loc_22><loc_49><loc_37></location>Now, we briefly describe the inverse geometric optimization technique [10, 11]. The procedure starts with a parameterization of the noise-free evolution. Let U 0 ( t ) be the solution to the time-dependent Schrodinger equation ˙ U 0 ( t ) = -iH 0 ( t ) U 0 ( t ) , where H 0 ( t ) is as shown in Eq. (1) with glyph[epsilon1] a , glyph[epsilon1] d = 0 . We parameterize U 0 ( t ) based on ZYZ decomposition, that is, an arbitrary single-qubit unitary operator can be written as exp( iβ ) R z ( ϕ ) R y ( θ ) R z ( γ ) , for some real numbers β, ϕ, γ ∈ [ -π, π ) and θ ∈ [ -π, π ] [37]. In our problem here, β = 0 because H 0 is traceless. Hence, we have</text>
<formula><location><page_2><loc_9><loc_18><loc_48><loc_21></location>U 0 ( t ) = [ cos( θ/ 2) e -iϕ/ 2 e -iγ/ 2 -sin( θ/ 2) e -iϕ/ 2 e iγ/ 2 sin( θ/ 2) e iϕ/ 2 e -iγ/ 2 cos( θ/ 2) e iϕ/ 2 e iγ/ 2 ] .</formula>
<text><location><page_2><loc_9><loc_15><loc_41><loc_17></location>As such, the Schrodinger equation is rewritten as</text>
<formula><location><page_2><loc_20><loc_12><loc_49><loc_14></location>˙ θ = Ωsin( φ -ϕ ) , (2a)</formula>
<formula><location><page_2><loc_20><loc_8><loc_49><loc_11></location>˙ γ = Ωcos( φ -ϕ ) / sin θ. (2c)</formula>
<formula><location><page_2><loc_20><loc_10><loc_49><loc_12></location>˙ ϕ = -Ωcos( φ -ϕ ) cot θ, (2b)</formula>
<text><location><page_2><loc_52><loc_78><loc_92><loc_93></location>We perform optimization over these dynamical angular variables in order to find an evolution trajectory that has the property of dynamically correcting errors on itself. In this geometric formulation of the control problem, the optimization objective consists of control target, robustness requirement, boundary conditions and certain practical considerations such as bounded control amplitude, all expressed in terms of θ, ϕ and γ . Once a robust evolution trajectory specified by the three angular variables is obtained, we can determine the control field by evaluating the inversion of Eq. (2), i.e., Ω =</text>
<formula><location><page_2><loc_52><loc_76><loc_78><loc_78></location>√ ˙ θ 2 + ˙ γ 2 sin 2 θ, φ = arcsin( ˙ θ/ Ω) + ϕ .</formula>
<text><location><page_2><loc_52><loc_50><loc_92><loc_76></location>Quantum gate and quantum state transfer.We first consider the control target of implementing a desired quantum gate or quantum state transfer. The key step is to effect the transformation operator to the toggling frame defined by U glyph[epsilon1] a ,glyph[epsilon1] d ( t ) = U 0 ( t ) U tog ( t ) , where U glyph[epsilon1] a ,glyph[epsilon1] d ( t ) represents the propagator in the presence of the noises. Through Dyson perturbative expansion [38], there is U tog ( t ) = 1 -∑ µ = a,d [ i ∫ t 0 dt 1 glyph[epsilon1] µ ( t 1 ) ˜ E µ ( t 1 ) + ∫ t 0 dt 1 ∫ t 1 0 dt 2 glyph[epsilon1] µ ( t 1 ) glyph[epsilon1] µ ( t 2 ) ˜ E µ ( t 1 ) ˜ E µ ( t 2 ) + · · · ] with 1 the identity operator and ˜ E µ ( t ) = U † 0 ( t ) E µ U 0 ( t ) , µ ∈ { a, d } . Substitute into the parameterized U 0 ( t ) , we obtain ˜ E a,x ( t ) = [ ˙ θ sin γ + (˙ γ sin 2 θ cos γ ) / 2] σ x / 2 , ˜ E a,y ( t ) = [ ˙ θ cos γ -( ˙ γ sin 2 θ sin γ ) / 2] σ y / 2 , ˜ E a,z ( t ) = ( ˙ γ sin 2 θ ) σ z / 2 ; ˜ E d,x ( t ) = ( -sin θ cos γ ) σ x / 2 , ˜ E d,y ( t ) = (sin θ sin γ ) σ y / 2 , ˜ E d,z ( t ) = (cos θ ) σ z / 2 . These formulas are then to be substituted into the Dyson series to evaluate the error terms.</text>
<text><location><page_2><loc_52><loc_36><loc_92><loc_50></location>For the quantum gate problem, we are given a target gate U and intend to find a robust implementing pulse. Suppose that the ideal evolution at time T satisfies U 0 ( T ) = U , then for a single realization of glyph[epsilon1] a ( t ) and glyph[epsilon1] d ( t ) , the gate fidelity reads F = ∣ ∣ ∣ Tr ( U † U glyph[epsilon1] a ,glyph[epsilon1] d ( T ) )∣ ∣ ∣ 2 / 4 = | Tr( U tog ( T )) | 2 / 4 . Taking the ensemble average of the noises and transferring to the frequency domain, the average gate infidelity defined by F avg = 1 -〈 F 〉 can be estimated to the second order approximation by the filter-function formalism [39, 40]</text>
<formula><location><page_2><loc_57><loc_30><loc_92><loc_34></location>F avg ≈ 1 2 π ∑ µ = a,d α = x,y,z ∫ ∞ -∞ dω ω 2 S µ ( ω ) | R µ,α ( ω ) | 2 , (3)</formula>
<text><location><page_2><loc_52><loc_8><loc_92><loc_28></location>in which R µ,α ( ω ) = -iω ∫ T 0 dt Tr [ ˜ E µ,α ( t ) σ α / 2] e iωt , and ∑ α | R d,α ( ω ) | 2 /ω 2 , ∑ α | R a,α ( ω ) | 2 / ( ω 2 Ω 2 max ) are the so called filter functions. This formula provides a simple quantitative means to evaluate the performance of a control protocol in the presence of time-dependent noises. It is thus natural to take F avg as our objective function. As a concrete example, we consider implementing a π rotational gate U = exp( -iπσ y / 2) . For this problem, at t = 0 , U 0 (0) equals to the identity, corresponding to the initial conditions θ (0) = 0 and ϕ (0) = -γ (0) (value not specified). The ending point conditions are θ ( T ) = π and ϕ ( T ) = γ ( T ) . The latter can be rewritten as a constraint for θ and γ by noting that from Eqs. (2b) and (2c) there is ˙ ϕ = -˙ γ cos( θ ) , hence one requires</text>
<figure>
<location><page_3><loc_9><loc_58><loc_49><loc_93></location>
<caption>FIG. 2. Geometric trajectories, control waveforms, FFs and noise spectra of different sequences for realizing a π rotational gate subject to time-dependent noise. (a 1 )-(a 3 ) For detuning noise, the noise strength is set as √ 〈 glyph[epsilon1] 2 d (0) 〉 = 0 . 03Ω max with Ω max / (2 π ) = 10 7 Hz, and the noise spectrum is ohmic. (b 1 )-(b 3 ) For amplitude noise, its strength is √ 〈 glyph[epsilon1] 2 a (0) 〉 = 0 . 03 , and the noise spectrum is two Lorentzian peaks added on top of 1 /f background ( λ 1 = λ 2 = 100 Hz, κ = 1 , A 1 = A 2 = 1 , B = 0 . 05 ). The solid lines in (a 2 ) and (b 2 ) represent the pulse amplitudes in the unit of Ω max , and the dashed lines in (b 2 ) are the pulse phases depicted by axis on the right. It can be seen that ROC FFs are suppressed at the characteristic frequencies, implying better noise filtering capability. Meanwhile, ROC control waveforms and geometric trajectories are much smoother.</caption>
</figure>
<text><location><page_3><loc_9><loc_29><loc_49><loc_36></location>the condition γ (0) + γ ( T ) + ∫ T 0 ˙ γ cos θdt = 0 to be satisfied. With the objective function and all the constraints, we search ROC pulses using the gradient-based algorithm [41]; see details in Supplemental Material [40].</text>
<text><location><page_3><loc_9><loc_14><loc_49><loc_29></location>For the quantum state transfer problem, without loss of generality, we suppose the initial state to be | 0 〉 . The target is an arbitrary state | ψ 〉 on the Bloch sphere. For one realization of the noise glyph[epsilon1] a ( t ) and glyph[epsilon1] d ( t ) , the state transfer fidelity reads F = |〈 ψ | U glyph[epsilon1] a ,glyph[epsilon1] d ( T ) | 0 〉| 2 . Suppose the ideal evolution U 0 ( T ) implements the desired state transfer. Again, we turn into the toggling frame and get F = |〈 0 | U tog ( T ) | 0 〉| 2 . Substitute into the perturbative expansion of U tog ( T ) , and take ensemble average of the noise Hamiltonian, it can be derived that the average state infidelity is</text>
<formula><location><page_3><loc_14><loc_8><loc_49><loc_13></location>F avg ≈ ∑ µ = a,d α = x,y,z 1 2 π ∫ ∞ -∞ dω ω 2 S µ ( ω ) | P µ,α ( ω ) | 2 , (4)</formula>
<text><location><page_3><loc_52><loc_79><loc_92><loc_93></location>where we define P µ,α = -iω ∫ T 0 dt 〈 0 | ˜ E µ,α ( t ) | 1 〉 e iωt , and ∑ α | P d,α ( ω ) | 2 /ω 2 , ∑ α | P a,α ( ω ) | 2 / ( ω 2 Ω 2 max ) are the filter functions [40]. Concretely, we consider preparing target state | ψ 〉 = | 1 〉 starting from | 0 〉 . This converts to the conditions θ (0) = 0 , θ ( T ) = π, ϕ (0) = ϕ ( T ) = 0 , and no requirement of γ is involved. Moreover, the relation between θ and γ , namely ˙ ϕ = -˙ γ cos( θ ) , requires the condition ∫ T 0 ˙ γ cos θdt = 0 to be satisfied. The optimization procedure is the same as that described in the gate problem.</text>
<text><location><page_3><loc_52><loc_11><loc_92><loc_79></location>As demonstration, we show the numerical simulation results of implementing a π rotational gate under realistic detuning or amplitude noise, as shown in in Fig. 2. We compare performances between primitive, typical composite pulses and robust optimal control pulses, for the same given noise spectrum. The primitive pulse is the elementary rectangular pulse of maximum Rabi frequency Ω max , which corresponds to the time-minimal control t min = 1 / (2Ω max ) . For detuning noise, as shown in Figs. 2(a 1 )-2(a 3 ), we consider ohmic spectrum with sharp cut-offs, i.e., S d ( ω ) ∝ ω, ω ∈ [ ω lc , ω uc ] , which describes a spin suffering bosonic environment [42]. The composite pulse we chosen is CORPSE [43], which is robust to detuning error to the first order. For amplitude noise, as shown in Figs. 2(b 1 )-2(b 3 ), we examine a noise spectrum of several Lorentzian peaks added on top of a broad 1 /f κ background S a ( ω ) ∝ ∑ k A k / ( λ 2 k +( ω -ω 0 ,k ) 2 ) + B/ω κ , which describes the random fluctuations in superconducting flux terms [44]. The composite pulse tested for this case is BB1 [43], which is robust to amplitude error to the second order. It can be seen that in each test example, the ROC filter function has sharp dips at the central frequencies of the imported noise spectrum, hence their frequency overlap is significantly suppressed; see Figs. 2(a 3 ) and 2(b 3 ). This feature implies that ROC has better performance in mitigating timedependent noises. We can verify this conclusion by computing their fidelities as follows. We calculate a single instance of noise perturbed evolution operator U glyph[epsilon1] α ( T ) and a single value for fidelity, and then take average over N = 150 noise realizations. For detuning noise with ohmic spectrum centered in the range [0 . 5Ω max , Ω max ] , as shown in the insert of Fig. 2(a 3 ), we obtain F ROC avg = 4 × 10 -4 , while F Primitive avg = 1 × 10 -3 and F CORPSE avg = 9 × 10 -3 . This result is consist with the conclusion that composite pulses are only robust to fluctuating noises up to as fast as around 10% of the Rabi frequency [21], yet our ROC pulse can still function for high-frequency noise. For amplitude noise with Lorentzian peaks centered at 0 . 2Ω max and 0 . 4Ω max (see the insert of Fig. 2(b 3 )), we obtain F Primitive avg = 2 × 10 -3 and F BB1 avg = 8 × 10 -3 , while our ROC pulse can decrease the infidelity to F ROC avg = 4 × 10 -6 . Another benefit of ROC pulse is that its shape can be made much smoother than CORPSE and BB1; see Figs. 2(a 2 ) and 2(b 2 ). This is particularly favorable for experiments, as real pulse generators have limited bandwidths. Accordingly, ROC produces smoother geometric evolution trajectories, as shown in Figs. 2(a 1 ) and 2(b 1 ).</text>
<text><location><page_3><loc_53><loc_8><loc_92><loc_10></location>More simulation results for state transfer from | 0 〉 to | 1 〉 ,</text>
<text><location><page_4><loc_9><loc_81><loc_49><loc_93></location>for the case when detuning and amplitude noise are simultaneously present, for other types of realistic noise models, and for varied characteristic frequency positions of the tested noise spectra are all put in the Supplemental Material [40]. These results reveal that, in general, ROC pulses offer fidelity improvement for almost an order of magnitude compared with composite pulses and primitive pulse, and meanwhile featuring smooth pulse shapes and geometric trajectories.</text>
<text><location><page_4><loc_9><loc_69><loc_49><loc_81></location>Resistance of T 1 , T 2 relaxation.When noises vary fast such that the Markovian approximation is valid, the controlled system dynamics can be described by the Bloch equation [45] ˙ x = ( H 0 ( t ) + γ 1 R T 1 + γ 2 R T 2 ) x , where x ≡ (1 / 2 , x 1 , x 2 , x 3 ) T is the vectorized representation of the system density matrix ρ = 1 / 2+ x 1 σ x + x 2 σ y + x 3 σ 3 ( x 2 1 + x 2 2 + x 2 3 ≤ 1 / 4) , H 0 ( t ) is the control Hamiltonian, γ 1 , 2 = 1 /T 1 , 2 are relaxation rates and</text>
<formula><location><page_4><loc_11><loc_61><loc_49><loc_68></location>R T 1 = 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 -1 , R T 2 = 0 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 0 (5)</formula>
<text><location><page_4><loc_9><loc_43><loc_49><loc_61></location>are their corresponding operators. Relaxation is an irreversible process, hence it is usually thought that the best strategy to alleviate effects of relaxation is to make the operation time as small as possible. Therefore, the primitive pulse sets a fundament limit hard to surpass by other pulses [11]. Here, we study this issue using inverse geometric optimization, which works as follows. We first parameterize the relaxation-free evolution with extended threedimensional rotations R z ( δ ) , R y ( η ) and R z ( ξ ) , namely V 0 ( t ) = ( 1 0 0 R z ( δ ) R y ( η ) R z ( ξ ) ) with δ, ξ ∈ [ -π, π ] , η ∈ [0 , π ] [40]. Thus, the Bloch equation is rewritten as</text>
<formula><location><page_4><loc_20><loc_40><loc_49><loc_42></location>˙ ξ = Ωcos( φ -δ ) / sin η, (6a)</formula>
<formula><location><page_4><loc_20><loc_36><loc_49><loc_38></location>˙ δ = -Ωcos( φ -δ ) / tan η. (6c)</formula>
<formula><location><page_4><loc_20><loc_38><loc_49><loc_40></location>˙ η = Ωsin( φ -δ ) , (6b)</formula>
<text><location><page_4><loc_9><loc_27><loc_49><loc_35></location>The actual evolution is then transformed to the toggling frame for conveniently displaying the perturbation effects due to T 1 and T 2 relaxation, i.e., V T 1 ,T 2 ( t ) = V 0 ( t ) V tog ( t ) ≈ V 0 ( t )( 1 4 + ∑ k ∫ t 0 dt 1 γ k ˜ R T k ( t 1 ) + · · · ) , where 1 4 is the 4dimensional identity, ˜ R T k ( t ) = V † 0 ( t ) R T k V 0 ( t ) .</text>
<text><location><page_4><loc_9><loc_20><loc_49><loc_27></location>Take the quantum state transfer problem as an example. Staring from state x (0) , the Euclidean distance between the actual state and the target state ¯ x , defined by F = | ¯ x -V T 1 ,T 2 ( T ) x (0) | 2 , can be expressed in terms of the angular variables as follows [40]</text>
<formula><location><page_4><loc_15><loc_15><loc_49><loc_19></location>F ≈ ∫ T 0 dt [ γ 2 1 (cos η -2) 2 + γ 2 2 sin 2 η ] / 4 . (7)</formula>
<text><location><page_4><loc_9><loc_9><loc_49><loc_15></location>As a concrete example, we consider quantum state transfer from the north pole x (0) = [1 / 2 , 0 , 0 , 1 / 2] T to the south pole x = [1 / 2 , 0 , 0 , -1 / 2] T . This gives the constraint conditions η (0) = 0 , η ( T ) = π and we set δ (0) = δ ( T ) = 0 ,</text>
<figure>
<location><page_4><loc_53><loc_71><loc_90><loc_93></location>
<caption>FIG. 3. Performance comparison of different sequences for realizing state transfer from the north pole to the south pole subject to both transverse and longitudinal relaxation. (a) State Euclidean distance vs different relaxation parameters, where we set γ 1 = 10 3 s -1 . Specifically, we show geometric trajectories and control waveforms (in the unit of Ω max ) for the case of γ 2 /γ 1 = 10 in (b) and (c), respectively.</caption>
</figure>
<text><location><page_4><loc_52><loc_33><loc_92><loc_56></location>while no requirement of ξ is involved. Besides, from Eqs. (6a) and (6c) we have ˙ δ = -˙ ξ cos η , thus the condition ∫ T 0 ˙ ξ cos ηdt = 0 should be satisfied. In our optimization, we also use the gradient-based algorithm to search robust ROC pulses. Results are summarized in Fig. 3, where we consider solid-state spin defect system with Ω max / (2 π ) = 10 7 Hz, and the relaxation parameters are typically chosen as γ 1 = 10 3 s -1 , γ 2 /γ 1 = 10 ∼ 100 [46]. From Fig. 3(a), we find that typical composite pules, including CORPSE and BB1, can not resist relaxation, as they result in much larger errors compared with the primitive pulse. On the other hand, ROC pulses can improve up to four times compared with primitive pulse for all the tested relaxation parameters. Meanwhile, the geometric trajectories and control waveforms for ROC pulses are smoother, as shown in Figs. 3(b) and 3(c), respectively.</text>
<text><location><page_4><loc_52><loc_15><loc_92><loc_33></location>Discussion and outlook.The task of mitigating timedependent noises is generally considered to be a thorny challenge and a long-term objective of quantum system engineering. The robust control method presented here has a critical advantage of flexibility as it is effective for a wide variety of noise environments, which is hence particularly applicable in reality since real experiments often involve complicated noise spectrum. Moreover, in the Markovian limit, the method is also effective in improving the state transfer fidelity against transverse and longitudinal relaxation effects. We hope the control examples tested here or other possible applications can soon find their experimental verifications.</text>
<text><location><page_4><loc_52><loc_9><loc_92><loc_15></location>For future work, we can combine inverse geometric engineering with other robust optimal control methods. For example, the technique developed in Ref. [47], which expresses Dyson perturbative terms based on Van Loan's integral ex-</text>
<text><location><page_5><loc_9><loc_81><loc_49><loc_93></location>pression, provides a rather convenient means to evaluate the perturbative impacts of the noises. We can also apply analytic expression of the filter function derivatives [33] to further improve the performance of our method, or attempt to derive exact analytical control fields [48, 49]. In addition, the method presented here can be easily extended to handle other robust quantum control tasks, such as quantum sensing under timedependent background noises [50].</text>
<text><location><page_5><loc_9><loc_54><loc_49><loc_81></location>Acknowledgments . We thank Ze Wu for helpful discussions. This work was supported by the National Natural Science Foundation of China (1212200199, 11975117, 92065111, 12075110, 11905099, 11875159, 11905111, and U1801661), National Key Research and Development Program of China (2019YFA0308100), Guangdong Basic and Applied Basic Research Foundation (2019A1515011383 and 2021B1515020070), Guangdong Provincial Key Laboratory (2019B121203002), Guangdong International Collaboration Program (2020A0505100001), Shenzhen Science and Technology Program (RCYX20200714114522109 and KQTD20200820113010023), China Postdoctoral Science Foundation (2021M691445), Science, Technology and Innovation Commission of Shenzhen Municipality (ZDSYS20190902092905285, KQTD20190929173815000 and JCYJ20200109140803865), and Pengcheng Scholars, Guangdong Innovative and Entrepreneurial Research Team Program (2019ZT08C044).</text>
<unordered_list>
<list_item><location><page_5><loc_11><loc_47><loc_25><loc_48></location>∗ ludw@sustech.edu.cn</list_item>
<list_item><location><page_5><loc_11><loc_46><loc_24><loc_47></location>† lij3@sustech.edu.cn</list_item>
<list_item><location><page_5><loc_10><loc_42><loc_49><loc_46></location>[1] D. Suter and G. A. ' Alvarez, Colloquium: Protecting quantum information against environmental noise, Rev. Mod. Phys. 88 , 041001 (2016).</list_item>
<list_item><location><page_5><loc_10><loc_39><loc_49><loc_42></location>[2] M. H. Levitt, Composite pulses, Prog. Nucl. Magn. Reson. Spectrosc. 18 , 61 (1986).</list_item>
<list_item><location><page_5><loc_10><loc_35><loc_49><loc_39></location>[3] H. K. Cummins, G. Llewellyn, and J. A. Jones, Tackling systematic errors in quantum logic gates with composite rotations, Phys. Rev. A 67 , 042308 (2003).</list_item>
<list_item><location><page_5><loc_10><loc_31><loc_49><loc_35></location>[4] K. R. Brown, A. W. Harrow, and I. L. Chuang, Arbitrarily accurate composite pulse sequences, Phys. Rev. A 70 , 052318 (2004).</list_item>
<list_item><location><page_5><loc_10><loc_28><loc_49><loc_31></location>[5] L. Viola, E. Knill, and S. Lloyd, Dynamical Decoupling of Open Quantum Systems, Phys. Rev. Lett. 82 , 2417 (1999).</list_item>
<list_item><location><page_5><loc_10><loc_24><loc_49><loc_28></location>[6] L. Viola and E. Knill, Robust Dynamical Decoupling of Quantum Systems with Bounded Controls, Phys. Rev. Lett. 90 , 037901 (2003).</list_item>
<list_item><location><page_5><loc_10><loc_22><loc_49><loc_25></location>[7] A. M. Souza, G. A. ' Alvarez, and D. Suter, Robust dynamical decoupling, Phil. Trans. R. Soc. A 370 , 4748 (2012).</list_item>
<list_item><location><page_5><loc_10><loc_18><loc_49><loc_22></location>[8] C. Chen, D. Dong, R. Long, I. R. Petersen, and H. A. Rabitz, Sampling-based learning control of inhomogeneous quantum ensembles, Phys. Rev. A 89 , 023402 (2014).</list_item>
<list_item><location><page_5><loc_10><loc_13><loc_49><loc_18></location>[9] D. Dong, M. A. Mabrok, I. R. Petersen, B. Qi, C. Chen, and H. Rabitz, Sampling-based learning control for quantum systems with uncertainties, IEEE Trans. Control Syst. Technol. 23 , 2155 (2015).</list_item>
<list_item><location><page_5><loc_9><loc_9><loc_49><loc_13></location>[10] D. Daems, A. Ruschhaupt, D. Sugny, and S. Gu'erin, Robust Quantum Control by a Single-Shot Shaped Pulse, Phys. Rev. Lett. 111 , 050404 (2013).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_5><loc_52><loc_89><loc_92><loc_93></location>[11] G. Dridi, K. Liu, and S. Gu'erin, Optimal Robust Quantum Control by Inverse Geometric Optimization, Phys. Rev. Lett. 125 , 250403 (2020).</list_item>
<list_item><location><page_5><loc_52><loc_85><loc_92><loc_89></location>[12] E. Barnes, X. Wang, and S. D. Sarma, Robust quantum control using smooth pulses and topological winding, Sci. Rep. 5 , 12685 (2015).</list_item>
<list_item><location><page_5><loc_52><loc_81><loc_92><loc_85></location>[13] J. Zeng, C. H. Yang, A. S. Dzurak, and E. Barnes, Geometric formalism for constructing arbitrary single-qubit dynamically corrected gates, Phys. Rev. A 99 , 052321 (2019).</list_item>
<list_item><location><page_5><loc_52><loc_77><loc_92><loc_81></location>[14] D. Buterakos, S. Das Sarma, and E. Barnes, Geometrical Formalism for Dynamically Corrected Gates in Multiqubit Systems, PRX Quantum 2 , 010341 (2021).</list_item>
<list_item><location><page_5><loc_52><loc_73><loc_92><loc_77></location>[15] U. Gungordu and J. P. Kestner, Analytically parametrized solutions for robust quantum control using smooth pulses, Phys. Rev. A 100 , 062310 (2019).</list_item>
<list_item><location><page_5><loc_52><loc_69><loc_92><loc_73></location>[16] E. Paladino, Y. M. Galperin, G. Falci, and B. L. Altshuler, 1 /f noise: Implications for solid-state quantum information, Rev. Mod. Phys. 86 , 361 (2014).</list_item>
<list_item><location><page_5><loc_52><loc_67><loc_92><loc_69></location>[17] J. Clarke and F. K. Wilhelm, Superconducting quantum bits, Nature 453 , 1031 (2008).</list_item>
<list_item><location><page_5><loc_52><loc_63><loc_92><loc_67></location>[18] P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson, and W. D. Oliver, A quantum engineer's guide to superconducting qubits, Appl. Phys. Rev. 6 , 021318 (2019).</list_item>
<list_item><location><page_5><loc_52><loc_60><loc_92><loc_63></location>[19] T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O'Brien, Quantum computers, Nature 464 , 45 (2010).</list_item>
<list_item><location><page_5><loc_52><loc_55><loc_92><loc_60></location>[20] F. K. Malinowski, F. Martins, L. Cywi'nski, M. S. Rudner, P. D. Nissen, S. Fallahi, G. C. Gardner, M. J. Manfra, C. M. Marcus, and F. Kuemmeth, Spectrum of the Nuclear Environment for GaAs Spin Qubits, Phys. Rev. Lett. 118 , 177702 (2017).</list_item>
<list_item><location><page_5><loc_52><loc_51><loc_92><loc_55></location>[21] C. Kabytayev, T. J. Green, K. Khodjasteh, M. J. Biercuk, L. Viola, and K. R. Brown, Robustness of composite pulses to timedependent control noise, Phys. Rev. A 90 , 012316 (2014).</list_item>
<list_item><location><page_5><loc_52><loc_47><loc_92><loc_51></location>[22] J. Zhang, A. M. Souza, F. D. Brandao, and D. Suter, Protected quantum computing: Interleaving gate operations with dynamical decoupling sequences, Phys. Rev. Lett. 112 , 050502 (2014).</list_item>
<list_item><location><page_5><loc_52><loc_44><loc_92><loc_47></location>[23] J. Werschnik and E. K. U. Gross, Quantum optimal control theory, J. Phys. B: At. Mol. Opt. Phys. 40 , R175 (2007).</list_item>
<list_item><location><page_5><loc_52><loc_40><loc_92><loc_44></location>[24] J. M. Martinis, S. Nam, J. Aumentado, K. M. Lang, and C. Urbina, Decoherence of a superconducting qubit due to bias noise, Phys. Rev. B 67 , 094510 (2003).</list_item>
<list_item><location><page_5><loc_52><loc_36><loc_92><loc_40></location>[25] A. G. Kofman and G. Kurizki, Unified theory of dynamically suppressed qubit decoherence in thermal baths, Phys. Rev. Lett. 93 , 130406 (2004).</list_item>
<list_item><location><page_5><loc_52><loc_34><loc_92><loc_36></location>[26] G. S. Uhrig, Keeping a quantum bit alive by optimized π -pulse sequences, Phys. Rev. Lett. 98 , 100504 (2007).</list_item>
<list_item><location><page_5><loc_52><loc_30><loc_92><loc_34></location>[27] G. Gordon, G. Kurizki, and D. A. Lidar, Optimal Dynamical Decoherence Control of a Qubit, Phys. Rev. Lett. 101 , 010403 (2008).</list_item>
<list_item><location><page_5><loc_52><loc_26><loc_92><loc_30></location>[28] M. J. Biercuk, H. Uys, A. P. VanDevender, N. Shiga, W. M. Itano, and J. J. Bollinger, Optimized dynamical decoupling in a model quantum memory, Nature 458 , 996 (2009).</list_item>
<list_item><location><page_5><loc_52><loc_22><loc_92><loc_26></location>[29] H. Uys, M. J. Biercuk, and J. J. Bollinger, Optimized Noise Filtration through Dynamical Decoupling, Phys. Rev. Lett. 103 , 040501 (2009).</list_item>
<list_item><location><page_5><loc_52><loc_18><loc_92><loc_22></location>[30] J. Clausen, G. Bensky, and G. Kurizki, Bath-Optimized Minimal-Energy Protection of Quantum Operations from Decoherence, Phys. Rev. Lett. 104 , 040401 (2010).</list_item>
<list_item><location><page_5><loc_52><loc_15><loc_92><loc_18></location>[31] C. Kabytayev, Quantum control for time-dependent noise (Ph.D. Thesis, Georgia Institute of Technology, 2015).</list_item>
<list_item><location><page_5><loc_52><loc_11><loc_92><loc_15></location>[32] M. Biercuk, A. Doherty, and H. Uys, Dynamical decoupling sequence construction as a filter-design problem, J. Phys. B: At. Mol. Opt. Phys. 44 , 154002 (2011).</list_item>
<list_item><location><page_5><loc_52><loc_9><loc_92><loc_11></location>[33] I. N. M. Le, J. D. Teske, T. Hangleiter, P. Cerfontaine, and H. Bluhm, Analytic filter-function derivatives for quantum op-</list_item>
</unordered_list>
<text><location><page_6><loc_12><loc_92><loc_43><loc_93></location>timal control, Phys. Rev. Applied 17 , 024006 (2022).</text>
<unordered_list>
<list_item><location><page_6><loc_9><loc_89><loc_49><loc_92></location>[34] C. Ferrie and O. Moussa, Robust and efficient in situ quantum control, Phys. Rev. A 91 , 052306 (2015).</list_item>
<list_item><location><page_6><loc_9><loc_85><loc_49><loc_89></location>[35] T. Yuge, S. Sasaki, and Y. Hirayama, Measurement of the noise spectrum using a multiple-pulse sequence, Phys. Rev. Lett. 107 , 170504 (2011).</list_item>
<list_item><location><page_6><loc_9><loc_81><loc_49><loc_86></location>[36] G. A. ' Alvarez and D. Suter, Measuring the spectrum of colored noise by dynamical decoupling, Phys. Rev. Lett. 107 , 230501 (2011).</list_item>
<list_item><location><page_6><loc_9><loc_77><loc_49><loc_81></location>[37] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2010).</list_item>
<list_item><location><page_6><loc_9><loc_75><loc_49><loc_77></location>[38] F. J. Dyson, The Radiation Theories of Tomonaga, Schwinger, and Feynman, Phys. Rev. 75 , 486 (1949).</list_item>
<list_item><location><page_6><loc_9><loc_71><loc_49><loc_75></location>[39] T. J. Green, J. Sastrawan, H. Uys, and M. J. Biercuk, Arbitrary quantum control of qubits in the presence of universal noise, New J. Phys. 15 , 095004 (2013).</list_item>
<list_item><location><page_6><loc_9><loc_69><loc_38><loc_71></location>[40] See Supplemental Material for more details.</list_item>
<list_item><location><page_6><loc_9><loc_64><loc_49><loc_69></location>[41] N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbruggen, and S. J. Glaser, Optimal control of coupled spin dynamics: design of nmr pulse sequences by gradient ascent algorithms, J. Magn. Reson. 172 , 296 (2005).</list_item>
<list_item><location><page_6><loc_9><loc_60><loc_49><loc_64></location>[42] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, Dynamics of the dissipative two-state system, Rev. Mod. Phys. 59 , 1 (1987).</list_item>
<list_item><location><page_6><loc_9><loc_56><loc_49><loc_60></location>[43] M. Bando, T. Ichikawa, Y. Kondo, and M. Nakahara, Concatenated composite pulses compensating simultaneous systematic errors, J. Phys. Soc. Japan 82 , 014004 (2012).</list_item>
<list_item><location><page_6><loc_9><loc_51><loc_49><loc_56></location>[44] F. Yan, S. Gustavsson, J. Bylander, X. Jin, F. Yoshihara, D. G. Cory, Y. Nakamura, T. P. Orlando, and W. D. Oliver, Rotatingframe relaxation as a noise spectrum analyser of a superconducting qubit undergoing driven evolution, Nat. Commun. 4 , 1</list_item>
</unordered_list>
<section_header_level_1><location><page_6><loc_55><loc_92><loc_59><loc_93></location>(2013).</section_header_level_1>
<unordered_list>
<list_item><location><page_6><loc_52><loc_89><loc_92><loc_92></location>[45] J. Jeener, Superoperators in magnetic resonance, Adv. Magn. Reson. 10 , 1 (1982).</list_item>
<list_item><location><page_6><loc_52><loc_84><loc_92><loc_89></location>[46] G. Wolfowicz, F. J. Heremans, C. P. Anderson, S. Kanai, H. Seo, A. Gali, G. Galli, and D. D. Awschalom, Quantum guidelines for solid-state spin defects, Nat. Rev. Mater. 6 , 906 (2021).</list_item>
<list_item><location><page_6><loc_52><loc_81><loc_92><loc_84></location>[47] H. Haas, D. Puzzuoli, F. Zhang, and D. G. Cory, Engineering effective Hamiltonians, New. J. Phys. 21 , 103011 (2019).</list_item>
<list_item><location><page_6><loc_52><loc_77><loc_92><loc_81></location>[48] E. Barnes, Analytically solvable two-level quantum systems and landau-zener interferometry, Phys. Rev. A 88 , 013818 (2013).</list_item>
<list_item><location><page_6><loc_52><loc_73><loc_92><loc_77></location>[49] E. Barnes and S. Das Sarma, Analytically Solvable Driven Time-Dependent Two-Level Quantum Systems, Phys. Rev. Lett. 109 , 060401 (2012).</list_item>
<list_item><location><page_6><loc_52><loc_71><loc_92><loc_73></location>[50] P. Titum, K. Schultz, A. Seif, G. Quiroz, and B. Clader, Optimal control for quantum detectors, Npj Quantum Inf. 7 , 1 (2021).</list_item>
<list_item><location><page_6><loc_52><loc_65><loc_92><loc_71></location>[51] N. Bar-Gill, L. M. Pham, C. Belthangady, D. Le Sage, P. Cappellaro, J. Maze, M. D. Lukin, A. Yacoby, and R. Walsworth, Suppression of spin-bath dynamics for improved coherence of multi-spin-qubit systems, Nat. Commun. 3 , 1 (2012).</list_item>
<list_item><location><page_6><loc_52><loc_59><loc_92><loc_65></location>[52] L. Hall, P. Kehayias, D. A. Simpson, A. Jarmola, A. Stacey, D. Budker, and L. C. L. Hollenberg, Detection of nanoscale electron spin resonance spectra demonstrated using nitrogenvacancy centre probes in diamond, Nat. Commun. 7 , 10211 (2015).</list_item>
<list_item><location><page_6><loc_52><loc_52><loc_92><loc_59></location>[53] K. W. Chan, W. Huang, C. H. Yang, J. C. C. Hwang, B. Hensen, T. Tanttu, F. E. Hudson, K. M. Itoh, A. Laucht, A. Morello, and A. S. Dzurak, Assessment of a Silicon Quantum Dot Spin Qubit Environment via Noise Spectroscopy, Phys. Rev. Applied 10 , 044017 (2018).</list_item>
</unordered_list>
<section_header_level_1><location><page_6><loc_40><loc_34><loc_61><loc_36></location>Supplementary Material</section_header_level_1>
<section_header_level_1><location><page_6><loc_22><loc_29><loc_79><loc_30></location>INVERSE GEOMETRIC OPTIMIZATION FOR SINGLE-QUBIT QUANTUM SYSTEM</section_header_level_1>
<section_header_level_1><location><page_6><loc_29><loc_25><loc_71><loc_26></location>Derivation of Eq. (2) of the Main Text: Evolution Parameterization</section_header_level_1>
<text><location><page_6><loc_10><loc_22><loc_56><loc_23></location>An arbitrary single-qubit noise-free evolution can be parameterized by</text>
<formula><location><page_6><loc_32><loc_17><loc_69><loc_20></location>U = [ cos( θ/ 2) e -iϕ/ 2 e -iγ/ 2 -sin( θ/ 2) e -iϕ/ 2 e iγ/ 2 sin( θ/ 2) e iϕ/ 2 e -iγ/ 2 cos( θ/ 2) e iϕ/ 2 e iγ/ 2 ] .</formula>
<text><location><page_6><loc_9><loc_13><loc_45><loc_15></location>Using this, the Schrodinger equation can be rewritten as</text>
<formula><location><page_6><loc_30><loc_8><loc_71><loc_12></location>[ ˙ U 11 ˙ U 12 ˙ U 21 ˙ U 22 ] = [ 0 ( -iu x -u y ) / 2 ( -iu x + u y ) / 2 0 ][ U 11 U 12 U 21 U 22 ]</formula>
<text><location><page_7><loc_9><loc_92><loc_14><loc_93></location>which is</text>
<formula><location><page_7><loc_25><loc_88><loc_92><loc_91></location>-˙ θ 2 sin( θ/ 2) -i ˙ ϕ 2 cos( θ/ 2) -i ˙ γ 2 cos( θ/ 2) = 1 2 ( -iu x -u y ) sin( θ/ 2) e iϕ , (S.1a)</formula>
<formula><location><page_7><loc_25><loc_85><loc_92><loc_88></location>-˙ θ 2 cos( θ/ 2) + i ˙ ϕ 2 sin( θ/ 2) -i ˙ γ 2 sin( θ/ 2) = 1 2 ( -iu x -u y ) cos( θ/ 2) e iϕ , (S.1b)</formula>
<formula><location><page_7><loc_27><loc_81><loc_92><loc_84></location>˙ θ 2 cos( θ/ 2) + i ˙ ϕ 2 sin( θ/ 2) -i ˙ γ 2 sin( θ/ 2) = 1 2 ( -iu x + u y ) cos( θ/ 2) e -iϕ , (S.1c)</formula>
<formula><location><page_7><loc_25><loc_78><loc_92><loc_81></location>-˙ θ 2 sin( θ/ 2) + i ˙ ϕ 2 cos( θ/ 2) + i ˙ γ 2 cos( θ/ 2) = 1 2 ( iu x -u y ) sin( θ/ 2) e -iϕ . (S.1d)</formula>
<text><location><page_7><loc_9><loc_76><loc_33><loc_77></location>From these equations, we can obtain</text>
<formula><location><page_7><loc_31><loc_74><loc_92><loc_75></location>˙ θ = u sin ϕ + u cos ϕ = Ωsin( φ ϕ ) , (S.2a)</formula>
<formula><location><page_7><loc_31><loc_71><loc_92><loc_73></location>˙ ϕ = -( u x cos ϕ + u y sin ϕ ) cot( θ ) = -Ωcos( φ -ϕ ) cot θ, (S.2b)</formula>
<formula><location><page_7><loc_34><loc_73><loc_57><loc_75></location>-x y -</formula>
<formula><location><page_7><loc_31><loc_69><loc_92><loc_71></location>˙ γ = ( u x cos ϕ + u y sin ϕ ) / sin( θ ) = Ω cos( φ -ϕ ) / sin θ. (S.2c)</formula>
<text><location><page_7><loc_9><loc_67><loc_64><loc_69></location>Once a robust evolution trajectory is obtained, we can determine the control fields by</text>
<formula><location><page_7><loc_34><loc_64><loc_92><loc_67></location>Ω( t ) = √ ˙ θ 2 + ˙ γ 2 sin 2 θ, φ ( t ) = arcsin( ˙ θ/ Ω) + ϕ. (S.3)</formula>
<section_header_level_1><location><page_7><loc_31><loc_60><loc_70><loc_61></location>Derivation of Eq. (3) of the Main Text: Average Gate Infidelity</section_header_level_1>
<text><location><page_7><loc_10><loc_57><loc_54><loc_58></location>For one realization of noise glyph[epsilon1] a ( t ) and glyph[epsilon1] d ( t ) , the real gate fidelity is</text>
<formula><location><page_7><loc_33><loc_53><loc_92><loc_56></location>F = ∣ ∣ ∣ Tr ( U † U glyph[epsilon1] a ,glyph[epsilon1] d ( T ) )∣ ∣ ∣ 2 / 4 = | Tr( U tog ( T )) | 2 / 4 . (S.4)</formula>
<text><location><page_7><loc_9><loc_47><loc_92><loc_52></location>It is convenient to write U tog ( T ) = exp {-i ∑ µ = a,d ∫ T 0 ε µ ( t ) ˜ E µ ( t ) dt } ≡ e -i a · σ / 2 = 1 cos( a/ 2) -i sin( a/ 2) a · σ /a , where σ = ( σ x , σ y , σ z ) , thus F = [1 + cos( a )] / 2 . For the first-order approximation, we obtain F ≈ 1 -a 2 / 4 . Take the ensemble average of the noise, we get the average gate infidelity</text>
<formula><location><page_7><loc_17><loc_36><loc_92><loc_45></location>F avg = 1 -〈 F 〉 ≈ ∑ µ = a,d [ ∫ T 0 dt 1 ∫ T 0 dt 2 glyph[epsilon1] µ ( t 1 ) glyph[epsilon1] µ ( t 2 ) ∑ α = x,y,z Tr [ ˜ E µ,α ( t 1 ) σ α / 2] Tr [ ˜ E µ,α ( t 2 ) σ α / 2] ] = 1 2 π ∑ µ = a,d α = x,y,z ∫ ∞ -∞ dω ω 2 S µ ( ω ) | R µ,α ( ω ) | 2 . (S.5)</formula>
<text><location><page_7><loc_9><loc_32><loc_42><loc_35></location>where R µ,α ( ω ) = -iω ∫ T 0 dt Tr [ ˜ E µ,α ( t ) σ α / 2] e iωt .</text>
<section_header_level_1><location><page_7><loc_28><loc_29><loc_73><loc_30></location>Derivation of Eq. (4) of the Main Text: Average State Transfer Infidelity</section_header_level_1>
<text><location><page_7><loc_10><loc_26><loc_54><loc_27></location>For one realization of noise glyph[epsilon1] a ( t ) and glyph[epsilon1] d ( t ) , the real state fidelity is</text>
<formula><location><page_7><loc_10><loc_19><loc_77><loc_25></location>F = |〈 ψ | U glyph[epsilon1] a ,glyph[epsilon1] d ( T ) | 0 〉| 2 = |〈 0 | U tog ( T ) | 0 〉| 2 = |〈 0 | ( 1 -∑ µ = a,d [ i ∫ T 0 dt 1 glyph[epsilon1] µ ( t 1 ) ˜ E µ ( t 1 ) + ∫ T 0 dt 1 ∫ t 1 0 dt 2 glyph[epsilon1] µ ( t 1 ) glyph[epsilon1] µ ( t 2 ) ˜ E µ ( t 1 ) ˜ E µ ( t 2 ) + · · · ]) | 0 〉| 2</formula>
<formula><location><page_7><loc_12><loc_15><loc_57><loc_19></location>≈ 1 + ∑ µ = a,d ∫ T 0 dt 1 ∫ T 0 dt 2 glyph[epsilon1] µ ( t 1 ) glyph[epsilon1] µ ( t 2 ) 〈 0 | ˜ E µ ( t 1 ) | 0 〉〈 0 | ˜ E µ ( t 2 ) | 0 〉 ∗</formula>
<formula><location><page_7><loc_14><loc_11><loc_90><loc_14></location>-∑ µ = a,d ∫ T 0 dt 1 ∫ t 1 0 dt 2 glyph[epsilon1] µ ( t 1 ) glyph[epsilon1] µ ( t 2 ) 〈 0 | ˜ E µ ( t 1 ) ˜ E µ ( t 2 ) | 0 〉 -∑ µ = a,d ∫ T 0 dt 1 ∫ t 1 0 dt 2 glyph[epsilon1] µ ( t 1 ) glyph[epsilon1] µ ( t 2 ) 〈 0 | ˜ E µ ( t 1 ) ˜ E µ ( t 2 ) | 0 〉 ∗ ,</formula>
<text><location><page_7><loc_89><loc_9><loc_92><loc_11></location>(S.6)</text>
<text><location><page_8><loc_9><loc_91><loc_86><loc_93></location>where we omit the cross terms containing glyph[epsilon1] µ 1 ( t 1 ) glyph[epsilon1] µ 2 ( t 2 ) with µ 1 = µ 2 . Insert | 0 〉〈 0 | + | 1 〉〈 1 | into the expression, then</text>
<text><location><page_8><loc_52><loc_91><loc_52><loc_93></location>glyph[negationslash]</text>
<formula><location><page_8><loc_27><loc_87><loc_92><loc_91></location>F ≈ 1 -∑ µ = a,d ∫ T 0 dt 1 ∫ T 0 dt 2 glyph[epsilon1] µ ( t 1 ) glyph[epsilon1] µ ( t 2 ) 〈 0 | ˜ E µ ( t 1 ) | 1 〉〈 1 | ˜ E µ ( t 2 ) | 0 〉 . (S.7)</formula>
<text><location><page_8><loc_9><loc_84><loc_31><loc_86></location>Thus the average state infidelity is</text>
<formula><location><page_8><loc_32><loc_79><loc_92><loc_83></location>F avg = 1 -〈 F 〉 ≈ ∑ µ = a,d α = x,y,z 1 2 π ∫ ∞ -∞ dω ω 2 S µ ( ω ) | P µ,α ( ω ) | 2 , (S.8)</formula>
<text><location><page_8><loc_9><loc_75><loc_38><loc_78></location>where P µ,α = -iω ∫ T 0 dt 〈 0 | ˜ E µ,α ( t ) | 1 〉 e iωt .</text>
<section_header_level_1><location><page_8><loc_41><loc_72><loc_60><loc_73></location>Gradient-based Optimization</section_header_level_1>
<text><location><page_8><loc_9><loc_65><loc_92><loc_70></location>To search robust trajectory, we apply gradient-based optimization. In our optimization, we discretize θ ( t ) and γ ( t ) as M -slice sequences θ [1] , ..., θ [ M ] and γ [1] , ..., γ [ M ] , respectively, with the time length of each slice τ = T/M . For quantum gate, the derivative of the average gate infidelity function reads</text>
<formula><location><page_8><loc_20><loc_54><loc_92><loc_64></location>∂ F avg ∂χ [ m ] = ∑ µ = a,d α = x,y,z 1 2 π ∞ ∫ -∞ dω ω 2 S µ ( ω ) Re { ∂R µ,α ( ω ) ∂χ [ m ] R ∗ µ,α ( ω ) } (S.9) = ∑ µ = a,d α = x,y,z 1 2 π ∞ ∫ -∞ dω ω 2 S µ ( ω ) Re { -iωτe iωmτ ∂ Tr [ ˜ E µ,α [ m ] σ α 2 ] ∂χ [ m ] R ∗ µ,α ( ω ) } , χ = θ, γ.</formula>
<text><location><page_8><loc_9><loc_52><loc_71><loc_53></location>Similarly, for quantum state transfer, the derivative of the average state infidelity function reads</text>
<formula><location><page_8><loc_20><loc_40><loc_92><loc_51></location>∂ F avg ∂χ [ m ] = ∑ µ = a,d α = x,y,z 1 2 π ∞ ∫ -∞ dω ω 2 S µ ( ω ) Re { ∂P µ,α ( ω ) ∂χ [ m ] P ∗ µ,α ( ω ) } (S.10) = ∑ µ = a,d α = x,y,z 1 2 π ∞ ∫ -∞ dω ω 2 S µ ( ω ) Re { -iωτe iωmτ ∂ Tr [ ˜ E µ,α [ m ] | 1 〉〈 0 | ] ∂χ [ m ] P ∗ µ,α ( ω ) } , χ = θ, γ.</formula>
<text><location><page_8><loc_9><loc_37><loc_86><loc_39></location>The above gradients are then used to update the trajectory by χ [ m ] ← χ [ m ] + l ∂ F avg ∂χ [ m ] , where l is appropriate step size.</text>
<text><location><page_8><loc_9><loc_9><loc_92><loc_37></location>We test our robust control method for realizing a π rotational gate or state transfer from | 0 〉 to | 1 〉 subject to various types of time-dependent noise, as shown in Figs. S1-S2 and Figs. S3-S4, respectively. For detuning noise, we compare the performance between three sequences, namely primitive, CORPSE and ROC pulses, see the results in Figs. S1-S4(a 1 )-(a 3 ),(b 1 )-(b 3 ),(c 1 )-(c 3 ). Specifically, we consider (i) ohmic spectrum with sharp cut-offs, i.e., S d ( ω ) ∝ ω, ω ∈ [ ω lc , ω uc ] , which describes a spin suffering bosonic environment [42]; (ii) single Lorentzian spectrum S d ( ω ) ∝ 1 / ( λ 2 +( ω -ω 0 ) 2 ) or multiple Lorentzian spectrum S d ( ω ) ∝ ∑ k A k / ( λ 2 k +( ω -ω 0 ,k ) 2 ) , which captures the solid-state spin environment of a spin bath as measured in, e.g., Refs. [51, 52]. For amplitude noise, we compare the performance between primitive, BB1 and ROC pulses, as shown in Figs. S1-S4(d 1 )-(d 3 ),(e 1 )-(e 3 ). Here we examine (i) a noise model corresponding to a strong and narrow Gaussian peak added on top of a broad 1 /f κ background with a roll-off to white noise, i.e., S a ( ω ) ∝ A exp[ -( ω -ω 0 ) 2 / (2 σ 2 )] + B/ω κ , ω < ω wc; S a ( ω ) = const. , ω ≥ ω wc. This type of noise spectrum was observed, for example, in a silicon quantum dot spin qubit due to the imperfect control apparatus [53]. (ii) several Lorentzian peaks added on top of a broad 1 /f κ background S a ( ω ) ∝ ∑ k A k / ( λ 2 k + ( ω -ω 0 ,k ) 2 ) + B/ω κ , which describes the random fluctuations in superconducting flux terms [44]. For simultaneous detuning noise and amplitude noise, for simplicity, we choose both of the noise spectrums as a single Lorentzian peak added on top of a broad 1 /f background, i.e., S µ ( ω ) ∝ A/ ( λ 2 +( ω -ω 0 ) 2 )+ B/ω . We compare the performance between primitive, reduced CinBB and ROC pulses. Reduced CinBB [43] is a concatenated composite pulse for suppressing both of the detuning and amplitude noises. Results are summarized in Figs. S1-S4(f 1 )-(f 3 ). We also list explicitly the parameters of the noise power density spectrums and the tested control sequences in Table. I. All the simulations reveal that our ROC method find high-quality, smooth and low-power robust pulses for resisting various realistic time-dependent noises. In the main text, we demonstrate several typical results to show the effectiveness of our robust control method.</text>
<figure>
<location><page_9><loc_13><loc_64><loc_87><loc_92></location>
<caption>FIG. S1. Geometric trajectories, control waveforms (in the unit of Ω max ), FFs and noise spectrums of different sequences for realizing a π rotation gate subject to high-frequency time-dependent (a 1 )-(a 3 ),(b 1 )-(b 3 ),(c 1 )-(c 3 ) detuning noise, (d 1 )-(d 3 ),(e 1 )-(e 3 ) amplitude noise, or (f 1 )-(f 3 ) both. For detuning noise, the strength is √ 〈 glyph[epsilon1] 2 d (0) 〉 = 0 . 03Ω max with Ω max / (2 π ) = 10 7 Hz, and the noise spectrums are ohmic (insert in (a 3 )), single Lorentzian (insert in (b 3 ), λ = 100 Hz) , or multiple Lorentzian (insert in (c 3 ), λ 1 = λ 2 = λ 3 = 100 Hz, A 1 = 0 . 8 , A 2 = 1 . 5 , A 3 = 0 . 6 ), respectively. For amplitude noise, the strength is √ 〈 glyph[epsilon1] 2 a (0) 〉 = 0 . 03 , and the noise spectrums are a Gaussian peak added on top of 1 /f background with a roll-off to white noise (insert in (d 3 ), σ = 5000 Hz, κ = 1 , A = 1 , B = 0 . 05 , ω wc = Ω max ), or two Lorentzian peaks added on top of 1 /f background (insert in (e 3 ), λ 1 = λ 2 = 100 Hz, κ = 1 , A 1 = A 2 = 1 , B = 0 . 05 ), respectively. For simultaneous detuning noise and amplitude noise, the noise spectrums are both chosen as a single Lorentzian peak added on top of 1 /f background (insert in (f 3 ), λ = 100 Hz, κ = 1 , A = 1 , B = 0 . 05 ) with the same strengths as above.</caption>
</figure>
<table>
<location><page_9><loc_9><loc_15><loc_91><loc_47></location>
<caption>TABLE I. Nose spectrum features and sequence parameters for realizing a π rotational gate or state transfer from | 0 〉 to | 1 〉 subject to timedependent detuning noise (the first sub table), amplitude noise (the second sub table) or both (the third sub table), respectively, where Ω max is the maximum rabi frequency, T p represents the length of primitive sequence.</caption>
</table>
<figure>
<location><page_10><loc_13><loc_64><loc_87><loc_93></location>
<caption>FIG. S2. Geometric trajectories, control waveforms, FFs and noise spectrums of different sequences for realizing a π rotation gate subject to low-frequency time-dependent (a 1 )-(a 3 ),(b 1 )-(b 3 ),(c 1 )-(c 3 ) detuning noise, (d 1 )-(d 3 ),(e 1 )-(e 3 ) amplitude noise, or (f 1 )-(f 3 ) both. For detuning noise, the noise spectrums are ohmic (insert in (a 3 )), single Lorentzian (insert in (b 3 ), λ = 100 Hz) , or multiple Lorentzian (insert in (c 3 )), respectively. For amplitude noise, the noise spectrums are a Gaussian peak added on top of 1 /f background with a roll-off to white noise (insert in (d 3 )), or two Lorentzian peaks added on top of 1 /f background (insert in (e 3 )), respectively. For simultaneous detuning noise and amplitude noise, the noise spectrums are both chosen as a single Lorentzian peak added on top of 1 /f background (insert in (f 3 )) with the same strengths as above. All the spectrum and control pulse parameters are the same with above case.</caption>
</figure>
<figure>
<location><page_10><loc_13><loc_21><loc_87><loc_50></location>
<caption>FIG. S3. Geometric trajectories, control waveforms, FFs and noise spectrums of different sequences for realizing state transfer from | 0 〉 to | 1 〉 subject to high-frequency time-dependent (a 1 )-(a 3 ),(b 1 )-(b 3 ),(c 1 )-(c 3 ) detuning noise, (d 1 )-(d 3 ),(e 1 )-(e 3 ) amplitude noise, or (f 1 )-(f 3 ) both. For detuning noise, the noise spectrums are ohmic (insert in (a 3 )), single Lorentzian (insert in (b 3 )) , or multiple Lorentzian (insert in (c 3 )), respectively. For amplitude noise, the noise spectrums are a Gaussian peak added on top of 1 /f background with a roll-off to white noise (insert in (d 3 )), or two Lorentzian peaks added on top of 1 /f background (insert in (e 3 )), respectively. For simultaneous detuning noise and amplitude noise, the noise spectrums are both chosen as a single Lorentzian peak added on top of 1 /f background (insert in (f 3 )) with the same strengths as above. All the spectrum and control pulse parameters are the same with that in quantum gate case.</caption>
</figure>
<figure>
<location><page_11><loc_13><loc_64><loc_87><loc_93></location>
<caption>FIG. S4. Geometric trajectories, control waveforms, FFs and noise spectrums of different sequences for realizing state transfer from | 0 〉 to | 1 〉 subject to low-frequency time-dependent (a 1 )-(a 3 ),(b 1 )-(b 3 ),(c 1 )-(c 3 ) detuning noise, (d 1 )-(d 3 ),(e 1 )-(e 3 ) amplitude noise, or (f 1 )-(f 3 ) both. For detuning noise, the noise spectrums are ohmic (insert in (a 3 )), single Lorentzian (insert in (b 3 )) , or multiple Lorentzian (insert in (c 3 )), respectively. For amplitude noise, the noise spectrums are a Gaussian peak added on top of 1 /f background with a roll-off to white noise (insert in (d 3 )), or two Lorentzian peaks added on top of 1 /f background (insert in (e 3 )), respectively. For simultaneous detuning noise and amplitude noise, the noise spectrums are both chosen as a single Lorentzian peak added on top of 1 /f background (insert in (f 3 )) with the same strengths as above. All the spectrum and control pulse parameters are the same with that in quantum gate case.</caption>
</figure>
<section_header_level_1><location><page_11><loc_21><loc_47><loc_80><loc_48></location>INVERSE GEOMETRIC OPTIMIZATION FOR TWO-LEVEL OPEN QUANTUM SYSTEM</section_header_level_1>
<section_header_level_1><location><page_11><loc_42><loc_44><loc_59><loc_45></location>Evolution Parameterization</section_header_level_1>
<text><location><page_11><loc_9><loc_37><loc_92><loc_41></location>For an isolated spin-1/2 ensemble, the density matrix can be conventionally expressed as ρ = 1 / 2 + x 1 σ x + x 2 σ y + x 3 σ 3 . When undergoing both transverse and longitudinal relaxation, and under Markovian approximation, the system dynamics under controls u x = Ω( t ) cos( φ ( t )) , u y = Ω( t ) sin( φ ( t )) can be described by the Bloch equations</text>
<formula><location><page_11><loc_39><loc_33><loc_92><loc_35></location>˙ x = ( H 0 ( t ) + γ 1 R T 1 + γ 2 R T 2 ) x, (S.11)</formula>
<text><location><page_11><loc_9><loc_29><loc_58><loc_31></location>where x = (1 / 2 , x 1 , x 2 , x 3 ) T , γ 1 , 2 = 1 /T 1 , 2 are relaxation parameters and</text>
<formula><location><page_11><loc_20><loc_20><loc_92><loc_27></location>H 0 ( t ) = 0 0 0 0 0 0 -Ω 0 u y 0 Ω 0 0 -u x 0 -u y u x 0 , R T 1 = 0 0 0 0 0 0 0 0 0 0 0 0 2 M 0 0 0 -1 , R T 2 = 0 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 0 . (S.12)</formula>
<text><location><page_11><loc_9><loc_17><loc_78><loc_19></location>For simplicity, we set the off-resonance frequency Ω 0 = 0 and the equilibrium state polarization M 0 = 1 .</text>
<text><location><page_11><loc_10><loc_15><loc_67><loc_17></location>To apply inverse geometric optimization, we introduce the three-dimensional rotations</text>
<formula><location><page_11><loc_18><loc_8><loc_92><loc_13></location>R z ( δ ) = cos δ -sin δ 0 sin δ cos δ 0 0 0 1 , R y ( η ) = cos η 0 sin η 0 1 0 -sin η 0 cos η , R z ( ξ ) = cos ξ -sin ξ 0 sin ξ cos ξ 0 0 0 1 . (S.13)</formula>
<text><location><page_12><loc_9><loc_68><loc_19><loc_70></location>This then gives</text>
<text><location><page_12><loc_9><loc_92><loc_72><loc_93></location>As such, the noise-free evolution of two-level open quantum system can the be parameterized by</text>
<formula><location><page_12><loc_15><loc_80><loc_92><loc_91></location>V 0 ( t ) = ( 1 0 0 R z ( δ ) R y ( η ) R z ( ξ ) ) (S.14) = 1 0 0 0 0 cos( δ ) cos( η ) cos( ξ ) -sin( δ ) sin( ξ ) -sin( δ ) cos( ξ ) -cos( δ ) cos( η ) sin( ξ ) cos( δ ) sin( η ) 0 sin( δ ) cos( η ) cos( ξ ) + cos( δ ) sin( ξ ) cos( δ ) cos( ξ ) -sin( δ ) cos( η ) sin( ξ ) sin( δ ) sin( η ) 0 -sin( η ) cos( ξ ) sin( η ) sin( ξ ) cos( η ) ,</formula>
<text><location><page_12><loc_9><loc_78><loc_30><loc_80></location>thus the Bloch equation becomes</text>
<formula><location><page_12><loc_32><loc_70><loc_92><loc_77></location>˙ V 0 ( t ) = H 0 ( t ) V 0 ( t ) , H 0 ( t ) = 0 0 0 0 0 0 0 u y 0 0 0 -u x 0 -u y u x 0 . (S.15)</formula>
<formula><location><page_12><loc_31><loc_64><loc_92><loc_67></location>˙ ξ = ( u x cos δ + u y sin δ ) / sin η = Ωcos( φ -δ ) / sin η, (S.16a) (S.16b)</formula>
<formula><location><page_12><loc_31><loc_61><loc_92><loc_63></location>˙ δ = -( u x cos δ + u y sin δ ) / tan η = -Ωcos( φ -δ ) / tan η. (S.16c)</formula>
<formula><location><page_12><loc_31><loc_63><loc_57><loc_65></location>˙ η = u y cos δ -u x sin δ = Ωsin( φ -δ ) ,</formula>
<text><location><page_12><loc_9><loc_59><loc_64><loc_60></location>Once a robust evolution trajectory is obtained, we can determine the control fields by</text>
<formula><location><page_12><loc_34><loc_55><loc_92><loc_58></location>Ω( t ) = √ ˙ η 2 + ˙ ξ 2 sin 2 η, φ ( t ) = arcsin ( ˙ η/ Ω) + δ. (S.17)</formula>
<section_header_level_1><location><page_12><loc_43><loc_51><loc_58><loc_52></location>State Transfer Infidelity</section_header_level_1>
<text><location><page_12><loc_10><loc_48><loc_76><loc_49></location>To characterize the distance between the actual state V T 1 ,T 2 ( T ) x (0) and the target state ¯ x , we define</text>
<formula><location><page_12><loc_88><loc_45><loc_92><loc_47></location>(S.18)</formula>
<formula><location><page_12><loc_11><loc_28><loc_90><loc_47></location>F = | ¯ x -V T 1 ,T 2 ( T ) x (0) | 2 = [¯ x -V T 1 ,T 2 ( T ) x (0)] T [¯ x -V T 1 ,T 2 ( T ) x (0)] = [¯ x -V 0 ( T ) V tog ( T ) x (0)] T [¯ x -V 0 ( T ) V tog ( T ) x (0)] ≈ V 0 ( T ) x (0) -V 0 ( T )( 1 + ∑ k =1 , 2 ∫ T 0 dtγ k ˜ R T k ( t )) x (0) T V 0 ( T ) x (0) -V 0 ( T )( 1 + ∑ k =1 , 2 ∫ T 0 dtγ k ˜ R T k ( t )) x (0) = ∑ k =1 , 2 ∫ T 0 dtγ k ˜ R T k ( t ) x (0) T ∑ j =1 , 2 ∫ T 0 dtγ j ˜ R T j ( t ) x (0) = ∑ k =1 , 2 [ ∫ T 0 dtγ k ˜ R T k ( t ) x (0) ] T [ ∫ T 0 dtγ k ˜ R T k ( t ) x (0) ] ,</formula>
<text><location><page_12><loc_9><loc_24><loc_54><loc_26></location>where one should notice that [ ˜ R T k x (0)] T [ ˜ R T j x (0)] = 0 when k = j .</text>
<text><location><page_12><loc_51><loc_24><loc_51><loc_26></location>glyph[negationslash]</text>
</document>
[
{
"title": "Quantum Control for Time-dependent Noise by Inverse Geometric Optimization",
"content": "Xiaodong Yang, 1, 2, 3 Xinfang Nie, 4, 1, 2, 3 Tao Xin, 1, 2, 3 Dawei Lu, 4, 1, 2, 3, ∗ and Jun Li 1, 2, 3, † 1 Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen, 518055, China 2 International Quantum Academy, Shenzhen, 518055, China 3 Guangdong Provincial Key Laboratory of Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen, 518055, China 4 Department of Physics, Southern University of Science and Technology, Shenzhen, 518055, China Quantum systems are exceedingly difficult to engineer because they are sensitive to various types of noises. In particular, time-dependent noises are frequently encountered in experiments but how to overcome them remains a challenging problem. In this work, we extend and apply the recently proposed robust control technique of inverse geometric optimization to time-dependent noises by working it in the filter-function formalism. The basic idea is to parameterize the control filter function geometrically and minimize its overlap with the noise spectral density. This then effectively reduces the noise susceptibility of the controlled system evolution. We show that the proposed method can produce high-quality robust pulses for realizing desired quantum evolutions under realistic noise models, and thus will find practical applications for current physical platforms. Introduction.The ability to precisely manipulate quantum systems against noise is central to practical quantum information processing [1]. There have been developed a variety of robust quantum control methods, such as composite pulses [2-4], dynamical decoupling [5-7], sampling-based learning control [8, 9], geometric-formalism-based pulse control [1015]. Many of these methods assume the considered noise to be quasi-static, i.e., slow enough compared to the operation time, which is however not always a valid noise model in reality. Actually, time-dependent noises are routinely encountered in experiments. For example, 1 /f type noise, which contains wide distribution of correlation times [16], is present in many solid-state qubit platforms such as superconducting qubits [17, 18] and semiconductor quantum dots [19, 20]. Therefore, in order to further enhance experimental control fidelities, it is of vital importance to develop robust quantum control for general time-dependent noises. Attempts to address errors induced by time-dependent noises in quantum system engineering are challenging. Results to date suggest that conventional methods usually have their limitations. For example, composite pulses, originally designed to tackle static, systematic errors, were found to be robust to fluctuating noises up to as fast as around 10% of the Rabi frequency [21]. Dynamical decoupling can protect quantum coherence in a fluctuating environment, but it requires rapid and strong control modulation, which might be problematic to realize experimentally. Moreover, how to incorporate dynamical decoupling into the task of realizing arbitrary quantum operations is still not fully clear [22]. Optimal control provides a flexible and generically applicable approach, in which the requirements of pulse smoothness and robustness can be added as optimization constraints [23]. Usually, the control variables to be optimized are temporal pulse parameters such as amplitudes and phases. Alternatively, optimization can be done in the dynamical variable space with a geometric flavor, as proposed and developed in Refs. [10-15], yet only static errors have been considered therein. In this work, we consider combing the geometric-based op- timal control method with the filter function (FF) formalism [24-26] to overcome these limitations for the purpose of resisting time-dependent noises. FFs were originally introduced to evaluate operational infidelities under stationary stochastic noises, and have proven very useful in quantum control, especially for designing dynamical decoupling sequences [27-32]. Recently, there have been studies on incorporating FF into gradient-based optimal control [33]. Here, we take the geometric approach, that is, we first parameterize the controlled system evolution trajectory with dynamical variables, which corresponds to a parameterized filter function in the frequency domain, and then minimize the overlap of the filter function and the noise spectral density; see Fig. 1 for an illustration of the basic idea. We give test examples of finding robust optimal control (ROC) pulses for producing target quantum gate and state transfer under realistic, experimentally relevant noise environments. It is found that our robust pulses outperform typical composite pulses in that their resultant FFs are suppressed at the characteristic frequencies of the considered noises, thus having much improved control fidelities. A separate section is devoted to treat the case of Markovian noise based directly on the Bloch equation, and the optimization results show that the T 1 and T 2 limit can be surpassed in the quantum state transfer task. Finally, discussions and implications are presented. Inverse geometric engineering.We consider a prototypical robust quantum control model, i.e., a resonantly controlled two-level system under time-dependent detuning noise and control amplitude noise. By convention, we parameterize the control field as Ω( t )[cos φ ( t ) , sin φ ( t )] ( t ∈ [0 , T ] ), with Ω( t ) ( | Ω( t ) | ≤ Ω max ) being the pulsed Rabi frequency and φ ( t ) ∈ [ -π, π ] the phase. Taking into account of noises, we have the following resonant frame Hamiltonian where glyph[epsilon1] a ( t ) , glyph[epsilon1] d ( t ) represent fluctuating noises on control amplitude and detuning, respectively, and we introduce E a ( t ) ≡ Ω[cos φσ x / 2 + sin φσ y / 2] and E d ≡ σ z / 2 as their corresponding noise operators. Physically, control amplitude noise is usually due to imperfect fabricated components, noisy electronics or varied fields [34], while detuning may originate from, e.g., random shifts in control driving frequency, or Overhauser effects on an electron spin by its surrounding nuclear spins [20]. In the following, we shall assume that glyph[epsilon1] a ( t ) , glyph[epsilon1] d ( t ) are mutually independent stationary Gaussian processes with zero means. Under this assumption, each noise is fully characterized in terms of its own power spectral density S µ ( ω ) = ∫ ∞ -∞ dte -iωt 〈 glyph[epsilon1] µ (0) glyph[epsilon1] µ ( t ) 〉 , µ ∈ { a, d } . For practical applications, S µ ( ω ) will be determined from noise spectroscopy measurements in real experiments [35, 36]. Now, we briefly describe the inverse geometric optimization technique [10, 11]. The procedure starts with a parameterization of the noise-free evolution. Let U 0 ( t ) be the solution to the time-dependent Schrodinger equation ˙ U 0 ( t ) = -iH 0 ( t ) U 0 ( t ) , where H 0 ( t ) is as shown in Eq. (1) with glyph[epsilon1] a , glyph[epsilon1] d = 0 . We parameterize U 0 ( t ) based on ZYZ decomposition, that is, an arbitrary single-qubit unitary operator can be written as exp( iβ ) R z ( ϕ ) R y ( θ ) R z ( γ ) , for some real numbers β, ϕ, γ ∈ [ -π, π ) and θ ∈ [ -π, π ] [37]. In our problem here, β = 0 because H 0 is traceless. Hence, we have As such, the Schrodinger equation is rewritten as We perform optimization over these dynamical angular variables in order to find an evolution trajectory that has the property of dynamically correcting errors on itself. In this geometric formulation of the control problem, the optimization objective consists of control target, robustness requirement, boundary conditions and certain practical considerations such as bounded control amplitude, all expressed in terms of θ, ϕ and γ . Once a robust evolution trajectory specified by the three angular variables is obtained, we can determine the control field by evaluating the inversion of Eq. (2), i.e., Ω = Quantum gate and quantum state transfer.We first consider the control target of implementing a desired quantum gate or quantum state transfer. The key step is to effect the transformation operator to the toggling frame defined by U glyph[epsilon1] a ,glyph[epsilon1] d ( t ) = U 0 ( t ) U tog ( t ) , where U glyph[epsilon1] a ,glyph[epsilon1] d ( t ) represents the propagator in the presence of the noises. Through Dyson perturbative expansion [38], there is U tog ( t ) = 1 -∑ µ = a,d [ i ∫ t 0 dt 1 glyph[epsilon1] µ ( t 1 ) ˜ E µ ( t 1 ) + ∫ t 0 dt 1 ∫ t 1 0 dt 2 glyph[epsilon1] µ ( t 1 ) glyph[epsilon1] µ ( t 2 ) ˜ E µ ( t 1 ) ˜ E µ ( t 2 ) + · · · ] with 1 the identity operator and ˜ E µ ( t ) = U † 0 ( t ) E µ U 0 ( t ) , µ ∈ { a, d } . Substitute into the parameterized U 0 ( t ) , we obtain ˜ E a,x ( t ) = [ ˙ θ sin γ + (˙ γ sin 2 θ cos γ ) / 2] σ x / 2 , ˜ E a,y ( t ) = [ ˙ θ cos γ -( ˙ γ sin 2 θ sin γ ) / 2] σ y / 2 , ˜ E a,z ( t ) = ( ˙ γ sin 2 θ ) σ z / 2 ; ˜ E d,x ( t ) = ( -sin θ cos γ ) σ x / 2 , ˜ E d,y ( t ) = (sin θ sin γ ) σ y / 2 , ˜ E d,z ( t ) = (cos θ ) σ z / 2 . These formulas are then to be substituted into the Dyson series to evaluate the error terms. For the quantum gate problem, we are given a target gate U and intend to find a robust implementing pulse. Suppose that the ideal evolution at time T satisfies U 0 ( T ) = U , then for a single realization of glyph[epsilon1] a ( t ) and glyph[epsilon1] d ( t ) , the gate fidelity reads F = ∣ ∣ ∣ Tr ( U † U glyph[epsilon1] a ,glyph[epsilon1] d ( T ) )∣ ∣ ∣ 2 / 4 = | Tr( U tog ( T )) | 2 / 4 . Taking the ensemble average of the noises and transferring to the frequency domain, the average gate infidelity defined by F avg = 1 -〈 F 〉 can be estimated to the second order approximation by the filter-function formalism [39, 40] in which R µ,α ( ω ) = -iω ∫ T 0 dt Tr [ ˜ E µ,α ( t ) σ α / 2] e iωt , and ∑ α | R d,α ( ω ) | 2 /ω 2 , ∑ α | R a,α ( ω ) | 2 / ( ω 2 Ω 2 max ) are the so called filter functions. This formula provides a simple quantitative means to evaluate the performance of a control protocol in the presence of time-dependent noises. It is thus natural to take F avg as our objective function. As a concrete example, we consider implementing a π rotational gate U = exp( -iπσ y / 2) . For this problem, at t = 0 , U 0 (0) equals to the identity, corresponding to the initial conditions θ (0) = 0 and ϕ (0) = -γ (0) (value not specified). The ending point conditions are θ ( T ) = π and ϕ ( T ) = γ ( T ) . The latter can be rewritten as a constraint for θ and γ by noting that from Eqs. (2b) and (2c) there is ˙ ϕ = -˙ γ cos( θ ) , hence one requires the condition γ (0) + γ ( T ) + ∫ T 0 ˙ γ cos θdt = 0 to be satisfied. With the objective function and all the constraints, we search ROC pulses using the gradient-based algorithm [41]; see details in Supplemental Material [40]. For the quantum state transfer problem, without loss of generality, we suppose the initial state to be | 0 〉 . The target is an arbitrary state | ψ 〉 on the Bloch sphere. For one realization of the noise glyph[epsilon1] a ( t ) and glyph[epsilon1] d ( t ) , the state transfer fidelity reads F = |〈 ψ | U glyph[epsilon1] a ,glyph[epsilon1] d ( T ) | 0 〉| 2 . Suppose the ideal evolution U 0 ( T ) implements the desired state transfer. Again, we turn into the toggling frame and get F = |〈 0 | U tog ( T ) | 0 〉| 2 . Substitute into the perturbative expansion of U tog ( T ) , and take ensemble average of the noise Hamiltonian, it can be derived that the average state infidelity is where we define P µ,α = -iω ∫ T 0 dt 〈 0 | ˜ E µ,α ( t ) | 1 〉 e iωt , and ∑ α | P d,α ( ω ) | 2 /ω 2 , ∑ α | P a,α ( ω ) | 2 / ( ω 2 Ω 2 max ) are the filter functions [40]. Concretely, we consider preparing target state | ψ 〉 = | 1 〉 starting from | 0 〉 . This converts to the conditions θ (0) = 0 , θ ( T ) = π, ϕ (0) = ϕ ( T ) = 0 , and no requirement of γ is involved. Moreover, the relation between θ and γ , namely ˙ ϕ = -˙ γ cos( θ ) , requires the condition ∫ T 0 ˙ γ cos θdt = 0 to be satisfied. The optimization procedure is the same as that described in the gate problem. As demonstration, we show the numerical simulation results of implementing a π rotational gate under realistic detuning or amplitude noise, as shown in in Fig. 2. We compare performances between primitive, typical composite pulses and robust optimal control pulses, for the same given noise spectrum. The primitive pulse is the elementary rectangular pulse of maximum Rabi frequency Ω max , which corresponds to the time-minimal control t min = 1 / (2Ω max ) . For detuning noise, as shown in Figs. 2(a 1 )-2(a 3 ), we consider ohmic spectrum with sharp cut-offs, i.e., S d ( ω ) ∝ ω, ω ∈ [ ω lc , ω uc ] , which describes a spin suffering bosonic environment [42]. The composite pulse we chosen is CORPSE [43], which is robust to detuning error to the first order. For amplitude noise, as shown in Figs. 2(b 1 )-2(b 3 ), we examine a noise spectrum of several Lorentzian peaks added on top of a broad 1 /f κ background S a ( ω ) ∝ ∑ k A k / ( λ 2 k +( ω -ω 0 ,k ) 2 ) + B/ω κ , which describes the random fluctuations in superconducting flux terms [44]. The composite pulse tested for this case is BB1 [43], which is robust to amplitude error to the second order. It can be seen that in each test example, the ROC filter function has sharp dips at the central frequencies of the imported noise spectrum, hence their frequency overlap is significantly suppressed; see Figs. 2(a 3 ) and 2(b 3 ). This feature implies that ROC has better performance in mitigating timedependent noises. We can verify this conclusion by computing their fidelities as follows. We calculate a single instance of noise perturbed evolution operator U glyph[epsilon1] α ( T ) and a single value for fidelity, and then take average over N = 150 noise realizations. For detuning noise with ohmic spectrum centered in the range [0 . 5Ω max , Ω max ] , as shown in the insert of Fig. 2(a 3 ), we obtain F ROC avg = 4 × 10 -4 , while F Primitive avg = 1 × 10 -3 and F CORPSE avg = 9 × 10 -3 . This result is consist with the conclusion that composite pulses are only robust to fluctuating noises up to as fast as around 10% of the Rabi frequency [21], yet our ROC pulse can still function for high-frequency noise. For amplitude noise with Lorentzian peaks centered at 0 . 2Ω max and 0 . 4Ω max (see the insert of Fig. 2(b 3 )), we obtain F Primitive avg = 2 × 10 -3 and F BB1 avg = 8 × 10 -3 , while our ROC pulse can decrease the infidelity to F ROC avg = 4 × 10 -6 . Another benefit of ROC pulse is that its shape can be made much smoother than CORPSE and BB1; see Figs. 2(a 2 ) and 2(b 2 ). This is particularly favorable for experiments, as real pulse generators have limited bandwidths. Accordingly, ROC produces smoother geometric evolution trajectories, as shown in Figs. 2(a 1 ) and 2(b 1 ). More simulation results for state transfer from | 0 〉 to | 1 〉 , for the case when detuning and amplitude noise are simultaneously present, for other types of realistic noise models, and for varied characteristic frequency positions of the tested noise spectra are all put in the Supplemental Material [40]. These results reveal that, in general, ROC pulses offer fidelity improvement for almost an order of magnitude compared with composite pulses and primitive pulse, and meanwhile featuring smooth pulse shapes and geometric trajectories. Resistance of T 1 , T 2 relaxation.When noises vary fast such that the Markovian approximation is valid, the controlled system dynamics can be described by the Bloch equation [45] ˙ x = ( H 0 ( t ) + γ 1 R T 1 + γ 2 R T 2 ) x , where x ≡ (1 / 2 , x 1 , x 2 , x 3 ) T is the vectorized representation of the system density matrix ρ = 1 / 2+ x 1 σ x + x 2 σ y + x 3 σ 3 ( x 2 1 + x 2 2 + x 2 3 ≤ 1 / 4) , H 0 ( t ) is the control Hamiltonian, γ 1 , 2 = 1 /T 1 , 2 are relaxation rates and are their corresponding operators. Relaxation is an irreversible process, hence it is usually thought that the best strategy to alleviate effects of relaxation is to make the operation time as small as possible. Therefore, the primitive pulse sets a fundament limit hard to surpass by other pulses [11]. Here, we study this issue using inverse geometric optimization, which works as follows. We first parameterize the relaxation-free evolution with extended threedimensional rotations R z ( δ ) , R y ( η ) and R z ( ξ ) , namely V 0 ( t ) = ( 1 0 0 R z ( δ ) R y ( η ) R z ( ξ ) ) with δ, ξ ∈ [ -π, π ] , η ∈ [0 , π ] [40]. Thus, the Bloch equation is rewritten as The actual evolution is then transformed to the toggling frame for conveniently displaying the perturbation effects due to T 1 and T 2 relaxation, i.e., V T 1 ,T 2 ( t ) = V 0 ( t ) V tog ( t ) ≈ V 0 ( t )( 1 4 + ∑ k ∫ t 0 dt 1 γ k ˜ R T k ( t 1 ) + · · · ) , where 1 4 is the 4dimensional identity, ˜ R T k ( t ) = V † 0 ( t ) R T k V 0 ( t ) . Take the quantum state transfer problem as an example. Staring from state x (0) , the Euclidean distance between the actual state and the target state ¯ x , defined by F = | ¯ x -V T 1 ,T 2 ( T ) x (0) | 2 , can be expressed in terms of the angular variables as follows [40] As a concrete example, we consider quantum state transfer from the north pole x (0) = [1 / 2 , 0 , 0 , 1 / 2] T to the south pole x = [1 / 2 , 0 , 0 , -1 / 2] T . This gives the constraint conditions η (0) = 0 , η ( T ) = π and we set δ (0) = δ ( T ) = 0 , while no requirement of ξ is involved. Besides, from Eqs. (6a) and (6c) we have ˙ δ = -˙ ξ cos η , thus the condition ∫ T 0 ˙ ξ cos ηdt = 0 should be satisfied. In our optimization, we also use the gradient-based algorithm to search robust ROC pulses. Results are summarized in Fig. 3, where we consider solid-state spin defect system with Ω max / (2 π ) = 10 7 Hz, and the relaxation parameters are typically chosen as γ 1 = 10 3 s -1 , γ 2 /γ 1 = 10 ∼ 100 [46]. From Fig. 3(a), we find that typical composite pules, including CORPSE and BB1, can not resist relaxation, as they result in much larger errors compared with the primitive pulse. On the other hand, ROC pulses can improve up to four times compared with primitive pulse for all the tested relaxation parameters. Meanwhile, the geometric trajectories and control waveforms for ROC pulses are smoother, as shown in Figs. 3(b) and 3(c), respectively. Discussion and outlook.The task of mitigating timedependent noises is generally considered to be a thorny challenge and a long-term objective of quantum system engineering. The robust control method presented here has a critical advantage of flexibility as it is effective for a wide variety of noise environments, which is hence particularly applicable in reality since real experiments often involve complicated noise spectrum. Moreover, in the Markovian limit, the method is also effective in improving the state transfer fidelity against transverse and longitudinal relaxation effects. We hope the control examples tested here or other possible applications can soon find their experimental verifications. For future work, we can combine inverse geometric engineering with other robust optimal control methods. For example, the technique developed in Ref. [47], which expresses Dyson perturbative terms based on Van Loan's integral ex- pression, provides a rather convenient means to evaluate the perturbative impacts of the noises. We can also apply analytic expression of the filter function derivatives [33] to further improve the performance of our method, or attempt to derive exact analytical control fields [48, 49]. In addition, the method presented here can be easily extended to handle other robust quantum control tasks, such as quantum sensing under timedependent background noises [50]. Acknowledgments . We thank Ze Wu for helpful discussions. This work was supported by the National Natural Science Foundation of China (1212200199, 11975117, 92065111, 12075110, 11905099, 11875159, 11905111, and U1801661), National Key Research and Development Program of China (2019YFA0308100), Guangdong Basic and Applied Basic Research Foundation (2019A1515011383 and 2021B1515020070), Guangdong Provincial Key Laboratory (2019B121203002), Guangdong International Collaboration Program (2020A0505100001), Shenzhen Science and Technology Program (RCYX20200714114522109 and KQTD20200820113010023), China Postdoctoral Science Foundation (2021M691445), Science, Technology and Innovation Commission of Shenzhen Municipality (ZDSYS20190902092905285, KQTD20190929173815000 and JCYJ20200109140803865), and Pengcheng Scholars, Guangdong Innovative and Entrepreneurial Research Team Program (2019ZT08C044). timal control, Phys. Rev. Applied 17 , 024006 (2022).",
"pages": [
1,
2,
3,
4,
5,
6
]
},
{
"title": "Derivation of Eq. (2) of the Main Text: Evolution Parameterization",
"content": "An arbitrary single-qubit noise-free evolution can be parameterized by Using this, the Schrodinger equation can be rewritten as which is From these equations, we can obtain Once a robust evolution trajectory is obtained, we can determine the control fields by",
"pages": [
6,
7
]
},
{
"title": "Derivation of Eq. (3) of the Main Text: Average Gate Infidelity",
"content": "For one realization of noise glyph[epsilon1] a ( t ) and glyph[epsilon1] d ( t ) , the real gate fidelity is It is convenient to write U tog ( T ) = exp {-i ∑ µ = a,d ∫ T 0 ε µ ( t ) ˜ E µ ( t ) dt } ≡ e -i a · σ / 2 = 1 cos( a/ 2) -i sin( a/ 2) a · σ /a , where σ = ( σ x , σ y , σ z ) , thus F = [1 + cos( a )] / 2 . For the first-order approximation, we obtain F ≈ 1 -a 2 / 4 . Take the ensemble average of the noise, we get the average gate infidelity where R µ,α ( ω ) = -iω ∫ T 0 dt Tr [ ˜ E µ,α ( t ) σ α / 2] e iωt .",
"pages": [
7
]
},
{
"title": "Derivation of Eq. (4) of the Main Text: Average State Transfer Infidelity",
"content": "For one realization of noise glyph[epsilon1] a ( t ) and glyph[epsilon1] d ( t ) , the real state fidelity is (S.6) where we omit the cross terms containing glyph[epsilon1] µ 1 ( t 1 ) glyph[epsilon1] µ 2 ( t 2 ) with µ 1 = µ 2 . Insert | 0 〉〈 0 | + | 1 〉〈 1 | into the expression, then glyph[negationslash] Thus the average state infidelity is where P µ,α = -iω ∫ T 0 dt 〈 0 | ˜ E µ,α ( t ) | 1 〉 e iωt .",
"pages": [
7,
8
]
},
{
"title": "Gradient-based Optimization",
"content": "To search robust trajectory, we apply gradient-based optimization. In our optimization, we discretize θ ( t ) and γ ( t ) as M -slice sequences θ [1] , ..., θ [ M ] and γ [1] , ..., γ [ M ] , respectively, with the time length of each slice τ = T/M . For quantum gate, the derivative of the average gate infidelity function reads Similarly, for quantum state transfer, the derivative of the average state infidelity function reads The above gradients are then used to update the trajectory by χ [ m ] ← χ [ m ] + l ∂ F avg ∂χ [ m ] , where l is appropriate step size. We test our robust control method for realizing a π rotational gate or state transfer from | 0 〉 to | 1 〉 subject to various types of time-dependent noise, as shown in Figs. S1-S2 and Figs. S3-S4, respectively. For detuning noise, we compare the performance between three sequences, namely primitive, CORPSE and ROC pulses, see the results in Figs. S1-S4(a 1 )-(a 3 ),(b 1 )-(b 3 ),(c 1 )-(c 3 ). Specifically, we consider (i) ohmic spectrum with sharp cut-offs, i.e., S d ( ω ) ∝ ω, ω ∈ [ ω lc , ω uc ] , which describes a spin suffering bosonic environment [42]; (ii) single Lorentzian spectrum S d ( ω ) ∝ 1 / ( λ 2 +( ω -ω 0 ) 2 ) or multiple Lorentzian spectrum S d ( ω ) ∝ ∑ k A k / ( λ 2 k +( ω -ω 0 ,k ) 2 ) , which captures the solid-state spin environment of a spin bath as measured in, e.g., Refs. [51, 52]. For amplitude noise, we compare the performance between primitive, BB1 and ROC pulses, as shown in Figs. S1-S4(d 1 )-(d 3 ),(e 1 )-(e 3 ). Here we examine (i) a noise model corresponding to a strong and narrow Gaussian peak added on top of a broad 1 /f κ background with a roll-off to white noise, i.e., S a ( ω ) ∝ A exp[ -( ω -ω 0 ) 2 / (2 σ 2 )] + B/ω κ , ω < ω wc; S a ( ω ) = const. , ω ≥ ω wc. This type of noise spectrum was observed, for example, in a silicon quantum dot spin qubit due to the imperfect control apparatus [53]. (ii) several Lorentzian peaks added on top of a broad 1 /f κ background S a ( ω ) ∝ ∑ k A k / ( λ 2 k + ( ω -ω 0 ,k ) 2 ) + B/ω κ , which describes the random fluctuations in superconducting flux terms [44]. For simultaneous detuning noise and amplitude noise, for simplicity, we choose both of the noise spectrums as a single Lorentzian peak added on top of a broad 1 /f background, i.e., S µ ( ω ) ∝ A/ ( λ 2 +( ω -ω 0 ) 2 )+ B/ω . We compare the performance between primitive, reduced CinBB and ROC pulses. Reduced CinBB [43] is a concatenated composite pulse for suppressing both of the detuning and amplitude noises. Results are summarized in Figs. S1-S4(f 1 )-(f 3 ). We also list explicitly the parameters of the noise power density spectrums and the tested control sequences in Table. I. All the simulations reveal that our ROC method find high-quality, smooth and low-power robust pulses for resisting various realistic time-dependent noises. In the main text, we demonstrate several typical results to show the effectiveness of our robust control method.",
"pages": [
8
]
},
{
"title": "Evolution Parameterization",
"content": "For an isolated spin-1/2 ensemble, the density matrix can be conventionally expressed as ρ = 1 / 2 + x 1 σ x + x 2 σ y + x 3 σ 3 . When undergoing both transverse and longitudinal relaxation, and under Markovian approximation, the system dynamics under controls u x = Ω( t ) cos( φ ( t )) , u y = Ω( t ) sin( φ ( t )) can be described by the Bloch equations where x = (1 / 2 , x 1 , x 2 , x 3 ) T , γ 1 , 2 = 1 /T 1 , 2 are relaxation parameters and For simplicity, we set the off-resonance frequency Ω 0 = 0 and the equilibrium state polarization M 0 = 1 . To apply inverse geometric optimization, we introduce the three-dimensional rotations This then gives As such, the noise-free evolution of two-level open quantum system can the be parameterized by thus the Bloch equation becomes Once a robust evolution trajectory is obtained, we can determine the control fields by",
"pages": [
11,
12
]
},
{
"title": "State Transfer Infidelity",
"content": "To characterize the distance between the actual state V T 1 ,T 2 ( T ) x (0) and the target state ¯ x , we define where one should notice that [ ˜ R T k x (0)] T [ ˜ R T j x (0)] = 0 when k = j . glyph[negationslash]",
"pages": [
12
]
}
]
2024Symm...16.1577D
https://arxiv.org/pdf/2302.11189.pdf
<document>
<section_header_level_1><location><page_1><loc_28><loc_89><loc_72><loc_91></location>Topological classes of BTZ black holes</section_header_level_1>
<text><location><page_1><loc_34><loc_86><loc_65><loc_87></location>Yongbin Du and Xiangdong Zhang ∗</text>
<text><location><page_1><loc_12><loc_80><loc_87><loc_83></location>Department of Physics, South China University of Technology, Guangzhou 510641, China (Dated: February 24, 2023)</text>
<section_header_level_1><location><page_1><loc_45><loc_77><loc_54><loc_79></location>Abstract</section_header_level_1>
<text><location><page_1><loc_12><loc_55><loc_88><loc_76></location>In the recent paper [Phys. Rev. Lett. 129, 191101 (2022)], the black holes were viewed as topological thermodynamic defects by using the generalized off-shell free energy. Their work indicates that all black hole solutions in the pure Einstein-Maxwell gravity theory could be classified into three different topological classes for four and higher spacetime dimensions. In this paper, we investigate the topological number of BTZ black holes with different charges ( Q ) and rotational ( J ) parameters. By using generalized free energy and Duan's φ -mapping topological current theory, we interestingly found only two topological classes for BTZ spacetime. Particularly, for Q = J = 0 BTZ black hole, there has only one zero point and therefore the total topological number is 1. While for rotating or charged cases, there are always two zero points and the global topological number is zero.</text>
<text><location><page_1><loc_12><loc_52><loc_23><loc_53></location>PACS numbers:</text>
<section_header_level_1><location><page_2><loc_12><loc_89><loc_32><loc_91></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_2><loc_12><loc_75><loc_88><loc_87></location>As one of the most fascinating objects in nature, the black hole has receiving increasingly attention both in theoretical and observational physics. On one hand, singularity theorem [1] reveals the intrinsic limitation of general relativity. On the other side, when taking quantum effect into consideration, people found that black hole is not black at all [2] and surprisingly sharing rich thermodynamical properties [3-5]. Many issues, such as information paradox [6], arise due to these remarkable discoveries.</text>
<text><location><page_2><loc_12><loc_62><loc_88><loc_74></location>Recently, by introducing Duan's topological current φ -mapping theory[7], Wei et. al. [8, 9] originally consider black hole as topological thermodynamic defects, and classify different solutions with their global topological charge. They divided black hole solutions into three different topological classes according to their different topological numbers. This method gives us an insight into black hole solutions and implies more inherent properties of this extraordinary thermodynamic system.</text>
<text><location><page_2><loc_12><loc_43><loc_88><loc_62></location>Inspired by this results, many works have been done to investigate the topological charge of different types of black hole. Since [8] only foucus on the static solutions, in [10], stationary black holes are explored. The topological number of Kerr and Kerr-Newman black hole are calculated. Furthermore, they calculate the singly-rotating black hole in higher dimensions [10]. Apart from the researches on black hole in general relativity, some researchers are also consider the black hole in modified gravity theories such as Gauss-Bonnet gravity [11] or Lovelock gravity [12] and so on [13-15]. The topological number are quite different in these cases, as provide us a new perspective to consider the difference between modified gravity theory and general relativity.</text>
<text><location><page_2><loc_12><loc_24><loc_88><loc_42></location>An interesting feature of these works is that whether we consider black hole solutions in general relativity or modified gravity or even high-dimensional extension, they can always be classified into three topological classes [10-12]. Moreover, dimension may play an improtant role in the amount of zero points and topological number [10, 11]. Hence extending the approach to lower dimensional case could provide us with a test to the seemingly universal phenomenon or provide a counterexample which become very relevant for the future study on this issue. To this aim, in this paper, we follow this newly-hewed path to study the topological number of the Banados-Teitelboim-Zanelli (BTZ) black hole [16] and trying to provide counterexample to the above observation.</text>
<text><location><page_2><loc_12><loc_7><loc_88><loc_23></location>The BTZ black hole is the black hole solution of 3-dimensional general relativity plus a negative cosmological constant and plays an important role in quantum theory of gravity. As a low-dimensional toy model, BTZ black hole have many interesting features. Locally, it has constant negative curvature and consequently no curvature singularity [17]. But it possess a horizon, and the Bekenstein's area law could also be applied in it. Thus it is a genuine black hole and the simply form may benefit quantum gravity or holographic priciple a lot. Many paper focus on this 3-dimensional black hole solution to find some unexpected result of common physical phenomenon, such as Penrose process [18, 19]. In this article we shall</text>
<text><location><page_3><loc_12><loc_85><loc_88><loc_91></location>concentrate on thermodynamics of BTZ black hole and then calculate the topological charge when the angular momentum and electric charge are taken different values. We would also like to give a simple comparison with the results of black hole in four dimensions [8-10].</text>
<text><location><page_3><loc_12><loc_74><loc_88><loc_84></location>This paper is organized as follows: In section 2, we briefly review some useful results of thermodynamics of BTZ black hole. Then we use the generalized free energy to establish a parameter space in section 3, and find the zero points with their topological charge. In section 4, we summarize our result, compare it with higher dimensional cases and make some comment on this new approach.</text>
<section_header_level_1><location><page_3><loc_12><loc_70><loc_60><loc_71></location>II. THERMODYNAMICS OF BTZ BLACK HOLE</section_header_level_1>
<text><location><page_3><loc_12><loc_57><loc_88><loc_67></location>In this section, we first give a brief review of thermodynamics of BTZ black hole. We use the generalized off-shell free energy [20] so that we could classify different black hole solutions. This means black holes with the same energy (and electric charge or angular momentum, if any) can be in different temperature. The metric line element for the BTZ black hole reads [16]</text>
<formula><location><page_3><loc_31><loc_51><loc_88><loc_55></location>ds 2 = -N 2 dt 2 + N -2 dr 2 + r 2 ( N φ dt + dφ ) 2 , (2.1)</formula>
<text><location><page_3><loc_12><loc_50><loc_83><loc_52></location>where the squared lapse N 2 ( r ) and the angular shift N φ ( r ) are respectively given by</text>
<formula><location><page_3><loc_38><loc_45><loc_88><loc_48></location>N 2 ( r ) = -M + r 2 L 2 + J 2 4 r 2 , (2.2)</formula>
<formula><location><page_3><loc_38><loc_41><loc_88><loc_45></location>N φ ( r ) = -J 2 r 2 (2.3)</formula>
<text><location><page_3><loc_12><loc_28><loc_88><loc_40></location>with -∞ < t < ∞ , 0 < r < ∞ , and 0 ≤ φ ≤ 2 π . M and J are respectively the mass and angular momentum of the black hole. L represents the radius of curvature of spacetime and satisfy L 2 = -1 / Λ with Λ being the negative cosmological constant. The metric is singular at the inner and outer horizons where N 2 ( r ) vanishes. Using gravitational path integral and saddle point approximation, the generalized off-shell free energy of BTZ black hole reads [16, 20]</text>
<formula><location><page_3><loc_40><loc_24><loc_88><loc_28></location>F = M -S τ + N φ ( r h ) J, (2.4)</formula>
<text><location><page_3><loc_12><loc_16><loc_88><loc_24></location>The parameter τ is an extra variable with the dimension of time. It varies freely and can be thought as the inverse temperature of the cavity enclosing the black hole. r h represents the radius of the outer horizon. The mass and entropy can be written as the function of r h and J</text>
<formula><location><page_3><loc_43><loc_11><loc_88><loc_15></location>M = r 2 h L 2 + J 2 4 r 2 h , (2.5)</formula>
<formula><location><page_3><loc_43><loc_9><loc_88><loc_11></location>S = 4 πr h . (2.6)</formula>
<text><location><page_4><loc_12><loc_83><loc_88><loc_91></location>We have to stress that L is no more a fixed value so that the modified Bekenstein-Smarr mass formula still holds in 3-dimensional black hole thermodynamics [21, 22]. However, as we will show in this paper, its value will not affect the topological charge of BTZ solution. With the help of (2.5) and (2.6), the generalized free energy (2.4) can be written as</text>
<formula><location><page_4><loc_40><loc_77><loc_88><loc_81></location>F = r 2 h L 2 -J 2 4 r 2 h -4 πr h τ . (2.7)</formula>
<text><location><page_4><loc_12><loc_74><loc_88><loc_76></location>It is off-shell except at τ = 1 /T which means the black hole is in the maximal mixed state.</text>
<section_header_level_1><location><page_4><loc_12><loc_70><loc_77><loc_71></location>III. TOPOLOGICAL CLASSES OF BTZ BLACK HOLE SOLUTIONS</section_header_level_1>
<text><location><page_4><loc_12><loc_55><loc_88><loc_67></location>When the generalized free energy is obtained, we could apply the method in [8] to estabilish a parameter space and find the zero point of the vector field in it. Profoundly, the zero points are exactly corresponding to the on-shell black hole solution. We can calculate the topological number of them by virtue of Duan's φ -mapping topological current theory [7]. The number can be seen as a characteristic value of the on-shell black hole solution. Following the spirit of [8], we define the vector as</text>
<formula><location><page_4><loc_36><loc_50><loc_88><loc_53></location>φ = ( φ r h , φ θ ) = ( ∂F ∂r h , -cot θ csc θ ) (3.1)</formula>
<text><location><page_4><loc_12><loc_40><loc_88><loc_48></location>with θ ∈ [0 , π ] for convenience. The vector field is on θ -r h space, and we can see that φ θ is divergent when θ = 0 , π , making the direction of vectors point vertically outward at this boundary. The zero point, corresponding to τ = 1 /T [9], can only be obtained when θ = π/ 2. Now we introduce the topological current as</text>
<formula><location><page_4><loc_33><loc_35><loc_88><loc_39></location>j µ = 1 2 π /epsilon1 µνρ /epsilon1 ab ∂ ν n a ∂ ρ n b , µ, ν, ρ = 0 , 1 , 2 (3.2)</formula>
<text><location><page_4><loc_12><loc_24><loc_88><loc_34></location>where ∂ ν = ( ∂/∂x ν ) and x ν = ( τ, r h , θ ). The unit vector is defined as n a = ( φ a / ‖ φ ‖ ) ( a = 1 , 2). The conservation law of the current, ∂ µ j µ = 0 can be easily obtained by the definition of j µ , and τ here serves as a time parameter of the topological defect. By using the Jacobi tensor /epsilon1 ab J µ ( φ/x ) = /epsilon1 µνρ ∂ ν φ a ∂ ρ φ b and the two-dimensional Laplacian Green function ∆ φ a ln ‖ φ ‖ = 2 πδ 2 ( φ ), the topological current can be written as</text>
<formula><location><page_4><loc_42><loc_19><loc_88><loc_23></location>j µ = δ 2 ( φ ) J µ ( φ x ) . (3.3)</formula>
<text><location><page_4><loc_12><loc_14><loc_88><loc_17></location>where j µ is nonzero only at φ a ( x i ) = 0, and we denote its i -th solution as /vectorx = /vectorz i . The density of the topological current reads [23]</text>
<formula><location><page_4><loc_40><loc_7><loc_88><loc_12></location>j 0 = N ∑ i =1 β i η i δ 2 ( /vectorx -/vectorz i ) , (3.4)</formula>
<text><location><page_5><loc_12><loc_83><loc_88><loc_91></location>where β i is Hopf index, which counts the number of the loops that φ a makes in the vector φ space when x µ goes around the zero point z i . Thus Hopf index is always positive. η i is the Brouwer degree and satisfy η i = sign ( J 0 ( φ/x ) z i ) = ± 1. Given a parameter region Σ, the corresponding topological number can be obtained as</text>
<formula><location><page_5><loc_36><loc_77><loc_88><loc_82></location>W = ∫ Σ j 0 d 2 x = N ∑ i =1 β i η i = N ∑ i =1 w i , (3.5)</formula>
<text><location><page_5><loc_73><loc_70><loc_73><loc_72></location>/negationslash</text>
<text><location><page_5><loc_12><loc_64><loc_88><loc_76></location>where w i is the winding number for the i -th zero point of φ contained in Σ and its value does not depend on the shape of the region where we perform the calculation. Usually, distinct zero points of the vector field are isolated, making Jacobian J 0 ( φ/x ) = 0. If Jacobian J 0 ( φ/x ) = 0, it means that the defect bifurcates [24]. Eq.(3.5) shows that in any given region, the global topological number is the sum of the winding number of each zero point which reflects the local property of the topological defect.</text>
<text><location><page_5><loc_12><loc_60><loc_88><loc_63></location>Based on the approach introduced above, we first investigate BTZ black hole with J = 0. In this case, the generalized free energy is</text>
<formula><location><page_5><loc_43><loc_55><loc_88><loc_59></location>F = r 2 h L 2 -4 πr h τ . (3.6)</formula>
<text><location><page_5><loc_12><loc_53><loc_35><loc_54></location>We define the vector field as</text>
<formula><location><page_5><loc_33><loc_49><loc_88><loc_52></location>φ = ( φ r h , φ θ ) = ( 2 r h L 2 -4 π τ , -cot θ csc θ ) . (3.7)</formula>
<text><location><page_5><loc_12><loc_47><loc_57><loc_48></location>By solving the equation φ = 0, we acquire the relation</text>
<formula><location><page_5><loc_42><loc_42><loc_88><loc_46></location>τ = 2 πL 2 r h , θ = π 2 . (3.8)</formula>
<text><location><page_5><loc_12><loc_29><loc_88><loc_41></location>We take L = 1 /r 0 and τ = 2 π/r 3 0 with r 0 an arbitrary positive constant, so there is one zero point in θ -r h plane at ( r h /r 0 , θ ) = (1 , π/ 2). We plot the unit vector field in Fig. 1. The loop surrounding the zero point sets the boundary of a given region so we can use Eq.(3.5) to acquire the winding number of point P . We find that w = 1. Since there is only one defect in the parameter space, it gives rise the global topological number W = 1 for BTZ black hole solution with J = 0.</text>
<text><location><page_5><loc_12><loc_24><loc_88><loc_28></location>Now we would further explore the rotating BTZ black hole. The generalized free energy is</text>
<formula><location><page_5><loc_40><loc_21><loc_88><loc_25></location>F = r 2 h L 2 -4 πr h τ -J 2 4 r 2 h . (3.9)</formula>
<text><location><page_5><loc_12><loc_19><loc_45><loc_20></location>As a result, we define the vector field as</text>
<formula><location><page_5><loc_30><loc_14><loc_88><loc_18></location>φ = ( φ r h , φ θ ) = ( 2 r h L 2 -4 π τ + J 2 2 r 3 h , -cot θ csc θ ) . (3.10)</formula>
<text><location><page_5><loc_12><loc_11><loc_51><loc_13></location>Solving the equation φ = 0, we get the relation</text>
<formula><location><page_5><loc_39><loc_6><loc_88><loc_10></location>τ = 8 πr 3 h 4 L -2 r 4 h + J 2 , θ = π 2 . (3.11)</formula>
<figure>
<location><page_6><loc_33><loc_66><loc_66><loc_91></location>
<caption>FIG. 1: Unit vector field in parameter space with J = 0, Q = 0 and L = 1 /r 0 solution as τ = 2 π/r 3 0 . P marked with black dot at ( r h /r 0 , θ ) = (1 , π/ 2) is the zero point of the vector field. The black contour is a closed loop enclosing the zero point and we can performing the calculation of Eq.(3.5) inside it. The shape and the size of the loop will not affect the winding number.</caption>
</figure>
<text><location><page_6><loc_12><loc_30><loc_88><loc_53></location>Upon the value of L and J are determined, we could acquire a curve in τ -r h plane. For instance, as shown in Fig. 2, when taking L = 1 /r 0 and J = r 3 0 , the curve meets a peak at ( r h , τ ) ≈ (0 . 93 r 0 , 5 . 06 /r 3 0 ) and rapidly goes down as r h gets larger. Consequently, the vector field possesses two zero points for small τ , in comtrast to one zero point for nonrotating BTZ black hole case. As τ = 4 π/ 5 r 3 0 , the two intersection points are respectively at r h = 0 . 5 r 0 and r h ≈ 2 . 482 r 0 . We illustrate the vector field and the zero points in Fig. 3. When τ = τ cri ≈ 5 . 06 /r 3 0 , the intersection points coincide, and for larger τ , annihilate. It is easy to check the critical point satisfy d 2 τ/dr 2 h < 0, which belongs to annihilation point. For τ < τ cri , we find that the winding number of the two zero points are w 1 = -1 and w 2 = 1. Thus the global topological number for BTZ rotating solution is W = w 1 + w 2 = 0, different from non-rotating case.</text>
<text><location><page_6><loc_12><loc_26><loc_88><loc_30></location>When taking electric charge into account, the squared lapse N 2 ( r ) and the angular shift N φ ( r ) are respectively given by</text>
<formula><location><page_6><loc_36><loc_21><loc_88><loc_24></location>N 2 ( r ) = -M + r 2 L 2 + 1 2 QA 0 ( r ) , (3.12)</formula>
<formula><location><page_6><loc_36><loc_19><loc_88><loc_20></location>N φ ( r ) = 0 (3.13)</formula>
<text><location><page_6><loc_12><loc_13><loc_88><loc_17></location>Q denotes the electric charge of the black hole, A 0 ( r ) is the only nonvanishing component of the electromagnetic vector potential and is taken to be A 0 ( r ) = -Q ln( r/r c ) with r c being</text>
<figure>
<location><page_7><loc_28><loc_70><loc_71><loc_91></location>
<caption>FIG. 2: Solution curves in τ -r h plane with different values of L and J . There is always one turning point in the curve as long as J = 0.</caption>
</figure>
<text><location><page_7><loc_43><loc_64><loc_43><loc_65></location>/negationslash</text>
<figure>
<location><page_7><loc_36><loc_41><loc_64><loc_62></location>
<caption>FIG. 3: Vector field in θ -r h plane with J = 0. We take τ = 4 π/ 5 r 3 0 . The points marked in black dot are zero points of the field. They are respectively at P 1 = ( r h /r 0 , θ ) = (0 . 5 , π/ 2) and P 2 = ( r h /r 0 , θ ) = (2 . 482 , π/ 2).</caption>
</figure>
<text><location><page_7><loc_48><loc_37><loc_48><loc_38></location>/negationslash</text>
<text><location><page_7><loc_12><loc_28><loc_88><loc_30></location>an arbitrary constant. the generalized off-shell free energy of BTZ black hole reads [16, 20]</text>
<formula><location><page_7><loc_36><loc_20><loc_88><loc_27></location>F = M -S τ -A 0 ( r h ) Q = = r 2 h L 2 -4 πr h τ + 1 2 Q 2 ln r h r c . (3.14)</formula>
<text><location><page_7><loc_12><loc_17><loc_86><loc_18></location>Following the same step, we get the on-shell solution curve in τ -r h plane, which satisfy</text>
<formula><location><page_7><loc_41><loc_12><loc_88><loc_16></location>τ = 8 πr 3 h 4 L -2 r 4 h + Q 2 r 2 h . (3.15)</formula>
<text><location><page_7><loc_12><loc_7><loc_88><loc_10></location>Interestingly, we find that the electric charge do not change the rough trend of the curve. As shown in Fig. 4, there is invaribly one anihilation point in τ -r h plane, and τ cri merely</text>
<figure>
<location><page_8><loc_28><loc_70><loc_71><loc_91></location>
<caption>FIG. 4: Solution curve in τ -r h plane with different value of L and Q . We find that non-trivial Q do not bring essential difference to the trend of the curve compared with rotational case.</caption>
</figure>
<text><location><page_8><loc_12><loc_38><loc_88><loc_61></location>gets smaller as the value of Q taken larger. There are two zero points as well. We calculate the winding number of them for a given τ and find w 1 = -1, w 2 = 1. See Fig. 5 as an example. The global topological number of charged BTZ black hole solution is W = 0, which is the same as the case of rotational BTZ black hole. Hence from the perspective of topological charge, the two kinds of 3-dimensional black hole are just the same. This conclusion is the same as four dimensional cases [10]. Yet the topological number of nonrotating and uncharged black hole are different, with W BTZ = 1 whereas W Schwarzchild = -1. Besides, in contrast to the charged Reissner-Nordstrom anti de-Sitter (RN-AdS) black hole in four dimension [8], three dimensional black hole solution evidently has fewer zero points. So the dimension of black hole of the same kind may have an unique influence on the quantity of defects and the global topological number.</text>
<section_header_level_1><location><page_8><loc_12><loc_33><loc_31><loc_35></location>IV. CONCLUSION</section_header_level_1>
<text><location><page_8><loc_12><loc_17><loc_88><loc_31></location>In this paper, we use the generalized free energy of BTZ black hole to define a vector field in a parameter space θ -r h . We find the zero point of the field and obtain the winding number by applying Duan's φ -mapping topological current theory. It is discovered that the unique black hole solution in three dimensional general relativity has one zero point for Q = J = 0 and two for rotating or charged cases. The global topological charges are one and zero respectively. We also find that the dimension of the AdS background would lead to a distinct amount of zero points.</text>
<text><location><page_8><loc_12><loc_8><loc_88><loc_16></location>The previous works [8, 10] indicate that all black hole solutions in the pure EinsteinMaxwell gravity theory should be classified into three different topological classes for four and higher spacetime dimensions. This observation is further enhanced in the modified gravity case [11, 12]. However, our investigation on BTZ black holes found only two topological</text>
<figure>
<location><page_9><loc_32><loc_65><loc_67><loc_91></location>
<caption>FIG. 5: Vector field in θ -r h plane with Q = 0. We also take τ = 4 π/ 5 r 3 0 . The zero points of the field are respectively at P 1 = ( r h /r 0 , θ ) = (0 . 104 , π/ 2) and P 2 = ( r h /r 0 , θ ) = (2 . 395 , π/ 2).</caption>
</figure>
<text><location><page_9><loc_46><loc_61><loc_46><loc_62></location>/negationslash</text>
<text><location><page_9><loc_12><loc_52><loc_88><loc_56></location>classes for BTZ spacetime. This means this feature is not universal and the spacetime dimension seems strongly relevant in the topological classification of black holes.</text>
<text><location><page_9><loc_12><loc_42><loc_88><loc_52></location>There are many issues that deserve further investigation. Generalize our results to KerrAdS and Kerr-dS and compare with the existing result will be interesting. Moreover, another interesting object is to investigate the topological number of the black hole solutions in the supergravity and modified gravity theories. We leave these interesting topics for future studies.</text>
<section_header_level_1><location><page_9><loc_14><loc_37><loc_30><loc_38></location>Acknowledgments</section_header_level_1>
<text><location><page_9><loc_12><loc_29><loc_88><loc_34></location>We would like to thank Prof. Pujian Mao for helpful discussion. This work is supported by NSFC with Grants No.12275087 and 'the Fundamental Research Funds for the Central Universities'.</text>
<unordered_list>
<list_item><location><page_9><loc_13><loc_12><loc_83><loc_13></location>[4] S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43, 199 (1975)</list_item>
<list_item><location><page_9><loc_13><loc_10><loc_71><loc_11></location>[5] J. D. Bekenstein, Black holes and entropy, Phys. Rev. D 7,2333 (1973).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_10><loc_13><loc_87><loc_88><loc_91></location>[6] S, W, Hawking, Breakdown of predictability in gravitational collapse, Physical Review D, 1976.</list_item>
<list_item><location><page_10><loc_13><loc_85><loc_87><loc_86></location>[7] Y. S. Duan, The structure of the topological current, Report No. SLAC-PUB-3301, (1984).</list_item>
<list_item><location><page_10><loc_13><loc_81><loc_88><loc_84></location>[8] S.-W. Wei, Y.-X. Liu and R.B. Mann, Black Hole Solutions as Topological Thermodynamic Defects, Phys. Rev. Lett. 129,191101 (2022).</list_item>
<list_item><location><page_10><loc_13><loc_76><loc_88><loc_80></location>[9] S.-W. Wei and Y.-X. Liu, Topology of black hole thermodynamics, Phys. Rev. D 105, 104003 (2022).</list_item>
<list_item><location><page_10><loc_12><loc_74><loc_75><loc_75></location>[10] D. Wu, Topological classes of rotating black holes, Phys. Rev. D 107 (2023)</list_item>
<list_item><location><page_10><loc_12><loc_70><loc_88><loc_73></location>[11] C.H. Liu and J. Wang, The topological nature of the Gauss-Bonnet black hole in AdS space, arXiv: 2211.05524</list_item>
<list_item><location><page_10><loc_12><loc_65><loc_88><loc_69></location>[12] N. C. Bai, L. Li and J. Tao, Topology of black hole thermodynamics in Lovelock gravity, [arXiv:2208.10177 [gr-qc]].</list_item>
<list_item><location><page_10><loc_12><loc_61><loc_88><loc_64></location>[13] P.K. Yerra, C. Bhamidipati and S. Mukherji, Topology of critical points and Hawking-Page transition, Phys. Rev. D 106 (2022) 064059.</list_item>
<list_item><location><page_10><loc_12><loc_56><loc_88><loc_60></location>[14] P.K. Yerra and C. Bhamidipati, Topology of Born-Infeld AdS black holes in 4D novel EinsteinGauss-Bonnet gravity, Phys. Lett. B 835 (2022) 137591.</list_item>
<list_item><location><page_10><loc_12><loc_52><loc_88><loc_56></location>[15] P.K. Yerra and C. Bhamidipati, Topology of black hole thermodynamics in Gauss-Bonnet gravity, Phys. Rev. D 105 (2022) 104053.</list_item>
<list_item><location><page_10><loc_12><loc_48><loc_88><loc_51></location>[16] M. Banados, C. Teitelboim, and J. Zanelli, Black Hole in Three-Dimensional Spacetime, Phys. Rev. Lett. 69 (1992)</list_item>
<list_item><location><page_10><loc_12><loc_46><loc_88><loc_47></location>[17] S. Carlip, The (2+1)-Dimensional Black Hole, arXiv, 10.1088/0264-9381/12/12/005[P]. 1999.</list_item>
<list_item><location><page_10><loc_12><loc_41><loc_88><loc_45></location>[18] X. Wu and X. Zhang, Collisional Penrose process of BTZ black holes, Phys. Rev. D 103 (2021) no.4, 044048</list_item>
<list_item><location><page_10><loc_12><loc_37><loc_88><loc_40></location>[19] X. Yuan, Y. Liu and X. Zhang, Collision of spinning particles near BTZ black holes, Chin. Phys. C 44 (2020) no.6, 065104</list_item>
<list_item><location><page_10><loc_12><loc_32><loc_88><loc_36></location>[20] J. W. York, Black-hole thermodynamics and the Euclidean Einstein action, Phys. Rev. D 33, 2092 (1986).</list_item>
<list_item><location><page_10><loc_12><loc_28><loc_88><loc_32></location>[21] S. Wang, S. Q. Wu, F. Xie and L. Dan, The First laws of thermodynamics of the (2+1)dimensional BTZ black holes and Kerr-de Sitter spacetimes, Chin. Phys. Lett. 23 (2006)</list_item>
<list_item><location><page_10><loc_12><loc_24><loc_88><loc_27></location>[22] E. A. Larranaga Rubio, On the first law of thermodynamics for (2+1) dimensional charged BTZ black hole and charged de Sitter space, Turkish journal of physics, 2008(1):32.</list_item>
<list_item><location><page_10><loc_12><loc_21><loc_72><loc_23></location>[23] J. A. Schouton, Tensor Analysis for Physicists Claredon,Oxford, (1951).</list_item>
<list_item><location><page_10><loc_12><loc_17><loc_88><loc_21></location>[24] L.-B. Fu, Y.-S. Duan, and H. Zhang, Evolution of the Chern-Simons vortices, Phys. Rev. D 61, 045004 (2000).</list_item>
</unordered_list>
</document>
[
{
"title": "Topological classes of BTZ black holes",
"content": "Yongbin Du and Xiangdong Zhang ∗ Department of Physics, South China University of Technology, Guangzhou 510641, China (Dated: February 24, 2023)",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "In the recent paper [Phys. Rev. Lett. 129, 191101 (2022)], the black holes were viewed as topological thermodynamic defects by using the generalized off-shell free energy. Their work indicates that all black hole solutions in the pure Einstein-Maxwell gravity theory could be classified into three different topological classes for four and higher spacetime dimensions. In this paper, we investigate the topological number of BTZ black holes with different charges ( Q ) and rotational ( J ) parameters. By using generalized free energy and Duan's φ -mapping topological current theory, we interestingly found only two topological classes for BTZ spacetime. Particularly, for Q = J = 0 BTZ black hole, there has only one zero point and therefore the total topological number is 1. While for rotating or charged cases, there are always two zero points and the global topological number is zero. PACS numbers:",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "As one of the most fascinating objects in nature, the black hole has receiving increasingly attention both in theoretical and observational physics. On one hand, singularity theorem [1] reveals the intrinsic limitation of general relativity. On the other side, when taking quantum effect into consideration, people found that black hole is not black at all [2] and surprisingly sharing rich thermodynamical properties [3-5]. Many issues, such as information paradox [6], arise due to these remarkable discoveries. Recently, by introducing Duan's topological current φ -mapping theory[7], Wei et. al. [8, 9] originally consider black hole as topological thermodynamic defects, and classify different solutions with their global topological charge. They divided black hole solutions into three different topological classes according to their different topological numbers. This method gives us an insight into black hole solutions and implies more inherent properties of this extraordinary thermodynamic system. Inspired by this results, many works have been done to investigate the topological charge of different types of black hole. Since [8] only foucus on the static solutions, in [10], stationary black holes are explored. The topological number of Kerr and Kerr-Newman black hole are calculated. Furthermore, they calculate the singly-rotating black hole in higher dimensions [10]. Apart from the researches on black hole in general relativity, some researchers are also consider the black hole in modified gravity theories such as Gauss-Bonnet gravity [11] or Lovelock gravity [12] and so on [13-15]. The topological number are quite different in these cases, as provide us a new perspective to consider the difference between modified gravity theory and general relativity. An interesting feature of these works is that whether we consider black hole solutions in general relativity or modified gravity or even high-dimensional extension, they can always be classified into three topological classes [10-12]. Moreover, dimension may play an improtant role in the amount of zero points and topological number [10, 11]. Hence extending the approach to lower dimensional case could provide us with a test to the seemingly universal phenomenon or provide a counterexample which become very relevant for the future study on this issue. To this aim, in this paper, we follow this newly-hewed path to study the topological number of the Banados-Teitelboim-Zanelli (BTZ) black hole [16] and trying to provide counterexample to the above observation. The BTZ black hole is the black hole solution of 3-dimensional general relativity plus a negative cosmological constant and plays an important role in quantum theory of gravity. As a low-dimensional toy model, BTZ black hole have many interesting features. Locally, it has constant negative curvature and consequently no curvature singularity [17]. But it possess a horizon, and the Bekenstein's area law could also be applied in it. Thus it is a genuine black hole and the simply form may benefit quantum gravity or holographic priciple a lot. Many paper focus on this 3-dimensional black hole solution to find some unexpected result of common physical phenomenon, such as Penrose process [18, 19]. In this article we shall concentrate on thermodynamics of BTZ black hole and then calculate the topological charge when the angular momentum and electric charge are taken different values. We would also like to give a simple comparison with the results of black hole in four dimensions [8-10]. This paper is organized as follows: In section 2, we briefly review some useful results of thermodynamics of BTZ black hole. Then we use the generalized free energy to establish a parameter space in section 3, and find the zero points with their topological charge. In section 4, we summarize our result, compare it with higher dimensional cases and make some comment on this new approach.",
"pages": [
2,
3
]
},
{
"title": "II. THERMODYNAMICS OF BTZ BLACK HOLE",
"content": "In this section, we first give a brief review of thermodynamics of BTZ black hole. We use the generalized off-shell free energy [20] so that we could classify different black hole solutions. This means black holes with the same energy (and electric charge or angular momentum, if any) can be in different temperature. The metric line element for the BTZ black hole reads [16] where the squared lapse N 2 ( r ) and the angular shift N φ ( r ) are respectively given by with -∞ < t < ∞ , 0 < r < ∞ , and 0 ≤ φ ≤ 2 π . M and J are respectively the mass and angular momentum of the black hole. L represents the radius of curvature of spacetime and satisfy L 2 = -1 / Λ with Λ being the negative cosmological constant. The metric is singular at the inner and outer horizons where N 2 ( r ) vanishes. Using gravitational path integral and saddle point approximation, the generalized off-shell free energy of BTZ black hole reads [16, 20] The parameter τ is an extra variable with the dimension of time. It varies freely and can be thought as the inverse temperature of the cavity enclosing the black hole. r h represents the radius of the outer horizon. The mass and entropy can be written as the function of r h and J We have to stress that L is no more a fixed value so that the modified Bekenstein-Smarr mass formula still holds in 3-dimensional black hole thermodynamics [21, 22]. However, as we will show in this paper, its value will not affect the topological charge of BTZ solution. With the help of (2.5) and (2.6), the generalized free energy (2.4) can be written as It is off-shell except at τ = 1 /T which means the black hole is in the maximal mixed state.",
"pages": [
3,
4
]
},
{
"title": "III. TOPOLOGICAL CLASSES OF BTZ BLACK HOLE SOLUTIONS",
"content": "When the generalized free energy is obtained, we could apply the method in [8] to estabilish a parameter space and find the zero point of the vector field in it. Profoundly, the zero points are exactly corresponding to the on-shell black hole solution. We can calculate the topological number of them by virtue of Duan's φ -mapping topological current theory [7]. The number can be seen as a characteristic value of the on-shell black hole solution. Following the spirit of [8], we define the vector as with θ ∈ [0 , π ] for convenience. The vector field is on θ -r h space, and we can see that φ θ is divergent when θ = 0 , π , making the direction of vectors point vertically outward at this boundary. The zero point, corresponding to τ = 1 /T [9], can only be obtained when θ = π/ 2. Now we introduce the topological current as where ∂ ν = ( ∂/∂x ν ) and x ν = ( τ, r h , θ ). The unit vector is defined as n a = ( φ a / ‖ φ ‖ ) ( a = 1 , 2). The conservation law of the current, ∂ µ j µ = 0 can be easily obtained by the definition of j µ , and τ here serves as a time parameter of the topological defect. By using the Jacobi tensor /epsilon1 ab J µ ( φ/x ) = /epsilon1 µνρ ∂ ν φ a ∂ ρ φ b and the two-dimensional Laplacian Green function ∆ φ a ln ‖ φ ‖ = 2 πδ 2 ( φ ), the topological current can be written as where j µ is nonzero only at φ a ( x i ) = 0, and we denote its i -th solution as /vectorx = /vectorz i . The density of the topological current reads [23] where β i is Hopf index, which counts the number of the loops that φ a makes in the vector φ space when x µ goes around the zero point z i . Thus Hopf index is always positive. η i is the Brouwer degree and satisfy η i = sign ( J 0 ( φ/x ) z i ) = ± 1. Given a parameter region Σ, the corresponding topological number can be obtained as /negationslash where w i is the winding number for the i -th zero point of φ contained in Σ and its value does not depend on the shape of the region where we perform the calculation. Usually, distinct zero points of the vector field are isolated, making Jacobian J 0 ( φ/x ) = 0. If Jacobian J 0 ( φ/x ) = 0, it means that the defect bifurcates [24]. Eq.(3.5) shows that in any given region, the global topological number is the sum of the winding number of each zero point which reflects the local property of the topological defect. Based on the approach introduced above, we first investigate BTZ black hole with J = 0. In this case, the generalized free energy is We define the vector field as By solving the equation φ = 0, we acquire the relation We take L = 1 /r 0 and τ = 2 π/r 3 0 with r 0 an arbitrary positive constant, so there is one zero point in θ -r h plane at ( r h /r 0 , θ ) = (1 , π/ 2). We plot the unit vector field in Fig. 1. The loop surrounding the zero point sets the boundary of a given region so we can use Eq.(3.5) to acquire the winding number of point P . We find that w = 1. Since there is only one defect in the parameter space, it gives rise the global topological number W = 1 for BTZ black hole solution with J = 0. Now we would further explore the rotating BTZ black hole. The generalized free energy is As a result, we define the vector field as Solving the equation φ = 0, we get the relation Upon the value of L and J are determined, we could acquire a curve in τ -r h plane. For instance, as shown in Fig. 2, when taking L = 1 /r 0 and J = r 3 0 , the curve meets a peak at ( r h , τ ) ≈ (0 . 93 r 0 , 5 . 06 /r 3 0 ) and rapidly goes down as r h gets larger. Consequently, the vector field possesses two zero points for small τ , in comtrast to one zero point for nonrotating BTZ black hole case. As τ = 4 π/ 5 r 3 0 , the two intersection points are respectively at r h = 0 . 5 r 0 and r h ≈ 2 . 482 r 0 . We illustrate the vector field and the zero points in Fig. 3. When τ = τ cri ≈ 5 . 06 /r 3 0 , the intersection points coincide, and for larger τ , annihilate. It is easy to check the critical point satisfy d 2 τ/dr 2 h < 0, which belongs to annihilation point. For τ < τ cri , we find that the winding number of the two zero points are w 1 = -1 and w 2 = 1. Thus the global topological number for BTZ rotating solution is W = w 1 + w 2 = 0, different from non-rotating case. When taking electric charge into account, the squared lapse N 2 ( r ) and the angular shift N φ ( r ) are respectively given by Q denotes the electric charge of the black hole, A 0 ( r ) is the only nonvanishing component of the electromagnetic vector potential and is taken to be A 0 ( r ) = -Q ln( r/r c ) with r c being /negationslash /negationslash an arbitrary constant. the generalized off-shell free energy of BTZ black hole reads [16, 20] Following the same step, we get the on-shell solution curve in τ -r h plane, which satisfy Interestingly, we find that the electric charge do not change the rough trend of the curve. As shown in Fig. 4, there is invaribly one anihilation point in τ -r h plane, and τ cri merely gets smaller as the value of Q taken larger. There are two zero points as well. We calculate the winding number of them for a given τ and find w 1 = -1, w 2 = 1. See Fig. 5 as an example. The global topological number of charged BTZ black hole solution is W = 0, which is the same as the case of rotational BTZ black hole. Hence from the perspective of topological charge, the two kinds of 3-dimensional black hole are just the same. This conclusion is the same as four dimensional cases [10]. Yet the topological number of nonrotating and uncharged black hole are different, with W BTZ = 1 whereas W Schwarzchild = -1. Besides, in contrast to the charged Reissner-Nordstrom anti de-Sitter (RN-AdS) black hole in four dimension [8], three dimensional black hole solution evidently has fewer zero points. So the dimension of black hole of the same kind may have an unique influence on the quantity of defects and the global topological number.",
"pages": [
4,
5,
6,
7,
8
]
},
{
"title": "IV. CONCLUSION",
"content": "In this paper, we use the generalized free energy of BTZ black hole to define a vector field in a parameter space θ -r h . We find the zero point of the field and obtain the winding number by applying Duan's φ -mapping topological current theory. It is discovered that the unique black hole solution in three dimensional general relativity has one zero point for Q = J = 0 and two for rotating or charged cases. The global topological charges are one and zero respectively. We also find that the dimension of the AdS background would lead to a distinct amount of zero points. The previous works [8, 10] indicate that all black hole solutions in the pure EinsteinMaxwell gravity theory should be classified into three different topological classes for four and higher spacetime dimensions. This observation is further enhanced in the modified gravity case [11, 12]. However, our investigation on BTZ black holes found only two topological /negationslash classes for BTZ spacetime. This means this feature is not universal and the spacetime dimension seems strongly relevant in the topological classification of black holes. There are many issues that deserve further investigation. Generalize our results to KerrAdS and Kerr-dS and compare with the existing result will be interesting. Moreover, another interesting object is to investigate the topological number of the black hole solutions in the supergravity and modified gravity theories. We leave these interesting topics for future studies.",
"pages": [
8,
9
]
},
{
"title": "Acknowledgments",
"content": "We would like to thank Prof. Pujian Mao for helpful discussion. This work is supported by NSFC with Grants No.12275087 and 'the Fundamental Research Funds for the Central Universities'.",
"pages": [
9
]
}
]
2024Univ...10..253B
https://arxiv.org/pdf/2304.05666.pdf
<document>
<section_header_level_1><location><page_1><loc_32><loc_92><loc_69><loc_93></location>Cosmic Strings from Thermal Inflation</section_header_level_1>
<text><location><page_1><loc_33><loc_89><loc_67><loc_90></location>Robert Brandenberger 1, ∗ and Aline Favero 1, †</text>
<text><location><page_1><loc_24><loc_86><loc_77><loc_88></location>1 Department of Physics, McGill University, Montr´eal, QC, H3A 2T8, Canada (Dated: April 13, 2023)</text>
<text><location><page_1><loc_18><loc_72><loc_83><loc_85></location>Thermal inflation was proposed as a mechanism to dilute the density of cosmological moduli. Thermal inflation is driven by a complex scalar field possessing a large vacuum expectation value and a very flat potential, called a 'flaton'. Such a model admits cosmic string solutions, and a network of such strings will inevitably form in the symmetry breaking phase transition at the end of the period of thermal inflation. We discuss the differences of these strings compared to the strings which form in the Abelian Higgs model. Specifically, we find that the upper bound on the symmetry breaking scale is parametrically lower than in the case of Abelian Higgs strings, and that the lower cutoff on the string loop distribution is determined by cusp annihilation rather than by gravitational radiation (for the value of the transition temperature proposed in the original work on thermal inflation).</text>
<section_header_level_1><location><page_1><loc_20><loc_68><loc_37><loc_69></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_47><loc_49><loc_66></location>Thermal inflation (not to be confused with warm inflation [1]) was proposed [2] as a mechanism to dilute the number density of unwanted moduli quanta. Moduli fields are predicted in many models beyond the particle physics Standard Model. In such models, moduli quanta are often left over from the Big Bang period, produced at the end of a phase of primordial inflation, or generated during a compactification phase transition. The proposal of [2] is to dilute the number density of moduli by invoking a period of inflation. This period should be sufficiently long to dilute the moduli, but short enough not to redshift the fluctuations which are generated in the primordial universe.</text>
<text><location><page_1><loc_9><loc_34><loc_49><loc_47></location>Thermal inflation is generated by adding a new matter sector involving a complex scalar field 1 φ with a symmetry breaking potential V ( φ ), and gauging the resulting U (1) symmetry (this gauging is done since there is evidence that global symmetries are inconsistent with quantum gravity [4]). Thermal inflation is assumed to occur in the radiation phase of cosmology. Thermal effects are assumed to trap φ at the symmetric point φ = 0. At a temperature T i given by</text>
<formula><location><page_1><loc_24><loc_31><loc_49><loc_33></location>V (0) = T 4 i , (1)</formula>
<text><location><page_1><loc_9><loc_26><loc_49><loc_30></location>the potential energy of φ begins to dominate and inflation begins. The coupling of φ to the thermal bath generates a finite temperature contribution to the effective potential</text>
<formula><location><page_1><loc_20><loc_23><loc_49><loc_25></location>V T ( φ ) = V ( φ ) + gT 2 φ 2 , (2)</formula>
<text><location><page_1><loc_9><loc_18><loc_49><loc_22></location>where g is the coupling constant describing the interactions between φ and the thermal bath. At a temperature T c when the positive contribution to the curvature of the</text>
<text><location><page_1><loc_52><loc_58><loc_92><loc_69></location>potential at φ = 0 equals the absolute value of the (negative) curvature coming from the bare potential V ( φ ), symmetry breaking sets in, φ rolls to the bottom of its potential and thermal inflation ends. In order to generate the required hierarchy between T c and T i , it was assumed that the bare potential V ( φ ) contains no quartic term. The hierarchy between T i and T c determines the number N of e-foldings of thermal inflation</text>
<formula><location><page_1><loc_68><loc_51><loc_92><loc_54></location>T c T i = e -N . (3)</formula>
<text><location><page_1><loc_52><loc_18><loc_92><loc_46></location>Since the vacuum manifold of φ is S 1 , cosmic string defects inevitably form in the phase transition which ends thermal inflation [5] (see [6] for reviews of the role of cosmic strings in cosmology). As the scale of symmetry breaking for thermal inflation is assumed to be of the order m 0 ∼ 10 2 -10 3 GeV one might - based on intuition from Abelian Higgs strings [7] - have expected the signatures of these strings to be negligible. As we show here, thermal inflation strings have different properties compared to strings formed in the standard Abelian Higgs model. For the same value of the symmetry breaking temperature, thermal inflation strings have a parametrically larger mass per unit length than regular strings. Comparing strings with the same mass per unit length µ , thermal inflation strings have a parametrically greater width than regular strings. These differences affect the distribution of string loops, and hence the observational consequences of the strings (see e.g. [8] for a short review) need to be revisited.</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_17></location>In this article we use natural units in which the speed of light, Planck's constant and Boltzmann's constant are set to 1. The Planck mass is denoted by m pl . The mass per unit length of a string will be denoted by µ , or often in terms of the dimensionless quantity Gµ , where G is Newton's gravitational constant.</text>
<section_header_level_1><location><page_2><loc_13><loc_92><loc_45><loc_93></location>II. THERMAL INFLATION STRINGS</section_header_level_1>
<text><location><page_2><loc_9><loc_87><loc_49><loc_90></location>The thermal inflation model [2] assumes a potential given by</text>
<formula><location><page_2><loc_15><loc_82><loc_49><loc_86></location>V ( φ ) = V 0 -m 2 0 | φ | 2 + ∞ ∑ n =1 λ n | φ | 2 n +4 m pl 2 n , (4)</formula>
<text><location><page_2><loc_9><loc_68><loc_49><loc_81></location>where V 0 is tuned such that the potential energy in the vacuum manifold vanishes. The λ n are dimensionless coupling constants, and the mass scale m 0 determines the (negative) curvature of the potential at the origin. We shall consider a simplified potential containing only the n = 1 term (the contributions from the terms with n > 1 are Planck suppressed for the questions we are asking, i.e. those involving small field values) 2 . Thus, we consider the potential</text>
<formula><location><page_2><loc_17><loc_65><loc_49><loc_66></location>V ( φ ) = V 0 -m 2 0 | φ | 2 + λ | φ | 6 m -2 pl . (5)</formula>
<text><location><page_2><loc_9><loc_61><loc_49><loc_63></location>This potential is to be compared with the potential for the Abelian Higgs model which is</text>
<formula><location><page_2><loc_18><loc_56><loc_49><loc_59></location>V AH ( φ ) = λ AH 4 ( | φ | 2 -η 2 ) 2 , (6)</formula>
<text><location><page_2><loc_9><loc_52><loc_49><loc_55></location>where η is the value of | φ | in the vacuum manifold, and λ AH is a dimensionless coupling constant.</text>
<text><location><page_2><loc_9><loc_47><loc_49><loc_52></location>As we will see below, the hierarchy between T i and T c increases as m 0 decreases. To obtain an e-folding number N ∼ 10 of thermal inflation a value of m 0 ∼ 10 2 -10 3 GeV was suggested [2].</text>
<text><location><page_2><loc_9><loc_44><loc_49><loc_46></location>From (5) it immediately follows that the value η of | φ | which minimizes the potential is given by</text>
<formula><location><page_2><loc_20><loc_39><loc_49><loc_42></location>η 2 = ( 1 3 ) 1 / 2 λ -1 / 2 m pl m 0 m 2 0 (7)</formula>
<text><location><page_2><loc_9><loc_29><loc_49><loc_38></location>which is parametrically larger by a factor of m pl /m 0 than what is obtained for an Abelian Higgs string given the same value of the curvature of the potential at φ = 0. For the value of m 0 indicated above, η is of the order of 10 10 GeV and not 10 2 GeV is it would be for an Abelian Higgs string with the same value of m 0 .</text>
<text><location><page_2><loc_9><loc_27><loc_49><loc_29></location>Demanding that the potential vanishes for | φ | = η yields</text>
<formula><location><page_2><loc_19><loc_22><loc_49><loc_25></location>V 0 = 2 3 ( 1 3 ) 1 / 2 λ -1 / 2 m 3 0 m pl (8)</formula>
<text><location><page_2><loc_9><loc_15><loc_49><loc_21></location>which is also parametrically larger by a factor of m pl /m 0 compared to the corresponding result for the Abelian Higgs string (taking coupling constants to be of the order 1). This leads to the fact that the temperature T i</text>
<text><location><page_2><loc_52><loc_89><loc_92><loc_93></location>corresponding to the onset of thermal inflation is parametrically larger than what one might have guessed from Abelian Higgs string intuition</text>
<formula><location><page_2><loc_63><loc_85><loc_92><loc_88></location>T i ∼ λ -1 / 8 ( m pl m 0 ) 1 / 4 m 0 . (9)</formula>
<text><location><page_2><loc_52><loc_82><loc_92><loc_84></location>On the other hand, the temperature at which the symmetry breaking phase transition takes place is given by</text>
<formula><location><page_2><loc_68><loc_79><loc_92><loc_80></location>T c ∼ m 0 , (10)</formula>
<text><location><page_2><loc_52><loc_72><loc_92><loc_78></location>setting g to be of the order 1. Comparing (9) and (10) we see that it is precisely the enhancement factor discussed above which allows for a period of thermal inflation to take place.</text>
<text><location><page_2><loc_52><loc_58><loc_92><loc_72></location>Let us now compare the width of a thermal inflation string with that of an Abelian Higgs string for the same value of the phase transition temperature T c . The width is determined by minimizing the sum of the potential and gradient energy terms. Increasing the width of the string costs potential energy while decreasing the width leads to an increase of the gradient energy. For a straight string centered at r = 0 (in polar coordinates), the field configuration of a string with winding number 1 can be written as</text>
<formula><location><page_2><loc_65><loc_55><loc_92><loc_56></location>φ ( r, θ ) = f ( r ) ηe iθ , (11)</formula>
<text><location><page_2><loc_52><loc_50><loc_92><loc_54></location>where the profile function f ( r ) increases from f (0) = 0 to f ( r ) = 1 for r > w . The potential energy µ p per unit length of the string can hence be estimated to be</text>
<formula><location><page_2><loc_66><loc_47><loc_92><loc_48></location>µ p ( w ) ∼ πw 2 V 0 . (12)</formula>
<text><location><page_2><loc_52><loc_37><loc_92><loc_46></location>The scalar field angular gradient energy for a local string is cancelled by the gauge fields beyond a radius r A which is set by the gauge field mass. For r < w the angular gradient energy decays since f ( r ) decays as r decreases. Hence, the mass per unit length µ a from gradients can be estimated as</text>
<formula><location><page_2><loc_62><loc_33><loc_92><loc_36></location>µ a ( w ) ∼ 2 πη 2 ∫ r A w 1 r f ( r ) 2 . (13)</formula>
<text><location><page_2><loc_52><loc_31><loc_73><loc_32></location>It then follows from (13) that</text>
<formula><location><page_2><loc_64><loc_27><loc_92><loc_30></location>∂ ∂w µ a ( w ) ∼ -2 πη 2 1 w . (14)</formula>
<text><location><page_2><loc_52><loc_22><loc_92><loc_26></location>Hence, by minimizing the sum of potential and angular gradient energy (to first approximation the radial gradient energy does not depend on w ) it follows that</text>
<formula><location><page_2><loc_67><loc_19><loc_92><loc_20></location>w ∼ V -1 / 2 0 η . (15)</formula>
<text><location><page_2><loc_52><loc_16><loc_78><loc_17></location>For Abelian Higgs strings this yields</text>
<formula><location><page_2><loc_65><loc_14><loc_92><loc_15></location>w AH ∼ λ -1 / 2 η -1 , (16)</formula>
<text><location><page_2><loc_52><loc_11><loc_76><loc_12></location>while for thermal inflation strings</text>
<formula><location><page_2><loc_68><loc_8><loc_92><loc_10></location>w ∼ m -1 0 . (17)</formula>
<text><location><page_3><loc_9><loc_82><loc_49><loc_93></location>In terms of the phase transition temperature, the widths of thermal inflation strings and Abelian Higgs strings are of the same order of magnitude. However, in terms of the mass per unit length there is a parametric difference, and this difference will have important implications for the string loop distribution. From energy equipartition, the energy per unit length µ of a string is given by</text>
<formula><location><page_3><loc_25><loc_79><loc_49><loc_81></location>µ ∼ w 2 V 0 . (18)</formula>
<text><location><page_3><loc_9><loc_77><loc_35><loc_78></location>For Abelian Higgs strings this yields</text>
<formula><location><page_3><loc_22><loc_74><loc_49><loc_75></location>µ AH ∼ η 2 ∼ T 2 c , (19)</formula>
<text><location><page_3><loc_9><loc_71><loc_33><loc_72></location>while for thermal inflation strings</text>
<formula><location><page_3><loc_22><loc_67><loc_49><loc_70></location>µ ∼ λ -1 / 2 m pl m 0 T 2 c . (20)</formula>
<text><location><page_3><loc_9><loc_64><loc_49><loc_66></location>Thus, for a fixed mass per unit length, a thermal inflation string has a width</text>
<formula><location><page_3><loc_21><loc_60><loc_49><loc_63></location>w ∼ λ -1 / 2 m pl √ µ µ -1 / 2 , (21)</formula>
<text><location><page_3><loc_9><loc_57><loc_36><loc_59></location>which is much greater than the width</text>
<formula><location><page_3><loc_21><loc_55><loc_49><loc_56></location>w AH ∼ λ -1 / 2 µ -1 / 2 . (22)</formula>
<text><location><page_3><loc_9><loc_52><loc_28><loc_53></location>of an Abelian Higgs string.</text>
<text><location><page_3><loc_9><loc_38><loc_49><loc_52></location>Comparing (19) and (20), we see that for fixed symmetry breaking scale of T c ∼ m 0 ∼ 10 2 GeV, Abelian Higgs strings would have a mass per unit length of Gµ AB ∼ 10 -34 which is many orders of magnitude smaller than the range of valiues of Gµ which can have interesting cosmological effects. In the case of thermal inflation strings, on the other hand, for the same value of T c we obtain Gµ ∼ 10 -17 which is now approaching the range which is of interest for string signals in cosmological observations.</text>
<section_header_level_1><location><page_3><loc_9><loc_34><loc_49><loc_35></location>III. THERMAL STRING LOOP DISTRIBUTION</section_header_level_1>
<text><location><page_3><loc_9><loc_26><loc_49><loc_31></location>The parametric enhancement of the width of a thermal inflation string compared to the width of an Abelian Higgs string with the same mass per unit length has important implications for the loop distribution.</text>
<text><location><page_3><loc_9><loc_9><loc_49><loc_26></location>The causality argument [5] which implies that the distribution of long strings (strings with a curvature radius comparable and greater than the Hubble radius) takes on a scaling solution where the number of long string segments crossing any given Hubble volume is independent of time applies equally to Abelian Higgs and thermal inflation strings. This scaling solution of the long string network is maintained by string loop production. Like for Abelian Higgs strings, we can assume that the one scale loop production model [9] also applies to thermal inflation strings, implying that at time t loops are produced with radius R = αt , where α is a constant which can be</text>
<text><location><page_3><loc_52><loc_86><loc_92><loc_93></location>normalized by string evolution simulations which yield α ∼ 10 -1 . Once produced, the number density n ( R,t ) of loops in the radius interval between R and R + dR redshifts. Thus at times t after the time t eq of equal matter and radiation we have</text>
<formula><location><page_3><loc_54><loc_81><loc_92><loc_85></location>n ( R,t ) dR = NR -2 t -2 dR αt > R > αt eq (23) n ( R,t ) dR = NR -5 / 2 t 1 / 2 eq t -2 dR αt eq > R > R co ,</formula>
<text><location><page_3><loc_52><loc_74><loc_92><loc_80></location>where N is a constant determined by the number of long string segments per Hubble volume. R co is a cutoff radius below which loops live for less than one Hubble time, and whose consequences for cosmology can be neglected.</text>
<text><location><page_3><loc_52><loc_66><loc_92><loc_74></location>For non-superconducting strings there are two main mechanisms by which string loops decay. The first is gravitational radiation: string loops have relativistic tension and hence oscillate and emit gravitational radiation. The power of gravitational radiation from a string loop of radius R is [10]</text>
<formula><location><page_3><loc_67><loc_63><loc_92><loc_65></location>P g = γGµ 2 , (24)</formula>
<text><location><page_3><loc_52><loc_59><loc_92><loc_62></location>where γ is a constant of the order 10 2 . Gravitational radiation implies that loops with radius R < R g where</text>
<formula><location><page_3><loc_68><loc_56><loc_92><loc_58></location>R g = γGµt (25)</formula>
<text><location><page_3><loc_52><loc_52><loc_92><loc_55></location>will live less than one Hubble expansion time, and hence their cosmological effects are negligible.</text>
<text><location><page_3><loc_52><loc_45><loc_92><loc_52></location>Cusp evaporation is a second decay mechanism [11]. A cusp is a point on the string which moves at the speed of light. Strings have finite width, and around the cusp point the string segments on either side of the cusp point overlap for a region of length 3</text>
<formula><location><page_3><loc_65><loc_42><loc_92><loc_44></location>l c ( R ) ∼ R 1 / 2 w 1 / 2 . (26)</formula>
<text><location><page_3><loc_52><loc_31><loc_92><loc_41></location>Locally the cusp region looks like a string-antistring pair, and there is no topology protecting the cusp region against annihilation into gauge and scalar field quanta. It can be proven that string loops described by the effective Nambu-Goto action have at least one cusp per oscillation time [13]. Hence, the power of the cusp annihilation process is</text>
<formula><location><page_3><loc_62><loc_27><loc_92><loc_30></location>P c ∼ 1 R l c ( R ) µ = ( w R ) 1 / 2 µ. (27)</formula>
<text><location><page_3><loc_52><loc_24><loc_92><loc_26></location>From the above equation it follows that, due to cusp evaporation, string loops with radius less than</text>
<formula><location><page_3><loc_67><loc_21><loc_92><loc_22></location>R c = w 1 / 3 t 2 / 3 (28)</formula>
<text><location><page_3><loc_52><loc_15><loc_92><loc_20></location>live for less than one Hubble expansion time. The cutoff radius R co in the loop distribution of (23) is the larger of R g and R c .</text>
<text><location><page_4><loc_9><loc_82><loc_49><loc_93></location>Comparing the strengths of gravitational radiation power (24) and cusp annihilation power (27) we see that the parametrically larger width of a thermal inflation string (for a given mass per unit length) will lead to a parametric amplification of the role of cusp annihilation compared to gravitational wave decay. We also see that the relative importance of cusp annihilation increases the lower the value of Gµ is.</text>
<text><location><page_4><loc_9><loc_79><loc_49><loc_81></location>The condition for cusp annihilation to dominate over gravitational radiation is R c > R g or</text>
<formula><location><page_4><loc_24><loc_76><loc_49><loc_77></location>w > ( γGµ ) 3 t . (29)</formula>
<text><location><page_4><loc_9><loc_73><loc_35><loc_75></location>For Abelian Higgs strings this yields</text>
<formula><location><page_4><loc_19><loc_69><loc_49><loc_72></location>( T m pl ) 2 > λ 1 / 2 γ 3 ( Gµ ) 7 / 2 , (30)</formula>
<text><location><page_4><loc_9><loc_67><loc_46><loc_68></location>or, expressed in terms of the critical temperature T c</text>
<formula><location><page_4><loc_20><loc_62><loc_49><loc_65></location>T T c > λ 1 / 4 γ 3 / 2 ( T m pl ) 5 / 2 . (31)</formula>
<text><location><page_4><loc_9><loc_60><loc_49><loc_61></location>On the other hand, for thermal inflation strings we obtain</text>
<formula><location><page_4><loc_22><loc_56><loc_49><loc_59></location>( T m pl ) 2 > γ 3 ( Gµ ) 4 , (32)</formula>
<text><location><page_4><loc_9><loc_53><loc_49><loc_54></location>or, after expressing µ in terms of the critical temperature</text>
<formula><location><page_4><loc_22><loc_49><loc_49><loc_52></location>T T c > γ 3 / 2 λ -1 T c m pl . (33)</formula>
<text><location><page_4><loc_9><loc_39><loc_49><loc_47></location>Comparing these expressions we see that, for a fixed value of the string tension, cusp annihilation is more important for thermal inflation strings than for Abelian Higgs strings. On the other hand, fixing T c we see that the importance of cusp annihilation is, maybe surprisingly, less than for Abelian Higgs strings.</text>
<text><location><page_4><loc_9><loc_25><loc_49><loc_39></location>Evaluating (31) and (33) at the temperature T eq ∼ 1eV of equal matter and radiation (the temperature relevant for cosmological signatures of strings), we see that for Abelian Higgs strings the cutoff in the loop distribution is determined by cusp annihilation for values T c < 10 10 GeV while for thermal inflation strings it is for values T c < 10 8 GeV. In particular, for the value m 0 ∼ 10 2 -10 3 GeV assumed in the original thermal inflation paper [2], we conclude that the cutoff in the loop distribution is given by the cusp annihilation process.</text>
<section_header_level_1><location><page_4><loc_10><loc_19><loc_48><loc_21></location>IV. CONSTRAINTS FROM COSMOLOGICAL OBSERVATIONS</section_header_level_1>
<text><location><page_4><loc_9><loc_9><loc_49><loc_17></location>In this section we study what constraints on the symmetry breaking scale m 0 of thermal inflation can be derived from cosmological observations. Cosmic strings leave behind interesting signals in many observational windows. In most cases, the effects are gravitational, and hence the magnitude of the string signal depends on Gµ .</text>
<text><location><page_4><loc_52><loc_85><loc_92><loc_93></location>In terms of Gµ , the strength of the signals will hence be the same for Abelian Higgs and thermal inflation strings. However, since the relation between T c and µ is different, the magnitude of the string signals as a function of T c will change. As the string network consists of both long strings and loops, each will induce specific signatures.</text>
<text><location><page_4><loc_52><loc_70><loc_92><loc_84></location>We first turn to signatures of the long string network. For example, long strings lead to line discontinuities in cosmic microwave background (CMB) anisotropy maps [14]. This is due to the fact that a long straight string produces a conical deformation of the metric with deficit angle proportional to Gµ [15].The magnitude of this signal depends on Gµ . The study of [16] shows that experiments with the specifications of the South Pole Telescope or the Atacama Cosmology Telescope can constrain the string tension to be</text>
<formula><location><page_4><loc_67><loc_68><loc_92><loc_70></location>Gµ < 10 -8 . (34)</formula>
<text><location><page_4><loc_52><loc_66><loc_70><loc_67></location>A slighty weaker bound of</text>
<formula><location><page_4><loc_68><loc_64><loc_92><loc_65></location>Gµ < 10 -7 (35)</formula>
<text><location><page_4><loc_52><loc_58><loc_92><loc_63></location>can be derived from the angular power spectrum of CMB anisotropies [17]. The resulting bound on T c for thermal inflation strings is parametrically stronger than for Abelian Higgs strings, namely</text>
<formula><location><page_4><loc_66><loc_54><loc_92><loc_57></location>T c m pl < λ 1 / 2 10 -7 . (36)</formula>
<text><location><page_4><loc_52><loc_50><loc_92><loc_53></location>This bound is obviously satisfied for the value T c ∼ m 0 = 10 2 -10 3 GeV assumed in [2].</text>
<text><location><page_4><loc_53><loc_49><loc_91><loc_50></location>For Abelian Higgs strings there is a tighter bound of</text>
<formula><location><page_4><loc_67><loc_47><loc_92><loc_48></location>Gµ < 10 -10 (37)</formula>
<text><location><page_4><loc_52><loc_33><loc_92><loc_46></location>which comes from the upper bound on the stochastic background of gravitational waves from pulsar timing array measurements [18]. This bound depends on having a scaling distribution of loops down to the gravitational radiation cutoff R g . This bound remains valid for thermal inflation strings since, as the discussion at the end of the previous section showed, cusp annihilation only changes the loop distribution for values of Gµ which are lower than the above bound.</text>
<text><location><page_4><loc_52><loc_16><loc_92><loc_33></location>Long strings moving through space produce overdense regions in their wake [19]. CMB photons passing through these wakes get absorbed at the 21-cm wavelength. Long cosmic strings hence lead to distinct signals in high redshift 21-cm surveys [20]: wedges of absorption in 21-cm redshift maps which are extended in the angular directions and narrow in redshift direction. The study of [21] shows that the string signal can be detected by surveys such as the MWA telescope down to a value of Gµ comparable to that of (34), and prospects indicate that with better analysis tools a significant improvement of this bound can be expected.</text>
<text><location><page_4><loc_52><loc_9><loc_92><loc_16></location>String wakes also lead to rectangles in the sky with induced CMB polarization (including a B-mode component) [22]. From the analysis of [23] it appears, however, that this signal is harder to extract from observations than the 21-cm signal. At lower redshifts, string</text>
<text><location><page_5><loc_9><loc_89><loc_49><loc_93></location>wakes also lead to planar overdensities of galaxies. These are, however, disrupted by the gravitational effects of the dominant source of fluctuations [24].</text>
<text><location><page_5><loc_9><loc_66><loc_49><loc_89></location>Turning now to signatures of cosmic string loops, the gravitational signals are the same for Abelian Higgs strings and thermal inflation strings given the same value of Gµ . String loops seed nonlinear structures by gravitational accretion [25]. Given the bounds on Gµ discussed above, strings can only play a subdominant role in explaining the nonlinear structures today. However, since strings form nonlinear seeds immediately after their formation, they will dominate the halo mass function at sufficiently early times [26]. They can provide seeds for intermediate and super-massive black holes at high redshifts [27]. It has recently been shown [28] that for superconducting cosmic strings the 'Direct Collapse Black Hole' criteria can be satisifed, and that such loops indeed could explain the origin of the observed high redshift super-massive black holes.</text>
<text><location><page_5><loc_9><loc_57><loc_49><loc_66></location>Since thermal inflation strings have a greater width than Abelian Higgs strings for a fixed value of T c , nongravitational signals from thermal inflation strings may differ from those of Abelian Higgs strings. Specifically, the flux of cosmic rays [29] due to cosmic strings will be larger [30].</text>
<unordered_list>
<list_item><location><page_5><loc_10><loc_48><loc_49><loc_51></location>[1] A. Berera, 'Warm inflation,' Phys. Rev. Lett. 75 , 3218-3221 (1995) doi:10.1103/PhysRevLett.75.3218 [arXiv:astro-ph/9509049 [astro-ph]].</list_item>
<list_item><location><page_5><loc_10><loc_42><loc_49><loc_47></location>[2] D. H. Lyth and E. D. Stewart, 'Thermal inflation and the moduli problem,' Phys. Rev. D 53 , 17841798 (1996) doi:10.1103/PhysRevD.53.1784 [arXiv:hepph/9510204 [hep-ph]].</list_item>
<list_item><location><page_5><loc_10><loc_37><loc_49><loc_42></location>[3] Y. B. Zeldovich, I. Y. Kobzarev and L. B. Okun, 'Cosmological Consequences of the Spontaneous Breakdown of Discrete Symmetry,' Zh. Eksp. Teor. Fiz. 67 , 3-11 (1974).</list_item>
<list_item><location><page_5><loc_10><loc_32><loc_49><loc_37></location>[4] T. Banks and N. Seiberg, 'Symmetries and Strings in Field Theory and Gravity,' Phys. Rev. D 83 , 084019 (2011) doi:10.1103/PhysRevD.83.084019 [arXiv:1011.5120 [hep-th]].</list_item>
<list_item><location><page_5><loc_10><loc_27><loc_49><loc_32></location>[5] T. W. B. Kibble, 'Phase Transitions In The Early Universe,' Acta Phys. Polon. B 13 , 723 (1982); T. W. B. Kibble, 'Some Implications Of A Cosmological Phase Transition,' Phys. Rept. 67 , 183 (1980).</list_item>
<list_item><location><page_5><loc_10><loc_23><loc_49><loc_26></location>[6] A. Vilenkin and E. P. S. Shellard, 'Cosmic Strings and Other Topological Defects,' (Cambridge Univ. Press, Cambridge, 2000);</list_item>
<list_item><location><page_5><loc_12><loc_19><loc_49><loc_22></location>M. B. Hindmarsh and T. W. B. Kibble, 'Cosmic strings,' Rept. Prog. Phys. 58 , 477 (1995) doi:10.1088/00344885/58/5/001 [hep-ph/9411342];</list_item>
<list_item><location><page_5><loc_12><loc_15><loc_49><loc_18></location>R. H. Brandenberger, 'Topological defects and structure formation,' Int. J. Mod. Phys. A 9 , 2117 (1994) doi:10.1142/S0217751X9400090X [astro-ph/9310041];</list_item>
<list_item><location><page_5><loc_12><loc_9><loc_49><loc_14></location>R. Durrer, M. Kunz and A. Melchiorri, 'Cosmic structure formation with topological defects,' Phys. Rept. 364 , 1 (2002) doi:10.1016/S0370-1573(02)00014-5 [astroph/0110348].</list_item>
</unordered_list>
<section_header_level_1><location><page_5><loc_56><loc_92><loc_88><loc_93></location>V. CONCLUSIONS AND DISCUSSION</section_header_level_1>
<text><location><page_5><loc_52><loc_80><loc_92><loc_90></location>We have pointed out that thermal inflation models lead to the production of a network of cosmic strings. These thermal inflation strings have different poperties compared to strings arising in the Abelian Higgs model. Specifically, for a fixed phase transition temperature, thermal inflation strings have a larger mass per unit length, and hence lead to larger gravitational effects.</text>
<text><location><page_5><loc_52><loc_72><loc_92><loc_79></location>For thermal inflation strings, the upper bound on the phase transition temperature from cosmological observations are parametrically more stringent than for Abelian Higgs strings. However, for the value of the symmetry breaking scale suggested in [2], the bounds are satisfied.</text>
<section_header_level_1><location><page_5><loc_65><loc_67><loc_79><loc_68></location>Acknowledgement</section_header_level_1>
<text><location><page_5><loc_52><loc_57><loc_92><loc_64></location>RB wishes to thank the Pauli Center and the Institutes of Theoretical Physics and of Particle- and Astrophysics of the ETH for hospitality. The research at McGill is supported in part by funds from NSERC and from the Canada Research Chair program.</text>
<unordered_list>
<list_item><location><page_5><loc_53><loc_48><loc_92><loc_51></location>[7] H. B. Nielsen and P. Olesen, 'Vortex Line Models for Dual Strings,' Nucl. Phys. B 61 , 45-61 (1973) doi:10.1016/0550-3213(73)90350-7</list_item>
<list_item><location><page_5><loc_53><loc_41><loc_92><loc_47></location>[8] R. H. Brandenberger, 'Searching for Cosmic Strings in New Observational Windows,' Nucl. Phys. B Proc. Suppl. 246-247 , 45-57 (2014) doi:10.1016/j.nuclphysbps.2013.10.064 [arXiv:1301.2856 [astro-ph.CO]].</list_item>
<list_item><location><page_5><loc_53><loc_28><loc_92><loc_41></location>[9] E. J. Copeland, T. W. B. Kibble and D. Austin, 'Scaling solutions in cosmic string networks,' Phys. Rev. D 45 , 1000 (1992). doi:10.1103/PhysRevD.45.R1000; L. Perivolaropoulos, 'COBE versus cosmic strings: An Analytical model,' Phys. Lett. B 298 , 305 (1993) doi:10.1016/0370-2693(93)91825-8 [hep-ph/9208247]; D. Austin, E. J. Copeland and T. W. B. Kibble, 'Evolution of cosmic string configurations,' Phys. Rev. D 48 , 5594 (1993) doi:10.1103/PhysRevD.48.5594 [hepph/9307325].</list_item>
<list_item><location><page_5><loc_52><loc_25><loc_92><loc_28></location>[10] T. Vachaspati and A. Vilenkin, 'Gravitational Radiation from Cosmic Strings,' Phys. Rev. D 31 , 3052 (1985).</list_item>
<list_item><location><page_5><loc_52><loc_21><loc_92><loc_25></location>[11] R. H. Brandenberger, 'On the Decay of Cosmic String Loops,' Nucl. Phys. B 293 , 812 (1987). doi:10.1016/0550-3213(87)90092-7</list_item>
<list_item><location><page_5><loc_52><loc_17><loc_92><loc_21></location>[12] J. J. Blanco-Pillado and K. D. Olum, 'The Form of cosmic string cusps,' Phys. Rev. D 59 , 063508 (1999) doi:10.1103/PhysRevD.59.063508 [gr-qc/9810005].</list_item>
<list_item><location><page_5><loc_52><loc_13><loc_92><loc_17></location>[13] T. W. B. Kibble and N. Turok, 'Selfintersection of Cosmic Strings,' Phys. Lett. 116B , 141 (1982). doi:10.1016/0370-2693(82)90993-5</list_item>
<list_item><location><page_5><loc_52><loc_9><loc_92><loc_13></location>[14] N. Kaiser and A. Stebbins, 'Microwave Anisotropy Due To Cosmic Strings', Nature 310 , 391 (1984); R. Moessner, L. Perivolaropoulos and R. H. Branden-</list_item>
<list_item><location><page_6><loc_12><loc_89><loc_49><loc_93></location>berger, 'A Cosmic string specific signature on the cosmic microwave background,' Astrophys. J. 425 , 365 (1994) [astro-ph/9310001].</list_item>
<list_item><location><page_6><loc_9><loc_85><loc_49><loc_89></location>[15] A. Vilenkin, 'Gravitational Field of Vacuum Domain Walls and Strings,' Phys. Rev. D 23 , 852-857 (1981) doi:10.1103/PhysRevD.23.852</list_item>
<list_item><location><page_6><loc_9><loc_79><loc_49><loc_85></location>[16] L. Hergt, A. Amara, R. Brandenberger, T. Kacprzak and A. Refregier, 'Searching for Cosmic Strings in CMB Anisotropy Maps using Wavelets and Curvelets,' JCAP 06 , 004 (2017) doi:10.1088/1475-7516/2017/06/004 [arXiv:1608.00004 [astro-ph.CO]];</list_item>
<list_item><location><page_6><loc_12><loc_71><loc_49><loc_78></location>J. D. McEwen, S. M. Feeney, H. V. Peiris, Y. Wiaux, C. Ringeval and F. R. Bouchet, 'Wavelet-Bayesian inference of cosmic strings embedded in the cosmic microwave background,' Mon. Not. Roy. Astron. Soc. 472 , no.4, 4081-4098 (2017) doi:10.1093/mnras/stx2268 [arXiv:1611.10347 [astro-ph.IM]].</list_item>
<list_item><location><page_6><loc_9><loc_64><loc_49><loc_70></location>[17] T. Charnock, A. Avgoustidis, E. J. Copeland and A. Moss, 'CMB constraints on cosmic strings and superstrings,' Phys. Rev. D 93 , no.12, 123503 (2016) doi:10.1103/PhysRevD.93.123503 [arXiv:1603.01275 [astro-ph.CO]];</list_item>
<list_item><location><page_6><loc_12><loc_59><loc_49><loc_64></location>C. Dvorkin, M. Wyman and W. Hu, 'Cosmic String constraints from WMAP and the South Pole Telescope', Phys. Rev. D 84 , 123519 (2011) [arXiv:1109.4947 [astro-ph.CO]];</list_item>
<list_item><location><page_6><loc_12><loc_54><loc_49><loc_59></location>P. A. R. Ade et al. [Planck Collaboration], 'Planck 2013 results. XXV. Searches for cosmic strings and other topological defects', Astron. Astrophys. 571 , A25 (2014) [arXiv:1303.5085 [astro-ph.CO]].</list_item>
<list_item><location><page_6><loc_9><loc_47><loc_49><loc_53></location>[18] J. J. Blanco-Pillado, K. D. Olum and X. Siemens, 'New limits on cosmic strings from gravitational wave observation,' Phys. Lett. B 778 , 392 (2018) doi:10.1016/j.physletb.2018.01.050 [arXiv:1709.02434 [astro-ph.CO]];</list_item>
<list_item><location><page_6><loc_12><loc_39><loc_49><loc_47></location>Z. Arzoumanian et al. [NANOGRAV Collaboration], 'The NANOGrav 11-year Data Set: Pulsar-timing Constraints On The Stochastic Gravitational-wave Background,' Astrophys. J. 859 , no. 1, 47 (2018) doi:10.3847/1538-4357/aabd3b [arXiv:1801.02617 [astroph.HE]].</list_item>
<list_item><location><page_6><loc_9><loc_36><loc_49><loc_39></location>[19] J. Silk and A. Vilenkin, 'Cosmic Strings And Galaxy Formation', Phys. Rev. Lett. 53 , 1700 (1984);</list_item>
<list_item><location><page_6><loc_12><loc_31><loc_49><loc_36></location>M. J. Rees, 'Baryon concentrations in string wakes at z glyph[greaterorsimilar] 200: implications for galaxy formation and largescale structure', Mon. Not. Roy. Astron. Soc. 222 , 27 (1986);</list_item>
<list_item><location><page_6><loc_12><loc_27><loc_49><loc_31></location>T. Vachaspati, 'Cosmic Strings and the Large-Scale Structure of the Universe', Phys. Rev. Lett. 57 , 1655 (1986);</list_item>
<list_item><location><page_6><loc_12><loc_23><loc_49><loc_27></location>A. Stebbins, S. Veeraraghavan, R. H. Brandenberger, J. Silk and N. Turok, 'Cosmic String Wakes', Astrophys. J. 322 , 1 (1987).</list_item>
<list_item><location><page_6><loc_9><loc_18><loc_49><loc_23></location>[20] R. H. Brandenberger, R. J. Danos, O. F. Hernandez and G. P. Holder, 'The 21 cm Signature of Cosmic String Wakes,' JCAP 1012 , 028 (2010) doi:10.1088/14757516/2010/12/028 [arXiv:1006.2514 [astro-ph.CO]].</list_item>
<list_item><location><page_6><loc_9><loc_11><loc_49><loc_18></location>[21] D. Maibach, R. Brandenberger, D. Crichton and A. Refregier, 'Extracting the signal of cosmic string wakes from 21-cm observations,' Phys. Rev. D 104 , no.12, 123535 (2021) doi:10.1103/PhysRevD.104.123535 [arXiv:2107.07289 [astro-ph.CO]].</list_item>
<list_item><location><page_6><loc_9><loc_10><loc_49><loc_11></location>[22] R. J. Danos, R. H. Brandenberger and G. Holder, 'A</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_6><loc_55><loc_88><loc_92><loc_93></location>Signature of Cosmic Strings Wakes in the CMB Polarization,' Phys. Rev. D 82 , 023513 (2010) doi:10.1103/PhysRevD.82.023513 [arXiv:1003.0905 [astro-ph.CO]].</list_item>
<list_item><location><page_6><loc_52><loc_81><loc_92><loc_88></location>[23] M. Blamart, H. Fronenberg and R. Brandenberger, 'Signal of cosmic strings in cross-correlation of 21cm redshift and CMB polarization maps,' JCAP 11 , 012 (2022) doi:10.1088/1475-7516/2022/11/012 [arXiv:2205.02725 [astro-ph.CO]].</list_item>
<list_item><location><page_6><loc_52><loc_73><loc_92><loc_81></location>[24] D. C. Neves da Cunha, J. Harnois-Deraps, R. Brandenberger, A. Amara and A. Refregier, 'Dark Matter Distribution Induced by a Cosmic String Wake in the Nonlinear Regime,' Phys. Rev. D 98 , no.8, 083015 (2018) doi:10.1103/PhysRevD.98.083015 [arXiv:1804.00083 [astro-ph.CO]].</list_item>
<list_item><location><page_6><loc_52><loc_59><loc_92><loc_73></location>[25] A. Vilenkin, 'Cosmological Density Fluctuations Produced by Vacuum Strings', Phys. Rev. Lett. 46 , 1169 (1981) Erratum: [Phys. Rev. Lett. 46 , 1496 (1981)]. doi:10.1103/PhysRevLett.46.1169, 10.1103/PhysRevLett.46.1496; N. Turok and R. H. Brandenberger, 'Cosmic Strings And The Formation Of Galaxies And Clusters Of Galaxies', Phys. Rev. D 33 , 2175 (1986); H. Sato, 'Galaxy Formation by Cosmic Strings', Prog. Theor. Phys. 75 , 1342 (1986); A. Stebbins, 'Cosmic Strings and Cold Matter', Ap. J.</list_item>
<list_item><location><page_6><loc_55><loc_58><loc_71><loc_59></location>(Lett.) 303 , L21 (1986).</list_item>
<list_item><location><page_6><loc_52><loc_55><loc_92><loc_57></location>[26] H. Jiao, R. Brandenberger and A. Refregier, in preparation.</list_item>
<list_item><location><page_6><loc_52><loc_48><loc_92><loc_55></location>[27] S. F. Bramberger, R. H. Brandenberger, P. Jreidini and J. Quintin, 'Cosmic String Loops as the Seeds of Super-Massive Black Holes,' JCAP 1506 , no. 06, 007 (2015) doi:10.1088/1475-7516/2015/06/007 [arXiv:1503.02317 [astro-ph.CO]];</list_item>
<list_item><location><page_6><loc_55><loc_43><loc_92><loc_48></location>B. Cyr, H. Jiao and R. Brandenberger, 'Massive black holes at high redshifts from superconducting cosmic strings,' Mon. Not. Roy. Astron. Soc. 517 , no.2, 2221-2230 (2022) doi:10.1093/mnras/stac1939</list_item>
<list_item><location><page_6><loc_55><loc_42><loc_77><loc_43></location>[arXiv:2202.01799 [astro-ph.CO]].</list_item>
<list_item><location><page_6><loc_52><loc_35><loc_92><loc_41></location>[28] B. Cyr, H. Jiao and R. Brandenberger, 'Massive black holes at high redshifts from superconducting cosmic strings,' Mon. Not. Roy. Astron. Soc. 517 , no.2, 2221-2230 (2022) doi:10.1093/mnras/stac1939 [arXiv:2202.01799 [astro-ph.CO]].</list_item>
<list_item><location><page_6><loc_52><loc_30><loc_92><loc_35></location>[29] J. H. MacGibbon and R. H. Brandenberger, 'Gammaray signatures from ordicosmic strings,' Phys. Rev. D 47 , 2283 (1993) doi:10.1103/PhysRevD.47.2283 [astroph/9206003];</list_item>
<list_item><location><page_6><loc_55><loc_25><loc_92><loc_30></location>J. H. MacGibbon and R. H. Brandenberger, 'Highenergy neutrino flux from ordinary cosmic strings,' Nucl. Phys. B 331 , 153 (1990); doi:10.1016/05503213(90)90020-E</list_item>
<list_item><location><page_6><loc_55><loc_19><loc_92><loc_24></location>C. T. Hill, D. N. Schramm and T. P. Walker, 'Ultrahigh-Energy Cosmic Rays from Superconducting Cosmic Strings,' Phys. Rev. D 36 , 1007 (1987); doi:10.1103/PhysRevD.36.1007;</list_item>
<list_item><location><page_6><loc_55><loc_13><loc_92><loc_19></location>U. F. Wichoski, J. H. MacGibbon and R. H. Brandenberger, 'High-energy neutrinos, photons and cosmic ray fluxes from VHS cosmic strings,' Phys. Rev. D 65 , 063005 (2002) doi:10.1103/PhysRevD.65.063005 [hep-ph/9805419].</list_item>
<list_item><location><page_6><loc_52><loc_11><loc_87><loc_12></location>[30] A. Favero and R. Brandenberger, in preparation.</list_item>
</document>
[
{
"title": "Cosmic Strings from Thermal Inflation",
"content": "Robert Brandenberger 1, ∗ and Aline Favero 1, † 1 Department of Physics, McGill University, Montr´eal, QC, H3A 2T8, Canada (Dated: April 13, 2023) Thermal inflation was proposed as a mechanism to dilute the density of cosmological moduli. Thermal inflation is driven by a complex scalar field possessing a large vacuum expectation value and a very flat potential, called a 'flaton'. Such a model admits cosmic string solutions, and a network of such strings will inevitably form in the symmetry breaking phase transition at the end of the period of thermal inflation. We discuss the differences of these strings compared to the strings which form in the Abelian Higgs model. Specifically, we find that the upper bound on the symmetry breaking scale is parametrically lower than in the case of Abelian Higgs strings, and that the lower cutoff on the string loop distribution is determined by cusp annihilation rather than by gravitational radiation (for the value of the transition temperature proposed in the original work on thermal inflation).",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "Thermal inflation (not to be confused with warm inflation [1]) was proposed [2] as a mechanism to dilute the number density of unwanted moduli quanta. Moduli fields are predicted in many models beyond the particle physics Standard Model. In such models, moduli quanta are often left over from the Big Bang period, produced at the end of a phase of primordial inflation, or generated during a compactification phase transition. The proposal of [2] is to dilute the number density of moduli by invoking a period of inflation. This period should be sufficiently long to dilute the moduli, but short enough not to redshift the fluctuations which are generated in the primordial universe. Thermal inflation is generated by adding a new matter sector involving a complex scalar field 1 φ with a symmetry breaking potential V ( φ ), and gauging the resulting U (1) symmetry (this gauging is done since there is evidence that global symmetries are inconsistent with quantum gravity [4]). Thermal inflation is assumed to occur in the radiation phase of cosmology. Thermal effects are assumed to trap φ at the symmetric point φ = 0. At a temperature T i given by the potential energy of φ begins to dominate and inflation begins. The coupling of φ to the thermal bath generates a finite temperature contribution to the effective potential where g is the coupling constant describing the interactions between φ and the thermal bath. At a temperature T c when the positive contribution to the curvature of the potential at φ = 0 equals the absolute value of the (negative) curvature coming from the bare potential V ( φ ), symmetry breaking sets in, φ rolls to the bottom of its potential and thermal inflation ends. In order to generate the required hierarchy between T c and T i , it was assumed that the bare potential V ( φ ) contains no quartic term. The hierarchy between T i and T c determines the number N of e-foldings of thermal inflation Since the vacuum manifold of φ is S 1 , cosmic string defects inevitably form in the phase transition which ends thermal inflation [5] (see [6] for reviews of the role of cosmic strings in cosmology). As the scale of symmetry breaking for thermal inflation is assumed to be of the order m 0 ∼ 10 2 -10 3 GeV one might - based on intuition from Abelian Higgs strings [7] - have expected the signatures of these strings to be negligible. As we show here, thermal inflation strings have different properties compared to strings formed in the standard Abelian Higgs model. For the same value of the symmetry breaking temperature, thermal inflation strings have a parametrically larger mass per unit length than regular strings. Comparing strings with the same mass per unit length µ , thermal inflation strings have a parametrically greater width than regular strings. These differences affect the distribution of string loops, and hence the observational consequences of the strings (see e.g. [8] for a short review) need to be revisited. In this article we use natural units in which the speed of light, Planck's constant and Boltzmann's constant are set to 1. The Planck mass is denoted by m pl . The mass per unit length of a string will be denoted by µ , or often in terms of the dimensionless quantity Gµ , where G is Newton's gravitational constant.",
"pages": [
1
]
},
{
"title": "II. THERMAL INFLATION STRINGS",
"content": "The thermal inflation model [2] assumes a potential given by where V 0 is tuned such that the potential energy in the vacuum manifold vanishes. The λ n are dimensionless coupling constants, and the mass scale m 0 determines the (negative) curvature of the potential at the origin. We shall consider a simplified potential containing only the n = 1 term (the contributions from the terms with n > 1 are Planck suppressed for the questions we are asking, i.e. those involving small field values) 2 . Thus, we consider the potential This potential is to be compared with the potential for the Abelian Higgs model which is where η is the value of | φ | in the vacuum manifold, and λ AH is a dimensionless coupling constant. As we will see below, the hierarchy between T i and T c increases as m 0 decreases. To obtain an e-folding number N ∼ 10 of thermal inflation a value of m 0 ∼ 10 2 -10 3 GeV was suggested [2]. From (5) it immediately follows that the value η of | φ | which minimizes the potential is given by which is parametrically larger by a factor of m pl /m 0 than what is obtained for an Abelian Higgs string given the same value of the curvature of the potential at φ = 0. For the value of m 0 indicated above, η is of the order of 10 10 GeV and not 10 2 GeV is it would be for an Abelian Higgs string with the same value of m 0 . Demanding that the potential vanishes for | φ | = η yields which is also parametrically larger by a factor of m pl /m 0 compared to the corresponding result for the Abelian Higgs string (taking coupling constants to be of the order 1). This leads to the fact that the temperature T i corresponding to the onset of thermal inflation is parametrically larger than what one might have guessed from Abelian Higgs string intuition On the other hand, the temperature at which the symmetry breaking phase transition takes place is given by setting g to be of the order 1. Comparing (9) and (10) we see that it is precisely the enhancement factor discussed above which allows for a period of thermal inflation to take place. Let us now compare the width of a thermal inflation string with that of an Abelian Higgs string for the same value of the phase transition temperature T c . The width is determined by minimizing the sum of the potential and gradient energy terms. Increasing the width of the string costs potential energy while decreasing the width leads to an increase of the gradient energy. For a straight string centered at r = 0 (in polar coordinates), the field configuration of a string with winding number 1 can be written as where the profile function f ( r ) increases from f (0) = 0 to f ( r ) = 1 for r > w . The potential energy µ p per unit length of the string can hence be estimated to be The scalar field angular gradient energy for a local string is cancelled by the gauge fields beyond a radius r A which is set by the gauge field mass. For r < w the angular gradient energy decays since f ( r ) decays as r decreases. Hence, the mass per unit length µ a from gradients can be estimated as It then follows from (13) that Hence, by minimizing the sum of potential and angular gradient energy (to first approximation the radial gradient energy does not depend on w ) it follows that For Abelian Higgs strings this yields while for thermal inflation strings In terms of the phase transition temperature, the widths of thermal inflation strings and Abelian Higgs strings are of the same order of magnitude. However, in terms of the mass per unit length there is a parametric difference, and this difference will have important implications for the string loop distribution. From energy equipartition, the energy per unit length µ of a string is given by For Abelian Higgs strings this yields while for thermal inflation strings Thus, for a fixed mass per unit length, a thermal inflation string has a width which is much greater than the width of an Abelian Higgs string. Comparing (19) and (20), we see that for fixed symmetry breaking scale of T c ∼ m 0 ∼ 10 2 GeV, Abelian Higgs strings would have a mass per unit length of Gµ AB ∼ 10 -34 which is many orders of magnitude smaller than the range of valiues of Gµ which can have interesting cosmological effects. In the case of thermal inflation strings, on the other hand, for the same value of T c we obtain Gµ ∼ 10 -17 which is now approaching the range which is of interest for string signals in cosmological observations.",
"pages": [
2,
3
]
},
{
"title": "III. THERMAL STRING LOOP DISTRIBUTION",
"content": "The parametric enhancement of the width of a thermal inflation string compared to the width of an Abelian Higgs string with the same mass per unit length has important implications for the loop distribution. The causality argument [5] which implies that the distribution of long strings (strings with a curvature radius comparable and greater than the Hubble radius) takes on a scaling solution where the number of long string segments crossing any given Hubble volume is independent of time applies equally to Abelian Higgs and thermal inflation strings. This scaling solution of the long string network is maintained by string loop production. Like for Abelian Higgs strings, we can assume that the one scale loop production model [9] also applies to thermal inflation strings, implying that at time t loops are produced with radius R = αt , where α is a constant which can be normalized by string evolution simulations which yield α ∼ 10 -1 . Once produced, the number density n ( R,t ) of loops in the radius interval between R and R + dR redshifts. Thus at times t after the time t eq of equal matter and radiation we have where N is a constant determined by the number of long string segments per Hubble volume. R co is a cutoff radius below which loops live for less than one Hubble time, and whose consequences for cosmology can be neglected. For non-superconducting strings there are two main mechanisms by which string loops decay. The first is gravitational radiation: string loops have relativistic tension and hence oscillate and emit gravitational radiation. The power of gravitational radiation from a string loop of radius R is [10] where γ is a constant of the order 10 2 . Gravitational radiation implies that loops with radius R < R g where will live less than one Hubble expansion time, and hence their cosmological effects are negligible. Cusp evaporation is a second decay mechanism [11]. A cusp is a point on the string which moves at the speed of light. Strings have finite width, and around the cusp point the string segments on either side of the cusp point overlap for a region of length 3 Locally the cusp region looks like a string-antistring pair, and there is no topology protecting the cusp region against annihilation into gauge and scalar field quanta. It can be proven that string loops described by the effective Nambu-Goto action have at least one cusp per oscillation time [13]. Hence, the power of the cusp annihilation process is From the above equation it follows that, due to cusp evaporation, string loops with radius less than live for less than one Hubble expansion time. The cutoff radius R co in the loop distribution of (23) is the larger of R g and R c . Comparing the strengths of gravitational radiation power (24) and cusp annihilation power (27) we see that the parametrically larger width of a thermal inflation string (for a given mass per unit length) will lead to a parametric amplification of the role of cusp annihilation compared to gravitational wave decay. We also see that the relative importance of cusp annihilation increases the lower the value of Gµ is. The condition for cusp annihilation to dominate over gravitational radiation is R c > R g or For Abelian Higgs strings this yields or, expressed in terms of the critical temperature T c On the other hand, for thermal inflation strings we obtain or, after expressing µ in terms of the critical temperature Comparing these expressions we see that, for a fixed value of the string tension, cusp annihilation is more important for thermal inflation strings than for Abelian Higgs strings. On the other hand, fixing T c we see that the importance of cusp annihilation is, maybe surprisingly, less than for Abelian Higgs strings. Evaluating (31) and (33) at the temperature T eq ∼ 1eV of equal matter and radiation (the temperature relevant for cosmological signatures of strings), we see that for Abelian Higgs strings the cutoff in the loop distribution is determined by cusp annihilation for values T c < 10 10 GeV while for thermal inflation strings it is for values T c < 10 8 GeV. In particular, for the value m 0 ∼ 10 2 -10 3 GeV assumed in the original thermal inflation paper [2], we conclude that the cutoff in the loop distribution is given by the cusp annihilation process.",
"pages": [
3,
4
]
},
{
"title": "IV. CONSTRAINTS FROM COSMOLOGICAL OBSERVATIONS",
"content": "In this section we study what constraints on the symmetry breaking scale m 0 of thermal inflation can be derived from cosmological observations. Cosmic strings leave behind interesting signals in many observational windows. In most cases, the effects are gravitational, and hence the magnitude of the string signal depends on Gµ . In terms of Gµ , the strength of the signals will hence be the same for Abelian Higgs and thermal inflation strings. However, since the relation between T c and µ is different, the magnitude of the string signals as a function of T c will change. As the string network consists of both long strings and loops, each will induce specific signatures. We first turn to signatures of the long string network. For example, long strings lead to line discontinuities in cosmic microwave background (CMB) anisotropy maps [14]. This is due to the fact that a long straight string produces a conical deformation of the metric with deficit angle proportional to Gµ [15].The magnitude of this signal depends on Gµ . The study of [16] shows that experiments with the specifications of the South Pole Telescope or the Atacama Cosmology Telescope can constrain the string tension to be A slighty weaker bound of can be derived from the angular power spectrum of CMB anisotropies [17]. The resulting bound on T c for thermal inflation strings is parametrically stronger than for Abelian Higgs strings, namely This bound is obviously satisfied for the value T c ∼ m 0 = 10 2 -10 3 GeV assumed in [2]. For Abelian Higgs strings there is a tighter bound of which comes from the upper bound on the stochastic background of gravitational waves from pulsar timing array measurements [18]. This bound depends on having a scaling distribution of loops down to the gravitational radiation cutoff R g . This bound remains valid for thermal inflation strings since, as the discussion at the end of the previous section showed, cusp annihilation only changes the loop distribution for values of Gµ which are lower than the above bound. Long strings moving through space produce overdense regions in their wake [19]. CMB photons passing through these wakes get absorbed at the 21-cm wavelength. Long cosmic strings hence lead to distinct signals in high redshift 21-cm surveys [20]: wedges of absorption in 21-cm redshift maps which are extended in the angular directions and narrow in redshift direction. The study of [21] shows that the string signal can be detected by surveys such as the MWA telescope down to a value of Gµ comparable to that of (34), and prospects indicate that with better analysis tools a significant improvement of this bound can be expected. String wakes also lead to rectangles in the sky with induced CMB polarization (including a B-mode component) [22]. From the analysis of [23] it appears, however, that this signal is harder to extract from observations than the 21-cm signal. At lower redshifts, string wakes also lead to planar overdensities of galaxies. These are, however, disrupted by the gravitational effects of the dominant source of fluctuations [24]. Turning now to signatures of cosmic string loops, the gravitational signals are the same for Abelian Higgs strings and thermal inflation strings given the same value of Gµ . String loops seed nonlinear structures by gravitational accretion [25]. Given the bounds on Gµ discussed above, strings can only play a subdominant role in explaining the nonlinear structures today. However, since strings form nonlinear seeds immediately after their formation, they will dominate the halo mass function at sufficiently early times [26]. They can provide seeds for intermediate and super-massive black holes at high redshifts [27]. It has recently been shown [28] that for superconducting cosmic strings the 'Direct Collapse Black Hole' criteria can be satisifed, and that such loops indeed could explain the origin of the observed high redshift super-massive black holes. Since thermal inflation strings have a greater width than Abelian Higgs strings for a fixed value of T c , nongravitational signals from thermal inflation strings may differ from those of Abelian Higgs strings. Specifically, the flux of cosmic rays [29] due to cosmic strings will be larger [30].",
"pages": [
4,
5
]
},
{
"title": "V. CONCLUSIONS AND DISCUSSION",
"content": "We have pointed out that thermal inflation models lead to the production of a network of cosmic strings. These thermal inflation strings have different poperties compared to strings arising in the Abelian Higgs model. Specifically, for a fixed phase transition temperature, thermal inflation strings have a larger mass per unit length, and hence lead to larger gravitational effects. For thermal inflation strings, the upper bound on the phase transition temperature from cosmological observations are parametrically more stringent than for Abelian Higgs strings. However, for the value of the symmetry breaking scale suggested in [2], the bounds are satisfied.",
"pages": [
5
]
},
{
"title": "Acknowledgement",
"content": "RB wishes to thank the Pauli Center and the Institutes of Theoretical Physics and of Particle- and Astrophysics of the ETH for hospitality. The research at McGill is supported in part by funds from NSERC and from the Canada Research Chair program.",
"pages": [
5
]
}
]
2024arXiv240109685A
https://arxiv.org/pdf/2401.09685.pdf
<document>
<section_header_level_1><location><page_1><loc_23><loc_78><loc_77><loc_81></location>Decades of Transformation: Evolution of the NASA Astrophysics Data System's Infrastructure</section_header_level_1>
<section_header_level_1><location><page_1><loc_23><loc_74><loc_37><loc_75></location>Alberto Accomazzi</section_header_level_1>
<text><location><page_1><loc_23><loc_70><loc_75><loc_73></location>Center for Astrophysics | Harvard & Smithsonian, Cambridge, MA, USA; aaccomazzi@cfa.harvard.edu</text>
<text><location><page_1><loc_23><loc_60><loc_79><loc_68></location>Abstract. The NASA Astrophysics Data System (ADS) is the primary Digital Library portal for researchers in astronomy and astrophysics. Over the past 30 years, the ADS has gone from being an astronomy-focused bibliographic database to an open digital library system supporting research in space and (soon) earth sciences. This paper describes the evolution of the ADS system, its capabilities, and the technological infrastructure underpinning it.</text>
<text><location><page_1><loc_23><loc_49><loc_79><loc_60></location>We give an overview of the ADS's original architecture, constructed primarily around simple database models. This bespoke system allowed for the e ffi cient indexing of metadata and citations, the digitization and archival of full-text articles, and the rapid development of discipline-specific capabilities running on commodity hardware. The move towards a cloud-based microservices architecture and an open-source search engine in the late 2010s marked a significant shift, bringing full-text search capabilities, a modern API, higher uptime, more reliable data retrieval, and integration of advanced visualizations and analytics.</text>
<text><location><page_1><loc_23><loc_41><loc_79><loc_49></location>Another crucial evolution came with the gradual and ongoing incorporation of Machine Learning and Natural Language Processing algorithms in our data pipelines. Originally used for information extraction and classification tasks, NLP and ML techniques are now being developed to improve metadata enrichment, search, notifications, and recommendations. we describe how these computational techniques are being embedded into our software infrastructure, the challenges faced, and the benefits reaped.</text>
<text><location><page_1><loc_23><loc_36><loc_79><loc_41></location>Finally, we conclude by describing the future prospects of ADS and its ongoing expansion, discussing the challenges of managing an interdisciplinary information system in the era of AI and Open Science, where information is abundant, technology is transformative, but their trustworthiness can be elusive.</text>
<section_header_level_1><location><page_1><loc_18><loc_29><loc_30><loc_30></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_18><loc_17><loc_79><loc_27></location>The ADS as it is known today consists of a literature centric system connected to a variety of research products and enhanced by a number of services. Originally, this functionality was implemented as a part of a larger networked system which was designed to allow for the discovery, retrieval, and analysis of astrophysical data (Murray et al. 1992). That larger system was abandoned in the mid-90s in favor of a distributed set of web-based archives, but the literature component (named the 'ADS Abstract Service') survived, and later thrived, on its own.</text>
<text><location><page_1><loc_18><loc_11><loc_79><loc_16></location>The early literature indexing system that was developed by a small team at the Smithsonian Astrophysical Observatory in the early nineties has grown to become a discovery system which is now an essential tool not only for astronomers but for an increasing number of researchers in the space sciences. Today, an expanded version</text>
<text><location><page_2><loc_18><loc_79><loc_79><loc_86></location>of ADS is becoming the centerpiece of NASA's open science initiatives which aim to promote transparency and access to the research e ff orts funded by the agency. This paper provides an historical overview of the growth of the ADS as a technological system, focusing on the inflection points in software development which informed its evolution.</text>
<section_header_level_1><location><page_2><loc_18><loc_74><loc_38><loc_75></location>2. Early Days: 1992-2007</section_header_level_1>
<text><location><page_2><loc_18><loc_58><loc_79><loc_72></location>Because of the requirements involved in gathering, processing, and indexing textual data, most of the software used to create and manage the ADS Abstract database and user interface had to be developed from scratch. Relational SQL-based databases were already popular in the early 1990s, but their limitations in handling text fields meant that we could not store or index the text of an average abstract in a record. Additionally, the costs associated with purchasing access licenses for the database were significant, so the decision was made to use a file-based system for storing bibliographic records using UNIX-based servers (Sun / SunOS). This also meant that we had to design a system for indexing and then retrieving the content, thus rolling our own search engine.</text>
<text><location><page_2><loc_18><loc_49><loc_79><loc_58></location>The advantage of using our own search system proved important when it came to enhancing the system with functionality provided by external services. In 1993, less than one year after the introduction of the Abstract Service, we implemented the capability to perform object searches via a federated query to the SIMBAD database (Eichhorn 1993), developed by the CDS in Strasbourg. To our knowledge, this was the first implementation of an inter-continental networked query as part of a search service.</text>
<section_header_level_1><location><page_2><loc_18><loc_45><loc_48><loc_47></location>2.1. The Era of Open Source Software</section_header_level_1>
<text><location><page_2><loc_18><loc_29><loc_79><loc_44></location>As for many similar computing-intensive projects, the lack of existing software meant that the team had to develop its own. The available tools consisted of programming languages and compilers that were either part of the UNIX operating system or were available as free software, which was starting to become popular in academic and research circles. Given our needs, we focused on developing the search engine in C and the ingest pipelines in PERL, which was then considered as the most flexible language available for processing text data. Workflows and pipelines were strung together using the components provided to us by the operating system itself: UNIX pipes, interprocess communication, shell error handling, and text processing tools such as look, sort, join, uniq.</text>
<text><location><page_2><loc_18><loc_11><loc_79><loc_29></location>The advent of the World Wide Web signified an inflection point for the project. As early as 1993 it became clear that the Web was going to be the best framework for delivering the services that we were developing, and the availability of free software implementing the HTTP protocol and web browsers lead to the rapid adoption of webbased technologies everywhere. Up until this point access to the ADS Abstract Service required a dedicated X-windows application that used a proprietary protocol to connect to the server hosting the database. Removing this last obstacle meant that the ADS infrastructure was able to run on commodity hardware, using open source code, and open source client-server applications. The hyperlinked nature of the web also meant that the ADS user interface could be quickly developed, enhanced, and deployed without the need for any software upgrade. Figure 1 shows the ADS abstract service query form circa 1994.</text>
<figure>
<location><page_3><loc_19><loc_30><loc_78><loc_86></location>
<caption>Figure 1. The beloved ADS abstract service query form, featuring over 100 search parameter settings in one single web page.</caption>
</figure>
<text><location><page_3><loc_18><loc_11><loc_79><loc_21></location>The introduction of the GNU software suite, the Linux operating system, and the larger open source ecosystem accelerated the development and availability of tools and libraries that became useful to the project, in particular the implementation of TIFF and PNG graphic formats for our full-text document handling, the Berkeley DB system for on-disk database implementations, and POSIX threads for parallelization of tasks. Adopting the technology stack built around the open source movement meant that we had full control of our destiny, and that we could develop quickly and independently.</text>
<section_header_level_1><location><page_4><loc_18><loc_85><loc_41><loc_86></location>2.2. The ADS Coming of Age</section_header_level_1>
<text><location><page_4><loc_18><loc_69><loc_79><loc_83></location>Having identified the environment and platform for our system, the ADS team, which at the time was composed of 4 FTEs, was able to develop components of the web-based Abstract Service and, starting in 1995, the Article Service, which provided access to the full-text scans of the major astronomical journals (Accomazzi et al. 1995). Shortly after that, in 1996, we introduced links between bibliographic records, datasets and catalogs (Accomazzi et al. 1997). In 1997 citation data which was originally purchased from the Institute for Scientific Information by the AAS was indexed in our system (Kurtz et al. 1996), allowing users to traverse the citation graph from paper to paper, as well as to sort any search results by citation counts.</text>
<text><location><page_4><loc_18><loc_56><loc_79><loc_69></location>By 1998 astronomers were hooked. A study by Kurtz (Kurtz et al. (1999)) showed that online readership through the ADS surpassed the worldwide print readership in all astronomy institutions. The availability of an increasing amount of full-text publications in the ADS and the ability for any scientist to find them and retrieve them easily and freely lead to the massive clearing of shelves from the o ffi ces of most astronomers worldwide. The embrace of the ADS platform by the community had a positive feedback e ff ect on the evolution of the system. Publishers started hearing from their editors and referees that if their content was not in ADS, it may as well not exist, which helped the project obtain metadata and back-records of historical content in the system.</text>
<text><location><page_4><loc_18><loc_42><loc_79><loc_55></location>The resounding success and adoption of the ADS in the late 1990s provided additional opportunities for the project to gain additional recognition and support from NASA, which led to an increase in sta ff (6 FTEs) and funds to continue the development and operations of the system. The following decade was one during which we developed system capabilities to ensure that our system was up-to-date, resilient and accessible. This led to the creation of a pipeline to automatically extract and ingest citations from the literature (Demleitner et al. 1999), a workflow to scan and digitize content on an on-going basis, and the development of a network of mirror sites hosted by collaborating institutions throughout the world (Accomazzi et al. 2000).</text>
<text><location><page_4><loc_18><loc_28><loc_79><loc_41></location>Additional features and services were also developed during the first decade of the 2000s. The myADS notification service was launched in 2003, providing customized digests of new publications based on the preferences of individual researchers. The original weekly updates of astronomy content from arXiv became a daily update of all the subject categories available in the preprint server. Customizations were added allowing users to save bibliographic articles in virtual private collections ('ADS libraries'), configure access to publisher content via their library's subscriptions (based on the OpenURL protocol), and customize their search experience through a set of preferences tied to their login account.</text>
<text><location><page_4><loc_18><loc_15><loc_79><loc_27></location>By 2007, the ADS Abstract and Article services provided the unparalleled capabilities of any research portal in any discipline. ADS usage in 2007 shows that there were over 20,000 registered users (with login accounts), and 6,000 subscribers to the myADS notification system. Monthly usage showed that we had more than 30,000 heavy users out of an overall total of 1.2M users. Given the number of professional astronomers on earth, it was clear that the system was being used not just by the totality of researchers in the discipline, but also by people outside of astronomy as well as amateurs and the general public (Henneken et al. 2009).</text>
<text><location><page_4><loc_18><loc_11><loc_80><loc_15></location>The system architecture grew from one which consisted of a single abstract database to a complex set of data products and relationships, as illustrated in Figure 2. Greater functionality brought along greater complexity of the code, which had been growing</text>
<text><location><page_5><loc_18><loc_80><loc_79><loc_86></location>organically for the previous 15 years. A census of our codebase conducted in 2007 showed that our search engine had grown to consist of 250K lines of C, while our indexing and application software consisted of another 250K lines of C, PERL and python 2 code. All of it custom-built and scarcely documented.</text>
<figure>
<location><page_5><loc_19><loc_47><loc_77><loc_77></location>
<caption>Figure 2. The ADS system architecture circa 2007. By this time the ADS system had grown into a complex set of custom-built modules and workflows.</caption>
</figure>
<section_header_level_1><location><page_5><loc_18><loc_35><loc_39><loc_37></location>3. ADS Reborn:2008-2020</section_header_level_1>
<text><location><page_5><loc_18><loc_23><loc_79><loc_33></location>In 2007, the departure of Guenther Eichhorn, project scientist and key developer of the ADS search engine, caused a major disruption in the small team of 6 employees, and forced the project to reconsider its technical infrastructure and future development strategies. It became immediately apparent that the di ffi culty of maintaining and enhancing undocumented legacy code was not a wise nor sustainable path for the project, and that digital library technologies had matured to the point that open source alternatives should be considered instead.</text>
<text><location><page_5><loc_18><loc_11><loc_79><loc_23></location>The task before us was not small: keep the ADS running 24 / 7 while rebuilding, more or less from scratch, a new system with equal or better performance and functionality. Drawing from our recent experience with system development, an additional self-imposed requirement was to develop such a system in the open, reusing as much as possible existing technologies and seeking strategic partnerships whenever appropriate. Thankfully NASA was receptive to our plea for help and supported the plan we submitted to the Astrophysics Archives review in 2008, which led to an incremental doubling in ADS sta ffi ng over the next decade.</text>
<section_header_level_1><location><page_6><loc_30><loc_88><loc_38><loc_90></location>Accomazzi</section_header_level_1>
<text><location><page_6><loc_18><loc_74><loc_79><loc_86></location>Starting in 2009, we were able to start hiring new developers and experiment with a variety of technologies to inform our future architecture. The initial phase of this effort involved leveraging the existing search engine to document its capabilities and explore the e ff ort required in implementing a new user interface. A pivotal shift occurred with the adoption of Apache Solr, a widely recognized open-source search platform, to manage the backend metadata indexing. This strategic change was implemented in 2013, marking a critical step in modernizing the system's architecture and setting the foundation for our search infrastructure.</text>
<text><location><page_6><loc_18><loc_62><loc_79><loc_74></location>Further advancement was made in 2015 with the development and launch of a custom JavaScript application, internally codenamed 'Bumblebee.' This application represented a major forward leap in terms of user interface and experience. It was during this phase that cloud computing was integrated into the system's architecture, signifying a move towards more scalable and flexible infrastructure. The cloud-based aspect of the architecture was particularly notable, with the JavaScript application utilizing a JSON API hosted in the cloud, thereby enhancing data accessibility and system responsiveness.</text>
<text><location><page_6><loc_18><loc_51><loc_79><loc_61></location>The subsequent phase focused on expanding the system's capabilities in order to reach feature parity with the ADS Classic system and to incorporate the evolving needs of users. One of the key enhancements was the integration of ORCID claiming, responding to the growing necessity for interoperability with academic and research author identification systems. Alongside this, considerable e ff orts were devoted to backend infrastructure development, ensuring the robustness and reliability required to support new functionalities and user demands.</text>
<text><location><page_6><loc_18><loc_37><loc_79><loc_50></location>By 2019, this comprehensive developmental journey culminated in achieving and even surpassing feature parity with the original ADS system. This milestone was not just about matching the existing functionality but also about providing an enhanced, modern platform that could support the evolving requirements of users in a more e ffi -cient, scalable, and sustainable way. The new system featured a search engine which integrated high-performance text search with citation and usage graphs; a well-structured and well-documented JSON API; and a modern user interface featuring a number of visualizations and analytics (Chyla et al. 2015). The corresponding architecture is illustrated in Figure 3.</text>
<section_header_level_1><location><page_6><loc_18><loc_31><loc_39><loc_32></location>4. Future ADS: 2021-2030</section_header_level_1>
<text><location><page_6><loc_18><loc_17><loc_79><loc_29></location>In 2019, the NASA Science Mission Directorate (SMD) published a white paper detailing a strategy for data and computing which included support for open science initiatives 1 . One of the goals of the strategic plan was to provide support for open science initiatives through the creation of an interdisciplinary literature portal that could be used to understand how NASA data is used and to provide services to the research community across the disciplines funded by NASA SMD. It became immediately clear that ADShad been fulfilling these goals for the Astrophysics community, so NASA selected the ADS team to expand its mission to include all five SMD disciplines: Astrophysics,</text>
<figure>
<location><page_7><loc_19><loc_62><loc_78><loc_86></location>
<caption>Figure 3. The current (and still evolving) ADS architecture consists of cloudbased microservices, a JSON API, and javascript user interface along with and onpremises pipelines which are still integrating some existing legacy services.</caption>
</figure>
<text><location><page_7><loc_18><loc_51><loc_79><loc_54></location>Planetary Science, Heliophysics, Earth Science, and Biological and Physical Sciences. The name of the expanded system is the NASA Science Explorer 2 , or SciX for short.</text>
<text><location><page_7><loc_18><loc_35><loc_79><loc_50></location>This expansion e ff ort is transformative for the project, which will go from being a literature database focusing on a single research community to a multidisciplinary platform used by a much larger community of space, earth, and biological scientists. While much of the system infrastructure can be scaled up for additional content and use, a lot of the curation workflows and associated pipelines will need to be adapted to deal with new data providers, a new set of science data archives, and a much larger community of researchers with varied cultural backgrounds and habits. Most importantly, the ADS has become an essential research tool thanks to its role at the center of a nexus of archives and information services in astronomy. Becoming part of similar ecosystems in new disciplines will be a major challenge.</text>
<text><location><page_7><loc_18><loc_20><loc_79><loc_35></location>From a technical perspective, one of the things we are betting on is that we can use emerging technologies - specifically AI and machine learning - to help us perform a lot of the activities that traditionally we have been doing through human curation, for example metadata aggregation and enrichment. As an example, we want to automatically assign concepts drawn from the Unified Astronomy Thesaurus to all the records in our astronomy collection, or extract planetary feature names from planetary science papers. And part of doing this as our contribution to open science e ff orts means that we are not only building and delivering an AI-enhanced service, but we will also be generating and sharing the underlying training datasets and open source code that we hope will be used by all of the communities that we serve.</text>
<text><location><page_7><loc_18><loc_15><loc_79><loc_19></location>ADS already has a long tradition of openness, and with the expansion into SciX we will redouble our e ff orts to share our e ff orts with the larger research community. In 2021 we created and released a custom language model built on the astronomical</text>
<text><location><page_8><loc_18><loc_74><loc_79><loc_86></location>literature called astroBERT (Grezes et al. 2021). We have contributed data sets for the 2022 and 2023 data challenges at the first and second Workshops on Information Extraction from Scholarly Papers 3 . In 2023 we have started working with the UniverseTBD collaboration 4 , which recently released the first fine-tuned version of the popular LLaMA-2 model, named AstroLLaMA (Dung Nguyen et al. 2023). We have also participated in the creation of a Large Language Model (LLM) developed by the NASA Science Mission Directorate in partnership with IBM which will be released in 2024.</text>
<text><location><page_8><loc_18><loc_57><loc_79><loc_74></location>The future is uncertain but most of us will agree that AI will play a major role in the development of digital scholarship. The technologies that are most likely to revolutionize the way we interact with the scientific literature are LLMs and Knowledge Graphs (KGs). While LLM-powered chatbots have gotten the most attention from the general public, the adaptability of LLMs make them general purpose tools for a variety of Natural Language Processing tasks, such as structured information extraction, named entity recognition, and metadata enrichment. Knowledge Graphs can be similarly used to support information retrieval, semantic search, and metadata normalization. For an example of how ADS started using LLMs and KGs in its pipelines, see Shapurian et al. (2023). We expect that this approach will be fundamental in developing SciX into a fully featured interdisciplinary system.</text>
<section_header_level_1><location><page_8><loc_18><loc_53><loc_29><loc_54></location>5. Discussion</section_header_level_1>
<text><location><page_8><loc_18><loc_39><loc_79><loc_51></location>Along with the promise of an exciting future, the latest AI technologies bring with them a lot of questions related to trust. Today's LLMs are essentially black boxes, deep networks of billions and trillions of parameters trained on data of various quality, often in non-transparent ways. These systems are really too large for anyone to inspect them in any detail, but they rather seem to work in mysterious, if not magical, ways. However, as scientists, we have all been trained to reject 'magic' and instead study complex systems in order to understand how they work, then modify their environment and behavior in order to control and adapt them to our needs.</text>
<text><location><page_8><loc_18><loc_20><loc_79><loc_39></location>This is the task before us: use the body of knowledge generated by the scientific process to create AI technologies that advance knowledge and insight into the physical world. The ADS team and the community at large have begun investigating the use of open source LLMs for information retrieval and reasoning (Blanco-Cuaresma et al. 2024; Ciuc˘a & Ting 2023). While strategies based on an Retrieval Augmented Generation approach seems the most promising right now, these are still early days and there is a lot to be explored. Given the increasing number of open-source LLMs being generated, one interesting scenario for future development may be one in which there are a few open source LLMs being fine-tuned for specific tasks and domains, using custom curated datasets. Under this scenario, ADS and SciX would be the authoritative source of data used to train LLMs and build KGs used in the earth and space sciences to ensure their trustworthiness and completeness.</text>
<text><location><page_8><loc_18><loc_17><loc_79><loc_20></location>ADS has been a transformative service for astronomers, and it's likely to be as transformative for a larger group of earth and space scientists in the near future. Its</text>
<text><location><page_9><loc_18><loc_77><loc_79><loc_86></location>success couldn't have been possible without the support of NASA and the existence of a larger ecosystem of open, interoperable information services within astronomy. The now universal support for open science initiatives gives us an opportunity to extend the ADS model to a wider set of disciplines, and the continued development of open source code and models means that there is still a bright future ahead for the scientific enterprise.</text>
<text><location><page_9><loc_18><loc_59><loc_79><loc_76></location>Acknowledgments. The ADS would not exist without NASA's continued support over the past 30 years. We are grateful to the agency for making it possible for the system to flourish and grow into its current form. The ADS team today is composed of 20 talented individuals 5 , soon to become 30 FTEs as part of our expansion. We have all benefited from the work of those who came before us and provided the vision and focus that made the ADS indispensable in its early days. There are too many names to mention, but three stand above the rest: Steve Murray, founder of the ADS and PI until his passing in 2015; Michael Kurtz, the ADS Project Scientist and visionary who is still contributing to our ongoing e ff orts; and Guenther Eichhorn, who managed the project until 2007 when he left the Center for Astrophysics. Today's ADS stands on the shoulders of these giants, without which none of this would have been possible.</text>
<section_header_level_1><location><page_9><loc_18><loc_55><loc_25><loc_56></location>References</section_header_level_1>
<text><location><page_9><loc_18><loc_51><loc_79><loc_53></location>Accomazzi, A., Eichhorn, G., Grant, C. S., Murray, S. S., & Kurtz, M. J. 1995, Vistas in Astronomy, 39, 63</text>
<text><location><page_9><loc_18><loc_47><loc_79><loc_51></location>Accomazzi, A., Eichhorn, G., Kurtz, M. J., Grant, C. S., & Murray, S. S. 1997, in Astronomical Data Analysis Software and Systems VI, edited by G. Hunt, & H. Payne, vol. 125 of Astronomical Society of the Pacific Conference Series, 357</text>
<text><location><page_9><loc_18><loc_45><loc_34><loc_47></location>-2000, A&AS, 143, 85.</text>
<text><location><page_9><loc_35><loc_45><loc_49><loc_47></location>astro-ph/0002105</text>
<text><location><page_9><loc_18><loc_40><loc_79><loc_45></location>Blanco-Cuaresma, S., Accomazzi, A., Kurtz, M. J., Henneken, E., Lockhart, K. E., Grezes, F., Allen, T., Shapurian, G., Grant, C. S., Thompson, D. M., Hostetler, T. W., Templeton, M. R., Chen, S., Koch, J., Jacovich, T., Chivvis, D., de Macedo Alves, F., Paquin, J.-C., Batlett, J., Polimera, M., & Jarmak, S. 2024, these proceedings</text>
<text><location><page_9><loc_18><loc_33><loc_79><loc_40></location>Chyla, R., Accomazzi, A., Holachek, A., Grant, C. S., Elliott, J., Henneken, E. A., Thompson, D. M., Kurtz, M. J., Murray, S. S., & Sudilovsky, V. 2015, in Astronomical Data Analysis Software an Systems XXIV (ADASS XXIV), edited by A. R. Taylor, & E. Rosolowsky, vol. 495 of Astronomical Society of the Pacific Conference Series, 401. 1503.05881</text>
<text><location><page_9><loc_18><loc_31><loc_79><loc_33></location>Ciuc˘a, I., & Ting, Y.-S. 2023, Research Notes of the American Astronomical Society, 7, 193. 2304.05406</text>
<text><location><page_9><loc_18><loc_26><loc_79><loc_30></location>Demleitner, M., Accomazzi, A., Eichhorn, G., Grant, C. S., Kurtz, M. J., & Murray, S. S. 1999, in American Astronomical Society Meeting Abstracts, vol. 195 of American Astronomical Society Meeting Abstracts, 82.09</text>
<text><location><page_9><loc_18><loc_20><loc_79><loc_26></location>Dung Nguyen, T., Ting, Y.-S., Ciuc˘a, I., O'Neill, C., Sun, Z.-C., Jabło'nska, M., Kruk, S., Perkowski, E., Miller, J., Li, J., Peek, J., Iyer, K., Ró˙za'nski, T., Khetarpal, P., Zaman, S., Brodrick, D., Rodríguez Méndez, S. J., Bui, T., Goodman, A., Accomazzi, A., Naiman, J., Cranney, J., Schawinski, K., & UniverseTBD 2023, arXiv e-prints, arXiv:2309.06126. 2309.06126</text>
<text><location><page_9><loc_18><loc_18><loc_42><loc_19></location>Eichhorn, G. 1993, JAAVSO, 22, 136</text>
<text><location><page_9><loc_18><loc_16><loc_79><loc_18></location>Grezes, F., Blanco-Cuaresma, S., Accomazzi, A., Kurtz, M. J., Shapurian, G., Henneken, E., Grant, C. S., Thompson, D. M., Chyla, R., McDonald, S., Hostetler, T. W., Templeton,</text>
<text><location><page_10><loc_22><loc_83><loc_79><loc_86></location>M. R., Lockhart, K. E., Martinovic, N., Chen, S., Tanner, C., & Protopapas, P. 2021, arXiv e-prints, arXiv:2112.00590. 2112.00590</text>
<text><location><page_10><loc_18><loc_81><loc_79><loc_83></location>Henneken, E. A., Kurtz, M. J., Accomazzi, A., Grant, C. S., Thompson, D., Bohlen, E., & Murray, S. S. 2009, Journal of Informetrics, 3, 1. 0808.0103</text>
<text><location><page_10><loc_18><loc_78><loc_79><loc_80></location>Kurtz, M. J., Eichhorn, G., Accomazzi, A., Grant, C. S., Demleitner, M., & Murray, S. S. 1999, D-Lib Magazine, 5</text>
<text><location><page_10><loc_18><loc_74><loc_79><loc_78></location>Kurtz, M. J., Eichhorn, G., Accomazzi, A., Grant, C. S., & Murray, S. S. 1996, in American Astronomical Society Meeting Abstracts, vol. 189 of American Astronomical Society Meeting Abstracts, 06.07</text>
<text><location><page_10><loc_18><loc_68><loc_79><loc_74></location>Murray, S. S., Brugel, E. W., Eichhorn, G., Farris, A., Good, J. C., Kurtz, M. J., Nousek, J. A., & Stoner, J. L. 1992, in European Southern Observatory Conference and Workshop Proceedings, vol. 43 of European Southern Observatory Conference and Workshop Proceedings, 387</text>
<text><location><page_10><loc_18><loc_66><loc_79><loc_68></location>Shapurian, G., Kurtz, M. J., & Accomazzi, A. 2023, arXiv e-prints, arXiv:2312.08579. 2312. 08579</text>
</document>
[
{
"title": "Alberto Accomazzi",
"content": "Center for Astrophysics | Harvard & Smithsonian, Cambridge, MA, USA; aaccomazzi@cfa.harvard.edu Abstract. The NASA Astrophysics Data System (ADS) is the primary Digital Library portal for researchers in astronomy and astrophysics. Over the past 30 years, the ADS has gone from being an astronomy-focused bibliographic database to an open digital library system supporting research in space and (soon) earth sciences. This paper describes the evolution of the ADS system, its capabilities, and the technological infrastructure underpinning it. We give an overview of the ADS's original architecture, constructed primarily around simple database models. This bespoke system allowed for the e ffi cient indexing of metadata and citations, the digitization and archival of full-text articles, and the rapid development of discipline-specific capabilities running on commodity hardware. The move towards a cloud-based microservices architecture and an open-source search engine in the late 2010s marked a significant shift, bringing full-text search capabilities, a modern API, higher uptime, more reliable data retrieval, and integration of advanced visualizations and analytics. Another crucial evolution came with the gradual and ongoing incorporation of Machine Learning and Natural Language Processing algorithms in our data pipelines. Originally used for information extraction and classification tasks, NLP and ML techniques are now being developed to improve metadata enrichment, search, notifications, and recommendations. we describe how these computational techniques are being embedded into our software infrastructure, the challenges faced, and the benefits reaped. Finally, we conclude by describing the future prospects of ADS and its ongoing expansion, discussing the challenges of managing an interdisciplinary information system in the era of AI and Open Science, where information is abundant, technology is transformative, but their trustworthiness can be elusive.",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "The ADS as it is known today consists of a literature centric system connected to a variety of research products and enhanced by a number of services. Originally, this functionality was implemented as a part of a larger networked system which was designed to allow for the discovery, retrieval, and analysis of astrophysical data (Murray et al. 1992). That larger system was abandoned in the mid-90s in favor of a distributed set of web-based archives, but the literature component (named the 'ADS Abstract Service') survived, and later thrived, on its own. The early literature indexing system that was developed by a small team at the Smithsonian Astrophysical Observatory in the early nineties has grown to become a discovery system which is now an essential tool not only for astronomers but for an increasing number of researchers in the space sciences. Today, an expanded version of ADS is becoming the centerpiece of NASA's open science initiatives which aim to promote transparency and access to the research e ff orts funded by the agency. This paper provides an historical overview of the growth of the ADS as a technological system, focusing on the inflection points in software development which informed its evolution.",
"pages": [
1,
2
]
},
{
"title": "2. Early Days: 1992-2007",
"content": "Because of the requirements involved in gathering, processing, and indexing textual data, most of the software used to create and manage the ADS Abstract database and user interface had to be developed from scratch. Relational SQL-based databases were already popular in the early 1990s, but their limitations in handling text fields meant that we could not store or index the text of an average abstract in a record. Additionally, the costs associated with purchasing access licenses for the database were significant, so the decision was made to use a file-based system for storing bibliographic records using UNIX-based servers (Sun / SunOS). This also meant that we had to design a system for indexing and then retrieving the content, thus rolling our own search engine. The advantage of using our own search system proved important when it came to enhancing the system with functionality provided by external services. In 1993, less than one year after the introduction of the Abstract Service, we implemented the capability to perform object searches via a federated query to the SIMBAD database (Eichhorn 1993), developed by the CDS in Strasbourg. To our knowledge, this was the first implementation of an inter-continental networked query as part of a search service.",
"pages": [
2
]
},
{
"title": "2.1. The Era of Open Source Software",
"content": "As for many similar computing-intensive projects, the lack of existing software meant that the team had to develop its own. The available tools consisted of programming languages and compilers that were either part of the UNIX operating system or were available as free software, which was starting to become popular in academic and research circles. Given our needs, we focused on developing the search engine in C and the ingest pipelines in PERL, which was then considered as the most flexible language available for processing text data. Workflows and pipelines were strung together using the components provided to us by the operating system itself: UNIX pipes, interprocess communication, shell error handling, and text processing tools such as look, sort, join, uniq. The advent of the World Wide Web signified an inflection point for the project. As early as 1993 it became clear that the Web was going to be the best framework for delivering the services that we were developing, and the availability of free software implementing the HTTP protocol and web browsers lead to the rapid adoption of webbased technologies everywhere. Up until this point access to the ADS Abstract Service required a dedicated X-windows application that used a proprietary protocol to connect to the server hosting the database. Removing this last obstacle meant that the ADS infrastructure was able to run on commodity hardware, using open source code, and open source client-server applications. The hyperlinked nature of the web also meant that the ADS user interface could be quickly developed, enhanced, and deployed without the need for any software upgrade. Figure 1 shows the ADS abstract service query form circa 1994. The introduction of the GNU software suite, the Linux operating system, and the larger open source ecosystem accelerated the development and availability of tools and libraries that became useful to the project, in particular the implementation of TIFF and PNG graphic formats for our full-text document handling, the Berkeley DB system for on-disk database implementations, and POSIX threads for parallelization of tasks. Adopting the technology stack built around the open source movement meant that we had full control of our destiny, and that we could develop quickly and independently.",
"pages": [
2,
3
]
},
{
"title": "2.2. The ADS Coming of Age",
"content": "Having identified the environment and platform for our system, the ADS team, which at the time was composed of 4 FTEs, was able to develop components of the web-based Abstract Service and, starting in 1995, the Article Service, which provided access to the full-text scans of the major astronomical journals (Accomazzi et al. 1995). Shortly after that, in 1996, we introduced links between bibliographic records, datasets and catalogs (Accomazzi et al. 1997). In 1997 citation data which was originally purchased from the Institute for Scientific Information by the AAS was indexed in our system (Kurtz et al. 1996), allowing users to traverse the citation graph from paper to paper, as well as to sort any search results by citation counts. By 1998 astronomers were hooked. A study by Kurtz (Kurtz et al. (1999)) showed that online readership through the ADS surpassed the worldwide print readership in all astronomy institutions. The availability of an increasing amount of full-text publications in the ADS and the ability for any scientist to find them and retrieve them easily and freely lead to the massive clearing of shelves from the o ffi ces of most astronomers worldwide. The embrace of the ADS platform by the community had a positive feedback e ff ect on the evolution of the system. Publishers started hearing from their editors and referees that if their content was not in ADS, it may as well not exist, which helped the project obtain metadata and back-records of historical content in the system. The resounding success and adoption of the ADS in the late 1990s provided additional opportunities for the project to gain additional recognition and support from NASA, which led to an increase in sta ff (6 FTEs) and funds to continue the development and operations of the system. The following decade was one during which we developed system capabilities to ensure that our system was up-to-date, resilient and accessible. This led to the creation of a pipeline to automatically extract and ingest citations from the literature (Demleitner et al. 1999), a workflow to scan and digitize content on an on-going basis, and the development of a network of mirror sites hosted by collaborating institutions throughout the world (Accomazzi et al. 2000). Additional features and services were also developed during the first decade of the 2000s. The myADS notification service was launched in 2003, providing customized digests of new publications based on the preferences of individual researchers. The original weekly updates of astronomy content from arXiv became a daily update of all the subject categories available in the preprint server. Customizations were added allowing users to save bibliographic articles in virtual private collections ('ADS libraries'), configure access to publisher content via their library's subscriptions (based on the OpenURL protocol), and customize their search experience through a set of preferences tied to their login account. By 2007, the ADS Abstract and Article services provided the unparalleled capabilities of any research portal in any discipline. ADS usage in 2007 shows that there were over 20,000 registered users (with login accounts), and 6,000 subscribers to the myADS notification system. Monthly usage showed that we had more than 30,000 heavy users out of an overall total of 1.2M users. Given the number of professional astronomers on earth, it was clear that the system was being used not just by the totality of researchers in the discipline, but also by people outside of astronomy as well as amateurs and the general public (Henneken et al. 2009). The system architecture grew from one which consisted of a single abstract database to a complex set of data products and relationships, as illustrated in Figure 2. Greater functionality brought along greater complexity of the code, which had been growing organically for the previous 15 years. A census of our codebase conducted in 2007 showed that our search engine had grown to consist of 250K lines of C, while our indexing and application software consisted of another 250K lines of C, PERL and python 2 code. All of it custom-built and scarcely documented.",
"pages": [
4,
5
]
},
{
"title": "3. ADS Reborn:2008-2020",
"content": "In 2007, the departure of Guenther Eichhorn, project scientist and key developer of the ADS search engine, caused a major disruption in the small team of 6 employees, and forced the project to reconsider its technical infrastructure and future development strategies. It became immediately apparent that the di ffi culty of maintaining and enhancing undocumented legacy code was not a wise nor sustainable path for the project, and that digital library technologies had matured to the point that open source alternatives should be considered instead. The task before us was not small: keep the ADS running 24 / 7 while rebuilding, more or less from scratch, a new system with equal or better performance and functionality. Drawing from our recent experience with system development, an additional self-imposed requirement was to develop such a system in the open, reusing as much as possible existing technologies and seeking strategic partnerships whenever appropriate. Thankfully NASA was receptive to our plea for help and supported the plan we submitted to the Astrophysics Archives review in 2008, which led to an incremental doubling in ADS sta ffi ng over the next decade.",
"pages": [
5
]
},
{
"title": "Accomazzi",
"content": "Starting in 2009, we were able to start hiring new developers and experiment with a variety of technologies to inform our future architecture. The initial phase of this effort involved leveraging the existing search engine to document its capabilities and explore the e ff ort required in implementing a new user interface. A pivotal shift occurred with the adoption of Apache Solr, a widely recognized open-source search platform, to manage the backend metadata indexing. This strategic change was implemented in 2013, marking a critical step in modernizing the system's architecture and setting the foundation for our search infrastructure. Further advancement was made in 2015 with the development and launch of a custom JavaScript application, internally codenamed 'Bumblebee.' This application represented a major forward leap in terms of user interface and experience. It was during this phase that cloud computing was integrated into the system's architecture, signifying a move towards more scalable and flexible infrastructure. The cloud-based aspect of the architecture was particularly notable, with the JavaScript application utilizing a JSON API hosted in the cloud, thereby enhancing data accessibility and system responsiveness. The subsequent phase focused on expanding the system's capabilities in order to reach feature parity with the ADS Classic system and to incorporate the evolving needs of users. One of the key enhancements was the integration of ORCID claiming, responding to the growing necessity for interoperability with academic and research author identification systems. Alongside this, considerable e ff orts were devoted to backend infrastructure development, ensuring the robustness and reliability required to support new functionalities and user demands. By 2019, this comprehensive developmental journey culminated in achieving and even surpassing feature parity with the original ADS system. This milestone was not just about matching the existing functionality but also about providing an enhanced, modern platform that could support the evolving requirements of users in a more e ffi -cient, scalable, and sustainable way. The new system featured a search engine which integrated high-performance text search with citation and usage graphs; a well-structured and well-documented JSON API; and a modern user interface featuring a number of visualizations and analytics (Chyla et al. 2015). The corresponding architecture is illustrated in Figure 3.",
"pages": [
6
]
},
{
"title": "4. Future ADS: 2021-2030",
"content": "In 2019, the NASA Science Mission Directorate (SMD) published a white paper detailing a strategy for data and computing which included support for open science initiatives 1 . One of the goals of the strategic plan was to provide support for open science initiatives through the creation of an interdisciplinary literature portal that could be used to understand how NASA data is used and to provide services to the research community across the disciplines funded by NASA SMD. It became immediately clear that ADShad been fulfilling these goals for the Astrophysics community, so NASA selected the ADS team to expand its mission to include all five SMD disciplines: Astrophysics, Planetary Science, Heliophysics, Earth Science, and Biological and Physical Sciences. The name of the expanded system is the NASA Science Explorer 2 , or SciX for short. This expansion e ff ort is transformative for the project, which will go from being a literature database focusing on a single research community to a multidisciplinary platform used by a much larger community of space, earth, and biological scientists. While much of the system infrastructure can be scaled up for additional content and use, a lot of the curation workflows and associated pipelines will need to be adapted to deal with new data providers, a new set of science data archives, and a much larger community of researchers with varied cultural backgrounds and habits. Most importantly, the ADS has become an essential research tool thanks to its role at the center of a nexus of archives and information services in astronomy. Becoming part of similar ecosystems in new disciplines will be a major challenge. From a technical perspective, one of the things we are betting on is that we can use emerging technologies - specifically AI and machine learning - to help us perform a lot of the activities that traditionally we have been doing through human curation, for example metadata aggregation and enrichment. As an example, we want to automatically assign concepts drawn from the Unified Astronomy Thesaurus to all the records in our astronomy collection, or extract planetary feature names from planetary science papers. And part of doing this as our contribution to open science e ff orts means that we are not only building and delivering an AI-enhanced service, but we will also be generating and sharing the underlying training datasets and open source code that we hope will be used by all of the communities that we serve. ADS already has a long tradition of openness, and with the expansion into SciX we will redouble our e ff orts to share our e ff orts with the larger research community. In 2021 we created and released a custom language model built on the astronomical literature called astroBERT (Grezes et al. 2021). We have contributed data sets for the 2022 and 2023 data challenges at the first and second Workshops on Information Extraction from Scholarly Papers 3 . In 2023 we have started working with the UniverseTBD collaboration 4 , which recently released the first fine-tuned version of the popular LLaMA-2 model, named AstroLLaMA (Dung Nguyen et al. 2023). We have also participated in the creation of a Large Language Model (LLM) developed by the NASA Science Mission Directorate in partnership with IBM which will be released in 2024. The future is uncertain but most of us will agree that AI will play a major role in the development of digital scholarship. The technologies that are most likely to revolutionize the way we interact with the scientific literature are LLMs and Knowledge Graphs (KGs). While LLM-powered chatbots have gotten the most attention from the general public, the adaptability of LLMs make them general purpose tools for a variety of Natural Language Processing tasks, such as structured information extraction, named entity recognition, and metadata enrichment. Knowledge Graphs can be similarly used to support information retrieval, semantic search, and metadata normalization. For an example of how ADS started using LLMs and KGs in its pipelines, see Shapurian et al. (2023). We expect that this approach will be fundamental in developing SciX into a fully featured interdisciplinary system.",
"pages": [
6,
7,
8
]
},
{
"title": "5. Discussion",
"content": "Along with the promise of an exciting future, the latest AI technologies bring with them a lot of questions related to trust. Today's LLMs are essentially black boxes, deep networks of billions and trillions of parameters trained on data of various quality, often in non-transparent ways. These systems are really too large for anyone to inspect them in any detail, but they rather seem to work in mysterious, if not magical, ways. However, as scientists, we have all been trained to reject 'magic' and instead study complex systems in order to understand how they work, then modify their environment and behavior in order to control and adapt them to our needs. This is the task before us: use the body of knowledge generated by the scientific process to create AI technologies that advance knowledge and insight into the physical world. The ADS team and the community at large have begun investigating the use of open source LLMs for information retrieval and reasoning (Blanco-Cuaresma et al. 2024; Ciuc˘a & Ting 2023). While strategies based on an Retrieval Augmented Generation approach seems the most promising right now, these are still early days and there is a lot to be explored. Given the increasing number of open-source LLMs being generated, one interesting scenario for future development may be one in which there are a few open source LLMs being fine-tuned for specific tasks and domains, using custom curated datasets. Under this scenario, ADS and SciX would be the authoritative source of data used to train LLMs and build KGs used in the earth and space sciences to ensure their trustworthiness and completeness. ADS has been a transformative service for astronomers, and it's likely to be as transformative for a larger group of earth and space scientists in the near future. Its success couldn't have been possible without the support of NASA and the existence of a larger ecosystem of open, interoperable information services within astronomy. The now universal support for open science initiatives gives us an opportunity to extend the ADS model to a wider set of disciplines, and the continued development of open source code and models means that there is still a bright future ahead for the scientific enterprise. Acknowledgments. The ADS would not exist without NASA's continued support over the past 30 years. We are grateful to the agency for making it possible for the system to flourish and grow into its current form. The ADS team today is composed of 20 talented individuals 5 , soon to become 30 FTEs as part of our expansion. We have all benefited from the work of those who came before us and provided the vision and focus that made the ADS indispensable in its early days. There are too many names to mention, but three stand above the rest: Steve Murray, founder of the ADS and PI until his passing in 2015; Michael Kurtz, the ADS Project Scientist and visionary who is still contributing to our ongoing e ff orts; and Guenther Eichhorn, who managed the project until 2007 when he left the Center for Astrophysics. Today's ADS stands on the shoulders of these giants, without which none of this would have been possible.",
"pages": [
8,
9
]
},
{
"title": "References",
"content": "Accomazzi, A., Eichhorn, G., Grant, C. S., Murray, S. S., & Kurtz, M. J. 1995, Vistas in Astronomy, 39, 63 Accomazzi, A., Eichhorn, G., Kurtz, M. J., Grant, C. S., & Murray, S. S. 1997, in Astronomical Data Analysis Software and Systems VI, edited by G. Hunt, & H. Payne, vol. 125 of Astronomical Society of the Pacific Conference Series, 357 -2000, A&AS, 143, 85. astro-ph/0002105 Blanco-Cuaresma, S., Accomazzi, A., Kurtz, M. J., Henneken, E., Lockhart, K. E., Grezes, F., Allen, T., Shapurian, G., Grant, C. S., Thompson, D. M., Hostetler, T. W., Templeton, M. R., Chen, S., Koch, J., Jacovich, T., Chivvis, D., de Macedo Alves, F., Paquin, J.-C., Batlett, J., Polimera, M., & Jarmak, S. 2024, these proceedings Chyla, R., Accomazzi, A., Holachek, A., Grant, C. S., Elliott, J., Henneken, E. A., Thompson, D. M., Kurtz, M. J., Murray, S. S., & Sudilovsky, V. 2015, in Astronomical Data Analysis Software an Systems XXIV (ADASS XXIV), edited by A. R. Taylor, & E. Rosolowsky, vol. 495 of Astronomical Society of the Pacific Conference Series, 401. 1503.05881 Ciuc˘a, I., & Ting, Y.-S. 2023, Research Notes of the American Astronomical Society, 7, 193. 2304.05406 Demleitner, M., Accomazzi, A., Eichhorn, G., Grant, C. S., Kurtz, M. J., & Murray, S. S. 1999, in American Astronomical Society Meeting Abstracts, vol. 195 of American Astronomical Society Meeting Abstracts, 82.09 Dung Nguyen, T., Ting, Y.-S., Ciuc˘a, I., O'Neill, C., Sun, Z.-C., Jabło'nska, M., Kruk, S., Perkowski, E., Miller, J., Li, J., Peek, J., Iyer, K., Ró˙za'nski, T., Khetarpal, P., Zaman, S., Brodrick, D., Rodríguez Méndez, S. J., Bui, T., Goodman, A., Accomazzi, A., Naiman, J., Cranney, J., Schawinski, K., & UniverseTBD 2023, arXiv e-prints, arXiv:2309.06126. 2309.06126 Eichhorn, G. 1993, JAAVSO, 22, 136 Grezes, F., Blanco-Cuaresma, S., Accomazzi, A., Kurtz, M. J., Shapurian, G., Henneken, E., Grant, C. S., Thompson, D. M., Chyla, R., McDonald, S., Hostetler, T. W., Templeton, M. R., Lockhart, K. E., Martinovic, N., Chen, S., Tanner, C., & Protopapas, P. 2021, arXiv e-prints, arXiv:2112.00590. 2112.00590 Henneken, E. A., Kurtz, M. J., Accomazzi, A., Grant, C. S., Thompson, D., Bohlen, E., & Murray, S. S. 2009, Journal of Informetrics, 3, 1. 0808.0103 Kurtz, M. J., Eichhorn, G., Accomazzi, A., Grant, C. S., Demleitner, M., & Murray, S. S. 1999, D-Lib Magazine, 5 Kurtz, M. J., Eichhorn, G., Accomazzi, A., Grant, C. S., & Murray, S. S. 1996, in American Astronomical Society Meeting Abstracts, vol. 189 of American Astronomical Society Meeting Abstracts, 06.07 Murray, S. S., Brugel, E. W., Eichhorn, G., Farris, A., Good, J. C., Kurtz, M. J., Nousek, J. A., & Stoner, J. L. 1992, in European Southern Observatory Conference and Workshop Proceedings, vol. 43 of European Southern Observatory Conference and Workshop Proceedings, 387 Shapurian, G., Kurtz, M. J., & Accomazzi, A. 2023, arXiv e-prints, arXiv:2312.08579. 2312. 08579",
"pages": [
9,
10
]
}
]
2024arXiv240117156J
https://arxiv.org/pdf/2401.17156.pdf
<document>
<section_header_level_1><location><page_1><loc_4><loc_84><loc_94><loc_90></location>Collapsing massive stars with self-gravity and their electromagnetic transients</section_header_level_1>
<text><location><page_1><loc_4><loc_75><loc_98><loc_77></location>Agnieszka Janiuk 1,a Narjes Shahamat, 2 and Dominika Kr'ol , 3</text>
<unordered_list>
<list_item><location><page_1><loc_4><loc_73><loc_67><loc_74></location>1 Center for Theoretical Physics, Polish Academy of Sciences</list_item>
</unordered_list>
<text><location><page_1><loc_5><loc_71><loc_52><loc_72></location>Al. Lotnik'ow 32/46, 02-668, Warsaw, Poland</text>
<unordered_list>
<list_item><location><page_1><loc_4><loc_69><loc_48><loc_71></location>2 Department of Physics, School of Science</list_item>
</unordered_list>
<text><location><page_1><loc_5><loc_67><loc_42><loc_69></location>Ferdowsi University, Mashhad, Iran</text>
<text><location><page_1><loc_4><loc_66><loc_59><loc_67></location>3 Astronomical Observatory, Jagiellonian University</text>
<text><location><page_1><loc_5><loc_64><loc_22><loc_65></location>Krak'ow, Poland</text>
<text><location><page_1><loc_4><loc_62><loc_25><loc_64></location>a agnes@cft.edu.pl</text>
<section_header_level_1><location><page_1><loc_4><loc_52><loc_19><loc_53></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_4><loc_11><loc_81><loc_52></location>We investigate the fate of a collapsing stellar core, which is the final state of evolution of a massive, rotating star of a Wolf-Rayet type. Such stars explode as type I b/c supernovae, which have been observed in association with long gamma ray bursts (GRBs). The core of the star is potentially forming a black hole, which is embedded in a dense, rotating, and possibly highly magnetized envelope. We study the process of collapse using General Relativistic MHD simulations, and we account for the growth of the black hole mass and its spin, as well as related evolution of the spacetime metric. We find that some particular configurations of the initial black hole spin, the content of angular momentum in the stellar core, and the magnetic field configuration and its strength, are favored for producing a bright electromagnetic transient (i.e., a gamma ray burst). On the other hand, most of the typical configurations studied in our models do not lead to a transient electromagnetic explosion and will end up in a direct collapse, accompanied by some residual variability induced by changing accretion rate. We also study the role of self-gravity in the stellar core and quantify the relative strength of the interfacial instabilities, such as Self-Gravity Interfacial (SGI) instability and Rayleigh-Taylor (RT), which may account for the production of an inhomogeneous structure, including spikes and bubbles, through the inner radii of the collapsing core (inside ∼ 200 r g ). We find that in self-gravitating collapsars the RT modes cannot grow efficiently. We also conclude that transonic shocks are formed in the collapsing envelope, but they are weaker in magnetized stars.</text>
<text><location><page_1><loc_4><loc_6><loc_81><loc_9></location>Keywords: Accretion-black hole physics - gravitation - magnetohydrodynamics - massive stars - gamma ray bursts</text>
<section_header_level_1><location><page_2><loc_4><loc_92><loc_32><loc_94></location>1 INTRODUCTION</section_header_level_1>
<text><location><page_2><loc_4><loc_84><loc_97><loc_91></location>Long gamma ray bursts (GRBs) originate from the collapse of massive, rotating stars. Some of the GRBs exhibit much stronger variability patterns in the prompt GRB emission than the usual stochastic variations. We discuss the mechanisms of these variations in the frame of self-gravitating collapsar model.</text>
<text><location><page_2><loc_4><loc_72><loc_97><loc_83></location>Our computations confirm that gravitational instability can account for flaring activity in GRBs and the variations in their prompt emission. Rapid variability detected in the brightest GRBs, most likely powered by spinning black holes, is consistent with the self-gravitating collapsar model, where the density inhomogeneities are formed. The transonic fshocks may also appear, but their effect should be weakened by magnetic field.</text>
<text><location><page_2><loc_4><loc_58><loc_97><loc_71></location>We calculate the time evolution of the collapsing massive star using the General Relativisitic Magneto-Hydrodynamic (GR MHD) scheme. We have developed a new version of the code HARM-METRIC, upgraded from that presented in Janiuk et al. (2018). The evolution of the space-time Kerr metric is accounted for by the increasing mass and changing spin of the black hole. We added also the new terms, that describe the self-gravity of the star and are changing at every time-step during dynamical simulation.</text>
<text><location><page_2><loc_4><loc_46><loc_97><loc_57></location>In our formulation, the black hole has been already formed in the centre of the collapsing stellar core and its initial mass in of 3 M ⊙ . Our computational grid size is of 1000 r g , which makes it smaller than a compact C-O core of a Wolf-Rayet star or a presupernova. Thereofre, our model is compact enough to address the problem of self-gravitating gas close to the horizon of a newly formed black hole, but we do not address any prior or ongoing supernova explosion.</text>
<text><location><page_2><loc_4><loc_28><loc_97><loc_46></location>Depending on the rotation of the star, the ultimate outcome might be either a direct collapse or the formation of a mini-disc inside the core, that is, a collapsar which may lead to an electromagnetic transient. At the onset of the GRB, the collapsar consists of a black hole, stellar envelope composed of accreting shells with decreasing density, and rotationally supported disc formed at the equatorial region. At any chosen radius above the horizon, the gas is subject to gravity force induced by the Kerr black hole, the centrifugal force due to envelope rotation, and in addition, it feels the perturbative force due to the self-gravity of the matter, enclosed within a given radius.</text>
<section_header_level_1><location><page_2><loc_4><loc_22><loc_55><loc_24></location>2 NUMERICAL CODE AND SETUP</section_header_level_1>
<text><location><page_2><loc_4><loc_14><loc_97><loc_21></location>We use the general relativistic MHD code called high-accuracy relativistic magnetohydrodynamics (HARM), originally published by Gammie et al. (2003) and further developed by various groups. Our code version, HARM-METRIC, includes the Kerr metric evolution, as first described in Janiuk et al. (2018).</text>
<text><location><page_2><loc_4><loc_6><loc_97><loc_13></location>The code introduces a conservative, shock-capturing scheme with low numerical viscosity to solve the hyperbolic system of partial differential equations of GR MHD. The numerical scheme uses the plasma energy-momentum tensor, with contributions from matter (gas) and electromagnetic field. For the GR MHD evolution, two</text>
<table>
<location><page_3><loc_24><loc_84><loc_76><loc_94></location>
<caption>Table 1. Conversion units between numerical code and physical scale of the collapsar.</caption>
</table>
<text><location><page_3><loc_4><loc_74><loc_97><loc_76></location>fundamental equations are solved for the mass and energy-momentum conservation.</text>
<formula><location><page_3><loc_4><loc_68><loc_97><loc_71></location>( ρu µ ) ; µ = 0; T µ ν ; µ = 0 . (1)</formula>
<formula><location><page_3><loc_4><loc_61><loc_97><loc_63></location>T µν ( m ) = ρhu µ u ν + pg µν . (2)</formula>
<formula><location><page_3><loc_4><loc_53><loc_97><loc_57></location>T µν ( em ) = b k b k hu µ u ν + 1 2 b k b k g µν -b µ b ν . (3)</formula>
<formula><location><page_3><loc_4><loc_46><loc_97><loc_49></location>T µν = T µν ( m ) + T µν ( em ) . (4)</formula>
<text><location><page_3><loc_4><loc_37><loc_97><loc_44></location>An additional constraint is given by the Equation of State (EOS). In the current project, we used analytic form of adiabatic EOS that relates gas pressure with density. This scales with the power of 4/3, as adequate for a relativistic gas of degenerate particles.</text>
<formula><location><page_3><loc_4><loc_30><loc_97><loc_34></location>p = Kρ γ ; γ = 4 3 (5)</formula>
<text><location><page_3><loc_4><loc_22><loc_97><loc_29></location>The HARM code works in dimensionless units of G = c = 1. Conversion coefficients can be found in 1, where the black hole of 3 Solar masses is assumed. Notice that in the plots below, we use geometric unit to express distance, while physical units are used to express time.</text>
<section_header_level_1><location><page_3><loc_4><loc_17><loc_32><loc_18></location>2.1 Initial conditions</section_header_level_1>
<text><location><page_3><loc_4><loc_6><loc_97><loc_15></location>Initial conditions for our collapsing stellar core are given by quasi-spherical distribution of gas endowed with small angular momentum, concentrated at the equatorial plane (Kr'ol and Janiuk, 2021). The distributions of density and radial velocity are obtained from the Bondi solution, integrated numerically below and above the sonic point. The sonic point is a parameter of our model, and here it is assumed at 80 r g .</text>
<text><location><page_3><loc_60><loc_2><loc_63><loc_4></location>➤</text>
<text><location><page_3><loc_60><loc_2><loc_61><loc_4></location>❙</text>
<text><location><page_3><loc_64><loc_2><loc_74><loc_4></location>➤ ➤ ➤❙</text>
<text><location><page_3><loc_77><loc_2><loc_83><loc_4></location>➥ ➥</text>
<text><location><page_3><loc_84><loc_2><loc_86><loc_4></location>?</text>
<text><location><page_3><loc_88><loc_2><loc_97><loc_4></location>❏ ■ ✖</text>
<text><location><page_4><loc_4><loc_91><loc_97><loc_94></location>Below this point, matter flows into black hole supersonically, and reaches the speed of light at the horizon.</text>
<text><location><page_4><loc_4><loc_77><loc_97><loc_90></location>We illustrate the initial condition in Figure 1, left panel. Density of the gas is normalized to physical units (given in cgs on the plot), assuming that the collapsing star has the initial mass of 25 Solar masses. This mass is enclosed within our computational domain with outer radius of a R out = 1000 r g . The plot shows only the innermost region, of 100 r g . Most mass of the core is located very near to the center, as it represents the evolved state of stellar evolution with a compact (iron) core formed.</text>
<text><location><page_4><loc_4><loc_67><loc_97><loc_76></location>In the initial conditions, we also introduce a small angular momentum imposed on the spherically distributed gas. The specific angular momentum is normalized by the parameter S , with respect to that at the innermost stable circular orbit (ISCO). In addition, the rotation velocity scales with the polar angle, to be maximal at the equator, θ = π/ 2.</text>
<formula><location><page_4><loc_4><loc_61><loc_97><loc_63></location>l = Sl isco r 2 sin 2 θ, (6)</formula>
<text><location><page_4><loc_4><loc_57><loc_9><loc_59></location>with</text>
<formula><location><page_4><loc_4><loc_50><loc_97><loc_56></location>l isco = u ϕ, isco = r 1 / 2 isco -2 a/r isco + a 2 /r 3 / 2 isco √ 1 -3 /r isco +2 a/r 3 / 2 isco . (7)</formula>
<text><location><page_4><loc_4><loc_41><loc_97><loc_48></location>Notice that the radius r ISCO in Kerr geometry depends on the black hole spin. In this proceeding, we show results obtained for the value of initial black hole spin a 0 = 0 . 5. We use several values of rotation parameter, as denoted on the plots in next sections.</text>
<text><location><page_4><loc_4><loc_31><loc_97><loc_40></location>After the onset of collapse, the rotation of gas induces formation of a mini-disk, i.e. toroidal structure, located at the equatorial plane. The density distribution becomes no longer spherical. Also, the radial velocity is decreased, as the gas is subject to a centrifugal barrier. Flow is falling into the black hole with supersonic speed from the poles, while at the equator the speed is subsonic.</text>
<text><location><page_4><loc_4><loc_21><loc_97><loc_30></location>Map on the Figure 1, right panel, shows the flow distribution at time t=0.089 s, for the model normalized with rotation parameter S=1.4. This means that the specific angular momentum is above critical value (S=1) which allows for the formation of rotationally supported torus. Sonic surface, Mach=1, is plotted with a solid line, and marks the location of a transonic shock at the equatorial region.</text>
<section_header_level_1><location><page_4><loc_4><loc_15><loc_78><loc_16></location>3 IMPACT OF SELF-GRAVITY ON THE COLLAPSE</section_header_level_1>
<text><location><page_4><loc_4><loc_6><loc_97><loc_13></location>In our new simulations, both the mass and angular momentum accreted onto the event horizon -and used to update the Kerr metric coefficients- are now modified by the perturbation acting on the metric in the region above the horizon due to the self-gravity force that the gas feels at a given distance from the horizon. These</text>
<text><location><page_5><loc_4><loc_85><loc_97><loc_94></location>perturbative terms are calculated from the stress-energy tensor. Therefore, in addition to the two equations governing the growth of black hole mass and spin via the mass and angular momentum transfer through the horizon, as given below, (Kr'ol and Janiuk, 2021), we now add perturbative terms to mass and angular momentum, computed at every radius above the event horizon.</text>
<formula><location><page_5><loc_4><loc_79><loc_97><loc_82></location>˙ M BH = ∫ dθdϕ √ -g T r t , (8)</formula>
<formula><location><page_5><loc_4><loc_72><loc_97><loc_75></location>˙ J = ∫ dθdϕ √ -g T r ϕ , (9)</formula>
<formula><location><page_5><loc_4><loc_64><loc_97><loc_68></location>δM BH ( t, r ) = 2 π ∫ r r hor T r t √ -gdθ, (10)</formula>
<formula><location><page_5><loc_4><loc_57><loc_97><loc_61></location>δJ ( t, r ) = 2 π ∫ r r hor T r ϕ √ -gdθ, (11)</formula>
<formula><location><page_5><loc_4><loc_50><loc_97><loc_54></location>δa = J + δJ ( r ) M BH + δM BH ( r ) -a i , (12)</formula>
<formula><location><page_5><loc_4><loc_45><loc_97><loc_47></location>a i = a i -1 +∆ a. (13)</formula>
<text><location><page_5><loc_4><loc_23><loc_97><loc_43></location>The terms computed in addition to mass and angular momentum changes (Janiuk et al., 2018) as these self-gravity perturbations, are integrated at each grid point in the radial direction and at each time. They affect the change of Kerr metric coefficients, which are sensitive to the mass and spin updates. The dimensionless black hole spin, a, evolves as a result of black hole mass and angular momentum changes due to accretion of mass under the horizon, and is additionally changed due to self-gravity of the collapsing core. The numerical method has been described in detail in Janiuk et al. (2023). Below, we compare the results of self-gravitating collapsar models to the runs without self-gravity, in order to emphasize the difference and to investigate the role of self-gravity in the collapsar physics.</text>
<text><location><page_5><loc_4><loc_6><loc_97><loc_23></location>As shown in Figure 2, the results are strongly sensitive to the adopted self-gravity effects, and also weakly sensitive to the rotation of the collapsing envelope. The latter is normalized with respect to the critical angular momentum, for which the flow is circularized at the innermost stable orbit, ISCO (Kr'ol and Janiuk, 2021). In addition, the rotation velocity scales with the polar angle, so that at the equator, the rotation of the star is maximal. We notice that the larger the initial rotation magnitude, the longer it takes for the black hole mass to evolve. The non-SG simulations end with very different final black hole mass, depending on the rotation parameter.</text>
<figure>
<location><page_6><loc_14><loc_69><loc_88><loc_88></location>
<caption>Figure 1. Left: Density distribution at the onset of core collapse. Arrows represent velocity field (normalized length). Density distribution and velocity field, after the onset of collapse. Model is parameterized with rotation parameter S=1.4. Thick white line represents sonic surface, Mach=1. Right: Density distribution at the onset of core collapse. Arrows represent velocity field.</caption>
</figure>
<figure>
<location><page_6><loc_8><loc_34><loc_92><loc_45></location>
<caption>Figure 2. Left: Black hole mass changing as function of time during the collapse. We start from 3 Solar mass black hole. Models including and excluding self-gravity are plotted by the thick and thin curves, respectively. Three color refer to different amount of angular momentum in the collapsing star: S=1.0 (blue), S=1.4 (red) and S=2.0 (green). Middle: Evolution of accretion rate through the horizon, for self-gravitating and non-selfgravitating collapsars, shown with thick and thin lines, respectively. The different colors refer to various amounts of angular momentum in the collapsar, same as in the left plot. Right: Evolution of the black hole dimensionless spin parameter, during the collaspe. We start from oderately spinning black hole with a=0.5. Models including and excluding selfgravity are plotted by the thick and thin curves, respectively. Three color refer to different amount of angular momentum in the collapsing star, same as in the left and middle plots.</caption>
</figure>
<text><location><page_7><loc_4><loc_81><loc_97><loc_94></location>In contrast, the self-gravity of the envelope can speed up the evolution of the collapsing stellar core significantly. Also, accretion rate and its fluctuations are of much higher amplitude when self-gravity effects taken into account. Without selfgravity, there are longer time intervals where there is considerably less fluctuation of the accretion rate; in this case, there exist only some small oscillations in the accretion rate during some time intervals (around 0.2 s for S=1.4, and 0.4-0.5 s for S=2).</text>
<section_header_level_1><location><page_7><loc_4><loc_76><loc_55><loc_77></location>3.1 Instabilities on the collapsing core</section_header_level_1>
<text><location><page_7><loc_4><loc_61><loc_97><loc_74></location>As an effect of self-gravity we observe density inhomogeneities and formation of the accretion shocks in all our models, regardless of the initial black hole spin, or rotation parameter of the collapsar. First, there appears an equatorial outflow of matter, which reaches radii of up to about 80 r g and is then stalled in the transonic shock. The small inhomogeneities in the pressure and density at the chosen time intervals, are visible in more detail in the plots below, in Figure 3 and in Figure 4, respectively.</text>
<text><location><page_7><loc_4><loc_49><loc_97><loc_60></location>We quantify the inhomogeneities in the collapsar by computing the radial derivatives of density an pressure at specific times, and locations. We identify the mechanism for their creation as the SGI instability (Self-Gravity Interfacial instability) and we compare its strength with another well-known hydrodynamical instability, the Rayleigh-Taylor (RT) instability. Their growth rates are given as below (Kifonidis et al., 2003; Hunter Jr et al., 1997).</text>
<formula><location><page_7><loc_4><loc_41><loc_97><loc_46></location>σ RT = √ -p ρ ∂lnρ ∂r ∂lnp ∂r , (14)</formula>
<formula><location><page_7><loc_4><loc_33><loc_97><loc_37></location>σ SGI = √ 2 πG ( ρ 2 -ρ 1 ) 2 ( ρ 2 + ρ 1 ) . (15)</formula>
<text><location><page_7><loc_4><loc_18><loc_97><loc_31></location>The RT and SGI instabilities result in very similar configurations at density snapshots. However, they have their own characteristics, which allows us to differentiate between them. As self-gravity has no 'preferred' direction, it is destabilizing across all density interfaces, while an interface is RT-unstable only if the heavy fluid is on top of the light fluid. It has also been confirmed that RT instability is characterized by dense spikes penetrating the tenuous fluid, whereas the SGI develops with tenuous spikes streaming into the denser fluid.</text>
<text><location><page_7><loc_4><loc_10><loc_97><loc_17></location>We find that SGI instability seems to dominate over RT instability and produces the inhomogeneities. In particular, we checked that the growth rates of RT, are having imaginary values, as computed at radii between 20 and 25 r g , around the mixing boundary.</text>
<text><location><page_7><loc_4><loc_6><loc_97><loc_9></location>Finally, we investigated the formation of transonic shocks in the collapsars. In Figure 5 we present radial profiles of Mach number at some specific time snapshots,</text>
<figure>
<location><page_8><loc_8><loc_79><loc_33><loc_92></location>
</figure>
<figure>
<location><page_8><loc_65><loc_79><loc_90><loc_92></location>
</figure>
<figure>
<location><page_8><loc_37><loc_79><loc_61><loc_92></location>
<caption>Figure 3. Left: Pressure profile for the model with rotation parameter S=2 and initial black hole spin A 0 = 0 . 5, taken at time t=0.118 s., at which largest accretion rate fluctuations appear. Middle: Pressure profile at time t=0.133, for the same model as in the left plot. Strong inhomogeneity regions are visible. Right: Pressure profile at later time of the simulation, for the same model as in the left and middle plots. Inhomogeneities are same and at this time accretion rate fluctuations are smoothed as well. The map is zoomed out to larger radius.</caption>
</figure>
<figure>
<location><page_8><loc_8><loc_44><loc_32><loc_58></location>
</figure>
<figure>
<location><page_8><loc_37><loc_44><loc_61><loc_58></location>
</figure>
<figure>
<location><page_8><loc_65><loc_44><loc_90><loc_58></location>
<caption>Figure 4. Left: Density profile at t=0.118, for the same model as above. Middle: Density profile at t=0.133, for the same model as above. Right: Density profile at late time, t=0.665, for the same model as above.</caption>
</figure>
<figure>
<location><page_8><loc_8><loc_16><loc_50><loc_31></location>
</figure>
<figure>
<location><page_8><loc_51><loc_16><loc_92><loc_31></location>
<caption>Figure 5. Left: Mach number profile at three different times, t=0.118, t=0.133, and t=0.148 s, for the model with S = 2 . 0. Right: Mach number profile at three different times, for the same model with magnetic field.</caption>
</figure>
<text><location><page_9><loc_4><loc_71><loc_97><loc_94></location>for models with S = 2 and a 0 = 0 . 5. The left panel shows the profiles in the selfgravitating case, while the plot in the right panel shows those of self-gravitating magnetized case (we introduced a weak vertical magnetic field in the initial condition). For the sake of more visibility, we provide zommed-in inset panels representing the inner regions. We observe the sonic front expansion, and also some transient shock formation during the collapse. At early times, the small transonic shocks appear around 100 r g and they present a moderate density contrast (pre-shock to post-shock density ratio R = ρ 1 /ρ 2 ∼ 10). Such shocks also appear at later times. Their formation is enhanced by the self-gravity effects. We find that magnetic field does not make any significant difference on the shock expansion timescales, but it affects the strength of the shock, consistently with previous studies (Komissarov, 1999).</text>
<section_header_level_1><location><page_9><loc_4><loc_61><loc_30><loc_62></location>4 CONCLUSIONS</section_header_level_1>
<text><location><page_9><loc_4><loc_48><loc_97><loc_59></location>In this work, we show numerical models of the collapsing stellar core where we account for the dynamical evolution of central black hole mass and its spin. The related coefficients of the Kerr space-time metric are evolved accordingly, at every time step. In addition, we calculate the self-gravity of the stellar envelope and we add the relevant perturbative terms to the dynamical evolution of the black hole spin parameter.</text>
<text><location><page_9><loc_4><loc_34><loc_97><loc_48></location>The last modification of the model turned out to have an impact on the global evolution of the collapsing star, and produces dramatic fluctuations in the accretion rate at the initial phase of collapse. More importantly, it also plays crucial role in development of the SGI interfacial instability in its specific regions. We identified inhomogeneities in density and pressure distributions which arise due to self-gravity, and we concluded that the SGI instability dominates over the RT, as its growth rate is positive in the regions of mixing boundaries.</text>
<section_header_level_1><location><page_9><loc_4><loc_24><loc_39><loc_26></location>ACKNOWLEDGEMENTS</section_header_level_1>
<text><location><page_9><loc_4><loc_6><loc_97><loc_23></location>The present work was supported by the grant DEC-2019/35/B/ST9/04000 from Polish National Science Center. We made use of computational resources of the PL-Grid infrastructure, under grant pglgrb6, and Warsaw University ICM. D. glyph[suppress]L. K. was supported by the Polish National Science Center Dec-2019/35/O/ST9/04054 and N. Sh. D. was supported by Iran National Science Foundation (INSF) under project number No.4013178 and also acknowledges Ferdowsi University of Mashhad (FUM), Iran, and the FUM Sci-HPC center. Prof. Shahram Abbassi also deserves gratitude for his accompaniment to N.Sh.D. in this project. A.J. acknowledges the Czech-Polish mobility program (M ˇ SMT 8J20PL037 and PPN/BCZ/2019/1/00069).</text>
<section_header_level_1><location><page_10><loc_4><loc_92><loc_24><loc_94></location>REFERENCES</section_header_level_1>
<text><location><page_10><loc_4><loc_86><loc_97><loc_91></location>Gammie, C. F., McKinney, J. C. and T'oth, G. (2003), HARM: A Numerical Scheme for General Relativistic Magnetohydrodynamics, Astrophysical Journal , 589 (1), pp. 444457, arXiv: astro-ph/0301509 .</text>
<text><location><page_10><loc_4><loc_81><loc_97><loc_85></location>Hunter Jr, J. H., Whitaker, R. W. and Lovelace, R. V. (1997), Kelvin-helmholtz and thermal-dynamic instabilities with self-gravity: a new gravitational interface instability, The Astrophysical Journal , 482 (2), p. 852.</text>
<text><location><page_10><loc_4><loc_75><loc_97><loc_80></location>Janiuk, A., Shahamat Dehsorkh, N. and Kr'ol, D. glyph[suppress]L. (2023), Self-gravitating collapsing star and black hole spin-up in long gamma ray bursts, Astronomy & Astrophysics , 677 , A19, arXiv: 2304.01342 .</text>
<text><location><page_10><loc_4><loc_69><loc_97><loc_74></location>Janiuk, A., Sukova, P. and Palit, I. (2018), Accretion in a Dynamical Spacetime and the Spinning Up of the Black Hole in the Gamma-Ray Burst Central Engine, Astrophysical Journal , 868 (1), 68, arXiv: 1810.05261 .</text>
<text><location><page_10><loc_4><loc_64><loc_97><loc_69></location>Kifonidis, K., Plewa, T., Janka, H.-T. and Muller, E. (2003), Non-spherical core collapse supernovae-i. neutrino-driven convection, rayleigh-taylor instabilities, and the formation and propagation of metal clumps, Astronomy & Astrophysics , 408 (2), pp. 621-649.</text>
<text><location><page_10><loc_4><loc_60><loc_97><loc_63></location>Komissarov, S. S. (1999), Numerical simulations of relativistic magnetized jets, MNRAS , 308 (4), pp. 1069-1076.</text>
<text><location><page_10><loc_4><loc_54><loc_97><loc_59></location>Kr'ol, D. glyph[suppress]L. and Janiuk, A. (2021), Accretion-induced Black Hole Spin-up Revised by Numerical General Relativistic MHD, Astrophysical Journal , 912 (2), 132, arXiv: 2104. 00741 .</text>
</document>
[
{
"title": "ABSTRACT",
"content": "We investigate the fate of a collapsing stellar core, which is the final state of evolution of a massive, rotating star of a Wolf-Rayet type. Such stars explode as type I b/c supernovae, which have been observed in association with long gamma ray bursts (GRBs). The core of the star is potentially forming a black hole, which is embedded in a dense, rotating, and possibly highly magnetized envelope. We study the process of collapse using General Relativistic MHD simulations, and we account for the growth of the black hole mass and its spin, as well as related evolution of the spacetime metric. We find that some particular configurations of the initial black hole spin, the content of angular momentum in the stellar core, and the magnetic field configuration and its strength, are favored for producing a bright electromagnetic transient (i.e., a gamma ray burst). On the other hand, most of the typical configurations studied in our models do not lead to a transient electromagnetic explosion and will end up in a direct collapse, accompanied by some residual variability induced by changing accretion rate. We also study the role of self-gravity in the stellar core and quantify the relative strength of the interfacial instabilities, such as Self-Gravity Interfacial (SGI) instability and Rayleigh-Taylor (RT), which may account for the production of an inhomogeneous structure, including spikes and bubbles, through the inner radii of the collapsing core (inside ∼ 200 r g ). We find that in self-gravitating collapsars the RT modes cannot grow efficiently. We also conclude that transonic shocks are formed in the collapsing envelope, but they are weaker in magnetized stars. Keywords: Accretion-black hole physics - gravitation - magnetohydrodynamics - massive stars - gamma ray bursts",
"pages": [
1
]
},
{
"title": "Collapsing massive stars with self-gravity and their electromagnetic transients",
"content": "Agnieszka Janiuk 1,a Narjes Shahamat, 2 and Dominika Kr'ol , 3 Al. Lotnik'ow 32/46, 02-668, Warsaw, Poland Ferdowsi University, Mashhad, Iran 3 Astronomical Observatory, Jagiellonian University Krak'ow, Poland a agnes@cft.edu.pl",
"pages": [
1
]
},
{
"title": "1 INTRODUCTION",
"content": "Long gamma ray bursts (GRBs) originate from the collapse of massive, rotating stars. Some of the GRBs exhibit much stronger variability patterns in the prompt GRB emission than the usual stochastic variations. We discuss the mechanisms of these variations in the frame of self-gravitating collapsar model. Our computations confirm that gravitational instability can account for flaring activity in GRBs and the variations in their prompt emission. Rapid variability detected in the brightest GRBs, most likely powered by spinning black holes, is consistent with the self-gravitating collapsar model, where the density inhomogeneities are formed. The transonic fshocks may also appear, but their effect should be weakened by magnetic field. We calculate the time evolution of the collapsing massive star using the General Relativisitic Magneto-Hydrodynamic (GR MHD) scheme. We have developed a new version of the code HARM-METRIC, upgraded from that presented in Janiuk et al. (2018). The evolution of the space-time Kerr metric is accounted for by the increasing mass and changing spin of the black hole. We added also the new terms, that describe the self-gravity of the star and are changing at every time-step during dynamical simulation. In our formulation, the black hole has been already formed in the centre of the collapsing stellar core and its initial mass in of 3 M ⊙ . Our computational grid size is of 1000 r g , which makes it smaller than a compact C-O core of a Wolf-Rayet star or a presupernova. Thereofre, our model is compact enough to address the problem of self-gravitating gas close to the horizon of a newly formed black hole, but we do not address any prior or ongoing supernova explosion. Depending on the rotation of the star, the ultimate outcome might be either a direct collapse or the formation of a mini-disc inside the core, that is, a collapsar which may lead to an electromagnetic transient. At the onset of the GRB, the collapsar consists of a black hole, stellar envelope composed of accreting shells with decreasing density, and rotationally supported disc formed at the equatorial region. At any chosen radius above the horizon, the gas is subject to gravity force induced by the Kerr black hole, the centrifugal force due to envelope rotation, and in addition, it feels the perturbative force due to the self-gravity of the matter, enclosed within a given radius.",
"pages": [
2
]
},
{
"title": "2 NUMERICAL CODE AND SETUP",
"content": "We use the general relativistic MHD code called high-accuracy relativistic magnetohydrodynamics (HARM), originally published by Gammie et al. (2003) and further developed by various groups. Our code version, HARM-METRIC, includes the Kerr metric evolution, as first described in Janiuk et al. (2018). The code introduces a conservative, shock-capturing scheme with low numerical viscosity to solve the hyperbolic system of partial differential equations of GR MHD. The numerical scheme uses the plasma energy-momentum tensor, with contributions from matter (gas) and electromagnetic field. For the GR MHD evolution, two fundamental equations are solved for the mass and energy-momentum conservation. An additional constraint is given by the Equation of State (EOS). In the current project, we used analytic form of adiabatic EOS that relates gas pressure with density. This scales with the power of 4/3, as adequate for a relativistic gas of degenerate particles. The HARM code works in dimensionless units of G = c = 1. Conversion coefficients can be found in 1, where the black hole of 3 Solar masses is assumed. Notice that in the plots below, we use geometric unit to express distance, while physical units are used to express time.",
"pages": [
2,
3
]
},
{
"title": "2.1 Initial conditions",
"content": "Initial conditions for our collapsing stellar core are given by quasi-spherical distribution of gas endowed with small angular momentum, concentrated at the equatorial plane (Kr'ol and Janiuk, 2021). The distributions of density and radial velocity are obtained from the Bondi solution, integrated numerically below and above the sonic point. The sonic point is a parameter of our model, and here it is assumed at 80 r g . ➤ ❙ ➤ ➤ ➤❙ ➥ ➥ ? ❏ ■ ✖ Below this point, matter flows into black hole supersonically, and reaches the speed of light at the horizon. We illustrate the initial condition in Figure 1, left panel. Density of the gas is normalized to physical units (given in cgs on the plot), assuming that the collapsing star has the initial mass of 25 Solar masses. This mass is enclosed within our computational domain with outer radius of a R out = 1000 r g . The plot shows only the innermost region, of 100 r g . Most mass of the core is located very near to the center, as it represents the evolved state of stellar evolution with a compact (iron) core formed. In the initial conditions, we also introduce a small angular momentum imposed on the spherically distributed gas. The specific angular momentum is normalized by the parameter S , with respect to that at the innermost stable circular orbit (ISCO). In addition, the rotation velocity scales with the polar angle, to be maximal at the equator, θ = π/ 2. with Notice that the radius r ISCO in Kerr geometry depends on the black hole spin. In this proceeding, we show results obtained for the value of initial black hole spin a 0 = 0 . 5. We use several values of rotation parameter, as denoted on the plots in next sections. After the onset of collapse, the rotation of gas induces formation of a mini-disk, i.e. toroidal structure, located at the equatorial plane. The density distribution becomes no longer spherical. Also, the radial velocity is decreased, as the gas is subject to a centrifugal barrier. Flow is falling into the black hole with supersonic speed from the poles, while at the equator the speed is subsonic. Map on the Figure 1, right panel, shows the flow distribution at time t=0.089 s, for the model normalized with rotation parameter S=1.4. This means that the specific angular momentum is above critical value (S=1) which allows for the formation of rotationally supported torus. Sonic surface, Mach=1, is plotted with a solid line, and marks the location of a transonic shock at the equatorial region.",
"pages": [
3,
4
]
},
{
"title": "3 IMPACT OF SELF-GRAVITY ON THE COLLAPSE",
"content": "In our new simulations, both the mass and angular momentum accreted onto the event horizon -and used to update the Kerr metric coefficients- are now modified by the perturbation acting on the metric in the region above the horizon due to the self-gravity force that the gas feels at a given distance from the horizon. These perturbative terms are calculated from the stress-energy tensor. Therefore, in addition to the two equations governing the growth of black hole mass and spin via the mass and angular momentum transfer through the horizon, as given below, (Kr'ol and Janiuk, 2021), we now add perturbative terms to mass and angular momentum, computed at every radius above the event horizon. The terms computed in addition to mass and angular momentum changes (Janiuk et al., 2018) as these self-gravity perturbations, are integrated at each grid point in the radial direction and at each time. They affect the change of Kerr metric coefficients, which are sensitive to the mass and spin updates. The dimensionless black hole spin, a, evolves as a result of black hole mass and angular momentum changes due to accretion of mass under the horizon, and is additionally changed due to self-gravity of the collapsing core. The numerical method has been described in detail in Janiuk et al. (2023). Below, we compare the results of self-gravitating collapsar models to the runs without self-gravity, in order to emphasize the difference and to investigate the role of self-gravity in the collapsar physics. As shown in Figure 2, the results are strongly sensitive to the adopted self-gravity effects, and also weakly sensitive to the rotation of the collapsing envelope. The latter is normalized with respect to the critical angular momentum, for which the flow is circularized at the innermost stable orbit, ISCO (Kr'ol and Janiuk, 2021). In addition, the rotation velocity scales with the polar angle, so that at the equator, the rotation of the star is maximal. We notice that the larger the initial rotation magnitude, the longer it takes for the black hole mass to evolve. The non-SG simulations end with very different final black hole mass, depending on the rotation parameter. In contrast, the self-gravity of the envelope can speed up the evolution of the collapsing stellar core significantly. Also, accretion rate and its fluctuations are of much higher amplitude when self-gravity effects taken into account. Without selfgravity, there are longer time intervals where there is considerably less fluctuation of the accretion rate; in this case, there exist only some small oscillations in the accretion rate during some time intervals (around 0.2 s for S=1.4, and 0.4-0.5 s for S=2).",
"pages": [
4,
5,
7
]
},
{
"title": "3.1 Instabilities on the collapsing core",
"content": "As an effect of self-gravity we observe density inhomogeneities and formation of the accretion shocks in all our models, regardless of the initial black hole spin, or rotation parameter of the collapsar. First, there appears an equatorial outflow of matter, which reaches radii of up to about 80 r g and is then stalled in the transonic shock. The small inhomogeneities in the pressure and density at the chosen time intervals, are visible in more detail in the plots below, in Figure 3 and in Figure 4, respectively. We quantify the inhomogeneities in the collapsar by computing the radial derivatives of density an pressure at specific times, and locations. We identify the mechanism for their creation as the SGI instability (Self-Gravity Interfacial instability) and we compare its strength with another well-known hydrodynamical instability, the Rayleigh-Taylor (RT) instability. Their growth rates are given as below (Kifonidis et al., 2003; Hunter Jr et al., 1997). The RT and SGI instabilities result in very similar configurations at density snapshots. However, they have their own characteristics, which allows us to differentiate between them. As self-gravity has no 'preferred' direction, it is destabilizing across all density interfaces, while an interface is RT-unstable only if the heavy fluid is on top of the light fluid. It has also been confirmed that RT instability is characterized by dense spikes penetrating the tenuous fluid, whereas the SGI develops with tenuous spikes streaming into the denser fluid. We find that SGI instability seems to dominate over RT instability and produces the inhomogeneities. In particular, we checked that the growth rates of RT, are having imaginary values, as computed at radii between 20 and 25 r g , around the mixing boundary. Finally, we investigated the formation of transonic shocks in the collapsars. In Figure 5 we present radial profiles of Mach number at some specific time snapshots, for models with S = 2 and a 0 = 0 . 5. The left panel shows the profiles in the selfgravitating case, while the plot in the right panel shows those of self-gravitating magnetized case (we introduced a weak vertical magnetic field in the initial condition). For the sake of more visibility, we provide zommed-in inset panels representing the inner regions. We observe the sonic front expansion, and also some transient shock formation during the collapse. At early times, the small transonic shocks appear around 100 r g and they present a moderate density contrast (pre-shock to post-shock density ratio R = ρ 1 /ρ 2 ∼ 10). Such shocks also appear at later times. Their formation is enhanced by the self-gravity effects. We find that magnetic field does not make any significant difference on the shock expansion timescales, but it affects the strength of the shock, consistently with previous studies (Komissarov, 1999).",
"pages": [
7,
9
]
},
{
"title": "4 CONCLUSIONS",
"content": "In this work, we show numerical models of the collapsing stellar core where we account for the dynamical evolution of central black hole mass and its spin. The related coefficients of the Kerr space-time metric are evolved accordingly, at every time step. In addition, we calculate the self-gravity of the stellar envelope and we add the relevant perturbative terms to the dynamical evolution of the black hole spin parameter. The last modification of the model turned out to have an impact on the global evolution of the collapsing star, and produces dramatic fluctuations in the accretion rate at the initial phase of collapse. More importantly, it also plays crucial role in development of the SGI interfacial instability in its specific regions. We identified inhomogeneities in density and pressure distributions which arise due to self-gravity, and we concluded that the SGI instability dominates over the RT, as its growth rate is positive in the regions of mixing boundaries.",
"pages": [
9
]
},
{
"title": "ACKNOWLEDGEMENTS",
"content": "The present work was supported by the grant DEC-2019/35/B/ST9/04000 from Polish National Science Center. We made use of computational resources of the PL-Grid infrastructure, under grant pglgrb6, and Warsaw University ICM. D. glyph[suppress]L. K. was supported by the Polish National Science Center Dec-2019/35/O/ST9/04054 and N. Sh. D. was supported by Iran National Science Foundation (INSF) under project number No.4013178 and also acknowledges Ferdowsi University of Mashhad (FUM), Iran, and the FUM Sci-HPC center. Prof. Shahram Abbassi also deserves gratitude for his accompaniment to N.Sh.D. in this project. A.J. acknowledges the Czech-Polish mobility program (M ˇ SMT 8J20PL037 and PPN/BCZ/2019/1/00069).",
"pages": [
9
]
},
{
"title": "REFERENCES",
"content": "Gammie, C. F., McKinney, J. C. and T'oth, G. (2003), HARM: A Numerical Scheme for General Relativistic Magnetohydrodynamics, Astrophysical Journal , 589 (1), pp. 444457, arXiv: astro-ph/0301509 . Hunter Jr, J. H., Whitaker, R. W. and Lovelace, R. V. (1997), Kelvin-helmholtz and thermal-dynamic instabilities with self-gravity: a new gravitational interface instability, The Astrophysical Journal , 482 (2), p. 852. Janiuk, A., Shahamat Dehsorkh, N. and Kr'ol, D. glyph[suppress]L. (2023), Self-gravitating collapsing star and black hole spin-up in long gamma ray bursts, Astronomy & Astrophysics , 677 , A19, arXiv: 2304.01342 . Janiuk, A., Sukova, P. and Palit, I. (2018), Accretion in a Dynamical Spacetime and the Spinning Up of the Black Hole in the Gamma-Ray Burst Central Engine, Astrophysical Journal , 868 (1), 68, arXiv: 1810.05261 . Kifonidis, K., Plewa, T., Janka, H.-T. and Muller, E. (2003), Non-spherical core collapse supernovae-i. neutrino-driven convection, rayleigh-taylor instabilities, and the formation and propagation of metal clumps, Astronomy & Astrophysics , 408 (2), pp. 621-649. Komissarov, S. S. (1999), Numerical simulations of relativistic magnetized jets, MNRAS , 308 (4), pp. 1069-1076. Kr'ol, D. glyph[suppress]L. and Janiuk, A. (2021), Accretion-induced Black Hole Spin-up Revised by Numerical General Relativistic MHD, Astrophysical Journal , 912 (2), 132, arXiv: 2104. 00741 .",
"pages": [
10
]
}
]
2024arXiv240117683Z
https://arxiv.org/pdf/2401.17683.pdf
<document>
<text><location><page_1><loc_14><loc_95><loc_21><loc_96></location>, 1-11 (2023)</text>
<section_header_level_1><location><page_1><loc_7><loc_86><loc_92><loc_90></location>Very blue-shifted broad H /u1D6FC in a low redshift Type-1.9 AGN: a disk emitter or a recoiling black hole scenario</section_header_level_1>
<text><location><page_1><loc_7><loc_82><loc_24><loc_84></location>Xue-Guang Zhang 1 /uni2605</text>
<text><location><page_1><loc_7><loc_81><loc_89><loc_82></location>1 Guangxi Key Laboratory for Relativistic Astrophysics, School of Physical Science and Technology, GuangXi University, 530004, Nanning, P. R. China</text>
<text><location><page_1><loc_7><loc_77><loc_16><loc_78></location>1 February 2024</text>
<section_header_level_1><location><page_1><loc_7><loc_73><loc_15><loc_74></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_7><loc_57><loc_92><loc_72></location>In this manuscript, very blue-shifted broad H /u1D6FC with shifted velocity ∼ 2200km/s is reported in the low redshift Type-1.9 AGN SDSS J1052+1036. Blue-shifted broad emission lines may arise due to the presence of a rotating gas disk around central black hole (BH), but may also be a signature of rare phenomena such as gravitational wave recoil of a supermassive BH (rSMBH) or the presence of a binary BH (BBH) system. Here, due to larger shifted velocity of stronger and wider blue-shifted broad H /u1D6FC , the BBH system is disfavoured. Meanwhile, if this object contained a rSMBH, intrinsic obscuration with E(B-V) ≤ 0.6 should lead to a detectable broad H /u1D6FD , indicating the rSMBH scenario not preferred. We find that the blue-shifted broad H /u1D6FC can be well explained by emission from an AGN disk, indicating that SDSS J1052+1036 is likely a disk-emitting AGN. In order to determine which scenario, a rSMBH or a disk emitter, is more preferred, a re-observed spectrum in 2025 can provide robust clues, with a disk emitter probably leading to clear variations of peak positions, peak separations and/or peak intensity ratios in broad H /u1D6FC , but with a rSMBH scenario probably leading to no variations of peak separations in broad H /u1D6FC .</text>
<text><location><page_1><loc_7><loc_54><loc_82><loc_56></location>Key words: galaxies:active - galaxies:nuclei - quasars:emission lines - quasars: individual (SDSS J1052+1036)</text>
<section_header_level_1><location><page_1><loc_7><loc_48><loc_21><loc_49></location>1 INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_7><loc_26><loc_48><loc_47></location>Shifted broad emission lines relative to stellar absorption features (or narrow emission lines) may be indicators of a gravitational wave recoiling supermassive black hole (rSMBH) in called offnucleus active galactic nuclei (AGN), due to gravitational wave carried off linear momentum leading central BH being kicked away from central region of AGN, as discussed in Bekenstein (1973); Madau & Quataert (2004); Merritt et al. (2006); Volonteri (2007); Blecha & Loeb (2008); Komossa & Merritt (2008a); Blecha et al. (2016). Broad emission lines from broad emission line regions (BLRs) bound to a rSMBH with a large kick velocity can lead to blue-shifted broad emission lines relative to narrow emission lines of AGN, due to no effects of a rSMBH on NLRs (narrow emission line regions). Until now, there are a few individual AGN and samples of AGN reported with blue-shifted broad emission lines, and expected rSMBH scenarios have been discussed in the literature.</text>
<text><location><page_1><loc_7><loc_10><loc_48><loc_26></location>Komossa et al. (2008) have reported SDSS J0927+2943 ( /u1D467 ∼ 0 . 713) with blue-shifted velocities 2650km/s in broad emission lines, to support a rSMBH scenario. However, Bogdanovic et al. (2009) have discussed a binary BH (BBH) system with mass ratio 0.1 also leading to the shifted features in SDSS J0927+2943. Shields et al. (2009) have reported blue-shifted velocity 3500km/s in broad H /u1D6FD in SDSS J1050, however, BLRs lying into central accretion disk (=disk emitter) would be preferred to explain the blue-shifted broad H /u1D6FD , rather than the rSMBH scenario. Steinhardt et al. (2012) have reported very blue-shifted broad emission lines in SDSS J0956+5128, however, either an extreme disk emitter or a rSMBH is not the preferred scenario to explain all of the observed features, especially</text>
<unordered_list>
<list_item><location><page_1><loc_7><loc_6><loc_36><loc_7></location>/uni2605 Corresponding author Email: xgzhang@gxu.edu.cn</list_item>
</unordered_list>
<text><location><page_1><loc_51><loc_33><loc_92><loc_50></location>the different profiles between broad Balmer lines and broad Mg /i.pc/i.pc line. Kim et al. (2017) have discussed the rSMBH candidate of CXO J1015+6259 ( /u1D467 ∼ 0 . 35) with blue-shifted velocity 175km/s in broad emission lines. Kalfountzou et al. (2017) have shown a rSMBH is one proposed scenario to explain the three strong emission-line nuclei with velocity offset 250km/s in SDSS J1056+5516 ( /u1D467 ∼ 0 . 256), as well as a triple BH accreting system. Kim et al. (2018) have applied an oscillating rSMBH scenario to explain the broad emission line variability properties in Mrk1018. Chiaberge et al. (2017, 2018); Morishita et al. (2022) have shown that the quasar 3C186 ( /u1D467 ∼ 1 . 07) have blue-shifted velocity 2140km/s in broad emission lines, consistent with expected results by a rSMBH.</text>
<text><location><page_1><loc_51><loc_13><loc_92><loc_33></location>Meanwhile, there are samples of AGN with blue-shifted broad emission lines. Eracleous et al. (2012) and followed in Runnoe et al. (2015, 2017) have reported a sample of tens of low redshift (z<0.7) SDSS quasars with blue-shifted velocities larger than 1000km/s in broad H /u1D6FD , and discussed that BBH systems should be preferred in a fraction of the candidates, after carefully checked changes of peak velocities through multi-epoch spectra. Lena et al. (2014) have shown 10 rSMBH candidates in nearby galaxies with small displacements betweencentral activity region and center of galaxy. Kim et al. (2016) have reported a sample of candidates with mean blue-shifted velocity about 150km/s for rSMBHs in SDSS quasars with redshift less than 0.25. Ward et al. (2021) have shown nine AGN that may be spatially offset from their host galaxies and are considered as candidates for rSMBHs.</text>
<text><location><page_1><loc_51><loc_6><loc_92><loc_13></location>Based on the reported candidates of AGN with blue-shifted broad emission lines, besides the rSMBH scenarios, either the BBH or disk emitter hypotheses can be applied. Moreover, as discussed in Komossa & Merritt (2008b); Shen et al. (2019), candidates of rSMBHs with large recoiling velocities at low redshift are ex-</text>
<figure>
<location><page_2><loc_9><loc_51><loc_91><loc_92></location>
<caption>Figure 1. Top region of the top panel shows the SSP method determined the best descriptions (solid red line) to the SDSS spectrum (solid dark green line) with emission lines being masked out. Solid blue line and solid pink line show the determined host galaxy contributions and the power law AGN continuum emissions, vertical dashed red line marks the blue-shifted broad H /u1D6FC . Bottom region of the top panel shows the pure line spectrum calculated by the SDSS spectrum minus the sum of the host galaxy contributions and the power law AGN continuum emissions. Top regions and bottom regions of the bottom panels show the best fitting results (solid red line) and the corresponding residuals (the line spectrum minus the best fitting results and then divided by uncertainties of the SDSS spectrum) to the absorption features (solid dark green line) around the Ca /i.pc/i.pc H+K (bottom left panel) and around the Mg /i.pc (bottom right panel). In the bottom region of each bottom panel, horizontal red dashed lines show residuals= ± 1, respectively.</caption>
</figure>
<text><location><page_2><loc_7><loc_25><loc_48><loc_36></location>tremely rare. Here, a candidate at redshift 0.088 is reported with blue-shifted velocity ∼ 2200km/s in broad H /u1D6FC in a Type-1.9 AGN SDSS J105232.97+103620.08 (=SDSS J1052+1036), with different scenarios discussed. This manuscript is organized as follows. Section 2 presents the spectroscopic results of the Type-1.9 AGN SDSS J1052+1036. Section 3 gives main discussions. Section 4 gives our final conclusions. And the cosmological parameters have been adopted as /u1D43B 0 = 70km · s -1 Mpc -1 , /uni03A9 /uni039B = 0 . 7 and /uni03A9 m = 0 . 3.</text>
<section_header_level_1><location><page_2><loc_7><loc_20><loc_20><loc_21></location>2 MAIN RESULTS</section_header_level_1>
<text><location><page_2><loc_7><loc_8><loc_48><loc_19></location>SDSS J1052+1036 is selected as the subject of this manuscript, due to its very blue-shifted broad H /u1D6FC , while studying properties of double-peaked narrow emission lines in low redshift ( /u1D467 < 0 . 35) SDSS quasars including some objects reported in the sample of Ge et al. (2012). SDSS J1052+1036 has its SDSS spectrum (platemjd-fiberid=1602-53117-0243) with signal-to-noise about 18 shown in top left panel of Fig. 1 with apparently shifted broad H /u1D6FC marked by vertical dashed red line.</text>
<text><location><page_2><loc_9><loc_6><loc_48><loc_7></location>In order to measure the emission lines as well as to measure the stel-</text>
<text><location><page_2><loc_51><loc_6><loc_92><loc_36></location>lar velocity dispersion, the commonly accepted SSP (Simple Stellar Population) method is applied to determine host galaxy contributions in SDSS J1052+1036. More detailed descriptions on the SSP method can be found in Bruzual & Charlot (2003); Kauffmann et al. (2003); Cid Fernandes et al. (2005); Cappellari (2017). The SSP method has also been applied in our previous papers Zhang (2021a,b,d, 2022a,b). Here, we briefly describe the SSP method. The 39 simple stellar population templates from Bruzual & Charlot (2003); Kauffmann et al. (2003) are applied to describe stellar lights, combined with a power law function to describe the AGN continuum. When the SSP method is applied, narrow emission lines are masked out by full width at zero intensity about 450km/s, and the spectrum with wavelength range from 6450Å to 6750Å are also masked out due to the strongly broad H /u1D6FC . Then, through the Levenberg-Marquardt least-squares minimization technique (the MPFIT package), SDSS spectrum in rest frame with emission lines being masked out can be well described. The best descriptions and the corresponding line spectrum (SDSS spectrum minus the best descriptions) are shown in the top panel of Fig. 1 with /u1D712 2 / /u1D451 /u1D45C /u1D453 ∼ 1 . 26 (the summed squared residuals divided by degree of freedom). Considering the totally obscured broad H /u1D6FD , the determined red power law continuum emissions was</text>
<figure>
<location><page_3><loc_7><loc_34><loc_91><loc_92></location>
<caption>Figure 2. Top left panel shows the best fitting results (solid red line) (top region) and the corresponding residuals (bottom region) to the emission lines around the H /u1D6FD (solid dark green line) by multiple Gaussian functions. In top region of the top left panel, solid pink lines show the three Gaussian components in the narrow H /u1D6FD , solid cyan lines show the six Gaussian components in the [O /i.pc/i.pc/i.pc ] doublet. Top right panel shows the best fitting results (solid red line) (top region) and the corresponding residuals (bottom region) to the emission lines around the H /u1D6FC (solid dark green line) by multiple Gaussian functions. In top region of the top right panel, solid pink lines show the three Gaussian components in the narrow H /u1D6FC , solid cyan lines shows the six Gaussian components in the [N /i.pc/i.pc ] doublet, dashed purple lines and dashed cyan lines show the Gaussian components in the [O /i.pc ] and [S /i.pc/i.pc ] doublets, thick dashed blue lines show the two broad Gaussian components in the broad H /u1D6FC . Middle left panel shows the best fitting results (solid red line) (top region) and the corresponding residuals (bottom region) to the emission lines around the H /u1D6FC , with the elliptical accretion disk model applied to describe the broad H /u1D6FC . In top region of the middle left panel, solid blue line shows the determined broad H /u1D6FC by the elliptical accretion disk model, dotted blue line and dot-dashed blue line represent the expected line profiles of the broad H /u1D6FC in Jul. 2025 with considering the standard elliptical accretion disk model applied with anti-clockwise rotation and clockwise processions respectively, dashed blue line shows the determined broad H /u1D6FC by a circular accretion disk model with /u1D452 = 0. Due to totally similar descriptions to the narrow emission lines, the Gaussian components are only shown in the top right panel for the narrow emission lines around the H /u1D6FC . Middle right panel shows the best fitting results (solid red line) (top region) and the corresponding residuals (bottom region) to the emission lines around the H /u1D6FC , with the circular accretion disk plus spiral arm model applied to describe the broad H /u1D6FC . In top region of the middle right panel, solid blue line shows the model determined broad H /u1D6FC , dotted blue line shows the expected line profile of the broad H /u1D6FC determined by the circular disk plus spiral arm model with different /u1D434 . Bottom panels show the best fitting results (solid red line) (top regions) and the corresponding residuals (bottom regions) to the emission lines around H /u1D6FC , with the elliptical accretion disk model with /u1D449 /u1D460 = -265km/s (bottom left panel) and with /u1D449 /u1D460 = 231km/s (bottom right panel) applied to describe the broad H /u1D6FC . In top regions of the bottom panels, dotted blue line and dot-dashed blue line represent the expected line profiles of broad H /u1D6FC in Jul. 2025 with considering the standard elliptical accretion disk model with /u1D449 /u1D460 = -265km/s (bottom left panel) and with /u1D449 /u1D460 = 231km/s (bottom right panel) applied with anti-clockwise rotation and clockwise processions respectively.</caption>
</figure>
<figure>
<location><page_4><loc_11><loc_71><loc_47><loc_91></location>
</figure>
<figure>
<location><page_4><loc_51><loc_71><loc_92><loc_91></location>
</figure>
<figure>
<location><page_4><loc_9><loc_47><loc_47><loc_67></location>
</figure>
<figure>
<location><page_4><loc_53><loc_47><loc_90><loc_67></location>
<caption>Figure 3. MCMC technique determined two-dimensional posterior distributions in contour of the model parameters in the standard elliptical accretion disk model applied to describe the broad H /u1D6FC . In each panel, sold circle plus error bars in red mark the positions of the accepted values and the corresponding 1 /u1D70E uncertainties of the model parameters. The number densities related to different colors are shown in color bar in the left region of each panel.</caption>
</figure>
<text><location><page_4><loc_7><loc_26><loc_48><loc_37></location>acceptable, due to seriously obscurations on central continuum emissions. Meanwhile, we measured the stellar velocity dispersion to be 113 ± 10km/s. Moreover, in order to show the stellar velocity dispersion, the bottom panels of Fig. 1 show the SSP method determined the best descriptions and the corresponding residuals (SDSS spectrum minus the best descriptions and then divided by the uncertainties of SDSS spectrum) to the absorption features around Ca /i.pc/i.pc H+K from 3880 to 4400Å and around Mg /i.pc from 5050 to 5300Å.</text>
<text><location><page_4><loc_7><loc_6><loc_48><loc_26></location>After subtractions of the host galaxy contributions, more apparent blue-shifted broad H /u1D6FC can be found. And the emission lines can be measured by multiple Gaussian functions, similar as what we have recently done in Zhang (2021a,b, 2022a,b,c). Considering the double-peaked features in the narrow emission lines (especially in the narrow Balmer lines, the [O /i.pc/i.pc/i.pc ] doublet and the [N /i.pc/i.pc ] doublet) in SDSS J1052+1036, three Gaussian functions are applied to describe each narrow emission line: two narrow Gaussian components for the double-peaked feature and one Gaussian component for the probably extended emissions underneath the double-peaked feature. Therefore, for the emission lines within the rest wavelength from 4830Å to 5020Å and from 6200Å to 6800Å, there are three Gaussian functions applied to describe the double-peaked narrow H /u1D6FD (H /u1D6FC ), one broad Gaussian function to describe the probable broad H /u1D6FD , two</text>
<text><location><page_4><loc_51><loc_25><loc_92><loc_37></location>broad Gaussian functions to describe the broad H /u1D6FC , six Gaussian functions to describe the [O /i.pc/i.pc/i.pc ] /u1D706 4959 , 5007Å doublet, six Gaussian functions to describe the [N /i.pc/i.pc ] /u1D706 6549 , 6585Å doublet, one Gaussian function to describe each line in the [O /i.pc ] /u1D706 6300 , 6363Å and the [S /i.pc/i.pc ] /u1D706 6716 , 6731Å doublets without apparent double-peaked features. When the functions above are applied, each Gaussian component has line intensity not smaller than zero, and the corresponding [O /i.pc/i.pc/i.pc ] ([N /i.pc/i.pc ]) components have the same redshift and the same line width and have the flux ratio to be fixed to the theoretical value 3.</text>
<text><location><page_4><loc_51><loc_6><loc_92><loc_24></location>Then, through the MPFIT package, the best fitting results (in top regions) and the corresponding residuals (in bottom regions) to the emission lines around H /u1D6FD and H /u1D6FC are shown in the top left panel and the top right panel of Fig. 2 with /u1D712 2 / /u1D451 /u1D45C /u1D453 ∼ 0 . 89and /u1D712 2 / /u1D451 /u1D45C /u1D453 ∼ 0 . 88, respectively. Based on the best fitting results, it is not necessary to consider broad Gaussian component in the H /u1D6FD , because the determined line width and line flux (around to zero) of the broad Gaussian component are smaller than their corresponding uncertainties, indicating not apparent broad H /u1D6FD in SDSS J1052+1036. Moreover, as shown in the top right panel of Fig. 2, each line in the [O /i.pc ] and [S /i.pc/i.pc ] doublets can be well described by one Gaussian component, and whether two or three Gaussian components applied to describe each line in the [O /i.pc ] and [S /i.pc/i.pc ] doublets have few effects on our discussed</text>
<figure>
<location><page_5><loc_9><loc_72><loc_49><loc_92></location>
</figure>
<figure>
<location><page_5><loc_52><loc_72><loc_93><loc_92></location>
</figure>
<figure>
<location><page_5><loc_9><loc_50><loc_48><loc_70></location>
</figure>
<figure>
<location><page_5><loc_52><loc_50><loc_91><loc_70></location>
</figure>
<figure>
<location><page_5><loc_10><loc_29><loc_48><loc_49></location>
<caption>Figure 4. MCMCtechnique determined two-dimensional posterior distributions in contour of the model parameters in the circular disk plus arm model applied to describe the broad H /u1D6FC . In each panel, sold circle plus error bars in red mark the positions of the accepted values and the corresponding 1 /u1D70E uncertainties of the model parameters. The number densities related to different colors are shown in color bar in the left region of each panel.</caption>
</figure>
<text><location><page_5><loc_7><loc_12><loc_48><loc_19></location>results on the broad H /u1D6FC . The parameters of the Gaussian components applied to describe the emission lines are listed in the Table 1. Based on the best descriptions to the stellar absorption features in Fig. 1 and the best fitting results to the broad H /u1D6FC in the top right panel of Fig. 2, about 2200km/s blue-shifted broad H /u1D6FC can be confirmed.</text>
<text><location><page_5><loc_7><loc_6><loc_48><loc_12></location>Besides the broad H /u1D6FC described by two Gaussian components, the blue-shifted broad H /u1D6FC can also be described by the known elliptical accretion disk model discussed in Eracleous et al. (1995), similar as what we have recently done on double-peaked broad</text>
<text><location><page_5><loc_51><loc_7><loc_92><loc_19></location>emission lines in Zhang (2021c, 2022a). The elliptical accretion disk model have seven model parameters, inner and out boundaries [ /u1D45F 0 , /u1D45F 1 ] in unit of /u1D445 /u1D43A (Schwarzschild radius), inclination angle /u1D456 of disk-like BLRs, eccentricity /u1D452 , orientation angle /u1D719 0 of elliptical rings, local broadening velocity /u1D70E /u1D43F in units of km / s, line emissivity slope /u1D45E ( /u1D453 /u1D45F ∝ /u1D45F -/u1D45E ). In order to obtain more reliable model parameters and corresponding uncertainties, the Maximum Likelihood method combining with the MCMC (Markov Chain Monte Carlo) technique (Foreman-Mackey et al. 2013) is applied. The</text>
<figure>
<location><page_6><loc_9><loc_76><loc_34><loc_92></location>
</figure>
<figure>
<location><page_6><loc_37><loc_76><loc_64><loc_92></location>
</figure>
<figure>
<location><page_6><loc_66><loc_76><loc_92><loc_92></location>
<caption>Figure 5. MCMCtechnique determined two-dimensional posterior distributions in contour of the model parameters in the pure symmetric circular disk model applied to describe the broad H /u1D6FC . In each panel, sold circle plus error bars in red mark the positions of the accepted values and the corresponding 1 /u1D70E uncertainties of the model parameters. The number densities related to different colors are shown in color bar in the left region of each panel.</caption>
</figure>
<text><location><page_6><loc_7><loc_41><loc_48><loc_66></location>evenly prior distributions of the seven model parameters are accepted with the following limitations, log ( /u1D45F 0 ) ∈ [ 1 , 3 ] , log ( /u1D45F 1 ) ∈ [ 2 , 6 ] ( /u1D45F 1 > /u1D45F 0 ), log ( sin ( /u1D456 )) ∈ [-3 , 0 ] , log ( /u1D45E ) ∈ [-1 , 1 ] , log ( /u1D70E /u1D43F ) ∈ [ 2 , 4 ] , log ( /u1D452 ) ∈ [-5 , 0 ] , log ( /u1D719 0 ) ∈ [-5 , log ( 2 × /u1D70B )] . The determined best fitting results and corresponding residuals to the emission lines around H /u1D6FC are shown in the top region and the bottom region of the middle left panel of Fig. 2 with /u1D712 2 / /u1D451 /u1D45C /u1D453 = 324 . 41 / 391 ∼ 0 . 83. And the model determined best descriptions to the broad H /u1D6FC are shown as solid blue line in the top region of the middle left panel of Fig. 2. The MCMC technique determined posterior distributions of the model parameters in the elliptical accretion disk model are shown in Fig. 3. The determined parameters and the corresponding 1 /u1D70E uncertainties are listed in the Table 2. Moreover, as discussed in Zhang (2022a), clean double-peaked broad line emission features can lead to solely determined model parameters in the elliptical accretion disk model. Therefore, there are no further discussions on whether is there solely determined model parameters, due to the apparent blue peak in broad H /u1D6FC in SDSS J1052+1036.</text>
<text><location><page_6><loc_7><loc_6><loc_48><loc_38></location>Meanwhile, as suggested in Eracleous et al. (2009); Storchi-Bergmann et al. (2003, 2017), rather than the elliptical accretion disk model, the spiral arm rotation is the preferred explanation for most disk emitter profile evolution. Therefore, the circular disk plus spiral arm model with 10 model parameters is also applied to describe the shifted broad H /u1D6FC of SDSS J1052+1036. Besides the model parameters ( /u1D452 = 0) applied in the elliptical accretion disk model, four additional model parameters are applied to describe structures of spiral arms, the azimuthal width /u1D6FF , the pitch angle /u1D45D and the innermost radius /u1D45F /u1D45A of the spiral arm, and the brightness contrast /u1D434 between the spiral arm and the underlying axisymmetric disk. Then, based on the new emissivity formula shown in Equation (2) in Storchi-Bergmann et al. (2003) and accepted /u1D45F /u1D45A = /u1D45F 0 , the best descriptions and the corresponding residuals to the emission lines around the H /u1D6FC are shown in the middle right panel of Fig. 2 with /u1D712 2 / /u1D451 /u1D45C /u1D453 = 287 . 88 / 389 ∼ 0 . 74. And the model determined best descriptions to the broad H /u1D6FC are shown as solid blue line in the top region of the middle right panel of Fig. 2. The MCMC technique determined posterior distributions of the model parameters in the circular disk plus arm model are shown in Fig. 4. The determined model parameters and the corresponding 1 /u1D70E uncertainties are also listed in the Table 2 for the circular disk plus arm model.</text>
<table>
<location><page_6><loc_51><loc_28><loc_91><loc_62></location>
<caption>Table 1. parameters of the emission line components described by Gaussian functions</caption>
</table>
<text><location><page_6><loc_51><loc_23><loc_92><loc_28></location>Notice: For the Gaussian components, the first column shows which line is measured, the Second, third, fourth columns show the measured line parameters: center wavelength /u1D706 0 in unit of Å, line width (second moment) /u1D70E in unit of Å and line flux in unit of 10 -17 erg / s / cm 2 .</text>
<section_header_level_1><location><page_6><loc_51><loc_19><loc_67><loc_20></location>3 MAIN DISCUSSIONS</section_header_level_1>
<section_header_level_1><location><page_6><loc_51><loc_17><loc_87><loc_18></location>3.1 A kpc-scale dual core system in SDSS J1052+1036?</section_header_level_1>
<text><location><page_6><loc_51><loc_6><loc_92><loc_16></location>Double-peaked [O /i.pc/i.pc/i.pc ] /u1D706 5007Å can be seen in the spectrum of SDSS J1052+1036, as well as shown in Ge et al. (2012), widely indicating a kpc-scale dual core system (Zhou et al. 2004; Xu & Komossa 2009; Fu et al. 2011; Wang et al. 2019). Based on the measured doublepeaked features in the [O /i.pc/i.pc/i.pc ] /u1D706 5007Å, the peak separation is about 350 ± 22km/s in SDSS J1052+1036, leading the broad emission lines from the assumed central two cores to have the same peak separa-</text>
<text><location><page_7><loc_13><loc_85><loc_42><loc_90></location>elliptical accretion disk model /u1D45F 0 = 38 ± 2, /u1D45F 1 = 490 ± 35, sin ( /u1D456 ) = 0 . 48 ± 0 . 04 /u1D45E = 0 . 17 ± 0 . 01, /u1D452 = 0 . 56 ± 0 . 03, /u1D70E /u1D43F = 850 ± 40km / s = ·</text>
<text><location><page_7><loc_24><loc_85><loc_31><loc_86></location>/u1D719 0 237 ± 5</text>
<section_header_level_1><location><page_7><loc_17><loc_82><loc_38><loc_83></location>the circular disk plus spiral arm model</section_header_level_1>
<text><location><page_7><loc_8><loc_78><loc_47><loc_82></location>/u1D45F 0 = 150 ± 60, /u1D45F 1 = 4500 ± 1300, sin ( /u1D456 ) = 0 . 44 ± 0 . 04, /u1D45E = 1 . 82 ± 0 . 21 /u1D70E /u1D43F = 270 ± 90km / s, /u1D719 0 = 179 ± 3 · , /u1D434 = 26 ± 7 /u1D6FF = 93 ± 10 · , /u1D45D = 28 ± 2 ·</text>
<text><location><page_7><loc_8><loc_73><loc_47><loc_77></location>the pure symmetric circular disk model /u1D45F 0 = 18 ± 1, /u1D45F 1 = ( 8 ± 3 ) × 10 5 , sin ( /u1D456 ) = 0 . 37 ± 0 . 01, /u1D45E = 7 . 48 ± 0 . 91 /u1D70E /u1D43F = 650 ± 103km / s, /u1D719 0 = 16 ± 3 ·</text>
<text><location><page_7><loc_13><loc_67><loc_42><loc_72></location>the elliptical accretion disk model with /u1D449 /u1D460 = -265km/s /u1D45F 0 = 31 ± 2, /u1D45F 1 = 246 ± 25, sin ( /u1D456 ) = 0 . 33 ± 0 . 03 /u1D45E = 0 . 66 ± 0 . 05, /u1D452 = 0 . 58 ± 0 . 03, /u1D70E /u1D43F = 916 ± 60km / s /u1D719 0 = 240 ± 6 ·</text>
<text><location><page_7><loc_13><loc_61><loc_42><loc_66></location>the elliptical accretion disk model with /u1D449 /u1D460 = 231km/s /u1D45F 0 = 44 ± 4, /u1D45F 1 = 381 ± 40, sin ( /u1D456 ) = 0 . 38 ± 0 . 04 /u1D45E = 1 . 26 ± 0 . 15, /u1D452 = 0 . 54 ± 0 . 03, /u1D70E /u1D43F = 532 ± 30km / s /u1D719 0 = 240 ± 6 ·</text>
<text><location><page_7><loc_7><loc_49><loc_48><loc_58></location>tion 350km/s. However, the peak separation about 4200 ± 580km/s between the blue-shifted broad component and the red-shifted broad component in the broad H /u1D6FC in SDSS J1052+1036 is about twelve times higher than the peak separation of the double-peaked narrow emission lines. Therefore, the shifted broad H /u1D6FC is not related to a kpc-scale dual core system expected by the double-peaked [O /i.pc/i.pc/i.pc ] in SDSS J1052+1036.</text>
<section_header_level_1><location><page_7><loc_7><loc_45><loc_31><loc_46></location>3.2 A rSMBH in SDSS J1052+1036?</section_header_level_1>
<text><location><page_7><loc_7><loc_31><loc_48><loc_44></location>One another explanation for the blue shifted broad H /u1D6FC in SDSS J1052+1036 is that it is a rSMBH, after considering materials in the BLRs being carried away with the rSMBH. Meanwhile, not a single but two broad Gaussian components in the broad H /u1D6FC in SDSS J1052+1036 are probably indicating asymmetric structures of the BLRs bound to the rSMBH. As discussed in Merritt et al. (2006); Gualandris & Merritt (2008); Komossa & Merritt (2008b), the materials in the BLRs can be bound to a rSMBH within a region with the radius /u1D45F /u1D458 given by</text>
<formula><location><page_7><loc_7><loc_28><loc_48><loc_30></location>/u1D45F /u1D458 ∼ 512 /u1D440 /u1D435 /u1D43B 10 8 M /circledot ( /u1D449 /u1D458 10 3 km / s ) -2 light -days (1)</formula>
<text><location><page_7><loc_7><loc_16><loc_48><loc_27></location>with /u1D440 /u1D435 /u1D43B and /u1D449 /u1D458 as the BH mass and the kick velocity of a rSMBH. Meanwhile, in order to support a rSMBH by blue-shifted broad emission lines, the blue-shifted broad emission component related to the emission materials bound to a rSMBH should be apparent enough, indicating almost all the materials in the original BLRs bound to the rSMBH. Therefore, we can expect that the estimated /u1D45F /u1D458 should be not smaller than the origin BLRs size /u1D445 /u1D435 /u1D43F /u1D445 /u1D460 which can be estimated by the continuum luminosity /u1D43F 5100 at 5100Å (Bentz et al. 2013),</text>
<formula><location><page_7><loc_7><loc_12><loc_48><loc_15></location>/u1D45F /u1D458 light -days ≥ /u1D445 /u1D435 /u1D43F /u1D445 /u1D460 light -days = 10 1 . 555 + 0 . 542 × log ( /u1D43F 5100 10 44 erg / s ) (2)</formula>
<text><location><page_7><loc_7><loc_6><loc_48><loc_12></location>In SDSS J1052+1036 with the well measured stellar velocity dispersion about 113 ± 10km/s, after considering the M-sigma relation discussed in Ferrarese & Merritt (2000); Gebhardt et al. (2000); Kormendy & Ho (2013); Batiste et al. (2017); Bennert et al. (2021)</text>
<text><location><page_7><loc_51><loc_83><loc_92><loc_93></location>for both quiescent and active galaxies and also as discussed in Di Matteo et al. (2005); Johansson et al. (2009) in galaxy merging systems, the BH mass can be estimated as 2 . 5 + 2 . 2 -1 . 3 × 10 7 M /circledot in SDSS J1052+1036, accepted Equation (7) in Kormendy & Ho (2013). Therefore, based on the equation above, we could have /u1D43F 5100 < 4 × 10 43 erg/s, which can lead to apparent blue-shifted broad H /u1D6FC totally related to an expected rSMBH.</text>
<text><location><page_7><loc_51><loc_67><loc_92><loc_83></location>Based on the determined continuum emissions in the top panel of Fig. 1, the observed continuum luminosity at 5100Å is about 6 . 3 × 10 42 erg/s in SDSS J1052+1036. Accepted the intrinsic /u1D43F 5100 should be smaller than 4 × 10 43 erg/s, the intrinsic obscuration should have E(B-V) ≤ 0.6. Then, accepted the intrinsic flux ratio 3.1 of broad H /u1D6FC to broad H /u1D6FD , the expected observed flux ratio of the broad H /u1D6FC to the broad H /u1D6FD should be smaller than 6.2, leading to a detectable blueshifted broad component in the H /u1D6FD in SDSS J1052+1036. Unfortunately, as shown in the top left panel of Fig. 2, there are no detectable broad components in the H /u1D6FD in SDSS J1052+1036. Therefore, the blue-shifted broad H /u1D6FC probably contains weak contributions from a rSMBH scenario in SDSS J1052+1036.</text>
<text><location><page_7><loc_51><loc_54><loc_92><loc_66></location>Unfortunately, the discussions above are not sufficient enough to totally disfavour the rSMBH scenario in SDSS J1052+1036, however, multi-epoch spectroscopic results should provide clear clues to support or to be against the rSMBH scenario. If the expected rSMBH in SDSS J1052+1036 moves rectilinearly (or moves curvilinearly as the case in Mrk 1018 in Kim et al. 2018), very tiny (or no) changes of peak separations between the blue-shifted component and the red-shifted component in the broad H /u1D6FC could be expected in SDSS J1052+1036.</text>
<section_header_level_1><location><page_7><loc_51><loc_50><loc_78><loc_51></location>3.3 A BBH system in SDSS J1052+1036?</section_header_level_1>
<text><location><page_7><loc_51><loc_41><loc_92><loc_49></location>If a BBH system was accepted in SDSS J1052+1036 with the estimated total BH mass 2 . 5 + 2 . 2 -1 . 3 × 10 7 M /circledot by the /u1D440 BH -/u1D70E relation, the two broad Gaussian components in the broad H /u1D6FC could be simply accepted to estimate the observational peak separation about /u1D449 /u1D45D , /u1D45C /u1D44F /u1D460 = 4200 ± 600km / s, leading the upper limit of the space separation /u1D446 of the central two BHs to be</text>
<formula><location><page_7><loc_51><loc_36><loc_92><loc_40></location>/u1D446 < /u1D43A × /u1D440 /u1D435 /u1D43B /u1D449 2 /u1D45D , /u1D45C /u1D44F /u1D460 ∼ 7 . 3 + 11 . 3 -4 . 6 light -days (3)</formula>
<text><location><page_7><loc_51><loc_27><loc_92><loc_36></location>Based on the measured luminosity 2 . 01 × 10 41 erg / s of the observed broad H /u1D6FC or the measured continuum luminosity 6 . 2 × 10 42 erg / s at 5100Å in the rest frame, the estimated BLRs size should be about 7light-days, after considering the correlation between broad H /u1D6FC luminosity and continuum luminosity discussed in Greene & Ho (2005) and the empirical R-L relation discussed in Bentz et al. (2013).</text>
<text><location><page_7><loc_51><loc_6><loc_92><loc_27></location>However, considering SDSS J1052+1036 as a Type-1.9 AGN, serious obscuration indicates the intrinsic BLRs size should be much larger than 7light-days. The BLRs size is similar as the upper limit of space separation of the central two BHs, strongly indicating the two BLRs probably totally mixed, leading to no apparent variability in the peak positions in the broad H /u1D6FC , as discussed in Shen & Loeb (2010). Moreover, under the assumption of a BBH system in SDSS J1052+1036, probable optical quasiperiodic oscillations (Graham et al. 2015a,b; Zheng et al. 2016; Zhang 2022d,e, 2023) should be detected. However, after checking long-term light curves from Catalina Sky Survey (Drake et al. 2009), All-Sky Automated Survey for Supernovae (Shappee et al. 2014; Kochanek et al. 2017) and Zwicky Transient Facility (Bellm et al. 2019; Dekany et al. 2020), there is no significant variability, which can not provide clues to support a BBH system in SDSS J1052+1036.</text>
<text><location><page_8><loc_7><loc_72><loc_48><loc_93></location>Meanwhile, under the assumption of a BBH system in SDSS J1052+1036, considering the stronger and wider broad blue-shifted component in the H /u1D6FC (H /u1D6FC /u1D435 ), the virial BH mass /u1D440 /u1D435 /u1D43B , /u1D435 related to the H /u1D6FC /u1D435 should be simply expected to be 6.4 times larger than the virial BH mass /u1D440 /u1D435 /u1D43B , /u1D445 related to the red-shifted broad component in the H /u1D6FC (H /u1D6FC /u1D445 ), accepted the virialization assumptions to the broad emission lines as discussed in Greene & Ho (2005); Peterson et al. (2004). Here, the factor 6.4 is simply calculated by ( 1036 . 8 144 . 6 ) 0 . 5 ( 35 . 12 22 . 72 ) 2 with 1036.8 and 144.6 (35.12 and 22.72) as the line fluxes (the line widths) of the H /u1D6FC /u1D435 and the H /u1D6FC /u1D445 in SDSS J1052+1036. Then, the H /u1D6FC /u1D435 should have 6.3 times smaller shifted velocity than that of the H /u1D6FC /u1D445 , which is against the measured results that the shifted velocity 2430 ± 50km/s of the H /u1D6FC /u1D435 is larger than the shifted velocity about 1760 ± 530km/s of the H /u1D6FC /u1D445 , indicating a BBH system is disfavoured in SDSS J1052+1036.</text>
<text><location><page_8><loc_7><loc_53><loc_48><loc_72></location>Furthermore, if we accepted the double-peaked narrow emission lines as signs of kpc-scale dual core systems and also accepted the blue-shifted broad H /u1D6FC related to a BBH system, there should be a rare close-pair binary in a triple BH system in SDSS J1052+1036, similar as those discussed in Hoffman & Loeb (2007); Deane et al. (2014). In such a rare close-pair binary in a triple BH system in SDSS J1052+1036, similar results can be expected that the shifted velocity of the H /u1D6FC /u1D435 should be quite smaller than that of the H /u1D6FC /u1D445 . However, whether the red-shifted (or the blue-shifted) narrow emission component in the double-peaked narrow H /u1D6FC is applied to trace the rotating velocity of the close-pair binary BH system in a triple BH system, larger shifted velocity of the H /u1D6FC /u1D435 can be determined than that of the H /u1D6FC /u1D445 . Therefore, a close-pair binary in a triple BH system is disfavoured in SDSS J1052+1036</text>
<section_header_level_1><location><page_8><loc_7><loc_48><loc_33><loc_49></location>3.4 A disk emitter in SDSS J1052+1036?</section_header_level_1>
<text><location><page_8><loc_7><loc_20><loc_48><loc_47></location>Based on the model parameters of the elliptical accretion disk model listed in the Table 2, the expected disk precession period should be about /u1D447 pre ∼ 1040 /u1D440 8 /u1D445 2 . 5 3 /u1D466 /u1D45F . Using the /u1D440 BH -/u1D70E determined BH mass /u1D440 8 ∼ 0 . 25 + 0 . 22 -0 . 13 in units of 10 8 M /circledot and /u1D445 3 as radius in units of 1000 /u1D445 /u1D43A , based on the determined /u1D45F 0 , /u1D45F 1 and /u1D45E , the flux weighted size of the emission regions for the broad H /u1D6FC to the central BH is about 248 /u1D445 /u1D43A , leading to an approximately estimated disk precession period of 8years. As well known, asymmetric structures in accretion disk model are key factors leading to apparent variabilities of the peak positions and the peak separations of the double-peaked broad emission lines due to pure disk precessions, which will provide clues to support a disk emitter in SDSS J1052+1036. If there should be a re-observed spectrum in Jul. 2025 (MJD ∼ 60858), based on the expected precession period of about 8years, the expected line profiles of the broad H /u1D6FC in SDSS J1052+1036 in 2025 are shown as dotted blue line and dot-dashed blue line in the top region of the middle left panel of Fig. 2 with considering the standard elliptical accretion disk model applied with anti-clockwise rotation and clockwise processions respectively.</text>
<text><location><page_8><loc_7><loc_6><loc_48><loc_20></location>Meanwhile, it is necessary to check whether a pure symmetric circular accretion disk model (with eccentricity to be zero) (without spiral arms) can be applied to describe the observed shifted broad H /u1D6FC in SDSS J1052+1036. For a circular accretion disk model with /u1D452 = 0, a similar fitting procedure is applied to describe the broad H /u1D6FC in SDSS J1052+1036, with the final determined fitting results to the broad H /u1D6FC shown as dashed blue line in the top region of the middle left panel of Fig. 2 with corresponding /u1D712 2 / /u1D451 /u1D45C /u1D453 = 469 . 37 / 392 ∼ 1 . 21. The MCMC technique determined posterior distributions of the model parameters in the pure symmetric circular disk model are shown in</text>
<text><location><page_8><loc_51><loc_74><loc_92><loc_93></location>Fig. 5. The determined model parameters and the corresponding 1 /u1D70E uncertainties are also listed in the Table 2 for the pure symmetric circular disk model. Based on the F-test technique similar as what we have recently done in Zhang (2022c), due to the different values of /u1D712 2 and /u1D451 /u1D45C /u1D453 for the different accretion disk models, the confidence level can be determined to be higher than 6 /u1D70E to support that the elliptical accretion disk model and the circular accretion disk plus arm model is preferred than the pure symmetric circular accretion disk model. Unfortunately, only through the single-epoch spectroscopic properties of SDSS J1052+1036, we can not find more clues to support that the pure symmetric circular disk model is totally disfavored in SDSS J1052+1036. Therefore, in the manuscript, the pure symmetric circular disk model is also accepted as a reasonable model to describe the shifted broad H /u1D6FC in SDSS J1052+1036.</text>
<text><location><page_8><loc_51><loc_29><loc_92><loc_73></location>Moreover, if accepted the double-peaked features in the [O /i.pc/i.pc/i.pc ] /u1D706 4959 , 5007Å doublet as signs of a kpc-scale dual core system, a rotating disk emitter (disk emission regions with a rotating velocity related to the orbital motions of central dual cores) contained in a dual core system could also be applied to describe the observed blue-shifted broad H /u1D6FC in SDSS J1052+1036. Considering the double-peaked features in the narrow H /u1D6FC to trace the rotating velocity /u1D449 /u1D460 of the disk emitter in a dual core system, the best fitting results and the corresponding residuals to the emission lines around H /u1D6FC can be re-determined with /u1D449 /u1D460 = -265km/s and with /u1D449 /u1D460 = 231km/s, and shown in the bottom left panel and the bottom right panel of Fig. 2 with corresponding /u1D712 2 / /u1D451 /u1D45C /u1D453 ∼ 1 . 03 and /u1D712 2 / /u1D451 /u1D45C /u1D453 ∼ 1 . 02, respectively. The model determined broad H /u1D6FC after considering /u1D449 /u1D460 are shown as solid blue lines in the top regions of the bottom panels of Fig. 2. Moreover, if considering the elliptical accretion disk model with /u1D449 /u1D460 in SDSS J1052+1036 , there are similar 1 /u1D70E uncertainties of the model parameters as those of the standard elliptical accretion disk model shown in Fig. 3. Therefore, we did not show the posterior distributions of the model parameters for the rotating elliptical disk models, but the model parameters and the corresponding 1 /u1D70E uncertainties are listed in the Table 2. Based on the determined model parameters listed in the Table 2 for the rotating elliptical accretion disk model with /u1D449 /u1D460 , disk precession periods can be estimated as 1.68years and 4.95years, with the central wavelengths of the blue-shifted component and the red-shifted component in the narrow H /u1D6FC applied to determine the /u1D449 /u1D460 . Then, the expected line profiles of the broad H /u1D6FC in SDSS J1052+1036 in Jul. 2025 are shown as dotted blue line and dot-dashed blue line in the top regions of the bottom left panel and the bottom right panel of Fig. 2 with considering the elliptical accretion disk model with /u1D449 /u1D460 = -265km/s and /u1D449 /u1D460 = 231km/s applied with anti-clockwise rotation and clockwise processions respectively.</text>
<text><location><page_8><loc_51><loc_8><loc_92><loc_28></location>Either a rotating elliptical disk emitter contained in a dual core system or a standard elliptical disk emitter can lead to apparent time dependent variations of the peak positions and the peak separations of the shifted broad H /u1D6FC in SDSS J1052+1036 as a disk emitter. Considering different disk precession periods determined by standard accretion disk model and/or rotating disk emitter, if there should be a re-observed spectrum in Jul. 2025, the expected broad H /u1D6FC in 2025 in SDSS J1052+1036 have quite different peak positions and different peak separations from the broad H /u1D6FC in the SDSS spectrum observed in MJD=53117. Therefore, a re-observed spectrum in 2025 should provide clues enough to confirm whether a disk emitter is preferred in SDSS J1052+1036. Unfortunately, unless there are detailed timedependent variabilities of the broad H /u1D6FC in SDSS J1052+1036, it is hard to distinguish a standard elliptical disk emitter from a rotating elliptical disk emitter in a kpc-scale dual core system.</text>
<text><location><page_8><loc_53><loc_6><loc_92><loc_7></location>Furthermore, if considering the circular disk plus spiral arm model</text>
<figure>
<location><page_9><loc_26><loc_66><loc_77><loc_90></location>
<caption>Figure 6. Expected line profiles of the shifted broad H /u1D6FC with only /u1D45F 0 and /u1D45F 1 changed in the different accretion disk models. As described in the legend, besides the solid dark green line representing the observed shifted broad H /u1D6FC described by two broad Gaussian functions, each other dashed line in different color represents the corresponding model expected line profile of the shifted broad H /u1D6FC in different epoch with only /u1D45F 0 and /u1D45F 1 changed.</caption>
</figure>
<text><location><page_9><loc_7><loc_29><loc_48><loc_57></location>in SDSS J1052+1036, the expected disk precession period should be about 2100years, due to the quite large flux weighted size /u1D445 3 ∼ 2 . 3 of the emission regions of the blue-shifted broad H /u1D6FC to the central BH. Meanwhile, the precessing spiral pattern could have precession period more than 100years, considering ten times of the dynamical time of accretion disk as discussed in Storchi-Bergmann et al. (2003). The large precession periods strongly indicates no apparent variabilities of profiles of the broad H /u1D6FC in SDSS J1052+1036, if considering the circular disk plus arm model to describe the shifted broad H /u1D6FC . However, as discussed in Storchi-Bergmann et al. (2003), the model parameters of /u1D434 , /u1D45E are varying in the circular disk plus spiral arm model. The varying parameter of /u1D434 can lead to apparent variability of the peak intensity ratio of the broad H /u1D6FC of SDSS J1052+1036. As an example, if /u1D434 was changed from 26 to 5 in 2025, quite stronger red peak could be possibly expected in the broad H /u1D6FC of SDSS J1052+1036, as shown in the top region of the middle right panel of Fig. 2. Similar results can be found, if the circular plus arm model discussed in a dual core system. Therefore, there are no plots or discussions for the corresponding results based on the circular plus arm contained in a dual core system.</text>
<text><location><page_9><loc_7><loc_16><loc_48><loc_29></location>Before the end of the subsection, an additional point should be noted. Discussions above on variations of the profiles of the shifted broad H /u1D6FC are due to pure disk precessions in SDSS J1052+1036 as a disk emitter, which can lead to apparent variations from the elliptical accretion disk model but no apparent variations from the circular disk model neither from the circular plus arm model. However, effects of variability of AGN activities should also have apparent effects on the variations of profiles of the shifted broad H /u1D6FC in SDSS J1052+1036 as a disk emitter, which will be discussed as follows.</text>
<text><location><page_9><loc_7><loc_6><loc_48><loc_16></location>Sizes /u1D445 /u1D435 /u1D43F /u1D445 /u1D460 of central BLRs of dozens of broad line AGN have been measured through the reverberation mapping technique (Blandford & McKee 1982; Peterson 1993; Peterson et al. 1999; Barth et al. 2015; Du et al. 2018; Shen et al. 2019b; Lu et al. 2023), leading to the known dependence of /u1D445 /u1D435 /u1D43F /u1D445 /u1D460 on AGN continuum luminosity (or on broad line luminosity) (Kaspi et al. 2000; Bentz et al. 2013). Therefore, stronger central AGN emissions can lead to deeper</text>
<text><location><page_9><loc_51><loc_23><loc_92><loc_57></location>ionization boundaries of central BLRs, in other words, stronger AGN emissions can lead to larger /u1D445 /u1D435 /u1D43F /u1D445 /u1D460 . Therefore, if there was apparent variability of AGN activities in SDSS J1052+1036, different inner radius and outer radius in different epochs could be expected from those listed in the Table 2 of the different accretion disk models for shifted broad H /u1D6FC shown in Fig. 2, leading to probably different line profiles. As an example, if we assumed that there was a spectrum observed again at any time but with the re-observed AGN continuum luminosity at 5100Å about 2 times of the AGN continuum luminosity shown in the top panel of Fig. 1, leading the expected /u1D445 /u1D435 /u1D43F /u1D445 /u1D460 to be 1.4 times larger than the /u1D445 /u1D435 /u1D43F /u1D445 /u1D460 for the shifted broad H /u1D6FC shown in Fig. 2. Then, the re-determined inner radius and outer radius of the different disk models should be approximately 1.4 times of the /u1D45F 0 and /u1D45F 1 listed in the Table 2. Based on the re-determined inner radius and outer radius and the other model parameters which remain unchanged, the model expected line profiles are shown in Fig. 6 only considering AGN variability, leading to apparent variations of the line profiles of the shifted broad H /u1D6FC by different accretion disk models (even the pure symmetric circular disk model) in different epochs, strongly different from the expected results by the rSMBH scenario. Therefore, besides the accretion disk precessions, strong AGN variations have also apparent effects on variations of the profiles of the shifted broad H /u1D6FC , which will provide further clues to support SDSS J1052+1036 as a disk emitter.</text>
<section_header_level_1><location><page_9><loc_51><loc_18><loc_80><loc_19></location>4 MAIN SUMMARY AND CONCLUSIONS</section_header_level_1>
<text><location><page_9><loc_51><loc_13><loc_92><loc_17></location>A rare low-redshift Type-1.9 AGN SDSS J1052+1036 but with very blue-shifted broad H /u1D6FC is reported in this manuscript. The main summary and conclusions are as follows.</text>
<unordered_list>
<list_item><location><page_9><loc_51><loc_6><loc_92><loc_12></location>· The very blue-shifted broad H /u1D6FC is not related to a kpc-scale dual core system in SDSS J1052+1036 with apparent double-peaked narrow emission lines, due to quite different peak separations of the double-peaked narrow emission lines from the peak separation of the</list_item>
</unordered_list>
<text><location><page_10><loc_7><loc_90><loc_48><loc_93></location>blue-shifted component and the red-shifted component in the broad H /u1D6FC .</text>
<unordered_list>
<list_item><location><page_10><loc_7><loc_86><loc_48><loc_90></location>· The very blue-shifted broad H /u1D6FC is not related to a sub-pc BBH system in SDSS J1052+1036, due to stronger and wider blue-shifted H /u1D6FC having larger shifted velocity.</list_item>
<list_item><location><page_10><loc_7><loc_79><loc_48><loc_86></location>· The very blue-shifted broad H /u1D6FC probably does not arise due to the rSMBH scenario, mainly due to rSMBH scenario expected obscuration having E(B-V) ≤ 0.6, leading to probably detectable broad components in the H /u1D6FD , against the spectroscopic results without broad H /u1D6FD in SDSS J1052+1036.</list_item>
<list_item><location><page_10><loc_7><loc_76><loc_48><loc_79></location>· The very blue-shifted broad H /u1D6FC can be well explained by a disk emitter in SDSS J1052+1036 without any caveats.</list_item>
<list_item><location><page_10><loc_7><loc_69><loc_48><loc_76></location>· Due to disk precessions of accretion disk models, the standard elliptical accretion disk model can lead to apparent variations of the profiles of the shifted broad H /u1D6FC in SDSS J1052+1036 as a disk emitter, however no apparent variations could be expected in recent years through the circular disk model or the circular plus arm model.</list_item>
<list_item><location><page_10><loc_7><loc_64><loc_48><loc_69></location>· If considering strong variations of AGN activities leading to variations of ionization boundaries, apparent variations in different epochs can be expected in the profiles of the shifted broad H /u1D6FC in SDSS J1052+1036 as a disk emitter.</list_item>
<list_item><location><page_10><loc_7><loc_58><loc_48><loc_64></location>· A re-observed spectrum in 2025 could provide robust clues to support a disk emitter in SDSS J1052+1036, if there were apparent variations of peak positions, peak separations and/or peak intensity ratios in the broad H /u1D6FC .</list_item>
</unordered_list>
<section_header_level_1><location><page_10><loc_7><loc_54><loc_25><loc_55></location>ACKNOWLEDGEMENTS</section_header_level_1>
<text><location><page_10><loc_7><loc_37><loc_50><loc_53></location>Zhang gratefully acknowledges the anonymous referee for giving us constructive comments and suggestions to greatly improve our paper. Zhang gratefully acknowledges the kind financial support from GuangXi University and the kind funding support NSFC-12173020 and NSFC-12373014. This research has made use of the data from the SDSS ( https://www.sdss.org/ ) funded by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation and the U.S. Department of Energy Office of Science. The research has made use of the MPFIT package https://pages.physics.wisc.edu/~craigm/idl/cmpfit.html , and of the emcee package https://pypi.org/project/emcee/ .</text>
<section_header_level_1><location><page_10><loc_7><loc_33><loc_22><loc_34></location>DATA AVAILABILITY</section_header_level_1>
<text><location><page_10><loc_7><loc_29><loc_48><loc_32></location>The data underlying this article will be shared on request to the corresponding author (xgzhang@gxu.edu.cn).</text>
<section_header_level_1><location><page_10><loc_7><loc_25><loc_17><loc_26></location>REFERENCES</section_header_level_1>
<text><location><page_10><loc_7><loc_6><loc_48><loc_24></location>Barth, A. J.; Bennert, V. N.; Canalizo, G.; Filippenko, A. V.; Gates, E. L., 2015, ApJS, 217, 26 Batiste, M.; Bentz, M. C.; Raimundo, S. I.; Vestergaard, M.; Onken, C. A., 2017, ApJL, 838, 10 Bekenstein, J. D. 1973, ApJ, 183, 657 Bellm E. C., Kulkarni, S. R.; Barlow, T., et al., 2019, PASP, 131, 068003 Blandford, R. D.; McKee, C. F., 1982, ApJ, 255, 419 Blecha, L., Loeb, A. 2008, MNRAS, 390, 1311 Blecha, L., Sijacki, D., Kelley, L. Z., et al. 2016, MNRAS, 456, 961 Bennert, V. N.; Treu, T.; Ding, X.; et al., 2021, ApJ, 921, 36 Bentz, M. C.; Denney, K. D.; Grier, C. J, et al., 2013, ApJ, 767, 149 Bogdanovic, T.; Eracleous, M.; Sigurdsson, S., 2009, ApJ, 697, 288 Bruzual, G.; Charlot, S. 2003, MNRAS, 344, 1000 Cappellari, M., 2017, MNRAS, 466, 798</text>
<text><location><page_10><loc_51><loc_88><loc_92><loc_93></location>Chiaberge, M.; Ely, J. C.; Meyer, E. T.; et al., 2017, A&A, 600, 57 Chiaberge, M.; Tremblay, G. R.; Capetti, A.; Norman, C., 2018, ApJ, 861, 56 Cid Fernandes, R.; Mateus, A.; Sodre, L.; Stasinska, G.; Gomes, J. M., 2005, MNRAS, 358, 363</text>
<text><location><page_10><loc_51><loc_82><loc_92><loc_88></location>Deane, R. P.; Paragi, Z.; Jarvis, M. J.; et al., 2014, Natur, 511, 57 Dekany, R.; Smith, R. M.; Riddle, R., et al., 2020, PASP, 132, 038001 Di Matteo, T.; Springel, V.; Hernquist, L., Natur, 433, 604 Drake, A. J.; Djorgovski, S. G.; Mahabal, A.; et al., 2009, ApJ, 696, 870 Du, P.; Zhang, Z.-X.; Wang, K.; Huang, Y.-K.; Zhang, Y.; et al., 2018, ApJ,</text>
<unordered_list>
<list_item><location><page_10><loc_54><loc_80><loc_57><loc_81></location>856, 6</list_item>
<list_item><location><page_10><loc_51><loc_78><loc_92><loc_80></location>Eracleous, M., Livio, M., Halpern, J. P., Storchi-Bergmann, T.,1995, ApJ, 438, 610</list_item>
</unordered_list>
<text><location><page_10><loc_51><loc_77><loc_87><loc_78></location>Eracleous, M.; Lewis, K. T.; Flohic, H. M. L. G., NewAR, 53, 133</text>
<text><location><page_10><loc_51><loc_75><loc_91><loc_76></location>Eracleous, M.; Boroson, T. A.; Halpern, J. P.; Liu, J., 2012, ApJS, 201, 23</text>
<text><location><page_10><loc_51><loc_74><loc_75><loc_75></location>Ferrarese, F.; Merritt, D., 2000, ApJL, 539, 9</text>
<unordered_list>
<list_item><location><page_10><loc_51><loc_71><loc_92><loc_74></location>Foreman-Mackey, D.; Hogg, D. W.; Lang, D.; Goodman, J., 2013, PASP, 125, 306</list_item>
<list_item><location><page_10><loc_51><loc_70><loc_82><loc_71></location>Fu, H.; Zhang, Z.; Assef, R. J., et al., 2011, ApJL, 740, 44</list_item>
<list_item><location><page_10><loc_51><loc_69><loc_85><loc_70></location>Ge, J.; Hu, C.; Wang, J.; Bai, J.; Zhang, S., 2012, ApJS, 201, 31</list_item>
<list_item><location><page_10><loc_51><loc_68><loc_85><loc_69></location>Gebhardt, K.; Bender, R.; Bower, G, et al., 2000, ApJL, 539, 13</list_item>
<list_item><location><page_10><loc_51><loc_66><loc_90><loc_67></location>Graham, M. J.; Djorgovski, S. G.; Stern, D., et al., 2015a, Natur, 518, 74</list_item>
</unordered_list>
<text><location><page_10><loc_51><loc_65><loc_92><loc_66></location>Graham, M. J., Djorgovski, S. G., Stern, D., et al., 2015b, MNRAS, 453, 1562</text>
<text><location><page_10><loc_51><loc_64><loc_75><loc_65></location>Greene, J. E.; Ho, L. C., 2005, ApJ, 630, 122</text>
<text><location><page_10><loc_51><loc_62><loc_77><loc_64></location>Gualandris, A.; Merritt, D., 2008, ApJ, 678, 180</text>
<unordered_list>
<list_item><location><page_10><loc_51><loc_61><loc_77><loc_62></location>Hoffman, L.; Loeb, A., 2007, MNRAS, 377, 957</list_item>
<list_item><location><page_10><loc_51><loc_60><loc_82><loc_61></location>Johansson, P.; Naab, T.; Burkert, A., 2009, ApJ, 690, 802</list_item>
<list_item><location><page_10><loc_51><loc_59><loc_87><loc_60></location>Kalfountzou, E.; Santos Lleo, M.; Trichas M., 2017, ApJL, 851, 15</list_item>
<list_item><location><page_10><loc_51><loc_56><loc_92><loc_58></location>Kaspi, S.; Smith, P. S.; Netzer, H.; Maoz, D.; Jannuzi, B. T.; Giveon, U., 2000, ApJ, 533, 631</list_item>
<list_item><location><page_10><loc_51><loc_53><loc_92><loc_56></location>Kauffmann, G.; Heckman, T. M.; Tremonti, C., et al. 2003, MNRAS, 346, 1055</list_item>
<list_item><location><page_10><loc_51><loc_52><loc_91><loc_53></location>Kim, D. C.; Evans, A. S.; Stierwalt, S.; Privon, G. C., 2016, ApJ, 824, 122</list_item>
</unordered_list>
<text><location><page_10><loc_51><loc_51><loc_92><loc_52></location>Kim, D. C.; Yoon, I.; Privon, G. C.; Evans, A. S.; Harvey, D.; Stierwalt, S.;</text>
<text><location><page_10><loc_54><loc_50><loc_70><loc_51></location>Kim, J. H., 2017, ApJ, 840, 71</text>
<unordered_list>
<list_item><location><page_10><loc_51><loc_48><loc_80><loc_50></location>Kim, D. C.; Yoon, I.; Evans, A. S., 2018, ApJ, 861, 51</list_item>
</unordered_list>
<text><location><page_10><loc_51><loc_47><loc_92><loc_48></location>Kochanek, C. S.; Shappee, B. J.; Stanek, K. Z.; et al., 2017, PASP, 129, 4502</text>
<text><location><page_10><loc_51><loc_46><loc_77><loc_47></location>Komossa, S., Merritt, D., 2008a, ApJL, 683, 21</text>
<text><location><page_10><loc_51><loc_45><loc_77><loc_46></location>Komossa, S., Merritt, D., 2008b, ApJL, 689, 89</text>
<text><location><page_10><loc_51><loc_43><loc_79><loc_44></location>Komossa, S., Zhou, H., Lu, H., 2008, ApJL, 678, 81</text>
<unordered_list>
<list_item><location><page_10><loc_51><loc_42><loc_78><loc_43></location>Kormendy, J.; Ho, L. C., 2013, ARA&A, 51, 511</list_item>
<list_item><location><page_10><loc_51><loc_39><loc_92><loc_42></location>Lena, D.; Robinson, A.; Marconi, A.; Axon, D. J.; Capetti, A.; Merritt, D.; Batcheldor, D., 2014, ApJ, 795, 146</list_item>
<list_item><location><page_10><loc_51><loc_37><loc_91><loc_39></location>Lu, K.-X.; Bai, J.-M.; Wang, J.-M.; Hu, C.; Li, Y.-R., 2023, ApJS, 263, 10 Madau, P., Quataert, E. 2004, ApJL, 606, 17</list_item>
<list_item><location><page_10><loc_51><loc_34><loc_92><loc_37></location>Merritt, D.; Storchi-Bergmann, T.; Robinson, A.; Batcheldor, D.; Axon, D.; Cid Fernandes, R., 2006, MNRAS, 367, 1746</list_item>
<list_item><location><page_10><loc_51><loc_32><loc_88><loc_34></location>Morishita, T.; Chiaberge, M.; Hilbert, B.; et al., 2022, ApJ, 931, 165 Peterson, B. M., 1993, PASP, 105, 247</list_item>
<list_item><location><page_10><loc_51><loc_29><loc_92><loc_32></location>Peterson, B. M.; Barth, A. J.; Berlind, P.; Bertram, R.; Bischoff, K.; et al., 1999, ApJ, 510, 659</list_item>
</unordered_list>
<text><location><page_10><loc_51><loc_28><loc_90><loc_29></location>Peterson, B. M.; Ferrarese, L.; Gilbert, K. M., et al., 2004, ApJ, 613, 682</text>
<unordered_list>
<list_item><location><page_10><loc_51><loc_27><loc_89><loc_28></location>Popovic, L. C.; Shapovalova, A. I.; Ilic, D., et al., 2014, A&A, 572, 66</list_item>
<list_item><location><page_10><loc_51><loc_25><loc_88><loc_26></location>Runnoe, J. C.; Eracleous, M.; Mathes, G.; et al., 2015, ApJS, 221, 7</list_item>
</unordered_list>
<text><location><page_10><loc_51><loc_24><loc_91><loc_25></location>Runnoe, J. C.; Eracleous, M.; Pennell, A.; et al., 2017, MNRAS, 468, 1683</text>
<unordered_list>
<list_item><location><page_10><loc_51><loc_21><loc_86><loc_24></location>Shappee, B. J.; Prieto, J. L.; Grupe, D.; et al., 2014, ApJ, 788, 48 Shen, Y.; Loeb, A., 2010, ApJ, 725, 249</list_item>
<list_item><location><page_10><loc_51><loc_20><loc_86><loc_21></location>Shen, Y.; Hwang, H.; Zakamska, N.; Liu, X., 2019, ApJL, 885, 4</list_item>
<list_item><location><page_10><loc_51><loc_18><loc_92><loc_20></location>Shen, Y.; Hall, P. B.; Horne, K.; Zhu, G.; McGreer, I.; et al., 2019b, ApJS, 241, 34</list_item>
<list_item><location><page_10><loc_51><loc_13><loc_92><loc_18></location>Shields, G. A.; Rosario, D. J.; Smith, K. L.; et al., 2009, ApJ, 707, 936 Steinhardt, C. L.; Schramm, M.; Silverman, J. D.; et al., 2012, ApJ, 759, 24 Storchi-Bergmann, T.; Nemmen da Silva, R., Eracleous, M., et al., 2003, ApJ, 598, 956</list_item>
</unordered_list>
<text><location><page_10><loc_51><loc_9><loc_92><loc_12></location>Storchi-Bergmann, T.; Schimoia, J. S.; Peterson, B. M.; Elvis, M.; Denney, K. D.; Eracleous, M.; Nemmen, R. S., 2017, ApJ, 835, 236 Volonteri, M., 2007, ApJL, 663, 5</text>
<text><location><page_10><loc_51><loc_7><loc_85><loc_9></location>Wang, M.; Luo, A.; Song, Y., et al., 2019, MNRAS, 482, 1889</text>
<text><location><page_10><loc_51><loc_6><loc_84><loc_7></location>Ward, C.; Gezari, S.; Frederick, S.; et al., 2021, ApJ, 913, 102</text>
<code><location><page_11><loc_7><loc_77><loc_45><loc_93></location>Xu, D.; Komossa, S., 2009, ApJL, 705, 20 Zhang, X. G., 2021d, MNRAS, 502, 2508 Zhang, X. G., 2021a, ApJ, 909, 16, ArXiv:2101.02465 Zhang, X. G., 2021b, ApJ, 919, 13, ArXiv:2107.09214 Zhang, X. G., 2021c, MNRAS Letter, 500, 57 Zhang, X. G., 2022a, ApJS, 260, 31 Zhang, X. G., 2022b, ApJS, 261, 23 Zhang, X. G., 2022c, ApJ, 937, 105, ArXiv:2209.02164 Zhang, X. G., 2022d, MNRAS, 512, 1003, arXiv:2202.11995 Zhang, X. G., 2022e, MNRAS, 516, 3650, arXiv:2209.01923 Zhang, X. G., 2023, MNRAS accepted, arXiv:2309.08078 Zheng, Z.; Butler, N. R.; Shen, Y.; et al., 2016, ApJ, 827, 56 Zhou, H., Wang, T., Zhang, X., Dong, X., Li, C. 2004, ApJL, 604, L33</code>
</document>
[
{
"title": "ABSTRACT",
"content": "In this manuscript, very blue-shifted broad H /u1D6FC with shifted velocity ∼ 2200km/s is reported in the low redshift Type-1.9 AGN SDSS J1052+1036. Blue-shifted broad emission lines may arise due to the presence of a rotating gas disk around central black hole (BH), but may also be a signature of rare phenomena such as gravitational wave recoil of a supermassive BH (rSMBH) or the presence of a binary BH (BBH) system. Here, due to larger shifted velocity of stronger and wider blue-shifted broad H /u1D6FC , the BBH system is disfavoured. Meanwhile, if this object contained a rSMBH, intrinsic obscuration with E(B-V) ≤ 0.6 should lead to a detectable broad H /u1D6FD , indicating the rSMBH scenario not preferred. We find that the blue-shifted broad H /u1D6FC can be well explained by emission from an AGN disk, indicating that SDSS J1052+1036 is likely a disk-emitting AGN. In order to determine which scenario, a rSMBH or a disk emitter, is more preferred, a re-observed spectrum in 2025 can provide robust clues, with a disk emitter probably leading to clear variations of peak positions, peak separations and/or peak intensity ratios in broad H /u1D6FC , but with a rSMBH scenario probably leading to no variations of peak separations in broad H /u1D6FC . Key words: galaxies:active - galaxies:nuclei - quasars:emission lines - quasars: individual (SDSS J1052+1036)",
"pages": [
1
]
},
{
"title": "Very blue-shifted broad H /u1D6FC in a low redshift Type-1.9 AGN: a disk emitter or a recoiling black hole scenario",
"content": "Xue-Guang Zhang 1 /uni2605 1 Guangxi Key Laboratory for Relativistic Astrophysics, School of Physical Science and Technology, GuangXi University, 530004, Nanning, P. R. China 1 February 2024",
"pages": [
1
]
},
{
"title": "1 INTRODUCTION",
"content": "Shifted broad emission lines relative to stellar absorption features (or narrow emission lines) may be indicators of a gravitational wave recoiling supermassive black hole (rSMBH) in called offnucleus active galactic nuclei (AGN), due to gravitational wave carried off linear momentum leading central BH being kicked away from central region of AGN, as discussed in Bekenstein (1973); Madau & Quataert (2004); Merritt et al. (2006); Volonteri (2007); Blecha & Loeb (2008); Komossa & Merritt (2008a); Blecha et al. (2016). Broad emission lines from broad emission line regions (BLRs) bound to a rSMBH with a large kick velocity can lead to blue-shifted broad emission lines relative to narrow emission lines of AGN, due to no effects of a rSMBH on NLRs (narrow emission line regions). Until now, there are a few individual AGN and samples of AGN reported with blue-shifted broad emission lines, and expected rSMBH scenarios have been discussed in the literature. Komossa et al. (2008) have reported SDSS J0927+2943 ( /u1D467 ∼ 0 . 713) with blue-shifted velocities 2650km/s in broad emission lines, to support a rSMBH scenario. However, Bogdanovic et al. (2009) have discussed a binary BH (BBH) system with mass ratio 0.1 also leading to the shifted features in SDSS J0927+2943. Shields et al. (2009) have reported blue-shifted velocity 3500km/s in broad H /u1D6FD in SDSS J1050, however, BLRs lying into central accretion disk (=disk emitter) would be preferred to explain the blue-shifted broad H /u1D6FD , rather than the rSMBH scenario. Steinhardt et al. (2012) have reported very blue-shifted broad emission lines in SDSS J0956+5128, however, either an extreme disk emitter or a rSMBH is not the preferred scenario to explain all of the observed features, especially the different profiles between broad Balmer lines and broad Mg /i.pc/i.pc line. Kim et al. (2017) have discussed the rSMBH candidate of CXO J1015+6259 ( /u1D467 ∼ 0 . 35) with blue-shifted velocity 175km/s in broad emission lines. Kalfountzou et al. (2017) have shown a rSMBH is one proposed scenario to explain the three strong emission-line nuclei with velocity offset 250km/s in SDSS J1056+5516 ( /u1D467 ∼ 0 . 256), as well as a triple BH accreting system. Kim et al. (2018) have applied an oscillating rSMBH scenario to explain the broad emission line variability properties in Mrk1018. Chiaberge et al. (2017, 2018); Morishita et al. (2022) have shown that the quasar 3C186 ( /u1D467 ∼ 1 . 07) have blue-shifted velocity 2140km/s in broad emission lines, consistent with expected results by a rSMBH. Meanwhile, there are samples of AGN with blue-shifted broad emission lines. Eracleous et al. (2012) and followed in Runnoe et al. (2015, 2017) have reported a sample of tens of low redshift (z<0.7) SDSS quasars with blue-shifted velocities larger than 1000km/s in broad H /u1D6FD , and discussed that BBH systems should be preferred in a fraction of the candidates, after carefully checked changes of peak velocities through multi-epoch spectra. Lena et al. (2014) have shown 10 rSMBH candidates in nearby galaxies with small displacements betweencentral activity region and center of galaxy. Kim et al. (2016) have reported a sample of candidates with mean blue-shifted velocity about 150km/s for rSMBHs in SDSS quasars with redshift less than 0.25. Ward et al. (2021) have shown nine AGN that may be spatially offset from their host galaxies and are considered as candidates for rSMBHs. Based on the reported candidates of AGN with blue-shifted broad emission lines, besides the rSMBH scenarios, either the BBH or disk emitter hypotheses can be applied. Moreover, as discussed in Komossa & Merritt (2008b); Shen et al. (2019), candidates of rSMBHs with large recoiling velocities at low redshift are ex- tremely rare. Here, a candidate at redshift 0.088 is reported with blue-shifted velocity ∼ 2200km/s in broad H /u1D6FC in a Type-1.9 AGN SDSS J105232.97+103620.08 (=SDSS J1052+1036), with different scenarios discussed. This manuscript is organized as follows. Section 2 presents the spectroscopic results of the Type-1.9 AGN SDSS J1052+1036. Section 3 gives main discussions. Section 4 gives our final conclusions. And the cosmological parameters have been adopted as /u1D43B 0 = 70km · s -1 Mpc -1 , /uni03A9 /uni039B = 0 . 7 and /uni03A9 m = 0 . 3.",
"pages": [
1,
2
]
},
{
"title": "2 MAIN RESULTS",
"content": "SDSS J1052+1036 is selected as the subject of this manuscript, due to its very blue-shifted broad H /u1D6FC , while studying properties of double-peaked narrow emission lines in low redshift ( /u1D467 < 0 . 35) SDSS quasars including some objects reported in the sample of Ge et al. (2012). SDSS J1052+1036 has its SDSS spectrum (platemjd-fiberid=1602-53117-0243) with signal-to-noise about 18 shown in top left panel of Fig. 1 with apparently shifted broad H /u1D6FC marked by vertical dashed red line. In order to measure the emission lines as well as to measure the stel- lar velocity dispersion, the commonly accepted SSP (Simple Stellar Population) method is applied to determine host galaxy contributions in SDSS J1052+1036. More detailed descriptions on the SSP method can be found in Bruzual & Charlot (2003); Kauffmann et al. (2003); Cid Fernandes et al. (2005); Cappellari (2017). The SSP method has also been applied in our previous papers Zhang (2021a,b,d, 2022a,b). Here, we briefly describe the SSP method. The 39 simple stellar population templates from Bruzual & Charlot (2003); Kauffmann et al. (2003) are applied to describe stellar lights, combined with a power law function to describe the AGN continuum. When the SSP method is applied, narrow emission lines are masked out by full width at zero intensity about 450km/s, and the spectrum with wavelength range from 6450Å to 6750Å are also masked out due to the strongly broad H /u1D6FC . Then, through the Levenberg-Marquardt least-squares minimization technique (the MPFIT package), SDSS spectrum in rest frame with emission lines being masked out can be well described. The best descriptions and the corresponding line spectrum (SDSS spectrum minus the best descriptions) are shown in the top panel of Fig. 1 with /u1D712 2 / /u1D451 /u1D45C /u1D453 ∼ 1 . 26 (the summed squared residuals divided by degree of freedom). Considering the totally obscured broad H /u1D6FD , the determined red power law continuum emissions was acceptable, due to seriously obscurations on central continuum emissions. Meanwhile, we measured the stellar velocity dispersion to be 113 ± 10km/s. Moreover, in order to show the stellar velocity dispersion, the bottom panels of Fig. 1 show the SSP method determined the best descriptions and the corresponding residuals (SDSS spectrum minus the best descriptions and then divided by the uncertainties of SDSS spectrum) to the absorption features around Ca /i.pc/i.pc H+K from 3880 to 4400Å and around Mg /i.pc from 5050 to 5300Å. After subtractions of the host galaxy contributions, more apparent blue-shifted broad H /u1D6FC can be found. And the emission lines can be measured by multiple Gaussian functions, similar as what we have recently done in Zhang (2021a,b, 2022a,b,c). Considering the double-peaked features in the narrow emission lines (especially in the narrow Balmer lines, the [O /i.pc/i.pc/i.pc ] doublet and the [N /i.pc/i.pc ] doublet) in SDSS J1052+1036, three Gaussian functions are applied to describe each narrow emission line: two narrow Gaussian components for the double-peaked feature and one Gaussian component for the probably extended emissions underneath the double-peaked feature. Therefore, for the emission lines within the rest wavelength from 4830Å to 5020Å and from 6200Å to 6800Å, there are three Gaussian functions applied to describe the double-peaked narrow H /u1D6FD (H /u1D6FC ), one broad Gaussian function to describe the probable broad H /u1D6FD , two broad Gaussian functions to describe the broad H /u1D6FC , six Gaussian functions to describe the [O /i.pc/i.pc/i.pc ] /u1D706 4959 , 5007Å doublet, six Gaussian functions to describe the [N /i.pc/i.pc ] /u1D706 6549 , 6585Å doublet, one Gaussian function to describe each line in the [O /i.pc ] /u1D706 6300 , 6363Å and the [S /i.pc/i.pc ] /u1D706 6716 , 6731Å doublets without apparent double-peaked features. When the functions above are applied, each Gaussian component has line intensity not smaller than zero, and the corresponding [O /i.pc/i.pc/i.pc ] ([N /i.pc/i.pc ]) components have the same redshift and the same line width and have the flux ratio to be fixed to the theoretical value 3. Then, through the MPFIT package, the best fitting results (in top regions) and the corresponding residuals (in bottom regions) to the emission lines around H /u1D6FD and H /u1D6FC are shown in the top left panel and the top right panel of Fig. 2 with /u1D712 2 / /u1D451 /u1D45C /u1D453 ∼ 0 . 89and /u1D712 2 / /u1D451 /u1D45C /u1D453 ∼ 0 . 88, respectively. Based on the best fitting results, it is not necessary to consider broad Gaussian component in the H /u1D6FD , because the determined line width and line flux (around to zero) of the broad Gaussian component are smaller than their corresponding uncertainties, indicating not apparent broad H /u1D6FD in SDSS J1052+1036. Moreover, as shown in the top right panel of Fig. 2, each line in the [O /i.pc ] and [S /i.pc/i.pc ] doublets can be well described by one Gaussian component, and whether two or three Gaussian components applied to describe each line in the [O /i.pc ] and [S /i.pc/i.pc ] doublets have few effects on our discussed results on the broad H /u1D6FC . The parameters of the Gaussian components applied to describe the emission lines are listed in the Table 1. Based on the best descriptions to the stellar absorption features in Fig. 1 and the best fitting results to the broad H /u1D6FC in the top right panel of Fig. 2, about 2200km/s blue-shifted broad H /u1D6FC can be confirmed. Besides the broad H /u1D6FC described by two Gaussian components, the blue-shifted broad H /u1D6FC can also be described by the known elliptical accretion disk model discussed in Eracleous et al. (1995), similar as what we have recently done on double-peaked broad emission lines in Zhang (2021c, 2022a). The elliptical accretion disk model have seven model parameters, inner and out boundaries [ /u1D45F 0 , /u1D45F 1 ] in unit of /u1D445 /u1D43A (Schwarzschild radius), inclination angle /u1D456 of disk-like BLRs, eccentricity /u1D452 , orientation angle /u1D719 0 of elliptical rings, local broadening velocity /u1D70E /u1D43F in units of km / s, line emissivity slope /u1D45E ( /u1D453 /u1D45F ∝ /u1D45F -/u1D45E ). In order to obtain more reliable model parameters and corresponding uncertainties, the Maximum Likelihood method combining with the MCMC (Markov Chain Monte Carlo) technique (Foreman-Mackey et al. 2013) is applied. The evenly prior distributions of the seven model parameters are accepted with the following limitations, log ( /u1D45F 0 ) ∈ [ 1 , 3 ] , log ( /u1D45F 1 ) ∈ [ 2 , 6 ] ( /u1D45F 1 > /u1D45F 0 ), log ( sin ( /u1D456 )) ∈ [-3 , 0 ] , log ( /u1D45E ) ∈ [-1 , 1 ] , log ( /u1D70E /u1D43F ) ∈ [ 2 , 4 ] , log ( /u1D452 ) ∈ [-5 , 0 ] , log ( /u1D719 0 ) ∈ [-5 , log ( 2 × /u1D70B )] . The determined best fitting results and corresponding residuals to the emission lines around H /u1D6FC are shown in the top region and the bottom region of the middle left panel of Fig. 2 with /u1D712 2 / /u1D451 /u1D45C /u1D453 = 324 . 41 / 391 ∼ 0 . 83. And the model determined best descriptions to the broad H /u1D6FC are shown as solid blue line in the top region of the middle left panel of Fig. 2. The MCMC technique determined posterior distributions of the model parameters in the elliptical accretion disk model are shown in Fig. 3. The determined parameters and the corresponding 1 /u1D70E uncertainties are listed in the Table 2. Moreover, as discussed in Zhang (2022a), clean double-peaked broad line emission features can lead to solely determined model parameters in the elliptical accretion disk model. Therefore, there are no further discussions on whether is there solely determined model parameters, due to the apparent blue peak in broad H /u1D6FC in SDSS J1052+1036. Meanwhile, as suggested in Eracleous et al. (2009); Storchi-Bergmann et al. (2003, 2017), rather than the elliptical accretion disk model, the spiral arm rotation is the preferred explanation for most disk emitter profile evolution. Therefore, the circular disk plus spiral arm model with 10 model parameters is also applied to describe the shifted broad H /u1D6FC of SDSS J1052+1036. Besides the model parameters ( /u1D452 = 0) applied in the elliptical accretion disk model, four additional model parameters are applied to describe structures of spiral arms, the azimuthal width /u1D6FF , the pitch angle /u1D45D and the innermost radius /u1D45F /u1D45A of the spiral arm, and the brightness contrast /u1D434 between the spiral arm and the underlying axisymmetric disk. Then, based on the new emissivity formula shown in Equation (2) in Storchi-Bergmann et al. (2003) and accepted /u1D45F /u1D45A = /u1D45F 0 , the best descriptions and the corresponding residuals to the emission lines around the H /u1D6FC are shown in the middle right panel of Fig. 2 with /u1D712 2 / /u1D451 /u1D45C /u1D453 = 287 . 88 / 389 ∼ 0 . 74. And the model determined best descriptions to the broad H /u1D6FC are shown as solid blue line in the top region of the middle right panel of Fig. 2. The MCMC technique determined posterior distributions of the model parameters in the circular disk plus arm model are shown in Fig. 4. The determined model parameters and the corresponding 1 /u1D70E uncertainties are also listed in the Table 2 for the circular disk plus arm model. Notice: For the Gaussian components, the first column shows which line is measured, the Second, third, fourth columns show the measured line parameters: center wavelength /u1D706 0 in unit of Å, line width (second moment) /u1D70E in unit of Å and line flux in unit of 10 -17 erg / s / cm 2 .",
"pages": [
2,
4,
5,
6
]
},
{
"title": "3.1 A kpc-scale dual core system in SDSS J1052+1036?",
"content": "Double-peaked [O /i.pc/i.pc/i.pc ] /u1D706 5007Å can be seen in the spectrum of SDSS J1052+1036, as well as shown in Ge et al. (2012), widely indicating a kpc-scale dual core system (Zhou et al. 2004; Xu & Komossa 2009; Fu et al. 2011; Wang et al. 2019). Based on the measured doublepeaked features in the [O /i.pc/i.pc/i.pc ] /u1D706 5007Å, the peak separation is about 350 ± 22km/s in SDSS J1052+1036, leading the broad emission lines from the assumed central two cores to have the same peak separa- elliptical accretion disk model /u1D45F 0 = 38 ± 2, /u1D45F 1 = 490 ± 35, sin ( /u1D456 ) = 0 . 48 ± 0 . 04 /u1D45E = 0 . 17 ± 0 . 01, /u1D452 = 0 . 56 ± 0 . 03, /u1D70E /u1D43F = 850 ± 40km / s = · /u1D719 0 237 ± 5",
"pages": [
6,
7
]
},
{
"title": "the circular disk plus spiral arm model",
"content": "/u1D45F 0 = 150 ± 60, /u1D45F 1 = 4500 ± 1300, sin ( /u1D456 ) = 0 . 44 ± 0 . 04, /u1D45E = 1 . 82 ± 0 . 21 /u1D70E /u1D43F = 270 ± 90km / s, /u1D719 0 = 179 ± 3 · , /u1D434 = 26 ± 7 /u1D6FF = 93 ± 10 · , /u1D45D = 28 ± 2 · the pure symmetric circular disk model /u1D45F 0 = 18 ± 1, /u1D45F 1 = ( 8 ± 3 ) × 10 5 , sin ( /u1D456 ) = 0 . 37 ± 0 . 01, /u1D45E = 7 . 48 ± 0 . 91 /u1D70E /u1D43F = 650 ± 103km / s, /u1D719 0 = 16 ± 3 · the elliptical accretion disk model with /u1D449 /u1D460 = -265km/s /u1D45F 0 = 31 ± 2, /u1D45F 1 = 246 ± 25, sin ( /u1D456 ) = 0 . 33 ± 0 . 03 /u1D45E = 0 . 66 ± 0 . 05, /u1D452 = 0 . 58 ± 0 . 03, /u1D70E /u1D43F = 916 ± 60km / s /u1D719 0 = 240 ± 6 · the elliptical accretion disk model with /u1D449 /u1D460 = 231km/s /u1D45F 0 = 44 ± 4, /u1D45F 1 = 381 ± 40, sin ( /u1D456 ) = 0 . 38 ± 0 . 04 /u1D45E = 1 . 26 ± 0 . 15, /u1D452 = 0 . 54 ± 0 . 03, /u1D70E /u1D43F = 532 ± 30km / s /u1D719 0 = 240 ± 6 · tion 350km/s. However, the peak separation about 4200 ± 580km/s between the blue-shifted broad component and the red-shifted broad component in the broad H /u1D6FC in SDSS J1052+1036 is about twelve times higher than the peak separation of the double-peaked narrow emission lines. Therefore, the shifted broad H /u1D6FC is not related to a kpc-scale dual core system expected by the double-peaked [O /i.pc/i.pc/i.pc ] in SDSS J1052+1036.",
"pages": [
7
]
},
{
"title": "3.2 A rSMBH in SDSS J1052+1036?",
"content": "One another explanation for the blue shifted broad H /u1D6FC in SDSS J1052+1036 is that it is a rSMBH, after considering materials in the BLRs being carried away with the rSMBH. Meanwhile, not a single but two broad Gaussian components in the broad H /u1D6FC in SDSS J1052+1036 are probably indicating asymmetric structures of the BLRs bound to the rSMBH. As discussed in Merritt et al. (2006); Gualandris & Merritt (2008); Komossa & Merritt (2008b), the materials in the BLRs can be bound to a rSMBH within a region with the radius /u1D45F /u1D458 given by with /u1D440 /u1D435 /u1D43B and /u1D449 /u1D458 as the BH mass and the kick velocity of a rSMBH. Meanwhile, in order to support a rSMBH by blue-shifted broad emission lines, the blue-shifted broad emission component related to the emission materials bound to a rSMBH should be apparent enough, indicating almost all the materials in the original BLRs bound to the rSMBH. Therefore, we can expect that the estimated /u1D45F /u1D458 should be not smaller than the origin BLRs size /u1D445 /u1D435 /u1D43F /u1D445 /u1D460 which can be estimated by the continuum luminosity /u1D43F 5100 at 5100Å (Bentz et al. 2013), In SDSS J1052+1036 with the well measured stellar velocity dispersion about 113 ± 10km/s, after considering the M-sigma relation discussed in Ferrarese & Merritt (2000); Gebhardt et al. (2000); Kormendy & Ho (2013); Batiste et al. (2017); Bennert et al. (2021) for both quiescent and active galaxies and also as discussed in Di Matteo et al. (2005); Johansson et al. (2009) in galaxy merging systems, the BH mass can be estimated as 2 . 5 + 2 . 2 -1 . 3 × 10 7 M /circledot in SDSS J1052+1036, accepted Equation (7) in Kormendy & Ho (2013). Therefore, based on the equation above, we could have /u1D43F 5100 < 4 × 10 43 erg/s, which can lead to apparent blue-shifted broad H /u1D6FC totally related to an expected rSMBH. Based on the determined continuum emissions in the top panel of Fig. 1, the observed continuum luminosity at 5100Å is about 6 . 3 × 10 42 erg/s in SDSS J1052+1036. Accepted the intrinsic /u1D43F 5100 should be smaller than 4 × 10 43 erg/s, the intrinsic obscuration should have E(B-V) ≤ 0.6. Then, accepted the intrinsic flux ratio 3.1 of broad H /u1D6FC to broad H /u1D6FD , the expected observed flux ratio of the broad H /u1D6FC to the broad H /u1D6FD should be smaller than 6.2, leading to a detectable blueshifted broad component in the H /u1D6FD in SDSS J1052+1036. Unfortunately, as shown in the top left panel of Fig. 2, there are no detectable broad components in the H /u1D6FD in SDSS J1052+1036. Therefore, the blue-shifted broad H /u1D6FC probably contains weak contributions from a rSMBH scenario in SDSS J1052+1036. Unfortunately, the discussions above are not sufficient enough to totally disfavour the rSMBH scenario in SDSS J1052+1036, however, multi-epoch spectroscopic results should provide clear clues to support or to be against the rSMBH scenario. If the expected rSMBH in SDSS J1052+1036 moves rectilinearly (or moves curvilinearly as the case in Mrk 1018 in Kim et al. 2018), very tiny (or no) changes of peak separations between the blue-shifted component and the red-shifted component in the broad H /u1D6FC could be expected in SDSS J1052+1036.",
"pages": [
7
]
},
{
"title": "3.3 A BBH system in SDSS J1052+1036?",
"content": "If a BBH system was accepted in SDSS J1052+1036 with the estimated total BH mass 2 . 5 + 2 . 2 -1 . 3 × 10 7 M /circledot by the /u1D440 BH -/u1D70E relation, the two broad Gaussian components in the broad H /u1D6FC could be simply accepted to estimate the observational peak separation about /u1D449 /u1D45D , /u1D45C /u1D44F /u1D460 = 4200 ± 600km / s, leading the upper limit of the space separation /u1D446 of the central two BHs to be Based on the measured luminosity 2 . 01 × 10 41 erg / s of the observed broad H /u1D6FC or the measured continuum luminosity 6 . 2 × 10 42 erg / s at 5100Å in the rest frame, the estimated BLRs size should be about 7light-days, after considering the correlation between broad H /u1D6FC luminosity and continuum luminosity discussed in Greene & Ho (2005) and the empirical R-L relation discussed in Bentz et al. (2013). However, considering SDSS J1052+1036 as a Type-1.9 AGN, serious obscuration indicates the intrinsic BLRs size should be much larger than 7light-days. The BLRs size is similar as the upper limit of space separation of the central two BHs, strongly indicating the two BLRs probably totally mixed, leading to no apparent variability in the peak positions in the broad H /u1D6FC , as discussed in Shen & Loeb (2010). Moreover, under the assumption of a BBH system in SDSS J1052+1036, probable optical quasiperiodic oscillations (Graham et al. 2015a,b; Zheng et al. 2016; Zhang 2022d,e, 2023) should be detected. However, after checking long-term light curves from Catalina Sky Survey (Drake et al. 2009), All-Sky Automated Survey for Supernovae (Shappee et al. 2014; Kochanek et al. 2017) and Zwicky Transient Facility (Bellm et al. 2019; Dekany et al. 2020), there is no significant variability, which can not provide clues to support a BBH system in SDSS J1052+1036. Meanwhile, under the assumption of a BBH system in SDSS J1052+1036, considering the stronger and wider broad blue-shifted component in the H /u1D6FC (H /u1D6FC /u1D435 ), the virial BH mass /u1D440 /u1D435 /u1D43B , /u1D435 related to the H /u1D6FC /u1D435 should be simply expected to be 6.4 times larger than the virial BH mass /u1D440 /u1D435 /u1D43B , /u1D445 related to the red-shifted broad component in the H /u1D6FC (H /u1D6FC /u1D445 ), accepted the virialization assumptions to the broad emission lines as discussed in Greene & Ho (2005); Peterson et al. (2004). Here, the factor 6.4 is simply calculated by ( 1036 . 8 144 . 6 ) 0 . 5 ( 35 . 12 22 . 72 ) 2 with 1036.8 and 144.6 (35.12 and 22.72) as the line fluxes (the line widths) of the H /u1D6FC /u1D435 and the H /u1D6FC /u1D445 in SDSS J1052+1036. Then, the H /u1D6FC /u1D435 should have 6.3 times smaller shifted velocity than that of the H /u1D6FC /u1D445 , which is against the measured results that the shifted velocity 2430 ± 50km/s of the H /u1D6FC /u1D435 is larger than the shifted velocity about 1760 ± 530km/s of the H /u1D6FC /u1D445 , indicating a BBH system is disfavoured in SDSS J1052+1036. Furthermore, if we accepted the double-peaked narrow emission lines as signs of kpc-scale dual core systems and also accepted the blue-shifted broad H /u1D6FC related to a BBH system, there should be a rare close-pair binary in a triple BH system in SDSS J1052+1036, similar as those discussed in Hoffman & Loeb (2007); Deane et al. (2014). In such a rare close-pair binary in a triple BH system in SDSS J1052+1036, similar results can be expected that the shifted velocity of the H /u1D6FC /u1D435 should be quite smaller than that of the H /u1D6FC /u1D445 . However, whether the red-shifted (or the blue-shifted) narrow emission component in the double-peaked narrow H /u1D6FC is applied to trace the rotating velocity of the close-pair binary BH system in a triple BH system, larger shifted velocity of the H /u1D6FC /u1D435 can be determined than that of the H /u1D6FC /u1D445 . Therefore, a close-pair binary in a triple BH system is disfavoured in SDSS J1052+1036",
"pages": [
7,
8
]
},
{
"title": "3.4 A disk emitter in SDSS J1052+1036?",
"content": "Based on the model parameters of the elliptical accretion disk model listed in the Table 2, the expected disk precession period should be about /u1D447 pre ∼ 1040 /u1D440 8 /u1D445 2 . 5 3 /u1D466 /u1D45F . Using the /u1D440 BH -/u1D70E determined BH mass /u1D440 8 ∼ 0 . 25 + 0 . 22 -0 . 13 in units of 10 8 M /circledot and /u1D445 3 as radius in units of 1000 /u1D445 /u1D43A , based on the determined /u1D45F 0 , /u1D45F 1 and /u1D45E , the flux weighted size of the emission regions for the broad H /u1D6FC to the central BH is about 248 /u1D445 /u1D43A , leading to an approximately estimated disk precession period of 8years. As well known, asymmetric structures in accretion disk model are key factors leading to apparent variabilities of the peak positions and the peak separations of the double-peaked broad emission lines due to pure disk precessions, which will provide clues to support a disk emitter in SDSS J1052+1036. If there should be a re-observed spectrum in Jul. 2025 (MJD ∼ 60858), based on the expected precession period of about 8years, the expected line profiles of the broad H /u1D6FC in SDSS J1052+1036 in 2025 are shown as dotted blue line and dot-dashed blue line in the top region of the middle left panel of Fig. 2 with considering the standard elliptical accretion disk model applied with anti-clockwise rotation and clockwise processions respectively. Meanwhile, it is necessary to check whether a pure symmetric circular accretion disk model (with eccentricity to be zero) (without spiral arms) can be applied to describe the observed shifted broad H /u1D6FC in SDSS J1052+1036. For a circular accretion disk model with /u1D452 = 0, a similar fitting procedure is applied to describe the broad H /u1D6FC in SDSS J1052+1036, with the final determined fitting results to the broad H /u1D6FC shown as dashed blue line in the top region of the middle left panel of Fig. 2 with corresponding /u1D712 2 / /u1D451 /u1D45C /u1D453 = 469 . 37 / 392 ∼ 1 . 21. The MCMC technique determined posterior distributions of the model parameters in the pure symmetric circular disk model are shown in Fig. 5. The determined model parameters and the corresponding 1 /u1D70E uncertainties are also listed in the Table 2 for the pure symmetric circular disk model. Based on the F-test technique similar as what we have recently done in Zhang (2022c), due to the different values of /u1D712 2 and /u1D451 /u1D45C /u1D453 for the different accretion disk models, the confidence level can be determined to be higher than 6 /u1D70E to support that the elliptical accretion disk model and the circular accretion disk plus arm model is preferred than the pure symmetric circular accretion disk model. Unfortunately, only through the single-epoch spectroscopic properties of SDSS J1052+1036, we can not find more clues to support that the pure symmetric circular disk model is totally disfavored in SDSS J1052+1036. Therefore, in the manuscript, the pure symmetric circular disk model is also accepted as a reasonable model to describe the shifted broad H /u1D6FC in SDSS J1052+1036. Moreover, if accepted the double-peaked features in the [O /i.pc/i.pc/i.pc ] /u1D706 4959 , 5007Å doublet as signs of a kpc-scale dual core system, a rotating disk emitter (disk emission regions with a rotating velocity related to the orbital motions of central dual cores) contained in a dual core system could also be applied to describe the observed blue-shifted broad H /u1D6FC in SDSS J1052+1036. Considering the double-peaked features in the narrow H /u1D6FC to trace the rotating velocity /u1D449 /u1D460 of the disk emitter in a dual core system, the best fitting results and the corresponding residuals to the emission lines around H /u1D6FC can be re-determined with /u1D449 /u1D460 = -265km/s and with /u1D449 /u1D460 = 231km/s, and shown in the bottom left panel and the bottom right panel of Fig. 2 with corresponding /u1D712 2 / /u1D451 /u1D45C /u1D453 ∼ 1 . 03 and /u1D712 2 / /u1D451 /u1D45C /u1D453 ∼ 1 . 02, respectively. The model determined broad H /u1D6FC after considering /u1D449 /u1D460 are shown as solid blue lines in the top regions of the bottom panels of Fig. 2. Moreover, if considering the elliptical accretion disk model with /u1D449 /u1D460 in SDSS J1052+1036 , there are similar 1 /u1D70E uncertainties of the model parameters as those of the standard elliptical accretion disk model shown in Fig. 3. Therefore, we did not show the posterior distributions of the model parameters for the rotating elliptical disk models, but the model parameters and the corresponding 1 /u1D70E uncertainties are listed in the Table 2. Based on the determined model parameters listed in the Table 2 for the rotating elliptical accretion disk model with /u1D449 /u1D460 , disk precession periods can be estimated as 1.68years and 4.95years, with the central wavelengths of the blue-shifted component and the red-shifted component in the narrow H /u1D6FC applied to determine the /u1D449 /u1D460 . Then, the expected line profiles of the broad H /u1D6FC in SDSS J1052+1036 in Jul. 2025 are shown as dotted blue line and dot-dashed blue line in the top regions of the bottom left panel and the bottom right panel of Fig. 2 with considering the elliptical accretion disk model with /u1D449 /u1D460 = -265km/s and /u1D449 /u1D460 = 231km/s applied with anti-clockwise rotation and clockwise processions respectively. Either a rotating elliptical disk emitter contained in a dual core system or a standard elliptical disk emitter can lead to apparent time dependent variations of the peak positions and the peak separations of the shifted broad H /u1D6FC in SDSS J1052+1036 as a disk emitter. Considering different disk precession periods determined by standard accretion disk model and/or rotating disk emitter, if there should be a re-observed spectrum in Jul. 2025, the expected broad H /u1D6FC in 2025 in SDSS J1052+1036 have quite different peak positions and different peak separations from the broad H /u1D6FC in the SDSS spectrum observed in MJD=53117. Therefore, a re-observed spectrum in 2025 should provide clues enough to confirm whether a disk emitter is preferred in SDSS J1052+1036. Unfortunately, unless there are detailed timedependent variabilities of the broad H /u1D6FC in SDSS J1052+1036, it is hard to distinguish a standard elliptical disk emitter from a rotating elliptical disk emitter in a kpc-scale dual core system. Furthermore, if considering the circular disk plus spiral arm model in SDSS J1052+1036, the expected disk precession period should be about 2100years, due to the quite large flux weighted size /u1D445 3 ∼ 2 . 3 of the emission regions of the blue-shifted broad H /u1D6FC to the central BH. Meanwhile, the precessing spiral pattern could have precession period more than 100years, considering ten times of the dynamical time of accretion disk as discussed in Storchi-Bergmann et al. (2003). The large precession periods strongly indicates no apparent variabilities of profiles of the broad H /u1D6FC in SDSS J1052+1036, if considering the circular disk plus arm model to describe the shifted broad H /u1D6FC . However, as discussed in Storchi-Bergmann et al. (2003), the model parameters of /u1D434 , /u1D45E are varying in the circular disk plus spiral arm model. The varying parameter of /u1D434 can lead to apparent variability of the peak intensity ratio of the broad H /u1D6FC of SDSS J1052+1036. As an example, if /u1D434 was changed from 26 to 5 in 2025, quite stronger red peak could be possibly expected in the broad H /u1D6FC of SDSS J1052+1036, as shown in the top region of the middle right panel of Fig. 2. Similar results can be found, if the circular plus arm model discussed in a dual core system. Therefore, there are no plots or discussions for the corresponding results based on the circular plus arm contained in a dual core system. Before the end of the subsection, an additional point should be noted. Discussions above on variations of the profiles of the shifted broad H /u1D6FC are due to pure disk precessions in SDSS J1052+1036 as a disk emitter, which can lead to apparent variations from the elliptical accretion disk model but no apparent variations from the circular disk model neither from the circular plus arm model. However, effects of variability of AGN activities should also have apparent effects on the variations of profiles of the shifted broad H /u1D6FC in SDSS J1052+1036 as a disk emitter, which will be discussed as follows. Sizes /u1D445 /u1D435 /u1D43F /u1D445 /u1D460 of central BLRs of dozens of broad line AGN have been measured through the reverberation mapping technique (Blandford & McKee 1982; Peterson 1993; Peterson et al. 1999; Barth et al. 2015; Du et al. 2018; Shen et al. 2019b; Lu et al. 2023), leading to the known dependence of /u1D445 /u1D435 /u1D43F /u1D445 /u1D460 on AGN continuum luminosity (or on broad line luminosity) (Kaspi et al. 2000; Bentz et al. 2013). Therefore, stronger central AGN emissions can lead to deeper ionization boundaries of central BLRs, in other words, stronger AGN emissions can lead to larger /u1D445 /u1D435 /u1D43F /u1D445 /u1D460 . Therefore, if there was apparent variability of AGN activities in SDSS J1052+1036, different inner radius and outer radius in different epochs could be expected from those listed in the Table 2 of the different accretion disk models for shifted broad H /u1D6FC shown in Fig. 2, leading to probably different line profiles. As an example, if we assumed that there was a spectrum observed again at any time but with the re-observed AGN continuum luminosity at 5100Å about 2 times of the AGN continuum luminosity shown in the top panel of Fig. 1, leading the expected /u1D445 /u1D435 /u1D43F /u1D445 /u1D460 to be 1.4 times larger than the /u1D445 /u1D435 /u1D43F /u1D445 /u1D460 for the shifted broad H /u1D6FC shown in Fig. 2. Then, the re-determined inner radius and outer radius of the different disk models should be approximately 1.4 times of the /u1D45F 0 and /u1D45F 1 listed in the Table 2. Based on the re-determined inner radius and outer radius and the other model parameters which remain unchanged, the model expected line profiles are shown in Fig. 6 only considering AGN variability, leading to apparent variations of the line profiles of the shifted broad H /u1D6FC by different accretion disk models (even the pure symmetric circular disk model) in different epochs, strongly different from the expected results by the rSMBH scenario. Therefore, besides the accretion disk precessions, strong AGN variations have also apparent effects on variations of the profiles of the shifted broad H /u1D6FC , which will provide further clues to support SDSS J1052+1036 as a disk emitter.",
"pages": [
8,
9
]
},
{
"title": "4 MAIN SUMMARY AND CONCLUSIONS",
"content": "A rare low-redshift Type-1.9 AGN SDSS J1052+1036 but with very blue-shifted broad H /u1D6FC is reported in this manuscript. The main summary and conclusions are as follows. blue-shifted component and the red-shifted component in the broad H /u1D6FC .",
"pages": [
9,
10
]
},
{
"title": "ACKNOWLEDGEMENTS",
"content": "Zhang gratefully acknowledges the anonymous referee for giving us constructive comments and suggestions to greatly improve our paper. Zhang gratefully acknowledges the kind financial support from GuangXi University and the kind funding support NSFC-12173020 and NSFC-12373014. This research has made use of the data from the SDSS ( https://www.sdss.org/ ) funded by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation and the U.S. Department of Energy Office of Science. The research has made use of the MPFIT package https://pages.physics.wisc.edu/~craigm/idl/cmpfit.html , and of the emcee package https://pypi.org/project/emcee/ .",
"pages": [
10
]
},
{
"title": "DATA AVAILABILITY",
"content": "The data underlying this article will be shared on request to the corresponding author (xgzhang@gxu.edu.cn).",
"pages": [
10
]
},
{
"title": "REFERENCES",
"content": "Barth, A. J.; Bennert, V. N.; Canalizo, G.; Filippenko, A. V.; Gates, E. L., 2015, ApJS, 217, 26 Batiste, M.; Bentz, M. C.; Raimundo, S. I.; Vestergaard, M.; Onken, C. A., 2017, ApJL, 838, 10 Bekenstein, J. D. 1973, ApJ, 183, 657 Bellm E. C., Kulkarni, S. R.; Barlow, T., et al., 2019, PASP, 131, 068003 Blandford, R. D.; McKee, C. F., 1982, ApJ, 255, 419 Blecha, L., Loeb, A. 2008, MNRAS, 390, 1311 Blecha, L., Sijacki, D., Kelley, L. Z., et al. 2016, MNRAS, 456, 961 Bennert, V. N.; Treu, T.; Ding, X.; et al., 2021, ApJ, 921, 36 Bentz, M. C.; Denney, K. D.; Grier, C. J, et al., 2013, ApJ, 767, 149 Bogdanovic, T.; Eracleous, M.; Sigurdsson, S., 2009, ApJ, 697, 288 Bruzual, G.; Charlot, S. 2003, MNRAS, 344, 1000 Cappellari, M., 2017, MNRAS, 466, 798 Chiaberge, M.; Ely, J. C.; Meyer, E. T.; et al., 2017, A&A, 600, 57 Chiaberge, M.; Tremblay, G. R.; Capetti, A.; Norman, C., 2018, ApJ, 861, 56 Cid Fernandes, R.; Mateus, A.; Sodre, L.; Stasinska, G.; Gomes, J. M., 2005, MNRAS, 358, 363 Deane, R. P.; Paragi, Z.; Jarvis, M. J.; et al., 2014, Natur, 511, 57 Dekany, R.; Smith, R. M.; Riddle, R., et al., 2020, PASP, 132, 038001 Di Matteo, T.; Springel, V.; Hernquist, L., Natur, 433, 604 Drake, A. J.; Djorgovski, S. G.; Mahabal, A.; et al., 2009, ApJ, 696, 870 Du, P.; Zhang, Z.-X.; Wang, K.; Huang, Y.-K.; Zhang, Y.; et al., 2018, ApJ, Eracleous, M.; Lewis, K. T.; Flohic, H. M. L. G., NewAR, 53, 133 Eracleous, M.; Boroson, T. A.; Halpern, J. P.; Liu, J., 2012, ApJS, 201, 23 Ferrarese, F.; Merritt, D., 2000, ApJL, 539, 9 Graham, M. J., Djorgovski, S. G., Stern, D., et al., 2015b, MNRAS, 453, 1562 Greene, J. E.; Ho, L. C., 2005, ApJ, 630, 122 Gualandris, A.; Merritt, D., 2008, ApJ, 678, 180 Kim, D. C.; Yoon, I.; Privon, G. C.; Evans, A. S.; Harvey, D.; Stierwalt, S.; Kim, J. H., 2017, ApJ, 840, 71 Kochanek, C. S.; Shappee, B. J.; Stanek, K. Z.; et al., 2017, PASP, 129, 4502 Komossa, S., Merritt, D., 2008a, ApJL, 683, 21 Komossa, S., Merritt, D., 2008b, ApJL, 689, 89 Komossa, S., Zhou, H., Lu, H., 2008, ApJL, 678, 81 Peterson, B. M.; Ferrarese, L.; Gilbert, K. M., et al., 2004, ApJ, 613, 682 Runnoe, J. C.; Eracleous, M.; Pennell, A.; et al., 2017, MNRAS, 468, 1683 Storchi-Bergmann, T.; Schimoia, J. S.; Peterson, B. M.; Elvis, M.; Denney, K. D.; Eracleous, M.; Nemmen, R. S., 2017, ApJ, 835, 236 Volonteri, M., 2007, ApJL, 663, 5 Wang, M.; Luo, A.; Song, Y., et al., 2019, MNRAS, 482, 1889 Ward, C.; Gezari, S.; Frederick, S.; et al., 2021, ApJ, 913, 102",
"pages": [
10
]
}
]
2024arXiv240202257S
https://arxiv.org/pdf/2402.02257.pdf
<document>
<section_header_level_1><location><page_1><loc_6><loc_79><loc_97><loc_84></location>Cooling and heating regions of Joule-Thomson expansion for AdS black holes: Maxwell-power-Yang-Mills and Kerr Sen black holes</section_header_level_1>
<text><location><page_1><loc_19><loc_76><loc_84><loc_77></location>Jafar Sadeghi /star 1 , Mohammad Reza Alipour /star 2 Saeed Noori Gashti † ,/star 3 ,</text>
<text><location><page_1><loc_38><loc_73><loc_64><loc_75></location>Mohammad Ali S. Afshar /star 4</text>
<text><location><page_1><loc_20><loc_66><loc_82><loc_72></location>/star Department of Physics, Faculty of Basic Sciences, University of Mazandaran P. O. Box 47416-95447, Babolsar, Iran † School of Physics, Damghan University, P. O. Box 3671641167, Damghan, Iran</text>
<section_header_level_1><location><page_1><loc_47><loc_59><loc_55><loc_61></location>Abstract</section_header_level_1>
<text><location><page_1><loc_10><loc_36><loc_93><loc_58></location>In this paper, we explore the Joule-Thomson expansion (JTE) process for the Einstein-Power-YoungMills (EPYM) and the AdS Kerr Sen (AKS) black holes. We study the effect of free parameters on the Joule-Thomson coefficient (JTC), the inversion curve, and the T min i /T c . The isenthalpic curves of the AKS black hole show cooling or heating behavior depending on the inversion curve, which is affected by the mass and the parameters b and a of the black hole. If we assume the parameter b to be zero, the results reduce to the Kerr-AdS black holes [1]. In [2, 3], for the Einstein-Power-Yang-Mills AdS black hole with q > 1 and n = 2, the T min i /T c is 1 / 2. But in this paper, for the AdS-Maxwell-power-Yang-Mills black hole, when q > 1, the T min i /T c is almost equal to 1 / 2 for the increase of Maxwell's charge C , and when q = 1 / 2, the T min i /T c is equal to 1 / 2 for all values of C . Also, when 1 / 2 < q < 1, the T min i /T c is close to the value of 1 / 2, and finally when 0 < q < 1 / 2, the values of T min i /T c move away from the value of 1 / 2, that is, they become smaller. For the AKS black hole, we found that for free parameters a = 0 . 00951 and b = 0 . 00475, the T min i /T c is almost 1 / 2. Finally, we compare our findings with others in the literature and summarize our results in Tables 1-5.</text>
<text><location><page_1><loc_10><loc_33><loc_78><loc_34></location>Keywords: JTE, AdS-Maxwell-power-Yang-Mills black holes, AdS Kerr Sen black holes</text>
<section_header_level_1><location><page_1><loc_5><loc_28><loc_17><loc_30></location>Contents</section_header_level_1>
<unordered_list>
<list_item><location><page_1><loc_5><loc_25><loc_20><loc_26></location>1 Introduction</list_item>
</unordered_list>
<text><location><page_1><loc_96><loc_25><loc_97><loc_26></location>2</text>
<unordered_list>
<list_item><location><page_1><loc_5><loc_21><loc_33><loc_23></location>2 Joule-Thomson expansion</list_item>
</unordered_list>
<text><location><page_1><loc_96><loc_21><loc_97><loc_23></location>3</text>
<unordered_list>
<list_item><location><page_1><loc_5><loc_18><loc_57><loc_19></location>3 Case I: AdS-Maxwell-power-Yang-Mills black holes</list_item>
</unordered_list>
<text><location><page_1><loc_96><loc_18><loc_97><loc_19></location>5</text>
<table>
<location><page_2><loc_5><loc_72><loc_98><loc_84></location>
</table>
<section_header_level_1><location><page_2><loc_5><loc_67><loc_27><loc_69></location>1 Introduction</section_header_level_1>
<text><location><page_2><loc_5><loc_34><loc_97><loc_65></location>The insights provided by astronomy, astrophysics, and experimental cosmology suggest that cosmic structures adhere to the same principles and foundations observed on Earth, albeit with minor variations and necessary simplifications. This structural and fractal similarity is a crucial tool for human ideation and cognition, enabling us to decode more aspects of this global pattern daily. Researchers leverage this understanding in hypothesizing theoretical models, which may not yet be empirically verifiable. They construct various models based on mathematical logic and physical laws, exploring all potential scenarios to predict the most plausible ideas consistent with the formation of the enigmatic, infinite universe. The existing black hole models, are actually one of the most obvious examples of this form of inference. Despite being one of the most elusive cosmic entities, they are studied based on this pattern. The comparison of a black hole's gravitational behavior with a thermodynamic ensemble in the 1970s gave rise to a significant branch of black hole physics, namely black hole thermodynamics. For instance, the four laws of black hole mechanics bear a striking resemblance to the laws of thermodynamics [4,5]. Similarly, a black hole's surface area is analogous to entropy in thermodynamics, and its surface gravity is comparable to temperature [4,5]. The introduction of Maldacena duality, also known as the AdS/CFT correspondence, established a deeper connection between black hole thermodynamics and this duality, offering profound insights into quantum gravity. In the context of AdS/CFT, a black hole's entropy in the bulk AdS space correlates with the entropy of the corresponding CFT on the boundary, known as the Bekenstein-Hawking entropy [6]. Moreover, the black hole's temperature is related to the CFT's temperature [6-9].</text>
<text><location><page_2><loc_5><loc_11><loc_97><loc_32></location>This correspondence provides a powerful tool to study the quantum aspects of gravity and black holes using the methods of quantum field theory. In AdS space, there is a Hawking-Page phase transition between a stable large black hole and a thermal gas [10]. This phase transition is a first-order transition that occurs when the temperature of the system reaches a critical value, where the free energy of the black hole becomes lower than that of the thermal gas [10]. This phase transition can be interpreted as a confinement deconfinement phase transition of a gauge field [11]. The Hawking-Page phase transition can be seen as a transition from a deconfined phase in the thermal gas to a confined phase in the black hole [11]. When the AdS black holes have electric charge, they exhibit rich phase structures that were studied by Chamblin et al [12,13]. They found that the phase transition behavior of charged AdS black holes resembles the liquid-gas phase transition in a van der Waals system [14]. In the extended phase space where the cosmological constant is treated as pressure [15], the P-V critical behavior of charged AdS black holes was investigated and it was shown that they have a similar analogy to the van der Waals liquid-gas system. In addition to the phase transition and critical phenom-</text>
<text><location><page_3><loc_5><loc_59><loc_97><loc_84></location>ena [16-23], the analogy between the black holes and the van der Waals system was also creatively applied to the well-known JTE process [24] recently. This means that the JTC can be used to study the thermodynamics of black holes as well. For example, the isenthalpic expansion process is the analogue of the JTE process for black holes, where the black hole mass is constant while the black hole pressure and volume are changed. The black hole pressure is related to the cosmological constant, and the black hole volume is related to the horizon radius.In this case, the inversion curve is the curve that separates the regions where the black hole temperature increases or decreases as the pressure decreases.One of the intriguing features of the inversion curves for black hole systems is that they have only positive slopes, unlike the van der Waals system, which has both positive and negative slopes. For example, for charged AdS black holes [24] and Kerr-AdS black holes [1], the isenthalpic expansion process and the inversion curve have been analyzed and observed that the inversion curve was found to have a positive slope, meaning that the black hole temperature always decreases as the pressure decreases. Then the analysis was generalized to other types of AdS black holes, such as quintessence charged AdS black holes [25], holographic superfluids [26], charged AdS black holes in f(R) gravity [27], AdS black holes with a global monopole [28], and AdS black holes in Lovelock gravity [29]. For further study, you can see also [2,30-38].</text>
<text><location><page_3><loc_5><loc_33><loc_97><loc_56></location>All the results showed that the inversion curves for all these black hole systems have only positive slopes. We are interested in exploring whether this feature is universal for all black hole systems, or whether there are other effects that can alter it. To this end, we will focus on two types of AdS black holes: AdS-Maxwell-power-YangMills and AdS Kerr Sen. These black holes have additional parameters that can affect their thermodynamic behavior and phase transitions. AdS-Maxwell-power-Yang-Mills black holes are black holes that have a nonlinear electromagnetic field, which is described by a power-law function of the field strength [39]. This field can be seen as a generalization of the Maxwell field, which is the standard model of electromagnetism. AdS Kerr Sen black holes are black holes that have electric charge and angular momentum in a low-energy limit of heterotic string theory [40]. This theory is a type of string theory that combines the features of bosonic and supersymmetric strings. These black holes have different properties and characteristics than the standard AdS black holes, such as the existence of a dilaton field, which is a scalar field that couples to the electromagnetic field and the curvature. We will investigate how these parameters affect the inversion curves and the JTE process for these black holes, and compare them with the previous results for other black hole systems.</text>
<text><location><page_3><loc_5><loc_22><loc_97><loc_31></location>The structure of this paper is as follows. In Sec.II, we give a brief overview of the JTE. In Sec.III and Sec.IV, We introduce briefly AdS-Maxwell-Power-Yang-Mills and the AdS Kerr Sen black holes and their thermodynamic properties.In Sec.V, we study the JTE process for these black holes, derive an explicit expression for the JTC, and analyze and discuss the effect of the parameters of each model on the inversion curves. In Sec.VI, we present our conclusion and discussion.</text>
<section_header_level_1><location><page_3><loc_5><loc_17><loc_45><loc_19></location>2 Joule-Thomson expansion</section_header_level_1>
<text><location><page_3><loc_5><loc_12><loc_97><loc_15></location>In classical thermodynamics, a throttling process or the JTE process, discovered in 1852, is a method of cooling or heating a system by changing its pressure and volume, without adding or removing heat. In this process, the</text>
<text><location><page_4><loc_5><loc_77><loc_97><loc_84></location>high-pressure gas passes through a porous plug into a region with a low pressure, while keeping the enthalpy constant. Since it is a constant-enthalpy process, it can be used to experimentally measure the lines of constant enthalpy (isenthalps) on the (p, T) diagram of a system. Combined with the specific heat capacity at constant pressure, it allows the complete measurement of the thermodynamic potential for the gas [41].</text>
<text><location><page_4><loc_5><loc_69><loc_97><loc_75></location>In this method, The main goal is to investigate the behavior of the coefficient that describes the temperature change during the expansion or compression of a system at constant enthalpy, which is denoted by µ and is known as the JTC,</text>
<text><location><page_4><loc_5><loc_58><loc_97><loc_65></location>If the above coefficient is positive, as a result of pressure reduction, the temperature will decrease. In other words, the expansion of the gas causes cooling and the compression of the gas under investigation causes it to heat up. In other words, the positive JTC indicates the same direction of temperature and pressure. Whereas, if the JTC is negative, a decrease in pressure causes an increase in temperature.</text>
<formula><location><page_4><loc_37><loc_65><loc_65><loc_70></location>µ = ( ∂T ∂P ) H = 1 C P [ T ( ∂V ∂T ) P -V ] .</formula>
<text><location><page_4><loc_5><loc_53><loc_97><loc_58></location>The JT inversion temperature,which determined by setting µ = 0, is the temperature at which the sign of the JTC changes. Most real gases have an inversion point. The temperature of this point depends on the gas pressure before expansion.</text>
<text><location><page_4><loc_5><loc_16><loc_97><loc_49></location>The important point is that if you plot the JTC on the (p,T) diagram, a closed parabolic curve is created. In simpler terms, the inversion temperature is placed on the boundary of the curve of temperature changes in terms of pressure. At this temperature, the JTC changes from negative to positive. At a given pressure, the isopressure-temperature line intersects the drawn curve at two different points. These two points are called low temperature and high temperature of inversion. In fact, at temperatures higher and lower than these two temperatures, the sign of the JTC is negative and between these two temperatures, the sign of the JTC is positive. According to the above, the interesting phenomenon in this process is that the (p, T) diagram has two regions: one where the gas cools down and one where the gas heats up. These regions are separated by the inversion curve,the curve that shows the points where the system temperature does not change during the expansion process, that divides the graph into two regions: the cooling region and the heating region. The cooling region is where the gas temperature decreases as the pressure decreases, and the heating region is where the gas temperature increases as the pressure decreases. The inversion curve depends on the type of the gas and its initial conditions. If with respect to zero coefficient for ideal gases, we choose the van der Waals system, which is a more realistic model than the ideal gas, and takes into account the finite size and the attractive forces of the molecules, we find that for the van der Waals system, the inversion curves have both positive and negative slopes, forming a circle in the pressure axis. The inversion curve for the van der Waals system has a negative slope in the low-pressure region, where the attractive forces dominate, and a positive slope in the high-pressure region, where the repulsive forces dominate.</text>
<formula><location><page_4><loc_45><loc_48><loc_57><loc_53></location>T i = V ( ∂T ∂V ) P .</formula>
<section_header_level_1><location><page_5><loc_5><loc_82><loc_80><loc_84></location>3 Case I: AdS-Maxwell-power-Yang-Mills black holes</section_header_level_1>
<text><location><page_5><loc_5><loc_30><loc_97><loc_80></location>The AMPYM black holes are rooted in the study of black hole solutions in the context of supergravity theories, especially in anti-de Sitter (AdS) space. The study of black holes in AdS space is important for various reasons, including its relevance to string theory and the AdS/CFT correspondence, which is a duality between gravitational theories in AdS space and field theories defined on its boundary. These black holes are solutions to the equations of motion of supergravity theories with additional matter fields, such as Maxwell fields and power-Yang-Mills fields, within the context of AdS space that arises from a generalization of the EinsteinMaxwell theory with a negative cosmological constant and a non-Abelian gauge field.The history of AMPYM black holes can be traced back to the discovery of the first black holes in Einstein-Yang-Mills theory, which were considered in the works of Yasskin [42] and Kasuya [43]. It should be noted that in the non-Abelian case there are various gauge groups, but to obtain black hole solutions it is necessary to choose a specific form for the gauge potential. One of the simplest forms that nevertheless allowed to derive interesting and important results is the so-called Wu-Yang ansatz, which leads to magnetic-type solutions. [44-47]. Of course, an interesting point that can be mentioned is that the primary black holes with non-Abelian fields, which were studied in the late 80s [48-51], were found to be unstable in the case of asymptotically flat geometry [52,53], but later black hole solutions were obtained in the AdS case, which were shown to be stable [54-57]. The AMPYM black holes have been extensively studied in the context of theoretical physics, particularly in the context of holography and its applications to understanding strongly coupled field theories. In recent years, the study of AMPYM black holes has continued to be an active area of research, with a focus on understanding their thermodynamic properties, phase transitions, and connections to gauge/gravity duality. These black holes have been studied as models for strongly coupled systems in the dual field theories, providing valuable insights into nonperturbative phenomena in quantum field theories. Researchers have investigated various aspects of AMPYM black holes, such as their stability, entropy, and critical behavior. Their thermodynamic properties have been of particular interest, as they exhibit rich phase structures and can undergo phase transitions similar to those observed in condensed matter systems. Furthermore, the holographic interpretation of these black holes has led to fruitful connections between gravitational physics and field theory, shedding light on fundamental questions in quantum gravity and strongly coupled systems [58-60]. This section provides a summary of the thermodynamics of N-dimensional Einstein-Maxwell-power-Yang-Mills gravity with a cosmological constant Λ. This theory of gravity is based on the following action, which we will explain in more details. So we will have,</text>
<formula><location><page_5><loc_20><loc_24><loc_97><loc_29></location>I = 1 2 ∫ dx N √ -g ( R -( N -1)( N -2)Λ 3 -F µν F µν -( Tr ( F ( a ) µν F ( a ) µν )) q ) . (1)</formula>
<text><location><page_5><loc_5><loc_19><loc_97><loc_23></location>The trace element is represented by Tr ( . ) = Σ ( N -1)( N -2) / 2 a =1 ( . ), with R as the Ricci scalar and q as a real positive parameter. The Yang-Mills and Maxwell fields are defined accordingly [4],</text>
<formula><location><page_5><loc_34><loc_14><loc_97><loc_18></location>F ( a ) µν = ∂ µ A ( a ) ν -∂ ν A ( a ) µ + 1 2 σ C ( a ) ( b )( c ) A b µ A c ν , (2)</formula>
<formula><location><page_5><loc_43><loc_10><loc_97><loc_13></location>F µν = ∂ µ A ν -∂ ν A µ , (3)</formula>
<text><location><page_6><loc_5><loc_76><loc_97><loc_84></location>C ( a ) ( b )( c ) represents the structure constants of the ( N -1)( N -1) / 2 parameter Lie group G , while σ denotes the coupling constant. A ( a ) µ refers to the SO ( N -1) gauge group Yang-Mills potentials, with A µ representing the conventional Maxwell potential [39]. The metric solution corresponding to the N -dimensional spherically symmetric line element is as follows [40],</text>
<formula><location><page_6><loc_36><loc_71><loc_97><loc_75></location>ds 2 = -f ( r ) dt 2 + 1 f ( r ) dr 2 + r 2 d Ω 2 n . (4)</formula>
<text><location><page_6><loc_69><loc_65><loc_69><loc_68></location>/negationslash</text>
<text><location><page_6><loc_5><loc_62><loc_97><loc_70></location>The d Ω 2 n denotes the volume of the unit n -sphere. In this study, we will direct our attention to the EinsteinMaxwell-power-Yang-Mills theory ( EMPYM ) with N (= n + 2) ≥ 4 and q = ( n + 1) / 4. The solution to the N -dimensional EMPYM black hole with a negative cosmological constant under the condition of q = n +1 4 is provided [39],</text>
<text><location><page_6><loc_90><loc_63><loc_90><loc_66></location>/negationslash</text>
<formula><location><page_6><loc_32><loc_52><loc_97><loc_61></location>f ( r ) = 1 -2 m r n -1 -Λ r 2 3 + 2( n -1) C 2 nr 2( n -1) + Q r 4 q -2 Q = [ n ( n -1) Q 2 1 ] q n (4 q -n -1) . (5)</formula>
<text><location><page_6><loc_5><loc_45><loc_97><loc_52></location>It is important to note that the parameter m represents the mass of the black hole, while C and Q 1 denote the charges of the Maxwell field and Yang-Mills field, respectively. In the extended phase space, the cosmological constant is considered as a thermodynamic pressure P = -Λ 8 π , in this case, the Hawking temperature, mass and entropy of the black hole are obtained as follows,</text>
<formula><location><page_6><loc_25><loc_39><loc_97><loc_43></location>T = -C 2 ( n -1) 2 2 πnr 2 n -1 + + 2 3 ( n +1) Pr + -Q ( -n +4 q -1) 4 πr 4 q -1 + + n -1 4 πr + , (6)</formula>
<formula><location><page_6><loc_27><loc_34><loc_97><loc_39></location>M = nω n 48 π ( 6 C 2 ( n -1) nr n -1 + +8 πPr n +1 + +3 Qr n -4 q +1 + +3 r n -1 + ) , (7)</formula>
<formula><location><page_6><loc_39><loc_27><loc_97><loc_33></location>S = ω n r n + 4 , ω n = 2 π n +1 2 Γ ( n +1 2 ) , (8)</formula>
<text><location><page_6><loc_5><loc_27><loc_79><loc_28></location>where r + and ω n are the horizon radius and the volume of the unit n -sphere respectively.</text>
<section_header_level_1><location><page_6><loc_5><loc_22><loc_56><loc_24></location>4 Case II: AdS-Kerr-Sen black hole</section_header_level_1>
<text><location><page_6><loc_5><loc_11><loc_97><loc_20></location>The Kerr-Sen-AdS black hole is a complex and fascinating topic in the field of theoretical physics. It is a type of rotating, charged black hole that emerges from heterotic string theory. It is a generalization of the Kerr-NewmanAdS black hole, which is a solution of the Einstein-Maxwell equations with a negative cosmological constant. The Kerr-Sen-AdS black hole also involves a dilaton and an axion field, which are scalar and pseudoscalar fields that appear in string theory [61]. The first exact solution of the Einstein field equations, known as the</text>
<text><location><page_7><loc_5><loc_55><loc_97><loc_84></location>Schwarzschild solution, was discovered by Karl Schwarzschild in 1916. This solution describes a simple, nonrotating, uncharged black hole. The next major advancement came in 1963, when Roy P. Kerr found a solution to the Einstein field equations that describes a rotating black hole. This was a significant development, as it is believed that most black holes in the universe are rotating. In 1965, Ezra Newman and his collaborators found a solution that describes a rotating, charged black hole, known as the Kerr-Newman black hole. This solution was later extended by Ashoke Sen to include a dilaton field and an axion field, resulting in the Kerr-Sen black hole. The AdS form of this black hole was first derived by Ashoke Sen in 1992, by applying a series of duality transformations to the Kerr-Newman-AdS black hole. Sen showed that the Kerr-Sen-AdS black hole retains some of the symmetries and properties of the Kerr-Newman-AdS black hole, such as the existence of an event horizon, an ergosphere, and a Penrose process. However, the Kerr-Sen-AdS black hole also has some unique features, such as the dependence of the mass and angular momentum on the dilaton charge, and the violation of the cosmic censorship conjecture for some values of the parameters [61]. The Kerr-Sen-AdS black hole solution is a specific example of a rotating black hole with additional fields, which is of particular interest due to the role of angular momentum and the presence of nontrivial matter content in the spacetime geometry and is a solution to the equations of motion of supergravity theories with additional matter fields, such as the Sen-type dilaton and antisymmetric tensor fields, within the context of AdS space.</text>
<text><location><page_7><loc_5><loc_17><loc_97><loc_55></location>The properties and thermodynamics of Kerr Sen-AdS black holes have been the subject of intense research, given their relevance to understanding the behavior of rotating black holes in the presence of nontrivial matter content and their implications for the AdS/CFT correspondence [62-72]. The study of Kerr Sen-AdS black holes has been motivated by theoretical developments in string theory, quantum gravity, and holography, as well as by their potential implications for gravity dualities and the behavior of strongly coupled systems in the dual field theories. This black hole has been studied from various perspectives, including its thermodynamic properties, stability, and connections to the dynamics of dual field theories. In recent years, there has been growing interest in exploring the dynamical behavior of Kerr Sen-AdS black holes, including their evolution, instability, and the connections to chaos and information loss puzzles. Researchers have investigated the behavior of these black holes under various perturbations and have sought to understand their implications for fundamental questions about black hole thermodynamics and the fate of information in quantum gravity. Moreover, the holographic interpretation of Kerr Sen-AdS black holes has led to fruitful connections between gravitational physics and field theory, shedding light on fundamental questions in quantum gravity and strongly coupled systems [62-72]. In summary, the history and ongoing research on Kerr Sen-AdS black holes represent a rich and interdisciplinary area of study that has fruitful connections to diverse fields of theoretical physics, including string theory, quantum gravity, holography, and nonlinear dynamics. The exploration of these black holes continues to be an exciting frontier for probing the fundamental nature of spacetime and its connections to quantum theory. In this section provides a short overview of the Kerr-Sen black hole and its generalization to the anti-de Sitter spacetimes. Sen [61] found a solution of the low-energy effective action of the heterotic string theory, which describes a charged rotating black hole, known as the Kerr-Sen black hole.The action is a modification of the general relativity action with additional fields from the heterotic string theory, given by [62,78],</text>
<formula><location><page_7><loc_30><loc_9><loc_97><loc_16></location>S = ∫ d 4 x √ -˜ ge -Φ [ R +( ∇ Φ) 2 -1 8 F 2 -1 12 H 2 ] , (9)</formula>
<text><location><page_8><loc_5><loc_79><loc_97><loc_84></location>where ˜ g is the determinant of the metric tensor g µν , R is the Ricci scalar, F = F µν F µν with F µν being the U (1) Maxwell field strength tensor, Φ is a scalar dilaton field, and H = H µνρ H µνρ is the field strength for the axion field. The action can be transformed to the Einstein frame by a conformal transformation of the metric:</text>
<text><location><page_8><loc_5><loc_72><loc_42><loc_74></location>The action in the Einstein frame is given by:</text>
<formula><location><page_8><loc_45><loc_72><loc_97><loc_77></location>ds 2 E = e -Φ d ˜ s 2 . (10)</formula>
<formula><location><page_8><loc_29><loc_67><loc_97><loc_72></location>S = ∫ d 4 x √ -g [ R -1 2 ( ∇ Φ) 2 -e -Φ 8 F 2 -e -2Φ 12 H 2 ] . (11)</formula>
<text><location><page_8><loc_5><loc_65><loc_82><loc_66></location>The metric of a Kerr-Sen-AdS black hole in Boyer-Lindquist coordinates is given by [62,78]:</text>
<formula><location><page_8><loc_15><loc_60><loc_97><loc_63></location>ds 2 = -∆ r ρ 2 ( dt -a sin 2 θ Ξ dφ ) 2 + ρ 2 ∆ r dr 2 + ρ 2 ∆ θ dθ 2 + sin 2 θ ∆ θ ρ 2 [ -( r 2 +2 br + a 2 ) Ξ dφ + adt ] 2 , (12)</formula>
<text><location><page_8><loc_5><loc_57><loc_10><loc_59></location>where</text>
<formula><location><page_8><loc_34><loc_43><loc_97><loc_56></location>ρ 2 = r 2 + a 2 cos 2 θ +2 br, ∆ = ( r 2 +2 br + a 2 )(1 + r 2 +2 br /lscript 2 ) -2 mr, Ξ = 1 -a 2 /lscript 2 , ∆ θ = 1 -a 2 /lscript 2 cos 2 θ. (13)</formula>
<text><location><page_8><loc_5><loc_27><loc_97><loc_42></location>The parameter b is the dyonic charge of the black holes and is expressed as b = q 2 / (2 m ), where q is the electric charge and m is the mass of the black holes. In the limit of /lscript → ∞ , the metric reduces to the usual KerrSen black holes. The non-rotating case ( a = 0) reduces to the Gibbons-Maeda-Garfinkle-Horowitz-Strominger (GMGHS) solution. Gibbons and Maeda [73,74] found the black hole and black brane solutions in the dilaton theory, and Garfinkle-Horowitz-Strominger obtained their charged version [75]. It is worth mentioning that the Kerr-Sen-AdS black holes in four dimensions have been explored from various aspects, such as the black hole shadows [76] and the phase space thermodynamics in the extended phase space [77]. The ADM mass M, the angular momentum J, and the charge Q in AdS spacetimes are related by,</text>
<formula><location><page_8><loc_47><loc_17><loc_97><loc_27></location>M = m Ξ 2 , J = ma Ξ 2 , Q = q Ξ . (14)</formula>
<text><location><page_8><loc_5><loc_14><loc_29><loc_16></location>Also, the entropy is given by,</text>
<formula><location><page_8><loc_40><loc_10><loc_97><loc_13></location>S = A 4 = π ( r 2 + +2 br + a 2 ) Ξ , (15)</formula>
<text><location><page_9><loc_5><loc_79><loc_97><loc_84></location>where r + is the radius of the event horizon, which is the largest root of ∆ = 0. When b=0, the thermodynamic quantities become the same as those of Kerr-AdS black holes. Kerr-AdS black holes are solutions of the Einstein field equations in four dimensions with negative cosmological constant and rotation.</text>
<section_header_level_1><location><page_9><loc_5><loc_74><loc_40><loc_75></location>5 Discussion and results</section_header_level_1>
<text><location><page_9><loc_5><loc_67><loc_97><loc_72></location>In this section, we investigate the JTE process for AdS black holes: AMPYM and AdS Kerr Sen and derive a clear formula for the JTC. We also examine how the charge and the parameters of AMPYM and Kerr-Sen theories affect the inversion curves. We also compare the inversion curves for different scenarios</text>
<section_header_level_1><location><page_9><loc_5><loc_62><loc_18><loc_64></location>5.1 Case I</section_header_level_1>
<text><location><page_9><loc_5><loc_47><loc_97><loc_61></location>In this section, we explore the JTE of black hole systems in the extended phase space, where the black hole mass M is the same as the enthalpy H. We compare this process with the JT process of van der Waals gases with fixed particle number, and we use the fixed charge Q for the black hole systems. We also assume that the other parameters are constant. Using [2,45] as a reference, and considering the mass of black holes, we can write the pressure P as a function of M and r + , the mass and the horizon radius of a black hole. By substituting this expression in the temperature formula, we can obtain the temperature as a function of M and r + as well. Furthermore, we can express the mass M and the temperature T in terms of the pressure P and the radius r + of a black hole. Therefore, we get,</text>
<formula><location><page_9><loc_27><loc_40><loc_97><loc_46></location>P ( M,r + ) = -3 ( 2 C 2 ( n -1) r -2 n + n + Qr -4 q + + 1 r 2 + ) 8 π + 6 Mr -n -1 + nω n . (16)</formula>
<text><location><page_9><loc_5><loc_38><loc_46><loc_39></location>Also, we have T in terms of M and r + as follows,</text>
<formula><location><page_9><loc_24><loc_32><loc_97><loc_36></location>T ( M,r + ) = -2 C 2 ( n -1) r 2 -2 n + +2 qQr 2 -4 q + +1 2 πr + + 4 M ( n +1) r -n + nω n . (17)</formula>
<text><location><page_9><loc_5><loc_20><loc_97><loc_31></location>By solving the above equations, one can obtain the function T(M, P), which is lengthy and will not be shown here. For particular free parameters for this model, the T(M, P) curve can be shown. According to the definition of the JTC µ = ( ∂T ∂P ) M , the inversion pressure and temperature between the cooling and heating regions are ( ˜ P -˜ T ), which are determined by µ = 0. Therefore, the most important thing is to find the function expression of µ . By setting µ = 0, one can obtain the inversion points ( ˜ P , ˜ T ) for different fixed enthalpy M. With respect to [24], the JTC is given by,</text>
<formula><location><page_9><loc_37><loc_14><loc_97><loc_19></location>µ = ( ∂T ∂P ) M = 1 C P [ T ( ∂V ∂T ) P -V ] . (18)</formula>
<text><location><page_9><loc_5><loc_10><loc_97><loc_14></location>This approach is elegant. However, in this paper, we will use more straightforward methods by applying only mass and temperature to derive the JTC µ . we can see that temperature is a function of pressure and radius,</text>
<text><location><page_10><loc_5><loc_82><loc_64><loc_84></location>and radius is a function of pressure and mass. So, the JTC is given by,</text>
<formula><location><page_10><loc_42><loc_76><loc_97><loc_81></location>µ = ( ∂T ∂P ) M = ∂ ,r + T ∂ ,r + P . (19)</formula>
<text><location><page_10><loc_5><loc_72><loc_85><loc_76></location>Now, using the relationship µ = ∂T ∂P | M = ∂T ∂r + ∂P ∂r + , we calculate Joule Thomson coefficient as follows,</text>
<formula><location><page_10><loc_8><loc_65><loc_97><loc_72></location>µ = 2 nr + ( ω n ( r 4 q + ( 2 C 2 (2 n 2 -3 n +1) r 2 + + r 2 n + ) +8 q 2 Qr 2 n +2 -2 qQr 2 n +2 + ) -8 πM ( n +1) r n +4 q +1 + ) 3 ( nω n ( r 4 q + (2 C 2 ( n -1) r 2 + + r 2 n + ) + 2 qQr 2 n +2 + ) -8 πM ( n +1) r n +4 q +1 ) . (20)</formula>
<formula><location><page_10><loc_45><loc_50><loc_97><loc_55></location>T i = V ( ∂T ∂V ) P , (21)</formula>
<text><location><page_10><loc_5><loc_54><loc_97><loc_65></location>When the value of the coefficient µ is positive during the expansion, it means that the temperature decreases and therefore it is called a cooling phenomenon. However, when µ is negative, the temperature increases, and this is called a heating process.Using various equations, we can write the mass M and the temperature T as functions of the pressure P and the radius r + , which are the properties of a black hole. For µ = 0, we can obtain the inversion temperature, in which the process of the temperature changes reverses. It can be obtained by the formula,</text>
<formula><location><page_10><loc_42><loc_47><loc_97><loc_50></location>V = ∂M ∂p = 1 6 nr n +1 + ω n . (22)</formula>
<text><location><page_10><loc_5><loc_43><loc_97><loc_46></location>At the inversion temperature, the value of µ is 0, and the inversion temperature is determined by the following equation:</text>
<formula><location><page_10><loc_43><loc_37><loc_97><loc_41></location>T i = V ∂T ∂V = V ∂T ∂r + ∂V ∂r + . (23)</formula>
<text><location><page_10><loc_5><loc_32><loc_97><loc_35></location>This is beneficial for identifying the areas of heating and cooling in the T -P plane. We calculate t using the equations (17), (20), (21), (22) and (23),</text>
<formula><location><page_10><loc_17><loc_26><loc_97><loc_31></location>T i = r + ( 6 C 2 ( n -1) 2 (2 n -1) r -2 n + n +8 π ( n +1) P i +3(1 -4 q ) Q ( n -4 q +1) r -4 q + + 3 -3 n r 2 + ) 12 π ( n +1) . (24)</formula>
<text><location><page_10><loc_5><loc_23><loc_35><loc_24></location>We can also have from relation (17),</text>
<formula><location><page_10><loc_22><loc_17><loc_97><loc_21></location>T i = -6 C 2 ( n -1) 2 r 1 -2 n + n +8 π ( n +1) r + P i +3 Q ( n -4 q +1) r 1 -4 q + + 3( n -1) r + 12 π . (25)</formula>
<text><location><page_10><loc_5><loc_10><loc_97><loc_16></location>Figure 1 displays the Hawking temperature as a function of the horizon. We keep the free parameters constant for each plot. In each subfigure, we can observe some zero points for different free parameters. These zero points correspond to the divergence points of the JTC, as we can easily see in figure 1. According to the radius</text>
<figure>
<location><page_11><loc_20><loc_32><loc_82><loc_84></location>
<caption>Figure 1: It shows the plot of ( T -r + ) for the AMPYM black hole with respect to free parameters that are determined in each plot</caption>
</figure>
<text><location><page_11><loc_5><loc_10><loc_97><loc_23></location>of the horizon and the values of free parameters of a black hole, for small values for the radius of the event horizon, our structural behavior is completely distinct, and for larger radii, the figures almost converge. We have plotted the isenthalpic curves and the inversion curve of the AMPYM black hole for various values of the free parameter in each plot of Figure 2. In every subfigure, two isenthalpic curves with different values of M, along with the corresponding inversion curve that occurs at the highest point of the isenthalpic curves. We denote the inversion temperature and pressure of each isenthalpic curve as T i and P i , respectively. The isenthalpic curves are divided into two regions by the inversion curve: for P < P i , the isenthalpic curve has</text>
<text><location><page_12><loc_5><loc_79><loc_97><loc_84></location>a positive slope, indicating that the black hole undergoes cooling during the expansion process. However, for P > P i , the isenthalpic curve has a negative slope, implying that the black hole experiences heating during the expansion process. This behavior is consistent for different values of the free parameter in Figures (2a-2b). We</text>
<figure>
<location><page_12><loc_20><loc_51><loc_82><loc_77></location>
<caption>Figure 2: Isenthalpic curves and inversion curve of the AMPYM black hole with respect to mentioned free parameters that determine in each plot.</caption>
</figure>
<text><location><page_12><loc_5><loc_29><loc_97><loc_43></location>do not present the explicit expressions of the solutions here, as they are too long and complex. However, we can illustrate the relation between the inversion temperature T i and the inversion pressure P i by using the equations given above. Figure 3 shows the inversion curves for different free parameters. The inversion curve has only one branch. The inversion temperature rises steadily with the inversion pressure, but the slope of the inversion curves becomes smaller. We can also notice some fine structures in the cases of subfigures 3. For low pressure, the inversion temperature varies with the free parameters that are specified in each plot. It was interesting for us to study the effect of parameters and we observed that the slope of the inversion curve increases with the changes in the various values of each of the parameters.</text>
<text><location><page_12><loc_5><loc_10><loc_97><loc_28></location>Before the end of this section, if we want to talk purely about the effect of Maxwell charges as a specific result of our work, we should say that in [3], the authors found that for q > 1 and n = 2 , the ratio T min i /T c is fully established, but in the present work, by adding the Maxwell charge to the mentioned black hole, we found that for C /greatermuch 1 the Yang-Mills parameter has no effect on the ratio, and in any case, the T min i /T c is equal to 1 / 2 , which has the same behavior as a four-dimensional charged black hole. For q > 1 , when C increases, the T min /T c approaches the value of 1 / 2 . However, for q < 1 , the conditions are slightly different from the previous situation, because when q = 1 / 2 , n = 2 , and Q 1 = 1 , the T min i /T c is equal to 1 / 2 for all values of 'C', which is the same as a charged black hole in four dimensions. Also, according to Table 1 to 3, it can be inferred that when 1 / 2 < q < 1 , with the increase of Maxwell charge, i.e. C , the T min i /T c only approaches the value of 1 / 2 and exhibits a more similar behavior to a four-dimensional charged black hole. If for 0 < q < 1 / 2 , the T min i /T c</text>
<figure>
<location><page_13><loc_20><loc_32><loc_82><loc_84></location>
<caption>Figure 3: It shows the plot of The inversion curve for the AMPYM black hole with respect to free parameters that are determined in each plot</caption>
</figure>
<text><location><page_13><loc_5><loc_16><loc_97><loc_23></location>becomes further or smaller than the value of 1 / 2 , which is contrary to a charged black hole in four dimensions. In general, when q > 1 , for the increase of Maxwell's charge C , the T min i /T c is almost equal to 1 / 2 and when q = 1 / 2 , for all values of the Maxwell's charge, the T min i /T c is equal to 1 / 2 . Also, when 1 < q < 1 / 2 , it is close to the value of 1 / 2 and finally when 1 / 2 < q < 0 , the values of the T min i /T c become smaller than of 1 / 2 .</text>
<section_header_level_1><location><page_14><loc_5><loc_82><loc_19><loc_84></location>5.2 Case II</section_header_level_1>
<text><location><page_14><loc_5><loc_76><loc_97><loc_81></location>With respect to [62, 78] and the Mass of black holes, We can express pressure P in terms of M and r + and replace it in the formula for the temperature, which will also become in terms of M and r + . We also rewrite the mass M and temperature T as a function of the pressure P and the radius r + of a black hole. So we will have,</text>
<formula><location><page_14><loc_31><loc_71><loc_97><loc_74></location>P ( M,r + ) = -3 ( a 2 + r + (2 b -2Ξ 2 M + r + )) 8 πr + (2 b + r + ) ( a 2 + r + (2 b + r + )) . (26)</formula>
<text><location><page_14><loc_5><loc_67><loc_59><loc_69></location>Also, the temperature in terms of M and r + is as follows [62, 78],</text>
<formula><location><page_14><loc_6><loc_57><loc_97><loc_66></location>X = a 4 ( b + r + ) + a 2 r + (4 b 2 +6 br + + r + (2 r + -Ξ 2 M )) + r 2 + (2 b + r + )(3 r + ( b -Ξ 2 M ) + 2 b ( b -Ξ 2 M ) + r 2 + ) Y = 2 πr + (2 b + r + )( a 2 +2 br + + r + )( a 2 + r + (2 b + r + )) T ( M,r + ) = -X Y . (27)</formula>
<text><location><page_14><loc_5><loc_55><loc_49><loc_57></location>Therefore, by using the equation (19) on can obtain,</text>
<formula><location><page_14><loc_6><loc_19><loc_96><loc_55></location>A = a 8 (2 b 2 +2 br + + r 2 + ) + a 6 r + ( 16 b 3 +2 b 2 (9 r + +2) + br + (2 e 2 M +8 r + +5) + r 2 + ( r + +2) ) + a 4 r 2 + [ 48 b 4 +4 b 3 (2Ξ 2 M +16 r + +5) + 4 b 2 r + (8Ξ 2 M +7 r + +8) + br 2 + (20Ξ 2 M +2 r + +19) -r 2 + ( Ξ 2 ( M -4 Mr + ) + ( r + -4) r + ) ] + a 2 r 3 + (2 b + r + ) [ 32 b 4 +4 b 3 (9 r + +4) + 2 b 2 r + (8Ξ 2 M +5 r + +11) + br + ( 4Ξ 2 M ( r + +1) + r + (11 -2 r + ) ) -( r + -2) r 3 + ] +(2 b +1) r 4 + (2 b + r + ) 2 [ 4 b 2 ( b -Ξ 2 M ) + r 2 + ( b -3Ξ 2 M ) + 4 br + ( b -Ξ 2 M ) ] B = [ 2 πr 2 + (2 b + r + ) 2 ( a 2 + r + +2 br + ) 2 ( a 2 + r + (2 b + r + )) 2 ] C = 3 [ a 4 ( b + r + ) + a 2 r + ( 4 b 2 +6 br + + r + (2 r + -Ξ 2 M ) ) + r 2 + (2 b + r + ) ( 3 r + ( b -Ξ 2 M ) + 2 b ( b -Ξ 2 M ) + r 2 + ) ] D = 4 πr 2 + (2 b + r + ) 2 ( a 2 + r + (2 b + r + ) ) 2 µ = A / B C / D .</formula>
<formula><location><page_14><loc_94><loc_19><loc_97><loc_20></location>(28)</formula>
<text><location><page_14><loc_5><loc_14><loc_97><loc_17></location>We express the mass M and temperature T in terms of the pressure P and the radius r + of a black hole, using different equations,</text>
<formula><location><page_14><loc_29><loc_10><loc_97><loc_14></location>M ( P, r + ) = ( a 2 + r + (2 b + r + )) (8 πpr + (2 b + r + ) + 3) 6Ξ 2 r + , (29)</formula>
<formula><location><page_15><loc_25><loc_80><loc_97><loc_85></location>T ( P, r + ) = a 2 ( 8 πpr 2 + -3 ) + r 2 + (8 πp (2 b + r + )(2 b +3 r + ) + 3) 12 πr + ( a 2 +2 br + + r + ) . (30)</formula>
<text><location><page_15><loc_5><loc_78><loc_53><loc_79></location>The V ( P ) for the AKS black hole is calculates as follows,</text>
<formula><location><page_15><loc_34><loc_73><loc_97><loc_77></location>V ( P ) = 4 π (2 b + r + ) ( a 2 + r + (2 b + r + )) 3Ξ 2 . (31)</formula>
<text><location><page_15><loc_5><loc_70><loc_61><loc_72></location>So, by using the equation (19), (21), (22) and (23) one can obtain,</text>
<formula><location><page_15><loc_17><loc_54><loc_97><loc_69></location>T i = (2 b + r + )( a 2 + r + (2 b + r + )) × [ a 4 (8 πP i r 2 + +3) + a 2 r + ( 32 πb 2 P i r + +4 b (32 πP i r 2 + +3) + 3(24 πP i r 3 + + r + +2) ) +16 π (2 b +1) P i r 4 + (4 b +3 r + ) ] / 12 πr 2 + ( a 2 +2 br + + r + ) 2 ( a 2 +(2 b + r + )(2 b +3 r + ) ) (32)</formula>
<text><location><page_15><loc_5><loc_51><loc_45><loc_53></location>Also, with respect to equation (30) we will have,</text>
<formula><location><page_15><loc_27><loc_46><loc_97><loc_51></location>T i = a 2 ( 8 πr 2 + P i -3 ) + r 2 + (8 πP (2 b + r + )(2 b +3 r + ) + 3) 12 πr + ( a 2 +2 br + + r + ) . (33)</formula>
<text><location><page_15><loc_5><loc_10><loc_97><loc_45></location>We have drawn the isenthalpic curves and the inversion curve of the AdS Kerr Sen black hole for various free parameter values in each plot of Figure 4. In every subfigure, three isenthalpic curves with different M are visible, along with the corresponding inversion curve that occurs at the highest point of the isenthalpic curves. We denote each isenthalpic curve's inversion temperature and pressure as T i and P i , respectively. The isenthalpic curves are divided into two regions by the inversion curve: for P < P i , the isenthalpic curve has a positive slope, indicating that the black hole undergoes cooling during the expansion process. With the changes in the parameters of the black hole, we found that the slope of the figures will change, and these changes are visible in each subfigure. Because an analytical solution is difficult to obtain, we used a numerical method to draw graphs. If we assume the parameter b to be zero, our equations and graphs will simplify to the Kerr-AdS black holes, whose results are thoroughly discussed in [1]. Due to the complexity of the calculations, we do not show the explicit expressions here. However, we can demonstrate the relation between the inversion temperature T i and the inversion pressure P i by using the equations given above. Figure (5a-5d) displays the inversion curves for different free parameters of the AKS black holes. The inversion curve has only one branch. The inversion temperature increases gradually with the inversion pressure, but the slope of the inversion curves becomes smaller. We can also observe some fine structures in the cases of subfigures 5. Inversion curves change with the free parameters that are specified in each plot. It was very difficult to find the analytical solution for drawing the graphs, so we used numerical solutions to draw them. Moreover, we note that due to the huge difference in the values of the horizontal and vertical graphs corresponding to the change of the free parameters, we have drawn the figures separately. In Figure 6, we plot the Hawking temperature as a function of the horizon, which is the</text>
<figure>
<location><page_16><loc_20><loc_32><loc_83><loc_84></location>
<caption>Figure 4: Isenthalpic curves and inversion curve of the AdS-Ker-Sen black hole with respect to mentioned free parameters that determine in each plot.</caption>
</figure>
<text><location><page_16><loc_5><loc_11><loc_97><loc_23></location>boundary of the black hole. The horizon can be affected by the presence of other fields or dimensions, which are represented by the free parameters in our model. We consider the free parameters constant for each plot, but we vary them across different plots to see how they influence the Hawking temperature. In each subfigure, we can observe some zero points, where the Hawking temperature becomes zero for different free parameters. This means that the black hole became extremal at these points. These zero points correspond to the divergence points of the JTC, which is a quantity that describes the temperature change when a gas expands or compresses at constant enthalpy. In the study of Kerr black hole [1], it was shown that the rotation parameter 'a' does not</text>
<figure>
<location><page_17><loc_20><loc_32><loc_83><loc_84></location>
<caption>Figure 5: It shows the plot of The inversion curve for the AdS-Ker-Sen black hole with respect to free parameters that are determined in each plot with M=20</caption>
</figure>
<text><location><page_17><loc_5><loc_11><loc_97><loc_23></location>have much effect on the Joule-Thomson representation. For example, it is practically eliminated in the ratio of the T min i /T C , and the value of this ratio is always a constant value of 1/2. In this paper, we found that the parameters related to AKS black hole, i.e. a and b, can play a vital role in the representation of the value of the T min i /T C . The value of this ratio increases with the reduction of these parameters. For this purpose, due to the structural complexities of this black hole, we could not use the analytical method to obtain the critical points of the black hole, so we resorted to the approximate method and the numerical calculations. In the [77], the value of critical temperature and critical pressure has been calculated. We calculated some values for this black hole</text>
<figure>
<location><page_18><loc_19><loc_32><loc_83><loc_84></location>
<caption>Figure 6: It shows the plot of ( T -r + ) for the AdS-Ker-Sen black hole with respect to free parameters that are determined in each plot</caption>
</figure>
<text><location><page_18><loc_5><loc_20><loc_97><loc_23></location>according to Table 4. As evident, regarding various values of free parameters, the T min i /T C is obtained, which shows the obvious effect of parameters a and b.</text>
<section_header_level_1><location><page_19><loc_5><loc_82><loc_25><loc_84></location>6 Conclusion</section_header_level_1>
<text><location><page_19><loc_5><loc_11><loc_97><loc_80></location>In this article, we investigated the behavior of two structurally different black holes under the JTE thermodynamic process. The JTE is a process that cools or heats a system by changing its pressure and volume at constant enthalpy, and its main goal is to study the behavior of the µ coefficient, which describes the change in temperature during the expansion or compression of a system. But before presenting the results, we must first explain the motivations that led us to choose these particular black holes. Recently, Einstein-Power-YoungMills black hole with AdS structure under JTE process was investigated in a research paper [2,3]. This raised the question in our mind that if a fields in the Maxwellian form are added to the elements in the action of the above article, can this change and addition of a special form of the field cause a different thermodynamic behavior in their JT representation? Similarly, in the past, the behavior of various forms of rotating black holes was also investigated in this thermodynamic process. But what effect can the addition of Sen-type dilaton fields and antisymmetric tensor fields have on its thermodynamic properties and JT representation? This was a question that could be a good motivation for our investigation. For this purpose, we plotted the isenthalpic curves and the inversion curves for each of the two black holes, for different values of the free parameters, which can be referred to the following results for each. For the AMPYM black hole, the Hawking temperature has zero points that depend on the free parameters and the horizon radius, and the JTC diverges at these points. The isenthalpic curves are divided into two regions by the inversion curve: for P < P i , the isenthalpic curve has a positive slope, indicating that the black hole is cooling during the expansion process, and for P > P i , the isenthalpic curve has a negative slope, which means that the black hole experiences heating during the expansion process. The inversion curves for different free parameters show that the inversion curve has only one branch and the inversion temperature increases steadily with the inversion pressure, but the slope of the inversion curves becomes smaller. Also, the graphs showed that for low pressure, the inversion temperature varies with the free parameters specified in each graph. Studying the effect of parameters was interesting for us because we observed that the slope of the inversion curve increases with changes in different values of each parameter. But if we want to talk about the effect of Maxwell's charges as a special result of our work, we must state that, we have found that the ratio of minimum to critical temperature for a 4-dimensional black hole with Yang-Mills hair depends on the values of q , M , and C . For q > 1 , the ratio is almost equal to 1 / 2 regardless of the Yang-Mills parameter when ' C /greatermuch 1' . For q = 1 / 2 , the ratio is exactly equal to 1 / 2 for all values of C . For 1 / 2 < q < 1 , the ratio approaches 1 / 2 as C increases. For 0 < q < 1 / 2 , the ratio deviates from 1 / 2 as C increases. These results show that the Maxwell charge can affect the thermodynamic behavior of the black hole and make it more or less similar to a 4-dimensional charged black hole. Also, for the AKS black hole, we found that its parameters, i.e. a and b, can play a vital role in the representation of the value of the ratio of T min i /T C , so that by reducing these parameters, the value of the ratio increases. The results are shown in Figure 7 and are summarized in Tables 1 to 5. For the AdS Kerr Sen black, It was very difficult to find the analytical solution for drawing the graphs, so we used numerical solutions to draw them. For P < P i , the isenthalpic curve has a positive slope, indicating that the black hole is cooling during the expansion process, but with the increases of the black hole parameters, we found that the slope of the figures will change. Also, if we assume the parameter b ( dyonic charge) to be zero, our equations and graphs will simplify to the Kerr-AdS black holes, whose results are thoroughly discussed in [1]. In general, it can be said about this black hole the isenthalpic curves of the</text>
<text><location><page_20><loc_5><loc_75><loc_97><loc_84></location>AKS black hole show cooling or heating behavior depending on the inversion curve, which is affected by the mass and the parameter b, a of the black hole. The inversion curve has a single branch with a positive slope that decreases with the free parameters, and the inversion temperature and pressure are related by some equations that are numerically solved. The Hawking temperature of the AKS black hole has zero points that depend on the free parameters and the horizon radius and lead the JTC to reflect divergence behavior.</text>
<figure>
<location><page_20><loc_15><loc_34><loc_87><loc_73></location>
<caption>Figure 7: It shows the plot of ( T min i T c ) in terms of C with respect to free parameters mentioned in the plot.</caption>
</figure>
<table>
<location><page_21><loc_10><loc_63><loc_92><loc_84></location>
<caption>Table 1: Summary of the results for the AMPYM black hole for q = 0 . 3</caption>
</table>
<table>
<location><page_21><loc_10><loc_38><loc_92><loc_59></location>
<caption>Table 2: Summary of the results for the AMPYM black hole for q = 0 . 5</caption>
</table>
<table>
<location><page_21><loc_10><loc_14><loc_92><loc_34></location>
<caption>Table 3: Summary of the results for the AMPYM black hole for q = 0 . 9</caption>
</table>
<table>
<location><page_22><loc_10><loc_62><loc_92><loc_80></location>
<caption>Table 4: Summary of the results for the AKS black hole</caption>
</table>
<table>
<location><page_22><loc_9><loc_18><loc_93><loc_50></location>
<caption>Table 5: Summary of the results for the AMPYM and AKS black holes compare with other results</caption>
</table>
<section_header_level_1><location><page_23><loc_5><loc_82><loc_20><loc_84></location>References</section_header_level_1>
<table>
<location><page_23><loc_5><loc_12><loc_98><loc_81></location>
</table>
<table>
<location><page_24><loc_5><loc_11><loc_98><loc_84></location>
</table>
<table>
<location><page_25><loc_5><loc_13><loc_98><loc_84></location>
</table>
<table>
<location><page_26><loc_5><loc_17><loc_98><loc_84></location>
</table>
<unordered_list>
<list_item><location><page_26><loc_5><loc_14><loc_97><loc_17></location>[59] Gogoi, Dhruba Jyoti, et al. 'Quasinormal Modes and Optical Properties of 4-D black holes in Einstein Power-Yang-Mills Gravity.' arXiv preprint arXiv:2306.14273 (2023).</list_item>
</unordered_list>
<table>
<location><page_27><loc_5><loc_12><loc_98><loc_84></location>
</table>
<table>
<location><page_28><loc_5><loc_46><loc_97><loc_84></location>
</table>
</document>
[
{
"title": "Cooling and heating regions of Joule-Thomson expansion for AdS black holes: Maxwell-power-Yang-Mills and Kerr Sen black holes",
"content": "Jafar Sadeghi /star 1 , Mohammad Reza Alipour /star 2 Saeed Noori Gashti † ,/star 3 , Mohammad Ali S. Afshar /star 4 /star Department of Physics, Faculty of Basic Sciences, University of Mazandaran P. O. Box 47416-95447, Babolsar, Iran † School of Physics, Damghan University, P. O. Box 3671641167, Damghan, Iran",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "In this paper, we explore the Joule-Thomson expansion (JTE) process for the Einstein-Power-YoungMills (EPYM) and the AdS Kerr Sen (AKS) black holes. We study the effect of free parameters on the Joule-Thomson coefficient (JTC), the inversion curve, and the T min i /T c . The isenthalpic curves of the AKS black hole show cooling or heating behavior depending on the inversion curve, which is affected by the mass and the parameters b and a of the black hole. If we assume the parameter b to be zero, the results reduce to the Kerr-AdS black holes [1]. In [2, 3], for the Einstein-Power-Yang-Mills AdS black hole with q > 1 and n = 2, the T min i /T c is 1 / 2. But in this paper, for the AdS-Maxwell-power-Yang-Mills black hole, when q > 1, the T min i /T c is almost equal to 1 / 2 for the increase of Maxwell's charge C , and when q = 1 / 2, the T min i /T c is equal to 1 / 2 for all values of C . Also, when 1 / 2 < q < 1, the T min i /T c is close to the value of 1 / 2, and finally when 0 < q < 1 / 2, the values of T min i /T c move away from the value of 1 / 2, that is, they become smaller. For the AKS black hole, we found that for free parameters a = 0 . 00951 and b = 0 . 00475, the T min i /T c is almost 1 / 2. Finally, we compare our findings with others in the literature and summarize our results in Tables 1-5. Keywords: JTE, AdS-Maxwell-power-Yang-Mills black holes, AdS Kerr Sen black holes",
"pages": [
1
]
},
{
"title": "Contents",
"content": "2 3 5",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "The insights provided by astronomy, astrophysics, and experimental cosmology suggest that cosmic structures adhere to the same principles and foundations observed on Earth, albeit with minor variations and necessary simplifications. This structural and fractal similarity is a crucial tool for human ideation and cognition, enabling us to decode more aspects of this global pattern daily. Researchers leverage this understanding in hypothesizing theoretical models, which may not yet be empirically verifiable. They construct various models based on mathematical logic and physical laws, exploring all potential scenarios to predict the most plausible ideas consistent with the formation of the enigmatic, infinite universe. The existing black hole models, are actually one of the most obvious examples of this form of inference. Despite being one of the most elusive cosmic entities, they are studied based on this pattern. The comparison of a black hole's gravitational behavior with a thermodynamic ensemble in the 1970s gave rise to a significant branch of black hole physics, namely black hole thermodynamics. For instance, the four laws of black hole mechanics bear a striking resemblance to the laws of thermodynamics [4,5]. Similarly, a black hole's surface area is analogous to entropy in thermodynamics, and its surface gravity is comparable to temperature [4,5]. The introduction of Maldacena duality, also known as the AdS/CFT correspondence, established a deeper connection between black hole thermodynamics and this duality, offering profound insights into quantum gravity. In the context of AdS/CFT, a black hole's entropy in the bulk AdS space correlates with the entropy of the corresponding CFT on the boundary, known as the Bekenstein-Hawking entropy [6]. Moreover, the black hole's temperature is related to the CFT's temperature [6-9]. This correspondence provides a powerful tool to study the quantum aspects of gravity and black holes using the methods of quantum field theory. In AdS space, there is a Hawking-Page phase transition between a stable large black hole and a thermal gas [10]. This phase transition is a first-order transition that occurs when the temperature of the system reaches a critical value, where the free energy of the black hole becomes lower than that of the thermal gas [10]. This phase transition can be interpreted as a confinement deconfinement phase transition of a gauge field [11]. The Hawking-Page phase transition can be seen as a transition from a deconfined phase in the thermal gas to a confined phase in the black hole [11]. When the AdS black holes have electric charge, they exhibit rich phase structures that were studied by Chamblin et al [12,13]. They found that the phase transition behavior of charged AdS black holes resembles the liquid-gas phase transition in a van der Waals system [14]. In the extended phase space where the cosmological constant is treated as pressure [15], the P-V critical behavior of charged AdS black holes was investigated and it was shown that they have a similar analogy to the van der Waals liquid-gas system. In addition to the phase transition and critical phenom- ena [16-23], the analogy between the black holes and the van der Waals system was also creatively applied to the well-known JTE process [24] recently. This means that the JTC can be used to study the thermodynamics of black holes as well. For example, the isenthalpic expansion process is the analogue of the JTE process for black holes, where the black hole mass is constant while the black hole pressure and volume are changed. The black hole pressure is related to the cosmological constant, and the black hole volume is related to the horizon radius.In this case, the inversion curve is the curve that separates the regions where the black hole temperature increases or decreases as the pressure decreases.One of the intriguing features of the inversion curves for black hole systems is that they have only positive slopes, unlike the van der Waals system, which has both positive and negative slopes. For example, for charged AdS black holes [24] and Kerr-AdS black holes [1], the isenthalpic expansion process and the inversion curve have been analyzed and observed that the inversion curve was found to have a positive slope, meaning that the black hole temperature always decreases as the pressure decreases. Then the analysis was generalized to other types of AdS black holes, such as quintessence charged AdS black holes [25], holographic superfluids [26], charged AdS black holes in f(R) gravity [27], AdS black holes with a global monopole [28], and AdS black holes in Lovelock gravity [29]. For further study, you can see also [2,30-38]. All the results showed that the inversion curves for all these black hole systems have only positive slopes. We are interested in exploring whether this feature is universal for all black hole systems, or whether there are other effects that can alter it. To this end, we will focus on two types of AdS black holes: AdS-Maxwell-power-YangMills and AdS Kerr Sen. These black holes have additional parameters that can affect their thermodynamic behavior and phase transitions. AdS-Maxwell-power-Yang-Mills black holes are black holes that have a nonlinear electromagnetic field, which is described by a power-law function of the field strength [39]. This field can be seen as a generalization of the Maxwell field, which is the standard model of electromagnetism. AdS Kerr Sen black holes are black holes that have electric charge and angular momentum in a low-energy limit of heterotic string theory [40]. This theory is a type of string theory that combines the features of bosonic and supersymmetric strings. These black holes have different properties and characteristics than the standard AdS black holes, such as the existence of a dilaton field, which is a scalar field that couples to the electromagnetic field and the curvature. We will investigate how these parameters affect the inversion curves and the JTE process for these black holes, and compare them with the previous results for other black hole systems. The structure of this paper is as follows. In Sec.II, we give a brief overview of the JTE. In Sec.III and Sec.IV, We introduce briefly AdS-Maxwell-Power-Yang-Mills and the AdS Kerr Sen black holes and their thermodynamic properties.In Sec.V, we study the JTE process for these black holes, derive an explicit expression for the JTC, and analyze and discuss the effect of the parameters of each model on the inversion curves. In Sec.VI, we present our conclusion and discussion.",
"pages": [
2,
3
]
},
{
"title": "2 Joule-Thomson expansion",
"content": "In classical thermodynamics, a throttling process or the JTE process, discovered in 1852, is a method of cooling or heating a system by changing its pressure and volume, without adding or removing heat. In this process, the high-pressure gas passes through a porous plug into a region with a low pressure, while keeping the enthalpy constant. Since it is a constant-enthalpy process, it can be used to experimentally measure the lines of constant enthalpy (isenthalps) on the (p, T) diagram of a system. Combined with the specific heat capacity at constant pressure, it allows the complete measurement of the thermodynamic potential for the gas [41]. In this method, The main goal is to investigate the behavior of the coefficient that describes the temperature change during the expansion or compression of a system at constant enthalpy, which is denoted by µ and is known as the JTC, If the above coefficient is positive, as a result of pressure reduction, the temperature will decrease. In other words, the expansion of the gas causes cooling and the compression of the gas under investigation causes it to heat up. In other words, the positive JTC indicates the same direction of temperature and pressure. Whereas, if the JTC is negative, a decrease in pressure causes an increase in temperature. The JT inversion temperature,which determined by setting µ = 0, is the temperature at which the sign of the JTC changes. Most real gases have an inversion point. The temperature of this point depends on the gas pressure before expansion. The important point is that if you plot the JTC on the (p,T) diagram, a closed parabolic curve is created. In simpler terms, the inversion temperature is placed on the boundary of the curve of temperature changes in terms of pressure. At this temperature, the JTC changes from negative to positive. At a given pressure, the isopressure-temperature line intersects the drawn curve at two different points. These two points are called low temperature and high temperature of inversion. In fact, at temperatures higher and lower than these two temperatures, the sign of the JTC is negative and between these two temperatures, the sign of the JTC is positive. According to the above, the interesting phenomenon in this process is that the (p, T) diagram has two regions: one where the gas cools down and one where the gas heats up. These regions are separated by the inversion curve,the curve that shows the points where the system temperature does not change during the expansion process, that divides the graph into two regions: the cooling region and the heating region. The cooling region is where the gas temperature decreases as the pressure decreases, and the heating region is where the gas temperature increases as the pressure decreases. The inversion curve depends on the type of the gas and its initial conditions. If with respect to zero coefficient for ideal gases, we choose the van der Waals system, which is a more realistic model than the ideal gas, and takes into account the finite size and the attractive forces of the molecules, we find that for the van der Waals system, the inversion curves have both positive and negative slopes, forming a circle in the pressure axis. The inversion curve for the van der Waals system has a negative slope in the low-pressure region, where the attractive forces dominate, and a positive slope in the high-pressure region, where the repulsive forces dominate.",
"pages": [
3,
4
]
},
{
"title": "3 Case I: AdS-Maxwell-power-Yang-Mills black holes",
"content": "The AMPYM black holes are rooted in the study of black hole solutions in the context of supergravity theories, especially in anti-de Sitter (AdS) space. The study of black holes in AdS space is important for various reasons, including its relevance to string theory and the AdS/CFT correspondence, which is a duality between gravitational theories in AdS space and field theories defined on its boundary. These black holes are solutions to the equations of motion of supergravity theories with additional matter fields, such as Maxwell fields and power-Yang-Mills fields, within the context of AdS space that arises from a generalization of the EinsteinMaxwell theory with a negative cosmological constant and a non-Abelian gauge field.The history of AMPYM black holes can be traced back to the discovery of the first black holes in Einstein-Yang-Mills theory, which were considered in the works of Yasskin [42] and Kasuya [43]. It should be noted that in the non-Abelian case there are various gauge groups, but to obtain black hole solutions it is necessary to choose a specific form for the gauge potential. One of the simplest forms that nevertheless allowed to derive interesting and important results is the so-called Wu-Yang ansatz, which leads to magnetic-type solutions. [44-47]. Of course, an interesting point that can be mentioned is that the primary black holes with non-Abelian fields, which were studied in the late 80s [48-51], were found to be unstable in the case of asymptotically flat geometry [52,53], but later black hole solutions were obtained in the AdS case, which were shown to be stable [54-57]. The AMPYM black holes have been extensively studied in the context of theoretical physics, particularly in the context of holography and its applications to understanding strongly coupled field theories. In recent years, the study of AMPYM black holes has continued to be an active area of research, with a focus on understanding their thermodynamic properties, phase transitions, and connections to gauge/gravity duality. These black holes have been studied as models for strongly coupled systems in the dual field theories, providing valuable insights into nonperturbative phenomena in quantum field theories. Researchers have investigated various aspects of AMPYM black holes, such as their stability, entropy, and critical behavior. Their thermodynamic properties have been of particular interest, as they exhibit rich phase structures and can undergo phase transitions similar to those observed in condensed matter systems. Furthermore, the holographic interpretation of these black holes has led to fruitful connections between gravitational physics and field theory, shedding light on fundamental questions in quantum gravity and strongly coupled systems [58-60]. This section provides a summary of the thermodynamics of N-dimensional Einstein-Maxwell-power-Yang-Mills gravity with a cosmological constant Λ. This theory of gravity is based on the following action, which we will explain in more details. So we will have, The trace element is represented by Tr ( . ) = Σ ( N -1)( N -2) / 2 a =1 ( . ), with R as the Ricci scalar and q as a real positive parameter. The Yang-Mills and Maxwell fields are defined accordingly [4], C ( a ) ( b )( c ) represents the structure constants of the ( N -1)( N -1) / 2 parameter Lie group G , while σ denotes the coupling constant. A ( a ) µ refers to the SO ( N -1) gauge group Yang-Mills potentials, with A µ representing the conventional Maxwell potential [39]. The metric solution corresponding to the N -dimensional spherically symmetric line element is as follows [40], /negationslash The d Ω 2 n denotes the volume of the unit n -sphere. In this study, we will direct our attention to the EinsteinMaxwell-power-Yang-Mills theory ( EMPYM ) with N (= n + 2) ≥ 4 and q = ( n + 1) / 4. The solution to the N -dimensional EMPYM black hole with a negative cosmological constant under the condition of q = n +1 4 is provided [39], /negationslash It is important to note that the parameter m represents the mass of the black hole, while C and Q 1 denote the charges of the Maxwell field and Yang-Mills field, respectively. In the extended phase space, the cosmological constant is considered as a thermodynamic pressure P = -Λ 8 π , in this case, the Hawking temperature, mass and entropy of the black hole are obtained as follows, where r + and ω n are the horizon radius and the volume of the unit n -sphere respectively.",
"pages": [
5,
6
]
},
{
"title": "4 Case II: AdS-Kerr-Sen black hole",
"content": "The Kerr-Sen-AdS black hole is a complex and fascinating topic in the field of theoretical physics. It is a type of rotating, charged black hole that emerges from heterotic string theory. It is a generalization of the Kerr-NewmanAdS black hole, which is a solution of the Einstein-Maxwell equations with a negative cosmological constant. The Kerr-Sen-AdS black hole also involves a dilaton and an axion field, which are scalar and pseudoscalar fields that appear in string theory [61]. The first exact solution of the Einstein field equations, known as the Schwarzschild solution, was discovered by Karl Schwarzschild in 1916. This solution describes a simple, nonrotating, uncharged black hole. The next major advancement came in 1963, when Roy P. Kerr found a solution to the Einstein field equations that describes a rotating black hole. This was a significant development, as it is believed that most black holes in the universe are rotating. In 1965, Ezra Newman and his collaborators found a solution that describes a rotating, charged black hole, known as the Kerr-Newman black hole. This solution was later extended by Ashoke Sen to include a dilaton field and an axion field, resulting in the Kerr-Sen black hole. The AdS form of this black hole was first derived by Ashoke Sen in 1992, by applying a series of duality transformations to the Kerr-Newman-AdS black hole. Sen showed that the Kerr-Sen-AdS black hole retains some of the symmetries and properties of the Kerr-Newman-AdS black hole, such as the existence of an event horizon, an ergosphere, and a Penrose process. However, the Kerr-Sen-AdS black hole also has some unique features, such as the dependence of the mass and angular momentum on the dilaton charge, and the violation of the cosmic censorship conjecture for some values of the parameters [61]. The Kerr-Sen-AdS black hole solution is a specific example of a rotating black hole with additional fields, which is of particular interest due to the role of angular momentum and the presence of nontrivial matter content in the spacetime geometry and is a solution to the equations of motion of supergravity theories with additional matter fields, such as the Sen-type dilaton and antisymmetric tensor fields, within the context of AdS space. The properties and thermodynamics of Kerr Sen-AdS black holes have been the subject of intense research, given their relevance to understanding the behavior of rotating black holes in the presence of nontrivial matter content and their implications for the AdS/CFT correspondence [62-72]. The study of Kerr Sen-AdS black holes has been motivated by theoretical developments in string theory, quantum gravity, and holography, as well as by their potential implications for gravity dualities and the behavior of strongly coupled systems in the dual field theories. This black hole has been studied from various perspectives, including its thermodynamic properties, stability, and connections to the dynamics of dual field theories. In recent years, there has been growing interest in exploring the dynamical behavior of Kerr Sen-AdS black holes, including their evolution, instability, and the connections to chaos and information loss puzzles. Researchers have investigated the behavior of these black holes under various perturbations and have sought to understand their implications for fundamental questions about black hole thermodynamics and the fate of information in quantum gravity. Moreover, the holographic interpretation of Kerr Sen-AdS black holes has led to fruitful connections between gravitational physics and field theory, shedding light on fundamental questions in quantum gravity and strongly coupled systems [62-72]. In summary, the history and ongoing research on Kerr Sen-AdS black holes represent a rich and interdisciplinary area of study that has fruitful connections to diverse fields of theoretical physics, including string theory, quantum gravity, holography, and nonlinear dynamics. The exploration of these black holes continues to be an exciting frontier for probing the fundamental nature of spacetime and its connections to quantum theory. In this section provides a short overview of the Kerr-Sen black hole and its generalization to the anti-de Sitter spacetimes. Sen [61] found a solution of the low-energy effective action of the heterotic string theory, which describes a charged rotating black hole, known as the Kerr-Sen black hole.The action is a modification of the general relativity action with additional fields from the heterotic string theory, given by [62,78], where ˜ g is the determinant of the metric tensor g µν , R is the Ricci scalar, F = F µν F µν with F µν being the U (1) Maxwell field strength tensor, Φ is a scalar dilaton field, and H = H µνρ H µνρ is the field strength for the axion field. The action can be transformed to the Einstein frame by a conformal transformation of the metric: The action in the Einstein frame is given by: The metric of a Kerr-Sen-AdS black hole in Boyer-Lindquist coordinates is given by [62,78]: where The parameter b is the dyonic charge of the black holes and is expressed as b = q 2 / (2 m ), where q is the electric charge and m is the mass of the black holes. In the limit of /lscript → ∞ , the metric reduces to the usual KerrSen black holes. The non-rotating case ( a = 0) reduces to the Gibbons-Maeda-Garfinkle-Horowitz-Strominger (GMGHS) solution. Gibbons and Maeda [73,74] found the black hole and black brane solutions in the dilaton theory, and Garfinkle-Horowitz-Strominger obtained their charged version [75]. It is worth mentioning that the Kerr-Sen-AdS black holes in four dimensions have been explored from various aspects, such as the black hole shadows [76] and the phase space thermodynamics in the extended phase space [77]. The ADM mass M, the angular momentum J, and the charge Q in AdS spacetimes are related by, Also, the entropy is given by, where r + is the radius of the event horizon, which is the largest root of ∆ = 0. When b=0, the thermodynamic quantities become the same as those of Kerr-AdS black holes. Kerr-AdS black holes are solutions of the Einstein field equations in four dimensions with negative cosmological constant and rotation.",
"pages": [
6,
7,
8,
9
]
},
{
"title": "5 Discussion and results",
"content": "In this section, we investigate the JTE process for AdS black holes: AMPYM and AdS Kerr Sen and derive a clear formula for the JTC. We also examine how the charge and the parameters of AMPYM and Kerr-Sen theories affect the inversion curves. We also compare the inversion curves for different scenarios",
"pages": [
9
]
},
{
"title": "5.1 Case I",
"content": "In this section, we explore the JTE of black hole systems in the extended phase space, where the black hole mass M is the same as the enthalpy H. We compare this process with the JT process of van der Waals gases with fixed particle number, and we use the fixed charge Q for the black hole systems. We also assume that the other parameters are constant. Using [2,45] as a reference, and considering the mass of black holes, we can write the pressure P as a function of M and r + , the mass and the horizon radius of a black hole. By substituting this expression in the temperature formula, we can obtain the temperature as a function of M and r + as well. Furthermore, we can express the mass M and the temperature T in terms of the pressure P and the radius r + of a black hole. Therefore, we get, Also, we have T in terms of M and r + as follows, By solving the above equations, one can obtain the function T(M, P), which is lengthy and will not be shown here. For particular free parameters for this model, the T(M, P) curve can be shown. According to the definition of the JTC µ = ( ∂T ∂P ) M , the inversion pressure and temperature between the cooling and heating regions are ( ˜ P -˜ T ), which are determined by µ = 0. Therefore, the most important thing is to find the function expression of µ . By setting µ = 0, one can obtain the inversion points ( ˜ P , ˜ T ) for different fixed enthalpy M. With respect to [24], the JTC is given by, This approach is elegant. However, in this paper, we will use more straightforward methods by applying only mass and temperature to derive the JTC µ . we can see that temperature is a function of pressure and radius, and radius is a function of pressure and mass. So, the JTC is given by, Now, using the relationship µ = ∂T ∂P | M = ∂T ∂r + ∂P ∂r + , we calculate Joule Thomson coefficient as follows, When the value of the coefficient µ is positive during the expansion, it means that the temperature decreases and therefore it is called a cooling phenomenon. However, when µ is negative, the temperature increases, and this is called a heating process.Using various equations, we can write the mass M and the temperature T as functions of the pressure P and the radius r + , which are the properties of a black hole. For µ = 0, we can obtain the inversion temperature, in which the process of the temperature changes reverses. It can be obtained by the formula, At the inversion temperature, the value of µ is 0, and the inversion temperature is determined by the following equation: This is beneficial for identifying the areas of heating and cooling in the T -P plane. We calculate t using the equations (17), (20), (21), (22) and (23), We can also have from relation (17), Figure 1 displays the Hawking temperature as a function of the horizon. We keep the free parameters constant for each plot. In each subfigure, we can observe some zero points for different free parameters. These zero points correspond to the divergence points of the JTC, as we can easily see in figure 1. According to the radius of the horizon and the values of free parameters of a black hole, for small values for the radius of the event horizon, our structural behavior is completely distinct, and for larger radii, the figures almost converge. We have plotted the isenthalpic curves and the inversion curve of the AMPYM black hole for various values of the free parameter in each plot of Figure 2. In every subfigure, two isenthalpic curves with different values of M, along with the corresponding inversion curve that occurs at the highest point of the isenthalpic curves. We denote the inversion temperature and pressure of each isenthalpic curve as T i and P i , respectively. The isenthalpic curves are divided into two regions by the inversion curve: for P < P i , the isenthalpic curve has a positive slope, indicating that the black hole undergoes cooling during the expansion process. However, for P > P i , the isenthalpic curve has a negative slope, implying that the black hole experiences heating during the expansion process. This behavior is consistent for different values of the free parameter in Figures (2a-2b). We do not present the explicit expressions of the solutions here, as they are too long and complex. However, we can illustrate the relation between the inversion temperature T i and the inversion pressure P i by using the equations given above. Figure 3 shows the inversion curves for different free parameters. The inversion curve has only one branch. The inversion temperature rises steadily with the inversion pressure, but the slope of the inversion curves becomes smaller. We can also notice some fine structures in the cases of subfigures 3. For low pressure, the inversion temperature varies with the free parameters that are specified in each plot. It was interesting for us to study the effect of parameters and we observed that the slope of the inversion curve increases with the changes in the various values of each of the parameters. Before the end of this section, if we want to talk purely about the effect of Maxwell charges as a specific result of our work, we should say that in [3], the authors found that for q > 1 and n = 2 , the ratio T min i /T c is fully established, but in the present work, by adding the Maxwell charge to the mentioned black hole, we found that for C /greatermuch 1 the Yang-Mills parameter has no effect on the ratio, and in any case, the T min i /T c is equal to 1 / 2 , which has the same behavior as a four-dimensional charged black hole. For q > 1 , when C increases, the T min /T c approaches the value of 1 / 2 . However, for q < 1 , the conditions are slightly different from the previous situation, because when q = 1 / 2 , n = 2 , and Q 1 = 1 , the T min i /T c is equal to 1 / 2 for all values of 'C', which is the same as a charged black hole in four dimensions. Also, according to Table 1 to 3, it can be inferred that when 1 / 2 < q < 1 , with the increase of Maxwell charge, i.e. C , the T min i /T c only approaches the value of 1 / 2 and exhibits a more similar behavior to a four-dimensional charged black hole. If for 0 < q < 1 / 2 , the T min i /T c becomes further or smaller than the value of 1 / 2 , which is contrary to a charged black hole in four dimensions. In general, when q > 1 , for the increase of Maxwell's charge C , the T min i /T c is almost equal to 1 / 2 and when q = 1 / 2 , for all values of the Maxwell's charge, the T min i /T c is equal to 1 / 2 . Also, when 1 < q < 1 / 2 , it is close to the value of 1 / 2 and finally when 1 / 2 < q < 0 , the values of the T min i /T c become smaller than of 1 / 2 .",
"pages": [
9,
10,
11,
12,
13
]
},
{
"title": "5.2 Case II",
"content": "With respect to [62, 78] and the Mass of black holes, We can express pressure P in terms of M and r + and replace it in the formula for the temperature, which will also become in terms of M and r + . We also rewrite the mass M and temperature T as a function of the pressure P and the radius r + of a black hole. So we will have, Also, the temperature in terms of M and r + is as follows [62, 78], Therefore, by using the equation (19) on can obtain, We express the mass M and temperature T in terms of the pressure P and the radius r + of a black hole, using different equations, The V ( P ) for the AKS black hole is calculates as follows, So, by using the equation (19), (21), (22) and (23) one can obtain, Also, with respect to equation (30) we will have, We have drawn the isenthalpic curves and the inversion curve of the AdS Kerr Sen black hole for various free parameter values in each plot of Figure 4. In every subfigure, three isenthalpic curves with different M are visible, along with the corresponding inversion curve that occurs at the highest point of the isenthalpic curves. We denote each isenthalpic curve's inversion temperature and pressure as T i and P i , respectively. The isenthalpic curves are divided into two regions by the inversion curve: for P < P i , the isenthalpic curve has a positive slope, indicating that the black hole undergoes cooling during the expansion process. With the changes in the parameters of the black hole, we found that the slope of the figures will change, and these changes are visible in each subfigure. Because an analytical solution is difficult to obtain, we used a numerical method to draw graphs. If we assume the parameter b to be zero, our equations and graphs will simplify to the Kerr-AdS black holes, whose results are thoroughly discussed in [1]. Due to the complexity of the calculations, we do not show the explicit expressions here. However, we can demonstrate the relation between the inversion temperature T i and the inversion pressure P i by using the equations given above. Figure (5a-5d) displays the inversion curves for different free parameters of the AKS black holes. The inversion curve has only one branch. The inversion temperature increases gradually with the inversion pressure, but the slope of the inversion curves becomes smaller. We can also observe some fine structures in the cases of subfigures 5. Inversion curves change with the free parameters that are specified in each plot. It was very difficult to find the analytical solution for drawing the graphs, so we used numerical solutions to draw them. Moreover, we note that due to the huge difference in the values of the horizontal and vertical graphs corresponding to the change of the free parameters, we have drawn the figures separately. In Figure 6, we plot the Hawking temperature as a function of the horizon, which is the boundary of the black hole. The horizon can be affected by the presence of other fields or dimensions, which are represented by the free parameters in our model. We consider the free parameters constant for each plot, but we vary them across different plots to see how they influence the Hawking temperature. In each subfigure, we can observe some zero points, where the Hawking temperature becomes zero for different free parameters. This means that the black hole became extremal at these points. These zero points correspond to the divergence points of the JTC, which is a quantity that describes the temperature change when a gas expands or compresses at constant enthalpy. In the study of Kerr black hole [1], it was shown that the rotation parameter 'a' does not have much effect on the Joule-Thomson representation. For example, it is practically eliminated in the ratio of the T min i /T C , and the value of this ratio is always a constant value of 1/2. In this paper, we found that the parameters related to AKS black hole, i.e. a and b, can play a vital role in the representation of the value of the T min i /T C . The value of this ratio increases with the reduction of these parameters. For this purpose, due to the structural complexities of this black hole, we could not use the analytical method to obtain the critical points of the black hole, so we resorted to the approximate method and the numerical calculations. In the [77], the value of critical temperature and critical pressure has been calculated. We calculated some values for this black hole according to Table 4. As evident, regarding various values of free parameters, the T min i /T C is obtained, which shows the obvious effect of parameters a and b.",
"pages": [
14,
15,
16,
17,
18
]
},
{
"title": "6 Conclusion",
"content": "In this article, we investigated the behavior of two structurally different black holes under the JTE thermodynamic process. The JTE is a process that cools or heats a system by changing its pressure and volume at constant enthalpy, and its main goal is to study the behavior of the µ coefficient, which describes the change in temperature during the expansion or compression of a system. But before presenting the results, we must first explain the motivations that led us to choose these particular black holes. Recently, Einstein-Power-YoungMills black hole with AdS structure under JTE process was investigated in a research paper [2,3]. This raised the question in our mind that if a fields in the Maxwellian form are added to the elements in the action of the above article, can this change and addition of a special form of the field cause a different thermodynamic behavior in their JT representation? Similarly, in the past, the behavior of various forms of rotating black holes was also investigated in this thermodynamic process. But what effect can the addition of Sen-type dilaton fields and antisymmetric tensor fields have on its thermodynamic properties and JT representation? This was a question that could be a good motivation for our investigation. For this purpose, we plotted the isenthalpic curves and the inversion curves for each of the two black holes, for different values of the free parameters, which can be referred to the following results for each. For the AMPYM black hole, the Hawking temperature has zero points that depend on the free parameters and the horizon radius, and the JTC diverges at these points. The isenthalpic curves are divided into two regions by the inversion curve: for P < P i , the isenthalpic curve has a positive slope, indicating that the black hole is cooling during the expansion process, and for P > P i , the isenthalpic curve has a negative slope, which means that the black hole experiences heating during the expansion process. The inversion curves for different free parameters show that the inversion curve has only one branch and the inversion temperature increases steadily with the inversion pressure, but the slope of the inversion curves becomes smaller. Also, the graphs showed that for low pressure, the inversion temperature varies with the free parameters specified in each graph. Studying the effect of parameters was interesting for us because we observed that the slope of the inversion curve increases with changes in different values of each parameter. But if we want to talk about the effect of Maxwell's charges as a special result of our work, we must state that, we have found that the ratio of minimum to critical temperature for a 4-dimensional black hole with Yang-Mills hair depends on the values of q , M , and C . For q > 1 , the ratio is almost equal to 1 / 2 regardless of the Yang-Mills parameter when ' C /greatermuch 1' . For q = 1 / 2 , the ratio is exactly equal to 1 / 2 for all values of C . For 1 / 2 < q < 1 , the ratio approaches 1 / 2 as C increases. For 0 < q < 1 / 2 , the ratio deviates from 1 / 2 as C increases. These results show that the Maxwell charge can affect the thermodynamic behavior of the black hole and make it more or less similar to a 4-dimensional charged black hole. Also, for the AKS black hole, we found that its parameters, i.e. a and b, can play a vital role in the representation of the value of the ratio of T min i /T C , so that by reducing these parameters, the value of the ratio increases. The results are shown in Figure 7 and are summarized in Tables 1 to 5. For the AdS Kerr Sen black, It was very difficult to find the analytical solution for drawing the graphs, so we used numerical solutions to draw them. For P < P i , the isenthalpic curve has a positive slope, indicating that the black hole is cooling during the expansion process, but with the increases of the black hole parameters, we found that the slope of the figures will change. Also, if we assume the parameter b ( dyonic charge) to be zero, our equations and graphs will simplify to the Kerr-AdS black holes, whose results are thoroughly discussed in [1]. In general, it can be said about this black hole the isenthalpic curves of the AKS black hole show cooling or heating behavior depending on the inversion curve, which is affected by the mass and the parameter b, a of the black hole. The inversion curve has a single branch with a positive slope that decreases with the free parameters, and the inversion temperature and pressure are related by some equations that are numerically solved. The Hawking temperature of the AKS black hole has zero points that depend on the free parameters and the horizon radius and lead the JTC to reflect divergence behavior.",
"pages": [
19,
20
]
}
]
2024arXiv240203351M
https://arxiv.org/pdf/2402.03351.pdf
<document>
<section_header_level_1><location><page_1><loc_14><loc_75><loc_81><loc_80></location>Single and entangled atomic systems in thermal bath and the Fulling-Davies-Unruh effect</section_header_level_1>
<text><location><page_1><loc_14><loc_66><loc_73><loc_68></location>Arnab Mukherjee, a, 1 Sunandan Gangopadhyay, b Archan. S. Majumdar c</text>
<text><location><page_1><loc_15><loc_63><loc_86><loc_65></location>Department of Astrophysics and High Energy Physics, S.N. Bose National Centre for Basic Sciences, JD Block, Sector-III, Salt Lake, Kolkata 700106, India</text>
<text><location><page_1><loc_15><loc_61><loc_79><loc_62></location>E-mail: arnab.mukherjee@bose.res.in , sunandan.gangopadhyay@gmail.com ,</text>
<text><location><page_1><loc_15><loc_59><loc_32><loc_60></location>archan@bose.res.in</text>
<text><location><page_1><loc_14><loc_29><loc_86><loc_57></location>Abstract: In this study, we revisit the Fulling-Davies-Unruh effect in the context of twolevel single and entangled atomic systems that are static in a thermal bath. We consider the interaction between the systems and a massless scalar field, covering the scenarios of free space as well as within a cavity. Through the calculation of atomic transition rates and comparing with the results of [ Phys. Rev. D 108 (2023) 085018 ], it is found that in free space there is an equivalence between the upward and downward transition rates of an uniformly accelerated atom with respect to an observer with that of a single atom which is static with respect to the observer and immersed in a thermal bath, as long as the temperature of the thermal bath matches the Unruh temperature. This equivalence between the upward and downward transition rates breaks down in the presence of a cavity. For two-atom systems, considering the initial state to be in a general pure entangled form, we find that in this case the equivalence between the upward and downward transition rates of the accelerated and static thermal bath scenarios holds only under specific limiting conditions in free space, but breaks down completely in a cavity setup. Though the ratio of the upward and downward transition rates in the thermal bath matches exactly with those of the accelerated systems in free space as well as inside the cavity.</text>
<section_header_level_1><location><page_2><loc_14><loc_85><loc_23><loc_87></location>Contents</section_header_level_1>
<table>
<location><page_2><loc_14><loc_43><loc_86><loc_83></location>
</table>
<section_header_level_1><location><page_2><loc_14><loc_37><loc_30><loc_38></location>1 Introduction</section_header_level_1>
<text><location><page_2><loc_14><loc_18><loc_86><loc_35></location>The physics of interaction of two-level atomic systems with quantum fields is endowed with rich and diverse properties [1-12]. Investigations into the radiative properties of a uniformly accelerated single atom [6-11] have been extended to scenarios involving multiple atoms interacting with the massless scalar field and the electromagnetic field [13-22]. A two-level atom serves as a detector, and coupled to an external field, can give rise to the phenomenon of the Fulling-Davies-Unruh (FDU) effect [23-26]. Emanating from the study of quantum field theory in curved background, the FDU effect reveals that a uniformly accelerated atom feels a thermal bath in the Minkowski vacuum, with the temperature T being related to the proper acceleration α as T = α/ 2 π . The FDU effect has a deep connection with black hole thermodynamics and the information loss paradox [25, 27-29].</text>
<text><location><page_2><loc_14><loc_14><loc_86><loc_17></location>It has been understood that the upward and downward transition rates of the accelerated detector, which in this case is the atom, is exactly the same as seen by a local inertial</text>
<text><location><page_3><loc_14><loc_67><loc_86><loc_90></location>observer and by an observer who is coaccelerating with the detector. This theoretical equivalence between the transition rates has been found to hold for a single atom in free space as well as in the setting of a single reflecting boundary [10]. A similar equivalence also holds at the level of transition rates for an entangled atom in free space [30] provided a thermal bath at the FDU temperature exists in the coaccelerated frame. Moreover, it has been recently shown [31] that the equivalence between the upward and downward transition rates in the context of an uniformly accelerating Unruh-DeWitt detector and a coaccelerating detector immersed in a thermal bath holds completely in free space as well as inside a cavity, for both the single and entangled atomic systems. However, it has also been found [30] that for an entangled atom in free space this theoretical equivalence between the transition rates breaks down completely when the coaccelerated frame scenario is replaced with a static atom immersed in a background thermal bath with temperature equal to the Unruh temperature.</text>
<text><location><page_3><loc_14><loc_51><loc_86><loc_67></location>The primary motivation of the present work is to establish further the status of equivalence stemming from the FDU effect. Though the equivalence dictated by the FDU effect is universally valid at the conceptual level, its revelation in real physical or experimental scenarios is contingent on the choice of set-ups or contexts. Context plays a very important role in manifestation of quantum features. Quantum contextuality is a well-studied subject with foundational implications (see, [32] for a recent review) as well as diverse applications [33], and entanglement is known to elucidate certain subtle features of contextuality. In this work we consider entangled atomic systems as a detector model to study whether the equivalence is manifested for such set-ups involving entanglement.</text>
<text><location><page_3><loc_14><loc_37><loc_86><loc_51></location>Moreover, currently there exists a wide upsurge of interest in quantum entanglement in relativistic settings, intertwining profound concepts from quantum field theory, information theory and gravitational physics [34-51]. Initially uncorrelated Unruh-DeWitt detectors may get entangled by interacting locally with the vacuum state of a quantum field [52]. Localized detectors can extract non-local correlations from the a quantum field through the process of entanglement harvesting [53-59]. Relativistic quantum information explores how entanglement is affected by not only Lorentz boosts [60], but by non-inertial effects as well [61, 62].</text>
<text><location><page_3><loc_14><loc_19><loc_86><loc_37></location>In practical scenarios, the degradation of entanglement due to uncontrolled coupling with external fields is a genuine concern, and extensive research has been undertaken to investigate the transition rates between the states of entangled atoms moving in various trajectories, yielding a rich paradigm of possibilities [63-71]. Since configurations in analogue cavity QED such as superconducting circuits [72] and laser-driven technologies [73-75] can achieve substantial accelerations, such systems are beginning to be used towards experimental evidence of theoretical results in relativistic quantum information. Reflecting boundaries have been shown to play a significant role in such studies on relativistic quantum phenomena in superconducting circuits [72, 76], as well as in secure quantum communication over long distances [77-80].</text>
<text><location><page_3><loc_14><loc_14><loc_86><loc_19></location>Due to the apparently contradictory nature regarding the status of equivalence between the transition rates as manifested from certain findings of [10, 30, 31], it becomes pertinent to explore further the implications of replacing the coaccelerated frame scenario with a</text>
<text><location><page_4><loc_14><loc_74><loc_86><loc_90></location>static frame immersed in a background thermal bath. The aim of our present study is a comprehensive investigation of the FDU phenomenon by computing the upward and downward transition rates in the context of a static Unruh-DeWitt detector immersed in a background thermal bath and comparing with those of an accelerating Unruh-DeWitt detector given in [31]. Our study pertains to single two-level atoms as well as an entangled two-atom systems either in free space, and confined in a cavity. For the case of two-atom systems, one of our main focus is on the role played by quantum entanglement on the transition rates in the presence of boundaries, which in turn, have a direct bearing on revelation of the FDU effect, as shown through our subsequent analysis.</text>
<text><location><page_4><loc_14><loc_56><loc_86><loc_74></location>From a fundamental perspective, the impact of cavity setup on atom-field interactions and radiative processes of entangled atoms are manifold [81-89]. It was observed [17, 21, 90] that reflecting boundaries strongly influence the resonance interaction energy of uniformly accelerated two-atom system. In [91], it was found that reflecting boundaries induce effects which lead to the violation of equivalence in an accelerating atom-mirror system in the generalized uncertainty principle framework. Reflecting boundaries have several other interesting consequences in the context of quantum entanglement [84-86, 88], and quantum thermodynamics [92]. This line of inquiry presents a possibility for advancing our understanding about the FDU effect within a cavity quantum electrodynamics (QED) framework, that is important both from fundamental and practical points of view.</text>
<text><location><page_4><loc_14><loc_44><loc_86><loc_56></location>The paper is organised as follows. In section 2, we analyze the transition rates for the cases when a single and two entangled static atomic systems interact with a massless scalar field in a background thermal bath in empty space and in the presence of a cavity, respectively. In section 3, we calculate a quantitative estimation of the upward transition rate of a single and two atom system inside a cavity by considering the Rubidium and Caesium atom. We conclude with a summary of our results in section 4. Throughout the paper, we take /planckover2pi1 = c = k B = 1 , where k B is the Boltzmann constant.</text>
<section_header_level_1><location><page_4><loc_14><loc_41><loc_76><loc_42></location>2 Interaction of the static atomic system with a thermal bath</section_header_level_1>
<text><location><page_4><loc_14><loc_34><loc_86><loc_39></location>In this section, we start by investigating the case when a static atomic system interacts with a massless scalar field in a thermal state at an arbitrary temperature T and undergoes transitions in between its lower and higher energy states.</text>
<section_header_level_1><location><page_4><loc_14><loc_31><loc_37><loc_33></location>2.1 Single atom system</section_header_level_1>
<text><location><page_4><loc_14><loc_22><loc_86><loc_30></location>In this analysis, we also consider a single atom (an Unruh-DeWitt detector) with two energy levels | g 〉 and | e 〉 with corresponding energy values -ω 0 / 2 and + ω 0 / 2 , remains static in a thermal state of a massless scalar field. Following the procedure given in the subsection (A.1), the transition probability from the initial state | i 〉 to the final state | f 〉 can be written as</text>
<formula><location><page_4><loc_39><loc_19><loc_86><loc_22></location>P β | i 〉→| f 〉 = λ 2 | m fi | 2 F β (∆ E ) (2.1)</formula>
<text><location><page_4><loc_14><loc_17><loc_85><loc_19></location>where ∆ E = E f -E i , m fi = 〈 f | m (0) | i 〉 and the response function F β (∆ E ) is defined as</text>
<formula><location><page_4><loc_27><loc_13><loc_86><loc_17></location>F β (∆ E ) = ∫ + ∞ -∞ dτ ∫ + ∞ -∞ dτ ' e -i ∆ E ( τ -τ ' ) G + β ( x ( τ ) , x ( τ ' )) (2.2)</formula>
<text><location><page_5><loc_14><loc_88><loc_19><loc_90></location>where</text>
<formula><location><page_5><loc_33><loc_85><loc_86><loc_89></location>G + β ( x ( τ ) , x ( τ ' )) = tr [ e -β H F φ ( x ( τ )) φ ( x ( τ ' )] tr [ e -β H F ] (2.3)</formula>
<text><location><page_5><loc_14><loc_79><loc_86><loc_84></location>is the positive frequency Wightman function of the massless scalar field in a thermal state at an arbitrary temperature T with H F = ∑ k ω k a † k a k [26].</text>
<text><location><page_5><loc_14><loc_77><loc_86><loc_80></location>Exploiting the time translational invariance property of the positive frequency Wightman function, the response function per unit proper time can be written as</text>
<formula><location><page_5><loc_30><loc_72><loc_86><loc_76></location>F β (∆ E ) = ∫ + ∞ -∞ d (∆ τ ) e -i ∆ E ∆ τ G + β ( x ( τ ) , x ( τ ' )) (2.4)</formula>
<text><location><page_5><loc_14><loc_67><loc_86><loc_71></location>where ∆ τ = τ -τ ' . Therefore, the transition probability per unit proper time from the initial state | i 〉 to the final state | f 〉 turns out to be</text>
<formula><location><page_5><loc_38><loc_64><loc_86><loc_66></location>R β | i 〉→| f 〉 = λ 2 | m fi | 2 F β (∆ E ) . (2.5)</formula>
<text><location><page_5><loc_14><loc_58><loc_86><loc_63></location>In the next subsections, we employ the above equations to study the transitions of a static single atom interacting with a thermal state of a massless scalar field in both empty space and a cavity.</text>
<section_header_level_1><location><page_5><loc_14><loc_55><loc_71><loc_56></location>2.1.1 Transition rates for single atom system in empty space</section_header_level_1>
<text><location><page_5><loc_14><loc_49><loc_86><loc_54></location>We initially take into account the transition rates of a single atom that is interacting with a massless thermal scalar field in the empty space. In the laboratory frame, atomic trajectory is given by</text>
<formula><location><page_5><loc_41><loc_47><loc_86><loc_49></location>t = τ, x = y = z = 0 (2.6)</formula>
<text><location><page_5><loc_14><loc_45><loc_50><loc_46></location>where τ denote the proper time of the atom.</text>
<text><location><page_5><loc_14><loc_41><loc_86><loc_44></location>The thermal Wightman function is given by [30] (see Appendix B for a detailed calculation)</text>
<formula><location><page_5><loc_14><loc_32><loc_89><loc_40></location>G + β ( x ( τ ) , x ( τ ' )) = -1 4 π 2 ∞ ∑ n = -∞ 1 ( t ( τ ) -t ( τ ' ) -inβ -iε ) 2 -( x ( τ ) -x ( τ ' )) 2 -( y ( τ ) -y ( τ ' )) 2 -( z ( τ ) -z ( τ ' )) 2 . (2.7)</formula>
<text><location><page_5><loc_14><loc_29><loc_77><loc_30></location>Substituting (2.6) in (2.7), the thermal Wightman function turns out to be [26]</text>
<formula><location><page_5><loc_30><loc_23><loc_86><loc_28></location>G + β ( x ( τ ) , x ( τ ' )) = -1 4 π 2 ∞ ∑ n = -∞ 1 (∆ τ -inβ -iε ) 2 . (2.8)</formula>
<text><location><page_5><loc_14><loc_18><loc_86><loc_22></location>Substituting the Wightman function into eq.(2.4) and eq.(2.5), the transition rate from the initial state | i 〉 to the final state | f 〉 becomes</text>
<formula><location><page_5><loc_23><loc_13><loc_86><loc_18></location>R β | i 〉→| f 〉 = -λ 2 | m fi | 2 4 π 2 ∞ ∑ n = -∞ ∫ + ∞ -∞ d (∆ τ ) e -i ∆ E ∆ τ 1 (∆ τ -inβ -iε ) 2 . (2.9)</formula>
<text><location><page_6><loc_14><loc_87><loc_86><loc_90></location>Simplifying the transition rates, eq.(2.9), by performing the contour integration [93] as shown in Appendix A of [31], we obtain</text>
<formula><location><page_6><loc_14><loc_81><loc_87><loc_85></location>R β | i 〉→| f 〉 = λ 2 | m fi | 2 | ∆ E | 2 π [ θ ( -∆ E ) ( 1 + 1 exp( | ∆ E | /T ) -1 ) + θ (∆ E ) ( 1 exp(∆ E/T ) -1 )] (2.10)</formula>
<text><location><page_6><loc_14><loc_79><loc_57><loc_80></location>where θ (∆ E ) is the Heaviside step function defined as</text>
<formula><location><page_6><loc_40><loc_73><loc_86><loc_77></location>θ (∆ E ) = { 1 , ∆ E > 0 0 , ∆ E < 0 . (2.11)</formula>
<text><location><page_6><loc_14><loc_58><loc_86><loc_72></location>Eq. (2.10) reveals that the two transition processes, namely, the upward and downward transition can take place even when the atom is static but placed inside a thermal bath. From the above equation, it may also be noted that the upward transition or excitation process is solely depends on the temperature of the thermal bath. Considering the initial state | i 〉 = | g 〉 , final state | f 〉 = | e 〉 and vice-versa and using the definition m eg = 〈 e | m (0) | g 〉 , we obtain | m ge | 2 = | m eg | 2 = 1 , and ∆ E = ω 0 for the transition g → e and ∆ E = -ω 0 for the transition e → g . Using the above results the upward and downward transition rate takes the form</text>
<formula><location><page_6><loc_34><loc_50><loc_86><loc_54></location>R β | e 〉→| g 〉 = λ 2 ω 0 2 π ( 1 + 1 exp( ω 0 /T ) -1 ) . (2.13)</formula>
<formula><location><page_6><loc_36><loc_54><loc_86><loc_58></location>R β | g 〉→| e 〉 = λ 2 ω 0 2 π ( 1 exp( ω 0 /T ) -1 ) (2.12)</formula>
<text><location><page_6><loc_14><loc_49><loc_53><loc_50></location>Taking the ratio of the above two results, we get</text>
<formula><location><page_6><loc_36><loc_43><loc_86><loc_48></location>R β | g 〉→| e 〉 R β | e 〉→| g 〉 ≡ R β up R β down = exp( -ω 0 /T ) . (2.14)</formula>
<text><location><page_6><loc_14><loc_23><loc_86><loc_42></location>Eq. (2.14) is the consequence of a universal relation called the principle of detailed balance [94]. From the above expressions two points can be noted. First, it is seen that the upward transition rate entirely depends on the temperature of the thermal state of the massless scalar field. At T = 0 , the upward transition rate vanishes. Secondly, if we take the thermal bath temperature in the static frame at T = α/ 2 π , then eqs.(2.12), (2.13), (A.6) and (A.7) clearly show that the upward and the downward transition rates of an uniformly accelerated atom seen by an instantaneously inertial observer (see Appendix A) and by a static observer in a thermal bath are identical. Further, from eq.(2.14) and eq.(A.8), it can be seen that the ratio in both cases also matches in the limit T = α/ 2 π . Therefore, for a single atom system in empty space the equivalence between the effect of uniform acceleration and the effect of thermal bath holds at the level of transition rates as well as their ratios.</text>
<section_header_level_1><location><page_6><loc_14><loc_20><loc_67><loc_21></location>2.1.2 Transition rates for single atom system in a cavity</section_header_level_1>
<text><location><page_6><loc_14><loc_13><loc_86><loc_19></location>We now consider that the a static atom is interacting with a massless thermal scalar field inside a cavity of length L as shown in Figure 1. Assuming the scalar field obeys the Dirichlet boundary condition φ | z =0 = φ | z = L = 0 , the Wightman function of the thermal</text>
<text><location><page_7><loc_14><loc_87><loc_86><loc_90></location>scalar field confined in the cavity of length L takes the form [95] (see Appendix C for a detailed calculation)</text>
<formula><location><page_7><loc_14><loc_70><loc_86><loc_85></location>G + β ( x ( τ ) , x ( τ ' )) = -1 4 π 2 ∞ ∑ m = -∞ ∞ ∑ n = -∞ [ 1 ( t ( τ ) -( τ ' ) -imβ -iε ) 2 -| ∆ x ⊥ | 2 -( z ( τ ) -z ( τ ' ) -2 nL ) 2 -1 ( t ( τ ) -t ( τ ' ) -imβ -iε ) 2 -| ∆ x ⊥ | 2 -( z ( τ ) + z ( τ ' ) -2 nL ) 2 ] (2.15) with | ∆ x ⊥ | 2 = √ ( x ( τ ) -x ( τ ' )) 2 +( y ( τ ) -y ( τ ' )) 2 .</formula>
<figure>
<location><page_7><loc_26><loc_58><loc_74><loc_69></location>
<caption>Figure 1 : Static atom confined in a cavity with a thermal bath at a temperature T .</caption>
</figure>
<text><location><page_7><loc_14><loc_51><loc_54><loc_52></location>Inside the cavity the atomic trajectory is given by</text>
<formula><location><page_7><loc_39><loc_48><loc_86><loc_49></location>t ( τ ) = τ, x = y = 0 , z = z 0 (2.16)</formula>
<text><location><page_7><loc_14><loc_45><loc_73><loc_47></location>Using the above trajectories in eq.(2.15), the Wightman function becomes</text>
<formula><location><page_7><loc_15><loc_39><loc_86><loc_44></location>G + β ( x ( τ ) , x ( τ ' )) = -1 4 π 2 ∞ ∑ m = -∞ ∞ ∑ n = -∞ [ 1 (∆ τ -imβ -iε ) 2 -d 2 1 -1 (∆ τ -imβ -iε ) 2 -d 2 2 ] (2.17)</formula>
<text><location><page_7><loc_14><loc_34><loc_86><loc_38></location>with d 1 = 2 nL, d 2 = 2 z 0 -2 nL . Using the Wightman function, eq. (2.17) into eq.(2.4), the transition rate from the initial state | i 〉 to the final state | f 〉 is given by</text>
<formula><location><page_7><loc_18><loc_25><loc_86><loc_34></location>R β | i 〉→| f 〉 = -λ 2 | m fi | 2 4 π 2 ∞ ∑ m = -∞ ∞ ∑ n = -∞ [∫ + ∞ -∞ d (∆ τ ) e -i ∆ E ∆ τ 1 (∆ τ -imβ -iε ) 2 -d 2 1 -∫ + ∞ -∞ d (∆ τ ) e -i ∆ E ∆ τ 1 (∆ τ -imβ -iε ) 2 -d 2 2 ] . (2.18)</formula>
<text><location><page_7><loc_14><loc_21><loc_86><loc_25></location>Simplifying the above equation by following the method of contour integral, rate of transition from the initial state | i 〉 to the final state | f 〉 can be written as</text>
<formula><location><page_7><loc_15><loc_13><loc_92><loc_21></location>R β | i 〉→| f 〉 = λ 2 | m fi | 2 [ θ ( -∆ E ) { | ∆ E | 2 π + q ( | ∆ E | , 2 L ) -r ( | ∆ E | , 2 z 0 , 2 L ) }( 1 + 1 exp(∆ E/T ) -1 ) + θ (∆ E ) { ∆ E 2 π + q (∆ E, 2 L ) -r (∆ E, 2 z 0 , 2 L ) }( 1 exp(∆ E/T ) -1 )] (2.19)</formula>
<text><location><page_8><loc_14><loc_88><loc_32><loc_90></location>where we have defined</text>
<formula><location><page_8><loc_36><loc_82><loc_86><loc_87></location>q (∆ E, 2 L ) = 2 ∞ ∑ n =1 p (∆ E, 2 nL ) (2.20)</formula>
<formula><location><page_8><loc_32><loc_78><loc_86><loc_83></location>r (∆ E, 2 z 0 , 2 L ) = ∞ ∑ n = -∞ p (∆ E, 2( z 0 -nL )) (2.21)</formula>
<text><location><page_8><loc_14><loc_76><loc_36><loc_77></location>with p (∆ E, z 0 ) is given by</text>
<formula><location><page_8><loc_40><loc_73><loc_86><loc_76></location>p (∆ E, z 0 ) = sin(∆ Ez 0 ) 2 πz 0 . (2.22)</formula>
<text><location><page_8><loc_14><loc_70><loc_86><loc_72></location>Hence, from the above result the upward and downward transition rates can be written as</text>
<formula><location><page_8><loc_20><loc_65><loc_86><loc_69></location>R β | g 〉→| e 〉 = λ 2 [ { ω 0 2 π + q ( ω 0 , 2 L ) -r ( ω 0 , 2 z 0 , 2 L ) } ( 1 exp( ω 0 /T ) -1 )] (2.23)</formula>
<formula><location><page_8><loc_17><loc_60><loc_86><loc_64></location>R β | e 〉→| g 〉 = λ 2 [ { ω 0 2 π + q ( ω 0 , 2 L ) -r ( ω 0 , 2 z 0 , 2 L ) } ( 1 + 1 exp( ω 0 /T ) -1 )] . (2.24)</formula>
<text><location><page_8><loc_14><loc_34><loc_86><loc_60></location>From the above analysis it follows that the transitions observed by an instantaneously inertial observer and a static observer in a thermal bath for both the upward and the downward transition rates when the atom is confined in a cavity are clearly distinct. We also observe that taking the thermal bath temperature in the static frame T = α/ 2 π , eqs. (2.23), (2.24), (A.10) and (A.11) indicate that there is a non-equivalence between the transition rates of a uniformly accelerated atom seen by an instantaneously inertial observer and a static atom seen by a static observer in a thermal bath inside the cavity. Though the upward and the downward transition rates of a static atom in a thermal bath are not same as those corresponding to a uniformly accelerated atom, it can be seen from eqs. (2.23), (2.24), (A.10) and (A.11), that at T = α/ 2 π , the ratio of the upward and the downward transition rates of a static atom in a thermal bath is identical with that of a uniformly accelerated atom (eq.(A.15)). From the above analysis, it is also observed that the ratio of eqs. (2.23), (2.24) is identical with the free space result (eq.(2.14)). This is an universal feature independent of the detector model and follows from the principle of detailed balance [94].</text>
<text><location><page_8><loc_14><loc_23><loc_86><loc_33></location>In order to describe the single boundary and free space scenarios, we now derive the limiting cases of these expressions. Taking the limit L → ∞ , we find that in eq.(s)(2.23, 2.24) only n = 0 term survives from the infinite summation and one can effectively reduce the cavity scenario to a situation where only one reflecting boundary exists. Hence, using this limit, the upward and downward transition rates in the presence of a single reflecting boundary turn out to be</text>
<formula><location><page_8><loc_28><loc_17><loc_86><loc_22></location>R β | g 〉→| e 〉 = λ 2 [ { ω 0 2 π -p ( ω 0 , 2 z 0 ) } ( 1 exp( ω 0 /T ) -1 )] (2.25)</formula>
<formula><location><page_8><loc_26><loc_13><loc_86><loc_17></location>R β | e 〉→| g 〉 = λ 2 [ { ω 0 2 π -p ( ω 0 , 2 z 0 ) } ( 1 + 1 exp( ω 0 /T ) -1 )] . (2.26)</formula>
<text><location><page_9><loc_14><loc_85><loc_86><loc_90></location>On the other hand, taking the limits L →∞ and z 0 →∞ together, eq.(s)(2.23, 2.24) lead to the expression for the upward and downward transition rates in the free space given by eq.(s)(2.12, 2.13).</text>
<text><location><page_9><loc_14><loc_65><loc_86><loc_84></location>We study the variation of the transition rate of a single two-level atom confined to a cavity, where the parameters are the atom's distance from the boundary ( z 0 ), the cavity's length ( L ), and the temperature of the thermal field ( T ). The atom's ground state energy level is | g 〉 , and its excited state energy level is | e 〉 . The findings are plotted below, where all physical quantities are expressed in dimensionless units. To fix the dimensionless parameters, we consider a single 87 Rb atom and take the atomic data from Ref. [96]. Recent time, it is observed that experimentally atomic excitations in nanoscale waveguides [97] can be achievable through some novel nanofabrication techniques [98, 99]. Following the Refs. [96, 98, 100], we choose ω 0 = 1 . 59 eV and L = 400 nm. The cavity effect becomes prominent when the all the parameters are comparable [101]. Due to this reason, we take T and z 0 in such a way so that ω 0 L , ω 0 z 0 , and T/ω 0 becomes comparable.</text>
<figure>
<location><page_9><loc_29><loc_45><loc_70><loc_64></location>
<caption>Figure 2 : Transition rate from | g 〉 → | e 〉 (per unit λ 2 ω 0 2 π ) versus separation between the two boundaries for a fixed value of T/ω 0 = 1 and ω 0 z 0 = 0 . 8 .</caption>
</figure>
<text><location><page_9><loc_14><loc_21><loc_86><loc_39></location>Figure 2 shows the variation of the transition rate from | g 〉 → | e 〉 (per unit λ 2 ω 0 2 π ) with respect to the cavity length for a fixed value of the distance of the atom from one boundary and the temperature of the thermal field. From the figure, it can be seen that for a fixed value of the initial atomic distance z 0 from one boundary, the transition rate initially very low due to the cavity effect and get enhanced after a certain cavity length and get saturated for large values of L ( ω 0 L >> ω 0 z 0 ). This is to be expected as extending the cavity length results in an increased number of field modes participating in the interaction between the atom and the scalar field, which raises the transition rate. When ω 0 L >> ω 0 z 0 , the cavity scenario reduces to the case of a single boundary, and hence, the upward transition rate saturates for large L .</text>
<text><location><page_9><loc_14><loc_14><loc_86><loc_21></location>Figure 3 shows the variation of the transition rate from | g 〉 → | e 〉 (per unit λ 2 ω 0 2 π ) with respect to the distance of the atom from one boundary for a fixed value of the length of the cavity and the temperature of the thermal field. From the figure, it is observed that for a fixed value of the cavity length L , when we increase the atomic distance from one</text>
<figure>
<location><page_10><loc_29><loc_72><loc_71><loc_90></location>
<caption>Figure 3 : Transition rate from | g 〉 → | e 〉 (per unit λ 2 ω 0 2 π ) versus distance of the atom from one boundary for a fixed value of T/ω 0 = 1 and ω 0 L = 3 . 2 .</caption>
</figure>
<text><location><page_10><loc_14><loc_56><loc_86><loc_64></location>boundary, the transition rate increases and at a certain value of z 0 it attains a maximum value and then it gets reduced by further increment of z 0 . The reason behind this is the following. Increasing the atomic distance from the boundary reduces the boundary effect on the number of field modes taking part in the interaction between the atom and the scalar field, which in turn increases the transition rate. This result is consistent with Figure 2.</text>
<figure>
<location><page_10><loc_29><loc_36><loc_71><loc_55></location>
<caption>Figure 4 : Transition rate from | g 〉 → | e 〉 (per unit λ 2 ω 0 2 π ) versus temperature for a fixed value of ω 0 L = 3 . 2 and ω 0 z 0 = 1 .</caption>
</figure>
<text><location><page_10><loc_14><loc_21><loc_86><loc_30></location>Figure 4 shows the variation of the transition rate from | g 〉 → | e 〉 (per unit λ 2 ω 0 2 π ) with respect to the temperature of the thermal bath for a fixed value of the cavity length and distance of the atom from one boundary. From the figure, it is observed that for a fixed value of the length of the cavity L and the atomic distance z 0 from one boundary, the transition rate increases when the temperature of the thermal bath is increased.</text>
<section_header_level_1><location><page_10><loc_14><loc_18><loc_35><loc_19></location>2.2 Two-atom system</section_header_level_1>
<text><location><page_10><loc_14><loc_13><loc_86><loc_17></location>In this subsection, we analyse the transition rates of a static two-atom system prepared in any generic pure entangled state | ψ 〉 that interacts with the massless scalar field in a thermal</text>
<text><location><page_11><loc_14><loc_87><loc_86><loc_90></location>state at an arbitrary temperature T . Following similar procedures as given in subsection A.2, the transition rate of a static two-atom system takes the form</text>
<formula><location><page_11><loc_17><loc_82><loc_86><loc_86></location>R β | ψ 〉→| E n 〉 = λ 2 [ | m ( A ) E n ψ | 2 F β AA (∆ E ) + m ( B ) E n ψ m ( A ) ∗ E n ψ F β AB (∆ E ) ] + A /harpoonleftright B terms (2.27)</formula>
<formula><location><page_11><loc_29><loc_77><loc_86><loc_81></location>F β ξξ ' (∆ E ) = ∫ + ∞ -∞ d (∆ τ ) e -i ∆ E ∆ τ G + β ( x ξ ( τ ) , x ξ ' ( τ ' )) (2.28)</formula>
<text><location><page_11><loc_14><loc_81><loc_19><loc_82></location>where</text>
<text><location><page_11><loc_14><loc_76><loc_46><loc_77></location>with ξ, ξ ' can be labeled by A or B , and</text>
<formula><location><page_11><loc_31><loc_72><loc_86><loc_75></location>G + β ( x ξ ( τ ) , x ξ ' ( τ ' )) = tr [ e -β H F φ ( x ξ ( τ )) φ ( x ξ ' ( τ ' )] tr [ e -β H F ] (2.29)</formula>
<text><location><page_11><loc_14><loc_65><loc_86><loc_71></location>is the positive frequency Wightman function of the scalar field in a thermal state at an arbitrary temperature T , and H F = ∑ k ω k a † k a k .</text>
<section_header_level_1><location><page_11><loc_14><loc_64><loc_69><loc_65></location>2.2.1 Transition rates for two-atom system in empty space</section_header_level_1>
<text><location><page_11><loc_14><loc_58><loc_86><loc_63></location>We take into account the transition rates of a stationary two-atom system which interacts with the massless scalar field in a thermal state at an arbitrary temperature T in the empty space. In the laboratory frame the trajectories of both the atoms read</text>
<formula><location><page_11><loc_29><loc_55><loc_86><loc_56></location>t A/B ( τ ) = τ, x A/B = 0 , y A/B = 0 , z A = 0 , z B = d. (2.30)</formula>
<text><location><page_11><loc_14><loc_50><loc_86><loc_54></location>Using the usual mode expansion of the scalar field operator in eq. (2.29), Wightman function becomes [30]</text>
<formula><location><page_11><loc_17><loc_42><loc_86><loc_49></location>G + β ( x ξ ( τ ) , x ξ ' ( τ ' )) = -1 4 π 2 ∞ ∑ m = -∞ 1 ( t ξ -t ' ξ ' -imβ -iε ) 2 -( x ξ -x ' ξ ' ) 2 -( y ξ -y ' ξ ' ) 2 -( z ξ -z ' ξ ' ) 2 . (2.31)</formula>
<text><location><page_11><loc_14><loc_41><loc_65><loc_42></location>Using eq.(s)(2.30, 2.31) the Wightman function turns out to be</text>
<formula><location><page_11><loc_28><loc_35><loc_86><loc_40></location>G + β ( x ξ ( τ ) , x ξ ' ( τ ' )) = -1 4 π 2 ∞ ∑ m = -∞ [ 1 (∆ τ -imβ -iε ) 2 ] (2.32)</formula>
<text><location><page_11><loc_14><loc_32><loc_40><loc_35></location>with ∆ τ = τ -τ ' for ξ = ξ ' , and</text>
<formula><location><page_11><loc_26><loc_27><loc_86><loc_32></location>G + β ( x ξ ( τ ) , x ξ ' ( τ ' )) = -1 4 π 2 ∞ ∑ m = -∞ [ 1 (∆ τ -imβ -iε ) 2 -d 2 ] (2.33)</formula>
<text><location><page_11><loc_14><loc_26><loc_22><loc_27></location>for ξ = ξ ' .</text>
<text><location><page_11><loc_14><loc_20><loc_86><loc_25></location>Substituting the Wightman functions, eq.(s) (2.32, 2.33) into eq.(A.18), the transition rate of the two-atom system from the initial state | ψ 〉 to the final state | E n 〉 can be rewritten as</text>
<text><location><page_11><loc_18><loc_25><loc_18><loc_27></location>/negationslash</text>
<formula><location><page_11><loc_18><loc_13><loc_86><loc_19></location>R β | ψ 〉→| E n 〉 = λ 2 [ | m ( A ) E n ψ | 2 F β AA (∆ E ) + | m ( B ) E n ψ | 2 F β BB (∆ E ) + m ( B ) E n ψ m ( A ) ∗ E n ψ F β AB (∆ E ) + m ( A ) E n ψ m ( B ) ∗ E n ψ F β BA (∆ E ) ] (2.34)</formula>
<text><location><page_12><loc_14><loc_88><loc_18><loc_90></location>with</text>
<formula><location><page_12><loc_21><loc_82><loc_86><loc_87></location>F β ξξ ' (∆ E ) = -1 4 π 2 ∞ ∑ m = -∞ ∫ + ∞ -∞ d (∆ τ ) e -i ∆ E ∆ τ [ 1 (∆ τ -imβ -iε ) 2 ] (2.35)</formula>
<text><location><page_12><loc_14><loc_80><loc_25><loc_82></location>for ξ = ξ ' and</text>
<formula><location><page_12><loc_19><loc_74><loc_86><loc_79></location>F β ξξ ' (∆ E ) = -1 4 π 2 ∞ ∑ m = -∞ ∫ + ∞ -∞ d (∆ τ ) e -i ∆ E ∆ τ [ 1 (∆ τ -imβ -iε ) 2 -d 2 ] (2.36)</formula>
<text><location><page_12><loc_14><loc_72><loc_22><loc_73></location>for ξ = ξ ' .</text>
<text><location><page_12><loc_14><loc_68><loc_86><loc_71></location>We may simplify the transition rates, eq.(2.34), by performing the above integrations using the contour integration technique</text>
<text><location><page_12><loc_18><loc_71><loc_18><loc_73></location>/negationslash</text>
<formula><location><page_12><loc_17><loc_59><loc_86><loc_67></location>R β | ψ 〉→| E n 〉 = λ 2 { θ ( -∆ E ) ( | ∆ E | 2 π + sin 2 θ sin( | ∆ E | d ) 2 πd )( 1 + 1 exp { ( | ∆ E | /T ) } -1 ) + θ (∆ E ) ( ∆ E 2 π + sin 2 θ sin(∆ Ed ) 2 πd )( 1 exp { (∆ E/T ) } -1 )} . (2.37)</formula>
<text><location><page_12><loc_14><loc_54><loc_86><loc_58></location>The above equation reveals that the two transition processes, namely, the downward and the upward transition can take place for the two-atom system with the upward transition rate</text>
<formula><location><page_12><loc_24><loc_49><loc_86><loc_53></location>R β | ψ 〉→| e A e B 〉 = λ 2 {( ω 0 2 π + sin 2 θ sin( ω 0 d ) 2 πd )( 1 exp { ( ω 0 /T ) } -1 )} (2.38)</formula>
<text><location><page_12><loc_14><loc_48><loc_41><loc_49></location>and the downward transition rate</text>
<formula><location><page_12><loc_19><loc_43><loc_86><loc_47></location>R β | ψ 〉→| g A g B 〉 = λ 2 {( ω 0 2 π + sin 2 θ sin( ω 0 d ) 2 πd )( 1 + 1 exp { ( ω 0 /T ) } -1 )} . (2.39)</formula>
<text><location><page_12><loc_14><loc_14><loc_86><loc_42></location>From the above analysis it is observed that if we compare the transition rates, eq.(s)(2.38, 2.39), with those of the uniformly accelerated two-atom system as seen by an instantaneously inertial observer, eq.(s)(A.21, A.22), we find that the transition rates of the static two-atom system immersed in a thermal bath as seen by a static observer are in general distinct from those of the two-atom system uniformly accelerated in the Minkowski vacuum even when the temperature of the thermal bath is taken to be the FDU temperature [30]. Although, here we would like to point out an additional feature. It is observed that in the limit αd << 1 , expanding eqs.(A.21) and (A.22) and keeping terms upto O ( α 2 d 2 ) gives the results eqs.(A.23) and (A.24). Now from these equations, it is seen that the leading term of the transition rates of uniformly accelerated two-atom system seen by a inertial observer in free space (eqs.(A.23) and (A.24)) matches with those of the static two-atom system seen by a static observer in thermal bath in free space (eqs.(2.38) and (2.39)) if we take the temperature of the thermal bath equal to the FDU temperature. This observation implies that in empty space there is an approximate equivalence between the upward and downward transition rates of the scenarios when two static atoms are placed in a thermal bath and two atoms are accelerating uniformly with respect to an inertial observer.</text>
<section_header_level_1><location><page_13><loc_14><loc_88><loc_65><loc_90></location>2.2.2 Transition rates for two-atom system in a cavity</section_header_level_1>
<text><location><page_13><loc_14><loc_80><loc_86><loc_87></location>Let us consider a static two-atom system interacting with a thermal state of a massless scalar field confined in a cavity of length L as shown in Figure 5. Assuming that the scalar field obeys the Dirichlet boundary condition φ | z =0 = φ | z = L = 0 , the thermal Wightman function inside the cavity takes the form [95]</text>
<formula><location><page_13><loc_15><loc_68><loc_86><loc_78></location>G + β ( x ξ ( τ ) , x ξ ' ( τ ' )) = -1 4 π 2 ∞ ∑ m = -∞ ∞ ∑ n = -∞ [ 1 ( t ξ ( τ ) -t ξ ' ( τ ' ) -imβ -iε ) 2 -| ∆ x ⊥ | 2 -( z ξ ( τ ) -z ξ ' ( τ ' ) -2 nL ) 2 -1 ( t ξ ( τ ) -t ξ ' ( τ ' ) -imβ -iε ) 2 -| ∆ x ⊥ | 2 -( z ξ ( τ ) + z ξ ' ( τ ' ) -2 nL ) 2 ] (2.40)</formula>
<text><location><page_13><loc_14><loc_62><loc_58><loc_66></location>with | ∆ x ⊥ | 2 = √ ( x ξ ( τ ) -x ξ ' ( τ ' )) 2 +( y ξ ( τ ) -y ξ ' ( τ ' )) 2 .</text>
<figure>
<location><page_13><loc_23><loc_48><loc_76><loc_61></location>
<caption>Figure 5 : Static two-atom confined in a cavity with a thermal bath at a temperature T .</caption>
</figure>
<text><location><page_13><loc_14><loc_41><loc_74><loc_43></location>In case of two atoms inside the cavity the atomic trajectories take the form</text>
<formula><location><page_13><loc_27><loc_37><loc_86><loc_39></location>t A/B ( τ ) = τ, x A/B = 0 , y A/B = 0 , z A = z 0 , z B = z 0 + d. (2.41)</formula>
<text><location><page_13><loc_14><loc_34><loc_70><loc_35></location>Using above trajectories in eq.(2.40), the Wightman function becomes</text>
<formula><location><page_13><loc_14><loc_27><loc_88><loc_32></location>G + β ( x ξ ( τ ) , x ξ ' ( τ ' )) = -1 4 π 2 ∞ ∑ m = -∞ ∞ ∑ n = -∞ [ 1 (∆ τ -imβ -iε ) 2 -d ' 2 1 -1 (∆ τ -imβ -iε ) 2 -d ' 2 2 ] (2.42)</formula>
<text><location><page_13><loc_14><loc_24><loc_52><loc_26></location>for ξ = ξ ' , with d ' 1 = 2 nL, d ' 2 = 2 z ξ -2 nL and</text>
<formula><location><page_13><loc_14><loc_18><loc_88><loc_23></location>G + β ( x ξ ( τ ) , x ξ ' ( τ ' )) = -1 4 π 2 ∞ ∑ m = -∞ ∞ ∑ n = -∞ [ 1 (∆ τ -imβ -iε ) 2 -d ' 2 3 -1 (∆ τ -imβ -iε ) 2 -d ' 2 4 ] (2.43)</formula>
<text><location><page_13><loc_18><loc_15><loc_18><loc_17></location>/negationslash</text>
<text><location><page_13><loc_14><loc_13><loc_86><loc_17></location>for ξ = ξ ' , with d ' 3 = d +2 nL (for ξ = A,ξ ' = B ), d ' 3 = d -2 nL (for ξ = B,ξ ' = A ) and d ' 4 = 2 z 0 + d -2 nL .</text>
<text><location><page_14><loc_14><loc_86><loc_86><loc_90></location>Using above Wightman functions, the rate of transition from the initial entangled state | ψ 〉 to the final separable state | E n 〉 can be written as</text>
<formula><location><page_14><loc_15><loc_79><loc_86><loc_86></location>R β | ψ 〉→| E n 〉 = λ 2 ∞ ∑ n = -∞ [ | m ( A ) E n ψ | 2 F β AA (∆ E ) + | m ( B ) E n ψ | 2 F β BB (∆ E ) + m ( B ) E n ψ m ( A ) ∗ E n ψ F β AB (∆ E ) + m ( A ) E n ψ m ( B ) ∗ E n ψ F β BA (∆ E ) (2.44)</formula>
<text><location><page_14><loc_14><loc_77><loc_18><loc_79></location>with</text>
<formula><location><page_14><loc_53><loc_78><loc_54><loc_82></location>]</formula>
<formula><location><page_14><loc_14><loc_72><loc_94><loc_77></location>F β ξξ ' (∆ E ) = -1 4 π 2 ∞ ∑ m = -∞ ∫ + ∞ -∞ d (∆ τ ) e -i ∆ E ∆ τ [ 1 (∆ τ -imβ -iε ) 2 -d ' 2 1 -1 (∆ τ -imβ -iε ) 2 -d ' 2 2 ] (2.45)</formula>
<text><location><page_14><loc_14><loc_70><loc_25><loc_71></location>for ξ = ξ ' and</text>
<formula><location><page_14><loc_14><loc_64><loc_94><loc_69></location>F β ξξ ' (∆ E ) = -1 4 π 2 ∞ ∑ m = -∞ ∫ + ∞ -∞ d (∆ τ ) e -i ∆ E ∆ τ [ 1 (∆ τ -imβ -iε ) 2 -d ' 2 3 -1 (∆ τ -imβ -iε ) 2 -d ' 2 4 ] (2.46)</formula>
<text><location><page_14><loc_14><loc_62><loc_22><loc_64></location>for ξ = ξ ' .</text>
<text><location><page_14><loc_14><loc_59><loc_86><loc_62></location>Eq.(2.44) can be further simplified by performing the above integrations using the contour integration procedure</text>
<text><location><page_14><loc_18><loc_61><loc_18><loc_63></location>/negationslash</text>
<formula><location><page_14><loc_14><loc_42><loc_95><loc_58></location>R β | ψ 〉→| E n 〉 = λ 2 { θ ( -∆ E ) ( | ∆ E | 2 π + q ( | ∆ E | , 2 L ) -cos 2 θ r ( | ∆ E | , 2 z 0 , 2 L ) -sin 2 θ s ( | ∆ E | , 2 z 0 , 2 d, 2 L ) + sin 2 θ t ( | ∆ E | , d, 2 L ) -sin 2 θ s ( | ∆ E | , 2 z 0 , d, 2 L ) )( 1 + 1 exp { ( | ∆ E | /T ) } -1 ) + θ (∆ E ) ( ∆ E 2 π + q (∆ E, 2 L ) -cos 2 θ r (∆ E, 2 z 0 , 2 L ) -sin 2 θ s (∆ E, 2 z 0 , 2 d, 2 L ) + sin 2 θ t (∆ E, d, 2 L ) -sin 2 θ s (∆ E, 2 z 0 , d, 2 L ) )( 1 exp { (∆ E/T ) } -1 )} (2.47)</formula>
<text><location><page_14><loc_14><loc_40><loc_32><loc_41></location>where we have defined</text>
<formula><location><page_14><loc_30><loc_34><loc_86><loc_39></location>s (∆ E, 2 z 0 , d, 2 L ) = ∞ ∑ n = -∞ p (∆ E, 2 z 0 + d -2 nL ) (2.48)</formula>
<text><location><page_14><loc_14><loc_29><loc_80><loc_30></location>and q (∆ E, 2 L ) , r (∆ E, 2 z 0 , 2 L ) , p (∆ E, 2 z 0 ) are given in eq.(s)(2.20, 2.21, 2.22).</text>
<formula><location><page_14><loc_34><loc_30><loc_86><loc_35></location>t (∆ E, d, 2 L ) = ∞ ∑ n = -∞ p (∆ E, d -2 nL ) (2.49)</formula>
<text><location><page_14><loc_14><loc_24><loc_86><loc_28></location>Similar to the previous case, the above equation also suggests that two transition process can take place for the two-atom system in presence of a reflecting boundary with the upward transition rate</text>
<formula><location><page_14><loc_15><loc_15><loc_86><loc_23></location>R β | ψ 〉→| e A e B 〉 = λ 2 {( ω 0 2 π + q ( ω 0 , 2 L ) -cos 2 θ r ( ω 0 , 2 z 0 , 2 L ) -sin 2 θ s ( ω 0 , 2 z 0 , 2 d, 2 L ) + sin 2 θ t ( ω 0 , d, 2 L ) -sin 2 θ s ( ω 0 , 2 z 0 , d, 2 L ) )( 1 exp { ( ω 0 /T ) } -1 )} (2.50)</formula>
<text><location><page_15><loc_14><loc_88><loc_41><loc_90></location>and the downward transition rate</text>
<formula><location><page_15><loc_14><loc_79><loc_87><loc_87></location>R β | ψ 〉→| g A g B 〉 = λ 2 {( ω 0 2 π + q ( ω 0 , 2 L ) -cos 2 θ r ( ω 0 , 2 z 0 , 2 L ) -sin 2 θ s ( ω 0 , 2 z 0 , 2 d, 2 L ) + sin 2 θ t ( ω 0 , d, 2 L ) -sin 2 θ s ( ω 0 , 2 z 0 , d, 2 L ) )( 1 + 1 exp { ( ω 0 /T ) } -1 )} . (2.51)</formula>
<text><location><page_15><loc_14><loc_56><loc_86><loc_77></location>From the above analysis, eqs (2.50), (2.51), (A.25) and (A.26) clearly display that transition rates of a uniformly accelerated two-atom seen by an instantaneously inertial observer and a static two-atom seen by a static observer in a thermal bath are non-identical inside the cavity even if we consider the temperature of the thermal bath to be the same as the FDU temperature. It may also be noted that the eq.(s)(2.50, 2.51) cannot be restored from the eq.(s)(A.25, A.26) even after taking the limit αd << 1 . Hence, this observation confirms that the equivalence between the transition rates no longer holds for the cases when a twoatom system uniformly accelerates and when a static two-atom system placed in a thermal bath. However, as the ratio of the eqs (2.50), (2.51), and (A.25), (A.26) matches exactly at T = α/ 2 π , so the effects of uniform acceleration and the effects of a thermal bath holds completely for the two-atom system interacts with the massless scalar field confined in a cavity.</text>
<text><location><page_15><loc_14><loc_49><loc_86><loc_56></location>To obtain the single mirror and free space scenarios, we now take the limiting cases of these expressions. Taking the limit L → ∞ , we find that eq.(s)(2.50, 2.51) reduce to the expression for the upward and the downward transition rate in the presence of a single reflecting boundary</text>
<formula><location><page_15><loc_17><loc_40><loc_86><loc_48></location>R β | ψ 〉→| e A e B 〉 = λ 2 {( ω 0 2 π -cos 2 θ p ( ω 0 , 2 z 0 ) -sin 2 θ p ( ω 0 , 2( z 0 + d )) + sin2 θ ( p ( ω 0 , d ) -p ( ω 0 , 2 z 0 + d ) ) )( 1 exp { ( ω 0 /T ) } -1 )} (2.52)</formula>
<formula><location><page_15><loc_15><loc_30><loc_86><loc_38></location>R β | ψ 〉→| g A g B 〉 = λ 2 {( ω 0 2 π -cos 2 θ p ( ω 0 , 2 z 0 ) -sin 2 θ p ( ω 0 , 2( z 0 + d )) + sin 2 θ ( p ( ω 0 , d ) -p ( ω 0 , 2 z 0 + d ) ) )( 1 + 1 exp { ( ω 0 /T ) } -1 )} . (2.53)</formula>
<text><location><page_15><loc_14><loc_26><loc_86><loc_30></location>Similarly, taking the limits L →∞ and z 0 →∞ , eq.(s)(2.50, 2.51) lead to the expressions for the upward and the downward transition rate in free space given by eq.(s)(2.38, 2.39).</text>
<text><location><page_15><loc_14><loc_14><loc_86><loc_26></location>We now study how the transition rate of an entangled two atom system from an initial entangled state | ψ 〉 to a product state with higher energy value | e A e B 〉 confined to a cavity varies with parameters such as thermal field temperature ( T ), cavity length ( L ), atomic distance from one boundary ( z 0 ), interatomic distance ( d ), and the entanglement parameter ( θ ). Following the single atom case, here also we present our findings in the following plots. In the following plots, we fix the parameters T/ω 0 , ω 0 L, ω 0 z 0 , and ω 0 d in such a way so that cavity effects remain significant.</text>
<figure>
<location><page_16><loc_29><loc_70><loc_71><loc_90></location>
<caption>Figure 6 : Transition rate (per unit λ 2 ω 0 2 π ) versus entanglement parameter for a fixed value of ω 0 z 0 = 18 , ω 0 L = 20 (without boundary), ω 0 z 0 = 1 . 4 , ω 0 L = 8 (single boundary), and ω 0 z 0 = 0 . 6 , ω 0 L = 3 . 4 (double boundary). All three cases T/ω 0 = 1 , ω 0 d = 0 . 2 .</caption>
</figure>
<text><location><page_16><loc_14><loc_18><loc_86><loc_60></location>In Figure 6, we show the behaviour of the transition rate with respect to the entanglement parameter for the cases where the atoms are in free space, in the vicinity of a single boundary and inside a cavity. From Figure 6, it can be seen that the transition rate | ψ 〉 → | e A e B 〉 (per unit λ 2 ω 0 2 π ) varies sinusoidally with the entanglement parameter θ . In free space, the transition rate increases (from the case corresponding to the zero initial entanglement product state) with increase in the entanglement parameter and it becomes maximum when the initial state is maximally entangled ( θ = π/ 4 super-radiant state) [102]. Further increment of the entanglement parameter decreases the transition rate and it vanishes at θ = 3 π/ 4 (maximally entangled sub-radiant state). In the vicinity of a single boundary, behaviour of the transition rate is quite similar to the free space scenario, with a slight shifting of the extremum points. Inside the cavity, the behaviour of the transition rate is also similar to the free space scenario. The peak value of transition rate inside the cavity is much smaller compared to the free space and single boundary cases. It can be noted that around θ = 3 π/ 4 , the values of the transition rate corresponding to cases of empty space, single boundary, and two boundaries, nearly vanish. At this point, it should be noted that increasing and decreasing the atom-boundary distance z 0 and the cavity length L the upward transition rate decreases for the value of θ = π/ 4 . In the thermal bath, we observe that the transition rate of the superradiant state, θ = π/ 4 of the two atoms inside the cavity does not vanish, in contrast to the result obtained for a co-accelerating frame in [31]. Here, the superradiance property of the state ( θ = π/ 4 ) remains intact. On the other hand, in the thermal bath, we observe that the transition rate of the sub-radiant state, θ = 3 π/ 4 of the two atoms inside the cavity vanishes. Therefore, entanglement of the initial state is preserved and from a quantum information theoretic viewpoint, this kind of an initial state may act as a good resource for performing various tasks.</text>
<text><location><page_16><loc_14><loc_14><loc_86><loc_17></location>In Figure 7, we show the behaviour of the transition rate with respect to the entanglement parameter inside a cavity for a fixed value of T/ω 0 = 1 , ω 0 d = 0 . 6 , ω 0 L = 3 . 6 .</text>
<figure>
<location><page_17><loc_29><loc_70><loc_71><loc_90></location>
<caption>Figure 7 : Transition rate (per unit λ 2 ω 0 2 π ) versus entanglement parameter for a fixed value of T/ω 0 = 1 , ω 0 d = 0 . 6 , ω 0 L = 3 . 6 .</caption>
</figure>
<figure>
<location><page_17><loc_29><loc_43><loc_71><loc_63></location>
<caption>Figure 8 : Transition rate (per unit λ 2 ω 0 2 π ) versus entanglement parameter for a fixed value of T/ω 0 = 1 , ω 0 z 0 = 1 . 6 , ω 0 L = 3 . 6 .</caption>
</figure>
<text><location><page_17><loc_14><loc_23><loc_86><loc_35></location>From Figure 7, it can be seen that inside the cavity at θ = π/ 4 , the upward transition rate | ψ 〉 → | e A e B 〉 (per unit λ 2 ω 0 2 π ) increases when the distance between one atom and the nearest boundary z 0 increases. As the length of the cavity and the inter-atomic distance is fixed therefore after increasing the distance between the nearest atom and the boundary, boundary effect reduces on the number of field modes taking part in the interaction between the atom and the scalar field, which in turn increases the upward transition rates of the system.</text>
<text><location><page_17><loc_14><loc_14><loc_86><loc_22></location>In Figure 8, we show the behaviour of the transition rate with respect to the entanglement parameter inside a cavity for a fixed value of T/ω 0 = 1 , ω 0 z 0 = 1 . 6 , ω 0 L = 3 . 6 . From Figure 8, it can be seen that inside the cavity at θ = π/ 4 , the upward transition rate | ψ 〉 → | e A e B 〉 (per unit λ 2 ω 0 2 π ) decreases when the inter-atomic distance d increases. As the length of the cavity and the distance of one atom from the nearest boundary is</text>
<figure>
<location><page_18><loc_29><loc_72><loc_71><loc_90></location>
<caption>Figure 10 : Transition rate from | ψ 〉 → | e A e B 〉 (per unit λ 2 ω 0 2 π ) versus separation between two boundaries for a fixed value of T/ω 0 = 1 , ω 0 d = 0 . 5 , and ω 0 L = 3 . 2 .</caption>
</figure>
<text><location><page_18><loc_14><loc_58><loc_86><loc_65></location>fixed therefore after increasing the inter-atomic distance the second atom moves toward the second boundary. Therefore, due to the boundary condition, number of field modes taking part in the inteaction between the atom and the scalar field reduces from the both end and as a result the transition rate of the system decreases.</text>
<figure>
<location><page_18><loc_29><loc_38><loc_70><loc_57></location>
<caption>Figure 9 : Transition rate from | ψ 〉 → | e A e B 〉 (per unit λ 2 ω 0 2 π ) versus separation between two boundaries for a fixed value of T/ω 0 = 1 , ω 0 d = 0 . 5 , and ω 0 z 0 = 1 . 6 .</caption>
</figure>
<text><location><page_18><loc_14><loc_14><loc_86><loc_32></location>Figure 9 shows the variation of the transition rate from | ψ 〉 → | e A e B 〉 (per unit λ 2 ω 0 2 π ) with respect to the length of the cavity for a fixed value of the distance of any one atom from one boundary and the interatomic distance. From the figure, it can be seen that for a fixed value of the initial atomic distance z 0 of any one atom from the nearest boundary, the transition rate get enhanced when the cavity length increases and attains a maximum value for large values of L ( ω 0 L >> ω 0 z 0 ). This behaviour is similar to that of the single atom case, as mentioned earlier. As more number of field modes take part in the interaction between the scalar field and the atoms due to the increased cavity length, the transition rate increases. When ω 0 L >> ω 0 z 0 , the cavity scenario reduces to a single boundary set up and hence the upward transition rate becomes neary constant.</text>
<figure>
<location><page_19><loc_29><loc_72><loc_71><loc_90></location>
<caption>Figure 11 : Transition rate from | ψ 〉 → | e A e B 〉 (per unit λ 2 ω 0 2 π ) versus separation between two boundaries, T/ω 0 = 1 , ω 0 d = 0 . 5 .</caption>
</figure>
<text><location><page_19><loc_14><loc_52><loc_86><loc_64></location>Figure 10 shows the variation of the transition rate from | ψ 〉 → | e A e B 〉 (per unit λ 2 ω 0 2 π ) with respect to the distance of any one atom from one boundary for a fixed value of the cavity length and the interatomic distance. From the figure, it is observed that for a fixed value of the cavity length L and the interatomic distance d , when we increase the atomic distance from one boundary, the transition rate for the two atom system also increases and at a certain value of z 0 it attains a maximum value and then it gets reduced by further increment of z 0 . The reason behind this variation is also similar to the single atom case.</text>
<text><location><page_19><loc_14><loc_37><loc_86><loc_52></location>Figure 11 shows the variation of the transition rate from | ψ 〉 → | e A e B 〉 (per unit λ 2 ω 0 2 π ) with respect to the interatomic distance for a fixed value of the cavity length and the distance of any one atom from one boundary. From the figure, it is observed that for a fixed value of the cavity length L and the distance of any one atom from one boundary z 0 , when we increase the interatomic distance, the transition rate for the two atom system decreases. Due to the increament of the interatomic distance the second atom moves toward the second boundary. Therefore, due the boundary effect the transition rate of the two atom system falls down.</text>
<figure>
<location><page_19><loc_30><loc_19><loc_70><loc_36></location>
<caption>Figure 12 : Transition rate from | ψ 〉 → | e A e B 〉 (per unit λ 2 ω 0 2 π ) versus acceleration, ω 0 d = 0 . 5 , ω 0 z 0 = 0 . 4 .</caption>
</figure>
<text><location><page_20><loc_14><loc_81><loc_86><loc_90></location>Figure 12 shows the variation of the transition rate from | ψ 〉 → | e A e B 〉 (per unit λ 2 ω 0 2 π ) with respect to the atomic acceleration for a fixed value of the cavity length. From the figure, it is observed that for a fixed value of the length of the cavity L , the atomic distance z 0 from one boundary, and the interatomic distance d , it is observed that the transition rate increases when the temperature of the thermal bath is increased.</text>
<section_header_level_1><location><page_20><loc_14><loc_78><loc_62><loc_79></location>3 Quantitative estimation of the transition rate</section_header_level_1>
<text><location><page_20><loc_14><loc_69><loc_86><loc_76></location>In this section, we now present a quantitative estimation of the upward transition rate of a single and two atom system inside a cavity by considering the Rubidium and Caesium atom. Following some recent works Ref. [97-101, 103], it has been seen that 87 Rb and 133 Cs widely used in the experimental setup.</text>
<section_header_level_1><location><page_20><loc_14><loc_66><loc_39><loc_68></location>3.1 Transition rate of 87 Rb</section_header_level_1>
<text><location><page_20><loc_14><loc_57><loc_86><loc_65></location>From the Ref. [96, 100], we have taken the transition energy and the cavity length for calculating the upward transition rate of 87 Rb . The values are /planckover2pi1 ω 0 = 1 . 59 eV and L = 400 nm respectively. Setting thermal bath temperature T = 20 , 000 K and the distance between the atom and the nearest boundary z 0 = 100 nm. Fixing all the physical quantity and calculating the dimensionless parameters, we get</text>
<formula><location><page_20><loc_28><loc_54><loc_86><loc_55></location>T/ω 0 = 1 , ω 0 L = 3 . 2 , ω 0 z 0 = 0 . 8 (3.1)</formula>
<text><location><page_20><loc_14><loc_49><loc_86><loc_52></location>Taking the coupling constant λ = 0 . 1 and using above values of the parameters, the upward transition rate for a single 87 Rb atom becomes</text>
<formula><location><page_20><loc_32><loc_45><loc_86><loc_48></location>R β | g 〉→| e 〉 = 1 . 45 × 10 -3 eV = 3 . 51 × 10 11 s -1 . (3.2)</formula>
<text><location><page_20><loc_14><loc_40><loc_86><loc_45></location>In case of two 87 Rb atoms, we additionally choose the inter-atomic distance d = 50 nm and the entanglement parameter θ = π/ 4 . Therefore, ω 0 d = 0 . 4 . Taking λ = 0 . 1 , the upward transition rate for two 87 Rb atoms becomes</text>
<formula><location><page_20><loc_31><loc_36><loc_86><loc_39></location>R β | ψ 〉→| e A e B 〉 = 3 . 87 × 10 -3 eV = 9 . 37 × 10 11 s -1 . (3.3)</formula>
<section_header_level_1><location><page_20><loc_14><loc_34><loc_40><loc_35></location>3.2 Transition rate of 133 Cs</section_header_level_1>
<text><location><page_20><loc_14><loc_24><loc_86><loc_33></location>From the Ref. [100, 104], we have taken the transition energy and the cavity length for calculating the upward transition rate of 133 Cs . The values are /planckover2pi1 ω 0 = 1 . 46 eV and L = 500 nm respectively. Setting thermal bath temperature T = 20 , 000 K and the distance between the atom and the nearest boundary z 0 = 150 nm. Fixing all the physical quantity and calculating the dimensionless parameters, we get</text>
<formula><location><page_20><loc_27><loc_21><loc_86><loc_23></location>T/ω 0 = 1 . 2 , ω 0 L = 3 . 7 , ω 0 z 0 = 1 . 1 (3.4)</formula>
<text><location><page_20><loc_14><loc_17><loc_86><loc_20></location>Taking the coupling constant λ = 0 . 1 and using above values of the parameters, the upward transition rate for a single 133 Cs atom becomes</text>
<formula><location><page_20><loc_32><loc_13><loc_86><loc_16></location>R β | g 〉→| e 〉 = 1 . 96 × 10 -3 eV = 4 . 74 × 10 11 s -1 . (3.5)</formula>
<text><location><page_21><loc_14><loc_85><loc_86><loc_90></location>In case of two 133 Cs atoms, we additionally choose the inter-atomic distance d = 75 nm and the entanglement parameter θ = π/ 4 . Therefore, ω 0 d = 0 . 4 . Taking λ = 0 . 1 , the upward transition rate for two 133 Cs atoms becomes</text>
<formula><location><page_21><loc_31><loc_81><loc_86><loc_84></location>R β | ψ 〉→| e A e B 〉 = 4 . 86 × 10 -3 eV = 1 . 18 × 10 12 s -1 . (3.6)</formula>
<section_header_level_1><location><page_21><loc_14><loc_79><loc_29><loc_80></location>4 Conclusions</section_header_level_1>
<text><location><page_21><loc_14><loc_64><loc_86><loc_77></location>In this work we take a fresh look at the FDU effect in the context of two-level single and entangled atomic systems that are static in a thermal bath. We investigated the question as to whether the FDU effect shows up or not for single atom and entangled two-atom detector systems in free-space and inside cavity setups inside a thermal bath. In order to investigate the actual equivalence in real physical systems, the first step is to identify choices of specific detectors in context of which one may hope to observe such equivalence. Phenomena such as the excitation rate of a detector are directly observable, and are therefore of substantial significance in empirically verifying deep physical concepts, such as the FDU effect.</text>
<text><location><page_21><loc_14><loc_46><loc_86><loc_63></location>Our study takes into consideration both single and entangled two-atom systems which can be located either in free space or inside cavities with reflecting boundary conditions. The two-atom system is taken to be initially prepared in a most general pure entangled state. The interactions of both systems with a massless scalar field in its vacuum state are studied in order to investigate the equivalence between the thermal bath and the uniform acceleration for the cases of the single and two-atom systems. For doing so, we have calculated the upward and downward transition rates of the single and entangled atomic systems immersed in a thermal bath in free space and inside cavities. We have compared these results with those of the results of accelerated single and entangled atomic systems in free space and inside a cavity [31] given in Appendix A.</text>
<text><location><page_21><loc_14><loc_18><loc_86><loc_45></location>From the results, it is observed that while both the upward and downward transitions occur for the single atom as well as the entangled two-atom system, the upward transition is significant as it solely arises due to the temperature of the thermal bath. The actual transition rate depends on the cavity length, the distance of an atom from one boundary, the temperature of the thermal bath, and the magnitude of initial atomic entanglement. From our analysis it is evident that the transition rate shows oscillatory behaviour with the entanglement parameter. Considering a small magnitude of initial entanglement, we find that increasing the entanglement parameter enhances the upward transition and downward transition rates in free space, whereas, both the transition rates get suppressed due to the decreased number of field modes participating in the interaction between the atom and the scalar field in the presence of the cavity. In the case when the initial entanglement parameter has the value θ = 3 π/ 4 , we observe that both the transition rates vanish, indicating that no transition occurs from the maximally entangled initial state to any higher or lower energetic product state. Hence, the entanglement of the sub-radiant initial state can be preserved, a result which may be of significance, since preservation of entanglement enables its use as a resource for performing various quantum information processing tasks.</text>
<text><location><page_21><loc_14><loc_14><loc_86><loc_17></location>Our investigation of the upward and downward transition rates enables us to conclude that in the case of single as well as two-atom systems, the physical state of the system</text>
<text><location><page_22><loc_14><loc_58><loc_86><loc_90></location>and the observer's reference frame both significantly influence the upward and downward transition rates of the systems. Whether the equivalence inherent in the FDU effect is manifested or not depends upon the interplay of the following physical conditions. In free space, the upward and downward transition rates of a static atom placed in a thermal bath match with those of an accelerated atom, if the temperature of the thermal bath is equal to the Unruh temperature T = α/ 2 π . Such kind of equivalence also holds at the level of transition rates when a single atom is placed in a coaccelerated frame in a thermal bath [31]. The picture for two atoms is more intricate where atomic entanglement plays the key role. In an earlier work, it is observed that the upward and downward transition rates of two entangled atoms accelerating through a cavity turns out to be similar to the transition rates of a coaccelerated observer in a thermal bath [31]. However, in this study it is seen that the upward and downward transition rates of a uniformly accelerated two-atom system with the transition rates when the system is static but immersed in a thermal bath, such equivalence between the transition rates holds only under specific limiting conditions in free space, but breaks down completely inside a cavity setup. However, it turns out that the ratio of the upward and downward transition rates inside the thermal bath in free space as well as inside a cavity match exactly with those of the accelerated system in free space and inside a cavity.</text>
<text><location><page_22><loc_14><loc_47><loc_86><loc_57></location>Apart from the computing of the transition rate in different scenario for single and two level system, in this study we provide a quantitative estimation of the transition rate inside the cavity setup. Following some recent experimental setup [97-101, 103], we fix the physical parameters for the single and two Rubidium ( 87 Rb ) and Cesium ( 133 Cs ) atoms respectively. From this estimation, it can be seen that within the cavity of order nanometer the transition rates of this atoms are very high.</text>
<text><location><page_22><loc_14><loc_27><loc_86><loc_46></location>Our present analysis serves to comprehensively reestablish that in general, there is no equivalence at the level of transition rates between the accelerating and the static thermal bath frame for the chosen single atom model inside a cavity and the entangled two-atom detector model as realized in both free-space and cavity setups. However, at the level of the ratio of the transition rates, there is a complete equivalence between the accelerating and the static thermal bath frame for both the single and the entangled two-atom detector model as realized in both free-space and cavity setups. Therefore, at this point we should emphasize though, that our present analysis does not impact in any way the conceptual validity of the FDU effect. Rather, its revelation through the manifestation of equivalence, is model dependent [94], as evident through comparison of earlier results involving co-accelerating observers [30, 31].</text>
<text><location><page_22><loc_14><loc_14><loc_86><loc_26></location>Finally, it may be noted that in our present study we have considered only the effect of varying magnitudes of initial atomic entanglement on the transition rates. The focus here is not to evaluate any effects of generation or degradation of entanglement that may arise from the dynamics. However, other phenomena such as generation or degradation of atomic entanglement, as well as atom-field entanglement may have interesting implications [105-109]. Evaluation of such effects could be undertaken through the master-equation approach [110-112] in future works.</text>
<section_header_level_1><location><page_23><loc_14><loc_88><loc_32><loc_90></location>Acknowledgement</section_header_level_1>
<text><location><page_23><loc_14><loc_84><loc_86><loc_87></location>AM and ASM acknowledges support from project no. DST/ICPS/QuEST/2019/Q79 of the Department of Science and Technology (DST), Government of India.</text>
<section_header_level_1><location><page_23><loc_14><loc_78><loc_86><loc_81></location>Appendix A Interaction of the accelerated atomic system with a massless scalar field</section_header_level_1>
<text><location><page_23><loc_14><loc_73><loc_86><loc_76></location>In this Appendix, we mainly review some key results of [31] which we use for the sake of comparison with the subsequent results of our present work.</text>
<section_header_level_1><location><page_23><loc_14><loc_70><loc_37><loc_71></location>A.1 Single atom system</section_header_level_1>
<text><location><page_23><loc_14><loc_58><loc_86><loc_69></location>Let us consider a single atom (an Unruh-DeWitt detector) with two energy levels | g 〉 and | e 〉 with corresponding energy values -ω 0 / 2 and + ω 0 / 2 , travelling along a stationary trajectory in a vacuum with massless scalar field fluctuations. In the laboratory frame, trajectories of the atom can be represented through x ( τ ) ≡ ( t ( τ ) , x ( τ )) . In the instantaneous inertial frame, the Hamiltonian describing the atom-field interaction in the interaction picture is given by [37]</text>
<formula><location><page_23><loc_41><loc_57><loc_86><loc_58></location>H int = λm ( τ ) φ ( x ( τ )) (A.1)</formula>
<text><location><page_23><loc_14><loc_48><loc_86><loc_56></location>where λ is a small coupling constant, m ( τ ) = e i H 0 τ m (0) e -i H 0 τ is the monopole operator at any proper time τ of a single atom, φ ( x ( τ )) is the massless quantum scalar field evaluated at the trajectory x ( τ ) with m (0) = | g 〉〈 e | + | e 〉〈 g | being the initial monopole operator and H 0 | e 〉 = ω 2 | e 〉 being the free Hamiltonian of a single atom respectively [92].</text>
<text><location><page_23><loc_14><loc_44><loc_86><loc_48></location>Using the formalism discussed in [26, 31], the rate of transition probability from the initial atomic state | i 〉 to the final atomic state | f 〉 turns out to be</text>
<formula><location><page_23><loc_39><loc_41><loc_86><loc_44></location>R | i 〉→| f 〉 = λ 2 | m fi | 2 F (∆ E ) (A.2)</formula>
<text><location><page_23><loc_14><loc_37><loc_86><loc_40></location>where ∆ E = E f -E i , m fi = 〈 f | m (0) | i 〉 , and the response function per unit proper time can be written as</text>
<formula><location><page_23><loc_31><loc_32><loc_86><loc_36></location>F (∆ E ) = ∫ + ∞ -∞ d (∆ τ ) e -i ∆ E ∆ τ G + ( x ( τ ) , x ( τ ' )) (A.3)</formula>
<text><location><page_23><loc_14><loc_29><loc_33><loc_32></location>where ∆ τ = τ -τ ' and</text>
<formula><location><page_23><loc_33><loc_26><loc_86><loc_28></location>G + ( x ( τ ) , x ( τ ' )) = 〈 0 M | φ ( x ( τ )) φ ( x ( τ ' )) | 0 M 〉 (A.4)</formula>
<text><location><page_23><loc_14><loc_18><loc_86><loc_25></location>is the positive frequency Wightman function of the massless scalar field [26]. In empty space, using the atomic trajectory in the laboratory frame t ( τ ) = 1 α sinh( ατ ) , x ( τ ) = 1 α cosh( ατ ) , y = z = 0 with α being the proper acceleration and τ being the proper time of the atom, the Wightman function becomes</text>
<formula><location><page_23><loc_31><loc_11><loc_86><loc_17></location>G + ( x ( τ ) , x ( τ ' )) = -α 2 16 π 2 1 sinh 2 [ 1 2 ( α ∆ τ -iε ) ] . (A.5)</formula>
<text><location><page_24><loc_14><loc_87><loc_86><loc_90></location>Substituting the above Wightman function into eq.(A.2), the upward and downward transition rate becomes (See [31] for a detailed calculation)</text>
<formula><location><page_24><loc_35><loc_81><loc_86><loc_85></location>R | g 〉→| e 〉 = λ 2 ω 0 2 π ( 1 exp(2 πω 0 /α ) -1 ) (A.6)</formula>
<formula><location><page_24><loc_33><loc_77><loc_86><loc_81></location>R | e 〉→| g 〉 = λ 2 ω 0 2 π ( 1 + 1 exp(2 πω 0 /α ) -1 ) . (A.7)</formula>
<text><location><page_24><loc_14><loc_72><loc_86><loc_77></location>From the above equations, it is clearly seen that in case of a single atom system the upward transition rate solely depends on the atomic acceleration. Taking the ratio of the above two results, we get</text>
<formula><location><page_24><loc_35><loc_69><loc_86><loc_72></location>R | g 〉→| e 〉 R | e 〉→| g 〉 ≡ R up R down = exp( -2 πω 0 /α ) . (A.8)</formula>
<figure>
<location><page_24><loc_29><loc_56><loc_70><loc_65></location>
<caption>Figure 13 : Uniformly accelerated atom confined in a cavity [31].</caption>
</figure>
<text><location><page_24><loc_14><loc_46><loc_86><loc_51></location>Now, inside a cavity of length L as shown in Figure 13, with the atomic trajectory being t ( τ ) = 1 α sinh( ατ ) , x ( τ ) = 1 α cosh( ατ ) , y = 0 , and z = z 0 , the Wightman function is given by [31]</text>
<formula><location><page_24><loc_14><loc_39><loc_91><loc_45></location>G + ( x ( τ ) , x ( τ ' )) = -α 2 16 π 2 ∞ ∑ n = -∞ [ 1 sinh 2 [ 1 2 ( α ∆ τ -iε ) ] -1 4 d 2 1 α 2 -1 sinh 2 [ 1 2 ( α ∆ τ -iε ) ] -1 4 d 2 2 α 2 ] (A.9)</formula>
<text><location><page_24><loc_14><loc_36><loc_86><loc_39></location>with d 1 = nL, d 2 = 2 z 0 -nL . Now using the process outlined in Appendix B of [31], the upward and downward transition rate inside the cavity turn out to be</text>
<formula><location><page_24><loc_16><loc_25><loc_86><loc_34></location>R | g 〉→| e 〉 = λ 2 [{ ω 0 2 π + f ( ω 0 , α, L 2 ) -h ( ω 0 , α, z 0 , L 2 )}( 1 exp { (2 πω 0 /α ) } -1 )] (A.10) R | e 〉→| g 〉 = λ 2 [{ ω 0 2 π + f ( ω 0 , α, L 2 ) -h ( ω 0 , α, z 0 , L 2 )}( 1 + 1 exp { (2 πω 0 /α ) } -1 )] (A.11)</formula>
<text><location><page_24><loc_14><loc_24><loc_32><loc_25></location>where we have defined</text>
<formula><location><page_24><loc_33><loc_18><loc_86><loc_23></location>f ( ∆ E, α, L 2 ) = 2 ∞ ∑ n =1 g ( ∆ E, α, nL 2 ) (A.12)</formula>
<formula><location><page_24><loc_30><loc_13><loc_86><loc_18></location>h ( ∆ E, α, z 0 , L 2 ) = ∞ ∑ n = -∞ g ( ∆ E, α, z 0 -nL 2 ) (A.13)</formula>
<text><location><page_25><loc_14><loc_88><loc_37><loc_90></location>with g (∆ E, α, z 0 ) defined as</text>
<formula><location><page_25><loc_34><loc_81><loc_86><loc_87></location>g (∆ E, α, z 0 ) = sin ( 2∆ E α sinh -1 ( αz 0 ) ) 4 πz 0 √ 1 + α 2 z 2 0 . (A.14)</formula>
<text><location><page_25><loc_14><loc_81><loc_75><loc_82></location>Taking the ratio of the eqs.(A.10) and (A.11), in cavity scenario, we also get</text>
<formula><location><page_25><loc_35><loc_76><loc_86><loc_80></location>R | g 〉→| e 〉 R | e 〉→| g 〉 ≡ R up R down = exp( -2 πω 0 /α ) . (A.15)</formula>
<section_header_level_1><location><page_25><loc_14><loc_74><loc_35><loc_75></location>A.2 Two-atom system</section_header_level_1>
<text><location><page_25><loc_14><loc_50><loc_86><loc_73></location>In this subsection, considering two identical atoms A and B , we assume that they are travelling synchronously along stationary trajectories in the vacuum of a massless scalar field. The interatomic distance is assumed to be constant and the proper times of the two atoms can be described by the same time τ [89]. In the laboratory frame, trajectories of the two atoms can be represented through x A ( τ ) and x B ( τ ) . Here we consider each atom as a two level system having energy levels | g 〉 and | e 〉 with corresponding energy values -ω 0 / 2 and + ω 0 / 2 . Therefore, the entire two-atom system can be described by the eigenstates of the initial atomic Hamiltonian , namely, | g A , g B 〉 , | g A , e B 〉 , | e A , g B 〉 , and | e A , e B 〉 with the corresponding energy eigenvalues -ω 0 , 0 , 0 , and ω 0 respectively. Here, as the eigenstates | g A , e B 〉 and | e A , g B 〉 are the degenerate eigenstates of the Hamiltonian with the energy value zero, therefore any linear combination of this eigenstates will also be an eigenstate of this system with same energy eigenvalue [67]. Hence, the most general quantum state of the two-atom system with zero energy value is given by [91]</text>
<formula><location><page_25><loc_37><loc_46><loc_86><loc_49></location>| ψ 〉 = sin θ | g A , e B 〉 +cos θ | e A , g B 〉 (A.16)</formula>
<text><location><page_25><loc_14><loc_39><loc_86><loc_46></location>where the entanglement parameter θ lies in the range 0 ≤ θ ≤ π . It may be noted that though the Bell-state basis is frequently used when working with entangled states, for our present purpose use of the Bell basis is inconvenient, since the Bell states are not eigenstates of the atomic Hamiltonian.</text>
<text><location><page_25><loc_14><loc_35><loc_86><loc_39></location>In the instantaneously inertial frame, the Hamiltonian describing the atom-field interaction is given by</text>
<formula><location><page_25><loc_38><loc_31><loc_86><loc_35></location>H int = λ ∑ ξ = A,B m ξ ( τ ) φ ( x ξ ( τ )) (A.17)</formula>
<text><location><page_25><loc_14><loc_30><loc_52><loc_31></location>where λ is a small atom-field coupling constant.</text>
<text><location><page_25><loc_14><loc_26><loc_86><loc_30></location>As a result of the atom-field interaction, the transition probability rate of the two-atom system from the initial state | χ 〉 to the final state | χ ' 〉 turns out to be</text>
<formula><location><page_25><loc_18><loc_21><loc_86><loc_25></location>R | χ 〉→| χ ' 〉 = λ 2 [ | m ( A ) χ ' χ | 2 F AA (∆ E ) + m ( B ) χ ' χ m ( A ) ∗ χ ' χ F AB (∆ E ) ] + A /harpoonleftright B terms (A.18)</formula>
<text><location><page_25><loc_14><loc_18><loc_86><loc_22></location>where m ( A ) χ ' χ = 〈 χ ' | m (0) ⊗ 1 B | χ 〉 , m ( B ) χ ' χ = 〈 χ ' | 1 A ⊗ m (0) | χ 〉 . The response function per unit proper time can be written as</text>
<formula><location><page_25><loc_29><loc_13><loc_86><loc_17></location>F ξξ ' (∆ E ) = ∫ + ∞ -∞ d (∆ τ ) e -i ∆ E ∆ τ G + ( x ξ ( τ ) , x ξ ' ( τ ' )) (A.19)</formula>
<text><location><page_26><loc_14><loc_87><loc_58><loc_90></location>where ∆ τ = τ -τ ' , ξ, ξ ' can be labeled by A or B , and</text>
<formula><location><page_26><loc_31><loc_85><loc_86><loc_87></location>G + ( x ξ ( τ ) , x ξ ' ( τ ' )) = 〈 0 M | φ ( x ξ ( τ )) φ ( x ξ ' ( τ ' )) | 0 M 〉 (A.20)</formula>
<text><location><page_26><loc_14><loc_83><loc_57><loc_84></location>is the Wightman function of the massless scalar field.</text>
<text><location><page_26><loc_14><loc_72><loc_86><loc_82></location>In empty space, using the trajectories of both the atoms in the laboratory frame t A ( τ ) = t B ( τ ) = 1 α sinh( ατ ) , x A ( τ ) = x B ( τ ) = 1 α cosh( ατ ) , y B = y A + d , and z A = z B = 0 , with d being the constant interatomic distance, α being the proper acceleration and τ being the proper time of the two-atom system, the transition rates of the two-atom system from the initial entangled state | ψ 〉 to the final product states | e A , e B 〉 and | g A , g B 〉 can be expressed as [31]</text>
<formula><location><page_26><loc_17><loc_63><loc_86><loc_71></location>R | ψ 〉→| e A , e B 〉 = λ 2 ω 0 2 π + sin 2 θ sin ( 2 ω 0 α sinh -1 ( 1 2 αd ) ) 2 πd √ 1 + 1 4 d 2 α 2 ( 1 exp { (2 πω 0 /α ) } -1 ) (A.21)</formula>
<formula><location><page_26><loc_15><loc_54><loc_86><loc_62></location>R | ψ 〉→| g A , g B 〉 = λ 2 ω 0 2 π + sin 2 θ sin ( 2 ω 0 α sinh -1 ( 1 2 αd ) ) 2 πd √ 1 + 1 4 d 2 α 2 ( 1 + 1 exp { (2 πω 0 /α ) } -1 ) . (A.22)</formula>
<text><location><page_26><loc_14><loc_53><loc_67><loc_55></location>In the limit αd << 1 , above equations upto O ( α 2 d 2 ) take the form</text>
<formula><location><page_26><loc_16><loc_45><loc_86><loc_53></location>R | ψ 〉→| e A , e B 〉 = λ 2 {( ω 0 2 π + sin 2 θ 2 πd [ sin( ω 0 d ) -1 8 { sin( ω 0 d ) + 1 3 ( ω 0 d ) cos( ω 0 d ) } α 2 d 2 ]) × ( 1 exp { (2 πω 0 /α ) } -1 )} (A.23)</formula>
<formula><location><page_26><loc_16><loc_35><loc_86><loc_43></location>R | ψ 〉→| g A , g B 〉 = λ 2 {( ω 0 2 π + sin 2 θ 2 πd [ sin( ω 0 d ) -1 8 { sin( ω 0 d ) + 1 3 ( ω 0 d ) cos( ω 0 d ) } α 2 d 2 ]) × ( 1 + 1 exp { (2 πω 0 /α ) } -1 )} . (A.24)</formula>
<text><location><page_26><loc_15><loc_34><loc_86><loc_35></location>Inside a cavity of length L as shown in Figure 14, using the atomic trajectories t A/B ( τ ) =</text>
<figure>
<location><page_26><loc_29><loc_20><loc_71><loc_32></location>
<caption>Figure 14 : Uniformly accelerated two-atom confined in a cavity [31].</caption>
</figure>
<text><location><page_26><loc_14><loc_14><loc_86><loc_16></location>1 α sinh( ατ ) , x A/B ( τ ) = 1 α cosh( ατ ) , y A/B = y 0 , and z A = z 0 , z B = z 0 + d , the transition</text>
<text><location><page_27><loc_14><loc_86><loc_86><loc_90></location>rate of the two-atom system from the initial entangled state | ψ 〉 to the final product state | e A , e B 〉 and | g A , g B 〉 inside the cavity can be expressed as [31]</text>
<formula><location><page_27><loc_14><loc_75><loc_90><loc_83></location>R | ψ 〉→| e A , e B 〉 = λ 2 {( ω 0 2 π + f ( ω 0 , α, L 2 ) -cos 2 θ h ( ω 0 , α, z 0 , L 2 ) -sin 2 θ m ( ω 0 , α, z 0 , d, L 2 ) + sin 2 θ ( n ( ω 0 , α, d 2 , L 2 ) -m ( ω 0 , α, z 0 , d 2 , L 2 ))( 1 exp { (2 πω 0 /α ) } -1 )} (A.25)</formula>
<formula><location><page_27><loc_14><loc_59><loc_92><loc_67></location>R | ψ 〉→| g A , g B 〉 = λ 2 {( ω 0 2 π + f ( ω 0 , α, L 2 ) -cos 2 θ h ( ω 0 , α, z 0 , L 2 ) -sin 2 θ m ( ω 0 , α, z 0 , d, L 2 ) + sin 2 θ ( n ( ω 0 , α, d 2 , L 2 ) -m ( ω 0 , α, z 0 , d 2 , L 2 ))( 1 + 1 exp { (2 πω 0 /α ) } -1 )} (A.26)</formula>
<text><location><page_27><loc_14><loc_54><loc_32><loc_55></location>where we have defined</text>
<formula><location><page_27><loc_28><loc_41><loc_86><loc_51></location>m ( ∆ E, α, z 0 , d, L 2 ) = ∞ ∑ n = -∞ g ( ∆ E, α, z 0 + d -nL 2 ) (A.27) n ( ∆ E, α, d 2 , L 2 ) = ∞ ∑ n = -∞ g ( ∆ E, α, d -nL 2 ) (A.28)</formula>
<text><location><page_27><loc_14><loc_35><loc_86><loc_39></location>with f ( ∆ E, α, L 2 ) , h ( ∆ E, α, z 0 , L 2 ) , g (∆ E, α, z 0 ) are given in eq.(s)(A.12, A.13, A.14).</text>
<section_header_level_1><location><page_27><loc_14><loc_29><loc_74><loc_30></location>Appendix B Derivation of the thermal Wightman function</section_header_level_1>
<text><location><page_27><loc_14><loc_22><loc_86><loc_25></location>In this Appendix, for the sake of completeness, we provide a complete derivation of the thermal Wightman function of the massless scalar field eq.(2.7).</text>
<text><location><page_27><loc_14><loc_20><loc_47><loc_22></location>Thermal Wightman function is defined as</text>
<formula><location><page_27><loc_32><loc_14><loc_86><loc_17></location>G + β ( x ( τ ) , x ( τ ' )) = tr [ e -β H F φ ( x ( τ )) φ ( x ( τ ' ))] tr [ e -β H F ] (B.1)</formula>
<text><location><page_28><loc_14><loc_86><loc_34><loc_90></location>where H F = ∑ k ω k a † k a k .</text>
<text><location><page_28><loc_14><loc_86><loc_22><loc_87></location>Therefore,</text>
<formula><location><page_28><loc_17><loc_60><loc_86><loc_83></location>G + β ( x ( τ ) , x ( τ ' )) = tr [ e -β H F φ ( x ( τ )) φ ( x ( τ ' ))] tr [ e -β H F ] = tr [ exp { -β ∑ k ω k a † k a k } φ ( x ( τ )) φ ( x ( τ ' )) ] / tr [ exp { -β ∑ k ω k a † k a k }] = ∞ ∑ ξ =0 〈 ξ | exp { -β ∑ k ω k a † k a k } φ ( x ( τ )) φ ( x ( τ ' )) | ξ 〉 / ∞ ∑ ξ =0 〈 ξ | exp { -β ∑ k ω k a † k a k } | ξ 〉 = ∞ ∑ ξ,σ =0 〈 ξ | exp { -β ∑ k ω k a † k a k } | σ 〉〈 σ | φ ( x ( τ )) φ ( x ( τ ' )) | ξ 〉 / ∞ ∑ ξ =0 e -βξω = [ ∞ ∑ σ =0 exp ( -βσω ) 〈 σ | φ ( x ( τ )) φ ( x ( τ ' )) | σ 〉 ]/ ∞ ∑ σ =0 exp ( -βσω ) . (B.2)</formula>
<text><location><page_28><loc_14><loc_57><loc_61><loc_58></location>Using the mode expansion of the massless scalar field [113]</text>
<formula><location><page_28><loc_26><loc_50><loc_86><loc_54></location>φ ( x ( τ )) = 1 (2 π ) 3 / 2 ∫ + ∞ -∞ d 3 k √ 2 ω k [ a k e -iω k t + i k · x + a † k e iω k t -i k · x ] (B.3)</formula>
<text><location><page_28><loc_14><loc_46><loc_38><loc_48></location>〈 σ | φ ( x ( τ )) φ ( x ( τ ' )) | σ 〉 becomes</text>
<formula><location><page_28><loc_14><loc_36><loc_92><loc_43></location>〈 σ | φ ( x ( τ )) φ ( x ( τ ' )) | σ 〉 = 1 (2 π ) 3 ∫ + ∞ -∞ d 3 k d 3 k ' √ 4 ω k ω k ' [ 〈 σ | a k a † k ' | σ 〉 e -i ( ω k t -ω k ' t ' )+ i ( k · x -k ' · x ' ) + 〈 σ | a † k a k ' | σ 〉 e i ( ω k t -ω k ' t ' ) -i ( k · x -k ' · x ' ) ] . (B.4)</formula>
<text><location><page_28><loc_14><loc_31><loc_48><loc_33></location>Now using the relation between a k and a † k</text>
<formula><location><page_28><loc_41><loc_24><loc_86><loc_28></location>[ a k , a † k ] = δ 3 ( k -k ' ) . (B.5)</formula>
<text><location><page_28><loc_14><loc_22><loc_60><loc_23></location>Using the above commutation relation eq. (B.4) becomes</text>
<formula><location><page_28><loc_14><loc_14><loc_86><loc_19></location>〈 σ | φ ( x ( τ )) φ ( x ( τ ' )) | σ 〉 = 1 (2 π ) 3 ∫ + ∞ -∞ d 3 k 2 ω k [ ( σ +1) e -iω k ( t -t ' )+ i k · ( x -x ' ) + σe iω k ( t -t ' ) -i k · ( x -x ' ) ] . (B.6)</formula>
<text><location><page_29><loc_14><loc_87><loc_86><loc_90></location>For the massless scalar field ω k = | k | ≡ k and defining t -t ' ≡ ∆ t and x -x ' ≡ ∆ x and putting these values in eq.(B.2), we get</text>
<formula><location><page_29><loc_14><loc_61><loc_89><loc_85></location>G + β ( x ( τ ) , x ( τ ' )) = 1 (2 π ) 3 ∫ + ∞ -∞ d 3 k 2 k [ ∞ ∑ σ =0 ( σ +1) e -βkσ e -ik ∆ t + i k · ∆ x + ∞ ∑ σ =1 σe -βkσ e ik ∆ t -i k · ∆ x ]/ ∞ ∑ σ =0 e -βkσ = 1 (2 π ) 3 ∫ + ∞ -∞ d 3 k 2 k [ e βk e βk -1 e -ik ∆ t + i k · ∆ x + 1 e βk -1 e ik ∆ t -i k · ∆ x ] = 1 4 π 2 [ 1 | ∆ x | ∫ + ∞ -∞ sin( k | ∆ x | ) e ik (∆ t -iε ) ( e βk -1) dk ] [ where ε → 0] = 1 4 π 2 [ π 2 β | ∆ x | { coth ( π ( | ∆ x | -∆ t + iε ) β ) +coth ( π ( | ∆ x | +∆ t -iε ) β )}] = -1 4 π 2 ∞ ∑ n = -∞ 1 ( t ( τ ) -t ( τ ' ) -inβ -iε ) 2 -( x ( τ ) -x ( τ ' )) 2 -( y ( τ ) -y ( τ ' )) 2 -( z ( τ ) -z ( τ ' )) 2 . (B.7)</formula>
<section_header_level_1><location><page_29><loc_14><loc_55><loc_86><loc_59></location>Appendix C Derivation of the thermal Wightman function inside a cavity</section_header_level_1>
<text><location><page_29><loc_14><loc_50><loc_86><loc_54></location>In this Appendix, we provide a complete derivation of the thermal Wightman function of the massless scalar field inside a cavity eq.(2.15).</text>
<text><location><page_29><loc_14><loc_49><loc_78><loc_50></location>Using the definition of the thermal Wightman function given in eq.(B.1), we get</text>
<formula><location><page_29><loc_20><loc_34><loc_86><loc_47></location>G + β ( x ( τ ) , x ( τ ' )) = tr [ e -β H F φ ( x ( τ )) φ ( x ( τ ' ))] tr [ e -β H F ] = tr [ exp { -β ∑ k ω k a † k a k } φ ( x ( τ )) φ ( x ( τ ' )) ] / tr [ exp { -β ∑ k ω k a † k a k }] = [ ∞ ∑ σ =0 exp ( -βσω ) 〈 σ | φ ( x ( τ )) φ ( x ( τ ' )) | σ 〉 ]/ ∞ ∑ σ =0 exp ( -βσω ) . (C.1)</formula>
<text><location><page_29><loc_14><loc_30><loc_86><loc_34></location>In the presence of a single reflecting boundary, the mode function of the massless scalar field operator obeys Dirichlet boundary condition φ | z = z 0 = 0 and takes the form</text>
<formula><location><page_29><loc_36><loc_27><loc_86><loc_29></location>f ( t, x, y, z ) ∼ sin( k z z ) e -ikt e ik y y + ik x x (C.2)</formula>
<text><location><page_29><loc_14><loc_23><loc_82><loc_26></location>Now, using this mode function and computing the term 〈 σ | φ ( x ( τ )) φ ( x ( τ ' )) | σ 〉 we get</text>
<formula><location><page_29><loc_15><loc_13><loc_86><loc_23></location>〈 σ | φ ( x ( τ )) φ ( x ( τ ' )) | σ 〉 = 1 (2 π ) 3 ∫ + ∞ -∞ d 3 k 2 k { cos[ k z ( z -z ' )] -cos[ k z ( z + z ' )] } [ ( σ +1) e -ik ( t -t ' )+ ik x ( x -x ' )+ ik y ( y -y ' ) + σe ik ( t -t ' ) -ik x ( x -x ' ) -ik y ( y -y ' ) ] . (C.3)</formula>
<text><location><page_30><loc_14><loc_84><loc_86><loc_90></location>Inside the cavity of length L , the mode function of the massless scalar field operator obeys Dirichlet boundary condition φ | z = z 0 = φ | z = L 0 . Using the mode function given in Ref.[114] and computing the term 〈 σ | φ ( x ( τ )) φ ( x ( τ ' )) | σ 〉 we get</text>
<formula><location><page_30><loc_14><loc_72><loc_93><loc_83></location>〈 σ | φ ( x ( τ )) φ ( x ( τ ' )) | σ 〉 = 1 (2 π ) 3 ∞ ∑ n = -∞ ∫ + ∞ -∞ d 3 k 2 k { cos[ k z ( z -z ' -2 nL )] -cos[ k z ( z + z ' -2 nL )] } [ ( σ +1) e -ik ∆ t + ik x ∆ x + ik y ∆ y + σe ik ∆ t -ik x ∆ x -ik y ∆ y ] . (C.4)</formula>
<text><location><page_30><loc_14><loc_69><loc_86><loc_72></location>Therefore, using eq.(C.4) into the last line of eq.(C.1) and computing the summation over σ , thermal Wightman function takes the form</text>
<formula><location><page_30><loc_14><loc_56><loc_94><loc_67></location>G + β ( x ( τ ) , x ( τ ' )) = 1 (2 π ) 3 ∞ ∑ n = -∞ ∫ + ∞ -∞ d 3 k 2 k { cos[ k z ( z -z ' -2 nL )] -cos[ k z ( z + z ' -2 nL )] } [ e βk e βk -1 e -ik ∆ t + ik x ∆ x + ik y ∆ y + 1 e βk -1 e ik ∆ t -ik x ∆ x -ik y ∆ y ] . (C.5)</formula>
<text><location><page_30><loc_14><loc_55><loc_85><loc_56></location>Now, after some algebraic manipulation, and solving the angular integrals, we finally get</text>
<formula><location><page_30><loc_15><loc_38><loc_86><loc_53></location>G + β ( x ( τ ) , x ( τ ' )) = -1 4 π 2 [ 1 | ∆ x 2 | ∫ + ∞ -∞ sin( k | ∆ x 2 | ) e ik (∆ t -iε ) ( e βk -1) dk -1 | ∆ x 1 | ∫ + ∞ -∞ sin( k | ∆ x 1 | ) e ik (∆ t -iε ) ( e βk -1) dk ] = -1 4 π 2 [ π 2 β | ∆ x 2 | { coth ( π ( | ∆ x 2 | -∆ t + iε ) β ) +coth ( π ( | ∆ x 2 | +∆ t -iε ) β )} -π 2 β | ∆ x 1 | { coth ( π ( | ∆ x 1 | -∆ t + iε ) β ) +coth ( π ( | ∆ x 1 | +∆ t -iε ) β )}] (C.6)</formula>
<text><location><page_30><loc_14><loc_30><loc_86><loc_36></location>where we consider ε is very small, | ∆ x 2 | = √ (∆ x ) 2 +(∆ y ) 2 +(∆ z -2 nL ) 2 and | ∆ x 1 | = √ (∆ x ) 2 +(∆ y ) 2 +( z + z ' -2 nL ) 2 . This equation can be recast as</text>
<formula><location><page_30><loc_16><loc_18><loc_86><loc_29></location>G + β ( x ( τ ) , x ( τ ' )) = -1 4 π 2 ∞ ∑ m = -∞ ∞ ∑ n = -∞ [ 1 ( t -t ' -imβ -iε ) 2 -( x -x ' ) 2 -( y -y ' ) 2 -( z -z ' -2 nL ) 2 -1 ( t -t ' -inβ -iε ) 2 -( x -x ' ) 2 -( y -y ' ) 2 -( z + z ' -2 nL ) 2 ] (C.7)</formula>
<text><location><page_30><loc_14><loc_14><loc_86><loc_17></location>This is the form of the thermal Wightman function inside the cavity which is used in eq.(2.15).</text>
<section_header_level_1><location><page_31><loc_14><loc_88><loc_25><loc_90></location>References</section_header_level_1>
<unordered_list>
<list_item><location><page_31><loc_16><loc_84><loc_82><loc_87></location>[1] M.B. Plenio, S.F. Huelga, A. Beige and P.L. Knight, Cavity-loss-induced generation of entangled atoms , Phys. Rev. A 59 (1999) 2468.</list_item>
<list_item><location><page_31><loc_16><loc_80><loc_81><loc_83></location>[2] L. Amico, R. Fazio, A. Osterloh and V. Vedral, Entanglement in many-body systems , Rev. Mod. Phys. 80 (2008) 517.</list_item>
<list_item><location><page_31><loc_16><loc_77><loc_86><loc_80></location>[3] Z. Ficek and R. Tanaś, Entanglement induced by spontaneous emission in spatially extended two-atom systems , Journal of Modern Optics 50 (2003) 2765.</list_item>
<list_item><location><page_31><loc_16><loc_73><loc_71><loc_76></location>[4] R. Tanaś and Z. Ficek, Entangling two atoms via spontaneous emission , Journal of Optics B: Quantum and Semiclassical Optics 6 (2004) S90.</list_item>
<list_item><location><page_31><loc_16><loc_69><loc_82><loc_72></location>[5] D.T. Alves and L.C.B. Crispino, Response rate of a uniformly accelerated source in the presence of boundaries , Phys. Rev. D 70 (2004) 107703.</list_item>
<list_item><location><page_31><loc_16><loc_65><loc_82><loc_68></location>[6] H. Yu and S. Lu, Spontaneous excitation of an accelerated atom in a spacetime with a reflecting plane boundary , Phys. Rev. D 72 (2005) 064022.</list_item>
<list_item><location><page_31><loc_16><loc_62><loc_84><loc_65></location>[7] Z. Zhu, H. Yu and S. Lu, Spontaneous excitation of an accelerated hydrogen atom coupled with electromagnetic vacuum fluctuations , Phys. Rev. D 73 (2006) 107501.</list_item>
<list_item><location><page_31><loc_16><loc_58><loc_79><loc_61></location>[8] H. Yu and Z. Zhu, Spontaneous absorption of an accelerated hydrogen atom near a conducting plane in vacuum , Phys. Rev. D 74 (2006) 044032.</list_item>
<list_item><location><page_31><loc_16><loc_54><loc_81><loc_57></location>[9] S.-Y. Lin and B.L. Hu, Accelerated detector-quantum field correlations: From vacuum fluctuations to radiation flux , Phys. Rev. D 73 (2006) 124018.</list_item>
<list_item><location><page_31><loc_15><loc_51><loc_86><loc_54></location>[10] Z. Zhu and H. Yu, Fulling-davies-unruh effect and spontaneous excitation of an accelerated atom interacting with a quantum scalar field , Physics Letters B 645 (2007) 459.</list_item>
<list_item><location><page_31><loc_15><loc_47><loc_83><loc_50></location>[11] W. Zhou and H. Yu, Spontaneous excitation of a uniformly accelerated atom coupled to vacuum dirac field fluctuations , Phys. Rev. A 86 (2012) 033841.</list_item>
<list_item><location><page_31><loc_15><loc_43><loc_85><loc_46></location>[12] Z. Zhi-Ying and Y. Hong-Wei, Accelerated multi-level atoms in an electromagnetic vacuum and fulling-davies-unruh effect , Chinese Physics Letters 25 (2008) 1575.</list_item>
<list_item><location><page_31><loc_15><loc_39><loc_85><loc_42></location>[13] A. Noto and R. Passante, van der waals interaction energy between two atoms moving with uniform acceleration , Phys. Rev. D 88 (2013) 025041.</list_item>
<list_item><location><page_31><loc_15><loc_36><loc_86><loc_39></location>[14] J. Marino, A. Noto and R. Passante, Thermal and nonthermal signatures of the unruh effect in casimir-polder forces , Phys. Rev. Lett. 113 (2014) 020403.</list_item>
<list_item><location><page_31><loc_15><loc_30><loc_86><loc_35></location>[15] L. Rizzuto, M. Lattuca, J. Marino, A. Noto, S. Spagnolo, W. Zhou et al., Nonthermal effects of acceleration in the resonance interaction between two uniformly accelerated atoms , Phys. Rev. A 94 (2016) 012121.</list_item>
<list_item><location><page_31><loc_15><loc_25><loc_86><loc_30></location>[16] W. Zhou, R. Passante and L. Rizzuto, Resonance interaction energy between two accelerated identical atoms in a coaccelerated frame and the unruh effect , Phys. Rev. D 94 (2016) 105025.</list_item>
<list_item><location><page_31><loc_15><loc_21><loc_85><loc_24></location>[17] G. Fiscelli, L. Rizzuto and R. Passante, Resonance energy transfer between two atoms in a conducting cylindrical waveguide , Phys. Rev. A 98 (2018) 013849.</list_item>
<list_item><location><page_31><loc_15><loc_18><loc_86><loc_21></location>[18] G. Menezes and N.F. Svaiter, Radiative processes of uniformly accelerated entangled atoms , Phys. Rev. A 93 (2016) 052117.</list_item>
<list_item><location><page_31><loc_15><loc_14><loc_85><loc_17></location>[19] G. Menezes, Radiative processes of two entangled atoms outside a schwarzschild black hole , Phys. Rev. D 94 (2016) 105008.</list_item>
</unordered_list>
<table>
<location><page_32><loc_15><loc_14><loc_87><loc_90></location>
</table>
<unordered_list>
<list_item><location><page_33><loc_15><loc_87><loc_82><loc_90></location>[40] J. Audretsch and R. Müller, Radiative energy shifts of an accelerated two-level system , Phys. Rev. A 52 (1995) 629.</list_item>
<list_item><location><page_33><loc_15><loc_83><loc_79><loc_86></location>[41] E. Hagley, X. Maître, G. Nogues, C. Wunderlich, M. Brune, J.M. Raimond et al., Generation of einstein-podolsky-rosen pairs of atoms , Phys. Rev. Lett. 79 (1997) 1.</list_item>
<list_item><location><page_33><loc_15><loc_79><loc_85><loc_82></location>[42] R. Passante, Radiative level shifts of an accelerated hydrogen atom and the unruh effect in quantum electrodynamics , Phys. Rev. A 57 (1998) 1590.</list_item>
<list_item><location><page_33><loc_15><loc_75><loc_85><loc_78></location>[43] D.A.T. Vanzella and G.E.A. Matsas, Decay of accelerated protons and the existence of the fulling-davies-unruh effect , Phys. Rev. Lett. 87 (2001) 151301.</list_item>
<list_item><location><page_33><loc_15><loc_72><loc_79><loc_74></location>[44] P.M. Alsing and G.J. Milburn, Teleportation with a uniformly accelerated partner , Phys. Rev. Lett. 91 (2003) 180404.</list_item>
<list_item><location><page_33><loc_15><loc_68><loc_80><loc_71></location>[45] P.M. Alsing, D. McMahon and G.J. Milburn, Teleportation in a non-inertial frame , Journal of Optics B: Quantum and Semiclassical Optics 6 (2004) S834.</list_item>
<list_item><location><page_33><loc_15><loc_64><loc_78><loc_67></location>[46] I. Fuentes-Schuller and R.B. Mann, Alice falls into a black hole: Entanglement in noninertial frames , Phys. Rev. Lett. 95 (2005) 120404.</list_item>
<list_item><location><page_33><loc_15><loc_60><loc_85><loc_63></location>[47] P.M. Alsing, I. Fuentes-Schuller, R.B. Mann and T.E. Tessier, Entanglement of dirac fields in noninertial frames , Phys. Rev. A 74 (2006) 032326.</list_item>
<list_item><location><page_33><loc_15><loc_56><loc_82><loc_59></location>[48] A. Bermudez and M.A. Martin-Delgado, Hyper-entanglement in a relativistic two-body system , Journal of Physics A: Mathematical and Theoretical 41 (2008) 485302.</list_item>
<list_item><location><page_33><loc_15><loc_52><loc_80><loc_55></location>[49] M.-R. Hwang, D. Park and E. Jung, Tripartite entanglement in a noninertial frame , Phys. Rev. A 83 (2011) 012111.</list_item>
<list_item><location><page_33><loc_15><loc_49><loc_71><loc_51></location>[50] M.-R. Hwang, E. Jung and D. Park, Three-tangle in non-inertial frame , Classical and Quantum Gravity 29 (2012) 224004.</list_item>
<list_item><location><page_33><loc_15><loc_45><loc_86><loc_48></location>[51] B. Richter and Y. Omar, Degradation of entanglement between two accelerated parties: Bell states under the unruh effect , Phys. Rev. A 92 (2015) 022334.</list_item>
<list_item><location><page_33><loc_15><loc_42><loc_79><loc_44></location>[52] B. Reznik, Entanglement from the vacuum , Foundations of Physics 33 (2003) 167.</list_item>
<list_item><location><page_33><loc_15><loc_39><loc_84><loc_42></location>[53] G. Salton, R.B. Mann and N.C. Menicucci, Acceleration-assisted entanglement harvesting and rangefinding , New Journal of Physics 17 (2015) 035001.</list_item>
<list_item><location><page_33><loc_15><loc_35><loc_82><loc_38></location>[54] A. Pozas-Kerstjens and E. Martín-Martínez, Harvesting correlations from the quantum vacuum , Phys. Rev. D 92 (2015) 064042.</list_item>
<list_item><location><page_33><loc_15><loc_31><loc_81><loc_34></location>[55] E. Martín-Martínez, A.R.H. Smith and D.R. Terno, Spacetime structure and vacuum entanglement , Phys. Rev. D 93 (2016) 044001.</list_item>
<list_item><location><page_33><loc_15><loc_27><loc_79><loc_30></location>[56] A. Pozas-Kerstjens and E. Martín-Martínez, Entanglement harvesting from the electromagnetic vacuum with hydrogenlike atoms , Phys. Rev. D 94 (2016) 064074.</list_item>
<list_item><location><page_33><loc_15><loc_22><loc_81><loc_26></location>[57] L.J. Henderson, R.A. Hennigar, R.B. Mann, A.R.H. Smith and J. Zhang, Harvesting entanglement from the black hole vacuum , Classical and Quantum Gravity 35 (2018) 21LT02.</list_item>
<list_item><location><page_33><loc_15><loc_18><loc_73><loc_21></location>[58] L.J. Henderson and N.C. Menicucci, Bandlimited entanglement harvesting , Phys. Rev. D 102 (2020) 125026.</list_item>
<list_item><location><page_33><loc_15><loc_14><loc_86><loc_17></location>[59] N. Stritzelberger, L.J. Henderson, V. Baccetti, N.C. Menicucci and A. Kempf, Entanglement harvesting with coherently delocalized matter , Phys. Rev. D 103 (2021) 016007.</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_34><loc_15><loc_87><loc_85><loc_90></location>[60] R. Chatterjee and A.S. Majumdar, Preservation of quantum coherence under lorentz boost for narrow uncertainty wave packets , Phys. Rev. A 96 (2017) 052301.</list_item>
<list_item><location><page_34><loc_15><loc_83><loc_70><loc_86></location>[61] A. Peres and D.R. Terno, Quantum information and relativity theory , Rev. Mod. Phys. 76 (2004) 93.</list_item>
<list_item><location><page_34><loc_15><loc_79><loc_84><loc_82></location>[62] R. Chatterjee and A.S. Majumdar, Bell-inequality violation by dynamical casimir photons in a superconducting microwave circuit , Phys. Rev. A 106 (2022) 042224.</list_item>
<list_item><location><page_34><loc_15><loc_76><loc_81><loc_79></location>[63] S. Massar and P. Spindel, Einstein-podolsky-rosen correlations between two uniformly accelerated oscillators , Phys. Rev. D 74 (2006) 085031.</list_item>
<list_item><location><page_34><loc_15><loc_70><loc_66><loc_75></location>[64] J. Franson, Generation of entanglement outside of the light cone , Journal of Modern Optics 55 (2008) 2117 [ https://doi.org/10.1080/09500340801983129 ].</list_item>
<list_item><location><page_34><loc_15><loc_67><loc_85><loc_69></location>[65] S.-Y. Lin and B.L. Hu, Temporal and spatial dependence of quantum entanglement from a field theory perspective , Phys. Rev. D 79 (2009) 085020.</list_item>
<list_item><location><page_34><loc_15><loc_63><loc_84><loc_66></location>[66] S.-Y. Lin and B.L. Hu, Entanglement creation between two causally disconnected objects , Phys. Rev. D 81 (2010) 045019.</list_item>
<list_item><location><page_34><loc_15><loc_59><loc_81><loc_62></location>[67] C. Rodríguez-Camargo, G. Menezes and N. Svaiter, Finite-time response function of uniformly accelerated entangled atoms , Annals of Physics 396 (2018) 266.</list_item>
<list_item><location><page_34><loc_15><loc_55><loc_80><loc_58></location>[68] J. Hu and H. Yu, Entanglement dynamics for uniformly accelerated two-level atoms , Phys. Rev. A 91 (2015) 012327.</list_item>
<list_item><location><page_34><loc_15><loc_52><loc_83><loc_54></location>[69] H. Cai and Z. Ren, Radiative properties of an inertial multilevel atom in a compactified minkowski spacetime , Classical and Quantum Gravity 36 (2019) 165001.</list_item>
<list_item><location><page_34><loc_15><loc_48><loc_82><loc_51></location>[70] G. Picanço, N.F. Svaiter and C.A. Zarro, Radiative processes of entangled detectors in rotating frames , Journal of High Energy Physics 2020 (2020) 25.</list_item>
<list_item><location><page_34><loc_15><loc_44><loc_83><loc_47></location>[71] W. Zhou and H. Yu, Radiation-reaction-induced transitions of two maximally entangled atoms in noninertial motion , Phys. Rev. D 101 (2020) 025009.</list_item>
<list_item><location><page_34><loc_15><loc_40><loc_85><loc_43></location>[72] S. Felicetti, C. Sabín, I. Fuentes, L. Lamata, G. Romero and E. Solano, Relativistic motion with superconducting qubits , Phys. Rev. B 92 (2015) 064501.</list_item>
<list_item><location><page_34><loc_15><loc_37><loc_75><loc_39></location>[73] G.A. Mourou, T. Tajima and S.V. Bulanov, Optics in the relativistic regime , Rev. Mod. Phys. 78 (2006) 309.</list_item>
<list_item><location><page_34><loc_15><loc_33><loc_86><loc_36></location>[74] U. Eichmann, T. Nubbemeyer, H. Rottke and W. Sandner, Acceleration of neutral atoms in strong short-pulse laser fields , Nature 461 (2009) 1261.</list_item>
<list_item><location><page_34><loc_15><loc_29><loc_83><loc_32></location>[75] C. Maher-McWilliams, P. Douglas and P.F. Barker, Laser-driven acceleration of neutral particles , Nature Photonics 6 (2012) 386.</list_item>
<list_item><location><page_34><loc_15><loc_25><loc_84><loc_28></location>[76] N. Friis, A.R. Lee, K. Truong, C. Sabín, E. Solano, G. Johansson et al., Relativistic quantum teleportation with superconducting circuits , Phys. Rev. Lett. 110 (2013) 113602.</list_item>
<list_item><location><page_34><loc_15><loc_22><loc_78><loc_24></location>[77] Z. Huang and H. Situ, Protection of quantum dialogue affected by quantum field , Quantum Information Processing 18 (2019) .</list_item>
<list_item><location><page_34><loc_15><loc_18><loc_79><loc_21></location>[78] Z. Huang and Z. He, Deterministic secure quantum communication under vacuum fluctuation , The European Physical Journal D 74 (2020) .</list_item>
<list_item><location><page_34><loc_15><loc_14><loc_85><loc_17></location>[79] M.R.R. Good, A. Lapponi, O. Luongo and S. Mancini, Quantum communication through a partially reflecting accelerating mirror , Phys. Rev. D 104 (2021) 105020.</list_item>
</unordered_list>
<table>
<location><page_35><loc_15><loc_14><loc_86><loc_90></location>
</table>
<table>
<location><page_36><loc_14><loc_25><loc_87><loc_90></location>
</table>
</document>
[
{
"title": "Single and entangled atomic systems in thermal bath and the Fulling-Davies-Unruh effect",
"content": "Arnab Mukherjee, a, 1 Sunandan Gangopadhyay, b Archan. S. Majumdar c Department of Astrophysics and High Energy Physics, S.N. Bose National Centre for Basic Sciences, JD Block, Sector-III, Salt Lake, Kolkata 700106, India E-mail: arnab.mukherjee@bose.res.in , sunandan.gangopadhyay@gmail.com , archan@bose.res.in Abstract: In this study, we revisit the Fulling-Davies-Unruh effect in the context of twolevel single and entangled atomic systems that are static in a thermal bath. We consider the interaction between the systems and a massless scalar field, covering the scenarios of free space as well as within a cavity. Through the calculation of atomic transition rates and comparing with the results of [ Phys. Rev. D 108 (2023) 085018 ], it is found that in free space there is an equivalence between the upward and downward transition rates of an uniformly accelerated atom with respect to an observer with that of a single atom which is static with respect to the observer and immersed in a thermal bath, as long as the temperature of the thermal bath matches the Unruh temperature. This equivalence between the upward and downward transition rates breaks down in the presence of a cavity. For two-atom systems, considering the initial state to be in a general pure entangled form, we find that in this case the equivalence between the upward and downward transition rates of the accelerated and static thermal bath scenarios holds only under specific limiting conditions in free space, but breaks down completely in a cavity setup. Though the ratio of the upward and downward transition rates in the thermal bath matches exactly with those of the accelerated systems in free space as well as inside the cavity.",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "The physics of interaction of two-level atomic systems with quantum fields is endowed with rich and diverse properties [1-12]. Investigations into the radiative properties of a uniformly accelerated single atom [6-11] have been extended to scenarios involving multiple atoms interacting with the massless scalar field and the electromagnetic field [13-22]. A two-level atom serves as a detector, and coupled to an external field, can give rise to the phenomenon of the Fulling-Davies-Unruh (FDU) effect [23-26]. Emanating from the study of quantum field theory in curved background, the FDU effect reveals that a uniformly accelerated atom feels a thermal bath in the Minkowski vacuum, with the temperature T being related to the proper acceleration α as T = α/ 2 π . The FDU effect has a deep connection with black hole thermodynamics and the information loss paradox [25, 27-29]. It has been understood that the upward and downward transition rates of the accelerated detector, which in this case is the atom, is exactly the same as seen by a local inertial observer and by an observer who is coaccelerating with the detector. This theoretical equivalence between the transition rates has been found to hold for a single atom in free space as well as in the setting of a single reflecting boundary [10]. A similar equivalence also holds at the level of transition rates for an entangled atom in free space [30] provided a thermal bath at the FDU temperature exists in the coaccelerated frame. Moreover, it has been recently shown [31] that the equivalence between the upward and downward transition rates in the context of an uniformly accelerating Unruh-DeWitt detector and a coaccelerating detector immersed in a thermal bath holds completely in free space as well as inside a cavity, for both the single and entangled atomic systems. However, it has also been found [30] that for an entangled atom in free space this theoretical equivalence between the transition rates breaks down completely when the coaccelerated frame scenario is replaced with a static atom immersed in a background thermal bath with temperature equal to the Unruh temperature. The primary motivation of the present work is to establish further the status of equivalence stemming from the FDU effect. Though the equivalence dictated by the FDU effect is universally valid at the conceptual level, its revelation in real physical or experimental scenarios is contingent on the choice of set-ups or contexts. Context plays a very important role in manifestation of quantum features. Quantum contextuality is a well-studied subject with foundational implications (see, [32] for a recent review) as well as diverse applications [33], and entanglement is known to elucidate certain subtle features of contextuality. In this work we consider entangled atomic systems as a detector model to study whether the equivalence is manifested for such set-ups involving entanglement. Moreover, currently there exists a wide upsurge of interest in quantum entanglement in relativistic settings, intertwining profound concepts from quantum field theory, information theory and gravitational physics [34-51]. Initially uncorrelated Unruh-DeWitt detectors may get entangled by interacting locally with the vacuum state of a quantum field [52]. Localized detectors can extract non-local correlations from the a quantum field through the process of entanglement harvesting [53-59]. Relativistic quantum information explores how entanglement is affected by not only Lorentz boosts [60], but by non-inertial effects as well [61, 62]. In practical scenarios, the degradation of entanglement due to uncontrolled coupling with external fields is a genuine concern, and extensive research has been undertaken to investigate the transition rates between the states of entangled atoms moving in various trajectories, yielding a rich paradigm of possibilities [63-71]. Since configurations in analogue cavity QED such as superconducting circuits [72] and laser-driven technologies [73-75] can achieve substantial accelerations, such systems are beginning to be used towards experimental evidence of theoretical results in relativistic quantum information. Reflecting boundaries have been shown to play a significant role in such studies on relativistic quantum phenomena in superconducting circuits [72, 76], as well as in secure quantum communication over long distances [77-80]. Due to the apparently contradictory nature regarding the status of equivalence between the transition rates as manifested from certain findings of [10, 30, 31], it becomes pertinent to explore further the implications of replacing the coaccelerated frame scenario with a static frame immersed in a background thermal bath. The aim of our present study is a comprehensive investigation of the FDU phenomenon by computing the upward and downward transition rates in the context of a static Unruh-DeWitt detector immersed in a background thermal bath and comparing with those of an accelerating Unruh-DeWitt detector given in [31]. Our study pertains to single two-level atoms as well as an entangled two-atom systems either in free space, and confined in a cavity. For the case of two-atom systems, one of our main focus is on the role played by quantum entanglement on the transition rates in the presence of boundaries, which in turn, have a direct bearing on revelation of the FDU effect, as shown through our subsequent analysis. From a fundamental perspective, the impact of cavity setup on atom-field interactions and radiative processes of entangled atoms are manifold [81-89]. It was observed [17, 21, 90] that reflecting boundaries strongly influence the resonance interaction energy of uniformly accelerated two-atom system. In [91], it was found that reflecting boundaries induce effects which lead to the violation of equivalence in an accelerating atom-mirror system in the generalized uncertainty principle framework. Reflecting boundaries have several other interesting consequences in the context of quantum entanglement [84-86, 88], and quantum thermodynamics [92]. This line of inquiry presents a possibility for advancing our understanding about the FDU effect within a cavity quantum electrodynamics (QED) framework, that is important both from fundamental and practical points of view. The paper is organised as follows. In section 2, we analyze the transition rates for the cases when a single and two entangled static atomic systems interact with a massless scalar field in a background thermal bath in empty space and in the presence of a cavity, respectively. In section 3, we calculate a quantitative estimation of the upward transition rate of a single and two atom system inside a cavity by considering the Rubidium and Caesium atom. We conclude with a summary of our results in section 4. Throughout the paper, we take /planckover2pi1 = c = k B = 1 , where k B is the Boltzmann constant.",
"pages": [
2,
3,
4
]
},
{
"title": "2 Interaction of the static atomic system with a thermal bath",
"content": "In this section, we start by investigating the case when a static atomic system interacts with a massless scalar field in a thermal state at an arbitrary temperature T and undergoes transitions in between its lower and higher energy states.",
"pages": [
4
]
},
{
"title": "2.1 Single atom system",
"content": "In this analysis, we also consider a single atom (an Unruh-DeWitt detector) with two energy levels | g 〉 and | e 〉 with corresponding energy values -ω 0 / 2 and + ω 0 / 2 , remains static in a thermal state of a massless scalar field. Following the procedure given in the subsection (A.1), the transition probability from the initial state | i 〉 to the final state | f 〉 can be written as where ∆ E = E f -E i , m fi = 〈 f | m (0) | i 〉 and the response function F β (∆ E ) is defined as where is the positive frequency Wightman function of the massless scalar field in a thermal state at an arbitrary temperature T with H F = ∑ k ω k a † k a k [26]. Exploiting the time translational invariance property of the positive frequency Wightman function, the response function per unit proper time can be written as where ∆ τ = τ -τ ' . Therefore, the transition probability per unit proper time from the initial state | i 〉 to the final state | f 〉 turns out to be In the next subsections, we employ the above equations to study the transitions of a static single atom interacting with a thermal state of a massless scalar field in both empty space and a cavity.",
"pages": [
4,
5
]
},
{
"title": "2.1.1 Transition rates for single atom system in empty space",
"content": "We initially take into account the transition rates of a single atom that is interacting with a massless thermal scalar field in the empty space. In the laboratory frame, atomic trajectory is given by where τ denote the proper time of the atom. The thermal Wightman function is given by [30] (see Appendix B for a detailed calculation) Substituting (2.6) in (2.7), the thermal Wightman function turns out to be [26] Substituting the Wightman function into eq.(2.4) and eq.(2.5), the transition rate from the initial state | i 〉 to the final state | f 〉 becomes Simplifying the transition rates, eq.(2.9), by performing the contour integration [93] as shown in Appendix A of [31], we obtain where θ (∆ E ) is the Heaviside step function defined as Eq. (2.10) reveals that the two transition processes, namely, the upward and downward transition can take place even when the atom is static but placed inside a thermal bath. From the above equation, it may also be noted that the upward transition or excitation process is solely depends on the temperature of the thermal bath. Considering the initial state | i 〉 = | g 〉 , final state | f 〉 = | e 〉 and vice-versa and using the definition m eg = 〈 e | m (0) | g 〉 , we obtain | m ge | 2 = | m eg | 2 = 1 , and ∆ E = ω 0 for the transition g → e and ∆ E = -ω 0 for the transition e → g . Using the above results the upward and downward transition rate takes the form Taking the ratio of the above two results, we get Eq. (2.14) is the consequence of a universal relation called the principle of detailed balance [94]. From the above expressions two points can be noted. First, it is seen that the upward transition rate entirely depends on the temperature of the thermal state of the massless scalar field. At T = 0 , the upward transition rate vanishes. Secondly, if we take the thermal bath temperature in the static frame at T = α/ 2 π , then eqs.(2.12), (2.13), (A.6) and (A.7) clearly show that the upward and the downward transition rates of an uniformly accelerated atom seen by an instantaneously inertial observer (see Appendix A) and by a static observer in a thermal bath are identical. Further, from eq.(2.14) and eq.(A.8), it can be seen that the ratio in both cases also matches in the limit T = α/ 2 π . Therefore, for a single atom system in empty space the equivalence between the effect of uniform acceleration and the effect of thermal bath holds at the level of transition rates as well as their ratios.",
"pages": [
5,
6
]
},
{
"title": "2.1.2 Transition rates for single atom system in a cavity",
"content": "We now consider that the a static atom is interacting with a massless thermal scalar field inside a cavity of length L as shown in Figure 1. Assuming the scalar field obeys the Dirichlet boundary condition φ | z =0 = φ | z = L = 0 , the Wightman function of the thermal scalar field confined in the cavity of length L takes the form [95] (see Appendix C for a detailed calculation) Inside the cavity the atomic trajectory is given by Using the above trajectories in eq.(2.15), the Wightman function becomes with d 1 = 2 nL, d 2 = 2 z 0 -2 nL . Using the Wightman function, eq. (2.17) into eq.(2.4), the transition rate from the initial state | i 〉 to the final state | f 〉 is given by Simplifying the above equation by following the method of contour integral, rate of transition from the initial state | i 〉 to the final state | f 〉 can be written as where we have defined with p (∆ E, z 0 ) is given by Hence, from the above result the upward and downward transition rates can be written as From the above analysis it follows that the transitions observed by an instantaneously inertial observer and a static observer in a thermal bath for both the upward and the downward transition rates when the atom is confined in a cavity are clearly distinct. We also observe that taking the thermal bath temperature in the static frame T = α/ 2 π , eqs. (2.23), (2.24), (A.10) and (A.11) indicate that there is a non-equivalence between the transition rates of a uniformly accelerated atom seen by an instantaneously inertial observer and a static atom seen by a static observer in a thermal bath inside the cavity. Though the upward and the downward transition rates of a static atom in a thermal bath are not same as those corresponding to a uniformly accelerated atom, it can be seen from eqs. (2.23), (2.24), (A.10) and (A.11), that at T = α/ 2 π , the ratio of the upward and the downward transition rates of a static atom in a thermal bath is identical with that of a uniformly accelerated atom (eq.(A.15)). From the above analysis, it is also observed that the ratio of eqs. (2.23), (2.24) is identical with the free space result (eq.(2.14)). This is an universal feature independent of the detector model and follows from the principle of detailed balance [94]. In order to describe the single boundary and free space scenarios, we now derive the limiting cases of these expressions. Taking the limit L → ∞ , we find that in eq.(s)(2.23, 2.24) only n = 0 term survives from the infinite summation and one can effectively reduce the cavity scenario to a situation where only one reflecting boundary exists. Hence, using this limit, the upward and downward transition rates in the presence of a single reflecting boundary turn out to be On the other hand, taking the limits L →∞ and z 0 →∞ together, eq.(s)(2.23, 2.24) lead to the expression for the upward and downward transition rates in the free space given by eq.(s)(2.12, 2.13). We study the variation of the transition rate of a single two-level atom confined to a cavity, where the parameters are the atom's distance from the boundary ( z 0 ), the cavity's length ( L ), and the temperature of the thermal field ( T ). The atom's ground state energy level is | g 〉 , and its excited state energy level is | e 〉 . The findings are plotted below, where all physical quantities are expressed in dimensionless units. To fix the dimensionless parameters, we consider a single 87 Rb atom and take the atomic data from Ref. [96]. Recent time, it is observed that experimentally atomic excitations in nanoscale waveguides [97] can be achievable through some novel nanofabrication techniques [98, 99]. Following the Refs. [96, 98, 100], we choose ω 0 = 1 . 59 eV and L = 400 nm. The cavity effect becomes prominent when the all the parameters are comparable [101]. Due to this reason, we take T and z 0 in such a way so that ω 0 L , ω 0 z 0 , and T/ω 0 becomes comparable. Figure 2 shows the variation of the transition rate from | g 〉 → | e 〉 (per unit λ 2 ω 0 2 π ) with respect to the cavity length for a fixed value of the distance of the atom from one boundary and the temperature of the thermal field. From the figure, it can be seen that for a fixed value of the initial atomic distance z 0 from one boundary, the transition rate initially very low due to the cavity effect and get enhanced after a certain cavity length and get saturated for large values of L ( ω 0 L >> ω 0 z 0 ). This is to be expected as extending the cavity length results in an increased number of field modes participating in the interaction between the atom and the scalar field, which raises the transition rate. When ω 0 L >> ω 0 z 0 , the cavity scenario reduces to the case of a single boundary, and hence, the upward transition rate saturates for large L . Figure 3 shows the variation of the transition rate from | g 〉 → | e 〉 (per unit λ 2 ω 0 2 π ) with respect to the distance of the atom from one boundary for a fixed value of the length of the cavity and the temperature of the thermal field. From the figure, it is observed that for a fixed value of the cavity length L , when we increase the atomic distance from one boundary, the transition rate increases and at a certain value of z 0 it attains a maximum value and then it gets reduced by further increment of z 0 . The reason behind this is the following. Increasing the atomic distance from the boundary reduces the boundary effect on the number of field modes taking part in the interaction between the atom and the scalar field, which in turn increases the transition rate. This result is consistent with Figure 2. Figure 4 shows the variation of the transition rate from | g 〉 → | e 〉 (per unit λ 2 ω 0 2 π ) with respect to the temperature of the thermal bath for a fixed value of the cavity length and distance of the atom from one boundary. From the figure, it is observed that for a fixed value of the length of the cavity L and the atomic distance z 0 from one boundary, the transition rate increases when the temperature of the thermal bath is increased.",
"pages": [
6,
7,
8,
9,
10
]
},
{
"title": "2.2 Two-atom system",
"content": "In this subsection, we analyse the transition rates of a static two-atom system prepared in any generic pure entangled state | ψ 〉 that interacts with the massless scalar field in a thermal state at an arbitrary temperature T . Following similar procedures as given in subsection A.2, the transition rate of a static two-atom system takes the form where with ξ, ξ ' can be labeled by A or B , and is the positive frequency Wightman function of the scalar field in a thermal state at an arbitrary temperature T , and H F = ∑ k ω k a † k a k .",
"pages": [
10,
11
]
},
{
"title": "2.2.1 Transition rates for two-atom system in empty space",
"content": "We take into account the transition rates of a stationary two-atom system which interacts with the massless scalar field in a thermal state at an arbitrary temperature T in the empty space. In the laboratory frame the trajectories of both the atoms read Using the usual mode expansion of the scalar field operator in eq. (2.29), Wightman function becomes [30] Using eq.(s)(2.30, 2.31) the Wightman function turns out to be with ∆ τ = τ -τ ' for ξ = ξ ' , and for ξ = ξ ' . Substituting the Wightman functions, eq.(s) (2.32, 2.33) into eq.(A.18), the transition rate of the two-atom system from the initial state | ψ 〉 to the final state | E n 〉 can be rewritten as /negationslash with for ξ = ξ ' and for ξ = ξ ' . We may simplify the transition rates, eq.(2.34), by performing the above integrations using the contour integration technique /negationslash The above equation reveals that the two transition processes, namely, the downward and the upward transition can take place for the two-atom system with the upward transition rate and the downward transition rate From the above analysis it is observed that if we compare the transition rates, eq.(s)(2.38, 2.39), with those of the uniformly accelerated two-atom system as seen by an instantaneously inertial observer, eq.(s)(A.21, A.22), we find that the transition rates of the static two-atom system immersed in a thermal bath as seen by a static observer are in general distinct from those of the two-atom system uniformly accelerated in the Minkowski vacuum even when the temperature of the thermal bath is taken to be the FDU temperature [30]. Although, here we would like to point out an additional feature. It is observed that in the limit αd << 1 , expanding eqs.(A.21) and (A.22) and keeping terms upto O ( α 2 d 2 ) gives the results eqs.(A.23) and (A.24). Now from these equations, it is seen that the leading term of the transition rates of uniformly accelerated two-atom system seen by a inertial observer in free space (eqs.(A.23) and (A.24)) matches with those of the static two-atom system seen by a static observer in thermal bath in free space (eqs.(2.38) and (2.39)) if we take the temperature of the thermal bath equal to the FDU temperature. This observation implies that in empty space there is an approximate equivalence between the upward and downward transition rates of the scenarios when two static atoms are placed in a thermal bath and two atoms are accelerating uniformly with respect to an inertial observer.",
"pages": [
11,
12
]
},
{
"title": "2.2.2 Transition rates for two-atom system in a cavity",
"content": "Let us consider a static two-atom system interacting with a thermal state of a massless scalar field confined in a cavity of length L as shown in Figure 5. Assuming that the scalar field obeys the Dirichlet boundary condition φ | z =0 = φ | z = L = 0 , the thermal Wightman function inside the cavity takes the form [95] with | ∆ x ⊥ | 2 = √ ( x ξ ( τ ) -x ξ ' ( τ ' )) 2 +( y ξ ( τ ) -y ξ ' ( τ ' )) 2 . In case of two atoms inside the cavity the atomic trajectories take the form Using above trajectories in eq.(2.40), the Wightman function becomes for ξ = ξ ' , with d ' 1 = 2 nL, d ' 2 = 2 z ξ -2 nL and /negationslash for ξ = ξ ' , with d ' 3 = d +2 nL (for ξ = A,ξ ' = B ), d ' 3 = d -2 nL (for ξ = B,ξ ' = A ) and d ' 4 = 2 z 0 + d -2 nL . Using above Wightman functions, the rate of transition from the initial entangled state | ψ 〉 to the final separable state | E n 〉 can be written as with for ξ = ξ ' and for ξ = ξ ' . Eq.(2.44) can be further simplified by performing the above integrations using the contour integration procedure /negationslash where we have defined and q (∆ E, 2 L ) , r (∆ E, 2 z 0 , 2 L ) , p (∆ E, 2 z 0 ) are given in eq.(s)(2.20, 2.21, 2.22). Similar to the previous case, the above equation also suggests that two transition process can take place for the two-atom system in presence of a reflecting boundary with the upward transition rate and the downward transition rate From the above analysis, eqs (2.50), (2.51), (A.25) and (A.26) clearly display that transition rates of a uniformly accelerated two-atom seen by an instantaneously inertial observer and a static two-atom seen by a static observer in a thermal bath are non-identical inside the cavity even if we consider the temperature of the thermal bath to be the same as the FDU temperature. It may also be noted that the eq.(s)(2.50, 2.51) cannot be restored from the eq.(s)(A.25, A.26) even after taking the limit αd << 1 . Hence, this observation confirms that the equivalence between the transition rates no longer holds for the cases when a twoatom system uniformly accelerates and when a static two-atom system placed in a thermal bath. However, as the ratio of the eqs (2.50), (2.51), and (A.25), (A.26) matches exactly at T = α/ 2 π , so the effects of uniform acceleration and the effects of a thermal bath holds completely for the two-atom system interacts with the massless scalar field confined in a cavity. To obtain the single mirror and free space scenarios, we now take the limiting cases of these expressions. Taking the limit L → ∞ , we find that eq.(s)(2.50, 2.51) reduce to the expression for the upward and the downward transition rate in the presence of a single reflecting boundary Similarly, taking the limits L →∞ and z 0 →∞ , eq.(s)(2.50, 2.51) lead to the expressions for the upward and the downward transition rate in free space given by eq.(s)(2.38, 2.39). We now study how the transition rate of an entangled two atom system from an initial entangled state | ψ 〉 to a product state with higher energy value | e A e B 〉 confined to a cavity varies with parameters such as thermal field temperature ( T ), cavity length ( L ), atomic distance from one boundary ( z 0 ), interatomic distance ( d ), and the entanglement parameter ( θ ). Following the single atom case, here also we present our findings in the following plots. In the following plots, we fix the parameters T/ω 0 , ω 0 L, ω 0 z 0 , and ω 0 d in such a way so that cavity effects remain significant. In Figure 6, we show the behaviour of the transition rate with respect to the entanglement parameter for the cases where the atoms are in free space, in the vicinity of a single boundary and inside a cavity. From Figure 6, it can be seen that the transition rate | ψ 〉 → | e A e B 〉 (per unit λ 2 ω 0 2 π ) varies sinusoidally with the entanglement parameter θ . In free space, the transition rate increases (from the case corresponding to the zero initial entanglement product state) with increase in the entanglement parameter and it becomes maximum when the initial state is maximally entangled ( θ = π/ 4 super-radiant state) [102]. Further increment of the entanglement parameter decreases the transition rate and it vanishes at θ = 3 π/ 4 (maximally entangled sub-radiant state). In the vicinity of a single boundary, behaviour of the transition rate is quite similar to the free space scenario, with a slight shifting of the extremum points. Inside the cavity, the behaviour of the transition rate is also similar to the free space scenario. The peak value of transition rate inside the cavity is much smaller compared to the free space and single boundary cases. It can be noted that around θ = 3 π/ 4 , the values of the transition rate corresponding to cases of empty space, single boundary, and two boundaries, nearly vanish. At this point, it should be noted that increasing and decreasing the atom-boundary distance z 0 and the cavity length L the upward transition rate decreases for the value of θ = π/ 4 . In the thermal bath, we observe that the transition rate of the superradiant state, θ = π/ 4 of the two atoms inside the cavity does not vanish, in contrast to the result obtained for a co-accelerating frame in [31]. Here, the superradiance property of the state ( θ = π/ 4 ) remains intact. On the other hand, in the thermal bath, we observe that the transition rate of the sub-radiant state, θ = 3 π/ 4 of the two atoms inside the cavity vanishes. Therefore, entanglement of the initial state is preserved and from a quantum information theoretic viewpoint, this kind of an initial state may act as a good resource for performing various tasks. In Figure 7, we show the behaviour of the transition rate with respect to the entanglement parameter inside a cavity for a fixed value of T/ω 0 = 1 , ω 0 d = 0 . 6 , ω 0 L = 3 . 6 . From Figure 7, it can be seen that inside the cavity at θ = π/ 4 , the upward transition rate | ψ 〉 → | e A e B 〉 (per unit λ 2 ω 0 2 π ) increases when the distance between one atom and the nearest boundary z 0 increases. As the length of the cavity and the inter-atomic distance is fixed therefore after increasing the distance between the nearest atom and the boundary, boundary effect reduces on the number of field modes taking part in the interaction between the atom and the scalar field, which in turn increases the upward transition rates of the system. In Figure 8, we show the behaviour of the transition rate with respect to the entanglement parameter inside a cavity for a fixed value of T/ω 0 = 1 , ω 0 z 0 = 1 . 6 , ω 0 L = 3 . 6 . From Figure 8, it can be seen that inside the cavity at θ = π/ 4 , the upward transition rate | ψ 〉 → | e A e B 〉 (per unit λ 2 ω 0 2 π ) decreases when the inter-atomic distance d increases. As the length of the cavity and the distance of one atom from the nearest boundary is fixed therefore after increasing the inter-atomic distance the second atom moves toward the second boundary. Therefore, due to the boundary condition, number of field modes taking part in the inteaction between the atom and the scalar field reduces from the both end and as a result the transition rate of the system decreases. Figure 9 shows the variation of the transition rate from | ψ 〉 → | e A e B 〉 (per unit λ 2 ω 0 2 π ) with respect to the length of the cavity for a fixed value of the distance of any one atom from one boundary and the interatomic distance. From the figure, it can be seen that for a fixed value of the initial atomic distance z 0 of any one atom from the nearest boundary, the transition rate get enhanced when the cavity length increases and attains a maximum value for large values of L ( ω 0 L >> ω 0 z 0 ). This behaviour is similar to that of the single atom case, as mentioned earlier. As more number of field modes take part in the interaction between the scalar field and the atoms due to the increased cavity length, the transition rate increases. When ω 0 L >> ω 0 z 0 , the cavity scenario reduces to a single boundary set up and hence the upward transition rate becomes neary constant. Figure 10 shows the variation of the transition rate from | ψ 〉 → | e A e B 〉 (per unit λ 2 ω 0 2 π ) with respect to the distance of any one atom from one boundary for a fixed value of the cavity length and the interatomic distance. From the figure, it is observed that for a fixed value of the cavity length L and the interatomic distance d , when we increase the atomic distance from one boundary, the transition rate for the two atom system also increases and at a certain value of z 0 it attains a maximum value and then it gets reduced by further increment of z 0 . The reason behind this variation is also similar to the single atom case. Figure 11 shows the variation of the transition rate from | ψ 〉 → | e A e B 〉 (per unit λ 2 ω 0 2 π ) with respect to the interatomic distance for a fixed value of the cavity length and the distance of any one atom from one boundary. From the figure, it is observed that for a fixed value of the cavity length L and the distance of any one atom from one boundary z 0 , when we increase the interatomic distance, the transition rate for the two atom system decreases. Due to the increament of the interatomic distance the second atom moves toward the second boundary. Therefore, due the boundary effect the transition rate of the two atom system falls down. Figure 12 shows the variation of the transition rate from | ψ 〉 → | e A e B 〉 (per unit λ 2 ω 0 2 π ) with respect to the atomic acceleration for a fixed value of the cavity length. From the figure, it is observed that for a fixed value of the length of the cavity L , the atomic distance z 0 from one boundary, and the interatomic distance d , it is observed that the transition rate increases when the temperature of the thermal bath is increased.",
"pages": [
13,
14,
15,
16,
17,
18,
19,
20
]
},
{
"title": "3 Quantitative estimation of the transition rate",
"content": "In this section, we now present a quantitative estimation of the upward transition rate of a single and two atom system inside a cavity by considering the Rubidium and Caesium atom. Following some recent works Ref. [97-101, 103], it has been seen that 87 Rb and 133 Cs widely used in the experimental setup.",
"pages": [
20
]
},
{
"title": "3.1 Transition rate of 87 Rb",
"content": "From the Ref. [96, 100], we have taken the transition energy and the cavity length for calculating the upward transition rate of 87 Rb . The values are /planckover2pi1 ω 0 = 1 . 59 eV and L = 400 nm respectively. Setting thermal bath temperature T = 20 , 000 K and the distance between the atom and the nearest boundary z 0 = 100 nm. Fixing all the physical quantity and calculating the dimensionless parameters, we get Taking the coupling constant λ = 0 . 1 and using above values of the parameters, the upward transition rate for a single 87 Rb atom becomes In case of two 87 Rb atoms, we additionally choose the inter-atomic distance d = 50 nm and the entanglement parameter θ = π/ 4 . Therefore, ω 0 d = 0 . 4 . Taking λ = 0 . 1 , the upward transition rate for two 87 Rb atoms becomes",
"pages": [
20
]
},
{
"title": "3.2 Transition rate of 133 Cs",
"content": "From the Ref. [100, 104], we have taken the transition energy and the cavity length for calculating the upward transition rate of 133 Cs . The values are /planckover2pi1 ω 0 = 1 . 46 eV and L = 500 nm respectively. Setting thermal bath temperature T = 20 , 000 K and the distance between the atom and the nearest boundary z 0 = 150 nm. Fixing all the physical quantity and calculating the dimensionless parameters, we get Taking the coupling constant λ = 0 . 1 and using above values of the parameters, the upward transition rate for a single 133 Cs atom becomes In case of two 133 Cs atoms, we additionally choose the inter-atomic distance d = 75 nm and the entanglement parameter θ = π/ 4 . Therefore, ω 0 d = 0 . 4 . Taking λ = 0 . 1 , the upward transition rate for two 133 Cs atoms becomes",
"pages": [
20,
21
]
},
{
"title": "4 Conclusions",
"content": "In this work we take a fresh look at the FDU effect in the context of two-level single and entangled atomic systems that are static in a thermal bath. We investigated the question as to whether the FDU effect shows up or not for single atom and entangled two-atom detector systems in free-space and inside cavity setups inside a thermal bath. In order to investigate the actual equivalence in real physical systems, the first step is to identify choices of specific detectors in context of which one may hope to observe such equivalence. Phenomena such as the excitation rate of a detector are directly observable, and are therefore of substantial significance in empirically verifying deep physical concepts, such as the FDU effect. Our study takes into consideration both single and entangled two-atom systems which can be located either in free space or inside cavities with reflecting boundary conditions. The two-atom system is taken to be initially prepared in a most general pure entangled state. The interactions of both systems with a massless scalar field in its vacuum state are studied in order to investigate the equivalence between the thermal bath and the uniform acceleration for the cases of the single and two-atom systems. For doing so, we have calculated the upward and downward transition rates of the single and entangled atomic systems immersed in a thermal bath in free space and inside cavities. We have compared these results with those of the results of accelerated single and entangled atomic systems in free space and inside a cavity [31] given in Appendix A. From the results, it is observed that while both the upward and downward transitions occur for the single atom as well as the entangled two-atom system, the upward transition is significant as it solely arises due to the temperature of the thermal bath. The actual transition rate depends on the cavity length, the distance of an atom from one boundary, the temperature of the thermal bath, and the magnitude of initial atomic entanglement. From our analysis it is evident that the transition rate shows oscillatory behaviour with the entanglement parameter. Considering a small magnitude of initial entanglement, we find that increasing the entanglement parameter enhances the upward transition and downward transition rates in free space, whereas, both the transition rates get suppressed due to the decreased number of field modes participating in the interaction between the atom and the scalar field in the presence of the cavity. In the case when the initial entanglement parameter has the value θ = 3 π/ 4 , we observe that both the transition rates vanish, indicating that no transition occurs from the maximally entangled initial state to any higher or lower energetic product state. Hence, the entanglement of the sub-radiant initial state can be preserved, a result which may be of significance, since preservation of entanglement enables its use as a resource for performing various quantum information processing tasks. Our investigation of the upward and downward transition rates enables us to conclude that in the case of single as well as two-atom systems, the physical state of the system and the observer's reference frame both significantly influence the upward and downward transition rates of the systems. Whether the equivalence inherent in the FDU effect is manifested or not depends upon the interplay of the following physical conditions. In free space, the upward and downward transition rates of a static atom placed in a thermal bath match with those of an accelerated atom, if the temperature of the thermal bath is equal to the Unruh temperature T = α/ 2 π . Such kind of equivalence also holds at the level of transition rates when a single atom is placed in a coaccelerated frame in a thermal bath [31]. The picture for two atoms is more intricate where atomic entanglement plays the key role. In an earlier work, it is observed that the upward and downward transition rates of two entangled atoms accelerating through a cavity turns out to be similar to the transition rates of a coaccelerated observer in a thermal bath [31]. However, in this study it is seen that the upward and downward transition rates of a uniformly accelerated two-atom system with the transition rates when the system is static but immersed in a thermal bath, such equivalence between the transition rates holds only under specific limiting conditions in free space, but breaks down completely inside a cavity setup. However, it turns out that the ratio of the upward and downward transition rates inside the thermal bath in free space as well as inside a cavity match exactly with those of the accelerated system in free space and inside a cavity. Apart from the computing of the transition rate in different scenario for single and two level system, in this study we provide a quantitative estimation of the transition rate inside the cavity setup. Following some recent experimental setup [97-101, 103], we fix the physical parameters for the single and two Rubidium ( 87 Rb ) and Cesium ( 133 Cs ) atoms respectively. From this estimation, it can be seen that within the cavity of order nanometer the transition rates of this atoms are very high. Our present analysis serves to comprehensively reestablish that in general, there is no equivalence at the level of transition rates between the accelerating and the static thermal bath frame for the chosen single atom model inside a cavity and the entangled two-atom detector model as realized in both free-space and cavity setups. However, at the level of the ratio of the transition rates, there is a complete equivalence between the accelerating and the static thermal bath frame for both the single and the entangled two-atom detector model as realized in both free-space and cavity setups. Therefore, at this point we should emphasize though, that our present analysis does not impact in any way the conceptual validity of the FDU effect. Rather, its revelation through the manifestation of equivalence, is model dependent [94], as evident through comparison of earlier results involving co-accelerating observers [30, 31]. Finally, it may be noted that in our present study we have considered only the effect of varying magnitudes of initial atomic entanglement on the transition rates. The focus here is not to evaluate any effects of generation or degradation of entanglement that may arise from the dynamics. However, other phenomena such as generation or degradation of atomic entanglement, as well as atom-field entanglement may have interesting implications [105-109]. Evaluation of such effects could be undertaken through the master-equation approach [110-112] in future works.",
"pages": [
21,
22
]
},
{
"title": "Acknowledgement",
"content": "AM and ASM acknowledges support from project no. DST/ICPS/QuEST/2019/Q79 of the Department of Science and Technology (DST), Government of India.",
"pages": [
23
]
},
{
"title": "Appendix A Interaction of the accelerated atomic system with a massless scalar field",
"content": "In this Appendix, we mainly review some key results of [31] which we use for the sake of comparison with the subsequent results of our present work.",
"pages": [
23
]
},
{
"title": "A.1 Single atom system",
"content": "Let us consider a single atom (an Unruh-DeWitt detector) with two energy levels | g 〉 and | e 〉 with corresponding energy values -ω 0 / 2 and + ω 0 / 2 , travelling along a stationary trajectory in a vacuum with massless scalar field fluctuations. In the laboratory frame, trajectories of the atom can be represented through x ( τ ) ≡ ( t ( τ ) , x ( τ )) . In the instantaneous inertial frame, the Hamiltonian describing the atom-field interaction in the interaction picture is given by [37] where λ is a small coupling constant, m ( τ ) = e i H 0 τ m (0) e -i H 0 τ is the monopole operator at any proper time τ of a single atom, φ ( x ( τ )) is the massless quantum scalar field evaluated at the trajectory x ( τ ) with m (0) = | g 〉〈 e | + | e 〉〈 g | being the initial monopole operator and H 0 | e 〉 = ω 2 | e 〉 being the free Hamiltonian of a single atom respectively [92]. Using the formalism discussed in [26, 31], the rate of transition probability from the initial atomic state | i 〉 to the final atomic state | f 〉 turns out to be where ∆ E = E f -E i , m fi = 〈 f | m (0) | i 〉 , and the response function per unit proper time can be written as where ∆ τ = τ -τ ' and is the positive frequency Wightman function of the massless scalar field [26]. In empty space, using the atomic trajectory in the laboratory frame t ( τ ) = 1 α sinh( ατ ) , x ( τ ) = 1 α cosh( ατ ) , y = z = 0 with α being the proper acceleration and τ being the proper time of the atom, the Wightman function becomes Substituting the above Wightman function into eq.(A.2), the upward and downward transition rate becomes (See [31] for a detailed calculation) From the above equations, it is clearly seen that in case of a single atom system the upward transition rate solely depends on the atomic acceleration. Taking the ratio of the above two results, we get Now, inside a cavity of length L as shown in Figure 13, with the atomic trajectory being t ( τ ) = 1 α sinh( ατ ) , x ( τ ) = 1 α cosh( ατ ) , y = 0 , and z = z 0 , the Wightman function is given by [31] with d 1 = nL, d 2 = 2 z 0 -nL . Now using the process outlined in Appendix B of [31], the upward and downward transition rate inside the cavity turn out to be where we have defined with g (∆ E, α, z 0 ) defined as Taking the ratio of the eqs.(A.10) and (A.11), in cavity scenario, we also get",
"pages": [
23,
24,
25
]
},
{
"title": "A.2 Two-atom system",
"content": "In this subsection, considering two identical atoms A and B , we assume that they are travelling synchronously along stationary trajectories in the vacuum of a massless scalar field. The interatomic distance is assumed to be constant and the proper times of the two atoms can be described by the same time τ [89]. In the laboratory frame, trajectories of the two atoms can be represented through x A ( τ ) and x B ( τ ) . Here we consider each atom as a two level system having energy levels | g 〉 and | e 〉 with corresponding energy values -ω 0 / 2 and + ω 0 / 2 . Therefore, the entire two-atom system can be described by the eigenstates of the initial atomic Hamiltonian , namely, | g A , g B 〉 , | g A , e B 〉 , | e A , g B 〉 , and | e A , e B 〉 with the corresponding energy eigenvalues -ω 0 , 0 , 0 , and ω 0 respectively. Here, as the eigenstates | g A , e B 〉 and | e A , g B 〉 are the degenerate eigenstates of the Hamiltonian with the energy value zero, therefore any linear combination of this eigenstates will also be an eigenstate of this system with same energy eigenvalue [67]. Hence, the most general quantum state of the two-atom system with zero energy value is given by [91] where the entanglement parameter θ lies in the range 0 ≤ θ ≤ π . It may be noted that though the Bell-state basis is frequently used when working with entangled states, for our present purpose use of the Bell basis is inconvenient, since the Bell states are not eigenstates of the atomic Hamiltonian. In the instantaneously inertial frame, the Hamiltonian describing the atom-field interaction is given by where λ is a small atom-field coupling constant. As a result of the atom-field interaction, the transition probability rate of the two-atom system from the initial state | χ 〉 to the final state | χ ' 〉 turns out to be where m ( A ) χ ' χ = 〈 χ ' | m (0) ⊗ 1 B | χ 〉 , m ( B ) χ ' χ = 〈 χ ' | 1 A ⊗ m (0) | χ 〉 . The response function per unit proper time can be written as where ∆ τ = τ -τ ' , ξ, ξ ' can be labeled by A or B , and is the Wightman function of the massless scalar field. In empty space, using the trajectories of both the atoms in the laboratory frame t A ( τ ) = t B ( τ ) = 1 α sinh( ατ ) , x A ( τ ) = x B ( τ ) = 1 α cosh( ατ ) , y B = y A + d , and z A = z B = 0 , with d being the constant interatomic distance, α being the proper acceleration and τ being the proper time of the two-atom system, the transition rates of the two-atom system from the initial entangled state | ψ 〉 to the final product states | e A , e B 〉 and | g A , g B 〉 can be expressed as [31] In the limit αd << 1 , above equations upto O ( α 2 d 2 ) take the form Inside a cavity of length L as shown in Figure 14, using the atomic trajectories t A/B ( τ ) = 1 α sinh( ατ ) , x A/B ( τ ) = 1 α cosh( ατ ) , y A/B = y 0 , and z A = z 0 , z B = z 0 + d , the transition rate of the two-atom system from the initial entangled state | ψ 〉 to the final product state | e A , e B 〉 and | g A , g B 〉 inside the cavity can be expressed as [31] where we have defined with f ( ∆ E, α, L 2 ) , h ( ∆ E, α, z 0 , L 2 ) , g (∆ E, α, z 0 ) are given in eq.(s)(A.12, A.13, A.14).",
"pages": [
25,
26,
27
]
},
{
"title": "Appendix B Derivation of the thermal Wightman function",
"content": "In this Appendix, for the sake of completeness, we provide a complete derivation of the thermal Wightman function of the massless scalar field eq.(2.7). Thermal Wightman function is defined as where H F = ∑ k ω k a † k a k . Therefore, Using the mode expansion of the massless scalar field [113] 〈 σ | φ ( x ( τ )) φ ( x ( τ ' )) | σ 〉 becomes Now using the relation between a k and a † k Using the above commutation relation eq. (B.4) becomes For the massless scalar field ω k = | k | ≡ k and defining t -t ' ≡ ∆ t and x -x ' ≡ ∆ x and putting these values in eq.(B.2), we get",
"pages": [
27,
28,
29
]
},
{
"title": "Appendix C Derivation of the thermal Wightman function inside a cavity",
"content": "In this Appendix, we provide a complete derivation of the thermal Wightman function of the massless scalar field inside a cavity eq.(2.15). Using the definition of the thermal Wightman function given in eq.(B.1), we get In the presence of a single reflecting boundary, the mode function of the massless scalar field operator obeys Dirichlet boundary condition φ | z = z 0 = 0 and takes the form Now, using this mode function and computing the term 〈 σ | φ ( x ( τ )) φ ( x ( τ ' )) | σ 〉 we get Inside the cavity of length L , the mode function of the massless scalar field operator obeys Dirichlet boundary condition φ | z = z 0 = φ | z = L 0 . Using the mode function given in Ref.[114] and computing the term 〈 σ | φ ( x ( τ )) φ ( x ( τ ' )) | σ 〉 we get Therefore, using eq.(C.4) into the last line of eq.(C.1) and computing the summation over σ , thermal Wightman function takes the form Now, after some algebraic manipulation, and solving the angular integrals, we finally get where we consider ε is very small, | ∆ x 2 | = √ (∆ x ) 2 +(∆ y ) 2 +(∆ z -2 nL ) 2 and | ∆ x 1 | = √ (∆ x ) 2 +(∆ y ) 2 +( z + z ' -2 nL ) 2 . This equation can be recast as This is the form of the thermal Wightman function inside the cavity which is used in eq.(2.15).",
"pages": [
29,
30
]
}
]
2024arXiv240205105V
https://arxiv.org/pdf/2402.05105.pdf
<document>
<section_header_level_1><location><page_1><loc_15><loc_79><loc_85><loc_85></location>On gauge transformations in twistless torsional Newton-Cartan geometry</section_header_level_1>
<text><location><page_1><loc_20><loc_75><loc_79><loc_77></location>Arian L. von Blanckenburg 1 ,a and Philip K. Schwartz 1 ,b</text>
<text><location><page_1><loc_28><loc_68><loc_72><loc_74></location>1 Institute for Theoretical Physics, Leibniz University Hannover, Appelstraße 2 , 30167 Hannover, Germany a a.von.blanckenburg@stud.uni-hannover.de b philip.schwartz@itp.uni-hannover.de</text>
<text><location><page_1><loc_20><loc_43><loc_81><loc_62></location>We observe that in type I twistless torsional Newton-Cartan (TTNC) geometry, one can always find (at least locally) a gauge transformation that transforms a specific locally Galilei-invariant function-that we dub the 'locally Galilei-invariant potential'-to zero, due to the corresponding equation for the gauge parameter taking the form of a Hamilton-Jacobi equation. In the case of type II TTNC geometry, the same gauge fixing may locally be performed by subleading spatial diffeomorphisms. We show (a) how this generalises a classical result in standard Newton-Cartan geometry, and (b) how it allows to parametrise the metric structure of a Galilei manifold as well as the gauge equivalence class of the Bargmann form of TTNC geometry in terms of just the space metric and a unit timelike vector field.</text>
<section_header_level_1><location><page_1><loc_15><loc_39><loc_31><loc_41></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_15><loc_26><loc_85><loc_37></location>Newton-Cartan gravity [ 1 -9 ], [ 10 , chapter 4 ], [ 11 ] is a differential-geometric reformulation of Newtonian gravity. Similar to General Relativity (GR), it describes gravity by a curved connection on the manifold of spacetime events. However, differently to the GR case, in Newton-Cartan gravity spacetime is endowed not with a Lorentzian metric (with which the connection is compatible), but with a specific notion of Galileirelativistic metric structure, ensuring local Galilei invariance of physical laws. If the connection is torsion-free, there is necessarily a notion of absolute time.</text>
<text><location><page_1><loc_15><loc_19><loc_85><loc_25></location>One can weaken the notion of absolute time, only demanding the existence of a concept of absolute simultaneity. The resulting geometry, in which compatible connections necessarily have non-vanishing torsion, is called twistless torsional NewtonCartan (TTNC) geometry [ 12 -15 ]. Already in standard Newton-Cartan geometry one can</text>
<text><location><page_2><loc_15><loc_77><loc_85><loc_90></location>introduce, in addition to the metric structure, an additional 'auxiliary' field related to the Bargmann algebra, the central extension of the Galilei algebra [ 16 -20 ]. In TTNC geometry, this field features more heavily. In the literature, it goes under several different names; we will call it a 'Bargmann form' (for more on this naming, see footnote 1 in section 2 ). There is a gauge freedom for the Bargmann form: certain transformations of it are to be considered gauge transformations, relating different equivalent descriptions of the same geometric situation. Depending on which transformations are considered gauge, one distinguishes between so-called type I and type II TTNC geometry [ 21 , 22 , 11 ].</text>
<text><location><page_2><loc_15><loc_56><loc_85><loc_76></location>The Bargmann form can be used to define a locally Galilei-invariant vector field-that is, a vector field invariantly determined by the metric structure and the Bargmann form. In the study of this vector field, a certain scalar function arises naturally, which we call the 'locally Galilei-invariant potential'. We will show in this paper that in both the type I and type II cases locally there exists a gauge transformation such that after performing this transformation, the locally Galilei-invariant potential vanishes. This (partial) gauge fixing is interesting for two reasons: on the one hand, we will show that it allows for a (local) parametrisation of TTNC geometries up to gauge by just the space metric and a unit timelike vector field. On the other hand, it also offers a natural generalisation to TTNC geometry of a classical result in standard Newton-Cartan geometry, namely the local existence of twist-free geodesic unit timelike vector fields [ 5 , thm. 3 . 6 ], [ 10 , prop. 4 . 3 . 7 ].</text>
<text><location><page_2><loc_15><loc_53><loc_85><loc_56></location>Throughout this paper we use the notation and conventions from [ 20 ]. In particular, the local representative of the Bargmann form is called a and not m as in other literature.</text>
<text><location><page_2><loc_15><loc_41><loc_85><loc_52></location>The structure of this paper is as follows. In section 2 we give a brief review of TTNC geometry and introduce the locally Galilei-invariant potential. We show how local gauge transformations can be used to 'gauge away' the locally Galilei-invariant potential in both type I and type II TTNC geometry in section 3 . In section 4 we apply this to parametrisation of TTNC geometries as mentioned above, and explain how it generalises the classical result. We conclude and discuss further research directions in section 5 .</text>
<section_header_level_1><location><page_2><loc_15><loc_37><loc_64><loc_38></location>2 Twistless torsional Newton-Cartan geometry</section_header_level_1>
<text><location><page_2><loc_15><loc_27><loc_85><loc_35></location>In this section, we briefly review TTNC geometry in a style similar to that of [ 7 -9 ], [ 10 , chapter 4 ], [ 20 ], i.e. from what might be called a relativist's point of view, and introduce the locally Galilei-invariant potential. For a somewhat more 'field-theoretic' perspective on the subject, we refer to the original literature on type I [ 12 -14 ] and type II TTNC geometry [ 15 , 21 , 22 ], and in particular the review [ 11 ].</text>
<text><location><page_2><loc_15><loc_20><loc_85><loc_26></location>The basic geometric structure in Newton-Cartan geometry is that of a Galilei manifold , i.e. a differentiable manifold M of dimension n + 1 ( n ≥ 1) with a nowhere-vanishing clock form τ ∈ Ω 1 ( M ) and a symmetric contravariant 2 -tensor field h , the space metric , which is positive semidefinite of rank n , such that the degenerate direction of h is</text>
<text><location><page_3><loc_15><loc_88><loc_61><loc_90></location>spanned by τ . The latter condition may be expressed as</text>
<formula><location><page_3><loc_46><loc_85><loc_85><loc_87></location>τµ h µν = 0 . ( 2 . 1 )</formula>
<text><location><page_3><loc_15><loc_67><loc_85><loc_84></location>Vectors in the kernel of τ are called spacelike , other vectors are called timelike . The integral of τ along any worldline (i.e. curve) in M is interpreted as the time elapsed along this worldline. If d τ = 0, the time between two events (i.e. points in M ) is (locally) independent of the connecting worldline chosen to measure it; one then speaks of a Galilei manifold with absolute time . If only the weaker condition τ ∧ d τ = 0 holds, such that by Frobenius' theorem the distribution ker τ of spacelike vectors is integrable (i.e. we have an absolute notion of simultaneity), then one says to be in the situation of twistless torsional Newton-Cartan (TTNC) geometry. In any case, h induces a positive definite bundle metric on the spacelike vectors. In the following, unless otherwise stated we will always assume the 'twistless torsion' condition τ ∧ d τ = 0.</text>
<text><location><page_3><loc_15><loc_64><loc_85><loc_67></location>A (local) Galilei frame on a Galilei manifold is a local frame ( v , e a ) of vector fields satisfying</text>
<formula><location><page_3><loc_40><loc_62><loc_85><loc_64></location>τ ( v ) = 1, h µν = δ ab e µ a e ν b . ( 2 . 2 )</formula>
<text><location><page_3><loc_15><loc_48><loc_85><loc_61></location>It follows that τ ( e a ) = 0, such that the dual frame of one-forms takes the form ( τ , e a ) . Changes from one choice of Galilei frame to another are called local Galilei transformations ; they are precisely given by basis change matrix functions with values in the (orthochronous) homogeneous Galilei group Gal = O ( n ) ⋉ R n ⊂ GL ( n + 1 ) . Therefore, Galilei frames may be understood as local sections of the Galilei frame bundle G ( M ) of ( M , τ , h ) , a principal Gal -bundle which is a reduction of the structure group of the general linear frame bundle of M . Local Galilei boosts , also called Milne boosts , are changes of the unit timelike frame vector field v alone, which are of the form</text>
<text><location><page_3><loc_44><loc_45><loc_44><loc_46></location>↦</text>
<formula><location><page_3><loc_43><loc_45><loc_85><loc_46></location>v → v ' = v -k a e a ( 2 . 3 )</formula>
<text><location><page_3><loc_15><loc_40><loc_85><loc_43></location>for an R n -valued boost velocity function k . The dual frame ( τ , e a ) then transforms according to</text>
<text><location><page_3><loc_46><loc_39><loc_46><loc_40></location>↦</text>
<formula><location><page_3><loc_41><loc_38><loc_85><loc_40></location>( τ , e a ) → ( τ , e a + k a τ ) . ( 2 . 4 )</formula>
<text><location><page_3><loc_15><loc_34><loc_85><loc_37></location>Note that we will not discuss connections compatible with Galilei structures in this section, but only later in section 4 . 2 .</text>
<text><location><page_3><loc_15><loc_26><loc_85><loc_34></location>There is an additional ingredient that is commonly viewed as part of the geometric structures defining a Newton-Cartan geometry, particularly in the TTNC case [ 16 -20 ]. Locally, it is given by the specification of a one-form a for each choice of local Galilei frame in such a way that it is invariant under spatial rotations of the frame and under local Galilei boosts ( 2 . 3 ) transforms according to</text>
<text><location><page_3><loc_41><loc_22><loc_41><loc_24></location>↦</text>
<formula><location><page_3><loc_40><loc_21><loc_85><loc_24></location>a → a + δ ab k a e b + 1 2 | k | 2 τ . ( 2 . 5 )</formula>
<text><location><page_4><loc_15><loc_87><loc_85><loc_90></location>Globally, it may be understood as follows [ 16 , 17 , 20 ]. We consider the representation ˙ ρ : Gal → GL ( R n + 1 ⊕ u ( 1 )) given by</text>
<formula><location><page_4><loc_25><loc_83><loc_85><loc_85></location>˙ ρ ( R , k ) ( y t , y a , i φ ) = ( y t , R a b y b + y t k a , i ( φ + 1 2 | k | 2 y t + kaR a b y b ) ) . ( 2 . 6 )</formula>
<text><location><page_4><loc_20><loc_70><loc_20><loc_71></location≯</text>
<text><location><page_4><loc_15><loc_66><loc_85><loc_81></location>On the Galilei frame bundle G ( M ) , we have a tensorial R n + 1 -valued one-form θ corresponding to the canonical solder form of G ( M ) × Gal R n + 1 . The global object locally represented by the form a from above is then a one-form a on G ( M ) that together with this θ yields a ˙ ρ -tensorial form ( θ , i a ) . The local form a is then the pullback of a along the local Galilei frame ( v , e a ) understood as a section of G ( M ) . The representation ˙ ρ arises in the study of the Bargmann group, whose Lie algebra is (for n = 2) the essentially unique non-trivial 1 -dimensional central extension of the inhomogeneous Galilei algebra. For that reason, we will call a field a as described here a Bargmann form for ( M , τ , h ) . 1</text>
<text><location><page_4><loc_15><loc_63><loc_85><loc_66></location>Using a Bargmann form a for ( M , τ , h ) , one can construct a unit timelike vector field ˆ v ∈ Γ ( TM ) that is locally Galilei-invariant , according to</text>
<formula><location><page_4><loc_43><loc_60><loc_85><loc_61></location>ˆ v µ : = v µ + h µν a ν , ( 2 . 7 )</formula>
<text><location><page_4><loc_15><loc_52><loc_85><loc_58></location>where a is the local representative of a with respect to v . 2 Invariance under spatial frame rotations is obvious; invariance under local Galilei boosts follows by direct computation. Note that ˆ v being locally Galilei-invariant means that it really is determined just by τ , h , and a .</text>
<text><location><page_4><loc_15><loc_47><loc_85><loc_52></location>Understanding the transformation from a general unit timelike vector field v to ˆ v as a local Galilei boost, we may compute the corresponding representative of the Bargmann form a with respect to ˆ v to be given by</text>
<formula><location><page_4><loc_40><loc_42><loc_85><loc_46></location>ˆ a = ( a ( v ) + 1 2 h ( a , a ) ) τ . ( 2 . 8 )</formula>
<text><location><page_4><loc_15><loc_36><loc_85><loc_41></location>By construction, we know that the right-hand side of ( 2 . 8 ) is locally Galilei-invariant; this may, however, also be checked by direct calculation. Note that ˆ a is proportional to τ , which may even be used to define ˆ v : one easily checks that, for τ , h , a given, ˆ v is the</text>
<text><location><page_5><loc_15><loc_87><loc_85><loc_90></location>unique unit timelike vector field for which the local representative of a is proportional to τ . The proportionality factor</text>
<formula><location><page_5><loc_42><loc_82><loc_85><loc_86></location>ˆ ϕ : = a ( v ) + 1 2 h ( a , a ) ( 2 . 9 )</formula>
<text><location><page_5><loc_15><loc_77><loc_85><loc_82></location>we call the locally Galilei-invariant potential . (Some motivation for why we choose this name will be given in footnote 7 in section 4 .) Note again that ˆ ϕ being locally Galilei-invariant means that it is fully determined by τ , h , and a .</text>
<text><location><page_5><loc_15><loc_57><loc_85><loc_77></location>In Newton-Cartan geometry, there are several different kinds of 'gauge transformations', i.e. transformations relating mathematical situations which are to be considered equivalent descriptions of 'the same' geometric situation. First, due to the differentialgeometric nature of the theory, there are diffeomorphisms: acting on all the involved objects with pushforward 3 by a diffeomorphism φ : M → N leads to a different mathematical description of the same geometric situation. Second, there are local Galilei transformations: these are just changes in the (arbitrary) choice of local Galilei frame ( v , e a ) , giving local descriptions of global objects living on G ( M ) ; all the true global geometric objects such as τ , h , a stay invariant. When we use the name gauge transformation in the following, we will always mean the third kind: certain transformations of the Bargmann form a . These come in two different flavours, dubbed type I and type II , distinguishing two different kinds of TTNC geometry.</text>
<text><location><page_5><loc_15><loc_52><loc_85><loc_56></location>In type I TTNC geometry, the Bargmann form is interpreted as arising from a 'gauging' of the central direction of the Bargmann algebra [ 18 , 19 ]. 4 The corresponding U ( 1 ) gauge transformations locally take the form</text>
<text><location><page_5><loc_47><loc_49><loc_47><loc_50></location>↦</text>
<formula><location><page_5><loc_45><loc_49><loc_85><loc_50></location>a → a + d χ ( 2 . 10 )</formula>
<text><location><page_5><loc_15><loc_45><loc_85><loc_48></location>for functions χ ∈ C ∞ ( M ) . Type I TTNC geometry naturally arises by null reduction of Lorentzian geometry [ 23 , 24 ].</text>
<text><location><page_5><loc_15><loc_38><loc_85><loc_44></location>Type II TTNC geometry arises instead from a formal expansion of Lorentzian geometry in c -1 , where c is the velocity of light [ 15 , 21 , 22 , 11 ]. 5 Here, a is the next-to-leadingorder part of the timelike Lorentzian coframe field. The gauge transformations it inherits from subleading c -2 -dependent diffeomorphisms take the form</text>
<text><location><page_5><loc_36><loc_35><loc_36><loc_37></location>↦</text>
<formula><location><page_5><loc_35><loc_35><loc_85><loc_37></location>a → a -L ζ τ = a -d τ ( ζ , · ) -d ( τ ( ζ )) ( 2 . 11 )</formula>
<text><location><page_6><loc_15><loc_85><loc_85><loc_90></location>for vector fields ζ ∈ Γ ( TM ) . Note that in the case of absolute time (d τ = 0), type I and type II gauge transformations coincide. Note also that the locally Galilei-invariant vector field ˆ v ( 2 . 7 ) is not gauge-invariant (in both the type I and type II cases).</text>
<section_header_level_1><location><page_6><loc_15><loc_81><loc_83><loc_82></location>3 Gauge transformations of the locally Galilei-invariant potential</section_header_level_1>
<text><location><page_6><loc_15><loc_74><loc_85><loc_79></location>In this section, we are going to argue that in both type I and type II TTNC geometry, there is always a local gauge transformation that transforms the locally Galilei-invariant potential ( 2 . 9 ) to zero.</text>
<section_header_level_1><location><page_6><loc_15><loc_71><loc_37><loc_72></location>3.1 Type I TTNC geometry</section_header_level_1>
<text><location><page_6><loc_15><loc_66><loc_85><loc_69></location>Under a type I TTNC gauge transformation ( 2 . 10 ) of a , the locally Galilei-invariant potential ( 2 . 9 ) transforms according to</text>
<text><location><page_6><loc_36><loc_63><loc_36><loc_64></location>↦</text>
<formula><location><page_6><loc_34><loc_56><loc_85><loc_65></location>ˆ ϕ → ( a + d χ )( v ) + 1 2 h ( a + d χ , a + d χ ) = ˆ ϕ + d χ ( v ) + h ( d χ , a ) + 1 2 h ( d χ , d χ ) = ˆ ϕ + d χ ( ˆ v ) + 1 2 h ( d χ , d χ ) . ( 3 . 1 )</formula>
<text><location><page_6><loc_15><loc_52><loc_85><loc_55></location>Therefore, we may transform ˆ ϕ to zero by performing a gauge transformation with the gauge parameter χ solving the PDE</text>
<formula><location><page_6><loc_38><loc_48><loc_85><loc_51></location>0 = ˆ ϕ + d χ ( ˆ v ) + 1 2 h ( d χ , d χ ) . ( 3 . 2 )</formula>
<text><location><page_6><loc_15><loc_37><loc_85><loc_47></location>This equation looks suspiciously like a Hamilton-Jacobi equation. And indeed it may be brought into the form of one: due to τ satisfying the 'twistless torsion' condition τ ∧ d τ = 0, there are local coordinates ( t , x a ) and a function f (an 'integrating factor') such that locally τ = f d t ; since τ is nowhere-vanishing, so is f . Together with τµ h µν = 0, this implies h = h ab ∂ a ⊗ ∂ b (using the notation ∂ a = ∂ ∂ x a ), and τ ( ˆ v ) = 1 implies ˆ v = 1 f ∂ t + ˆ v a ∂ a . Hence ( 3 . 2 ) takes the coordinate form</text>
<formula><location><page_6><loc_34><loc_33><loc_85><loc_36></location>0 = ˆ ϕ + 1 f ∂ t χ + ˆ v a ∂ a χ + 1 2 h ab ∂ a χ∂ b χ , ( 3 . 3 a)</formula>
<text><location><page_6><loc_15><loc_30><loc_27><loc_31></location>or equivalently</text>
<formula><location><page_6><loc_34><loc_26><loc_85><loc_29></location>0 = f ˆ ϕ + ∂ t χ + f ˆ v a ∂ a χ + 1 2 f h ab ∂ a χ∂ b χ . ( 3 . 3 b)</formula>
<text><location><page_6><loc_15><loc_24><loc_51><loc_25></location>This is precisely a Hamilton-Jacobi equation</text>
<formula><location><page_6><loc_27><loc_19><loc_85><loc_23></location>0 = H ( t , ⃗ x , ∂χ ( t , ⃗ x ) ∂ ⃗ x ) + ∂χ ( t , ⃗ x ) ∂ t ( 3 . 4 a)</formula>
<text><location><page_7><loc_15><loc_88><loc_29><loc_90></location>with Hamiltonian</text>
<formula><location><page_7><loc_21><loc_84><loc_85><loc_87></location>H ( t , ⃗ x , ⃗ p ) = 1 2 f ( t , ⃗ x ) h ab ( t , ⃗ x ) pap b + f ( t , ⃗ x ) ˆ v a ( t , ⃗ x ) pa + f ( t , ⃗ x ) ˆ ϕ ( t , ⃗ x ) . ( 3 . 4 b)</formula>
<text><location><page_7><loc_15><loc_75><loc_85><loc_83></location>The Hamiltonian being sufficiently regular (which is the case if τ , h , a and v are sufficiently regular), the corresponding Hamiltonian equations of motion admit (local) solutions, which is equivalent to the Hamilton-Jacobi equation admitting a local solution. Thus, we can always perform a local gauge transformation such that after the transformation we have ˆ ϕ = 0.</text>
<section_header_level_1><location><page_7><loc_15><loc_71><loc_38><loc_73></location>3.2 Type II TTNC geometry</section_header_level_1>
<text><location><page_7><loc_15><loc_65><loc_85><loc_70></location>Under a type II subleading diffeomorphism ( 2 . 11 ) parametrised by a vector field ζ = -χ v + λ with λ spacelike, the locally Galilei-invariant potential ˆ ϕ ( 2 . 9 ) transforms as</text>
<text><location><page_7><loc_19><loc_62><loc_19><loc_64></location>↦</text>
<formula><location><page_7><loc_17><loc_57><loc_85><loc_64></location>ˆ ϕ → ( a -d τ ( ζ , · ) + d χ )( v ) + 1 2 h ( a -d τ ( ζ , · ) + d χ , a -d τ ( ζ , · ) + d χ ) = ˆ ϕ + d χ ( ˆ v ) + 1 2 h ( d χ , d χ ) -d τ ( ζ , ˆ v ) -h ( d τ ( ζ , · ) , d χ ) + 1 2 h ( d τ ( ζ , · ) , d τ ( ζ , · )) . ( 3 . 5 )</formula>
<text><location><page_7><loc_17><loc_54><loc_57><loc_55></location>Setting the spacelike part λ to zero, this becomes</text>
<text><location><page_7><loc_17><loc_50><loc_17><loc_52></location>↦</text>
<formula><location><page_7><loc_15><loc_48><loc_85><loc_53></location>ˆ ϕ → ˆ ϕ + d χ ( ˆ v ) + 1 2 h ( d χ , d χ ) + χ d τ ( v , ˆ v ) + χ h ( d τ ( v , · ) , d χ ) + 1 2 χ 2 h ( d τ ( v , · ) , d τ ( v , · )) . ( 3 . 6 )</formula>
<text><location><page_7><loc_15><loc_39><loc_85><loc_47></location>Even when imposing the twistless torsion condition τ ∧ d τ = 0, the terms involving h ( d τ ( v , · ) , · ) will in general not vanish. Hence, the equation for gauging ˆ ϕ to zero involves terms linear and quadratic in χ (non-differentiated); in particular, differently to the type I case, it is not a Hamilton-Jacobi equation and so we cannot easily argue for it to have a solution.</text>
<text><location><page_7><loc_15><loc_33><loc_85><loc_38></location>Instead, however, we now consider the transformation ( 3 . 5 ) for χ = 0, i.e. with ζ = λ spacelike. As in the type I case we use the twistless torsion condition τ ∧ d τ = 0 to locally write τ = f d t , implying d τ = ∂ a f d x a ∧ d t . We then have</text>
<formula><location><page_7><loc_33><loc_29><loc_85><loc_32></location>d τ ( ζ , · ) = d τ ( λ , · ) = λ a ∂ a f d t = λ a ∂ a f f τ , ( 3 . 7 )</formula>
<text><location><page_7><loc_15><loc_26><loc_67><loc_28></location>such that the transformation behaviour ( 3 . 5 ) of ˆ ϕ takes the form</text>
<text><location><page_7><loc_29><loc_23><loc_29><loc_24></location>↦</text>
<formula><location><page_7><loc_27><loc_22><loc_85><loc_25></location>ˆ ϕ → ˆ ϕ -d τ ( λ , ˆ v ) + 1 2 h ( d τ ( λ , · ) , d τ ( λ , · )) = ˆ ϕ -λ a ∂ a f f . ( 3 . 8 )</formula>
<text><location><page_8><loc_15><loc_87><loc_85><loc_90></location>Therefore, in order to transform ˆ ϕ to zero by a subleading spacelike diffeomorphism, the parametrising spacelike vector field λ has to satisfy the equation</text>
<formula><location><page_8><loc_44><loc_84><loc_85><loc_85></location>ˆ ϕ = λ a ∂ a ln f . ( 3 . 9 )</formula>
<text><location><page_8><loc_51><loc_81><loc_51><loc_82></location≯</text>
<text><location><page_8><loc_15><loc_81><loc_82><loc_82></location>If f is not locally spatially constant, i.e. ∂ a f = 0, this equation has a local solution.</text>
<text><location><page_8><loc_15><loc_76><loc_85><loc_80></location>If instead f is locally spatially constant in some region, this means that d τ = 0 there, such that type I and type II gauge transformations coincide and we can again argue as in section 3 . 1 for the existence of a gauge transformation transforming ˆ ϕ to zero.</text>
<text><location><page_8><loc_15><loc_71><loc_85><loc_75></location>Thus, we have seen that also in the case of type II TTNC geometry one can always locally find a gauge transformation (here meaning a type II subleading diffeomorphism) transforming ˆ ϕ to zero.</text>
<section_header_level_1><location><page_8><loc_15><loc_66><loc_68><loc_68></location>4 Application: parametrisation of TTNC geometries</section_header_level_1>
<text><location><page_8><loc_15><loc_60><loc_85><loc_65></location>In this section, we are going to apply the special gauge transformation derived in the previous section in order to parametrise TTNC geometries by just the space metric and a unit timelike vector field.</text>
<text><location><page_8><loc_17><loc_58><loc_78><loc_60></location>First, let us restate the result of the previous section in the following form:</text>
<text><location><page_8><loc_15><loc_50><loc_85><loc_57></location>Theorem 1 . Let ( M , τ , h ) be a Galilei manifold satisfying the twistless torsion condition τ ∧ d τ = 0 , and let ˜ a be a Bargmann form for it. Then locally there exist a gauge-equivalent Bargmann form a and a unit timelike vector field v such that the local representative of a with respect to v is a = 0 .</text>
<text><location><page_8><loc_15><loc_46><loc_85><loc_49></location>In the type I case, this implies in particular that the local representative ˜ a of ˜ a with respect to v is closed, since it differs from a by a type I gauge transformation.</text>
<text><location><page_8><loc_15><loc_35><loc_85><loc_45></location>Proof of theorem 1 . According to the previous section, there exists a local gauge transformation such that the locally Galilei-invariant potential of the gauge-transformed Bargmann form a vanishes, ˆ ϕ = 0. Then taking v to be the locally Galilei-invariant vector field determined by a , by construction we have a = ˆ ϕτ = 0. (Concretely, this means that choosing any unit timelike vector vield v ' , we define v µ = v ' µ + h µν a ' ν where a ' is the local representative of a with respect to v ' .)</text>
<text><location><page_8><loc_15><loc_27><loc_85><loc_33></location>Note that the statement of theorem 1 really is a reformulation of the result of section 3 : if after the gauge transformation a has representative a = 0 with respect to v , this representative is proportional to τ , which implies that the proportionality factor is the locally Galilei-invariant potential ˆ ϕ , i.e. in this case ˆ ϕ = 0.</text>
<section_header_level_1><location><page_8><loc_15><loc_23><loc_60><loc_25></location>4.1 Parametrisation of TTNC geometries up to gauge</section_header_level_1>
<text><location><page_8><loc_15><loc_19><loc_85><loc_22></location>Theorem 1 implies in particular that, fixing τ and h , if we let v vary over the set of all unit timelike vector fields and consider, for each v , the Bargmann form a which is</text>
<text><location><page_9><loc_15><loc_87><loc_85><loc_90></location>defined by its representative with respect to v vanishing, we will (locally) parametrise all gauge equivalence classes of Bargmann forms for ( M , τ , h ) .</text>
<text><location><page_9><loc_15><loc_78><loc_85><loc_86></location>We can even strip τ from the prescribed data, as follows: given a Galilei manifold ( M , τ , h ) and a unit timelike vector field v , we have that y : = h + v ⊗ v is non-degenerate. Conversely, starting with just a (positive semidefinite) symmetric contravariant 2 -tensor field h of rank n and a vector field v such that y = h + v ⊗ v is non-degenerate, it is easy to see that the conditions</text>
<formula><location><page_9><loc_41><loc_76><loc_85><loc_77></location>τ ( v ) = 1 , τµ h µν = 0 ( 4 . 1 )</formula>
<text><location><page_9><loc_15><loc_69><loc_85><loc_74></location>then determine a unique one-form τ . We can then also express the twistless torsion condition for the τ thus determined in terms of h and v : the condition τ ∧ d τ = 0 is equivalent to h µρ h σν ∂ [ µ τ ν ] = 0, which can be rewritten as</text>
<formula><location><page_9><loc_40><loc_66><loc_85><loc_68></location>( y -1 ) µν v ν h λ [ ρ ∂ λ h σ ] µ = 0 , ( 4 . 2 )</formula>
<text><location><page_9><loc_15><loc_64><loc_49><loc_66></location>where y -1 is the inverse of y = h + v ⊗ v .</text>
<text><location><page_9><loc_15><loc_59><loc_85><loc_64></location>Combined with theorem 1 , this shows that we may (locally) parametrise all the basic geometric objects defining a TTNC geometry up to (type I or type II) gauge transformations by just h and v :</text>
<text><location><page_9><loc_15><loc_46><loc_85><loc_58></location>Theorem 2 . Let M be an ( n + 1 ) -dimensional differentiable manifold, h a positive semidefinite symmetric contravariant 2 -tensor field of rank n on M, and v a vector field on M such that (i) y = h + v ⊗ v is non-degenerate and (ii) ( 4 . 2 ) holds. Consider the unique one-form τ ∈ Ω 1 ( M ) satisfying ( 4 . 1 ) . This makes ( M , τ , h ) into a Galilei manifold with v a unit timelike vector field, and satisfies the twistless torsion condition τ ∧ d τ = 0 . Furthermore, we may define a Bargmann form a for ( M , τ , h ) by demanding that its local representative with respect to v vanish, a = 0 .</text>
<text><location><page_9><loc_15><loc_43><loc_85><loc_46></location>Locally, any twistless torsional Galilei manifold with Bargmann form is of this form, up to gauge transformations of the Bargmann form.</text>
<section_header_level_1><location><page_9><loc_15><loc_39><loc_49><loc_41></location>4.2 Interlude: compatible connections</section_header_level_1>
<text><location><page_9><loc_15><loc_33><loc_85><loc_38></location>Now we are going to discuss, in addition to the basic metric structure of a Galilei manifold (and possibly a Bargmann form for it), compatible connections. For details on this material we refer to [ 19 , 20 ]; a more general discussion may be found in [ 25 ].</text>
<text><location><page_9><loc_15><loc_26><loc_85><loc_33></location>A connection compatible with the structure of a Galilei manifold ( M , τ , h ) is called a Galilei connection . Phrased in terms of a covariant derivative operator, this amounts to a covariant derivative ∇ on the tangent bundle TM compatible with τ and h , i.e. satisfying</text>
<formula><location><page_9><loc_42><loc_25><loc_85><loc_26></location>∇ τ = 0 , ∇ h = 0 . ( 4 . 3 )</formula>
<text><location><page_9><loc_15><loc_19><loc_85><loc_24></location>Compatibility with τ implies that the torsion T of any Galilei connection satisfies τρ T ρ µν = ( d τ ) µν . A Galilei connection on ( M , τ , h ) may also be understood as a principal connection ω on the Galilei frame bundle G ( M ) of ( M , τ , h ) . Its local connection</text>
<text><location><page_10><loc_15><loc_85><loc_85><loc_90></location>form with respect to a Galilei frame ( v , e a ) is then a locally defined one-form ( ω a b , ϖ a ) taking values in the Galilei algebra gal = so ( n ) i R n , with rotational part ω a b and boost part ϖ a .</text>
<text><location><page_10><loc_15><loc_77><loc_85><loc_85></location>Due to the degeneracy of the metric structure, differently to the pseudo-Riemannian case Galilei connections are not uniquely determined by their torsion. Instead, with respect to a choice of unit timelike vector field v , a Galilei connection ∇ is uniquely determined by its torsion T and its Newton-Coriolis form Ω with respect to v . The latter is a two-form that may be written as</text>
<formula><location><page_10><loc_42><loc_73><loc_85><loc_75></location>Ω µν = 2 ( ∇ [ µ v ρ ) h ν ] ρ , ( 4 . 4 )</formula>
<text><location><page_10><loc_15><loc_65><loc_85><loc_72></location>where h µν are the components of the covariant space metric with respect to v , which is defined by h µν = h νµ , h µν v ν = 0, h µν h νρ = δ ρ µ -v ρ τµ . Alternatively, extending v to a local Galilei frame ( v , e a ) , Ω can be written in terms of the boost part of the local connection form and the dual frame as</text>
<formula><location><page_10><loc_43><loc_62><loc_85><loc_64></location>Ω = δ ab ϖ a ∧ e b . ( 4 . 5 )</formula>
<text><location><page_10><loc_15><loc_54><loc_85><loc_61></location>Conversely, choosing an arbitrary tensor field T subject to the constraints T ρ µν = -T ρ νµ and τρ T ρ µν = ( d τ ) µν , a unit timelike vector field v , and an arbitrary two-form Ω , there is a (unique) Galilei connection whose torsion is T and whose Newton-Coriolis form with respect to v is Ω .</text>
<text><location><page_10><loc_15><loc_41><loc_85><loc_54></location>By definition, together with the tensorial R n + 1 -valued one-form θ on G ( M ) which corresponds to the canonical solder form of G ( M ) × Gal R n + 1 , any Bargmann form a combines to a tensorial ( R n + 1 ⊕ u ( 1 )) -valued form ( θ , i a ) on G ( M ) . The exterior covariant derivative d ω ( θ , i a ) of this tensorial form with respect to a Galilei connection ω is the so-called extended torsion of ω with respect to a . Note that the extended torsion depends only 6 on d a , such that it is invariant under type I gauge transformations. The local representative of the extended torsion with respect to a local Galilei frame σ = ( v , e a ) may be seen to be given by</text>
<formula><location><page_10><loc_37><loc_38><loc_85><loc_40></location>σ ∗ d ω ( θ , i a ) = ( T A , i ( d a + Ω )) : ( 4 . 6 )</formula>
<text><location><page_10><loc_15><loc_28><loc_85><loc_36></location>it consists of the frame components T A of the torsion and the so-called mass torsion d a + Ω . In particular, we see that knowing a Galilei connection's extended torsion with respect to some Bargmann form a amounts to knowing its torsion and Newton-Coriolis form. Hence, fixing a Bargmann form (or a type I gauge equivalence class of Bargmann forms), Galilei connections are uniquely determined by their extended torsion .</text>
<text><location><page_10><loc_15><loc_23><loc_85><loc_28></location>In standard Newton-Cartan gravity, Newtonian gravity is encoded by so-called Newtonian connections. These are torsion-free Galilei connections (on a Galilei manifold with absolute time) whose curvature tensor satisfies the symmetry condition</text>
<text><location><page_11><loc_15><loc_85><loc_85><loc_90></location>R µ ν ρ σ = R ν µ σ ρ (where the third index was raised with h ). This is equivalent to the Newton-Coriolis form (with respect to any v ) being closed, d Ω = 0 (this equivalence is not obvious, but requires a somewhat lengthy calculation). 7</text>
<text><location><page_11><loc_15><loc_78><loc_85><loc_85></location>By the Poincaré lemma, a torsion-free Galilei connection being Newtonian is locally equivalent to the Newton-Coriolis form being exact, Ω = -d a for some one-form a . This means that locally, Newtonian connections are precisely those Galilei connections whose extended torsion with respect to some Bargmann form vanishes.</text>
<section_header_level_1><location><page_11><loc_15><loc_75><loc_66><loc_76></location>4.3 Parametrisation of standard Newton-Cartan geometry</section_header_level_1>
<text><location><page_11><loc_15><loc_70><loc_85><loc_73></location>We can now see how our results imply the following classical result from standard Newton-Cartan geometry [ 5 , thm. 3 . 6 ], [ 10 , prop. 4 . 3 . 7 ]:</text>
<text><location><page_11><loc_15><loc_64><loc_85><loc_69></location>Corollary 3 . Let ( M , τ , h ) be a Galilei manifold with absolute time, d τ = 0 , and let ω be a Newtonian connection on it. Then locally there exists a unit timelike vector field v such that the Newton-Coriolis form of ω with respect to v vanishes.</text>
<text><location><page_11><loc_15><loc_56><loc_85><loc_63></location>Proof. Locally there exists a Bargmann form ˜ a such that ω is the (unique) Galilei connection whose extended torsion with respect to ˜ a vanishes. By theorem 1 , locally there exists a unit timelike vector field v such that the local representative ˜ a of ˜ a with respect to v is closed. Hence, the extended torsion vanishing implies 0 = d˜ a + Ω = Ω .</text>
<text><location><page_11><loc_15><loc_48><loc_85><loc_55></location>We see that theorem 1 is the natural generalisation of this result to the TTNC case. Note that while Malament's proof of corollary 3 [ 10 , prop. 4 . 3 . 7 ] is very different in spirit from our approach, the original proof by Dombrowski and Horneffer [ 5 , thm. 3 . 6 ] is very close (at least to this special case of our gauge fixing): it also reduces</text>
<text><location><page_11><loc_17><loc_19><loc_85><loc_30></location>Due to ∇ being Newtonian, we may (at least locally) demand it to be the unique extended-torsionfree Galilei connection with respect to a , which as an equation takes the form Ω = -d a and therefore determines a up to gauge transformations. In the case discussed above, we have Ω = τ ∧ d ϕ = -d ( ϕτ ) , so we may take a = ϕτ . So in this special situation and for this choice of a , the proportionality factor between a and τ plays the role of the Newtonian potential, which is why we call the locally Galileiinvariant quantity ˆ ϕ from ( 2 . 9 ) the locally Galilei-invariant potential . Note however that after performing a gauge transformation, the new ˆ ϕ will no longer be equal to the Newtonian potential ϕ with respect to v .</text>
<text><location><page_12><loc_15><loc_87><loc_85><loc_90></location>the statement (which is given in an appreciably different formulation) to solving a Hamilton-Jacobi equation. 8</text>
<text><location><page_12><loc_15><loc_82><loc_85><loc_86></location>Using corollary 3 , we may (locally) parametrise, similar to theorem 2 , absolute-time Galilei manifolds with Newtonian connections in terms of just h and v (also expressing the condition d τ = 0 in terms of these):</text>
<text><location><page_12><loc_15><loc_69><loc_85><loc_80></location>Theorem 4 . Let M be an ( n + 1 ) -dimensional differentiable manifold, h a positive semidefinite symmetric contravariant 2 -tensor field of rank n on M, and v a vector field on M such that (i) y = h + v ⊗ v is non-degenerate and (ii) ∂ [ µ (( y -1 ) ν ] ρ v ρ ) = 0 holds. Consider the unique one-form τ ∈ Ω 1 ( M ) satisfying ( 4 . 1 ) . This makes ( M , τ , h ) into a Galilei manifold with absolute time, and v a unit timelike vector field. Furthermore, we may define a Newtonian connection on ( M , τ , h ) by demanding that its torsion and Newton-Coriolis form with respect to v vanish, Ω = 0 .</text>
<text><location><page_12><loc_17><loc_67><loc_82><loc_68></location>Locally, any Galilei manifold with absolute time and Newtonian connection is of this form.</text>
<section_header_level_1><location><page_12><loc_15><loc_63><loc_29><loc_65></location>5 Conclusion</section_header_level_1>
<text><location><page_12><loc_15><loc_51><loc_85><loc_61></location>In this paper, we have shown that in twistless torsional Newton-Cartan geometry (both its type I and type II incarnations), there always exists a local gauge transformation transforming the locally Galilei-invariant potential to zero. Equivalently, there exist (locally) a gauge transformation and a unit timelike vector field such that the representative of the gauge-transformed Bargmann form with respect to this vector field vanishes.</text>
<text><location><page_12><loc_15><loc_45><loc_85><loc_51></location>This generalises the classical result in standard Newton-Cartan geometry that for any Newtonian connection, locally there exists a unit timelike vector field that is geodesic and twist-free with respect to the connection, i.e. such that the Newton-Coriolis form of the connection with respect to this vector field vanishes.</text>
<text><location><page_12><loc_15><loc_40><loc_85><loc_44></location>The above result also allowed us to argue that all of the geometric data determining a TTNC geometry up to gauge may be parametrised by just the space metric and a unit timelike vector field.</text>
<text><location><page_12><loc_15><loc_24><loc_85><loc_39></location>This possibility of 're-packaging' of all the geometric data in the form of two contravariant tensor fields h µν and v µ enables the following interesting application. The field equations of standard Newton-Cartan gravity, as well as those of TTNC (type II) gravity that arise from expanding the Einstein equations in c -1 [ 15 ], are symmetric covariant 2 -tensor equations. This means that h µν and v µ together fully encode the possible configurations of the system under consideration (TTNC geometry), and their combination h µν + v µ v ν is non-degenerate and (tensorially) dual to the type of the field equations. This allows to study application of the canonical variational completion formalism [ 26 , 27 ] to TTNC gravity. If this application turned out successful, this would offer a new route</text>
<text><location><page_13><loc_15><loc_83><loc_85><loc_90></location>towards action formulations of TTNC gravity, starting from the theory (defined by its geometry and field equations) alone, thus complementing existing approaches based on expansions of GR or on symmetry considerations [ 21 , 22 ]. We plan to pursue this route in future work.</text>
<section_header_level_1><location><page_13><loc_15><loc_79><loc_35><loc_81></location>Acknowledgements</section_header_level_1>
<text><location><page_13><loc_15><loc_76><loc_64><loc_77></location>We wish to thank Domenico Giulini for helpful discussions.</text>
<section_header_level_1><location><page_13><loc_15><loc_71><loc_26><loc_73></location>References</section_header_level_1>
<unordered_list>
<list_item><location><page_13><loc_16><loc_63><loc_85><loc_70></location>[ 1 ] E. Cartan, 'Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie)', Ann. Sci. Éc. Norm. Supér. 40 , 325 -412 ( 1923 ); 'Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie) (Suite)', ibid. 41 , 1 -25 ( 1924 ).</list_item>
<list_item><location><page_13><loc_16><loc_58><loc_85><loc_62></location>[ 2 ] K. Friedrichs, 'Eine invariante Formulierung des Newtonschen Gravitationsgesetzes und des Grenzüberganges vom Einsteinschen zum Newtonschen Gesetz', Math. Ann. 98 , 566 -575 ( 1928 ).</list_item>
<list_item><location><page_13><loc_16><loc_54><loc_85><loc_57></location>[ 3 ] A. Trautman, 'Sur la théorie newtonienne de la gravitation', C. R. Acad. Sci. 257 , 617 -620 ( 1963 ).</list_item>
<list_item><location><page_13><loc_16><loc_48><loc_85><loc_53></location>[ 4 ] A. Trautman, 'Foundations and current problems of general relativity', in Lectures on General Relativity , edited by S. Deser and K. W. Ford (Prentice-Hall, Englewood Cliffs, N. J., 1965 ), pp. 1 -248 .</list_item>
<list_item><location><page_13><loc_16><loc_44><loc_85><loc_47></location>[ 5 ] H. D. Dombrowski and K. Horneffer, 'Die Differentialgeometrie des Galileischen Relativitätsprinzips', Math. Z. 86 , 291 -311 ( 1964 ).</list_item>
<list_item><location><page_13><loc_16><loc_39><loc_85><loc_43></location>[ 6 ] H. P. Künzle, 'Galilei and Lorentz structures on space-time : comparison of the corresponding geometry and physics', Ann. Inst. Henri Poincaré 17 , 337 -362 ( 1972 ).</list_item>
<list_item><location><page_13><loc_16><loc_35><loc_85><loc_38></location>[ 7 ] H. P. Künzle, 'Covariant Newtonian limit of Lorentz space-times', Gen. Relativ. Gravit. 7 , 445 -457 ( 1976 ).</list_item>
<list_item><location><page_13><loc_16><loc_27><loc_85><loc_34></location>[ 8 ] J. Ehlers, 'Über den Newtonschen Grenzwert der Einsteinschen Gravitationstheorie', in Grundlagenprobleme der modernen Physik: Festschrift für Peter Mittelstaedt zum 50 . Geburtstag , edited by J. Nitsch, J. Pfarr and E.-W. Stachow (Bibliographisches Institut, Mannheim, Wien, Zürich, 1981 ), pp. 65 -84 , republished as [ 9 ].</list_item>
<list_item><location><page_13><loc_16><loc_24><loc_85><loc_27></location>[ 9 ] J. Ehlers, 'On the Newtonian limit of Einstein's theory of gravitation', Gen. Relativ. Grav. 51 , 163 ( 2019 ), republication of original article [ 8 ] as 'Golden Oldie'.</list_item>
</unordered_list>
<table>
<location><page_14><loc_15><loc_20><loc_85><loc_90></location>
</table>
<unordered_list>
<list_item><location><page_15><loc_15><loc_85><loc_85><loc_90></location>[ 24 ] B. Julia and H. Nicolai, 'Null-Killing vector dimensional reduction and Galilean geometrodynamics', Nucl. Phys. B 439 , 291 -323 ( 1995 ), arXiv: hep-th/9412002 [hep-th] .</list_item>
<list_item><location><page_15><loc_15><loc_79><loc_85><loc_84></location>[ 25 ] E. A. Bergshoeff, K. van Helden, J. Lahnsteiner, L. Romano and J. Rosseel, 'Generalized Newton-Cartan geometries for particles and strings', Class. Quantum Gravity 40 , 075010 ( 2023 ), arXiv: 2207.00363 [hep-th] .</list_item>
<list_item><location><page_15><loc_15><loc_75><loc_85><loc_78></location>[ 26 ] N. Voicu and D. Krupka, 'Canonical variational completion of differential equations', J. Math. Phys. 56 , 043507 ( 2015 ), arXiv: 1406.6646 [math-ph] .</list_item>
<list_item><location><page_15><loc_15><loc_70><loc_85><loc_75></location>[ 27 ] M. Hohmann, C. Pfeifer and N. Voicu, 'Canonical variational completion and 4 D Gauss-Bonnet gravity', Eur. Phys. J. Plus 136 , 180 ( 2021 ), arXiv: 2009.05459 [gr-qc] .</list_item>
</unordered_list>
</document>
[
{
"title": "On gauge transformations in twistless torsional Newton-Cartan geometry",
"content": "Arian L. von Blanckenburg 1 ,a and Philip K. Schwartz 1 ,b 1 Institute for Theoretical Physics, Leibniz University Hannover, Appelstraße 2 , 30167 Hannover, Germany a a.von.blanckenburg@stud.uni-hannover.de b philip.schwartz@itp.uni-hannover.de We observe that in type I twistless torsional Newton-Cartan (TTNC) geometry, one can always find (at least locally) a gauge transformation that transforms a specific locally Galilei-invariant function-that we dub the 'locally Galilei-invariant potential'-to zero, due to the corresponding equation for the gauge parameter taking the form of a Hamilton-Jacobi equation. In the case of type II TTNC geometry, the same gauge fixing may locally be performed by subleading spatial diffeomorphisms. We show (a) how this generalises a classical result in standard Newton-Cartan geometry, and (b) how it allows to parametrise the metric structure of a Galilei manifold as well as the gauge equivalence class of the Bargmann form of TTNC geometry in terms of just the space metric and a unit timelike vector field.",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "Newton-Cartan gravity [ 1 -9 ], [ 10 , chapter 4 ], [ 11 ] is a differential-geometric reformulation of Newtonian gravity. Similar to General Relativity (GR), it describes gravity by a curved connection on the manifold of spacetime events. However, differently to the GR case, in Newton-Cartan gravity spacetime is endowed not with a Lorentzian metric (with which the connection is compatible), but with a specific notion of Galileirelativistic metric structure, ensuring local Galilei invariance of physical laws. If the connection is torsion-free, there is necessarily a notion of absolute time. One can weaken the notion of absolute time, only demanding the existence of a concept of absolute simultaneity. The resulting geometry, in which compatible connections necessarily have non-vanishing torsion, is called twistless torsional NewtonCartan (TTNC) geometry [ 12 -15 ]. Already in standard Newton-Cartan geometry one can introduce, in addition to the metric structure, an additional 'auxiliary' field related to the Bargmann algebra, the central extension of the Galilei algebra [ 16 -20 ]. In TTNC geometry, this field features more heavily. In the literature, it goes under several different names; we will call it a 'Bargmann form' (for more on this naming, see footnote 1 in section 2 ). There is a gauge freedom for the Bargmann form: certain transformations of it are to be considered gauge transformations, relating different equivalent descriptions of the same geometric situation. Depending on which transformations are considered gauge, one distinguishes between so-called type I and type II TTNC geometry [ 21 , 22 , 11 ]. The Bargmann form can be used to define a locally Galilei-invariant vector field-that is, a vector field invariantly determined by the metric structure and the Bargmann form. In the study of this vector field, a certain scalar function arises naturally, which we call the 'locally Galilei-invariant potential'. We will show in this paper that in both the type I and type II cases locally there exists a gauge transformation such that after performing this transformation, the locally Galilei-invariant potential vanishes. This (partial) gauge fixing is interesting for two reasons: on the one hand, we will show that it allows for a (local) parametrisation of TTNC geometries up to gauge by just the space metric and a unit timelike vector field. On the other hand, it also offers a natural generalisation to TTNC geometry of a classical result in standard Newton-Cartan geometry, namely the local existence of twist-free geodesic unit timelike vector fields [ 5 , thm. 3 . 6 ], [ 10 , prop. 4 . 3 . 7 ]. Throughout this paper we use the notation and conventions from [ 20 ]. In particular, the local representative of the Bargmann form is called a and not m as in other literature. The structure of this paper is as follows. In section 2 we give a brief review of TTNC geometry and introduce the locally Galilei-invariant potential. We show how local gauge transformations can be used to 'gauge away' the locally Galilei-invariant potential in both type I and type II TTNC geometry in section 3 . In section 4 we apply this to parametrisation of TTNC geometries as mentioned above, and explain how it generalises the classical result. We conclude and discuss further research directions in section 5 .",
"pages": [
1,
2
]
},
{
"title": "2 Twistless torsional Newton-Cartan geometry",
"content": "In this section, we briefly review TTNC geometry in a style similar to that of [ 7 -9 ], [ 10 , chapter 4 ], [ 20 ], i.e. from what might be called a relativist's point of view, and introduce the locally Galilei-invariant potential. For a somewhat more 'field-theoretic' perspective on the subject, we refer to the original literature on type I [ 12 -14 ] and type II TTNC geometry [ 15 , 21 , 22 ], and in particular the review [ 11 ]. The basic geometric structure in Newton-Cartan geometry is that of a Galilei manifold , i.e. a differentiable manifold M of dimension n + 1 ( n ≥ 1) with a nowhere-vanishing clock form τ ∈ Ω 1 ( M ) and a symmetric contravariant 2 -tensor field h , the space metric , which is positive semidefinite of rank n , such that the degenerate direction of h is spanned by τ . The latter condition may be expressed as Vectors in the kernel of τ are called spacelike , other vectors are called timelike . The integral of τ along any worldline (i.e. curve) in M is interpreted as the time elapsed along this worldline. If d τ = 0, the time between two events (i.e. points in M ) is (locally) independent of the connecting worldline chosen to measure it; one then speaks of a Galilei manifold with absolute time . If only the weaker condition τ ∧ d τ = 0 holds, such that by Frobenius' theorem the distribution ker τ of spacelike vectors is integrable (i.e. we have an absolute notion of simultaneity), then one says to be in the situation of twistless torsional Newton-Cartan (TTNC) geometry. In any case, h induces a positive definite bundle metric on the spacelike vectors. In the following, unless otherwise stated we will always assume the 'twistless torsion' condition τ ∧ d τ = 0. A (local) Galilei frame on a Galilei manifold is a local frame ( v , e a ) of vector fields satisfying It follows that τ ( e a ) = 0, such that the dual frame of one-forms takes the form ( τ , e a ) . Changes from one choice of Galilei frame to another are called local Galilei transformations ; they are precisely given by basis change matrix functions with values in the (orthochronous) homogeneous Galilei group Gal = O ( n ) ⋉ R n ⊂ GL ( n + 1 ) . Therefore, Galilei frames may be understood as local sections of the Galilei frame bundle G ( M ) of ( M , τ , h ) , a principal Gal -bundle which is a reduction of the structure group of the general linear frame bundle of M . Local Galilei boosts , also called Milne boosts , are changes of the unit timelike frame vector field v alone, which are of the form ↦ for an R n -valued boost velocity function k . The dual frame ( τ , e a ) then transforms according to ↦ Note that we will not discuss connections compatible with Galilei structures in this section, but only later in section 4 . 2 . There is an additional ingredient that is commonly viewed as part of the geometric structures defining a Newton-Cartan geometry, particularly in the TTNC case [ 16 -20 ]. Locally, it is given by the specification of a one-form a for each choice of local Galilei frame in such a way that it is invariant under spatial rotations of the frame and under local Galilei boosts ( 2 . 3 ) transforms according to ↦ Globally, it may be understood as follows [ 16 , 17 , 20 ]. We consider the representation ˙ ρ : Gal → GL ( R n + 1 ⊕ u ( 1 )) given by ̸ On the Galilei frame bundle G ( M ) , we have a tensorial R n + 1 -valued one-form θ corresponding to the canonical solder form of G ( M ) × Gal R n + 1 . The global object locally represented by the form a from above is then a one-form a on G ( M ) that together with this θ yields a ˙ ρ -tensorial form ( θ , i a ) . The local form a is then the pullback of a along the local Galilei frame ( v , e a ) understood as a section of G ( M ) . The representation ˙ ρ arises in the study of the Bargmann group, whose Lie algebra is (for n = 2) the essentially unique non-trivial 1 -dimensional central extension of the inhomogeneous Galilei algebra. For that reason, we will call a field a as described here a Bargmann form for ( M , τ , h ) . 1 Using a Bargmann form a for ( M , τ , h ) , one can construct a unit timelike vector field ˆ v ∈ Γ ( TM ) that is locally Galilei-invariant , according to where a is the local representative of a with respect to v . 2 Invariance under spatial frame rotations is obvious; invariance under local Galilei boosts follows by direct computation. Note that ˆ v being locally Galilei-invariant means that it really is determined just by τ , h , and a . Understanding the transformation from a general unit timelike vector field v to ˆ v as a local Galilei boost, we may compute the corresponding representative of the Bargmann form a with respect to ˆ v to be given by By construction, we know that the right-hand side of ( 2 . 8 ) is locally Galilei-invariant; this may, however, also be checked by direct calculation. Note that ˆ a is proportional to τ , which may even be used to define ˆ v : one easily checks that, for τ , h , a given, ˆ v is the unique unit timelike vector field for which the local representative of a is proportional to τ . The proportionality factor we call the locally Galilei-invariant potential . (Some motivation for why we choose this name will be given in footnote 7 in section 4 .) Note again that ˆ ϕ being locally Galilei-invariant means that it is fully determined by τ , h , and a . In Newton-Cartan geometry, there are several different kinds of 'gauge transformations', i.e. transformations relating mathematical situations which are to be considered equivalent descriptions of 'the same' geometric situation. First, due to the differentialgeometric nature of the theory, there are diffeomorphisms: acting on all the involved objects with pushforward 3 by a diffeomorphism φ : M → N leads to a different mathematical description of the same geometric situation. Second, there are local Galilei transformations: these are just changes in the (arbitrary) choice of local Galilei frame ( v , e a ) , giving local descriptions of global objects living on G ( M ) ; all the true global geometric objects such as τ , h , a stay invariant. When we use the name gauge transformation in the following, we will always mean the third kind: certain transformations of the Bargmann form a . These come in two different flavours, dubbed type I and type II , distinguishing two different kinds of TTNC geometry. In type I TTNC geometry, the Bargmann form is interpreted as arising from a 'gauging' of the central direction of the Bargmann algebra [ 18 , 19 ]. 4 The corresponding U ( 1 ) gauge transformations locally take the form ↦ for functions χ ∈ C ∞ ( M ) . Type I TTNC geometry naturally arises by null reduction of Lorentzian geometry [ 23 , 24 ]. Type II TTNC geometry arises instead from a formal expansion of Lorentzian geometry in c -1 , where c is the velocity of light [ 15 , 21 , 22 , 11 ]. 5 Here, a is the next-to-leadingorder part of the timelike Lorentzian coframe field. The gauge transformations it inherits from subleading c -2 -dependent diffeomorphisms take the form ↦ for vector fields ζ ∈ Γ ( TM ) . Note that in the case of absolute time (d τ = 0), type I and type II gauge transformations coincide. Note also that the locally Galilei-invariant vector field ˆ v ( 2 . 7 ) is not gauge-invariant (in both the type I and type II cases).",
"pages": [
2,
3,
4,
5,
6
]
},
{
"title": "3 Gauge transformations of the locally Galilei-invariant potential",
"content": "In this section, we are going to argue that in both type I and type II TTNC geometry, there is always a local gauge transformation that transforms the locally Galilei-invariant potential ( 2 . 9 ) to zero.",
"pages": [
6
]
},
{
"title": "3.1 Type I TTNC geometry",
"content": "Under a type I TTNC gauge transformation ( 2 . 10 ) of a , the locally Galilei-invariant potential ( 2 . 9 ) transforms according to ↦ Therefore, we may transform ˆ ϕ to zero by performing a gauge transformation with the gauge parameter χ solving the PDE This equation looks suspiciously like a Hamilton-Jacobi equation. And indeed it may be brought into the form of one: due to τ satisfying the 'twistless torsion' condition τ ∧ d τ = 0, there are local coordinates ( t , x a ) and a function f (an 'integrating factor') such that locally τ = f d t ; since τ is nowhere-vanishing, so is f . Together with τµ h µν = 0, this implies h = h ab ∂ a ⊗ ∂ b (using the notation ∂ a = ∂ ∂ x a ), and τ ( ˆ v ) = 1 implies ˆ v = 1 f ∂ t + ˆ v a ∂ a . Hence ( 3 . 2 ) takes the coordinate form or equivalently This is precisely a Hamilton-Jacobi equation with Hamiltonian The Hamiltonian being sufficiently regular (which is the case if τ , h , a and v are sufficiently regular), the corresponding Hamiltonian equations of motion admit (local) solutions, which is equivalent to the Hamilton-Jacobi equation admitting a local solution. Thus, we can always perform a local gauge transformation such that after the transformation we have ˆ ϕ = 0.",
"pages": [
6,
7
]
},
{
"title": "3.2 Type II TTNC geometry",
"content": "Under a type II subleading diffeomorphism ( 2 . 11 ) parametrised by a vector field ζ = -χ v + λ with λ spacelike, the locally Galilei-invariant potential ˆ ϕ ( 2 . 9 ) transforms as ↦ Setting the spacelike part λ to zero, this becomes ↦ Even when imposing the twistless torsion condition τ ∧ d τ = 0, the terms involving h ( d τ ( v , · ) , · ) will in general not vanish. Hence, the equation for gauging ˆ ϕ to zero involves terms linear and quadratic in χ (non-differentiated); in particular, differently to the type I case, it is not a Hamilton-Jacobi equation and so we cannot easily argue for it to have a solution. Instead, however, we now consider the transformation ( 3 . 5 ) for χ = 0, i.e. with ζ = λ spacelike. As in the type I case we use the twistless torsion condition τ ∧ d τ = 0 to locally write τ = f d t , implying d τ = ∂ a f d x a ∧ d t . We then have such that the transformation behaviour ( 3 . 5 ) of ˆ ϕ takes the form ↦ Therefore, in order to transform ˆ ϕ to zero by a subleading spacelike diffeomorphism, the parametrising spacelike vector field λ has to satisfy the equation ̸ If f is not locally spatially constant, i.e. ∂ a f = 0, this equation has a local solution. If instead f is locally spatially constant in some region, this means that d τ = 0 there, such that type I and type II gauge transformations coincide and we can again argue as in section 3 . 1 for the existence of a gauge transformation transforming ˆ ϕ to zero. Thus, we have seen that also in the case of type II TTNC geometry one can always locally find a gauge transformation (here meaning a type II subleading diffeomorphism) transforming ˆ ϕ to zero.",
"pages": [
7,
8
]
},
{
"title": "4 Application: parametrisation of TTNC geometries",
"content": "In this section, we are going to apply the special gauge transformation derived in the previous section in order to parametrise TTNC geometries by just the space metric and a unit timelike vector field. First, let us restate the result of the previous section in the following form: Theorem 1 . Let ( M , τ , h ) be a Galilei manifold satisfying the twistless torsion condition τ ∧ d τ = 0 , and let ˜ a be a Bargmann form for it. Then locally there exist a gauge-equivalent Bargmann form a and a unit timelike vector field v such that the local representative of a with respect to v is a = 0 . In the type I case, this implies in particular that the local representative ˜ a of ˜ a with respect to v is closed, since it differs from a by a type I gauge transformation. Proof of theorem 1 . According to the previous section, there exists a local gauge transformation such that the locally Galilei-invariant potential of the gauge-transformed Bargmann form a vanishes, ˆ ϕ = 0. Then taking v to be the locally Galilei-invariant vector field determined by a , by construction we have a = ˆ ϕτ = 0. (Concretely, this means that choosing any unit timelike vector vield v ' , we define v µ = v ' µ + h µν a ' ν where a ' is the local representative of a with respect to v ' .) Note that the statement of theorem 1 really is a reformulation of the result of section 3 : if after the gauge transformation a has representative a = 0 with respect to v , this representative is proportional to τ , which implies that the proportionality factor is the locally Galilei-invariant potential ˆ ϕ , i.e. in this case ˆ ϕ = 0.",
"pages": [
8
]
},
{
"title": "4.1 Parametrisation of TTNC geometries up to gauge",
"content": "Theorem 1 implies in particular that, fixing τ and h , if we let v vary over the set of all unit timelike vector fields and consider, for each v , the Bargmann form a which is defined by its representative with respect to v vanishing, we will (locally) parametrise all gauge equivalence classes of Bargmann forms for ( M , τ , h ) . We can even strip τ from the prescribed data, as follows: given a Galilei manifold ( M , τ , h ) and a unit timelike vector field v , we have that y : = h + v ⊗ v is non-degenerate. Conversely, starting with just a (positive semidefinite) symmetric contravariant 2 -tensor field h of rank n and a vector field v such that y = h + v ⊗ v is non-degenerate, it is easy to see that the conditions then determine a unique one-form τ . We can then also express the twistless torsion condition for the τ thus determined in terms of h and v : the condition τ ∧ d τ = 0 is equivalent to h µρ h σν ∂ [ µ τ ν ] = 0, which can be rewritten as where y -1 is the inverse of y = h + v ⊗ v . Combined with theorem 1 , this shows that we may (locally) parametrise all the basic geometric objects defining a TTNC geometry up to (type I or type II) gauge transformations by just h and v : Theorem 2 . Let M be an ( n + 1 ) -dimensional differentiable manifold, h a positive semidefinite symmetric contravariant 2 -tensor field of rank n on M, and v a vector field on M such that (i) y = h + v ⊗ v is non-degenerate and (ii) ( 4 . 2 ) holds. Consider the unique one-form τ ∈ Ω 1 ( M ) satisfying ( 4 . 1 ) . This makes ( M , τ , h ) into a Galilei manifold with v a unit timelike vector field, and satisfies the twistless torsion condition τ ∧ d τ = 0 . Furthermore, we may define a Bargmann form a for ( M , τ , h ) by demanding that its local representative with respect to v vanish, a = 0 . Locally, any twistless torsional Galilei manifold with Bargmann form is of this form, up to gauge transformations of the Bargmann form.",
"pages": [
8,
9
]
},
{
"title": "4.2 Interlude: compatible connections",
"content": "Now we are going to discuss, in addition to the basic metric structure of a Galilei manifold (and possibly a Bargmann form for it), compatible connections. For details on this material we refer to [ 19 , 20 ]; a more general discussion may be found in [ 25 ]. A connection compatible with the structure of a Galilei manifold ( M , τ , h ) is called a Galilei connection . Phrased in terms of a covariant derivative operator, this amounts to a covariant derivative ∇ on the tangent bundle TM compatible with τ and h , i.e. satisfying Compatibility with τ implies that the torsion T of any Galilei connection satisfies τρ T ρ µν = ( d τ ) µν . A Galilei connection on ( M , τ , h ) may also be understood as a principal connection ω on the Galilei frame bundle G ( M ) of ( M , τ , h ) . Its local connection form with respect to a Galilei frame ( v , e a ) is then a locally defined one-form ( ω a b , ϖ a ) taking values in the Galilei algebra gal = so ( n ) i R n , with rotational part ω a b and boost part ϖ a . Due to the degeneracy of the metric structure, differently to the pseudo-Riemannian case Galilei connections are not uniquely determined by their torsion. Instead, with respect to a choice of unit timelike vector field v , a Galilei connection ∇ is uniquely determined by its torsion T and its Newton-Coriolis form Ω with respect to v . The latter is a two-form that may be written as where h µν are the components of the covariant space metric with respect to v , which is defined by h µν = h νµ , h µν v ν = 0, h µν h νρ = δ ρ µ -v ρ τµ . Alternatively, extending v to a local Galilei frame ( v , e a ) , Ω can be written in terms of the boost part of the local connection form and the dual frame as Conversely, choosing an arbitrary tensor field T subject to the constraints T ρ µν = -T ρ νµ and τρ T ρ µν = ( d τ ) µν , a unit timelike vector field v , and an arbitrary two-form Ω , there is a (unique) Galilei connection whose torsion is T and whose Newton-Coriolis form with respect to v is Ω . By definition, together with the tensorial R n + 1 -valued one-form θ on G ( M ) which corresponds to the canonical solder form of G ( M ) × Gal R n + 1 , any Bargmann form a combines to a tensorial ( R n + 1 ⊕ u ( 1 )) -valued form ( θ , i a ) on G ( M ) . The exterior covariant derivative d ω ( θ , i a ) of this tensorial form with respect to a Galilei connection ω is the so-called extended torsion of ω with respect to a . Note that the extended torsion depends only 6 on d a , such that it is invariant under type I gauge transformations. The local representative of the extended torsion with respect to a local Galilei frame σ = ( v , e a ) may be seen to be given by it consists of the frame components T A of the torsion and the so-called mass torsion d a + Ω . In particular, we see that knowing a Galilei connection's extended torsion with respect to some Bargmann form a amounts to knowing its torsion and Newton-Coriolis form. Hence, fixing a Bargmann form (or a type I gauge equivalence class of Bargmann forms), Galilei connections are uniquely determined by their extended torsion . In standard Newton-Cartan gravity, Newtonian gravity is encoded by so-called Newtonian connections. These are torsion-free Galilei connections (on a Galilei manifold with absolute time) whose curvature tensor satisfies the symmetry condition R µ ν ρ σ = R ν µ σ ρ (where the third index was raised with h ). This is equivalent to the Newton-Coriolis form (with respect to any v ) being closed, d Ω = 0 (this equivalence is not obvious, but requires a somewhat lengthy calculation). 7 By the Poincaré lemma, a torsion-free Galilei connection being Newtonian is locally equivalent to the Newton-Coriolis form being exact, Ω = -d a for some one-form a . This means that locally, Newtonian connections are precisely those Galilei connections whose extended torsion with respect to some Bargmann form vanishes.",
"pages": [
9,
10,
11
]
},
{
"title": "4.3 Parametrisation of standard Newton-Cartan geometry",
"content": "We can now see how our results imply the following classical result from standard Newton-Cartan geometry [ 5 , thm. 3 . 6 ], [ 10 , prop. 4 . 3 . 7 ]: Corollary 3 . Let ( M , τ , h ) be a Galilei manifold with absolute time, d τ = 0 , and let ω be a Newtonian connection on it. Then locally there exists a unit timelike vector field v such that the Newton-Coriolis form of ω with respect to v vanishes. Proof. Locally there exists a Bargmann form ˜ a such that ω is the (unique) Galilei connection whose extended torsion with respect to ˜ a vanishes. By theorem 1 , locally there exists a unit timelike vector field v such that the local representative ˜ a of ˜ a with respect to v is closed. Hence, the extended torsion vanishing implies 0 = d˜ a + Ω = Ω . We see that theorem 1 is the natural generalisation of this result to the TTNC case. Note that while Malament's proof of corollary 3 [ 10 , prop. 4 . 3 . 7 ] is very different in spirit from our approach, the original proof by Dombrowski and Horneffer [ 5 , thm. 3 . 6 ] is very close (at least to this special case of our gauge fixing): it also reduces Due to ∇ being Newtonian, we may (at least locally) demand it to be the unique extended-torsionfree Galilei connection with respect to a , which as an equation takes the form Ω = -d a and therefore determines a up to gauge transformations. In the case discussed above, we have Ω = τ ∧ d ϕ = -d ( ϕτ ) , so we may take a = ϕτ . So in this special situation and for this choice of a , the proportionality factor between a and τ plays the role of the Newtonian potential, which is why we call the locally Galileiinvariant quantity ˆ ϕ from ( 2 . 9 ) the locally Galilei-invariant potential . Note however that after performing a gauge transformation, the new ˆ ϕ will no longer be equal to the Newtonian potential ϕ with respect to v . the statement (which is given in an appreciably different formulation) to solving a Hamilton-Jacobi equation. 8 Using corollary 3 , we may (locally) parametrise, similar to theorem 2 , absolute-time Galilei manifolds with Newtonian connections in terms of just h and v (also expressing the condition d τ = 0 in terms of these): Theorem 4 . Let M be an ( n + 1 ) -dimensional differentiable manifold, h a positive semidefinite symmetric contravariant 2 -tensor field of rank n on M, and v a vector field on M such that (i) y = h + v ⊗ v is non-degenerate and (ii) ∂ [ µ (( y -1 ) ν ] ρ v ρ ) = 0 holds. Consider the unique one-form τ ∈ Ω 1 ( M ) satisfying ( 4 . 1 ) . This makes ( M , τ , h ) into a Galilei manifold with absolute time, and v a unit timelike vector field. Furthermore, we may define a Newtonian connection on ( M , τ , h ) by demanding that its torsion and Newton-Coriolis form with respect to v vanish, Ω = 0 . Locally, any Galilei manifold with absolute time and Newtonian connection is of this form.",
"pages": [
11,
12
]
},
{
"title": "5 Conclusion",
"content": "In this paper, we have shown that in twistless torsional Newton-Cartan geometry (both its type I and type II incarnations), there always exists a local gauge transformation transforming the locally Galilei-invariant potential to zero. Equivalently, there exist (locally) a gauge transformation and a unit timelike vector field such that the representative of the gauge-transformed Bargmann form with respect to this vector field vanishes. This generalises the classical result in standard Newton-Cartan geometry that for any Newtonian connection, locally there exists a unit timelike vector field that is geodesic and twist-free with respect to the connection, i.e. such that the Newton-Coriolis form of the connection with respect to this vector field vanishes. The above result also allowed us to argue that all of the geometric data determining a TTNC geometry up to gauge may be parametrised by just the space metric and a unit timelike vector field. This possibility of 're-packaging' of all the geometric data in the form of two contravariant tensor fields h µν and v µ enables the following interesting application. The field equations of standard Newton-Cartan gravity, as well as those of TTNC (type II) gravity that arise from expanding the Einstein equations in c -1 [ 15 ], are symmetric covariant 2 -tensor equations. This means that h µν and v µ together fully encode the possible configurations of the system under consideration (TTNC geometry), and their combination h µν + v µ v ν is non-degenerate and (tensorially) dual to the type of the field equations. This allows to study application of the canonical variational completion formalism [ 26 , 27 ] to TTNC gravity. If this application turned out successful, this would offer a new route towards action formulations of TTNC gravity, starting from the theory (defined by its geometry and field equations) alone, thus complementing existing approaches based on expansions of GR or on symmetry considerations [ 21 , 22 ]. We plan to pursue this route in future work.",
"pages": [
12,
13
]
},
{
"title": "Acknowledgements",
"content": "We wish to thank Domenico Giulini for helpful discussions.",
"pages": [
13
]
}
]
2024arXiv240212328V
https://arxiv.org/pdf/2402.12328.pdf
<document>
<section_header_level_1><location><page_1><loc_9><loc_92><loc_91><loc_93></location>Observations of compact stars and fermion-boson stars with a quartic self-interaction</section_header_level_1>
<text><location><page_1><loc_28><loc_87><loc_73><loc_90></location>Susana Valdez-Alvarado, 1, ∗ Alejandro Cruz-Osorio, 2, † J. M. D'avila, 1, ‡ L. Arturo Ure˜na-L'opez, 3, § and Ricardo Becerril 4, ¶</text>
<text><location><page_1><loc_12><loc_79><loc_89><loc_87></location>1 Facultad de Ciencias de la Universidad Aut'onoma del Estado de M'exico (UAEM'ex.), Instituto Literario No. 100, C.P. 50000, Toluca, Estado de M'exico, M'exico 2 Instituto de Astronom'ıa, Universidad Nacional Aut'onoma de M'exico, AP 70-264, Ciudad de M'exico 04510, M'exico 3 Departamento de F'ısica, DCI, Campus Le'on, Universidad de Guanajuato, 37150, Le'on, Guanajuato, M'exico. 4 Instituto de F'ısica y Matem'aticas, Universidad Michoacana de San Nicol'as de Hidalgo. Edif. C-3, 58040 Morelia, Michoac'an, M'exico</text>
<text><location><page_1><loc_41><loc_78><loc_59><loc_79></location>(Dated: February 20, 2024)</text>
<text><location><page_1><loc_18><loc_65><loc_83><loc_77></location>We investigated the possibility that compact stars could be described by a fermion-boson star with a quartic self-interaction in the boson sector. Specifically, by varying the polytropic constant K and adiabatic index Γ in the polytropic equation of state, the boson mass µ , and the selfinteraction parameter Λ, we construct equilibrium configurations of these mixed-stars with total mass compatible with the mass constraints obtained from observational data of the collaborations NICE, NICER/XMN-Newton, and LIGO. Our work confirms that the addition of a boson sector eases the comparison of neutron star models with gravitational events related to compact objects and that in such a case observations may have preference for a positive self-interaction in the boson sector.</text>
<section_header_level_1><location><page_1><loc_20><loc_61><loc_37><loc_62></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_49><loc_49><loc_59></location>Objects composed of a combination of fermions and bosons were first studied by Henriques and Liddle in [13] (see also [4, 5]), and their properties have been further explored in recent work [6-8]. These objects are known as fermion-boson stars (FBS) and are solutions of the Einstein-Klein-Gordon-hydrodynamic (EKGH) system of equations[9].</text>
<text><location><page_1><loc_9><loc_35><loc_49><loc_49></location>The boson sector of FBS consists of a massive scalar field that is minimally coupled to gravity, with a boson mass µ and a self-interaction parameter Λ in the scalar potential [10-13]. The fermion sector is usually modeled with a polytropic equation of state (EOS) of the form P ( ρ ) = Kρ Γ [7], where K and Γ take values that correspond to masses and compactness in the range of neutron stars (NS). The stability of FBS has been verified for a wide range of parameters, allowing for a variety of masses and sizes in the resulting objects [7, 8].</text>
<text><location><page_1><loc_9><loc_19><loc_49><loc_35></location>On the other hand, the successful detection of gravitational waves by the LIGO and Virgo scientific collaborations (LVC) has significantly widened the window to observe our universe and to understand gravity itself, but some of the gravitational wave events have brought new theoretical challenges and opportunities. The GW190814 event [14], for example, involved a black hole (BH) with a mass of 22 . 2 M ⊙ -24 . 3 M ⊙ and a spin of ≤ 0 . 07, accompanied by a compact object with a mass of 2 . 5 M ⊙ -2 . 67 M ⊙ . This binary source of a gravitational wave has the smallest mass ratio ever measured for a similar system, at</text>
<text><location><page_1><loc_52><loc_57><loc_92><loc_63></location>0 . 112 +0 . 008 -0 . 009 . The nature of the second object is unknown, and there are no constraints on its radius, so it could be either a very light black hole or a very heavy neutron star (NS), which would make it the heaviest NS ever observed.</text>
<text><location><page_1><loc_52><loc_39><loc_92><loc_56></location>The peculiarities of events GW170817 and GW190814 have been used to set limits on the maximum mass and radii of generic neutron stars [15-17], which in turn have been used to constrain the parameters of polytropic fluids in models [18]. More recent constraints on the properties of neutron stars have been found from observations of the NICER telescope [19-21]. In [22], it was already suggested that an FBS, with a boson mass of the order of µ ∼ 10 -9 eV, could help explain the constitution of the self-gravitating objects behind the occurrence of these events, without self-interaction in the boson sector (i.e. Λ = 0).</text>
<text><location><page_1><loc_52><loc_12><loc_92><loc_39></location>In this paper, we investigate the possibility that the secondary component of the event GW190814 is a FBS with a quartic self-interaction in the boson sector. For this, we have kept in mind four facts: (a) the mass of the secondary object for the LIGO gravitational wave signal GW190814 lies in the interval [2 . 5 , 2 . 67] M ⊙ ; (b) for Rezzolla - Nathanail simulations for neutron stars, its mass is located in the interval [2 . 0 , 2 . 3] M ⊙ ; (c) for the NICER collaborations its corresponding mass interval is [1 . 95 , 2 . 2] M ⊙ for PSRJ0030+0451 and [1 . 2 , 1 . 65] M ⊙ for PSJ0740+6620 ; (d) ultra-light bosons have been proposed as candidates to be dark matter in our universe [23-34], and then these new events could give hints about their possible existence. We constructed mixed star configurations selecting Γ and K in such a way that our numerical results yield total masses that match the bounds mentioned in (a), (b), and (c) above. Furthermore, we also consider the constraint of the sound speed c s < 1 (in natural units) in the fluid sector of the FBS.</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_11></location>This paper is structured as follows. Section II outlines the equations of motion of the coupled Einstein-</text>
<text><location><page_2><loc_9><loc_83><loc_49><loc_93></location>Klein-Gordon-hydrodynamic system and the boundary conditions that allow us to construct equilibrium configurations of FBS with a given quartic self-interaction in the boson sector. We also discuss the general numerical results in the case of FBS with total masses and radii similar to those of NS and the influence of the boson self-interaction in those quantities.</text>
<text><location><page_2><loc_9><loc_74><loc_49><loc_83></location>In Sec. III we present the comparison of our models with observational data of selected gravitational events, and discuss the general properties of the boson sector that can help the fluid star to be in more agreement with the data. Finally, in Sec. IV, we provide a summary of our results and final remarks about our study.</text>
<section_header_level_1><location><page_2><loc_12><loc_70><loc_45><loc_71></location>II. EQUILIBRIUM CONFIGURATIONS</section_header_level_1>
<text><location><page_2><loc_9><loc_58><loc_49><loc_68></location>We model the boson sector of the FBS of interest with a complex scalar field ϕ that has a scalar potential V ( ϕ ). The fermion sector is represented by a perfect fluid with rest-mass density ρ , pressure P , internal energy ϵ and 4-velocity u µ . The (relativistic) equations of motion for this system, expressed in geometrical units with G = c = ℏ = 1, are as follows:</text>
<formula><location><page_2><loc_10><loc_51><loc_49><loc_57></location>G µν = 8 π ( T ( ϕ ) µν + T ( f ) µν ) , □ 2 ϕ -ϕ dV ( ϕ ) d ( | ϕ | 2 ) = 0 , (1a) ∇ µ T ( f ) µν = 0 , ∇ µ ( ρu µ ) = 0 , (1b)</formula>
<text><location><page_2><loc_52><loc_90><loc_92><loc_93></location>where the stress-energy tensors of the bosonic and fermionic components are, respectively,</text>
<formula><location><page_2><loc_55><loc_85><loc_92><loc_88></location>T ϕ µν = 1 2 ( ∂ µ ϕ∂ ν ϕ ∗ ) -1 2 g µν ( ∂ α ϕ ∗ ∂ α ϕ +2 V ) , (1c)</formula>
<formula><location><page_2><loc_55><loc_83><loc_92><loc_85></location>T f µν = [ ρ (1 + ϵ ) + P ] u µ u ν + Pg µν . (1d)</formula>
<text><location><page_2><loc_52><loc_78><loc_92><loc_81></location>The boson sector is endowed with a quartic scalar potential of the form</text>
<formula><location><page_2><loc_63><loc_73><loc_92><loc_77></location>V ( ϕ ) = µ 2 2 | ϕ | 2 + λ 4 | ϕ | 4 , (2)</formula>
<text><location><page_2><loc_52><loc_67><loc_92><loc_71></location>that represents an ensemble of boson particles with mass µ and a self-interaction parameter λ , which may be either positive or negative.</text>
<text><location><page_2><loc_52><loc_52><loc_92><loc_67></location>We are interested in equilibrium configurations, so we assume a static and spherically symmetric metric, with the line element ds 2 = -α 2 ( r ) dt 2 + a 2 ( r ) dr 2 + r 2 d Ω 2 . The scalar field is expressed in the standard harmonic form ϕ ( t, r ) = ϕ ( r ) e -iωt , with ω being the characteristic frequency of the solution, while all fermionic variables depend only on the radial coordinate r . To simplify the equations, we introduce a set of new variables: Ω = ω/µ , √ 4 πϕ → ϕ , Λ = λ/ (4 πµ 2 ), 4 πρ → ρ and 4 πP → P , with which Eqs. (1) explicitly become,</text>
<formula><location><page_2><loc_25><loc_45><loc_92><loc_48></location>a ' = a 2 ( 1 -a 2 r + a 2 r [( Ω 2 α 2 +1+ Λ 2 ϕ 2 ) µ 2 ϕ 2 + Φ 2 a 2 +2 ρ (1 + ϵ ) ]) , (3a)</formula>
<formula><location><page_2><loc_25><loc_42><loc_92><loc_45></location>α ' = α 2 ( a 2 -1 r + a 2 r [( Ω 2 α 2 -1 -Λ 2 ϕ 2 ) µ 2 ϕ 2 + Φ 2 a 2 +2 P ]) , (3b)</formula>
<text><location><page_2><loc_9><loc_35><loc_49><loc_37></location>for the metric variables α and a , whereas for the field and fluid variables we get</text>
<formula><location><page_2><loc_9><loc_25><loc_49><loc_34></location>ϕ ' = Φ , (3c) Φ ' = ( 1 -Ω 2 α 2 +Λ ϕ 2 ) a 2 µ 2 ϕ -( 2 r + α ' α -a ' a ) Φ , (3d) P ' = -α ' α [ ρ (1 + ϵ ) + P ] . (3e)</formula>
<text><location><page_2><loc_9><loc_20><loc_49><loc_25></location>Here, a prime denotes derivatives with respect to the radial coordinate r and the internal energy is defined as ϵ = P/ ( ρ (Γ -1)).</text>
<text><location><page_2><loc_9><loc_16><loc_49><loc_20></location>The corresponding boundary conditions that guaranty regularity at the origin and asymptotic flatness at infinity are</text>
<formula><location><page_2><loc_13><loc_13><loc_49><loc_15></location>a (0) = 1 , α (0) = 1 lim r →∞ a ( r ) = 1 , (4a)</formula>
<formula><location><page_2><loc_13><loc_8><loc_49><loc_10></location>ρ (0) = ρ 0 , P (0) = Kρ Γ 0 , lim r →∞ P ( r ) = 0 . (4c)</formula>
<formula><location><page_2><loc_13><loc_10><loc_49><loc_13></location>ϕ (0) = ϕ 0 , Φ(0) = 0 , lim r →∞ ϕ ( r ) = 0 , (4b)</formula>
<text><location><page_2><loc_52><loc_35><loc_92><loc_37></location>The total mass of the equilibrium configurations, M T is obtained using the Schwarzschild definition,</text>
<formula><location><page_2><loc_62><loc_30><loc_92><loc_33></location>M T = lim r →∞ r 2 [ 1 -1 a 2 ( r ) ] . (5)</formula>
<text><location><page_2><loc_52><loc_20><loc_92><loc_29></location>The numerical results from all the equations are dimensionless and given in code units. We use the standard convention for NS studies, where M T = 1 is equal to one solar mass M ⊙ in physical units. To obtain the total physical mass M and the radius 99% R 99 of the equilibrium configurations, we use the following expressions,</text>
<formula><location><page_2><loc_59><loc_15><loc_92><loc_17></location>M = 1 M ⊙ M T , R 99 ≃ 1 . 47 km r 99 , (6)</formula>
<text><location><page_2><loc_52><loc_8><loc_92><loc_14></location>where r 99 is the value of r containing 99% of the total mass M T but given in code units. A typical solution of a NS is then obtained without a scalar field ( ϕ 0 = 0) and for appropriate values of fluid quantities ρ 0 , K and Γ.</text>
<text><location><page_3><loc_9><loc_85><loc_49><loc_93></location>For the mixed case of a FBS, we only need to solve the equations of motion (3) for different values of the central field density ϕ 0 , the boson mass µ and the selfinteracting parameter Λ. In particular, the boson mass µ in geometrical units is related to its physical value µ B by</text>
<formula><location><page_3><loc_20><loc_82><loc_49><loc_84></location>µ = 0 . 75( µ B c 2 / 10 -10 eV) . (7)</formula>
<text><location><page_3><loc_9><loc_72><loc_49><loc_81></location>We solved the equations of motion of FBS for three pairs of values Γ and K namely { (Γ, K ) } = { (2 . 8, 5 . 6 × 10 4 ), (2 . 85, 7 × 10 4 ), (2 . 9, 9 × 10 4 ) } ; for each single pair we considering the following intervals for their free parameters: 0 ≤ ϕ 0 ≤ 0 . 2, 0 . 006 ≤ ρ 0 ≤ 0 . 06 and -50 ≤ Λ ≤ 50 1 .</text>
<text><location><page_3><loc_9><loc_61><loc_49><loc_72></location>The general behavior of the resulting configurations, in terms of their rescaled physical total mass M and the 99% radius R 99 , is shown in Fig. 1 for the cases Λ = 0 , 50 , -50 with Γ = 2 . 8 and K = 5 . 6 × 10 4 . In all cases, the mass of the boson is µ = 1, which according to Eq. (7) means that µ B c 2 ≃ 1 . 33 × 10 -10 eV. This is the expected mass for boson stars with a total mass of the order of solar masses and a radius in the range of tens of kilometers [35].</text>
<text><location><page_3><loc_9><loc_47><loc_49><loc_61></location>Each of the curves in the graphs represents a family of equilibrium configurations with a fixed value of ϕ 0 , and for which the central mass density ρ 0 is changed from bottom to top in each curve. Furthermore, as indicated in the graphs, the total area covered by the set of curves is limited by the extreme values of the central field ϕ 0 , whose corresponding curves have been highlighted. It should be noted that the curve ϕ 0 = 0 is purely fermionic and represents the standard NS for the chosen fluid parameters.</text>
<text><location><page_3><loc_9><loc_35><loc_49><loc_46></location>One last important feature is the stability of the equilibrium configurations. As previously studied in [6], see also [8], there is a standard method for determining the stability of FBS in terms of their number of bosons and fermions, which is slow and long. However, given that our FBS are fermion-dominated, we can take a shortcut by following the standard method for fermionic (fluid) stars.</text>
<text><location><page_3><loc_9><loc_23><loc_49><loc_35></location>For a given ϕ 0 , we increase the central density ρ 0 until the maximum mass of the star is reached. All configurations with a central density higher than this should be unstable. It has been demonstrated in [7] that this shortened method is consistent with the standard one for the small values of ϕ 0 discussed here. For the sake of completeness, the configurations at the stability limit are also shown in Fig. 1.</text>
<text><location><page_3><loc_9><loc_16><loc_49><loc_23></location>When Λ = 0, the total mass and radius of stable FBS are in the ranges of 0 . 73 < M/M ⊙ < 3 and 10 < R 99 / km < 22, respectively. The presence of the self-interaction parameter increases the area of the region covered by the curves for Λ = 50 and decreases it</text>
<figure>
<location><page_3><loc_51><loc_22><loc_93><loc_94></location>
<caption>FIG. 1. The top, middle and bottom panels of the figure show the equilibrium configurations of FBS for different values of the self-interaction parameter Λ = 0 , 50 , -50, respectively, in terms of their physical total mass M and 99% radius R 99 . The cases Λ = 50 , -50 are superimposed on the case Λ = 0 to compare their coverage of the parameter space. For reference, the configurations corresponding to the special values for the variation of the central field ϕ 0 = 0 , 0 . 1 , 0 . 2 are highlighted in each plot, while the black line represents the stability boundary. Further details can be found in the text.</caption>
</figure>
<text><location><page_4><loc_9><loc_85><loc_49><loc_93></location>for Λ = -50, while the mass and radius ranges remain similar for the three cases shown in Fig. 1. There is some overlap of the regions in the different cases, which implies that the self-interaction parameter has some degree of degeneracy with the central density for certain values of the mass and radius of the configurations.</text>
<text><location><page_4><loc_9><loc_73><loc_49><loc_84></location>For a positive self-interaction parameter, the mass of a boson star M B is given by M B ≲ 0 . 165Λ 1 / 2 M ⊙ /µ [10] in the so-called Thomas-Fermi limit Λ ≫ 1, and then M B ≲ 8 M ⊙ for Λ = 50. This shows that the boson part of the FBS can contribute more to the mass of the star for the same value of ϕ 0 , and then the bundle of curves in the middle panel of Fig. 1 can cover a larger portion of the parameter space.</text>
<text><location><page_4><loc_9><loc_63><loc_49><loc_73></location>On the contrary, for a negative self-interaction parameter the upper bound on the total mass in the ThomasFermi limit | Λ | ≫ 1 is M B < 1 . 4 | Λ | -1 / 2 M ⊙ /µ [36], so that M B < 0 . 2 M ⊙ for Λ = -50. Being the mass contribution of the boson star this small, a narrow bundle of curves close to the anchor fermion star is to be expected, as the one shown in the bottom panel of Fig. 1.</text>
<section_header_level_1><location><page_4><loc_11><loc_58><loc_47><loc_59></location>III. COMPARISON WITH OBSERVATIONS</section_header_level_1>
<text><location><page_4><loc_9><loc_49><loc_49><loc_56></location>Our goal in this section is to find regions in this parameter space for which the values of the total mass M and radius R 99 , of the resultant configurations, are compatible with those of NS as inferred from theoretical models and the NICER's observations.</text>
<text><location><page_4><loc_9><loc_29><loc_49><loc_49></location>Figures 2 and 3 display the masses and radii of FBS obtained in physical units, superimposed on the properties of objects detected by different observational events. The green and red regions represent the density distributions determined by the NICER Collaboration for PSR J0740 + 6620 [19, 20] and PSRJ0030 + 0451 [21], respectively. The yellow and cyan regions indicate the constraints on NS mass values obtained from simulations of a large number of polytropic models conducted by Rezzolla et al. [15] and Nathanail et al. [16] respectively. Whereas the green band represents the overlap region between both simulations. The pink region corresponds to the mass associated with the second compact object detected in GW190814 [14] by LIGO, whose source is unknown.</text>
<text><location><page_4><loc_9><loc_13><loc_49><loc_29></location>The case with fixed value Λ = 10 is presented in Fig. 2 in terms of the central field strength ϕ 0 . In all instances, the value ϕ 0 = 0 corresponds to the NS (white curve), which we also referred to before as the anchor fermion star, constructed from three sets of values of the fluid parameters (Γ , K ): (2 . 80 , 5 . 6 × 10 4 ) (top row), (2 . 85 , 7 × 10 4 ) (middle row) and (2 . 90 , 9 × 10 4 ) (bottom row), as indicated in each graph. Likewise, three values of the boson mass were considered for the solution of the boson part of the star: µ = 0 . 5 (left column), µ = 1 . 0 (middle column) and µ = 2 . 0 (right column).</text>
<text><location><page_4><loc_9><loc_9><loc_49><loc_13></location>As noted in [22], as we increase the value of ϕ 0 , we can cover a larger area and agree with the regions suggested by the different observations. Notice that the NS pro-</text>
<text><location><page_4><loc_52><loc_78><loc_92><loc_93></location>ile mainly determines the mass range of the whole set of equilibrium configurations, while the scalar field helps to extend the range in the size of the FBS, as already pointed out in the discussion of Figs. 1. In particular, the variation on the field strength ϕ 0 seems to give the required degree of freedom (width) to cover the confidence regions of PSRJ0030 + 0451 and PSRJ0740 + 6620. It can be seen that the best options appear to be those with (Γ , K ) = (2 . 85 , 7 × 10 4 ) , (2 . 90 , 9 × 10 4 ) and with the boson mass µ of the order of unity ( µ B c 2 ≃ 10 -10 eV).</text>
<text><location><page_4><loc_52><loc_67><loc_92><loc_79></location>In Fig. 3, we present the masses and radii of FBS obtained with the central value of the scalar field fixed at ϕ 0 = 0 . 1, and for different values of the self-interaction parameter Λ in the range ( -50 , 50). Given that the central scalar field is non-zero, the anchor configuration in these cases is not an NS but one FBS with Λ = 0, and then configurations with Λ > 0 (Λ < 0) have smaller radii (larger radii) in comparison to the anchor FBS.</text>
<text><location><page_4><loc_52><loc_58><loc_92><loc_67></location>In contrast to the cases in Fig. 2, the best scenarios seem to be those in the column µ = 2, for the three sets of values of the fluid parameters. The values of Λ provide the extra width in the parameter space to cover the density regions suggested by the observations, since Λ can also influence the mass and radius of the configurations.</text>
<text><location><page_4><loc_52><loc_47><loc_92><loc_58></location>It is clear from the curves in Figs. 2 and 3 that the masses and radii associated with the numerical configurations are capable of crossing all the areas suggested by the observations. To illustrate this in more detail, we have selected three cases with Λ = 10, ϕ 0 = 0 . 1125, K = 5 . 6 × 10 4 , Γ = 2 . 8, and µ = 1, for which the corresponding three FBS configurations have masses and radii that match the NICER and LIGO data.</text>
<text><location><page_4><loc_52><loc_28><loc_92><loc_46></location>These cases are represented by the red stars in Fig. 2, and their internal morphology is presented in Fig. 4. The gray color map represents the boson density, whereas the rest-mass density of fluid is represented in the blue-red scale. In general, we see that the contribution of the boson sector decreases as the configuration is more massive and compact, as a result of the concentration induced by the gravitational force of the fluid component, which is also the dominant one. The central sound speed of the FBS is also shown, which was calculated using c 2 s = Pγ ( γ -1) / [ Pγ -ρ ( γ -1)]. Even for the most extreme case with M = 2 . 56 M ⊙ , the sound speed remains below unity.</text>
<text><location><page_4><loc_52><loc_19><loc_92><loc_27></location>As mentioned above, the secondary component of the recently reported event GW190814 is of unknown nature, but its extreme mass indicates that it could be a light BH or a very heavy NS. However, our numerical results in Figs. 2 and 3 make it plausible to identify this secondary component with an FBS with a quartic self-interaction.</text>
<text><location><page_4><loc_52><loc_8><loc_92><loc_19></location>To verify this more carefully, we have performed a detailed survey in the parameter space { Γ , K, µ, Λ } to determine the regions corresponding to an FBS with a fixed total mass of the order of M ≃ 2 . 16 M ⊙ . The results are shown in Fig. 5, for the total radius R as a function of the self-interaction parameter Λ (top panel) and as a function of the central field density ϕ 0 (bottom panel). The</text>
<text><location><page_5><loc_52><loc_92><loc_53><loc_93></location>φ</text>
<text><location><page_5><loc_90><loc_91><loc_92><loc_92></location>20</text>
<figure>
<location><page_5><loc_9><loc_35><loc_91><loc_93></location>
<caption>FIG. 2. We show a comparison between plots of mixed stars masses vs radii (blue-white curves) with Λ = 10 and the observational data obtained from NICER (red and green regions) and LIGO (pink horizontal band) collaborations as well as from simulations performed by Rezzolla (green-yellow band) and Nathanail (cyan band). Each curve was constructed numerically taking a fixed value for ϕ 0 and varying the central density ρ 0 in the range [0 . 006 , 0 . 06]. Each row corresponds to a given set of fluid parameters (Γ , N ) as indicated on the labels, and each column gives the results for a fixed value of the boson mass µ . See the text for more details.</caption>
</figure>
<text><location><page_5><loc_9><loc_19><loc_49><loc_22></location>values of the polytropic parameters are the same as those considered in Figs. 2 and 3 above.</text>
<text><location><page_5><loc_9><loc_9><loc_49><loc_19></location>In general, we note that the presence of the scalar field ϕ , in terms of its central density or its self-interaction, reduces the size of the FBS, with the total radius reaching values on the order of R 99 ≃ 12 km. The most extreme cases are reached for the case µ = 2, together with the highest values of ϕ 0 and Λ. For these extreme cases shown in Fig. 5, the compactness of the FBS is of the</text>
<text><location><page_5><loc_52><loc_19><loc_92><loc_22></location>order of C = 0 . 22, which is approximately 16% higher than the anchor NS alone.</text>
<text><location><page_5><loc_52><loc_8><loc_92><loc_17></location>Another quantity of interest is the central sound speed c s of the FBS with M = 2 . 16 M ⊙ , which we show in Fig. 6. Similarly to the case of the compactness of the star, we see that the presence of the scalar field ϕ and its self-interaction Λ increases the value of c s , although in all cases the top sound speed is of the order of c s ≃ 0 . 9.</text>
<text><location><page_6><loc_90><loc_91><loc_92><loc_92></location>50</text>
<figure>
<location><page_6><loc_9><loc_34><loc_91><loc_93></location>
<caption>FIG. 3. Comparison between families of mass-radius diagrams of mixed stars (multicolored curves) with ϕ 0 = 0 . 1 and the observational data obtained from NICER (red and green regions) and LIGO (pink horizontal band) collaborations as well as from simulations performed by Rezzolla (green-yellow band) and Nathanail (cyan band). Each curve was constructed numerically taking a fixed value for Λ and varying ρ 0 in the range [0 . 006 , 0 . 06]. Each row corresponds to a given set of fluid parameters (Γ , N ) as indicated on the labels, and each column gives the results for a fixed value of the boson mass µ . See the text for more details.</caption>
</figure>
<section_header_level_1><location><page_6><loc_10><loc_21><loc_47><loc_22></location>IV. CONCLUSIONS AND FINAL REMARKS</section_header_level_1>
<text><location><page_6><loc_9><loc_9><loc_49><loc_19></location>We have constructed FBS configurations with a selfinteraction term in the boson sector and considered different values of their physical parameters to match the data for NS obtained from NICER, Rezzolla, and Nathanail collaborations as well as from the LIGO gravitational wave signal GW190814. For the fluid parameters, we picked values that yielded NS masses in the</text>
<text><location><page_6><loc_52><loc_16><loc_92><loc_22></location>vicinity of the values reported for different observed objects, and then varied the boson parameters, mass and self-interaction, to find their influence in the mass, size, and morphology of the corresponding FBS.</text>
<text><location><page_6><loc_52><loc_9><loc_92><loc_16></location>In general terms, the variation of the boson parameters allows our models to cover a wider area in the parameter space of the observed objects, which means that the addition of a boson sector to a given NS configuration could help explain the presence of different types of object. Ac-</text>
<figure>
<location><page_7><loc_11><loc_68><loc_90><loc_93></location>
<caption>FIG. 4. Two dimensional morphology of representative fermion-boson stars to model three observational constraints. The left panel shows an FBS with mass M = 1 . 4 M ⊙ and size R 99 = 12 . 9 km , which meets the NICER observational constraints for the pulsar PSRJ0030+0451 . The middle panel is an FBS with mass M = 2 . 1 M ⊙ and radius R = 13 . 3 km to model the pulsar PSRJ0740+6620 , satisfying the mass estimated by theory and NICER observations. The right panel shows the morphology of the FBS to model the event GW190814 with mass M = 2 . 56 M ⊙ and radius R 99 = 12 . 1 km . The fermion component is represented by the rest-mass density, and the boson component by the scalar field density, both in logarithmic scale. The three FBS are modeled by the same family of solutions (see the red stars in Fig. 2), by fixing the self-interaction parameter Λ = 10, the central strength of the scalar field ϕ 0 = 0 . 1125, the equations for the state parameters Γ = 2 . 8 and K new = 5 . 6 × 10 4 , and the mass of the boson µ = 1. See the text for more details.</caption>
</figure>
<figure>
<location><page_7><loc_10><loc_26><loc_91><loc_52></location>
<caption>FIG. 5. The size of the FBS as function of the self-interaction parameter (top panel) and the scalar field strength (bottom panel) for star with total mass M = 2 . 16 M ⊙ . Changes in EoS are represented by black, blue, and red colors, while line styles show the effects of varying the boson mass µ . See the text for more details.</caption>
</figure>
<text><location><page_7><loc_9><loc_10><loc_49><loc_18></location>cording to our results, the best options seem to have a boson mass of order µ ≃ 1 in our units, which means that the total mass of the corresponding boson star is of the same order of magnitude as that of the selected NS (or anchor NS, as we call it in the main text). Similarly, there is a preference for positive values of the self-interaction</text>
<text><location><page_7><loc_52><loc_16><loc_92><loc_18></location>parameter, as a negative value of it makes it difficult to reconcile our theoretical results with observations.</text>
<text><location><page_7><loc_52><loc_9><loc_92><loc_14></location>Our work confirms and extends the results presented in [22], in that the addition of a boson sector eases the comparison of NS models with different observations. We have already mentioned the preference for the value</text>
<text><location><page_8><loc_10><loc_81><loc_12><loc_81></location>c</text>
<figure>
<location><page_8><loc_11><loc_68><loc_50><loc_93></location>
<caption>FIG. 6. Sound speed at the center of the FBS as function of the self-interaction parameter (top panel), and as function of the amplitude at the star center. In all models, we fixed the mass of the FBS to M = 2 . 16 M ⊙ , the corresponding sizes are reported in Fig. 5. The colors of the lines represent the three equations of states (Γ = 2 . 8 , K = 5 . 6 × 10 4 ), (Γ = 2 . 85 , K = 7 × 10 4 ), and (Γ = 2 . 9 , K = 9 × 10 4 ), respectively. The line styles correspond to different choices for the boson mass, µ = 0 . 5 , 1 . 0 , 1 . 5 , 2 . 0. See the text for more details.</caption>
</figure>
<text><location><page_8><loc_53><loc_91><loc_53><loc_92></location>1</text>
<text><location><page_8><loc_53><loc_91><loc_54><loc_92></location>.</text>
<text><location><page_8><loc_54><loc_91><loc_55><loc_92></location>00</text>
<text><location><page_8><loc_53><loc_88><loc_53><loc_89></location>0</text>
<text><location><page_8><loc_53><loc_88><loc_54><loc_89></location>.</text>
<text><location><page_8><loc_54><loc_88><loc_55><loc_89></location>95</text>
<text><location><page_8><loc_53><loc_85><loc_53><loc_86></location>0</text>
<text><location><page_8><loc_53><loc_85><loc_54><loc_86></location>.</text>
<text><location><page_8><loc_54><loc_85><loc_55><loc_86></location>90</text>
<text><location><page_8><loc_53><loc_82><loc_53><loc_83></location>0</text>
<text><location><page_8><loc_53><loc_82><loc_54><loc_83></location>.</text>
<text><location><page_8><loc_54><loc_82><loc_55><loc_83></location>85</text>
<text><location><page_8><loc_53><loc_79><loc_53><loc_80></location>0</text>
<text><location><page_8><loc_53><loc_79><loc_54><loc_80></location>.</text>
<text><location><page_8><loc_54><loc_79><loc_55><loc_80></location>80</text>
<text><location><page_8><loc_53><loc_76><loc_53><loc_77></location>0</text>
<text><location><page_8><loc_53><loc_76><loc_54><loc_77></location>.</text>
<text><location><page_8><loc_54><loc_76><loc_55><loc_77></location>75</text>
<text><location><page_8><loc_53><loc_73><loc_53><loc_74></location>0</text>
<text><location><page_8><loc_53><loc_73><loc_54><loc_74></location>.</text>
<text><location><page_8><loc_54><loc_73><loc_55><loc_74></location>70</text>
<text><location><page_8><loc_53><loc_70><loc_53><loc_71></location>0</text>
<text><location><page_8><loc_53><loc_70><loc_54><loc_71></location>.</text>
<text><location><page_8><loc_54><loc_70><loc_55><loc_71></location>65</text>
<text><location><page_8><loc_55><loc_69><loc_56><loc_70></location>0</text>
<text><location><page_8><loc_56><loc_69><loc_56><loc_70></location>.</text>
<text><location><page_8><loc_56><loc_69><loc_58><loc_70></location>00</text>
<text><location><page_8><loc_61><loc_69><loc_61><loc_70></location>0</text>
<text><location><page_8><loc_61><loc_69><loc_62><loc_70></location>.</text>
<text><location><page_8><loc_62><loc_69><loc_63><loc_70></location>02</text>
<text><location><page_8><loc_66><loc_69><loc_67><loc_70></location>0</text>
<text><location><page_8><loc_67><loc_69><loc_67><loc_70></location>.</text>
<text><location><page_8><loc_67><loc_69><loc_69><loc_70></location>04</text>
<text><location><page_8><loc_71><loc_69><loc_72><loc_70></location>0</text>
<text><location><page_8><loc_72><loc_69><loc_73><loc_70></location>.</text>
<text><location><page_8><loc_73><loc_69><loc_74><loc_70></location>06</text>
<text><location><page_8><loc_77><loc_69><loc_78><loc_70></location>0</text>
<text><location><page_8><loc_78><loc_69><loc_78><loc_70></location>.</text>
<text><location><page_8><loc_78><loc_69><loc_79><loc_70></location>08</text>
<text><location><page_8><loc_82><loc_69><loc_83><loc_70></location>0</text>
<text><location><page_8><loc_83><loc_69><loc_83><loc_70></location>.</text>
<text><location><page_8><loc_83><loc_69><loc_85><loc_70></location>10</text>
<text><location><page_8><loc_88><loc_69><loc_88><loc_70></location>0</text>
<text><location><page_8><loc_88><loc_69><loc_89><loc_70></location>.</text>
<text><location><page_8><loc_89><loc_69><loc_90><loc_70></location>12</text>
<text><location><page_8><loc_72><loc_68><loc_73><loc_69></location>φ</text>
<text><location><page_8><loc_73><loc_68><loc_73><loc_68></location>0</text>
<text><location><page_8><loc_9><loc_48><loc_49><loc_57></location>µ ≃ 1, which is not surprising given the typical masses and sizes of compact objects suggested by the observations we used. The interesting new result is that the self-interaction, the next-to-leading order parameter in the description of the boson sector, in particular for the scalar field potential, should be positive.</text>
<text><location><page_8><loc_9><loc_35><loc_49><loc_47></location>Some other comments are in turn. Our anchor NS was modeled as a polytropic fluid, for which the parameter values were selected according to previous studies of NS [18]. However, our study could also consider other types of equation of state for the fermion sector, such as those chosen in [22], for which we would expect the same qualitative results obtained so far: preference of data for µ ≃ 1 and Λ ≳ 0.</text>
<text><location><page_8><loc_9><loc_18><loc_49><loc_35></location>Ultralight bosons have been proposed as candidates for dark matter in our universe, and a boson with mass µ B c 2 ∼ 10 -10 eV would certainly qualify as ultralight, although such a mass would be 10 orders of magnitude larger than the preferred astrophysical value of µ B c 2 ∼ 10 -22 eV [31, 34]. Our new result Λ ≳ 0, if taken at face value, would mean that there is an extra constraint on dark matter models with axion-like bosons (with negative self-interaction), at least for the cases for which µ B c 2 ∼ 10 -10 eV, as there are differentiable consequences at astrophysical and cosmological scales for the formation of cosmic structure from the sign of the self-</text>
<text><location><page_8><loc_52><loc_55><loc_77><loc_57></location>in the potential [37-40].</text>
<text><location><page_8><loc_52><loc_38><loc_92><loc_55></location>As already noted in [22], for comparison with observations, we just took the confidence regions obtained by other groups that considered models quite different from ours to fit the data. In this respect, our constraints should be considered conservative and more like a proofof-concept for FBS with a self-interaction. A separate study with a direct comparison with raw data from the different observations would be necessary to have a more accurate judgment of the appropriateness of FBS and the role of their self-interaction in each of the reported gravitational events. This is work in progress, which we expect to report elsewhere.</text>
<section_header_level_1><location><page_8><loc_62><loc_34><loc_82><loc_35></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_8><loc_52><loc_18><loc_92><loc_32></location>S.V.A and J.M.D. acknowledge support from Programa de Desarrollo Profesional Docente de la Secretaria de Educaci'on P'ublica (PRODEP-SEP), under No. UAEM-CA-14 project. R.B. acknowledges partial support from C.I.C-UMSNH. LAU-L. acknowledges partial support from the Programa para el Desarrollo Profesional Docente; Direcci'on de Apoyo a la Investigaci'on y al Posgrado, Universidad de Guanajuato; CONACyT M'exico under Grants No. 286897, No. 297771, No. 304001; and the Instituto Avanzado de Cosmolog'ıa Collaboration.</text>
<text><location><page_8><loc_51><loc_81><loc_52><loc_82></location>s</text>
<text><location><page_8><loc_51><loc_81><loc_52><loc_81></location>c</text>
<text><location><page_8><loc_60><loc_90><loc_61><loc_91></location>µ</text>
<text><location><page_8><loc_62><loc_90><loc_63><loc_91></location>= 0</text>
<text><location><page_8><loc_63><loc_90><loc_64><loc_91></location>.</text>
<text><location><page_8><loc_64><loc_90><loc_64><loc_91></location>5</text>
<text><location><page_8><loc_60><loc_88><loc_61><loc_89></location>µ</text>
<text><location><page_8><loc_62><loc_88><loc_63><loc_89></location>= 1</text>
<text><location><page_8><loc_63><loc_88><loc_64><loc_89></location>.</text>
<text><location><page_8><loc_64><loc_88><loc_64><loc_89></location>0</text>
<text><location><page_8><loc_58><loc_84><loc_61><loc_85></location>Γ = 2</text>
<text><location><page_8><loc_61><loc_84><loc_62><loc_85></location>.</text>
<text><location><page_8><loc_62><loc_84><loc_63><loc_85></location>80</text>
<text><location><page_8><loc_58><loc_82><loc_61><loc_83></location>Γ = 2</text>
<text><location><page_8><loc_61><loc_82><loc_62><loc_83></location>.</text>
<text><location><page_8><loc_62><loc_82><loc_63><loc_83></location>85</text>
<text><location><page_8><loc_58><loc_81><loc_61><loc_82></location>Γ = 2</text>
<text><location><page_8><loc_61><loc_81><loc_62><loc_82></location>.</text>
<text><location><page_8><loc_62><loc_81><loc_63><loc_82></location>90</text>
<text><location><page_8><loc_65><loc_92><loc_70><loc_93></location>Λ = 10</text>
<text><location><page_8><loc_71><loc_92><loc_73><loc_93></location>M</text>
<text><location><page_8><loc_73><loc_92><loc_75><loc_93></location>= 2</text>
<text><location><page_8><loc_75><loc_92><loc_76><loc_93></location>.</text>
<text><location><page_8><loc_76><loc_92><loc_77><loc_93></location>16</text>
<text><location><page_8><loc_78><loc_92><loc_79><loc_93></location>M</text>
<text><location><page_8><loc_70><loc_90><loc_71><loc_91></location>µ</text>
<text><location><page_8><loc_71><loc_90><loc_73><loc_91></location>= 1</text>
<text><location><page_8><loc_73><loc_90><loc_73><loc_91></location>.</text>
<text><location><page_8><loc_73><loc_90><loc_74><loc_91></location>5</text>
<text><location><page_8><loc_70><loc_88><loc_71><loc_89></location>µ</text>
<text><location><page_8><loc_71><loc_88><loc_73><loc_89></location>= 2</text>
<text><location><page_8><loc_73><loc_88><loc_73><loc_89></location>.</text>
<text><location><page_8><loc_73><loc_88><loc_74><loc_89></location>0</text>
<text><location><page_8><loc_79><loc_91><loc_80><loc_92></location>/circledot</text>
<unordered_list>
<list_item><location><page_9><loc_10><loc_91><loc_49><loc_93></location>[3] A. B. Henriques, A. R. Liddle, and R. G. Moorhouse, Nucl. Phys. B337 , 737 (1990).</list_item>
<list_item><location><page_9><loc_10><loc_88><loc_49><loc_90></location>[4] A. B. Henriques and L. E. Mendes, Astrophys. Space Sci. 300 , 367 (2005), arXiv:astro-ph/0301015.</list_item>
<list_item><location><page_9><loc_10><loc_85><loc_49><loc_88></location>[5] L. M. Lopes and A. B. Henriques, Phys. Lett. B285 , 80 (1992).</list_item>
<list_item><location><page_9><loc_10><loc_81><loc_49><loc_85></location>[6] S. Valdez-Alvarado, C. Palenzuela, D. Alic, and L. A. Ure˜na-L'opez, Phys. Rev. D87 , 084040 (2013), arXiv:1210.2299 [gr-qc].</list_item>
<list_item><location><page_9><loc_10><loc_77><loc_49><loc_81></location>[7] S. Valdez-Alvarado, R. Becerril, and L. A. Ure˜na L'opez, Phys. Rev. D 102 , 064038 (2020), arXiv:2001.11009 [grqc].</list_item>
<list_item><location><page_9><loc_10><loc_73><loc_49><loc_77></location>[8] F. Di Giovanni, S. Fakhry, N. Sanchis-Gual, J. C. Degollado, and J. A. Font, Phys. Rev. D 102 , 084063 (2020), arXiv:2006.08583 [gr-qc].</list_item>
<list_item><location><page_9><loc_10><loc_71><loc_49><loc_73></location>[9] S. L. Liebling and C. Palenzuela, Living Reviews in Relativity 20 , 5 (2017).</list_item>
<list_item><location><page_9><loc_9><loc_68><loc_49><loc_70></location>[10] M. Colpi, S. Shapiro, and I. Wasserman, Phys.Rev.Lett. 57 , 2485 (1986).</list_item>
<list_item><location><page_9><loc_9><loc_66><loc_49><loc_68></location>[11] J. Balakrishna, E. Seidel, and W.-M. Suen, Phys. Rev. D 58 , 104004 (1998), arXiv:gr-qc/9712064 [gr-qc].</list_item>
<list_item><location><page_9><loc_9><loc_63><loc_49><loc_65></location>[12] F. S. Guzman and L. A. Urena-Lopez, Phys. Rev. D69 , 124033 (2004), arXiv:gr-qc/0404014 [gr-qc].</list_item>
<list_item><location><page_9><loc_9><loc_60><loc_49><loc_63></location>[13] F. E. Schunck and E. W. Mielke, Classical and Quantum Gravity 20 , R301 (2003), arXiv:0801.0307 [astro-ph].</list_item>
<list_item><location><page_9><loc_9><loc_55><loc_49><loc_60></location>[14] R. Abbott, T. D. Abbott, S. Abraham, F. Acernese, K. Ackley, C. Adams, LIGO Scientific Collaboration, and Virgo Collaboration, Astroph. J. Lett. 896 , L44 (2020), arXiv:2006.12611 [astro-ph.HE].</list_item>
<list_item><location><page_9><loc_9><loc_52><loc_49><loc_55></location>[15] L. Rezzolla, E. R. Most, and L. R. Weih, Astroph. J. Lett. 852 , L25 (2018), arXiv:1711.00314 [astro-ph.HE].</list_item>
<list_item><location><page_9><loc_9><loc_50><loc_49><loc_52></location>[16] A. Nathanail, E. R. Most, and L. Rezzolla, Astroph. J. Lett. 908 , L28 (2021), arXiv:2101.01735 [astro-ph.HE].</list_item>
<list_item><location><page_9><loc_9><loc_46><loc_49><loc_49></location>[17] E. R. Most, L. R. Weih, L. Rezzolla, and J. Schaffner-Bielich, Phys. Rev. Lett. 120 , 261103 (2018), arXiv:1803.00549 [gr-qc].</list_item>
<list_item><location><page_9><loc_9><loc_42><loc_49><loc_45></location>[18] G. Arroyo-Ch'avez, A. Cruz-Osorio, F. D. Lora-Clavijo, C. Campuzano Vargas, and L. A. Garc'ıa Mora, Astrophys. Space Sci. 365 , 43 (2020), arXiv:2002.08879 [gr-qc].</list_item>
<list_item><location><page_9><loc_9><loc_29><loc_49><loc_41></location>[19] T. E. Riley, A. L. Watts, P. S. Ray, S. Bogdanov, S. Guillot, S. M. Morsink, A. V. Bilous, Z. Arzoumanian, D. Choudhury, J. S. Deneva, K. C. Gendreau, A. K. Harding, W. C. G. Ho, J. M. Lattimer, M. Loewenstein, R. M. Ludlam, C. B. Markwardt, T. Okajima, C. Prescod-Weinstein, R. A. Remillard, M. T. Wolff, E. Fonseca, H. T. Cromartie, M. Kerr, T. T. Pennucci, A. Parthasarathy, S. Ransom, I. Stairs, L. Guillemot, and I. Cognard, Astroph. J. Lett. 918 , L27 (2021), arXiv:2105.06980 [astro-ph.HE].</list_item>
<list_item><location><page_9><loc_9><loc_23><loc_49><loc_28></location>[20] G. Raaijmakers, S. K. Greif, K. Hebeler, T. Hinderer, S. Nissanke, A. Schwenk, T. E. Riley, A. L. Watts, J. M. Lattimer, and W. C. G. Ho, Astroph. J. Lett. 918 , L29 (2021), arXiv:2105.06981 [astro-ph.HE].</list_item>
<list_item><location><page_9><loc_9><loc_19><loc_49><loc_23></location>[21] M. C. Miller, F. K. Lamb, A. J. Dittmann, S. Bogdanov, Z. Arzoumanian, K. C. Gendreau, S. Guillot, A. K. Harding, W. C. G. Ho, J. M. Lattimer, R. M. Lud-</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_9><loc_55><loc_88><loc_92><loc_93></location>lam, S. Mahmoodifar, S. M. Morsink, P. S. Ray, T. E. Strohmayer, K. S. Wood, T. Enoto, R. Foster, T. Okajima, G. Prigozhin, and Y. Soong, Astroph. J. Lett. 887 , L24 (2019), arXiv:1912.05705 [astro-ph.HE].</list_item>
<list_item><location><page_9><loc_52><loc_84><loc_92><loc_88></location>[22] F. Di Giovanni, N. Sanchis-Gual, P. Cerd'a-Dur'an, and J. A. Font, Phys. Rev. D 105 , 063005 (2022), arXiv:2110.11997 [gr-qc].</list_item>
<list_item><location><page_9><loc_52><loc_80><loc_92><loc_84></location>[23] T. Matos, F. S. Guzman, and L. A. UrenaLopez, Class.Quant.Grav. 17 , 1707 (2000), arXiv:astroph/9908152 [astro-ph].</list_item>
<list_item><location><page_9><loc_52><loc_77><loc_92><loc_80></location>[24] T. Matos and L. A. Urena-Lopez, Class.Quant.Grav. 17 , L75 (2000), arXiv:astro-ph/0004332 [astro-ph].</list_item>
<list_item><location><page_9><loc_52><loc_75><loc_92><loc_77></location>[25] W. Hu, R. Barkana, and A. Gruzinov, Phys. Rev. Lett. 85 , 1158 (2000).</list_item>
<list_item><location><page_9><loc_52><loc_72><loc_92><loc_74></location>[26] A. Arbey, J. Lesgourgues, and P. Salati, Phys. Rev. D 64 , 123528 (2001).</list_item>
<list_item><location><page_9><loc_52><loc_71><loc_91><loc_72></location>[27] J.-w. Lee and I.-g. Koh, Phys. Rev. D 53 , 2236 (1996).</list_item>
<list_item><location><page_9><loc_52><loc_68><loc_92><loc_70></location>[28] T. Matos and L. A. Urena-Lopez, Int.J.Mod.Phys. D13 , 2287 (2004), arXiv:astro-ph/0406194 [astro-ph].</list_item>
<list_item><location><page_9><loc_52><loc_66><loc_92><loc_68></location>[29] T. Matos, J. A. Vazquez, and J. Magana, (2008), arXiv:0806.0683 [astro-ph].</list_item>
<list_item><location><page_9><loc_52><loc_62><loc_92><loc_65></location>[30] J. Magana, T. Matos, A. Suarez, and F. J. SanchezSalcedo, JCAP 1210 , 003 (2012), arXiv:1204.5255 [astroph.CO].</list_item>
<list_item><location><page_9><loc_52><loc_58><loc_92><loc_61></location>[31] L. Hui, J. P. Ostriker, S. Tremaine, and E. Witten, Phys. Rev. D95 , 043541 (2017), arXiv:1610.08297 [astroph.CO].</list_item>
<list_item><location><page_9><loc_52><loc_55><loc_92><loc_57></location>[32] L. A. Ure˜na-L'opez, Frontiers in Astronomy and Space Sciences 6 , 47 (2019).</list_item>
<list_item><location><page_9><loc_52><loc_52><loc_92><loc_55></location>[33] L. Hui, Annual Review of Astronomy and Astrophysics 59 , 247 (2021), arXiv:2101.11735 [astro-ph.CO].</list_item>
<list_item><location><page_9><loc_52><loc_47><loc_92><loc_52></location>[34] T. Matos, L. A. Ure˜na-L'opez, and J.-W. Lee, 'Short review of the main achievements of the scalar field, fuzzy, ultralight, wave, bec dark matter model,' (2023), arXiv:2312.00254 [astro-ph.CO].</list_item>
<list_item><location><page_9><loc_52><loc_44><loc_92><loc_47></location>[35] S. L. Liebling and C. Palenzuela, Living Reviews in Relativity 26 (2023), 10.1007/s41114-023-00043-4.</list_item>
<list_item><location><page_9><loc_52><loc_42><loc_92><loc_44></location>[36] P.-H. Chavanis, Phys. Rev. D 84 , 043531 (2011), arXiv:1103.2050 [astro-ph.CO].</list_item>
<list_item><location><page_9><loc_52><loc_38><loc_92><loc_41></location>[37] S. G. Medell'ın-Gonz'alez, L. A. Ure˜na-L'opez, and A. X. Gonz'alez-Morales, Phys. Rev. D 103 , 083509 (2021), arXiv:2010.13998 [astro-ph.CO].</list_item>
<list_item><location><page_9><loc_52><loc_33><loc_92><loc_37></location>[38] F. X. Linares Cede˜no, A. X. Gonz'alez-Morales, and L. A. Ure˜na-L'opez, Journal of Cosmology and Astroparticle Physics 2021 , 051 (2021), arXiv:2006.05037 [astroph.CO].</list_item>
<list_item><location><page_9><loc_52><loc_29><loc_92><loc_32></location>[39] F. X. L. Cede˜no, A. X. Gonz'alez-Morales, and L. A. Ure˜na-L'opez, Phys. Rev. D 96 , 061301 (2017), arXiv:1703.10180 [gr-qc].</list_item>
<list_item><location><page_9><loc_52><loc_21><loc_92><loc_28></location>[40] P. Mocz, A. Fialkov, M. Vogelsberger, M. BoylanKolchin, P.-H. Chavanis, M. A. Amin, S. Bose, T. Dome, L. Hernquist, L. Lancaster, M. Notis, C. Painter, V. H. Robles, and J. Zavala, Monthly Notices of the Royal Astronomical Society 521 , 2608 (2023), arXiv:2301.10266 [astro-ph.CO].</list_item>
</document>
[
{
"title": "Observations of compact stars and fermion-boson stars with a quartic self-interaction",
"content": "Susana Valdez-Alvarado, 1, ∗ Alejandro Cruz-Osorio, 2, † J. M. D'avila, 1, ‡ L. Arturo Ure˜na-L'opez, 3, § and Ricardo Becerril 4, ¶ 1 Facultad de Ciencias de la Universidad Aut'onoma del Estado de M'exico (UAEM'ex.), Instituto Literario No. 100, C.P. 50000, Toluca, Estado de M'exico, M'exico 2 Instituto de Astronom'ıa, Universidad Nacional Aut'onoma de M'exico, AP 70-264, Ciudad de M'exico 04510, M'exico 3 Departamento de F'ısica, DCI, Campus Le'on, Universidad de Guanajuato, 37150, Le'on, Guanajuato, M'exico. 4 Instituto de F'ısica y Matem'aticas, Universidad Michoacana de San Nicol'as de Hidalgo. Edif. C-3, 58040 Morelia, Michoac'an, M'exico (Dated: February 20, 2024) We investigated the possibility that compact stars could be described by a fermion-boson star with a quartic self-interaction in the boson sector. Specifically, by varying the polytropic constant K and adiabatic index Γ in the polytropic equation of state, the boson mass µ , and the selfinteraction parameter Λ, we construct equilibrium configurations of these mixed-stars with total mass compatible with the mass constraints obtained from observational data of the collaborations NICE, NICER/XMN-Newton, and LIGO. Our work confirms that the addition of a boson sector eases the comparison of neutron star models with gravitational events related to compact objects and that in such a case observations may have preference for a positive self-interaction in the boson sector.",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "Objects composed of a combination of fermions and bosons were first studied by Henriques and Liddle in [13] (see also [4, 5]), and their properties have been further explored in recent work [6-8]. These objects are known as fermion-boson stars (FBS) and are solutions of the Einstein-Klein-Gordon-hydrodynamic (EKGH) system of equations[9]. The boson sector of FBS consists of a massive scalar field that is minimally coupled to gravity, with a boson mass µ and a self-interaction parameter Λ in the scalar potential [10-13]. The fermion sector is usually modeled with a polytropic equation of state (EOS) of the form P ( ρ ) = Kρ Γ [7], where K and Γ take values that correspond to masses and compactness in the range of neutron stars (NS). The stability of FBS has been verified for a wide range of parameters, allowing for a variety of masses and sizes in the resulting objects [7, 8]. On the other hand, the successful detection of gravitational waves by the LIGO and Virgo scientific collaborations (LVC) has significantly widened the window to observe our universe and to understand gravity itself, but some of the gravitational wave events have brought new theoretical challenges and opportunities. The GW190814 event [14], for example, involved a black hole (BH) with a mass of 22 . 2 M ⊙ -24 . 3 M ⊙ and a spin of ≤ 0 . 07, accompanied by a compact object with a mass of 2 . 5 M ⊙ -2 . 67 M ⊙ . This binary source of a gravitational wave has the smallest mass ratio ever measured for a similar system, at 0 . 112 +0 . 008 -0 . 009 . The nature of the second object is unknown, and there are no constraints on its radius, so it could be either a very light black hole or a very heavy neutron star (NS), which would make it the heaviest NS ever observed. The peculiarities of events GW170817 and GW190814 have been used to set limits on the maximum mass and radii of generic neutron stars [15-17], which in turn have been used to constrain the parameters of polytropic fluids in models [18]. More recent constraints on the properties of neutron stars have been found from observations of the NICER telescope [19-21]. In [22], it was already suggested that an FBS, with a boson mass of the order of µ ∼ 10 -9 eV, could help explain the constitution of the self-gravitating objects behind the occurrence of these events, without self-interaction in the boson sector (i.e. Λ = 0). In this paper, we investigate the possibility that the secondary component of the event GW190814 is a FBS with a quartic self-interaction in the boson sector. For this, we have kept in mind four facts: (a) the mass of the secondary object for the LIGO gravitational wave signal GW190814 lies in the interval [2 . 5 , 2 . 67] M ⊙ ; (b) for Rezzolla - Nathanail simulations for neutron stars, its mass is located in the interval [2 . 0 , 2 . 3] M ⊙ ; (c) for the NICER collaborations its corresponding mass interval is [1 . 95 , 2 . 2] M ⊙ for PSRJ0030+0451 and [1 . 2 , 1 . 65] M ⊙ for PSJ0740+6620 ; (d) ultra-light bosons have been proposed as candidates to be dark matter in our universe [23-34], and then these new events could give hints about their possible existence. We constructed mixed star configurations selecting Γ and K in such a way that our numerical results yield total masses that match the bounds mentioned in (a), (b), and (c) above. Furthermore, we also consider the constraint of the sound speed c s < 1 (in natural units) in the fluid sector of the FBS. This paper is structured as follows. Section II outlines the equations of motion of the coupled Einstein- Klein-Gordon-hydrodynamic system and the boundary conditions that allow us to construct equilibrium configurations of FBS with a given quartic self-interaction in the boson sector. We also discuss the general numerical results in the case of FBS with total masses and radii similar to those of NS and the influence of the boson self-interaction in those quantities. In Sec. III we present the comparison of our models with observational data of selected gravitational events, and discuss the general properties of the boson sector that can help the fluid star to be in more agreement with the data. Finally, in Sec. IV, we provide a summary of our results and final remarks about our study.",
"pages": [
1,
2
]
},
{
"title": "II. EQUILIBRIUM CONFIGURATIONS",
"content": "We model the boson sector of the FBS of interest with a complex scalar field ϕ that has a scalar potential V ( ϕ ). The fermion sector is represented by a perfect fluid with rest-mass density ρ , pressure P , internal energy ϵ and 4-velocity u µ . The (relativistic) equations of motion for this system, expressed in geometrical units with G = c = ℏ = 1, are as follows: where the stress-energy tensors of the bosonic and fermionic components are, respectively, The boson sector is endowed with a quartic scalar potential of the form that represents an ensemble of boson particles with mass µ and a self-interaction parameter λ , which may be either positive or negative. We are interested in equilibrium configurations, so we assume a static and spherically symmetric metric, with the line element ds 2 = -α 2 ( r ) dt 2 + a 2 ( r ) dr 2 + r 2 d Ω 2 . The scalar field is expressed in the standard harmonic form ϕ ( t, r ) = ϕ ( r ) e -iωt , with ω being the characteristic frequency of the solution, while all fermionic variables depend only on the radial coordinate r . To simplify the equations, we introduce a set of new variables: Ω = ω/µ , √ 4 πϕ → ϕ , Λ = λ/ (4 πµ 2 ), 4 πρ → ρ and 4 πP → P , with which Eqs. (1) explicitly become, for the metric variables α and a , whereas for the field and fluid variables we get Here, a prime denotes derivatives with respect to the radial coordinate r and the internal energy is defined as ϵ = P/ ( ρ (Γ -1)). The corresponding boundary conditions that guaranty regularity at the origin and asymptotic flatness at infinity are The total mass of the equilibrium configurations, M T is obtained using the Schwarzschild definition, The numerical results from all the equations are dimensionless and given in code units. We use the standard convention for NS studies, where M T = 1 is equal to one solar mass M ⊙ in physical units. To obtain the total physical mass M and the radius 99% R 99 of the equilibrium configurations, we use the following expressions, where r 99 is the value of r containing 99% of the total mass M T but given in code units. A typical solution of a NS is then obtained without a scalar field ( ϕ 0 = 0) and for appropriate values of fluid quantities ρ 0 , K and Γ. For the mixed case of a FBS, we only need to solve the equations of motion (3) for different values of the central field density ϕ 0 , the boson mass µ and the selfinteracting parameter Λ. In particular, the boson mass µ in geometrical units is related to its physical value µ B by We solved the equations of motion of FBS for three pairs of values Γ and K namely { (Γ, K ) } = { (2 . 8, 5 . 6 × 10 4 ), (2 . 85, 7 × 10 4 ), (2 . 9, 9 × 10 4 ) } ; for each single pair we considering the following intervals for their free parameters: 0 ≤ ϕ 0 ≤ 0 . 2, 0 . 006 ≤ ρ 0 ≤ 0 . 06 and -50 ≤ Λ ≤ 50 1 . The general behavior of the resulting configurations, in terms of their rescaled physical total mass M and the 99% radius R 99 , is shown in Fig. 1 for the cases Λ = 0 , 50 , -50 with Γ = 2 . 8 and K = 5 . 6 × 10 4 . In all cases, the mass of the boson is µ = 1, which according to Eq. (7) means that µ B c 2 ≃ 1 . 33 × 10 -10 eV. This is the expected mass for boson stars with a total mass of the order of solar masses and a radius in the range of tens of kilometers [35]. Each of the curves in the graphs represents a family of equilibrium configurations with a fixed value of ϕ 0 , and for which the central mass density ρ 0 is changed from bottom to top in each curve. Furthermore, as indicated in the graphs, the total area covered by the set of curves is limited by the extreme values of the central field ϕ 0 , whose corresponding curves have been highlighted. It should be noted that the curve ϕ 0 = 0 is purely fermionic and represents the standard NS for the chosen fluid parameters. One last important feature is the stability of the equilibrium configurations. As previously studied in [6], see also [8], there is a standard method for determining the stability of FBS in terms of their number of bosons and fermions, which is slow and long. However, given that our FBS are fermion-dominated, we can take a shortcut by following the standard method for fermionic (fluid) stars. For a given ϕ 0 , we increase the central density ρ 0 until the maximum mass of the star is reached. All configurations with a central density higher than this should be unstable. It has been demonstrated in [7] that this shortened method is consistent with the standard one for the small values of ϕ 0 discussed here. For the sake of completeness, the configurations at the stability limit are also shown in Fig. 1. When Λ = 0, the total mass and radius of stable FBS are in the ranges of 0 . 73 < M/M ⊙ < 3 and 10 < R 99 / km < 22, respectively. The presence of the self-interaction parameter increases the area of the region covered by the curves for Λ = 50 and decreases it for Λ = -50, while the mass and radius ranges remain similar for the three cases shown in Fig. 1. There is some overlap of the regions in the different cases, which implies that the self-interaction parameter has some degree of degeneracy with the central density for certain values of the mass and radius of the configurations. For a positive self-interaction parameter, the mass of a boson star M B is given by M B ≲ 0 . 165Λ 1 / 2 M ⊙ /µ [10] in the so-called Thomas-Fermi limit Λ ≫ 1, and then M B ≲ 8 M ⊙ for Λ = 50. This shows that the boson part of the FBS can contribute more to the mass of the star for the same value of ϕ 0 , and then the bundle of curves in the middle panel of Fig. 1 can cover a larger portion of the parameter space. On the contrary, for a negative self-interaction parameter the upper bound on the total mass in the ThomasFermi limit | Λ | ≫ 1 is M B < 1 . 4 | Λ | -1 / 2 M ⊙ /µ [36], so that M B < 0 . 2 M ⊙ for Λ = -50. Being the mass contribution of the boson star this small, a narrow bundle of curves close to the anchor fermion star is to be expected, as the one shown in the bottom panel of Fig. 1.",
"pages": [
2,
3,
4
]
},
{
"title": "III. COMPARISON WITH OBSERVATIONS",
"content": "Our goal in this section is to find regions in this parameter space for which the values of the total mass M and radius R 99 , of the resultant configurations, are compatible with those of NS as inferred from theoretical models and the NICER's observations. Figures 2 and 3 display the masses and radii of FBS obtained in physical units, superimposed on the properties of objects detected by different observational events. The green and red regions represent the density distributions determined by the NICER Collaboration for PSR J0740 + 6620 [19, 20] and PSRJ0030 + 0451 [21], respectively. The yellow and cyan regions indicate the constraints on NS mass values obtained from simulations of a large number of polytropic models conducted by Rezzolla et al. [15] and Nathanail et al. [16] respectively. Whereas the green band represents the overlap region between both simulations. The pink region corresponds to the mass associated with the second compact object detected in GW190814 [14] by LIGO, whose source is unknown. The case with fixed value Λ = 10 is presented in Fig. 2 in terms of the central field strength ϕ 0 . In all instances, the value ϕ 0 = 0 corresponds to the NS (white curve), which we also referred to before as the anchor fermion star, constructed from three sets of values of the fluid parameters (Γ , K ): (2 . 80 , 5 . 6 × 10 4 ) (top row), (2 . 85 , 7 × 10 4 ) (middle row) and (2 . 90 , 9 × 10 4 ) (bottom row), as indicated in each graph. Likewise, three values of the boson mass were considered for the solution of the boson part of the star: µ = 0 . 5 (left column), µ = 1 . 0 (middle column) and µ = 2 . 0 (right column). As noted in [22], as we increase the value of ϕ 0 , we can cover a larger area and agree with the regions suggested by the different observations. Notice that the NS pro- ile mainly determines the mass range of the whole set of equilibrium configurations, while the scalar field helps to extend the range in the size of the FBS, as already pointed out in the discussion of Figs. 1. In particular, the variation on the field strength ϕ 0 seems to give the required degree of freedom (width) to cover the confidence regions of PSRJ0030 + 0451 and PSRJ0740 + 6620. It can be seen that the best options appear to be those with (Γ , K ) = (2 . 85 , 7 × 10 4 ) , (2 . 90 , 9 × 10 4 ) and with the boson mass µ of the order of unity ( µ B c 2 ≃ 10 -10 eV). In Fig. 3, we present the masses and radii of FBS obtained with the central value of the scalar field fixed at ϕ 0 = 0 . 1, and for different values of the self-interaction parameter Λ in the range ( -50 , 50). Given that the central scalar field is non-zero, the anchor configuration in these cases is not an NS but one FBS with Λ = 0, and then configurations with Λ > 0 (Λ < 0) have smaller radii (larger radii) in comparison to the anchor FBS. In contrast to the cases in Fig. 2, the best scenarios seem to be those in the column µ = 2, for the three sets of values of the fluid parameters. The values of Λ provide the extra width in the parameter space to cover the density regions suggested by the observations, since Λ can also influence the mass and radius of the configurations. It is clear from the curves in Figs. 2 and 3 that the masses and radii associated with the numerical configurations are capable of crossing all the areas suggested by the observations. To illustrate this in more detail, we have selected three cases with Λ = 10, ϕ 0 = 0 . 1125, K = 5 . 6 × 10 4 , Γ = 2 . 8, and µ = 1, for which the corresponding three FBS configurations have masses and radii that match the NICER and LIGO data. These cases are represented by the red stars in Fig. 2, and their internal morphology is presented in Fig. 4. The gray color map represents the boson density, whereas the rest-mass density of fluid is represented in the blue-red scale. In general, we see that the contribution of the boson sector decreases as the configuration is more massive and compact, as a result of the concentration induced by the gravitational force of the fluid component, which is also the dominant one. The central sound speed of the FBS is also shown, which was calculated using c 2 s = Pγ ( γ -1) / [ Pγ -ρ ( γ -1)]. Even for the most extreme case with M = 2 . 56 M ⊙ , the sound speed remains below unity. As mentioned above, the secondary component of the recently reported event GW190814 is of unknown nature, but its extreme mass indicates that it could be a light BH or a very heavy NS. However, our numerical results in Figs. 2 and 3 make it plausible to identify this secondary component with an FBS with a quartic self-interaction. To verify this more carefully, we have performed a detailed survey in the parameter space { Γ , K, µ, Λ } to determine the regions corresponding to an FBS with a fixed total mass of the order of M ≃ 2 . 16 M ⊙ . The results are shown in Fig. 5, for the total radius R as a function of the self-interaction parameter Λ (top panel) and as a function of the central field density ϕ 0 (bottom panel). The φ 20 values of the polytropic parameters are the same as those considered in Figs. 2 and 3 above. In general, we note that the presence of the scalar field ϕ , in terms of its central density or its self-interaction, reduces the size of the FBS, with the total radius reaching values on the order of R 99 ≃ 12 km. The most extreme cases are reached for the case µ = 2, together with the highest values of ϕ 0 and Λ. For these extreme cases shown in Fig. 5, the compactness of the FBS is of the order of C = 0 . 22, which is approximately 16% higher than the anchor NS alone. Another quantity of interest is the central sound speed c s of the FBS with M = 2 . 16 M ⊙ , which we show in Fig. 6. Similarly to the case of the compactness of the star, we see that the presence of the scalar field ϕ and its self-interaction Λ increases the value of c s , although in all cases the top sound speed is of the order of c s ≃ 0 . 9. 50",
"pages": [
4,
5,
6
]
},
{
"title": "IV. CONCLUSIONS AND FINAL REMARKS",
"content": "We have constructed FBS configurations with a selfinteraction term in the boson sector and considered different values of their physical parameters to match the data for NS obtained from NICER, Rezzolla, and Nathanail collaborations as well as from the LIGO gravitational wave signal GW190814. For the fluid parameters, we picked values that yielded NS masses in the vicinity of the values reported for different observed objects, and then varied the boson parameters, mass and self-interaction, to find their influence in the mass, size, and morphology of the corresponding FBS. In general terms, the variation of the boson parameters allows our models to cover a wider area in the parameter space of the observed objects, which means that the addition of a boson sector to a given NS configuration could help explain the presence of different types of object. Ac- cording to our results, the best options seem to have a boson mass of order µ ≃ 1 in our units, which means that the total mass of the corresponding boson star is of the same order of magnitude as that of the selected NS (or anchor NS, as we call it in the main text). Similarly, there is a preference for positive values of the self-interaction parameter, as a negative value of it makes it difficult to reconcile our theoretical results with observations. Our work confirms and extends the results presented in [22], in that the addition of a boson sector eases the comparison of NS models with different observations. We have already mentioned the preference for the value c 1 . 00 0 . 95 0 . 90 0 . 85 0 . 80 0 . 75 0 . 70 0 . 65 0 . 00 0 . 02 0 . 04 0 . 06 0 . 08 0 . 10 0 . 12 φ 0 µ ≃ 1, which is not surprising given the typical masses and sizes of compact objects suggested by the observations we used. The interesting new result is that the self-interaction, the next-to-leading order parameter in the description of the boson sector, in particular for the scalar field potential, should be positive. Some other comments are in turn. Our anchor NS was modeled as a polytropic fluid, for which the parameter values were selected according to previous studies of NS [18]. However, our study could also consider other types of equation of state for the fermion sector, such as those chosen in [22], for which we would expect the same qualitative results obtained so far: preference of data for µ ≃ 1 and Λ ≳ 0. Ultralight bosons have been proposed as candidates for dark matter in our universe, and a boson with mass µ B c 2 ∼ 10 -10 eV would certainly qualify as ultralight, although such a mass would be 10 orders of magnitude larger than the preferred astrophysical value of µ B c 2 ∼ 10 -22 eV [31, 34]. Our new result Λ ≳ 0, if taken at face value, would mean that there is an extra constraint on dark matter models with axion-like bosons (with negative self-interaction), at least for the cases for which µ B c 2 ∼ 10 -10 eV, as there are differentiable consequences at astrophysical and cosmological scales for the formation of cosmic structure from the sign of the self- in the potential [37-40]. As already noted in [22], for comparison with observations, we just took the confidence regions obtained by other groups that considered models quite different from ours to fit the data. In this respect, our constraints should be considered conservative and more like a proofof-concept for FBS with a self-interaction. A separate study with a direct comparison with raw data from the different observations would be necessary to have a more accurate judgment of the appropriateness of FBS and the role of their self-interaction in each of the reported gravitational events. This is work in progress, which we expect to report elsewhere.",
"pages": [
6,
7,
8
]
},
{
"title": "ACKNOWLEDGMENTS",
"content": "S.V.A and J.M.D. acknowledge support from Programa de Desarrollo Profesional Docente de la Secretaria de Educaci'on P'ublica (PRODEP-SEP), under No. UAEM-CA-14 project. R.B. acknowledges partial support from C.I.C-UMSNH. LAU-L. acknowledges partial support from the Programa para el Desarrollo Profesional Docente; Direcci'on de Apoyo a la Investigaci'on y al Posgrado, Universidad de Guanajuato; CONACyT M'exico under Grants No. 286897, No. 297771, No. 304001; and the Instituto Avanzado de Cosmolog'ıa Collaboration. s c µ = 0 . 5 µ = 1 . 0 Γ = 2 . 80 Γ = 2 . 85 Γ = 2 . 90 Λ = 10 M = 2 . 16 M µ = 1 . 5 µ = 2 . 0 /circledot",
"pages": [
8
]
}
]
2024arXiv240212578W
https://arxiv.org/pdf/2402.12578.pdf
<document>
<section_header_level_1><location><page_1><loc_20><loc_90><loc_80><loc_92></location>Evidence for Planck Luminosity Bound in Quantum Gravity</section_header_level_1>
<text><location><page_1><loc_42><loc_87><loc_58><loc_88></location>Wolfgang Wieland 1</text>
<text><location><page_1><loc_18><loc_83><loc_82><loc_86></location>Institute for Quantum Gravity, Theoretical Physics III, Department of Physics Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7, 91052 Erlangen, Germany</text>
<text><location><page_1><loc_44><loc_81><loc_56><loc_82></location>19 February 2024</text>
<section_header_level_1><location><page_1><loc_12><loc_73><loc_20><loc_75></location>Abstract</section_header_level_1>
<text><location><page_1><loc_12><loc_63><loc_88><loc_72></location>Recently, we introduced a non-perturbative quantization of impulsive gravitational null initial data. In this note, we investigate an immediate physical implication of the model. One of the quantum numbers is the total luminosity carried to infinity. We show that a transition happens when the luminosity reaches the Planck power L P . Below L P , the spectrum of the radiated power is discrete. Above the Planck power, the spectrum is continuous and contains caustics that can spoil the semi-classical interpretation of the resulting quantum states of geometry.</text>
<section_header_level_1><location><page_1><loc_12><loc_59><loc_27><loc_60></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_12><loc_56><loc_50><loc_57></location>It has been often argued that the Planck power</text>
<formula><location><page_1><loc_35><loc_52><loc_88><loc_55></location>L P = m P c 2 t P = c 5 /G ≈ 3 , 63 × 10 52 W (1)</formula>
<text><location><page_1><loc_12><loc_28><loc_88><loc_50></location>places an upper bound on the gravitational wave luminosity, see e.g. [1, 2], but also [3] for more critical remarks. The perhaps simplest argument in favour of this idea can be found in Misner, Thorne, Wheeler [1]. Take the virial theorem, i.e. E kin ∼ Mω 2 R 2 ∼ GM 2 /R , and the quadrupole formula for the gravitational wave luminosity, i.e. L GW ∼ L -1 P ( MR 2 ω 3 ) 2 , where M is the mass of the system, R denotes its spatial extension and ω is the frequency at which the system oscillates. Since the emission can only happen when R is greater than the Schwarzschild radius 2 GM/c 2 , we obtain L GW ≲ L P . Such a simple bound can not exist in higher dimensions. In D > 4 spacetime dimensions, the Planck power depends not only on G and c , but also on ℏ , which can not appear at the classical level. It is no surprise therefore that it is in D = 4 alone that we have an equation for the peak luminosity of two coalescing black holes, which depends only on dimensionless observables, such as the mass ratio η = m 1 m 2 / ( m 1 + m 2 ) 2 and dimensionless spin components [4, 5]. In higher dimensions, we need an additional 1length scale [6]. This makes it doubtful that there is a universal bound on the gravitational wave luminosity when D > 4 .</text>
<text><location><page_1><loc_12><loc_17><loc_88><loc_28></location>Recently, we introduced a non-perturbative quantization of impulsive gravitational null initial data in D = 4 , see [7]. The proposal relies on the geometry of light-like boundaries, which simplifies the construction of gauge-invariant observables [8-11]. In this note, we investigate the role of the Planck power L P in the model. The analysis is based on a combination of nonperturbative and semi-classical techniques. The bound appears upon adding a parity-violating γ -term [12-14] to the gravitational action in the bulk, which, by coincidence or not, only appears in D = 4 . The resulting γ -Hilbert-Palatini action is</text>
<formula><location><page_1><loc_28><loc_12><loc_88><loc_16></location>S [ A,e ] = 1 16 πG ∫ M 4 [ 1 2 ϵ αβ α ' β ' -1 γ δ [ α α ' δ β ] β ' ] e α ∧ e β ∧ F α ' β ' , (2)</formula>
<text><location><page_2><loc_12><loc_86><loc_88><loc_92></location>where F = d A + 1 2 [ A,A ] is the curvature of the so (1 , 3) connection, e α is the co-tetrad and γ is the Barbero-Immirzi parameter [15, 16]. Since L P → ∞ as G → 0 , we also note that it seems implausible to find a bound on the gravitational wave luminosity from a perturbative quantisation, where we have a formal perturbative expansion with respect to κ = √ 8 πG/c 3 .</text>
<section_header_level_1><location><page_2><loc_12><loc_82><loc_42><loc_83></location>2. Phase space of impulsive data</section_header_level_1>
<text><location><page_2><loc_12><loc_75><loc_88><loc_80></location>In the following, we consider the phase space of a pulse of gravitational null initial data on a null boundary N 3 ≃ [ -1 , 1] × S 2 . Since the boundary is null, we can introduce a co-dyad ( e 1 a , e 2 a ) ∈ Ω 1 ( N 3 : R 2 ) , intrinsic to N 3 , that diagonalizes the signature (0++) metric</text>
<formula><location><page_2><loc_40><loc_73><loc_88><loc_74></location>q ab := φ ∗ N 3 g ab = δ ij e i a e j b , (3)</formula>
<text><location><page_2><loc_12><loc_67><loc_88><loc_72></location>where δ ij is the Kronecker delta and φ ∗ N 3 g ab is the pull-back of the spacetime metric g ab in M 4 to the boundary N 3 ⊂ ∂ M 4 . Any such co-dyad can be parametrized by a conformal factor Ω and an SL (2 , R ) holonomy S ,</text>
<formula><location><page_2><loc_43><loc_65><loc_88><loc_67></location>e i a = Ω S i m · e m a , (4)</formula>
<text><location><page_2><loc_12><loc_60><loc_88><loc_64></location>where · e m a is a fiducial dyad, which we keep fixed once and for all. A possible choice is · e 1 a = ∂ a ϑ , · e 2 a = sin( ϑ ) ∂ a φ , where ( ϑ, φ ) are standard spherical coordinates that are Lie dragged along the null generators.</text>
<text><location><page_2><loc_12><loc_55><loc_88><loc_59></location>By introducing a time coordinate U : N 3 → [ -1 , 1] , we extend the angular coordinates ( ϑ, φ ) into a three-dimensional coordinate system of N 3 . The resulting vector field ∂ U is null. We assume it to be future pointing. The boundary conditions are</text>
<formula><location><page_2><loc_36><loc_51><loc_88><loc_53></location>for ∂ N 3 = C + ∪ C -1 -: U ∣ ∣ C ± = ± 1 . (5)</formula>
<text><location><page_2><loc_12><loc_41><loc_88><loc_50></location>In the absence of additional structure, such as a preferred foliation, symmetries or asymptotic boundary conditions, there is no natural torsionless and metric compatible covariant derivative in T N 3 , see e.g. [17]. A natural derivative D a exists on the extended vector bundle ⋃ p ∈ N 3 { p }× T p M 4 , where it is induced from the bulk, i.e. D a = φ ∗ N 3 ∇ a , given the usual Levi-Civita covariant derivative ∇ a in ( M 4 , g ab ) . The corresponding connection, which depends on both the extrinsic and intrinsic geometry of N 3 , is the metric analogue of the self-dual Ashtekar connection [18].</text>
<text><location><page_2><loc_12><loc_37><loc_88><loc_40></location>There are infinitely many clock variables U that satisfy the boundary condition (5). To pick a unique representative, we impose the gauge condition</text>
<formula><location><page_2><loc_39><loc_33><loc_88><loc_36></location>∂ a U D a ∂ U = -Ω -1 d d U [Ω] ∂ U . (6)</formula>
<text><location><page_2><loc_12><loc_28><loc_88><loc_32></location>Upon choosing this gauge, the residual constraints simplify [11]. We are left with a transport equation for the SL (2 , R ) holonomy and the Raychaudhuri equation [19] for Ω 2 . The Rachaudhuri equation becomes</text>
<formula><location><page_2><loc_42><loc_23><loc_88><loc_26></location>d 2 d U 2 Ω 2 = -2 σ ¯ σ Ω 2 , (7)</formula>
<text><location><page_2><loc_12><loc_20><loc_88><loc_22></location>where σ is the shear of the null generators ∂ U of N 3 . It defines a transport equation for the SL (2 , R ) holonomy,</text>
<formula><location><page_2><loc_30><loc_15><loc_88><loc_18></location>d d U H = ( σ ¯ X + ¯ σX ) H, H ( U = -1 , ϑ, φ ) = 1 , (8)</formula>
<text><location><page_2><loc_12><loc_11><loc_88><loc_14></location>where we introduced a decomposition of sl (2 , R ) into translational components X and ¯ X and a U (1) generator J with sl (2 , R ) commutation relations</text>
<formula><location><page_2><loc_33><loc_9><loc_88><loc_10></location>[ J, X ] = -2i X, [ J, ¯ X ] = +2i ¯ X, [ X, ¯ X ] = i J. (9)</formula>
<text><location><page_3><loc_12><loc_90><loc_82><loc_92></location>The SL (2 , R ) element S ( U , ϑ, φ ) that parametrizes the co-dyads in (4) is then given by</text>
<formula><location><page_3><loc_44><loc_87><loc_88><loc_89></location>S = e ∆ J HS -, (10)</formula>
<text><location><page_3><loc_12><loc_80><loc_88><loc_86></location>where ∆ : N 3 → [0 , 2 π )mod2 π is an unspecified U (1) angle with boundary condition ∆( U = -1 , ϑ, φ ) = 0 . In addition, S ( U = -1 , ϑ, φ ) = S -( ϑ, φ ) . In the interior of N 3 , the value of ∆ can be gauged to zero. At the upper boundary C + it can not. The boundary data S -( ϑ, φ ) and ∆ + ( ϑ, φ ) = ∆( U = 1 , ϑ, φ ) are an example of gravitational edge modes [20-30].</text>
<text><location><page_3><loc_12><loc_73><loc_88><loc_79></location>The shear σ : N 3 → C is unconstrained. It determines the free radiative data along N 3 . To describe the quantum geometry of a single pulse of radiation, we consider only those configurations on phase space, where σ is constant along the null generators of N 3 . We can then integrate the Raychaudhuri equation and obtain</text>
<formula><location><page_3><loc_14><loc_68><loc_88><loc_72></location>Ω 2 ( U , ϑ, φ ) = E + ( ϑ, φ ) + E -( ϑ, φ ) 2 cos ( √ 2 σ ¯ σ U ) cos ( √ 2 σ ¯ σ ) + E + ( ϑ, φ ) -E -( ϑ, φ ) 2 sin ( √ 2 σ ¯ σ U ) sin ( √ 2 σ ¯ σ ) , (11)</formula>
<text><location><page_3><loc_12><loc_65><loc_71><loc_66></location>where E ± are free corner data at the initial and final cross section of N 3 .</text>
<text><location><page_3><loc_12><loc_60><loc_88><loc_64></location>The action (2) determines the symplectic structure for the initial data on N 3 . Upon taking the pull-back to the solution space of the transport equations, we obtain canonical commutation relations [7]. The only non-vanishing brackets are</text>
<formula><location><page_3><loc_37><loc_57><loc_88><loc_59></location>{ a ( z ) , ¯ a ( z ' ) } = i δ (2) ( z | z ' ) , (12a)</formula>
<formula><location><page_3><loc_37><loc_55><loc_88><loc_57></location>{ b ( z ) , ¯ b ( z ' ) } = i δ (2) ( z | z ' ) , (12b)</formula>
<formula><location><page_3><loc_37><loc_53><loc_88><loc_54></location>{ c ( z ) , ¯ c ( z ' ) } = 2i δ (2) ( z | z ' ) L ( z ) , (12c)</formula>
<formula><location><page_3><loc_37><loc_51><loc_88><loc_52></location>{ L ( z ) , c ( z ' ) } = -i δ (2) ( z | z ' ) c ( z ) , (12d)</formula>
<formula><location><page_3><loc_37><loc_48><loc_88><loc_50></location>{ L ( z ) , ¯ c ( z ' ) } = +i δ (2) ( z | z ' ) ¯ c ( z ) , (12e)</formula>
<formula><location><page_3><loc_36><loc_43><loc_88><loc_44></location>{ c ( z ) , U ( z ' ) } = XU ( z ) δ (2) ( z | z ' ) , (13a)</formula>
<formula><location><page_3><loc_36><loc_40><loc_88><loc_42></location>{ ¯ c ( z ) , U ( z ' ) } = ¯ XU ( z ) δ (2) ( z | z ' ) , (13b)</formula>
<formula><location><page_3><loc_35><loc_37><loc_88><loc_40></location>{ L ( z ) , U ( z ' ) } = -1 2 JU ( z ) δ (2) ( z | z ' ) . (13c)</formula>
<text><location><page_3><loc_12><loc_33><loc_88><loc_36></location>The relationship between the canonical variables and the geometry of the impulsive boundary data is determined by two sets of equations. First of all, we have</text>
<formula><location><page_3><loc_31><loc_30><loc_88><loc_32></location>U = e γ ln ( tan ( √ 2 σ ¯ σ ) / √ 2 σ ¯ σ ) J S -, (14)</formula>
<formula><location><page_3><loc_31><loc_26><loc_88><loc_30></location>a = √ E + √ 8 πγG ch ( 2 √ σ ¯ σ ) e -i [ ∆ + +2 γ ln ( cos ( √ 2 σ ¯ σ ))] , (15)</formula>
<formula><location><page_3><loc_31><loc_22><loc_88><loc_26></location>b = √ E + √ 8 πγG sh ( 2 √ σ ¯ σ ) e i [ ∆ + + ϕ +2 γ ln ( sin ( √ 2 σ ¯ σ ) √ 2 σ ¯ σ )] , (16)</formula>
<text><location><page_3><loc_12><loc_14><loc_88><loc_20></location>where ∆ + ( ϑ, φ ) = ∆( U = 1 , ϑ, φ ) and ϕ : σ = | σ | e i ϕ are U (1) angles. The SL (2 , R ) element U determines additional corner data that parametrize the shape degrees of freedom of the signature (0++) metric at the boundary. The overall scale of the boundary metric is set by the U (1) generator L and the norm of the oscillator variables a and b . We obtain</text>
<formula><location><page_3><loc_39><loc_11><loc_88><loc_12></location>E -+ E + = 16 πγG ( L + a ¯ a ) , (17)</formula>
<formula><location><page_3><loc_39><loc_9><loc_88><loc_11></location>E --E + = 16 πγG ( L + b ¯ b ) . (18)</formula>
<text><location><page_3><loc_12><loc_46><loc_28><loc_47></location>with z = ( ϑ, φ ) and</text>
<text><location><page_4><loc_12><loc_90><loc_57><loc_92></location>The quotient of the two oscillators determines the shear</text>
<formula><location><page_4><loc_42><loc_86><loc_88><loc_89></location>th ( 2 √ σ ¯ σ ) = √ ¯ bb ¯ aa . (19)</formula>
<text><location><page_4><loc_12><loc_82><loc_88><loc_84></location>Finally, there is one residual pair of second-class constraints, imposing recurrence relations for physical states,</text>
<formula><location><page_4><loc_31><loc_78><loc_88><loc_81></location>c ¯ a ¯ b = -√ 2 γ ( L +¯ aa ) √ ¯ aa ¯ bb tan ( √ 2 σ ¯ σ ) -i¯ aa ¯ bb, (20)</formula>
<formula><location><page_4><loc_31><loc_76><loc_88><loc_78></location>¯ cab = -√ 2 γ ( L +¯ aa ) √ ¯ aa ¯ bb tan ( √ 2 σ ¯ σ ) +i¯ aa ¯ bb. (21)</formula>
<text><location><page_4><loc_12><loc_71><loc_88><loc_74></location>The constraints are second-class. At the quantum level, only one of them can be imposed strongly. The other maps the physical Hilbert space into its orthogonal complement.</text>
<section_header_level_1><location><page_4><loc_12><loc_68><loc_32><loc_69></location>3. Critical luminosity</section_header_level_1>
<text><location><page_4><loc_12><loc_60><loc_88><loc_67></location>Upon quantizing the oscillators a, ¯ a , and b, ¯ b and the SL (2 , R ) × sl (2 , R ) variables ( U, L, c, ¯ c ) , we obtain a kinematical Hilbert space. Physical states lie in the kernel of one of the constraints, e.g. (20). The kinematical Hilbert space carries a unitary representation of SL (2 , R ) . The representations are characterized by the value of the Casimir. At the classical level,</text>
<formula><location><page_4><loc_14><loc_56><loc_88><loc_59></location>L 2 -c ¯ c = 1 8 πG [ 1 4 γ 2 ( E --E + ) 2 -1 γ 2 sh 2 ( 2 √ σ ¯ σ ) E + E --1 2 ( E + + E -) 2 tan 2 ( √ 2 σ ¯ σ ) ] . (22)</formula>
<text><location><page_4><loc_25><loc_51><loc_25><loc_53></location≯</text>
<text><location><page_4><loc_12><loc_48><loc_88><loc_54></location>When L 2 > c ¯ c , the spectrum of the Casimir is discrete. When L 2 < c ¯ c , it is continuous. For σ = 0 and E + = E -, the Casimir is positive, i.e. L 2 > c ¯ c . As we increase σ , we will reach a critical value σ crit , where the sign will flip. Expanding the Casimir (22) for small shear, we obtain</text>
<formula><location><page_4><loc_30><loc_45><loc_88><loc_48></location>| σ crit . | 2 = 1 4 ( E --E + ) 2 γ 2 ( E + + E -) 2 +4 E + E -+ O ( | σ crit | 3 ) . (23)</formula>
<text><location><page_4><loc_12><loc_36><loc_88><loc_44></location>In a neighbourhood of future null infinity I + , we have an asymptotic expansion with respect to an affine radial Bondi coordinate r [31-33]. The shear of the ingoing null generators vanishes as O ( r -1 ) . The area density Ω 2 blows up as O ( r 2 ) . We can thus use (23) to evaluate σ crit in the asymptotic limit, in which we take N 3 to future null infinity I + . Below, we have a pictorial representation of the resulting geometry.</text>
<figure>
<location><page_4><loc_39><loc_23><loc_64><loc_35></location>
</figure>
<text><location><page_4><loc_12><loc_18><loc_88><loc_20></location>Impulsive data at future null infinity. In the shaded region N 3 , the time derivative of the asymptotic shear σ (0) ( u, ϑ, φ ) is constant in u , everywhere else ˙ σ (0) = 0 .</text>
<text><location><page_4><loc_12><loc_11><loc_88><loc_16></location>If we consider such boundary data at future null infinity I + , we can embed them into Bondi coordinates ( u, r, ϑ, ϕ ) . The pulse starts at an asymptotic Bondi time u -( ϑ, φ ) and terminates at u + ( ϑ, φ ) with total duration (∆ u )( ϑ, φ ) = u + ( ϑ, φ ) -u -( ϑ, φ ) . To leading order in the</text>
<text><location><page_5><loc_12><loc_89><loc_88><loc_92></location>1 /r -expansion, the map between the boundary intrinsic time coordinate U : N 3 → [ -1 , 1] and the asymptotic Bondi time u is a mere angle-dependent dilation,</text>
<formula><location><page_5><loc_38><loc_84><loc_88><loc_87></location>∂ a U = (∆ u )( ϑ, φ ) 2 ∂ a u + O ( r -1 ) . (24)</formula>
<text><location><page_5><loc_12><loc_79><loc_88><loc_83></location>Upon introducing an adapted Newman-Penrose tetrad ( k a , ℓ a , m a , ¯ m a ) , see [34], where k a = ∂ a r and ℓ a = ∂ a u + O ( r -1 ) , and both k a = -∇ a u and ℓ a are surface orthogonal, we obtain the asymptotic expansion</text>
<formula><location><page_5><loc_32><loc_74><loc_88><loc_77></location>σ ( ℓ ) = m a m b ∇ a ℓ b = -˙ σ (0) ( u, ϑ, φ ) r + O ( r -2 ) , (25)</formula>
<formula><location><page_5><loc_32><loc_71><loc_88><loc_74></location>σ ( k ) = m a m b ∇ a k b = σ (0) ( u, ϑ, φ ) r 2 + O ( r -3 ) . (26)</formula>
<text><location><page_5><loc_12><loc_67><loc_88><loc_69></location>The map (24) between the two clock variables implies a relationship between the asymptotic Bondi shear σ (0) and a family of radiative data { σ r } at the abstract boundary N 3 ,</text>
<formula><location><page_5><loc_33><loc_62><loc_88><loc_65></location>σ r ( ϑ, φ ) = (∆ u )( ϑ, φ ) 2 ˙ σ (0) ( ϑ, φ ) r + O ( r -2 ) . (27)</formula>
<text><location><page_5><loc_12><loc_56><loc_88><loc_61></location>This equation allows us to relate the asymptotic Bondi shear to the critical shear (23), where the Casimir changes its sign. Using the standard round metric at future null infinity I + , we then also have</text>
<formula><location><page_5><loc_31><loc_53><loc_88><loc_55></location>E + ( ϑ, φ ) = E -( ϑ, φ ) + O ( r 1 ) = r 2 + O ( r 1 ) , (28)</formula>
<formula><location><page_5><loc_31><loc_51><loc_88><loc_53></location>E -( ϑ, φ ) -E + ( ϑ, φ ) = -4 r (∆ u )( ϑ, φ ) + O ( r 0 ) . (29)</formula>
<text><location><page_5><loc_12><loc_48><loc_87><loc_50></location>We insert the expansion back into (23) and obtain the critical shear of the null generators ∂ a U</text>
<formula><location><page_5><loc_35><loc_44><loc_88><loc_47></location>| σ crit. | 2 = 1 4 (∆ u ) 2 ( ϑ, φ ) γ 2 +1 1 r 2 + O ( r -3 ) . (30)</formula>
<text><location><page_5><loc_12><loc_40><loc_88><loc_42></location>We translate this value back into the asymptotic Bondi frame. Equation (27) implies that σ crit. corresponds to a critical value for the asymptotic shear given by</text>
<formula><location><page_5><loc_39><loc_35><loc_88><loc_38></location>| ˙ σ (0) crit. | = 1 √ γ 2 +1 + O ( r -1 ) . (31)</formula>
<text><location><page_5><loc_12><loc_31><loc_88><loc_34></location>Using the Bondi mass loss formula, we infer a critical value for the luminosity of the gravitational wave pulse,</text>
<formula><location><page_5><loc_34><loc_27><loc_88><loc_31></location>L crit. = c 5 4 πG ∮ S 2 d 2 Ω | ˙ σ (0) crit. | 2 = L P γ 2 +1 . (32)</formula>
<text><location><page_5><loc_12><loc_22><loc_88><loc_27></location>If L 2 -c ¯ c > 0 , the impulsive wave will have a luminosity smaller than L crit. In this regime, each light ray carries a discrete unitary representation of SL (2 , R ) . The fundamental operators are</text>
<formula><location><page_5><loc_26><loc_19><loc_88><loc_21></location>c † | N,n a , n b , n c ⟩ = i √ ( N + n c )( n c -N +1) | N,n a , n b , n c +1 ⟩ , (33)</formula>
<formula><location><page_5><loc_26><loc_16><loc_88><loc_18></location>( L 2 -1 2 ( cc † + c † c ) ) | N,n a , n b , n c ⟩ = N ( N -1) | N,n a , n b , n c ⟩ , (34)</formula>
<formula><location><page_5><loc_26><loc_12><loc_88><loc_14></location>b † | N,n a , n b , n c ⟩ = √ n b +1 | N,n a , n b +1 , n c ⟩ , (36)</formula>
<formula><location><page_5><loc_26><loc_14><loc_88><loc_16></location>a † | N,n a , n b , n c ⟩ = √ n a +1 | N,n a +1 , n b , n c ⟩ , (35)</formula>
<text><location><page_6><loc_12><loc_87><loc_88><loc_92></location>where n a and n b are integers, n c = N,N +1 , . . . is integer or half-integer and N = 1 , 3 2 , . . . . Physical states are annihilated by the constraint (20). For the discrete series representations of SL (2 , R ) , a unique solution can be found by a linear combination of states</text>
<formula><location><page_6><loc_34><loc_85><loc_88><loc_86></location>{ | N,n + r, r, N + m -r ⟩ : r = 0 , . . . , m } . (37)</formula>
<text><location><page_6><loc_12><loc_81><loc_88><loc_84></location>For the continuous series representations of SL (2 , R ) , L 2 -c ¯ c < 0 . The spectrum of the Casimir is continuous, for any s ∈ R , we have</text>
<formula><location><page_6><loc_26><loc_74><loc_88><loc_80></location>c † | s, n a , n b , n c ⟩ = 1 2i (i s -2 n c -1) | s, n a , n b , n c +1 ⟩ , ( L 2 -1 2 ( cc † + c † c ) ) | s, n a , n b , n c ⟩ = -1 4 ( s 2 +1) | s, n a , n b , n c ⟩ . (38)</formula>
<text><location><page_6><loc_12><loc_53><loc_88><loc_73></location>In this regime, the impulsive wave will have a luminosity bigger than L crit. The spectrum of the Casimir is continuous, and the operator L is no longer bounded from below. This has important consequences. The recurrence relations (20) will not terminate, physical states will be superpositions of kinematical states, where the quantum numbers n a and n b will become arbitrarily large. Therefore, the shear σ ¯ σ will be unbounded from above, see (19). If we take the Born rule and compute the probabilities for σ ¯ σ to take a certain value, there will always be a chance that an observer obtains √ 2 σ ¯ σ > π/ 2 . In this case, the profile of the area density (11) will pass through a caustic, where Ω 2 = 0 . When there is a caustic, we violate the implicit assumption that we are in a smooth asymptotic region, in which Ω 2 = r 2 + O ( r ) , as r → ∞ . This can be avoided only when L 2 > c ¯ c , i.e. when the luminosity of the gravitational wave pulse is below L crit. . Then, the physical states are built from superpositions of the discrete series unitary representations of SL (2 , R ) and for any physical state the quantum numbers n a and n b will be bounded from above.</text>
<section_header_level_1><location><page_6><loc_12><loc_49><loc_37><loc_50></location>4. Outlook and Conclusion</section_header_level_1>
<text><location><page_6><loc_42><loc_44><loc_42><loc_45></location≯</text>
<text><location><page_6><loc_12><loc_16><loc_88><loc_48></location>The critical luminosity (32) separates the continuous spectrum from the discrete eigenvalues of the SL (2 , R ) Casimir. The bound depends on the Barbero-Immirzi parameter γ . This is a common feature in D = 4 . When γ = 0 , the boundary charges are a mixture of electric and magnetic contributions that otherwise vanish in the γ →∞ limit [7, 11, 35]. The spectrum of the charges is determined by their algebraic properties alone, but the map between the charges and the physical observables depends on γ . In this way, the spectrum of physical observables can depend on γ . This is analogous to how the θ -angle in quantum electrodynamics enters the Dirac quantization condition between magnetic and electric charges [36]. In loop quantum gravity, this effect is responsible for the quantization of geometric observables, such as area, angles, volumes and length [37-41]. Such a fundamental quantum discreteness of geometry affects other physical observables. It creates a fundamental bound on the energy density of matter [42, 43] and perhaps also acceleration [44]. Here, we found a similar bound on the gravitational wave luminosity (32). The analysis is based on a non-perturbative quantization of radiative data at finite distance. The result is only partial, because there is an implicit assumption: the validity of the classical asymptotic 1 /r -expansion when applied to the spectrum of gravitational observables at finite null boundaries. If the luminosity exceeds the critical luminosity (32), this assumption may no longer be valid due to the possible creation of caustics. What we have shown so far is only a first step. A more refined investigation will follow to understand the significance of the Planck power for the spectrum of the gravitational wave luminosity in non-perturbative quantum gravity.</text>
<section_header_level_1><location><page_6><loc_12><loc_13><loc_22><loc_14></location>References</section_header_level_1>
<text><location><page_6><loc_13><loc_9><loc_82><loc_11></location>[1] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation . W. H. Freeman, San Francisco, 1973.</text>
<unordered_list>
<list_item><location><page_7><loc_13><loc_89><loc_80><loc_92></location>[2] V. Cardoso, T. Ikeda, C. J. Moore, and C.-M. Yoo, 'Remarks on the maximum luminosity,' Phys. Rev. D 97 (2018), no. 8, 084013, arXiv:1803.03271 .</list_item>
<list_item><location><page_7><loc_13><loc_85><loc_88><loc_88></location>[3] A. Jowsey and M. Visser, 'Reconsidering maximum luminosity,' Int. J. Mod. Phys. D 30 (2021), no. 14, 2142026, arXiv:2105.06650 .</list_item>
<list_item><location><page_7><loc_13><loc_80><loc_86><loc_85></location>[4] D. Keitel et al. , 'The most powerful astrophysical events: Gravitational-wave peak luminosity of binary black holes as predicted by numerical relativity,' Phys. Rev. D 96 (2017), no. 2, 024006, arXiv:1612.09566 .</list_item>
<list_item><location><page_7><loc_13><loc_75><loc_87><loc_80></location>[5] F. Zappa, S. Bernuzzi, D. Radice, A. Perego, and T. Dietrich, 'Gravitational-Wave Luminosity of Binary Neutron Stars Mergers,' Phys. Rev. Lett. 120 (Mar, 2018) 111101, arXiv:1712.04267 .</list_item>
<list_item><location><page_7><loc_13><loc_72><loc_87><loc_75></location>[6] V. Cardoso, O. J. C. Dias, and J. P. S. Lemos, 'Gravitational radiation in D-dimensional space-times,' Phys. Rev. D 67 (2003) 064026, arXiv:hep-th/0212168 .</list_item>
<list_item><location><page_7><loc_13><loc_70><loc_74><loc_71></location>[7] W. Wieland, 'Quantum geometry of the null cone,' arXiv:2401.17491 .</list_item>
<list_item><location><page_7><loc_13><loc_67><loc_87><loc_70></location>[8] M. P. Reisenberger, 'The symplectic 2-form for gravity in terms of free null initial data,' Class. Quant. Grav. 30 (2013) 155022, arXiv:1211.3880 .</list_item>
<list_item><location><page_7><loc_13><loc_63><loc_84><loc_66></location>[9] M. P. Reisenberger, 'The Poisson brackets of free null initial data for vacuum general relativity,' Class. Quant. Grav. 35 (2018), no. 18, 185012, arXiv:1804.10284 .</list_item>
<list_item><location><page_7><loc_12><loc_60><loc_87><loc_63></location>[10] L. Ciambelli, L. Freidel, and R. G. Leigh, 'Null Raychaudhuri: Canonical Structure and the Dressing Time,' arXiv:2309.03932 .</list_item>
<list_item><location><page_7><loc_12><loc_57><loc_85><loc_59></location>[11] W. Wieland, 'Gravitational SL(2, R ) algebra on the light cone,' JHEP 07 (2021) 057, arXiv:2104.05803 .</list_item>
<list_item><location><page_7><loc_12><loc_53><loc_88><loc_56></location>[12] J. Samuel, 'A Lagrangian basis for Ashtekar's formulation of canonical gravity,' Pramana 28 (1987) L429-L432, doi:10.1007/BF02847105 .</list_item>
<list_item><location><page_7><loc_12><loc_48><loc_81><loc_53></location>[13] T. Jacobson and L. Smolin, 'The Left-Handed Spin Connection as a Variable for Canonical Gravity,' Phys. Lett. B 196 (1987) 39-42, doi:10.1016/0370-2693(87)91672-8 .</list_item>
<list_item><location><page_7><loc_12><loc_45><loc_84><loc_48></location>[14] S. Holst, 'Barbero's Hamiltonian derived from a generalized Hilbert-Palatini action,' Phys. Rev. D 53 (November, 1996) 5966-5969, arXiv:gr-qc/9511026 .</list_item>
<list_item><location><page_7><loc_12><loc_42><loc_86><loc_44></location>[15] J. F. Barbero G., 'Real Ashtekar variables for Lorentzian signature space times,' Phys. Rev. D 51 (1995) 5507-5510, arXiv:gr-qc/9410014 .</list_item>
<list_item><location><page_7><loc_12><loc_38><loc_86><loc_41></location>[16] G. Immirzi, 'Real and complex connections for canonical gravity,' Class. Quant. Grav. 14 (1997) L177-L181, arXiv:gr-qc/9612030 .</list_item>
<list_item><location><page_7><loc_12><loc_35><loc_79><loc_38></location>[17] A. Ashtekar and S. Speziale, 'Horizons and Null Infinity: A Fugue in 4 voices,' arXiv:2401.15618 .</list_item>
<list_item><location><page_7><loc_12><loc_31><loc_85><loc_34></location>[18] A. Ashtekar, 'New Variables for Classical and Quantum Gravity,' Phys. Rev. Lett. 57 (1986) 2244-2247, doi:10.1103/PhysRevLett.57.2244 .</list_item>
<list_item><location><page_7><loc_12><loc_28><loc_79><loc_31></location>[19] A. Raychaudhuri, 'Relativistic Cosmology. I,' Phys. Rev. 98 (1955) 1123-1126, doi:10.1103/PhysRev.98.1123 .</list_item>
<list_item><location><page_7><loc_12><loc_25><loc_87><loc_27></location>[20] A. P. Balachandran, L. Chandar, and A. Momen, 'Edge states in gravity and black hole physics,' Nucl. Phys. B 461 (1996) 581-596, arXiv:gr-qc/9412019 .</list_item>
<list_item><location><page_7><loc_12><loc_21><loc_86><loc_24></location>[21] S. Carlip, 'Statistical mechanics of the (2+1)-dimensional black hole,' Phys. Rev. D 51 (1995) 632-637, arXiv:gr-qc/9409052 .</list_item>
<list_item><location><page_7><loc_12><loc_18><loc_76><loc_21></location>[22] G. Barnich and C. Troessaert, 'BMS charge algebra,' JHEP 12 (2011) 105, arXiv:1106.0213 .</list_item>
<list_item><location><page_7><loc_12><loc_14><loc_86><loc_17></location>[23] W. Donnelly and L. Freidel, 'Local subsystems in gauge theory and gravity,' JHEP 09 (2016) 102, arXiv:1601.04744 .</list_item>
<list_item><location><page_7><loc_12><loc_11><loc_84><loc_14></location>[24] S. Carrozza and P. A. Hoehn, 'Edge modes as reference frames and boundary actions from post-selection,' JHEP 02 (2022) 172, arXiv:2109.06184 .</list_item>
<list_item><location><page_7><loc_12><loc_9><loc_87><loc_10></location>[25] L. Freidel, R. Oliveri, D. Pranzetti, and S. Speziale, 'Extended corner symmetry, charge</list_item>
</unordered_list>
<text><location><page_8><loc_16><loc_90><loc_77><loc_92></location>bracket and Einstein's equations,' JHEP 09 (2021) 083, arXiv:2104.12881 .</text>
<unordered_list>
<list_item><location><page_8><loc_12><loc_87><loc_88><loc_90></location>[26] L. Freidel, M. Geiller, and D. Pranzetti, 'Edge modes of gravity. Part I. Corner potentials and charges,' JHEP 11 (2020) 026, arXiv:2006.12527 .</list_item>
<list_item><location><page_8><loc_12><loc_84><loc_86><loc_86></location>[27] L. Freidel, M. Geiller, and D. Pranzetti, 'Edge modes of gravity. Part II. Corner metric and Lorentz charges,' JHEP 11 (2020) 027, arXiv:2007.03563 .</list_item>
<list_item><location><page_8><loc_12><loc_79><loc_87><loc_83></location>[28] L. Freidel, M. Geiller, and W. Wieland, 'Corner symmetry and quantum geometry,' in Handbook of Quantum Gravity , L. M. Cosimo Bambi and I. Shapiro, eds. Springer, 2023. arXiv:2302.12799 .</list_item>
<list_item><location><page_8><loc_12><loc_75><loc_82><loc_78></location>[29] W. Wieland, 'Null infinity as an open Hamiltonian system,' JHEP 04 (2021) 095, arXiv:2012.01889 .</list_item>
<list_item><location><page_8><loc_12><loc_72><loc_83><loc_75></location>[30] W. Wieland, 'Discrete gravity as a topological field theory with light-like curvature defects,' JHEP 05 (2017) 142, arXiv:1611.02784 .</list_item>
<list_item><location><page_8><loc_12><loc_65><loc_87><loc_71></location>[31] H. Bondi, M. G. J. van der Burg, and A. W. K. Metzner, 'Gravitational waves in general relativity, VII. Waves from axi-symmetric isolated system,' Proc. of the Royal Soc. Lond. A: Mathematical, Physical and Engineering Sciences 269 (1962), no. 1336, 21-52, doi:10.1098/rspa.1962.0161 .</list_item>
<list_item><location><page_8><loc_12><loc_60><loc_86><loc_65></location>[32] R. Sachs, 'Gravitational waves in general relativity VIII. Waves in asymptotically flat space-time,' Proceedings of the Royal Society London A 270 (1962), no. 1340, 103-126, doi:10.1098/rspa.1962.0206 .</list_item>
<list_item><location><page_8><loc_12><loc_57><loc_84><loc_60></location>[33] H. Bondi, 'Gravitational Waves in General Relativity,' Nature 186 (1960), no. 4724, 535-535, doi:10.1038/186535a0 .</list_item>
<list_item><location><page_8><loc_12><loc_52><loc_85><loc_56></location>[34] E. Newman and R. Penrose, 'An Approach to Gravitational Radiation by a Method of Spin Coefficients,' Journal of Mathematical Physics 3 (1962), no. 3, 566-578, doi:10.1063/1.1724257 .</list_item>
<list_item><location><page_8><loc_12><loc_48><loc_87><loc_51></location>[35] W. Wieland, 'Fock representation of gravitational boundary modes and the discreteness of the area spectrum,' Ann. Henri Poincaré 18 (2017) 3695-3717, arXiv:1706.00479 .</list_item>
<list_item><location><page_8><loc_12><loc_45><loc_73><loc_48></location>[36] E. Witten, 'Dyons of Charge eθ/ 2 π ,' Phys. Lett. B 86 (1979) 283-287, doi:10.1016/0370-2693(79)90838-4 .</list_item>
<list_item><location><page_8><loc_12><loc_42><loc_87><loc_44></location>[37] C. Rovelli and L. Smolin, 'Discreteness of area and volume in quantum gravity,' Nuclear Physics B 442 (1995), no. 3, 593-619, arXiv:gr-qc/9411005 .</list_item>
<list_item><location><page_8><loc_12><loc_38><loc_84><loc_41></location>[38] A. Ashtekar and J. Lewandowski, 'Quantum theory of geometry I.: Area operators,' Class. Quant. Grav. 14 (1997) A55-A82, arXiv:gr-qc/9602046 .</list_item>
<list_item><location><page_8><loc_12><loc_33><loc_87><loc_38></location>[39] A. Ashtekar and J. Lewandowski, 'Quantum Theory of Geometry II: Volume operators,' Advances in Mathematical and Theoretical Physics 1 (1997) 388-429, arXiv:gr-qc/9711031 .</list_item>
<list_item><location><page_8><loc_12><loc_30><loc_87><loc_33></location>[40] E. Bianchi, 'The Length operator in Loop Quantum Gravity,' Nucl. Phys. B 807 (2009) 591-624, arXiv:0806.4710 .</list_item>
<list_item><location><page_8><loc_12><loc_27><loc_87><loc_29></location>[41] E. Bianchi and H. M. Haggard, 'Discreteness of the volume of space from Bohr-Sommerfeld quantization,' Phys. Rev. Lett. 107 (2011) 011301, arXiv:1102.5439 .</list_item>
<list_item><location><page_8><loc_12><loc_23><loc_87><loc_26></location>[42] A. Ashtekar, T. Pawlowski, and P. Singh, 'Quantum nature of the big bang,' Phys. Rev. Lett. 96 (2006) 141301, arXiv:gr-qc/0602086 .</list_item>
<list_item><location><page_8><loc_12><loc_20><loc_86><loc_23></location>[43] A. Ashtekar, T. Pawlowski, and P. Singh, 'Quantum Nature of the Big Bang: Improved dynamics,' Phys. Rev. D 74 (2006) 084003, arXiv:gr-qc/0607039 .</list_item>
<list_item><location><page_8><loc_12><loc_15><loc_85><loc_19></location>[44] C. Rovelli and F. Vidotto, 'Evidence for Maximal Acceleration and Singularity Resolution in Covariant Loop Quantum Gravity,' Phys. Rev. Lett. 111 (2013) 091303, arXiv:1307.3228 .</list_item>
</unordered_list>
</document>
[
{
"title": "Evidence for Planck Luminosity Bound in Quantum Gravity",
"content": "Wolfgang Wieland 1 Institute for Quantum Gravity, Theoretical Physics III, Department of Physics Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7, 91052 Erlangen, Germany 19 February 2024",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "Recently, we introduced a non-perturbative quantization of impulsive gravitational null initial data. In this note, we investigate an immediate physical implication of the model. One of the quantum numbers is the total luminosity carried to infinity. We show that a transition happens when the luminosity reaches the Planck power L P . Below L P , the spectrum of the radiated power is discrete. Above the Planck power, the spectrum is continuous and contains caustics that can spoil the semi-classical interpretation of the resulting quantum states of geometry.",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "It has been often argued that the Planck power places an upper bound on the gravitational wave luminosity, see e.g. [1, 2], but also [3] for more critical remarks. The perhaps simplest argument in favour of this idea can be found in Misner, Thorne, Wheeler [1]. Take the virial theorem, i.e. E kin ∼ Mω 2 R 2 ∼ GM 2 /R , and the quadrupole formula for the gravitational wave luminosity, i.e. L GW ∼ L -1 P ( MR 2 ω 3 ) 2 , where M is the mass of the system, R denotes its spatial extension and ω is the frequency at which the system oscillates. Since the emission can only happen when R is greater than the Schwarzschild radius 2 GM/c 2 , we obtain L GW ≲ L P . Such a simple bound can not exist in higher dimensions. In D > 4 spacetime dimensions, the Planck power depends not only on G and c , but also on ℏ , which can not appear at the classical level. It is no surprise therefore that it is in D = 4 alone that we have an equation for the peak luminosity of two coalescing black holes, which depends only on dimensionless observables, such as the mass ratio η = m 1 m 2 / ( m 1 + m 2 ) 2 and dimensionless spin components [4, 5]. In higher dimensions, we need an additional 1length scale [6]. This makes it doubtful that there is a universal bound on the gravitational wave luminosity when D > 4 . Recently, we introduced a non-perturbative quantization of impulsive gravitational null initial data in D = 4 , see [7]. The proposal relies on the geometry of light-like boundaries, which simplifies the construction of gauge-invariant observables [8-11]. In this note, we investigate the role of the Planck power L P in the model. The analysis is based on a combination of nonperturbative and semi-classical techniques. The bound appears upon adding a parity-violating γ -term [12-14] to the gravitational action in the bulk, which, by coincidence or not, only appears in D = 4 . The resulting γ -Hilbert-Palatini action is where F = d A + 1 2 [ A,A ] is the curvature of the so (1 , 3) connection, e α is the co-tetrad and γ is the Barbero-Immirzi parameter [15, 16]. Since L P → ∞ as G → 0 , we also note that it seems implausible to find a bound on the gravitational wave luminosity from a perturbative quantisation, where we have a formal perturbative expansion with respect to κ = √ 8 πG/c 3 .",
"pages": [
1,
2
]
},
{
"title": "2. Phase space of impulsive data",
"content": "In the following, we consider the phase space of a pulse of gravitational null initial data on a null boundary N 3 ≃ [ -1 , 1] × S 2 . Since the boundary is null, we can introduce a co-dyad ( e 1 a , e 2 a ) ∈ Ω 1 ( N 3 : R 2 ) , intrinsic to N 3 , that diagonalizes the signature (0++) metric where δ ij is the Kronecker delta and φ ∗ N 3 g ab is the pull-back of the spacetime metric g ab in M 4 to the boundary N 3 ⊂ ∂ M 4 . Any such co-dyad can be parametrized by a conformal factor Ω and an SL (2 , R ) holonomy S , where · e m a is a fiducial dyad, which we keep fixed once and for all. A possible choice is · e 1 a = ∂ a ϑ , · e 2 a = sin( ϑ ) ∂ a φ , where ( ϑ, φ ) are standard spherical coordinates that are Lie dragged along the null generators. By introducing a time coordinate U : N 3 → [ -1 , 1] , we extend the angular coordinates ( ϑ, φ ) into a three-dimensional coordinate system of N 3 . The resulting vector field ∂ U is null. We assume it to be future pointing. The boundary conditions are In the absence of additional structure, such as a preferred foliation, symmetries or asymptotic boundary conditions, there is no natural torsionless and metric compatible covariant derivative in T N 3 , see e.g. [17]. A natural derivative D a exists on the extended vector bundle ⋃ p ∈ N 3 { p }× T p M 4 , where it is induced from the bulk, i.e. D a = φ ∗ N 3 ∇ a , given the usual Levi-Civita covariant derivative ∇ a in ( M 4 , g ab ) . The corresponding connection, which depends on both the extrinsic and intrinsic geometry of N 3 , is the metric analogue of the self-dual Ashtekar connection [18]. There are infinitely many clock variables U that satisfy the boundary condition (5). To pick a unique representative, we impose the gauge condition Upon choosing this gauge, the residual constraints simplify [11]. We are left with a transport equation for the SL (2 , R ) holonomy and the Raychaudhuri equation [19] for Ω 2 . The Rachaudhuri equation becomes where σ is the shear of the null generators ∂ U of N 3 . It defines a transport equation for the SL (2 , R ) holonomy, where we introduced a decomposition of sl (2 , R ) into translational components X and ¯ X and a U (1) generator J with sl (2 , R ) commutation relations The SL (2 , R ) element S ( U , ϑ, φ ) that parametrizes the co-dyads in (4) is then given by where ∆ : N 3 → [0 , 2 π )mod2 π is an unspecified U (1) angle with boundary condition ∆( U = -1 , ϑ, φ ) = 0 . In addition, S ( U = -1 , ϑ, φ ) = S -( ϑ, φ ) . In the interior of N 3 , the value of ∆ can be gauged to zero. At the upper boundary C + it can not. The boundary data S -( ϑ, φ ) and ∆ + ( ϑ, φ ) = ∆( U = 1 , ϑ, φ ) are an example of gravitational edge modes [20-30]. The shear σ : N 3 → C is unconstrained. It determines the free radiative data along N 3 . To describe the quantum geometry of a single pulse of radiation, we consider only those configurations on phase space, where σ is constant along the null generators of N 3 . We can then integrate the Raychaudhuri equation and obtain where E ± are free corner data at the initial and final cross section of N 3 . The action (2) determines the symplectic structure for the initial data on N 3 . Upon taking the pull-back to the solution space of the transport equations, we obtain canonical commutation relations [7]. The only non-vanishing brackets are The relationship between the canonical variables and the geometry of the impulsive boundary data is determined by two sets of equations. First of all, we have where ∆ + ( ϑ, φ ) = ∆( U = 1 , ϑ, φ ) and ϕ : σ = | σ | e i ϕ are U (1) angles. The SL (2 , R ) element U determines additional corner data that parametrize the shape degrees of freedom of the signature (0++) metric at the boundary. The overall scale of the boundary metric is set by the U (1) generator L and the norm of the oscillator variables a and b . We obtain with z = ( ϑ, φ ) and The quotient of the two oscillators determines the shear Finally, there is one residual pair of second-class constraints, imposing recurrence relations for physical states, The constraints are second-class. At the quantum level, only one of them can be imposed strongly. The other maps the physical Hilbert space into its orthogonal complement.",
"pages": [
2,
3,
4
]
},
{
"title": "3. Critical luminosity",
"content": "Upon quantizing the oscillators a, ¯ a , and b, ¯ b and the SL (2 , R ) × sl (2 , R ) variables ( U, L, c, ¯ c ) , we obtain a kinematical Hilbert space. Physical states lie in the kernel of one of the constraints, e.g. (20). The kinematical Hilbert space carries a unitary representation of SL (2 , R ) . The representations are characterized by the value of the Casimir. At the classical level, ̸ When L 2 > c ¯ c , the spectrum of the Casimir is discrete. When L 2 < c ¯ c , it is continuous. For σ = 0 and E + = E -, the Casimir is positive, i.e. L 2 > c ¯ c . As we increase σ , we will reach a critical value σ crit , where the sign will flip. Expanding the Casimir (22) for small shear, we obtain In a neighbourhood of future null infinity I + , we have an asymptotic expansion with respect to an affine radial Bondi coordinate r [31-33]. The shear of the ingoing null generators vanishes as O ( r -1 ) . The area density Ω 2 blows up as O ( r 2 ) . We can thus use (23) to evaluate σ crit in the asymptotic limit, in which we take N 3 to future null infinity I + . Below, we have a pictorial representation of the resulting geometry. Impulsive data at future null infinity. In the shaded region N 3 , the time derivative of the asymptotic shear σ (0) ( u, ϑ, φ ) is constant in u , everywhere else ˙ σ (0) = 0 . If we consider such boundary data at future null infinity I + , we can embed them into Bondi coordinates ( u, r, ϑ, ϕ ) . The pulse starts at an asymptotic Bondi time u -( ϑ, φ ) and terminates at u + ( ϑ, φ ) with total duration (∆ u )( ϑ, φ ) = u + ( ϑ, φ ) -u -( ϑ, φ ) . To leading order in the 1 /r -expansion, the map between the boundary intrinsic time coordinate U : N 3 → [ -1 , 1] and the asymptotic Bondi time u is a mere angle-dependent dilation, Upon introducing an adapted Newman-Penrose tetrad ( k a , ℓ a , m a , ¯ m a ) , see [34], where k a = ∂ a r and ℓ a = ∂ a u + O ( r -1 ) , and both k a = -∇ a u and ℓ a are surface orthogonal, we obtain the asymptotic expansion The map (24) between the two clock variables implies a relationship between the asymptotic Bondi shear σ (0) and a family of radiative data { σ r } at the abstract boundary N 3 , This equation allows us to relate the asymptotic Bondi shear to the critical shear (23), where the Casimir changes its sign. Using the standard round metric at future null infinity I + , we then also have We insert the expansion back into (23) and obtain the critical shear of the null generators ∂ a U We translate this value back into the asymptotic Bondi frame. Equation (27) implies that σ crit. corresponds to a critical value for the asymptotic shear given by Using the Bondi mass loss formula, we infer a critical value for the luminosity of the gravitational wave pulse, If L 2 -c ¯ c > 0 , the impulsive wave will have a luminosity smaller than L crit. In this regime, each light ray carries a discrete unitary representation of SL (2 , R ) . The fundamental operators are where n a and n b are integers, n c = N,N +1 , . . . is integer or half-integer and N = 1 , 3 2 , . . . . Physical states are annihilated by the constraint (20). For the discrete series representations of SL (2 , R ) , a unique solution can be found by a linear combination of states For the continuous series representations of SL (2 , R ) , L 2 -c ¯ c < 0 . The spectrum of the Casimir is continuous, for any s ∈ R , we have In this regime, the impulsive wave will have a luminosity bigger than L crit. The spectrum of the Casimir is continuous, and the operator L is no longer bounded from below. This has important consequences. The recurrence relations (20) will not terminate, physical states will be superpositions of kinematical states, where the quantum numbers n a and n b will become arbitrarily large. Therefore, the shear σ ¯ σ will be unbounded from above, see (19). If we take the Born rule and compute the probabilities for σ ¯ σ to take a certain value, there will always be a chance that an observer obtains √ 2 σ ¯ σ > π/ 2 . In this case, the profile of the area density (11) will pass through a caustic, where Ω 2 = 0 . When there is a caustic, we violate the implicit assumption that we are in a smooth asymptotic region, in which Ω 2 = r 2 + O ( r ) , as r → ∞ . This can be avoided only when L 2 > c ¯ c , i.e. when the luminosity of the gravitational wave pulse is below L crit. . Then, the physical states are built from superpositions of the discrete series unitary representations of SL (2 , R ) and for any physical state the quantum numbers n a and n b will be bounded from above.",
"pages": [
4,
5,
6
]
},
{
"title": "4. Outlook and Conclusion",
"content": "̸ The critical luminosity (32) separates the continuous spectrum from the discrete eigenvalues of the SL (2 , R ) Casimir. The bound depends on the Barbero-Immirzi parameter γ . This is a common feature in D = 4 . When γ = 0 , the boundary charges are a mixture of electric and magnetic contributions that otherwise vanish in the γ →∞ limit [7, 11, 35]. The spectrum of the charges is determined by their algebraic properties alone, but the map between the charges and the physical observables depends on γ . In this way, the spectrum of physical observables can depend on γ . This is analogous to how the θ -angle in quantum electrodynamics enters the Dirac quantization condition between magnetic and electric charges [36]. In loop quantum gravity, this effect is responsible for the quantization of geometric observables, such as area, angles, volumes and length [37-41]. Such a fundamental quantum discreteness of geometry affects other physical observables. It creates a fundamental bound on the energy density of matter [42, 43] and perhaps also acceleration [44]. Here, we found a similar bound on the gravitational wave luminosity (32). The analysis is based on a non-perturbative quantization of radiative data at finite distance. The result is only partial, because there is an implicit assumption: the validity of the classical asymptotic 1 /r -expansion when applied to the spectrum of gravitational observables at finite null boundaries. If the luminosity exceeds the critical luminosity (32), this assumption may no longer be valid due to the possible creation of caustics. What we have shown so far is only a first step. A more refined investigation will follow to understand the significance of the Planck power for the spectrum of the gravitational wave luminosity in non-perturbative quantum gravity.",
"pages": [
6
]
},
{
"title": "References",
"content": "[1] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation . W. H. Freeman, San Francisco, 1973. bracket and Einstein's equations,' JHEP 09 (2021) 083, arXiv:2104.12881 .",
"pages": [
6,
8
]
}
]
2024arXiv240216533H
https://arxiv.org/pdf/2402.16533.pdf
<document>
<section_header_level_1><location><page_1><loc_13><loc_88><loc_87><loc_90></location>Visualization of frequency structures in gravitational wave signals</section_header_level_1>
<text><location><page_1><loc_19><loc_83><loc_81><loc_85></location>CHAD HENSHAW, MEGAN AROGETI, ALICE HERANVAL, LAURA CADONATI</text>
<text><location><page_1><loc_25><loc_81><loc_75><loc_82></location>School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA</text>
<text><location><page_1><loc_43><loc_78><loc_57><loc_79></location>February 27, 2024</text>
<section_header_level_1><location><page_1><loc_47><loc_75><loc_53><loc_76></location>Abstract</section_header_level_1>
<text><location><page_1><loc_12><loc_60><loc_88><loc_73></location>The gravitational wave signals produced by the coalescence of compact binaries progress through three stages: inspiral, merger, and postmerger. The evolution of their frequency follows a slow build up during the inspiral that peaks at merger, forming the characteristic 'chirp' pattern in the signal's time-frequency map. Herein we introduce a framework for localizing further characteristic structures in the time-frequency space of gravitational wave signals using the continuous wavelet transform. We consider two example cases where there are specific patterns in the postmerger stage of the signal that are rich with information on the physical nature of the source: highly-inclined black hole binaries with asymmetric mass ratio, and neutron star binaries with postmerger remnant oscillations. It is demonstrated that the choice of quality factor Q plays a central role in distinguishing the postmerger features from that of the inspiral, with black hole systems preferring lower Q and neutron star systems preferring higher Q. Furthermore, we demonstrate the use of chirplets as the wavelet transform basis, which allow for manipulation of structure in the time-frequency map.</text>
<section_header_level_1><location><page_1><loc_20><loc_55><loc_36><loc_57></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_26><loc_48><loc_53></location>Gravitational waves (GWs) from compact binary coalescences (CBCs) exhibit distinctive characteristics in the evolution of their frequency. To date, the LIGO, Virgo, and KAGRA scientific collaborations [1, 2, 3], which now comprise the LIGO-Virgo-KAGRA Collaboration (LVK), have detected 90+ signals from CBC systems [4, 5, 6, 7]. Starting at some initial separation distance, the two objects in the binary orbit a common center of mass, and as time progresses their separation distance will decrease as energy is lost through gravitational radiation in the form of GWs. As the separation distance decreases, the orbital period decreases, thus the orbital frequency increases, and thus the GW frequency also increases ( fgw ≈ 2 f orb . ) . During this stage of the merger event, known as the inspiral, the GW amplitude also increases, peaking at the time of merger, and therefore the frequency evolution during the inspiral is referred to as a 'chirp.' At lowest post-Newtonian (PN) order, the GW frequency during the inspiral evolves as ˙ f ( t ) ∝ M 5 / 3 f 11 / 3 [8]</text>
<text><location><page_1><loc_18><loc_25><loc_26><loc_26></location>[ ] 3 / 5</text>
<text><location><page_1><loc_8><loc_12><loc_48><loc_26></location>where M = M q ( 1 + q ) 2 is the 'chirp mass,' M = m 1 + m 2 is the total mass, and q = m 1 / m 2 , m 1 ≥ m 2 is the mass ratio. The inclusion of higher order terms also sees contribution from the mass ratio, the spin angular momenta of the compact objects, and their tidal deformabilities in the case of objects with matter like neutron stars (NSs). Near the point of merger this approximation breaks down, and a fully relativistic evaluation is needed to describe the merger. The characterization of the postmerger stage varies depending on the nature of the</text>
<text><location><page_1><loc_52><loc_49><loc_92><loc_57></location>compact objects in the binary. If one of the two objects is a black hole (BH), then the merger constitutes the formation of a distorted common horizon that encapsulates both objects, which then 'rings down' in the postmerger to spherical symmetry.</text>
<text><location><page_1><loc_52><loc_19><loc_92><loc_48></location>The frequency evolution of a GW signal can reveal much about the nature of the originating system, as GWs fundamentally encode the properties of their source. Time-frequency analysis sees wide use in the gravitational wave community, over a variety of applications. One of the primary applications for time-frequency mapping in gravitational wave data analysis is the unmodeled search for transient signals. This effort does not assume anything about the shape of the gravitational wave signal, and instead relies on time-frequency analysis utilizing the multi-resolution 'Q-scan,' introduced in [9, 10]. If the detector noise is assumed to be stationary and Gaussian, then the presence of any other component - like a gravitational wave signal, or other noise transient - will show up in the time-frequency map as excess power [11]. This excess power method has been widely used in all-sky searches of gravitational wave data [12, 13, 14, 15], which incorporate both online [16] and offline follow-up [17, 18] algorithms. Time-frequency analysis is also commonly used in the identification of noise transients in LVK data for detector characterization [17, 19].</text>
<text><location><page_1><loc_52><loc_12><loc_92><loc_17></location>Beyond these applications, time-frequency analysis can also be used to understand specific frequency characteristics in gravitational waves that are directly connected to the physics of their originating system. In this document we will</text>
<text><location><page_2><loc_8><loc_70><loc_48><loc_88></location>review two examples of systems that contain distinctive frequency structure in the post-merger gravitational wave signal: binary neutron star (BNS) systems, and highly-inclined binary black hole (BBH) systems with asymmetric mass ratio. By applying targeted time-frequency analysis using the continuous wavelet transform (CWT) to synthetic representations of these signals, we will demonstrate that these features may be isolated from the primary frequency spike due to the inspiral. This enables routes to further analysis such as parameterization for theoretical exploration, or detection using stacking methods, both of which will be discussed in subsequent publications.</text>
<text><location><page_2><loc_8><loc_47><loc_48><loc_69></location>In a recent Numerical Relativity (NR) study [20], it was demonstrated that GW signals from BBH systems with asymmetric mass ratio q > 1 exhibit additional frequency peaks after merger when viewed from a highly-inclined angle. This study further demonstrates that these frequency peaks are correlated to regions of tight curvature on the horizon of the distorted final BH. Such signals are therefore of immediate interest, as if this correlation can be understood then there is a possibility of inferring the final black hole horizon geometry from the gravitational wave signals that we receive on Earth. Additionally, as these postmerger frequency peaks can be easily mistaken for spurious transients (glitches) in detector data [21], understanding how to differentiate them as a physical effect is important for the realistic detection of intermediate-mass black hole (IMBH) binary systems.</text>
<text><location><page_2><loc_8><loc_12><loc_48><loc_46></location>In the case of a BNS system, the postmerger gravitational wave signal is highly variable depending on several factors. Neutron stars are highly dense, compact stellar objects where the interior structure is dictated by an equation of state (EoS), which provides a relationship between the NS interior pressure, density, and temperature. The exact nature of these conditions in the interior of neutron stars is unknown[22, 23], but gravitational waves signals from BNS mergers offer a means by which the EoS may be studied. The EoS is encoded not only in the inspiral stage of a BNS merger [24, 25], but also in the postmerger stage. Depending on the component masses and the EoS, a BNS merger could result in i) a prompt collapse to a black hole, ii) a hypermassive or supramassive neutron star that undergoes a delayed collapse to a black hole or iii) a stable neutron star [26, 27, 28]. In general, oscillations of this postmerger remnant can emit a high-frequency gravitational wave signal, the specific characteristics of which are dependent on the remnant properties. In the case of a delayed collapse to a black hole, the post-merger gravitational wave signal could have a duration on the order of 10 -100 [ms], with frequencies between 1 -4 [kHz]. If detected, this postmerger signal could provide a window to study a higher-mass regime than allowed by the inspiral signal [27, 28].</text>
<text><location><page_2><loc_52><loc_76><loc_92><loc_88></location>This document is organized as follows. In Sec.(II), we will briefly review the fundamentals of time-frequency analysis, and in Sec.(III) we introduce an implementation of the CWT method. In Sec.(IV), we will then apply the CWT to select simulated BBH and BNS postmerger waveforms to demonstrate its use for feature discrimination. Finally, we conclude in Sec.(V) with a discussion on further applications and routes of analysis.</text>
<section_header_level_1><location><page_2><loc_52><loc_73><loc_91><loc_74></location>II. PRECEPTS OF TIME-FREQUENCY ANALYSIS</section_header_level_1>
<text><location><page_2><loc_52><loc_67><loc_92><loc_71></location>The Fourier transform takes a continuous time series x ( t ) and decomposes it into its constituent frequencies, rendering a continuous frequency series ˆ x ( f ) :</text>
<formula><location><page_2><loc_61><loc_61><loc_92><loc_64></location>ˆ x ( f ) = ∫ ∞ -∞ x ( t ) exp [ -i 2 π f t ] dt . (1)</formula>
<text><location><page_2><loc_52><loc_53><loc_92><loc_60></location>Considering the product between the time series x ( t ) and the complex sinusoid exp [ -i 2 π f t ] in Eq.(1), the Fourier transform is a check for similarity. The sinusoid can be thought of as an analyzing function g ( t ) , and thus the integral in Eq.(1) is the inner product:</text>
<formula><location><page_2><loc_64><loc_47><loc_92><loc_50></location>⟨ x , g ⟩ = ∫ ∞ -∞ x ( t ) g ∗ ( t ) dt , (2)</formula>
<text><location><page_2><loc_52><loc_37><loc_92><loc_46></location>where g ∗ ( t ) denotes the complex conjugate of g ( t ) 1 . The resultant complex function ˆ x ( f ) describes the amplitude and phase of the component of x ( t ) that matches the frequency f of the sinusoid, which spans a continuous domain. This frequency domain representation ˆ x ( f ) can likewise be converted back to the time domain by the inverse Fourier transform:</text>
<formula><location><page_2><loc_62><loc_31><loc_92><loc_34></location>x ( t ) = ∫ ∞ -∞ ˆ x ( f ) exp [ i 2 π f t ] d f , (3)</formula>
<text><location><page_2><loc_52><loc_15><loc_92><loc_30></location>such that x ( t ) and ˆ x ( f ) form a Fourier pair. Note that in performing the conversion from x ( t ) to ˆ x ( f ) , all information about the signal's time evolution is forsaken. Therefore given a signal x ( t ) , the Fourier transform can describe the signal's spectrum (the strength of its frequency components), but it does not describe when in time the signal reaches those frequencies. To understand simultaneously the signal's frequency evolution, and its strength per frequency, one needs to compute a two-dimensional function X ( t , f ) that preserves both the time and frequency information. One of the primary</text>
<text><location><page_3><loc_8><loc_86><loc_48><loc_88></location>methods for time-frequency analysis is the short-time Fourier transform, first devised by Dennis Gabor in 1946 [29]:</text>
<formula><location><page_3><loc_14><loc_80><loc_48><loc_83></location>X ( τ , f ) = ∫ ∞ -∞ x ( t ) w ∗ ( t -τ ) exp [ -i 2 π f t ] dt , (4)</formula>
<text><location><page_3><loc_8><loc_49><loc_48><loc_79></location>where w ( τ ) is a window function centered on τ = 0; i.e. a function that is only non-zero within some chosen interval. Note that Eq.(4) is exactly a FT as in Eq.(1), where the effective time series x ( t ) is now the product between the signal and the time-shifted window w ( t -τ ) . One can also think of the inner product picture, where the analyzing function is now the product between the window and the sinusoid. Under either interpretation, the introduction of the time-shift changes the Fourier transform into a convolution , in which the analyzing function 'slides' along the signal, and the overlap X ( τ , f ) is computed at each (continuous) time τ and (continuous) frequency f . Because the window has a finite width, this is effectively computing the FT of the signal over a short time segment, then stacking those segments next to each other to build a continuous two-dimensional time-frequency representation column by column. Alternatively in the inner product picture, the convolution produces a new time series of coefficients at the central frequency of the analyzing function, essentially comprising one row in the resultant two-dimensional time-frequency map.</text>
<text><location><page_3><loc_8><loc_31><loc_48><loc_49></location>There are several different types of window functions that may be used with the STFT, but one of the drawbacks of this method is that the window function has a fixed width. Regardless of the specific function, the window width is related to the time-resolution of the resultant spectrogram. The larger the window duration, the poorer one's time resolution, but the finer the frequency resolution becomes. Because the width of the window is fixed, the STFT is limited in its ability to adapt to signals with rapid changes in frequency evolution. In contrast, the CWT method discussed below allows for variable window widths, and therefore has greater flexibility in adapting to the signal.</text>
<section_header_level_1><location><page_3><loc_9><loc_28><loc_48><loc_29></location>III. THE CONTINUOUS WAVELET TRANSFORM</section_header_level_1>
<text><location><page_3><loc_8><loc_17><loc_48><loc_26></location>The continuous wavelet transform (CWT) is a process that scans a type of filter called a wavelet - an oscillating function modulated by some envelope - across a signal to probe its time and frequency space. For more detail on wavelets see Appendix Sec.(AII). The CWT is defined by the following expression:</text>
<formula><location><page_3><loc_15><loc_12><loc_48><loc_15></location>T ( a , b ) = ∫ ∞ -∞ x ( t ) [ w ( a ) ψ ∗ ( t -b a )] dt , (5)</formula>
<text><location><page_3><loc_52><loc_57><loc_92><loc_88></location>where x ( t ) represents the signal, w ( a ) is a weighting function 2 , and ψ ∗ is the complex conjugate of the 'mother wavelet' - the wavelet being scanned across the signal. The scanning occurs by scaling the wavelet by a dilation parameter a , and shifting its time localization by a translation parameter b . As with the STFT, the time shifting makes Eq.(5) a convolution between the scaled wavelet function - i.e. the analyzing function - and the signal. The CWT coefficients T ( a , b ) may then be computed in the same manner as previously, but now instead of passing explicit frequencies to define our filter we pass an array of discrete wavelet scales. In practice the signal data is a digital time series with sampling rate fs and time steps ∆ b = f -1 s . For each scale a , we fast Fourier transform (FFT) both the wavelet and the signal, take their product in Fourier space, then inverse FFT the result to produce a time series array of coefficients per a . 3 The resultant two-dimensional map is a scalogram - as it maps the scaled wavelets to their inner product with the signal at central times b . The wavelet scales a may then be related to their central frequencies, and in this way the CWT builds the time-frequency map row by row.</text>
<text><location><page_3><loc_52><loc_28><loc_92><loc_56></location>Alternatively, one may also imagine the CWT process as a point-wise inner product between the wavelet and the signal. For each time step b , the wavelet's scale is varied by the dilation parameter a . These parameters therefore become coordinates that map the signal's time-frequency representation. Under this interpretation, the CWT is often referred to as a 'mathematical microscope,' a metaphor that may be further extended by considering the wavelet form chosen for the operation as a type of lens. The dilation parameter a therefore describes the width of the lens' aperture, and the translation parameter b describes its location within the signal domain. Regardless of the interpretation, T ( a , b ) is a measure of the similarity between the wavelet and the signal at the point ( a , b ) . In this way the CWT can be tuned to identify specific substructures in the time-frequency space of a signal, by choosing a mother wavelet that resembles those features. Let's now consider the example of the scaled and translated Morlet-Gabor (MG) wavelet, given in the time and frequency domains by:</text>
<formula><location><page_3><loc_53><loc_18><loc_92><loc_25></location>ψ ( t ; f 0 , a , b ) = = π -1 4 a -1 / 2 exp [ -1 2 ( t -b a ) 2 ] exp [ i 2 π f 0 ( t -b a )] . (6)</formula>
<text><location><page_4><loc_10><loc_87><loc_10><loc_88></location>˜</text>
<text><location><page_4><loc_9><loc_87><loc_11><loc_88></location>ψ</text>
<text><location><page_4><loc_11><loc_87><loc_11><loc_88></location>(</text>
<text><location><page_4><loc_12><loc_87><loc_12><loc_88></location>f</text>
<text><location><page_4><loc_12><loc_87><loc_13><loc_88></location>;</text>
<text><location><page_4><loc_13><loc_87><loc_14><loc_88></location>f</text>
<text><location><page_4><loc_14><loc_87><loc_14><loc_88></location>0</text>
<text><location><page_4><loc_14><loc_87><loc_15><loc_88></location>,</text>
<text><location><page_4><loc_15><loc_87><loc_16><loc_88></location>a</text>
<text><location><page_4><loc_16><loc_87><loc_16><loc_88></location>,</text>
<text><location><page_4><loc_16><loc_87><loc_17><loc_88></location>b</text>
<text><location><page_4><loc_17><loc_87><loc_19><loc_88></location>) =</text>
<formula><location><page_4><loc_10><loc_83><loc_48><loc_87></location>= √ 2 π 1 4 a 1 / 2 exp [ -i 2 π f b ] exp [ -1 2 ( 2 π f 0 -2 π af ) 2 ] . (7)</formula>
<text><location><page_4><loc_8><loc_70><loc_48><loc_82></location>This wavelet has unit energy, central time τ = b , central frequency φ = f 0 / a , characteristic duration σ t = a / √ 2, and bandwidth σ f = 1 / 2 √ 2 π a ; see Appendix Sec.(AII) for derivations. Note that the duration-bandwidth product is σ t σ f = 1 / 4 π , so MG wavelets are maximally compact in time-frequency space; i.e. they reach the minimum timefrequency area allowed by the Heisenberg-Gabor uncertainty principle. The quality factor Q is then given by:</text>
<formula><location><page_4><loc_22><loc_64><loc_48><loc_67></location>Q ≡ φ σ f = 2 √ 2 π f 0 , (8)</formula>
<text><location><page_4><loc_8><loc_46><loc_48><loc_62></location>which we see does not depend on the wavelet scale; Q depends only on the starting frequency of the sinusoid in the mother wavelet. The time resolution is proportional to the wavelet scale, and the frequency resolution is inversely proportional to the wavelet scale. Given the relationship between wavelet scale and central frequency, this means that passing a linearly spaced array of wavelet scales into the CWT will yield better time localization at higher frequencies, at the cost of frequency resolution. As a corollary, the CWT will yield better frequency resolution at lower frequencies, and the cost of time localization.</text>
<text><location><page_4><loc_8><loc_22><loc_48><loc_46></location>For this reason the CWT is well suited for analyzing gravitational waves produced by CBC events that have rapidly changing frequency structure in the post-merger. The frequency evolution of such signals is characterized by a slow build at low frequency during the inspiral which peaks at merger. However for specific cases the post-merger signal can contain more complex structure in frequency evolution, which occur at higher overall frequency. Such cases are naturally suited for analysis with the CWT, as the slowly evolving behavior of the inspiral is less sensitive to the lower time resolution inherent at lower frequency, but the rapid changes in the post-merger that occur at higher frequency may be resolved. In Sec.(IV) below, we investigate two such cases: oscillation frequencies in the post-merger BNS signal, and frequency peaks in the post-merger of highly-inclined asymmetric mass ratio BBH signals.</text>
<section_header_level_1><location><page_4><loc_20><loc_18><loc_37><loc_19></location>IV. APPLICATIONS</section_header_level_1>
<text><location><page_4><loc_8><loc_12><loc_48><loc_16></location>When applied to a gravitational wave signal, the CWT produces a time-frequency map, or scalogram, that captures the primary aspects of the signal's frequency evolution. Shown in</text>
<figure>
<location><page_4><loc_52><loc_69><loc_93><loc_88></location>
<caption>Figure 1: A time-frequency map of GW150914, the first detected gravitational wave signal, created using the CWT with Morlet-Gabor wavelets at Q = 6 . 0 and logarithmic frequency spacing.</caption>
</figure>
<text><location><page_4><loc_52><loc_48><loc_92><loc_59></location>Fig.(1) is an example scalogram for GW150914, the first gravitational wave signal detected by the LIGO and Virgo Collaborations [30, 4]. This image does not display the entire inspiral portion of the signal, but one can see that over the course of 0.2 seconds the frequency rapidly and smoothly evolves monotonically, increasing by a factor of ∼ 10 before reaching the merger.</text>
<text><location><page_4><loc_52><loc_30><loc_92><loc_48></location>To produce the image in Fig.(1), the data for GW150914 was obtained using the the CWT was run using Morlet-Gabor wavelets at Q = 6 . 0, and the resultant scalogram was normalized by dividing all coefficient values by the largest value in the map. This makes the units in the colorbar of Fig.(1) somewhat arbitrary, and as such this scaling method makes the analysis less useful for differentiating the signal from noise when compared to the Q-scan method that searches for excess coherent power [9, 10, 11]. However, the consistent normalization facilitates exploration into how adjusting the parameters of the CWT analysis affects the structures in the timefrequency map for a freespace gravitational wave.</text>
<section_header_level_1><location><page_4><loc_52><loc_26><loc_68><loc_28></location>i. BBH Postmerger</section_header_level_1>
<text><location><page_4><loc_52><loc_12><loc_92><loc_25></location>Beyond the primary frequency peak denoting the inspiralmerger transition, gravitational waves from binary black hole systems with asymmetric mass ratio q > 1 exhibit additional frequency peaks after merger when viewed from a highlyinclined angle. These frequency structures were first studied in [20], where it was demonstrated that there is a correlation between the post-merger signal and the curvature of the final black hole horizon. Such signals are therefore of immediate interest, as if this correlation can be understood then there is a</text>
<figure>
<location><page_5><loc_9><loc_69><loc_49><loc_88></location>
<caption>Figure 2: Frequency structure evolution in the noiseless, freespace gravitational waveform GT0568 ( q = 10 . 0 , | S 1 | = | S 2 | = 0 ) , scaled to M = 80 M ⊙ at d L = 500 [Mpc], oriented at inclination ι = π / 2 and phase φ = 0 , and generated from a starting frequency of 30.0 [Hz]. The black vertical line denotes the time location of the maximum waveform amplitude, which is used as an approximation for the time of merger.</caption>
</figure>
<text><location><page_5><loc_8><loc_50><loc_48><loc_53></location>possibility of inferring the final black hole horizon geometry from the gravitational wave signals that we receive on Earth.</text>
<text><location><page_5><loc_8><loc_21><loc_48><loc_49></location>As a first step towards investigating this possibility, it is critical to understand how to map the time-frequency space of these waveforms. For this purpose, we use the CWT to probe frequency structure. Shown in Fig.(2) is the time-frequency map of the NR waveform GT0568 [31, 32], which is a zerospin quasicircular BBH merger with mass ratio q = 10, that includes spherical harmonic modes up to l = 8. One can see that in contrast to the relatively simple structure in Fig.(1), which is consistent with an equal mass ratio ( q ≈ 1 ) system, the mass asymmetry in GT0568 introduces a more complex frequency structure when viewed from the edge-on inclination of π / 2 at a phase angle of φ = 0. The waveform is shifted in time to locate the peak waveform amplitude at t = 0 . 0, which we use as an approximation for the time of merger. One can see that after the initial frequency spike from the inspiral-merger transition, there is an additional frequency spike in the postmerger. See Sec.(V) for further discussion on the physical interpretation of this feature, which we refer to as the doublechirp pattern.</text>
<text><location><page_5><loc_8><loc_12><loc_48><loc_20></location>The scalogram in Fig.(2) was made using the CWT with the Morlet wavelet at Q = 5 . 0. As discussed in Sec.(II) above, the quality factor Q not only defines the mother frequency for the wavelet, it is also related to the time and frequency resolutions of the CWT. For lower values of Q, the time resolution will be enhanced but the frequency resolution will be</text>
<figure>
<location><page_5><loc_52><loc_70><loc_93><loc_88></location>
<caption>Figure 3: Quality factor variation for the CWT time-frequency map with the Morlet wavelet applied to the asymmetric mass ratio BBH waveform GT0568, viewed at edge-on inclination. From left to right, Q = 1 . 0 , 8 . 0 , 16 . 0 .</caption>
</figure>
<text><location><page_5><loc_52><loc_33><loc_92><loc_60></location>diminished, and vice versa. Towards the goal of identifying the postmerger frequency peak from that of the inspiral, one should then use a lower value of Q , to better separate these two features in time. However, in doing so the frequency content becomes largely indistinguishable, and is can be difficult to discern that the time-frequency map even represents a gravitational wave signal. On the other hand, at higher values of Q , the frequency values are more strongly highlighted but the time localization becomes smeared such that the double-chirp pattern is indistinguishable from a single peak. An example of this effect is shown in Fig.(3), where we display CWT scalograms with Q = 1 . 0 , 8 . 0 , 16 . 0. In testing a variety of options with the CWT over several example waveforms with this characteristic, values of Q ∈ [ 4 . 0 , 6 . 0 ] , corresponding to mother frequencies of f 0 ∈ [ 0 . 45 , 0 . 68 ] have been the most useful for analyzing the double chirp pattern. This range is consistent with the analysis in[20], where a mother frequency of f 0 = 0 . 4 is used.</text>
<text><location><page_5><loc_52><loc_12><loc_92><loc_32></location>One can see that the scalogram in Fig.(2) reveals an undulating frequency structure during the inspiral, with a series of peaks that occur at higher and higher frequency as the system approaches merger. This is a consequence of the high mass ratio of the system; the asymmetry modulates the waveform, with each orbit effectively sweeping a burst of GW radiation across the observer's line of sight. One can view this frequency structure as a series of consecutive up-chirps and down-chirps, and considering the inner product perspective on CWT described in Sec.(III), one is provoked to wonder what the time-frequency representation looks like using a different wavelet basis. To this end we can also use chirplets in the CWT process: Morlet-Gabor wavelets that have their own frequency evolution. Further details on the chirplet basis are</text>
<figure>
<location><page_6><loc_9><loc_69><loc_49><loc_88></location>
<caption>Figure 5: Frequency structure evolution in the postmerger of the noiseless, freespace gravitational waveform GT0568 ( q = 10 . 0 , | S 1 | = | S 2 | = 0 ) , scaled to M = 80 M ⊙ at d L = 500 [Mpc], oriented at inclination ι = π / 2 and phase φ = 0 , and generated from a starting frequency of 30.0 [Hz]. The black vertical line denotes the time location of the maximum waveform amplitude, which is used as an approximation for the time of merger. A chirplet basis with d = 0 . 1 has been used for the CWT analysis, with Q = 6 . 0 . The red line traces the brightest pixel at each time step, yielding an effective one-dimensional f ( t ) .</caption>
</figure>
<figure>
<location><page_6><loc_52><loc_69><loc_93><loc_88></location>
<caption>Figure 4: Frequency structure evolution in the noiseless, freespace gravitational waveform GT0568 ( q = 10 . 0 , | S 1 | = | S 2 | = 0 ) , scaled to M = 80 M ⊙ at d L = 500 [Mpc], oriented at inclination ι = π / 2 and phase φ = 0 , and generated from a starting frequency of 30.0 [Hz]. The black vertical line denotes the time location of the maximum waveform amplitude, which is used as an approximation for the time of merger. A chirplet basis with d = 0 . 1 has been used for the CWT analysis, with Q = 5 . 0 .</caption>
</figure>
<section_header_level_1><location><page_6><loc_8><loc_49><loc_31><loc_50></location>discussed in Appendix Sec.(AIII).</section_header_level_1>
<text><location><page_6><loc_8><loc_20><loc_48><loc_48></location>In Fig.(4) one can see that in the chirplet basis, the CWT scalogram has several differences when compared to its MG wavelet counterpart in Fig.(2). As discussed in Appendix Sec.(AIII), the chirplet basis rotates burst-like signals in timefrequency space. Using a basis with an upward chirp rate (up-chirps) causes bursts to rotate counter-clockwise in timefrequency space, and using a basis with a downward chirp rate (down-chirps) causes bursts to rotate clockwise in timefrequency space. Additionally, for a basis of either up-chirps or down-chirps, bursts get stretched in frequency. Here we see that with an up-chirp CWT basis, each inspiral burst appears tilted counter-clockwise, and stretched in frequency. In a down-chirp CWT basis, each inspiral burst appears tilted clockwise, and is likewise stretched in frequency, although we omit the figure. The frequency-stretching effect can be mitigated to some degree by increasing Q for the CWT analysis, but this does yield poorer time localization, and going too high on Q can make the postmerger chirp feature indistinguishable from the merger peak.</text>
<text><location><page_6><loc_8><loc_12><loc_48><loc_19></location>Shown in Fig.(5) is a zoomed-in view of the postmerger of GT0568, rendered using CWT with a chirplet basis of d = 0 . 1 at Q = 6 . 0. Additionally, in this figure we trace an effective one-dimensional f ( t ) by highlighting the brightest pixel in the time-frequency map (shown in red). It is for this reason that</text>
<text><location><page_6><loc_52><loc_36><loc_92><loc_49></location>the chirplet basis is most useful when applied to this class of signal; one can see that there are clear boundaries that denote the inspiral peak, the postmerger peak, and the end of the frequency evolution. These boundaries allow one to clearly define the postmerger region of the waveform based on its time-frequency map, and also the existence of the postmerger frequency peak. Further discussion of this method and its use towards parameterizing the double-chirp pattern will be explored in a subsequent publication.</text>
<section_header_level_1><location><page_6><loc_52><loc_32><loc_68><loc_33></location>ii. BNS Postmerger</section_header_level_1>
<text><location><page_6><loc_52><loc_12><loc_92><loc_31></location>In the case of a BNS merger, oscillations of the postmerger remnant can emit a rich, high frequency gravitational wave signal. Given a postmerger remnant of a hypermassive or supramassive neutron star, the post-merger gravitational wave signal could be milliseconds long with high frequencies between 1-4 kHz [27, 28]. With a confident detection, this postmerger signal could provide a new probe to study extremely dense nuclear matter given the the quasi-universal relations between spectral features of the signal and properties of the neutron star [22, 33]. For example, [22, 34] found that there is a relationship between the peak frequency of the BNS postmerger signal and the radius of non-rotating NSs. This radius can then be used to help constrain the NS EoS through</text>
<figure>
<location><page_7><loc_9><loc_69><loc_49><loc_88></location>
<caption>Figure 7: Quality factor variation for the CWT time-frequency map with the Morlet wavelet applied to the BNS waveform DD2. From left to right, Q = 16 . 0 , 32 . 0 , 64 . 0 .</caption>
</figure>
<figure>
<location><page_7><loc_52><loc_70><loc_93><loc_88></location>
<caption>Figure 6: Frequency structure evolution in the noiseless, freespace BNS gravitational waveform, DD2 with component masses m 1 = 1 . 35 M ⊙ and m 2 = 1 . 35 M ⊙ and a peak postmerger frequency f peak ∼ 2500 [ Hz ] . The black vertical line denotes the time location of the maximum waveform amplitude, which is used as an approximation for the time of merger.</caption>
</figure>
<text><location><page_7><loc_8><loc_34><loc_48><loc_52></location>the one-to-one mapping from the mass-radius relationship of neutron stars to the EoS [22]. Following a detection, one way to pinpoint the peak frequency of the postmerger signal is through a time-frequency map of the waveform. To visualize the BNS postmerger signal in the time-frequency plane, we use CWT. Shown in Fig.(6) is the time-frequency map of the postmerger waveform for the DD2 EoS [35] with m 1 = 1 . 35 M ⊙ and m 2 = 1 . 35 M ⊙ . One can see that following the initial frequency increase characteristic of a compact binary coalescence, there is excess power at approximately 2500 [ Hz ] . See Sec.V for additional discussion on the physical interpretation of this high frequency signature.</text>
<text><location><page_7><loc_8><loc_12><loc_48><loc_32></location>The time-frequency map in Fig.(6) was made with the Morlet wavelet at Q = 32 . 0. As explained in Sec.III, at higher Q values, the frequency resolution is enhanced, but at the cost of diminished time localization. Given that peak frequency of a BNS postmerger signal could be used to help constrain the NS EoS, a higher value of Q should be used to get a more confident measure of f peak . In Fig.(7), CWT was tested with varied Q values of Q ∈ [ 16 . 0 , 32 . 0 , 64 . 0 ] , corresponding to mother frequencies of f 0 ∈ [ 1 . 80 , 3 . 60 , 7 . 20 ] . One can see that at lower Q the postmerger oscillation is smeared in frequency, but as Q increases it becomes more localized. However the localization of the merger is also smeared in time as Q becomes large. For this reason Q = 32 . 0 is preferred to preserve both characteristics.</text>
<section_header_level_1><location><page_7><loc_65><loc_60><loc_79><loc_62></location>V. DISCUSSION</section_header_level_1>
<text><location><page_7><loc_52><loc_29><loc_92><loc_58></location>The time-frequency maps above highlight the ability of the CWT to distinguish specific features in the frequency evolution of gravitational wave signals. In particular we have examined the postmerger waveforms from both BBH and BNS systems, where characteristic features are connected to the physics of their originating systems. The CWT method is highly flexible, and can be tuned to the specific problem at hand by varying chiefly the quality factor Q , and also the wavelet basis. The quality factor determines one's ability to localize features in either time or frequency, with simultaneous localization in both being limited by the HeisenbergGabor uncertainty relation. A lower value Q will improve time localization at the cost of frequency localization, and vice versa for a higher value of Q . In the case of BBH systems with asymmetric mass ratio, where frequency features in the postmerger are evident when viewed from an edge-on inclination, a lower Q is preferred. This allows one to differentiate the postmerger frequency peak from that of the inspiral; the exact frequency content of the postmerger peak is of secondary importance to its location in time.</text>
<text><location><page_7><loc_52><loc_12><loc_92><loc_28></location>The BNS postmerger case has the opposite requirements, as we are more interested in the frequency content than in the time localization. Here we therefore want a higher value of Q , to better isolate the frequency of the postmerger remnant oscillations. High frequency localization of the postmerger peak frequency would give way to constraints on the NS EoS through the empirical relationship between f peak and the radius of a non-rotating NS. As such, exact time localization is sacrificed for the purpose of determining the precise frequency content of the BNS postmerger signal. This work primarily examined noiseless, freespace gravitational waves, and</text>
<text><location><page_8><loc_8><loc_70><loc_48><loc_88></location>demonstrated how CWT can be utilized to extract important frequency features embedded in the BNS postmerger signal. Given estimated merger rates, current and future detector sensitivities, and predicted waveform morphologies, the probability of near-future detection is low [26, 23, 36]. Given the low probability of detection, studies such as [36] and [37] have explored combining observations using Bayesian statistics to increase the likelihood of detection. An additional stacking method that can be used to boost the detection sensitivity is to combine TF maps from multiple postmerger candidates created using CWT. This method will be detailed in a subsequent paper.</text>
<text><location><page_8><loc_8><loc_43><loc_48><loc_70></location>In addition to varying the quality factor, one can also use CWT with different bases of wavelets. Many applications in gravitational wave analysis use the Morlet-Gabor wavelet, as it is maximally compact in time-frequency space. Modifying the MG wavelet into a chirplet largely preserves this benefit, but offers an additional axis of exploration in the space of the signal's time-frequency representation. In the BBH case, where one is primarily interested in the separation between the inspiral and postmerger chirps, the chirplet basis allows one to clearly identify boundaries that denote the start and end of the postmerger stage of the signal. This enables the development of a consistent statistic that tracks the existence of the postmerger chirp over the angular variations of the observer orientation, which can then be used to predict the direction of maximal emission based on the intrinsic parameters of the waveform. This parameterization process and its application towards inferring the geometry of the final black hole horizon will be detailed in a subsequent publication.</text>
<text><location><page_8><loc_8><loc_17><loc_48><loc_42></location>Beyond the specific applications to the BBH and BNS postmerger waveforms, there are other potential uses for the CWT framework. In this work, we chiefly considered freespace gravitational waves - simulated waveforms absent the realities of sensitivity and noise that must be considered for detectors like LIGO, Virgo, and KAGRA. In practice the multiresolution Q-scan [9, 10], which performs a CWT-like process over multiple quality factors, is very effective in distinguishing a gravitational wave signal (or other transient) from the background detector noise. However this method constitutes a more general approach, and lacks the specificity of time/frequency localization offered by the CWT framework. We speculate that a variation of the Q-scan algorithm and excess power method [11] that incorporates the capabilities of CWT could be developed to search for specific features in localized time-frequency tiles. For now we leave this possibility to future work.</text>
<text><location><page_8><loc_8><loc_12><loc_48><loc_16></location>Additionally, the flexibility of CWT to incorporate further wavelets beyond the sine-Gaussian MG wavelets allows for the creation of time-frequency maps that have a fundamen-</text>
<text><location><page_8><loc_52><loc_63><loc_92><loc_88></location>tally different structure. Herein we have explored the use of chirplets, which rotate structures in time-frequency space, for the BBH postmerger case. Chirplets, which add the chirp rate d to the MG wavelet, therefore effectively add an extra axis to the space of possible time-frequency analyses on gravitational wave data. Included with this work is an example script for running CWT with chirplets on previous catalog events, which may be found at github.com/chadhenshaw/gw cwt ; see Appendix Sec.(AIV) for further details. Additionally there are also many other wavelet bases that may be of interest; for example the 'Mexican hat' wavelet described in [38] may be useful for analysis of spurious noise transients in LVK data, as it bears similar time-frequency morphology to 'blip' glitches. In a similar vein, exponential shapelets [39] may be useful for transients that feature damped oscillations, or even for black hole ringdown. We leave the implementation and exploration of such methods and applications to future work.</text>
<section_header_level_1><location><page_8><loc_62><loc_55><loc_81><loc_57></location>ACKNOWLEDGEMENTS</section_header_level_1>
<text><location><page_8><loc_52><loc_12><loc_92><loc_54></location>We thank Meg Millhouse and James Clark for helpful discussions and technical advice. This research has made use of data, software and/or web tools obtained from the Gravitational Wave Open Science Center (https://www.gwopenscience.org/), a service of LIGO Laboratory, the LIGO Scientific Collaboration and the Virgo Collaboration. LIGO Laboratory and Advanced LIGO are funded by the United States National Science Foundation (NSF) as well as the Science and Technology Facilities Council (STFC) of the United Kingdom, the Max-Planck-Society (MPS), and the State of Niedersachsen/Germany for support of the construction of Advanced LIGO and construction and operation of the GEO600 detector. Additional support for Advanced LIGO was provided by the Australian Research Council. Virgo is funded, through the European Gravitational Observatory (EGO), by the French Centre National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale di Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions by institutions from Belgium, Germany, Greece, Hungary, Ireland, Japan, Monaco, Poland, Portugal, Spain. KAGRA is supported by the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) in Japan, and is hosted by the Institute for Cosmic Ray Research (ICRR), the University of Tokyo, and co-hosted by High Energy Accelerator Research Organization (KEK) and the National Astronomical Observatory of Japan (NAOJ). This material is based upon work supported by the LIGO Laboratory which is a major facility fully funded by the NSF. This work was supported by NSF grants</text>
<text><location><page_9><loc_8><loc_86><loc_48><loc_88></location>PHY-1809572 & PHY-2110481. This paper has LIGO document number P2400016.</text>
<section_header_level_1><location><page_9><loc_22><loc_79><loc_35><loc_80></location>A. APPENDIX</section_header_level_1>
<text><location><page_9><loc_8><loc_74><loc_48><loc_77></location>For a function g ( x ) , with power distribution | g ( x ) | 2 , its normalized n 'th central moment is given by:</text>
<formula><location><page_9><loc_20><loc_68><loc_48><loc_72></location>mn == ∫ ∞ -∞ x n | g ( x ) | 2 dx ∫ ∞ -∞ | g ( x ) | 2 dx , (A1)</formula>
<text><location><page_9><loc_8><loc_61><loc_48><loc_67></location>where n = 1 corresponds to the mean value, and n = 2 to the variance. If g ( x ) is a time series g ( t ) with corresponding Fourier transform ˆ g ( f ) , its central time τ , and a central frequency φ are given by the first central moments:</text>
<formula><location><page_9><loc_22><loc_55><loc_48><loc_59></location>τ = ∫ ∞ -∞ t | g ( t ) | 2 dt ∫ ∞ -∞ | g ( t ) | 2 dt , (A2)</formula>
<formula><location><page_9><loc_21><loc_51><loc_48><loc_54></location>φ = ∫ ∞ -∞ f | ˆ g ( f ) | 2 d f ∫ ∞ -∞ | ˆ g ( f ) | 2 d f . (A3)</formula>
<text><location><page_9><loc_8><loc_47><loc_48><loc_49></location>The second central moments then give the characteristic duration σ t and bandwidth σ f :</text>
<formula><location><page_9><loc_19><loc_40><loc_48><loc_44></location>σ 2 t = ∫ ∞ -∞ ( t -τ ) 2 | g ( t ) | 2 dt ∫ ∞ -∞ | g ( t ) | 2 dt (A4)</formula>
<formula><location><page_9><loc_19><loc_36><loc_48><loc_40></location>σ 2 f = ∫ ∞ -∞ ( f -φ ) 2 | ˜ g ( f ) | 2 d f ∫ ∞ -∞ | ˜ g ( f ) | 2 d f (A5)</formula>
<section_header_level_1><location><page_9><loc_21><loc_33><loc_36><loc_34></location>AII. WAVELETS</section_header_level_1>
<text><location><page_9><loc_8><loc_29><loc_48><loc_31></location>A wavelet is a localized wave-like function ψ ( t ) that satisfies three criteria. First, a wavelet must have finite energy E :</text>
<formula><location><page_9><loc_20><loc_23><loc_48><loc_26></location>E = ∫ ∞ -∞ | ψ ( t ) | 2 dt < ∞ . (A6)</formula>
<text><location><page_9><loc_8><loc_12><loc_48><loc_22></location>Note that here the vertical brackets represent the modulus operator - i.e if ψ ( t ) is a complex function, then | ψ ( t ) | 2 = ψ ∗ ( t ) ψ ( t ) , where ψ ∗ ( t ) is the complex conjugate of ψ ( t ) . In practice, it is common for the wavelet to further be normalized such that it has unit energy. Secondly, given the Fourier transform of the wavelet ˆ ψ ( f ) , the wavelet must have no zerofrequency component:</text>
<formula><location><page_9><loc_63><loc_83><loc_92><loc_86></location>Cg = ∫ ∞ 0 | ˆ ψ ( f ) | 2 f d f < ∞ . (A7)</formula>
<text><location><page_9><loc_52><loc_68><loc_92><loc_81></location>This criterion is referred to as the admissibility condition , and Cg is called the admissibility constant , whose value depends on the specific wavelet. The final criterion applies to complex wavelets - their Fourier transform must be real, and must be zero for negative frequencies. Any wave-like function that satisfies these three criteria is a wavelet, and may be used in the continuous wavelet transform. For gravitational wave analysis, the most common such function is the Morlet wavelet:</text>
<formula><location><page_9><loc_52><loc_61><loc_94><loc_65></location>ψ ( t , f 0 ) = π -1 4 exp [ -t 2 2 ] ( exp [ i 2 π f 0 t ] -exp [ -( 2 π f 0 ) 2 / 2 ]) . (A8)</formula>
<text><location><page_9><loc_52><loc_47><loc_92><loc_59></location>Here the second exponential term within the parentheses is referred to as the correction term, as it ensures the wavelet satisfies the second criterion by giving it zero mean. However in practice this term is discarded, as it becomes negligible for f 0 ≫ 0 4 . We will still refer to the resultant approximation as a wavelet, as it only violates the admissibility condition for low frequencies. This approximation is often referred to as the Gabor wavelet, or the Morlet-Gabor (MG) wavelet:</text>
<formula><location><page_9><loc_56><loc_42><loc_92><loc_45></location>ψ ( t , f 0 ) = π -1 4 exp [ -t 2 2 ] exp [ i 2 π f 0 t ] (A9)</formula>
<formula><location><page_9><loc_55><loc_38><loc_92><loc_41></location>˜ ψ ( f , f 0 ) = π 1 4 √ 2exp [ -1 2 ( 2 π f -2 π f 0 ) 2 ] . (A10)</formula>
<text><location><page_9><loc_52><loc_15><loc_92><loc_37></location>Note that the MG wavelet has three components: an amplitude π -1 4 , a Gaussian window exp [ -t 2 2 ] with unit standard deviation, and a complex sinusoid exp [ i 2 π f 0 t ] with frequency f 0. Its central time is identically zero, and its central frequency is f 0. Its characteristic duration is σ t = 1 / √ 2, and its bandwidth is 1 / 2 √ 2 π . The duration-bandwidth product of the MG wavelet is thus σ t σ f = 1 / 4 π , which is the minimum value in the Heisenberg-Gabor uncertainty principle. In this way, MG wavelets are maximally compact in time-frequency space, which makes them ideal for time-frequency analysis methods like the continuous wavelet transform (Eq.(5)). In this method, the wavelet's scale is adjusted by the dilation parameter a , and it's time localization is shifted by the translation parameter b :</text>
<formula><location><page_10><loc_13><loc_79><loc_48><loc_86></location>ψ ( t ; f 0 , a , b ) = π -1 4 a -1 / 2 exp [ -1 2 ( t -b a ) 2 ] · · exp [ i 2 π f 0 ( t -b a )] . (A11)</formula>
<text><location><page_10><loc_10><loc_76><loc_43><loc_77></location>The wavelet's energy is normalized at every scale:</text>
<formula><location><page_10><loc_11><loc_68><loc_48><loc_74></location>∫ ∞ -∞ | ψ ( t ) | 2 dt = 1 a √ π ∫ ∞ -∞ exp [ -( t -b a ) 2 ] dt = 1 . (A12)</formula>
<text><location><page_10><loc_8><loc_63><loc_48><loc_67></location>This normalization is maintained in the frequency domain; first we compute the Fourier transform of the dilated and translated MG wavelet:</text>
<formula><location><page_10><loc_9><loc_52><loc_48><loc_60></location>ˆ ψ ( f ) = ∫ ∞ -∞ ψ ( t ) exp [ -2 π i f t ] dt = √ 2 π 1 4 a 1 / 2 exp [ -i 2 π f b ] exp [ -1 2 ( 2 π f 0 -2 π af ) 2 ] , (A13)</formula>
<text><location><page_10><loc_8><loc_47><loc_48><loc_51></location>Note that in the untranslated, non-dilated case ( b = 0 , a = 1 ) , one recovers exactly the Fourier transform of the MG wavelet in Eq.(A10). The energy is then:</text>
<formula><location><page_10><loc_22><loc_41><loc_48><loc_44></location>∫ ∞ -∞ | ˜ ψ ( f ) | 2 d f = 1 , (A14)</formula>
<text><location><page_10><loc_8><loc_37><loc_48><loc_40></location>The denominators in Eqs.(A2-A5) for the MG wavelet are thus unity, and we can proceed to calculate the central time:</text>
<formula><location><page_10><loc_21><loc_32><loc_48><loc_35></location>τ = ∫ ∞ -∞ t | ψ ( t ) | 2 dt = b , (A15)</formula>
<text><location><page_10><loc_8><loc_28><loc_48><loc_31></location>which returns as the translation parameter b , as expected. Now we calculate the central frequency:</text>
<formula><location><page_10><loc_19><loc_23><loc_48><loc_26></location>φ = ∫ ∞ -∞ f | ˜ ψ ( f ) | 2 d f = f 0 a . (A16)</formula>
<text><location><page_10><loc_8><loc_17><loc_48><loc_22></location>Note that in the case of unit scale, i.e. a = 1, we recover the central frequency as exactly f 0, as one expects. Next we will calculate the characteristic duration:</text>
<formula><location><page_10><loc_17><loc_12><loc_48><loc_15></location>σ 2 t = ∫ ∞ -∞ ( t -τ ) 2 | ψ ( t ) | 2 dt = a √ 2 , (A17)</formula>
<text><location><page_10><loc_52><loc_82><loc_92><loc_88></location>which gives an indication of how localized the wavelet is in the time domain. We see that this is directly proportional to the dilation parameter; thus at larger scales the wavelet is more spread out in time. The bandwidth is then:</text>
<formula><location><page_10><loc_56><loc_77><loc_92><loc_80></location>σ 2 f = ∫ ∞ -∞ ( f -φ ) 2 | ˜ ψ ( f ) | 2 d f = 1 2 √ 2 π a . (A18)</formula>
<text><location><page_10><loc_52><loc_67><loc_92><loc_75></location>We obtain a bandwidth - and thus a frequency resolution - that is inversely proportional to the wavelet scale. Therefore at larger scales the wavelet has a narrower bandwidth, as one would expect - more frequencies are encapsulated by the larger wavelet. The quality factor Q is then defined as the ratio of central frequency to bandwidth:</text>
<formula><location><page_10><loc_65><loc_61><loc_92><loc_64></location>Q ≡ φ σ f = 2 √ 2 π f 0 , (A19)</formula>
<text><location><page_10><loc_52><loc_49><loc_92><loc_59></location>which we see does not depend on the wavelet scale; Q depends only on the starting frequency of the sinusoid. We also see that the duration-bandwidth product is still σ t σ f = 1 / 4 π , so dilated MG wavelets remain maximally compact in timefrequency space. Now considering that the quality factor does not depend on wavelet scale, we can rewrite the equation for the dilated and translated Morlet wavelet as:</text>
<formula><location><page_10><loc_53><loc_39><loc_92><loc_46></location>ψ ( t ; Q , a , b ) = = π -1 4 a -1 / 2 exp [ -1 2 ( t -b a ) 2 ] exp [ i Q √ 2 ( t -b a )] , (A20)</formula>
<text><location><page_10><loc_52><loc_34><loc_92><loc_37></location>where we see that for a given dilation factor a we have the effective sinusoid frequency:</text>
<formula><location><page_10><loc_65><loc_29><loc_92><loc_32></location>f e f f = Q 2 √ 2 π a = f 0 a , (A21)</formula>
<text><location><page_10><loc_52><loc_23><loc_92><loc_27></location>which is equivalent to the central frequency φ . When running CWT, it is this effective frequency that you are interrogating when convolving the wavelet with the signal.</text>
<section_header_level_1><location><page_10><loc_64><loc_19><loc_79><loc_20></location>AIII. CHIRPLETS</section_header_level_1>
<text><location><page_10><loc_52><loc_12><loc_92><loc_17></location>The central frequency of the MG wavelet in Sec.(AII) above is stationary, but may be modified to evolve in time by including a chirp rate parameter d (following the notation of [40]) such that the central sinusoid frequency f 0 becomes:</text>
<formula><location><page_11><loc_23><loc_84><loc_48><loc_86></location>f 0 ( t ) = f 0 + d ∗ t , (A22)</formula>
<text><location><page_11><loc_10><loc_82><loc_42><loc_83></location>and the mother wavelet, now a chirplet , becomes:</text>
<formula><location><page_11><loc_9><loc_61><loc_48><loc_79></location>ψ ( t ; f 0 , d , a , b ) = π -1 4 exp [ -1 2 ( t -b a ) 2 ] ∗ ∗ exp [ i 2 π ( f 0 ( t -b a ) + d 2 ( t -b a ) 2 )] , ψ ( t ; Q , a , b ) = π -1 4 exp [ -1 2 ( t -b a ) 2 ] ∗ ∗ exp [ i ( Q √ 2 ( t -b a ) + π d ( t -b a ) 2 )] . (A23)</formula>
<text><location><page_11><loc_8><loc_42><loc_48><loc_60></location>Note that here the chirp parameter d is a fixed quantity, and as such in the CWT process becomes another parametric input for the mother wavelet. A positive value of d > 0 corresponds to an up-chirp , where the wavelet frequency increases over time, and a negative value of d > 0 corresponds to a downchirp , where the wavelet frequency decreases over time. Note that if d = 0 we recover exactly the Morlet-Gabor wavelet. Also note that this parameterization of the chirp rate is different than that in e.g.[41], where the chirp parameter β = 2 d is used. Below we create a variation of Fig.2 from that work, using the CWT process to create the time-frequency map of different chirplets.</text>
<text><location><page_11><loc_8><loc_22><loc_48><loc_42></location>Fig.(8) was constructed by creating a series of ten chirplet signals with different parameters, running CWT on each signal, then displaying together all ten time-frequency maps. Each chirplet can be thought of as a short burst-like signal. The middle panel shows the representation of the ten chirplets using the MG wavelet (i.e. d = 0 . 0) basis with CWT. Here one can see that the time-frequency representation of each burst is an ellipse, as as the chirp rate becomes more positive (negative), the ellipse is rotated clockwise (counter-clockwise). The left and right panels show the same analysis, but with a chirplet basis at d = -04 and d = 0 . 4 respectively. One can see that in this basis, the corresponding chirplet signal is counter-rotated in time-frequency space.</text>
<text><location><page_11><loc_8><loc_12><loc_48><loc_22></location>Consider first the right panel in Fig.(8). The right-most chirplet signal ( d = 0 . 4 ) , which is tilted clockwise in the middle panel, has been rotated counter-clockwise to a vertical position. This is because the underlying chirp rate of the CWT basis matches the chirp rate of the signal, and as such its frequency structure has been localized in time. The remaining bursts, which all have a chirp rate less than that of the CWT</text>
<figure>
<location><page_11><loc_52><loc_70><loc_93><loc_88></location>
<caption>Figure 8: Time-frequency representation of different chirplets, created using different chirplet bases with CWT. From left to right, the CWT process was used with a chirp rate of d =[ -0 . 4 , 0 . 0 , 0 . 4 ] , with Q = 64 in each case. Each panel contains ten different chirplet signals. The bottom row in each panel contains chirplets at f 0 = 300 [Hz] and Q = 100 , and the top row contains chirplets at f 0 = 800 [Hz] and Q = 80 . From left to right, the chirplets in each panel have chirp rates of d =[ -0 . 4 , -0 . 2 , 0 . 0 , 0 . 2 , 0 . 4 ] .</caption>
</figure>
<text><location><page_11><loc_52><loc_43><loc_92><loc_53></location>basis, have also been rotated counter-clockwise, and their frequency localization has been stretched in accordance with the wider bandwidth of the CWT basis. The same principle applies to the left panel in Fig.(8), but with the opposite sense. The bursts with chirp rates greater than that of the underlying CWT basis are now rotated clockwise, countering their natural rotation in time-frequency space.</text>
<section_header_level_1><location><page_11><loc_52><loc_39><loc_91><loc_40></location>AIV. NUMERICAL IMPLEMENTATION OF CWT</section_header_level_1>
<text><location><page_11><loc_52><loc_16><loc_92><loc_37></location>To create the time-frequency maps herein, we use a modified version of a previous Python implementation of CWT [42]. The modified code, which we refer to as gw cwt , is available at github.com/chadhenshaw/gw cwt along with example scripts. The primary function in this package is build cwt , which accepts as input a timeseries array and corresponding uniformly-sampled timestamps, runs the cwt function, and returns a dictionary containing properties of the resultant timefrequency map. The cwt function, which is a time domain convolution, is implemented as a product between signal and wavelet in the frequency domain. At each wavelet scale, the timeseries and wavelet are converted to the frequency domain by FFT, and the inverse FFT of their product is then computed to complete the convolution.</text>
<text><location><page_11><loc_52><loc_12><loc_92><loc_16></location>The build cwt function also contains a number of options that the user may specify to customize their analysis, with the primary variables being the quality factor Q , and the chirp</text>
<text><location><page_12><loc_8><loc_63><loc_48><loc_88></location>rate d . The number of frequencies interrogated is determined by the option n conv , which is the number of convolutions. Increasing this setting samples more frequencies within the given range at the cost of compute time. Given a range of frequencies, wavelet scales are computed as a = f 0 fs / f e f f , where fs is the sampling rate of the timeseries. Additionally, the Nyquist frequency fs / 2 determines the upper bound on the signal frequency one can interrogate. By default, the program will create a linearly-spaced array of wavelet scales, which corresponds to a logarithmically-spaced array of frequencies. The user may instead request a linearly-spaced frequency array, in which case the program will create the corresponding logarithmically-spaced array of wavelet scales. Alternatively for each case, the user may specify the spacing between frequencies or scales as d f or da , which supersedes the number of convolutions. Finally, the user may instead input directly an array of frequencies to compute over.</text>
<text><location><page_12><loc_8><loc_43><loc_48><loc_62></location>Included also in the gw cwt package is the script cwt catalog.py , which simplifies the process of running CWT on GW data. Within this program, the get data function pulls gravitational wave data from the Pythonbased pycbc.catalog [43]. The data are downloaded and stored in a local hdf5 compressed file, the organizational structure of which resembles a Python dictionary. The run cwt function then computes the time-frequency map for the downloaded event data, given user input of CWT parameters, and appends the results to the hdf5 file. Finally, plot cwt is a function for plotting the time-frequency map. Examples and further documentation are available at github.com/chadhenshaw/gw cwt .</text>
<section_header_level_1><location><page_12><loc_23><loc_38><loc_34><loc_39></location>REFERENCES</section_header_level_1>
<unordered_list>
<list_item><location><page_12><loc_9><loc_31><loc_48><loc_36></location>[1] J Aasi et al. 'Advanced LIGO'. In: Classical and Quantum Gravity 32.7 (2015), p. 074001. DOI: 10 . 1088 / 0264 - 9381 / 32 / 7 / 074001 . URL: https : //doi.org/10.1088/0264-9381/32/7/074001 .</list_item>
<list_item><location><page_12><loc_9><loc_23><loc_48><loc_30></location>[2] F. Acernese et al. 'Advanced Virgo: A secondgeneration interferometric gravitational wave detector'. In: Classical and Quantum Gravity 32.2 (2015). ISSN: 13616382. DOI: 10.1088/0264-9381/32/2/ 024001 . arXiv: 1408.3978 .</list_item>
<list_item><location><page_12><loc_9><loc_15><loc_48><loc_22></location>[3] T. Akutsu et al. 'Overview of KAGRA: Detector design and construction history'. In: Progress of Theoretical and Experimental Physics 2021.5 (2021). ISSN: 20503911. DOI: 10 . 1093 / ptep / ptaa125 . arXiv: 2005.05574 .</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_12><loc_52><loc_78><loc_92><loc_88></location>[4] B. P. Abbott et al. 'GWTC-1: A Gravitational-Wave Transient Catalog of Compact Binary Mergers Observed by LIGO and Virgo during the First and Second Observing Runs'. In: Physical Review X 9.3 (2019), p. 31040. ISSN: 21603308. DOI: 10.1103/PhysRevX. 9.031040 . arXiv: 1811.12907 . URL: https://doi. org/10.1103/PhysRevX.9.031040 .</list_item>
<list_item><location><page_12><loc_52><loc_68><loc_92><loc_77></location>[5] R. Abbott et al. 'GWTC-2: Compact Binary Coalescences Observed by LIGO and Virgo during the First Half of the Third Observing Run'. In: Phys. Rev. X 11 (2 2021), p. 021053. DOI: 10.1103/PhysRevX.11. 021053 . URL: https://link.aps.org/doi/10. 1103/PhysRevX.11.021053 .</list_item>
<list_item><location><page_12><loc_52><loc_60><loc_92><loc_68></location>[6] The LIGO Scientific Collaboration et al. GWTC-2.1: Deep Extended Catalog of Compact Binary Coalescences Observed by LIGO and Virgo During the First Half of the Third Observing Run . 2021. arXiv: 2108. 01045 [gr-qc] .</list_item>
<list_item><location><page_12><loc_52><loc_54><loc_92><loc_59></location>[7] The LIGO Scientific Collaboration et al. GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo During the Second Part of the Third Observing Run . 2021. arXiv: 2111.03606 [gr-qc] .</list_item>
<list_item><location><page_12><loc_52><loc_41><loc_92><loc_53></location>[8] Curt Cutler and Ianna E. Flanagan. 'Gravitational waves from merging compact binaries: How accurately can one extract the binary's parameters from the inspiral waveform?' In: Physical Review D 49.6 (1994), pp. 2658-2697. ISSN: 05562821. DOI: 10 . 1103 / PhysRevD.49.2658 . arXiv: 9402014 [gr-qc] . URL: https : / / link . aps . org / doi / 10 . 1103 / PhysRevD.49.2658 .</list_item>
<list_item><location><page_12><loc_52><loc_32><loc_92><loc_40></location>[9] S. Chatterji, L. Blackburn, G. Martin, and E. Katsavounidis. 'Multiresolution techniques for the detection of gravitational-wave bursts'. In: Classical and Quantum Gravity 21.20 SPEC. ISS. (2004). ISSN: 02649381. DOI: 10.1088/0264-9381/21/20/024 . arXiv: 0412119 [gr-qc] .</list_item>
<list_item><location><page_12><loc_52><loc_24><loc_92><loc_31></location>[10] Shourov Keith Chatterji. 'The search for gravitational wave bursts in data from the second LIGO science run'. In: Thesis (Ph. D.)-Massachusetts Institute of Technology, Dept. of Physics (2005). URL: https://dspace. mit.edu/handle/1721.1/34388 .</list_item>
<list_item><location><page_12><loc_52><loc_13><loc_92><loc_23></location>[11] W. G. Anderson, P. R. Brady, J. D.E. Creighton, and E. E. Flanagan. 'Excess power statistic for detection of burst sources of gravitational radiation'. In: Physical Review D - Particles, Fields, Gravitation and Cosmology 63.4 (2001), pp. 420031-4200320. ISSN: 05562821. DOI: 10.1103/PhysRevD.63.042003 . arXiv: 0008066 [gr-qc] .</list_item>
<list_item><location><page_13><loc_8><loc_82><loc_48><loc_88></location>[12] Enrico Calloni et al. 'All-sky search for gravitationalwave bursts in the first joint LIGO-GEO-Virgo run.' In: Physical Review D, Particles, Fields, Gravitation, and Cosmology 102001.81 (2010), pp. 1-20.</list_item>
<list_item><location><page_13><loc_8><loc_74><loc_48><loc_82></location>[13] B. P. Abbott et al. 'All-sky search for short gravitational-wave bursts in the first Advanced LIGO run'. In: Physical Review D 95.4 (2017), pp. 1-14. ISSN: 24700029. DOI: 10 . 1103 / PhysRevD . 95 . 042003 . arXiv: 1611.02972 .</list_item>
<list_item><location><page_13><loc_8><loc_63><loc_48><loc_74></location>[14] B. P. Abbott et al. 'All-sky search for short gravitational-wave bursts in the second Advanced LIGO and Advanced Virgo run'. In: Physical Review D 100.2 (2019), p. 24017. ISSN: 24700029. DOI: 10. 1103/PhysRevD.100.024017 . arXiv: 1905.03457 . URL: https://doi.org/10.1103/PhysRevD.100. 024017 .</list_item>
<list_item><location><page_13><loc_8><loc_52><loc_48><loc_62></location>[15] R. Abbott et al. 'All-sky search for short gravitationalwave bursts in the third Advanced LIGO and Advanced Virgo run'. In: Physical Review D 104.12 (2021), p. 122004. ISSN: 24700029. DOI: 10.1103/ PhysRevD.104.122004 . arXiv: 2107.03701 . URL: https://doi.org/10.1103/PhysRevD.104. 122004 .</list_item>
<list_item><location><page_13><loc_8><loc_44><loc_48><loc_51></location>[16] S. Klimenko et al. 'Method for detection and reconstruction of gravitational wave transients with networks of advanced detectors'. In: Physical Review D 93.4 (2016), pp. 1-10. ISSN: 24700029. DOI: 10 . 1103 / PhysRevD.93.042004 . arXiv: 1511.05999 .</list_item>
<list_item><location><page_13><loc_8><loc_35><loc_48><loc_43></location>[17] Florent Robinet et al. 'Omicron: A tool to characterize transient noise in gravitational-wave detectors'. In: SoftwareX 12 (2020), p. 100620. ISSN: 23527110. DOI: 10.1016/j.softx.2020.100620 . arXiv: 2007. 11374 . URL: https : / / doi . org / 10 . 1016 / j . softx.2020.100620 .</list_item>
<list_item><location><page_13><loc_8><loc_24><loc_48><loc_34></location>[18] Neil J Cornish et al. 'BayesWave analysis pipeline in the era of gravitational wave observations'. In: Physical Review D 103.4 (2021), pp. 1-19. ISSN: 24700029. DOI: 10 . 1103 / PhysRevD . 103 . 044006 . arXiv: 2011.09494 . URL: http://arxiv.org/abs/2011. 09494http://dx.doi.org/10.1103/PhysRevD. 103.044006 .</list_item>
<list_item><location><page_13><loc_8><loc_16><loc_48><loc_23></location>[19] M. Zevin et al. 'Gravity Spy: Integrating advanced LIGO detector characterization, machine learning, and citizen science'. In: Classical and Quantum Gravity 34.6 (2017). ISSN: 13616382. DOI: 10.1088/13616382/aa5cea . arXiv: 1611.04596 .</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_13><loc_52><loc_79><loc_92><loc_88></location>[20] Juan Calderon Bustillo et al. 'Post-merger chirps from binary black holes as probes of the final black-hole horizon'. In: Communications Physics 3.1 (2020), pp. 1-7. ISSN: 23993650. DOI: 10.1038/s42005020-00446-7 . arXiv: 1906.01153 . URL: http:// dx.doi.org/10.1038/s42005-020-00446-7 .</list_item>
<list_item><location><page_13><loc_52><loc_73><loc_92><loc_79></location>[21] Kritti Sharma, Koustav Chandra, and Archana Pai. 'Fishing massive black hole binaries with THAMES'. In: (2022), pp. 1-22. arXiv: 2208.02545 . URL: http: //arxiv.org/abs/2208.02545 .</list_item>
<list_item><location><page_13><loc_52><loc_65><loc_92><loc_72></location>[22] A. Bauswein, H. T. Janka, K. Hebeler, and A. Schwenk. 'Equation-of-state dependence of the gravitationalwave signal from the ring-down phase of neutron-star mergers'. In: Physical Review D 86.063001 (2012), p. 9.</list_item>
<list_item><location><page_13><loc_52><loc_55><loc_92><loc_64></location>[23] A. Torres-Rivas, K. Chatziioannou, A. Bauswein, and J. A. Clark. 'Observing the post-merger signal of GW170817-like events with improved gravitationalwave detectors'. In: Phys. Rev. D 99.4 (2019), p. 044014. DOI: 10.1103/PhysRevD.99.044014 . arXiv: 1811.08931 [gr-qc] .</list_item>
<list_item><location><page_13><loc_52><loc_46><loc_92><loc_54></location>[24] K. Chatziioannou, C-J Haster, and A. Zimmerman. 'Measuring the neutron star tidal deformability with equation-of-state-independent relations and gravitational waves'. In: Phys. Rev. D 97.10 (2018), p. 104036. DOI: 10.1103/PhysRevD.97.104036 . arXiv: 1804.03221 [gr-qc] .</list_item>
<list_item><location><page_13><loc_52><loc_38><loc_92><loc_45></location>[25] LIGO Scientific Collaboration and Virgo Collaboration. 'GW170817: Measurements of neutron star radii and equation of state'. In: Phys. Rev. Lett. 121.16 (2018), p. 161101. DOI: 10 . 1103 / PhysRevLett . 121.161101 . arXiv: 1805.11581 [gr-qc] .</list_item>
<list_item><location><page_13><loc_52><loc_28><loc_92><loc_37></location>[26] J. A. Clark, A. Bauswein, N. Stergioulas, and D. Shoemaker. 'Observing gravitational waves from the postmerger phase of binary neutron star coalescence'. In: Classical and Quantum Gravity 33.8 (2016). ISSN: 13616382. DOI: 10 . 1088 / 0264 - 9381 / 33 / 8 / 085003 . arXiv: 1509.08522 .</list_item>
<list_item><location><page_13><loc_52><loc_19><loc_92><loc_27></location>[27] A. Torres-Rivas, K. Chatziioannou, A. Bauswein, and J. A. Clark. 'Observing the post-merger signal of GW170817-like events with improved gravitationalwave detectors'. In: Phys. Rev. D 99.4 (2019), p. 044014. DOI: 10.1103/PhysRevD.99.044014 . arXiv: 1811.08931 [gr-qc] .</list_item>
<list_item><location><page_13><loc_52><loc_14><loc_92><loc_18></location>[28] K. Chatziioannou et al. 'Inferring the post-merger gravitational wave emission from binary neutron star coalescences'. In: Phys. Rev. D 96.12 (2017),</list_item>
<list_item><location><page_14><loc_13><loc_86><loc_48><loc_88></location>p. 124035. DOI: 10.1103/PhysRevD.96.124035 . arXiv: 1711.00040 [gr-qc] .</list_item>
<list_item><location><page_14><loc_8><loc_77><loc_48><loc_85></location>[29] D. Gabor. 'Theory of communication. Part 1: The analysis of information'. In: Journal of the Institution of Electrical Engineers - Part III: Radio and Communication Engineering 93.26 (1946), pp. 429-441. DOI: 10.1049/ji-3-2.1946.0074 .</list_item>
<list_item><location><page_14><loc_8><loc_69><loc_48><loc_77></location>[30] B. P. Abbott et al. 'Observation of gravitational waves from a binary black hole merger'. In: Physical Review Letters 116.6 (2016), pp. 1-16. ISSN: 10797114. DOI: 10.1103/PhysRevLett.116.061102 . arXiv: 1602. 03837 .</list_item>
<list_item><location><page_14><loc_8><loc_58><loc_48><loc_69></location>[31] Karan Jani et al. 'Georgia tech catalog of gravitational waveforms'. In: Classical and Quantum Gravity 33.20 (2016), pp. 1-8. ISSN: 13616382. DOI: 10 . 1088 / 0264-9381/33/20/204001 . arXiv: 1605.03204 . URL: http://arxiv.org/abs/1605.03204http: //dx.doi.org/10.1088/0264-9381/33/20/ 204001 .</list_item>
<list_item><location><page_14><loc_8><loc_52><loc_48><loc_57></location>[32] Deborah Ferguson et al. 'Second MAYA Catalog of Binary Black Hole Numerical Relativity Waveforms'. In: (2023), pp. 1-11. arXiv: 2309.00262 . URL: http: //arxiv.org/abs/2309.00262 .</list_item>
<list_item><location><page_14><loc_8><loc_39><loc_48><loc_51></location>[33] M. Breschi, S. Bernuzzi, D. Godzieba, A. Perego, and D. Radice. 'Constraints on the Maximum Densities of Neutron Stars from Postmerger Gravitational Waves with Third-Generation Observations'. In: Phys. Rev. Lett. 128 (16 2022), p. 161102. DOI: 10.1103/ PhysRevLett.128.161102 . URL: https://link. aps . org / doi / 10 . 1103 / PhysRevLett . 128 . 161102 .</list_item>
<list_item><location><page_14><loc_8><loc_30><loc_48><loc_38></location>[34] A. Bauswein and H.-T. Janka. 'Measuring NeutronStar Properties via Gravitational Waves from NeutronStar Mergers'. In: Phys. Rev. Lett. 108 (1 2012), p. 011101. DOI: 10 . 1103 / PhysRevLett . 108 . 011101 . URL: https://link.aps.org/doi/10. 1103/PhysRevLett.108.011101 .</list_item>
<list_item><location><page_14><loc_8><loc_19><loc_48><loc_29></location>[35] Alexander W. Criswell et al. 'Hierarchical Bayesian method for constraining the neutron star equation of state with an ensemble of binary neutron star postmerger remnants'. In: Physical Review D 107.4 (2023), p. 43021. ISSN: 24700029. DOI: 10.1103/PhysRevD. 107.043021 . URL: https://doi.org/10.1103/ PhysRevD.107.043021 .</list_item>
<list_item><location><page_14><loc_8><loc_14><loc_48><loc_18></location>[36] H. Yang et al. 'Gravitational wave spectroscopy of binary neutron star merger remnants with mode stacking'. In: Phys. Rev. D 97 (2 2018), p. 024049. DOI: 10.</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_14><loc_56><loc_86><loc_92><loc_88></location>1103/PhysRevD.97.024049 . URL: https://link. aps.org/doi/10.1103/PhysRevD.97.024049 .</list_item>
<list_item><location><page_14><loc_52><loc_76><loc_92><loc_85></location>[37] A. Criswell et al. 'Hierarchical Bayesian method for constraining the neutron star equation of state with an ensemble of binary neutron star postmerger remnants'. In: Phys. Rev. D 107.4 (2023), p. 043021. DOI: 10. 1103/PhysRevD.107.043021 . arXiv: 2211.05250 [astro-ph.HE] .</list_item>
<list_item><location><page_14><loc_52><loc_69><loc_92><loc_75></location>[38] P. S. Addison, J. N. Watson, and T. Feng. 'Lowoscillation complex wavelets'. In: Journal of Sound and Vibration 254.4 (2002), pp. 733-762. ISSN: 0022460X. DOI: 10.1006/jsvi.2001.4119 .</list_item>
<list_item><location><page_14><loc_52><loc_60><loc_92><loc_69></location>[39] Joel Berge, Richard Massey, Quentin Baghi, and Pierre Touboul. 'Exponential shapelets: Basis functions for data analysis of isolated features'. In: Monthly Notices of the Royal Astronomical Society 486.1 (2019), pp. 544-559. ISSN: 13652966. DOI: 10.1093/mnras/ stz787 . arXiv: 1903.05837 .</list_item>
<list_item><location><page_14><loc_52><loc_50><loc_92><loc_59></location>[40] Satya Mohapatra, Zachary Nemtzow, ' Eric ChassandeMottin, and Laura Cadonati. 'Performance of a Chirplet-based analysis for gravitational-waves from binary black-hole mergers'. In: Journal of Physics: Conference Series 363.1 (2012). ISSN: 17426596. DOI: 10.1088/1742-6596/363/1/012031 .</list_item>
<list_item><location><page_14><loc_52><loc_39><loc_92><loc_49></location>[41] Margaret Millhouse, Neil J. Cornish, and Tyson Littenberg. 'Bayesian reconstruction of gravitational wave bursts using chirplets'. In: Physical Review D 97.10 (2018), p. 104057. ISSN: 24700029. DOI: 10.1103/ PhysRevD.97.104057 . arXiv: 1804.03239 . URL: https : / / doi . org / 10 . 1103 / PhysRevD . 97 . 104057 .</list_item>
<list_item><location><page_14><loc_52><loc_36><loc_92><loc_38></location>[42] pyCWT : Continuous Wavelet Transform library for Python . https://github.com/Unidata/pyCWT .</list_item>
<list_item><location><page_14><loc_52><loc_29><loc_92><loc_35></location>[43] Alex Nitz et al. gwastro/pycbc: v2.3.2 release of PyCBC . Version v2.3.2. Nov. 2023. DOI: 10 . 5281 / zenodo.10137381 . URL: https://doi.org/10. 5281/zenodo.10137381 .</list_item>
</document>
[
{
"title": "Visualization of frequency structures in gravitational wave signals",
"content": "CHAD HENSHAW, MEGAN AROGETI, ALICE HERANVAL, LAURA CADONATI School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA February 27, 2024",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "The gravitational wave signals produced by the coalescence of compact binaries progress through three stages: inspiral, merger, and postmerger. The evolution of their frequency follows a slow build up during the inspiral that peaks at merger, forming the characteristic 'chirp' pattern in the signal's time-frequency map. Herein we introduce a framework for localizing further characteristic structures in the time-frequency space of gravitational wave signals using the continuous wavelet transform. We consider two example cases where there are specific patterns in the postmerger stage of the signal that are rich with information on the physical nature of the source: highly-inclined black hole binaries with asymmetric mass ratio, and neutron star binaries with postmerger remnant oscillations. It is demonstrated that the choice of quality factor Q plays a central role in distinguishing the postmerger features from that of the inspiral, with black hole systems preferring lower Q and neutron star systems preferring higher Q. Furthermore, we demonstrate the use of chirplets as the wavelet transform basis, which allow for manipulation of structure in the time-frequency map.",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "Gravitational waves (GWs) from compact binary coalescences (CBCs) exhibit distinctive characteristics in the evolution of their frequency. To date, the LIGO, Virgo, and KAGRA scientific collaborations [1, 2, 3], which now comprise the LIGO-Virgo-KAGRA Collaboration (LVK), have detected 90+ signals from CBC systems [4, 5, 6, 7]. Starting at some initial separation distance, the two objects in the binary orbit a common center of mass, and as time progresses their separation distance will decrease as energy is lost through gravitational radiation in the form of GWs. As the separation distance decreases, the orbital period decreases, thus the orbital frequency increases, and thus the GW frequency also increases ( fgw ≈ 2 f orb . ) . During this stage of the merger event, known as the inspiral, the GW amplitude also increases, peaking at the time of merger, and therefore the frequency evolution during the inspiral is referred to as a 'chirp.' At lowest post-Newtonian (PN) order, the GW frequency during the inspiral evolves as ˙ f ( t ) ∝ M 5 / 3 f 11 / 3 [8] [ ] 3 / 5 where M = M q ( 1 + q ) 2 is the 'chirp mass,' M = m 1 + m 2 is the total mass, and q = m 1 / m 2 , m 1 ≥ m 2 is the mass ratio. The inclusion of higher order terms also sees contribution from the mass ratio, the spin angular momenta of the compact objects, and their tidal deformabilities in the case of objects with matter like neutron stars (NSs). Near the point of merger this approximation breaks down, and a fully relativistic evaluation is needed to describe the merger. The characterization of the postmerger stage varies depending on the nature of the compact objects in the binary. If one of the two objects is a black hole (BH), then the merger constitutes the formation of a distorted common horizon that encapsulates both objects, which then 'rings down' in the postmerger to spherical symmetry. The frequency evolution of a GW signal can reveal much about the nature of the originating system, as GWs fundamentally encode the properties of their source. Time-frequency analysis sees wide use in the gravitational wave community, over a variety of applications. One of the primary applications for time-frequency mapping in gravitational wave data analysis is the unmodeled search for transient signals. This effort does not assume anything about the shape of the gravitational wave signal, and instead relies on time-frequency analysis utilizing the multi-resolution 'Q-scan,' introduced in [9, 10]. If the detector noise is assumed to be stationary and Gaussian, then the presence of any other component - like a gravitational wave signal, or other noise transient - will show up in the time-frequency map as excess power [11]. This excess power method has been widely used in all-sky searches of gravitational wave data [12, 13, 14, 15], which incorporate both online [16] and offline follow-up [17, 18] algorithms. Time-frequency analysis is also commonly used in the identification of noise transients in LVK data for detector characterization [17, 19]. Beyond these applications, time-frequency analysis can also be used to understand specific frequency characteristics in gravitational waves that are directly connected to the physics of their originating system. In this document we will review two examples of systems that contain distinctive frequency structure in the post-merger gravitational wave signal: binary neutron star (BNS) systems, and highly-inclined binary black hole (BBH) systems with asymmetric mass ratio. By applying targeted time-frequency analysis using the continuous wavelet transform (CWT) to synthetic representations of these signals, we will demonstrate that these features may be isolated from the primary frequency spike due to the inspiral. This enables routes to further analysis such as parameterization for theoretical exploration, or detection using stacking methods, both of which will be discussed in subsequent publications. In a recent Numerical Relativity (NR) study [20], it was demonstrated that GW signals from BBH systems with asymmetric mass ratio q > 1 exhibit additional frequency peaks after merger when viewed from a highly-inclined angle. This study further demonstrates that these frequency peaks are correlated to regions of tight curvature on the horizon of the distorted final BH. Such signals are therefore of immediate interest, as if this correlation can be understood then there is a possibility of inferring the final black hole horizon geometry from the gravitational wave signals that we receive on Earth. Additionally, as these postmerger frequency peaks can be easily mistaken for spurious transients (glitches) in detector data [21], understanding how to differentiate them as a physical effect is important for the realistic detection of intermediate-mass black hole (IMBH) binary systems. In the case of a BNS system, the postmerger gravitational wave signal is highly variable depending on several factors. Neutron stars are highly dense, compact stellar objects where the interior structure is dictated by an equation of state (EoS), which provides a relationship between the NS interior pressure, density, and temperature. The exact nature of these conditions in the interior of neutron stars is unknown[22, 23], but gravitational waves signals from BNS mergers offer a means by which the EoS may be studied. The EoS is encoded not only in the inspiral stage of a BNS merger [24, 25], but also in the postmerger stage. Depending on the component masses and the EoS, a BNS merger could result in i) a prompt collapse to a black hole, ii) a hypermassive or supramassive neutron star that undergoes a delayed collapse to a black hole or iii) a stable neutron star [26, 27, 28]. In general, oscillations of this postmerger remnant can emit a high-frequency gravitational wave signal, the specific characteristics of which are dependent on the remnant properties. In the case of a delayed collapse to a black hole, the post-merger gravitational wave signal could have a duration on the order of 10 -100 [ms], with frequencies between 1 -4 [kHz]. If detected, this postmerger signal could provide a window to study a higher-mass regime than allowed by the inspiral signal [27, 28]. This document is organized as follows. In Sec.(II), we will briefly review the fundamentals of time-frequency analysis, and in Sec.(III) we introduce an implementation of the CWT method. In Sec.(IV), we will then apply the CWT to select simulated BBH and BNS postmerger waveforms to demonstrate its use for feature discrimination. Finally, we conclude in Sec.(V) with a discussion on further applications and routes of analysis.",
"pages": [
1,
2
]
},
{
"title": "II. PRECEPTS OF TIME-FREQUENCY ANALYSIS",
"content": "The Fourier transform takes a continuous time series x ( t ) and decomposes it into its constituent frequencies, rendering a continuous frequency series ˆ x ( f ) : Considering the product between the time series x ( t ) and the complex sinusoid exp [ -i 2 π f t ] in Eq.(1), the Fourier transform is a check for similarity. The sinusoid can be thought of as an analyzing function g ( t ) , and thus the integral in Eq.(1) is the inner product: where g ∗ ( t ) denotes the complex conjugate of g ( t ) 1 . The resultant complex function ˆ x ( f ) describes the amplitude and phase of the component of x ( t ) that matches the frequency f of the sinusoid, which spans a continuous domain. This frequency domain representation ˆ x ( f ) can likewise be converted back to the time domain by the inverse Fourier transform: such that x ( t ) and ˆ x ( f ) form a Fourier pair. Note that in performing the conversion from x ( t ) to ˆ x ( f ) , all information about the signal's time evolution is forsaken. Therefore given a signal x ( t ) , the Fourier transform can describe the signal's spectrum (the strength of its frequency components), but it does not describe when in time the signal reaches those frequencies. To understand simultaneously the signal's frequency evolution, and its strength per frequency, one needs to compute a two-dimensional function X ( t , f ) that preserves both the time and frequency information. One of the primary methods for time-frequency analysis is the short-time Fourier transform, first devised by Dennis Gabor in 1946 [29]: where w ( τ ) is a window function centered on τ = 0; i.e. a function that is only non-zero within some chosen interval. Note that Eq.(4) is exactly a FT as in Eq.(1), where the effective time series x ( t ) is now the product between the signal and the time-shifted window w ( t -τ ) . One can also think of the inner product picture, where the analyzing function is now the product between the window and the sinusoid. Under either interpretation, the introduction of the time-shift changes the Fourier transform into a convolution , in which the analyzing function 'slides' along the signal, and the overlap X ( τ , f ) is computed at each (continuous) time τ and (continuous) frequency f . Because the window has a finite width, this is effectively computing the FT of the signal over a short time segment, then stacking those segments next to each other to build a continuous two-dimensional time-frequency representation column by column. Alternatively in the inner product picture, the convolution produces a new time series of coefficients at the central frequency of the analyzing function, essentially comprising one row in the resultant two-dimensional time-frequency map. There are several different types of window functions that may be used with the STFT, but one of the drawbacks of this method is that the window function has a fixed width. Regardless of the specific function, the window width is related to the time-resolution of the resultant spectrogram. The larger the window duration, the poorer one's time resolution, but the finer the frequency resolution becomes. Because the width of the window is fixed, the STFT is limited in its ability to adapt to signals with rapid changes in frequency evolution. In contrast, the CWT method discussed below allows for variable window widths, and therefore has greater flexibility in adapting to the signal.",
"pages": [
2,
3
]
},
{
"title": "III. THE CONTINUOUS WAVELET TRANSFORM",
"content": "The continuous wavelet transform (CWT) is a process that scans a type of filter called a wavelet - an oscillating function modulated by some envelope - across a signal to probe its time and frequency space. For more detail on wavelets see Appendix Sec.(AII). The CWT is defined by the following expression: where x ( t ) represents the signal, w ( a ) is a weighting function 2 , and ψ ∗ is the complex conjugate of the 'mother wavelet' - the wavelet being scanned across the signal. The scanning occurs by scaling the wavelet by a dilation parameter a , and shifting its time localization by a translation parameter b . As with the STFT, the time shifting makes Eq.(5) a convolution between the scaled wavelet function - i.e. the analyzing function - and the signal. The CWT coefficients T ( a , b ) may then be computed in the same manner as previously, but now instead of passing explicit frequencies to define our filter we pass an array of discrete wavelet scales. In practice the signal data is a digital time series with sampling rate fs and time steps ∆ b = f -1 s . For each scale a , we fast Fourier transform (FFT) both the wavelet and the signal, take their product in Fourier space, then inverse FFT the result to produce a time series array of coefficients per a . 3 The resultant two-dimensional map is a scalogram - as it maps the scaled wavelets to their inner product with the signal at central times b . The wavelet scales a may then be related to their central frequencies, and in this way the CWT builds the time-frequency map row by row. Alternatively, one may also imagine the CWT process as a point-wise inner product between the wavelet and the signal. For each time step b , the wavelet's scale is varied by the dilation parameter a . These parameters therefore become coordinates that map the signal's time-frequency representation. Under this interpretation, the CWT is often referred to as a 'mathematical microscope,' a metaphor that may be further extended by considering the wavelet form chosen for the operation as a type of lens. The dilation parameter a therefore describes the width of the lens' aperture, and the translation parameter b describes its location within the signal domain. Regardless of the interpretation, T ( a , b ) is a measure of the similarity between the wavelet and the signal at the point ( a , b ) . In this way the CWT can be tuned to identify specific substructures in the time-frequency space of a signal, by choosing a mother wavelet that resembles those features. Let's now consider the example of the scaled and translated Morlet-Gabor (MG) wavelet, given in the time and frequency domains by: ˜ ψ ( f ; f 0 , a , b ) = This wavelet has unit energy, central time τ = b , central frequency φ = f 0 / a , characteristic duration σ t = a / √ 2, and bandwidth σ f = 1 / 2 √ 2 π a ; see Appendix Sec.(AII) for derivations. Note that the duration-bandwidth product is σ t σ f = 1 / 4 π , so MG wavelets are maximally compact in time-frequency space; i.e. they reach the minimum timefrequency area allowed by the Heisenberg-Gabor uncertainty principle. The quality factor Q is then given by: which we see does not depend on the wavelet scale; Q depends only on the starting frequency of the sinusoid in the mother wavelet. The time resolution is proportional to the wavelet scale, and the frequency resolution is inversely proportional to the wavelet scale. Given the relationship between wavelet scale and central frequency, this means that passing a linearly spaced array of wavelet scales into the CWT will yield better time localization at higher frequencies, at the cost of frequency resolution. As a corollary, the CWT will yield better frequency resolution at lower frequencies, and the cost of time localization. For this reason the CWT is well suited for analyzing gravitational waves produced by CBC events that have rapidly changing frequency structure in the post-merger. The frequency evolution of such signals is characterized by a slow build at low frequency during the inspiral which peaks at merger. However for specific cases the post-merger signal can contain more complex structure in frequency evolution, which occur at higher overall frequency. Such cases are naturally suited for analysis with the CWT, as the slowly evolving behavior of the inspiral is less sensitive to the lower time resolution inherent at lower frequency, but the rapid changes in the post-merger that occur at higher frequency may be resolved. In Sec.(IV) below, we investigate two such cases: oscillation frequencies in the post-merger BNS signal, and frequency peaks in the post-merger of highly-inclined asymmetric mass ratio BBH signals.",
"pages": [
3,
4
]
},
{
"title": "IV. APPLICATIONS",
"content": "When applied to a gravitational wave signal, the CWT produces a time-frequency map, or scalogram, that captures the primary aspects of the signal's frequency evolution. Shown in Fig.(1) is an example scalogram for GW150914, the first gravitational wave signal detected by the LIGO and Virgo Collaborations [30, 4]. This image does not display the entire inspiral portion of the signal, but one can see that over the course of 0.2 seconds the frequency rapidly and smoothly evolves monotonically, increasing by a factor of ∼ 10 before reaching the merger. To produce the image in Fig.(1), the data for GW150914 was obtained using the the CWT was run using Morlet-Gabor wavelets at Q = 6 . 0, and the resultant scalogram was normalized by dividing all coefficient values by the largest value in the map. This makes the units in the colorbar of Fig.(1) somewhat arbitrary, and as such this scaling method makes the analysis less useful for differentiating the signal from noise when compared to the Q-scan method that searches for excess coherent power [9, 10, 11]. However, the consistent normalization facilitates exploration into how adjusting the parameters of the CWT analysis affects the structures in the timefrequency map for a freespace gravitational wave.",
"pages": [
4
]
},
{
"title": "i. BBH Postmerger",
"content": "Beyond the primary frequency peak denoting the inspiralmerger transition, gravitational waves from binary black hole systems with asymmetric mass ratio q > 1 exhibit additional frequency peaks after merger when viewed from a highlyinclined angle. These frequency structures were first studied in [20], where it was demonstrated that there is a correlation between the post-merger signal and the curvature of the final black hole horizon. Such signals are therefore of immediate interest, as if this correlation can be understood then there is a possibility of inferring the final black hole horizon geometry from the gravitational wave signals that we receive on Earth. As a first step towards investigating this possibility, it is critical to understand how to map the time-frequency space of these waveforms. For this purpose, we use the CWT to probe frequency structure. Shown in Fig.(2) is the time-frequency map of the NR waveform GT0568 [31, 32], which is a zerospin quasicircular BBH merger with mass ratio q = 10, that includes spherical harmonic modes up to l = 8. One can see that in contrast to the relatively simple structure in Fig.(1), which is consistent with an equal mass ratio ( q ≈ 1 ) system, the mass asymmetry in GT0568 introduces a more complex frequency structure when viewed from the edge-on inclination of π / 2 at a phase angle of φ = 0. The waveform is shifted in time to locate the peak waveform amplitude at t = 0 . 0, which we use as an approximation for the time of merger. One can see that after the initial frequency spike from the inspiral-merger transition, there is an additional frequency spike in the postmerger. See Sec.(V) for further discussion on the physical interpretation of this feature, which we refer to as the doublechirp pattern. The scalogram in Fig.(2) was made using the CWT with the Morlet wavelet at Q = 5 . 0. As discussed in Sec.(II) above, the quality factor Q not only defines the mother frequency for the wavelet, it is also related to the time and frequency resolutions of the CWT. For lower values of Q, the time resolution will be enhanced but the frequency resolution will be diminished, and vice versa. Towards the goal of identifying the postmerger frequency peak from that of the inspiral, one should then use a lower value of Q , to better separate these two features in time. However, in doing so the frequency content becomes largely indistinguishable, and is can be difficult to discern that the time-frequency map even represents a gravitational wave signal. On the other hand, at higher values of Q , the frequency values are more strongly highlighted but the time localization becomes smeared such that the double-chirp pattern is indistinguishable from a single peak. An example of this effect is shown in Fig.(3), where we display CWT scalograms with Q = 1 . 0 , 8 . 0 , 16 . 0. In testing a variety of options with the CWT over several example waveforms with this characteristic, values of Q ∈ [ 4 . 0 , 6 . 0 ] , corresponding to mother frequencies of f 0 ∈ [ 0 . 45 , 0 . 68 ] have been the most useful for analyzing the double chirp pattern. This range is consistent with the analysis in[20], where a mother frequency of f 0 = 0 . 4 is used. One can see that the scalogram in Fig.(2) reveals an undulating frequency structure during the inspiral, with a series of peaks that occur at higher and higher frequency as the system approaches merger. This is a consequence of the high mass ratio of the system; the asymmetry modulates the waveform, with each orbit effectively sweeping a burst of GW radiation across the observer's line of sight. One can view this frequency structure as a series of consecutive up-chirps and down-chirps, and considering the inner product perspective on CWT described in Sec.(III), one is provoked to wonder what the time-frequency representation looks like using a different wavelet basis. To this end we can also use chirplets in the CWT process: Morlet-Gabor wavelets that have their own frequency evolution. Further details on the chirplet basis are",
"pages": [
4,
5
]
},
{
"title": "discussed in Appendix Sec.(AIII).",
"content": "In Fig.(4) one can see that in the chirplet basis, the CWT scalogram has several differences when compared to its MG wavelet counterpart in Fig.(2). As discussed in Appendix Sec.(AIII), the chirplet basis rotates burst-like signals in timefrequency space. Using a basis with an upward chirp rate (up-chirps) causes bursts to rotate counter-clockwise in timefrequency space, and using a basis with a downward chirp rate (down-chirps) causes bursts to rotate clockwise in timefrequency space. Additionally, for a basis of either up-chirps or down-chirps, bursts get stretched in frequency. Here we see that with an up-chirp CWT basis, each inspiral burst appears tilted counter-clockwise, and stretched in frequency. In a down-chirp CWT basis, each inspiral burst appears tilted clockwise, and is likewise stretched in frequency, although we omit the figure. The frequency-stretching effect can be mitigated to some degree by increasing Q for the CWT analysis, but this does yield poorer time localization, and going too high on Q can make the postmerger chirp feature indistinguishable from the merger peak. Shown in Fig.(5) is a zoomed-in view of the postmerger of GT0568, rendered using CWT with a chirplet basis of d = 0 . 1 at Q = 6 . 0. Additionally, in this figure we trace an effective one-dimensional f ( t ) by highlighting the brightest pixel in the time-frequency map (shown in red). It is for this reason that the chirplet basis is most useful when applied to this class of signal; one can see that there are clear boundaries that denote the inspiral peak, the postmerger peak, and the end of the frequency evolution. These boundaries allow one to clearly define the postmerger region of the waveform based on its time-frequency map, and also the existence of the postmerger frequency peak. Further discussion of this method and its use towards parameterizing the double-chirp pattern will be explored in a subsequent publication.",
"pages": [
6
]
},
{
"title": "ii. BNS Postmerger",
"content": "In the case of a BNS merger, oscillations of the postmerger remnant can emit a rich, high frequency gravitational wave signal. Given a postmerger remnant of a hypermassive or supramassive neutron star, the post-merger gravitational wave signal could be milliseconds long with high frequencies between 1-4 kHz [27, 28]. With a confident detection, this postmerger signal could provide a new probe to study extremely dense nuclear matter given the the quasi-universal relations between spectral features of the signal and properties of the neutron star [22, 33]. For example, [22, 34] found that there is a relationship between the peak frequency of the BNS postmerger signal and the radius of non-rotating NSs. This radius can then be used to help constrain the NS EoS through the one-to-one mapping from the mass-radius relationship of neutron stars to the EoS [22]. Following a detection, one way to pinpoint the peak frequency of the postmerger signal is through a time-frequency map of the waveform. To visualize the BNS postmerger signal in the time-frequency plane, we use CWT. Shown in Fig.(6) is the time-frequency map of the postmerger waveform for the DD2 EoS [35] with m 1 = 1 . 35 M ⊙ and m 2 = 1 . 35 M ⊙ . One can see that following the initial frequency increase characteristic of a compact binary coalescence, there is excess power at approximately 2500 [ Hz ] . See Sec.V for additional discussion on the physical interpretation of this high frequency signature. The time-frequency map in Fig.(6) was made with the Morlet wavelet at Q = 32 . 0. As explained in Sec.III, at higher Q values, the frequency resolution is enhanced, but at the cost of diminished time localization. Given that peak frequency of a BNS postmerger signal could be used to help constrain the NS EoS, a higher value of Q should be used to get a more confident measure of f peak . In Fig.(7), CWT was tested with varied Q values of Q ∈ [ 16 . 0 , 32 . 0 , 64 . 0 ] , corresponding to mother frequencies of f 0 ∈ [ 1 . 80 , 3 . 60 , 7 . 20 ] . One can see that at lower Q the postmerger oscillation is smeared in frequency, but as Q increases it becomes more localized. However the localization of the merger is also smeared in time as Q becomes large. For this reason Q = 32 . 0 is preferred to preserve both characteristics.",
"pages": [
6,
7
]
},
{
"title": "V. DISCUSSION",
"content": "The time-frequency maps above highlight the ability of the CWT to distinguish specific features in the frequency evolution of gravitational wave signals. In particular we have examined the postmerger waveforms from both BBH and BNS systems, where characteristic features are connected to the physics of their originating systems. The CWT method is highly flexible, and can be tuned to the specific problem at hand by varying chiefly the quality factor Q , and also the wavelet basis. The quality factor determines one's ability to localize features in either time or frequency, with simultaneous localization in both being limited by the HeisenbergGabor uncertainty relation. A lower value Q will improve time localization at the cost of frequency localization, and vice versa for a higher value of Q . In the case of BBH systems with asymmetric mass ratio, where frequency features in the postmerger are evident when viewed from an edge-on inclination, a lower Q is preferred. This allows one to differentiate the postmerger frequency peak from that of the inspiral; the exact frequency content of the postmerger peak is of secondary importance to its location in time. The BNS postmerger case has the opposite requirements, as we are more interested in the frequency content than in the time localization. Here we therefore want a higher value of Q , to better isolate the frequency of the postmerger remnant oscillations. High frequency localization of the postmerger peak frequency would give way to constraints on the NS EoS through the empirical relationship between f peak and the radius of a non-rotating NS. As such, exact time localization is sacrificed for the purpose of determining the precise frequency content of the BNS postmerger signal. This work primarily examined noiseless, freespace gravitational waves, and demonstrated how CWT can be utilized to extract important frequency features embedded in the BNS postmerger signal. Given estimated merger rates, current and future detector sensitivities, and predicted waveform morphologies, the probability of near-future detection is low [26, 23, 36]. Given the low probability of detection, studies such as [36] and [37] have explored combining observations using Bayesian statistics to increase the likelihood of detection. An additional stacking method that can be used to boost the detection sensitivity is to combine TF maps from multiple postmerger candidates created using CWT. This method will be detailed in a subsequent paper. In addition to varying the quality factor, one can also use CWT with different bases of wavelets. Many applications in gravitational wave analysis use the Morlet-Gabor wavelet, as it is maximally compact in time-frequency space. Modifying the MG wavelet into a chirplet largely preserves this benefit, but offers an additional axis of exploration in the space of the signal's time-frequency representation. In the BBH case, where one is primarily interested in the separation between the inspiral and postmerger chirps, the chirplet basis allows one to clearly identify boundaries that denote the start and end of the postmerger stage of the signal. This enables the development of a consistent statistic that tracks the existence of the postmerger chirp over the angular variations of the observer orientation, which can then be used to predict the direction of maximal emission based on the intrinsic parameters of the waveform. This parameterization process and its application towards inferring the geometry of the final black hole horizon will be detailed in a subsequent publication. Beyond the specific applications to the BBH and BNS postmerger waveforms, there are other potential uses for the CWT framework. In this work, we chiefly considered freespace gravitational waves - simulated waveforms absent the realities of sensitivity and noise that must be considered for detectors like LIGO, Virgo, and KAGRA. In practice the multiresolution Q-scan [9, 10], which performs a CWT-like process over multiple quality factors, is very effective in distinguishing a gravitational wave signal (or other transient) from the background detector noise. However this method constitutes a more general approach, and lacks the specificity of time/frequency localization offered by the CWT framework. We speculate that a variation of the Q-scan algorithm and excess power method [11] that incorporates the capabilities of CWT could be developed to search for specific features in localized time-frequency tiles. For now we leave this possibility to future work. Additionally, the flexibility of CWT to incorporate further wavelets beyond the sine-Gaussian MG wavelets allows for the creation of time-frequency maps that have a fundamen- tally different structure. Herein we have explored the use of chirplets, which rotate structures in time-frequency space, for the BBH postmerger case. Chirplets, which add the chirp rate d to the MG wavelet, therefore effectively add an extra axis to the space of possible time-frequency analyses on gravitational wave data. Included with this work is an example script for running CWT with chirplets on previous catalog events, which may be found at github.com/chadhenshaw/gw cwt ; see Appendix Sec.(AIV) for further details. Additionally there are also many other wavelet bases that may be of interest; for example the 'Mexican hat' wavelet described in [38] may be useful for analysis of spurious noise transients in LVK data, as it bears similar time-frequency morphology to 'blip' glitches. In a similar vein, exponential shapelets [39] may be useful for transients that feature damped oscillations, or even for black hole ringdown. We leave the implementation and exploration of such methods and applications to future work.",
"pages": [
7,
8
]
},
{
"title": "ACKNOWLEDGEMENTS",
"content": "We thank Meg Millhouse and James Clark for helpful discussions and technical advice. This research has made use of data, software and/or web tools obtained from the Gravitational Wave Open Science Center (https://www.gwopenscience.org/), a service of LIGO Laboratory, the LIGO Scientific Collaboration and the Virgo Collaboration. LIGO Laboratory and Advanced LIGO are funded by the United States National Science Foundation (NSF) as well as the Science and Technology Facilities Council (STFC) of the United Kingdom, the Max-Planck-Society (MPS), and the State of Niedersachsen/Germany for support of the construction of Advanced LIGO and construction and operation of the GEO600 detector. Additional support for Advanced LIGO was provided by the Australian Research Council. Virgo is funded, through the European Gravitational Observatory (EGO), by the French Centre National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale di Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions by institutions from Belgium, Germany, Greece, Hungary, Ireland, Japan, Monaco, Poland, Portugal, Spain. KAGRA is supported by the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) in Japan, and is hosted by the Institute for Cosmic Ray Research (ICRR), the University of Tokyo, and co-hosted by High Energy Accelerator Research Organization (KEK) and the National Astronomical Observatory of Japan (NAOJ). This material is based upon work supported by the LIGO Laboratory which is a major facility fully funded by the NSF. This work was supported by NSF grants PHY-1809572 & PHY-2110481. This paper has LIGO document number P2400016.",
"pages": [
8,
9
]
},
{
"title": "A. APPENDIX",
"content": "For a function g ( x ) , with power distribution | g ( x ) | 2 , its normalized n 'th central moment is given by: where n = 1 corresponds to the mean value, and n = 2 to the variance. If g ( x ) is a time series g ( t ) with corresponding Fourier transform ˆ g ( f ) , its central time τ , and a central frequency φ are given by the first central moments: The second central moments then give the characteristic duration σ t and bandwidth σ f :",
"pages": [
9
]
},
{
"title": "AII. WAVELETS",
"content": "A wavelet is a localized wave-like function ψ ( t ) that satisfies three criteria. First, a wavelet must have finite energy E : Note that here the vertical brackets represent the modulus operator - i.e if ψ ( t ) is a complex function, then | ψ ( t ) | 2 = ψ ∗ ( t ) ψ ( t ) , where ψ ∗ ( t ) is the complex conjugate of ψ ( t ) . In practice, it is common for the wavelet to further be normalized such that it has unit energy. Secondly, given the Fourier transform of the wavelet ˆ ψ ( f ) , the wavelet must have no zerofrequency component: This criterion is referred to as the admissibility condition , and Cg is called the admissibility constant , whose value depends on the specific wavelet. The final criterion applies to complex wavelets - their Fourier transform must be real, and must be zero for negative frequencies. Any wave-like function that satisfies these three criteria is a wavelet, and may be used in the continuous wavelet transform. For gravitational wave analysis, the most common such function is the Morlet wavelet: Here the second exponential term within the parentheses is referred to as the correction term, as it ensures the wavelet satisfies the second criterion by giving it zero mean. However in practice this term is discarded, as it becomes negligible for f 0 ≫ 0 4 . We will still refer to the resultant approximation as a wavelet, as it only violates the admissibility condition for low frequencies. This approximation is often referred to as the Gabor wavelet, or the Morlet-Gabor (MG) wavelet: Note that the MG wavelet has three components: an amplitude π -1 4 , a Gaussian window exp [ -t 2 2 ] with unit standard deviation, and a complex sinusoid exp [ i 2 π f 0 t ] with frequency f 0. Its central time is identically zero, and its central frequency is f 0. Its characteristic duration is σ t = 1 / √ 2, and its bandwidth is 1 / 2 √ 2 π . The duration-bandwidth product of the MG wavelet is thus σ t σ f = 1 / 4 π , which is the minimum value in the Heisenberg-Gabor uncertainty principle. In this way, MG wavelets are maximally compact in time-frequency space, which makes them ideal for time-frequency analysis methods like the continuous wavelet transform (Eq.(5)). In this method, the wavelet's scale is adjusted by the dilation parameter a , and it's time localization is shifted by the translation parameter b : The wavelet's energy is normalized at every scale: This normalization is maintained in the frequency domain; first we compute the Fourier transform of the dilated and translated MG wavelet: Note that in the untranslated, non-dilated case ( b = 0 , a = 1 ) , one recovers exactly the Fourier transform of the MG wavelet in Eq.(A10). The energy is then: The denominators in Eqs.(A2-A5) for the MG wavelet are thus unity, and we can proceed to calculate the central time: which returns as the translation parameter b , as expected. Now we calculate the central frequency: Note that in the case of unit scale, i.e. a = 1, we recover the central frequency as exactly f 0, as one expects. Next we will calculate the characteristic duration: which gives an indication of how localized the wavelet is in the time domain. We see that this is directly proportional to the dilation parameter; thus at larger scales the wavelet is more spread out in time. The bandwidth is then: We obtain a bandwidth - and thus a frequency resolution - that is inversely proportional to the wavelet scale. Therefore at larger scales the wavelet has a narrower bandwidth, as one would expect - more frequencies are encapsulated by the larger wavelet. The quality factor Q is then defined as the ratio of central frequency to bandwidth: which we see does not depend on the wavelet scale; Q depends only on the starting frequency of the sinusoid. We also see that the duration-bandwidth product is still σ t σ f = 1 / 4 π , so dilated MG wavelets remain maximally compact in timefrequency space. Now considering that the quality factor does not depend on wavelet scale, we can rewrite the equation for the dilated and translated Morlet wavelet as: where we see that for a given dilation factor a we have the effective sinusoid frequency: which is equivalent to the central frequency φ . When running CWT, it is this effective frequency that you are interrogating when convolving the wavelet with the signal.",
"pages": [
9,
10
]
},
{
"title": "AIII. CHIRPLETS",
"content": "The central frequency of the MG wavelet in Sec.(AII) above is stationary, but may be modified to evolve in time by including a chirp rate parameter d (following the notation of [40]) such that the central sinusoid frequency f 0 becomes: and the mother wavelet, now a chirplet , becomes: Note that here the chirp parameter d is a fixed quantity, and as such in the CWT process becomes another parametric input for the mother wavelet. A positive value of d > 0 corresponds to an up-chirp , where the wavelet frequency increases over time, and a negative value of d > 0 corresponds to a downchirp , where the wavelet frequency decreases over time. Note that if d = 0 we recover exactly the Morlet-Gabor wavelet. Also note that this parameterization of the chirp rate is different than that in e.g.[41], where the chirp parameter β = 2 d is used. Below we create a variation of Fig.2 from that work, using the CWT process to create the time-frequency map of different chirplets. Fig.(8) was constructed by creating a series of ten chirplet signals with different parameters, running CWT on each signal, then displaying together all ten time-frequency maps. Each chirplet can be thought of as a short burst-like signal. The middle panel shows the representation of the ten chirplets using the MG wavelet (i.e. d = 0 . 0) basis with CWT. Here one can see that the time-frequency representation of each burst is an ellipse, as as the chirp rate becomes more positive (negative), the ellipse is rotated clockwise (counter-clockwise). The left and right panels show the same analysis, but with a chirplet basis at d = -04 and d = 0 . 4 respectively. One can see that in this basis, the corresponding chirplet signal is counter-rotated in time-frequency space. Consider first the right panel in Fig.(8). The right-most chirplet signal ( d = 0 . 4 ) , which is tilted clockwise in the middle panel, has been rotated counter-clockwise to a vertical position. This is because the underlying chirp rate of the CWT basis matches the chirp rate of the signal, and as such its frequency structure has been localized in time. The remaining bursts, which all have a chirp rate less than that of the CWT basis, have also been rotated counter-clockwise, and their frequency localization has been stretched in accordance with the wider bandwidth of the CWT basis. The same principle applies to the left panel in Fig.(8), but with the opposite sense. The bursts with chirp rates greater than that of the underlying CWT basis are now rotated clockwise, countering their natural rotation in time-frequency space.",
"pages": [
10,
11
]
},
{
"title": "AIV. NUMERICAL IMPLEMENTATION OF CWT",
"content": "To create the time-frequency maps herein, we use a modified version of a previous Python implementation of CWT [42]. The modified code, which we refer to as gw cwt , is available at github.com/chadhenshaw/gw cwt along with example scripts. The primary function in this package is build cwt , which accepts as input a timeseries array and corresponding uniformly-sampled timestamps, runs the cwt function, and returns a dictionary containing properties of the resultant timefrequency map. The cwt function, which is a time domain convolution, is implemented as a product between signal and wavelet in the frequency domain. At each wavelet scale, the timeseries and wavelet are converted to the frequency domain by FFT, and the inverse FFT of their product is then computed to complete the convolution. The build cwt function also contains a number of options that the user may specify to customize their analysis, with the primary variables being the quality factor Q , and the chirp rate d . The number of frequencies interrogated is determined by the option n conv , which is the number of convolutions. Increasing this setting samples more frequencies within the given range at the cost of compute time. Given a range of frequencies, wavelet scales are computed as a = f 0 fs / f e f f , where fs is the sampling rate of the timeseries. Additionally, the Nyquist frequency fs / 2 determines the upper bound on the signal frequency one can interrogate. By default, the program will create a linearly-spaced array of wavelet scales, which corresponds to a logarithmically-spaced array of frequencies. The user may instead request a linearly-spaced frequency array, in which case the program will create the corresponding logarithmically-spaced array of wavelet scales. Alternatively for each case, the user may specify the spacing between frequencies or scales as d f or da , which supersedes the number of convolutions. Finally, the user may instead input directly an array of frequencies to compute over. Included also in the gw cwt package is the script cwt catalog.py , which simplifies the process of running CWT on GW data. Within this program, the get data function pulls gravitational wave data from the Pythonbased pycbc.catalog [43]. The data are downloaded and stored in a local hdf5 compressed file, the organizational structure of which resembles a Python dictionary. The run cwt function then computes the time-frequency map for the downloaded event data, given user input of CWT parameters, and appends the results to the hdf5 file. Finally, plot cwt is a function for plotting the time-frequency map. Examples and further documentation are available at github.com/chadhenshaw/gw cwt .",
"pages": [
11,
12
]
}
]
2024arXiv240302471L
https://arxiv.org/pdf/2403.02471.pdf
<document>
<section_header_level_1><location><page_1><loc_16><loc_79><loc_85><loc_84></location>Anisotropic examples of inflation-generating initial conditions for the big bang</section_header_level_1>
<text><location><page_1><loc_34><loc_75><loc_67><loc_77></location>Eric Ling ∗ 1 and Annachiara Piubello † 2</text>
<text><location><page_1><loc_25><loc_69><loc_75><loc_74></location>1 Copenhagen Centre for Geometry and Topology (GeoTop), 2</text>
<text><location><page_1><loc_19><loc_68><loc_82><loc_72></location>Department of Mathematical Sciences, University of Copenhagen, Denmark Institute of Mathematics, University of Potsdam, Germany.</text>
<section_header_level_1><location><page_1><loc_47><loc_62><loc_54><loc_64></location>Abstract</section_header_level_1>
<text><location><page_1><loc_19><loc_47><loc_81><loc_62></location>The inflationary scenario, which states that the early universe underwent a brief but dramatic period of accelerated spatial expansion, has become the current paradigm of early universe cosmology. Although inflationary cosmology has its many successes, it does not (as of yet) have the status of an established physical theory. In this paper, we provide mathematical support for the inflationary scenario in a class of anisotropic spacetimes by generalizing the work in [21]. These anisotropic spacetimes satisfy certain initial conditions so that they are perfectly isotropic at the big bang but become less isotropic as time progresses. The resulting inflationary eras are a consequence of the initial conditions which force the energy-momentum tensor to be dominated by a cosmological constant at the big bang.</text>
<section_header_level_1><location><page_1><loc_15><loc_43><loc_25><loc_45></location>Contents</section_header_level_1>
<table>
<location><page_1><loc_15><loc_17><loc_86><loc_41></location>
</table>
<section_header_level_1><location><page_2><loc_15><loc_87><loc_33><loc_89></location>1 Introduction</section_header_level_1>
<section_header_level_1><location><page_2><loc_15><loc_84><loc_36><loc_86></location>1.1 Cosmic inflation</section_header_level_1>
<text><location><page_2><loc_15><loc_80><loc_85><loc_83></location>The inflationary scenario has become the current paradigm of early universe cosmology. Roughly, it states the following.</text>
<text><location><page_2><loc_15><loc_76><loc_86><loc_79></location>The inflationary scenario: In the early universe, before the radiation-dominated era, there was a brief but dramatic period of accelerated spatial expansion.</text>
<text><location><page_2><loc_15><loc_65><loc_86><loc_75></location>The inflationary scenario was proposed in the late 1970s and early 80s [13,18,29] as a solution to some problems in the standard big bang model, e.g., the flatness and horizon problems. It was soon realized that inflation can provide a framework for generating the seeds of the large-scale structures in our universe [24]. Observations of the anisotropies in the CMB radiation performed by COBE, WMAP, and most recently by Planck [8] support these claims.</text>
<text><location><page_2><loc_15><loc_54><loc_86><loc_64></location>Given the many successes of the inflationary scenario, it is perhaps not too surprising that most papers on early universe cosmology give the impression that inflation has been firmly established and observationally proven. However there are many inflationary models that can be in agreement with observation [23]. In fact, any theory which predicts an almost flat universe with a nearly scale-invariant curvature power spectrum, small tensor-to-scalar ratio, and small Gaussian fluctuations would be in agreement with current data, e.g., [5].</text>
<text><location><page_2><loc_15><loc_48><loc_86><loc_54></location>Moreover, although phenomenologically successful, current realizations of inflationary models suffer from conceptual problems, perhaps none more so than the problem of initial conditions [4, 17]. In fact there are conflicting opinions on the naturalness of initial conditions for inflation [14,16].</text>
<text><location><page_2><loc_15><loc_37><loc_86><loc_47></location>Most papers on initial conditions for inflation begin in an inhomogeneous universe with an energy-momentum tensor dominated by an inflaton scalar field in a slow-roll potential and see if the resulting dynamics can produce an inflationary era followed by a homogeneous universe. This is not our approach. Our approach is purely geometrical. Quantities of interest are described solely in terms of a special unit timelike vector field u (whose integral curves represent the comoving observers in the universe) and the spacetime metric g .</text>
<text><location><page_2><loc_15><loc_24><loc_86><loc_37></location>In this paper we provide mathematical support for the inflationary scenario. In section 3, we show that inflation arises for a class of anisotropic spacetimes from special geometrical initial conditions. Our initial conditions are stated informally in section 1.3 and formally in section 2. These anisotropic spacetimes are examples of our main result, Theorem 2.4, which concludes from the special initial conditions that the Ricci tensor (and hence also the energy-momentum tensor) is dominated by a cosmological constant at the big bang. Theorem 2.4 is a generalization of the main result in [21]. In fact, a major inspiration for this paper was to find anisotropic examples of the main result in [21].</text>
<text><location><page_2><loc_15><loc_18><loc_86><loc_23></location>The benefit of our geometrical approach is its conceptual clarity: we will describe precisely which comoving observers experience inflation and how fast they are accelerating solely in terms of the unit timelike vector field u and spacetime metric g .</text>
<text><location><page_2><loc_15><loc_12><loc_86><loc_18></location>Our geometrical initial conditions can be thought of as a certain type of fine-tuning condition for the big bang. As briefly reviewed in the next section, a Boltzmannian viewpoint on the arrow of time suggests that some type of fine-tuning initial condition for the big bang should exist.</text>
<section_header_level_1><location><page_3><loc_15><loc_87><loc_50><loc_89></location>1.2 Inflation and the arrow of time</section_header_level_1>
<text><location><page_3><loc_15><loc_74><loc_86><loc_86></location>An obvious feature of our universe is the existence of an arrow of time. We observe certain processes in our everyday experience, but we hardly ever observe those same processes time-reversed. A vase shatters into a multitude of pieces, but we never observe these pieces spontaneously arranging themselves perfectly together into a vase. The second law of thermodynamics is postulated to explain the arrow of time, and a modern Boltzmannian mindset of the second law leads to the conclusion that the universe began with special, non-generic, fine-tuned initial conditions.</text>
<text><location><page_3><loc_15><loc_66><loc_86><loc_74></location>It was Penrose who originally argued [27] that the overall arrow of time we observe is linked to special initial conditions for the universe that are drastically far away from the dynamical trend towards gravitational collapse. He calculated that the entropy of the radiation-dominated early universe is around 30 orders of magnitude smaller than the Beckenstein-Hawking entropy of its corresponding black hole state. See also [1,6].</text>
<text><location><page_3><loc_15><loc_49><loc_86><loc_66></location>With this understanding, the homogeneous and isotropic assumptions of the standard FLRW models of cosmology are a reasonable choice of initial conditions as they match exceedingly well with current observations. However, some inflationary cosmologists instinctively take a different perspective. They seek an explanation for the large-scale isotropy of the universe from dynamical processes during inflation. But special initial conditions - by their nature - go against dynamical trends. That is, the creation of special initial conditions from dynamics beginning with generic initial conditions seems contradictory. Summarizing, some inflationary cosmologists seek generic initial conditions for the universe, but those who adopt a Boltzmannian point of view of the second law of thermodynamics (as we do) argue that initial conditions should instead be special in order to explain the arrow of time.</text>
<text><location><page_3><loc_15><loc_32><loc_86><loc_48></location>While inflationary theory alone may not suffice to explain the large-scale isotropy of the universe, it still has many successes and remains the prevailing paradigm in early universe cosmology. The simplest way to generate inflation is to introduce an inflaton scalar field in a slow-roll potential - a methodology that is somewhat adhoc since it simply postulates the existence of a scalar field for which we have no direct evidence for. So a natural inquiry is to ask if there is other evidence to support the inflationary scenario. A primary motivation of this paper is to demonstrate that there is mathematical evidence in support of the inflationary scenario. We will see that inflation is inevitable provided certain geometrical initial conditions are assumed at the big bang, and, as discussed in this section, some degree of fine-tuning in the initial conditions is anticipated.</text>
<section_header_level_1><location><page_3><loc_15><loc_28><loc_65><loc_29></location>1.3 Geometrical initial conditions for the big bang</section_header_level_1>
<text><location><page_3><loc_15><loc_22><loc_86><loc_27></location>In this section we describe, informally, the primary geometrical initial conditions we will be considering in our main result, Theorem 2.4. These initial conditions are supposed to mimic - without assuming isotropy - the geometrical properties at the big bang.</text>
<text><location><page_3><loc_15><loc_12><loc_86><loc_22></location>Let's first clarify what we mean by the 'big bang' as there are conflicting view points in the literature. For us the big bang refers to a time when the scale factor limits to zero. For example, if the scale factor is a ( τ ) = τ (as in the Milne model), then the big bang corresponds to τ = 0. If the scale factor is a ( τ ) = e τ (as in the flat de Sitter model), then the big bang corresponds to τ = -∞ .</text>
<text><location><page_4><loc_15><loc_77><loc_86><loc_89></location>To motivate the type of geometrical initial conditions we will be considering, we focus on scale factor perturbations of the Milne model, which have been dubbed 'Milne-like spacetimes' in [10]. These models were extensively studied in [20], detailing possible applications to fundamental problems in cosmology. See also [7,22,25]. They are k = -1 FLRW spacetimes whose scale factor satisfies a ( τ ) ≈ τ for τ near τ = 0. (An inflating example would be a ( τ ) = sinh( τ ).) Interestingly, for Milne-like spacetimes, the big bang appears as a coordinate singularity, and so they extend into a larger spacetime.</text>
<figure>
<location><page_4><loc_15><loc_57><loc_51><loc_74></location>
<caption>Figure 1: A Milne-like spacetime represented in two different coordinate systems. On the left, standard comoving coordinates are used; the metric is degenerate at τ = 0. On the right, conformal Minkowskian coordinates are used; the metric is nondegenerate at τ = 0 which corresponds to the lightcone at O . The black lines depict the comoving observers.</caption>
</figure>
<text><location><page_4><loc_26><loc_56><loc_27><loc_58></location>-</text>
<text><location><page_4><loc_60><loc_57><loc_61><loc_58></location>g</text>
<text><location><page_4><loc_62><loc_57><loc_64><loc_58></location>= Ω</text>
<text><location><page_4><loc_64><loc_57><loc_65><loc_58></location>2</text>
<text><location><page_4><loc_65><loc_55><loc_66><loc_58></location>(</text>
<text><location><page_4><loc_66><loc_57><loc_66><loc_58></location>τ</text>
<text><location><page_4><loc_66><loc_57><loc_67><loc_58></location>)[</text>
<text><location><page_4><loc_67><loc_56><loc_68><loc_58></location>-</text>
<text><location><page_4><loc_68><loc_57><loc_69><loc_58></location>dt</text>
<text><location><page_4><loc_70><loc_73><loc_70><loc_74></location>t</text>
<text><location><page_4><loc_70><loc_61><loc_71><loc_63></location>O</text>
<text><location><page_4><loc_69><loc_57><loc_70><loc_58></location>2</text>
<text><location><page_4><loc_70><loc_57><loc_71><loc_58></location>+</text>
<text><location><page_4><loc_72><loc_57><loc_73><loc_58></location>dx</text>
<text><location><page_4><loc_73><loc_57><loc_74><loc_58></location>2</text>
<text><location><page_4><loc_74><loc_57><loc_75><loc_58></location>+</text>
<text><location><page_4><loc_75><loc_57><loc_77><loc_58></location>dy</text>
<text><location><page_4><loc_77><loc_57><loc_77><loc_58></location>2</text>
<text><location><page_4><loc_78><loc_57><loc_79><loc_58></location>+</text>
<text><location><page_4><loc_79><loc_57><loc_80><loc_58></location>dz</text>
<text><location><page_4><loc_80><loc_57><loc_81><loc_58></location>2</text>
<text><location><page_4><loc_81><loc_57><loc_81><loc_58></location>]</text>
<text><location><page_4><loc_15><loc_34><loc_86><loc_45></location>Recall that the comoving observers are the integral curves of the vector field u given by u = ∂ τ in comoving coordinates. As illustrated in Figure 1, the comoving observers for a Milne-like spacetime all emanate from a single point O in the extended spacetime, which is just the origin (0 , 0 , 0 , 0) in the conformal Minkowskian coordinates ( t, x, y, z ). We refer to this property as ' O being an origin point for u ,' see Definition 2.1. The existence of an origin point O for u is a highly fine-tuned and non-generic assumption. Recall that some fine-tuning is to be expected from the discussion on the arrow of time in section 1.2.</text>
<text><location><page_4><loc_15><loc_20><loc_86><loc_33></location>An origin point O for u is the first main assumption in Theorem 2.4. The other main assumption is that the energy-momentum tensor T approaches that of a perfect fluid at O . See Definition 2.2. This assumption is more physically convincing than assuming that T is exactly a perfect fluid (as in the FLRW models) since we expect small deviations from perfect isotropy in our universe. Therefore the perspective taken here is that the universe began in a state of perfect isotropy at the big bang. This is the crux of Definition 2.2. Moreover, this perspective is reinforced in our examples since the shear vanishes towards the big bang, see eq. (3.23).</text>
<text><location><page_4><loc_15><loc_12><loc_86><loc_20></location>An 'origin point O for u ' and ' T approaching a perfect fluid at O ' are the two primary assumptions in Theorem 2.4. There are three other assumptions that are purely technical. The conclusion of Theorem 2.4 is that the energy-momentum tensor is precisely given by a cosmological constant at O . This fact will be used in section 3.2 to prove the existence of inflationary eras in our anisotropic examples.</text>
<text><location><page_4><loc_83><loc_67><loc_84><loc_68></location>τ</text>
<text><location><page_4><loc_84><loc_67><loc_91><loc_68></location>= constant</text>
<text><location><page_4><loc_85><loc_62><loc_85><loc_63></location>x</text>
<text><location><page_4><loc_85><loc_62><loc_86><loc_63></location>i</text>
<section_header_level_1><location><page_5><loc_15><loc_87><loc_40><loc_89></location>2 The main theorem</section_header_level_1>
<text><location><page_5><loc_15><loc_79><loc_86><loc_86></location>The initial conditions stated informally in section 1.3 will be stated formally in this section. Our main result, Theorem 2.4, is a generalization of the main result (Theorem 2.2) in [21]. Anisotropic examples of our main theorem are provided in section 3.1, and we prove the existence of inflationary eras for these examples in section 3.2.</text>
<text><location><page_5><loc_15><loc_67><loc_86><loc_79></location>We set our conventions. Our definition of a spacetime ( M,g ) will follow [19]. (Except that, for simplicity, we will assume that all spacetimes are four-dimensional.) The manifold M is always assumed to be smooth. A C k spacetime is one where the metric g is C k , that is, its components g µν = g ( ∂ µ , ∂ ν ) are C k functions with respect to any coordinates ( x 0 , . . . , x 3 ). A continuous spacetime is one where the metric is continuous, that is, its components are continuous functions with respect to any coordinates. Our definitions of timelike curves and the timelike future I + will also follow [19].</text>
<text><location><page_5><loc_15><loc_64><loc_86><loc_67></location>Let ( M,g ) be a C k spacetime. A C 0 spacetime ( M ext , g ext ) is said to be a continuous spacetime extension of ( M,g ) provided there is an isometric embedding</text>
<formula><location><page_5><loc_41><loc_60><loc_59><loc_62></location>( M,g ) ↪ → ( M ext , g ext )</formula>
<text><location><page_5><loc_15><loc_52><loc_86><loc_59></location>preserving time orientations such that M ⊂ M ext is a proper subset. ( M is in fact an open submanifold of M ext since they are both four-dimensional.) Note that we are identifying M with its image under the embedding. We remark that g ext is C 2 in the examples constructed in the next section.</text>
<text><location><page_5><loc_15><loc_39><loc_86><loc_50></location>Definition 2.1 (Origin point) . Let ( M ext , g ext ) be a continuous spacetime extension of a C k spacetime ( M,g ). Let u be a unit future directed timelike vector field on M . We say that a point O is an origin point for u if O ∈ M ext \ M and O is a past endpoint for each integral curve of u , and each extended integral curve is C 1 at O . (Clearly this implies O lies in the closure M within M ext .) In other words, O is an origin point for u if each integral curve of u , parameterized as γ : (0 , b ) → M , satisfies</text>
<unordered_list>
<list_item><location><page_5><loc_17><loc_36><loc_30><loc_39></location>(i) lim τ → 0 γ ( τ ) = O ,</list_item>
<list_item><location><page_5><loc_16><loc_30><loc_46><loc_36></location>(ii) ˜ γ ' (0) exists and ˜ γ ' (0) = lim τ → 0 γ ' ( τ ),</list_item>
</unordered_list>
<text><location><page_5><loc_15><loc_21><loc_86><loc_30></location>˜ Remarks. Definition 2.1 is supposed to model the behavior of the comoving observers in Figure 1 (right). It is essentially the same as assumption (b) in [21, Thm. 2.2]. Actually, Definition 2.1 is slightly stronger; we assume this stronger assumption since it's easier to state and all the examples in section 3 will satisfy it.</text>
<text><location><page_5><loc_15><loc_27><loc_86><loc_32></location>where ˜ γ : [0 , b ) → M ext is the extended curve defined by ˜ γ (0) = O and ˜ γ ( τ ) = γ ( τ ) for τ > 0. Continuity of the metric implies γ ' (0) is a unit future directed timelike vector.</text>
<text><location><page_5><loc_15><loc_10><loc_86><loc_20></location>We recall some terminology from section 2 of [21]. Let O ∈ M ext \ M be an origin point for u . A C k function f : M → R extends continuously to M ∪ {O} if there is a continuous function ˜ f : M ∪ {O} → R such that ˜ f | M = f . In this case, we call ˜ f the continuous extension of f . A C k tensor T defined on M extends continuously to M ∪ {O} if there is a coordinate neighborhood U of O with coordinates ( x 0 , . . . , x 3 ) such that each of the</text>
<text><location><page_6><loc_15><loc_78><loc_86><loc_89></location>components of T extends continuously to ( U ∩ M ) ∪{O} . (This definition does not depend on the choice of coordinate system by the usual transformation law for tensor components.) This defines a continuous tensor ˜ T on M ∪{O} , called the continuous extension of T , which satisfies ˜ T | M = T . For example, the metric tensor g extends continuously to M ∪ {O} (by definition of a continuous extension). Trivially, if T is a smooth tensor defined on all of M ext , then clearly T | M extends continuously to M ∪ {O} .</text>
<text><location><page_6><loc_15><loc_71><loc_86><loc_78></location>Definition 2.2 (Limiting to a perfect fluid near O ) . Let ( M,g ) be a C 2 spacetime, and let O ∈ M ext \ M be an origin point for u . Let T be the energy-momentum tensor on M (i.e., T = 1 8 π G in suitable units where G = Ric -1 2 Rg is the Einstein tensor). Let ρ 0 , p 0 ∈ R . We say that T limits to a perfect fluid ( u, ρ 0 , p 0 ) at O if</text>
<unordered_list>
<list_item><location><page_6><loc_16><loc_63><loc_86><loc_70></location>(i) ρ := T ( u, u ) extends continuously to M ∪ {O} and ˜ ρ ( O ) = ρ 0 , (ii) for any unit spacelike vector field e on M , which is orthogonal to u , the function p e := T ( e, e ) extends continuously to M ∪ {O} and p e ( O ) = p 0 ,</list_item>
<list_item><location><page_6><loc_16><loc_59><loc_86><loc_66></location>˜ (iii) T -T perfect extends continuously to M ∪ {O} and its continuous extension is zero at O , where T perfect is the tensor on M given by</list_item>
</unordered_list>
<formula><location><page_6><loc_39><loc_56><loc_66><loc_58></location>T perfect = ( ρ 0 + p 0 ) u ∗ ⊗ u ∗ + p 0 g,</formula>
<text><location><page_6><loc_19><loc_53><loc_67><loc_56></location>where u ∗ = g ( u, · ) is the one-form metrically equivalent to u .</text>
<text><location><page_6><loc_15><loc_46><loc_86><loc_53></location>Remark. Definition 2.2 relaxes the requirement that T is identically a perfect fluid in assumption (a) of [21, Thm. 2.2]. Moreover, it's more physically convincing: FLRW models have perfect fluid energy-momentum tensors, and we expect that an FLRW model approximates our universe better as we go back in time towards the big bang.</text>
<text><location><page_6><loc_18><loc_44><loc_70><loc_45></location>Lastly, we require a mild, technical timelike convexity assumption:</text>
<text><location><page_6><loc_15><loc_35><loc_86><loc_43></location>Definition 2.3 (Locally timelike convex near O ) . Let O ∈ M ext \ M be an origin point for u . Let γ : (0 , b ) → M be an integral curve of u . We say M is locally timelike convex about γ near O if there is an ε > 0 and a coordinate neighborhood U ⊂ M ext centered at O with coordinates ( x 0 , . . . , x 3 ) satisfying</text>
<unordered_list>
<list_item><location><page_6><loc_17><loc_33><loc_58><loc_35></location>(i) g µν ( O ) = η µν and | g µν ( p ) -η µν | < ε for all p ∈ U ,</list_item>
<list_item><location><page_6><loc_16><loc_27><loc_31><loc_32></location>(ii) ∂ 0 | O = ˜ γ ' (0), (iii) I + η ε ( O , U ) ⊂ M ,</list_item>
</unordered_list>
<text><location><page_6><loc_15><loc_25><loc_79><loc_27></location>where η ε is the narrow Minkowskian metric given by η ε = -ε 2 -ε ( dx 0 ) 2 + δ ij dx i dx j .</text>
<text><location><page_6><loc_15><loc_13><loc_86><loc_24></location>Remarks. In [21, Thm. 2.2], it was assumed that the manifold M satisfies M = I + ( O , M ext ). Definition 2.3 relaxes this requirement and is a much weaker assumption. It will hold for the examples constructed in section 3. Also, conditions (i) and (ii) in Definition 2.3 will always be satisfied by continuity of the metric and applying the Gram-Schmidt orthogonalization process appropriately. The heart of Definition 2.3 is condition (iii) and is the motivation for the terminology 'timelike convex near O .'</text>
<text><location><page_6><loc_18><loc_12><loc_79><loc_13></location>We are now ready to state our main theorem which generalizes [21, Thm. 2.2].</text>
<text><location><page_7><loc_15><loc_86><loc_85><loc_89></location>Theorem 2.4. Let ( M ext , g ext ) be a continuous spacetime extension of a C 2 spacetime ( M,g ) . Let u be a unit future directed timelike vector field on M . Assume the following.</text>
<unordered_list>
<list_item><location><page_7><loc_16><loc_82><loc_50><loc_85></location>(a) O ∈ M ext \ M is an origin point for u .</list_item>
<list_item><location><page_7><loc_16><loc_80><loc_81><loc_82></location>(b) The energy-momentum tensor T on M limits to a perfect fluid ( u, ρ 0 , p 0 ) at O .</list_item>
<list_item><location><page_7><loc_16><loc_77><loc_78><loc_79></location>(c) M is locally timelike convex about γ near O for some integral curve γ of u .</list_item>
<list_item><location><page_7><loc_16><loc_74><loc_68><loc_77></location>(d) The Ricci tensor Ric on M extends continuously to M ∪{O} .</list_item>
<list_item><location><page_7><loc_16><loc_72><loc_47><loc_74></location>(e) ( M ext , g ext ) is strongly causal at O .</list_item>
</unordered_list>
<text><location><page_7><loc_15><loc_70><loc_19><loc_72></location>Then</text>
<text><location><page_7><loc_15><loc_65><loc_61><loc_68></location>Moreover, the continuous extension of Ric at O is given by</text>
<formula><location><page_7><loc_46><loc_68><loc_54><loc_70></location>ρ 0 = -p 0 .</formula>
<text><location><page_7><loc_15><loc_54><loc_86><loc_66></location>˜ Ric | O = 8 πρ 0 g ext | O . Remark. Assumptions (a), (b), and (c) are Definitions 2.1, 2.2, and 2.3, respectively. Assumption (d) will be satisfied whenever ( M ext , g ext ) is a C 2 extension of ( M,g ), which is the case for the examples constructed in the next section. Assumption (e) is a technical assumption needed for the proof; it's satisfied, for example, whenever M ext is a subset of a globally hyperbolic spacetime.</text>
<text><location><page_7><loc_15><loc_50><loc_58><loc_52></location>Proof. Seeking a contradiction, assume ρ 0 = -p 0 . Then</text>
<text><location><page_7><loc_48><loc_50><loc_48><loc_52></location>/negationslash</text>
<text><location><page_7><loc_15><loc_30><loc_86><loc_42></location>By assumption (b), T perfect -T extends continuously to M ∪ {O} , and its continuous extension is zero at O . Also T extends continuously to M ∪ {O} by assumption (d). Therefore u ∗ ⊗ u ∗ extends continuously to M ∪{O} . As in the proof of [21, Thm. 2.2], this implies that the vector field u extends continuously to M ∪ {O} . However, assumptions (c) and (e) prove that u does not extend continuously. Heuristically, this can be seen in Figure 1 (right). Rigorously, this follows from an analogous contradiction argument used in the proof of [21, Thm. 2.2]. Thus we have ρ 0 = -p 0 .</text>
<formula><location><page_7><loc_33><loc_41><loc_68><loc_50></location>u ∗ ⊗ u ∗ = 1 ρ 0 + p 0 ( T perfect -p 0 g ) = 1 ρ 0 + p 0 ( ( T perfect -T ) + T -p 0 g ) .</formula>
<text><location><page_7><loc_18><loc_28><loc_73><loc_30></location>Next we prove that Ric | O = 8 πρ 0 g ext | O . The Einstein equations imply</text>
<formula><location><page_7><loc_32><loc_19><loc_69><loc_31></location>˜ Ric = 8 πT + 1 2 Rg = 8 π ( T -T perfect ) + 8 πT perfect + 1 2 Rg = 8 π ( T -T perfect ) + 8 πT perfect -4 π (tr T ) g.</formula>
<text><location><page_7><loc_15><loc_13><loc_86><loc_19></location>Since ρ 0 = -p 0 , we have T perfect = -ρ 0 g , and so T perfect extends continuously to M ∪ {O} . Also tr T extends continuously to M ∪{O} , and its continuous extension is -ρ 0 +3 p 0 = -4 ρ 0 at O . Therefore evaluating the above expression at O gives</text>
<formula><location><page_7><loc_29><loc_8><loc_71><loc_13></location>˜ Ric | O = 0 -8 πρ 0 g ext | O +16 πρ 0 g ext | O = 8 πρ 0 g ext | O .</formula>
<section_header_level_1><location><page_8><loc_15><loc_87><loc_68><loc_89></location>3 Anisotropic examples of the main theorem</section_header_level_1>
<text><location><page_8><loc_15><loc_64><loc_86><loc_86></location>In section 3.1 we construct explicit examples of spacetimes satisfying the hypotheses of Theorem 2.4. Clearly any Milne-like spacetime with a C 2 spacetime extension will satisfy the hypotheses of the theorem. But the goal of this section is to construct anisotropic examples as well, i.e., examples that are not FLRW spacetimes. (Recall Milne-like spacetimes are k = -1 FLRW spacetimes and hence are isotropic.) Briefly, to achieve this, we generalize Milne-like spacetimes in the following way: In spherical coordinates ( t, r, θ, ϕ ), the comoving observers in a Milne-like spacetime are parameterized by the curves t = µr for 1 < µ ≤ ∞ , see Figure 1. ( µ = ∞ corresponds to the comoving observer traveling along r = 0.) In our anisotropic examples, we stipulate that the comoving observers follow the trajectories t = µf ( r ), where f ( r ) ≈ r for r small. Like Milne-like spacetimes, the metric is still conformally flat and the conformal factor is a function of the foliation of the spacelike hypersurfaces orthogonal to the comoving observers, i.e., the conformal factor is a function of the rest spaces of u .</text>
<text><location><page_8><loc_15><loc_54><loc_86><loc_63></location>In section 3.2, we use the conclusion of Theorem 2.4 (that the Ricci tensor, and hence also the energy-momentum tensor, is dominated by a cosmological constant) to show that those comoving observers with µ -value greater than some critical number µ crit will experience inflationary eras, lending support to the inflationary scenario. Our analysis depends on investigating the terms in the Raychaudhuri equation as they approach the origin point O .</text>
<section_header_level_1><location><page_8><loc_15><loc_51><loc_33><loc_53></location>3.1 The examples</section_header_level_1>
<text><location><page_8><loc_15><loc_47><loc_86><loc_50></location>In this section we construct explicit examples of spacetimes satisfying the hypotheses of Theorem 2.4. Our examples will depend on only two functions f ( r ) and Φ( ζ ).</text>
<text><location><page_8><loc_15><loc_42><loc_86><loc_47></location>Let f ( r ) be a smooth positive function on [0 , ∞ ) satisfying f ( r ) = r + O ( r 3 ) as r → 0 and f ' ( r ) ≥ 1 for all r ≥ 0. 1 A simple example of such a function is f ( r ) = sinh( r ). Our manifold of interest is</text>
<formula><location><page_8><loc_28><loc_37><loc_85><loc_42></location>M := { ( t, x, y, z ) | t > f ( r ) , where r = √ x 2 + y 2 + z 2 } , (3.1)</formula>
<text><location><page_8><loc_15><loc_36><loc_35><loc_38></location>equipped with the metric</text>
<formula><location><page_8><loc_36><loc_33><loc_85><loc_35></location>g = e 2Φ( ζ ) [ -dt 2 + dx 2 + dy 2 + dz 2 ] (3.2)</formula>
<text><location><page_8><loc_15><loc_31><loc_74><loc_32></location>for some arbitrary smooth 2 function Φ( ζ ) on R . Here ζ = ζ ( t, r ) is given by</text>
<formula><location><page_8><loc_40><loc_25><loc_85><loc_30></location>ζ ( t, r ) = t 2 2 -∫ r 0 f ( s ) f ' ( s ) ds. (3.3)</formula>
<text><location><page_8><loc_15><loc_21><loc_86><loc_25></location>The spacetime extension ( M ext , g ext ) of ( M,g ) is simply defined by extending g to all R 4 ≈ M ext . In fact the metric is C 2 on M ext , which follows from the assumptions on f ( r ).</text>
<text><location><page_8><loc_15><loc_16><loc_86><loc_20></location>Remark. The simple case f ( r ) = r corresponds to (a subclass of) Milne-like spacetimes [20]. This follows since the conformal factor is a function of t 2 -r 2 .</text>
<figure>
<location><page_9><loc_15><loc_72><loc_51><loc_87></location>
<caption>M := { ( t, x, y, z ) | t > r } , g = e 2Φ( τ ( t,r )) η with τ ( t, r ) = t 2 2 -r 2 2 .</caption>
</figure>
<figure>
<location><page_9><loc_52><loc_72><loc_88><loc_87></location>
<caption>M := { ( t, x, y, z ) | t > f ( r ) } , g = e 2Φ( ζ ( t,r )) η with ζ ( t, r ) = t 2 2 -∫ r 0 f ( s ) f ' ( s ) ds.</caption>
</figure>
<paragraph><location><page_9><loc_15><loc_56><loc_86><loc_64></location>Figure 2: On the left is a Milne-like spacetime represented in conformal Minkowskian coordinates. On the right, the anisotropic examples constructed in this section. They are constructed to look like a Milne-like spacetime around the origin O . The comoving observers (i.e., the integral curves of u ) still emanate from the origin, and the manifold is still foliated by slices orthogonal to the comoving observers.</paragraph>
<text><location><page_9><loc_15><loc_50><loc_85><loc_53></location>The unit future directed timelike vector field u (whose integral curves are the comoving observers) will be given by normalized gradient of ζ :</text>
<formula><location><page_9><loc_28><loc_42><loc_85><loc_49></location>u := -∇ ζ |∇ ζ | g = e -Φ( ζ ) √ t 2 f ' ( r ) 2 -f ( r ) 2 ( t f ' ( r ) ∂ t + f ( r ) ∂ r ) . (3.4)</formula>
<formula><location><page_9><loc_42><loc_32><loc_85><loc_38></location>r ↦→ ( µf ( r ) , r, θ 0 , ϕ 0 ) (3.5)</formula>
<text><location><page_9><loc_15><loc_37><loc_86><loc_43></location>By construction the integral curves of u emanate from the origin O = (0 , 0 , 0 , 0) in ( t, x, y, z )-coordinates. Each integral curve of u follows the trajectory of the curve t = µf ( r ) for 1 < µ ≤ ∞ (with µ = ∞ corresponding to r = 0). To see this, recognize that the curve</text>
<text><location><page_9><loc_15><loc_28><loc_86><loc_33></location>in ( t, r, θ, ϕ )-coordinates has tangent vector parallel to u . By rewriting these curves in ( t, x, y, z )-coordinates, it's clear that they extend as C 1 curves through the origin O . Thus O is an origin point for u . Hence part (a) of Theorem 2.4 is verified.</text>
<text><location><page_9><loc_15><loc_22><loc_86><loc_28></location>Now we verify properties (b) through (e) of Theorem 2.4. Property (c) is evidently satisfied; simply consider the integral curve along the t -axis given by r = 0. Property (e) holds since M ext is conformal to Minkowski spacetime. Property (d) holds since the metric is C 2 on all of M ext .</text>
<text><location><page_9><loc_15><loc_15><loc_86><loc_21></location>The remainder of this section will be dedicated to proving property (b), namely, that the energy-momentum tensor converges to that of a perfect fluid. However, to gain control over the terms appearing in the energy-momentum tensor, we found it easier to work with the following subset of our original manifold:</text>
<formula><location><page_9><loc_25><loc_9><loc_85><loc_14></location>M ε := { ( t, x, y, z ) | t > (1 + ε ) f ( r ) , where r = √ x 2 + y 2 + z 2 } , (3.6)</formula>
<text><location><page_10><loc_15><loc_85><loc_86><loc_89></location>where ε > 0 is arbitrary. Note that M ε approaches M as ε → 0. Moreover, for any ε > 0, we see that M ε also satisfies properties (a) and (c) - (e) of Theorem 2.4.</text>
<text><location><page_10><loc_15><loc_82><loc_85><loc_85></location>Now we prove property (b) of Theorem 2.4 with M ε playing the role of M in statement of Theorem 2.4. And this will hold for any ε > 0. The following fact will be used.</text>
<text><location><page_10><loc_15><loc_79><loc_48><loc_80></location>Fact: We have the following bound on M ε :</text>
<text><location><page_10><loc_15><loc_71><loc_67><loc_72></location>Proof of fact. Since M ε is only defined for t > (1 + ε ) f ( r ), we have</text>
<formula><location><page_10><loc_38><loc_71><loc_85><loc_78></location>∣ ∣ ∣ ∣ tf ( r ) t 2 f ' ( r ) 2 -f ( r ) 2 ∣ ∣ ∣ ∣ < 1 + ε 2 ε + ε 2 . (3.7)</formula>
<formula><location><page_10><loc_27><loc_56><loc_74><loc_69></location>∣ ∣ ∣ ∣ tf ( r ) t 2 f ' ( r ) 2 -f ( r ) 2 ∣ ∣ ∣ ∣ = f ( r ) tf ' ( r ) 2 ( 1 + f ( r ) 2 t 2 f ' ( r ) 2 -f ( r ) 2 ) < 1 (1 + ε ) f ' ( r ) 2 ( 1 + 1 (1 + ε ) 2 f ' ( r ) 2 -1 ) ≤ 1 1 + ε ( 1 + 1 (1 + ε ) 2 -1 ) ,</formula>
<text><location><page_10><loc_15><loc_53><loc_65><loc_56></location>where we used the positivity of f ( r ) and the fact that f ' ( r ) ≥ 1.</text>
<text><location><page_10><loc_15><loc_49><loc_86><loc_52></location>We start by showing property (i) in Definition 2.2. From conformal geometry, the Ricci tensor is given by</text>
<formula><location><page_10><loc_29><loc_44><loc_85><loc_49></location>R αβ = -2(Hess Φ) αβ +2 ∇ α Φ ∇ β Φ -( /square Φ+2 | d Φ | 2 η ) η αβ . (3.8)</formula>
<text><location><page_10><loc_15><loc_41><loc_86><loc_45></location>Here, all operators on the right-hand side are taken with respect to the Minkowski metric η . Using (3.8), we have</text>
<formula><location><page_10><loc_27><loc_34><loc_74><loc_40></location>8 πT αβ = R αβ -1 2 Rg αβ = -2(Hess Φ) αβ +2 ∇ α Φ ∇ β Φ+ /square Φ η αβ +2 | d Φ | 2 η η αβ ,</formula>
<text><location><page_10><loc_15><loc_32><loc_43><loc_34></location>Straightforward computation shows</text>
<formula><location><page_10><loc_17><loc_24><loc_85><loc_30></location>ρ = T ( u, u ) = e -2Φ( ζ ) 8 π [( 2 + 4 f ( r ) rf ' ( r ) -2 t 2 f ( r ) f '' ( r ) t 2 f ' ( r ) 2 -f ( r ) 2 ) Φ ' ( ζ ) + 3( t 2 f ' ( r ) 2 -f ( r ) 2 ) f ' ( r ) 2 Φ ' ( ζ ) 2 ] . (3.9)</formula>
<text><location><page_10><loc_15><loc_17><loc_86><loc_23></location>We are interested in showing that ρ extends continuously to M ε ∪ {O} and finding its limit at the origin O . Both the first and second terms will contribute to ˜ ρ ( O ) since, for small r , we have:</text>
<formula><location><page_10><loc_41><loc_15><loc_59><loc_19></location>2 + 4 f ( r ) rf ' ( r ) = 6 + O ( r 2 ) .</formula>
<text><location><page_10><loc_15><loc_11><loc_85><loc_14></location>Additionally, by utilizing (3.7), we see that the third term vanishes at the origin O . Finally, the fourth term in (3.9) also vanishes since f ' ( r ) = 1 + O ( r 2 ).</text>
<text><location><page_11><loc_18><loc_87><loc_54><loc_89></location>Hence, ρ extends continously to the origin and</text>
<text><location><page_11><loc_15><loc_80><loc_77><loc_82></location>This shows property (i) in Definition 2.2. To show (ii), consider the vector field</text>
<formula><location><page_11><loc_41><loc_80><loc_85><loc_86></location>˜ ρ ( O ) = 3 4 π e -2Φ(0) Φ ' (0) . (3.10)</formula>
<formula><location><page_11><loc_33><loc_73><loc_68><loc_79></location>v = e -Φ( ζ ) √ t 2 f ' ( r ) 2 -f ( r ) 2 ( f ( r ) ∂ t + t f ' ( r ) ∂ r ) .</formula>
<text><location><page_11><loc_15><loc_69><loc_86><loc_74></location>By construction v is unit spacelike and orthogonal to u . Let e θ , e ϕ be the standard orthonormal vectors on the sphere so that { u, v, e θ , e ϕ } forms an orthonormal basis on M ε (modulo some spherical coordinate singularities). Straightforward computations show</text>
<formula><location><page_11><loc_16><loc_57><loc_64><loc_67></location>p v = T ( v, v ) = e -2Φ( ζ ) 8 π [( -2 -4 f ( r ) rf ' ( r ) -2 f ( r ) 3 f '' ( r ) f ' ( r ) 2 ( t 2 f ' ( r ) 2 -f ( r ) 2 ) ) Φ ' ( ζ ) -t 2 f ' ( r ) 2 -f ( r ) 2 f ' ( r ) 2 (2Φ '' ( ζ ) + Φ ' ( ζ ) 2 ) ] ,</formula>
<text><location><page_11><loc_15><loc_49><loc_23><loc_51></location>p e ϕ = p e θ .</text>
<formula><location><page_11><loc_16><loc_51><loc_85><loc_57></location>p e θ = T ( e θ , e θ ) = e -2Φ( ζ ) 8 π [( -4 -2 f ( r ) rf ' ( r ) + 2 f ( r ) f '' ( r ) f ' ( r ) 2 ) Φ ' ( ζ ) -t 2 f ' ( r ) 2 -f ( r ) 2 f ' ( r ) 2 (2Φ '' ( ζ ) + Φ ' ( ζ ) 2 ) ] ,</formula>
<text><location><page_11><loc_15><loc_46><loc_81><loc_48></location>Using (3.7) again, we see that these functions extend continuously to M ε ∪ {O} and</text>
<text><location><page_11><loc_15><loc_40><loc_85><loc_41></location>To finish our analysis, we need to compute all the cross terms of T . These cross terms are</text>
<formula><location><page_11><loc_31><loc_39><loc_69><loc_45></location>˜ p v ( O ) = ˜ p e θ ( O ) = ˜ p e ϕ ( O ) = -3 4 π e -2Φ(0) Φ ' (0) .</formula>
<formula><location><page_11><loc_28><loc_32><loc_73><loc_38></location>T ( u, v ) = -2 e -2Φ( ζ ) 8 π tf ( r ) 2 f '' ( r ) f ' ( r ) ( t 2 f ' ( r ) 2 -f ( r ) 2 ) Φ ' ( ζ ) , T ( u, e θ ) = T ( u, e ϕ ) = T ( v, e θ ) = T ( v, e ϕ ) = T ( e θ , e ϕ ) = 0 .</formula>
<text><location><page_11><loc_15><loc_29><loc_39><loc_31></location>Moreover, using (3.7), we have</text>
<formula><location><page_11><loc_45><loc_27><loc_85><loc_29></location>T ( u, v ) → 0 . (3.11)</formula>
<text><location><page_11><loc_15><loc_19><loc_86><loc_25></location>Let e 0 , e 1 , e 2 , e 3 denote u, v, e θ , e ϕ respectively. If e is any unit spacelike vector field orthogonal to u , then it can be written as e = ∑ 3 i =1 a i e i with ∑ 3 i =1 a 2 i = 1. Then vanishing of the cross terms implies</text>
<text><location><page_11><loc_15><loc_24><loc_38><loc_27></location>as we approach O within M ε .</text>
<text><location><page_11><loc_15><loc_14><loc_29><loc_15></location>and so in the limit</text>
<formula><location><page_11><loc_39><loc_15><loc_62><loc_20></location>p e = T ( e, e ) = 3 ∑ i =1 a 2 i T ( e i , e i ) ,</formula>
<formula><location><page_11><loc_40><loc_8><loc_85><loc_14></location>˜ p e ( O ) = -3 4 π e -2Φ(0) Φ ' (0) . (3.12)</formula>
<text><location><page_12><loc_15><loc_85><loc_86><loc_89></location>It is only left to show property (iii) in Definition 2.2, i.e., that T -T perfect extends continuously to M ε ∪ {O} and is zero at O . Recall that</text>
<formula><location><page_12><loc_37><loc_82><loc_63><loc_84></location>T perfect = ( ρ 0 + p 0 ) u ∗ ⊗ u ∗ + p 0 g,</formula>
<text><location><page_12><loc_15><loc_80><loc_52><loc_81></location>where ρ 0 and p 0 are given by (3.10) and (3.12).</text>
<text><location><page_12><loc_18><loc_77><loc_79><loc_79></location>We work in ( t, x, y, z )-coordinates as they clearly cover the origin O . We have</text>
<text><location><page_12><loc_15><loc_56><loc_19><loc_58></location>Then</text>
<formula><location><page_12><loc_20><loc_57><loc_80><loc_77></location>∂ t = e Φ( ζ ) √ t 2 f ' ( r ) 2 -f ( r ) 2 ( tf ' ( r ) u + f ( r ) v ) ∂ x = e Φ( ζ ) [ sin( θ ) cos( ϕ ) √ t 2 f ' ( r ) 2 -f ( r ) 2 ( tf ' ( r ) u + f ( r ) v ) +cos( θ ) cos( ϕ ) e θ -sin( ϕ ) e ϕ ] ∂ y = e Φ( ζ ) [ sin( θ ) sin( ϕ ) √ t 2 f ' ( r ) 2 -f ( r ) 2 ( tf ' ( r ) u + f ( r ) v ) +cos( θ ) sin( ϕ ) e θ +cos( ϕ ) e ϕ ] ∂ z = e Φ( ζ ) [ cos( θ ) √ t 2 f ' ( r ) 2 -f ( r ) 2 ( tf ' ( r ) u + f ( r ) v ) -sin( θ ) e θ ] .</formula>
<formula><location><page_12><loc_23><loc_50><loc_77><loc_55></location>T ( ∂ t , ∂ t ) = e 2Φ( ζ ) t 2 f ' ( r ) 2 -f ( r ) 2 [ t 2 f ' ( r ) 2 ρ +2 tf ( r ) f ' ( r ) T ( u, v ) + f ( r ) 2 p v ] .</formula>
<text><location><page_12><loc_15><loc_49><loc_30><loc_51></location>On the other hand,</text>
<formula><location><page_12><loc_21><loc_39><loc_80><loc_48></location>T perfect ( ∂ t , ∂ t ) = e 2Φ( ζ ) t 2 f ' ( r ) 2 -f ( r ) 2 [ ( ρ 0 + p 0 ) t 2 f ' ( r ) 2 + p 0 ( -t 2 f ' ( r ) 2 + f ( r ) 2 ) ] = e 2Φ( ζ ) t 2 f ' ( r ) 2 -f ( r ) 2 [ ρ 0 t 2 f ' ( r ) 2 + p 0 f ( r ) 2 ] .</formula>
<text><location><page_12><loc_15><loc_38><loc_22><loc_39></location>Therefore</text>
<formula><location><page_12><loc_22><loc_30><loc_79><loc_37></location>T ( ∂ t , ∂ t ) -T perfect ( ∂ t , ∂ t ) = e 2Φ( ζ ) t 2 f ' ( r ) 2 -f ( r ) 2 [ t 2 f ' ( r ) 2 ( ρ -¯ ρ ) + 2 tf ( r ) f ' ( r ) T ( u, v ) + f ( r ) 2 ( p v -¯ p ) ] .</formula>
<text><location><page_12><loc_18><loc_29><loc_59><loc_30></location>Combining (3.7), (3.10), (3.11), and (3.12), we obtain</text>
<formula><location><page_12><loc_38><loc_25><loc_62><loc_27></location>T ( ∂ t , ∂ t ) -T perfect ( ∂ t , ∂ t ) → 0 ,</formula>
<text><location><page_12><loc_15><loc_19><loc_86><loc_24></location>as we approach the origin O within M ε . In a similar manner, all other components of T -T perfect in ( t, x, y, z )-coordinates also converge to 0. Consequently, assumption (b) in Theorem 2.4 is satisfied as well. Hence the conclusions of the Theorem 2.4 hold:</text>
<formula><location><page_12><loc_26><loc_15><loc_85><loc_19></location>ρ 0 = -p 0 = 3 4 π e -2Φ(0) Φ ' (0) and Ric | O = 8 πρ 0 g | O . (3.13)</formula>
<text><location><page_12><loc_15><loc_11><loc_86><loc_14></location>Remark. We emphasize that we have applied Theorem 2.4 to the spacetime M ε and not M ; this is sufficient for the analysis in the next section.</text>
<section_header_level_1><location><page_13><loc_15><loc_87><loc_64><loc_89></location>3.2 Existence of inflationary eras in the examples</section_header_level_1>
<text><location><page_13><loc_15><loc_78><loc_86><loc_86></location>In this section we show how the conclusion of our main result, Theorem 2.4, proves the existence of inflationary eras for the examples constructed in section 3.1. A majority of the analysis in this section was outlined in section 3 of [21]. (At the time of writing [21], we had not yet found anisotropic examples of our main theorem which is a main inspiration for writing this paper.)</text>
<text><location><page_13><loc_15><loc_74><loc_86><loc_77></location>To gain some familiarity with the problem at hand, let's consider the FLRW setting. Friedmann's second equations is</text>
<text><location><page_13><loc_15><loc_67><loc_22><loc_68></location>Therefore</text>
<formula><location><page_13><loc_38><loc_68><loc_85><loc_73></location>3 a '' ( τ ) a ( τ ) = -4 π ( ρ ( τ ) + 3 p ( τ ) ) . (3.14)</formula>
<formula><location><page_13><loc_29><loc_64><loc_85><loc_67></location>ρ (0) = -p (0) > 0 = ⇒ a '' ( τ ) > 0 for τ near τ = 0 . (3.15)</formula>
<text><location><page_13><loc_15><loc_59><loc_86><loc_64></location>The assumption in (3.15) is what we mean by 'the cosmological constant appears as an initial condition.' It holds for a class of Milne-like spacetimes, see [21, eq. (1.11)]. In fact, our main result, Theorem 2.4, is essentially an anisotropoic generalization of this.</text>
<text><location><page_13><loc_15><loc_41><loc_86><loc_59></location>In this section, we generalize (3.15) to our anisotropic examples. Specifically what we demonstrate is the following. Let ( M,g ) be the spacetime defined by equations (3.1), (3.2), and (3.3). Let γ ( τ ) denote a comoving observer in M (i.e., γ is an integral curve of u ). Here τ is the proper time of the comoving observer, and we fix it so that γ (0) = O . We will define a 'generalized scale factor' a ( τ ) associated with γ ( τ ) and show that this generalized scale factor is accelerating, a '' ( τ ) > 0, for proper times τ near τ = 0 (i.e., near the big bang). However, we only prove that some comoving observers experience inflation. Recall that the comoving observers follow the trajectories t = µf ( r ), see (3.5). We find that only those comoving observers with a µ -value above a certain threshold µ crit experience an inflationary era. This threshold is given by (3.27); it's completely determined by the functions f ( r ) and Φ( ζ ) appearing in the previous section, and hence depends solely on the spacetime metric.</text>
<text><location><page_13><loc_15><loc_32><loc_86><loc_39></location>Remark. Throughout this section, we have in mind a fixed comoving observer. Since the comoving observers travel along the trajectories t = µf ( r ), we can assume any fixed comoving observer is contained in some M ε (see (3.6)) by choosing ε > 0 small enough. Therefore the bound (3.7) can be utilized.</text>
<text><location><page_13><loc_15><loc_20><loc_86><loc_30></location>Recall u is given by (3.4). By construction u is orthogonal to the spacelike hypersurfaces of constant ζ . In Figure 1 (right), one should image that the spacelike hypersurfaces τ = constant are replaced with ζ = constant. In the terminology of [26, p. 359], u is 'synchronizable,' but it is not necessarily 'proper time synchronizable.' The latter occurs if and only if u is geodesic which occurs if and only if f ( r ) = r , see eq. (3.24). (Recall f ( r ) = r corresponds to a Milne-like spacetime, and we know u is geodesic in this case.)</text>
<text><location><page_13><loc_15><loc_15><loc_86><loc_20></location>Set H = 1 3 div u so that H coincides with the mean curvature of the spacelike hypersurfaces orthogonal to u , i.e., H is one-third the trace of the second fundamental form K . 3 Let τ denote the proper time of the flow lines of u (i.e., the proper time of the comoving</text>
<figure>
<location><page_14><loc_36><loc_74><loc_64><loc_89></location>
<caption>Figure 3: We prove the existence of inflationary eras along a fixed comoving observer by computing the terms on the right-hand side of the Raychaudhuri equation (3.19) in a small neighborhood in M about the origin.</caption>
</figure>
<text><location><page_14><loc_15><loc_61><loc_86><loc_66></location>observers). If c ( r ) denotes the curve r ↦→ ( µf ( r ) , r, θ 0 , ϕ 0 ) along the trajectory t = µf ( r ), then the proper time τ is simply</text>
<text><location><page_14><loc_15><loc_55><loc_69><loc_56></location>When c ( r ) is reparameterized by τ , it yields a comoving observer γ ( τ</text>
<formula><location><page_14><loc_26><loc_56><loc_85><loc_61></location>τ ( r ) = ∫ r 0 √ -g ( c ' ( s ) , c ' ( s ) ) ds = ∫ r 0 e Φ( ζ ) √ µ 2 f ' ( s ) 2 -1 ds. (3.16)</formula>
<text><location><page_14><loc_18><loc_53><loc_80><loc_54></location>Along each comoving observer γ ( τ ), we define a generalized scale factor a ( τ ) by</text>
<text><location><page_14><loc_69><loc_54><loc_81><loc_56></location>). 4</text>
<formula><location><page_14><loc_47><loc_49><loc_85><loc_52></location>a ' a = H. (3.17)</formula>
<text><location><page_14><loc_15><loc_42><loc_86><loc_48></location>We have H ( τ ) ≈ 1 τ for τ small along each comoving observer γ ( τ ), see eq. (3.21) below. Since a ( τ ) = exp( ∫ τ τ 0 H ) for some arbitrary time τ 0 , it follows that</text>
<formula><location><page_14><loc_41><loc_40><loc_85><loc_43></location>a ( τ ) → 0 as τ → 0 (3.18)</formula>
<text><location><page_14><loc_15><loc_35><loc_86><loc_40></location>along each comoving observer. Recall that, for us, the big bang corresponds to the time when the scale factor limits to 0. Therefore (3.18) suggests that the origin point O represents the big bang in these models.</text>
<text><location><page_14><loc_15><loc_30><loc_86><loc_35></location>For FLRW spacetimes, Friedmann's second equation (3.14) is used to analyze the acceleration of the scale factor. In the anisotropic setting, the generalization of Friedmann's second equation is the Raychaudhuri equation [15, eq. (4.26)],</text>
<formula><location><page_14><loc_36><loc_25><loc_85><loc_29></location>3 a '' a = -Ric( u, u ) -2 σ 2 +div( ∇ u u ) . (3.19)</formula>
<text><location><page_14><loc_15><loc_23><loc_65><loc_25></location>(The vorticity term vanishes since u is hypersurface orthogonal.)</text>
<text><location><page_14><loc_15><loc_17><loc_86><loc_23></location>Our goal is to compute all the terms on the right-hand side of (3.19) for points along a comoving observer near O (see Figure 3). First, using the conclusions of Theorem 2.4 and eq. (3.13), sufficiently close to the origin O , we have</text>
<formula><location><page_15><loc_36><loc_85><loc_85><loc_87></location>-Ric( u, u ) ≈ 8 πρ 0 = 6 e -2Φ(0) Φ ' (0) . (3.20)</formula>
<text><location><page_15><loc_15><loc_81><loc_86><loc_85></location>The ≈ in the above expression is understood in the following way: -Ric( u, u ) can be made arbitrarily close to 8 πρ 0 by choosing points in M arbitrarily close to O .</text>
<text><location><page_15><loc_15><loc_77><loc_86><loc_82></location>The shear term is defined by 2 σ 2 = ∑ 3 i,j =1 σ ( e i , e j ) σ ( e i , e j ) where { e 1 , e 2 , e 3 } is an orthonormal basis spanning u ⊥ and</text>
<formula><location><page_15><loc_39><loc_74><loc_61><loc_77></location>σ ( e i , e j ) = K ( e i , e j ) -Hδ ij ,</formula>
<text><location><page_15><loc_15><loc_69><loc_86><loc_74></location>where K ( X,Y ) = g ( ∇ X u, Y ) is the second fundamental form of the hypersurfaces orthogonal to u . (Recall H = 1 3 tr K .) Choosing the orthonormal basis { v, e θ , e ϕ } from the previous section, the only nonvanishing terms for K are</text>
<text><location><page_15><loc_15><loc_58><loc_82><loc_69></location>K ( v, v ) = e -Φ( ζ ) √ t 2 f ' ( r ) 2 -f ( r ) 2 [ f ' ( r ) -t 2 f ( r ) f ' ( r ) f '' ( r ) t 2 f ' ( r ) 2 -f ( r ) 2 +Φ ' ( ζ ) t 2 f ' ( r ) 2 -f ( r ) 2 f ' ( r ) ] K ( e θ , e θ ) = K ( e ϕ e ϕ ) = e -Φ( ζ ) √ t 2 f ' ( r ) 2 -f ( r ) 2 [ f ( r ) r +Φ ' ( ζ ) t 2 f ' ( r ) 2 -f ( r ) 2 f ' ( r ) ] . Therefore the mean curvature H is</text>
<text><location><page_15><loc_15><loc_47><loc_85><loc_53></location>Along t = µf ( r ), we have H ∣ ∣ t = µf ( r ) = e -Φ(0) r √ µ 2 -1 + O ( r ) . Using (3.16), we reparameterize in terms of τ giving</text>
<formula><location><page_15><loc_16><loc_51><loc_84><loc_58></location>3 H = e -Φ( ζ ) √ t 2 f ' ( r ) 2 -f ( r ) 2 [ f ' ( r ) + 2 f ( r ) r -t 2 f ( r ) f ' ( r ) f '' ( r ) t 2 f ' ( r ) 2 -f ( r ) 2 +3Φ ' ( ζ ) t 2 f ' ( r ) 2 -f ( r ) 2 f ' ( r ) ] .</formula>
<formula><location><page_15><loc_43><loc_45><loc_85><loc_49></location>H | γ ( τ ) = 1 τ + o (1) . (3.21)</formula>
<text><location><page_15><loc_18><loc_44><loc_42><loc_45></location>Direct computation shows that</text>
<formula><location><page_15><loc_26><loc_38><loc_85><loc_43></location>2 σ 2 = 2 e -2Φ( ζ ) 3[ t 2 f ' ( r ) 2 -f ( r ) 2 ] [ f ' ( r ) -f ( r ) r -t 2 f ( r ) f ' ( r ) f '' ( r ) t 2 f ' ( r ) 2 -f ( r ) 2 ] 2 . (3.22)</formula>
<text><location><page_15><loc_15><loc_35><loc_74><loc_38></location>For small r , we have f ' ( r ) -f ( r ) r = O ( r 2 ) which combined with (3.7) yields</text>
<formula><location><page_15><loc_46><loc_33><loc_85><loc_35></location>2 σ 2 → 0 . (3.23)</formula>
<text><location><page_15><loc_15><loc_27><loc_86><loc_33></location>In other words 2 σ 2 extends continuously to M ε ∪ {O} and takes on the value 0 at O . Geometrically, this 'isotropization' effect is a consequence of the u -orthogonal hypersurfaces becoming more hyperbolic as we approach the origin O .</text>
<text><location><page_15><loc_18><loc_25><loc_63><loc_27></location>The last term in (3.19) to compute is div( ∇ u u ). We have</text>
<text><location><page_15><loc_15><loc_19><loc_20><loc_20></location>Hence</text>
<formula><location><page_15><loc_31><loc_20><loc_85><loc_25></location>∇ u u = -tf ( r ) 2 f '' ( r ) e -2Φ( ζ ) ( t 2 f ' ( r ) 2 -f ( r ) 2 ) 2 ( f ( r ) ∂ t + tf ' ( r ) ∂ r ) . (3.24)</formula>
<formula><location><page_15><loc_18><loc_10><loc_83><loc_19></location>div( ∇ u u ) = -f ( r ) e -2Φ( ζ ) ( t 2 f ' ( r ) 2 -f ( r ) 2 ) 2 [ 2 t 2 f ' ( r ) 2 f '' ( r ) + t 2 f ( r ) f '' ( r ) 2 + t 2 f ( r ) f ' ( r ) f ''' ( r ) + f ( r ) 2 f '' ( r ) + 2 t 2 f ( r ) f ' ( r ) f '' ( r ) r -4 t 4 f ( r ) f ' ( r ) 2 f '' ( r ) 2 t 2 f ' ( r ) 2 -f ( r ) 2 ] .</formula>
<text><location><page_16><loc_15><loc_86><loc_64><loc_89></location>Evaluating along t = µf ( r ) and taking the limit r → 0, we find</text>
<text><location><page_16><loc_15><loc_77><loc_86><loc_81></location>Using (3.20), (3.23), and (3.25), the Raychaudhuri equation (3.19), for points along the comoving observer sufficiently close to the origin O , becomes</text>
<formula><location><page_16><loc_28><loc_80><loc_85><loc_87></location>div( ∇ u u ) ∣ ∣ t = µf ( r ) = -e -2Φ(0) [ f ''' (0)(5 µ 2 +1) ( µ 2 -1) 2 + O ( r ) ] . (3.25)</formula>
<formula><location><page_16><loc_30><loc_70><loc_85><loc_78></location>3 a '' a ∣ ∣ ∣ ∣ t = µf ( r ) ≈ 6 e -2Φ(0) [ Φ ' (0) -1 6 f ''' (0)(5 µ 2 +1) ( µ 2 -1) 2 ] . (3.26)</formula>
<text><location><page_16><loc_15><loc_58><loc_86><loc_66></location>From (3.26) we can determine which comoving observers experience an inflationary era, i.e., which comoving observers experience a '' ( τ ) > 0 arbitrarily close to τ = 0. Assuming Φ ' (0) > 0 (which is equivalent to ˜ ρ ( O ) > 0), it's precisely those comoving observers with µ -values satisfying</text>
<text><location><page_16><loc_15><loc_66><loc_86><loc_72></location>Similar to (3.20), the ≈ symbol in the above expression is understood in the following way: 3( a '' / a ) can be made arbitrarily close to the right-hand side of (3.26) by choosing points along t = µf ( r ) that are sufficiently close to the origin O .</text>
<formula><location><page_16><loc_24><loc_54><loc_85><loc_60></location>µ > µ crit := √ 12Φ ' (0) + 5 f ''' (0) + √ 144Φ ' (0) f ''' (0) + 25 f ''' (0) 2 12Φ ' (0) . (3.27)</formula>
<text><location><page_16><loc_15><loc_50><loc_86><loc_53></location>Moreover, we see that if f ''' (0) = 0 and Φ ' (0) > 0, then all the comoving observers experience an inflationary era. This reproduces the results for Milne-like spacetimes, see (3.15).</text>
<section_header_level_1><location><page_16><loc_15><loc_47><loc_50><loc_48></location>3.3 Remarks on proving anisotropy</section_header_level_1>
<text><location><page_16><loc_15><loc_40><loc_86><loc_45></location>In this section we show that the examples constructed in section 3.1 are generally anisotropic. Although this is heuristically evident, a formal mathematical proof is not immediately clear.</text>
<text><location><page_16><loc_15><loc_34><loc_86><loc_40></location>First, the definition of an 'isotropic spacetime' is not consistent throughout the literature. See [2] and [28] and references therein. We will adopt the definition in [26, Ch. 12] since, as discussed in [2], this definition is the optimal one as it implies that the spacetime is isometric to a subset of an FLRW spacetime, see [26, Prop. 12.6] and [2, Thm. 2.1].</text>
<text><location><page_16><loc_15><loc_11><loc_86><loc_33></location>Therefore any spacetime that is not isometric to a subset of an FLRW model is anisotropic according to [26]. For the examples constructed in section 3.1, if f ( r ) = r then they are isometric to a subclass of Milne-like spacetimes which are a subclass of k = -1 FLRW models, and hence they are isotropic. Moreover, regardless of the form of f ( r ), if the conformal factor is identically 1, then the spacetime is isometric to a subset of Minkowski spacetime which is clearly isotropic. (This shows that it is not sufficient to simply recognize that the shear term (3.22) is nonzero. However, in this case, the vector field defining the comoving observers changes.) This suggests that if f ( r ) is not identically r and the conformal factor is not constant, then the resulting spacetime is not a subset of an FLRW spacetime and hence is anisotropic. We believe such a statement can be proven rigorously. However, in this section, we will content ourselves with the following algorithm: Pick functions f ( r ) and Φ( ζ ). The steps below show how to verify that the corresponding spacetime ( M,g ) from section 3.1 is anisotropic.</text>
<text><location><page_17><loc_15><loc_74><loc_86><loc_89></location>Seeking a contradiction, suppose ( M,g ) is in fact isometric to a subset of an FLRW spacetime. Since FLRW spacetimes satisfy the Einstein equations with a perfect fluid [26, Thm. 12.11], there is a unit future directed timelike vector field ˜ u on M such that T is a perfect fluid with respect to ˜ u . There exist functions a , b , c , d such that ˜ u = au + bv + ce θ + de ϕ , where { u, v, e θ , e ϕ } is the orthonormal frame constructed in section 3.1. Consider the unit spacelike vectors orthogonal to u</text>
<text><location><page_17><loc_15><loc_64><loc_86><loc_68></location>From section 3.1, we know how T acts on the orthonormal frame { u, v, e θ , e ϕ } , and so we know how T acts on { u, v, e θ , e ϕ } .</text>
<formula><location><page_17><loc_29><loc_67><loc_72><loc_76></location>˜ ˜ v = bu + av a 2 -b 2 , ˜ e θ = cu + ae θ a 2 -c 2 , ˜ e ϕ = du + ae ϕ a 2 -d 2 .</formula>
<text><location><page_17><loc_15><loc_61><loc_86><loc_67></location>˜ ˜ ˜ ˜ Fix a point p 0 ∈ M given by ( t 0 , r 0 , θ 0 , ϕ 0 ). At p 0 , the following equations set up an overdetermined system for ( a, b, c, d ) at p 0 .</text>
<formula><location><page_17><loc_37><loc_49><loc_63><loc_60></location>-a 2 + b 2 + c 2 + d 2 = -1 T ( ˜ u, ˜ v ) = 0 T ( ˜ u, ˜ e θ ) = 0 T ( ˜ u, ˜ e ϕ ) = 0 T ( v, v ) = T ( e θ , e θ ) = T ( e ϕ , e ϕ ) .</formula>
<text><location><page_17><loc_15><loc_39><loc_86><loc_52></location>˜ ˜ ˜ ˜ ˜ ˜ For most choices of f ( r ) and Φ( ζ ), this system does not have any solutions, giving a contradiction. However, even if there are solutions, one can still obtain a contradiction by other means, e.g., showing that the orthogonal subspace to ˜ u does not have constant sectional curvature. Lastly, we remark that the point p 0 must lie away from r = 0. Indeed, points along r = 0 will past the above tests. This is due to the spacetime being spherically symmetric and hence spatially isotropic precisely at points along r = 0.</text>
<text><location><page_17><loc_15><loc_31><loc_86><loc_38></location>We remark that if Φ is constant or f ( r ) = r , then ( M,g ) passes the above tests. In the first case, the metric is homothetic to the Minkowski metric (and hence isometric to a subset of a k = 0 FLRW spacetime), and in the second case, the spacetime is given by a Milne-like spacetime and hence is isometric to a k = -1 FLRW spacetime.</text>
<section_header_level_1><location><page_17><loc_15><loc_28><loc_45><loc_29></location>4 Summary and outlook</section_header_level_1>
<text><location><page_17><loc_15><loc_12><loc_86><loc_26></location>The inflationary scenario has become the current paradigm of early universe cosmology. Roughly, it states that scale factor underwent a brief but dramatic period of acceleration after the big bang but before the radiation dominated era. Although inflationary theory has many successes (e.g., solutions to the horizon and flatness problems along with providing a framework for generating the seeds of large-scale structures in our universe), it does not carry the status of an established physical theory. In this work, we provide mathematical support for the inflationary scenario by showing that a class of anisotropic spacetimes experience inflationary eras after the big bang.</text>
<text><location><page_18><loc_15><loc_78><loc_86><loc_89></location>Our main result, Theorem 2.4, says that if the universe began with special initial conditions at the big bang, then the energy-momentum tensor was dominated by a cosmological constant at the big bang. These special initial conditions are (1) the existence of an origin point O for a unit timelike vector field u (whose integral curves represent the comoving observers in the universe) and (2) the energy-momentum tensor approaches a perfect fluid at O . An informal discussion of these special initial conditions is given in section 1.3.</text>
<text><location><page_18><loc_15><loc_67><loc_86><loc_78></location>In section 3.1, we construct anisotropic spacetimes which satisfy the hypotheses of Theorem 2.4. These examples can be thought of as 'quasi Milne-like spacetimes.' In section 3.2, we define a generalized scale factor a ( τ ) along each comoving observer ( τ denotes the proper time of the comoving observer), and we show that a ( τ ) → 0 as τ → 0, see (3.18). Consequently, we associate τ = 0 (and hence also the origin point O ) with the big bang. Lastly, we describe which comoving observers experience inflation, a '' ( τ ) > 0, immediately after the big bang τ = 0. See equations (3.26) and (3.27).</text>
<text><location><page_18><loc_15><loc_58><loc_86><loc_66></location>Our examples exhibit isotropization towards the past and, in fact, are perfectly isotropic at the big bang O , see (3.23). Our isotropization-towards-the-past result is consistent with a universe starting from special initial conditions. This is unlike results related to the cosmic no-hair conjecture (see Wald's original paper [30] or some more recent work, e.g., [3]), where isotropization occurs towards the future.</text>
<text><location><page_18><loc_15><loc_51><loc_86><loc_58></location>A limitation of our approach is that we only show accelerated expansion immediately after the big bang. For example reheating does not appear in our analysis. For an analysis of the physics after the accelerated expansion, our geometrical initial conditions should be supplemented with, for example, appropriate scalar field matter models.</text>
<text><location><page_18><loc_15><loc_33><loc_86><loc_51></location>We believe that differential geometry (and geometric analysis in particular) has a role to play in the investigation of initial conditions for the big bang. The work presented in this paper should be thought of as a 'proof of concept' of this proposal. Our work can be generalized in many ways. In particular, although our examples are not necessarily isotropic, they are still spherically symmetric. So a natural generalization is to reproduce the analysis in sections 3.1 and 3.2 with non-spherically symmetric spacetimes. Also, our examples are anisotropic versions of k = -1 FLRW spacetimes. What about k = 0 FLRW spacetimes? In this case one would want to apply [12, Thm. 5.2] or a suitable generalization thereof. Lastly, it remains to be seen if the results in [11] can be used to generate comoving observers with an origin point O .</text>
<section_header_level_1><location><page_18><loc_15><loc_30><loc_36><loc_31></location>Acknowledgments</section_header_level_1>
<text><location><page_18><loc_15><loc_20><loc_86><loc_28></location>Eric Ling was supported by Carlsberg Foundation CF21-0680 and Danmarks Grundforskningsfond CPH-GEOTOP-DNRF151. Annachiara Piubello was supported by the DFG Project ME 3816/3-1, part of the SPP2026. We thank Jerome Quintin for helpful comments on an earlier draft and are grateful to the Minkowski institute where this project began to take shape.</text>
<section_header_level_1><location><page_19><loc_15><loc_87><loc_27><loc_89></location>References</section_header_level_1>
<unordered_list>
<list_item><location><page_19><loc_16><loc_82><loc_85><loc_86></location>[1] Andreas Albrecht, Cosmic inflation and the arrow of time , 2003, arXiv:astro-ph/0210527v3.</list_item>
<list_item><location><page_19><loc_16><loc_78><loc_86><loc_81></location>[2] Rodrigo Avalos, On the rigidity of cosmological space-times , Letters of Math Phys. 113 (2023), no. 98.</list_item>
<list_item><location><page_19><loc_16><loc_74><loc_86><loc_77></location>[3] Ferez Azhar and David I. Kaiser, Flows into de Sitter space from anisotropic initial conditions: An effective field theory approach , Phys. Rev. D 107 (2023), no. 4, 043506.</list_item>
<list_item><location><page_19><loc_16><loc_70><loc_86><loc_73></location>[4] Robert Brandenberger, Initial conditions for inflation - A short review , Int. J. Mod. Phys. D 26 (2016), no. 01, 1740002.</list_item>
<list_item><location><page_19><loc_16><loc_66><loc_86><loc_69></location>[5] Robert Brandenberger and Patrick Peter, Bouncing Cosmologies: Progress and Problems , Found. Phys. 47 (2017), no. 6, 797-850.</list_item>
<list_item><location><page_19><loc_16><loc_62><loc_85><loc_65></location>[6] Sean M. Carroll and Jennifer Chen, Spontaneous inflation and the origin of the arrow of time , 2004, arXiv:hep-th/0410270.</list_item>
<list_item><location><page_19><loc_16><loc_58><loc_85><loc_61></location>[7] Sidney Coleman and Frank De Luccia, Gravitational effects on and of vacuum decay , Phys. Rev. D 21 (1980), 3305-3315.</list_item>
<list_item><location><page_19><loc_16><loc_54><loc_86><loc_57></location>[8] Planck Collaboration, Planck 2018 results. X. Constraints on inflation , Astron. Astrophys. 641 (2020), A10.</list_item>
<list_item><location><page_19><loc_16><loc_50><loc_85><loc_53></location>[9] George F.R. Ellis, Relativistic cosmology , Proc. Int. Sch. Phys. Fermi 47 (1971), 104182.</list_item>
<list_item><location><page_19><loc_15><loc_45><loc_85><loc_49></location>[10] Gregory Galloway and Eric Ling, Some Remarks on the C 0 -(in)extendibility of Spacetimes , Annales Henri Poincare 18 (2017), no. 10, 3427-3447.</list_item>
<list_item><location><page_19><loc_15><loc_41><loc_85><loc_44></location>[11] Ya Gao and Jing Mao, Inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space R n +1 1 , 2021, arXiv:2104.10600v5.</list_item>
<list_item><location><page_19><loc_15><loc_36><loc_86><loc_40></location>[12] Ghazal Geshnizjani, Eric Ling, and Jerome Quintin, On the initial singularity and extendibility of flat quasi-de Sitter spacetimes , Journal of High Energy Physics 2023 (2023), no. 10.</list_item>
<list_item><location><page_19><loc_15><loc_31><loc_86><loc_34></location>[13] Alan H. Guth, The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems , Phys. Rev. D 23 (1981), 347-356.</list_item>
<list_item><location><page_19><loc_15><loc_27><loc_86><loc_30></location>[14] Alan H. Guth, David I. Kaiser, and Yasunori Nomura, Inflationary paradigm after planck 2013 , Physics Letters B 733 (2014), 112-119.</list_item>
<list_item><location><page_19><loc_15><loc_23><loc_86><loc_26></location>[15] Stephen W. Hawking and George F. R. Ellis, The Large Scale Structure of Space-Time , Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2023.</list_item>
<list_item><location><page_19><loc_15><loc_19><loc_86><loc_22></location>[16] Anna Ijjas, Paul J. Steinhardt, and Abraham Loeb, Inflationary paradigm in trouble after Planck2013 , Phys. Lett. B 723 (2013), 261-266.</list_item>
<list_item><location><page_19><loc_15><loc_15><loc_86><loc_18></location>[17] Andrei Linde, On the problem of initial conditions for inflation , Foundations of Physics 48 (2018), no. 10, 1246-1260.</list_item>
</unordered_list>
<table>
<location><page_20><loc_15><loc_34><loc_86><loc_89></location>
</table>
</document>
[
{
"title": "Anisotropic examples of inflation-generating initial conditions for the big bang",
"content": "Eric Ling ∗ 1 and Annachiara Piubello † 2 1 Copenhagen Centre for Geometry and Topology (GeoTop), 2 Department of Mathematical Sciences, University of Copenhagen, Denmark Institute of Mathematics, University of Potsdam, Germany.",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "The inflationary scenario, which states that the early universe underwent a brief but dramatic period of accelerated spatial expansion, has become the current paradigm of early universe cosmology. Although inflationary cosmology has its many successes, it does not (as of yet) have the status of an established physical theory. In this paper, we provide mathematical support for the inflationary scenario in a class of anisotropic spacetimes by generalizing the work in [21]. These anisotropic spacetimes satisfy certain initial conditions so that they are perfectly isotropic at the big bang but become less isotropic as time progresses. The resulting inflationary eras are a consequence of the initial conditions which force the energy-momentum tensor to be dominated by a cosmological constant at the big bang.",
"pages": [
1
]
},
{
"title": "1.1 Cosmic inflation",
"content": "The inflationary scenario has become the current paradigm of early universe cosmology. Roughly, it states the following. The inflationary scenario: In the early universe, before the radiation-dominated era, there was a brief but dramatic period of accelerated spatial expansion. The inflationary scenario was proposed in the late 1970s and early 80s [13,18,29] as a solution to some problems in the standard big bang model, e.g., the flatness and horizon problems. It was soon realized that inflation can provide a framework for generating the seeds of the large-scale structures in our universe [24]. Observations of the anisotropies in the CMB radiation performed by COBE, WMAP, and most recently by Planck [8] support these claims. Given the many successes of the inflationary scenario, it is perhaps not too surprising that most papers on early universe cosmology give the impression that inflation has been firmly established and observationally proven. However there are many inflationary models that can be in agreement with observation [23]. In fact, any theory which predicts an almost flat universe with a nearly scale-invariant curvature power spectrum, small tensor-to-scalar ratio, and small Gaussian fluctuations would be in agreement with current data, e.g., [5]. Moreover, although phenomenologically successful, current realizations of inflationary models suffer from conceptual problems, perhaps none more so than the problem of initial conditions [4, 17]. In fact there are conflicting opinions on the naturalness of initial conditions for inflation [14,16]. Most papers on initial conditions for inflation begin in an inhomogeneous universe with an energy-momentum tensor dominated by an inflaton scalar field in a slow-roll potential and see if the resulting dynamics can produce an inflationary era followed by a homogeneous universe. This is not our approach. Our approach is purely geometrical. Quantities of interest are described solely in terms of a special unit timelike vector field u (whose integral curves represent the comoving observers in the universe) and the spacetime metric g . In this paper we provide mathematical support for the inflationary scenario. In section 3, we show that inflation arises for a class of anisotropic spacetimes from special geometrical initial conditions. Our initial conditions are stated informally in section 1.3 and formally in section 2. These anisotropic spacetimes are examples of our main result, Theorem 2.4, which concludes from the special initial conditions that the Ricci tensor (and hence also the energy-momentum tensor) is dominated by a cosmological constant at the big bang. Theorem 2.4 is a generalization of the main result in [21]. In fact, a major inspiration for this paper was to find anisotropic examples of the main result in [21]. The benefit of our geometrical approach is its conceptual clarity: we will describe precisely which comoving observers experience inflation and how fast they are accelerating solely in terms of the unit timelike vector field u and spacetime metric g . Our geometrical initial conditions can be thought of as a certain type of fine-tuning condition for the big bang. As briefly reviewed in the next section, a Boltzmannian viewpoint on the arrow of time suggests that some type of fine-tuning initial condition for the big bang should exist.",
"pages": [
2
]
},
{
"title": "1.2 Inflation and the arrow of time",
"content": "An obvious feature of our universe is the existence of an arrow of time. We observe certain processes in our everyday experience, but we hardly ever observe those same processes time-reversed. A vase shatters into a multitude of pieces, but we never observe these pieces spontaneously arranging themselves perfectly together into a vase. The second law of thermodynamics is postulated to explain the arrow of time, and a modern Boltzmannian mindset of the second law leads to the conclusion that the universe began with special, non-generic, fine-tuned initial conditions. It was Penrose who originally argued [27] that the overall arrow of time we observe is linked to special initial conditions for the universe that are drastically far away from the dynamical trend towards gravitational collapse. He calculated that the entropy of the radiation-dominated early universe is around 30 orders of magnitude smaller than the Beckenstein-Hawking entropy of its corresponding black hole state. See also [1,6]. With this understanding, the homogeneous and isotropic assumptions of the standard FLRW models of cosmology are a reasonable choice of initial conditions as they match exceedingly well with current observations. However, some inflationary cosmologists instinctively take a different perspective. They seek an explanation for the large-scale isotropy of the universe from dynamical processes during inflation. But special initial conditions - by their nature - go against dynamical trends. That is, the creation of special initial conditions from dynamics beginning with generic initial conditions seems contradictory. Summarizing, some inflationary cosmologists seek generic initial conditions for the universe, but those who adopt a Boltzmannian point of view of the second law of thermodynamics (as we do) argue that initial conditions should instead be special in order to explain the arrow of time. While inflationary theory alone may not suffice to explain the large-scale isotropy of the universe, it still has many successes and remains the prevailing paradigm in early universe cosmology. The simplest way to generate inflation is to introduce an inflaton scalar field in a slow-roll potential - a methodology that is somewhat adhoc since it simply postulates the existence of a scalar field for which we have no direct evidence for. So a natural inquiry is to ask if there is other evidence to support the inflationary scenario. A primary motivation of this paper is to demonstrate that there is mathematical evidence in support of the inflationary scenario. We will see that inflation is inevitable provided certain geometrical initial conditions are assumed at the big bang, and, as discussed in this section, some degree of fine-tuning in the initial conditions is anticipated.",
"pages": [
3
]
},
{
"title": "1.3 Geometrical initial conditions for the big bang",
"content": "In this section we describe, informally, the primary geometrical initial conditions we will be considering in our main result, Theorem 2.4. These initial conditions are supposed to mimic - without assuming isotropy - the geometrical properties at the big bang. Let's first clarify what we mean by the 'big bang' as there are conflicting view points in the literature. For us the big bang refers to a time when the scale factor limits to zero. For example, if the scale factor is a ( τ ) = τ (as in the Milne model), then the big bang corresponds to τ = 0. If the scale factor is a ( τ ) = e τ (as in the flat de Sitter model), then the big bang corresponds to τ = -∞ . To motivate the type of geometrical initial conditions we will be considering, we focus on scale factor perturbations of the Milne model, which have been dubbed 'Milne-like spacetimes' in [10]. These models were extensively studied in [20], detailing possible applications to fundamental problems in cosmology. See also [7,22,25]. They are k = -1 FLRW spacetimes whose scale factor satisfies a ( τ ) ≈ τ for τ near τ = 0. (An inflating example would be a ( τ ) = sinh( τ ).) Interestingly, for Milne-like spacetimes, the big bang appears as a coordinate singularity, and so they extend into a larger spacetime. - g = Ω 2 ( τ )[ - dt t O 2 + dx 2 + dy 2 + dz 2 ] Recall that the comoving observers are the integral curves of the vector field u given by u = ∂ τ in comoving coordinates. As illustrated in Figure 1, the comoving observers for a Milne-like spacetime all emanate from a single point O in the extended spacetime, which is just the origin (0 , 0 , 0 , 0) in the conformal Minkowskian coordinates ( t, x, y, z ). We refer to this property as ' O being an origin point for u ,' see Definition 2.1. The existence of an origin point O for u is a highly fine-tuned and non-generic assumption. Recall that some fine-tuning is to be expected from the discussion on the arrow of time in section 1.2. An origin point O for u is the first main assumption in Theorem 2.4. The other main assumption is that the energy-momentum tensor T approaches that of a perfect fluid at O . See Definition 2.2. This assumption is more physically convincing than assuming that T is exactly a perfect fluid (as in the FLRW models) since we expect small deviations from perfect isotropy in our universe. Therefore the perspective taken here is that the universe began in a state of perfect isotropy at the big bang. This is the crux of Definition 2.2. Moreover, this perspective is reinforced in our examples since the shear vanishes towards the big bang, see eq. (3.23). An 'origin point O for u ' and ' T approaching a perfect fluid at O ' are the two primary assumptions in Theorem 2.4. There are three other assumptions that are purely technical. The conclusion of Theorem 2.4 is that the energy-momentum tensor is precisely given by a cosmological constant at O . This fact will be used in section 3.2 to prove the existence of inflationary eras in our anisotropic examples. τ = constant x i",
"pages": [
3,
4
]
},
{
"title": "2 The main theorem",
"content": "The initial conditions stated informally in section 1.3 will be stated formally in this section. Our main result, Theorem 2.4, is a generalization of the main result (Theorem 2.2) in [21]. Anisotropic examples of our main theorem are provided in section 3.1, and we prove the existence of inflationary eras for these examples in section 3.2. We set our conventions. Our definition of a spacetime ( M,g ) will follow [19]. (Except that, for simplicity, we will assume that all spacetimes are four-dimensional.) The manifold M is always assumed to be smooth. A C k spacetime is one where the metric g is C k , that is, its components g µν = g ( ∂ µ , ∂ ν ) are C k functions with respect to any coordinates ( x 0 , . . . , x 3 ). A continuous spacetime is one where the metric is continuous, that is, its components are continuous functions with respect to any coordinates. Our definitions of timelike curves and the timelike future I + will also follow [19]. Let ( M,g ) be a C k spacetime. A C 0 spacetime ( M ext , g ext ) is said to be a continuous spacetime extension of ( M,g ) provided there is an isometric embedding preserving time orientations such that M ⊂ M ext is a proper subset. ( M is in fact an open submanifold of M ext since they are both four-dimensional.) Note that we are identifying M with its image under the embedding. We remark that g ext is C 2 in the examples constructed in the next section. Definition 2.1 (Origin point) . Let ( M ext , g ext ) be a continuous spacetime extension of a C k spacetime ( M,g ). Let u be a unit future directed timelike vector field on M . We say that a point O is an origin point for u if O ∈ M ext \\ M and O is a past endpoint for each integral curve of u , and each extended integral curve is C 1 at O . (Clearly this implies O lies in the closure M within M ext .) In other words, O is an origin point for u if each integral curve of u , parameterized as γ : (0 , b ) → M , satisfies ˜ Remarks. Definition 2.1 is supposed to model the behavior of the comoving observers in Figure 1 (right). It is essentially the same as assumption (b) in [21, Thm. 2.2]. Actually, Definition 2.1 is slightly stronger; we assume this stronger assumption since it's easier to state and all the examples in section 3 will satisfy it. where ˜ γ : [0 , b ) → M ext is the extended curve defined by ˜ γ (0) = O and ˜ γ ( τ ) = γ ( τ ) for τ > 0. Continuity of the metric implies γ ' (0) is a unit future directed timelike vector. We recall some terminology from section 2 of [21]. Let O ∈ M ext \\ M be an origin point for u . A C k function f : M → R extends continuously to M ∪ {O} if there is a continuous function ˜ f : M ∪ {O} → R such that ˜ f | M = f . In this case, we call ˜ f the continuous extension of f . A C k tensor T defined on M extends continuously to M ∪ {O} if there is a coordinate neighborhood U of O with coordinates ( x 0 , . . . , x 3 ) such that each of the components of T extends continuously to ( U ∩ M ) ∪{O} . (This definition does not depend on the choice of coordinate system by the usual transformation law for tensor components.) This defines a continuous tensor ˜ T on M ∪{O} , called the continuous extension of T , which satisfies ˜ T | M = T . For example, the metric tensor g extends continuously to M ∪ {O} (by definition of a continuous extension). Trivially, if T is a smooth tensor defined on all of M ext , then clearly T | M extends continuously to M ∪ {O} . Definition 2.2 (Limiting to a perfect fluid near O ) . Let ( M,g ) be a C 2 spacetime, and let O ∈ M ext \\ M be an origin point for u . Let T be the energy-momentum tensor on M (i.e., T = 1 8 π G in suitable units where G = Ric -1 2 Rg is the Einstein tensor). Let ρ 0 , p 0 ∈ R . We say that T limits to a perfect fluid ( u, ρ 0 , p 0 ) at O if where u ∗ = g ( u, · ) is the one-form metrically equivalent to u . Remark. Definition 2.2 relaxes the requirement that T is identically a perfect fluid in assumption (a) of [21, Thm. 2.2]. Moreover, it's more physically convincing: FLRW models have perfect fluid energy-momentum tensors, and we expect that an FLRW model approximates our universe better as we go back in time towards the big bang. Lastly, we require a mild, technical timelike convexity assumption: Definition 2.3 (Locally timelike convex near O ) . Let O ∈ M ext \\ M be an origin point for u . Let γ : (0 , b ) → M be an integral curve of u . We say M is locally timelike convex about γ near O if there is an ε > 0 and a coordinate neighborhood U ⊂ M ext centered at O with coordinates ( x 0 , . . . , x 3 ) satisfying where η ε is the narrow Minkowskian metric given by η ε = -ε 2 -ε ( dx 0 ) 2 + δ ij dx i dx j . Remarks. In [21, Thm. 2.2], it was assumed that the manifold M satisfies M = I + ( O , M ext ). Definition 2.3 relaxes this requirement and is a much weaker assumption. It will hold for the examples constructed in section 3. Also, conditions (i) and (ii) in Definition 2.3 will always be satisfied by continuity of the metric and applying the Gram-Schmidt orthogonalization process appropriately. The heart of Definition 2.3 is condition (iii) and is the motivation for the terminology 'timelike convex near O .' We are now ready to state our main theorem which generalizes [21, Thm. 2.2]. Theorem 2.4. Let ( M ext , g ext ) be a continuous spacetime extension of a C 2 spacetime ( M,g ) . Let u be a unit future directed timelike vector field on M . Assume the following. Then Moreover, the continuous extension of Ric at O is given by ˜ Ric | O = 8 πρ 0 g ext | O . Remark. Assumptions (a), (b), and (c) are Definitions 2.1, 2.2, and 2.3, respectively. Assumption (d) will be satisfied whenever ( M ext , g ext ) is a C 2 extension of ( M,g ), which is the case for the examples constructed in the next section. Assumption (e) is a technical assumption needed for the proof; it's satisfied, for example, whenever M ext is a subset of a globally hyperbolic spacetime. Proof. Seeking a contradiction, assume ρ 0 = -p 0 . Then /negationslash By assumption (b), T perfect -T extends continuously to M ∪ {O} , and its continuous extension is zero at O . Also T extends continuously to M ∪ {O} by assumption (d). Therefore u ∗ ⊗ u ∗ extends continuously to M ∪{O} . As in the proof of [21, Thm. 2.2], this implies that the vector field u extends continuously to M ∪ {O} . However, assumptions (c) and (e) prove that u does not extend continuously. Heuristically, this can be seen in Figure 1 (right). Rigorously, this follows from an analogous contradiction argument used in the proof of [21, Thm. 2.2]. Thus we have ρ 0 = -p 0 . Next we prove that Ric | O = 8 πρ 0 g ext | O . The Einstein equations imply Since ρ 0 = -p 0 , we have T perfect = -ρ 0 g , and so T perfect extends continuously to M ∪ {O} . Also tr T extends continuously to M ∪{O} , and its continuous extension is -ρ 0 +3 p 0 = -4 ρ 0 at O . Therefore evaluating the above expression at O gives",
"pages": [
5,
6,
7
]
},
{
"title": "3 Anisotropic examples of the main theorem",
"content": "In section 3.1 we construct explicit examples of spacetimes satisfying the hypotheses of Theorem 2.4. Clearly any Milne-like spacetime with a C 2 spacetime extension will satisfy the hypotheses of the theorem. But the goal of this section is to construct anisotropic examples as well, i.e., examples that are not FLRW spacetimes. (Recall Milne-like spacetimes are k = -1 FLRW spacetimes and hence are isotropic.) Briefly, to achieve this, we generalize Milne-like spacetimes in the following way: In spherical coordinates ( t, r, θ, ϕ ), the comoving observers in a Milne-like spacetime are parameterized by the curves t = µr for 1 < µ ≤ ∞ , see Figure 1. ( µ = ∞ corresponds to the comoving observer traveling along r = 0.) In our anisotropic examples, we stipulate that the comoving observers follow the trajectories t = µf ( r ), where f ( r ) ≈ r for r small. Like Milne-like spacetimes, the metric is still conformally flat and the conformal factor is a function of the foliation of the spacelike hypersurfaces orthogonal to the comoving observers, i.e., the conformal factor is a function of the rest spaces of u . In section 3.2, we use the conclusion of Theorem 2.4 (that the Ricci tensor, and hence also the energy-momentum tensor, is dominated by a cosmological constant) to show that those comoving observers with µ -value greater than some critical number µ crit will experience inflationary eras, lending support to the inflationary scenario. Our analysis depends on investigating the terms in the Raychaudhuri equation as they approach the origin point O .",
"pages": [
8
]
},
{
"title": "3.1 The examples",
"content": "In this section we construct explicit examples of spacetimes satisfying the hypotheses of Theorem 2.4. Our examples will depend on only two functions f ( r ) and Φ( ζ ). Let f ( r ) be a smooth positive function on [0 , ∞ ) satisfying f ( r ) = r + O ( r 3 ) as r → 0 and f ' ( r ) ≥ 1 for all r ≥ 0. 1 A simple example of such a function is f ( r ) = sinh( r ). Our manifold of interest is equipped with the metric for some arbitrary smooth 2 function Φ( ζ ) on R . Here ζ = ζ ( t, r ) is given by The spacetime extension ( M ext , g ext ) of ( M,g ) is simply defined by extending g to all R 4 ≈ M ext . In fact the metric is C 2 on M ext , which follows from the assumptions on f ( r ). Remark. The simple case f ( r ) = r corresponds to (a subclass of) Milne-like spacetimes [20]. This follows since the conformal factor is a function of t 2 -r 2 . The unit future directed timelike vector field u (whose integral curves are the comoving observers) will be given by normalized gradient of ζ : By construction the integral curves of u emanate from the origin O = (0 , 0 , 0 , 0) in ( t, x, y, z )-coordinates. Each integral curve of u follows the trajectory of the curve t = µf ( r ) for 1 < µ ≤ ∞ (with µ = ∞ corresponding to r = 0). To see this, recognize that the curve in ( t, r, θ, ϕ )-coordinates has tangent vector parallel to u . By rewriting these curves in ( t, x, y, z )-coordinates, it's clear that they extend as C 1 curves through the origin O . Thus O is an origin point for u . Hence part (a) of Theorem 2.4 is verified. Now we verify properties (b) through (e) of Theorem 2.4. Property (c) is evidently satisfied; simply consider the integral curve along the t -axis given by r = 0. Property (e) holds since M ext is conformal to Minkowski spacetime. Property (d) holds since the metric is C 2 on all of M ext . The remainder of this section will be dedicated to proving property (b), namely, that the energy-momentum tensor converges to that of a perfect fluid. However, to gain control over the terms appearing in the energy-momentum tensor, we found it easier to work with the following subset of our original manifold: where ε > 0 is arbitrary. Note that M ε approaches M as ε → 0. Moreover, for any ε > 0, we see that M ε also satisfies properties (a) and (c) - (e) of Theorem 2.4. Now we prove property (b) of Theorem 2.4 with M ε playing the role of M in statement of Theorem 2.4. And this will hold for any ε > 0. The following fact will be used. Fact: We have the following bound on M ε : Proof of fact. Since M ε is only defined for t > (1 + ε ) f ( r ), we have where we used the positivity of f ( r ) and the fact that f ' ( r ) ≥ 1. We start by showing property (i) in Definition 2.2. From conformal geometry, the Ricci tensor is given by Here, all operators on the right-hand side are taken with respect to the Minkowski metric η . Using (3.8), we have Straightforward computation shows We are interested in showing that ρ extends continuously to M ε ∪ {O} and finding its limit at the origin O . Both the first and second terms will contribute to ˜ ρ ( O ) since, for small r , we have: Additionally, by utilizing (3.7), we see that the third term vanishes at the origin O . Finally, the fourth term in (3.9) also vanishes since f ' ( r ) = 1 + O ( r 2 ). Hence, ρ extends continously to the origin and This shows property (i) in Definition 2.2. To show (ii), consider the vector field By construction v is unit spacelike and orthogonal to u . Let e θ , e ϕ be the standard orthonormal vectors on the sphere so that { u, v, e θ , e ϕ } forms an orthonormal basis on M ε (modulo some spherical coordinate singularities). Straightforward computations show p e ϕ = p e θ . Using (3.7) again, we see that these functions extend continuously to M ε ∪ {O} and To finish our analysis, we need to compute all the cross terms of T . These cross terms are Moreover, using (3.7), we have Let e 0 , e 1 , e 2 , e 3 denote u, v, e θ , e ϕ respectively. If e is any unit spacelike vector field orthogonal to u , then it can be written as e = ∑ 3 i =1 a i e i with ∑ 3 i =1 a 2 i = 1. Then vanishing of the cross terms implies as we approach O within M ε . and so in the limit It is only left to show property (iii) in Definition 2.2, i.e., that T -T perfect extends continuously to M ε ∪ {O} and is zero at O . Recall that where ρ 0 and p 0 are given by (3.10) and (3.12). We work in ( t, x, y, z )-coordinates as they clearly cover the origin O . We have Then On the other hand, Therefore Combining (3.7), (3.10), (3.11), and (3.12), we obtain as we approach the origin O within M ε . In a similar manner, all other components of T -T perfect in ( t, x, y, z )-coordinates also converge to 0. Consequently, assumption (b) in Theorem 2.4 is satisfied as well. Hence the conclusions of the Theorem 2.4 hold: Remark. We emphasize that we have applied Theorem 2.4 to the spacetime M ε and not M ; this is sufficient for the analysis in the next section.",
"pages": [
8,
9,
10,
11,
12
]
},
{
"title": "3.2 Existence of inflationary eras in the examples",
"content": "In this section we show how the conclusion of our main result, Theorem 2.4, proves the existence of inflationary eras for the examples constructed in section 3.1. A majority of the analysis in this section was outlined in section 3 of [21]. (At the time of writing [21], we had not yet found anisotropic examples of our main theorem which is a main inspiration for writing this paper.) To gain some familiarity with the problem at hand, let's consider the FLRW setting. Friedmann's second equations is Therefore The assumption in (3.15) is what we mean by 'the cosmological constant appears as an initial condition.' It holds for a class of Milne-like spacetimes, see [21, eq. (1.11)]. In fact, our main result, Theorem 2.4, is essentially an anisotropoic generalization of this. In this section, we generalize (3.15) to our anisotropic examples. Specifically what we demonstrate is the following. Let ( M,g ) be the spacetime defined by equations (3.1), (3.2), and (3.3). Let γ ( τ ) denote a comoving observer in M (i.e., γ is an integral curve of u ). Here τ is the proper time of the comoving observer, and we fix it so that γ (0) = O . We will define a 'generalized scale factor' a ( τ ) associated with γ ( τ ) and show that this generalized scale factor is accelerating, a '' ( τ ) > 0, for proper times τ near τ = 0 (i.e., near the big bang). However, we only prove that some comoving observers experience inflation. Recall that the comoving observers follow the trajectories t = µf ( r ), see (3.5). We find that only those comoving observers with a µ -value above a certain threshold µ crit experience an inflationary era. This threshold is given by (3.27); it's completely determined by the functions f ( r ) and Φ( ζ ) appearing in the previous section, and hence depends solely on the spacetime metric. Remark. Throughout this section, we have in mind a fixed comoving observer. Since the comoving observers travel along the trajectories t = µf ( r ), we can assume any fixed comoving observer is contained in some M ε (see (3.6)) by choosing ε > 0 small enough. Therefore the bound (3.7) can be utilized. Recall u is given by (3.4). By construction u is orthogonal to the spacelike hypersurfaces of constant ζ . In Figure 1 (right), one should image that the spacelike hypersurfaces τ = constant are replaced with ζ = constant. In the terminology of [26, p. 359], u is 'synchronizable,' but it is not necessarily 'proper time synchronizable.' The latter occurs if and only if u is geodesic which occurs if and only if f ( r ) = r , see eq. (3.24). (Recall f ( r ) = r corresponds to a Milne-like spacetime, and we know u is geodesic in this case.) Set H = 1 3 div u so that H coincides with the mean curvature of the spacelike hypersurfaces orthogonal to u , i.e., H is one-third the trace of the second fundamental form K . 3 Let τ denote the proper time of the flow lines of u (i.e., the proper time of the comoving observers). If c ( r ) denotes the curve r ↦→ ( µf ( r ) , r, θ 0 , ϕ 0 ) along the trajectory t = µf ( r ), then the proper time τ is simply When c ( r ) is reparameterized by τ , it yields a comoving observer γ ( τ Along each comoving observer γ ( τ ), we define a generalized scale factor a ( τ ) by ). 4 We have H ( τ ) ≈ 1 τ for τ small along each comoving observer γ ( τ ), see eq. (3.21) below. Since a ( τ ) = exp( ∫ τ τ 0 H ) for some arbitrary time τ 0 , it follows that along each comoving observer. Recall that, for us, the big bang corresponds to the time when the scale factor limits to 0. Therefore (3.18) suggests that the origin point O represents the big bang in these models. For FLRW spacetimes, Friedmann's second equation (3.14) is used to analyze the acceleration of the scale factor. In the anisotropic setting, the generalization of Friedmann's second equation is the Raychaudhuri equation [15, eq. (4.26)], (The vorticity term vanishes since u is hypersurface orthogonal.) Our goal is to compute all the terms on the right-hand side of (3.19) for points along a comoving observer near O (see Figure 3). First, using the conclusions of Theorem 2.4 and eq. (3.13), sufficiently close to the origin O , we have The ≈ in the above expression is understood in the following way: -Ric( u, u ) can be made arbitrarily close to 8 πρ 0 by choosing points in M arbitrarily close to O . The shear term is defined by 2 σ 2 = ∑ 3 i,j =1 σ ( e i , e j ) σ ( e i , e j ) where { e 1 , e 2 , e 3 } is an orthonormal basis spanning u ⊥ and where K ( X,Y ) = g ( ∇ X u, Y ) is the second fundamental form of the hypersurfaces orthogonal to u . (Recall H = 1 3 tr K .) Choosing the orthonormal basis { v, e θ , e ϕ } from the previous section, the only nonvanishing terms for K are K ( v, v ) = e -Φ( ζ ) √ t 2 f ' ( r ) 2 -f ( r ) 2 [ f ' ( r ) -t 2 f ( r ) f ' ( r ) f '' ( r ) t 2 f ' ( r ) 2 -f ( r ) 2 +Φ ' ( ζ ) t 2 f ' ( r ) 2 -f ( r ) 2 f ' ( r ) ] K ( e θ , e θ ) = K ( e ϕ e ϕ ) = e -Φ( ζ ) √ t 2 f ' ( r ) 2 -f ( r ) 2 [ f ( r ) r +Φ ' ( ζ ) t 2 f ' ( r ) 2 -f ( r ) 2 f ' ( r ) ] . Therefore the mean curvature H is Along t = µf ( r ), we have H ∣ ∣ t = µf ( r ) = e -Φ(0) r √ µ 2 -1 + O ( r ) . Using (3.16), we reparameterize in terms of τ giving Direct computation shows that For small r , we have f ' ( r ) -f ( r ) r = O ( r 2 ) which combined with (3.7) yields In other words 2 σ 2 extends continuously to M ε ∪ {O} and takes on the value 0 at O . Geometrically, this 'isotropization' effect is a consequence of the u -orthogonal hypersurfaces becoming more hyperbolic as we approach the origin O . The last term in (3.19) to compute is div( ∇ u u ). We have Hence Evaluating along t = µf ( r ) and taking the limit r → 0, we find Using (3.20), (3.23), and (3.25), the Raychaudhuri equation (3.19), for points along the comoving observer sufficiently close to the origin O , becomes From (3.26) we can determine which comoving observers experience an inflationary era, i.e., which comoving observers experience a '' ( τ ) > 0 arbitrarily close to τ = 0. Assuming Φ ' (0) > 0 (which is equivalent to ˜ ρ ( O ) > 0), it's precisely those comoving observers with µ -values satisfying Similar to (3.20), the ≈ symbol in the above expression is understood in the following way: 3( a '' / a ) can be made arbitrarily close to the right-hand side of (3.26) by choosing points along t = µf ( r ) that are sufficiently close to the origin O . Moreover, we see that if f ''' (0) = 0 and Φ ' (0) > 0, then all the comoving observers experience an inflationary era. This reproduces the results for Milne-like spacetimes, see (3.15).",
"pages": [
13,
14,
15,
16
]
},
{
"title": "3.3 Remarks on proving anisotropy",
"content": "In this section we show that the examples constructed in section 3.1 are generally anisotropic. Although this is heuristically evident, a formal mathematical proof is not immediately clear. First, the definition of an 'isotropic spacetime' is not consistent throughout the literature. See [2] and [28] and references therein. We will adopt the definition in [26, Ch. 12] since, as discussed in [2], this definition is the optimal one as it implies that the spacetime is isometric to a subset of an FLRW spacetime, see [26, Prop. 12.6] and [2, Thm. 2.1]. Therefore any spacetime that is not isometric to a subset of an FLRW model is anisotropic according to [26]. For the examples constructed in section 3.1, if f ( r ) = r then they are isometric to a subclass of Milne-like spacetimes which are a subclass of k = -1 FLRW models, and hence they are isotropic. Moreover, regardless of the form of f ( r ), if the conformal factor is identically 1, then the spacetime is isometric to a subset of Minkowski spacetime which is clearly isotropic. (This shows that it is not sufficient to simply recognize that the shear term (3.22) is nonzero. However, in this case, the vector field defining the comoving observers changes.) This suggests that if f ( r ) is not identically r and the conformal factor is not constant, then the resulting spacetime is not a subset of an FLRW spacetime and hence is anisotropic. We believe such a statement can be proven rigorously. However, in this section, we will content ourselves with the following algorithm: Pick functions f ( r ) and Φ( ζ ). The steps below show how to verify that the corresponding spacetime ( M,g ) from section 3.1 is anisotropic. Seeking a contradiction, suppose ( M,g ) is in fact isometric to a subset of an FLRW spacetime. Since FLRW spacetimes satisfy the Einstein equations with a perfect fluid [26, Thm. 12.11], there is a unit future directed timelike vector field ˜ u on M such that T is a perfect fluid with respect to ˜ u . There exist functions a , b , c , d such that ˜ u = au + bv + ce θ + de ϕ , where { u, v, e θ , e ϕ } is the orthonormal frame constructed in section 3.1. Consider the unit spacelike vectors orthogonal to u From section 3.1, we know how T acts on the orthonormal frame { u, v, e θ , e ϕ } , and so we know how T acts on { u, v, e θ , e ϕ } . ˜ ˜ ˜ ˜ Fix a point p 0 ∈ M given by ( t 0 , r 0 , θ 0 , ϕ 0 ). At p 0 , the following equations set up an overdetermined system for ( a, b, c, d ) at p 0 . ˜ ˜ ˜ ˜ ˜ ˜ For most choices of f ( r ) and Φ( ζ ), this system does not have any solutions, giving a contradiction. However, even if there are solutions, one can still obtain a contradiction by other means, e.g., showing that the orthogonal subspace to ˜ u does not have constant sectional curvature. Lastly, we remark that the point p 0 must lie away from r = 0. Indeed, points along r = 0 will past the above tests. This is due to the spacetime being spherically symmetric and hence spatially isotropic precisely at points along r = 0. We remark that if Φ is constant or f ( r ) = r , then ( M,g ) passes the above tests. In the first case, the metric is homothetic to the Minkowski metric (and hence isometric to a subset of a k = 0 FLRW spacetime), and in the second case, the spacetime is given by a Milne-like spacetime and hence is isometric to a k = -1 FLRW spacetime.",
"pages": [
16,
17
]
},
{
"title": "4 Summary and outlook",
"content": "The inflationary scenario has become the current paradigm of early universe cosmology. Roughly, it states that scale factor underwent a brief but dramatic period of acceleration after the big bang but before the radiation dominated era. Although inflationary theory has many successes (e.g., solutions to the horizon and flatness problems along with providing a framework for generating the seeds of large-scale structures in our universe), it does not carry the status of an established physical theory. In this work, we provide mathematical support for the inflationary scenario by showing that a class of anisotropic spacetimes experience inflationary eras after the big bang. Our main result, Theorem 2.4, says that if the universe began with special initial conditions at the big bang, then the energy-momentum tensor was dominated by a cosmological constant at the big bang. These special initial conditions are (1) the existence of an origin point O for a unit timelike vector field u (whose integral curves represent the comoving observers in the universe) and (2) the energy-momentum tensor approaches a perfect fluid at O . An informal discussion of these special initial conditions is given in section 1.3. In section 3.1, we construct anisotropic spacetimes which satisfy the hypotheses of Theorem 2.4. These examples can be thought of as 'quasi Milne-like spacetimes.' In section 3.2, we define a generalized scale factor a ( τ ) along each comoving observer ( τ denotes the proper time of the comoving observer), and we show that a ( τ ) → 0 as τ → 0, see (3.18). Consequently, we associate τ = 0 (and hence also the origin point O ) with the big bang. Lastly, we describe which comoving observers experience inflation, a '' ( τ ) > 0, immediately after the big bang τ = 0. See equations (3.26) and (3.27). Our examples exhibit isotropization towards the past and, in fact, are perfectly isotropic at the big bang O , see (3.23). Our isotropization-towards-the-past result is consistent with a universe starting from special initial conditions. This is unlike results related to the cosmic no-hair conjecture (see Wald's original paper [30] or some more recent work, e.g., [3]), where isotropization occurs towards the future. A limitation of our approach is that we only show accelerated expansion immediately after the big bang. For example reheating does not appear in our analysis. For an analysis of the physics after the accelerated expansion, our geometrical initial conditions should be supplemented with, for example, appropriate scalar field matter models. We believe that differential geometry (and geometric analysis in particular) has a role to play in the investigation of initial conditions for the big bang. The work presented in this paper should be thought of as a 'proof of concept' of this proposal. Our work can be generalized in many ways. In particular, although our examples are not necessarily isotropic, they are still spherically symmetric. So a natural generalization is to reproduce the analysis in sections 3.1 and 3.2 with non-spherically symmetric spacetimes. Also, our examples are anisotropic versions of k = -1 FLRW spacetimes. What about k = 0 FLRW spacetimes? In this case one would want to apply [12, Thm. 5.2] or a suitable generalization thereof. Lastly, it remains to be seen if the results in [11] can be used to generate comoving observers with an origin point O .",
"pages": [
17,
18
]
},
{
"title": "Acknowledgments",
"content": "Eric Ling was supported by Carlsberg Foundation CF21-0680 and Danmarks Grundforskningsfond CPH-GEOTOP-DNRF151. Annachiara Piubello was supported by the DFG Project ME 3816/3-1, part of the SPP2026. We thank Jerome Quintin for helpful comments on an earlier draft and are grateful to the Minkowski institute where this project began to take shape.",
"pages": [
18
]
}
]
2024arXiv240302540A
https://arxiv.org/pdf/2403.02540.pdf
<document>
<section_header_level_1><location><page_1><loc_20><loc_85><loc_77><loc_90></location>Surface gravity analysis in Gauss-Bonnet and Barrow black holes</section_header_level_1>
<text><location><page_1><loc_37><loc_82><loc_61><loc_83></location>Everton M. C. Abreu 1, 2, 3, ∗</text>
<text><location><page_1><loc_21><loc_69><loc_77><loc_81></location>1 Departamento de F'ısica, Universidade Federal Rural do Rio de Janeiro, 23890-971, Serop'edica, RJ, Brazil 2 Departamento de F'ısica, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora, MG, Brazil 3 Programa de P'os-Gradua¸c˜ao Interdisciplinar em F'ısica Aplicada, Instituto de F'ısica, Universidade Federal do Rio de Janeiro, 21941-972, Rio de Janeiro, RJ, Brazil</text>
<text><location><page_1><loc_40><loc_67><loc_58><loc_69></location>(Dated: March 6, 2024)</text>
<section_header_level_1><location><page_1><loc_44><loc_65><loc_53><loc_67></location>Abstract</section_header_level_1>
<text><location><page_1><loc_12><loc_43><loc_86><loc_64></location>We have different definitions of the surface gravity (SG) of a horizon since we can say we have distinct classifications of horizons. The SG has an underlying role in the laws of black hole (BH) thermodynamics, being constant in the event horizon. The SG also acts in the emission of Hawking radiation being connected to its temperature. Concerning this last issue, the quantum features that permeate Hawking radiation provide us a direct indication that a BH has its temperature directly connected to its area and that its entropy is proportional to the horizon area. In this work we analyzed some aspects of event horizons. Analyzing how the SG can be classically defined for stationary BHs together with the radial pressure computation. So, the SG, through the laws of BH mechanics is connected to the real thermodynamical temperature of a thermal spectrum. We discussed these subjects in two different BHs scenarios, the five dimensional Gauss-Bonnet one and the recently developed Barrow entropy construction. We discussed how the quantum fluctuations affect these both quantities.</text>
<text><location><page_1><loc_12><loc_40><loc_44><loc_41></location>PACS numbers: 04.50.Gh, 04.70.Dy, 04.70.-s</text>
<text><location><page_1><loc_12><loc_38><loc_69><loc_39></location>Keywords: Black holes event horizon, surface gravity, Barrow black hole entropy</text>
<section_header_level_1><location><page_2><loc_12><loc_88><loc_32><loc_90></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_2><loc_12><loc_76><loc_86><loc_86></location>There is a historically well established connection between black hole (BH) evolvement and thermodynamics, and it has been intensely investigated until now since the works of Bekenstein and Hawking [1, 2]. At the beginning, this issue was constructed using the concepts of purely classical evolution [3]. The posterior disclosure of the semiclassical Hawking radiation effect reordered BH thermodynamics in a stable theoretical position. For a review the interested reader can see [4].</text>
<text><location><page_2><loc_12><loc_55><loc_86><loc_76></location>One of the consequences of this investigation is that the surface gravity (SG) that appears in the laws of BH mechanics is connected to the actual thermodynamic temperature of a thermal spectrum. The existence of the Hawking radiation indicates that the development of an astrophysical BH during its lifetime will, in general, ever be completely static. Namely, we have two distinct situations. In one circumstance the BH will be evolving by amassing matter (accretion), from either close matter sources or CMB. Or it will be evaporating, although slowly, through the emission of Hawking radiation. Considering very small BHs it is assumed that their development is hugely dynamic with a great radiation flux. To arrange for a BH whose magnitude is neither growing nor declining, it would demand a fine equilibrium between the volume of matter being amassed by accretion and the quantity of energy being lost through Hawking radiation [5]. We can also ask what can we say about SG in both situations.</text>
<text><location><page_2><loc_12><loc_39><loc_86><loc_54></location>Most computations in the literature concerning the Hawking radiation effect consider both a quasi-stationary and a quasi-equilibrium expansion. As a matter of fact, many calculations are based on the data of an exactly static background spacetime. Besides, they neglect any backreaction from the radiation energy flux. The thermodynamical analogy between BH and thermal systems implies that we should equate both the classical SG and the Hawking radiation temperature. The SG can be determined classically by using the geometric properties of the fundamental metric. This same quantity can be demonstrated as being proportional to the temperature of the quantum fields in the Hawking effect.</text>
<text><location><page_2><loc_12><loc_12><loc_86><loc_39></location>As we mentioned before, the SG is defined classically in terms of the geometric features of the fundamental metric. Although the same quantity appears as the temperature of quantum fields concerning the Hawking effect. It is determined by considering purely geometric quantities like a Killing field, which means that, in standard terms, it is only defined for precisely static scenarios. It is not even described concerning quasi-static situations, where one hopes that the evolution is a quasi-equilibrium picture, and the Hawking radiation is thermal. In this way, the standard description of the SG is not even true in a spacetime that considers gravitational radiation anywhere in the spacetime. In fact, the SG has importance and, at the same time, distinct roles in BH mechanics. Concerning the classical BH evolution law and considering the so-called zeroth law, this evolution law keeps true independently of any quantum effects and hence, it is essentially a classical effect. The SG acts as the constant of proportionality between the BH mass change and this change occurs in the area. It is important for the definition of this zeroth law that the mass of a BH be well defined either. So, the definition of SG is closely associated with our choice of quasi-local mass. On the other hand, the SG acts in the</text>
<text><location><page_3><loc_12><loc_86><loc_86><loc_90></location>emission of Hawking radiation. A suitable generalization of the SG is expected to act in dynamical Hawking radiation, even in non-equilibrium processes [5].</text>
<text><location><page_3><loc_12><loc_74><loc_86><loc_86></location>A well known definition of the SG is in terms of a Killing horizon which, for instance, is the way the SG is computed for a Schwarzschild BH (the interested reader can find a good review of Killing horizons properties in [6]). This works well in a stationary scenario but spoils in a complete dynamical picture, where no such Killing horizon exists. One way of knowing how the SG works for BHs that are in progress, either by amassing matter or by emitting Hawking radiation was given in [5]. However, concerning modified gravitational models, we can say that horizons can be non-Killing or non-null [7].</text>
<text><location><page_3><loc_12><loc_58><loc_86><loc_74></location>In stationary spacetimes, as the scenario we will analyze here, the BH event horizon is typically a Killing horizon for a convenient chosen Killing vector. On the other hand, for more general, non-stationary spacetimes event horizons, the SG can be determined concerning the null generators of the horizon. To consider these more general cases, we have no Killing vector field to use to fix the normalization of the SG. As the generator of the horizon can be defined only on the event horizon, there is no standard way of fixing this normalization by compelling a condition off the horizon. Recently, big interest has focused on local definitions of horizons [8]. Since the SG is expected to act in the ruling of the quantity of Hawking radiation, it is interesting to analyze SG definitions.</text>
<text><location><page_3><loc_12><loc_41><loc_86><loc_58></location>The Gauss-Bonnet (GB) term in D ( > 4) dimensions is the lowest order correction of the Lanczos-Lovelock (LL) gravity [9, 10]. In both theories the field equations derived from the respective Lagrangians are quasi linear since the initial value problem keeps well defined. Concerning the horizon thermodynamics described through the Einstein's gravity, it is still true in LL gravity models [11]. The interest in D > 4 GB theories arises from the fact that in D = 4 it is a pure divergence term, which does not occur in higher dimensions. The other point of interest is that several string theoretical models have GBtype elements as corrections, what is accomplished after proper field redefinitions [12]. The absence of manifest general covariance is not a big problem since this problem also exists for the Einstein-Hilbert action.</text>
<text><location><page_3><loc_12><loc_30><loc_86><loc_40></location>In this work, we analyzed some different structures for the SG. Besides, the GB one mentioned above, another one appeared recently in the literature, the Barrow toy model [13] for BHs, considering a fractal spacetime geometry caused by the quantum fluctuations [14-17]. This fractallity feature is reflected in the ∆-parameter present in the definition of the entropy 1 . In this paper we make some considerations and compute the SG as a function of this ∆-exponent and the respective radial pressure.</text>
<text><location><page_3><loc_12><loc_21><loc_86><loc_30></location>The distribution of the subjects obeys the following points: in section 2 we considered a 5D Gauss-Bonnet (GB) BH and we calculated its SG and radial pressure. In section 3 we analyzed Barrow's BH theory and we computed these both quantities to analyze the role of the geometry on both structures. The final analysis and remarks are described at the conclusion section.</text>
<section_header_level_1><location><page_4><loc_12><loc_88><loc_77><loc_90></location>2. THE SURFACE GRAVITY OF GAUSS-BONNET BLACK HOLES</section_header_level_1>
<text><location><page_4><loc_12><loc_72><loc_86><loc_86></location>As explained above, the GB formulation of gravity is a natural generalization of Einstein theory of relativity. Since it is the zero-torsion most general theory of gravity, it heads us to the stable second-order equations of motion in higher dimensions. It was proposed firstly by C. Lanczos [9] and confirmed latter by D. Lovelock in [10]. The theory includes Einstein's concept of gravity and therefore, GB gravity is more appropriate than other higher-curvature gravitation theories. Both GB and Einstein terms are within Lovelock's Lagrangian framework. GB gravitational formulation is the most direct nontrivial generalization of Einstein theory of gravitation.</text>
<text><location><page_4><loc_12><loc_62><loc_86><loc_72></location>Having said that, we know that we have in the literature, strong theoretical evidences that there is a strong connection between thermodynamics and the gravitational horizon dynamics. Alas, this is still not deeply understood. On the other hand, we know that the Hawking radiation implies that the evolution of an astrophysical BH during its lifetime will almost never be precisely static [5] or the BH will accretes matter or it will be evaporating via Hawking radiation.</text>
<text><location><page_4><loc_12><loc_58><loc_86><loc_61></location>Let us begin with the case of a 5D GB BH explored in [20], where the GB Lagrangian L in D dimensions [21, 22] is given by</text>
<formula><location><page_4><loc_27><loc_52><loc_86><loc_57></location>16 π L = R + α GB ( R 2 -4 R ab R ab + R abcd R abcd ) , (1)</formula>
<text><location><page_4><loc_12><loc_46><loc_86><loc_53></location>where R is the D dimensional Ricci scalar and the Lagrangian in Eq. (1) can be obtained from superstring theory in the low energy limit, α GB is considered as the inverse string tension and it is positive definite [12]. The field equations for the semiclassical action obtained using the Lagrangian in Eq. (1) is</text>
<formula><location><page_4><loc_38><loc_43><loc_86><loc_44></location>G ab + α GB H ab = 8 π T ab (2)</formula>
<text><location><page_4><loc_12><loc_38><loc_38><loc_41></location>where G ab = R ab -1 2 g ab R and</text>
<formula><location><page_4><loc_20><loc_33><loc_86><loc_38></location>H ab = 2 ( RR ab -2 R ac R c b -2 R cd R acbd + R cdm a R bcdm ) -1 2 g ab L , (3)</formula>
<formula><location><page_4><loc_33><loc_27><loc_86><loc_30></location>ds 2 = -f ( r ) dt 2 + dr 2 f ( r ) + r 2 d Ω 2 D -2 , (4)</formula>
<text><location><page_4><loc_12><loc_30><loc_86><loc_33></location>where L is given in Eq. (1). The static, spherically symmetric BH solutions in this theory have the form</text>
<text><location><page_4><loc_12><loc_24><loc_69><loc_26></location>where the last term is relative to the angular part of the metric and</text>
<formula><location><page_4><loc_33><loc_18><loc_86><loc_23></location>f ( r ) = 1 + r 2 2 α [ 1 -√ 1 + 4 αω r D -1 ] , (5)</formula>
<text><location><page_4><loc_12><loc_14><loc_86><loc_17></location>where α = ( D -3)( D -4) α GB and ω is related to the ADM mass M through the equation</text>
<formula><location><page_4><loc_39><loc_10><loc_86><loc_14></location>ω = 16 π ( D -2) V D -2 M , (6)</formula>
<text><location><page_5><loc_12><loc_86><loc_86><loc_90></location>where V D -2 is the volume of unit ( D -2)-sphere. The Hawking temperature T and entropy S for this spacetime are, respectively,</text>
<formula><location><page_5><loc_29><loc_80><loc_86><loc_85></location>T = D -3 4 πr + [ r 2 + r 2 + +2 α + α ( D -5 D -3 ) 1 r 2 + +2 α ] (7)</formula>
<text><location><page_5><loc_12><loc_77><loc_15><loc_79></location>and</text>
<formula><location><page_5><loc_30><loc_70><loc_86><loc_76></location>S = A 4 [ 1 + 2 α ( D -2 D -4 )( A Σ D -2 ) -2 D -2 ] , (8)</formula>
<text><location><page_5><loc_12><loc_66><loc_86><loc_69></location>where A = V D -2 r D -2 + is the horizon area. We can also realize that D = 4 is a divergent point in GB gravity.</text>
<text><location><page_5><loc_12><loc_59><loc_86><loc_66></location>Several authors dedicated to calculate the entropy of GB theory [19]. Therefore we can see, as well known, that in such general theories, different from Einstein's gravity, the horizon entropy is not proportional to the area, it only occurs when α = 0 in Eq. (8) [20]. The locus of the horizon is found as roots of q ( r + ) = 0, where</text>
<formula><location><page_5><loc_37><loc_55><loc_86><loc_57></location>q ( r ) = R D -3 + αr D -5 -ω (9)</formula>
<text><location><page_5><loc_12><loc_52><loc_61><loc_54></location>and for the horizon to exist at all, one must also have that</text>
<formula><location><page_5><loc_43><loc_48><loc_86><loc_51></location>r 2 + + 2 α ≥ 0 . (10)</formula>
<text><location><page_5><loc_14><loc_46><loc_56><loc_47></location>In this generalized GB BH, let us compute the SG</text>
<text><location><page_5><loc_12><loc_36><loc_45><loc_38></location>where the derivative of f ( r ) is given by</text>
<formula><location><page_5><loc_35><loc_36><loc_86><loc_44></location>κ = 1 2 df ( r ) dr ∣ ∣ ∣ ∣ ∣ r = r + = 1 2 f ' ( r ) ∣ ∣ ∣ r = r + , (11)</formula>
<formula><location><page_5><loc_21><loc_32><loc_51><loc_35></location>f ( r ) = 1 + r 2 1 1 + Ar 1 -D</formula>
<formula><location><page_5><loc_17><loc_24><loc_86><loc_34></location>2 α [ -√ ] = ⇒ f ' ( r ) = r α { 1 -√ 1 + Ar 1 -D + r 2 [ A 2 (1 -D ) r -D ( 1 + Ar 1 -D ) -1 2 ]} . (12)</formula>
<text><location><page_5><loc_67><loc_12><loc_67><loc_15></location>/negationslash</text>
<text><location><page_5><loc_12><loc_12><loc_86><loc_24></location>where A = 4 αω . We can see in Eq. (12) the SG in Eq. (11) is directly dependent of the dimensionality of the geometry. Besides, it is a function of the ADM mass via the A term. We can also see that for D = 3 or D = 4 we have an infinite value for SG since α → 0, which means form Eq. (17) that r + ≥ 0, which, from Eq. (7), results in a plausible value for the Hawking temperature and from Eq. (8), the Bekenstein-Hawking entropy is recovered. For r + → 0, i.e., very small BHs, for a D = 3 GB BH, from Eq. (7) we confirm the result that BHs are very hot at the event horizon. Since the entropy</text>
<text><location><page_6><loc_14><loc_85><loc_47><loc_86></location>The Einstein equation for this metric is</text>
<text><location><page_6><loc_12><loc_85><loc_86><loc_90></location>is not dependent directly of r + , it keeps the Bekenstein-Hawking form when D = 3 and α → 0.</text>
<formula><location><page_6><loc_35><loc_79><loc_86><loc_84></location>rf ' ( r ) -( 1 -f ( r ) ) = 8 π P r 2 , (13)</formula>
<text><location><page_6><loc_12><loc_75><loc_86><loc_80></location>where P = P ( r ) is the radial pressure. But we know that f ( r ) is zero at the horizon, i.e., in r = r + and we have that f ( r + ) = 0, hence, using this information in Eq. (13) we have that</text>
<formula><location><page_6><loc_38><loc_72><loc_86><loc_74></location>r + f ' ( r + ) -1 = 8 πP r 2 + , (14)</formula>
<text><location><page_6><loc_12><loc_69><loc_86><loc_72></location>where we are using that G = c = /planckover2pi1 = 1. Hence, from the above equations we can write the radial pressure as</text>
<formula><location><page_6><loc_35><loc_64><loc_86><loc_69></location>P ( r + ) = 1 8 πr 2 + ( r + f ' ( r + ) -1 ) , (15)</formula>
<text><location><page_6><loc_12><loc_63><loc_63><loc_64></location>and from Eq. (11) the final expression for the SG is given by</text>
<formula><location><page_6><loc_13><loc_56><loc_86><loc_62></location>κ = 1 2 f ' ( r + ) = r + 2 α { 1 -√ 1 + Ar 1 -D + + r + 2 [ A 2 (1 -D ) r -D + ( 1 + Ar 1 -D + ) -1 2 ]} . (16)</formula>
<text><location><page_6><loc_12><loc_44><loc_86><loc_56></location>The zeroth law of BH thermodynamics asserts the κ is constant in the event horizon, which means that we have two alternatives. Firstly all the terms in Eq. (16) are constant, obviously. The other one is that there is a compensation such that the final result keeps constant. Let us consider the first alternative as the correct one, So, ω in Eq. (6) is also constant which means that the mass M is constant and all the terms in Eq. (16) are constants. Hence, since the horizon radius r + is constant, from Eq. (15) we can see that the radial pressure is also constant in the event horizon.</text>
<section_header_level_1><location><page_6><loc_12><loc_39><loc_39><loc_40></location>3. BARROW BLACK HOLE</section_header_level_1>
<text><location><page_6><loc_12><loc_23><loc_86><loc_37></location>Barrow introduced recently in 2020 [13], based on the Covid-19 shape of the virus, which shows a fractal structure, i.e., a fractal framework for the horizon of BHs with finite volume and infinite, or not, area. As a toy model, it can show effects of quantum gravity spacetime foam, with important consequences concerning the entropy of BHs and the Universe. The so-called Barrow-parameter, as will be shown just below, is confined within 0 ≤ ∆ ≤ 1, where ∆ = 0, geometrically speaking, responds for a smooth spacetime structure and ∆ = 1 shows a most intricate geometry. As demonstrated in [13], the Hawking lifetime of BHs is also shortened, but we do not discuss this issue here.</text>
<text><location><page_6><loc_14><loc_22><loc_63><loc_23></location>Now, let us analyze the Barrow BH with fractal geometry</text>
<formula><location><page_6><loc_41><loc_15><loc_86><loc_20></location>S = ( A H 4 A Pl ) 1+ ∆ 2 , (17)</formula>
<text><location><page_6><loc_12><loc_11><loc_86><loc_14></location>where A Pl = /lscript 2 P is the Planck area, which is equal to 1 in relativistic dimensions. Thermodynamically speaking, for ∆ = 0 we have the Bekenstein-Hawking entropy and for ∆ = 1,</text>
<text><location><page_6><loc_79><loc_87><loc_79><loc_90></location>/negationslash</text>
<text><location><page_7><loc_12><loc_86><loc_86><loc_90></location>the entropy for the most intricate geometry. For A H = 4 πr 2 + we can write the entropy derivative in a convenient form such that</text>
<formula><location><page_7><loc_33><loc_81><loc_86><loc_85></location>dS = 2 ( 1 + ∆ 2 ) π 1+ ∆ 2 r ∆ + ( r + dr + ) , (18)</formula>
<text><location><page_7><loc_12><loc_78><loc_84><loc_79></location>and the Einstein equations, with an 'Ad hoc' dr + -term in each factor of the equation,</text>
<formula><location><page_7><loc_33><loc_73><loc_86><loc_76></location>1 2 f ' ( r + ) r + dr + -1 2 dr + = 4 πP r 2 + dr + , (19)</formula>
<text><location><page_7><loc_12><loc_66><loc_86><loc_71></location>and we will use this equation including the radial pressure in a moment. We know that the periodicity in Euclidean time permits us to connect the horizon to the temperature such that</text>
<formula><location><page_7><loc_39><loc_63><loc_86><loc_67></location>k B T = κ 2 π = f ' ( r + ) 4 π . (20)</formula>
<text><location><page_7><loc_14><loc_61><loc_41><loc_63></location>Let us use the first law equation</text>
<formula><location><page_7><loc_39><loc_57><loc_86><loc_59></location>k B T dS -dE = P dV , (21)</formula>
<text><location><page_7><loc_12><loc_55><loc_78><loc_56></location>which is analogous to Eq. (19) [23] and, using also Eqs. (18)-(21) we have that</text>
<formula><location><page_7><loc_36><loc_48><loc_86><loc_53></location>f ' ( r + ) = 2 κ ( 1 + ∆ 2 ) π ∆ / 2 r ∆ + (22)</formula>
<text><location><page_7><loc_12><loc_47><loc_43><loc_48></location>and hence, the SG for Barrow's BH is</text>
<formula><location><page_7><loc_38><loc_39><loc_86><loc_45></location>κ Barrow = f ' ( r + ) r -∆ + 2 ( 1 + ∆ 2 ) π ∆ / 2 . (23)</formula>
<text><location><page_7><loc_12><loc_37><loc_79><loc_39></location>Let us now analyze the geometrical features coming from Barrow structure. For</text>
<formula><location><page_7><loc_30><loc_32><loc_68><loc_36></location>∆ = 0 = ⇒ κ Barrow = κ = 1 2 f ' ( r + ) ,</formula>
<text><location><page_7><loc_12><loc_30><loc_43><loc_31></location>as expected. For ∆ = 1 we have that</text>
<formula><location><page_7><loc_40><loc_24><loc_86><loc_28></location>κ Barrow = 1 3 √ π f ' ( r + ) r + (24)</formula>
<text><location><page_7><loc_12><loc_20><loc_86><loc_23></location>where the presence of r + shows that in a fractal spacetime the SG expression is dependent of the radius, what does not happen in the flat spacetime.</text>
<text><location><page_7><loc_14><loc_18><loc_55><loc_19></location>From Eq. (20) the temperature of the horizon is</text>
<formula><location><page_7><loc_39><loc_13><loc_86><loc_16></location>T Barrow = 1 2 πk B κ Barrow , (25)</formula>
<text><location><page_8><loc_12><loc_86><loc_86><loc_90></location>where T Barrow is also an effective temperature. It differs by the same factor from the thermodynamical temperature</text>
<formula><location><page_8><loc_40><loc_79><loc_86><loc_85></location>t Barrow = T Barrow √ 1 -2 E r , (26)</formula>
<text><location><page_8><loc_12><loc_77><loc_48><loc_79></location>measured by constant-radius detectors [24].</text>
<text><location><page_8><loc_12><loc_74><loc_86><loc_77></location>It is well known that the SG is also constant for stationary BH and, in this way, linking both zero laws. in this case, from Eq. (23)</text>
<formula><location><page_8><loc_31><loc_66><loc_86><loc_72></location>df dr ∣ ∣ r = r + = 2 κ Barrow ( 1 + ∆ 2 ) π ∆ / 2 r ∆ ∣ ∣ ∣ r = r + , (27)</formula>
<text><location><page_8><loc_12><loc_66><loc_47><loc_70></location>∣ which allows us to write a general relation</text>
<formula><location><page_8><loc_33><loc_60><loc_86><loc_65></location>df ( r + ) dr = 2 κ Barrow ( 1 + ∆ 2 ) π ∆ / 2 r ∆ + , (28)</formula>
<formula><location><page_8><loc_32><loc_55><loc_86><loc_60></location>f ( r + ) = 2 ∆+1 κ Barrow ( 1 + ∆ 2 ) π ∆ / 2 r ∆+1 + + Λ o , (29)</formula>
<text><location><page_8><loc_24><loc_57><loc_26><loc_59></location>=</text>
<text><location><page_8><loc_25><loc_56><loc_27><loc_59></location>⇒</text>
<text><location><page_8><loc_12><loc_54><loc_70><loc_56></location>where Λ o is a constant that depends on the initial conditions of f ( r ).</text>
<text><location><page_8><loc_14><loc_53><loc_71><loc_54></location>Hence, from Eqs. (13) and (28) we can calculate the radial pressure</text>
<formula><location><page_8><loc_32><loc_47><loc_86><loc_51></location>P ( r ) = 1 8 π ∆+2 ∆+1 1 r 2 ( r ∆+1 -∆+1 ∆+2 ) , (30)</formula>
<text><location><page_8><loc_12><loc_37><loc_86><loc_45></location>as a function of the radius and the ∆-parameter. We can see clearly that the radial pressure depends directly on the structure of the spacetime represented by the ∆-parameter in Barrows's formulation. Moreover we can see that for r →∞ the radial pressure goes to zero which is coherent to the fact that as the radius strongly grows, the radial pressure diminishes.</text>
<section_header_level_1><location><page_8><loc_12><loc_32><loc_72><loc_33></location>4. CONCLUSIONS, FINAL REMARKS AND PERSPECTIVES</section_header_level_1>
<text><location><page_8><loc_12><loc_21><loc_86><loc_30></location>In this work our objective was to show precisely that using the recently developed Barrow's formulation of BHs, we can see that some aspects of the event horizon are affected by the quantum fluctuations that dictate the geometry. In some papers published by us before [15], the thermodynamics of BHs were discussed, but some issues relative exactly to the event horizon was now computed.</text>
<text><location><page_8><loc_12><loc_11><loc_86><loc_21></location>We have analyzed the SG issue concerning two popular BH formalisms, the GB and the Barrow toy model, which shows that the geometry affects directly the entropy of the BH. Our main motivation concerning SG is because it is a relevant subject since it is equivalent to the thermodynamic temperature in the investigation of BH thermodynamics laws, more specific, the zeroth-law where a direct association between SG and the thermodynamic temperature appears.</text>
<text><location><page_9><loc_12><loc_76><loc_86><loc_90></location>We calculated the expressions of the SG and radial pressure for both models. We saw that both are dependent of the dimensionality and the structure of the spacetime, in the Barrow's case. In GB case, we saw that the SG remains constant since all the terms are constant or if there is a compensation procedure concerning accretion and radiation processes, and consequently the zeroth-law can be obeyed. By analyzing the final expressions of SG, using particular values for some parameters connected to the dimensionality and the ADM mass, The results recover the well known results that shows that the event horizon is a very hot geometrical place.</text>
<text><location><page_9><loc_12><loc_67><loc_86><loc_76></location>In Barrow's case, the final expressions of both SG and radial pressure show a strong dependence of geometrical structure through the presence of ∆ since we have to consider ∆ as a constant term because of the zeroth-law. So, an analysis like the one made in [14, 16] is not possible directly, but maybe a compensating factor relative to accretion and radiation simultaneous effects can be constructed.</text>
<text><location><page_9><loc_12><loc_58><loc_86><loc_67></location>Another perspective is relative to how can the expressions obtained here affects the other laws of BH thermodynamics since we have modified the SG as well as the entropy. The thermodynamical potentials can also be studied. Other gravitation models can also be analyzed, of course. On the other hand, we can also include numerical values in our results and obtain a value for ∆, for example.</text>
<unordered_list>
<list_item><location><page_9><loc_13><loc_51><loc_74><loc_52></location>[1] J. D. Bekenstein, Phys. Rev. D 7 (1973) 2333; Phys. Rev. D 9 (1974) 3292.</list_item>
<list_item><location><page_9><loc_13><loc_49><loc_84><loc_50></location>[2] S. W. Hawking, Commun. Math. Phys. 43 (1975) 199; Phys. Rev. Lett. 26 (1971) 1344.</list_item>
<list_item><location><page_9><loc_13><loc_47><loc_82><loc_49></location>[3] J. M. Bardeen, B. Carter and S. W. Hawking, Commun. Math. Phys. 31 (1973) 161.</list_item>
<list_item><location><page_9><loc_13><loc_44><loc_86><loc_47></location>[4] T. Padmanabhan, Mod. Phys. Lett. A 30 (2015) 1540007; Phys. Rept. 406 (2005) 49; Mod. Phys. Lett. A 17 (2002) 923.</list_item>
<list_item><location><page_9><loc_13><loc_42><loc_71><loc_43></location>[5] A. B. Nielsen and J. H. Yoon, Class. Quantum Grav. 25 (2008) 085010.</list_item>
<list_item><location><page_9><loc_13><loc_40><loc_65><loc_41></location>[6] S. Sarkar and S. Bhattacharya, Phys. Rev. D 87 (2013) 049023.</list_item>
<list_item><location><page_9><loc_13><loc_38><loc_77><loc_39></location>[7] B. Cropp, S. Liberati and M. Visser, Class. Quantum Grav. 30 (2013) 125001.</list_item>
<list_item><location><page_9><loc_13><loc_36><loc_15><loc_38></location>[8]</list_item>
<list_item><location><page_9><loc_16><loc_34><loc_60><loc_38></location>S. A. Hayward, Phys. Rev. D 49 (1994) 6467; A. Ashtekar and B. Krishnan, Liv. Rev. Rel. 7 (2004) 10;</list_item>
<list_item><location><page_9><loc_16><loc_33><loc_66><loc_34></location>A. B. Nielsen, 'Black holes as local horizons,' arXiv: 0711.0313.</list_item>
<list_item><location><page_9><loc_13><loc_31><loc_65><loc_32></location>[9] C. Lanczos, Z. Phys. 73 (19320 147; Ann. Math. 39 (1938) 842.</list_item>
<list_item><location><page_9><loc_12><loc_27><loc_86><loc_30></location>[10] D. Lovelock, J. Math. Phys. 12 (1971) 498; J. Math. Phys. 13 (1972) 874, the interested reader can see N. Deruelle and J. Madore, arXiv: gr-qc/0305004, for a review.</list_item>
<list_item><location><page_9><loc_12><loc_25><loc_77><loc_27></location>[11] A. Paranjape, S. Sarkar, and T. Padmanabhan, Phys. Rev.D 74 (2006) 104015;</list_item>
<list_item><location><page_9><loc_16><loc_23><loc_73><loc_25></location>A. Mukhopadhyay and T. Padmanabhan, Phys. Rev. D 74 (2006) 124023.</list_item>
<list_item><location><page_9><loc_12><loc_22><loc_49><loc_23></location>[12] B. Zwiebach, Phys. Lett. B 156 (1985) 315;</list_item>
<list_item><location><page_9><loc_16><loc_20><loc_65><loc_21></location>D. G. Boulware and S. Deser, Phys. Rev. Lett. 55 (1985) 2656.</list_item>
<list_item><location><page_9><loc_12><loc_18><loc_50><loc_19></location>[13] J. Barrow, Phys. Lett. B 808 (2020) 135643.</list_item>
<list_item><location><page_9><loc_12><loc_14><loc_86><loc_18></location>[14] E. M. C. Abreu, 'Barrow black hole variable parameter model and the information theory,' arXiv 2402.15922.</list_item>
<list_item><location><page_9><loc_12><loc_13><loc_86><loc_14></location>[15] E. M. C. Abreu, J. Ananias Neto and E. M. Barboza, Europhysics Lett. 130 (2020) 40005;</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_10><loc_16><loc_86><loc_86><loc_88></location>E. M. C. Abreu and J. Ananias Neto, Phys. Lett. B 810 (2020) 135805; Eur. Phys. J. C 80</list_item>
<list_item><location><page_10><loc_16><loc_85><loc_51><loc_90></location>arXiv: 2005.11609, (2020) 776; Phys. Lett. B 807 (2020) 135602.</list_item>
<list_item><location><page_10><loc_12><loc_81><loc_86><loc_84></location>[16] S. Basilakos, A. Lymperis, M. Petronikolou and E. N. Saridakis, 'Barrow holographic dark enrgy with varying exponet,' arXiv2312.15767.</list_item>
<list_item><location><page_10><loc_12><loc_77><loc_86><loc_80></location>[17] J. D. Barrow, S. Basilakos and E. N. Saridakis, 'Big Bang Nucleosynthesis constraints on Barrow entropy,' arXiv: 2010.00986 [gr-qc];</list_item>
<list_item><location><page_10><loc_16><loc_75><loc_60><loc_77></location>E. N. Saridakis, JCAP 07 (2020) 031, arXiv: 2006.01105;</list_item>
<list_item><location><page_10><loc_16><loc_74><loc_85><loc_75></location>F. K. Anagnostopoulos, S. Basilakos and E. N. Saridakis, Eur. Phys. J. C 80 (2020) 826;</list_item>
<list_item><location><page_10><loc_16><loc_70><loc_86><loc_73></location>E. N. Saridakis and S. Basilakos, 'The generalized second law of thermodynamics with Barrow entropy,' arXiv: 2005.08258 [gr-qc];</list_item>
<list_item><location><page_10><loc_16><loc_68><loc_69><loc_70></location>E. N. Saridakis, Phys. Rev. D 102 (2020) 123525, arXiv: 2005.04115.</list_item>
<list_item><location><page_10><loc_12><loc_61><loc_86><loc_68></location>[18] S. Nojiri, S. D. Odintsov and V. Faraoni, Phys. Rev. D 105 (2022) 044042, arXiv: 2201.02424; Phys. Rev. D 104 (2021) 084030, arXiv: 2109.05315; Int. J. Geom. Meth. Phys. 19 (2022) 2250210, arXiv: 2207.07905; S. Nojiri, S. D. Odintsov and T. Paul, Phys. Lett. B 831 (2022) 137189, arXiv: 2205.08876;</list_item>
<list_item><location><page_10><loc_16><loc_59><loc_81><loc_60></location>S. D. Odintsov and T. Paul, Phys. Dark Univ. 38 (2023) 101159, arXiv: 2212.05531.</list_item>
<list_item><location><page_10><loc_12><loc_57><loc_65><loc_59></location>[19] T. Jacobson and R. C. Myers, Phys. Rev. Lett. 70 (1993) 3684;</list_item>
<list_item><location><page_10><loc_16><loc_55><loc_62><loc_57></location>R. C. Myers and J. Z. Simon, Phys. Rev. D 38 (1988) 2434;</list_item>
<list_item><location><page_10><loc_16><loc_54><loc_73><loc_55></location>R.-G. Cai, Phys. Rev. D 65 (2002) 084014; Phys. Lett. B 582 (2004) 237;</list_item>
<list_item><location><page_10><loc_16><loc_52><loc_73><loc_53></location>S. Nojiri, S. D. Odintsov, and S. Ogushi, Phys. Rev. D 65 (2002) 023521;</list_item>
<list_item><location><page_10><loc_16><loc_50><loc_74><loc_51></location>S. Nojiri and S. D. Odintsov, Phys. Lett. B 521 (2001) 87; 542 (2002) 301;</list_item>
<list_item><location><page_10><loc_16><loc_48><loc_71><loc_49></location>M. Cvetic, S. Nojiri, and S. D. Odintsov, Nucl. Phys. B 628 (2002) 295;</list_item>
<list_item><location><page_10><loc_16><loc_46><loc_77><loc_48></location>T. Clunan, S. F. Ross, and D. J. Smith, Class. Quantum Grav. 21 (2004) 3447;</list_item>
<list_item><location><page_10><loc_16><loc_44><loc_52><loc_46></location>I. P. Neupane, Phys. Rev. D 67 (2003) 061501;</list_item>
<list_item><location><page_10><loc_16><loc_43><loc_64><loc_44></location>Y. M. Cho and I. P. Neupane, Phys. Rev. D 66 (2002) 024044;</list_item>
<list_item><location><page_10><loc_16><loc_41><loc_69><loc_42></location>N. Deruelle, J. Katz, and S. Ogushi, Class. Q. Grav. 21 (2004) 1971;</list_item>
<list_item><location><page_10><loc_16><loc_39><loc_60><loc_40></location>G. Kofinas and R. Olea, Phys. Rev. D 74 (2006) 084035;</list_item>
<list_item><location><page_10><loc_16><loc_37><loc_54><loc_39></location>R.-G. Cai and S. Pyo Kim, JHEP 02 (2005) 050;</list_item>
<list_item><location><page_10><loc_16><loc_35><loc_58><loc_37></location>M. Akbar and R.-G. Cai, Phys. Lett. B 635, 7 (2006).</list_item>
<list_item><location><page_10><loc_12><loc_34><loc_78><loc_35></location>[20] D. Kothawala, T. Padmanabhan and S. Sarkar, Phys. Rev. D 78 (2008) 104018.</list_item>
<list_item><location><page_10><loc_12><loc_32><loc_65><loc_33></location>[21] T. Jacobson and R. C. Myers, Phys. Rev. Lett. 70 (1993) 3684.</list_item>
<list_item><location><page_10><loc_12><loc_30><loc_62><loc_31></location>[22] R. C. Myers and J. Z. Simon, Phys. Rev. D 38 (1988) 2434.</list_item>
<list_item><location><page_10><loc_12><loc_28><loc_77><loc_29></location>[23] A. Paranjape, S. Sarkar and T. Padmanabhan, Phys. Rev. D 74 (2006) 104015.</list_item>
<list_item><location><page_10><loc_12><loc_24><loc_86><loc_28></location>[24] N. D. Birrel and P. C. W. Davies, 'Quantum fields in curved space,' Cambridge University Press (CUP), 1992.</list_item>
</unordered_list>
</document>
[
{
"title": "Surface gravity analysis in Gauss-Bonnet and Barrow black holes",
"content": "Everton M. C. Abreu 1, 2, 3, ∗ 1 Departamento de F'ısica, Universidade Federal Rural do Rio de Janeiro, 23890-971, Serop'edica, RJ, Brazil 2 Departamento de F'ısica, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora, MG, Brazil 3 Programa de P'os-Gradua¸c˜ao Interdisciplinar em F'ısica Aplicada, Instituto de F'ısica, Universidade Federal do Rio de Janeiro, 21941-972, Rio de Janeiro, RJ, Brazil (Dated: March 6, 2024)",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "We have different definitions of the surface gravity (SG) of a horizon since we can say we have distinct classifications of horizons. The SG has an underlying role in the laws of black hole (BH) thermodynamics, being constant in the event horizon. The SG also acts in the emission of Hawking radiation being connected to its temperature. Concerning this last issue, the quantum features that permeate Hawking radiation provide us a direct indication that a BH has its temperature directly connected to its area and that its entropy is proportional to the horizon area. In this work we analyzed some aspects of event horizons. Analyzing how the SG can be classically defined for stationary BHs together with the radial pressure computation. So, the SG, through the laws of BH mechanics is connected to the real thermodynamical temperature of a thermal spectrum. We discussed these subjects in two different BHs scenarios, the five dimensional Gauss-Bonnet one and the recently developed Barrow entropy construction. We discussed how the quantum fluctuations affect these both quantities. PACS numbers: 04.50.Gh, 04.70.Dy, 04.70.-s Keywords: Black holes event horizon, surface gravity, Barrow black hole entropy",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "There is a historically well established connection between black hole (BH) evolvement and thermodynamics, and it has been intensely investigated until now since the works of Bekenstein and Hawking [1, 2]. At the beginning, this issue was constructed using the concepts of purely classical evolution [3]. The posterior disclosure of the semiclassical Hawking radiation effect reordered BH thermodynamics in a stable theoretical position. For a review the interested reader can see [4]. One of the consequences of this investigation is that the surface gravity (SG) that appears in the laws of BH mechanics is connected to the actual thermodynamic temperature of a thermal spectrum. The existence of the Hawking radiation indicates that the development of an astrophysical BH during its lifetime will, in general, ever be completely static. Namely, we have two distinct situations. In one circumstance the BH will be evolving by amassing matter (accretion), from either close matter sources or CMB. Or it will be evaporating, although slowly, through the emission of Hawking radiation. Considering very small BHs it is assumed that their development is hugely dynamic with a great radiation flux. To arrange for a BH whose magnitude is neither growing nor declining, it would demand a fine equilibrium between the volume of matter being amassed by accretion and the quantity of energy being lost through Hawking radiation [5]. We can also ask what can we say about SG in both situations. Most computations in the literature concerning the Hawking radiation effect consider both a quasi-stationary and a quasi-equilibrium expansion. As a matter of fact, many calculations are based on the data of an exactly static background spacetime. Besides, they neglect any backreaction from the radiation energy flux. The thermodynamical analogy between BH and thermal systems implies that we should equate both the classical SG and the Hawking radiation temperature. The SG can be determined classically by using the geometric properties of the fundamental metric. This same quantity can be demonstrated as being proportional to the temperature of the quantum fields in the Hawking effect. As we mentioned before, the SG is defined classically in terms of the geometric features of the fundamental metric. Although the same quantity appears as the temperature of quantum fields concerning the Hawking effect. It is determined by considering purely geometric quantities like a Killing field, which means that, in standard terms, it is only defined for precisely static scenarios. It is not even described concerning quasi-static situations, where one hopes that the evolution is a quasi-equilibrium picture, and the Hawking radiation is thermal. In this way, the standard description of the SG is not even true in a spacetime that considers gravitational radiation anywhere in the spacetime. In fact, the SG has importance and, at the same time, distinct roles in BH mechanics. Concerning the classical BH evolution law and considering the so-called zeroth law, this evolution law keeps true independently of any quantum effects and hence, it is essentially a classical effect. The SG acts as the constant of proportionality between the BH mass change and this change occurs in the area. It is important for the definition of this zeroth law that the mass of a BH be well defined either. So, the definition of SG is closely associated with our choice of quasi-local mass. On the other hand, the SG acts in the emission of Hawking radiation. A suitable generalization of the SG is expected to act in dynamical Hawking radiation, even in non-equilibrium processes [5]. A well known definition of the SG is in terms of a Killing horizon which, for instance, is the way the SG is computed for a Schwarzschild BH (the interested reader can find a good review of Killing horizons properties in [6]). This works well in a stationary scenario but spoils in a complete dynamical picture, where no such Killing horizon exists. One way of knowing how the SG works for BHs that are in progress, either by amassing matter or by emitting Hawking radiation was given in [5]. However, concerning modified gravitational models, we can say that horizons can be non-Killing or non-null [7]. In stationary spacetimes, as the scenario we will analyze here, the BH event horizon is typically a Killing horizon for a convenient chosen Killing vector. On the other hand, for more general, non-stationary spacetimes event horizons, the SG can be determined concerning the null generators of the horizon. To consider these more general cases, we have no Killing vector field to use to fix the normalization of the SG. As the generator of the horizon can be defined only on the event horizon, there is no standard way of fixing this normalization by compelling a condition off the horizon. Recently, big interest has focused on local definitions of horizons [8]. Since the SG is expected to act in the ruling of the quantity of Hawking radiation, it is interesting to analyze SG definitions. The Gauss-Bonnet (GB) term in D ( > 4) dimensions is the lowest order correction of the Lanczos-Lovelock (LL) gravity [9, 10]. In both theories the field equations derived from the respective Lagrangians are quasi linear since the initial value problem keeps well defined. Concerning the horizon thermodynamics described through the Einstein's gravity, it is still true in LL gravity models [11]. The interest in D > 4 GB theories arises from the fact that in D = 4 it is a pure divergence term, which does not occur in higher dimensions. The other point of interest is that several string theoretical models have GBtype elements as corrections, what is accomplished after proper field redefinitions [12]. The absence of manifest general covariance is not a big problem since this problem also exists for the Einstein-Hilbert action. In this work, we analyzed some different structures for the SG. Besides, the GB one mentioned above, another one appeared recently in the literature, the Barrow toy model [13] for BHs, considering a fractal spacetime geometry caused by the quantum fluctuations [14-17]. This fractallity feature is reflected in the ∆-parameter present in the definition of the entropy 1 . In this paper we make some considerations and compute the SG as a function of this ∆-exponent and the respective radial pressure. The distribution of the subjects obeys the following points: in section 2 we considered a 5D Gauss-Bonnet (GB) BH and we calculated its SG and radial pressure. In section 3 we analyzed Barrow's BH theory and we computed these both quantities to analyze the role of the geometry on both structures. The final analysis and remarks are described at the conclusion section.",
"pages": [
2,
3
]
},
{
"title": "2. THE SURFACE GRAVITY OF GAUSS-BONNET BLACK HOLES",
"content": "As explained above, the GB formulation of gravity is a natural generalization of Einstein theory of relativity. Since it is the zero-torsion most general theory of gravity, it heads us to the stable second-order equations of motion in higher dimensions. It was proposed firstly by C. Lanczos [9] and confirmed latter by D. Lovelock in [10]. The theory includes Einstein's concept of gravity and therefore, GB gravity is more appropriate than other higher-curvature gravitation theories. Both GB and Einstein terms are within Lovelock's Lagrangian framework. GB gravitational formulation is the most direct nontrivial generalization of Einstein theory of gravitation. Having said that, we know that we have in the literature, strong theoretical evidences that there is a strong connection between thermodynamics and the gravitational horizon dynamics. Alas, this is still not deeply understood. On the other hand, we know that the Hawking radiation implies that the evolution of an astrophysical BH during its lifetime will almost never be precisely static [5] or the BH will accretes matter or it will be evaporating via Hawking radiation. Let us begin with the case of a 5D GB BH explored in [20], where the GB Lagrangian L in D dimensions [21, 22] is given by where R is the D dimensional Ricci scalar and the Lagrangian in Eq. (1) can be obtained from superstring theory in the low energy limit, α GB is considered as the inverse string tension and it is positive definite [12]. The field equations for the semiclassical action obtained using the Lagrangian in Eq. (1) is where G ab = R ab -1 2 g ab R and where L is given in Eq. (1). The static, spherically symmetric BH solutions in this theory have the form where the last term is relative to the angular part of the metric and where α = ( D -3)( D -4) α GB and ω is related to the ADM mass M through the equation where V D -2 is the volume of unit ( D -2)-sphere. The Hawking temperature T and entropy S for this spacetime are, respectively, and where A = V D -2 r D -2 + is the horizon area. We can also realize that D = 4 is a divergent point in GB gravity. Several authors dedicated to calculate the entropy of GB theory [19]. Therefore we can see, as well known, that in such general theories, different from Einstein's gravity, the horizon entropy is not proportional to the area, it only occurs when α = 0 in Eq. (8) [20]. The locus of the horizon is found as roots of q ( r + ) = 0, where and for the horizon to exist at all, one must also have that In this generalized GB BH, let us compute the SG where the derivative of f ( r ) is given by /negationslash where A = 4 αω . We can see in Eq. (12) the SG in Eq. (11) is directly dependent of the dimensionality of the geometry. Besides, it is a function of the ADM mass via the A term. We can also see that for D = 3 or D = 4 we have an infinite value for SG since α → 0, which means form Eq. (17) that r + ≥ 0, which, from Eq. (7), results in a plausible value for the Hawking temperature and from Eq. (8), the Bekenstein-Hawking entropy is recovered. For r + → 0, i.e., very small BHs, for a D = 3 GB BH, from Eq. (7) we confirm the result that BHs are very hot at the event horizon. Since the entropy The Einstein equation for this metric is is not dependent directly of r + , it keeps the Bekenstein-Hawking form when D = 3 and α → 0. where P = P ( r ) is the radial pressure. But we know that f ( r ) is zero at the horizon, i.e., in r = r + and we have that f ( r + ) = 0, hence, using this information in Eq. (13) we have that where we are using that G = c = /planckover2pi1 = 1. Hence, from the above equations we can write the radial pressure as and from Eq. (11) the final expression for the SG is given by The zeroth law of BH thermodynamics asserts the κ is constant in the event horizon, which means that we have two alternatives. Firstly all the terms in Eq. (16) are constant, obviously. The other one is that there is a compensation such that the final result keeps constant. Let us consider the first alternative as the correct one, So, ω in Eq. (6) is also constant which means that the mass M is constant and all the terms in Eq. (16) are constants. Hence, since the horizon radius r + is constant, from Eq. (15) we can see that the radial pressure is also constant in the event horizon.",
"pages": [
4,
5,
6
]
},
{
"title": "3. BARROW BLACK HOLE",
"content": "Barrow introduced recently in 2020 [13], based on the Covid-19 shape of the virus, which shows a fractal structure, i.e., a fractal framework for the horizon of BHs with finite volume and infinite, or not, area. As a toy model, it can show effects of quantum gravity spacetime foam, with important consequences concerning the entropy of BHs and the Universe. The so-called Barrow-parameter, as will be shown just below, is confined within 0 ≤ ∆ ≤ 1, where ∆ = 0, geometrically speaking, responds for a smooth spacetime structure and ∆ = 1 shows a most intricate geometry. As demonstrated in [13], the Hawking lifetime of BHs is also shortened, but we do not discuss this issue here. Now, let us analyze the Barrow BH with fractal geometry where A Pl = /lscript 2 P is the Planck area, which is equal to 1 in relativistic dimensions. Thermodynamically speaking, for ∆ = 0 we have the Bekenstein-Hawking entropy and for ∆ = 1, /negationslash the entropy for the most intricate geometry. For A H = 4 πr 2 + we can write the entropy derivative in a convenient form such that and the Einstein equations, with an 'Ad hoc' dr + -term in each factor of the equation, and we will use this equation including the radial pressure in a moment. We know that the periodicity in Euclidean time permits us to connect the horizon to the temperature such that Let us use the first law equation which is analogous to Eq. (19) [23] and, using also Eqs. (18)-(21) we have that and hence, the SG for Barrow's BH is Let us now analyze the geometrical features coming from Barrow structure. For as expected. For ∆ = 1 we have that where the presence of r + shows that in a fractal spacetime the SG expression is dependent of the radius, what does not happen in the flat spacetime. From Eq. (20) the temperature of the horizon is where T Barrow is also an effective temperature. It differs by the same factor from the thermodynamical temperature measured by constant-radius detectors [24]. It is well known that the SG is also constant for stationary BH and, in this way, linking both zero laws. in this case, from Eq. (23) ∣ which allows us to write a general relation = ⇒ where Λ o is a constant that depends on the initial conditions of f ( r ). Hence, from Eqs. (13) and (28) we can calculate the radial pressure as a function of the radius and the ∆-parameter. We can see clearly that the radial pressure depends directly on the structure of the spacetime represented by the ∆-parameter in Barrows's formulation. Moreover we can see that for r →∞ the radial pressure goes to zero which is coherent to the fact that as the radius strongly grows, the radial pressure diminishes.",
"pages": [
6,
7,
8
]
},
{
"title": "4. CONCLUSIONS, FINAL REMARKS AND PERSPECTIVES",
"content": "In this work our objective was to show precisely that using the recently developed Barrow's formulation of BHs, we can see that some aspects of the event horizon are affected by the quantum fluctuations that dictate the geometry. In some papers published by us before [15], the thermodynamics of BHs were discussed, but some issues relative exactly to the event horizon was now computed. We have analyzed the SG issue concerning two popular BH formalisms, the GB and the Barrow toy model, which shows that the geometry affects directly the entropy of the BH. Our main motivation concerning SG is because it is a relevant subject since it is equivalent to the thermodynamic temperature in the investigation of BH thermodynamics laws, more specific, the zeroth-law where a direct association between SG and the thermodynamic temperature appears. We calculated the expressions of the SG and radial pressure for both models. We saw that both are dependent of the dimensionality and the structure of the spacetime, in the Barrow's case. In GB case, we saw that the SG remains constant since all the terms are constant or if there is a compensation procedure concerning accretion and radiation processes, and consequently the zeroth-law can be obeyed. By analyzing the final expressions of SG, using particular values for some parameters connected to the dimensionality and the ADM mass, The results recover the well known results that shows that the event horizon is a very hot geometrical place. In Barrow's case, the final expressions of both SG and radial pressure show a strong dependence of geometrical structure through the presence of ∆ since we have to consider ∆ as a constant term because of the zeroth-law. So, an analysis like the one made in [14, 16] is not possible directly, but maybe a compensating factor relative to accretion and radiation simultaneous effects can be constructed. Another perspective is relative to how can the expressions obtained here affects the other laws of BH thermodynamics since we have modified the SG as well as the entropy. The thermodynamical potentials can also be studied. Other gravitation models can also be analyzed, of course. On the other hand, we can also include numerical values in our results and obtain a value for ∆, for example.",
"pages": [
8,
9
]
}
]
2024arXiv240319942L
https://arxiv.org/pdf/2403.19942.pdf
<document>
<section_header_level_1><location><page_1><loc_9><loc_92><loc_92><loc_93></location>Research on high-frequency quasi-periodic oscillations in black bounce-type spacetime</section_header_level_1>
<text><location><page_1><loc_23><loc_88><loc_77><loc_90></location>Jianbo Lu, 1, ∗ Shining Yang, 1, † Yuying Zhang, 1 Liu Yang, 1 and Mou Xu 1</text>
<text><location><page_1><loc_22><loc_86><loc_78><loc_87></location>1 Department of Physics, Liaoning Normal University, Dalian 116029, P. R. China</text>
<text><location><page_1><loc_18><loc_51><loc_83><loc_84></location>This paper investigates the high frequency quasi-periodic oscillations (HFQPOs) phenomenon around the black bounce-type (BBT) spacetime using the resonance models. We calculated the location of the innermost stable circular orbit (ISCO) for different types of celestial bodies, and derived the expression for the epicyclic frequencies of test particles. The results show that the BBT spacetime possesses unique observational characteristics, where the ordering of epicyclic frequencies varies with the regularization parameter a , enabling the excitation of low-order resonances and producing stronger observational signals. Using parametric and forced resonance models, we compared theoretical results with the observed 3:2 twin-peak HFQPOs in microquasars (GRO 1655-40, XTE 1550-564, GRS 1915 + 105 ), analyzed the formation mechanisms of HFQPOs, constrained the parameters of the BBT model, and explored the possible types of celestial objects corresponding to microquasars. The study indicates that, certain parametric resonance conditions (e.g., n = 1 , 2) lead to traversable wormhole models in BBT that closely align with observations. And forced resonance corresponding to BH or wormhole models can be verified through observations. These results deviate from the data fits of the original black-bounce model. It is found that the oscillatory behavior of three types of microquasars can also be explained by particle oscillations generated in BBT theory, providing evidence for exploring the existence of wormholes, under the assumptions of parametric resonance and forced resonance.</text>
<text><location><page_1><loc_18><loc_48><loc_27><loc_49></location>PACS numbers:</text>
<text><location><page_1><loc_18><loc_46><loc_60><loc_47></location>Keywords: black bounce-type; quasi-periodic oscillations; wormholes.</text>
<section_header_level_1><location><page_1><loc_44><loc_39><loc_57><loc_40></location>I. Introduction</section_header_level_1>
<text><location><page_1><loc_9><loc_20><loc_92><loc_36></location>It is widely known that General Relativity (GR) predicts the existence of black holes (BH). In recent years, the study of BH physics has made significant progress, including the discovery of gravitational waves [1] and the imaging of black hole shadows [2, 3]. These observational findings either indirectly or directly confirm the predictions of BH in the universe. However, the predictions of GR regarding BH as being subject to inevitable spacetime singularities result in the eventual breakdown of classical physical laws. Although people have hoped to resolve this issue within the framework of quantum gravity, a reliable theory of quantum gravity remains elusive as of today. Physicists have thus endeavored to tackle this problem through diverse approaches, suggesting notions such as regular black holes [4-20] and singularity-free gravitational collapse models [21-26].</text>
<text><location><page_1><loc_10><loc_18><loc_92><loc_19></location>The idea of regular BH was initially introduced by Bardeen in 1968 [4]. Simpson and Visser proposed a space-</text>
<text><location><page_2><loc_9><loc_79><loc_92><loc_93></location>time metric, known as the black bounce [27], which built upon this idea. By introducing a length scale parameter l to regularize central singularities, this metric offers a comprehensive characterization of various objects including Schwarzschild solution, regular BH, and traversable wormholes. It provides a straightforward method for demonstrating the impacts of quantum gravity [28]. Numerous authors have investigated the physical characteristics of the black bounce metric and its varieties, encompassing various topics such as quasi-periodic oscillations (QPOs), gravitational lensing effects, quasi-normal mode frequencies, shadows, and accretion disks [28-40]. However, research has uncovered inconsistencies between the black bounce model and certain observations [28].</text>
<text><location><page_2><loc_9><loc_40><loc_92><loc_78></location>In addition to BH, wormholes are another significant theoretical prediction of GR. However, in General Relativity, the formation of a wormhole requires the existence of exotic matter that violates the null energy condition [41-43]. Exotic matter is commonly rationalized as quantum fields possessing negative energy density within the framework of quantum gravity physics. Although there is currently no astronomical observation that confirms the existence of wormholes, recent research in wormhole physics has been dedicated to exploring observable signals, which are based on theoretical studies [44-48]. Several studies suggest that visible indications nearby wormholes might comprise induced gravitational lensing [49-52], shadows [53-56], and accretion disk radiation [57, 58]. The exploration of various effects induced by BH and wormholes offers a theoretical foundation for differentiating various types of celestial objects in observations, while also enabling a comprehensive analysis of the central objects' properties. Reference [49] differentiates between Schwarzschild BH and Ellis wormholes through an analysis of Einstein rings and gravitational lensing. Reference [59] employs the kinematic displacement of photon frequencies to differentiate between BH and wormholes. Reference [60] examines the variation in accretion mass among rotating wormholes and Kerr BH with equivalent mass and accretion rate, revealing that the emission spectra from accretion disks can be utilized to discern the geometric shape of wormholes. In this paper, we aim to explore the distinctive features induced by BH and wormholes in the context of black bounce-type (BBT) geometry, utilizing the high-frequency quasi-periodic oscillations (HFQPOs) method. Our aim is to establish a theoretical framework to account for potential observational disparities between the two, and to facilitate the exploration of various compact celestial bodies and their discernment in observations.</text>
<text><location><page_2><loc_9><loc_20><loc_92><loc_39></location>Quasi-periodic oscillations (QPOs), as one of the powerful tools for testing gravitational theories, have been extensively studied by researchers [61-70]. QPOs correspond to peaks observed in the radio-to-X-ray bands of the electromagnetic spectrum emitted by compact objects, as stated in reference [71]. Based on their observed oscillation frequencies, these oscillations are categorized into low-frequency QPOs and high-frequency QPOs. By analyzing the spectra of QPOs [59, 71-74], scientists can extract certain physical information about the central celestial object. Although the specific causes of QPOs are not fully understood, it is often believed that they are induced by precession and resonance phenomena related to the effects of GR [75-77]. In this paper, we apply observations of microquasars to constrain and explore the BBT theoretical model, and investigate the potential physical mechanisms underlying the generation of QPOs.</text>
<text><location><page_2><loc_9><loc_9><loc_92><loc_19></location>The structure of this paper is as follows. Section II briefly introduces the BBT theory [34], and shows the action for the BBT spacetime. In section III, the stable circular orbit regions and the innermost stable circular orbit (ISCO) are investigated for various celestial bodies in BBT spacetime. Section IV centers on particles that experience oscillatory motion around the central celestial object on stable circular orbits, and we compute their inherent radial and azimuthal epicyclic angular frequencies. Furthermore, utilizing models such as parametric resonance and forced resonance in</text>
<text><location><page_3><loc_9><loc_81><loc_92><loc_93></location>HFQPOs, we conduct an analysis of the resonance locations for various types of celestial bodies in BBT spacetime, under different ratios of intrinsic radial and azimuthal epicyclic angular frequencies. In section V of this paper, we employ two different resonance models to fit observational data and impose constraints on the parameter a in the black bounce-type spacetime. In addition, we explore the feasibility of examining various celestial bodies in BBT by using three distinct sets of microquasar oscillation data, and examine the potential physical mechanisms that give rise to HFQPOs. The sixth section concludes the paper.</text>
<section_header_level_1><location><page_3><loc_38><loc_76><loc_63><loc_77></location>II. A black bounce-type metric</section_header_level_1>
<text><location><page_3><loc_10><loc_72><loc_81><loc_73></location>Considering a static spherically symmetric spacetime geometry, its metric can be expressed as [34]:</text>
<formula><location><page_3><loc_36><loc_69><loc_92><loc_70></location>dS 2 = -A ( x ) dt 2 + B ( x ) dx 2 + r 2 ( x ) d Ω 2 , (1)</formula>
<text><location><page_3><loc_9><loc_61><loc_92><loc_67></location>where A ( x ), B ( x ) and r ( x ) are three unspecified functions, the domain of the radial coordinate is x ∈ ( -∞ , + ∞ ), and d Ω 2 = dθ 2 +sin 2 θdϕ 2 describes the line element of a two-dimensional sphere. For the BBT geometry that we are investigating, proposed by Lobo et al. in reference [34], the metric functions can be written as:</text>
<formula><location><page_3><loc_31><loc_57><loc_92><loc_60></location>A ( x ) = B -1 ( x ) = 1 -2 Mx 2 ( x 2 + a 2 ) 3 / 2 ; r 2 ( x ) = x 2 + a 2 , (2)</formula>
<text><location><page_3><loc_9><loc_39><loc_92><loc_55></location>where a and M are two constant parameters. Based on the Fan-Wang mass function [78], Ref.[34] indicates that solution (2) can be as a special case appeared in a class of general metric function: A ( x ) = B -1 ( x ) = 1 -2 m ( x ) Σ( x ) , with m ( x ) = M Σ( x ) x k ( x 2 n + a 2 n ) ( k +1) / (2 n ) and n = 1 and k = 2. For taking other values of constant parameters (e.g. n = 1 and k = 0), expressions (2) will reduce to black bounce model [27]: A ( x ) = B -1 ( x ) = 1 -2 M ( x 2 + a 2 ) 1 / 2 . It is important to provide an explicit form for the action of system that corresponds to solution (2) of the gravitational field equation, which can uplift the status of BBT metric from ad-hoc mathematical model to an exact solution of gravitational theory. Following the method in Ref.[79], the BBT solution (1) with signature ( -, + , + , +) can be given by the following action [80]:</text>
<formula><location><page_3><loc_29><loc_33><loc_92><loc_35></location>S = ∫ d 4 x √ -g [ R -2 κ 2 ( g µν ∂ µ ϕ∂ ν ϕ + V ( ϕ )) -2 κ 2 L ( F ) ] , (3)</formula>
<text><location><page_3><loc_9><loc_30><loc_12><loc_31></location>with</text>
<formula><location><page_3><loc_34><loc_26><loc_92><loc_29></location>V ( ϕ ) = 4 M cos 5 ( ϕκ ) ( 7 sin 2 ( ϕκ ) -8 cos 2 ( ϕκ ) ) 35 κ 2 | q 3 | , (4)</formula>
<formula><location><page_3><loc_38><loc_20><loc_92><loc_24></location>L ( F ) = 4 4 √ 2 F 5 / 4 M (91 -75 √ 2 Fq ) 35 κ 2 √ | q | . (5)</formula>
<text><location><page_3><loc_9><loc_9><loc_92><loc_18></location>Here the parameter a = q is the magnetic charge, and R is the Ricci scalar, g is the determinant of the metric, ϕ is a non-canonical phantom field, κ 2 = 8 πG with the gravitational constant G , V ( ϕ ) is the potential of ϕ , L ( F ) is the Lagrangian for a nonlinear electromagnetic field F µν with F = F µν F µν / 4 = q 2 / 2( a 2 + x 2 ) 2 . Obviously, the action (3) denotes a gravitational system, at which Einstein's gravitational field minimally coupled with a self-interacting phantom scalar field combined with a nonlinear electrodynamics field. It is well known, the phantom field as a famous</text>
<text><location><page_4><loc_9><loc_81><loc_92><loc_93></location>dark energy candidate with the equation of state w < -1, has been wildly applied to interpret the late accelerating expansion of universe. Also, phantom could appear in string theory in the form of negative tension branes, which play an important role in string dualities [79, 81, 82]. In fact, in the framework of GR, one of the necessary conditions for forming a wormhole is that one needs to introduce an amount of exotic matter that violates the null energy condition [83], e.g. the phantom field. A plenty of wormhole solutions with various kinds of phantom matter were proposed [79, 84-87].</text>
<text><location><page_4><loc_9><loc_44><loc_92><loc_80></location>BBT solution has some attractive properties. For example, (I) it is a simple one-parameter extension of the Schwarzschild metric; (II) It is a candidate of regular BH geometry in the framework of GR, then avoiding the singularity of spacetime of BH. In contrast to singular black holes, the BBT metric restores the integrity of spacetime geodesics, because the area of the two-dimensional sphere S = 4 πr 2 (0) = 4 πa 2 is finite at x = 0. The bouncing nature of the radial function can be interpreted as a signal of the existence of a wormhole throat, at which point spacetime is divided into two asymptotically flat regions: x -∈ ( -∞ , 0) , x + ∈ (0 , + ∞ ). Clearly, when a → 0, the wormhole throat vanishes, and the above metric degenerates into the form of a Schwarzschild BH, i.e., A ( x ) ≈ 1 -2 M/r . In the asymptotic limits x → ±∞ and x → 0, metric (2) corresponds to the forms of Schwarzschild solution and de Sitter solution, respectively, ensuring that the curvature scalar does not diverge; (III) BBT as a simple model and a unified treatment of distinct kinds of geometries, it smoothly interpolates between some typical BHs and traversable WH. It can be seen that the above static spherically symmetric metric (2) can describe Schwarzschild BH, double-horizon regular BH, extreme BH, and traversable wormholes for different values of parameter a . Specifically, when a = 0 and M > 0, it is equal to Schwarzschild BH; when 0 < a/M < 4 √ 3 / 9, it describes a regular BH with two horizons; when a/M = 4 √ 3 / 9, it corresponds to an extreme black hole; and when a/M > 4 √ 3 / 9, it represents a traversable wormhole [88]. This paper considers the relevant properties of the BBT theoretical model in conjunction with observational data, given the inconsistencies between the black bounce model and certain observational data [28] and the intriguing properties of the spherical BBT spacetime metric mentioned above.</text>
<section_header_level_1><location><page_4><loc_15><loc_39><loc_86><loc_40></location>III. Stable circular orbits and ISCOs for different types of celestial bodies in BBT spacetime</section_header_level_1>
<text><location><page_4><loc_10><loc_35><loc_66><loc_37></location>In BBT spacetime, the motion of particles follows the following equation [72]:</text>
<formula><location><page_4><loc_21><loc_31><loc_92><loc_34></location>n = 1 2 [ -1 A ( x ) ( ∂S ∂t ) 2 + 1 B ( x ) ( ∂S ∂x ) 2 + 1 r 2 ( x ) ( ∂S ∂θ ) 2 + 1 r 2 ( x ) sin 2 θ ( ∂S ∂ϕ ) 2 ] , (6)</formula>
<text><location><page_4><loc_9><loc_18><loc_92><loc_30></location>here S is the action function, which can be related with the 4-momentum of particle: p µ ≡ ∂S/∂x µ . p µ is defined as p µ = dx µ /dλ with the affine parameter λ . For n = 0, Eq.(6) corresponds to the motion of massless particles (e.g., photons), while n = -1 / 2 corresponds to the case of massive particles. We set θ = π/ 2 (the equatorial plane) without any loss of generality. In the BBT geometry, a thin accretion flow is assumed to move along a Keplerian stable circular orbit, which in this case is represented by ˙ θ = 0. Here θ represents the angular coordinate and the 'dot' denotes the derivative with respect to proper time. The timelike geodesic equations for massive particles are expressed as follows:</text>
<formula><location><page_4><loc_45><loc_14><loc_92><loc_17></location>˙ t = dt dλ = E A ( x ) (7)</formula>
<formula><location><page_4><loc_39><loc_9><loc_92><loc_12></location>dx dλ = √ E 2 -A ( x ) ( J 2 r 2 ( x ) +1 ) (8)</formula>
<formula><location><page_5><loc_44><loc_91><loc_92><loc_93></location>˙ ϕ = dϕ dλ = J r 2 ( x ) . (9)</formula>
<text><location><page_5><loc_9><loc_86><loc_92><loc_89></location>Here, E and J stand for energy and angular momentum, respectively. By utilizing equation (8), we derive the effective potential V eff for the movement of massive particles on the equatorial plane:</text>
<formula><location><page_5><loc_41><loc_82><loc_92><loc_85></location>V eff = A ( x ) ( 1 + J 2 r 2 ( x ) ) . (10)</formula>
<text><location><page_5><loc_9><loc_79><loc_51><loc_80></location>Using the circular orbit condition dV eff /dx = 0, we obtain:</text>
<formula><location><page_5><loc_20><loc_76><loc_92><loc_78></location>2 a 4 M -Mx 4 + J 2 x 2 ( -3 M + √ a 2 + x 2 ) + a 2 ( Mx 2 + J 2 ( 2 M + √ a 2 + x 2 )) = 0 . (11)</formula>
<text><location><page_5><loc_9><loc_66><loc_92><loc_74></location>In general, people can derive the radial coordinate position of a circular orbit based on equation (11). However, for the BBT metric under consideration, we cannot directly obtain an analytical expression for the circular orbit position using equation (11). Through observation, it is evident that equation (11) is quadratic in relation to a particular angular momentum J , thus enabling the determination of circular orbits through the following relationship:</text>
<formula><location><page_5><loc_30><loc_61><loc_92><loc_65></location>J c ± = ± √ M ( -2 a 4 -a 2 x 2 + x 4 ) x 2 ( -3 M + √ a 2 + x 2 ) + a 2 ( 2 M + √ a 2 + x 2 ) . (12)</formula>
<text><location><page_5><loc_9><loc_56><loc_92><loc_60></location>J c + , J c -represents the angular momentum in two possible directions when particles perform circular motion around the central celestial object in the equatorial plane. And the energy of particles on circular orbits is:</text>
<formula><location><page_5><loc_26><loc_51><loc_92><loc_55></location>E = -2 Mx 2 + ( a 2 + x 2 ) 3 / 2 √ ( a 2 + x 2 ) 3 / 2 ( x 2 ( -3 M + √ a 2 + x 2 ) + a 2 ( 2 M + √ a 2 + x 2 )) . (13)</formula>
<text><location><page_5><loc_9><loc_39><loc_92><loc_49></location>In the context of BBT geometry, it is clear that the angular momentum (12) is symmetric with respect to the radial coordinate x . For the purpose of this paper, we have chosen the case of x ≥ 0 for discussion. In order to provide significance to equation (12), it is necessary to impose limitations on the domain of the radial coordinate x and establish the area where particles have the ability to execute circular orbit motion. Calculations reveal that when a ≥ 4 √ 3 M/ 9, the circular orbit interval exists within:</text>
<formula><location><page_5><loc_47><loc_36><loc_92><loc_38></location>| x | ≥ √ 2 a. (14)</formula>
<text><location><page_5><loc_9><loc_33><loc_54><loc_35></location>And when a < 4 √ 3 M/ 9, the circular orbit region is confined to:</text>
<formula><location><page_5><loc_32><loc_28><loc_92><loc_32></location>| x | > √ -a 2 +3 M 2 -10 3 √ 2 a 2 M 2 +9 3 √ 2 M 4 p + p 3 √ 2 , (15)</formula>
<text><location><page_5><loc_9><loc_24><loc_58><loc_27></location>where p = ( 25 a 4 M 2 -90 a 2 M 4 +54 M 6 +5 √ 5 √ 5 a 8 M 4 -4 a 6 M 6 ) 1 / 3 .</text>
<text><location><page_5><loc_9><loc_14><loc_92><loc_23></location>Next, we analyze the stability of circular orbits. Clearly, when dJ c + /dx ≥ 0, it corresponds to stable circular orbits where the angular momentum J c + has a local extremum, namely dJ c + /dx = 0 corresponding to the ISCO. ISCO serves as the inner boundary of the accretion disk and the starting point of electromagnetic radiation, making it crucial in the study of accretion disks around compact objects [89-93]. For the BBT model, we derive using dJ c + /dx ≥ 0, the following:</text>
<formula><location><page_5><loc_21><loc_9><loc_92><loc_13></location>4 a 6 x + x 7 -6 x 5 √ a 2 + x 2 + a 4 ( 9 x 3 -16 x √ a 2 + x 2 ) + a 2 ( 6 x 5 +8 x 3 √ a 2 + x 2 ) 2 √ a 2 + x 2 √ -2 a 4 -a 2 x 2 + x 4 ( x 2 ( -3 + √ a 2 + x 2 ) + a 2 ( 2 + √ a 2 + x 2 )) 3 / 2 ≥ 0 . (16)</formula>
<text><location><page_6><loc_9><loc_77><loc_92><loc_93></location>Equation (16) indicates that the position of stable circular orbits x varies with different values of a , which corresponds to different types of celestial bodies. We establish the relationship between them through numerical calculations (as shown in Figure 1). Without loss of generality, we set M = 1 in this paper. From Figure 1, it can be observed that: when a ≤ 4 √ 3 / 9, there exists an ISCO around celestial bodies (Schwarzschild BH, regular BH, extremal BH). When a > 4 √ 3 / 9, celestial bodies (traversable wormholes) have two ISCOs (for 4 √ 3 / 9 < a ≲ 1 . 050), or one ISCO (for a > 1 . 050). It should be noted that in the case of two ISCOs, there is an unstable circular orbit region between them, where there is a 'vacuum' annular region between the accreting matter around celestial bodies, similar in nature to the Janis-Newman-Winicour spacetime [94].</text>
<figure>
<location><page_6><loc_32><loc_52><loc_69><loc_75></location>
<caption>FIG. 1: Variations of ISCO ( dJ c + /dx = 0) and stable circular orbit positions ( dJ c + /dx ≥ 0) relative to the parameter a for different types of celestial bodies in BBT spacetime.</caption>
</figure>
<text><location><page_6><loc_9><loc_29><loc_92><loc_43></location>Furthermore, Figure 1 reveals that the expression dJ c + /dx > 0 is consistently held when the value of a is larger (e.g., a ≳ 1 . 050 ), thereby indicating our inability to determine the position of ISCO through calculation dJ c + /dx = 0. Since all circular orbits that correspond to dJ c + /dx > 0 are stable, we can calculate the position of ISCO by intersecting J c + with the x -axis. Figure 2 (bottom right) illustrates the variation of J c + relative to x when a ≳ 1 . 050. For instance, consider a = 1 . 2 and 1.5. In addition, to offer a more intuitive depiction of the ISCO properties corresponding to different types of celestial bodies, we also plot J c + and J ex in Figure 2 for specific values of a (the intersection of the two represents ISCO). The expression for J ex can be derived from d 2 V eff /dx 2 = 0:</text>
<formula><location><page_6><loc_32><loc_24><loc_92><loc_29></location>J ex = √ -2 a 4 -3 a 2 x 2 +5 x 4 √ a 2 ( 2 + √ a 2 + x 2 ) + x 2 ( -9 + 4 √ a 2 + x 2 ) . (17)</formula>
<text><location><page_6><loc_9><loc_15><loc_92><loc_23></location>From Figure 2 (top left), it becomes evident that for Schwarzschild BH ( a = 0), regular BH (e.g., considering a = 0 . 5), extreme BH ( a = 4 √ 3 / 9), there exists a single intersection point in their respective J c + versus J ex graphs. If we label the position of this intersection point as x ISCO , then the regions corresponding to stable circular orbits are represented as x ≥ x ISCO , while the unstable circular orbit regions are x < x ISCO .</text>
<text><location><page_6><loc_9><loc_9><loc_92><loc_14></location>For other plots in Figure 2, we show the stable circular orbits for wormholes. Concretely, (1) for the case of a traversable wormhole with two or one photon sphere (4 √ 3 / 9 < a ≤ 2 √ 5 / 5), e.g., taking a = 0 . 8 , 2 √ 5 / 5, as observed in Figure 2 (top right), the plot is divided into two segments, which means there exist two regions of stable circular</text>
<figure>
<location><page_7><loc_13><loc_54><loc_88><loc_93></location>
<caption>FIG. 2: Variation curves of J c + and J ex relative to x for various types of celestial bodies (with varying values of a ), where solid lines represent J c + and dashed lines represent J ex .</caption>
</figure>
<text><location><page_7><loc_9><loc_25><loc_92><loc_46></location>orbits. For the left part of this picture, we can derive the position of the ISCO using the following general relation: x = √ 2 a , which is located at the intersection of the solid line and the horizontal axis. And for the right part, the intersection points of J c + and J ex represent the position: x ISCO . Then x > x ISCO describes stable circular orbits, while x < x ISCO denotes unstable circular orbits; (2) For the case of traversable wormholes with a single photon sphere (2 √ 5 / 5 < a ≲ 1 . 050), the stable circular orbits are also divided into two segments when we set a = 1 as an example, as shown in Figure 2 (bottom left). But unlike the top-right case, the curves of J c + and J ex are continuous for bottom-left picture. The stable circular orbits correspond to the position intervals of √ 2 ≤ x ≲ 2 . 491 and x ≳ 4 . 203, respectively. The interval region between J c + and J ex intersections (2 . 491 ≲ x ≲ 4 . 203) corresponds to unstable circular orbits; (3) When a is taken the larger values, such as a = 1 . 2 or 1 . 5, the stable circular orbit region becomes continuous, and the position of ISCO is given by the intersection of J c + and the x -axis: x = 6 √ 2 / 5 and 3 √ 2 / 2.</text>
<text><location><page_7><loc_9><loc_10><loc_92><loc_24></location>In order to demonstrate properties of the effective potential associated with various types of celestial bodies, we utilize equation (10) to graph the variation curve of the effective potential with respect to the radial coordinate in Figure 3 (left). In order to distinguish the effective potential images of various celestial bodies, a constant parameter J = 3 . 6 is designated. Figure 3 (right) presents a close-up and enlarged image of the effective potential that is specifically targeted. In Figure 3 (top), the positions of the event horizons for Schwarzschild BH ( a = 0), regular BH ( a = 0 . 5), extreme BH ( a = 4 √ 3 / 9) are represented by black, red, and green dashed lines, respectively. When a > 4 √ 3 / 9, the BBT spacetime describes the traversable wormholes, where the event horizon is not present. Furthermore,</text>
<text><location><page_8><loc_9><loc_90><loc_92><loc_93></location>in the zoomed-in view, we use circular and square markers to indicate the positions of stable and unstable circular orbits located outside the event horizons for various celestial bodies, respectively.</text>
<text><location><page_8><loc_31><loc_87><loc_32><loc_88></location>v</text>
<figure>
<location><page_8><loc_13><loc_50><loc_88><loc_88></location>
<caption>FIG. 3: The left pictures shows the effective potential for test particles with J = 3 . 6, where a = 0 corresponds to the Schwarzschild BH, 0 < a < 4 √ 3 / 9 describes the regular BH, a = 4 √ 3 / 9 corresponds to the extremal BH, and a > 4 √ 3 / 9 represents the traversable wormhole. The positions of the event horizons for a =0, a = 0 . 5, and a ≤ 4 √ 3 / 9 are represented by black, red, and green dashed lines, respectively. The right sides are the zoomed-in view of the left pictures, where the dots represent the positions of stable circular orbits, and the squares represent the positions of unstable circular orbits for various celestial bodies.</caption>
</figure>
<text><location><page_8><loc_9><loc_20><loc_92><loc_34></location>Notably, assuming that a particle with an angular momentum of J = 3 . 6, it could own stable circular orbits for all cases of a considered in this paper. Furthermore, consider the particle coming from infinity, there exist unstable circular orbits in the cases of a ≤ 2 √ 5 / 5. For the case of unstable circular orbits, if a ≤ 4 √ 3 / 9, an inward perturbation causes the particle to fall into the black hole and be captured, while an outward perturbation results in the particle flying off to infinity; If a > 4 √ 3 / 9 (e.g. a = 0 . 8, a = 2 √ 5 / 5), an outward perturbation likewise causes the particle to fly off to infinity, but an inward perturbation could not make the particle to fall into the wormhole. In contrast, when dr/dλ = 0, it will return.</text>
<section_header_level_1><location><page_9><loc_9><loc_92><loc_93><loc_93></location>IV. Resonance frequency and resonance position of particles around different types of celestial bodies in BBT</section_header_level_1>
<section_header_level_1><location><page_9><loc_33><loc_88><loc_68><loc_89></location>A. Angular frequency of oscillating particles</section_header_level_1>
<text><location><page_9><loc_9><loc_69><loc_92><loc_85></location>In this section, we explore the frequency of oscillation of test particles around various celestial bodies in BBT spacetime, on stable circular orbits. If the moving particle assumes a slight deviation from the minimum of the effective potential, it follows that the particle will oscillate on a stable circular orbit, thereby achieving epicyclic motion that is controlled by linear harmonic oscillation. Taking into account x = x c + δx , where x c represents the radial coordinate at the minimum of the effective potential, and δx describes the radial perturbation displacement it is a small quantity. On the equatorial plane, the transverse displacement in the presence of a small perturbation δθ is represented as θ = π/ 2 + δθ . Under linear perturbations, the equations governing the particle's epicyclic motion around a stable circular orbit in the radial and latitudinal directions may be represented as follows:</text>
<formula><location><page_9><loc_39><loc_65><loc_92><loc_67></location>δ x + ω 2 x δx = 0 , δ ¨ θ + ω 2 θ δθ = 0 . (18)</formula>
<text><location><page_9><loc_9><loc_58><loc_92><loc_63></location>Here, the 'dot' denotes the derivative with respect to the particle's proper time τ , and ω x (or ω θ ) represents the radial (or latitudinal) angular frequency of the particle undergoing oscillatory motion at the circular orbit position. Considering the Hamiltonian:</text>
<formula><location><page_9><loc_37><loc_54><loc_92><loc_57></location>H = H dyn + H pot = 1 2 g αβ p α p β + m 2 2 , (19)</formula>
<text><location><page_9><loc_9><loc_51><loc_13><loc_52></location>where</text>
<formula><location><page_9><loc_41><loc_47><loc_92><loc_49></location>H dyn = 1 2 ( g xx p 2 x + g θθ p 2 θ ) , (20)</formula>
<formula><location><page_9><loc_39><loc_41><loc_92><loc_44></location>H pot = 1 2 ( g tt E 2 + g ϕϕ J 2 +1 ) , (21)</formula>
<text><location><page_9><loc_9><loc_34><loc_92><loc_40></location>correspond to the kinetic and potential energy parts of the Hamiltonian. Here p x = ∂S ∂x = √ E 2 A ( x ) 2 -1 A ( x ) [ J 2 r ( x ) 2 +1], and p θ = ∂S ∂θ = 0. The angular frequencies ω x 2 and ω θ 2 for the radial and latitudinal epicyclic motion, respectively, can be calculated using the following relationships:</text>
<formula><location><page_9><loc_44><loc_30><loc_92><loc_33></location>ω 2 x = 1 g xx ∂ 2 H pot ∂x 2 , (22)</formula>
<formula><location><page_9><loc_44><loc_24><loc_92><loc_27></location>ω 2 θ = 1 g θθ ∂ 2 H pot ∂θ 2 . (23)</formula>
<text><location><page_9><loc_9><loc_22><loc_56><loc_23></location>For the BBT model studied in this paper, we derive the following:</text>
<formula><location><page_9><loc_22><loc_16><loc_92><loc_21></location>ω x = √ √ √ √ x 6 ( -6 + √ a 2 + x 2 ) +4 a 4 x 2 ( -4 + √ a 2 + x 2 ) + a 2 x 4 ( 8 + 5 √ a 2 + x 2 ) ( a 2 + x 2 ) 7 / 2 ( x 2 ( -3 + √ a 2 + x 2 ) + a 2 ( 2 + √ a 2 + x 2 )) . (24)</formula>
<formula><location><page_9><loc_29><loc_10><loc_92><loc_13></location>ω θ = √ -2 a 2 + x 2 ( a 2 + x 2 ) ( x 2 ( -3 + √ a 2 + x 2 ) + a 2 ( 2 + √ a 2 + x 2 )) . (25)</formula>
<text><location><page_10><loc_9><loc_92><loc_66><loc_93></location>The angular frequency of particle's orbital (or vertical) motion is represented as:</text>
<formula><location><page_10><loc_44><loc_88><loc_92><loc_90></location>ω ϕ = dϕ dt = J g ϕϕ . (26)</formula>
<text><location><page_10><loc_9><loc_74><loc_92><loc_86></location>Clearly, in spherically symmetric spacetimes, we have ω θ = ω ϕ . The primary sources of the QPO phenomenon are considered to be orbital precession and epicyclic motion. Models, such as orbital precession models and resonance models, can be constructed to investigate the behavior of QPOs in celestial bodies. Resonant behavior frequently manifests in accretion disks, enabling researchers to glean valuable insights about the central object and its associated accretion disk by examining QPO phenomena occurring around various celestial bodies. This includes possible excitation of resonance modes, the locations where resonance occurs, and peak frequencies, among other factors [94].</text>
<text><location><page_10><loc_9><loc_26><loc_92><loc_73></location>People typically assume that the epicyclic motion may be caused by the motion of accretion flows inside the accretion disk. Considering that the ISCO serves as the inner boundary of the accretion disk, in our study, we consider the physically meaningful range of the radial coprdinate: x ≥ x ISCO . For the Schwarzschild black hole, the computed values for x ISCO = 6 are obtained. As seen in Figure 4 (first and second rows), for the Schwarzschild BH ( a = 0), regular BH (e.g., with a = 0 . 5 chosen), extremal BH ( a = 4 √ 3 / 9), traversable wormhole with double photon spheres (e.g., with a = 0 . 8 chosen), and traversable wormhole with a single photon sphere ( a = 2 √ 5 / 5). The latitudinal epicyclic angular frequencies of particles undergoing simple harmonic motion in the x ≥ x ISCO region are always greater than the radial epicyclic angular frequencies. For the cases of a = 0 . 8 and a = 2 √ 5 / 5, we plot pictures by only choosing the right segment of stable circular orbits, as shown in Fig. 2 (top right). This choice would not change the conclusions in the BBT spacetime presented below. In the BBT spacetime, the trends of ω x and ω θ Figure 4 (first and second rows), are similar to the Schwarzschild black-hole case, i.e., ω θ monotonically decreases with increasing radial coordinate x and ω x exhibits a single-peaked structure. However, for the traversable wormhole with a single photon sphere ( a > 2 √ 5 / 5), as seen in Figure 4 (third and fourth rows), the ω x and ω θ patterns in the BBT model are notably different from the Schwarzschild BH case. Contrary to the black-bounce results reported in reference [28], we observe the presence of ω x ≥ ω θ in the BBT model. Furthermore, when a = 1, the resulting accretion disk has a ring-like structure, leading to a more complex shapes for ω x and ω θ that need to be represented using piecewise functions. In this case, the region of stable circular orbit is x ISCO1 ≤ x < x orange and x ISCO2 ≤ x , while the region of unstable circular orbits corresponds to x orange ≤ x < x ISCO2 . When a = 1 . 2 and 1.5, it differs from the conclusions presented for the case a ≤ 2 √ 5 / 5 shown in Figure 4 (first and second rows): ω x exhibits a decaying mode with increasing x value, and ω θ has a single-peaked structure. Finally, comparing Figure 4 (first and second rows) and Figure 4 (third and fourth rows), we observe that the case of traversable wormholes with a single photon sphere corresponds to larger angular frequency values.</text>
<section_header_level_1><location><page_10><loc_26><loc_22><loc_74><loc_23></location>B. Study of resonance positions based on the HFQPOs model</section_header_level_1>
<text><location><page_10><loc_9><loc_9><loc_92><loc_19></location>A wealth of observational evidence suggests that in low-mass X -ray binaries (LMXBs) containing black holes, the double peaks of HFQPOs are often observed with a fixed ratio of high-peak and low-peak frequencies, typically in a 3:2 ratio ( ν u : ν l ) [95]. Speculation exists that the phenomenon may be caused by a resonance, which is produced by an oscillatory mechanism within the accretion disk. In the preceding sections, we examined the characteristics of oscillation frequencies at circular orbits for various types of celestial bodies in the uncoupled scenario, where</text>
<figure>
<location><page_11><loc_13><loc_19><loc_88><loc_93></location>
<caption>FIG. 4: Variation of the particle's radial and latitudinal epicyclic motion frequencies relative to the radial coordinate x for different values of a corresponding to different types of celestial bodies. In the figure, the solid black and red lines represent the radial and latitudinal epicyclic motion angular frequencies for the Schwarzschild BH, while the dashed green and blue lines depict the radial and latitudinal epicyclic motion angular frequencies in the BBT geometry. The gray line denotes the location of the ISCO in the BBT geometry, and the orange line represents the position of the innermost unstable circular orbit.</caption>
</figure>
<text><location><page_12><loc_9><loc_70><loc_92><loc_93></location>perturbations δx and δθ were not linked. However, in many specific scenarios, it is often assumed that there may be dissipation, pressure effects, or the influence of forces such as viscosity and magnetic fields inside the accretion disk, as suggested in references [96-98]. This requires taking into account the coupling between δx and δθ , which implies including associated nonlinear terms in the perturbation equations. Due to the existing constraints on the investigation of accretion disk physics, it is a challenging task to offer a universal mathematical equation to characterize the perturbation behavior. A more practical approach is to establish models by taking into account specific physical circumstances in order to discuss the problem. Various theoretical models have been proposed to explain the observed QPOs phenomenon, including the parametric resonance model, the forced resonance model, the Keplerian resonance model, the non-axisymmetric disk oscillation model, and the relativistic precession model [96]. Here, we explore the parametric resonance model and the forced resonance model, which are frequently encountered in the study of black hole physics and epicyclic motion. Considering the perturbation equations:</text>
<formula><location><page_12><loc_26><loc_64><loc_92><loc_66></location>δ x + ω 2 x δx = ω 2 x F x ( δx, δθ, δ ˙ x, δ ˙ θ ) , δ ¨ θ + ω 2 θ δθ = ω 2 θ F θ ( δx, δθ, δ ˙ x, δ ˙ θ ) , (27)</formula>
<text><location><page_12><loc_9><loc_57><loc_92><loc_62></location>where F x and F θ represent two undetermined functions corresponding to the coupling effects caused by perturbation terms. In the parametric resonance model [99], it is assumed that F x = 0 , F θ = hδθδx , and h is constant. In this case, equation (27) becomes:</text>
<formula><location><page_12><loc_33><loc_53><loc_92><loc_55></location>δ x + ω 2 x δx = 0 , δ ¨ θ + ω 2 θ [1 + h cos ( ω x t )] δθ = 0 . (28)</formula>
<text><location><page_12><loc_9><loc_50><loc_80><loc_51></location>According to equation (28), parametric resonance occurs when the following conditions are satisfied:</text>
<formula><location><page_12><loc_39><loc_46><loc_92><loc_49></location>ω x ω θ = ν x ν θ = 2 n , ( n = 1 , 2 , 3 . . . ) . (29)</formula>
<text><location><page_12><loc_9><loc_28><loc_92><loc_44></location>Here ν x = ω x / 2 π, ν θ = ω θ / 2 π , and n denotes positive integers. Clearly, as the resonance parametric n decreases, the resonance phenomenon becomes more pronounced [97]. In the case of a BBT spacetime, when a ≤ 2 √ 5 / 5, we have ω θ > ω x , which prevents the lowest-order resonance parameters ( n = 1 , n = 2) from being excited. This means that for the central celestial bodies (including BH and wormhole) corresponding to this situation, the minimum value of the resonance parametric n can only be 3. However, for larger values of a ( a > 2 √ 5 / 5), because the relationship between the radial and latitudinal epicyclic oscillation frequency values is uncertain (i.e., ω θ > ω x , ω θ < ω x , and ω θ = ω x can all occur), this suggests that low-order resonance parameters ( n = 1 , n = 2) can be excited in such celestial bodies. This is different from what is implied in the case of a ≤ 2 √ 5 / 5.</text>
<text><location><page_12><loc_9><loc_9><loc_92><loc_27></location>By selecting specific values of n in the resonance model (e.g., for cases where resonance is more pronounced: n = 1 , 2 , 3), we plotted the variation of resonance positions x with respect to the parameter a in Figure 5. From Figure 5, we can visually observe the positions where resonance occurs for particles around different types of celestial bodies in the BBT spacetime. Specifically, when n = 1 , 2 (corresponding to ω θ : ω x = 1 : 2 and ω θ : ω x = 1 : 1), these two resonance behaviors can only occur in traversable wormholes with larger throats ( a > 2 √ 5 / 5), and the positions of resonance occurrence move farther away from the center of the radial coordinate as a increases. When n = 3 (corresponding to ω θ : ω x = 3 : 2 ), this resonance mode requires a ≲ 1 . 534. Additionally, for the case of 0 ≤ a ≤ 2 √ 5 / 5, the positions of resonance occurrence move closer to the center of the radial coordinate as a increases. In the case of 2 √ 5 / 5 ≤ a ≲ 1 . 534, we observed that resonance phenomena corresponding to the same value of a can</text>
<text><location><page_13><loc_9><loc_88><loc_92><loc_93></location>occur at two different positions. The specific reason for this phenomenon is not yet clear, and it may be caused by the unique ring-like structure of the accretion disk around BBT wormholes or different physical processes inside the accretion disk, which requires further exploration.</text>
<figure>
<location><page_13><loc_27><loc_49><loc_74><loc_86></location>
<caption>FIG. 5: The positions of resonance phenomena for different types of celestial bodies (with different values of a ) in the parametric resonance model. The dashed line and dotted dashed line correspond to positions on the y -axis labeled as a = 2 √ 5 / 5 and a ≈ 1 . 534, respectively.</caption>
</figure>
<text><location><page_13><loc_9><loc_31><loc_92><loc_39></location>In practical studies of resonance problems, it is often assumed that factors such as viscous or magnetic stresses in the accretion flow lead to the appearance of non-zero forcing terms [96, 100]. Based on this, researchers have established the forced resonance model. In this model, the perturbation equations, which include non-zero forcing terms, can be written as:</text>
<formula><location><page_13><loc_39><loc_27><loc_92><loc_29></location>δ ¨ θ + ω 2 θ δθ = -ω 2 θ δxδθ + F θ ( δθ ) , (30)</formula>
<text><location><page_13><loc_9><loc_22><loc_92><loc_25></location>where δx = B cos ( ω x t ) , F θ correspond to the nonlinear terms related to δθ . When the relationship between the epicyclic frequencies satisfies the following equation:</text>
<formula><location><page_13><loc_45><loc_18><loc_92><loc_21></location>ω θ ω x = ν θ ν x = p q , (31)</formula>
<text><location><page_13><loc_9><loc_9><loc_92><loc_16></location>resonance will be activated. In the equation, p and q are small natural numbers. From Figure 4, we can see that in the case of a ≤ 2 √ 5 / 5, we have ω θ > ω x in the BBT spacetime, which requires p/q > 1. In this case, prominent resonance phenomena can occur in situations where the frequency ratio is ω θ : ω x = p : q = 2 : 1 or 3 : 1 (resonance phenomena for p : q = 3 : 2 are the same as parameter resonance for n = 3, and we won't go into detail here). However, when</text>
<text><location><page_14><loc_9><loc_88><loc_92><loc_94></location>a > 2 √ 5 / 5, as we can see from Figure 4, both ω θ > ω x and ω θ ≤ ω x can occur, indicating that situations with frequency ratios of p : q = 1 : 2 , p : q = 1 : 3, or p : q = 2 : 3 can induce resonance phenomena. We plotted the variation of resonance positions x with respect to the parameter a in the forced resonance model in Figure 6.</text>
<figure>
<location><page_14><loc_27><loc_49><loc_74><loc_86></location>
<caption>FIG. 6: Positions of resonance phenomena for different types of celestial bodies (with different values of a ) in the forced resonance model. The dashed line corresponds to positions on the y -axis labeled as a = 2 √ 5 / 5.</caption>
</figure>
<text><location><page_14><loc_9><loc_26><loc_92><loc_41></location>From Figure 6, we can visually see that in the BBT spacetime, when p : q = 1 : 2 , 1 : 3 or 2 : 3 (corresponding to ω θ : ω x = 1 : 2 , 1 : 3 or 2 : 3 ), these three cases of resonance phenomena can only occur in traversable wormholes with larger throats ( a > 2 √ 5 / 5). When p : q = 2 : 1 or 3 : 1, the occurrence of these two resonance modes requires either a ≲ 1 . 271 or a ≲ 1 . 140. For the cases of 2 √ 5 / 5 ≤ a ≲ 1 . 271 (orange curve) and 2 √ 5 / 5 ≤ a ≲ 1 . 140 (blue curve) in the forced resonance model, similar conclusions to those in the parameter resonance model (as shown in Figure 5 for the 2 √ 5 / 5 ≤ a ≲ 1 . 534 case) can be drawn. This means that for the same parametric value a , the same type of vibration can occur at different positions in the accretion disk.</text>
<section_header_level_1><location><page_14><loc_9><loc_22><loc_94><loc_23></location>V. Fitting observed data to constrain BBT model and exploring potential mechanisms for producing HFQPOs</section_header_level_1>
<text><location><page_14><loc_9><loc_9><loc_92><loc_19></location>It is a well-known fact that in the data of experimentally observed HFQPOs, the high and low frequencies in the double peaks frequently exhibit a fixed ratio of 3:2. Reference [28] studied particles oscillating around a central celestial body in the black-bounce spacetime with resonance models, and discovered that achieving the 3:2 structure observed in microquasars, such as GRO 1655-40, XTE 1550-564, and GRS 1915+105, is not possible. In this section, we explore the frequencies of epicyclic motion of oscillating particles in the BBT geometry and compare our findings with the 3:2</text>
<text><location><page_15><loc_9><loc_88><loc_92><loc_93></location>pattern in HFQPOs observed in microquasars. We investigate the various celestial bodies that microquasars could correspond to and assess the potential mechanisms responsible for producing HFQPOs. In addition, we further limit the BBT theoretical model by fitting it with microquasar data.</text>
<text><location><page_15><loc_9><loc_81><loc_92><loc_87></location>In order to establish a connection between the theoretical values of the epicyclic motion angular frequencies ω for particle's local motion and the observed values, we use the redshift factor to transform equations (24) and (25) as follows:</text>
<formula><location><page_15><loc_46><loc_77><loc_92><loc_80></location>¯ ω = ω x,θ -g tt E , (32)</formula>
<text><location><page_15><loc_9><loc_72><loc_92><loc_75></location>the expression for E can be found in equation (13). To ensure that the physical quantities in the theoretical model have the same dimensions as the corresponding observed quantities, we define:</text>
<formula><location><page_15><loc_41><loc_68><loc_92><loc_71></location>ν i = 1 2 π c 3 GM ¯ ω i , ( i = x, θ ) (33)</formula>
<text><location><page_15><loc_9><loc_65><loc_81><loc_66></location>where c is the speed of light, G is the gravitational constant, and M is the mass of the celestial body.</text>
<section_header_level_1><location><page_15><loc_23><loc_60><loc_77><loc_61></location>A. Studying on the resonance positions based on the HFQPOs model</section_header_level_1>
<text><location><page_15><loc_9><loc_50><loc_92><loc_58></location>We consider the observational data of HFQPOs from three sets of microquasars (as listed in Table 1) [101, 102], which are labeled as GRO 1655-40, XTE 1550-564, and GRS 1915+105. The specific data includes the high and low frequencies in the HFOPOs double peaks, the mass of the central celestial body M/M ⊙ , and its spin ξ . Next, we will apply the observational data listed in Table 1 to constrain and analyze the BBT theory.</text>
<table>
<location><page_15><loc_32><loc_38><loc_68><loc_49></location>
<caption>TABLE I: Observational HFQPOs data for three sets of microquasars.</caption>
</table>
<text><location><page_15><loc_9><loc_9><loc_92><loc_32></location>Firstly, let's consider the popular parametric resonance model. In Figure 7, we calculate the resonance frequencies for particles in the BBT spacetime when they oscillate around different types of central celestial bodies. To ensure that the parametric n can achieve the observed result of ν u : ν l = 3 : 2 for different values of n (e.g., n = 1 , 2 , 3 ), we need to consider the possible correspondence between the observed high and low frequencies of the double peaks and the theoretical epicyclic frequencies. In fact, through calculations, it can be found that for a given n value, the ratio of radial to azimuthal frequencies will be determined, and as a result, the resonance positions and the results of applying observational data to constrain the theoretical model will remain unchanged. As an example, in this paper, we consider the following cases for discussion: when n = 1 , ν u = 3 ν θ , ν l = ν x ; when n = 2, ν u = 3 ν θ , ν l = 2 ν x ; when n = 3 , ν u = ν θ , ν l = ν x . In addition, the three-sets observational HFQPOs data form microquasars listed in Table 1 are plotted in Figure 7, and which are compared with the theoretical values calculated by using the BBT model. We find that under different values of n , in order for the theoretical model to pass the experimental observations of</text>
<text><location><page_16><loc_9><loc_90><loc_92><loc_93></location>microquasars, the constraints on the model parameter a/M with respect to the observational data need to satisfy the results shown in Table 2.</text>
<figure>
<location><page_16><loc_13><loc_52><loc_88><loc_89></location>
<caption>FIG. 7: Variation of particle oscillation frequencies relative to the mass of the central celestial body in the BBT spacetime at the resonance point ν u : ν l = 3 : 2 with taking different values of a/M in the parametric resonance model. Here, resonance parameters n = 1 , 2 , 3 correspond to the top, bottom left, and bottom right, respectively. The observational data for three sets of microquasars are also displayed in the figure.</caption>
</figure>
<table>
<location><page_16><loc_33><loc_31><loc_68><loc_40></location>
<caption>TABLE II: Constraints on the BBT model parameter a/M from the observational data of three sets of microquasars under different resonance parameter values n in the parameter resonance model.</caption>
</table>
<text><location><page_16><loc_9><loc_9><loc_92><loc_23></location>From Figure 7, we can see that the oscillation frequencies of particles located on stable circular orbits in the BBT spacetime can closely match the observational data of the three microquasars when the resonance parameter is set to n = 1 or n = 2 (e.g., when n = 1 , a/M = 3 . 5, and when n = 2 , a/M = 3). This indicates that the observed resonance phenomena can also be generated by particles oscillating around a central celestial body as a wormhole ( a/M > 4 √ 3 / 9) in the BBT spacetime. However, when n = 3, the BBT model deviates significantly from the observational data. Table 2 presents the constraint results of fitting the observational data under assumptions for different frequency ratio to the model parameter a/M . Obviously, for the cases of n = 1 and n = 2, the fitting</text>
<text><location><page_17><loc_9><loc_85><loc_92><loc_93></location>results suggest that the central celestial body corresponds to a wormhole. Furthermore, from Table 2, it can be found that the constraint value of the model parameter a/M for n = 1 is greater than the fitting value of a/M for n = 2. Combining Table 2 and Figure 5, we can conclude that for both n = 1 and n = 2, the resonance occurs near the throat of the wormhole, making QPOs phenomena a tool for probing strong gravity effects.</text>
<section_header_level_1><location><page_17><loc_25><loc_81><loc_75><loc_82></location>B. Data fitting based on the forced resonance model and results</section_header_level_1>
<text><location><page_17><loc_9><loc_57><loc_92><loc_78></location>For models focusing on the relationship between the radial and latitudinal oscillation frequencies, there are typically two types: the parametric resonance model and the forced resonance model. In this section, to analyze other potential mechanisms for generating HFQPOs in the BBT spacetime, we apply observational data to constrain and test the theoretical model based on the forced resonance hypothesis. Similarly, in order to ensure the double peak structure of v u : ν l = 3 : 2 under different p : q ratios in the forced resonance model, we consider the following theoretical expressions for ν u and ν l . For example, when p : q = 2 : 1 , ν u = ν θ + ν x , ν l = ν θ ; when p : q = 3 : 1 , ν u = ν θ , ν l = ν θ -ν x ; when p : q = 1 : 2 , ν u = ν θ + ν x , ν l = ν x ; when p : q = 1 : 3 , ν u = ν x , ν l = ν x -ν θ ; when p : q = 3 : 2 , ν u = ν x , ν l = ν θ . In Figure 8, we compare the theoretically calculated frequency values based on the forced resonance in the BBT spacetime with the observational data from microquasars. We also use astronomical experimental data to constrain the model parameter a/M (results are shown in Table 3).</text>
<figure>
<location><page_17><loc_12><loc_30><loc_89><loc_56></location>
<caption>FIG. 8: Variations of particle oscillation frequencies relative to the mass of the central body in BBT spacetime at ν u : ν l = 3 : 2 within the forced resonance model, where the different values of a/M are taken. We also compare these theoretical results with observational data.</caption>
</figure>
<text><location><page_17><loc_9><loc_9><loc_92><loc_21></location>Based on the constraints provided by fitting the microquasars data (Table 3), we find that the resonant phenomena excited in the BBT theory can be explained through the forced resonance model. Specifically, we observe that: for the frequency ratio p : q = 2 : 1 , black hole in BBT spacetime ( a/M ≤ 4 √ 3 / 9) can be tested against the observational data from XTE 1550-564 and GRS 1915+105. For the case of p : q = 3 : 1, the quasi-periodic oscillations of particles around black holes in BBT spacetime align with the observations of microquasar GRS 1915+105. Moreover, in the BBT wormhole spacetime ( a/M > 4 √ 3 / 9), for cases of taking some specific values of p : q listed in Table 3, the BBT</text>
<table>
<location><page_18><loc_31><loc_81><loc_69><loc_94></location>
<caption>TABLE III: Constraints on the BBT model parameter a/M from the three-sets observational microquasars data under the forced resonance models.</caption>
</table>
<text><location><page_18><loc_9><loc_67><loc_92><loc_73></location>model can meet the requirements tested by the observations of the three types of microquasars. This suggests that the observed oscillatory behavior in these three microquasar classes can be explained by particle oscillations occurring in the BBT wormhole spacetime.</text>
<section_header_level_1><location><page_18><loc_44><loc_62><loc_57><loc_63></location>VI. Conclusion</section_header_level_1>
<text><location><page_18><loc_9><loc_47><loc_92><loc_59></location>Regular black holes were proposed as a solution to the spacetime singularity problem in gravitational physics. The BBT spacetime metric, as proposed by Lobo et al., has the capability to describe various objects such as Schwarzschild BH, regular BH, extremal BH, and traversable wormhole, depending on the varying values of the model parameter a . Following the method shown in Ref.[79, 80], it is found that the BBT solution can be obtained by Einstein's theory of general relativity sourced by a combination of a minimally coupled self-interacting phantom scalar field with a nonzero potential and a nonlinear electromagnetic field.</text>
<text><location><page_18><loc_9><loc_15><loc_92><loc_46></location>In the BBT spacetime studied in this paper, we explored the regions of stable circular orbits and investigated the locations of the ISCOs for various celestial bodies. Research indicates that for both regular black holes and extremal black holes, only a single ISCO exists. In contrast, traversable wormholes can exhibit either one or two ISCOs, depending on the size of the throat. Furthermore, as QPOs are potent tools for testing gravitational theories, our research concentration was placed on particles oscillating on stable circular orbits around central bodies. We investigated the properties of the angular frequencies of their radial and latitudinal epicyclics. It is shown that particles surrounding various types of celestial bodies display unique frequency oscillation characteristics. When the BBT spacetime describes black holes and wormholes with single or double photon spheres (0 ≤ a ≤ 2 √ 5 / 5), particles in the region of x ≥ x ISCO demonstrate a higher radial epicycle frequency than their latitudinal epicycle frequency. The epicycle frequency characteristics in these scenarios resemble those of Schwarzschild black hole, wherein the latitudinal frequencies decrease monotonically with increasing radial coordinates x and possess a single-peaked structure. In contrast, for wormholes with a single photon sphere ( a > 2 √ 5 / 5), there is a result where radial epicycle frequencies are greater than latitudinal epicycle frequencies (in contrast to the results in black hole spacetime), and the epicycle frequency differ significantly from those in Schwarzschild BH, this means that lower-order resonance parameters can be excited, resulting in stronger observational signals.</text>
<text><location><page_18><loc_9><loc_11><loc_92><loc_14></location>The research on the phenomenon of HFQPOs generated by particles around wormholes using microquasar data is still limited. This paper conducts a theoretical study by fitting observational data within the framework of spacetime</text>
<text><location><page_19><loc_9><loc_72><loc_92><loc_93></location>metrics capable of describing both black holes and wormholes simultaneously. Using two resonance models, we offer numerical calculations of resonance occurrence positions in the BBT spacetime for various celestial bodies (differing in a -values) in relation to their corresponding frequencies. Furthermore, we investigate the possibility of utilizing the oscillation data from three microquasars to assess the feasibility of testing the BBT model. The research reveals that the resonance positions move away from the central origin as the value of a increases when the resonance parametric n = 1 or 2, for the case of a > 2 √ 5 / 5. Conversely, in the case of 0 ≤ a ≤ 2 √ 5 / 5, the resonance positions shift closer to the central origin as the value of parameter a increases. Moreover, the research suggests that when parametric resonance is triggered (e.g., n = 1 or 2 ), the observable aligns closely with the traversable wormhole model in the BBT spacetime ( a > 4 √ 3 / 9). And in the forced resonance models, black hole or wormhole models can be tested through observations at different frequency ratios in the radial and latitudinal directions.</text>
<text><location><page_19><loc_9><loc_57><loc_92><loc_71></location>Finally, we used observational data to constrain the regularization parameter a/M in the BBT spacetime (results detailed in Tables 2 and 3) and analyzed the possible mechanisms for the generation of HFQPOs. The study found that, unlike the black bounce spacetime, which cannot be tested by microquasar observation data under the resonance model [28], in the BBT spacetime, the oscillatory behavior of three types of microquasars can also be explained by the particle oscillation phenomenon that occurs in the BBT spacetime under the parameter resonance and forced resonance models. This means that the BBT model improves the poor fit between the black-bounce spacetime and microquasar observational data, while also providing a basis for exploring the existence of wormholes.</text>
<text><location><page_19><loc_9><loc_53><loc_92><loc_56></location>Acknowledgments The research work is supported by the National Natural Science Foundation of China (12175095,12075109 and 11865012), and supported by LiaoNing Revitalization Talents Program (XLYC2007047).</text>
<unordered_list>
<list_item><location><page_19><loc_10><loc_45><loc_50><loc_46></location>[1] B. P. Abbott et al., Phys. Rev. Lett. 116 (2016), 061102.</list_item>
<list_item><location><page_19><loc_10><loc_43><loc_45><loc_44></location>[2] K. Akiyama et al., Astrophys. J. 875 (2019), L1.</list_item>
<list_item><location><page_19><loc_10><loc_41><loc_45><loc_42></location>[3] K. Akiyama et al., Astrophys. J. 875 (2019), L4.</list_item>
<list_item><location><page_19><loc_10><loc_39><loc_74><loc_40></location>[4] J. M. Bardeen, in Proceedings of International Conference GR5, 1968, Tbilisi, USSR, p. 174.</list_item>
<list_item><location><page_19><loc_10><loc_37><loc_46><loc_38></location>[5] S. A. Hayward, Phys. Rev. Lett. 96(2006), 031103.</list_item>
<list_item><location><page_19><loc_10><loc_35><loc_57><loc_36></location>[6] T. A. Roman, P. G. Bergmann, Phys. Rev. D 28 (1983), 1265-1277.</list_item>
<list_item><location><page_19><loc_10><loc_33><loc_58><loc_34></location>[7] J. T. S. S. Junior, M. E. Rodrigues, Eur. Phys. J. C 83(2023), 475 .</list_item>
<list_item><location><page_19><loc_10><loc_31><loc_37><loc_32></location>[8] V. P. Frolov, JHEP 1405(2014), 049.</list_item>
<list_item><location><page_19><loc_10><loc_29><loc_43><loc_30></location>[9] V. P. Frolov, Phys. Rev. D 94 (2016), 104056.</list_item>
<list_item><location><page_19><loc_10><loc_27><loc_51><loc_28></location>[10] V. P. Frolov, A. Zelnikov, Phys. Rev. D 95 (2017), 124028.</list_item>
<list_item><location><page_19><loc_10><loc_25><loc_73><loc_26></location>[11] R. Carballo-Rubio, F. Di Filippo, S. Liberati, C. Pacilio, M. Visser, JHEP 1807 (2018), 023.</list_item>
<list_item><location><page_19><loc_10><loc_23><loc_71><loc_24></location>[12] R. Carballo-Rubio, F. Di Filippo, S. Liberati, M. Visser, Phy. Rev. D. 98 (2018), 124009.</list_item>
<list_item><location><page_19><loc_10><loc_21><loc_58><loc_22></location>[13] Y. Li, Y.G. Miao, Eur. Phys. J. C 82, 503 (2022), [arXiv:2110.14201].</list_item>
<list_item><location><page_19><loc_10><loc_19><loc_67><loc_20></location>[14] C. Lan, Y.G Miao, Y.X. Zang, Eur. Phys. J. C 82, 231 (2022), [arXiv:2109.13556].</list_item>
<list_item><location><page_19><loc_10><loc_18><loc_68><loc_19></location>[15] R.G. Cai, T. Chen, S.J. Wang, X.Y. Yang, JCAP 03 (2023) 043, [arXiv:2210.02078].</list_item>
<list_item><location><page_19><loc_10><loc_16><loc_65><loc_17></location>[16] Y. Ling, M.H. Wu, Class. Quantum Grav. 40 075009 (2023), [arXiv:2109.05974].</list_item>
<list_item><location><page_19><loc_10><loc_14><loc_51><loc_15></location>[17] W. Zeng, Y. Ling, Q.Q. Jiang, G.P. Li, [arXiv:2308.00976].</list_item>
<list_item><location><page_19><loc_10><loc_12><loc_47><loc_13></location>[18] W.D. Guo, S.W. Wei, Y.Y. Liu, [arXiv:2203.13477].</list_item>
<list_item><location><page_19><loc_10><loc_10><loc_69><loc_11></location>[19] S.J. Yang, Y.P. Zhang, S.W. Wei, Y.X. Liu, JHEP 04, 066 (2022), [arXiv:2201.03381].</list_item>
<list_item><location><page_20><loc_10><loc_92><loc_72><loc_93></location>[20] C. Lan, Y.G. Miao, Y.X. Zang, Chin. Phys. C 47, no.5,052001 (2023), [arXiv:2206.08694].</list_item>
<list_item><location><page_20><loc_10><loc_90><loc_49><loc_91></location>[21] R. Torres, F. Fayos, Phys. Lett. B 733 (2014), 169-175.</list_item>
<list_item><location><page_20><loc_10><loc_88><loc_41><loc_89></location>[22] R. Torres, Phys. Lett. B 733 (2014), 21-24.</list_item>
<list_item><location><page_20><loc_10><loc_86><loc_48><loc_87></location>[23] M. R. Mbonye, D. Kazanas, IJPD17 (2008), 165-177.</list_item>
<list_item><location><page_20><loc_10><loc_84><loc_34><loc_85></location>[24] K. Jusufi, Universe 9 (2023), 41.</list_item>
<list_item><location><page_20><loc_10><loc_82><loc_56><loc_83></location>[25] P. Bin'etruy, A. Helou, F. Lamy, Phys. Rev. D 98 (2018), 064058.</list_item>
<list_item><location><page_20><loc_10><loc_80><loc_72><loc_81></location>[26] S.W. Wei, Y.X. Liu, R. B. Mann, Phys. Rev. Lett. 129, 191101 (2022), [arXiv:2208.01932].</list_item>
<list_item><location><page_20><loc_10><loc_78><loc_43><loc_79></location>[27] A. Simpson, M. Visser, JCAP 02 (2019), 042.</list_item>
<list_item><location><page_20><loc_10><loc_76><loc_42><loc_77></location>[28] Z. Stuchl /acute.ts1 ık, J. Vrba, Universe 7 (2021), 279.</list_item>
<list_item><location><page_20><loc_10><loc_74><loc_65><loc_75></location>[29] E. Franzin, S. Liberati, J. Mazza, A. Simpson, M. Visser, JCAP 07 (2021), 036.</list_item>
<list_item><location><page_20><loc_10><loc_72><loc_45><loc_73></location>[30] N. Tsukamoto, Phys. Rev. D 104 (2021), 064022.</list_item>
<list_item><location><page_20><loc_10><loc_70><loc_67><loc_71></location>[31] K. A. Bronnikov, R. A. Konoplya, T. D. Pappas, Phys. Rev. D 103 (2021), 124062.</list_item>
<list_item><location><page_20><loc_10><loc_68><loc_67><loc_69></location>[32] R. Shaikh, K. Pal, T. Sarkar, Mon. Not. Roy. Astron. Soc. 506 (2021), 1229-1236 .</list_item>
<list_item><location><page_20><loc_10><loc_66><loc_45><loc_67></location>[33] N. Tsukamoto, Phys. Rev. D 103 (2021), 024033.</list_item>
<list_item><location><page_20><loc_10><loc_64><loc_84><loc_65></location>[34] F. S. N. Lobo, M. E. Rodrigues, M. V. d. S. Silva, A. Simpson, M. Visser, Phys. Rev. D 103 (2021), 084052.</list_item>
<list_item><location><page_20><loc_10><loc_62><loc_69><loc_64></location>[35] M. Guerrero, G. J. Olmo, D. Rubiera-Garcia, D. S. C. G /acute.ts1 omez, JCAP 08 (2021), 036.</list_item>
<list_item><location><page_20><loc_10><loc_60><loc_47><loc_61></location>[36] H. Huang, J. Yang, Phys. Rev. D 100(2019), 124063.</list_item>
<list_item><location><page_20><loc_10><loc_58><loc_44><loc_59></location>[37] Z. Xu, M. Tang, Eur. Phys. J. C 81 (2021), 863.</list_item>
<list_item><location><page_20><loc_10><loc_56><loc_67><loc_57></location>[38] N. Chatzifotis, E. Papantonopoulos, C. Vlachos, Phys. Rev. D 105 (2022), 064025.</list_item>
<list_item><location><page_20><loc_10><loc_54><loc_43><loc_55></location>[39] Y. Guo, Y. G. Miao, arXiv:2112.01747 [gr-qc].</list_item>
<list_item><location><page_20><loc_10><loc_52><loc_62><loc_53></location>[40] J. Barrientos, A. Cisterna, N. Mora, A. Vigan'o, arXiv:2202.06706 [hep-th].</list_item>
<list_item><location><page_20><loc_10><loc_50><loc_50><loc_51></location>[41] D. Hochberg, M. Visser, Phys. Rev. Lett. 81(1998), 746.</list_item>
<list_item><location><page_20><loc_10><loc_48><loc_50><loc_49></location>[42] D. Hochberg, M. Visser, Phys. Rev. D 58 (1998), 044021.</list_item>
<list_item><location><page_20><loc_10><loc_46><loc_39><loc_47></location>[43] E. Teo, Phys. Rev. D 58 (1998), 024014.</list_item>
<list_item><location><page_20><loc_10><loc_44><loc_41><loc_45></location>[44] C. Bambi, Phys. Rev. D 87 (2013), 107501.</list_item>
<list_item><location><page_20><loc_10><loc_42><loc_51><loc_43></location>[45] D. C. Dai, D. Stojkovic, Phys. Rev. D 100 (2019), 083513.</list_item>
<list_item><location><page_20><loc_10><loc_40><loc_57><loc_41></location>[46] D. C. Dai, D. Minic, D. Stojkovic, Eur. Phys. J. C 80 (2020), 1103.</list_item>
<list_item><location><page_20><loc_10><loc_38><loc_80><loc_39></location>[47] J.B. Lu, M. Xu, J. Guo, R.N. Li, General Relativity and Gravitation (2024) 56:37 [arXiv:2403.01828].</list_item>
<list_item><location><page_20><loc_10><loc_36><loc_80><loc_37></location>[48] J.B. Lu, S.N. Yang, Y. Liu, Y.Y. Zhang, Y. Liu, Eur. Phys. J. Plus (2024) 139:274 [arXiv:2402.17498].</list_item>
<list_item><location><page_20><loc_10><loc_34><loc_60><loc_35></location>[49] N. Tsukamoto, T. Harada, K. Yajima, Phys. Rev. D 86 (2012), 104062.</list_item>
<list_item><location><page_20><loc_10><loc_33><loc_52><loc_34></location>[50] N. Tsukamoto, T. Harada, Phys. Rev. D 95 (2017), 024030.</list_item>
<list_item><location><page_20><loc_10><loc_31><loc_53><loc_32></location>[51] R. Takahashi, H. Asada, Astrophys. J. Lett. 768 (2013), L16.</list_item>
<list_item><location><page_20><loc_10><loc_29><loc_41><loc_30></location>[52] V. Perlick, Phys. Rev. D 69 (2004), 064017.</list_item>
<list_item><location><page_20><loc_10><loc_27><loc_62><loc_28></location>[53] H. Huang, J. Kunz, Y. Jinbo, Z. Cong, Phys. Rev. D 107 (2023), 104060.</list_item>
<list_item><location><page_20><loc_10><loc_25><loc_82><loc_26></location>[54] F. Rahaman, K. N. Singh, Ra. Shaikh, T. Manna, S. Aktar, Class. Quantum Grav. 38 (2021), 215007.</list_item>
<list_item><location><page_20><loc_10><loc_23><loc_68><loc_24></location>[55] M. S. Churilova, R. A. Konoplya, Z. Stuchlik, A. Zhidenko, JCAP 10 (2021), 010.</list_item>
<list_item><location><page_20><loc_10><loc_21><loc_53><loc_22></location>[56] S. Kasuya, M. Kobayashi, Phys. Rev. D 103 (2021), 104050.</list_item>
<list_item><location><page_20><loc_10><loc_19><loc_60><loc_20></location>[57] E. Deligianni, B. Kleihaus, J. Kunz, Phys. Rev. D 104 (2021), 064043.</list_item>
<list_item><location><page_20><loc_10><loc_17><loc_54><loc_18></location>[58] T. Harko, Z. Kovacs, F. Lobo, Phys. Rev. D 79(2009), 064001.</list_item>
<list_item><location><page_20><loc_10><loc_15><loc_53><loc_16></location>[59] H. Wei, T. Jun, Z. TongJie, Phys.Rev.D. 104(2021), 124063.</list_item>
<list_item><location><page_20><loc_10><loc_13><loc_58><loc_14></location>[60] T. Harko, Z. Kov'acs, F. S. N. Lobo, Phys.Rev.D 79 (2009), 064001.</list_item>
<list_item><location><page_20><loc_10><loc_11><loc_46><loc_12></location>[61] L. Stella, M. Vietri, Phys. Rev. Lett. 82 (1999), 17.</list_item>
<list_item><location><page_20><loc_10><loc_9><loc_54><loc_10></location>[62] Z. Stuchlik, A. Kotrlova, Gen. Relativ. Gravit. 41(2009), 1305 .</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_21><loc_10><loc_92><loc_61><loc_93></location>[63] A. Aliev, G. Esmer, P. Talazan,Class. Quantum Grav. 30 (2013), 045010.</list_item>
<list_item><location><page_21><loc_10><loc_90><loc_49><loc_91></location>[64] T. Johannsen, D. Psaltis, Astrophys. J. 726 (2011), 11.</list_item>
<list_item><location><page_21><loc_10><loc_88><loc_68><loc_89></location>[65] A. Maselli, L. Gualtieri, P. Pani, L. Stella, V. Ferrari, Astrophys. J. 801 (2015), 115.</list_item>
<list_item><location><page_21><loc_10><loc_86><loc_48><loc_87></location>[66] F. Vincent, Class. Quantum Grav. 31 (2014), 025010.</list_item>
<list_item><location><page_21><loc_10><loc_84><loc_63><loc_85></location>[67] A. Maselli, P. Pani, L. Gualtieri, V. Ferrari, Phys. Rev. D 92 (2015),083014.</list_item>
<list_item><location><page_21><loc_10><loc_82><loc_44><loc_83></location>[68] C. Bambi, J. C. Astropart, Phys. 09 (2012) 014.</list_item>
<list_item><location><page_21><loc_10><loc_80><loc_64><loc_81></location>[69] Z. Wang, S. Chen, J. Jing, Eur. Phys. J. C (2022) 82:528, [arXiv:2112.02895].</list_item>
<list_item><location><page_21><loc_10><loc_78><loc_58><loc_79></location>[70] S. Chen, Z. Wang, J. Jing, JCAP 06, 043 (2021), [arXiv:2103.11788].</list_item>
<list_item><location><page_21><loc_10><loc_76><loc_68><loc_77></location>[71] J. Rayimbaev, K. F. Dialektopoulos, F. Sarikulov, Eur. Phys. J. C 83 (2023), 572.</list_item>
<list_item><location><page_21><loc_10><loc_74><loc_59><loc_75></location>[72] A. Tursunov, Z. Stuchl'ık, M. Koloˇs, Phys. Rev. D 93 (2016), 084012.</list_item>
<list_item><location><page_21><loc_10><loc_72><loc_47><loc_73></location>[73] L. Stella, M. Vietri, Phys. Rev. Lett. 82 (1999), 17.</list_item>
<list_item><location><page_21><loc_10><loc_70><loc_56><loc_71></location>[74] K. L. Smith, C. R. Tandon, R. V. Wagoner, ApJ 906 (2021), 92.</list_item>
<list_item><location><page_21><loc_10><loc_68><loc_75><loc_69></location>[75] J. Horak, M. Abramowicz, V. Karas, W. Kluzniak, Publ.Astron.Soc.Jap. 56 (2004) 819-822.</list_item>
<list_item><location><page_21><loc_10><loc_66><loc_84><loc_67></location>[76] J. Horak, Astronomical Notes Astronomische Nachrichten, 326 (9):824-829 (2005), [arXiv:astro-ph/0408092].</list_item>
<list_item><location><page_21><loc_10><loc_64><loc_35><loc_65></location>[77] P. Rebusco, PASJ, 56 (2004), 553.</list_item>
<list_item><location><page_21><loc_10><loc_62><loc_60><loc_63></location>[78] Z.Y Fan, X. Wang, Phys. Rev. D 94, 124027 (2016), [arXiv:1610.02636].</list_item>
<list_item><location><page_21><loc_10><loc_60><loc_68><loc_61></location>[79] K. A. Bronnikov, R. K. Walia, Phys. Rev. D 15, 044039 (2022), [arXiv:2112.13198].</list_item>
<list_item><location><page_21><loc_10><loc_58><loc_72><loc_59></location>[80] M. E. Rodrigues, M. V. de S. Silva Phys. Rev. D 107 (2023) 4, 044064, [arXiv:2302.10772].</list_item>
<list_item><location><page_21><loc_10><loc_56><loc_35><loc_57></location>[81] C. M. Hull, JHEP 11, 017 (1998).</list_item>
<list_item><location><page_21><loc_10><loc_54><loc_44><loc_55></location>[82] T. Okuda, T. Takayanagi, JHEP 03, 062 (2006).</list_item>
<list_item><location><page_21><loc_10><loc_52><loc_69><loc_53></location>[83] M. Visser, Lorenzian Wormholes: From Einstein to Hawking (AIP, New York, 1995).</list_item>
<list_item><location><page_21><loc_10><loc_50><loc_41><loc_51></location>[84] H. G. Ellis, J. Math. Phys. 14, 104 (1973).</list_item>
<list_item><location><page_21><loc_10><loc_48><loc_46><loc_49></location>[85] K. A. Bronnikov, Acta Phys. Pol. B 4, 251 (1973).</list_item>
<list_item><location><page_21><loc_10><loc_46><loc_79><loc_47></location>[86] T. Karakasis, E. Papantonopoulos, C. Vlachos, Phys. Rev. D 105, 024006 (2022), [arXiv:2107.09713].</list_item>
<list_item><location><page_21><loc_10><loc_44><loc_60><loc_45></location>[87] K. A. Bronnikov, Phys. Rev. D 106, 064029 (2022), [arXiv:2206.09227].</list_item>
<list_item><location><page_21><loc_10><loc_42><loc_82><loc_43></location>[88] M. Guerrero, G. J. Olmo, D. Rubiera-Garcia, D. S'aez-Chill'on G'omez, Phys. Rev. D 105 (2022), 084057</list_item>
<list_item><location><page_21><loc_10><loc_40><loc_62><loc_41></location>[89] S.Y. Hu, C. Deng, S. Guo, X. Wu, E. Liang, Eur. Phys. J. C (2023) 83:264.</list_item>
<list_item><location><page_21><loc_10><loc_38><loc_67><loc_39></location>[90] I.D. Novikov, K.S. Thorne, in Black Holes, (Gordon and Breach, New York, 1973).</list_item>
<list_item><location><page_21><loc_10><loc_36><loc_70><loc_37></location>[91] P. Bambhaniya, K. Saurabh, K. Jusufi, P. S. Joshi, Phys. Rev. D 105 (2022), 023021.</list_item>
<list_item><location><page_21><loc_10><loc_34><loc_69><loc_35></location>[92] A. Ditta, G. Mustafa, G. Abbas, F. Atamurotov, K. Jusufi, JCAP 08 (2023), 002.</list_item>
<list_item><location><page_21><loc_10><loc_33><loc_52><loc_34></location>[93] K. Hioki, U. Miyamoto, Phys. Rev. D 107 (2023), 044042.</list_item>
<list_item><location><page_21><loc_10><loc_31><loc_76><loc_32></location>[94] E. Deligianni, J. Kunz, P. Nedkova, S. Yazadjiev, R. Zheleva, Phys. Rev. D 104 (2021), 024048.</list_item>
<list_item><location><page_21><loc_10><loc_29><loc_64><loc_30></location>[95] M. Koloˇs, Z. Stuchl'ık, A. Tursunov, Class. Quantum Grav. 32 (2015) 165009.</list_item>
<list_item><location><page_21><loc_10><loc_27><loc_33><loc_28></location>[96] I. Banerjee, [arXiv:2203.10890].</list_item>
<list_item><location><page_21><loc_10><loc_25><loc_46><loc_26></location>[97] P. Rebusco, Publ. Astron. Soc. Jpn. 56(2004), 553.</list_item>
<list_item><location><page_21><loc_10><loc_23><loc_72><loc_24></location>[98] J. Hork, M. Abramowicz, V. Karas, W. Kluzniak, Publ. Astron. Soc. Jpn. 56 (2004), 819.</list_item>
<list_item><location><page_21><loc_10><loc_21><loc_86><loc_22></location>[99] M. A. Abramowicz, V. Karas, W. Kluzniak, W. H. Lee, P. Rebusco, Publ. Astron. Soc. Jap. 55 (2003) 466-467.</list_item>
<list_item><location><page_21><loc_9><loc_18><loc_62><loc_20></location>[100] M. A. Abramowicz, W. Klu /acute.ts1 zniak, Z. Stuchl /acute.ts1 ık, Astro-ph. 436 (2005), 1-8.</list_item>
<list_item><location><page_21><loc_9><loc_17><loc_74><loc_18></location>[101] R. Shafee, J. E. McClintock, R. Narayan, S. W. Davis, Astrophys. J. Lett. 636 (2006), 113-6.</list_item>
<list_item><location><page_21><loc_9><loc_15><loc_68><loc_16></location>[102] R. A. Remillard, J. E. Mcclintock, Annu. Rev. Astron. Astrophys. 44 (2006), 49-92.</list_item>
</unordered_list>
</document>
[
{
"title": "Research on high-frequency quasi-periodic oscillations in black bounce-type spacetime",
"content": "Jianbo Lu, 1, ∗ Shining Yang, 1, † Yuying Zhang, 1 Liu Yang, 1 and Mou Xu 1 1 Department of Physics, Liaoning Normal University, Dalian 116029, P. R. China This paper investigates the high frequency quasi-periodic oscillations (HFQPOs) phenomenon around the black bounce-type (BBT) spacetime using the resonance models. We calculated the location of the innermost stable circular orbit (ISCO) for different types of celestial bodies, and derived the expression for the epicyclic frequencies of test particles. The results show that the BBT spacetime possesses unique observational characteristics, where the ordering of epicyclic frequencies varies with the regularization parameter a , enabling the excitation of low-order resonances and producing stronger observational signals. Using parametric and forced resonance models, we compared theoretical results with the observed 3:2 twin-peak HFQPOs in microquasars (GRO 1655-40, XTE 1550-564, GRS 1915 + 105 ), analyzed the formation mechanisms of HFQPOs, constrained the parameters of the BBT model, and explored the possible types of celestial objects corresponding to microquasars. The study indicates that, certain parametric resonance conditions (e.g., n = 1 , 2) lead to traversable wormhole models in BBT that closely align with observations. And forced resonance corresponding to BH or wormhole models can be verified through observations. These results deviate from the data fits of the original black-bounce model. It is found that the oscillatory behavior of three types of microquasars can also be explained by particle oscillations generated in BBT theory, providing evidence for exploring the existence of wormholes, under the assumptions of parametric resonance and forced resonance. PACS numbers: Keywords: black bounce-type; quasi-periodic oscillations; wormholes.",
"pages": [
1
]
},
{
"title": "I. Introduction",
"content": "It is widely known that General Relativity (GR) predicts the existence of black holes (BH). In recent years, the study of BH physics has made significant progress, including the discovery of gravitational waves [1] and the imaging of black hole shadows [2, 3]. These observational findings either indirectly or directly confirm the predictions of BH in the universe. However, the predictions of GR regarding BH as being subject to inevitable spacetime singularities result in the eventual breakdown of classical physical laws. Although people have hoped to resolve this issue within the framework of quantum gravity, a reliable theory of quantum gravity remains elusive as of today. Physicists have thus endeavored to tackle this problem through diverse approaches, suggesting notions such as regular black holes [4-20] and singularity-free gravitational collapse models [21-26]. The idea of regular BH was initially introduced by Bardeen in 1968 [4]. Simpson and Visser proposed a space- time metric, known as the black bounce [27], which built upon this idea. By introducing a length scale parameter l to regularize central singularities, this metric offers a comprehensive characterization of various objects including Schwarzschild solution, regular BH, and traversable wormholes. It provides a straightforward method for demonstrating the impacts of quantum gravity [28]. Numerous authors have investigated the physical characteristics of the black bounce metric and its varieties, encompassing various topics such as quasi-periodic oscillations (QPOs), gravitational lensing effects, quasi-normal mode frequencies, shadows, and accretion disks [28-40]. However, research has uncovered inconsistencies between the black bounce model and certain observations [28]. In addition to BH, wormholes are another significant theoretical prediction of GR. However, in General Relativity, the formation of a wormhole requires the existence of exotic matter that violates the null energy condition [41-43]. Exotic matter is commonly rationalized as quantum fields possessing negative energy density within the framework of quantum gravity physics. Although there is currently no astronomical observation that confirms the existence of wormholes, recent research in wormhole physics has been dedicated to exploring observable signals, which are based on theoretical studies [44-48]. Several studies suggest that visible indications nearby wormholes might comprise induced gravitational lensing [49-52], shadows [53-56], and accretion disk radiation [57, 58]. The exploration of various effects induced by BH and wormholes offers a theoretical foundation for differentiating various types of celestial objects in observations, while also enabling a comprehensive analysis of the central objects' properties. Reference [49] differentiates between Schwarzschild BH and Ellis wormholes through an analysis of Einstein rings and gravitational lensing. Reference [59] employs the kinematic displacement of photon frequencies to differentiate between BH and wormholes. Reference [60] examines the variation in accretion mass among rotating wormholes and Kerr BH with equivalent mass and accretion rate, revealing that the emission spectra from accretion disks can be utilized to discern the geometric shape of wormholes. In this paper, we aim to explore the distinctive features induced by BH and wormholes in the context of black bounce-type (BBT) geometry, utilizing the high-frequency quasi-periodic oscillations (HFQPOs) method. Our aim is to establish a theoretical framework to account for potential observational disparities between the two, and to facilitate the exploration of various compact celestial bodies and their discernment in observations. Quasi-periodic oscillations (QPOs), as one of the powerful tools for testing gravitational theories, have been extensively studied by researchers [61-70]. QPOs correspond to peaks observed in the radio-to-X-ray bands of the electromagnetic spectrum emitted by compact objects, as stated in reference [71]. Based on their observed oscillation frequencies, these oscillations are categorized into low-frequency QPOs and high-frequency QPOs. By analyzing the spectra of QPOs [59, 71-74], scientists can extract certain physical information about the central celestial object. Although the specific causes of QPOs are not fully understood, it is often believed that they are induced by precession and resonance phenomena related to the effects of GR [75-77]. In this paper, we apply observations of microquasars to constrain and explore the BBT theoretical model, and investigate the potential physical mechanisms underlying the generation of QPOs. The structure of this paper is as follows. Section II briefly introduces the BBT theory [34], and shows the action for the BBT spacetime. In section III, the stable circular orbit regions and the innermost stable circular orbit (ISCO) are investigated for various celestial bodies in BBT spacetime. Section IV centers on particles that experience oscillatory motion around the central celestial object on stable circular orbits, and we compute their inherent radial and azimuthal epicyclic angular frequencies. Furthermore, utilizing models such as parametric resonance and forced resonance in HFQPOs, we conduct an analysis of the resonance locations for various types of celestial bodies in BBT spacetime, under different ratios of intrinsic radial and azimuthal epicyclic angular frequencies. In section V of this paper, we employ two different resonance models to fit observational data and impose constraints on the parameter a in the black bounce-type spacetime. In addition, we explore the feasibility of examining various celestial bodies in BBT by using three distinct sets of microquasar oscillation data, and examine the potential physical mechanisms that give rise to HFQPOs. The sixth section concludes the paper.",
"pages": [
1,
2,
3
]
},
{
"title": "II. A black bounce-type metric",
"content": "Considering a static spherically symmetric spacetime geometry, its metric can be expressed as [34]: where A ( x ), B ( x ) and r ( x ) are three unspecified functions, the domain of the radial coordinate is x ∈ ( -∞ , + ∞ ), and d Ω 2 = dθ 2 +sin 2 θdϕ 2 describes the line element of a two-dimensional sphere. For the BBT geometry that we are investigating, proposed by Lobo et al. in reference [34], the metric functions can be written as: where a and M are two constant parameters. Based on the Fan-Wang mass function [78], Ref.[34] indicates that solution (2) can be as a special case appeared in a class of general metric function: A ( x ) = B -1 ( x ) = 1 -2 m ( x ) Σ( x ) , with m ( x ) = M Σ( x ) x k ( x 2 n + a 2 n ) ( k +1) / (2 n ) and n = 1 and k = 2. For taking other values of constant parameters (e.g. n = 1 and k = 0), expressions (2) will reduce to black bounce model [27]: A ( x ) = B -1 ( x ) = 1 -2 M ( x 2 + a 2 ) 1 / 2 . It is important to provide an explicit form for the action of system that corresponds to solution (2) of the gravitational field equation, which can uplift the status of BBT metric from ad-hoc mathematical model to an exact solution of gravitational theory. Following the method in Ref.[79], the BBT solution (1) with signature ( -, + , + , +) can be given by the following action [80]: with Here the parameter a = q is the magnetic charge, and R is the Ricci scalar, g is the determinant of the metric, ϕ is a non-canonical phantom field, κ 2 = 8 πG with the gravitational constant G , V ( ϕ ) is the potential of ϕ , L ( F ) is the Lagrangian for a nonlinear electromagnetic field F µν with F = F µν F µν / 4 = q 2 / 2( a 2 + x 2 ) 2 . Obviously, the action (3) denotes a gravitational system, at which Einstein's gravitational field minimally coupled with a self-interacting phantom scalar field combined with a nonlinear electrodynamics field. It is well known, the phantom field as a famous dark energy candidate with the equation of state w < -1, has been wildly applied to interpret the late accelerating expansion of universe. Also, phantom could appear in string theory in the form of negative tension branes, which play an important role in string dualities [79, 81, 82]. In fact, in the framework of GR, one of the necessary conditions for forming a wormhole is that one needs to introduce an amount of exotic matter that violates the null energy condition [83], e.g. the phantom field. A plenty of wormhole solutions with various kinds of phantom matter were proposed [79, 84-87]. BBT solution has some attractive properties. For example, (I) it is a simple one-parameter extension of the Schwarzschild metric; (II) It is a candidate of regular BH geometry in the framework of GR, then avoiding the singularity of spacetime of BH. In contrast to singular black holes, the BBT metric restores the integrity of spacetime geodesics, because the area of the two-dimensional sphere S = 4 πr 2 (0) = 4 πa 2 is finite at x = 0. The bouncing nature of the radial function can be interpreted as a signal of the existence of a wormhole throat, at which point spacetime is divided into two asymptotically flat regions: x -∈ ( -∞ , 0) , x + ∈ (0 , + ∞ ). Clearly, when a → 0, the wormhole throat vanishes, and the above metric degenerates into the form of a Schwarzschild BH, i.e., A ( x ) ≈ 1 -2 M/r . In the asymptotic limits x → ±∞ and x → 0, metric (2) corresponds to the forms of Schwarzschild solution and de Sitter solution, respectively, ensuring that the curvature scalar does not diverge; (III) BBT as a simple model and a unified treatment of distinct kinds of geometries, it smoothly interpolates between some typical BHs and traversable WH. It can be seen that the above static spherically symmetric metric (2) can describe Schwarzschild BH, double-horizon regular BH, extreme BH, and traversable wormholes for different values of parameter a . Specifically, when a = 0 and M > 0, it is equal to Schwarzschild BH; when 0 < a/M < 4 √ 3 / 9, it describes a regular BH with two horizons; when a/M = 4 √ 3 / 9, it corresponds to an extreme black hole; and when a/M > 4 √ 3 / 9, it represents a traversable wormhole [88]. This paper considers the relevant properties of the BBT theoretical model in conjunction with observational data, given the inconsistencies between the black bounce model and certain observational data [28] and the intriguing properties of the spherical BBT spacetime metric mentioned above.",
"pages": [
3,
4
]
},
{
"title": "III. Stable circular orbits and ISCOs for different types of celestial bodies in BBT spacetime",
"content": "In BBT spacetime, the motion of particles follows the following equation [72]: here S is the action function, which can be related with the 4-momentum of particle: p µ ≡ ∂S/∂x µ . p µ is defined as p µ = dx µ /dλ with the affine parameter λ . For n = 0, Eq.(6) corresponds to the motion of massless particles (e.g., photons), while n = -1 / 2 corresponds to the case of massive particles. We set θ = π/ 2 (the equatorial plane) without any loss of generality. In the BBT geometry, a thin accretion flow is assumed to move along a Keplerian stable circular orbit, which in this case is represented by ˙ θ = 0. Here θ represents the angular coordinate and the 'dot' denotes the derivative with respect to proper time. The timelike geodesic equations for massive particles are expressed as follows: Here, E and J stand for energy and angular momentum, respectively. By utilizing equation (8), we derive the effective potential V eff for the movement of massive particles on the equatorial plane: Using the circular orbit condition dV eff /dx = 0, we obtain: In general, people can derive the radial coordinate position of a circular orbit based on equation (11). However, for the BBT metric under consideration, we cannot directly obtain an analytical expression for the circular orbit position using equation (11). Through observation, it is evident that equation (11) is quadratic in relation to a particular angular momentum J , thus enabling the determination of circular orbits through the following relationship: J c + , J c -represents the angular momentum in two possible directions when particles perform circular motion around the central celestial object in the equatorial plane. And the energy of particles on circular orbits is: In the context of BBT geometry, it is clear that the angular momentum (12) is symmetric with respect to the radial coordinate x . For the purpose of this paper, we have chosen the case of x ≥ 0 for discussion. In order to provide significance to equation (12), it is necessary to impose limitations on the domain of the radial coordinate x and establish the area where particles have the ability to execute circular orbit motion. Calculations reveal that when a ≥ 4 √ 3 M/ 9, the circular orbit interval exists within: And when a < 4 √ 3 M/ 9, the circular orbit region is confined to: where p = ( 25 a 4 M 2 -90 a 2 M 4 +54 M 6 +5 √ 5 √ 5 a 8 M 4 -4 a 6 M 6 ) 1 / 3 . Next, we analyze the stability of circular orbits. Clearly, when dJ c + /dx ≥ 0, it corresponds to stable circular orbits where the angular momentum J c + has a local extremum, namely dJ c + /dx = 0 corresponding to the ISCO. ISCO serves as the inner boundary of the accretion disk and the starting point of electromagnetic radiation, making it crucial in the study of accretion disks around compact objects [89-93]. For the BBT model, we derive using dJ c + /dx ≥ 0, the following: Equation (16) indicates that the position of stable circular orbits x varies with different values of a , which corresponds to different types of celestial bodies. We establish the relationship between them through numerical calculations (as shown in Figure 1). Without loss of generality, we set M = 1 in this paper. From Figure 1, it can be observed that: when a ≤ 4 √ 3 / 9, there exists an ISCO around celestial bodies (Schwarzschild BH, regular BH, extremal BH). When a > 4 √ 3 / 9, celestial bodies (traversable wormholes) have two ISCOs (for 4 √ 3 / 9 < a ≲ 1 . 050), or one ISCO (for a > 1 . 050). It should be noted that in the case of two ISCOs, there is an unstable circular orbit region between them, where there is a 'vacuum' annular region between the accreting matter around celestial bodies, similar in nature to the Janis-Newman-Winicour spacetime [94]. Furthermore, Figure 1 reveals that the expression dJ c + /dx > 0 is consistently held when the value of a is larger (e.g., a ≳ 1 . 050 ), thereby indicating our inability to determine the position of ISCO through calculation dJ c + /dx = 0. Since all circular orbits that correspond to dJ c + /dx > 0 are stable, we can calculate the position of ISCO by intersecting J c + with the x -axis. Figure 2 (bottom right) illustrates the variation of J c + relative to x when a ≳ 1 . 050. For instance, consider a = 1 . 2 and 1.5. In addition, to offer a more intuitive depiction of the ISCO properties corresponding to different types of celestial bodies, we also plot J c + and J ex in Figure 2 for specific values of a (the intersection of the two represents ISCO). The expression for J ex can be derived from d 2 V eff /dx 2 = 0: From Figure 2 (top left), it becomes evident that for Schwarzschild BH ( a = 0), regular BH (e.g., considering a = 0 . 5), extreme BH ( a = 4 √ 3 / 9), there exists a single intersection point in their respective J c + versus J ex graphs. If we label the position of this intersection point as x ISCO , then the regions corresponding to stable circular orbits are represented as x ≥ x ISCO , while the unstable circular orbit regions are x < x ISCO . For other plots in Figure 2, we show the stable circular orbits for wormholes. Concretely, (1) for the case of a traversable wormhole with two or one photon sphere (4 √ 3 / 9 < a ≤ 2 √ 5 / 5), e.g., taking a = 0 . 8 , 2 √ 5 / 5, as observed in Figure 2 (top right), the plot is divided into two segments, which means there exist two regions of stable circular orbits. For the left part of this picture, we can derive the position of the ISCO using the following general relation: x = √ 2 a , which is located at the intersection of the solid line and the horizontal axis. And for the right part, the intersection points of J c + and J ex represent the position: x ISCO . Then x > x ISCO describes stable circular orbits, while x < x ISCO denotes unstable circular orbits; (2) For the case of traversable wormholes with a single photon sphere (2 √ 5 / 5 < a ≲ 1 . 050), the stable circular orbits are also divided into two segments when we set a = 1 as an example, as shown in Figure 2 (bottom left). But unlike the top-right case, the curves of J c + and J ex are continuous for bottom-left picture. The stable circular orbits correspond to the position intervals of √ 2 ≤ x ≲ 2 . 491 and x ≳ 4 . 203, respectively. The interval region between J c + and J ex intersections (2 . 491 ≲ x ≲ 4 . 203) corresponds to unstable circular orbits; (3) When a is taken the larger values, such as a = 1 . 2 or 1 . 5, the stable circular orbit region becomes continuous, and the position of ISCO is given by the intersection of J c + and the x -axis: x = 6 √ 2 / 5 and 3 √ 2 / 2. In order to demonstrate properties of the effective potential associated with various types of celestial bodies, we utilize equation (10) to graph the variation curve of the effective potential with respect to the radial coordinate in Figure 3 (left). In order to distinguish the effective potential images of various celestial bodies, a constant parameter J = 3 . 6 is designated. Figure 3 (right) presents a close-up and enlarged image of the effective potential that is specifically targeted. In Figure 3 (top), the positions of the event horizons for Schwarzschild BH ( a = 0), regular BH ( a = 0 . 5), extreme BH ( a = 4 √ 3 / 9) are represented by black, red, and green dashed lines, respectively. When a > 4 √ 3 / 9, the BBT spacetime describes the traversable wormholes, where the event horizon is not present. Furthermore, in the zoomed-in view, we use circular and square markers to indicate the positions of stable and unstable circular orbits located outside the event horizons for various celestial bodies, respectively. v Notably, assuming that a particle with an angular momentum of J = 3 . 6, it could own stable circular orbits for all cases of a considered in this paper. Furthermore, consider the particle coming from infinity, there exist unstable circular orbits in the cases of a ≤ 2 √ 5 / 5. For the case of unstable circular orbits, if a ≤ 4 √ 3 / 9, an inward perturbation causes the particle to fall into the black hole and be captured, while an outward perturbation results in the particle flying off to infinity; If a > 4 √ 3 / 9 (e.g. a = 0 . 8, a = 2 √ 5 / 5), an outward perturbation likewise causes the particle to fly off to infinity, but an inward perturbation could not make the particle to fall into the wormhole. In contrast, when dr/dλ = 0, it will return.",
"pages": [
4,
5,
6,
7,
8
]
},
{
"title": "A. Angular frequency of oscillating particles",
"content": "In this section, we explore the frequency of oscillation of test particles around various celestial bodies in BBT spacetime, on stable circular orbits. If the moving particle assumes a slight deviation from the minimum of the effective potential, it follows that the particle will oscillate on a stable circular orbit, thereby achieving epicyclic motion that is controlled by linear harmonic oscillation. Taking into account x = x c + δx , where x c represents the radial coordinate at the minimum of the effective potential, and δx describes the radial perturbation displacement it is a small quantity. On the equatorial plane, the transverse displacement in the presence of a small perturbation δθ is represented as θ = π/ 2 + δθ . Under linear perturbations, the equations governing the particle's epicyclic motion around a stable circular orbit in the radial and latitudinal directions may be represented as follows: Here, the 'dot' denotes the derivative with respect to the particle's proper time τ , and ω x (or ω θ ) represents the radial (or latitudinal) angular frequency of the particle undergoing oscillatory motion at the circular orbit position. Considering the Hamiltonian: where correspond to the kinetic and potential energy parts of the Hamiltonian. Here p x = ∂S ∂x = √ E 2 A ( x ) 2 -1 A ( x ) [ J 2 r ( x ) 2 +1], and p θ = ∂S ∂θ = 0. The angular frequencies ω x 2 and ω θ 2 for the radial and latitudinal epicyclic motion, respectively, can be calculated using the following relationships: For the BBT model studied in this paper, we derive the following: The angular frequency of particle's orbital (or vertical) motion is represented as: Clearly, in spherically symmetric spacetimes, we have ω θ = ω ϕ . The primary sources of the QPO phenomenon are considered to be orbital precession and epicyclic motion. Models, such as orbital precession models and resonance models, can be constructed to investigate the behavior of QPOs in celestial bodies. Resonant behavior frequently manifests in accretion disks, enabling researchers to glean valuable insights about the central object and its associated accretion disk by examining QPO phenomena occurring around various celestial bodies. This includes possible excitation of resonance modes, the locations where resonance occurs, and peak frequencies, among other factors [94]. People typically assume that the epicyclic motion may be caused by the motion of accretion flows inside the accretion disk. Considering that the ISCO serves as the inner boundary of the accretion disk, in our study, we consider the physically meaningful range of the radial coprdinate: x ≥ x ISCO . For the Schwarzschild black hole, the computed values for x ISCO = 6 are obtained. As seen in Figure 4 (first and second rows), for the Schwarzschild BH ( a = 0), regular BH (e.g., with a = 0 . 5 chosen), extremal BH ( a = 4 √ 3 / 9), traversable wormhole with double photon spheres (e.g., with a = 0 . 8 chosen), and traversable wormhole with a single photon sphere ( a = 2 √ 5 / 5). The latitudinal epicyclic angular frequencies of particles undergoing simple harmonic motion in the x ≥ x ISCO region are always greater than the radial epicyclic angular frequencies. For the cases of a = 0 . 8 and a = 2 √ 5 / 5, we plot pictures by only choosing the right segment of stable circular orbits, as shown in Fig. 2 (top right). This choice would not change the conclusions in the BBT spacetime presented below. In the BBT spacetime, the trends of ω x and ω θ Figure 4 (first and second rows), are similar to the Schwarzschild black-hole case, i.e., ω θ monotonically decreases with increasing radial coordinate x and ω x exhibits a single-peaked structure. However, for the traversable wormhole with a single photon sphere ( a > 2 √ 5 / 5), as seen in Figure 4 (third and fourth rows), the ω x and ω θ patterns in the BBT model are notably different from the Schwarzschild BH case. Contrary to the black-bounce results reported in reference [28], we observe the presence of ω x ≥ ω θ in the BBT model. Furthermore, when a = 1, the resulting accretion disk has a ring-like structure, leading to a more complex shapes for ω x and ω θ that need to be represented using piecewise functions. In this case, the region of stable circular orbit is x ISCO1 ≤ x < x orange and x ISCO2 ≤ x , while the region of unstable circular orbits corresponds to x orange ≤ x < x ISCO2 . When a = 1 . 2 and 1.5, it differs from the conclusions presented for the case a ≤ 2 √ 5 / 5 shown in Figure 4 (first and second rows): ω x exhibits a decaying mode with increasing x value, and ω θ has a single-peaked structure. Finally, comparing Figure 4 (first and second rows) and Figure 4 (third and fourth rows), we observe that the case of traversable wormholes with a single photon sphere corresponds to larger angular frequency values.",
"pages": [
9,
10
]
},
{
"title": "B. Study of resonance positions based on the HFQPOs model",
"content": "A wealth of observational evidence suggests that in low-mass X -ray binaries (LMXBs) containing black holes, the double peaks of HFQPOs are often observed with a fixed ratio of high-peak and low-peak frequencies, typically in a 3:2 ratio ( ν u : ν l ) [95]. Speculation exists that the phenomenon may be caused by a resonance, which is produced by an oscillatory mechanism within the accretion disk. In the preceding sections, we examined the characteristics of oscillation frequencies at circular orbits for various types of celestial bodies in the uncoupled scenario, where perturbations δx and δθ were not linked. However, in many specific scenarios, it is often assumed that there may be dissipation, pressure effects, or the influence of forces such as viscosity and magnetic fields inside the accretion disk, as suggested in references [96-98]. This requires taking into account the coupling between δx and δθ , which implies including associated nonlinear terms in the perturbation equations. Due to the existing constraints on the investigation of accretion disk physics, it is a challenging task to offer a universal mathematical equation to characterize the perturbation behavior. A more practical approach is to establish models by taking into account specific physical circumstances in order to discuss the problem. Various theoretical models have been proposed to explain the observed QPOs phenomenon, including the parametric resonance model, the forced resonance model, the Keplerian resonance model, the non-axisymmetric disk oscillation model, and the relativistic precession model [96]. Here, we explore the parametric resonance model and the forced resonance model, which are frequently encountered in the study of black hole physics and epicyclic motion. Considering the perturbation equations: where F x and F θ represent two undetermined functions corresponding to the coupling effects caused by perturbation terms. In the parametric resonance model [99], it is assumed that F x = 0 , F θ = hδθδx , and h is constant. In this case, equation (27) becomes: According to equation (28), parametric resonance occurs when the following conditions are satisfied: Here ν x = ω x / 2 π, ν θ = ω θ / 2 π , and n denotes positive integers. Clearly, as the resonance parametric n decreases, the resonance phenomenon becomes more pronounced [97]. In the case of a BBT spacetime, when a ≤ 2 √ 5 / 5, we have ω θ > ω x , which prevents the lowest-order resonance parameters ( n = 1 , n = 2) from being excited. This means that for the central celestial bodies (including BH and wormhole) corresponding to this situation, the minimum value of the resonance parametric n can only be 3. However, for larger values of a ( a > 2 √ 5 / 5), because the relationship between the radial and latitudinal epicyclic oscillation frequency values is uncertain (i.e., ω θ > ω x , ω θ < ω x , and ω θ = ω x can all occur), this suggests that low-order resonance parameters ( n = 1 , n = 2) can be excited in such celestial bodies. This is different from what is implied in the case of a ≤ 2 √ 5 / 5. By selecting specific values of n in the resonance model (e.g., for cases where resonance is more pronounced: n = 1 , 2 , 3), we plotted the variation of resonance positions x with respect to the parameter a in Figure 5. From Figure 5, we can visually observe the positions where resonance occurs for particles around different types of celestial bodies in the BBT spacetime. Specifically, when n = 1 , 2 (corresponding to ω θ : ω x = 1 : 2 and ω θ : ω x = 1 : 1), these two resonance behaviors can only occur in traversable wormholes with larger throats ( a > 2 √ 5 / 5), and the positions of resonance occurrence move farther away from the center of the radial coordinate as a increases. When n = 3 (corresponding to ω θ : ω x = 3 : 2 ), this resonance mode requires a ≲ 1 . 534. Additionally, for the case of 0 ≤ a ≤ 2 √ 5 / 5, the positions of resonance occurrence move closer to the center of the radial coordinate as a increases. In the case of 2 √ 5 / 5 ≤ a ≲ 1 . 534, we observed that resonance phenomena corresponding to the same value of a can occur at two different positions. The specific reason for this phenomenon is not yet clear, and it may be caused by the unique ring-like structure of the accretion disk around BBT wormholes or different physical processes inside the accretion disk, which requires further exploration. In practical studies of resonance problems, it is often assumed that factors such as viscous or magnetic stresses in the accretion flow lead to the appearance of non-zero forcing terms [96, 100]. Based on this, researchers have established the forced resonance model. In this model, the perturbation equations, which include non-zero forcing terms, can be written as: where δx = B cos ( ω x t ) , F θ correspond to the nonlinear terms related to δθ . When the relationship between the epicyclic frequencies satisfies the following equation: resonance will be activated. In the equation, p and q are small natural numbers. From Figure 4, we can see that in the case of a ≤ 2 √ 5 / 5, we have ω θ > ω x in the BBT spacetime, which requires p/q > 1. In this case, prominent resonance phenomena can occur in situations where the frequency ratio is ω θ : ω x = p : q = 2 : 1 or 3 : 1 (resonance phenomena for p : q = 3 : 2 are the same as parameter resonance for n = 3, and we won't go into detail here). However, when a > 2 √ 5 / 5, as we can see from Figure 4, both ω θ > ω x and ω θ ≤ ω x can occur, indicating that situations with frequency ratios of p : q = 1 : 2 , p : q = 1 : 3, or p : q = 2 : 3 can induce resonance phenomena. We plotted the variation of resonance positions x with respect to the parameter a in the forced resonance model in Figure 6. From Figure 6, we can visually see that in the BBT spacetime, when p : q = 1 : 2 , 1 : 3 or 2 : 3 (corresponding to ω θ : ω x = 1 : 2 , 1 : 3 or 2 : 3 ), these three cases of resonance phenomena can only occur in traversable wormholes with larger throats ( a > 2 √ 5 / 5). When p : q = 2 : 1 or 3 : 1, the occurrence of these two resonance modes requires either a ≲ 1 . 271 or a ≲ 1 . 140. For the cases of 2 √ 5 / 5 ≤ a ≲ 1 . 271 (orange curve) and 2 √ 5 / 5 ≤ a ≲ 1 . 140 (blue curve) in the forced resonance model, similar conclusions to those in the parameter resonance model (as shown in Figure 5 for the 2 √ 5 / 5 ≤ a ≲ 1 . 534 case) can be drawn. This means that for the same parametric value a , the same type of vibration can occur at different positions in the accretion disk.",
"pages": [
10,
12,
13,
14
]
},
{
"title": "V. Fitting observed data to constrain BBT model and exploring potential mechanisms for producing HFQPOs",
"content": "It is a well-known fact that in the data of experimentally observed HFQPOs, the high and low frequencies in the double peaks frequently exhibit a fixed ratio of 3:2. Reference [28] studied particles oscillating around a central celestial body in the black-bounce spacetime with resonance models, and discovered that achieving the 3:2 structure observed in microquasars, such as GRO 1655-40, XTE 1550-564, and GRS 1915+105, is not possible. In this section, we explore the frequencies of epicyclic motion of oscillating particles in the BBT geometry and compare our findings with the 3:2 pattern in HFQPOs observed in microquasars. We investigate the various celestial bodies that microquasars could correspond to and assess the potential mechanisms responsible for producing HFQPOs. In addition, we further limit the BBT theoretical model by fitting it with microquasar data. In order to establish a connection between the theoretical values of the epicyclic motion angular frequencies ω for particle's local motion and the observed values, we use the redshift factor to transform equations (24) and (25) as follows: the expression for E can be found in equation (13). To ensure that the physical quantities in the theoretical model have the same dimensions as the corresponding observed quantities, we define: where c is the speed of light, G is the gravitational constant, and M is the mass of the celestial body.",
"pages": [
14,
15
]
},
{
"title": "A. Studying on the resonance positions based on the HFQPOs model",
"content": "We consider the observational data of HFQPOs from three sets of microquasars (as listed in Table 1) [101, 102], which are labeled as GRO 1655-40, XTE 1550-564, and GRS 1915+105. The specific data includes the high and low frequencies in the HFOPOs double peaks, the mass of the central celestial body M/M ⊙ , and its spin ξ . Next, we will apply the observational data listed in Table 1 to constrain and analyze the BBT theory. Firstly, let's consider the popular parametric resonance model. In Figure 7, we calculate the resonance frequencies for particles in the BBT spacetime when they oscillate around different types of central celestial bodies. To ensure that the parametric n can achieve the observed result of ν u : ν l = 3 : 2 for different values of n (e.g., n = 1 , 2 , 3 ), we need to consider the possible correspondence between the observed high and low frequencies of the double peaks and the theoretical epicyclic frequencies. In fact, through calculations, it can be found that for a given n value, the ratio of radial to azimuthal frequencies will be determined, and as a result, the resonance positions and the results of applying observational data to constrain the theoretical model will remain unchanged. As an example, in this paper, we consider the following cases for discussion: when n = 1 , ν u = 3 ν θ , ν l = ν x ; when n = 2, ν u = 3 ν θ , ν l = 2 ν x ; when n = 3 , ν u = ν θ , ν l = ν x . In addition, the three-sets observational HFQPOs data form microquasars listed in Table 1 are plotted in Figure 7, and which are compared with the theoretical values calculated by using the BBT model. We find that under different values of n , in order for the theoretical model to pass the experimental observations of microquasars, the constraints on the model parameter a/M with respect to the observational data need to satisfy the results shown in Table 2. From Figure 7, we can see that the oscillation frequencies of particles located on stable circular orbits in the BBT spacetime can closely match the observational data of the three microquasars when the resonance parameter is set to n = 1 or n = 2 (e.g., when n = 1 , a/M = 3 . 5, and when n = 2 , a/M = 3). This indicates that the observed resonance phenomena can also be generated by particles oscillating around a central celestial body as a wormhole ( a/M > 4 √ 3 / 9) in the BBT spacetime. However, when n = 3, the BBT model deviates significantly from the observational data. Table 2 presents the constraint results of fitting the observational data under assumptions for different frequency ratio to the model parameter a/M . Obviously, for the cases of n = 1 and n = 2, the fitting results suggest that the central celestial body corresponds to a wormhole. Furthermore, from Table 2, it can be found that the constraint value of the model parameter a/M for n = 1 is greater than the fitting value of a/M for n = 2. Combining Table 2 and Figure 5, we can conclude that for both n = 1 and n = 2, the resonance occurs near the throat of the wormhole, making QPOs phenomena a tool for probing strong gravity effects.",
"pages": [
15,
16,
17
]
},
{
"title": "B. Data fitting based on the forced resonance model and results",
"content": "For models focusing on the relationship between the radial and latitudinal oscillation frequencies, there are typically two types: the parametric resonance model and the forced resonance model. In this section, to analyze other potential mechanisms for generating HFQPOs in the BBT spacetime, we apply observational data to constrain and test the theoretical model based on the forced resonance hypothesis. Similarly, in order to ensure the double peak structure of v u : ν l = 3 : 2 under different p : q ratios in the forced resonance model, we consider the following theoretical expressions for ν u and ν l . For example, when p : q = 2 : 1 , ν u = ν θ + ν x , ν l = ν θ ; when p : q = 3 : 1 , ν u = ν θ , ν l = ν θ -ν x ; when p : q = 1 : 2 , ν u = ν θ + ν x , ν l = ν x ; when p : q = 1 : 3 , ν u = ν x , ν l = ν x -ν θ ; when p : q = 3 : 2 , ν u = ν x , ν l = ν θ . In Figure 8, we compare the theoretically calculated frequency values based on the forced resonance in the BBT spacetime with the observational data from microquasars. We also use astronomical experimental data to constrain the model parameter a/M (results are shown in Table 3). Based on the constraints provided by fitting the microquasars data (Table 3), we find that the resonant phenomena excited in the BBT theory can be explained through the forced resonance model. Specifically, we observe that: for the frequency ratio p : q = 2 : 1 , black hole in BBT spacetime ( a/M ≤ 4 √ 3 / 9) can be tested against the observational data from XTE 1550-564 and GRS 1915+105. For the case of p : q = 3 : 1, the quasi-periodic oscillations of particles around black holes in BBT spacetime align with the observations of microquasar GRS 1915+105. Moreover, in the BBT wormhole spacetime ( a/M > 4 √ 3 / 9), for cases of taking some specific values of p : q listed in Table 3, the BBT model can meet the requirements tested by the observations of the three types of microquasars. This suggests that the observed oscillatory behavior in these three microquasar classes can be explained by particle oscillations occurring in the BBT wormhole spacetime.",
"pages": [
17,
18
]
},
{
"title": "VI. Conclusion",
"content": "Regular black holes were proposed as a solution to the spacetime singularity problem in gravitational physics. The BBT spacetime metric, as proposed by Lobo et al., has the capability to describe various objects such as Schwarzschild BH, regular BH, extremal BH, and traversable wormhole, depending on the varying values of the model parameter a . Following the method shown in Ref.[79, 80], it is found that the BBT solution can be obtained by Einstein's theory of general relativity sourced by a combination of a minimally coupled self-interacting phantom scalar field with a nonzero potential and a nonlinear electromagnetic field. In the BBT spacetime studied in this paper, we explored the regions of stable circular orbits and investigated the locations of the ISCOs for various celestial bodies. Research indicates that for both regular black holes and extremal black holes, only a single ISCO exists. In contrast, traversable wormholes can exhibit either one or two ISCOs, depending on the size of the throat. Furthermore, as QPOs are potent tools for testing gravitational theories, our research concentration was placed on particles oscillating on stable circular orbits around central bodies. We investigated the properties of the angular frequencies of their radial and latitudinal epicyclics. It is shown that particles surrounding various types of celestial bodies display unique frequency oscillation characteristics. When the BBT spacetime describes black holes and wormholes with single or double photon spheres (0 ≤ a ≤ 2 √ 5 / 5), particles in the region of x ≥ x ISCO demonstrate a higher radial epicycle frequency than their latitudinal epicycle frequency. The epicycle frequency characteristics in these scenarios resemble those of Schwarzschild black hole, wherein the latitudinal frequencies decrease monotonically with increasing radial coordinates x and possess a single-peaked structure. In contrast, for wormholes with a single photon sphere ( a > 2 √ 5 / 5), there is a result where radial epicycle frequencies are greater than latitudinal epicycle frequencies (in contrast to the results in black hole spacetime), and the epicycle frequency differ significantly from those in Schwarzschild BH, this means that lower-order resonance parameters can be excited, resulting in stronger observational signals. The research on the phenomenon of HFQPOs generated by particles around wormholes using microquasar data is still limited. This paper conducts a theoretical study by fitting observational data within the framework of spacetime metrics capable of describing both black holes and wormholes simultaneously. Using two resonance models, we offer numerical calculations of resonance occurrence positions in the BBT spacetime for various celestial bodies (differing in a -values) in relation to their corresponding frequencies. Furthermore, we investigate the possibility of utilizing the oscillation data from three microquasars to assess the feasibility of testing the BBT model. The research reveals that the resonance positions move away from the central origin as the value of a increases when the resonance parametric n = 1 or 2, for the case of a > 2 √ 5 / 5. Conversely, in the case of 0 ≤ a ≤ 2 √ 5 / 5, the resonance positions shift closer to the central origin as the value of parameter a increases. Moreover, the research suggests that when parametric resonance is triggered (e.g., n = 1 or 2 ), the observable aligns closely with the traversable wormhole model in the BBT spacetime ( a > 4 √ 3 / 9). And in the forced resonance models, black hole or wormhole models can be tested through observations at different frequency ratios in the radial and latitudinal directions. Finally, we used observational data to constrain the regularization parameter a/M in the BBT spacetime (results detailed in Tables 2 and 3) and analyzed the possible mechanisms for the generation of HFQPOs. The study found that, unlike the black bounce spacetime, which cannot be tested by microquasar observation data under the resonance model [28], in the BBT spacetime, the oscillatory behavior of three types of microquasars can also be explained by the particle oscillation phenomenon that occurs in the BBT spacetime under the parameter resonance and forced resonance models. This means that the BBT model improves the poor fit between the black-bounce spacetime and microquasar observational data, while also providing a basis for exploring the existence of wormholes. Acknowledgments The research work is supported by the National Natural Science Foundation of China (12175095,12075109 and 11865012), and supported by LiaoNing Revitalization Talents Program (XLYC2007047).",
"pages": [
18,
19
]
}
]
2024arXiv240505377M
https://arxiv.org/pdf/2405.05377.pdf
<document>
<section_header_level_1><location><page_1><loc_17><loc_84><loc_83><loc_91></location>Transverse expansion of the metric at null hypersurfaces I. Uniqueness and application to Killing horizons</section_header_level_1>
<text><location><page_1><loc_22><loc_78><loc_78><loc_82></location>Marc Mars ∗ and Gabriel S'anchez-P'erez † Departamento de F'ısica Fundamental, Universidad de Salamanca Plaza de la Merced s/n, 37008 Salamanca, Spain</text>
<text><location><page_1><loc_44><loc_75><loc_56><loc_76></location>May 10, 2024</text>
<section_header_level_1><location><page_1><loc_46><loc_69><loc_54><loc_70></location>Abstract</section_header_level_1>
<text><location><page_1><loc_17><loc_46><loc_83><loc_68></location>This is the first in a series of two papers where we analyze the transverse expansion of the metric on a general null hypersurface. In this paper we obtain general geometric identities relating the transverse derivatives of the ambient Ricci tensor and the transverse expansion of the metric at the null hypersurface. We also explore the case where the hypersurface exhibits a generalized symmetry generator, namely a privileged vector field in the ambient space which, at the hypersurface, is null and tangent (including the possibility of zeroes). This covers the Killing, homothetic, or conformal horizon cases, and, more generally, any situation where detailed information on the deformation tensor of the symmetry generator is available. Our approach is entirely covariant, independent on any field equations, and does not make any assumptions regarding the topology or dimension of the null hypersurface. As an application we prove that the full transverse expansion of the spacetime metric at a nondegenerate Killing horizon (also allowing for bifurcation surfaces) is uniquely determined in terms of abstract data on the horizon and the tower of derivatives of the ambient Ricci tensor at the horizon. In particular, the transverse expansion of the metric in Λ-vacuum spacetimes admitting a non-degenerate horizon is uniquely determined in terms of abstract data at the horizon.</text>
<section_header_level_1><location><page_1><loc_12><loc_42><loc_31><loc_43></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_12><loc_14><loc_88><loc_40></location>The initial value problem of the Einstein field equations plays a central role in General Relativity. The fundamental breakthrough of Choquet-Buhat established that by specifying an abstract n -dimensional Riemannian manifold (Σ , h ) along with a symmetric, two covariant tensor field K subject to constraint equations, an ( n +1)-dimensional spacetime ( M , g ) solution of the Einstein equations exists where (Σ , h, K ) is embedded [4, 6, 35]. Another important initial value problem in general relativity is the characteristic one, where the data is posed on a pair of null hypersurfaces that intersect transversely. Rendall's work [34] demonstrates that proving the existence of solutions to the Einstein equations in a neighborhood of the intersecting surface only requires prescribing the full spacetime metric (but not its transverse derivatives) on the initial hypersurfaces. In [18] Luk extends Rendall's result when the intersection is a two-sphere by proving that the spacetime exists on a neighbourhood of the full two null hypersurfaces. In a recent work [10] the authors show that the same result holds regardless the dimension and topology of the intersection. Various other approaches to this problem can be found in [8, 3]. An initial value problem is also known to be well-posed when the data is given on the future null cone of a point [5]. Recent research [30, 29] has also approached the characteristic problem from a more geometric perspective, incorporating transverse derivatives of the metric as initial data (and thus adding the corresponding constraint equations).</text>
<text><location><page_2><loc_12><loc_81><loc_88><loc_91></location>In this paper we analyze the situation when initial data is posed on a single null hypersurface. From a causal perspective, the absence of a second null hypersurface implies that extra information coming from the past can potentially influence the solution, hence spoiling uniqueness. However, in cases where the solution exhibits some symmetry, such as Killing vectors, initial data on a single null hypersurface may suffice to establish a unique solution to the Einstein equations. This issue has been extensively studied in recent years, for example when the hypersurface is a homothetic or a Killing horizon, as we review next.</text>
<text><location><page_2><loc_12><loc_53><loc_88><loc_79></location>In [11] C. Fefferman and R. Graham introduced the so-called ambient metric , inspired by the idea that the light-cone of a point in Minkowski spacetime of dimension n + 2 encodes the conformal structure of the n -sphere. In order to generalize this idea to an arbitrary conformal class, their goal was to construct a homothetic horizon from the given conformal class, and embed it into an ambient manifold Ricci flat to infinite order. Fefferman and Graham proved that in the cases where the dimension of the conformal class is odd, the Einstein equations uniquely determine the full transverse expansion of the ambient metric along a transverse null coordinate at the horizon. However, in even dimensions, additional data is required to fully determine the expansion and, in addition there appears a so-called obstruction tensor whose vanishing or not determines whether the ambient metric can be Ricci flat to all orders at the horizon or not. When the obstruction tensor does not vanish, one can still construct an ambient metric which is Ricci flat to infinite order at the horizon by including log terms in the expansion. This gives rise to the so-called 'generalized ambient metric' which exhibits only finite differentiability at the horizon. In a remarkable work [36] the authors prove that this generalized ambient metric is not only Ricci flat to infinite order but in fact exactly Ricci flat. It is important to note that in both scenarios, the data is not constrained by any equations, unlike in the spacelike and characteristic Cauchy problems.</text>
<text><location><page_2><loc_12><loc_32><loc_88><loc_51></location>Another example where the curvature conditions of the ambient spacetime fix the transverse expansion is a non-degenerate Killing horizon. In [32] Moncrief shows that the transverse expansion of any 3-dimensional, non-degenerate Killing horizon on a Ricci flat spacetime can be described by six functions at the horizon. This analysis was done in a particular coordinate system, which makes it difficult to determine whether two spacetimes are isometric to infinite order at the horizon or not. In a recent work [16], the authors study this problem from a geometric perspective and establish that, for Ricci flat spacetimes, the complete expansion along a null transverse vector field is determined by 'non-degenerate Killing horizon data', namely a triple ( H , σ , V ) where σ is a Riemannian metric and V is a non-trivial Killing vector field V on ( H , σ ) with constant norm. Although this analysis characterizes Killing horizons geometrically, it excludes the possibility of V having zeros (and consequently bifurcation surfaces on the horizon) and the field equations are restricted to vacuum with Λ = 0.</text>
<text><location><page_2><loc_12><loc_9><loc_88><loc_31></location>The last example we want to illustrate is the degenerate Killing horizon case. Given a degenerate Killing horizon, its so-called near horizon limit [17] is characterized by a function F , a one-form ω , and a metric h on the cross-sections. In a recent work [14] the authors focus on establishing a uniqueness theorem for extremal Schwarzschild-de Sitter spacetime. For a specific value of the mass parameter, this spacetime admits a degenerate Killing horizon with compact, maximally symmetric cross-sections, thus possessing an associated near horizon limit. By staticity and compactness arguments, it can be proven that the one-form ω vanishes [9, 2, 37], simplifying the data to the function F and the metric h . By solving the Λ-vacuum equations order by order, which translates into elliptic equations for the transverse expansion, the authors show that all the transverse derivatives of the metric at the horizon can be computed from the data ( h, F ) and that they agree either with those of extremal Schwarzschild-de Sitter or with its near horizon geometry (Nariai). Consequently, in the case of a real analytic spacetime, they can prove that in a neighbourhood of the horizon the spacetime is isometric to one of these two solutions. This analysis has been extended to electrovacuum in [13].</text>
<text><location><page_3><loc_12><loc_76><loc_88><loc_90></location>Motivated by the previous examples, in this paper we prove general identities that relate the transverse derivatives of the ambient Ricci tensor with the transverse expansion of the metric at an arbitrary null hypersurface. Our analysis is coordinate free, does not require any field equations and holds regardless the signature of the ambient metric 1 , the dimension or the topology of the hypersurface. Furthermore, we analyze the case where the ambient space admits a preferred vector field η that is null and tangent to the hypersurface (e.g. a Killing or a homothetic vector field) and whose deformation tensor L η g is known. Such vectors are called 'symmetry generators'. Moreover, our analysis includes the possibility of η vanishing on subsets of H with empty interior (e.g. a bifurcate Killing horizon).</text>
<text><location><page_3><loc_12><loc_52><loc_88><loc_74></location>To work with abstract null hypersurfaces we employ the so-called hypersurface data formalism [31, 26, 27], as it allows one to study hypersurfaces in a detached way, irrespective of their causal character. At the core of this formalism lies the concept of metric hypersurface data , which comprises an abstract manifold H along with a (0,2)-symmetric tensor field γ , a one-form /lscript , and a scalar function /lscript (2) . When H happens to be embedded in an ambient manifold ( M , g ), the tensor γ agrees with the first fundamental form of H , while the one-form /lscript and the scalar /lscript (2) capture the transverse-tangent and transverse-transverse components of the ambient metric, respectively. From { γ , /lscript , /lscript (2) } one can also introduce contravariant data on H that consists of a (2,0) symmetric tensor field P , a vector n and a scalar n (2) . In addition, metric hypersurface data {H , γ , /lscript , /lscript (2) } is subject to a gauge freedom that encompasses, at the abstract level, the multiplicity of choices of transverse vector ξ along H . Metric hypersurface data is also endowed with a torsion-free connection · ∇ constructed solely in terms of γ , /lscript and /lscript (2) . The arbitrary causal character of H allows the analysis of abstract null geometry, even though neither a metric nor its Levi-Civita connection are present in the null case.</text>
<text><location><page_3><loc_12><loc_31><loc_88><loc_51></location>Let us denote by 2 Y ( k ) the pullback of the k -th Lie derivative of g along a transverse vector ξ at H . The collection { Y ( k ) } k ≥ 1 will be called transverse or asymptotic expansion . Our approach to determine this expansion at a general null hypersurface using the hypersurface data formalism is as follows. We begin by establishing a general identity that relates the m -th Lie derivative of the curvature and Ricci tensors along any vector ξ with the Lie derivatives of a connection along ξ . For the Levi-Civita connection this results in an identity linking the m -th Lie derivative of the Ricci tensor with the Lie derivatives of the metric. By applying this general identity to a null hypersurface H with a transverse vector ξ we can explicitly obtain identities for the leading order terms of the (i) completely transverse, (ii) transverse-tangent, and (iii) completely tangent components of the m -th Lie derivative of the ambient Ricci tensor along ξ (Corollary 4.26). A key property of the result is its geometric nature, as the identities depend solely on metric hypersurface data and { Y ( k ) } k ≥ 1 once the transverse vector ξ is extended off the hypersurface geodesically.</text>
<text><location><page_3><loc_12><loc_12><loc_88><loc_29></location>The identities (i)-(iii) described in the previous paragraph are of a very distinct nature. Firstly, the completely tangential component of the m -th derivative of the Ricci tensor depends algebraically on the trace of the tensor Y ( m +2) w.r.t P , that we denote by tr P Y ( m +2) . Secondly, its transverse-tangent components depend algebraically on the one-form Y ( m +2) ( n, · ), where n is the null generator of the hypersurface. And finally, the identity for the fully tangential components of the m -th derivative of the Ricci depends on Y ( m +1) via a transport equation along the null generator of H , but also on the scalar tr P Y ( m +1) and the one form Y ( m +1) ( n, · ) and its derivatives. Since these two last objects depend algebraically on the derivatives of the Ricci tensor, identity (iii) allows us to know the evolution of the transverse expansion on any null hypersurface along the null direction provided the Ricci tensor is known at all orders on H . Observe that, for a fixed m , the leading order terms of the identity (iii) carry at most m +1</text>
<text><location><page_4><loc_12><loc_89><loc_88><loc_91></location>transverse derivatives of the metric, whereas the leading order terms in identities (i) and (ii) always depend on m +2 transverse derivatives of g at H .</text>
<text><location><page_4><loc_12><loc_70><loc_88><loc_87></location>When a symmetry generator η is present, it is possible to transfer information from its deformation tensor L η g into the identity (iii). This gives rise to a new identity called 'generalized master equation of order m ' (cf. (129)) due to its close relationship to the 'generalized master equation' found in [19] for the case m = 1. Its key property is that its dependence on Y ( m +1) is not via a transport equation anymore. In addition, it also depends on the one-form Y ( m +1) ( n, · ) and the scalar tr P Y ( m +1) . This identity will allow us to identify the minimum amount of data at H in order to determine the full transverse expansion in terms of the Ricci tensor and its derivatives at the hypersurface. Informally, the idea is that, at each order, from the identities (i) and (ii) of order m -1 one can obtain algebraically tr P Y ( m +1) as well as Y ( m +1) ( n, · ). By introducing such tr P Y ( m +1) and Y ( m +1) ( n, · ) into the generalized master equation of order m , the remaining components of Y ( m +1) can then be obtained algebraically.</text>
<text><location><page_4><loc_12><loc_56><loc_88><loc_68></location>In the last part of the paper we apply the generalized master equation of order m to the Killing horizon case. For non-degenerate Killing horizons, we are able to show that the full transverse expansion is uniquely determined from abstract data at the horizon as well as on the tower of derivatives of the ambient Ricci tensor at H (Theorem 6.15). This extends the main result of [16] in several directions. Firstly because our approach allows for much more general field equations besides vacuum with Λ = 0. Secondly because we are allowing zeroes of η , which includes the possibility of the horizon having bifurcation surfaces. And finally, because our result extends to arbitrary ambient signature.</text>
<text><location><page_4><loc_12><loc_34><loc_88><loc_54></location>The collection of transverse derivatives of the ambient Ricci tensor at H can be thought at least in two different ways. One possibility is to provide such collection as prescribed data on the hypersurface, e.g. given by some external matter field. Another option is to think of the collection as functional relations between metric hypersurface data and the transverse expansion at the horizon. The easiest example that illustrates the second viewpoint are the Λ-vacuum equations, as in this case the Ricci tensor is proportional to the metric, so its transverse derivatives will depend on the expansion itself. This property is captured in our Definition 6.16, and allows us to prove that when the ambient Ricci tensor fulfills this property, the full expansion at H is uniquely determined from abstract data at the horizon (Theorem 6.17). When applied to the vacuum equations with cosmological constant, our theorem provides a characterization of all possible analytic Λ-vacuum manifolds in the vicinity of non-degenerate horizons. As a direct consequence, we establish a uniqueness result for non-extremal Schwarzschild-de Sitter spacetime (Proposition 6.22), complementing the main result of [14].</text>
<text><location><page_4><loc_12><loc_11><loc_88><loc_32></location>The paper is organized as follows. In Section 2, we provide a self-contained overview of the fundamental concepts of hypersurface data formalism. In particular, we recall the fact that the fully tangential components of the ambient Ricci tensor at a null hypersurface are given in terms of hypersurface data. In Section 3 we proceed to calculate the remaining components of the Ricci tensor, namely its fully transverse and transverse-tangent components. Once the full Ricci tensor is expressed in terms of metric data and transverse derivatives of the metric on the null hypersurface, in Section 4 we determine the connection between higher order derivatives of the ambient Ricci tensor and higher order derivatives of the metric. Section 5 is devoted to the analysis of the algebraic identities obtained when a general symmetry generator is present on H . Finally, in Section 6 we focus our attention to non-degenerate Killing horizons and we prove that the asymptotic expansion on any non-degenerate Killing horizon is uniquely determined in terms of abstract data at the horizon. Additionally, the paper includes an appendix (Appendix A) where we derive several identities involving the pull-back of arbitrary covariant tensors and their derivatives on the null hypersurface. These play a key role in our analysis in Section 4.</text>
<section_header_level_1><location><page_5><loc_12><loc_90><loc_43><loc_92></location>Notation and conventions</section_header_level_1>
<text><location><page_5><loc_12><loc_59><loc_88><loc_89></location>Throughout this paper ( M , g ) denotes an arbitrary smooth d -dimensional semi-Riemannian manifold of any signature ( p, q ) with both p and q different from zero. When we specifically need this signature to be Lorentzian, we will say so explicitly. We employ both index-free and abstract index notation at our convenience. Ambient indices are denoted with Greek letters, abstract indices on a hypersurface are written in lowercase Latin letters, and abstract indices at cross-sections of a hypersurface are expressed in uppercase Latin letters. As usual, square brackets enclosing indices denote antisymmetrization and parenthesis are for symmetrization. The symmetrized tensor product is denoted with ⊗ s . By F ( M ), X ( M ) and X /star ( M ) we denote respectively the ring of smooth functions, the F ( M )-module of smooth vector fields and the F ( M )-module of smooth one-forms on M . The subset F /star ( M ) ⊂ F ( M ) consists of the nowhere vanishing functions on M . The pullback of a function f via a diffeomorphism Φ will be denoted by Φ /star f or simply by f depending on the context. Given a diffeomorphism Φ and a vector field X , we define Φ /star X := (Φ -1 ) /star X . A ( p, q )-tensor refers to a tensor field p times contravariant and q times covariant. Given any pair of (2 , 0) and (0 , 2) tensors A ab and B cd we denote tr A B := A ab B ab . We employ the symbol ∇ for the Levi-Civita connection of g . Throughout this paper we use the notation L ( m ) X T to denote the m -th Lie derivative of the tensor T along X , and X ( m ) ( f ) for the m -th directional derivative of the function f along X . When m = 1 we also write L X T and X ( f ), respectively, and when m = 0 they are just the identity operators. All manifolds are assumed to be connected and smooth.</text>
<section_header_level_1><location><page_5><loc_12><loc_55><loc_63><loc_56></location>2 Review of hypersurface data formalism</section_header_level_1>
<text><location><page_5><loc_12><loc_46><loc_88><loc_53></location>This section is devoted to review the basic notions of the so-called hypersurface data formalism . Further details can be found in [26, 27, 31]. The usefulness of this formalism has been recently demonstrated in the context of matching of spacetimes [20, 21, 22] and in solving the characteristic problem of general relativity from an abstract point of view [30, 29]. Let us start by introducing the necessary objects for this paper.</text>
<text><location><page_5><loc_12><loc_40><loc_88><loc_44></location>Definition 2.1. Let H be a d -dimensional manifold, γ a symmetric (0,2)-tensor field, /lscript a oneform and /lscript (2) a scalar function on H . We say that {H , γ , /lscript , /lscript (2) } defines a metric hypersurface data set provided that (0,2) symmetric tensor A | p on T p H× R defined by</text>
<formula><location><page_5><loc_25><loc_37><loc_88><loc_39></location>A| p (( W,a ) , ( V, b )) := γ | p ( W,V ) + a /lscript | p ( V ) + b /lscript | p ( W ) + ab/lscript (2) | p (1)</formula>
<text><location><page_5><loc_12><loc_33><loc_88><loc_36></location>is non-degenerate at every p ∈ H . A five-tuple {H , γ , /lscript , /lscript (2) , Y } , where Y is a (0,2) symmetric tensor field on H , is called hypersurface data.</text>
<text><location><page_5><loc_12><loc_27><loc_88><loc_32></location>The non-degeneracy of A allows us to introduce its 'inverse' A /sharp by A /sharp ( A (( V, a ) , · ) , · ) = ( V, a ) for every ( V, a ) ∈ X ( H ) ⊗F ( H ). From A /sharp one can define a (2,0) symmetric tensor field P , a vector n and a scalar n (2) on H by the decomposition</text>
<formula><location><page_5><loc_13><loc_24><loc_88><loc_25></location>A /sharp (( α , a ) , ( β , b )) = P ( α , β ) + an ( β ) + bn ( α ) + abn (2) ∀ ( α , a ) , ( β , b ) ∈ X /star ( H ) × F ( H ) , (2)</formula>
<text><location><page_5><loc_12><loc_21><loc_49><loc_22></location>Equivalently, P , n and n (2) can be defined by</text>
<formula><location><page_5><loc_24><loc_18><loc_49><loc_19></location>γ ab n b + n (2) /lscript a = 0 , (3)</formula>
<formula><location><page_5><loc_23><loc_16><loc_49><loc_17></location>/lscript a n a + n (2) /lscript (2) = 1 . (4)</formula>
<formula><location><page_5><loc_61><loc_18><loc_88><loc_19></location>P ab /lscript b + /lscript (2) n a = 0 , (5)</formula>
<formula><location><page_5><loc_62><loc_16><loc_88><loc_17></location>P ac γ cb + /lscript b n a = δ a b . (6)</formula>
<text><location><page_5><loc_12><loc_9><loc_88><loc_14></location>Despite its name, the notion of hypersurface data does not view H as a hypersurface of another ambient manifold. The connection between Definition 2.1 and the standard definition of a hypersurface is as follows.</text>
<text><location><page_6><loc_12><loc_87><loc_89><loc_92></location>Definition 2.2. Metric hypersurface data {H , γ , /lscript , /lscript (2) } is (Φ , ξ ) -embedded in a semi-Riemannian manifold ( M , g ) if there exists an embedding Φ : H ↪ →M and a vector field ξ along Φ( H ) everywhere transversal to Φ( H ) , called rigging, such that</text>
<formula><location><page_6><loc_30><loc_84><loc_88><loc_86></location>Φ /star ( g ) = γ , Φ /star ( g ( ξ, · )) = /lscript , Φ /star ( g ( ξ, ξ )) = /lscript (2) . (7)</formula>
<text><location><page_6><loc_12><loc_81><loc_72><loc_83></location>Hypersurface data {H , γ , /lscript , /lscript (2) , Y } is embedded provided that, in addition,</text>
<formula><location><page_6><loc_44><loc_77><loc_88><loc_80></location>1 2 Φ /star ( L ξ g ) = Y . (8)</formula>
<text><location><page_6><loc_12><loc_67><loc_88><loc_76></location>In the context of embedded (metric) hypersurface data, γ being degenerate is equivalent to Φ( H ) being an embedded null hypersurface. It is easy to show [27] that the degeneracy of γ is equivalent to n (2) = 0. Hence, (metric) hypersurface data satisfying n (2) = 0 are called null (metric) hypersurface data . A cross-section (or simply a section) S of H is an embedded hypersurface S ↪ →H with the property that every integral curve of n crosses S exactly once. From now on we restrict ourselves to the null case.</text>
<text><location><page_6><loc_12><loc_63><loc_62><loc_65></location>Given null metric hypersurface data we define the tensor field</text>
<formula><location><page_6><loc_45><loc_59><loc_88><loc_62></location>U := 1 2 L n γ . (9)</formula>
<text><location><page_6><loc_12><loc_54><loc_88><loc_58></location>When the data is embedded U coincides with the second fundamental form of Φ( H ) w.r.t the unique normal one-form ν satisfying ν ( ξ ) = 1 (see [26]). It is also convenient to introduce the tensors</text>
<formula><location><page_6><loc_32><loc_51><loc_33><loc_52></location>2</formula>
<formula><location><page_6><loc_27><loc_52><loc_88><loc_54></location>F := 1 d /lscript , (10) Π := Y + F . (11)</formula>
<text><location><page_6><loc_12><loc_43><loc_88><loc_50></location>Let {H , γ , /lscript , /lscript (2) } be null metric hypersurface data (Φ , ξ )-embedded in ( M , g ). By the transversality of ξ , given a (local) basis { e a } of H , the set { ̂ e a := Φ /star e a , ξ } is a (local) basis of Φ( H ) with dual basis { θ a , ν } . Raising the indices we can introduce ν := g /sharp ( ν , · ) and θ a := g /sharp ( θ a , · ), which are given in terms of { ξ, e a } by</text>
<text><location><page_6><loc_12><loc_37><loc_72><loc_42></location>̂ ̂ From (2) the inverse metric g αβ at H can be written in the basis { ξ, e a } as</text>
<formula><location><page_6><loc_37><loc_40><loc_88><loc_45></location>̂ ν = n a e a , θ a = P ab e b + n a ξ. (12)</formula>
<formula><location><page_6><loc_36><loc_34><loc_88><loc_39></location>̂ g αβ H = P ab e α a e β b + n a e α a ξ β + n b e β b ξ α . (13)</formula>
<text><location><page_6><loc_12><loc_28><loc_88><loc_36></location>̂ ̂ ̂ ̂ In the embedded picture the notion of rigging vector is non-unique, since given a rigging ξ any other vector of the form ξ ' = z ( ξ + Φ /star V ) with ( z, V ) ∈ F /star ( H ) × X ( H ) is also transverse to Φ( H ). Translating this into the abstract setting we have the following definition.</text>
<text><location><page_6><loc_12><loc_24><loc_88><loc_27></location>Definition 2.3. Let {H , γ , /lscript , /lscript (2) , Y } be hypersurface data and ( z, V ) ∈ F /star ( H ) × X ( H ) . We define the gauge transformed hypersurface data with gauge parameters ( z, V ) by</text>
<formula><location><page_6><loc_35><loc_21><loc_88><loc_23></location>G ( z,V ) ( γ ) := γ , (14)</formula>
<formula><location><page_6><loc_35><loc_19><loc_88><loc_21></location>G ( z,V ) ( /lscript ) := z ( /lscript + γ ( V, · )) , (15)</formula>
<formula><location><page_6><loc_34><loc_14><loc_88><loc_17></location>G ( z,V ) ( Y ) := z Y + /lscript ⊗ s dz + 1 2 L zV γ . (17)</formula>
<formula><location><page_6><loc_33><loc_15><loc_88><loc_19></location>G ( z,V ) ( /lscript (2) ) := z 2 ( /lscript (2) +2 /lscript ( V ) + γ ( V, V ) ) , (16)</formula>
<text><location><page_6><loc_12><loc_11><loc_80><loc_12></location>Transformations (14)-(17) induce the corresponding transformations on P and n [26]</text>
<formula><location><page_7><loc_73><loc_90><loc_88><loc_92></location>-1 (19)</formula>
<formula><location><page_7><loc_20><loc_90><loc_76><loc_91></location>G ( z,V ) ( P ) = P -2 V ⊗ s n, (18) G ( z,V ) ( n ) = z n.</formula>
<text><location><page_7><loc_12><loc_86><loc_88><loc_88></location>The set of gauge transformations defines a group whose composition law and the inverse element are [26]</text>
<formula><location><page_7><loc_16><loc_82><loc_49><loc_84></location>G ( z 1 ,V 1 ) · G ( z 2 ,V 2 ) = G ( z 1 z 2 ,V 2 + z -1 2 V 1 ) (20)</formula>
<formula><location><page_7><loc_61><loc_82><loc_88><loc_84></location>G -1 ( z,V ) = G ( z -1 , -zV ) . (21)</formula>
<text><location><page_7><loc_12><loc_78><loc_88><loc_81></location>As expected from the geometric interpretation of U as the second fundamental form of Φ( H ) w.r.t ν := Φ /star n , its gauge transformation is given by [26]</text>
<formula><location><page_7><loc_42><loc_75><loc_88><loc_77></location>G ( z,V ) ( U ) = z -1 U . (22)</formula>
<text><location><page_7><loc_12><loc_72><loc_60><loc_74></location>A consequence of this together with (18) and U ( n, · ) = 0 is</text>
<formula><location><page_7><loc_42><loc_69><loc_88><loc_71></location>tr P ' U ' = z -1 tr P U . (23)</formula>
<text><location><page_7><loc_12><loc_65><loc_88><loc_68></location>Given null metric hypersurface data {H , γ , /lscript , /lscript (2) } it is possible to define a torsion-free connection · ∇ on H by means of [27]</text>
<formula><location><page_7><loc_21><loc_62><loc_41><loc_64></location>· ∇ γ = -/lscript U -/lscript U ,</formula>
<formula><location><page_7><loc_22><loc_62><loc_49><loc_63></location>a bc c ab b ac (24)</formula>
<formula><location><page_7><loc_61><loc_62><loc_88><loc_64></location>· ∇ a /lscript b = F ab -/lscript (2) U ab . (25)</formula>
<text><location><page_7><loc_12><loc_59><loc_88><loc_61></location>When the data is embedded in ( M , g ), · ∇ is related with the Levi-Civita connection ∇ of g by</text>
<formula><location><page_7><loc_24><loc_56><loc_88><loc_58></location>∇ Φ /star X Φ /star Y H = Φ /star · ∇ X Y -Y ( X,Y ) ν -U ( X,Y ) ξ ∀ X,Y ∈ X ( H ) . (26)</formula>
<text><location><page_7><loc_12><loc_51><loc_88><loc_54></location>Unless otherwise indicated, scalar functions related by Φ /star are denoted with the same symbol. The action of · ∇ on the contravariant data { P, n } is given by [27]</text>
<formula><location><page_7><loc_32><loc_48><loc_88><loc_50></location>· ∇ c n b = s c n b + P ba U ca , (27)</formula>
<formula><location><page_7><loc_31><loc_43><loc_88><loc_48></location>· ∇ c P ab = -( n a P bd + n b P ad ) F cd -n a n b ( d/lscript (2) ) c , (28)</formula>
<text><location><page_7><loc_12><loc_43><loc_60><loc_44></location>where s := F ( n, · ). A direct consequence of (26) and (27) is</text>
<formula><location><page_7><loc_43><loc_40><loc_88><loc_42></location>∇ ν ν H = -Y ( n, n ) ν. (29)</formula>
<text><location><page_7><loc_12><loc_35><loc_88><loc_38></location>It is therefore natural to define the surface gravity of n by κ n := -Y ( n, n ), whose gauge transformation law follows directly from (19) and (17) and is</text>
<formula><location><page_7><loc_39><loc_32><loc_88><loc_34></location>κ ' n = z -1 κ n -z -1 n (log | z | ) . (30)</formula>
<text><location><page_7><loc_12><loc_28><loc_88><loc_31></location>Applying the Cartan identity L n = ι n · d + d · ι n to the one-form /lscript and using 2 F = d /lscript and /lscript ( n ) = 1 one has</text>
<formula><location><page_7><loc_46><loc_27><loc_88><loc_28></location>L n /lscript = 2 s . (31)</formula>
<text><location><page_7><loc_12><loc_24><loc_67><loc_26></location>Another consequence of (27) is that for any one-form θ it holds [23]</text>
<formula><location><page_7><loc_28><loc_19><loc_88><loc_24></location>2 n b · ∇ ( a θ b ) = L n θ a + · ∇ a ( θ ( n ) ) -2 ( θ ( n )s a + P bc U ab θ c ) . (32)</formula>
<text><location><page_7><loc_12><loc_17><loc_88><loc_20></location>For future use we need to know the commutator [ P ab , L n ] acting on a 2-covariant, symmetric tensor field T ab .</text>
<text><location><page_7><loc_12><loc_14><loc_61><loc_15></location>Lemma 2.4. Let T ab a (0,2) symmetric tensor field. Then,</text>
<formula><location><page_7><loc_24><loc_11><loc_88><loc_13></location>P ab L n T ab = L n (tr P T ) + 4 P ( t , s ) + 2 P ac P bd U cd T ab + n ( /lscript (2) ) t ( n ) , (33)</formula>
<text><location><page_7><loc_12><loc_8><loc_27><loc_10></location>where t := T ( n, · ) .</text>
<text><location><page_8><loc_12><loc_89><loc_88><loc_91></location>Proof. The result follows at once from the following expression of the Lie derivative of P along n [28]</text>
<formula><location><page_8><loc_25><loc_84><loc_88><loc_89></location>L n P ab = -2s c ( P ac n b + P bc n a ) -2 P ac P bd U cd -n a n b n ( /lscript (2) ) . (34)</formula>
<text><location><page_8><loc_12><loc_82><loc_76><loc_83></location>Later we will also need the ∇ -derivative of ξ along tangent directions to H [26]</text>
<text><location><page_8><loc_12><loc_76><loc_83><loc_77></location>where r := Y ( n, · ). The gauge transformation law of the one-form Π ( · , n ) = r -s is [30]</text>
<formula><location><page_8><loc_34><loc_71><loc_88><loc_75></location>G ( z,V ) ( r -s ) = r -s -U ( V, · ) + d log | z | . (36)</formula>
<formula><location><page_8><loc_31><loc_75><loc_88><loc_81></location>̂ e µ a ∇ µ ξ β H = (r -s) a ξ β + P cd Π ac ̂ e β d + 1 2 ν β · ∇ a /lscript (2) , (35)</formula>
<text><location><page_8><loc_12><loc_67><loc_88><loc_72></location>As proven in [26], the completely tangential components of the ambient Riemann tensor, as well as its 3-tangential, 1-transverse components can be written in terms of hypersurface data as</text>
<formula><location><page_8><loc_29><loc_65><loc_88><loc_68></location>R αβµν ξ α e β b e µ c e ν d H = A bcd , R αβµν e α a e β b e µ c e ν d H = B abcd , (37)</formula>
<text><location><page_8><loc_12><loc_64><loc_51><loc_65></location>where A and B are the tensors on H defined by</text>
<formula><location><page_8><loc_28><loc_61><loc_88><loc_63></location>A bcd := 2 · ∇ [ d F c ] b +2 · ∇ [ d Y c ] b +U b [ d · ∇ c ] /lscript (2) +2Y b [ d (r -s) c ] , (38)</formula>
<formula><location><page_8><loc_27><loc_58><loc_88><loc_60></location>B abcd := γ af · R f bcd +2 /lscript a · ∇ [ d U c ] b +2U a [ d Y c ] b +2U b [ c Π d ] a , (39)</formula>
<text><location><page_8><loc_12><loc_54><loc_88><loc_57></location>and · R f bcd is the curvature of · ∇ . It follows from (37) that all the tangential components of the ambient Ricci tensor can be written in terms of hypersurface data as [30]</text>
<formula><location><page_8><loc_32><loc_51><loc_68><loc_53></location>g αβ R αµβν e µ a e ν b H = B acbd P cd -( A bac + A abc ) n c .</formula>
<text><location><page_8><loc_12><loc_49><loc_79><loc_50></location>The RHS defines a tensor on H called constraint tensor R . Its explicit form is [23]</text>
<formula><location><page_8><loc_22><loc_41><loc_88><loc_48></location>R ab = · R ( ab ) -2 L n Y ab -(2 κ n +tr P U )Y ab + · ∇ ( a ( s b ) +2r b ) ) -2r a r b +4r ( a s b ) -s a s b -(tr P Y )U ab +2 P cd U d ( a ( 2Y b ) c +F b ) c ) , (40)</formula>
<text><location><page_8><loc_12><loc_34><loc_88><loc_42></location>where · R ab is the Ricci tensor of the connection · ∇ . The tensor R is abstract in the sense that it does not require the data to be embedded in any ambient manifold. Note that all the dependence on the tensor Y in (40) is explicit. Below we shall need the explicit form of the contraction of R ab with n a and P ab . The former was obtained in [23] and relies on the following general identity, also derived in [23]</text>
<formula><location><page_8><loc_22><loc_29><loc_88><loc_34></location>· R ( ab ) n a = 1 2 L n s b -2 P ac U ab s c + P ac · ∇ c U ab -· ∇ b ( tr P U ) + ( tr P U ) s b . (41)</formula>
<text><location><page_8><loc_12><loc_28><loc_22><loc_30></location>The result is</text>
<formula><location><page_8><loc_13><loc_26><loc_88><loc_28></location>R ab n a = -L n (r b -s b ) -· ∇ b κ n -(tr P U )(r b -s b ) -· ∇ b (tr P U ) + P cd · ∇ c U bd -2 P cd U bd s c . (42)</formula>
<text><location><page_8><loc_12><loc_23><loc_40><loc_24></location>Another contraction with n b gives</text>
<formula><location><page_8><loc_29><loc_20><loc_88><loc_22></location>R ab n a n b = -n (tr P U ) + (tr P U ) κ n -P ab P cd U ac U bd , (43)</formula>
<text><location><page_8><loc_12><loc_16><loc_88><loc_19></location>which is the abstract version of the Raychaudhuri equation. For the trace of (40) with respect to P we simply need to use (33) and get</text>
<formula><location><page_8><loc_23><loc_11><loc_88><loc_16></location>tr P R = tr P · R -2 L n (tr P Y ) -2 ( κ n +tr P U ) tr P Y +div P ( s +2 r ) -2 P ( r , r ) -4 P ( r , s ) -P ( s , s ) + 2 κ n n ( /lscript (2) ) , (44)</formula>
<text><location><page_8><loc_12><loc_8><loc_32><loc_10></location>where div P t := P ab · ∇ a t b .</text>
<section_header_level_1><location><page_9><loc_12><loc_90><loc_62><loc_92></location>3 Full Riemann tensor at a hypersurface</section_header_level_1>
<text><location><page_9><loc_12><loc_82><loc_88><loc_89></location>While the components of the form R αβµν ξ α ̂ e β b ̂ e µ c ̂ e ν d and R αβµν ̂ e α a ̂ e β b ̂ e µ c ̂ e ν d can be written solely in terms of hypersurface data, the remaining ones, i.e. R αβµν ξ α ̂ e β b ξ µ ̂ e ν d , cannot, because in addition they require second order transverse derivatives of the metric. In order to compute them we use a lemma from [25].</text>
<text><location><page_9><loc_12><loc_80><loc_82><loc_81></location>Lemma 3.1. Let ( M , g ) be a semi-Riemannian manifold and ξ, X, Y ∈ X ( M ) . Then,</text>
<formula><location><page_9><loc_22><loc_74><loc_88><loc_78></location>Riem( ξ, X, Y, ξ ) = 1 2 ( L (2) ξ g ) ( X,Y ) -1 2 ( L ∇ ξ ξ g ) ( X,Y ) -g ( ∇ X ξ, ∇ Y ξ ) . (45)</formula>
<text><location><page_9><loc_12><loc_71><loc_88><loc_74></location>This lemma allows us to write the remaining components of the ambient Riemann tensor in terms of second transverse derivatives of the metric and hypersurface data.</text>
<text><location><page_9><loc_12><loc_67><loc_88><loc_70></location>Proposition 3.2. Let {H , γ , /lscript , /lscript (2) , Y } be null hypersurface data (Φ , ξ ) -embedded in ( M , g ) and define β ∈ F ( H ) and T ∈ X ( H ) by the decomposition ∇ ξ ξ H = βξ +Φ /star T . Then,</text>
<formula><location><page_9><loc_14><loc_61><loc_88><loc_66></location>Φ /star (Riem( ξ, · , · , ξ )) = 1 2 Φ /star ( L (2) ξ g ) -β Y -dβ ⊗ s /lscript -1 2 L T γ -Π · Π -( r -s ) ⊗ s d/lscript (2) , (46)</formula>
<text><location><page_9><loc_12><loc_54><loc_88><loc_58></location>Proof. Let { e a } be a (local) basis of X ( H ) and ̂ e a := Φ /star e a . From (35) together with (7) and recalling that ν is null, normal to Φ( H ) and satisfies g ( ξ, ν ) = 1, it follows</text>
<text><location><page_9><loc_12><loc_57><loc_48><loc_62></location>where ( Π · Π ) ( X,Y ) := P ( Π ( X, · ) , Π ( Y, · )) .</text>
<formula><location><page_9><loc_21><loc_47><loc_79><loc_54></location>g αβ ∇ ̂ e a ξ α ∇ ̂ e b ξ β = /lscript (2) (r -s) a (r -s) b +2(r -s) ( a | ( /lscript d P fd Π | b ) f + 1 2 · ∇ | b ) /lscript (2) ) + P cd P fg Π ac Π bf γ dg = P cd Π ac Π bd +(r -s) ( a · ∇ b ) /lscript (2) ,</formula>
<text><location><page_9><loc_12><loc_43><loc_88><loc_46></location>where in the second equality we used /lscript d P fd = -/lscript (2) n f . Equation (46) follows from (45) after using</text>
<formula><location><page_9><loc_26><loc_36><loc_88><loc_42></location>Φ /star ( L ∇ ξ ξ g ) = Φ /star ( L βξ g ) + Φ /star ( L Φ /star T g ) = Φ /star ( β L ξ g ) + Φ /star (2 dβ ⊗ s g ( ξ, · )) + Φ /star ( L Φ /star T g ) = 2 β Y +2 dβ ⊗ s /lscript + L T γ . (47)</formula>
<text><location><page_9><loc_12><loc_29><loc_88><loc_32></location>Let X,Y ∈ X ( H ). Since ∇ Φ /star X ξ and ∇ Φ /star Y ξ only depend on ξ along Φ( H ) and Riem is a tensor, it follows that Riem( ξ, Φ /star X, Φ /star Y, ξ ) + g ( ∇ Φ /star X ξ, ∇ Φ /star Y ξ ) only depends on ξ along Φ( H ). Thus,</text>
<text><location><page_9><loc_12><loc_27><loc_34><loc_28></location>by equation (45) the tensor</text>
<text><location><page_9><loc_12><loc_22><loc_83><loc_24></location>is independent of how one extends ξ off Φ( H ), so by (47) it follows that the tensor field</text>
<formula><location><page_9><loc_39><loc_23><loc_61><loc_27></location>1 2 Φ /star ( L (2) ξ g ) -1 2 Φ /star ( L ∇ ξ ξ g )</formula>
<formula><location><page_9><loc_35><loc_16><loc_65><loc_21></location>1 2 Φ /star ( L (2) ξ g ) -β Y -dβ ⊗ s /lscript -1 2 L T γ</formula>
<text><location><page_9><loc_12><loc_14><loc_88><loc_17></location>does not depend on the extension of ξ off Φ( H ). This suggests extending the definition of hypersurface data set as follows.</text>
<text><location><page_9><loc_12><loc_10><loc_88><loc_13></location>Definition 3.3. A septuple {H , γ , /lscript , /lscript (2) , Y , Z (2) } defines an extended hypersurface data set provided {H , γ , /lscript , /lscript (2) , Y } is hypersurface data and Z (2) is a (0,2) symmetric tensor field on H .</text>
<text><location><page_10><loc_12><loc_87><loc_88><loc_92></location>Definition 3.4. An extended hypersurface data set {H , γ , /lscript , /lscript (2) , Y , Z (2) } is said to be (Φ , ξ ) -embedded in a semi-Riemannian manifold ( M , g ) provided {H , γ , /lscript , /lscript (2) , Y } is (Φ , ξ ) -embedded in the sense of Definition 2.2 and, in addition,</text>
<formula><location><page_10><loc_22><loc_82><loc_78><loc_86></location>Z (2) = 1 2 Φ /star ( L (2) ξ g ) -1 2 β Φ /star ( L ξ g ) -dβ ⊗ s Φ /star ( g ( ξ, · )) -1 2 Φ /star ( L Φ /star T g ) ,</formula>
<text><location><page_10><loc_12><loc_79><loc_88><loc_82></location>where ( β, T ) ∈ F ( H ) × X ( H ) are defined by ∇ ξ ξ H = βξ + Φ /star T and ξ is any extension of the rigging off Φ( H ) .</text>
<text><location><page_10><loc_12><loc_77><loc_51><loc_78></location>In terms of Z (2) , equation (46) gets rewritten as</text>
<formula><location><page_10><loc_29><loc_74><loc_88><loc_76></location>Φ /star (Riem( ξ, · , · , ξ )) = Z (2) -Π · Π -( r -s ) ⊗ s d/lscript (2) . (48)</formula>
<text><location><page_10><loc_12><loc_70><loc_88><loc_74></location>Once the full Riemann tensor at H is computable from hypersurface data and Z (2) , one can write down explicitly the full Ricci tensor on the abstract null hypersurface.</text>
<text><location><page_10><loc_12><loc_64><loc_88><loc_70></location>Proposition 3.5. Let D = {H , γ , /lscript , /lscript (2) , Y , Z (2) } be extended null hypersurface data (Φ , ξ ) -embedded in ( M , g ) . Let Ric be the Ricci tensor of g and ˙ R := Φ /star ( Ric( ξ, · ) ) , R := Φ /star ( Ric( ξ, ξ ) ) . Then,</text>
<formula><location><page_10><loc_22><loc_59><loc_88><loc_65></location>R = -tr P Z (2) +tr P ( Π · Π ) + P ( r -s , d/lscript (2) ) , (49) ˙ R c = -P ab A abc +z (2) c -P ab Π da Π cb n d + 1 2 κ n · ∇ c /lscript (2) -1 2 n ( /lscript (2) )(r -s) c , (50)</formula>
<text><location><page_10><loc_12><loc_57><loc_53><loc_59></location>where A abc is defined in (38) and z (2) := Z (2) ( n, · ) .</text>
<text><location><page_10><loc_12><loc_51><loc_88><loc_56></location>Proof. Let { e a } be a (local) basis of X ( H ) and ̂ e a := Φ /star e a . From the definition of the Ricci tensor, the symmetries of the Riemann tensor and equations (13) and (48) it follows</text>
<formula><location><page_10><loc_27><loc_47><loc_73><loc_52></location>Ric( ξ, ξ ) H = ( P ab ̂ e α a ̂ e β b + n a ̂ e α a ξ β + n b ̂ e β b ξ α ) R αµβν ξ µ ξ ν H = -P ab Z (2) ab + P ab P cd Π ca Π db + P ab (r -s) a · ∇ b /lscript (2) .</formula>
<text><location><page_10><loc_12><loc_45><loc_65><loc_47></location>In the same way, using the first equation in (37), given X ∈ X ( H )</text>
<formula><location><page_10><loc_17><loc_39><loc_83><loc_45></location>Ric( ξ, Φ /star X ) H = ( P ab ̂ e α a ̂ e β b + n a ̂ e α a ξ β + n a ̂ e β a ξ α ) R αµβν ξ µ (Φ /star X ) ν H = -P ab A abc X c +Z (2) ab n a X b -P cd Π ac Π bd n a X b -n a X b (r -s) ( a · ∇ b ) /lscript (2) .</formula>
<section_header_level_1><location><page_10><loc_12><loc_34><loc_45><loc_35></location>4 Higher order derivatives</section_header_level_1>
<text><location><page_10><loc_12><loc_27><loc_88><loc_32></location>In this section we compute the derivatives L ( m ) ξ Ric on H in terms of transverse derivatives of g on H up to order m +1, i.e. making L ( m +2) ξ g and L ( m +1) ξ g explicit. In order to simplify the notation let us introduce the tensors</text>
<text><location><page_10><loc_12><loc_21><loc_20><loc_23></location>as well as</text>
<formula><location><page_10><loc_22><loc_22><loc_78><loc_27></location>Y ( m ) := 1 2 Φ /star ( L ( m ) ξ g ) , r ( m ) := Y ( m ) ( n, · ) , κ ( m ) := -Y ( m ) ( n, n ) ,</formula>
<formula><location><page_10><loc_14><loc_17><loc_86><loc_21></location>R ( m ) := Φ /star ( L ( m -1) ξ Ric ) , ˙ R ( m ) := Φ /star ( L ( m -1) ξ Ric( ξ, · ) ) , R ( m ) := Φ /star ( L ( m -1) ξ Ric( ξ, ξ ) ) .</formula>
<text><location><page_10><loc_12><loc_15><loc_88><loc_18></location>Observe that Y (1) , r (1) and κ (1) agree with Y , r and κ n , respectively. In what follows we refer to the collection of tensors { Y , Y (2) , ... } as the transverse or asymptotic expansion .</text>
<text><location><page_10><loc_12><loc_8><loc_88><loc_13></location>In the following lemma we recall a well-known identity for derivatives of products of any two objects S and T . Note than when S and T are tensors, the expression also holds when contractions are allowed.</text>
<text><location><page_11><loc_12><loc_89><loc_88><loc_91></location>Lemma 4.1. Let S and T be two objects, S /circleasterisk T any product of them and D any derivative operator. Then,</text>
<formula><location><page_11><loc_31><loc_84><loc_88><loc_89></location>D ( m ) ( S /circleasterisk T ) = m ∑ i =0 ( m i ) ( D ( i ) S ) /circleasterisk ( D ( m -i ) T ) . (51)</formula>
<text><location><page_11><loc_12><loc_82><loc_86><loc_84></location>Given any vector field ξ we introduce the tensor Σ[ ξ ] := L ξ ∇ , or in abstract index notation</text>
<formula><location><page_11><loc_29><loc_78><loc_88><loc_81></location>Σ[ ξ ] α µν = 1 2 g αβ ( ∇ µ K [ ξ ] νβ + ∇ ν K [ ξ ] µβ -∇ β K [ ξ ] µν ) , (52)</formula>
<text><location><page_11><loc_12><loc_71><loc_88><loc_77></location>where K [ ξ ] is the so-called deformation tensor of ξ , defined by K [ ξ ] := L ξ g . In order not to overload the notation in this section we will simply use the symbols Σ and K for Σ[ ξ ] and K [ ξ ], respectively. In later sections, we will come back to the notation K [ ξ ] because deformation tensors of more than one vector field will occur.</text>
<text><location><page_11><loc_12><loc_66><loc_88><loc_69></location>To compute L ( m ) ξ Ric up to order m +1 we use the following classical result [38] relating the Lie derivative of the curvature with the Lie derivative of the connection</text>
<formula><location><page_11><loc_30><loc_63><loc_88><loc_65></location>L ξ R µ ανβ = H γρ νβ ∇ γ Σ µ αρ , H γρ νβ := δ γ ν δ ρ β -δ γ β δ ρ ν . (53)</formula>
<text><location><page_11><loc_12><loc_60><loc_39><loc_62></location>For future convenience we define</text>
<formula><location><page_11><loc_23><loc_56><loc_88><loc_62></location>̂ Σ ναβ := g µν Σ µ αβ = 1 2 ( ∇ α K βν + ∇ β K αν -∇ ν K αβ ) =: F ρλγ ναβ ∇ ρ K λγ , (54)</formula>
<text><location><page_11><loc_12><loc_54><loc_42><loc_55></location>where we have introduced the tensor</text>
<formula><location><page_11><loc_34><loc_47><loc_66><loc_54></location>F ρλγ ναβ := 1 2 ( δ ρ α δ λ β δ γ ν + δ ρ β δ λ α δ γ ν -δ ρ ν δ λ α δ γ β ) . ̂</formula>
<text><location><page_11><loc_12><loc_46><loc_88><loc_52></location>The hat in ̂ Σ ναβ is not really necessary because Σ ναβ is just Σ ν αβ with the index lowered. However the distinction will be necessary later for the Lie derivative of ̂ Σ ναβ and Σ ν αβ .</text>
<text><location><page_11><loc_12><loc_42><loc_88><loc_45></location>Remark 4.2. The notations H γρ νβ and F ρλγ ναβ are unambiguous since we shall never lower/raise their indices. We stick to this rule for any tensor written with indices on top of each other.</text>
<text><location><page_11><loc_12><loc_33><loc_88><loc_40></location>The idea now is to apply the operator L ( m -1) ξ to (53) and express the result by making L ( m +1) ξ g and L ( m +2) ξ g explicit. In order to do that we need to commute L ( m -1) ξ and ∇ when they act on a ( q, p ) tensor A α 1 ··· α q β 1 ··· β p . We introduce the notation A ( m ) := L ( m -1) ξ A , m ≥ 1. The commutator is found explicitly in the following proposition.</text>
<text><location><page_11><loc_12><loc_29><loc_87><loc_32></location>Proposition 4.3. Let ξ ∈ X ( M ) and m ≥ 1 be a integer. Then, given any ( p, q ) tensor A α 1 ··· α q β 1 ··· β p the following identity holds</text>
<formula><location><page_11><loc_13><loc_18><loc_87><loc_28></location>L ( m ) ξ ∇ γ A α 1 ··· α q β 1 ··· β p = ∇ γ A ( m +1) α 1 ··· α q β 1 ··· β p + m -1 ∑ k =0 ( m k +1 ) q ∑ j =1 A ( m -k ) α 1 ··· α j -1 σα j +1 ··· α q β 1 ··· β p Σ ( k +1) α j σγ -p A ( m -k ) α 1 ··· α q β 1 ··· β i -1 σβ i +1 ··· β p Σ ( k +1) σ β i γ ) .</formula>
<formula><location><page_11><loc_55><loc_17><loc_58><loc_22></location>∑ i =1</formula>
<text><location><page_11><loc_12><loc_15><loc_53><loc_17></location>Proof. The case m = 1 is the classical identity [38]</text>
<formula><location><page_11><loc_15><loc_9><loc_85><loc_14></location>L ξ ∇ γ A α 1 ··· α q β 1 ··· β p = ∇ γ L ξ A α 1 ··· α q β 1 ··· β p + q ∑ j =1 A α 1 ··· α j -1 σα j +1 ··· α q β 1 ··· β p Σ α j σγ -p ∑ i =1 A α 1 ··· α q β 1 ··· β i -1 σβ i +1 ··· β p Σ σ β i γ .</formula>
<text><location><page_12><loc_12><loc_89><loc_88><loc_91></location>We prove the result by induction, so let us assume that the claim is true up to some m ≥ 1 and show that it is then true for m +1 also. We compute, using the induction hypothesis,</text>
<formula><location><page_12><loc_14><loc_85><loc_15><loc_87></location>L</formula>
<text><location><page_12><loc_12><loc_56><loc_88><loc_59></location>In the last term we rename k as k -1 and split the second sum in two parts and the last sum also in two parts,</text>
<formula><location><page_12><loc_15><loc_59><loc_86><loc_88></location>( m +1) ξ ∇ γ A α 1 ··· α q β 1 ··· β p = L ξ ( L ( m ) ξ ∇ γ A α 1 ··· α q β 1 ··· β p ) = L ξ ∇ γ A ( m +1) α 1 ··· α q β 1 ··· β p + m -1 ∑ k =0 ( m k +1 ) q ∑ j =1 A ( m -k ) α 1 ··· σα q β 1 ··· β p Σ ( k +1) α j σγ -p ∑ i =1 A ( m -k ) α 1 ··· α q β 1 ··· σ ··· β p Σ ( k +1) σ β i γ )) = ∇ γ L ξ A ( m +1) α 1 ··· α q β 1 ··· β p + q ∑ j =1 A ( m +1) α 1 ··· σ ··· α q β 1 ··· β p Σ α j σγ -p ∑ i =1 A ( m +1) α 1 ··· α q β 1 ··· β i -1 σβ i +1 ··· β p Σ σ β i γ + m -1 ∑ k =0 ( m k +1 ) q ∑ j =1 A ( m -k +1) α 1 ··· σ ··· α q β 1 ··· β p Σ ( k +1) α j σγ -p ∑ i =1 A ( m -k +1) α 1 ··· α q β 1 ··· σ ··· β p Σ ( k +1) σ β i γ + m -1 ∑ k =0 ( m k +1 ) q ∑ j =1 A ( m -k ) α 1 ··· σ ··· α q β 1 ··· β p Σ ( k +2) α j σγ -p ∑ i =1 A ( m -k ) α 1 ··· α q β 1 ··· σ ··· β p Σ ( k +2) σ β i γ .</formula>
<formula><location><page_12><loc_12><loc_38><loc_97><loc_56></location>L ( m +1) ξ ∇ γ A α 1 ··· α q β 1 ··· β p = ∇ γ L ξ A ( m +1) α 1 ··· α q β 1 ··· β p + ( 1 + ( m 1 )) q ∑ j =1 A ( m +1) α 1 ··· α q β 1 ··· β p Σ α j σγ -p ∑ i =1 A ( m +1) α 1 ··· α q β 1 ··· β p Σ σ β i γ + m -1 ∑ k =1 (( m k +1 ) + ( m k )) q ∑ j =1 A ( m -k +1) α 1 ··· α q β 1 ··· β p Σ ( k +1) α j σγ -p ∑ i =1 A ( m -k +1) α 1 ··· α q β 1 ··· β p Σ ( k +1) σ β i γ + ( m m ) q ∑ j =1 A α 1 ··· α q β 1 ··· β p Σ ( m +1) α j σγ -p ∑ i =1 A α 1 ··· α q β 1 ··· β p Σ ( m +1) σ β i γ .</formula>
<text><location><page_12><loc_12><loc_28><loc_88><loc_36></location>Once we know how to commute L ( m -1) ξ and ∇ when acting on an arbitrary tensor field, we can apply the result to equation (53) to compute the explicit expression of the tensor L ( m ) ξ R µ ανβ . This will be used below to compute the derivatives L ( m ) ξ Ric on a null hypersurface H in terms of transverse derivatives of g on H up to order m +1.</text>
<text><location><page_12><loc_12><loc_35><loc_82><loc_39></location>Using the binomial identity ( m i +1 ) + ( m i ) = ( m +1 i +1 ) , the proposition follows by induction.</text>
<text><location><page_12><loc_12><loc_26><loc_63><loc_27></location>Proposition 4.4. Let ξ ∈ X ( M ) and m ≥ 2 an integer. Then,</text>
<formula><location><page_12><loc_22><loc_19><loc_88><loc_25></location>L ( m ) ξ R µ ανβ = H γρ νβ ( ∇ γ Σ ( m ) µ αρ + m -2 ∑ k =0 ( m k +1 ) Σ ( m -k -1) σ αρ Σ ( k +1) µ σγ ) (55)</formula>
<text><location><page_12><loc_12><loc_17><loc_15><loc_19></location>and</text>
<formula><location><page_12><loc_22><loc_13><loc_88><loc_18></location>L ( m ) ξ R αβ = H γρ µβ ( ∇ γ Σ ( m ) µ αρ + m -2 ∑ k =0 ( m k +1 ) Σ ( m -k -1) σ αρ Σ ( k +1) µ σγ ) . (56)</formula>
<text><location><page_13><loc_12><loc_90><loc_59><loc_92></location>Proof. Applying L ( m -1) ξ to (53) and using Proposition 4.3,</text>
<formula><location><page_13><loc_12><loc_80><loc_89><loc_89></location>L ( m ) ξ R µ ανβ = H γρ νβ ( ∇ γ Σ ( m ) µ αρ + m -2 ∑ k =0 ( m -1 k +1 ) ( Σ ( m -k -1) σ αρ Σ ( k +1) µ σγ -Σ ( m -k -1) µ σα Σ ( k +1) σ ργ -Σ ( m -k -1) µ ρσ Σ ( k +1) σ αγ )) .</formula>
<text><location><page_13><loc_12><loc_77><loc_88><loc_80></location>Since H γρ νβ is antisymmetric in γ, ρ its contraction with the third term vanishes. Renaming k ' = m -2 -k in the last term the sum simplifies to</text>
<formula><location><page_13><loc_13><loc_66><loc_87><loc_76></location>H γρ νβ m -2 ∑ k =0 ( m -1 k +1 ) ( Σ ( m -k -1) σ ρα Σ ( k +1) µ σγ -Σ ( m -k -1) µ ρσ Σ ( k +1) σ αγ ) = H γρ νβ ( m -2 ∑ k =0 ( m -1 k +1 ) Σ ( m -k -1) σ ρα Σ ( k +1) µ σγ -m -2 ∑ k ' =0 ( m -1 m -1 -k ' ) Σ ( k ' +1) µ ρσ Σ ( m -k ' -1) σ αγ ) .</formula>
<text><location><page_13><loc_12><loc_60><loc_88><loc_66></location>From the antisymmetry of H γρ νβ , the symmetry of Σ ( k +1) µ ρσ and the combinatorial properties ( m -1 m -1 -k ' ) = ( m k ' ) and ( m -1 k +1 ) + ( m -1 k ) = ( m k +1 ) , (55) follows. Equation (56) is immediate since the Lie derivative and the trace commute.</text>
<text><location><page_13><loc_12><loc_51><loc_88><loc_59></location>Identity (56) constitutes the exact relation between the m -th Lie derivative of the Ricci tensor with the Lie derivatives of the tensor Σ. Before restricting it to a null hypersurface H with rigging ξ and computing the leading order terms, we shall establish a property that will play a key role in Section 6, namely that when the rigging ξ is extended off Φ( H ) by ∇ ξ ξ = 0, the tensors L ( m ) ξ R αβ on H are geometrical in the following sense.</text>
<text><location><page_13><loc_12><loc_47><loc_88><loc_50></location>Definition 4.5. Let {H , γ , /lscript , /lscript (2) } be null metric hypersurface data (Φ , ξ ) -embedded in ( M , g ) . Let T and T be (0 , p ) tensor fields on H and M , respectively.</text>
<unordered_list>
<list_item><location><page_13><loc_15><loc_43><loc_88><loc_46></location>· We say that T is H -geometrical provided that it depends at most on null metric data { γ , /lscript , /lscript (2) } and on the tensors { Y ( k ) } k ≥ 1 .</list_item>
<list_item><location><page_13><loc_15><loc_38><loc_87><loc_41></location>· We say that T is geometrical provided that the pullback of arbitrary contractions of T with ξ (including no contraction) into H is H -geometrical.</list_item>
</unordered_list>
<text><location><page_13><loc_12><loc_32><loc_88><loc_37></location>In order to prove that L ( m ) ξ R αβ is geometrical for any natural number m we first explore some general properties of geometrical objects and also establish that several building-block tensors that appear in the argument are indeed geometrical. We start with the latter.</text>
<text><location><page_13><loc_12><loc_28><loc_88><loc_31></location>Lemma 4.6. Let {H , γ , /lscript , /lscript (2) } be null metric hypersurface data (Φ , ξ ) -embedded in ( M , g ) and extend ξ off Φ( H ) by ∇ ξ ξ = 0 . Then the tensor ∇ α ξ β is geometrical.</text>
<text><location><page_13><loc_12><loc_20><loc_88><loc_27></location>Proof. Since ξ α ∇ α ξ β = 0 it suffices to check that ̂ e α a ̂ e β b ∇ α ξ β and ̂ e α a ξ β ∇ α ξ β only depend on hypersurface data. Using ̂ e µ a ∇ µ ξ β H = (r -s) a ξ β + P cd Π ac ̂ e β d + 1 2 ν β · ∇ a /lscript (2) (cf. (35)) and Definition 2.2 a straightforward computation shows that</text>
<text><location><page_13><loc_12><loc_15><loc_35><loc_16></location>Hence, ∇ α ξ β is geometrical.</text>
<formula><location><page_13><loc_33><loc_14><loc_68><loc_20></location>̂ e α a ̂ e β b ∇ α ξ β = Π ab , ̂ e α a ξ β ∇ α ξ β = 1 2 · ∇ a /lscript (2) .</formula>
<text><location><page_13><loc_12><loc_9><loc_88><loc_13></location>Next we show that when ∇ ξ ξ = 0 the tensor K ( m ) µν is geometrical for all m ≥ 0. As a preliminary step we first compute in full generality the contraction ξ α K ( m ) αβ on H .</text>
<text><location><page_14><loc_12><loc_87><loc_88><loc_91></location>Proposition 4.7. Let H be a null hypersurface (Φ , ξ ) -embedded in ( M , g ) and let Z be a vector field along Φ( H ) . Extend ξ arbitrarily off Φ( H ) and define a ξ := ∇ ξ ξ . Then for every m ∈ N ∪ { 0 } ,</text>
<formula><location><page_14><loc_25><loc_81><loc_88><loc_86></location>K ( m +1) ( ξ, ξ ) H = 2 m ∑ i =0 ( m i ) ( K ( m -i ) )( L ( i ) ξ a ξ , ξ ) , (57)</formula>
<formula><location><page_14><loc_24><loc_76><loc_88><loc_82></location>K ( m +1) ( ξ, Z ) H = 1 2 Z ( K ( m ) ( ξ, ξ ) ) + m ∑ i =0 ( m i ) ( K ( m -i ) )( L ( i ) ξ a ξ , Z ) . (58)</formula>
<text><location><page_14><loc_12><loc_73><loc_88><loc_76></location>Proof. We first prove (58) and then we show that (57) follows from (58). Let us extend Z off H by L ξ Z = 0 (at the end we prove that the result is independent of the extension). First,</text>
<formula><location><page_14><loc_25><loc_67><loc_75><loc_72></location>( L ξ g ) ( ξ, Z ) = g ( ∇ ξ ξ, Z ) + g ( ξ, ∇ Z ξ ) = g ( a ξ , Z ) + 1 2 Z ( g ( ξ, ξ )) .</formula>
<text><location><page_14><loc_12><loc_66><loc_36><loc_68></location>Applying L ( m ) ξ to both sides,</text>
<formula><location><page_14><loc_22><loc_60><loc_78><loc_65></location>( L ( m +1) ξ g ) ( ξ, Z ) = m ∑ i =0 ( m i ) ( L ( m -i ) ξ g )( L ( i ) ξ a ξ , Z ) + 1 2 ξ ( m ) ( Z ( g ( ξ, ξ ))) ,</formula>
<text><location><page_14><loc_12><loc_57><loc_88><loc_60></location>which becomes (58) after commuting ξ ( m ) Z = Zξ ( m ) (since [ ξ, Z ] = 0). Obviously equation (58) is independent of the extension of Z off H . To show (57) just use (58) with Z = ξ .</text>
<text><location><page_14><loc_12><loc_52><loc_88><loc_55></location>Corollary 4.8. Let {H , γ , /lscript , /lscript (2) } be null metric hypersurface data (Φ , ξ ) -embedded in ( M , g ) and extend ξ off Φ( H ) by ∇ ξ ξ = 0 . Then K ( m ) ( ξ, · ) = 0 for all m ≥ 1 .</text>
<text><location><page_14><loc_12><loc_48><loc_88><loc_51></location>Lemma 4.9. Let {H , γ , /lscript , /lscript (2) } be null metric hypersurface data (Φ , ξ ) -embedded in ( M , g ) and extend ξ off Φ( H ) by ∇ ξ ξ = 0 . Then the tensor K ( m ) µν is geometrical for all m ≥ 0 .</text>
<text><location><page_14><loc_12><loc_41><loc_88><loc_47></location>Proof. The tensor K (0) µν := g µν is clearly geometrical by the definition of embedded metric hypersurface data. By Corollary 4.8 and recalling Φ /star K ( m ) = 2 Y ( m ) , the tensor K ( m ) µν for m ≥ 1 is geometrical as well.</text>
<text><location><page_14><loc_12><loc_36><loc_88><loc_40></location>Lemma 4.10. Let {H , γ , /lscript , /lscript (2) } be null metric hypersurface data (Φ , ξ ) -embedded in ( M , g ) and extend ξ off Φ( H ) by ∇ ξ ξ = 0 . Let T and S be any two geometrical objects. Then, the following properties are true.</text>
<unordered_list>
<list_item><location><page_14><loc_15><loc_33><loc_34><loc_34></location>1. T ⊗ S is geometrical.</list_item>
<list_item><location><page_14><loc_15><loc_31><loc_47><loc_32></location>2. Any trace of T w.r.t g is geometrical.</list_item>
<list_item><location><page_14><loc_15><loc_27><loc_64><loc_29></location>3. Any trace of T w.r.t L ( m ) ξ g µν is geometrical for any m ≥ 1 .</list_item>
</unordered_list>
<text><location><page_14><loc_12><loc_24><loc_74><loc_26></location>As a consequence, any trace of T ⊗ S w.r.t g or L ( m ) ξ g µν is also geometrical.</text>
<text><location><page_14><loc_12><loc_16><loc_88><loc_23></location>Proof. The first property is obvious. The second one follows from g αβ H = P ab ̂ e α a ̂ e β b + n a ̂ e α a ξ β + n b ̂ e β b ξ α (cf. (13)). We prove the third property by induction. First observe that for m ≥ 1</text>
<formula><location><page_14><loc_19><loc_13><loc_88><loc_19></location>L ( m ) ξ ( g αµ g µβ ) = 0 = ⇒ L ( m ) ξ g αβ = -g βν m ∑ i =1 ( m i ) ( L ( m -i ) ξ g αµ ) L ( i ) ξ g µν . (59)</formula>
<text><location><page_14><loc_12><loc_8><loc_88><loc_13></location>Particularizing for m = 1 gives L ξ g αβ = -g βν g αν K µν . By Lemma 4.9 and items 1. and 2. of this lemma, item 3. follows for m = 1. Assume it is true up to some integer m -1. Expression (59) together with Lemma 4.9 then show that it is also true for m .</text>
<text><location><page_15><loc_12><loc_88><loc_88><loc_92></location>Lemma 4.11. Let {H , γ , /lscript , /lscript (2) } be null metric hypersurface data (Φ , ξ ) -embedded in ( M , g ) and extend ξ off Φ( H ) by ∇ ξ ξ = 0 . Then the tensor ∇ ρ K ( m ) µν is geometrical for all m ≥ 0 .</text>
<text><location><page_15><loc_12><loc_84><loc_88><loc_87></location>Proof. The argument relies on the following well-known relation between ∇ and L ξ acting on a (0,2) symmetric tensor T</text>
<formula><location><page_15><loc_38><loc_82><loc_88><loc_84></location>∇ ξ T λγ = L ξ T λγ -2 T µ ( λ ∇ γ ) ξ µ . (60)</formula>
<text><location><page_15><loc_12><loc_80><loc_39><loc_81></location>Particularizing to T = K ( m ) gives</text>
<formula><location><page_15><loc_23><loc_74><loc_77><loc_79></location>ξ ρ ∇ ρ K ( m ) µν = ( L ξ K ( m ) ) µν -2 K ( m ) ρ ( µ ∇ ν ) ξ ρ = K ( m +1) µν -2 g ρε K ( m ) ρ ( µ ∇ ν ) ξ ε .</formula>
<text><location><page_15><loc_12><loc_61><loc_88><loc_70></location>Proposition 4.12. Let {H , γ , /lscript , /lscript (2) } be null metric hypersurface data (Φ , ξ ) -embedded in ( M , g ) and extend ξ off Φ( H ) by ∇ ξ ξ = 0 . Then, the tensors ̂ Σ ( m ) ναβ and Σ ( m ) ναβ for all m ≥ 1 are geometrical. Proof. Recalling definition (54), namely ̂ Σ ναβ := g νµ Σ µ αβ , and applying identity (51) gives</text>
<text><location><page_15><loc_12><loc_70><loc_88><loc_75></location>The first term is geometrical by Lemma 4.9, and the second one is also geometrical as a consequence of Lemmas 4.6, 4.9 and 4.10. Next we show that the tensor ̂ Σ ( m ) is geometrical.</text>
<formula><location><page_15><loc_14><loc_53><loc_88><loc_62></location>Σ ( m ) ναβ = g µν Σ ( m ) µ αβ = g µν L ( m -1) ξ ( g µρ ̂ Σ ραβ ) = g µν m -1 ∑ i =0 ( m -1 i ) ( L ( i ) ξ g µρ ) ̂ Σ ( m -i ) ραβ . (61) ̂</formula>
<formula><location><page_15><loc_18><loc_35><loc_82><loc_46></location>̂ Σ ( m +1) ναβ = L ( m ) ξ ̂ Σ ναβ = F ργλ ναβ ∇ ρ K ( m +1) λγ -2 m -1 ∑ k =0 ( m k +1 ) F ργλ ναβ K ( m -k ) σ ( λ Σ ( k +1) σ γ ) ρ = F ργλ ναβ ∇ ρ K ( m +1) λγ -2 m -1 ∑ k =0 ( m k +1 ) F ργλ ναβ g σε K ( m -k ) σ ( λ Σ ( k +1) ε | γ ) ρ .</formula>
<text><location><page_15><loc_12><loc_46><loc_88><loc_56></location>So, by Lemma 4.10 if Σ ( k ) ναβ is geometrical for all k ≤ m , then Σ ( m ) ναβ is geometrical as well. It only remains to prove that ̂ Σ ( m ) ναβ is geometrical for all m ≥ 1. We establish this by an induction argument. From ̂ Σ ναβ = F ρλγ ναβ ∇ ρ K λγ and Lemma 4.11 it follows that the tensor ̂ Σ ναβ is geometrical. Assume ̂ Σ ( k ) ναβ (and hence also Σ ( k ) ναβ ) is geometrical for all k ≤ m . Applying Proposition 4.3 to A = K and taking into account L ξ F = 0 it follows</text>
<text><location><page_15><loc_12><loc_31><loc_88><loc_35></location>Using Lemmas 4.10 and 4.11 and the fact that every Σ ( k ) ναβ for k ≤ m is geometrical, we conclude that ̂ Σ ( m +1) ναβ (and thus Σ ( m +1) ναβ ) is geometrical as well.</text>
<text><location><page_15><loc_12><loc_24><loc_88><loc_29></location>That R αβ is geometrical when ∇ ξ ξ = 0 is immediate from (40), (49) and (50) because in this case Z (2) = Y (2) (by Definition 3.4). In order to prove that L ( m ) ξ R αβ is geometrical for every m ≥ 1 when ∇ ξ ξ = 0 it is convenient to first rewrite (56) making the tensor H explicit, namely</text>
<formula><location><page_15><loc_16><loc_14><loc_88><loc_23></location>L ( m ) ξ R αβ = ∇ µ Σ ( m ) µ αβ -∇ β Σ ( m ) µ αµ + m -2 ∑ k =0 ( m k +1 ) g σρ g µν Σ ( m -k -1) ραβ Σ ( k +1) νσµ -m -2 ∑ k =0 ( m k +1 ) g σρ g µν Σ ( m -k -1) ραµ Σ ( k +1) νσβ . (62)</formula>
<text><location><page_15><loc_12><loc_5><loc_88><loc_13></location>By Lemma 4.10 and Proposition 4.12 all the terms in the two sums are geometrical, so it suffices to show that ∇ µ Σ ( m ) µ αβ and ∇ β Σ ( m ) µ αµ are also geometrical. In order to do that we shall contract both tensors with two tangent vectors ̂ e α a ̂ e β b , one tangent and one transverse 15</text>
<text><location><page_16><loc_12><loc_85><loc_88><loc_92></location>̂ e α a ξ β , and two transverse vectors ξ α ξ β and check that in all cases the result only depends on metric data and the expansion { Y ( k ) } . These computations rely on general expressions for the pullback of ambient tensor fields into an arbitrary null hypersurface H . These are computed in full generality in Appendix A. The following notation is used.</text>
<text><location><page_16><loc_12><loc_78><loc_88><loc_84></location>Notation 4.13. Given a (0 , p ) tensor field T α 1 ··· α p on M we use the standard notation T a 1 ··· a p to denote the pullback of T to H . Moreover, we introduce the notation ( i ) T α 1 ··· α i -1 α i +1 ··· α p for ξ α i T α 1 ··· α p and ( i ) T a 1 ··· a i -1 a i +1 ··· a p for the pullback of ( i ) T α 1 ··· α i -1 α i +1 ··· α p to H . Several numbers between parenthesis denote contractions with ξ on those slots.</text>
<text><location><page_16><loc_12><loc_74><loc_88><loc_77></location>Let start by analyzing ∇ µ Σ ( m ) µ αβ . Firstly, from Proposition A.2 and the fact that Σ ( m ) ναβ is geometrical, it follows that</text>
<formula><location><page_16><loc_15><loc_68><loc_85><loc_73></location>e α a e β b ∇ µ Σ ( m ) µ αβ = n c ( L ξ Σ ( m ) ) cab + H -geometrical terms = n c Σ ( m +1) cab + H -geo. terms ,</formula>
<formula><location><page_16><loc_25><loc_62><loc_75><loc_67></location>̂ ξ α e β b ∇ µ Σ ( m ) µ αβ = g µν e β b ∇ µ (2) Σ ( m ) νβ -g µν g ρα e β b Σ ( m ) ναβ ∇ µ ξ ρ .</formula>
<text><location><page_16><loc_12><loc_64><loc_88><loc_72></location>̂ ̂ and hence by Proposition 4.12 ̂ e α a ̂ e β b ∇ µ Σ ( m ) µ αβ only depends on metric data and { Y ( k ) } . Secondly, its contraction with ξ α e β b yields</text>
<text><location><page_16><loc_12><loc_58><loc_88><loc_64></location>̂ ̂ ̂ By Lemma 4.10 and Proposition 4.12 the second term is H -geometrical. Moreover, by Proposition A.2,</text>
<formula><location><page_16><loc_15><loc_53><loc_85><loc_58></location>g µν e β b ∇ µ (2) Σ ( m ) νβ = n c ( L ξ (2) Σ ( m ) ) cb + H -geo. terms = (2) Σ ( m +1) cb n c + H -geo. terms ,</formula>
<formula><location><page_16><loc_16><loc_45><loc_84><loc_47></location>ξ α ξ β ∇ µ Σ ( m ) µ αβ = g µν ∇ µ (2 , 3) Σ ( m ) ν -g µν g βρ (2) Σ ( m ) νβ ∇ µ ξ ρ -g µν g ρα (3) Σ ( m ) να ∇ µ ξ ρ .</formula>
<text><location><page_16><loc_12><loc_47><loc_88><loc_57></location>̂ where we used L ξ ( ( i ) T ( m ) ) = ( i ) T ( m +1) since obviously L ξ ξ = 0. Again Prop. 4.12 shows that ξ α ̂ e β b ∇ µ Σ ( m ) µ αβ (and by symmetry also ξ β ̂ e α b ∇ µ Σ ( m ) µ αβ ) only depends on metric data and { Y ( k ) } . Finally, the contraction with ξ α ξ β is</text>
<text><location><page_16><loc_12><loc_43><loc_36><loc_44></location>By Prop. A.2 the first term is</text>
<formula><location><page_16><loc_30><loc_40><loc_70><loc_41></location>g µν ∇ µ (2 , 3) Σ ( m ) ν = (2 , 3) Σ ( m +1) c n c + H -geo. terms ,</formula>
<text><location><page_16><loc_12><loc_26><loc_88><loc_38></location>so by Prop. 4.12 it is H -geometrical. The second and third terms are H -geometrical as well by Lemmas 4.6 and 4.10 and Prop. 4.12. Hence ∇ µ Σ ( m ) µ αβ is geometrical. For the tensor ∇ β Σ ( m ) µ αµ we introduce T ( m ) α := Σ ( m ) µ αµ , which is geometrical by Lemma 4.10. The contractions ̂ e α a ̂ e β b ∇ β T α , ̂ e α a ξ β ∇ β T α and ξ α ̂ e β b ∇ β T α are automatically H -geometrical after using identities (156)-(158) in Appendix A and L ξ T ( m ) = T ( m +1) . Finally, ξ α ξ β ∇ β T ( m ) α = L ξ ( (1) T ( m ) ) = (1) T ( m +1) is also H -geometrical, and hence ∇ β Σ ( m ) µ αµ is geometrical. Thus, the following result has been proved.</text>
<text><location><page_16><loc_12><loc_21><loc_88><loc_26></location>Proposition 4.14. Let {H , γ , /lscript , /lscript (2) } be null metric hypersurface data (Φ , ξ ) -embedded in ( M , g ) and extend ξ off Φ( H ) by ∇ ξ ξ = 0 . Then, the tensor L ( m ) ξ R αβ is geometrical for every m ≥ 0 .</text>
<section_header_level_1><location><page_16><loc_12><loc_17><loc_52><loc_19></location>4.1 Analysis of the leading order terms</section_header_level_1>
<text><location><page_16><loc_12><loc_9><loc_88><loc_17></location>In this subsection we compute the leading order terms of the tensor L ( m ) R αβ on an arbitrary null hypersurface H . As we shall see, it turns out that the leading order terms of ξ α L ( m ) R αβ involve m +2 transverse derivatives of the metric, whereas the fully tangential components of L ( m ) R αβ depend at most on m + 1 transverse derivatives. In order to write down equalities only to the leading order it is useful introduce the following notation.</text>
<text><location><page_17><loc_12><loc_86><loc_88><loc_91></location>Notation 4.15. Let ( M , g ) be a semi-Riemannian manifold and T, S be two tensor fields involving g and its derivatives. The notation T [ m ] = S means that the tensor T -S does not depend on derivatives of g of order m or higher.</text>
<text><location><page_17><loc_12><loc_82><loc_88><loc_85></location>The only terms that have a chance to carry m +1 and m +2 derivatives of the metric in identity (62) are the ones of the form ∇ γ Σ ( m ) µ αβ . Thus, with the notation above,</text>
<formula><location><page_17><loc_29><loc_79><loc_88><loc_81></location>L ( m ) ξ R αβ [ m +1] = ∇ µ Σ ( m ) µ αβ -∇ β Σ ( m ) µ µα , m ≥ 1 . (63)</formula>
<text><location><page_17><loc_12><loc_76><loc_64><loc_77></location>Observe that equation (51) together with L ξ g µν = -K µν implies</text>
<text><location><page_17><loc_12><loc_70><loc_34><loc_71></location>Lowering the index µ gives</text>
<formula><location><page_17><loc_25><loc_70><loc_88><loc_77></location>Σ ( m ) µ αβ = L ( m -1) ξ ( g µν ̂ Σ ναβ ) [ m ] = g µν ̂ Σ ( m ) ναβ -( m -1) K µν ̂ Σ ( m -1) ναβ . (64)</formula>
<formula><location><page_17><loc_26><loc_66><loc_88><loc_71></location>Σ ( m ) µαβ [ m ] = ̂ Σ ( m ) µαβ -( m -1) g νρ K µν ̂ Σ ( m -1) ραβ , Σ ( m ) µαβ [ m +1] = ̂ Σ ( m ) µαβ , (65)</formula>
<text><location><page_17><loc_12><loc_63><loc_26><loc_65></location>and applying L ξ ,</text>
<formula><location><page_17><loc_22><loc_60><loc_88><loc_65></location>L ξ Σ ( m ) µαβ [ m +1] = ̂ Σ ( m +1) µαβ -( m -1) g νρ K µν ̂ Σ ( m ) ραβ , L ξ Σ ( m ) µαβ [ m +2] = ̂ Σ ( m +1) µαβ . (66)</formula>
<text><location><page_17><loc_12><loc_57><loc_60><loc_58></location>Lemma 4.16. Let ξ ∈ X ( M ) and m ≥ 1 an integer. Then,</text>
<formula><location><page_17><loc_32><loc_53><loc_88><loc_58></location>̂ Σ ( m ) ναβ [ m ] = F ρλγ ναβ ∇ ρ K ( m ) λγ -g σε K σν F ρλγ εαβ ∇ ρ K ( m -1) λγ , (67)</formula>
<formula><location><page_17><loc_29><loc_51><loc_88><loc_53></location>Σ ( m ) µ αβ [ m ] = g µν F ρλγ ναβ ∇ ρ K ( m ) λγ -m K µν F ρλγ ναβ ∇ ρ K ( m -1) λγ , (68)</formula>
<text><location><page_17><loc_12><loc_47><loc_71><loc_49></location>Proof. By Proposition 4.3 and (54), together with L ξ F ρλγ ναβ = 0, it follows</text>
<text><location><page_17><loc_12><loc_37><loc_27><loc_38></location>which simplifies to</text>
<formula><location><page_17><loc_15><loc_38><loc_85><loc_48></location>L ( m -1) ξ ̂ Σ ναβ = F ρλγ ναβ ∇ ρ K ( m ) λγ -m -2 ∑ k =0 ( m -1 k +1 ) ( K ( m -1 -k ) σ ( β Σ ( k +1) σ ν ) α + K ( m -1 -k ) σ ( α Σ ( k +1) σ ν ) β -K ( m -1 -k ) σ ( α Σ ( k +1) σ β ) ν ) ,</formula>
<formula><location><page_17><loc_25><loc_30><loc_75><loc_37></location>L ( m -1) ξ ̂ Σ ναβ = F ρλγ ναβ ∇ ρ K ( m ) λγ -m -2 ∑ k =0 ( m -1 k +1 ) K ( m -1 -k ) σν Σ ( k +1) σ αβ</formula>
<text><location><page_17><loc_12><loc_28><loc_56><loc_30></location>because of the symmetries of K ( i ) αβ and Σ ( i ) σ µν . Hence,</text>
<text><location><page_17><loc_12><loc_20><loc_88><loc_26></location>Combining (64) and (70) it follows Σ ( m -1) σ αβ [ m ] = g σν ̂ Σ ( m -1) ναβ [ m ] = g σε F ρλγ εαβ ∇ ρ K ( m -1) λγ , which inserted into (69) gives</text>
<formula><location><page_17><loc_12><loc_25><loc_88><loc_29></location>̂ Σ ( m ) ναβ [ m ] = F ρλγ ναβ ∇ ρ K ( m ) λγ -K σν Σ ( m -1) σ αβ , (69) ̂ Σ ( m -1) ναβ [ m ] = F ρλγ ναβ ∇ ρ K ( m -1) λγ . (70)</formula>
<formula><location><page_17><loc_31><loc_16><loc_69><loc_21></location>̂ Σ ( m ) ναβ [ m ] = F ρλγ ναβ ∇ ρ K ( m ) λγ -g σε K σν F ρλγ εαβ ∇ ρ K ( m -1) λγ .</formula>
<text><location><page_17><loc_12><loc_14><loc_67><loc_15></location>This establishes (67). Introducing (70) and (67) into (64) gives (68).</text>
<text><location><page_17><loc_12><loc_11><loc_67><loc_12></location>We quote the following immediate consequence for future reference.</text>
<text><location><page_17><loc_12><loc_8><loc_62><loc_10></location>Corollary 4.17. Let ξ ∈ X ( M ) and m ≥ 0 an integer. Then,</text>
<formula><location><page_18><loc_21><loc_90><loc_22><loc_94></location>̂</formula>
<formula><location><page_18><loc_21><loc_89><loc_88><loc_92></location>Σ ( m ) ναβ [ m +1] = F ρλγ ναβ ∇ ρ K ( m ) λγ , (71) Σ ( m ) µ αβ [ m +1] = g µν F ρλγ ναβ ∇ ρ K ( m ) λγ . (72)</formula>
<text><location><page_18><loc_12><loc_85><loc_88><loc_88></location>Before computing R ( m ) , ˙ R ( m ) and R ( m ) on H it is important to make the following observation based on Proposition 4.7.</text>
<text><location><page_18><loc_12><loc_76><loc_88><loc_84></location>Remark 4.18. By Proposition 4.7 the derivatives K ( m ) ( ξ, · ) are given in terms of transverse derivatives of a ξ := ∇ ξ ξ on Φ( H ) , metric hypersurface data as well as on the tensors { Y , ..., Y ( m -1) } . Hence, when computing R ( m ) , ˙ R ( m ) and R ( m ) , terms of the form K ( m ) ( ξ, · ) can always be replaced by lower order terms, i.e. terms that depend on metric hypersurface data and { Y , ..., Y ( m -1) } .</text>
<text><location><page_18><loc_12><loc_66><loc_88><loc_75></location>Notation 4.19. Let ( M , g ) be a semi-Riemannian manifold, H an embedded hypersurface and T, S two tensors involving g and its derivatives. We introduce the notation T ( m ) = S to denote that the tensor ( T -S ) | H does not depend on transverse derivatives of g at H of order m or higher. ̂</text>
<text><location><page_18><loc_12><loc_61><loc_88><loc_69></location>In the next lemma and following proposition we compute the pullback of Σ ( m ) as well as several contractions of the tensors Σ ( m ) and ̂ Σ ( m ) that we shall use below. The computation relies on the general identities for the pullback of ambient tensor fields to arbitrary null hypersurfaces computed in Appendix A.</text>
<text><location><page_18><loc_12><loc_57><loc_88><loc_60></location>Lemma 4.20. Let H be a null hypersurface (Φ , ξ ) -embedded in a semi-Riemannian manifold ( M , g ) . Extend ξ arbitrarily off Φ( H ) and let { e a } be a local basis on H and e a := Φ /star e a . Then,</text>
<formula><location><page_18><loc_19><loc_53><loc_41><loc_56></location>ξ λ ∇ ρ K ( m ) λγ ( m +1) = 0 , (73)</formula>
<formula><location><page_18><loc_16><loc_51><loc_41><loc_53></location>ξ γ e ρ a e λ b ∇ ρ K ( m ) λγ ( m +1) = 0 , (75)</formula>
<formula><location><page_18><loc_43><loc_50><loc_88><loc_58></location>̂ ̂ e ρ a ̂ e λ b ̂ e γ c ∇ ρ K ( m +1) λγ ( m +1) = 2 · ∇ a Y ( m +1) bc +4Y a ( b r ( m +1) c ) , (74) e ρ a e λ b e γ c ∇ ρ K ( m ) λγ ( m +1) = 0 , (76)</formula>
<text><location><page_18><loc_12><loc_35><loc_88><loc_47></location>̂ ̂ ̂ Proof. Equations (75) and (76) are particular cases of (73) and (74), respectively. Moreover, the validity of (73) and (74) implies at once (79) and (80) because g σγ can be decomposed using (13). Hence it suffices to prove the equations in the first and third lines. In order to prove (73) we contract ξ λ ∇ ρ K ( m ) λγ with (i) ξ ρ ξ γ , (ii) ξ ρ ̂ e γ c , (iii) ̂ e ρ c ξ γ and (iv) ̂ e ρ c ̂ e γ d and show that all of them are at most of order m . Let us start with (i). Applying (60) for T = K ( m ) gives</text>
<formula><location><page_18><loc_12><loc_45><loc_41><loc_53></location>̂ ̂ ̂ e γ a ̂ e λ b ξ ρ ∇ ρ K ( m ) λγ ( m +1) = 2Y ( m +1) ab , (77) g λγ e ρ a ∇ ρ K ( m ) λγ ( m +1) = 0 , (79)</formula>
<formula><location><page_18><loc_52><loc_45><loc_88><loc_52></location>̂ ̂ ̂ ̂ e λ a ̂ e γ b g σρ ∇ ρ K ( m ) λγ ( m +1) = 2 ν σ Y ( m +1) ab , (78) e ρ a e λ b g σγ ∇ ρ K ( m ) λγ ( m +1) = 0 . (80)</formula>
<formula><location><page_18><loc_21><loc_31><loc_79><loc_36></location>ξ ρ ξ γ ξ λ ∇ ρ K ( m ) λγ = ξ λ ξ γ ( L ξ K ( m ) λγ -2 K ( m ) µ ( λ ∇ γ ) ξ µ ) ( m +1) = ξ λ ξ γ K ( m +1) λγ ( m +1) = 0 ,</formula>
<formula><location><page_18><loc_22><loc_24><loc_78><loc_30></location>̂ e γ c ξ ρ ξ λ ∇ ρ K ( m ) λγ = e γ c ξ ρ ∇ ρ (1) K ( m ) γ -e γ c ξ ρ K ( m ) λγ ∇ ρ ξ λ ( m +1) = e γ c ξ ρ ∇ ρ (1) K ( m ) γ .</formula>
<text><location><page_18><loc_12><loc_28><loc_88><loc_32></location>where in the second equality we used that K ( m ) is of order m and in the last one Remark 4.18. For (ii) we contract ξ λ ∇ ρ K ( m ) λγ with ξ ρ e γ c and use again that K ( m ) is of order m ,</text>
<text><location><page_18><loc_12><loc_16><loc_88><loc_26></location>̂ ̂ ̂ ̂ By Remark 4.18 the tensor (1) K ( m ) is of order m -1. Applying (157) for T = (1) K ( m ) all terms in the right hand side are at most of order m , so ̂ e γ c ξ ρ ξ λ ∇ ρ K ( m ) λγ ( m +1) = 0. For item (iii) we contract ξ λ ∇ ρ K ( m ) λγ with e ρ c ξ γ , namely</text>
<text><location><page_18><loc_12><loc_7><loc_88><loc_15></location>̂ ̂ because (1) K ( m ) is at most of order m -1. In order to prove (iv), i.e. ̂ e ρ c ̂ e γ d ξ λ ∇ ρ K ( m ) λγ ( m +1) = 0, we use identity (158) with T = K ( m ) . Since all the terms in the right hand side are at most of</text>
<formula><location><page_18><loc_27><loc_11><loc_74><loc_19></location>̂ e ρ c ξ γ ξ λ ∇ ρ K ( m ) λγ = e ρ c ∇ ρ ( K ( m ) ( ξ, ξ ) ) -2 ξ γ K ( m ) λγ ∇ ρ ξ λ ( m +1) = 0 ,</formula>
<text><location><page_19><loc_12><loc_83><loc_88><loc_92></location>order m , ̂ e ρ c ̂ e γ d ξ λ ∇ ρ K ( m ) λγ ( m +1) = 0. This establishes (73). Equation (74) follows from (156) with T = K ( m +1) after recalling that ( i ) K ( m +1) is at most of order m and that Φ /star K ( m +1) = 2 Y ( m +1) . It only remains to prove (77) and (78). For the first one we insert (13) into ̂ e λ a ̂ e γ b g σρ ∇ ρ K ( m ) λγ and get</text>
<text><location><page_19><loc_12><loc_75><loc_88><loc_83></location>̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ where equation (76) has been used in the first and second terms. This expression shows that (78) follows from (77). To establish the latter we use (157) with T = K ( m ) because only the first term in the right hand side is of order m +1.</text>
<formula><location><page_19><loc_17><loc_79><loc_83><loc_84></location>e λ a e γ b g σρ ∇ ρ K ( m ) λγ = e λ a e γ b ( P cd e σ c e ρ d + ξ σ e ρ c n c + ξ ρ ν σ ) ∇ ρ K ( m ) λγ ( m +1) = e λ a e γ b ξ ρ ν σ ∇ ρ K ( m ) λγ ,</formula>
<text><location><page_19><loc_12><loc_71><loc_88><loc_74></location>Proposition 4.21. Let H be a null hypersurface (Φ , ξ ) -embedded in a semi-Riemannian manifold ( M , g ) and extend ξ arbitrarily off Φ( H ) . Then,</text>
<formula><location><page_19><loc_12><loc_62><loc_88><loc_69></location>̂ Σ ( m ) abc ( m +1) = 0 , (1) ̂ Σ ( m ) ab ( m +1) = -Y ( m +1) ab , (2) ̂ Σ ( m ) ab ( m +1) = Y ( m +1) ab , (3) ̂ Σ ( m ) ab ( m +1) = Y ( m +1) ab , ξ α ξ β Σ ( m ) µ αβ ( m +1) = 0 , Σ ( m ) µ µa ( m +1) = 0 , ξ α Σ ( m ) µ µα ( m +1) = tr P Y ( m +1) .</formula>
<formula><location><page_19><loc_20><loc_68><loc_80><loc_73></location>̂ Σ ( m +1) cab ( m +1) = · ∇ a Y ( m +1) bc + · ∇ b Y ( m +1) ac -· ∇ c Y ( m +1) ab +2r ( m +1) c Y ab +2r c Y ( m +1) ab ,</formula>
<text><location><page_19><loc_12><loc_58><loc_74><loc_60></location>Proof. Consider equation (67) for m +1 and contract it with e ν c e α a e β b , namely</text>
<text><location><page_19><loc_12><loc_50><loc_88><loc_57></location>̂ ̂ ̂ ̂ ̂ ̂ Defining F def cab := 1 2 ( δ d a δ e b δ f c + δ d b δ e a δ f c -δ d c δ e a δ f b ) and using equation (74) the first term takes the form</text>
<formula><location><page_19><loc_24><loc_55><loc_76><loc_60></location>̂ ̂ ̂ ̂ Σ ( m +1) cab [ m +1] = e ν c e α a e β b F ρλγ ναβ ∇ ρ K ( m +1) λγ -e ν c e α a e β b g σε K σν F ρλγ εαβ ∇ ρ K ( m ) λγ .</formula>
<formula><location><page_19><loc_20><loc_45><loc_80><loc_50></location>F def cab ( ∇K ( m +1) ) def ( m +1) = 2 F def cab ( · ∇ d Y ( m +1) ef +2Y d ( e r ( m +1) f ) ) ( m +1) = · ∇ a Y ( m +1) bc + · ∇ b Y ( m +1) ac -· ∇ c Y ( m +1) ab +2r ( m +1) c Y ab .</formula>
<text><location><page_19><loc_12><loc_42><loc_45><loc_44></location>Making F ρλγ εαβ explicit the second term is</text>
<formula><location><page_19><loc_12><loc_35><loc_81><loc_41></location>̂ e ν c ̂ e α a ̂ e β b g σε K σν F ρλγ εαβ ∇ ρ K ( m ) λγ = 1 2 ( ̂ e ρ a ̂ e λ b g σγ + ̂ e λ a ̂ e ρ b g σγ -̂ e λ a ̂ e γ b g σρ ) K σν ̂ e ν c ∇ ρ K ( m ) λγ . Applying (80) to the first and second terms and (78) in the last term,</formula>
<formula><location><page_19><loc_21><loc_32><loc_79><loc_34></location>e ν c e α a e β b g σε K σν F ρλγ εαβ ∇ ρ K ( m ) λγ ( m +1) = -K σν e ν c n d e σ d Y ( m +1) ab ( m +1) = -2r c Y ( m +1) ab ,</formula>
<formula><location><page_19><loc_40><loc_24><loc_88><loc_29></location>̂ Σ ( m ) ναβ [ m +1] = F ρλγ ναβ ∇ ρ K ( m ) λγ . (81)</formula>
<text><location><page_19><loc_12><loc_28><loc_88><loc_33></location>̂ ̂ ̂ ̂ ̂ because Φ /star K = 2 Y . Hence the equation of the first line is established. The equations of the second line follow from (71), namely</text>
<text><location><page_19><loc_12><loc_17><loc_88><loc_26></location>Contracting this with ̂ e ν c ̂ e α a ̂ e β b and using (76) gives ̂ Σ ( m ) cab ( m +1) = 0, and contracting it with ξ ν e α a ̂ e β b gives ̂</text>
<formula><location><page_19><loc_29><loc_15><loc_71><loc_20></location>(1) Σ ( m ) ab [ m +1] = 1 2 ( ξ γ e ρ a e λ b + ξ γ e λ a e γ b -ξ ρ e λ a e γ b ) ∇ ρ K ( m ) λγ .</formula>
<formula><location><page_19><loc_29><loc_5><loc_71><loc_13></location>̂ ̂ (2) ̂ Σ ( m ) ab [ m +1] = 1 2 ( ξ ρ ̂ e γ a ̂ e λ b + ξ λ ̂ e γ a ̂ e ρ b -ξ λ ̂ e ρ a ̂ e γ b ) ∇ ρ K ( m ) λγ .</formula>
<text><location><page_19><loc_12><loc_12><loc_88><loc_19></location>̂ ̂ ̂ ̂ ̂ ̂ Applying (75) to the first and second terms and (77) to the last one yields (1) ̂ Σ ( m ) ab ( m +1) = -Y ( m +1) ab . In a similar way, contracting (81) with e ν a ξ α e β b gives</text>
<text><location><page_20><loc_12><loc_86><loc_88><loc_94></location>By (75) the only term that contributes is the first one, which after using (77) is (2) ̂ Σ ( m ) ab ( m +1) = Y ( m +1) ab . By the symmetry of ̂ Σ ( m ) , (3) ̂ Σ ( m ) ab ( m +1) = Y ( m +1) ab . To prove the third line consider (72), namely</text>
<formula><location><page_20><loc_38><loc_84><loc_88><loc_86></location>Σ ( m ) µ αβ [ m +1] = g µν F ρλγ ναβ ∇ ρ K ( m ) λγ . (82)</formula>
<text><location><page_20><loc_12><loc_81><loc_34><loc_83></location>The contraction with ξ α ξ β ,</text>
<formula><location><page_20><loc_29><loc_76><loc_88><loc_81></location>ξ α ξ β Σ ( m ) µ αβ [ m +1] = 1 2 ( 2 ξ λ ξ ρ g µγ -ξ γ ξ λ g µρ ) ∇ ρ K ( m ) λγ , (83)</formula>
<text><location><page_20><loc_12><loc_74><loc_75><loc_76></location>yields ξ α ξ β Σ ( m ) µ αβ ( m +1) = 0 after using (73). Taking trace in µ, α in (82) gives</text>
<formula><location><page_20><loc_15><loc_68><loc_88><loc_73></location>Σ ( m ) µ µβ [ m +1] = g µν F ρλγ νµβ ∇ ρ K ( m ) λγ = 1 2 ( g ργ δ λ β + g λγ δ ρ β -g λρ δ γ β ) ∇ ρ K ( m ) λγ = 1 2 g λγ ∇ β K ( m ) λγ , (84)</formula>
<text><location><page_20><loc_12><loc_61><loc_88><loc_68></location>where in the last equality we used the symmetry of K ( m ) . Contracting with ̂ e β a and using equation (79) yields ̂ e β a Σ ( m ) µ µβ ( m +1) = 0. Finally, from (60) with T = K ( m ) , ∇ ξ K ( m ) [ m +1] = K ( m +1) , (85)</text>
<text><location><page_20><loc_12><loc_55><loc_88><loc_60></location>so contracting (84) with ξ β gives ξ β Σ ( m ) µ µβ ( m +1) = 1 2 g λγ K ( m +1) λγ . Using (13) and the fact that (1) K ( m +1) ( m +1) = 0, the last equation of the third line follows.</text>
<text><location><page_20><loc_12><loc_48><loc_88><loc_54></location>By virtue of (63) the last ingredient to compute the tensors R ( m ) , ˙ R ( m ) and R ( m ) on H is being able to calculate the pullback both of a divergence and of a ambient tensor field. These are computed in full generality in Propositions A.1 and A.2, and as a consequence we have the following three propositions.</text>
<text><location><page_20><loc_12><loc_44><loc_88><loc_47></location>Proposition 4.22. Let H be a null hypersurface (Φ , ξ ) -embedded in ( M , g ) and extended ξ arbitrarily off Φ( H ) . Then for every m ≥ 0 ,</text>
<formula><location><page_20><loc_39><loc_40><loc_88><loc_42></location>R ( m +1) ( m +2) = -tr P Y ( m +2) . (86)</formula>
<text><location><page_20><loc_12><loc_36><loc_88><loc_39></location>Proof. The case m = 0 has already been established in (49). In order to prove the identity for m ≥ 1 we contract the general expression (63) with ξ α ξ β so that</text>
<formula><location><page_20><loc_27><loc_32><loc_88><loc_34></location>ξ α ξ β L ( m ) ξ Ric αβ [ m +2] = ξ α ξ β ∇ µ Σ ( m ) µ αβ -ξ α ξ β ∇ β Σ ( m ) µ µα (87)</formula>
<formula><location><page_20><loc_39><loc_27><loc_88><loc_32></location>[ m +2] = ∇ µ ( ξ α ξ β Σ ( m ) µ αβ ) -∇ ξ ( ξ α Σ ( m ) µ µα ) , (88)</formula>
<text><location><page_20><loc_12><loc_25><loc_88><loc_28></location>where in the second line we used that Σ ( m ) involves up to m +1 derivatives of g . For the same reason, by (83) it follows</text>
<formula><location><page_20><loc_16><loc_18><loc_84><loc_24></location>∇ µ ( ξ α ξ β Σ ( m ) µ αβ ) [ m +2] = ξ λ ξ ρ g µγ ∇ µ ∇ ρ K ( m ) λγ -1 2 ξ γ ξ λ g µρ ∇ µ ∇ ρ K ( m ) λγ [ m +2] = ξ λ g µγ ∇ µ ∇ ξ K ( m ) λγ -ξ λ g µγ ∇ ρ K ( m ) λγ ∇ µ ξ ρ -1 2 ξ γ ξ λ g µρ ∇ µ ∇ ρ K ( m ) λγ .</formula>
<text><location><page_20><loc_12><loc_12><loc_88><loc_16></location>From the expression of g µρ in (13) and the fact that ∇K ( m ) is at most order m +1, the only term that has a chance of carrying m +2 transverse derivatives of g is the first one. Using (85) and recalling (73),</text>
<formula><location><page_20><loc_31><loc_9><loc_69><loc_12></location>ξ λ g µγ ∇ µ ∇ ξ K ( m ) λγ [ m +2] = ξ λ g µγ ∇ µ K ( m +1) λγ ( m +2) = 0 .</formula>
<text><location><page_21><loc_12><loc_90><loc_34><loc_91></location>Finally from (84) it follows</text>
<formula><location><page_21><loc_19><loc_84><loc_81><loc_89></location>∇ ξ ( ξ α Σ ( m ) µ µα ) [ m +2] = 1 2 g λγ ∇ ξ ( ξ µ ∇ µ K ( m ) λγ ) ( m +2) = 1 2 g λγ K ( m +2) λγ ( m +2) = tr P Y ( m +2) ,</formula>
<text><location><page_21><loc_12><loc_80><loc_88><loc_85></location>where in the second equality we used ∇ ξ K ( m ) [ m +1] = K ( m +1) and ∇ ξ K ( m +1) [ m +2] = K ( m +2) (see (85)) and in the third one we inserted (13) and used the fact that (1) K ( m +2) is at most of order m +1 (see Remark 4.18).</text>
<text><location><page_21><loc_12><loc_75><loc_88><loc_78></location>Proposition 4.23. Let H be a null hypersurface (Φ , ξ ) -embedded in ( M , g ) and extend ξ arbitrarily off Φ( H ) . Then for any m ≥ 0 ,</text>
<formula><location><page_21><loc_42><loc_72><loc_88><loc_74></location>˙ R ( m +1) ( m +2) = r ( m +2) . (89)</formula>
<text><location><page_21><loc_12><loc_69><loc_86><loc_71></location>Proof. The case m = 0 is (50). From (63) and the fact that Σ ( m ) is at most of order m +1,</text>
<formula><location><page_21><loc_20><loc_62><loc_88><loc_68></location>̂ e α a ξ β L ( m ) ξ R αβ [ m +1] = e α a ξ β ∇ µ Σ ( m ) µ αβ -e α a ξ β ∇ β Σ ( m ) µ µα (90)</formula>
<formula><location><page_21><loc_31><loc_61><loc_88><loc_67></location>̂ ̂ [ m +1] = e α a ∇ µ ( ξ β Σ ( m ) µ αβ ) -e α a Σ ( m ) µ αβ ∇ µ ξ β -e α a ∇ ξ ( Σ ( m ) µ µα ) , (91)</formula>
<text><location><page_21><loc_12><loc_59><loc_88><loc_64></location>̂ ̂ ̂ so the only terms capable of containing m + 2 transverse derivatives are the first and third ones. By Proposition A.2 applied to (3) Σ ( m ) µ α and L ξ ξ β = 0 it follows</text>
<formula><location><page_21><loc_34><loc_53><loc_66><loc_58></location>e α a ∇ µ ( ξ β Σ ( m ) µ αβ ) ( m +2) = n b e α a e µ b ξ β L ξ Σ ( m ) µαβ</formula>
<text><location><page_21><loc_12><loc_51><loc_88><loc_57></location>̂ ̂ ̂ because the rest of the terms are of (transverse) order m + 1 or below. Using the second equation in (66) and Proposition 4.21 we obtain</text>
<text><location><page_21><loc_12><loc_44><loc_49><loc_49></location>̂ Finally, by (157) with T = Σ ( m ) µ µα it follows</text>
<formula><location><page_21><loc_28><loc_44><loc_72><loc_52></location>e α a ∇ µ ( ξ β Σ ( m ) µ αβ ) ( m +2) = n b ̂ e α a ̂ e µ b ξ β ̂ Σ ( m +1) µαβ ( m +2) = r ( m +2) a .</formula>
<text><location><page_21><loc_12><loc_38><loc_52><loc_39></location>where the last equality we used Proposition 4.21.</text>
<formula><location><page_21><loc_26><loc_37><loc_74><loc_43></location>̂ e α a ∇ ξ ( Σ ( m ) µ µα ) ( m +2) = ̂ e α a L ξ Σ ( m ) µ µα ( m +2) = Σ ( m +1) µ µa ( m +2) = 0 ,</formula>
<text><location><page_21><loc_12><loc_34><loc_88><loc_37></location>Proposition 4.24. Let H be a null hypersurface (Φ , ξ ) -embedded in ( M , g ) and extended ξ arbitrarily off Φ( H ) . Let m ≥ 1 be an integer. Then,</text>
<formula><location><page_21><loc_17><loc_28><loc_88><loc_32></location>R ( m +1) ab ( m +1) = -2 L n Y ( m +1) ab -(2( m +1) κ n +tr P U ) Y ( m +1) ab -(tr P Y ( m +1) )U ab +4 P cd U c ( a Y ( m +1) b ) d +4(s -r) ( a r ( m +1) b ) +2 · ∇ ( a r ( m +1) b ) -2 κ ( m +1) Y ab . (92)</formula>
<text><location><page_21><loc_12><loc_25><loc_47><loc_26></location>Proof. By the general formula (63) we have</text>
<formula><location><page_21><loc_31><loc_21><loc_88><loc_24></location>R ( m +1) ab [ m +1] = e α a e β b ∇ µ Σ ( m ) µ αβ -e α a e β b ∇ β Σ ( m ) µ µα , (93)</formula>
<text><location><page_21><loc_12><loc_17><loc_88><loc_23></location>̂ ̂ ̂ ̂ so we only need to compute each of these terms. For the first one we use Proposition A.2 applied to Σ ( m ) , namely</text>
<formula><location><page_21><loc_15><loc_9><loc_85><loc_16></location>(div Σ ( m ) ) ab ( m +1) = P bc · ∇ b Σ ( m ) cab + n c ( L ξ Σ ( m ) ) cab + n c · ∇ c (1) Σ ( m ) ab + ( 2 κ n +tr P U ) (1) Σ ( m ) ab +(tr P Y -n ( /lscript (2) )) n c Σ ( m ) cab -2 P dc (r + s) d Σ ( m ) cab +2 P dc Y d ( a | Σ ( m ) c | b ) f n f +2 P dc U d ( a | (2) Σ ( m ) c | b ) -2(r -s) ( a | (2) Σ ( m ) c | b ) n c -2 V c ( a | n d Σ ( m ) dc | b ) +2r ( a | (1) Σ ( m ) c | b ) n c .</formula>
<text><location><page_22><loc_12><loc_85><loc_88><loc_94></location>From the second equation in (65), Σ ( m ) αβγ ( m +1) = ̂ Σ ( m ) αβγ , and thus by Proposition 4.21 it follows Σ ( m ) abc ( m +1) = 0, (1) Σ ( m ) ab ( m +1) = -Y ( m +1) ab and (2) Σ ( m ) ab ( m +1) = Y ( m +1) ab . Hence the expression for (div Σ ( m ) ) ab simplifies to</text>
<formula><location><page_22><loc_21><loc_80><loc_88><loc_85></location>(div Σ ( m ) ) ab ( m +1) = n c ( L ξ Σ ( m ) ) cab -n c · ∇ c Y ( m +1) ab -( 2 κ n +tr P U ) Y ( m +1) ab +2 P dc U d ( a | Y ( m +1) c | b ) -2(r -s) ( a | Y ( m +1) c | b ) n c -2r ( a | Y ( m +1) c | b ) n c . (94)</formula>
<text><location><page_22><loc_12><loc_74><loc_88><loc_79></location>In order to compute the term n c ( L ξ Σ ( m ) ) cab we contract the first equation in (66) with ̂ e µ c ̂ e α a ̂ e β b , which gives ( L ξ Σ ( m ) ) cab [ m +1] = e µ c e α a e β b ̂ Σ ( m +1) µαβ -( m -1) e µ c e α a e β b g νρ K µν ̂ Σ ( m ) ραβ .</text>
<text><location><page_22><loc_12><loc_69><loc_88><loc_75></location>̂ ̂ ̂ ̂ ̂ ̂ The first term is given by the first line in Proposition 4.21, and, for the second one, using (13) as well as ̂ Σ ( m ) abc ( m +1) = 0 and (1) ̂ Σ ( m ) ab ( m +1) = -Y ( m +1) ab one gets</text>
<text><location><page_22><loc_12><loc_64><loc_38><loc_65></location>Combining everything it follows</text>
<formula><location><page_22><loc_17><loc_61><loc_83><loc_63></location>( L ξ Σ ( m ) ) cab ( m +1) = · ∇ a Y ( m +1) bc + · ∇ b Y ( m +1) ac -· ∇ c Y ( m +1) ab +2r ( m +1) c Y ab +2 m r c Y ( m +1) ab ,</formula>
<formula><location><page_22><loc_35><loc_63><loc_65><loc_71></location>̂ e µ c ̂ e α a ̂ e β b g νρ K µν ̂ Σ ( m ) ραβ ( m +1) = -2r c Y ( m +1) ab .</formula>
<text><location><page_22><loc_12><loc_59><loc_55><loc_60></location>and after contracting with n c equation (94) becomes</text>
<formula><location><page_22><loc_12><loc_53><loc_88><loc_58></location>(div Σ ( m ) ) ab ( m +1) = 2 n c · ∇ ( a Y ( m +1) b ) c -2 n c · ∇ c Y ( m +1) ab -2 κ ( m +1) Y ab -( 2( m +1) κ n +tr P U ) Y ( m +1) ab +2 P dc U d ( a Y ( m +1) b ) c +2(s -2r) ( a r ( m +1) b ) .</formula>
<text><location><page_22><loc_12><loc_51><loc_17><loc_52></location>Using</text>
<formula><location><page_22><loc_13><loc_43><loc_87><loc_50></location>2 n c · ∇ ( a Y ( m +1) b ) c -2 n c · ∇ c Y ( m +1) ab = 2 · ∇ ( a r ( m +1) b ) -2Y ( m +1) c ( a · ∇ b ) n c -2 L n Y ( m +1) ab +4Y ( m +1) c ( a · ∇ b ) n c = 2 · ∇ ( a r ( m +1) b ) -2 L n Y ( m +1) ab +2Y ( m +1) c ( a · ∇ b ) n c = 2 · ∇ ( a r ( m +1) b ) -2 L n Y ( m +1) ab +2Y ( m +1) c ( a U b ) d P cd +2r ( m +1) ( a s b ) ,</formula>
<text><location><page_22><loc_12><loc_40><loc_79><loc_42></location>where the third equality follows from (27), the expression for (div Σ ( m ) ) ab is finally</text>
<formula><location><page_22><loc_16><loc_35><loc_88><loc_40></location>(div Σ ( m ) ) ab ( m +1) = -2 L n Y ( m +1) ab -( 2( m +1) κ n +tr P U ) Y ( m +1) ab +4 P cd Y ( m +1) c ( a U b ) d +4(s -r) ( a r ( m +1) b ) +2 · ∇ ( a r ( m +1) b ) -2 κ ( m +1) Y ab . (95)</formula>
<text><location><page_22><loc_12><loc_26><loc_88><loc_34></location>To compute the second term in (93) we use equation (156) for T = Σ ( m ) µ µα . Since by Proposition 4.21 Σ ( m ) µ µa ( m +1) = 0 and ξ α Σ ( m ) µ µα ( m +1) = tr P Y ( m +1) , we find that ̂ e α a ̂ e β b ∇ β Σ ( m ) µ µα = ( tr P Y ( m +1) ) U ab . Inserting this and (95) into (93) proves the Proposition.</text>
<text><location><page_22><loc_12><loc_23><loc_88><loc_27></location>Proposition 4.24 is interesting because it allows one to determine the evolution of the transverse expansion of the metric along the null generator in any null hypersurface. We next compute two contractions of (92) as well as its trace w.r.t P .</text>
<text><location><page_22><loc_12><loc_19><loc_88><loc_22></location>Corollary 4.25. Let H be a null hypersurface (Φ , ξ ) -embedded in ( M , g ) and extend ξ arbitrarily off Φ( H ) . Let Ric be the Ricci tensor of g . Then,</text>
<formula><location><page_22><loc_17><loc_13><loc_88><loc_18></location>R ( m +1) ab n b ( m +1) = -L n r ( m +1) a -( 2 mκ n +tr P U ) r ( m +1) a -· ∇ a κ ( m +1) , (96)</formula>
<formula><location><page_22><loc_16><loc_6><loc_88><loc_13></location>P ab R ( m +1) ab ( m +1) = -2 L n ( tr P Y ( m +1) ) -2(( m +1) κ n +tr P U ) tr P Y ( m +1) +2 κ ( m +1) ( n ( /lscript (2) ) -tr P Y ) -4 P ( r + s , r ( m +1) ) +2div P r ( m +1) . (98)</formula>
<formula><location><page_22><loc_15><loc_11><loc_88><loc_16></location>R ( m +1) ab n a n b ( m +1) = ( 2 mκ n +tr P U ) κ ( m +1) , (97)</formula>
<text><location><page_23><loc_12><loc_90><loc_56><loc_92></location>Proof. To prove (96) we contract (92) with n b and use</text>
<formula><location><page_23><loc_22><loc_82><loc_78><loc_89></location>2 n b · ∇ ( a r ( m +1) b ) = -· ∇ a κ ( m +1) -r ( m +1) b · ∇ a n b + n b · ∇ b r ( m +1) a = -· ∇ a κ ( m +1) -2r ( m +1) b · ∇ a n b + L n r ( m +1) a = -· ∇ a κ ( m +1) +2 κ ( m +1) s a -2 P bc U ac r ( m +1) b + L n r ( m +1) a ,</formula>
<text><location><page_23><loc_12><loc_78><loc_88><loc_81></location>where the second equality follows from (27). Contracting (96) with n a (97) follows at once. For the last one apply P ab to (92) and use (33) with T = Y ( m +1) .</text>
<text><location><page_23><loc_12><loc_72><loc_88><loc_76></location>Later on we shall need to use the 'complete' identities (86), (89) and (92), i.e. including a term that gathers all the lower order terms. By Proposition 4.14 these terms are H -geometrical when ∇ ξ ξ = 0.</text>
<text><location><page_23><loc_12><loc_68><loc_88><loc_71></location>Corollary 4.26. Let {H , γ , /lscript , /lscript (2) } be null metric hypersurface data (Φ , ξ ) -embedded in ( M , g ) and extended ξ arbitrarily off Φ( H ) . Let m ≥ 1 be an integer. Then,</text>
<formula><location><page_23><loc_15><loc_65><loc_88><loc_66></location>R ( m ) = -tr P Y ( m +1) + O ( m ) ( Y ≤ m ) , (99) ˙ R ( m ) a = r ( m +1) a + O ( m ) a ( Y ≤ m ) , (100)</formula>
<formula><location><page_23><loc_18><loc_56><loc_88><loc_63></location>R ( m +1) ab = -2 L n Y ( m +1) ab -(2( m +1) κ n +tr P U ) Y ( m +1) ab -(tr P Y ( m +1) )U ab +4 P cd U c ( a Y ( m +1) b ) d +4(s -r) ( a r ( m +1) b ) +2 · ∇ ( a r ( m +1) b ) -2 κ ( m +1) Y ab + O ( m ) ab ( Y ≤ m ) , (101)</formula>
<text><location><page_23><loc_12><loc_50><loc_88><loc_56></location>where O ( m ) , O ( m ) a and O ( m ) ab are, respectively, a scalar, a one-form and a (0,2) symmetric tensor on H with the property that when ∇ ξ ξ = 0 they only depend on null metric data { γ , /lscript , /lscript (2) } and on the tensors { Y , ..., Y ( m ) } .</text>
<section_header_level_1><location><page_23><loc_12><loc_46><loc_68><loc_48></location>5 Deformation tensor and algebraic identities</section_header_level_1>
<text><location><page_23><loc_12><loc_31><loc_88><loc_45></location>In some situations one has a privileged vector field η on ( M , g ) whose deformation tensor K [ η ] := L η g is known. This is in general a very valuable information that one may want to incorporate into the identities of Section 4. For instance, in [19] it is shown that the expression of the constraint tensor in (40) can be rewritten so that the dependence on the tensor Y is algebraic instead of via a transport equation. The corresponding identity was called the generalized master equation . The aim of this section is to extend the same idea to the higher order derivatives of the Ricci tensor, i.e. to combine identity (101) with information on K [ η ] so that the dependence on Y ( m +1) becomes algebraic. Let us start by reviewing a result from [19].</text>
<text><location><page_23><loc_12><loc_25><loc_88><loc_30></location>Lemma 5.1. Let {H , γ , /lscript , /lscript (2) } be null metric hypersurface data (Φ , ξ ) -embedded in ( M , g ) and let η ∈ X ( M ) be such that η | Φ( H ) is tangent to Φ( H ) . Denote by ¯ η the vector field on H satisfying Φ /star (¯ η ) = η | Φ( H ) . Then,</text>
<formula><location><page_23><loc_41><loc_22><loc_88><loc_24></location>[ η, ξ ] H = A η ξ +Φ /star ( X η ) , (102)</formula>
<text><location><page_23><loc_12><loc_19><loc_17><loc_20></location>where</text>
<formula><location><page_23><loc_38><loc_18><loc_88><loc_19></location>A η := -K [ η ]( ξ, ν ) + ( L ¯ η /lscript )( n ) , (103)</formula>
<text><location><page_23><loc_12><loc_15><loc_62><loc_17></location>and the components of X η in any basis { e a } of H are given by</text>
<formula><location><page_23><loc_30><loc_9><loc_88><loc_14></location>X a η := 1 2 K [ η ] ( ξ, n a ξ -2 θ a ) + 1 2 n a ¯ η ( /lscript (2) ) + P ab L ¯ η /lscript b , (104)</formula>
<text><location><page_23><loc_12><loc_8><loc_34><loc_10></location>where θ a = P ab Φ /star e b + n a ξ .</text>
<text><location><page_24><loc_65><loc_89><loc_65><loc_90></location>/negationslash</text>
<text><location><page_24><loc_12><loc_87><loc_88><loc_91></location>From now on we assume η | H to be null and tangent to Φ( H ), so there must exist a function α ∈ F ( H ) such that η | H = αν (in principle we do not assume α = 0, thus allowing η to have zeros). The gauge behaviour of the scalars α and K [ η ]( ξ, ν ) are as follows [19].</text>
<text><location><page_24><loc_12><loc_83><loc_88><loc_86></location>Lemma 5.2. Let ( z, V ) be a gauge element and denote with a prime the gauge-transformed objects. Then,</text>
<formula><location><page_24><loc_35><loc_81><loc_88><loc_83></location>α ' = zα, K [ η ]( ξ ' , ν ' ) = K [ η ]( ξ, ν ) . (105)</formula>
<text><location><page_24><loc_12><loc_77><loc_88><loc_80></location>From this it is immediate to check that the transformation of n ( α ) and that of A η are then given by</text>
<formula><location><page_24><loc_18><loc_74><loc_78><loc_76></location>n ' ( α ' ) = n ( α ) + n (log | z | ) α, (106) A = A η + n (log | z | ) α.</formula>
<formula><location><page_24><loc_62><loc_74><loc_88><loc_76></location>' η (107)</formula>
<text><location><page_24><loc_12><loc_72><loc_64><loc_73></location>In particular, n ( α ) is gauge-invariant at the points where α = 0.</text>
<text><location><page_24><loc_12><loc_69><loc_65><loc_70></location>Following [19], let us introduce the scalar function κ by means of</text>
<formula><location><page_24><loc_43><loc_66><loc_88><loc_67></location>κ := n ( α ) + ακ n . (108)</formula>
<text><location><page_24><loc_12><loc_57><loc_88><loc_64></location>From equation (30) and Lemma 5.2 it follows that κ is gauge invariant. This function extends the standard notion of surface gravity. Indeed, at the points where the vector field η does not vanish, its surface gravity ˜ κ is the scalar on H defined by ∇ η η H = κη. (109)</text>
<text><location><page_24><loc_14><loc_53><loc_14><loc_54></location>/negationslash</text>
<text><location><page_24><loc_12><loc_47><loc_69><loc_52></location>Using the identity L [ X,Y ] = [ L X , L Y ] applied to the metric g it follows</text>
<text><location><page_24><loc_12><loc_51><loc_88><loc_58></location>˜ Inserting η | H = αν into (109) and using (26) it follows that ˜ κ = κ on the subset of H where ¯ η = 0. Note however that κ is well defined and smooth everywhere on H .</text>
<formula><location><page_24><loc_40><loc_47><loc_88><loc_48></location>L η L ξ g = L ξ L η g -L [ ξ,η ] g. (110)</formula>
<text><location><page_24><loc_12><loc_44><loc_65><loc_46></location>Introducing (102) and pulling back this equation into H gives [19]</text>
<formula><location><page_24><loc_30><loc_39><loc_88><loc_44></location>L ¯ η Y = A η Y + /lscript ⊗ s dA η + 1 2 L X η γ + 1 2 Φ /star ( L ξ K [ η ] ) . (111)</formula>
<text><location><page_24><loc_12><loc_37><loc_88><loc_39></location>Inserting ¯ η = αn and recalling (31) one has L ¯ η /lscript = L αn /lscript = 2 α s + dα , so equations (103), (104) and (111) become</text>
<formula><location><page_24><loc_26><loc_34><loc_88><loc_35></location>A η = -K [ η ]( ξ, ν ) + n ( α ) , (112)</formula>
<formula><location><page_24><loc_26><loc_29><loc_88><loc_34></location>X a η = 1 2 K [ η ] ( ξ, n a ξ -2 θ a ) + 1 2 αn ( /lscript (2) ) n a +2 αP ab s b + P ab · ∇ b α, (113)</formula>
<text><location><page_24><loc_12><loc_22><loc_88><loc_26></location>The generalized master equation [19] relates the constraint tensor R ab , the metric hypersurface data and information on the deformation of η codified via the tensorial quantities w , p , q and I defined by</text>
<formula><location><page_24><loc_24><loc_26><loc_88><loc_30></location>α L n Y = A η Y -2 dα ⊗ s r + /lscript ⊗ s dA η + 1 2 L X η γ + 1 2 Φ /star ( L ξ K [ η ] ) . (114)</formula>
<formula><location><page_24><loc_15><loc_16><loc_88><loc_21></location>w := K [ η ]( ξ, ν ) , p := K [ η ]( ξ, ξ ) , q := Φ /star ( K [ η ]( ξ, · ) ) , I := 1 2 Φ /star ( L ξ K [ η ] ) . (115)</formula>
<text><location><page_24><loc_12><loc_16><loc_70><loc_17></location>The equation is obtained by inserting (114) into (40). The result is [19]</text>
<formula><location><page_24><loc_17><loc_8><loc_88><loc_15></location>α R ab = -( 2 κ + α tr P U -2 w ) Y ab +2 αP cd U d ( a ( 2Y b ) c +F b ) c ) -2 I ab + ( p -α (tr P Y ) -αn ( /lscript (2) ) ) U ab -2 α · ∇ ( a (s -r) b ) -4(s -r) ( a · ∇ b ) α -2 α (s -r) a (s -r) b -2 · ∇ a · ∇ b α -α · ∇ ( a s b ) + α s a s b +2 · ∇ ( a q b ) + α · R ( ab ) . (116)</formula>
<text><location><page_25><loc_12><loc_89><loc_88><loc_91></location>As already mentioned before, the dependence on Y in this relation is purely algebraic. The contraction of (116) with n is [19]</text>
<formula><location><page_25><loc_21><loc_81><loc_88><loc_88></location>α R ab n b = ( w -α tr P U ) r a -· ∇ a κ -I ab n b + P cd · ∇ c ( α U ad ) + α (tr P U )s a -α · ∇ a tr P U + 1 2 ( · ∇ n q a + · ∇ a w -w s a -P bc U ca q b ) , (117)</formula>
<text><location><page_25><loc_12><loc_80><loc_36><loc_81></location>and contracting again with n ,</text>
<formula><location><page_25><loc_38><loc_77><loc_88><loc_78></location>n ( κ ) = n ( w ) + κ n w + I ( n, n ) . (118)</formula>
<text><location><page_25><loc_12><loc_74><loc_81><loc_75></location>In terms of (115) the function A η and the vector X η in (112)-(113) can be written as</text>
<formula><location><page_25><loc_22><loc_68><loc_88><loc_73></location>A η = n ( α ) -w , X a η = 1 2 ( αn ( /lscript (2) ) -p ) n a + P ab ( 2 α s b + · ∇ b α -q b ) . (119)</formula>
<text><location><page_25><loc_12><loc_67><loc_52><loc_68></location>Hence, relation (102) becomes, after using (108),</text>
<formula><location><page_25><loc_19><loc_60><loc_88><loc_66></location>L ξ η µ H = ( w + ακ n -κ ) ξ µ + 1 2 ( p -αn ( /lscript (2) ) ) ν µ + P ab ( q b -2 α s b -· ∇ b α ) ̂ e µ a . (120)</formula>
<text><location><page_25><loc_12><loc_57><loc_88><loc_62></location>The idea now is to repeat this process with the identity (101), i.e. to replace the derivative L n Y ( m +1) by derivatives of K [ η ], Y ( m +1) itself and lower order terms. As usual, let us define K [ η ] ( m ) := L ( m -1) ξ K [ η ] and</text>
<formula><location><page_25><loc_16><loc_51><loc_88><loc_56></location>w ( m -1) := K [ η ] ( m ) ( ξ, ν ) , p ( m -1) := K [ η ] ( m ) ( ξ, ξ ) , q ( m -1) := Φ /star ( K [ η ] ( m ) ( ξ, · ) ) , (121)</formula>
<text><location><page_25><loc_12><loc_44><loc_88><loc_49></location>Observe that w (0) = w , p (0) = p , q (0) = q and I (1) = I . The rule of thumb is that the number inside the parenthesis denotes the number of transverse derivatives applied to K [ η ]. Following the notation of Section 4 we introduce the tensor Σ[ η ] := L η ∇ .</text>
<formula><location><page_25><loc_40><loc_48><loc_88><loc_52></location>I ( m ) := 1 2 Φ /star ( K [ η ] ( m +1) ) . (122)</formula>
<text><location><page_25><loc_12><loc_38><loc_88><loc_43></location>Proposition 5.3. Let {H , γ , /lscript , /lscript (2) } be null metric hypersurface data ( Φ , ξ )-embedded in ( M , g ) and extend ξ off H arbitrarily. Let η be a vector field on ( M , g ) satisfying that η | Φ( H ) = αν for some α ∈ F ( H ) . Then, for any integer m ≥ 1</text>
<formula><location><page_25><loc_32><loc_33><loc_88><loc_38></location>L ¯ η Y ( m ) = I ( m ) + m ( n ( α ) -w ) Y ( m ) + P ( m ) , (123)</formula>
<formula><location><page_25><loc_28><loc_26><loc_88><loc_31></location>P ( m ) := m L X η Y ( m -1) -1 2 m ∑ i =2 ( m i ) Φ /star ( L L ( i ) ξ η L ( m -i ) ξ g ) , (124)</formula>
<text><location><page_25><loc_12><loc_29><loc_88><loc_34></location>where P ( m ) is a tensor that depends on { Y , ..., Y ( m -1) } and {L ξ η, ..., L ( m ) ξ η } ∣ ∣ H and it is given explicitly by</text>
<text><location><page_25><loc_12><loc_22><loc_71><loc_26></location>and X a η = 1 2 ( αn ( /lscript (2) ) -p ) n a + P ab ( 2 α s b + · ∇ b α -q b ) . As a consequence,</text>
<text><location><page_25><loc_12><loc_18><loc_57><loc_19></location>Proof. We first show by induction the following relation</text>
<formula><location><page_25><loc_26><loc_18><loc_88><loc_23></location>α L n Y ( m ) = I ( m ) + m ( n ( α ) -w ) Y ( m ) -2 dα ⊗ s r ( m ) + P ( m ) . (125)</formula>
<formula><location><page_25><loc_32><loc_12><loc_88><loc_17></location>L η L ( m ) ξ g = L ( m ) ξ L η g -m ∑ i =1 ( m i ) L L ( i ) ξ η L ( m -i ) ξ g. (126)</formula>
<text><location><page_26><loc_12><loc_89><loc_88><loc_91></location>For m = 1 it holds (cf. (110)). Let us assume (126) is true up to some m ≥ 1 and show that it is then true for m +1 also. We compute</text>
<formula><location><page_26><loc_13><loc_69><loc_87><loc_88></location>L η L ( m +1) ξ g = L η L ξ L ( m ) ξ g = L ξ L η L ( m ) ξ g + L [ η,ξ ] L ( m ) ξ g = L ( m +1) ξ L η g -m ∑ i =1 ( m i ) L ξ L L ( i ) ξ η L ( m -i ) ξ g -L L ξ η L ( m ) ξ g = L ( m +1) ξ L η g -m ∑ i =1 ( m i ) L L ( i ) ξ η L ( m +1 -i ) ξ g -m ∑ i =1 ( m i ) L L ( i +1) ξ η L ( m -i ) ξ g -L L ξ η L ( m ) ξ g = L ( m +1) ξ L η g -m ∑ i =1 ( m i ) L L ( i ) ξ η L ( m +1 -i ) ξ g -m ∑ i =0 ( m i ) L L ( i +1) ξ η L ( m -i ) ξ g,</formula>
<text><location><page_26><loc_12><loc_64><loc_88><loc_69></location>where in the third line we introduced the induction hypothesis. Renaming i ↦-→ i -1 in the second sum and using the binomial identity ( m i -1 ) + ( m i ) = ( m +1 i ) it follows</text>
<formula><location><page_26><loc_23><loc_56><loc_77><loc_66></location>L η L ( m +1) ξ g = L ( m +1) ξ L η g -m ∑ i =1 ( m +1 i ) L L ( i ) ξ η L ( m +1 -i ) ξ g -L L ( m +1) ξ η g = L ( m +1) ξ L η g -m +1 ∑ i =1 ( m +1 i ) L L ( i ) ξ η L ( m +1 -i ) ξ g.</formula>
<text><location><page_26><loc_12><loc_53><loc_88><loc_56></location>This establishes (126) for m ≥ 1. The first term in the sum can be computed by means of (102). Pulling (126) back into H and using definition (122) gives</text>
<formula><location><page_26><loc_18><loc_48><loc_82><loc_53></location>2 L ¯ η Y ( m ) = 2 I ( m ) +2 mA η Y ( m ) +2 m L X η Y ( m -1) -m ∑ i =2 ( m i ) Φ /star ( L L ( i ) ξ η L ( m -i ) ξ g ) .</formula>
<text><location><page_26><loc_12><loc_46><loc_44><loc_48></location>Relation (123) follows after using (119).</text>
<text><location><page_26><loc_12><loc_44><loc_70><loc_45></location>An immediate consequence of the previous Proposition is the following.</text>
<text><location><page_26><loc_12><loc_38><loc_88><loc_43></location>Corollary 5.4. Let {H , γ , /lscript , /lscript (2) } be null metric hypersurface data ( Φ , ξ )-embedded in ( M , g ) and extend ξ off H arbitrarily. Let η be a vector field on ( M , g ) satisfying that η | Φ( H ) = αν for some function α ∈ F ( H ) . Then, for any integer m ≥ 1</text>
<text><location><page_26><loc_12><loc_33><loc_27><loc_34></location>As a consequence,</text>
<formula><location><page_26><loc_35><loc_33><loc_88><loc_38></location>L ¯ η Y ( m ) ( m ) = I ( m ) + m ( n ( α ) -w ) Y ( m ) . (127)</formula>
<formula><location><page_26><loc_28><loc_28><loc_88><loc_32></location>α L n Y ( m ) ( m ) = I ( m ) + m ( n ( α ) -w ) Y ( m ) -2 dα ⊗ s r ( m ) . (128)</formula>
<text><location><page_26><loc_12><loc_26><loc_88><loc_29></location>Inserting (125) into (101) we arrive at the main result of this section, namely the generalized master equation of order m> 1,</text>
<formula><location><page_26><loc_18><loc_18><loc_88><loc_25></location>α R ( m +1) ab = -(2( m +1)( κ -w ) + α tr P U ) Y ( m +1) ab -2 I ( m +1) ab -α ( tr P Y ( m +1) ) U ab +4 αP cd U c ( a Y ( m +1) b ) d +4 α (s -r) ( a r ( m +1) b ) -4r ( m +1) ( a · ∇ b ) α +2 α · ∇ ( a r ( m +1) b ) -2 ακ ( m +1) Y ab + α O ( m ) ab + P ( m ) , (129)</formula>
<text><location><page_26><loc_12><loc_8><loc_88><loc_17></location>where recall that when ∇ ξ ξ = 0 the tensor O ( m ) depends only on metric hypersurface data and { Y , ..., Y ( m ) } (see Corollary 4.26). The tensor P ( m ) also depends on { Y , ..., Y ( m ) } and in addition on the vectors L ξ η, ..., L ( m +1) ξ η on Φ( H ). Its key property is that it vanishes when X η = 0 and L ( i ) ξ η H = 0 for all i = 2 , ..., m +1. Finally we prove an interesting property of the vector field L ξ η that will play a key role in the next section.</text>
<text><location><page_27><loc_12><loc_84><loc_88><loc_92></location>Lemma 5.5. Let {H , γ , /lscript , /lscript (2) } be null metric hypersurface data (Φ , ξ ) -embedded in ( M , g ) and extend ξ off Φ( H ) by ∇ ξ ξ = 0 . Let η ∈ X ( M ) be such that η | Φ( H ) is null and tangent to Φ( H ) and such that its deformation tensor vanishes to all orders on Φ( H ) . If L ξ η H = hξ for some function h ∈ F ( H ) , then L ( k ) ξ η H = 0 for any integer k ≥ 2 .</text>
<text><location><page_27><loc_12><loc_82><loc_51><loc_83></location>Proof. Let m ≥ 1. By Proposition 4.3 it follows</text>
<formula><location><page_27><loc_16><loc_76><loc_88><loc_81></location>L ( m ) ξ ∇ γ ξ α = ∇ γ L ( m ) ξ ξ β + m -1 ∑ k =0 ( m k +1 ) ( L ( m -k -1) ξ ξ σ )Σ[ ξ ] ( k +1) α σγ = ξ σ Σ[ ξ ] ( m ) α σγ . (130)</formula>
<text><location><page_27><loc_12><loc_74><loc_58><loc_75></location>Moreover, equation (51) and the fact that ∇ ξ ξ = 0 imply</text>
<formula><location><page_27><loc_38><loc_71><loc_88><loc_73></location>0 = L ( m ) ξ ∇ ξ ξ α = ξ γ L ( m ) ξ ∇ γ ξ α . (131)</formula>
<text><location><page_27><loc_12><loc_68><loc_39><loc_69></location>Combining (130) and (131) gives</text>
<formula><location><page_27><loc_38><loc_65><loc_88><loc_67></location>ξ γ ξ σ Σ[ ξ ] ( i ) α γσ H = 0 ∀ i ≥ 0 . (132)</formula>
<text><location><page_27><loc_12><loc_62><loc_52><loc_64></location>The definition of Σ[ η ] entails the general identity</text>
<formula><location><page_27><loc_33><loc_57><loc_88><loc_62></location>-∇ ξ L η ξ = Σ[ η ]( ξ, ξ ) + ∇ L η ξ ξ -L η ( ∇ ξ ξ ) . (133)</formula>
<text><location><page_27><loc_12><loc_57><loc_39><loc_58></location>This together with ∇ ξ ξ = 0 gives</text>
<formula><location><page_27><loc_26><loc_54><loc_88><loc_56></location>L (2) ξ η = -L ξ L η ξ = -∇ ξ L η ξ + ∇ L η ξ ξ = Σ[ η ]( ξ, ξ ) -2 ∇ L ξ η ξ. (134)</formula>
<text><location><page_27><loc_12><loc_49><loc_88><loc_52></location>Since L ξ η H = hξ and Σ[ η ] vanishes on Φ( H ), it follows L (2) ξ η H = 0. Applying L ( m ) ξ to (134) and using (51) and (130) gives</text>
<formula><location><page_27><loc_24><loc_43><loc_88><loc_48></location>L ( m +2) ξ η = Σ[ η ] ( m +1) ( ξ, ξ ) -2 m ∑ k =0 ( m k ) ( L ( k +1) ξ η γ ) ξ σ Σ[ ξ ] ( m -k ) σγ . (135)</formula>
<text><location><page_27><loc_12><loc_37><loc_88><loc_42></location>We prove the statement by induction. Let m ≥ 1, assume L ( i ) ξ η H = 0 for all i = 2 , ..., m +1 and let us prove that L ( m +2) ξ η H = 0. Since Σ[ η ] ( k ) ( ξ, ξ ) H = 0 for all k ≥ 1 equation (135) on Φ( H ) reads</text>
<formula><location><page_27><loc_27><loc_35><loc_73><loc_37></location>L ( m +2) ξ η H = -2( L ξ η γ ) ξ σ Σ[ ξ ] ( m ) σγ H = -2 hξ γ ξ σ Σ[ ξ ] ( m ) σγ H = 0 ,</formula>
<text><location><page_27><loc_12><loc_33><loc_45><loc_34></location>where in the last equality we used (132).</text>
<text><location><page_27><loc_12><loc_21><loc_88><loc_32></location>Remark 5.6. It is remarkable that Lemma 5.5 is insensitive to the extension of h off Φ( H ) . At first sight one could think that this is not possible and that counterexamples are easy to construct. For instance, assume that L ξ η has the form hξ not just on H but everywhere. Clearly the lemma can only hold true if ξ ( h ) = 0 . So, the conditions of the lemma must somehow ensure that this property holds. And indeed this can be proved, as we show next. Assume L ξ η = hξ everywhere, ∇ ξ ξ = 0 and that the deformation tensor of η vanishes. Then, equation (133) reads</text>
<formula><location><page_27><loc_18><loc_16><loc_82><loc_21></location>-∇ ξ L η ξ = Σ[ η ]( ξ, ξ ) + ∇ L η ξ ξ -L η ( ∇ ξ ξ ) = -h ∇ ξ ξ = 0 = ⇒ ξ ( h ) ξ = 0 .</formula>
<text><location><page_27><loc_12><loc_14><loc_88><loc_17></location>Since ξ is a rigging of Φ( H ) there exists a neighbourhood U ⊂ M of Φ( H ) in which ξ = 0 , and hence ξ ( h ) = 0 on U .</text>
<text><location><page_27><loc_81><loc_16><loc_81><loc_17></location>/negationslash</text>
<text><location><page_27><loc_12><loc_8><loc_88><loc_13></location>When the deformation tensor of η vanishes in a neighbourhood of Φ( H ) (and not only to infinite order on Φ( H )) and h is assumed to be constant along ξ , then in fact L ξ η = hξ holds in a neighbourhood of Φ( H ), as we prove next.</text>
<text><location><page_28><loc_12><loc_87><loc_88><loc_91></location>Lemma 5.7. Let ( M , g ) admit a Killing vector η and consider an embedded hypersurface Φ : H ↪ → M with rigging ξ . Extend ξ off Φ( H ) by means of ∇ ξ ξ = 0 and assume L ξ η H = hξ with ξ ( h ) = 0 . Then, L ξ η = hξ in a neighbourhood of Φ( H ) .</text>
<text><location><page_28><loc_12><loc_84><loc_86><loc_85></location>Proof. Define ζ := L ξ η -cξ . Equation (133) together with the fact that ξ is geodesic yields</text>
<formula><location><page_28><loc_39><loc_81><loc_61><loc_82></location>∇ ξ ζ + ∇ ζ ξ -Σ[ η ]( ξ, ξ ) = 0 .</formula>
<text><location><page_28><loc_12><loc_76><loc_88><loc_80></location>Since η is a Killing the last term vanishes and then we have a linear homogeneous transport equation for ζ . Since ζ H = 0 we conclude ζ = 0 in a neighbourhood of Φ( H ).</text>
<section_header_level_1><location><page_28><loc_12><loc_72><loc_72><loc_74></location>6 Application to non-degenerate Killing horizons</section_header_level_1>
<text><location><page_28><loc_12><loc_55><loc_88><loc_71></location>In this section we study the case when ( M , g ) admits a Killing vector η (and therefore K [ η ] = L η g = 0). First, we review some well known properties of Killing horizons and particularize the identities of Section 5 to the present case. This will lead to a natural definition of 'abstract Killing horizon data' as well as its embedded counterpart. We will prove that, fixing the extension of the rigging to being geodesic, the transverse expansion at any non-degenerate Killing horizon is uniquely determined in terms of its abstract Killing horizon data and the ambient Ricci tensor to infinite order on the horizon. Moreover, when the Ricci tensor fulfills a so-called hierarchical dependence, the transverse expansion only depends on abstract Killing horizon data. Finally, we apply this result to characterize Λ-vacuum manifolds near non-degenerate horizons.</text>
<text><location><page_28><loc_21><loc_49><loc_21><loc_51></location>/negationslash</text>
<text><location><page_28><loc_12><loc_43><loc_88><loc_54></location>A Killing horizon of η is an embedded hypersurface Φ : H ↪ → M to which η is tangent, null and nowhere zero. We say that the Killing horizon is non-degenerate when its surface gravity satisfies ˜ κ = 0 at some point, and we say it is degenerate when ˜ κ = 0 everywhere. In the literature it is common to define the notion of 'non-degenerate' by the requirement that ˜ κ = 0 everywhere. We prefer the weaker definition above because then a Killing horizon is either degenerate or non-degenerate. In many relevant cases of interest (e.g. Λ-vacuum spacetimes) both definitions turn out to be equivalent.</text>
<text><location><page_28><loc_85><loc_48><loc_85><loc_49></location>/negationslash</text>
<text><location><page_28><loc_12><loc_37><loc_88><loc_41></location>A well-known property of Killing horizons is that they are totally geodesic, i.e. U = 0, which follows at once from Φ /star K [ η ] = 2 α U . As a consequence, given any X ∈ X ( H ), equations (26) and (27) give</text>
<formula><location><page_28><loc_24><loc_31><loc_88><loc_36></location>∇ Φ /star X η = ∇ Φ /star X ( αν ) = α ∇ Φ /star X ν + X ( α ) ν = ( α ( s -r ) + dα ) ( X ) ν. (136)</formula>
<text><location><page_28><loc_12><loc_31><loc_45><loc_32></location>Moreover equations (40)-(43) simplify to</text>
<formula><location><page_28><loc_18><loc_25><loc_88><loc_30></location>R ab = · R ( ab ) -2 L n Y ab -2 κ n Y ab + · ∇ ( a ( s b ) +2r b ) ) -2r a r b +4r ( a s b ) -s a s b , (137) R ab n a = L n (s b -r b ) -· ∇ b κ n , (138)</formula>
<formula><location><page_28><loc_17><loc_23><loc_88><loc_25></location>R ab n a n b = 0 , (139)</formula>
<text><location><page_28><loc_12><loc_21><loc_73><loc_22></location>while, using that the tensors (115) all vanish, equations (116)-(117) become</text>
<formula><location><page_28><loc_23><loc_15><loc_88><loc_19></location>α R ab = -2 κ Y ab -2 α · ∇ ( a (s -r) b ) -4(s -r) ( a · ∇ b ) α -2 α (s -r) a (s -r) b -2 · ∇ a · ∇ b α -α · ∇ ( a s b ) + α s a s b + α · R ( ab ) , (140)</formula>
<formula><location><page_28><loc_21><loc_13><loc_88><loc_15></location>α R ab n b = -· ∇ a κ. (141)</formula>
<text><location><page_28><loc_12><loc_8><loc_88><loc_11></location>In addition, equation (118) gives n ( κ ) = 0, so the surface gravity is constant along the null generator of the horizon. Hence, L n dκ = d ( n ( κ )) = 0 as well, so dκ is also Lie-constant along</text>
<text><location><page_29><loc_12><loc_85><loc_88><loc_91></location>the null generators. For bifurcate horizons (141) implies dκ = 0 on the bifurcation surface (and hence everywhere), which recovers the well-known fact that κ is constant for bifurcate horizons [15]. Moreover, when H is geodesically complete and admits a section, dκ must vanish everywhere [33].</text>
<text><location><page_29><loc_12><loc_82><loc_64><loc_84></location>Particularizing (129) for m> 1 to the case U = 0 and K [ η ] = 0,</text>
<formula><location><page_29><loc_24><loc_76><loc_88><loc_81></location>α R ( m +1) ab = -2( m +1) κ Y ( m +1) ab +4 α (s -r) ( a r ( m +1) b ) -4r ( m +1) ( a · ∇ b ) α +2 α · ∇ ( a r ( m +1) b ) -2 ακ ( m +1) Y ab + α O ( m ) ab + P ( m ) . (142)</formula>
<text><location><page_29><loc_12><loc_74><loc_62><loc_75></location>In view of relation (136) it is useful to introduce the one-form</text>
<formula><location><page_29><loc_43><loc_71><loc_88><loc_72></location>τ := α ( s -r ) + dα (143)</formula>
<text><location><page_29><loc_12><loc_67><loc_88><loc_69></location>which will play an important role below. Let us discuss some of its basic properties. Note first that for any tangent vector X ∈ X ( H ) it holds</text>
<formula><location><page_29><loc_43><loc_64><loc_88><loc_65></location>∇ Φ /star X η = τ ( X ) ν. (144)</formula>
<text><location><page_29><loc_52><loc_59><loc_52><loc_61></location>/negationslash</text>
<text><location><page_29><loc_12><loc_58><loc_88><loc_62></location>This one-form extends the commonly used one-form /pi1 defined by ∇ Φ /star X η = /pi1 ( X ) η (see for instance [1, 12]). Indeed, at the points where α = 0 the one-form τ agrees with α /pi1 . Note however that τ is well-defined and smooth everywhere on H .</text>
<text><location><page_29><loc_12><loc_55><loc_50><loc_57></location>Proposition 6.1. Let τ be as in (143) . Then,</text>
<formula><location><page_29><loc_15><loc_53><loc_84><loc_54></location>1. G ( z,V ) τ = z τ , 2. τ ( n ) = κ , 3. α L n τ = n ( α ) τ -τ ( n ) dα .</formula>
<text><location><page_29><loc_12><loc_45><loc_88><loc_52></location>Proof. The gauge transformation of τ follows from (36) and Lemma 5.2. Using (108) and r ( n ) = -κ n the second property is immediate. For the third one we use that η is a Killing vector, so L η ∇ = 0 and thus 0 = L η ∇ X η -∇ X L η η -∇ L η X η for every X ∈ X ( M ). When X is tangent to H the vector L η X on H is also tangent (because η is tangent as well). Then,</text>
<formula><location><page_29><loc_16><loc_40><loc_84><loc_45></location>0 = L η ∇ X η -∇ L η X η = L ¯ η ( τ ( X ) ) ν + τ ( X ) L η ν -τ ( L ¯ η X ) ν = ( L ¯ η τ -n ( α ) τ ) ( X ) ν,</formula>
<text><location><page_29><loc_12><loc_38><loc_88><loc_41></location>where we used L η ν = L αν ν = -ν ( α ) ν = -n ( α ) ν . Then L ¯ η τ = n ( α ) τ , so item 3. follows after using L ¯ η τ = L αn τ = α L n τ + τ ( n ) dα .</text>
<text><location><page_29><loc_36><loc_33><loc_36><loc_34></location>/negationslash</text>
<text><location><page_29><loc_12><loc_20><loc_88><loc_37></location>In order to construct the abstract notion of 'Killing horizon data' we want to find the minimum amount of data on the horizon that allows for the determination of the full transverse expansion. As we will see below, when κ = 0 everywhere, the required information involves the null metric hypersurface data, the one-form τ and the function α . The abstract conditions we need to incorporate into the definition must be such that, once the data is embedded, the corresponding horizon satisfies (i) Φ /star ( L η g ) = 0, (ii) the one-form τ satisfies item 3. of Proposition 6.1, (iii) the set of zeros of α has empty interior and (iv) that one-form α -1 ( τ -dα ) extends smoothly to all H . Item (iii) is necessary because Killing vectors that vanish on any hypersurface are necessarily identically zero. In terms of η this means that α vanishing on any open subset of H would only be compatible with η being identically vanishing. Item (iv) is necessary because for actual Killing horizons, the following equality holds (cf. (143))</text>
<formula><location><page_29><loc_42><loc_17><loc_59><loc_19></location>α -1 ( τ -dα ) = s -r ,</formula>
<text><location><page_29><loc_12><loc_8><loc_88><loc_16></location>and the right hand side is smooth everywhere on H . Note that the condition that the zeroes of α have empty interior means, in particular, that the extension of α -1 ( τ -dα ) is necessarily unique. Note also than when α has no zeroes, condition (iv) is automatically satisfied. As we explained at the beginning of this section, condition (i) gives U = 0. This discussion motivates the following definition.</text>
<text><location><page_30><loc_12><loc_85><loc_88><loc_92></location>Definition 6.2. We say {H , γ , /lscript , /lscript (2) , τ , α } is abstract Killing horizon data (AKH data) provided that (i) {H , γ , /lscript , /lscript (2) } is null metric hypersurface data satisfying U = 0 , (ii) α is a smooth function such that the set { α = 0 } has empty interior and (iii) τ is a one-form such that α L n τ = n ( α ) τ -τ ( n ) dα and the one-form α -1 ( τ -dα ) extends smoothly to all H .</text>
<text><location><page_30><loc_12><loc_75><loc_88><loc_84></location>Remark 6.3. It is worth comparing Definition 6.2 with the notions of abstract Killing horizons of order zero (AKH 0 ) and one (AKH 1 ) introduced in [24]. The main difference is that the definitions in [24] involve full hypersurface data (i.e. involve the tensor Y ) while the definition above makes no reference to Y . In fact, we want to use Definition 6.2 in combination with the field equations to construct Y in such a way that the data corresponds to a Killing horizon. This is why we have added the term 'data' in the definition of 'AKH data'.</text>
<text><location><page_30><loc_12><loc_71><loc_88><loc_74></location>Next we extend the notion of gauge transformation to the context of AKH data motivated by Lemma 5.2 and Proposition 6.1.</text>
<text><location><page_30><loc_12><loc_65><loc_88><loc_70></location>Definition 6.4. Let K = {H , γ , /lscript , /lscript (2) , τ , α } be AKH data and ( z, V ) gauge parameters. We define the gauge-transformed data by G ( z,V ) K := {H , γ ' , /lscript ' , /lscript (2) ' , z τ , zα } , where { γ ' , /lscript ' , /lscript (2) ' } are given by (14) -(16) .</text>
<text><location><page_30><loc_12><loc_59><loc_88><loc_64></location>The condition U = 0 of Definition 6.2 only guarantees that the pullback of the deformation tensor vanishes on H . To capture the full information about the deformation tensor we need to restrict ourselves to the embedded case.</text>
<text><location><page_30><loc_12><loc_50><loc_88><loc_58></location>Definition 6.5. Let K = {H , γ , /lscript , /lscript (2) , τ , α } be AKH data and define ¯ η := αn . We say that K is (Φ , ξ ) -embedded in ( M , g ) if (i) {H , γ , /lscript , /lscript (2) } is (Φ , ξ ) -embedded in ( M , g ) as in Def. 2.2 and (ii) ∇ Φ /star X Φ /star ¯ η = τ ( X ) ν for every X ∈ X ( H ) . Moreover, we say that K is an (Φ , ξ ) -embedded Killing horizon data (EKH data) if, additionally, (iii) there exist an extension η of Φ /star ¯ η such that its deformation tensor K [ η ] := L η g vanishes to all orders at Φ( H ) .</text>
<text><location><page_30><loc_12><loc_43><loc_88><loc_49></location>Remark 6.6. It is easy to check that if K = {H , γ , /lscript , /lscript (2) , τ , α } is (Φ , ξ ) -embedded in ( M , g ) , then G ( z,V ) K is (Φ , ξ ' ) -embedded in ( M , g ) with ξ ' = z ( ξ +Φ /star V ) . Moreover, since the property of K [ η ] vanishing to all orders on Φ( H ) is independent of ξ | Φ( H ) and its extension off Φ( H ) , it follows that if K is (Φ , ξ ) -EKH data, then G ( z,V ) K is (Φ , ξ ' ) -EKH data.</text>
<text><location><page_30><loc_12><loc_31><loc_88><loc_42></location>Remark 6.7. The definition of AKH data allows us to define a smooth one-form r := s -α -1 ( τ -dα ) and a scalar k n := α -1 ( τ ( n ) -n ( α )) . When the data happens to be embedded, the one-form r and the function k n agree with r and κ n , respectively. This is because from equation (136) and condition (ii) of Definition 6.5 it follows α ( r -r ) = 0 , and since the interior of the zeroes of α is empty, then r = r . Hence in the embedded case we shall not distinguish between r and r anymore. Observe also that this fact together with (108) imply that the surface gravity κ of the hypersurface is given by κ = n ( α ) + ακ n = τ ( n ) .</text>
<text><location><page_30><loc_12><loc_27><loc_88><loc_30></location>Example 6.8. Consider the d -dimensional Schwarzschild-de Sitter spacetime ( M , g ) . In ingoing Eddington-Finkelstein coordinates { v, r } the metric g is</text>
<formula><location><page_30><loc_29><loc_22><loc_71><loc_26></location>g = -( 1 -2 M r d -3 -Λ d -1 r 2 ) dv 2 +2 dvdr + r 2 γ S d -2 ,</formula>
<text><location><page_30><loc_12><loc_10><loc_88><loc_21></location>where γ S d -2 is the d -2 dimensional spherical metric. When M and Λ are both positive, and M sufficiently small, the polynomial G ( r ) := -r d -3 g vv = r d -3 -2 M -Λ d -1 r d -1 admits precisely two positive roots r + 0 ≥ R := √ d -3 Λ ≥ r -0 . Since ∂ r g vv ∣ ∣ r = r ± 0 = -2 M ( d -3) ( r ± 0 ) d -2 + 2Λ d -1 r ± 0 = Λ r ± 0 -d -3 r ± 0 it follows that when r + 0 > R > r -0 the two null hypersurfaces H ± := { r = r ± 0 } are non-degenerate Killing horizons with Killing η = ∂ v . H + is called cosmological horizon, and H -is called event horizon. When r + 0 = r -0 = R , H + = H -is a degenerate Killing horizon. Let us compute the</text>
<text><location><page_31><loc_12><loc_88><loc_88><loc_92></location>induced hypersurface data of any of the two horizons (we use r 0 to denote at once r + 0 and r -0 ). Choosing ξ = ∂ r as the rigging (observe ∇ ξ ξ = 0 ) it follows</text>
<formula><location><page_31><loc_15><loc_83><loc_85><loc_88></location>γ = r 2 0 γ S d -2 , /lscript = dv, /lscript (2) = 0 , Y = ( -( d -3) M r d -2 0 + Λ r 0 d -1 ) dv 2 + r 0 γ S d -2 .</formula>
<text><location><page_31><loc_12><loc_80><loc_71><loc_82></location>It follows at once that /lscript ( η ) = 1 , s = 1 2 d /lscript ( η, · ) = 0 , so by (143) with α = 1</text>
<formula><location><page_31><loc_32><loc_75><loc_68><loc_80></location>τ = -r = -Y ( η, · ) = ( ( d -3) M r d -2 0 -Λ r 0 d -1 ) dv</formula>
<text><location><page_31><loc_63><loc_73><loc_63><loc_74></location>/negationslash</text>
<text><location><page_31><loc_77><loc_73><loc_77><loc_74></location>/negationslash</text>
<text><location><page_31><loc_12><loc_69><loc_88><loc_74></location>and thus κ = τ ( η ) = ( d -3) M r d -2 0 -Λ r 0 d -1 = d -3 2 r 0 -Λ r 0 2 . When r 0 = R it follows κ = 0 . In this case K SdS := {H , γ , /lscript , /lscript (2) , τ , α = 1 } is EKH in Schwarzschild-de Sitter spacetime with κ = 0 everywhere.</text>
<text><location><page_31><loc_85><loc_71><loc_85><loc_72></location>/negationslash</text>
<text><location><page_31><loc_12><loc_63><loc_88><loc_68></location>In Proposition 6.11 we find necessary conditions for two AKH data to be embeddable in isometric semi-Riemannian manifolds. The idea is to use this information to then define a notion of isometry at the AKH data level. First we introduce some notation.</text>
<text><location><page_31><loc_12><loc_59><loc_88><loc_62></location>Notation 6.9. Let K ' = {H ' , γ ' , /lscript ' , /lscript (2) ' , ω ' , α ' } be AKH data and χ : H -→ H ' a diffeomorphism. We define</text>
<formula><location><page_31><loc_22><loc_56><loc_88><loc_58></location>χ /star K ' = χ /star {H ' , γ ' , /lscript ' , /lscript (2) ' , τ ' , α ' } := {H , χ /star γ ' , χ /star /lscript ' , χ /star /lscript (2) ' , χ /star τ ' , χ /star α ' } . (145)</formula>
<text><location><page_31><loc_12><loc_54><loc_88><loc_55></location>This notation will also be used for subsets of the data, e.g. χ /star { γ ' , /lscript ' , /lscript (2) ' } := { χ /star γ ' , χ /star /lscript ' , χ /star /lscript (2) ' } .</text>
<text><location><page_31><loc_12><loc_50><loc_88><loc_53></location>Lemma 6.10. Let K ' = {H ' , γ ' , /lscript ' , /lscript (2) ' , ω ' , α ' } be AKH data and χ : H -→ H ' a diffeomorphism. Then χ /star K ' is AKH data.</text>
<text><location><page_31><loc_12><loc_38><loc_88><loc_48></location>Proof. Firstly, from Definition 2.1 and the fact that χ is a diffeomorphism, it follows that {H , γ := χ /star γ ' , /lscript := χ /star /lscript ' , /lscript (2) := χ /star /lscript (2) ' } is metric hypersurface data. Moreover, from (3)(6) having a unique solution for { P, n, n (2) } given { γ , /lscript , /lscript (2) } , it follows that P = χ /star P ' , n = χ /star n ' and n (2) = χ /star n (2) ' . Thus, the causal character of { γ , /lscript , /lscript (2) } is the same as the one of { γ ' , /lscript ' , /lscript (2) ' } , and in particular if { γ ' , /lscript ' , /lscript (2) ' } is null, so it is { γ , /lscript , /lscript (2) } . Since χ /star L χ /star n γ ' = L n γ it also follows U = 0. Moreover, since χ is a diffeomorphism, α = χ /star α ' is smooth and has empty interior. Finally,</text>
<formula><location><page_31><loc_21><loc_32><loc_79><loc_37></location>0 = χ /star ( α ' L χ /star n τ ' -( χ /star n )( α ' ) τ ' + τ ' ( χ /star n ) dα ' ) = α L n τ -n ( α ) τ + τ ( n ) dα</formula>
<text><location><page_31><loc_12><loc_29><loc_88><loc_34></location>and χ /star ( α '-1 ( τ ' -dα ' ) ) = α -1 ( τ -dα ) extends smoothly to all H . Hence {H , γ , /lscript , /lscript (2) , τ , α } is AKH data.</text>
<text><location><page_31><loc_12><loc_22><loc_88><loc_29></location>Proposition 6.11. Let K = {H , γ , /lscript , /lscript (2) , ω , α } be AKH data (Φ , ξ ) -embedded in ( M , g ) and let K ' = {H ' , γ ' , /lscript ' , /lscript (2) ' , ω ' , α ' } be AKH data (Φ ' , ξ ' ) -embedded in ( M ' , g ' ) . Let ¯ η := αn and ¯ η ' := α ' n ' . Assume there exists an isometry ϕ : ( M , g ) -→ ( M ' , g ' ) such that ϕ (Φ( H )) = Φ ' ( H ' ) and ϕ /star Φ /star ¯ η = Φ ' /star ¯ η ' . Then, there exist a diffeomorphism χ : H -→ H ' and gauge parameters ( z, V ) such that</text>
<formula><location><page_31><loc_29><loc_18><loc_88><loc_20></location>χ /star {H ' , γ ' , /lscript ' , /lscript (2) ' , τ ' , α ' } = G ( z,V ) {H , γ , /lscript , /lscript (2) , τ , α } . (146)</formula>
<text><location><page_31><loc_12><loc_11><loc_88><loc_17></location>Proof. First we prove that the vector field ϕ /star ξ ' is everywhere transverse to Φ( H ). Observe that since the vector field ν ' is null and normal to Φ ' ( H ' ) and ϕ is an isometry, the vector ϕ /star ν ' is also null and normal to Φ( H ), so in particular it is proportional to ν with non-zero proportionality factor. Then, from g ' ( ξ ' , ν ' ) = 1 it follows</text>
<formula><location><page_31><loc_24><loc_8><loc_76><loc_10></location>1 = ( ϕ /star g ' )( ϕ /star ξ ' , ϕ /star ν ' ) = g ( ϕ /star ξ ' , ϕ /star ν ' ) = ⇒ g ( ϕ /star ξ, ν ) = 0 .</formula>
<text><location><page_31><loc_73><loc_8><loc_73><loc_10></location>/negationslash</text>
<text><location><page_32><loc_12><loc_85><loc_88><loc_92></location>This proves that ϕ /star ξ ' is everywhere transverse to Φ( H ), so there must exist a function z ∈ F /star ( H ) and a vector field V ∈ X ( H ) such that ϕ /star ξ ' = z ( ξ + Φ /star V ) . Since Φ( H ) and Φ ' ( H ' ) are diffeomorphic via ϕ and both Φ and Φ ' are embeddings, there exists a diffeomorphism χ making the following diagram commutative.</text>
<formula><location><page_32><loc_45><loc_79><loc_56><loc_85></location>H H ' M M ' χ Φ Φ ' ϕ</formula>
<text><location><page_32><loc_12><loc_76><loc_17><loc_78></location>Then,</text>
<formula><location><page_32><loc_36><loc_75><loc_60><loc_75></location>/star ' /star ' /star ' /star /star ' /star</formula>
<text><location><page_32><loc_12><loc_64><loc_88><loc_69></location>Hence χ /star { γ ' , /lscript ' , /lscript (2) ' } = G ( z,V ) { γ , /lscript , /lscript (2) } (see Def. 2.2), so χ /star ν ' = z -1 ν and χ /star α ' = zα because ϕ /star Φ /star ¯ η = Φ ' /star ¯ η ' . Finally, from condition (ii) in Definition 6.5 together with ϕ /star Φ /star ¯ η = Φ ' /star ¯ η ' and the fact that ϕ is an isometry, it follows</text>
<formula><location><page_32><loc_12><loc_67><loc_88><loc_75></location>χ γ = χ Φ g = Φ ϕ g = Φ g = γ , χ /star /lscript ' = χ /star Φ ' /star ( g ' ( ξ ' , · ) ) = Φ /star ϕ /star ( g ' ( ξ ' , · ) ) = Φ /star ( g ( z ( ξ +Φ /star V ) , · ) ) = z ( /lscript + γ ( V, · ) ) χ /star /lscript (2) ' = χ /star Φ ' /star ( g ' ( ξ ' , ξ ' ) ) = Φ /star ϕ /star ( g ' ( ξ ' , ξ ' ) ) = Φ /star ( g ( ϕ /star ξ ' , ϕ /star ξ ' ) ) = z 2 ( /lscript (2) +2 /lscript ( V ) + γ ( V, V ) ) .</formula>
<text><location><page_32><loc_12><loc_59><loc_86><loc_64></location>ϕ /star ( ∇ ' Φ ' /star X ' Φ ' /star ¯ η ' ) = ∇ Φ /star χ /star X ' Φ /star ¯ η = ⇒ χ /star τ ' ⊗ χ /star ν ' = τ ⊗ ν ⇐⇒ χ /star τ ' = z τ . Taking into account Definition 6.4, (146) is established.</text>
<text><location><page_32><loc_12><loc_55><loc_88><loc_58></location>This proposition motivates the following natural notion of isometry in the context of AKH data.</text>
<text><location><page_32><loc_12><loc_49><loc_88><loc_54></location>Definition 6.12. Let K = {H , γ , /lscript , /lscript (2) , τ , α } and K ' = {H ' , γ ' , /lscript ' , /lscript (2) ' , τ ' , α ' } be two AKH data. We say K and K ' are isometric provided that there exists a diffeomorphism χ : H -→ H ' and gauge parameters ( z, V ) such that</text>
<formula><location><page_32><loc_29><loc_47><loc_88><loc_48></location>χ /star {H ' , γ ' , /lscript ' , /lscript (2) ' , τ ' , α ' } = G ( z,V ) {H , γ , /lscript , /lscript (2) , τ , α } . (147)</formula>
<text><location><page_32><loc_12><loc_44><loc_60><loc_46></location>Lemma 6.10 guarantees that Definition 6.12 is well defined.</text>
<text><location><page_32><loc_68><loc_41><loc_68><loc_43></location>/negationslash</text>
<text><location><page_32><loc_12><loc_18><loc_88><loc_43></location>Our next aim is to show that given EKH data K satisfying τ ( n ) = 0 everywhere, the full asymptotic expansion { Y ( k ) } k ≥ 1 is uniquely determined in terms of K and the set { R ( m ) αβ } m ≥ 1 . As we shall see, to prove the uniqueness part of such statement we need to be able to extend the rigging vector ξ such that the tensor L ( m ) ξ R αβ is geometrical (in the sense of Definition 4.5) for every m ≥ 0. By identities (99), (100) and (142), the extension of ξ must be such that the tensors O ( m ) , O ( m ) a , O ( m ) ab and P ( m ) ab are H -geometrical for every m ≥ 0. In Section 4 we have proved that by extending ξ off Φ( H ) by means of ∇ ξ ξ = 0 the tensors O ( m ) , O ( m ) a and O ( m ) ab are H -geometrical. However, this is not sufficient to guarantee that P ( m ) ab is H -geometrical, because this tensor also depends on X η and L ( i ) ξ η ∣ ∣ H for i ≥ 2 (see the comment below equation (129)). A natural way to make this dependence disappear is to ensure that X η = 0 and L ( i ) ξ η H = 0 for every i ≥ 2. Our strategy is as follows. In Lemma 6.13 we show that given AKH data embedded on an ambient manifold with rigging ξ , one can always choose the gauge such that L ξ η is proportional to ξ on Φ( H ). With this choice X η automatically vanishes (cf. (102)). By combining this result with Lemma 5.5 we will be able to prove that L ( i ) η H = 0 for every i ≥ 2 as well.</text>
<text><location><page_32><loc_64><loc_18><loc_64><loc_18></location>ξ</text>
<text><location><page_32><loc_12><loc_15><loc_73><loc_16></location>Particularizing equation (120) to the Killing horizon case, namely K [ η ] = 0,</text>
<formula><location><page_32><loc_27><loc_8><loc_88><loc_14></location>L ξ η µ H = ( ακ n -κ ) ξ µ -α 2 n ( /lscript (2) ) ν µ -P ab ( 2 α s b + · ∇ b α ) ̂ e µ a . (148)</formula>
<text><location><page_32><loc_12><loc_8><loc_88><loc_10></location>In the following lemma we show that there exists a choice of gauge in which L ξ η H = ( ακ n -κ ) ξ .</text>
<text><location><page_33><loc_17><loc_89><loc_17><loc_90></location>/negationslash</text>
<text><location><page_33><loc_12><loc_85><loc_88><loc_92></location>Lemma 6.13. Let K = {H , γ , /lscript , /lscript (2) , τ , α } be AKH data (Φ , ξ ) -embedded in ( M , g ) satisfying τ ( n ) = 0 everywhere on H . Assume there exists an extension η of ¯ η := αn off Φ( H ) satisfying K [ η ]( ξ, · ) = 0 on Φ( H ) . Then there exists a family of gauges satisfying /lscript = κ -1 τ and /lscript (2) = 0 . Moreover, any element of the family satisfies</text>
<formula><location><page_33><loc_43><loc_82><loc_88><loc_84></location>L ξ η H = ( ακ n -κ ) ξ, (149)</formula>
<text><location><page_33><loc_12><loc_76><loc_88><loc_81></location>and the whole family can be generated from any element by the action of the subgroup of transformations {G ( z, 0) } (i.e. this subgroup acts transitively on the family). Any element of this family will be said to be written in an ' η -gauge'.</text>
<text><location><page_33><loc_12><loc_68><loc_88><loc_75></location>Proof. A vector V ∈ X ( H ) is defined uniquely in terms of the one-form w := γ ( V, · ) and the scalar function f := /lscript ( V ). From (6) and (3) it follows P ab w a w b = P ab γ ac γ bd V c V d = ( δ b c -n b /lscript c ) γ bd V c V d = γ cd V c V d , so in terms of f and w the gauge transformations of /lscript and /lscript (2) in (15)-(16) read</text>
<formula><location><page_33><loc_25><loc_66><loc_80><loc_68></location>/lscript ' = z ( /lscript + w ) , (150) /lscript (2) ' = z 2 /lscript (2) +2 f + P ( w , w ) .</formula>
<text><location><page_33><loc_12><loc_51><loc_88><loc_65></location>From the transformations of τ and κ , namely τ ' = z τ and κ ' = κ , it is straightforward to check that by choosing w := κ -1 τ -/lscript and f := -/lscript (2) -1 2 κ -2 P ( τ , τ ), the gauge-transformed data satisfies (i) /lscript ' = κ -1 τ ' and (ii) /lscript (2) ' = 0. Moreover, by the transformations (150)-(151) and those of τ and κ , it is clear that any additional transformation G ( z ' ,V ' ) will preserve properties (i) and (ii) if and only if V ' = 0. Thus, the whole family is generated by applying G ( z, 0) (with z ∈ F /star ( H ) arbitrary) to any element of the family. To prove that in this class of gauges expression (148) simplifies to (149) it suffices to show that P ab ( 2 α s b + · ∇ b α ) = 0. Writing item 3. of Proposition 6.1 in terms of κ τ = /lscript gives</text>
<formula><location><page_33><loc_62><loc_64><loc_88><loc_69></location>( ) (151)</formula>
<formula><location><page_33><loc_13><loc_47><loc_87><loc_52></location>α L n τ = αn ( κ ) /lscript + ακ L n /lscript = n ( α ) κ /lscript -κdα = ⇒ κ ( α L n /lscript + dα ) = ( κn ( α ) -αn ( κ ) ) /lscript .</formula>
<text><location><page_33><loc_12><loc_40><loc_88><loc_43></location>Observe that the EKH data in Example 6.8 is written in an η -gauge. By combining Lemmas 5.5 and 6.13 and Remark 6.6 we arrive at the following.</text>
<text><location><page_33><loc_12><loc_42><loc_88><loc_48></location>Using L n /lscript = 2 s (cf. (31)) it follows that the one-form 2 α s + dα is proportional to /lscript , and since /lscript (2) = 0 then P ab ( 2 α s b + · ∇ b α ) = 0 as a consequence of (5).</text>
<text><location><page_33><loc_12><loc_32><loc_88><loc_39></location>Corollary 6.14. Let {H , γ , /lscript , /lscript (2) , τ , α } be (Φ , ξ ) -EKH in ( M , g ) . Then there exists a choice of ξ on Φ( H ) in which L ξ η H = ( ακ n -κ ) ξ . Such gauge is unique once n is fixed. Moreover, extending ξ off Φ( H ) by ∇ ξ ξ = 0 , then L ( k ) ξ η H = 0 for all k ≥ 2 . In particular, all the tensors P ( m ) in (124) vanish.</text>
<text><location><page_33><loc_45><loc_28><loc_45><loc_29></location>/negationslash</text>
<text><location><page_33><loc_12><loc_26><loc_88><loc_31></location>We are ready to show one of the main results of this section, namely that the full transverse expansion of an EKH data satisfying κ = 0 everywhere is uniquely determined in terms of AKH data and the collection { R ( m ) αβ } m ≥ 1 .</text>
<text><location><page_33><loc_85><loc_23><loc_85><loc_25></location>/negationslash</text>
<text><location><page_33><loc_12><loc_20><loc_88><loc_25></location>Theorem 6.15. Let K = {H , γ , /lscript , /lscript (2) , τ , α } be (Φ , ξ ) -EKH in any ( M , g ) . Assume τ ( n ) = 0 everywhere on H . Then, when the data is written in an η -gauge and ξ is extended geodesically, the full transverse expansion { Y ( m ) } m ≥ 1 only depends on K and { R ( m ) αβ } m ≥ 1 .</text>
<text><location><page_33><loc_12><loc_8><loc_88><loc_19></location>Proof. Let us write the data in any of the η -gauges of Lemma 6.13, which fixes the vector ξ up to a multiplicative non-vanishing function ξ ↦-→ zξ , and extend ξ off Φ( H ) by means of ∇ ξ ξ = 0. We want to prove that the tensors { Y ( k ) } k ≥ 1 only depend on K and { R ( m ) αβ } m ≥ 1 , and thus they are insensitive to the particular manifold they are embedded in, provided their tensors R ( m ) αβ agree on Φ( H ). With this choice of gauge and extension of ξ the tensors O ( m ) , O ( m ) a and O ( m ) ab only depend on metric data and { Y , ..., Y ( m ) } (see Corollary 4.26) and the tensor</text>
<text><location><page_34><loc_12><loc_87><loc_88><loc_92></location>P ( m ) vanishes for every m ≥ 1 (see Corollary 6.14). These two facts will be used repeatedly throughout the proof. By Remark 6.7 we can identify the one-form r := s -α -1 ( τ -dα ) with r := Y ( n, · ) and the scalar τ ( n ) with κ . Therefore, equation (140) can be rewritten as</text>
<formula><location><page_34><loc_40><loc_84><loc_88><loc_86></location>α R ab = -2 κ Y ab + C ab , (152)</formula>
<text><location><page_34><loc_14><loc_80><loc_14><loc_81></location>/negationslash</text>
<text><location><page_34><loc_12><loc_75><loc_88><loc_83></location>where C ab is a tensor that only depends on AKH data { γ , /lscript , /lscript (2) , τ , α } . This proves that when κ = 0 everywhere on H , the full tensor Y is determined in terms of AKH data and the tensor R ab . Therefore, since in equations (99)-(100) for m = 0 the lower order terms only involve the tensor Y and metric data, it follows that the scalar tr P Y (2) and the one-form r (2) only depend on AKH data and the tensor R αβ on H . Hence, equation (142) for m = 1 reads</text>
<formula><location><page_34><loc_40><loc_72><loc_88><loc_74></location>α R (2) ab = -4 κ Y (2) ab + C (2) ab , (153)</formula>
<text><location><page_34><loc_62><loc_69><loc_62><loc_71></location>/negationslash</text>
<text><location><page_34><loc_12><loc_62><loc_88><loc_71></location>where C (2) ab only depends on AKH data and R αβ . When κ = 0 everywhere this shows that the tensor Y (2) ab is uniquely determined from AKH data and the tensors R αβ and R (2) ab on H . Iterating this process by means of equations (99), (100) and (142) one obtains the full transverse expansion { Y ( k ) } k ≥ 1 , and by Corollaries 6.14 and 4.26 this expansion only depends on AKH data and { R ( m ) αβ } m ≥ 1 , and not on the particular ( M , g ) where K is embedded.</text>
<text><location><page_34><loc_75><loc_60><loc_75><loc_61></location>/negationslash</text>
<text><location><page_34><loc_12><loc_34><loc_88><loc_61></location>This theorem shows that the asymptotic expansion of an EKH data satisfying κ = 0 everywhere only depends on the abstract data K and the tensors { R ( m ) αβ } m ≥ 1 , and thus it is insensitive to the particular ( M , g ) they may be embedded in. The collection { R ( m ) αβ } m ≥ 1 can be thought at least in two different ways. One possibility is to provide each R ( m ) αβ as prescribed data on the null hypersurface, e.g. by some external matter field. Another option is to provide R ( m ) αβ as a functional relation between the abstract data K and the transverse expansion { Y (1) , ..., Y ( m ) } . The simplest example of the second viewpoint is a d -dimensional manifold ( M , g ) satisfying the Λ-vacuum equations, where R ( m +1) αβ = λ K ( m ) αβ and λ = 2Λ d -2 . Then, R = λ γ , ˙ R = λ /lscript , R = λ/lscript (2) and R ( m +1) = 2 λ Y ( m ) , ˙ R ( m +1) = λ Φ /star ( K ( m ) ( ξ, · ) ) and R ( m +1) = λ Φ /star ( K ( m ) ( ξ, ξ ) ) for every m ≥ 1. Since by Remark 4.18 the tensor K ( m ) ( ξ, · ) on Φ( H ) depends on { Y (1) , ..., Y ( m -1) } , one concludes that the tensor R ( m +1) αβ on Φ( H ) depends at most on AKH data and { Y , ..., Y ( m ) } . In general, when the functional relations are such that each R ( m ) αβ depends on low enough transverse derivatives of the metric, i.e. such that the LHS in equations (99)-(101) depend on derivatives that we already have under control, the proof of Theorem 6.15 shows that the transverse expansion only depends on the abstract data. Let us make this property precise.</text>
<text><location><page_34><loc_12><loc_28><loc_88><loc_33></location>Definition 6.16. Let K = {H , γ , /lscript , /lscript (2) , τ , α } be AKH data (Φ , ξ ) -embedded in ( M , g ) , and extend ξ off Φ( H ) by ∇ ξ ξ = 0 . We say that the Ricci tensor of g is hierarchical on Φ( H ) provided that R (1) only depends on K and, for every m ≥ 1 ,</text>
<unordered_list>
<list_item><location><page_34><loc_14><loc_26><loc_75><loc_27></location>(i) R ( m +1) depends (at most) on K , { Y , ..., Y ( m ) } , tr P Y ( m +1) and r ( m +1) .</list_item>
<list_item><location><page_34><loc_13><loc_23><loc_64><loc_25></location>(ii) ˙ R ( m ) and R ( m ) depend (at most) on K and { Y , ..., Y ( m ) } .</list_item>
</unordered_list>
<text><location><page_34><loc_12><loc_19><loc_88><loc_22></location>When a Ricci tensor is hierarchical on Φ( H ) we shall refer to its particular dependence stated in (i) and (ii) by its 'hierarchical dependence'.</text>
<text><location><page_34><loc_12><loc_8><loc_88><loc_18></location>Recall that when ξ is extended geodesically the tensor R ( m ) αβ is geometrical for every m ≥ 1 (cf. Proposition 4.14). As noted above, the canonical example is the Λ-vacuum equations, since in this case the tensors R ( m +1) , ˙ R ( m ) and R ( m ) only depend on AKH data and { Y (1) , ..., Y ( m ) } . In fact, R (1) ab = λγ ab , ˙ R a = λ/lscript a , R = λ/lscript (2) and R ( m +1) = 2 λ Y ( m ) , ˙ R ( m ) a = 0 and R ( m ) = 0 for every m ≥ 1 (see Corollary 4.8). An immediate consequence of the proof of Theorem 6.15 and Definition 6.16 is the following.</text>
<text><location><page_35><loc_17><loc_44><loc_17><loc_45></location>/negationslash</text>
<text><location><page_35><loc_12><loc_85><loc_88><loc_92></location>Theorem 6.17. Let K = {H , γ , /lscript , /lscript (2) , τ , α } be (Φ , ξ ) -EKH in any ( M , g ) . Assume τ ( n ) = 0 everywhere on H and that the field equations satisfied by g are such that its Ricci tensor is hierarchical on Φ( H ) . Then, when the data is written in an η -gauge and ξ is extended geodesically, the full transverse expansion { Y ( m ) } m ≥ 1 only depends on K .</text>
<text><location><page_35><loc_86><loc_90><loc_86><loc_91></location>/negationslash</text>
<text><location><page_35><loc_12><loc_67><loc_88><loc_84></location>This theorem generalizes the recent work [16] in several directions. The main theorem in [16] proves that for spacetimes admitting a non-degenerate Killing horizon H and being Ricci flat to infinite order at H , the full asymptotic expansion of the metric along certain privileged transverse vector at the horizon can be determined geometrically from a so-called 'non-degenerate Killing horizon data'. Theorem 6.17 is more general firstly because we are allowing zeroes of η on H (while Killing horizons by definition only include points where η is non-zero), secondly because our hierarchical property includes more field equations besides vacuum with vanishing Λ, and finally because we have extended the result to arbitrary signature (provided it admits degenerate hypersurfaces). We recover the result in [16] simply by imposing that R αβ is zero to all orders on the horizon. As an interesting corollary of Theorem 6.17 we extend the geometric uniqueness of the transverse expansion to the case of asymptotic Λ-vacuum spacetimes.</text>
<text><location><page_35><loc_56><loc_63><loc_56><loc_64></location>/negationslash</text>
<text><location><page_35><loc_12><loc_60><loc_88><loc_66></location>Corollary 6.18. Let K = {H , γ , /lscript , /lscript (2) , τ , α } be (Φ , ξ ) -EKH in any ( M , g ) satisfying the Λ -vacuum equations to infinite order on H . Assume τ ( n ) = 0 at least at one point p ∈ H and that K is written in an η -gauge and with ξ extended geodesically off Φ( H ) . Then the full transverse expansion { Y ( m ) } m ≥ 1 is uniquely determined in terms of K .</text>
<text><location><page_35><loc_76><loc_56><loc_76><loc_57></location>/negationslash</text>
<text><location><page_35><loc_12><loc_53><loc_88><loc_59></location>Proof. Since ( M , g ) satisfies the Λ-vacuum equations to infinite order on H it follows R ab n b = 0, and thus from (141) one has dκ = 0, so κ = τ ( n ) is constant on H . Since κ = 0 at least at one point and H is assumed to be connected, we conclude κ = 0 everywhere on H , and hence Theorem 6.17 applies.</text>
<text><location><page_35><loc_63><loc_45><loc_63><loc_47></location>/negationslash</text>
<text><location><page_35><loc_61><loc_54><loc_61><loc_55></location>/negationslash</text>
<text><location><page_35><loc_12><loc_39><loc_88><loc_51></location>Let us perform a detailed comparison between our AKH data and the 'non-degenerate Killing horizon data' in [16], which is a triple ( H , σ , V ) where σ is a Riemannian metric, V is a nowhere vanishing Killing vector of ( H , σ ) with constant (non-zero) norm. In order to compare both objects at the same footing we restrict our AKH data to α = 0, γ semi-positive definite, τ ( n ) = 0 at some point and vacuum, which in particular implies τ ( n ) must be a nonzero constant (see the proof of Corollary 6.18). Let us see the equivalence between AKH data and 'non-degenerate Killing horizon data'. In one direction, given {H , γ , /lscript , /lscript (2) , τ , α } the vector V and the metric σ can be defined as follows:</text>
<formula><location><page_35><loc_37><loc_36><loc_63><loc_38></location>V := αn, σ := γ + α -2 τ ⊗ τ .</formula>
<text><location><page_35><loc_12><loc_30><loc_88><loc_36></location>Clearly V is nowhere vanishing and σ ( V , V ) = ( τ ( n ) ) 2 is a non-vanishing constant, so σ is a gauge-invariant Riemannian metric on H (see Lemma 5.2 and Proposition 6.1). Moreover, since L V τ = n ( α ) τ , it follows</text>
<formula><location><page_35><loc_15><loc_25><loc_85><loc_30></location>L V σ = 2 α U + L V ( α -2 ) τ ⊗ τ +2 α -2 L V τ ⊗ s τ = ( -2 α -2 n ( α ) + 2 α -2 n ( α ) ) τ ⊗ τ = 0 .</formula>
<text><location><page_35><loc_12><loc_24><loc_83><loc_26></location>Conversely, given a 'non-degenerate Killing horizon data' ( H , σ , V ) as in [16] we define</text>
<formula><location><page_35><loc_21><loc_20><loc_79><loc_23></location>α := 1 , τ := 1 σ ( V , V ) σ ( V , · ) , γ := σ -1 σ ( V , V ) σ ( V , · ) ⊗ σ ( V , · )</formula>
<formula><location><page_35><loc_34><loc_15><loc_63><loc_22></location>√ /lscript (2) := 0 , /lscript := 1 σ ( V , V ) σ ( V , · ) .</formula>
<text><location><page_35><loc_12><loc_8><loc_88><loc_15></location>Since /lscript ( V ) = 1 and γ ( V , · ) = 0 it follows at once that { γ , /lscript , /lscript (2) } is null metric hypersurface data and n = V . Moreover, from L V σ = 0 one gets U = 1 2 L n γ = 0 and α L n τ = 0 = n ( α ) τ -τ ( n ) dα . In addition τ ( n ) is a non-zero constant and the one-form α -1 ( τ -dα ) extends smoothly to all H (because α = 1). Hence {H , γ , /lscript , /lscript (2) , τ , α } fulfills the conditions of Definition 6.2 with</text>
<text><location><page_36><loc_12><loc_90><loc_36><loc_91></location>α = 0 and τ ( n ) = const. = 0.</text>
<text><location><page_36><loc_34><loc_85><loc_34><loc_87></location>/negationslash</text>
<text><location><page_36><loc_14><loc_90><loc_14><loc_91></location>/negationslash</text>
<text><location><page_36><loc_33><loc_90><loc_33><loc_91></location>/negationslash</text>
<text><location><page_36><loc_12><loc_79><loc_88><loc_88></location>In [16] the choice of the rigging ξ is such that L ξ η = 0. This is possible only when α is nowhere vanishing. Indeed, when α = 0 one can exploit the freedom in Lemma 6.13 to set α = 1, and thus κ = κ n (cf. (108)) and L ξ η H = 0 (see (149)). Lemma 5.7 then shows that L ξ η = 0 whenever η is a Killing. However, if ¯ η admits zeroes on H this choice of gauge is not possible anymore because the properties α = 0 or α = 0 at a point are gauge invariant, so if α = 0 in some gauge it is impossible to make α = 1 by a gauge transformation.</text>
<text><location><page_36><loc_17><loc_71><loc_17><loc_72></location>/negationslash</text>
<text><location><page_36><loc_32><loc_80><loc_32><loc_82></location>/negationslash</text>
<text><location><page_36><loc_12><loc_65><loc_88><loc_77></location>In Proposition 6.11 we proved that two AKH data K and K ' embedded in isometric manifolds ( M , g ) and ( M ' , g ' ) are necessarily isometric (in the sense of Definition 6.12). Our next aim is to prove a kind of converse, namely that isometric EKH data K and K ' both satisfying τ ( n ) = 0 imply that the respective manifolds they are embedded in are isometric to infinite order. The proof involves two steps. Firstly, we construct and map to each other suitable neighbourhoods of Φ( H ) and Φ ' ( H ' ), and secondly we prove that within an appropriate gauge and extension of ξ , the two asymptotic expansions agree. We accomplish the first task in the following proposition.</text>
<text><location><page_36><loc_12><loc_56><loc_88><loc_64></location>Proposition 6.19. Let Φ : H ↪ → M and Φ ' : H ' ↪ → M ' be two embedded hypersurfaces in ambient manifolds ( M , g ) and ( M ' , g ' ) and let ξ , ξ ' be respectively riggings of Φ( H ) , Φ ' ( H ' ) extended geodesically. Assume that there exists a diffeomorphism χ : H -→ H ' . Then, there exist open neighbourhoods U ⊂ M and U ' ⊂ M ' of Φ( H ) and Φ ' ( H ' ) and a unique diffeomorphism Ψ : U -→ U ' satisfying Ψ /star ξ = ξ ' and Φ ' · χ = Ψ · Φ .</text>
<text><location><page_36><loc_12><loc_44><loc_88><loc_55></location>Proof. Pick a neighbourhood U ⊂ M of Φ( H ) small enough so that the integral curves of ξ do not intersect each other and intersect Φ( H ) precisely once. Construct a neighbourhood U ' of Φ ' ( H ' ) similarly. Then, given a point q ∈ U there exist a unique p ∈ H such that the integral curve σ ( τ ) of ξ through p reaches q at a finite τ q , i.e. σ ( τ q ) = q . Now consider the point p ' := χ ( p ) ∈ H ' and the integral curve σ ' of ξ ' through p ' . We define Ψ( q ) := σ ' ( τ q ). It is clear that Ψ is a diffeomorphism when U and U ' are small enough, and by construction it satisfies Ψ /star ξ = ξ ' and Φ ' · χ = Ψ · Φ.</text>
<text><location><page_36><loc_12><loc_37><loc_88><loc_43></location>Once we know how to identify neighbourhoods of Φ( H ) and Φ ' ( H ' ), we need to show that the two asymptotic expansions agree. The following theorem shows that a sufficient condition is that the field equations on ( M , g ) and ( M ' , g ' ) are such that their Ricci tensors satisfy the same hierarchical dependence on the horizons.</text>
<text><location><page_36><loc_84><loc_32><loc_84><loc_34></location>/negationslash</text>
<text><location><page_36><loc_12><loc_25><loc_88><loc_36></location>Theorem 6.20. Let K = {H , γ , /lscript , /lscript (2) , τ , α } and K ' = {H ' , γ ' , /lscript ' , /lscript (2) ' , τ ' , α ' } be two isometric EKH in respective ambient manifolds ( M , g ) and ( M ' , g ' ) and satisfying τ ( n ) = 0 everywhere on H . Assume the field equations on ( M , g ) and ( M ' , g ' ) are such that their Ricci tensors satisfy the same hierarchical dependence on Φ( H ) and Φ ' ( H ' ) , respectively. Let K ' be written in an η -gauge and K in the gauge in which K = χ /star K ' and extend the riggings geodesically. Then, there exist neighbourhoods U ⊂ M and U ' ⊂ M ' of Φ( H ) and Φ ' ( H ' ) and a diffeomorphism Ψ : U -→ U ' such that</text>
<formula><location><page_36><loc_44><loc_21><loc_88><loc_23></location>Ψ /star L ( i ) ξ ' g ' H = L ( i ) ξ g (154)</formula>
<text><location><page_36><loc_12><loc_18><loc_30><loc_20></location>for every i ∈ N ∪ { 0 } .</text>
<text><location><page_36><loc_12><loc_8><loc_88><loc_17></location>Proof. By Proposition 6.19 there exists a unique diffeomorphism Ψ : U -→ U ' satisfying Ψ /star ξ = ξ ' and Φ ' · χ = Ψ · Φ. Let us prove that Ψ /star L ( i ) ξ ' g ' H = L ( i ) ξ g for every i ≥ 0. The case i = 0 is immediate because γ = χ /star γ ' , /lscript = χ /star /lscript ' and /lscript (2) = χ /star /lscript (2) ' (see Definition 2.2). Proving the case i ≥ 1 amounts to show Ψ /star ( L ( i ) ξ ' g ' ) ( ξ ' , · ) = ( L ( i ) ξ g ) ( ξ, · ) and χ /star Y ( i ) ' = Y ( i ) . The former is a direct consequence of Corollary 4.8 because they both vanish. Let us prove the latter. Observe</text>
<text><location><page_37><loc_12><loc_87><loc_88><loc_92></location>that since K ' is written in an η -gauge and K = χ /star K ' , then /lscript = χ /star /lscript ' = χ /star ( κ '-1 τ ' ) = κ -1 τ and /lscript (2) = χ /star /lscript (2) ' = 0, so K is also written in an η -gauge. Equation (140) and its primed version read</text>
<formula><location><page_37><loc_29><loc_85><loc_71><loc_87></location>-2 κ Y ab + C ab = α R ab , -2 κ ' Y ' ab + C ab = α ' R ' ab ,</formula>
<text><location><page_37><loc_12><loc_76><loc_88><loc_84></location>where C ab and C ' ab only depend on AKH data (see Corollaries 6.14 and 4.26) and thus agree. Since the dependence of R ab on K is the same as the dependence of R ' ab on K ' , taking the pullback χ /star of the second equation and subtracting the first one gives χ /star Y ' = Y . A similar argument applied to (99)-(100) for m = 0 proves χ /star ( tr P ' Y (2) ' ) = tr P Y (2) and χ /star r (2) ' = r (2) . By iterating this process one shows that χ /star Y ( i ) ' = Y ( i ) for every i ≥ 1.</text>
<text><location><page_37><loc_12><loc_71><loc_88><loc_75></location>An example of two manifolds satisfying the same hierarchical dependence is two spacetimes ( M , g ) and ( M ' , g ' ) such that R αβ = λg αβ and R ' αβ = λg ' αβ . This leads to the following immediate corollary.</text>
<text><location><page_37><loc_42><loc_65><loc_42><loc_66></location>/negationslash</text>
<text><location><page_37><loc_12><loc_60><loc_88><loc_70></location>Corollary 6.21. Let K = {H , γ , /lscript , /lscript (2) , τ , α } and K ' = {H ' , γ ' , /lscript ' , /lscript (2) ' , τ ' , α ' } be two isometric EKH in respective Λ -vacuum ambient manifolds ( M , g ) and ( M ' , g ' ) with the same cosmological constant. Assume τ ( n ) = 0 everywhere on H . Let K ' be written in an η -gauge and K in the gauge in which K = χ /star K ' and extend the riggings geodesically. Then, there exist neighbourhoods U ⊂ M and U ' ⊂ M ' of Φ( H ) and Φ ' ( H ' ) and a diffeomorphism Ψ : U -→ U ' such that</text>
<formula><location><page_37><loc_44><loc_58><loc_88><loc_60></location>Ψ /star L ( i ) ξ ' g ' H = L ( i ) ξ g (155)</formula>
<text><location><page_37><loc_12><loc_56><loc_30><loc_58></location>for every i ∈ N ∪ { 0 } .</text>
<text><location><page_37><loc_12><loc_49><loc_88><loc_55></location>A consequence of this corollary is that every analytic Λ-vacuum manifold admitting a nondegenerate Killing horizon (possibly with bifurcation surfaces) is characterized near the horizon by its AKH data. The question of whether an AKH data gives rise to a Λ-vacuum manifold to infinite order will be analyzed in an forthcoming paper.</text>
<text><location><page_37><loc_12><loc_38><loc_88><loc_47></location>A direct application of Corollary 6.21 is that if a spacetime admits an EKH isometric to the non-extremal Schwarzschild-de Sitter data introduced in Example 6.8, then the spacetime is isometric to infinite order (i.e. (155) holds) to non-extremal Schwarzschild-de Sitter. Moreover, when the spacetime is real analytic it is necessarily isometric to Schwarzschild-de Sitter spacetime (at least in a neighbourhood of the horizon). This result complements the main result of [14], where the extremal case was considered.</text>
<text><location><page_37><loc_27><loc_31><loc_27><loc_32></location>/negationslash</text>
<text><location><page_37><loc_83><loc_31><loc_83><loc_32></location>/negationslash</text>
<text><location><page_37><loc_12><loc_29><loc_88><loc_37></location>Proposition 6.22. Let ( M , g ) be a d ≥ 4 -dimensional spacetime satisfying the vacuum Einstein equations with cosmological constant Λ > 0 and admitting a non-degenerate Killing horizon Φ : H ↪ → M with spherical cross-sections. Let η be the corresponding Killing vector with surface gravity κ = 0 (necessarily constant). Assume Φ /star g = r 2 0 γ S d -2 for some 0 < r 0 = d -3 Λ and τ ∧ d τ = 0 , where τ is defined in (143) . Let M := 1 2 r d -3 0 -Λ 2( d -1) r d -1 0 .</text>
<unordered_list>
<list_item><location><page_37><loc_15><loc_24><loc_88><loc_27></location>1. If r 2 0 > d -3 Λ , ( M , g ) is isometric to infinite order to non-extremal Schwarzschild-de Sitter spacetime with mass M at its cosmological horizon .</list_item>
<list_item><location><page_37><loc_15><loc_20><loc_88><loc_23></location>2. If r 2 0 < d -3 Λ , ( M , g ) is isometric to infinite order to non-extremal Schwarzschild-de Sitter spacetime with mass M at its event horizon .</list_item>
</unordered_list>
<text><location><page_37><loc_12><loc_16><loc_88><loc_19></location>Moreover, if ( M , g ) is analytic, then it is isometric to Schwarzschild-de Sitter spacetime in a neighbourhood of its cosmological horizon (case 1.) or its event horizon (case 2.).</text>
<text><location><page_37><loc_28><loc_10><loc_28><loc_11></location>/negationslash</text>
<text><location><page_37><loc_12><loc_9><loc_88><loc_15></location>Proof. We denote all the objects referring to Schwarzschild-de Sitter described in Example 6.8 with the label 'SdS'. Let us scale the Killing vector η by the constant κ SdS /κ , where κ SdS := d -3 2 r 0 -Λ r 0 2 = 0. Then, the surface gravity of (the re-scaled) η is precisely κ SdS . Since on the horizon the Killing field is nowhere vanishing we can define n := η and thus α = 1.</text>
<text><location><page_38><loc_12><loc_82><loc_88><loc_91></location>From item 3. of Proposition 6.1 it follows L n τ = 0. Let us pick any section S ⊂ H and solve the differential equation n ( u ) = 1 with initial condition u | S = 0. This gives a global function u ∈ F ( H ) that defines a foliation {S u } u ∈ R . Let us decompose the cotangent space T /star H by T /star H = span { du } ⊕ 〈 n 〉 ⊥ , where as usual 〈 n 〉 ⊥ is the set of covectors that annihilate n . Therefore the one-form τ can be uniquely decomposed as τ = Adu + b with A ∈ F ( H ) and b ∈ T /star H satisfying b ( n ) = 0. Condition τ ( n ) = κ gives A = κ , and condition τ ∧ d τ = 0 gives</text>
<formula><location><page_38><loc_28><loc_77><loc_72><loc_82></location>( κdu + b ) ∧ d b = 0 = ⇒ κdu ∧ d b + b ∧ d b = 0 .</formula>
<text><location><page_38><loc_12><loc_64><loc_88><loc_70></location>Let us define u ' := u + κ -1 B (hence τ = κdu ' ). Since b ( n ) = 0 then n ( B ) = 0, so n ( u ' ) = n ( u ) = 1. Consider the new foliation defined by u ' , {S u ' } u ' ∈ R . The remaining gauge freedom can be fixed by requiring ξ to be null and orthogonal to that foliation, which in terms of the abstract data implies /lscript (2) = 0 and /lscript = du ' . Then, the tuple</text>
<text><location><page_38><loc_12><loc_69><loc_88><loc_79></location>Since L n τ = L n ( κdu + b ) = L n b = 0 then from the Cartan identity d b ( n, · ) = L n b -d ( b ( n )) = 0. Contracting the equation above with n yields κd b = 0. Since κ is a nonzero constant, the one-form b is closed, and hence exact because R × S d -2 is simply connected for d ≥ 4. Let B ∈ F ( H ) be such that b = dB , so τ = κd ( u + κ -1 B ) .</text>
<formula><location><page_38><loc_27><loc_61><loc_73><loc_63></location>K := {H , γ = r 2 0 γ S d -2 , /lscript = du ' , /lscript (2) = 0 , τ = κdu ' , α = 1 }</formula>
<text><location><page_38><loc_81><loc_58><loc_81><loc_60></location>/negationslash</text>
<text><location><page_38><loc_12><loc_36><loc_88><loc_60></location>fulfills all the conditions of Definition 6.2, so K is AKH data satisfying τ ( n ) = κ = 0 everywhere on H . Moreover, K is written in an η -gauge. The last step is to construct the diffeomorphism χ : H -→ H SdS . Choose any isometry φ that maps the section { u ' = 0 } of H with the section { v = 0 } of H SdS . Consider any point p ∈ H and let σ ( u ' ) be the integral curve of n through p . This curve intersects S u ' =0 at a single point q . Now consider the integral curve σ SdS ( v ) of n SdS through φ ( q ). We define χ ( p ) := σ SdS ( u ' ( p )). That χ is a diffeomorphism follows from the product topology of H and H SdS and because n and n SdS are both smooth and globally defined. Since φ is an isometry and the integral curves of n and n SdS are identified it follows χ /star γ SdS = γ . With the definition M := 1 2 r d -3 0 -Λ 2( d -1) r d -1 0 it is clear that K is isometric (as in Definition 6.12) either to K + SdS or K -SdS . In order to distinguish the two cases recall that the equation 1 -2 M r 0 d -3 -Λ d -1 r 0 2 = 0 admits exactly two solutions r 0 + > √ d -3 Λ > r 0 -. Therefore, when r 2 0 > d -3 Λ then r 0 = r + 0 (and we are in the cosmological horizon case) and when r 2 0 < d -3 Λ then r 0 = r -0 (which is the event horizon case). The Proposition follows after using Corollary 6.21.</text>
<text><location><page_38><loc_12><loc_16><loc_88><loc_35></location>The case r 2 0 = d -3 Λ is excluded in Proposition 6.22 because the horizon described in Example 6.8 is necessarily degenerate when r 2 0 = d -3 Λ . A result analogous to the one in Proposition 6.22 can be formulated with Λ = 0 and Schwarzschild spacetime, or Λ < 0 and Schwarzschild-anti de Sitter spacetime. It is worth emphasizing that, in contrast to the degenerate case treated in [14], it is not necessary to require neither Λ > 0 nor staticity nor compact cross sections for the uniqueness argument to work. The reason is that, for non-degenerate horizons, the equations that allow us to obtain recursively the asymptotic expansion are all algebraic. In the degenerate (also called extremal) case, once r ( m +1) has been replaced in (142) using (100), the leading order in identity (142) is Y ( m ) because κ = 0. The dependence between R ( m ) and Y ( m ) is not algebraic anymore, but via a partial differential equation on the cross-sections of H . In order for such equations to determine uniquely the expansion it becomes sufficient to require Λ > 0, staticity and that the cross-sections are maximally symmetric [14].</text>
<text><location><page_38><loc_12><loc_9><loc_88><loc_15></location>Remarkably, when the ( M , g ) in Proposition 6.22 is analytic it necessarily admits a static Killing vector, since Schwarzschild-de Sitter does. It would be interesting to relate this to the recent work [7] where it is shown that for bifurcate Killing horizons the condition η being static is equivalent to the torsion one-form being closed on the bifurcation surface.</text>
<section_header_level_1><location><page_39><loc_12><loc_90><loc_35><loc_92></location>Acknowledgements</section_header_level_1>
<text><location><page_39><loc_12><loc_83><loc_88><loc_89></location>This work has been supported by Projects PID2021-122938NB-I00 (Spanish Ministerio de Ciencia e Innovaci'on and FEDER 'A way of making Europe') and SA096P20 (JCyL). G. S'anchezP'erez also acknowledges support of the PhD. grant FPU20/03751 from Spanish Ministerio de Universidades.</text>
<section_header_level_1><location><page_39><loc_12><loc_79><loc_64><loc_80></location>A Some pullbacks into a null hypersurface</section_header_level_1>
<text><location><page_39><loc_12><loc_69><loc_88><loc_77></location>In this appendix we compute the pullback of several derivatives of ambient tensors into a null hypersurface that we shall need in the main text of this paper. Recall that given a (0 , p ) tensor field T α 1 ··· α p on M we use the standard notation T a 1 ··· a p to denote the pullback of T to H . Moreover, we also use the notation ( i,j ) T a 1 ··· a p as the pullback to H of the contraction of T with ξ in the i-th and j-th slots.</text>
<text><location><page_39><loc_12><loc_65><loc_88><loc_68></location>Proposition A.1. Let ( M , g ) be a semi-Riemannian manifold and Φ : H ↪ →M a smooth null hypersurface with rigging ξ . Let T be a (0 , p ) -tensor on M . Then,</text>
<formula><location><page_39><loc_16><loc_46><loc_88><loc_64></location>Φ /star ( ∇ T ) ba 1 ··· a p = · ∇ b T a 1 ··· a p + p ∑ i =1 Y ba i T a 1 ··· a i -1 ca i +1 ··· a p n c + p ∑ i =1 U ba i ( i ) T a 1 ··· a p , (156) Φ /star ( (1) ∇ T ) a 1 ··· a p = ( L ξ T ) a 1 ··· a p -p ∑ i =1 (r -s) a i ( i ) T a i ··· a p -p ∑ i =1 V b a i T a i ··· a i -1 ba i +1 ··· a p , (157) Φ /star ( (2) ∇ T ) a 1 ··· a p = · ∇ a 1 (1) T a 2 ··· a p + p ∑ i =2 Y a 1 a i (1) T a 2 ··· a i -1 ba i +1 ··· a p n b + p ∑ i =2 U a 1 a i (1 ,i ) T a 2 ··· a p -(r -s) a 1 (1) T a 2 ··· a p -V b a 1 T ba 2 ··· a p , (158)</formula>
<text><location><page_39><loc_12><loc_42><loc_40><loc_45></location>where V b a := P cb Π ac + 1 2 ( d/lscript (2) ) a n b .</text>
<text><location><page_39><loc_12><loc_35><loc_88><loc_41></location>Proof. Let { e a } be a local basis of H and ̂ e a := Φ /star ( e a ). To prove the first identity we contract ∇ β T α 1 ··· α p with ̂ e β b ̂ e α 1 a 1 · · · ̂ e α p a p and use (26), namely</text>
<text><location><page_39><loc_12><loc_32><loc_59><loc_33></location>For the second one we use the relation between L ξ and ∇</text>
<formula><location><page_39><loc_34><loc_32><loc_66><loc_38></location>∇ ̂ e β b ̂ e α a = ( · ∇ e b e a ) c ̂ e α c -Y ba n c ̂ e α c -U ba ξ α .</formula>
<formula><location><page_39><loc_29><loc_26><loc_71><loc_31></location>∇ ξ T α 1 ··· α p = L ξ T α 1 ··· α p -p ∑ i =1 T α 1 ··· α i -1 µα i +1 ··· α p ∇ α i ξ µ ,</formula>
<text><location><page_39><loc_12><loc_20><loc_88><loc_25></location>contract it with ̂ e α 1 a 1 · · · ̂ e α p a p and employ (35). Finally, to prove the last one just contract the relation</text>
<formula><location><page_39><loc_31><loc_19><loc_69><loc_21></location>ξ µ ∇ α 1 T µα 2 ··· α p = ∇ α 1 (1) T α 2 ··· α p -T µα 2 ··· α p ∇ α 1 ξ µ .</formula>
<text><location><page_39><loc_12><loc_16><loc_44><loc_18></location>with e α 1 a 1 · · · e α p a p and use (156) and (35).</text>
<text><location><page_39><loc_12><loc_12><loc_88><loc_18></location>̂ ̂ Proposition A.2. Let ( M , g ) be a semi-Riemannian manifold and Φ : H ↪ → M a smooth null hypersurface with rigging ξ . Let T be a (0 , p + 1) -tensor on M and denote by div T the</text>
<text><location><page_40><loc_12><loc_90><loc_67><loc_92></location>p -covariant tensor defined by (div T ) α 1 ··· α p := g µν ∇ µ T να 1 ··· α p . Then,</text>
<formula><location><page_40><loc_13><loc_70><loc_86><loc_89></location>Φ /star (div T ) a 1 ··· a p = P bc · ∇ b T ca 1 ··· a p + n b ( L ξ T ) ba 1 ··· a p + n c · ∇ c (1) T a 1 ··· a p + ( 2 κ n +tr P U ) (1) T a 1 ··· a p +(tr P Y -n ( /lscript (2) )) n c T ca 1 ··· a p -2 P ac (r + s) a T ca 1 ··· a p + p ∑ i =1 P bc Y ba i T ca 1 ··· a i -1 da i +1 ··· a p n d + p ∑ i =1 P bc U ba i ( i ) T ca 1 ··· a i -1 a i +1 ··· a p -p ∑ i =1 (r -s) a i n b ( i ) T ba 1 ··· a i -1 a i +1 ··· a p -p ∑ i =1 V c a i n b T ba 1 ··· a i -1 ca i +1 ··· a p + p ∑ i =1 r a i (1) T a 1 ··· a i -1 ca i +1 ··· a p n c .</formula>
<text><location><page_40><loc_12><loc_68><loc_27><loc_70></location>Proof. From (13),</text>
<text><location><page_40><loc_12><loc_60><loc_35><loc_61></location>Using (156), the first term is</text>
<formula><location><page_40><loc_20><loc_60><loc_81><loc_67></location>̂ e α 1 a 1 · · · ̂ e α p a p ∇ µ T µ α 1 ··· α p = ̂ e α 1 a 1 · · · ̂ e α p a p g µν ∇ µ T να 1 ··· α p = ̂ e α 1 a 1 · · · ̂ e α p a p ( P bc ̂ e µ b ̂ e ν c + ξ µ n b ̂ e ν b + ξ ν n b ̂ e µ b ) ∇ µ T να 1 ··· α p .</formula>
<formula><location><page_40><loc_23><loc_49><loc_77><loc_59></location>P bc · ∇ b T ca 1 ··· a p +(tr P Y ) T da 1 ··· a p n d + p ∑ i =1 P bc Y ba i T ca 1 ··· a i -1 da i +1 ··· a p n d +(tr P U ) (1) T a 1 ··· a p + p ∑ i =1 P bc U ba i ( i ) T ca 1 ··· a i -1 a i +1 ··· a p .</formula>
<text><location><page_40><loc_12><loc_45><loc_88><loc_49></location>From (157), V c b n b = P ac (r + s) a + 1 2 n ( /lscript (2) ) n c and r ( n ) = -κ n , s ( n ) = 0 the second term becomes</text>
<formula><location><page_40><loc_26><loc_39><loc_76><loc_44></location>n b ( L ξ T ) ba 1 ··· a p + κ n (1) T a 1 ··· a p -p ∑ i =1 (r -s) a i n b ( i ) T ba 1 ··· a i -1 a i +1 ··· a p</formula>
<formula><location><page_40><loc_24><loc_34><loc_76><loc_40></location>-( P ac (r + s) a + 1 2 n ( /lscript (2) ) n c ) T ca 1 ··· a p -p ∑ i =1 V c a i n b T ba 1 ··· a i -1 ca i +1 ··· a p</formula>
<text><location><page_40><loc_12><loc_33><loc_54><loc_34></location>Finally, using (158) and U ( n, · ) = 0 the last term is</text>
<formula><location><page_40><loc_12><loc_26><loc_90><loc_32></location>n c · ∇ c (1) T a 1 ··· a p + p ∑ i =1 r a i (1) T a 1 ··· a i -1 ca i +1 ··· a p n c + κ n (1) T a 1 ··· a p -( P ac (r + s) a + 1 2 n ( /lscript (2) ) n c ) T ca 1 ··· a p .</formula>
<text><location><page_40><loc_12><loc_25><loc_44><loc_26></location>Combining the three the result follows.</text>
<section_header_level_1><location><page_40><loc_12><loc_21><loc_25><loc_22></location>References</section_header_level_1>
<unordered_list>
<list_item><location><page_40><loc_13><loc_16><loc_88><loc_19></location>[1] Ashtekar, A., Beetle, C., and Lewandowski, J. Geometry of generic isolated horizons. Classical and Quantum Gravity 19 , 6 (2002), 1195.</list_item>
<list_item><location><page_40><loc_13><loc_10><loc_88><loc_15></location>[2] Bahuaud, E., Gunasekaran, S., Kunduri, H. K., and Woolgar, E. Static nearhorizon geometries and rigidity of quasi-einstein manifolds. Letters in Mathematical Physics 112 , 6 (2022), 116.</list_item>
</unordered_list>
<table>
<location><page_41><loc_13><loc_9><loc_88><loc_92></location>
</table>
<table>
<location><page_42><loc_12><loc_11><loc_88><loc_91></location>
</table>
</document>
[
{
"title": "Transverse expansion of the metric at null hypersurfaces I. Uniqueness and application to Killing horizons",
"content": "Marc Mars ∗ and Gabriel S'anchez-P'erez † Departamento de F'ısica Fundamental, Universidad de Salamanca Plaza de la Merced s/n, 37008 Salamanca, Spain May 10, 2024",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "This is the first in a series of two papers where we analyze the transverse expansion of the metric on a general null hypersurface. In this paper we obtain general geometric identities relating the transverse derivatives of the ambient Ricci tensor and the transverse expansion of the metric at the null hypersurface. We also explore the case where the hypersurface exhibits a generalized symmetry generator, namely a privileged vector field in the ambient space which, at the hypersurface, is null and tangent (including the possibility of zeroes). This covers the Killing, homothetic, or conformal horizon cases, and, more generally, any situation where detailed information on the deformation tensor of the symmetry generator is available. Our approach is entirely covariant, independent on any field equations, and does not make any assumptions regarding the topology or dimension of the null hypersurface. As an application we prove that the full transverse expansion of the spacetime metric at a nondegenerate Killing horizon (also allowing for bifurcation surfaces) is uniquely determined in terms of abstract data on the horizon and the tower of derivatives of the ambient Ricci tensor at the horizon. In particular, the transverse expansion of the metric in Λ-vacuum spacetimes admitting a non-degenerate horizon is uniquely determined in terms of abstract data at the horizon.",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "The initial value problem of the Einstein field equations plays a central role in General Relativity. The fundamental breakthrough of Choquet-Buhat established that by specifying an abstract n -dimensional Riemannian manifold (Σ , h ) along with a symmetric, two covariant tensor field K subject to constraint equations, an ( n +1)-dimensional spacetime ( M , g ) solution of the Einstein equations exists where (Σ , h, K ) is embedded [4, 6, 35]. Another important initial value problem in general relativity is the characteristic one, where the data is posed on a pair of null hypersurfaces that intersect transversely. Rendall's work [34] demonstrates that proving the existence of solutions to the Einstein equations in a neighborhood of the intersecting surface only requires prescribing the full spacetime metric (but not its transverse derivatives) on the initial hypersurfaces. In [18] Luk extends Rendall's result when the intersection is a two-sphere by proving that the spacetime exists on a neighbourhood of the full two null hypersurfaces. In a recent work [10] the authors show that the same result holds regardless the dimension and topology of the intersection. Various other approaches to this problem can be found in [8, 3]. An initial value problem is also known to be well-posed when the data is given on the future null cone of a point [5]. Recent research [30, 29] has also approached the characteristic problem from a more geometric perspective, incorporating transverse derivatives of the metric as initial data (and thus adding the corresponding constraint equations). In this paper we analyze the situation when initial data is posed on a single null hypersurface. From a causal perspective, the absence of a second null hypersurface implies that extra information coming from the past can potentially influence the solution, hence spoiling uniqueness. However, in cases where the solution exhibits some symmetry, such as Killing vectors, initial data on a single null hypersurface may suffice to establish a unique solution to the Einstein equations. This issue has been extensively studied in recent years, for example when the hypersurface is a homothetic or a Killing horizon, as we review next. In [11] C. Fefferman and R. Graham introduced the so-called ambient metric , inspired by the idea that the light-cone of a point in Minkowski spacetime of dimension n + 2 encodes the conformal structure of the n -sphere. In order to generalize this idea to an arbitrary conformal class, their goal was to construct a homothetic horizon from the given conformal class, and embed it into an ambient manifold Ricci flat to infinite order. Fefferman and Graham proved that in the cases where the dimension of the conformal class is odd, the Einstein equations uniquely determine the full transverse expansion of the ambient metric along a transverse null coordinate at the horizon. However, in even dimensions, additional data is required to fully determine the expansion and, in addition there appears a so-called obstruction tensor whose vanishing or not determines whether the ambient metric can be Ricci flat to all orders at the horizon or not. When the obstruction tensor does not vanish, one can still construct an ambient metric which is Ricci flat to infinite order at the horizon by including log terms in the expansion. This gives rise to the so-called 'generalized ambient metric' which exhibits only finite differentiability at the horizon. In a remarkable work [36] the authors prove that this generalized ambient metric is not only Ricci flat to infinite order but in fact exactly Ricci flat. It is important to note that in both scenarios, the data is not constrained by any equations, unlike in the spacelike and characteristic Cauchy problems. Another example where the curvature conditions of the ambient spacetime fix the transverse expansion is a non-degenerate Killing horizon. In [32] Moncrief shows that the transverse expansion of any 3-dimensional, non-degenerate Killing horizon on a Ricci flat spacetime can be described by six functions at the horizon. This analysis was done in a particular coordinate system, which makes it difficult to determine whether two spacetimes are isometric to infinite order at the horizon or not. In a recent work [16], the authors study this problem from a geometric perspective and establish that, for Ricci flat spacetimes, the complete expansion along a null transverse vector field is determined by 'non-degenerate Killing horizon data', namely a triple ( H , σ , V ) where σ is a Riemannian metric and V is a non-trivial Killing vector field V on ( H , σ ) with constant norm. Although this analysis characterizes Killing horizons geometrically, it excludes the possibility of V having zeros (and consequently bifurcation surfaces on the horizon) and the field equations are restricted to vacuum with Λ = 0. The last example we want to illustrate is the degenerate Killing horizon case. Given a degenerate Killing horizon, its so-called near horizon limit [17] is characterized by a function F , a one-form ω , and a metric h on the cross-sections. In a recent work [14] the authors focus on establishing a uniqueness theorem for extremal Schwarzschild-de Sitter spacetime. For a specific value of the mass parameter, this spacetime admits a degenerate Killing horizon with compact, maximally symmetric cross-sections, thus possessing an associated near horizon limit. By staticity and compactness arguments, it can be proven that the one-form ω vanishes [9, 2, 37], simplifying the data to the function F and the metric h . By solving the Λ-vacuum equations order by order, which translates into elliptic equations for the transverse expansion, the authors show that all the transverse derivatives of the metric at the horizon can be computed from the data ( h, F ) and that they agree either with those of extremal Schwarzschild-de Sitter or with its near horizon geometry (Nariai). Consequently, in the case of a real analytic spacetime, they can prove that in a neighbourhood of the horizon the spacetime is isometric to one of these two solutions. This analysis has been extended to electrovacuum in [13]. Motivated by the previous examples, in this paper we prove general identities that relate the transverse derivatives of the ambient Ricci tensor with the transverse expansion of the metric at an arbitrary null hypersurface. Our analysis is coordinate free, does not require any field equations and holds regardless the signature of the ambient metric 1 , the dimension or the topology of the hypersurface. Furthermore, we analyze the case where the ambient space admits a preferred vector field η that is null and tangent to the hypersurface (e.g. a Killing or a homothetic vector field) and whose deformation tensor L η g is known. Such vectors are called 'symmetry generators'. Moreover, our analysis includes the possibility of η vanishing on subsets of H with empty interior (e.g. a bifurcate Killing horizon). To work with abstract null hypersurfaces we employ the so-called hypersurface data formalism [31, 26, 27], as it allows one to study hypersurfaces in a detached way, irrespective of their causal character. At the core of this formalism lies the concept of metric hypersurface data , which comprises an abstract manifold H along with a (0,2)-symmetric tensor field γ , a one-form /lscript , and a scalar function /lscript (2) . When H happens to be embedded in an ambient manifold ( M , g ), the tensor γ agrees with the first fundamental form of H , while the one-form /lscript and the scalar /lscript (2) capture the transverse-tangent and transverse-transverse components of the ambient metric, respectively. From { γ , /lscript , /lscript (2) } one can also introduce contravariant data on H that consists of a (2,0) symmetric tensor field P , a vector n and a scalar n (2) . In addition, metric hypersurface data {H , γ , /lscript , /lscript (2) } is subject to a gauge freedom that encompasses, at the abstract level, the multiplicity of choices of transverse vector ξ along H . Metric hypersurface data is also endowed with a torsion-free connection · ∇ constructed solely in terms of γ , /lscript and /lscript (2) . The arbitrary causal character of H allows the analysis of abstract null geometry, even though neither a metric nor its Levi-Civita connection are present in the null case. Let us denote by 2 Y ( k ) the pullback of the k -th Lie derivative of g along a transverse vector ξ at H . The collection { Y ( k ) } k ≥ 1 will be called transverse or asymptotic expansion . Our approach to determine this expansion at a general null hypersurface using the hypersurface data formalism is as follows. We begin by establishing a general identity that relates the m -th Lie derivative of the curvature and Ricci tensors along any vector ξ with the Lie derivatives of a connection along ξ . For the Levi-Civita connection this results in an identity linking the m -th Lie derivative of the Ricci tensor with the Lie derivatives of the metric. By applying this general identity to a null hypersurface H with a transverse vector ξ we can explicitly obtain identities for the leading order terms of the (i) completely transverse, (ii) transverse-tangent, and (iii) completely tangent components of the m -th Lie derivative of the ambient Ricci tensor along ξ (Corollary 4.26). A key property of the result is its geometric nature, as the identities depend solely on metric hypersurface data and { Y ( k ) } k ≥ 1 once the transverse vector ξ is extended off the hypersurface geodesically. The identities (i)-(iii) described in the previous paragraph are of a very distinct nature. Firstly, the completely tangential component of the m -th derivative of the Ricci tensor depends algebraically on the trace of the tensor Y ( m +2) w.r.t P , that we denote by tr P Y ( m +2) . Secondly, its transverse-tangent components depend algebraically on the one-form Y ( m +2) ( n, · ), where n is the null generator of the hypersurface. And finally, the identity for the fully tangential components of the m -th derivative of the Ricci depends on Y ( m +1) via a transport equation along the null generator of H , but also on the scalar tr P Y ( m +1) and the one form Y ( m +1) ( n, · ) and its derivatives. Since these two last objects depend algebraically on the derivatives of the Ricci tensor, identity (iii) allows us to know the evolution of the transverse expansion on any null hypersurface along the null direction provided the Ricci tensor is known at all orders on H . Observe that, for a fixed m , the leading order terms of the identity (iii) carry at most m +1 transverse derivatives of the metric, whereas the leading order terms in identities (i) and (ii) always depend on m +2 transverse derivatives of g at H . When a symmetry generator η is present, it is possible to transfer information from its deformation tensor L η g into the identity (iii). This gives rise to a new identity called 'generalized master equation of order m ' (cf. (129)) due to its close relationship to the 'generalized master equation' found in [19] for the case m = 1. Its key property is that its dependence on Y ( m +1) is not via a transport equation anymore. In addition, it also depends on the one-form Y ( m +1) ( n, · ) and the scalar tr P Y ( m +1) . This identity will allow us to identify the minimum amount of data at H in order to determine the full transverse expansion in terms of the Ricci tensor and its derivatives at the hypersurface. Informally, the idea is that, at each order, from the identities (i) and (ii) of order m -1 one can obtain algebraically tr P Y ( m +1) as well as Y ( m +1) ( n, · ). By introducing such tr P Y ( m +1) and Y ( m +1) ( n, · ) into the generalized master equation of order m , the remaining components of Y ( m +1) can then be obtained algebraically. In the last part of the paper we apply the generalized master equation of order m to the Killing horizon case. For non-degenerate Killing horizons, we are able to show that the full transverse expansion is uniquely determined from abstract data at the horizon as well as on the tower of derivatives of the ambient Ricci tensor at H (Theorem 6.15). This extends the main result of [16] in several directions. Firstly because our approach allows for much more general field equations besides vacuum with Λ = 0. Secondly because we are allowing zeroes of η , which includes the possibility of the horizon having bifurcation surfaces. And finally, because our result extends to arbitrary ambient signature. The collection of transverse derivatives of the ambient Ricci tensor at H can be thought at least in two different ways. One possibility is to provide such collection as prescribed data on the hypersurface, e.g. given by some external matter field. Another option is to think of the collection as functional relations between metric hypersurface data and the transverse expansion at the horizon. The easiest example that illustrates the second viewpoint are the Λ-vacuum equations, as in this case the Ricci tensor is proportional to the metric, so its transverse derivatives will depend on the expansion itself. This property is captured in our Definition 6.16, and allows us to prove that when the ambient Ricci tensor fulfills this property, the full expansion at H is uniquely determined from abstract data at the horizon (Theorem 6.17). When applied to the vacuum equations with cosmological constant, our theorem provides a characterization of all possible analytic Λ-vacuum manifolds in the vicinity of non-degenerate horizons. As a direct consequence, we establish a uniqueness result for non-extremal Schwarzschild-de Sitter spacetime (Proposition 6.22), complementing the main result of [14]. The paper is organized as follows. In Section 2, we provide a self-contained overview of the fundamental concepts of hypersurface data formalism. In particular, we recall the fact that the fully tangential components of the ambient Ricci tensor at a null hypersurface are given in terms of hypersurface data. In Section 3 we proceed to calculate the remaining components of the Ricci tensor, namely its fully transverse and transverse-tangent components. Once the full Ricci tensor is expressed in terms of metric data and transverse derivatives of the metric on the null hypersurface, in Section 4 we determine the connection between higher order derivatives of the ambient Ricci tensor and higher order derivatives of the metric. Section 5 is devoted to the analysis of the algebraic identities obtained when a general symmetry generator is present on H . Finally, in Section 6 we focus our attention to non-degenerate Killing horizons and we prove that the asymptotic expansion on any non-degenerate Killing horizon is uniquely determined in terms of abstract data at the horizon. Additionally, the paper includes an appendix (Appendix A) where we derive several identities involving the pull-back of arbitrary covariant tensors and their derivatives on the null hypersurface. These play a key role in our analysis in Section 4.",
"pages": [
1,
2,
3,
4
]
},
{
"title": "Notation and conventions",
"content": "Throughout this paper ( M , g ) denotes an arbitrary smooth d -dimensional semi-Riemannian manifold of any signature ( p, q ) with both p and q different from zero. When we specifically need this signature to be Lorentzian, we will say so explicitly. We employ both index-free and abstract index notation at our convenience. Ambient indices are denoted with Greek letters, abstract indices on a hypersurface are written in lowercase Latin letters, and abstract indices at cross-sections of a hypersurface are expressed in uppercase Latin letters. As usual, square brackets enclosing indices denote antisymmetrization and parenthesis are for symmetrization. The symmetrized tensor product is denoted with ⊗ s . By F ( M ), X ( M ) and X /star ( M ) we denote respectively the ring of smooth functions, the F ( M )-module of smooth vector fields and the F ( M )-module of smooth one-forms on M . The subset F /star ( M ) ⊂ F ( M ) consists of the nowhere vanishing functions on M . The pullback of a function f via a diffeomorphism Φ will be denoted by Φ /star f or simply by f depending on the context. Given a diffeomorphism Φ and a vector field X , we define Φ /star X := (Φ -1 ) /star X . A ( p, q )-tensor refers to a tensor field p times contravariant and q times covariant. Given any pair of (2 , 0) and (0 , 2) tensors A ab and B cd we denote tr A B := A ab B ab . We employ the symbol ∇ for the Levi-Civita connection of g . Throughout this paper we use the notation L ( m ) X T to denote the m -th Lie derivative of the tensor T along X , and X ( m ) ( f ) for the m -th directional derivative of the function f along X . When m = 1 we also write L X T and X ( f ), respectively, and when m = 0 they are just the identity operators. All manifolds are assumed to be connected and smooth.",
"pages": [
5
]
},
{
"title": "2 Review of hypersurface data formalism",
"content": "This section is devoted to review the basic notions of the so-called hypersurface data formalism . Further details can be found in [26, 27, 31]. The usefulness of this formalism has been recently demonstrated in the context of matching of spacetimes [20, 21, 22] and in solving the characteristic problem of general relativity from an abstract point of view [30, 29]. Let us start by introducing the necessary objects for this paper. Definition 2.1. Let H be a d -dimensional manifold, γ a symmetric (0,2)-tensor field, /lscript a oneform and /lscript (2) a scalar function on H . We say that {H , γ , /lscript , /lscript (2) } defines a metric hypersurface data set provided that (0,2) symmetric tensor A | p on T p H× R defined by is non-degenerate at every p ∈ H . A five-tuple {H , γ , /lscript , /lscript (2) , Y } , where Y is a (0,2) symmetric tensor field on H , is called hypersurface data. The non-degeneracy of A allows us to introduce its 'inverse' A /sharp by A /sharp ( A (( V, a ) , · ) , · ) = ( V, a ) for every ( V, a ) ∈ X ( H ) ⊗F ( H ). From A /sharp one can define a (2,0) symmetric tensor field P , a vector n and a scalar n (2) on H by the decomposition Equivalently, P , n and n (2) can be defined by Despite its name, the notion of hypersurface data does not view H as a hypersurface of another ambient manifold. The connection between Definition 2.1 and the standard definition of a hypersurface is as follows. Definition 2.2. Metric hypersurface data {H , γ , /lscript , /lscript (2) } is (Φ , ξ ) -embedded in a semi-Riemannian manifold ( M , g ) if there exists an embedding Φ : H ↪ →M and a vector field ξ along Φ( H ) everywhere transversal to Φ( H ) , called rigging, such that Hypersurface data {H , γ , /lscript , /lscript (2) , Y } is embedded provided that, in addition, In the context of embedded (metric) hypersurface data, γ being degenerate is equivalent to Φ( H ) being an embedded null hypersurface. It is easy to show [27] that the degeneracy of γ is equivalent to n (2) = 0. Hence, (metric) hypersurface data satisfying n (2) = 0 are called null (metric) hypersurface data . A cross-section (or simply a section) S of H is an embedded hypersurface S ↪ →H with the property that every integral curve of n crosses S exactly once. From now on we restrict ourselves to the null case. Given null metric hypersurface data we define the tensor field When the data is embedded U coincides with the second fundamental form of Φ( H ) w.r.t the unique normal one-form ν satisfying ν ( ξ ) = 1 (see [26]). It is also convenient to introduce the tensors Let {H , γ , /lscript , /lscript (2) } be null metric hypersurface data (Φ , ξ )-embedded in ( M , g ). By the transversality of ξ , given a (local) basis { e a } of H , the set { ̂ e a := Φ /star e a , ξ } is a (local) basis of Φ( H ) with dual basis { θ a , ν } . Raising the indices we can introduce ν := g /sharp ( ν , · ) and θ a := g /sharp ( θ a , · ), which are given in terms of { ξ, e a } by ̂ ̂ From (2) the inverse metric g αβ at H can be written in the basis { ξ, e a } as ̂ ̂ ̂ ̂ In the embedded picture the notion of rigging vector is non-unique, since given a rigging ξ any other vector of the form ξ ' = z ( ξ + Φ /star V ) with ( z, V ) ∈ F /star ( H ) × X ( H ) is also transverse to Φ( H ). Translating this into the abstract setting we have the following definition. Definition 2.3. Let {H , γ , /lscript , /lscript (2) , Y } be hypersurface data and ( z, V ) ∈ F /star ( H ) × X ( H ) . We define the gauge transformed hypersurface data with gauge parameters ( z, V ) by Transformations (14)-(17) induce the corresponding transformations on P and n [26] The set of gauge transformations defines a group whose composition law and the inverse element are [26] As expected from the geometric interpretation of U as the second fundamental form of Φ( H ) w.r.t ν := Φ /star n , its gauge transformation is given by [26] A consequence of this together with (18) and U ( n, · ) = 0 is Given null metric hypersurface data {H , γ , /lscript , /lscript (2) } it is possible to define a torsion-free connection · ∇ on H by means of [27] When the data is embedded in ( M , g ), · ∇ is related with the Levi-Civita connection ∇ of g by Unless otherwise indicated, scalar functions related by Φ /star are denoted with the same symbol. The action of · ∇ on the contravariant data { P, n } is given by [27] where s := F ( n, · ). A direct consequence of (26) and (27) is It is therefore natural to define the surface gravity of n by κ n := -Y ( n, n ), whose gauge transformation law follows directly from (19) and (17) and is Applying the Cartan identity L n = ι n · d + d · ι n to the one-form /lscript and using 2 F = d /lscript and /lscript ( n ) = 1 one has Another consequence of (27) is that for any one-form θ it holds [23] For future use we need to know the commutator [ P ab , L n ] acting on a 2-covariant, symmetric tensor field T ab . Lemma 2.4. Let T ab a (0,2) symmetric tensor field. Then, where t := T ( n, · ) . Proof. The result follows at once from the following expression of the Lie derivative of P along n [28] Later we will also need the ∇ -derivative of ξ along tangent directions to H [26] where r := Y ( n, · ). The gauge transformation law of the one-form Π ( · , n ) = r -s is [30] As proven in [26], the completely tangential components of the ambient Riemann tensor, as well as its 3-tangential, 1-transverse components can be written in terms of hypersurface data as where A and B are the tensors on H defined by and · R f bcd is the curvature of · ∇ . It follows from (37) that all the tangential components of the ambient Ricci tensor can be written in terms of hypersurface data as [30] The RHS defines a tensor on H called constraint tensor R . Its explicit form is [23] where · R ab is the Ricci tensor of the connection · ∇ . The tensor R is abstract in the sense that it does not require the data to be embedded in any ambient manifold. Note that all the dependence on the tensor Y in (40) is explicit. Below we shall need the explicit form of the contraction of R ab with n a and P ab . The former was obtained in [23] and relies on the following general identity, also derived in [23] The result is Another contraction with n b gives which is the abstract version of the Raychaudhuri equation. For the trace of (40) with respect to P we simply need to use (33) and get where div P t := P ab · ∇ a t b .",
"pages": [
5,
6,
7,
8
]
},
{
"title": "3 Full Riemann tensor at a hypersurface",
"content": "While the components of the form R αβµν ξ α ̂ e β b ̂ e µ c ̂ e ν d and R αβµν ̂ e α a ̂ e β b ̂ e µ c ̂ e ν d can be written solely in terms of hypersurface data, the remaining ones, i.e. R αβµν ξ α ̂ e β b ξ µ ̂ e ν d , cannot, because in addition they require second order transverse derivatives of the metric. In order to compute them we use a lemma from [25]. Lemma 3.1. Let ( M , g ) be a semi-Riemannian manifold and ξ, X, Y ∈ X ( M ) . Then, This lemma allows us to write the remaining components of the ambient Riemann tensor in terms of second transverse derivatives of the metric and hypersurface data. Proposition 3.2. Let {H , γ , /lscript , /lscript (2) , Y } be null hypersurface data (Φ , ξ ) -embedded in ( M , g ) and define β ∈ F ( H ) and T ∈ X ( H ) by the decomposition ∇ ξ ξ H = βξ +Φ /star T . Then, Proof. Let { e a } be a (local) basis of X ( H ) and ̂ e a := Φ /star e a . From (35) together with (7) and recalling that ν is null, normal to Φ( H ) and satisfies g ( ξ, ν ) = 1, it follows where ( Π · Π ) ( X,Y ) := P ( Π ( X, · ) , Π ( Y, · )) . where in the second equality we used /lscript d P fd = -/lscript (2) n f . Equation (46) follows from (45) after using Let X,Y ∈ X ( H ). Since ∇ Φ /star X ξ and ∇ Φ /star Y ξ only depend on ξ along Φ( H ) and Riem is a tensor, it follows that Riem( ξ, Φ /star X, Φ /star Y, ξ ) + g ( ∇ Φ /star X ξ, ∇ Φ /star Y ξ ) only depends on ξ along Φ( H ). Thus, by equation (45) the tensor is independent of how one extends ξ off Φ( H ), so by (47) it follows that the tensor field does not depend on the extension of ξ off Φ( H ). This suggests extending the definition of hypersurface data set as follows. Definition 3.3. A septuple {H , γ , /lscript , /lscript (2) , Y , Z (2) } defines an extended hypersurface data set provided {H , γ , /lscript , /lscript (2) , Y } is hypersurface data and Z (2) is a (0,2) symmetric tensor field on H . Definition 3.4. An extended hypersurface data set {H , γ , /lscript , /lscript (2) , Y , Z (2) } is said to be (Φ , ξ ) -embedded in a semi-Riemannian manifold ( M , g ) provided {H , γ , /lscript , /lscript (2) , Y } is (Φ , ξ ) -embedded in the sense of Definition 2.2 and, in addition, where ( β, T ) ∈ F ( H ) × X ( H ) are defined by ∇ ξ ξ H = βξ + Φ /star T and ξ is any extension of the rigging off Φ( H ) . In terms of Z (2) , equation (46) gets rewritten as Once the full Riemann tensor at H is computable from hypersurface data and Z (2) , one can write down explicitly the full Ricci tensor on the abstract null hypersurface. Proposition 3.5. Let D = {H , γ , /lscript , /lscript (2) , Y , Z (2) } be extended null hypersurface data (Φ , ξ ) -embedded in ( M , g ) . Let Ric be the Ricci tensor of g and ˙ R := Φ /star ( Ric( ξ, · ) ) , R := Φ /star ( Ric( ξ, ξ ) ) . Then, where A abc is defined in (38) and z (2) := Z (2) ( n, · ) . Proof. Let { e a } be a (local) basis of X ( H ) and ̂ e a := Φ /star e a . From the definition of the Ricci tensor, the symmetries of the Riemann tensor and equations (13) and (48) it follows In the same way, using the first equation in (37), given X ∈ X ( H )",
"pages": [
9,
10
]
},
{
"title": "4 Higher order derivatives",
"content": "In this section we compute the derivatives L ( m ) ξ Ric on H in terms of transverse derivatives of g on H up to order m +1, i.e. making L ( m +2) ξ g and L ( m +1) ξ g explicit. In order to simplify the notation let us introduce the tensors as well as Observe that Y (1) , r (1) and κ (1) agree with Y , r and κ n , respectively. In what follows we refer to the collection of tensors { Y , Y (2) , ... } as the transverse or asymptotic expansion . In the following lemma we recall a well-known identity for derivatives of products of any two objects S and T . Note than when S and T are tensors, the expression also holds when contractions are allowed. Lemma 4.1. Let S and T be two objects, S /circleasterisk T any product of them and D any derivative operator. Then, Given any vector field ξ we introduce the tensor Σ[ ξ ] := L ξ ∇ , or in abstract index notation where K [ ξ ] is the so-called deformation tensor of ξ , defined by K [ ξ ] := L ξ g . In order not to overload the notation in this section we will simply use the symbols Σ and K for Σ[ ξ ] and K [ ξ ], respectively. In later sections, we will come back to the notation K [ ξ ] because deformation tensors of more than one vector field will occur. To compute L ( m ) ξ Ric up to order m +1 we use the following classical result [38] relating the Lie derivative of the curvature with the Lie derivative of the connection For future convenience we define where we have introduced the tensor The hat in ̂ Σ ναβ is not really necessary because Σ ναβ is just Σ ν αβ with the index lowered. However the distinction will be necessary later for the Lie derivative of ̂ Σ ναβ and Σ ν αβ . Remark 4.2. The notations H γρ νβ and F ρλγ ναβ are unambiguous since we shall never lower/raise their indices. We stick to this rule for any tensor written with indices on top of each other. The idea now is to apply the operator L ( m -1) ξ to (53) and express the result by making L ( m +1) ξ g and L ( m +2) ξ g explicit. In order to do that we need to commute L ( m -1) ξ and ∇ when they act on a ( q, p ) tensor A α 1 ··· α q β 1 ··· β p . We introduce the notation A ( m ) := L ( m -1) ξ A , m ≥ 1. The commutator is found explicitly in the following proposition. Proposition 4.3. Let ξ ∈ X ( M ) and m ≥ 1 be a integer. Then, given any ( p, q ) tensor A α 1 ··· α q β 1 ··· β p the following identity holds Proof. The case m = 1 is the classical identity [38] We prove the result by induction, so let us assume that the claim is true up to some m ≥ 1 and show that it is then true for m +1 also. We compute, using the induction hypothesis, In the last term we rename k as k -1 and split the second sum in two parts and the last sum also in two parts, Once we know how to commute L ( m -1) ξ and ∇ when acting on an arbitrary tensor field, we can apply the result to equation (53) to compute the explicit expression of the tensor L ( m ) ξ R µ ανβ . This will be used below to compute the derivatives L ( m ) ξ Ric on a null hypersurface H in terms of transverse derivatives of g on H up to order m +1. Using the binomial identity ( m i +1 ) + ( m i ) = ( m +1 i +1 ) , the proposition follows by induction. Proposition 4.4. Let ξ ∈ X ( M ) and m ≥ 2 an integer. Then, and Proof. Applying L ( m -1) ξ to (53) and using Proposition 4.3, Since H γρ νβ is antisymmetric in γ, ρ its contraction with the third term vanishes. Renaming k ' = m -2 -k in the last term the sum simplifies to From the antisymmetry of H γρ νβ , the symmetry of Σ ( k +1) µ ρσ and the combinatorial properties ( m -1 m -1 -k ' ) = ( m k ' ) and ( m -1 k +1 ) + ( m -1 k ) = ( m k +1 ) , (55) follows. Equation (56) is immediate since the Lie derivative and the trace commute. Identity (56) constitutes the exact relation between the m -th Lie derivative of the Ricci tensor with the Lie derivatives of the tensor Σ. Before restricting it to a null hypersurface H with rigging ξ and computing the leading order terms, we shall establish a property that will play a key role in Section 6, namely that when the rigging ξ is extended off Φ( H ) by ∇ ξ ξ = 0, the tensors L ( m ) ξ R αβ on H are geometrical in the following sense. Definition 4.5. Let {H , γ , /lscript , /lscript (2) } be null metric hypersurface data (Φ , ξ ) -embedded in ( M , g ) . Let T and T be (0 , p ) tensor fields on H and M , respectively. In order to prove that L ( m ) ξ R αβ is geometrical for any natural number m we first explore some general properties of geometrical objects and also establish that several building-block tensors that appear in the argument are indeed geometrical. We start with the latter. Lemma 4.6. Let {H , γ , /lscript , /lscript (2) } be null metric hypersurface data (Φ , ξ ) -embedded in ( M , g ) and extend ξ off Φ( H ) by ∇ ξ ξ = 0 . Then the tensor ∇ α ξ β is geometrical. Proof. Since ξ α ∇ α ξ β = 0 it suffices to check that ̂ e α a ̂ e β b ∇ α ξ β and ̂ e α a ξ β ∇ α ξ β only depend on hypersurface data. Using ̂ e µ a ∇ µ ξ β H = (r -s) a ξ β + P cd Π ac ̂ e β d + 1 2 ν β · ∇ a /lscript (2) (cf. (35)) and Definition 2.2 a straightforward computation shows that Hence, ∇ α ξ β is geometrical. Next we show that when ∇ ξ ξ = 0 the tensor K ( m ) µν is geometrical for all m ≥ 0. As a preliminary step we first compute in full generality the contraction ξ α K ( m ) αβ on H . Proposition 4.7. Let H be a null hypersurface (Φ , ξ ) -embedded in ( M , g ) and let Z be a vector field along Φ( H ) . Extend ξ arbitrarily off Φ( H ) and define a ξ := ∇ ξ ξ . Then for every m ∈ N ∪ { 0 } , Proof. We first prove (58) and then we show that (57) follows from (58). Let us extend Z off H by L ξ Z = 0 (at the end we prove that the result is independent of the extension). First, Applying L ( m ) ξ to both sides, which becomes (58) after commuting ξ ( m ) Z = Zξ ( m ) (since [ ξ, Z ] = 0). Obviously equation (58) is independent of the extension of Z off H . To show (57) just use (58) with Z = ξ . Corollary 4.8. Let {H , γ , /lscript , /lscript (2) } be null metric hypersurface data (Φ , ξ ) -embedded in ( M , g ) and extend ξ off Φ( H ) by ∇ ξ ξ = 0 . Then K ( m ) ( ξ, · ) = 0 for all m ≥ 1 . Lemma 4.9. Let {H , γ , /lscript , /lscript (2) } be null metric hypersurface data (Φ , ξ ) -embedded in ( M , g ) and extend ξ off Φ( H ) by ∇ ξ ξ = 0 . Then the tensor K ( m ) µν is geometrical for all m ≥ 0 . Proof. The tensor K (0) µν := g µν is clearly geometrical by the definition of embedded metric hypersurface data. By Corollary 4.8 and recalling Φ /star K ( m ) = 2 Y ( m ) , the tensor K ( m ) µν for m ≥ 1 is geometrical as well. Lemma 4.10. Let {H , γ , /lscript , /lscript (2) } be null metric hypersurface data (Φ , ξ ) -embedded in ( M , g ) and extend ξ off Φ( H ) by ∇ ξ ξ = 0 . Let T and S be any two geometrical objects. Then, the following properties are true. As a consequence, any trace of T ⊗ S w.r.t g or L ( m ) ξ g µν is also geometrical. Proof. The first property is obvious. The second one follows from g αβ H = P ab ̂ e α a ̂ e β b + n a ̂ e α a ξ β + n b ̂ e β b ξ α (cf. (13)). We prove the third property by induction. First observe that for m ≥ 1 Particularizing for m = 1 gives L ξ g αβ = -g βν g αν K µν . By Lemma 4.9 and items 1. and 2. of this lemma, item 3. follows for m = 1. Assume it is true up to some integer m -1. Expression (59) together with Lemma 4.9 then show that it is also true for m . Lemma 4.11. Let {H , γ , /lscript , /lscript (2) } be null metric hypersurface data (Φ , ξ ) -embedded in ( M , g ) and extend ξ off Φ( H ) by ∇ ξ ξ = 0 . Then the tensor ∇ ρ K ( m ) µν is geometrical for all m ≥ 0 . Proof. The argument relies on the following well-known relation between ∇ and L ξ acting on a (0,2) symmetric tensor T Particularizing to T = K ( m ) gives Proposition 4.12. Let {H , γ , /lscript , /lscript (2) } be null metric hypersurface data (Φ , ξ ) -embedded in ( M , g ) and extend ξ off Φ( H ) by ∇ ξ ξ = 0 . Then, the tensors ̂ Σ ( m ) ναβ and Σ ( m ) ναβ for all m ≥ 1 are geometrical. Proof. Recalling definition (54), namely ̂ Σ ναβ := g νµ Σ µ αβ , and applying identity (51) gives The first term is geometrical by Lemma 4.9, and the second one is also geometrical as a consequence of Lemmas 4.6, 4.9 and 4.10. Next we show that the tensor ̂ Σ ( m ) is geometrical. So, by Lemma 4.10 if Σ ( k ) ναβ is geometrical for all k ≤ m , then Σ ( m ) ναβ is geometrical as well. It only remains to prove that ̂ Σ ( m ) ναβ is geometrical for all m ≥ 1. We establish this by an induction argument. From ̂ Σ ναβ = F ρλγ ναβ ∇ ρ K λγ and Lemma 4.11 it follows that the tensor ̂ Σ ναβ is geometrical. Assume ̂ Σ ( k ) ναβ (and hence also Σ ( k ) ναβ ) is geometrical for all k ≤ m . Applying Proposition 4.3 to A = K and taking into account L ξ F = 0 it follows Using Lemmas 4.10 and 4.11 and the fact that every Σ ( k ) ναβ for k ≤ m is geometrical, we conclude that ̂ Σ ( m +1) ναβ (and thus Σ ( m +1) ναβ ) is geometrical as well. That R αβ is geometrical when ∇ ξ ξ = 0 is immediate from (40), (49) and (50) because in this case Z (2) = Y (2) (by Definition 3.4). In order to prove that L ( m ) ξ R αβ is geometrical for every m ≥ 1 when ∇ ξ ξ = 0 it is convenient to first rewrite (56) making the tensor H explicit, namely By Lemma 4.10 and Proposition 4.12 all the terms in the two sums are geometrical, so it suffices to show that ∇ µ Σ ( m ) µ αβ and ∇ β Σ ( m ) µ αµ are also geometrical. In order to do that we shall contract both tensors with two tangent vectors ̂ e α a ̂ e β b , one tangent and one transverse 15 ̂ e α a ξ β , and two transverse vectors ξ α ξ β and check that in all cases the result only depends on metric data and the expansion { Y ( k ) } . These computations rely on general expressions for the pullback of ambient tensor fields into an arbitrary null hypersurface H . These are computed in full generality in Appendix A. The following notation is used. Notation 4.13. Given a (0 , p ) tensor field T α 1 ··· α p on M we use the standard notation T a 1 ··· a p to denote the pullback of T to H . Moreover, we introduce the notation ( i ) T α 1 ··· α i -1 α i +1 ··· α p for ξ α i T α 1 ··· α p and ( i ) T a 1 ··· a i -1 a i +1 ··· a p for the pullback of ( i ) T α 1 ··· α i -1 α i +1 ··· α p to H . Several numbers between parenthesis denote contractions with ξ on those slots. Let start by analyzing ∇ µ Σ ( m ) µ αβ . Firstly, from Proposition A.2 and the fact that Σ ( m ) ναβ is geometrical, it follows that ̂ ̂ and hence by Proposition 4.12 ̂ e α a ̂ e β b ∇ µ Σ ( m ) µ αβ only depends on metric data and { Y ( k ) } . Secondly, its contraction with ξ α e β b yields ̂ ̂ ̂ By Lemma 4.10 and Proposition 4.12 the second term is H -geometrical. Moreover, by Proposition A.2, ̂ where we used L ξ ( ( i ) T ( m ) ) = ( i ) T ( m +1) since obviously L ξ ξ = 0. Again Prop. 4.12 shows that ξ α ̂ e β b ∇ µ Σ ( m ) µ αβ (and by symmetry also ξ β ̂ e α b ∇ µ Σ ( m ) µ αβ ) only depends on metric data and { Y ( k ) } . Finally, the contraction with ξ α ξ β is By Prop. A.2 the first term is so by Prop. 4.12 it is H -geometrical. The second and third terms are H -geometrical as well by Lemmas 4.6 and 4.10 and Prop. 4.12. Hence ∇ µ Σ ( m ) µ αβ is geometrical. For the tensor ∇ β Σ ( m ) µ αµ we introduce T ( m ) α := Σ ( m ) µ αµ , which is geometrical by Lemma 4.10. The contractions ̂ e α a ̂ e β b ∇ β T α , ̂ e α a ξ β ∇ β T α and ξ α ̂ e β b ∇ β T α are automatically H -geometrical after using identities (156)-(158) in Appendix A and L ξ T ( m ) = T ( m +1) . Finally, ξ α ξ β ∇ β T ( m ) α = L ξ ( (1) T ( m ) ) = (1) T ( m +1) is also H -geometrical, and hence ∇ β Σ ( m ) µ αµ is geometrical. Thus, the following result has been proved. Proposition 4.14. Let {H , γ , /lscript , /lscript (2) } be null metric hypersurface data (Φ , ξ ) -embedded in ( M , g ) and extend ξ off Φ( H ) by ∇ ξ ξ = 0 . Then, the tensor L ( m ) ξ R αβ is geometrical for every m ≥ 0 .",
"pages": [
10,
11,
12,
13,
14,
15,
16
]
},
{
"title": "4.1 Analysis of the leading order terms",
"content": "In this subsection we compute the leading order terms of the tensor L ( m ) R αβ on an arbitrary null hypersurface H . As we shall see, it turns out that the leading order terms of ξ α L ( m ) R αβ involve m +2 transverse derivatives of the metric, whereas the fully tangential components of L ( m ) R αβ depend at most on m + 1 transverse derivatives. In order to write down equalities only to the leading order it is useful introduce the following notation. Notation 4.15. Let ( M , g ) be a semi-Riemannian manifold and T, S be two tensor fields involving g and its derivatives. The notation T [ m ] = S means that the tensor T -S does not depend on derivatives of g of order m or higher. The only terms that have a chance to carry m +1 and m +2 derivatives of the metric in identity (62) are the ones of the form ∇ γ Σ ( m ) µ αβ . Thus, with the notation above, Observe that equation (51) together with L ξ g µν = -K µν implies Lowering the index µ gives and applying L ξ , Lemma 4.16. Let ξ ∈ X ( M ) and m ≥ 1 an integer. Then, Proof. By Proposition 4.3 and (54), together with L ξ F ρλγ ναβ = 0, it follows which simplifies to because of the symmetries of K ( i ) αβ and Σ ( i ) σ µν . Hence, Combining (64) and (70) it follows Σ ( m -1) σ αβ [ m ] = g σν ̂ Σ ( m -1) ναβ [ m ] = g σε F ρλγ εαβ ∇ ρ K ( m -1) λγ , which inserted into (69) gives This establishes (67). Introducing (70) and (67) into (64) gives (68). We quote the following immediate consequence for future reference. Corollary 4.17. Let ξ ∈ X ( M ) and m ≥ 0 an integer. Then, Before computing R ( m ) , ˙ R ( m ) and R ( m ) on H it is important to make the following observation based on Proposition 4.7. Remark 4.18. By Proposition 4.7 the derivatives K ( m ) ( ξ, · ) are given in terms of transverse derivatives of a ξ := ∇ ξ ξ on Φ( H ) , metric hypersurface data as well as on the tensors { Y , ..., Y ( m -1) } . Hence, when computing R ( m ) , ˙ R ( m ) and R ( m ) , terms of the form K ( m ) ( ξ, · ) can always be replaced by lower order terms, i.e. terms that depend on metric hypersurface data and { Y , ..., Y ( m -1) } . Notation 4.19. Let ( M , g ) be a semi-Riemannian manifold, H an embedded hypersurface and T, S two tensors involving g and its derivatives. We introduce the notation T ( m ) = S to denote that the tensor ( T -S ) | H does not depend on transverse derivatives of g at H of order m or higher. ̂ In the next lemma and following proposition we compute the pullback of Σ ( m ) as well as several contractions of the tensors Σ ( m ) and ̂ Σ ( m ) that we shall use below. The computation relies on the general identities for the pullback of ambient tensor fields to arbitrary null hypersurfaces computed in Appendix A. Lemma 4.20. Let H be a null hypersurface (Φ , ξ ) -embedded in a semi-Riemannian manifold ( M , g ) . Extend ξ arbitrarily off Φ( H ) and let { e a } be a local basis on H and e a := Φ /star e a . Then, ̂ ̂ ̂ Proof. Equations (75) and (76) are particular cases of (73) and (74), respectively. Moreover, the validity of (73) and (74) implies at once (79) and (80) because g σγ can be decomposed using (13). Hence it suffices to prove the equations in the first and third lines. In order to prove (73) we contract ξ λ ∇ ρ K ( m ) λγ with (i) ξ ρ ξ γ , (ii) ξ ρ ̂ e γ c , (iii) ̂ e ρ c ξ γ and (iv) ̂ e ρ c ̂ e γ d and show that all of them are at most of order m . Let us start with (i). Applying (60) for T = K ( m ) gives where in the second equality we used that K ( m ) is of order m and in the last one Remark 4.18. For (ii) we contract ξ λ ∇ ρ K ( m ) λγ with ξ ρ e γ c and use again that K ( m ) is of order m , ̂ ̂ ̂ ̂ By Remark 4.18 the tensor (1) K ( m ) is of order m -1. Applying (157) for T = (1) K ( m ) all terms in the right hand side are at most of order m , so ̂ e γ c ξ ρ ξ λ ∇ ρ K ( m ) λγ ( m +1) = 0. For item (iii) we contract ξ λ ∇ ρ K ( m ) λγ with e ρ c ξ γ , namely ̂ ̂ because (1) K ( m ) is at most of order m -1. In order to prove (iv), i.e. ̂ e ρ c ̂ e γ d ξ λ ∇ ρ K ( m ) λγ ( m +1) = 0, we use identity (158) with T = K ( m ) . Since all the terms in the right hand side are at most of order m , ̂ e ρ c ̂ e γ d ξ λ ∇ ρ K ( m ) λγ ( m +1) = 0. This establishes (73). Equation (74) follows from (156) with T = K ( m +1) after recalling that ( i ) K ( m +1) is at most of order m and that Φ /star K ( m +1) = 2 Y ( m +1) . It only remains to prove (77) and (78). For the first one we insert (13) into ̂ e λ a ̂ e γ b g σρ ∇ ρ K ( m ) λγ and get ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ where equation (76) has been used in the first and second terms. This expression shows that (78) follows from (77). To establish the latter we use (157) with T = K ( m ) because only the first term in the right hand side is of order m +1. Proposition 4.21. Let H be a null hypersurface (Φ , ξ ) -embedded in a semi-Riemannian manifold ( M , g ) and extend ξ arbitrarily off Φ( H ) . Then, Proof. Consider equation (67) for m +1 and contract it with e ν c e α a e β b , namely ̂ ̂ ̂ ̂ ̂ ̂ Defining F def cab := 1 2 ( δ d a δ e b δ f c + δ d b δ e a δ f c -δ d c δ e a δ f b ) and using equation (74) the first term takes the form Making F ρλγ εαβ explicit the second term is ̂ ̂ ̂ ̂ ̂ because Φ /star K = 2 Y . Hence the equation of the first line is established. The equations of the second line follow from (71), namely Contracting this with ̂ e ν c ̂ e α a ̂ e β b and using (76) gives ̂ Σ ( m ) cab ( m +1) = 0, and contracting it with ξ ν e α a ̂ e β b gives ̂ ̂ ̂ ̂ ̂ ̂ ̂ Applying (75) to the first and second terms and (77) to the last one yields (1) ̂ Σ ( m ) ab ( m +1) = -Y ( m +1) ab . In a similar way, contracting (81) with e ν a ξ α e β b gives By (75) the only term that contributes is the first one, which after using (77) is (2) ̂ Σ ( m ) ab ( m +1) = Y ( m +1) ab . By the symmetry of ̂ Σ ( m ) , (3) ̂ Σ ( m ) ab ( m +1) = Y ( m +1) ab . To prove the third line consider (72), namely The contraction with ξ α ξ β , yields ξ α ξ β Σ ( m ) µ αβ ( m +1) = 0 after using (73). Taking trace in µ, α in (82) gives where in the last equality we used the symmetry of K ( m ) . Contracting with ̂ e β a and using equation (79) yields ̂ e β a Σ ( m ) µ µβ ( m +1) = 0. Finally, from (60) with T = K ( m ) , ∇ ξ K ( m ) [ m +1] = K ( m +1) , (85) so contracting (84) with ξ β gives ξ β Σ ( m ) µ µβ ( m +1) = 1 2 g λγ K ( m +1) λγ . Using (13) and the fact that (1) K ( m +1) ( m +1) = 0, the last equation of the third line follows. By virtue of (63) the last ingredient to compute the tensors R ( m ) , ˙ R ( m ) and R ( m ) on H is being able to calculate the pullback both of a divergence and of a ambient tensor field. These are computed in full generality in Propositions A.1 and A.2, and as a consequence we have the following three propositions. Proposition 4.22. Let H be a null hypersurface (Φ , ξ ) -embedded in ( M , g ) and extended ξ arbitrarily off Φ( H ) . Then for every m ≥ 0 , Proof. The case m = 0 has already been established in (49). In order to prove the identity for m ≥ 1 we contract the general expression (63) with ξ α ξ β so that where in the second line we used that Σ ( m ) involves up to m +1 derivatives of g . For the same reason, by (83) it follows From the expression of g µρ in (13) and the fact that ∇K ( m ) is at most order m +1, the only term that has a chance of carrying m +2 transverse derivatives of g is the first one. Using (85) and recalling (73), Finally from (84) it follows where in the second equality we used ∇ ξ K ( m ) [ m +1] = K ( m +1) and ∇ ξ K ( m +1) [ m +2] = K ( m +2) (see (85)) and in the third one we inserted (13) and used the fact that (1) K ( m +2) is at most of order m +1 (see Remark 4.18). Proposition 4.23. Let H be a null hypersurface (Φ , ξ ) -embedded in ( M , g ) and extend ξ arbitrarily off Φ( H ) . Then for any m ≥ 0 , Proof. The case m = 0 is (50). From (63) and the fact that Σ ( m ) is at most of order m +1, ̂ ̂ ̂ so the only terms capable of containing m + 2 transverse derivatives are the first and third ones. By Proposition A.2 applied to (3) Σ ( m ) µ α and L ξ ξ β = 0 it follows ̂ ̂ ̂ because the rest of the terms are of (transverse) order m + 1 or below. Using the second equation in (66) and Proposition 4.21 we obtain ̂ Finally, by (157) with T = Σ ( m ) µ µα it follows where the last equality we used Proposition 4.21. Proposition 4.24. Let H be a null hypersurface (Φ , ξ ) -embedded in ( M , g ) and extended ξ arbitrarily off Φ( H ) . Let m ≥ 1 be an integer. Then, Proof. By the general formula (63) we have ̂ ̂ ̂ ̂ so we only need to compute each of these terms. For the first one we use Proposition A.2 applied to Σ ( m ) , namely From the second equation in (65), Σ ( m ) αβγ ( m +1) = ̂ Σ ( m ) αβγ , and thus by Proposition 4.21 it follows Σ ( m ) abc ( m +1) = 0, (1) Σ ( m ) ab ( m +1) = -Y ( m +1) ab and (2) Σ ( m ) ab ( m +1) = Y ( m +1) ab . Hence the expression for (div Σ ( m ) ) ab simplifies to In order to compute the term n c ( L ξ Σ ( m ) ) cab we contract the first equation in (66) with ̂ e µ c ̂ e α a ̂ e β b , which gives ( L ξ Σ ( m ) ) cab [ m +1] = e µ c e α a e β b ̂ Σ ( m +1) µαβ -( m -1) e µ c e α a e β b g νρ K µν ̂ Σ ( m ) ραβ . ̂ ̂ ̂ ̂ ̂ ̂ The first term is given by the first line in Proposition 4.21, and, for the second one, using (13) as well as ̂ Σ ( m ) abc ( m +1) = 0 and (1) ̂ Σ ( m ) ab ( m +1) = -Y ( m +1) ab one gets Combining everything it follows and after contracting with n c equation (94) becomes Using where the third equality follows from (27), the expression for (div Σ ( m ) ) ab is finally To compute the second term in (93) we use equation (156) for T = Σ ( m ) µ µα . Since by Proposition 4.21 Σ ( m ) µ µa ( m +1) = 0 and ξ α Σ ( m ) µ µα ( m +1) = tr P Y ( m +1) , we find that ̂ e α a ̂ e β b ∇ β Σ ( m ) µ µα = ( tr P Y ( m +1) ) U ab . Inserting this and (95) into (93) proves the Proposition. Proposition 4.24 is interesting because it allows one to determine the evolution of the transverse expansion of the metric along the null generator in any null hypersurface. We next compute two contractions of (92) as well as its trace w.r.t P . Corollary 4.25. Let H be a null hypersurface (Φ , ξ ) -embedded in ( M , g ) and extend ξ arbitrarily off Φ( H ) . Let Ric be the Ricci tensor of g . Then, Proof. To prove (96) we contract (92) with n b and use where the second equality follows from (27). Contracting (96) with n a (97) follows at once. For the last one apply P ab to (92) and use (33) with T = Y ( m +1) . Later on we shall need to use the 'complete' identities (86), (89) and (92), i.e. including a term that gathers all the lower order terms. By Proposition 4.14 these terms are H -geometrical when ∇ ξ ξ = 0. Corollary 4.26. Let {H , γ , /lscript , /lscript (2) } be null metric hypersurface data (Φ , ξ ) -embedded in ( M , g ) and extended ξ arbitrarily off Φ( H ) . Let m ≥ 1 be an integer. Then, where O ( m ) , O ( m ) a and O ( m ) ab are, respectively, a scalar, a one-form and a (0,2) symmetric tensor on H with the property that when ∇ ξ ξ = 0 they only depend on null metric data { γ , /lscript , /lscript (2) } and on the tensors { Y , ..., Y ( m ) } .",
"pages": [
16,
17,
18,
19,
20,
21,
22,
23
]
},
{
"title": "5 Deformation tensor and algebraic identities",
"content": "In some situations one has a privileged vector field η on ( M , g ) whose deformation tensor K [ η ] := L η g is known. This is in general a very valuable information that one may want to incorporate into the identities of Section 4. For instance, in [19] it is shown that the expression of the constraint tensor in (40) can be rewritten so that the dependence on the tensor Y is algebraic instead of via a transport equation. The corresponding identity was called the generalized master equation . The aim of this section is to extend the same idea to the higher order derivatives of the Ricci tensor, i.e. to combine identity (101) with information on K [ η ] so that the dependence on Y ( m +1) becomes algebraic. Let us start by reviewing a result from [19]. Lemma 5.1. Let {H , γ , /lscript , /lscript (2) } be null metric hypersurface data (Φ , ξ ) -embedded in ( M , g ) and let η ∈ X ( M ) be such that η | Φ( H ) is tangent to Φ( H ) . Denote by ¯ η the vector field on H satisfying Φ /star (¯ η ) = η | Φ( H ) . Then, where and the components of X η in any basis { e a } of H are given by where θ a = P ab Φ /star e b + n a ξ . /negationslash From now on we assume η | H to be null and tangent to Φ( H ), so there must exist a function α ∈ F ( H ) such that η | H = αν (in principle we do not assume α = 0, thus allowing η to have zeros). The gauge behaviour of the scalars α and K [ η ]( ξ, ν ) are as follows [19]. Lemma 5.2. Let ( z, V ) be a gauge element and denote with a prime the gauge-transformed objects. Then, From this it is immediate to check that the transformation of n ( α ) and that of A η are then given by In particular, n ( α ) is gauge-invariant at the points where α = 0. Following [19], let us introduce the scalar function κ by means of From equation (30) and Lemma 5.2 it follows that κ is gauge invariant. This function extends the standard notion of surface gravity. Indeed, at the points where the vector field η does not vanish, its surface gravity ˜ κ is the scalar on H defined by ∇ η η H = κη. (109) /negationslash Using the identity L [ X,Y ] = [ L X , L Y ] applied to the metric g it follows ˜ Inserting η | H = αν into (109) and using (26) it follows that ˜ κ = κ on the subset of H where ¯ η = 0. Note however that κ is well defined and smooth everywhere on H . Introducing (102) and pulling back this equation into H gives [19] Inserting ¯ η = αn and recalling (31) one has L ¯ η /lscript = L αn /lscript = 2 α s + dα , so equations (103), (104) and (111) become The generalized master equation [19] relates the constraint tensor R ab , the metric hypersurface data and information on the deformation of η codified via the tensorial quantities w , p , q and I defined by The equation is obtained by inserting (114) into (40). The result is [19] As already mentioned before, the dependence on Y in this relation is purely algebraic. The contraction of (116) with n is [19] and contracting again with n , In terms of (115) the function A η and the vector X η in (112)-(113) can be written as Hence, relation (102) becomes, after using (108), The idea now is to repeat this process with the identity (101), i.e. to replace the derivative L n Y ( m +1) by derivatives of K [ η ], Y ( m +1) itself and lower order terms. As usual, let us define K [ η ] ( m ) := L ( m -1) ξ K [ η ] and Observe that w (0) = w , p (0) = p , q (0) = q and I (1) = I . The rule of thumb is that the number inside the parenthesis denotes the number of transverse derivatives applied to K [ η ]. Following the notation of Section 4 we introduce the tensor Σ[ η ] := L η ∇ . Proposition 5.3. Let {H , γ , /lscript , /lscript (2) } be null metric hypersurface data ( Φ , ξ )-embedded in ( M , g ) and extend ξ off H arbitrarily. Let η be a vector field on ( M , g ) satisfying that η | Φ( H ) = αν for some α ∈ F ( H ) . Then, for any integer m ≥ 1 where P ( m ) is a tensor that depends on { Y , ..., Y ( m -1) } and {L ξ η, ..., L ( m ) ξ η } ∣ ∣ H and it is given explicitly by and X a η = 1 2 ( αn ( /lscript (2) ) -p ) n a + P ab ( 2 α s b + · ∇ b α -q b ) . As a consequence, Proof. We first show by induction the following relation For m = 1 it holds (cf. (110)). Let us assume (126) is true up to some m ≥ 1 and show that it is then true for m +1 also. We compute where in the third line we introduced the induction hypothesis. Renaming i ↦-→ i -1 in the second sum and using the binomial identity ( m i -1 ) + ( m i ) = ( m +1 i ) it follows This establishes (126) for m ≥ 1. The first term in the sum can be computed by means of (102). Pulling (126) back into H and using definition (122) gives Relation (123) follows after using (119). An immediate consequence of the previous Proposition is the following. Corollary 5.4. Let {H , γ , /lscript , /lscript (2) } be null metric hypersurface data ( Φ , ξ )-embedded in ( M , g ) and extend ξ off H arbitrarily. Let η be a vector field on ( M , g ) satisfying that η | Φ( H ) = αν for some function α ∈ F ( H ) . Then, for any integer m ≥ 1 As a consequence, Inserting (125) into (101) we arrive at the main result of this section, namely the generalized master equation of order m> 1, where recall that when ∇ ξ ξ = 0 the tensor O ( m ) depends only on metric hypersurface data and { Y , ..., Y ( m ) } (see Corollary 4.26). The tensor P ( m ) also depends on { Y , ..., Y ( m ) } and in addition on the vectors L ξ η, ..., L ( m +1) ξ η on Φ( H ). Its key property is that it vanishes when X η = 0 and L ( i ) ξ η H = 0 for all i = 2 , ..., m +1. Finally we prove an interesting property of the vector field L ξ η that will play a key role in the next section. Lemma 5.5. Let {H , γ , /lscript , /lscript (2) } be null metric hypersurface data (Φ , ξ ) -embedded in ( M , g ) and extend ξ off Φ( H ) by ∇ ξ ξ = 0 . Let η ∈ X ( M ) be such that η | Φ( H ) is null and tangent to Φ( H ) and such that its deformation tensor vanishes to all orders on Φ( H ) . If L ξ η H = hξ for some function h ∈ F ( H ) , then L ( k ) ξ η H = 0 for any integer k ≥ 2 . Proof. Let m ≥ 1. By Proposition 4.3 it follows Moreover, equation (51) and the fact that ∇ ξ ξ = 0 imply Combining (130) and (131) gives The definition of Σ[ η ] entails the general identity This together with ∇ ξ ξ = 0 gives Since L ξ η H = hξ and Σ[ η ] vanishes on Φ( H ), it follows L (2) ξ η H = 0. Applying L ( m ) ξ to (134) and using (51) and (130) gives We prove the statement by induction. Let m ≥ 1, assume L ( i ) ξ η H = 0 for all i = 2 , ..., m +1 and let us prove that L ( m +2) ξ η H = 0. Since Σ[ η ] ( k ) ( ξ, ξ ) H = 0 for all k ≥ 1 equation (135) on Φ( H ) reads where in the last equality we used (132). Remark 5.6. It is remarkable that Lemma 5.5 is insensitive to the extension of h off Φ( H ) . At first sight one could think that this is not possible and that counterexamples are easy to construct. For instance, assume that L ξ η has the form hξ not just on H but everywhere. Clearly the lemma can only hold true if ξ ( h ) = 0 . So, the conditions of the lemma must somehow ensure that this property holds. And indeed this can be proved, as we show next. Assume L ξ η = hξ everywhere, ∇ ξ ξ = 0 and that the deformation tensor of η vanishes. Then, equation (133) reads Since ξ is a rigging of Φ( H ) there exists a neighbourhood U ⊂ M of Φ( H ) in which ξ = 0 , and hence ξ ( h ) = 0 on U . /negationslash When the deformation tensor of η vanishes in a neighbourhood of Φ( H ) (and not only to infinite order on Φ( H )) and h is assumed to be constant along ξ , then in fact L ξ η = hξ holds in a neighbourhood of Φ( H ), as we prove next. Lemma 5.7. Let ( M , g ) admit a Killing vector η and consider an embedded hypersurface Φ : H ↪ → M with rigging ξ . Extend ξ off Φ( H ) by means of ∇ ξ ξ = 0 and assume L ξ η H = hξ with ξ ( h ) = 0 . Then, L ξ η = hξ in a neighbourhood of Φ( H ) . Proof. Define ζ := L ξ η -cξ . Equation (133) together with the fact that ξ is geodesic yields Since η is a Killing the last term vanishes and then we have a linear homogeneous transport equation for ζ . Since ζ H = 0 we conclude ζ = 0 in a neighbourhood of Φ( H ).",
"pages": [
23,
24,
25,
26,
27,
28
]
},
{
"title": "6 Application to non-degenerate Killing horizons",
"content": "In this section we study the case when ( M , g ) admits a Killing vector η (and therefore K [ η ] = L η g = 0). First, we review some well known properties of Killing horizons and particularize the identities of Section 5 to the present case. This will lead to a natural definition of 'abstract Killing horizon data' as well as its embedded counterpart. We will prove that, fixing the extension of the rigging to being geodesic, the transverse expansion at any non-degenerate Killing horizon is uniquely determined in terms of its abstract Killing horizon data and the ambient Ricci tensor to infinite order on the horizon. Moreover, when the Ricci tensor fulfills a so-called hierarchical dependence, the transverse expansion only depends on abstract Killing horizon data. Finally, we apply this result to characterize Λ-vacuum manifolds near non-degenerate horizons. /negationslash A Killing horizon of η is an embedded hypersurface Φ : H ↪ → M to which η is tangent, null and nowhere zero. We say that the Killing horizon is non-degenerate when its surface gravity satisfies ˜ κ = 0 at some point, and we say it is degenerate when ˜ κ = 0 everywhere. In the literature it is common to define the notion of 'non-degenerate' by the requirement that ˜ κ = 0 everywhere. We prefer the weaker definition above because then a Killing horizon is either degenerate or non-degenerate. In many relevant cases of interest (e.g. Λ-vacuum spacetimes) both definitions turn out to be equivalent. /negationslash A well-known property of Killing horizons is that they are totally geodesic, i.e. U = 0, which follows at once from Φ /star K [ η ] = 2 α U . As a consequence, given any X ∈ X ( H ), equations (26) and (27) give Moreover equations (40)-(43) simplify to while, using that the tensors (115) all vanish, equations (116)-(117) become In addition, equation (118) gives n ( κ ) = 0, so the surface gravity is constant along the null generator of the horizon. Hence, L n dκ = d ( n ( κ )) = 0 as well, so dκ is also Lie-constant along the null generators. For bifurcate horizons (141) implies dκ = 0 on the bifurcation surface (and hence everywhere), which recovers the well-known fact that κ is constant for bifurcate horizons [15]. Moreover, when H is geodesically complete and admits a section, dκ must vanish everywhere [33]. Particularizing (129) for m> 1 to the case U = 0 and K [ η ] = 0, In view of relation (136) it is useful to introduce the one-form which will play an important role below. Let us discuss some of its basic properties. Note first that for any tangent vector X ∈ X ( H ) it holds /negationslash This one-form extends the commonly used one-form /pi1 defined by ∇ Φ /star X η = /pi1 ( X ) η (see for instance [1, 12]). Indeed, at the points where α = 0 the one-form τ agrees with α /pi1 . Note however that τ is well-defined and smooth everywhere on H . Proposition 6.1. Let τ be as in (143) . Then, Proof. The gauge transformation of τ follows from (36) and Lemma 5.2. Using (108) and r ( n ) = -κ n the second property is immediate. For the third one we use that η is a Killing vector, so L η ∇ = 0 and thus 0 = L η ∇ X η -∇ X L η η -∇ L η X η for every X ∈ X ( M ). When X is tangent to H the vector L η X on H is also tangent (because η is tangent as well). Then, where we used L η ν = L αν ν = -ν ( α ) ν = -n ( α ) ν . Then L ¯ η τ = n ( α ) τ , so item 3. follows after using L ¯ η τ = L αn τ = α L n τ + τ ( n ) dα . /negationslash In order to construct the abstract notion of 'Killing horizon data' we want to find the minimum amount of data on the horizon that allows for the determination of the full transverse expansion. As we will see below, when κ = 0 everywhere, the required information involves the null metric hypersurface data, the one-form τ and the function α . The abstract conditions we need to incorporate into the definition must be such that, once the data is embedded, the corresponding horizon satisfies (i) Φ /star ( L η g ) = 0, (ii) the one-form τ satisfies item 3. of Proposition 6.1, (iii) the set of zeros of α has empty interior and (iv) that one-form α -1 ( τ -dα ) extends smoothly to all H . Item (iii) is necessary because Killing vectors that vanish on any hypersurface are necessarily identically zero. In terms of η this means that α vanishing on any open subset of H would only be compatible with η being identically vanishing. Item (iv) is necessary because for actual Killing horizons, the following equality holds (cf. (143)) and the right hand side is smooth everywhere on H . Note that the condition that the zeroes of α have empty interior means, in particular, that the extension of α -1 ( τ -dα ) is necessarily unique. Note also than when α has no zeroes, condition (iv) is automatically satisfied. As we explained at the beginning of this section, condition (i) gives U = 0. This discussion motivates the following definition. Definition 6.2. We say {H , γ , /lscript , /lscript (2) , τ , α } is abstract Killing horizon data (AKH data) provided that (i) {H , γ , /lscript , /lscript (2) } is null metric hypersurface data satisfying U = 0 , (ii) α is a smooth function such that the set { α = 0 } has empty interior and (iii) τ is a one-form such that α L n τ = n ( α ) τ -τ ( n ) dα and the one-form α -1 ( τ -dα ) extends smoothly to all H . Remark 6.3. It is worth comparing Definition 6.2 with the notions of abstract Killing horizons of order zero (AKH 0 ) and one (AKH 1 ) introduced in [24]. The main difference is that the definitions in [24] involve full hypersurface data (i.e. involve the tensor Y ) while the definition above makes no reference to Y . In fact, we want to use Definition 6.2 in combination with the field equations to construct Y in such a way that the data corresponds to a Killing horizon. This is why we have added the term 'data' in the definition of 'AKH data'. Next we extend the notion of gauge transformation to the context of AKH data motivated by Lemma 5.2 and Proposition 6.1. Definition 6.4. Let K = {H , γ , /lscript , /lscript (2) , τ , α } be AKH data and ( z, V ) gauge parameters. We define the gauge-transformed data by G ( z,V ) K := {H , γ ' , /lscript ' , /lscript (2) ' , z τ , zα } , where { γ ' , /lscript ' , /lscript (2) ' } are given by (14) -(16) . The condition U = 0 of Definition 6.2 only guarantees that the pullback of the deformation tensor vanishes on H . To capture the full information about the deformation tensor we need to restrict ourselves to the embedded case. Definition 6.5. Let K = {H , γ , /lscript , /lscript (2) , τ , α } be AKH data and define ¯ η := αn . We say that K is (Φ , ξ ) -embedded in ( M , g ) if (i) {H , γ , /lscript , /lscript (2) } is (Φ , ξ ) -embedded in ( M , g ) as in Def. 2.2 and (ii) ∇ Φ /star X Φ /star ¯ η = τ ( X ) ν for every X ∈ X ( H ) . Moreover, we say that K is an (Φ , ξ ) -embedded Killing horizon data (EKH data) if, additionally, (iii) there exist an extension η of Φ /star ¯ η such that its deformation tensor K [ η ] := L η g vanishes to all orders at Φ( H ) . Remark 6.6. It is easy to check that if K = {H , γ , /lscript , /lscript (2) , τ , α } is (Φ , ξ ) -embedded in ( M , g ) , then G ( z,V ) K is (Φ , ξ ' ) -embedded in ( M , g ) with ξ ' = z ( ξ +Φ /star V ) . Moreover, since the property of K [ η ] vanishing to all orders on Φ( H ) is independent of ξ | Φ( H ) and its extension off Φ( H ) , it follows that if K is (Φ , ξ ) -EKH data, then G ( z,V ) K is (Φ , ξ ' ) -EKH data. Remark 6.7. The definition of AKH data allows us to define a smooth one-form r := s -α -1 ( τ -dα ) and a scalar k n := α -1 ( τ ( n ) -n ( α )) . When the data happens to be embedded, the one-form r and the function k n agree with r and κ n , respectively. This is because from equation (136) and condition (ii) of Definition 6.5 it follows α ( r -r ) = 0 , and since the interior of the zeroes of α is empty, then r = r . Hence in the embedded case we shall not distinguish between r and r anymore. Observe also that this fact together with (108) imply that the surface gravity κ of the hypersurface is given by κ = n ( α ) + ακ n = τ ( n ) . Example 6.8. Consider the d -dimensional Schwarzschild-de Sitter spacetime ( M , g ) . In ingoing Eddington-Finkelstein coordinates { v, r } the metric g is where γ S d -2 is the d -2 dimensional spherical metric. When M and Λ are both positive, and M sufficiently small, the polynomial G ( r ) := -r d -3 g vv = r d -3 -2 M -Λ d -1 r d -1 admits precisely two positive roots r + 0 ≥ R := √ d -3 Λ ≥ r -0 . Since ∂ r g vv ∣ ∣ r = r ± 0 = -2 M ( d -3) ( r ± 0 ) d -2 + 2Λ d -1 r ± 0 = Λ r ± 0 -d -3 r ± 0 it follows that when r + 0 > R > r -0 the two null hypersurfaces H ± := { r = r ± 0 } are non-degenerate Killing horizons with Killing η = ∂ v . H + is called cosmological horizon, and H -is called event horizon. When r + 0 = r -0 = R , H + = H -is a degenerate Killing horizon. Let us compute the induced hypersurface data of any of the two horizons (we use r 0 to denote at once r + 0 and r -0 ). Choosing ξ = ∂ r as the rigging (observe ∇ ξ ξ = 0 ) it follows It follows at once that /lscript ( η ) = 1 , s = 1 2 d /lscript ( η, · ) = 0 , so by (143) with α = 1 /negationslash /negationslash and thus κ = τ ( η ) = ( d -3) M r d -2 0 -Λ r 0 d -1 = d -3 2 r 0 -Λ r 0 2 . When r 0 = R it follows κ = 0 . In this case K SdS := {H , γ , /lscript , /lscript (2) , τ , α = 1 } is EKH in Schwarzschild-de Sitter spacetime with κ = 0 everywhere. /negationslash In Proposition 6.11 we find necessary conditions for two AKH data to be embeddable in isometric semi-Riemannian manifolds. The idea is to use this information to then define a notion of isometry at the AKH data level. First we introduce some notation. Notation 6.9. Let K ' = {H ' , γ ' , /lscript ' , /lscript (2) ' , ω ' , α ' } be AKH data and χ : H -→ H ' a diffeomorphism. We define This notation will also be used for subsets of the data, e.g. χ /star { γ ' , /lscript ' , /lscript (2) ' } := { χ /star γ ' , χ /star /lscript ' , χ /star /lscript (2) ' } . Lemma 6.10. Let K ' = {H ' , γ ' , /lscript ' , /lscript (2) ' , ω ' , α ' } be AKH data and χ : H -→ H ' a diffeomorphism. Then χ /star K ' is AKH data. Proof. Firstly, from Definition 2.1 and the fact that χ is a diffeomorphism, it follows that {H , γ := χ /star γ ' , /lscript := χ /star /lscript ' , /lscript (2) := χ /star /lscript (2) ' } is metric hypersurface data. Moreover, from (3)(6) having a unique solution for { P, n, n (2) } given { γ , /lscript , /lscript (2) } , it follows that P = χ /star P ' , n = χ /star n ' and n (2) = χ /star n (2) ' . Thus, the causal character of { γ , /lscript , /lscript (2) } is the same as the one of { γ ' , /lscript ' , /lscript (2) ' } , and in particular if { γ ' , /lscript ' , /lscript (2) ' } is null, so it is { γ , /lscript , /lscript (2) } . Since χ /star L χ /star n γ ' = L n γ it also follows U = 0. Moreover, since χ is a diffeomorphism, α = χ /star α ' is smooth and has empty interior. Finally, and χ /star ( α '-1 ( τ ' -dα ' ) ) = α -1 ( τ -dα ) extends smoothly to all H . Hence {H , γ , /lscript , /lscript (2) , τ , α } is AKH data. Proposition 6.11. Let K = {H , γ , /lscript , /lscript (2) , ω , α } be AKH data (Φ , ξ ) -embedded in ( M , g ) and let K ' = {H ' , γ ' , /lscript ' , /lscript (2) ' , ω ' , α ' } be AKH data (Φ ' , ξ ' ) -embedded in ( M ' , g ' ) . Let ¯ η := αn and ¯ η ' := α ' n ' . Assume there exists an isometry ϕ : ( M , g ) -→ ( M ' , g ' ) such that ϕ (Φ( H )) = Φ ' ( H ' ) and ϕ /star Φ /star ¯ η = Φ ' /star ¯ η ' . Then, there exist a diffeomorphism χ : H -→ H ' and gauge parameters ( z, V ) such that Proof. First we prove that the vector field ϕ /star ξ ' is everywhere transverse to Φ( H ). Observe that since the vector field ν ' is null and normal to Φ ' ( H ' ) and ϕ is an isometry, the vector ϕ /star ν ' is also null and normal to Φ( H ), so in particular it is proportional to ν with non-zero proportionality factor. Then, from g ' ( ξ ' , ν ' ) = 1 it follows /negationslash This proves that ϕ /star ξ ' is everywhere transverse to Φ( H ), so there must exist a function z ∈ F /star ( H ) and a vector field V ∈ X ( H ) such that ϕ /star ξ ' = z ( ξ + Φ /star V ) . Since Φ( H ) and Φ ' ( H ' ) are diffeomorphic via ϕ and both Φ and Φ ' are embeddings, there exists a diffeomorphism χ making the following diagram commutative. Then, Hence χ /star { γ ' , /lscript ' , /lscript (2) ' } = G ( z,V ) { γ , /lscript , /lscript (2) } (see Def. 2.2), so χ /star ν ' = z -1 ν and χ /star α ' = zα because ϕ /star Φ /star ¯ η = Φ ' /star ¯ η ' . Finally, from condition (ii) in Definition 6.5 together with ϕ /star Φ /star ¯ η = Φ ' /star ¯ η ' and the fact that ϕ is an isometry, it follows ϕ /star ( ∇ ' Φ ' /star X ' Φ ' /star ¯ η ' ) = ∇ Φ /star χ /star X ' Φ /star ¯ η = ⇒ χ /star τ ' ⊗ χ /star ν ' = τ ⊗ ν ⇐⇒ χ /star τ ' = z τ . Taking into account Definition 6.4, (146) is established. This proposition motivates the following natural notion of isometry in the context of AKH data. Definition 6.12. Let K = {H , γ , /lscript , /lscript (2) , τ , α } and K ' = {H ' , γ ' , /lscript ' , /lscript (2) ' , τ ' , α ' } be two AKH data. We say K and K ' are isometric provided that there exists a diffeomorphism χ : H -→ H ' and gauge parameters ( z, V ) such that Lemma 6.10 guarantees that Definition 6.12 is well defined. /negationslash Our next aim is to show that given EKH data K satisfying τ ( n ) = 0 everywhere, the full asymptotic expansion { Y ( k ) } k ≥ 1 is uniquely determined in terms of K and the set { R ( m ) αβ } m ≥ 1 . As we shall see, to prove the uniqueness part of such statement we need to be able to extend the rigging vector ξ such that the tensor L ( m ) ξ R αβ is geometrical (in the sense of Definition 4.5) for every m ≥ 0. By identities (99), (100) and (142), the extension of ξ must be such that the tensors O ( m ) , O ( m ) a , O ( m ) ab and P ( m ) ab are H -geometrical for every m ≥ 0. In Section 4 we have proved that by extending ξ off Φ( H ) by means of ∇ ξ ξ = 0 the tensors O ( m ) , O ( m ) a and O ( m ) ab are H -geometrical. However, this is not sufficient to guarantee that P ( m ) ab is H -geometrical, because this tensor also depends on X η and L ( i ) ξ η ∣ ∣ H for i ≥ 2 (see the comment below equation (129)). A natural way to make this dependence disappear is to ensure that X η = 0 and L ( i ) ξ η H = 0 for every i ≥ 2. Our strategy is as follows. In Lemma 6.13 we show that given AKH data embedded on an ambient manifold with rigging ξ , one can always choose the gauge such that L ξ η is proportional to ξ on Φ( H ). With this choice X η automatically vanishes (cf. (102)). By combining this result with Lemma 5.5 we will be able to prove that L ( i ) η H = 0 for every i ≥ 2 as well. ξ Particularizing equation (120) to the Killing horizon case, namely K [ η ] = 0, In the following lemma we show that there exists a choice of gauge in which L ξ η H = ( ακ n -κ ) ξ . /negationslash Lemma 6.13. Let K = {H , γ , /lscript , /lscript (2) , τ , α } be AKH data (Φ , ξ ) -embedded in ( M , g ) satisfying τ ( n ) = 0 everywhere on H . Assume there exists an extension η of ¯ η := αn off Φ( H ) satisfying K [ η ]( ξ, · ) = 0 on Φ( H ) . Then there exists a family of gauges satisfying /lscript = κ -1 τ and /lscript (2) = 0 . Moreover, any element of the family satisfies and the whole family can be generated from any element by the action of the subgroup of transformations {G ( z, 0) } (i.e. this subgroup acts transitively on the family). Any element of this family will be said to be written in an ' η -gauge'. Proof. A vector V ∈ X ( H ) is defined uniquely in terms of the one-form w := γ ( V, · ) and the scalar function f := /lscript ( V ). From (6) and (3) it follows P ab w a w b = P ab γ ac γ bd V c V d = ( δ b c -n b /lscript c ) γ bd V c V d = γ cd V c V d , so in terms of f and w the gauge transformations of /lscript and /lscript (2) in (15)-(16) read From the transformations of τ and κ , namely τ ' = z τ and κ ' = κ , it is straightforward to check that by choosing w := κ -1 τ -/lscript and f := -/lscript (2) -1 2 κ -2 P ( τ , τ ), the gauge-transformed data satisfies (i) /lscript ' = κ -1 τ ' and (ii) /lscript (2) ' = 0. Moreover, by the transformations (150)-(151) and those of τ and κ , it is clear that any additional transformation G ( z ' ,V ' ) will preserve properties (i) and (ii) if and only if V ' = 0. Thus, the whole family is generated by applying G ( z, 0) (with z ∈ F /star ( H ) arbitrary) to any element of the family. To prove that in this class of gauges expression (148) simplifies to (149) it suffices to show that P ab ( 2 α s b + · ∇ b α ) = 0. Writing item 3. of Proposition 6.1 in terms of κ τ = /lscript gives Observe that the EKH data in Example 6.8 is written in an η -gauge. By combining Lemmas 5.5 and 6.13 and Remark 6.6 we arrive at the following. Using L n /lscript = 2 s (cf. (31)) it follows that the one-form 2 α s + dα is proportional to /lscript , and since /lscript (2) = 0 then P ab ( 2 α s b + · ∇ b α ) = 0 as a consequence of (5). Corollary 6.14. Let {H , γ , /lscript , /lscript (2) , τ , α } be (Φ , ξ ) -EKH in ( M , g ) . Then there exists a choice of ξ on Φ( H ) in which L ξ η H = ( ακ n -κ ) ξ . Such gauge is unique once n is fixed. Moreover, extending ξ off Φ( H ) by ∇ ξ ξ = 0 , then L ( k ) ξ η H = 0 for all k ≥ 2 . In particular, all the tensors P ( m ) in (124) vanish. /negationslash We are ready to show one of the main results of this section, namely that the full transverse expansion of an EKH data satisfying κ = 0 everywhere is uniquely determined in terms of AKH data and the collection { R ( m ) αβ } m ≥ 1 . /negationslash Theorem 6.15. Let K = {H , γ , /lscript , /lscript (2) , τ , α } be (Φ , ξ ) -EKH in any ( M , g ) . Assume τ ( n ) = 0 everywhere on H . Then, when the data is written in an η -gauge and ξ is extended geodesically, the full transverse expansion { Y ( m ) } m ≥ 1 only depends on K and { R ( m ) αβ } m ≥ 1 . Proof. Let us write the data in any of the η -gauges of Lemma 6.13, which fixes the vector ξ up to a multiplicative non-vanishing function ξ ↦-→ zξ , and extend ξ off Φ( H ) by means of ∇ ξ ξ = 0. We want to prove that the tensors { Y ( k ) } k ≥ 1 only depend on K and { R ( m ) αβ } m ≥ 1 , and thus they are insensitive to the particular manifold they are embedded in, provided their tensors R ( m ) αβ agree on Φ( H ). With this choice of gauge and extension of ξ the tensors O ( m ) , O ( m ) a and O ( m ) ab only depend on metric data and { Y , ..., Y ( m ) } (see Corollary 4.26) and the tensor P ( m ) vanishes for every m ≥ 1 (see Corollary 6.14). These two facts will be used repeatedly throughout the proof. By Remark 6.7 we can identify the one-form r := s -α -1 ( τ -dα ) with r := Y ( n, · ) and the scalar τ ( n ) with κ . Therefore, equation (140) can be rewritten as /negationslash where C ab is a tensor that only depends on AKH data { γ , /lscript , /lscript (2) , τ , α } . This proves that when κ = 0 everywhere on H , the full tensor Y is determined in terms of AKH data and the tensor R ab . Therefore, since in equations (99)-(100) for m = 0 the lower order terms only involve the tensor Y and metric data, it follows that the scalar tr P Y (2) and the one-form r (2) only depend on AKH data and the tensor R αβ on H . Hence, equation (142) for m = 1 reads /negationslash where C (2) ab only depends on AKH data and R αβ . When κ = 0 everywhere this shows that the tensor Y (2) ab is uniquely determined from AKH data and the tensors R αβ and R (2) ab on H . Iterating this process by means of equations (99), (100) and (142) one obtains the full transverse expansion { Y ( k ) } k ≥ 1 , and by Corollaries 6.14 and 4.26 this expansion only depends on AKH data and { R ( m ) αβ } m ≥ 1 , and not on the particular ( M , g ) where K is embedded. /negationslash This theorem shows that the asymptotic expansion of an EKH data satisfying κ = 0 everywhere only depends on the abstract data K and the tensors { R ( m ) αβ } m ≥ 1 , and thus it is insensitive to the particular ( M , g ) they may be embedded in. The collection { R ( m ) αβ } m ≥ 1 can be thought at least in two different ways. One possibility is to provide each R ( m ) αβ as prescribed data on the null hypersurface, e.g. by some external matter field. Another option is to provide R ( m ) αβ as a functional relation between the abstract data K and the transverse expansion { Y (1) , ..., Y ( m ) } . The simplest example of the second viewpoint is a d -dimensional manifold ( M , g ) satisfying the Λ-vacuum equations, where R ( m +1) αβ = λ K ( m ) αβ and λ = 2Λ d -2 . Then, R = λ γ , ˙ R = λ /lscript , R = λ/lscript (2) and R ( m +1) = 2 λ Y ( m ) , ˙ R ( m +1) = λ Φ /star ( K ( m ) ( ξ, · ) ) and R ( m +1) = λ Φ /star ( K ( m ) ( ξ, ξ ) ) for every m ≥ 1. Since by Remark 4.18 the tensor K ( m ) ( ξ, · ) on Φ( H ) depends on { Y (1) , ..., Y ( m -1) } , one concludes that the tensor R ( m +1) αβ on Φ( H ) depends at most on AKH data and { Y , ..., Y ( m ) } . In general, when the functional relations are such that each R ( m ) αβ depends on low enough transverse derivatives of the metric, i.e. such that the LHS in equations (99)-(101) depend on derivatives that we already have under control, the proof of Theorem 6.15 shows that the transverse expansion only depends on the abstract data. Let us make this property precise. Definition 6.16. Let K = {H , γ , /lscript , /lscript (2) , τ , α } be AKH data (Φ , ξ ) -embedded in ( M , g ) , and extend ξ off Φ( H ) by ∇ ξ ξ = 0 . We say that the Ricci tensor of g is hierarchical on Φ( H ) provided that R (1) only depends on K and, for every m ≥ 1 , When a Ricci tensor is hierarchical on Φ( H ) we shall refer to its particular dependence stated in (i) and (ii) by its 'hierarchical dependence'. Recall that when ξ is extended geodesically the tensor R ( m ) αβ is geometrical for every m ≥ 1 (cf. Proposition 4.14). As noted above, the canonical example is the Λ-vacuum equations, since in this case the tensors R ( m +1) , ˙ R ( m ) and R ( m ) only depend on AKH data and { Y (1) , ..., Y ( m ) } . In fact, R (1) ab = λγ ab , ˙ R a = λ/lscript a , R = λ/lscript (2) and R ( m +1) = 2 λ Y ( m ) , ˙ R ( m ) a = 0 and R ( m ) = 0 for every m ≥ 1 (see Corollary 4.8). An immediate consequence of the proof of Theorem 6.15 and Definition 6.16 is the following. /negationslash Theorem 6.17. Let K = {H , γ , /lscript , /lscript (2) , τ , α } be (Φ , ξ ) -EKH in any ( M , g ) . Assume τ ( n ) = 0 everywhere on H and that the field equations satisfied by g are such that its Ricci tensor is hierarchical on Φ( H ) . Then, when the data is written in an η -gauge and ξ is extended geodesically, the full transverse expansion { Y ( m ) } m ≥ 1 only depends on K . /negationslash This theorem generalizes the recent work [16] in several directions. The main theorem in [16] proves that for spacetimes admitting a non-degenerate Killing horizon H and being Ricci flat to infinite order at H , the full asymptotic expansion of the metric along certain privileged transverse vector at the horizon can be determined geometrically from a so-called 'non-degenerate Killing horizon data'. Theorem 6.17 is more general firstly because we are allowing zeroes of η on H (while Killing horizons by definition only include points where η is non-zero), secondly because our hierarchical property includes more field equations besides vacuum with vanishing Λ, and finally because we have extended the result to arbitrary signature (provided it admits degenerate hypersurfaces). We recover the result in [16] simply by imposing that R αβ is zero to all orders on the horizon. As an interesting corollary of Theorem 6.17 we extend the geometric uniqueness of the transverse expansion to the case of asymptotic Λ-vacuum spacetimes. /negationslash Corollary 6.18. Let K = {H , γ , /lscript , /lscript (2) , τ , α } be (Φ , ξ ) -EKH in any ( M , g ) satisfying the Λ -vacuum equations to infinite order on H . Assume τ ( n ) = 0 at least at one point p ∈ H and that K is written in an η -gauge and with ξ extended geodesically off Φ( H ) . Then the full transverse expansion { Y ( m ) } m ≥ 1 is uniquely determined in terms of K . /negationslash Proof. Since ( M , g ) satisfies the Λ-vacuum equations to infinite order on H it follows R ab n b = 0, and thus from (141) one has dκ = 0, so κ = τ ( n ) is constant on H . Since κ = 0 at least at one point and H is assumed to be connected, we conclude κ = 0 everywhere on H , and hence Theorem 6.17 applies. /negationslash /negationslash Let us perform a detailed comparison between our AKH data and the 'non-degenerate Killing horizon data' in [16], which is a triple ( H , σ , V ) where σ is a Riemannian metric, V is a nowhere vanishing Killing vector of ( H , σ ) with constant (non-zero) norm. In order to compare both objects at the same footing we restrict our AKH data to α = 0, γ semi-positive definite, τ ( n ) = 0 at some point and vacuum, which in particular implies τ ( n ) must be a nonzero constant (see the proof of Corollary 6.18). Let us see the equivalence between AKH data and 'non-degenerate Killing horizon data'. In one direction, given {H , γ , /lscript , /lscript (2) , τ , α } the vector V and the metric σ can be defined as follows: Clearly V is nowhere vanishing and σ ( V , V ) = ( τ ( n ) ) 2 is a non-vanishing constant, so σ is a gauge-invariant Riemannian metric on H (see Lemma 5.2 and Proposition 6.1). Moreover, since L V τ = n ( α ) τ , it follows Conversely, given a 'non-degenerate Killing horizon data' ( H , σ , V ) as in [16] we define Since /lscript ( V ) = 1 and γ ( V , · ) = 0 it follows at once that { γ , /lscript , /lscript (2) } is null metric hypersurface data and n = V . Moreover, from L V σ = 0 one gets U = 1 2 L n γ = 0 and α L n τ = 0 = n ( α ) τ -τ ( n ) dα . In addition τ ( n ) is a non-zero constant and the one-form α -1 ( τ -dα ) extends smoothly to all H (because α = 1). Hence {H , γ , /lscript , /lscript (2) , τ , α } fulfills the conditions of Definition 6.2 with α = 0 and τ ( n ) = const. = 0. /negationslash /negationslash /negationslash In [16] the choice of the rigging ξ is such that L ξ η = 0. This is possible only when α is nowhere vanishing. Indeed, when α = 0 one can exploit the freedom in Lemma 6.13 to set α = 1, and thus κ = κ n (cf. (108)) and L ξ η H = 0 (see (149)). Lemma 5.7 then shows that L ξ η = 0 whenever η is a Killing. However, if ¯ η admits zeroes on H this choice of gauge is not possible anymore because the properties α = 0 or α = 0 at a point are gauge invariant, so if α = 0 in some gauge it is impossible to make α = 1 by a gauge transformation. /negationslash /negationslash In Proposition 6.11 we proved that two AKH data K and K ' embedded in isometric manifolds ( M , g ) and ( M ' , g ' ) are necessarily isometric (in the sense of Definition 6.12). Our next aim is to prove a kind of converse, namely that isometric EKH data K and K ' both satisfying τ ( n ) = 0 imply that the respective manifolds they are embedded in are isometric to infinite order. The proof involves two steps. Firstly, we construct and map to each other suitable neighbourhoods of Φ( H ) and Φ ' ( H ' ), and secondly we prove that within an appropriate gauge and extension of ξ , the two asymptotic expansions agree. We accomplish the first task in the following proposition. Proposition 6.19. Let Φ : H ↪ → M and Φ ' : H ' ↪ → M ' be two embedded hypersurfaces in ambient manifolds ( M , g ) and ( M ' , g ' ) and let ξ , ξ ' be respectively riggings of Φ( H ) , Φ ' ( H ' ) extended geodesically. Assume that there exists a diffeomorphism χ : H -→ H ' . Then, there exist open neighbourhoods U ⊂ M and U ' ⊂ M ' of Φ( H ) and Φ ' ( H ' ) and a unique diffeomorphism Ψ : U -→ U ' satisfying Ψ /star ξ = ξ ' and Φ ' · χ = Ψ · Φ . Proof. Pick a neighbourhood U ⊂ M of Φ( H ) small enough so that the integral curves of ξ do not intersect each other and intersect Φ( H ) precisely once. Construct a neighbourhood U ' of Φ ' ( H ' ) similarly. Then, given a point q ∈ U there exist a unique p ∈ H such that the integral curve σ ( τ ) of ξ through p reaches q at a finite τ q , i.e. σ ( τ q ) = q . Now consider the point p ' := χ ( p ) ∈ H ' and the integral curve σ ' of ξ ' through p ' . We define Ψ( q ) := σ ' ( τ q ). It is clear that Ψ is a diffeomorphism when U and U ' are small enough, and by construction it satisfies Ψ /star ξ = ξ ' and Φ ' · χ = Ψ · Φ. Once we know how to identify neighbourhoods of Φ( H ) and Φ ' ( H ' ), we need to show that the two asymptotic expansions agree. The following theorem shows that a sufficient condition is that the field equations on ( M , g ) and ( M ' , g ' ) are such that their Ricci tensors satisfy the same hierarchical dependence on the horizons. /negationslash Theorem 6.20. Let K = {H , γ , /lscript , /lscript (2) , τ , α } and K ' = {H ' , γ ' , /lscript ' , /lscript (2) ' , τ ' , α ' } be two isometric EKH in respective ambient manifolds ( M , g ) and ( M ' , g ' ) and satisfying τ ( n ) = 0 everywhere on H . Assume the field equations on ( M , g ) and ( M ' , g ' ) are such that their Ricci tensors satisfy the same hierarchical dependence on Φ( H ) and Φ ' ( H ' ) , respectively. Let K ' be written in an η -gauge and K in the gauge in which K = χ /star K ' and extend the riggings geodesically. Then, there exist neighbourhoods U ⊂ M and U ' ⊂ M ' of Φ( H ) and Φ ' ( H ' ) and a diffeomorphism Ψ : U -→ U ' such that for every i ∈ N ∪ { 0 } . Proof. By Proposition 6.19 there exists a unique diffeomorphism Ψ : U -→ U ' satisfying Ψ /star ξ = ξ ' and Φ ' · χ = Ψ · Φ. Let us prove that Ψ /star L ( i ) ξ ' g ' H = L ( i ) ξ g for every i ≥ 0. The case i = 0 is immediate because γ = χ /star γ ' , /lscript = χ /star /lscript ' and /lscript (2) = χ /star /lscript (2) ' (see Definition 2.2). Proving the case i ≥ 1 amounts to show Ψ /star ( L ( i ) ξ ' g ' ) ( ξ ' , · ) = ( L ( i ) ξ g ) ( ξ, · ) and χ /star Y ( i ) ' = Y ( i ) . The former is a direct consequence of Corollary 4.8 because they both vanish. Let us prove the latter. Observe that since K ' is written in an η -gauge and K = χ /star K ' , then /lscript = χ /star /lscript ' = χ /star ( κ '-1 τ ' ) = κ -1 τ and /lscript (2) = χ /star /lscript (2) ' = 0, so K is also written in an η -gauge. Equation (140) and its primed version read where C ab and C ' ab only depend on AKH data (see Corollaries 6.14 and 4.26) and thus agree. Since the dependence of R ab on K is the same as the dependence of R ' ab on K ' , taking the pullback χ /star of the second equation and subtracting the first one gives χ /star Y ' = Y . A similar argument applied to (99)-(100) for m = 0 proves χ /star ( tr P ' Y (2) ' ) = tr P Y (2) and χ /star r (2) ' = r (2) . By iterating this process one shows that χ /star Y ( i ) ' = Y ( i ) for every i ≥ 1. An example of two manifolds satisfying the same hierarchical dependence is two spacetimes ( M , g ) and ( M ' , g ' ) such that R αβ = λg αβ and R ' αβ = λg ' αβ . This leads to the following immediate corollary. /negationslash Corollary 6.21. Let K = {H , γ , /lscript , /lscript (2) , τ , α } and K ' = {H ' , γ ' , /lscript ' , /lscript (2) ' , τ ' , α ' } be two isometric EKH in respective Λ -vacuum ambient manifolds ( M , g ) and ( M ' , g ' ) with the same cosmological constant. Assume τ ( n ) = 0 everywhere on H . Let K ' be written in an η -gauge and K in the gauge in which K = χ /star K ' and extend the riggings geodesically. Then, there exist neighbourhoods U ⊂ M and U ' ⊂ M ' of Φ( H ) and Φ ' ( H ' ) and a diffeomorphism Ψ : U -→ U ' such that for every i ∈ N ∪ { 0 } . A consequence of this corollary is that every analytic Λ-vacuum manifold admitting a nondegenerate Killing horizon (possibly with bifurcation surfaces) is characterized near the horizon by its AKH data. The question of whether an AKH data gives rise to a Λ-vacuum manifold to infinite order will be analyzed in an forthcoming paper. A direct application of Corollary 6.21 is that if a spacetime admits an EKH isometric to the non-extremal Schwarzschild-de Sitter data introduced in Example 6.8, then the spacetime is isometric to infinite order (i.e. (155) holds) to non-extremal Schwarzschild-de Sitter. Moreover, when the spacetime is real analytic it is necessarily isometric to Schwarzschild-de Sitter spacetime (at least in a neighbourhood of the horizon). This result complements the main result of [14], where the extremal case was considered. /negationslash /negationslash Proposition 6.22. Let ( M , g ) be a d ≥ 4 -dimensional spacetime satisfying the vacuum Einstein equations with cosmological constant Λ > 0 and admitting a non-degenerate Killing horizon Φ : H ↪ → M with spherical cross-sections. Let η be the corresponding Killing vector with surface gravity κ = 0 (necessarily constant). Assume Φ /star g = r 2 0 γ S d -2 for some 0 < r 0 = d -3 Λ and τ ∧ d τ = 0 , where τ is defined in (143) . Let M := 1 2 r d -3 0 -Λ 2( d -1) r d -1 0 . Moreover, if ( M , g ) is analytic, then it is isometric to Schwarzschild-de Sitter spacetime in a neighbourhood of its cosmological horizon (case 1.) or its event horizon (case 2.). /negationslash Proof. We denote all the objects referring to Schwarzschild-de Sitter described in Example 6.8 with the label 'SdS'. Let us scale the Killing vector η by the constant κ SdS /κ , where κ SdS := d -3 2 r 0 -Λ r 0 2 = 0. Then, the surface gravity of (the re-scaled) η is precisely κ SdS . Since on the horizon the Killing field is nowhere vanishing we can define n := η and thus α = 1. From item 3. of Proposition 6.1 it follows L n τ = 0. Let us pick any section S ⊂ H and solve the differential equation n ( u ) = 1 with initial condition u | S = 0. This gives a global function u ∈ F ( H ) that defines a foliation {S u } u ∈ R . Let us decompose the cotangent space T /star H by T /star H = span { du } ⊕ 〈 n 〉 ⊥ , where as usual 〈 n 〉 ⊥ is the set of covectors that annihilate n . Therefore the one-form τ can be uniquely decomposed as τ = Adu + b with A ∈ F ( H ) and b ∈ T /star H satisfying b ( n ) = 0. Condition τ ( n ) = κ gives A = κ , and condition τ ∧ d τ = 0 gives Let us define u ' := u + κ -1 B (hence τ = κdu ' ). Since b ( n ) = 0 then n ( B ) = 0, so n ( u ' ) = n ( u ) = 1. Consider the new foliation defined by u ' , {S u ' } u ' ∈ R . The remaining gauge freedom can be fixed by requiring ξ to be null and orthogonal to that foliation, which in terms of the abstract data implies /lscript (2) = 0 and /lscript = du ' . Then, the tuple Since L n τ = L n ( κdu + b ) = L n b = 0 then from the Cartan identity d b ( n, · ) = L n b -d ( b ( n )) = 0. Contracting the equation above with n yields κd b = 0. Since κ is a nonzero constant, the one-form b is closed, and hence exact because R × S d -2 is simply connected for d ≥ 4. Let B ∈ F ( H ) be such that b = dB , so τ = κd ( u + κ -1 B ) . /negationslash fulfills all the conditions of Definition 6.2, so K is AKH data satisfying τ ( n ) = κ = 0 everywhere on H . Moreover, K is written in an η -gauge. The last step is to construct the diffeomorphism χ : H -→ H SdS . Choose any isometry φ that maps the section { u ' = 0 } of H with the section { v = 0 } of H SdS . Consider any point p ∈ H and let σ ( u ' ) be the integral curve of n through p . This curve intersects S u ' =0 at a single point q . Now consider the integral curve σ SdS ( v ) of n SdS through φ ( q ). We define χ ( p ) := σ SdS ( u ' ( p )). That χ is a diffeomorphism follows from the product topology of H and H SdS and because n and n SdS are both smooth and globally defined. Since φ is an isometry and the integral curves of n and n SdS are identified it follows χ /star γ SdS = γ . With the definition M := 1 2 r d -3 0 -Λ 2( d -1) r d -1 0 it is clear that K is isometric (as in Definition 6.12) either to K + SdS or K -SdS . In order to distinguish the two cases recall that the equation 1 -2 M r 0 d -3 -Λ d -1 r 0 2 = 0 admits exactly two solutions r 0 + > √ d -3 Λ > r 0 -. Therefore, when r 2 0 > d -3 Λ then r 0 = r + 0 (and we are in the cosmological horizon case) and when r 2 0 < d -3 Λ then r 0 = r -0 (which is the event horizon case). The Proposition follows after using Corollary 6.21. The case r 2 0 = d -3 Λ is excluded in Proposition 6.22 because the horizon described in Example 6.8 is necessarily degenerate when r 2 0 = d -3 Λ . A result analogous to the one in Proposition 6.22 can be formulated with Λ = 0 and Schwarzschild spacetime, or Λ < 0 and Schwarzschild-anti de Sitter spacetime. It is worth emphasizing that, in contrast to the degenerate case treated in [14], it is not necessary to require neither Λ > 0 nor staticity nor compact cross sections for the uniqueness argument to work. The reason is that, for non-degenerate horizons, the equations that allow us to obtain recursively the asymptotic expansion are all algebraic. In the degenerate (also called extremal) case, once r ( m +1) has been replaced in (142) using (100), the leading order in identity (142) is Y ( m ) because κ = 0. The dependence between R ( m ) and Y ( m ) is not algebraic anymore, but via a partial differential equation on the cross-sections of H . In order for such equations to determine uniquely the expansion it becomes sufficient to require Λ > 0, staticity and that the cross-sections are maximally symmetric [14]. Remarkably, when the ( M , g ) in Proposition 6.22 is analytic it necessarily admits a static Killing vector, since Schwarzschild-de Sitter does. It would be interesting to relate this to the recent work [7] where it is shown that for bifurcate Killing horizons the condition η being static is equivalent to the torsion one-form being closed on the bifurcation surface.",
"pages": [
28,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38
]
},
{
"title": "Acknowledgements",
"content": "This work has been supported by Projects PID2021-122938NB-I00 (Spanish Ministerio de Ciencia e Innovaci'on and FEDER 'A way of making Europe') and SA096P20 (JCyL). G. S'anchezP'erez also acknowledges support of the PhD. grant FPU20/03751 from Spanish Ministerio de Universidades.",
"pages": [
39
]
},
{
"title": "A Some pullbacks into a null hypersurface",
"content": "In this appendix we compute the pullback of several derivatives of ambient tensors into a null hypersurface that we shall need in the main text of this paper. Recall that given a (0 , p ) tensor field T α 1 ··· α p on M we use the standard notation T a 1 ··· a p to denote the pullback of T to H . Moreover, we also use the notation ( i,j ) T a 1 ··· a p as the pullback to H of the contraction of T with ξ in the i-th and j-th slots. Proposition A.1. Let ( M , g ) be a semi-Riemannian manifold and Φ : H ↪ →M a smooth null hypersurface with rigging ξ . Let T be a (0 , p ) -tensor on M . Then, where V b a := P cb Π ac + 1 2 ( d/lscript (2) ) a n b . Proof. Let { e a } be a local basis of H and ̂ e a := Φ /star ( e a ). To prove the first identity we contract ∇ β T α 1 ··· α p with ̂ e β b ̂ e α 1 a 1 · · · ̂ e α p a p and use (26), namely For the second one we use the relation between L ξ and ∇ contract it with ̂ e α 1 a 1 · · · ̂ e α p a p and employ (35). Finally, to prove the last one just contract the relation with e α 1 a 1 · · · e α p a p and use (156) and (35). ̂ ̂ Proposition A.2. Let ( M , g ) be a semi-Riemannian manifold and Φ : H ↪ → M a smooth null hypersurface with rigging ξ . Let T be a (0 , p + 1) -tensor on M and denote by div T the p -covariant tensor defined by (div T ) α 1 ··· α p := g µν ∇ µ T να 1 ··· α p . Then, Proof. From (13), Using (156), the first term is From (157), V c b n b = P ac (r + s) a + 1 2 n ( /lscript (2) ) n c and r ( n ) = -κ n , s ( n ) = 0 the second term becomes Finally, using (158) and U ( n, · ) = 0 the last term is Combining the three the result follows.",
"pages": [
39,
40
]
}
]
2024arXiv240601770D
https://arxiv.org/pdf/2406.01770.pdf
<document>
<section_header_level_1><location><page_1><loc_36><loc_88><loc_60><loc_90></location>DOCTORAL THESIS</section_header_level_1>
<section_header_level_1><location><page_1><loc_19><loc_74><loc_77><loc_81></location>Thermodynamical Aspects of Some Cosmological Models</section_header_level_1>
<text><location><page_1><loc_39><loc_58><loc_57><loc_65></location>By Tanima Duary Roll No.: 14IP021</text>
<text><location><page_1><loc_30><loc_54><loc_66><loc_56></location>Supervisor: Prof. Narayan Banerjee</text>
<text><location><page_1><loc_31><loc_51><loc_65><loc_52></location>Department of Physical Sciences</text>
<text><location><page_1><loc_17><loc_48><loc_80><loc_50></location>Indian Institute of Science Education and Research Kolkata</text>
<figure>
<location><page_1><loc_39><loc_26><loc_57><loc_41></location>
</figure>
<text><location><page_1><loc_15><loc_15><loc_81><loc_21></location>A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Physical Sciences at Indian Institute of Science Education and Research Kolkata</text>
<text><location><page_1><loc_42><loc_10><loc_54><loc_12></location>April, 2024</text>
<section_header_level_1><location><page_3><loc_14><loc_81><loc_65><loc_84></location>Declaration by the student</section_header_level_1>
<text><location><page_3><loc_14><loc_63><loc_82><loc_79></location>I, Ms. Tanima Duary , Registration No. 14IP021 dated 24th July 2014 , a student of the Department of Physical Sciences of the Integrated PhD Programme of Indian Institute of Science Education and Research Kolkata (IISER Kolkata), hereby declare that this thesis is my own work and, to the best of my knowledge, it neither contains materials previously published or written by any other person, nor has it been submitted for any degree/diploma or any other academic award anywhere before. I have used the originality checking service to prevent inappropriate copying.</text>
<text><location><page_3><loc_14><loc_53><loc_82><loc_60></location>I also declare that all copyrighted material incorporated into this thesis is in compliance with the Indian Copyright Act, 1957 (amended in 2012) and that I have received written permission from the copyright owners for my use of their work.</text>
<text><location><page_3><loc_14><loc_46><loc_82><loc_50></location>I hereby grant permission to IISER Kolkata to store the thesis in a database which can be accessed by others.</text>
<text><location><page_3><loc_14><loc_36><loc_19><loc_37></location>Date:</text>
<text><location><page_3><loc_69><loc_29><loc_82><loc_30></location>Tanima Duary</text>
<text><location><page_3><loc_30><loc_22><loc_82><loc_28></location>Department of Physical Sciences Indian Institute of Science Education and Research Kolkata Mohanpur 741246, West Bengal, India</text>
<section_header_level_1><location><page_5><loc_14><loc_71><loc_73><loc_74></location>Certificate from the Supervisor</section_header_level_1>
<text><location><page_5><loc_14><loc_51><loc_82><loc_70></location>This is to certify that the thesis entitled 'Thermodynamical Aspects of Some Cosmological Models' submitted by Ms. Tanima Duary , Registration No. 14IP021 dated 24th July 2014 , a student of the Department of Physical Sciences of the Integrated PhD Programme of IISER Kolkata, is based upon her own research work under my supervision. I also certify, to the best of my knowledge, that neither the thesis nor any part of it has been submitted for any degree/diploma or any other academic award anywhere before. In my opinion, the thesis fulfils the requirement for the award of the degree of Doctor of Philosophy.</text>
<text><location><page_5><loc_14><loc_37><loc_19><loc_39></location>Date:</text>
<text><location><page_5><loc_61><loc_30><loc_82><loc_32></location>Prof. Narayan Banerjee</text>
<text><location><page_5><loc_30><loc_22><loc_82><loc_29></location>Professor Department of Physical Sciences Indian Institute of Science Education and Research Kolkata Mohanpur 741246, West Bengal, India</text>
<section_header_level_1><location><page_7><loc_14><loc_87><loc_52><loc_90></location>Acknowledgements</section_header_level_1>
<text><location><page_7><loc_14><loc_76><loc_82><loc_86></location>I extend my heartfelt gratitude to all those who played important roles in making this journey possible. Without their unwavering support and encouragement, the completion of this thesis would not have been possible. I want to take this moment to express my deep appreciation for their contributions.</text>
<text><location><page_7><loc_14><loc_42><loc_82><loc_75></location>First and foremost, I wish to express my sincere and deepest gratitude to my supervisor, Prof. Narayan Banerjee, for being a guiding light throughout mydoctoral journey. He charted the course of my research with his visionary guidance and profound wisdom. I am truly indebted to him for his boundless patience and efforts, which have not only made this thesis possible but also enriched my understanding of the subject. I express my gratitude to him for maintaining faith in my abilities even during times when I doubted myself. Having the privilege of working under his guidance has been a rare and invaluable opportunity. I am grateful for the incredible teachings, enlightening discussions, and invaluable advice that have consistently motivated me on this scientific expedition. I am thankful for the time and effort he dedicated to reviewing drafts, providing constructive feedback, and offering valuable suggestions that significantly enhanced the overall quality of this thesis. I extend my heartfelt thanks to him for providing guidance and support when I felt directionless, helping me discover the strength to overcome challenges.</text>
<text><location><page_7><loc_14><loc_22><loc_82><loc_41></location>I am profoundly grateful to Dr. Ananda Dasgupta, whose exceptional teaching prowess and constant support have left an indelible mark on my academic journey. His teaching style, characterized by clarity, enthusiasm, and a genuine passion for the subject matter, has made learning not just a scholarly pursuit but also an enriching experience. His wisdom and insight have been a source of inspiration, instilling in me a resilience that extends beyond academic pursuits. His impact on my education and personal growth is immeasurable, and I am privileged to have had the opportunity to learn under his tutelage.</text>
<text><location><page_7><loc_14><loc_9><loc_82><loc_21></location>I extend my gratitude to the members of my research progress committee, Dr. Golam Mortuza Hossain and Prof. Rajesh Kumble Nayak, for their valuable insights and recommendations. I express my gratefulness to Dr. Koushik Dutta for his encouragement and I am thankful for providing me with the opportunity to serve as a volunteer at the 32nd IAGRG conference in 2022. This experience has been enriching. I am thankful to all the faculty</text>
<text><location><page_8><loc_18><loc_88><loc_72><loc_90></location>members of Department of Physical Sciences, IISER Kolkata.</text>
<text><location><page_8><loc_18><loc_72><loc_86><loc_87></location>I would like to express my heartfelt thanks to Sangita di, Munna da, and Ipsita di from the departmental office for their invaluable assistance with all official matters. My sincere appreciation goes to the librarian and Assistant Librarian at IISER-K Library for their cooperative support. Gratitude also to the individuals from DOAA, DOSA, DORD, CCC, SAC, and the Medical unit for their assistance. I would like to specially thank Ms. Saberi Roy Choudhury and Mr. Arun Dutta for helping us in CSIR fellowship matters.</text>
<text><location><page_8><loc_18><loc_53><loc_86><loc_71></location>In the close-knit community of IISER Kolkata, Shibendu, Prashanti, Purba, Shreya, Basabendra, and Sourav have been more than friends; they are cherished companions who have shared in the joys and difficulties of academic pursuits. Their friendship has added richness to my experience at IISER Kolkata, and I am truly fortunate to have them as the closest allies. With deep reverence, I extend my regards and profound gratitude to my beloved friend, the late Subhadip Roy. Each passing day is a poignant reminder of his absence, and not a moment goes by when I do not miss the warmth of his friendship.</text>
<text><location><page_8><loc_18><loc_28><loc_86><loc_51></location>I would like to thank my seniors- Shantanu da, Subhajit da, Chiranjeeb da, Ankan da, Soumya da, Sachin da, Avijit da, Srijita di and Anushree di for the precious pieces of advice they generously shared. I would also like to thank Sampurna, Debraj, Medha, Siddhartha, Abhirup, Toushik da, Priyanka di, Diganta, Budhaditya, Poulomi, Saikat, Arkayan, Amulya, Chiranjit, Swarup, Brotoraj, Soumya, Debajyoti, Narayan, Arnab, Samit, Roshan, Kakali, Madhura, Soumi, Ananya, Fareeha, Debanajana and Lucky for making this journey memorable. I wish to thank Branali, Rajrupa, Poulami, Beetihotra, Navonil, Sayantan, and Siddharth. It was fun working with you side by side at the IAGRG conference. I am thankful to all my friends from Scottish Church College.</text>
<text><location><page_8><loc_18><loc_21><loc_86><loc_27></location>I am grateful to my grandparents (dadu, dida), maternal uncle (mama), aunty (mami), sister Sushree, and little brother Sashreek, for the affection they have bestowed upon me.</text>
<text><location><page_8><loc_18><loc_10><loc_86><loc_20></location>I express my deepest gratitude to my parents, whose love, encouragement, and support which has been a steadfast anchor during both the peaks and valleys of my journey. Their encouragement and presence have been invaluable, providing solace in challenging times and magnifying the joy in moments of success. Their sacrifices, guidance, and belief in my potential</text>
<text><location><page_9><loc_14><loc_67><loc_82><loc_90></location>have shaped the person I am today. I am truly blessed to have such caring and supportive parents by my side. I want to extend my heartfelt acknowledgment to my little brother, Moni, whose presence and enthusiasm have brought joy and inspiration to my life. His infectious energy, and genuine camaraderie have been a constant source of encouragement throughout my journey. I am grateful for the special bond we share, and I appreciate the positive impact he has had on my life. I express my gratitude to my dearest friend, Aritra, for being by my side through thick and thin. In the symphony of life, grateful for the serendipitous notes that brought him into the melody of my journey. Across the tapestry of time and the vastness of space, we share an epoch and a planet, a harmonious rhyme.</text>
<text><location><page_9><loc_14><loc_60><loc_82><loc_65></location>I extend my sincere gratitude to the Council of Scientific and Industrial Research, India, for granting financial assistance through the CSIR-NET fellowship (Award No. 09/921(0171)/2017-EMR-I).</text>
<text><location><page_9><loc_14><loc_53><loc_82><loc_58></location>I would like to express my appreciation for the enchanting campus of IISER Kolkata, characterized by its captivating beauty, vibrant sunset skies, and picturesque walking paths.</text>
<text><location><page_11><loc_19><loc_59><loc_78><loc_62></location>To Moni and my adoring parents...</text>
<section_header_level_1><location><page_13><loc_14><loc_82><loc_30><loc_85></location>Abstract</section_header_level_1>
<text><location><page_13><loc_14><loc_51><loc_82><loc_80></location>This thesis is focused on the thermodynamic analysis of cosmological models, specially the models that explain late-time cosmic acceleration. The cosmological principle says that the universe exhibits spatial homogeneity and isotropy. To describe it we consider the Friedmann-Lemaître-RobertsonWalker (FLRW) metric. A thorough evaluation of the feasibility of the models was conducted through the application of the Generalized Second Law (GSL). This law says that the overall entropy, i.e., the sum of the entropy of the horizon and the fluid enclosed within the horizon, should never decrease. Considering the dynamic nature of the universe, our methodology focused on the apparent horizon, instead of the event horizon. Within this framework, we have considered a condition of thermodynamic equilibrium between the apparent horizon and the fluid contained within it. In this state of equilibrium, we have considered the Hayward-Kodama temperature as the temperature associated with the apparent horizon.</text>
<text><location><page_13><loc_14><loc_34><loc_82><loc_48></location>The first chapter contains concise overview of cosmology. Chapter 2 goes deeper into the thermodynamics applied to cosmology. It focuses more on the Generalized Second Law of Thermodynamics and explains it in more detail. Furthermore, we go into extensive details regarding Hayward-Kodama temperature. This chapter also contains a detail discussion about apparent horizon. The conditions required for thermodynamic stability has been discussed in this chapter.</text>
<text><location><page_13><loc_14><loc_8><loc_82><loc_31></location>In chapter 3, we conduct a thermodynamic comparison between quintessence models involving thawing and freezing scenarios. We have considered an ansatz on the energy density of the scalar field, which is picked up from the literature. The motivation for picking the ansatz was that, by choosing values of just one parameter, we can get either thawing or freezing behaviour. Both of these models are observed to violate the Generalized Second Law of Thermodynamics. Nevertheless, in the case of freezing models, there is still a possible way to resolve this, as this violation occurs in the distant past, deep within the radiation-dominated era, a period where a conventional scalar field model combined with pressureless matter is not an accurate representation of the matter content. In contrast, the</text>
<text><location><page_14><loc_18><loc_84><loc_86><loc_90></location>thawing model exhibits a violation of GSL, manifesting as a finite future breakdown. Therefore, we conclude that the freezing models are favoured compared to the thawing ones on the considerations of GSL viability.</text>
<text><location><page_14><loc_18><loc_67><loc_86><loc_81></location>In chapter 4, we scrutinize Brans-Dicke cosmological models within the context of a spatially isotropic and homogeneous universe, evaluating their compatibility with the GSL. Our investigation is carried out within the Einstein frame. We find that in dust era, these models exhibit thermodynamic feasibility when the Brans-Dicke parameter ω assumes negative values. This range has strong alignment with the range that is required for the recent observations of the cosmic acceleration.</text>
<text><location><page_14><loc_18><loc_39><loc_86><loc_64></location>In chapter 5, we explore the thermodynamic viability of a selection of dark energy models, which have been reconstructed using the cosmological jerk parameter. Our investigation involves the adoption of models previously documented in the literature. These models are categorized into two groups based on the presence or absence of interactions in the dark sector. We employ the GSL as a diagnostic tool for our analysis. In an attempt to capture the dynamic nature of spacetime, we replace the Hawking temperature with the Hayward-Kodama temperature. Our results indicate that, dependent on the chosen parametrization ansatz for jerk, the total entropy exhibits a time-increasing trend. This suggests the potential existence of viable models within this framework. This trend persists even when there is interaction in the dark sector.</text>
<text><location><page_14><loc_18><loc_11><loc_86><loc_36></location>In chapter 6, we have considered a model in spatially flat FRW spacetime, that mimics the characteristics of Λ CDM model and checked the thermodynamic stability. In this chapter also we have utilized the Hayward-Kodama temperature as the temperature of the apparent horizon. Assuming the thermal equilibrium between the apparent horizon and the fluid inside the horizon, we investigated the thermodynamic stability of the matter composition within the universe and found out that it lacks the thermodynamic stability. We found out an interesting result while calculating the heat capacity at constant volume ( C V ). It is shown that the transition from the decelerated to the accelerated cosmic expansion is a second-order thermodynamic phase transition, while the deceleration parameter q serves as the order parameter.</text>
<text><location><page_15><loc_14><loc_84><loc_82><loc_90></location>In chapter 7, we reach the epilogue of this thesis, where we not only provide our final conclusions but also briefly discussed the aspects of the work presented in this dissertation and future prospects.</text>
<section_header_level_1><location><page_17><loc_14><loc_74><loc_28><loc_77></location>Preface</section_header_level_1>
<text><location><page_17><loc_14><loc_58><loc_82><loc_73></location>The research presented in this dissertation was conducted at the Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata, India. The initial chapter encapsulates a preamble to the realm of cosmology, centering upon the different models to explain cosmic acceleration. The next chapter (chapter 2) focuses on the intricate interrelation linking cosmology and the principles of thermodynamics. The succeeding chapters are grounded on the subsequent scholarly articles:</text>
<unordered_list>
<list_item><location><page_17><loc_17><loc_53><loc_82><loc_56></location>· Chapter 3 : Thermodynamics of Thawing and Freezing Quintessence Models</list_item>
<list_item><location><page_17><loc_19><loc_46><loc_82><loc_52></location>Tanima Duary , Ananda Dasgupta and Narayan Banerjee , 'Thawing and Freezing Quintessence Models: A thermodynamic Consideration" , Eur.Phys.J.C 79 (2019) 11, 888, arXiv:1906.10408.</list_item>
<list_item><location><page_17><loc_17><loc_39><loc_82><loc_44></location>· Chapter 4 : Thermodynamics of Brans-Dicke Cosmology Tanima Duary , Narayan Banerjee, Brans-Dicke cosmology: thermodynamic viability , Eur.Phys.J.Plus 135 (2020) 1, 4, arXiv:1910.10931.</list_item>
<list_item><location><page_17><loc_17><loc_33><loc_82><loc_37></location>· Chapter 5 : Thermodynamic Analysis of Cosmological models reconstructed from jerk parameter</list_item>
<list_item><location><page_17><loc_19><loc_29><loc_82><loc_33></location>Tanima Duary , Narayan Banerjee, Cosmological models reconstructed from jerk: A thermodynamic analysis , New Astron. 92 (2022) 101726.</list_item>
<list_item><location><page_17><loc_17><loc_24><loc_82><loc_27></location>· Chapter 6 : A Possible Thermodynamic Phase Transition: Signature flip of the Deceleration Parameter</list_item>
<list_item><location><page_17><loc_19><loc_17><loc_82><loc_23></location>Tanima Duary , Narayan Banerjee and Ananda Dasgupta Signature flip in deceleration parameter: A thermodynamic phase transition? , Eur.Phys.J.C 83 (2023) 9, 815, arXiv:2303.14031v2.</list_item>
</unordered_list>
<section_header_level_1><location><page_19><loc_14><loc_81><loc_31><loc_84></location>Contents</section_header_level_1>
<table>
<location><page_19><loc_14><loc_11><loc_82><loc_75></location>
</table>
<table>
<location><page_20><loc_18><loc_11><loc_86><loc_90></location>
</table>
<text><location><page_20><loc_18><loc_10><loc_19><loc_11></location>7</text>
<text><location><page_20><loc_21><loc_10><loc_45><loc_11></location>Conclusions and Outlook</text>
<text><location><page_21><loc_79><loc_88><loc_82><loc_90></location>119</text>
<section_header_level_1><location><page_23><loc_14><loc_82><loc_47><loc_85></location>List of Acronyms</section_header_level_1>
<text><location><page_23><loc_19><loc_74><loc_26><loc_76></location>Λ CDM</text>
<text><location><page_23><loc_34><loc_74><loc_58><loc_76></location>L ambda C old D ark M atter</text>
<text><location><page_23><loc_19><loc_72><loc_22><loc_73></location>BD</text>
<text><location><page_23><loc_34><loc_72><loc_45><loc_73></location>B rans- D icke</text>
<text><location><page_23><loc_19><loc_69><loc_25><loc_71></location>CMBR</text>
<text><location><page_23><loc_34><loc_69><loc_72><loc_71></location>C osmic M icrowave B ackground R adiation</text>
<text><location><page_23><loc_19><loc_67><loc_23><loc_68></location>FRW</text>
<text><location><page_23><loc_34><loc_67><loc_61><loc_68></location>F riedmann- R obertson- W alker</text>
<text><location><page_23><loc_19><loc_65><loc_23><loc_66></location>GTR</text>
<text><location><page_23><loc_34><loc_65><loc_60><loc_66></location>G eneral T heory of R elativity</text>
<text><location><page_23><loc_19><loc_62><loc_24><loc_64></location>GSLT</text>
<text><location><page_23><loc_34><loc_62><loc_75><loc_64></location>G eneralised S econd L aw of T hermodynamics</text>
<text><location><page_23><loc_19><loc_60><loc_31><loc_61></location>KGequation</text>
<text><location><page_23><loc_34><loc_60><loc_55><loc_61></location>K lein G ordon equation</text>
<text><location><page_23><loc_19><loc_57><loc_32><loc_59></location>EoS parameter</text>
<text><location><page_23><loc_34><loc_57><loc_59><loc_59></location>E quation o f S tate parameter</text>
<text><location><page_23><loc_19><loc_55><loc_23><loc_56></location>BDT</text>
<text><location><page_23><loc_34><loc_55><loc_52><loc_56></location>B rans- D icke T heory</text>
<text><location><page_23><loc_19><loc_53><loc_28><loc_54></location>NMCSTT</text>
<text><location><page_23><loc_34><loc_53><loc_77><loc_54></location>N on- M inimally C oupled S calar- T ensor T heories</text>
<text><location><page_23><loc_19><loc_50><loc_23><loc_52></location>DDE</text>
<text><location><page_23><loc_34><loc_50><loc_56><loc_52></location>Dy namical D ark E nergy</text>
<text><location><page_23><loc_19><loc_48><loc_29><loc_49></location>CPL model</text>
<text><location><page_23><loc_34><loc_48><loc_66><loc_49></location>C hevallier - P olarski- L inder model.</text>
<text><location><page_23><loc_19><loc_46><loc_24><loc_47></location>CMB</text>
<text><location><page_23><loc_34><loc_46><loc_63><loc_47></location>C osmic M icrowave B ackground.</text>
<text><location><page_23><loc_19><loc_43><loc_23><loc_45></location>HDE</text>
<text><location><page_23><loc_34><loc_43><loc_57><loc_45></location>H olographic D ark E nergy.</text>
<section_header_level_1><location><page_25><loc_14><loc_81><loc_42><loc_84></location>List of Figures</section_header_level_1>
<table>
<location><page_25><loc_17><loc_9><loc_82><loc_75></location>
</table>
<section_header_level_1><location><page_26><loc_18><loc_93><loc_22><loc_94></location>xxvi</section_header_level_1>
<table>
<location><page_26><loc_21><loc_82><loc_86><loc_90></location>
</table>
<section_header_level_1><location><page_27><loc_14><loc_62><loc_51><loc_65></location>List of Publications</section_header_level_1>
<section_header_level_1><location><page_27><loc_14><loc_57><loc_55><loc_58></location>This thesis is based on the following works:</section_header_level_1>
<unordered_list>
<list_item><location><page_27><loc_14><loc_49><loc_82><loc_55></location>1. Tanima Duary , Ananda Dasgupta and Narayan Banerjee , Thawing and Freezing Quintessence Models: A thermodynamic Consideration , Eur.Phys.J.C 79 (2019) 11, 888, arXiv:1906.10408.</list_item>
<list_item><location><page_27><loc_14><loc_44><loc_82><loc_47></location>2. Tanima Duary , Narayan Banerjee, Brans-Dicke cosmology: thermodynamic viability , Eur.Phys.J.Plus 135 (2020) 1, 4, arXiv:1910.10931.</list_item>
<list_item><location><page_27><loc_14><loc_38><loc_82><loc_42></location>3. Tanima Duary , Narayan Banerjee, Cosmological models reconstructed from jerk: A thermodynamic analysis , New Astron. 92 (2022) 101726.</list_item>
<list_item><location><page_27><loc_14><loc_31><loc_82><loc_36></location>4. Tanima Duary , Narayan Banerjee and Ananda Dasgupta Signature flip in deceleration parameter: A thermodynamic phase transition? , Eur.Phys.J.C 83 (2023) 9, 815, arXiv:2303.14031v2.</list_item>
</unordered_list>
<section_header_level_1><location><page_29><loc_14><loc_82><loc_30><loc_84></location>Chapter 1</section_header_level_1>
<section_header_level_1><location><page_29><loc_14><loc_75><loc_38><loc_78></location>Introduction</section_header_level_1>
<section_header_level_1><location><page_29><loc_14><loc_67><loc_46><loc_69></location>1.1 Concise Overview:</section_header_level_1>
<section_header_level_1><location><page_29><loc_14><loc_63><loc_57><loc_65></location>1.1.1 Brief chronological background:</section_header_level_1>
<text><location><page_29><loc_14><loc_39><loc_82><loc_61></location>Einstein's General Theory of Relativity is a revolutionary framework in physics that transformed our understanding of gravity and hence, the evolution of the universe. Proposed by Albert Einstein in 1915 [1], it represents one of the most significant intellectual achievements in human history. At its core, the theory suggests that gravity is not a force as traditionally understood but rather a curvature in the fabric of spacetime caused by the presence of mass and energy. According to General Relativity, massive objects such as stars and planets distort the geometry of spacetime, causing objects to follow curved paths. The theory provides a new mathematical description of gravity, utilizing a set of equations that relate the distribution of matter and energy to the curvature of spacetime.</text>
<text><location><page_29><loc_14><loc_10><loc_82><loc_37></location>General Theory of Relativity provides a theoretical framework for examining the structure and dynamics of the entire universe, and Einstein devised this framework in 1917 [2]. At that time, physicists believed the cosmos was stationary and not evolving, a celestial clockwork mechanism that would run forever. However, this prevailing view was refuted by Einstein's equations in general relativity, which indicated that the universe could either expand or contract. To account for this, Einstein introduced a modification to his equations by including a cosmological constant ( Λ ), which acted as a repulsive force to counterbalance gravity and maintain a stable universe. Willem de Sitter proposed the maximally symmetric vacuum solution of Einstein's field equations with a positive cosmological constant Λ [3, 4]. Karl Schwarzschild derived a solution to Einstein's equations that described the spacetime curvature around a massive object, such as a star or a black hole [5]. This solution</text>
<text><location><page_30><loc_18><loc_82><loc_86><loc_90></location>laid the foundation for our understanding of gravitational collapse and singularities. In 1919, Arthur Eddington led an expedition to observe a solar eclipse and confirmed Einstein's prediction that light rays would bend in the presence of massive objects [6].</text>
<text><location><page_30><loc_18><loc_62><loc_86><loc_80></location>In 1922, Alexander Friedmann introduced the idea of an expanding universe in his paper "On the Curvature of Space" [7]. He derived a set of mathematical solutions to Einstein's equations that described a dynamic cosmos, where the matter was in motion and the universe itself was evolving over time. Friedmann's work laid the groundwork for modern cosmology by challenging the prevailing belief in a static universe and introducing the concept of cosmic expansion. It provided a theoretical basis for the subsequent development of the Big Bang model and set the stage for further investigations into the nature, origin, and evolution of the universe.</text>
<text><location><page_30><loc_18><loc_38><loc_86><loc_60></location>In 1929, Edwin Hubble made a significant discovery that revolutionized our understanding of the universe. He studied distant galaxies and their light spectra, which contained characteristic features called spectral lines [8]. These lines could be used to determine the motion of an object. By examining the spectra of galaxies, Hubble noticed a peculiar pattern: the spectral lines were consistently shifted towards longer wavelengths, known as a redshift. Based on the known Doppler effect, i.e., the redshift of light, Hubble concluded that the galaxies were moving away from each other. Moreover, he observed that the more distant a galaxy was, the greater its redshift. This finding implied that if galaxies were moving away from us, then at some point in the past, they must have been closer together.</text>
<text><location><page_30><loc_18><loc_9><loc_86><loc_36></location>Based on Hubble's discovery, Georges Lemaître suggested that the universe began from an extremely hot and dense state, which he referred to as the primeval atom or the cosmic egg . He theorized that this primordial atom contained all the matter and energy in the universe and exploded in an event he called the explosion of the primeval atom . This explosion marked the beginning of the universe, a moment commonly referred to as the Big Bang . Though he proposed the model in 1931 [9], this gained wider recognition in the scientific community later, with additional evidence supporting the Big Bang model emerged, such as the discovery of the cosmic microwave background radiation, which is considered a remnant of the early stages of the universe. In 1948, Alpher, Bethe, Gamow, and Herman developed the first detailed model of Big Bang nucleosynthesis, which provided an explanation for the formation of light elements, such as hydrogen and helium, in</text>
<text><location><page_31><loc_14><loc_75><loc_82><loc_90></location>the early universe [10-12]. Alpher, Gamow, and Herman's model proposed that in the first few minutes after the Big Bang, the universe was extremely hot and dense, with temperatures of billions of degrees. During this time, the conditions were favorable for nuclear reactions to take place. According to their calculations, the extreme temperatures and densities allowed for the fusion of protons and neutrons to form light elements, primarily hydrogen and helium.</text>
<text><location><page_31><loc_14><loc_36><loc_82><loc_74></location>In 1965, Arno Penzias and Robert Wilson made a groundbreaking discovery known as the cosmic microwave background radiation (CMB) or the Penzias-Wilson radiation [13]. Penzias and Wilson were astronomers working at Bell Labs in New Jersey, conducting experiments using a large horn antenna originally built for satellite communication. However, no matter what they did, they encountered a persistent noise signal in their measurements. They could not eliminate this noise, which seemed to be coming from every direction in the sky. It was unknown to Penzias and Wilson that their observations were coinciding with the work of Robert Dicke, Jim Peebles, and others at Princeton University [14, 15]. Theoretical work at Princeton had predicted the existence of a faint radiation left over from the early stages of the universe, known as the cosmic microwave background. When Penzias and Wilson learned about the Princeton team's prediction, they realized that the noise they were observing matched the characteristics of the cosmic microwave background radiation (CMB). This radiation is the remnant of the intense heat and energy that filled the universe shortly after the Big Bang. The discovery of the CMB [16] provided strong evidence in support of the Big Bang model.</text>
<text><location><page_31><loc_14><loc_10><loc_82><loc_35></location>According to the Big Bang model, the universe expanded rapidly from an extremely hot and dense state. The observations indicated that different regions of the universe that were now far apart appeared to have the same temperature, properties and flat. To address this issue, Alan Guth(1981) proposed the concept of cosmic inflation [17] and suggested that a brief period of exponential expansion occurred in the very early universe, causing it to grow exponentially larger in size. This rapid expansion would have smoothed out the initial irregularities and made the universe homogeneous and isotropic, providing an explanation for the observed uniformity across vast regions of space. The cosmic inflation also indicates that quantum fluctuations during inflation could lead to the formation of tiny density perturbations in the early universe. These fluctuations served as seeds for the later formation of</text>
<text><location><page_32><loc_18><loc_77><loc_86><loc_90></location>galaxies, clusters of galaxies, and other cosmic structures. Guth's work had a profound impact on our understanding of the universe. Subsequent observations, such as the measurement of the cosmic microwave background radiation by the Cosmic Background Explore(COBE) [18] and The Wilkinson Microwave Anisotropy Probe(WMAP) satellites [19], provided strong evidence in support of cosmic inflation and moreover the Big Bang model.</text>
<text><location><page_32><loc_18><loc_36><loc_86><loc_76></location>The discovery of cosmic acceleration in late 1990s posed a threat to our understanding of the universe. Two independent teams of astronomers, the High-Z Supernova Search Team (Schmidt et al ., 1998) [20] and the Supernova Cosmology Project (Perlmutter et al ., 1999) [193] provided supporting data that Hubble expansion is accelerating over time, while studying distant supernovae, specifically Type Ia supernovae(SNIa) as standard candles to measure the correlation between distance and redshift. This discovery was contrary to the prevailing belief that the gravitational pull of matter should be causing the expansion to decelerate. Adam G. Riess and his colleagues analysed the data provided by the previously mentioned groups in the article titled "Observational Evidence from Supernovae for an Accelerating universe and a Cosmological Constant" [192] and concluded that the expansion of the universe was accelerating rather than slowing down and proposed the presence of a cosmological constant (i.e., the vacuum energy density) in the equations of general relativity. The cosmological constant acts as a repulsive force, causing the expansion of the universe to accelerate over time. This discovery of an accelerating universe and the need for dark energy revolutionized cosmology and deeply influenced our comprehension of the basic nature of the universe.</text>
<text><location><page_32><loc_18><loc_10><loc_86><loc_35></location>Later, WMAP team discovered that approximately 71.4% of the universe consists of dark energy [21]. Planck space observatory, designed to map the anisotropies of the cosmic microwave background (CMB) at infrared and microwave frequencies, revealed that the ordinary matter makes up only about 4-5% only, with dark matter(25-26%) and dark energy(69-70%) accounting for the remaining 95% [22]. This dominating dark energy is thought to exert a repulsive gravity that counteracts the attractive force of gravity, causing galaxies and other cosmic structures to move away from each other at an accelerating rate. There are several theories and hypotheses regarding the source of this exotic dark energy, but no definitive explanation has been established. Such theories include two most talked about categories: (i) cosmological constant, and (ii) quintessence field.</text>
<text><location><page_33><loc_14><loc_69><loc_82><loc_90></location>The other way of explaining this accelerated expansion is to modify the theory of gravity. Some eminent researchers of this era have proposed that our understanding of gravity on cosmological scales may need modification. Modified theories of gravity, such as the f(R) theory, Brans-Dicke theory attempt to explain the observed cosmic acceleration without invoking dark energy. These theories suggest that gravity behaves differently on large scales and can account for the accelerated expansion. But there is no certain observational evidence to choose one theoretical model. There are reverse engineering techniques, known as the Reconstruction of dark energy models from observational datas to understand the nature of dark energy.</text>
<text><location><page_33><loc_14><loc_58><loc_82><loc_68></location>However, theorists use several methods to compare the available theories of dark energy in literature and try to find out which one is preferable. In this thesis, we have used the Generalised Second Law of Thermodynamics to study different cosmological models. Before delving into the specifics of that topic, initially, a concise aspect related to cosmology is explored here.</text>
<section_header_level_1><location><page_33><loc_14><loc_53><loc_69><loc_54></location>1.1.2 Geometry and Dynamics of the Spacetime:</section_header_level_1>
<text><location><page_33><loc_14><loc_30><loc_82><loc_51></location>In 1915, Einstein's work on a new theory of gravity did not just result in a different way to think about forces or gravitational fields. It led to a big change in how we see space and time, shaking up our understanding in a major way. Einstein recognized that the empirical observation of all objects falling with the same acceleration in a gravitational field naturally pointed towards an interpretation of gravity based on the curvature of spacetime. According to the General theory of relativity, gravity is manifested by the interaction of spacetime curvature and matter [23-31]. Einstein's field equations give the mathematical manifestation of this theory. By varying Einstein-Hilbert action with respect to g µν ,</text>
<formula><location><page_33><loc_32><loc_25><loc_82><loc_29></location>S EH = ∫ √ -g ( R 16 π G + L m ) d 4 x , (1.1)</formula>
<text><location><page_33><loc_14><loc_22><loc_46><loc_23></location>we get the Einstein's field equation,</text>
<formula><location><page_33><loc_34><loc_17><loc_82><loc_20></location>G µν ≡ R µν -1 2 g µν R = 8 π GT µν . (1.2)</formula>
<text><location><page_33><loc_14><loc_9><loc_82><loc_15></location>In the above equation, g µν denotes the metric tensor and g ≡ det g µν ; g µν is symmetric and throughout the thesis we follow the signature convention (,+,+,+). Here, R is the scalar curvature, known as Ricci scalar and defined as,</text>
<text><location><page_34><loc_18><loc_75><loc_86><loc_90></location>R = g µν R µν . R µν is the Ricci tensor that captures the local curvature of spacetime at each point. G ≈ 6.7 × 10 -11 m 3 kg -1 s -2 represents the Newtonian constant of gravitation. In the action (1.1), the Lagrangian density is divided in two parts: (i) gravitational part: R and (ii) matter part: L m . The volume element is √ -gd 4 x . Equation (1.2) signifies the connection between geometry, contributed by G µν and matter, contributed by T µν . Energy-momentum tensor or stress-energy tensor T µν is defined as,</text>
<formula><location><page_34><loc_41><loc_70><loc_86><loc_73></location>T µν = -2 ∂ L m ∂ g µν + g µν L m , (1.3)</formula>
<text><location><page_34><loc_18><loc_65><loc_86><loc_68></location>describes the distribution of mass, energy, and momentum in spacetime and depends on the specific physical system under consideration.</text>
<section_header_level_1><location><page_34><loc_18><loc_60><loc_38><loc_61></location>1.1.3 Cosmology:</section_header_level_1>
<text><location><page_34><loc_18><loc_44><loc_86><loc_58></location>The foundation of the contemporary cosmological model lies in the Cosmological Principle, asserting that the universe exhibits spatial homogeneity i.e., translationally invariant (Copernican principle) and isotropy i.e., rotationally invariant at scales much larger than the galaxy cluster [26, 32]. This suggests that the space is maximally symmetric. Thus it becomes feasible to depict the universe using a lucid geometric framework by taking 1+3 foliation of the spacetime. The invariant interval [33],</text>
<formula><location><page_34><loc_44><loc_40><loc_86><loc_42></location>ds 2 = g µν dx µ dx ν , (1.4)</formula>
<text><location><page_34><loc_18><loc_30><loc_86><loc_38></location>does not depend on the choice of co-ordinates x µ s ( x 0 denotes cosmic time t and x 1 , x 2 , x 3 are space co-ordinates). The curvature of the spatial part can be of three forms: (i) positive curvature, (ii) zero curvature or (iii) negative curvature. A possible way to represent the metric in 1+3 foliation is,</text>
<formula><location><page_34><loc_41><loc_26><loc_86><loc_28></location>ds 2 = -dt 2 + a 2 ( t ) h ij x i x j , (1.5)</formula>
<text><location><page_34><loc_18><loc_22><loc_76><loc_23></location>where a ( t ) is known as the cosmic scale factor. The 3-metric h ij is,</text>
<formula><location><page_34><loc_23><loc_13><loc_86><loc_20></location>h ij = δ ij + k x i x j 1 -k | x | 2 , k = 0 flat + 1 positive curvature -1 negative curvature . (1.6)</formula>
<text><location><page_34><loc_18><loc_9><loc_63><loc_10></location>In the above equation k is the curvature parameter.</text>
<section_header_level_1><location><page_35><loc_17><loc_88><loc_66><loc_90></location>· Friedmann-Robertson-Walker metric (FRW metric):</section_header_level_1>
<text><location><page_35><loc_14><loc_83><loc_82><loc_86></location>From the above-written metric in the maximally symmetric spacetime, we can write the FRW metric [9, 34-39],</text>
<formula><location><page_35><loc_25><loc_77><loc_82><loc_81></location>ds 2 = -dt 2 + a 2 ( t ) ( dr 2 1 -kr 2 + r 2 d θ 2 + r 2 sin 2 θ d ϕ 2 ) , (1.7)</formula>
<text><location><page_35><loc_14><loc_65><loc_82><loc_75></location>in the comoving spatial coordinates ( r , θ , ϕ ). The scale factor a ( t ) is the length scale of the universe serves as a gauge for understanding how physical distances evolve over time, where the coordinate distances, denoted as r , are considered fixed by definition. The present value of the cosmic scale factor a ( t ) is often considered as, a 0 = a ( t 0 ) = 1.</text>
<section_header_level_1><location><page_35><loc_17><loc_62><loc_41><loc_64></location>· Cosmological Redshift:</section_header_level_1>
<text><location><page_35><loc_14><loc_44><loc_82><loc_60></location>The primary and critical information about the cosmic scale factor, a ( t ) , is obtained from the observation of redshifts in the frequency of light emitted by distant celestial sources. As the universe expands, the wavelengths ( λ ) of photons traveling through space also stretch, leading to a redshift in their observed frequency. This is a key observational evidence of the expanding universe. Astronomer Edwin Hubble interpreted this redshift as due to a Doppler effect and therefore, ascribed a recessional velocity v to the galaxy, which is related to the shift in wavelength by the Doppler formula,</text>
<formula><location><page_35><loc_43><loc_39><loc_82><loc_42></location>v c = ∆ λ λ ≡ z , (1.8)</formula>
<text><location><page_35><loc_14><loc_36><loc_43><loc_37></location>where z is known as the redshift.</text>
<text><location><page_35><loc_14><loc_23><loc_82><loc_35></location>The cosmological redshift behaves symmetrically between the receiver (observer) and the emitter (distant galaxy). In other words, if we observe a redshift in the light coming from a distant galaxy, the light sent from the Earth to that galaxy would also experience a redshift when observed from the distant galaxy's perspective. According to Hubble's Law , the recessional velocity increases proportionally with the distance d of the galaxy [40], i.e.,</text>
<formula><location><page_35><loc_44><loc_17><loc_82><loc_21></location>v ∝ d , ∴ v = Hd , (1.9)</formula>
<text><location><page_35><loc_14><loc_8><loc_82><loc_14></location>here the proportionality constant H is known as Hubble constant . To provide a more accurate historical account, credit for this fundamental discovery, known as Hubble's law, should potentially be shared with G. Lemaître</text>
<text><location><page_36><loc_18><loc_62><loc_86><loc_90></location>as well. In many cosmological models, the Hubble constant H , is a timedependent function, and therefore, nomenclatured as Hubble parameter. However, in the given equation, H represents the present-day value of the Hubble constant, often denoted as H 0 . Mega-parsecs (Mpc) are a common unit of measurement for galactic distances. The term parsec is derived from parallax of one arcsecond and is based on the phenomenon of parallax, which is the apparent shift in the position of a nearby object when observed from different vantage points. One parsec is defined as the distance at which an object shows a parallax angle of one arcsecond, or approximately 3.26 lightyears (about 3.086 × 10 13 kilometers or 1.917 × 10 13 miles). Due to the old trigonometric technique of determining star distances, this unit was created. The unit in which H is measured, is kmMpc -1 s -1 . At current epoch its value is, H 0 ≈ 70kmMpc -1 s -1 .</text>
<text><location><page_36><loc_21><loc_60><loc_82><loc_62></location>In view of FRW cosmology, redshift parameter z can be expressed as,</text>
<formula><location><page_36><loc_46><loc_56><loc_86><loc_59></location>z = a 0 a ( t ) -1, (1.10)</formula>
<text><location><page_36><loc_18><loc_52><loc_56><loc_54></location>and the Hubble parameter is expressed as,</text>
<formula><location><page_36><loc_48><loc_47><loc_86><loc_50></location>H = ˙ a ( t ) a ( t ) . (1.11)</formula>
<text><location><page_36><loc_18><loc_42><loc_86><loc_45></location>An overdot signifies a derivative of the variable with respect to cosmic time ( t ).</text>
<section_header_level_1><location><page_36><loc_21><loc_38><loc_37><loc_39></location>· A Perfect Fluid:</section_header_level_1>
<text><location><page_36><loc_18><loc_24><loc_86><loc_36></location>We consider that the universe is composed of a fluid. By definition, a perfect fluid is characterized by the property that a comoving observer perceives the fluid surrounding them as isotropic. In case of a perfect fluid, the heat conduction, viscosity or other transport or dissipative processes are considered negligible. One can write the energy-momentum tensor for perfect fluid in comoving coordinates as,</text>
<formula><location><page_36><loc_40><loc_20><loc_86><loc_21></location>T µν = ( ρ + p ) u µ u ν + pg µν , (1.12)</formula>
<text><location><page_36><loc_18><loc_12><loc_86><loc_17></location>which acts as a source of spacetime curvature in the Einstein equation. In above equation u µ represents the 4-velocity, in comoving frame its componennts are,</text>
<formula><location><page_36><loc_45><loc_9><loc_86><loc_11></location>u µ = ( 1, 0, 0, 0 ) . (1.13)</formula>
<text><location><page_37><loc_14><loc_86><loc_82><loc_90></location>In equation (1.12), ρ and p are the energy density and pressure of the perfect fluid respectively. The continuity equation is then expressed as,</text>
<formula><location><page_37><loc_43><loc_82><loc_82><loc_84></location>∇ ν T µν = 0, (1.14)</formula>
<text><location><page_37><loc_14><loc_74><loc_82><loc_80></location>where ∇ ν is the covariant derivative and is often indicated by the symbol ;. The energy density ( ρ ) and the pressure are related by the equation of state (EoS),</text>
<formula><location><page_37><loc_45><loc_72><loc_82><loc_73></location>p = w ρ , (1.15)</formula>
<text><location><page_37><loc_14><loc_67><loc_82><loc_70></location>where ω is called the equation of state parameter. Here are some examples of perfect fluid:</text>
<unordered_list>
<list_item><location><page_37><loc_17><loc_57><loc_82><loc_65></location>1. Dust: In case of dust particles, the pressure is zero, as in this case the kinetic energy is negligible compared to the rest energy. Therefore, p = 0 and thus w = 0. Dust is widely accepted as an accurate representation of baryonic matter (and cold dark matter) in the present epoch.</list_item>
<list_item><location><page_37><loc_17><loc_51><loc_82><loc_55></location>2. Radiation: For electro-magnetic radiation, the equation of state is, p = ρ 3 . Therefore, the EoS parameter is w = 1/3.</list_item>
<list_item><location><page_37><loc_17><loc_46><loc_82><loc_50></location>3. Cosmological constant: For Λ , EoS is, p = -ρ . Therefore, EoS parameter is w = -1.</list_item>
</unordered_list>
<text><location><page_37><loc_14><loc_43><loc_77><loc_44></location>For the fluids mentioned above, we see that EoS parameter is constant.</text>
<section_header_level_1><location><page_37><loc_17><loc_39><loc_45><loc_40></location>· The Einstein Field Equations:</section_header_level_1>
<text><location><page_37><loc_14><loc_35><loc_81><loc_37></location>In FRW cosmology, the Einstein equations give two independent equations,</text>
<formula><location><page_37><loc_45><loc_30><loc_60><loc_34></location>˙ a 2 2 + k 2 = 8 π G ρ ,</formula>
<formula><location><page_37><loc_35><loc_26><loc_62><loc_30></location>2 ( a a ) + ( k + ˙ a 2 a 2 ) = -8 π Gp .</formula>
<formula><location><page_37><loc_45><loc_27><loc_82><loc_32></location>a a 3 (1.16) (1.17)</formula>
<text><location><page_37><loc_17><loc_23><loc_48><loc_24></location>The fluid conservation equation is,</text>
<formula><location><page_37><loc_40><loc_19><loc_82><loc_20></location>˙ ρ + 3 H ( ρ + p ) = 0. (1.18)</formula>
<text><location><page_37><loc_14><loc_13><loc_82><loc_17></location>Due to the Bianchi identities ( G µν ; ν = 0), the conservation equation is not independent of Einstein equations and can be derived from them.</text>
<unordered_list>
<list_item><location><page_37><loc_17><loc_9><loc_54><loc_11></location>· Some Important Kinematic Quantities:</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_38><loc_18><loc_86><loc_86><loc_90></location>1. Deceleration Parameter: The deceleration parameter, denoted by q , measures the rate of change of expansion rate and defined as,</list_item>
</unordered_list>
<formula><location><page_38><loc_40><loc_80><loc_86><loc_84></location>q ≡ -a / a ( ˙ a / a ) 2 = -1 aH 2 d 2 a dt 2 . (1.19)</formula>
<text><location><page_38><loc_18><loc_64><loc_86><loc_78></location>(i) If q > 0: The universe is decelerating. The expansion rate is slowing down over time. Normal baryonic matter shows q > 0. (ii) If q < 0: The universe is accelerating. The expansion rate is increasing over time. Observational evidence suggests that our universe is currently in an accelerated expansion phase. This discovery led to the proposal of the concept of dark energy which gives q < 0. (iii) If q = 0: The universe is coasting. The expansion rate remains constant over time.</text>
<unordered_list>
<list_item><location><page_38><loc_18><loc_60><loc_86><loc_63></location>2. Jerk Parameter: Jerk parameter is related to the third derivative of the scale factor with respect to cosmic time and defined as,</list_item>
</unordered_list>
<formula><location><page_38><loc_46><loc_54><loc_86><loc_58></location>j ≡ -1 aH 3 d 3 a dt 3 . (1.20)</formula>
<text><location><page_38><loc_18><loc_28><loc_86><loc_53></location>The behaviour of the jerk parameter can give insights into the presence of dark energy, as it plays a crucial role in the accelerated expansion of the universe. The expansion of the universe is currently accelerating, and this acceleration is attributed to dark energy, represented by the cosmological constant ( Λ ). The value of the jerk parameter for Λ CDMmodel is j Λ CDM = -1. A value of j not equal to -1 could suggest the presence of additional forces or components in the universe that contribute to its acceleration and whose effects are not fully captured by the cosmological constant. It is to be noted that there is another parameter, known as the snap parameter, s = 1 aH 4 d 4 a dt 4 , related to fourth order derivative of the scale factor. However, in the aspects of observational cosmology, the evolution of q holds physical significance. Therefore, we only narrow our analysis to the jerk parameter.</text>
<unordered_list>
<list_item><location><page_38><loc_18><loc_22><loc_86><loc_27></location>3. Critical Density & Density Parameter: The critical density ( ρ c ) is a critical value of density that determines the geometry of the universe. It is defined as,</list_item>
</unordered_list>
<formula><location><page_38><loc_46><loc_18><loc_86><loc_22></location>ρ c ( t ) ≡ 3 H 2 8 π G . (1.21)</formula>
<text><location><page_38><loc_18><loc_14><loc_86><loc_18></location>The critical density represents the dividing line between an open, flat, or closed universe. Ω represents the ratio of the average density of the universe</text>
<text><location><page_39><loc_14><loc_88><loc_47><loc_90></location>to the critical density. It is defined as,</text>
<formula><location><page_39><loc_45><loc_83><loc_82><loc_86></location>Ω = ρ ρ c . (1.22)</formula>
<text><location><page_39><loc_14><loc_78><loc_82><loc_81></location>The spatial shape and curvature of the cosmos are determined by the value of the density parameter in the following way:</text>
<unordered_list>
<list_item><location><page_39><loc_14><loc_76><loc_73><loc_77></location>(i) Ω < 1 ( ⇒ k = -1 ) : implies the universe has an open geometry.</list_item>
<list_item><location><page_39><loc_14><loc_74><loc_61><loc_75></location>(ii) Ω = 1 ( ⇒ k = 0 ) : the universe has a flat geometry.</list_item>
<list_item><location><page_39><loc_14><loc_67><loc_82><loc_73></location>(iii) Ω > 1 ( ⇒ k = + 1 ) :the universe has a closed geometry. To gain a deeper understanding of the fundamental principles underlying modern cosmology, readers may refer to [41-47].</list_item>
</unordered_list>
<section_header_level_1><location><page_39><loc_14><loc_61><loc_78><loc_63></location>1.2 Thermal Evolution of the Universe in Brief:</section_header_level_1>
<text><location><page_39><loc_14><loc_52><loc_82><loc_59></location>Understanding the thermal evolution history is crucial in cosmology as it provides insights into the behaviour of different constituents of the universe and the processes that shaped its current state. Here is an overview of the key stages in the thermal evolution of the universe:</text>
<unordered_list>
<list_item><location><page_39><loc_17><loc_40><loc_82><loc_50></location>1. Planck Era : (Time: 0 to 10 -43 seconds after the Big Bang): Temperature Range: > 1.416 × 10 32 K. During this incredibly brief epoch, the universe was in a state of extremely high energy and temperature. In the Planck epoch, it is presumed that the characteristics of cosmology and physics were primarily governed by the quantum effects of gravity.</list_item>
<list_item><location><page_39><loc_17><loc_32><loc_82><loc_38></location>2. Grand Unification Era : (Time: 10 -43 seconds to 10 -36 seconds): Temperature Range: 1.416 × 10 32 K to 10 27 K. The three forces of the Standard Model remained unified (gravity excluded).</list_item>
<list_item><location><page_39><loc_17><loc_16><loc_82><loc_31></location>3. Inflationary epoch and Electroweak Era : (Time:10 -36 seconds to 10 -12 seconds): Temperature Range: 10 27 K to 10 15 K. Cosmic inflation stretched the fabric of space by approximately a factor of 10 26 within a time span ranging from 10 -36 to 10 -32 seconds. The electroweak force (combining electromagnetism and weak nuclear force) separated from the strong force, and elementary particles gained mass through interactions with the Higgs field.</list_item>
<list_item><location><page_39><loc_17><loc_11><loc_82><loc_15></location>4. Quark-Gluon Plasma Era : (Time: 10 -12 seconds to 10 -6 seconds): Temperature Range: 10 15 K to 10 12 K. The temperature was still extremely</list_item>
</unordered_list>
<text><location><page_40><loc_23><loc_84><loc_86><loc_90></location>high, causing quarks and gluons to roam in what is known as a quarkgluon plasma. As the universe expands and cools, this state transitioned to confinement, forming protons and neutrons.</text>
<unordered_list>
<list_item><location><page_40><loc_21><loc_70><loc_86><loc_82></location>5. Nucleosynthesis Era : (Time: 10 seconds to 10 3 seconds): Temperature Range: 10 10 K to 10 9 K. The universe had cooled enough for nuclear reactions to take place. These reactions, known as nucleosynthesis, created the primordial nuclei of hydrogen, helium, and trace amounts of lithium and beryllium. Most of the helium in the universe was formed during this phase.</list_item>
<list_item><location><page_40><loc_21><loc_50><loc_86><loc_68></location>6. Recombination Era : (Time: ≈ 380, 000 years): Temperature Range: ≈ 3000 K. At this point, the temperature was low enough for protons and electrons to combine and form neutral hydrogen atoms through a process called recombination. The positively charged protons captured negatively charged electrons to create stable neutral atoms. This transition led to a significant decrease in the scattering of photons by charged particles, making the universe transparent to light. As a result, the photons that were once tightly coupled to matter were released, filling space and creating the Cosmic Microwave Background (CMB) radiation.</list_item>
<list_item><location><page_40><loc_21><loc_19><loc_86><loc_48></location>7. Dark Ages : (Time: ≈ 380, 000 years to ≈ 150 million years): Temperature Range: Cooling from ≈ 3000 K to a few K. Following recombination and decoupling, the universe entered a transparent state. However, the process of hydrogen clouds collapsing to form stars and galaxies was extremely slow, leading to a lack of new sources of light. Consequently, the only remaining photons (electromagnetic radiation or light ) in the universe were those released during decoupling, which are now observable as the cosmic microwave background. Additionally, occasional 21 cm radio emissions emitted by hydrogen atoms contributed to the available light. Initially, the decoupled photons filled the cosmos with a brilliant pale orange glow, gradually redshifting to wavelengths that are not visible after approximately 3 million years. As a result, the universe ended up without visible light. During this era, matter started dominating over radiation.</list_item>
<list_item><location><page_40><loc_21><loc_9><loc_86><loc_17></location>8. Formation of the First Stars and Galaxies : (Time: ≈ 150 million years and onward): Temperature Range: Cooling from a few K to thousands of K. The first stars and galaxies began to form, releasing energy and ionizing the surrounding gas.</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_41><loc_17><loc_77><loc_82><loc_90></location>9. Present-Day universe : (Time: ≈ 13.8 billion years): Temperature Range: ≈ 2.73 K (Temperature of the Cosmic Microwave Background Radiation). The universe continues to expand and cool. Its average temperature is now very low, and most of the thermal radiation is in the form of the CMB. Now the universe is dominated by the dark energy and causes acceleration in the expansion.</list_item>
</unordered_list>
<section_header_level_1><location><page_41><loc_14><loc_69><loc_82><loc_74></location>1.3 Possible ways to Elucidate the Cosmic Acceleration:</section_header_level_1>
<text><location><page_41><loc_14><loc_42><loc_82><loc_66></location>The discovery of the late-time accelerated expansion of the universe occurred in 1998 when two separate groups, the Supernova Cosmology Project [193] and the High-Z Supernova Search Team [20, 48, 192], independently observed supernovae with redshifts z < 1. Type Ia supernovae, known for their consistent intrinsic brightness, serve as reliable standard candles, easily distinguishable over varying distances. These supernovae are thought to occur when a white dwarf star in a binary system accumulates enough mass from its companion, triggering a thermonuclear explosion. As the universe expands, the separation between the observer and celestial objects increases. Consequently, the emitted photons experience redshift. By analyzing the observed brightness of these objects and the redshift of the photons, researchers can gauge the rate of the expansion of the universe.</text>
<text><location><page_41><loc_14><loc_17><loc_82><loc_40></location>The brightness of a supernova is represented by its absolute magnitude. It can be utilized in calculating the cosmic luminosity distance ( d L ). The connection between the apparent luminosity ( f ) and the intrinsic luminosity ( L ) is expressed as, f = L 4 π d 2 L . Measurements of the luminosity distance ( d L ) for supernovae are recorded at various redshifts in the format of distance modulus ( µ B ). This modulus is defined as the disparity between the apparent magnitude ( m B ) and the absolute magnitude ( M B ) within the wavelength of the blue line of the observed spectrum as, µ B = m B -M B = 5 log 10 d L 1Mpc + 25. In terms of H 0 and q 0 , i.e., the e present values of the Hubble parameter and the deceleration parameter respectively, we can write, d L = cz H 0 ( 1 + z 2 [ 1 -q 0 ] + O ( z 2 ) ) .</text>
<text><location><page_41><loc_14><loc_10><loc_82><loc_16></location>Both supernova groups gauged the luminosity distances and witnessed the fading of supernovae. The determined luminosity distances surpassed their anticipated values, signifying a negative q 0 . This affirmation supported</text>
<text><location><page_42><loc_18><loc_86><loc_86><loc_90></location>the conclusion that light sources are moving away from each other at an accelerated pace.</text>
<text><location><page_42><loc_18><loc_70><loc_86><loc_85></location>This observed accelerated expansion of the universe is a perplexing phenomenon that cannot be fully accounted for by the known components of the universe. All known forms of matter and energy that we encounter in the universe, such as ordinary matter, radiation (including light), and the dark matter, contribute to slowing down the expansion of the universe. These fluids follow the strong energy condition, ρ + 3 p > 0. Now if we combine the equations, (1.16) and (1.17), we can write,</text>
<formula><location><page_42><loc_43><loc_65><loc_86><loc_69></location>a a = -4 π G 3 ( ρ + 3 p ) . (1.23)</formula>
<text><location><page_42><loc_18><loc_49><loc_86><loc_64></location>From the definition of deceleration parameter (1.19), q = -1 aH 2 d 2 a dt 2 and the equation (1.23), we see that for these fluids q > 0, i.e., the decelerating cosmic expansion . However, rather surprisingly, the universe has undergone two separate periods of accelerated expansion: one marked by early exponential inflation and the other by late-time cosmic acceleration. In the interim, these two phases of rapid expansion were separated by a period characterized by a decelerated expansion.</text>
<text><location><page_42><loc_18><loc_33><loc_86><loc_48></location>As a result, scientists have put forward various theoretical models to try to elucidate the acceleration in the expansion history in two distinct times, (i)inflation and (ii)late-time cosmic acceleration. Nevertheless, none of these models has achieved widespread acceptance as flawless or has been strongly supported by irrefutable observational evidence. Additionally, these models have not yet provided a direct and conclusive method for detecting the underlying cause of cosmic acceleration.</text>
<section_header_level_1><location><page_42><loc_18><loc_26><loc_35><loc_28></location>1.3.1 Inflation:</section_header_level_1>
<text><location><page_42><loc_18><loc_17><loc_86><loc_25></location>Inflation is a suggested paradigm in cosmology that proposes a rapid exponential expansion of the universe during its early epoch ( ∼ 10 -36 -10 -32 seconds). This model was developed to address several outstanding problems and observed phenomena in the standard Big Bang model.</text>
<text><location><page_42><loc_18><loc_10><loc_86><loc_15></location>The concept of inflation was first proposed in the early 1980s by Alan Guth [17] and later refined by other physicists, (see [49-55]). The key idea behind inflation is that in the tiny fraction of a second after the Big Bang,</text>
<text><location><page_43><loc_14><loc_80><loc_82><loc_90></location>the universe underwent an exponential expansion, increasing its size by an enormous factor. This rapid expansion allowed the universe to smooth out irregularities and achieve a high degree of homogeneity and isotropy, which explains why the cosmic microwave background radiation appears so uniform in all directions. (For introductions to inflation, see [56]).</text>
<text><location><page_43><loc_17><loc_77><loc_70><loc_78></location>Inflation addresses several important cosmological puzzles:</text>
<unordered_list>
<list_item><location><page_43><loc_14><loc_60><loc_82><loc_76></location>1. The Horizon Problem: The universe appears to have a very uniform temperature on large scales, even regions that are too distant to have ever been in causal contact. Without inflation, there would not have been enough time for these regions to reach thermal equilibrium and have the same temperature. Inflation solves this by stretching these regions out of causal contact, so they can reach the same temperature before inflation ends. (See [57], for a comprehensive discussion on to solve the horizon problem without invoking inflation. )</list_item>
<list_item><location><page_43><loc_14><loc_53><loc_82><loc_59></location>2. The Flatness Problem: The universe appears to be spatially flat on cosmological scales. Inflation can flatten and stretch the geometry of space, making it very close to flat, k = 0 in equation (1.6).</list_item>
<list_item><location><page_43><loc_14><loc_43><loc_82><loc_53></location>3. The Monopole Problem: Some particle physics theories predict the existence of magnetic monopoles, which are highly magnetically charged particles. However, these monopoles are not observed in the universe in the abundance predicted. Inflation dilutes their numbers, making them incredibly rare and explaining their absence.</list_item>
</unordered_list>
<text><location><page_43><loc_14><loc_29><loc_82><loc_41></location>Inflationary theory suggests that the exponential expansion was driven by a scalar field, often called the inflaton field . As the universe expanded and cooled, this field settled into its lowest energy state, ending the inflationary phase. The energy released during the decay of the inflaton field reheated the universe, initiating the subsequent hot phase and the standard cosmological evolution that followed.</text>
<text><location><page_43><loc_14><loc_14><loc_82><loc_28></location>While inflation has been highly successful in explaining various observed features of the universe, it remains a theoretical framework. Direct evidence for inflation is still a subject of intense research and observational efforts, such as studying the cosmic microwave background radiation and the largescale structure of the universe. Confirming inflation would be a profound confirmation of our understanding of the early history of the universe and the processes that shaped its current state.</text>
<section_header_level_1><location><page_44><loc_18><loc_88><loc_60><loc_90></location>1.3.2 Late-time Cosmic Acceleration:</section_header_level_1>
<text><location><page_44><loc_18><loc_70><loc_86><loc_86></location>Late-time cosmic acceleration was discovered in the late 1990s from observational data of type IA supernova. In addition to using type-Ia supernovae [58-60], Baryon Acoustic Oscillations (BAO) [61, 62], the WMAP [63], Planck [22, 64-66] satellite missions, and the Dark Energy Survey (DES) [67] data also provide substantial support for the existence of a seamless transition. This transition involves the universe transitioning from a prior phase of decelerated expansion to its current state of accelerated expansion, occurring at an intermediary redshift of approximately z ≈ 0.5 [68-75].</text>
<text><location><page_44><loc_18><loc_57><loc_86><loc_69></location>There are two potential explanations to find out a way to unravel the mystery. One of those is that gravity acts in a manner unlike from our current understanding, hence the theory of gravity needs to be modified. The other one is that the cosmos contains an exotic element with unconventional gravitational attributes, leading to an apparent negative pressure effect. These two main broad categories are discussed below.</text>
<section_header_level_1><location><page_44><loc_18><loc_52><loc_53><loc_54></location>1.3.2.1 Modified Theories of Gravity:</section_header_level_1>
<text><location><page_44><loc_18><loc_13><loc_86><loc_51></location>Modified gravity theory refers to a class of theoretical frameworks that propose modifications or extensions to Einstein's general theory of relativity (GR). The idea is to account for the cosmic acceleration without invoking any exotic fluid component, such as dark energy. Not only cosmic acceleration, but the theories also attempt to give insight into some other problems such as addressing the nature of dark matter, resolving singularities etc. Additionally, they aim to provide a semi-classical description of gravitational interactions, where quantum effects are taken into account through an effective action. There are many different proposals for modified gravity, each with its own set of mathematical equations and predictions [76-82]. Some of the well-known examples of modified gravity theories include: f(R) Gravity [83-91], scalar-tensor theories [92-106], Scalar-Einstein-Gauss-Bonnet Gravity [107-109], Galileon Gravity [110-112], TeVeS (Tensor-Vector-Scalar) theory [113-115], f(T) Gravity [116], Horndeski Gravity [117-119], Dilaton Gravity [120, 121], Dvali-Gabadadze-Porrati (DGP) Model(braneworld scenario) [122, 123], Chameleon Gravity [124-127] and some extended theories [128, 129]. Among these, f(R) gravity and scalar-tensor theories will be discussed here.</text>
<text><location><page_45><loc_14><loc_75><loc_82><loc_90></location>1. f ( R ) Theories: In the realm of gravitational theories, a straightforward modification to General Relativity (GR) is presented through f ( R ) gravity. This modification centers on the spacetime action and its transformation. Rather than adhering to the conventional Ricci scalar R , f ( R ) gravity introduces a more flexible approach, incorporating an analytical function, f = f ( R ) . As a result, the action governing f ( R ) gravity can be succinctly represented as,</text>
<formula><location><page_45><loc_34><loc_72><loc_82><loc_75></location>S = ∫ √ -g ( f ( R ) 16 π G + L m ) d 4 x . (1.24)</formula>
<text><location><page_45><loc_14><loc_65><loc_82><loc_71></location>By taking variation of the action given in equation (1.24) with respect to the metric tensor g µν , we arrive at the field equations, which are expressed as,</text>
<formula><location><page_45><loc_30><loc_60><loc_82><loc_63></location>f ' R µν -f 2 g µν -( ∇ µ ∇ ν -g µν □ ) f ' = T µν . (1.25)</formula>
<text><location><page_45><loc_14><loc_53><loc_82><loc_59></location>Here superscript ' implies the derivative with respect to R and the symbol □ represents the d'Alembertian operator and given by, □ = g µν ∇ µ ∇ ν . In terms of Einstein tensor this equation can be written as,</text>
<formula><location><page_45><loc_36><loc_48><loc_82><loc_51></location>G ab = 1 f ' ( R ) ( T µν + T eff µν ) , (1.26)</formula>
<text><location><page_45><loc_14><loc_39><loc_82><loc_46></location>where the effective stress-energy tensor is T eff µν = [ f -Rf ' 2 g µν + ( ∇ µ ∇ ν -g µν □ ) f ' ] . The source of this term is entirely geometric in nature.</text>
<text><location><page_45><loc_14><loc_33><loc_82><loc_39></location>The inclusion of f ' in the denominator within Equation (1.26) signifies the existence of a non-minimal coupling, which in turn renders the effective gravitational constant a variable in these theories.</text>
<text><location><page_45><loc_14><loc_9><loc_82><loc_32></location>These f ( R ) gravity theories have the capability to elucidate phases of cosmic acceleration without the requirement for exotic or unconventional matter constituents. This is achievable through the careful selection of the function f . These f ( R ) models, characterized by f ( R ) ∼ R 2 have proven their capability in generating scenarios of early inflation. The models featuring f ( R ) ∼ 1/ R n , where n is positive, are put forward to account for the accelerated expansion of the universe during its later phases. A thought-provoking discourse of how this theory harmonizes with the established cosmological model to explicate cosmic acceleration can be encountered in the extensive review [130]. To know more detail about f ( R ) theories of gravity, we refer to the articles [84, 90, 131-144]</text>
<section_header_level_1><location><page_46><loc_21><loc_88><loc_43><loc_90></location>2. Scalar-tensor theories:</section_header_level_1>
<text><location><page_46><loc_18><loc_73><loc_86><loc_87></location>In gravitation and cosmology, scalar-tensor theories are rooted in the concept of a non-minimal coupling between the scalar field and the spacetime geometry. The origins of these theories can be traced back to the pioneering work of Jordan [145] and were subsequently advanced by Brans and Dicke [146]. The generality of their findings was later augmented by the insights of Nordvedt [147] and Wagoner [148]. The action to represent a broad category of Non-Minimally Coupled Scalar-Tensor Theories (NMCSTT) is,</text>
<formula><location><page_46><loc_28><loc_68><loc_86><loc_71></location>S = 1 2 ∫ √ -g [( f ( ϕ ) R -ω ( ϕ ) ϕ ∂µϕ∂ µ ϕ ) + L m ] d 4 x . (1.27)</formula>
<text><location><page_46><loc_18><loc_54><loc_86><loc_66></location>Within this framework, the scalar field ϕ is coupled to the Ricci scalar R in a non-minimal manner, leading to a consequential adjustment in the gravitational coupling strength. The field equations governing the NMCSTT encompassed within the action (1.27) are derived through variations in both the metric g µν and the scalar field ϕ . These equations can be expressed as follows:</text>
<formula><location><page_46><loc_19><loc_48><loc_86><loc_53></location>f ( R µν -1 2 g µν R ) + g µν □ f -∇ ν ∇ µ f -ω ϕ ∂µϕ∂νϕ + 1 2 g µν ω ϕ ∂µϕ∂ µ ϕ = T µν , (1.28)</formula>
<text><location><page_46><loc_18><loc_45><loc_21><loc_47></location>and</text>
<formula><location><page_46><loc_33><loc_42><loc_86><loc_46></location>R df d ϕ + 2 ω ϕ □ ϕ + ( 1 ϕ d ω d ϕ -ω ϕ 2 ) ∂ a ϕ∂ a ϕ = 0. (1.29)</formula>
<text><location><page_46><loc_18><loc_37><loc_86><loc_41></location>Here the effective gravitational coupling is G eff = f ( ϕ ) -1 . As a consequence, it is presumed that f ( ϕ ) is positive to guarantee a positive coupling.</text>
<text><location><page_46><loc_18><loc_25><loc_86><loc_37></location>Brans-Dicke (BD) theory serves as the quintessential example within this classification. Diverse scalar-tensor theories are born from the overarching action presented in equation (1.27) each distinguished by their unique choices of the functions f ( ϕ ) and ω ( ϕ ) . For comprehensive discussions on scalar-tensor theories, one may follow the articles [149-151]. Here we shall discuss about the Brans-Dicke theory.</text>
<section_header_level_1><location><page_46><loc_21><loc_22><loc_41><loc_23></location>· Brans-Dicke theory</section_header_level_1>
<text><location><page_46><loc_18><loc_8><loc_86><loc_20></location>Brans-Dicke theory stands as one of the most widely discussed theoretical framework in the field of theoretical physics and cosmology that extends Einstein's theory of general relativity. Proposed by Carl H. Brans and Robert H. Dicke in 1961 [146], this theory introduces a scalar field, known as the Brans-Dicke field, alongside the familiar metric tensor used in general relativity. This addition was motivated by the desire to explore a broader range</text>
<text><location><page_47><loc_14><loc_86><loc_82><loc_90></location>of gravitational theories that could accommodate variations in the strength of gravity and address certain shortcomings of general relativity.</text>
<text><location><page_47><loc_14><loc_82><loc_82><loc_85></location>Brans-Dicke action follows from equation (1.27) by the choice f ( ϕ ) = ϕ and ω = a constant (dimensionless). Therefore, now the action looks like,</text>
<formula><location><page_47><loc_25><loc_76><loc_82><loc_80></location>S = 1 16 π G 0 ∫ √ -g [( ϕ R -ω ϕ ∂µϕ∂ µ ϕ ) + L m ] d 4 x . (1.30)</formula>
<text><location><page_47><loc_17><loc_73><loc_60><loc_75></location>Upon taking the conformal transformation [152],</text>
<formula><location><page_47><loc_43><loc_69><loc_82><loc_71></location>˜ g µν = ϕ g µν , (1.31)</formula>
<text><location><page_47><loc_14><loc_65><loc_32><loc_67></location>the action becomes,</text>
<formula><location><page_47><loc_19><loc_60><loc_82><loc_64></location>¯ S = 1 16 π G 0 ∫ √ -¯ g [ ϕ 0 ( ¯ R -2 ω + 3 2 ψ , αψ , β ¯ g αβ ) + ¯ L M ] d 4 x , (1.32)</formula>
<text><location><page_47><loc_14><loc_56><loc_45><loc_58></location>where ψ = ln ( ϕ ϕ 0 ) . ϕ 0 is a constant.</text>
<text><location><page_47><loc_14><loc_50><loc_82><loc_56></location>This version is popularly known as BD theory in Einstein frame . Then the field equations assume a significantly simplified form, which can be expressed as follows:</text>
<formula><location><page_47><loc_29><loc_43><loc_82><loc_47></location>G αβ = T αβ + 2 ω + 3 2 ( ψ , αψ , β -1 2 g αβ ψ , µ ψ , µ ) . (1.33)</formula>
<text><location><page_47><loc_14><loc_36><loc_82><loc_42></location>This is written in the unit of 8 π G 0 = 1. The equation of motion for the scalar field ψ , derived by varying the action (1.32) with respect to ψ , results in the expression,</text>
<formula><location><page_47><loc_42><loc_30><loc_82><loc_34></location>2 ψ = T 2 ω + 3 , (1.34)</formula>
<text><location><page_47><loc_14><loc_26><loc_82><loc_29></location>where T represents the trace of the energy-momentum tensor for the matter sector.</text>
<text><location><page_47><loc_14><loc_9><loc_82><loc_25></location>The Brans-Dicke theory has been subjected to various experimental tests to determine its validity compared to General Relativity. These tests include measurements of the deflection of light by massive objects, the Nordtvedt effect, precision tests in the solar system, and cosmological observations. Observational constraints limit ω to a very large value [153-155]. The BransDicke (BD) theory was initially suggested to converge into the General Theory of Relativity (GTR) in the limit as ω approaches infinity, a notion initially posited in [41]. Subsequently, it has been elucidated in the literature that the</text>
<text><location><page_48><loc_65><loc_85><loc_65><loc_86></location≯</text>
<table>
<location><page_48><loc_18><loc_64><loc_86><loc_90></location>
<caption>TABLE 1.1: Exact solutions of BD theories in cosmology that have been studied in the existing literature.</caption>
</table>
<text><location><page_48><loc_52><loc_80><loc_52><loc_81></location≯</text>
<text><location><page_48><loc_18><loc_28><loc_86><loc_55></location>inclusion of the trace of the energy-momentum tensor of matter distribution imposes certain constraints on this proposition, as expounded in studies [92, 156]. The static spherically symmetric vacuum solutions of this theory can be found in articles [157]. To know more aspects of these solutions, readers may refer to [158-171]. Numerous spatially homogeneous and isotropic cosmological solutions within the framework of Brans-Dicke (BD) theory can be found in the literature. A few such examples are listed in Table 1.1. BD theory provides solutions to a variety of cosmological quandaries. It offers solutions to issues like the graceful exit problem during inflationary phases [177, 178] and the late-stage accelerated expansion of the universe, all without requiring the introduction of dark energy [100]. The monograph [150] by Faraoni provides an extensive exploration of exact cosmological solutions within BD theory.</text>
<section_header_level_1><location><page_48><loc_18><loc_24><loc_45><loc_25></location>1.3.2.2 Dark Energy Models</section_header_level_1>
<text><location><page_48><loc_18><loc_12><loc_86><loc_22></location>Dark energy models seek to elucidate the cosmic acceleration within the framework of General Relativity (GR) [185-191]. Equation (1.23) illustrates that an accelerated expansion can arise when a constituent within the energy sector exerts substantial negative pressure. This exotic component, commonly referred to as dark energy, instigates acceleration due to its negative</text>
<text><location><page_49><loc_14><loc_75><loc_82><loc_90></location>pressure, distinct from the fluid pressure attributed to particle motion. To facilitate analysis, we introduce dimensionless representations of energy densities by scaling them with the critical density ( ρ c ), defined in equation (1.21) and density parameter ( Ω i ), defined in equation, (1.22). Subscript ' i ' implies either dust ( i = m ), radiation ( i = r ) or dark energy ( i = d ). Now, the Hubble parameter H , scaled by its present value H 0 , can be written in terms of density parameters as (for spatially flat models),</text>
<formula><location><page_49><loc_36><loc_69><loc_82><loc_73></location>h 2 ≡ H 2 H 2 0 = Ω m + Ω r + Ω d . (1.35)</formula>
<text><location><page_49><loc_14><loc_62><loc_82><loc_68></location>Dust matter, a significant component in the energy budget, is pressureless ( p m = 0), thus yielding w m ≡ pm ρ m = 0 as its equation of state parameter. The dark energy equation of state parameter ( w d ) is defined as, w d ≡ p d / ρ d .</text>
<text><location><page_49><loc_14><loc_54><loc_82><loc_62></location>We denote the density parameters at the present epoch, as Ω m 0 , Ω r 0 , Ω d 0 . For non-pressured matter, Ω m = Ω m 0 ( 1 + z ) 3 ; for radiation, Ω r = Ω r 0 ( 1 + z ) 4 , using the standard scaling of the scale factor with a 0 = 1. The composite model's effective equation of state (EoS) is expressed as:</text>
<formula><location><page_49><loc_34><loc_50><loc_82><loc_51></location>w e f f = ( p r + p d ) / ( ρ m + ρ r + ρ d ) . (1.36)</formula>
<text><location><page_49><loc_14><loc_44><loc_82><loc_47></location>To achieve an accelerated universe, w e f f must be less than -1/3, as indicated by equation (1.23).</text>
<text><location><page_49><loc_14><loc_37><loc_82><loc_43></location>Given the negligible contribution of radiation in the late-time universe compared to other components, we can estimate q for a spatially flat universe composed of pressureless matter and dark energy as follows,</text>
<formula><location><page_49><loc_40><loc_32><loc_82><loc_35></location>q ≈ 1 2 ( 1 + 3 w d Ω d ) . (1.37)</formula>
<text><location><page_49><loc_14><loc_22><loc_82><loc_31></location>Hence, the condition for late-time cosmic acceleration is w d < -1 3 Ω -1 d . Recent cosmological observations data suggests Ω d 0 = Ω d | z = 0 ≈ 0.7. Consequently, the value of the dark energy equation of state (DE EoS) parameter at the present epoch should be w d 0 = w d | z = 0 ≲ -0.5.</text>
<text><location><page_49><loc_14><loc_14><loc_82><loc_22></location>Various theoretical frameworks exist for explaining dark energy, yet none have gained universal acceptance, each exhibiting its own drawbacks and limitations. The most straightforward model of dark energy is the cosmological constant Λ , characterized by an equation of state w = -1.</text>
<text><location><page_49><loc_17><loc_12><loc_82><loc_13></location>Below, we delve into an array of diverse models that seek to understand</text>
<text><location><page_50><loc_18><loc_84><loc_86><loc_90></location>the nature of the exotic dark energy. These models offer distinct perspectives and approaches, each contributing to our ongoing quest to unravel the mysteries of cosmic acceleration.</text>
<section_header_level_1><location><page_50><loc_21><loc_80><loc_44><loc_82></location>· Cosmological constant</section_header_level_1>
<text><location><page_50><loc_18><loc_62><loc_86><loc_78></location>In 1917, Einstein introduced the cosmological constant, denoted by Λ , into his equations of general relativity. Einstein added Λ to his equations to achieve a static solution for the universe. At the time, the prevailing view was that the universe was static and unchanging. However, when Edwin Hubble discovered the expansion of the universe in the late 1920s, Einstein famously referred to the inclusion of the cosmological constant as his greatest blunder [2]. He removed the cosmological constant from his equations because it was no longer needed to describe a static universe.</text>
<text><location><page_50><loc_18><loc_47><loc_86><loc_61></location>The cosmological constant made a comeback as vacuum energy in the context of inflation [17]. In the late 20th century, the cosmological constant again resurged in cosmology to explain the observed accelerated expansion of the universe [20, 192, 193]. This acceleration was discovered in the late 1990s through observations of distant supernovae and is attributed to a mysterious dark energy that permeates the universe. One may refer to [185, 187, 194-198], to learn more about Λ as a representative of this dark energy.</text>
<text><location><page_50><loc_18><loc_40><loc_86><loc_46></location>In a universe where the cosmological constant dominates, the solution corresponds to an exponential rate of expansion, where the scale factor a ( t ) grows exponentially with time (see reference [199]),</text>
<formula><location><page_50><loc_38><loc_35><loc_86><loc_38></location>a ( t ) ∝ exp ( √ Λ 3 t ) = exp ( Ht ) . (1.38)</formula>
<text><location><page_50><loc_18><loc_24><loc_86><loc_33></location>The Λ CDM model is the prevailing cosmological framework, consisting of cold dark matter (CDM) with no pressure and the cosmological constant serving as dark energy. It is commonly referred to as the standard cosmological model. The energy density tied to Λ is defined as,</text>
<formula><location><page_50><loc_47><loc_19><loc_86><loc_23></location>ρ Λ = Λ 8 π G . (1.39)</formula>
<text><location><page_50><loc_18><loc_10><loc_86><loc_18></location>To maintain ρ Λ as a constant by definition, it necessitates that the pressure associated with Λ , must be, p Λ = -ρ Λ . Therefore, the cosmological constant exhibits an effective negative pressure, with the equation of state parameter w = -1.</text>
<text><location><page_51><loc_14><loc_73><loc_82><loc_90></location>In a universe exhibiting matter dominance, the scale factor evolves according to the relationship a ( t ) ∝ t 2/3 . As the universe enters a phase where cosmological constant holds sway, with EoS parameter w = -1, the scale factor approaches an asymptotic form expressed in equation (1.38). When we consider a universe with a spatially flat geometry, accommodating both matter and cosmological constant, the solution seamlessly integrates the properties of these two components across different cosmic eras and as given by [196],</text>
<formula><location><page_51><loc_29><loc_63><loc_82><loc_71></location>a ( t ) = ( Ω m Ω vac ) 1/3 ( sinh [ 3 √ Ω vac H 0 t /2 ] ) 2/3 = ( Ω m Ω vac ) 1/3 sinh 2/3 ( t / t 0 ) . (1.40)</formula>
<text><location><page_51><loc_14><loc_52><loc_82><loc_62></location>This solution adeptly mirrors the evolving characteristics of matter and vacuum energy as we traverse from earlier to later cosmic eras. It essentially embodies the principles of the Λ CDM model. Various facets of the cosmological constant model have been thoroughly examined and elaborated upon by Carroll [194] and extensively discussed by Padmanabhan [187, 200].</text>
<text><location><page_51><loc_14><loc_22><loc_82><loc_51></location>The cosmological constant model stands as the favoured choice in light of compelling observational support. Nevertheless, it is essential to acknowledge the inherent complexities of this model. Notably, the only plausible candidate to account for the cosmological constant is the energy density inherent in the vacuum. The notable hurdle lies in the staggering disparity between the energy density's observed value ( ρ obs Λ ) and its theoretically calculated counterpart ( ρ theory Λ ). This incongruity is striking, with the ratio ρ obs Λ ρ theory Λ hovering around the minuscule figure of 10 -120 . This enigma is fundamentally recognized as the fine-tuning dilemma inherent to the cosmological constant model. Recent efforts have been directed towards achieving a reduced cosmological constant in the current cosmic era. A concise overview of various models featuring a cosmological constant, diminishing with time, can be found in table 1.2, which is adapted from [201] (see also [185]).</text>
<text><location><page_51><loc_64><loc_18><loc_64><loc_20></location≯</text>
<section_header_level_1><location><page_51><loc_17><loc_18><loc_70><loc_20></location>· Constant dark energy equation of state model ( w d = -1 )</section_header_level_1>
<text><location><page_51><loc_14><loc_10><loc_82><loc_16></location>In the pursuit of understanding dark energy through phenomenological analysis, cosmologists often opt for a constant value when characterizing the dark energy Equation of State (EoS). Importantly, this constant is not limited</text>
<table>
<location><page_52><loc_26><loc_52><loc_78><loc_90></location>
<caption>TABLE 1.2: List of some phenomenological Λ models.</caption>
</table>
<text><location><page_52><loc_18><loc_39><loc_86><loc_45></location>to the value of -1, providing a degree of flexibility in the modeling process. Dark energy models characterized by this constant EoS are referred to as Quiessence models [247].</text>
<text><location><page_52><loc_18><loc_20><loc_86><loc_39></location>To illustrate, consider a scenario where the universe consists of Cold Dark Matter (CDM) and Dark Energy (DE), with the latter being defined by a constant EoS, w d . This framework is commonly referred to as the w CDMmodel. Its purpose is to scrutinize observational data for any potential deviations from the established Λ CDMmodel. Within the context of the w CDMmodel, it is noteworthy that the dark energy density is no longer fixed but instead exhibits an evolution with changing redshift. This dynamic characteristic allows for a more comprehensive examination of the behaviour of dark energy over cosmic history.</text>
<section_header_level_1><location><page_52><loc_21><loc_16><loc_63><loc_18></location>· Variable dark energy equation of state model</section_header_level_1>
<text><location><page_52><loc_18><loc_8><loc_86><loc_14></location>To tackle the cosmic coincidence problem, i.e., the mystery of why the energy densities of dark matter and dark energy in the universe are of the same order of magnitude in the present cosmic epoch and gain insights into</text>
<text><location><page_53><loc_14><loc_73><loc_82><loc_90></location>the evolving nature of dark energy, researchers have turned their attention to models that exhibit dynamic behaviour over cosmic history. These models operate on the assumption that the equation of state for dark energy ( w d ) has undergone temporal changes throughout the evolution of the universe. This paradigm shift has given rise to a multitude of dynamical dark energy (DDE) models, each characterized by a time-dependent EoS parameter. Among these models, the Chevallier-Polarski-Linder (CPL) model, [248, 249], has gained significant prominence. It is defined by the functional form,</text>
<formula><location><page_53><loc_37><loc_68><loc_82><loc_72></location>w CPL ( z ) = w 0 + w a z 1 + z , (1.41)</formula>
<text><location><page_53><loc_14><loc_44><loc_82><loc_67></location>where w 0 and w a are real numbers. This model is widely regarded as the most popular DDE model. The conventional approach often involves parameterizing our limited understanding of how dark energy behaves. This methodology found extensive application within the Dark Energy Task Force [250]. It has served as a practical benchmark for evaluating and contrasting the performance of various techniques aimed at investigating dark energy dynamics (see, for instance, [251]). The simplicity of the CPL parameterization belies its rich characteristics, as thoroughly explored by Linder [257]. Of particular note are the two parameters it introduces: w 0 , serving as a representation of the equation of state's present condition, and w a , which encapsulates its broader temporal dynamics.</text>
<text><location><page_53><loc_14><loc_38><loc_82><loc_43></location>An interesting proposition, detailed in references [249, 257], posits that the optimal way to characterize w a in relation to the derivative of w is through the following expression:</text>
<formula><location><page_53><loc_41><loc_35><loc_82><loc_37></location>w a = -2 w ' | z = 1 . (1.42)</formula>
<text><location><page_53><loc_14><loc_24><loc_82><loc_34></location>In this equation, w ' represents the derivative of w , defined as w ' ≡ dw d ln a , with " a " denoting the scale factor. While it may not encompass the full spectrum of potential dynamics [257-260], the CPL parameterization appears to strike a favorable balance for conducting a model-independent analysis. Other models with EoS dependent on z have been listed in table 1.3.</text>
<section_header_level_1><location><page_53><loc_17><loc_20><loc_47><loc_22></location>· Quintessence scalar field model</section_header_level_1>
<text><location><page_53><loc_14><loc_8><loc_82><loc_18></location>The quintessence scalar field model is a captivating and influential theoretical framework in the field of cosmology. The concept of a quintessence scalar field was initially presented by Ratra and Peebles [261] and independently by Wetterich [238] in order to facilitate the inflationary paradigm. In trying to grasp how the universe works on a big scale, the quintessence scalar</text>
<table>
<location><page_54><loc_18><loc_69><loc_88><loc_90></location>
<caption>TABLE 1.3: Various dark energy models with evolving equations of state, w ( z ) , investigated in the literature other than the CPL parametrization.</caption>
</table>
<text><location><page_54><loc_18><loc_33><loc_86><loc_58></location>field model is different from the usual cosmological constant model. It gives a dynamic and evolving explanation for what is causing the universe to expand. At its core, the quintessence scalar field model introduces a new dynamic component to the cosmic energy budget in the form of a scalar field, often denoted as Φ , that permeates the universe. This scalar field possesses unique properties, including a potential energy function V ( Φ ) and negative pressure, allowing it to act as a driving force for the accelerated expansion of the cosmos. Φ slowly rolls down to its potential V ( Φ ) , which leads to the dominant potential term over the kinetic term. Unlike the cosmological constant associated with the Λ CDM model, which represents a static form of dark energy, quintessence offers a dynamic alternative that evolves over time. The relevant action for quintessence field Φ is,</text>
<formula><location><page_54><loc_29><loc_28><loc_86><loc_32></location>S f ield = ∫ d 4 x √ -g [ R 16 π G -1 2 g µν ∂µ Φ ∂ν Φ -V ( Φ ) ] . (1.43)</formula>
<text><location><page_54><loc_18><loc_25><loc_80><loc_27></location>Stress-energy tensor of the quintessence field is given by the equation,</text>
<formula><location><page_54><loc_30><loc_20><loc_86><loc_23></location>T Φ µν = [ ∂µ Φ ∂ν Φ -1 2 g µν g αβ ∂α Φ ∂ β Φ -2 g µν V ( Φ ) ] . (1.44)</formula>
<text><location><page_54><loc_18><loc_15><loc_86><loc_19></location>From the above equation, we can write the quintessence field energy density ( ρ Φ ) and pressure ( p Φ ) and respectively given by,</text>
<formula><location><page_54><loc_39><loc_10><loc_86><loc_13></location>ρ Φ ≡ T 0 ( q ) 0 = 1 2 ( ˙ Φ ) 2 + V ( Φ ) , (1.45)</formula>
<text><location><page_55><loc_14><loc_88><loc_18><loc_90></location>and</text>
<formula><location><page_55><loc_28><loc_85><loc_82><loc_88></location>p Φ ≡ T 1 ( q ) 1 = T 2 ( q ) 2 = T 3 ( q ) 3 = 1 2 ( ˙ Φ ) 2 -V ( Φ ) . (1.46)</formula>
<text><location><page_55><loc_17><loc_83><loc_54><loc_85></location>The Klein-Gordon(KG) equation for Φ is,</text>
<formula><location><page_55><loc_38><loc_79><loc_82><loc_81></location>¨ Φ + 3 H ˙ Φ + V ' ( Φ ) = 0. (1.47)</formula>
<text><location><page_55><loc_17><loc_75><loc_45><loc_77></location>The EoS parameter is therefore,</text>
<formula><location><page_55><loc_37><loc_70><loc_82><loc_73></location>w Φ = p Φ ρ Φ = ˙ Φ 2 -2 V ( Φ ) ˙ Φ 2 + 2 V ( Φ ) . (1.48)</formula>
<text><location><page_55><loc_14><loc_60><loc_82><loc_68></location>Therefore, the range of evolution of w Φ is, -1 < w Φ < 1. Quintessence models are categorized into three distinct classes based on the characteristics of the potential energy function V ( Φ ) . This classification is determined by the specific nature of the potential of the scalar field.</text>
<unordered_list>
<list_item><location><page_55><loc_17><loc_54><loc_82><loc_58></location>· V ( Φ ) << ˙ Φ 2 , w Φ ≈ 1 = ⇒ ρ Φ ∼ a -6 , resembling stiff matter, and it does not play a role in contributing to dark energy.</list_item>
<list_item><location><page_55><loc_17><loc_49><loc_82><loc_53></location>· V ( Φ ) >> ˙ Φ 2 , w Φ ≈ -1 = ⇒ ρ Φ ∼ constant, resembling cosmological constant.</list_item>
<list_item><location><page_55><loc_17><loc_44><loc_82><loc_47></location>· -1 < w Φ < 1 = ⇒ ρ Φ ∼ a -m , leads to accelerated cosmic expansion for the range 0 < m < 2 [189].</list_item>
</unordered_list>
<text><location><page_55><loc_14><loc_32><loc_82><loc_41></location>The classification of quintessence fields into thawing and freezing models is a significant concept in the study of dark energy and its role in the evolution of the universe. These models are categorized based on the behaviour of the effective EoS parameter, w of the quintessence field as the universe expands over time [274].</text>
<section_header_level_1><location><page_55><loc_17><loc_28><loc_35><loc_30></location>· Thawing Models:</section_header_level_1>
<text><location><page_55><loc_14><loc_8><loc_82><loc_27></location>Thawing models are characterized by a quintessence field with an effective EoS parameter, w that starts out as almost a constant close to -1 (similar to the cosmological constant), which corresponds to a vacuum energy with w = -1. Over time, the equation of state parameter thaws or gradually evolves into a more dynamic, time-dependent value that deviates from -1. Thawing models are particularly interesting because they represent quintessence fields that initially behave like a cosmological constant but then change their behaviour, which can have implications for the evolution of the universe.</text>
<table>
<location><page_56><loc_26><loc_46><loc_78><loc_90></location>
<caption>TABLE 1.4: Various quintessence scalar field potentials V ( Φ ) explored in prior research.</caption>
</table>
<section_header_level_1><location><page_56><loc_21><loc_36><loc_39><loc_37></location>· Freezing Models:</section_header_level_1>
<text><location><page_56><loc_18><loc_17><loc_86><loc_34></location>In contrast, freezing models are characterized by a quintessence field for which the effective EoS parameter, w settles down to a constant value close to -1, but this happens relatively late in the evolution of the universe. Initially, the quintessence field may have a dynamic behaviour, but it eventually reaches a stage where its equation of state parameter becomes nearly indistinguishable from that of a cosmological constant. One important subclass of freezing models is known as trackers , which are of particular interest [265, 275, 276].</text>
<text><location><page_56><loc_18><loc_11><loc_86><loc_17></location>In tracker models, the energy density of the scalar field evolves almost parallel to the energy density of dark matter for most of cosmic history without dominating dark matter. However, it eventually freezes to a value greater</text>
<text><location><page_57><loc_14><loc_80><loc_82><loc_90></location>than the dark matter density at a later stage. Thakur, Nautiyal, Sen, and Seshadri [402] have undertaken a comparison between a thawing model and freezing models that exhibit tracking behaviour. Their investigation revolves around the compatibility of these models with empirical data and offers valuable insights into their observational viability.</text>
<text><location><page_57><loc_14><loc_60><loc_82><loc_79></location>It is worth noting that not all quintessence fields neatly fit into either of these two categories, and there can be variations and more complex behaviours. The classification into thawing and freezing models is primarily a way to categorize and study the different possible behaviours of quintessence fields in the context of cosmic expansion and their ability to address issues such as the coincidence problem [277], which relates to why dark energy and dark matter densities are of similar magnitude in the present universe. Both thawing and freezing models have been extensively studied in cosmology to understand their implications for the evolution of the universe.</text>
<text><location><page_57><loc_14><loc_50><loc_82><loc_60></location>Table 1.4 provides an exhaustive compilation of the various scalar field potentials that have been explored within the context of quintessence models. To explore further into the vast body of research about late-time cosmic acceleration, which includes various versions of quintessence potentials, one may refer to [247, 262, 263, 265, 274, 275, 277-287].</text>
<section_header_level_1><location><page_57><loc_17><loc_46><loc_38><loc_48></location>· Phantom field model</section_header_level_1>
<text><location><page_57><loc_14><loc_30><loc_82><loc_44></location>The idea of the phantom field in the realm of dark energy was initially introduced by Caldwell [288]. What sets the phantom field apart from the quintessence field is its distinctive feature of possessing negative kinetic energy( X ). Therefore, the Lagrangian density for phantom field ϕ is, L p = -X -V ( ϕ ) = -∂µϕ∂ µ ϕ /2 -V ( ϕ ) (with metric signature + ---). The corresponding action governing the behaviour of the phantom field can be expressed as follows,</text>
<formula><location><page_57><loc_35><loc_26><loc_82><loc_28></location>S = ∫ d 4 x √ -g [ -X -V ( ϕ )] . (1.49)</formula>
<text><location><page_57><loc_14><loc_22><loc_79><loc_23></location>The phantom field energy density and pressure are respectively given as,</text>
<formula><location><page_57><loc_40><loc_16><loc_82><loc_20></location>ρ p = -˙ ϕ 2 2 + V ( ϕ ) , (1.50)</formula>
<text><location><page_57><loc_14><loc_13><loc_18><loc_15></location>and</text>
<formula><location><page_57><loc_40><loc_10><loc_82><loc_14></location>p p = -˙ ϕ 2 2 -V ( ϕ ) . (1.51)</formula>
<text><location><page_58><loc_18><loc_88><loc_43><loc_90></location>The equations of motion are,</text>
<formula><location><page_58><loc_44><loc_84><loc_86><loc_86></location>¨ ϕ + 3 H ˙ ϕ = + V , ϕ . (1.52)</formula>
<text><location><page_58><loc_18><loc_80><loc_82><loc_82></location>The equation of state parameter of dark energy for the phantom field is,</text>
<formula><location><page_58><loc_41><loc_75><loc_86><loc_78></location>w de = p p ρ p = ˙ ϕ 2 + 2 V ( ϕ ) ˙ ϕ 2 -2 V ( ϕ ) . (1.53)</formula>
<text><location><page_58><loc_18><loc_61><loc_86><loc_73></location>For w de to be less than -1, V ( ϕ ) >> ˙ ϕ 2 . A phantom field, propelled by its negative kinetic energy will tend to run up potential energy. The consequence is an extremely swift expansion of the universe, reaching infinite extent within a finite time. This phenomenon is termed the Big Rip . This is characterized by the infinite growth of both the volume and the expansion rate.</text>
<text><location><page_58><loc_18><loc_41><loc_86><loc_60></location>A potential for the phantom field featuring a maximum value(for example V ( ϕ ) = V 0 [ cosh ( αϕ m pl )] -1 , where α is a constant) has the capability to avert the occurrence of the Big Rip. For ˙ ϕ = 0, the field eventually reaches its maximum position, following a damped oscillatory phase, causing the equation of state parameter, w de , to attain a value of -1. Consequently, this behaviour can reinstate the scenario of the cosmological constant. Quintom models is referred to the scalar field models, wherein the behaviour of the equation of state (EoS) parameter mirrors that of the phantom field [289292].</text>
<section_header_level_1><location><page_58><loc_21><loc_37><loc_42><loc_39></location>· Chaplygin Gas model</section_header_level_1>
<text><location><page_58><loc_18><loc_27><loc_86><loc_35></location>The Chaplygin gas model, named after the Russian mathematician Sergei Alekseevich Chaplygin, is a theoretical model that was originally introduced by Chaplygin in 1904 in aerodynamics to describe the lift force on an object moving through a gas.</text>
<text><location><page_58><loc_18><loc_21><loc_86><loc_27></location>This model was reintroduced in cosmology by Kamenshchik, Moschella and Pasquir [293] in the context of cosmic acceleration. The Chaplygin gas model is characterized by an equation of state given by:</text>
<formula><location><page_58><loc_48><loc_16><loc_86><loc_19></location>p = -A ρ , (1.54)</formula>
<text><location><page_58><loc_18><loc_10><loc_86><loc_14></location>where A is a positive constant and p and ρ are respectively pressure and energy density in a comoving frame with ρ > 0. This equation</text>
<text><location><page_59><loc_14><loc_86><loc_82><loc_90></location>The Chaplygin gas model provides a smooth transition between different phases of cosmic evolution. Specifically, it can smoothly interpolate between:</text>
<unordered_list>
<list_item><location><page_59><loc_17><loc_83><loc_58><loc_86></location>· A dust dominated phase where ρ ∼ √ Ba -3 .</list_item>
<list_item><location><page_59><loc_17><loc_79><loc_60><loc_81></location>· A de Sitter phase, where the pressure p ∼ -ρ .</list_item>
<list_item><location><page_59><loc_17><loc_74><loc_82><loc_78></location>· An intermediate regime with the equation of state for stiff matter the, where p = ρ .</list_item>
</unordered_list>
<text><location><page_59><loc_14><loc_69><loc_82><loc_72></location>In this model, once a universe undergoing expansion enters a phase of acceleration, it is incapable of reverting to a state of deceleration.</text>
<text><location><page_59><loc_14><loc_54><loc_82><loc_68></location>This model has been generalized by Bento, Bertolami and Sen [294]. The authors proposed that the change in the characteristics of the elusive energy density could be governed by alterations in the equation of state of the underlying fluid, rather than relying on adjustments to the potential. This approach offers a means to circumvent the intricate fine-tuning problem. This is achieved by introducing a more flexible equation of state that allows for a wider range of behaviours. The generalized equation of state is given as:</text>
<formula><location><page_59><loc_42><loc_48><loc_82><loc_52></location>p = -( 1 ρ ) α , (1.55)</formula>
<text><location><page_59><loc_14><loc_41><loc_82><loc_47></location>where α is the parameter with 0 < α < 1, that determines the behaviour of the gas. The energy density ρ evolves with the scale factor a of the universe as,</text>
<formula><location><page_59><loc_38><loc_37><loc_82><loc_41></location>ρ = ( A + B a 3 ( 1 + α ) ) 1 1 + α , (1.56)</formula>
<text><location><page_59><loc_76><loc_32><loc_76><loc_34></location≯</text>
<text><location><page_59><loc_14><loc_32><loc_82><loc_36></location>where B is the integration constant. In this scenario, instead of stiff matter, the intermediate phase is soft matter with equation of state, p = αρ ( α = 1).</text>
<text><location><page_59><loc_14><loc_17><loc_82><loc_32></location>This model has been explored in cosmology as a way to unify dark matter and dark energy within a single framework and has connections to brane theory and supersymmetry in theoretical physics [295]. While the Chaplygin gas models have been invalidated by data related to temperature fluctuations in the cosmic microwave background [296, 297], there is a limited range of parameter values, specifically 0 ≤ α ≤ 0.2, within which the generalized Chaplygin gas models are still considered plausible [296].</text>
<section_header_level_1><location><page_59><loc_17><loc_14><loc_34><loc_15></location>· K-essence model</section_header_level_1>
<text><location><page_59><loc_14><loc_8><loc_82><loc_12></location>The K-essence model is a theoretical framework in cosmology that introduces a scalar field with a non-standard kinetic term to explain the dynamics of</text>
<text><location><page_60><loc_18><loc_69><loc_86><loc_90></location>dark energy and, in some cases, an early inflation as well. It offers an alternative to more traditional scalar field models, such as quintessence, where the potential energy of the scalar field plays a central role. In the K-essence model, the kinetic energy of the scalar field dominates, hence the name Kessence (kinetic essence). At its inception, it was introduced in the context of inflation [298, 299]. Later, Chiba et al. introduced it when considering latetime acceleration [300]. The generalized k-essence for dark energy was proposed by Armendariz-Picon et al. [301, 302]. In this model, the Lagrangian density is written in the form of pressure as P ( ϕ , X ) , where X = -1 2 ( ∇ ϕ ) 2 is the kinetic energy for the scalar field ϕ . Therefore, the action is given as,</text>
<formula><location><page_60><loc_41><loc_65><loc_86><loc_67></location>S = ∫ d 4 x √ -gP ( ϕ , X ) . (1.57)</formula>
<text><location><page_60><loc_18><loc_61><loc_46><loc_62></location>The energy density is given by,</text>
<formula><location><page_60><loc_45><loc_56><loc_86><loc_59></location>ρ = 2 X ∂ P ∂ X -P . (1.58)</formula>
<text><location><page_60><loc_21><loc_53><loc_60><loc_54></location>Therefore, EoS parameter can be written as,</text>
<formula><location><page_60><loc_45><loc_47><loc_86><loc_51></location>w = P 2 X ∂ P ∂ X -P . (1.59)</formula>
<text><location><page_60><loc_18><loc_39><loc_86><loc_45></location>For 2 X ∂ P ∂ X = 0, this model reduces to cosmological constant model with w = -1 [303]. More detailed knowledge about the K-essence model can be found in the references [304-307].</text>
<section_header_level_1><location><page_60><loc_21><loc_36><loc_41><loc_37></location>· Tachyon field model</section_header_level_1>
<text><location><page_60><loc_18><loc_9><loc_86><loc_34></location>Tachyons are hypothetical particles that are often characterized by having imaginary mass. During the decay process of D-branes, a state emerges that behaves like a gas lacking pressure but possessing finite energy density. It resembles classical dust [308-315]. Tachyons exhibit an equation of state (EoS) parameter that smoothly varies within the range of -1 < w < 0. This has piqued the interest of cosmologists, prompting them to consider tachyons as a plausible contender of dark energy [316]. For a comprehensive analysis of tachyonic dark energy models, with a focus on their ability to generate latetime cosmic acceleration, refer to [317-322]. The tachyon's squared mass is intrinsically negative and stabilizes at the peaks of its associated scalar field potential. This state experiences infinitesimal perturbations, ultimately resulting in a tachyon condensation process characterized by a descent from</text>
<text><location><page_61><loc_14><loc_86><loc_82><loc_90></location>the peaks, leading to the attainment of a real mass. The action for tachyon field ( ψ ) with potential V ( ψ ) can be expressed as follows:</text>
<formula><location><page_61><loc_30><loc_82><loc_82><loc_84></location>S = -∫ d 4 xV ( ψ ) √ -det ( g µν + ∂µψ∂νψ ) . (1.60)</formula>
<text><location><page_61><loc_14><loc_78><loc_45><loc_79></location>The wave equation for this field is,</text>
<formula><location><page_61><loc_36><loc_73><loc_82><loc_76></location>¨ ψ 1 -˙ ψ 2 + 3 H ˙ ψ + 1 V dV d ψ = 0. (1.61)</formula>
<text><location><page_61><loc_14><loc_69><loc_66><loc_71></location>The energy density and pressure are respectively given as,</text>
<formula><location><page_61><loc_38><loc_64><loc_82><loc_67></location>ρψ = V ( ψ ) √ 1 -˙ ψ 2 , (1.62)</formula>
<formula><location><page_61><loc_38><loc_61><loc_82><loc_63></location>p ψ = -V ( ψ ) √ 1 -˙ ψ 2 . (1.63)</formula>
<text><location><page_61><loc_14><loc_57><loc_45><loc_59></location>We can write the EoS parameter as,</text>
<formula><location><page_61><loc_40><loc_52><loc_82><loc_56></location>w = p ψ ρψ = ˙ ψ 2 -1. (1.64)</formula>
<text><location><page_61><loc_14><loc_47><loc_82><loc_50></location>Therefore, the criterion for the universe to experience accelerated expansion is when ˙ ψ 2 < 2/3.</text>
<section_header_level_1><location><page_61><loc_17><loc_43><loc_42><loc_44></location>· Holographic Dark Energy</section_header_level_1>
<text><location><page_61><loc_14><loc_14><loc_82><loc_41></location>The holographic dark energy (HDE) draws its inspiration from the holographic principle, a concept rooted in quantum gravity theory. This model aims to shed light on the mysterious nature of dark energy and its role in the accelerated expansion of the universe by establishing a connection between dark energy and the information content residing on the boundary. The holographic principle, originally proposed by 't Hooft [323] and Susskind [324], suggests that the physical properties and degrees of freedom within a given region of space can be entirely encoded on its boundary rather than within the volume itself. This concept is closely linked to the notion of entropy and has its roots in the work of Bekenstein on black hole entropy bounds [325, 326]. According to this bound, there exists a connection between the shortdistance ultraviolet (UV) cut-off and the long-distance infrared (IR) cut-off due to the constraint that the total quantum zero-point energy of a system</text>
<text><location><page_62><loc_18><loc_86><loc_86><loc_90></location>should not exceed the mass of black holes of the same size [327]. This constraint can be expressed as:</text>
<formula><location><page_62><loc_46><loc_82><loc_86><loc_84></location>L 3 ρ Λ ≤ LM 2 p , (1.65)</formula>
<text><location><page_62><loc_18><loc_70><loc_86><loc_80></location>where M p = ( 8 π G ) -1/2 is the reduced Planck mass, ρ Λ represents the quantum zero-point energy density determined by the UV cut-off, and L is the length scale of the system size. The IR cut-off is the length for which this inequality saturates. The largest allowable value for L is the one that makes this inequality reach its limit. Therefore,</text>
<formula><location><page_62><loc_44><loc_66><loc_86><loc_68></location>ρ Λ = 3 C 2 M 2 p L -2 . (1.66)</formula>
<text><location><page_62><loc_18><loc_58><loc_86><loc_63></location>In the context of dark energy, the holographic principle is applied by introducing the holographic energy density ( ρ H ) with the following expression [328],</text>
<formula><location><page_62><loc_45><loc_55><loc_86><loc_57></location>ρ H = 3 c 2 M 2 p L -2 . (1.67)</formula>
<text><location><page_62><loc_18><loc_40><loc_86><loc_54></location>Here system size is the size of the observable universe and C 2 is a dimensionless coupling parameter. The cosmic horizon serves as the IR (infrared) cut-off. Similar concepts and ideas were explored in references [329, 330]. Various approaches can be found in the literature, each with its own choice of the IR cut-off length scale. Such examples include the particle horizon [331, 332], the future event horizon [328, 333-338] and the Hubble horizon [339-341] as IR cut-off.</text>
<section_header_level_1><location><page_62><loc_18><loc_34><loc_52><loc_36></location>1.4 Outline of the thesis</section_header_level_1>
<text><location><page_62><loc_18><loc_9><loc_86><loc_32></location>The primary focus of this thesis centers on delving into the thermodynamic characteristics of diverse cosmological models. An assessment of the models' viability has been carried out by applying the Generalized Second Law (GSL), which asserts that the total entropy comprising both the horizon and the encompassed fluid must never decrease. Given the evolving nature of the universe, our approach has involved working with the apparent horizon. We have assumed a state of thermodynamic equilibrium between this apparent horizon and the fluid inside the horizon. In this equilibrium state, we have adopted the Hayward-Kodama temperature as the horizon temperature. Chapters 3-6 contain the main research work. Chapters 3,4 and 5 are focused on GSL test. Chapter 6 is focused on the stability analysis of a</text>
<text><location><page_63><loc_14><loc_88><loc_33><loc_90></location>cosmological model.</text>
<text><location><page_63><loc_14><loc_75><loc_82><loc_85></location>In the second chapter, we have discussed about apparent horizon, blackhole thermodynamics and their analogies in the dynamical apparent horizon in cosmology. A thorough description of how to calculate the rate of change of entropy has been prescribed. Also, we have touched upon the subject of the thermodynamic stability of the cosmological model.</text>
<text><location><page_63><loc_14><loc_31><loc_82><loc_72></location>In the third chapter, our study conducts a thorough comparative analysis between thawing and freezing models, particularly with regard to their adherence to the Generalized Second Law (GSL) of thermodynamics. In our comprehensive evaluation of the total entropy ( S tot), we incorporate both the entropy of the horizon and the entropy stemming from the matter enclosed within the horizon. To facilitate this investigation, we employ a simple ansatz proposed by Carvalho et al. [287] to model the dynamic evolution of the energy density within the quintessence field, allowing us to pinpoint the parameter range ( α ) associated with thawing and freezing behaviours. Our findings reveal a common inconsistency with the GSL in both type of models. In the context of freezing models, this GSL breakdown is traced back to a remote past, corresponding to a redshift of approximately z ∼ 10 4 . During this distant epoch, a quintessence model along with cold dark matter fails to adequately account for the evolution of the universe, as the dominant contribution comes from radiation distribution. This suggests that the GSL breakdown may not hold true under these circumstances. Conversely, for thawing models, this unusual violation of the GSL is anticipated to manifest in a finite future. The key implication here is that freezing models appear to enjoy better thermodynamic favourability when compared with their thawing counterparts.</text>
<text><location><page_63><loc_14><loc_9><loc_82><loc_28></location>In the fourth chapter, we explore GSL to models of the universe filled with radiation and dust, assuming the universe is flat, homogeneous and isotropic in the framework of Brans-Dicke theory in Einstein frame . When it comes to a universe dominated by radiation, the solutions in Brans-Dicke theory with a positive value for the BD parameter ω , do not follow GSL. But when ω has a negative value within a certain range, it does match the thermodynamic requirements. And that is exciting because for cosmic acceleration, as per observation, one needs this negative ω value. Now, if we switch our focus to a universe dominated by dust (like galaxies and matter), the model does</text>
<text><location><page_64><loc_18><loc_84><loc_86><loc_90></location>satisfy GSL when ω is a small negative number within a specific range. This range of ω significantly overlaps with the range necessary for achieving accelerated expansion without the need for any exotic matter.</text>
<text><location><page_64><loc_18><loc_41><loc_86><loc_83></location>In the fifth chapter, we studied the thermodynamic viability of some dark energy models reconstructed through the cosmological jerk parameter. As the deceleration parameter, q evolves, we become interested in the next-order derivative called the jerk parameter , represented as j . It tells us how q changes over time. In this study, we select some of these models from existing literature and evaluate them in terms of their thermodynamic viability. By reconstructing the jerk parameter, it is entirely possible to find models that satisfy the laws of thermodynamics. Among the four models tested for a non-interacting scenario, only one (model IV), which has an inverse relationship with (1 + z ), shows a decrease in entropy in the future (at z < 0). This decrease is particularly significant near the present epoch ( z = 0). All the other models, including one that allows for interaction in the dark sector, pass the GSL. Model I, lacking an explicit dependence on z , satisfies the GSL but undergoes a sudden entropy surge in the future ( z < 0). However, any analysis in terms of z is not very sound for negative values of z . Models II and III, dependent on ( 1 + z ) and ( 1 + z ) 2 respectively, as well as Model V with interaction in the matter sector, exhibit favorable behaviour across a broad parameter range within a 3 σ confidence interval. Overall, from the past to the present epoch, all models demonstrate satisfactory behaviour.</text>
<text><location><page_64><loc_18><loc_9><loc_86><loc_38></location>In the sixth chapter, we conducted a thermodynamic stability analysis on a model designed to mimic the Λ CDMmodel for the current state of the universe. In this analysis, we took into account the evolving horizon and considered the Hayward-Kodama temperature as the temperature of the horizon. In thermodynamics, the stability of an equilibrium system hinges on having a positive thermal capacity and compressibility. This principle also applies to the matter content within a cosmological system. However, in our current scenario, the specific heat capacity ( C V ) turns out to be negative. This suggests that the model is likely to exhibit thermodynamic instability. The significant outcome of our investigation is that the matter content undergoes a phase transition as the universe transitions from a decelerated to an accelerated state of expansion. This phase transition is notably a second-order transition, evident by the discontinuity in C V . The deceleration parameter, q serves as the order parameter in this context. This feature goes missing if</text>
<text><location><page_65><loc_14><loc_86><loc_82><loc_90></location>we consider Hawking temperature, which neglects the fact that the apparent horizon is evolving over time.</text>
<text><location><page_65><loc_14><loc_82><loc_82><loc_85></location>The final chapter, chapter 7, concludes the research in this thesis and provides some future aspects.</text>
<section_header_level_1><location><page_67><loc_14><loc_82><loc_30><loc_84></location>Chapter 2</section_header_level_1>
<section_header_level_1><location><page_67><loc_14><loc_71><loc_76><loc_78></location>Cosmological Apparent Horizon and Thermodynamics</section_header_level_1>
<section_header_level_1><location><page_67><loc_14><loc_63><loc_64><loc_65></location>2.1 Cosmological Apparent Horizon</section_header_level_1>
<text><location><page_67><loc_14><loc_53><loc_82><loc_61></location>We shall discuss the cosmological apparent horizon by drawing an analogy with black hole apparent horizon. Therefore, let us first take some notes on black hole apparent horizon and then delve into the apparent horizon in FRW cosmologies.</text>
<section_header_level_1><location><page_67><loc_14><loc_48><loc_42><loc_50></location>2.1.1 Apparent Horizon</section_header_level_1>
<text><location><page_67><loc_14><loc_28><loc_82><loc_46></location>In simple words, a horizon can be described as "a frontier between things observable and things unobservable" [342]. The key feature of a black hole spacetime is the presence of an event horizon, a boundary that separates the black hole from the external observers and conceals internal events. This hypersurface consists of a congruence of null geodesics, or null generators, which are crucial in comprehending the behavior of the horizon. To grasp the overall behavior of the horizon, it becomes essential to study how these generators behave. In this thesis, this topic shall be briefly discussed. For more details, we refer to[404].</text>
<section_header_level_1><location><page_67><loc_14><loc_24><loc_50><loc_25></location>2.1.1.1 Congruence of Null Geodesics</section_header_level_1>
<section_header_level_1><location><page_67><loc_17><loc_20><loc_62><loc_22></location>· Affinely parametrized null geodesic congruence:</section_header_level_1>
<text><location><page_67><loc_14><loc_8><loc_82><loc_18></location>A null geodesic refers to a path followed by a massless particle (such as a photon, which has zero rest mass) in the spacetime described by relativistic theory of gravity. Therefore, on spacetime manifold, its tangent, denoted by l µ is null,i.e., l µ l µ = 0. The geodesic equation it satisfies is given by the equation,</text>
<formula><location><page_68><loc_46><loc_86><loc_86><loc_87></location>l µ ; ν l ν = λ l µ , (2.1)</formula>
<text><location><page_68><loc_18><loc_77><loc_86><loc_84></location>here λ serves as a parameter along the curve. The geodesic equation signifies that the tangent is parallelly transported to itself as one follows the geodesic. The choice of the parameter λ allows for simplification of the geodesic equation to a more convenient form without loss of generality as,</text>
<formula><location><page_68><loc_47><loc_73><loc_86><loc_74></location>l µ ; ν l ν = 0. (2.2)</formula>
<text><location><page_68><loc_21><loc_69><loc_82><loc_70></location>In terms of Christoffel symbols, the above equation can be written as,</text>
<formula><location><page_68><loc_42><loc_63><loc_86><loc_67></location>dx α d λ 2 + Γ α µν dx µ d λ dx ν d λ = 0. (2.3)</formula>
<text><location><page_68><loc_18><loc_58><loc_86><loc_62></location>The variable λ is therefore an affine parameter of the affinely parametrized geodesic equation(2.3).</text>
<text><location><page_68><loc_18><loc_37><loc_86><loc_58></location>Consider an open region M in the spacetime manifold. A congruence of curves represents a family of curves in which every point within M is traversed by one and only one curve from the family. The tangents to these curves define a vector field on M and conversely, any continuous vector field within M generates a congruence of curves with the tangents of the field. When the field of tangents is smooth, we refer to the congruence as smooth. In particular, we can focus on a smooth congruence of null geodesics whose tangents are represented by λ within the open region M . To mark different geodesics within the congruence in M , we consider another parameter ζ . We now define the deviation vector with components η µ ≡ ∂ x µ ∂ζ .</text>
<text><location><page_68><loc_18><loc_20><loc_86><loc_36></location>Though by construction, we see l µ ηµ = 0, but that does not mean η µ is orthogonal to the curve, since l µ is a null vector. However, we can limit to deviation vectors that are considered equivalent if they vary solely by a component along l µ . The tangent space, comprising all vectors that are orthogonal in this manner to l µ , forms a 2-dimensional vector space. We can also explore its dual space and the set of tensors constructed using these vectors. The geodesic deviation vector follows the geodesic deviation equation, given by,</text>
<formula><location><page_68><loc_41><loc_18><loc_86><loc_20></location>( D λ ) 2 η µ = -R µ ναβ u ν η α u β , (2.4)</formula>
<text><location><page_68><loc_18><loc_8><loc_86><loc_17></location>where R µ ναβ is the Riemann tensor. D λ is covariant derivative operator, D λ η µ = d d λ η µ + Γ µ αβ dx α d λ η β . This mathematical equation expresses how the behaviour of geodesics, rather the deviations of nearby geodesics is impacted by the spacetime curvature.</text>
<text><location><page_69><loc_14><loc_84><loc_82><loc_90></location>We can obtain the famous Raychaudhuri equation for null geodesics by deriving the rate of change of the gradient l µ ; µ along the geodesic. Let us consider a tensor field [24, 347],</text>
<formula><location><page_69><loc_43><loc_82><loc_82><loc_83></location>B µν ≡ l µ ; ν . (2.5)</formula>
<text><location><page_69><loc_14><loc_76><loc_82><loc_80></location>B µν satisfies l µ ; µ η ν = B µ ν η µ . B µν is orthogonal to l µ , therefore, B µν l µ = B µν l ν = 0. So it has components only along the transverse of l µ .</text>
<text><location><page_69><loc_17><loc_74><loc_78><loc_76></location>Now the spatial metric in 2-space orthogonal to l µ can be defined as,</text>
<formula><location><page_69><loc_38><loc_70><loc_82><loc_72></location>h µν ≡ g µν + l µ n ν + l ν n µ , (2.6)</formula>
<text><location><page_69><loc_14><loc_60><loc_82><loc_68></location>where n α is another null vector and normalized as l α n α = -1. The selection of n α is not unique. The only fixed quantity is the null congruence with tangent l α . However, geometric and physically relevant quantities remain independent of the choice of n α .</text>
<text><location><page_69><loc_17><loc_58><loc_41><loc_60></location>The trace of the tensor B µν ,</text>
<formula><location><page_69><loc_37><loc_54><loc_82><loc_56></location>θ ≡ B µ µ = g µν B µν = l µ ; µ , (2.7)</formula>
<text><location><page_69><loc_14><loc_48><loc_82><loc_52></location>is called expansion of the affinely-parametrized congruence. The expansion tensor is defined as,</text>
<formula><location><page_69><loc_43><loc_45><loc_82><loc_48></location>θ αβ ≡ θ 2 h αβ . (2.8)</formula>
<text><location><page_69><loc_14><loc_40><loc_82><loc_44></location>The transverse tensor ˜ B αβ can be written as sum of symmetric and antisymmetric parts,</text>
<formula><location><page_69><loc_40><loc_38><loc_82><loc_40></location>˜ B αβ = ˜ B ( αβ ) + ˜ B [ αβ ] . (2.9)</formula>
<text><location><page_69><loc_14><loc_35><loc_69><loc_37></location>It can be further decomposed into trace and traceless parts as,</text>
<formula><location><page_69><loc_36><loc_30><loc_82><loc_33></location>˜ B αβ = ( θ 2 h αβ + σ αβ ) + ω αβ . (2.10)</formula>
<text><location><page_69><loc_14><loc_26><loc_63><loc_28></location>In the above equation, σ αβ is known as the shear tensor,</text>
<formula><location><page_69><loc_40><loc_21><loc_82><loc_24></location>σ αβ ≡ ˜ B ( αβ ) -θ 2 h αβ , (2.11)</formula>
<text><location><page_69><loc_14><loc_18><loc_50><loc_20></location>and ω αβ is known as the vorticity tensor,</text>
<formula><location><page_69><loc_43><loc_14><loc_82><loc_16></location>ω αβ ≡ ˜ B [ αβ ] . (2.12)</formula>
<text><location><page_70><loc_18><loc_88><loc_49><loc_90></location>The shear and vorticity scalars are,</text>
<formula><location><page_70><loc_37><loc_84><loc_86><loc_86></location>σ 2 = σ αβ σ αβ , ω 2 = ω αβ ω αβ . (2.13)</formula>
<text><location><page_70><loc_18><loc_78><loc_86><loc_82></location>The Raychaudhuri equation that governs the expansion along affinely parametrized null geodesic,</text>
<formula><location><page_70><loc_38><loc_73><loc_86><loc_76></location>d θ d λ = -θ 2 2 -σ 2 -ω 2 -R αβ l α l β . (2.14)</formula>
<text><location><page_70><loc_18><loc_57><loc_86><loc_71></location>The other choices of n α do not affect this equation. The term d θ d λ < 0 implies that expansion decreases with the evolution of congruence, which means null rays will be focused, and d θ d λ > 0 implies that expansion increases with the evolution of congruence, which means null rays will be defocused. Therefore, this equation illustrates how the focusing or defocusing of null rays occurs due to the combined effects of expansion, shear, vorticity, and matter.</text>
<section_header_level_1><location><page_70><loc_21><loc_53><loc_70><loc_54></location>· Non-affinely parametrized null geodesic congruence:</section_header_level_1>
<text><location><page_70><loc_18><loc_47><loc_86><loc_51></location>The geodesic equation for non-affinely parametrized null geodesic congruence is,</text>
<formula><location><page_70><loc_46><loc_45><loc_86><loc_47></location>l µ ; ν l ν = κ l µ . (2.15)</formula>
<text><location><page_70><loc_18><loc_42><loc_50><loc_43></location>The expansion scalar takes the form,</text>
<formula><location><page_70><loc_47><loc_38><loc_86><loc_40></location>θ = l µ ; µ -κ . (2.16)</formula>
<text><location><page_70><loc_18><loc_34><loc_77><loc_36></location>The Raychaudhuri equation transforms into the modified form as,</text>
<formula><location><page_70><loc_37><loc_29><loc_86><loc_32></location>d θ d λ = κθ -θ 2 2 -σ 2 -ω 2 -R αβ l α l β . (2.17)</formula>
<text><location><page_70><loc_18><loc_15><loc_86><loc_27></location>A compact and orientable surface possesses two distinct directions perpendicular to it, representing outgoing and ingoing null rays. When a spherical symmetry is present, it naturally prompts an examination of congruences formed by radial outgoing and ingoing null geodesics with their respective tangent fields l α and n α . For non-affinely parametrized congruence of null geodesics, the expansion of null rays l α is given by the following equation,</text>
<formula><location><page_70><loc_34><loc_9><loc_86><loc_13></location>θ l ≡ h αβ l β ; α = [ g αβ + l α n β + l β n α ( l µ n ν g µν ) ] l β ; α . (2.18)</formula>
<text><location><page_71><loc_14><loc_86><loc_82><loc_90></location>In the above mathematical expression, h αβ serves as a projection metric onto 2-D hypersurface.</text>
<section_header_level_1><location><page_71><loc_14><loc_82><loc_58><loc_83></location>2.1.1.2 Definitions pertain to closed 2-surfaces</section_header_level_1>
<text><location><page_71><loc_14><loc_76><loc_82><loc_80></location>In the subsequent definitions, θ l and θ n represent the outgoing and ingoing future directed null geodesic congruences respectively.</text>
<unordered_list>
<list_item><location><page_71><loc_17><loc_71><loc_82><loc_75></location>· Normal Surface : Future-directed outgoing null ray is diverging but ingoing null ray is converging,i.e., θ l > 0 and θ n < 0.</list_item>
<list_item><location><page_71><loc_17><loc_66><loc_82><loc_69></location>· Trapped surface : Both the future-directed null rays are converging,i.e., θ l < 0 and θ n < 0.</list_item>
<list_item><location><page_71><loc_17><loc_62><loc_70><loc_64></location>· Marginally outer trapped surface(MOTS) : θ l = 0 and θ n < 0.</list_item>
<list_item><location><page_71><loc_17><loc_57><loc_82><loc_61></location>· Anti-trapped surface : Both the future-directed null rays are diverging,i.e., θ l > 0 and θ n > 0.</list_item>
<list_item><location><page_71><loc_17><loc_54><loc_43><loc_55></location>· Untrapped surface : θ l θ n < 0.</list_item>
<list_item><location><page_71><loc_17><loc_50><loc_73><loc_52></location>· Marginally outer trapped tube : 3-D surface foliated by MOTSs.</list_item>
</unordered_list>
<text><location><page_71><loc_14><loc_43><loc_82><loc_49></location>The closure of a surface, often a three-surface that is foliated by marginal surfaces, is what is known as a future apparent horizon. The future apparent horizon is characterized by specific conditions,</text>
<formula><location><page_71><loc_45><loc_36><loc_82><loc_40></location>θ l = 0, θ n < 0. (2.19)</formula>
<text><location><page_71><loc_14><loc_24><loc_82><loc_34></location>Therefore, on the apparent horizon, the future-directed outgoing null geodesics stop transmitting outward. This differs from the event horizon in the sense that, it is a quasi-local horizon. It is to be noted that, the inner trapping horizon can be obtained by switching l α , n α and reversing the inequality sign.</text>
<section_header_level_1><location><page_71><loc_14><loc_20><loc_58><loc_21></location>2.1.1.3 Apparent Horizon in FRW Cosmologies</section_header_level_1>
<text><location><page_71><loc_14><loc_8><loc_82><loc_18></location>The use of notation and terminology can become perplexing when transitioning between black hole horizons and cosmological horizons. It also confuses readers when shifting from observations made by external observers to those positioned within a horizon's domain. For an expanding FLRW space and observer inside the apparent horizon, the notations modify as,</text>
<unordered_list>
<list_item><location><page_72><loc_21><loc_86><loc_86><loc_90></location>· Normal Surface : Outgoing null ray is diverging but ingoing null ray is converging,i.e., θ l > 0 and θ n < 0.</list_item>
<list_item><location><page_72><loc_21><loc_83><loc_85><loc_84></location>· Trapped surface : Both the null rays are diverging,i.e., θ l > 0 and θ n > 0.</list_item>
<list_item><location><page_72><loc_21><loc_79><loc_67><loc_81></location>· Marginally trapped (past) surface : θ l > 0 and θ n = 0.</list_item>
</unordered_list>
<text><location><page_72><loc_18><loc_74><loc_86><loc_78></location>The apparent horizon for expanding FRW cosmologies is characterized by the conditions,</text>
<formula><location><page_72><loc_49><loc_66><loc_86><loc_70></location>θ l > 0, θ n = 0. (2.20)</formula>
<text><location><page_72><loc_18><loc_60><loc_86><loc_63></location>In comoving co-ordinates, the tangent of outgoing and ingoing radial null geodesics are,</text>
<formula><location><page_72><loc_40><loc_49><loc_86><loc_59></location>l α = ( 1, √ 1 -kr 2 a ( t ) , 0, 0 ) , n α = ( 1, -√ 1 -kr 2 a ( t ) , 0, 0 ) , (2.21)</formula>
<text><location><page_72><loc_18><loc_46><loc_85><loc_47></location>respectively. Therefore, the covariant derivatives of these null geodesics are,</text>
<formula><location><page_72><loc_32><loc_20><loc_86><loc_44></location>∇ α l α = 1 √ -g ∂µ (√ -gl µ ) = ( 1 -kr 2 ) 1/2 a 3 r 2 ( 3 a 2 ˙ ar 2 ( 1 -kr 2 ) 1/2 + 2 a 2 r ) = 3 H + 2 ar √ 1 -kr 2 , (2.22) ∇ α n α = 1 √ -g ∂µ (√ -gn µ ) = ( 1 -kr 2 ) 1/2 a 3 r 2 ( 3 a 2 ˙ ar 2 ( 1 -kr 2 ) 1/2 -2 a 2 r ) = 3 H -2 ar √ 1 -kr 2 . (2.23)</formula>
<text><location><page_72><loc_18><loc_15><loc_86><loc_18></location>Using the equation (2.18), we get the expansions of these null geodesic congruences as,</text>
<formula><location><page_72><loc_30><loc_8><loc_86><loc_14></location>θ l = 2 ( ˙ ar + √ 1 -kr 2 ) ar = 2 ( H + 1 R √ 1 -kR 2 a 2 ) , (2.24)</formula>
<formula><location><page_73><loc_26><loc_84><loc_82><loc_89></location>θ n = 2 ( ˙ ar -√ 1 -kr 2 ) ar = 2 ( H -1 R √ 1 -kR 2 a 2 ) . (2.25)</formula>
<text><location><page_73><loc_14><loc_79><loc_82><loc_82></location>Now, from the definition (2.20), we get the cosmological apparent horizon is located at,</text>
<formula><location><page_73><loc_42><loc_75><loc_82><loc_79></location>r h = 1 √ ˙ a 2 + k . (2.26)</formula>
<text><location><page_73><loc_14><loc_73><loc_82><loc_75></location>In terms of proper radius R ≡ ar , the above equation(2.26) can be written as,</text>
<formula><location><page_73><loc_40><loc_68><loc_82><loc_71></location>R h = 1 √ H 2 + k / a 2 . (2.27)</formula>
<text><location><page_73><loc_57><loc_48><loc_57><loc_49></location≯</text>
<text><location><page_73><loc_14><loc_47><loc_82><loc_66></location>It is important to highlight that the definition of the apparent horizon relies exclusively on null geodesic congruences and their expansions, without any consideration of the global causal structure. At times, there may be a temptation to approximate the location of apparent horizons by speculating where the outward radial null rays come to a halt, essentially setting l r = 0. This simplified approach may yield accurate results on occasion, especially with spherically symmetric metrics in Painlevé-Gullstrand coordinates [343]. But it should not be regarded as a substitute for the proper procedure, which entails identifying surfaces where θ l = 0 and θ n = 0.</text>
<text><location><page_73><loc_14><loc_28><loc_82><loc_46></location>If we were to apply equation(2.21) and assume that n 1 is equal to zero, it would lead to an incorrect inference, indicating the absence of apparent or trapping horizons in FRW cosmologies with k = 0 or k = -1. It would yield an inaccurate value for R h in the case of k = 1 in FRW cosmologies. Clearly, this is not the case. With the exception of the scenario where H = 0, apparent horizons consistently exist and can be determined using equation(2.27). The event horizon may not exist in some cases. Therefore, the radial null geodesic congruences within FRW cosmologies provide a case that contradicts this approach.</text>
<text><location><page_73><loc_14><loc_18><loc_82><loc_26></location>Unlike the event and particle horizons, the apparent horizon is typically not a null surface. Similar to horizons in flat space, the cosmological apparent horizon is observer-dependent. It operates as a spherical barrier encircling the observer and veiling information.</text>
<section_header_level_1><location><page_74><loc_18><loc_88><loc_48><loc_90></location>2.2 Thermodynamics</section_header_level_1>
<text><location><page_74><loc_18><loc_76><loc_86><loc_86></location>This section commences with an exploration of the four laws governing the mechanics of black holes and their intricate relationship with the four fundamental laws of thermodynamics. Subsequently, we will delve into the extension of this connection to include the apparent horizon within the context of the FRW cosmological framework.</text>
<section_header_level_1><location><page_74><loc_18><loc_71><loc_56><loc_73></location>2.2.1 Black hole thermodynamics</section_header_level_1>
<text><location><page_74><loc_18><loc_57><loc_86><loc_69></location>Black hole thermodynamics is a fascinating field that explores the connection between black holes, which are objects with extremely strong gravitational forces, and the laws of thermodynamics, which describe the behavior of energy and entropy in various physical systems. This connection was first suggested by Jacob Bekenstein in 1973 [325] and later developed by Stephen Hawking and others.</text>
<text><location><page_74><loc_18><loc_34><loc_86><loc_57></location>In 1973, Jim Bardeen, Brandon Carter, and Stephen Hawking formulated a set of four laws governing the behaviour of black holes [345]. These laws of black-hole mechanics bear a striking resemblance to the four laws of thermodynamics. While this analogy was at first perceived to be purely formal and coincidental, it soon became clear that black holes do indeed behave as thermodynamic systems. The crucial step in this realization was Hawking's remarkable discovery of 1974 [344] that quantum processes allow a black hole to emit a thermal flux of particles. It is thus possible for a black hole to be in thermal equilibrium with other thermodynamic systems. The laws of black-hole mechanics, therefore, are nothing but a description of the thermodynamics of black holes.</text>
<text><location><page_74><loc_21><loc_32><loc_70><loc_33></location>The four laws of black hole mechanics are [24, 345-347],</text>
<unordered_list>
<list_item><location><page_74><loc_21><loc_26><loc_86><loc_30></location>· Zeroth Law : The surface gravity( κ ) of a stationary black hole is uniform over the entire event horizon (see [345, 348, 349]).</list_item>
<list_item><location><page_74><loc_21><loc_14><loc_86><loc_24></location>· First Law : We consider a quasi-static process during which a stationary black hole of mass M , angular momentum J , and surface area A is taken to a new stationary black hole with parameters M + δ M , J + δ J , and A + δ A . The first law of black-hole mechanics states that the changes in mass, angular momentum, and surface area are related by,</list_item>
</unordered_list>
<formula><location><page_74><loc_45><loc_10><loc_86><loc_13></location>δ M = κ 8 π δ A + Ω H δ J . (2.28)</formula>
<unordered_list>
<list_item><location><page_75><loc_17><loc_86><loc_82><loc_90></location>· Second Law : It states that if the null energy condition is satisfied, then the surface area of a black hole can never decrease [350], i.e., δ A ≥ 0.</list_item>
<list_item><location><page_75><loc_17><loc_76><loc_82><loc_84></location>· Third Law : It states that if the stress-energy tensor is bounded and satisfies the weak energy condition, then the surface gravity of a black hole cannot be reduced to zero within a finite advanced time. A precise formulation of this law was given by Werner Israel in 1986.</list_item>
</unordered_list>
<text><location><page_75><loc_17><loc_73><loc_61><loc_74></location>These statements are quoted from reference [347].</text>
<text><location><page_75><loc_14><loc_58><loc_82><loc_72></location>The intriguing connection between the four fundamental laws governing black hole mechanics and the well-established principles of thermodynamics has not gone unnoticed. In this relationship, the parameter κ takes on the role of temperature, while A mirrors the concept of entropy, and M serves as an analogy to internal energy. The discovery of Hawking radiation evinced that black holes indeed possess a distinct temperature, which correlates directly with their surface gravity,</text>
<formula><location><page_75><loc_44><loc_55><loc_82><loc_58></location>T = ¯ h 2 π κ . (2.29)</formula>
<text><location><page_75><loc_14><loc_42><loc_82><loc_54></location>In a fascinating parallel, the zeroth law can be regarded as a specific manifestation of the analogous thermodynamic concept, signifying that a system achieves thermal equilibrium when it maintains a consistent temperature throughout. Similarly, when we view the first law through the lens of its thermodynamic counterpart, it logically implies that the entropy associated with a black hole should be defined as follows [351, 352],</text>
<formula><location><page_75><loc_44><loc_37><loc_82><loc_40></location>S = 1 4¯ h A . (2.30)</formula>
<text><location><page_75><loc_14><loc_10><loc_82><loc_35></location>The second law, likewise, emerges as a specific instantiation of its thermodynamic counterpart, asserting the principle that the entropy of an isolated system never experiences a decrease. It is pertinent to acknowledge that Hawking radiation [344] leads to a reduction in the black hole's surface area, a seeming contradiction to the area theorem due to the non-compliance of the radiation's stress-energy tensor with the null energy condition. However, it is important to emphasize that the process of black hole evaporation [353] remains in harmony with the generalized second law, which dictates that the overall entropy, encompassing the entropies of both radiation and black holes, remains conserved. We shall discuss the generalized second law of thermodynamics later in more detail. The revelation that black holes adhere to thermodynamic principles unveils a profound interplay between</text>
<table>
<location><page_76><loc_18><loc_68><loc_87><loc_90></location>
<caption>TABLE 2.1: Laws of Black Holes and Thermodynamics</caption>
</table>
<text><location><page_76><loc_18><loc_53><loc_86><loc_61></location>seemingly distinct realms of science, namely, gravitation, quantum mechanics, and thermodynamics. Remarkably, this intricate relationship continues to be a subject of active investigation and comprehension in contemporary scientific exploration.</text>
<section_header_level_1><location><page_76><loc_18><loc_46><loc_86><loc_50></location>2.2.2 Thermodynamics of Apparent Horizons in FLRW Space</section_header_level_1>
<text><location><page_76><loc_18><loc_17><loc_86><loc_44></location>At its inception, black hole thermodynamics primarily centered around stationary event horizons. However, its evolution has led to a more comprehensive exploration, now encompassing various horizon types like apparent, trapping, isolated, dynamical, and slowly evolving horizons. Early on, it was recognized that cosmological horizons, with their origins traced back to the static event horizon of de Sitter space [354], possess thermodynamic properties. A substantial amount of research is dedicated to investigating the thermodynamic characteristics of de Sitter spacetime (for thermodynamics applied in de Sitter space see references [354-358]). Additionally, endeavors have been undertaken to broaden these inquiries beyond the confines of de Sitter space [359-361]. Researchers in this field have put forth the argument that this thermodynamic framework is also applicable to FLRW apparent horizons (such as in [362, 363]).</text>
<text><location><page_76><loc_18><loc_11><loc_86><loc_17></location>Alarge number of investigations adapted the originally formulated thermodynamic equations, tailored for the de Sitter event horizon and the apparent horizon, to suit the dynamic nature of the non-static apparent horizon</text>
<text><location><page_77><loc_14><loc_75><loc_82><loc_90></location>within FLRW space. It is imperative to emphasize that this apparent horizon does not necessarily align with the event horizon, which may not even have a presence in this specific context. The apparent horizon is frequently regarded as a causal boundary linked to gravitational temperature, entropy, and surface gravity within the evolving spacetime framework. These intriguing concepts have been thoroughly explored in a body of research [362, 364368, 414].</text>
<text><location><page_77><loc_14><loc_67><loc_82><loc_75></location>In contrast, previous investigations [364, 366, 369-371] contend that the application of thermodynamics to the event horizon of FRW space encounters challenges in establishing a coherent framework except however in de Sitter space.</text>
<text><location><page_77><loc_14><loc_52><loc_82><loc_66></location>Moreover, there have been concerted attempts to calculate the Hawking radiation emanating from the apparent horizon within the framework of FLRW cosmology. The studies conducted by Jiang and Zhu [372, 373] and the work by Medved [374] and Cai [375], have delved into this area. Various techniques, including the Hamilton-Jacobi method [343, 376, 377], along with the Parikh-Wilczek approach, initially crafted for the analysis of black hole horizons [378].</text>
<text><location><page_77><loc_14><loc_9><loc_82><loc_51></location>There exists an extensive corpus of literature dedicated to exploring the thermodynamic aspects of FRW cosmologies [363, 379-384]. For a comprehensive assessment of the thermodynamic attributes associated with the FLRW apparent horizon, one may refer to Reference [379]. In classical terms, surface gravity finds its definition grounded in the geometric characteristics of the metric tensor. However, it takes on a significant role in the realm of black hole thermodynamics, acting as the constant factor that relates changes in black hole mass (representing internal energy) to variations in the area of the event horizon (directly proportional to entropy)(see section 2.2.1 in this chapter). The ongoing debate surrounding the appropriate definition of black hole mass in non-trivial backgrounds, as delineated in the review [385], naturally extends to the definition of surface gravity. Moreover, surface gravity also emerges in semiclassical contexts, as it essentially corresponds, with minor numerical adjustments, to the Hawking temperature of a black hole. The conventional characterization of surface gravity pertains to a Killing horizon in the context of stationary spacetimes [24, 346]. In static and stationary scenarios, a timelike Killing vector field exists beyond the horizon and becomes null at the horizon itself. In these situations, the definitions of surface gravity align, and they are familiar concepts derived from the analysis of Kerr-Newman black holes in GTR. The surface gravity ( κ ) is defined in</text>
<text><location><page_78><loc_18><loc_88><loc_72><loc_90></location>relation to the Killing vector ( ξ a ) using the equation [24, 346]:</text>
<formula><location><page_78><loc_46><loc_84><loc_86><loc_86></location>ξ a ξ b ; a = κξ b . (2.31)</formula>
<text><location><page_78><loc_18><loc_61><loc_86><loc_82></location>In dynamic situations, the presence of a timelike Killing vector field is absent. Hence the applicability of Killing horizons does not extend to general non-stationary contexts, particularly when dealing with quasi-local horizons rather than event or Killing horizons. In such situations, a suitable notion of surface gravity becomes essential. Multiple definitions of surface gravity can be found in the literature, and it is important to recognize that these definitions are not synonymous. Within the framework of spherical symmetry, the Kodama vector successfully replaces a Killing vector in an evolving system, ultimately leading to the generation of a conserved current and surface gravity.</text>
<text><location><page_78><loc_18><loc_46><loc_86><loc_60></location>The Kodama vector [387] broadens the applicability of a Killing vector field to spacetimes that lack one, making it a viable substitute for a Killing vector in the thermodynamics associated with evolving horizons. It is essential to emphasize that the Kodama vector is only defined within the context of spherical symmetry (an approach to generalization in non-spherical symmetric spacetimes can be found in reference [386]). The spacetime metric can be represented as,</text>
<formula><location><page_78><loc_40><loc_44><loc_86><loc_46></location>dS 2 = h ab dx a dx b + R 2 d Ω 2 ( 2 ) . (2.32)</formula>
<text><location><page_78><loc_18><loc_39><loc_86><loc_42></location>Here a , b = 0, 1 and R is the areal radius. Let us denote the volume of 2-metric h ab by ϵ ab [24]. The definition of Kodama vector [387] is,</text>
<formula><location><page_78><loc_46><loc_34><loc_86><loc_36></location>K a ≡ ϵ ab R ; b , (2.33)</formula>
<text><location><page_78><loc_18><loc_23><loc_86><loc_33></location>with K θ = K ϕ = 0. The Kodama vector lies in the (t,R) plane that is orthogonal to the 2-spheres of symmetry. The antisymmetric nature of ϵ ab and the symmetric nature of R ; a R ; b makes K a R ; a = ϵ ab R ; a R ; b vanish. In a static spacetime, the Kodama vector is parallel to the timelike Killing vector. The Kodama vector exhibits zero divergence [387, 388],</text>
<formula><location><page_78><loc_48><loc_19><loc_86><loc_21></location>K a ; a = 0. (2.34)</formula>
<text><location><page_78><loc_18><loc_15><loc_69><loc_17></location>Consequently, the Kodama energy current( J a ), defined as,</text>
<formula><location><page_78><loc_47><loc_11><loc_86><loc_13></location>J a ≡ G ab K b , (2.35)</formula>
<text><location><page_79><loc_14><loc_88><loc_52><loc_90></location>becomes conserved covariantly. Therefore,</text>
<formula><location><page_79><loc_45><loc_84><loc_82><loc_86></location>J a ; a = 0. (2.36)</formula>
<text><location><page_79><loc_14><loc_76><loc_82><loc_82></location>This conservation occurs even if there is no timelike killing vector. This remarkable attribute is at times humorously coined as the Kodama miracle [387, 388].</text>
<text><location><page_79><loc_14><loc_68><loc_82><loc_75></location>The Hayward proposition for surface gravity, tailored for spherical symmetry [414], employs the Kodama vector K a , which is consistently applicable in scenarios involving spherical symmetry. This future-directed vector satisfies,</text>
<formula><location><page_79><loc_42><loc_65><loc_82><loc_67></location>( K a T ab ) ; b = 0. (2.37)</formula>
<text><location><page_79><loc_14><loc_60><loc_82><loc_64></location>The Hayward concept of surface gravity ( κ ko ) associated with a trapping horizon is expressed as follows:</text>
<formula><location><page_79><loc_35><loc_55><loc_82><loc_58></location>1 2 g ab K c ( K a ; c -K c ; a ) = κ ko K b . (2.38)</formula>
<text><location><page_79><loc_14><loc_51><loc_82><loc_54></location>This definition stands out due to the uniqueness of the Kodama vector. An equivalent expression to equation (2.38) is,</text>
<formula><location><page_79><loc_34><loc_45><loc_82><loc_49></location>κ ko = 1 2 √ -h ∂α ( √ -hh αβ ∂ν R ) , (2.39)</formula>
<text><location><page_79><loc_14><loc_40><loc_82><loc_44></location>where h denotes the determinant of the 2-metric h ab . This surface gravity is popularly known as the Hayward-Kodama surface gravity.</text>
<text><location><page_79><loc_14><loc_36><loc_82><loc_39></location>In FRW cosmology, this Hayward-Kodama surface gravity, given in equation (2.39), becomes,</text>
<formula><location><page_79><loc_34><loc_31><loc_82><loc_34></location>κ ko = -1 2 H ( ˙ H + 2 H 2 + k a 2 ) . (2.40)</formula>
<text><location><page_79><loc_14><loc_27><loc_72><loc_29></location>For spatially flat FRW metric ( k = 0) this equation takes the form,</text>
<formula><location><page_79><loc_38><loc_22><loc_82><loc_25></location>κ ko = -1 2 H ( ˙ H + 2 H 2 ) . (2.41)</formula>
<text><location><page_79><loc_14><loc_17><loc_82><loc_21></location>The Hayward-Kodama dynamic surface gravity ((2.40) and (2.41)) becomes null when the scale factor( a ( t ) ) exhibits the specific condition,</text>
<formula><location><page_79><loc_38><loc_13><loc_82><loc_15></location>a ( t ) = √ α t 2 + β t + γ , (2.42)</formula>
<text><location><page_80><loc_18><loc_86><loc_86><loc_90></location>where α , β , γ are constants. One such example would be the universe purely dominated by radiation.</text>
<text><location><page_80><loc_18><loc_82><loc_86><loc_85></location>Thus, based on this surface gravity, we can derive the Hayward-Kodama temperature for spatially flat FRW cosmologies in the following manner:</text>
<formula><location><page_80><loc_46><loc_73><loc_86><loc_80></location>T = | κ ko | 2 π = 2 H 2 + ˙ H 4 π H . (2.43)</formula>
<section_header_level_1><location><page_80><loc_21><loc_70><loc_54><loc_71></location>· HKtemperature in de Sitter space</section_header_level_1>
<text><location><page_80><loc_18><loc_67><loc_81><loc_68></location>In Schwarzschild-like coordinates, the spherically symmetric metric is,</text>
<formula><location><page_80><loc_33><loc_62><loc_86><loc_64></location>dS 2 = -A ( t , R ) dt 2 + B ( t , R ) dR 2 + R 2 d Ω 2 ( 2 ) . (2.44)</formula>
<text><location><page_80><loc_18><loc_59><loc_68><loc_60></location>In this metric, the kodama vector is written as [387, 389],</text>
<formula><location><page_80><loc_42><loc_53><loc_86><loc_57></location>K a = -1 √ AB ( ∂ ∂ t ) a . (2.45)</formula>
<text><location><page_80><loc_18><loc_50><loc_80><loc_52></location>In static Schwarzschild-like coordinates, de Sitter metric is written as,</text>
<formula><location><page_80><loc_31><loc_45><loc_86><loc_49></location>dS 2 = -( 1 -H 2 R 2 ) dT 2 + dR 2 1 -H 2 R 2 + R 2 d Ω 2 ( 2 ) . (2.46)</formula>
<text><location><page_80><loc_18><loc_42><loc_80><loc_43></location>Therefore, from (2.45) and (2.46), the kodama vector can be written as,</text>
<formula><location><page_80><loc_46><loc_37><loc_86><loc_40></location>K a = ( ∂ ∂ t ) a . (2.47)</formula>
<text><location><page_80><loc_18><loc_25><loc_86><loc_35></location>It is important to observe that the Kodama vector aligns with the timelike Killing vector in de Sitter space. The surface gravity induced by this KillingKodama field is, κ = H = √ Λ 3 . As a consequence, the temperature corresponding to this scenario is, T = H 2 π , which nothing but the famous Hawking temperature [344].</text>
<section_header_level_1><location><page_80><loc_18><loc_20><loc_76><loc_21></location>2.2.3 Generalized Second Law of Thermodynamics</section_header_level_1>
<text><location><page_80><loc_18><loc_8><loc_86><loc_18></location>Bekenstein proposed generalization of second law of black hole thermodynamics [325, 390, 391]. He formulated the GSL as, "The sum of the blackhole entropy and the common (ordinary) entropy in the black-hole exterior never decreases". Bekenstein delved into the realm of black hole physics through the lens of information theory [392-396]. He provided an argument rooted</text>
<text><location><page_81><loc_14><loc_65><loc_82><loc_90></location>in information theory that lends support to the credibility of the generalized second law. An illustrative instance of the connection between an increase in information and a reduction in entropy can be observed in a common scenario. Consider an ideal gas confined within a container undergoing an isothermal compression process. As the compression takes place, the entropy of the gas notably diminishes, a widely acknowledged fact. However, simultaneously, our knowledge regarding the internal arrangement of the gas increases. Following the compression, the gas molecules become more tightly concentrated, leading to a heightened precision in our awareness of their positions compared to the state before compression. Therefore, though the entropy of the system decreases, the total entropy, i.e., the sum of system's entropy and the entropy of the surrounding does not decrease.</text>
<text><location><page_81><loc_14><loc_38><loc_82><loc_63></location>In case of black hole, consider a scenario in which an entity carrying a certain measure of conventional entropy descends into a black hole. During this process, the entropy within the observable universe diminishes. This situation appears to challenge the second law of thermodynamics, as an external observer cannot directly confirm whether the overall entropy of the entire universe remains unchanged during this event. Nonetheless, according to insights from the literature, we understand that the black hole's area compensates for the vanishing entity by undergoing an irreversible increase. As a result, it appears reasonable to hypothesize that the second law remains intact, albeit in a more generalized formulation: the aggregate entropy within the external vicinity of the black hole, along with the entropy possessed by the black hole itself, consistently maintains a non-decreasing trend.</text>
<text><location><page_81><loc_14><loc_16><loc_82><loc_37></location>Researchers have expanded the scope of the Generalized Second Law (GSL) to realms beyond the domain of black hole physics. In earlier research, a second law of thermodynamics applicable to the de Sitter horizon was established by Gibbons and Hawking [354], and this principle was revisited in [397]. Davies [369] explored the event horizon within the context of FLRW space, particularly in the context of General Relativity with a perfect fluid acting as the source. In FRW spacetime, we consider apparent horizon to work with and hence the GSL shapes as: in any physical process, the combined entropy of matter and the horizon must remain constant or increase, i.e.,</text>
<formula><location><page_81><loc_35><loc_14><loc_82><loc_16></location>δ S tot = δ S matter + δ S h ≥ 0, (2.48)</formula>
<text><location><page_81><loc_14><loc_9><loc_82><loc_13></location>where S tot denotes the total entropy and S matter , S h denotes entropy of matter bounded by the horizon and apparent horizon respectively.</text>
<text><location><page_82><loc_21><loc_88><loc_80><loc_90></location>For spatially flat FLRW metric, the field equations are (1.16)-(1.17),</text>
<formula><location><page_82><loc_42><loc_81><loc_86><loc_85></location>H 2 = ˙ a 2 a 2 = 8 π G 3 ρ , (2.49)</formula>
<formula><location><page_82><loc_42><loc_77><loc_86><loc_81></location>2 a a + ( ˙ a a ) 2 = -8 π Gp . (2.50)</formula>
<text><location><page_82><loc_18><loc_72><loc_86><loc_76></location>The entropy of the apparent horizon is assumed to be proportional to area( A ), and given by,</text>
<formula><location><page_82><loc_34><loc_67><loc_86><loc_71></location>S h = A 4 G = 2 π A (in geometrized unit). (2.51)</formula>
<text><location><page_82><loc_18><loc_64><loc_58><loc_66></location>The area of the apparent horizon is given by,</text>
<formula><location><page_82><loc_47><loc_56><loc_86><loc_62></location>A = 4 π R 2 h = 4 π H 2 , (2.52)</formula>
<text><location><page_82><loc_18><loc_51><loc_86><loc_55></location>where we have substituted R h from equation (2.27), and put k = 0. Therefore, the rate of change of entropy of the apparent horizon is given by,</text>
<formula><location><page_82><loc_45><loc_46><loc_86><loc_50></location>˙ S h = -16 π 2 ˙ H H 3 . (2.53)</formula>
<text><location><page_82><loc_18><loc_41><loc_86><loc_44></location>For the fluid inside the horizon, first law of thermodynamics applied to a hydrostatic system looks like,</text>
<formula><location><page_82><loc_43><loc_36><loc_86><loc_38></location>TdS in = dU + pdV , (2.54)</formula>
<text><location><page_82><loc_18><loc_29><loc_86><loc_34></location>where S in , U and V denote the entropy, the internal energy and the volume of the fluid inside the horizon respectively. V is bounded by the apparent horizon,</text>
<formula><location><page_82><loc_47><loc_20><loc_86><loc_27></location>V = 4 3 π R 3 h = 4 3 π 1 H 3 . (2.55)</formula>
<text><location><page_82><loc_21><loc_17><loc_70><loc_19></location>Rate of change of entropy of fluid inside the horizon is,</text>
<formula><location><page_82><loc_40><loc_9><loc_86><loc_16></location>˙ S in = 1 T [ ( ρ + p ) ˙ V + ˙ ρ V ] = 1 T ( ρ + p )( ˙ V -3 HV ) . (2.56)</formula>
<text><location><page_83><loc_14><loc_85><loc_82><loc_90></location>Now inserting T from equation (2.43) and V from equation (2.55) in (2.56), one obtains the expression of ˙ S in as,</text>
<formula><location><page_83><loc_34><loc_80><loc_82><loc_84></location>˙ S in = 16 π 2 ˙ H H 3 ( 1 + ˙ H 2 H 2 + ˙ H ) . (2.57)</formula>
<text><location><page_83><loc_17><loc_77><loc_59><loc_79></location>Therefore rate of change of the total entropy is,</text>
<formula><location><page_83><loc_35><loc_69><loc_82><loc_75></location>˙ S tot = ˙ S h + ˙ S in = 16 π 2 ˙ H 2 H 3 ( 1 2 H 2 + ˙ H ) . (2.58)</formula>
<text><location><page_83><loc_14><loc_63><loc_82><loc_67></location>Therefore we can find ˙ S for different cosmological models if we know the expression of scale factor a ( t ) .</text>
<section_header_level_1><location><page_83><loc_14><loc_58><loc_50><loc_60></location>2.2.4 Thermodynamic Stability</section_header_level_1>
<text><location><page_83><loc_14><loc_38><loc_82><loc_57></location>In a hydrodynamic system, a stable equilibrium can be achieved by minimizing the thermodynamic potentials i.e., the internal energy, enthalpy, Helmholtz free energy, and Gibbs free energy. On the other hand, the entropy principle, which says that in any process total entropy is a non-decreasing function, leads to an increase in entropy as the system approaches equilibrium, characterized by the fields becoming time-independent. Therefore, to achieve thermal equilibrium, total entropy is to be maximized. In other words, entropy must be a concave function if the system is to be in stable equilibrium [398-401]. To achieve concavity, the hessian matrix defined as,</text>
<formula><location><page_83><loc_39><loc_29><loc_82><loc_37></location>W = ∂ 2 S in ∂ U 2 ∂ 2 S in ∂ U ∂ V ∂ 2 S in ∂ V ∂ U ∂ 2 S in ∂ V 2 , (2.59)</formula>
<text><location><page_83><loc_14><loc_22><loc_82><loc_28></location>has to be semi-negative definite. Therefore, all the k th order principle minors of the matrix W are ≤ 0 if k is odd and ≥ 0 if k is even. Hence, the thermodynamic stability requires that the conditions</text>
<formula><location><page_83><loc_52><loc_17><loc_63><loc_20></location>i ) ∂ 2 S in 2 ≤ 0,</formula>
<formula><location><page_83><loc_34><loc_12><loc_63><loc_16></location>ii ) ∂ 2 S in ∂ U 2 ∂ 2 S in ∂ V 2 -( ∂ 2 S in ∂ U ∂ V ) 2 ≥ 0,</formula>
<formula><location><page_83><loc_33><loc_13><loc_82><loc_19></location>( ∂ U (2.60) ( (2.61)</formula>
<text><location><page_84><loc_21><loc_88><loc_40><loc_90></location>are satisfied together.</text>
<text><location><page_84><loc_18><loc_77><loc_86><loc_85></location>Using this we explore the thermodynamic stability of a cosmological model that mimics a Λ CDM model. We also find that the transition from the decelerated to the accelerated expansion of the universe is a second-order thermodynamic phase transition for the matter content of the universe.</text>
<section_header_level_1><location><page_85><loc_14><loc_82><loc_30><loc_84></location>Chapter 3</section_header_level_1>
<section_header_level_1><location><page_85><loc_14><loc_71><loc_79><loc_78></location>Thermodynamics of Thawing and Freezing Quintessence Models</section_header_level_1>
<section_header_level_1><location><page_85><loc_14><loc_63><loc_38><loc_65></location>3.1 Introduction:</section_header_level_1>
<text><location><page_85><loc_14><loc_53><loc_82><loc_61></location>In chapter 1, we discussed different dark energy models. In this chapter, we shall consider the two broad classes of quintessence fields as dark energy models and try to understand their thermodynamic behaviour in the context of the Generalized Second Law of Thermodynamics (GSL).</text>
<text><location><page_85><loc_14><loc_29><loc_82><loc_52></location>The quintessence can be broadly categorized into two main groups known as thawing and freezing models. The behaviour of these models is distinguished by the way the effective equation of state parameter ( w ), evolves over time. The thawing model exhibits an effective EoS parameter ( w ) that initially behaves as almost constant and close to -1. However, as the universe evolves, w undergoes a transformation, transitioning into an evolving state. On the other hand, the freezing model behaves differently. In this model, from an evolving state at the initial stage, w freeze at a constant value -1 in late-time evolution. Among the freezing models, one particularly interesting group is known as the trackers . To get a concise and comprehensible overview of these models refer to [274].</text>
<text><location><page_85><loc_14><loc_15><loc_82><loc_29></location>The comparative studies of these models have been done concerning how well they align with observational data [402] and from the viewpoint of stability [403]. The cluster number count comparison of these models has been studied [408]. However the outcome does not provide solid evidence to favor any of these models. In this chapter, we have done comparative studies of thawing and freezing models from the perspective of the generalized second law(GSL) of thermodynamics.</text>
<text><location><page_85><loc_14><loc_10><loc_82><loc_14></location>We employ a straightforward definition of freezing and thawing models and graph the rate of total entropy change during evolution. The expectation</text>
<text><location><page_86><loc_18><loc_73><loc_86><loc_90></location>is that this rate remains positive, as per the GSL where total entropy never decreases. Surprisingly, the outcomes demonstrate that both freezing and thawing models encounter a violation of the GSL. We conducted tests using a pure quintessence, followed by a quintessence along with a cold dark matter (CDM), yielding similar results, highlighting the inherent thermodynamic non-compliance of the quintessence field. However, freezing models possess an advantage over thawing models, as the GSL breakdown occurs significantly further back in the past.</text>
<section_header_level_1><location><page_86><loc_18><loc_68><loc_54><loc_69></location>3.2 Quintessence Models:</section_header_level_1>
<unordered_list>
<list_item><location><page_86><loc_21><loc_60><loc_86><loc_65></location>· Action and Stress-Energy Tensor: We consider that the universe consists of a perfect fluid and the dark energy contribution comes from a scalar field Φ with potential V ( Φ ) . Therefore, the action takes the form,</list_item>
</unordered_list>
<formula><location><page_86><loc_27><loc_54><loc_86><loc_58></location>S f ield = ∫ d 4 x √ -g [ R 16 π G + L m -g µν 2 ∂µ Φ ∂ν Φ -V ( Φ ) ] , (3.1)</formula>
<text><location><page_86><loc_23><loc_51><loc_61><loc_52></location>The Einstein field equation takes the form,</text>
<formula><location><page_86><loc_43><loc_47><loc_86><loc_49></location>G µν = 8 π G ( T ( m ) µν + T ( q ) µν ) . (3.2)</formula>
<text><location><page_86><loc_23><loc_38><loc_86><loc_44></location>Throughout this chapter, quantities with the superscripts m and q denote the quantities related to matter and the quintessence field respectively.</text>
<text><location><page_86><loc_23><loc_36><loc_79><loc_37></location>Stress-energy tensor of the matter part is given by the equation,</text>
<formula><location><page_86><loc_42><loc_32><loc_86><loc_34></location>T ( m ) µν = ( ρ + p ) u µ u ν + pg µν , (3.3)</formula>
<text><location><page_86><loc_23><loc_26><loc_86><loc_29></location>where ρ , p are the density and pressure of the fluid respectively. Stressenergy tensor of the quintessence field is given by the equation,</text>
<formula><location><page_86><loc_32><loc_20><loc_86><loc_24></location>T ( q ) µν = [ ∂µ Φ ∂ν Φ -1 2 g µν g αβ ∂α Φ ∂ β Φ -2 g µν V ( Φ ) ] . (3.4)</formula>
<section_header_level_1><location><page_86><loc_21><loc_17><loc_48><loc_18></location>· Metric and Field equations:</section_header_level_1>
<text><location><page_86><loc_23><loc_10><loc_86><loc_16></location>According to the cosmological principle , the universe is spatially isotropic, homogeneous in the large-scale. Here we assume that the universe is spatially flat. Therefore, we consider the spacetime metric is</text>
<text><location><page_87><loc_19><loc_86><loc_82><loc_90></location>given by spatially flat (k=0) Friedmann-Robertson-Walker metric(FRW metric),</text>
<formula><location><page_87><loc_30><loc_82><loc_82><loc_84></location>d s 2 = -d t 2 + a ( t ) 2 [ d r 2 + r 2 d θ 2 + r 2 sin 2 θ d ϕ 2 ] . (3.5)</formula>
<text><location><page_87><loc_19><loc_76><loc_82><loc_80></location>Here Φ is solely a function of cosmic time t . Now the field equations can be explicitly written as,</text>
<formula><location><page_87><loc_37><loc_71><loc_82><loc_74></location>3 H 2 = 3 ( ˙ a a ) 2 = 8 π G ( ρ + ρ Φ ) , (3.6)</formula>
<formula><location><page_87><loc_37><loc_64><loc_82><loc_68></location>2 a a + ( ˙ a a ) 2 = -8 π G ( p + p Φ ) . (3.7)</formula>
<text><location><page_87><loc_19><loc_59><loc_82><loc_63></location>In equations (3.6) and (3.7), ρ Φ , p Φ represent energy density and pressure due to the quintessence field.</text>
<formula><location><page_87><loc_38><loc_54><loc_82><loc_57></location>ρ Φ ≡ T 0 ( q ) 0 = 1 2 ( ˙ Φ ) 2 + V ( Φ ) , (3.8)</formula>
<text><location><page_87><loc_19><loc_51><loc_23><loc_53></location>and</text>
<formula><location><page_87><loc_31><loc_48><loc_82><loc_52></location>p Φ ≡ T 1 ( q ) 1 = T 2 ( q ) 2 = T 3 ( q ) 3 = 1 2 ( ˙ Φ ) 2 -V ( Φ ) . (3.9)</formula>
<text><location><page_87><loc_19><loc_43><loc_82><loc_47></location>The conservation of the energy-momentum tensors of the cosmic fluid leads to,</text>
<formula><location><page_87><loc_42><loc_39><loc_82><loc_40></location>˙ ρ + 3 H ( ρ + p ) = 0, (3.10)</formula>
<text><location><page_87><loc_19><loc_33><loc_82><loc_36></location>and the conservation related to the quintessence field leads to the KleinGordon (KG) equation for Φ as follows,</text>
<formula><location><page_87><loc_40><loc_28><loc_82><loc_30></location>¨ Φ + 3 H ˙ Φ + V ' ( Φ ) = 0. (3.11)</formula>
<text><location><page_87><loc_19><loc_19><loc_82><loc_26></location>These equations, (3.10) and (3.11), are not independent from the field equations (3.6)and (3.7). One of the equations (3.10) and (3.11), along with equations (3.6)and (3.7), will yield the other because of Bianchi identity.</text>
<section_header_level_1><location><page_88><loc_18><loc_88><loc_86><loc_90></location>3.3 Generalized Second Law of Thermodynamics:</section_header_level_1>
<text><location><page_88><loc_18><loc_78><loc_86><loc_86></location>According to Generalized Second Law (GSL) of thermodynamics, the total entropy of the universe, i.e., sum of the horizon entropy and entropy of the fluid inside the horizon does not decrease with time [325, 390, 391]. In mathematical expression,</text>
<formula><location><page_88><loc_43><loc_74><loc_86><loc_76></location>˙ S tot = ˙ S h + ˙ S in > 0, (3.12)</formula>
<text><location><page_88><loc_18><loc_66><loc_86><loc_72></location>where total entropy is denoted by S tot , entropy of the horizon and fluid inside the horizon is denoted respectively by S h and S in . An overhead dot indicates derivative with respect to the cosmic time t .</text>
<text><location><page_88><loc_18><loc_55><loc_86><loc_65></location>When studying the kinematics of the universe, it is more sensible to focus on the entropy of the dynamic apparent horizon instead of the teleological event horizon. We have already discussed about the apparent horizon in a concise manner in chapter 2. The following equation represents the entropy of the apparent horizon,</text>
<formula><location><page_88><loc_48><loc_52><loc_86><loc_55></location>S h = A 4 G , (3.13)</formula>
<text><location><page_88><loc_18><loc_45><loc_86><loc_51></location>where A = 4 π R 2 h is the area of the apparent horizon. In section 2.1.1, we have computed the apparent horizon radius R h . In case a spatially flat FRW spacetime (putting k = 0), we get,</text>
<formula><location><page_88><loc_48><loc_40><loc_86><loc_43></location>R h = 1 H . (3.14)</formula>
<text><location><page_88><loc_18><loc_37><loc_68><loc_39></location>Hence, the rate at which the horizon entropy changes is,</text>
<formula><location><page_88><loc_44><loc_32><loc_86><loc_35></location>˙ S h = -2 π G ( ˙ H H 3 ) . (3.15)</formula>
<text><location><page_88><loc_18><loc_27><loc_86><loc_30></location>The mathematical equation that expresses the 1st law of thermodynamics as applied to the matter content by the horizon is,</text>
<formula><location><page_88><loc_41><loc_23><loc_86><loc_24></location>T in d S in = d E in + p tot d V h , (3.16)</formula>
<text><location><page_88><loc_18><loc_15><loc_86><loc_21></location>where V h represents the volume and heres V h = 4 3 π R 3 h . From the above equation(3.16), we can write the rate at which the entropy of the fluid within the horizon changes as,</text>
<formula><location><page_88><loc_37><loc_9><loc_86><loc_13></location>˙ S in = 1 T in [( ρ tot + p tot ) ˙ V h + ˙ ρ tot V h ] , (3.17)</formula>
<text><location><page_89><loc_14><loc_75><loc_82><loc_90></location>where ρ tot and p tot denotes respectively the total energy density and pressure. We assume that the fluid within the horizon is in a state of thermal equilibrium with the horizon. Therefore, the temperature ( T in ) will be equivalent to the apparent horizon temperature ( T h ). Here, we have considered the Hayward-Kodama temperature as the temperature of the dynamic apparent horizon [380, 404-407]. The reason is discussed in chapter 2. The temperature of the dynamic apparent horizon is,</text>
<formula><location><page_89><loc_41><loc_70><loc_82><loc_74></location>T h = 2 H 2 + ˙ H 4 π H . (3.18)</formula>
<text><location><page_89><loc_17><loc_67><loc_78><loc_68></location>Now putting the temperature (3.18) in the equation (3.17), we obtain,</text>
<formula><location><page_89><loc_33><loc_61><loc_82><loc_65></location>˙ S in = ( ρ Φ + p Φ ) T in 4 π R 2 h [ ˙ R h -HR h ] . (3.19)</formula>
<text><location><page_89><loc_14><loc_56><loc_82><loc_60></location>This equation can further be simplified using the fields equations(3.6)and(3.7) and written as,</text>
<formula><location><page_89><loc_34><loc_51><loc_82><loc_55></location>˙ S in = 2 π G ( ˙ H H 3 )( 1 + ˙ H 2 H 2 + ˙ H ) . (3.20)</formula>
<text><location><page_89><loc_14><loc_46><loc_82><loc_49></location>Summing up the equations (3.15) and (3.20), we can express the rate of change of the total entropy as,</text>
<formula><location><page_89><loc_31><loc_40><loc_82><loc_44></location>˙ S tot = ˙ S h + ˙ S in = 2 π G ( ˙ H 2 H 3 )( 1 2 H 2 + ˙ H ) . (3.21)</formula>
<section_header_level_1><location><page_89><loc_14><loc_36><loc_49><loc_38></location>3.4 APure Quintessence:</section_header_level_1>
<text><location><page_89><loc_14><loc_24><loc_82><loc_33></location>At first, only pure quintessence model is taken into account. This implies that the fluid inside the horizon is considered to be the quintessence field only, devoid of baryonic and dark matter( p = ρ = 0). Therefore there are no contributions of energy density and pressure of these matters in the field equation (3.6)and(3.7).</text>
<text><location><page_89><loc_14><loc_17><loc_82><loc_23></location>At this point, we have two equations to determine three unknowns: a , ϕ , V , since the Klein-Gordon (KG) equation is not an independent equation. To complete the system of equations, we adopt the ansatz for ρ proposed by</text>
<text><location><page_90><loc_18><loc_86><loc_86><loc_90></location>Carvalho et al [287]. The ansatz says the divergence of the logarithm of energy density is power law dependent on scale factor,</text>
<formula><location><page_90><loc_44><loc_81><loc_86><loc_84></location>1 ρ Φ ∂ρ Φ ∂ a = -λ a 1 -2 α , (3.22)</formula>
<text><location><page_90><loc_18><loc_73><loc_86><loc_79></location>where λ and α are parameters. λ is chosen to be positive, but the parameter α can take both positive or negative values. Integrating the above equation, we obtain,</text>
<formula><location><page_90><loc_37><loc_68><loc_86><loc_71></location>ρ Φ ( a ) = ρ Φ ,0 exp [ -λ 2 α ( a 2 α -1 ) ] , (3.23)</formula>
<text><location><page_90><loc_18><loc_57><loc_86><loc_67></location>where the present value of the scale factor is taken to be 1. Since λ is positive, the energy density of the quintessence field decreases with the evolution. The quintessence energy density reduces to a power law, ρ Φ ( a ) ∝ a -λ in the limit α → 0. Using equation (3.11), the potential V is obtained as (for more details see [287]),</text>
<formula><location><page_90><loc_21><loc_50><loc_86><loc_54></location>V ( Φ ) = [ 1 -λ 6 ( 1 + α √ σ Φ ) 2 ] ρ Φ ,0 exp [ -λ √ σ ( Φ + α √ σ Φ 2 2 ) ] . (3.24)</formula>
<text><location><page_90><loc_18><loc_44><loc_86><loc_48></location>The equation of state parameter (EoS) for the scalar-field is defined by w Φ ≡ p Φ ρ Φ . In this model, we get w Φ in terms of scale factor as,</text>
<formula><location><page_90><loc_44><loc_39><loc_86><loc_42></location>w Φ = -1 + λ 3 a 2 α . (3.25)</formula>
<text><location><page_90><loc_69><loc_36><loc_69><loc_37></location≯</text>
<text><location><page_90><loc_18><loc_31><loc_86><loc_37></location>The above EoS is not time-independent quantity (for α = 0). The dependency of time in w Φ comes through the scale factor a . It is to be noticed that, it reduces to a constant, w Φ = -1 + λ 3 in the limit α → 0.</text>
<text><location><page_90><loc_18><loc_12><loc_86><loc_29></location>Now let us see the behaviour of evolution of w Φ pictorially. We define a function N as N = ln ( a a 0 ) , and plot w Φ against N (fig:3.1(a) & 3.1(b)). From the definition we see that, here N = 0 implies the present epoch, positive and negative N imply the future and past respectively. In the fig:3.1(a), we plot w Φ vs N for positive values of the parameter α , which shows thawing behaviour. And freezing behaviour is shown in the fig:3.1(b), where we plot w Φ vs N for positive values of the parameter α . Since λ is positive, w Φ ⩾ -1 for all values of a irrespective of the sign of α [287].</text>
<text><location><page_90><loc_18><loc_10><loc_86><loc_12></location>The equation (3.22) serves as a comprehensive ansatz, encompassing both</text>
<figure>
<location><page_91><loc_28><loc_70><loc_68><loc_90></location>
</figure>
<text><location><page_91><loc_47><loc_70><loc_49><loc_71></location>(a)</text>
<figure>
<location><page_91><loc_28><loc_49><loc_68><loc_69></location>
<caption>FIGURE 3.1: w Φ is plotted as a function of N . (a) For positive values of α , EoS increases from -1, i.e., it exhibits a thawing behaviour. (b) For negative values of α , EoS > -1 decreases to a more negative value, i.e., it exhibits a freezing behaviour.</caption>
</figure>
<text><location><page_91><loc_14><loc_24><loc_82><loc_38></location>thawing and freezing behaviours through the parameter α . For α > 0, the scalar field exhibits a thawing behaviour, where the equation of state parameter w Φ starts with a nearly flat value close to -1 in the past and gradually transitions to less negative values. Conversely, for α < 0, the scalar field demonstrates a freezing behaviour, wherein w Φ decreases towards more and more negative values and eventually settles into a plateau near -1 in the future.</text>
<text><location><page_91><loc_14><loc_17><loc_82><loc_23></location>It is essential to clarify that we have adopted the ansatz (3.22) from the work by Carvalho et al [287], and we have employed the parameter values α , λ , and others from the reference [408].</text>
<text><location><page_91><loc_14><loc_13><loc_82><loc_16></location>Now, substituting ρ Φ from equation (3.23) in the Friedmann equation (3.6) with ρ = 0 we obtain the solution for the scale factor as,</text>
<formula><location><page_91><loc_29><loc_8><loc_82><loc_11></location>1 2 α [ Γ ( 0, -λ 2 α ) -Γ ( 0, -λ 4 α a 2 α ) ] = γ ( t -t 0 ) , (3.26)</formula>
<text><location><page_92><loc_18><loc_82><loc_86><loc_90></location>where Γ ( a , x ) is the well-known upper incomplete gamma function and is defined as, Γ ( a , x ) = ∫ ∞ x z a -1 exp ( -z ) dz . And γ = √ 8 π G 3 ρ Φ ,0 exp ( λ 4 α ) is a constant term. While the solution may be intricate, it serves the purpose of assessing the thermodynamic feasibility of the model.</text>
<text><location><page_92><loc_18><loc_77><loc_86><loc_81></location>By employing the solution (3.26), it becomes possible to express H and ˙ H in terms of a . Consequently, the equation (3.21) takes the form,</text>
<formula><location><page_92><loc_24><loc_71><loc_86><loc_75></location>˙ S tot = πλ 2 G ( 1 √ 8 π G 3 ρ Φ ,0 exp ( -λ 4 α ( a 2 α -1 ) ) )( a 4 α 4 -λ a 2 α ) . (3.27)</formula>
<text><location><page_92><loc_18><loc_65><loc_86><loc_69></location>In order to assess the thermodynamic feasibility of the model, we proceed to plot ˙ S tot as a function of the cosmic e-folding factor N .</text>
<text><location><page_92><loc_52><loc_63><loc_52><loc_64></location></text>
<figure>
<location><page_92><loc_32><loc_44><loc_72><loc_64></location>
</figure>
<text><location><page_92><loc_52><loc_42><loc_52><loc_43></location></text>
<figure>
<location><page_92><loc_32><loc_23><loc_72><loc_43></location>
<caption>FIGURE 3.2: ˙ Stot is plotted as a function of N . (a) in thawing scenario (for α = 0.3, 0.5 and 0.9) (b) in freezing scenario (for α = -0.3, -0.5 and -0.9 )</caption>
</figure>
<text><location><page_92><loc_18><loc_8><loc_86><loc_14></location>As shown in Fig-3.2(a), in the case of thawing quintessence ( α > 0), the total entropy experiences an initial increase up to a certain time, after which it deviates from obeying the Generalized Second Law (GSL). Notably, ˙ S tot rises</text>
<text><location><page_93><loc_14><loc_67><loc_82><loc_90></location>sharply to an infinitely large value and then abruptly drops to an infinitely large negative value. This pattern is similar for all permissible values of α , with only the range of N indicating the onset of this abnormal behaviour varies. On the other hand, freezing quintessence ( α < 0) demonstrates an opposing behaviour. It satisfies the GSL for the future, as the net entropy increases and eventually stabilizes at a constant value when ˙ S tot approaches zero (as observed in Fig-3.2(b)). However, this convergence is not as rapid as depicted in the figure but is likely to occur asymptotically, as indicated by the zoomed-in version in the inset. Nevertheless, the model fails to comply with GSL in the past. A discontinuity in ˙ S tot is evident, and it takes on negative values, signifying a decrease in S tot during that period.</text>
<text><location><page_93><loc_14><loc_52><loc_82><loc_66></location>The parameter λ , which governs the rate at which the dark energy density ρ Φ declines (as shown in equation 3.22), possesses a small positive value as indicated in a prior study [408]. Throughout all the figures, we maintain a consistent value of λ = 0.06. We have also tested the model using significantly lower ( λ = 0.01) and higher ( λ = 0.1) values. However, since these variations do not significantly impact the qualitative characteristics, we have omitted them to avoid redundant information.</text>
<section_header_level_1><location><page_93><loc_14><loc_46><loc_69><loc_48></location>3.5 Quintessence with cold dark matter:</section_header_level_1>
<text><location><page_93><loc_14><loc_36><loc_82><loc_44></location>In this section, we consider a more realistic model, incorporating a pressureless fluid ( p = 0) comprising baryonic matter and cold dark matter in addition to the quintessence matter. Consequently, the field equations (3.6, 3.7) will have the input p = 0.</text>
<text><location><page_93><loc_17><loc_34><loc_56><loc_36></location>Directly integrating equation (3.10) leads to,</text>
<formula><location><page_93><loc_45><loc_29><loc_82><loc_32></location>ρ = ρ 0 a 3 , (3.28)</formula>
<text><location><page_93><loc_14><loc_20><loc_82><loc_28></location>where ρ 0 is the energy density of matter at the present epoch. Since the KleinGordon equation is not an independent equation, we are left with two equations (3.6, 3.7), to solve for three variables: a , Φ , V . To close the system of equations, we employ the same ansatz (3.22) as used in the previous section.</text>
<text><location><page_94><loc_18><loc_86><loc_85><loc_90></location>The expression for the deceleration parameter q , defined by q ≡ -˙ H + H 2 H 2 is given by,</text>
<formula><location><page_94><loc_31><loc_76><loc_86><loc_84></location>q ( a ) = 3 ( -1 + λ 3 a 2 α ) + 1 + ρ m ,0 a 3 ρ Φ ,0 exp [ -λ 2 α ( a 2 α -1 ) ] 2 ( 1 + ρ m ,0 a 3 ρ Φ ,0 exp [ -λ 2 α ( a 2 α -1 ) ] ) . (3.29)</formula>
<text><location><page_94><loc_18><loc_60><loc_86><loc_74></location>Fig-3.3(a) reveals an intriguing possibility for a thawing model that the current accelerated expansion of the universe may only be a temporary phenomenon. According to the plot, the universe is expected to transit back to a decelerated phase in a finite future. This characteristic has been observed previously such as by Carvalho et al [287] and Devi et al [408]. However, in the freezing model, the universe eventually settles into a phase of accelerated expansion after passing through the decelerated phase.</text>
<text><location><page_94><loc_18><loc_52><loc_86><loc_57></location>It is important to note that these features have already been welldocumented in the literature. However, the primary objective of this study is to assess the thermodynamic viability of the model.</text>
<text><location><page_94><loc_18><loc_32><loc_86><loc_51></location>In this analysis, we assume that the fluid inside the horizon is in thermal equilibrium with the apparent horizon. To evaluate the rate of change of the total entropy, we use the Hayward-Kodama temperature, as given in equation (3.18), following the same approach as in the previous section. Now, the total energy density is represented as ρ tot = ρ + ρ Φ . The general form of the rate of change of the total entropy remains unchanged, as expressed in equation (3.21). However, due to the modification of the Friedmann equations for the contribution of cold dark matter, the explicit form of ˙ S tot can be obtained as,</text>
<formula><location><page_94><loc_28><loc_21><loc_86><loc_31></location>˙ S tot = 3 √ 3 π 2 √ 2 G 3/2 [ ρ Φ ,0 exp [ -λ 2 α ( a 2 α -1 )]( λ 3 a 2 α ) + ρ m ,0 a 3 ] 2 [ ρ Φ ,0 exp [ -λ 2 α ( a 2 α -1 )] + ρ m ,0 a 3 ] 3/2 × 1 [ ρ Φ ,0 exp [ -λ 2 α ( a 2 α -1 )]( 4 -λ a 2 α ) + ρ m ,0 a 3 ] . (3.30)</formula>
<text><location><page_94><loc_18><loc_10><loc_86><loc_20></location>The qualitative characteristics remain similar to those observed in the case of pure quintessence. As demonstrated in Figure 4(a), the thawing models ( α > 0) will eventually violate the Generalized Second Law (GSL) in some finite future, since ˙ S tot becomes negative at a certain value of N , indicating a decrease in entropy. On the other hand, the freezing models ( α < 0) adhere</text>
<figure>
<location><page_95><loc_28><loc_72><loc_68><loc_90></location>
</figure>
<text><location><page_95><loc_47><loc_70><loc_49><loc_71></location>(a)</text>
<figure>
<location><page_95><loc_28><loc_49><loc_68><loc_69></location>
<caption>FIGURE 3.3: q is plotted as a function of N . (a) In Thawing scenario, transient acceleration is a possibility,(b) Freezing quintessence leads the universe into an eternal accelerating phase.</caption>
</figure>
<text><location><page_95><loc_14><loc_30><loc_82><loc_38></location>to GSL in the future. Despite gradually decreasing, ˙ S tot remains positive (as seen in Figure 4(b)), signifying that the entropy is continuously increasing and will eventually stabilize at a constant value in the future, as illustrated in Figure 4(b).</text>
<text><location><page_95><loc_14><loc_11><loc_82><loc_30></location>However, there is a concern in connection with the past evolution. The rate of entropy change becomes negative at a certain finite time in the past. Nonetheless, this issue can be mitigated by adjusting the parameter α . For instance, by choosing α = -0.3, the discrepancy is observed at z ∼ 10 4 (see Figure 5(a)), which occurs before the onset of matter domination over radiation, rendering this system of equations inapplicable to describe the dynamics of the universe at that stage. Similarly, for α = -0.1, this discrepancy is observed at z ∼ 10 12 , which is far beyond the regime of quintessence along with CDM (see Figure 5(b)).</text>
<text><location><page_95><loc_14><loc_9><loc_82><loc_11></location>Upon careful inspection of Eq. (3.21), it becomes evident that the term 2 H 2 +</text>
<text><location><page_96><loc_18><loc_86><loc_86><loc_90></location>˙ H determines the thermodynamic viability of the models. If ˙ H + 2 H 2 < 0 (i.e., q ≥ 1), the model fails to meet the thermodynamic requirements.</text>
<text><location><page_96><loc_18><loc_82><loc_86><loc_85></location>Thus, it is evident that freezing models exhibit stronger thermodynamic viability, at least within the relevant period.</text>
<text><location><page_96><loc_18><loc_71><loc_86><loc_81></location>Similar to the previous section, in this analysis, we have maintained the same value of λ = 0.06 for the figures. The values of λ = 0.01 and 0.1 have been excluded since they do not alter the qualitative aspects. The only noticeable difference is a slight shift in epochs, such as the onset of GSL violation for different values of λ .</text>
<text><location><page_96><loc_52><loc_69><loc_53><loc_70></location></text>
<figure>
<location><page_96><loc_32><loc_50><loc_72><loc_70></location>
</figure>
<text><location><page_96><loc_52><loc_48><loc_52><loc_49></location></text>
<figure>
<location><page_96><loc_32><loc_29><loc_72><loc_49></location>
<caption>FIGURE 3.4: ˙ Stot is plotted as a function of N , (a) in thawing scenario (for α = 0.3, 0.5 and 0.9) (b) in freezing scenario (for α = -0.3, -0.5 and -0.9. )</caption>
</figure>
<section_header_level_1><location><page_96><loc_18><loc_14><loc_59><loc_16></location>3.6 Summary and Discussion:</section_header_level_1>
<text><location><page_96><loc_18><loc_9><loc_86><loc_12></location>In this study, we conduct a comparison between thawing and freezing models regarding their adherence to the Generalized Second Law (GSL) of</text>
<figure>
<location><page_97><loc_28><loc_70><loc_68><loc_90></location>
</figure>
<text><location><page_97><loc_65><loc_90><loc_66><loc_90></location></text>
<text><location><page_97><loc_65><loc_68><loc_66><loc_69></location></text>
<figure>
<location><page_97><loc_28><loc_48><loc_68><loc_68></location>
<caption>FIGURE 3.5: ˙ Stot is plotted as a function of N in Freezing scenario for (a) α = -0.3 and (b) α = -0.1.</caption>
</figure>
<text><location><page_97><loc_14><loc_29><loc_82><loc_41></location>thermodynamics. To assess the total entropy ( S tot), we combine the horizon entropy with the entropy of the matter enclosed within the horizon. We adopt a simple ansatz [287] to model the evolution of the energy density of the quintessence field. By doing so, we can readily identify the range of parameter values ( α ) that correspond to the thawing and freezing behaviour of the field.</text>
<text><location><page_97><loc_14><loc_9><loc_82><loc_26></location>We find that both types of models exhibit an incompatibility with GSL. There are instances where the entropy ( S ) actually decreases and does so at a rapid pace. For the freezing models, this breakdown in GSL can occur in a distant past, corresponding to a redshift of z ∼ 10 4 . At such a distant past, a quintessence model in conjunction with cold dark matter does not adequately account for the evolution of the universe, and a dominant contribution from a radiation distribution would be necessary. Hence, this breakdown of GSL may not hold true in such a scenario.</text>
<text><location><page_98><loc_18><loc_84><loc_86><loc_87></location>On the other hand, for the thawing models, this pathological breakdown of GSL is predicted to happen in a finite future.</text>
<text><location><page_98><loc_18><loc_77><loc_86><loc_83></location>Thus the major implication here is that the freezing models seem to be more favorable compared to the thawing models from the perspective of thermodynamic viability.</text>
<section_header_level_1><location><page_99><loc_14><loc_82><loc_30><loc_84></location>Chapter 4</section_header_level_1>
<section_header_level_1><location><page_99><loc_14><loc_71><loc_77><loc_78></location>Thermodynamics of Brans-Dicke Cosmology</section_header_level_1>
<section_header_level_1><location><page_99><loc_14><loc_63><loc_38><loc_65></location>4.1 Introduction:</section_header_level_1>
<text><location><page_99><loc_14><loc_10><loc_82><loc_61></location>In this chapter, we discuss the thermodynamic feasibility, especially the viability of GSL in Brans-Dicke Cosmology. As we have discussed in chapter 1, the Brans-Dicke theory (BDT) of gravity [146] is widely recognized and frequently discussed due to its potential in addressing a lot of cosmological issues. Two notable examples include extended inflation [177, 178], which solves the problem of graceful exit in the inflationary scenario, and the potential for driving the late-time acceleration of the universe even in the absence of dark energy [100]. However, BDT has its limitations. One limitation is that the characteristic coupling constant of the theory, denoted as ω , needs to have a very high value to be consistent with local astronomical tests [409]. On the other hand, for successfully addressing cosmological problems, a small value of ω is required. Despite these challenges, the theory continues to attract attention due to its relevance in various real cosmological problems and its formal resemblance to other gravity theories with nonminimal coupling. To make the equations more manageable, we examine them in what is known as the Einstein frame, achieved through a suitable conformal transformation [152]. This results in a scenario where the evolution of matter distribution becomes intertwined with the BD scalar field. Consequently, the Bianchi identity enforces the conservation of both the BD field and the matter distribution as a whole. This characteristic aligns with the notion that there is no inherent reason to assume that dark matter and dark energy evolve independently, without any non-gravitational interaction between them [410]. Some studies have already explored situations in which the BD scalar field interacts with matter [411, 412].</text>
<text><location><page_100><loc_18><loc_58><loc_86><loc_87></location>The thermodynamic viability during the matter-dominated phase of the universe, encompassing the transition from decelerated to accelerated expansion is the primary motivation. However, for a comprehensive analysis, the radiation-dominated phase is also taken into consideration. A similar thermodynamic analysis of cosmological models was carried out by Bhattacharya and Debnath [413], albeit in a different context. They examined accelerated expansion in an extended version of BDT, introducing a potential and allowing the BD parameter ω to be a function of the scalar field. In contrast, the present study focuses on the original BDT, capable of driving accelerated expansion on its own. The results of this investigation are highly promising. Specifically, the models, especially those for a universe containing a pressureless fluid, align with exclusively negative values of the parameter ω , which fall within the range where BDT can independently facilitate accelerated expansion [100].</text>
<section_header_level_1><location><page_100><loc_18><loc_53><loc_75><loc_55></location>4.2 Brans-Dicke Theory in Einstein frame</section_header_level_1>
<unordered_list>
<list_item><location><page_100><loc_21><loc_47><loc_86><loc_51></location>· Action: The action in the BD theory in the Jordan frame is given by [146],</list_item>
</unordered_list>
<formula><location><page_100><loc_33><loc_43><loc_86><loc_47></location>S = 1 16 π G 0 ∫ √ -g [ ϕ R -ω ϕ , αϕ , α ϕ + L M ] d 4 x . (4.1)</formula>
<text><location><page_100><loc_23><loc_35><loc_86><loc_42></location>The action describes the dynamics of gravity and the Brans-Dicke scalar field, as well as their interaction with matter. In this equation, ϕ is the BDscalar field. The second term represents the kinetic term of the scalar field. Now let us employ the following conformal transformation [152],</text>
<formula><location><page_100><loc_49><loc_31><loc_86><loc_32></location>¯ g µν = ϕ g µν . (4.2)</formula>
<text><location><page_100><loc_23><loc_26><loc_50><loc_28></location>Then the action takes the form,</text>
<formula><location><page_100><loc_26><loc_21><loc_86><loc_25></location>¯ S = 1 16 π G 0 ∫ √ -¯ g [ ϕ 0 ( ¯ R -2 ω + 3 2 ψ , αψ , β ¯ g αβ ) + ¯ L M ] d 4 x , (4.3)</formula>
<text><location><page_100><loc_23><loc_15><loc_86><loc_19></location>where ψ = ln ( ϕ ϕ 0 ) and ϕ 0 is a constant. The overhead bar will be omitted henceforth, as the rest of the work is in the revised frame.</text>
<text><location><page_101><loc_19><loc_86><loc_82><loc_90></location>In this modified version, commonly referred to as the Einstein frame , the field equations take on a notably simplified appearance as follows,</text>
<formula><location><page_101><loc_32><loc_81><loc_82><loc_84></location>G αβ = T αβ + 2 ω + 3 2 ( ψ , αψ , β -1 2 g αβ ψ , µ ψ , µ ) . (4.4)</formula>
<text><location><page_101><loc_19><loc_67><loc_82><loc_79></location>This is expressed using units where the quantity 8 π G 0 is set to unity. This conformal transformation reconfigures the mathematical representation of the theory, leading to a formulation in which the complexity of the field equations is significantly reduced. As a result, the Einstein frame becomes a preferred framework for analysis due to its enhanced mathematical tractability.</text>
<text><location><page_101><loc_19><loc_62><loc_82><loc_66></location>The equation governing the evolution of the scalar field ψ , derived by varying the action (4.3) with respect to ψ , results in the expression,</text>
<formula><location><page_101><loc_44><loc_56><loc_82><loc_61></location>2 ψ = T 2 ω + 3 , (4.5)</formula>
<text><location><page_101><loc_19><loc_48><loc_82><loc_56></location>where T represents the trace of the energy-momentum tensor for the matter sector. Let us now consider that the universe is filled with perfect fluid. Therefore, the energy-momentum tensor is given by the equation (1.12). For the sake of clarity, lets rewrite it down here,</text>
<formula><location><page_101><loc_39><loc_44><loc_82><loc_45></location>T αβ = ( ρ + p ) u α u β + pg αβ , (4.6)</formula>
<text><location><page_101><loc_19><loc_35><loc_82><loc_41></location>where ρ , p are the energy density and pressure of the fluid respectively and u α is the unit timelike vector, u µ u µ = -1. In a comoving coordinate system, v µ = δ µ 0 .</text>
<text><location><page_101><loc_19><loc_33><loc_80><loc_35></location>It is important to note that ρ and p are presented in the revised units.</text>
<unordered_list>
<list_item><location><page_101><loc_17><loc_28><loc_82><loc_31></location>· The Metric and The Field Equations: We consider a a spatially flat, homogeneous and isotropic FRW cosmology. The metric is given by,</list_item>
</unordered_list>
<formula><location><page_101><loc_34><loc_24><loc_82><loc_26></location>d s 2 = -d t 2 + a ( t ) 2 [ d x 2 + d y 2 + d z 2 ] . (4.7)</formula>
<text><location><page_101><loc_19><loc_15><loc_82><loc_21></location>Here a is the scale factor that characterizes the change in the size of the universe over time. The form that the Einstein-Brans-Dicke field equations assume in the version obtained through conformal transformation</text>
<text><location><page_102><loc_23><loc_88><loc_58><loc_90></location>is as follows, as outlined by Dicke [152],</text>
<formula><location><page_102><loc_47><loc_83><loc_86><loc_86></location>3 ( ˙ a a ) 2 = ρ + ρψ , (4.8)</formula>
<formula><location><page_102><loc_42><loc_75><loc_86><loc_79></location>2 ( a a ) + ( ˙ a a ) 2 = -p -p ψ . (4.9)</formula>
<text><location><page_102><loc_23><loc_67><loc_86><loc_73></location>In these equations, the energy density and pressure of the scalar field is represented by ρψ and p ψ . The connection between the energy density and pressure of the scalar field is expressed as,</text>
<formula><location><page_102><loc_45><loc_62><loc_64><loc_65></location>ρψ = p ψ = 2 ω + 3 4 ˙ ψ 2 .</formula>
<text><location><page_102><loc_23><loc_56><loc_86><loc_60></location>The wave equation that describes the behavior and evolution of the scalar field ψ is given by,</text>
<formula><location><page_102><loc_46><loc_51><loc_86><loc_54></location>¨ ψ + 3 ˙ a a ˙ ψ = -T 2 ω + 3 . (4.10)</formula>
<text><location><page_102><loc_23><loc_35><loc_86><loc_49></location>In this version, the equations take on a formal similarity to those describing a scenario with two types of matter components: one corresponds to a fluid, while the other represents a massless scalar field denoted as ψ . But the fact is that those do not evolve independently. Hence, we do not have separate conservation equations for these two matters. When considering both the fluid and scalar field together, the continuity equation assumes a form,</text>
<formula><location><page_102><loc_46><loc_30><loc_86><loc_33></location>˙ ρ t = -3 ˙ a a ( ρ t + p t ) . (4.11)</formula>
<text><location><page_102><loc_23><loc_21><loc_86><loc_29></location>The subscript t indicates the total quantity, which is composed of the combined contributions from the fluid component and the scalar field component. This equation is not self-standing; instead, it derives from the Bianchi identities.</text>
<section_header_level_1><location><page_102><loc_18><loc_15><loc_62><loc_17></location>4.3 Thermodynamic Quantities:</section_header_level_1>
<text><location><page_102><loc_18><loc_9><loc_86><loc_13></location>In this chapter also, we check the viability of the model against the Generalised Second Law of Thermodynamics (GSL), according to which the total</text>
<text><location><page_103><loc_14><loc_88><loc_73><loc_90></location>entropy of the universe can not decrease with time [325, 390, 391].</text>
<text><location><page_103><loc_14><loc_77><loc_82><loc_87></location>We assume that the fluid achieves thermal equilibrium with the horizon. In this scenario, the temperature of the fluid inside the horizon ( T in ) is the same as the temperature of the dynamical apparent horizon ( T h ). In the current section, we have adopted Hayward-Kodama temperature [379, 404, 414, 415] as the temperature of the apparent horizon, which is expressed as,</text>
<formula><location><page_103><loc_41><loc_72><loc_82><loc_76></location>T h = 2 H 2 + ˙ H 4 π H . (4.12)</formula>
<text><location><page_103><loc_14><loc_69><loc_81><loc_71></location>In the unit of 8 π G 0 = 1, the entropy of the apparent horizon becomes [362],</text>
<formula><location><page_103><loc_43><loc_65><loc_82><loc_67></location>S h = 2 π A . (4.13)</formula>
<text><location><page_103><loc_17><loc_61><loc_82><loc_63></location>In flat FRW spacetime, the area A is related to Hubble parameter [362] as,</text>
<formula><location><page_103><loc_44><loc_56><loc_82><loc_59></location>A = 4 π H 2 . (4.14)</formula>
<text><location><page_103><loc_17><loc_53><loc_58><loc_55></location>Therefore, rate of change in horizon entropy is</text>
<formula><location><page_103><loc_41><loc_48><loc_82><loc_51></location>˙ S h = -16 π 2 ˙ H H 3 . (4.15)</formula>
<text><location><page_103><loc_14><loc_43><loc_82><loc_46></location>The rate of change of entropy of the fluid confined within the horizon can be expressed as,</text>
<formula><location><page_103><loc_35><loc_39><loc_82><loc_43></location>˙ S in = 1 T in [( ρ t + p t ) ˙ V h + ˙ ρ t V h ] , (4.16)</formula>
<text><location><page_103><loc_14><loc_36><loc_72><loc_38></location>where in flat FRW spacetime the volume confined is, V h = 4 3 π 1 H 3 .</text>
<text><location><page_103><loc_14><loc_30><loc_82><loc_36></location>By utilizing the equation labeled as Eq. (4.12) along with the field equations provided in Eq. (4.16), we derive the expression for the rate of entropy change within the horizon as represented by the equation,</text>
<formula><location><page_103><loc_34><loc_25><loc_82><loc_28></location>˙ S in = 16 π 2 ˙ H H 3 ( 1 + ˙ H 2 H 2 + ˙ H ) . (4.17)</formula>
<text><location><page_103><loc_14><loc_17><loc_82><loc_23></location>Combining the equations referenced as Eq. (4.15) and Eq. (4.17), we arrive at the expression for the rate of total entropy change, which can be represented as,</text>
<formula><location><page_103><loc_36><loc_14><loc_82><loc_17></location>˙ S t = 16 π 2 ˙ H 2 H 3 ( 1 2 H 2 + ˙ H ) . (4.18)</formula>
<text><location><page_103><loc_14><loc_9><loc_82><loc_13></location>We have discussed this in detail in Chapter 2. Here we are rewriting these equations in the unit 8 π G 0 = 1, because throughout this chapter we have</text>
<text><location><page_104><loc_18><loc_88><loc_49><loc_90></location>worked out everything in this unit.</text>
<text><location><page_104><loc_18><loc_79><loc_86><loc_87></location>In the following step, we will examine two distinct epochs in cosmic evolution, namely the radiation era and the dust era. We will proceed to analyze how the Generalized Second Law (GSL) of thermodynamics is upheld during these specific periods of the evolution of the universe.</text>
<text><location><page_104><loc_18><loc_66><loc_86><loc_78></location>It is worth noting that research by Mimoso and Diego [416] highlights the difficulty of achieving thermal equilibrium between radiation and the cosmic horizon. This challenge stems from Wien's law, which consistently predicts a wavelength larger than the horizon radius at all times. However, there is a potential for nonrelativistic particles to reach equilibrium based on their masses.</text>
<section_header_level_1><location><page_104><loc_18><loc_60><loc_43><loc_62></location>4.4 Radiation Era:</section_header_level_1>
<text><location><page_104><loc_18><loc_48><loc_86><loc_58></location>In the scenario of radiation-dominated era, the equation of state of the fluid is characterized by the expression p = 1 3 ρ . As a result of this equation of state, the trace of the stress-energy tensor becomes null. As a consequence, the wave equation labeled as (4.10) can be readily integrated, leading to the derivation of the subsequent relationship:</text>
<formula><location><page_104><loc_48><loc_44><loc_86><loc_47></location>˙ ψ = α a 3 , (4.19)</formula>
<text><location><page_104><loc_18><loc_41><loc_49><loc_42></location>where α is an integration constant.</text>
<text><location><page_104><loc_18><loc_36><loc_86><loc_40></location>Utilizing the field equations (4.8) and (4.9), and incorporating the equation of state along with equation (4.19), it becomes possible to arrive at,</text>
<formula><location><page_104><loc_46><loc_29><loc_86><loc_33></location>a + ˙ a 2 a = -β a 5 , (4.20)</formula>
<text><location><page_104><loc_18><loc_26><loc_65><loc_28></location>where β is a constant, specifically defined as 2 ω + 3 12 α 2 .</text>
<text><location><page_104><loc_21><loc_24><loc_70><loc_25></location>Upon integrating the aforementioned equation, we get,</text>
<formula><location><page_104><loc_46><loc_18><loc_86><loc_22></location>˙ a 2 = σ a 2 + β a 4 , (4.21)</formula>
<text><location><page_104><loc_18><loc_15><loc_49><loc_17></location>where σ is the integration constant.</text>
<text><location><page_104><loc_18><loc_11><loc_86><loc_15></location>The characteristics of the solutions to this equation exhibit variation contingent upon the positive or negative nature of both σ and β . It is important</text>
<text><location><page_105><loc_14><loc_86><loc_82><loc_90></location>to observe, as indicated in equation (4.21), that both σ and β cannot take on negative values concurrently.</text>
<section_header_level_1><location><page_105><loc_17><loc_82><loc_52><loc_84></location>· Case-I : when both σ and β positive</section_header_level_1>
<text><location><page_105><loc_14><loc_77><loc_82><loc_80></location>In this particular scenario, the solution to the wave equation (4.10) can be expressed as follows,</text>
<formula><location><page_105><loc_30><loc_70><loc_82><loc_75></location>ψ + ψ 0 = α 2 √ β ln ∣ ∣ ∣ ∣ ∣ √ a 2 + β / σ -√ β / σ √ a 2 + β / σ + √ β / σ ∣ ∣ ∣ ∣ ∣ . (4.22)</formula>
<text><location><page_105><loc_14><loc_65><loc_82><loc_68></location>By performing the integration of equation (4.21), we derive the connection that establishes the relationship between the scale factor and time as,</text>
<formula><location><page_105><loc_21><loc_60><loc_82><loc_63></location>t + t 0 = 1 2 √ σ a √ a 2 + β / σ -1 2 β σ 3/2 ln ∣ ∣ ∣ ∣ a + √ a 2 + β / σ ∣ ∣ ∣ ∣ . (4.23)</formula>
<text><location><page_105><loc_14><loc_54><loc_82><loc_58></location>This equation is involved and it is difficult to write a in terms of t explicitly. Asubscript zero, as usual, indicates the present value of the quantity.</text>
<text><location><page_105><loc_14><loc_44><loc_82><loc_53></location>These solutions for the scale factor have previously been established and documented in existing literature [417, 418]. Our motive here is not to find the solution but to see how that goes with GSL. Upon substituting the expression of the Hubble parameter ( H ) and the second derivative of the scale factor (a ) into the definition of the deceleration parameter, the resulting outcome is,</text>
<formula><location><page_105><loc_39><loc_38><loc_82><loc_41></location>q = 1 + β β + σ a 2 > 0. (4.24)</formula>
<text><location><page_105><loc_14><loc_26><loc_82><loc_36></location>Consequently, this particular model gives rise to a universe that consistently experiences deceleration over time. This implies that the expansion of the universe slows down progressively as time advances. By utilizing equations (4.18),(4.20)and(4.21), we derive the expression for the rate of total entropy change as follows,</text>
<formula><location><page_105><loc_33><loc_20><loc_82><loc_25></location>˙ S t = -64 π 2 √ σ β a 3 ( a 2 + 3 β /2 σ ) 2 ( a 2 + β / σ ) 3/2 . (4.25)</formula>
<text><location><page_105><loc_14><loc_8><loc_82><loc_18></location>Given that both β and σ possess positive values, it becomes evident that the rate of change of total entropy ( ˙ S t ) is negative. This scenario unequivocally indicates that the test of the Generalized Second Law (GSL) is not satisfied in this case. In essence, the observed decrease in total entropy contradicts the requirements of the GSL, highlighting a failure of the test under</text>
<text><location><page_106><loc_18><loc_88><loc_33><loc_90></location>these conditions.</text>
<section_header_level_1><location><page_106><loc_21><loc_84><loc_58><loc_86></location>· Case-II : σ is positive but β is negative</section_header_level_1>
<text><location><page_106><loc_18><loc_77><loc_86><loc_82></location>When ω is less than -3/2, the value of β becomes negative. We express this negative value of β as -γ 2 , where γ is a real number. Consequently, equation (4.21) transforms into the following form,</text>
<formula><location><page_106><loc_45><loc_71><loc_86><loc_75></location>˙ a 2 = σ a 2 -γ 2 a 4 . (4.26)</formula>
<text><location><page_106><loc_18><loc_55><loc_86><loc_70></location>The equation presented above indicates the presence of a rebound effect at the value of the scale factor a = γ / √ σ . Moreover, at this bounce point, it is necessary for the proper volume ( a 3 ) to reach a minimum since the second derivative of the scale factor (a ) is positive, given by a = -β a 5 . It is worth noting that the total density and pressure also remain finite at this juncture. Consequently, this rebound mechanism seems to circumvent the singularity commonly associated with the Big Bang model.</text>
<text><location><page_106><loc_18><loc_47><loc_86><loc_55></location>However, there is an important consideration to make. While the bounce occurs at a = γ / √ σ , the equation (4.26) imposes a constraint, disallowing smaller values of a . This limitation introduces a discontinuity in the model at this rebound point.</text>
<text><location><page_106><loc_18><loc_43><loc_86><loc_46></location>Under these circumstances, the solution to the wave equation (4.10) is as follows,</text>
<formula><location><page_106><loc_37><loc_37><loc_86><loc_41></location>ψ + ψ 0 = α γ arctan ( √ σ a 2 γ 2 -1 ) . (4.27)</formula>
<text><location><page_106><loc_18><loc_33><loc_78><loc_34></location>In this scenario, the solution for the scale factor can be expressed as,</text>
<formula><location><page_106><loc_24><loc_28><loc_86><loc_31></location>t + t 0 = 1 2 √ σ a √ a 2 -γ 2 / σ + 1 2 γ 2 σ 3/2 ln ∣ ∣ ∣ ∣ a + √ a 2 -γ 2 / σ ∣ ∣ ∣ ∣ . (4.28)</formula>
<text><location><page_106><loc_18><loc_24><loc_72><loc_26></location>Hence, the deceleration parameter can be calculated to yield,</text>
<formula><location><page_106><loc_44><loc_19><loc_86><loc_22></location>q = -2 γ 2 / σ -a 2 a 2 -γ 2 / σ . (4.29)</formula>
<text><location><page_106><loc_18><loc_9><loc_86><loc_17></location>Consequently, an accelerated phase emerges within the range of γ 2 / σ < a 2 < 2 γ 2 / σ . Once the square of the scale factor surpasses 2 γ 2 / σ , the universe transitions into a phase of decelerated expansion. Importantly, it is crucial to observe that the validity of the model breaks down when the square of the</text>
<text><location><page_107><loc_14><loc_86><loc_82><loc_90></location>scale factor is smaller than γ 2 / σ . This particular range of values is where the model does not hold true.</text>
<text><location><page_107><loc_14><loc_82><loc_82><loc_85></location>By utilizing equations (4.18) and (4.26), we derive the rate of change of total entropy as,</text>
<formula><location><page_107><loc_34><loc_75><loc_82><loc_80></location>˙ S t = 64 π 2 √ σ γ 2 a 3 ( a 2 -3 γ 2 /2 σ ) 2 ( a 2 -γ 2 / σ ) 3/2 . (4.30)</formula>
<text><location><page_107><loc_14><loc_61><loc_82><loc_75></location>Given that the scale factor ( a ) is greater than or equal to γ / √ σ , it follows that the rate of total entropy change remains positive. Consequently, the Generalized Second Law (GSL) is upheld in both the phases of acceleration and deceleration. This implies that the increase in total entropy aligns with the criteria of the GSL throughout these distinct phases of the evolution of the universe.</text>
<section_header_level_1><location><page_107><loc_17><loc_58><loc_55><loc_59></location>· Case-III : σ is negative but β is positive</section_header_level_1>
<text><location><page_107><loc_14><loc_50><loc_82><loc_56></location>Let us look at the case where σ is negative and is given by σ = -λ 2 . As a result, equation (4.21) undergoes a modification, transforming into the expression,</text>
<formula><location><page_107><loc_42><loc_46><loc_82><loc_50></location>˙ a 2 = β -λ 2 a 2 a 4 . (4.31)</formula>
<text><location><page_107><loc_14><loc_44><loc_77><loc_46></location>The equation that dictates the variation in the scalar field is as follows,</text>
<formula><location><page_107><loc_29><loc_38><loc_82><loc_43></location>ψ + ψ 0 = α 2 √ β ln ∣ ∣ ∣ ∣ ∣ √ β / λ 2 -a 2 -√ β / λ 2 √ β / λ 2 -a 2 + √ β / λ 2 ∣ ∣ ∣ ∣ ∣ . (4.32)</formula>
<text><location><page_107><loc_14><loc_33><loc_82><loc_36></location>This equation is derived by integrating the wave equation utilizing equation (4.19).</text>
<text><location><page_107><loc_14><loc_29><loc_82><loc_32></location>The solution for the scale factor can be obtained by a straightforward integration of equation (4.31) as,</text>
<formula><location><page_107><loc_25><loc_23><loc_82><loc_27></location>t + t 0 = -1 2 λ a √ β / λ 2 -a 2 + β 2 λ 3 arcsin ( a √ β / λ 2 ) . (4.33)</formula>
<text><location><page_107><loc_14><loc_8><loc_82><loc_20></location>According to equation (4.31), it is evident that a is not greater than √ β / λ . When a reaches a = √ β / λ , the rate of change ˙ a becomes zero, and the second derivative a equals -λ 5 / β 5/2 . However, the validity of the model does not extend beyond √ β / λ , rendering this point not a true maximum for the scale factor. In reality, the model lacks definition beyond this threshold, and its validity ceases. Due to the absence of a lower bound limit on a , it will</text>
<text><location><page_108><loc_18><loc_86><loc_86><loc_90></location>inevitably approach a singularity when traced back in time. The deceleration parameter is obtained as,</text>
<formula><location><page_108><loc_45><loc_82><loc_86><loc_86></location>q = 2 β -λ 2 a 2 β -λ 2 a 2 . (4.34)</formula>
<text><location><page_108><loc_18><loc_77><loc_86><loc_81></location>As a is not greater than √ β / λ , it follows that q takes on a positive value. Consequently, this model results in a deceleration phase.</text>
<text><location><page_108><loc_18><loc_73><loc_86><loc_77></location>Through the utilization of equations (4.18) and (4.31), the rate of change of overall entropy is obtained as,</text>
<formula><location><page_108><loc_37><loc_67><loc_86><loc_71></location>˙ S t = -64 π 2 λ β a 3 ( 3 β /2 λ 2 -a 2 ) 2 ( β / λ 2 -a 2 ) 3/2 . (4.35)</formula>
<text><location><page_108><loc_18><loc_55><loc_86><loc_65></location>The rate at which the total entropy changes experiences a decrease, as indicated by the fact that a is not greater than √ β / λ . Consequently, this particular model does not align with the Generalized Second Law (GSL) of thermodynamics, which asserts that the total entropy of a closed system either remains constant or increases over time.</text>
<section_header_level_1><location><page_108><loc_18><loc_49><loc_37><loc_51></location>4.5 Dust Era:</section_header_level_1>
<text><location><page_108><loc_18><loc_46><loc_83><loc_47></location>When the fluid is a pressureless dust, the field equations are expressed as,</text>
<formula><location><page_108><loc_44><loc_40><loc_86><loc_44></location>3 ( ˙ a a ) 2 = ρ + ρψ , (4.36)</formula>
<formula><location><page_108><loc_42><loc_33><loc_86><loc_37></location>2 ( a a ) + ( ˙ a a ) 2 = -p ψ . (4.37)</formula>
<text><location><page_108><loc_21><loc_30><loc_79><loc_31></location>Substituting p = 0 into the wave equation (4.10) yields the result,</text>
<formula><location><page_108><loc_43><loc_24><loc_86><loc_28></location>¨ ψ + 3 ˙ a a ˙ ψ = ρ 2 ω + 3 . (4.38)</formula>
<text><location><page_108><loc_18><loc_16><loc_86><loc_24></location>After taking the derivative of the field Equation (4.36) and subsequently employing the field Equations (4.36) through (4.37), along with the wave equation (4.38), we arrive at a revised form of the continuity equation, which can be expressed as,</text>
<formula><location><page_108><loc_45><loc_12><loc_86><loc_16></location>˙ ρ + 3 ˙ a a ρ = -ρ ˙ ψ 2 . (4.39)</formula>
<text><location><page_109><loc_14><loc_82><loc_82><loc_90></location>The presence of a non-zero term on the right-hand side of the aforementioned equation indicates that the energy densities of two distinct fluids, namely, dust and scalar field, do not independently maintain conservation. However, they do adhere to a conservation equation, expressed as,</text>
<formula><location><page_109><loc_40><loc_77><loc_82><loc_80></location>˙ ρ t = -3 ˙ a a ( ρ t + p t ) . (4.40)</formula>
<text><location><page_109><loc_14><loc_69><loc_82><loc_75></location>This signifies that their combined energy conservation is satisfied. Obtaining this overall conservation equation from the field equations is a straightforward process. Upon integrating equation (4.39), the result is as follows,</text>
<formula><location><page_109><loc_41><loc_64><loc_82><loc_67></location>ρ = ρ 0 exp [ -ψ 2 ] a 3 . (4.41)</formula>
<text><location><page_109><loc_14><loc_59><loc_82><loc_62></location>By employing the wave equation to eliminate ρ , the integration of a composite set of field equations leads to,</text>
<formula><location><page_109><loc_39><loc_53><loc_82><loc_57></location>a 2 ˙ a = 2 ω + 3 2 a 3 ˙ ψ + ξ . (4.42)</formula>
<text><location><page_109><loc_14><loc_46><loc_82><loc_52></location>In the above equation, we have introduced ξ as an integration constant. By plugging in the expression for ˙ ψ from the preceding equation into Equation (4.37), we derive,</text>
<formula><location><page_109><loc_32><loc_41><loc_82><loc_45></location>2a = -χ ˙ a 2 a + 2 ξ 2 ω + 3 ˙ a a 3 -ξ 2 2 ω + 3 1 a 5 . (4.43)</formula>
<text><location><page_109><loc_14><loc_31><loc_82><loc_39></location>Here, χ = 2 ω + 4 2 ω + 3 stands as a constant linked to the Brans-Dicke parameter ω . To continue our analysis, we can assume, without losing generality, that the constant of integration ξ is set to zero. Consequently, Equation (4.43) transforms into,</text>
<formula><location><page_109><loc_43><loc_28><loc_82><loc_31></location>2a = -χ ˙ a 2 a . (4.44)</formula>
<text><location><page_109><loc_17><loc_26><loc_66><loc_27></location>Now, the equation is amenable to integration, yielding,</text>
<formula><location><page_109><loc_43><loc_22><loc_82><loc_24></location>˙ a = µ a -χ /2 . (4.45)</formula>
<text><location><page_109><loc_14><loc_18><loc_75><loc_19></location>Here, µ represents a constant arising from the process of integration.</text>
<text><location><page_109><loc_17><loc_16><loc_58><loc_17></location>Next, by utilizing Equation (4.42), we acquire,</text>
<formula><location><page_109><loc_39><loc_10><loc_82><loc_14></location>˙ ψ = 2 µ 2 ω + 3 a -( χ 2 + 1 ) . (4.46)</formula>
<text><location><page_110><loc_18><loc_86><loc_86><loc_90></location>Integrating the aforementioned pair of equations leads to the solution for the system, expressed as,</text>
<formula><location><page_110><loc_42><loc_80><loc_86><loc_84></location>t + t 0 = 1 µ ( χ 2 + 1 ) a χ 2 + 1 , (4.47)</formula>
<formula><location><page_110><loc_41><loc_78><loc_86><loc_80></location>ψ + ψ 0 = ln ( a 2 2 ω + 3 ) . (4.48)</formula>
<text><location><page_110><loc_18><loc_70><loc_86><loc_76></location>Here, t 0 and ψ 0 represent integration constants. Now, by employing the definition of the deceleration parameter, equation (4.44), and the definition of χ , we deduce the expression for the deceleration parameter as follows,</text>
<formula><location><page_110><loc_47><loc_65><loc_86><loc_68></location>q = ω + 2 2 ω + 3 . (4.49)</formula>
<text><location><page_110><loc_18><loc_55><loc_86><loc_63></location>By observing equation (4.47), it becomes evident that for the purpose of modeling an expanding universe, both µ and χ 2 + 1 need to take on positive values. The requirement for χ 2 + 1 to be positive can be satisfied through two distinct approaches, and we will consider each of these methods separately.</text>
<section_header_level_1><location><page_110><loc_21><loc_52><loc_74><loc_54></location>· 1st way to achieve χ 2 + 1 to be positive and explore GSL:</section_header_level_1>
<text><location><page_110><loc_18><loc_41><loc_86><loc_51></location>The expression χ 2 + 1 = 3 ω + 5 2 ω + 3 must be greater than zero for our analysis. In the initial scenario, we examine the conditions under which both 3 ω + 5 and 2 ω + 3 are positive, leading to ω > -3 2 . As a result, we can conclude that the expansion of the universe is accompanied by a decelerating motion in this case.</text>
<text><location><page_110><loc_18><loc_30><loc_86><loc_38></location>Subsequently, our task involves examining the Generalized Second Law (GSL) within this context. By utilizing equation (4.18), we are able to determine the rate at which the total entropy changes. This equation provides insights into how the total entropy of a system evolves over time,</text>
<formula><location><page_110><loc_38><loc_25><loc_86><loc_29></location>˙ S t = 16 π 2 ( χ 2 + 1 ) 3 ( 2 2 -χ ) t . (4.50)</formula>
<text><location><page_110><loc_18><loc_16><loc_86><loc_24></location>In order for the quantity ˙ S t to exhibit a positive value, it is necessary that the value of ω exceeds -1. This condition on the Brans-Dicke parameter ensures that the total entropy of the system is increasing over time, contributing to a consistent interpretation of the physical processes at play.</text>
<section_header_level_1><location><page_110><loc_21><loc_13><loc_75><loc_15></location>· 2nd way to achieve χ 2 + 1 to be positive and explore GSL:</section_header_level_1>
<text><location><page_110><loc_18><loc_8><loc_86><loc_12></location>Another approach to satisfy the condition χ 2 + 1 > 0 is when both 3 ω + 5 and 2 ω + 3 are negative, corresponding to ω < -5/3. In this scenario,</text>
<text><location><page_111><loc_14><loc_80><loc_82><loc_90></location>cosmic acceleration can be achieved within the range of -2 < ω < -5/3. However, it is important to note that for ω < -2, the expansion of the universe remains decelerated, as implied by equation (4.49). The limitations on the potential values of ω align consistently with those deduced by Banerjee and Pavón [100].</text>
<text><location><page_111><loc_14><loc_60><loc_82><loc_77></location>From equation (4.50), it becomes apparent that in order for the rate of change ˙ S t to exhibit a positive value, the parameter χ needs to be smaller than 2. Given that ω is constrained to be less than -5/3 in this scenario, it is assured that χ is indeed less than 2. As a result of this, we can confidently conclude that ˙ S t > 0, indicating an increase in the total entropy. Consequently, this particular model adheres to the requirements of the Generalized Second Law (GSL). This alignment signifies that the model upholds a fundamental principle of thermodynamics and consistent physical behavior.</text>
<section_header_level_1><location><page_111><loc_14><loc_55><loc_55><loc_57></location>4.6 Summary and Discussion:</section_header_level_1>
<text><location><page_111><loc_14><loc_34><loc_82><loc_53></location>This research delves into the thermodynamic characteristics of cosmological models dominated by radiation and dust within the framework of BransDicke theory. Specifically, we consider a spatially flat, homogeneous, and isotropic universe in what is referred to as the Einstein frame , which is the conformally transformed version of Brans-Dicke theory in the so-called Jordan frame . Our focus centers on investigating the validity of the generalized second law of thermodynamics, posited by Bekenstein in the early 1970s [325, 390, 391], which asserts that the combined entropy of the universe, the sum of matter entropy and horizon entropy, cannot decrease with time.</text>
<text><location><page_111><loc_14><loc_21><loc_82><loc_31></location>Our findings indicate that for a radiation-dominated universe, the solutions derived from Brans-Dicke theory with a positively definite parameter ω fail to uphold the principles of the generalized second law. However, intriguingly, when certain ranges of negative ω values are considered, the model indeed aligns with the requirements of thermodynamics.</text>
<text><location><page_111><loc_14><loc_11><loc_82><loc_19></location>For a universe dominated by dust, the model does satisfy the generalized second law for specific small negative values of ω . Notably, this range of -2 < ω < -5 3 substantially overlaps with the parameter values required for an accelerated expansion without the need for exotic forms of matter [100].</text>
<text><location><page_112><loc_18><loc_80><loc_86><loc_90></location>Perhaps the most intriguing revelation from this investigation is that Brans-Dicke theory finds thermodynamic support precisely within the parameter range of ω that is associated with driving the alleged accelerated expansion of the universe, as opposed to general cases of decelerated expansion.</text>
<section_header_level_1><location><page_113><loc_14><loc_82><loc_30><loc_84></location>Chapter 5</section_header_level_1>
<section_header_level_1><location><page_113><loc_14><loc_66><loc_82><loc_78></location>Thermodynamic Analysis of Cosmological models reconstructed from jerk parameter</section_header_level_1>
<section_header_level_1><location><page_113><loc_14><loc_58><loc_38><loc_60></location>5.1 Introduction:</section_header_level_1>
<text><location><page_113><loc_14><loc_29><loc_82><loc_56></location>The progress in observational cosmology yielded a remarkable outcome, indicating that the current universe is expanding at an accelerated rate [20, 58, 59, 63, 66, 192, 193, 419]. However, the force responsible for countering the attractive nature of ordinary matter and fueling this acceleration remains an enigma. The absence of any theoretical inclination towards any of the proposed agents, referred to as dark energy, has prompted a different perspective on the issue in recent times. Instead of constructing models based on the matter sector and deriving conclusions about the evolution of the universe, this alternative approach formulates the evolution history based on observations and endeavors to deduce the plausible matter sector from that vantage point. This approach, initially proposed by Ellis and Madsen [420] long ago, has gained attention as none of the suggested candidates decisively emerges as the dark energy.</text>
<text><location><page_113><loc_14><loc_10><loc_82><loc_26></location>Numerous endeavors have been made towards reconstructing cosmological models from physical quantities, such as the dark energy equation of state parameter [274, 279, 280, 421-427], the quintessence potential [428-430], and others. Additionally, attempts have been made to construct models based on kinematical quantities like the deceleration parameter q = -a a ˙ a 2 [431-435], and the jerk parameter j = -1 aH 3 d 3 a d t 3 [436-440], where a denotes the scale factor and H = ˙ a a represents the Hubble parameter. In many of these investigations concerning kinematical quantities, q or j is expressed</text>
<text><location><page_114><loc_18><loc_80><loc_86><loc_90></location>parametrically, and the parameters are estimated using observational data. Among these cosmographic quantities, the Hubble parameter H has been an observable and is known to evolve over time. With the evolution of the deceleration parameter q itself being observable, attention has turned to the third-order derivative, namely the jerk parameter j .</text>
<text><location><page_114><loc_18><loc_46><loc_86><loc_77></location>Beyond depicting the evolutionary trajectory, a cosmological model must account for structure formation in terms of matter perturbation growth and must adhere to some fitness test, such as thermodynamic viability. The motivation behind this study is to assess the thermodynamic viability of some specific models reconstructed from the jerk parameter. We focus on two sets of reconstructed jerk models existing in the literature: one without any interaction in the matter sector [439] and the other allowing for the possibility of interaction [440]. The rationale for selecting these models lies in the fact that they deviate from the usual anstaz considered in parametrically reconstructing j . Typically, the ansatz is designed such that the jerk parameter j , as a function of the redshift parameter z , equals -1 at z = 0, which aligns the model with the present-day Λ CDM model. However, the models in [439] and [440] permit any value of j ( z = 0 ) initially, to be determined by actual observational datasets. Hence, these models are more versatile in that sense.</text>
<text><location><page_114><loc_18><loc_28><loc_86><loc_43></location>Given the reconstructed j , it is possible to find the first derivative of the Hubble parameter and then the Hubble parameter. This, in turn, allows estimation of the rate of change of total entropy, enabling an assessment of whether entropy increases and thereby verifying adherence to the Generalized Second Law (GSL), which asserts that the total entropy of the universe cannot decrease. The total entropy is a combination of the entropy of the fluid enclosed by the horizon and that of the horizon itself [325, 390, 391].</text>
<text><location><page_114><loc_18><loc_11><loc_86><loc_26></location>Acritical assumption in these calculations is that cosmic matter is in thermal equilibrium with the horizon, with the temperature of the latter determined by the Hayward-Kodama temperature [404, 414, 415], which we have taken in all the previous chapters. This thermal equilibrium becomes questionable if the matter sector involves a radiation component, as recently demonstrated by Mimoso and Pavón [416]. Importantly, the reconstructed models examined in this study lack a radiation component; they consist of</text>
<text><location><page_115><loc_14><loc_84><loc_82><loc_90></location>either cold dark matter or a similar matter component where thermal equilibrium holds, alongside a dark energy component driving the acceleration, for which equilibrium is approximately valid [416].</text>
<section_header_level_1><location><page_115><loc_14><loc_78><loc_36><loc_80></location>5.2 Kinematics:</section_header_level_1>
<text><location><page_115><loc_14><loc_68><loc_82><loc_76></location>We examine a cosmological scenario characterized by a spatially flat, homogeneous, and isotropic FRW model. The metric is given by the equation(3.5). Although the pertinent kinematic quantities are introduced and discussed in detail in Chapter 1, let us reiterate the definitions here for the sake of clarity.</text>
<unordered_list>
<list_item><location><page_115><loc_17><loc_65><loc_36><loc_66></location>· Hubble parameter:</list_item>
<list_item><location><page_115><loc_17><loc_59><loc_41><loc_60></location>· Deceleration parameter:</list_item>
<list_item><location><page_115><loc_17><loc_53><loc_33><loc_54></location>· Jerk parameter:</list_item>
</unordered_list>
<formula><location><page_115><loc_44><loc_49><loc_57><loc_53></location>j ≡ -1 aH 3 d 3 a dt 3 .</formula>
<text><location><page_115><loc_14><loc_36><loc_82><loc_48></location>As usual the dot denotes differentiation in relation to cosmic time t , which is the argument of all functions. We will now express these quantities in terms of a dimensionless parameter, specifically the redshift parameter denoted as z , defined by 1 + z ≡ a 0 a . The subscript 0 designates the current value of the quantity. The expressions for the deceleration parameter and the jerk parameter as functions of z are as follows,</text>
<formula><location><page_115><loc_26><loc_30><loc_82><loc_34></location>q ( z ) = -1 + 1 2 ( 1 + z ) [ H 2 ( z )] ' H 2 ( z ) , (5.1)</formula>
<formula><location><page_115><loc_26><loc_26><loc_82><loc_30></location>j ( z ) = -1 +( 1 + z ) [ H 2 ( z )] ' H 2 ( z ) -1 2 ( 1 + z ) 2 [ H 2 ( z )] '' H 2 ( z ) . (5.2)</formula>
<text><location><page_115><loc_14><loc_21><loc_82><loc_24></location>In the entirety of this article, the symbol prime is employed to signify differentiation with respect to the redshift parameter z .</text>
<section_header_level_1><location><page_115><loc_14><loc_15><loc_74><loc_17></location>5.3 Fundamental Thermodynamic Principle</section_header_level_1>
<text><location><page_115><loc_14><loc_10><loc_82><loc_13></location>As previously stated in the introduction, we will assess the thermodynamic viability of the models by subjecting them to a test based on the Generalized</text>
<formula><location><page_115><loc_47><loc_62><loc_54><loc_65></location>H ≡ ˙ a a .</formula>
<formula><location><page_115><loc_46><loc_56><loc_55><loc_59></location>q ≡ -aa ˙ a 2 .</formula>
<text><location><page_116><loc_18><loc_77><loc_86><loc_90></location>Second Law of Thermodynamics (GSL) [325, 390, 391]. This principle asserts that the overall entropy of the universe remains constant or increases over time. The collective entropy comprises the entropy associated with the horizon and the entropy of the fluid enclosed by the horizon. In our analysis, we make the assumption that the fluid components are in a state of thermal equilibrium with the horizon.</text>
<section_header_level_1><location><page_116><loc_21><loc_74><loc_62><loc_76></location>· Revisiting Hayward-Kodama Temperature:</section_header_level_1>
<text><location><page_116><loc_18><loc_64><loc_86><loc_72></location>Complete analytic description of Hyaward-Kodama temperature has been provided in chapter 2, section 2.2.2. It seems requied to revisit the fact in brief, why we consider Hayward-Kodama temperature instead of Hawking temperature as the equilibrium temperature.</text>
<text><location><page_116><loc_18><loc_60><loc_86><loc_64></location>Within a spatially flat FRW cosmological framework, the surface gravity pertaining to the apparent horizon is given by,</text>
<formula><location><page_116><loc_42><loc_55><loc_86><loc_58></location>κ ko = -1 2 H ( ˙ H + 2 H 2 ) , (5.3)</formula>
<text><location><page_116><loc_18><loc_50><loc_86><loc_53></location>where κ ko signifies the surface gravity. Thus, based on the preceding analysis, we can determine the Hayward-Kodama temperature as follows,</text>
<formula><location><page_116><loc_46><loc_41><loc_86><loc_48></location>T = | κ ko | 2 π = 2 H 2 + ˙ H 4 π H . (5.4)</formula>
<text><location><page_116><loc_18><loc_23><loc_86><loc_39></location>Since our analysis will focus on a spacetime that changes with time, it's important to carefully consider which temperature assumption to use. In this context, we opt for the Hayward-Kodama temperature instead of the Hawking temperature. Moreover, we shall direct our attention to the concept of an apparent horizon, as distinct from an event horizon, for our subsequent discourse. This temperature is particularly relevant when analyzing spacetimes with evolving geometry and kinematics, as it takes into account the dynamic aspects of the system.</text>
<text><location><page_116><loc_18><loc_17><loc_86><loc_22></location>In thermal equilibrium, the horizon temperature and the temperature of the fluid within the enclosed region are equal. Therefore, temperature of the enclosed fluid is also the Hayward-Kodama temperature.</text>
<text><location><page_116><loc_18><loc_8><loc_86><loc_16></location>To delve deeper into the theoretical underpinnings of these concepts and their implications, it is beneficial to refer to the existing literature. Relevant works, such as the monograph by Faraoni [404], offer comprehensive discussions on spacetime thermodynamics, temperature concepts, and horizon</text>
<text><location><page_117><loc_14><loc_82><loc_82><loc_90></location>properties. This reference provides a solid foundation for understanding the motivations and applications of employing the Hayward-Kodama temperature in our analysis and the significance of utilizing the apparent horizon in the context of thermal equilibrium scenarios.</text>
<text><location><page_117><loc_14><loc_75><loc_82><loc_81></location>In the preceding chapter (Chapter 2), we computed the rate of change in the total entropy. In the current chapter, we are merely restating the expression for subsequent calculations. The rate of change of the total entropy is,</text>
<formula><location><page_117><loc_35><loc_70><loc_61><loc_74></location>˙ S tot = 16 π 2 ( ˙ H 2 H 3 )( 1 2 H 2 + ˙ H ) .</formula>
<text><location><page_117><loc_14><loc_65><loc_82><loc_68></location>To facilitate a dimensionless representation, we express the Hubble parameter in the following manner:</text>
<text><location><page_117><loc_14><loc_58><loc_82><loc_64></location>We introduce the dimensionless Hubble parameter h ( z ) , which is defined as the ratio of the Hubble parameter at a specific redshift z to the current value of the Hubble parameter H 0 . Mathematically, this can be written as:</text>
<formula><location><page_117><loc_42><loc_53><loc_54><loc_56></location>h ( z ) = H ( z ) H 0 .</formula>
<text><location><page_117><loc_14><loc_35><loc_82><loc_51></location>This dimensionless form of the Hubble parameter offers a normalized perspective on the rate of expansion of the universe at different cosmic epochs. By comparing the Hubble parameter at a particular redshift to its present-day value, we gain insight into the relative pace of cosmic expansion across different eras. This representation aids in simplifying calculations and analyses by removing the explicit dependence on the absolute scale of the Hubble parameter, focusing instead on the relative changes with respect to the current state of the universe.</text>
<text><location><page_117><loc_17><loc_33><loc_82><loc_34></location>Therefore, now we write the rate of change of entropy in terms of h(z) as,</text>
<formula><location><page_117><loc_32><loc_24><loc_82><loc_29></location>˙ S tot = 4 π 2 H 0 ( 1 + z ) 2 [ ( h 2 ) ' ] 2 h 3 [ 2 h 2 -1 2 ( 1 + z )( h 2 ) ' ] . (5.5)</formula>
<text><location><page_117><loc_14><loc_17><loc_82><loc_22></location>Consequently, armed with an understanding of the progression of h and its rate of change with respect to z , it becomes feasible to assess the adherence to the Generalized Second Law (GSL) for a specific model.</text>
<section_header_level_1><location><page_118><loc_18><loc_85><loc_86><loc_90></location>5.4 Cosmological Models Devoid of an Interactions in the Matter Sector</section_header_level_1>
<text><location><page_118><loc_18><loc_75><loc_86><loc_83></location>The first subsection delves into an in-depth discussion regarding the specific model that has been selected for examination. Subsequently, the second subsection is dedicated to an exploration of how the Generalized Second Law (GSL) is assessed within the context of these models.</text>
<section_header_level_1><location><page_118><loc_18><loc_70><loc_37><loc_72></location>5.4.1 The model</section_header_level_1>
<text><location><page_118><loc_18><loc_50><loc_86><loc_68></location>We consider existing dark energy models sourced from existing literature, where the reconstruction is accomplished using the jerk parameter j . Specifically, these models are adapted from the work presented in [439], which we shall brielfly summarize here. What distinguishes these models is that, unlike certain other methods, they don't require a pre-established condition, such as j ( z = 0 ) = -1, that accords with the Λ CDM model. These models do not rely on any specific theory of gravity as a foundational assumption. We select all four jerk parameter parametrizations outlined in [439], as our starting point. These are as follows,</text>
<formula><location><page_118><loc_39><loc_45><loc_86><loc_48></location>I. j ( z ) = -1 + j 1 1 h 2 ( z ) , (5.6)</formula>
<formula><location><page_118><loc_39><loc_41><loc_86><loc_44></location>II. j ( z ) = -1 + j 1 ( 1 + z ) h 2 ( z ) , (5.7)</formula>
<formula><location><page_118><loc_38><loc_37><loc_86><loc_40></location>III. j ( z ) = -1 + j 1 ( 1 + z ) 2 h 2 ( z ) , (5.8)</formula>
<formula><location><page_118><loc_38><loc_33><loc_86><loc_36></location>IV. j ( z ) = -1 + j 1 1 ( 1 + z ) h 2 ( z ) . (5.9)</formula>
<text><location><page_118><loc_18><loc_10><loc_86><loc_31></location>Here, j 1 is a constant value to be determined by observational data. The selection of these parametrizations initially appears arbitrary. However, there are specific reasons underlying these choices. In Model I, the dependence on z is not explicitly present; the variation of j is exclusively managed by the function h ( z ) . Models II and III, on the other hand, exemplify instances where j exhibits a direct proportionality to uncomplicated positive powers of ( 1 + z ) . In the fourth model, however, the relationship takes a form where j is inversely proportional to ( 1 + z ) . The rationale underlying these chosen parameterizations becomes evident when the respective parameters are derived from datasets, a process undertaken in [439]. The fundamental notion</text>
<text><location><page_119><loc_14><loc_82><loc_82><loc_90></location>driving this selection is to opt for straightforward functions of z . This approach aligns with the principles often used for parameterizations of quantities with more tangible physical significance, such as the equation of state parameter. These equations can be integrated to yield,</text>
<formula><location><page_119><loc_31><loc_77><loc_82><loc_80></location>I. h 2 ( z ) = c 1 ( 1 + z ) 3 + c 2 + 2 3 j 1 ln ( 1 + z ) , (5.10)</formula>
<formula><location><page_119><loc_30><loc_74><loc_82><loc_76></location>II. h 2 ( z ) = c 1 ( 1 + z ) 3 + c 2 + j 1 ( 1 + z ) , (5.11)</formula>
<formula><location><page_119><loc_29><loc_72><loc_82><loc_74></location>III. h 2 ( z ) = c 1 ( 1 + z ) 3 + c 2 + j 1 ( 1 + z ) 2 , (5.12)</formula>
<formula><location><page_119><loc_29><loc_68><loc_82><loc_71></location>IV. h 2 ( z ) = c 1 ( 1 + z ) 3 + c 2 -j 1 1 2 ( 1 + z ) , (5.13)</formula>
<text><location><page_119><loc_14><loc_58><loc_82><loc_66></location>correspondingly. Here, c 1 and c 2 denote integration constants. Nevertheless, it is important to note that these constants are subject to limitations imposed by h 0 ( z ) = 1, a condition derived from the definition of h . Consequently, c 2 can be substituted using the subsequent relationships,</text>
<formula><location><page_119><loc_41><loc_54><loc_82><loc_56></location>I. c 2 = 1 -c 1 , (5.14)</formula>
<formula><location><page_119><loc_40><loc_52><loc_82><loc_53></location>II. c 2 = 1 -j 1 -c 1 , (5.15)</formula>
<formula><location><page_119><loc_39><loc_49><loc_82><loc_51></location>III. c 2 = 1 -j 1 -c 1 , (5.16)</formula>
<formula><location><page_119><loc_39><loc_45><loc_82><loc_49></location>IV. c 2 = 1 + j 1 2 -c 1 . (5.17)</formula>
<text><location><page_119><loc_14><loc_23><loc_82><loc_44></location>The values of the two distinct parameters, namely c 1 and j 1 , are determined through the utilization of observational data. This calibration is conducted using combinations of datasets originating from sources such as SNe (Supernovae), OHD (Observational Hubble Data), BAO (Baryon Acoustic Oscillations), and CMBShift (Cosmic Microwave Background Shift), as meticulously detailed in [439]. These calibrated values are presented in Table 1. It is noteworthy that c 1 serves as Ω m 0 , which signifies the matter density parameter at the current epoch. The determination of both c 1 and j 1 through empirical data plays a pivotal role in establishing the foundation for the subsequent analyses and interpretations of the chosen dark energy models.</text>
<text><location><page_119><loc_14><loc_8><loc_82><loc_21></location>The equations (5.10-5.13) readily reveal that upon establishing the correspondence between c 1 and the current matter density parameter Ω m 0 , the initial term in all four models inherently represents the independent evolution of pressureless cold matter. This distinctive feature substantiates the characterization of this particular group as non-interacting models, as highlighted in [439]. It is worth emphasizing that both Model I and Model</text>
<text><location><page_120><loc_18><loc_84><loc_86><loc_90></location>IV encounter challenges concerning their future evolution. This predicament arises due to the presence of singularities at z = -1, a fact discerned from the expressions in equations (5.10) and (5.13).</text>
<text><location><page_120><loc_18><loc_69><loc_86><loc_83></location>An intriguing observation stems from equation (5.12)is that the behavior of the so-called dark energy in Model III appears reminiscent of the spatial curvature term found in the Friedmann equations. However, it is crucial to recognize that the reconstruction detailed in [439] exclusively takes into account a spatially flat metric as depicted in equation (3.5). Consequently, this similarity is merely a coincidental resemblance rather than a substantial connection.</text>
<table>
<location><page_120><loc_27><loc_56><loc_77><loc_66></location>
</table>
<text><location><page_120><loc_51><loc_55><loc_53><loc_56></location>(a)</text>
<table>
<location><page_120><loc_27><loc_39><loc_77><loc_50></location>
<caption>TABLE 5.1: Outcomes of the statistical analysis of observational data provided for two distinct datasets: (a) SNe+OHD+BAO data and (b) SNe+OHD+BAO+CMBShift data. These tabulated results have been extracted from [439].</caption>
</table>
<text><location><page_120><loc_18><loc_21><loc_86><loc_29></location>While not within the scope of this current study, it is worth noting that the outcomes of the best fits along with their associated 1 σ error margins, as presented in Table 5.1, were attained through a minimization process using χ 2 in [439].</text>
<section_header_level_1><location><page_120><loc_18><loc_16><loc_54><loc_18></location>5.4.2 Thermodynamic Analysis</section_header_level_1>
<text><location><page_120><loc_18><loc_11><loc_86><loc_15></location>By substituting the solutions for h ( z ) as provided by equations (5.10)-(5.13) into equation (5.5), we are able to derive the rate of change of total entropy</text>
<text><location><page_121><loc_14><loc_88><loc_34><loc_90></location>for various models as,</text>
<formula><location><page_121><loc_16><loc_74><loc_84><loc_86></location>I. ˙ S tot = 4 π 2 H 0 ( 1 + z ) 2 [ 3 c 1 ( 1 + z ) 2 + 2 3 j 1 1 ( 1 + z ) ] 2 [ c 1 ( 1 + z ) 3 + c 2 + 2 3 j 1 ln ( 1 + z ) ] 3/2 × 1 2 [ c 1 ( 1 + z ) 3 + c 2 + 2 3 j 1 ln ( 1 + z ) ] -( 1 + z ) 2 [ 3 c 1 ( 1 + z ) 2 + 2 3 j 1 1 ( 1 + z ) ] , (5.18)</formula>
<formula><location><page_121><loc_15><loc_62><loc_82><loc_73></location>II. ˙ S tot = 4 π 2 H 0 ( 1 + z ) 2 [ 3 c 1 ( 1 + z ) 2 + j 1 ] 2 [ c 1 ( 1 + z ) 3 + c 2 + j 1 ( 1 + z )] 3/2 × 1 [ 2 { c 1 ( 1 + z ) 3 + c 2 + j 1 ( 1 + z ) } -( 1 + z ) 2 { 3 c 1 ( 1 + z ) 2 + j 1 } ] , (5.19)</formula>
<formula><location><page_121><loc_14><loc_50><loc_86><loc_62></location>III. ˙ S tot = 4 π 2 H 0 ( 1 + z ) 2 [ 3 c 1 ( 1 + z ) 2 + 2 j 1 ( 1 + z ) ] 2 [ c 1 ( 1 + z ) 3 + c 2 + j 1 ( 1 + z ) 2 ] 3/2 × 1 [ 2 { c 1 ( 1 + z ) 3 + c 2 + j 1 ( 1 + z ) 2 } -( 1 + z ) 2 { 3 c 1 ( 1 + z ) 2 + 2 j 1 ( 1 + z ) } ] , (5.20)</formula>
<formula><location><page_121><loc_14><loc_37><loc_82><loc_50></location>IV. ˙ S tot = 4 π 2 H 0 ( 1 + z ) 2 [ 3 c 1 ( 1 + z ) 2 + j 1 2 1 ( 1 + z ) 2 ] 2 [ c 1 ( 1 + z ) 3 + c 2 -j 1 2 1 ( 1 + z ) ] 3/2 × 1 [ 2 { c 1 ( 1 + z ) 3 + c 2 -j 1 2 1 ( 1 + z ) } -( 1 + z ) 2 { 3 c 1 ( 1 + z ) 2 + j 1 2 1 ( 1 + z ) 2 }] . (5.21)</formula>
<text><location><page_121><loc_14><loc_14><loc_82><loc_34></location>Utilizing the values of c 1 and j 1 extracted from Table I, we generate plots that illustrate the relationship between ˙ S tot and the redshift z . For these plots, we adopt a value of H 0 at 70 km s -1 Mpc -1 . It is important to note that the plots are presented on an arbitrary scale, with the primary focus directed towards conveying the qualitative characteristics of the entropy alteration rates. In the depicted plots (with the exception of Model-IV for SNe+OHD+BAO data), a consistent methodology is employed. The thick solid line represents the ˙ S tot values corresponding to the best-fit c 1 and j 1 parameters. This selection aligns with the contour plots outlined in [439], capturing a 3 σ confidence level.</text>
<text><location><page_121><loc_17><loc_9><loc_82><loc_11></location>Furthermore, within these plots, we maintain the best-fit c 1 value while</text>
<text><location><page_122><loc_18><loc_79><loc_86><loc_90></location>systematically varying the j 1 parameter. This involves selecting the farthest permissible values of j 1 within the contours established at the 3 σ confidence level, as detailed in [439]. The same strategy is then applied when keeping c 1 fixed at its best-fit value ± 3 σ , while exploring the furthest allowable values for j 1 .</text>
<text><location><page_122><loc_18><loc_69><loc_86><loc_77></location>In summary, these plots elucidate the variations in ˙ S tot against the redshift z using specific parameter values. The methodology employed ensures a comprehensive exploration of the parameter space, offering insights into the qualitative trends of entropy change rates for distinct models.</text>
<text><location><page_122><loc_18><loc_58><loc_86><loc_68></location>For Model-IV with the SNe+OHD+BAO data, an examination of the contour plot in [439] unveils a notable observation: it is unfeasible to encompass all the data points within the 3 σ confidence level. In response, we adopt a distinct approach to selecting the ( c 1 , j 1 ) values. Within this context, the more specific procedure is outlined below.</text>
<unordered_list>
<list_item><location><page_122><loc_21><loc_53><loc_86><loc_56></location>1. The boldest line depicted in the plot corresponds to the best-fit ( c 1 , j 1 ) value.</list_item>
<list_item><location><page_122><loc_21><loc_45><loc_86><loc_51></location>2. With c 1 set at its best-fit value, we identify the farthest permissible j 1 values as indicated by the contour plot featuring the 2 σ confidence level.</list_item>
<list_item><location><page_122><loc_21><loc_40><loc_86><loc_43></location>3. Maintaining c 1 at its best-fit value but now at -3 σ , we choose the outermost allowed j 1 values, adhering to the 3 σ confidence level.</list_item>
<list_item><location><page_122><loc_21><loc_32><loc_86><loc_38></location>4. Subsequently, we set c 1 at its best-fit value + 3 σ and identify two j 1 values, one corresponding to the 3 σ confidence level and the other to the 2 σ confidence level.</list_item>
</unordered_list>
<text><location><page_122><loc_18><loc_20><loc_86><loc_30></location>By adopting this methodology, we assemble a total of seven ( c 1 , j 1 ) value pairs, encompassing both the best-fit values and the permissible range according to the confidence levels. This approach offers a comprehensive perspective on the rate of entropy change within the region defined by the ( c 1 , j 1 ) contour plots.</text>
<text><location><page_122><loc_18><loc_10><loc_86><loc_20></location>Figures (5.1), (5.2), and (5.3) distinctly illustrate that the rate of entropy change remains consistently positive. At most, it can approach zero at some point in the future ( z < 0). In the case of Model I, there is a notable sharp surge in entropy in the future, observed for both data set combinations. Despite this surge, ˙ S tot maintains a non-negative trajectory (Figure (5.1)).</text>
<figure>
<location><page_123><loc_28><loc_75><loc_68><loc_90></location>
</figure>
<figure>
<location><page_123><loc_28><loc_59><loc_68><loc_74></location>
<caption>FIGURE 5.1: ˙ Stot vs z for Model-I. (a) SNe+OHD+BAO data, (b) SNe+OHD+BAO+CMBShift data. The parameter values are mentioned in the order ( c 1, j 1 ) .</caption>
</figure>
<text><location><page_123><loc_14><loc_37><loc_82><loc_49></location>However, Model IV does not meet this test of compatibility with thermodynamic principles. For certain parameter selections within the permissible range, ˙ S tot exhibits negative values in a future that is very close to z = 0, particularly when the CMBShift data is not incorporated (Figure (5.4)). This indicates a violation of the expected thermodynamic behavior and suggests a lack of compatibility with established principles.</text>
<text><location><page_123><loc_14><loc_18><loc_82><loc_34></location>Both Model I and Model IV exhibit singular behavior in the rate of entropy change as the redshift approaches z → -1. This outcome arises from the chosen framework outlined in [439], evident from the expressions derived for h 2 as presented in equations (5.10) and (5.13). These equations incorporate terms like ln ( 1 + z ) and 1 1 + z , leading to the observed singularities. However, it is important to note that z = -1 corresponds to a → ∞ , thereby signifying that these singularities do not manifest in any finite future. Consequently, there is no need for concern in this regard.</text>
<text><location><page_123><loc_14><loc_10><loc_82><loc_18></location>The sudden increase in ˙ S tot observed within the domain 0 > z > -1 for Model I is not a consequence of the chosen framework itself. Instead, this phenomenon emerges due to the specific model parameters that have been determined based on the available datasets.</text>
<figure>
<location><page_124><loc_32><loc_75><loc_72><loc_90></location>
</figure>
<figure>
<location><page_124><loc_32><loc_58><loc_72><loc_74></location>
<caption>FIGURE 5.2: ˙ Stot vs z for Model-II. (a) SNe+OHD+BAO data, (b) SNe+OHD+BAO+CMBShift data. The parameter values are mentioned in the order ( c 1, j 1 ) .</caption>
</figure>
<section_header_level_1><location><page_124><loc_18><loc_44><loc_86><loc_49></location>5.5 Model Incorporating Interaction in the Matter Sector</section_header_level_1>
<text><location><page_124><loc_18><loc_34><loc_86><loc_42></location>The first subsection provides a detailed description of the chosen model which allows an interaction in the matter sector which is given in [440]. Following that, the second subsection focuses on the evaluation of the Generalized Second Law (GSL) within the framework of these models.</text>
<section_header_level_1><location><page_124><loc_18><loc_29><loc_37><loc_31></location>5.5.1 The model</section_header_level_1>
<text><location><page_124><loc_18><loc_11><loc_86><loc_28></location>The initial terms in equations (5.10-5.13) experience changes in proportion to ( 1 + z ) 3 . This characteristic strongly indicates the presence of cold dark matter (CDM) in all the models, which exhibits no interaction with other components in the matter sector. However, there exists a specific model that deviates from this non-interacting CDM constraint, as detailed in the reference [440]. This model introduces a relaxation to the non-interacting CDM scenario by assuming that j is a slowly varying parameter. Consequently, the differential equation (5.2) can be integrated under the assumption that j</text>
<figure>
<location><page_125><loc_29><loc_75><loc_68><loc_90></location>
</figure>
<figure>
<location><page_125><loc_28><loc_59><loc_68><loc_74></location>
<caption>FIGURE 5.3: ˙ Stot vs z for Model-III. (a) SNe+OHD+BAO data, (b) SNe+OHD+BAO+CMBShift data. The parameter values are mentioned in the order ( c 1, j 1 ) .</caption>
</figure>
<figure>
<location><page_125><loc_29><loc_36><loc_68><loc_51></location>
</figure>
<figure>
<location><page_125><loc_28><loc_19><loc_68><loc_35></location>
<caption>FIGURE 5.4: ˙ Stot vs z for Model-IV. (a) SNe+OHD+BAO data, (b) SNe+OHD+BAO+CMBShift data. The parameter values are mentioned in the order ( c 1, j 1 ) .</caption>
</figure>
<text><location><page_126><loc_18><loc_88><loc_71><loc_90></location>remains constant. This integration results in the expression,</text>
<formula><location><page_126><loc_28><loc_84><loc_86><loc_87></location>h 2 ( z ) = A ( 1 + z ) 3 + √ 9 -8 ( 1 + j ) 2 +( 1 -A )( 1 + z ) 3 -√ 9 -8 ( 1 + j ) 2 . (5.22)</formula>
<text><location><page_126><loc_18><loc_71><loc_86><loc_81></location>In this context, A represents a dimensionless constant coefficient (potentially corresponding to Ω m 0 ), while the jerk parameter j takes on the role of a model parameter. Through a statistical analysis of various combinations of SNe+OHD+BAO data sets, the subsequent outcomes for the parameters A and j have been derived (for further insights, refer to [440])as,</text>
<formula><location><page_126><loc_43><loc_63><loc_86><loc_67></location>A = 0.286 ± 0.015, j = -1.027 ± 0.037. (5.23)</formula>
<text><location><page_126><loc_18><loc_38><loc_86><loc_60></location>The best fit parameter values are acquired through a conventional minimization of χ 2 , and these values are presented at a 1 σ confidence interval. This specific reconstruction process is elaborated upon in [440]. We will refer to this as model V . Importantly, it should be acknowledged that when j equals -1, the model effectively simplifies to the well-known Λ CDMmodel. The initial term bears a striking resemblance to the evolution pattern of pressureless matter, albeit with subtle distinctions. These deviations from the conventional CDM behavior can be interpreted as resulting from an interaction with another constituent within the matter sector. This underlying concept forms the basis for characterizing this model as an interacting model in [440].</text>
<section_header_level_1><location><page_126><loc_18><loc_33><loc_54><loc_34></location>5.5.2 Thermodynamic Analysis</section_header_level_1>
<text><location><page_126><loc_18><loc_27><loc_86><loc_32></location>Utilizing the derived solution for h ( z ) as presented in equation (5.22), we can calculate the rate of change of total entropy for model V by applying equation (5.5), yielding:</text>
<formula><location><page_126><loc_18><loc_13><loc_89><loc_26></location>V. ˙ Stot = 8 π 2 H 0 ( 1 + z ) 2 × [ A ( 3 + √ 9 -8 ( 1 + j ) 2 ) ( 1 + z ) 1 + √ 9 -8 ( 1 + j ) 2 +( 1 -A ) ( 3 -√ 9 -8 ( 1 + j ) 2 ) ( 1 + z ) 1 -√ 9 -8 ( 1 + j ) 2 ] 2 [ A ( 1 + z ) 3 + √ 9 -8 ( 1 + j ) 2 +( 1 -A ) ( 1 + z ) 3 -√ 9 -8 ( 1 + j ) 2 ] 3/2 × [ 4 { A ( 1 + z ) 3 + √ 9 -8 ( 1 + j ) 2 +( 1 -A ) ( 1 + z ) 3 -√ 9 -8 ( 1 + j ) 2 } -( 1 + z ) { A ( 3 + √ 9 -8 ( 1 + j ) 2 ) ( 1 + z ) 1 + √ 9 -8 ( 1 + j ) 2 +( 1 -A ) ( 3 -√ 9 -8 ( 1 + j ) 2 ) ( 1 + z ) 1 -√ 9 -8 ( 1 + j ) 2 }] -1 .</formula>
<text><location><page_127><loc_14><loc_73><loc_82><loc_90></location>Using the aforementioned optimal parameter values for A and j , which were determined in [440], we proceed to generate a plot of the rate of entropy change against redshift ( z ), complete with error bars, as illustrated in Figure (5.5). Employing a comparable approach as outlined in Section 5.4.2 for models I, II, and III, we select the seven sets of ( A , j ) values. This strategy allows for a comprehensive analysis of ˙ S tot, revealing that it remains consistently positive across the entire range. Furthermore, the Generalized Second Law (GSL) is upheld with considerable fidelity.</text>
<figure>
<location><page_127><loc_28><loc_58><loc_68><loc_72></location>
<caption>FIGURE 5.5: ˙ Stot vs z for Model-V. for SNe+OHD+BAO data. The parameter values are mentioned in the order ( A , j ) .</caption>
</figure>
<section_header_level_1><location><page_127><loc_14><loc_47><loc_55><loc_49></location>5.6 Summary and Discussion</section_header_level_1>
<text><location><page_127><loc_14><loc_35><loc_82><loc_45></location>Given the absence of a universally accepted candidate for dark energy or a specific modified theory of gravity that can fully explain the universe's supposed accelerated expansion, a new approach has surfaced. This approach involves reconstructing models to match observed characteristics of the universe based on the kinematical aspects.</text>
<text><location><page_127><loc_14><loc_18><loc_82><loc_34></location>However, it is essential for these reconstructed models to withstand certain critical evaluations. Among these tests, the examination of thermodynamic consistency is of utmost significance. Fortunately, it is noteworthy that the assessment of the Generalized Second Law (GSL) of thermodynamics can be effectively conducted by utilizing kinematical quantities such as the Hubble parameter and its corresponding derivative, as outlined in equation (5.5). This thermodynamic viability analysis ensures that the reconstructed models align with the principles of thermodynamics.</text>
<text><location><page_127><loc_14><loc_9><loc_82><loc_17></location>Our investigation reveals that it is indeed feasible to identify cosmological models that uphold thermodynamic consistency through the reconstruction of the jerk parameter. Notably, within the subset of four models that pertain to the non-interacting scenario, only one-specifically, model IV - exhibits</text>
<text><location><page_128><loc_18><loc_67><loc_86><loc_90></location>a decline in entropy within the future epoch ( z < 0) due to its inverse dependence on ( 1 + z ) . This phenomenon occurs in close proximity to the present era ( z = 0), as depicted in figure (5.4(a)). On the other hand, all the remaining models, including the one that entertains the prospect of interaction within the dark sector (Model V), satisfactorily adhere to the Generalized Second Law (GSL) of thermodynamics. Model I, while it does conform to the GSL, displays an abrupt surge in entropy within the future ( z < 0). Models II and III exhibit well-behaved attributes in all respects, as does Model V, which permits an interaction within the dark sector. The graphical representations encompass a broad parameter space, encompassing the scope of variation within the 3 σ range.</text>
<text><location><page_128><loc_18><loc_54><loc_86><loc_66></location>Nonetheless, it is important to highlight that the assessment of entropy evolution is predominantly centered around the redshift z = 0, and this evaluation is particularly reliable for values of z greater than zero. The anomalies and irregularities we identify predominantly manifest for values of z that are less than zero. Until the current epoch, all the models exhibit commendable behavior without any significant issues.</text>
<text><location><page_128><loc_18><loc_37><loc_86><loc_53></location>It is worth highlighting that this study operates under the implicit assumption that the temperature of the horizon coincides with that of the fluid. However, this assumption might not hold true in cases involving a radiation distribution. Nevertheless, the scope of this investigation is limited to scenarios involving non-relativistic matter, with models I to IV encompassing a pressureless fluid and model V involving a similar component. In the case of the former, the equality of temperatures is indeed accurate, and even for the latter, it remains reasonable at the very least, as discussed in reference [416].</text>
<section_header_level_1><location><page_129><loc_14><loc_82><loc_30><loc_84></location>Chapter 6</section_header_level_1>
<section_header_level_1><location><page_129><loc_14><loc_66><loc_79><loc_78></location>APossible Thermodynamic Phase Transition: Signature flip of the Deceleration Parameter</section_header_level_1>
<section_header_level_1><location><page_129><loc_14><loc_58><loc_38><loc_60></location>6.1 Introduction:</section_header_level_1>
<text><location><page_129><loc_14><loc_38><loc_82><loc_56></location>In this chapter, we delve into a comprehensive examination of thermodynamics for an accelerating universe. The scale factor is modeled using a hyperbolic function ( a ∼ sinh 2 3 ( t / t 0 ) ). This chosen mathematical representation closely emulates the characteristic behaviour of a Λ CDM model. A notable distinction from the preceding chapters lies in the fact that our current study encompasses not only the application of the Generalized Second Law (GSL) test but also extends to an exploration of the thermodynamic stability of the system. This entails a deeper understanding of how the system evolves and behaves from a thermodynamic perspective.</text>
<text><location><page_129><loc_14><loc_23><loc_82><loc_37></location>Our calculation has led to an exciting result. Through careful examination, it has emerged that despite the entropy ( S ) retaining its continuity, a discontinuity becomes evident in the thermal capacity at constant volume ( C V ) at a specific value of redshift ( z ). This notable occurrence coincides with the point at which the cosmological evolution undergoes a shift from a decelerated state to an accelerated state of expansion. This discontinuity indicates a second-order phase transition.</text>
<text><location><page_129><loc_14><loc_14><loc_82><loc_22></location>The deceleration parameter ( q ) plays a crucial role similar to an order parameter. This parameter encapsulates the essence of the transition between the two different expansion states. Moreover, the discontinuity seen in C V is characterized by an order of unity.</text>
<text><location><page_129><loc_14><loc_10><loc_82><loc_14></location>Essentially, our analysis highlights a striking connection between the thermodynamic features of the system and the evolution of the universe. The</text>
<text><location><page_130><loc_18><loc_75><loc_86><loc_90></location>significant change in thermal capacity at a particular redshift indicates a transformative phase transition, supported by its alignment with an order parameter and the inherent order of the discontinuity. This finding deepens our understanding of the complex interplay between cosmological dynamics and thermodynamic behavior, bringing together different ideas into a cohesive framework that enhances our comprehension of the evolution of the universe.</text>
<text><location><page_130><loc_18><loc_63><loc_86><loc_75></location>As in the preceding chapters, we continue to center our analysis on the apparent horizon. There is a notable difference between the context of a stationary black hole and the evolving nature of the apparent horizon. This distinctive feature serves as the motivation for replacing the Hawking temperature with the Hayward-Kodama temperature [414, 415] to designate the temperature of the horizon.</text>
<text><location><page_130><loc_18><loc_50><loc_86><loc_62></location>It is worth highlighting that recent investigations into evolving black holes within a de-Sitter spacetime have indicated the presence of a secondorder phase transition [441-445]. This observation adds further credence to the potential significance of the connection we have identified between the signature flip in the deceleration parameter ( q ) and the occurrence of a second-order phase transition.</text>
<text><location><page_130><loc_18><loc_31><loc_86><loc_49></location>The stability of the model's thermodynamics can be determined by examining the characteristics of the second-order derivatives pertaining to the system's internal entropy. For examples that demonstrate this approach, readers may refer to the references provided in [446] and [447]. The primary impetus behind the current study is to initially explore the thermodynamic stability of a cosmological model designed to replicate the late-stage evolution characteristics of a Λ CDMmodel. The methodology employed involves investigating the concavity of the entropy function associated with the matter content present in the universe.</text>
<text><location><page_130><loc_18><loc_14><loc_86><loc_30></location>To undertake this investigation, we delve into the intricacies of the behaviour of entropy and its relationship with the matter content. The key technique revolves around scrutinizing the concavity properties of the entropy function. This analysis hinges on the properties of the Hessian matrix, which encompasses the second-order derivatives of the entropy. For a more comprehensive understanding, we refer to Chapter (2), Section 2.2.4. To provide a concrete example of this approach within the realm of cosmology, reference can be made to the work conducted by Bhandari, Haldar, and</text>
<text><location><page_131><loc_14><loc_82><loc_82><loc_90></location>Chakraborty [448]. This framework allows us to establish a connection between the thermodynamic stability of the cosmological model and the underlying properties of the entropy function, thereby offering insights into the system's overall behaviour and evolution.</text>
<section_header_level_1><location><page_131><loc_14><loc_73><loc_82><loc_78></location>6.2 A Model Mimicking the Characteristics of Λ CDM</section_header_level_1>
<text><location><page_131><loc_14><loc_65><loc_82><loc_71></location>We consider a universe that is spatially flat, homogeneous, and isotropic, as described by the FRW metric presented in equation(3.5). Let us reiterate the metric formulation here,</text>
<formula><location><page_131><loc_34><loc_61><loc_82><loc_63></location>ds 2 = -dt 2 + a 2 ( t )[ dr 2 + r 2 d Ω 2 ] , (6.1)</formula>
<text><location><page_131><loc_14><loc_55><loc_82><loc_59></location>a ( t ) denotes the scale factor here. The Einstein field equations associated with this context can be expressed as follows:</text>
<formula><location><page_131><loc_41><loc_51><loc_82><loc_53></location>3 H 2 = ρ , (6.2)</formula>
<formula><location><page_131><loc_41><loc_49><loc_82><loc_50></location>2 ˙ H = -( ρ + p ) . (6.3)</formula>
<text><location><page_131><loc_14><loc_30><loc_82><loc_46></location>In these equations, ρ and p symbolize the total energy density and pressure attributed to the matter constituents. The parameter H = ˙ a a corresponds to the Hubble parameter, while a dot positioned above a variable signifies its derivative with respect to cosmic time t . We adopt units where c = 1 and 8 π G = 1. In this context, the radius of the apparent horizon, denoted as R ah , is defined by the equation g µν R ah , µ R ah , ν = 0. For a universe with spatial flatness ( k = 0), the radius of the apparent horizon is given by R ah = 1 H , as detailed in the work by Faraoni [404].</text>
<text><location><page_131><loc_14><loc_26><loc_82><loc_29></location>We adopt a straightforward assumption for the scale factor, given by the expression:</text>
<formula><location><page_131><loc_40><loc_22><loc_82><loc_26></location>a a 0 ∼ sinh 2/3 ( t / t 0 ) sinh 2/3 ( 1 ) . (6.4)</formula>
<text><location><page_131><loc_14><loc_8><loc_82><loc_20></location>This choice results in an expansion that is accelerated during later periods while a decelerated expansion in the earlier era is dominated by matter. Here, it is important to note that we consider a = a 0 at the time t = t 0 , and we have set t 0 = 1. Remarkably, this particular ansatz (6.4) effectively captures the behaviour akin to the Λ CDM model during the later stages, which is currently favored as the model for our present universe [187].</text>
<figure>
<location><page_132><loc_32><loc_72><loc_72><loc_90></location>
<caption>FIGURE 6.1: q plotted as a function of z</caption>
</figure>
<text><location><page_132><loc_18><loc_57><loc_86><loc_67></location>The equation (6.4) can be utilized to express the cosmic time t / t 0 in terms of the redshift ( z ), wherein z is defined as 1 + z = a 0 a . This relationship can be written as t / t 0 = arcsinh ( ( 1 1 + z ) 3/2 sinh ( 1 ) ) , effectively establishing a connection between time and redshift, where a 0 signifies the present value of the scale factor.</text>
<text><location><page_132><loc_18><loc_53><loc_86><loc_56></location>The Hubble parameter can be expressed in terms of the redshift z as follows,</text>
<formula><location><page_132><loc_39><loc_41><loc_86><loc_51></location>H = 2 3 coth ( t / t 0 ) = 2 csch ( 1 ) 3 √ 1 + sinh 2 ( 1 ) ( 1 + z ) 3 ( 1 1 + z ) 3/2 . (6.5)</formula>
<text><location><page_132><loc_18><loc_35><loc_86><loc_39></location>The deceleration parameter, which is defined as q = -[ 1 + ˙ H H 2 ] , takes the form:</text>
<formula><location><page_132><loc_39><loc_30><loc_86><loc_35></location>q ( z ) = -1 + 3 2 ( 1 + sinh 2 ( 1 ) ( 1 + z ) 3 ) , (6.6)</formula>
<text><location><page_132><loc_18><loc_28><loc_55><loc_29></location>when expressed in terms of the redshift z .</text>
<text><location><page_132><loc_18><loc_19><loc_86><loc_27></location>In Figure (1), we observe the variation of the deceleration parameter ( q ) with respect to the redshift ( z ). Notably, at a specific redshift value of approximately z = -1 + 2 1/3 sinh 2/3 ( 1 ) ≃ 0.403 the evolution of the universe transitions from a decelerating phase to an accelerating one.</text>
<section_header_level_1><location><page_133><loc_14><loc_88><loc_56><loc_90></location>6.3 Thermodynamic approach</section_header_level_1>
<unordered_list>
<list_item><location><page_133><loc_17><loc_74><loc_82><loc_86></location>· GSL: To perform the Generalized Second Law (GSL) test, it is essential to determine the rate at which the total energy changes within the model under consideration. We already provided a calculation method for the evaluation of the rate of change of total entropy, specifically in terms of the Hubble parameter and its derivative, within the framework of FRW cosmology in chapter 2.</list_item>
</unordered_list>
<text><location><page_133><loc_19><loc_48><loc_82><loc_72></location>We have assumed that the fluid within the horizon is in a state of thermodynamic equilibrium with the horizon itself. The conclusions drawn from the research conducted by Mimoso and Pavón [416] provide insights into the topic at hand. Their study establishes that achieving thermal equilibrium between radiation and the cosmic horizon remains an elusive endeavor. This challenge arises due to Wien's law, which consistently yields a wavelength exceeding the horizon radius across all temporal phases. However, it is possible for nonrelativistic particles to reach equilibrium, depending on their individual masses. In this regard, the notion of thermal equilibrium between dark energy and the horizon, advocated by various scholars, such as in [371, 449-454], finds justifiable ground.</text>
<text><location><page_133><loc_19><loc_34><loc_82><loc_46></location>In the present investigation, it is pertinent to note that we have excluded any radiation component from consideration. Consequently, the assumption of establishing thermodynamic equilibrium between the horizon and the fluid content remains valid and applicable in our study. This rationalizes our approach and lends support to the foundational assumptions guiding our analysis.</text>
<text><location><page_133><loc_19><loc_27><loc_82><loc_33></location>We consider the temperature at equilibrium to be determined by the Hayward-Kodama temperature formula, as established in the references by [414, 415]:</text>
<formula><location><page_133><loc_44><loc_24><loc_82><loc_27></location>T = 2 H 2 + ˙ H 4 π H . (6.7)</formula>
<text><location><page_133><loc_19><loc_10><loc_82><loc_22></location>It is worth noting that the temperature becomes zero when the scale factor takes the specific form of a ( t ) = √ α t 2 + β t + γ . Consequently, during a phase dominated solely by radiation (for which a ( t ) ∝ t 1/2 ), equation (6.7) results in a temperature of zero. However, there is no cause for concern in the context of our present study, as we do not engage with radiation-related considerations in any manner.</text>
<text><location><page_134><loc_23><loc_86><loc_86><loc_90></location>In Chapter 2, we derived the expression for the rate of change of the total entropy, which can be written as follows,</text>
<formula><location><page_134><loc_43><loc_78><loc_86><loc_84></location>˙ S = ˙ S h + ˙ S in = 16 π 2 ˙ H 2 H 3 ( 1 2 H 2 + ˙ H ) . (6.8)</formula>
<unordered_list>
<list_item><location><page_134><loc_21><loc_61><loc_86><loc_75></location>· Thermodynamic Stability Conditions: To ensure thermodynamic stability, it is essential to maximize the entropy of the fluid contained within the horizon. This requirement can be translated into a condition involving the Hessian matrix of entropy, as explained in references [398-401]. Specifically, all the principle minors of order k in the matrix follow the pattern where they are ≤ 0 when k is odd and ≥ 0 when k is even.</list_item>
</unordered_list>
<text><location><page_134><loc_23><loc_56><loc_86><loc_60></location>The Hessian matrix, denoted as W , pertains to the entropy S in can be written as follows,</text>
<formula><location><page_134><loc_45><loc_52><loc_86><loc_56></location>W = [ S in UU S in UV S in VU S in VV ] . (6.9)</formula>
<text><location><page_134><loc_23><loc_45><loc_86><loc_51></location>In the above matrix, a subscript indicates a partial derivative with respect to the specific variable ( U , V ). Hence, if we express the Hessian matrix explicitly using the partial derivative notation, we obtain,</text>
<formula><location><page_134><loc_45><loc_35><loc_86><loc_43></location>W = ∂ 2 S in ∂ U 2 ∂ 2 S in ∂ U ∂ V ∂ 2 S in ∂ V ∂ U ∂ 2 S in ∂ V 2 . (6.10)</formula>
<text><location><page_134><loc_23><loc_30><loc_86><loc_33></location>Therefore, in order to ensure thermodynamic stability, it becomes imperative the simultaneous satisfaction of the following conditions,</text>
<formula><location><page_134><loc_42><loc_25><loc_86><loc_27></location>( i ) S in UU ≤ 0, (6.11)</formula>
<formula><location><page_134><loc_41><loc_22><loc_86><loc_25></location>( ii ) S in UU S in VV -S 2 in UV ≥ 0. (6.12)</formula>
<text><location><page_134><loc_23><loc_19><loc_81><loc_20></location>In terms of thermodynamic parameters, we get the conditions as,</text>
<formula><location><page_134><loc_47><loc_13><loc_86><loc_16></location>S in UU = -1 T 2 C V , (6.13)</formula>
<text><location><page_134><loc_23><loc_10><loc_26><loc_11></location>and</text>
<formula><location><page_135><loc_34><loc_85><loc_82><loc_88></location>S in UU S in VV -S 2 in UV = 1 C V T 3 V β T = α . (6.14)</formula>
<text><location><page_135><loc_19><loc_78><loc_82><loc_83></location>For the sake of conciseness, the second expression is denoted as α . In these equations, T represents temperature, C V stands for heat capacity at constant volume, and β T indicates isothermal compressibility.</text>
<text><location><page_135><loc_19><loc_66><loc_82><loc_74></location>Heat capacity at constant volume ( C V ) is a thermodynamic property that quantifies the amount of heat energy required to produce a unit change in its temperature while keeping its volume constant. It is defined as,</text>
<formula><location><page_135><loc_42><loc_63><loc_82><loc_66></location>C V = T ( ∂ S in ∂ T ) V . (6.15)</formula>
<text><location><page_135><loc_19><loc_53><loc_82><loc_61></location>Heat capacity at constant pressure ( C P ) is a thermodynamic property that measures the amount of heat energy required to produce a unit change in its temperature while allowing it to expand or contract under constant pressure conditions. It is defined as,</text>
<formula><location><page_135><loc_42><loc_48><loc_82><loc_52></location>C P = T ( ∂ S in ∂ T ) P . (6.16)</formula>
<text><location><page_135><loc_19><loc_36><loc_82><loc_46></location>Isothermal compressibility ( β T ) is a thermodynamic property that quantifies a substance's responsiveness to changes in pressure while its temperature is held constant. It measures the fractional change in volume of a substance in response to a unit change in pressure, keeping the temperature constant. Isothermal compressibility is defined as,</text>
<formula><location><page_135><loc_42><loc_30><loc_82><loc_33></location>β T = -1 V ( ∂ V ∂ P ) T . (6.17)</formula>
<section_header_level_1><location><page_135><loc_14><loc_25><loc_72><loc_27></location>6.4 Thermodynamic analysis of the model</section_header_level_1>
<text><location><page_135><loc_14><loc_15><loc_82><loc_23></location>We will examine whether the model adheres to the GSL and assess its thermodynamic stability. We will explore the characteristics exhibited by various thermodynamic parameters. We substitute the expression of H (as given in equation (6.5)) into equation (6.8), resulting in the following expression,</text>
<formula><location><page_135><loc_28><loc_9><loc_82><loc_13></location>˙ S = 108 π 2 csch 4 ( t ) coth 3 ( t ) 1 ( 4 coth 2 ( t ) -3 csch 2 ( t ) ) . (6.18)</formula>
<text><location><page_136><loc_18><loc_84><loc_86><loc_90></location>Based on the aforementioned mathematical expression, it is evident that entropy increases with time. Consequently, the GSL remains valid for the model.</text>
<text><location><page_136><loc_18><loc_80><loc_86><loc_83></location>When considering the fluid confined within the horizon, the expression of Gibbs' law takes the form:</text>
<formula><location><page_136><loc_43><loc_75><loc_86><loc_77></location>TdS in = dU + pdV . (6.19)</formula>
<text><location><page_136><loc_18><loc_66><loc_86><loc_74></location>This equation serves as a cornerstone in comprehending the thermodynamic processes occurring within the confines of the event horizon. By utilizing the equation (6.19), one can compute the heat capacities and isothermal compressibility for the material enclosed within the event horizon.</text>
<formula><location><page_136><loc_39><loc_51><loc_86><loc_62></location>C V = V ( ∂ρ ∂ T ) V = 32 π 2 ˙ H 2 H 2 ˙ H + H H -˙ H 2 = 144 π 2 -2 +( 1 + z ) 3 csch 2 ( 1 ) , (6.20)</formula>
<formula><location><page_136><loc_37><loc_34><loc_86><loc_47></location>C P = V ( ∂ρ ∂ T ) P +( ρ + P ) ( ∂ V ∂ T ) P = 32 π 2 H 2 ˙ H + ˙ H 2 H 2 ( 2 H 2 ˙ H + H H -˙ H 2 ) = -72 π 2 sinh 2 ( 1 ) ( 1 + z ) 3 + sinh 2 ( 1 ) , (6.21)</formula>
<formula><location><page_136><loc_35><loc_22><loc_86><loc_30></location>β T = 3 ˙ H ( 2 H 2 + ˙ H ) 2 ( H 2 + ˙ H )( 2 H 2 ˙ H + H H -˙ H 2 ) = -27 4 4 +( 1 + z ) 3 csch 2 ( 1 ) ( -2 +( 1 + z ) 3 csch 2 ( 1 ) ) . (6.22)</formula>
<text><location><page_136><loc_21><loc_19><loc_22><loc_20></location>.</text>
<text><location><page_136><loc_18><loc_8><loc_86><loc_18></location>Figures (6.2) and (6.3) illustrate the behaviour of C V and C P , respectively, with respect to the redshift z within the range of low values (0 ≤ z ≤ 1). Utilizing the expressions for C V and β T derived from equations (6.20) and (6.22) in equations (6.13) and (6.14), it is possible to generate plots for S in UU and α , as depicted in figures (6.4) and (6.5), respectively. It is evident that</text>
<figure>
<location><page_137><loc_30><loc_74><loc_66><loc_90></location>
<caption>FIGURE 6.2: CV plotted as a function of z</caption>
</figure>
<figure>
<location><page_137><loc_30><loc_54><loc_66><loc_70></location>
<caption>FIGURE 6.3: CP plotted as a function of z</caption>
</figure>
<text><location><page_137><loc_14><loc_43><loc_82><loc_48></location>the two conditions (6.11) and (6.12) are not simultaneously met within the specified low redshift range (0 ≤ z ≤ 1). Consequently, the model does not exhibit thermodynamic stability within this particular redshift range.</text>
<text><location><page_137><loc_14><loc_26><loc_82><loc_40></location>The significant observation gleaned from figure (6.2) is that heat capacity at constant volume ( C V ) displays a noteworthy behaviour - specifically, it exhibits a divergence or discontinuity at a particular redshift value of z = -1 + 2 1/3 sinh 2/3 ( 1 ) ≃ 0.403. This redshift value corresponds to the critical point where the expansion of the universe transitions from a decelerated phase to an accelerated one. Remarkably, this transition holds the characteristics of a thermodynamic phase transition.</text>
<text><location><page_137><loc_14><loc_11><loc_82><loc_23></location>The transition from decelerated to accelerated expansion, it turns out, aligns with a distinct thermodynamic phase transition. Notably, the entropy ( S mboxin ) does not exhibit any corresponding discontinuity within the specified redshift range; rather, the discontinuity manifests in the behaviour of C V . Consequently, this phase transition is unequivocally identified as a second-order phase transition.</text>
<figure>
<location><page_138><loc_34><loc_73><loc_70><loc_90></location>
<caption>FIGURE 6.4: S in UU plotted as a function of z</caption>
</figure>
<figure>
<location><page_138><loc_34><loc_53><loc_70><loc_69></location>
<caption>FIGURE 6.5: α plotted as a function of z</caption>
</figure>
<text><location><page_138><loc_18><loc_34><loc_86><loc_48></location>It is noteworthy to highlight that C V assumes a negative value for the present universe, specifically for z > -1 + 2 1/3 sinh 2/3 ( 1 ) ≃ 0.403. Nevertheless, the presence of a negative heat capacity in gravitational systems is not a surprising phenomenon (for a comprehensive overview, we refer to [455]). A study conducted by Luongo and Quevedo [456] yielded a significant finding that in the context of a currently accelerating universe, a negative value for C V becomes a requisite.</text>
<text><location><page_138><loc_18><loc_27><loc_86><loc_31></location>By substituting equation (6.6) into equation (6.20), it becomes possible to express C V in terms of the deceleration parameter ( q ) as follows,</text>
<formula><location><page_138><loc_44><loc_22><loc_86><loc_26></location>C V = 24 π 2 1 -2 q q . (6.23)</formula>
<text><location><page_138><loc_18><loc_10><loc_86><loc_20></location>Therefore, it becomes evident that the origin of the discontinuity in C V arises from the presence of the deceleration parameter ( q ) in the denominator with a positive exponent of 1. As a consequence, the deceleration parameter ( q ) serves as the order parameter, and the observed discontinuity is characterized by an order of unity.</text>
<section_header_level_1><location><page_139><loc_14><loc_88><loc_55><loc_90></location>6.5 Summary and Discussion</section_header_level_1>
<text><location><page_139><loc_14><loc_72><loc_82><loc_86></location>The focus of the thermodynamic stability analysis was directed toward a model designed to emulate the behaviour of the Λ CDMmodel, which faithfully represents the present universe. Given the dynamic nature of the evolving horizon, the Hayward-Kodama temperature was chosen as the appropriate measure of the temperature of the horizon. The analysis revealed that the thermal capacity exhibited a negative value, implying that the cosmic matter contained within the horizon lacks thermodynamic stability.</text>
<text><location><page_139><loc_14><loc_59><loc_82><loc_71></location>The significance of this study lies in the profound result indicating that the matter content experiences a phase transition precisely at the point where the universe undergoes a transition from decelerated expansion to an accelerated one. Remarkably, this phase transition follows the characteristics of a second-order transition, as evidenced by the discontinuity in the heat capacity ( C V ). The deceleration parameter q serves as the order parameter.</text>
<text><location><page_139><loc_14><loc_46><loc_82><loc_58></location>Interestingly, the earlier studies failed to detect the second-order phase transition at the onset of the accelerated expansion of the universe could potentially be attributed to the utilization of the Hawking temperature of the horizon. This approach overlooked the fact that the apparent horizon is in a state of evolution, which the Hayward-Kodama temperature accurately addresses.</text>
<text><location><page_139><loc_14><loc_33><loc_82><loc_43></location>Pavón and Wang [457] showed that the dark matter and dark energy could potentially evolve independently. Therefore, they may not be in thermal equilibrium with each other. However, it is important to note that our approach considers a composite fluid in which distinct sectors are not explicitly differentiated, only the evolution history holds significance.</text>
<section_header_level_1><location><page_141><loc_14><loc_82><loc_30><loc_84></location>Chapter 7</section_header_level_1>
<section_header_level_1><location><page_141><loc_14><loc_75><loc_63><loc_78></location>Conclusions and Outlook</section_header_level_1>
<text><location><page_141><loc_14><loc_39><loc_82><loc_70></location>This dissertation centers on the exploration of thermodynamic analyses in the context of cosmology. This thermodynamic exploration involves two main aspects: assessment through the Generalized Second Law (GSL)(Chapters 3,4,5) and the analysis of thermodynamic stability (Chapter6). For the purpose of our research, we have made the assumption that the fluid contained within the horizon is in a state of thermodynamic equilibrium with the horizon itself. Thermal equilibrium between radiation and the cosmic horizon is impossible due to Wien's law, which consistently yields a wavelength larger than the horizon radius over all time periods [416]. Nevertheless, nonrelativistic particles can achieve equilibrium at a certain expansion point based on particle mass. Given this, the notion of thermal equilibrium between dark energy and the horizon, proposed by various researchers, has a valid basis. Our study excludes radiation (except for a part of chapter 4), affirming the validity of assuming thermodynamic equilibrium between the horizon and fluid content.</text>
<text><location><page_141><loc_14><loc_23><loc_82><loc_37></location>As the universe is evolving, we have considered the dynamic apparent horizon, instead of the event horizon to work with. The temperature of the apparent horizon is considered to be the Hayward-Kodama temperature. Hayward proposed the definition of this temperature linked through an alternative definition of surface gravity that applies to dynamic, spherically symmetric spacetimes, which relies on the Kodama vector. This temperature, T = 2 H 2 + ˙ H 4 π H serves as the equilibrium temperature [414, 415].</text>
<text><location><page_141><loc_14><loc_10><loc_82><loc_22></location>Initially, the generalised second law of thermodynamics was proposed by Bekenstein in the early 1970s [325, 390, 391]. This principle asserts that the total entropy of the universe, the sum of matter entropy and horizon entropy, must never diminish as time progresses. Therefore, we have subjected various cosmological models to scrutiny using the GSL criterion, identifying which models successfully meet the test requirements or determining the</text>
<text><location><page_142><loc_18><loc_88><loc_53><loc_90></location>constraints needed for passing the test.</text>
<text><location><page_142><loc_18><loc_79><loc_86><loc_87></location>Our next course of action involved conducting a thermodynamic stability analysis on a model. This involved utilizing the property of the concavity of the entropy, wherein we examined the Hessian matrix of entropy to ascertain whether it exhibited semi-negative definiteness.</text>
<text><location><page_142><loc_18><loc_33><loc_86><loc_77></location>In chapter 3, we have undertaken a comparative analysis between thawing and freezing models, focusing on their adherence to the fundamental principles of thermodynamics, specifically the GSL. Our work involves the assessment of the total entropy ( S tot), accomplished by adding up the entropy of the cosmic horizon with that of the enclosed matter within said horizon. To facilitate this study, we employ a straightforward ansatz, 1 ρ Φ ∂ρ Φ ∂ a = -λ a 1 -2 α , proposed by Carvalho et al [287], to formulate the evolution of the energy density of the quintessence field. The ansatz proposes that the power-law dependence of the scalar field is reflected in the divergence of the logarithm of energy density. This approach enables us to effectively delineate the parameter space ( α ) that corresponds to the distinct behaviors of thawing and freezing of the field. The findings reveal an intriguing incongruity between both model categories and the GSL. Notably, there are instances in which the entropy ( S ) experiences a decrease, and this descent occurs at an accelerated rate. For the freezing models, this contravention of the GSL is observed in a remote cosmic past, specifically during an epoch characterized by a redshift of z ∼ 10 4 . During this epoch, a cosmological model that combines quintessence and cold dark matter fails to satisfactorily elucidate the cosmic evolution, necessitating a dominant contribution from a radiation distribution. Consequently, the applicability of the GSL seems questionable in such a scenario.</text>
<text><location><page_142><loc_18><loc_24><loc_86><loc_32></location>Conversely, in the context of thawing models, our analysis predicts an anomalous breakdown of the GSL within a finite future. This observation underscores a pivotal implication: freezing models exhibit a thermodynamically more tenable model when compared to their thawing counterparts.</text>
<text><location><page_142><loc_18><loc_13><loc_86><loc_23></location>In chapter 4, our research thoroughly investigates the thermodynamic properties of cosmological models governed by radiation and dust dominance. These models are analyzed in the framework of the Brans-Dicke theory. In particular, we delve into the properties of a spatially flat, homogenous, and isotropic universe within the Einstein frame . This frame represents</text>
<text><location><page_143><loc_14><loc_73><loc_82><loc_90></location>the conformally transformed version of the Brans-Dicke theory. In a universe dominated by radiation, the solutions obtained from Brans-Dicke theory, with a positive parameter ω , fail to satisfy the principles of the generalized second law. However, remarkably, when specific ranges of negative ω values are considered, the model harmonizes effectively with the requisites of thermodynamics. This finding is very promising because negative values of the parameter ω have been strongly associated with the phenomenon of accelerated cosmic expansion.</text>
<text><location><page_143><loc_14><loc_58><loc_82><loc_72></location>Shifting the focus to a universe governed predominantly by dust, the model successfully upholds the principles of the generalized second law, but notably, only for certain small negative values of ω . Particularly noteworthy is the fact that this range, characterized by -2 < ω < -5 3 , extensively overlaps with the parameter values necessary to elucidate an accelerated expansion of the universe, all without necessitating the introduction of exotic forms of matter [100].</text>
<text><location><page_143><loc_14><loc_43><loc_82><loc_57></location>Chapter 5 is about the thermodynamic assessment of the models reconstructed from the jerk parameter, proposed by Mukherjee et al. [439, 440]. Encouragingly, the assessment of the Generalized Second Law (GSL) of thermodynamics can be effectively conducted using kinematical quantities like the Hubble parameter and its derivative, as described in equation (3.21). This analysis of thermodynamic viability ensures that the reconstructed models adhere to thermodynamic principles, strengthening their credibility.</text>
<text><location><page_143><loc_14><loc_19><loc_82><loc_41></location>Our investigation reveals the feasibility of identifying cosmological models that maintain thermodynamic consistency through the reconstruction of the jerk parameter. Notably, among the subset of four non-interacting models, model IV-exhibits a future decline in entropy ( z < 0) due to its inverse dependence on ( 1 + z ) . This behavior occurs in proximity to the present era ( z = 0). However, all other models, including the one involving interaction within the dark sector (Model V), adhere to the Generalized Second Law (GSL) of thermodynamics. Model I, while in line with the GSL, demonstrates a significant and sudden entropy increase in the future ( z < 0). Models II and III exhibit consistent behavior, as does Model V, which allows interaction within the dark sector.</text>
<text><location><page_143><loc_14><loc_11><loc_82><loc_17></location>Chapter 6 deviates slightly from the preceding chapters in that it encompasses a dual focus. Here, our exploration goes beyond solely examining the feasibility of the Generalized Second Law (GSL). Instead, we make an</text>
<text><location><page_144><loc_18><loc_86><loc_86><loc_90></location>attempt to analyze the thermodynamic stability of an accelerating cosmological model.</text>
<text><location><page_144><loc_18><loc_68><loc_86><loc_85></location>The thermodynamic stability analysis focused on a model crafted to mimic the behavior of the Λ CDM model, favoured model for the portrayal of the current state of the universe. In light of the dynamic nature of the evolving horizon, the Hayward-Kodama temperature emerged as the suitable parameter to gauge the temperature of the horizon. The findings of the analysis brought forth a significant revelation, the thermal capacity exhibited a negative value, indicating an absence of thermodynamic stability within the cosmic matter confined by the horizon.</text>
<text><location><page_144><loc_18><loc_50><loc_86><loc_67></location>The interesting outcome lies in the fact that the thermodynamic phase transition of matter content is intimately connected with the nature of the cosmic evolution. The results suggest a thermodynamic phase transition that aligns perfectly with the shift from a decelerated cosmic expansion to an accelerated one. Notably, this phase transition demonstrates the characteristics of a second-order transition, as evident from the discontinuity observed in the heat capacity ( C V ). The deceleration parameter ( q ) serves as the order parameter in this intricate connection.</text>
<text><location><page_144><loc_18><loc_35><loc_86><loc_49></location>The aim of this investigation was merely to align the observed value of z with q = 0, but rather to delve into the qualitative nature of the thermodynamic aspect accompanying the pivotal shift in q . Apparently, the prior investigations failed to detect the second-order phase transition at the onset of the accelerated expansion of the universe because the dynamical nature of the horizon had been ignored and Hawking temperature was relied upon. The use of Hayward-Kodama temperature brings out the remarkable feature.</text>
<section_header_level_1><location><page_144><loc_18><loc_29><loc_40><loc_31></location>Future Prospects</section_header_level_1>
<text><location><page_144><loc_18><loc_9><loc_86><loc_27></location>In this thesis, we have explored the thermodynamics of some cosmological models, especially those that give rise to a late-time cosmic acceleration. We have examined the viability of these models via GSL. We have considered the thermodynamic equilibrium between the apparent horizon and the matter bounded by the horizon. Haward-Kodama (HK) temperature has been considered as the horizon temperature. It will be interesting to proceed with the thermodynamic analysis by considering another horizon temperature such as Cai-Kim (CK) temperature [375, 458] instead of the Hayward-Kodama temperature. One can also do a comparative analysis between the outcomes</text>
<text><location><page_145><loc_14><loc_77><loc_82><loc_90></location>of using the HK temperature and CK temperature. One may take the thermodynamic analysis to the next step and explore GSL in non-equilibrium thermodynamics. As shown in the article [457], dark matter and dark energy may evolve independently and therefore they may not be in thermal equilibrium; hence it will be worthwhile to consider these two fluids separately and then proceed with the thermodynamic analysis.</text>
<text><location><page_145><loc_14><loc_66><loc_82><loc_76></location>Thermodynamic stability analysis can also be conducted for other models. It will be interesting to see what thermodynamic features these models reveal. We have tried to see if the transition from decelerated to accelerated cosmic expansion is connected to the phase transition without considering any model. But we have not progressed much in this project.</text>
<section_header_level_1><location><page_147><loc_14><loc_81><loc_40><loc_84></location>Bibliography</section_header_level_1>
<unordered_list>
<list_item><location><page_147><loc_16><loc_72><loc_82><loc_76></location>1. Einstein, A. The Field Equations of Gravitation. Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ) 1915, 844-847 (1915).</list_item>
<list_item><location><page_147><loc_16><loc_65><loc_82><loc_71></location>2. Einstein, A. Cosmological Considerations in the General Theory of Relativity. Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ) 1917, 142152 (1917).</list_item>
<list_item><location><page_147><loc_16><loc_58><loc_82><loc_64></location>3. De Sitter, W. On the Relativity of Inertia. Remarks Concerning Einstein?s Latest Hypothesis. Philosophical Problems in Science 63, 205-222 (2017).</list_item>
<list_item><location><page_147><loc_16><loc_54><loc_82><loc_57></location>4. De Sitter, W. Einstein's theory of gravitation and its astronomical consequences, Third Paper. Mon. Not. Roy. Astron. Soc. 78, 3-28 (1917).</list_item>
<list_item><location><page_147><loc_16><loc_47><loc_82><loc_52></location>5. Schwarzschild, K. On the gravitational field of a mass point according to Einstein's theory. Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ) 1916, 189-196. arXiv: physics/9905030 (1916).</list_item>
<list_item><location><page_147><loc_16><loc_37><loc_82><loc_45></location>6. Dyson, F. W., Eddington, A. S. & Davidson, C. A Determination of the Deflection of Light by the Sun's Gravitational Field, from Observations Made at the Total Eclipse of May 29, 1919. Phil. Trans. Roy. Soc. Lond. A 220, 291-333 (1920).</list_item>
<list_item><location><page_147><loc_16><loc_35><loc_80><loc_36></location>7. Friedman, A. On the Curvature of space. Z. Phys. 10, 377-386 (1922).</list_item>
<list_item><location><page_147><loc_16><loc_30><loc_82><loc_33></location>8. Hubble, E. A relation between distance and radial velocity among extra-galactic nebulae. Proc. Nat. Acad. Sci. 15, 168-173 (1929).</list_item>
<list_item><location><page_147><loc_16><loc_23><loc_82><loc_29></location>9. Lemaitre, G. A homogeneous universe of constant mass and increasing radius accounting for the radial velocity of extra-galactic nebulae. Mon. Not. Roy. Astron. Soc. 91, 483-490 (1931).</list_item>
<list_item><location><page_147><loc_15><loc_18><loc_82><loc_22></location>10. Alpher, R. A., Bethe, H. & Gamow, G. The origin of chemical elements. Phys. Rev. 73, 803-804 (1948).</list_item>
<list_item><location><page_147><loc_15><loc_13><loc_82><loc_17></location>11. Alpher, R. A. & Herman, R. Evolution of the Universe. Nature 162, 774775 (1948).</list_item>
<list_item><location><page_147><loc_15><loc_8><loc_82><loc_12></location>12. Alpher, R. A. & Herman, R. C. On the Relative Abundance of the Elements. Phys. Rev. 74, 1737-1742 (1948).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_148><loc_19><loc_86><loc_86><loc_90></location>13. Penzias, A. A. & Wilson, R. W. A Measurement of excess antenna temperature at 4080-Mc/s. Astrophys. J. 142, 419-421 (1965).</list_item>
<list_item><location><page_148><loc_19><loc_81><loc_86><loc_85></location>14. Dicke, R. H. The Measurement of Thermal Radiation at Microwave Frequencies. Rev. Sci. Instrum. 17, 268-275 (1946).</list_item>
<list_item><location><page_148><loc_19><loc_74><loc_86><loc_80></location>15. Peebles, P. J. & Dicke, R. H. Cosmology and the Radioactive Decay Ages of Terrestrial Rocks and Meteorites. Phys. Rev. 128, 2006-2011 (1962).</list_item>
<list_item><location><page_148><loc_19><loc_69><loc_86><loc_73></location>16. Dicke, R. H., Peebles, P. J. E., Roll, P. G. & Wilkinson, D. T. Cosmic Black-Body Radiation. Astrophys. J. 142, 414-419 (1965).</list_item>
<list_item><location><page_148><loc_19><loc_62><loc_86><loc_68></location>17. Guth, A. H. The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems. Phys. Rev. D 23 (eds Fang, L.-Z. & Ruffini, R.) 347-356 (1981).</list_item>
<list_item><location><page_148><loc_19><loc_55><loc_86><loc_61></location>18. Salopek, D. S. Consequences of the COBE satellite results for the inflationary scenario. Phys. Rev. Lett. 69, 3602-3605. https://link.aps. org/doi/10.1103/PhysRevLett.69.3602 (25 1992).</list_item>
<list_item><location><page_148><loc_19><loc_48><loc_86><loc_54></location>19. Peiris, H. V. et al. First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Implications for inflation. Astrophys. J. Suppl. 148, 213-231. arXiv: astro-ph/0302225 (2003).</list_item>
<list_item><location><page_148><loc_19><loc_41><loc_86><loc_47></location>20. Schmidt, B. P. et al. The High Z supernova search: Measuring cosmic deceleration and global curvature of the universe using type Ia supernovae. Astrophys. J. 507, 46-63. arXiv: astro-ph/9805200 (1998).</list_item>
<list_item><location><page_148><loc_19><loc_34><loc_86><loc_40></location>21. Nolta, M. R. et al. First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Dark energy induced correlation with radio sources. Astrophys. J. 608, 10-15. arXiv: astro-ph/0305097 (2004).</list_item>
<list_item><location><page_148><loc_19><loc_30><loc_86><loc_33></location>22. Ade, P. A. R. et al. Planck 2015 results. XIII. Cosmological parameters. Astron. Astrophys. 594, A13. arXiv: 1502.01589 [astro-ph.CO] (2016).</list_item>
<list_item><location><page_148><loc_19><loc_25><loc_86><loc_28></location>23. Misner, C. W., Thorne, K. & Wheeler, J. Gravitation ISBN: 978-0-71670344-0, 978-0-691-17779-3 (W. H. Freeman, San Francisco, 1973).</list_item>
<list_item><location><page_148><loc_19><loc_22><loc_86><loc_23></location>24. Wald, R. M. General Relativity (Chicago Univ. Pr., Chicago, USA, 1984).</list_item>
<list_item><location><page_148><loc_19><loc_17><loc_86><loc_21></location>25. Carroll, S. M. Lecture notes on general relativity. arXiv: gr-qc/9712019 [gr-qc] (1997).</list_item>
<list_item><location><page_148><loc_19><loc_10><loc_86><loc_16></location>26. Carroll, S. M. Spacetime and Geometry: An Introduction to General Relativity ISBN: 978-0-8053-8732-2, 978-1-108-48839-6, 978-1-108-77555-7 (Cambridge University Press, July 2019).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_149><loc_15><loc_86><loc_82><loc_90></location>27. Narlikar, J. V. AnIntroduction to Relativity (Cambridge University Press, Jan. 2010).</list_item>
<list_item><location><page_149><loc_15><loc_81><loc_82><loc_85></location>28. D'Inverno, R. Introducing Einstein's Relativity ISBN: 9780198596868 (Clarendon Press, 1992).</list_item>
<list_item><location><page_149><loc_15><loc_76><loc_82><loc_80></location>29. Hooft, G. Introduction to General Relativity ISBN: 9781589490000 (Rinton Press, 2001).</list_item>
<list_item><location><page_149><loc_15><loc_71><loc_82><loc_75></location>30. Schutz, B. F. A first course in general relativity (Cambridge University Press, 1985).</list_item>
<list_item><location><page_149><loc_15><loc_64><loc_82><loc_70></location>31. Hartle, J. Gravity: An Introduction to Einstein's General Relativity ISBN: 9780805386622. https : / / books . google . co . in / books ? id = ZHgpAQAAMAAJ (Addison-Wesley, 2003).</list_item>
<list_item><location><page_149><loc_15><loc_60><loc_82><loc_63></location>32. Blau, M. Lecture notes on general relativity (Albert Einstein Center for Fundamental Physics Bern, 2011).</list_item>
<list_item><location><page_149><loc_15><loc_50><loc_82><loc_58></location>33. Landau, L. & Lifshitz, E. The Classical Theory of Fields (Fourth Edition) Fourth Edition, 259-294. ISBN: 978-0-08-025072-4. https://www. sciencedirect.com/science/article/pii/B9780080250724500186 (Pergamon, Amsterdam, 1975).</list_item>
<list_item><location><page_149><loc_15><loc_44><loc_82><loc_49></location>34. Friedman, A. Über die Krümmung des Raumes. Zeitschrift für Physik 10, 377-386. https : / / api . semanticscholar . org / CorpusID : 125190902 (1922).</list_item>
<list_item><location><page_149><loc_15><loc_37><loc_82><loc_42></location>35. Friedmann, A. Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes. Zeitschrift fur Physik 21, 326-332 (Dec. 1924).</list_item>
<list_item><location><page_149><loc_15><loc_25><loc_82><loc_35></location>36. Lemaître, G. Un Univers homogène de masse constante et de rayon croissant rendant compte de la vitesse radiale des nébuleuses extragalactiques. Annales de la Société; Scientifique de Bruxelles 47. (Translated and Reprinted in Mon. Not. Roy. Astron. Soc. 91 , 483-490 (1931)), 49-59 (Jan. 1927).</list_item>
<list_item><location><page_149><loc_15><loc_20><loc_82><loc_24></location>37. Lemaître, G. L'Univers en expansion. Annales de la Société Scientifique de Bruxelles 53, 51 (Jan. 1933).</list_item>
<list_item><location><page_149><loc_15><loc_16><loc_82><loc_19></location>38. Robertson, H. P. Kinematics and World-Structure. Astrophys. J. 82, 284301 (1935).</list_item>
<list_item><location><page_149><loc_15><loc_11><loc_82><loc_14></location>39. Walker, A. G. On Milne's Theory of World-Structure. Proceedings of the London Mathematical Society 42, 90-127 (Jan. 1937).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_150><loc_19><loc_86><loc_86><loc_90></location>40. Hubble, E. & Humason, M. L. The Velocity-Distance Relation among Extra-Galactic Nebulae. Astrophys. J. 74, 43-80 (1931).</list_item>
<list_item><location><page_150><loc_19><loc_79><loc_86><loc_85></location>41. Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity ISBN: 978-0-471-92567-5, 978-0-471-92567-5 (John Wiley and Sons, New York, 1972).</list_item>
<list_item><location><page_150><loc_19><loc_70><loc_86><loc_78></location>42. Hawking, S. W. & Ellis, G. F. R. The Large Scale Structure of Space-Time ISBN: 978-1-00-925316-1, 978-1-00-925315-4, 978-0-521-20016-5, 978-0521-09906-6, 978-0-511-82630-6, 978-0-521-09906-6 (Cambridge University Press, Feb. 2023).</list_item>
<list_item><location><page_150><loc_19><loc_65><loc_86><loc_69></location>43. Raychaudhuri, A. K., Banerji, S. & Banerjee, A. General relativity, astrophysics, and cosmology (1992).</list_item>
<list_item><location><page_150><loc_19><loc_60><loc_86><loc_64></location>44. Kolb, E. W. & Turner, M. S. The Early Universe ISBN: 978-0-201-62674-2 (1990).</list_item>
<list_item><location><page_150><loc_19><loc_55><loc_86><loc_59></location>45. Padmanabhan, T. Theoretical astrophysics: volume 3, galaxies and cosmology (Cambridge University Press, 2000).</list_item>
<list_item><location><page_150><loc_19><loc_53><loc_68><loc_54></location>46. Rich, J. Fundamentals of cosmology (Springer, 2009).</list_item>
<list_item><location><page_150><loc_19><loc_48><loc_86><loc_51></location>47. Liddle, A. An Introduction to Modern Cosmology (John Wiley & Sons Ltd, 2015).</list_item>
<list_item><location><page_150><loc_19><loc_43><loc_86><loc_46></location>48. Riess, A. G. et al. BV RI light curves for 22 type Ia supernovae. Astron. J. 117, 707-724. arXiv: astro-ph/9810291 (1999).</list_item>
<list_item><location><page_150><loc_19><loc_38><loc_86><loc_42></location>49. Barrow, J. D. & Turner, M. S. Inflation in the Universe. Nature 292, 3538 (1981).</list_item>
<list_item><location><page_150><loc_19><loc_29><loc_86><loc_37></location>50. Linde, A. D. A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems. Phys. Lett. B 108 (eds Fang, L.-Z. & Ruffini, R.) 389-393 (1982).</list_item>
<list_item><location><page_150><loc_19><loc_22><loc_86><loc_28></location>51. Turner, M. S. A GRACEFUL END TO INFLATION in 2nd Moriond Astrophysics Meeting: Recent Developments in Observational and Theoretical Cosmology (1982), 69-90.</list_item>
<list_item><location><page_150><loc_19><loc_17><loc_86><loc_21></location>52. Albrecht, A., Steinhardt, P. J., Turner, M. S. & Wilczek, F. Reheating an Inflationary Universe. Phys. Rev. Lett. 48, 1437 (1982).</list_item>
<list_item><location><page_150><loc_19><loc_12><loc_86><loc_16></location>53. Guth, A. H. & Pi, S. Y. Fluctuations in the New Inflationary Universe. Phys. Rev. Lett. 49, 1110-1113 (1982).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_151><loc_15><loc_84><loc_82><loc_90></location>54. Starobinsky, A. A. Dynamics of Phase Transition in the New Inflationary Universe Scenario and Generation of Perturbations. Phys. Lett. B 117, 175-178 (1982).</list_item>
<list_item><location><page_151><loc_15><loc_77><loc_82><loc_83></location>55. Kofman, L. A., Linde, A. D. & Starobinsky, A. A. Inflationary Universe Generated by the Combined Action of a Scalar Field and Gravitational Vacuum Polarization. Phys. Lett. B 157, 361-367 (1985).</list_item>
<list_item><location><page_151><loc_15><loc_72><loc_82><loc_76></location>56. Ryden, B. Introduction to cosmology ISBN: 978-1-107-15483-4, 978-1-31688984-8, 978-1-316-65108-7 (Cambridge University Press, 1970).</list_item>
<list_item><location><page_151><loc_15><loc_67><loc_82><loc_71></location>57. Plebanski, J. & Krasinski, A. An introduction to general relativity and cosmology (2006).</list_item>
<list_item><location><page_151><loc_15><loc_62><loc_82><loc_66></location>58. Tonry, J. L. et al. Cosmological results from high-z supernovae. Astrophys. J. 594, 1-24. arXiv: astro-ph/0305008 (2003).</list_item>
<list_item><location><page_151><loc_15><loc_53><loc_82><loc_61></location>59. Suzuki, N. et al. The Hubble Space Telescope Cluster Supernova Survey: V. Improving the Dark Energy Constraints Above z>1 and Building an Early-Type-Hosted Supernova Sample. Astrophys. J. 746, 85. arXiv: 1105.3470 [astro-ph.CO] (2012).</list_item>
<list_item><location><page_151><loc_15><loc_46><loc_82><loc_52></location>60. Betoule, M. et al. Improved cosmological constraints from a joint analysis of the SDSS-II and SNLS supernova samples. Astron. Astrophys. 568, A22. arXiv: 1401.4064 [astro-ph.CO] (2014).</list_item>
<list_item><location><page_151><loc_15><loc_39><loc_82><loc_45></location>61. Eisenstein, D. J. et al. Detection of the Baryon Acoustic Peak in the Large-Scale Correlation Function of SDSS Luminous Red Galaxies. Astrophys. J. 633, 560-574. arXiv: astro-ph/0501171 (2005).</list_item>
<list_item><location><page_151><loc_15><loc_34><loc_82><loc_38></location>62. Eisenstein, D. J. Dark energy and cosmic sound. New Astron. Rev. 49, 360-365 (2005).</list_item>
<list_item><location><page_151><loc_15><loc_27><loc_82><loc_33></location>63. Barris, B. J. et al. 23 High redshift supernovae from the IFA Deep Survey: Doubling the SN sample at z > 0.7. Astrophys. J. 602, 571-594. arXiv: astro-ph/0310843 (2004).</list_item>
<list_item><location><page_151><loc_15><loc_20><loc_82><loc_26></location>64. Aghanim, N. et al. Planck 2018 results. VI. Cosmological parameters. Astron. Astrophys. 641. [Erratum: Astron.Astrophys. 652, C4 (2021)], A6. arXiv: 1807.06209 [astro-ph.CO] (2020).</list_item>
<list_item><location><page_151><loc_15><loc_13><loc_82><loc_19></location>65. Allen, S. W. et al. Improved constraints on dark energy from Chandra X-ray observations of the largest relaxed galaxy clusters. Mon. Not. Roy. Astron. Soc. 383, 879-896. arXiv: 0706.0033 [astro-ph] (2008).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_152><loc_19><loc_84><loc_86><loc_90></location>66. Hicken, M. et al. Improved Dark Energy Constraints from 100 New CfA Supernova Type Ia Light Curves. Astrophys. J. 700, 1097-1140. arXiv: 0901.4804 [astro-ph.CO] (2009).</list_item>
<list_item><location><page_152><loc_19><loc_75><loc_86><loc_83></location>67. Abbott, T. M. C. et al. First Cosmology Results using Type Ia Supernovae from the Dark Energy Survey: Constraints on Cosmological Parameters. Astrophys. J. Lett. 872, L30. arXiv: 1811.02374 [astro-ph.CO] (2019).</list_item>
<list_item><location><page_152><loc_19><loc_66><loc_86><loc_74></location>68. Riess, A. G. et al. Type Ia supernova discoveries at z > 1 from the Hubble Space Telescope: Evidence for past deceleration and constraints on dark energy evolution. Astrophys. J. 607, 665-687. arXiv: astro- ph/ 0402512 (2004).</list_item>
<list_item><location><page_152><loc_19><loc_57><loc_86><loc_65></location>69. Riess, A. G. et al. New Hubble Space Telescope Discoveries of Type Ia Supernovae at z ≥ 1: Narrowing Constraints on the Early Behavior of Dark Energy. Astrophys. J. 659, 98-121. arXiv: astro-ph/0611572 (2007).</list_item>
<list_item><location><page_152><loc_19><loc_50><loc_86><loc_55></location>70. Turner, M. S. & Riess, A. G. Do Type Ia Supernovae Provide Direct Evidence for Past Deceleration of the Universe? Astrophys. J. 569, 18. https://dx.doi.org/10.1086/338580 (2002).</list_item>
<list_item><location><page_152><loc_19><loc_43><loc_86><loc_49></location>71. Cunha, J. V. & Lima, J. A. S. Transition Redshift: New Kinematic Constraints from Supernovae. Mon. Not. Roy. Astron. Soc. 390, 210-217. arXiv: 0805.1261 [astro-ph] (2008).</list_item>
<list_item><location><page_152><loc_19><loc_36><loc_86><loc_42></location>72. Cunha, J. V. Kinematic Constraints to the Transition Redshift from SNe Ia Union Data. Phys. Rev. D 79, 047301. arXiv: 0811.2379 [astro-ph] (2009).</list_item>
<list_item><location><page_152><loc_19><loc_29><loc_86><loc_35></location>73. Guimaraes, A. C. C., Cunha, J. V. & Lima, J. A. S. Bayesian Analysis and Constraints on Kinematic Models from Union SNIa. JCAP 10, 010. arXiv: 0904.3550 [astro-ph.CO] (2009).</list_item>
<list_item><location><page_152><loc_19><loc_22><loc_86><loc_28></location>74. Lu, J., Xu, L. & Liu, M. Constraints on kinematic models from the latest observational data. Phys. Lett. B 699, 246-250. arXiv: 1105 . 1871 [astro-ph.CO] (2011).</list_item>
<list_item><location><page_152><loc_19><loc_15><loc_86><loc_21></location>75. Cattoen, C. & Visser, M. The Hubble series: Convergence properties and redshift variables. Class. Quant. Grav. 24, 5985-5998. arXiv: 0710. 1887 [gr-qc] (2007).</list_item>
<list_item><location><page_152><loc_19><loc_8><loc_86><loc_14></location>76. Tsujikawa, S. in Lectures on Cosmology: Accelerated Expansion of the Universe (ed Wolschin, G.) 99-145 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2010).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_153><loc_15><loc_86><loc_82><loc_90></location>77. Sotiriou, T. P. & Faraoni, V. f(R) Theories Of Gravity. Rev. Mod. Phys. 82, 451-497. arXiv: 0805.1726 [gr-qc] (2010).</list_item>
<list_item><location><page_153><loc_15><loc_81><loc_82><loc_85></location>78. De Felice, A. & Tsujikawa, S. f(R) theories. Living Rev. Rel. 13, 3. arXiv: 1002.4928 [gr-qc] (2010).</list_item>
<list_item><location><page_153><loc_15><loc_76><loc_82><loc_80></location>79. Maartens, R. & Koyama, K. Brane-World Gravity. Living Rev. Rel. 13, 5. arXiv: 1004.3962 [hep-th] (2010).</list_item>
<list_item><location><page_153><loc_15><loc_69><loc_82><loc_75></location>80. Clifton, T., Ferreira, P. G., Padilla, A. & Skordis, C. Modified Gravity and Cosmology. Phys. Rept. 513, 1-189. arXiv: 1106 . 2476 [astro-ph.CO] (2012).</list_item>
<list_item><location><page_153><loc_15><loc_62><loc_82><loc_68></location>81. Joyce, A., Jain, B., Khoury, J. & Trodden, M. Beyond the Cosmological Standard Model. Phys. Rept. 568, 1-98. arXiv: 1407 . 0059 [astro-ph.CO] (2015).</list_item>
<list_item><location><page_153><loc_15><loc_55><loc_82><loc_61></location>82. Koyama, K. Cosmological tests of modified gravity. Reports on Progress in Physics 79, 046902. https://dx.doi.org/10.1088/0034-4885/79/ 4/046902 (2016).</list_item>
<list_item><location><page_153><loc_15><loc_50><loc_82><loc_54></location>83. Capozziello, S. Curvature quintessence. Int. J. Mod. Phys. D 11, 483492. arXiv: gr-qc/0201033 (2002).</list_item>
<list_item><location><page_153><loc_15><loc_44><loc_82><loc_49></location>84. Carroll, S. M., Duvvuri, V., Trodden, M. & Turner, M. S. Is cosmic speed - up due to new gravitational physics? Phys. Rev. D 70, 043528. arXiv: astro-ph/0306438 (2004).</list_item>
<list_item><location><page_153><loc_15><loc_39><loc_82><loc_42></location>85. Carroll, S. M. et al. The Cosmology of generalized modified gravity models. Phys. Rev. D 71, 063513. arXiv: astro-ph/0410031 (2005).</list_item>
<list_item><location><page_153><loc_15><loc_32><loc_82><loc_37></location>86. Nojiri, S. & Odintsov, S. D. Modified gravity with negative and positive powers of the curvature: Unification of the inflation and of the cosmic acceleration. Phys. Rev. D 68, 123512. arXiv: hep-th/0307288 (2003).</list_item>
<list_item><location><page_153><loc_15><loc_25><loc_82><loc_30></location>87. Nojiri, S. & Odintsov, S. D. Modified f(R) gravity consistent with realistic cosmology: From matter dominated epoch to dark energy universe. Phys. Rev. D 74, 086005. arXiv: hep-th/0608008 (2006).</list_item>
<list_item><location><page_153><loc_15><loc_18><loc_82><loc_23></location>88. Nojiri, S. & Odintsov, S. D. Newton law corrections and instabilities in f(R) gravity with the effective cosmological constant epoch. Phys. Lett. B 652, 343-348. arXiv: 0706.1378 [hep-th] (2007).</list_item>
<list_item><location><page_153><loc_15><loc_11><loc_82><loc_16></location>89. Nojiri, S. & Odintsov, S. D. Unifying inflation with LambdaCDM epoch in modified f(R) gravity consistent with Solar System tests. Phys. Lett. B 657, 238-245. arXiv: 0707.1941 [hep-th] (2007).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_154><loc_19><loc_84><loc_86><loc_90></location>90. Vollick, D. N. 1/R Curvature corrections as the source of the cosmological acceleration. Phys. Rev. D 68, 063510. arXiv: astro-ph/0306630 (2003).</list_item>
<list_item><location><page_154><loc_19><loc_77><loc_86><loc_83></location>91. Mena, O., Santiago, J. & Weller, J. Constraining inverse curvature gravity with supernovae. Phys. Rev. Lett. 96, 041103. arXiv: astro - ph / 0510453 (2006).</list_item>
<list_item><location><page_154><loc_19><loc_72><loc_86><loc_76></location>92. Banerjee, N. & Sen, S. Does Brans-Dicke theory always yield general relativity in the infinite omega limit? Phys. Rev. D 56, 1334-1337 (1997).</list_item>
<list_item><location><page_154><loc_19><loc_67><loc_86><loc_71></location>93. Amendola, L. Scaling solutions in general nonminimal coupling theories. Phys. Rev. D 60, 043501. arXiv: astro-ph/9904120 (1999).</list_item>
<list_item><location><page_154><loc_19><loc_62><loc_86><loc_66></location>94. Uzan, J.-P. Cosmological scaling solutions of nonminimally coupled scalar fields. Phys. Rev. D 59, 123510. arXiv: gr-qc/9903004 (1999).</list_item>
<list_item><location><page_154><loc_19><loc_57><loc_86><loc_61></location>95. Chiba, T. Quintessence, the gravitational constant, and gravity. Phys. Rev. D 60, 083508. arXiv: gr-qc/9903094 (1999).</list_item>
<list_item><location><page_154><loc_19><loc_53><loc_86><loc_56></location>96. Amendola, L. Coupled quintessence. Phys. Rev. D 62, 043511. arXiv: astro-ph/9908023 (2000).</list_item>
<list_item><location><page_154><loc_19><loc_48><loc_86><loc_51></location>97. Bartolo, N. & Pietroni, M. Scalar tensor gravity and quintessence. Phys. Rev. D 61, 023518. arXiv: hep-ph/9908521 (2000).</list_item>
<list_item><location><page_154><loc_19><loc_43><loc_86><loc_46></location>98. Perrotta, F., Baccigalupi, C. & Matarrese, S. Extended quintessence. Phys. Rev. D 61, 023507. arXiv: astro-ph/9906066 (1999).</list_item>
</unordered_list>
<text><location><page_154><loc_19><loc_40><loc_22><loc_42></location>99.</text>
<text><location><page_154><loc_24><loc_40><loc_86><loc_42></location>Riazuelo, A. & Uzan, J.-P. Cosmological observations in scalar - tensor</text>
<text><location><page_154><loc_24><loc_38><loc_35><loc_39></location>quintessence.</text>
<text><location><page_154><loc_36><loc_38><loc_46><loc_39></location>Phys. Rev. D</text>
<text><location><page_154><loc_47><loc_38><loc_49><loc_39></location>66,</text>
<text><location><page_154><loc_50><loc_38><loc_62><loc_39></location>023525. arXiv:</text>
<text><location><page_154><loc_63><loc_38><loc_79><loc_39></location>astro-ph/0107386</text>
<text><location><page_154><loc_80><loc_38><loc_86><loc_39></location>(2002).</text>
<unordered_list>
<list_item><location><page_154><loc_18><loc_33><loc_86><loc_37></location>100. Banerjee, N. & Pavon, D. Cosmic acceleration without quintessence. Phys. Rev. D 63, 043504. arXiv: gr-qc/0012048 (2001).</list_item>
<list_item><location><page_154><loc_18><loc_28><loc_86><loc_32></location>101. Banerjee, N. & Pavon, D. A Quintessence scalar field in Brans-Dicke theory. Class. Quant. Grav. 18, 593. arXiv: gr-qc/0012098 (2001).</list_item>
<list_item><location><page_154><loc_18><loc_24><loc_86><loc_27></location>102. Sen, S. & Sen, A. A. Late time acceleration in Brans-Dicke cosmology. Phys. Rev. D 63, 124006. arXiv: gr-qc/0010092 (2001).</list_item>
<list_item><location><page_154><loc_18><loc_17><loc_86><loc_22></location>103. Mota, D. F. & Barrow, J. D. Local and global variations of the fine structure constant. Mon. Not. Roy. Astron. Soc. 349, 291. arXiv: astro-ph/ 0309273 (2004).</list_item>
<list_item><location><page_154><loc_18><loc_12><loc_86><loc_15></location>104. Mota, D. F. & Barrow, J. D. Varying alpha in a more realistic Universe. Phys. Lett. B 581, 141-146. arXiv: astro-ph/0306047 (2004).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_155><loc_14><loc_84><loc_82><loc_90></location>105. Mukherjee, P. & Chakrabarti, S. Exact solutions and accelerating universe in modified Brans-Dicke theories. Eur. Phys. J. C 79, 681. arXiv: 1908.01564 [gr-qc] (2019).</list_item>
<list_item><location><page_155><loc_14><loc_79><loc_82><loc_83></location>106. Bergmann, P. G. Comments on the scalar tensor theory. Int. J. Theor. Phys. 1, 25-36 (1968).</list_item>
<list_item><location><page_155><loc_14><loc_74><loc_82><loc_78></location>107. Nojiri, S., Odintsov, S. D. & Sasaki, M. Gauss-Bonnet dark energy. Phys. Rev. D 71, 123509. arXiv: hep-th/0504052 (2005).</list_item>
<list_item><location><page_155><loc_14><loc_67><loc_82><loc_73></location>108. Nojiri, S. & Odintsov, S. D. Modified Gauss-Bonnet theory as gravitational alternative for dark energy. Phys. Lett. B 631, 1-6. arXiv: hepth/0508049 (2005).</list_item>
<list_item><location><page_155><loc_14><loc_60><loc_82><loc_66></location>109. Nojiri, S., Odintsov, S. D. & Sami, M. Dark energy cosmology from higher-order, string-inspired gravity and its reconstruction. Phys. Rev. D 74, 046004. arXiv: hep-th/0605039 (2006).</list_item>
<list_item><location><page_155><loc_14><loc_53><loc_82><loc_59></location>110. Nicolis, A., Rattazzi, R. & Trincherini, E. The Galileon as a local modification of gravity. Phys. Rev. D 79, 064036. arXiv: 0811.2197 [hep-th] (2009).</list_item>
<list_item><location><page_155><loc_14><loc_48><loc_82><loc_52></location>111. Babichev, E. Plane waves in the generalized Galileon theory. Phys. Rev. D 86, 084037. arXiv: 1207.4764 [gr-qc] (2012).</list_item>
<list_item><location><page_155><loc_14><loc_41><loc_82><loc_47></location>112. Gao, X. & Steer, D. A. Inflation and primordial non-Gaussianities of 'generalized Galileons'. JCAP 12, 019. arXiv: 1107.2642 [astro-ph.CO] (2011).</list_item>
<list_item><location><page_155><loc_14><loc_37><loc_82><loc_40></location>113. Park, D. H. Geometric construction of static dyons in scalar-vectortensor theory. J. Korean Phys. Soc. 37, 177-182 (2000).</list_item>
<list_item><location><page_155><loc_14><loc_32><loc_82><loc_35></location>114. Cho, Y. M. Effective Couplings in Kaluza-Klein Unification. Phys. Lett. B 199, 358-362 (1987).</list_item>
<list_item><location><page_155><loc_14><loc_27><loc_82><loc_30></location>115. Park, D. H. General BPS solution in Einstein-Maxwell-dilaton theory. J. Korean Phys. Soc. 31, 894-897 (1997).</list_item>
<list_item><location><page_155><loc_14><loc_18><loc_82><loc_26></location>116. Cai, Y.-F., Capozziello, S., Laurentis, M. D. & Saridakis, E. N. f(T) teleparallel gravity and cosmology. Reports on Progress in Physics 79, 106901. https://dx.doi.org/10.1088/0034-4885/79/10/106901 (2016).</list_item>
<list_item><location><page_155><loc_14><loc_13><loc_82><loc_16></location>117. Horndeski, G. W. Second-order scalar-tensor field equations in a fourdimensional space. Int. J. Theor. Phys. 10, 363-384 (1974).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_156><loc_18><loc_84><loc_86><loc_90></location>118. Barrow, J. D., Thorsrud, M. & Yamamoto, K. Cosmologies in Horndeski's second-order vector-tensor theory. JHEP 02, 146. arXiv: 1211. 5403 [gr-qc] (2013).</list_item>
<list_item><location><page_156><loc_18><loc_77><loc_86><loc_83></location>119. Koyama, K., Niz, G. & Tasinato, G. Effective theory for the Vainshtein mechanism from the Horndeski action. Phys. Rev. D 88, 021502. arXiv: 1305.0279 [hep-th] (2013).</list_item>
<list_item><location><page_156><loc_18><loc_72><loc_86><loc_76></location>120. Fujii, Y. Dilaton and possible non-newtonian gravity. Nature 234, 5-7 (1971).</list_item>
<list_item><location><page_156><loc_18><loc_67><loc_86><loc_71></location>121. Sugimoto, D. Astrophysical test for dilaton theory of non-newtonian gravity. Prog. Theor. Phys. 48, 699-700 (1972).</list_item>
<list_item><location><page_156><loc_18><loc_60><loc_86><loc_66></location>122. Dvali, G. R., Gabadadze, G. & Porrati, M. 4-D gravity on a brane in 5-D Minkowski space. Phys. Lett. B 485, 208-214. arXiv: hep-th/0005016 (2000).</list_item>
<list_item><location><page_156><loc_18><loc_53><loc_86><loc_59></location>123. Deffayet, C., Dvali, G. R. & Gabadadze, G. Accelerated universe from gravity leaking to extra dimensions. Phys. Rev. D 65, 044023. arXiv: astro-ph/0105068 (2002).</list_item>
<list_item><location><page_156><loc_18><loc_48><loc_86><loc_52></location>124. Khoury, J. & Weltman, A. Chameleon cosmology. Phys. Rev. D 69, 044026. arXiv: astro-ph/0309411 (2004).</list_item>
<list_item><location><page_156><loc_18><loc_41><loc_86><loc_47></location>125. Brax, P., van de Bruck, C., Davis, A.-C., Khoury, J. & Weltman, A. Detecting dark energy in orbit: The cosmological chameleon. Phys. Rev. D 70, 123518. arXiv: astro-ph/0408415 (2004).</list_item>
<list_item><location><page_156><loc_18><loc_37><loc_86><loc_40></location>126. Waterhouse, T. P. An Introduction to Chameleon Gravity. arXiv: astroph/0611816 (Nov. 2006).</list_item>
<list_item><location><page_156><loc_18><loc_30><loc_86><loc_35></location>127. Brax, P., van de Bruck, C., Davis, A.-C. & Shaw, D. J. f(R) Gravity and Chameleon Theories. Phys. Rev. D 78, 104021. arXiv: 0806.3415 [astro-ph] (2008).</list_item>
<list_item><location><page_156><loc_18><loc_20><loc_86><loc_28></location>128. Bamba, K., Hossain, M. W., Myrzakulov, R., Nojiri, S. & Sami, M. Cosmological investigations of (extended) nonlinear massive gravity schemes with nonminimal coupling. Phys. Rev. D 89, 083518. arXiv: 1309.6413 [hep-th] (2014).</list_item>
<list_item><location><page_156><loc_18><loc_13><loc_86><loc_19></location>129. Hossain, M. W., Myrzakulov, R., Sami, M. & Saridakis, E. N. Variable gravity: A suitable framework for quintessential inflation. Phys. Rev. D 90, 023512. arXiv: 1402.6661 [gr-qc] (2014).</list_item>
<list_item><location><page_156><loc_18><loc_9><loc_86><loc_12></location>130. Capozziello, S., De Laurentis, M. & Faraoni, V. A Bird's eye view of f(R)-gravity. Open Astron. J. 3, 49. arXiv: 0909.4672 [gr-qc] (2010).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_157><loc_14><loc_84><loc_82><loc_90></location>131. Starobinsky, A. A. A New Type of Isotropic Cosmological Models Without Singularity. Phys. Lett. B 91 (eds Khalatnikov, I. M. & Mineev, V. P.) 99-102 (1980).</list_item>
<list_item><location><page_157><loc_14><loc_79><loc_82><loc_83></location>132. Kerner, R. Cosmology without singularity and nonlinear gravitational Lagrangians. Gen. Rel. Grav. 14, 453-469 (1982).</list_item>
<list_item><location><page_157><loc_14><loc_72><loc_82><loc_78></location>133. Duruisseau, J. P. & Kerner, R. The Effective Gravitational Lagrangian and the Energy Momentum Tensor in the Inflationary Universe. Class. Quant. Grav. 3, 817-824 (1986).</list_item>
<list_item><location><page_157><loc_14><loc_65><loc_82><loc_71></location>134. Capozziello, S., Cardone, V. F., Carloni, S. & Troisi, A. Curvature quintessence matched with observational data. Int. J. Mod. Phys. D 12, 1969-1982. arXiv: astro-ph/0307018 (2003).</list_item>
<list_item><location><page_157><loc_14><loc_58><loc_82><loc_64></location>135. Nojiri, S. & Odintsov, S. D. Modified gravity with in R terms and cosmic acceleration. Gen. Rel. Grav. 36, 1765-1780. arXiv: hep-th/0308176 (2004).</list_item>
<list_item><location><page_157><loc_14><loc_51><loc_82><loc_57></location>136. Das, S., Banerjee, N. & Dadhich, N. Curvature driven acceleration : a utopia or a reality ? Class. Quant. Grav. 23, 4159-4166. arXiv: astroph/0505096 (2006).</list_item>
<list_item><location><page_157><loc_14><loc_44><loc_82><loc_50></location>137. Bean, R., Bernat, D., Pogosian, L., Silvestri, A. & Trodden, M. Dynamics of Linear Perturbations in f(R) Gravity. Phys. Rev. D 75, 064020. arXiv: astro-ph/0611321 (2007).</list_item>
<list_item><location><page_157><loc_14><loc_37><loc_82><loc_43></location>138. Pogosian, L. & Silvestri, A. The pattern of growth in viable f(R) cosmologies. Phys. Rev. D 77. [Erratum: Phys.Rev.D 81, 049901 (2010)], 023503. arXiv: 0709.0296 [astro-ph] (2008).</list_item>
<list_item><location><page_157><loc_14><loc_30><loc_82><loc_36></location>139. Evans, J. D., Hall, L. M. H. & Caillol, P. Standard Cosmological Evolution in a Wide Range of f(R) Models. Phys. Rev. D 77, 083514. arXiv: 0711.3695 [astro-ph] (2008).</list_item>
<list_item><location><page_157><loc_14><loc_23><loc_82><loc_29></location>140. Mukherjee, A. & Banerjee, N. Acceleration of the Universe in f ( R ) Gravity Models. Astrophys. Space Sci. 352, 893-898. arXiv: 1405.6788 [gr-qc] (2014).</list_item>
<list_item><location><page_157><loc_14><loc_16><loc_82><loc_22></location>141. Amendola, L., Polarski, D. & Tsujikawa, S. Are f(R) dark energy models cosmologically viable ? Phys. Rev. Lett. 98, 131302. arXiv: astro-ph/ 0603703 (2007).</list_item>
<list_item><location><page_157><loc_14><loc_9><loc_82><loc_15></location>142. Amendola, L., Gannouji, R., Polarski, D. & Tsujikawa, S. Conditions for the cosmological viability of f(R) dark energy models. Phys. Rev. D 75, 083504. arXiv: gr-qc/0612180 (2007).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_158><loc_18><loc_84><loc_86><loc_90></location>143. De Felice, A., Mukherjee, P. & Wang, Y. Observational Bounds on Modified Gravity Models. Phys. Rev. D 77, 024017. arXiv: 0706.1197 [astro-ph] (2008).</list_item>
<list_item><location><page_158><loc_18><loc_77><loc_86><loc_83></location>144. Capozziello, S. & Salzano, V. Cosmography and large scale structure by f(R) gravity: new results. Adv. Astron. 2009, 217420. arXiv: 0902.0088 [astro-ph.CO] (2009).</list_item>
<list_item><location><page_158><loc_18><loc_72><loc_86><loc_76></location>145. Jordan, P. Formation of the Stars and Development of the Universe. Nature 164, 637-640 (1949).</list_item>
<list_item><location><page_158><loc_18><loc_67><loc_86><loc_71></location>146. Brans, C. & Dicke, R. H. Mach's principle and a relativistic theory of gravitation. Phys. Rev. 124, 925-935 (1961).</list_item>
<list_item><location><page_158><loc_18><loc_60><loc_86><loc_66></location>147. Nordtvedt Jr., K. PostNewtonian metric for a general class of scalar tensor gravitational theories and observational consequences. Astrophys. J. 161, 1059-1067 (1970).</list_item>
<list_item><location><page_158><loc_18><loc_55><loc_86><loc_59></location>148. Wagoner, R. V. Scalar tensor theory and gravitational waves. Phys. Rev. D 1, 3209-3216 (1970).</list_item>
<list_item><location><page_158><loc_18><loc_48><loc_86><loc_54></location>149. Fujii, Y. & Maeda, K. The scalar-tensor theory of gravitation ISBN: 978-0521-03752-5, 978-0-521-81159-0, 978-0-511-02988-2 (Cambridge University Press, July 2007).</list_item>
<list_item><location><page_158><loc_18><loc_44><loc_86><loc_47></location>150. Faraoni, V. Cosmology in scalar tensor gravity ISBN: 978-1-4020-1988-3 (2004).</list_item>
<list_item><location><page_158><loc_18><loc_39><loc_86><loc_42></location>151. Quiros, I. Selected topics in scalar-tensor theories and beyond. Int. J. Mod. Phys. D 28, 1930012. arXiv: 1901.08690 [gr-qc] (2019).</list_item>
<list_item><location><page_158><loc_18><loc_34><loc_86><loc_37></location>152. Dicke, R. H. Mach's principle and invariance under transformation of units. Phys. Rev. 125, 2163-2167 (1962).</list_item>
<list_item><location><page_158><loc_18><loc_27><loc_86><loc_33></location>153. Will, C. M. Theory and Experiment in Gravitational Physics ISBN: 9781-108-67982-4, 978-1-107-11744-0 (Cambridge University Press, Sept. 2018).</list_item>
<list_item><location><page_158><loc_18><loc_20><loc_86><loc_26></location>154. Reasenberg, R. D. et al. Viking relativity experiment: Verification of signal retardation by solar gravity. Astrophys. J. Lett. 234, L219-L221 (1979).</list_item>
<list_item><location><page_158><loc_18><loc_15><loc_86><loc_19></location>155. Bertotti, B., Iess, L. & Tortora, P. A test of general relativity using radio links with the Cassini spacecraft. Nature 425, 374-376 (2003).</list_item>
<list_item><location><page_158><loc_18><loc_10><loc_86><loc_14></location>156. Faraoni, V. Illusions of general relativity in Brans-Dicke gravity. Phys. Rev. D 59, 084021. arXiv: gr-qc/9902083 (1999).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_159><loc_14><loc_86><loc_82><loc_90></location>157. Brans, C. H. Mach's Principle and a Relativistic Theory of Gravitation. II. Phys. Rev. 125, 2194-2201 (1962).</list_item>
<list_item><location><page_159><loc_14><loc_79><loc_82><loc_85></location>158. Bhadra, A. & Nandi, K. K. Brans type II-IV solutions in the Einstein frame and physical interpretation of constants in the solutions. Mod. Phys. Lett. A 16, 2079-2087 (2001).</list_item>
<list_item><location><page_159><loc_14><loc_72><loc_82><loc_78></location>159. Bhadra, A. & Sarkar, K. On static spherically symmetric solutions of the vacuum Brans-Dicke theory. Gen. Rel. Grav. 37, 2189-2199. arXiv: gr-qc/0505141 (2005).</list_item>
<list_item><location><page_159><loc_14><loc_65><loc_82><loc_71></location>160. Campanelli, M. & Lousto, C. O. Are black holes in Brans-Dicke theory precisely the same as a general relativity? Int. J. Mod. Phys. D 2, 451462. arXiv: gr-qc/9301013 (1993).</list_item>
<list_item><location><page_159><loc_14><loc_60><loc_82><loc_64></location>161. Ruffini, R. & Wheeler, J. A. Introducing the black hole. Phys. Today 24, 30 (1971).</list_item>
<list_item><location><page_159><loc_14><loc_53><loc_82><loc_59></location>162. Faraoni, V., Hammad, F. & Belknap-Keet, S. D. Revisiting the Brans solutions of scalar-tensor gravity. Phys. Rev. D 94, 104019. arXiv: 1609. 02783 [gr-qc] (2016).</list_item>
<list_item><location><page_159><loc_14><loc_46><loc_82><loc_52></location>163. Bhadra, A. & Nandi, K. K. On the equivalence of the Buchdahl and the Janis-Newman-Winnicour solutions. Int. J. Mod. Phys. A 16, 4543-4545 (2001).</list_item>
<list_item><location><page_159><loc_14><loc_41><loc_82><loc_45></location>164. Agnese, A. G. & La Camera, M. Wormholes in the Brans-Dicke theory of gravitation. Phys. Rev. D 51, 2011-2013 (1995).</list_item>
<list_item><location><page_159><loc_14><loc_37><loc_82><loc_40></location>165. Nandi, K. K., Islam, A. & Evans, J. Brans wormholes. Phys. Rev. D 55, 2497-2500. arXiv: 0906.0436 [gr-qc] (1997).</list_item>
<list_item><location><page_159><loc_14><loc_30><loc_82><loc_35></location>166. Anchordoqui, L. A., Grunfeld, A. G. & Torres, D. F. Vacuum static Brans-Dicke wormhole. Grav. Cosmol. 4, 287-290. arXiv: gr-qc/9707025 (1998).</list_item>
<list_item><location><page_159><loc_14><loc_23><loc_82><loc_28></location>167. He, F. & Kim, S.-W. New Brans-Dicke wormholes. Phys. Rev. D 65, 084022. https://link.aps.org/doi/10.1103/PhysRevD.65.084022 (8 2002).</list_item>
<list_item><location><page_159><loc_14><loc_16><loc_82><loc_21></location>168. Bhadra, A., Simaciu, I., Nandi, K. K. & Zhang, Y.-Z. Comment on 'New Brans-Dicke wormholes'. Phys. Rev. D 71, 128501. arXiv: gr-qc/ 0406014 (2005).</list_item>
<list_item><location><page_159><loc_14><loc_9><loc_82><loc_14></location>169. Nandi, K. K. & Zhang, Y.-Z. On traversable Lorentzian wormholes in the vacuum low energy effective string theory in Einstein and Jordan frames. Phys. Rev. D 70, 044040. arXiv: gr-qc/0405051 (2004).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_160><loc_18><loc_84><loc_86><loc_90></location>170. Bronnikov, K. A., Clement, G., Constantinidis, C. P. & Fabris, J. C. Cold scalar tensor black holes: Causal structure, geodesics, stability. Grav. Cosmol. 4, 128-138. arXiv: gr-qc/9804064 (1998).</list_item>
<list_item><location><page_160><loc_18><loc_77><loc_86><loc_83></location>171. Nandi, K. K., Bhadra, A., Alsing, P. M. & Nayak, T. B. Tidal forces in cold black hole space-times. Int. J. Mod. Phys. D 10, 529-538. arXiv: grqc/0008025 (2001).</list_item>
<list_item><location><page_160><loc_18><loc_72><loc_86><loc_76></location>172. O' Hanlon, J. & Tupper, B. O. J. Vacuum-field solutions in the BransDicke theory. Nuovo Cim. B 7, 305-312 (1972).</list_item>
<list_item><location><page_160><loc_18><loc_65><loc_86><loc_71></location>173. Nariai, H. On the Green's Function in an Expanding Universe and Its Role in the Problem of Mach's Principle. Progress of Theoretical Physics 40, 49-59 (July 1968).</list_item>
<list_item><location><page_160><loc_18><loc_58><loc_86><loc_64></location>174. Gurevich, L., Finkelstein, A. & Ruban, V. On the problem of the initial state in the isotropic scalar-tensor cosmology of Brans-Dicke. Astrophysics and Space Science 22, 231-242 (1973).</list_item>
<list_item><location><page_160><loc_18><loc_53><loc_86><loc_57></location>175. Lorenz-Petzold, D. Exact perfect fluid solutions in the Brans-Dicketheory. Astrophysics and space science 98, 249-254 (1984).</list_item>
<list_item><location><page_160><loc_18><loc_48><loc_86><loc_52></location>176. Morganstern, R. E. Exact Solutions to Radiation-Filled Brans-Dicke Cosmologies. Phys. Rev. D 4, 282-286 (1971).</list_item>
<list_item><location><page_160><loc_18><loc_41><loc_86><loc_47></location>177. Mathiazhagan, C. & Johri, V. B. An inflationary universe in BransDicke theory: a hopeful sign of theoretical estimation of the gravitational constant. Class. Quant. Grav. 1, L29-L32 (1984).</list_item>
<list_item><location><page_160><loc_18><loc_37><loc_86><loc_40></location>178. La, D. & Steinhardt, P. J. Extended Inflationary Cosmology. Phys. Rev. Lett. 62. [Erratum: Phys.Rev.Lett. 62, 1066 (1989)], 376 (1989).</list_item>
<list_item><location><page_160><loc_18><loc_30><loc_86><loc_35></location>179. Romero, C. & Barros, A. Brans-Dicke cosmology and the cosmological constant: the spectrum of vacuum solutions. Astrophys. Space Sci. 192, 263-274 (1992).</list_item>
<list_item><location><page_160><loc_18><loc_25><loc_86><loc_28></location>180. Liddle, A. R. & Lyth, D. H. The Cold dark matter density perturbation. Phys. Rept. 231, 1-105. arXiv: astro-ph/9303019 (1993).</list_item>
<list_item><location><page_160><loc_18><loc_20><loc_86><loc_23></location>181. Barrow, J. D. Scalar-tensor cosmologies. Phys. Rev. D 47, 5329-5335. https://link.aps.org/doi/10.1103/PhysRevD.47.5329 (12 1993).</list_item>
<list_item><location><page_160><loc_18><loc_15><loc_86><loc_19></location>182. Dehnen, H & Obregen, O. Exact Cosmological Solutions in Brans and Dicke s Scalar-Tensor Theory, I. Astrophys. Space Sci. 14, 454-459 (1971).</list_item>
<list_item><location><page_160><loc_18><loc_10><loc_86><loc_14></location>183. Levin, J. J. & Freese, K. Curvature and flatness in a Brans-Dicke universe. Nucl. Phys. B 421, 635-661. arXiv: gr-qc/9312025 (1994).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_161><loc_14><loc_86><loc_82><loc_90></location>184. Mimoso, J. P. & Wands, D. Massless fields in scalar - tensor cosmologies. Phys. Rev. D 51, 477-489. arXiv: gr-qc/9405025 (1995).</list_item>
<list_item><location><page_161><loc_14><loc_79><loc_82><loc_85></location>185. Sahni, V. & Starobinsky, A. A. The Case for a positive cosmological Lambda term. Int. J. Mod. Phys. D 9, 373-444. arXiv: astro-ph/9904398 (2000).</list_item>
<list_item><location><page_161><loc_14><loc_74><loc_82><loc_78></location>186. Sahni, V. Dark matter and dark energy. Lect. Notes Phys. 653 (ed Papantonopoulos, E.) 141-180. arXiv: astro-ph/0403324 (2004).</list_item>
<list_item><location><page_161><loc_14><loc_69><loc_82><loc_73></location>187. Padmanabhan, T. Cosmological constant: The Weight of the vacuum. Phys. Rept. 380, 235-320. arXiv: hep-th/0212290 (2003).</list_item>
<list_item><location><page_161><loc_14><loc_62><loc_82><loc_68></location>188. Padmanabhan, T. Darker side of the universe ... and the crying need for some bright ideas! in 29th International Cosmic Ray Conference (Oct. 2005). arXiv: astro-ph/0510492 .</list_item>
<list_item><location><page_161><loc_14><loc_57><loc_82><loc_61></location>189. Copeland, E. J., Sami, M. & Tsujikawa, S. Dynamics of dark energy. Int. J. Mod. Phys. D 15, 1753-1936. arXiv: hep-th/0603057 (2006).</list_item>
<list_item><location><page_161><loc_14><loc_53><loc_82><loc_56></location>190. Ruiz-Lapuente, P. Dark energy, gravitation and supernovae. Class. Quant. Grav. 24, R91-R111. arXiv: 0704.1058 [gr-qc] (2007).</list_item>
<list_item><location><page_161><loc_14><loc_48><loc_82><loc_51></location>191. Durrer, R. & Maartens, R. Dark Energy and Dark Gravity. Gen. Rel. Grav. 40, 301-328. arXiv: 0711.0077 [astro-ph] (2008).</list_item>
<list_item><location><page_161><loc_14><loc_41><loc_82><loc_46></location>192. Riess, A. G. et al. Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J. 116, 1009-1038. arXiv: astro-ph/9805201 (1998).</list_item>
<list_item><location><page_161><loc_14><loc_36><loc_82><loc_40></location>193. Perlmutter, S. et al. Measurements of Ω and Λ from 42 high redshift supernovae. Astrophys. J. 517, 565-586. arXiv: astro-ph/9812133 (1999).</list_item>
<list_item><location><page_161><loc_14><loc_31><loc_82><loc_35></location>194. Carroll, S. M. The Cosmological constant. Living Rev. Rel. 4, 1. arXiv: astro-ph/0004075 (2001).</list_item>
<list_item><location><page_161><loc_14><loc_26><loc_82><loc_30></location>195. Peebles, P. J. E. & Ratra, B. The Cosmological Constant and Dark Energy. Rev. Mod. Phys. 75, 559-606. arXiv: astro-ph/0207347 (2003).</list_item>
<list_item><location><page_161><loc_14><loc_19><loc_82><loc_25></location>196. Frieman, J., Turner, M. & Huterer, D. Dark Energy and the Accelerating Universe. Ann. Rev. Astron. Astrophys. 46, 385-432. arXiv: 0803.0982 [astro-ph] (2008).</list_item>
<list_item><location><page_161><loc_14><loc_12><loc_82><loc_18></location>197. Amendola, L., Kainulainen, K., Marra, V. & Quartin, M. Large-scale inhomogeneities may improve the cosmic concordance of supernovae. Phys. Rev. Lett. 105, 121302. arXiv: 1002.1232 [astro-ph.CO] (2010).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_162><loc_18><loc_84><loc_86><loc_90></location>198. Mehrabi, A. Growth of perturbations in dark energy parametrization scenarios. Phys. Rev. D 97, 083522. arXiv: 1804.09886 [astro-ph.CO] (2018).</list_item>
<list_item><location><page_162><loc_18><loc_79><loc_86><loc_83></location>199. Liddle, A. R. & Lyth, D. H. Cosmological inflation and large scale structure ISBN: 978-0-521-57598-0, 978-0-521-82849-9 (2000).</list_item>
<list_item><location><page_162><loc_18><loc_74><loc_86><loc_78></location>200. Padmanabhan, T. Dark energy: The Cosmological challenge of the millennium. Curr. Sci. 88, 1057. arXiv: astro-ph/0411044 (2005).</list_item>
<list_item><location><page_162><loc_18><loc_67><loc_86><loc_73></location>201. Overduin, J. M. & Cooperstock, F. I. Evolution of the scale factor with a variable cosmological term. Phys. Rev. D 58, 043506. arXiv: astroph/9805260 (1998).</list_item>
<list_item><location><page_162><loc_18><loc_63><loc_86><loc_66></location>202. End¯o, M. & Fukui, T. The cosmological term and a modified BransDicke cosmology. General Relativity and Gravitation 8, 833-839 (1977).</list_item>
<list_item><location><page_162><loc_18><loc_56><loc_86><loc_61></location>203. Canuto, V., Hsieh, S. H. & Adams, P. J. Scale-Covariant Theory of Gravitation and Astrophysical Applications. Phys. Rev. Lett. 39, 429432 (1977).</list_item>
<list_item><location><page_162><loc_18><loc_51><loc_86><loc_54></location>204. Bertolami, O. Time Dependent Cosmological Term. Nuovo Cim. B 93, 36-42 (1986).</list_item>
<list_item><location><page_162><loc_18><loc_46><loc_86><loc_50></location>205. Berman, M. S. & Som, M. Brans-Dicke models with time-dependent cosmological term. Int.J.Theor.Phys. 29, 1411-1414 (1990).</list_item>
<list_item><location><page_162><loc_18><loc_41><loc_86><loc_45></location>206. Beesham, A. Bianchi type I cosmological models with variable G and Lambda. Gen. Rel. Grav. 26, 159-165 (1994).</list_item>
<list_item><location><page_162><loc_18><loc_36><loc_86><loc_40></location>207. Lopez, J. L. & Nanopoulos, D. V. A New cosmological constant model. Mod. Phys. Lett. A 11, 1-7. arXiv: hep-ph/9501293 (1996).</list_item>
<list_item><location><page_162><loc_18><loc_32><loc_86><loc_35></location>208. Kalligas, D, Wesson, P & Everitt, C. Flat FRW models with variable G and Λ . Gen. Rel. Grav. 24, 351-357 (1992).</list_item>
<list_item><location><page_162><loc_18><loc_25><loc_86><loc_30></location>209. Kalligas, D., Wesson, P. S. & Everitt, C. W. F. Bianchi type I cosmological models with variable G and Lambda: A Comment. Gen. Rel. Grav. 27, 645-650 (1995).</list_item>
<list_item><location><page_162><loc_18><loc_20><loc_86><loc_23></location>210. Beesham, A. Cosmological models with a variable cosmological term and bulk viscous models. Phys. Rev. D 48, 3539-3543 (1993).</list_item>
<list_item><location><page_162><loc_18><loc_15><loc_86><loc_19></location>211. Spindel, P. & Brout, R. Entropy production from vacuum decay. Phys. Lett. B 320, 241-244. arXiv: gr-qc/9310023 (1994).</list_item>
<list_item><location><page_162><loc_18><loc_8><loc_86><loc_14></location>212. Özer, M. & Taha, M. A possible solution to the main cosmological problems. Physics Letters B 171, 363-365. ISSN: 0370-2693. https://www. sciencedirect.com/science/article/pii/0370269386914218 (1986).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_163><loc_14><loc_84><loc_82><loc_90></location>213. Özer, M. & Taha, M. A model of the universe free of cosmological problems. Nuclear Physics B 287, 776-796. ISSN: 0550-3213. https://www. sciencedirect.com/science/article/pii/0550321387901283 (1987).</list_item>
<list_item><location><page_163><loc_14><loc_79><loc_82><loc_83></location>214. Abdel-Rahman, A. M. M. Singularity - free decaying vacuum cosmologies. Phys. Rev. D 45, 3497-3511 (1992).</list_item>
<list_item><location><page_163><loc_14><loc_72><loc_82><loc_78></location>215. Chen, W. & Wu, Y. S. Implications of a cosmological constant varying as R**(-2). Phys. Rev. D 41. [Erratum: Phys.Rev.D 45, 4728 (1992)], 695698 (1990).</list_item>
<list_item><location><page_163><loc_14><loc_61><loc_82><loc_71></location>216. Gott J. Richard, I. & Rees, M. J. Astronomical constraints on a stringdominated universe. Mon. Not. Roy. Astron. Soc. 227, 453-459. ISSN: 0035-8711. eprint: https : / / academic . oup . com / mnras / article pdf/227/2/453/2877800/mnras227-0453.pdf . https://doi.org/ 10.1093/mnras/227.2.453 (July 1987).</list_item>
<list_item><location><page_163><loc_14><loc_56><loc_82><loc_60></location>217. Abdussattar & Vishwakarma, R. G. Some FRW models with variable G and Lambda. Class. Quant. Grav. 14, 945-953 (1997).</list_item>
<list_item><location><page_163><loc_14><loc_51><loc_82><loc_55></location>218. Olson, T. S. & Jordan, T. F. Ages of the Universe for Decreasing Cosmological Constants. Phys. Rev. D 35, 3258-3260 (1987).</list_item>
<list_item><location><page_163><loc_14><loc_46><loc_82><loc_50></location>219. Pavon, D. Nonequilibrium fluctuations in cosmic vacuum decay. Phys. Rev. D 43, 375-378 (1991).</list_item>
<list_item><location><page_163><loc_14><loc_41><loc_82><loc_45></location>220. Sahni, V., Feldman, H. & Stebbins, A. Loitering universe. Astrophys. J. 385, 1-8 (1992).</list_item>
<list_item><location><page_163><loc_14><loc_37><loc_82><loc_40></location>221. Maia, M. D. & Silva, G. S. Geometrical constraints on the cosmological constant. Phys. Rev. D 50, 7233-7238. arXiv: gr-qc/9401005 (1994).</list_item>
<list_item><location><page_163><loc_14><loc_32><loc_82><loc_35></location>222. Matyjasek, J. Cosmological models with a time dependent Lambda term. Phys. Rev. D 51, 4154-4159 (1995).</list_item>
<list_item><location><page_163><loc_14><loc_27><loc_82><loc_31></location>223. Silveira, V. & Waga, I. Decaying Lambda cosmologies and power spectrum. Phys. Rev. D 50, 4890-4894 (1994).</list_item>
<list_item><location><page_163><loc_14><loc_20><loc_82><loc_26></location>224. Silveira, V. & Waga, I. Cosmological properties of a class of Lambda decaying cosmologies. Phys. Rev. D 56, 4625-4632. arXiv: astro-ph/ 9703185 (1997).</list_item>
<list_item><location><page_163><loc_14><loc_13><loc_82><loc_19></location>225. John, M. V. & Joseph, K. B. A Low matter density decaying vacuum cosmology from complex metric. Class. Quant. Grav. 14, 1115. arXiv: gr-qc/0007052 (1997).</list_item>
<list_item><location><page_163><loc_14><loc_8><loc_82><loc_12></location>226. Turner, M. S. & White, M. J. CDM models with a smooth component. Phys. Rev. D 56, R4439. arXiv: astro-ph/9701138 (1997).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_164><loc_18><loc_84><loc_86><loc_90></location>227. Wang, L.-M. & Steinhardt, P. J. Cluster abundance constraints on quintessence models. Astrophys. J. 508, 483-490. arXiv: astro - ph / 9804015 (1998).</list_item>
<list_item><location><page_164><loc_18><loc_77><loc_86><loc_83></location>228. Caldwell, R. R. & Steinhardt, P. J. The Imprint of gravitational waves in models dominated by a dynamical cosmic scalar field. Phys. Rev. D 57, 6057-6064. arXiv: astro-ph/9710062 (1998).</list_item>
<list_item><location><page_164><loc_18><loc_70><loc_86><loc_76></location>229. Caldwell, R. R., Dave, R. & Steinhardt, P. J. Cosmological imprint of an energy component with general equation of state. Phys. Rev. Lett. 80, 1582-1585. arXiv: astro-ph/9708069 (1998).</list_item>
<list_item><location><page_164><loc_18><loc_65><loc_86><loc_69></location>230. Hoyle, F, Burbidge, G & Narlikar, J. On the Hubble constant and the cosmological constant. Mon. Not. Roy. Astron. Soc. 286, 173-182 (1997).</list_item>
<list_item><location><page_164><loc_18><loc_58><loc_86><loc_64></location>231. Hu, W., Eisenstein, D. J., Tegmark, M. & White, M. J. Observationally determining the properties of dark matter. Phys. Rev. D 59, 023512. arXiv: astro-ph/9806362 (1999).</list_item>
<list_item><location><page_164><loc_18><loc_53><loc_86><loc_57></location>232. Rajeev, S. G. Why Is the Cosmological Constant Small? Phys. Lett. B 125, 144-146 (1983).</list_item>
<list_item><location><page_164><loc_18><loc_48><loc_86><loc_52></location>233. Kazanas, D. Dynamics of the Universe and Spontaneous Symmetry Breaking. Astrophys. J. Lett. 241, L59-L63 (1980).</list_item>
<list_item><location><page_164><loc_18><loc_41><loc_86><loc_47></location>234. Copeland, E. J., Liddle, A. R. & Wands, D. Exponential potentials and cosmological scaling solutions. Phys. Rev. D 57, 4686-4690. arXiv: grqc/9711068 (1998).</list_item>
<list_item><location><page_164><loc_18><loc_37><loc_86><loc_40></location>235. Ferreira, P. G. & Joyce, M. Structure formation with a selftuning scalar field. Phys. Rev. Lett. 79, 4740-4743. arXiv: astro-ph/9707286 (1997).</list_item>
<list_item><location><page_164><loc_18><loc_32><loc_86><loc_35></location>236. Lima, J. A. S. & Carvalho, J. C. Dirac's cosmology with varying cosmological constant. Gen. Rel. Grav. 26, 909-916 (1994).</list_item>
<list_item><location><page_164><loc_18><loc_25><loc_86><loc_31></location>237. Wetterich, C. The Cosmon model for an asymptotically vanishing time dependent cosmological 'constant'. Astron. Astrophys. 301, 321-328. arXiv: hep-th/9408025 (1995).</list_item>
<list_item><location><page_164><loc_18><loc_20><loc_86><loc_24></location>238. Wetterich, C. Cosmology and the Fate of Dilatation Symmetry. Nucl. Phys. B 302, 668-696. arXiv: 1711.03844 [hep-th] (1988).</list_item>
<list_item><location><page_164><loc_18><loc_13><loc_86><loc_19></location>239. Waga, I. Decaying vacuum flat cosmological models: Expressions for some observable quantities and their properties. Astrophys. J. 414, 436448 (1993).</list_item>
<list_item><location><page_164><loc_18><loc_8><loc_86><loc_12></location>240. Salim, J. M. & Waga, I. Thermodynamic constraints on a time dependent Lambda model. Class. Quant. Grav. 10, 1767-1774 (1993).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_165><loc_14><loc_84><loc_82><loc_90></location>241. Carvalho, J. C., Lima, J. A. S. & Waga, I. On the cosmological consequences of a time dependent lambda term. Phys. Rev. D 46, 2404-2407 (1992).</list_item>
<list_item><location><page_165><loc_14><loc_79><loc_82><loc_83></location>242. Arbab, A. I. & Abdel-Rahman, A. M. M. Nonsingular cosmology with a time dependent cosmological term. Phys. Rev. D 50, 7725-7728 (1994).</list_item>
<list_item><location><page_165><loc_14><loc_74><loc_82><loc_78></location>243. Lima, J. A. S. & Maia, J. M. F. Deflationary cosmology with decaying vacuum energy density. Phys. Rev. D 49, 5597-5600 (1994).</list_item>
<list_item><location><page_165><loc_14><loc_67><loc_82><loc_73></location>244. Lima, J. A. S. & Trodden, M. Decaying vacuum energy and deflationary cosmology in open and closed universes. Phys. Rev. D 53, 4280-4286. arXiv: astro-ph/9508049 (1996).</list_item>
<list_item><location><page_165><loc_14><loc_63><loc_82><loc_66></location>245. Hiscock, W. A. Quantum Instabilities and the Cosmological Constant. Phys. Lett. B 166, 285-288 (1986).</list_item>
<list_item><location><page_165><loc_14><loc_58><loc_82><loc_61></location>246. Reuter, M. & Wetterich, C. Time Evolution of the Cosmological 'Constant'. Phys. Lett. B 188, 38-43 (1987).</list_item>
<list_item><location><page_165><loc_14><loc_53><loc_82><loc_57></location>247. Sahni, V. & Starobinsky, A. Reconstructing Dark Energy. Int. J. Mod. Phys. D 15, 2105-2132. arXiv: astro-ph/0610026 (2006).</list_item>
<list_item><location><page_165><loc_14><loc_48><loc_82><loc_52></location>248. Chevallier, M. & Polarski, D. Accelerating universes with scaling dark matter. Int. J. Mod. Phys. D 10, 213-224. arXiv: gr-qc/0009008 (2001).</list_item>
<list_item><location><page_165><loc_14><loc_43><loc_82><loc_47></location>249. Linder, E. V. Exploring the expansion history of the universe. Phys. Rev. Lett. 90, 091301. arXiv: astro-ph/0208512 (2003).</list_item>
<list_item><location><page_165><loc_14><loc_39><loc_82><loc_42></location>250. Albrecht, A. et al. Report of the Dark Energy Task Force. arXiv: astroph/0609591 (Sept. 2006).</list_item>
<list_item><location><page_165><loc_14><loc_32><loc_82><loc_37></location>251. Virey, J.-M. & Ealet, A. Sensitivity and figures of merit for dark energy supernovae surveys. Astron. Astrophys. 464, 837. arXiv: astroph/0607589 (2007).</list_item>
<list_item><location><page_165><loc_14><loc_25><loc_82><loc_30></location>252. Jassal, H. K., Bagla, J. S. & Padmanabhan, T. WMAP constraints on low redshift evolution of dark energy. Mon. Not. Roy. Astron. Soc. 356, L11L16. arXiv: astro-ph/0404378 (2005).</list_item>
<list_item><location><page_165><loc_14><loc_18><loc_82><loc_23></location>253. Jassal, H. K., Bagla, J. S. & Padmanabhan, T. Understanding the origin of CMB constraints on Dark Energy. Mon. Not. Roy. Astron. Soc. 405, 2639-2650. arXiv: astro-ph/0601389 (2010).</list_item>
<list_item><location><page_165><loc_14><loc_13><loc_82><loc_17></location>254. Barboza Jr., E. M. & Alcaniz, J. S. A parametric model for dark energy. Phys. Lett. B 666, 415-419. arXiv: 0805.1713 [astro-ph] (2008).</list_item>
<list_item><location><page_165><loc_14><loc_8><loc_82><loc_12></location>255. Wetterich, C. Phenomenological parameterization of quintessence. Phys. Lett. B 594, 17-22. arXiv: astro-ph/0403289 (2004).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_166><loc_18><loc_84><loc_86><loc_90></location>256. Ma, J.-Z. & Zhang, X. Probing the dynamics of dark energy with novel parametrizations. Phys. Lett. B 699, 233-238. arXiv: 1102.2671 [astro-ph.CO] (2011).</list_item>
<list_item><location><page_166><loc_18><loc_77><loc_86><loc_83></location>257. Linder, E. V. The Dynamics of Quintessence, The Quintessence of Dynamics. Gen. Rel. Grav. 40, 329-356. arXiv: 0704 . 2064 [astro-ph] (2008).</list_item>
<list_item><location><page_166><loc_18><loc_70><loc_86><loc_76></location>258. Wang, Y. & Mukherjee, P. Model - independent constraints on dark energy density from flux - averaging analysis of type Ia supernova data. Astrophys. J. 606, 654-663. arXiv: astro-ph/0312192 (2004).</list_item>
<list_item><location><page_166><loc_18><loc_63><loc_86><loc_69></location>259. Wang, Y. & Tegmark, M. New dark energy constraints from supernovae, microwave background and galaxy clustering. Phys. Rev. Lett. 92, 241302. arXiv: astro-ph/0403292 (2004).</list_item>
<list_item><location><page_166><loc_18><loc_56><loc_86><loc_62></location>260. Wang, Y. & Freese, K. Probing dark energy using its density instead of its equation of state. Phys. Lett. B 632, 449-452. arXiv: astro-ph/ 0402208 (2006).</list_item>
<list_item><location><page_166><loc_18><loc_51><loc_86><loc_55></location>261. Ratra, B. & Peebles, P. J. E. Cosmological Consequences of a Rolling Homogeneous Scalar Field. Phys. Rev. D 37, 3406 (1988).</list_item>
<list_item><location><page_166><loc_18><loc_44><loc_86><loc_50></location>262. Urena-Lopez, L. A. & Matos, T. A New cosmological tracker solution for quintessence. Phys. Rev. D 62, 081302. arXiv: astro- ph/0003364 (2000).</list_item>
<list_item><location><page_166><loc_18><loc_39><loc_86><loc_43></location>263. Sen, A. A. & Sethi, S. Quintessence model with double exponential potential. Phys. Lett. B 532, 159-165. arXiv: gr-qc/0111082 (2002).</list_item>
<list_item><location><page_166><loc_18><loc_32><loc_86><loc_38></location>264. Barreiro, T., Copeland, E. J. & Nunes, N. J. Quintessence arising from exponential potentials. Phys. Rev. D 61, 127301. arXiv: astro - ph / 9910214 (2000).</list_item>
<list_item><location><page_166><loc_18><loc_25><loc_86><loc_31></location>265. Zlatev, I., Wang, L.-M. & Steinhardt, P. J. Quintessence, cosmic coincidence, and the cosmological constant. Phys. Rev. Lett. 82, 896-899. arXiv: astro-ph/9807002 (1999).</list_item>
<list_item><location><page_166><loc_18><loc_18><loc_86><loc_24></location>266. Sahni, V. & Wang, L.-M. A New cosmological model of quintessence and dark matter. Phys. Rev. D 62, 103517. arXiv: astro-ph/9910097 (2000).</list_item>
<list_item><location><page_166><loc_18><loc_13><loc_86><loc_17></location>267. Kim, J. E. Axion and almost massless quark as ingredients of quintessence. JHEP 05, 022. arXiv: hep-ph/9811509 (1999).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_167><loc_14><loc_84><loc_82><loc_90></location>268. Frieman, J. A., Hill, C. T., Stebbins, A. & Waga, I. Cosmology with ultralight pseudo Nambu-Goldstone bosons. Phys. Rev. Lett. 75, 2077-2080. arXiv: astro-ph/9505060 (1995).</list_item>
<list_item><location><page_167><loc_14><loc_79><loc_82><loc_83></location>269. Brax, P. & Martin, J. The Robustness of quintessence. Phys. Rev. D 61, 103502. arXiv: astro-ph/9912046 (2000).</list_item>
<list_item><location><page_167><loc_14><loc_74><loc_82><loc_78></location>270. Brax, P. & Martin, J. Quintessence and supergravity. Phys. Lett. B 468, 40-45. arXiv: astro-ph/9905040 (1999).</list_item>
<list_item><location><page_167><loc_14><loc_67><loc_82><loc_73></location>271. Albrecht, A. & Skordis, C. Phenomenology of a realistic accelerating universe using only Planck scale physics. Phys. Rev. Lett. 84, 2076-2079. arXiv: astro-ph/9908085 (2000).</list_item>
<list_item><location><page_167><loc_14><loc_60><loc_82><loc_66></location>272. Dodelson, S., Kaplinghat, M. & Stewart, E. Solving the Coincidence Problem : Tracking Oscillating Energy. Phys. Rev. Lett. 85, 5276-5279. arXiv: astro-ph/0002360 (2000).</list_item>
<list_item><location><page_167><loc_14><loc_53><loc_82><loc_59></location>273. Banerjee, N. & Das, S. Acceleration of the Universe with a simple trigonometric potential. Gen. Rel. Grav. 37, 1695-1703. arXiv: astroph/0505121 (2005).</list_item>
<list_item><location><page_167><loc_14><loc_48><loc_82><loc_52></location>274. Scherrer, R. J. & Sen, A. A. Thawing quintessence with a nearly flat potential. Phys. Rev. D 77, 083515. arXiv: 0712.3450 [astro-ph] (2008).</list_item>
<list_item><location><page_167><loc_14><loc_44><loc_82><loc_47></location>275. Steinhardt, P. J., Wang, L.-M. & Zlatev, I. Cosmological tracking solutions. Phys. Rev. D 59, 123504. arXiv: astro-ph/9812313 (1999).</list_item>
<list_item><location><page_167><loc_14><loc_39><loc_82><loc_42></location>276. Johri, V. B. The Genesis of cosmological tracker fields. Phys. Rev. D 63, 103504. arXiv: astro-ph/0005608 (2001).</list_item>
<list_item><location><page_167><loc_14><loc_32><loc_82><loc_37></location>277. Wang, L.-M., Caldwell, R. R., Ostriker, J. P. & Steinhardt, P. J. Cosmic concordance and quintessence. Astrophys. J. 530, 17-35. arXiv: astroph/9901388 (2000).</list_item>
<list_item><location><page_167><loc_14><loc_25><loc_82><loc_30></location>278. Zlatev, I. & Steinhardt, P. J. A Tracker solution to the cold dark matter cosmic coincidence problem. Phys. Lett. B 459, 570-574. arXiv: astroph/9906481 (1999).</list_item>
<list_item><location><page_167><loc_14><loc_18><loc_82><loc_23></location>279. Sahlen, M., Liddle, A. R. & Parkinson, D. Quintessence reconstructed: New constraints and tracker viability. Phys. Rev. D 75, 023502. arXiv: astro-ph/0610812 (2007).</list_item>
<list_item><location><page_167><loc_14><loc_11><loc_82><loc_16></location>280. Scherrer, R. J. & Sen, A. A. Phantom Dark Energy Models with a Nearly Flat Potential. Phys. Rev. D 78, 067303. arXiv: 0808.1880 [astro-ph] (2008).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_168><loc_18><loc_84><loc_86><loc_90></location>281. Gupta, G., Rangarajan, R. & Sen, A. A. Thawing quintessence from the inflationary epoch to today. Phys. Rev. D 92, 123003. arXiv: 1412.6915 [astro-ph.CO] (2015).</list_item>
<list_item><location><page_168><loc_18><loc_77><loc_86><loc_83></location>282. Dalmazi, D., de Souza Dutra, A. & Abreu, E. M. C. Generalizing the Soldering procedure. Phys. Rev. D 74. [Erratum: Phys.Rev.D 79, 109902 (2009)], 025015. arXiv: hep-th/0607102 (2006).</list_item>
<list_item><location><page_168><loc_18><loc_70><loc_86><loc_76></location>283. Chiba, T., De Felice, A. & Tsujikawa, S. Observational constraints on quintessence: thawing, tracker, and scaling models. Phys. Rev. D 87, 083505. arXiv: 1210.3859 [astro-ph.CO] (2013).</list_item>
<list_item><location><page_168><loc_18><loc_63><loc_86><loc_69></location>284. Pantazis, G., Nesseris, S. & Perivolaropoulos, L. Comparison of thawing and freezing dark energy parametrizations. Phys. Rev. D 93, 103503. arXiv: 1603.02164 [astro-ph.CO] (2016).</list_item>
<list_item><location><page_168><loc_18><loc_58><loc_86><loc_62></location>285. Roy, N. & Banerjee, N. Tracking quintessence: a dynamical systems study. Gen. Rel. Grav. 46, 1651. arXiv: 1312.2670 [gr-qc] (2014).</list_item>
<list_item><location><page_168><loc_18><loc_53><loc_86><loc_57></location>286. Roy, N. & Banerjee, N. Quintessence Scalar Field: A Dynamical Systems Study. Eur. Phys. J. Plus 129, 162. arXiv: 1402.6821 [gr-qc] (2014).</list_item>
<list_item><location><page_168><loc_18><loc_46><loc_86><loc_52></location>287. Carvalho, F. C., Alcaniz, J. S., Lima, J. A. S. & Silva, R. Scalar-fielddominated cosmology with a transient accelerating phase. Phys. Rev. Lett. 97, 081301. arXiv: astro-ph/0608439 (2006).</list_item>
<list_item><location><page_168><loc_18><loc_41><loc_86><loc_45></location>288. Caldwell, R. R. A Phantom menace? Phys. Lett. B 545, 23-29. arXiv: astro-ph/9908168 (2002).</list_item>
<list_item><location><page_168><loc_18><loc_34><loc_86><loc_40></location>289. Feng, B., Wang, X.-L. & Zhang, X.-M. Dark energy constraints from the cosmic age and supernova. Phys. Lett. B 607, 35-41. arXiv: astroph/0404224 (2005).</list_item>
<list_item><location><page_168><loc_18><loc_30><loc_86><loc_33></location>290. Cai, Y.-f., Li, H., Piao, Y.-S. & Zhang, X.-m. Cosmic Duality in Quintom Universe. Phys. Lett. B 646, 141-144. arXiv: gr-qc/0609039 (2007).</list_item>
<list_item><location><page_168><loc_18><loc_25><loc_86><loc_28></location>291. Cai, Y.-f. et al. AString-Inspired Quintom Model Of Dark Energy. Phys. Lett. B 651, 1-7. arXiv: hep-th/0701016 (2007).</list_item>
<list_item><location><page_168><loc_18><loc_18><loc_86><loc_23></location>292. Cai, Y.-F. & Wang, J. Dark Energy Model with Spinor Matter and Its Quintom Scenario. Class. Quant. Grav. 25, 165014. arXiv: 0806.3890 [hep-th] (2008).</list_item>
<list_item><location><page_168><loc_18><loc_13><loc_86><loc_16></location>293. Kamenshchik, A. Y., Moschella, U. & Pasquier, V. An Alternative to quintessence. Phys. Lett. B 511, 265-268. arXiv: gr-qc/0103004 (2001).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_169><loc_14><loc_84><loc_82><loc_90></location>294. Bento, M. C., Bertolami, O. & Sen, A. A. Generalized Chaplygin gas, accelerated expansion and dark energy matter unification. Phys. Rev. D 66, 043507. arXiv: gr-qc/0202064 (2002).</list_item>
<list_item><location><page_169><loc_14><loc_77><loc_82><loc_83></location>295. Jackiw, R. A Particle field theorist's lectures on supersymmetric, nonAbelian fluid mechanics and d-branes. arXiv: physics/0010042 (Oct. 2000).</list_item>
<list_item><location><page_169><loc_14><loc_70><loc_82><loc_76></location>296. Amendola, L., Finelli, F., Burigana, C. & Carturan, D. WMAP and the generalized Chaplygin gas. JCAP 07, 005. arXiv: astro-ph/0304325 (2003).</list_item>
<list_item><location><page_169><loc_14><loc_63><loc_82><loc_69></location>297. Bento, M. d. C., Bertolami, O. & Sen, A. A. Generalized Chaplygin gas and CMBR constraints. Phys. Rev. D 67, 063003. arXiv: astro-ph/ 0210468 (2003).</list_item>
<list_item><location><page_169><loc_14><loc_58><loc_82><loc_62></location>298. Armendariz-Picon, C., Damour, T. & Mukhanov, V. F. k - inflation. Phys. Lett. B 458, 209-218. arXiv: hep-th/9904075 (1999).</list_item>
<list_item><location><page_169><loc_14><loc_53><loc_82><loc_57></location>299. Garriga, J. & Mukhanov, V. F. Perturbations in k-inflation. Phys. Lett. B 458, 219-225. arXiv: hep-th/9904176 (1999).</list_item>
<list_item><location><page_169><loc_14><loc_48><loc_82><loc_52></location>300. Chiba, T., Okabe, T. & Yamaguchi, M. Kinetically driven quintessence. Phys. Rev. D 62, 023511. arXiv: astro-ph/9912463 (2000).</list_item>
<list_item><location><page_169><loc_14><loc_39><loc_82><loc_47></location>301. Armendariz-Picon, C., Mukhanov, V. F. & Steinhardt, P. J. A Dynamical solution to the problem of a small cosmological constant and late time cosmic acceleration. Phys. Rev. Lett. 85, 4438-4441. arXiv: astro-ph/ 0004134 (2000).</list_item>
<list_item><location><page_169><loc_14><loc_34><loc_82><loc_38></location>302. Armendariz-Picon, C., Mukhanov, V. F. & Steinhardt, P. J. Essentials of k essence. Phys. Rev. D 63, 103510. arXiv: astro-ph/0006373 (2001).</list_item>
<list_item><location><page_169><loc_14><loc_30><loc_82><loc_33></location>303. Arkani-Hamed, N., Creminelli, P., Mukohyama, S. & Zaldarriaga, M. Ghost inflation. JCAP 04, 001. arXiv: hep-th/0312100 (2004).</list_item>
<list_item><location><page_169><loc_14><loc_23><loc_82><loc_28></location>304. Barger, V. D. & Marfatia, D. Supernova data may be unable to distinguish between quintessence and k-essence. Phys. Lett. B 498, 67-73. arXiv: astro-ph/0009256 (2001).</list_item>
<list_item><location><page_169><loc_14><loc_18><loc_82><loc_21></location>305. Li, M. & Zhang, X. k-essential leptogenesis. Phys. Lett. B 573, 20-26. arXiv: hep-ph/0209093 (2003).</list_item>
<list_item><location><page_169><loc_14><loc_11><loc_82><loc_16></location>306. Malquarti, M., Copeland, E. J., Liddle, A. R. & Trodden, M. A New view of k-essence. Phys. Rev. D 67, 123503. arXiv: astro-ph/0302279 (2003).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_170><loc_18><loc_84><loc_86><loc_90></location>307. Malquarti, M., Copeland, E. J. & Liddle, A. R. K-essence and the coincidence problem. Phys. Rev. D 68, 023512. arXiv: astro-ph/0304277 (2003).</list_item>
<list_item><location><page_170><loc_18><loc_79><loc_86><loc_83></location>308. Sen, A. Descent relations among bosonic D-branes. Int. J. Mod. Phys. A 14, 4061-4078. arXiv: hep-th/9902105 (1999).</list_item>
<list_item><location><page_170><loc_18><loc_76><loc_84><loc_78></location>309. Sen, A. Tachyon matter. JHEP 07, 065. arXiv: hep-th/0203265 (2002).</list_item>
<list_item><location><page_170><loc_18><loc_74><loc_84><loc_75></location>310. Sen, A. Rolling tachyon. JHEP 04, 048. arXiv: hep-th/0203211 (2002).</list_item>
<list_item><location><page_170><loc_18><loc_67><loc_86><loc_72></location>311. Garousi, M. R. Tachyon couplings on nonBPS D-branes and DiracBorn-Infeld action. Nucl. Phys. B 584, 284-299. arXiv: hep-th/0003122 (2000).</list_item>
<list_item><location><page_170><loc_18><loc_60><loc_86><loc_66></location>312. Bergshoeff, E. A., de Roo, M., de Wit, T. C., Eyras, E. & Panda, S. T duality and actions for nonBPS D-branes. JHEP 05, 009. arXiv: hepth/0003221 (2000).</list_item>
<list_item><location><page_170><loc_18><loc_55><loc_86><loc_59></location>313. Kluson, J. Proposal for nonBPS D-brane action. Phys. Rev. D 62, 126003. arXiv: hep-th/0004106 (2000).</list_item>
<list_item><location><page_170><loc_18><loc_50><loc_86><loc_54></location>314. Garousi, M. R. On shell S matrix and tachyonic effective actions. Nucl. Phys. B 647, 117-130. arXiv: hep-th/0209068 (2002).</list_item>
<list_item><location><page_170><loc_18><loc_45><loc_86><loc_49></location>315. Garousi, M. R. Slowly varying tachyon and tachyon potential. JHEP 05, 058. arXiv: hep-th/0304145 (2003).</list_item>
<list_item><location><page_170><loc_18><loc_41><loc_86><loc_44></location>316. Gibbons, G. W. Cosmological evolution of the rolling tachyon. Phys. Lett. B 537, 1-4. arXiv: hep-th/0204008 (2002).</list_item>
<list_item><location><page_170><loc_18><loc_34><loc_86><loc_39></location>317. Padmanabhan, T. Accelerated expansion of the universe driven by tachyonic matter. Phys. Rev. D 66, 021301. arXiv: hep - th / 0204150 (2002).</list_item>
<list_item><location><page_170><loc_18><loc_27><loc_86><loc_33></location>318. Bagla, J. S., Jassal, H. K. & Padmanabhan, T. Cosmology with tachyon field as dark energy. Phys. Rev. D 67, 063504. arXiv: astro-ph/0212198 (2003).</list_item>
<list_item><location><page_170><loc_18><loc_20><loc_86><loc_26></location>319. Abramo, L. R. W. & Finelli, F. Cosmological dynamics of the tachyon with an inverse power-law potential. Phys. Lett. B 575, 165-171. arXiv: astro-ph/0307208 (2003).</list_item>
<list_item><location><page_170><loc_18><loc_15><loc_86><loc_19></location>320. Aguirregabiria, J. M. & Lazkoz, R. Tracking solutions in tachyon cosmology. Phys. Rev. D 69, 123502. arXiv: hep-th/0402190 (2004).</list_item>
<list_item><location><page_170><loc_18><loc_8><loc_86><loc_14></location>321. Guo, Z.-K. & Zhang, Y.-Z. Cosmological scaling solutions of the tachyon with multiple inverse square potentials. JCAP 08, 010. arXiv: hep-th/0403151 (2004).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_171><loc_14><loc_84><loc_82><loc_90></location>322. Copeland, E. J., Garousi, M. R., Sami, M. & Tsujikawa, S. What is needed of a tachyon if it is to be the dark energy? Phys. Rev. D 71, 043003. arXiv: hep-th/0411192 (2005).</list_item>
<list_item><location><page_171><loc_14><loc_79><loc_82><loc_83></location>323. 't Hooft, G. Dimensional reduction in quantum gravity. Conf. Proc. C 930308, 284-296. arXiv: gr-qc/9310026 (1993).</list_item>
<list_item><location><page_171><loc_14><loc_74><loc_82><loc_78></location>324. Susskind, L. The World as a hologram. J. Math. Phys. 36, 6377-6396. arXiv: hep-th/9409089 (1995).</list_item>
<list_item><location><page_171><loc_14><loc_69><loc_82><loc_73></location>325. Bekenstein, J. D. Black holes and entropy. Phys. Rev. D 7, 2333-2346 (1973).</list_item>
<list_item><location><page_171><loc_14><loc_64><loc_82><loc_68></location>326. Bekenstein, J. D. Entropy bounds and black hole remnants. Phys. Rev. D 49, 1912-1921. arXiv: gr-qc/9307035 (1994).</list_item>
<list_item><location><page_171><loc_14><loc_57><loc_82><loc_63></location>327. Cohen, A. G., Kaplan, D. B. & Nelson, A. E. Effective field theory, black holes, and the cosmological constant. Phys. Rev. Lett. 82, 4971-4974. arXiv: hep-th/9803132 (1999).</list_item>
<list_item><location><page_171><loc_14><loc_53><loc_82><loc_56></location>328. Li, M. A Model of holographic dark energy. Phys. Lett. B 603, 1. arXiv: hep-th/0403127 (2004).</list_item>
<list_item><location><page_171><loc_14><loc_46><loc_82><loc_51></location>329. Horava, P. & Minic, D. Probable values of the cosmological constant in a holographic theory. Phys. Rev. Lett. 85, 1610-1613. arXiv: hep-th/ 0001145 (2000).</list_item>
<list_item><location><page_171><loc_14><loc_41><loc_82><loc_44></location>330. Thomas, S. D. Holography stabilizes the vacuum energy. Phys. Rev. Lett. 89, 081301 (2002).</list_item>
<list_item><location><page_171><loc_14><loc_36><loc_82><loc_39></location>331. Fischler, W. & Susskind, L. Holography and cosmology. arXiv: hepth/9806039 (June 1998).</list_item>
<list_item><location><page_171><loc_14><loc_29><loc_82><loc_35></location>332. Cataldo, M., Cruz, N., del Campo, S. & Lepe, S. Holographic principle and the dominant energy condition for Kasner type metrics. Phys. Lett. B 509, 138-142. arXiv: gr-qc/0104028 (2001).</list_item>
<list_item><location><page_171><loc_14><loc_22><loc_82><loc_28></location>333. Guberina, B., Horvat, R. & Nikolic, H. Generalized holographic dark energy and the IR cutoff problem. Phys. Rev. D 72, 125011. arXiv: astro-ph/0507666 (2005).</list_item>
<list_item><location><page_171><loc_14><loc_15><loc_82><loc_21></location>334. Wang, B., Gong, Y.-g. & Abdalla, E. Transition of the dark energy equation of state in an interacting holographic dark energy model. Phys. Lett. B 624, 141-146. arXiv: hep-th/0506069 (2005).</list_item>
<list_item><location><page_171><loc_14><loc_10><loc_82><loc_14></location>335. Huang, Q.-G. & Li, M. The Holographic dark energy in a non-flat universe. JCAP 08, 013. arXiv: astro-ph/0404229 (2004).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_172><loc_18><loc_86><loc_86><loc_90></location>336. Gong, Y.-g., Wang, B. & Zhang, Y.-Z. The Holographic dark energy revisited. Phys. Rev. D 72, 043510. arXiv: hep-th/0412218 (2005).</list_item>
<list_item><location><page_172><loc_18><loc_77><loc_86><loc_85></location>337. Nojiri, S. & Odintsov, S. D. Unifying phantom inflation with late-time acceleration: Scalar phantom-non-phantom transition model and generalized holographic dark energy. Gen. Rel. Grav. 38, 1285-1304. arXiv: hep-th/0506212 (2006).</list_item>
<list_item><location><page_172><loc_18><loc_70><loc_86><loc_76></location>338. Wang, B., Lin, C.-Y. & Abdalla, E. Constraints on the interacting holographic dark energy model. Phys. Lett. B 637, 357-361. arXiv: hep-th/ 0509107 (2006).</list_item>
<list_item><location><page_172><loc_18><loc_65><loc_86><loc_69></location>339. Pavon, D. & Zimdahl, W. Holographic dark energy and cosmic coincidence. Phys. Lett. B 628, 206-210. arXiv: gr-qc/0505020 (2005).</list_item>
<list_item><location><page_172><loc_18><loc_60><loc_86><loc_64></location>340. Zimdahl, W. & Pavon, D. Interacting holographic dark energy. Class. Quant. Grav. 24, 5461-5478. arXiv: astro-ph/0606555 (2007).</list_item>
<list_item><location><page_172><loc_18><loc_55><loc_86><loc_59></location>341. Xu, L. Holographic Dark Energy Model with Hubble Horizon as an IR Cut-off. JCAP 09, 016. arXiv: 0907.1709 [astro-ph.CO] (2009).</list_item>
<list_item><location><page_172><loc_18><loc_50><loc_86><loc_54></location>342. Rindler, W. Visual horizons in world-models. Mon. Not. Roy. Astron. Soc. 116. (Reprinted in Gen. Rel. Gravit. 34 , 133 (2002)), 662-677 (1956).</list_item>
<list_item><location><page_172><loc_18><loc_46><loc_86><loc_49></location>343. Nielsen, A. B. & Visser, M. Production and decay of evolving horizons. Class. Quant. Grav. 23, 4637-4658. arXiv: gr-qc/0510083 (2006).</list_item>
<list_item><location><page_172><loc_18><loc_43><loc_80><loc_44></location>344. Hawking, S. W. Black hole explosions. Nature 248, 30-31 (1974).</list_item>
<list_item><location><page_172><loc_18><loc_38><loc_86><loc_42></location>345. Bardeen, J. M., Carter, B. & Hawking, S. W. The Four laws of black hole mechanics. Commun. Math. Phys. 31, 161-170 (1973).</list_item>
<list_item><location><page_172><loc_18><loc_33><loc_86><loc_37></location>346. Wald, R. M. The thermodynamics of black holes. Living Rev. Rel. 4, 6. arXiv: gr-qc/9912119 (2001).</list_item>
<list_item><location><page_172><loc_18><loc_28><loc_86><loc_32></location>347. Poisson, E. ARelativist's Toolkit: The Mathematics of Black-Hole Mechanics (Cambridge University Press, Dec. 2009).</list_item>
<list_item><location><page_172><loc_18><loc_21><loc_86><loc_27></location>348. Carter, B. Black holes equilibrium states in Les Houches Summer School of Theoretical Physics: Black Holes Reprinted in Gen Relativ Gravit 42, 653-744 (2010) (1973), 57-214.</list_item>
<list_item><location><page_172><loc_18><loc_17><loc_86><loc_20></location>349. Hawking, S. W. The event horizon in Les Houches Summer School of Theoretical Physics: Black Holes (1973), 1-56.</list_item>
<list_item><location><page_172><loc_18><loc_12><loc_86><loc_15></location>350. Hawking, S. W. Black holes in general relativity. Commun. Math. Phys. 25, 152-166 (1972).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_173><loc_14><loc_84><loc_82><loc_90></location>351. Hawking, S. W. Particle Creation by Black Holes. Commun. Math. Phys. 43 (eds Gibbons, G. W. & Hawking, S. W.) [Erratum: Commun.Math.Phys. 46, 206 (1976)], 199-220 (1975).</list_item>
<list_item><location><page_173><loc_14><loc_79><loc_82><loc_83></location>352. Majumdar, P. Black hole entropy and quantum gravity. Indian J. Phys. B 73, 147. arXiv: gr-qc/9807045 (1999).</list_item>
<list_item><location><page_173><loc_14><loc_74><loc_82><loc_78></location>353. Susskind, L. & Lindesay, J. An introduction to black holes, information and the string theory revolution: The holographic universe (2005).</list_item>
<list_item><location><page_173><loc_14><loc_69><loc_82><loc_73></location>354. Gibbons, G. W. & Hawking, S. W. Cosmological Event Horizons, Thermodynamics, and Particle Creation. Phys. Rev. D 15, 2738-2751 (1977).</list_item>
<list_item><location><page_173><loc_14><loc_62><loc_82><loc_68></location>355. Cai, R.-G. Cardy-Verlinde formula and thermodynamics of black holes in de Sitter spaces. Nucl. Phys. B 628, 375-386. arXiv: hep-th/0112253 (2002).</list_item>
<list_item><location><page_173><loc_14><loc_57><loc_82><loc_61></location>356. Frolov, A. V. & Kofman, L. Inflation and de Sitter thermodynamics. JCAP 05, 009. arXiv: hep-th/0212327 (2003).</list_item>
<list_item><location><page_173><loc_14><loc_53><loc_82><loc_56></location>357. Gibbons, G. W. & Hawking, S. W. Action Integrals and Partition Functions in Quantum Gravity. Phys. Rev. D 15, 2752-2756 (1977).</list_item>
<list_item><location><page_173><loc_14><loc_48><loc_82><loc_51></location>358. Redmount, I. H. & Ruiz Ruiz, F. Thermal Equilibrium in de Sitter Space. Phys. Rev. D 39, 2289 (1989).</list_item>
<list_item><location><page_173><loc_14><loc_43><loc_82><loc_46></location>359. Davies, P. C. W. Cosmological Horizons and the Generalized Second Law of Thermodynamics. Class. Quant. Grav. 4, L225 (1987).</list_item>
<list_item><location><page_173><loc_14><loc_38><loc_82><loc_42></location>360. Davies, P. C. W., Ford, L. H. & Page, D. N. Gravitational entropy: Beyond the black hole. Phys. Rev. D 34, 1700-1707 (1986).</list_item>
<list_item><location><page_173><loc_14><loc_33><loc_82><loc_37></location>361. Calcagni, G. de Sitter thermodynamics and the braneworld. JHEP 09, 060. arXiv: hep-th/0507125 (2005).</list_item>
<list_item><location><page_173><loc_14><loc_28><loc_82><loc_32></location>362. Bak, D. & Rey, S.-J. Cosmic holography. Class. Quant. Grav. 17, L83. arXiv: hep-th/9902173 (2000).</list_item>
<list_item><location><page_173><loc_14><loc_21><loc_82><loc_27></location>363. Akbar, M. & Cai, R.-G. Thermodynamic Behavior of Friedmann Equations at Apparent Horizon of FRW Universe. Phys. Rev. D 75, 084003. arXiv: hep-th/0609128 (2007).</list_item>
<list_item><location><page_173><loc_14><loc_17><loc_82><loc_20></location>364. Bousso, R. Cosmology and the S-matrix. Phys. Rev. D 71, 064024. arXiv: hep-th/0412197 (2005).</list_item>
<list_item><location><page_173><loc_14><loc_14><loc_82><loc_15></location>365. Collins, W. Mechanics of apparent horizons. Phys. Rev. D 45, 495 (1992).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_174><loc_18><loc_82><loc_86><loc_90></location>366. Ghersi, J. T. G., Geshnizjani, G., Piazza, F. & Shandera, S. Eternal inflation and a thermodynamic treatment of Einstein's equations. JCAP 2011, 005. https://dx.doi.org/10.1088/1475-7516/2011/06/005 (2011).</list_item>
<list_item><location><page_174><loc_18><loc_77><loc_86><loc_80></location>367. Hayward, S. A., Mukohyama, S. & Ashworth, M. C. Dynamic black hole entropy. Phys. Lett. A 256, 347-350. arXiv: gr-qc/9810006 (1999).</list_item>
<list_item><location><page_174><loc_18><loc_70><loc_86><loc_76></location>368. Nielsen, A. B. & Yeom, D.-h. Spherically symmetric trapping horizons, the Misner-Sharp mass and black hole evaporation. Int. J. Mod. Phys. A 24, 5261-5285. arXiv: 0804.4435 [gr-qc] (2009).</list_item>
<list_item><location><page_174><loc_18><loc_65><loc_86><loc_69></location>369. Davies, P. C. W. Cosmological Horizons and Entropy. Class. Quant. Grav. 5, 1349 (1988).</list_item>
<list_item><location><page_174><loc_18><loc_60><loc_86><loc_64></location>370. Frolov, A. V. & Kofman, L. Inflation and de Sitter thermodynamics. JCAP 05, 009. arXiv: hep-th/0212327 (2003).</list_item>
<list_item><location><page_174><loc_18><loc_53><loc_86><loc_59></location>371. Wang, B., Gong, Y. & Abdalla, E. Thermodynamics of an accelerated expanding universe. Phys. Rev. D 74, 083520. arXiv: gr-qc/0511051 (2006).</list_item>
<list_item><location><page_174><loc_18><loc_46><loc_86><loc_52></location>372. Jiang, K.-X., Feng, T. & Peng, D.-T. Hawking radiation of apparent horizon in a FRW universe as tunneling beyond semiclassical approximation. Int. J. Theor. Phys. 48, 2112-2121 (2009).</list_item>
<list_item><location><page_174><loc_18><loc_39><loc_86><loc_45></location>373. Zhu, T. & Ren, J.-R. Corrections to Hawking-like Radiation for a Friedmann-Robertson-Walker Universe. Eur. Phys. J. C 62, 413-418. arXiv: 0811.4074 [hep-th] (2009).</list_item>
<list_item><location><page_174><loc_18><loc_34><loc_86><loc_38></location>374. Medved, A. J. M. Radiation via tunneling from a de Sitter cosmological horizon. Phys. Rev. D 66, 124009. arXiv: hep-th/0207247 (2002).</list_item>
<list_item><location><page_174><loc_18><loc_27><loc_86><loc_33></location>375. Cai, R.-G., Cao, L.-M. & Hu, Y.-P. Hawking Radiation of Apparent Horizon in a FRW Universe. Class. Quant. Grav. 26, 155018. arXiv: 0809. 1554 [hep-th] (2009).</list_item>
<list_item><location><page_174><loc_18><loc_20><loc_86><loc_26></location>376. Angheben, M., Nadalini, M., Vanzo, L. & Zerbini, S. Hawking radiation as tunneling for extremal and rotating black holes. JHEP 05, 014. arXiv: hep-th/0503081 (2005).</list_item>
<list_item><location><page_174><loc_18><loc_16><loc_86><loc_19></location>377. Visser, M. Essential and inessential features of Hawking radiation. Int. J. Mod. Phys. D 12, 649-661. arXiv: hep-th/0106111 (2003).</list_item>
<list_item><location><page_174><loc_18><loc_11><loc_86><loc_14></location>378. Parikh, M. K. & Wilczek, F. Hawking radiation as tunneling. Phys. Rev. Lett. 85, 5042-5045. arXiv: hep-th/9907001 (2000).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_175><loc_14><loc_82><loc_82><loc_90></location>379. Di Criscienzo, R., Hayward, S. A., Nadalini, M., Vanzo, L. & Zerbini, S. Hamilton-Jacobi tunneling method for dynamical horizons in different coordinate gauges. Class. Quant. Grav. 27, 015006. arXiv: 0906.1725 [gr-qc] (2010).</list_item>
<list_item><location><page_175><loc_14><loc_75><loc_82><loc_80></location>380. Cai, R.-G. & Kim, S. P. First law of thermodynamics and Friedmann equations of Friedmann-Robertson-Walker universe. JHEP 02, 050. arXiv: hep-th/0501055 (2005).</list_item>
<list_item><location><page_175><loc_14><loc_70><loc_82><loc_73></location>381. Verlinde, E. P. On the holographic principle in a radiation dominated universe. arXiv: hep-th/0008140 (Aug. 2000).</list_item>
<list_item><location><page_175><loc_14><loc_63><loc_82><loc_69></location>382. Cai, R.-G. & Myung, Y. S. Holography in radiation dominated universe with a positive cosmological constant. Phys. Rev. D 67, 124021. arXiv: hep-th/0210272 (2003).</list_item>
<list_item><location><page_175><loc_14><loc_58><loc_82><loc_62></location>383. Pavon, D., Bafaluy, J. & Jou, D. Causal Friedmann-Robertson-Walker cosmology. Class. Quant. Grav. 8, 347-360 (1991).</list_item>
<list_item><location><page_175><loc_14><loc_51><loc_82><loc_57></location>384. Akbar, M. & Cai, R.-G. Friedmann equations of FRW universe in scalartensor gravity, f(R) gravity and first law of thermodynamics. Phys. Lett. B 635, 7-10. arXiv: hep-th/0602156 (2006).</list_item>
<list_item><location><page_175><loc_14><loc_46><loc_82><loc_50></location>385. Szabados, L. B. Quasi-Local Energy-Momentum and Angular Momentum in GR: A Review Article. Living Rev. Rel. 7, 4 (2004).</list_item>
<list_item><location><page_175><loc_14><loc_41><loc_82><loc_45></location>386. Tung, R.-S. Stationary untrapped boundary conditions in general relativity. Class. Quant. Grav. 25, 085005. arXiv: 0710.4299 [gr-qc] (2008).</list_item>
<list_item><location><page_175><loc_14><loc_34><loc_82><loc_40></location>387. Kodama, H. Conserved Energy Flux for the Spherically Symmetric System and the Back Reaction Problem in the Black Hole Evaporation. Prog. Theor. Phys. 63, 1217 (1980).</list_item>
<list_item><location><page_175><loc_14><loc_27><loc_82><loc_33></location>388. Abreu, G. & Visser, M. Kodama time: Geometrically preferred foliations of spherically symmetric spacetimes. Phys. Rev. D 82, 044027. arXiv: 1004.1456 [gr-qc] (2010).</list_item>
<list_item><location><page_175><loc_14><loc_20><loc_82><loc_26></location>389. Racz, I. On the use of the Kodama vector field in spherically symmetric dynamical problems. Class. Quant. Grav. 23, 115-124. arXiv: gr- qc/ 0511052 (2006).</list_item>
<list_item><location><page_175><loc_14><loc_16><loc_82><loc_19></location>390. Bekenstein, J. D. Black holes and the second law. Lett. Nuovo Cim. 4, 737-740 (1972).</list_item>
<list_item><location><page_175><loc_14><loc_11><loc_82><loc_14></location>391. Bekenstein, J. D. Generalized second law of thermodynamics in black hole physics. Phys. Rev. D 9, 3292-3300 (1974).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_176><loc_18><loc_86><loc_86><loc_90></location>392. Shannon, C. E. A mathematical theory of communication. The Bell system technical journal 27, 379-423 (1948).</list_item>
<list_item><location><page_176><loc_18><loc_81><loc_86><loc_85></location>393. Shannon, C. E. & Weaver, W. The mathematical theory of communication. University of Illinois. Urbana (1949).</list_item>
<list_item><location><page_176><loc_18><loc_76><loc_86><loc_80></location>394. Tribus, M. & McIrvine, E. C. Energy and information. Scientific American 225, 179-190 (1971).</list_item>
<list_item><location><page_176><loc_18><loc_69><loc_86><loc_75></location>395. Jaynes, E. T. Information Theory and Statistical Mechanics. Phys. Rev. 106, 620-630. https://link.aps.org/doi/10.1103/PhysRev.106.620 (4 1957).</list_item>
<list_item><location><page_176><loc_18><loc_62><loc_86><loc_68></location>396. Jaynes, E. T. Information Theory and Statistical Mechanics. II. Phys. Rev. 108, 171-190. https://link.aps.org/doi/10.1103/PhysRev. 108.171 (2 1957).</list_item>
<list_item><location><page_176><loc_18><loc_57><loc_86><loc_61></location>397. Mottola, E. Thermodynamic Instability of de Sitter Space. Phys. Rev. D 33, 1616-1621 (1986).</list_item>
<list_item><location><page_176><loc_18><loc_53><loc_86><loc_56></location>398. Callen, H. B. Thermodynamics and an Introduction to Thermostatistics (American Association of Physics Teachers, 1998).</list_item>
<list_item><location><page_176><loc_18><loc_48><loc_86><loc_51></location>399. Kubo, R. Thermodynamics: an advanced course with problems and solutions (North-Holland Publishing Company, 1968).</list_item>
<list_item><location><page_176><loc_18><loc_43><loc_86><loc_46></location>400. Carter, A. H. Classical and statistical thermodynamics (American Association of Physics Teachers, 2000).</list_item>
<list_item><location><page_176><loc_18><loc_40><loc_70><loc_42></location>401. Muller, I. Thermodynamics (Pitman Publishing, 1985).</list_item>
<list_item><location><page_176><loc_18><loc_33><loc_86><loc_39></location>402. Thakur, S., Nautiyal, A., Sen, A. A. & Seshadri, T. R. Thawing Versus. Tracker Behaviour: Observational Evidence. Mon. Not. Roy. Astron. Soc. 427, 988-993. arXiv: 1204.2617 [astro-ph.CO] (2012).</list_item>
<list_item><location><page_176><loc_18><loc_26><loc_86><loc_32></location>403. Chakraborty, A., Banerjee, N. & Ghosh, A. Thawing versus tracker solutions: a dynamical systems approach. Gen. Rel. Grav. 51, 5. arXiv: 1811.00736 [gr-qc] (2019).</list_item>
<list_item><location><page_176><loc_18><loc_21><loc_86><loc_25></location>404. Faraoni, V. Cosmological and Black Hole Apparent Horizons ISBN: 978-3319-19239-0, 978-3-319-19240-6 (2015).</list_item>
<list_item><location><page_176><loc_18><loc_17><loc_86><loc_20></location>405. Helou, A. Dynamics of the Cosmological Apparent Horizon: Surface Gravity & Temperature. arXiv: 1502.04235 [gr-qc] (Feb. 2015).</list_item>
<list_item><location><page_176><loc_18><loc_10><loc_86><loc_15></location>406. Rani, S., Jawad, A., Nawaz, T. & Manzoor, R. Thermodynamics in modified Brans-Dicke gravity with entropy corrections. Eur. Phys. J. C 78, 58 (2018).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_177><loc_14><loc_84><loc_82><loc_90></location>407. Di Criscienzo, R., Nadalini, M., Vanzo, L., Zerbini, S. & Zoccatelli, G. On the Hawking radiation as tunneling for a class of dynamical black holes. Phys. Lett. B 657, 107-111. arXiv: 0707.4425 [hep-th] (2007).</list_item>
<list_item><location><page_177><loc_14><loc_77><loc_82><loc_83></location>408. Devi, N. C., Gonzalez, J. E. & Alcaniz, J. S. Constraining thawing and freezing models with cluster number counts. JCAP 06, 055. arXiv: 1402.2590 [astro-ph.CO] (2014).</list_item>
<list_item><location><page_177><loc_14><loc_72><loc_82><loc_76></location>409. Will, C. M. Theory and Experiment in Gravitational Physics (Cambridge University Press, 1993).</list_item>
<list_item><location><page_177><loc_14><loc_63><loc_82><loc_71></location>410. Wang, B., Abdalla, E., Atrio-Barandela, F. & Pavon, D. Dark Matter and Dark Energy Interactions: Theoretical Challenges, Cosmological Implications and Observational Signatures. Rept. Prog. Phys. 79, 096901. arXiv: 1603.08299 [astro-ph.CO] (2016).</list_item>
<list_item><location><page_177><loc_14><loc_58><loc_82><loc_62></location>411. Das, S. & Banerjee, N. An Interacting scalar field and the recent cosmic acceleration. Gen. Rel. Grav. 38, 785-794. arXiv: gr-qc/0507115 (2006).</list_item>
<list_item><location><page_177><loc_14><loc_51><loc_82><loc_57></location>412. Banerjee, N. & Das, S. A Late time acceleration of the universe with two scalar fields: Many possibilities. Mod. Phys. Lett. A 21, 2663-2670. arXiv: gr-qc/0605110 (2006).</list_item>
<list_item><location><page_177><loc_14><loc_44><loc_82><loc_50></location>413. Bhattacharya, S. & Debnath, U. Brans-Dicke Theory and Thermodynamical Laws on Apparent and Event Horizons. Can. J. Phys. 89, 883889. arXiv: 1006.2609 [gr-qc] (2011).</list_item>
<list_item><location><page_177><loc_14><loc_37><loc_82><loc_43></location>414. Hayward, S. A. Unified first law of black hole dynamics and relativistic thermodynamics. Class. Quant. Grav. 15, 3147-3162. arXiv: gr-qc/ 9710089 (1998).</list_item>
<list_item><location><page_177><loc_14><loc_30><loc_82><loc_36></location>415. Hayward, S. A., Di Criscienzo, R., Vanzo, L., Nadalini, M. & Zerbini, S. Local Hawking temperature for dynamical black holes. Class. Quant. Grav. 26, 062001. arXiv: 0806.0014 [gr-qc] (2009).</list_item>
<list_item><location><page_177><loc_14><loc_23><loc_82><loc_29></location>416. Mimoso, J. P. & Pavón, D. Considerations on the thermal equilibrium between matter and the cosmic horizon. Phys. Rev. D 94, 103507. arXiv: 1610.07788 [gr-qc] (2016).</list_item>
<list_item><location><page_177><loc_14><loc_16><loc_82><loc_22></location>417. Banerjee, A, Banerjee, N & Santos, N. Anisotropic cosmological model in Nordtvedt's scalar-tensor theory of gravitation. J. Math. Phys 26, 3125-3130 (1985).</list_item>
<list_item><location><page_177><loc_14><loc_11><loc_82><loc_15></location>418. Banerjee, N. An isotropic cosmological model in general scalar tensor theory. Pramana 24, 701-706 (1985).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_178><loc_18><loc_82><loc_86><loc_90></location>419. Knop, R. A. et al. New constraints on Omega(M), Omega(lambda), and w from an independent set of eleven high-redshift supernovae observed with HST. Astrophys. J. 598, 102. arXiv: astro-ph/0309368 (2003).</list_item>
<list_item><location><page_178><loc_18><loc_77><loc_86><loc_80></location>420. Ellis, G. F. R. & Madsen, M. S. Exact scalar field cosmologies. Class. Quant. Grav. 8, 667-676 (1991).</list_item>
<list_item><location><page_178><loc_18><loc_70><loc_86><loc_76></location>421. Saini, T. D., Raychaudhury, S., Sahni, V. & Starobinsky, A. A. Reconstructing the cosmic equation of state from supernova distances. Phys. Rev. Lett. 85, 1162-1165. arXiv: astro-ph/9910231 (2000).</list_item>
<list_item><location><page_178><loc_18><loc_63><loc_86><loc_69></location>422. Sahlen, M., Liddle, A. R. & Parkinson, D. Direct reconstruction of the quintessence potential. Phys. Rev. D 72, 083511. arXiv: astro - ph / 0506696 (2005).</list_item>
<list_item><location><page_178><loc_18><loc_56><loc_86><loc_62></location>423. Holsclaw, T. et al. Nonparametric Reconstruction of the Dark Energy Equation of State. Phys. Rev. D 82, 103502. arXiv: 1009 . 5443 [astro-ph.CO] (2010).</list_item>
<list_item><location><page_178><loc_18><loc_49><loc_86><loc_55></location>424. Holsclaw, T. et al. Nonparametric Dark Energy Reconstruction from Supernova Data. Phys. Rev. Lett. 105, 241302. arXiv: 1011 . 3079 [astro-ph.CO] (2010).</list_item>
<list_item><location><page_178><loc_18><loc_42><loc_86><loc_48></location>425. Holsclaw, T. et al. Nonparametric Reconstruction of the Dark Energy Equation of State from Diverse Data Sets. Phys. Rev. D 84, 083501. arXiv: 1104.2041 [astro-ph.CO] (2011).</list_item>
<list_item><location><page_178><loc_18><loc_35><loc_86><loc_41></location>426. Crittenden, R. G., Zhao, G.-B., Pogosian, L., Samushia, L. & Zhang, X. Fables of reconstruction: controlling bias in the dark energy equation of state. JCAP 02, 048. arXiv: 1112.1693 [astro-ph.CO] (2012).</list_item>
<list_item><location><page_178><loc_18><loc_28><loc_86><loc_34></location>427. Nair, R., Jhingan, S. & Jain, D. Exploring scalar field dynamics with Gaussian processes. JCAP 01, 005. arXiv: 1306.0606 [astro-ph.CO] (2014).</list_item>
<list_item><location><page_178><loc_18><loc_21><loc_86><loc_27></location>428. Starobinsky, A. A. How to determine an effective potential for a variable cosmological term. JETP Lett. 68, 757-763. arXiv: astro - ph / 9810431 (1998).</list_item>
<list_item><location><page_178><loc_18><loc_14><loc_86><loc_20></location>429. Huterer, D. & Turner, M. S. Prospects for probing the dark energy via supernova distance measurements. Phys. Rev. D 60, 081301. arXiv: astro-ph/9808133 (1999).</list_item>
<list_item><location><page_178><loc_18><loc_9><loc_86><loc_13></location>430. Huterer, D. & Turner, M. S. Probing the dark energy: Methods and strategies. Phys. Rev. D 64, 123527. arXiv: astro-ph/0012510 (2001).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_179><loc_14><loc_84><loc_82><loc_90></location>431. Gong, Y.-G. & Wang, A. Reconstruction of the deceleration parameter and the equation of state of dark energy. Phys. Rev. D 75, 043520. arXiv: astro-ph/0612196 (2007).</list_item>
<list_item><location><page_179><loc_14><loc_75><loc_82><loc_83></location>432. Yu-Ting, W., Li-Xin, X., Jian-Bo, L. & Yuan-Xing, G. Reconstructing dark energy potentials from parameterized deceleration parameters. Chinese Physics B 19, 019801. https://dx.doi.org/10.1088/16741056/19/1/019801 (2010).</list_item>
<list_item><location><page_179><loc_14><loc_68><loc_82><loc_73></location>433. Mamon, A. A. & Das, S. A parametric reconstruction of the deceleration parameter. Eur. Phys. J. C 77, 495. arXiv: 1610.07337 [gr-qc] (2017).</list_item>
<list_item><location><page_179><loc_14><loc_59><loc_82><loc_66></location>434. Naik, D. M., Kavya, N. S., Sudharani, L. & Venkatesha, V. Modelindependent cosmological insights from three newly reconstructed deceleration parameters with observational data. Phys. Lett. B 844, 138117 (2023).</list_item>
<list_item><location><page_179><loc_14><loc_52><loc_82><loc_57></location>435. Xu, L.-I., Zhang, C.-W., Chang, B.-R. & Liu, H.-Y. Constraints to deceleration parameters by recent cosmic observations. Mod. Phys. Lett. A 23, 1939-1948. arXiv: astro-ph/0701519 (2008).</list_item>
<list_item><location><page_179><loc_14><loc_47><loc_82><loc_50></location>436. Luongo, O. Dark energy from a positive jerk parameter. Mod. Phys. Lett. A 28, 1350080 (2013).</list_item>
<list_item><location><page_179><loc_14><loc_40><loc_82><loc_46></location>437. Rapetti, D., Allen, S. W., Amin, M. A. & Blandford, R. D. A kinematical approach to dark energy studies. Mon. Not. Roy. Astron. Soc. 375, 15101520. arXiv: astro-ph/0605683 (2007).</list_item>
<list_item><location><page_179><loc_14><loc_31><loc_82><loc_39></location>438. Zhai, Z.-X., Zhang, M.-J., Zhang, Z.-S., Liu, X.-M. & Zhang, T.-J. Reconstruction and constraining of the jerk parameter from OHD and SNe Ia observations. Phys. Lett. B 727, 8-20. arXiv: 1303.1620 [astro-ph.CO] (2013).</list_item>
<list_item><location><page_179><loc_14><loc_24><loc_82><loc_29></location>439. Mukherjee, A. & Banerjee, N. Parametric reconstruction of the cosmological jerk from diverse observational data sets. Phys. Rev. D 93, 043002. arXiv: 1601.05172 [gr-qc] (2016).</list_item>
<list_item><location><page_179><loc_14><loc_17><loc_82><loc_22></location>440. Mukherjee, A. & Banerjee, N. In search of the dark matter dark energy interaction: a kinematic approach. Class. Quant. Grav. 34, 035016. arXiv: 1610.04419 [astro-ph.CO] (2017).</list_item>
<list_item><location><page_179><loc_14><loc_10><loc_82><loc_15></location>441. Zhang, Y., Ma, Y.-b., Du, Y.-Z., Li, H.-F. & Zhang, L.-C. Phase Transition and Entropic Force in Reissner-Nordström-de Sitter Spacetime. Adv. High Energy Phys. 2022, 7376502 (2022).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_180><loc_18><loc_84><loc_86><loc_90></location>442. Zhao, R. & Zhang, L.-C. Entropy of higher-dimensional charged de Sitter black holes and phase transition. Commun. Theor. Phys. 70, 578. arXiv: 1710.07225 [gr-qc] (2018).</list_item>
<list_item><location><page_180><loc_18><loc_77><loc_86><loc_83></location>443. Ali, M. S. & Ghosh, S. G. Thermodynamics and phase transition of rotating regular-de Sitter black holes. Eur. Phys. J. Plus 137, 486. arXiv: 1906.11284 [gr-qc] (2022).</list_item>
<list_item><location><page_180><loc_18><loc_70><loc_86><loc_76></location>444. Zhang, Y., Wang, W.-q., Ma, Y.-b. & Wang, J. Phase Transition and Entropy Force between Two Horizons in (n+2)-Dimensional de Sitter Space. Adv. High Energy Phys. 2020, 7263059 (2020).</list_item>
<list_item><location><page_180><loc_18><loc_65><loc_86><loc_69></location>445. Ma, Y.-B. et al. Entropy of the electrically charged hairy black holes. Eur. Phys. J. C 78, 763 (2018).</list_item>
<list_item><location><page_180><loc_18><loc_60><loc_86><loc_64></location>446. Ferreira, P. C. & Pavon, D. Thermodynamics of nonsingular bouncing universes. Eur. Phys. J. C 76, 37. arXiv: 1509.03725 [gr-qc] (2016).</list_item>
<list_item><location><page_180><loc_18><loc_53><loc_86><loc_59></location>447. Mukherjee, P. & Banerjee, N. Nonparametric reconstruction of interaction in the cosmic dark sector. Phys. Rev. D 103, 123530. arXiv: 2105. 09995 [astro-ph.CO] (2021).</list_item>
<list_item><location><page_180><loc_18><loc_48><loc_86><loc_52></location>448. Bhandari, P., Haldar, S. & Chakraborty, S. Interacting dark energy model and thermal stability. Eur. Phys. J. C 77, 840 (2017).</list_item>
<list_item><location><page_180><loc_18><loc_44><loc_86><loc_47></location>449. Gong, Y., Wang, B. & Wang, A. Thermodynamical properties of the Universe with dark energy. JCAP 01, 024. arXiv: gr-qc/0610151 (2007).</list_item>
<list_item><location><page_180><loc_18><loc_39><loc_86><loc_42></location>450. Izquierdo, G. & Pavon, D. Dark energy and the generalized second law. Phys. Lett. B 633, 420-426. arXiv: astro-ph/0505601 (2006).</list_item>
<list_item><location><page_180><loc_18><loc_34><loc_86><loc_37></location>451. Gong, Y., Wang, B. & Wang, A. On thermodynamical properties of dark energy. Phys. Rev. D 75, 123516. arXiv: gr-qc/0611155 (2007).</list_item>
<list_item><location><page_180><loc_18><loc_27><loc_86><loc_33></location>452. Zhang, Y., Yi, Z.-L., Zhang, T.-J. & Liu, W. Thermodynamical properties of dark energy with the equation of state omega=omega(0)+omega(1)z. Phys. Rev. D 77, 023502. arXiv: 0709.2745 [astro-ph] (2008).</list_item>
<list_item><location><page_180><loc_18><loc_22><loc_86><loc_26></location>453. Pereira, S. H. & Lima, J. A. S. On Phantom Thermodynamics. Phys. Lett. B 669, 266-270. arXiv: 0806.0682 [gr-qc] (2008).</list_item>
<list_item><location><page_180><loc_18><loc_15><loc_86><loc_21></location>454. Mohseni Sadjadi, H. & Jamil, M. Generalized second law of thermodynamics for FRW cosmology with logarithmic correction. EPL 92, 69001. arXiv: 1002.3588 [gr-qc] (2010).</list_item>
<list_item><location><page_180><loc_18><loc_10><loc_86><loc_14></location>455. Padmanabhan, T. Statistical Mechanics of Gravitating Systems. Phys. Rept. 188, 285 (1990).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_181><loc_14><loc_84><loc_82><loc_90></location>456. Luongo, O. & Quevedo, H. Cosmographic study of the universe's specific heat: A landscape for Cosmology? Gen. Rel. Grav. 46, 1649. arXiv: 1211.0626 [gr-qc] (2014).</list_item>
<list_item><location><page_181><loc_14><loc_79><loc_82><loc_83></location>457. Pavon, D. & Wang, B. Le Chatelier-Braun principle in cosmological physics. Gen. Rel. Grav. 41, 1-5. arXiv: 0712.0565 [gr-qc] (2009).</list_item>
<list_item><location><page_181><loc_14><loc_72><loc_82><loc_78></location>458. Cai, R.-G. & Kim, S. P. First law of thermodynamics and Friedmann equations of Friedmann-Robertson-Walker universe. JHEP 02, 050. arXiv: hep-th/0501055 (2005).</list_item>
</document>
[
{
"title": "Thermodynamical Aspects of Some Cosmological Models",
"content": "By Tanima Duary Roll No.: 14IP021 Supervisor: Prof. Narayan Banerjee Department of Physical Sciences Indian Institute of Science Education and Research Kolkata A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Physical Sciences at Indian Institute of Science Education and Research Kolkata April, 2024",
"pages": [
1
]
},
{
"title": "Declaration by the student",
"content": "I, Ms. Tanima Duary , Registration No. 14IP021 dated 24th July 2014 , a student of the Department of Physical Sciences of the Integrated PhD Programme of Indian Institute of Science Education and Research Kolkata (IISER Kolkata), hereby declare that this thesis is my own work and, to the best of my knowledge, it neither contains materials previously published or written by any other person, nor has it been submitted for any degree/diploma or any other academic award anywhere before. I have used the originality checking service to prevent inappropriate copying. I also declare that all copyrighted material incorporated into this thesis is in compliance with the Indian Copyright Act, 1957 (amended in 2012) and that I have received written permission from the copyright owners for my use of their work. I hereby grant permission to IISER Kolkata to store the thesis in a database which can be accessed by others. Date: Tanima Duary Department of Physical Sciences Indian Institute of Science Education and Research Kolkata Mohanpur 741246, West Bengal, India",
"pages": [
3
]
},
{
"title": "Certificate from the Supervisor",
"content": "This is to certify that the thesis entitled 'Thermodynamical Aspects of Some Cosmological Models' submitted by Ms. Tanima Duary , Registration No. 14IP021 dated 24th July 2014 , a student of the Department of Physical Sciences of the Integrated PhD Programme of IISER Kolkata, is based upon her own research work under my supervision. I also certify, to the best of my knowledge, that neither the thesis nor any part of it has been submitted for any degree/diploma or any other academic award anywhere before. In my opinion, the thesis fulfils the requirement for the award of the degree of Doctor of Philosophy. Date: Prof. Narayan Banerjee Professor Department of Physical Sciences Indian Institute of Science Education and Research Kolkata Mohanpur 741246, West Bengal, India",
"pages": [
5
]
},
{
"title": "Acknowledgements",
"content": "I extend my heartfelt gratitude to all those who played important roles in making this journey possible. Without their unwavering support and encouragement, the completion of this thesis would not have been possible. I want to take this moment to express my deep appreciation for their contributions. First and foremost, I wish to express my sincere and deepest gratitude to my supervisor, Prof. Narayan Banerjee, for being a guiding light throughout mydoctoral journey. He charted the course of my research with his visionary guidance and profound wisdom. I am truly indebted to him for his boundless patience and efforts, which have not only made this thesis possible but also enriched my understanding of the subject. I express my gratitude to him for maintaining faith in my abilities even during times when I doubted myself. Having the privilege of working under his guidance has been a rare and invaluable opportunity. I am grateful for the incredible teachings, enlightening discussions, and invaluable advice that have consistently motivated me on this scientific expedition. I am thankful for the time and effort he dedicated to reviewing drafts, providing constructive feedback, and offering valuable suggestions that significantly enhanced the overall quality of this thesis. I extend my heartfelt thanks to him for providing guidance and support when I felt directionless, helping me discover the strength to overcome challenges. I am profoundly grateful to Dr. Ananda Dasgupta, whose exceptional teaching prowess and constant support have left an indelible mark on my academic journey. His teaching style, characterized by clarity, enthusiasm, and a genuine passion for the subject matter, has made learning not just a scholarly pursuit but also an enriching experience. His wisdom and insight have been a source of inspiration, instilling in me a resilience that extends beyond academic pursuits. His impact on my education and personal growth is immeasurable, and I am privileged to have had the opportunity to learn under his tutelage. I extend my gratitude to the members of my research progress committee, Dr. Golam Mortuza Hossain and Prof. Rajesh Kumble Nayak, for their valuable insights and recommendations. I express my gratefulness to Dr. Koushik Dutta for his encouragement and I am thankful for providing me with the opportunity to serve as a volunteer at the 32nd IAGRG conference in 2022. This experience has been enriching. I am thankful to all the faculty members of Department of Physical Sciences, IISER Kolkata. I would like to express my heartfelt thanks to Sangita di, Munna da, and Ipsita di from the departmental office for their invaluable assistance with all official matters. My sincere appreciation goes to the librarian and Assistant Librarian at IISER-K Library for their cooperative support. Gratitude also to the individuals from DOAA, DOSA, DORD, CCC, SAC, and the Medical unit for their assistance. I would like to specially thank Ms. Saberi Roy Choudhury and Mr. Arun Dutta for helping us in CSIR fellowship matters. In the close-knit community of IISER Kolkata, Shibendu, Prashanti, Purba, Shreya, Basabendra, and Sourav have been more than friends; they are cherished companions who have shared in the joys and difficulties of academic pursuits. Their friendship has added richness to my experience at IISER Kolkata, and I am truly fortunate to have them as the closest allies. With deep reverence, I extend my regards and profound gratitude to my beloved friend, the late Subhadip Roy. Each passing day is a poignant reminder of his absence, and not a moment goes by when I do not miss the warmth of his friendship. I would like to thank my seniors- Shantanu da, Subhajit da, Chiranjeeb da, Ankan da, Soumya da, Sachin da, Avijit da, Srijita di and Anushree di for the precious pieces of advice they generously shared. I would also like to thank Sampurna, Debraj, Medha, Siddhartha, Abhirup, Toushik da, Priyanka di, Diganta, Budhaditya, Poulomi, Saikat, Arkayan, Amulya, Chiranjit, Swarup, Brotoraj, Soumya, Debajyoti, Narayan, Arnab, Samit, Roshan, Kakali, Madhura, Soumi, Ananya, Fareeha, Debanajana and Lucky for making this journey memorable. I wish to thank Branali, Rajrupa, Poulami, Beetihotra, Navonil, Sayantan, and Siddharth. It was fun working with you side by side at the IAGRG conference. I am thankful to all my friends from Scottish Church College. I am grateful to my grandparents (dadu, dida), maternal uncle (mama), aunty (mami), sister Sushree, and little brother Sashreek, for the affection they have bestowed upon me. I express my deepest gratitude to my parents, whose love, encouragement, and support which has been a steadfast anchor during both the peaks and valleys of my journey. Their encouragement and presence have been invaluable, providing solace in challenging times and magnifying the joy in moments of success. Their sacrifices, guidance, and belief in my potential have shaped the person I am today. I am truly blessed to have such caring and supportive parents by my side. I want to extend my heartfelt acknowledgment to my little brother, Moni, whose presence and enthusiasm have brought joy and inspiration to my life. His infectious energy, and genuine camaraderie have been a constant source of encouragement throughout my journey. I am grateful for the special bond we share, and I appreciate the positive impact he has had on my life. I express my gratitude to my dearest friend, Aritra, for being by my side through thick and thin. In the symphony of life, grateful for the serendipitous notes that brought him into the melody of my journey. Across the tapestry of time and the vastness of space, we share an epoch and a planet, a harmonious rhyme. I extend my sincere gratitude to the Council of Scientific and Industrial Research, India, for granting financial assistance through the CSIR-NET fellowship (Award No. 09/921(0171)/2017-EMR-I). I would like to express my appreciation for the enchanting campus of IISER Kolkata, characterized by its captivating beauty, vibrant sunset skies, and picturesque walking paths. To Moni and my adoring parents...",
"pages": [
7,
8,
9,
11
]
},
{
"title": "Abstract",
"content": "This thesis is focused on the thermodynamic analysis of cosmological models, specially the models that explain late-time cosmic acceleration. The cosmological principle says that the universe exhibits spatial homogeneity and isotropy. To describe it we consider the Friedmann-Lemaître-RobertsonWalker (FLRW) metric. A thorough evaluation of the feasibility of the models was conducted through the application of the Generalized Second Law (GSL). This law says that the overall entropy, i.e., the sum of the entropy of the horizon and the fluid enclosed within the horizon, should never decrease. Considering the dynamic nature of the universe, our methodology focused on the apparent horizon, instead of the event horizon. Within this framework, we have considered a condition of thermodynamic equilibrium between the apparent horizon and the fluid contained within it. In this state of equilibrium, we have considered the Hayward-Kodama temperature as the temperature associated with the apparent horizon. The first chapter contains concise overview of cosmology. Chapter 2 goes deeper into the thermodynamics applied to cosmology. It focuses more on the Generalized Second Law of Thermodynamics and explains it in more detail. Furthermore, we go into extensive details regarding Hayward-Kodama temperature. This chapter also contains a detail discussion about apparent horizon. The conditions required for thermodynamic stability has been discussed in this chapter. In chapter 3, we conduct a thermodynamic comparison between quintessence models involving thawing and freezing scenarios. We have considered an ansatz on the energy density of the scalar field, which is picked up from the literature. The motivation for picking the ansatz was that, by choosing values of just one parameter, we can get either thawing or freezing behaviour. Both of these models are observed to violate the Generalized Second Law of Thermodynamics. Nevertheless, in the case of freezing models, there is still a possible way to resolve this, as this violation occurs in the distant past, deep within the radiation-dominated era, a period where a conventional scalar field model combined with pressureless matter is not an accurate representation of the matter content. In contrast, the thawing model exhibits a violation of GSL, manifesting as a finite future breakdown. Therefore, we conclude that the freezing models are favoured compared to the thawing ones on the considerations of GSL viability. In chapter 4, we scrutinize Brans-Dicke cosmological models within the context of a spatially isotropic and homogeneous universe, evaluating their compatibility with the GSL. Our investigation is carried out within the Einstein frame. We find that in dust era, these models exhibit thermodynamic feasibility when the Brans-Dicke parameter ω assumes negative values. This range has strong alignment with the range that is required for the recent observations of the cosmic acceleration. In chapter 5, we explore the thermodynamic viability of a selection of dark energy models, which have been reconstructed using the cosmological jerk parameter. Our investigation involves the adoption of models previously documented in the literature. These models are categorized into two groups based on the presence or absence of interactions in the dark sector. We employ the GSL as a diagnostic tool for our analysis. In an attempt to capture the dynamic nature of spacetime, we replace the Hawking temperature with the Hayward-Kodama temperature. Our results indicate that, dependent on the chosen parametrization ansatz for jerk, the total entropy exhibits a time-increasing trend. This suggests the potential existence of viable models within this framework. This trend persists even when there is interaction in the dark sector. In chapter 6, we have considered a model in spatially flat FRW spacetime, that mimics the characteristics of Λ CDM model and checked the thermodynamic stability. In this chapter also we have utilized the Hayward-Kodama temperature as the temperature of the apparent horizon. Assuming the thermal equilibrium between the apparent horizon and the fluid inside the horizon, we investigated the thermodynamic stability of the matter composition within the universe and found out that it lacks the thermodynamic stability. We found out an interesting result while calculating the heat capacity at constant volume ( C V ). It is shown that the transition from the decelerated to the accelerated cosmic expansion is a second-order thermodynamic phase transition, while the deceleration parameter q serves as the order parameter. In chapter 7, we reach the epilogue of this thesis, where we not only provide our final conclusions but also briefly discussed the aspects of the work presented in this dissertation and future prospects.",
"pages": [
13,
14,
15
]
},
{
"title": "Preface",
"content": "The research presented in this dissertation was conducted at the Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata, India. The initial chapter encapsulates a preamble to the realm of cosmology, centering upon the different models to explain cosmic acceleration. The next chapter (chapter 2) focuses on the intricate interrelation linking cosmology and the principles of thermodynamics. The succeeding chapters are grounded on the subsequent scholarly articles:",
"pages": [
17
]
},
{
"title": "Contents",
"content": "7 Conclusions and Outlook 119",
"pages": [
20,
21
]
},
{
"title": "List of Acronyms",
"content": "Λ CDM L ambda C old D ark M atter BD B rans- D icke CMBR C osmic M icrowave B ackground R adiation FRW F riedmann- R obertson- W alker GTR G eneral T heory of R elativity GSLT G eneralised S econd L aw of T hermodynamics KGequation K lein G ordon equation EoS parameter E quation o f S tate parameter BDT B rans- D icke T heory NMCSTT N on- M inimally C oupled S calar- T ensor T heories DDE Dy namical D ark E nergy CPL model C hevallier - P olarski- L inder model. CMB C osmic M icrowave B ackground. HDE H olographic D ark E nergy.",
"pages": [
23
]
},
{
"title": "1.1.1 Brief chronological background:",
"content": "Einstein's General Theory of Relativity is a revolutionary framework in physics that transformed our understanding of gravity and hence, the evolution of the universe. Proposed by Albert Einstein in 1915 [1], it represents one of the most significant intellectual achievements in human history. At its core, the theory suggests that gravity is not a force as traditionally understood but rather a curvature in the fabric of spacetime caused by the presence of mass and energy. According to General Relativity, massive objects such as stars and planets distort the geometry of spacetime, causing objects to follow curved paths. The theory provides a new mathematical description of gravity, utilizing a set of equations that relate the distribution of matter and energy to the curvature of spacetime. General Theory of Relativity provides a theoretical framework for examining the structure and dynamics of the entire universe, and Einstein devised this framework in 1917 [2]. At that time, physicists believed the cosmos was stationary and not evolving, a celestial clockwork mechanism that would run forever. However, this prevailing view was refuted by Einstein's equations in general relativity, which indicated that the universe could either expand or contract. To account for this, Einstein introduced a modification to his equations by including a cosmological constant ( Λ ), which acted as a repulsive force to counterbalance gravity and maintain a stable universe. Willem de Sitter proposed the maximally symmetric vacuum solution of Einstein's field equations with a positive cosmological constant Λ [3, 4]. Karl Schwarzschild derived a solution to Einstein's equations that described the spacetime curvature around a massive object, such as a star or a black hole [5]. This solution laid the foundation for our understanding of gravitational collapse and singularities. In 1919, Arthur Eddington led an expedition to observe a solar eclipse and confirmed Einstein's prediction that light rays would bend in the presence of massive objects [6]. In 1922, Alexander Friedmann introduced the idea of an expanding universe in his paper \"On the Curvature of Space\" [7]. He derived a set of mathematical solutions to Einstein's equations that described a dynamic cosmos, where the matter was in motion and the universe itself was evolving over time. Friedmann's work laid the groundwork for modern cosmology by challenging the prevailing belief in a static universe and introducing the concept of cosmic expansion. It provided a theoretical basis for the subsequent development of the Big Bang model and set the stage for further investigations into the nature, origin, and evolution of the universe. In 1929, Edwin Hubble made a significant discovery that revolutionized our understanding of the universe. He studied distant galaxies and their light spectra, which contained characteristic features called spectral lines [8]. These lines could be used to determine the motion of an object. By examining the spectra of galaxies, Hubble noticed a peculiar pattern: the spectral lines were consistently shifted towards longer wavelengths, known as a redshift. Based on the known Doppler effect, i.e., the redshift of light, Hubble concluded that the galaxies were moving away from each other. Moreover, he observed that the more distant a galaxy was, the greater its redshift. This finding implied that if galaxies were moving away from us, then at some point in the past, they must have been closer together. Based on Hubble's discovery, Georges Lemaître suggested that the universe began from an extremely hot and dense state, which he referred to as the primeval atom or the cosmic egg . He theorized that this primordial atom contained all the matter and energy in the universe and exploded in an event he called the explosion of the primeval atom . This explosion marked the beginning of the universe, a moment commonly referred to as the Big Bang . Though he proposed the model in 1931 [9], this gained wider recognition in the scientific community later, with additional evidence supporting the Big Bang model emerged, such as the discovery of the cosmic microwave background radiation, which is considered a remnant of the early stages of the universe. In 1948, Alpher, Bethe, Gamow, and Herman developed the first detailed model of Big Bang nucleosynthesis, which provided an explanation for the formation of light elements, such as hydrogen and helium, in the early universe [10-12]. Alpher, Gamow, and Herman's model proposed that in the first few minutes after the Big Bang, the universe was extremely hot and dense, with temperatures of billions of degrees. During this time, the conditions were favorable for nuclear reactions to take place. According to their calculations, the extreme temperatures and densities allowed for the fusion of protons and neutrons to form light elements, primarily hydrogen and helium. In 1965, Arno Penzias and Robert Wilson made a groundbreaking discovery known as the cosmic microwave background radiation (CMB) or the Penzias-Wilson radiation [13]. Penzias and Wilson were astronomers working at Bell Labs in New Jersey, conducting experiments using a large horn antenna originally built for satellite communication. However, no matter what they did, they encountered a persistent noise signal in their measurements. They could not eliminate this noise, which seemed to be coming from every direction in the sky. It was unknown to Penzias and Wilson that their observations were coinciding with the work of Robert Dicke, Jim Peebles, and others at Princeton University [14, 15]. Theoretical work at Princeton had predicted the existence of a faint radiation left over from the early stages of the universe, known as the cosmic microwave background. When Penzias and Wilson learned about the Princeton team's prediction, they realized that the noise they were observing matched the characteristics of the cosmic microwave background radiation (CMB). This radiation is the remnant of the intense heat and energy that filled the universe shortly after the Big Bang. The discovery of the CMB [16] provided strong evidence in support of the Big Bang model. According to the Big Bang model, the universe expanded rapidly from an extremely hot and dense state. The observations indicated that different regions of the universe that were now far apart appeared to have the same temperature, properties and flat. To address this issue, Alan Guth(1981) proposed the concept of cosmic inflation [17] and suggested that a brief period of exponential expansion occurred in the very early universe, causing it to grow exponentially larger in size. This rapid expansion would have smoothed out the initial irregularities and made the universe homogeneous and isotropic, providing an explanation for the observed uniformity across vast regions of space. The cosmic inflation also indicates that quantum fluctuations during inflation could lead to the formation of tiny density perturbations in the early universe. These fluctuations served as seeds for the later formation of galaxies, clusters of galaxies, and other cosmic structures. Guth's work had a profound impact on our understanding of the universe. Subsequent observations, such as the measurement of the cosmic microwave background radiation by the Cosmic Background Explore(COBE) [18] and The Wilkinson Microwave Anisotropy Probe(WMAP) satellites [19], provided strong evidence in support of cosmic inflation and moreover the Big Bang model. The discovery of cosmic acceleration in late 1990s posed a threat to our understanding of the universe. Two independent teams of astronomers, the High-Z Supernova Search Team (Schmidt et al ., 1998) [20] and the Supernova Cosmology Project (Perlmutter et al ., 1999) [193] provided supporting data that Hubble expansion is accelerating over time, while studying distant supernovae, specifically Type Ia supernovae(SNIa) as standard candles to measure the correlation between distance and redshift. This discovery was contrary to the prevailing belief that the gravitational pull of matter should be causing the expansion to decelerate. Adam G. Riess and his colleagues analysed the data provided by the previously mentioned groups in the article titled \"Observational Evidence from Supernovae for an Accelerating universe and a Cosmological Constant\" [192] and concluded that the expansion of the universe was accelerating rather than slowing down and proposed the presence of a cosmological constant (i.e., the vacuum energy density) in the equations of general relativity. The cosmological constant acts as a repulsive force, causing the expansion of the universe to accelerate over time. This discovery of an accelerating universe and the need for dark energy revolutionized cosmology and deeply influenced our comprehension of the basic nature of the universe. Later, WMAP team discovered that approximately 71.4% of the universe consists of dark energy [21]. Planck space observatory, designed to map the anisotropies of the cosmic microwave background (CMB) at infrared and microwave frequencies, revealed that the ordinary matter makes up only about 4-5% only, with dark matter(25-26%) and dark energy(69-70%) accounting for the remaining 95% [22]. This dominating dark energy is thought to exert a repulsive gravity that counteracts the attractive force of gravity, causing galaxies and other cosmic structures to move away from each other at an accelerating rate. There are several theories and hypotheses regarding the source of this exotic dark energy, but no definitive explanation has been established. Such theories include two most talked about categories: (i) cosmological constant, and (ii) quintessence field. The other way of explaining this accelerated expansion is to modify the theory of gravity. Some eminent researchers of this era have proposed that our understanding of gravity on cosmological scales may need modification. Modified theories of gravity, such as the f(R) theory, Brans-Dicke theory attempt to explain the observed cosmic acceleration without invoking dark energy. These theories suggest that gravity behaves differently on large scales and can account for the accelerated expansion. But there is no certain observational evidence to choose one theoretical model. There are reverse engineering techniques, known as the Reconstruction of dark energy models from observational datas to understand the nature of dark energy. However, theorists use several methods to compare the available theories of dark energy in literature and try to find out which one is preferable. In this thesis, we have used the Generalised Second Law of Thermodynamics to study different cosmological models. Before delving into the specifics of that topic, initially, a concise aspect related to cosmology is explored here.",
"pages": [
29,
30,
31,
32,
33
]
},
{
"title": "1.1.2 Geometry and Dynamics of the Spacetime:",
"content": "In 1915, Einstein's work on a new theory of gravity did not just result in a different way to think about forces or gravitational fields. It led to a big change in how we see space and time, shaking up our understanding in a major way. Einstein recognized that the empirical observation of all objects falling with the same acceleration in a gravitational field naturally pointed towards an interpretation of gravity based on the curvature of spacetime. According to the General theory of relativity, gravity is manifested by the interaction of spacetime curvature and matter [23-31]. Einstein's field equations give the mathematical manifestation of this theory. By varying Einstein-Hilbert action with respect to g µν , we get the Einstein's field equation, In the above equation, g µν denotes the metric tensor and g ≡ det g µν ; g µν is symmetric and throughout the thesis we follow the signature convention (,+,+,+). Here, R is the scalar curvature, known as Ricci scalar and defined as, R = g µν R µν . R µν is the Ricci tensor that captures the local curvature of spacetime at each point. G ≈ 6.7 × 10 -11 m 3 kg -1 s -2 represents the Newtonian constant of gravitation. In the action (1.1), the Lagrangian density is divided in two parts: (i) gravitational part: R and (ii) matter part: L m . The volume element is √ -gd 4 x . Equation (1.2) signifies the connection between geometry, contributed by G µν and matter, contributed by T µν . Energy-momentum tensor or stress-energy tensor T µν is defined as, describes the distribution of mass, energy, and momentum in spacetime and depends on the specific physical system under consideration.",
"pages": [
33,
34
]
},
{
"title": "1.1.3 Cosmology:",
"content": "The foundation of the contemporary cosmological model lies in the Cosmological Principle, asserting that the universe exhibits spatial homogeneity i.e., translationally invariant (Copernican principle) and isotropy i.e., rotationally invariant at scales much larger than the galaxy cluster [26, 32]. This suggests that the space is maximally symmetric. Thus it becomes feasible to depict the universe using a lucid geometric framework by taking 1+3 foliation of the spacetime. The invariant interval [33], does not depend on the choice of co-ordinates x µ s ( x 0 denotes cosmic time t and x 1 , x 2 , x 3 are space co-ordinates). The curvature of the spatial part can be of three forms: (i) positive curvature, (ii) zero curvature or (iii) negative curvature. A possible way to represent the metric in 1+3 foliation is, where a ( t ) is known as the cosmic scale factor. The 3-metric h ij is, In the above equation k is the curvature parameter.",
"pages": [
34
]
},
{
"title": "· Friedmann-Robertson-Walker metric (FRW metric):",
"content": "From the above-written metric in the maximally symmetric spacetime, we can write the FRW metric [9, 34-39], in the comoving spatial coordinates ( r , θ , ϕ ). The scale factor a ( t ) is the length scale of the universe serves as a gauge for understanding how physical distances evolve over time, where the coordinate distances, denoted as r , are considered fixed by definition. The present value of the cosmic scale factor a ( t ) is often considered as, a 0 = a ( t 0 ) = 1.",
"pages": [
35
]
},
{
"title": "· Cosmological Redshift:",
"content": "The primary and critical information about the cosmic scale factor, a ( t ) , is obtained from the observation of redshifts in the frequency of light emitted by distant celestial sources. As the universe expands, the wavelengths ( λ ) of photons traveling through space also stretch, leading to a redshift in their observed frequency. This is a key observational evidence of the expanding universe. Astronomer Edwin Hubble interpreted this redshift as due to a Doppler effect and therefore, ascribed a recessional velocity v to the galaxy, which is related to the shift in wavelength by the Doppler formula, where z is known as the redshift. The cosmological redshift behaves symmetrically between the receiver (observer) and the emitter (distant galaxy). In other words, if we observe a redshift in the light coming from a distant galaxy, the light sent from the Earth to that galaxy would also experience a redshift when observed from the distant galaxy's perspective. According to Hubble's Law , the recessional velocity increases proportionally with the distance d of the galaxy [40], i.e., here the proportionality constant H is known as Hubble constant . To provide a more accurate historical account, credit for this fundamental discovery, known as Hubble's law, should potentially be shared with G. Lemaître as well. In many cosmological models, the Hubble constant H , is a timedependent function, and therefore, nomenclatured as Hubble parameter. However, in the given equation, H represents the present-day value of the Hubble constant, often denoted as H 0 . Mega-parsecs (Mpc) are a common unit of measurement for galactic distances. The term parsec is derived from parallax of one arcsecond and is based on the phenomenon of parallax, which is the apparent shift in the position of a nearby object when observed from different vantage points. One parsec is defined as the distance at which an object shows a parallax angle of one arcsecond, or approximately 3.26 lightyears (about 3.086 × 10 13 kilometers or 1.917 × 10 13 miles). Due to the old trigonometric technique of determining star distances, this unit was created. The unit in which H is measured, is kmMpc -1 s -1 . At current epoch its value is, H 0 ≈ 70kmMpc -1 s -1 . In view of FRW cosmology, redshift parameter z can be expressed as, and the Hubble parameter is expressed as, An overdot signifies a derivative of the variable with respect to cosmic time ( t ).",
"pages": [
35,
36
]
},
{
"title": "· A Perfect Fluid:",
"content": "We consider that the universe is composed of a fluid. By definition, a perfect fluid is characterized by the property that a comoving observer perceives the fluid surrounding them as isotropic. In case of a perfect fluid, the heat conduction, viscosity or other transport or dissipative processes are considered negligible. One can write the energy-momentum tensor for perfect fluid in comoving coordinates as, which acts as a source of spacetime curvature in the Einstein equation. In above equation u µ represents the 4-velocity, in comoving frame its componennts are, In equation (1.12), ρ and p are the energy density and pressure of the perfect fluid respectively. The continuity equation is then expressed as, where ∇ ν is the covariant derivative and is often indicated by the symbol ;. The energy density ( ρ ) and the pressure are related by the equation of state (EoS), where ω is called the equation of state parameter. Here are some examples of perfect fluid: For the fluids mentioned above, we see that EoS parameter is constant.",
"pages": [
36,
37
]
},
{
"title": "· The Einstein Field Equations:",
"content": "In FRW cosmology, the Einstein equations give two independent equations, The fluid conservation equation is, Due to the Bianchi identities ( G µν ; ν = 0), the conservation equation is not independent of Einstein equations and can be derived from them. (i) If q > 0: The universe is decelerating. The expansion rate is slowing down over time. Normal baryonic matter shows q > 0. (ii) If q < 0: The universe is accelerating. The expansion rate is increasing over time. Observational evidence suggests that our universe is currently in an accelerated expansion phase. This discovery led to the proposal of the concept of dark energy which gives q < 0. (iii) If q = 0: The universe is coasting. The expansion rate remains constant over time. The behaviour of the jerk parameter can give insights into the presence of dark energy, as it plays a crucial role in the accelerated expansion of the universe. The expansion of the universe is currently accelerating, and this acceleration is attributed to dark energy, represented by the cosmological constant ( Λ ). The value of the jerk parameter for Λ CDMmodel is j Λ CDM = -1. A value of j not equal to -1 could suggest the presence of additional forces or components in the universe that contribute to its acceleration and whose effects are not fully captured by the cosmological constant. It is to be noted that there is another parameter, known as the snap parameter, s = 1 aH 4 d 4 a dt 4 , related to fourth order derivative of the scale factor. However, in the aspects of observational cosmology, the evolution of q holds physical significance. Therefore, we only narrow our analysis to the jerk parameter. The critical density represents the dividing line between an open, flat, or closed universe. Ω represents the ratio of the average density of the universe to the critical density. It is defined as, The spatial shape and curvature of the cosmos are determined by the value of the density parameter in the following way:",
"pages": [
37,
38,
39
]
},
{
"title": "1.2 Thermal Evolution of the Universe in Brief:",
"content": "Understanding the thermal evolution history is crucial in cosmology as it provides insights into the behaviour of different constituents of the universe and the processes that shaped its current state. Here is an overview of the key stages in the thermal evolution of the universe: high, causing quarks and gluons to roam in what is known as a quarkgluon plasma. As the universe expands and cools, this state transitioned to confinement, forming protons and neutrons.",
"pages": [
39,
40
]
},
{
"title": "1.3 Possible ways to Elucidate the Cosmic Acceleration:",
"content": "The discovery of the late-time accelerated expansion of the universe occurred in 1998 when two separate groups, the Supernova Cosmology Project [193] and the High-Z Supernova Search Team [20, 48, 192], independently observed supernovae with redshifts z < 1. Type Ia supernovae, known for their consistent intrinsic brightness, serve as reliable standard candles, easily distinguishable over varying distances. These supernovae are thought to occur when a white dwarf star in a binary system accumulates enough mass from its companion, triggering a thermonuclear explosion. As the universe expands, the separation between the observer and celestial objects increases. Consequently, the emitted photons experience redshift. By analyzing the observed brightness of these objects and the redshift of the photons, researchers can gauge the rate of the expansion of the universe. The brightness of a supernova is represented by its absolute magnitude. It can be utilized in calculating the cosmic luminosity distance ( d L ). The connection between the apparent luminosity ( f ) and the intrinsic luminosity ( L ) is expressed as, f = L 4 π d 2 L . Measurements of the luminosity distance ( d L ) for supernovae are recorded at various redshifts in the format of distance modulus ( µ B ). This modulus is defined as the disparity between the apparent magnitude ( m B ) and the absolute magnitude ( M B ) within the wavelength of the blue line of the observed spectrum as, µ B = m B -M B = 5 log 10 d L 1Mpc + 25. In terms of H 0 and q 0 , i.e., the e present values of the Hubble parameter and the deceleration parameter respectively, we can write, d L = cz H 0 ( 1 + z 2 [ 1 -q 0 ] + O ( z 2 ) ) . Both supernova groups gauged the luminosity distances and witnessed the fading of supernovae. The determined luminosity distances surpassed their anticipated values, signifying a negative q 0 . This affirmation supported the conclusion that light sources are moving away from each other at an accelerated pace. This observed accelerated expansion of the universe is a perplexing phenomenon that cannot be fully accounted for by the known components of the universe. All known forms of matter and energy that we encounter in the universe, such as ordinary matter, radiation (including light), and the dark matter, contribute to slowing down the expansion of the universe. These fluids follow the strong energy condition, ρ + 3 p > 0. Now if we combine the equations, (1.16) and (1.17), we can write, From the definition of deceleration parameter (1.19), q = -1 aH 2 d 2 a dt 2 and the equation (1.23), we see that for these fluids q > 0, i.e., the decelerating cosmic expansion . However, rather surprisingly, the universe has undergone two separate periods of accelerated expansion: one marked by early exponential inflation and the other by late-time cosmic acceleration. In the interim, these two phases of rapid expansion were separated by a period characterized by a decelerated expansion. As a result, scientists have put forward various theoretical models to try to elucidate the acceleration in the expansion history in two distinct times, (i)inflation and (ii)late-time cosmic acceleration. Nevertheless, none of these models has achieved widespread acceptance as flawless or has been strongly supported by irrefutable observational evidence. Additionally, these models have not yet provided a direct and conclusive method for detecting the underlying cause of cosmic acceleration.",
"pages": [
41,
42
]
},
{
"title": "1.3.1 Inflation:",
"content": "Inflation is a suggested paradigm in cosmology that proposes a rapid exponential expansion of the universe during its early epoch ( ∼ 10 -36 -10 -32 seconds). This model was developed to address several outstanding problems and observed phenomena in the standard Big Bang model. The concept of inflation was first proposed in the early 1980s by Alan Guth [17] and later refined by other physicists, (see [49-55]). The key idea behind inflation is that in the tiny fraction of a second after the Big Bang, the universe underwent an exponential expansion, increasing its size by an enormous factor. This rapid expansion allowed the universe to smooth out irregularities and achieve a high degree of homogeneity and isotropy, which explains why the cosmic microwave background radiation appears so uniform in all directions. (For introductions to inflation, see [56]). Inflation addresses several important cosmological puzzles: Inflationary theory suggests that the exponential expansion was driven by a scalar field, often called the inflaton field . As the universe expanded and cooled, this field settled into its lowest energy state, ending the inflationary phase. The energy released during the decay of the inflaton field reheated the universe, initiating the subsequent hot phase and the standard cosmological evolution that followed. While inflation has been highly successful in explaining various observed features of the universe, it remains a theoretical framework. Direct evidence for inflation is still a subject of intense research and observational efforts, such as studying the cosmic microwave background radiation and the largescale structure of the universe. Confirming inflation would be a profound confirmation of our understanding of the early history of the universe and the processes that shaped its current state.",
"pages": [
42,
43
]
},
{
"title": "1.3.2 Late-time Cosmic Acceleration:",
"content": "Late-time cosmic acceleration was discovered in the late 1990s from observational data of type IA supernova. In addition to using type-Ia supernovae [58-60], Baryon Acoustic Oscillations (BAO) [61, 62], the WMAP [63], Planck [22, 64-66] satellite missions, and the Dark Energy Survey (DES) [67] data also provide substantial support for the existence of a seamless transition. This transition involves the universe transitioning from a prior phase of decelerated expansion to its current state of accelerated expansion, occurring at an intermediary redshift of approximately z ≈ 0.5 [68-75]. There are two potential explanations to find out a way to unravel the mystery. One of those is that gravity acts in a manner unlike from our current understanding, hence the theory of gravity needs to be modified. The other one is that the cosmos contains an exotic element with unconventional gravitational attributes, leading to an apparent negative pressure effect. These two main broad categories are discussed below.",
"pages": [
44
]
},
{
"title": "1.3.2.1 Modified Theories of Gravity:",
"content": "Modified gravity theory refers to a class of theoretical frameworks that propose modifications or extensions to Einstein's general theory of relativity (GR). The idea is to account for the cosmic acceleration without invoking any exotic fluid component, such as dark energy. Not only cosmic acceleration, but the theories also attempt to give insight into some other problems such as addressing the nature of dark matter, resolving singularities etc. Additionally, they aim to provide a semi-classical description of gravitational interactions, where quantum effects are taken into account through an effective action. There are many different proposals for modified gravity, each with its own set of mathematical equations and predictions [76-82]. Some of the well-known examples of modified gravity theories include: f(R) Gravity [83-91], scalar-tensor theories [92-106], Scalar-Einstein-Gauss-Bonnet Gravity [107-109], Galileon Gravity [110-112], TeVeS (Tensor-Vector-Scalar) theory [113-115], f(T) Gravity [116], Horndeski Gravity [117-119], Dilaton Gravity [120, 121], Dvali-Gabadadze-Porrati (DGP) Model(braneworld scenario) [122, 123], Chameleon Gravity [124-127] and some extended theories [128, 129]. Among these, f(R) gravity and scalar-tensor theories will be discussed here. 1. f ( R ) Theories: In the realm of gravitational theories, a straightforward modification to General Relativity (GR) is presented through f ( R ) gravity. This modification centers on the spacetime action and its transformation. Rather than adhering to the conventional Ricci scalar R , f ( R ) gravity introduces a more flexible approach, incorporating an analytical function, f = f ( R ) . As a result, the action governing f ( R ) gravity can be succinctly represented as, By taking variation of the action given in equation (1.24) with respect to the metric tensor g µν , we arrive at the field equations, which are expressed as, Here superscript ' implies the derivative with respect to R and the symbol □ represents the d'Alembertian operator and given by, □ = g µν ∇ µ ∇ ν . In terms of Einstein tensor this equation can be written as, where the effective stress-energy tensor is T eff µν = [ f -Rf ' 2 g µν + ( ∇ µ ∇ ν -g µν □ ) f ' ] . The source of this term is entirely geometric in nature. The inclusion of f ' in the denominator within Equation (1.26) signifies the existence of a non-minimal coupling, which in turn renders the effective gravitational constant a variable in these theories. These f ( R ) gravity theories have the capability to elucidate phases of cosmic acceleration without the requirement for exotic or unconventional matter constituents. This is achievable through the careful selection of the function f . These f ( R ) models, characterized by f ( R ) ∼ R 2 have proven their capability in generating scenarios of early inflation. The models featuring f ( R ) ∼ 1/ R n , where n is positive, are put forward to account for the accelerated expansion of the universe during its later phases. A thought-provoking discourse of how this theory harmonizes with the established cosmological model to explicate cosmic acceleration can be encountered in the extensive review [130]. To know more detail about f ( R ) theories of gravity, we refer to the articles [84, 90, 131-144]",
"pages": [
44,
45
]
},
{
"title": "2. Scalar-tensor theories:",
"content": "In gravitation and cosmology, scalar-tensor theories are rooted in the concept of a non-minimal coupling between the scalar field and the spacetime geometry. The origins of these theories can be traced back to the pioneering work of Jordan [145] and were subsequently advanced by Brans and Dicke [146]. The generality of their findings was later augmented by the insights of Nordvedt [147] and Wagoner [148]. The action to represent a broad category of Non-Minimally Coupled Scalar-Tensor Theories (NMCSTT) is, Within this framework, the scalar field ϕ is coupled to the Ricci scalar R in a non-minimal manner, leading to a consequential adjustment in the gravitational coupling strength. The field equations governing the NMCSTT encompassed within the action (1.27) are derived through variations in both the metric g µν and the scalar field ϕ . These equations can be expressed as follows: and Here the effective gravitational coupling is G eff = f ( ϕ ) -1 . As a consequence, it is presumed that f ( ϕ ) is positive to guarantee a positive coupling. Brans-Dicke (BD) theory serves as the quintessential example within this classification. Diverse scalar-tensor theories are born from the overarching action presented in equation (1.27) each distinguished by their unique choices of the functions f ( ϕ ) and ω ( ϕ ) . For comprehensive discussions on scalar-tensor theories, one may follow the articles [149-151]. Here we shall discuss about the Brans-Dicke theory.",
"pages": [
46
]
},
{
"title": "· Brans-Dicke theory",
"content": "Brans-Dicke theory stands as one of the most widely discussed theoretical framework in the field of theoretical physics and cosmology that extends Einstein's theory of general relativity. Proposed by Carl H. Brans and Robert H. Dicke in 1961 [146], this theory introduces a scalar field, known as the Brans-Dicke field, alongside the familiar metric tensor used in general relativity. This addition was motivated by the desire to explore a broader range of gravitational theories that could accommodate variations in the strength of gravity and address certain shortcomings of general relativity. Brans-Dicke action follows from equation (1.27) by the choice f ( ϕ ) = ϕ and ω = a constant (dimensionless). Therefore, now the action looks like, Upon taking the conformal transformation [152], the action becomes, where ψ = ln ( ϕ ϕ 0 ) . ϕ 0 is a constant. This version is popularly known as BD theory in Einstein frame . Then the field equations assume a significantly simplified form, which can be expressed as follows: This is written in the unit of 8 π G 0 = 1. The equation of motion for the scalar field ψ , derived by varying the action (1.32) with respect to ψ , results in the expression, where T represents the trace of the energy-momentum tensor for the matter sector. The Brans-Dicke theory has been subjected to various experimental tests to determine its validity compared to General Relativity. These tests include measurements of the deflection of light by massive objects, the Nordtvedt effect, precision tests in the solar system, and cosmological observations. Observational constraints limit ω to a very large value [153-155]. The BransDicke (BD) theory was initially suggested to converge into the General Theory of Relativity (GTR) in the limit as ω approaches infinity, a notion initially posited in [41]. Subsequently, it has been elucidated in the literature that the ̸ ̸ inclusion of the trace of the energy-momentum tensor of matter distribution imposes certain constraints on this proposition, as expounded in studies [92, 156]. The static spherically symmetric vacuum solutions of this theory can be found in articles [157]. To know more aspects of these solutions, readers may refer to [158-171]. Numerous spatially homogeneous and isotropic cosmological solutions within the framework of Brans-Dicke (BD) theory can be found in the literature. A few such examples are listed in Table 1.1. BD theory provides solutions to a variety of cosmological quandaries. It offers solutions to issues like the graceful exit problem during inflationary phases [177, 178] and the late-stage accelerated expansion of the universe, all without requiring the introduction of dark energy [100]. The monograph [150] by Faraoni provides an extensive exploration of exact cosmological solutions within BD theory.",
"pages": [
46,
47,
48
]
},
{
"title": "1.3.2.2 Dark Energy Models",
"content": "Dark energy models seek to elucidate the cosmic acceleration within the framework of General Relativity (GR) [185-191]. Equation (1.23) illustrates that an accelerated expansion can arise when a constituent within the energy sector exerts substantial negative pressure. This exotic component, commonly referred to as dark energy, instigates acceleration due to its negative pressure, distinct from the fluid pressure attributed to particle motion. To facilitate analysis, we introduce dimensionless representations of energy densities by scaling them with the critical density ( ρ c ), defined in equation (1.21) and density parameter ( Ω i ), defined in equation, (1.22). Subscript ' i ' implies either dust ( i = m ), radiation ( i = r ) or dark energy ( i = d ). Now, the Hubble parameter H , scaled by its present value H 0 , can be written in terms of density parameters as (for spatially flat models), Dust matter, a significant component in the energy budget, is pressureless ( p m = 0), thus yielding w m ≡ pm ρ m = 0 as its equation of state parameter. The dark energy equation of state parameter ( w d ) is defined as, w d ≡ p d / ρ d . We denote the density parameters at the present epoch, as Ω m 0 , Ω r 0 , Ω d 0 . For non-pressured matter, Ω m = Ω m 0 ( 1 + z ) 3 ; for radiation, Ω r = Ω r 0 ( 1 + z ) 4 , using the standard scaling of the scale factor with a 0 = 1. The composite model's effective equation of state (EoS) is expressed as: To achieve an accelerated universe, w e f f must be less than -1/3, as indicated by equation (1.23). Given the negligible contribution of radiation in the late-time universe compared to other components, we can estimate q for a spatially flat universe composed of pressureless matter and dark energy as follows, Hence, the condition for late-time cosmic acceleration is w d < -1 3 Ω -1 d . Recent cosmological observations data suggests Ω d 0 = Ω d | z = 0 ≈ 0.7. Consequently, the value of the dark energy equation of state (DE EoS) parameter at the present epoch should be w d 0 = w d | z = 0 ≲ -0.5. Various theoretical frameworks exist for explaining dark energy, yet none have gained universal acceptance, each exhibiting its own drawbacks and limitations. The most straightforward model of dark energy is the cosmological constant Λ , characterized by an equation of state w = -1. Below, we delve into an array of diverse models that seek to understand the nature of the exotic dark energy. These models offer distinct perspectives and approaches, each contributing to our ongoing quest to unravel the mysteries of cosmic acceleration.",
"pages": [
48,
49,
50
]
},
{
"title": "· Cosmological constant",
"content": "In 1917, Einstein introduced the cosmological constant, denoted by Λ , into his equations of general relativity. Einstein added Λ to his equations to achieve a static solution for the universe. At the time, the prevailing view was that the universe was static and unchanging. However, when Edwin Hubble discovered the expansion of the universe in the late 1920s, Einstein famously referred to the inclusion of the cosmological constant as his greatest blunder [2]. He removed the cosmological constant from his equations because it was no longer needed to describe a static universe. The cosmological constant made a comeback as vacuum energy in the context of inflation [17]. In the late 20th century, the cosmological constant again resurged in cosmology to explain the observed accelerated expansion of the universe [20, 192, 193]. This acceleration was discovered in the late 1990s through observations of distant supernovae and is attributed to a mysterious dark energy that permeates the universe. One may refer to [185, 187, 194-198], to learn more about Λ as a representative of this dark energy. In a universe where the cosmological constant dominates, the solution corresponds to an exponential rate of expansion, where the scale factor a ( t ) grows exponentially with time (see reference [199]), The Λ CDM model is the prevailing cosmological framework, consisting of cold dark matter (CDM) with no pressure and the cosmological constant serving as dark energy. It is commonly referred to as the standard cosmological model. The energy density tied to Λ is defined as, To maintain ρ Λ as a constant by definition, it necessitates that the pressure associated with Λ , must be, p Λ = -ρ Λ . Therefore, the cosmological constant exhibits an effective negative pressure, with the equation of state parameter w = -1. In a universe exhibiting matter dominance, the scale factor evolves according to the relationship a ( t ) ∝ t 2/3 . As the universe enters a phase where cosmological constant holds sway, with EoS parameter w = -1, the scale factor approaches an asymptotic form expressed in equation (1.38). When we consider a universe with a spatially flat geometry, accommodating both matter and cosmological constant, the solution seamlessly integrates the properties of these two components across different cosmic eras and as given by [196], This solution adeptly mirrors the evolving characteristics of matter and vacuum energy as we traverse from earlier to later cosmic eras. It essentially embodies the principles of the Λ CDM model. Various facets of the cosmological constant model have been thoroughly examined and elaborated upon by Carroll [194] and extensively discussed by Padmanabhan [187, 200]. The cosmological constant model stands as the favoured choice in light of compelling observational support. Nevertheless, it is essential to acknowledge the inherent complexities of this model. Notably, the only plausible candidate to account for the cosmological constant is the energy density inherent in the vacuum. The notable hurdle lies in the staggering disparity between the energy density's observed value ( ρ obs Λ ) and its theoretically calculated counterpart ( ρ theory Λ ). This incongruity is striking, with the ratio ρ obs Λ ρ theory Λ hovering around the minuscule figure of 10 -120 . This enigma is fundamentally recognized as the fine-tuning dilemma inherent to the cosmological constant model. Recent efforts have been directed towards achieving a reduced cosmological constant in the current cosmic era. A concise overview of various models featuring a cosmological constant, diminishing with time, can be found in table 1.2, which is adapted from [201] (see also [185]). ̸",
"pages": [
50,
51
]
},
{
"title": "· Constant dark energy equation of state model ( w d = -1 )",
"content": "In the pursuit of understanding dark energy through phenomenological analysis, cosmologists often opt for a constant value when characterizing the dark energy Equation of State (EoS). Importantly, this constant is not limited to the value of -1, providing a degree of flexibility in the modeling process. Dark energy models characterized by this constant EoS are referred to as Quiessence models [247]. To illustrate, consider a scenario where the universe consists of Cold Dark Matter (CDM) and Dark Energy (DE), with the latter being defined by a constant EoS, w d . This framework is commonly referred to as the w CDMmodel. Its purpose is to scrutinize observational data for any potential deviations from the established Λ CDMmodel. Within the context of the w CDMmodel, it is noteworthy that the dark energy density is no longer fixed but instead exhibits an evolution with changing redshift. This dynamic characteristic allows for a more comprehensive examination of the behaviour of dark energy over cosmic history.",
"pages": [
51,
52
]
},
{
"title": "· Variable dark energy equation of state model",
"content": "To tackle the cosmic coincidence problem, i.e., the mystery of why the energy densities of dark matter and dark energy in the universe are of the same order of magnitude in the present cosmic epoch and gain insights into the evolving nature of dark energy, researchers have turned their attention to models that exhibit dynamic behaviour over cosmic history. These models operate on the assumption that the equation of state for dark energy ( w d ) has undergone temporal changes throughout the evolution of the universe. This paradigm shift has given rise to a multitude of dynamical dark energy (DDE) models, each characterized by a time-dependent EoS parameter. Among these models, the Chevallier-Polarski-Linder (CPL) model, [248, 249], has gained significant prominence. It is defined by the functional form, where w 0 and w a are real numbers. This model is widely regarded as the most popular DDE model. The conventional approach often involves parameterizing our limited understanding of how dark energy behaves. This methodology found extensive application within the Dark Energy Task Force [250]. It has served as a practical benchmark for evaluating and contrasting the performance of various techniques aimed at investigating dark energy dynamics (see, for instance, [251]). The simplicity of the CPL parameterization belies its rich characteristics, as thoroughly explored by Linder [257]. Of particular note are the two parameters it introduces: w 0 , serving as a representation of the equation of state's present condition, and w a , which encapsulates its broader temporal dynamics. An interesting proposition, detailed in references [249, 257], posits that the optimal way to characterize w a in relation to the derivative of w is through the following expression: In this equation, w ' represents the derivative of w , defined as w ' ≡ dw d ln a , with \" a \" denoting the scale factor. While it may not encompass the full spectrum of potential dynamics [257-260], the CPL parameterization appears to strike a favorable balance for conducting a model-independent analysis. Other models with EoS dependent on z have been listed in table 1.3.",
"pages": [
52,
53
]
},
{
"title": "· Quintessence scalar field model",
"content": "The quintessence scalar field model is a captivating and influential theoretical framework in the field of cosmology. The concept of a quintessence scalar field was initially presented by Ratra and Peebles [261] and independently by Wetterich [238] in order to facilitate the inflationary paradigm. In trying to grasp how the universe works on a big scale, the quintessence scalar field model is different from the usual cosmological constant model. It gives a dynamic and evolving explanation for what is causing the universe to expand. At its core, the quintessence scalar field model introduces a new dynamic component to the cosmic energy budget in the form of a scalar field, often denoted as Φ , that permeates the universe. This scalar field possesses unique properties, including a potential energy function V ( Φ ) and negative pressure, allowing it to act as a driving force for the accelerated expansion of the cosmos. Φ slowly rolls down to its potential V ( Φ ) , which leads to the dominant potential term over the kinetic term. Unlike the cosmological constant associated with the Λ CDM model, which represents a static form of dark energy, quintessence offers a dynamic alternative that evolves over time. The relevant action for quintessence field Φ is, Stress-energy tensor of the quintessence field is given by the equation, From the above equation, we can write the quintessence field energy density ( ρ Φ ) and pressure ( p Φ ) and respectively given by, and The Klein-Gordon(KG) equation for Φ is, The EoS parameter is therefore, Therefore, the range of evolution of w Φ is, -1 < w Φ < 1. Quintessence models are categorized into three distinct classes based on the characteristics of the potential energy function V ( Φ ) . This classification is determined by the specific nature of the potential of the scalar field. The classification of quintessence fields into thawing and freezing models is a significant concept in the study of dark energy and its role in the evolution of the universe. These models are categorized based on the behaviour of the effective EoS parameter, w of the quintessence field as the universe expands over time [274].",
"pages": [
53,
54,
55
]
},
{
"title": "· Thawing Models:",
"content": "Thawing models are characterized by a quintessence field with an effective EoS parameter, w that starts out as almost a constant close to -1 (similar to the cosmological constant), which corresponds to a vacuum energy with w = -1. Over time, the equation of state parameter thaws or gradually evolves into a more dynamic, time-dependent value that deviates from -1. Thawing models are particularly interesting because they represent quintessence fields that initially behave like a cosmological constant but then change their behaviour, which can have implications for the evolution of the universe.",
"pages": [
55
]
},
{
"title": "· Freezing Models:",
"content": "In contrast, freezing models are characterized by a quintessence field for which the effective EoS parameter, w settles down to a constant value close to -1, but this happens relatively late in the evolution of the universe. Initially, the quintessence field may have a dynamic behaviour, but it eventually reaches a stage where its equation of state parameter becomes nearly indistinguishable from that of a cosmological constant. One important subclass of freezing models is known as trackers , which are of particular interest [265, 275, 276]. In tracker models, the energy density of the scalar field evolves almost parallel to the energy density of dark matter for most of cosmic history without dominating dark matter. However, it eventually freezes to a value greater than the dark matter density at a later stage. Thakur, Nautiyal, Sen, and Seshadri [402] have undertaken a comparison between a thawing model and freezing models that exhibit tracking behaviour. Their investigation revolves around the compatibility of these models with empirical data and offers valuable insights into their observational viability. It is worth noting that not all quintessence fields neatly fit into either of these two categories, and there can be variations and more complex behaviours. The classification into thawing and freezing models is primarily a way to categorize and study the different possible behaviours of quintessence fields in the context of cosmic expansion and their ability to address issues such as the coincidence problem [277], which relates to why dark energy and dark matter densities are of similar magnitude in the present universe. Both thawing and freezing models have been extensively studied in cosmology to understand their implications for the evolution of the universe. Table 1.4 provides an exhaustive compilation of the various scalar field potentials that have been explored within the context of quintessence models. To explore further into the vast body of research about late-time cosmic acceleration, which includes various versions of quintessence potentials, one may refer to [247, 262, 263, 265, 274, 275, 277-287].",
"pages": [
56,
57
]
},
{
"title": "· Phantom field model",
"content": "The idea of the phantom field in the realm of dark energy was initially introduced by Caldwell [288]. What sets the phantom field apart from the quintessence field is its distinctive feature of possessing negative kinetic energy( X ). Therefore, the Lagrangian density for phantom field ϕ is, L p = -X -V ( ϕ ) = -∂µϕ∂ µ ϕ /2 -V ( ϕ ) (with metric signature + ---). The corresponding action governing the behaviour of the phantom field can be expressed as follows, The phantom field energy density and pressure are respectively given as, and The equations of motion are, The equation of state parameter of dark energy for the phantom field is, For w de to be less than -1, V ( ϕ ) >> ˙ ϕ 2 . A phantom field, propelled by its negative kinetic energy will tend to run up potential energy. The consequence is an extremely swift expansion of the universe, reaching infinite extent within a finite time. This phenomenon is termed the Big Rip . This is characterized by the infinite growth of both the volume and the expansion rate. A potential for the phantom field featuring a maximum value(for example V ( ϕ ) = V 0 [ cosh ( αϕ m pl )] -1 , where α is a constant) has the capability to avert the occurrence of the Big Rip. For ˙ ϕ = 0, the field eventually reaches its maximum position, following a damped oscillatory phase, causing the equation of state parameter, w de , to attain a value of -1. Consequently, this behaviour can reinstate the scenario of the cosmological constant. Quintom models is referred to the scalar field models, wherein the behaviour of the equation of state (EoS) parameter mirrors that of the phantom field [289292].",
"pages": [
57,
58
]
},
{
"title": "· Chaplygin Gas model",
"content": "The Chaplygin gas model, named after the Russian mathematician Sergei Alekseevich Chaplygin, is a theoretical model that was originally introduced by Chaplygin in 1904 in aerodynamics to describe the lift force on an object moving through a gas. This model was reintroduced in cosmology by Kamenshchik, Moschella and Pasquir [293] in the context of cosmic acceleration. The Chaplygin gas model is characterized by an equation of state given by: where A is a positive constant and p and ρ are respectively pressure and energy density in a comoving frame with ρ > 0. This equation The Chaplygin gas model provides a smooth transition between different phases of cosmic evolution. Specifically, it can smoothly interpolate between: In this model, once a universe undergoing expansion enters a phase of acceleration, it is incapable of reverting to a state of deceleration. This model has been generalized by Bento, Bertolami and Sen [294]. The authors proposed that the change in the characteristics of the elusive energy density could be governed by alterations in the equation of state of the underlying fluid, rather than relying on adjustments to the potential. This approach offers a means to circumvent the intricate fine-tuning problem. This is achieved by introducing a more flexible equation of state that allows for a wider range of behaviours. The generalized equation of state is given as: where α is the parameter with 0 < α < 1, that determines the behaviour of the gas. The energy density ρ evolves with the scale factor a of the universe as, ̸ where B is the integration constant. In this scenario, instead of stiff matter, the intermediate phase is soft matter with equation of state, p = αρ ( α = 1). This model has been explored in cosmology as a way to unify dark matter and dark energy within a single framework and has connections to brane theory and supersymmetry in theoretical physics [295]. While the Chaplygin gas models have been invalidated by data related to temperature fluctuations in the cosmic microwave background [296, 297], there is a limited range of parameter values, specifically 0 ≤ α ≤ 0.2, within which the generalized Chaplygin gas models are still considered plausible [296].",
"pages": [
58,
59
]
},
{
"title": "· K-essence model",
"content": "The K-essence model is a theoretical framework in cosmology that introduces a scalar field with a non-standard kinetic term to explain the dynamics of dark energy and, in some cases, an early inflation as well. It offers an alternative to more traditional scalar field models, such as quintessence, where the potential energy of the scalar field plays a central role. In the K-essence model, the kinetic energy of the scalar field dominates, hence the name Kessence (kinetic essence). At its inception, it was introduced in the context of inflation [298, 299]. Later, Chiba et al. introduced it when considering latetime acceleration [300]. The generalized k-essence for dark energy was proposed by Armendariz-Picon et al. [301, 302]. In this model, the Lagrangian density is written in the form of pressure as P ( ϕ , X ) , where X = -1 2 ( ∇ ϕ ) 2 is the kinetic energy for the scalar field ϕ . Therefore, the action is given as, The energy density is given by, Therefore, EoS parameter can be written as, For 2 X ∂ P ∂ X = 0, this model reduces to cosmological constant model with w = -1 [303]. More detailed knowledge about the K-essence model can be found in the references [304-307].",
"pages": [
59,
60
]
},
{
"title": "· Tachyon field model",
"content": "Tachyons are hypothetical particles that are often characterized by having imaginary mass. During the decay process of D-branes, a state emerges that behaves like a gas lacking pressure but possessing finite energy density. It resembles classical dust [308-315]. Tachyons exhibit an equation of state (EoS) parameter that smoothly varies within the range of -1 < w < 0. This has piqued the interest of cosmologists, prompting them to consider tachyons as a plausible contender of dark energy [316]. For a comprehensive analysis of tachyonic dark energy models, with a focus on their ability to generate latetime cosmic acceleration, refer to [317-322]. The tachyon's squared mass is intrinsically negative and stabilizes at the peaks of its associated scalar field potential. This state experiences infinitesimal perturbations, ultimately resulting in a tachyon condensation process characterized by a descent from the peaks, leading to the attainment of a real mass. The action for tachyon field ( ψ ) with potential V ( ψ ) can be expressed as follows: The wave equation for this field is, The energy density and pressure are respectively given as, We can write the EoS parameter as, Therefore, the criterion for the universe to experience accelerated expansion is when ˙ ψ 2 < 2/3.",
"pages": [
60,
61
]
},
{
"title": "· Holographic Dark Energy",
"content": "The holographic dark energy (HDE) draws its inspiration from the holographic principle, a concept rooted in quantum gravity theory. This model aims to shed light on the mysterious nature of dark energy and its role in the accelerated expansion of the universe by establishing a connection between dark energy and the information content residing on the boundary. The holographic principle, originally proposed by 't Hooft [323] and Susskind [324], suggests that the physical properties and degrees of freedom within a given region of space can be entirely encoded on its boundary rather than within the volume itself. This concept is closely linked to the notion of entropy and has its roots in the work of Bekenstein on black hole entropy bounds [325, 326]. According to this bound, there exists a connection between the shortdistance ultraviolet (UV) cut-off and the long-distance infrared (IR) cut-off due to the constraint that the total quantum zero-point energy of a system should not exceed the mass of black holes of the same size [327]. This constraint can be expressed as: where M p = ( 8 π G ) -1/2 is the reduced Planck mass, ρ Λ represents the quantum zero-point energy density determined by the UV cut-off, and L is the length scale of the system size. The IR cut-off is the length for which this inequality saturates. The largest allowable value for L is the one that makes this inequality reach its limit. Therefore, In the context of dark energy, the holographic principle is applied by introducing the holographic energy density ( ρ H ) with the following expression [328], Here system size is the size of the observable universe and C 2 is a dimensionless coupling parameter. The cosmic horizon serves as the IR (infrared) cut-off. Similar concepts and ideas were explored in references [329, 330]. Various approaches can be found in the literature, each with its own choice of the IR cut-off length scale. Such examples include the particle horizon [331, 332], the future event horizon [328, 333-338] and the Hubble horizon [339-341] as IR cut-off.",
"pages": [
61,
62
]
},
{
"title": "1.4 Outline of the thesis",
"content": "The primary focus of this thesis centers on delving into the thermodynamic characteristics of diverse cosmological models. An assessment of the models' viability has been carried out by applying the Generalized Second Law (GSL), which asserts that the total entropy comprising both the horizon and the encompassed fluid must never decrease. Given the evolving nature of the universe, our approach has involved working with the apparent horizon. We have assumed a state of thermodynamic equilibrium between this apparent horizon and the fluid inside the horizon. In this equilibrium state, we have adopted the Hayward-Kodama temperature as the horizon temperature. Chapters 3-6 contain the main research work. Chapters 3,4 and 5 are focused on GSL test. Chapter 6 is focused on the stability analysis of a cosmological model. In the second chapter, we have discussed about apparent horizon, blackhole thermodynamics and their analogies in the dynamical apparent horizon in cosmology. A thorough description of how to calculate the rate of change of entropy has been prescribed. Also, we have touched upon the subject of the thermodynamic stability of the cosmological model. In the third chapter, our study conducts a thorough comparative analysis between thawing and freezing models, particularly with regard to their adherence to the Generalized Second Law (GSL) of thermodynamics. In our comprehensive evaluation of the total entropy ( S tot), we incorporate both the entropy of the horizon and the entropy stemming from the matter enclosed within the horizon. To facilitate this investigation, we employ a simple ansatz proposed by Carvalho et al. [287] to model the dynamic evolution of the energy density within the quintessence field, allowing us to pinpoint the parameter range ( α ) associated with thawing and freezing behaviours. Our findings reveal a common inconsistency with the GSL in both type of models. In the context of freezing models, this GSL breakdown is traced back to a remote past, corresponding to a redshift of approximately z ∼ 10 4 . During this distant epoch, a quintessence model along with cold dark matter fails to adequately account for the evolution of the universe, as the dominant contribution comes from radiation distribution. This suggests that the GSL breakdown may not hold true under these circumstances. Conversely, for thawing models, this unusual violation of the GSL is anticipated to manifest in a finite future. The key implication here is that freezing models appear to enjoy better thermodynamic favourability when compared with their thawing counterparts. In the fourth chapter, we explore GSL to models of the universe filled with radiation and dust, assuming the universe is flat, homogeneous and isotropic in the framework of Brans-Dicke theory in Einstein frame . When it comes to a universe dominated by radiation, the solutions in Brans-Dicke theory with a positive value for the BD parameter ω , do not follow GSL. But when ω has a negative value within a certain range, it does match the thermodynamic requirements. And that is exciting because for cosmic acceleration, as per observation, one needs this negative ω value. Now, if we switch our focus to a universe dominated by dust (like galaxies and matter), the model does satisfy GSL when ω is a small negative number within a specific range. This range of ω significantly overlaps with the range necessary for achieving accelerated expansion without the need for any exotic matter. In the fifth chapter, we studied the thermodynamic viability of some dark energy models reconstructed through the cosmological jerk parameter. As the deceleration parameter, q evolves, we become interested in the next-order derivative called the jerk parameter , represented as j . It tells us how q changes over time. In this study, we select some of these models from existing literature and evaluate them in terms of their thermodynamic viability. By reconstructing the jerk parameter, it is entirely possible to find models that satisfy the laws of thermodynamics. Among the four models tested for a non-interacting scenario, only one (model IV), which has an inverse relationship with (1 + z ), shows a decrease in entropy in the future (at z < 0). This decrease is particularly significant near the present epoch ( z = 0). All the other models, including one that allows for interaction in the dark sector, pass the GSL. Model I, lacking an explicit dependence on z , satisfies the GSL but undergoes a sudden entropy surge in the future ( z < 0). However, any analysis in terms of z is not very sound for negative values of z . Models II and III, dependent on ( 1 + z ) and ( 1 + z ) 2 respectively, as well as Model V with interaction in the matter sector, exhibit favorable behaviour across a broad parameter range within a 3 σ confidence interval. Overall, from the past to the present epoch, all models demonstrate satisfactory behaviour. In the sixth chapter, we conducted a thermodynamic stability analysis on a model designed to mimic the Λ CDMmodel for the current state of the universe. In this analysis, we took into account the evolving horizon and considered the Hayward-Kodama temperature as the temperature of the horizon. In thermodynamics, the stability of an equilibrium system hinges on having a positive thermal capacity and compressibility. This principle also applies to the matter content within a cosmological system. However, in our current scenario, the specific heat capacity ( C V ) turns out to be negative. This suggests that the model is likely to exhibit thermodynamic instability. The significant outcome of our investigation is that the matter content undergoes a phase transition as the universe transitions from a decelerated to an accelerated state of expansion. This phase transition is notably a second-order transition, evident by the discontinuity in C V . The deceleration parameter, q serves as the order parameter in this context. This feature goes missing if we consider Hawking temperature, which neglects the fact that the apparent horizon is evolving over time. The final chapter, chapter 7, concludes the research in this thesis and provides some future aspects.",
"pages": [
62,
63,
64,
65
]
},
{
"title": "2.1 Cosmological Apparent Horizon",
"content": "We shall discuss the cosmological apparent horizon by drawing an analogy with black hole apparent horizon. Therefore, let us first take some notes on black hole apparent horizon and then delve into the apparent horizon in FRW cosmologies.",
"pages": [
67
]
},
{
"title": "2.1.1 Apparent Horizon",
"content": "In simple words, a horizon can be described as \"a frontier between things observable and things unobservable\" [342]. The key feature of a black hole spacetime is the presence of an event horizon, a boundary that separates the black hole from the external observers and conceals internal events. This hypersurface consists of a congruence of null geodesics, or null generators, which are crucial in comprehending the behavior of the horizon. To grasp the overall behavior of the horizon, it becomes essential to study how these generators behave. In this thesis, this topic shall be briefly discussed. For more details, we refer to[404].",
"pages": [
67
]
},
{
"title": "· Affinely parametrized null geodesic congruence:",
"content": "A null geodesic refers to a path followed by a massless particle (such as a photon, which has zero rest mass) in the spacetime described by relativistic theory of gravity. Therefore, on spacetime manifold, its tangent, denoted by l µ is null,i.e., l µ l µ = 0. The geodesic equation it satisfies is given by the equation, here λ serves as a parameter along the curve. The geodesic equation signifies that the tangent is parallelly transported to itself as one follows the geodesic. The choice of the parameter λ allows for simplification of the geodesic equation to a more convenient form without loss of generality as, In terms of Christoffel symbols, the above equation can be written as, The variable λ is therefore an affine parameter of the affinely parametrized geodesic equation(2.3). Consider an open region M in the spacetime manifold. A congruence of curves represents a family of curves in which every point within M is traversed by one and only one curve from the family. The tangents to these curves define a vector field on M and conversely, any continuous vector field within M generates a congruence of curves with the tangents of the field. When the field of tangents is smooth, we refer to the congruence as smooth. In particular, we can focus on a smooth congruence of null geodesics whose tangents are represented by λ within the open region M . To mark different geodesics within the congruence in M , we consider another parameter ζ . We now define the deviation vector with components η µ ≡ ∂ x µ ∂ζ . Though by construction, we see l µ ηµ = 0, but that does not mean η µ is orthogonal to the curve, since l µ is a null vector. However, we can limit to deviation vectors that are considered equivalent if they vary solely by a component along l µ . The tangent space, comprising all vectors that are orthogonal in this manner to l µ , forms a 2-dimensional vector space. We can also explore its dual space and the set of tensors constructed using these vectors. The geodesic deviation vector follows the geodesic deviation equation, given by, where R µ ναβ is the Riemann tensor. D λ is covariant derivative operator, D λ η µ = d d λ η µ + Γ µ αβ dx α d λ η β . This mathematical equation expresses how the behaviour of geodesics, rather the deviations of nearby geodesics is impacted by the spacetime curvature. We can obtain the famous Raychaudhuri equation for null geodesics by deriving the rate of change of the gradient l µ ; µ along the geodesic. Let us consider a tensor field [24, 347], B µν satisfies l µ ; µ η ν = B µ ν η µ . B µν is orthogonal to l µ , therefore, B µν l µ = B µν l ν = 0. So it has components only along the transverse of l µ . Now the spatial metric in 2-space orthogonal to l µ can be defined as, where n α is another null vector and normalized as l α n α = -1. The selection of n α is not unique. The only fixed quantity is the null congruence with tangent l α . However, geometric and physically relevant quantities remain independent of the choice of n α . The trace of the tensor B µν , is called expansion of the affinely-parametrized congruence. The expansion tensor is defined as, The transverse tensor ˜ B αβ can be written as sum of symmetric and antisymmetric parts, It can be further decomposed into trace and traceless parts as, In the above equation, σ αβ is known as the shear tensor, and ω αβ is known as the vorticity tensor, The shear and vorticity scalars are, The Raychaudhuri equation that governs the expansion along affinely parametrized null geodesic, The other choices of n α do not affect this equation. The term d θ d λ < 0 implies that expansion decreases with the evolution of congruence, which means null rays will be focused, and d θ d λ > 0 implies that expansion increases with the evolution of congruence, which means null rays will be defocused. Therefore, this equation illustrates how the focusing or defocusing of null rays occurs due to the combined effects of expansion, shear, vorticity, and matter.",
"pages": [
67,
68,
69,
70
]
},
{
"title": "· Non-affinely parametrized null geodesic congruence:",
"content": "The geodesic equation for non-affinely parametrized null geodesic congruence is, The expansion scalar takes the form, The Raychaudhuri equation transforms into the modified form as, A compact and orientable surface possesses two distinct directions perpendicular to it, representing outgoing and ingoing null rays. When a spherical symmetry is present, it naturally prompts an examination of congruences formed by radial outgoing and ingoing null geodesics with their respective tangent fields l α and n α . For non-affinely parametrized congruence of null geodesics, the expansion of null rays l α is given by the following equation, In the above mathematical expression, h αβ serves as a projection metric onto 2-D hypersurface.",
"pages": [
70,
71
]
},
{
"title": "2.1.1.2 Definitions pertain to closed 2-surfaces",
"content": "In the subsequent definitions, θ l and θ n represent the outgoing and ingoing future directed null geodesic congruences respectively. The closure of a surface, often a three-surface that is foliated by marginal surfaces, is what is known as a future apparent horizon. The future apparent horizon is characterized by specific conditions, Therefore, on the apparent horizon, the future-directed outgoing null geodesics stop transmitting outward. This differs from the event horizon in the sense that, it is a quasi-local horizon. It is to be noted that, the inner trapping horizon can be obtained by switching l α , n α and reversing the inequality sign.",
"pages": [
71
]
},
{
"title": "2.1.1.3 Apparent Horizon in FRW Cosmologies",
"content": "The use of notation and terminology can become perplexing when transitioning between black hole horizons and cosmological horizons. It also confuses readers when shifting from observations made by external observers to those positioned within a horizon's domain. For an expanding FLRW space and observer inside the apparent horizon, the notations modify as, The apparent horizon for expanding FRW cosmologies is characterized by the conditions, In comoving co-ordinates, the tangent of outgoing and ingoing radial null geodesics are, respectively. Therefore, the covariant derivatives of these null geodesics are, Using the equation (2.18), we get the expansions of these null geodesic congruences as, Now, from the definition (2.20), we get the cosmological apparent horizon is located at, In terms of proper radius R ≡ ar , the above equation(2.26) can be written as, ̸ It is important to highlight that the definition of the apparent horizon relies exclusively on null geodesic congruences and their expansions, without any consideration of the global causal structure. At times, there may be a temptation to approximate the location of apparent horizons by speculating where the outward radial null rays come to a halt, essentially setting l r = 0. This simplified approach may yield accurate results on occasion, especially with spherically symmetric metrics in Painlevé-Gullstrand coordinates [343]. But it should not be regarded as a substitute for the proper procedure, which entails identifying surfaces where θ l = 0 and θ n = 0. If we were to apply equation(2.21) and assume that n 1 is equal to zero, it would lead to an incorrect inference, indicating the absence of apparent or trapping horizons in FRW cosmologies with k = 0 or k = -1. It would yield an inaccurate value for R h in the case of k = 1 in FRW cosmologies. Clearly, this is not the case. With the exception of the scenario where H = 0, apparent horizons consistently exist and can be determined using equation(2.27). The event horizon may not exist in some cases. Therefore, the radial null geodesic congruences within FRW cosmologies provide a case that contradicts this approach. Unlike the event and particle horizons, the apparent horizon is typically not a null surface. Similar to horizons in flat space, the cosmological apparent horizon is observer-dependent. It operates as a spherical barrier encircling the observer and veiling information.",
"pages": [
71,
72,
73
]
},
{
"title": "2.2 Thermodynamics",
"content": "This section commences with an exploration of the four laws governing the mechanics of black holes and their intricate relationship with the four fundamental laws of thermodynamics. Subsequently, we will delve into the extension of this connection to include the apparent horizon within the context of the FRW cosmological framework.",
"pages": [
74
]
},
{
"title": "2.2.1 Black hole thermodynamics",
"content": "Black hole thermodynamics is a fascinating field that explores the connection between black holes, which are objects with extremely strong gravitational forces, and the laws of thermodynamics, which describe the behavior of energy and entropy in various physical systems. This connection was first suggested by Jacob Bekenstein in 1973 [325] and later developed by Stephen Hawking and others. In 1973, Jim Bardeen, Brandon Carter, and Stephen Hawking formulated a set of four laws governing the behaviour of black holes [345]. These laws of black-hole mechanics bear a striking resemblance to the four laws of thermodynamics. While this analogy was at first perceived to be purely formal and coincidental, it soon became clear that black holes do indeed behave as thermodynamic systems. The crucial step in this realization was Hawking's remarkable discovery of 1974 [344] that quantum processes allow a black hole to emit a thermal flux of particles. It is thus possible for a black hole to be in thermal equilibrium with other thermodynamic systems. The laws of black-hole mechanics, therefore, are nothing but a description of the thermodynamics of black holes. The four laws of black hole mechanics are [24, 345-347], These statements are quoted from reference [347]. The intriguing connection between the four fundamental laws governing black hole mechanics and the well-established principles of thermodynamics has not gone unnoticed. In this relationship, the parameter κ takes on the role of temperature, while A mirrors the concept of entropy, and M serves as an analogy to internal energy. The discovery of Hawking radiation evinced that black holes indeed possess a distinct temperature, which correlates directly with their surface gravity, In a fascinating parallel, the zeroth law can be regarded as a specific manifestation of the analogous thermodynamic concept, signifying that a system achieves thermal equilibrium when it maintains a consistent temperature throughout. Similarly, when we view the first law through the lens of its thermodynamic counterpart, it logically implies that the entropy associated with a black hole should be defined as follows [351, 352], The second law, likewise, emerges as a specific instantiation of its thermodynamic counterpart, asserting the principle that the entropy of an isolated system never experiences a decrease. It is pertinent to acknowledge that Hawking radiation [344] leads to a reduction in the black hole's surface area, a seeming contradiction to the area theorem due to the non-compliance of the radiation's stress-energy tensor with the null energy condition. However, it is important to emphasize that the process of black hole evaporation [353] remains in harmony with the generalized second law, which dictates that the overall entropy, encompassing the entropies of both radiation and black holes, remains conserved. We shall discuss the generalized second law of thermodynamics later in more detail. The revelation that black holes adhere to thermodynamic principles unveils a profound interplay between seemingly distinct realms of science, namely, gravitation, quantum mechanics, and thermodynamics. Remarkably, this intricate relationship continues to be a subject of active investigation and comprehension in contemporary scientific exploration.",
"pages": [
74,
75,
76
]
},
{
"title": "2.2.2 Thermodynamics of Apparent Horizons in FLRW Space",
"content": "At its inception, black hole thermodynamics primarily centered around stationary event horizons. However, its evolution has led to a more comprehensive exploration, now encompassing various horizon types like apparent, trapping, isolated, dynamical, and slowly evolving horizons. Early on, it was recognized that cosmological horizons, with their origins traced back to the static event horizon of de Sitter space [354], possess thermodynamic properties. A substantial amount of research is dedicated to investigating the thermodynamic characteristics of de Sitter spacetime (for thermodynamics applied in de Sitter space see references [354-358]). Additionally, endeavors have been undertaken to broaden these inquiries beyond the confines of de Sitter space [359-361]. Researchers in this field have put forth the argument that this thermodynamic framework is also applicable to FLRW apparent horizons (such as in [362, 363]). Alarge number of investigations adapted the originally formulated thermodynamic equations, tailored for the de Sitter event horizon and the apparent horizon, to suit the dynamic nature of the non-static apparent horizon within FLRW space. It is imperative to emphasize that this apparent horizon does not necessarily align with the event horizon, which may not even have a presence in this specific context. The apparent horizon is frequently regarded as a causal boundary linked to gravitational temperature, entropy, and surface gravity within the evolving spacetime framework. These intriguing concepts have been thoroughly explored in a body of research [362, 364368, 414]. In contrast, previous investigations [364, 366, 369-371] contend that the application of thermodynamics to the event horizon of FRW space encounters challenges in establishing a coherent framework except however in de Sitter space. Moreover, there have been concerted attempts to calculate the Hawking radiation emanating from the apparent horizon within the framework of FLRW cosmology. The studies conducted by Jiang and Zhu [372, 373] and the work by Medved [374] and Cai [375], have delved into this area. Various techniques, including the Hamilton-Jacobi method [343, 376, 377], along with the Parikh-Wilczek approach, initially crafted for the analysis of black hole horizons [378]. There exists an extensive corpus of literature dedicated to exploring the thermodynamic aspects of FRW cosmologies [363, 379-384]. For a comprehensive assessment of the thermodynamic attributes associated with the FLRW apparent horizon, one may refer to Reference [379]. In classical terms, surface gravity finds its definition grounded in the geometric characteristics of the metric tensor. However, it takes on a significant role in the realm of black hole thermodynamics, acting as the constant factor that relates changes in black hole mass (representing internal energy) to variations in the area of the event horizon (directly proportional to entropy)(see section 2.2.1 in this chapter). The ongoing debate surrounding the appropriate definition of black hole mass in non-trivial backgrounds, as delineated in the review [385], naturally extends to the definition of surface gravity. Moreover, surface gravity also emerges in semiclassical contexts, as it essentially corresponds, with minor numerical adjustments, to the Hawking temperature of a black hole. The conventional characterization of surface gravity pertains to a Killing horizon in the context of stationary spacetimes [24, 346]. In static and stationary scenarios, a timelike Killing vector field exists beyond the horizon and becomes null at the horizon itself. In these situations, the definitions of surface gravity align, and they are familiar concepts derived from the analysis of Kerr-Newman black holes in GTR. The surface gravity ( κ ) is defined in relation to the Killing vector ( ξ a ) using the equation [24, 346]: In dynamic situations, the presence of a timelike Killing vector field is absent. Hence the applicability of Killing horizons does not extend to general non-stationary contexts, particularly when dealing with quasi-local horizons rather than event or Killing horizons. In such situations, a suitable notion of surface gravity becomes essential. Multiple definitions of surface gravity can be found in the literature, and it is important to recognize that these definitions are not synonymous. Within the framework of spherical symmetry, the Kodama vector successfully replaces a Killing vector in an evolving system, ultimately leading to the generation of a conserved current and surface gravity. The Kodama vector [387] broadens the applicability of a Killing vector field to spacetimes that lack one, making it a viable substitute for a Killing vector in the thermodynamics associated with evolving horizons. It is essential to emphasize that the Kodama vector is only defined within the context of spherical symmetry (an approach to generalization in non-spherical symmetric spacetimes can be found in reference [386]). The spacetime metric can be represented as, Here a , b = 0, 1 and R is the areal radius. Let us denote the volume of 2-metric h ab by ϵ ab [24]. The definition of Kodama vector [387] is, with K θ = K ϕ = 0. The Kodama vector lies in the (t,R) plane that is orthogonal to the 2-spheres of symmetry. The antisymmetric nature of ϵ ab and the symmetric nature of R ; a R ; b makes K a R ; a = ϵ ab R ; a R ; b vanish. In a static spacetime, the Kodama vector is parallel to the timelike Killing vector. The Kodama vector exhibits zero divergence [387, 388], Consequently, the Kodama energy current( J a ), defined as, becomes conserved covariantly. Therefore, This conservation occurs even if there is no timelike killing vector. This remarkable attribute is at times humorously coined as the Kodama miracle [387, 388]. The Hayward proposition for surface gravity, tailored for spherical symmetry [414], employs the Kodama vector K a , which is consistently applicable in scenarios involving spherical symmetry. This future-directed vector satisfies, The Hayward concept of surface gravity ( κ ko ) associated with a trapping horizon is expressed as follows: This definition stands out due to the uniqueness of the Kodama vector. An equivalent expression to equation (2.38) is, where h denotes the determinant of the 2-metric h ab . This surface gravity is popularly known as the Hayward-Kodama surface gravity. In FRW cosmology, this Hayward-Kodama surface gravity, given in equation (2.39), becomes, For spatially flat FRW metric ( k = 0) this equation takes the form, The Hayward-Kodama dynamic surface gravity ((2.40) and (2.41)) becomes null when the scale factor( a ( t ) ) exhibits the specific condition, where α , β , γ are constants. One such example would be the universe purely dominated by radiation. Thus, based on this surface gravity, we can derive the Hayward-Kodama temperature for spatially flat FRW cosmologies in the following manner:",
"pages": [
76,
77,
78,
79,
80
]
},
{
"title": "· HKtemperature in de Sitter space",
"content": "In Schwarzschild-like coordinates, the spherically symmetric metric is, In this metric, the kodama vector is written as [387, 389], In static Schwarzschild-like coordinates, de Sitter metric is written as, Therefore, from (2.45) and (2.46), the kodama vector can be written as, It is important to observe that the Kodama vector aligns with the timelike Killing vector in de Sitter space. The surface gravity induced by this KillingKodama field is, κ = H = √ Λ 3 . As a consequence, the temperature corresponding to this scenario is, T = H 2 π , which nothing but the famous Hawking temperature [344].",
"pages": [
80
]
},
{
"title": "2.2.3 Generalized Second Law of Thermodynamics",
"content": "Bekenstein proposed generalization of second law of black hole thermodynamics [325, 390, 391]. He formulated the GSL as, \"The sum of the blackhole entropy and the common (ordinary) entropy in the black-hole exterior never decreases\". Bekenstein delved into the realm of black hole physics through the lens of information theory [392-396]. He provided an argument rooted in information theory that lends support to the credibility of the generalized second law. An illustrative instance of the connection between an increase in information and a reduction in entropy can be observed in a common scenario. Consider an ideal gas confined within a container undergoing an isothermal compression process. As the compression takes place, the entropy of the gas notably diminishes, a widely acknowledged fact. However, simultaneously, our knowledge regarding the internal arrangement of the gas increases. Following the compression, the gas molecules become more tightly concentrated, leading to a heightened precision in our awareness of their positions compared to the state before compression. Therefore, though the entropy of the system decreases, the total entropy, i.e., the sum of system's entropy and the entropy of the surrounding does not decrease. In case of black hole, consider a scenario in which an entity carrying a certain measure of conventional entropy descends into a black hole. During this process, the entropy within the observable universe diminishes. This situation appears to challenge the second law of thermodynamics, as an external observer cannot directly confirm whether the overall entropy of the entire universe remains unchanged during this event. Nonetheless, according to insights from the literature, we understand that the black hole's area compensates for the vanishing entity by undergoing an irreversible increase. As a result, it appears reasonable to hypothesize that the second law remains intact, albeit in a more generalized formulation: the aggregate entropy within the external vicinity of the black hole, along with the entropy possessed by the black hole itself, consistently maintains a non-decreasing trend. Researchers have expanded the scope of the Generalized Second Law (GSL) to realms beyond the domain of black hole physics. In earlier research, a second law of thermodynamics applicable to the de Sitter horizon was established by Gibbons and Hawking [354], and this principle was revisited in [397]. Davies [369] explored the event horizon within the context of FLRW space, particularly in the context of General Relativity with a perfect fluid acting as the source. In FRW spacetime, we consider apparent horizon to work with and hence the GSL shapes as: in any physical process, the combined entropy of matter and the horizon must remain constant or increase, i.e., where S tot denotes the total entropy and S matter , S h denotes entropy of matter bounded by the horizon and apparent horizon respectively. For spatially flat FLRW metric, the field equations are (1.16)-(1.17), The entropy of the apparent horizon is assumed to be proportional to area( A ), and given by, The area of the apparent horizon is given by, where we have substituted R h from equation (2.27), and put k = 0. Therefore, the rate of change of entropy of the apparent horizon is given by, For the fluid inside the horizon, first law of thermodynamics applied to a hydrostatic system looks like, where S in , U and V denote the entropy, the internal energy and the volume of the fluid inside the horizon respectively. V is bounded by the apparent horizon, Rate of change of entropy of fluid inside the horizon is, Now inserting T from equation (2.43) and V from equation (2.55) in (2.56), one obtains the expression of ˙ S in as, Therefore rate of change of the total entropy is, Therefore we can find ˙ S for different cosmological models if we know the expression of scale factor a ( t ) .",
"pages": [
80,
81,
82,
83
]
},
{
"title": "2.2.4 Thermodynamic Stability",
"content": "In a hydrodynamic system, a stable equilibrium can be achieved by minimizing the thermodynamic potentials i.e., the internal energy, enthalpy, Helmholtz free energy, and Gibbs free energy. On the other hand, the entropy principle, which says that in any process total entropy is a non-decreasing function, leads to an increase in entropy as the system approaches equilibrium, characterized by the fields becoming time-independent. Therefore, to achieve thermal equilibrium, total entropy is to be maximized. In other words, entropy must be a concave function if the system is to be in stable equilibrium [398-401]. To achieve concavity, the hessian matrix defined as, has to be semi-negative definite. Therefore, all the k th order principle minors of the matrix W are ≤ 0 if k is odd and ≥ 0 if k is even. Hence, the thermodynamic stability requires that the conditions are satisfied together. Using this we explore the thermodynamic stability of a cosmological model that mimics a Λ CDM model. We also find that the transition from the decelerated to the accelerated expansion of the universe is a second-order thermodynamic phase transition for the matter content of the universe.",
"pages": [
83,
84
]
},
{
"title": "3.1 Introduction:",
"content": "In chapter 1, we discussed different dark energy models. In this chapter, we shall consider the two broad classes of quintessence fields as dark energy models and try to understand their thermodynamic behaviour in the context of the Generalized Second Law of Thermodynamics (GSL). The quintessence can be broadly categorized into two main groups known as thawing and freezing models. The behaviour of these models is distinguished by the way the effective equation of state parameter ( w ), evolves over time. The thawing model exhibits an effective EoS parameter ( w ) that initially behaves as almost constant and close to -1. However, as the universe evolves, w undergoes a transformation, transitioning into an evolving state. On the other hand, the freezing model behaves differently. In this model, from an evolving state at the initial stage, w freeze at a constant value -1 in late-time evolution. Among the freezing models, one particularly interesting group is known as the trackers . To get a concise and comprehensible overview of these models refer to [274]. The comparative studies of these models have been done concerning how well they align with observational data [402] and from the viewpoint of stability [403]. The cluster number count comparison of these models has been studied [408]. However the outcome does not provide solid evidence to favor any of these models. In this chapter, we have done comparative studies of thawing and freezing models from the perspective of the generalized second law(GSL) of thermodynamics. We employ a straightforward definition of freezing and thawing models and graph the rate of total entropy change during evolution. The expectation is that this rate remains positive, as per the GSL where total entropy never decreases. Surprisingly, the outcomes demonstrate that both freezing and thawing models encounter a violation of the GSL. We conducted tests using a pure quintessence, followed by a quintessence along with a cold dark matter (CDM), yielding similar results, highlighting the inherent thermodynamic non-compliance of the quintessence field. However, freezing models possess an advantage over thawing models, as the GSL breakdown occurs significantly further back in the past.",
"pages": [
85,
86
]
},
{
"title": "3.2 Quintessence Models:",
"content": "The Einstein field equation takes the form, Throughout this chapter, quantities with the superscripts m and q denote the quantities related to matter and the quintessence field respectively. Stress-energy tensor of the matter part is given by the equation, where ρ , p are the density and pressure of the fluid respectively. Stressenergy tensor of the quintessence field is given by the equation,",
"pages": [
86
]
},
{
"title": "· Metric and Field equations:",
"content": "According to the cosmological principle , the universe is spatially isotropic, homogeneous in the large-scale. Here we assume that the universe is spatially flat. Therefore, we consider the spacetime metric is given by spatially flat (k=0) Friedmann-Robertson-Walker metric(FRW metric), Here Φ is solely a function of cosmic time t . Now the field equations can be explicitly written as, In equations (3.6) and (3.7), ρ Φ , p Φ represent energy density and pressure due to the quintessence field. and The conservation of the energy-momentum tensors of the cosmic fluid leads to, and the conservation related to the quintessence field leads to the KleinGordon (KG) equation for Φ as follows, These equations, (3.10) and (3.11), are not independent from the field equations (3.6)and (3.7). One of the equations (3.10) and (3.11), along with equations (3.6)and (3.7), will yield the other because of Bianchi identity.",
"pages": [
86,
87
]
},
{
"title": "3.3 Generalized Second Law of Thermodynamics:",
"content": "According to Generalized Second Law (GSL) of thermodynamics, the total entropy of the universe, i.e., sum of the horizon entropy and entropy of the fluid inside the horizon does not decrease with time [325, 390, 391]. In mathematical expression, where total entropy is denoted by S tot , entropy of the horizon and fluid inside the horizon is denoted respectively by S h and S in . An overhead dot indicates derivative with respect to the cosmic time t . When studying the kinematics of the universe, it is more sensible to focus on the entropy of the dynamic apparent horizon instead of the teleological event horizon. We have already discussed about the apparent horizon in a concise manner in chapter 2. The following equation represents the entropy of the apparent horizon, where A = 4 π R 2 h is the area of the apparent horizon. In section 2.1.1, we have computed the apparent horizon radius R h . In case a spatially flat FRW spacetime (putting k = 0), we get, Hence, the rate at which the horizon entropy changes is, The mathematical equation that expresses the 1st law of thermodynamics as applied to the matter content by the horizon is, where V h represents the volume and heres V h = 4 3 π R 3 h . From the above equation(3.16), we can write the rate at which the entropy of the fluid within the horizon changes as, where ρ tot and p tot denotes respectively the total energy density and pressure. We assume that the fluid within the horizon is in a state of thermal equilibrium with the horizon. Therefore, the temperature ( T in ) will be equivalent to the apparent horizon temperature ( T h ). Here, we have considered the Hayward-Kodama temperature as the temperature of the dynamic apparent horizon [380, 404-407]. The reason is discussed in chapter 2. The temperature of the dynamic apparent horizon is, Now putting the temperature (3.18) in the equation (3.17), we obtain, This equation can further be simplified using the fields equations(3.6)and(3.7) and written as, Summing up the equations (3.15) and (3.20), we can express the rate of change of the total entropy as,",
"pages": [
88,
89
]
},
{
"title": "3.4 APure Quintessence:",
"content": "At first, only pure quintessence model is taken into account. This implies that the fluid inside the horizon is considered to be the quintessence field only, devoid of baryonic and dark matter( p = ρ = 0). Therefore there are no contributions of energy density and pressure of these matters in the field equation (3.6)and(3.7). At this point, we have two equations to determine three unknowns: a , ϕ , V , since the Klein-Gordon (KG) equation is not an independent equation. To complete the system of equations, we adopt the ansatz for ρ proposed by Carvalho et al [287]. The ansatz says the divergence of the logarithm of energy density is power law dependent on scale factor, where λ and α are parameters. λ is chosen to be positive, but the parameter α can take both positive or negative values. Integrating the above equation, we obtain, where the present value of the scale factor is taken to be 1. Since λ is positive, the energy density of the quintessence field decreases with the evolution. The quintessence energy density reduces to a power law, ρ Φ ( a ) ∝ a -λ in the limit α → 0. Using equation (3.11), the potential V is obtained as (for more details see [287]), The equation of state parameter (EoS) for the scalar-field is defined by w Φ ≡ p Φ ρ Φ . In this model, we get w Φ in terms of scale factor as, ̸ The above EoS is not time-independent quantity (for α = 0). The dependency of time in w Φ comes through the scale factor a . It is to be noticed that, it reduces to a constant, w Φ = -1 + λ 3 in the limit α → 0. Now let us see the behaviour of evolution of w Φ pictorially. We define a function N as N = ln ( a a 0 ) , and plot w Φ against N (fig:3.1(a) & 3.1(b)). From the definition we see that, here N = 0 implies the present epoch, positive and negative N imply the future and past respectively. In the fig:3.1(a), we plot w Φ vs N for positive values of the parameter α , which shows thawing behaviour. And freezing behaviour is shown in the fig:3.1(b), where we plot w Φ vs N for positive values of the parameter α . Since λ is positive, w Φ ⩾ -1 for all values of a irrespective of the sign of α [287]. The equation (3.22) serves as a comprehensive ansatz, encompassing both (a) thawing and freezing behaviours through the parameter α . For α > 0, the scalar field exhibits a thawing behaviour, where the equation of state parameter w Φ starts with a nearly flat value close to -1 in the past and gradually transitions to less negative values. Conversely, for α < 0, the scalar field demonstrates a freezing behaviour, wherein w Φ decreases towards more and more negative values and eventually settles into a plateau near -1 in the future. It is essential to clarify that we have adopted the ansatz (3.22) from the work by Carvalho et al [287], and we have employed the parameter values α , λ , and others from the reference [408]. Now, substituting ρ Φ from equation (3.23) in the Friedmann equation (3.6) with ρ = 0 we obtain the solution for the scale factor as, where Γ ( a , x ) is the well-known upper incomplete gamma function and is defined as, Γ ( a , x ) = ∫ ∞ x z a -1 exp ( -z ) dz . And γ = √ 8 π G 3 ρ Φ ,0 exp ( λ 4 α ) is a constant term. While the solution may be intricate, it serves the purpose of assessing the thermodynamic feasibility of the model. By employing the solution (3.26), it becomes possible to express H and ˙ H in terms of a . Consequently, the equation (3.21) takes the form, In order to assess the thermodynamic feasibility of the model, we proceed to plot ˙ S tot as a function of the cosmic e-folding factor N . As shown in Fig-3.2(a), in the case of thawing quintessence ( α > 0), the total entropy experiences an initial increase up to a certain time, after which it deviates from obeying the Generalized Second Law (GSL). Notably, ˙ S tot rises sharply to an infinitely large value and then abruptly drops to an infinitely large negative value. This pattern is similar for all permissible values of α , with only the range of N indicating the onset of this abnormal behaviour varies. On the other hand, freezing quintessence ( α < 0) demonstrates an opposing behaviour. It satisfies the GSL for the future, as the net entropy increases and eventually stabilizes at a constant value when ˙ S tot approaches zero (as observed in Fig-3.2(b)). However, this convergence is not as rapid as depicted in the figure but is likely to occur asymptotically, as indicated by the zoomed-in version in the inset. Nevertheless, the model fails to comply with GSL in the past. A discontinuity in ˙ S tot is evident, and it takes on negative values, signifying a decrease in S tot during that period. The parameter λ , which governs the rate at which the dark energy density ρ Φ declines (as shown in equation 3.22), possesses a small positive value as indicated in a prior study [408]. Throughout all the figures, we maintain a consistent value of λ = 0.06. We have also tested the model using significantly lower ( λ = 0.01) and higher ( λ = 0.1) values. However, since these variations do not significantly impact the qualitative characteristics, we have omitted them to avoid redundant information.",
"pages": [
89,
90,
91,
92,
93
]
},
{
"title": "3.5 Quintessence with cold dark matter:",
"content": "In this section, we consider a more realistic model, incorporating a pressureless fluid ( p = 0) comprising baryonic matter and cold dark matter in addition to the quintessence matter. Consequently, the field equations (3.6, 3.7) will have the input p = 0. Directly integrating equation (3.10) leads to, where ρ 0 is the energy density of matter at the present epoch. Since the KleinGordon equation is not an independent equation, we are left with two equations (3.6, 3.7), to solve for three variables: a , Φ , V . To close the system of equations, we employ the same ansatz (3.22) as used in the previous section. The expression for the deceleration parameter q , defined by q ≡ -˙ H + H 2 H 2 is given by, Fig-3.3(a) reveals an intriguing possibility for a thawing model that the current accelerated expansion of the universe may only be a temporary phenomenon. According to the plot, the universe is expected to transit back to a decelerated phase in a finite future. This characteristic has been observed previously such as by Carvalho et al [287] and Devi et al [408]. However, in the freezing model, the universe eventually settles into a phase of accelerated expansion after passing through the decelerated phase. It is important to note that these features have already been welldocumented in the literature. However, the primary objective of this study is to assess the thermodynamic viability of the model. In this analysis, we assume that the fluid inside the horizon is in thermal equilibrium with the apparent horizon. To evaluate the rate of change of the total entropy, we use the Hayward-Kodama temperature, as given in equation (3.18), following the same approach as in the previous section. Now, the total energy density is represented as ρ tot = ρ + ρ Φ . The general form of the rate of change of the total entropy remains unchanged, as expressed in equation (3.21). However, due to the modification of the Friedmann equations for the contribution of cold dark matter, the explicit form of ˙ S tot can be obtained as, The qualitative characteristics remain similar to those observed in the case of pure quintessence. As demonstrated in Figure 4(a), the thawing models ( α > 0) will eventually violate the Generalized Second Law (GSL) in some finite future, since ˙ S tot becomes negative at a certain value of N , indicating a decrease in entropy. On the other hand, the freezing models ( α < 0) adhere (a) to GSL in the future. Despite gradually decreasing, ˙ S tot remains positive (as seen in Figure 4(b)), signifying that the entropy is continuously increasing and will eventually stabilize at a constant value in the future, as illustrated in Figure 4(b). However, there is a concern in connection with the past evolution. The rate of entropy change becomes negative at a certain finite time in the past. Nonetheless, this issue can be mitigated by adjusting the parameter α . For instance, by choosing α = -0.3, the discrepancy is observed at z ∼ 10 4 (see Figure 5(a)), which occurs before the onset of matter domination over radiation, rendering this system of equations inapplicable to describe the dynamics of the universe at that stage. Similarly, for α = -0.1, this discrepancy is observed at z ∼ 10 12 , which is far beyond the regime of quintessence along with CDM (see Figure 5(b)). Upon careful inspection of Eq. (3.21), it becomes evident that the term 2 H 2 + ˙ H determines the thermodynamic viability of the models. If ˙ H + 2 H 2 < 0 (i.e., q ≥ 1), the model fails to meet the thermodynamic requirements. Thus, it is evident that freezing models exhibit stronger thermodynamic viability, at least within the relevant period. Similar to the previous section, in this analysis, we have maintained the same value of λ = 0.06 for the figures. The values of λ = 0.01 and 0.1 have been excluded since they do not alter the qualitative aspects. The only noticeable difference is a slight shift in epochs, such as the onset of GSL violation for different values of λ . ",
"pages": [
93,
94,
95,
96
]
},
{
"title": "3.6 Summary and Discussion:",
"content": "In this study, we conduct a comparison between thawing and freezing models regarding their adherence to the Generalized Second Law (GSL) of thermodynamics. To assess the total entropy ( S tot), we combine the horizon entropy with the entropy of the matter enclosed within the horizon. We adopt a simple ansatz [287] to model the evolution of the energy density of the quintessence field. By doing so, we can readily identify the range of parameter values ( α ) that correspond to the thawing and freezing behaviour of the field. We find that both types of models exhibit an incompatibility with GSL. There are instances where the entropy ( S ) actually decreases and does so at a rapid pace. For the freezing models, this breakdown in GSL can occur in a distant past, corresponding to a redshift of z ∼ 10 4 . At such a distant past, a quintessence model in conjunction with cold dark matter does not adequately account for the evolution of the universe, and a dominant contribution from a radiation distribution would be necessary. Hence, this breakdown of GSL may not hold true in such a scenario. On the other hand, for the thawing models, this pathological breakdown of GSL is predicted to happen in a finite future. Thus the major implication here is that the freezing models seem to be more favorable compared to the thawing models from the perspective of thermodynamic viability.",
"pages": [
96,
97,
98
]
},
{
"title": "4.1 Introduction:",
"content": "In this chapter, we discuss the thermodynamic feasibility, especially the viability of GSL in Brans-Dicke Cosmology. As we have discussed in chapter 1, the Brans-Dicke theory (BDT) of gravity [146] is widely recognized and frequently discussed due to its potential in addressing a lot of cosmological issues. Two notable examples include extended inflation [177, 178], which solves the problem of graceful exit in the inflationary scenario, and the potential for driving the late-time acceleration of the universe even in the absence of dark energy [100]. However, BDT has its limitations. One limitation is that the characteristic coupling constant of the theory, denoted as ω , needs to have a very high value to be consistent with local astronomical tests [409]. On the other hand, for successfully addressing cosmological problems, a small value of ω is required. Despite these challenges, the theory continues to attract attention due to its relevance in various real cosmological problems and its formal resemblance to other gravity theories with nonminimal coupling. To make the equations more manageable, we examine them in what is known as the Einstein frame, achieved through a suitable conformal transformation [152]. This results in a scenario where the evolution of matter distribution becomes intertwined with the BD scalar field. Consequently, the Bianchi identity enforces the conservation of both the BD field and the matter distribution as a whole. This characteristic aligns with the notion that there is no inherent reason to assume that dark matter and dark energy evolve independently, without any non-gravitational interaction between them [410]. Some studies have already explored situations in which the BD scalar field interacts with matter [411, 412]. The thermodynamic viability during the matter-dominated phase of the universe, encompassing the transition from decelerated to accelerated expansion is the primary motivation. However, for a comprehensive analysis, the radiation-dominated phase is also taken into consideration. A similar thermodynamic analysis of cosmological models was carried out by Bhattacharya and Debnath [413], albeit in a different context. They examined accelerated expansion in an extended version of BDT, introducing a potential and allowing the BD parameter ω to be a function of the scalar field. In contrast, the present study focuses on the original BDT, capable of driving accelerated expansion on its own. The results of this investigation are highly promising. Specifically, the models, especially those for a universe containing a pressureless fluid, align with exclusively negative values of the parameter ω , which fall within the range where BDT can independently facilitate accelerated expansion [100].",
"pages": [
99,
100
]
},
{
"title": "4.2 Brans-Dicke Theory in Einstein frame",
"content": "The action describes the dynamics of gravity and the Brans-Dicke scalar field, as well as their interaction with matter. In this equation, ϕ is the BDscalar field. The second term represents the kinetic term of the scalar field. Now let us employ the following conformal transformation [152], Then the action takes the form, where ψ = ln ( ϕ ϕ 0 ) and ϕ 0 is a constant. The overhead bar will be omitted henceforth, as the rest of the work is in the revised frame. In this modified version, commonly referred to as the Einstein frame , the field equations take on a notably simplified appearance as follows, This is expressed using units where the quantity 8 π G 0 is set to unity. This conformal transformation reconfigures the mathematical representation of the theory, leading to a formulation in which the complexity of the field equations is significantly reduced. As a result, the Einstein frame becomes a preferred framework for analysis due to its enhanced mathematical tractability. The equation governing the evolution of the scalar field ψ , derived by varying the action (4.3) with respect to ψ , results in the expression, where T represents the trace of the energy-momentum tensor for the matter sector. Let us now consider that the universe is filled with perfect fluid. Therefore, the energy-momentum tensor is given by the equation (1.12). For the sake of clarity, lets rewrite it down here, where ρ , p are the energy density and pressure of the fluid respectively and u α is the unit timelike vector, u µ u µ = -1. In a comoving coordinate system, v µ = δ µ 0 . It is important to note that ρ and p are presented in the revised units. Here a is the scale factor that characterizes the change in the size of the universe over time. The form that the Einstein-Brans-Dicke field equations assume in the version obtained through conformal transformation is as follows, as outlined by Dicke [152], In these equations, the energy density and pressure of the scalar field is represented by ρψ and p ψ . The connection between the energy density and pressure of the scalar field is expressed as, The wave equation that describes the behavior and evolution of the scalar field ψ is given by, In this version, the equations take on a formal similarity to those describing a scenario with two types of matter components: one corresponds to a fluid, while the other represents a massless scalar field denoted as ψ . But the fact is that those do not evolve independently. Hence, we do not have separate conservation equations for these two matters. When considering both the fluid and scalar field together, the continuity equation assumes a form, The subscript t indicates the total quantity, which is composed of the combined contributions from the fluid component and the scalar field component. This equation is not self-standing; instead, it derives from the Bianchi identities.",
"pages": [
100,
101,
102
]
},
{
"title": "4.3 Thermodynamic Quantities:",
"content": "In this chapter also, we check the viability of the model against the Generalised Second Law of Thermodynamics (GSL), according to which the total entropy of the universe can not decrease with time [325, 390, 391]. We assume that the fluid achieves thermal equilibrium with the horizon. In this scenario, the temperature of the fluid inside the horizon ( T in ) is the same as the temperature of the dynamical apparent horizon ( T h ). In the current section, we have adopted Hayward-Kodama temperature [379, 404, 414, 415] as the temperature of the apparent horizon, which is expressed as, In the unit of 8 π G 0 = 1, the entropy of the apparent horizon becomes [362], In flat FRW spacetime, the area A is related to Hubble parameter [362] as, Therefore, rate of change in horizon entropy is The rate of change of entropy of the fluid confined within the horizon can be expressed as, where in flat FRW spacetime the volume confined is, V h = 4 3 π 1 H 3 . By utilizing the equation labeled as Eq. (4.12) along with the field equations provided in Eq. (4.16), we derive the expression for the rate of entropy change within the horizon as represented by the equation, Combining the equations referenced as Eq. (4.15) and Eq. (4.17), we arrive at the expression for the rate of total entropy change, which can be represented as, We have discussed this in detail in Chapter 2. Here we are rewriting these equations in the unit 8 π G 0 = 1, because throughout this chapter we have worked out everything in this unit. In the following step, we will examine two distinct epochs in cosmic evolution, namely the radiation era and the dust era. We will proceed to analyze how the Generalized Second Law (GSL) of thermodynamics is upheld during these specific periods of the evolution of the universe. It is worth noting that research by Mimoso and Diego [416] highlights the difficulty of achieving thermal equilibrium between radiation and the cosmic horizon. This challenge stems from Wien's law, which consistently predicts a wavelength larger than the horizon radius at all times. However, there is a potential for nonrelativistic particles to reach equilibrium based on their masses.",
"pages": [
102,
103,
104
]
},
{
"title": "4.4 Radiation Era:",
"content": "In the scenario of radiation-dominated era, the equation of state of the fluid is characterized by the expression p = 1 3 ρ . As a result of this equation of state, the trace of the stress-energy tensor becomes null. As a consequence, the wave equation labeled as (4.10) can be readily integrated, leading to the derivation of the subsequent relationship: where α is an integration constant. Utilizing the field equations (4.8) and (4.9), and incorporating the equation of state along with equation (4.19), it becomes possible to arrive at, where β is a constant, specifically defined as 2 ω + 3 12 α 2 . Upon integrating the aforementioned equation, we get, where σ is the integration constant. The characteristics of the solutions to this equation exhibit variation contingent upon the positive or negative nature of both σ and β . It is important to observe, as indicated in equation (4.21), that both σ and β cannot take on negative values concurrently.",
"pages": [
104,
105
]
},
{
"title": "· Case-I : when both σ and β positive",
"content": "In this particular scenario, the solution to the wave equation (4.10) can be expressed as follows, By performing the integration of equation (4.21), we derive the connection that establishes the relationship between the scale factor and time as, This equation is involved and it is difficult to write a in terms of t explicitly. Asubscript zero, as usual, indicates the present value of the quantity. These solutions for the scale factor have previously been established and documented in existing literature [417, 418]. Our motive here is not to find the solution but to see how that goes with GSL. Upon substituting the expression of the Hubble parameter ( H ) and the second derivative of the scale factor (a ) into the definition of the deceleration parameter, the resulting outcome is, Consequently, this particular model gives rise to a universe that consistently experiences deceleration over time. This implies that the expansion of the universe slows down progressively as time advances. By utilizing equations (4.18),(4.20)and(4.21), we derive the expression for the rate of total entropy change as follows, Given that both β and σ possess positive values, it becomes evident that the rate of change of total entropy ( ˙ S t ) is negative. This scenario unequivocally indicates that the test of the Generalized Second Law (GSL) is not satisfied in this case. In essence, the observed decrease in total entropy contradicts the requirements of the GSL, highlighting a failure of the test under these conditions.",
"pages": [
105,
106
]
},
{
"title": "· Case-II : σ is positive but β is negative",
"content": "When ω is less than -3/2, the value of β becomes negative. We express this negative value of β as -γ 2 , where γ is a real number. Consequently, equation (4.21) transforms into the following form, The equation presented above indicates the presence of a rebound effect at the value of the scale factor a = γ / √ σ . Moreover, at this bounce point, it is necessary for the proper volume ( a 3 ) to reach a minimum since the second derivative of the scale factor (a ) is positive, given by a = -β a 5 . It is worth noting that the total density and pressure also remain finite at this juncture. Consequently, this rebound mechanism seems to circumvent the singularity commonly associated with the Big Bang model. However, there is an important consideration to make. While the bounce occurs at a = γ / √ σ , the equation (4.26) imposes a constraint, disallowing smaller values of a . This limitation introduces a discontinuity in the model at this rebound point. Under these circumstances, the solution to the wave equation (4.10) is as follows, In this scenario, the solution for the scale factor can be expressed as, Hence, the deceleration parameter can be calculated to yield, Consequently, an accelerated phase emerges within the range of γ 2 / σ < a 2 < 2 γ 2 / σ . Once the square of the scale factor surpasses 2 γ 2 / σ , the universe transitions into a phase of decelerated expansion. Importantly, it is crucial to observe that the validity of the model breaks down when the square of the scale factor is smaller than γ 2 / σ . This particular range of values is where the model does not hold true. By utilizing equations (4.18) and (4.26), we derive the rate of change of total entropy as, Given that the scale factor ( a ) is greater than or equal to γ / √ σ , it follows that the rate of total entropy change remains positive. Consequently, the Generalized Second Law (GSL) is upheld in both the phases of acceleration and deceleration. This implies that the increase in total entropy aligns with the criteria of the GSL throughout these distinct phases of the evolution of the universe.",
"pages": [
106,
107
]
},
{
"title": "· Case-III : σ is negative but β is positive",
"content": "Let us look at the case where σ is negative and is given by σ = -λ 2 . As a result, equation (4.21) undergoes a modification, transforming into the expression, The equation that dictates the variation in the scalar field is as follows, This equation is derived by integrating the wave equation utilizing equation (4.19). The solution for the scale factor can be obtained by a straightforward integration of equation (4.31) as, According to equation (4.31), it is evident that a is not greater than √ β / λ . When a reaches a = √ β / λ , the rate of change ˙ a becomes zero, and the second derivative a equals -λ 5 / β 5/2 . However, the validity of the model does not extend beyond √ β / λ , rendering this point not a true maximum for the scale factor. In reality, the model lacks definition beyond this threshold, and its validity ceases. Due to the absence of a lower bound limit on a , it will inevitably approach a singularity when traced back in time. The deceleration parameter is obtained as, As a is not greater than √ β / λ , it follows that q takes on a positive value. Consequently, this model results in a deceleration phase. Through the utilization of equations (4.18) and (4.31), the rate of change of overall entropy is obtained as, The rate at which the total entropy changes experiences a decrease, as indicated by the fact that a is not greater than √ β / λ . Consequently, this particular model does not align with the Generalized Second Law (GSL) of thermodynamics, which asserts that the total entropy of a closed system either remains constant or increases over time.",
"pages": [
107,
108
]
},
{
"title": "4.5 Dust Era:",
"content": "When the fluid is a pressureless dust, the field equations are expressed as, Substituting p = 0 into the wave equation (4.10) yields the result, After taking the derivative of the field Equation (4.36) and subsequently employing the field Equations (4.36) through (4.37), along with the wave equation (4.38), we arrive at a revised form of the continuity equation, which can be expressed as, The presence of a non-zero term on the right-hand side of the aforementioned equation indicates that the energy densities of two distinct fluids, namely, dust and scalar field, do not independently maintain conservation. However, they do adhere to a conservation equation, expressed as, This signifies that their combined energy conservation is satisfied. Obtaining this overall conservation equation from the field equations is a straightforward process. Upon integrating equation (4.39), the result is as follows, By employing the wave equation to eliminate ρ , the integration of a composite set of field equations leads to, In the above equation, we have introduced ξ as an integration constant. By plugging in the expression for ˙ ψ from the preceding equation into Equation (4.37), we derive, Here, χ = 2 ω + 4 2 ω + 3 stands as a constant linked to the Brans-Dicke parameter ω . To continue our analysis, we can assume, without losing generality, that the constant of integration ξ is set to zero. Consequently, Equation (4.43) transforms into, Now, the equation is amenable to integration, yielding, Here, µ represents a constant arising from the process of integration. Next, by utilizing Equation (4.42), we acquire, Integrating the aforementioned pair of equations leads to the solution for the system, expressed as, Here, t 0 and ψ 0 represent integration constants. Now, by employing the definition of the deceleration parameter, equation (4.44), and the definition of χ , we deduce the expression for the deceleration parameter as follows, By observing equation (4.47), it becomes evident that for the purpose of modeling an expanding universe, both µ and χ 2 + 1 need to take on positive values. The requirement for χ 2 + 1 to be positive can be satisfied through two distinct approaches, and we will consider each of these methods separately.",
"pages": [
108,
109,
110
]
},
{
"title": "· 1st way to achieve χ 2 + 1 to be positive and explore GSL:",
"content": "The expression χ 2 + 1 = 3 ω + 5 2 ω + 3 must be greater than zero for our analysis. In the initial scenario, we examine the conditions under which both 3 ω + 5 and 2 ω + 3 are positive, leading to ω > -3 2 . As a result, we can conclude that the expansion of the universe is accompanied by a decelerating motion in this case. Subsequently, our task involves examining the Generalized Second Law (GSL) within this context. By utilizing equation (4.18), we are able to determine the rate at which the total entropy changes. This equation provides insights into how the total entropy of a system evolves over time, In order for the quantity ˙ S t to exhibit a positive value, it is necessary that the value of ω exceeds -1. This condition on the Brans-Dicke parameter ensures that the total entropy of the system is increasing over time, contributing to a consistent interpretation of the physical processes at play.",
"pages": [
110
]
},
{
"title": "· 2nd way to achieve χ 2 + 1 to be positive and explore GSL:",
"content": "Another approach to satisfy the condition χ 2 + 1 > 0 is when both 3 ω + 5 and 2 ω + 3 are negative, corresponding to ω < -5/3. In this scenario, cosmic acceleration can be achieved within the range of -2 < ω < -5/3. However, it is important to note that for ω < -2, the expansion of the universe remains decelerated, as implied by equation (4.49). The limitations on the potential values of ω align consistently with those deduced by Banerjee and Pavón [100]. From equation (4.50), it becomes apparent that in order for the rate of change ˙ S t to exhibit a positive value, the parameter χ needs to be smaller than 2. Given that ω is constrained to be less than -5/3 in this scenario, it is assured that χ is indeed less than 2. As a result of this, we can confidently conclude that ˙ S t > 0, indicating an increase in the total entropy. Consequently, this particular model adheres to the requirements of the Generalized Second Law (GSL). This alignment signifies that the model upholds a fundamental principle of thermodynamics and consistent physical behavior.",
"pages": [
110,
111
]
},
{
"title": "4.6 Summary and Discussion:",
"content": "This research delves into the thermodynamic characteristics of cosmological models dominated by radiation and dust within the framework of BransDicke theory. Specifically, we consider a spatially flat, homogeneous, and isotropic universe in what is referred to as the Einstein frame , which is the conformally transformed version of Brans-Dicke theory in the so-called Jordan frame . Our focus centers on investigating the validity of the generalized second law of thermodynamics, posited by Bekenstein in the early 1970s [325, 390, 391], which asserts that the combined entropy of the universe, the sum of matter entropy and horizon entropy, cannot decrease with time. Our findings indicate that for a radiation-dominated universe, the solutions derived from Brans-Dicke theory with a positively definite parameter ω fail to uphold the principles of the generalized second law. However, intriguingly, when certain ranges of negative ω values are considered, the model indeed aligns with the requirements of thermodynamics. For a universe dominated by dust, the model does satisfy the generalized second law for specific small negative values of ω . Notably, this range of -2 < ω < -5 3 substantially overlaps with the parameter values required for an accelerated expansion without the need for exotic forms of matter [100]. Perhaps the most intriguing revelation from this investigation is that Brans-Dicke theory finds thermodynamic support precisely within the parameter range of ω that is associated with driving the alleged accelerated expansion of the universe, as opposed to general cases of decelerated expansion.",
"pages": [
111,
112
]
},
{
"title": "5.1 Introduction:",
"content": "The progress in observational cosmology yielded a remarkable outcome, indicating that the current universe is expanding at an accelerated rate [20, 58, 59, 63, 66, 192, 193, 419]. However, the force responsible for countering the attractive nature of ordinary matter and fueling this acceleration remains an enigma. The absence of any theoretical inclination towards any of the proposed agents, referred to as dark energy, has prompted a different perspective on the issue in recent times. Instead of constructing models based on the matter sector and deriving conclusions about the evolution of the universe, this alternative approach formulates the evolution history based on observations and endeavors to deduce the plausible matter sector from that vantage point. This approach, initially proposed by Ellis and Madsen [420] long ago, has gained attention as none of the suggested candidates decisively emerges as the dark energy. Numerous endeavors have been made towards reconstructing cosmological models from physical quantities, such as the dark energy equation of state parameter [274, 279, 280, 421-427], the quintessence potential [428-430], and others. Additionally, attempts have been made to construct models based on kinematical quantities like the deceleration parameter q = -a a ˙ a 2 [431-435], and the jerk parameter j = -1 aH 3 d 3 a d t 3 [436-440], where a denotes the scale factor and H = ˙ a a represents the Hubble parameter. In many of these investigations concerning kinematical quantities, q or j is expressed parametrically, and the parameters are estimated using observational data. Among these cosmographic quantities, the Hubble parameter H has been an observable and is known to evolve over time. With the evolution of the deceleration parameter q itself being observable, attention has turned to the third-order derivative, namely the jerk parameter j . Beyond depicting the evolutionary trajectory, a cosmological model must account for structure formation in terms of matter perturbation growth and must adhere to some fitness test, such as thermodynamic viability. The motivation behind this study is to assess the thermodynamic viability of some specific models reconstructed from the jerk parameter. We focus on two sets of reconstructed jerk models existing in the literature: one without any interaction in the matter sector [439] and the other allowing for the possibility of interaction [440]. The rationale for selecting these models lies in the fact that they deviate from the usual anstaz considered in parametrically reconstructing j . Typically, the ansatz is designed such that the jerk parameter j , as a function of the redshift parameter z , equals -1 at z = 0, which aligns the model with the present-day Λ CDM model. However, the models in [439] and [440] permit any value of j ( z = 0 ) initially, to be determined by actual observational datasets. Hence, these models are more versatile in that sense. Given the reconstructed j , it is possible to find the first derivative of the Hubble parameter and then the Hubble parameter. This, in turn, allows estimation of the rate of change of total entropy, enabling an assessment of whether entropy increases and thereby verifying adherence to the Generalized Second Law (GSL), which asserts that the total entropy of the universe cannot decrease. The total entropy is a combination of the entropy of the fluid enclosed by the horizon and that of the horizon itself [325, 390, 391]. Acritical assumption in these calculations is that cosmic matter is in thermal equilibrium with the horizon, with the temperature of the latter determined by the Hayward-Kodama temperature [404, 414, 415], which we have taken in all the previous chapters. This thermal equilibrium becomes questionable if the matter sector involves a radiation component, as recently demonstrated by Mimoso and Pavón [416]. Importantly, the reconstructed models examined in this study lack a radiation component; they consist of either cold dark matter or a similar matter component where thermal equilibrium holds, alongside a dark energy component driving the acceleration, for which equilibrium is approximately valid [416].",
"pages": [
113,
114,
115
]
},
{
"title": "5.2 Kinematics:",
"content": "We examine a cosmological scenario characterized by a spatially flat, homogeneous, and isotropic FRW model. The metric is given by the equation(3.5). Although the pertinent kinematic quantities are introduced and discussed in detail in Chapter 1, let us reiterate the definitions here for the sake of clarity. As usual the dot denotes differentiation in relation to cosmic time t , which is the argument of all functions. We will now express these quantities in terms of a dimensionless parameter, specifically the redshift parameter denoted as z , defined by 1 + z ≡ a 0 a . The subscript 0 designates the current value of the quantity. The expressions for the deceleration parameter and the jerk parameter as functions of z are as follows, In the entirety of this article, the symbol prime is employed to signify differentiation with respect to the redshift parameter z .",
"pages": [
115
]
},
{
"title": "5.3 Fundamental Thermodynamic Principle",
"content": "As previously stated in the introduction, we will assess the thermodynamic viability of the models by subjecting them to a test based on the Generalized Second Law of Thermodynamics (GSL) [325, 390, 391]. This principle asserts that the overall entropy of the universe remains constant or increases over time. The collective entropy comprises the entropy associated with the horizon and the entropy of the fluid enclosed by the horizon. In our analysis, we make the assumption that the fluid components are in a state of thermal equilibrium with the horizon.",
"pages": [
115,
116
]
},
{
"title": "· Revisiting Hayward-Kodama Temperature:",
"content": "Complete analytic description of Hyaward-Kodama temperature has been provided in chapter 2, section 2.2.2. It seems requied to revisit the fact in brief, why we consider Hayward-Kodama temperature instead of Hawking temperature as the equilibrium temperature. Within a spatially flat FRW cosmological framework, the surface gravity pertaining to the apparent horizon is given by, where κ ko signifies the surface gravity. Thus, based on the preceding analysis, we can determine the Hayward-Kodama temperature as follows, Since our analysis will focus on a spacetime that changes with time, it's important to carefully consider which temperature assumption to use. In this context, we opt for the Hayward-Kodama temperature instead of the Hawking temperature. Moreover, we shall direct our attention to the concept of an apparent horizon, as distinct from an event horizon, for our subsequent discourse. This temperature is particularly relevant when analyzing spacetimes with evolving geometry and kinematics, as it takes into account the dynamic aspects of the system. In thermal equilibrium, the horizon temperature and the temperature of the fluid within the enclosed region are equal. Therefore, temperature of the enclosed fluid is also the Hayward-Kodama temperature. To delve deeper into the theoretical underpinnings of these concepts and their implications, it is beneficial to refer to the existing literature. Relevant works, such as the monograph by Faraoni [404], offer comprehensive discussions on spacetime thermodynamics, temperature concepts, and horizon properties. This reference provides a solid foundation for understanding the motivations and applications of employing the Hayward-Kodama temperature in our analysis and the significance of utilizing the apparent horizon in the context of thermal equilibrium scenarios. In the preceding chapter (Chapter 2), we computed the rate of change in the total entropy. In the current chapter, we are merely restating the expression for subsequent calculations. The rate of change of the total entropy is, To facilitate a dimensionless representation, we express the Hubble parameter in the following manner: We introduce the dimensionless Hubble parameter h ( z ) , which is defined as the ratio of the Hubble parameter at a specific redshift z to the current value of the Hubble parameter H 0 . Mathematically, this can be written as: This dimensionless form of the Hubble parameter offers a normalized perspective on the rate of expansion of the universe at different cosmic epochs. By comparing the Hubble parameter at a particular redshift to its present-day value, we gain insight into the relative pace of cosmic expansion across different eras. This representation aids in simplifying calculations and analyses by removing the explicit dependence on the absolute scale of the Hubble parameter, focusing instead on the relative changes with respect to the current state of the universe. Therefore, now we write the rate of change of entropy in terms of h(z) as, Consequently, armed with an understanding of the progression of h and its rate of change with respect to z , it becomes feasible to assess the adherence to the Generalized Second Law (GSL) for a specific model.",
"pages": [
116,
117
]
},
{
"title": "5.4 Cosmological Models Devoid of an Interactions in the Matter Sector",
"content": "The first subsection delves into an in-depth discussion regarding the specific model that has been selected for examination. Subsequently, the second subsection is dedicated to an exploration of how the Generalized Second Law (GSL) is assessed within the context of these models.",
"pages": [
118
]
},
{
"title": "5.4.1 The model",
"content": "We consider existing dark energy models sourced from existing literature, where the reconstruction is accomplished using the jerk parameter j . Specifically, these models are adapted from the work presented in [439], which we shall brielfly summarize here. What distinguishes these models is that, unlike certain other methods, they don't require a pre-established condition, such as j ( z = 0 ) = -1, that accords with the Λ CDM model. These models do not rely on any specific theory of gravity as a foundational assumption. We select all four jerk parameter parametrizations outlined in [439], as our starting point. These are as follows, Here, j 1 is a constant value to be determined by observational data. The selection of these parametrizations initially appears arbitrary. However, there are specific reasons underlying these choices. In Model I, the dependence on z is not explicitly present; the variation of j is exclusively managed by the function h ( z ) . Models II and III, on the other hand, exemplify instances where j exhibits a direct proportionality to uncomplicated positive powers of ( 1 + z ) . In the fourth model, however, the relationship takes a form where j is inversely proportional to ( 1 + z ) . The rationale underlying these chosen parameterizations becomes evident when the respective parameters are derived from datasets, a process undertaken in [439]. The fundamental notion driving this selection is to opt for straightforward functions of z . This approach aligns with the principles often used for parameterizations of quantities with more tangible physical significance, such as the equation of state parameter. These equations can be integrated to yield, correspondingly. Here, c 1 and c 2 denote integration constants. Nevertheless, it is important to note that these constants are subject to limitations imposed by h 0 ( z ) = 1, a condition derived from the definition of h . Consequently, c 2 can be substituted using the subsequent relationships, The values of the two distinct parameters, namely c 1 and j 1 , are determined through the utilization of observational data. This calibration is conducted using combinations of datasets originating from sources such as SNe (Supernovae), OHD (Observational Hubble Data), BAO (Baryon Acoustic Oscillations), and CMBShift (Cosmic Microwave Background Shift), as meticulously detailed in [439]. These calibrated values are presented in Table 1. It is noteworthy that c 1 serves as Ω m 0 , which signifies the matter density parameter at the current epoch. The determination of both c 1 and j 1 through empirical data plays a pivotal role in establishing the foundation for the subsequent analyses and interpretations of the chosen dark energy models. The equations (5.10-5.13) readily reveal that upon establishing the correspondence between c 1 and the current matter density parameter Ω m 0 , the initial term in all four models inherently represents the independent evolution of pressureless cold matter. This distinctive feature substantiates the characterization of this particular group as non-interacting models, as highlighted in [439]. It is worth emphasizing that both Model I and Model IV encounter challenges concerning their future evolution. This predicament arises due to the presence of singularities at z = -1, a fact discerned from the expressions in equations (5.10) and (5.13). An intriguing observation stems from equation (5.12)is that the behavior of the so-called dark energy in Model III appears reminiscent of the spatial curvature term found in the Friedmann equations. However, it is crucial to recognize that the reconstruction detailed in [439] exclusively takes into account a spatially flat metric as depicted in equation (3.5). Consequently, this similarity is merely a coincidental resemblance rather than a substantial connection. (a) While not within the scope of this current study, it is worth noting that the outcomes of the best fits along with their associated 1 σ error margins, as presented in Table 5.1, were attained through a minimization process using χ 2 in [439].",
"pages": [
118,
119,
120
]
},
{
"title": "5.4.2 Thermodynamic Analysis",
"content": "By substituting the solutions for h ( z ) as provided by equations (5.10)-(5.13) into equation (5.5), we are able to derive the rate of change of total entropy for various models as, Utilizing the values of c 1 and j 1 extracted from Table I, we generate plots that illustrate the relationship between ˙ S tot and the redshift z . For these plots, we adopt a value of H 0 at 70 km s -1 Mpc -1 . It is important to note that the plots are presented on an arbitrary scale, with the primary focus directed towards conveying the qualitative characteristics of the entropy alteration rates. In the depicted plots (with the exception of Model-IV for SNe+OHD+BAO data), a consistent methodology is employed. The thick solid line represents the ˙ S tot values corresponding to the best-fit c 1 and j 1 parameters. This selection aligns with the contour plots outlined in [439], capturing a 3 σ confidence level. Furthermore, within these plots, we maintain the best-fit c 1 value while systematically varying the j 1 parameter. This involves selecting the farthest permissible values of j 1 within the contours established at the 3 σ confidence level, as detailed in [439]. The same strategy is then applied when keeping c 1 fixed at its best-fit value ± 3 σ , while exploring the furthest allowable values for j 1 . In summary, these plots elucidate the variations in ˙ S tot against the redshift z using specific parameter values. The methodology employed ensures a comprehensive exploration of the parameter space, offering insights into the qualitative trends of entropy change rates for distinct models. For Model-IV with the SNe+OHD+BAO data, an examination of the contour plot in [439] unveils a notable observation: it is unfeasible to encompass all the data points within the 3 σ confidence level. In response, we adopt a distinct approach to selecting the ( c 1 , j 1 ) values. Within this context, the more specific procedure is outlined below. By adopting this methodology, we assemble a total of seven ( c 1 , j 1 ) value pairs, encompassing both the best-fit values and the permissible range according to the confidence levels. This approach offers a comprehensive perspective on the rate of entropy change within the region defined by the ( c 1 , j 1 ) contour plots. Figures (5.1), (5.2), and (5.3) distinctly illustrate that the rate of entropy change remains consistently positive. At most, it can approach zero at some point in the future ( z < 0). In the case of Model I, there is a notable sharp surge in entropy in the future, observed for both data set combinations. Despite this surge, ˙ S tot maintains a non-negative trajectory (Figure (5.1)). However, Model IV does not meet this test of compatibility with thermodynamic principles. For certain parameter selections within the permissible range, ˙ S tot exhibits negative values in a future that is very close to z = 0, particularly when the CMBShift data is not incorporated (Figure (5.4)). This indicates a violation of the expected thermodynamic behavior and suggests a lack of compatibility with established principles. Both Model I and Model IV exhibit singular behavior in the rate of entropy change as the redshift approaches z → -1. This outcome arises from the chosen framework outlined in [439], evident from the expressions derived for h 2 as presented in equations (5.10) and (5.13). These equations incorporate terms like ln ( 1 + z ) and 1 1 + z , leading to the observed singularities. However, it is important to note that z = -1 corresponds to a → ∞ , thereby signifying that these singularities do not manifest in any finite future. Consequently, there is no need for concern in this regard. The sudden increase in ˙ S tot observed within the domain 0 > z > -1 for Model I is not a consequence of the chosen framework itself. Instead, this phenomenon emerges due to the specific model parameters that have been determined based on the available datasets.",
"pages": [
120,
121,
122,
123
]
},
{
"title": "5.5 Model Incorporating Interaction in the Matter Sector",
"content": "The first subsection provides a detailed description of the chosen model which allows an interaction in the matter sector which is given in [440]. Following that, the second subsection focuses on the evaluation of the Generalized Second Law (GSL) within the framework of these models.",
"pages": [
124
]
},
{
"title": "5.5.1 The model",
"content": "The initial terms in equations (5.10-5.13) experience changes in proportion to ( 1 + z ) 3 . This characteristic strongly indicates the presence of cold dark matter (CDM) in all the models, which exhibits no interaction with other components in the matter sector. However, there exists a specific model that deviates from this non-interacting CDM constraint, as detailed in the reference [440]. This model introduces a relaxation to the non-interacting CDM scenario by assuming that j is a slowly varying parameter. Consequently, the differential equation (5.2) can be integrated under the assumption that j remains constant. This integration results in the expression, In this context, A represents a dimensionless constant coefficient (potentially corresponding to Ω m 0 ), while the jerk parameter j takes on the role of a model parameter. Through a statistical analysis of various combinations of SNe+OHD+BAO data sets, the subsequent outcomes for the parameters A and j have been derived (for further insights, refer to [440])as, The best fit parameter values are acquired through a conventional minimization of χ 2 , and these values are presented at a 1 σ confidence interval. This specific reconstruction process is elaborated upon in [440]. We will refer to this as model V . Importantly, it should be acknowledged that when j equals -1, the model effectively simplifies to the well-known Λ CDMmodel. The initial term bears a striking resemblance to the evolution pattern of pressureless matter, albeit with subtle distinctions. These deviations from the conventional CDM behavior can be interpreted as resulting from an interaction with another constituent within the matter sector. This underlying concept forms the basis for characterizing this model as an interacting model in [440].",
"pages": [
124,
126
]
},
{
"title": "5.5.2 Thermodynamic Analysis",
"content": "Utilizing the derived solution for h ( z ) as presented in equation (5.22), we can calculate the rate of change of total entropy for model V by applying equation (5.5), yielding: Using the aforementioned optimal parameter values for A and j , which were determined in [440], we proceed to generate a plot of the rate of entropy change against redshift ( z ), complete with error bars, as illustrated in Figure (5.5). Employing a comparable approach as outlined in Section 5.4.2 for models I, II, and III, we select the seven sets of ( A , j ) values. This strategy allows for a comprehensive analysis of ˙ S tot, revealing that it remains consistently positive across the entire range. Furthermore, the Generalized Second Law (GSL) is upheld with considerable fidelity.",
"pages": [
126,
127
]
},
{
"title": "5.6 Summary and Discussion",
"content": "Given the absence of a universally accepted candidate for dark energy or a specific modified theory of gravity that can fully explain the universe's supposed accelerated expansion, a new approach has surfaced. This approach involves reconstructing models to match observed characteristics of the universe based on the kinematical aspects. However, it is essential for these reconstructed models to withstand certain critical evaluations. Among these tests, the examination of thermodynamic consistency is of utmost significance. Fortunately, it is noteworthy that the assessment of the Generalized Second Law (GSL) of thermodynamics can be effectively conducted by utilizing kinematical quantities such as the Hubble parameter and its corresponding derivative, as outlined in equation (5.5). This thermodynamic viability analysis ensures that the reconstructed models align with the principles of thermodynamics. Our investigation reveals that it is indeed feasible to identify cosmological models that uphold thermodynamic consistency through the reconstruction of the jerk parameter. Notably, within the subset of four models that pertain to the non-interacting scenario, only one-specifically, model IV - exhibits a decline in entropy within the future epoch ( z < 0) due to its inverse dependence on ( 1 + z ) . This phenomenon occurs in close proximity to the present era ( z = 0), as depicted in figure (5.4(a)). On the other hand, all the remaining models, including the one that entertains the prospect of interaction within the dark sector (Model V), satisfactorily adhere to the Generalized Second Law (GSL) of thermodynamics. Model I, while it does conform to the GSL, displays an abrupt surge in entropy within the future ( z < 0). Models II and III exhibit well-behaved attributes in all respects, as does Model V, which permits an interaction within the dark sector. The graphical representations encompass a broad parameter space, encompassing the scope of variation within the 3 σ range. Nonetheless, it is important to highlight that the assessment of entropy evolution is predominantly centered around the redshift z = 0, and this evaluation is particularly reliable for values of z greater than zero. The anomalies and irregularities we identify predominantly manifest for values of z that are less than zero. Until the current epoch, all the models exhibit commendable behavior without any significant issues. It is worth highlighting that this study operates under the implicit assumption that the temperature of the horizon coincides with that of the fluid. However, this assumption might not hold true in cases involving a radiation distribution. Nevertheless, the scope of this investigation is limited to scenarios involving non-relativistic matter, with models I to IV encompassing a pressureless fluid and model V involving a similar component. In the case of the former, the equality of temperatures is indeed accurate, and even for the latter, it remains reasonable at the very least, as discussed in reference [416].",
"pages": [
127,
128
]
},
{
"title": "6.1 Introduction:",
"content": "In this chapter, we delve into a comprehensive examination of thermodynamics for an accelerating universe. The scale factor is modeled using a hyperbolic function ( a ∼ sinh 2 3 ( t / t 0 ) ). This chosen mathematical representation closely emulates the characteristic behaviour of a Λ CDM model. A notable distinction from the preceding chapters lies in the fact that our current study encompasses not only the application of the Generalized Second Law (GSL) test but also extends to an exploration of the thermodynamic stability of the system. This entails a deeper understanding of how the system evolves and behaves from a thermodynamic perspective. Our calculation has led to an exciting result. Through careful examination, it has emerged that despite the entropy ( S ) retaining its continuity, a discontinuity becomes evident in the thermal capacity at constant volume ( C V ) at a specific value of redshift ( z ). This notable occurrence coincides with the point at which the cosmological evolution undergoes a shift from a decelerated state to an accelerated state of expansion. This discontinuity indicates a second-order phase transition. The deceleration parameter ( q ) plays a crucial role similar to an order parameter. This parameter encapsulates the essence of the transition between the two different expansion states. Moreover, the discontinuity seen in C V is characterized by an order of unity. Essentially, our analysis highlights a striking connection between the thermodynamic features of the system and the evolution of the universe. The significant change in thermal capacity at a particular redshift indicates a transformative phase transition, supported by its alignment with an order parameter and the inherent order of the discontinuity. This finding deepens our understanding of the complex interplay between cosmological dynamics and thermodynamic behavior, bringing together different ideas into a cohesive framework that enhances our comprehension of the evolution of the universe. As in the preceding chapters, we continue to center our analysis on the apparent horizon. There is a notable difference between the context of a stationary black hole and the evolving nature of the apparent horizon. This distinctive feature serves as the motivation for replacing the Hawking temperature with the Hayward-Kodama temperature [414, 415] to designate the temperature of the horizon. It is worth highlighting that recent investigations into evolving black holes within a de-Sitter spacetime have indicated the presence of a secondorder phase transition [441-445]. This observation adds further credence to the potential significance of the connection we have identified between the signature flip in the deceleration parameter ( q ) and the occurrence of a second-order phase transition. The stability of the model's thermodynamics can be determined by examining the characteristics of the second-order derivatives pertaining to the system's internal entropy. For examples that demonstrate this approach, readers may refer to the references provided in [446] and [447]. The primary impetus behind the current study is to initially explore the thermodynamic stability of a cosmological model designed to replicate the late-stage evolution characteristics of a Λ CDMmodel. The methodology employed involves investigating the concavity of the entropy function associated with the matter content present in the universe. To undertake this investigation, we delve into the intricacies of the behaviour of entropy and its relationship with the matter content. The key technique revolves around scrutinizing the concavity properties of the entropy function. This analysis hinges on the properties of the Hessian matrix, which encompasses the second-order derivatives of the entropy. For a more comprehensive understanding, we refer to Chapter (2), Section 2.2.4. To provide a concrete example of this approach within the realm of cosmology, reference can be made to the work conducted by Bhandari, Haldar, and Chakraborty [448]. This framework allows us to establish a connection between the thermodynamic stability of the cosmological model and the underlying properties of the entropy function, thereby offering insights into the system's overall behaviour and evolution.",
"pages": [
129,
130,
131
]
},
{
"title": "6.2 A Model Mimicking the Characteristics of Λ CDM",
"content": "We consider a universe that is spatially flat, homogeneous, and isotropic, as described by the FRW metric presented in equation(3.5). Let us reiterate the metric formulation here, a ( t ) denotes the scale factor here. The Einstein field equations associated with this context can be expressed as follows: In these equations, ρ and p symbolize the total energy density and pressure attributed to the matter constituents. The parameter H = ˙ a a corresponds to the Hubble parameter, while a dot positioned above a variable signifies its derivative with respect to cosmic time t . We adopt units where c = 1 and 8 π G = 1. In this context, the radius of the apparent horizon, denoted as R ah , is defined by the equation g µν R ah , µ R ah , ν = 0. For a universe with spatial flatness ( k = 0), the radius of the apparent horizon is given by R ah = 1 H , as detailed in the work by Faraoni [404]. We adopt a straightforward assumption for the scale factor, given by the expression: This choice results in an expansion that is accelerated during later periods while a decelerated expansion in the earlier era is dominated by matter. Here, it is important to note that we consider a = a 0 at the time t = t 0 , and we have set t 0 = 1. Remarkably, this particular ansatz (6.4) effectively captures the behaviour akin to the Λ CDM model during the later stages, which is currently favored as the model for our present universe [187]. The equation (6.4) can be utilized to express the cosmic time t / t 0 in terms of the redshift ( z ), wherein z is defined as 1 + z = a 0 a . This relationship can be written as t / t 0 = arcsinh ( ( 1 1 + z ) 3/2 sinh ( 1 ) ) , effectively establishing a connection between time and redshift, where a 0 signifies the present value of the scale factor. The Hubble parameter can be expressed in terms of the redshift z as follows, The deceleration parameter, which is defined as q = -[ 1 + ˙ H H 2 ] , takes the form: when expressed in terms of the redshift z . In Figure (1), we observe the variation of the deceleration parameter ( q ) with respect to the redshift ( z ). Notably, at a specific redshift value of approximately z = -1 + 2 1/3 sinh 2/3 ( 1 ) ≃ 0.403 the evolution of the universe transitions from a decelerating phase to an accelerating one.",
"pages": [
131,
132
]
},
{
"title": "6.3 Thermodynamic approach",
"content": "We have assumed that the fluid within the horizon is in a state of thermodynamic equilibrium with the horizon itself. The conclusions drawn from the research conducted by Mimoso and Pavón [416] provide insights into the topic at hand. Their study establishes that achieving thermal equilibrium between radiation and the cosmic horizon remains an elusive endeavor. This challenge arises due to Wien's law, which consistently yields a wavelength exceeding the horizon radius across all temporal phases. However, it is possible for nonrelativistic particles to reach equilibrium, depending on their individual masses. In this regard, the notion of thermal equilibrium between dark energy and the horizon, advocated by various scholars, such as in [371, 449-454], finds justifiable ground. In the present investigation, it is pertinent to note that we have excluded any radiation component from consideration. Consequently, the assumption of establishing thermodynamic equilibrium between the horizon and the fluid content remains valid and applicable in our study. This rationalizes our approach and lends support to the foundational assumptions guiding our analysis. We consider the temperature at equilibrium to be determined by the Hayward-Kodama temperature formula, as established in the references by [414, 415]: It is worth noting that the temperature becomes zero when the scale factor takes the specific form of a ( t ) = √ α t 2 + β t + γ . Consequently, during a phase dominated solely by radiation (for which a ( t ) ∝ t 1/2 ), equation (6.7) results in a temperature of zero. However, there is no cause for concern in the context of our present study, as we do not engage with radiation-related considerations in any manner. In Chapter 2, we derived the expression for the rate of change of the total entropy, which can be written as follows, The Hessian matrix, denoted as W , pertains to the entropy S in can be written as follows, In the above matrix, a subscript indicates a partial derivative with respect to the specific variable ( U , V ). Hence, if we express the Hessian matrix explicitly using the partial derivative notation, we obtain, Therefore, in order to ensure thermodynamic stability, it becomes imperative the simultaneous satisfaction of the following conditions, In terms of thermodynamic parameters, we get the conditions as, and For the sake of conciseness, the second expression is denoted as α . In these equations, T represents temperature, C V stands for heat capacity at constant volume, and β T indicates isothermal compressibility. Heat capacity at constant volume ( C V ) is a thermodynamic property that quantifies the amount of heat energy required to produce a unit change in its temperature while keeping its volume constant. It is defined as, Heat capacity at constant pressure ( C P ) is a thermodynamic property that measures the amount of heat energy required to produce a unit change in its temperature while allowing it to expand or contract under constant pressure conditions. It is defined as, Isothermal compressibility ( β T ) is a thermodynamic property that quantifies a substance's responsiveness to changes in pressure while its temperature is held constant. It measures the fractional change in volume of a substance in response to a unit change in pressure, keeping the temperature constant. Isothermal compressibility is defined as,",
"pages": [
133,
134,
135
]
},
{
"title": "6.4 Thermodynamic analysis of the model",
"content": "We will examine whether the model adheres to the GSL and assess its thermodynamic stability. We will explore the characteristics exhibited by various thermodynamic parameters. We substitute the expression of H (as given in equation (6.5)) into equation (6.8), resulting in the following expression, Based on the aforementioned mathematical expression, it is evident that entropy increases with time. Consequently, the GSL remains valid for the model. When considering the fluid confined within the horizon, the expression of Gibbs' law takes the form: This equation serves as a cornerstone in comprehending the thermodynamic processes occurring within the confines of the event horizon. By utilizing the equation (6.19), one can compute the heat capacities and isothermal compressibility for the material enclosed within the event horizon. . Figures (6.2) and (6.3) illustrate the behaviour of C V and C P , respectively, with respect to the redshift z within the range of low values (0 ≤ z ≤ 1). Utilizing the expressions for C V and β T derived from equations (6.20) and (6.22) in equations (6.13) and (6.14), it is possible to generate plots for S in UU and α , as depicted in figures (6.4) and (6.5), respectively. It is evident that the two conditions (6.11) and (6.12) are not simultaneously met within the specified low redshift range (0 ≤ z ≤ 1). Consequently, the model does not exhibit thermodynamic stability within this particular redshift range. The significant observation gleaned from figure (6.2) is that heat capacity at constant volume ( C V ) displays a noteworthy behaviour - specifically, it exhibits a divergence or discontinuity at a particular redshift value of z = -1 + 2 1/3 sinh 2/3 ( 1 ) ≃ 0.403. This redshift value corresponds to the critical point where the expansion of the universe transitions from a decelerated phase to an accelerated one. Remarkably, this transition holds the characteristics of a thermodynamic phase transition. The transition from decelerated to accelerated expansion, it turns out, aligns with a distinct thermodynamic phase transition. Notably, the entropy ( S mboxin ) does not exhibit any corresponding discontinuity within the specified redshift range; rather, the discontinuity manifests in the behaviour of C V . Consequently, this phase transition is unequivocally identified as a second-order phase transition. It is noteworthy to highlight that C V assumes a negative value for the present universe, specifically for z > -1 + 2 1/3 sinh 2/3 ( 1 ) ≃ 0.403. Nevertheless, the presence of a negative heat capacity in gravitational systems is not a surprising phenomenon (for a comprehensive overview, we refer to [455]). A study conducted by Luongo and Quevedo [456] yielded a significant finding that in the context of a currently accelerating universe, a negative value for C V becomes a requisite. By substituting equation (6.6) into equation (6.20), it becomes possible to express C V in terms of the deceleration parameter ( q ) as follows, Therefore, it becomes evident that the origin of the discontinuity in C V arises from the presence of the deceleration parameter ( q ) in the denominator with a positive exponent of 1. As a consequence, the deceleration parameter ( q ) serves as the order parameter, and the observed discontinuity is characterized by an order of unity.",
"pages": [
135,
136,
137,
138
]
},
{
"title": "6.5 Summary and Discussion",
"content": "The focus of the thermodynamic stability analysis was directed toward a model designed to emulate the behaviour of the Λ CDMmodel, which faithfully represents the present universe. Given the dynamic nature of the evolving horizon, the Hayward-Kodama temperature was chosen as the appropriate measure of the temperature of the horizon. The analysis revealed that the thermal capacity exhibited a negative value, implying that the cosmic matter contained within the horizon lacks thermodynamic stability. The significance of this study lies in the profound result indicating that the matter content experiences a phase transition precisely at the point where the universe undergoes a transition from decelerated expansion to an accelerated one. Remarkably, this phase transition follows the characteristics of a second-order transition, as evidenced by the discontinuity in the heat capacity ( C V ). The deceleration parameter q serves as the order parameter. Interestingly, the earlier studies failed to detect the second-order phase transition at the onset of the accelerated expansion of the universe could potentially be attributed to the utilization of the Hawking temperature of the horizon. This approach overlooked the fact that the apparent horizon is in a state of evolution, which the Hayward-Kodama temperature accurately addresses. Pavón and Wang [457] showed that the dark matter and dark energy could potentially evolve independently. Therefore, they may not be in thermal equilibrium with each other. However, it is important to note that our approach considers a composite fluid in which distinct sectors are not explicitly differentiated, only the evolution history holds significance.",
"pages": [
139
]
},
{
"title": "Conclusions and Outlook",
"content": "This dissertation centers on the exploration of thermodynamic analyses in the context of cosmology. This thermodynamic exploration involves two main aspects: assessment through the Generalized Second Law (GSL)(Chapters 3,4,5) and the analysis of thermodynamic stability (Chapter6). For the purpose of our research, we have made the assumption that the fluid contained within the horizon is in a state of thermodynamic equilibrium with the horizon itself. Thermal equilibrium between radiation and the cosmic horizon is impossible due to Wien's law, which consistently yields a wavelength larger than the horizon radius over all time periods [416]. Nevertheless, nonrelativistic particles can achieve equilibrium at a certain expansion point based on particle mass. Given this, the notion of thermal equilibrium between dark energy and the horizon, proposed by various researchers, has a valid basis. Our study excludes radiation (except for a part of chapter 4), affirming the validity of assuming thermodynamic equilibrium between the horizon and fluid content. As the universe is evolving, we have considered the dynamic apparent horizon, instead of the event horizon to work with. The temperature of the apparent horizon is considered to be the Hayward-Kodama temperature. Hayward proposed the definition of this temperature linked through an alternative definition of surface gravity that applies to dynamic, spherically symmetric spacetimes, which relies on the Kodama vector. This temperature, T = 2 H 2 + ˙ H 4 π H serves as the equilibrium temperature [414, 415]. Initially, the generalised second law of thermodynamics was proposed by Bekenstein in the early 1970s [325, 390, 391]. This principle asserts that the total entropy of the universe, the sum of matter entropy and horizon entropy, must never diminish as time progresses. Therefore, we have subjected various cosmological models to scrutiny using the GSL criterion, identifying which models successfully meet the test requirements or determining the constraints needed for passing the test. Our next course of action involved conducting a thermodynamic stability analysis on a model. This involved utilizing the property of the concavity of the entropy, wherein we examined the Hessian matrix of entropy to ascertain whether it exhibited semi-negative definiteness. In chapter 3, we have undertaken a comparative analysis between thawing and freezing models, focusing on their adherence to the fundamental principles of thermodynamics, specifically the GSL. Our work involves the assessment of the total entropy ( S tot), accomplished by adding up the entropy of the cosmic horizon with that of the enclosed matter within said horizon. To facilitate this study, we employ a straightforward ansatz, 1 ρ Φ ∂ρ Φ ∂ a = -λ a 1 -2 α , proposed by Carvalho et al [287], to formulate the evolution of the energy density of the quintessence field. The ansatz proposes that the power-law dependence of the scalar field is reflected in the divergence of the logarithm of energy density. This approach enables us to effectively delineate the parameter space ( α ) that corresponds to the distinct behaviors of thawing and freezing of the field. The findings reveal an intriguing incongruity between both model categories and the GSL. Notably, there are instances in which the entropy ( S ) experiences a decrease, and this descent occurs at an accelerated rate. For the freezing models, this contravention of the GSL is observed in a remote cosmic past, specifically during an epoch characterized by a redshift of z ∼ 10 4 . During this epoch, a cosmological model that combines quintessence and cold dark matter fails to satisfactorily elucidate the cosmic evolution, necessitating a dominant contribution from a radiation distribution. Consequently, the applicability of the GSL seems questionable in such a scenario. Conversely, in the context of thawing models, our analysis predicts an anomalous breakdown of the GSL within a finite future. This observation underscores a pivotal implication: freezing models exhibit a thermodynamically more tenable model when compared to their thawing counterparts. In chapter 4, our research thoroughly investigates the thermodynamic properties of cosmological models governed by radiation and dust dominance. These models are analyzed in the framework of the Brans-Dicke theory. In particular, we delve into the properties of a spatially flat, homogenous, and isotropic universe within the Einstein frame . This frame represents the conformally transformed version of the Brans-Dicke theory. In a universe dominated by radiation, the solutions obtained from Brans-Dicke theory, with a positive parameter ω , fail to satisfy the principles of the generalized second law. However, remarkably, when specific ranges of negative ω values are considered, the model harmonizes effectively with the requisites of thermodynamics. This finding is very promising because negative values of the parameter ω have been strongly associated with the phenomenon of accelerated cosmic expansion. Shifting the focus to a universe governed predominantly by dust, the model successfully upholds the principles of the generalized second law, but notably, only for certain small negative values of ω . Particularly noteworthy is the fact that this range, characterized by -2 < ω < -5 3 , extensively overlaps with the parameter values necessary to elucidate an accelerated expansion of the universe, all without necessitating the introduction of exotic forms of matter [100]. Chapter 5 is about the thermodynamic assessment of the models reconstructed from the jerk parameter, proposed by Mukherjee et al. [439, 440]. Encouragingly, the assessment of the Generalized Second Law (GSL) of thermodynamics can be effectively conducted using kinematical quantities like the Hubble parameter and its derivative, as described in equation (3.21). This analysis of thermodynamic viability ensures that the reconstructed models adhere to thermodynamic principles, strengthening their credibility. Our investigation reveals the feasibility of identifying cosmological models that maintain thermodynamic consistency through the reconstruction of the jerk parameter. Notably, among the subset of four non-interacting models, model IV-exhibits a future decline in entropy ( z < 0) due to its inverse dependence on ( 1 + z ) . This behavior occurs in proximity to the present era ( z = 0). However, all other models, including the one involving interaction within the dark sector (Model V), adhere to the Generalized Second Law (GSL) of thermodynamics. Model I, while in line with the GSL, demonstrates a significant and sudden entropy increase in the future ( z < 0). Models II and III exhibit consistent behavior, as does Model V, which allows interaction within the dark sector. Chapter 6 deviates slightly from the preceding chapters in that it encompasses a dual focus. Here, our exploration goes beyond solely examining the feasibility of the Generalized Second Law (GSL). Instead, we make an attempt to analyze the thermodynamic stability of an accelerating cosmological model. The thermodynamic stability analysis focused on a model crafted to mimic the behavior of the Λ CDM model, favoured model for the portrayal of the current state of the universe. In light of the dynamic nature of the evolving horizon, the Hayward-Kodama temperature emerged as the suitable parameter to gauge the temperature of the horizon. The findings of the analysis brought forth a significant revelation, the thermal capacity exhibited a negative value, indicating an absence of thermodynamic stability within the cosmic matter confined by the horizon. The interesting outcome lies in the fact that the thermodynamic phase transition of matter content is intimately connected with the nature of the cosmic evolution. The results suggest a thermodynamic phase transition that aligns perfectly with the shift from a decelerated cosmic expansion to an accelerated one. Notably, this phase transition demonstrates the characteristics of a second-order transition, as evident from the discontinuity observed in the heat capacity ( C V ). The deceleration parameter ( q ) serves as the order parameter in this intricate connection. The aim of this investigation was merely to align the observed value of z with q = 0, but rather to delve into the qualitative nature of the thermodynamic aspect accompanying the pivotal shift in q . Apparently, the prior investigations failed to detect the second-order phase transition at the onset of the accelerated expansion of the universe because the dynamical nature of the horizon had been ignored and Hawking temperature was relied upon. The use of Hayward-Kodama temperature brings out the remarkable feature.",
"pages": [
141,
142,
143,
144
]
},
{
"title": "Future Prospects",
"content": "In this thesis, we have explored the thermodynamics of some cosmological models, especially those that give rise to a late-time cosmic acceleration. We have examined the viability of these models via GSL. We have considered the thermodynamic equilibrium between the apparent horizon and the matter bounded by the horizon. Haward-Kodama (HK) temperature has been considered as the horizon temperature. It will be interesting to proceed with the thermodynamic analysis by considering another horizon temperature such as Cai-Kim (CK) temperature [375, 458] instead of the Hayward-Kodama temperature. One can also do a comparative analysis between the outcomes of using the HK temperature and CK temperature. One may take the thermodynamic analysis to the next step and explore GSL in non-equilibrium thermodynamics. As shown in the article [457], dark matter and dark energy may evolve independently and therefore they may not be in thermal equilibrium; hence it will be worthwhile to consider these two fluids separately and then proceed with the thermodynamic analysis. Thermodynamic stability analysis can also be conducted for other models. It will be interesting to see what thermodynamic features these models reveal. We have tried to see if the transition from decelerated to accelerated cosmic expansion is connected to the phase transition without considering any model. But we have not progressed much in this project.",
"pages": [
144,
145
]
},
{
"title": "Bibliography",
"content": "99. Riazuelo, A. & Uzan, J.-P. Cosmological observations in scalar - tensor quintessence. Phys. Rev. D 66, 023525. arXiv: astro-ph/0107386 (2002).",
"pages": [
154
]
}
]
2024arXiv240608078L
https://arxiv.org/pdf/2406.08078.pdf
<document>
<section_header_level_1><location><page_1><loc_9><loc_90><loc_92><loc_93></location>Analyzing the effect of higher dimensions on the black hole silhouette, deflection angles, and PINN approximated quasinormal modes</section_header_level_1>
<text><location><page_1><loc_17><loc_87><loc_83><loc_89></location>Nikko John Leo S. Lobos, 1, ∗ Anele M. Ncube, 2, † Reggie C. Pantig, 3, ‡ and Alan S. Cornell 2, §</text>
<text><location><page_1><loc_24><loc_85><loc_77><loc_87></location>1 Mathematics and Physics Department, Technological Institute of the Philippines,</text>
<text><location><page_1><loc_33><loc_84><loc_67><loc_85></location>363 Casal St, Quiapo, Manila, 1001 Metro Manila</text>
<text><location><page_1><loc_17><loc_81><loc_83><loc_84></location>2 Department of Physics, University of Johannesburg, PO Box 524, Auckland Park 2006, South Africa 3 Physics Department, Map´ua University, 658 Muralla St., Intramuros, Manila 1002, Philippines</text>
<text><location><page_1><loc_18><loc_67><loc_83><loc_80></location>We investigate the impact of higher dimensions on the properties of Schwarzschild-Tangherlini black holes, focusing on the photonsphere, black hole shadow, deflection angles, and quasinormal modes (QNMs). We find that these properties diminish as the dimensionality ( n ) of the black hole increases. Analysis of the shadow radius measured by the Event Horizon Telescope suggests non-integer dimensions around n ≶ 4 . We derive an analytic formula for the weak field deflection angle, highlighting the need for advanced sensitive detection devices to observe lensed images influenced by higher dimensions. Our study of QNMs using physics-informed neural networks and the WKB method reveals a convergence towards known relationships between QNM frequencies and photon-sphere orbit frequencies. Despite the energetic nature of perturbing fields in higher dimensions, their damping increases. This suggests a complex interplay between dimensionality and the dynamics of black hole phenomena.</text>
<text><location><page_1><loc_18><loc_64><loc_72><loc_66></location>PACS numbers: 95.30.Sf, 04.70.-s, 97.60.Lf, 04.50.+h Keywords: Supermassive black holes; black hole shadow, deflection angle, quasinormal modes</text>
<section_header_level_1><location><page_1><loc_43><loc_57><loc_58><loc_58></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_40><loc_92><loc_55></location>Black holes, enigmatic cosmic entities born from the gravitational collapse of massive stars, have long captivated the imagination of scientists and the general public alike. These celestial phenomena serve as unique laboratories, pushing the boundaries of our understanding of gravity and the very fabric of spacetime. Black holes become crucial probes for testing the predictions of Einstein's general theory of relativity, unraveling the mysteries of galactic evolution, and offering insights into the larger cosmic landscape [1, 2]. In this pursuit, black holes are indispensable tools for verifying the robustness of general relativity in extreme conditions. The Event Horizon Telescope's (EHT) imaging of the supermassive black hole in the M87 galaxy in 2019 provides a compelling visual confirmation of theoretical predictions concerning event horizons and photonspheres [3-5], which affirms the seminal works of Synge, Luminet, and Falcke [6-8]. Another success immediately followed in 2022, as the black hole at the heart of our galaxy was confirmed [9, 10]. In addition to the shadow detection, gravitational waves were detected by LIGO and Virgo from black hole mergers has offered unprecedented evidence supporting general relativity's predictions about spacetime dynamics [11-13], and opening a new era in astronomy.</text>
<text><location><page_1><loc_9><loc_16><loc_92><loc_39></location>As such, in exploring black holes beyond the confines of our familiar four-dimensional spacetime, we are compelled by several reasons: (1) Theoretical frameworks like string theory propose the existence of extra spatial dimensions beyond the familiar three [14, 15]. Black holes in higher dimensions emerge naturally from these theories, and are often invoked in discussions of quantum gravity (See Ref. [16] for a comprehensive review). Studying Schwarzschild black holes in extra-dimensions helps probe the quantum nature of spacetime and gravity, potentially providing clues about the microscopic structure of black holes; (2) Higher-dimensional gravity theories, such as Kaluza-Klein theory, aim to unify gravity with other fundamental forces. Black holes in higher dimensions are key in testing and understanding these unified theories; (3) Some theoretical models involve the concept of higher-dimensional spacetimes with branes representing our visible universe. Black holes in these scenarios can have unique properties and effects that differ from their counterparts in standard four-dimensional spacetime; and (4) Studying black holes in higher dimensions is not only motivated by theoretical physics but also by mathematical interest. Higher-dimensional spacetimes can reveal new insights and challenges in understanding geometry and general relativity. These motivations, led to several studies present in the literature [16-27] as the scientific community tries to uncover the effects of higher dimensions on certain aspects of the black hole geometry. Investigating black holes in extra dimensions may help identify unique signatures that could be observed or tested through astrophysical observations such as in the black hole shadow [28, 29], gravitational lensing and optical properties [30, 31], quasinormal modes [32, 33], thermodynamics [34-36], etc [37-41].</text>
<text><location><page_1><loc_9><loc_9><loc_10><loc_10></location>§</text>
<text><location><page_2><loc_9><loc_85><loc_92><loc_93></location>It is understood that the higher dimensionality of black holes is theoretically and mathematically appealing. With the recent constraints for the radius of the invisible shadow provided by the EHT, our aim in this paper is to constrain the dimensionality of the black hole and whether some of the uncertainties in the shadow radius may be caused by the higher dimensionality of the black hole. Furthermore, we explore the deflection angles within the weak and strong field regimes, checking higher dimensional signatures. To study the weak field deflection angle, we utilize the Gauss-Bonnet theorem to include the finite distance and time-like particles [42-46], to give us the general case.</text>
<text><location><page_2><loc_9><loc_66><loc_92><loc_84></location>The strong deflection angle, commonly denoted as the angle of light bending in the proximity of a massive object, encapsulates the profound gravitational influence on the trajectory of light rays. Unlike weak deflection, characterized by minimal light deviation, strong deflection occurs when light traverses in close proximity to the gravitational source, resulting in significant bending of light rays. This phenomenon manifests observable signatures that offer invaluable insights into the nature and characteristics of the intervening mass [47-51]. Note that the works of Bozza [52] and Tsukamoto [53] have investigated the calculation of the strong deflection angle in the vicinity of a black hole. Tsukamoto, in particular, extends Bozza's methodology by introducing a novel term, denoted as z , enabling an analytical approximation for the deflection angle equation. The outcomes exhibit consistency, with negligible discrepancies [53]. Tsukamoto's approach, and its subsequent refinement, broaden the applicability of the strong deflection angle formalism to encompass various black hole models, thus facilitating investigations into more intricate black hole configurations [54-56]. Furthermore, numerous insightful findings have been presented concerning the behavior of light under such conditions [31, 54, 57-63]. For instance, within the strong field regime, the influence of charged matter and dark matter on the deflection angle becomes more discernible in comparison to weak deflection scenarios [64].</text>
<text><location><page_2><loc_9><loc_31><loc_92><loc_66></location>The formulation of theories elucidating the strong deflection angle within higher-dimensional black hole spacetimes holds profound implications for our understanding of black hole physics [65]. It not only sheds light on the behavior of gravity within unconventional higher-dimensional contexts but also elucidates fundamental principles underlying black hole theory [66]. The exploration of how gravitational lensing attributes are influenced by higher dimensions possesses the potential to refine our conceptual frameworks regarding black hole formation, evolution, and the interpretation of observational data pertaining to astrophysical phenomena involving black holes. As we shall discuss, the photonsphere plays an important role in forming the black hole shadows, as it dictates the size and shape of the black hole shadow. However, complementary information can be extracted by studying quasinormal modes (QNMs) of black holes. QNMs are characteristic oscillations of black holes that arise when the black hole is perturbed. These perturbations can be thought of as the 'ringing' of the black hole, and they decay over time due to the emission of gravitational waves. QNMs are defined by complex frequencies, where the real part represents the oscillation frequency and the imaginary part denotes the damping rate. The link between photon orbits and QNMs is established through the concept that the real part of the quasinormal mode frequencies is related to the orbital frequency of the photonsphere. In contrast, the imaginary part is associated with the Lyapunov exponent, which characterizes the stability of the photon orbits. This relationship was demonstrated by Cardoso et al. [67], who showed that for high-frequency (eikonal) QNMs, the frequencies are directly tied to the properties of the photonsphere. The oscillation frequencies of QNMs therefore correspond to the orbital frequencies of photons in the photonsphere, and the decay rates correspond to the stability properties of these orbits. As noted in Refs. [68, 69] some care is required when calculating QNMs in extra-dimensional spacetimes and presupposing this correspondence in the eikonal regime, which has been demonstrated to be invalid whenever the BH perturbation equations have effective potentials that are not 'well-behaved' from the stand-point of applicability of the WKB method. Inferring from the effective potentials of asymptotically flat Schwarzschild-Tangherlini BHs, which have a single extremum in the domain r ∈ [2 M, ∞ ) , the WKB method (and, in turn, the correspondence) is guaranteed to be applicable in this case. But in order to move toward a full analysis of the phenomena, we consider here the behavior of the deflection angle in the presence of a higher-dimensional black hole, and how to calculate QNMs in such a space in a new robust neural network manner.</text>
<text><location><page_2><loc_9><loc_22><loc_92><loc_31></location>We organize the paper as follows: In Sect. II, we provide a quick review of the Schwarzschild-Tanghelini black hole, along with examining the higher dimension parameter n to the photon orbit, and the black hole shadow. We also provide constraints to n in light of EHT observations. Next, we study the weak and strong deflection angles of this higher-dimensional black hole in Sect. III. Finally, we employed machine learning methods to calculate the QNMs in Sect. IV using PINN approximations. We give concluding remarks and future research prospects in Sect. V. Throughout this paper, we used ( -, + , + , +) metric signature and geometrized units by considering G = c = 1 .</text>
<section_header_level_1><location><page_2><loc_22><loc_18><loc_79><loc_19></location>II. SHADOW OF A SCHWARZSCHILD BLACK HOLE IN HIGHER DIMENSIONS</section_header_level_1>
<text><location><page_2><loc_9><loc_11><loc_92><loc_16></location>The Schwarzschild-Tangherlini metric extends the Schwarzschild solution to higher dimensions in the context of general relativity. It provides a solution to the Einstein field equations for a spherically symmetric, non-rotating black hole in a spacetime with n dimensions, where n is greater than 4 . For its full review, see Ref. [16], where the metric of the Schwarzschild black hole in extra-dimensions (Schwarzschild-Tangherlini black hole) is given by</text>
<formula><location><page_2><loc_37><loc_8><loc_64><loc_10></location>ds 2 = -A ( r ) dt 2 + B ( r ) dr 2 + r 2 d Ω 2 n -2 ,</formula>
<text><location><page_3><loc_9><loc_92><loc_41><loc_93></location>where A ( r ) , B ( r ) , and d Ω 2 n -2 are expressed by</text>
<formula><location><page_3><loc_32><loc_81><loc_92><loc_91></location>A ( r ) = 1 -2 M r n -3 , B ( r ) = A ( r ) -1 , d Ω 2 n -2 = dθ 2 1 +sin 2 ( θ 1 ) dθ 2 2 + ... + n -3 ∏ i =1 sin 2 ( θ i ) dθ 2 n -2 , (2)</formula>
<text><location><page_3><loc_9><loc_78><loc_92><loc_80></location>respectively. As one considers only the pure Schwarzschild field and its spherical symmetry, it can be noted from Ref. [70] that dθ 1 = dθ = 0 , dθ 2 = dϕ , and dθ n -2 = 0 . Furthermore, M admits the dimensionality n through</text>
<formula><location><page_3><loc_44><loc_73><loc_92><loc_76></location>M = 8 πm ( n -2)Ω n -2 , (3)</formula>
<text><location><page_3><loc_9><loc_71><loc_52><loc_72></location>where Ω n -2 is the area of the ( n -2) -dimensional unit sphere:</text>
<formula><location><page_3><loc_44><loc_66><loc_92><loc_70></location>Ω n -2 = 2 π ( n -1 2 ) Γ ( n -1 2 ) , (4)</formula>
<text><location><page_3><loc_9><loc_64><loc_29><loc_65></location>and m is the black hole mass.</text>
<text><location><page_3><loc_9><loc_58><loc_92><loc_63></location>The photonsphere ( r ps ) is a region around a black hole where photons can theoretically orbit the black hole in a circular path. It is well known that for the Schwarzschild case, r ps = 3 m . This section shows how the number of dimensions n affects the photonsphere, as we apply the formalism developed in Ref. [71]. The results here will be important in the next section.</text>
<text><location><page_3><loc_10><loc_57><loc_33><loc_58></location>We take the standard Lagrangian</text>
<formula><location><page_3><loc_38><loc_53><loc_92><loc_55></location>L = 1 2 ( -A ( r ) ˙ t 2 + B ( r ) ˙ r 2 + r 2 ˙ ϕ 2 ) . (5)</formula>
<text><location><page_3><loc_9><loc_50><loc_57><loc_52></location>Applying the variational principle gives us the two constants of motion:</text>
<formula><location><page_3><loc_40><loc_46><loc_92><loc_49></location>E = A ( r ) dt dλ , L = r 2 dϕ dλ , (6)</formula>
<text><location><page_3><loc_9><loc_44><loc_40><loc_45></location>where the impact parameter can be defined as</text>
<formula><location><page_3><loc_44><loc_40><loc_92><loc_43></location>b ≡ L E = r 2 A ( r ) dϕ dt . (7)</formula>
<text><location><page_3><loc_9><loc_38><loc_49><loc_39></location>Light rays obey g µν ˙ x µ ˙ x ν = 0 , leading to an orbit equation</text>
<formula><location><page_3><loc_40><loc_33><loc_92><loc_36></location>( dr dϕ ) 2 = r 2 B ( r ) ( h ( r ) 2 b 2 -1 ) , (8)</formula>
<text><location><page_3><loc_9><loc_31><loc_37><loc_32></location>where the function h ( r ) is defined as [71]</text>
<formula><location><page_3><loc_45><loc_26><loc_92><loc_29></location>h ( r ) 2 = r 2 A ( r ) . (9)</formula>
<text><location><page_3><loc_9><loc_22><loc_92><loc_25></location>The location of the photonsphere can be found by either satisfying the condition dr/dϕ = d 2 r/dϕ 2 = 0 , or using h ' ( r ) = 0 , which can lead to</text>
<formula><location><page_3><loc_43><loc_20><loc_92><loc_21></location>A ' ( r ) r 2 -2 rA ( r ) = 0 . (10)</formula>
<text><location><page_3><loc_9><loc_17><loc_48><loc_18></location>The analytical formula for the photonsphere radius is then</text>
<formula><location><page_3><loc_41><loc_13><loc_92><loc_16></location>r ps = [ 1 M ( n -1) ] -1 n -3 , (11)</formula>
<text><location><page_3><loc_9><loc_9><loc_92><loc_11></location>where we plot this in Fig. 1. The effect of increasing the number of dimensions n decreases the radius of the photonsphere, when normalized to the mass of the black hole.</text>
<figure>
<location><page_4><loc_27><loc_66><loc_74><loc_93></location>
<caption>FIG. 1. Photonsphere behavior as a function of the number of dimensions.</caption>
</figure>
<text><location><page_4><loc_9><loc_52><loc_92><loc_58></location>The presence of the photonsphere plays a crucial role in shaping the appearance of the black hole silhouette and contributes to the distinctive features observed in images, such as the shadow or the bright ring surrounding it. After obtaining the location of the photonsphere through the analytic equation (11), a small perturbation can either make the photons escape or spiral toward the black hole. For escaped photons, these will contribute to the black hole silhouette observed at r = r obs , which can backward trace the photons' path to study the black hole shadow formation. For such an observer [71],</text>
<formula><location><page_4><loc_34><loc_46><loc_92><loc_50></location>tan( α sh ) = lim ∆ x → 0 ∆ y ∆ x = ( r 2 B ( r ) ) 1 / 2 dϕ dr ∣ ∣ ∣ ∣ r = r obs , (12)</formula>
<text><location><page_4><loc_9><loc_44><loc_27><loc_45></location>which can be simplified as</text>
<formula><location><page_4><loc_43><loc_39><loc_92><loc_42></location>sin 2 ( α sh ) = b 2 crit h ( r obs ) 2 . (13)</formula>
<text><location><page_4><loc_9><loc_36><loc_91><loc_38></location>Here, the critical impact parameter, which is a function of r ps , can be determined from the condition d 2 r/dϕ 2 = 0 [72]:</text>
<formula><location><page_4><loc_24><loc_31><loc_92><loc_35></location>b 2 crit = h ( r ) B ' ( r ) r 2 -2 B ( r ) r ( h ( r ) B ' ( r ) r 2 -2 h ( r ) B ( r ) r -2 h ' ( r ) B ( r ) r 2 ) ∣ ∣ ∣ ∣ r = r ps . (14)</formula>
<text><location><page_4><loc_9><loc_28><loc_45><loc_30></location>With the help of Eq. (11), the shadow radius is then</text>
<formula><location><page_4><loc_35><loc_20><loc_92><loc_27></location>R sh = b crit √ A ( r obs ) (15) = √ √ √ √ M 2 n -3 ( -1 + n ) -1+ n n -3 n -3 ( 1 -2 M r n -3 obs ) . (16)</formula>
<text><location><page_4><loc_9><loc_9><loc_92><loc_18></location>We plot the above equation to see the effect of the observer's location for different dimensions n (see the left panel of Fig. 2). Here, we can drastically see the decrease in the shadow radius when the observer is so far away from the black hole. Interestingly, if the observer is near the Schwarzschild event horizon n = 4 , there are points where n = 5 , and n = 6 that cannot be distinguished from n = 4 . Finally, on the right panel, we attempted to constrain n for at least 3 σ (99 . 7%) confidence level shadow radius observed by the EHT. Looking at Refs. [3, 9, 73, 74], such a level reveals the lower and upper bounds as 3 . 871 M ≤ R sh ≤ 5 . 898 M , and 2 . 546 M ≤ R sh ≤ 7 . 846 M for Sgr. A* and M87*, respectively. These bounds were also used recently in Refs. [75, 76].</text>
<figure>
<location><page_5><loc_11><loc_71><loc_90><loc_93></location>
<caption>FIG. 2. Left: Dependence of shadow radius to the observer's location for different values of n . Right: Shadow constraints in n with a simple parameter estimation using the EHT data. The blue horizontal dotted line is the Schwarzschild shadow R sh /M = 3 √ 3 .</caption>
</figure>
<section_header_level_1><location><page_5><loc_25><loc_62><loc_76><loc_63></location>III. PARTICLE DEFLECTION IN WEAK AND STRONG FIELD REGIME</section_header_level_1>
<section_header_level_1><location><page_5><loc_41><loc_59><loc_59><loc_60></location>A. Weak deflection angle</section_header_level_1>
<text><location><page_5><loc_9><loc_54><loc_92><loc_56></location>In this section, we aim to analyze the effect of the number of dimensions on the weak deflection angle. To do so, we use the Gauss-Bonnet theorem stating that [77, 78]</text>
<formula><location><page_5><loc_34><loc_48><loc_92><loc_52></location>∫∫ D KdS + N ∑ i =1 ∫ ∂D a κ g dℓ + N ∑ i =1 Θ i = 2 πχ ( D ) . (17)</formula>
<text><location><page_5><loc_9><loc_41><loc_92><loc_47></location>Here, K is the Gaussian curvature, dS is the area measure, Θ i , and κ g is the jump angles and geodesic curvature of ∂D , respectively, and dℓ is the arc length measure. The application to null geodesics in the equatorial plane implies that the Euler characteristics should be χ ( D ) = 1 . If the integral is evaluated over the infinite area surface bounded by the light ray, it was shown in Ref. [43] that the above reduces to</text>
<formula><location><page_5><loc_37><loc_36><loc_92><loc_40></location>ˆ α = ϕ RS +Ψ R -Ψ S = -∫∫ ∞ R □ ∞ S KdS, (18)</formula>
<text><location><page_5><loc_9><loc_29><loc_92><loc_35></location>where ˆ α is the weak deflection angle. In the above formula, ϕ RS = ϕ R -ϕ S is the azimuthal separation angle between the source S and receiver R, ϕ R and ϕ S are the positional angles, and ∞ R □ ∞ S is the integration domain. While this study can use the above formula, we prefer to use its extension to the non-asymptotically flat case given in Ref. [46], where one uses the path in the photonsphere orbit instead of the path at infinity. Eq. (18) can be reformulated as:</text>
<formula><location><page_5><loc_41><loc_24><loc_92><loc_28></location>ˆ α = ∫∫ R r ps □ S r ps KdS + ϕ RS . (19)</formula>
<text><location><page_5><loc_9><loc_20><loc_92><loc_23></location>Before using the above equation, we are interested in the deflection angle of massive particles. Such a general case reduces to a special case of photon deflection when v = 1 . As such, we need the Jacobi metric stating that</text>
<formula><location><page_5><loc_30><loc_15><loc_92><loc_18></location>dl 2 = g ij dx i dx j = ( E 2 -µ 2 A ( r )) ( B ( r ) A ( r ) dr 2 + r 2 A ( r ) dϕ 2 ) , (20)</formula>
<text><location><page_5><loc_9><loc_13><loc_53><loc_14></location>where E is the energy per unit mass ( µ ) of the massive particle:</text>
<formula><location><page_5><loc_45><loc_8><loc_92><loc_11></location>E = µ √ 1 -v 2 . (21)</formula>
<text><location><page_6><loc_9><loc_90><loc_92><loc_93></location>It is then useful to define another constant quantity in terms of the impact parameter b , which is the angular momentum per unit mass:</text>
<formula><location><page_6><loc_45><loc_86><loc_92><loc_89></location>J = µvb √ 1 -v 2 , (22)</formula>
<text><location><page_6><loc_9><loc_84><loc_49><loc_85></location>and with E and J , we can define the impact parameter as</text>
<formula><location><page_6><loc_47><loc_80><loc_92><loc_83></location>b = J vE . (23)</formula>
<text><location><page_6><loc_9><loc_76><loc_92><loc_79></location>Using the line element for time-like particles ds 2 = g µν dx µ dx ν = -1 and defining u = 1 /r , the orbit equation can be derived as</text>
<formula><location><page_6><loc_29><loc_72><loc_92><loc_75></location>F ( u ) ≡ ( du dϕ ) 2 = 1 A ( u ) B ( u ) [ ( 1 vb ) 2 -A ( u ) ( 1 J 2 + u 2 ) ] , (24)</formula>
<text><location><page_6><loc_9><loc_69><loc_34><loc_70></location>which, in our case, yields ( p = n -3 )</text>
<formula><location><page_6><loc_33><loc_64><loc_92><loc_67></location>F ( u ) = 1 v 2 b 2 + ( 1 J 2 + u 2 ) [ 2 Mu p ( u 2 J 2 +1 ) J 2 ] . (25)</formula>
<text><location><page_6><loc_9><loc_61><loc_68><loc_62></location>Next, by an iterative method, the goal is to find u as a function of ϕ , which we find as</text>
<formula><location><page_6><loc_38><loc_57><loc_92><loc_60></location>u ( ϕ ) = sin( ϕ ) b + 1 + v 2 cos 2 ( ϕ ) b 2 v 2 M. (26)</formula>
<text><location><page_6><loc_9><loc_55><loc_38><loc_56></location>From here, we obtain the solution for ϕ as</text>
<formula><location><page_6><loc_36><loc_50><loc_92><loc_54></location>ϕ = arcsin( bu ) + M [ v 2 ( b 2 u 2 -1 ] -1 ) bv 2 √ 1 -b 2 u 2 . (27)</formula>
<text><location><page_6><loc_9><loc_47><loc_82><loc_49></location>This is the expression for the positional angle ϕ S , which implies that u → u S . Note also that ϕ R = π -ϕ S . The Gaussian curvature can be derived using</text>
<formula><location><page_6><loc_40><loc_42><loc_92><loc_46></location>K = -1 √ g [ ∂ ∂r ( √ g g rr Γ ϕ rϕ )] , (28)</formula>
<text><location><page_6><loc_9><loc_39><loc_59><loc_41></location>since Γ ϕ rr = 0 from Eq. (20). Furthermore, the determinant of Eq. (20) is</text>
<formula><location><page_6><loc_40><loc_35><loc_92><loc_38></location>g = B ( r ) r 2 A ( r ) 2 ( E 2 -µ 2 A ( r )) 2 . (29)</formula>
<text><location><page_6><loc_9><loc_33><loc_47><loc_34></location>With the analytical solution to r ps , it is easy to see that</text>
<formula><location><page_6><loc_42><loc_28><loc_92><loc_31></location>[∫ K √ gdr ]∣ ∣ ∣ ∣ r = r ps = 0 , (30)</formula>
<text><location><page_6><loc_9><loc_26><loc_17><loc_27></location>which yields</text>
<formula><location><page_6><loc_28><loc_17><loc_92><loc_24></location>∫ r ( ϕ ) r ps K √ gdr = -A ( r ) ( E 2 -A ( r ) ) (2 r ) -E 2 r 2 A ( r ) ' 2 rA ( r ) ( E 2 -A ( r )) √ B ( r ) ∣ ∣ ∣ ∣ r = r ( ϕ ) = -1 + M [ -1 + ( p +1) E 2 ] E 2 -1 [ b -p ( 1 sin( ϕ ) ) -p ] , (31)</formula>
<text><location><page_6><loc_9><loc_14><loc_75><loc_15></location>where the prime denotes differentiation with respect to r . The weak deflection angle is then [46],</text>
<formula><location><page_6><loc_27><loc_9><loc_92><loc_13></location>ˆ α = ∫ ϕ R ϕ S [ -A ( r ) ( E 2 -A ( r ) ) (2 r ) -E 2 r 2 A ( r ) ' 2 rA ( r ) ( E 2 -A ( r )) √ B ( r ) ∣ ∣ ∣ ∣ r = r ( ϕ ) ] dϕ + ϕ RS . (32)</formula>
<text><location><page_7><loc_9><loc_92><loc_33><loc_93></location>Using Eq. (26) in Eq. (31), we find</text>
<formula><location><page_7><loc_9><loc_83><loc_95><loc_91></location>[∫ K √ gdr ]∣ ∣ ∣ ∣ r = r ϕ = -ϕ RS -( M [ ( p +1) E 2 -1 ] E 2 -1 ) b -p cos( ϕ ) ( sin 2 ( ϕ ) ) 1 2 -p 2 ( csc 1 -p ( ϕ ) ) 2 F 1 ( 1 2 , 1 2 -p 2 ; 3 2 ; cos 2 ( ϕ ) )∣ ∣ ∣ ∣ ϕ R ϕ S = -ϕ RS + ( 2 M [ ( p +1) E 2 -1 ] E 2 -1 ) b -p cos( ϕ ) ( sin 2 ( ϕ ) ) 1 2 -p 2 ( csc 1 -p ( ϕ ) ) 2 F 1 ( 1 2 , 1 2 -p 2 ; 3 2 ; cos 2 ( ϕ ) ) (33)</formula>
<text><location><page_7><loc_9><loc_79><loc_92><loc_81></location>since ϕ S has the same form as Eq. (27), while keeping in mind that the definite integral is already evaluated. Then, we find Eq. (32) as</text>
<formula><location><page_7><loc_17><loc_74><loc_92><loc_78></location>ˆ α = ( 2 M [ ( p +1) E 2 -1 ] E 2 -1 ) b -p cos( ϕ ) ( sin 2 ( ϕ ) ) 1 2 -p 2 ( csc 1 -p ( ϕ ) ) 2 F 1 ( 1 2 , 1 2 -p 2 ; 3 2 ; cos 2 ( ϕ ) ) . (34)</formula>
<text><location><page_7><loc_9><loc_72><loc_88><loc_73></location>Finally, we need to express the equation above in terms of finite distance u S and u R . We use the following relations:</text>
<formula><location><page_7><loc_27><loc_59><loc_92><loc_71></location>cos( ϕ ) = √ 1 -b 2 u 2 -Mu [ v 2 ( b 2 u 2 -1 ) -1 ] v 2 √ (1 -b 2 u 2 ) , ( sin 2 ( ϕ ) ) 1 2 -p 2 = bu ( bu ) p -M ( b 2 u 2 ) 1 2 -p 2 [ v 2 ( b 2 u 2 -1 ) -1 ] ( p -1) uv 2 b 2 , csc 1 -p ( ϕ ) = ( bu ) p bu + [ v 2 ( b 2 u 2 -1 ) -1 ] ( p -1) ( 1 bu ) -p M b 3 u 2 v 2 . (35)</formula>
<text><location><page_7><loc_9><loc_56><loc_92><loc_58></location>After approximating Eq. (34) again, the general analytic expression for the deflection angle of the Schwarzschild-Tangherlini black hole with finite distance as</text>
<formula><location><page_7><loc_11><loc_51><loc_92><loc_55></location>ˆ α = ( 1 + p v 2 ) Mb -p [ √ 1 -b 2 u 2 S 2 F 1 ( 1 2 , 1 2 -p 2 ; 3 2 ; -b 2 u 2 S +1 ) + √ -b 2 u 2 R +1 2 F 1 ( 1 2 , 1 2 -p 2 ; 3 2 ; -b 2 u 2 R +1 )] , (36)</formula>
<text><location><page_7><loc_9><loc_49><loc_46><loc_50></location>which is written in terms of a hypergeometric function.</text>
<text><location><page_7><loc_9><loc_46><loc_92><loc_49></location>One can easily check that if p = 1 (where n = 4 ), the equation above reduces to the timelike expression in finite distance as</text>
<formula><location><page_7><loc_33><loc_42><loc_92><loc_45></location>ˆ α n =4 timelike = ( v 2 +1 ) m b v 2 [ √ 1 -b 2 u 2 S + √ 1 -b 2 u 2 R ] . (37)</formula>
<text><location><page_7><loc_9><loc_40><loc_54><loc_41></location>If we assume that u S = u R and both are approximately zero, then</text>
<formula><location><page_7><loc_42><loc_36><loc_92><loc_39></location>ˆ α n =4 timelike = 2 ( v 2 +1 ) m b v 2 . (38)</formula>
<text><location><page_7><loc_9><loc_34><loc_31><loc_35></location>Finally, for photons where v = 1 ,</text>
<formula><location><page_7><loc_45><loc_30><loc_92><loc_33></location>ˆ α n =4 photon = 4 m b . (39)</formula>
<text><location><page_7><loc_9><loc_28><loc_92><loc_29></location>We observe that when n is odd, ˆ α contains a factor involving a hypergeometric function. For instance, when n = 5 ( p = 2 ),</text>
<formula><location><page_7><loc_11><loc_23><loc_92><loc_26></location>ˆ α n =5 timelike = ( v 2 +2 ) M b 2 v 2 [ √ 1 -b 2 u 2 S 2 F 1 ( -1 2 , 1 2 ; 3 2 ; -b 2 u 2 S +1 ) + √ -b 2 u 2 R +1 2 F 1 ( -1 2 , 1 2 ; 3 2 ; -b 2 u 2 R +1 )] . (40)</formula>
<text><location><page_7><loc_9><loc_21><loc_90><loc_22></location>When n is even, such as when p = 3 , we observe that ˆ α does not contain a factor involving a hypergeometric function:</text>
<formula><location><page_7><loc_24><loc_17><loc_92><loc_20></location>ˆ α n =6 timelike = ( v 2 +3 ) M 3 b 3 v 2 [ √ 1 -b 2 u 2 S ( 2 + b 2 u 2 S ) + √ 1 -b 2 u 2 R ( 2 + b 2 u 2 R ) ] . (41)</formula>
<text><location><page_7><loc_9><loc_9><loc_92><loc_16></location>We plot Eq. (36), shown in Fig. 3. First, we observe that the deflection angle caused by time-like particles gives a slightly higher value of α than photon deflection. Second, our observation is that as the number of dimensions increases, α decreases. It only means that if we use the phenomenon of deflection angle to probe the existence of higher dimensions in black holes, more sensitive devices are needed. Finally, the effect of finite distance indicates that α slightly increases as the impact parameter b/M becomes greater than the observer distance from the black hole.</text>
<figure>
<location><page_8><loc_11><loc_71><loc_49><loc_93></location>
<caption>FIG. 3. The behavior of α (in µ as) for various dimensions n . Left panel: u/M = 0 . Right panel: u/M = 1 . 85 × 10 -11 , corresponding to the reciprocal of our distance from M87* (also represented by the red vertical dotted line).</caption>
</figure>
<text><location><page_8><loc_57><loc_72><loc_57><loc_73></location>0</text>
<text><location><page_8><loc_62><loc_72><loc_62><loc_73></location>2</text>
<text><location><page_8><loc_67><loc_72><loc_68><loc_73></location>4</text>
<text><location><page_8><loc_72><loc_72><loc_73><loc_73></location>6</text>
<text><location><page_8><loc_77><loc_72><loc_78><loc_73></location>8</text>
<text><location><page_8><loc_82><loc_72><loc_83><loc_73></location>10</text>
<text><location><page_8><loc_87><loc_72><loc_89><loc_73></location>12</text>
<text><location><page_8><loc_70><loc_71><loc_73><loc_72></location>log10 (</text>
<text><location><page_8><loc_73><loc_71><loc_73><loc_72></location>b</text>
<text><location><page_8><loc_73><loc_71><loc_74><loc_72></location>/</text>
<text><location><page_8><loc_74><loc_71><loc_74><loc_72></location>M</text>
<text><location><page_8><loc_74><loc_71><loc_75><loc_72></location>)</text>
<section_header_level_1><location><page_8><loc_41><loc_62><loc_60><loc_63></location>B. Strong Deflection Angle</section_header_level_1>
<text><location><page_8><loc_9><loc_54><loc_92><loc_60></location>Photons slightly greater than the photonsphere radius will undergo the phenomenon of strong deflection angle. In this section, we calculate and showcase the deflection of light as it approaches the photonsphere in the strong field limit. Using the approach of Tsukamoto in Ref. [53], the deflection angle is derived using the light trajectory shown in Eq. (8) but, in this section we express it as,</text>
<formula><location><page_8><loc_44><loc_50><loc_92><loc_53></location>( dr dϕ ) 2 = R ( r ) r 2 B ( r ) , (42)</formula>
<text><location><page_8><loc_9><loc_47><loc_13><loc_48></location>where</text>
<formula><location><page_8><loc_43><loc_43><loc_92><loc_46></location>R ( r ) = A ( r 0 ) r 2 A ( r ) r 2 0 -1 . (43)</formula>
<text><location><page_8><loc_9><loc_39><loc_92><loc_42></location>Note that the A ( r ) is the metric function defined by Eq. (2), while A ( r 0 ) is the metric function evaluated at distance r 0 . The solution of Eq. (42) yields the strong deflection angle α ( r 0 ) as shown in Ref. [52, 53, 79],</text>
<formula><location><page_8><loc_40><loc_32><loc_92><loc_37></location>α ( r 0 ) = I ( r 0 ) -π = 2 ∫ ∞ r 0 dr √ R ( r ) C ( r ) B ( r ) -π. (44)</formula>
<text><location><page_8><loc_9><loc_27><loc_92><loc_30></location>In order to evaluate the integral in Eq. (44), we expand over r = r 0 . This yields a regular integral κ R and a diverging integral κ D , and by introducing a new variable, z, defined as,</text>
<formula><location><page_8><loc_46><loc_24><loc_92><loc_27></location>z ≡ 1 -r 0 r , (45)</formula>
<text><location><page_8><loc_9><loc_21><loc_23><loc_23></location>I ( r 0 ) is expressed as,</text>
<formula><location><page_8><loc_32><loc_17><loc_92><loc_20></location>I ( r 0 ) = ∫ 1 0 κ ( z, r 0 ) dz = ∫ 1 0 κ D ( z, r 0 ) + κ R ( z, r 0 ) dz, (46)</formula>
<text><location><page_8><loc_9><loc_13><loc_92><loc_15></location>where κ ( z, r 0 ) is expressed as the sum of the diverging integral, κ D , and regular integral, κ R . The details of the expansion of Eq. (44) was shown in Refs. [52, 53]. As a result the strong deflection angle is expressed as,</text>
<formula><location><page_8><loc_30><loc_8><loc_92><loc_11></location>ˆ α str = -¯ a log ( b 0 b crit -1 ) + ¯ b + O ( b 0 b c -1 ) log ( b 0 b c -1 ) , (47)</formula>
<text><location><page_8><loc_52><loc_84><loc_53><loc_84></location>)</text>
<text><location><page_8><loc_52><loc_83><loc_53><loc_84></location>(</text>
<text><location><page_8><loc_52><loc_83><loc_53><loc_83></location>0</text>
<text><location><page_8><loc_52><loc_83><loc_53><loc_83></location>1</text>
<text><location><page_8><loc_52><loc_82><loc_53><loc_83></location>g</text>
<text><location><page_8><loc_52><loc_82><loc_53><loc_82></location>o</text>
<text><location><page_8><loc_52><loc_82><loc_53><loc_82></location>l</text>
<text><location><page_8><loc_54><loc_89><loc_55><loc_90></location>0</text>
<text><location><page_8><loc_54><loc_86><loc_55><loc_87></location>10</text>
<text><location><page_8><loc_54><loc_83><loc_55><loc_84></location>20</text>
<text><location><page_8><loc_54><loc_80><loc_55><loc_81></location>30</text>
<text><location><page_8><loc_54><loc_76><loc_55><loc_77></location>40</text>
<text><location><page_8><loc_54><loc_73><loc_55><loc_74></location>50</text>
<text><location><page_8><loc_59><loc_80><loc_60><loc_81></location>n</text>
<text><location><page_8><loc_60><loc_80><loc_62><loc_81></location>=4,</text>
<text><location><page_8><loc_62><loc_80><loc_63><loc_81></location>v</text>
<text><location><page_8><loc_63><loc_80><loc_65><loc_81></location>=1</text>
<text><location><page_8><loc_59><loc_79><loc_60><loc_80></location>n</text>
<text><location><page_8><loc_60><loc_79><loc_62><loc_80></location>=5</text>
<text><location><page_8><loc_59><loc_78><loc_60><loc_78></location>n</text>
<text><location><page_8><loc_60><loc_78><loc_62><loc_78></location>=6</text>
<text><location><page_8><loc_59><loc_76><loc_60><loc_77></location>n</text>
<text><location><page_8><loc_60><loc_76><loc_62><loc_77></location>=7</text>
<text><location><page_8><loc_59><loc_75><loc_60><loc_76></location>n</text>
<text><location><page_8><loc_60><loc_75><loc_62><loc_76></location>=8</text>
<text><location><page_8><loc_59><loc_74><loc_60><loc_75></location>n</text>
<text><location><page_8><loc_60><loc_74><loc_62><loc_75></location>=4,</text>
<text><location><page_8><loc_62><loc_74><loc_63><loc_75></location>v</text>
<text><location><page_8><loc_63><loc_74><loc_66><loc_75></location>=0.75</text>
<text><location><page_9><loc_9><loc_89><loc_92><loc_93></location>where ¯ a and ¯ b are coefficients and b 0 and b crit are the impact parameter evaluated at the closest approach, r 0 , and critical impact parameter, respectively. The first term in Eq. (47) is the result of the diverging integral and the second term is the result of the regular integral. The coefficients ¯ a and ¯ b are expressed as [53],</text>
<formula><location><page_9><loc_40><loc_84><loc_92><loc_87></location>¯ a = √ 2 B ( r ps ) A ( r ps ) 2 A ( r ps ) -A '' ( r ps ) r 2 ps , (48)</formula>
<text><location><page_9><loc_9><loc_81><loc_11><loc_82></location>and</text>
<formula><location><page_9><loc_34><loc_77><loc_92><loc_80></location>¯ b = ¯ a log [ r ps ( 2 r 2 ps -A '' ( r ps ) A ( r ps ) )] + I R ( r ps ) -π, (49)</formula>
<text><location><page_9><loc_9><loc_73><loc_92><loc_75></location>where A ( r ps ) is metric function evaluated at the photonsphere, and I R is the regular integral evaluated from 0 to 1. The double prime in Eq. (48) and Eq. (49) correspond to the second derivative with respect to r evaluated over r ps .</text>
<text><location><page_9><loc_10><loc_71><loc_60><loc_72></location>Applying Eq. (48) to the black hole metric would yield the coefficient ¯ a ,</text>
<formula><location><page_9><loc_41><loc_66><loc_92><loc_70></location>¯ a = r p/ 2 ps √ ( p 2 + p -2) M + r p ps , (50)</formula>
<text><location><page_9><loc_9><loc_61><loc_92><loc_65></location>where p = n -3 . When p = 1 , we have the Schwarzschild result of a = 1 . A pattern was observed from ¯ a which can be simplified as ¯ a = √ p/p . As we increase the dimension of the black hole the value of ¯ a decreases exponentially. The argument of the natural logarithmic term of Eq. (49) becomes,</text>
<formula><location><page_9><loc_40><loc_58><loc_92><loc_59></location>¯ b = ¯ a ln [2 p +4] + I R ( r ps ) -π, (51)</formula>
<text><location><page_9><loc_9><loc_55><loc_92><loc_56></location>and we retrieve the Schwarschild result when p = 1 (which is 6 as in Ref. [52, 53]). The regular integral I R is defined as,</text>
<formula><location><page_9><loc_37><loc_50><loc_92><loc_54></location>I R ( r 0 ) ≡ ∫ 1 0 f R ( z, r 0 ) -f D ( z, r 0 ) dz, (52)</formula>
<text><location><page_9><loc_9><loc_48><loc_76><loc_49></location>where the f R ( z, r 0 ) was generated from the expansion of the trajectory in Eq. (42), which gives us</text>
<formula><location><page_9><loc_42><loc_43><loc_92><loc_47></location>f R ( z, r 0 ) = 2 r 0 √ G ( z, r 0 ) , (53)</formula>
<text><location><page_9><loc_9><loc_39><loc_92><loc_42></location>where G ( z, r 0 ) = RCA (1 -z ) 4 . Notice that C and A are the metric functions for which the position r is expressed in terms of z and r 0 , while R is shown in Eq. (43). The generated expression from Eq. (53) is,</text>
<formula><location><page_9><loc_39><loc_34><loc_92><loc_38></location>f R ( z, r ps ) = 2 r ps √ ∑ m m =2 c m ( r ps ) z m , (54)</formula>
<text><location><page_9><loc_9><loc_32><loc_61><loc_33></location>when we evaluate r 0 = r ps . On the other hand the f D ( z, r ps ) is expressed as,</text>
<formula><location><page_9><loc_43><loc_27><loc_92><loc_31></location>f D ( z, r ps ) = 2 r ps √ c 2 z 2 , (55)</formula>
<text><location><page_9><loc_9><loc_24><loc_92><loc_26></location>where the c 's are coefficients of the new variable z . For the equations in (50), (51), and by evaluating the integral in Eq. (52), our results are summarized in Table I.</text>
<table>
<location><page_9><loc_21><loc_13><loc_79><loc_22></location>
<caption>TABLE I. The table shows the numerical values of the essential parts of the strong deflection angle. Note that n are the dimensions, ¯ a is the coefficient, 2( n -3) + 4 is the argument of the logarithmic term, and I R is the regular integral.</caption>
</table>
<text><location><page_10><loc_9><loc_88><loc_92><loc_93></location>The results in Table I are consistent with results when using the Schwarzschild metric where n = 4 as shown in Refs. [52, 53, 58, 65, 80] with an extension to higher dimension black holes. As the dimensionality increases, the coefficient ¯ a decreases which significantly affects ¯ b and leaves the I R to dominate the expression. The increasing value of I R greatly influences the decrease of the deflection angle for higher dimensions.</text>
<text><location><page_10><loc_9><loc_80><loc_92><loc_87></location>As shown in Fig. 4, the strong deflection angle at higher dimensions exhibits the same behavior but in decreasing values [31] as dimensionality increases. These decreased values can be attributed to the increase in the number of degrees of freedom in the higher dimensional spaces [16]. Investigating the existence of higher dimensions from n = 4 to n = 7 is theoretically possible since, at this region, the strong deflection angle is relatively large compared to the weak deflection angle. From n = 8 , the strong deflection becomes significantly small and would require ultrasensitive devices to probe.</text>
<figure>
<location><page_10><loc_25><loc_51><loc_75><loc_79></location>
<caption>FIG. 4. The behavior of ˆ α str for dimensions n = 4 to n = 8 .</caption>
</figure>
<section_header_level_1><location><page_10><loc_40><loc_42><loc_61><loc_43></location>IV. QUASINORMAL MODES</section_header_level_1>
<text><location><page_10><loc_9><loc_36><loc_92><loc_40></location>In light of recent observations of gravitational waves [11], the study of QNMs also becomes an interesting avenue for the study of extra-dimensional models [81]. To complement the study of photonspheres above, we now look at how to calculate QNMs in this spacetime with a neural network approach.</text>
<text><location><page_10><loc_9><loc_33><loc_92><loc_36></location>The use of neural networks, detailed below, is for its robustness to a range of partial differential equations, as we have here, with a consideration of different dimensionalities. In this section we shall rewrite the metric as [82]:</text>
<formula><location><page_10><loc_37><loc_30><loc_92><loc_32></location>ds 2 = f ( r ) dt 2 -f -1 ( r ) dr 2 -r 2 d Ω 2 n -2 , (56)</formula>
<text><location><page_10><loc_9><loc_27><loc_92><loc_30></location>where f ( r ) = 1 -r 3 -n when working in the 2 M = 1 units. The equations describing the perturbation of this metric to produce damped sinusoids, the QNMs, are given as [82, 83]:</text>
<formula><location><page_10><loc_24><loc_23><loc_92><loc_26></location>ψ '' + { ω 2 -f ( r ) [ ℓ ( ℓ + n -3) r 2 + ( n -2)( n -4) 4 r 2 + (1 -j 2 )( n -2) 2 4 r n -1 ]} ψ, (57)</formula>
<text><location><page_10><loc_9><loc_12><loc_92><loc_21></location>where the prime denotes derivatives with respect to the tortoise co-ordinate x . Here j (i.e. spin of perturbing field) is assigned 0 , 2 , 2 / ( n -2) for massless scalar, gravitational vector and electromagnetic vector perturbations, respectively. The QNMs, represented here by the perturbation quantities ψ , are solutions to the Schrodinger-like eigenvalue problem given in Eq. (57). and are indexed by the spin j , multipole number ℓ and overtone number N (capitalised to distinguish it from n denoting spacetime dimensions), with the least-damped mode being N = 0 . The eigenvalues of Eq. (57) are the frequencies of the damped sinusoids denoted as ω = ω Re -iω Im , with ω Re signifying the physical oscillation frequency and ω Im being associated with the damping rate. These frequencies are important for probing the parameters of the perturbed source.</text>
<text><location><page_10><loc_9><loc_9><loc_92><loc_11></location>In the astrophysical setting, the asymptotic behavior of physically allowed modes constitutes only in-going waves to the black hole horizon and only outgoing waves to spatial infinity, in the radial domain of Eq. (57), where the radial and</text>
<figure>
<location><page_11><loc_10><loc_77><loc_89><loc_93></location>
<caption>FIG. 5. The effective potentials V ( r ) for perturbations of a Schwarzschild-Tangherlini BH in n = 5 , 6 and 7 dimensions. They are 'WKB-well-behaved'; that is, they have a single extremum and they decay at the event horizon r = 1 and at spatial infinity.</caption>
</figure>
<text><location><page_11><loc_9><loc_67><loc_92><loc_69></location>tortoise co-ordinates ( r ∈ [1 , ∞ ) and x ∈ ( -∞ , ∞ ) , respectively) are related through the differential equation dr/dx = f ( r ) . Factoring out the asymptotic behavior, ψ ( r ) can be expressed in the form [82]:</text>
<formula><location><page_11><loc_34><loc_61><loc_92><loc_65></location>ψ ( r ) = ( r -1 r ) iω/ ( n -3) e iωr χ ( r ) , if n even, ( r -1 r +1 ) iω/ ( n -3) e iωr χ ( r ) , if n odd. (58)</formula>
<text><location><page_11><loc_9><loc_54><loc_92><loc_60></location>Substituting the asymptotic behavior in the perturbation equation and transforming to a finite co-ordinate ξ = 1 -(1 /r ) or ξ = 1 /r , both of which yield a finite domain ξ ∈ [0 , 1] , the differential equation can be written in the form implicitly incorporating the physical behavior. For example, when working with ξ = 1 /r and n = 5 , 6 and 7 , for which some QNMs are computed using PINNs, the perturbation equations are expressed as:</text>
<formula><location><page_11><loc_14><loc_50><loc_92><loc_53></location>-(( -1 + ξ )( -((1 + ξ )(3 + 4 ℓ (2 + ℓ ) -9( -1 + j 2 ) ξ 2 )) + 6 iξ (1 + ξ )(2 + ξ ) ω +(2 + ξ ) 2 (3 + ξ ) ω 2 ) χ ) +4( -1 + ξ 2 )(( -2 ξ +4 ξ 3 -i ( -2 + ξ 2 (3 + ξ )) ω ) χ ' + ξ 2 ( -1 + ξ 2 ) χ '' )) = 0 , for n = 5 , (59)</formula>
<formula><location><page_11><loc_13><loc_42><loc_92><loc_47></location>( -1 + ξ )(9(1 + x + ξ 2 )(2 + ℓ (3 + ℓ ) -4( -1 + j 2 ) ξ 3 ) -6 iξ (1 + ξ + ξ 2 )(1 + 2 ξ (3 + ξ )) ω -(6 + ξ (2 + ξ )(3 + ξ ))(1 + ξ (4 + ξ )) ω 2 ) χ +3( -1 + ξ 3 )(3 ξ ( -2 + 5 ξ 3 ) χ ' -2 i ( -3 + ξ 2 +4 ξ 3 + ξ 4 ) ωχ ' +3 ξ 2 ( -1 + ξ 3 ) χ '' ) = 0 , for n = 6 , (60)</formula>
<formula><location><page_11><loc_16><loc_36><loc_92><loc_40></location>((1 + ξ 2 )(15 + 4 ℓ (4 + ℓ ) -25( -1 + j 2 ) ξ 4 ) -4 iξ (1 + 7 ξ 2 +6 ξ 4 ) ω -(4 + 15 ξ 2 +9 ξ 4 ) ω 2 ) χ +4(1 + ξ 2 )(( -2 ξ +6 ξ 5 -i ( -2 + ξ 2 +3 ξ 4 ) ω ) χ ' + ξ 2 ( -1 + ξ 4 )) χ '' ) = 0 , for n = 7 . (61)</formula>
<text><location><page_11><loc_9><loc_15><loc_92><loc_35></location>The corresponding equations in terms of ξ = 1 -(1 /r ) are similarly second-order, linear, homogeneous differential equations. They are solved here using physics-informed neural networks (PINNs) and the sixth-order WKB method (for comparison between two methods), where the latter is a method that was developed by Ref. [83]. The applicability of the WKB method in the Schwarzschild-Tangherlini case can be inferred from the related effective potential, which should be a potential barrier with a single extremum point between the event horizon and spatial infinity boundaries where the potential decays. This is seen in Fig IV, for various perturbing fields including gravitational vector perturbations ( j = 2 ), for ℓ = 2 and n = 5 , 6 , and 7 . The PINN algorithm is a different approximation method; it is machine learning-based and uses highly parameterized ansatzes to approximate the solutions of a differential equation. The approximation is facilitated by an optimization algorithm in which the deviation of the approximate function from the target functions, quantified using the loss function, is minimized by backpropagating the derivatives of the loss function with respect to the 'ansatz' parameters to take iterative steps towards the loss's global minimum. Let ˆ χ ( ξ, θ ) be the output value of the neural network (the approximation of the solutions to Eqs. (59) - (61)), then ξ is set as an input node to pass into the neural network some data sampled from ξ ∈ [0 , 1] . The parameters to be optimized are denoted by θ . As such, the neural network is represented in general as [84]:</text>
<formula><location><page_11><loc_33><loc_13><loc_92><loc_14></location>N ℓ ( ξ, θ ) = σ ( N ℓ -1 ( ξ ) W ℓ + b ℓ ) , for 1 ≤ ℓ ≤ L, (62)</formula>
<text><location><page_11><loc_9><loc_9><loc_92><loc_11></location>where the neural network output N ℓ = L ( ξ, θ ) = ˆ χ ( ξ, θ ) is a composite function of recursive linear transformations of the hidden layers N L -1 , N L -2 , ... N 1 as given within the parentheses of the right-hand side of Eq. (60). Here W ℓ and b ℓ are</text>
<table>
<location><page_12><loc_13><loc_57><loc_87><loc_93></location>
<caption>TABLE II. PINN approximations of the QNMs frequencies ( ω PINN ) for n = 5 , 6 and 7 Schwarzschild-Tangherlini black holes perturbed by fields of various spins j . We compare the PINN approximations with values obtained using separate methods (i.e. the continued fraction method for n = 5 , see Ref. [82], and the 6-th order WKB method for n = 6 and 7 ). The values in the parentheses are percentage deviations from comparing with the reference values.</caption>
</table>
<text><location><page_12><loc_9><loc_39><loc_92><loc_47></location>the weight matrices and bias vectors, respectively, which we represent collectively with the term θ . To enhance function approximation, an activation function σ is applied to each hidden layer ℓ ∈ [1 , L -1] . We use PINNs here to approximate ω for n = 5 , 6 and 7 Schwarzschild-Tangherlini black holes using a basic deep neural network set-up consisting of two fully connected layers and the adaptive moment estimation (ADAM) algorithm as the optimizer. Table II lists the approximate QNM frequencies generated by PINNs whose loss function we actively tuned toward N = 0 QNMs. The loss function minimized by the ADAM optimizer is given as:</text>
<formula><location><page_12><loc_39><loc_35><loc_92><loc_38></location>L = ⟨ [ D (ˆ χ )] 2 ⟩ + 1 ⟨ ˆ χ 2 ⟩ + 1 ⟨ ˆ ω k Re ⟩ , (63)</formula>
<text><location><page_12><loc_9><loc_17><loc_92><loc_34></location>which consists of the differential equation residual D ( χ ) , where D is the differential operator in Eqs. (59)-(61). The extra two terms constrain the space of eigenvalues, to be learned during optimisation, to within the fundamental mode values, which facilitates convergence to a single solution. Note that for the n = 6 and 7 , for various multipole numbers, the determination of the optimal choice of change of variable (i.e. ξ = 1 /r or ξ = 1 -(1 /r ) ) and choosing between k = 0 , 2 , 4 in the third term is a heuristic process and there is no one optimal set of parameters. We compared the n = 5 QNMs with Ref. [82] to estimate the accuracy of PINNs, given the high level of accuracy of the continued fraction method used in Ref. [82]. As shown in the n = 5 case in Table II, the PINN approximations generally differ from the CFM by ≪ 0 . 1% . When comparing with the sixth-order WKB method for n = 6 and 7 the percentages in the parentheses quantify the consistency of the QNMs between the WKB method and PINNs, where neither method is assumed the more accurate than the other. As can be seen, the QNMs agree more with increasing ℓ , which is when the WKB method is expected to improve in accuracy. It could be deduced from this behaviour that we obtain higher accuracy computations of QNMs from PINNs than the sixth-order WKB method for lower multipole numbers.</text>
<text><location><page_12><loc_9><loc_13><loc_92><loc_17></location>We are interested in the correspondence between the dynamics of the photon sphere of a Schwarzschild-Tangherlini black hole and their QNMs produced in the wake of perturbations by various test fields. The correspondence was determined by Ref. [67, 68] to be:</text>
<formula><location><page_12><loc_38><loc_8><loc_92><loc_11></location>ω N = Ω c ℓ -i ( N + 1 2 ) | λ | , ℓ ≪ N, (64)</formula>
<figure>
<location><page_13><loc_11><loc_45><loc_90><loc_93></location>
<caption>FIG. 6. The approach of the fundamental mode and spin 2 / ( n -2) QNMs towards the eikonal limit relation ω ℓ →∞ = Ω c ℓ -i ( N +1 / 2) | λ | (represented by the red curve).</caption>
</figure>
<text><location><page_13><loc_9><loc_20><loc_92><loc_33></location>where Ω c and λ are the orbital frequency and the Lyapunov exponent (i.e. the instability time scale), respectively, at the black hole's photon sphere. Note that Eqn (64) does not apply in all QNM computations, such as when considering gravitational perturbations of arbitrary asymptotically de Sitter BHs [68, 69]. However, we can still invoke the correspondence in the asymptotically flat Schwarzschild-Tangherlini case, which is demonstrated in Fig IV. For varying ℓ and n , QNMs plotted in Fig IV are obtained using the WKB method, where in the eikonal limit the tendency towards the properties of the photonsphere (i.e. Eqn. (64)) is seen. Physically, in such cases, massless perturbing fields have sufficient energy to orbit near the black hole in the location of the unstable circular geodesic from where they are either drawn inwards or escape outward to spatial infinity. As such the attenuation of QNMs in the temporal domain, quantified by ω Im , is explained by the leaking of particles from the unstable orbits in a time scale given by the Lyapunov exponent.</text>
<text><location><page_13><loc_9><loc_9><loc_92><loc_18></location>Additionally, the effect of dimensionality can be seen in QNMs as is done with the strong deflection angle, for example. For an increase in the number of dimensions, the imaginary part of ω is higher for comparable physical oscillation frequencies across different values of n (see Table II). In other words, there is an increased rate of leaking of particles (quantified by the Lyapunov exponent) at the unstable orbits, for higher dimensions. Regarding ω Re , Konoplya [83] showed that, for massless scalar fields and a given ℓ , the values lie on strict line; that is, they are directly proportional to r 0 n , where r 0 = 2 M determines the units of ω . Here, we see similar behavior across the different perturbing fields considered (whose QNMs are in Table II) with ω Re increasing with kn , with k as some constant of proportionality.</text>
<section_header_level_1><location><page_14><loc_44><loc_92><loc_57><loc_93></location>V. CONCLUSION</section_header_level_1>
<text><location><page_14><loc_9><loc_76><loc_92><loc_90></location>In this paper we have studied the effect of the number of dimension n on the black hole's photonsphere, shadow, deflection angle both in the weak and strong regimes, and quasinormal modes. First, we have seen how the photonsphere r ps decreases with increasing n . Results indicate that the rate at which r ps changes relative to n becomes smaller as n increases. As the shadow radius R sh is related to r ps , we have seen the same behavior. For instance, increasing n results in decreased R sh . Interestingly, while r ps /M = 2 when n = 5 , we have seen that R sh is smaller than the photonsphere. Such a peculiar case indicates the existence of extra dimensions if detected experimentally. Speculatively, if fractal dimensions are permitted in nature, the uncertainties in EHT's R sh measurement might indicate an upper bound in n as 3 . 84 and 3 . 94 for M87* and Sgr. A*, respectively. Meanwhile, the lower bound for n are 4 . 51 and 4 . 16 for M87* and Sgr. A*, respectively. This result implies that n greater than 5 and beyond are forbidden as far as the EHT data is concerned. Nevertheless, the theoretical study of higher dimensions must not be underestimated.</text>
<text><location><page_14><loc_9><loc_55><loc_92><loc_75></location>We then analyzed how the deflection angle changes with n by deriving the most general formula for the weak deflection angle α ; not only accommodating the finite distance of the source and observers but also valid for both time-like and null particles. As higher dimensionality is included, the formula necessitates the existence of a hypergeometric function, making the calculation of α more precise. If α is due to the time-like particles, we have seen how it deviates from the null result, depending on its speed v . Finally, the results imply that probing the existence of higher dimensions using weak deflection angle (as well as the lens images that it can cause) requires ultrasensitivity as increased n results in small α . We also examined the strong deflection angle that occurs near the r ps . Dimensionality and gravitational lensing are correlated, as can be inferred from the plot in Fig. 4 that the strong deflection angle around a higher dimensional black hole decreases as the dimension, n , increases. The degrees of freedom rise in proportion to dimensionality, which impacts the gravitational interaction surrounding the black hole. Such a decrease in the strong deflection angle raises the possibility that the gravitational lensing phenomenon is dampened due to higher dimensions leading to more difficulty of measurement. Investigating the precise mechanisms underlying this relationship may yield important new information about how gravity behaves in higher-dimensional regions and how this may affect astrophysical occurrences. Since the strong deflection exhibits large values as it approaches the photon sphere, higher dimension can possibly detected when investigated at these regions.</text>
<text><location><page_14><loc_9><loc_44><loc_92><loc_55></location>Given the increasing difficulty in measuring gravitational phenomenon as the number of dimensions increases, QNMs in higher dimensions offer another means to probe the photonsphere and related properties about the effect of SchwarzschildTangherlini black holes on interacting fields. This is made possible by the known relation between QNMs and the dynamics of the photonsphere in the eikonal limit, which are described by the orbital frequency and Lyapunov exponent. With the decrease of the black hole photon-sphere, shadow and defection angle, there is an increase in both the physical oscillation frequency and damping time-scale of QNMs (such as is seen in Table II). Knowing the correspondence between QNMs and the parameters of the photonsphere in the eikonal limit, QNMs could offer a useful indirect probe of the optical properties of the black hole which may not be accessible through direct measurement.</text>
<text><location><page_14><loc_9><loc_41><loc_92><loc_43></location>As a final concluding thought, the possibility of extending this study to incorporate black hole charge Q , or, in a more general sense, the spin parameter a , would allow for greater comparison with physically observable phenomena.</text>
<section_header_level_1><location><page_14><loc_42><loc_37><loc_59><loc_38></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_14><loc_9><loc_26><loc_92><loc_35></location>R. P. would like to acknowledge networking support of the COST Action CA18108 - Quantum gravity phenomenology in the multi-messenger approach (QG-MM), COST Action CA21106 - COSMIC WISPers in the Dark Universe: Theory, astrophysics and experiments (CosmicWISPers), the COST Action CA22113 - Fundamental challenges in theoretical physics (THEORY-CHALLENGES), and the COST Action CA21136 - Addressing observational tensions in cosmology with systematics and fundamental physics (CosmoVerse). A. S. C. is supported in part by the National Research Foundation (NRF) of South Africa. A. N. is supported by an SA-CERN Excellence Bursary through iThemba LABS.</text>
<unordered_list>
<list_item><location><page_14><loc_10><loc_19><loc_39><loc_20></location>[1] R. Penrose, Phys. Rev. Lett. 14 , 57 (1965).</list_item>
<list_item><location><page_14><loc_10><loc_18><loc_37><loc_19></location>[2] S. W. Hawking, Nature 248 , 30 (1974).</list_item>
<list_item><location><page_14><loc_10><loc_17><loc_71><loc_18></location>[3] K. Akiyama et al. (Event Horizon Telescope), Astrophys. J. Lett. 875 , L1 (2019), 1906.11238.</list_item>
<list_item><location><page_14><loc_10><loc_15><loc_71><loc_16></location>[4] K. Akiyama et al. (Event Horizon Telescope), Astrophys. J. Lett. 875 , L4 (2019), 1906.11241.</list_item>
<list_item><location><page_14><loc_10><loc_14><loc_72><loc_15></location>[5] K. Akiyama et al. (Event Horizon Telescope), Astrophys. J. Lett. 930 , L17 (2022), 2311.09484.</list_item>
<list_item><location><page_14><loc_10><loc_13><loc_49><loc_14></location>[6] J. L. Synge, Mon. Not. Roy. Astron. Soc. 131 , 463 (1966).</list_item>
<list_item><location><page_14><loc_10><loc_11><loc_43><loc_12></location>[7] J. P. Luminet, Astron. Astrophys. 75 , 228 (1979).</list_item>
<list_item><location><page_14><loc_10><loc_10><loc_68><loc_11></location>[8] H. Falcke, F. Melia, and E. Agol, Astrophys. J. Lett. 528 , L13 (2000), astro-ph/9912263.</list_item>
<list_item><location><page_14><loc_10><loc_9><loc_72><loc_10></location>[9] K. Akiyama et al. (Event Horizon Telescope), Astrophys. J. Lett. 930 , L12 (2022), 2311.08680.</list_item>
<list_item><location><page_15><loc_9><loc_92><loc_72><loc_93></location>[10] K. Akiyama et al. (Event Horizon Telescope), Astrophys. J. Lett. 930 , L14 (2022), 2311.09479.</list_item>
<list_item><location><page_15><loc_9><loc_91><loc_72><loc_92></location>[11] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. Lett. 116 , 061102 (2016), 1602.03837.</list_item>
<list_item><location><page_15><loc_9><loc_89><loc_72><loc_90></location>[12] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. Lett. 119 , 161101 (2017), 1710.05832.</list_item>
<list_item><location><page_15><loc_9><loc_88><loc_75><loc_89></location>[13] R. Abbott et al. (KAGRA, LIGO Scientific, VIRGO), Phys. Rev. D 106 , 042003 (2022), 2204.04523.</list_item>
<list_item><location><page_15><loc_9><loc_87><loc_62><loc_88></location>[14] L. Randall and R. Sundrum, Phys. Rev. Lett. 83 , 4690 (1999), hep-th/9906064.</list_item>
<list_item><location><page_15><loc_9><loc_85><loc_60><loc_86></location>[15] J. M. Maldacena, Adv. Theor. Math. Phys. 2 , 231 (1998), hep-th/9711200.</list_item>
<list_item><location><page_15><loc_9><loc_84><loc_57><loc_85></location>[16] R. Emparan and H. S. Reall, Living Rev. Rel. 11 , 6 (2008), 0801.3471.</list_item>
<list_item><location><page_15><loc_9><loc_83><loc_50><loc_84></location>[17] R. C. Myers and M. J. Perry, Annals Phys. 172 , 304 (1986).</list_item>
<list_item><location><page_15><loc_9><loc_81><loc_63><loc_82></location>[18] C. Cadeau and E. Woolgar, Class. Quant. Grav. 18 , 527 (2001), gr-qc/0011029.</list_item>
<list_item><location><page_15><loc_9><loc_80><loc_49><loc_81></location>[19] Y.-G. Shen and Z.-Q. Tan, Phys. Lett. A 142 , 341 (1989).</list_item>
<list_item><location><page_15><loc_9><loc_79><loc_50><loc_80></location>[20] B. R. Iyer and C. V. Vishveshwara, Pramana 32 , 749 (1989).</list_item>
<list_item><location><page_15><loc_9><loc_77><loc_52><loc_78></location>[21] A. Chodos and S. L. Detweiler, Gen. Rel. Grav. 14 , 879 (1982).</list_item>
<list_item><location><page_15><loc_9><loc_76><loc_40><loc_77></location>[22] D.-Y. Xu, Class. Quant. Grav. 5 , 871 (1988).</list_item>
<list_item><location><page_15><loc_9><loc_75><loc_57><loc_76></location>[23] X.-C. Cai and Y.-G. Miao, Eur. Phys. J. C 81 , 559 (2021), 2011.05542.</list_item>
<list_item><location><page_15><loc_9><loc_73><loc_41><loc_74></location>[24] T. Q. Do, EPJ Web Conf. 206 , 08002 (2019).</list_item>
<list_item><location><page_15><loc_9><loc_72><loc_71><loc_73></location>[25] Z. W. Feng, H. L. Li, X. T. Zu, and S. Z. Yang, Eur. Phys. J. C 76 , 212 (2016), 1604.04702.</list_item>
<list_item><location><page_15><loc_9><loc_71><loc_49><loc_72></location>[26] Y. S. Myung, Phys. Rev. D 88 , 084006 (2013), 1308.3907.</list_item>
<list_item><location><page_15><loc_9><loc_69><loc_62><loc_70></location>[27] B. C. Paul, Nonsingular Black Holes in Higher dimensions (2023), 2301.00358.</list_item>
<list_item><location><page_15><loc_9><loc_68><loc_77><loc_69></location>[28] U. Papnoi, F. Atamurotov, S. G. Ghosh, and B. Ahmedov, Phys. Rev. D 90 , 024073 (2014), 1407.0834.</list_item>
<list_item><location><page_15><loc_9><loc_67><loc_58><loc_68></location>[29] B. P. Singh and S. G. Ghosh, Annals Phys. 395 , 127 (2018), 1707.07125.</list_item>
<list_item><location><page_15><loc_9><loc_66><loc_89><loc_67></location>[30] A. Abdujabbarov, F. Atamurotov, N. Dadhich, B. Ahmedov, and Z. Stuchl'ık, Eur. Phys. J. C 75 , 399 (2015), 1508.00331.</list_item>
<list_item><location><page_15><loc_9><loc_64><loc_89><loc_65></location>[31] A. Belhaj, M. Benali, A. El Balali, H. El Moumni, and S. E. Ennadifi, Class. Quant. Grav. 37 , 215004 (2020), 2006.01078.</list_item>
<list_item><location><page_15><loc_9><loc_63><loc_51><loc_64></location>[32] J. Matyjasek, Phys. Rev. D 104 , 084066 (2021), 2107.04815.</list_item>
<list_item><location><page_15><loc_9><loc_62><loc_61><loc_63></location>[33] Z. Yan, C. Wu, and W. Guo, Nucl. Phys. B 961 , 115217 (2020), 2012.00320.</list_item>
<list_item><location><page_15><loc_9><loc_60><loc_61><loc_61></location>[34] R. Andr'e and J. P. S. Lemos, Phys. Rev. D 103 , 064069 (2021), 2101.11010.</list_item>
<list_item><location><page_15><loc_9><loc_59><loc_60><loc_60></location>[35] S. Chen, B. Wang, and R. Su, Phys. Rev. D 77 , 124011 (2008), 0801.2053.</list_item>
<list_item><location><page_15><loc_9><loc_58><loc_47><loc_59></location>[36] S. Barman, Eur. Phys. J. C 80 , 50 (2020), 1907.09228.</list_item>
<list_item><location><page_15><loc_9><loc_56><loc_49><loc_57></location>[37] M. S. Fox, J. Math. Phys. 60 , 102502 (2019), 1907.07622.</list_item>
<list_item><location><page_15><loc_9><loc_55><loc_63><loc_56></location>[38] G. Kunstatter and H. Maeda, Class. Quant. Grav. 31 , 115009 (2014), 1311.4888.</list_item>
<list_item><location><page_15><loc_9><loc_54><loc_65><loc_55></location>[39] A. Ishibashi and H. Kodama, Prog. Theor. Phys. 110 , 901 (2003), hep-th/0305185.</list_item>
<list_item><location><page_15><loc_9><loc_52><loc_56><loc_53></location>[40] B. Ahmedov, O. Rahimov, and B. Toshmatov, Universe 7 , 307 (2021).</list_item>
<list_item><location><page_15><loc_9><loc_51><loc_42><loc_52></location>[41] K. Lake, JCAP 10 , 007 (2003), gr-qc/0306073.</list_item>
<list_item><location><page_15><loc_9><loc_50><loc_66><loc_51></location>[42] G. W. Gibbons and M. C. Werner, Class. Quant. Grav. 25 , 235009 (2008), 0807.0854.</list_item>
<list_item><location><page_15><loc_9><loc_48><loc_79><loc_49></location>[43] A. Ishihara, Y. Suzuki, T. Ono, T. Kitamura, and H. Asada, Phys. Rev. D 94 , 084015 (2016), 1604.08308.</list_item>
<list_item><location><page_15><loc_9><loc_47><loc_59><loc_48></location>[44] Z. Li, G. He, and T. Zhou, Phys. Rev. D 101 , 044001 (2020), 1908.01647.</list_item>
<list_item><location><page_15><loc_9><loc_46><loc_55><loc_47></location>[45] Z. Li and A. Ovgun, Phys. Rev. D 101 , 024040 (2020), 2001.02074.</list_item>
<list_item><location><page_15><loc_9><loc_44><loc_54><loc_45></location>[46] Z. Li, G. Zhang, and A. Ovgun, Phys. Rev. D 101 , 124058 (2020).</list_item>
<list_item><location><page_15><loc_9><loc_43><loc_67><loc_44></location>[47] K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D 62 , 084003 (2000), astro-ph/9904193.</list_item>
<list_item><location><page_15><loc_9><loc_41><loc_55><loc_42></location>[48] K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D 65 , 103004 (2002).</list_item>
<list_item><location><page_15><loc_9><loc_40><loc_51><loc_41></location>[49] K. S. Virbhadra, Phys. Rev. D 79 , 083004 (2009), 0810.2109.</list_item>
<list_item><location><page_15><loc_9><loc_39><loc_53><loc_40></location>[50] K. S. Virbhadra, Phys. Rev. D 106 , 064038 (2022), 2204.01879.</list_item>
<list_item><location><page_15><loc_9><loc_38><loc_62><loc_39></location>[51] K. S. Virbhadra and C. R. Keeton, Phys. Rev. D 77 , 124014 (2008), 0710.2333.</list_item>
<list_item><location><page_15><loc_9><loc_36><loc_50><loc_37></location>[52] V. Bozza, Phys. Rev. D 66 , 103001 (2002), gr-qc/0208075.</list_item>
<list_item><location><page_15><loc_9><loc_35><loc_51><loc_36></location>[53] N. Tsukamoto, Phys. Rev. D 95 , 064035 (2017), 1612.08251.</list_item>
<list_item><location><page_15><loc_9><loc_34><loc_52><loc_35></location>[54] N. Tsukamoto, Phys. Rev. D 104 , 064022 (2021), 2105.14336.</list_item>
<list_item><location><page_15><loc_9><loc_32><loc_52><loc_33></location>[55] N. Tsukamoto, Phys. Rev. D 102 , 104029 (2020), 2008.12244.</list_item>
<list_item><location><page_15><loc_9><loc_31><loc_52><loc_32></location>[56] N. Tsukamoto, Phys. Rev. D 105 , 064013 (2022), 2111.13501.</list_item>
<list_item><location><page_15><loc_9><loc_30><loc_58><loc_31></location>[57] C.-Y. Wang, Y.-F. Shen, and Y. Xie, JCAP 04 , 022 (2019), 1902.03789.</list_item>
<list_item><location><page_15><loc_9><loc_28><loc_79><loc_29></location>[58] J. Chagoya, C. Ortiz, B. Rodr'ıguez, and A. A. Roque, Class. Quant. Grav. 38 , 075026 (2021), 2007.09473.</list_item>
<list_item><location><page_15><loc_9><loc_27><loc_50><loc_28></location>[59] N. Tsukamoto, Eur. Phys. J. C 83 , 284 (2023), 2211.04239.</list_item>
<list_item><location><page_15><loc_9><loc_26><loc_61><loc_27></location>[60] G. Abbas, A. Mahmood, and M. Zubair, Phys. Dark Univ. 31 , 100750 (2021).</list_item>
<list_item><location><page_15><loc_9><loc_24><loc_53><loc_25></location>[61] K. S. Virbhadra, Phys. Rev. D 109 , 124004 (2024), 2402.17190.</list_item>
<list_item><location><page_15><loc_9><loc_23><loc_64><loc_24></location>[62] T. Hsieh, D.-S. Lee, and C.-Y. Lin, Phys. Rev. D 104 , 104013 (2021), 2108.05006.</list_item>
<list_item><location><page_15><loc_9><loc_22><loc_64><loc_23></location>[63] T. Hsieh, D.-S. Lee, and C.-Y. Lin, Phys. Rev. D 103 , 104063 (2021), 2101.09008.</list_item>
<list_item><location><page_15><loc_9><loc_20><loc_62><loc_21></location>[64] N. J. L. S. Lobos and R. C. Pantig, MDPI Physics 4 , 1318 (2022), 2208.00618.</list_item>
<list_item><location><page_15><loc_9><loc_19><loc_77><loc_20></location>[65] N. Tsukamoto, T. Kitamura, K. Nakajima, and H. Asada, Phys. Rev. D 90 , 064043 (2014), 1402.6823.</list_item>
<list_item><location><page_15><loc_9><loc_18><loc_70><loc_19></location>[66] A. Friedlander, N. Song, and A. C. Vincent, Phys. Rev. D 108 , 043523 (2023), 2306.01520.</list_item>
<list_item><location><page_15><loc_9><loc_16><loc_83><loc_17></location>[67] V. Cardoso, A. S. Miranda, E. Berti, H. Witek, and V. T. Zanchin, Phys. Rev. D 79 , 064016 (2009), 0812.1806.</list_item>
<list_item><location><page_15><loc_9><loc_14><loc_92><loc_16></location>[68] R. Konoplya, Physics Letters B 838 , 137674 (2023), ISSN 0370-2693, URL https://www.sciencedirect.com/science/ article/pii/S0370269323000084 .</list_item>
<list_item><location><page_15><loc_9><loc_11><loc_92><loc_13></location>[69] R. Konoplya and Z. Stuchlk, Physics Letters B 771 , 597 (2017), ISSN 0370-2693, URL https://www.sciencedirect. com/science/article/pii/S0370269317304823 .</list_item>
<list_item><location><page_15><loc_9><loc_10><loc_41><loc_11></location>[70] F. R. Tangherlini, Nuovo Cim. 27 , 636 (1963).</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_16><loc_9><loc_92><loc_56><loc_93></location>[71] V. Perlick and O. Y. Tsupko, Phys. Rept. 947 , 1 (2022), 2105.07101.</list_item>
<list_item><location><page_16><loc_9><loc_90><loc_60><loc_92></location>[72] R. C. Pantig and A. Ovgun, Annals Phys. 448 , 169197 (2023), 2206.02161.</list_item>
<list_item><location><page_16><loc_9><loc_89><loc_74><loc_90></location>[73] P. Kocherlakota et al. (Event Horizon Telescope), Phys. Rev. D 103 , 104047 (2021), 2105.09343.</list_item>
<list_item><location><page_16><loc_9><loc_88><loc_57><loc_89></location>[74] S. Vagnozzi et al., Class. Quant. Grav. 40 , 165007 (2023), 2205.07787.</list_item>
<list_item><location><page_16><loc_9><loc_86><loc_71><loc_88></location>[75] A. Ovgun, L. J. F. Sese, and R. C. Pantig, Annalen Phys. 2023 , 2300390 (2023), 2309.07442.</list_item>
<list_item><location><page_16><loc_9><loc_85><loc_52><loc_86></location>[76] R. C. Pantig, Phys. Dark Univ. 45 , 101550 (2024), 2405.07531.</list_item>
<list_item><location><page_16><loc_9><loc_82><loc_92><loc_85></location>[77] M. P. Do Carmo, Differential geometry of curves and surfaces: revised and updated second edition (Courier Dover Publications, 2016).</list_item>
<list_item><location><page_16><loc_9><loc_81><loc_76><loc_82></location>[78] W. Klingenberg, A course in differential geometry , vol. 51 (Springer Science & Business Media, 2013).</list_item>
<list_item><location><page_16><loc_9><loc_80><loc_77><loc_81></location>[79] K. S. Virbhadra, D. Narasimha, and S. M. Chitre, Astron. Astrophys. 337 , 1 (1998), astro-ph/9801174.</list_item>
<list_item><location><page_16><loc_9><loc_78><loc_52><loc_79></location>[80] N. Tsukamoto, Phys. Rev. D 104 , 124016 (2021), 2107.07146.</list_item>
<list_item><location><page_16><loc_9><loc_77><loc_85><loc_78></location>[81] A. Chrysostomou, A. Cornell, A. Deandrea, E. Ligout, and D. Tsimpis, Eur. Phys. J. C 83 , 325 (2023), 2211.08489.</list_item>
<list_item><location><page_16><loc_9><loc_76><loc_42><loc_77></location>[82] J. Matyjasek, Phys. Rev. D 104 , 084066 (2021).</list_item>
<list_item><location><page_16><loc_9><loc_74><loc_54><loc_75></location>[83] R. A. Konoplya, Phys. Rev. D 68 , 024018 (2003), gr-qc/0303052.</list_item>
<list_item><location><page_16><loc_9><loc_73><loc_60><loc_74></location>[84] L. Lu, X. Meng, Z. Mao, and G. E. Karniadakis, SIAM Rev. 63 , 208 (2021).</list_item>
</unordered_list>
</document>
[
{
"title": "Analyzing the effect of higher dimensions on the black hole silhouette, deflection angles, and PINN approximated quasinormal modes",
"content": "Nikko John Leo S. Lobos, 1, ∗ Anele M. Ncube, 2, † Reggie C. Pantig, 3, ‡ and Alan S. Cornell 2, § 1 Mathematics and Physics Department, Technological Institute of the Philippines, 363 Casal St, Quiapo, Manila, 1001 Metro Manila 2 Department of Physics, University of Johannesburg, PO Box 524, Auckland Park 2006, South Africa 3 Physics Department, Map´ua University, 658 Muralla St., Intramuros, Manila 1002, Philippines We investigate the impact of higher dimensions on the properties of Schwarzschild-Tangherlini black holes, focusing on the photonsphere, black hole shadow, deflection angles, and quasinormal modes (QNMs). We find that these properties diminish as the dimensionality ( n ) of the black hole increases. Analysis of the shadow radius measured by the Event Horizon Telescope suggests non-integer dimensions around n ≶ 4 . We derive an analytic formula for the weak field deflection angle, highlighting the need for advanced sensitive detection devices to observe lensed images influenced by higher dimensions. Our study of QNMs using physics-informed neural networks and the WKB method reveals a convergence towards known relationships between QNM frequencies and photon-sphere orbit frequencies. Despite the energetic nature of perturbing fields in higher dimensions, their damping increases. This suggests a complex interplay between dimensionality and the dynamics of black hole phenomena. PACS numbers: 95.30.Sf, 04.70.-s, 97.60.Lf, 04.50.+h Keywords: Supermassive black holes; black hole shadow, deflection angle, quasinormal modes",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "Black holes, enigmatic cosmic entities born from the gravitational collapse of massive stars, have long captivated the imagination of scientists and the general public alike. These celestial phenomena serve as unique laboratories, pushing the boundaries of our understanding of gravity and the very fabric of spacetime. Black holes become crucial probes for testing the predictions of Einstein's general theory of relativity, unraveling the mysteries of galactic evolution, and offering insights into the larger cosmic landscape [1, 2]. In this pursuit, black holes are indispensable tools for verifying the robustness of general relativity in extreme conditions. The Event Horizon Telescope's (EHT) imaging of the supermassive black hole in the M87 galaxy in 2019 provides a compelling visual confirmation of theoretical predictions concerning event horizons and photonspheres [3-5], which affirms the seminal works of Synge, Luminet, and Falcke [6-8]. Another success immediately followed in 2022, as the black hole at the heart of our galaxy was confirmed [9, 10]. In addition to the shadow detection, gravitational waves were detected by LIGO and Virgo from black hole mergers has offered unprecedented evidence supporting general relativity's predictions about spacetime dynamics [11-13], and opening a new era in astronomy. As such, in exploring black holes beyond the confines of our familiar four-dimensional spacetime, we are compelled by several reasons: (1) Theoretical frameworks like string theory propose the existence of extra spatial dimensions beyond the familiar three [14, 15]. Black holes in higher dimensions emerge naturally from these theories, and are often invoked in discussions of quantum gravity (See Ref. [16] for a comprehensive review). Studying Schwarzschild black holes in extra-dimensions helps probe the quantum nature of spacetime and gravity, potentially providing clues about the microscopic structure of black holes; (2) Higher-dimensional gravity theories, such as Kaluza-Klein theory, aim to unify gravity with other fundamental forces. Black holes in higher dimensions are key in testing and understanding these unified theories; (3) Some theoretical models involve the concept of higher-dimensional spacetimes with branes representing our visible universe. Black holes in these scenarios can have unique properties and effects that differ from their counterparts in standard four-dimensional spacetime; and (4) Studying black holes in higher dimensions is not only motivated by theoretical physics but also by mathematical interest. Higher-dimensional spacetimes can reveal new insights and challenges in understanding geometry and general relativity. These motivations, led to several studies present in the literature [16-27] as the scientific community tries to uncover the effects of higher dimensions on certain aspects of the black hole geometry. Investigating black holes in extra dimensions may help identify unique signatures that could be observed or tested through astrophysical observations such as in the black hole shadow [28, 29], gravitational lensing and optical properties [30, 31], quasinormal modes [32, 33], thermodynamics [34-36], etc [37-41]. § It is understood that the higher dimensionality of black holes is theoretically and mathematically appealing. With the recent constraints for the radius of the invisible shadow provided by the EHT, our aim in this paper is to constrain the dimensionality of the black hole and whether some of the uncertainties in the shadow radius may be caused by the higher dimensionality of the black hole. Furthermore, we explore the deflection angles within the weak and strong field regimes, checking higher dimensional signatures. To study the weak field deflection angle, we utilize the Gauss-Bonnet theorem to include the finite distance and time-like particles [42-46], to give us the general case. The strong deflection angle, commonly denoted as the angle of light bending in the proximity of a massive object, encapsulates the profound gravitational influence on the trajectory of light rays. Unlike weak deflection, characterized by minimal light deviation, strong deflection occurs when light traverses in close proximity to the gravitational source, resulting in significant bending of light rays. This phenomenon manifests observable signatures that offer invaluable insights into the nature and characteristics of the intervening mass [47-51]. Note that the works of Bozza [52] and Tsukamoto [53] have investigated the calculation of the strong deflection angle in the vicinity of a black hole. Tsukamoto, in particular, extends Bozza's methodology by introducing a novel term, denoted as z , enabling an analytical approximation for the deflection angle equation. The outcomes exhibit consistency, with negligible discrepancies [53]. Tsukamoto's approach, and its subsequent refinement, broaden the applicability of the strong deflection angle formalism to encompass various black hole models, thus facilitating investigations into more intricate black hole configurations [54-56]. Furthermore, numerous insightful findings have been presented concerning the behavior of light under such conditions [31, 54, 57-63]. For instance, within the strong field regime, the influence of charged matter and dark matter on the deflection angle becomes more discernible in comparison to weak deflection scenarios [64]. The formulation of theories elucidating the strong deflection angle within higher-dimensional black hole spacetimes holds profound implications for our understanding of black hole physics [65]. It not only sheds light on the behavior of gravity within unconventional higher-dimensional contexts but also elucidates fundamental principles underlying black hole theory [66]. The exploration of how gravitational lensing attributes are influenced by higher dimensions possesses the potential to refine our conceptual frameworks regarding black hole formation, evolution, and the interpretation of observational data pertaining to astrophysical phenomena involving black holes. As we shall discuss, the photonsphere plays an important role in forming the black hole shadows, as it dictates the size and shape of the black hole shadow. However, complementary information can be extracted by studying quasinormal modes (QNMs) of black holes. QNMs are characteristic oscillations of black holes that arise when the black hole is perturbed. These perturbations can be thought of as the 'ringing' of the black hole, and they decay over time due to the emission of gravitational waves. QNMs are defined by complex frequencies, where the real part represents the oscillation frequency and the imaginary part denotes the damping rate. The link between photon orbits and QNMs is established through the concept that the real part of the quasinormal mode frequencies is related to the orbital frequency of the photonsphere. In contrast, the imaginary part is associated with the Lyapunov exponent, which characterizes the stability of the photon orbits. This relationship was demonstrated by Cardoso et al. [67], who showed that for high-frequency (eikonal) QNMs, the frequencies are directly tied to the properties of the photonsphere. The oscillation frequencies of QNMs therefore correspond to the orbital frequencies of photons in the photonsphere, and the decay rates correspond to the stability properties of these orbits. As noted in Refs. [68, 69] some care is required when calculating QNMs in extra-dimensional spacetimes and presupposing this correspondence in the eikonal regime, which has been demonstrated to be invalid whenever the BH perturbation equations have effective potentials that are not 'well-behaved' from the stand-point of applicability of the WKB method. Inferring from the effective potentials of asymptotically flat Schwarzschild-Tangherlini BHs, which have a single extremum in the domain r ∈ [2 M, ∞ ) , the WKB method (and, in turn, the correspondence) is guaranteed to be applicable in this case. But in order to move toward a full analysis of the phenomena, we consider here the behavior of the deflection angle in the presence of a higher-dimensional black hole, and how to calculate QNMs in such a space in a new robust neural network manner. We organize the paper as follows: In Sect. II, we provide a quick review of the Schwarzschild-Tanghelini black hole, along with examining the higher dimension parameter n to the photon orbit, and the black hole shadow. We also provide constraints to n in light of EHT observations. Next, we study the weak and strong deflection angles of this higher-dimensional black hole in Sect. III. Finally, we employed machine learning methods to calculate the QNMs in Sect. IV using PINN approximations. We give concluding remarks and future research prospects in Sect. V. Throughout this paper, we used ( -, + , + , +) metric signature and geometrized units by considering G = c = 1 .",
"pages": [
1,
2
]
},
{
"title": "II. SHADOW OF A SCHWARZSCHILD BLACK HOLE IN HIGHER DIMENSIONS",
"content": "The Schwarzschild-Tangherlini metric extends the Schwarzschild solution to higher dimensions in the context of general relativity. It provides a solution to the Einstein field equations for a spherically symmetric, non-rotating black hole in a spacetime with n dimensions, where n is greater than 4 . For its full review, see Ref. [16], where the metric of the Schwarzschild black hole in extra-dimensions (Schwarzschild-Tangherlini black hole) is given by where A ( r ) , B ( r ) , and d Ω 2 n -2 are expressed by respectively. As one considers only the pure Schwarzschild field and its spherical symmetry, it can be noted from Ref. [70] that dθ 1 = dθ = 0 , dθ 2 = dϕ , and dθ n -2 = 0 . Furthermore, M admits the dimensionality n through where Ω n -2 is the area of the ( n -2) -dimensional unit sphere: and m is the black hole mass. The photonsphere ( r ps ) is a region around a black hole where photons can theoretically orbit the black hole in a circular path. It is well known that for the Schwarzschild case, r ps = 3 m . This section shows how the number of dimensions n affects the photonsphere, as we apply the formalism developed in Ref. [71]. The results here will be important in the next section. We take the standard Lagrangian Applying the variational principle gives us the two constants of motion: where the impact parameter can be defined as Light rays obey g µν ˙ x µ ˙ x ν = 0 , leading to an orbit equation where the function h ( r ) is defined as [71] The location of the photonsphere can be found by either satisfying the condition dr/dϕ = d 2 r/dϕ 2 = 0 , or using h ' ( r ) = 0 , which can lead to The analytical formula for the photonsphere radius is then where we plot this in Fig. 1. The effect of increasing the number of dimensions n decreases the radius of the photonsphere, when normalized to the mass of the black hole. The presence of the photonsphere plays a crucial role in shaping the appearance of the black hole silhouette and contributes to the distinctive features observed in images, such as the shadow or the bright ring surrounding it. After obtaining the location of the photonsphere through the analytic equation (11), a small perturbation can either make the photons escape or spiral toward the black hole. For escaped photons, these will contribute to the black hole silhouette observed at r = r obs , which can backward trace the photons' path to study the black hole shadow formation. For such an observer [71], which can be simplified as Here, the critical impact parameter, which is a function of r ps , can be determined from the condition d 2 r/dϕ 2 = 0 [72]: With the help of Eq. (11), the shadow radius is then We plot the above equation to see the effect of the observer's location for different dimensions n (see the left panel of Fig. 2). Here, we can drastically see the decrease in the shadow radius when the observer is so far away from the black hole. Interestingly, if the observer is near the Schwarzschild event horizon n = 4 , there are points where n = 5 , and n = 6 that cannot be distinguished from n = 4 . Finally, on the right panel, we attempted to constrain n for at least 3 σ (99 . 7%) confidence level shadow radius observed by the EHT. Looking at Refs. [3, 9, 73, 74], such a level reveals the lower and upper bounds as 3 . 871 M ≤ R sh ≤ 5 . 898 M , and 2 . 546 M ≤ R sh ≤ 7 . 846 M for Sgr. A* and M87*, respectively. These bounds were also used recently in Refs. [75, 76].",
"pages": [
2,
3,
4
]
},
{
"title": "A. Weak deflection angle",
"content": "In this section, we aim to analyze the effect of the number of dimensions on the weak deflection angle. To do so, we use the Gauss-Bonnet theorem stating that [77, 78] Here, K is the Gaussian curvature, dS is the area measure, Θ i , and κ g is the jump angles and geodesic curvature of ∂D , respectively, and dℓ is the arc length measure. The application to null geodesics in the equatorial plane implies that the Euler characteristics should be χ ( D ) = 1 . If the integral is evaluated over the infinite area surface bounded by the light ray, it was shown in Ref. [43] that the above reduces to where ˆ α is the weak deflection angle. In the above formula, ϕ RS = ϕ R -ϕ S is the azimuthal separation angle between the source S and receiver R, ϕ R and ϕ S are the positional angles, and ∞ R □ ∞ S is the integration domain. While this study can use the above formula, we prefer to use its extension to the non-asymptotically flat case given in Ref. [46], where one uses the path in the photonsphere orbit instead of the path at infinity. Eq. (18) can be reformulated as: Before using the above equation, we are interested in the deflection angle of massive particles. Such a general case reduces to a special case of photon deflection when v = 1 . As such, we need the Jacobi metric stating that where E is the energy per unit mass ( µ ) of the massive particle: It is then useful to define another constant quantity in terms of the impact parameter b , which is the angular momentum per unit mass: and with E and J , we can define the impact parameter as Using the line element for time-like particles ds 2 = g µν dx µ dx ν = -1 and defining u = 1 /r , the orbit equation can be derived as which, in our case, yields ( p = n -3 ) Next, by an iterative method, the goal is to find u as a function of ϕ , which we find as From here, we obtain the solution for ϕ as This is the expression for the positional angle ϕ S , which implies that u → u S . Note also that ϕ R = π -ϕ S . The Gaussian curvature can be derived using since Γ ϕ rr = 0 from Eq. (20). Furthermore, the determinant of Eq. (20) is With the analytical solution to r ps , it is easy to see that which yields where the prime denotes differentiation with respect to r . The weak deflection angle is then [46], Using Eq. (26) in Eq. (31), we find since ϕ S has the same form as Eq. (27), while keeping in mind that the definite integral is already evaluated. Then, we find Eq. (32) as Finally, we need to express the equation above in terms of finite distance u S and u R . We use the following relations: After approximating Eq. (34) again, the general analytic expression for the deflection angle of the Schwarzschild-Tangherlini black hole with finite distance as which is written in terms of a hypergeometric function. One can easily check that if p = 1 (where n = 4 ), the equation above reduces to the timelike expression in finite distance as If we assume that u S = u R and both are approximately zero, then Finally, for photons where v = 1 , We observe that when n is odd, ˆ α contains a factor involving a hypergeometric function. For instance, when n = 5 ( p = 2 ), When n is even, such as when p = 3 , we observe that ˆ α does not contain a factor involving a hypergeometric function: We plot Eq. (36), shown in Fig. 3. First, we observe that the deflection angle caused by time-like particles gives a slightly higher value of α than photon deflection. Second, our observation is that as the number of dimensions increases, α decreases. It only means that if we use the phenomenon of deflection angle to probe the existence of higher dimensions in black holes, more sensitive devices are needed. Finally, the effect of finite distance indicates that α slightly increases as the impact parameter b/M becomes greater than the observer distance from the black hole. 0 2 4 6 8 10 12 log10 ( b / M )",
"pages": [
5,
6,
7,
8
]
},
{
"title": "B. Strong Deflection Angle",
"content": "Photons slightly greater than the photonsphere radius will undergo the phenomenon of strong deflection angle. In this section, we calculate and showcase the deflection of light as it approaches the photonsphere in the strong field limit. Using the approach of Tsukamoto in Ref. [53], the deflection angle is derived using the light trajectory shown in Eq. (8) but, in this section we express it as, where Note that the A ( r ) is the metric function defined by Eq. (2), while A ( r 0 ) is the metric function evaluated at distance r 0 . The solution of Eq. (42) yields the strong deflection angle α ( r 0 ) as shown in Ref. [52, 53, 79], In order to evaluate the integral in Eq. (44), we expand over r = r 0 . This yields a regular integral κ R and a diverging integral κ D , and by introducing a new variable, z, defined as, I ( r 0 ) is expressed as, where κ ( z, r 0 ) is expressed as the sum of the diverging integral, κ D , and regular integral, κ R . The details of the expansion of Eq. (44) was shown in Refs. [52, 53]. As a result the strong deflection angle is expressed as, ) ( 0 1 g o l 0 10 20 30 40 50 n =4, v =1 n =5 n =6 n =7 n =8 n =4, v =0.75 where ¯ a and ¯ b are coefficients and b 0 and b crit are the impact parameter evaluated at the closest approach, r 0 , and critical impact parameter, respectively. The first term in Eq. (47) is the result of the diverging integral and the second term is the result of the regular integral. The coefficients ¯ a and ¯ b are expressed as [53], and where A ( r ps ) is metric function evaluated at the photonsphere, and I R is the regular integral evaluated from 0 to 1. The double prime in Eq. (48) and Eq. (49) correspond to the second derivative with respect to r evaluated over r ps . Applying Eq. (48) to the black hole metric would yield the coefficient ¯ a , where p = n -3 . When p = 1 , we have the Schwarzschild result of a = 1 . A pattern was observed from ¯ a which can be simplified as ¯ a = √ p/p . As we increase the dimension of the black hole the value of ¯ a decreases exponentially. The argument of the natural logarithmic term of Eq. (49) becomes, and we retrieve the Schwarschild result when p = 1 (which is 6 as in Ref. [52, 53]). The regular integral I R is defined as, where the f R ( z, r 0 ) was generated from the expansion of the trajectory in Eq. (42), which gives us where G ( z, r 0 ) = RCA (1 -z ) 4 . Notice that C and A are the metric functions for which the position r is expressed in terms of z and r 0 , while R is shown in Eq. (43). The generated expression from Eq. (53) is, when we evaluate r 0 = r ps . On the other hand the f D ( z, r ps ) is expressed as, where the c 's are coefficients of the new variable z . For the equations in (50), (51), and by evaluating the integral in Eq. (52), our results are summarized in Table I. The results in Table I are consistent with results when using the Schwarzschild metric where n = 4 as shown in Refs. [52, 53, 58, 65, 80] with an extension to higher dimension black holes. As the dimensionality increases, the coefficient ¯ a decreases which significantly affects ¯ b and leaves the I R to dominate the expression. The increasing value of I R greatly influences the decrease of the deflection angle for higher dimensions. As shown in Fig. 4, the strong deflection angle at higher dimensions exhibits the same behavior but in decreasing values [31] as dimensionality increases. These decreased values can be attributed to the increase in the number of degrees of freedom in the higher dimensional spaces [16]. Investigating the existence of higher dimensions from n = 4 to n = 7 is theoretically possible since, at this region, the strong deflection angle is relatively large compared to the weak deflection angle. From n = 8 , the strong deflection becomes significantly small and would require ultrasensitive devices to probe.",
"pages": [
8,
9,
10
]
},
{
"title": "IV. QUASINORMAL MODES",
"content": "In light of recent observations of gravitational waves [11], the study of QNMs also becomes an interesting avenue for the study of extra-dimensional models [81]. To complement the study of photonspheres above, we now look at how to calculate QNMs in this spacetime with a neural network approach. The use of neural networks, detailed below, is for its robustness to a range of partial differential equations, as we have here, with a consideration of different dimensionalities. In this section we shall rewrite the metric as [82]: where f ( r ) = 1 -r 3 -n when working in the 2 M = 1 units. The equations describing the perturbation of this metric to produce damped sinusoids, the QNMs, are given as [82, 83]: where the prime denotes derivatives with respect to the tortoise co-ordinate x . Here j (i.e. spin of perturbing field) is assigned 0 , 2 , 2 / ( n -2) for massless scalar, gravitational vector and electromagnetic vector perturbations, respectively. The QNMs, represented here by the perturbation quantities ψ , are solutions to the Schrodinger-like eigenvalue problem given in Eq. (57). and are indexed by the spin j , multipole number ℓ and overtone number N (capitalised to distinguish it from n denoting spacetime dimensions), with the least-damped mode being N = 0 . The eigenvalues of Eq. (57) are the frequencies of the damped sinusoids denoted as ω = ω Re -iω Im , with ω Re signifying the physical oscillation frequency and ω Im being associated with the damping rate. These frequencies are important for probing the parameters of the perturbed source. In the astrophysical setting, the asymptotic behavior of physically allowed modes constitutes only in-going waves to the black hole horizon and only outgoing waves to spatial infinity, in the radial domain of Eq. (57), where the radial and tortoise co-ordinates ( r ∈ [1 , ∞ ) and x ∈ ( -∞ , ∞ ) , respectively) are related through the differential equation dr/dx = f ( r ) . Factoring out the asymptotic behavior, ψ ( r ) can be expressed in the form [82]: Substituting the asymptotic behavior in the perturbation equation and transforming to a finite co-ordinate ξ = 1 -(1 /r ) or ξ = 1 /r , both of which yield a finite domain ξ ∈ [0 , 1] , the differential equation can be written in the form implicitly incorporating the physical behavior. For example, when working with ξ = 1 /r and n = 5 , 6 and 7 , for which some QNMs are computed using PINNs, the perturbation equations are expressed as: The corresponding equations in terms of ξ = 1 -(1 /r ) are similarly second-order, linear, homogeneous differential equations. They are solved here using physics-informed neural networks (PINNs) and the sixth-order WKB method (for comparison between two methods), where the latter is a method that was developed by Ref. [83]. The applicability of the WKB method in the Schwarzschild-Tangherlini case can be inferred from the related effective potential, which should be a potential barrier with a single extremum point between the event horizon and spatial infinity boundaries where the potential decays. This is seen in Fig IV, for various perturbing fields including gravitational vector perturbations ( j = 2 ), for ℓ = 2 and n = 5 , 6 , and 7 . The PINN algorithm is a different approximation method; it is machine learning-based and uses highly parameterized ansatzes to approximate the solutions of a differential equation. The approximation is facilitated by an optimization algorithm in which the deviation of the approximate function from the target functions, quantified using the loss function, is minimized by backpropagating the derivatives of the loss function with respect to the 'ansatz' parameters to take iterative steps towards the loss's global minimum. Let ˆ χ ( ξ, θ ) be the output value of the neural network (the approximation of the solutions to Eqs. (59) - (61)), then ξ is set as an input node to pass into the neural network some data sampled from ξ ∈ [0 , 1] . The parameters to be optimized are denoted by θ . As such, the neural network is represented in general as [84]: where the neural network output N ℓ = L ( ξ, θ ) = ˆ χ ( ξ, θ ) is a composite function of recursive linear transformations of the hidden layers N L -1 , N L -2 , ... N 1 as given within the parentheses of the right-hand side of Eq. (60). Here W ℓ and b ℓ are the weight matrices and bias vectors, respectively, which we represent collectively with the term θ . To enhance function approximation, an activation function σ is applied to each hidden layer ℓ ∈ [1 , L -1] . We use PINNs here to approximate ω for n = 5 , 6 and 7 Schwarzschild-Tangherlini black holes using a basic deep neural network set-up consisting of two fully connected layers and the adaptive moment estimation (ADAM) algorithm as the optimizer. Table II lists the approximate QNM frequencies generated by PINNs whose loss function we actively tuned toward N = 0 QNMs. The loss function minimized by the ADAM optimizer is given as: which consists of the differential equation residual D ( χ ) , where D is the differential operator in Eqs. (59)-(61). The extra two terms constrain the space of eigenvalues, to be learned during optimisation, to within the fundamental mode values, which facilitates convergence to a single solution. Note that for the n = 6 and 7 , for various multipole numbers, the determination of the optimal choice of change of variable (i.e. ξ = 1 /r or ξ = 1 -(1 /r ) ) and choosing between k = 0 , 2 , 4 in the third term is a heuristic process and there is no one optimal set of parameters. We compared the n = 5 QNMs with Ref. [82] to estimate the accuracy of PINNs, given the high level of accuracy of the continued fraction method used in Ref. [82]. As shown in the n = 5 case in Table II, the PINN approximations generally differ from the CFM by ≪ 0 . 1% . When comparing with the sixth-order WKB method for n = 6 and 7 the percentages in the parentheses quantify the consistency of the QNMs between the WKB method and PINNs, where neither method is assumed the more accurate than the other. As can be seen, the QNMs agree more with increasing ℓ , which is when the WKB method is expected to improve in accuracy. It could be deduced from this behaviour that we obtain higher accuracy computations of QNMs from PINNs than the sixth-order WKB method for lower multipole numbers. We are interested in the correspondence between the dynamics of the photon sphere of a Schwarzschild-Tangherlini black hole and their QNMs produced in the wake of perturbations by various test fields. The correspondence was determined by Ref. [67, 68] to be: where Ω c and λ are the orbital frequency and the Lyapunov exponent (i.e. the instability time scale), respectively, at the black hole's photon sphere. Note that Eqn (64) does not apply in all QNM computations, such as when considering gravitational perturbations of arbitrary asymptotically de Sitter BHs [68, 69]. However, we can still invoke the correspondence in the asymptotically flat Schwarzschild-Tangherlini case, which is demonstrated in Fig IV. For varying ℓ and n , QNMs plotted in Fig IV are obtained using the WKB method, where in the eikonal limit the tendency towards the properties of the photonsphere (i.e. Eqn. (64)) is seen. Physically, in such cases, massless perturbing fields have sufficient energy to orbit near the black hole in the location of the unstable circular geodesic from where they are either drawn inwards or escape outward to spatial infinity. As such the attenuation of QNMs in the temporal domain, quantified by ω Im , is explained by the leaking of particles from the unstable orbits in a time scale given by the Lyapunov exponent. Additionally, the effect of dimensionality can be seen in QNMs as is done with the strong deflection angle, for example. For an increase in the number of dimensions, the imaginary part of ω is higher for comparable physical oscillation frequencies across different values of n (see Table II). In other words, there is an increased rate of leaking of particles (quantified by the Lyapunov exponent) at the unstable orbits, for higher dimensions. Regarding ω Re , Konoplya [83] showed that, for massless scalar fields and a given ℓ , the values lie on strict line; that is, they are directly proportional to r 0 n , where r 0 = 2 M determines the units of ω . Here, we see similar behavior across the different perturbing fields considered (whose QNMs are in Table II) with ω Re increasing with kn , with k as some constant of proportionality.",
"pages": [
10,
11,
12,
13
]
},
{
"title": "V. CONCLUSION",
"content": "In this paper we have studied the effect of the number of dimension n on the black hole's photonsphere, shadow, deflection angle both in the weak and strong regimes, and quasinormal modes. First, we have seen how the photonsphere r ps decreases with increasing n . Results indicate that the rate at which r ps changes relative to n becomes smaller as n increases. As the shadow radius R sh is related to r ps , we have seen the same behavior. For instance, increasing n results in decreased R sh . Interestingly, while r ps /M = 2 when n = 5 , we have seen that R sh is smaller than the photonsphere. Such a peculiar case indicates the existence of extra dimensions if detected experimentally. Speculatively, if fractal dimensions are permitted in nature, the uncertainties in EHT's R sh measurement might indicate an upper bound in n as 3 . 84 and 3 . 94 for M87* and Sgr. A*, respectively. Meanwhile, the lower bound for n are 4 . 51 and 4 . 16 for M87* and Sgr. A*, respectively. This result implies that n greater than 5 and beyond are forbidden as far as the EHT data is concerned. Nevertheless, the theoretical study of higher dimensions must not be underestimated. We then analyzed how the deflection angle changes with n by deriving the most general formula for the weak deflection angle α ; not only accommodating the finite distance of the source and observers but also valid for both time-like and null particles. As higher dimensionality is included, the formula necessitates the existence of a hypergeometric function, making the calculation of α more precise. If α is due to the time-like particles, we have seen how it deviates from the null result, depending on its speed v . Finally, the results imply that probing the existence of higher dimensions using weak deflection angle (as well as the lens images that it can cause) requires ultrasensitivity as increased n results in small α . We also examined the strong deflection angle that occurs near the r ps . Dimensionality and gravitational lensing are correlated, as can be inferred from the plot in Fig. 4 that the strong deflection angle around a higher dimensional black hole decreases as the dimension, n , increases. The degrees of freedom rise in proportion to dimensionality, which impacts the gravitational interaction surrounding the black hole. Such a decrease in the strong deflection angle raises the possibility that the gravitational lensing phenomenon is dampened due to higher dimensions leading to more difficulty of measurement. Investigating the precise mechanisms underlying this relationship may yield important new information about how gravity behaves in higher-dimensional regions and how this may affect astrophysical occurrences. Since the strong deflection exhibits large values as it approaches the photon sphere, higher dimension can possibly detected when investigated at these regions. Given the increasing difficulty in measuring gravitational phenomenon as the number of dimensions increases, QNMs in higher dimensions offer another means to probe the photonsphere and related properties about the effect of SchwarzschildTangherlini black holes on interacting fields. This is made possible by the known relation between QNMs and the dynamics of the photonsphere in the eikonal limit, which are described by the orbital frequency and Lyapunov exponent. With the decrease of the black hole photon-sphere, shadow and defection angle, there is an increase in both the physical oscillation frequency and damping time-scale of QNMs (such as is seen in Table II). Knowing the correspondence between QNMs and the parameters of the photonsphere in the eikonal limit, QNMs could offer a useful indirect probe of the optical properties of the black hole which may not be accessible through direct measurement. As a final concluding thought, the possibility of extending this study to incorporate black hole charge Q , or, in a more general sense, the spin parameter a , would allow for greater comparison with physically observable phenomena.",
"pages": [
14
]
},
{
"title": "ACKNOWLEDGMENTS",
"content": "R. P. would like to acknowledge networking support of the COST Action CA18108 - Quantum gravity phenomenology in the multi-messenger approach (QG-MM), COST Action CA21106 - COSMIC WISPers in the Dark Universe: Theory, astrophysics and experiments (CosmicWISPers), the COST Action CA22113 - Fundamental challenges in theoretical physics (THEORY-CHALLENGES), and the COST Action CA21136 - Addressing observational tensions in cosmology with systematics and fundamental physics (CosmoVerse). A. S. C. is supported in part by the National Research Foundation (NRF) of South Africa. A. N. is supported by an SA-CERN Excellence Bursary through iThemba LABS.",
"pages": [
14
]
}
]
2024arXiv240609540D
https://arxiv.org/pdf/2406.09540.pdf
<document>
<section_header_level_1><location><page_1><loc_18><loc_83><loc_82><loc_88></location>Metric-affine cosmological models and the inverse problem of the calculus of variations. Part II: Variational bootstrapping of the Λ CDM model</section_header_level_1>
<text><location><page_1><loc_43><loc_80><loc_57><loc_82></location>Ludovic Ducobu ∗</text>
<text><location><page_1><loc_26><loc_77><loc_74><loc_80></location>Department of Mathematics and Computer Science, Transilvania University of Brasov, Brasov, Romania and Nuclear and Subnuclear Physics, University of Mons, Mons, Belgium</text>
<section_header_level_1><location><page_1><loc_44><loc_74><loc_56><loc_75></location>Nicoleta Voicu †</section_header_level_1>
<text><location><page_1><loc_30><loc_70><loc_70><loc_74></location>Department of Mathematics and Computer Science, Transilvania University of Brasov, Brasov, Romania and Lepage Research Institute, Presov, Slovakia</text>
<text><location><page_1><loc_17><loc_62><loc_84><loc_69></location>The method of variational bootstrapping, based on canonical variational completion, allows one to construct a Lagrangian for a physical theory depending on two sets of field variables, starting from a guess of the field equations for only one such set. This setup is particularly appealing for the construction of modified theories of gravity since one can take lessons from GR for the 'educated guess' of the metric field equations; the field equations for the other fields are then fixed by the obtained Lagrangian (up to terms completely independent from the metric tensor).</text>
<text><location><page_1><loc_17><loc_57><loc_84><loc_61></location>In the present paper, we explore the applications of variational bootstrapping in the context of metric-affine theories of gravity. We find those metric-affine models which are, in a variational sense, closest to the Λ CDM model of cosmology. The method then allows to find 'corrected' metric equations and to 'bootstrap' the connection field equations.</text>
<text><location><page_1><loc_17><loc_53><loc_84><loc_56></location>The Lagrangians obtained via this method, though imposing some restricting criteria, encompass a wide variety of metric-affine models. In particular, they allow for quadratic metric-affine theories only if these avoid non-linear terms in the curvature tensor.</text>
<section_header_level_1><location><page_1><loc_44><loc_47><loc_56><loc_48></location>I. Introduction</section_header_level_1>
<text><location><page_1><loc_15><loc_38><loc_85><loc_45></location>Despite its successes, it is notorious that General Relativity (GR) struggles at the smallest scales - due to an incompatibility with quantum theory - and at the largest scales - where it fails to correctly reproduce the cosmological behavior of the universe. For the latter problem (which is the one we focus on, in the present paper), a simplest way of correcting the GR model is to amend Einstein's equation</text>
<formula><location><page_1><loc_44><loc_35><loc_85><loc_38></location>G µν = 8 π G c 4 T µν (1)</formula>
<text><location><page_1><loc_15><loc_26><loc_85><loc_34></location>by adding entities known as dark matter and dark energy. This paradigm is known as the Λ CDM model of cosmology [1-5] and is, to this date, the best model available to fit cosmological observations. Nevertheless, the Λ CDM model is challenged, on the one hand, by the so-called H 0 -σ 8 tension [6] and on the other hand, by the fact that the very physical origin of dark energy and dark matter physical origin still eludes our understanding. These puzzles legitimize the quest for a more general (classical) gravitational theory.</text>
<text><location><page_1><loc_15><loc_24><loc_85><loc_26></location>Such generalizations of GR concern either the geometric structure of spacetime (i.e., the kinematical level), or the field equations (in other words, the dynamical level), or both.</text>
<text><location><page_2><loc_15><loc_82><loc_85><loc_88></location>At the kinematical level, one of the simplest extension is metric-affine geometry, which allows for more freedom by relaxing the GR hypotheses on the spacetime connection (metricity, absence of torsion), [7-9]. In metric-affine gravity models, one can thus encode the dark sector into the 'enhanced' geometry of spacetime 1 .</text>
<text><location><page_2><loc_15><loc_61><loc_85><loc_81></location>At the dynamical level, one usually discusses modified theories of gravity by first postulating a Lagrangian, from which the field equations are derived. Even though this procedure is automatically justified as long as one expects a fundamental physical theory to admit a Lagrangian, ultimately what one typically wants to control are the properties of the field equations , rather than the ones of the Lagrangian. For this reason, one of us proposed in [11], together with D. Krupka, a systematic method, called canonical variational completion , allowing to go the other way around: starting with an approximate form (an 'educated guess') of the field equations, one can canonically construct a Lagrangian and subsequently, the 'corrected' field equations. The obtained field equations are the closest ones to our original guess, in the sense that the correction terms to be added specifically measure the 'obstruction from variationality' of this guess. The method of canonical variational completion was then upgraded in the first part of this work [12], to accommodate theories that involve more than one set of dynamical variables, e.g., a metric and a connection. We showed that from an 'educated guess' of the field equations with respect to one variable (e.g., the metric) only, one can still canonically 'bootstrap' a Lagrangian, up to boundary terms and to terms that are completely independent of the respective variable.</text>
<text><location><page_2><loc_15><loc_47><loc_85><loc_59></location>In the present paper, we apply the method proposed in [12] to select those metric-affine Lagrangians producing the 'closest to Λ CDM' metric equations. Our 'educated guess' (or 'seed') for the metric equations is a system that formally resembles the Einstein equations with a cosmological constant (encoding dark energy, w = -1 ) and a term describing dark matter (with w = 0 ). To obtain it, we first consider the form of general tensor fields fitting the dark energy or dark matter equation of state (EOS) under the assumption of cosmological symmetry. Next, we step back from cosmological symmetry and consider this expression as valid in full generality; this correspond to the idea of finding the most minimal extension of the Λ CDM equations. We then bootstrap this guess under the following hypotheses:</text>
<unordered_list>
<list_item><location><page_2><loc_18><loc_41><loc_85><loc_47></location>· The dark sector has a geometric (more precisely, metric-affine) origin. That is, the cosmological 'constant' is actually a scalar function depending on the metric and on the connection components; similarly, the dark matter term can be completely expressed in terms of the metric and of the connection.</list_item>
<list_item><location><page_2><loc_18><loc_37><loc_85><loc_40></location>· Minimal coupling: In the seed equations, the dark sector terms are of order at most one. This ensures that the variationally completed equations are of order at most two, both in the metric and the connection.</list_item>
<list_item><location><page_2><loc_18><loc_33><loc_85><loc_36></location>· The metric tensor is used to build a volume form on spacetime and for raising/lowering indices of tensors.</list_item>
<list_item><location><page_2><loc_18><loc_30><loc_85><loc_32></location>· Each of the dark sector terms can be expressed as a finite sum of homogeneous expressions (not necessarily of positive degrees) in the metric.</list_item>
</unordered_list>
<text><location><page_2><loc_15><loc_24><loc_85><loc_29></location>The second and third hypotheses above ensure on the one hand, that the obtained Lagrangians are generally covariant and, on the other hand, that all their non-boundary terms can be found from our 'educated guess' of the metric equations only. This way, no input on the connection equations is necessary; these can be automatically 'bootstrapped' from the obtained Lagrangian.</text>
<text><location><page_3><loc_15><loc_82><loc_85><loc_88></location>We find that a number of the existing metric-affine models fall into this class, e.g. , those whose Lagrangian is the Ricci scalar of the independent connection and a subclass of quadratic metric-affine theories avoiding non-linear terms involving the curvature tensor of the independent connection.</text>
<text><location><page_3><loc_15><loc_63><loc_85><loc_81></location>The paper is structured as follows : In sections II and III, we fix our notations for the metricaffine geometry and briefly review some properties of the Λ CDM model that are necessary for our discussion. In section IV, we consider metric-affine theories of gravity based on modifications implemented directly at the level of GR field equations with aim of obtaining an 'educated guess' for the (metric) field equation in a metric-affine extension of GR. Section V contains our main results. In section V A, we remind the basics of the technique of variational bootstrapping presented in [12]. In section V B, we explicit our hypothesis in the construction of our model before performing, in section V C, the variational bootstrapping of the equations discussed in section IV - hence obtaining a Lagrangian for the full theory. Then, in section V D, we study the field equations associated to the Lagrangian obtained in section V C. In section V E, we then comment on the obtained equations and on their link with previously formulated metric-affine models of gravity. Finally, in section VI, we present our conclusions and discuss further directions of research.</text>
<section_header_level_1><location><page_3><loc_42><loc_59><loc_58><loc_60></location>II. Geometric setup</section_header_level_1>
<text><location><page_3><loc_15><loc_47><loc_85><loc_57></location>In the following, we model spacetime by means of a triple ( M, Γ , g ) , where M is an arbitrary 4-dimensional differentiable manifold, Γ is a generic linear connection on M and g a metric with Lorentzian signature. Local coordinates on M will be denoted by x µ and the local basis vectors on TM , by ∂ µ ; when writing connection coefficients, the second lower index refers to the 'derivative index' meaning that, for example, given a vector field V = V µ ∂ µ one has that ∇ µ V ν := ( ∇ ∂ µ V ) ν = ∂ µ V ν +Γ ν ρµ V ρ . We will denote by R the curvature tensor of Γ :</text>
<formula><location><page_3><loc_33><loc_44><loc_85><loc_46></location>R ( V, W ) Z := ∇ V ∇ W Z -∇ W ∇ V Z -∇ [ V,W ] Z, (2)</formula>
<text><location><page_3><loc_15><loc_41><loc_58><loc_43></location>where ∇ is the covariant derivative associated to Γ and by:</text>
<formula><location><page_3><loc_33><loc_39><loc_85><loc_41></location>R ρ σµν = ∂ µ Γ ρ σν -∂ ν Γ ρ σµ +Γ ρ αµ Γ α σν -Γ ρ βν Γ β σµ , (3)</formula>
<text><location><page_3><loc_15><loc_37><loc_75><loc_38></location>its components in a coordinate basis. Similarly, T will designate the torsion of Γ :</text>
<formula><location><page_3><loc_36><loc_33><loc_85><loc_35></location>T ( V, W ) := ∇ V W -∇ W V -[ V, W ] , (4)</formula>
<text><location><page_3><loc_15><loc_31><loc_44><loc_33></location>with components in a coordinate basis:</text>
<formula><location><page_3><loc_39><loc_28><loc_85><loc_30></location>T ρ µν = Γ ρ νµ -Γ ρ µν = -2 Γ ρ [ µν ] , (5)</formula>
<text><location><page_3><loc_15><loc_26><loc_48><loc_27></location>and by Q the non-metricity tensor defined as</text>
<formula><location><page_3><loc_47><loc_22><loc_85><loc_24></location>Q := ∇ g , (6)</formula>
<text><location><page_3><loc_15><loc_20><loc_32><loc_22></location>with local components:</text>
<formula><location><page_3><loc_33><loc_17><loc_85><loc_19></location>Q µνρ = ∇ ρ g µν := ∂ ρ g µν -Γ α µρ g αν -Γ β νρ g µβ . (7)</formula>
<text><location><page_4><loc_15><loc_85><loc_85><loc_88></location>Our metric-affine setup can equivalently (and more conveniently) be defined as the triple ( M, L , g ) , where the distortion of Γ :</text>
<formula><location><page_4><loc_46><loc_82><loc_85><loc_84></location>L = Γ -· Γ (8)</formula>
<text><location><page_4><loc_15><loc_79><loc_85><loc_82></location>is a tensor field of type (1 , 2) . The symbol · designates the Levi-Civita connection associated to g and all its related quantities; for instance:</text>
<formula><location><page_4><loc_29><loc_76><loc_85><loc_78></location>· Γ ρ µν := 1 2 g ρα ( ∂ ν g αµ + ∂ µ g να -∂ α g µν ) , Γ ρ µν = · Γ ρ µν + L ρ µν . (9)</formula>
<text><location><page_4><loc_15><loc_71><loc_85><loc_75></location>In the following, for conciseness, we will express our relations in terms of the distortion tensor L . If needed, all the relations can be translated in terms of the torsion T (respectively, the contortion K ) and non-metricity Q (respectively, disformation D ), using the relations below:</text>
<formula><location><page_4><loc_42><loc_69><loc_85><loc_70></location>L ρ µν := K ρ µν + D ρ µν , (10)</formula>
<text><location><page_4><loc_15><loc_67><loc_20><loc_68></location>where:</text>
<formula><location><page_4><loc_35><loc_64><loc_85><loc_66></location>D ρ µν := -1 2 g ρα ( Q αµν + Q ναµ -Q µνα ) , (11)</formula>
<formula><location><page_4><loc_31><loc_58><loc_85><loc_62></location>K ρ µν := -1 2 ( T ρ µν -g να g ρβ T α βµ + g µα g ρβ T α νβ ) . (12)</formula>
<text><location><page_4><loc_15><loc_58><loc_64><loc_59></location>For the sake of completeness, we also present the inverse relations:</text>
<formula><location><page_4><loc_32><loc_53><loc_85><loc_57></location>K ρ µν = L ρ [ µν ] + g ρβ ( g µα L α [ νβ ] + g να L α [ µβ ] ) , (13)</formula>
<formula><location><page_4><loc_32><loc_49><loc_85><loc_53></location>D ρ µν = L ρ ( µν ) -g ρβ ( g µα L α [ νβ ] + g να L α [ µβ ] ) , (14)</formula>
<formula><location><page_4><loc_39><loc_47><loc_85><loc_49></location>T ρ µν = -2 L ρ [ µν ] = -2 K ρ [ µν ] , (15)</formula>
<formula><location><page_4><loc_34><loc_44><loc_85><loc_46></location>Q ρµν = -2 g ( ρ | α L α | µ ) ν = -2 g ( ρ | α D α | µ ) ν , (16)</formula>
<text><location><page_4><loc_15><loc_40><loc_85><loc_44></location>where, as usual, square brackets (resp. round brackets) placed around a group of indices denote antisymmetrization (resp. symmetrization) over those indices, with exception of indices placed between vertical bars.</text>
<section_header_level_1><location><page_4><loc_35><loc_36><loc_65><loc_37></location>III. Brief review of the Λ CDM model</section_header_level_1>
<text><location><page_4><loc_15><loc_29><loc_85><loc_34></location>This section briefly presents the restrictions imposed by cosmological symmetry upon tensors on the spacetime manifold and the geometric setup of the Λ CDM model; for more detail see [4, 5]. Even though most of the material of this section is not new, this allow us to clearly fix our notations and to raise an important - yet rarely emphasized - problem.</text>
<section_header_level_1><location><page_4><loc_33><loc_25><loc_67><loc_26></location>A. Cosmologically symmetric tensor fields</section_header_level_1>
<text><location><page_4><loc_15><loc_18><loc_85><loc_23></location>Cosmological symmetry, understood as invariance under the six Killing vector fields defining it, imposes quite strong constraints on tensor fields of any rank, which we briefly present below; we direct the curious reader to Appendix A for more details and computations. In the following, we denote by ( x µ ) = ( t, r, θ, ϕ ) , a set of local spherical coordinates over M adapted to the symmetry.</text>
<text><location><page_5><loc_15><loc_81><loc_85><loc_85></location>A given vector field V = V µ ∂ µ on M is cosmologically symmetric if and only if its components in the coordinate basis { ∂ µ } satisfy</text>
<formula><location><page_5><loc_42><loc_77><loc_85><loc_81></location>( V µ ) = ( V t ( t ) , 0 , 0 , 0 ) . (17)</formula>
<formula><location><page_5><loc_43><loc_76><loc_58><loc_77></location>2. (0 , 2) -tensor fields</formula>
<text><location><page_5><loc_15><loc_70><loc_85><loc_74></location>Similarly, a given (0 , 2) -tensor field T = T µν d x µ ⊗ d x ν will be cosmologically symmetric if and only if its components in the coordinate basis { ∂ µ } satisfy</text>
<formula><location><page_5><loc_30><loc_66><loc_85><loc_70></location>( T µν ) = diag ( T tt ( t ) , T r ( t ) 1 -kr 2 , r 2 T r ( t ) , r 2 sin( θ ) 2 T r ( t ) ) , (18)</formula>
<text><location><page_5><loc_15><loc_58><loc_85><loc_66></location>where k ∈ { -1 , 0 , 1 } is the sign of the curvature of the spatial slices. It is worth noting that, in deducing (18), one does not need to assume that T is symmetric; its antisymmetric part vanishes as a consequences of cosmological symmetry. From this relation, we naturally recover that the components of the most general cosmologically symmetric metric with Lorentz signature g = g µν d x µ ⊗ d x ν can be written as</text>
<formula><location><page_5><loc_29><loc_54><loc_85><loc_58></location>( g µν ) = diag ( -N ( t ) 2 , a ( t ) 2 1 -kr 2 , r 2 a ( t ) 2 , r 2 sin( θ ) 2 a ( t ) 2 ) . (19)</formula>
<formula><location><page_5><loc_32><loc_51><loc_68><loc_52></location>3. Differential 1-forms and (1 , 1) -type tensor fields</formula>
<text><location><page_5><loc_15><loc_47><loc_85><loc_49></location>Using the musical isomorphisms offered by the metric, i.e. if we use the metric to raise or lower indices, we also get that V µ := g µν V ν and T µ ν := g µα T αν must be such that</text>
<formula><location><page_5><loc_42><loc_44><loc_85><loc_46></location>( V µ ) = ( θ V ( t ) , 0 , 0 , 0) , (20)</formula>
<text><location><page_5><loc_15><loc_41><loc_48><loc_43></location>where we defined θ V ( t ) := -N ( t ) 2 V t ( t ) , and</text>
<formula><location><page_5><loc_34><loc_37><loc_85><loc_41></location>( T µ ν ) = diag ( -ρ T ( t ) , P T ( t ) , P T ( t ) , P T ( t )) , (21)</formula>
<text><location><page_5><loc_15><loc_37><loc_62><loc_38></location>where we defined ρ T ( t ) := T tt ( t ) /N ( t ) 2 and P T ( t ) := T r ( t ) /a ( t ) 2 .</text>
<text><location><page_5><loc_15><loc_31><loc_85><loc_37></location>Since the metric provides an isomorphism between TM and T ∗ M , we can conclude that (20) gives us the most general expression for a differential 1-form and (21) the most general expression for a (1 , 1) -tensor field respecting cosmological symmetry; these can also be obtained by direct computation.</text>
<section_header_level_1><location><page_5><loc_36><loc_28><loc_64><loc_29></location>B. The Λ CDM model in a nutshell</section_header_level_1>
<text><location><page_5><loc_15><loc_20><loc_85><loc_26></location>From the theoretical perspective, the Λ CDM model of cosmology assumes that the universe behaves, at the kinematical level, according to the tools of pseudo-Riemaniann geometry ( i.e. a metric-affine setup where T ≡ 0 & Q ≡ 0 ) and fixes the dynamics of the universe (the evolution of the scale factor a ( t ) ) by means of the following modified Einstein equation [4, 5]</text>
<formula><location><page_5><loc_38><loc_18><loc_85><loc_19></location>· G µν +Λ g µν = κT (m) µν + κT (DM) µν , (22)</formula>
<text><location><page_6><loc_15><loc_82><loc_85><loc_88></location>where · G = · G µν d x µ ⊗ d x ν is the Einstein tensor, Λ is the cosmological constant, κ = 8 π G /c 4 , T (m) = T (m) µν d x µ ⊗ d x ν is the energy-momentum tensor of ordinary matter (including barionic matter and electromagnetic radiation) and T (DM) = T (DM) µν d x µ ⊗ d x ν is the energy-momentum tensor of dark matter.</text>
<text><location><page_6><loc_17><loc_80><loc_43><loc_82></location>Matter is modeled as a perfect fluid</text>
<formula><location><page_6><loc_39><loc_78><loc_85><loc_79></location>T (m) µν = Pg µν +( ρ + P ) u µ u ν , (23)</formula>
<text><location><page_6><loc_15><loc_72><loc_85><loc_76></location>where g µν is given by (19), ρ is the energy density of the fluid, P its pressure and u = u µ ∂ µ is the average 4-velocity of the fluid which is taken to be normalized ( g ( u, u ) = -1 ) and such that the fluid is comoving with inertial observers; i.e. one has</text>
<formula><location><page_6><loc_41><loc_70><loc_85><loc_71></location>( u µ ) = (1 /N ( t ) , 0 , 0 , 0) . (24)</formula>
<text><location><page_6><loc_15><loc_66><loc_85><loc_68></location>To complete this description, properties of matter are encoded by means of an equation of state (EOS) of the form</text>
<formula><location><page_6><loc_47><loc_63><loc_85><loc_64></location>P = wρ, (25)</formula>
<text><location><page_6><loc_15><loc_57><loc_85><loc_62></location>where w ∈ R is a constant. For ordinary (dust) matter one has w dust = 0 ( i.e. vanishing pressure) while for electromagnetic radiation (photons) w EM = 1 / 3 to ensure T (EM) µ µ = 0 . If we raise the first index of T (m) , we then get</text>
<formula><location><page_6><loc_35><loc_52><loc_85><loc_56></location>( T (m) µ ν ) = diag ( -ρ ( t ) , P ( t ) , P ( t ) , P ( t )) (26)</formula>
<formula><location><page_6><loc_43><loc_51><loc_85><loc_53></location>= ρ ( t ) diag ( -1 , w, w, w ) . (27)</formula>
<text><location><page_6><loc_15><loc_47><loc_85><loc_51></location>In the Λ CDM model, one considers the dark matter sector to be composed of 'cold dark matter'. This cold dark matter represents, in this model, some exotic non-relativistic particles described by a perfect fluid energy-momentum tensor (23) for which w DM = 0 . So</text>
<formula><location><page_6><loc_36><loc_42><loc_85><loc_45></location>( T (DM) µ ν ) = -ρ DM ( t ) diag (1 , 0 , 0 , 0) . (28)</formula>
<text><location><page_6><loc_15><loc_38><loc_85><loc_42></location>The term Λ g models dark energy. Following the usual reasoning, the term κ T (DE) := -Λ g can be interpreted as a perfect fluid energy-momentum tensor for which ρ DE = Λ /κ and w DE = -1 , leading to</text>
<formula><location><page_6><loc_37><loc_33><loc_85><loc_36></location>( T (DE) µ ν ) = -ρ DE diag (1 , 1 , 1 , 1) . (29)</formula>
<section_header_level_1><location><page_6><loc_36><loc_30><loc_64><loc_31></location>C. The cosmological smokescreen</section_header_level_1>
<text><location><page_6><loc_17><loc_27><loc_79><loc_28></location>From the computations of section III A, one can make the following observations [4]:</text>
<unordered_list>
<list_item><location><page_6><loc_17><loc_22><loc_85><loc_25></location>1. Comparing (24) and (17) we see that, when required to obey cosmological symmetry, any timelike vector field will look like the 4 -velocity of an isotropic observer (and then also like the average 4 -velocity of a perfect fluid).</list_item>
<list_item><location><page_6><loc_17><loc_18><loc_85><loc_20></location>2. Comparing (26) and (21) reveals that any type (1 , 1) -tensor field which obeys cosmological symmetry must look like a perfect fluid energy-momentum tensor.</list_item>
</unordered_list>
<text><location><page_7><loc_15><loc_55><loc_19><loc_56></location>where</text>
<formula><location><page_7><loc_31><loc_52><loc_85><loc_54></location>Θ (D) = Θ (D) ( g , ∂ g , · · · , ∂ · · · ∂ g ; L , ∂ L , · · · , ∂ · · · ∂ L ) (32)</formula>
<text><location><page_7><loc_15><loc_49><loc_85><loc_51></location>is a symmetric tensor field of type (0 , 2) (or (2 , 0) ) depending on the metric g , the distortion tensor components and a finite number of derivatives thereof.</text>
<text><location><page_7><loc_15><loc_42><loc_85><loc_48></location>In the particular case of cosmological symmetry, we want Θ (D) to geometrically encode the dark sector contributions, in other words, to be the geometric counterpart of T (D) . To stay as close as possible to the Λ CDM model, we may want to regard dark matter and dark energy as distinct effects. 2 This corresponds to the idea of writting (31) with</text>
<formula><location><page_7><loc_41><loc_40><loc_85><loc_41></location>Θ (D) := Θ (DM) + Θ (DE) . (33)</formula>
<text><location><page_7><loc_15><loc_33><loc_85><loc_39></location>In this case, the success of the Λ CDM model should be seen as constraining the form of Θ (DM) and Θ (DE) . In other words, one should make sure that, under cosmological symmetry, the form of Θ (DM) and Θ (DE) are compatible with the dark-matter and dark-energy EOS from the Λ CDM model</text>
<formula><location><page_7><loc_41><loc_30><loc_85><loc_32></location>w (DM) = 0 , w (DE) = -1 . (34)</formula>
<text><location><page_7><loc_15><loc_29><loc_46><loc_30></location>The first relation in (34) then implies that</text>
<formula><location><page_7><loc_45><loc_26><loc_85><loc_28></location>Θ (DM) µν · = V µ V ν , (35)</formula>
<text><location><page_7><loc_15><loc_84><loc_85><loc_88></location>These remarks, while supporting the parametrization of the energy-momentum tensor of ordinary matter via (23) subject to an EOS (25) with w ∈ { 0 , 1 / 3 } , can cast serious doubts on the interpretation of (28) and (29). If we further group together the 'dark contributions' as</text>
<formula><location><page_7><loc_32><loc_81><loc_85><loc_83></location>T (D) µν := T (DE) µν + T (DM) µν = -Λ /κ g µν + ρ DM u µ u ν (30)</formula>
<text><location><page_7><loc_15><loc_77><loc_85><loc_80></location>and raise the first index, we recover the general expression (21) with associated 'density' ρ D := ρ DM + ρ DE = ρ DM +Λ /κ and 'pressure' P D := P DM + P DE = -Λ /κ .</text>
<text><location><page_7><loc_15><loc_72><loc_85><loc_78></location>A conservative interpretation of the Λ CDM model then seems to only indicate that there is a missing piece in Einstein equation · G = κ T (m) , but does not allow to draw any stringent conclusion regarding the origin of the extra piece T (D) . It thus looks reasonable to consider the dark sector as a whole, as in (30), freed from the classical dark-matter+dark-energy interpretation.</text>
<section_header_level_1><location><page_7><loc_35><loc_68><loc_65><loc_69></location>IV. Seed for modified field equations</section_header_level_1>
<text><location><page_7><loc_15><loc_62><loc_85><loc_66></location>We will now discuss the construction of an 'educated guess' for a minimal metric-affine extension of the dynamics of the Λ CDM model. Therefore, for the rest of this paper, we consider the geometric setup as given by a triple ( M, L , g ) as presented in Section II.</text>
<text><location><page_7><loc_15><loc_60><loc_85><loc_62></location>In any metric-affine theory of gravity employing the metric and distortion tensor components as dynamical variables, the evolution equations for the metric can be cast in the form</text>
<formula><location><page_7><loc_43><loc_57><loc_57><loc_59></location>· G T (m) Θ (D)</formula>
<formula><location><page_7><loc_45><loc_57><loc_85><loc_58></location>= κ + , (31)</formula>
<text><location><page_8><loc_15><loc_84><loc_85><loc_88></location>where we use the symbol · = do denote an equality holding in a cosmologicaly symmetric situation. With the standard Λ CDM-interpretation, one would have V µ := √ ρ (DM) u ν g µν . Similarly, the second relation in (34) implies</text>
<formula><location><page_8><loc_44><loc_80><loc_85><loc_82></location>Θ (DE) µν · = -Λ g µν , (36)</formula>
<text><location><page_8><loc_15><loc_77><loc_85><loc_79></location>where Λ is a priori a scalar function (generally, non-constant) and the minus sign is purely conventional.</text>
<text><location><page_8><loc_15><loc_71><loc_85><loc_77></location>As already stated, our aim in this paper is to consider a minimal metric-affine extension of the Λ CDM model. This corresponds to the idea that the degrees of freedom describing dark matter and dark energy in a cosmologically symmetric situation are the only ones necessary to desribe the dynamics of these entities in all situations.</text>
<text><location><page_8><loc_15><loc_67><loc_85><loc_71></location>In other words, stepping now aside from the cosmological symmetry context, under this 'minimality' hypothesis, no extra quantities or terms appear, hence we can simply replace · = by a strict equality. This gives</text>
<formula><location><page_8><loc_44><loc_64><loc_85><loc_66></location>Θ (DM) := V ⊗ V , (37)</formula>
<text><location><page_8><loc_15><loc_60><loc_85><loc_63></location>with V = g µν V µ d x ν for a given vector field V = V µ ∂ µ obtained from the quantities defining the metric-affine geometry; that is V = V ( g , ∂ g , · · · , ∂ · · · ∂ g ; L , ∂ L , · · · , ∂ · · · ∂ L ) , and</text>
<formula><location><page_8><loc_45><loc_57><loc_85><loc_59></location>Θ (DE) := -Λ g , (38)</formula>
<text><location><page_8><loc_15><loc_55><loc_68><loc_57></location>where Λ = Λ( g , ∂ g , · · · , ∂ · · · ∂ g ; L , ∂ L , · · · , ∂ · · · ∂ L ) is a scalar function.</text>
<text><location><page_8><loc_15><loc_51><loc_85><loc_54></location>To summarize: staying as conservative as possible with respect to the dynamics of the Λ CDM model, one can consider as our 'educated guess' an equation of the form</text>
<formula><location><page_8><loc_40><loc_48><loc_85><loc_50></location>· G = κ T (m) -Λ g + V ⊗ V , (39)</formula>
<text><location><page_8><loc_15><loc_45><loc_85><loc_47></location>where Λ is a scalar function and V (accordingly, V ) is a 1-form (accordingly, a vector field) depending on g , L and their derivatives up to a finite order r .</text>
<text><location><page_8><loc_15><loc_42><loc_85><loc_44></location>Yet, obviously, if we were to stop here, the prescription (39) would be incomplete, for at least two reasons:</text>
<unordered_list>
<list_item><location><page_8><loc_17><loc_35><loc_85><loc_41></location>1. In a metric-affine theory of gravity, one has two sets of field equations, one for the metric and one for the connection (or the distortion). The above prescription provides an 'educated guess' for the metric equations only, it makes no claim on the expected form of the connection equation.</list_item>
<list_item><location><page_8><loc_17><loc_31><loc_85><loc_34></location>2. The above postulated equations are not necessarily variational - i.e., they generally do not arise from an action principle.</list_item>
</unordered_list>
<text><location><page_8><loc_15><loc_28><loc_85><loc_30></location>A solution to both the above problems is given by an algorithm introduced in [12], which we present in the next section.</text>
<section_header_level_1><location><page_8><loc_34><loc_24><loc_66><loc_25></location>V. Variational Bootstrapping of Λ CDM</section_header_level_1>
<text><location><page_8><loc_15><loc_18><loc_85><loc_22></location>The variational bootstrapping method [12] allows one to determine a Lagrangian whose EulerLagrange equations are the closest to an 'educated guess' for a subset of the field equations. After briefly reviewing the algorithm, we apply it to (39).</text>
<section_header_level_1><location><page_9><loc_32><loc_87><loc_68><loc_88></location>A. User's guide on Variational Bootstrapping</section_header_level_1>
<text><location><page_9><loc_15><loc_81><loc_85><loc_85></location>Let us start by presenting a minimalist guide for the method of variational bootstrapping, focussing on computational aspects. For complete and mathematically rigorous details, we refer the reader to [12].</text>
<text><location><page_9><loc_15><loc_77><loc_85><loc_81></location>Assume we want to build a theory involving two distinct sets of dynamical variables, y A and z I both depending on the independent variables x µ . Generaly speaking, the dynamics of the system must be described by a system of PDEs of an a priori given order</text>
<formula><location><page_9><loc_23><loc_70><loc_85><loc_76></location>{ Y A ( x µ ; y B , ∂ α y B , · · · , ∂ α 1 · · · ∂ α r y B ; z J , ∂ β z J , · · · , ∂ β 1 · · · ∂ β s z J ) = 0 Z I ( x µ ; y B , ∂ α y B , · · · , ∂ α 1 · · · ∂ α r y B ; z J , ∂ β z J , · · · , ∂ β 1 · · · ∂ β s z J ) = 0 , (40)</formula>
<text><location><page_9><loc_15><loc_67><loc_85><loc_70></location>Assume now that we have some insight (an 'educated guess') regarding the form of the y A -equation only and that this insight is of the form</text>
<text><location><page_9><loc_15><loc_69><loc_78><loc_71></location>with as many Y A -equations as y A -variables and as many Z I -equations as z I -variables.</text>
<formula><location><page_9><loc_25><loc_62><loc_85><loc_67></location>/star Y A ( x µ ; y B , ∂ α y B , · · · , ∂ α 1 · · · ∂ α r y B ; z J , ∂ β z J , · · · , ∂ β 1 · · · ∂ β s z J ) = 0 , (41)</formula>
<text><location><page_9><loc_15><loc_62><loc_44><loc_64></location>with as many equations as y A -variables.</text>
<text><location><page_9><loc_15><loc_55><loc_85><loc_62></location>Variational bootstrapping then allows one to find a canonical correction term to be added to (41) in such a way that the corrected y A -equations are derived from a Lagrangian; moreover, this Lagrangian is uniquely determined up to boundary terms and to terms that do not involve y A . This will allow us, under certain circumstances (to be detailed below) to also 'bootstrap' the z I -equation. The Lagrangian density of the said Lagrangian is given 3 by</text>
<formula><location><page_9><loc_18><loc_50><loc_85><loc_55></location>L y = y A ∫ 1 0 /star Y A ( x µ ; uy B , u∂ α y B , · · · , u∂ α 1 · · · ∂ α r y B ; z J , ∂ β z J , · · · , ∂ β 1 · · · ∂ β s z J ) d u, (42)</formula>
<text><location><page_9><loc_15><loc_48><loc_85><loc_51></location>where the y B variables and all their derivatives are scaled by the same u factor in the integrand and a sum over index A is understood.</text>
<text><location><page_9><loc_15><loc_40><loc_85><loc_48></location>In cases where our guessed equations (41) can be obtained from a variational principle, any Lagrangian density producing (41) as y A -part of the Euler-Lagrange equations must be of the form (42) up to Lagrangian densities completely independent of the y A -variables ( i.e. which do not contribute to the y A -equations) and total divergences (which do not contribute to the field equations at all), see [12]. The form of the y A -independent densities is not constrained by the above procedure and should thus be found by different means 4 .</text>
<text><location><page_9><loc_15><loc_36><loc_85><loc_40></location>But what if the equation (41) we start with is not variational ( i.e. what if it does not appear as the y A -equation of any Lagrangian) ? In that case, the idea behind variational bootstrapping is to still compute the Lagrangian density (42). This does two things for us:</text>
<unordered_list>
<list_item><location><page_9><loc_17><loc_34><loc_71><loc_35></location>1. It provides a Lagrangian density that will 'correct the y A -equations' 5 ,</list_item>
<list_item><location><page_9><loc_17><loc_32><loc_81><loc_33></location>2. It constrains the acceptable form (up to y A -independent terms) for the z I -equations.</list_item>
</unordered_list>
<text><location><page_10><loc_15><loc_85><loc_85><loc_88></location>In that case, the y A -part of the Euler-Lagrange equations associated with (42) will differ from (41) but (42) will provide a canonical (and 'minimal') Lagrangian based on the guess (41).</text>
<text><location><page_10><loc_15><loc_75><loc_85><loc_83></location>In the case of modified gravity theories based on GR, the field variables are usually the metric g and, generally speaking, another set of tensorial field variables, say ψ A . Our knowledges on metric-based approaches to gravity provides us with an 'educated guess' for the metric equation but not necessarilly for the dynamics of the ψ A fields. 6 One can then to apply variational bootstrapping with respect to the metric to obtain a Lagrangian density based on this guessed equation and then deduce the ψ A dynamics from it.</text>
<text><location><page_10><loc_17><loc_73><loc_61><loc_74></location>In this case, assuming our guess is schematically of the form</text>
<formula><location><page_10><loc_39><loc_69><loc_85><loc_73></location>· G µν = T µν ( g , ∂ g ; ψ A , ∂ ψ A ) , (43)</formula>
<text><location><page_10><loc_15><loc_66><loc_85><loc_70></location>where T µν ( g , ∂ g ; ψ A , ∂ ψ A ) stands for the energy-momentum tensor associated with the extra fields ψ A , the procedure prescribes to compute 7</text>
<formula><location><page_10><loc_30><loc_57><loc_85><loc_66></location>L g = g µν ∫ 1 0 · G µν ( u g , u∂ g , u∂∂ g ) √ -det( ug ) d u -g µν ∫ 1 0 T µν ( u g , u∂ g ; ψ A , ∂ ψ A ) √ -det( ug ) d u, =: L EH + L ψ . (44)</formula>
<text><location><page_10><loc_15><loc_43><loc_85><loc_57></location>Here again, this procedure provides the 'best' Lagrangian density related to our initial guess (43) up to terms that would not give any contribution to the metric equation; i.e. Lagrangian densities built without using the metric g . Such Lagrangian densities are quite uncommon for physical theories as one usually uses the metric in the construction of the Lagrangian either to manipulate indices or to define a volume element which is used in the construction of the action. They are, nevertheless, not ruled out a priori by the above construction. In [12], we classified all possible natural Lagrangians independent of the metric in the context of a metricaffine setup in 4 dimensions. It turns out there is a limited amount of these which are obtained from specific 4 -forms constructed solely from the components of the distorsion tensor L and their first derivatives, see [12] for a complete classification.</text>
<section_header_level_1><location><page_10><loc_40><loc_40><loc_60><loc_40></location>B. General requirements</section_header_level_1>
<text><location><page_10><loc_15><loc_32><loc_85><loc_38></location>Let us now apply the above algorithm to our 'educated guess' (41). To proceed with the construction of the Lagrangian density, we first need to refine the requirements on our field equations and on the quantities Λ and V appearing in (37)-(38). In what follows, we make the three following assumptions:</text>
<unordered_list>
<list_item><location><page_10><loc_17><loc_29><loc_85><loc_32></location>1. The quantities Λ and V must depend on g , L , and their first order derivatives ∂ g , ∂ L only and in a generaly covariant way, more precisely: 8</list_item>
</unordered_list>
<formula><location><page_10><loc_39><loc_26><loc_85><loc_28></location>Λ = Λ( g , L , · ∇ L ) , V = V ( g , L , · ∇ L ) . (45)</formula>
<formula><location><page_10><loc_31><loc_15><loc_69><loc_17></location>∇ α L ρ µν = · ∇ α L ρ µν + L ρ βα L β µν -L β µα L ρ βν -L β να L ρ µβ .</formula>
<text><location><page_10><loc_16><loc_14><loc_55><loc_15></location>We can then write Λ = Λ( g , L , · ∇ L ) without loss of generality.</text>
<text><location><page_11><loc_17><loc_30><loc_37><loc_31></location>From equation (48), we get</text>
<formula><location><page_11><loc_37><loc_23><loc_85><loc_27></location>L Λ = 4 √ -| g | ∫ 1 0 u Λ( u g , L , · ∇ L ) d u. (51)</formula>
<formula><location><page_11><loc_31><loc_48><loc_85><loc_53></location>L EH = g µν ∫ 1 0 · G µν ( u g , u∂ g , u∂∂ g ) √ -det( ug ) d u, (47)</formula>
<formula><location><page_11><loc_32><loc_43><loc_85><loc_48></location>L Λ = g µν ∫ 1 0 Λ( u g , L , · ∇ L ) u -1 g µν √ -det( ug ) d u, (48)</formula>
<formula><location><page_11><loc_28><loc_38><loc_85><loc_42></location>L V = -g µν ∫ 1 0 V µ ( u g , L , · ∇ L ) V ν ( u g , L , · ∇ L ) √ -det( ug ) d u. (49)</formula>
<text><location><page_11><loc_17><loc_37><loc_78><loc_38></location>Using the relations from Appendix B, following the usual computation, we get that</text>
<text><location><page_11><loc_15><loc_30><loc_28><loc_32></location>with | g | := det( g ) .</text>
<text><location><page_11><loc_17><loc_21><loc_71><loc_23></location>Under our hypothesis 3, we can explicitly compute L Λ ; to this aim, denote</text>
<formula><location><page_11><loc_37><loc_17><loc_85><loc_21></location>Λ( g , L , · ∇ L ) = N ∑ k = M Λ ( k ) ( g , L , · ∇ L ) , (52)</formula>
<formula><location><page_11><loc_37><loc_31><loc_85><loc_35></location>L EH = g µν · G µν √ -| g | = -· R √ -| g | , (50)</formula>
<text><location><page_11><loc_19><loc_84><loc_85><loc_88></location>In particular, this ensures that the guessed equations (41) are generally covariant and that the Euler-Lagrange equations of the Lagrangian obtained by variational bootstrapping are at most of second order in both g and L .</text>
<unordered_list>
<list_item><location><page_11><loc_17><loc_76><loc_85><loc_83></location>2. In densitizing the seed equations ('educated guess'), the volume form used is the Riemannian one /epsilon1 g = √ -det( g )d x and indices are lowered/raised by g . An immediate consequence of this assumption is that all Lagrangian terms will exhibit a nontrivial dependence in g - and hence the full Lagrangian can be recovered, up to boundary terms, by variational boostrapping with respect to the metric.</list_item>
<list_item><location><page_11><loc_17><loc_74><loc_42><loc_75></location>3. We can write Λ as a finite sum</list_item>
</unordered_list>
<formula><location><page_11><loc_47><loc_69><loc_57><loc_73></location>Λ = ∑ k Λ ( k ) ,</formula>
<text><location><page_11><loc_19><loc_67><loc_85><loc_69></location>were each Λ ( k ) is a homogeneous function of degree k in g and its derivative; and the same property holds for V .</text>
<section_header_level_1><location><page_11><loc_39><loc_63><loc_61><loc_64></location>C. Bootstrapping of Λ CDM</section_header_level_1>
<text><location><page_11><loc_15><loc_58><loc_85><loc_61></location>We want to construct here the Lagrangian density obtained by variational bootstrapping of (39). According to (44) and the above hypotheses, we thus need to compute</text>
<formula><location><page_11><loc_42><loc_55><loc_85><loc_57></location>L g = L EH + L Λ + L V , (46)</formula>
<text><location><page_11><loc_15><loc_54><loc_19><loc_55></location>where</text>
<text><location><page_12><loc_15><loc_86><loc_67><loc_88></location>where M,N ∈ Z and each Λ ( k ) is homogneous of degree k in g , that is:</text>
<formula><location><page_12><loc_37><loc_83><loc_85><loc_85></location>Λ ( k ) ( u g , L , · ∇ L ) = u k Λ ( k ) ( g , L , · ∇ L ) . (53)</formula>
<text><location><page_12><loc_17><loc_81><loc_36><loc_82></location>In this case, (51) becomes</text>
<formula><location><page_12><loc_37><loc_75><loc_85><loc_79></location>L Λ = 4 √ -| g | N ∑ k = M Λ ( k ) ( g , L , · ∇ L ) k +2 . (54)</formula>
<text><location><page_12><loc_17><loc_73><loc_51><loc_74></location>Following the same line, (49) can be written as</text>
<formula><location><page_12><loc_29><loc_68><loc_85><loc_72></location>L V = -g µν √ -| g | ∫ 1 0 u 2 V µ ( u g , L , · ∇ L ) V ν ( u g , L , · ∇ L ) d u. (55)</formula>
<text><location><page_12><loc_17><loc_65><loc_75><loc_67></location>Here again, according to hypothesis 3, we want to parametrize V µ ( g , L , · ∇ L ) as</text>
<formula><location><page_12><loc_37><loc_60><loc_85><loc_64></location>V µ ( g , L , · ∇ L ) = J ∑ k = I V µ ( k ) ( g , L , · ∇ L ) , (56)</formula>
<text><location><page_12><loc_15><loc_58><loc_49><loc_60></location>where I, J ∈ Z with the defining property that</text>
<formula><location><page_12><loc_37><loc_55><loc_85><loc_57></location>V µ ( k ) ( u g , L , · ∇ L ) = u k V µ ( k ) ( g , L , · ∇ L ) . (57)</formula>
<text><location><page_12><loc_17><loc_53><loc_33><loc_54></location>Using (56), (55) gives</text>
<formula><location><page_12><loc_30><loc_47><loc_85><loc_51></location>L V = -√ -| g | g µν J ∑ k = I J ∑ l = I V µ ( k ) ( g , L , · ∇ L ) V ν ( l ) ( g , L , · ∇ L ) k + l +3 . (58)</formula>
<section_header_level_1><location><page_12><loc_33><loc_43><loc_67><loc_44></location>D. Variationally completed field equations</section_header_level_1>
<text><location><page_12><loc_15><loc_39><loc_85><loc_41></location>In order to determine the field equations associated to the Lagrangian density (46), we will separately compute the variation of each one of the terms in the summations of (54) and (58).</text>
<section_header_level_1><location><page_12><loc_40><loc_35><loc_60><loc_36></location>1. Variation with respect to g</section_header_level_1>
<text><location><page_12><loc_17><loc_32><loc_76><loc_33></location>Using the relations from Appendix B, as expected, the variation of (50) will give</text>
<formula><location><page_12><loc_19><loc_27><loc_85><loc_30></location>δ g L EH = · G µν δg µν √ -| g | + · ∇ µ ( g αµ δ g · Γ β αβ -g αβ δ g · Γ µ αβ ) √ -| g | /similarequal √ -| g | · G µν δg µν , (59)</formula>
<text><location><page_12><loc_15><loc_23><loc_85><loc_25></location>Let us now come to the variation of (54) with respect to the metric. To this purpose, it is sufficient to consider</text>
<text><location><page_12><loc_15><loc_25><loc_50><loc_27></location>where /similarequal means equality up to boundary terms.</text>
<formula><location><page_12><loc_16><loc_18><loc_85><loc_22></location>δ g ( Λ ( k ) √ -| g | ) = ( ∂ Λ ( k ) ∂g µν δg µν + ( ∂ Λ ( k ) ) α σβ ρ δ g ( · ∇ α L ρ σβ ) ) √ -| g | + 1 2 Λ ( k ) g µν δg µν √ -| g | , (60)</formula>
<text><location><page_13><loc_15><loc_84><loc_85><loc_88></location>where, to shorten the notation, we have defined ( ∂ Λ ( k ) ) α σβ ρ := ∂ Λ ( k ) ∂ ( · ∇ α L ρ σβ ) . Using the relations from Appendix B (in particular (B13) and (B15)), we get that</text>
<text><location><page_13><loc_15><loc_78><loc_19><loc_79></location>where</text>
<formula><location><page_13><loc_36><loc_79><loc_85><loc_83></location>δ g ( Λ ( k ) √ -| g | ) /similarequal √ -| g | T ( µν ) ( k ) δg µν , (61)</formula>
<formula><location><page_13><loc_27><loc_70><loc_85><loc_77></location>T µν ( k ) := 1 2 Λ ( k ) g µν + ∂ Λ ( k ) ∂g µν -· ∇ α [ L /ceilingleft ν | σβ ( ∂ Λ ( k ) ) | α | σβ ρ g ρ | µ /floorright -{ ( ∂ Λ ( k ) ) /ceilingleft ν | | α | β ρ g σ | µ /floorright + ( ∂ Λ ( k ) ) /ceilingleft ν | σ | α | ρ g β | µ /floorright } L ρ σβ ] , (62)</formula>
<text><location><page_13><loc_15><loc_69><loc_67><loc_71></location>and, for a generic tensor field with components S ρµν , we have denoted:</text>
<formula><location><page_13><loc_37><loc_66><loc_85><loc_68></location>S /ceilingleft ρµν /floorright := 1 2 ( S ρµν -S νρµ + S µνρ ) . (63)</formula>
<text><location><page_13><loc_15><loc_62><loc_85><loc_65></location>To compute the variation of L V with respect to the metric, we can reuse the results of the previous computation. Indeed, if we introduce</text>
<formula><location><page_13><loc_43><loc_59><loc_85><loc_61></location>V ( kl ) := g µν V µ ( k ) V ν ( l ) , (64)</formula>
<text><location><page_13><loc_15><loc_57><loc_28><loc_58></location>we can write that</text>
<formula><location><page_13><loc_36><loc_52><loc_85><loc_56></location>δ g ( V ( kl ) √ -| g | ) /similarequal √ -| g | T ( µν ) ( kl ) δg µν , (65)</formula>
<text><location><page_13><loc_15><loc_48><loc_85><loc_51></location>Combining (59), (61) and (65), we get that the metric field equation for lagrangian (46), in the absence of matter source, is</text>
<text><location><page_13><loc_15><loc_51><loc_84><loc_53></location>where the expression for T µν ( kl ) is obtained by performing the replacement Λ ( k ) → V ( kl ) in (62).</text>
<formula><location><page_13><loc_32><loc_43><loc_85><loc_47></location>· G = -4 N ∑ k = M 1 k +2 T ( k ) + J ∑ k = I J ∑ l = I 1 k + l +3 T ( kl ) ; (66)</formula>
<text><location><page_13><loc_15><loc_40><loc_58><loc_42></location>where T ( k ) = T ( k ) ( µν ) d x µ ⊗ d x ν and T ( kl ) = T ( kl ) ( µν ) d x µ ⊗ d x ν .</text>
<section_header_level_1><location><page_13><loc_39><loc_36><loc_61><loc_37></location>2. Variation with respect to L</section_header_level_1>
<text><location><page_13><loc_15><loc_31><loc_85><loc_35></location>As already mentioned, even though our initial guess was only related to the metric equation, variational bootstrapping allows us to completely determine the Lagrangian, up to boundary terms 9 . In particular the distortion field equation is also completely determined.</text>
<text><location><page_13><loc_15><loc_28><loc_85><loc_30></location>Let us now compute these field equations. Since L EH is independent of L , only L Λ and L V will contribute.</text>
<text><location><page_13><loc_17><loc_26><loc_28><loc_28></location>For L Λ , we find</text>
<formula><location><page_13><loc_28><loc_21><loc_85><loc_25></location>δ L ( Λ ( k ) √ -| g | ) = δ L Λ ( k ) √ -| g | /similarequal √ -| g | ( Ψ ( k ) ) σβ ρ δL ρ σβ , (67)</formula>
<text><location><page_14><loc_15><loc_87><loc_19><loc_88></location>where</text>
<formula><location><page_14><loc_35><loc_81><loc_85><loc_86></location>( Ψ ( k ) ) σβ ρ := ∂ Λ ( k ) ∂L ρ σβ -· ∇ α [ ( ∂ Λ ( k ) ) α σβ ρ ] . (68)</formula>
<text><location><page_14><loc_15><loc_79><loc_85><loc_81></location>Here again, for L V , one can reuse the previous result by introducing V ( kl ) from (64) to obtain that</text>
<formula><location><page_14><loc_33><loc_74><loc_85><loc_78></location>δ L ( V ( kl ) √ -| g | ) /similarequal √ -| g | ( Ψ ( kl ) ) σβ ρ δL ρ σβ , (69)</formula>
<text><location><page_14><loc_15><loc_70><loc_79><loc_74></location>where the expression of ( Ψ ( kl ) ) ρ is obtained via the replacement Λ ( k ) → V ( kl ) in (68). This finally gives us a distortion field equation of the form</text>
<text><location><page_14><loc_39><loc_74><loc_40><loc_74></location>σβ</text>
<formula><location><page_14><loc_33><loc_65><loc_85><loc_70></location>4 N ∑ k = M 1 k +2 Ψ ( k ) -J ∑ k = I J ∑ l = I 1 k + l +3 Ψ ( kl ) = 0 , (70)</formula>
<text><location><page_14><loc_15><loc_61><loc_72><loc_65></location>where Ψ ( k ) := ( Ψ ( k ) ) σβ ρ d x ρ ⊗ ∂ σ ⊗ ∂ β and Ψ ( kl ) := ( Ψ ( kl ) ) σβ ρ d x ρ ⊗ ∂ σ ⊗ ∂ β .</text>
<section_header_level_1><location><page_14><loc_38><loc_59><loc_62><loc_60></location>E. Analysis of the equations</section_header_level_1>
<text><location><page_14><loc_15><loc_50><loc_85><loc_57></location>Before closing our discussion, let us comment on the level of generality of the obtained expressions and on how these relates to previously formulated models in metric-affine gravity. Because of the freedom of choice in Λ and /vector V , our Lagrangian (46) still encompasses many of the models present in the literature. For a state of the art on metric-affine theories of gravity, we redirect the reader to [9] while, for a review on historical developments, one can see [7, 14].</text>
<text><location><page_14><loc_15><loc_46><loc_85><loc_50></location>The most straightforward example is that of a metric-affine theory whose Lagrangian is the Ricci scalar R of the independent connection Γ , see e.g. [15, 16]. Because of the well-known geometric identity</text>
<text><location><page_14><loc_15><loc_41><loc_22><loc_42></location>the choise</text>
<formula><location><page_14><loc_36><loc_42><loc_85><loc_45></location>R ≡ · R +2 ( L µ ν [ µ | L νρ | ρ ] + · ∇ [ µ | L µν | ν ] ) , (71)</formula>
<formula><location><page_14><loc_33><loc_36><loc_85><loc_40></location>Λ( g , L , · ∇ L ) = -1 2 ( L µ ν [ µ | L νρ | ρ ] + · ∇ [ µ | L µν | ν ] ) , (72)</formula>
<text><location><page_14><loc_15><loc_33><loc_85><loc_36></location>where each term is ( -1) -homogeneous with respect to g , and /vector V ( g , L , · ∇ L ) = /vector 0 leads to the desired Lagrangian.</text>
<text><location><page_14><loc_15><loc_24><loc_85><loc_33></location>A more interesting example is that of quadratic metric-affine theories [17, 18], i.e. theories whose Lagrangian contains terms quadratic in the curvature R , torsion T and/or non-metricity Q of Γ . In this case, one can easily see that appropriate choices of Λ and /vector V can generate all the terms under consideration in [17] or [18] except those who are quadratic in R -e.g. the term proportional to the coefficient z 4 in equation (3.1) of [17]. This comes from our hypothesis 1 about the form of Λ and /vector V and the fact that our Lagrangian contains only a term linear in · R .</text>
<text><location><page_14><loc_15><loc_20><loc_85><loc_24></location>Following the same line, our procedure then also selects a restrictive class of quadratic gauge theories of gravity within a metric-affine framework as it constrain the dependence in R (Compare with chapter 9 of [7]).</text>
<text><location><page_14><loc_15><loc_18><loc_85><loc_20></location>Finally, let us mention that, while we here considered theories based on a fully general metricaffine setup, many models studied in the literature start with some restriction on the connection</text>
<text><location><page_15><loc_52><loc_16><loc_54><loc_16></location>M</text>
<text><location><page_15><loc_15><loc_81><loc_85><loc_88></location>Γ ; typically Q ≡ 0 , as in Einstein-Cartan gravity or Poincaré gauge gravity, see respectively chapters 4 and 5 of [7]. Naturally, this would not affect the behavior of our procedure and the conclusion would remain the same 10 : Variational bootstrapping of the Λ CDM equations is compatible with quadratic metric-affine theories of gravity except for terms that are non-linear in the curvature tensor R .</text>
<text><location><page_15><loc_15><loc_73><loc_85><loc_80></location>As a final comment, let us emphasize that the particularity of the method outlined here in this business of finding Lagrangians is that it directly relates the notion of 'minimality' to the variational structure of the theory. In other words, variational bootstrapping allows us to select metric affine models which are minimal extensions of the Λ CDM model in a variational sense and this, as we just saw, allows for a more restrictive selection of Lagrangians.</text>
<section_header_level_1><location><page_15><loc_44><loc_69><loc_56><loc_70></location>VI. Conclusion</section_header_level_1>
<text><location><page_15><loc_15><loc_58><loc_85><loc_67></location>In this article, we studied the application of variational bootstrapping, first discussed in [12], to metric-affine theories of gravity. Through a carreful examination of the Λ CDM model of cosmology, we proposed a minimal modification of GR's field equation, based on that model, in the context of a metric-affine theory. Our hypotheses for that construction are explicited in section IV (see also section V B). Then, variational bootstrapping allowed us to construct a Lagrangian for the full theory, leading to 'corrected' metric equations (66) but also giving access to field equations for the distortion tensor (independent connection) (70).</text>
<text><location><page_15><loc_15><loc_49><loc_85><loc_58></location>We showed in section V E how our model related to previously formulated ones. We saw that the notion of 'variationaly minimal extension' outlined in this paper gives a more restrictive criterion to select Lagrangians than the ones usually consiered in the literature when considering e.g. quadratic metric-affine theories. Naturally, this does not rule out the other existing models per se ! It should nevertheless be regarded as an interesting property since, in modified gravity in general (and in metric-affine gauge theories of gravity in particular), one usually struggles to single out a theory exihibiting a desired behaviour.</text>
<text><location><page_15><loc_15><loc_43><loc_85><loc_48></location>This investigation demonstrates that variational bootstrapping is a powerful tool to construct theoretical models based on phenomenological requirements. The most remarkable feature of the method is that it allows to obtain a sensible Lagrangain for the theory under study by only guessing a part of the system's dynamics.</text>
<text><location><page_15><loc_15><loc_35><loc_85><loc_42></location>An important direction to extend the present work is to investigate solutions to the field equations (66) ∧ (70). In particular, it would be interesting to see if a well-chosen form of the scalar function Λ and vector field V would allow, in a cosmological setup, to dynamically produce a behavior of the scale factor a ( t ) compatible with current universe's acceleration and/or with inflation.</text>
<text><location><page_15><loc_15><loc_31><loc_85><loc_35></location>In addition to that, it would also be necessary to study the behavior of compact objects (black holes, neutron stars) in our theory to see how these differ from GR's ones and to which extend this remains compatible with modern observations.</text>
<text><location><page_15><loc_15><loc_27><loc_85><loc_31></location>Another possible extension includes the application of our method to metric-affine theories using a tetrad and a spin connection as dynamical variables and the study of the conditions under which this approach is equivalent to the one presented here.</text>
<text><location><page_15><loc_42><loc_19><loc_43><loc_20></location>Y</text>
<text><location><page_15><loc_43><loc_19><loc_48><loc_20></location>= Met(</text>
<text><location><page_15><loc_48><loc_19><loc_49><loc_20></location>M</text>
<text><location><page_15><loc_49><loc_19><loc_50><loc_20></location>)</text>
<text><location><page_15><loc_50><loc_20><loc_51><loc_20></location>×</text>
<text><location><page_15><loc_51><loc_19><loc_53><loc_20></location>M</text>
<text><location><page_15><loc_53><loc_19><loc_56><loc_20></location>Tor(</text>
<text><location><page_15><loc_56><loc_19><loc_57><loc_20></location>M</text>
<text><location><page_15><loc_57><loc_19><loc_58><loc_20></location>)</text>
<text><location><page_15><loc_58><loc_19><loc_58><loc_20></location>,</text>
<text><location><page_15><loc_16><loc_17><loc_85><loc_19></location>where Tor( M ) denotes the bundle of torsion i.e. the bunble of type (1 , 2) tensor fields antisymmetric over their two lower indices - instead of</text>
<text><location><page_15><loc_43><loc_16><loc_44><loc_17></location>Y</text>
<text><location><page_15><loc_44><loc_16><loc_49><loc_17></location>= Met(</text>
<text><location><page_15><loc_49><loc_16><loc_50><loc_17></location>M</text>
<text><location><page_15><loc_50><loc_16><loc_51><loc_17></location>)</text>
<text><location><page_15><loc_51><loc_16><loc_52><loc_17></location>×</text>
<text><location><page_15><loc_54><loc_16><loc_55><loc_17></location>T</text>
<text><location><page_15><loc_55><loc_16><loc_56><loc_17></location>1</text>
<text><location><page_15><loc_55><loc_16><loc_55><loc_16></location>2</text>
<text><location><page_15><loc_56><loc_16><loc_57><loc_17></location>M</text>
<text><location><page_15><loc_16><loc_13><loc_85><loc_15></location>as in this paper, see [12]. Nevertheless, since we are interested in variational bootstrapping with respect to the metric anyway, the computation and conclusions would perform exactly the same as in our case.</text>
<text><location><page_16><loc_17><loc_87><loc_56><loc_88></location>We leave these and further questions for future works.</text>
<section_header_level_1><location><page_16><loc_43><loc_82><loc_57><loc_83></location>Acknowledgements</section_header_level_1>
<text><location><page_16><loc_15><loc_73><loc_85><loc_80></location>L.D. acknowledges support from the Transilvania Fellowship Program for Postdoctoral Research/Young Researchers (September 2022) and from a Postdoctoral sojourn grant from Complexys institute from University of Mons (Belgium) and also acknowledges networking support by the COST Action CA18108 in the early developments of the present project. N.V. acknowledges networking support by the COST Action 21136.</text>
<unordered_list>
<list_item><location><page_16><loc_16><loc_61><loc_100><loc_67></location>[1] A. G. Riess, A. V. Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P. M. Garnavich, R. L. Gilliland, C. J. Hogan, S. Jha, R. P. Kirshner, B. Leibundgut, M. M. Phillips, D. Reiss, B. P. Schmidt, R. A. Schommer, R. C. Smith, J. Spyromilio, C. Stubbs, N. B. Suntzeff, and J. Tonry, 'Observational evidence from supernovae for an accelerating universe and a cosmological constant,' The Astronomical Jo https://dx.doi.org/10.1086/300499 .</list_item>
<list_item><location><page_16><loc_16><loc_60><loc_81><loc_61></location>[2] S. Perlmutter, G. Aldering, G. Goldhaber, R. A. Knop, P. Nugent, P. G. Castro, S. Deustua,</list_item>
<list_item><location><page_16><loc_81><loc_54><loc_92><loc_55></location>(jun, 1999) 565.</list_item>
<list_item><location><page_16><loc_18><loc_53><loc_84><loc_60></location>S. Fabbro, A. Goobar, D. E. Groom, I. M. Hook, A. G. Kim, M. Y. Kim, J. C. Lee, N. J. Nunes, R. Pain, C. R. Pennypacker, R. Quimby, C. Lidman, R. S. Ellis, M. Irwin, R. G. McMahon, P. Ruiz-Lapuente, N. Walton, B. Schaefer, B. J. Boyle, A. V. Filippenko, T. Matheson, A. S. Fruchter, N. Panagia, H. J. M. Newberg, W. J. Couch, and T. S. C. Project, 'Measurements of ω and λ from 42 high-redshift supernovae,' The Astrophysical Journal 517 https://dx.doi.org/10.1086/307221 .</list_item>
<list_item><location><page_16><loc_16><loc_49><loc_82><loc_52></location>[3] Planck Collaboration, N. Aghanim et al. , 'Planck 2018 results. VI. Cosmological parameters,' Astron. Astrophys. 641 (2020) A6, arXiv:1807.06209 [astro-ph.CO] . [Erratum: Astron.Astrophys. 652, C4 (2021)].</list_item>
<list_item><location><page_16><loc_16><loc_46><loc_83><loc_49></location>[4] S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity . Wiley, 1972. https://books.google.lu/books?id=XLbvAAAAMAAJ .</list_item>
<list_item><location><page_16><loc_16><loc_43><loc_83><loc_46></location>[5] N. Deruelle and J.-P. Uzan, '19 The Lambda-CDM model of the hot Big Bang,' in Relativity in Modern Physics . Oxford University Press, 08, 2018. https://doi.org/10.1093/oso/9780198786399.003.0059 .</list_item>
<list_item><location><page_16><loc_16><loc_39><loc_70><loc_42></location>[6] J. L. Bernal, L. Verde, and A. G. Riess, 'The trouble with H 0 ,' Journal of Cosmology and Astroparticle Physics 2016 https://dx.doi.org/10.1088/1475-7516/2016/10/019</list_item>
</unordered_list>
<text><location><page_16><loc_56><loc_39><loc_81><loc_41></location>(oct, 2016) 019. .</text>
<unordered_list>
<list_item><location><page_16><loc_16><loc_33><loc_82><loc_39></location>[7] M. Blagojevic and F. W. Hehl, 'Gauge Theories of Gravitation,' arXiv:1210.3775 [gr-qc] . [8] F. W. Hehl, J. McCrea, E. W. Mielke, and Y. Ne'eman, 'Metric-affine gauge theory of gravity: field equations, noether identities, world spinors, and breaking of dilation invariance,' Physics Reports 258 (1995) no. 1, 1-171. https://www.sciencedirect.com/science/article/pii/037015739400111F .</list_item>
<list_item><location><page_16><loc_16><loc_31><loc_49><loc_32></location>[9] CANTATA Collaboration, Y. Akrami et al. ,</list_item>
<list_item><location><page_16><loc_18><loc_29><loc_78><loc_31></location>Modified Gravity and Cosmology: An Update by the CANTATA Network . Springer, 2021. arXiv:2105.12582 [gr-qc] .</list_item>
<list_item><location><page_16><loc_15><loc_28><loc_30><loc_29></location>[10] G. W. Horndeski,</list_item>
<list_item><location><page_16><loc_15><loc_24><loc_100><loc_27></location>'Second-Order Scalar-Tensor Field Equations in a Four-Dimensional Space,' International Journal of Theoretical Physics [11] N. Voicu and D. Krupka, 'Canonical variational completion of differential equations,' Journal of Mathematical Physics 56 (Apr., 2015) .</list_item>
<list_item><location><page_16><loc_15><loc_20><loc_80><loc_24></location>[12] L. Ducobu and N. Voicu, 'Metric-affine cosmological models and the inverse problem of the calculus of variations. Part 1: variational bootstrapping - the method,' arXiv:2403.15564 [math-ph] .</list_item>
<list_item><location><page_16><loc_15><loc_18><loc_62><loc_20></location>[13] F. Zwicky, 'Die Rotverschiebung von extragalaktischen Nebeln,' Helv. Phys. Acta 6 (1933) 110-127.</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_17><loc_15><loc_85><loc_79><loc_88></location>[14] D. Puetzfeld, 'Status of non-Riemannian cosmology,' New Astron. Rev. 49 (2005) 59-64, arXiv:gr-qc/0404119 .</list_item>
<list_item><location><page_17><loc_15><loc_83><loc_70><loc_85></location>[15] K. Shimada, K. Aoki, and K.-i. Maeda, 'Metric-affine Gravity and Inflation,' Phys. Rev. D 99 (2019) no. 10, 104020, arXiv:1812.03420 [gr-qc] .</list_item>
<list_item><location><page_17><loc_15><loc_80><loc_65><loc_83></location>[16] D. Iosifidis, 'Cosmological Hyperfluids, Torsion and Non-metricity,' Eur. Phys. J. C 80 (2020) no. 11, 1042, arXiv:2003.07384 [gr-qc] .</list_item>
<list_item><location><page_17><loc_15><loc_77><loc_100><loc_80></location>[17] Y. N. Obukhov, E. J. Vlachynsky, W. Esser, and F. W. Hehl, 'Effective einstein theory from metric-affine gravity models via irreducible decompositions,' Phys. Rev. D 56 (Dec, 1997) https://link.aps.org/doi/10.1103/PhysRevD.56.7769 .</list_item>
<list_item><location><page_17><loc_15><loc_73><loc_82><loc_76></location>[18] V. Vitagliano, T. P. Sotiriou, and S. Liberati, 'The dynamics of metric-affine gravity,' Annals Phys. 326 (2011) 1259-1273, arXiv:1008.0171 [gr-qc] . [Erratum: Annals Phys. 329, 186-187 (2013)].</list_item>
<list_item><location><page_17><loc_15><loc_70><loc_64><loc_73></location>[19] M. Hohmann, 'Metric-affine Geometries With Spherical Symmetry,' Symmetry 12 (2020) no. 3, 453, arXiv:1912.12906 [math-ph] .</list_item>
</unordered_list>
<section_header_level_1><location><page_17><loc_39><loc_67><loc_61><loc_68></location>A. Cosmological symmetry</section_header_level_1>
<section_header_level_1><location><page_17><loc_44><loc_64><loc_56><loc_65></location>1. Symmetry</section_header_level_1>
<text><location><page_17><loc_15><loc_59><loc_85><loc_62></location>As discussed in [19], when saying that a given tensor field T presents a symmetry, one means that our manifold is equipped with a certain group action</text>
<formula><location><page_17><loc_30><loc_56><loc_70><loc_59></location>Φ : G × M → M : ( g, x ) ↦→ Φ( g, x ) =: Φ g ( x ) =: Φ x ( g ) ,</formula>
<text><location><page_17><loc_15><loc_55><loc_85><loc_56></location>smooth over its second variable, and that T is invariant under this group action in the sense that</text>
<formula><location><page_17><loc_42><loc_52><loc_85><loc_54></location>∀ g ∈ G, Φ ∗ g ( T ) = T . (A1)</formula>
<text><location><page_17><loc_15><loc_47><loc_85><loc_52></location>When G is a Lie group, it is usually more convenient to study this notion of symmetry in terms of the infinitesimal action of the symmetry group. Let us denote by g the Lie algebra of G and by X ξ the generating vector field for a given ξ ∈ g 11 , if a tensor T is symmetric in the sense of (A1) then</text>
<formula><location><page_17><loc_43><loc_44><loc_85><loc_46></location>∀ ξ ∈ g , L X ξ T = 0 , (A2)</formula>
<text><location><page_17><loc_15><loc_43><loc_41><loc_44></location>where L denotes the Lie derivative.</text>
<text><location><page_17><loc_15><loc_39><loc_85><loc_43></location>If G is connected, conditions (A1) and (A2) are equivalent and one can then derive the conditions for a tensor field T to present a given symmetry via (A2). It is the condition that we will use in the following.</text>
<text><location><page_17><loc_15><loc_35><loc_85><loc_39></location>Following this definition of symmetry, a metric-affine spacetime will be symmetric under a given group G provided both the metric g and the distortion L are invariant under the action of G .</text>
<section_header_level_1><location><page_17><loc_34><loc_31><loc_66><loc_32></location>2. Generators of cosmological symmetry</section_header_level_1>
<text><location><page_17><loc_15><loc_26><loc_85><loc_29></location>Cosmological symmetry refers to the idea that, following the cosmological principle, no observer has a prefered place in the universe so that spacetime must be spatially homogeneous and</text>
<formula><location><page_17><loc_43><loc_19><loc_57><loc_21></location>C p X ξ : t ↦→ Φ(exp( ξt ) , p ) ,</formula>
<text><location><page_18><loc_15><loc_81><loc_85><loc_88></location>isotropic. More precisely here, we then expect our setup to be invariant under both the 'spatial' rotation group SO (3) (providing isotropy, at least around one point) and a group representing 'spatial translations' (ensuring isotropy around actually any point and hence homogeneity). One thus obtains 6 generators of cosmological symmetry X i ( i = 1 , · · · , 6) . In local spherical coordinates ( x µ ) = ( t, r, θ, ϕ ) on M , SO (3) -generators are described as:</text>
<formula><location><page_18><loc_24><loc_77><loc_85><loc_80></location>X 1 = sin( ϕ ) ∂ θ + cos( ϕ ) tan( θ ) ∂ ϕ , X 2 = -cos( ϕ ) ∂ θ + sin( ϕ ) tan( θ ) ∂ ϕ , X 3 = -∂ ϕ , (A3)</formula>
<text><location><page_18><loc_15><loc_75><loc_52><loc_76></location>as they satisfy the commutation relation of so (3) :</text>
<formula><location><page_18><loc_43><loc_72><loc_85><loc_73></location>[ X i , X j ] = /epsilon1 ijk X k , (A4)</formula>
<text><location><page_18><loc_15><loc_69><loc_56><loc_71></location>where [ · , · ] denotes the commutator of two vector fields.</text>
<text><location><page_18><loc_17><loc_68><loc_45><loc_69></location>Generators of (quasi-)translations are:</text>
<formula><location><page_18><loc_25><loc_63><loc_85><loc_67></location>X 4 = √ 1 -kr 2 ( sin( θ ) cos( ϕ ) ∂ r + 1 r cos( θ ) cos( ϕ ) ∂ θ -1 r sin( ϕ ) sin( θ ) ∂ ϕ ) , (A5)</formula>
<formula><location><page_18><loc_25><loc_56><loc_85><loc_60></location>X 6 = √ 1 -kr 2 ( cos( θ ) ∂ r -1 r sin( θ ) ∂ θ ) , (A7)</formula>
<formula><location><page_18><loc_25><loc_60><loc_85><loc_64></location>X 5 = √ 1 -kr 2 ( sin( θ ) sin( ϕ ) ∂ r + 1 r cos( θ ) sin( ϕ ) ∂ θ + 1 r cos( ϕ ) sin( θ ) ∂ ϕ ) , (A6)</formula>
<text><location><page_18><loc_15><loc_50><loc_85><loc_56></location>where k ∈ { -1 , 0 , 1 } is the sign of the spatial curvature. When k = 0 (flat spatial geometry), X 1 , X 2 and X 3 correspond to the expression, in spherical coordinates, of the usual generators of spatial rotation around respectivelly the x -, y - and z -axis of an orthonornmal cartesian frame, while X 4 , X 5 and X 6 correspond to the usual generators of spatial translation along respectivelly the x -, y - and z -axis.</text>
<text><location><page_18><loc_17><loc_47><loc_85><loc_48></location>According to (A2), a given tensor field T will be cosmologically symmetric provided it satisfies</text>
<formula><location><page_18><loc_41><loc_44><loc_85><loc_46></location>L X i T = 0 , ∀ i = 1 , · · · , 6 . (A8)</formula>
<section_header_level_1><location><page_18><loc_41><loc_41><loc_59><loc_42></location>B. Technical identities</section_header_level_1>
<text><location><page_18><loc_15><loc_36><loc_85><loc_39></location>In this section, we report for completeness some important relations used in the computations of this paper.</text>
<section_header_level_1><location><page_18><loc_39><loc_32><loc_61><loc_33></location>1. Rescaling of the metric</section_header_level_1>
<text><location><page_18><loc_15><loc_28><loc_85><loc_30></location>To perform the variational completion/bootstrapping of our equations, we must know the behavior of the different terms in the equations under the homothety</text>
<formula><location><page_18><loc_31><loc_23><loc_85><loc_25></location>χ u : ( g µν , g µν,ρ , g µν,ρσ ) ↦→ ( u g µν , u g µν,ρ , u g µν,ρσ ) . (B1)</formula>
<text><location><page_18><loc_17><loc_22><loc_47><loc_23></location>In this respect, one immediately obtains:</text>
<formula><location><page_18><loc_42><loc_17><loc_85><loc_19></location>det( g ) χ u -→ u 4 det( g ) , (B2)</formula>
<formula><location><page_19><loc_44><loc_86><loc_85><loc_88></location>g µν χ u -→ u -1 g µν , (B3)</formula>
<formula><location><page_19><loc_42><loc_81><loc_85><loc_84></location>· R ρ σµν χ u -→ · R ρ σµν , (B5)</formula>
<formula><location><page_19><loc_44><loc_84><loc_85><loc_86></location>· Γ ρ µν χ u -→ · Γ ρ µν , (B4)</formula>
<formula><location><page_19><loc_44><loc_79><loc_85><loc_82></location>· R µν χ u -→ · R µν , (B6)</formula>
<formula><location><page_19><loc_43><loc_75><loc_85><loc_77></location>· G µν χ u -→ · G µν , (B8)</formula>
<formula><location><page_19><loc_45><loc_77><loc_85><loc_80></location>· R χ u -→ u -1 · R , (B7)</formula>
<formula><location><page_19><loc_43><loc_73><loc_85><loc_75></location>· G µν χ u -→ u -2 · G µν . (B9)</formula>
<section_header_level_1><location><page_19><loc_40><loc_70><loc_60><loc_71></location>2. Calculus of variations</section_header_level_1>
<text><location><page_19><loc_17><loc_67><loc_85><loc_68></location>To compute the field equations associated to our Lagrangian, we need the following relations.</text>
<text><location><page_19><loc_36><loc_63><loc_64><loc_64></location>a. Variations with respect to the metric g</text>
<text><location><page_19><loc_17><loc_60><loc_63><loc_61></location>When computing variation with respect to the metric, one has</text>
<formula><location><page_19><loc_27><loc_56><loc_60><loc_57></location>δg µν = g µα g νβ δg αβ , δg µν = g µα g νβ δg αβ ,</formula>
<formula><location><page_19><loc_32><loc_42><loc_85><loc_57></location>--(B10) δ g √ -| g | = 1 2 √ -| g | g µν δg µν , (B11) δ g · Γ ρ µν = 1 2 g ρα ( · ∇ ν δg αµ + · ∇ µ δg να -· ∇ α δg µν ) (B12) = g ρα · ∇ /ceilingleft µ δg να /floorright , (B13) δ g · R ρ σµν = · ∇ µ δ g · Γ ρ σν -· ∇ ν δ g · Γ ρ σµ , (B14) δ g ( · ∇ α L ρ σβ ) = δ g · Γ ρ µα L µ σβ -δ g · Γ µ σα L ρ µβ -δ g · Γ µ βα L ρ σµ . (B15)</formula>
<text><location><page_19><loc_34><loc_39><loc_66><loc_40></location>b. Variations with respect to the distortion L</text>
<text><location><page_19><loc_17><loc_36><loc_65><loc_37></location>When computing variation with respect to the distortion, we get:</text>
<formula><location><page_19><loc_26><loc_27><loc_85><loc_34></location>δ L ( · ∇ α L ρ µν ) = δ L ( ∂ α L ρ µν + · Γ ρ βα L β µν -· Γ β µα L ρ βν -· Γ β να L ρ µβ ) (B16) = ∂ α ( δL ρ µν ) + · Γ ρ βα δL β µν -· Γ β µα δL ρ βν -· Γ β να δL ρ µβ (B17) · (B18)</formula>
<formula><location><page_19><loc_37><loc_25><loc_47><loc_29></location>= ∇ α ( δL ρ µν ) .</formula>
</document>
[
{
"title": "Metric-affine cosmological models and the inverse problem of the calculus of variations. Part II: Variational bootstrapping of the Λ CDM model",
"content": "Ludovic Ducobu ∗ Department of Mathematics and Computer Science, Transilvania University of Brasov, Brasov, Romania and Nuclear and Subnuclear Physics, University of Mons, Mons, Belgium",
"pages": [
1
]
},
{
"title": "Nicoleta Voicu †",
"content": "Department of Mathematics and Computer Science, Transilvania University of Brasov, Brasov, Romania and Lepage Research Institute, Presov, Slovakia The method of variational bootstrapping, based on canonical variational completion, allows one to construct a Lagrangian for a physical theory depending on two sets of field variables, starting from a guess of the field equations for only one such set. This setup is particularly appealing for the construction of modified theories of gravity since one can take lessons from GR for the 'educated guess' of the metric field equations; the field equations for the other fields are then fixed by the obtained Lagrangian (up to terms completely independent from the metric tensor). In the present paper, we explore the applications of variational bootstrapping in the context of metric-affine theories of gravity. We find those metric-affine models which are, in a variational sense, closest to the Λ CDM model of cosmology. The method then allows to find 'corrected' metric equations and to 'bootstrap' the connection field equations. The Lagrangians obtained via this method, though imposing some restricting criteria, encompass a wide variety of metric-affine models. In particular, they allow for quadratic metric-affine theories only if these avoid non-linear terms in the curvature tensor.",
"pages": [
1
]
},
{
"title": "I. Introduction",
"content": "Despite its successes, it is notorious that General Relativity (GR) struggles at the smallest scales - due to an incompatibility with quantum theory - and at the largest scales - where it fails to correctly reproduce the cosmological behavior of the universe. For the latter problem (which is the one we focus on, in the present paper), a simplest way of correcting the GR model is to amend Einstein's equation by adding entities known as dark matter and dark energy. This paradigm is known as the Λ CDM model of cosmology [1-5] and is, to this date, the best model available to fit cosmological observations. Nevertheless, the Λ CDM model is challenged, on the one hand, by the so-called H 0 -σ 8 tension [6] and on the other hand, by the fact that the very physical origin of dark energy and dark matter physical origin still eludes our understanding. These puzzles legitimize the quest for a more general (classical) gravitational theory. Such generalizations of GR concern either the geometric structure of spacetime (i.e., the kinematical level), or the field equations (in other words, the dynamical level), or both. At the kinematical level, one of the simplest extension is metric-affine geometry, which allows for more freedom by relaxing the GR hypotheses on the spacetime connection (metricity, absence of torsion), [7-9]. In metric-affine gravity models, one can thus encode the dark sector into the 'enhanced' geometry of spacetime 1 . At the dynamical level, one usually discusses modified theories of gravity by first postulating a Lagrangian, from which the field equations are derived. Even though this procedure is automatically justified as long as one expects a fundamental physical theory to admit a Lagrangian, ultimately what one typically wants to control are the properties of the field equations , rather than the ones of the Lagrangian. For this reason, one of us proposed in [11], together with D. Krupka, a systematic method, called canonical variational completion , allowing to go the other way around: starting with an approximate form (an 'educated guess') of the field equations, one can canonically construct a Lagrangian and subsequently, the 'corrected' field equations. The obtained field equations are the closest ones to our original guess, in the sense that the correction terms to be added specifically measure the 'obstruction from variationality' of this guess. The method of canonical variational completion was then upgraded in the first part of this work [12], to accommodate theories that involve more than one set of dynamical variables, e.g., a metric and a connection. We showed that from an 'educated guess' of the field equations with respect to one variable (e.g., the metric) only, one can still canonically 'bootstrap' a Lagrangian, up to boundary terms and to terms that are completely independent of the respective variable. In the present paper, we apply the method proposed in [12] to select those metric-affine Lagrangians producing the 'closest to Λ CDM' metric equations. Our 'educated guess' (or 'seed') for the metric equations is a system that formally resembles the Einstein equations with a cosmological constant (encoding dark energy, w = -1 ) and a term describing dark matter (with w = 0 ). To obtain it, we first consider the form of general tensor fields fitting the dark energy or dark matter equation of state (EOS) under the assumption of cosmological symmetry. Next, we step back from cosmological symmetry and consider this expression as valid in full generality; this correspond to the idea of finding the most minimal extension of the Λ CDM equations. We then bootstrap this guess under the following hypotheses: The second and third hypotheses above ensure on the one hand, that the obtained Lagrangians are generally covariant and, on the other hand, that all their non-boundary terms can be found from our 'educated guess' of the metric equations only. This way, no input on the connection equations is necessary; these can be automatically 'bootstrapped' from the obtained Lagrangian. We find that a number of the existing metric-affine models fall into this class, e.g. , those whose Lagrangian is the Ricci scalar of the independent connection and a subclass of quadratic metric-affine theories avoiding non-linear terms involving the curvature tensor of the independent connection. The paper is structured as follows : In sections II and III, we fix our notations for the metricaffine geometry and briefly review some properties of the Λ CDM model that are necessary for our discussion. In section IV, we consider metric-affine theories of gravity based on modifications implemented directly at the level of GR field equations with aim of obtaining an 'educated guess' for the (metric) field equation in a metric-affine extension of GR. Section V contains our main results. In section V A, we remind the basics of the technique of variational bootstrapping presented in [12]. In section V B, we explicit our hypothesis in the construction of our model before performing, in section V C, the variational bootstrapping of the equations discussed in section IV - hence obtaining a Lagrangian for the full theory. Then, in section V D, we study the field equations associated to the Lagrangian obtained in section V C. In section V E, we then comment on the obtained equations and on their link with previously formulated metric-affine models of gravity. Finally, in section VI, we present our conclusions and discuss further directions of research.",
"pages": [
1,
2,
3
]
},
{
"title": "II. Geometric setup",
"content": "In the following, we model spacetime by means of a triple ( M, Γ , g ) , where M is an arbitrary 4-dimensional differentiable manifold, Γ is a generic linear connection on M and g a metric with Lorentzian signature. Local coordinates on M will be denoted by x µ and the local basis vectors on TM , by ∂ µ ; when writing connection coefficients, the second lower index refers to the 'derivative index' meaning that, for example, given a vector field V = V µ ∂ µ one has that ∇ µ V ν := ( ∇ ∂ µ V ) ν = ∂ µ V ν +Γ ν ρµ V ρ . We will denote by R the curvature tensor of Γ : where ∇ is the covariant derivative associated to Γ and by: its components in a coordinate basis. Similarly, T will designate the torsion of Γ : with components in a coordinate basis: and by Q the non-metricity tensor defined as with local components: Our metric-affine setup can equivalently (and more conveniently) be defined as the triple ( M, L , g ) , where the distortion of Γ : is a tensor field of type (1 , 2) . The symbol · designates the Levi-Civita connection associated to g and all its related quantities; for instance: In the following, for conciseness, we will express our relations in terms of the distortion tensor L . If needed, all the relations can be translated in terms of the torsion T (respectively, the contortion K ) and non-metricity Q (respectively, disformation D ), using the relations below: where: For the sake of completeness, we also present the inverse relations: where, as usual, square brackets (resp. round brackets) placed around a group of indices denote antisymmetrization (resp. symmetrization) over those indices, with exception of indices placed between vertical bars.",
"pages": [
3,
4
]
},
{
"title": "III. Brief review of the Λ CDM model",
"content": "This section briefly presents the restrictions imposed by cosmological symmetry upon tensors on the spacetime manifold and the geometric setup of the Λ CDM model; for more detail see [4, 5]. Even though most of the material of this section is not new, this allow us to clearly fix our notations and to raise an important - yet rarely emphasized - problem.",
"pages": [
4
]
},
{
"title": "A. Cosmologically symmetric tensor fields",
"content": "Cosmological symmetry, understood as invariance under the six Killing vector fields defining it, imposes quite strong constraints on tensor fields of any rank, which we briefly present below; we direct the curious reader to Appendix A for more details and computations. In the following, we denote by ( x µ ) = ( t, r, θ, ϕ ) , a set of local spherical coordinates over M adapted to the symmetry. A given vector field V = V µ ∂ µ on M is cosmologically symmetric if and only if its components in the coordinate basis { ∂ µ } satisfy Similarly, a given (0 , 2) -tensor field T = T µν d x µ ⊗ d x ν will be cosmologically symmetric if and only if its components in the coordinate basis { ∂ µ } satisfy where k ∈ { -1 , 0 , 1 } is the sign of the curvature of the spatial slices. It is worth noting that, in deducing (18), one does not need to assume that T is symmetric; its antisymmetric part vanishes as a consequences of cosmological symmetry. From this relation, we naturally recover that the components of the most general cosmologically symmetric metric with Lorentz signature g = g µν d x µ ⊗ d x ν can be written as Using the musical isomorphisms offered by the metric, i.e. if we use the metric to raise or lower indices, we also get that V µ := g µν V ν and T µ ν := g µα T αν must be such that where we defined θ V ( t ) := -N ( t ) 2 V t ( t ) , and where we defined ρ T ( t ) := T tt ( t ) /N ( t ) 2 and P T ( t ) := T r ( t ) /a ( t ) 2 . Since the metric provides an isomorphism between TM and T ∗ M , we can conclude that (20) gives us the most general expression for a differential 1-form and (21) the most general expression for a (1 , 1) -tensor field respecting cosmological symmetry; these can also be obtained by direct computation.",
"pages": [
4,
5
]
},
{
"title": "B. The Λ CDM model in a nutshell",
"content": "From the theoretical perspective, the Λ CDM model of cosmology assumes that the universe behaves, at the kinematical level, according to the tools of pseudo-Riemaniann geometry ( i.e. a metric-affine setup where T ≡ 0 & Q ≡ 0 ) and fixes the dynamics of the universe (the evolution of the scale factor a ( t ) ) by means of the following modified Einstein equation [4, 5] where · G = · G µν d x µ ⊗ d x ν is the Einstein tensor, Λ is the cosmological constant, κ = 8 π G /c 4 , T (m) = T (m) µν d x µ ⊗ d x ν is the energy-momentum tensor of ordinary matter (including barionic matter and electromagnetic radiation) and T (DM) = T (DM) µν d x µ ⊗ d x ν is the energy-momentum tensor of dark matter. Matter is modeled as a perfect fluid where g µν is given by (19), ρ is the energy density of the fluid, P its pressure and u = u µ ∂ µ is the average 4-velocity of the fluid which is taken to be normalized ( g ( u, u ) = -1 ) and such that the fluid is comoving with inertial observers; i.e. one has To complete this description, properties of matter are encoded by means of an equation of state (EOS) of the form where w ∈ R is a constant. For ordinary (dust) matter one has w dust = 0 ( i.e. vanishing pressure) while for electromagnetic radiation (photons) w EM = 1 / 3 to ensure T (EM) µ µ = 0 . If we raise the first index of T (m) , we then get In the Λ CDM model, one considers the dark matter sector to be composed of 'cold dark matter'. This cold dark matter represents, in this model, some exotic non-relativistic particles described by a perfect fluid energy-momentum tensor (23) for which w DM = 0 . So The term Λ g models dark energy. Following the usual reasoning, the term κ T (DE) := -Λ g can be interpreted as a perfect fluid energy-momentum tensor for which ρ DE = Λ /κ and w DE = -1 , leading to",
"pages": [
5,
6
]
},
{
"title": "C. The cosmological smokescreen",
"content": "From the computations of section III A, one can make the following observations [4]: where is a symmetric tensor field of type (0 , 2) (or (2 , 0) ) depending on the metric g , the distortion tensor components and a finite number of derivatives thereof. In the particular case of cosmological symmetry, we want Θ (D) to geometrically encode the dark sector contributions, in other words, to be the geometric counterpart of T (D) . To stay as close as possible to the Λ CDM model, we may want to regard dark matter and dark energy as distinct effects. 2 This corresponds to the idea of writting (31) with In this case, the success of the Λ CDM model should be seen as constraining the form of Θ (DM) and Θ (DE) . In other words, one should make sure that, under cosmological symmetry, the form of Θ (DM) and Θ (DE) are compatible with the dark-matter and dark-energy EOS from the Λ CDM model The first relation in (34) then implies that These remarks, while supporting the parametrization of the energy-momentum tensor of ordinary matter via (23) subject to an EOS (25) with w ∈ { 0 , 1 / 3 } , can cast serious doubts on the interpretation of (28) and (29). If we further group together the 'dark contributions' as and raise the first index, we recover the general expression (21) with associated 'density' ρ D := ρ DM + ρ DE = ρ DM +Λ /κ and 'pressure' P D := P DM + P DE = -Λ /κ . A conservative interpretation of the Λ CDM model then seems to only indicate that there is a missing piece in Einstein equation · G = κ T (m) , but does not allow to draw any stringent conclusion regarding the origin of the extra piece T (D) . It thus looks reasonable to consider the dark sector as a whole, as in (30), freed from the classical dark-matter+dark-energy interpretation.",
"pages": [
6,
7
]
},
{
"title": "IV. Seed for modified field equations",
"content": "We will now discuss the construction of an 'educated guess' for a minimal metric-affine extension of the dynamics of the Λ CDM model. Therefore, for the rest of this paper, we consider the geometric setup as given by a triple ( M, L , g ) as presented in Section II. In any metric-affine theory of gravity employing the metric and distortion tensor components as dynamical variables, the evolution equations for the metric can be cast in the form where we use the symbol · = do denote an equality holding in a cosmologicaly symmetric situation. With the standard Λ CDM-interpretation, one would have V µ := √ ρ (DM) u ν g µν . Similarly, the second relation in (34) implies where Λ is a priori a scalar function (generally, non-constant) and the minus sign is purely conventional. As already stated, our aim in this paper is to consider a minimal metric-affine extension of the Λ CDM model. This corresponds to the idea that the degrees of freedom describing dark matter and dark energy in a cosmologically symmetric situation are the only ones necessary to desribe the dynamics of these entities in all situations. In other words, stepping now aside from the cosmological symmetry context, under this 'minimality' hypothesis, no extra quantities or terms appear, hence we can simply replace · = by a strict equality. This gives with V = g µν V µ d x ν for a given vector field V = V µ ∂ µ obtained from the quantities defining the metric-affine geometry; that is V = V ( g , ∂ g , · · · , ∂ · · · ∂ g ; L , ∂ L , · · · , ∂ · · · ∂ L ) , and where Λ = Λ( g , ∂ g , · · · , ∂ · · · ∂ g ; L , ∂ L , · · · , ∂ · · · ∂ L ) is a scalar function. To summarize: staying as conservative as possible with respect to the dynamics of the Λ CDM model, one can consider as our 'educated guess' an equation of the form where Λ is a scalar function and V (accordingly, V ) is a 1-form (accordingly, a vector field) depending on g , L and their derivatives up to a finite order r . Yet, obviously, if we were to stop here, the prescription (39) would be incomplete, for at least two reasons: A solution to both the above problems is given by an algorithm introduced in [12], which we present in the next section.",
"pages": [
7,
8
]
},
{
"title": "V. Variational Bootstrapping of Λ CDM",
"content": "The variational bootstrapping method [12] allows one to determine a Lagrangian whose EulerLagrange equations are the closest to an 'educated guess' for a subset of the field equations. After briefly reviewing the algorithm, we apply it to (39).",
"pages": [
8
]
},
{
"title": "A. User's guide on Variational Bootstrapping",
"content": "Let us start by presenting a minimalist guide for the method of variational bootstrapping, focussing on computational aspects. For complete and mathematically rigorous details, we refer the reader to [12]. Assume we want to build a theory involving two distinct sets of dynamical variables, y A and z I both depending on the independent variables x µ . Generaly speaking, the dynamics of the system must be described by a system of PDEs of an a priori given order Assume now that we have some insight (an 'educated guess') regarding the form of the y A -equation only and that this insight is of the form with as many Y A -equations as y A -variables and as many Z I -equations as z I -variables. with as many equations as y A -variables. Variational bootstrapping then allows one to find a canonical correction term to be added to (41) in such a way that the corrected y A -equations are derived from a Lagrangian; moreover, this Lagrangian is uniquely determined up to boundary terms and to terms that do not involve y A . This will allow us, under certain circumstances (to be detailed below) to also 'bootstrap' the z I -equation. The Lagrangian density of the said Lagrangian is given 3 by where the y B variables and all their derivatives are scaled by the same u factor in the integrand and a sum over index A is understood. In cases where our guessed equations (41) can be obtained from a variational principle, any Lagrangian density producing (41) as y A -part of the Euler-Lagrange equations must be of the form (42) up to Lagrangian densities completely independent of the y A -variables ( i.e. which do not contribute to the y A -equations) and total divergences (which do not contribute to the field equations at all), see [12]. The form of the y A -independent densities is not constrained by the above procedure and should thus be found by different means 4 . But what if the equation (41) we start with is not variational ( i.e. what if it does not appear as the y A -equation of any Lagrangian) ? In that case, the idea behind variational bootstrapping is to still compute the Lagrangian density (42). This does two things for us: In that case, the y A -part of the Euler-Lagrange equations associated with (42) will differ from (41) but (42) will provide a canonical (and 'minimal') Lagrangian based on the guess (41). In the case of modified gravity theories based on GR, the field variables are usually the metric g and, generally speaking, another set of tensorial field variables, say ψ A . Our knowledges on metric-based approaches to gravity provides us with an 'educated guess' for the metric equation but not necessarilly for the dynamics of the ψ A fields. 6 One can then to apply variational bootstrapping with respect to the metric to obtain a Lagrangian density based on this guessed equation and then deduce the ψ A dynamics from it. In this case, assuming our guess is schematically of the form where T µν ( g , ∂ g ; ψ A , ∂ ψ A ) stands for the energy-momentum tensor associated with the extra fields ψ A , the procedure prescribes to compute 7 Here again, this procedure provides the 'best' Lagrangian density related to our initial guess (43) up to terms that would not give any contribution to the metric equation; i.e. Lagrangian densities built without using the metric g . Such Lagrangian densities are quite uncommon for physical theories as one usually uses the metric in the construction of the Lagrangian either to manipulate indices or to define a volume element which is used in the construction of the action. They are, nevertheless, not ruled out a priori by the above construction. In [12], we classified all possible natural Lagrangians independent of the metric in the context of a metricaffine setup in 4 dimensions. It turns out there is a limited amount of these which are obtained from specific 4 -forms constructed solely from the components of the distorsion tensor L and their first derivatives, see [12] for a complete classification.",
"pages": [
9,
10
]
},
{
"title": "B. General requirements",
"content": "Let us now apply the above algorithm to our 'educated guess' (41). To proceed with the construction of the Lagrangian density, we first need to refine the requirements on our field equations and on the quantities Λ and V appearing in (37)-(38). In what follows, we make the three following assumptions: We can then write Λ = Λ( g , L , · ∇ L ) without loss of generality. From equation (48), we get Using the relations from Appendix B, following the usual computation, we get that with | g | := det( g ) . Under our hypothesis 3, we can explicitly compute L Λ ; to this aim, denote In particular, this ensures that the guessed equations (41) are generally covariant and that the Euler-Lagrange equations of the Lagrangian obtained by variational bootstrapping are at most of second order in both g and L . were each Λ ( k ) is a homogeneous function of degree k in g and its derivative; and the same property holds for V .",
"pages": [
10,
11
]
},
{
"title": "C. Bootstrapping of Λ CDM",
"content": "We want to construct here the Lagrangian density obtained by variational bootstrapping of (39). According to (44) and the above hypotheses, we thus need to compute where where M,N ∈ Z and each Λ ( k ) is homogneous of degree k in g , that is: In this case, (51) becomes Following the same line, (49) can be written as Here again, according to hypothesis 3, we want to parametrize V µ ( g , L , · ∇ L ) as where I, J ∈ Z with the defining property that Using (56), (55) gives",
"pages": [
11,
12
]
},
{
"title": "D. Variationally completed field equations",
"content": "In order to determine the field equations associated to the Lagrangian density (46), we will separately compute the variation of each one of the terms in the summations of (54) and (58).",
"pages": [
12
]
},
{
"title": "1. Variation with respect to g",
"content": "Using the relations from Appendix B, as expected, the variation of (50) will give Let us now come to the variation of (54) with respect to the metric. To this purpose, it is sufficient to consider where /similarequal means equality up to boundary terms. where, to shorten the notation, we have defined ( ∂ Λ ( k ) ) α σβ ρ := ∂ Λ ( k ) ∂ ( · ∇ α L ρ σβ ) . Using the relations from Appendix B (in particular (B13) and (B15)), we get that where and, for a generic tensor field with components S ρµν , we have denoted: To compute the variation of L V with respect to the metric, we can reuse the results of the previous computation. Indeed, if we introduce we can write that Combining (59), (61) and (65), we get that the metric field equation for lagrangian (46), in the absence of matter source, is where the expression for T µν ( kl ) is obtained by performing the replacement Λ ( k ) → V ( kl ) in (62). where T ( k ) = T ( k ) ( µν ) d x µ ⊗ d x ν and T ( kl ) = T ( kl ) ( µν ) d x µ ⊗ d x ν .",
"pages": [
12,
13
]
},
{
"title": "2. Variation with respect to L",
"content": "As already mentioned, even though our initial guess was only related to the metric equation, variational bootstrapping allows us to completely determine the Lagrangian, up to boundary terms 9 . In particular the distortion field equation is also completely determined. Let us now compute these field equations. Since L EH is independent of L , only L Λ and L V will contribute. For L Λ , we find where Here again, for L V , one can reuse the previous result by introducing V ( kl ) from (64) to obtain that where the expression of ( Ψ ( kl ) ) ρ is obtained via the replacement Λ ( k ) → V ( kl ) in (68). This finally gives us a distortion field equation of the form σβ where Ψ ( k ) := ( Ψ ( k ) ) σβ ρ d x ρ ⊗ ∂ σ ⊗ ∂ β and Ψ ( kl ) := ( Ψ ( kl ) ) σβ ρ d x ρ ⊗ ∂ σ ⊗ ∂ β .",
"pages": [
13,
14
]
},
{
"title": "E. Analysis of the equations",
"content": "Before closing our discussion, let us comment on the level of generality of the obtained expressions and on how these relates to previously formulated models in metric-affine gravity. Because of the freedom of choice in Λ and /vector V , our Lagrangian (46) still encompasses many of the models present in the literature. For a state of the art on metric-affine theories of gravity, we redirect the reader to [9] while, for a review on historical developments, one can see [7, 14]. The most straightforward example is that of a metric-affine theory whose Lagrangian is the Ricci scalar R of the independent connection Γ , see e.g. [15, 16]. Because of the well-known geometric identity the choise where each term is ( -1) -homogeneous with respect to g , and /vector V ( g , L , · ∇ L ) = /vector 0 leads to the desired Lagrangian. A more interesting example is that of quadratic metric-affine theories [17, 18], i.e. theories whose Lagrangian contains terms quadratic in the curvature R , torsion T and/or non-metricity Q of Γ . In this case, one can easily see that appropriate choices of Λ and /vector V can generate all the terms under consideration in [17] or [18] except those who are quadratic in R -e.g. the term proportional to the coefficient z 4 in equation (3.1) of [17]. This comes from our hypothesis 1 about the form of Λ and /vector V and the fact that our Lagrangian contains only a term linear in · R . Following the same line, our procedure then also selects a restrictive class of quadratic gauge theories of gravity within a metric-affine framework as it constrain the dependence in R (Compare with chapter 9 of [7]). Finally, let us mention that, while we here considered theories based on a fully general metricaffine setup, many models studied in the literature start with some restriction on the connection M Γ ; typically Q ≡ 0 , as in Einstein-Cartan gravity or Poincaré gauge gravity, see respectively chapters 4 and 5 of [7]. Naturally, this would not affect the behavior of our procedure and the conclusion would remain the same 10 : Variational bootstrapping of the Λ CDM equations is compatible with quadratic metric-affine theories of gravity except for terms that are non-linear in the curvature tensor R . As a final comment, let us emphasize that the particularity of the method outlined here in this business of finding Lagrangians is that it directly relates the notion of 'minimality' to the variational structure of the theory. In other words, variational bootstrapping allows us to select metric affine models which are minimal extensions of the Λ CDM model in a variational sense and this, as we just saw, allows for a more restrictive selection of Lagrangians.",
"pages": [
14,
15
]
},
{
"title": "VI. Conclusion",
"content": "In this article, we studied the application of variational bootstrapping, first discussed in [12], to metric-affine theories of gravity. Through a carreful examination of the Λ CDM model of cosmology, we proposed a minimal modification of GR's field equation, based on that model, in the context of a metric-affine theory. Our hypotheses for that construction are explicited in section IV (see also section V B). Then, variational bootstrapping allowed us to construct a Lagrangian for the full theory, leading to 'corrected' metric equations (66) but also giving access to field equations for the distortion tensor (independent connection) (70). We showed in section V E how our model related to previously formulated ones. We saw that the notion of 'variationaly minimal extension' outlined in this paper gives a more restrictive criterion to select Lagrangians than the ones usually consiered in the literature when considering e.g. quadratic metric-affine theories. Naturally, this does not rule out the other existing models per se ! It should nevertheless be regarded as an interesting property since, in modified gravity in general (and in metric-affine gauge theories of gravity in particular), one usually struggles to single out a theory exihibiting a desired behaviour. This investigation demonstrates that variational bootstrapping is a powerful tool to construct theoretical models based on phenomenological requirements. The most remarkable feature of the method is that it allows to obtain a sensible Lagrangain for the theory under study by only guessing a part of the system's dynamics. An important direction to extend the present work is to investigate solutions to the field equations (66) ∧ (70). In particular, it would be interesting to see if a well-chosen form of the scalar function Λ and vector field V would allow, in a cosmological setup, to dynamically produce a behavior of the scale factor a ( t ) compatible with current universe's acceleration and/or with inflation. In addition to that, it would also be necessary to study the behavior of compact objects (black holes, neutron stars) in our theory to see how these differ from GR's ones and to which extend this remains compatible with modern observations. Another possible extension includes the application of our method to metric-affine theories using a tetrad and a spin connection as dynamical variables and the study of the conditions under which this approach is equivalent to the one presented here. Y = Met( M ) × M Tor( M ) , where Tor( M ) denotes the bundle of torsion i.e. the bunble of type (1 , 2) tensor fields antisymmetric over their two lower indices - instead of Y = Met( M ) × T 1 2 M as in this paper, see [12]. Nevertheless, since we are interested in variational bootstrapping with respect to the metric anyway, the computation and conclusions would perform exactly the same as in our case. We leave these and further questions for future works.",
"pages": [
15,
16
]
},
{
"title": "Acknowledgements",
"content": "L.D. acknowledges support from the Transilvania Fellowship Program for Postdoctoral Research/Young Researchers (September 2022) and from a Postdoctoral sojourn grant from Complexys institute from University of Mons (Belgium) and also acknowledges networking support by the COST Action CA18108 in the early developments of the present project. N.V. acknowledges networking support by the COST Action 21136. (oct, 2016) 019. .",
"pages": [
16
]
},
{
"title": "1. Symmetry",
"content": "As discussed in [19], when saying that a given tensor field T presents a symmetry, one means that our manifold is equipped with a certain group action smooth over its second variable, and that T is invariant under this group action in the sense that When G is a Lie group, it is usually more convenient to study this notion of symmetry in terms of the infinitesimal action of the symmetry group. Let us denote by g the Lie algebra of G and by X ξ the generating vector field for a given ξ ∈ g 11 , if a tensor T is symmetric in the sense of (A1) then where L denotes the Lie derivative. If G is connected, conditions (A1) and (A2) are equivalent and one can then derive the conditions for a tensor field T to present a given symmetry via (A2). It is the condition that we will use in the following. Following this definition of symmetry, a metric-affine spacetime will be symmetric under a given group G provided both the metric g and the distortion L are invariant under the action of G .",
"pages": [
17
]
},
{
"title": "2. Generators of cosmological symmetry",
"content": "Cosmological symmetry refers to the idea that, following the cosmological principle, no observer has a prefered place in the universe so that spacetime must be spatially homogeneous and isotropic. More precisely here, we then expect our setup to be invariant under both the 'spatial' rotation group SO (3) (providing isotropy, at least around one point) and a group representing 'spatial translations' (ensuring isotropy around actually any point and hence homogeneity). One thus obtains 6 generators of cosmological symmetry X i ( i = 1 , · · · , 6) . In local spherical coordinates ( x µ ) = ( t, r, θ, ϕ ) on M , SO (3) -generators are described as: as they satisfy the commutation relation of so (3) : where [ · , · ] denotes the commutator of two vector fields. Generators of (quasi-)translations are: where k ∈ { -1 , 0 , 1 } is the sign of the spatial curvature. When k = 0 (flat spatial geometry), X 1 , X 2 and X 3 correspond to the expression, in spherical coordinates, of the usual generators of spatial rotation around respectivelly the x -, y - and z -axis of an orthonornmal cartesian frame, while X 4 , X 5 and X 6 correspond to the usual generators of spatial translation along respectivelly the x -, y - and z -axis. According to (A2), a given tensor field T will be cosmologically symmetric provided it satisfies",
"pages": [
17,
18
]
},
{
"title": "B. Technical identities",
"content": "In this section, we report for completeness some important relations used in the computations of this paper.",
"pages": [
18
]
},
{
"title": "1. Rescaling of the metric",
"content": "To perform the variational completion/bootstrapping of our equations, we must know the behavior of the different terms in the equations under the homothety In this respect, one immediately obtains:",
"pages": [
18
]
},
{
"title": "2. Calculus of variations",
"content": "To compute the field equations associated to our Lagrangian, we need the following relations. a. Variations with respect to the metric g When computing variation with respect to the metric, one has b. Variations with respect to the distortion L When computing variation with respect to the distortion, we get:",
"pages": [
19
]
}
]
2024arXiv240612789N
https://arxiv.org/pdf/2406.12789.pdf
<document>
<section_header_level_1><location><page_1><loc_8><loc_86><loc_92><loc_89></location>Extracting overlapping gravitational-wave signals of Galactic compact binaries: a mini review</section_header_level_1>
<section_header_level_1><location><page_1><loc_42><loc_82><loc_58><loc_83></location>Rui Niu a,b , Wen Zhao a,b</section_header_level_1>
<text><location><page_1><loc_13><loc_79><loc_87><loc_81></location>a Department of Astronomy, University of Science and Technology of China, Chinese Academy of Sciences, , Hefei, 230026, Anhui, China b School of Astronomy and Space Sciences, University of Science and Technology of China, , Hefei, 230026, Anhui, China</text>
<section_header_level_1><location><page_1><loc_6><loc_72><loc_13><loc_73></location>Abstract</section_header_level_1>
<text><location><page_1><loc_6><loc_53><loc_94><loc_71></location>Gravitational wave (GW) observations have provided a novel tool to explore the universe. In the near future, space-borne detectors will further open the window of low-frequency GW band where abundant sources exist and invaluable information for astrophysics, cosmology, and fundamental physics can be revealed. However, there are various new challenges in data analyses for spaceborne detectors coming with the abundance of GW signals. For example, there are Galactic compact binaries (GCBs) with an overwhelming number that can produce continuous GW signals existing the entire mission time of detectors. The enormous overlapping GCB signals tangle and correlate with each other, and blend with other types of sources together in the observed data. Extracting source information from overlapping signals is one of the key problems for data analyses of space-borne detectors. In the paper, we present a review of currently available solutions for extracting overlapping GCB signals as thoroughly as possible aiming at promoting more interest in this question and inspiring further improvements. Current solutions can be roughly categorized by two classes, iterative subtraction and global fitting. There are diverse implementations of both strategies with enhancements focusing on di ff erent aspects. Meanwhile, the hybrid approach and the machine learning technique are also used in recent years. In the last, we also present an introduction of the stochastic foreground formed by unresolvable faint GCBs about its separation from extra-galactic backgrounds and its utility in exploring the properties of the Galaxy.</text>
<section_header_level_1><location><page_1><loc_6><loc_49><loc_17><loc_50></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_6><loc_35><loc_48><loc_47></location>Direct detections of gravitational waves (GWs) by groundbased detectors have ushered in a new era of GW astronomy [1-4]. Fruitful results have been accomplished based on detected events and researches with GWs are explosively growing in recent years. These detections have initiated a paradigm shift in studies of gravity, astrophysics, and cosmology. In the near future, space-borne detectors including LISA[5], Taiji[6], and TianQin[7] will open the new window of low-frequency GW band.</text>
<text><location><page_1><loc_6><loc_13><loc_48><loc_34></location>One significant di ff erence between current ground-based detectors and future space-borne detectors is that data given by current ground-based detectors are noise-dominant whereas signals are dominant for space-borne detectors. For current ground-based detectors, the compact binaries will quickly merge after entering the sensitive band, and the transient signals are sparsely distributed in time [1-4]. The situations of signal overlapping are expected to be rare. As reported in the work [8] where the probability of signal overlapping and how severe overlapping can induce significant bias in parameter estimation for second-generation ground-based detectors are thoroughly investigated, it is unlikely to observe overlapping signals by current existing detectors. Whereas, for future space-borne detectors, the enormous sources can persist in the sensitive band during the whole mission period [5-7]. Their GW signals can</text>
<text><location><page_1><loc_9><loc_9><loc_37><loc_10></location>Email address: wzhao7@ustc.edu.cn (Wen Zhao)</text>
<text><location><page_1><loc_52><loc_12><loc_94><loc_50></location>be heavily overlapping both in the time and frequency domain, which brings new challenges in extracting physical information from the data. Similar issues of signal overlapping also arise in third-generation ground-based detectors. Due to the exquisite sensitivities especially the improvement in low frequency, the visible duration and signal number are both significantly increased. It is unlikely to observe signals without overlapping for third-generation detectors, and the events with very close merger time which may su ff er from significant bias in parameter estimation can be common [9, 10]. There are extensive works about parameter inference techniques and impacts on researches of scientific problems for overlapping signals in thirdgeneration detectors. For examples, the traditional Bayesian inference framework with strategies of hierarchical subtraction and joint estimation for overlapping signals is investigated in the work [11], and a joint parameter estimation analyzing two overlapping signals simultaneously using normalizing flows is demonstrated in [12]. In the work [13], impacts of signal overlapping on testing General Relativity are elaborated. In works [14, 15], the impacts of unresolvable overlapping foreground and subtraction residuals induced by parameter estimation bias due to signal overlapping on detecting cosmological stochastic gravitational wave background with third-generation detectors are investigated. A review for signal overlapping problems focusing on third-generation ground-based detectors can be found in [16]. In the rest of this paper, we mainly focus on researches targeting space-borne detectors.</text>
<text><location><page_1><loc_53><loc_9><loc_94><loc_10></location>There are diverse types of sources presented in the space-</text>
<text><location><page_2><loc_6><loc_70><loc_48><loc_90></location>borne detector sensitive band [17], such as massive black hole binaries (MBHBs), extreme mass ratio inspirals (EMRIs), and Galactic compact binaries (GCBs), etc. GCBs are likely to be the most numerous type of GW sources detected in future space GW observation. Tens of millions of such sources may exist in the space-borne detector band. Among them, ten of thousands of bright ones are expected to be independently resolvable, while others will constitute a stochastic foreground as confusion noise [18-21]. Identification and subtraction of the resolvable GCBs are not only important for extracting other kinds of sources but also can be assistance in researches about stellar and Galactic astrophysics with the advantage that the information carried by GWs will not be a ff ected by crowded matters in the Galaxy [22-27].</text>
<text><location><page_2><loc_6><loc_52><loc_48><loc_70></location>Although all sources including MBHBs and EMRIs will be blended together in data-stream from space-borne detectors and people may ultimately need algorithms that can separate or simultaneously fit all di ff erent sources, GCBs may be the kind that is most heavily overlapping due to their overwhelming amount and the feature of long-living. Therefore, the discussions dedicated to GCBs may be the foundation and starting point of the ultimate full algorithms for separating and fitting all overlapping sources. In this paper, we will focus on GCBs and present a review of current solutions to tackle enormous overlapping GCB sources as comprehensively as possible, aiming at paving for further improvements in the problems of signal overlapping for data analyses of space-borne detectors.</text>
<text><location><page_2><loc_6><loc_21><loc_48><loc_52></location>Current solutions of extracting GCB signals mainly work around the simulated data sets including the earlier Mock LISA Data Challenges (MLDCs) [28-34], the recent resurrected LISA Data Challenges (LDCs) [35], and the Taiji Data Challenges (TDCs) [36]. These data sets are released to encourage various e ff orts for tackling unsolved problems in data analyses of space-borne detectors. These challenges involve data sets not only dedicated to GCBs, but also blended with various other possible sources, and the complexity has been increasing progressively. In the MLDC1, isolated GCBs and moderately overlapping signals with dozens of GCBs are concerned [2830]. The MLDC2 considers the full population with 26 million GCBs, and includes two data sets where the MLDC2.1 only contains signals of GCBs while MLDC2.2 blends signals of GCBs with MBHBs and EMRIs [31, 32]. The data set of GCBs in MLDC3 contains ∼ 60 million binaries, which is descended from MLDC2.1 with the improvement of realism by considering two di ff erent kinds of GCBs, binaries with two detached components and binaries with interacted components [33, 34]. The MLDC4 is the descendant of MLDC2.2, which blends different types of sources concerned separately in sub-challenges of MLDC3 into a single data stream [33].</text>
<text><location><page_2><loc_6><loc_12><loc_48><loc_21></location>The new LDCs 1 [35] are resumed in recent years with the new design [5] of LISA. Currently, three challenges have been released, LDC1 Radler , LDC2a Sangria , and LDC2b Spritz . The LDC1 considers various types of possible LISA sources separately like MLDC3, and includes six subchallenges where LDC1-4 is dedicated to GCBs and contains</text>
<text><location><page_2><loc_52><loc_76><loc_94><loc_90></location>26 million signals. The LDC2a is the updated challenge similar to MLDC4, which mixes di ff erent types of sources. The LDC2b has the improvement of considering the realistic instrumental and environmental noise, including gaps, glitches, and non-stationary noise. While the GW sources considered in LDC2b are relatively simple, where only MBHBs and verification GCBs are contained. The Taiji project also releases the data challenge [36] for the configuration of the Taiji detector, which includes sub-challenges concerning various types of GWsources separately and the mixture of these sources.</text>
<text><location><page_2><loc_52><loc_14><loc_94><loc_76></location>Numerous endeavors have been made in previous works to address the problem of signal overlapping. The ideas for this problem can be roughly categorized into two groups, the iterative subtraction strategy and the global fitting strategy. The iterative subtraction strategy searches the maximum likelihood estimation for a single source in each step, and this procedure will be performed iteratively with the remaining data after subtracting the identified signals [37-41]. In contrast, the global fitting strategy fits all signals simultaneously in the full Bayesian approach with sophisticated Markov chain Monte Carlo (MCMC) sampling algorithms to obtain the joint posterior distributions [42-49]. Both two strategies are adopted and implemented in diverse full-scale and end-to-end pipelines for extracting overlapping signals of GCBs. The iterative subtraction solutions can extract source candidates quickly but su ff er from the correlations among overlapping signals and the contamination of accumulating signal residuals left by each subtraction iteration. The global fitting solutions employ the full Bayesian approach to analyze all sources in a band simultaneously, which can better deal with the source correlations and residual contamination, but with the price of extremely massive demand for computational resources. The hybrid approach combining the maximum likelihood estimation and the Bayesian parameter estimation with MCMC is also proposed for combining the strengths of the two strategies while evading their drawbacks [50-52]. The flourishing machine learning algorithms provide new avenues to solve parameter estimation problems. After training with simulated data, the algorithms can generate posteriors directly without enormously evaluating the computationally expensive likelihood, which can complete the parameter estimation nearly in real time. The techniques have been successfully used in the parameter estimation for current ground-based detector data [53-58]. Machine learning techniques are also considered in the problem of overlapping GCBs, which are expected to be prospective powerful tools in future data analyses of space-borne detectors [59]. The resolvable sources are only a small fraction of the whole GCB population, the remaining faint GCBs can form a stochastic foreground. On the one hand, this Galactic foreground plays the role of noise deteriorating the signal-to-noise ratio (SNR) of individual sources and blending with stochastic signals from extra-galactic sources [60-69]. On the other hand, this foreground also contains information on the GCBpopulation which is correlated to properties of the Galaxy, and o ff ers a unique tool to study Galactic astrophysics [70-72].</text>
<text><location><page_2><loc_52><loc_9><loc_94><loc_13></location>The rest of this paper is organized as follows. In the next section, we briefly introduce the characterizations of GW signals from GCBs and how detectors respond to them. Section</text>
<text><location><page_3><loc_6><loc_68><loc_48><loc_90></location>3 is the main part of this paper where we present a detailed review of e ff orts on extracting overlapping GCB signals reported in recent years. After briefly summarizing earlier works around MLDC in Section 3.1, we first introduce a typical implementation of the iterative subtraction scheme and two variants focusing respectively on reducing inaccurate subtraction contamination and improving search e ffi ciency in Section 3.2. Then, we elucidate basic conceptions of the global fitting scheme and introduce two independent implementations of this strategy in Section 3.3. The hybrid Bayesian approach combining the maximum likelihood estimation and the MCMC sampling, and a preliminary attempt of utilizing machine learning techniques in solving the problem of overlapping GCB signals are introduced next in Section 3.4 and 3.5. The discussions about unresolvable GCBs are given in Section 4. The final summary is presented in Section. 5.</text>
<section_header_level_1><location><page_3><loc_6><loc_64><loc_27><loc_65></location>2. Galactic compact binaries</section_header_level_1>
<text><location><page_3><loc_6><loc_25><loc_48><loc_62></location>According to population models [18-21], there are tens of millions of compact binaries in the Galaxy that are slowly inspiraling towards each other with emissions of GWs in the mHz band and might be the type of most numerous sources observed by the space-borne detectors. An illustration of the sky distribution of a simulated GCB population taken from the training dataset of LDC2a [35] is presented in Figure 1. Among the population, tens of thousands of GCBs are expected to be individually resolvable through the four-year observation time of space-borne detectors. Current electromagnetic observations have identified about dozens of GCBs 2 [73], while hundreds are predicted to be detected by future observations [20]. These known sources are referred to as verification binaries and the loud ones are guaranteed to be detectable by space-borne GW detectors. The loud verification binaries are expected to be quickly identified by just weeks integration time of observation, thus o ff ering an important tool for functional tests and performance monitoring of the instruments [74]. Most GCBs are binary white dwarfs including detached binaries, as well as interacted binaries that have reached the Roche lobe overflow and started mass transfer [75]. While a small fraction of GCBs may involve with neutron stars or black holes. A summary of expectation of GCBs in the space-borne detector band is present in Table 1 which is cited from [17]. More details about population synthesis simulations or formation scenarios can be found in reviews [17, 76].</text>
<text><location><page_3><loc_6><loc_18><loc_48><loc_25></location>GCBs in the band of space-borne detectors are far from merger, and will stay in the inspiral phase of slow chirping during the entire observation period. GWs radiated from GCBs can be well described by the quasi-monochromatic waveform which takes the form of [77, 78]</text>
<formula><location><page_3><loc_17><loc_14><loc_48><loc_17></location>h + ( t ) = A (1 + cos 2 ι ) cos Φ ( t ) , h × ( t ) = -2 A cos ι sin Φ ( t ) . (1)</formula>
<text><location><page_3><loc_52><loc_76><loc_94><loc_90></location>Here A is the amplitude, ι is the inclination of binary orbit, and Φ is the GW phase which can be expressed as Φ ( t ) = ϕ 0 + 2 π R f ( t ' ) dt ' with the arbitrary initial phase ϕ 0 and the frequency evolution f ( t ). Since the frequency evolution is extremely slow for GCBs in the early inspiral stage, f ( t ) can be characterized by the central frequency f 0 and the first derivative ˙ f . The evolution of amplitude is usually neglected and A is considered as a constant. The e ff ects of mass transfer between binaries can be encoded into the parameter ˙ f [79]. The phase evolution can be written as</text>
<formula><location><page_3><loc_64><loc_74><loc_94><loc_75></location>Φ ( t ) = 2 π f 0 t + π ˙ f t 2 + ϕ 0 . (2)</formula>
<text><location><page_3><loc_52><loc_70><loc_94><loc_73></location>The total matric perturbation can be assembled by the sum of two polarizations as</text>
<formula><location><page_3><loc_66><loc_68><loc_94><loc_69></location>h TT = ϵ + h + + ϵ × h × , (3)</formula>
<text><location><page_3><loc_52><loc_61><loc_94><loc_67></location>where ϵ + and ϵ × denote the polarization tensors which depend on the source location ( β, λ ) and the polarization angle ψ . The response to GWs of a single laser link of space-borne detectors is given by [80-82]</text>
<formula><location><page_3><loc_52><loc_56><loc_94><loc_60></location>ysr = 1 2(1 -ˆ k · ˆ n sr ) ˆ n sr · [ h TT ( t -L -ˆ k · p s ) -h TT ( t -ˆ k · p r )] · ˆ n sr , (4)</formula>
<text><location><page_3><loc_52><loc_45><loc_94><loc_56></location>where the subscripts s , r denote the laser sender node and the receiver node respectively, ˆ k denotes the unit vector of the GW propagation direction, ˆ n sr denotes the unit vector along the direction of the link, p s and p r are the vectors of positions of the sender node and receiver node in the heliocentric coordinate. Together with the parameters describing GW strains of GCBs as shown in Equation 1 and 2, there are 8 parameters ( A , f 0 , ˙ f , ι, λ, β, ψ, ϕ 0) to fully characterize a signal from GCB.</text>
<text><location><page_3><loc_52><loc_16><loc_94><loc_45></location>Since the noise behavior can be conveniently characterized by the power spectral density (PSD) in the frequency domain if the noise is stationary and Gaussian, data analyses of GWs are often performed in the frequency domain. The method proposed in [78] can compute the Fourier transformation of the response quickly and accurately by heterodyning the response signal with a carrier wave of the frequency f 0. By multiplying with the carrier wave, the response signal can be decomposed into the slow part and the fast part. The Fourier transformation of the fast part can be obtained analytically. The slow part is transformed through fast Fourier transformation numerically, whereas the number of time samples is significantly reduced. For a signal extending within the band of [ f 0 , (1 + η ) f 0], the heterodyning operation can shift the required Nyquist frequency from 2(1 + η ) f 0 to 2 η f 0. Since the GCBs signals are quasimonochromatic with η ∼ 10 -6 [21, 83], the number of samples can be much less than the original time samples when numerically computing the fast Fourier transformation for the slow part. There is open source code GBGPU [77, 84] that can be used to obtain the response signals of GBCs in practice.</text>
<text><location><page_3><loc_52><loc_9><loc_94><loc_16></location>The space-borne detectors are unequal arm interferometers where the laser frequency noise will experience di ff erent time delays when traveling along di ff erent arms and cannot be canceled out by itself at the photodetector like ground-based detectors. In order to suppress the laser frequency noise which can</text>
<text><location><page_4><loc_6><loc_82><loc_48><loc_90></location>be stronger than GW signals a few orders, the technique called time delay interferometer (TDI) where the observables are created by time-shifting and combining single link responses has to be used to construct artificial equal arm interferometers [82, 85, 86]. The 1.5th generation (or 1st generation in some literatures) TDI observable X is given by [87]</text>
<formula><location><page_4><loc_10><loc_77><loc_48><loc_80></location>X 1 . 5 = y 13 + D 13 y 31 + D 13 D 31 y 12 + D 13 D 31 D 12 y 21 -y 12 -D 12 y 21 -D 12 D 21 y 13 -D 12 D 21 D 13 y 31 , (5)</formula>
<text><location><page_4><loc_6><loc_55><loc_48><loc_76></location>where Dij denotes the delay operator defined as Dijysr = ysr ( t -Lij ) and Lij is the arm-length between the node i and node j of the constellation. (In the construction of TDI combinations, the non-commutativity and variations of arm-length may be taken into consideration, whereas when actually computing the detector responses, the approximation of rigid constellation where all arm-length is equal and constant is usually adopted [83, 87, 88]. Therefore, we follow this convention and use Lij to denote the arm-length when writing TDI combinations, while using L in other places.) The other two observables Y , Z can be obtained similarly by cyclic permutation of indices. The 2nd generation TDI incorporates that the delay operators are non-commutative for forward and inverse delay of a link due to the rotation of the constellation by compensating more virtual loops in two arms as [87]</text>
<formula><location><page_4><loc_6><loc_42><loc_50><loc_53></location>X 2 . 0 = y 13 + D 13 y 31 + D 13 D 31 y 12 + D 13 D 31 D 12 y 21 + D 13 D 31 D 12 D 21 y 12 + D 13 D 31 D 12 D 21 D 12 y 21 + D 13 D 31 D 12 D 21 D 12 D 21 y 13 + D 13 D 31 D 12 D 21 D 12 D 21 D 13 y 31 -y 12 -D 12 y 21 -D 12 D 21 y 13 -D 12 D 21 D 13 y 31 -D 12 D 21 D 13 D 31 y 13 -D 12 D 21 D 13 D 31 D 13 y 31 -D 12 D 21 D 13 D 31 D 13 D 31 y 12 -D 12 D 21 D 13 D 31 D 13 D 31 D 12 y 21 . (6)</formula>
<text><location><page_4><loc_6><loc_34><loc_48><loc_42></location>The constructions of TDI observable are not unique and have abundant forms. Di ff erent constructions can have di ff erent sensitivities to GWs of di ff erent polarizations and propagation directions [89, 90]. The X , Y , Z channels are correlated, and the independent TDI channels can be obtained through the combination of</text>
<formula><location><page_4><loc_20><loc_24><loc_48><loc_33></location>A = 1 √ 2 ( Z -X ) , E = 1 √ 6 ( X -2 Y + Z ) , T = 1 √ 3 ( X + Y + Z ) . (7)</formula>
<text><location><page_4><loc_6><loc_9><loc_48><loc_23></location>In the low-frequency limit where f < (1 / 2 π L ) with L denoting the arm-length of the detector, signals in the A channel are mainly contributed by the plus polarization, the E channel can be approximated to the cross polarization, and the T channel is approximated to the breath polarization but which is absent in General Relativity. Thus, in data analyses with low-frequency approximation, only A and E channels are considered usually [51, 91, 92]. We illustrate the time domain signal in the A channel of a typical GCB in Figure 2 and the frequency domain signals of the GCB population in Figure 3.</text>
<table>
<location><page_4><loc_57><loc_77><loc_88><loc_88></location>
<caption>Table 1: Expected numbers for the total and detectable sources of various GCB types in the Galaxy. This table is cited from [17].</caption>
</table>
<figure>
<location><page_4><loc_52><loc_53><loc_91><loc_65></location>
<caption>Figure 1: Illustration for sky distribution of GCBs . The simulated GCB catalog is from LDC2a [35]. The SNRs are calculated with 1-year observation and the sources are displayed in the Galactic coordinate.</caption>
</figure>
<figure>
<location><page_4><loc_53><loc_22><loc_92><loc_39></location>
<caption>Figure 2: Time domain signal of a typical GCB source. In the mHz band, the GCBs are at the early inspiral stage of their orbital evolution where the radiated GWs are quasi-monochromatic. However, the motion of detectors can endow signals with annual modulation which depends on the sky locations of the sources. The signal shown here is the A channel of 1.5th generation TDI combination for the responses of LISA to a verification source AM CVn [73]. The signal is generated by lisaanalysistools [93] and fastlisaresponse [94].</caption>
</figure>
<figure>
<location><page_5><loc_8><loc_73><loc_47><loc_90></location>
<caption>Figure 3: Signals of the simulated GCB population in frequency domain. The GCB catalog is from the training dataset of LDC2a [35]. The displayed signals are responses in the A channel of 1.5th generation TDI combination of LISA generated by gbgpu [77, 95]. The orange lines denote the sources that have SNR ≥ 8 for 1-year observation and are expected to be resolvable individually. The total contribution of remaining faint sources is denoted by gray lines which can form a stochastic foreground as confusion noise. The noise of the instrument is indicated by the blue line, which is given by the noise model SciRDv1 [87, 96].</caption>
</figure>
<section_header_level_1><location><page_5><loc_6><loc_57><loc_43><loc_58></location>3. Solutions for extracting overlapping GCB signals</section_header_level_1>
<text><location><page_5><loc_6><loc_46><loc_48><loc_55></location>This section provides a detailed review of currently available solutions for extracting overlapping GCB signals. We mainly focus on recent works, early e ff orts with MLDCs are briefly summarized in Section 3.1, and diverse innovations and new implementations in recent years are introduced in the subsequent sections. A summary of solutions mentioned here and corresponding references are presented in Table 2.</text>
<section_header_level_1><location><page_5><loc_6><loc_43><loc_21><loc_44></location>3.1. Early researches</section_header_level_1>
<text><location><page_5><loc_6><loc_31><loc_48><loc_42></location>Problems of extracting overlapping GCB signals have been discussed for more than two decades. The iterative subtraction strategy was proposed at the beginning of this century [97], where the brightest sources are iteratively identified and subtracted. The global fitting strategy can be traced back to the early researches [42, 43] which employs the trans-dimensional MCMC sampling algorithm to simultaneously infer the source number, joint posteriors of source parameters, and noise levels.</text>
<text><location><page_5><loc_6><loc_9><loc_48><loc_30></location>Diverse solutions for overlapping GCB signals are explosively presented working around the MLDCs. For example, the blocked annealed metropolis (BAM) algorithm developed in [98-100] is a quasi-Bayesian approach with a F -statistic likelihood and a customized MCMC sampling strategy. The subsequent work [101] extends the BAM algorithm through introducing parallel tempering, the reversible jump MCMC (RJMCMC), and the fast-slow decomposition waveform model, which is the basis of the recent full-scale global fitting pipeline GBMCMC introduced in Section 3.3. The search methods using F -statistic and various optimal algorithms [102-107] are also the foundation of diverse iterative subtraction solutions in recent years. Additionally, various intelligent ideas besides the above two schemes are also widely discussed, including a tomographic approach [108, 109], genetic searches [110], the</text>
<text><location><page_5><loc_52><loc_85><loc_94><loc_90></location>two-stage strategy [111], and a sophisticated MCMC walking strategy [112, 113], etc. More details about early e ff orts can be found in reports of each MLDC [30, 32-34], and a comprehensive review [114].</text>
<text><location><page_5><loc_52><loc_80><loc_94><loc_84></location>Based on early e ff orts, various enhancements, new methods and implementations have emerged with the resurrected LDCs in recent years, which will be detailed in the following.</text>
<section_header_level_1><location><page_5><loc_52><loc_76><loc_94><loc_79></location>3.2. Iterative substraction using F -statistic and particle swarm optimization</section_header_level_1>
<text><location><page_5><loc_52><loc_57><loc_94><loc_76></location>One of widely considered strategies for extracting overlapping signals is iterative subtraction where the search algorithm for one individual source will be run to identify the brightest source, then the identified signal will be subtracted from the data, and this procedure will be iteratively executed with the residual data until some stopping criteria are satisfied. The idea of iterative identification and subtraction for overlapping signals in mHz GW band is proposed early in [97], and is implemented extensively in previous works. In this review, we present an introduction of a typical iterative subtraction scheme developed in recent years [37-39], which employs F -statistic to construct likelihood and uses particle swarm optimization (PSO) to search the optimal estimations.</text>
<text><location><page_5><loc_52><loc_40><loc_94><loc_57></location>The framework developed in [37] is referred to as Galactic Binary Separation by Iterative Extraction and Validation using Extended Range (GBSIEVER). In the iterative subtraction scheme, the single brightest source needs to be identified in each iteration. GBSIEVER implements this through maximum likelihood estimation where the likelihood is constructed by F -statistic. The data from detectors are composed of noise n ( t ) and GW signals h ( t ), which can be expressed as d ( t ) = n ( t ) + h ( t ). Assuming the noise n ( t ) is Gaussian and stationary, the probability of a realization of d ( t ) given a specific GWsignal h ( t ) described by a set of parameters θ can be written by</text>
<formula><location><page_5><loc_56><loc_36><loc_94><loc_39></location>ln p ( d ( t ) | h ( t , θ )) = -1 2 ⟨ d ( t ) -h ( t , θ ) , d ( t ) -h ( t , θ ) ⟩ , (8)</formula>
<text><location><page_5><loc_52><loc_33><loc_94><loc_35></location>where the angle brackets denote the noise weighted inner product defined as</text>
<formula><location><page_5><loc_60><loc_29><loc_94><loc_32></location>⟨ a ( t ) , b ( t ) ⟩ = 4Re Z ∞ 0 ˜ a ( f ) ˜ b ∗ ( f ) Sn ( f ) d f . (9)</formula>
<text><location><page_5><loc_52><loc_18><loc_94><loc_28></location>Here, ˜ a ( f ) and ˜ b ( f ) is the Fourier transformation of time series a ( t ) and b ( t ), Sn ( f ) denotes the PSD of the noise. To identify the signal in data through maximum likelihood estimation, one needs to explore the parameter space of θ to find the parameters ˆ θ MLE where the probability (Equation 8) has the maximum value. Drop the term independent with the source parameters, the likelihood can be written by</text>
<formula><location><page_5><loc_65><loc_14><loc_94><loc_17></location>ln Λ = ⟨ d , h ⟩ -1 2 ⟨ h , h ⟩ . (10)</formula>
<text><location><page_5><loc_52><loc_9><loc_94><loc_13></location>For multiple independent measurements of the same signal, such as the independent TDI channels or di ff erent detectors, the final log-likelihood is the sum of log-likelihoods for each</text>
<text><location><page_6><loc_6><loc_87><loc_48><loc_90></location>independent measurement. The estimation of source parameters ˆ θ MLE is given by</text>
<formula><location><page_6><loc_22><loc_83><loc_48><loc_86></location>∂ ln Λ ∂ θ GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> θ = ˆ θ MLE = 0 . (11)</formula>
<text><location><page_6><loc_6><loc_80><loc_48><loc_82></location>Here, we use the hat to denote parameters with the maximum likelihood value.</text>
<text><location><page_6><loc_6><loc_65><loc_48><loc_79></location>Exploring the high-dimension parameter space to solve Equation 11 is computationally expensive. In the framework GBSIEVER, the reduced likelihood is constructed through F -statistic, where only four intrinsic parameters are searched and the extrinsic parameters can be obtained analytically. The full set of parameters to describe GCB signals is separated into intrinsic parameters { f 0 , ˙ f , λ, β } and extrinsic parameters {A , ι, ψ, ϕ 0 } . The GW signal can be decomposed by a linear combination of templates only depending on intrinsic parameters as</text>
<formula><location><page_6><loc_20><loc_64><loc_48><loc_65></location>h ( t , θ ) = Σ i aiXi ( t , κ ) , (12)</formula>
<text><location><page_6><loc_6><loc_56><loc_48><loc_63></location>where κ denotes a set of intrinsic parameters, ai is a reparametrization of extrinsic parameters, and basis templates Xi ( t , κ ) can be defined by the GW signals at 4 sets of specific extrinsic parameters. Using this decomposition, we can construct the vector U which has the element of</text>
<formula><location><page_6><loc_21><loc_54><loc_48><loc_55></location>Ui = ⟨ d ( t ) , Xi ( t , κ ) ⟩ , (13)</formula>
<text><location><page_6><loc_6><loc_51><loc_35><loc_52></location>and the matrix W which has the element of</text>
<formula><location><page_6><loc_19><loc_48><loc_48><loc_49></location>Wij = ⟨ Xi ( t , κ ) , Xj ( t , κ ) ⟩ . (14)</formula>
<text><location><page_6><loc_6><loc_44><loc_48><loc_47></location>Using vector U and matrix W together with the vector A consisting of ai , the likelihood Equation 10 can be rewritten as</text>
<formula><location><page_6><loc_20><loc_40><loc_48><loc_43></location>ln Λ = AU -1 2 AWA T . (15)</formula>
<text><location><page_6><loc_6><loc_37><loc_48><loc_39></location>Maximizing the original likelihood ln Λ can be substituted by maximizing the F -statistic defined as</text>
<formula><location><page_6><loc_21><loc_34><loc_48><loc_36></location>F ( κ ) = U T W -1 U , (16)</formula>
<text><location><page_6><loc_6><loc_29><loc_48><loc_33></location>from which one can obtain the maximum likelihood estimation for intrinsic parameters ˆ κ . The estimation of ˆ a can be obtained analytically through</text>
<formula><location><page_6><loc_23><loc_26><loc_48><loc_28></location>A T = W -1 U . (17)</formula>
<text><location><page_6><loc_6><loc_22><loc_48><loc_25></location>The full set of parameters with the maximum likelihood can be recovered through ˆ κ and ˆ a .</text>
<text><location><page_6><loc_6><loc_9><loc_48><loc_22></location>The remaining question is how to e ffi ciently explore the intrinsic parameter space to maximize F ( κ ). In the framework of GBSIEVER, PSO is performed to accomplish this task. PSO is proposed by Kennedy and Eberhart [115] and widely used in various optimal problems. PSO is inspired by the social behavior of organisms like bird flocks or fish schools. It utilizes a population (referred to as the swarm) of candidate solutions (referred to as particles) which are moved by the guidance of own experience of each particle and the collective knowledge</text>
<text><location><page_6><loc_52><loc_85><loc_94><loc_90></location>of the entire swarm to find the optimal solution in the parameter space. The interaction among particles endows the PSO method with more capability for global searching and against trapping in local optima.</text>
<text><location><page_6><loc_52><loc_80><loc_94><loc_84></location>Initially, the particles are randomly drawn in the parameter space and assigned with random velocities. The subsequent movement of particles is guided by [40]</text>
<formula><location><page_6><loc_58><loc_76><loc_94><loc_79></location>v t + 1 i = ω v t i + c 1 r 1 GLYPH<16> P t i -x t i GLYPH<17> + c 2 r 2 GLYPH<16> G t -x t i GLYPH<17> , x t + 1 i = x t i + v t + 1 i . (18)</formula>
<text><location><page_6><loc_52><loc_59><loc_94><loc_75></location>In the above equations, x t i and v t i denote the position and velocity of i -th particle at the iteration step of t . P t i is the best position of the i -th particle found in previous iterations, which represents the experience of individual particles, and G t is the best position found by the entire swarm in previous iterations, which represents the collective knowledge of the swarm. The terms corresponding to P t i and G t are usually called the personal term and the social term respectively in literature. ω , c 1, and c 2 are constant coe ffi cients that need to be tuned according to the specific problem through experimental runs or empirical knowledge.</text>
<text><location><page_6><loc_52><loc_31><loc_94><loc_59></location>The process of optimization can be viewed as two phases, the exploration phase where the particles explore the parameter space expansively and quickly to find better locations, and the exploitation phase where the particles have converged within a promising region and updates of better position will be relatively slow. ω is called the inertia weight controlling the balance of exploration and exploitation in the optimization process. A lower inertia weight favors exploitation and allows particles to quickly converge toward promising regions. Conversely, a higher inertia weight can help particles resist attractions of previous best positions of personal and social terms, explore the parameter space more expansively, and potentially escape from local optima. The coe ffi cients c 1 and c 2 are called acceleration coe ffi cients controlling the balance between the personal term and social term of the particles in their movement. The setting of these coe ffi cients has significant impacts on the performance and e ffi ciency of PSO. A typical configuration in the context of GWdata analyses can be found in [116], which is also adopted by GBSIEVER. After su ffi cient iterations, the particles are expected to converge to the optimal solution.</text>
<text><location><page_6><loc_52><loc_11><loc_94><loc_30></location>Some practical techniques are used in GBSIEVER to improve search e ffi ciency and eliminate false candidates. As introduced in Section 2, GCB signals have the characterization of narrowband. Therefore, in practice, it is usually to divide the whole frequency band into small bins. Independent analyses are performed in di ff erent frequency bins, and the edge e ff ects are carefully addressed in the meanwhile. Furthermore, another practical technique in GBSIEVER is a special downsampling operation which can reduce the number of samples thus relieve the computational cost in likelihood evaluation. To eliminate spurious sources, GBSIEVER employs a cross-validation procedure to compare the extracted candidates in two independent runs. Only the candidates with similar estimation results in two independent runs are considered to be genuine.</text>
<text><location><page_6><loc_53><loc_9><loc_94><loc_10></location>The performance of GBSIEVER is verified with the dataset</text>
<text><location><page_7><loc_1><loc_66><loc_2><loc_67></location>10</text>
<text><location><page_7><loc_2><loc_66><loc_3><loc_67></location>-2</text>
<figure>
<location><page_7><loc_7><loc_65><loc_47><loc_90></location>
<caption>Figure 4: The residual after subtracting identified signals given by GBSIEVER. Utilizing the iterative subtraction method with the F -statistic likelihood and the PSO algorithm, GBSIEVER can successfully identify O (10 4 ) GCB signals. This figure shows the residual of the A channel of the TDI combination of simulated LISA dataset LDC1-4 which is used to verify the performance of GBSIEVER. This figure is cited from [37].</caption>
</figure>
<text><location><page_7><loc_6><loc_40><loc_48><loc_54></location>of LDC1-4 and modified MLDC3.1, where it is demonstrated that O (10 4 ) GCBs can be successfully identified [37]. The residual after subtracting identified signals is shown in Figure 4. In the subsequent work [38], the framework GBSIEVER is extended by incorporating the network of detectors. Furthermore, in the work[39], the fact that the data collection is gradually incremental in real observations is considered. The search results obtained in ahead short period observations can be used to reduce the parameter space in following searches, which can help enhance the e ffi ciency of the algorithm.</text>
<text><location><page_7><loc_6><loc_31><loc_48><loc_40></location>Although the framework GBSIEVER has been demonstrated to be capable of extracting a large population of overlapping GCB signals, there are various aspects which can be further improved. In the following, other two implementations of iterative subtraction focusing on reliving inaccurate contamination and enhancing search e ffi ciency will be introduced.</text>
<section_header_level_1><location><page_7><loc_6><loc_28><loc_47><loc_29></location>3.2.1. Local maxima particle swarm optimization algorithm</section_header_level_1>
<text><location><page_7><loc_6><loc_9><loc_48><loc_27></location>In the iterative subtraction scheme, the inevitable errors of parameter estimation in each iteration will leave residual signals that can continuously accumulate and contaminate remaining data as noise. The work [40] improves the previous GBSIEVER framework and develops the new approach of local maxima particle swarm optimization algorithm with a special strategy of creating voids referred to as LMPSO-CV for dealing with inaccurate subtraction contamination, especially for the low SNR sources. LMPSO-CV starts with the remaining data assuming all sources with SNR > 15 have been identified and subtructed. The algorithm aims at identifying all local maxima of F -statistic in the parameter space, and the source parameters will be extracted from these local maxima.</text>
<text><location><page_7><loc_52><loc_86><loc_94><loc_90></location>In LMPSO-CV, to identify the local maxima, the configuration of PSO is adjusted by setting c 1 = ω = 0. The equations guiding the movement of particles turn into the form of [40]</text>
<formula><location><page_7><loc_65><loc_81><loc_94><loc_85></location>v t + 1 i = c 2 r 2 GLYPH<16> G t -x t i GLYPH<17> , x t + 1 i = x t i + v t + 1 i . (19)</formula>
<text><location><page_7><loc_52><loc_71><loc_94><loc_80></location>As mentioned in Section 3.2, a lower inertia weight favors the exploitation where particles tend to move within a local promising region, and emphasizing the global term will guide particles to move toward the same position. It can be expected that the setting of entirely dropping the inertia weight and the individual term will extremely improve the convergence towards local maxima.</text>
<text><location><page_7><loc_52><loc_63><loc_94><loc_70></location>Once a local maximum is identified, to avoid the search algorithm picking the same or too close local maxima multiple times, a void in the parameter space will be created and excluded in subsequent searches. The void is modeled by a spheroid as</text>
<formula><location><page_7><loc_67><loc_61><loc_94><loc_63></location>x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 , (20)</formula>
<text><location><page_7><loc_52><loc_52><loc_94><loc_60></location>where x , y , z correspond to the source parameter f 0, β , λ . The size of the spheroid is determined by the frequency resolution and the degeneracy of β and λ through experimental computations. In the process of searching for local maxima, if a particle moves into voids, it will be placed at a new random position outside voids in the next iteration to avoid redundancy.</text>
<text><location><page_7><loc_52><loc_28><loc_94><loc_51></location>The search of local maxima and creation of voids will be performed repeatedly until the termination rules designed for identifying all local maxima beyond the threshold are satisfied. The source parameters are extracted among the identified voids. As can be seen from Figure 5, there are enormous local maxima in likelihood surfaces corresponding to false signals which need to be eliminated when extracting real candidates. The extraction consists of three parts. The first part aims at removing local maxima induced by degeneracy noise of individual signals. From the observation of experimental evaluation of the likelihood for an individual signal as shown in the top row of Figure 5, the local maxima corresponding to false signals have smaller likelihood values than the real signal. Therefore, the local maxima are sorted in descending order, and the extraction starts with the local maxima of the highest F -statistic value which is assumed to be a real signal. Then, the quantities of ∆ F defined as [40]</text>
<formula><location><page_7><loc_58><loc_23><loc_94><loc_27></location>∆ F ( θ i ) = F ( θ i , d ( t )) - F θ i , i -1 X m = 1 h ( t , θ m ) (21)</formula>
<text><location><page_7><loc_52><loc_9><loc_94><loc_22></location>are computed for subsequent local maxima. In the above equation, the first term F ( θ i , d ( t )) is the F -statistic value of the i -th local maximum with parameter θ i with respect to the raw data, which is same to the F -statistic value used in the search process. The second term F GLYPH<16> θ i , P i -1 m = 1 h ( t , θ m ) GLYPH<17> is the F -statistic value of the i -th local maximum with respect to the summation of candidate signals previous extracted, which characterizes the degeneracy with previous extracted signals. If the F -statistic value of the i -th local maximum is mainly contributed by the</text>
<text><location><page_8><loc_6><loc_83><loc_48><loc_90></location>degeneracy with previous extracted signals. The quantity of ∆ F is expected to be small. Conversely, large ∆ F indicates the local maximum is likely to be a real signal. The local maxima that satisfy the criterion of ∆ F > 53 will be retained and passed into the second part for further removing false signals.</text>
<text><location><page_8><loc_6><loc_63><loc_49><loc_83></location>The second part of signal extraction and false signal elimination is based on the astrophysical properties of GCBs. The second part requires a new independent search of local maxima which is similar to the process discussed above except using a di ff erent range for ˙ f . In the first search, the search range for ˙ f is [ -10 -16 , 10 -15 ] when f ≤ 4 mHz and [ -10 -14 , 10 -13 ] when f > 4 mHz. Whereas, in the second search, the range of ˙ f is determined by the mass parameters of binaries. Since the majority of GCBs are consisted of binary white dwarfs which have the mass ranging from 0 . 1 to 1 . 4 M ⊙ . In the second search, the values of ˙ f are constrained within the range corresponding to the mass range of white dwarfs. After two independent searches, the candidate signals need to be compared through the correlation coe ffi cient defined as</text>
<formula><location><page_8><loc_17><loc_58><loc_48><loc_62></location>R ( θ , θ ' ) = C ( θ , θ ' ) [ C ( θ , θ ) C ( θ ' , θ ' )] 1 2 , (22)</formula>
<text><location><page_8><loc_6><loc_45><loc_48><loc_57></location>where C ( θ , θ ' ) = ⟨ h ( θ , h ( θ ' ) ⟩ . This correlation coe ffi cient quantifies how similarity between two signals identified in two independent searches. Only candidates with enough large R are considered as reliable sources and are retained for further processing. Another characterization of GCBs is the concentration in the Galactic disk. Therefore, one can simply remove candidates with Galactic latitude outside of [ -0 . 5rad , 0 . 5rad] to enhance the reliability.</text>
<text><location><page_8><loc_6><loc_32><loc_48><loc_45></location>The third part of signal extraction focuses on the false candidates induced by overlapping signal degeneracy noise. In the experimental computation of the likelihood surface for the situation of overlapping signals as shown in the middle and bottom rows of Figure 5, it is found that the F -statistic values of local maxima corresponding to false candidates can be even larger than real signals. To remove false candidates of this kind, a similar process of the first part needs to be performed again but in ascending order of local maxima.</text>
<text><location><page_8><loc_6><loc_9><loc_48><loc_32></location>The process of LMPSO-CV can be summarized in two steps. In the first step, the stochastic optimization algorithm is tuned to specializing in finding local maxima. A void in parameter space will be created to avoid redundant search once a local maximum is identified. The second step extracts signals and eliminates false candidates from the identified local maxima, which consists of three parts: removing false candidates induced by degeneracy of individual signals, applying the astrophysical properties of GCBs, and removing false candidates induced by degeneracy of overlapping signals. LMPSO-CV mainly focuses on the sources with SNR between 7 and 15. Since the LMPSOCV approach searches all local maxima simultaneously and filters out false candidates afterward, rather than iteratively performing search and subtraction for a single signal, it can better deal with inaccurate subtraction contamination in the traditional iteration subtraction scheme.</text>
<section_header_level_1><location><page_8><loc_52><loc_88><loc_94><loc_90></location>3.2.2. Combination of coarse template search and fine PSO search</section_header_level_1>
<text><location><page_8><loc_52><loc_79><loc_94><loc_87></location>Another implementation of the iterative subtraction scheme focuses on improving search e ffi ciency by the strategy of combination with the coarse search based on templates for quickly identifying rough parameters of candidates and the fine search based on F -statistic and PSO for detailed exploring small promising regions [41].</text>
<text><location><page_8><loc_52><loc_69><loc_94><loc_78></location>The strategy is composed of two steps. In the first step, candidates are searched by a template bank. A stochastic template bank [117] is constructed, where the places of template parameters are randomly drawn in the parameter space and additional pruning operation is performed to remove templates with too small separation. The number of templates can be approximated by [41]</text>
<formula><location><page_8><loc_60><loc_65><loc_94><loc_68></location>N ( η, m ∗ , S n ) ≈ V S n Vn ln 1 1 -η ! m -n / 2 ∗ , (23)</formula>
<text><location><page_8><loc_52><loc_33><loc_94><loc_63></location>where n denotes the dimension of the parameter space S n , η denotes the desired level of coverage confidence, m ∗ is the mismatch criterion, Vn is the volume of unit sphere in n -dimension, and V S n is the volume of the whole parameter space. The construction of the template bank starts with randomly generating N waveforms. Then, the KDtree (k-dimensional tree) algorithm [118, 119] is used to find the nearest neighbor for each waveform. If the mismatch between two waveforms is less than m ∗ , one of them will be removed. The above two procedures are performed repeatedly until the total number of templates gets to stable. The F -statistic likelihood is also used in the coarse search, the threshold for coarse search is set to F th coarse = 15 which corresponds to SNR ≈ 5 . 1. The templates whose F -statistic values exceed the threshold are recorded as candidates. Due to the annual orbital motion of detectors, modulation is imposed on the quasi-monochromatic signals of GCBs. The modulation can induce sidebands around the central frequency f 0 of GCB signals. A clustering operation will be performed to gather candidates with close central frequency. Among all candidates within the extended Doppler window, only the sources with the largest F -statistic values are retained for fine search in the next step.</text>
<text><location><page_8><loc_52><loc_25><loc_94><loc_32></location>In the second step, the parameters of candidates identified by the template search are accurately estimated using PSO, which is similar to the procedure discussed before, while the search ranges for the central frequency are tuned according to results given by coarse search.</text>
<text><location><page_8><loc_52><loc_15><loc_94><loc_25></location>The performance of this methodology is demonstrated with MLDC3.1. It is shown that O (10 4 ) sources can be successfully identified, and nearly 90 percent of them are well aligned with injected signals. This method performs a coarse template search in the first step, which can provide priori information guiding the PSO search thus enhancing the e ffi ciency of the fine parameter estimation.</text>
<section_header_level_1><location><page_8><loc_52><loc_13><loc_93><loc_14></location>3.3. Global fitting with trans-dimensional Bayesian inference</section_header_level_1>
<text><location><page_8><loc_52><loc_9><loc_94><loc_12></location>The iterative subtraction method has advantages of rapidity and e ffi ciency but also su ff ers from problems of correla-</text>
<figure>
<location><page_9><loc_11><loc_61><loc_48><loc_75></location>
<caption>Figure 5: The likelihood surfaces for cases of an individual signal and overlapping signals. These figures show the value of the F -statistic likelihood in the parameter space of ( f 0 , λ, β ). The red crosses denote the injected values, and the colorbars indicate the F -statistic values. The top row is for the case of an individual signal, the middle and bottom rows are for the cases of two and three overlapping signals. For the overlapping signals with enough small frequency intervals, there will be local maxima in likelihood surfaces corresponding to false candidates induced by correlations among overlapping signals, which may have higher likelihood values than the real candidates. These figures are cited from [40].</caption>
</figure>
<figure>
<location><page_9><loc_53><loc_61><loc_90><loc_75></location>
</figure>
<figure>
<location><page_9><loc_11><loc_46><loc_48><loc_60></location>
</figure>
<figure>
<location><page_9><loc_11><loc_31><loc_47><loc_45></location>
</figure>
<text><location><page_9><loc_54><loc_44><loc_55><loc_45></location>1.5</text>
<text><location><page_9><loc_55><loc_42><loc_55><loc_43></location>1</text>
<text><location><page_9><loc_54><loc_40><loc_55><loc_41></location>0.5</text>
<text><location><page_9><loc_55><loc_38><loc_55><loc_39></location>0</text>
<text><location><page_9><loc_54><loc_36><loc_55><loc_37></location>-0.5</text>
<text><location><page_9><loc_55><loc_35><loc_55><loc_35></location>-1</text>
<text><location><page_9><loc_54><loc_33><loc_55><loc_33></location>-1.5</text>
<text><location><page_9><loc_54><loc_32><loc_75><loc_32></location>2.0903 2.0904 2.0905 2.0906 2.0907 2.0908 2.0909</text>
<text><location><page_9><loc_76><loc_32><loc_78><loc_32></location>2.091</text>
<text><location><page_9><loc_78><loc_32><loc_87><loc_32></location>2.0911 2.0912 2.0913</text>
<text><location><page_9><loc_69><loc_31><loc_74><loc_32></location>Frequency</text>
<text><location><page_9><loc_86><loc_31><loc_87><loc_31></location>10</text>
<text><location><page_9><loc_87><loc_31><loc_87><loc_32></location>-3</text>
<text><location><page_9><loc_54><loc_59><loc_55><loc_60></location>1.5</text>
<text><location><page_9><loc_88><loc_59><loc_90><loc_59></location>9000</text>
<text><location><page_9><loc_88><loc_57><loc_90><loc_58></location>8000</text>
<text><location><page_9><loc_88><loc_56><loc_90><loc_57></location>7000</text>
<text><location><page_9><loc_88><loc_55><loc_90><loc_55></location>6000</text>
<text><location><page_9><loc_88><loc_54><loc_90><loc_54></location>5000</text>
<text><location><page_9><loc_88><loc_52><loc_90><loc_53></location>4000</text>
<text><location><page_9><loc_88><loc_51><loc_90><loc_52></location>3000</text>
<text><location><page_9><loc_88><loc_50><loc_90><loc_50></location>2000</text>
<text><location><page_9><loc_88><loc_49><loc_90><loc_49></location>1000</text>
<text><location><page_9><loc_88><loc_44><loc_90><loc_44></location>9000</text>
<text><location><page_9><loc_88><loc_42><loc_90><loc_43></location>8000</text>
<text><location><page_9><loc_88><loc_41><loc_90><loc_42></location>7000</text>
<text><location><page_9><loc_88><loc_40><loc_90><loc_40></location>6000</text>
<text><location><page_9><loc_88><loc_39><loc_90><loc_39></location>5000</text>
<text><location><page_9><loc_88><loc_37><loc_90><loc_38></location>4000</text>
<text><location><page_9><loc_88><loc_36><loc_90><loc_37></location>3000</text>
<text><location><page_9><loc_88><loc_35><loc_90><loc_35></location>2000</text>
<text><location><page_9><loc_88><loc_34><loc_90><loc_34></location>1000</text>
<text><location><page_9><loc_53><loc_53><loc_54><loc_55></location>Latitude</text>
<text><location><page_9><loc_53><loc_38><loc_54><loc_40></location>Latitude</text>
<text><location><page_9><loc_55><loc_57><loc_55><loc_58></location>1</text>
<text><location><page_9><loc_54><loc_55><loc_55><loc_56></location>0.5</text>
<text><location><page_9><loc_55><loc_53><loc_55><loc_54></location>0</text>
<text><location><page_9><loc_54><loc_52><loc_55><loc_52></location>-0.5</text>
<text><location><page_9><loc_55><loc_50><loc_55><loc_50></location>-1</text>
<text><location><page_9><loc_54><loc_48><loc_55><loc_48></location>-1.5</text>
<text><location><page_9><loc_54><loc_47><loc_75><loc_47></location>2.0903 2.0904 2.0905 2.0906 2.0907 2.0908 2.0909</text>
<text><location><page_9><loc_76><loc_47><loc_78><loc_47></location>2.091</text>
<text><location><page_9><loc_78><loc_47><loc_87><loc_47></location>2.0911 2.0912 2.0913</text>
<text><location><page_9><loc_69><loc_46><loc_74><loc_47></location>Frequency</text>
<text><location><page_9><loc_86><loc_46><loc_87><loc_47></location>10</text>
<text><location><page_9><loc_87><loc_46><loc_87><loc_47></location>-3</text>
<text><location><page_10><loc_6><loc_70><loc_48><loc_90></location>tions among overlapping signals and contamination of inaccurate subtraction. In each step of iterative subtraction, errors in parameter estimation are unavoidable, which can yield signal residuals left behind the subtraction. The residuals will contaminate remaining data and accumulate along with iterations. Furthermore, in each iteration only the parameter values with maximum likelihood are used in subtraction, the uncertainty of parameter fitting will propagate alone iterations. The uncertainty of parameter estimation for identified candidates is di ffi -cult to ascertain. Additionally, GCB signals are heavily overlapping in both time and frequency, which can induce high correlations among signals. The correlations may lead to bias in the parameter estimation of each iteration, and this will further intensify the problem of inaccurate subtraction contamination.</text>
<text><location><page_10><loc_6><loc_56><loc_48><loc_70></location>To alleviate these problems, a commonly used operation is dividing the whole frequency band into small bins and performing analyses independently in di ff erent bins while carefully addressing signals residing at edges, which can reduce the number of iterations and reduce the accumulation of inaccurate subtraction contamination. Besides, as introduced in Section 3.2.1, the method of modifying the particle movement rule in PSO aiming at identifying all local maxima on the likelihood surface simultaneously is also an e ff ective method to address inaccurate subtraction contamination for the low SNR sources.</text>
<text><location><page_10><loc_6><loc_39><loc_48><loc_56></location>The global fitting is another route that can e ff ectively deal with the problems of overlapping signal correlations. In the iterative subtraction scheme, only parameters for one source are estimated in each step. In contrast, parameters for all sources together with the source number are estimated simultaneously in the global fitting scheme. The global fitting with full Bayesian parameter estimation can obtain joint posterior distributions rather than just the maximum likelihood estimation, thus the uncertainty of fitting can be read out from posteriors straightforwardly. Since all sources are fitted simultaneously, the correlations of overlapping signals are taken into account and can be reflected in joint posteriors of multiple overlapping signals.</text>
<text><location><page_10><loc_6><loc_27><loc_48><loc_38></location>As mentioned in 3.1, the strategy of global fitting and the Bayesian approach have been proposed and implemented early in various works focusing on MLDCs [42, 43, 98-101]. Further improvements like incorporating the reality that the data collection is time evolving, various sophisticated methods for increasing the e ffi ciency of MCMC sampling, and new implementations of RJMCMC have been continuously presented in recent years [44-46, 48, 49].</text>
<section_header_level_1><location><page_10><loc_6><loc_24><loc_18><loc_25></location>3.3.1. RJMCMC</section_header_level_1>
<text><location><page_10><loc_6><loc_9><loc_48><loc_23></location>For the Bayesian inference of global fitting, One of the key di ff erences with the Bayesian parameter estimation commonly used in data analyses of current ground-based detectors is that the number of sources is uncertain and has to be inferred from the given data. A single GW source signal h ( t , θ ) in the likelihood shown in Equation 8 turns to a summation of multiple signals P k h ( t , θ k ) where the number of source is uncertain. The sampling algorithms to solve inference problems with uncertain dimensionality are referred to as trans-dimensional MCMC or RJMCMC.</text>
<text><location><page_10><loc_52><loc_86><loc_94><loc_90></location>In the parameter estimation problem of the Bayesian framework, the estimation results are represented by posteriors through the Bayes theorem</text>
<formula><location><page_10><loc_63><loc_82><loc_94><loc_85></location>p ( θ n | d ( t )) = π ( θ n ) p ( d ( t ) | θ n ) Z , (24)</formula>
<text><location><page_10><loc_52><loc_61><loc_94><loc_81></location>where d ( t ) denotes observed data, θ n represents parameters with the dimensionality of n , π ( θ n ) is the prior representing the knowledge before observations, p ( d ( t ) | θ n ) is the likelihood representing the probability of noise realization that can just present the time series d ( t ) observed on detectors when adding the GW signals described by parameters θ n and depending on the models of noise behavior and GW signals, Z is the normalization factor called as evidence. Since the dimension of parameters θ n for describing GW signals is usually high, it is impractical to compute posterior on a grid of the parameter space. Stochastic sampling algorithms are usually employed to perform random walking in the parameter space, and use the density distributions of random samples to approximate the probability distributions of posteriors.</text>
<text><location><page_10><loc_52><loc_51><loc_94><loc_61></location>The Metropolis-Hasting MCMC (MHMCMC) [120, 121] is one of the well known and widely used algorithms to perform sampling for fixed dimensional problems. After setting the initial state of a random walker, the subsequent movements are guided by the rules introduced below. If current state is θ i n , a new state can be drawn from a proposal distribution q ( θ i + 1 n | θ i n ). Then the acceptance probability of the new state is evaluated by</text>
<formula><location><page_10><loc_61><loc_47><loc_94><loc_50></location>α = min " 1 , p ( θ i + 1 n | d ) q ( θ i n | θ i + 1 n ) p ( θ i n | d ) q ( θ i + 1 n | θ i n ) # . (25)</formula>
<text><location><page_10><loc_52><loc_33><loc_94><loc_45></location>The probability of whether actually moving from current state θ i n to new proposed state θ i + 1 n is determined by α . It can be proven that after su ffi cient walking, the equilibrium distribution of the walker will converge to the target posterior distribution p ( θ n | d ). Although, in practical problems, the MHMCMC usually needs to be modified in various aspects to increase sampling e ffi ciency or avoid bias, the above description presents the most basic conception of the MCMC sampling algorithm for fixed dimensional problems.</text>
<text><location><page_10><loc_52><loc_24><loc_94><loc_33></location>In problems of uncertain dimensionality, to allow the walker to jump between di ff erent parameter spaces, the RJMCMC algorithm draws new proposals by a di ff erent procedure [122, 123]. Firstly, a random vector u with the dimension of r is generated from a chosen probability distribution g ( u ). Then, the proposal state is generated through a transformation</text>
<formula><location><page_10><loc_68><loc_21><loc_94><loc_23></location>θ i + 1 n ' = f ( θ i n , u ) . (26)</formula>
<text><location><page_10><loc_52><loc_19><loc_87><loc_20></location>The corresponding inverse transformation is given by</text>
<formula><location><page_10><loc_67><loc_16><loc_94><loc_18></location>θ i n = f -1 ( θ i + 1 n ' , u ' ) , (27)</formula>
<text><location><page_10><loc_52><loc_9><loc_94><loc_15></location>where u ' is a vector with size r ' and satisfying n + r = n ' + r ' . The only requirement for the transformation is that f and f -1 need to be di ff erentiable. The dimensions of parameter space n and n ' do not need to be the same, and even the parameterization</text>
<text><location><page_11><loc_6><loc_87><loc_48><loc_90></location>associated with θ i n and θ i + 1 n ' can be di ff erent. The acceptance ratio for the new proposed state is given by</text>
<formula><location><page_11><loc_17><loc_83><loc_48><loc_86></location>α = min " 1 , p ( θ i + 1 n ' | d ) g ' ( u ' ) p ( θ i n | d ) g ( u ) | J | # . (28)</formula>
<text><location><page_11><loc_6><loc_81><loc_32><loc_82></location>The term J is the Jacobian defined by</text>
<formula><location><page_11><loc_22><loc_76><loc_48><loc_79></location>J = GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> ∂ ( θ i + 1 n ' , u ' ) ∂ ( θ i n , u ) GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> (29)</formula>
<text><location><page_11><loc_6><loc_54><loc_48><loc_75></location>to account for the change of volume element under the transformation of parameter spaces. In general, the Jacobian can be di ffi cult to compute. However, in the situation of global fitting for GCB signals, where it only involves adding or removing GCB signals with the same set of parameters, the Jacobian can be easily computed by the ratio of prior volumes of parameter spaces before and after the transformation. The walker of RJMCMC can freely explore the entire possible parameter space including not only parameter spaces of individual sources but also parameter spaces for all possible source numbers. The estimation of source number can be naturally obtained through comparing the iteration numbers of the walker lingered in the di ff erent parameter spaces, and the Bayesian factor for the assumptions of di ff erent source numbers can be given by the ratio of the iteration numbers in di ff erent parameter spaces.</text>
<text><location><page_11><loc_6><loc_18><loc_48><loc_53></location>Although the movement rule of RJMCMC can be explicitly presented as above, the implementation of the RJMCMC algorithm in practice to solve the problem of overlapping GCB signals is very challenging. On the one hand, since all sources are simultaneously fitted, the parameter spaces are extremely high dimensional. Even the whole frequency band is usually divided into small bins and di ff erent bins can be analyzed independently after carefully addressing edge e ff ects. The source number in a single small bin can still be considerable. For example, the default setting of the prior range for the source number in each bin is [0 , 30] in the work [44]. As mentioned in Section 2, a GCB signal is characterized by 8 parameters. The maximal dimension of parameter space can be 240. And the posterior can have the feature of multimodality, which requires the sampler has the ability to deal with complicated likelihood surfaces in high-dimensional parameter spaces. On the other hand, the uncertainty of source number requires the sampler to jump between di ff erent parameter spaces, which further extremely expands the space needed to explore. Furthermore, since the likelihood in vast regions may be very small. Only if the proposal is enough close to the true values, it can likely be accepted. Generating proposals entirely uniformly may lead extremely low acceptance rate, especially for between-model proposals. One need to design good strategies for random walking to increase the sampling e ffi ciency.</text>
<section_header_level_1><location><page_11><loc_6><loc_15><loc_26><loc_16></location>3.3.2. Global fitting piplines</section_header_level_1>
<text><location><page_11><loc_6><loc_9><loc_48><loc_15></location>Currently, available full-scale and end-to-end global fitting pipelines for GCBs include GBMCMC [44-46] and Eryn [48, 49]. Various intelligence methods are utilized to overcome di ffi culties in the global fitting with RJMCMC. For example,</text>
<text><location><page_11><loc_52><loc_85><loc_94><loc_90></location>the method of parallel tempering [124-126] is used in both GBMCMC and Eryn. In parallel tempering, multiple walkers are randomly moving in parallel at di ff erent temperatures T . The likelihood is modified as</text>
<formula><location><page_11><loc_65><loc_82><loc_94><loc_83></location>pT ( θ | d ) ∝ π ( θ ) p ( d | θ ) 1 / T , (30)</formula>
<text><location><page_11><loc_52><loc_65><loc_94><loc_81></location>where π ( θ ) is the prior and p ( d | θ ) is the original likelihood function. When T = 1, pT ( θ | d ) returns to the target posterior distribution. When T → ∞ , pT ( θ | d ) approaches to the prior distribution. Higher temperatures correspond to more exploration where walkers can escape from local optima more easily, while lower temperatures correspond to more exploitation where walkers can sample the small promising regions in more detail. During the random walking, state exchanges are attempted between walkers at di ff erent temperatures. The probability of acceptance of exchange proposals is determined by the ratio</text>
<formula><location><page_11><loc_63><loc_62><loc_94><loc_65></location>α = min " 1 , pT i ( θ i ) pT j ( θ j ) pT i ( θ j ) pT j ( θ i ) # . (31)</formula>
<text><location><page_11><loc_52><loc_43><loc_94><loc_62></location>The detailed balance is maintained under this state exchange between walkers, which ensures that the equilibrium distribution of walkers can converge to the target posterior distributions. The temperature ladder needs to be chosen for maximizing the information flow among walkers at di ff erent temperatures. Ideally, one may expect an equal acceptance ratio between every pair of walkers with neighboring temperatures. The GBMCMC and Eryn adopt the scheme presented in [124] to set the temperature ladder, where the temperature space is dynamically adjusted in the initial burn-in phase to obtain a stable configuration for the rest of random walking. The parallel tempering mechanism can help samplers e ffi ciently explore the complicated likelihood surface with high multimodality.</text>
<text><location><page_11><loc_52><loc_22><loc_94><loc_43></location>The strategies for drawing proposals play a crucial role in the sampling e ffi ciency. Customized proposal distributions are developed in GBMCMC to enhance the sampling e ffi ciency. Ideal proposal distributions would be identical to posterior distributions or likelihood functions. The likelihood of multiple overlapping signals consists of three parts, the correlation between each signal and noise, the correlation of noise itself, and the cross-correlations among overlapping signals. As discussed in [44], high correlations among overlapping signals are relatively rare. Therefore, one can enhance proposal distributions focusing on individual signals. Although the cross-correlations of overlapping signals are neglected when designing the proposal distributions, the sampler still explores the joint parameter space of multiple overlapping signals which incorporates cross-correlations of overlapping signals.</text>
<text><location><page_11><loc_52><loc_9><loc_94><loc_22></location>The design of proposal distribution in GBMCMC utilizes the F -statistic, the feature of multimodality due to orbital motion of detectors, and posteriors from former epochs of observation. The method of F -statistic is widely used in various implementations of iterative subtraction as presented in previous sections, it can also be used here to construct the proposal distribution for a single source. GBMCMC uses the F -statistic to build proposals for parameters ( f 0, λ , β ) which are precomputed on a grid with spaces determined by estimation of the Fisher matrix. The</text>
<text><location><page_12><loc_6><loc_58><loc_48><loc_90></location>values of F -statistic can be approximated to the original likelihood values, thus the F -statistic proposals are expected to have a high accepted rate. As mentioned in Section 3.2.2, the modulation induced by orbital motion can cause sidebands around the central frequency f 0 of quasi-monochromatic GCB signals. The likelihood surface have the feature of multiple modes around f 0. GBMCMC designs a dedicated proposal to update the frequency parameter by shifting f 0 with modulation frequency f m as f 0 → f 0 + nf m, where f m = 1 / year. The multimodal proposal can address this known degeneracy and help to improve the convergence of the sampler. In reality, the data collection is gradually incremental. It is impractical to wait until after the finish of entire observation to analyze the obtained data. Data analyses must be performed accompanying with data accumulation. Therefore, one can utilize posteriors obtained from former periods of observation to construct posterior-based proposal distributions in parameter estimation with more observed data, which can significantly improve the sampler convergence. The results of GBMCMC with simulated GCB dataset are demonstrated in Figure 6 where the posteriors of sky location of identified GCBs are plotted. It can be seen that with the incremental data, the GCBs can be better localized and the structure of the Galaxy is clearer indicated.</text>
<text><location><page_12><loc_6><loc_21><loc_48><loc_57></location>Another independent implementation of RJMCMC referred to as Eryn [48, 49] which adopts di ff erent sophisticated mechanisms in stochastic sampling to overcome poor convergence in RJMCMC. Eryn is based on the ensemble sampler emcee [127] which uses multiple interacting walkers exploring the parameter space simultaneously with the so-called stretch-move proposal to enhance the e ffi ciency and convergence of the sampling. Whereas, in order to address the trans-dimensional movement and the heavily multimodal likelihood surface of overlapping GCBs, Eryn extends the stretch-move proposal in origin emcee sampler to the group proposal which sets a stationary group of walkers and makes the state updates likely to happen within one same mode in the likelihood surface. Furthermore, Eryn build e ffi cient proposals through a data-driven approach, where proposals can be drawn from a fitted distribution constructed by posteriors obtained in burn-in runs with residual data after subtracting bright sources. To overcome the problem of low acceptance rate, Eryn implements two mechanisms called delayed rejection [128, 129] and multiple try metropolis [130-133]. However, due to the high computational requirement of delayed rejection, although this mechanism is implemented in Eryn, it is not currently used in solving the problem of global fitting for GCBs. Another important feature of Eryn is the utilization of GPU which reduces the wall-time to perform the whole analysis by parallel computing on contemporary computational hardware.</text>
<text><location><page_12><loc_6><loc_9><loc_48><loc_20></location>The ultimate goal of global fitting is simultaneously analyzing all kinds of sources contained in data including MBHBs, EMRIs, and even unmodeled signals, etc. In the work [45] and [49], GBMCMC and Eryn are incorporated within global fitting pipelines that can handle blended data of MBHBs and GCBs. Global fitting for multiple source types can be realized by the blocked Gibbs scheme considering that correlations among different source types are relatively small. Parameter estimations</text>
<text><location><page_12><loc_52><loc_80><loc_94><loc_90></location>for di ff erent source types can be implemented in separate modules, and these modules are assembled by a wheel update strategy where one can perform updates in one module conditioning on fixed other modules and all modules are updated periodically. Figure 7 shows the results given by a global fitting pipeline incorporating GCBs and MBHBs where the fitting of GCBs is accomplished by Eryn.</text>
<text><location><page_12><loc_52><loc_55><loc_94><loc_80></location>In a short summary, fitting overlapping GCB signals through the full-Bayesian approach can be realized by RJMCMC. Practical implementations of RJMCMC have to overcome various di ffi culties such as the heavy multimodality of likelihood surface in vast parameter space, low acceptance rate of trans-dimensional proposals, and poor convergence of random walker, etc. which requires elaborate MCMC algorithms. Prototype global fitting pipelines for GCBs have been accomplished by GBMCMC and Eryn which can well address simulated data and are successfully incorporated within the global fitting pipelines for blended data of MBHBs and GCBs. The global fitting with RJMCMC is a full Bayesian approach and can provide joint posteriors of overlapping signals, which can well account for correlations among overlapping GCBs and avoid inaccurate subtraction contamination in the iterative subtraction scheme. Whereas the full Bayesian approach requires massive computational cost and complicated MCMC algorithms which are di ffi cult to implement.</text>
<section_header_level_1><location><page_12><loc_52><loc_52><loc_73><loc_53></location>3.4. Hybrid Bayesian approach</section_header_level_1>
<text><location><page_12><loc_52><loc_26><loc_94><loc_52></location>The iterative subtraction strategy o ff ers solutions with high e ffi ciency where the computation burden is modest and analysis can be finished in a relatively short time. Whereas, in each iteration, only the maximum likelihood estimation for a single source is extracted, the correlation of overlapping signals can not be well accounted for. The inaccuracy of estimation in each iteration will contaminate the remaining data. And uncertainty analyses for the identified signals are di ffi cult. The full Bayesian approach with RJMCMC where overlapping signals are fitted simultaneously can e ff ectively overcome these di ffi culties, but with the cost of extremely massive demand on computational resources. In the works [50-52], a hybrid approach where the maximum likelihood estimation is performed first to find the approximate values of signal parameters, and then the MCMC sampling is used to obtain posteriors of signal parameters is proposed aiming at combining the advantages of the maximum likelihood estimation and the Bayesian parameter estimation while evading their drawbacks.</text>
<text><location><page_12><loc_52><loc_9><loc_94><loc_26></location>Similar to the solutions of iterative subtraction introduced previously, the hybrid Bayesian approach also needs to identify the maxima of likelihood in the first step. However, the methods used for searching maxima are di ff erent. In this hybrid Bayesian approach, the o ff -the-shelve algorithms of the di ff erential evolution (DE) [134, 135] and the sequential least squared programming (SLSP) [136] implemented in the scipy library [137] are adopted for searching the maxima in the likelihood surface. The DE method is used to identify the candidates at first through the iterative subtraction scheme, where only one signal is fitted each time the fitting is repeatedly performed with the remaining data after subtracting the best-fit signal. Then, in</text>
<figure>
<location><page_13><loc_9><loc_61><loc_91><loc_90></location>
<caption>Figure 6: Posteriors of sky location of identified GCBs in the Galactic coordinate given by GBMCMC . The subplots correspond to the observations of 1.5 (top left), 3 (top right), 6 (bottom left), and 12 (bottom right) months. These figures are cited from [46]</caption>
</figure>
<text><location><page_13><loc_50><loc_56><loc_50><loc_57></location>.</text>
<figure>
<location><page_13><loc_10><loc_16><loc_87><loc_51></location>
<caption>Figure 7: Results given by a global fitting pipeline incorporating GCBs and MBHBs. The top and bottom rows show the results of training and hidden datasets in the LDC2a respectively. The left and middle columns present the injection data, the residual after subtracting identified signals, and the fitting of noise curves. The identified signals are illustrated in the right column. This figure is cited from [49].</caption>
</figure>
<text><location><page_14><loc_6><loc_85><loc_48><loc_90></location>order to address the correlation of overlapping signals, using all found candidates as the start, the global optimizations are performed by the SLSP method to search the maximum likelihood in the joint parameter space of all candidates.</text>
<text><location><page_14><loc_6><loc_53><loc_48><loc_84></location>The DE algorithm is a simple and e ffi cient method for optimization problems that searches the optima by manipulating a population of potential solutions similar to PSO introduced in Section 3.2. The DE algorithm is initialized by a population of candidate solutions drawn randomly within the search space. Then, in following iterations, the algorithm generates new potential solutions by combining and mutating existing individuals in the population. There are various strategies [135] for generating new solutions and updating populations, which may suit di ff erent problems and need to be determined through experimental runs. The populations will be continuously updated by replacing individuals with better fitness and eventually converge to the optima. The SLSQ method is a gradient-based optimization method suits for problems with smooth and continuously di ff erentiable likelihoods. After the iterative subtraction search with the DE algorithm, the SLSQ method starts a global search with the initial values given by the DE algorithm, and iteratively steps towards a better solution which incorporates correlations of overlapping signals by the line search in a direction found through derivatives of the likelihood surface. As examples, the true and recovered signals on a small frequency band are illustrated in Figure 8.</text>
<text><location><page_14><loc_6><loc_42><loc_48><loc_53></location>In the second step, The MCMC sampling algorithm is performed to obtain the posterior distribution for each identified signal. The MCMC algorithm for individual sources is similar to methods widely used for parameter estimation in data analysis of ground-based detectors [138, 139], except that the data used in likelihood evaluation are the remains after the subtraction of identified signals in the first step excluding the signals to be analyzed, which can be expressed by</text>
<formula><location><page_14><loc_14><loc_38><loc_48><loc_41></location>d ( i ) posterior = d original -X ˆ θ ∈ ˆ θ recovered h ( ˆ θ ) + h ( ˆ θ i ) , (32)</formula>
<text><location><page_14><loc_6><loc_22><loc_48><loc_36></location>where d ( i ) posterior is the data used in likelihood evaluation for the source i , and ˆ θ i denotes the parameters of maximum likelihood estimation obtained in the first step for the source i . One drawback is that the above MCMC sampling method for individual signals cannot account for the correlation of overlapping signals, which may lead to overoptimistic posterior distributions. To address this, some level of residual signals are intentionally left in the data used for estimating the noise characterization. The partial residual data used to obtain the noise PSD can be expressed as</text>
<formula><location><page_14><loc_15><loc_17><loc_48><loc_21></location>d partial = d original -s partial X ˆ θ ∈ ˆ θ recovered h ( ˆ θ ) , (33)</formula>
<text><location><page_14><loc_6><loc_12><loc_48><loc_16></location>where s partial is a factor from the the range of [0 , 1] which has to be determined experientially and is set to s partial = 0 . 7 in the work [52].</text>
<text><location><page_14><loc_6><loc_9><loc_48><loc_12></location>Various methods are used to improve the e ffi ciency of MCMCsampling in the second step, for example, constraining</text>
<text><location><page_14><loc_52><loc_73><loc_94><loc_90></location>the search space only within the promising region [52], or using Gaussian progress regression to model the likelihood [51]. The posterior distributions are typically concentrated within small regions of parameter space. To relieve the computational burden, one can only explore the reduced parameter space based on the maximum likelihood estimation obtained in the first step. The Fisher matrix is used to determine the boundaries of the reduced parameter space. The Fisher matrix is widely used in forecast works [140, 141], which is an approximation of the Bayesian method under the assumption of high SNR and can provide estimations for parameter measurement uncertainty. The Fisher matrix is given by</text>
<formula><location><page_14><loc_65><loc_69><loc_94><loc_72></location>Fij = * ∂ h ( ˆ θ ) ∂θ i , ∂ h ( ˆ θ ) ∂θ j + , (34)</formula>
<text><location><page_14><loc_52><loc_52><loc_94><loc_68></location>where ˆ θ is the maximum likelihood estimation obtained in the first step, and the angle brackets are the inner product defined as Equation 9. The inverse of the Fisher matrix presents the estimation of the covariance matrix of parameter measurement. The boundaries of reduced parameter space for the MCMC sampling are determined by the variance of parameters estimated through the Fisher matrix. Typically, 3σ regions are su ffi cient, while the practical settings need to be adjusted according to the features of posterior distributions in experimental runs, and the tolerance of computational burden or the desired coverage of the parameter space.</text>
<text><location><page_14><loc_52><loc_21><loc_94><loc_52></location>Additionally, in the procedure of MCMC sampling, the likelihood has to be evaluated a huge number of times. The computational cost for likelihood evaluation is one of the main barriers to MCMC sampling. In the work [52], the contemporary computational hardware is used to perform the likelihood evaluation. The likelihood can be computed on GPU in massively parallel, which can significantly reduce the wall-time required for finishing the MCMC sampling [77, 94]. On the other hand, as shown in the work [51] the likelihood can be modeled by the Gaussian process regression, where the likelihood is modeled by a joint Gaussian distribution whose mean vector and covariance matrix are determined by training samples [142]. The computation of Gaussian distributions can be much faster than the likelihood defined through the inner product in Equation 8. In the work [51], 1000 random samples are first drawn for training the Gaussian process regression model, and 500 samples are used for verification. After this, The likelihood is replaced by the Gaussian process regression model in following MCMC sampling, which can reduce the computational cost of the entire sampling process. The training and evaluating of the Gaussian process regression model are implemented through the scikit-learn package [143].</text>
<text><location><page_14><loc_52><loc_9><loc_94><loc_20></location>The hybrid Bayesian approach combines the maximum likelihood estimation and the MCMC sampling. A search of the iterative subtraction scheme is first performed to identify potential candidates which are used as the start in the subsequent global optimization in the joint parameter space for all overlapping signals. Then the MCMC sampling algorithm is performed for each identified signal to obtain the posterior distributions. This hybrid Bayesian approach uses the strategy of iter-</text>
<text><location><page_15><loc_6><loc_73><loc_48><loc_90></location>ative subtraction to determine the source number and the point estimation for source parameters, which can avoid the computationally expensive trans-dimensional MCMC sampling. Meanwhile, after the iterative subtraction search, the source parameters are again globally optimized around the identified candidates, which can rectify errors induced by inaccurate subtraction contamination and correlations of overlapping signals. However, since the MCMC sampling is performed for individual signals, only the marginalized posteriors can be obtained for each source. Incorporating additional uncertainty induced by signal overlapping requires tuning the algorithm configuration empirically.</text>
<section_header_level_1><location><page_15><loc_6><loc_70><loc_38><loc_71></location>3.5. Utilization of machine learning techniques</section_header_level_1>
<text><location><page_15><loc_6><loc_51><loc_48><loc_69></location>The development of machine learning techniques has led to revolutions in many fields including the data analysis of GWs. Various machine learning algorithms have been successfully used in tasks of signal identification and classification [144-146], parameter estimation [53-58], noise reduction [147], waveform modeling [148, 149], etc. Comprehensive reviews can be found in [150, 151]. Utilizing machine learning techniques in data analyses of space-borne detectors is also actively discussed [152-156]. A preliminary attempt of using the machine learning method to extract overlapping GCB signals is presented in the work [59] where normalizing flows are used to build proposals for improving the convergence of MCMC sampling and o ff er a new method for sharing analysis results.</text>
<text><location><page_15><loc_6><loc_39><loc_48><loc_50></location>Normalizing flows are a class of machine learning algorithms that can model complicated distributions and are widely used in density estimation problems [157-160]. Normalizing flows model a probability distribution through an invertible and di ff erentiable transformation f : X → Y that can map the simple and tractable base probability distribution P X ( x ) to the complicated target probability distribution P Y ( y ). The target distribution can be given by</text>
<formula><location><page_15><loc_16><loc_36><loc_48><loc_38></location>P Y ( y ) = P X GLYPH<16> f -1 ( y ) GLYPH<17> | det J f -1 ( y ) | (35)</formula>
<text><location><page_15><loc_6><loc_9><loc_48><loc_35></location>where | det J f -1 ( y ) | is the determinant of the Jacobian of the transformation f -1 accounting for the change of volume elements of parameter spaces. The transformation is parameterized by neural networks and can be composed of a sequence of transformations. Various kinds of tranformations are developed, a comprehensive review can be found in [158]. In the training procedure, the random samples are drawn from the known target distribution, and the neural network is trained by mapping these samples into the simple base distribution. In the inference procedure, the above process is performed inversely, the target distribution is obtained by transforming the random samples drawn from the base distribution through the trained neural network. Drawing samples from the base distribution and computing the transformation can usually be much faster than computing the likelihood defined by the inner product. Once the training procedure is finished, normalizing flows can generate posterior distribution more e ffi ciently than the MCMC sampling.</text>
<text><location><page_15><loc_52><loc_72><loc_94><loc_90></location>In the work [59], the neural density estimation with normalizing flows is used in three di ff erent aspects. In the first, the normalizing flows are used to build physical priors for amplitude and sky location based on population models. The spatial distribution of GCBs mainly concentrates on the Galactic disk, rather than uniformly distributes on the whole sky. The amplitude depends on the binary masses, distance, and orbital period which can also provided by population synthesis models [21, 161-163]. Utilizing available information can help to constrain the parameter space and improve the e ffi ciency of MCMC sampling. Training with the simulated GCB catalog in the LDC2a [35], the physical priors for amplitude and sky location can be constructed through normalizing flows.</text>
<text><location><page_15><loc_52><loc_35><loc_94><loc_72></location>In the second aspect, normalizing flows are used to construct proposal distributions from results given by former epoch observations. Constructing proposal distributions from available samples to improve sampling e ffi ciency has been proposed earlier in [164] where the density fit method of kernel density estimation (KDE) is used to build proposal distributions. However, the KDE method has drawbacks for high-dimensional problems where one needs to divide parameters into di ff erent groups and build KDE proposal distributions in low-dimensional parameter subspaces. Normalizing flows provide an alternative way to build proposal distributions from available samples with its capability of modeling complicated distributions. Available samples used to build proposal distributions can be obtained in two cases. As mentioned before, the data are incrementally collected, and it is unpractical to analyze data waiting until the end of space-borne detector missions. In reality, data analyses need to be repeatedly performed with growing data. It is essential to fully utilize the results obtained previously when analyzing updated data. Normalizing flows can fit the posterior distributions obtained in previous analyses, and be used to draw proposals in subsequent analyses, which is expected to have high acceptance rate and low autocorrelation of chains. Meanwhile, for the trans-dimensional MCMC, by utilizing the capability of normalizing flows to model complicated distribution, the proposal distributions can be constructed through candidates identified by an ahead iterative subtraction procedure.</text>
<text><location><page_15><loc_52><loc_14><loc_94><loc_35></location>Thirdly, the result posteriors can be published through normalizing flows. As mentioned in Section 2, the number of resolvable GCBs is expected to be ∼ O (10 4 ). Moreover, due to the overlap among signals, parameters of di ff erent sources may have correlations which can not represented by marginalized posteriors for individual sources. One may need a joint posterior for multiple sources. Using samples to represent posteriors may have higher demands for data transfer and storage, and will be inconvenient when sharing the analysis results with the community. Normalizing flows o ff er an e ff ective way to model complicated distributions. Therefore, the results sharing and data product publishing can be in the form of normalizing flow models trained by original posterior samples, from which users can generate samples of the identical distribution for any number needed.</text>
<text><location><page_15><loc_52><loc_9><loc_94><loc_13></location>In summary, normalizing flows o ff er a tool to fit arbitrary complicated distributions, which can be used to construct physical priors from simulated GCBs catalogs, build proposal dis-</text>
<figure>
<location><page_16><loc_8><loc_73><loc_92><loc_90></location>
<caption>Figure 8: Recovered GCBs by the hybrid Bayesian approach on four frequency segments. The top panel shows original data, injected signals, and recovered signals in the A TDI channel. The bottom panel illustrates the true and recovered values for the amplitude parameter of each source. The red vertical lines denote the boundaries of frequency segments which are divided for the convenience of extraction of GCBs considering the quasi-monochromatic feature of GCB signals. This figure is cited from [52].</caption>
</figure>
<text><location><page_16><loc_6><loc_54><loc_48><loc_64></location>tributions from available samples, and as a new representation of posteriors to share with the community substituting posterior samples. Although a full-scale and end-to-end search pipeline for GCBs based on machine learning techniques is not available up to now. Related algorithms are demonstrated to be e ff ective and are being incorporated into global fit pipelines as reported in the roadmap mentioned in [59].</text>
<section_header_level_1><location><page_16><loc_6><loc_51><loc_23><loc_52></location>4. Unresolvable sources</section_header_level_1>
<text><location><page_16><loc_6><loc_38><loc_48><loc_49></location>As mentioned in Section 2, the individually resolvable GCBs are only a small fraction of the total sources. The remains will form a stochastic foreground and contribute to the confusion noise in the data of space-borne detectors. On the one hand, the unresolvable GCBs play the role of noise and a ff ect the observations of other types of sources. On the other hand, the unresolvable sources can still provide invaluable information about the Galaxy.</text>
<section_header_level_1><location><page_16><loc_6><loc_35><loc_42><loc_36></location>4.1. Seperation of the stochastic Galactic foreground</section_header_level_1>
<text><location><page_16><loc_6><loc_15><loc_48><loc_35></location>The Galactic confusion noise will degrade the sensitivity of detectors and reduce the SNR of other sources [60-63]. Additionally, due to the orbital motion of detectors, the pointing of constellations will constantly change relative to the Galactic center where GCBs are concentratedly distributed, which influences the response to the population of unresolvable GCBs and induces the modulation of the Galactic confusion noise. Therefore the Galactic confusion noise is nonstationary, adding more challenges in data analyses of space-borne detectors [165, 166]. Furthermore, the foreground of unresolvable GCBs will blend together with the extra-galactic stochastic background of astrophysical or cosmological origin. It is important to separate or subtract the Galactic foreground for studies of other stochastic GWbackground signals [64-69].</text>
<text><location><page_16><loc_6><loc_9><loc_48><loc_15></location>The Galactic stochastic foreground can be separated from other components of stochastic signals or noise thanks to its distinct spectral shape and the modulation induced by the orbital motion of detectors. For the separation, one needs to first</text>
<text><location><page_16><loc_52><loc_53><loc_94><loc_64></location>properly model the di ff erent components. The stochastic signals can be characterized by the cross spectrum ⟨ ˜ si ( f ) ˜ s ∗ j ( f ) ⟩ where i and j denotes di ff erent TDI channels. Here the angle brackets denote the ensemble average. The total stochastic signals may be contributed by three components including the confusion foreground originating from unresolvable GCBs, the stochastic background of extra-galactic origin, and the instrument noise as</text>
<formula><location><page_16><loc_58><loc_51><loc_94><loc_52></location>⟨ ˜ si ˜ s ∗ j ⟩ = ⟨ ˜ hi ˜ h ∗ j ⟩ gal + ⟨ ˜ hi ˜ h ∗ j ⟩ extra-gal + ⟨ ˜ ni ˜ n ∗ j ⟩ ins . (36)</formula>
<text><location><page_16><loc_52><loc_46><loc_94><loc_50></location>There are various models developed to describe these components. For example, a typical model for extra-galactic stochastic background reads</text>
<formula><location><page_16><loc_60><loc_42><loc_94><loc_45></location>⟨ ˜ hi ˜ h ∗ j ⟩ extra-gal = 3 H 2 0 A ∗ 4 π 2 f 3 f 1mHz ! m Rij , (37)</formula>
<text><location><page_16><loc_52><loc_31><loc_94><loc_41></location>where Rij is the response functions of detectors for di ff erent channels averaged over sky locations and polarizations, H 0 is the Hubble constant, m is the spectral index, and A ∗ is the amplitude at the reference frequency 1 mHz. This power-law model is widely used in both stochastic signals of cosmological origin [167] and astrophysical origin [168]. More models for extragalactic stochastic background can be found in [64].</text>
<text><location><page_16><loc_52><loc_17><loc_94><loc_31></location>Modeling instrument noise can be extremely complicated involving detailed studies about the electronic systems, optical systems, and space environments of detectors, etc. However, in discussions of data analyses or science problems, a simplified noise model can be used, which groups noise into two components, Interferometry Metrology System (IMS) noise and acceleration noise, characterized by two quantities S IMS and S acc respectively [87, 169]. Detailed derivation for noise spectral densities and correlations of di ff erent channels for di ff erent TDI generations or levels of approximation can be found in [64, 87].</text>
<text><location><page_16><loc_52><loc_13><loc_94><loc_17></location>For modeling the stochastic Galactic foreground, one simplified method is utilizing the analytic fitting [170-172] which reads</text>
<formula><location><page_16><loc_52><loc_9><loc_94><loc_12></location>⟨ ˜ hi ˜ h ∗ j ⟩ gal = B ∗ f -7 / 3 exp h -f α -β f sin( κ f ) in 1 + tanh[ γ ( fk -f )] o Rij , (38)</formula>
<table>
<location><page_17><loc_8><loc_33><loc_92><loc_76></location>
<caption>Table 2: Acollection of currently available solutions for extracting GCB signals. Relevant codes if open source are listed in the tablenote for convenience of reference. Most equations appearing in the corresponding chapters can be found in the references listed above.</caption>
</table>
<unordered_list>
<list_item><location><page_17><loc_10><loc_27><loc_44><loc_28></location>1 https://github.com/tlittenberg/ldasoft</list_item>
<list_item><location><page_17><loc_10><loc_26><loc_41><loc_27></location>2 https://github.com/mikekatz04/Eryn</list_item>
<list_item><location><page_17><loc_10><loc_24><loc_43><loc_25></location>3 https://github.com/stefanstrub/LDC-GB</list_item>
</unordered_list>
<text><location><page_18><loc_6><loc_80><loc_48><loc_90></location>where fitting parameter B ∗ controls the overall amplitude, fk controls the position of the knee-like feature in the spectrum of Galactic foreground, together with α , β , γ , and κ describe the spectral shape of Galactic confusion foreground. In practice, one can only vary the overall amplitude in parameter estimation while taking the values of other parameters fitted through simulations of GCB population [64].</text>
<text><location><page_18><loc_6><loc_59><loc_48><loc_80></location>However, in this simplified model, the features of anisotropy and non-stationary are averaged and not incorporated. To fully exploit the distinctive features of the Galactic confusion foreground, a numerical model [66] can be used. In this numerical model, the modulated spectrum of the Galactic foreground is characterized by 17 Fourier coe ffi cients [66, 173, 174] which are treated as free parameters varied in parameter estimation, and their prior ranges are obtained through multiple runs of GCB population simulation. The simulations of GCB population can be based on the Galaxy model constrained by the bright and individually resolvable GCBs [175]. To properly incorporate the modulation induced by detector orbital motion, the entire data are divided into segments of week-long during which the responses of detectors to Galactic confusion foreground are considered to have no appreciable changes.</text>
<text><location><page_18><loc_6><loc_55><loc_48><loc_59></location>After properly modeling all components of stochastic signals, one can use Bayesian inference to obtain estimations of free parameters in models. The likelihood is given by</text>
<formula><location><page_18><loc_13><loc_50><loc_48><loc_53></location>p ( s | θ ) = Y n , d 1 (2 π ) N / 2 | Cd , i j | exp sd , iC -1 d , i j sd , j ! , (39)</formula>
<text><location><page_18><loc_6><loc_31><loc_48><loc_49></location>which is the probability of occurrence of observed data if the stochastic signals and noise are governed by the chosen models with parameters θ . Here, s denotes the entire observed data, θ denotes parameters required to describe models for all components of stochastic signals, n labels the data samples, d labels di ff erent segments used for addressing the modulation of Galactic confusion foreground, i and j represent the di ff erent TDI channels. Cd , i j is the correlation matrix depending on chosen models for each components in stochastic signals. The product needs to run all data samples and all segments. Posterior distributions for model parameters can be obtained through MCMC sampling, from which one can get the separation of di ff erent components of stochastic signals and noise.</text>
<text><location><page_18><loc_6><loc_9><loc_48><loc_30></location>Except for the stochastic background signals of extra-galactic origin, subtracting Galactic confusion foreground also benefits observations of individual sources as presented in [176] where the technique referred to as dictionary learning is used to reconstruct MBHB signals with low-SNR. The basic idea of dictionary learning [177] is representing a target signal h ( t ) through a set of basis elements called dictionary D and a sparse coe ffi -cient vector α as h ∼ D α . The dictionary can be predefined by training dataset created through signals of individual MBHBs without noise, and the coe ffi cient vector α for a given MBHB signal is searched in the presence of noise. Their combination provides a reconstruction of the signal hidden in the noisy data. As demonstrated in the reference [176], low-SNR massive black binaries can by successfully separated in the presence of Galactic noise through this method.</text>
<section_header_level_1><location><page_18><loc_52><loc_89><loc_75><loc_90></location>4.2. Tools for studying the Galaxy</section_header_level_1>
<text><location><page_18><loc_52><loc_69><loc_94><loc_88></location>The old stellar population is one of the excellent tools to trace the dynamical evolution of the Galaxy [178]. However, these sources are usually dim and di ffi cult to be observed in the electromagnetic band. GCBs make up the majority of the total old stellar population and their GW signals are not a ff ected by crowded matters in the Galaxy. GW Observations of GCBs o ff er a unique tool to study the Galaxy. Although, it has been demonstrated that the individual resolvable GCBs which have better measurement and localization can already trace the structure of the Galaxy [26, 27, 179]. These resolvable sources are usually more massive or nearby, which may require careful notice of potential bias [162]. The stochastic foreground containing the contributions of the full GCB population can also reveal various properties of the Galaxy [70-72].</text>
<text><location><page_18><loc_52><loc_36><loc_94><loc_69></location>In one aspect, one can extract information about the Galaxy from the spherical harmonic decomposition of the Galactic confusion foreground [70]. The angular power spectrum of the foreground that encodes information of the GCB spatial distribution can be obtained from the observed cross spctrum of di ff erent TDI channels by the framework presented in [180]. It is pointed out that in [70] the scale height of GCB distribution can be e ff ectively constrained using the hexadecapole moment of the spherical harmonic expansion. Although the constraint provided by this methodology has limited sensitivity comparing to the method with resolvable GCBs, this approach may be less a ff ected by the observation bias and provides a complementary way to measure the structure of the Galaxy. Another approach for obtaining information about the Galaxy is through the spectral shape and amplitude of the Galactic confusion foreground [72]. This methodology is similar to fitting Galactic confusion foreground with the analytic model of Equation 38 mentioned in the last section. Whereas the fitting parameters can be mapped to the physical parameters describing properties of the Galaxy. Thus, one can obtain information of the Galaxy like the total stellar mass as discussed in [72] through the similar parameter estimation procedure discussed in the last section from the Galactic confusion foreground.</text>
<section_header_level_1><location><page_18><loc_52><loc_33><loc_61><loc_34></location>5. Summary</section_header_level_1>
<text><location><page_18><loc_52><loc_16><loc_94><loc_32></location>Space-borne detectors will open the windows of the low GW band in the near future, which can provide new tools to explore the Universe. In contrast to ground-based detectors whose data is noise-dominant, data from space-borne detectors will be signal-dominant where signals are more crowded and various sources are tangled together. The abundance of sources can present plentiful invaluable information on the one hand, but also play the role of noise to disturb source detections and measurements on the other hand. The heavily overlapping signals pose new challenges for the data analyses of space-borne detectors.</text>
<text><location><page_18><loc_52><loc_9><loc_94><loc_16></location>Among various source types targeted by space-borne detectors, the vast population of GCBs is likely the type having the most number to be detected. It is expected that there are tens of millions GCBs in the mHz band, while tens of thousands massive or nearby sources among them are individually resolvable</text>
<text><location><page_19><loc_6><loc_69><loc_48><loc_90></location>and the remains will form a stochastic foreground contributing to the confusion noise. The GWs from GCBs are continuous signals and have the feature of quasi-monochrome. In the time domain, GCB signals with overwhelming numbers exist in the data of space-borne detectors simultaneously during the entire mission period. In the frequency domain, although a single GCB signal is only extended within a narrow band, due to their vast number, GCB signals can still heavily overlap in frequency. GCBs are likely the type that has the most heavy overlap and correlation. Separation and extraction of overlapping signals focusing on GCBs may be the foundation of the ultimate data analysis pipelines for globally separating all kinds of sources. Therefore, in the paper, we present a comprehensive review of precious e ff orts dedicated to the separation and extraction of GCB signals.</text>
<text><location><page_19><loc_6><loc_41><loc_48><loc_69></location>Current solutions for separating overlapping GCB signals can be mainly categorized into two classes, iterative subtraction and global fitting. The strategy of iterative subtraction searches for the optimal fitting for just one signal in each iteration and the same procedure will be performed repeatedly with the remaining data after the subtraction of identified signal. A typical iterative subtraction solution referred to as GBSIEVER employs the F -statistic likelihood and PSO to search the optimal fitting in each step. To address the problem of inaccurate subtraction contamination for low SNR sources, a di ff erent implementation of iterative subtraction uses a tuned particle movement rule of PSO aiming at identifying all local maxima in the likelihood surface. Then, the local maxima corresponding to false candidates induced by degeneracy of individual signals and overlapping signals are removed. Astrophysical properties of GCBs are incorporated to further filter out the low credible candidates. Another implementation of iterative subtraction focuses on improving the e ffi ciency in the PSO searching. Before the fine PSO search procedure, a template search is performed to obtain the coarse estimation of existing signals.</text>
<text><location><page_19><loc_6><loc_21><loc_48><loc_40></location>Another strategy is global fitting which estimates the parameters of all sources together with the source number simultaneously through the full Bayesian approach. The full Bayesian approach can provide the joint posterior distributions for multiple overlapping signals rather than the only point estimation given in the iterative subtraction scheme, which can well describe the correlation among overlapping signals and provide a straightforward way for uncertain analysis and model selection. Posterior estimation over the parameter space with an uncertain dimension is performed through RJMCMC. To overcome the poor convergence in the trans-dimensional MCMC sampling, various tuned proposal distributions according to the features of GCB signals and sophisticated random sampling mechanisms are adopted in di ff erent implementations.</text>
<text><location><page_19><loc_6><loc_9><loc_48><loc_20></location>Besides typical solutions of above two schemes, a hybrid approach that combines the point estimation of maximum likelihood and Bayesian posterior estimation is also proposed. A procedure of iterative subtraction is first performed to identify potential candidates, and a global optimization of all identified signals is performed again to relieve errors induced by correlations of overlapping signals. Then, the MCMC sampling is performed for each found signal to obtain the marginalized poste-</text>
<text><location><page_19><loc_52><loc_82><loc_94><loc_90></location>rior distributions of individual sources. Machine learning techniques are considered as very promising tools in future GW data analyses. Neural density estimation methods are investigated for assistance in drawing proposals and sharing the posterior results with the advantage of normalizing flows at e ffi ciently modeling complicated distributions.</text>
<text><location><page_19><loc_52><loc_58><loc_94><loc_81></location>The resolvable GCBs are only a small fraction of the entire population. The remaining sources will form a stochastic foreground contributing to the confusion noise. The Galactic confusion foreground on the one hand is a type of noise a ff ecting the observation of other sources, on the other hand provides a powerful tool to research the structure of the Galaxy with the advantage that GWs will not be suppressed or obscured by crowded matters in the Galaxy. Thanks to the features of the di ff erent spectral shape and the time-evolving modulation, the stochastic Galactic foreground is distinguishable with the instrument noise and the extra-galactic astrophysical or cosmological original background. And the distinct spectral shape of Galactic foreground can also be used in the measurement of properties of the Galaxy. The anisotropy of the Galactic foreground is associated with the stellar population distribution of the Galaxy, thus the Galactic structure can be constrained through the spherical harmonics decomposition of the Galactic foreground.</text>
<text><location><page_19><loc_52><loc_32><loc_94><loc_57></location>For future works, although there are already diverse end-toend prototype pipelines for extracting overlapping GCBs, various problems still need to be addressed for actually handling real data. For example, in current solutions the noises are usually assumed to be stationary and Gaussian, while in reality there is slow drift of instrument noise in the long period and glitches in the short period which are neither stationary nor Gaussian [181, 182]. Better noise modeling may required for future works. Besides, gaps will exist unavoidably in the data stream due to various reasons including scheduled maintenance or unplanned random events [183, 184]. Future pipelines may need to incorporate processing of various potential defects existing in data. Furthermore, the data from space-borne detectors blend all sources of di ff erent types, the ultimate pipelines need to be capable of addressing di ff erent kinds of overlapping signals. New independent implementations or enchantments of current pipelines are anticipated for preparation of the launch of future space-borne GW observation missions.</text>
<section_header_level_1><location><page_19><loc_52><loc_28><loc_67><loc_29></location>6. Acknowledgments</section_header_level_1>
<text><location><page_19><loc_52><loc_9><loc_94><loc_26></location>The authors have no conflict of interest. W. Z. is supported by the National Key R&D Program of China (Grant No. 2022YFC2204602 and 2021YFC2203102), Strategic Priority Research Program of the Chinese Academy of Science (Grant No. XDB0550300), the National Natural Science Foundation of China (Grant No. 12325301 and 12273035), the Science Research Grants from the China Manned Space Project (Grant No.CMS-CSST-2021-B01), the 111 Project for 'Observational and Theoretical Research on Dark Matter and Dark Energy' (Grant No. B23042) and Cyrus Chun Ying Tang Foundations. R. N. is supported in part by the National Key Research and Development Program of China Grant No.2022YFC2807303.</text>
<section_header_level_1><location><page_20><loc_6><loc_89><loc_14><loc_90></location>References</section_header_level_1>
<unordered_list>
<list_item><location><page_20><loc_8><loc_84><loc_48><loc_88></location>[1] the LVK Collaboration, GWTC-1: A gravitational-wave transient catalog of compact binary mergers observed by LIGO and virgo during the first and second observing runs, Physical Review X 9 (2019) 031040. doi: 10.1103/physrevx.9.031040 .</list_item>
<list_item><location><page_20><loc_8><loc_80><loc_48><loc_83></location>[2] the LVK Collaboration, GWTC-2: Compact binary coalescences observed by ligo and virgo during the first half of the third observing run (2020). arXiv:2010.14527 .</list_item>
<list_item><location><page_20><loc_8><loc_77><loc_48><loc_80></location>[3] the LVK Collaboration, GWTC-2.1: Deep extended catalog of compact binary coalescences observed by ligo and virgo during the first half of the third observing run (2021). arXiv:2108.01045 .</list_item>
<list_item><location><page_20><loc_8><loc_74><loc_48><loc_77></location>[4] the LVK Collaboration, GWTC-3: Compact binary coalescences observed by ligo and virgo during the second part of the third observing run (2021). arXiv:2111.03606 .</list_item>
<list_item><location><page_20><loc_8><loc_71><loc_48><loc_73></location>[5] P. Amaro-Seoane, H. Audley, S. Babak, et al., Laser interferometer space antenna (2017). arXiv:1702.00786 .</list_item>
<list_item><location><page_20><loc_8><loc_68><loc_48><loc_71></location>[6] W.-H. Ruan, Z.-K. Guo, R.-G. Cai, et al., Taiji program: Gravitationalwave sources, International Journal of Modern Physics A 35 (2020) 2050075. doi: 10.1142/s0217751x2050075x .</list_item>
<list_item><location><page_20><loc_8><loc_65><loc_48><loc_68></location>[7] J. Luo, L.-S. Chen, H.-Z. Duan, et al., TianQin: a space-borne gravitational wave detector, Classical and Quantum Gravity 33 (2016) 035010. doi: 10.1088/0264-9381/33/3/035010 .</list_item>
<list_item><location><page_20><loc_8><loc_61><loc_48><loc_64></location>[8] P. Relton, V. Raymond, Parameter estimation bias from overlapping binary black hole events in second generation interferometers, Physical Review D 104 (2021) 084039. doi: 10.1103/physrevd.104.084039 .</list_item>
<list_item><location><page_20><loc_8><loc_57><loc_48><loc_61></location>[9] A. Samajdar, J. Janquart, C. V. D. Broeck, et al., Biases in parameter estimation from overlapping gravitational-wave signals in the third-generation detector era, Physical Review D 104 (2021) 044003. doi: 10.1103/physrevd.104.044003 .</list_item>
<list_item><location><page_20><loc_7><loc_53><loc_48><loc_56></location>[10] E. Pizzati, S. Sachdev, A. Gupta, et al., Toward inference of overlapping gravitational-wave signals, Physical Review D 105 (2022) 104016. doi: 10.1103/physrevd.105.104016 .</list_item>
<list_item><location><page_20><loc_7><loc_50><loc_48><loc_53></location>[11] J. Janquart, T. Baka, A. Samajdar, et al., Parameter estimation methods for analyzing overlapping gravitational wave signals in the thirdgeneration detector era (2022). arXiv:2211.01304 .</list_item>
<list_item><location><page_20><loc_7><loc_47><loc_48><loc_50></location>[12] J. Langendor ff , A. Kolmus, J. Janquart, et al., Normalizing flows as an avenue to study overlapping gravitational wave signals (2022). arXiv:2211.15097 .</list_item>
<list_item><location><page_20><loc_7><loc_43><loc_48><loc_46></location>[13] Y. Dang, Z. Wang, D. Liang, et al., Impact of overlapping signals on parameterized post-newtonian coe ffi cients in tests of gravity (2023). arXiv:2311.16184 .</list_item>
<list_item><location><page_20><loc_7><loc_39><loc_48><loc_43></location>[14] Y. Himemoto, A. Nishizawa, A. Taruya, Impacts of overlapping gravitational-wave signals on the parameter estimation: Toward the search for cosmological backgrounds, Physical Review D 104 (2021) 044010. doi: 10.1103/physrevd.104.044010 .</list_item>
<list_item><location><page_20><loc_7><loc_34><loc_48><loc_38></location>[15] H. Zhong, R. Ormiston, V. Mandic, Detecting cosmological gravitational wave background after removal of compact binary coalescences in future gravitational wave detectors, Physical Review D 107 (2023) 064048. doi: 10.1103/physrevd.107.064048 .</list_item>
<list_item><location><page_20><loc_7><loc_30><loc_48><loc_34></location>[16] F. Badaracco, B. Banerjee, M. Branchesi, et al., Blind source separation in 3rd generation gravitational-wave detectors, New Astronomy Reviews 99 (2024): 101707 99 (2024) 101707. doi: 10.1016/j.newar. 2024.101707 . arXiv:2409.06458 .</list_item>
<list_item><location><page_20><loc_7><loc_26><loc_48><loc_29></location>[17] P. Amaro-Seoane, J. Andrews, M. A. Sedda, et al., Astrophysics with the laser interferometer space antenna, Living Reviews in Relativity 26 (2023). doi: 10.1007/s41114-022-00041-y .</list_item>
<list_item><location><page_20><loc_7><loc_23><loc_48><loc_26></location>[18] K. Belczynski, M. Benacquista, T. Bulik, Double compact objects as low-frequency gravitational wave sources, The Astrophysical Journal 725 (2010) 816-823. doi: 10.1088/0004-637x/725/1/816 .</list_item>
<list_item><location><page_20><loc_7><loc_18><loc_48><loc_23></location>[19] A. J. Ruiter, K. Belczynski, M. Benacquista, et al., The lisa gravitational wave foreground: A study of double white dwarfs, The Astrophysical Journal 717 (2010) 1006-1021. doi: 10.1088/0004-637x/717/2/ 1006 .</list_item>
<list_item><location><page_20><loc_7><loc_14><loc_48><loc_18></location>[20] V. Korol, E. M. Rossi, P. J. Groot, et al., Prospects for detection of detached double white dwarf binaries with gaia, lsst and lisa, Monthly Notices of the Royal Astronomical Society 470 (2017) 1894-1910. doi: 10.1093/mnras/stx1285 .</list_item>
<list_item><location><page_20><loc_7><loc_9><loc_48><loc_14></location>[21] G. Nelemans, L. R. Yungelson, S. F. Portegies Zwart, The gravitational wave signal from the galactic disk population of binaries containing two compact objects, Astronomy and Astrophysics 375 (2001) 890-898. doi: 10.1051/0004-6361:20010683 .</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_20><loc_52><loc_87><loc_94><loc_90></location>[22] G. Nelemans, The galactic gravitational wave foreground, Classical and Quantum Gravity 26 (2009) 094030. doi: 10.1088/0264-9381/26/9/ 094030 .</list_item>
<list_item><location><page_20><loc_52><loc_84><loc_94><loc_86></location>[23] T. R. Marsh, Double white dwarfs and lisa, Classical and Quantum Gravity 28 (2011) 094019. doi: 10.1088/0264-9381/28/9/094019 .</list_item>
<list_item><location><page_20><loc_52><loc_82><loc_94><loc_84></location>[24] P. Amaro-Seoane, S. Aoudia, S. Babak, et al., elisa: Astrophysics and cosmology in the millihertz regime (2012). arXiv:1201.3621 .</list_item>
<list_item><location><page_20><loc_52><loc_79><loc_94><loc_82></location>[25] P. Amaro-Seoane, S. Aoudia, S. Babak, et al., Low-frequency gravitational-wave science with elisa / ngo, Classical and Quantum Gravity 29 (2012) 124016. doi: 10.1088/0264-9381/29/12/124016 .</list_item>
<list_item><location><page_20><loc_52><loc_74><loc_94><loc_79></location>[26] V. Korol, E. M. Rossi, E. Barausse, A multimessenger study of the milky way's stellar disc and bulge with lisa,gaia, and lsst, Monthly Notices of the Royal Astronomical Society 483 (2018) 5518-5533. doi: 10.1093/ mnras/sty3440 .</list_item>
<list_item><location><page_20><loc_52><loc_70><loc_94><loc_74></location>[27] M. J. C. Wilhelm, V. Korol, E. M. Rossi, et al., The milky way's bar structural properties from gravitational waves, Monthly Notices of the Royal Astronomical Society 500 (2020) 4958-4971. doi: 10.1093/ mnras/staa3457 .</list_item>
<list_item><location><page_20><loc_52><loc_66><loc_94><loc_70></location>[28] K. A. Arnaud, S. Babak, J. G. Baker, et al., An overview of the mock lisa data challenges, in: AIP Conference Proceedings, AIP, 2006. doi: 10. 1063/1.2405108 .</list_item>
<list_item><location><page_20><loc_52><loc_63><loc_94><loc_66></location>[29] K. A. Arnaud, S. Babak, J. G. Baker, et al., A how-to for the mock lisa data challenges, AIPConf.Proc.873:625-632,2006 (2006). doi: 10. 1063/1.2405109 . arXiv:gr-qc/0609106 .</list_item>
<list_item><location><page_20><loc_52><loc_60><loc_94><loc_63></location>[30] K. A. Arnaud, G. Auger, S. Babak, et al., Report on the first round of the mock lisa data challenges, Classical and Quantum Gravity 24 (2007) S529-S539. doi: 10.1088/0264-9381/24/19/s16 .</list_item>
<list_item><location><page_20><loc_52><loc_56><loc_94><loc_59></location>[31] K. A. Arnaud, S. Babak, J. G. Baker, et al., An overview of the second round of the mock lisa data challenges, Classical and Quantum Gravity 24 (2007) S551-S564. doi: 10.1088/0264-9381/24/19/s18 .</list_item>
<list_item><location><page_20><loc_52><loc_53><loc_94><loc_56></location>[32] S. Babak, J. G. Baker, M. J. Benacquista, et al., Report on the second mock lisa data challenge, Classical and Quantum Gravity 25 (2008) 114037. doi: 10.1088/0264-9381/25/11/114037 .</list_item>
<list_item><location><page_20><loc_52><loc_50><loc_94><loc_53></location>[33] S. Babak, J. G. Baker, M. J. Benacquista, et al., The mock lisa data challenges: from challenge 3 to challenge 4, Classical and Quantum Gravity 27 (2010) 084009. doi: 10.1088/0264-9381/27/8/084009 .</list_item>
<list_item><location><page_20><loc_52><loc_46><loc_94><loc_49></location>[34] S. Babak, J. G. Baker, M. J. Benacquista, et al., The mock lisa data challenges: from challenge 1b to challenge 3, Classical and Quantum Gravity 25 (2008) 184026. doi: 10.1088/0264-9381/25/18/184026 .</list_item>
<list_item><location><page_20><loc_52><loc_45><loc_90><loc_46></location>[35] Q. Baghi, The lisa data challenges (2022). arXiv:2204.12142 .</list_item>
<list_item><location><page_20><loc_52><loc_43><loc_94><loc_45></location>[36] Z. Ren, T. Zhao, Z. Cao, et al., Taiji data challenge for exploring gravitational wave universe (2023). arXiv:2301.02967 .</list_item>
<list_item><location><page_20><loc_52><loc_38><loc_94><loc_43></location>[37] X.-H. Zhang, S. D. Mohanty, X.-B. Zou, et al., Resolving galactic binaries in LISA data using particle swarm optimization and crossvalidation, Physical Review D 104 (2021) 024023. doi: 10.1103/ physrevd.104.024023 .</list_item>
<list_item><location><page_20><loc_52><loc_35><loc_94><loc_38></location>[38] X.-H. Zhang, S.-D. Zhao, S. D. Mohanty, et al., Resolving galactic binaries using a network of space-borne gravitational wave detectors (2022). arXiv:2206.12083 .</list_item>
<list_item><location><page_20><loc_52><loc_31><loc_94><loc_35></location>[39] P. Gao, X.-L. Fan, Z.-J. Cao, et al., Fast resolving galactic binaries in lisa data and its ability to study the milky way, Phys. Rev. D 107, 123029, 2023 107 (2023) 123029. doi: 10.1103/physrevd.107. 123029 . arXiv:2309.06037 .</list_item>
<list_item><location><page_20><loc_52><loc_27><loc_94><loc_30></location>[40] P. Gao, X. Fan, Z. Cao, Simultaneously search for multi-target galactic binary gravitational waves in reduced parameter space with lmpso-cv (2024). doi: 10.48550/ARXIV.2401.09300 . arXiv:2401.09300 .</list_item>
<list_item><location><page_20><loc_52><loc_25><loc_94><loc_27></location>[41] Y. Lu, E.-K. Li, Y.-M. Hu, et al., An implementation of galactic white dwarf binary data analysis for mldc-3.1 (2022). arXiv:2205.02384 .</list_item>
<list_item><location><page_20><loc_52><loc_22><loc_94><loc_25></location>[42] R. Umstatter, N. Christensen, M. Hendry, et al., Lisa source confusion: identification and characterization of signals, Classical and Quantum Gravity 22 (2005) S901-S911. doi: 10.1088/0264-9381/22/18/s04 .</list_item>
<list_item><location><page_20><loc_52><loc_18><loc_94><loc_21></location>[43] R. Umstatter, N. Christensen, M. Hendry, et al., Bayesian modeling of source confusion in lisa data, Physical Review D 72 (2005) 022001. doi: 10.1103/physrevd.72.022001 .</list_item>
<list_item><location><page_20><loc_52><loc_15><loc_94><loc_18></location>[44] T. B. Littenberg, N. J. Cornish, K. Lackeos, et al., Global analysis of the gravitational wave signal from galactic binaries, Physical Review D 101 (2020) 123021. doi: 10.1103/physrevd.101.123021 .</list_item>
<list_item><location><page_20><loc_52><loc_13><loc_94><loc_14></location>[45] T. B. Littenberg, N. J. Cornish, Prototype global analysis of lisa data with multiple source types (2023). arXiv:2301.03673 .</list_item>
<list_item><location><page_20><loc_52><loc_10><loc_94><loc_12></location>[46] K. Lackeos, T. B. Littenberg, N. J. Cornish, et al., The lisa data challenge radler analysis and time-dependent ultra-compact binary cata-</list_item>
</unordered_list>
<table>
<location><page_21><loc_7><loc_11><loc_49><loc_90></location>
</table>
<text><location><page_21><loc_55><loc_89><loc_73><loc_90></location>doi: 10.1093/mnras/stab2479 .</text>
<table>
<location><page_21><loc_52><loc_11><loc_94><loc_89></location>
</table>
<table>
<location><page_22><loc_6><loc_11><loc_49><loc_90></location>
</table>
<unordered_list>
<list_item><location><page_22><loc_6><loc_10><loc_48><loc_11></location>[113] M. Trias, A. Vecchio, J. Veitch, Markov chain monte carlo</list_item>
</unordered_list>
<table>
<location><page_22><loc_51><loc_11><loc_94><loc_90></location>
</table>
<unordered_list>
<list_item><location><page_22><loc_52><loc_10><loc_94><loc_11></location>[135] J. Qiang, C. Mitchell, A unified di ff erential evolution algorithm for</list_item>
</unordered_list>
<table>
<location><page_23><loc_6><loc_11><loc_49><loc_90></location>
</table>
<table>
<location><page_23><loc_51><loc_11><loc_94><loc_90></location>
</table>
<unordered_list>
<list_item><location><page_23><loc_52><loc_10><loc_94><loc_11></location>[180] A. Taruya, H. Kudoh, Probing anisotropies of gravitational-wave back-</list_item>
</unordered_list>
<text><location><page_24><loc_10><loc_87><loc_48><loc_90></location>grounds with a space-based interferometer. ii. perturbative reconstruction of a low-frequency skymap, Physical Review D 72 (2005) 104015. doi: 10.1103/physrevd.72.104015 .</text>
<unordered_list>
<list_item><location><page_24><loc_6><loc_83><loc_48><loc_86></location>[181] A. Spadaro, R. Buscicchio, D. Vetrugno, et al., Glitch systematics on the observation of massive black-hole binaries with lisa, Physical Review D 108 (2023) 123029. doi: 10.1103/physrevd.108.123029 .</list_item>
<list_item><location><page_24><loc_6><loc_79><loc_48><loc_83></location>[182] Q. Baghi, N. Korsakova, J. Slutsky, et al., Detection and characterization of instrumental transients in lisa pathfinder and their projection to lisa, Physical Review D 105 (2022) 042002. doi: 10.1103/physrevd.105. 042002 .</list_item>
<list_item><location><page_24><loc_6><loc_77><loc_48><loc_79></location>[183] L. Wang, H.-Y. Chen, X. Lyu, et al., Window and inpainting: dealing with data gaps for tianqin (2024). arXiv:2405.14274 .</list_item>
<list_item><location><page_24><loc_6><loc_72><loc_48><loc_76></location>[184] Q. Baghi, J. I. Thorpe, J. Slutsky, et al., Gravitational-wave parameter estimation with gaps in LISA: A bayesian data augmentation method, Physical Review D 100 (2019) 022003. doi: 10.1103/physrevd.100. 022003 .</list_item>
</unordered_list>
</document>
[
{
"title": "Rui Niu a,b , Wen Zhao a,b",
"content": "a Department of Astronomy, University of Science and Technology of China, Chinese Academy of Sciences, , Hefei, 230026, Anhui, China b School of Astronomy and Space Sciences, University of Science and Technology of China, , Hefei, 230026, Anhui, China",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "Gravitational wave (GW) observations have provided a novel tool to explore the universe. In the near future, space-borne detectors will further open the window of low-frequency GW band where abundant sources exist and invaluable information for astrophysics, cosmology, and fundamental physics can be revealed. However, there are various new challenges in data analyses for spaceborne detectors coming with the abundance of GW signals. For example, there are Galactic compact binaries (GCBs) with an overwhelming number that can produce continuous GW signals existing the entire mission time of detectors. The enormous overlapping GCB signals tangle and correlate with each other, and blend with other types of sources together in the observed data. Extracting source information from overlapping signals is one of the key problems for data analyses of space-borne detectors. In the paper, we present a review of currently available solutions for extracting overlapping GCB signals as thoroughly as possible aiming at promoting more interest in this question and inspiring further improvements. Current solutions can be roughly categorized by two classes, iterative subtraction and global fitting. There are diverse implementations of both strategies with enhancements focusing on di ff erent aspects. Meanwhile, the hybrid approach and the machine learning technique are also used in recent years. In the last, we also present an introduction of the stochastic foreground formed by unresolvable faint GCBs about its separation from extra-galactic backgrounds and its utility in exploring the properties of the Galaxy.",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "Direct detections of gravitational waves (GWs) by groundbased detectors have ushered in a new era of GW astronomy [1-4]. Fruitful results have been accomplished based on detected events and researches with GWs are explosively growing in recent years. These detections have initiated a paradigm shift in studies of gravity, astrophysics, and cosmology. In the near future, space-borne detectors including LISA[5], Taiji[6], and TianQin[7] will open the new window of low-frequency GW band. One significant di ff erence between current ground-based detectors and future space-borne detectors is that data given by current ground-based detectors are noise-dominant whereas signals are dominant for space-borne detectors. For current ground-based detectors, the compact binaries will quickly merge after entering the sensitive band, and the transient signals are sparsely distributed in time [1-4]. The situations of signal overlapping are expected to be rare. As reported in the work [8] where the probability of signal overlapping and how severe overlapping can induce significant bias in parameter estimation for second-generation ground-based detectors are thoroughly investigated, it is unlikely to observe overlapping signals by current existing detectors. Whereas, for future space-borne detectors, the enormous sources can persist in the sensitive band during the whole mission period [5-7]. Their GW signals can Email address: wzhao7@ustc.edu.cn (Wen Zhao) be heavily overlapping both in the time and frequency domain, which brings new challenges in extracting physical information from the data. Similar issues of signal overlapping also arise in third-generation ground-based detectors. Due to the exquisite sensitivities especially the improvement in low frequency, the visible duration and signal number are both significantly increased. It is unlikely to observe signals without overlapping for third-generation detectors, and the events with very close merger time which may su ff er from significant bias in parameter estimation can be common [9, 10]. There are extensive works about parameter inference techniques and impacts on researches of scientific problems for overlapping signals in thirdgeneration detectors. For examples, the traditional Bayesian inference framework with strategies of hierarchical subtraction and joint estimation for overlapping signals is investigated in the work [11], and a joint parameter estimation analyzing two overlapping signals simultaneously using normalizing flows is demonstrated in [12]. In the work [13], impacts of signal overlapping on testing General Relativity are elaborated. In works [14, 15], the impacts of unresolvable overlapping foreground and subtraction residuals induced by parameter estimation bias due to signal overlapping on detecting cosmological stochastic gravitational wave background with third-generation detectors are investigated. A review for signal overlapping problems focusing on third-generation ground-based detectors can be found in [16]. In the rest of this paper, we mainly focus on researches targeting space-borne detectors. There are diverse types of sources presented in the space- borne detector sensitive band [17], such as massive black hole binaries (MBHBs), extreme mass ratio inspirals (EMRIs), and Galactic compact binaries (GCBs), etc. GCBs are likely to be the most numerous type of GW sources detected in future space GW observation. Tens of millions of such sources may exist in the space-borne detector band. Among them, ten of thousands of bright ones are expected to be independently resolvable, while others will constitute a stochastic foreground as confusion noise [18-21]. Identification and subtraction of the resolvable GCBs are not only important for extracting other kinds of sources but also can be assistance in researches about stellar and Galactic astrophysics with the advantage that the information carried by GWs will not be a ff ected by crowded matters in the Galaxy [22-27]. Although all sources including MBHBs and EMRIs will be blended together in data-stream from space-borne detectors and people may ultimately need algorithms that can separate or simultaneously fit all di ff erent sources, GCBs may be the kind that is most heavily overlapping due to their overwhelming amount and the feature of long-living. Therefore, the discussions dedicated to GCBs may be the foundation and starting point of the ultimate full algorithms for separating and fitting all overlapping sources. In this paper, we will focus on GCBs and present a review of current solutions to tackle enormous overlapping GCB sources as comprehensively as possible, aiming at paving for further improvements in the problems of signal overlapping for data analyses of space-borne detectors. Current solutions of extracting GCB signals mainly work around the simulated data sets including the earlier Mock LISA Data Challenges (MLDCs) [28-34], the recent resurrected LISA Data Challenges (LDCs) [35], and the Taiji Data Challenges (TDCs) [36]. These data sets are released to encourage various e ff orts for tackling unsolved problems in data analyses of space-borne detectors. These challenges involve data sets not only dedicated to GCBs, but also blended with various other possible sources, and the complexity has been increasing progressively. In the MLDC1, isolated GCBs and moderately overlapping signals with dozens of GCBs are concerned [2830]. The MLDC2 considers the full population with 26 million GCBs, and includes two data sets where the MLDC2.1 only contains signals of GCBs while MLDC2.2 blends signals of GCBs with MBHBs and EMRIs [31, 32]. The data set of GCBs in MLDC3 contains ∼ 60 million binaries, which is descended from MLDC2.1 with the improvement of realism by considering two di ff erent kinds of GCBs, binaries with two detached components and binaries with interacted components [33, 34]. The MLDC4 is the descendant of MLDC2.2, which blends different types of sources concerned separately in sub-challenges of MLDC3 into a single data stream [33]. The new LDCs 1 [35] are resumed in recent years with the new design [5] of LISA. Currently, three challenges have been released, LDC1 Radler , LDC2a Sangria , and LDC2b Spritz . The LDC1 considers various types of possible LISA sources separately like MLDC3, and includes six subchallenges where LDC1-4 is dedicated to GCBs and contains 26 million signals. The LDC2a is the updated challenge similar to MLDC4, which mixes di ff erent types of sources. The LDC2b has the improvement of considering the realistic instrumental and environmental noise, including gaps, glitches, and non-stationary noise. While the GW sources considered in LDC2b are relatively simple, where only MBHBs and verification GCBs are contained. The Taiji project also releases the data challenge [36] for the configuration of the Taiji detector, which includes sub-challenges concerning various types of GWsources separately and the mixture of these sources. Numerous endeavors have been made in previous works to address the problem of signal overlapping. The ideas for this problem can be roughly categorized into two groups, the iterative subtraction strategy and the global fitting strategy. The iterative subtraction strategy searches the maximum likelihood estimation for a single source in each step, and this procedure will be performed iteratively with the remaining data after subtracting the identified signals [37-41]. In contrast, the global fitting strategy fits all signals simultaneously in the full Bayesian approach with sophisticated Markov chain Monte Carlo (MCMC) sampling algorithms to obtain the joint posterior distributions [42-49]. Both two strategies are adopted and implemented in diverse full-scale and end-to-end pipelines for extracting overlapping signals of GCBs. The iterative subtraction solutions can extract source candidates quickly but su ff er from the correlations among overlapping signals and the contamination of accumulating signal residuals left by each subtraction iteration. The global fitting solutions employ the full Bayesian approach to analyze all sources in a band simultaneously, which can better deal with the source correlations and residual contamination, but with the price of extremely massive demand for computational resources. The hybrid approach combining the maximum likelihood estimation and the Bayesian parameter estimation with MCMC is also proposed for combining the strengths of the two strategies while evading their drawbacks [50-52]. The flourishing machine learning algorithms provide new avenues to solve parameter estimation problems. After training with simulated data, the algorithms can generate posteriors directly without enormously evaluating the computationally expensive likelihood, which can complete the parameter estimation nearly in real time. The techniques have been successfully used in the parameter estimation for current ground-based detector data [53-58]. Machine learning techniques are also considered in the problem of overlapping GCBs, which are expected to be prospective powerful tools in future data analyses of space-borne detectors [59]. The resolvable sources are only a small fraction of the whole GCB population, the remaining faint GCBs can form a stochastic foreground. On the one hand, this Galactic foreground plays the role of noise deteriorating the signal-to-noise ratio (SNR) of individual sources and blending with stochastic signals from extra-galactic sources [60-69]. On the other hand, this foreground also contains information on the GCBpopulation which is correlated to properties of the Galaxy, and o ff ers a unique tool to study Galactic astrophysics [70-72]. The rest of this paper is organized as follows. In the next section, we briefly introduce the characterizations of GW signals from GCBs and how detectors respond to them. Section 3 is the main part of this paper where we present a detailed review of e ff orts on extracting overlapping GCB signals reported in recent years. After briefly summarizing earlier works around MLDC in Section 3.1, we first introduce a typical implementation of the iterative subtraction scheme and two variants focusing respectively on reducing inaccurate subtraction contamination and improving search e ffi ciency in Section 3.2. Then, we elucidate basic conceptions of the global fitting scheme and introduce two independent implementations of this strategy in Section 3.3. The hybrid Bayesian approach combining the maximum likelihood estimation and the MCMC sampling, and a preliminary attempt of utilizing machine learning techniques in solving the problem of overlapping GCB signals are introduced next in Section 3.4 and 3.5. The discussions about unresolvable GCBs are given in Section 4. The final summary is presented in Section. 5.",
"pages": [
1,
2,
3
]
},
{
"title": "2. Galactic compact binaries",
"content": "According to population models [18-21], there are tens of millions of compact binaries in the Galaxy that are slowly inspiraling towards each other with emissions of GWs in the mHz band and might be the type of most numerous sources observed by the space-borne detectors. An illustration of the sky distribution of a simulated GCB population taken from the training dataset of LDC2a [35] is presented in Figure 1. Among the population, tens of thousands of GCBs are expected to be individually resolvable through the four-year observation time of space-borne detectors. Current electromagnetic observations have identified about dozens of GCBs 2 [73], while hundreds are predicted to be detected by future observations [20]. These known sources are referred to as verification binaries and the loud ones are guaranteed to be detectable by space-borne GW detectors. The loud verification binaries are expected to be quickly identified by just weeks integration time of observation, thus o ff ering an important tool for functional tests and performance monitoring of the instruments [74]. Most GCBs are binary white dwarfs including detached binaries, as well as interacted binaries that have reached the Roche lobe overflow and started mass transfer [75]. While a small fraction of GCBs may involve with neutron stars or black holes. A summary of expectation of GCBs in the space-borne detector band is present in Table 1 which is cited from [17]. More details about population synthesis simulations or formation scenarios can be found in reviews [17, 76]. GCBs in the band of space-borne detectors are far from merger, and will stay in the inspiral phase of slow chirping during the entire observation period. GWs radiated from GCBs can be well described by the quasi-monochromatic waveform which takes the form of [77, 78] Here A is the amplitude, ι is the inclination of binary orbit, and Φ is the GW phase which can be expressed as Φ ( t ) = ϕ 0 + 2 π R f ( t ' ) dt ' with the arbitrary initial phase ϕ 0 and the frequency evolution f ( t ). Since the frequency evolution is extremely slow for GCBs in the early inspiral stage, f ( t ) can be characterized by the central frequency f 0 and the first derivative ˙ f . The evolution of amplitude is usually neglected and A is considered as a constant. The e ff ects of mass transfer between binaries can be encoded into the parameter ˙ f [79]. The phase evolution can be written as The total matric perturbation can be assembled by the sum of two polarizations as where ϵ + and ϵ × denote the polarization tensors which depend on the source location ( β, λ ) and the polarization angle ψ . The response to GWs of a single laser link of space-borne detectors is given by [80-82] where the subscripts s , r denote the laser sender node and the receiver node respectively, ˆ k denotes the unit vector of the GW propagation direction, ˆ n sr denotes the unit vector along the direction of the link, p s and p r are the vectors of positions of the sender node and receiver node in the heliocentric coordinate. Together with the parameters describing GW strains of GCBs as shown in Equation 1 and 2, there are 8 parameters ( A , f 0 , ˙ f , ι, λ, β, ψ, ϕ 0) to fully characterize a signal from GCB. Since the noise behavior can be conveniently characterized by the power spectral density (PSD) in the frequency domain if the noise is stationary and Gaussian, data analyses of GWs are often performed in the frequency domain. The method proposed in [78] can compute the Fourier transformation of the response quickly and accurately by heterodyning the response signal with a carrier wave of the frequency f 0. By multiplying with the carrier wave, the response signal can be decomposed into the slow part and the fast part. The Fourier transformation of the fast part can be obtained analytically. The slow part is transformed through fast Fourier transformation numerically, whereas the number of time samples is significantly reduced. For a signal extending within the band of [ f 0 , (1 + η ) f 0], the heterodyning operation can shift the required Nyquist frequency from 2(1 + η ) f 0 to 2 η f 0. Since the GCBs signals are quasimonochromatic with η ∼ 10 -6 [21, 83], the number of samples can be much less than the original time samples when numerically computing the fast Fourier transformation for the slow part. There is open source code GBGPU [77, 84] that can be used to obtain the response signals of GBCs in practice. The space-borne detectors are unequal arm interferometers where the laser frequency noise will experience di ff erent time delays when traveling along di ff erent arms and cannot be canceled out by itself at the photodetector like ground-based detectors. In order to suppress the laser frequency noise which can be stronger than GW signals a few orders, the technique called time delay interferometer (TDI) where the observables are created by time-shifting and combining single link responses has to be used to construct artificial equal arm interferometers [82, 85, 86]. The 1.5th generation (or 1st generation in some literatures) TDI observable X is given by [87] where Dij denotes the delay operator defined as Dijysr = ysr ( t -Lij ) and Lij is the arm-length between the node i and node j of the constellation. (In the construction of TDI combinations, the non-commutativity and variations of arm-length may be taken into consideration, whereas when actually computing the detector responses, the approximation of rigid constellation where all arm-length is equal and constant is usually adopted [83, 87, 88]. Therefore, we follow this convention and use Lij to denote the arm-length when writing TDI combinations, while using L in other places.) The other two observables Y , Z can be obtained similarly by cyclic permutation of indices. The 2nd generation TDI incorporates that the delay operators are non-commutative for forward and inverse delay of a link due to the rotation of the constellation by compensating more virtual loops in two arms as [87] The constructions of TDI observable are not unique and have abundant forms. Di ff erent constructions can have di ff erent sensitivities to GWs of di ff erent polarizations and propagation directions [89, 90]. The X , Y , Z channels are correlated, and the independent TDI channels can be obtained through the combination of In the low-frequency limit where f < (1 / 2 π L ) with L denoting the arm-length of the detector, signals in the A channel are mainly contributed by the plus polarization, the E channel can be approximated to the cross polarization, and the T channel is approximated to the breath polarization but which is absent in General Relativity. Thus, in data analyses with low-frequency approximation, only A and E channels are considered usually [51, 91, 92]. We illustrate the time domain signal in the A channel of a typical GCB in Figure 2 and the frequency domain signals of the GCB population in Figure 3.",
"pages": [
3,
4
]
},
{
"title": "3. Solutions for extracting overlapping GCB signals",
"content": "This section provides a detailed review of currently available solutions for extracting overlapping GCB signals. We mainly focus on recent works, early e ff orts with MLDCs are briefly summarized in Section 3.1, and diverse innovations and new implementations in recent years are introduced in the subsequent sections. A summary of solutions mentioned here and corresponding references are presented in Table 2.",
"pages": [
5
]
},
{
"title": "3.1. Early researches",
"content": "Problems of extracting overlapping GCB signals have been discussed for more than two decades. The iterative subtraction strategy was proposed at the beginning of this century [97], where the brightest sources are iteratively identified and subtracted. The global fitting strategy can be traced back to the early researches [42, 43] which employs the trans-dimensional MCMC sampling algorithm to simultaneously infer the source number, joint posteriors of source parameters, and noise levels. Diverse solutions for overlapping GCB signals are explosively presented working around the MLDCs. For example, the blocked annealed metropolis (BAM) algorithm developed in [98-100] is a quasi-Bayesian approach with a F -statistic likelihood and a customized MCMC sampling strategy. The subsequent work [101] extends the BAM algorithm through introducing parallel tempering, the reversible jump MCMC (RJMCMC), and the fast-slow decomposition waveform model, which is the basis of the recent full-scale global fitting pipeline GBMCMC introduced in Section 3.3. The search methods using F -statistic and various optimal algorithms [102-107] are also the foundation of diverse iterative subtraction solutions in recent years. Additionally, various intelligent ideas besides the above two schemes are also widely discussed, including a tomographic approach [108, 109], genetic searches [110], the two-stage strategy [111], and a sophisticated MCMC walking strategy [112, 113], etc. More details about early e ff orts can be found in reports of each MLDC [30, 32-34], and a comprehensive review [114]. Based on early e ff orts, various enhancements, new methods and implementations have emerged with the resurrected LDCs in recent years, which will be detailed in the following.",
"pages": [
5
]
},
{
"title": "3.2. Iterative substraction using F -statistic and particle swarm optimization",
"content": "One of widely considered strategies for extracting overlapping signals is iterative subtraction where the search algorithm for one individual source will be run to identify the brightest source, then the identified signal will be subtracted from the data, and this procedure will be iteratively executed with the residual data until some stopping criteria are satisfied. The idea of iterative identification and subtraction for overlapping signals in mHz GW band is proposed early in [97], and is implemented extensively in previous works. In this review, we present an introduction of a typical iterative subtraction scheme developed in recent years [37-39], which employs F -statistic to construct likelihood and uses particle swarm optimization (PSO) to search the optimal estimations. The framework developed in [37] is referred to as Galactic Binary Separation by Iterative Extraction and Validation using Extended Range (GBSIEVER). In the iterative subtraction scheme, the single brightest source needs to be identified in each iteration. GBSIEVER implements this through maximum likelihood estimation where the likelihood is constructed by F -statistic. The data from detectors are composed of noise n ( t ) and GW signals h ( t ), which can be expressed as d ( t ) = n ( t ) + h ( t ). Assuming the noise n ( t ) is Gaussian and stationary, the probability of a realization of d ( t ) given a specific GWsignal h ( t ) described by a set of parameters θ can be written by where the angle brackets denote the noise weighted inner product defined as Here, ˜ a ( f ) and ˜ b ( f ) is the Fourier transformation of time series a ( t ) and b ( t ), Sn ( f ) denotes the PSD of the noise. To identify the signal in data through maximum likelihood estimation, one needs to explore the parameter space of θ to find the parameters ˆ θ MLE where the probability (Equation 8) has the maximum value. Drop the term independent with the source parameters, the likelihood can be written by For multiple independent measurements of the same signal, such as the independent TDI channels or di ff erent detectors, the final log-likelihood is the sum of log-likelihoods for each independent measurement. The estimation of source parameters ˆ θ MLE is given by Here, we use the hat to denote parameters with the maximum likelihood value. Exploring the high-dimension parameter space to solve Equation 11 is computationally expensive. In the framework GBSIEVER, the reduced likelihood is constructed through F -statistic, where only four intrinsic parameters are searched and the extrinsic parameters can be obtained analytically. The full set of parameters to describe GCB signals is separated into intrinsic parameters { f 0 , ˙ f , λ, β } and extrinsic parameters {A , ι, ψ, ϕ 0 } . The GW signal can be decomposed by a linear combination of templates only depending on intrinsic parameters as where κ denotes a set of intrinsic parameters, ai is a reparametrization of extrinsic parameters, and basis templates Xi ( t , κ ) can be defined by the GW signals at 4 sets of specific extrinsic parameters. Using this decomposition, we can construct the vector U which has the element of and the matrix W which has the element of Using vector U and matrix W together with the vector A consisting of ai , the likelihood Equation 10 can be rewritten as Maximizing the original likelihood ln Λ can be substituted by maximizing the F -statistic defined as from which one can obtain the maximum likelihood estimation for intrinsic parameters ˆ κ . The estimation of ˆ a can be obtained analytically through The full set of parameters with the maximum likelihood can be recovered through ˆ κ and ˆ a . The remaining question is how to e ffi ciently explore the intrinsic parameter space to maximize F ( κ ). In the framework of GBSIEVER, PSO is performed to accomplish this task. PSO is proposed by Kennedy and Eberhart [115] and widely used in various optimal problems. PSO is inspired by the social behavior of organisms like bird flocks or fish schools. It utilizes a population (referred to as the swarm) of candidate solutions (referred to as particles) which are moved by the guidance of own experience of each particle and the collective knowledge of the entire swarm to find the optimal solution in the parameter space. The interaction among particles endows the PSO method with more capability for global searching and against trapping in local optima. Initially, the particles are randomly drawn in the parameter space and assigned with random velocities. The subsequent movement of particles is guided by [40] In the above equations, x t i and v t i denote the position and velocity of i -th particle at the iteration step of t . P t i is the best position of the i -th particle found in previous iterations, which represents the experience of individual particles, and G t is the best position found by the entire swarm in previous iterations, which represents the collective knowledge of the swarm. The terms corresponding to P t i and G t are usually called the personal term and the social term respectively in literature. ω , c 1, and c 2 are constant coe ffi cients that need to be tuned according to the specific problem through experimental runs or empirical knowledge. The process of optimization can be viewed as two phases, the exploration phase where the particles explore the parameter space expansively and quickly to find better locations, and the exploitation phase where the particles have converged within a promising region and updates of better position will be relatively slow. ω is called the inertia weight controlling the balance of exploration and exploitation in the optimization process. A lower inertia weight favors exploitation and allows particles to quickly converge toward promising regions. Conversely, a higher inertia weight can help particles resist attractions of previous best positions of personal and social terms, explore the parameter space more expansively, and potentially escape from local optima. The coe ffi cients c 1 and c 2 are called acceleration coe ffi cients controlling the balance between the personal term and social term of the particles in their movement. The setting of these coe ffi cients has significant impacts on the performance and e ffi ciency of PSO. A typical configuration in the context of GWdata analyses can be found in [116], which is also adopted by GBSIEVER. After su ffi cient iterations, the particles are expected to converge to the optimal solution. Some practical techniques are used in GBSIEVER to improve search e ffi ciency and eliminate false candidates. As introduced in Section 2, GCB signals have the characterization of narrowband. Therefore, in practice, it is usually to divide the whole frequency band into small bins. Independent analyses are performed in di ff erent frequency bins, and the edge e ff ects are carefully addressed in the meanwhile. Furthermore, another practical technique in GBSIEVER is a special downsampling operation which can reduce the number of samples thus relieve the computational cost in likelihood evaluation. To eliminate spurious sources, GBSIEVER employs a cross-validation procedure to compare the extracted candidates in two independent runs. Only the candidates with similar estimation results in two independent runs are considered to be genuine. The performance of GBSIEVER is verified with the dataset 10 -2 of LDC1-4 and modified MLDC3.1, where it is demonstrated that O (10 4 ) GCBs can be successfully identified [37]. The residual after subtracting identified signals is shown in Figure 4. In the subsequent work [38], the framework GBSIEVER is extended by incorporating the network of detectors. Furthermore, in the work[39], the fact that the data collection is gradually incremental in real observations is considered. The search results obtained in ahead short period observations can be used to reduce the parameter space in following searches, which can help enhance the e ffi ciency of the algorithm. Although the framework GBSIEVER has been demonstrated to be capable of extracting a large population of overlapping GCB signals, there are various aspects which can be further improved. In the following, other two implementations of iterative subtraction focusing on reliving inaccurate contamination and enhancing search e ffi ciency will be introduced.",
"pages": [
5,
6,
7
]
},
{
"title": "3.2.1. Local maxima particle swarm optimization algorithm",
"content": "In the iterative subtraction scheme, the inevitable errors of parameter estimation in each iteration will leave residual signals that can continuously accumulate and contaminate remaining data as noise. The work [40] improves the previous GBSIEVER framework and develops the new approach of local maxima particle swarm optimization algorithm with a special strategy of creating voids referred to as LMPSO-CV for dealing with inaccurate subtraction contamination, especially for the low SNR sources. LMPSO-CV starts with the remaining data assuming all sources with SNR > 15 have been identified and subtructed. The algorithm aims at identifying all local maxima of F -statistic in the parameter space, and the source parameters will be extracted from these local maxima. In LMPSO-CV, to identify the local maxima, the configuration of PSO is adjusted by setting c 1 = ω = 0. The equations guiding the movement of particles turn into the form of [40] As mentioned in Section 3.2, a lower inertia weight favors the exploitation where particles tend to move within a local promising region, and emphasizing the global term will guide particles to move toward the same position. It can be expected that the setting of entirely dropping the inertia weight and the individual term will extremely improve the convergence towards local maxima. Once a local maximum is identified, to avoid the search algorithm picking the same or too close local maxima multiple times, a void in the parameter space will be created and excluded in subsequent searches. The void is modeled by a spheroid as where x , y , z correspond to the source parameter f 0, β , λ . The size of the spheroid is determined by the frequency resolution and the degeneracy of β and λ through experimental computations. In the process of searching for local maxima, if a particle moves into voids, it will be placed at a new random position outside voids in the next iteration to avoid redundancy. The search of local maxima and creation of voids will be performed repeatedly until the termination rules designed for identifying all local maxima beyond the threshold are satisfied. The source parameters are extracted among the identified voids. As can be seen from Figure 5, there are enormous local maxima in likelihood surfaces corresponding to false signals which need to be eliminated when extracting real candidates. The extraction consists of three parts. The first part aims at removing local maxima induced by degeneracy noise of individual signals. From the observation of experimental evaluation of the likelihood for an individual signal as shown in the top row of Figure 5, the local maxima corresponding to false signals have smaller likelihood values than the real signal. Therefore, the local maxima are sorted in descending order, and the extraction starts with the local maxima of the highest F -statistic value which is assumed to be a real signal. Then, the quantities of ∆ F defined as [40] are computed for subsequent local maxima. In the above equation, the first term F ( θ i , d ( t )) is the F -statistic value of the i -th local maximum with parameter θ i with respect to the raw data, which is same to the F -statistic value used in the search process. The second term F GLYPH<16> θ i , P i -1 m = 1 h ( t , θ m ) GLYPH<17> is the F -statistic value of the i -th local maximum with respect to the summation of candidate signals previous extracted, which characterizes the degeneracy with previous extracted signals. If the F -statistic value of the i -th local maximum is mainly contributed by the degeneracy with previous extracted signals. The quantity of ∆ F is expected to be small. Conversely, large ∆ F indicates the local maximum is likely to be a real signal. The local maxima that satisfy the criterion of ∆ F > 53 will be retained and passed into the second part for further removing false signals. The second part of signal extraction and false signal elimination is based on the astrophysical properties of GCBs. The second part requires a new independent search of local maxima which is similar to the process discussed above except using a di ff erent range for ˙ f . In the first search, the search range for ˙ f is [ -10 -16 , 10 -15 ] when f ≤ 4 mHz and [ -10 -14 , 10 -13 ] when f > 4 mHz. Whereas, in the second search, the range of ˙ f is determined by the mass parameters of binaries. Since the majority of GCBs are consisted of binary white dwarfs which have the mass ranging from 0 . 1 to 1 . 4 M ⊙ . In the second search, the values of ˙ f are constrained within the range corresponding to the mass range of white dwarfs. After two independent searches, the candidate signals need to be compared through the correlation coe ffi cient defined as where C ( θ , θ ' ) = ⟨ h ( θ , h ( θ ' ) ⟩ . This correlation coe ffi cient quantifies how similarity between two signals identified in two independent searches. Only candidates with enough large R are considered as reliable sources and are retained for further processing. Another characterization of GCBs is the concentration in the Galactic disk. Therefore, one can simply remove candidates with Galactic latitude outside of [ -0 . 5rad , 0 . 5rad] to enhance the reliability. The third part of signal extraction focuses on the false candidates induced by overlapping signal degeneracy noise. In the experimental computation of the likelihood surface for the situation of overlapping signals as shown in the middle and bottom rows of Figure 5, it is found that the F -statistic values of local maxima corresponding to false candidates can be even larger than real signals. To remove false candidates of this kind, a similar process of the first part needs to be performed again but in ascending order of local maxima. The process of LMPSO-CV can be summarized in two steps. In the first step, the stochastic optimization algorithm is tuned to specializing in finding local maxima. A void in parameter space will be created to avoid redundant search once a local maximum is identified. The second step extracts signals and eliminates false candidates from the identified local maxima, which consists of three parts: removing false candidates induced by degeneracy of individual signals, applying the astrophysical properties of GCBs, and removing false candidates induced by degeneracy of overlapping signals. LMPSO-CV mainly focuses on the sources with SNR between 7 and 15. Since the LMPSOCV approach searches all local maxima simultaneously and filters out false candidates afterward, rather than iteratively performing search and subtraction for a single signal, it can better deal with inaccurate subtraction contamination in the traditional iteration subtraction scheme.",
"pages": [
7,
8
]
},
{
"title": "3.2.2. Combination of coarse template search and fine PSO search",
"content": "Another implementation of the iterative subtraction scheme focuses on improving search e ffi ciency by the strategy of combination with the coarse search based on templates for quickly identifying rough parameters of candidates and the fine search based on F -statistic and PSO for detailed exploring small promising regions [41]. The strategy is composed of two steps. In the first step, candidates are searched by a template bank. A stochastic template bank [117] is constructed, where the places of template parameters are randomly drawn in the parameter space and additional pruning operation is performed to remove templates with too small separation. The number of templates can be approximated by [41] where n denotes the dimension of the parameter space S n , η denotes the desired level of coverage confidence, m ∗ is the mismatch criterion, Vn is the volume of unit sphere in n -dimension, and V S n is the volume of the whole parameter space. The construction of the template bank starts with randomly generating N waveforms. Then, the KDtree (k-dimensional tree) algorithm [118, 119] is used to find the nearest neighbor for each waveform. If the mismatch between two waveforms is less than m ∗ , one of them will be removed. The above two procedures are performed repeatedly until the total number of templates gets to stable. The F -statistic likelihood is also used in the coarse search, the threshold for coarse search is set to F th coarse = 15 which corresponds to SNR ≈ 5 . 1. The templates whose F -statistic values exceed the threshold are recorded as candidates. Due to the annual orbital motion of detectors, modulation is imposed on the quasi-monochromatic signals of GCBs. The modulation can induce sidebands around the central frequency f 0 of GCB signals. A clustering operation will be performed to gather candidates with close central frequency. Among all candidates within the extended Doppler window, only the sources with the largest F -statistic values are retained for fine search in the next step. In the second step, the parameters of candidates identified by the template search are accurately estimated using PSO, which is similar to the procedure discussed before, while the search ranges for the central frequency are tuned according to results given by coarse search. The performance of this methodology is demonstrated with MLDC3.1. It is shown that O (10 4 ) sources can be successfully identified, and nearly 90 percent of them are well aligned with injected signals. This method performs a coarse template search in the first step, which can provide priori information guiding the PSO search thus enhancing the e ffi ciency of the fine parameter estimation.",
"pages": [
8
]
},
{
"title": "3.3. Global fitting with trans-dimensional Bayesian inference",
"content": "The iterative subtraction method has advantages of rapidity and e ffi ciency but also su ff ers from problems of correla- 1.5 1 0.5 0 -0.5 -1 -1.5 2.0903 2.0904 2.0905 2.0906 2.0907 2.0908 2.0909 2.091 2.0911 2.0912 2.0913 Frequency 10 -3 1.5 9000 8000 7000 6000 5000 4000 3000 2000 1000 9000 8000 7000 6000 5000 4000 3000 2000 1000 Latitude Latitude 1 0.5 0 -0.5 -1 -1.5 2.0903 2.0904 2.0905 2.0906 2.0907 2.0908 2.0909 2.091 2.0911 2.0912 2.0913 Frequency 10 -3 tions among overlapping signals and contamination of inaccurate subtraction. In each step of iterative subtraction, errors in parameter estimation are unavoidable, which can yield signal residuals left behind the subtraction. The residuals will contaminate remaining data and accumulate along with iterations. Furthermore, in each iteration only the parameter values with maximum likelihood are used in subtraction, the uncertainty of parameter fitting will propagate alone iterations. The uncertainty of parameter estimation for identified candidates is di ffi -cult to ascertain. Additionally, GCB signals are heavily overlapping in both time and frequency, which can induce high correlations among signals. The correlations may lead to bias in the parameter estimation of each iteration, and this will further intensify the problem of inaccurate subtraction contamination. To alleviate these problems, a commonly used operation is dividing the whole frequency band into small bins and performing analyses independently in di ff erent bins while carefully addressing signals residing at edges, which can reduce the number of iterations and reduce the accumulation of inaccurate subtraction contamination. Besides, as introduced in Section 3.2.1, the method of modifying the particle movement rule in PSO aiming at identifying all local maxima on the likelihood surface simultaneously is also an e ff ective method to address inaccurate subtraction contamination for the low SNR sources. The global fitting is another route that can e ff ectively deal with the problems of overlapping signal correlations. In the iterative subtraction scheme, only parameters for one source are estimated in each step. In contrast, parameters for all sources together with the source number are estimated simultaneously in the global fitting scheme. The global fitting with full Bayesian parameter estimation can obtain joint posterior distributions rather than just the maximum likelihood estimation, thus the uncertainty of fitting can be read out from posteriors straightforwardly. Since all sources are fitted simultaneously, the correlations of overlapping signals are taken into account and can be reflected in joint posteriors of multiple overlapping signals. As mentioned in 3.1, the strategy of global fitting and the Bayesian approach have been proposed and implemented early in various works focusing on MLDCs [42, 43, 98-101]. Further improvements like incorporating the reality that the data collection is time evolving, various sophisticated methods for increasing the e ffi ciency of MCMC sampling, and new implementations of RJMCMC have been continuously presented in recent years [44-46, 48, 49].",
"pages": [
8,
9,
10
]
},
{
"title": "3.3.1. RJMCMC",
"content": "For the Bayesian inference of global fitting, One of the key di ff erences with the Bayesian parameter estimation commonly used in data analyses of current ground-based detectors is that the number of sources is uncertain and has to be inferred from the given data. A single GW source signal h ( t , θ ) in the likelihood shown in Equation 8 turns to a summation of multiple signals P k h ( t , θ k ) where the number of source is uncertain. The sampling algorithms to solve inference problems with uncertain dimensionality are referred to as trans-dimensional MCMC or RJMCMC. In the parameter estimation problem of the Bayesian framework, the estimation results are represented by posteriors through the Bayes theorem where d ( t ) denotes observed data, θ n represents parameters with the dimensionality of n , π ( θ n ) is the prior representing the knowledge before observations, p ( d ( t ) | θ n ) is the likelihood representing the probability of noise realization that can just present the time series d ( t ) observed on detectors when adding the GW signals described by parameters θ n and depending on the models of noise behavior and GW signals, Z is the normalization factor called as evidence. Since the dimension of parameters θ n for describing GW signals is usually high, it is impractical to compute posterior on a grid of the parameter space. Stochastic sampling algorithms are usually employed to perform random walking in the parameter space, and use the density distributions of random samples to approximate the probability distributions of posteriors. The Metropolis-Hasting MCMC (MHMCMC) [120, 121] is one of the well known and widely used algorithms to perform sampling for fixed dimensional problems. After setting the initial state of a random walker, the subsequent movements are guided by the rules introduced below. If current state is θ i n , a new state can be drawn from a proposal distribution q ( θ i + 1 n | θ i n ). Then the acceptance probability of the new state is evaluated by The probability of whether actually moving from current state θ i n to new proposed state θ i + 1 n is determined by α . It can be proven that after su ffi cient walking, the equilibrium distribution of the walker will converge to the target posterior distribution p ( θ n | d ). Although, in practical problems, the MHMCMC usually needs to be modified in various aspects to increase sampling e ffi ciency or avoid bias, the above description presents the most basic conception of the MCMC sampling algorithm for fixed dimensional problems. In problems of uncertain dimensionality, to allow the walker to jump between di ff erent parameter spaces, the RJMCMC algorithm draws new proposals by a di ff erent procedure [122, 123]. Firstly, a random vector u with the dimension of r is generated from a chosen probability distribution g ( u ). Then, the proposal state is generated through a transformation The corresponding inverse transformation is given by where u ' is a vector with size r ' and satisfying n + r = n ' + r ' . The only requirement for the transformation is that f and f -1 need to be di ff erentiable. The dimensions of parameter space n and n ' do not need to be the same, and even the parameterization associated with θ i n and θ i + 1 n ' can be di ff erent. The acceptance ratio for the new proposed state is given by The term J is the Jacobian defined by to account for the change of volume element under the transformation of parameter spaces. In general, the Jacobian can be di ffi cult to compute. However, in the situation of global fitting for GCB signals, where it only involves adding or removing GCB signals with the same set of parameters, the Jacobian can be easily computed by the ratio of prior volumes of parameter spaces before and after the transformation. The walker of RJMCMC can freely explore the entire possible parameter space including not only parameter spaces of individual sources but also parameter spaces for all possible source numbers. The estimation of source number can be naturally obtained through comparing the iteration numbers of the walker lingered in the di ff erent parameter spaces, and the Bayesian factor for the assumptions of di ff erent source numbers can be given by the ratio of the iteration numbers in di ff erent parameter spaces. Although the movement rule of RJMCMC can be explicitly presented as above, the implementation of the RJMCMC algorithm in practice to solve the problem of overlapping GCB signals is very challenging. On the one hand, since all sources are simultaneously fitted, the parameter spaces are extremely high dimensional. Even the whole frequency band is usually divided into small bins and di ff erent bins can be analyzed independently after carefully addressing edge e ff ects. The source number in a single small bin can still be considerable. For example, the default setting of the prior range for the source number in each bin is [0 , 30] in the work [44]. As mentioned in Section 2, a GCB signal is characterized by 8 parameters. The maximal dimension of parameter space can be 240. And the posterior can have the feature of multimodality, which requires the sampler has the ability to deal with complicated likelihood surfaces in high-dimensional parameter spaces. On the other hand, the uncertainty of source number requires the sampler to jump between di ff erent parameter spaces, which further extremely expands the space needed to explore. Furthermore, since the likelihood in vast regions may be very small. Only if the proposal is enough close to the true values, it can likely be accepted. Generating proposals entirely uniformly may lead extremely low acceptance rate, especially for between-model proposals. One need to design good strategies for random walking to increase the sampling e ffi ciency.",
"pages": [
10,
11
]
},
{
"title": "3.3.2. Global fitting piplines",
"content": "Currently, available full-scale and end-to-end global fitting pipelines for GCBs include GBMCMC [44-46] and Eryn [48, 49]. Various intelligence methods are utilized to overcome di ffi culties in the global fitting with RJMCMC. For example, the method of parallel tempering [124-126] is used in both GBMCMC and Eryn. In parallel tempering, multiple walkers are randomly moving in parallel at di ff erent temperatures T . The likelihood is modified as where π ( θ ) is the prior and p ( d | θ ) is the original likelihood function. When T = 1, pT ( θ | d ) returns to the target posterior distribution. When T → ∞ , pT ( θ | d ) approaches to the prior distribution. Higher temperatures correspond to more exploration where walkers can escape from local optima more easily, while lower temperatures correspond to more exploitation where walkers can sample the small promising regions in more detail. During the random walking, state exchanges are attempted between walkers at di ff erent temperatures. The probability of acceptance of exchange proposals is determined by the ratio The detailed balance is maintained under this state exchange between walkers, which ensures that the equilibrium distribution of walkers can converge to the target posterior distributions. The temperature ladder needs to be chosen for maximizing the information flow among walkers at di ff erent temperatures. Ideally, one may expect an equal acceptance ratio between every pair of walkers with neighboring temperatures. The GBMCMC and Eryn adopt the scheme presented in [124] to set the temperature ladder, where the temperature space is dynamically adjusted in the initial burn-in phase to obtain a stable configuration for the rest of random walking. The parallel tempering mechanism can help samplers e ffi ciently explore the complicated likelihood surface with high multimodality. The strategies for drawing proposals play a crucial role in the sampling e ffi ciency. Customized proposal distributions are developed in GBMCMC to enhance the sampling e ffi ciency. Ideal proposal distributions would be identical to posterior distributions or likelihood functions. The likelihood of multiple overlapping signals consists of three parts, the correlation between each signal and noise, the correlation of noise itself, and the cross-correlations among overlapping signals. As discussed in [44], high correlations among overlapping signals are relatively rare. Therefore, one can enhance proposal distributions focusing on individual signals. Although the cross-correlations of overlapping signals are neglected when designing the proposal distributions, the sampler still explores the joint parameter space of multiple overlapping signals which incorporates cross-correlations of overlapping signals. The design of proposal distribution in GBMCMC utilizes the F -statistic, the feature of multimodality due to orbital motion of detectors, and posteriors from former epochs of observation. The method of F -statistic is widely used in various implementations of iterative subtraction as presented in previous sections, it can also be used here to construct the proposal distribution for a single source. GBMCMC uses the F -statistic to build proposals for parameters ( f 0, λ , β ) which are precomputed on a grid with spaces determined by estimation of the Fisher matrix. The values of F -statistic can be approximated to the original likelihood values, thus the F -statistic proposals are expected to have a high accepted rate. As mentioned in Section 3.2.2, the modulation induced by orbital motion can cause sidebands around the central frequency f 0 of quasi-monochromatic GCB signals. The likelihood surface have the feature of multiple modes around f 0. GBMCMC designs a dedicated proposal to update the frequency parameter by shifting f 0 with modulation frequency f m as f 0 → f 0 + nf m, where f m = 1 / year. The multimodal proposal can address this known degeneracy and help to improve the convergence of the sampler. In reality, the data collection is gradually incremental. It is impractical to wait until after the finish of entire observation to analyze the obtained data. Data analyses must be performed accompanying with data accumulation. Therefore, one can utilize posteriors obtained from former periods of observation to construct posterior-based proposal distributions in parameter estimation with more observed data, which can significantly improve the sampler convergence. The results of GBMCMC with simulated GCB dataset are demonstrated in Figure 6 where the posteriors of sky location of identified GCBs are plotted. It can be seen that with the incremental data, the GCBs can be better localized and the structure of the Galaxy is clearer indicated. Another independent implementation of RJMCMC referred to as Eryn [48, 49] which adopts di ff erent sophisticated mechanisms in stochastic sampling to overcome poor convergence in RJMCMC. Eryn is based on the ensemble sampler emcee [127] which uses multiple interacting walkers exploring the parameter space simultaneously with the so-called stretch-move proposal to enhance the e ffi ciency and convergence of the sampling. Whereas, in order to address the trans-dimensional movement and the heavily multimodal likelihood surface of overlapping GCBs, Eryn extends the stretch-move proposal in origin emcee sampler to the group proposal which sets a stationary group of walkers and makes the state updates likely to happen within one same mode in the likelihood surface. Furthermore, Eryn build e ffi cient proposals through a data-driven approach, where proposals can be drawn from a fitted distribution constructed by posteriors obtained in burn-in runs with residual data after subtracting bright sources. To overcome the problem of low acceptance rate, Eryn implements two mechanisms called delayed rejection [128, 129] and multiple try metropolis [130-133]. However, due to the high computational requirement of delayed rejection, although this mechanism is implemented in Eryn, it is not currently used in solving the problem of global fitting for GCBs. Another important feature of Eryn is the utilization of GPU which reduces the wall-time to perform the whole analysis by parallel computing on contemporary computational hardware. The ultimate goal of global fitting is simultaneously analyzing all kinds of sources contained in data including MBHBs, EMRIs, and even unmodeled signals, etc. In the work [45] and [49], GBMCMC and Eryn are incorporated within global fitting pipelines that can handle blended data of MBHBs and GCBs. Global fitting for multiple source types can be realized by the blocked Gibbs scheme considering that correlations among different source types are relatively small. Parameter estimations for di ff erent source types can be implemented in separate modules, and these modules are assembled by a wheel update strategy where one can perform updates in one module conditioning on fixed other modules and all modules are updated periodically. Figure 7 shows the results given by a global fitting pipeline incorporating GCBs and MBHBs where the fitting of GCBs is accomplished by Eryn. In a short summary, fitting overlapping GCB signals through the full-Bayesian approach can be realized by RJMCMC. Practical implementations of RJMCMC have to overcome various di ffi culties such as the heavy multimodality of likelihood surface in vast parameter space, low acceptance rate of trans-dimensional proposals, and poor convergence of random walker, etc. which requires elaborate MCMC algorithms. Prototype global fitting pipelines for GCBs have been accomplished by GBMCMC and Eryn which can well address simulated data and are successfully incorporated within the global fitting pipelines for blended data of MBHBs and GCBs. The global fitting with RJMCMC is a full Bayesian approach and can provide joint posteriors of overlapping signals, which can well account for correlations among overlapping GCBs and avoid inaccurate subtraction contamination in the iterative subtraction scheme. Whereas the full Bayesian approach requires massive computational cost and complicated MCMC algorithms which are di ffi cult to implement.",
"pages": [
11,
12
]
},
{
"title": "3.4. Hybrid Bayesian approach",
"content": "The iterative subtraction strategy o ff ers solutions with high e ffi ciency where the computation burden is modest and analysis can be finished in a relatively short time. Whereas, in each iteration, only the maximum likelihood estimation for a single source is extracted, the correlation of overlapping signals can not be well accounted for. The inaccuracy of estimation in each iteration will contaminate the remaining data. And uncertainty analyses for the identified signals are di ffi cult. The full Bayesian approach with RJMCMC where overlapping signals are fitted simultaneously can e ff ectively overcome these di ffi culties, but with the cost of extremely massive demand on computational resources. In the works [50-52], a hybrid approach where the maximum likelihood estimation is performed first to find the approximate values of signal parameters, and then the MCMC sampling is used to obtain posteriors of signal parameters is proposed aiming at combining the advantages of the maximum likelihood estimation and the Bayesian parameter estimation while evading their drawbacks. Similar to the solutions of iterative subtraction introduced previously, the hybrid Bayesian approach also needs to identify the maxima of likelihood in the first step. However, the methods used for searching maxima are di ff erent. In this hybrid Bayesian approach, the o ff -the-shelve algorithms of the di ff erential evolution (DE) [134, 135] and the sequential least squared programming (SLSP) [136] implemented in the scipy library [137] are adopted for searching the maxima in the likelihood surface. The DE method is used to identify the candidates at first through the iterative subtraction scheme, where only one signal is fitted each time the fitting is repeatedly performed with the remaining data after subtracting the best-fit signal. Then, in . order to address the correlation of overlapping signals, using all found candidates as the start, the global optimizations are performed by the SLSP method to search the maximum likelihood in the joint parameter space of all candidates. The DE algorithm is a simple and e ffi cient method for optimization problems that searches the optima by manipulating a population of potential solutions similar to PSO introduced in Section 3.2. The DE algorithm is initialized by a population of candidate solutions drawn randomly within the search space. Then, in following iterations, the algorithm generates new potential solutions by combining and mutating existing individuals in the population. There are various strategies [135] for generating new solutions and updating populations, which may suit di ff erent problems and need to be determined through experimental runs. The populations will be continuously updated by replacing individuals with better fitness and eventually converge to the optima. The SLSQ method is a gradient-based optimization method suits for problems with smooth and continuously di ff erentiable likelihoods. After the iterative subtraction search with the DE algorithm, the SLSQ method starts a global search with the initial values given by the DE algorithm, and iteratively steps towards a better solution which incorporates correlations of overlapping signals by the line search in a direction found through derivatives of the likelihood surface. As examples, the true and recovered signals on a small frequency band are illustrated in Figure 8. In the second step, The MCMC sampling algorithm is performed to obtain the posterior distribution for each identified signal. The MCMC algorithm for individual sources is similar to methods widely used for parameter estimation in data analysis of ground-based detectors [138, 139], except that the data used in likelihood evaluation are the remains after the subtraction of identified signals in the first step excluding the signals to be analyzed, which can be expressed by where d ( i ) posterior is the data used in likelihood evaluation for the source i , and ˆ θ i denotes the parameters of maximum likelihood estimation obtained in the first step for the source i . One drawback is that the above MCMC sampling method for individual signals cannot account for the correlation of overlapping signals, which may lead to overoptimistic posterior distributions. To address this, some level of residual signals are intentionally left in the data used for estimating the noise characterization. The partial residual data used to obtain the noise PSD can be expressed as where s partial is a factor from the the range of [0 , 1] which has to be determined experientially and is set to s partial = 0 . 7 in the work [52]. Various methods are used to improve the e ffi ciency of MCMCsampling in the second step, for example, constraining the search space only within the promising region [52], or using Gaussian progress regression to model the likelihood [51]. The posterior distributions are typically concentrated within small regions of parameter space. To relieve the computational burden, one can only explore the reduced parameter space based on the maximum likelihood estimation obtained in the first step. The Fisher matrix is used to determine the boundaries of the reduced parameter space. The Fisher matrix is widely used in forecast works [140, 141], which is an approximation of the Bayesian method under the assumption of high SNR and can provide estimations for parameter measurement uncertainty. The Fisher matrix is given by where ˆ θ is the maximum likelihood estimation obtained in the first step, and the angle brackets are the inner product defined as Equation 9. The inverse of the Fisher matrix presents the estimation of the covariance matrix of parameter measurement. The boundaries of reduced parameter space for the MCMC sampling are determined by the variance of parameters estimated through the Fisher matrix. Typically, 3σ regions are su ffi cient, while the practical settings need to be adjusted according to the features of posterior distributions in experimental runs, and the tolerance of computational burden or the desired coverage of the parameter space. Additionally, in the procedure of MCMC sampling, the likelihood has to be evaluated a huge number of times. The computational cost for likelihood evaluation is one of the main barriers to MCMC sampling. In the work [52], the contemporary computational hardware is used to perform the likelihood evaluation. The likelihood can be computed on GPU in massively parallel, which can significantly reduce the wall-time required for finishing the MCMC sampling [77, 94]. On the other hand, as shown in the work [51] the likelihood can be modeled by the Gaussian process regression, where the likelihood is modeled by a joint Gaussian distribution whose mean vector and covariance matrix are determined by training samples [142]. The computation of Gaussian distributions can be much faster than the likelihood defined through the inner product in Equation 8. In the work [51], 1000 random samples are first drawn for training the Gaussian process regression model, and 500 samples are used for verification. After this, The likelihood is replaced by the Gaussian process regression model in following MCMC sampling, which can reduce the computational cost of the entire sampling process. The training and evaluating of the Gaussian process regression model are implemented through the scikit-learn package [143]. The hybrid Bayesian approach combines the maximum likelihood estimation and the MCMC sampling. A search of the iterative subtraction scheme is first performed to identify potential candidates which are used as the start in the subsequent global optimization in the joint parameter space for all overlapping signals. Then the MCMC sampling algorithm is performed for each identified signal to obtain the posterior distributions. This hybrid Bayesian approach uses the strategy of iter- ative subtraction to determine the source number and the point estimation for source parameters, which can avoid the computationally expensive trans-dimensional MCMC sampling. Meanwhile, after the iterative subtraction search, the source parameters are again globally optimized around the identified candidates, which can rectify errors induced by inaccurate subtraction contamination and correlations of overlapping signals. However, since the MCMC sampling is performed for individual signals, only the marginalized posteriors can be obtained for each source. Incorporating additional uncertainty induced by signal overlapping requires tuning the algorithm configuration empirically.",
"pages": [
12,
13,
14,
15
]
},
{
"title": "3.5. Utilization of machine learning techniques",
"content": "The development of machine learning techniques has led to revolutions in many fields including the data analysis of GWs. Various machine learning algorithms have been successfully used in tasks of signal identification and classification [144-146], parameter estimation [53-58], noise reduction [147], waveform modeling [148, 149], etc. Comprehensive reviews can be found in [150, 151]. Utilizing machine learning techniques in data analyses of space-borne detectors is also actively discussed [152-156]. A preliminary attempt of using the machine learning method to extract overlapping GCB signals is presented in the work [59] where normalizing flows are used to build proposals for improving the convergence of MCMC sampling and o ff er a new method for sharing analysis results. Normalizing flows are a class of machine learning algorithms that can model complicated distributions and are widely used in density estimation problems [157-160]. Normalizing flows model a probability distribution through an invertible and di ff erentiable transformation f : X → Y that can map the simple and tractable base probability distribution P X ( x ) to the complicated target probability distribution P Y ( y ). The target distribution can be given by where | det J f -1 ( y ) | is the determinant of the Jacobian of the transformation f -1 accounting for the change of volume elements of parameter spaces. The transformation is parameterized by neural networks and can be composed of a sequence of transformations. Various kinds of tranformations are developed, a comprehensive review can be found in [158]. In the training procedure, the random samples are drawn from the known target distribution, and the neural network is trained by mapping these samples into the simple base distribution. In the inference procedure, the above process is performed inversely, the target distribution is obtained by transforming the random samples drawn from the base distribution through the trained neural network. Drawing samples from the base distribution and computing the transformation can usually be much faster than computing the likelihood defined by the inner product. Once the training procedure is finished, normalizing flows can generate posterior distribution more e ffi ciently than the MCMC sampling. In the work [59], the neural density estimation with normalizing flows is used in three di ff erent aspects. In the first, the normalizing flows are used to build physical priors for amplitude and sky location based on population models. The spatial distribution of GCBs mainly concentrates on the Galactic disk, rather than uniformly distributes on the whole sky. The amplitude depends on the binary masses, distance, and orbital period which can also provided by population synthesis models [21, 161-163]. Utilizing available information can help to constrain the parameter space and improve the e ffi ciency of MCMC sampling. Training with the simulated GCB catalog in the LDC2a [35], the physical priors for amplitude and sky location can be constructed through normalizing flows. In the second aspect, normalizing flows are used to construct proposal distributions from results given by former epoch observations. Constructing proposal distributions from available samples to improve sampling e ffi ciency has been proposed earlier in [164] where the density fit method of kernel density estimation (KDE) is used to build proposal distributions. However, the KDE method has drawbacks for high-dimensional problems where one needs to divide parameters into di ff erent groups and build KDE proposal distributions in low-dimensional parameter subspaces. Normalizing flows provide an alternative way to build proposal distributions from available samples with its capability of modeling complicated distributions. Available samples used to build proposal distributions can be obtained in two cases. As mentioned before, the data are incrementally collected, and it is unpractical to analyze data waiting until the end of space-borne detector missions. In reality, data analyses need to be repeatedly performed with growing data. It is essential to fully utilize the results obtained previously when analyzing updated data. Normalizing flows can fit the posterior distributions obtained in previous analyses, and be used to draw proposals in subsequent analyses, which is expected to have high acceptance rate and low autocorrelation of chains. Meanwhile, for the trans-dimensional MCMC, by utilizing the capability of normalizing flows to model complicated distribution, the proposal distributions can be constructed through candidates identified by an ahead iterative subtraction procedure. Thirdly, the result posteriors can be published through normalizing flows. As mentioned in Section 2, the number of resolvable GCBs is expected to be ∼ O (10 4 ). Moreover, due to the overlap among signals, parameters of di ff erent sources may have correlations which can not represented by marginalized posteriors for individual sources. One may need a joint posterior for multiple sources. Using samples to represent posteriors may have higher demands for data transfer and storage, and will be inconvenient when sharing the analysis results with the community. Normalizing flows o ff er an e ff ective way to model complicated distributions. Therefore, the results sharing and data product publishing can be in the form of normalizing flow models trained by original posterior samples, from which users can generate samples of the identical distribution for any number needed. In summary, normalizing flows o ff er a tool to fit arbitrary complicated distributions, which can be used to construct physical priors from simulated GCBs catalogs, build proposal dis- tributions from available samples, and as a new representation of posteriors to share with the community substituting posterior samples. Although a full-scale and end-to-end search pipeline for GCBs based on machine learning techniques is not available up to now. Related algorithms are demonstrated to be e ff ective and are being incorporated into global fit pipelines as reported in the roadmap mentioned in [59].",
"pages": [
15,
16
]
},
{
"title": "4. Unresolvable sources",
"content": "As mentioned in Section 2, the individually resolvable GCBs are only a small fraction of the total sources. The remains will form a stochastic foreground and contribute to the confusion noise in the data of space-borne detectors. On the one hand, the unresolvable GCBs play the role of noise and a ff ect the observations of other types of sources. On the other hand, the unresolvable sources can still provide invaluable information about the Galaxy.",
"pages": [
16
]
},
{
"title": "4.1. Seperation of the stochastic Galactic foreground",
"content": "The Galactic confusion noise will degrade the sensitivity of detectors and reduce the SNR of other sources [60-63]. Additionally, due to the orbital motion of detectors, the pointing of constellations will constantly change relative to the Galactic center where GCBs are concentratedly distributed, which influences the response to the population of unresolvable GCBs and induces the modulation of the Galactic confusion noise. Therefore the Galactic confusion noise is nonstationary, adding more challenges in data analyses of space-borne detectors [165, 166]. Furthermore, the foreground of unresolvable GCBs will blend together with the extra-galactic stochastic background of astrophysical or cosmological origin. It is important to separate or subtract the Galactic foreground for studies of other stochastic GWbackground signals [64-69]. The Galactic stochastic foreground can be separated from other components of stochastic signals or noise thanks to its distinct spectral shape and the modulation induced by the orbital motion of detectors. For the separation, one needs to first properly model the di ff erent components. The stochastic signals can be characterized by the cross spectrum ⟨ ˜ si ( f ) ˜ s ∗ j ( f ) ⟩ where i and j denotes di ff erent TDI channels. Here the angle brackets denote the ensemble average. The total stochastic signals may be contributed by three components including the confusion foreground originating from unresolvable GCBs, the stochastic background of extra-galactic origin, and the instrument noise as There are various models developed to describe these components. For example, a typical model for extra-galactic stochastic background reads where Rij is the response functions of detectors for di ff erent channels averaged over sky locations and polarizations, H 0 is the Hubble constant, m is the spectral index, and A ∗ is the amplitude at the reference frequency 1 mHz. This power-law model is widely used in both stochastic signals of cosmological origin [167] and astrophysical origin [168]. More models for extragalactic stochastic background can be found in [64]. Modeling instrument noise can be extremely complicated involving detailed studies about the electronic systems, optical systems, and space environments of detectors, etc. However, in discussions of data analyses or science problems, a simplified noise model can be used, which groups noise into two components, Interferometry Metrology System (IMS) noise and acceleration noise, characterized by two quantities S IMS and S acc respectively [87, 169]. Detailed derivation for noise spectral densities and correlations of di ff erent channels for di ff erent TDI generations or levels of approximation can be found in [64, 87]. For modeling the stochastic Galactic foreground, one simplified method is utilizing the analytic fitting [170-172] which reads where fitting parameter B ∗ controls the overall amplitude, fk controls the position of the knee-like feature in the spectrum of Galactic foreground, together with α , β , γ , and κ describe the spectral shape of Galactic confusion foreground. In practice, one can only vary the overall amplitude in parameter estimation while taking the values of other parameters fitted through simulations of GCB population [64]. However, in this simplified model, the features of anisotropy and non-stationary are averaged and not incorporated. To fully exploit the distinctive features of the Galactic confusion foreground, a numerical model [66] can be used. In this numerical model, the modulated spectrum of the Galactic foreground is characterized by 17 Fourier coe ffi cients [66, 173, 174] which are treated as free parameters varied in parameter estimation, and their prior ranges are obtained through multiple runs of GCB population simulation. The simulations of GCB population can be based on the Galaxy model constrained by the bright and individually resolvable GCBs [175]. To properly incorporate the modulation induced by detector orbital motion, the entire data are divided into segments of week-long during which the responses of detectors to Galactic confusion foreground are considered to have no appreciable changes. After properly modeling all components of stochastic signals, one can use Bayesian inference to obtain estimations of free parameters in models. The likelihood is given by which is the probability of occurrence of observed data if the stochastic signals and noise are governed by the chosen models with parameters θ . Here, s denotes the entire observed data, θ denotes parameters required to describe models for all components of stochastic signals, n labels the data samples, d labels di ff erent segments used for addressing the modulation of Galactic confusion foreground, i and j represent the di ff erent TDI channels. Cd , i j is the correlation matrix depending on chosen models for each components in stochastic signals. The product needs to run all data samples and all segments. Posterior distributions for model parameters can be obtained through MCMC sampling, from which one can get the separation of di ff erent components of stochastic signals and noise. Except for the stochastic background signals of extra-galactic origin, subtracting Galactic confusion foreground also benefits observations of individual sources as presented in [176] where the technique referred to as dictionary learning is used to reconstruct MBHB signals with low-SNR. The basic idea of dictionary learning [177] is representing a target signal h ( t ) through a set of basis elements called dictionary D and a sparse coe ffi -cient vector α as h ∼ D α . The dictionary can be predefined by training dataset created through signals of individual MBHBs without noise, and the coe ffi cient vector α for a given MBHB signal is searched in the presence of noise. Their combination provides a reconstruction of the signal hidden in the noisy data. As demonstrated in the reference [176], low-SNR massive black binaries can by successfully separated in the presence of Galactic noise through this method.",
"pages": [
16,
18
]
},
{
"title": "4.2. Tools for studying the Galaxy",
"content": "The old stellar population is one of the excellent tools to trace the dynamical evolution of the Galaxy [178]. However, these sources are usually dim and di ffi cult to be observed in the electromagnetic band. GCBs make up the majority of the total old stellar population and their GW signals are not a ff ected by crowded matters in the Galaxy. GW Observations of GCBs o ff er a unique tool to study the Galaxy. Although, it has been demonstrated that the individual resolvable GCBs which have better measurement and localization can already trace the structure of the Galaxy [26, 27, 179]. These resolvable sources are usually more massive or nearby, which may require careful notice of potential bias [162]. The stochastic foreground containing the contributions of the full GCB population can also reveal various properties of the Galaxy [70-72]. In one aspect, one can extract information about the Galaxy from the spherical harmonic decomposition of the Galactic confusion foreground [70]. The angular power spectrum of the foreground that encodes information of the GCB spatial distribution can be obtained from the observed cross spctrum of di ff erent TDI channels by the framework presented in [180]. It is pointed out that in [70] the scale height of GCB distribution can be e ff ectively constrained using the hexadecapole moment of the spherical harmonic expansion. Although the constraint provided by this methodology has limited sensitivity comparing to the method with resolvable GCBs, this approach may be less a ff ected by the observation bias and provides a complementary way to measure the structure of the Galaxy. Another approach for obtaining information about the Galaxy is through the spectral shape and amplitude of the Galactic confusion foreground [72]. This methodology is similar to fitting Galactic confusion foreground with the analytic model of Equation 38 mentioned in the last section. Whereas the fitting parameters can be mapped to the physical parameters describing properties of the Galaxy. Thus, one can obtain information of the Galaxy like the total stellar mass as discussed in [72] through the similar parameter estimation procedure discussed in the last section from the Galactic confusion foreground.",
"pages": [
18
]
},
{
"title": "5. Summary",
"content": "Space-borne detectors will open the windows of the low GW band in the near future, which can provide new tools to explore the Universe. In contrast to ground-based detectors whose data is noise-dominant, data from space-borne detectors will be signal-dominant where signals are more crowded and various sources are tangled together. The abundance of sources can present plentiful invaluable information on the one hand, but also play the role of noise to disturb source detections and measurements on the other hand. The heavily overlapping signals pose new challenges for the data analyses of space-borne detectors. Among various source types targeted by space-borne detectors, the vast population of GCBs is likely the type having the most number to be detected. It is expected that there are tens of millions GCBs in the mHz band, while tens of thousands massive or nearby sources among them are individually resolvable and the remains will form a stochastic foreground contributing to the confusion noise. The GWs from GCBs are continuous signals and have the feature of quasi-monochrome. In the time domain, GCB signals with overwhelming numbers exist in the data of space-borne detectors simultaneously during the entire mission period. In the frequency domain, although a single GCB signal is only extended within a narrow band, due to their vast number, GCB signals can still heavily overlap in frequency. GCBs are likely the type that has the most heavy overlap and correlation. Separation and extraction of overlapping signals focusing on GCBs may be the foundation of the ultimate data analysis pipelines for globally separating all kinds of sources. Therefore, in the paper, we present a comprehensive review of precious e ff orts dedicated to the separation and extraction of GCB signals. Current solutions for separating overlapping GCB signals can be mainly categorized into two classes, iterative subtraction and global fitting. The strategy of iterative subtraction searches for the optimal fitting for just one signal in each iteration and the same procedure will be performed repeatedly with the remaining data after the subtraction of identified signal. A typical iterative subtraction solution referred to as GBSIEVER employs the F -statistic likelihood and PSO to search the optimal fitting in each step. To address the problem of inaccurate subtraction contamination for low SNR sources, a di ff erent implementation of iterative subtraction uses a tuned particle movement rule of PSO aiming at identifying all local maxima in the likelihood surface. Then, the local maxima corresponding to false candidates induced by degeneracy of individual signals and overlapping signals are removed. Astrophysical properties of GCBs are incorporated to further filter out the low credible candidates. Another implementation of iterative subtraction focuses on improving the e ffi ciency in the PSO searching. Before the fine PSO search procedure, a template search is performed to obtain the coarse estimation of existing signals. Another strategy is global fitting which estimates the parameters of all sources together with the source number simultaneously through the full Bayesian approach. The full Bayesian approach can provide the joint posterior distributions for multiple overlapping signals rather than the only point estimation given in the iterative subtraction scheme, which can well describe the correlation among overlapping signals and provide a straightforward way for uncertain analysis and model selection. Posterior estimation over the parameter space with an uncertain dimension is performed through RJMCMC. To overcome the poor convergence in the trans-dimensional MCMC sampling, various tuned proposal distributions according to the features of GCB signals and sophisticated random sampling mechanisms are adopted in di ff erent implementations. Besides typical solutions of above two schemes, a hybrid approach that combines the point estimation of maximum likelihood and Bayesian posterior estimation is also proposed. A procedure of iterative subtraction is first performed to identify potential candidates, and a global optimization of all identified signals is performed again to relieve errors induced by correlations of overlapping signals. Then, the MCMC sampling is performed for each found signal to obtain the marginalized poste- rior distributions of individual sources. Machine learning techniques are considered as very promising tools in future GW data analyses. Neural density estimation methods are investigated for assistance in drawing proposals and sharing the posterior results with the advantage of normalizing flows at e ffi ciently modeling complicated distributions. The resolvable GCBs are only a small fraction of the entire population. The remaining sources will form a stochastic foreground contributing to the confusion noise. The Galactic confusion foreground on the one hand is a type of noise a ff ecting the observation of other sources, on the other hand provides a powerful tool to research the structure of the Galaxy with the advantage that GWs will not be suppressed or obscured by crowded matters in the Galaxy. Thanks to the features of the di ff erent spectral shape and the time-evolving modulation, the stochastic Galactic foreground is distinguishable with the instrument noise and the extra-galactic astrophysical or cosmological original background. And the distinct spectral shape of Galactic foreground can also be used in the measurement of properties of the Galaxy. The anisotropy of the Galactic foreground is associated with the stellar population distribution of the Galaxy, thus the Galactic structure can be constrained through the spherical harmonics decomposition of the Galactic foreground. For future works, although there are already diverse end-toend prototype pipelines for extracting overlapping GCBs, various problems still need to be addressed for actually handling real data. For example, in current solutions the noises are usually assumed to be stationary and Gaussian, while in reality there is slow drift of instrument noise in the long period and glitches in the short period which are neither stationary nor Gaussian [181, 182]. Better noise modeling may required for future works. Besides, gaps will exist unavoidably in the data stream due to various reasons including scheduled maintenance or unplanned random events [183, 184]. Future pipelines may need to incorporate processing of various potential defects existing in data. Furthermore, the data from space-borne detectors blend all sources of di ff erent types, the ultimate pipelines need to be capable of addressing di ff erent kinds of overlapping signals. New independent implementations or enchantments of current pipelines are anticipated for preparation of the launch of future space-borne GW observation missions.",
"pages": [
18,
19
]
},
{
"title": "6. Acknowledgments",
"content": "The authors have no conflict of interest. W. Z. is supported by the National Key R&D Program of China (Grant No. 2022YFC2204602 and 2021YFC2203102), Strategic Priority Research Program of the Chinese Academy of Science (Grant No. XDB0550300), the National Natural Science Foundation of China (Grant No. 12325301 and 12273035), the Science Research Grants from the China Manned Space Project (Grant No.CMS-CSST-2021-B01), the 111 Project for 'Observational and Theoretical Research on Dark Matter and Dark Energy' (Grant No. B23042) and Cyrus Chun Ying Tang Foundations. R. N. is supported in part by the National Key Research and Development Program of China Grant No.2022YFC2807303.",
"pages": [
19
]
},
{
"title": "References",
"content": "doi: 10.1093/mnras/stab2479 . grounds with a space-based interferometer. ii. perturbative reconstruction of a low-frequency skymap, Physical Review D 72 (2005) 104015. doi: 10.1103/physrevd.72.104015 .",
"pages": [
21,
24
]
}
]
2024arXiv240615435C
https://arxiv.org/pdf/2406.15435.pdf
<document>
<section_header_level_1><location><page_1><loc_16><loc_75><loc_84><loc_76></location>Optimizing blind reconstruction of CMB B-modes for future experiments</section_header_level_1>
<section_header_level_1><location><page_1><loc_46><loc_71><loc_54><loc_72></location>A. Carones</section_header_level_1>
<text><location><page_1><loc_21><loc_68><loc_79><loc_70></location>Dipartimento di Fisica, Universit'a di Roma Tor Vergata & INFN Sez. Roma 2, via della Ricerca Scientifica 1, I-00133, Roma, Italy</text>
<text><location><page_1><loc_19><loc_55><loc_81><loc_66></location>The detection of primordial polarization B modes of the Cosmic Microwave Background (CMB) requires exquisite control of Galactic foreground contamination. The Needlet Internal Linear Combination (NILC) method has proven effective in reconstructing CMB B modes without suffering from mis-modeling errors of Galactic emission. However, with the most complex foreground models, residual Galactic contamination from NILC is proved to bias, especially at large angular scales, the recovered CMB B modes from simulated data of future CMB experiments. We therefore present two new extensions of NILC, Multi-Clustering NILC (MC-NILC) and optimized constrained Moment ILC (ocMILC), which allow to enhance foreground subtraction in the reconstructed CMB signal.</text>
<section_header_level_1><location><page_1><loc_12><loc_51><loc_51><loc_52></location>1 Blind reconstruction of CMB B modes</section_header_level_1>
<text><location><page_1><loc_12><loc_36><loc_88><loc_49></location>The detection of primordial polarization B modes of the Cosmic Microwave Background (CMB), targeted by upcoming CMB experiments, is hampered by the contamination of Galactic foregrounds. This points to the need for robust component separation techniques to be applied to future microwave observations. The reference model-independent ( i.e. blind) cleaning technique is the Needlet Internal Linear Combination 1 (NILC). In the NILC pipeline, input multi-frequency B -mode maps B i are first decomposed into needlet coefficients β i j through a deconvolution for a needlet kernel ψ j : β i j (ˆ γ ) = ψ j ⊛ B i (ˆ γ ). At each needlet scale j , sampling a specific range of angular scales, needlet maps are then linearly combined with frequency-dependent weights w i j :</text>
<formula><location><page_1><loc_38><loc_31><loc_88><loc_35></location>β NILC j (ˆ γ ) = N ν ∑ i =1 w i j (ˆ γ ) · β i j (ˆ γ ) , (1)</formula>
<text><location><page_1><loc_12><loc_19><loc_88><loc_30></location>so as to recover a blackbody solution β NILC j with minimum variance. The derivation of NILC weights, w j (ˆ γ ) = [ A T CMB C -1 j (ˆ γ ) A CMB ] -1 A T CMB C -1 j (ˆ γ ), requires only knowledge of the CMB spectral energy distribution (SED), A CMB , and the computation of the input needlet covariance matrix: C ik j (ˆ γ ) = ⟨ β i j (ˆ γ ) · β k j (ˆ γ ) ⟩ , with ⟨⟩ a local sample average within Gaussian domains, whose width varies with the needlet scale j . The final B -mode CMB map, ˜ B CMB , is reconstructed through an inverse needlet transform of NILC solutions at the different needlet scales:</text>
<formula><location><page_1><loc_24><loc_15><loc_88><loc_18></location>˜ B CMB (ˆ γ ) = ∑ j ψ j ⊛ β NILC j (ˆ γ ) = B CMB (ˆ γ ) + B fres (ˆ γ ) + B nres (ˆ γ ) , (2)</formula>
<text><location><page_1><loc_12><loc_8><loc_88><loc_14></location>and includes residual contamination by foregrounds ( B fres ) and instrumental noise ( B nres ). The contribution of B nres to the output angular power spectrum can be removed, whereas that of B fres , if relevant at the reionization peak ( ℓ ≲ 10) or at the recombination bump ( ℓ ∼ 80), may bias the estimate of the tensor-to-scalar ratio.</text>
<figure>
<location><page_2><loc_12><loc_75><loc_86><loc_91></location>
<caption>Figure 1 - Angular power spectrum (averaged over 50 simulations) of foreground (solid) and noise (dashed) residuals in reconstructed CMB B -mode maps compared against a reference range of tensor primordial CMB spectra (gray shaded area). Left: results from the application of NILC (orange), MC-NILC with ideal clusters (green), and MC-NILC with real clusters (red) on LiteBIRD sky simulations, with spectra computed in a sky fraction f sky = 50%. Left: results from the application of NILC (orange) and ocMILC (brown) on PICO sky simulations with spectra computed in a sky fraction f sky = 70%. Power spectra are binned with ∆ ℓ = 4.</caption>
</figure>
<section_header_level_1><location><page_2><loc_12><loc_62><loc_57><loc_63></location>2 Optimizing blind CMB component separation</section_header_level_1>
<text><location><page_2><loc_12><loc_44><loc_88><loc_60></location>We consider simulations of LiteBIRD 2 and PICO 3 satellite experiments, targeting a sensitivity on the tensor-to-scalar ratio of σ ( r ) = 0 . 001 and 0 . 0002, respectively. Galactic foreground emission is simulated with the PySM python package 4 assuming the dust d1 model (modified blackbody SED) and synchrotron s1 model (power-law SED), both featuring spatial variations of the corresponding spectral parameters. After proper masking, at large angular scales, the NILC foreground residuals still have power comparable to that of the targeted amplitude of tensor modes, as shown by solid orange lines in the left ( LiteBIRD ) and right ( PICO ) panels of Fig. 1. As demonstrated in 5 and 6 , these residuals would produce a non-negligible bias on the estimated tensor-to-scalar ratio. We thus present two alternative optimizations of NILC: Multi-Clustering NILC 5 (MC-NILC) and optimized constrained moment ILC 6 (ocMILC).</text>
<text><location><page_2><loc_12><loc_28><loc_88><loc_44></location>The MC-NILC technique implements variance minimization in an optimized sky partition with each patch including only pixels with similar spectral properties of the input B -mode foregrounds. This optimization allows variance minimization to handle a simpler Galactic emission in each region. Foreground spectral properties are blindly estimated by computing the ratio ˜ B hf fgds / ˜ B cf fgds of templates of B -mode foreground emission, one at a high-frequency channel, while the other at a central 'CMB' frequency 5 . We report results for an ideal case, where the templates correspond to the input foreground maps, and for a realistic one, with templates derived from input noisy data by applying the Generalized Needlet ILC technique 7 (GNILC). MC-NILC pipeline has been validated on LiteBIRD sky simulations and is proved to significantly reduce Galactic contamination, especially at large angular scales, as shown in the left panel of Fig. 1.</text>
<text><location><page_2><loc_12><loc_11><loc_88><loc_27></location>The ocMILC technique, instead, enhances foreground subtraction by deprojecting some moments of the polarized foreground emission 8 through the linear combination in Eq. 1. This deprojection, already introduced in 9 , is optimized in ocMILC. First, the optimal number of moments to deproject across the sky and needlet scales is derived through a blind diagnostic of input foreground complexity. The optimal set of moments and the amount of deprojection of each moment is then derived by considering the combination leading to the minimum denoised output variance w j T ( C j -N j ) w j , with N j the input noise needlet covariance 6 . The effectiveness of the ocMILC pipeline in reducing Galactic contamination at all angular scales, with only a moderate increase of noise residuals, is assessed on PICO sky simulations and shown in the right panel of Fig. 1.</text>
<text><location><page_2><loc_12><loc_8><loc_88><loc_11></location>Both the MC-NILC and ocMILC methods are robust to a variety of different foreground models and proved to provide unbiased estimates of the tensor-to-scalar ratio with associated</text>
<text><location><page_3><loc_12><loc_90><loc_74><loc_91></location>uncertainties within the targeted sensitivity of the considered experiments 5 , 6 .</text>
<section_header_level_1><location><page_3><loc_12><loc_86><loc_22><loc_88></location>References</section_header_level_1>
<unordered_list>
<list_item><location><page_3><loc_14><loc_84><loc_55><loc_85></location>1. J.Delabrouille et al., A&A 493 , 835-857 (2009).</list_item>
<list_item><location><page_3><loc_14><loc_82><loc_57><loc_83></location>2. LiteBIRD Collaboration, PTEP 4 , 042F01 (2023).</list_item>
<list_item><location><page_3><loc_14><loc_80><loc_50><loc_82></location>3. R.Aurlien et al., JCAP 2023 , 034 (2023).</list_item>
<list_item><location><page_3><loc_14><loc_79><loc_72><loc_80></location>4. A.Zonca et al., The Journal of Open Source Software 6 , 3783 (2021).</list_item>
<list_item><location><page_3><loc_14><loc_77><loc_56><loc_78></location>5. A.Carones et al., MNRAS 525 , 3117-3135 (2023).</list_item>
<list_item><location><page_3><loc_14><loc_76><loc_75><loc_77></location>6. A.Carones and M.Remazeilles, arXiv e-prints : 2402.17579[astro-ph.CO].</list_item>
<list_item><location><page_3><loc_14><loc_74><loc_55><loc_75></location>7. M.Remazeilles, et al., MNRAS 418 , 467 (2011).</list_item>
<list_item><location><page_3><loc_14><loc_72><loc_51><loc_73></location>8. J.Chluba et al., MNRAS 472 , 1195 (2017).</list_item>
<list_item><location><page_3><loc_14><loc_71><loc_55><loc_72></location>9. M.Remazeilles et al., MNRAS 503 , 2478 (2021).</list_item>
</document>
[
{
"title": "A. Carones",
"content": "Dipartimento di Fisica, Universit'a di Roma Tor Vergata & INFN Sez. Roma 2, via della Ricerca Scientifica 1, I-00133, Roma, Italy The detection of primordial polarization B modes of the Cosmic Microwave Background (CMB) requires exquisite control of Galactic foreground contamination. The Needlet Internal Linear Combination (NILC) method has proven effective in reconstructing CMB B modes without suffering from mis-modeling errors of Galactic emission. However, with the most complex foreground models, residual Galactic contamination from NILC is proved to bias, especially at large angular scales, the recovered CMB B modes from simulated data of future CMB experiments. We therefore present two new extensions of NILC, Multi-Clustering NILC (MC-NILC) and optimized constrained Moment ILC (ocMILC), which allow to enhance foreground subtraction in the reconstructed CMB signal.",
"pages": [
1
]
},
{
"title": "1 Blind reconstruction of CMB B modes",
"content": "The detection of primordial polarization B modes of the Cosmic Microwave Background (CMB), targeted by upcoming CMB experiments, is hampered by the contamination of Galactic foregrounds. This points to the need for robust component separation techniques to be applied to future microwave observations. The reference model-independent ( i.e. blind) cleaning technique is the Needlet Internal Linear Combination 1 (NILC). In the NILC pipeline, input multi-frequency B -mode maps B i are first decomposed into needlet coefficients β i j through a deconvolution for a needlet kernel ψ j : β i j (ˆ γ ) = ψ j ⊛ B i (ˆ γ ). At each needlet scale j , sampling a specific range of angular scales, needlet maps are then linearly combined with frequency-dependent weights w i j : so as to recover a blackbody solution β NILC j with minimum variance. The derivation of NILC weights, w j (ˆ γ ) = [ A T CMB C -1 j (ˆ γ ) A CMB ] -1 A T CMB C -1 j (ˆ γ ), requires only knowledge of the CMB spectral energy distribution (SED), A CMB , and the computation of the input needlet covariance matrix: C ik j (ˆ γ ) = ⟨ β i j (ˆ γ ) · β k j (ˆ γ ) ⟩ , with ⟨⟩ a local sample average within Gaussian domains, whose width varies with the needlet scale j . The final B -mode CMB map, ˜ B CMB , is reconstructed through an inverse needlet transform of NILC solutions at the different needlet scales: and includes residual contamination by foregrounds ( B fres ) and instrumental noise ( B nres ). The contribution of B nres to the output angular power spectrum can be removed, whereas that of B fres , if relevant at the reionization peak ( ℓ ≲ 10) or at the recombination bump ( ℓ ∼ 80), may bias the estimate of the tensor-to-scalar ratio.",
"pages": [
1
]
},
{
"title": "2 Optimizing blind CMB component separation",
"content": "We consider simulations of LiteBIRD 2 and PICO 3 satellite experiments, targeting a sensitivity on the tensor-to-scalar ratio of σ ( r ) = 0 . 001 and 0 . 0002, respectively. Galactic foreground emission is simulated with the PySM python package 4 assuming the dust d1 model (modified blackbody SED) and synchrotron s1 model (power-law SED), both featuring spatial variations of the corresponding spectral parameters. After proper masking, at large angular scales, the NILC foreground residuals still have power comparable to that of the targeted amplitude of tensor modes, as shown by solid orange lines in the left ( LiteBIRD ) and right ( PICO ) panels of Fig. 1. As demonstrated in 5 and 6 , these residuals would produce a non-negligible bias on the estimated tensor-to-scalar ratio. We thus present two alternative optimizations of NILC: Multi-Clustering NILC 5 (MC-NILC) and optimized constrained moment ILC 6 (ocMILC). The MC-NILC technique implements variance minimization in an optimized sky partition with each patch including only pixels with similar spectral properties of the input B -mode foregrounds. This optimization allows variance minimization to handle a simpler Galactic emission in each region. Foreground spectral properties are blindly estimated by computing the ratio ˜ B hf fgds / ˜ B cf fgds of templates of B -mode foreground emission, one at a high-frequency channel, while the other at a central 'CMB' frequency 5 . We report results for an ideal case, where the templates correspond to the input foreground maps, and for a realistic one, with templates derived from input noisy data by applying the Generalized Needlet ILC technique 7 (GNILC). MC-NILC pipeline has been validated on LiteBIRD sky simulations and is proved to significantly reduce Galactic contamination, especially at large angular scales, as shown in the left panel of Fig. 1. The ocMILC technique, instead, enhances foreground subtraction by deprojecting some moments of the polarized foreground emission 8 through the linear combination in Eq. 1. This deprojection, already introduced in 9 , is optimized in ocMILC. First, the optimal number of moments to deproject across the sky and needlet scales is derived through a blind diagnostic of input foreground complexity. The optimal set of moments and the amount of deprojection of each moment is then derived by considering the combination leading to the minimum denoised output variance w j T ( C j -N j ) w j , with N j the input noise needlet covariance 6 . The effectiveness of the ocMILC pipeline in reducing Galactic contamination at all angular scales, with only a moderate increase of noise residuals, is assessed on PICO sky simulations and shown in the right panel of Fig. 1. Both the MC-NILC and ocMILC methods are robust to a variety of different foreground models and proved to provide unbiased estimates of the tensor-to-scalar ratio with associated uncertainties within the targeted sensitivity of the considered experiments 5 , 6 .",
"pages": [
2,
3
]
}
]
2024arXiv240702313M
https://arxiv.org/pdf/2407.02313.pdf
<document>
<text><location><page_1><loc_20><loc_87><loc_21><loc_88></location>1</text>
<section_header_level_1><location><page_1><loc_26><loc_92><loc_75><loc_93></location>A Casimir-like probe for 4D Einstein-Gauss-Bonnet gravity</section_header_level_1>
<text><location><page_1><loc_38><loc_89><loc_62><loc_90></location>Syed Masood 1 ∗ and Said Mikki 1 , 2†</text>
<text><location><page_1><loc_21><loc_82><loc_80><loc_88></location>Zhejiang University/University of Illinois at Urbana-Champaign Institute (the ZJU-UIUC Institute), Zhejiang University, 718 East Haizhou Road, Haining 314400 , China. and 2 Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana IL 61801 , USA</text>
<text><location><page_1><loc_44><loc_81><loc_57><loc_82></location>(Dated: July 3, 2024)</text>
<text><location><page_1><loc_18><loc_61><loc_83><loc_79></location>Virtual transitions in a Casimir-like configuration are utilized to probe quantum aspects of the recently proposed four-dimensional Einstein-Gauss-Bonnet (4D EGB) gravity. This study employs a quantum optics-based approach, wherein an Unruh-DeWitt detector (modeled as a two-level atom) follows a radial timelike geodesic, falling freely into an uncharged, nonrotating black hole described by 4D EGB gravity, becoming thermalized in the usual Unruh manner. The black hole, asymptotically Minkowskian, is enclosed by a Casimir boundary proximate to its horizon, serving as a source for accelerated field modes that interact with the infalling detector. Observations are conducted by an asymptotic infinity observer, assuming a Boulware field state. Our numerical analysis reveals that, unlike in Einstein gravity, black holes in 4D EGB gravity can either enhance or suppress the intensity of acceleration radiation, contingent upon the Gauss-Bonnet coupling parameter α . Specifically, we observe radiation enhancement for negative α and suppression for positive α . These findings offer substantial insights into quantifying the influence of higher-curvature contributions on the behavior of quantum fields in black hole geometries within a 4D spacetime.</text>
<section_header_level_1><location><page_1><loc_22><loc_57><loc_36><loc_58></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_27><loc_49><loc_54></location>Considerable efforts have been made over the past few decades to uncover the deep connection between quantum mechanics, gravity, and thermodynamics [1, 2]. Among these endeavors, the discovery of Hawking radiation from black holes [3] and the Unruh effect for accelerated observers in flat Minkowski spacetime [4, 5] stand out as pivotal. Another significant phenomenon is Parker's idea of particle emission due to the expansion of the Universe [2]. In all these cases, the quantum state of the field is altered by a dynamic background spacetime geometry or the state of motion, resulting in the creation of real particles-an effect arising from the violation of Poincaré invariance [1]. This is similar to the dynamical Casimir effect (DCE) [6, 7], where accelerated plates or boundaries induce the quantum vacuum to radiate particles. Consequently, this scenario fosters a rich intersection of quantum fields, boundaries, and spacetime geometries [8-11].</text>
<text><location><page_1><loc_9><loc_15><loc_49><loc_26></location>With the advent of precise experimental and observational setups, it has become possible over the decades to test Einstein's general relativity (GR) in extreme gravity regimes. So far, GR has consistently matched observational data, with milestone achievements including gravitational wave detection [12, 13], black hole shadows [14], and neutron star mergers [15]. However, physicists have long recognized that GR</text>
<text><location><page_1><loc_52><loc_50><loc_92><loc_58></location>cannot address certain fundamental issues in the Universe, such as the existence of singularities, cosmological acceleration, dark matter, and a consistent merger of quantum mechanics and gravity. Thus, it is evident that a framework beyond GR is needed to resolve these challenges [16-18].</text>
<text><location><page_1><loc_52><loc_10><loc_92><loc_49></location>Several alternatives to GR predict additional highercurvature contributions to the gravitational action. A significant framework within this class of models originates from the works of Lanczos [19] and Lovelock [20, 21], leading to the well-known Einstein-Gauss-Bonnet (EGB) theory. It has been established that EGB gravity does not introduce modifications to gravitational dynamics unless coupled with additional field degrees of freedom or in spacetime dimensions D ≥ 5. One example of such additional fields is the dilaton field [22-25]. In addition to this, EGB gavity theories yield equations of motion that are quadratic in metric tensor. This quadratic nature is a unique feature of EGB gravity among all other alternatives to GR. The interesting coincidence is that the low energy effective descriptions of heterotic string theories also posit quadratic contributions to the dynamics of Einstein gravity [26-28]. It may be noted that the quadratic nature of equations of motion suffice to get rid of Ostrogradsky instability [29] and thus guarantees physicality of the dynamics. Furthermore, EGB gravity theories are characterized by equations of motion that are quadratic in the metric tensor. This quadratic nature distinguishes EGB gravity from other alternatives to GR. An intriguing coincidence arises in that the low-energy effective descriptions of heterotic string theo-</text>
<text><location><page_2><loc_9><loc_87><loc_49><loc_93></location>es also incorporate quadratic contributions to the dynamics of Einstein gravity [26-28]. Importantly, the quadratic form of the equations of motion resolves the Ostrogradsky instability [29], ensuring the physical viability of the theory.</text>
<text><location><page_2><loc_9><loc_55><loc_49><loc_86></location>Recently, Glavan and Lin [30] addressed the question of Gauss-Bonnet (GB) contributions in 4-dimensional spacetime geometry by proposing a specific rescaling of the GB coupling parameter α → α / ( D -4 ) , where D denotes the spacetime dimensionality. This rescaling ensures a well-defined limit as D → 4. The resulting model maintains quadratic behavior to prevent Ostrogradsky instability, yet it departs from the implications of the well-known Lovelock theorem [19-21]. It is noteworthy that no additional field coupling is required in this model. As a new phenomenological competitor to Einstein's General Relativity (GR), this model has sparked rigorous debates over the years. Some investigations include consistency checks [31-33], studies of black hole shadows and quasinormal modes [34-36], analysis of geodesics [37], particle accelerator models [38], and a wide array of thermodynamic analyses [39-47]. A comprehensive overview of 4D-EGB gravity, covering its various aspects, can be found in a review article by Fernandes et al. [48].</text>
<text><location><page_2><loc_9><loc_10><loc_49><loc_55></location>Recognizing the significance of the findings in Ref. [30], we are driven to investigate the potential quantum radiative signatures of 4D EGB gravity using elements from quantum optics and Casimir physics. Our approach involves a quantum optical cavity positioned with one end near a black hole horizon and the other at asymptotic infinity. Within this setup, a two-level Unruh-DeWitt detector (an atom) falls freely towards the black hole. Virtual transitions arising from the interaction between the detector and the field lead to acceleration radiation, which carries distinct imprints of the underlying gravitational background. Such a setup has been discussed in Ref. [8], where it was demonstrated that, under appropriate initial conditions, a detector near a Schwarzschild black hole emits radiation with a thermal spectrum. This unique radiative emission, known as Horizon Brightened Acceleration Radiation (HBAR), occurs when the detector is in free fall towards the black hole. This concept has been further explored in various contexts, revealing profound connections between the equivalence principle, quantum optics, and the HawkingUnruh effect [8, 49-52]. It also underscores connections to the Dynamical Casimir Effect (DCE) and moving mirror models [53, 54], frequently employed in studying quantum field behavior in curved spacetimes. But while the original work in Ref. [8] considers detectors moving along timelike geodesics, subsequent studies have shown that similar phenomena can occur for detectors following null geodesics [55]. This novel</text>
<text><location><page_2><loc_52><loc_88><loc_92><loc_93></location>radiative emission phenomenon can be attributed to the nearhorizon physics and conformal quantum mechanics of black holes [56-61].</text>
<text><location><page_2><loc_52><loc_75><loc_92><loc_88></location>Given that quantum field dynamics can elucidate the nature of underlying spacetime geometry [1, 2], we view the aforementioned setup as a potential avenue to probe 4D EGB gravity at a deeper level. Through numerical analysis, we demonstrate that 4D EGB gravity can imprint distinct features on the radiation spectrum compared to Einstein's GR, encompassing both negative and positive values of the Gauss-Bonnet coupling parameter.</text>
<text><location><page_2><loc_52><loc_62><loc_92><loc_74></location>The structure of the paper is as follows. The next Sec. II introduces the basics of 4D EGB black hole geometry, accompanied by discussions on the wave equation and the vacuum field state. In Sec. III, we compute the excitation probability or the detector response function of the falling detector. Sec. IV explores possible interpretations of our numerical findings. Finally, conclusions are drawn in Sec. V.</text>
<section_header_level_1><location><page_2><loc_54><loc_56><loc_89><loc_59></location>II. CONCEPTUAL ASPECTS: OUR SPACETIME GEOMETRY AND THE CHOICE OF FIELD MODES</section_header_level_1>
<text><location><page_2><loc_52><loc_51><loc_92><loc_54></location>The static, spherically symmetric metric of an uncharged and nonrotating black hole in 4D EGB gravity is given by [48]</text>
<formula><location><page_2><loc_54><loc_47><loc_92><loc_50></location>ds 2 = -f ( r ) d t 2 + 1 f ( r ) d r 2 + r 2 ( d θ 2 + sin 2 θ d φ 2 ) , (1)</formula>
<text><location><page_2><loc_52><loc_44><loc_56><loc_46></location>where 1</text>
<formula><location><page_2><loc_59><loc_40><loc_92><loc_43></location>f ( r ) = 1 + r 2 2 α ( 1 ± √ 1 + 8 α M r 3 ) , (2)</formula>
<text><location><page_2><loc_52><loc_32><loc_92><loc_38></location>where ± sign inside brackets denotes Gauss-Bonnet (GB) and GR branches, respectively. Here, we focus solely on the GR branch, as the GB branch is deemed unphysical [48]. To determine the event horizon radius, we set</text>
<formula><location><page_2><loc_58><loc_27><loc_92><loc_30></location>f ( r ) = 1 + r 2 2 α ( 1 -√ 1 + 8 α M r 3 ) = 0 , (3)</formula>
<text><location><page_2><loc_52><loc_24><loc_60><loc_25></location>which yields</text>
<formula><location><page_2><loc_64><loc_21><loc_92><loc_23></location>r ± = M 2 ± √ M 2 -α , (4)</formula>
<text><location><page_2><loc_52><loc_15><loc_92><loc_19></location>of which the one with plus sign is the real exterior horizon of the black hole. Thus, our event horizon is located at r g = r ± = M 2 + √ M 2 -α . The parameter α can take both positive</text>
<text><location><page_3><loc_9><loc_83><loc_49><loc_93></location>and negative values within the range -32 M 2 ≤ α ≤ 4 M 2 , as indicated in Refs. [34, 37] (also see [30]). It is evident that a positive GB coupling constant α decreases the black hole horizon radius, whereas a negative α increases it. The limit α = 0 corresponds to the Schwarzschild black hole in GR. These relationships are illustrated graphically in Fig. 1.</text>
<text><location><page_3><loc_9><loc_66><loc_49><loc_83></location>We also note that r + r -= α , and as r → 0, the metric components remain finite. This can be observed from Eq. (2), where lim r → 0 f ( r ) = 1. However, the finiteness of the metric components does not guarantee the absence of singularities due to the fact that the Ricci scalar R and the Kretschmann scalar R µνσδ R µνσδ vary as R ∝ r -3 / 2 and R µνσδ R µνσδ ∝ r -3 , respectively. It should be noted that for the Schwarzschild case, the Kretschmann scalar near r = 0 varies as r -6 , indicating that the GB contribution significantly weakens the singularity by several orders of magnitude [48].</text>
<section_header_level_1><location><page_3><loc_21><loc_61><loc_37><loc_62></location>A. Detector trajectories</section_header_level_1>
<text><location><page_3><loc_9><loc_51><loc_49><loc_59></location>In this section, we analyze the geodesics of the detector to compute both the coordinate time and proper (conformal) time that describe the timelike trajectory of the infalling (massive) detector. Generally, for a given Christoffel connection Γ µ ρσ , the complete geodesic equations are expressed as [62]</text>
<formula><location><page_3><loc_20><loc_47><loc_49><loc_50></location>d 2 x µ d τ 2 + Γ µ ρσ d x ρ d τ d x σ d τ = 0 . (5)</formula>
<text><location><page_3><loc_9><loc_38><loc_49><loc_46></location>Our spacetime geometry of interest exhibits spherical symmetry, and we restrict our analysis to the radial motion of the detector in the equatorial plane. Therefore, we set θ = π / 2, which implies ˙ θ = 0 and ˙ φ = 0. Consequently, the following conservation equations hold:</text>
<formula><location><page_3><loc_10><loc_33><loc_49><loc_36></location>( d r d τ ) 2 = E 2 -f ( r ) , ( d r d t ) 2 = [ f ( r ) E ] 2 [ E 2 -f ( r ) ] . (6)</formula>
<text><location><page_3><loc_9><loc_20><loc_49><loc_32></location>Note that E is a constant representing the specific energy of the detector. It is determined by the initial boundary conditions of the geodesic motion, given by E 2 = f ( r ) ∣ ∣ max . Since we assume that the detector started its motion from asymptotic infinity, where the spacetime is asymptotically Minkowski flat ( r → ∞ implies f ( r ) ∣ ∣ max = 1), these constraints from the above equations lead to</text>
<formula><location><page_3><loc_13><loc_16><loc_49><loc_19></location>( d r d τ ) 2 = 1 -f ( r ) , ( d r d t ) 2 = f 2 ( r )[ 1 -f ( r )] . (7)</formula>
<text><location><page_3><loc_9><loc_9><loc_49><loc_15></location>It should be emphasized that E , which is related to the maximum of f ( r ) , is the same for both GR and 4D EGB gravity. This value of E corresponds to asymptotic infinity, where both GRand 4D EGB theories reproduce flat Minkowski geometry.</text>
<text><location><page_3><loc_52><loc_88><loc_92><loc_93></location>Now, integrating Eq. (7) along the radial trajectories from some arbitrary initial point r i to a final point r f (where r i > r f ), we obtain</text>
<formula><location><page_3><loc_54><loc_85><loc_92><loc_88></location>τ = -∫ r f r i d r √ 1 -f ( r ) , t = -∫ r f r i d r f ( r ) √ 1 -f ( r ) . (8)</formula>
<text><location><page_3><loc_52><loc_80><loc_92><loc_83></location>We now substitute Eq. (2) into Eq. (8) in order to compute τ , resulting in</text>
<formula><location><page_3><loc_52><loc_70><loc_92><loc_79></location>τ = 2 r √ √ 1 + 8 α M r 3 -1tan -1 √ √ 1 + 8 α M r 3 -1 √ 2 -2 √ 2 r 3 √ r 2 (√ 1 + 8 α M r 3 -1 ) α + τ 0 . (9)</formula>
<text><location><page_3><loc_52><loc_58><loc_92><loc_68></location>Here, τ 0 serves as an integration constant, the insignificance of which we establish for the final detector response, as detailed in Sec. III. However, the complexity of the integral for t precludes straightforward analytical computation. Consequently, we resort to numerical methods and present the outcomes in Sec. III. Fig. 2 illustrates the plots of τ and t .</text>
<text><location><page_3><loc_52><loc_39><loc_92><loc_58></location>The plots clearly illustrate that t and τ exhibit typical Schwarzschild-like behavior. Specifically, t , which represents the time measured by an asymptotic observer, diverges as the detector approaches the black hole horizon, located at zero on the rescaled radial coordinate r -r g. This divergence signifies that, from the perspective of this observer, the detector never actually crosses the horizon. In contrast, τ remains finite at the horizon r -r g, indicating that from the detector's own frame of reference, it crosses the horizon in a finite amount of proper time. This disparity highlights the causal structure of black hole horizons and is recognized as gravitational time dilation.</text>
<text><location><page_3><loc_52><loc_29><loc_92><loc_39></location>Furthermore, in 4D EGB gravity, the coupling parameter α influences the behavior of t and τ . For positive α , which reduces the black hole size as discussed in Sec. II, it takes longer for the detector to approach the horizon as α increases. Conversely, for negative α , which inflates the black hole size, the situation is reversed.</text>
<section_header_level_1><location><page_3><loc_62><loc_24><loc_81><loc_25></location>B. Defining the vacuum state</section_header_level_1>
<text><location><page_3><loc_52><loc_9><loc_92><loc_22></location>The response function, or excitation probability, to be calculated in Sec. III, quantifies the detector-field coupling. To achieve this, we must obtain the appropriate field mode by solving the wave equation on the specified spacetime background. Here, we consider the simplest test field: a massless spin-0 Klein-Gordon field, minimally coupled to the spacetime geometry, described by ∇ µ ∇ µ Φ = 0 [1]. Given the spherical symmetry of the spacetime and the presence of a</text>
<figure>
<location><page_4><loc_15><loc_69><loc_50><loc_94></location>
<caption>Figure 1. Impact of GB coupling α on (a) f ( r ) and (b) horizon radius r + to show distinct nature of α for its positive and negative values.</caption>
</figure>
<text><location><page_4><loc_9><loc_54><loc_49><loc_62></location>timelike Killing vector ∂ t , we have Φ = 1 r Y l ( θ , φ ) ψ ( t , r ) , with Y l denoting spherical harmonics and l representing the multipole number. The radial part of the solution, after neglecting the angular dependence ( l = 0 ) , satisfies the following Schrödinger-like wave equation</text>
<formula><location><page_4><loc_15><loc_49><loc_49><loc_53></location>( -∂ 2 ∂ t 2 + ∂ 2 ∂ r 2 ∗ ) ψ ( t , r ) = V ( r ) ψ ( t , r ) . (10)</formula>
<text><location><page_4><loc_9><loc_43><loc_49><loc_48></location>Here, r ∗ denotes the Regge-Wheeler tortoise coordinate, a useful parameter for describing the propagation of test fields in black hole geometries, defined by [62]</text>
<formula><location><page_4><loc_18><loc_39><loc_49><loc_42></location>r ∗ = ∫ d r f ( r ) (11)</formula>
<formula><location><page_4><loc_20><loc_35><loc_49><loc_39></location>= ∫ d r 1 + r 2 2 α ( 1 -√ 1 + 8 α M r 3 ) , (12)</formula>
<text><location><page_4><loc_9><loc_18><loc_49><loc_33></location>where we utilized Eq. (2). Additionally, V ( r ) represents the effective potential experienced by the field, often describing scattering effects in black hole spacetimes [63]. However, given our focus on the simplest scenario possible, as also demonstrated in Refs. [8, 64], V ( r ) can be neglected. One approach to achieve this is by assuming that the frequency ν of the field mode is sufficiently large, enabling it to surmount the potential barrier imposed by the spacetime. Consequently, the field mode simplifies to</text>
<formula><location><page_4><loc_20><loc_15><loc_49><loc_16></location>ψ ( t , r ) = exp [ i ν ( t -r ∗ )] . (13)</formula>
<text><location><page_4><loc_9><loc_9><loc_49><loc_13></location>This represents a normalized outgoing field mode with frequency ν , as observed by an asymptotic infinity observer, qualifying as a Boulware field state. The ingoing field modes</text>
<text><location><page_4><loc_52><loc_59><loc_92><loc_62></location>generated propagate towards the boundary at the black hole horizon and are lost.</text>
<text><location><page_4><loc_52><loc_21><loc_92><loc_59></location>The Boulware field mode described above is an approximate field state obtained by neglecting V ( r ) and assuming ν to be very large. This assumption serves as one of the initial conditions required for the existence of HBAR emission [8]. Generally, in the context of black holes, multiple vacuum states are utilized due to the absence of a unique vacuum state in curved spacetime. This leads to various notions of vacuum states, such as the Unruh vacuum, Hartle-Hawking vacuum [1, 2], and others. In principle, there should be an infinite number of possible vacuum states due to the violation of Poincaré invariance in curved spacetimes [1]. In contrast, for Minkowski space, where the field satisfies Poincaré invariance, the vacuum state remains same for all inertial observers. In our scenario, the choice of the Boulware vacuum state arises because the observations are made by an asymptotic observer, for whom the Boulware field state is most appropriate. In this context, no Hawking radiation is detected by the observer. Moreover, the black hole is assumed to be entirely enclosed by a Casimir boundary, which effectively prevents any potential Hawking quanta from mixing with HBAR flux [8]. This distinction ensures that HBAR emission is fundamentally different from Hawking radiation.</text>
<text><location><page_4><loc_85><loc_18><loc_85><loc_19></location≯</text>
<text><location><page_4><loc_52><loc_9><loc_92><loc_20></location>Additionally, we have excluded l = 0 modes for simplicity. However, considering a smaller ν such that V ( r ) = 0 would lead to the emergence of scattering effects, potentially necessitating the inclusion of greybody factors [65]. Nevertheless, we argue that such inclusions would lead to the deviation from the primal essence of HBAR emission, which occurs under specific boundary conditions as emphasized in Refs. [8, 50].</text>
<figure>
<location><page_5><loc_15><loc_72><loc_50><loc_94></location>
</figure>
<figure>
<location><page_5><loc_15><loc_48><loc_50><loc_70></location>
</figure>
<figure>
<location><page_5><loc_52><loc_72><loc_86><loc_94></location>
</figure>
<figure>
<location><page_5><loc_52><loc_48><loc_86><loc_70></location>
<caption>Figure 2. The radial dependence of coordinate time t and proper time τ illustrates the influence of the Gauss-Bonnet coupling parameter α . Panels (a) and (c) depict the proper time for positive and negative α , respectively. Similarly, panels (b) and (d) show the coordinate time for positive and negative α , respectively.</caption>
</figure>
<section_header_level_1><location><page_5><loc_19><loc_36><loc_39><loc_37></location>III. DETECTOR RESPONSE</section_header_level_1>
<text><location><page_5><loc_9><loc_26><loc_49><loc_34></location>As discussed in the preceding section, the field is in a Boulware vacuum state, ensuring that no Hawking radiation is observed by the asymptotic observer. By neglecting the angular dependence of the field modes, the detector-field interaction Hamiltonian can be expressed as follows [8]:</text>
<formula><location><page_5><loc_10><loc_21><loc_49><loc_25></location>ˆ H ( τ ) = ¯ hg [ ˆ a νψ [ t ( τ ) , r ( τ )] + H . C . ][ ˆ σ ( τ ) e -i ωτ + H . C . ] . (14)</formula>
<text><location><page_5><loc_9><loc_12><loc_49><loc_20></location>Here, ˆ a ν is the annihilation operator for the field modes, ˆ σ is the detector lowering operator, and H . C . denotes the Hermitian conjugate. Here, g is a detector-field coupling parameter indicating the strength of the interaction and can be taken as a constant for a massless Klein-Gordon field (spin-0).</text>
<text><location><page_5><loc_9><loc_9><loc_49><loc_11></location>Assuming that the detector is initially in the ground state | b ⟩ , the probability that it transitions to an excited state | a ⟩</text>
<text><location><page_5><loc_52><loc_35><loc_92><loc_37></location>with the emission of a field quantum of frequency ν is given by</text>
<formula><location><page_5><loc_59><loc_26><loc_92><loc_29></location>Γ exc = 1 ¯ h 2 ∣ ∣ ∣ ∣ ∫ d τ ⟨ 1 ν , a | H ( τ ) | 0 , b ⟩ ∣ ∣ ∣ ∣ 2 . (15)</formula>
<text><location><page_5><loc_52><loc_9><loc_92><loc_20></location>Utilizing time-dependent perturbation theory, such a process is typically prohibited in quantum optics due to energy conservation principles. However, in non-inertial frames influenced by acceleration and gravity, these virtual processes can occur owing to counter-rotating terms in the Hamiltonian [50], as exemplified by the Unruh effect [4]. By employing Eq. (14) and performing some additional straightforward compu-</text>
<text><location><page_6><loc_52><loc_92><loc_77><loc_93></location>tations, Eq. (15) can be reexpressed as</text>
<formula><location><page_6><loc_60><loc_83><loc_92><loc_91></location>Γ exc = g 2 ∣ ∣ ∣ ∣ ∫ d τ ψ ∗ ( t ( τ ) , r ( τ )) e i ωτ ∣ ∣ ∣ ∣ 2 = g 2 ∣ ∣ ∣ ∣ ∫ d r ( d τ d r ) ψ ∗ ( r ) e i ωτ ∣ ∣ ∣ ∣ 2 . (16)</formula>
<text><location><page_6><loc_52><loc_81><loc_73><loc_82></location>Simplifying further, we arrive at</text>
<formula><location><page_6><loc_9><loc_64><loc_94><loc_75></location>Γ exc = g 2 ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ r g ∫ ∞ d r exp [ i ν { t ( r ) -r ∗ ( r ) } ] 1 √ r 2 2 α (√ 1 + 8 α M r 3 -1 ) exp i ω 2 r √ √ 1 + 8 α M r 3 -1tan -1 √ √ 1 + 8 α M r 3 -1 √ 2 -2 √ 2 r 3 √ r 2 (√ 1 + 8 α M r 3 -1 ) α ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ 2 , (17)</formula>
<text><location><page_6><loc_9><loc_57><loc_92><loc_62></location>which results in a complex expression involving nested integrals with respect to t ( r ) and r ∗ . It's important to note that the limits of integration correspond to the detector's trajectory from r = ∞ to r = r g, the horizon of the black hole. Thus, from Eqs. (8) and (12), we derive:</text>
<formula><location><page_6><loc_19><loc_52><loc_92><loc_56></location>t ( r ) = -∫ r g ∞ d r [ 1 + r 2 2 α ( 1 -√ 1 + 8 α M r 3 )] √ r 2 2 α (√ 1 + 8 α M r 3 -1 ) , r ∗ = ∫ ∞ r g d r 1 + r 2 2 α ( 1 -√ 1 + 8 α M r 3 ) . (18)</formula>
<text><location><page_6><loc_9><loc_47><loc_92><loc_49></location>Consider now the substitution r = r g z , where d r = r gd z . Using this transformation of variables, we may rewrite t ( r ) in Eq. (18) as follows:</text>
<formula><location><page_6><loc_27><loc_40><loc_92><loc_45></location>t ( z ) = -∫ 1 ∞ d zr g [ 1 + r 2 g z 2 2 α ( 1 -√ 1 + 8 α M r 3 g z 3 )] √ r 2 g z 2 2 α ( √ 1 + 8 α M r 3 g z 3 -1 ) . (19)</formula>
<text><location><page_6><loc_9><loc_37><loc_54><loc_38></location>A further substitution of the form x = z -1, such that z = x + 1, yields</text>
<formula><location><page_6><loc_23><loc_30><loc_92><loc_35></location>t ( x ) = ∫ ∞ 0 d xr g [ 1 + r 2 g ( x + 1 ) 2 2 α ( 1 -√ 1 + 8 α M r 3 g ( x + 1 ) 3 )] √ r 2 g ( x + 1 ) 2 2 α ( √ 1 + 8 α M r 3 g ( x + 1 ) 3 -1 ) . (20)</formula>
<text><location><page_6><loc_9><loc_27><loc_44><loc_28></location>One can follow a similar calculation for r ∗ , arriving at</text>
<formula><location><page_6><loc_34><loc_20><loc_92><loc_25></location>r ∗ ( x ) = ∫ ∞ 0 d xr g 1 + [ r g ( x + 1 ) ] 2 2 α ( 1 -√ 1 + 8 α M [ r g ( x + 1 ) ] 3 ) . (21)</formula>
<text><location><page_6><loc_9><loc_17><loc_88><loc_18></location>After deploying all the relevant quantities in Eq. (17), we derive the following final expression for the detector excitation:</text>
<formula><location><page_6><loc_24><loc_9><loc_76><loc_15></location>Γ exc = g 2 r 2 g ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∫ ∞ 0 d x exp [ i ν { t ( x ) -r ∗ ( x ) } ] G √ ( r g [ x + 1 ]) 2 2 α (√ 1 + 8 α M ( r g [ x + 1 ]) 3 -1 ) ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ 2 ,</formula>
<text><location><page_7><loc_9><loc_92><loc_13><loc_93></location>where</text>
<formula><location><page_7><loc_17><loc_80><loc_92><loc_91></location>G = exp i ω 2 r g ( x + 1 ) √ √ 1 + 8 α M ( r g [ x + 1 ]) 3 -1tan -1 √ √ 1 + 8 α M ( r g [ x + 1 ]) 3 -1 √ 2 -2 √ 2 [ r g ( x + 1 ) ] 3 √ ( r g [ x + 1 ]) 2 ( √ 1 + 8 α M ( r g [ x + 1 ]) 3 -1 ) α . (22)</formula>
<text><location><page_7><loc_9><loc_63><loc_49><loc_75></location>This represents the primary outcome of our investigation. The numerical integral in (22) is notably intricate, demanding a careful approach for its accurate computation. To achieve that, in what follows we deploy the numerical integration capabilities of the Mathematica symbolic math package for performing all required calculations. The figures presented in Fig. 3 were generated using optimized settings.</text>
<section_header_level_1><location><page_7><loc_17><loc_59><loc_41><loc_60></location>IV. RESULTS AND DISCUSSIONS</section_header_level_1>
<text><location><page_7><loc_9><loc_27><loc_49><loc_56></location>Based on the preceding analysis, the two-level UnruhDeWitt detector, operating in the Boulware vacuum state, registers detections while in free fall (inertial). This observation appears to challenge established field-theoretic concepts associated with the Hawking-Unruh effect. Specifically, there is no emission of Hawking radiation in the Boulware state as observed from asymptotic infinity, nor does the Unruh effect manifest for inertial detectors in the Minkowski vacuum. However, HBAR emission from detectors operates on different principles [8, 50]. While it shares similarities with Hawking radiation, such as the thermal nature of the emitted flux and the associated Bekenstein-Hawking entropy-area correspondence, there are also distinct characteristics. Notably, HBAR emission involves the evolution of field modes in pure states and includes phase correlations between them. These aspects naturally relate to the black hole information paradox [66, 67].</text>
<text><location><page_7><loc_9><loc_8><loc_49><loc_27></location>In Fig. 3, we present the detector excitation probability, Γ exc, plotted as a function of the emitted radiation frequency, ν . The impact of the GB coupling parameter, α , is depicted in Figs. 3(a) and 3(b) for positive and negative values of α , respectively. Fig. 3(c) illustrates how the detector transition frequency, ω , influences Γ exc, while Fig. 3(d), shown on a loglog scale, highlights the behavior of Γ exc near the origin and its convergence at higher frequencies. It is important to note that our interpretations and analyses are based on numerical estimations detailed in the preceding sections. These figures provide a comprehensive view of the radiative characteristics</text>
<text><location><page_7><loc_52><loc_72><loc_92><loc_75></location>under consideration, elucidating the role of α and the detector's transition frequency in shaping Γ exc.</text>
<text><location><page_7><loc_52><loc_35><loc_92><loc_71></location>From all plots, one of the prominent features observed is the thermal nature of the HBAR radiation flux, characterized by a Bose-Einstein (BE) distribution. This observation leads us to conclude that 4D Einstein-Gauss-Bonnet (EGB) gravity does not alter the thermal nature of the flux, consistent with earlier findings [8, 50, 55, 58, 59] in the context of Einstein gravity. This characteristic mirrors the thermal emission observed in Hawking radiation from pure black holes with asymptotically flat geometries. It is noteworthy that for the so-called 'dirty' black holes beyond the Kerr-Newman family, such as in the de Sitter case, there exists the possibility of observing a nonthermal spectrum [64, 68-70]. The detector excitation probability Γ exc, as observed in Fig. 3(a), decreases with increasing positive values of α and increases with negative values of α . As previously discussed, positive α reduces the size of the black hole horizon [see Fig. 1(b)], leading to the conclusion that smaller black holes emit less radiation flux compared to larger ones. This reasoning can similarly be applied to negative values of α . It is crucial to emphasize that in the limit α → 0, depicted in Fig. 3(a), the scenario converges to that of the pure Schwarzschild black hole.</text>
<text><location><page_7><loc_52><loc_14><loc_92><loc_35></location>The attenuation and augmentation of particle production can be conceptually grasped as follows. Particles generated within black hole spacetimes, as in Hawking radiation, experience backreaction due to the gravitational tidal forces exerted by the black hole. This backreaction diminishes the intensity of the radiation. Tidal effects in black holes stem from their surface gravities, which are directly related to their horizon radii. Specifically, for a black hole with a horizon radius rg , the surface gravity varies inversely with the square of rg . This relationship implies that larger black holes have smaller surface gravities and correspondingly weaker tidal effects, and conversely, smaller black holes exhibit stronger tidal effects.</text>
<text><location><page_7><loc_52><loc_9><loc_92><loc_14></location>In the context of positive α , the black hole horizon size decreases monotonically, leading to stronger surface gravity and tidal effects compared to a Schwarzschild black hole ( α = 0).</text>
<figure>
<location><page_8><loc_52><loc_48><loc_86><loc_69></location>
<caption>Figure 3. Impact of Gauss-Bonnet coupling α on detector response Γ exc :(a) positive α , and (b) negative α , for a fixed black hole mass M .(c) indicates impact of detector transition frequency ω , and (d) is the log-log plot of Γ exc to emphasize the BE-type nature of the spectrum. We chose M = 1, ω = 50 and g = 1.</caption>
</figure>
<text><location><page_8><loc_9><loc_19><loc_49><loc_37></location>Consequently, particles generated within such spacetimes experience heightened backreaction, impeding their propagation. This results in fewer particles escaping to asymptotic infinity, thereby reducing the intensity of the particle spectrum, as illustrated in Fig. 3(a). Conversely, for negative values of α , the black hole horizon radius increases, indicating reduced backreaction and tidal forces. This condition allows more particles to escape from the black hole spacetime, leading to an enhancement in the radiation flux, as depicted in Fig. 3(b). This constitutes the primary finding of our study, distinguishing 4D EGB gravity from Einstein GR.</text>
<text><location><page_8><loc_9><loc_10><loc_49><loc_18></location>To investigate the influence of the detector transition frequency ω on the radiation intensity, we plot Γ exc against ω in Fig. 3(c). The graphs clearly demonstrate a decrease in radiation intensity as ω increases. This behavior aligns with the principles of the Unruh effect [4] and can be understood in</text>
<text><location><page_8><loc_52><loc_33><loc_92><loc_37></location>terms of energy conservation: higher detector transition frequencies require more energy to excite the detector, resulting in a lower excitation probability, and vice versa.</text>
<text><location><page_8><loc_52><loc_21><loc_92><loc_32></location>Furthermore, to gain insight into the spectrum's behavior at low and high frequencies ( ν ), we reexamined Γ exc from Fig. 3(a) using a log-log scale. It is evident that the spectrum exhibits a finite Bose-Einstein (BE)-type distribution near the origin where ν → 0. As ν increases, the spectrum converges and exhibits a thermal tail, characteristic of a BE or Planckian distribution.</text>
<text><location><page_8><loc_52><loc_10><loc_92><loc_20></location>In the meantime, it is pertinent to consider the testable implications of this study. We can draw insights from analog gravity systems [71-73], which have been actively utilized over the last few decades to explore quantum fieldtheoretic phenomena in curved spacetimes. These systems have provided valuable analogs for understanding effects such</text>
<text><location><page_9><loc_9><loc_69><loc_49><loc_93></location>as Hawking radiation [74], the Unruh effect [75], and Parker particle generation in expanding spacetimes [76]. Most of these setups, whether based on condensed matter or quantum optical systems, are closely tied to Casimir physics involving moving boundaries. This connection is particularly relevant to our study, which is largely inspired by these concepts. Recently, condensed matter systems have been utilized to explore phenomena extending beyond particle generation in exotic backgrounds, including applications to fluid/gravity correspondence [77]. Looking forward, there is potential for future tabletop experiments to simulate black hole horizons resembling those in 4D EGB gravity [30] or to investigate HBAR radiation scenarios [8], where our findings could offer valuable insights.</text>
<section_header_level_1><location><page_9><loc_17><loc_62><loc_41><loc_63></location>V. CONCLUSION AND OUTLOOK</section_header_level_1>
<text><location><page_9><loc_9><loc_29><loc_49><loc_60></location>The exploration of theories beyond Einstein gravity has evolved in parallel with general relativity (GR) itself. These modified or extended gravity theories aim to tackle fundamental cosmological issues such as cosmic acceleration, singularities, and dark matter. Among these models, Einstein-GaussBonnet (EGB) gravity stands out, predicting higher-curvature corrections to the Einstein-Hilbert action. These corrections arise either in higher dimensions or through additional field couplings to the gravitational action. Interestingly, these contributions also emerge in the low-energy effective description of heterotic string theory. The 4D EGB theory represents a novel gravitational model that has sparked intense debate since its inception several years ago. This model predicts the presence of a Gauss-Bonnet (GB) term within 4D spacetime, which would otherwise not contribute to the latter's geometry. Its ability to provide a nontrivial contribution is achieved through a redefinition (rescalling) of the GB parameter [30]. Importantly, this theory circumvents the Lovelock theorem</text>
<unordered_list>
<list_item><location><page_9><loc_10><loc_11><loc_37><loc_12></location>[4] W. G. Unruh, Phys. Rev. D 14 , 870 (1976).</list_item>
</unordered_list>
<text><location><page_9><loc_52><loc_88><loc_92><loc_93></location>and sidesteps Ostrogadsky instability, ensuring that the resulting gravitational dynamics remain quadratic. The theory has been scrutinized across various phenomenological fronts.</text>
<text><location><page_9><loc_52><loc_75><loc_92><loc_88></location>In this paper, we examined the quantum radiative properties of a nonrotating, uncharged black hole in 4D EinsteinGauss-Bonnet (EGB) gravity using a Casimir-type configuration. The black hole, surrounded by a reflecting mirror, induced accelerated field modes from the Boulware vacuum state. We analyzed the interaction of these field modes with a freely falling two-level Unruh-DeWitt detector, which exhibited characteristic clicking behavior akin to the Unruh effect.</text>
<text><location><page_9><loc_52><loc_43><loc_92><loc_74></location>The spectrum detected by the detector follows a BoseEinstein (BE) distribution, with a notable dependence on the GB parameter α . By examining both positive and negative values of α , we studied their influence on the radiation intensity emitted by the detector. We observed that radiation intensity diminishes when α is positive. This reduction is attributed to the shrinking of the black hole size caused by positive α . Conversely, for negative α , we observed an increase in radiation intensity. The reduction or augmentation of the radiation flux is examined in relation to a pure Schwarzschild black hole, where the limit α → 0 is considered. Additionally, we observed that the transition frequency of the detector reduces the profile of particle creation due to the high energy needed for its excitation, consistent with the standard predictions of the Unruh effect. Finally, the spectrum is finite near the origin and monotonically converges at the high end of the frequency ranges, yielding the distinctive thermal tail characteristic of a Bose-Einstein or Planckian distribution.</text>
<text><location><page_9><loc_52><loc_29><loc_92><loc_43></location>Our work provides an opportunity to explore various aspects of 4D EGB gravity by incorporating different energymatter distributions around the simplest black hole model possible. Moreover, exploring other types of detector-field couplings could yield valuable insights into the nature of field configurations within the context of 4D EGB gravity. These and similar questions constitute promising extensions of this work, which we plan to pursue in the future.</text>
<unordered_list>
<list_item><location><page_9><loc_53><loc_21><loc_92><loc_23></location>[5] L. C. B. Crispino, A. Higuchi, and G. E. A. Matsas, Rev. Mod. Phys. 80 , 787 (2008).</list_item>
<list_item><location><page_9><loc_53><loc_19><loc_92><loc_20></location>[6] G. T. Moore, Journal of Mathematical Physics 11 , 2679 (1970).</list_item>
<list_item><location><page_9><loc_53><loc_18><loc_81><loc_19></location>[7] V. V. Dodonov, MDPI Physics 2 , 67 (2020).</list_item>
<list_item><location><page_9><loc_53><loc_13><loc_92><loc_17></location>[8] M. O. Scully, S. Fulling, D. Lee, D. N. Page, W. Schleich, and A. Svidzinsky, Proc. Nat. Acad. Sci. 115 , 8131 (2018), arXiv:1709.00481 [quant-ph].</list_item>
<list_item><location><page_9><loc_53><loc_10><loc_92><loc_12></location>[9] A. A. Svidzinsky, J. S. Ben-Benjamin, S. A. Fulling, and D. N. Page, Phys. Rev. Lett. 121 , 071301 (2018).</list_item>
<list_item><location><page_10><loc_9><loc_90><loc_49><loc_93></location>[10] R. Lopp, E. Martin-Martinez, and D. N. Page, Class. Quant. Grav. 35 , 224001 (2018), arXiv:1806.10158 [quant-ph].</list_item>
<list_item><location><page_10><loc_9><loc_87><loc_49><loc_90></location>[11] S. Masood A. S. Bukhari and L.-G. Wang, Front. Phys. (Beijing) 19 , 54203 (2024), arXiv:2307.12222 [gr-qc].</list_item>
<list_item><location><page_10><loc_9><loc_84><loc_49><loc_87></location>[12] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. Lett. 116 , 061102 (2016), arXiv:1602.03837 [gr-qc].</list_item>
<list_item><location><page_10><loc_9><loc_79><loc_49><loc_83></location>[13] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. Lett. 116 , 221101 (2016), [Erratum: Phys.Rev.Lett. 121, 129902 (2018)], arXiv:1602.03841 [gr-qc].</list_item>
<list_item><location><page_10><loc_9><loc_76><loc_49><loc_79></location>[14] K. Akiyama et al. (Event Horizon Telescope), Astrophys. J. Lett. 875 , L1 (2019), arXiv:1906.11238 [astro-ph.GA].</list_item>
<list_item><location><page_10><loc_9><loc_74><loc_44><loc_76></location>[15] E. M. Sänger et al. , (2024), arXiv:2406.03568 [gr-qc].</list_item>
<list_item><location><page_10><loc_9><loc_71><loc_49><loc_74></location>[16] S. Capozziello and M. De Laurentis, Phys. Rept. 509 , 167 (2011), arXiv:1108.6266 [gr-qc].</list_item>
<list_item><location><page_10><loc_9><loc_68><loc_49><loc_71></location>[17] E. Berti et al. , Class. Quant. Grav. 32 , 243001 (2015), arXiv:1501.07274 [gr-qc].</list_item>
<list_item><location><page_10><loc_9><loc_65><loc_49><loc_68></location>[18] S. Shankaranarayanan and J. P. Johnson, Gen. Rel. Grav. 54 , 44 (2022), arXiv:2204.06533 [gr-qc].</list_item>
<list_item><location><page_10><loc_9><loc_63><loc_37><loc_64></location>[19] C. Lanczos, Annals Math. 39 , 842 (1938).</list_item>
<list_item><location><page_10><loc_9><loc_62><loc_38><loc_63></location>[20] D. Lovelock, J. Math. Phys. 12 , 498 (1971).</list_item>
<list_item><location><page_10><loc_9><loc_60><loc_38><loc_61></location>[21] D. Lovelock, J. Math. Phys. 13 , 874 (1972).</list_item>
<list_item><location><page_10><loc_9><loc_55><loc_49><loc_60></location>[22] J. L. Blázquez-Salcedo, C. F. B. Macedo, V. Cardoso, V. Ferrari, L. Gualtieri, F. S. Khoo, J. Kunz, and P. Pani, Phys. Rev. D 94 , 104024 (2016), arXiv:1609.01286 [gr-qc].</list_item>
<list_item><location><page_10><loc_9><loc_52><loc_49><loc_55></location>[23] R. A. Konoplya, T. Pappas, and A. Zhidenko, Phys. Rev. D 101 , 044054 (2020), arXiv:1907.10112 [gr-qc].</list_item>
<list_item><location><page_10><loc_9><loc_49><loc_49><loc_52></location>[24] A. Maselli, L. Gualtieri, P. Pani, L. Stella, and V. Ferrari, Astrophys. J. 801 , 115 (2015), arXiv:1412.3473 [astro-ph.HE].</list_item>
<list_item><location><page_10><loc_9><loc_46><loc_49><loc_49></location>[25] D. Ayzenberg, K. Yagi, and N. Yunes, Phys. Rev. D 89 , 044023 (2014), arXiv:1310.6392 [gr-qc].</list_item>
<list_item><location><page_10><loc_9><loc_44><loc_38><loc_45></location>[26] B. Zwiebach, Phys. Lett. B 156 , 315 (1985).</list_item>
<list_item><location><page_10><loc_9><loc_43><loc_44><loc_44></location>[27] D. J. Gross and E. Witten, Nucl. Phys. B 277 , 1 (1986).</list_item>
<list_item><location><page_10><loc_9><loc_41><loc_46><loc_42></location>[28] D. J. Gross and J. H. Sloan, Nucl. Phys. B 291 , 41 (1987).</list_item>
<list_item><location><page_10><loc_9><loc_40><loc_48><loc_41></location>[29] M. Ostrogradsky, Mem. Acad. St. Petersbourg 6 , 385 (1850).</list_item>
<list_item><location><page_10><loc_9><loc_36><loc_49><loc_39></location>[30] D. Glavan and C. Lin, Phys. Rev. Lett. 124 , 081301 (2020), arXiv:1905.03601 [gr-qc].</list_item>
<list_item><location><page_10><loc_9><loc_33><loc_49><loc_36></location>[31] H. Lu and Y. Pang, Phys. Lett. B 809 , 135717 (2020), arXiv:2003.11552 [gr-qc].</list_item>
<list_item><location><page_10><loc_9><loc_30><loc_49><loc_33></location>[32] R. A. Hennigar, D. Kubizˇnák, R. B. Mann, and C. Pollack, JHEP 07 , 027 (2020), arXiv:2004.09472 [gr-qc].</list_item>
<list_item><location><page_10><loc_9><loc_27><loc_49><loc_30></location>[33] M. Gürses, T. c. ¸Si¸sman, and B. Tekin, Eur. Phys. J. C 80 , 647 (2020), arXiv:2004.03390 [gr-qc].</list_item>
<list_item><location><page_10><loc_9><loc_24><loc_49><loc_26></location>[34] R. A. Konoplya and A. F. Zinhailo, Eur. Phys. J. C 80 , 1049 (2020), arXiv:2003.01188 [gr-qc].</list_item>
<list_item><location><page_10><loc_9><loc_21><loc_49><loc_23></location>[35] R. Kumar and S. G. Ghosh, JCAP 07 , 053 (2020), arXiv:2003.08927 [gr-qc].</list_item>
<list_item><location><page_10><loc_9><loc_17><loc_49><loc_20></location>[36] S.-W. Wei and Y.-X. Liu, Eur. Phys. J. Plus 136 , 436 (2021), arXiv:2003.07769 [gr-qc].</list_item>
<list_item><location><page_10><loc_9><loc_14><loc_49><loc_17></location>[37] M. Guo and P.-C. Li, Eur. Phys. J. C 80 , 588 (2020), arXiv:2003.02523 [gr-qc].</list_item>
<list_item><location><page_10><loc_9><loc_10><loc_49><loc_14></location>[38] A. Naveena Kumara, C. L. A. Rizwan, K. Hegde, M. S. Ali, and K. M. Ajith, Annals Phys. 434 , 168599 (2021), arXiv:2004.04521 [gr-qc].</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_10><loc_52><loc_90><loc_92><loc_93></location>[39] S.-W. Wei and Y.-X. Liu, Phys. Rev. D 101 , 104018 (2020), arXiv:2003.14275 [gr-qc].</list_item>
<list_item><location><page_10><loc_52><loc_87><loc_92><loc_90></location>[40] S. A. Hosseini Mansoori, Phys. Dark Univ. 31 , 100776 (2021), arXiv:2003.13382 [gr-qc].</list_item>
<list_item><location><page_10><loc_52><loc_84><loc_92><loc_87></location>[41] B. Eslam Panah, K. Jafarzade, and S. H. Hendi, Nucl. Phys. B 961 , 115269 (2020), arXiv:2004.04058 [hep-th].</list_item>
<list_item><location><page_10><loc_52><loc_81><loc_92><loc_83></location>[42] K. Hegde, A. Naveena Kumara, C. L. A. Rizwan, A. K. M., and M. S. Ali, (2020), arXiv:2003.08778 [gr-qc].</list_item>
<list_item><location><page_10><loc_52><loc_76><loc_92><loc_80></location>[43] K. Hegde, A. Naveena Kumara, C. L. A. Rizwan, M. S. Ali, and K. M. Ajith, Annals Phys. 429 , 168461 (2021), arXiv:2007.10259 [gr-qc].</list_item>
<list_item><location><page_10><loc_52><loc_73><loc_92><loc_76></location>[44] A. Kumar, S. G. Ghosh, and A. Beesham, Eur. Phys. J. Plus 139 , 439 (2024).</list_item>
<list_item><location><page_10><loc_52><loc_70><loc_92><loc_72></location>[45] A. Kumar and S. G. Ghosh, Nucl. Phys. B 987 , 116089 (2023), arXiv:2302.02133 [gr-qc].</list_item>
<list_item><location><page_10><loc_52><loc_67><loc_92><loc_69></location>[46] S. G. Ghosh, A. Kumar, and D. V. Singh, Phys. Dark Univ. 30 , 100660 (2020).</list_item>
<list_item><location><page_10><loc_52><loc_65><loc_90><loc_66></location>[47] S. Masood and S. Mikki, (2024), arXiv:2406.05820 [gr-qc].</list_item>
<list_item><location><page_10><loc_52><loc_60><loc_92><loc_64></location>[48] P. G. S. Fernandes, P. Carrilho, T. Clifton, and D. J. Mulryne, Class. Quant. Grav. 39 , 063001 (2022), arXiv:2202.13908 [grqc].</list_item>
<list_item><location><page_10><loc_52><loc_57><loc_92><loc_60></location>[49] S. A. Fulling and J. H. Wilson, Phys. Scripta 94 , 014004 (2019), arXiv:1805.01013 [quant-ph].</list_item>
<list_item><location><page_10><loc_52><loc_54><loc_92><loc_56></location>[50] J. S. Ben-Benjamin et al. , Int. J. Mod. Phys. A 34 , 1941005 (2019), arXiv:1906.01729 [quant-ph].</list_item>
<list_item><location><page_10><loc_52><loc_51><loc_92><loc_53></location>[51] R. Chatterjee, S. Gangopadhyay, and A. S. Majumdar, Phys. Rev. D 104 , 124001 (2021), arXiv:2104.10531 [quant-ph].</list_item>
<list_item><location><page_10><loc_52><loc_48><loc_92><loc_50></location>[52] S. Sen, R. Mandal, and S. Gangopadhyay, Phys. Rev. D 105 , 085007 (2022), arXiv:2202.00671 [hep-th].</list_item>
<list_item><location><page_10><loc_52><loc_40><loc_92><loc_47></location>[53] P. R. Anderson, M. R. R. Good, and C. R. Evans, in 14th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories , Vol. 2 (2017) pp. 1701-1704, arXiv:1507.03489 [gr-qc].</list_item>
<list_item><location><page_10><loc_52><loc_32><loc_92><loc_39></location>[54] M. R. R. Good, P. R. Anderson, and C. R. Evans, in 14th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories , Vol. 2 (2017) pp. 1705-1708, arXiv:1507.05048 [gr-qc].</list_item>
<list_item><location><page_10><loc_52><loc_29><loc_92><loc_31></location>[55] K. Chakraborty and B. R. Majhi, Phys. Rev. D 100 , 045004 (2019), arXiv:1905.10554 [gr-qc].</list_item>
<list_item><location><page_10><loc_52><loc_25><loc_92><loc_28></location>[56] H. E. Camblong, A. Chakraborty, and C. R. Ordonez, Phys. Rev. D 102 , 085010 (2020), arXiv:2009.06580 [gr-qc].</list_item>
<list_item><location><page_10><loc_52><loc_21><loc_92><loc_25></location>[57] A. Azizi, H. E. Camblong, A. Chakraborty, C. R. Ordonez, and M. O. Scully, Phys. Rev. D 104 , 065006 (2021), arXiv:2011.08368 [gr-qc].</list_item>
<list_item><location><page_10><loc_52><loc_16><loc_92><loc_20></location>[58] A. Azizi, H. E. Camblong, A. Chakraborty, C. R. Ordonez, and M. O. Scully, Phys. Rev. D 104 (2021), 10.1103/PhysRevD.104.084086, arXiv:2108.07570 [gr-qc].</list_item>
<list_item><location><page_10><loc_52><loc_11><loc_92><loc_15></location>[59] A. Azizi, H. E. Camblong, A. Chakraborty, C. R. Ordonez, and M. O. Scully, Phys. Rev. D 104 (2021), 10.1103/PhysRevD.104.084085, arXiv:2108.07572 [gr-qc].</list_item>
<list_item><location><page_11><loc_9><loc_90><loc_49><loc_93></location>[60] S. Sen, R. Mandal, and S. Gangopadhyay, Phys. Rev. D 106 , 025004 (2022), arXiv:2205.11260 [gr-qc].</list_item>
<list_item><location><page_11><loc_9><loc_87><loc_49><loc_90></location>[61] S. Sen, R. Mandal, and S. Gangopadhyay, Eur. Phys. J. Plus 138 , 855 (2023), arXiv:2301.04834 [gr-qc].</list_item>
<list_item><location><page_11><loc_9><loc_84><loc_49><loc_87></location>[62] S. Chandrasekhar, The mathematical theory of black holes (Oxford University Press, 1992).</list_item>
<list_item><location><page_11><loc_9><loc_79><loc_49><loc_83></location>[63] J. A. H. Futterman, F. A. Handler, and R. A. Matzner, Scattering from Black Holes , Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2012).</list_item>
<list_item><location><page_11><loc_9><loc_76><loc_49><loc_79></location>[64] S. M. A. S. Bukhari, I. A. Bhat, C. Xu, and L.-G. Wang, Phys. Rev. D 107 , 105017 (2023), arXiv:2211.08793 [gr-qc].</list_item>
<list_item><location><page_11><loc_9><loc_73><loc_49><loc_76></location>[65] I. Sakalli and S. Kanzi, Turk. J. Phys. 46 , 51 (2022), arXiv:2205.01771 [hep-th].</list_item>
<list_item><location><page_11><loc_9><loc_71><loc_39><loc_72></location>[66] S. W. Hawking, Phys. Rev. D 14 , 2460 (1976).</list_item>
<list_item><location><page_11><loc_9><loc_67><loc_49><loc_71></location>[67] A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, Rev. Mod. Phys. 93 , 035002 (2021), arXiv:2006.06872 [hep-th].</list_item>
<list_item><location><page_11><loc_9><loc_63><loc_49><loc_66></location>[68] D. Kastor and J. H. Traschen, Class. Quant. Grav. 13 , 2753 (1996), arXiv:gr-qc/9311025.</list_item>
<list_item><location><page_11><loc_9><loc_62><loc_46><loc_63></location>[69] M. Visser, JHEP 07 , 009 (2015), arXiv:1409.7754 [gr-qc].</list_item>
<list_item><location><page_11><loc_9><loc_59><loc_49><loc_61></location>[70] Y. Qiu and J. Traschen, Class. Quant. Grav. 37 , 135012 (2020), arXiv:1908.02737 [hep-th].</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_11><loc_52><loc_90><loc_92><loc_93></location>[71] C. Barcelo, S. Liberati, and M. Visser, Living Rev. Rel. 8 , 12 (2005), arXiv:gr-qc/0505065.</list_item>
<list_item><location><page_11><loc_52><loc_87><loc_92><loc_90></location>[72] S. L. Braunstein, M. Faizal, L. M. Krauss, F. Marino, and N. A. Shah, (2023), 10.1038/s42254-023-00630-y.</list_item>
<list_item><location><page_11><loc_52><loc_84><loc_92><loc_87></location>[73] M. J. Jacquet, S. Weinfurtner, and F. König, Phil. Trans. Roy. Soc. Lond. A 378 , 20190239 (2020), arXiv:2005.04027 [gr-qc].</list_item>
<list_item><location><page_11><loc_52><loc_79><loc_92><loc_83></location>[74] S. Weinfurtner, E. W. Tedford, M. C. J. Penrice, W. G. Unruh, and G. A. Lawrence, Phys. Rev. Lett. 106 , 021302 (2011), arXiv:1008.1911 [gr-qc].</list_item>
<list_item><location><page_11><loc_52><loc_76><loc_92><loc_79></location>[75] J. Hu, L. Feng, Z. Zhang, and C. Chin, Nature Phys. 15 , 785 (2019), arXiv:1807.07504 [physics.atom-ph].</list_item>
<list_item><location><page_11><loc_52><loc_70><loc_92><loc_76></location>[76] J. Steinhauer, M. Abuzarli, T. Aladjidi, T. Bienaimé, C. Piekarski, W. Liu, E. Giacobino, A. Bramati, and Q. Glorieux, Nature Commun. 13 , 2890 (2022), arXiv:2102.08279 [cond-mat.quant-gas].</list_item>
<list_item><location><page_11><loc_52><loc_63><loc_92><loc_69></location>[77] V. E. Hubeny, S. Minwalla, and M. Rangamani, in Theoretical Advanced Study Institute in Elementary Particle Physics: String theory and its Applications: From meV to the Planck Scale (2012) pp. 348-383, arXiv:1107.5780 [hep-th].</list_item>
</document>
[
{
"title": "ABSTRACT",
"content": "1",
"pages": [
1
]
},
{
"title": "A Casimir-like probe for 4D Einstein-Gauss-Bonnet gravity",
"content": "Syed Masood 1 ∗ and Said Mikki 1 , 2† Zhejiang University/University of Illinois at Urbana-Champaign Institute (the ZJU-UIUC Institute), Zhejiang University, 718 East Haizhou Road, Haining 314400 , China. and 2 Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana IL 61801 , USA (Dated: July 3, 2024) Virtual transitions in a Casimir-like configuration are utilized to probe quantum aspects of the recently proposed four-dimensional Einstein-Gauss-Bonnet (4D EGB) gravity. This study employs a quantum optics-based approach, wherein an Unruh-DeWitt detector (modeled as a two-level atom) follows a radial timelike geodesic, falling freely into an uncharged, nonrotating black hole described by 4D EGB gravity, becoming thermalized in the usual Unruh manner. The black hole, asymptotically Minkowskian, is enclosed by a Casimir boundary proximate to its horizon, serving as a source for accelerated field modes that interact with the infalling detector. Observations are conducted by an asymptotic infinity observer, assuming a Boulware field state. Our numerical analysis reveals that, unlike in Einstein gravity, black holes in 4D EGB gravity can either enhance or suppress the intensity of acceleration radiation, contingent upon the Gauss-Bonnet coupling parameter α . Specifically, we observe radiation enhancement for negative α and suppression for positive α . These findings offer substantial insights into quantifying the influence of higher-curvature contributions on the behavior of quantum fields in black hole geometries within a 4D spacetime.",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "Considerable efforts have been made over the past few decades to uncover the deep connection between quantum mechanics, gravity, and thermodynamics [1, 2]. Among these endeavors, the discovery of Hawking radiation from black holes [3] and the Unruh effect for accelerated observers in flat Minkowski spacetime [4, 5] stand out as pivotal. Another significant phenomenon is Parker's idea of particle emission due to the expansion of the Universe [2]. In all these cases, the quantum state of the field is altered by a dynamic background spacetime geometry or the state of motion, resulting in the creation of real particles-an effect arising from the violation of Poincaré invariance [1]. This is similar to the dynamical Casimir effect (DCE) [6, 7], where accelerated plates or boundaries induce the quantum vacuum to radiate particles. Consequently, this scenario fosters a rich intersection of quantum fields, boundaries, and spacetime geometries [8-11]. With the advent of precise experimental and observational setups, it has become possible over the decades to test Einstein's general relativity (GR) in extreme gravity regimes. So far, GR has consistently matched observational data, with milestone achievements including gravitational wave detection [12, 13], black hole shadows [14], and neutron star mergers [15]. However, physicists have long recognized that GR cannot address certain fundamental issues in the Universe, such as the existence of singularities, cosmological acceleration, dark matter, and a consistent merger of quantum mechanics and gravity. Thus, it is evident that a framework beyond GR is needed to resolve these challenges [16-18]. Several alternatives to GR predict additional highercurvature contributions to the gravitational action. A significant framework within this class of models originates from the works of Lanczos [19] and Lovelock [20, 21], leading to the well-known Einstein-Gauss-Bonnet (EGB) theory. It has been established that EGB gravity does not introduce modifications to gravitational dynamics unless coupled with additional field degrees of freedom or in spacetime dimensions D ≥ 5. One example of such additional fields is the dilaton field [22-25]. In addition to this, EGB gavity theories yield equations of motion that are quadratic in metric tensor. This quadratic nature is a unique feature of EGB gravity among all other alternatives to GR. The interesting coincidence is that the low energy effective descriptions of heterotic string theories also posit quadratic contributions to the dynamics of Einstein gravity [26-28]. It may be noted that the quadratic nature of equations of motion suffice to get rid of Ostrogradsky instability [29] and thus guarantees physicality of the dynamics. Furthermore, EGB gravity theories are characterized by equations of motion that are quadratic in the metric tensor. This quadratic nature distinguishes EGB gravity from other alternatives to GR. An intriguing coincidence arises in that the low-energy effective descriptions of heterotic string theo- es also incorporate quadratic contributions to the dynamics of Einstein gravity [26-28]. Importantly, the quadratic form of the equations of motion resolves the Ostrogradsky instability [29], ensuring the physical viability of the theory. Recently, Glavan and Lin [30] addressed the question of Gauss-Bonnet (GB) contributions in 4-dimensional spacetime geometry by proposing a specific rescaling of the GB coupling parameter α → α / ( D -4 ) , where D denotes the spacetime dimensionality. This rescaling ensures a well-defined limit as D → 4. The resulting model maintains quadratic behavior to prevent Ostrogradsky instability, yet it departs from the implications of the well-known Lovelock theorem [19-21]. It is noteworthy that no additional field coupling is required in this model. As a new phenomenological competitor to Einstein's General Relativity (GR), this model has sparked rigorous debates over the years. Some investigations include consistency checks [31-33], studies of black hole shadows and quasinormal modes [34-36], analysis of geodesics [37], particle accelerator models [38], and a wide array of thermodynamic analyses [39-47]. A comprehensive overview of 4D-EGB gravity, covering its various aspects, can be found in a review article by Fernandes et al. [48]. Recognizing the significance of the findings in Ref. [30], we are driven to investigate the potential quantum radiative signatures of 4D EGB gravity using elements from quantum optics and Casimir physics. Our approach involves a quantum optical cavity positioned with one end near a black hole horizon and the other at asymptotic infinity. Within this setup, a two-level Unruh-DeWitt detector (an atom) falls freely towards the black hole. Virtual transitions arising from the interaction between the detector and the field lead to acceleration radiation, which carries distinct imprints of the underlying gravitational background. Such a setup has been discussed in Ref. [8], where it was demonstrated that, under appropriate initial conditions, a detector near a Schwarzschild black hole emits radiation with a thermal spectrum. This unique radiative emission, known as Horizon Brightened Acceleration Radiation (HBAR), occurs when the detector is in free fall towards the black hole. This concept has been further explored in various contexts, revealing profound connections between the equivalence principle, quantum optics, and the HawkingUnruh effect [8, 49-52]. It also underscores connections to the Dynamical Casimir Effect (DCE) and moving mirror models [53, 54], frequently employed in studying quantum field behavior in curved spacetimes. But while the original work in Ref. [8] considers detectors moving along timelike geodesics, subsequent studies have shown that similar phenomena can occur for detectors following null geodesics [55]. This novel radiative emission phenomenon can be attributed to the nearhorizon physics and conformal quantum mechanics of black holes [56-61]. Given that quantum field dynamics can elucidate the nature of underlying spacetime geometry [1, 2], we view the aforementioned setup as a potential avenue to probe 4D EGB gravity at a deeper level. Through numerical analysis, we demonstrate that 4D EGB gravity can imprint distinct features on the radiation spectrum compared to Einstein's GR, encompassing both negative and positive values of the Gauss-Bonnet coupling parameter. The structure of the paper is as follows. The next Sec. II introduces the basics of 4D EGB black hole geometry, accompanied by discussions on the wave equation and the vacuum field state. In Sec. III, we compute the excitation probability or the detector response function of the falling detector. Sec. IV explores possible interpretations of our numerical findings. Finally, conclusions are drawn in Sec. V.",
"pages": [
1,
2
]
},
{
"title": "II. CONCEPTUAL ASPECTS: OUR SPACETIME GEOMETRY AND THE CHOICE OF FIELD MODES",
"content": "The static, spherically symmetric metric of an uncharged and nonrotating black hole in 4D EGB gravity is given by [48] where 1 where ± sign inside brackets denotes Gauss-Bonnet (GB) and GR branches, respectively. Here, we focus solely on the GR branch, as the GB branch is deemed unphysical [48]. To determine the event horizon radius, we set which yields of which the one with plus sign is the real exterior horizon of the black hole. Thus, our event horizon is located at r g = r ± = M 2 + √ M 2 -α . The parameter α can take both positive and negative values within the range -32 M 2 ≤ α ≤ 4 M 2 , as indicated in Refs. [34, 37] (also see [30]). It is evident that a positive GB coupling constant α decreases the black hole horizon radius, whereas a negative α increases it. The limit α = 0 corresponds to the Schwarzschild black hole in GR. These relationships are illustrated graphically in Fig. 1. We also note that r + r -= α , and as r → 0, the metric components remain finite. This can be observed from Eq. (2), where lim r → 0 f ( r ) = 1. However, the finiteness of the metric components does not guarantee the absence of singularities due to the fact that the Ricci scalar R and the Kretschmann scalar R µνσδ R µνσδ vary as R ∝ r -3 / 2 and R µνσδ R µνσδ ∝ r -3 , respectively. It should be noted that for the Schwarzschild case, the Kretschmann scalar near r = 0 varies as r -6 , indicating that the GB contribution significantly weakens the singularity by several orders of magnitude [48].",
"pages": [
2,
3
]
},
{
"title": "A. Detector trajectories",
"content": "In this section, we analyze the geodesics of the detector to compute both the coordinate time and proper (conformal) time that describe the timelike trajectory of the infalling (massive) detector. Generally, for a given Christoffel connection Γ µ ρσ , the complete geodesic equations are expressed as [62] Our spacetime geometry of interest exhibits spherical symmetry, and we restrict our analysis to the radial motion of the detector in the equatorial plane. Therefore, we set θ = π / 2, which implies ˙ θ = 0 and ˙ φ = 0. Consequently, the following conservation equations hold: Note that E is a constant representing the specific energy of the detector. It is determined by the initial boundary conditions of the geodesic motion, given by E 2 = f ( r ) ∣ ∣ max . Since we assume that the detector started its motion from asymptotic infinity, where the spacetime is asymptotically Minkowski flat ( r → ∞ implies f ( r ) ∣ ∣ max = 1), these constraints from the above equations lead to It should be emphasized that E , which is related to the maximum of f ( r ) , is the same for both GR and 4D EGB gravity. This value of E corresponds to asymptotic infinity, where both GRand 4D EGB theories reproduce flat Minkowski geometry. Now, integrating Eq. (7) along the radial trajectories from some arbitrary initial point r i to a final point r f (where r i > r f ), we obtain We now substitute Eq. (2) into Eq. (8) in order to compute τ , resulting in Here, τ 0 serves as an integration constant, the insignificance of which we establish for the final detector response, as detailed in Sec. III. However, the complexity of the integral for t precludes straightforward analytical computation. Consequently, we resort to numerical methods and present the outcomes in Sec. III. Fig. 2 illustrates the plots of τ and t . The plots clearly illustrate that t and τ exhibit typical Schwarzschild-like behavior. Specifically, t , which represents the time measured by an asymptotic observer, diverges as the detector approaches the black hole horizon, located at zero on the rescaled radial coordinate r -r g. This divergence signifies that, from the perspective of this observer, the detector never actually crosses the horizon. In contrast, τ remains finite at the horizon r -r g, indicating that from the detector's own frame of reference, it crosses the horizon in a finite amount of proper time. This disparity highlights the causal structure of black hole horizons and is recognized as gravitational time dilation. Furthermore, in 4D EGB gravity, the coupling parameter α influences the behavior of t and τ . For positive α , which reduces the black hole size as discussed in Sec. II, it takes longer for the detector to approach the horizon as α increases. Conversely, for negative α , which inflates the black hole size, the situation is reversed.",
"pages": [
3
]
},
{
"title": "B. Defining the vacuum state",
"content": "The response function, or excitation probability, to be calculated in Sec. III, quantifies the detector-field coupling. To achieve this, we must obtain the appropriate field mode by solving the wave equation on the specified spacetime background. Here, we consider the simplest test field: a massless spin-0 Klein-Gordon field, minimally coupled to the spacetime geometry, described by ∇ µ ∇ µ Φ = 0 [1]. Given the spherical symmetry of the spacetime and the presence of a timelike Killing vector ∂ t , we have Φ = 1 r Y l ( θ , φ ) ψ ( t , r ) , with Y l denoting spherical harmonics and l representing the multipole number. The radial part of the solution, after neglecting the angular dependence ( l = 0 ) , satisfies the following Schrödinger-like wave equation Here, r ∗ denotes the Regge-Wheeler tortoise coordinate, a useful parameter for describing the propagation of test fields in black hole geometries, defined by [62] where we utilized Eq. (2). Additionally, V ( r ) represents the effective potential experienced by the field, often describing scattering effects in black hole spacetimes [63]. However, given our focus on the simplest scenario possible, as also demonstrated in Refs. [8, 64], V ( r ) can be neglected. One approach to achieve this is by assuming that the frequency ν of the field mode is sufficiently large, enabling it to surmount the potential barrier imposed by the spacetime. Consequently, the field mode simplifies to This represents a normalized outgoing field mode with frequency ν , as observed by an asymptotic infinity observer, qualifying as a Boulware field state. The ingoing field modes generated propagate towards the boundary at the black hole horizon and are lost. The Boulware field mode described above is an approximate field state obtained by neglecting V ( r ) and assuming ν to be very large. This assumption serves as one of the initial conditions required for the existence of HBAR emission [8]. Generally, in the context of black holes, multiple vacuum states are utilized due to the absence of a unique vacuum state in curved spacetime. This leads to various notions of vacuum states, such as the Unruh vacuum, Hartle-Hawking vacuum [1, 2], and others. In principle, there should be an infinite number of possible vacuum states due to the violation of Poincaré invariance in curved spacetimes [1]. In contrast, for Minkowski space, where the field satisfies Poincaré invariance, the vacuum state remains same for all inertial observers. In our scenario, the choice of the Boulware vacuum state arises because the observations are made by an asymptotic observer, for whom the Boulware field state is most appropriate. In this context, no Hawking radiation is detected by the observer. Moreover, the black hole is assumed to be entirely enclosed by a Casimir boundary, which effectively prevents any potential Hawking quanta from mixing with HBAR flux [8]. This distinction ensures that HBAR emission is fundamentally different from Hawking radiation. ̸ Additionally, we have excluded l = 0 modes for simplicity. However, considering a smaller ν such that V ( r ) = 0 would lead to the emergence of scattering effects, potentially necessitating the inclusion of greybody factors [65]. Nevertheless, we argue that such inclusions would lead to the deviation from the primal essence of HBAR emission, which occurs under specific boundary conditions as emphasized in Refs. [8, 50].",
"pages": [
3,
4
]
},
{
"title": "III. DETECTOR RESPONSE",
"content": "As discussed in the preceding section, the field is in a Boulware vacuum state, ensuring that no Hawking radiation is observed by the asymptotic observer. By neglecting the angular dependence of the field modes, the detector-field interaction Hamiltonian can be expressed as follows [8]: Here, ˆ a ν is the annihilation operator for the field modes, ˆ σ is the detector lowering operator, and H . C . denotes the Hermitian conjugate. Here, g is a detector-field coupling parameter indicating the strength of the interaction and can be taken as a constant for a massless Klein-Gordon field (spin-0). Assuming that the detector is initially in the ground state | b ⟩ , the probability that it transitions to an excited state | a ⟩ with the emission of a field quantum of frequency ν is given by Utilizing time-dependent perturbation theory, such a process is typically prohibited in quantum optics due to energy conservation principles. However, in non-inertial frames influenced by acceleration and gravity, these virtual processes can occur owing to counter-rotating terms in the Hamiltonian [50], as exemplified by the Unruh effect [4]. By employing Eq. (14) and performing some additional straightforward compu- tations, Eq. (15) can be reexpressed as Simplifying further, we arrive at which results in a complex expression involving nested integrals with respect to t ( r ) and r ∗ . It's important to note that the limits of integration correspond to the detector's trajectory from r = ∞ to r = r g, the horizon of the black hole. Thus, from Eqs. (8) and (12), we derive: Consider now the substitution r = r g z , where d r = r gd z . Using this transformation of variables, we may rewrite t ( r ) in Eq. (18) as follows: A further substitution of the form x = z -1, such that z = x + 1, yields One can follow a similar calculation for r ∗ , arriving at After deploying all the relevant quantities in Eq. (17), we derive the following final expression for the detector excitation: where This represents the primary outcome of our investigation. The numerical integral in (22) is notably intricate, demanding a careful approach for its accurate computation. To achieve that, in what follows we deploy the numerical integration capabilities of the Mathematica symbolic math package for performing all required calculations. The figures presented in Fig. 3 were generated using optimized settings.",
"pages": [
5,
6,
7
]
},
{
"title": "IV. RESULTS AND DISCUSSIONS",
"content": "Based on the preceding analysis, the two-level UnruhDeWitt detector, operating in the Boulware vacuum state, registers detections while in free fall (inertial). This observation appears to challenge established field-theoretic concepts associated with the Hawking-Unruh effect. Specifically, there is no emission of Hawking radiation in the Boulware state as observed from asymptotic infinity, nor does the Unruh effect manifest for inertial detectors in the Minkowski vacuum. However, HBAR emission from detectors operates on different principles [8, 50]. While it shares similarities with Hawking radiation, such as the thermal nature of the emitted flux and the associated Bekenstein-Hawking entropy-area correspondence, there are also distinct characteristics. Notably, HBAR emission involves the evolution of field modes in pure states and includes phase correlations between them. These aspects naturally relate to the black hole information paradox [66, 67]. In Fig. 3, we present the detector excitation probability, Γ exc, plotted as a function of the emitted radiation frequency, ν . The impact of the GB coupling parameter, α , is depicted in Figs. 3(a) and 3(b) for positive and negative values of α , respectively. Fig. 3(c) illustrates how the detector transition frequency, ω , influences Γ exc, while Fig. 3(d), shown on a loglog scale, highlights the behavior of Γ exc near the origin and its convergence at higher frequencies. It is important to note that our interpretations and analyses are based on numerical estimations detailed in the preceding sections. These figures provide a comprehensive view of the radiative characteristics under consideration, elucidating the role of α and the detector's transition frequency in shaping Γ exc. From all plots, one of the prominent features observed is the thermal nature of the HBAR radiation flux, characterized by a Bose-Einstein (BE) distribution. This observation leads us to conclude that 4D Einstein-Gauss-Bonnet (EGB) gravity does not alter the thermal nature of the flux, consistent with earlier findings [8, 50, 55, 58, 59] in the context of Einstein gravity. This characteristic mirrors the thermal emission observed in Hawking radiation from pure black holes with asymptotically flat geometries. It is noteworthy that for the so-called 'dirty' black holes beyond the Kerr-Newman family, such as in the de Sitter case, there exists the possibility of observing a nonthermal spectrum [64, 68-70]. The detector excitation probability Γ exc, as observed in Fig. 3(a), decreases with increasing positive values of α and increases with negative values of α . As previously discussed, positive α reduces the size of the black hole horizon [see Fig. 1(b)], leading to the conclusion that smaller black holes emit less radiation flux compared to larger ones. This reasoning can similarly be applied to negative values of α . It is crucial to emphasize that in the limit α → 0, depicted in Fig. 3(a), the scenario converges to that of the pure Schwarzschild black hole. The attenuation and augmentation of particle production can be conceptually grasped as follows. Particles generated within black hole spacetimes, as in Hawking radiation, experience backreaction due to the gravitational tidal forces exerted by the black hole. This backreaction diminishes the intensity of the radiation. Tidal effects in black holes stem from their surface gravities, which are directly related to their horizon radii. Specifically, for a black hole with a horizon radius rg , the surface gravity varies inversely with the square of rg . This relationship implies that larger black holes have smaller surface gravities and correspondingly weaker tidal effects, and conversely, smaller black holes exhibit stronger tidal effects. In the context of positive α , the black hole horizon size decreases monotonically, leading to stronger surface gravity and tidal effects compared to a Schwarzschild black hole ( α = 0). Consequently, particles generated within such spacetimes experience heightened backreaction, impeding their propagation. This results in fewer particles escaping to asymptotic infinity, thereby reducing the intensity of the particle spectrum, as illustrated in Fig. 3(a). Conversely, for negative values of α , the black hole horizon radius increases, indicating reduced backreaction and tidal forces. This condition allows more particles to escape from the black hole spacetime, leading to an enhancement in the radiation flux, as depicted in Fig. 3(b). This constitutes the primary finding of our study, distinguishing 4D EGB gravity from Einstein GR. To investigate the influence of the detector transition frequency ω on the radiation intensity, we plot Γ exc against ω in Fig. 3(c). The graphs clearly demonstrate a decrease in radiation intensity as ω increases. This behavior aligns with the principles of the Unruh effect [4] and can be understood in terms of energy conservation: higher detector transition frequencies require more energy to excite the detector, resulting in a lower excitation probability, and vice versa. Furthermore, to gain insight into the spectrum's behavior at low and high frequencies ( ν ), we reexamined Γ exc from Fig. 3(a) using a log-log scale. It is evident that the spectrum exhibits a finite Bose-Einstein (BE)-type distribution near the origin where ν → 0. As ν increases, the spectrum converges and exhibits a thermal tail, characteristic of a BE or Planckian distribution. In the meantime, it is pertinent to consider the testable implications of this study. We can draw insights from analog gravity systems [71-73], which have been actively utilized over the last few decades to explore quantum fieldtheoretic phenomena in curved spacetimes. These systems have provided valuable analogs for understanding effects such as Hawking radiation [74], the Unruh effect [75], and Parker particle generation in expanding spacetimes [76]. Most of these setups, whether based on condensed matter or quantum optical systems, are closely tied to Casimir physics involving moving boundaries. This connection is particularly relevant to our study, which is largely inspired by these concepts. Recently, condensed matter systems have been utilized to explore phenomena extending beyond particle generation in exotic backgrounds, including applications to fluid/gravity correspondence [77]. Looking forward, there is potential for future tabletop experiments to simulate black hole horizons resembling those in 4D EGB gravity [30] or to investigate HBAR radiation scenarios [8], where our findings could offer valuable insights.",
"pages": [
7,
8,
9
]
},
{
"title": "V. CONCLUSION AND OUTLOOK",
"content": "The exploration of theories beyond Einstein gravity has evolved in parallel with general relativity (GR) itself. These modified or extended gravity theories aim to tackle fundamental cosmological issues such as cosmic acceleration, singularities, and dark matter. Among these models, Einstein-GaussBonnet (EGB) gravity stands out, predicting higher-curvature corrections to the Einstein-Hilbert action. These corrections arise either in higher dimensions or through additional field couplings to the gravitational action. Interestingly, these contributions also emerge in the low-energy effective description of heterotic string theory. The 4D EGB theory represents a novel gravitational model that has sparked intense debate since its inception several years ago. This model predicts the presence of a Gauss-Bonnet (GB) term within 4D spacetime, which would otherwise not contribute to the latter's geometry. Its ability to provide a nontrivial contribution is achieved through a redefinition (rescalling) of the GB parameter [30]. Importantly, this theory circumvents the Lovelock theorem and sidesteps Ostrogadsky instability, ensuring that the resulting gravitational dynamics remain quadratic. The theory has been scrutinized across various phenomenological fronts. In this paper, we examined the quantum radiative properties of a nonrotating, uncharged black hole in 4D EinsteinGauss-Bonnet (EGB) gravity using a Casimir-type configuration. The black hole, surrounded by a reflecting mirror, induced accelerated field modes from the Boulware vacuum state. We analyzed the interaction of these field modes with a freely falling two-level Unruh-DeWitt detector, which exhibited characteristic clicking behavior akin to the Unruh effect. The spectrum detected by the detector follows a BoseEinstein (BE) distribution, with a notable dependence on the GB parameter α . By examining both positive and negative values of α , we studied their influence on the radiation intensity emitted by the detector. We observed that radiation intensity diminishes when α is positive. This reduction is attributed to the shrinking of the black hole size caused by positive α . Conversely, for negative α , we observed an increase in radiation intensity. The reduction or augmentation of the radiation flux is examined in relation to a pure Schwarzschild black hole, where the limit α → 0 is considered. Additionally, we observed that the transition frequency of the detector reduces the profile of particle creation due to the high energy needed for its excitation, consistent with the standard predictions of the Unruh effect. Finally, the spectrum is finite near the origin and monotonically converges at the high end of the frequency ranges, yielding the distinctive thermal tail characteristic of a Bose-Einstein or Planckian distribution. Our work provides an opportunity to explore various aspects of 4D EGB gravity by incorporating different energymatter distributions around the simplest black hole model possible. Moreover, exploring other types of detector-field couplings could yield valuable insights into the nature of field configurations within the context of 4D EGB gravity. These and similar questions constitute promising extensions of this work, which we plan to pursue in the future.",
"pages": [
9
]
}
]
2024arXiv240721482B
https://arxiv.org/pdf/2407.21482.pdf
<document>
<section_header_level_1><location><page_1><loc_20><loc_79><loc_80><loc_83></location>Measurements of Gravitational Attractions at small Accelerations</section_header_level_1>
<text><location><page_1><loc_23><loc_73><loc_77><loc_77></location>W. Bartel 1 † C. W. Elvers 1 L. Jonsson 2 G. Kempf 1 H. Krause 3 B. Loehr 1 E. Lohrmann 3 , H. Meyer 4 † P. Steffen 1 E. Wuensch 1</text>
<text><location><page_1><loc_24><loc_64><loc_76><loc_71></location>1 Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany 2 Lund University, Sweden 3 Universitat Hamburg 4 Bergische Universitat, Wuppertal †</text>
<text><location><page_1><loc_47><loc_63><loc_54><loc_65></location>deceased</text>
<section_header_level_1><location><page_1><loc_46><loc_42><loc_54><loc_43></location>Abstract</section_header_level_1>
<text><location><page_1><loc_20><loc_28><loc_80><loc_41></location>Gravitational interactions were studied by measuring the influence of small external field masses on a microwave resonator. It consisted of two spherical mirrors, which acted as independent pendulums individually suspended by strings. Two identical field masses were moved along the axis of the resonator symmetrically and periodically between a near and a far position. Their gravitational interaction altered the distance between the mirrors, changing the resonance frequency, which was measured and found consistent with Newton's law of gravity. The acceleration of a single mirror caused by the two field masses at the closest position varied from 5 . 4 · 10 -12 m / s 2 to 259 · 10 -12 m / s 2 .</text>
<section_header_level_1><location><page_2><loc_20><loc_86><loc_35><loc_88></location>1 Introduction</section_header_level_1>
<text><location><page_2><loc_20><loc_74><loc_80><loc_85></location>In the 1930s, observations were made by several astronomers that the galaxies contain more mass than expected from Newton's gravitational law, based on the visible mass. The first publication of such an observation was made by K. Lundmark [1]. Later the term 'dark matter' was coined. The present predominant hypothesis is that dark matter is some kind of stable, heavy, neutral, weakly interacting particle. However, in spite of many researches to find or identify the particles of dark matter, nothing has been found so far [2], [4]. An alternative to dark matter is the MOND-model that assumes a modified Newton's law of gravity [3].</text>
<text><location><page_2><loc_20><loc_71><loc_80><loc_74></location>Acompilation of astronomical data, by [5], shows a clear deviation from Newton's predictions, based on the baryonic matter, for accelerations less than 10 -10 m / s 2 .</text>
<text><location><page_2><loc_20><loc_68><loc_80><loc_71></location>This experiment aims to to test Newton's gravitational law at accelerations below 10 -10 m / s 2 on Earth, relying solely on gravitational interactions.</text>
<text><location><page_2><loc_20><loc_64><loc_80><loc_68></location>Small field masses were used to accelerate the masses of two pendulums perpendicular to the Earth's acceleration. The measured movements of the pendulums are proportional to the acceleration caused by the field masses.</text>
<text><location><page_2><loc_20><loc_58><loc_80><loc_64></location>Results from measurements with field masses ranging from 9 kg to 1 kg have been published previously [6] using an earlier version of the experiment. The results did not show any deviation from Newton's law of gravitation. A similar result was obtained using an experimental setup of the Cavendish-type [7].</text>
<text><location><page_2><loc_20><loc_53><loc_80><loc_58></location>This paper presents results from an improved experimental setup using field masses from 3 kg to 0.1 kg, which extend the range of accelerations to significantly smaller values, measured with considerably higher precision. The measurements were carried out in the years 2020 to 2022.</text>
<section_header_level_1><location><page_2><loc_20><loc_49><loc_42><loc_50></location>2 Experimental Setup</section_header_level_1>
<text><location><page_2><loc_20><loc_40><loc_80><loc_47></location>Figure 1 shows the main components of the experiment: a system of two pendulums that form a microwave resonator, similar to a Fabry-Perot interferometer. The introduction of external field masses causes a tiny acceleration of the pendulums in the direction of the resonator axis. The resulting change of the resonator length caused a measurable change in the resonance frequency.</text>
<text><location><page_2><loc_20><loc_38><loc_80><loc_40></location>The resonator was located inside a vacuum vessel. A transport system moved the field masses between positions A and B.</text>
<text><location><page_2><loc_20><loc_32><loc_80><loc_37></location>An earlier version of the apparatus had been built and operated at Wuppertal University for a precision measurement of the gravitational constant: [8], [9], [10], [11], where details can be found. It was later transferred to DESY to an experimental hall of the PETRA accelerator.</text>
<text><location><page_2><loc_20><loc_28><loc_80><loc_32></location>After an initial measurement period, which resulted in a publication [6], it was relocated to an underground experimental hall of the HERA accelerator, and the setup was improved significantly.</text>
<section_header_level_1><location><page_2><loc_20><loc_24><loc_36><loc_26></location>2.1 The Resonator</section_header_level_1>
<text><location><page_2><loc_20><loc_18><loc_80><loc_23></location>The central part of the experiment was a microwave resonator. Figure 2 shows a schematic view. It consisted of two mirrors with spherical surfaces and a cylindrical center piece in between, which suppresses higher order resonance modes. The mirrors and the center piece were suspended individually by thin strings of about 2.7 m in</text>
<text><location><page_3><loc_20><loc_84><loc_80><loc_88></location>length from the upper lid of the vacuum vessel. Each mirror acted as a mechanical pendulum with a period of 3.278 s. The resonator was operated at a resonance frequency of 23.2699 GHz.</text>
<text><location><page_3><loc_20><loc_78><loc_80><loc_83></location>Stochastic movements of the pendulums were damped by permanent magnets positioned below the mirrors in order to optimize the performance of the resonator. The center of gravity of each mirror coincided with the vertex of the mirror. The distance between the two vertices was b 0 = 0 . 2400 m.</text>
<text><location><page_3><loc_20><loc_74><loc_80><loc_78></location>Radio frequency power was fed into the resonator through a small hole in the center of one mirror, and was similarly extracted on the opposite side. It was rectified and read out.</text>
<section_header_level_1><location><page_3><loc_20><loc_70><loc_40><loc_72></location>2.2 The Vacuum Vessel</section_header_level_1>
<text><location><page_3><loc_20><loc_65><loc_80><loc_69></location>The vacuum vessel was a cylindrical stainless-steel tank with a removable lid on top. It was supported at the upper surface from a massive concrete structure. Ground vibrations were suppressed by a mechanical damping system.</text>
<text><location><page_3><loc_20><loc_54><loc_80><loc_65></location>The vacuum vessel was thermally shielded with a layer of Styrofoam. A removable, thermally insulating hood covered the top of the concrete structure, maintaining the lid's temperature variation at about 0.1 degrees Celsius within a 12-hour period. Over longer time periods the temperature changes were about a factor of 10 higher, especially between summer and winter. At the height of the resonator the vacuum vessel was shielded against magnetic fields by a µ -metal cylinder. The lid of the vessel was leveled along and transverse to the resonator axis with a precision of a few micro rad. Variations of the lid's inclination occurred up to 10 µ rad.</text>
<text><location><page_3><loc_20><loc_51><loc_80><loc_54></location>Environmental changes of air pressure and temperature led to a drift of the resonance frequency of up to 10 kHz within a week.</text>
<section_header_level_1><location><page_3><loc_20><loc_48><loc_59><loc_49></location>2.3 The Transport System of the Field Masses</section_header_level_1>
<text><location><page_3><loc_20><loc_41><loc_80><loc_47></location>Two granite optical benches were located on opposite sides of the vacuum vessel along the centerline of the resonator. Both benches supported a transport system on which field masses were moved between a far and a near position (see figure 1). The distance between the two positions was always 2.683 m on either side.</text>
<text><location><page_3><loc_20><loc_36><loc_80><loc_41></location>The construction of the transport system allowed for a change in the near position on both benches. This flexibility enabled different gravitational accelerations on the mirrors with given field masses (for details, see section 4.3). Several spheres of different materials but with the same diameter of 0.127 m were used as field masses.</text>
<section_header_level_1><location><page_3><loc_20><loc_32><loc_35><loc_34></location>2.4 Improvements</section_header_level_1>
<text><location><page_3><loc_20><loc_29><loc_80><loc_31></location>After the relocation to an underground experimental hall of the HERA accelerator the experimental setup was significantly improved:</text>
<unordered_list>
<list_item><location><page_3><loc_22><loc_25><loc_80><loc_28></location>· Stability of the lid inclination through the damped suspension of the vacuum vessel,</list_item>
<list_item><location><page_3><loc_22><loc_22><loc_80><loc_25></location>· Temperature stability of the lid was maintained down to 0.1 degrees Celsius over several hours.</list_item>
<list_item><location><page_3><loc_22><loc_20><loc_80><loc_22></location>· Stable vacuum conditions were maintained in the vessel at about 2 · 10 -5 mbar.</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_4><loc_22><loc_85><loc_80><loc_88></location>· An improved method was implemented for determination of the near positions of the field masses.</list_item>
<list_item><location><page_4><loc_22><loc_80><loc_80><loc_85></location>· The data acquisition was significantly improved by the installation of a recursive algorithm. This resulted in a more stable data collection process independent of external large distortions, like earthquakes.</list_item>
</unordered_list>
<section_header_level_1><location><page_4><loc_20><loc_76><loc_68><loc_78></location>3 Predictions from Newton's Law of Gravitation</section_header_level_1>
<text><location><page_4><loc_20><loc_71><loc_80><loc_75></location>The size of the deflection of a pendulum is determined by the equilibrium between the gravitational attraction by the field masses and the restoring force of the Earth's gravitation.</text>
<text><location><page_4><loc_20><loc_68><loc_80><loc_71></location>For point masses, the gravitational force F between a field mass m and a pendulum with mass M at a distance of r is given by:</text>
<formula><location><page_4><loc_45><loc_65><loc_80><loc_68></location>F = G · m r 2 · M (1)</formula>
<text><location><page_4><loc_20><loc_63><loc_59><loc_64></location>The acceleration of the pendulum by the field mass is</text>
<formula><location><page_4><loc_44><loc_60><loc_80><loc_62></location>a(m , r) = G · m r 2 (2)</formula>
<text><location><page_4><loc_20><loc_53><loc_80><loc_59></location>Two equal field masses were positioned symmetrically w.r.t. the resonator. They were alternated periodically between a near (r n ) and a far (r f ) position. Assuming point masses for field masses and mirrors, the change of the resonator length between these two positions is described by the formula</text>
<formula><location><page_4><loc_26><loc_49><loc_80><loc_52></location>db point = 2 G ω 2 0 · m · 1 r 2 n -1 (r n +b 0 ) 2 - 1 r 2 f -1 (r f +b 0 ) 2 (3)</formula>
<text><location><page_4><loc_20><loc_46><loc_80><loc_49></location>where ω 2 0 = g / L is the frequency of a pendulum with the length L and g is the local gravitational field of Earth. In terms of acceleration one has</text>
<formula><location><page_4><loc_22><loc_41><loc_80><loc_44></location>db point = 2 ω 2 0 a(m , r n ) -a(m , r n +b 0 ) - a(m , r f ) -a(m , r f +b 0 ) (4)</formula>
<text><location><page_4><loc_20><loc_35><loc_80><loc_40></location>The field masses were homogeneous spheres, treated as point masses. However, the mirrors have a complicated geometric structure composed of different materials. The acceleration of the extended mirrors were calculated by numerical integration over the shapes of different densities:</text>
<formula><location><page_4><loc_37><loc_31><loc_80><loc_34></location>a(m , r) ext = G · 1 M · ∫ V ρ (V) · m r(V) 2 dV , (5)</formula>
<text><location><page_4><loc_20><loc_27><loc_80><loc_30></location>where ρ (V) is the density of the volume elements, and r(V) is the distance of field mass and volume element projected onto the resonator axis.</text>
<text><location><page_4><loc_22><loc_26><loc_63><loc_27></location>Using a(m , r) ext in formula 4 gives the predicted result:</text>
<formula><location><page_4><loc_31><loc_18><loc_80><loc_24></location>db pred = 2 ω 2 0 a ( m,r n ) ext -a ( m,r n + b 0 ) ext - a ( m,r f ) ext -a ( m,r f + b 0 ) ext (6)</formula>
<text><location><page_5><loc_20><loc_82><loc_80><loc_88></location>The ratio of db pred for extended and point masses ranges from 0.96 to 0.99. The contribution of the far position to db pred is about 1%. It was neglected. Therefore the relevant acceleration on a single mirror by the two field masses in the near position is described by</text>
<formula><location><page_5><loc_27><loc_79><loc_80><loc_81></location>a pred = g L · db pred 2 and a meas = g L · db meas 2 , resp. (7)</formula>
<text><location><page_5><loc_22><loc_77><loc_65><loc_78></location>For the actual calculations the following values were used:</text>
<formula><location><page_5><loc_36><loc_73><loc_57><loc_76></location>G = 66 . 7428 · 10 -12 m 3 kg s 2</formula>
<formula><location><page_5><loc_37><loc_72><loc_64><loc_73></location>g = 9 . 8138 m / s [13]</formula>
<formula><location><page_5><loc_50><loc_72><loc_64><loc_75></location>[12] 2</formula>
<section_header_level_1><location><page_5><loc_20><loc_68><loc_44><loc_69></location>4 Measuring Procedure</section_header_level_1>
<text><location><page_5><loc_20><loc_63><loc_80><loc_66></location>The aim of the experiment is to measure the gravitational effect of the two field masses on the mirrors of the resonator. This is determined from the resonance frequencies of near and far position data.</text>
<section_header_level_1><location><page_5><loc_20><loc_59><loc_48><loc_60></location>4.1 Frequencies of the Resonator</section_header_level_1>
<text><location><page_5><loc_20><loc_56><loc_80><loc_58></location>According to [14], the resonance frequencies of a cylindrical resonator with spherical mirrors are given by:</text>
<formula><location><page_5><loc_30><loc_51><loc_80><loc_54></location>f = c 2b · { q + n π arcos ( 1 -b 0 R ) + N 8 π 2 Rk + O ( 10 -4 ) } (8)</formula>
<text><location><page_5><loc_20><loc_49><loc_24><loc_50></location>where:</text>
<unordered_list>
<list_item><location><page_5><loc_22><loc_47><loc_43><loc_49></location>· q = number of axial knots.</list_item>
<list_item><location><page_5><loc_22><loc_46><loc_44><loc_47></location>· p = number of radial knots.</list_item>
<list_item><location><page_5><loc_22><loc_44><loc_48><loc_45></location>· m = number of azimuthal knots.</list_item>
<list_item><location><page_5><loc_22><loc_42><loc_54><loc_44></location>· N = 2p 2 +2pm - m 2 +2p -2 + m ± 4m.</list_item>
<list_item><location><page_5><loc_22><loc_40><loc_35><loc_42></location>· n = 2p + m + 1.</list_item>
<list_item><location><page_5><loc_22><loc_39><loc_52><loc_40></location>· R is the curvature of spherical mirrors.</list_item>
<list_item><location><page_5><loc_22><loc_37><loc_38><loc_39></location>· c the speed of light.</list_item>
<list_item><location><page_5><loc_22><loc_35><loc_44><loc_37></location>· k = f / c the wave number.</list_item>
</unordered_list>
<text><location><page_5><loc_20><loc_31><loc_80><loc_35></location>Our experiment was performed at a resonance frequency with p = m = 0. The parameters q, p, m have been determined by a simultaneous fit of formula 8 to a series of resonances including the chosen one, where 37 axial knots were determined.</text>
<text><location><page_5><loc_20><loc_28><loc_80><loc_30></location>The relation between db and its corresponding change in resonance frequency df is given by:</text>
<formula><location><page_5><loc_43><loc_27><loc_80><loc_28></location>db = β · df with (9)</formula>
<formula><location><page_5><loc_33><loc_23><loc_80><loc_26></location>β = -b f ( 1 -n · c 2 π · f √ 1 2Rb 0 -b 2 0 + O ( 10 -4 ) ) . (10)</formula>
<text><location><page_5><loc_20><loc_18><loc_86><loc_22></location>For the used resonance frequency of 23.2699 GHz it results in β = 10 . 371 · 10 -12 m / Hz. It has been used to convert measured and predicted frequency differences into accelerations according to equations 7.</text>
<section_header_level_1><location><page_6><loc_20><loc_86><loc_59><loc_88></location>4.2 Determination of the Resonance Frequency</section_header_level_1>
<text><location><page_6><loc_20><loc_80><loc_80><loc_85></location>The resonance frequency, f r , was determined from measurements of the resonator amplitudes at five equidistant frequencies centered close to the resonance frequency. These frequencies cover 30% of the resonance width at half maximum of the resonance shape (about 130 kHz).</text>
<text><location><page_6><loc_22><loc_79><loc_67><loc_80></location>A Lorentz curve is expected to describe the resonance shape:</text>
<formula><location><page_6><loc_32><loc_75><loc_80><loc_77></location>U(f i ) = U max · 1 1 + 4((f i -f r ) / f w ) 2 for i = 1 - 5 (11)</formula>
<text><location><page_6><loc_20><loc_69><loc_80><loc_73></location>The maximum amplitude U max , the width of the Lorentz curve f w , and the resonance frequency f r could be obtained from a fit of equation 11 to the 5 amplitude measurements U(f i ).</text>
<text><location><page_6><loc_20><loc_67><loc_80><loc_69></location>However, the inverted Lorentz curve is a parabola that was used for a much faster fit:</text>
<formula><location><page_6><loc_31><loc_64><loc_80><loc_65></location>U(f i ) -1 = a + b · f i + c · f 2 i for i = 1 - 5 , with (12)</formula>
<formula><location><page_6><loc_35><loc_61><loc_80><loc_64></location>f r = -b 2 c (13)</formula>
<text><location><page_6><loc_20><loc_57><loc_80><loc_60></location>The parabola fit was sufficiently fast for an online determination, and allowed an immediate adjustments of the measurement range to a drifting resonance frequency.</text>
<text><location><page_6><loc_20><loc_55><loc_80><loc_57></location>The resolution σ (U(f i )) was about 0 . 05 mV. The resolution of the resonance frequency benefited from the improved data acquisition system (see section 2.4).</text>
<section_header_level_1><location><page_6><loc_20><loc_51><loc_35><loc_53></location>4.3 Field Masses</section_header_level_1>
<text><location><page_6><loc_20><loc_48><loc_80><loc_50></location>The used field masses were spheres of different materials and mass values, each with a diameter of 0.127 m, as specified in the table below.</text>
<table>
<location><page_6><loc_29><loc_41><loc_70><loc_47></location>
</table>
<text><location><page_6><loc_20><loc_36><loc_80><loc_39></location>Measurements were performed with different configurations of field masses and distances between the gravity centers of the field mass and the closer mirror as specified in the table below:</text>
<table>
<location><page_6><loc_22><loc_32><loc_78><loc_35></location>
</table>
<text><location><page_6><loc_20><loc_26><loc_80><loc_31></location>The positions of the two field masses were changed every half an hour from their far positions to their near positions and vice versa. About 5000 measurements of the resonance frequency were registered per half hour period. The typical duration of data collection for each configuration ranged from 500 to 1000 hours.</text>
<section_header_level_1><location><page_6><loc_20><loc_22><loc_41><loc_23></location>5 Systematic Effects</section_header_level_1>
<text><location><page_6><loc_20><loc_18><loc_80><loc_20></location>Two categories of systematic uncertainties affect the comparison of the results with the expectations from Newton's Law of gravitation.</text>
<text><location><page_7><loc_20><loc_85><loc_80><loc_88></location>The first category consists of effects that might or do influence the determination of the resonance frequency:</text>
<unordered_list>
<list_item><location><page_7><loc_22><loc_80><loc_80><loc_84></location>· The frequency stability of the radio frequency generator output was compared to a Rubidium Standard frequency. No difference was found that would influence the resonator frequency determinations.</list_item>
<list_item><location><page_7><loc_22><loc_74><loc_80><loc_80></location>· The removable cover of the vacuum vessel behaved like a membrane. External influences, such as temperature and pressure, led to changes of the resonance frequency. In section 7.1 it is described how this was treated in the analyses of the data.</list_item>
<list_item><location><page_7><loc_22><loc_69><loc_80><loc_74></location>· Measurements with no movement of the field masses were performed to test whether there is a bias in the measurement procedure and the analysis methods. The result df = -0 . 058 ± 0 . 063 Hz is compatible with zero.</list_item>
</unordered_list>
<text><location><page_7><loc_20><loc_66><loc_80><loc_68></location>The second category of uncertainties is associated with the calculation of expectations from Newton's law of gravity.</text>
<unordered_list>
<list_item><location><page_7><loc_22><loc_63><loc_80><loc_65></location>· A possible asymmetry of the resonator position in the experimental setup was determined using two methods:</list_item>
<list_item><location><page_7><loc_25><loc_59><loc_80><loc_62></location>-Measurements starting with one field masses in the near position and the other in the far position. These positions were exchanged every 0.5 h.</list_item>
<list_item><location><page_7><loc_25><loc_55><loc_80><loc_59></location>-Measurements with only one single field mass on one side, alternating between the far and near position. A second set of measurements were performed with the same field mass on the opposite side.</list_item>
</unordered_list>
<text><location><page_7><loc_24><loc_49><loc_80><loc_54></location>Both methods resulted in an asymmetric position of the resonator of about 5 mm. The asymmetry effect almost completely cancels when two field masses are used. The remaining deviation amounts to a negligible 0.03 % change of df.</text>
<unordered_list>
<list_item><location><page_7><loc_22><loc_38><loc_80><loc_48></location>· The distances between the centers of gravity of the field masses to the centers of gravity of the mirrors play a critical role in the calculation of the expectation from Newton's law. The distances in the near position could not be measured directly because the resonator was located in the vacuum vessel between the field-masses. Instead, the distance, D, between the centers of the two field masses in the near position was measured to be D = 1 . 4270 ± 0 . 0005 m, resulting in</list_item>
</unordered_list>
<formula><location><page_7><loc_41><loc_36><loc_80><loc_38></location>r n = 1 2 (D -b 0 ) = 0 . 5935 m . (14)</formula>
<unordered_list>
<list_item><location><page_7><loc_22><loc_31><loc_80><loc_35></location>· The difference between the near and far position of the field masses was always 2 . 683 ± 0 . 003 m. The contribution of the masses in the far position was about 1 % of the total df. Therefore the uncertainty of the far position is negligible.</list_item>
<list_item><location><page_7><loc_22><loc_25><loc_80><loc_30></location>· The field masses were measured with an electronic scale with an uncertainty of 0.1 g. Small differences between the field mass pairs cancel completely in the calculation of the prediction of df if the average of the two field masses are used.</list_item>
<list_item><location><page_7><loc_22><loc_20><loc_80><loc_24></location>· The horizontal and vertical alignments of the field masses transport systems at the near positions deviate by at most 2 mm from the axis of the resonator. This has a negligible effect on the determination of df.</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_8><loc_22><loc_82><loc_80><loc_88></location>· In the calculation of the expectation for df the square of the pendulum frequency enters, determined as ω 2 0 = g / L. The length of the pendulum has been measured to be 2 . 671 ± 0 . 001 m. The resulting relative uncertainty in the calculation of the predicted df can be neglected.</list_item>
<list_item><location><page_8><loc_22><loc_77><loc_80><loc_82></location>· The conversion factor from db to df is approximated to very high accuracy by β = b / f (described by equation 10). A fluctuation of the resonance frequency of ± 10 kHz leads to a negligible change of about 10 -6 · β .</list_item>
</unordered_list>
<section_header_level_1><location><page_8><loc_20><loc_73><loc_33><loc_75></location>6 Raw Data</section_header_level_1>
<text><location><page_8><loc_20><loc_61><loc_80><loc_72></location>The mirrors are permanently excited by vibrations of the ground floor resulting in oscillations of the resonance frequency. These vibrations added to the gravitational effects of the field masses in near and far positions. Figure 3 shows the originally determined resonance frequencies over a time interval of 40 s. The frequencies given on the y-axis are obtained after subtraction of a constant offset. Single frequencies are shown as dots. The pendulum period of approximately 3 s is clearly visible, as well as the data taking period of about 0.3 s. The bandwidth of the frequency oscillation was about 600 Hz.</text>
<text><location><page_8><loc_20><loc_51><loc_80><loc_61></location>Oscillations were filtered by averaging the resonance frequencies over time intervals of 30 to 60 seconds (depending on the analysis methods described in section 7.1). These averages are shown as black dots in figure 4 for a time interval of 2.5 hours. They show a slow drift of the frequency (see section 2.2). The short time fluctuations form a band with a standard deviation of about 5 Hz. A frequency step occurs between 1/2-hour periods, when the field masses alternate between near and far position.</text>
<text><location><page_8><loc_20><loc_48><loc_80><loc_51></location>Only the averaged and offset subtracted frequencies were used in the further analyses of the data.</text>
<section_header_level_1><location><page_8><loc_20><loc_44><loc_37><loc_46></location>7 Data Analyses</section_header_level_1>
<text><location><page_8><loc_20><loc_40><loc_80><loc_43></location>Four data analyses were performed separately for the different configurations. In general the least square method was used for all fits, unless otherwise stated.</text>
<section_header_level_1><location><page_8><loc_20><loc_37><loc_46><loc_38></location>7.1 Different Analysis Methods</section_header_level_1>
<text><location><page_8><loc_20><loc_33><loc_80><loc_36></location>Independent methods (m = a,b,c,d) were developed for the analyses of the data. All analyses begin with the data selection:</text>
<text><location><page_8><loc_20><loc_31><loc_80><loc_33></location>Strong temporary distortions like earthquakes etc. were eliminated by visual inspection (a, b) or by a programmed procedure (c, d).</text>
<text><location><page_8><loc_20><loc_26><loc_80><loc_30></location>The methods used time intervals of 1 to 10 hours for the determination of initial results of df and their statistical uncertainties, taking into account the slow frequency drift.</text>
<text><location><page_8><loc_22><loc_25><loc_52><loc_26></location>The details of the different methods are:</text>
<unordered_list>
<list_item><location><page_8><loc_22><loc_19><loc_80><loc_24></location>· Method a: Time intervals of 10 hours were used. The slow frequency drift is described by a 5th order polynomial, that was fit to the frequencies of the periods with field masses in the far position. The residuals of the fit for near and far periods are two nearly Gaussian distributions: one for the near position</list_item>
</unordered_list>
<text><location><page_9><loc_24><loc_85><loc_80><loc_88></location>data and one for the far position data. df is the difference of the two means. The uncertainty of df follows from the statistical uncertainty of the two means.</text>
<text><location><page_9><loc_24><loc_76><loc_80><loc_84></location>As an example, figure 5 shows distributions of the residuals of a 5th order polynomial fit to a selected 10-hour data-taking period, one for the field masses in the near position(dotted histogram) and one for the far position (solid histogram). The Gaussian fits of the histograms resulted in χ 2 / ndf close to one. These demonstrate also the accurate description of the slow drift by the 5th order polynomial fit.</text>
<unordered_list>
<list_item><location><page_9><loc_22><loc_72><loc_80><loc_76></location>· Method b: A parabola plus a rectangular function was fit to the resonance frequencies for 3 consecutive periods. The fit step-size of the rectangular function df and its statistical uncertainty were obtained from the fit.</list_item>
<list_item><location><page_9><loc_22><loc_61><loc_80><loc_71></location>· Method c: As shown in figure 4, a parabola was fit to the frequencies of 3 consecutive periods with the same position of the field masses. The green line of the parabola describes the slow frequency drift with sufficient accuracy. The values of the parabola have been subtracted from the data. A rectangular function were fit to the resulting residuals of 1 hours data giving the step-size df and its statistical uncertainty. The fit result is shown as red line on top of the parabola in the figure.</list_item>
<list_item><location><page_9><loc_22><loc_54><loc_80><loc_61></location>· Method d: A 5th order polynomial plus a rectangular wave were fit to all frequencies of a time intervals of 10 hours. The parameters of the polynomial describe the slow frequency drift. The rectangular wave changes its sign at every alternation between near and far position. The average step-size of the rectangular wave df and its statistical uncertainty was obtained from the fit.</list_item>
</unordered_list>
<text><location><page_9><loc_20><loc_47><loc_80><loc_53></location>For each method the initial results were combined to averages df m ± σ df m for each configuration of field masses and near distance. Table 1 shows the results of the different methods in columns 2 to 5 together with the predicted acceleration in column 1.</text>
<section_header_level_1><location><page_9><loc_20><loc_44><loc_52><loc_45></location>7.2 Combination of the four Methods</section_header_level_1>
<text><location><page_9><loc_20><loc_40><loc_80><loc_43></location>The mean value df of the methods as well as its statistical uncertainty σ stat were computed as weighted averages of the df m and the σ df m :</text>
<formula><location><page_9><loc_33><loc_36><loc_80><loc_39></location>df = ( ∑ m df m · ( σ df m ) -2 )/( ∑ m ( σ df m ) -2 ) (15)</formula>
<formula><location><page_9><loc_32><loc_31><loc_80><loc_35></location>σ stat = ( ∑ m σ df m · ( σ df m ) -2 )/( ∑ m ( σ df m ) -2 ) (16)</formula>
<text><location><page_9><loc_20><loc_28><loc_80><loc_30></location>Figure 6 shows the differences (df m ± σ df m ) -df for the different accelerations, demonstrating good agreement of the results from different analysis methods.</text>
<text><location><page_9><loc_20><loc_22><loc_80><loc_28></location>The standard deviation of df describes the spread of the method results df m around the mean df. It is used as additional systematic uncertainty, σ sys , for the difference between the four methods. Table 1 presents df, as well as σ stat and σ sys in columns 6.</text>
<text><location><page_9><loc_22><loc_21><loc_43><loc_22></location>The combined uncertainty is</text>
<formula><location><page_9><loc_43><loc_18><loc_80><loc_19></location>δ df = √ σ 2 stat + σ 2 sys . (17)</formula>
<section_header_level_1><location><page_10><loc_20><loc_86><loc_30><loc_88></location>8 Results</section_header_level_1>
<text><location><page_10><loc_20><loc_77><loc_80><loc_85></location>Table 2 gives df ± δ df in column 1 for the different accelerations. The results for df have uncertainties that are nearly the same for the different accelerations: the mean uncertainty is δ df = 0 . 074 Hz with a rather small standard deviation of 0.016 Hz. At the lowest acceleration df could be determined with a significance of 3 . 5 · δ df. Therefore measurements at lower acceleration would have given only insignificant results.</text>
<text><location><page_10><loc_20><loc_74><loc_80><loc_76></location>The predictions of Newton's law, df N ± δ df N , are displayed in table 2 column 3. The uncertainties δ df N were determined from the uncertainty of r n .</text>
<text><location><page_10><loc_20><loc_69><loc_80><loc_73></location>The differences of measurement results and predictions, ∆df ± δ ∆df, are given in table 2 columns 5 for the different accelerations. Measurements and predictions agree well within the uncertainties. This is shown in in figure 7 by the red symbols.</text>
<text><location><page_10><loc_20><loc_64><loc_80><loc_69></location>Also shown, as black symbols, are the results of an earlier publication [6]. These were determined from the information taken from [15] on which the publication was based. It demonstrates the large improvement in accuracy and the extended acceleration range that has been obtained since the previous publication.</text>
<text><location><page_10><loc_20><loc_48><loc_80><loc_63></location>The accelerations on a single mirror were calculated using equations 7 and 9 for the measured accelerations and the Newtonian predictions and their corresponding uncertainties. The results are shown in table 2 in columns 2 and 4. In figure 8 the measured accelerations are displayed versus the predictions from Newton's Law. The data are in good agreement with the predictions. The latter are indicated by the solid line. Also shown are results from reference [7]. They agree well with the data of this experiment. The numerical values plotted here are not given in [7] but were provided by the authors as private communication. For comparison figure 8 contains in addition the astronomical data provided by S. S. McGaugh [5,16]. For these data the predictions for the expected accelerations are based on the known baryonic matter.</text>
<section_header_level_1><location><page_10><loc_20><loc_44><loc_33><loc_46></location>9 Discussion</section_header_level_1>
<text><location><page_10><loc_20><loc_36><loc_80><loc_43></location>The results of this experiment agree well with Newton's Law for accelerations down to 10 -12 m / s 2 . This is not the case for astronomical data where a significant discrepancy is observed for accelerations < 10 -10 m / s 2 as shown in figure 8. This does not indicate that measurements on Earth contradict models and theories which explain the astronomical data.</text>
<text><location><page_10><loc_20><loc_26><loc_80><loc_36></location>Dark matter models explain data from regions in universe which are not dominated by gravitational effects of baryonic matter. This is not the case on Earth where gravitational effects are dominated by baryonic matter. In figure 8 the predicted accelerations for astronomical data are calculated for baryonic matter only. Dark matter causes additional gravitational acceleration. In the region of small baryonic acceleration, the presence of dark matter leads to higher predictions for the accelerations and may lead to an agreement with Newton's Law.</text>
<text><location><page_10><loc_20><loc_20><loc_80><loc_26></location>The MOND model has been formulated for regions where the gravitational potential is very small which practically eliminates external forces. This condition is not fulfilled on Earth. Therefore, the results of this experiments can neither prove nor disprove models and theories which are based on MOND.</text>
<section_header_level_1><location><page_11><loc_20><loc_86><loc_41><loc_88></location>10 Acknowledgment</section_header_level_1>
<text><location><page_11><loc_20><loc_73><loc_80><loc_85></location>We thank the DESY Directorate and the IT-division for their constant support. We are grateful for the assistance of the technical groups of DESY for their help and advice on many technical questions. Additionally we acknowledge the contributions of A. Brudgam, S. Fleig, Y. Holler, T. Kulper, S. Karstensen, and U. Packeiser in setting up the experiment. We express our gratitude for the fruitful discussions with the late N. Klein, W. Buchmuller, S. Glazov, C. Niebuhr, M. Takahashi, K. Schmidt-Hoberg, and A. Ringwald. We thank S. S. McGaugh for providing the astronomical data and S. Little and M. Little for providing the numerical values of their results.</text>
<section_header_level_1><location><page_12><loc_20><loc_86><loc_30><loc_88></location>References</section_header_level_1>
<unordered_list>
<list_item><location><page_12><loc_20><loc_81><loc_80><loc_85></location>[1] K. Lundmark, Uber die Bestimmung der Entfernungen, Dimensionen, Massen und Dichtigkeit fur die nachstgelegenen anagalacktischen Sternsysteme , Meddelanden fran Lunds Astronomiska Observatorium Series I. 125: 1 - 13, 1930</list_item>
<list_item><location><page_12><loc_20><loc_79><loc_67><loc_80></location>[2] L. Baudis, The Search for Dark Matter , arXiv:1801.08128v1</list_item>
<list_item><location><page_12><loc_20><loc_76><loc_80><loc_78></location>[3] M. Milgrom, A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis , Astrophys.J.270(1983)365</list_item>
<list_item><location><page_12><loc_20><loc_73><loc_80><loc_75></location>[4] M. J. Zurowski, Univ. Toronto , 38th International Cosmic Ray Conference, Dark Matter at ICRC 2023 arXiv:2309.12983v1</list_item>
<list_item><location><page_12><loc_20><loc_69><loc_80><loc_72></location>[5] S. S. McGaugh, F. Lelli, J. M. Schombert, The Radial Acceleration Relation in Rotationally Supported Galaxies , PRL 17, 201101</list_item>
<list_item><location><page_12><loc_20><loc_66><loc_80><loc_68></location>[6] H. Meyer et al, Test of the Law of Gravitation at small Accelerations , Gen.Rel.Grav. 44(2012) 2537-2545</list_item>
<list_item><location><page_12><loc_20><loc_61><loc_80><loc_65></location>[7] S. Little and M. Little, Laboratory test of Newton's law of gravity for small accelerations , Class. Quant. Grav. 31 (2014) no.19, 195008, doi:10.1088/0264-9381/31/19/195008</list_item>
<list_item><location><page_12><loc_20><loc_58><loc_80><loc_60></location>[8] J. Schurr, A new method to test the gravitational law of Newton , Dissertation, Bergische Universitat Wuppertal, 1992</list_item>
<list_item><location><page_12><loc_20><loc_55><loc_80><loc_57></location>[9] H. Walesch, Test des Newtonschen Gravitationsgesetzes und die prazise Bestimmung von G , Dissertation, Bergische Universitat Wuppertal, 1995</list_item>
<list_item><location><page_12><loc_20><loc_50><loc_80><loc_54></location>[10] A. Schuhmacher, Systematische Untersuchungen zur Messung der Newtonschen Gravitationskonstanten mit einem Pendelresonator , Dissertation, Bergische Universitat Wuppertal, 1999</list_item>
<list_item><location><page_12><loc_20><loc_46><loc_80><loc_49></location>[11] U. Kleinevoss, Bestimmung der Newtonschen Gravitationskonstanten G , Dissertation, Bergische Universitat Wuppertal, 2001</list_item>
<list_item><location><page_12><loc_20><loc_43><loc_80><loc_46></location>[12] E. Tiesinga, CODATA recommended values of the fundamental physical constants: 2018 , Rev.Mod.Phys.93,0250</list_item>
<list_item><location><page_12><loc_20><loc_41><loc_73><loc_42></location>[13] PTB National Metrology Institute of Germany, https://www.ptb.de</list_item>
<list_item><location><page_12><loc_20><loc_37><loc_80><loc_40></location>[14] Kwai-Man Luk and Ping-Kong Yu, Complex-source-point-theory of Gaussian Beams and Resonators , IEE Procedings, Vol 132, Issue 2, 105 - 113, doi:10.1049/ip-j.1985.0021</list_item>
<list_item><location><page_12><loc_20><loc_33><loc_80><loc_36></location>[15] S. Schubert, An experimental Test of Newton's Law of Gravitation for small Accelerations , Dissertation, Universitat Hamburg, 2011</list_item>
<list_item><location><page_12><loc_20><loc_28><loc_68><loc_32></location>[16] S. S. McGaugh et al, http://astroweb.case.edu/SPARC/ , binned data: http://astroweb.case.edu/SPARC/RARbins.mrt , SPARC, May 2020 12</list_item>
</unordered_list>
<table>
<location><page_13><loc_20><loc_60><loc_80><loc_88></location>
<caption>Table 1: Results and uncertainties of the 4 methods are listed in columns 2 to 5 for the different a pred . The combined results with statistical and systematic uncertainties are listed in column 6.</caption>
</table>
<table>
<location><page_14><loc_29><loc_67><loc_71><loc_95></location>
<caption>Table 2: The combined results for df and the corresponding accelerations, a meas , the prediction, df N , and their corresponding acceleration a pred , and their difference ∆df are listed, together with the corresponding uncertainties.</caption>
</table>
<figure>
<location><page_15><loc_15><loc_66><loc_84><loc_85></location>
<caption>Figure 1: Detector scheme: two mirror pendulums which are be pulled apart by the gravitational forces of field masses on both sides. The field masses change their positions between A and B.</caption>
</figure>
<figure>
<location><page_15><loc_18><loc_31><loc_84><loc_54></location>
<caption>Figure 2: Resonator scheme: the two cylindrical mirror pendulums (A, B) form a microwave resonator together with a cylindrical mode filter (C) in between. The mode filter suppresses higher mode resonances. The radio frequency power is fed into the resonator via the wave guide (D) . The resonator amplitude is measured at the opposite side (E). Oscillations of the pendulums are damped by independently adjustable magnets (H, K) on a movable table (G).</caption>
</figure>
<figure>
<location><page_16><loc_15><loc_67><loc_85><loc_92></location>
<caption>Figure 3: The points show the resonance frequencies for a time period of about 40 sec, after a fixed offset was subtracted. They were connected by straight lines to demonstrate clearly the effect of pendulum oscillations.</caption>
</figure>
<figure>
<location><page_16><loc_15><loc_33><loc_85><loc_58></location>
<caption>Figure 4: The points show the means of the resonance frequencies, averaged over 30 sec, for a time interval of 2.5 h. A fixed offset was subtracted from the frequencies as well as for the horizontal time axis. The data were derived from measurements with 2.924 kg field masses. A slow drift of the frequency of about 150 Hz is observed within the 2.5 h time interval, as described by the green line. Every half hour, a frequency step is observed originating from the position change of the field masses, described by the red line.</caption>
</figure>
<figure>
<location><page_17><loc_20><loc_65><loc_79><loc_86></location>
<caption>Figure 5: Shown are two histograms of the residuals of a 5th order polynomial fit to a selected 10-hour data-taking period, one for the field masses in the near position(dashed histogram), one for the far position(solid histogram). Gaussian curves were fitted to the histograms in blue and red. The resulting values of χ 2 / ndf are close to one, demonstrating the goodness of the Gaussian fit. These also demonstrate the accurate description of the slow frequency drift by the 5th order polynomial fit.</caption>
</figure>
<figure>
<location><page_17><loc_15><loc_24><loc_85><loc_51></location>
<caption>Figure 6: The differences ∆df m = (df m ± δ df m ) -df of the different methods are shown for the predicted accelerations. They are presented slightly displaced for better visibility.</caption>
</figure>
<figure>
<location><page_18><loc_15><loc_45><loc_85><loc_72></location>
<caption>Figure 7: Differences, ∆df = df -df N , of average results and predictions are shown for the predicted accelerations. The vertical error bars include the uncertainties of measurement and prediction. The red symbols show the results of this experiment. The black symbols show the results derived from the previous publication.</caption>
</figure>
<figure>
<location><page_19><loc_16><loc_42><loc_83><loc_81></location>
<caption>Figure 8: Comparison of measured and predicted acceleration of this experiment (red dots). The relative uncertainty of the predicted acceleration is about 0.14%. The black line indicates the equality of the two accelerations. The black symbols show the results from a previous publication [7]. The results of this experiment are compared to astronomical measurements [16] (blue squares); vertical axis: radial accelerations by galaxies measured at the boundaries; horizontal axis: predicted acceleration by the baryonic mass according to Newton's law. The uncertainties of the astronomical data are within the size of the symbols.</caption>
</figure>
</document>
[
{
"title": "Measurements of Gravitational Attractions at small Accelerations",
"content": "W. Bartel 1 † C. W. Elvers 1 L. Jonsson 2 G. Kempf 1 H. Krause 3 B. Loehr 1 E. Lohrmann 3 , H. Meyer 4 † P. Steffen 1 E. Wuensch 1 1 Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany 2 Lund University, Sweden 3 Universitat Hamburg 4 Bergische Universitat, Wuppertal † deceased",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "Gravitational interactions were studied by measuring the influence of small external field masses on a microwave resonator. It consisted of two spherical mirrors, which acted as independent pendulums individually suspended by strings. Two identical field masses were moved along the axis of the resonator symmetrically and periodically between a near and a far position. Their gravitational interaction altered the distance between the mirrors, changing the resonance frequency, which was measured and found consistent with Newton's law of gravity. The acceleration of a single mirror caused by the two field masses at the closest position varied from 5 . 4 · 10 -12 m / s 2 to 259 · 10 -12 m / s 2 .",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "In the 1930s, observations were made by several astronomers that the galaxies contain more mass than expected from Newton's gravitational law, based on the visible mass. The first publication of such an observation was made by K. Lundmark [1]. Later the term 'dark matter' was coined. The present predominant hypothesis is that dark matter is some kind of stable, heavy, neutral, weakly interacting particle. However, in spite of many researches to find or identify the particles of dark matter, nothing has been found so far [2], [4]. An alternative to dark matter is the MOND-model that assumes a modified Newton's law of gravity [3]. Acompilation of astronomical data, by [5], shows a clear deviation from Newton's predictions, based on the baryonic matter, for accelerations less than 10 -10 m / s 2 . This experiment aims to to test Newton's gravitational law at accelerations below 10 -10 m / s 2 on Earth, relying solely on gravitational interactions. Small field masses were used to accelerate the masses of two pendulums perpendicular to the Earth's acceleration. The measured movements of the pendulums are proportional to the acceleration caused by the field masses. Results from measurements with field masses ranging from 9 kg to 1 kg have been published previously [6] using an earlier version of the experiment. The results did not show any deviation from Newton's law of gravitation. A similar result was obtained using an experimental setup of the Cavendish-type [7]. This paper presents results from an improved experimental setup using field masses from 3 kg to 0.1 kg, which extend the range of accelerations to significantly smaller values, measured with considerably higher precision. The measurements were carried out in the years 2020 to 2022.",
"pages": [
2
]
},
{
"title": "2 Experimental Setup",
"content": "Figure 1 shows the main components of the experiment: a system of two pendulums that form a microwave resonator, similar to a Fabry-Perot interferometer. The introduction of external field masses causes a tiny acceleration of the pendulums in the direction of the resonator axis. The resulting change of the resonator length caused a measurable change in the resonance frequency. The resonator was located inside a vacuum vessel. A transport system moved the field masses between positions A and B. An earlier version of the apparatus had been built and operated at Wuppertal University for a precision measurement of the gravitational constant: [8], [9], [10], [11], where details can be found. It was later transferred to DESY to an experimental hall of the PETRA accelerator. After an initial measurement period, which resulted in a publication [6], it was relocated to an underground experimental hall of the HERA accelerator, and the setup was improved significantly.",
"pages": [
2
]
},
{
"title": "2.1 The Resonator",
"content": "The central part of the experiment was a microwave resonator. Figure 2 shows a schematic view. It consisted of two mirrors with spherical surfaces and a cylindrical center piece in between, which suppresses higher order resonance modes. The mirrors and the center piece were suspended individually by thin strings of about 2.7 m in length from the upper lid of the vacuum vessel. Each mirror acted as a mechanical pendulum with a period of 3.278 s. The resonator was operated at a resonance frequency of 23.2699 GHz. Stochastic movements of the pendulums were damped by permanent magnets positioned below the mirrors in order to optimize the performance of the resonator. The center of gravity of each mirror coincided with the vertex of the mirror. The distance between the two vertices was b 0 = 0 . 2400 m. Radio frequency power was fed into the resonator through a small hole in the center of one mirror, and was similarly extracted on the opposite side. It was rectified and read out.",
"pages": [
2,
3
]
},
{
"title": "2.2 The Vacuum Vessel",
"content": "The vacuum vessel was a cylindrical stainless-steel tank with a removable lid on top. It was supported at the upper surface from a massive concrete structure. Ground vibrations were suppressed by a mechanical damping system. The vacuum vessel was thermally shielded with a layer of Styrofoam. A removable, thermally insulating hood covered the top of the concrete structure, maintaining the lid's temperature variation at about 0.1 degrees Celsius within a 12-hour period. Over longer time periods the temperature changes were about a factor of 10 higher, especially between summer and winter. At the height of the resonator the vacuum vessel was shielded against magnetic fields by a µ -metal cylinder. The lid of the vessel was leveled along and transverse to the resonator axis with a precision of a few micro rad. Variations of the lid's inclination occurred up to 10 µ rad. Environmental changes of air pressure and temperature led to a drift of the resonance frequency of up to 10 kHz within a week.",
"pages": [
3
]
},
{
"title": "2.3 The Transport System of the Field Masses",
"content": "Two granite optical benches were located on opposite sides of the vacuum vessel along the centerline of the resonator. Both benches supported a transport system on which field masses were moved between a far and a near position (see figure 1). The distance between the two positions was always 2.683 m on either side. The construction of the transport system allowed for a change in the near position on both benches. This flexibility enabled different gravitational accelerations on the mirrors with given field masses (for details, see section 4.3). Several spheres of different materials but with the same diameter of 0.127 m were used as field masses.",
"pages": [
3
]
},
{
"title": "2.4 Improvements",
"content": "After the relocation to an underground experimental hall of the HERA accelerator the experimental setup was significantly improved:",
"pages": [
3
]
},
{
"title": "3 Predictions from Newton's Law of Gravitation",
"content": "The size of the deflection of a pendulum is determined by the equilibrium between the gravitational attraction by the field masses and the restoring force of the Earth's gravitation. For point masses, the gravitational force F between a field mass m and a pendulum with mass M at a distance of r is given by: The acceleration of the pendulum by the field mass is Two equal field masses were positioned symmetrically w.r.t. the resonator. They were alternated periodically between a near (r n ) and a far (r f ) position. Assuming point masses for field masses and mirrors, the change of the resonator length between these two positions is described by the formula where ω 2 0 = g / L is the frequency of a pendulum with the length L and g is the local gravitational field of Earth. In terms of acceleration one has The field masses were homogeneous spheres, treated as point masses. However, the mirrors have a complicated geometric structure composed of different materials. The acceleration of the extended mirrors were calculated by numerical integration over the shapes of different densities: where ρ (V) is the density of the volume elements, and r(V) is the distance of field mass and volume element projected onto the resonator axis. Using a(m , r) ext in formula 4 gives the predicted result: The ratio of db pred for extended and point masses ranges from 0.96 to 0.99. The contribution of the far position to db pred is about 1%. It was neglected. Therefore the relevant acceleration on a single mirror by the two field masses in the near position is described by For the actual calculations the following values were used:",
"pages": [
4,
5
]
},
{
"title": "4 Measuring Procedure",
"content": "The aim of the experiment is to measure the gravitational effect of the two field masses on the mirrors of the resonator. This is determined from the resonance frequencies of near and far position data.",
"pages": [
5
]
},
{
"title": "4.1 Frequencies of the Resonator",
"content": "According to [14], the resonance frequencies of a cylindrical resonator with spherical mirrors are given by: where: Our experiment was performed at a resonance frequency with p = m = 0. The parameters q, p, m have been determined by a simultaneous fit of formula 8 to a series of resonances including the chosen one, where 37 axial knots were determined. The relation between db and its corresponding change in resonance frequency df is given by: For the used resonance frequency of 23.2699 GHz it results in β = 10 . 371 · 10 -12 m / Hz. It has been used to convert measured and predicted frequency differences into accelerations according to equations 7.",
"pages": [
5
]
},
{
"title": "4.2 Determination of the Resonance Frequency",
"content": "The resonance frequency, f r , was determined from measurements of the resonator amplitudes at five equidistant frequencies centered close to the resonance frequency. These frequencies cover 30% of the resonance width at half maximum of the resonance shape (about 130 kHz). A Lorentz curve is expected to describe the resonance shape: The maximum amplitude U max , the width of the Lorentz curve f w , and the resonance frequency f r could be obtained from a fit of equation 11 to the 5 amplitude measurements U(f i ). However, the inverted Lorentz curve is a parabola that was used for a much faster fit: The parabola fit was sufficiently fast for an online determination, and allowed an immediate adjustments of the measurement range to a drifting resonance frequency. The resolution σ (U(f i )) was about 0 . 05 mV. The resolution of the resonance frequency benefited from the improved data acquisition system (see section 2.4).",
"pages": [
6
]
},
{
"title": "4.3 Field Masses",
"content": "The used field masses were spheres of different materials and mass values, each with a diameter of 0.127 m, as specified in the table below. Measurements were performed with different configurations of field masses and distances between the gravity centers of the field mass and the closer mirror as specified in the table below: The positions of the two field masses were changed every half an hour from their far positions to their near positions and vice versa. About 5000 measurements of the resonance frequency were registered per half hour period. The typical duration of data collection for each configuration ranged from 500 to 1000 hours.",
"pages": [
6
]
},
{
"title": "5 Systematic Effects",
"content": "Two categories of systematic uncertainties affect the comparison of the results with the expectations from Newton's Law of gravitation. The first category consists of effects that might or do influence the determination of the resonance frequency: The second category of uncertainties is associated with the calculation of expectations from Newton's law of gravity. Both methods resulted in an asymmetric position of the resonator of about 5 mm. The asymmetry effect almost completely cancels when two field masses are used. The remaining deviation amounts to a negligible 0.03 % change of df.",
"pages": [
6,
7
]
},
{
"title": "6 Raw Data",
"content": "The mirrors are permanently excited by vibrations of the ground floor resulting in oscillations of the resonance frequency. These vibrations added to the gravitational effects of the field masses in near and far positions. Figure 3 shows the originally determined resonance frequencies over a time interval of 40 s. The frequencies given on the y-axis are obtained after subtraction of a constant offset. Single frequencies are shown as dots. The pendulum period of approximately 3 s is clearly visible, as well as the data taking period of about 0.3 s. The bandwidth of the frequency oscillation was about 600 Hz. Oscillations were filtered by averaging the resonance frequencies over time intervals of 30 to 60 seconds (depending on the analysis methods described in section 7.1). These averages are shown as black dots in figure 4 for a time interval of 2.5 hours. They show a slow drift of the frequency (see section 2.2). The short time fluctuations form a band with a standard deviation of about 5 Hz. A frequency step occurs between 1/2-hour periods, when the field masses alternate between near and far position. Only the averaged and offset subtracted frequencies were used in the further analyses of the data.",
"pages": [
8
]
},
{
"title": "7 Data Analyses",
"content": "Four data analyses were performed separately for the different configurations. In general the least square method was used for all fits, unless otherwise stated.",
"pages": [
8
]
},
{
"title": "7.1 Different Analysis Methods",
"content": "Independent methods (m = a,b,c,d) were developed for the analyses of the data. All analyses begin with the data selection: Strong temporary distortions like earthquakes etc. were eliminated by visual inspection (a, b) or by a programmed procedure (c, d). The methods used time intervals of 1 to 10 hours for the determination of initial results of df and their statistical uncertainties, taking into account the slow frequency drift. The details of the different methods are: data and one for the far position data. df is the difference of the two means. The uncertainty of df follows from the statistical uncertainty of the two means. As an example, figure 5 shows distributions of the residuals of a 5th order polynomial fit to a selected 10-hour data-taking period, one for the field masses in the near position(dotted histogram) and one for the far position (solid histogram). The Gaussian fits of the histograms resulted in χ 2 / ndf close to one. These demonstrate also the accurate description of the slow drift by the 5th order polynomial fit. For each method the initial results were combined to averages df m ± σ df m for each configuration of field masses and near distance. Table 1 shows the results of the different methods in columns 2 to 5 together with the predicted acceleration in column 1.",
"pages": [
8,
9
]
},
{
"title": "7.2 Combination of the four Methods",
"content": "The mean value df of the methods as well as its statistical uncertainty σ stat were computed as weighted averages of the df m and the σ df m : Figure 6 shows the differences (df m ± σ df m ) -df for the different accelerations, demonstrating good agreement of the results from different analysis methods. The standard deviation of df describes the spread of the method results df m around the mean df. It is used as additional systematic uncertainty, σ sys , for the difference between the four methods. Table 1 presents df, as well as σ stat and σ sys in columns 6. The combined uncertainty is",
"pages": [
9
]
},
{
"title": "8 Results",
"content": "Table 2 gives df ± δ df in column 1 for the different accelerations. The results for df have uncertainties that are nearly the same for the different accelerations: the mean uncertainty is δ df = 0 . 074 Hz with a rather small standard deviation of 0.016 Hz. At the lowest acceleration df could be determined with a significance of 3 . 5 · δ df. Therefore measurements at lower acceleration would have given only insignificant results. The predictions of Newton's law, df N ± δ df N , are displayed in table 2 column 3. The uncertainties δ df N were determined from the uncertainty of r n . The differences of measurement results and predictions, ∆df ± δ ∆df, are given in table 2 columns 5 for the different accelerations. Measurements and predictions agree well within the uncertainties. This is shown in in figure 7 by the red symbols. Also shown, as black symbols, are the results of an earlier publication [6]. These were determined from the information taken from [15] on which the publication was based. It demonstrates the large improvement in accuracy and the extended acceleration range that has been obtained since the previous publication. The accelerations on a single mirror were calculated using equations 7 and 9 for the measured accelerations and the Newtonian predictions and their corresponding uncertainties. The results are shown in table 2 in columns 2 and 4. In figure 8 the measured accelerations are displayed versus the predictions from Newton's Law. The data are in good agreement with the predictions. The latter are indicated by the solid line. Also shown are results from reference [7]. They agree well with the data of this experiment. The numerical values plotted here are not given in [7] but were provided by the authors as private communication. For comparison figure 8 contains in addition the astronomical data provided by S. S. McGaugh [5,16]. For these data the predictions for the expected accelerations are based on the known baryonic matter.",
"pages": [
10
]
},
{
"title": "9 Discussion",
"content": "The results of this experiment agree well with Newton's Law for accelerations down to 10 -12 m / s 2 . This is not the case for astronomical data where a significant discrepancy is observed for accelerations < 10 -10 m / s 2 as shown in figure 8. This does not indicate that measurements on Earth contradict models and theories which explain the astronomical data. Dark matter models explain data from regions in universe which are not dominated by gravitational effects of baryonic matter. This is not the case on Earth where gravitational effects are dominated by baryonic matter. In figure 8 the predicted accelerations for astronomical data are calculated for baryonic matter only. Dark matter causes additional gravitational acceleration. In the region of small baryonic acceleration, the presence of dark matter leads to higher predictions for the accelerations and may lead to an agreement with Newton's Law. The MOND model has been formulated for regions where the gravitational potential is very small which practically eliminates external forces. This condition is not fulfilled on Earth. Therefore, the results of this experiments can neither prove nor disprove models and theories which are based on MOND.",
"pages": [
10
]
},
{
"title": "10 Acknowledgment",
"content": "We thank the DESY Directorate and the IT-division for their constant support. We are grateful for the assistance of the technical groups of DESY for their help and advice on many technical questions. Additionally we acknowledge the contributions of A. Brudgam, S. Fleig, Y. Holler, T. Kulper, S. Karstensen, and U. Packeiser in setting up the experiment. We express our gratitude for the fruitful discussions with the late N. Klein, W. Buchmuller, S. Glazov, C. Niebuhr, M. Takahashi, K. Schmidt-Hoberg, and A. Ringwald. We thank S. S. McGaugh for providing the astronomical data and S. Little and M. Little for providing the numerical values of their results.",
"pages": [
11
]
}
]
2024arXiv240721532M
https://arxiv.org/pdf/2407.21532.pdf
<document>
<section_header_level_1><location><page_1><loc_29><loc_86><loc_65><loc_88></location>On Extended d-D Kappa Distribution</section_header_level_1>
<section_header_level_1><location><page_1><loc_40><loc_80><loc_54><loc_82></location>Arak M. Mathai</section_header_level_1>
<text><location><page_1><loc_21><loc_74><loc_73><loc_79></location>Department of Mathematics and Statistics, McGill University, Montreal, Canada a.mathai@mcgill.ca</text>
<text><location><page_1><loc_45><loc_71><loc_49><loc_73></location>and</text>
<text><location><page_1><loc_15><loc_63><loc_79><loc_70></location>Hans J. Haubold Office for Outer Space Affairs, United Nations, Vienna International Centre, Vienna, Austria hans.haubold@gmail.com</text>
<text><location><page_1><loc_12><loc_48><loc_82><loc_61></location>Abstract. Thermal Doppler broadening of spectral profiles for particle populations in the absence or presence of potential fields are described by kappa distributions. The kappa distribution provides a replacement for the Maxwell-Boltzmann distribution which can be considered as a generalization for describing systems characterized by local correlations among their particles as found in space and astrophysical plasmas. This paper presents all special cases of kappa distributions as members of a general pathway family of densities introduced by Mathai.</text>
<text><location><page_1><loc_12><loc_45><loc_73><loc_47></location>Key words: atomic data - line: profiles - methods: analytical - plasmas</text>
<section_header_level_1><location><page_1><loc_12><loc_40><loc_27><loc_42></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_12><loc_23><loc_82><loc_39></location>A particle is in thermal equilibrium when the exchange of heat or entropy in the system stops. The kappa index controls the exchange of entropy and it is also a measure of departure from the equilibrium state. The particle velocity distribution can be given in terms of a kappa distribution. In the case of a collisional plasma, where no local correlations among particles exist, the system is stabilized into a MaxwellBoltzmann distribution. The kappa index is inversely proportional to the correlation between the energies of two particles. The d-D kappa distribution describes the particle velocity with dimensionality d . Livadiotis (2018), Livadiotis and McComas (2023), and Cuesta (2024) take this particle velocity density as the following:</text>
<formula><location><page_1><loc_26><loc_17><loc_82><loc_21></location>f ( X ; θ, k d ) = c d [1 + 1 k d -d 2 ( X -µ ) ' ( X -µ ) θ 2 ] -( k d +1) (1 . 1)</formula>
<text><location><page_2><loc_18><loc_81><loc_88><loc_88></location>where X is a d × 1 velocity vector, a prime denotes the transpose and µ = E [ X ] = < X > is the expected value of X , which is also a location parameter vector, where E [( · )] denotes the expected value with respect to the density f ( X ; θ, k d ), and the normalizing constant c d is the following:</text>
<formula><location><page_2><loc_39><loc_75><loc_88><loc_79></location>c d = Γ( k d +1) Γ( k d -d 2 +1)[ π ( k d -d 2 ) θ 2 ] d 2 (1 . 2)</formula>
<text><location><page_2><loc_18><loc_68><loc_88><loc_74></location>and θ = √ k B T/m is the thermal speed of the particle with mass m and temperature T expressed in speed units. If k d -d 2 is invariant over the dimensionality d then let k o be this constant value. Then, k d = k o + d 2 . Then, the normalizing constant becomes</text>
<formula><location><page_2><loc_43><loc_62><loc_88><loc_66></location>c d = Γ( k o +1+ d 2 ) Γ( k o +1)[ πk o θ 2 ] d 2 (1 . 3)</formula>
<text><location><page_2><loc_18><loc_59><loc_39><loc_61></location>and the functional part is</text>
<formula><location><page_2><loc_21><loc_54><loc_88><loc_58></location>[1 + 1 k d -d 2 ( X -µ ) ' ( X -µ ) θ 2 ] -( k d +1) = [1 + 1 k o ( X -µ ) ' ( X -µ ) θ 2 ] -( k o +1+ d 2 ) (1 . 4)</formula>
<text><location><page_2><loc_18><loc_45><loc_88><loc_52></location>Since k d -d 2 = k o is invariant k 3 -3 2 = k d -d 2 , k 3 = k d -( d -3) 2 . If θ depends on the kappa index, then θ is of the form θ k . Two forms of θ k are usually taken. One is θ 2 k = ( k 3 -3 2 ) k 3 ( d -1) 2 θ 2 and the other is θ 2 k,d = k o k d θ 2 = ( k d -d 2 ) k d θ 2 . If θ k is used then (1.4) becomes</text>
<formula><location><page_2><loc_38><loc_41><loc_88><loc_45></location>[1 + 1 k d ( X -µ ) ' ( X -µ ) θ 2 k ] -( k d + 1 2 ( d -1)) . (1 . 5)</formula>
<text><location><page_2><loc_18><loc_39><loc_62><loc_40></location>On the other hand, if θ k,d is used then (1.4) becomes</text>
<formula><location><page_2><loc_40><loc_34><loc_88><loc_37></location>[1 + 1 k d ( X -µ ) ' ( X -µ ) θ 2 k,d ] -( k d +1) . (1 . 6)</formula>
<text><location><page_2><loc_18><loc_27><loc_88><loc_32></location>Thus, one can write the kappa density in (1.1) in a number of different ways. One set of representations will be for d = 1 , 2 , 3, another set will be with θ replaced by θ k or θ k,d . When k o →∞ , (1.1) will go to the density</text>
<formula><location><page_2><loc_41><loc_23><loc_88><loc_25></location>f 1 ( X ; θ, k d ) = c ∗ e -( X -µ ) ' ( X -µ ) θ 2 (1 . 7)</formula>
<text><location><page_2><loc_18><loc_18><loc_88><loc_21></location>where c ∗ = ( πθ 2 ) -d 2 . This density (1.6) is a multivariate real Gaussian and also becomes as d -D Maxwellian distribution.</text>
<section_header_level_1><location><page_3><loc_12><loc_86><loc_45><loc_88></location>2. A Pathway Family of Densities</section_header_level_1>
<text><location><page_3><loc_12><loc_58><loc_82><loc_85></location>All the forms of the kappa distribution considered in Section 1 are special cases of a general pathway family of densities introduced in Mathai (2005). Let X be a p × 1 real vector random variable with the covariance matrix Σ = Cov( X ) = E [( X -E ( X ))( X -E ( X )) ' ] where a prime denotes the transpose and E ( · ) is the expected value of ( · ) = < ( · ) > with respect to the density of X . The p components of X may be correlated among themselves. One can get rid of the effect of correlations by taking Y = Σ -1 2 ( X -µ ) where Σ 1 2 is the real positive definite square root of the positive definite matrix Σ > O . Then the covariance matrix in Y is the identity matrix, free of all correlations. The Euclidean distance of Y from the origin is also ( X -µ ) ' Σ -1 ( X -µ ), µ = E ( X ). Also, ( X -µ ) ' Σ -1 ( X -µ ) = a positive constant is an offset ellipsoid and it is also known in statistical literature as the ellipsoid of concentration. If the components of X are to be simply weighted, rather than removing the effect of correlations, then one can replace Σ -1 in the quadratic form by a real positive definite matrix A > O and then we may consider ( X -µ ) ' A ( X -µ ). Consider the following density:</text>
<formula><location><page_3><loc_15><loc_55><loc_82><loc_57></location>g ( X )d X = c [( X -µ ) ' Σ -1 ( X -µ )] γ [1 + a (( X -µ ) ' Σ -1 ( X -µ )) δ ] -η d X (2 . 1)</formula>
<text><location><page_3><loc_12><loc_50><loc_82><loc_54></location>for a > 0 , δ > 0 , η > 0 , γ > 0, all real scalar parameters, and c is the normalizing constant, which can be evaluated as</text>
<formula><location><page_3><loc_26><loc_45><loc_82><loc_50></location>c = δ Γ( p δ )Γ( η ) a γ δ + p 2 δ | Σ | 1 2 π p 2 Γ( γ δ + p 2 δ )Γ( η -γ δ -p 2 δ ) , η -γ δ -p 2 δ > 0 (2 . 2)</formula>
<text><location><page_3><loc_12><loc_35><loc_82><loc_44></location>see the derivation of c in the Appendix. Consider δ = 1 , Σ = I (identity matrix), a = 1 θ 2 ( k d -d 2 ) , η = k d + 1 , γ = 0. Then (2.1) reduces to (1.1), the density given by Livadiotis (2018). The general representation in (2.1) and (2.2) has many advantages. This (2.1) is a member of the pathway family of distributions defined in Mathai (2005). Let a = b α -a o , b > 0, α > a o and η = ρ ( α -a o ) , ρ > 0. Then, (2.1) becomes</text>
<formula><location><page_3><loc_12><loc_29><loc_82><loc_34></location>g 1 ( X )d X = c 1 [( X -µ ) ' Σ -1 ( X -µ )] γ [1 + b α -a o (( X -µ ) ' Σ -1 ( X -µ )) δ ] -ρ ( α -a 0 ) d X. (2 . 3)</formula>
<text><location><page_3><loc_12><loc_27><loc_67><loc_28></location>Note that when α → a o from the right, g 1 ( X ) goes to the density</text>
<formula><location><page_3><loc_22><loc_24><loc_82><loc_26></location>g 2 ( X )d X = c 2 [( X -µ ) ' Σ -1 ( X -µ )] γ e -bρ [( X -µ ) ' Σ -1 ( X -µ )] δ d X (2 . 4)</formula>
<text><location><page_3><loc_12><loc_21><loc_17><loc_23></location>where</text>
<formula><location><page_3><loc_30><loc_17><loc_82><loc_22></location>c 2 = δ Γ( p 2 )( bρ ) γ δ + p 2 δ π p 2 | Σ | 1 2 Γ( γ δ + p 2 δ ) , δ > 0 , b > 0 , ρ > 0 . (2 . 5)</formula>
<text><location><page_4><loc_18><loc_73><loc_88><loc_88></location>The density in (2.4) can be taken as a power-transformed real multivariate MaxwellBoltzmann density. For δ = 1, (2.4) is a form of multivariate Maxwell-Botlzmann density. Hence, if (2.4) is the stable density in a physical system, then the unstable neighborhood and transitional stages are determined by (2.3) for various values of the pathway parameter α . We may also note that for α = k d , a o = d 2 , b = 1 θ 2 , δ = 1 , γ = 0 , ρ = k d +1 k d -d 2 one has (1.1) also. This density in (2.3) has another advantage. For α < a o we can write α -a o = -( a o -α ) , a o -α > 0 so that the density in (2.3) switches into a type-1 beta form of the density given by</text>
<formula><location><page_4><loc_18><loc_66><loc_88><loc_71></location>g 3 ( X )d X = c 3 [( X -µ ) ' Σ -1 ( X -µ )] γ [1 -b a o -α (( X -µ ) ' Σ -1 ( X µ )) δ ] ρ ( a o -α ) d X,α < a o (2 . 6)</formula>
<text><location><page_4><loc_18><loc_62><loc_88><loc_66></location>where [1 -b a o -α (( X -µ ) ' Σ -1 ( X -µ )) δ ] > 0 so that the density is defined within the ellipsoid</text>
<text><location><page_4><loc_18><loc_57><loc_21><loc_59></location>and</text>
<formula><location><page_4><loc_29><loc_59><loc_77><loc_63></location>( X -µ ) ' Σ -1 ( X -µ ) = ( a o -α b ) 1 δ for α < a o , b > 0 , δ > 0</formula>
<formula><location><page_4><loc_30><loc_53><loc_88><loc_57></location>c 3 = δ Γ( p δ )Γ(1 + ρ ( a o -α ) + γ δ + p 2 δ )( b a o -α ) γ δ + p 2 δ | Σ | 1 2 π p 2 Γ( γ δ + p 2 δ )Γ(1 + ρ ( a o -α )) , α < a o . (2 . 7)</formula>
<text><location><page_4><loc_18><loc_37><loc_90><loc_52></location>This normalizing constant is evaluated by going through the procedure in Appendix A1 and the final integral is evaluated by using a type-1 beta integral. Thus, g 1 , g 2 , g 3 belong to the pathway family of densities. If the power-transformed Maxwell-Boltzmann density in (2.4) is the stable density in a physical system then the unstable neighborhoods and the transitional stages are given by g 1 in (2.3) and g 3 in (2.6). One can switch among a generalized type-1 beta family, a type-2 beta family and a gamma family or Maxwell-Boltzmann family of distributions through the pathway parameter α . In the model in (2.3) one can identify α with k d and a o with d 2 if convenient.</text>
<section_header_level_1><location><page_4><loc_18><loc_35><loc_78><loc_36></location>3. Livadiotis' d -D Density Through an entropy Optimization</section_header_level_1>
<text><location><page_4><loc_18><loc_28><loc_88><loc_33></location>Let X be a p × 1 real vector random variable. Let f ( X ) be a density function, that is, f ( X ) ≥ 0 for all X and ∫ X f ( X )d X = 1 where f ( X ) is a real-valued scalar function of X . Consider Mathai's entropy for the vector random variable, namely,</text>
<formula><location><page_4><loc_39><loc_22><loc_88><loc_27></location>M α ( f ) = ∫ X [ f ( X )] 1+ ao -α η d X -1 α -a o (3 . 1)</formula>
<text><location><page_4><loc_18><loc_18><loc_88><loc_21></location>for a o a fixed quantity or anchoring point, α is the pathway parameter and the deviation of α from a o is measured in η > 0 units. Then, when α → a o , we can see</text>
<text><location><page_5><loc_12><loc_75><loc_82><loc_88></location>that (3.1) reduces to Shannon's entropy S ( f ) = -K ∫ X f ( X ) ln f ( X )d X where K is a constant. Shannon's entropy is for the scalar variable case and the corresponding real vector-variate form is denoted here as S ( f ). Let Σ > O be the covariance matrix of X . Consider the ellipsoid of concentration ( X -µ ) ' Σ -1 ( X µ ) = a positive constant, where µ is a p × 1 location parameter vector. We will set moment-type constraints on the ellipsoid of concentration and then optimize (3.1). Consider the following constraints:</text>
<formula><location><page_5><loc_13><loc_70><loc_82><loc_74></location>E [( X -µ ) ' Σ -1 ( X -µ )] γ ( ao -α η ) = fixed , E [( X -µ ) ' Σ -1 ( X -µ )] γ ( ao -α η )+ δ = fixed (3 . 2)</formula>
<text><location><page_5><loc_12><loc_65><loc_82><loc_70></location>for some parameters γ > 0 , η > 0 , δ > 0. If we use calculus of variation for the optimization, then the Euler equation becomes the following where λ 1 and λ 2 are Lagrangian multipliers:</text>
<formula><location><page_5><loc_12><loc_60><loc_82><loc_63></location>∂ ∂f [ f 1+ ao -α η -λ 1 [( X -µ ) ' Σ -1 ( X -µ )] γ ( ao -α η ) f + λ 2 [( X -µ ) ' Σ -1 ( X -µ )] γ ( ao -α η )+ δ f ] = 0 .</formula>
<text><location><page_5><loc_12><loc_57><loc_50><loc_58></location>This gives the solution for f as the following:</text>
<formula><location><page_5><loc_20><loc_52><loc_82><loc_55></location>f = ν 1 [( X -µ ) ' Σ -1 ( X -µ )] γ [1 -λ 2 λ 1 [( X -µ ) ' Σ -1 ( X -µ )] δ ] η ao -α (3 . 3)</formula>
<text><location><page_5><loc_12><loc_45><loc_82><loc_50></location>where ν 1 , λ 1 , λ 2 are constants. Let λ 2 λ 1 = b ( a o -α ) , b > 0 and let ν 1 be the normalizing constant to make (3.3) a density. Then, for u = ( X -µ ) ' Σ -1 ( X -µ ) we have the following densities from (3.3):</text>
<formula><location><page_5><loc_29><loc_41><loc_82><loc_43></location>f 1 ( X ) = ν 1 u γ [1 -b ( a o -α ) u δ ] η a 0 -α , α < a o (3.4)</formula>
<formula><location><page_5><loc_29><loc_39><loc_82><loc_41></location>f 2 ( X ) = ν 2 u γ [1 + b ( α -a o ) u δ ] -η α -ao , α > a o (3.5)</formula>
<formula><location><page_5><loc_29><loc_36><loc_82><loc_38></location>f 3 ( X ) = ν 3 u γ e -bηu δ , α → a o (3.6)</formula>
<text><location><page_5><loc_12><loc_31><loc_82><loc_34></location>where in (3.4) we need an additional condition 1 -b ( a o -α ) u δ > 0 in order to make (3.4) a density. Note that in the limiting form we have the following properties:</text>
<formula><location><page_5><loc_20><loc_23><loc_82><loc_30></location>e -bηu δ = lim α → a o [1 -b ( a o -α ) u δ ] η ao -α = lim α → a o [1 + b ( α -a o ) u δ ] -η α -ao = lim α → a o [1 -b a o -α u δ ] η ( a o -α ) = lim α → a o [1 + b α -a o u δ ] -η ( α -a o ) (3.7)</formula>
<text><location><page_5><loc_12><loc_17><loc_82><loc_21></location>Hence, one can take any one of the formats in (3.7) in the limiting case. Livadiotis' density in (1.1) is available from (3.5) by taking b ( α -a o ) = 1 θ ( k d -d 2 ) and η α -a o = k d +1.</text>
<text><location><page_6><loc_18><loc_79><loc_88><loc_88></location>For γ = 0 , δ = 1 , η = 1 , a o = 1 , b = 1, equations (3.4),(3.5)(3.6) give a real multivariate version of Tsallis statistics of non-extensive statistical mechanics. For δ = 1 , η = 1 , a o = 1 , b = 1 , δ = 1, (3.5) and (3.6) can be taken as a multivariate extension of superstatistics. Matrix-variate versions of Tsallis statistics and superstatistics can also be defined.</text>
<section_header_level_1><location><page_6><loc_18><loc_76><loc_73><loc_77></location>4. A Matrix-variate generalization of Livadiotis' Density</section_header_level_1>
<text><location><page_6><loc_18><loc_62><loc_88><loc_75></location>Let Y = ( y ij ) be a p × q, p ≤ q and of rank p matrix with distinct real scalar variables y ij 's as elements. Let g ( Y ) be a real-valued scalar function of Y such that g ( Y ) ≥ 0 for all Y and ∫ Y g ( Y )d Y = 1 so that g ( Y ) is a density function where d Y = ∧ p i =1 ∧ q j =1 d y ij = the wedge product of all distinct differentials in Y . Let M be a p × q, p ≤ q parameter matrix. Let A > O be p × p and B > O be q × q positive definite constant matrices. Let A 1 2 and B 1 2 be the positive definite square roots of A and B respectively. Let</text>
<formula><location><page_6><loc_28><loc_59><loc_88><loc_61></location>U = A 1 2 ( Y -M ) B 1 2 , V = UU ' = A 1 2 ( Y -M ) B ( Y -M ) ' A 1 2 (4 . 1)</formula>
<text><location><page_6><loc_18><loc_40><loc_88><loc_58></location>Then, the determinant of V , namely | V | , can be interpreted in different ways. | V | is the product of the eigenvalues of the matrix V . Let U 1 , ..., U p be the p linearly independent rows of U , the p × q, p ≤ q and of rank p matrix U . Then, U j , j = 1 , ..., p can be taken as p linearly independent points in a q -dimensional Euclidean space with p ≤ q . Then, | V | 1 2 = | UU ' | 1 2 = the volume of the p -parallelotope generated in the convex hull of the p linearly independent points, taken in the order. Thus, | V | is also the square of the volume content of this parallelotope. Consider Mathis's entropy in (3.1) with f ( X ) replaced by g ( Y ) where Y is now p × q, p ≤ q matrix of rank p . Consider the optimization of (3.1) with the real-valued scalar function g ( Y ) under the following constraints:</text>
<formula><location><page_6><loc_32><loc_36><loc_88><loc_39></location>E [ | V | γ ( ao -α η ) = fixed , E [ | V | γ ( ao -α η ) [tr( V )] δ = fixed (4 . 2)</formula>
<text><location><page_6><loc_18><loc_34><loc_88><loc_35></location>Then, proceeding as in Section 3, we can end up with the following pathway densities:</text>
<formula><location><page_6><loc_33><loc_30><loc_88><loc_33></location>g 1 ( Y ) = C 1 | V | γ [1 -b ( a o -α )(tr( V ))] η ao -α , α < a o (4.3)</formula>
<formula><location><page_6><loc_31><loc_28><loc_88><loc_30></location>g + 2( Y ) = C 2 | V | γ [1 + b ( α -a o )(tr( V ))] -η α -ao , α > a o (4.4)</formula>
<formula><location><page_6><loc_33><loc_26><loc_88><loc_27></location>g 3 ( Y ) = | V | γ e -bη tr( V ) (4.5)</formula>
<text><location><page_6><loc_18><loc_21><loc_88><loc_24></location>where C 1 , C 2 , C 3 are the normalizing constants and in (4.3) the additional condition needed is 1 -b ( a o -α )tr( V ) > 0 to make (4.3) a density, where</text>
<formula><location><page_6><loc_40><loc_18><loc_88><loc_20></location>V = A 1 2 ( Y -M ) B ( Y -M ) ' A 1 2 . (4 . 6)</formula>
<text><location><page_7><loc_12><loc_84><loc_82><loc_88></location>The normalizing constants are evaluated in Appendix A2 by using the following steps:</text>
<text><location><page_7><loc_12><loc_80><loc_15><loc_81></location>and</text>
<formula><location><page_7><loc_30><loc_82><loc_64><loc_84></location>U = A 1 2 ( Y -M ) B 1 2 ⇒ d U = | A | q 2 | B | p 2 d Y</formula>
<formula><location><page_7><loc_32><loc_76><loc_82><loc_80></location>V = UU ' ⇒ d U = π pq 2 Γ p ( q 2 ) | V | q 2 -p +1 2 d V ( i )</formula>
<text><location><page_7><loc_12><loc_72><loc_82><loc_75></location>where Γ p ( α ) is a real matrix-variate gamma function, associated with a real matrixvariate gamma integral, and it is the following:</text>
<formula><location><page_7><loc_22><loc_67><loc_82><loc_71></location>Γ p ( α ) = ∫ S>O | S | α -p +1 2 e -tr( S ) d S, /Rfractur ( α ) > p -1 2 (ii)</formula>
<formula><location><page_7><loc_28><loc_63><loc_82><loc_67></location>= π p ( p -1) 4 Γ( α )Γ( α -1 2 ) ... Γ( α -p -1 2 ) , /Rfractur ( α ) > p -1 2 (iii)</formula>
<text><location><page_7><loc_12><loc_59><loc_82><loc_62></location>where /Rfractur ( · ) denotes the real part of ( · ). Note that (4.5) is a real rectangular matrixvariate Maxwell-Boltzmann density.</text>
<text><location><page_7><loc_12><loc_43><loc_82><loc_58></location>Note 4.1. Results parallel to all the results in Sections 3 and 4 are also available in the complex domain. Corresponding physics can also be dealt with in the complex domain. A real rectangular matrix-variate version of Tsallis statistics is available from (4.3)-(4.5) for γ + q 2 = p +1 2 , b = 1 , α = 1 , a o = 1 , η = 1. Corresponding results in the complex domain can also be worked out. For b = 1 , α = 1 , η = 1, (4.4) and (4.5) give a real rectangular matrix-variate version of superstatistics. Corresponding versions in the complex domain can also be worked out. Such rectangular or square matrix-variate versions may not be available in the literature.</text>
<section_header_level_1><location><page_7><loc_41><loc_40><loc_53><loc_42></location>APPENDIX</section_header_level_1>
<text><location><page_7><loc_12><loc_36><loc_55><loc_38></location>A1: Derivation of the normalizing constant in (2.2)</text>
<text><location><page_7><loc_15><loc_33><loc_73><loc_35></location>Let X be a p × 1 vector random variable and µ = E [ X ] , Σ = Cov( X ).</text>
<formula><location><page_7><loc_19><loc_28><loc_75><loc_32></location>1 c = ∫ X [( X -µ ) ' Σ -1 ( X -µ )] γ [1 + a (( X -µ ) ' Σ -1 ( X -µ )) δ ] -η d X.</formula>
<text><location><page_7><loc_12><loc_26><loc_36><loc_27></location>Consider the transformation</text>
<formula><location><page_7><loc_32><loc_23><loc_62><loc_24></location>X = Σ -1 2 ( X -µ ) ⇒ d Z = | Σ | -1 2 d X</formula>
<text><location><page_7><loc_12><loc_18><loc_82><loc_21></location>see Mathai (1997). Let u = ZZ ' . Then writing Z uniquely as a product of a unique lower triangular matrix and a unique semi-orthonormal matrix and then integrating</text>
<text><location><page_8><loc_18><loc_84><loc_88><loc_88></location>out the differential element over a Stiefel manifold, we have a relation between d u and d Z , that is</text>
<formula><location><page_8><loc_43><loc_80><loc_88><loc_84></location>d Z = π q 2 Γ( p 2 ) u p 2 -1 d u, u > 0 ( A 1 )</formula>
<text><location><page_8><loc_18><loc_76><loc_88><loc_79></location>see Mathai (1997) for the details. Observe that u is real scalar whereas Z is a p × 1 vector. Then</text>
<formula><location><page_8><loc_28><loc_66><loc_79><loc_75></location>∫ X [( X -µ ) ' Σ -1 ( X -µ )] γ [1 + a (( X -µ ) ' Σ -1 ( X -µ )) δ ] -η d X = | Σ | 1 2 π p 2 Γ( p 2 ) ∫ ∞ u =0 u γ + p 2 -1 (1 + au δ ) -η d u.</formula>
<text><location><page_8><loc_18><loc_61><loc_88><loc_65></location>Now, let v = u δ , u > 0 , δ > 0 ⇒ d u = 1 δ v 1 δ -1 d v . Integrating out by using a type-2 beta integral we have</text>
<formula><location><page_8><loc_31><loc_56><loc_88><loc_60></location>1 c = | Σ | 1 2 π p 2 Γ( p 2 ) Γ( γ δ + p 2 δ )Γ( η -γ δ -p 2 δ ) δa γ δ + p 2 δ Γ( η ) , η -γ δ -p 2 δ > 0 ( A 2 )</formula>
<text><location><page_8><loc_18><loc_53><loc_49><loc_54></location>which gives the normalizing constant.</text>
<section_header_level_1><location><page_8><loc_18><loc_50><loc_69><loc_51></location>A2 Derivation of the normalizing constant C in (4.4)</section_header_level_1>
<text><location><page_8><loc_18><loc_47><loc_88><loc_48></location>Let Y be p × q, p ≤ q and of rank p matrix of real scalar random variables as elements.</text>
<formula><location><page_8><loc_18><loc_42><loc_88><loc_45></location>1 C = ∫ Y [ A 1 2 ( Y -M ) B ( Y -M ) ' A 1 2 ] γ [1+ b ( α -a o )tr[( A 1 2 ( Y -M ) B ( Y -M ) ' A 1 2 )] -η α -ao d Y,</formula>
<text><location><page_8><loc_18><loc_30><loc_88><loc_40></location>for α > a o . Let U = A 1 2 ( Y -M ) B 1 2 ⇒ d U = | A | q 2 | B | p 2 d Y where Y is p × q, p ≤ q and of rank p , M is a p × q, p ≤ q parameter matrix, A > O,B > O are p × p and q × q constant positive definite matrices, see Mathai (1997) for the Jacobian in this transformation. Let V = UU ' ⇒ d U = π pq 2 Γ p ( q 2 ) | V | q 2 -p +1 2 d V , see Mathai (1997) for details. Then</text>
<formula><location><page_8><loc_36><loc_21><loc_70><loc_27></location>1 C = | A | -q 2 | B | -p 2 π pq 2 Γ p ( q 2 ) ∫ V | V | γ + q 2 -p +1 2 × [1 + b ( α -a )tr( V )] -η α -ao d V, α > a o</formula>
<text><location><page_8><loc_21><loc_18><loc_71><loc_19></location>We can replace one factor by an equivalent integral, that is,</text>
<formula><location><page_9><loc_18><loc_81><loc_76><loc_85></location>[1 + b ( α -a o )tr( V )] -η α -ao ≡ 1 Γ( η α -a o ) ∫ ∞ z =0 z η α -ao -1 e -z (1+ b ( α -a o )tr( V )) d z.</formula>
<formula><location><page_9><loc_23><loc_75><loc_70><loc_80></location>1 C = | A | -q 2 | B | -p 2 π pq 2 Γ p ( q 2 ) e -z [ ∫ V | V | γ + q 2 -p +1 2 e -z ( α -a o )tr( V ) ]d V</formula>
<text><location><page_9><loc_12><loc_73><loc_79><loc_75></location>Evaluate the V-integral by using a real matrix-variate gamma integral. We have</text>
<formula><location><page_9><loc_33><loc_69><loc_35><loc_70></location>pq</formula>
<formula><location><page_9><loc_34><loc_57><loc_35><loc_58></location>2</formula>
<formula><location><page_9><loc_17><loc_58><loc_77><loc_70></location>1 C = | A | -q 2 | B | -p 2 π 2 Γ p ( q 2 ) [ b ( α -a o )] -p ( γ + q 2 ) Γ p ( γ + q 2 ) × ∫ ∞ z =0 z η α -ao -p ( γ + q 2 ) -1 e -z d z = | A | -q 2 | B | -p 2 π pq 2 Γ p ( q ) Γ p ( γ + q 2 )[ b ( α -a o )] -p ( γ + q 2 ) Γ( η α -a o -p ( γ + q 2 ))</formula>
<text><location><page_9><loc_12><loc_54><loc_61><loc_56></location>for η α -a o -p ( γ + q 2 ) > 0. This completes the computations.</text>
<section_header_level_1><location><page_9><loc_42><loc_51><loc_52><loc_53></location>References</section_header_level_1>
<text><location><page_9><loc_12><loc_39><loc_82><loc_48></location>M.E. Cuesta, A.T. Cummings, G. Livadiotis, D.J. McComas, C.M.S. Cohen, L.Y. Khoo, T. Sharma, M.M. Shen, R. Bandyopadhyay, J.S. Rankins, J.R. Szalay, H.A. Farooki, Z. Xu, G.D. Muro, M.L. Stevens, and S.D. Bale (2024): Observations of Kappa Distributions in Solar Energetic Protons and Derived Thermodynamic Properties, https://arxiv.org/abs/2407.20343.</text>
<unordered_list>
<list_item><location><page_9><loc_12><loc_33><loc_82><loc_38></location>G. Livadiotis (2018): Thermal Doppler broadening of spectral emissions in space and astrophysical plasmas. The Astrophysical Journal Supplement Series , 239:25 , https://doi.org/10.3847/1538-4365/aae835.</list_item>
<list_item><location><page_9><loc_12><loc_28><loc_82><loc_31></location>G. Livadiotis and D.J. McComas (2023): Entropy defect in thermodynamics. scientific reports , 13:9033 , https://doi.org/10.1038/s41598-023-36080-w.</list_item>
<list_item><location><page_9><loc_12><loc_24><loc_82><loc_27></location>A.M. Mathai (1997): Jacobians of Matrix Transformations and Functions of Matrix Argument , World Scientific Publishing, New York.</list_item>
<list_item><location><page_9><loc_12><loc_19><loc_82><loc_23></location>A.M. Mathai (2005): A pathway to matrix-variate gamma and normal densities. Linear Algebra and its Applications , 396 , 317-328.</list_item>
</unordered_list>
</document>
[
{
"title": "Arak M. Mathai",
"content": "Department of Mathematics and Statistics, McGill University, Montreal, Canada a.mathai@mcgill.ca and Hans J. Haubold Office for Outer Space Affairs, United Nations, Vienna International Centre, Vienna, Austria hans.haubold@gmail.com Abstract. Thermal Doppler broadening of spectral profiles for particle populations in the absence or presence of potential fields are described by kappa distributions. The kappa distribution provides a replacement for the Maxwell-Boltzmann distribution which can be considered as a generalization for describing systems characterized by local correlations among their particles as found in space and astrophysical plasmas. This paper presents all special cases of kappa distributions as members of a general pathway family of densities introduced by Mathai. Key words: atomic data - line: profiles - methods: analytical - plasmas",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "A particle is in thermal equilibrium when the exchange of heat or entropy in the system stops. The kappa index controls the exchange of entropy and it is also a measure of departure from the equilibrium state. The particle velocity distribution can be given in terms of a kappa distribution. In the case of a collisional plasma, where no local correlations among particles exist, the system is stabilized into a MaxwellBoltzmann distribution. The kappa index is inversely proportional to the correlation between the energies of two particles. The d-D kappa distribution describes the particle velocity with dimensionality d . Livadiotis (2018), Livadiotis and McComas (2023), and Cuesta (2024) take this particle velocity density as the following: where X is a d × 1 velocity vector, a prime denotes the transpose and µ = E [ X ] = < X > is the expected value of X , which is also a location parameter vector, where E [( · )] denotes the expected value with respect to the density f ( X ; θ, k d ), and the normalizing constant c d is the following: and θ = √ k B T/m is the thermal speed of the particle with mass m and temperature T expressed in speed units. If k d -d 2 is invariant over the dimensionality d then let k o be this constant value. Then, k d = k o + d 2 . Then, the normalizing constant becomes and the functional part is Since k d -d 2 = k o is invariant k 3 -3 2 = k d -d 2 , k 3 = k d -( d -3) 2 . If θ depends on the kappa index, then θ is of the form θ k . Two forms of θ k are usually taken. One is θ 2 k = ( k 3 -3 2 ) k 3 ( d -1) 2 θ 2 and the other is θ 2 k,d = k o k d θ 2 = ( k d -d 2 ) k d θ 2 . If θ k is used then (1.4) becomes On the other hand, if θ k,d is used then (1.4) becomes Thus, one can write the kappa density in (1.1) in a number of different ways. One set of representations will be for d = 1 , 2 , 3, another set will be with θ replaced by θ k or θ k,d . When k o →∞ , (1.1) will go to the density where c ∗ = ( πθ 2 ) -d 2 . This density (1.6) is a multivariate real Gaussian and also becomes as d -D Maxwellian distribution.",
"pages": [
1,
2
]
},
{
"title": "2. A Pathway Family of Densities",
"content": "All the forms of the kappa distribution considered in Section 1 are special cases of a general pathway family of densities introduced in Mathai (2005). Let X be a p × 1 real vector random variable with the covariance matrix Σ = Cov( X ) = E [( X -E ( X ))( X -E ( X )) ' ] where a prime denotes the transpose and E ( · ) is the expected value of ( · ) = < ( · ) > with respect to the density of X . The p components of X may be correlated among themselves. One can get rid of the effect of correlations by taking Y = Σ -1 2 ( X -µ ) where Σ 1 2 is the real positive definite square root of the positive definite matrix Σ > O . Then the covariance matrix in Y is the identity matrix, free of all correlations. The Euclidean distance of Y from the origin is also ( X -µ ) ' Σ -1 ( X -µ ), µ = E ( X ). Also, ( X -µ ) ' Σ -1 ( X -µ ) = a positive constant is an offset ellipsoid and it is also known in statistical literature as the ellipsoid of concentration. If the components of X are to be simply weighted, rather than removing the effect of correlations, then one can replace Σ -1 in the quadratic form by a real positive definite matrix A > O and then we may consider ( X -µ ) ' A ( X -µ ). Consider the following density: for a > 0 , δ > 0 , η > 0 , γ > 0, all real scalar parameters, and c is the normalizing constant, which can be evaluated as see the derivation of c in the Appendix. Consider δ = 1 , Σ = I (identity matrix), a = 1 θ 2 ( k d -d 2 ) , η = k d + 1 , γ = 0. Then (2.1) reduces to (1.1), the density given by Livadiotis (2018). The general representation in (2.1) and (2.2) has many advantages. This (2.1) is a member of the pathway family of distributions defined in Mathai (2005). Let a = b α -a o , b > 0, α > a o and η = ρ ( α -a o ) , ρ > 0. Then, (2.1) becomes Note that when α → a o from the right, g 1 ( X ) goes to the density where The density in (2.4) can be taken as a power-transformed real multivariate MaxwellBoltzmann density. For δ = 1, (2.4) is a form of multivariate Maxwell-Botlzmann density. Hence, if (2.4) is the stable density in a physical system, then the unstable neighborhood and transitional stages are determined by (2.3) for various values of the pathway parameter α . We may also note that for α = k d , a o = d 2 , b = 1 θ 2 , δ = 1 , γ = 0 , ρ = k d +1 k d -d 2 one has (1.1) also. This density in (2.3) has another advantage. For α < a o we can write α -a o = -( a o -α ) , a o -α > 0 so that the density in (2.3) switches into a type-1 beta form of the density given by where [1 -b a o -α (( X -µ ) ' Σ -1 ( X -µ )) δ ] > 0 so that the density is defined within the ellipsoid and This normalizing constant is evaluated by going through the procedure in Appendix A1 and the final integral is evaluated by using a type-1 beta integral. Thus, g 1 , g 2 , g 3 belong to the pathway family of densities. If the power-transformed Maxwell-Boltzmann density in (2.4) is the stable density in a physical system then the unstable neighborhoods and the transitional stages are given by g 1 in (2.3) and g 3 in (2.6). One can switch among a generalized type-1 beta family, a type-2 beta family and a gamma family or Maxwell-Boltzmann family of distributions through the pathway parameter α . In the model in (2.3) one can identify α with k d and a o with d 2 if convenient.",
"pages": [
3,
4
]
},
{
"title": "3. Livadiotis' d -D Density Through an entropy Optimization",
"content": "Let X be a p × 1 real vector random variable. Let f ( X ) be a density function, that is, f ( X ) ≥ 0 for all X and ∫ X f ( X )d X = 1 where f ( X ) is a real-valued scalar function of X . Consider Mathai's entropy for the vector random variable, namely, for a o a fixed quantity or anchoring point, α is the pathway parameter and the deviation of α from a o is measured in η > 0 units. Then, when α → a o , we can see that (3.1) reduces to Shannon's entropy S ( f ) = -K ∫ X f ( X ) ln f ( X )d X where K is a constant. Shannon's entropy is for the scalar variable case and the corresponding real vector-variate form is denoted here as S ( f ). Let Σ > O be the covariance matrix of X . Consider the ellipsoid of concentration ( X -µ ) ' Σ -1 ( X µ ) = a positive constant, where µ is a p × 1 location parameter vector. We will set moment-type constraints on the ellipsoid of concentration and then optimize (3.1). Consider the following constraints: for some parameters γ > 0 , η > 0 , δ > 0. If we use calculus of variation for the optimization, then the Euler equation becomes the following where λ 1 and λ 2 are Lagrangian multipliers: This gives the solution for f as the following: where ν 1 , λ 1 , λ 2 are constants. Let λ 2 λ 1 = b ( a o -α ) , b > 0 and let ν 1 be the normalizing constant to make (3.3) a density. Then, for u = ( X -µ ) ' Σ -1 ( X -µ ) we have the following densities from (3.3): where in (3.4) we need an additional condition 1 -b ( a o -α ) u δ > 0 in order to make (3.4) a density. Note that in the limiting form we have the following properties: Hence, one can take any one of the formats in (3.7) in the limiting case. Livadiotis' density in (1.1) is available from (3.5) by taking b ( α -a o ) = 1 θ ( k d -d 2 ) and η α -a o = k d +1. For γ = 0 , δ = 1 , η = 1 , a o = 1 , b = 1, equations (3.4),(3.5)(3.6) give a real multivariate version of Tsallis statistics of non-extensive statistical mechanics. For δ = 1 , η = 1 , a o = 1 , b = 1 , δ = 1, (3.5) and (3.6) can be taken as a multivariate extension of superstatistics. Matrix-variate versions of Tsallis statistics and superstatistics can also be defined.",
"pages": [
4,
5,
6
]
},
{
"title": "4. A Matrix-variate generalization of Livadiotis' Density",
"content": "Let Y = ( y ij ) be a p × q, p ≤ q and of rank p matrix with distinct real scalar variables y ij 's as elements. Let g ( Y ) be a real-valued scalar function of Y such that g ( Y ) ≥ 0 for all Y and ∫ Y g ( Y )d Y = 1 so that g ( Y ) is a density function where d Y = ∧ p i =1 ∧ q j =1 d y ij = the wedge product of all distinct differentials in Y . Let M be a p × q, p ≤ q parameter matrix. Let A > O be p × p and B > O be q × q positive definite constant matrices. Let A 1 2 and B 1 2 be the positive definite square roots of A and B respectively. Let Then, the determinant of V , namely | V | , can be interpreted in different ways. | V | is the product of the eigenvalues of the matrix V . Let U 1 , ..., U p be the p linearly independent rows of U , the p × q, p ≤ q and of rank p matrix U . Then, U j , j = 1 , ..., p can be taken as p linearly independent points in a q -dimensional Euclidean space with p ≤ q . Then, | V | 1 2 = | UU ' | 1 2 = the volume of the p -parallelotope generated in the convex hull of the p linearly independent points, taken in the order. Thus, | V | is also the square of the volume content of this parallelotope. Consider Mathis's entropy in (3.1) with f ( X ) replaced by g ( Y ) where Y is now p × q, p ≤ q matrix of rank p . Consider the optimization of (3.1) with the real-valued scalar function g ( Y ) under the following constraints: Then, proceeding as in Section 3, we can end up with the following pathway densities: where C 1 , C 2 , C 3 are the normalizing constants and in (4.3) the additional condition needed is 1 -b ( a o -α )tr( V ) > 0 to make (4.3) a density, where The normalizing constants are evaluated in Appendix A2 by using the following steps: and where Γ p ( α ) is a real matrix-variate gamma function, associated with a real matrixvariate gamma integral, and it is the following: where /Rfractur ( · ) denotes the real part of ( · ). Note that (4.5) is a real rectangular matrixvariate Maxwell-Boltzmann density. Note 4.1. Results parallel to all the results in Sections 3 and 4 are also available in the complex domain. Corresponding physics can also be dealt with in the complex domain. A real rectangular matrix-variate version of Tsallis statistics is available from (4.3)-(4.5) for γ + q 2 = p +1 2 , b = 1 , α = 1 , a o = 1 , η = 1. Corresponding results in the complex domain can also be worked out. For b = 1 , α = 1 , η = 1, (4.4) and (4.5) give a real rectangular matrix-variate version of superstatistics. Corresponding versions in the complex domain can also be worked out. Such rectangular or square matrix-variate versions may not be available in the literature.",
"pages": [
6,
7
]
},
{
"title": "APPENDIX",
"content": "A1: Derivation of the normalizing constant in (2.2) Let X be a p × 1 vector random variable and µ = E [ X ] , Σ = Cov( X ). Consider the transformation see Mathai (1997). Let u = ZZ ' . Then writing Z uniquely as a product of a unique lower triangular matrix and a unique semi-orthonormal matrix and then integrating out the differential element over a Stiefel manifold, we have a relation between d u and d Z , that is see Mathai (1997) for the details. Observe that u is real scalar whereas Z is a p × 1 vector. Then Now, let v = u δ , u > 0 , δ > 0 ⇒ d u = 1 δ v 1 δ -1 d v . Integrating out by using a type-2 beta integral we have which gives the normalizing constant.",
"pages": [
7,
8
]
},
{
"title": "A2 Derivation of the normalizing constant C in (4.4)",
"content": "Let Y be p × q, p ≤ q and of rank p matrix of real scalar random variables as elements. for α > a o . Let U = A 1 2 ( Y -M ) B 1 2 ⇒ d U = | A | q 2 | B | p 2 d Y where Y is p × q, p ≤ q and of rank p , M is a p × q, p ≤ q parameter matrix, A > O,B > O are p × p and q × q constant positive definite matrices, see Mathai (1997) for the Jacobian in this transformation. Let V = UU ' ⇒ d U = π pq 2 Γ p ( q 2 ) | V | q 2 -p +1 2 d V , see Mathai (1997) for details. Then We can replace one factor by an equivalent integral, that is, Evaluate the V-integral by using a real matrix-variate gamma integral. We have for η α -a o -p ( γ + q 2 ) > 0. This completes the computations.",
"pages": [
8,
9
]
},
{
"title": "References",
"content": "M.E. Cuesta, A.T. Cummings, G. Livadiotis, D.J. McComas, C.M.S. Cohen, L.Y. Khoo, T. Sharma, M.M. Shen, R. Bandyopadhyay, J.S. Rankins, J.R. Szalay, H.A. Farooki, Z. Xu, G.D. Muro, M.L. Stevens, and S.D. Bale (2024): Observations of Kappa Distributions in Solar Energetic Protons and Derived Thermodynamic Properties, https://arxiv.org/abs/2407.20343.",
"pages": [
9
]
}
]
2024arXiv240807711R
https://arxiv.org/pdf/2408.07711.pdf
<document>
<section_header_level_1><location><page_1><loc_21><loc_86><loc_79><loc_88></location>Cosmological functions and their relation.</section_header_level_1>
<text><location><page_1><loc_25><loc_81><loc_75><loc_83></location>Ricardo Rosas-Rodr'ıguez and Iv'an Cortes-Cruz</text>
<text><location><page_1><loc_35><loc_77><loc_66><loc_81></location>Universidad Tecnol'ogica de la Mixteca Instituto de F'ısica y Matem'aticas</text>
<text><location><page_1><loc_39><loc_74><loc_61><loc_76></location>16 de agosto de 2024</text>
<text><location><page_1><loc_10><loc_59><loc_91><loc_70></location>Abstract. In the non-metric gravity proposed by K. Krasnov, this is a theory in which the scalar constraint of the Ashtekar/acute.ts1s formalism is modified in such way that the cosmological constant is replaced by a cosmological function that depends of the canonical variables A a i and E i a , the General Relativity (GR) is the particular case when the cosmological function is a constant. Some years ago inspired by this theory Rosas-Rodriguez proposed two cosmological functions, one for the Ashtekar/acute.ts1s formalism and the other for the ADM formalism. In this paper we show that this cosmological functions are related thought the three-dimensional Ricci scalar.</text>
<section_header_level_1><location><page_1><loc_10><loc_54><loc_33><loc_56></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_10><loc_23><loc_91><loc_52></location>It is well know that today there are many problems to face off in General Relativity (GR) [1, 2, 3], for example, we don't know why the observed cosmological constant is so different from the one that is expected to arise as the vacuum energy of quantum fields. For this (and others) reason modified gravity theories have become popular recently [4, 5]. In 2008, K. Krosnov introduced a new class of modified gravity theories with the name of non-metric gravity [6]. The canonical description of this class of theories shows that as in the usual GR-Pleba'nsky-Asthekar case the phase space is parametrized by the canonically conjugate pairs ( A i a , ˜ E ai ), where A i a is an su(2) -connection, ˜ E ai is its canonically momentum and the theory is fully constrained [7]. Thus at canonical level the only modification with respect to the usual GR is that the cosmological constant gets replaced by a non-trivial (and arbitrary) function φ : T ∗ ( Α ) → R , where Α denotes the space of all the su(2) -connections, of the canonical variables. Inspired by this new class of theories Rosas-Rodr'ıguez around of 2014 proposed a cosmological function for the Ashtekar's formalism that seems to simplify the way what we look for states that solve the constrictions of the theory [8]. A few years ago was proposed a new cosmological function for the ADM formalism which turns out to have the same role that the function for the Ashtekar's formalism [9]. What we do here is give a relation between the proposed cosmological functions. As we will see these cosmological functions are related through the spacial Ricci scalar.</text>
<section_header_level_1><location><page_1><loc_10><loc_18><loc_44><loc_20></location>2. Non-Metric Gravity</section_header_level_1>
<text><location><page_1><loc_10><loc_10><loc_91><loc_17></location>It is well known that Ashtekar's constrain algebra arises as a result of the 3+1 decomposition of the Pleba'nski formulation [10] of GR. Here we describe the spacetime covariant theory that leads to a kind of modified constraint algebra with respect to the algebra of the Ashtekar's formalism. This theory was proposed in [11] and its action has the form</text>
<formula><location><page_1><loc_29><loc_6><loc_91><loc_10></location>S = ∫ Σ B i ∧ F i ( A ) + 1 2 (Ψ ij -1 3 δ ij φ (Ψ)) B i ∧ B j , (1)</formula>
<text><location><page_2><loc_10><loc_90><loc_91><loc_93></location>where F i is the curvature associate to the su(2) -connection A i , B i is a 2-form with values in su(2) and Ψ ij is the Lagrange multiplier field, which is required to be symmetric and traceless.</text>
<text><location><page_2><loc_12><loc_86><loc_76><loc_88></location>The 3 + 1 decomposition proceeds as in [12] and is given explicit in [7] by</text>
<formula><location><page_2><loc_18><loc_82><loc_91><loc_86></location>S = ∫ Σ ∫ R d 3 xdt [ ˜ E a i ˙ A i a + A i 0 ∆ a ˜ E a i +( δ ij N + 1 2 /epsilon1 ijk N k ) ˜ Ψ ij -N φ (Ψ)det ˜ E ai ] , (2)</formula>
<text><location><page_2><loc_10><loc_75><loc_91><loc_83></location>˜ ˜ ˜ where ˜ Ψ ij := F i ab /epsilon1 jkl ˜ E a k ˜ E b l , N ˜ k is the shift vector, N ˜ is the lapse function, and ˜ E ai is the momentum conjugate to A ai . Thus, the constraints of the theory can be written as</text>
<formula><location><page_2><loc_43><loc_72><loc_91><loc_74></location>Γ i = ∆ a ˜ E a i ≈ 0 , (3)</formula>
<formula><location><page_2><loc_43><loc_69><loc_91><loc_71></location>ς a = ˜ E b i F i ab ≈ 0 , (4)</formula>
<formula><location><page_2><loc_33><loc_65><loc_91><loc_69></location>Η = 1 2 /epsilon1 ijk ˜ E a i ˜ E b j F ab k -φ (Ψ)det ˜ E ai ≈ 0 . (5)</formula>
<text><location><page_2><loc_10><loc_58><loc_91><loc_65></location>These constraints still form a set of first class, an explicit proof is given in [7, 13], which means that the count of the number of physical degrees of freedom (DOF) is unchanged and the theory still propagates two DOF. Note that the cosmological function φ depends of the (symmetric) tensor</text>
<formula><location><page_2><loc_35><loc_55><loc_91><loc_59></location>Ψ ij := 1 det ˜ E ai ( F i ab /epsilon1 jkl ˜ E a k ˜ E b l ) tr -free , (6)</formula>
<text><location><page_2><loc_10><loc_53><loc_45><loc_55></location>where tr-free denotes the trace-free part.</text>
<text><location><page_2><loc_10><loc_45><loc_91><loc_51></location>Now, we should expect to get also a kind of modified constraint algebra with respect to the constraint algebra of the ADM formalism extended to triads, after all the ADM and Ashtekar/acute.ts1s formalism are related through a canonical transformation. This means that we should have constraints of the form</text>
<formula><location><page_2><loc_42><loc_43><loc_91><loc_45></location>L i := /epsilon1 ijk p a j e ak ≈ 0 , (7)</formula>
<formula><location><page_2><loc_41><loc_40><loc_91><loc_42></location>Η a := -e ai D b p bi ≈ 0 , (8)</formula>
<formula><location><page_2><loc_27><loc_36><loc_91><loc_40></location>Η := p ij p ij -1 2 ( p i i ) 2 -4 (3) e 2 (3) R -8 (3) e 2 , Λ(Ψ) ≈ 0 , (9)</formula>
<text><location><page_2><loc_10><loc_35><loc_53><loc_36></location>where Λ(Ψ) now denote the cosmological function.</text>
<text><location><page_2><loc_10><loc_26><loc_91><loc_33></location>Inspired by this class of modified theories, Rosas-Rodr'ıguez proposed a few years ago an couple of cosmological functions that appears to avoid some of the problems related to solving the constraints in GR at classical and quantum levels. This cosmological are the subject of the following section.</text>
<section_header_level_1><location><page_2><loc_10><loc_21><loc_48><loc_23></location>3. Cosmological functions</section_header_level_1>
<text><location><page_2><loc_10><loc_13><loc_91><loc_20></location>Note that the cosmological function given by Krasnov φ (Ψ) are generic, there is not a specific form for these today (however, see [14] ). But these cosmological functions gave the idea to Rosas-Rodriguez [8] to suggest the following ansatz as cosmological function in the Astekar's formalism</text>
<formula><location><page_2><loc_38><loc_10><loc_91><loc_13></location>ϕ := 1 2 det ˜ E ai /epsilon1 ijk ˜ E i a ˜ E j b F abk . (10)</formula>
<text><location><page_2><loc_10><loc_6><loc_91><loc_10></location>Note that is a scalar function. With this function the Hamiltonian constraint is satisfies automatically at classical and quantum levels. Note that this cosmological function is not a member</text>
<text><location><page_3><loc_10><loc_91><loc_31><loc_93></location>of the Krasnov's theories.</text>
<text><location><page_3><loc_10><loc_86><loc_91><loc_91></location>Thus, the problem of find physical states Ψ[ A ] ∈ Η phy , where Η phy denotes the physical states space, is reduced. There is no problem with the Hamiltonian constraint since any state that satisfies the Gauss and vectorial constraints,</text>
<formula><location><page_3><loc_42><loc_82><loc_91><loc_85></location>̂ Γ i Ψ = 0 , ̂ Η a Ψ = 0 , (11)</formula>
<text><location><page_3><loc_10><loc_78><loc_91><loc_83></location>is also a solution to this (Hamiltonian) constraint. For example, the Chern-Simons state is still a solution to these constrains but is no longer necessary verify that is also a solution to the Hamiltonian constraint.</text>
<text><location><page_3><loc_10><loc_73><loc_91><loc_76></location>Now, it suggest something analogous in the ADM formalism [15, 16], see [9]. Thus, the cosmological function for the ADM formalism extended to triads is given by</text>
<formula><location><page_3><loc_37><loc_68><loc_91><loc_72></location>Λ = p ij p ij -1 2 ( p i i ) 2 -4 e 2(3) R 8 e 2 . (12)</formula>
<text><location><page_3><loc_10><loc_64><loc_91><loc_68></location>This cosmological function also solves the Hamiltonian constraint in this formalism and is a scalar function again.</text>
<section_header_level_1><location><page_3><loc_10><loc_60><loc_72><loc_62></location>4. Relation between cosmological function</section_header_level_1>
<text><location><page_3><loc_10><loc_55><loc_91><loc_58></location>Now, the core of this work is to show that the cosmological function above are related through the spacial Ricci scalar and this is what we are going to do in this section.</text>
<text><location><page_3><loc_10><loc_48><loc_91><loc_53></location>Indeed, the first thing that we have to do is to rewrite the Hamiltonian constraint of the Ashtekar's formalism. We know that the ADM formalism in triads and the Ashtekar's formalism are related trough a canonical transformation 1 [17, 18, 19] given as</text>
<formula><location><page_3><loc_42><loc_45><loc_91><loc_47></location>K ai := i ( A ai -Γ ai ) , (13)</formula>
<text><location><page_3><loc_10><loc_41><loc_90><loc_44></location>where K ai is an auxiliary field related to the extrinsic curvature, A ai is the Asthekar's connection and Γ ai is the unique free-torsion spin-connection defined to annihilate the field ˜ E ai :</text>
<formula><location><page_3><loc_28><loc_38><loc_91><loc_40></location>∆ a ˜ E bi := ∂ a ˜ E ai +Γ b ac ˜ E ci -Γ c ac ˜ E bi + /epsilon1 ijk Γ aj ˜ E b i = 0 . (14)</formula>
<text><location><page_3><loc_10><loc_36><loc_18><loc_37></location>Note that</text>
<formula><location><page_3><loc_37><loc_34><loc_91><loc_36></location>∆ a ˜ E ai = ∂ a ˜ E ai + /epsilon1 ijk Γ aj ˜ E a k = 0 (15)</formula>
<text><location><page_3><loc_10><loc_32><loc_56><loc_34></location>Now, the curvature associate to A ai can be written as</text>
<formula><location><page_3><loc_19><loc_23><loc_81><loc_32></location>F ab i = 2 ∂ [ a (Γ b ] i -iK b ] i ) + /epsilon1 ijk ( Γ j a Γ k b -i Γ j a K k b -iK j a Γ k b -K j a K k b ) = 2 ∂ [ a Γ b ] i + /epsilon1 ijk Γ j a Γ k b -2 i∂ [ a K b ] i -/epsilon1 ijk K j a K k b -i/epsilon1 ijk ( Γ j a K k b -Γ j b K k a ) = f abi -2 i ( ∂ [ a K b ] i + /epsilon1 ijk Γ j [ a K k b ] ) -/epsilon1 ijk K j a K k b .</formula>
<text><location><page_3><loc_10><loc_22><loc_14><loc_23></location>Thus</text>
<formula><location><page_3><loc_35><loc_20><loc_91><loc_22></location>F i ab = f i ab -2 i ∆ [ a K i b ] -/epsilon1 ijk K aj K bk , (16)</formula>
<text><location><page_3><loc_10><loc_16><loc_91><loc_19></location>where f abi is the curvature associate to Γ ai defined by: f abi := 2 ∂ [ a Γ b ] i + /epsilon1 ijk Γ j a Γ k b . It follows that the Hamiltonian constraint Eq. (5) can be rewritten as</text>
<formula><location><page_3><loc_20><loc_8><loc_80><loc_15></location>Η = 1 2 /epsilon1 ijk ˜ E a i ˜ E b j ( f abk -2 i ∆ [ a K b ] k -/epsilon1 klm K l a K m b ) -ϕ det ˜ E ai = 1 2 /epsilon1 ijk ˜ E a i ˜ E b j f abk -i/epsilon1 ijk ˜ E a i ˜ E b j ∆ [ a K b ] k -˜ E a i ˜ E b j K [ i a K j ] b -ϕ det ˜ E ai ≈ 0 .</formula>
<text><location><page_4><loc_10><loc_91><loc_36><loc_93></location>Ignoring surface terms we have</text>
<formula><location><page_4><loc_29><loc_87><loc_91><loc_90></location>Η = 1 2 /epsilon1 ijk ˜ E a i ˜ E b j f abk -˜ E a i ˜ E b j K [ i a K j ] b -ϕ det ˜ E ai ≈ 0 (17)</formula>
<text><location><page_4><loc_10><loc_83><loc_91><loc_86></location>Something analogous was done by P. Peldan in [16]. Now we define a version undensized of the variable ˜ E ai and its inverse, i.e.,</text>
<formula><location><page_4><loc_29><loc_78><loc_91><loc_82></location>˜ E ai =: ee ai e bi e ai = δ a b e := det e ai = √ det ˜ E ai . (18)</formula>
<text><location><page_4><loc_10><loc_74><loc_91><loc_77></location>With this relation in hand we can write the scalar constraint Eq.(17) in terms of triads e ai and its determinant. It follows that</text>
<formula><location><page_4><loc_30><loc_70><loc_70><loc_73></location>Η = 1 2 /epsilon1 ijk e 2 e a i e b j f abk -e 2 e a i e b j K [ i a K j ] b -ϕe 2 ≈ 0 .</formula>
<text><location><page_4><loc_10><loc_67><loc_80><loc_69></location>Thus, the cosmological function for the Ashtekar's formalis can now be written as</text>
<formula><location><page_4><loc_36><loc_62><loc_91><loc_66></location>ϕ := 1 2 /epsilon1 ijk e a i e b j f abk -e a i e b j K [ i a K j ] b . (19)</formula>
<text><location><page_4><loc_10><loc_60><loc_18><loc_61></location>Note that</text>
<formula><location><page_4><loc_41><loc_58><loc_91><loc_60></location>/epsilon1 ijk e a i e b j f abk = e a i e b j Ω ij ab , (20)</formula>
<text><location><page_4><loc_10><loc_54><loc_91><loc_57></location>where Ω ij ab = -Ω ji ab is a Lorentz connection. It is well know that the spacial Ricci scalar can be related to this connection by the following equation</text>
<formula><location><page_4><loc_43><loc_50><loc_91><loc_52></location>(3) R := e a i e b j Ω ij ab . (21)</formula>
<text><location><page_4><loc_10><loc_47><loc_65><loc_49></location>With this the cosmological function Eq. (22) can be rewritten as</text>
<formula><location><page_4><loc_39><loc_43><loc_91><loc_46></location>ϕ := 1 2 (3) R -e a i e b j K [ i a K j ] b . (22)</formula>
<text><location><page_4><loc_10><loc_39><loc_91><loc_42></location>This is a more familiar equation with relation to the cosmological function for the ADM formalism, see Eq. (12). Indeed, we can always solve for the curvature in Eq. (12) to get</text>
<formula><location><page_4><loc_34><loc_34><loc_91><loc_38></location>(3) R = -2Λ + 1 4 e 2 ( p ij p ij -1 2 ( p i i ) 2 ) . (23)</formula>
<text><location><page_4><loc_10><loc_31><loc_40><loc_33></location>From Eqs. (23) and (22) we obtain</text>
<formula><location><page_4><loc_28><loc_26><loc_72><loc_30></location>ϕ = 1 2 [ -2Λ + 1 4 e 2 ( p ij p ij -1 2 ( p i i ) 2 )] -e a i e b j K [ i a K j ] b .</formula>
<text><location><page_4><loc_10><loc_24><loc_14><loc_25></location>Thus</text>
<formula><location><page_4><loc_30><loc_21><loc_91><loc_25></location>ϕ = -Λ+ 1 8 e 2 ( p ij p ij -1 2 ( p i i ) 2 ) -e a i e b j K [ i a K j ] b . (24)</formula>
<text><location><page_4><loc_10><loc_15><loc_91><loc_20></location>The above equation represents the first relation between the two proposed cosmological functions. What follows is to write this equation more compactly. Now, note that we can define the field K ai as [16, 20]</text>
<formula><location><page_4><loc_39><loc_12><loc_91><loc_16></location>K i a = 1 2 e ( p i a -1 2 ( p k k ) e i a ) . (25)</formula>
<text><location><page_4><loc_10><loc_10><loc_43><loc_12></location>We can always define p ij = p ai e j a , then,</text>
<formula><location><page_4><loc_22><loc_7><loc_91><loc_8></location>p ij p ij = p ai e j a p bi e b j = p ai p bi δ b a = p ai p ai , p := p ai e ai = p ij e a j e ai = p i i . (26)</formula>
<text><location><page_5><loc_10><loc_90><loc_91><loc_93></location>From Eqs. (25) and (26) is straightforward to obtain an explicit expression for e a i e b j K [ i a K j ] b . Thus, on one hand we have</text>
<formula><location><page_5><loc_12><loc_85><loc_87><loc_89></location>e a i e b j K i a K j b = e a i e b j 1 e 2 ( p i a -1 pe i a )( p j b -1 pe j b ) = 1 e 2 [ e a i e b j p i a p j b -p ( e j b p i a e a i e b j + e i a p j b e a i e b j</formula>
<formula><location><page_5><loc_16><loc_81><loc_88><loc_88></location>4 2 2 4 2 ) + 1 4 p 2 e a i e b j e i a e j b ] = 1 4 e 2 [ p i i p j j -p 2 ( p i i δ j j + p j j δ i i ) + 1 4 p 2 δ i i δ j j ] = 1 4 e 2 ( p 2 -3 p 2 + 9 4 p 2 ) .</formula>
<text><location><page_5><loc_10><loc_79><loc_12><loc_80></location>So</text>
<formula><location><page_5><loc_41><loc_76><loc_91><loc_79></location>e a i e b j K i a K j b = 1 16 e 2 p 2 . (27)</formula>
<text><location><page_5><loc_10><loc_74><loc_33><loc_76></location>On the other hand we have</text>
<formula><location><page_5><loc_17><loc_66><loc_83><loc_74></location>e a i e b j K j a K i b = 1 4 e 2 [ e a i e b j p j a p i b -p 2 e a i e b j ( p j a e i b + p i b e j a ) + 1 4 p 2 e a i e b j e j a e i b ] = 1 4 e 2 [ p j i p i j -p 2 ( p j i δ i j + p i j δ j i ) + 1 4 p 2 δ ii ] = 1 4 e 2 ( p ij p ij -p 2 + 3 4 p 2 ) .</formula>
<text><location><page_5><loc_10><loc_64><loc_12><loc_65></location>So</text>
<formula><location><page_5><loc_36><loc_61><loc_91><loc_65></location>e a i e b j K j a K i b = 1 4 e 2 ( p ij p ij -1 4 p 2 ) . (28)</formula>
<text><location><page_5><loc_10><loc_59><loc_22><loc_61></location>It follows that</text>
<formula><location><page_5><loc_17><loc_55><loc_83><loc_59></location>e a i e b j K [ i a K j ] b = 1 2 [ e a i e b j K i a K j b -e a i e b j K j a K i b ] = 1 2 [ 1 16 e 2 p 2 -1 4 e 2 ( p ij p ij -1 4 p 2 )] .</formula>
<text><location><page_5><loc_10><loc_52><loc_18><loc_54></location>Therefore</text>
<formula><location><page_5><loc_34><loc_49><loc_91><loc_53></location>e a i e b j K [ i a K j ] b = -1 8 e 2 ( p ij p ij -1 2 ( p k k ) 2 ) . (29)</formula>
<text><location><page_5><loc_10><loc_47><loc_33><loc_49></location>As an consequence we have</text>
<formula><location><page_5><loc_26><loc_43><loc_75><loc_47></location>ϕ = -Λ+ 1 8 e 2 ( p ij p ij -1 2 ( p i i ) 2 ) + 1 8 e 2 ( p ij p ij -1 2 ( p i i ) 2 ) .</formula>
<text><location><page_5><loc_10><loc_40><loc_77><loc_42></location>Hence the cosmological function in the Ashteka's formalism can be written as</text>
<formula><location><page_5><loc_36><loc_36><loc_91><loc_40></location>ϕ = -Λ+ 1 4 e 2 ( p ij p ij -1 2 ( p i i ) 2 ) . (30)</formula>
<text><location><page_5><loc_10><loc_32><loc_77><loc_35></location>This is a more compact relation between the proposed cosmological functions. But, note that from Eq. (12) we have that</text>
<formula><location><page_5><loc_36><loc_27><loc_91><loc_30></location>p ij p ij -1 2 ( p i i ) 2 = 8 e 2 Λ+4 e 2 (3) R. (31)</formula>
<text><location><page_5><loc_10><loc_25><loc_26><loc_26></location>Then, we can write</text>
<formula><location><page_5><loc_28><loc_20><loc_72><loc_24></location>ϕ = -Λ+ 1 4 e 2 ( 8 e 2 Λ+4 e 2 (3) R ) = -Λ+2Λ+ (3) R.</formula>
<text><location><page_5><loc_10><loc_18><loc_43><loc_19></location>Thus we have a more compact relation</text>
<formula><location><page_5><loc_44><loc_15><loc_91><loc_16></location>ϕ = Λ + (3) R, (32)</formula>
<text><location><page_5><loc_10><loc_10><loc_91><loc_13></location>Indeed, the above equation represents the most compact relation between the prosed cosmological functions that we found.</text>
<text><location><page_5><loc_10><loc_6><loc_90><loc_10></location>Note that the cosmological functions in both formalism are the same up to the spacial Ricci scalar.</text>
<text><location><page_6><loc_10><loc_90><loc_90><loc_93></location>We also found a way to relate this cosmological functions through the extrinsic curvature. To do this, we begin from the fact that the equation Eq. (30) can be written as</text>
<formula><location><page_6><loc_37><loc_86><loc_91><loc_90></location>ϕ = -Λ+ 1 4 e 2 ( p ai p ai -1 2 p 2 ) . (33)</formula>
<text><location><page_6><loc_10><loc_83><loc_55><loc_85></location>Then, from Eq. (25) we have p ai = 2 K ai e + 1 2 pe ai , so</text>
<formula><location><page_6><loc_22><loc_79><loc_91><loc_83></location>p ai p ai = 4 e 2 K ai K ai +2 p eK ai e ai + p 2 4 δ i i = 4 K ai K ai +2 p eK + 3 4 p 2 , (34)</formula>
<text><location><page_6><loc_10><loc_77><loc_40><loc_79></location>where we have defined K := K ai e ai .</text>
<text><location><page_6><loc_10><loc_75><loc_46><loc_77></location>Now an alternative definition to K i a is [21]:</text>
<formula><location><page_6><loc_41><loc_73><loc_91><loc_74></location>K i a := K ab e bi + J ab e bi , (35)</formula>
<text><location><page_6><loc_10><loc_67><loc_91><loc_72></location>where K ab = K ( ab ) is the extrinsic curvature and J ab = J [ ab ] . Because of the ADM formalism extended to triads the annihilation of J ab is equivalent to that the internal constraint is satisfied [see Eq. (7)]. Hence,</text>
<formula><location><page_6><loc_28><loc_64><loc_72><loc_66></location>K ai K ai = K ab K ac δ b c +2 e -1 K ab J ac δ b c + e -2 J ab J ac δ b c .</formula>
<text><location><page_6><loc_10><loc_62><loc_12><loc_63></location>So</text>
<text><location><page_6><loc_10><loc_58><loc_22><loc_59></location>It follows that</text>
<text><location><page_6><loc_10><loc_52><loc_23><loc_54></location>Next, note that</text>
<text><location><page_6><loc_10><loc_48><loc_15><loc_49></location>Hence</text>
<text><location><page_6><loc_10><loc_43><loc_14><loc_45></location>Then</text>
<text><location><page_6><loc_10><loc_38><loc_12><loc_39></location>So</text>
<text><location><page_6><loc_10><loc_34><loc_14><loc_35></location>Thus</text>
<formula><location><page_6><loc_31><loc_60><loc_91><loc_62></location>K ai K ai = K ab K ab +2 e -1 K ab J ab + e -2 J ab J ab . (36)</formula>
<formula><location><page_6><loc_23><loc_54><loc_77><loc_58></location>p ai p ai = 4 e 2 ( K ab K ab +2 e -1 K ab J ab + e -2 J ab J ab ) + 2 p eK + 3 4 p 2 .</formula>
<formula><location><page_6><loc_36><loc_49><loc_64><loc_53></location>p := e ai p ai = 2 K ai e ai e + p 2 e ai e ai .</formula>
<formula><location><page_6><loc_38><loc_45><loc_62><loc_48></location>K = -1 4 e p ⇒ K 2 = 1 16 e 2 p 2 .</formula>
<formula><location><page_6><loc_17><loc_39><loc_84><loc_43></location>p ai p ai = 4 e 2 ( K ab K ab +2 e -1 K ab J ab + e -2 J ab J ab ) +2 e ( -4 eK ) K + 3 4 (16 e 2 K 2 ) .</formula>
<formula><location><page_6><loc_25><loc_35><loc_91><loc_38></location>p ai p ai = 4 e 2 ( K ab K ab +2 e -1 K ab J ab + e -2 J ab J ab ) +4 e 2 K 2 . (37)</formula>
<formula><location><page_6><loc_16><loc_30><loc_85><loc_34></location>ϕ = -Λ + 1 4 e 2 [ 4 e 2 ( K ab K ab +2 e -1 K ab J ab + e -2 J ab J ab ) +4 e 2 K 2 -1 2 (16 e 2 K 2 ) ] ,</formula>
<text><location><page_6><loc_10><loc_28><loc_13><loc_29></location>and</text>
<formula><location><page_6><loc_21><loc_25><loc_91><loc_29></location>ϕ = -Λ + 1 4 e 2 [ 4 e 2 ( K ab K ab +2 e -1 K ab J ab + e -2 J ab J ab ) -4 e 2 K 2 ] . (38)</formula>
<text><location><page_6><loc_10><loc_23><loc_18><loc_25></location>Therefore</text>
<formula><location><page_6><loc_39><loc_22><loc_91><loc_23></location>ϕ ≈ -Λ + K ab K ab -K 2 , (39)</formula>
<text><location><page_6><loc_10><loc_18><loc_91><loc_21></location>where ≈ denote modulo the internal constriction. But when we do an restriction to X phy. we obtain that</text>
<formula><location><page_6><loc_41><loc_16><loc_91><loc_18></location>K ab K ab -K 2 ≈ (3) R. (40)</formula>
<text><location><page_6><loc_10><loc_14><loc_21><loc_15></location>In conclusion</text>
<formula><location><page_6><loc_43><loc_12><loc_91><loc_14></location>ϕ ≈ -Λ+ (3) R. (41)</formula>
<text><location><page_6><loc_10><loc_8><loc_90><loc_11></location>This represents the most compact relation between the proposed cosmological functions when we restrict our attention to the physical phase space X phy .</text>
<section_header_level_1><location><page_7><loc_10><loc_91><loc_45><loc_93></location>5. Concluding Remarks</section_header_level_1>
<text><location><page_7><loc_10><loc_81><loc_91><loc_90></location>Firs of all we need to say that the relations given by Eqs. (32) and (41) do not have to be equal, we find Eq. (32) considering the whole phase space, i.e. Met (Σ), while one finds Eq. (41) just when we restrict our attention to the physical phase space X phy. . Indeed, what led us to Eq. (41) was that we were looking for a more geometrical form of relate the proposed cosmological functions.</text>
<text><location><page_7><loc_10><loc_68><loc_91><loc_79></location>Although the ansatz Λ and ϕ , Eqs. (12) and (10) respectively, are obtained directly by one simple computation, the interpretation for these cosmological functions is not clear for us. We have to admit that at this moment is not clear why the cosmological function are related through the three-dimensional Ricci scalar. Perhaps a more geometric interpretation can be given from this relation in terms of extrinsic curvature, it is well know that the extrinsic curvature only makes sense when we have a manifold embedded on another of higher dimension and this is what we have in GR, see Eq. (39).</text>
<text><location><page_7><loc_10><loc_63><loc_91><loc_66></location>We hope to have a more precise interpretation of this cosmological function and their relation in the coming years.</text>
<section_header_level_1><location><page_7><loc_10><loc_58><loc_26><loc_60></location>Referencias</section_header_level_1>
<unordered_list>
<list_item><location><page_7><loc_11><loc_55><loc_84><loc_56></location>[1] R. M. Wald, General Relativity , The University of Chicago Press, Chicago (1984).</list_item>
<list_item><location><page_7><loc_11><loc_50><loc_91><loc_53></location>[2] J. Baez and J. P. Muniainn, Gauge Fields, Knots and Gravity , World Scientific, Singapore (1994).</list_item>
<list_item><location><page_7><loc_11><loc_46><loc_91><loc_49></location>[3] R. Gambini and J. Pullin, Loops, Knots, Gauge Theories and Quantum Gravity , Cambridge University Press, Cambridge, UK (2000).</list_item>
<list_item><location><page_7><loc_11><loc_41><loc_91><loc_44></location>[4] R. H. Dicke, 'Republication of The Theoretical significance of experimental relativity', Gen. Rel. and Grav. , 51 , 1-31, (2019).</list_item>
<list_item><location><page_7><loc_11><loc_38><loc_91><loc_40></location>[5] P. D. Mannheim, 'Linear potentials an galactic rotation curves', Ast. J. , 419 , 150 (1993).</list_item>
<list_item><location><page_7><loc_11><loc_34><loc_91><loc_37></location>[6] K. Krasnov, 'Non-metric Gravity: I. Field Equations', Class. Quantum Grav . 25 025001, (2008)</list_item>
<list_item><location><page_7><loc_11><loc_29><loc_91><loc_32></location>[7] K. Krasnov, 'On Deformations of Ashtekar's Constraint Algebra', Phys. Rev. Lett. 100 , 081102. (2007).</list_item>
<list_item><location><page_7><loc_11><loc_25><loc_91><loc_28></location>[8] R. Rosas-Rodr'ıguez, 'Hamiltonian Constrain of Gravity with Cosmological Function', AIP. Conf. Proc. 1548 , 191 (2013).</list_item>
<list_item><location><page_7><loc_11><loc_20><loc_91><loc_23></location>[9] R. Rosas-Rodr'ıguez, 'Cosmological Functions in ADM and Ashtekar's Representations of Gravity', en preparaci'on.</list_item>
<list_item><location><page_7><loc_10><loc_15><loc_91><loc_19></location>[10] J. F. Plebanski, 'On the Separation of Einsteinian Substructures', J. Math. Phys. 18 , 2511 (1977).</list_item>
<list_item><location><page_7><loc_10><loc_11><loc_91><loc_14></location>[11] K. Krasnov, 'Renormalizable Non-Metric Quantum Gravity?', arXiv:hep-th/0611182 (2007).</list_item>
<list_item><location><page_7><loc_10><loc_6><loc_91><loc_10></location>[12] R. Capovilla, J. Dell, T. Jacobson and L. Mason, 'Selfdual 2-forms and Gravity', Class. Quantum Grav. 8 , 41 (1990).</list_item>
</unordered_list>
<table>
<location><page_8><loc_9><loc_48><loc_91><loc_93></location>
</table>
</document>
[
{
"title": "Cosmological functions and their relation.",
"content": "Ricardo Rosas-Rodr'ıguez and Iv'an Cortes-Cruz Universidad Tecnol'ogica de la Mixteca Instituto de F'ısica y Matem'aticas 16 de agosto de 2024 Abstract. In the non-metric gravity proposed by K. Krasnov, this is a theory in which the scalar constraint of the Ashtekar/acute.ts1s formalism is modified in such way that the cosmological constant is replaced by a cosmological function that depends of the canonical variables A a i and E i a , the General Relativity (GR) is the particular case when the cosmological function is a constant. Some years ago inspired by this theory Rosas-Rodriguez proposed two cosmological functions, one for the Ashtekar/acute.ts1s formalism and the other for the ADM formalism. In this paper we show that this cosmological functions are related thought the three-dimensional Ricci scalar.",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "It is well know that today there are many problems to face off in General Relativity (GR) [1, 2, 3], for example, we don't know why the observed cosmological constant is so different from the one that is expected to arise as the vacuum energy of quantum fields. For this (and others) reason modified gravity theories have become popular recently [4, 5]. In 2008, K. Krosnov introduced a new class of modified gravity theories with the name of non-metric gravity [6]. The canonical description of this class of theories shows that as in the usual GR-Pleba'nsky-Asthekar case the phase space is parametrized by the canonically conjugate pairs ( A i a , ˜ E ai ), where A i a is an su(2) -connection, ˜ E ai is its canonically momentum and the theory is fully constrained [7]. Thus at canonical level the only modification with respect to the usual GR is that the cosmological constant gets replaced by a non-trivial (and arbitrary) function φ : T ∗ ( Α ) → R , where Α denotes the space of all the su(2) -connections, of the canonical variables. Inspired by this new class of theories Rosas-Rodr'ıguez around of 2014 proposed a cosmological function for the Ashtekar's formalism that seems to simplify the way what we look for states that solve the constrictions of the theory [8]. A few years ago was proposed a new cosmological function for the ADM formalism which turns out to have the same role that the function for the Ashtekar's formalism [9]. What we do here is give a relation between the proposed cosmological functions. As we will see these cosmological functions are related through the spacial Ricci scalar.",
"pages": [
1
]
},
{
"title": "2. Non-Metric Gravity",
"content": "It is well known that Ashtekar's constrain algebra arises as a result of the 3+1 decomposition of the Pleba'nski formulation [10] of GR. Here we describe the spacetime covariant theory that leads to a kind of modified constraint algebra with respect to the algebra of the Ashtekar's formalism. This theory was proposed in [11] and its action has the form where F i is the curvature associate to the su(2) -connection A i , B i is a 2-form with values in su(2) and Ψ ij is the Lagrange multiplier field, which is required to be symmetric and traceless. The 3 + 1 decomposition proceeds as in [12] and is given explicit in [7] by ˜ ˜ ˜ where ˜ Ψ ij := F i ab /epsilon1 jkl ˜ E a k ˜ E b l , N ˜ k is the shift vector, N ˜ is the lapse function, and ˜ E ai is the momentum conjugate to A ai . Thus, the constraints of the theory can be written as These constraints still form a set of first class, an explicit proof is given in [7, 13], which means that the count of the number of physical degrees of freedom (DOF) is unchanged and the theory still propagates two DOF. Note that the cosmological function φ depends of the (symmetric) tensor where tr-free denotes the trace-free part. Now, we should expect to get also a kind of modified constraint algebra with respect to the constraint algebra of the ADM formalism extended to triads, after all the ADM and Ashtekar/acute.ts1s formalism are related through a canonical transformation. This means that we should have constraints of the form where Λ(Ψ) now denote the cosmological function. Inspired by this class of modified theories, Rosas-Rodr'ıguez proposed a few years ago an couple of cosmological functions that appears to avoid some of the problems related to solving the constraints in GR at classical and quantum levels. This cosmological are the subject of the following section.",
"pages": [
1,
2
]
},
{
"title": "3. Cosmological functions",
"content": "Note that the cosmological function given by Krasnov φ (Ψ) are generic, there is not a specific form for these today (however, see [14] ). But these cosmological functions gave the idea to Rosas-Rodriguez [8] to suggest the following ansatz as cosmological function in the Astekar's formalism Note that is a scalar function. With this function the Hamiltonian constraint is satisfies automatically at classical and quantum levels. Note that this cosmological function is not a member of the Krasnov's theories. Thus, the problem of find physical states Ψ[ A ] ∈ Η phy , where Η phy denotes the physical states space, is reduced. There is no problem with the Hamiltonian constraint since any state that satisfies the Gauss and vectorial constraints, is also a solution to this (Hamiltonian) constraint. For example, the Chern-Simons state is still a solution to these constrains but is no longer necessary verify that is also a solution to the Hamiltonian constraint. Now, it suggest something analogous in the ADM formalism [15, 16], see [9]. Thus, the cosmological function for the ADM formalism extended to triads is given by This cosmological function also solves the Hamiltonian constraint in this formalism and is a scalar function again.",
"pages": [
2,
3
]
},
{
"title": "4. Relation between cosmological function",
"content": "Now, the core of this work is to show that the cosmological function above are related through the spacial Ricci scalar and this is what we are going to do in this section. Indeed, the first thing that we have to do is to rewrite the Hamiltonian constraint of the Ashtekar's formalism. We know that the ADM formalism in triads and the Ashtekar's formalism are related trough a canonical transformation 1 [17, 18, 19] given as where K ai is an auxiliary field related to the extrinsic curvature, A ai is the Asthekar's connection and Γ ai is the unique free-torsion spin-connection defined to annihilate the field ˜ E ai : Note that Now, the curvature associate to A ai can be written as Thus where f abi is the curvature associate to Γ ai defined by: f abi := 2 ∂ [ a Γ b ] i + /epsilon1 ijk Γ j a Γ k b . It follows that the Hamiltonian constraint Eq. (5) can be rewritten as Ignoring surface terms we have Something analogous was done by P. Peldan in [16]. Now we define a version undensized of the variable ˜ E ai and its inverse, i.e., With this relation in hand we can write the scalar constraint Eq.(17) in terms of triads e ai and its determinant. It follows that Thus, the cosmological function for the Ashtekar's formalis can now be written as Note that where Ω ij ab = -Ω ji ab is a Lorentz connection. It is well know that the spacial Ricci scalar can be related to this connection by the following equation With this the cosmological function Eq. (22) can be rewritten as This is a more familiar equation with relation to the cosmological function for the ADM formalism, see Eq. (12). Indeed, we can always solve for the curvature in Eq. (12) to get From Eqs. (23) and (22) we obtain Thus The above equation represents the first relation between the two proposed cosmological functions. What follows is to write this equation more compactly. Now, note that we can define the field K ai as [16, 20] We can always define p ij = p ai e j a , then, From Eqs. (25) and (26) is straightforward to obtain an explicit expression for e a i e b j K [ i a K j ] b . Thus, on one hand we have So On the other hand we have So It follows that Therefore As an consequence we have Hence the cosmological function in the Ashteka's formalism can be written as This is a more compact relation between the proposed cosmological functions. But, note that from Eq. (12) we have that Then, we can write Thus we have a more compact relation Indeed, the above equation represents the most compact relation between the prosed cosmological functions that we found. Note that the cosmological functions in both formalism are the same up to the spacial Ricci scalar. We also found a way to relate this cosmological functions through the extrinsic curvature. To do this, we begin from the fact that the equation Eq. (30) can be written as Then, from Eq. (25) we have p ai = 2 K ai e + 1 2 pe ai , so where we have defined K := K ai e ai . Now an alternative definition to K i a is [21]: where K ab = K ( ab ) is the extrinsic curvature and J ab = J [ ab ] . Because of the ADM formalism extended to triads the annihilation of J ab is equivalent to that the internal constraint is satisfied [see Eq. (7)]. Hence, So It follows that Next, note that Hence Then So Thus and Therefore where ≈ denote modulo the internal constriction. But when we do an restriction to X phy. we obtain that In conclusion This represents the most compact relation between the proposed cosmological functions when we restrict our attention to the physical phase space X phy .",
"pages": [
3,
4,
5,
6
]
},
{
"title": "5. Concluding Remarks",
"content": "Firs of all we need to say that the relations given by Eqs. (32) and (41) do not have to be equal, we find Eq. (32) considering the whole phase space, i.e. Met (Σ), while one finds Eq. (41) just when we restrict our attention to the physical phase space X phy. . Indeed, what led us to Eq. (41) was that we were looking for a more geometrical form of relate the proposed cosmological functions. Although the ansatz Λ and ϕ , Eqs. (12) and (10) respectively, are obtained directly by one simple computation, the interpretation for these cosmological functions is not clear for us. We have to admit that at this moment is not clear why the cosmological function are related through the three-dimensional Ricci scalar. Perhaps a more geometric interpretation can be given from this relation in terms of extrinsic curvature, it is well know that the extrinsic curvature only makes sense when we have a manifold embedded on another of higher dimension and this is what we have in GR, see Eq. (39). We hope to have a more precise interpretation of this cosmological function and their relation in the coming years.",
"pages": [
7
]
}
]
2024arXiv240913967S
https://arxiv.org/pdf/2409.13967.pdf
<document>
<section_header_level_1><location><page_1><loc_11><loc_87><loc_84><loc_91></location>Weakly magnetized black holes in Einstein-ModMax theory</section_header_level_1>
<text><location><page_1><loc_26><loc_71><loc_68><loc_76></location>Haryanto M. Siahaan ∗ Universitas Katolik Parahyangan, Jalan Ciumbuleuit 94, Bandung 40141, Indonesia</text>
<section_header_level_1><location><page_1><loc_43><loc_68><loc_51><loc_69></location>Abstract</section_header_level_1>
<text><location><page_1><loc_14><loc_53><loc_81><loc_66></location>Theories of non-linear electrodynamics inherently describe deviations from Maxwell's theory in the strong field regime. Among these, ModMax electrodynamics stands out as a unique one-parameter generalization of Maxwell's theory that preserves both conformal invariance and electromagnetic duality. In this paper, we investigate the extension of Wald's magnetization within the framework of Einstein-ModMax theory, concentrating on static charged and accelerating black holes. Additionally, we examine the influence of external magnetic fields on the motion of charged test particles in the vicinity of a charged black hole.</text>
<section_header_level_1><location><page_2><loc_9><loc_89><loc_31><loc_91></location>1 Introduction</section_header_level_1>
<text><location><page_2><loc_9><loc_73><loc_85><loc_87></location>Black holes are an inevitable implication of Einstein's theory of relativity, predicting their existence long before they were finally observed about a century later [1]. In Einstein-Maxwell theory, black holes can possess electric charge in addition to mass and rotational parameters. One of the most renowned exact solutions describing a rotating and electrically charged black hole in this theory is the Kerr-Newman solution [2]. Beyond mass, rotation, and charge, exact solutions in Einstein-Maxwell theory can also include NUT and acceleration parameters. The Plebanski-Demianski solution [3] is one of the most comprehensive black hole solutions in Einstein-Maxwell theory, encompassing multiple parameters.</text>
<text><location><page_2><loc_9><loc_56><loc_85><loc_73></location>Astrophysical black holes are often surrounded by external magnetic fields. Recent astronomical observations have revealed strong, organized magnetic fields spiraling from the edge of the supermassive black hole Sagittarius A* [4]. Previous studies by the Event Horizon Telescope Collaboration have also detected strong external magnetic fields around black holes at the centers of galaxies, inferred through the observed light polarization [5, 6]. The direction of circular light polarization, whether clockwise or counterclockwise, as it travels, provides information about the magnetic field and the types of high-energy particles surrounding the black hole. These findings have renewed interest in applying models of magnetized black holes to study the motion of objects around them.</text>
<text><location><page_2><loc_9><loc_42><loc_85><loc_56></location>In Einstein-Maxwell theory, there are two common approaches to modeling the interaction between a black hole and an external magnetic field. The first approach treats the external magnetic field as a perturbation in spacetime, constructing the associated vector potentials based on the existing Killing vectors of the spacetime [7]. The second approach involves using the Ernst transformation on a seed solution [8], which can be any member of the KerrNewman-Taub-NUT family, to incorporate the influence of the magnetic field on spacetime curvature [9, 10]. This latter approach can be used to investigate how the external magnetic field modifies the null geodesics around a black hole.</text>
<text><location><page_2><loc_9><loc_22><loc_85><loc_42></location>In recent years, there has been growing interest in a particular type of non-linear electrodynamics known as modified Maxwell (ModMax) theory [11]. The appeal of studying nonlinear electrodynamics lies in its potential to address the issue of field singularities. Various aspects of ModMax electrodynamics and the corresponding black hole solutions have been extensively investigated, as seen in works such as [12, 13, 11, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. In ModMax theory, the non-linear parameter effectively 'screens' the charges through an exponential factor. Interestingly, the exact solution for a static charged black hole in EinsteinModMax theory closely resembles the well-known Reissner-Nordstrom solution [16]. Aspects of a dyonic Einstein-ModMax black hole, such as shadow, lensing, and quasinormal modes, have been investigated in [24], with the potential for these phenomena to be detected as astronomical observational equipment continues to improve.</text>
<text><location><page_2><loc_9><loc_14><loc_85><loc_21></location>With the concepts of magnetized black holes and Einstein-ModMax theory in mind, a natural question arises: what if we consider magnetization within the framework of EinsteinModMax theory, at least at the perturbative level as introduced by Wald [7]? This approach is simpler compared to the more complex task of constructing an exact magnetized black</text>
<text><location><page_3><loc_9><loc_72><loc_85><loc_90></location>hole solution in Einstein-ModMax theory, as done by Ernst in the Einstein-Maxwell setup [8]. If a magnetized black hole in Einstein-ModMax theory can be constructed following Wald's method, we can study the motions of charged timelike objects to understand how the external magnetic field and non-linear parameter induce deviations. This paper aims to do precisely that: construct a magnetized black hole in Einstein-ModMax theory and examine the resulting motions of charged timelike objects. To the best of our knowledge, this is the first study to apply Wald's magnetization in the context of Einstein gravity with non-linear electrodynamics. Note that, there exist a recent work reported in [25] where the authors constructed electromagnetized black holes and vortex-like backgrounds within the framework of the ModMax theory.</text>
<text><location><page_3><loc_9><loc_61><loc_85><loc_72></location>The organization of this paper is as follows. In the next section, we provide a brief review of Einstein-ModMax theory and the corresponding static charged black hole solution. Section 3 discusses the construction of a magnetized black hole in Einstein-ModMax theory. In section 4, we examine the motions of charged timelike objects in the magnetized charged black hole. Finally, we present our conclusions. In this paper, we consider the natural units where c = /planckover2pi1 = k B = G 4 = 1.</text>
<section_header_level_1><location><page_3><loc_9><loc_57><loc_81><loc_58></location>2 Charged black holes in Einstein-ModMax theory</section_header_level_1>
<text><location><page_3><loc_9><loc_53><loc_63><loc_55></location>The ModMax theory [11, 13] is described by Lagrangian density</text>
<formula><location><page_3><loc_30><loc_47><loc_85><loc_52></location>L MM = -1 2 ( s cosh v -√ s 2 + p 2 sinh v ) (2.1)</formula>
<text><location><page_3><loc_9><loc_46><loc_67><loc_48></location>where s and p are the invariants of the electromagnetic fields, namely</text>
<formula><location><page_3><loc_34><loc_42><loc_85><loc_45></location>s = 1 2 F µν F µν , p = 1 2 F µν ˜ F µν . (2.2)</formula>
<text><location><page_3><loc_9><loc_28><loc_85><loc_41></location>In the equations above, the field strength tensor is defined as F µν = ∂ µ A ν -∂ ν A µ and its dual as ˜ F µν = 1 2 ε µναβ F αβ where ε 0123 = √ -g . In differential form notation, we have ˜ F = /star F , where /star denotes the Hodge dual star operator. The parameter v denotes the non-linear parameter of the theory, with the standard Maxwell theory recovered when v = 0. It has been shown that the condition v ≥ 0 must be imposed to ensure causality [12]. Furthermore, since the ordinary Maxwell theory describes our nature extremely well, we can expect the parameter v to be extremely small.</text>
<text><location><page_3><loc_12><loc_26><loc_85><loc_28></location>Following [17, 19], we introduce the two-form for the 'material' field strength as follows</text>
<formula><location><page_3><loc_39><loc_21><loc_85><loc_25></location>E = 2 ( f s F + f p ˜ F ) (2.3)</formula>
<text><location><page_3><loc_9><loc_17><loc_85><loc_21></location>where F and ˜ F are the two-forms for the Maxwell field-strength tensor and its dual, respectively. The functions f s and f p are defined as</text>
<formula><location><page_3><loc_35><loc_13><loc_85><loc_16></location>f s = ∂ L MM ∂s , f p = ∂ L MM ∂p . (2.4)</formula>
<text><location><page_4><loc_9><loc_87><loc_85><loc_90></location>In terms of the functions above, the electromagnetic stress-energy tensor in ModMax theory can be written as</text>
<text><location><page_4><loc_9><loc_80><loc_85><loc_84></location>Furthermore, the electric charge inside a closed two dimensional spacelike surface Σ can be obtained from the integral</text>
<formula><location><page_4><loc_33><loc_82><loc_85><loc_87></location>T µν = 1 4 π ( s g µν -2 F µκ F νλ g κλ ) f s . (2.5)</formula>
<formula><location><page_4><loc_40><loc_76><loc_85><loc_80></location>Q e = 1 4 π ∫ Σ /star E . (2.6)</formula>
<text><location><page_4><loc_12><loc_75><loc_71><loc_76></location>Now let us consider an action for Einstein-ModMax theory [16, 17, 19]</text>
<formula><location><page_4><loc_33><loc_69><loc_85><loc_74></location>S = 1 16 π ∫ d 4 x √ -g ( R -4 L MM ) . (2.7)</formula>
<formula><location><page_4><loc_38><loc_63><loc_85><loc_67></location>R µν -1 2 g µν = 8 πT µν , (2.8)</formula>
<text><location><page_4><loc_9><loc_66><loc_85><loc_70></location>From the action above, the corresponding equations of motion in Einstein-ModMax theory are</text>
<text><location><page_4><loc_9><loc_61><loc_12><loc_63></location>and</text>
<formula><location><page_4><loc_42><loc_59><loc_85><loc_61></location>∇ µ E µν = 0 . (2.9)</formula>
<text><location><page_4><loc_9><loc_50><loc_85><loc_59></location>Note that the last equation represents a generalization of the source-free condition ∇ µ F µν = 0 found in ordinary Einstein-Maxwell theory. Despite the complexity of the corresponding equations of motion, it turns out that one of the simplest static black hole solutions describing a collapsed charged mass closely resembles the well-known Reissner-Nordstrom solution. The metric reads [16]</text>
<formula><location><page_4><loc_12><loc_42><loc_85><loc_49></location>ds 2 = -( 1 -2 M r + e -v Q 2 r 2 ) dt 2 + dr 2 ( 1 -2 M r + e -v Q 2 r 2 ) + r 2 ( dθ 2 +sin 2 θdφ 2 ) , (2.10)</formula>
<text><location><page_4><loc_9><loc_42><loc_54><loc_43></location>whereas the corresponding vector potential is given by</text>
<formula><location><page_4><loc_39><loc_37><loc_85><loc_41></location>A µ dx µ = -e -v Q r dt . (2.11)</formula>
<text><location><page_4><loc_9><loc_34><loc_85><loc_37></location>In equations above, M and Q represent the black hole mass and electric charge, respectively, whereas v denotes the non-linear parameter of the ModMax theory.</text>
<text><location><page_4><loc_9><loc_28><loc_85><loc_33></location>For the solution described by equations (2.10) and (2.11), the non-vanishing components of the field strength tensor and its dual are given by F = -e -v Q r 2 dt ∧ dr and ˜ F = e -v Q sin θdt ∧ dr , respectively. Accordingly, by using eq. (2.2), we can find</text>
<formula><location><page_4><loc_41><loc_23><loc_85><loc_27></location>s = -e -2 v Q 2 2 r 4 , (2.12)</formula>
<text><location><page_4><loc_9><loc_21><loc_77><loc_23></location>and p = 0. Furthermore, the material tensor (2.3) for this solution can be read as</text>
<formula><location><page_4><loc_41><loc_17><loc_85><loc_21></location>E = Q r 2 dt ∧ dr , (2.13)</formula>
<text><location><page_4><loc_9><loc_13><loc_85><loc_16></location>and /star E = Q sin θdθ ∧ dφ . Therefore, the electric charge of the black hole can be computed using equation (2.6), yielding Q e = Q .</text>
<section_header_level_1><location><page_5><loc_9><loc_89><loc_84><loc_91></location>3 Weakly magnetized black holes in ModMax theory</section_header_level_1>
<text><location><page_5><loc_9><loc_78><loc_85><loc_87></location>In [7], magnetized black holes are studied by introducing external perturbative magnetic fields that are proportional to the Killing vectors of the spacetime. The Killing vectors ζ µ satisfy the Killing equations ∇ ν ζ µ + ∇ µ ζ ν = 0. Consequently, one can construct a field-strength tensor based on these Killing vectors, given by F µν = ∇ µ ζ ν - ∇ ν ζ µ . This construction ensures that the source-free condition</text>
<formula><location><page_5><loc_42><loc_74><loc_85><loc_77></location>∇ µ F µν = 0 (3.1)</formula>
<text><location><page_5><loc_9><loc_64><loc_85><loc_73></location>is satisfied. For the Reissner-Nordstrom-ModMax solution reviewed in the previous section, there are two Killing vectors associated with the stationary and axial symmetries of the spacetime, i.e. ζ µ ( t ) = [1 , 0 , 0 , 0] and ζ µ ( φ ) = [0 , 0 , 0 , 1], respectively. For each Killing vector, we can associate a field strength tensor, namely F ( t ) ∝ d ζ ( t ) and F ( φ ) ∝ d ζ ( φ ) , along with their corresponding dual tensors.</text>
<text><location><page_5><loc_9><loc_44><loc_85><loc_64></location>Let us now apply Wald's prescription to magnetize a black hole in Einstein-ModMax theory, and consider the generalized approach introduced by [26]. The process begins similarly by introducing vector potentials based on the Killing vectors of the spacetime. However, in contrast to the Einstein-Maxwell theory, where vector potentials must satisfy the source-free condition given by equation (3.1), in Einstein-ModMax theory, the vector potentials must fulfill the generalized source-free condition as specified in equation (2.9). Consequently, the vector potential for a weakly magnetized black hole in Einstein-ModMax theory is governed by more stringent equations. For instance, while in ordinary Einstein-Maxwell theory, one can use the vector potential A µ = [0 , 0 , 0 , 1] to construct the external magnetic field in describing the magnetized Kerr black hole [7], this approach is not applicable in the Einstein-ModMax theory for the Kerr solution due to the requirements of equation (2.9).</text>
<text><location><page_5><loc_9><loc_35><loc_85><loc_44></location>Continuing from the previous discussion, for the magnetized black hole in EinsteinModMax theory, rather than directly using a constant test field proportional to the Killing vector, we adopt a more general approach based on the ansatz for vector potentials introduced by Azreg-A¨ınou in [26]. This approach allows us to explore a broader class of vector potentials,</text>
<formula><location><page_5><loc_38><loc_33><loc_85><loc_35></location>A µ ( t ) = [ C t ( r ) , 0 , 0 , 0] , (3.2)</formula>
<formula><location><page_5><loc_38><loc_29><loc_85><loc_31></location>A µ ( φ ) = [0 , 0 , 0 , C φ ( r )] . (3.3)</formula>
<text><location><page_5><loc_9><loc_20><loc_85><loc_28></location>Thus, the test field corresponding to ζ µ ( t ) can be computed as F ( t ) = d A ( t ) , while the test field associated with ζ µ ( φ ) is given by F ( φ ) = d A ( φ ) . Furthermore, the material tensor can be defined associated to each Killing vector as well, i.e. E ( t ) and E ( φ ) . For the test field constructed from ζ ( t ) , the condition d /star E ( t ) = 0 results in an equation C t ( r ), namely</text>
<formula><location><page_5><loc_18><loc_13><loc_85><loc_19></location>r 2 e v ( r 2 -2 Mr + e -v Q 2 ) d 2 C t dr 2 -2 re v ( r 2 -e -v Q 2 ) dC t dr -2 Q 2 C t = 0 . (3.4)</formula>
<text><location><page_5><loc_9><loc_31><loc_12><loc_32></location>and</text>
<text><location><page_6><loc_9><loc_87><loc_85><loc_90></location>A general solution to this equation can be derived, resembling the solution for a weakly magnetized Reissner-Nordstrom black hole discussed in [26]. Specifically, it takes the form</text>
<formula><location><page_6><loc_41><loc_82><loc_85><loc_86></location>C t = C 1 rg tt + C 2 g tt (3.5)</formula>
<text><location><page_6><loc_9><loc_76><loc_85><loc_80></location>for the metric function as appeared in eq. (2.10). Finally, motivated by the vector solution in the non-magnetized case (2.11), we can take C 2 = 0 and C 1 = -e -v Q .</text>
<text><location><page_6><loc_9><loc_73><loc_85><loc_77></location>On the other hand, for the test field associated to the axial Killing vector, equation d /star E ( φ ) gives</text>
<formula><location><page_6><loc_11><loc_66><loc_85><loc_72></location>-r 2 e v ( r 2 -2 Mr + e -v Q 2 ) d 2 C t dr 2 -2 re v ( 2 r 2 + e -v Q 2 -3 Mr ) dC φ dr +2 Q 2 C φ = 0 , (3.6)</formula>
<text><location><page_6><loc_9><loc_65><loc_45><loc_67></location>whose general solution can be expressed as</text>
<formula><location><page_6><loc_12><loc_58><loc_85><loc_64></location>C φ = C 3 r 2 ( e v r 2 -Q 2 ) + C 4 r 2 ( ( Q 2 -e v r 2 ) tanh -1 ( ( M -r ) Z e v M 2 -Q 2 ) -( M + r ) Z ) (3.7)</formula>
<text><location><page_6><loc_9><loc_46><loc_85><loc_59></location>with Z = √ e 2 v M 2 -e v Q 2 . Indeed, taking the limit v → 0 reduces the last equation to the axial component of the vector potential for a magnetized Reissner-Nordstrom black hole, as discussed in [26]. To prevent singularities in the F ( φ ) component at the horizon, we set C 4 = 0. Additionally, to ensure a constant magnetic field as r → ∞ , similar to the magnetized Reissner-Nordstrom black hole, we set C 3 = e -v B/ 2. In summary, the vector components for the test field associated with the Killing vectors of a magnetized, static, electrically charged black hole in ModMax theory are given by</text>
<formula><location><page_6><loc_37><loc_40><loc_85><loc_44></location>A µ ( t ) = [ -e -v Q r , 0 , 0 , 0 ] , (3.8)</formula>
<text><location><page_6><loc_9><loc_38><loc_12><loc_39></location>and</text>
<formula><location><page_6><loc_33><loc_33><loc_85><loc_38></location>A µ ( φ ) = [ 0 , 0 , 0 , B 2 ( 1 -e -v Q 2 r 2 )] , (3.9)</formula>
<text><location><page_6><loc_9><loc_23><loc_85><loc_33></location>which reduce to the vector field in the magnetized Reissner-Nordstrom after setting v = 0. Note that for an electrically neutral black hole, the external magnetic field vector potential is identical to that in ordinary Einstein-Maxwell theory [7]. In other words, for the magnetized Schwarzschild black hole solution in Einstein-ModMax theory, it mirrors the solution found in the standard Einstein-Maxwell theory. This implies that the non-linearity of the theory becomes relevant only when an electric charge is present.</text>
<text><location><page_6><loc_9><loc_15><loc_85><loc_22></location>Let us examine the applicability of Wald's magnetization in the context of accelerating black holes within Einstein-ModMax theory. As detailed in [19], solutions for accelerating black holes in Einstein-ModMax theory have been explored. Note that Wald's approach to magnetization for accelerating black holes appears to be absent from the existing literature.</text>
<text><location><page_7><loc_9><loc_87><loc_85><loc_90></location>Here let us just consider to magnetize a neutral accelerating black hole where the line element can be expressed as [27, 28]</text>
<formula><location><page_7><loc_26><loc_81><loc_85><loc_86></location>ds 2 = 1 Ω 2 [ -Gdt 2 + dr 2 G + r 2 dθ 2 H + r 2 H sin 2 θdφ 2 ] , (3.10)</formula>
<text><location><page_7><loc_9><loc_63><loc_85><loc_81></location>where Ω = 1 + αr cos θ , G = ( r 2 -2 Mr ) (1 -α 2 r 2 ), and H = 1 + 2 Mα cos θ . In these equations, M and α represent the mass and acceleration parameters, respectively. The external magnetic field potentials can be constructed as constants multiplied by the corresponding Killing vectors ζ ( t ) and ζ ( φ ), with the generalized source-free condition (2.9) remaining satisfied. This setup represents an accelerating black hole immersed in an external magnetic field according to the Einstein-ModMax theory. However, if one introduces an electric charge to the accelerating black hole as presented in [19], the generalized source-free condition (2.9) no longer holds. This situation mirrors the case in ordinary Einstein-Maxwell theory, where the vector potentials A µ ( t ) = [1 , 0 , 0 , 0] or A µ ( φ ) = [0 , 0 , 0 , 1] also fail to satisfy the source-free condition (3.1).</text>
<text><location><page_7><loc_9><loc_59><loc_85><loc_63></location>Furthermore, we may consider magnetizing the Taub-NUT or Kerr black hole in EinsteinModMax theory. To achieve this, we can employ the general vector potential</text>
<formula><location><page_7><loc_39><loc_56><loc_85><loc_58></location>A µ = C 1 ζ µ ( t ) + C 2 ζ µ ( φ ) (3.11)</formula>
<text><location><page_7><loc_9><loc_42><loc_85><loc_55></location>where C 1 and C 2 are constants. Nevertheless, despite originating from relatively simple Einstein vacuum solutions, the generalized source-free condition (2.9) remains unsatisfied even for the vector potential given by (3.11). In contrast, for the magnetized Kerr solution in Einstein-Maxwell theory, the vector potential (3.11) satisfies the source-free condition (3.1). Moreover, when considering Kerr or Taub-NUT spacetimes, employing the vector potentials (3.2) and (3.3) that could potentially satisfy (2.9) leads to a highly complex differential equation for the vector function.</text>
<section_header_level_1><location><page_7><loc_9><loc_35><loc_85><loc_39></location>4 Motions of a test charged timelike object under the influence of external magnetic fields</section_header_level_1>
<text><location><page_7><loc_9><loc_22><loc_85><loc_33></location>To explore the astronomical implications of our previous studies, we now examine a charged timelike object as a perturbation in the spacetime of a charged, static ModMax black hole immersed in an external magnetic field described by the line element (2.10). To satisfy the generalized source-free condition (2.9), one must use either (3.8) or (3.9), but not both simultaneously. For our purposes, we will focus on the external electromagnetic potential given by</text>
<text><location><page_7><loc_9><loc_17><loc_69><loc_18></location>The Lagrangian for a charged timelike object with unit mass is given by</text>
<formula><location><page_7><loc_32><loc_17><loc_85><loc_22></location>A µ dx µ = B 2 sin 2 θ ( r 2 -e -v Q 2 ) dφ. (4.1)</formula>
<formula><location><page_7><loc_37><loc_12><loc_85><loc_16></location>L = 1 2 g µν ˙ x µ ˙ x ν + qA µ ˙ x µ , (4.2)</formula>
<text><location><page_8><loc_9><loc_83><loc_85><loc_90></location>where q represents the charge per unit mass of the test object. Here, the notation 'dot' denotes differentiation with respect to the affine parameter σ . The equatorial plane is defined by θ = 0, and the presence of equatorial motions can be analyzed as follows. On the equator, we have</text>
<formula><location><page_8><loc_40><loc_80><loc_85><loc_84></location>∂ L ∂θ -d dσ ∂ L ∂ ˙ θ = 0 . (4.3)</formula>
<text><location><page_8><loc_9><loc_78><loc_76><loc_79></location>Moreover, it can be verified for the metric (2.10) and vector potential (4.1) that</text>
<formula><location><page_8><loc_31><loc_74><loc_85><loc_76></location>∂ θ g αβ | θ = π/ 2 = 0 , ∂ x A α | θ = π/ 2 = π/ 2 . (4.4)</formula>
<text><location><page_8><loc_9><loc_71><loc_40><loc_73></location>It yields the equation (4.3) reduces to</text>
<formula><location><page_8><loc_42><loc_66><loc_85><loc_70></location>d dσ ∂ L ∂ ˙ θ = 0 . (4.5)</formula>
<text><location><page_8><loc_9><loc_58><loc_85><loc_65></location>It is evident that the final equation is met, as we are examining an object with no momentum and acceleration in the θ direction, i.e. ˙ θ = 0 and ¨ θ = 0 at the equator. Consequently, we can infer the possibility of equatorial motion in the magnetized spacetime under consideration, with further deliberation on the circular motion forthcoming.</text>
<text><location><page_8><loc_9><loc_52><loc_85><loc_57></location>In this magnetized spacetime, which has both stationary and axial symmetry, a test object has two conserved quantities, namely the energy and angular momentum. The energy E is given by</text>
<formula><location><page_8><loc_39><loc_49><loc_85><loc_52></location>E = -∂ L ∂ ˙ t = -g tt ˙ t , (4.6)</formula>
<text><location><page_8><loc_9><loc_43><loc_85><loc_48></location>where ˙ t and ˙ φ represent the derivatives of the object's coordinate time and azimuthal angle, respectively, and A t is the time component of the electromagnetic potential. The angular momentum L is given by</text>
<formula><location><page_8><loc_37><loc_39><loc_85><loc_43></location>L = ∂ L ∂ ˙ φ = g φφ ˙ φ -qA φ . (4.7)</formula>
<text><location><page_8><loc_9><loc_31><loc_85><loc_38></location>Equatorial motions demand ˙ θ = 0, thence we can obtain a general equation from the metric, given by g tt ˙ t 2 + g φφ ˙ φ 2 + 1 = -g rr ˙ r 2 . To study the circular motions, we can introduce an effective potential V eff = -˙ r 2 . By plugging in the conserved quantities in eqs. (4.6) and (4.7), we obtain the expression for the effective potential as</text>
<formula><location><page_8><loc_24><loc_23><loc_85><loc_29></location>V eff = 1 r 2 { ∆ r r 2 [ L + qB 2 ( r 2 -e -v Q 2 ) ] 2 +∆ r -r 2 E 2 } (4.8)</formula>
<text><location><page_8><loc_9><loc_14><loc_85><loc_23></location>where ∆ r = r 2 -2 Mr + e -v Q 2 . In equations above, E and L denote the energy and angular momentum, respectively, of the test object. It is important to note that, in the effective potential, the magnetic field parameter affects only charged particles. This contrasts with cases of strongly magnetized spacetimes, as discussed in [8, 9, 10], where even the geodesics of null objects are influenced by the presence of an external magnetic field.</text>
<text><location><page_9><loc_9><loc_78><loc_85><loc_90></location>In the presence of a magnetic field in spacetime, the effective potential increases without bound as the radius becomes larger. This behavior aligns with the scenario of a magnetized spacetime, where the magnetic field extends even to asymptotic regions. Additionally, since the magnetic field couples to the particle's charge in the effective action, the singular values of the effective potential do not apply to neutral timelike objects. To determine the innermost stable circular orbits (ISCO) on the equatorial plane for a timelike object, we must solve the following system of equations simultaneously to find the ISCO radius</text>
<formula><location><page_9><loc_32><loc_73><loc_85><loc_77></location>V eff = 0 , dV eff dr = 0 , d 2 V eff dr 2 = 0 . (4.9)</formula>
<text><location><page_9><loc_9><loc_54><loc_85><loc_72></location>By solving the three equations simultaneously, we determine the parameters E , L , and r which represent the energy, angular momentum, and ISCO radius of a charged timelike object in circular motion. Notably, even though the magnetized black hole under consideration is static, the angular momentum L can take both positive and negative real values. This phenomenon was also observed in [29], where positive angular momentum corresponds to an Anti-Larmor orbit and negative angular momentum represents a Larmor orbit. Larmor motion is associated with the Lorentz force directing the object towards the black hole, while Anti-Larmor motion is directed away. The numerical results presented in Figs. 4.1 and 4.2 illustrate these characteristics, showing that the ISCO radii for Larmor orbits tend to be smaller compared to those for Anti-Larmor orbits as the external magnetic field increases.</text>
<figure>
<location><page_9><loc_32><loc_31><loc_63><loc_52></location>
<caption>Figure 4.1: The ISCO radius of a charged timelike object in the presence of an external magnetic field is shown here. We consider parameters Q = 0 . 2 M and q = 0 . 1. The black curves indicate Anti-Larmor orbits, while the blue curves represent Larmor orbits. The cases with v = 0, v = 0 . 1, and v = 0 . 2 are depicted by solid, dashed, and dash-dotted curves, respectively.</caption>
</figure>
<text><location><page_9><loc_9><loc_13><loc_85><loc_18></location>The impact of the non-linear parameter v on circular motions is evident from the numerical examples for ISCO radii in Fig. 4.1 and the effective potential in Fig. 4.3. It is observed that larger values of v correspond to a lower peak of the local maxima in the effective po-</text>
<figure>
<location><page_10><loc_32><loc_68><loc_63><loc_90></location>
<caption>Figure 4.2: The absolute values of angular momentum for Larmor and Anti-Larmor motions for the test object illustrated in the plots in Fig. 4.1. Each object corresponds to a specific curve type and color.</caption>
</figure>
<figure>
<location><page_10><loc_32><loc_35><loc_63><loc_59></location>
<caption>Figure 4.3: Effective potential for a charged test object with a charge-to-mass ratio q = 0 . 1, energy E = 0 . 9, black hole charge Q = 0 . 5 M , and magnetic field strength BM = 0 . 1 in Einstein-ModMax theory, for various non-linear parameters v . The black, blue, and red curves correspond to v = 0 . 1, v = 0 . 2, and v = 0 . 3, respectively.</caption>
</figure>
<text><location><page_10><loc_9><loc_20><loc_85><loc_23></location>tential. This indicates a screening effect due to the v parameter, meaning that the effective charge of the black hole decreases as v increases.</text>
<section_header_level_1><location><page_11><loc_9><loc_89><loc_29><loc_91></location>5 Conclusion</section_header_level_1>
<text><location><page_11><loc_9><loc_74><loc_85><loc_87></location>In this work, we have demonstrated how to magnetize a black hole within the framework of Einstein-ModMax theory. We follow Wald's prescription by constructing the external magnetic field vector potential using the Killing vectors of the spacetime. Due to the complexity of the generalized source-free condition in Einstein-ModMax theory, not all magnetized black hole constructions from Einstein-Maxwell theory can be replicated in the ModMax context. For instance, the magnetized Kerr black hole in ModMax theory cannot be obtained using the ζ ( φ ) Killing vector of the spacetime.</text>
<text><location><page_11><loc_9><loc_58><loc_85><loc_74></location>Nevertheless, we can achieve a magnetized static charged black hole in Einstein-ModMax theory even using the generalized prescription introduced in [26]. As expected, the non-linear parameter in the vector potential solutions continues to play a role in screening the black hole's charge. To differentiate the motions of test charged objects for various non-linear parameter examples, section 4 presents numerical results related to the circular motions of the objects and the corresponding effective potentials. It was found that the general properties of the motions of charged objects in the magnetized ModMax black hole spacetime are similar to those in the magnetized black hole in Einstein-Maxwell theory [30], with some discrepancies arising from the screening effect on the effective black hole charge.</text>
<text><location><page_11><loc_9><loc_40><loc_85><loc_58></location>For future work, we plan to study the generalized Wald magnetization of black holes in other non-linear Einstein-Maxwell theories, such as the Born-Infeld, Euler-Heisenberg, and Ayon-Beato solutions [31, 32, 33]. Additionally, during the preparation of this work, we discovered that equatorial circular motion for a timelike test object cannot exist in an accelerating spacetime. This is due to the presence of momentum and acceleration perpendicular to the equatorial plane for this motion, similar to the problem of circular motion in Taub-NUT spacetime as described in [34]. In light of this issue, we are interested in investigating whether an external magnetic field can enable stable circular motion in accelerating or Taub-NUT spacetime for a charged timelike object. We will address this problem in our future research.</text>
<section_header_level_1><location><page_11><loc_9><loc_35><loc_34><loc_37></location>Acknowledgement</section_header_level_1>
<text><location><page_11><loc_9><loc_32><loc_48><loc_33></location>This work is supported by Kemendikbudristek.</text>
<section_header_level_1><location><page_11><loc_9><loc_27><loc_24><loc_28></location>References</section_header_level_1>
<unordered_list>
<list_item><location><page_11><loc_10><loc_23><loc_80><loc_25></location>[1] K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 875 (2019), L1</list_item>
<list_item><location><page_11><loc_10><loc_18><loc_85><loc_22></location>[2] I. D. Novikov and V. P. Frolov, 'PHYSICS OF BLACK HOLES,' Kluwer Academic, (1989)</list_item>
<list_item><location><page_11><loc_10><loc_15><loc_69><loc_17></location>[3] J. F. Plebanski and M. Demianski, Annals Phys. 98 (1976), 98-127</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_12><loc_10><loc_85><loc_85><loc_90></location>[4] T. H. T. Collaboration, K. Akiyama, A. Alberdi, W. Alef, J. C. Algaba, R. Anantua, K. Asada, R. Azulay, U. Bach and A. K. Baczko, et al. Astrophys. J. Lett. 964 (2024) no.2, L25</list_item>
<list_item><location><page_12><loc_10><loc_82><loc_85><loc_84></location>[5] K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 910 (2021) no.1, L12</list_item>
<list_item><location><page_12><loc_10><loc_79><loc_85><loc_81></location>[6] K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 910 (2021) no.1, L13</list_item>
<list_item><location><page_12><loc_10><loc_76><loc_53><loc_78></location>[7] R. M. Wald, Phys. Rev. D 10 (1974), 1680-1685</list_item>
<list_item><location><page_12><loc_10><loc_73><loc_54><loc_74></location>[8] F. J. Ernst, J. Math. Phys. 17 (1976) no.1, 54-56</list_item>
<list_item><location><page_12><loc_10><loc_70><loc_54><loc_71></location>[9] H. M. Siahaan, Phys. Lett. B 820 (2021), 136568</list_item>
<list_item><location><page_12><loc_9><loc_67><loc_73><loc_68></location>[10] M. Ghezelbash and H. M. Siahaan, Eur. Phys. J. C 81 (2021) no.7, 621</list_item>
<list_item><location><page_12><loc_9><loc_64><loc_78><loc_65></location>[11] I. Bandos, K. Lechner, D. Sorokin and P. K. Townsend, JHEP 03 (2021), 022</list_item>
<list_item><location><page_12><loc_9><loc_61><loc_60><loc_62></location>[12] D. P. Sorokin, Fortsch. Phys. 70 (2022) no.7-8, 2200092</list_item>
<list_item><location><page_12><loc_9><loc_58><loc_55><loc_59></location>[13] B. P. Kosyakov, Phys. Lett. B 810 (2020), 135840</list_item>
<list_item><location><page_12><loc_9><loc_54><loc_78><loc_56></location>[14] I. Bandos, K. Lechner, D. Sorokin and P. K. Townsend, JHEP 10 (2021), 031</list_item>
<list_item><location><page_12><loc_9><loc_51><loc_53><loc_53></location>[15] S. I. Kruglov, Phys. Lett. B 822 (2021), 136633</list_item>
<list_item><location><page_12><loc_9><loc_46><loc_85><loc_50></location>[16] D. Flores-Alfonso, B. A. Gonz'alez-Morales, R. Linares and M. Maceda, Phys. Lett. B 812 (2021), 136011</list_item>
<list_item><location><page_12><loc_9><loc_43><loc_83><loc_45></location>[17] A. Ballon Bordo, D. Kubizˇn'ak and T. R. Perche, Phys. Lett. B 817 (2021), 136312</list_item>
<list_item><location><page_12><loc_9><loc_40><loc_82><loc_42></location>[18] D. Kubiznak, T. Tahamtan and O. Svitek, Phys. Rev. D 105 (2022) no.10, 104064</list_item>
<list_item><location><page_12><loc_9><loc_37><loc_85><loc_39></location>[19] J. Barrientos, A. Cisterna, D. Kubiznak and J. Oliva, Phys. Lett. B 834 (2022), 137447</list_item>
<list_item><location><page_12><loc_9><loc_34><loc_65><loc_36></location>[20] H. M. Siahaan, Int. J. Mod. Phys. D 32 (2023) no.15, 2350099</list_item>
<list_item><location><page_12><loc_9><loc_31><loc_65><loc_33></location>[21] H. M. Siahaan, Commun. Theor. Phys. 76 (2024) no.6, 065402</list_item>
<list_item><location><page_12><loc_9><loc_28><loc_79><loc_29></location>[22] B. Eslam Panah, B. Hazarika and P. Phukon, PTEP 2024 (2024) no.8, 083E02</list_item>
<list_item><location><page_12><loc_9><loc_25><loc_55><loc_26></location>[23] B. Eslam Panah, PTEP 2024 (2024) no.2, 023E01</list_item>
<list_item><location><page_12><loc_9><loc_20><loc_85><loc_24></location>[24] R. C. Pantig, L. Mastrototaro, G. Lambiase and A. Ovgun, Eur. Phys. J. C 82 (2022) no.12, 1155</list_item>
<list_item><location><page_12><loc_9><loc_17><loc_83><loc_18></location>[25] J. Barrientos, A. Cisterna, M. Hassaine and K. Pallikaris, [arXiv:2409.12336 [gr-qc]].</list_item>
<list_item><location><page_12><loc_9><loc_14><loc_57><loc_15></location>[26] M. Azreg-A¨ınou, Eur. Phys. J. C 76 (2016) no.7, 414</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_13><loc_9><loc_89><loc_66><loc_90></location>[27] K. Hong and E. Teo, Class. Quant. Grav. 20 (2003), 3269-3277</list_item>
<list_item><location><page_13><loc_9><loc_84><loc_85><loc_87></location>[28] J. B. Griffiths and J. Podolsky, 'Exact Space-Times in Einstein's General Relativity,' Cambridge University Press, 2009</list_item>
<list_item><location><page_13><loc_9><loc_81><loc_79><loc_82></location>[29] A. N. Aliev and N. Ozdemir, Mon. Not. Roy. Astron. Soc. 336 (2002), 241-248</list_item>
<list_item><location><page_13><loc_9><loc_76><loc_85><loc_79></location>[30] S. Shaymatov, B. Narzilloev, A. Abdujabbarov and C. Bambi, Phys. Rev. D 103 (2021) no.12, 124066</list_item>
<list_item><location><page_13><loc_9><loc_73><loc_72><loc_74></location>[31] R. G. Cai, D. W. Pang and A. Wang, Phys. Rev. D 70 (2004), 124034</list_item>
<list_item><location><page_13><loc_9><loc_70><loc_63><loc_71></location>[32] T. Damour and R. Ruffini, Phys. Rev. D 14 (1976), 332-334</list_item>
<list_item><location><page_13><loc_9><loc_65><loc_85><loc_68></location>[33] H. S. Ramadhan, M. F. Ishlah, F. P. Pratama and I. Alfredo, Eur. Phys. J. C 83 (2023) no.6, 465</list_item>
<list_item><location><page_13><loc_9><loc_62><loc_74><loc_63></location>[34] P. Jefremov and V. Perlick, Class. Quant. Grav. 33 (2016) no.24, 245014</list_item>
</unordered_list>
<figure>
<location><page_14><loc_5><loc_21><loc_93><loc_79></location>
</figure>
<figure>
<location><page_15><loc_5><loc_21><loc_93><loc_79></location>
</figure>
</document>
[
{
"title": "Weakly magnetized black holes in Einstein-ModMax theory",
"content": "Haryanto M. Siahaan ∗ Universitas Katolik Parahyangan, Jalan Ciumbuleuit 94, Bandung 40141, Indonesia",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "Theories of non-linear electrodynamics inherently describe deviations from Maxwell's theory in the strong field regime. Among these, ModMax electrodynamics stands out as a unique one-parameter generalization of Maxwell's theory that preserves both conformal invariance and electromagnetic duality. In this paper, we investigate the extension of Wald's magnetization within the framework of Einstein-ModMax theory, concentrating on static charged and accelerating black holes. Additionally, we examine the influence of external magnetic fields on the motion of charged test particles in the vicinity of a charged black hole.",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "Black holes are an inevitable implication of Einstein's theory of relativity, predicting their existence long before they were finally observed about a century later [1]. In Einstein-Maxwell theory, black holes can possess electric charge in addition to mass and rotational parameters. One of the most renowned exact solutions describing a rotating and electrically charged black hole in this theory is the Kerr-Newman solution [2]. Beyond mass, rotation, and charge, exact solutions in Einstein-Maxwell theory can also include NUT and acceleration parameters. The Plebanski-Demianski solution [3] is one of the most comprehensive black hole solutions in Einstein-Maxwell theory, encompassing multiple parameters. Astrophysical black holes are often surrounded by external magnetic fields. Recent astronomical observations have revealed strong, organized magnetic fields spiraling from the edge of the supermassive black hole Sagittarius A* [4]. Previous studies by the Event Horizon Telescope Collaboration have also detected strong external magnetic fields around black holes at the centers of galaxies, inferred through the observed light polarization [5, 6]. The direction of circular light polarization, whether clockwise or counterclockwise, as it travels, provides information about the magnetic field and the types of high-energy particles surrounding the black hole. These findings have renewed interest in applying models of magnetized black holes to study the motion of objects around them. In Einstein-Maxwell theory, there are two common approaches to modeling the interaction between a black hole and an external magnetic field. The first approach treats the external magnetic field as a perturbation in spacetime, constructing the associated vector potentials based on the existing Killing vectors of the spacetime [7]. The second approach involves using the Ernst transformation on a seed solution [8], which can be any member of the KerrNewman-Taub-NUT family, to incorporate the influence of the magnetic field on spacetime curvature [9, 10]. This latter approach can be used to investigate how the external magnetic field modifies the null geodesics around a black hole. In recent years, there has been growing interest in a particular type of non-linear electrodynamics known as modified Maxwell (ModMax) theory [11]. The appeal of studying nonlinear electrodynamics lies in its potential to address the issue of field singularities. Various aspects of ModMax electrodynamics and the corresponding black hole solutions have been extensively investigated, as seen in works such as [12, 13, 11, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. In ModMax theory, the non-linear parameter effectively 'screens' the charges through an exponential factor. Interestingly, the exact solution for a static charged black hole in EinsteinModMax theory closely resembles the well-known Reissner-Nordstrom solution [16]. Aspects of a dyonic Einstein-ModMax black hole, such as shadow, lensing, and quasinormal modes, have been investigated in [24], with the potential for these phenomena to be detected as astronomical observational equipment continues to improve. With the concepts of magnetized black holes and Einstein-ModMax theory in mind, a natural question arises: what if we consider magnetization within the framework of EinsteinModMax theory, at least at the perturbative level as introduced by Wald [7]? This approach is simpler compared to the more complex task of constructing an exact magnetized black hole solution in Einstein-ModMax theory, as done by Ernst in the Einstein-Maxwell setup [8]. If a magnetized black hole in Einstein-ModMax theory can be constructed following Wald's method, we can study the motions of charged timelike objects to understand how the external magnetic field and non-linear parameter induce deviations. This paper aims to do precisely that: construct a magnetized black hole in Einstein-ModMax theory and examine the resulting motions of charged timelike objects. To the best of our knowledge, this is the first study to apply Wald's magnetization in the context of Einstein gravity with non-linear electrodynamics. Note that, there exist a recent work reported in [25] where the authors constructed electromagnetized black holes and vortex-like backgrounds within the framework of the ModMax theory. The organization of this paper is as follows. In the next section, we provide a brief review of Einstein-ModMax theory and the corresponding static charged black hole solution. Section 3 discusses the construction of a magnetized black hole in Einstein-ModMax theory. In section 4, we examine the motions of charged timelike objects in the magnetized charged black hole. Finally, we present our conclusions. In this paper, we consider the natural units where c = /planckover2pi1 = k B = G 4 = 1.",
"pages": [
2,
3
]
},
{
"title": "2 Charged black holes in Einstein-ModMax theory",
"content": "The ModMax theory [11, 13] is described by Lagrangian density where s and p are the invariants of the electromagnetic fields, namely In the equations above, the field strength tensor is defined as F µν = ∂ µ A ν -∂ ν A µ and its dual as ˜ F µν = 1 2 ε µναβ F αβ where ε 0123 = √ -g . In differential form notation, we have ˜ F = /star F , where /star denotes the Hodge dual star operator. The parameter v denotes the non-linear parameter of the theory, with the standard Maxwell theory recovered when v = 0. It has been shown that the condition v ≥ 0 must be imposed to ensure causality [12]. Furthermore, since the ordinary Maxwell theory describes our nature extremely well, we can expect the parameter v to be extremely small. Following [17, 19], we introduce the two-form for the 'material' field strength as follows where F and ˜ F are the two-forms for the Maxwell field-strength tensor and its dual, respectively. The functions f s and f p are defined as In terms of the functions above, the electromagnetic stress-energy tensor in ModMax theory can be written as Furthermore, the electric charge inside a closed two dimensional spacelike surface Σ can be obtained from the integral Now let us consider an action for Einstein-ModMax theory [16, 17, 19] From the action above, the corresponding equations of motion in Einstein-ModMax theory are and Note that the last equation represents a generalization of the source-free condition ∇ µ F µν = 0 found in ordinary Einstein-Maxwell theory. Despite the complexity of the corresponding equations of motion, it turns out that one of the simplest static black hole solutions describing a collapsed charged mass closely resembles the well-known Reissner-Nordstrom solution. The metric reads [16] whereas the corresponding vector potential is given by In equations above, M and Q represent the black hole mass and electric charge, respectively, whereas v denotes the non-linear parameter of the ModMax theory. For the solution described by equations (2.10) and (2.11), the non-vanishing components of the field strength tensor and its dual are given by F = -e -v Q r 2 dt ∧ dr and ˜ F = e -v Q sin θdt ∧ dr , respectively. Accordingly, by using eq. (2.2), we can find and p = 0. Furthermore, the material tensor (2.3) for this solution can be read as and /star E = Q sin θdθ ∧ dφ . Therefore, the electric charge of the black hole can be computed using equation (2.6), yielding Q e = Q .",
"pages": [
3,
4
]
},
{
"title": "3 Weakly magnetized black holes in ModMax theory",
"content": "In [7], magnetized black holes are studied by introducing external perturbative magnetic fields that are proportional to the Killing vectors of the spacetime. The Killing vectors ζ µ satisfy the Killing equations ∇ ν ζ µ + ∇ µ ζ ν = 0. Consequently, one can construct a field-strength tensor based on these Killing vectors, given by F µν = ∇ µ ζ ν - ∇ ν ζ µ . This construction ensures that the source-free condition is satisfied. For the Reissner-Nordstrom-ModMax solution reviewed in the previous section, there are two Killing vectors associated with the stationary and axial symmetries of the spacetime, i.e. ζ µ ( t ) = [1 , 0 , 0 , 0] and ζ µ ( φ ) = [0 , 0 , 0 , 1], respectively. For each Killing vector, we can associate a field strength tensor, namely F ( t ) ∝ d ζ ( t ) and F ( φ ) ∝ d ζ ( φ ) , along with their corresponding dual tensors. Let us now apply Wald's prescription to magnetize a black hole in Einstein-ModMax theory, and consider the generalized approach introduced by [26]. The process begins similarly by introducing vector potentials based on the Killing vectors of the spacetime. However, in contrast to the Einstein-Maxwell theory, where vector potentials must satisfy the source-free condition given by equation (3.1), in Einstein-ModMax theory, the vector potentials must fulfill the generalized source-free condition as specified in equation (2.9). Consequently, the vector potential for a weakly magnetized black hole in Einstein-ModMax theory is governed by more stringent equations. For instance, while in ordinary Einstein-Maxwell theory, one can use the vector potential A µ = [0 , 0 , 0 , 1] to construct the external magnetic field in describing the magnetized Kerr black hole [7], this approach is not applicable in the Einstein-ModMax theory for the Kerr solution due to the requirements of equation (2.9). Continuing from the previous discussion, for the magnetized black hole in EinsteinModMax theory, rather than directly using a constant test field proportional to the Killing vector, we adopt a more general approach based on the ansatz for vector potentials introduced by Azreg-A¨ınou in [26]. This approach allows us to explore a broader class of vector potentials, Thus, the test field corresponding to ζ µ ( t ) can be computed as F ( t ) = d A ( t ) , while the test field associated with ζ µ ( φ ) is given by F ( φ ) = d A ( φ ) . Furthermore, the material tensor can be defined associated to each Killing vector as well, i.e. E ( t ) and E ( φ ) . For the test field constructed from ζ ( t ) , the condition d /star E ( t ) = 0 results in an equation C t ( r ), namely and A general solution to this equation can be derived, resembling the solution for a weakly magnetized Reissner-Nordstrom black hole discussed in [26]. Specifically, it takes the form for the metric function as appeared in eq. (2.10). Finally, motivated by the vector solution in the non-magnetized case (2.11), we can take C 2 = 0 and C 1 = -e -v Q . On the other hand, for the test field associated to the axial Killing vector, equation d /star E ( φ ) gives whose general solution can be expressed as with Z = √ e 2 v M 2 -e v Q 2 . Indeed, taking the limit v → 0 reduces the last equation to the axial component of the vector potential for a magnetized Reissner-Nordstrom black hole, as discussed in [26]. To prevent singularities in the F ( φ ) component at the horizon, we set C 4 = 0. Additionally, to ensure a constant magnetic field as r → ∞ , similar to the magnetized Reissner-Nordstrom black hole, we set C 3 = e -v B/ 2. In summary, the vector components for the test field associated with the Killing vectors of a magnetized, static, electrically charged black hole in ModMax theory are given by and which reduce to the vector field in the magnetized Reissner-Nordstrom after setting v = 0. Note that for an electrically neutral black hole, the external magnetic field vector potential is identical to that in ordinary Einstein-Maxwell theory [7]. In other words, for the magnetized Schwarzschild black hole solution in Einstein-ModMax theory, it mirrors the solution found in the standard Einstein-Maxwell theory. This implies that the non-linearity of the theory becomes relevant only when an electric charge is present. Let us examine the applicability of Wald's magnetization in the context of accelerating black holes within Einstein-ModMax theory. As detailed in [19], solutions for accelerating black holes in Einstein-ModMax theory have been explored. Note that Wald's approach to magnetization for accelerating black holes appears to be absent from the existing literature. Here let us just consider to magnetize a neutral accelerating black hole where the line element can be expressed as [27, 28] where Ω = 1 + αr cos θ , G = ( r 2 -2 Mr ) (1 -α 2 r 2 ), and H = 1 + 2 Mα cos θ . In these equations, M and α represent the mass and acceleration parameters, respectively. The external magnetic field potentials can be constructed as constants multiplied by the corresponding Killing vectors ζ ( t ) and ζ ( φ ), with the generalized source-free condition (2.9) remaining satisfied. This setup represents an accelerating black hole immersed in an external magnetic field according to the Einstein-ModMax theory. However, if one introduces an electric charge to the accelerating black hole as presented in [19], the generalized source-free condition (2.9) no longer holds. This situation mirrors the case in ordinary Einstein-Maxwell theory, where the vector potentials A µ ( t ) = [1 , 0 , 0 , 0] or A µ ( φ ) = [0 , 0 , 0 , 1] also fail to satisfy the source-free condition (3.1). Furthermore, we may consider magnetizing the Taub-NUT or Kerr black hole in EinsteinModMax theory. To achieve this, we can employ the general vector potential where C 1 and C 2 are constants. Nevertheless, despite originating from relatively simple Einstein vacuum solutions, the generalized source-free condition (2.9) remains unsatisfied even for the vector potential given by (3.11). In contrast, for the magnetized Kerr solution in Einstein-Maxwell theory, the vector potential (3.11) satisfies the source-free condition (3.1). Moreover, when considering Kerr or Taub-NUT spacetimes, employing the vector potentials (3.2) and (3.3) that could potentially satisfy (2.9) leads to a highly complex differential equation for the vector function.",
"pages": [
5,
6,
7
]
},
{
"title": "4 Motions of a test charged timelike object under the influence of external magnetic fields",
"content": "To explore the astronomical implications of our previous studies, we now examine a charged timelike object as a perturbation in the spacetime of a charged, static ModMax black hole immersed in an external magnetic field described by the line element (2.10). To satisfy the generalized source-free condition (2.9), one must use either (3.8) or (3.9), but not both simultaneously. For our purposes, we will focus on the external electromagnetic potential given by The Lagrangian for a charged timelike object with unit mass is given by where q represents the charge per unit mass of the test object. Here, the notation 'dot' denotes differentiation with respect to the affine parameter σ . The equatorial plane is defined by θ = 0, and the presence of equatorial motions can be analyzed as follows. On the equator, we have Moreover, it can be verified for the metric (2.10) and vector potential (4.1) that It yields the equation (4.3) reduces to It is evident that the final equation is met, as we are examining an object with no momentum and acceleration in the θ direction, i.e. ˙ θ = 0 and ¨ θ = 0 at the equator. Consequently, we can infer the possibility of equatorial motion in the magnetized spacetime under consideration, with further deliberation on the circular motion forthcoming. In this magnetized spacetime, which has both stationary and axial symmetry, a test object has two conserved quantities, namely the energy and angular momentum. The energy E is given by where ˙ t and ˙ φ represent the derivatives of the object's coordinate time and azimuthal angle, respectively, and A t is the time component of the electromagnetic potential. The angular momentum L is given by Equatorial motions demand ˙ θ = 0, thence we can obtain a general equation from the metric, given by g tt ˙ t 2 + g φφ ˙ φ 2 + 1 = -g rr ˙ r 2 . To study the circular motions, we can introduce an effective potential V eff = -˙ r 2 . By plugging in the conserved quantities in eqs. (4.6) and (4.7), we obtain the expression for the effective potential as where ∆ r = r 2 -2 Mr + e -v Q 2 . In equations above, E and L denote the energy and angular momentum, respectively, of the test object. It is important to note that, in the effective potential, the magnetic field parameter affects only charged particles. This contrasts with cases of strongly magnetized spacetimes, as discussed in [8, 9, 10], where even the geodesics of null objects are influenced by the presence of an external magnetic field. In the presence of a magnetic field in spacetime, the effective potential increases without bound as the radius becomes larger. This behavior aligns with the scenario of a magnetized spacetime, where the magnetic field extends even to asymptotic regions. Additionally, since the magnetic field couples to the particle's charge in the effective action, the singular values of the effective potential do not apply to neutral timelike objects. To determine the innermost stable circular orbits (ISCO) on the equatorial plane for a timelike object, we must solve the following system of equations simultaneously to find the ISCO radius By solving the three equations simultaneously, we determine the parameters E , L , and r which represent the energy, angular momentum, and ISCO radius of a charged timelike object in circular motion. Notably, even though the magnetized black hole under consideration is static, the angular momentum L can take both positive and negative real values. This phenomenon was also observed in [29], where positive angular momentum corresponds to an Anti-Larmor orbit and negative angular momentum represents a Larmor orbit. Larmor motion is associated with the Lorentz force directing the object towards the black hole, while Anti-Larmor motion is directed away. The numerical results presented in Figs. 4.1 and 4.2 illustrate these characteristics, showing that the ISCO radii for Larmor orbits tend to be smaller compared to those for Anti-Larmor orbits as the external magnetic field increases. The impact of the non-linear parameter v on circular motions is evident from the numerical examples for ISCO radii in Fig. 4.1 and the effective potential in Fig. 4.3. It is observed that larger values of v correspond to a lower peak of the local maxima in the effective po- tential. This indicates a screening effect due to the v parameter, meaning that the effective charge of the black hole decreases as v increases.",
"pages": [
7,
8,
9,
10
]
},
{
"title": "5 Conclusion",
"content": "In this work, we have demonstrated how to magnetize a black hole within the framework of Einstein-ModMax theory. We follow Wald's prescription by constructing the external magnetic field vector potential using the Killing vectors of the spacetime. Due to the complexity of the generalized source-free condition in Einstein-ModMax theory, not all magnetized black hole constructions from Einstein-Maxwell theory can be replicated in the ModMax context. For instance, the magnetized Kerr black hole in ModMax theory cannot be obtained using the ζ ( φ ) Killing vector of the spacetime. Nevertheless, we can achieve a magnetized static charged black hole in Einstein-ModMax theory even using the generalized prescription introduced in [26]. As expected, the non-linear parameter in the vector potential solutions continues to play a role in screening the black hole's charge. To differentiate the motions of test charged objects for various non-linear parameter examples, section 4 presents numerical results related to the circular motions of the objects and the corresponding effective potentials. It was found that the general properties of the motions of charged objects in the magnetized ModMax black hole spacetime are similar to those in the magnetized black hole in Einstein-Maxwell theory [30], with some discrepancies arising from the screening effect on the effective black hole charge. For future work, we plan to study the generalized Wald magnetization of black holes in other non-linear Einstein-Maxwell theories, such as the Born-Infeld, Euler-Heisenberg, and Ayon-Beato solutions [31, 32, 33]. Additionally, during the preparation of this work, we discovered that equatorial circular motion for a timelike test object cannot exist in an accelerating spacetime. This is due to the presence of momentum and acceleration perpendicular to the equatorial plane for this motion, similar to the problem of circular motion in Taub-NUT spacetime as described in [34]. In light of this issue, we are interested in investigating whether an external magnetic field can enable stable circular motion in accelerating or Taub-NUT spacetime for a charged timelike object. We will address this problem in our future research.",
"pages": [
11
]
},
{
"title": "Acknowledgement",
"content": "This work is supported by Kemendikbudristek.",
"pages": [
11
]
}
]
2024arXiv241000099J
https://arxiv.org/pdf/2410.00099.pdf
<document>
<section_header_level_1><location><page_1><loc_12><loc_82><loc_88><loc_85></location>Reconstructing the Assembly of Massive Galaxies. III: Quiescent Galaxies Loose Angular Momentum as They Evolve in a Mass-dependent Fashion.</section_header_level_1>
<text><location><page_1><loc_37><loc_80><loc_63><loc_81></location>Zhiyuan Ji 1 and Mauro Giavalisco 2</text>
<text><location><page_1><loc_19><loc_76><loc_80><loc_79></location>1 Steward Observatory, University of Arizona, 933 N. Cherry Avenue, Tucson, AZ 85721, USA 2 University of Massachusetts Amherst, 710 North Pleasant Street, Amherst, MA 01003-9305, USA</text>
<section_header_level_1><location><page_1><loc_45><loc_73><loc_55><loc_74></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_44><loc_86><loc_72></location>We study the evolution of stellar kinematics of a sample of 952 massive quiescent galaxies with M ∗ > 10 10 . 5 M ⊙ at 0 . 6 < z < 1. Utilizing spatially integrated spectroscopy from the LEGA-C survey, we focus on the relationship between the observed integrated stellar velocity dispersion ( σ ' star ) and the morphological axial ratio ( q ), and its variation with the stellar age and mass of quiescent galaxies. For the youngest quiescent galaxies, regardless of stellar mass, σ ' star decreases with increasing q , a trend that is consistent with a system having significant rotation and hence suggests that massive galaxies still retain significant amount of angular momentum in the aftermath of quenching. As they continue to evolve, the variation of the σ ' star -q relationship depends on stellar mass. For quiescent galaxies with M ∗ < 10 11 . 3 M ⊙ , σ ' star decreases with q in all stellar-age bins, suggesting that the quiescent populations of this mass regime retain significant rotation even long time after quenching. In contrast, for more massive quiescent galaxies with M ∗ > 10 11 . 3 M ⊙ , the relationship between σ ' star and q becomes significantly flattened with increasing stellar age. This indicates that, as the very massive galaxy populations continue to evolve after quenching, angular momentum gradually reduces, which eventually transforms them into velocity-dispersion supported systems. We suggest that incoherent, continuous merging and accretion events onto the galaxies are the main drivers of the observed mass-dependent, posting-quenching dynamical evolution, because more massive galaxies are more likely to undergo such interactions. We are witnessing the early formation epoch of fast and slow rotators at z ∼ 0 . 8, when the Universe was only half of its age nowadays.</text>
<text><location><page_1><loc_14><loc_38><loc_86><loc_41></location>Keywords: Galaxy formation(595); Galaxy evolution(594); Galaxy structure(622); High-redshift galaxies(734)</text>
<section_header_level_1><location><page_1><loc_20><loc_35><loc_36><loc_36></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_17><loc_48><loc_34></location>Early type/quiescent galaxies that no longer actively form stars dominate the cosmic stellar-mass budget in the present-day Universe (Muzzin et al. 2013). At redshift z ∼ 0, integral-field spectroscopic (IFS) observations revealed a bimodal distribution in the stellar kinematics of massive quiescent galaxies (Emsellem et al. 2004, 2007; Cappellari et al. 2007). Two classes - fast and slow rotators - are identified to have distinct V/σ , i.e. the ratio of the ordered ( V ) to random ( σ ) motions in a stellar system. Relative to fast rotators, slow rotators have lower V/σ , and they generally are more mas-</text>
<text><location><page_1><loc_8><loc_12><loc_22><loc_12></location>zhiyuanji@arizona.edu</text>
<text><location><page_1><loc_52><loc_24><loc_92><loc_36></location>with stellar masses of M ∗ ≳ 10 11 . 3 M ⊙ and weakly triaxial (Cappellari 2016). Constraining the pathway to establishing the observed kinematical dichotomy at z ∼ 0 is the key for understanding the assembly of massive galaxies, which requires us to push the study of stellar kinematics in quiescent galaxies towards higher redshifts, i.e. closer to the epoch when the dichotomy was emerging.</text>
<text><location><page_1><loc_52><loc_10><loc_92><loc_23></location>For quiescent galaxies at high redshifts, however, measuring stellar kinematics with spatially resolved spectroscopy is very challenging, owing to their compact morphologies (e.g. van der Wel et al. 2014; Ji et al. 2024), and the lack of strong emission lines. Prior to the launch of James Webb Space Telescope (JWST, Gardner et al. 2023), such measurements only exist for a handful of rare, extremely bright quiescent galaxies whose observed fluxes are highly magnified due to strong grav-</text>
<text><location><page_2><loc_8><loc_70><loc_48><loc_91></location>itational lensing (Newman et al. 2015; Toft et al. 2017; Newman et al. 2018). The immense gain of JWST in sensitivity and angular resolution at IR wavelengths now enables spatially resolved spectroscopy of more general populations (e.g., unlensed) of high-z quiescent galaxies (D'Eugenio et al. 2023). However, such observations are still time-consuming, requiring ≳ 10 hours on-source exposure with NIRSpec/IFS (Jakobsen et al. 2022) for a single fairly bright ( K < 22 . 5 mag, M ∗ ∼ 10 11 M ⊙ ) quiescent galaxy (Nanayakkara et al. 2022). This makes it not possible - even with JWST - to measure stellar kinematics with spatially resolved spectroscopy in statistically large samples of high-z quiescent galaxies on a rapid timescale.</text>
<text><location><page_2><loc_8><loc_42><loc_48><loc_69></location>Yet, notwithstanding the very limited sample size of high-z quiescent galaxies with robust measures of stellar kinematics, the findings from existing studies are somewhat surprising. All systems that have been studied show rapid rotation (Newman et al. 2015; Toft et al. 2017; Newman et al. 2018; D'Eugenio et al. 2023), despite that their large stellar mass, i.e. typically ≳ 10 11 . 3 M ⊙ , suggests that they should be the progenitors of z ∼ 0 slow rotators. If those systems are good representative of the underlying population of high-z massive quiescent galaxies, the implication will be profound: Significant dynamical transformations, particularly the loss of angular momentum, must happen after the quenching of massive galaxies. Unfortunately, such an implication can be fraught with systematic errors, considering that the current sample size is rather small and the sample selection function can be complex for observations on target basis.</text>
<text><location><page_2><loc_8><loc_18><loc_48><loc_41></location>In this work, instead of relying on spatially resolved spectroscopy, we investigate the dynamical transformation of quiescent galaxies using spatially integrated/unresolved stellar kinematics. In such a way we are able to conduct the analysis with a statistically significant sample of ≈ 1000 massive quiescent galaxies at z ∼ 0 . 8, about half the Hubble time of the Universe today. In particular, we focus on the stellar-age dependence of the empirical relationship between σ ' star 1 , i.e. the observed integrated stellar velocity dispersion (after taking into account the instrumental resolution), and q , i.e. the ratio of the semi-minor to semi-major axes of the morphology of galaxies which is a sensitive probe of inclination. The idea is illustrated in Figure 1 and described in detail in what follows.</text>
<text><location><page_2><loc_52><loc_67><loc_92><loc_91></location>For spatially integrated spectroscopy, both random and ordered (if present) motions contribute to the broadening of intrinsic stellar templates. Therefore, σ ' star equals to the square root of the quadratic sum of the intrinsically random motion ( σ ) and the contribution from projected rotation along the line of sight ( σ rotation ). For a system with significant rotation, because σ rotation decreases with increasing q (from edge-on to face-on), σ ' star decreases with increasing q (black solid line in Figure 1). In contrast, for a system dominated by random motion, because the contribution from σ rotation to σ ' star is negligible compared to σ , a much weaker relationship between σ ' star and q is expected. A very similar idea 2 has been discussed and utilized in an earlier study by Belli et al. (2017) of a much smaller sample of 24 quiescent galaxies at z ∼ 2.</text>
<text><location><page_2><loc_52><loc_40><loc_92><loc_66></location>With spectral energy distribution (SED) modeling growing in sophistication and accuracy, statistically reconstructing high-fidelity star formation histories (SFHs) is becoming possible for high-z massive galaxies when high-quality, panchromatic data are available. The flexibility of the SFH treatment in SED modeling ensures a much less biased, if at all, inference of physical parameters (Carnall et al. 2019; Leja et al. 2019). Built upon this latest development in SED modeling, in the first two papers of this series (Ji & Giavalisco 2022a,b), we have utilized the fully Bayesian SED fitting code Prospector (Johnson et al. 2021) to reconstruct the nonparametric SFH of massive galaxies at z ∼ 2. Combining together the SFHs and morphological analysis, we were able to reconstruct the timing sequence of the morphological transformation of massive galaxies as they evolve from the main sequence to quiescence.</text>
<text><location><page_2><loc_52><loc_20><loc_92><loc_40></location>In this third paper, we focus on the dynamical transformation of quiescent galaxies in approximately the last half of the Hubble time. With robust stellar-age estimates from SED fitting, we study the dependence of the relationship between σ ' star and q on the stellar age of galaxies. Any significant change of the σ ' star vs. q relationship with stellar age is an indication of strong evolution in the dynamical state of massive galaxies after they quench. The redshift range considered here is 0 . 6 < z < 1, where statistically significant samples of quiescent galaxies with unresolved stellar kinematics are available. Throughout this paper, we adopt the AB magnitude system and the ΛCDM cosmology</text>
<figure>
<location><page_3><loc_22><loc_64><loc_75><loc_91></location>
<caption>Figure 1. Illustration of the main idea of this work (see Section 1 for details). We investigate the dynamical transformation of massive quiescent galaxies at high redshifts with spatially integrated/unresolved stellar kinematics by studying the empirical relationship between σ ' star ( y -axis) and q ( x -axis).</caption>
</figure>
<text><location><page_3><loc_8><loc_54><loc_48><loc_57></location>with Planck Collaboration et al. 2020 parameters, i.e., Ω m = 0 . 315 and h = H 0 / (100 km s -1 Mpc -1 ) = 0 . 673.</text>
<section_header_level_1><location><page_3><loc_21><loc_51><loc_35><loc_52></location>2. THE SAMPLE</section_header_level_1>
<text><location><page_3><loc_8><loc_27><loc_48><loc_50></location>The parent sample considered in this study comes from the Large Early Galaxy Astrophysics Census (LEGA-C, van der Wel et al. 2016; Straatman et al. 2018), the latest and final Data Release 3 (van der Wel et al. 2021). The LEGA-C survey is an ESO/Very Large Telescope public survey that observed with deep spectroscopy (median S/N ∼ 15 at 4000 ˚ A) for a sample of ∼ 3500 galaxies at 0 . 6 < z < 1, selected using the K -band flux from the COSMOS/UltraVISTA survey (Muzzin et al. 2013). Here we only focus on the galaxies with M ∗ > 10 10 . 5 M ⊙ , to ensure (1) good stellar mass completeness (see Figure A1 of van der Wel et al. 2021) and (2) that the environmental effects - external to the host halo of a galaxy - on the evolution of galaxies are minor (e.g. Ji et al. 2018).</text>
<text><location><page_3><loc_8><loc_9><loc_48><loc_26></location>We refine the sample selection using the flags from the LEGA-C data release. We require FLAG MORPH = 0, to ensure that during observations the light through the slit is from a single galaxy with a regular morphology, meaning that mergers and galaxies whose LEGA-C spectra are contaminated by adjacent galaxies are excluded from the sample. We also require FLAG SPEC = 0, to exclude the galaxies with clear AGN presence identified by either IR or X-ray observations. These two constraints together ensure the high-quality spectral measures, and eliminate the cases when the interpretation</text>
<text><location><page_3><loc_52><loc_48><loc_92><loc_57></location>of the dynamical measures becomes complicated. We cross match the LEGA-C catalog with the photometric catalog of COSMOS2020 (Weaver et al. 2022), and finally select quiescent galaxies using the UVJ criteria of Muzzin et al. (2013). Our final sample contains 952 UVJ-selected quiescent galaxies.</text>
<section_header_level_1><location><page_3><loc_64><loc_45><loc_80><loc_46></location>3. MEASUREMENTS</section_header_level_1>
<text><location><page_3><loc_66><loc_43><loc_78><loc_44></location>3.1. σ ' star and q</text>
<text><location><page_3><loc_52><loc_35><loc_92><loc_42></location>The measurements of unresolved stellar kinematics σ ' star and morphological axis ratio q are taken directly from the LEGA-C data release (Bezanson et al. 2018a; van der Wel et al. 2021). We refer readers to those references for technical details.</text>
<text><location><page_3><loc_52><loc_13><loc_92><loc_34></location>Briefly, q is derived from the HST/ACS I 814 imaging in the COSMOS field, following van der Wel et al. (2012) who used the Galfit package (Peng et al. 2010) to model the 2D light distribution of galaxies assuming a single S'ersic profile. σ ' star were measured with the pPFX package (Cappellari 2017) by fitting the observed spatially integrated spectra with the combination of (1) high-resolution (R = 10000) theoretical single stellar templates and emission lines at the instrumental resolution, (2) a 3rd-order multiplicative polynomial and (3) an additive polynomial. The unresolved stellar velocity dispersion σ ' star and gas velocity dispersion are estimated independently by broadening the templates with Gaussian kernels.</text>
<text><location><page_3><loc_61><loc_10><loc_84><loc_11></location>3.2. SED fitting with Prospector</text>
<figure>
<location><page_4><loc_11><loc_68><loc_50><loc_91></location>
</figure>
<figure>
<location><page_4><loc_51><loc_68><loc_89><loc_91></location>
<caption>Figure 2. Left: Distribution of the final sample of 952 massive quiescent galaxies in the plane of SFR vs M ∗ . The black solid line shows the star-forming main sequence of Leja et al. (2022), and the black dashed lines mark the range of ± 0 . 3 dex. Right: Distribution of the final sample in the plane of H δ A vs. D N 4000. The background grey contours show the distribution of all galaxies with M ∗ > 10 10 . 5 M ⊙ from the LEGA-C survey. Each one of the quiescent galaxies in our final sample is color coded according to its mass-weighted stellar age derived from SED fitting assuming non-parametric SFH. Our quiescent sample selected via UVJ technique also occupies the region of the parameter space of quiescent galaxies in these two planes.</caption>
</figure>
<text><location><page_4><loc_8><loc_27><loc_48><loc_57></location>The properties of the stellar-populations of the sample galaxies are derived by fitting the multi-band photometry from the COSMOS2020 catalog with the fully Bayesian code Prospector (Johnson et al. 2021). Each one of the sample galaxies has ≈ 40 band photometry that densely samples the rest-frame UV-to-NIR wavelengths. Compared to previous COSMOS catalogs, COSMOS2020 includes the new, significantly deeper optical and NIR imaging from the Subaru/HSC and VISTA/VIRCAM surveys (Weaver et al. 2022). Two catalogs using different aperture photometric methods are available in the COSMOS2020 release, namely the CLASSIC and FARMER catalogs. By default, we use the former where aperture-matched photometry was carried out following Laigle et al. (2016). We note, however, that the difference between the two photometric catalogs is negligible for galaxies in the magnitude range ( K < 21 . 5 mag) considered here (Figure 8 and 9 in Weaver et al. 2022).</text>
<text><location><page_4><loc_8><loc_10><loc_48><loc_27></location>The basic setups of our Prospector fitting are essentially the same as those in the first two papers of this series (Ji & Giavalisco 2022a,b). We adopt the Flexible Stellar Population Synthesis (FSPS) code (Conroy et al. 2009; Conroy & Gunn 2010) where the stellar isochrone libraries MIST (Choi et al. 2016; Dotter 2016) and the stellar spectral libraries MILES (Falc'onBarroso et al. 2011) are used. We assume the Kroupa (2001) initial mass function and the Byler et al. (2017) nebular emission model. We assume the Calzetti et al. 2000 dust attenuation law and fit the V-band dust op-</text>
<text><location><page_4><loc_52><loc_46><loc_92><loc_57></location>ical depth with a uniform prior τ V ∈ (0 , 2). We fix the redshift to the spectroscopically-measured values from LEGA-C, and set the stellar metallicity as a free parameter with a uniform prior in the logarithmic space log( Z ∗ /Z ⊙ ) ∈ ( -2 , 0 . 19), where the upper limit of the prior is chosen because it is the highest metallicity that the MILES library covers.</text>
<text><location><page_4><loc_52><loc_29><loc_92><loc_46></location>We use the nonparametric form of SFH that is critical for unbiased inference of stellar-population properties (e.g. Leja et al. 2019). Specifically, we use a piecewise step function composed of nine lookback time bins, where the star formation rate (SFR) is constant within each bin. We fix the first two bins as 0 -30 and 30 -100 Myr to capture recent episodes of star formation. We also fix the last bin as 0.9 t H -t H where t H is the Hubble Time of observation. The remaining six bins are evenly spaced in the logarithmic lookback time between 100 Myr and 0.9 t H .</text>
<text><location><page_4><loc_52><loc_10><loc_92><loc_29></location>To ensure the convergence of nonparametric SFH reconstructions and reasonable uncertainty estimations (e.g. Carnall et al. 2019; Leja et al. 2019), we adopt the Dirichlet prior (Leja et al. 2017) during the Prospector SED fitting. This prior has been demonstrated to be able to recover the diverse shape of SFHs (Leja et al. 2019). Moreover, using the synthetic observations of simulated galaxies that have similar data quality like the ones we use here for the LEGA-C galaxies, Ji & Giavalisco (2022a, see their Appendix A) demonstrated that the Dirichlet prior can better recover the stellar age of high-z quiescent galaxies compared to</text>
<text><location><page_5><loc_8><loc_87><loc_48><loc_91></location>other commonly-used priors, such as the continuity one, which is commonly adopted to measure the SFH of starforming galaxies.</text>
<text><location><page_5><loc_8><loc_71><loc_48><loc_87></location>In Figure 2, we show the distributions of the sample galaxies in the planes of SFR vs. M ∗ , and of H δ A (Worthey et al. 1994; Worthey & Ottaviani 1997) vs. D N 4000 (4000 ˚ A break, Balogh et al. 1999). From the left panel of the Figure it is immediately clear that the UVJ-selected quiescent galaxies also occupy the parameter space of galaxies below the star-forming main sequence, i.e. with depressed SFR at any given stellar mass. This shows very good consistency among different selection methods of quiescent galaxies.</text>
<text><location><page_5><loc_8><loc_51><loc_48><loc_71></location>In the right panel of Figure 2, each one of the galaxies is color coded according to mass-weighted stellar age. It has been extensively shown that H δ A and D N 4000 are sensitive diagnostics of galaxy's stellar age (e.g. Kauffmann et al. 2003). As the Figure shows, galaxies with lower SFR (at fixed M ∗ ), larger D N 4000 and larger (negative) H δ A also have larger ages (older stellar populations) from our Prospector fitting, demonstrating the robustness of our stellar-age inference. Because the main conclusion of this study depends on the stellar-age measures, in Appendix A we conduct a number of further tests on the robustness of the age inference. We conclude that the our stellar-age measures are robust.</text>
<section_header_level_1><location><page_5><loc_23><loc_48><loc_33><loc_50></location>4. RESULTS</section_header_level_1>
<text><location><page_5><loc_8><loc_35><loc_48><loc_48></location>We now present the relationship between σ ' star and q , i.e. the core of this study. We first divide the sample into the low-mass and high-mass subsamples using M ∗ = 10 11 . 3 M ⊙ , i.e. the characteristic mass to separate fast and slow rotators at z ∼ 0 (Cappellari 2016). We then further divide each subsample into three subgroups using the 33th- and 67th- percentiles of the stellar-age distribution of the entire quiescent sample.</text>
<section_header_level_1><location><page_5><loc_20><loc_33><loc_36><loc_34></location>4.1. The median trend</section_header_level_1>
<text><location><page_5><loc_8><loc_15><loc_48><loc_32></location>We measure the median relationship between σ ' star and q of each one of the subgroups using the Locally Weighted Scatterplot Smoothing (LOWESS 3 ) method that fits a smoothed curve to data points through a nonparametric approach, i.e. the process does not require to assume any specific functional form. We estimate the uncertainty of the median relationship via Monte Carlo simulations. In particular, we use Gaussian distributions to resample the individual σ ' star and q measures with their corresponding uncertainties. We then use LOWESS to measure the median σ ' star vs. q rela-</text>
<text><location><page_5><loc_52><loc_87><loc_92><loc_91></location>of the resampled data points. We repeat these 1000 times, and use the range between 16th- and 84thpercentiles as 1σ uncertainty.</text>
<text><location><page_5><loc_52><loc_74><loc_92><loc_87></location>To begin, as the first column of Figure 3 shows, in the youngest age bin, σ ' star decreases with increasing q , which is observed in both mass bins. This suggests that massive galaxies - regardless of their masses - still retain significant rotation in the aftermath of quenching. As they continue evolving and become older (the second and third columns of Figure 3), the relationship between σ ' star and q starts to differ in the two mass bins.</text>
<text><location><page_5><loc_52><loc_60><loc_92><loc_74></location>For low-mass quiescent galaxies with M ∗ < 10 11 . 3 M ⊙ (the first row of Figure 3), σ ' star decreases with q in all age bins, suggesting quiescent galaxies in this mass regime continue to retain significant rotation even long time after quenching. Overall, the median relationships between σ ' star and q are statistically consistent with each other for all age bins within the uncertainties, although there is some evidence that younger galaxies have a steeper relationship than the older ones.</text>
<text><location><page_5><loc_52><loc_48><loc_92><loc_60></location>For high-mass quiescent galaxies with M ∗ > 10 11 . 3 M ⊙ (the second row of Figure 3), the decreasing trend of σ ' star with q becomes significantly flattened as they become older. This shows that post-quenching dynamical transformations (1) happen in these very massive systems, which significantly reduce the amount of rotation and (2) these transformations are more profound in more massive quiescent galaxies.</text>
<text><location><page_5><loc_52><loc_21><loc_92><loc_47></location>In the analysis above we divided the low- and highmass subsamples into only three age bins. We continue to study the variation of the σ ' star -q relationship by dividing the subsamples into more age bins. Specifically, instead of binning the subsamples using arbitrary bins of stellar age, we first sort stellar ages of individual galaxies into an increasing order. And starting from the first 30% of the sorted subsamples, we measure the Spearman's rank correlation coefficient ρ . Then, we keep adding older quiescent galaxies into the correlation test, and study the change of ρ as a function of the maximum age of the galaxies included in the measure. The uncertainty of ρ calculated in this way is estimated by bootstrapping the sample galaxies 1000 times, and during each bootstrapping iteration we also resample the values of σ ' star and q with their corresponding measurement uncertainties using Gaussian distributions.</text>
<text><location><page_5><loc_52><loc_10><loc_92><loc_21></location>In Figure 4, ρ is plotted against the maximum massweighted age of the quiescent galaxies included to the Spearman's rank correlation test. For galaxies that are freshly quenched, i.e. with relatively young stellar ages, we find a strong negative ( ρ ∼ -0 . 5, i.e. a decreasing trend) correlation between σ ' star and q , regardless of stellar mass. As older quiescent galaxies are added to the</text>
<figure>
<location><page_6><loc_9><loc_74><loc_32><loc_92></location>
</figure>
<figure>
<location><page_6><loc_33><loc_74><loc_52><loc_92></location>
</figure>
<figure>
<location><page_6><loc_52><loc_74><loc_71><loc_92></location>
</figure>
<figure>
<location><page_6><loc_72><loc_74><loc_91><loc_92></location>
</figure>
<figure>
<location><page_6><loc_9><loc_55><loc_32><loc_73></location>
</figure>
<figure>
<location><page_6><loc_33><loc_55><loc_51><loc_73></location>
</figure>
<figure>
<location><page_6><loc_52><loc_55><loc_71><loc_73></location>
</figure>
<figure>
<location><page_6><loc_72><loc_55><loc_91><loc_73></location>
<caption>Figure 3. σ ' star vs. q . The first and second rows show the results of the quiescent galaxies with M ∗ < 10 11 . 3 M ⊙ and M ∗ > 10 11 . 3 M ⊙ , respectively. For each stellar-mass bin, galaxies are further divided into three age bins, each of which roughly contains 1/3 of the entire sample. In each panel the solid line and shaded region show the median trend and the corresponding 1σ uncertainty derived using the LOWESS algorithm. The dashed line shows the best-fit from our toy model (see Section 4.2). In the fourth column, all LOWESS median trends are shown together.</caption>
</figure>
<text><location><page_6><loc_8><loc_35><loc_48><loc_46></location>correlation test, ρ gradually changes from -0 . 5 to -0 . 3 for the high-mass ( > 10 11 . 3 M ⊙ ) quiescent populations, while it remains approximately unchanged for the lowmass ones. This shows that the variation of the σ ' star vs. q relationship with age is significantly stronger in more massive quiescent galaxies, confirming the conclusion reached above based on Figure 3.</text>
<section_header_level_1><location><page_6><loc_11><loc_32><loc_45><loc_34></location>4.2. A toy model for the σ ' star vs. q relationship</section_header_level_1>
<text><location><page_6><loc_8><loc_27><loc_48><loc_31></location>We now introduce a simple toy model, in an attempt to quantify the contribution from rotation to the observed relationship between σ ' star and q .</text>
<text><location><page_6><loc_8><loc_22><loc_48><loc_27></location>As detailed already in Section 1, both random and ordered motions contribute to σ ' star , which can be expressed as</text>
<formula><location><page_6><loc_19><loc_16><loc_48><loc_21></location>( σ ' star ) 2 = σ 2 + σ 2 rotation = σ 2 ( 1 + γ 2 ( V σ ) 2 sin 2 i ) . (1)</formula>
<text><location><page_6><loc_8><loc_9><loc_48><loc_15></location>In the above equation we have used σ rotation = γV sin i , which describes velocity dispersion observed through a slit (i.e. unresolved spectroscopy) due to a purely rotating disk, where V is the rotational velocity, i is the</text>
<text><location><page_6><loc_52><loc_35><loc_92><loc_46></location>inclination and γ is the conversion factor. Because to our knowledge there is no direct, statistical estimate of γ at high redshifts, we decide to fix γ = 0 . 7 which is the median value from Cappellari et al. (2013) who measured it using the spatially resolved stellar kinematics of z ∼ 0 early type galaxies from ATLAS 3D . The inclination can be estimated using q as</text>
<formula><location><page_6><loc_66><loc_30><loc_92><loc_33></location>sin i = √ 1 -q 1 -q z (2)</formula>
<text><location><page_6><loc_52><loc_12><loc_92><loc_28></location>where q z is the thickness of a disk which - following Belli et al. (2017) - we fix to be q z = 0 . 2, namely about the minimum axis ratio observed in large extragalactic imaging surveys. The remaining unknowns in Equation 1 are σ and V (or V/σ ) that we attempt to constrain by fitting Equation 1 to the observed σ ' star vs. q relationship. To estimate the uncertainties of the fitted parameters, we use the same Monte Carlo method mentioned above by resampling the σ ' star and q measures using their uncertainties.</text>
<text><location><page_6><loc_52><loc_9><loc_92><loc_12></location>The best-fit relationships are plotted as dashed lines in Figure 3. The observations can be reproduced very well</text>
<text><location><page_7><loc_8><loc_89><loc_48><loc_91></location>with the toy model. The best-fit models are in excellent agreement with the LOWESS median trends.</text>
<text><location><page_7><loc_8><loc_70><loc_48><loc_88></location>In Figure 5, the inferred V/σ is plotted as a function of stellar age. Regardless of stellar age, the V/σ of the quiescent galaxies with M ∗ < 10 11 . 3 M ⊙ is greater than 1. As the population evolves, the V/σ of these systems decreases from 2 . 3 ± 0 . 3 to 1 . 7 ± 0 . 4, implying that they remain rotationally supported, with V/σ ∼ 1 . 7, even at least 7 Gyr after their formation 4 . In contrast, during the same cosmic time, the V/σ of the highermass quiescent galaxies with M ∗ > 10 11 . 3 M ⊙ monotonically decreases, from 1 . 6 ± 0 . 3 (rotationally supported) to 1 . 0 ± 0 . 3 (velocity dispersion supported), with increasing stellar age.</text>
<text><location><page_7><loc_8><loc_24><loc_48><loc_69></location>Before moving forward, we note several caveats of the V/σ inferred from our simple model. The misalignment between the slit and the kinematic major axis of galaxies has not been taken into account in our analysis. Neglecting this effect, however - given the large sample size of this study - should lead to an equal/similar systematic error in all subgroups and hence may not cause any substantial impacts on our conclusions, which only rely on a differential comparison. Also, the uncertainty on the value of γ in Equation 1 affects the inferred V/σ , a situation which, unfortunately, cannot be addressed at the moment. However, we note that the γ of gas kinematics has been statistically estimated at high redshifts and found to have a typical value of 0 . 6 -1 (e.g., Weiner et al. 2006). Thus, while the adopted γ = 0 . 7 is bracket by the range determined from high-z gas kinematics, quantitatively the inferred V/σ presented in Figure 5 should be taken with caution. Qualitatively, however, our conclusions about the mass-dependent evolution of V/σ with stellar age should stand despite the over-simplified model. Finally, we also clarify that, when we state, e.g., that the quiescent population of a given stellar age is rotationally supported ( V/σ > 1), we do not mean that each one of the quiescent galaxies of that age bin retains significant rotation or that it has a disk, since the spatially integrated spectroscopy does not allow us to constrain that. Instead, what we really mean is that the quiescent population of that age on average should have significant rotation.</text>
<text><location><page_7><loc_8><loc_18><loc_48><loc_24></location>With the aforementioned caveats in mind, we now compare the inferred V/σ of this work with previous studies of the stellar kinematics in quiescent galaxies. Bezanson et al. (2018b) pioneered a LEGA-C study of</text>
<text><location><page_7><loc_52><loc_65><loc_92><loc_92></location>the dynamical transformation of z ∼ 0 . 8 massive quiescent galaxies using a small (relative to this study) sample of ∼ 100 galaxies whose major axes are overall aligned with the slit ( | PA | < 45 · ), which allows spatially resolved analysis of stellar kinematics. They found that the most massive ( > 10 11 . 3 M ⊙ ) quiescent galaxies show much less rotation compared to less massive systems. In broad agreement 5 with Bezanson et al. (2018b), our model suggests that quiescent populations of M ∗ > 10 11 . 3 M ⊙ are significantly less rotationally supported compared to the lower-mass ones, with V/σ = 1 . 2 ± 0 . 1 for the high-mass subsample compared to V/σ = 2 . 0 ± 0 . 2 for the low-mass one. The ≈ 10 × larger in sample size of this study allows us to further group galaxies according to their stellar ages, adding a new piece of information regarding the dynamical transformation of quiescent galaxies as they evolve.</text>
<text><location><page_7><loc_52><loc_26><loc_92><loc_65></location>We also compare our inferred V/σ with the very limited number of direct V/σ measures at higher redshifts z ∼ 2. In Figure 5, we show the results from Newman et al. (2018) who measured the stellar kinematics in three strongly lensed massive quiescent galaxies with M ∗ ≳ 10 11 . 3 M ⊙ at z ∼ 2 using spatially resolved spectroscopy. On average, those z ∼ 2 quiescent systems have even higher V/σ than that inferred for the youngest z ∼ 0 . 8 quiescent populations of similar masses ( ≳ 10 11 . 3 M ⊙ ). Note that the median stellar age of the youngest bin of our z ∼ 0 . 8 sample is ∼ 3 -4 Gyr which is longer than the Hubble time of z ∼ 2 (i.e. ∼ 3 Gyr). Thus, those z ∼ 2 quiescent galaxies must be - on average - younger (i.e. more freshly quenched) than the youngest quiescent populations considered in this study. If the dynamical state of the three lensed quiescent systems considered by Newman et al. (2018) is representative of the stellar kinematics of the entire quiescent populations at z ∼ 2 of comparable stellar mass, the implication is that the loss of angular momentum continuously and gradually happens after the cessation of star formation (quenching) in very massive ( M ∗ ≳ 10 11 . 3 M ⊙ ) galaxies, transforming them from fast rotators right after quenching to slow rotators in at least ∼ 7 Gyr after their quenching.</text>
<section_header_level_1><location><page_7><loc_59><loc_23><loc_85><loc_24></location>5. DISCUSSION AND SUMMARY</section_header_level_1>
<text><location><page_7><loc_52><loc_17><loc_92><loc_22></location>To summarize, we studied the relationship between the dynamical transformation and quenching using the unresolved stellar kinematics of a sample of 952 massive</text>
<figure>
<location><page_8><loc_9><loc_70><loc_47><loc_92></location>
<caption>Figure 4. Change of the Spearman's rank correlation coefficient as older quiescent galaxies are included to the correlation test (dashed lines). The x -axis shows the maximum mass-weighted age of the quiescent galaxies included in the test. The dark and light shaded regions mark the 0.5- and 1σ uncertainties.</caption>
</figure>
<text><location><page_8><loc_8><loc_49><loc_48><loc_58></location>( > 10 10 . 5 M ⊙ ) quiescent galaxies at 0 . 6 < z < 1 from the LEGA-C survey. Using the SED fitting code Prospector , we robustly measured the stellar-population properties of the sample galaxies. We focused on the variation of the relationship between σ ' star and q as a function of stellar age, and of stellar mass.</text>
<text><location><page_8><loc_8><loc_30><loc_48><loc_49></location>We found a decreasing trend of σ ' star with q for the youngest quiescent galaxies of all masses. The implication is that freshly quenched galaxies, regardless of stellar mass, still have significant rotation. This is strong evidence that the occurrence of quenching in itself at high redshift does not fully transform massive galaxies into dispersion supported systems as it takes place or immediately after. Based on what we have found in our recent study (Ji & Giavalisco 2022b), however, it is very likely that quenching happens in close temporal proximity to whatever mechanism alters the inner structure of galaxies by building dense central stellar cores.</text>
<text><location><page_8><loc_8><loc_10><loc_48><loc_30></location>We found that the post-quenching dynamical transformation of quiescent galaxies depends on stellar mass, adding an important piece of information regarding the early formation epoch of fast and slow rotators. We remind that M ∗ = 10 11 . 3 M ⊙ is the characteristic mass that separates the fast and slow rotators at z ∼ 0 (Cappellari 2016). For very massive galaxies, with M ∗ > 10 11 . 3 M ⊙ , at z ∼ 0 . 8, we observe that the incidence of rotational support gradually reduces, as the galaxies become older. Using a simple toy model, we infer these very massive systems transform from being rotationally supported in the aftermath of quenching, with V/σ ∼ 1 . 6, to being velocity-dispersion supported ≈ 7</text>
<text><location><page_8><loc_52><loc_82><loc_92><loc_92></location>Gyr after their formation, with V/σ ∼ 1 . 0. In contrast, lower-mass quiescent populations with M ∗ < 10 11 . 3 M ⊙ show a much weaker post-quenching dynamical evolution. Even 7 Gyr after their formation, these lowermass quiescent systems still retain significant rotation with V/σ ∼ 1 . 7.</text>
<text><location><page_8><loc_52><loc_51><loc_92><loc_82></location>Our findings are consistent with the picture that quiescent galaxies at the high-mass end formed in dense environments, presumably in the regions with large overdensities of the primordial density field. After quenching, these very massive galaxies continue changing their dynamical states via continuous gas accretion or inchoerent merging with other adjacent galaxies through dynamic friction. These very massive systems will eventually evolve into slow rotators seen at z ∼ 0, because multiple incoherent merging episodes can cause significant loss of angular momentum (e.g. Emsellem et al. 2011). For lower-mass quiescent galaxies, however, they very likely formed in regions with, comparatively speaking, smaller overdensities, meaning that they have shallower gravitational wells such that the frequency of merging events with other smaller galaxies is much lower than that in more massive halos. Consequently, lower-mass quiescent galaxies can retain significant rotations long time after their formation, i.e. they will evolve into fast rotators.</text>
<text><location><page_8><loc_52><loc_29><loc_92><loc_51></location>The mass-dependent evolution of massive quiescent galaxies has been revealed in both their morphological (e.g. van der Wel et al. 2014; Ji & Giavalisco 2022a; Ji et al. 2024) and chemical properties (e.g. Kriek et al. 2019; Jafariyazani et al. 2020; Cheng et al. 2024; Beverage et al. 2024) in the high-z Universe. The purely empirical study presented here, which uses unresolved spectroscopy, provides robust evidence that, statistically, the evolution of the dynamical properties (angular momentum in particular) of quiescent galaxies also depends on stellar mass. In other words, we are witnessing the early build-up of the populations of fast and slow rotators at z ∼ 0 . 8, when the Universe was only half of its age nowadays.</text>
<text><location><page_8><loc_52><loc_13><loc_92><loc_29></location>Undoubtedly, future spatially resolved spectroscopy of individual targets is needed to fully characterize the dispersion and intrinsic distribution of the dynamical state of massive quiescent galaxies and their evolution across cosmic time, including the mechanisms responsible for the dichotomy of stellar kinematics found in z ∼ 0 early type galaxies. Yet, results of the empirical study presented here, and its simplicity, provides robust evidence that the formation of the dichotomy was well underway at half the Hubble time.</text>
<figure>
<location><page_9><loc_22><loc_68><loc_78><loc_91></location>
<caption>Figure 5. V/σ as a function of stellar age. Blue ( M ∗ < 10 11 . 3 M ⊙ ) and red ( M ∗ > 10 11 . 3 M ⊙ ) squares show the V/σ , inferred from our toy model (Section 4.2), of z ∼ 0 . 8 quiescent populations of different ages. The horizontal dashed line marks V/σ = 1. In the left-most grey shaded region, we also plot three strongly lensed massive ( ≳ 10 11 . 3 M ⊙ ) quiescent galaxies at z ∼ 2 with direct, spatially resolved measures of stellar kinematics from Newman et al. (2018).</caption>
</figure>
<unordered_list>
<list_item><location><page_9><loc_50><loc_57><loc_92><loc_58></location>This work was completed in part with resources pro1</list_item>
<list_item><location><page_9><loc_50><loc_55><loc_92><loc_57></location>vided by the Green High Performance Computing 2</list_item>
<list_item><location><page_9><loc_50><loc_54><loc_92><loc_55></location>Cluster (GHPCC) of the University of Massachusetts 3</list_item>
<list_item><location><page_9><loc_50><loc_52><loc_59><loc_53></location>Amherst. 4</list_item>
</unordered_list>
<text><location><page_9><loc_52><loc_45><loc_92><loc_51></location>Software: Prospector (Johnson et al. 2021), FSPS (Conroy et al. 2009; Conroy & Gunn 2010), MIST (Choi et al. 2016; Dotter 2016), MILES (Falc'on-Barroso et al. 2011), GALFIT (Peng et al. 2010)</text>
<section_header_level_1><location><page_9><loc_46><loc_41><loc_54><loc_42></location>APPENDIX</section_header_level_1>
<section_header_level_1><location><page_9><loc_22><loc_38><loc_78><loc_39></location>A. TESTING THE ROBUSTNESS OF THE STELLAR-AGE INFERENCE</section_header_level_1>
<text><location><page_9><loc_8><loc_28><loc_92><loc_37></location>Here we present and discuss in detail about our tests on the stellar-age measures from our Prospector fitting. To begin, we compare the best-fit SED models predicted by Prospector with the observed LEGA-C spectra. Specifically, we compare the median observed and predicted spectra of each one of the subgroups presented in the main text (Section 4). We remind that our Prospector fitting only used photometric data ( ≈ 40 bands), i.e. the spectra were not used in the SED modeling. This comparison thus allows us to have a direct and broad view on the quality of our SED fitting.</text>
<text><location><page_9><loc_8><loc_9><loc_92><loc_28></location>As Figure 6 shows, the median spectra of all subgroups dim at the rest-frame UV wavelengths, and show strong stellar absorption features without any strong emission lines over rest-frame 3800 -5200 ˚ A, which demonstrates, once again, the effectiveness of the UVJ technique in identifying quiescent galaxies. The the best-fit spectra from SED modeling are in excellent agreement with the observed ones, within the uncertainties, for all subgroups. Moreover, as the right-most panel of Figure 6 shows, galaxies with older stellar ages - inferred from Prospector - also have redder observed spectra, even though the spectral information was not included during the SED fitting procedure. This agreement suggests that the stellar-age inference from our SED fitting procedure is robust. We stress, however, that the conclusion above does not at all imply that the spectral information is not needed for SED fitting. In fact, robust measurements of metallicity and elemental abundance are only possible with spectra. What we really mean is that the stellar age of high-z quiescent galaxies can be inferred robustly when densely-sampled, panchromatic photometry is available. Similar conclusions were also reached by Ji & Giavalisco (2022a) using synthetic galaxies from cosmological simulations.</text>
<figure>
<location><page_10><loc_10><loc_77><loc_32><loc_92></location>
</figure>
<text><location><page_10><loc_21><loc_76><loc_25><loc_77></location>rest [Å]</text>
<figure>
<location><page_10><loc_10><loc_59><loc_32><loc_75></location>
</figure>
<figure>
<location><page_10><loc_33><loc_59><loc_51><loc_75></location>
</figure>
<figure>
<location><page_10><loc_52><loc_59><loc_71><loc_75></location>
</figure>
<figure>
<location><page_10><loc_71><loc_59><loc_90><loc_75></location>
<caption>Figure 6. Comparison between the median observed (LEGA-C) and predicted ( Prospector fitting with photometry only) spectra of each one of the subgroups discussed in the main text. The sample division here is the same as used in Figure 3. In each panel, we also show the 1σ range (standard deviation) of the observed spectrum as shaded region. In the right-most panel, the median observed spectra of all stellar age bins are shown together.</caption>
</figure>
<text><location><page_10><loc_8><loc_32><loc_92><loc_49></location>Despite the numerous advantages of fitting photometric and spectral data simultaneously (e.g., Tacchella et al. 2022), here we want to highlight one potential, serious systematics: the non-trivial aperture matching that is required when combining photometry and spectroscopy. To tackle this, most studies simply rescale (or perform the fit with the rescaling factor as a free parameter) the observed spectra to the same flux level of photometry. The big assumption behind such a procedure is that there is no strong color/stellar-population variation between the photometric aperture and the spectral slit. This assumption can be problematic given that color gradients have been clearly observed in high-z massive quiescent galaxies (e.g. Suess et al. 2020; Ji et al. 2024). Ideally, in order to simultaneously fitting photometry and spectroscopy, one needs forward modeling the instrumental effects to properly account for e.g. the mismatch of apertures, which however is beyond the scope of this work. The fact that the predicted spectra using photometry alone are in very good agreement with the observed ones provides confidence that the measures of stellar age of the sample galaxies are robust.</text>
<text><location><page_10><loc_8><loc_25><loc_92><loc_32></location>We continue our tests on the stellar-age measures by comparing the age-color relationship from our Prospector fitting with earlier studies. Whitaker et al. (2013) stacked the 3D-HST grism spectra of massive quiescent galaxies at z ∼ 2, and fit the stacked spectral features with a solar-metallicity, single stellar population model. They found that quiescent galaxies having bluer rest-frame ( U -V ) and ( V -J ) colors are younger than those having the redder colors. We made a similar comparison and found excellent qualitative agreement, as we show in Figure 7.</text>
<text><location><page_10><loc_8><loc_14><loc_92><loc_24></location>Similarly, Belli et al. (2019) derived and calibrated the relationship between stellar age and rest-frame UVJ colors using a sample of 24 quiescent galaxies at 1 . 5 < z < 2 . 5 with deep rest-optical spectroscopy. The stellar ages of their sample galaxies were estimated by modeling spectra and photometry combined. To assess the consistency between their results and ours, we compare the relationship between stellar age and UVJ colors of our measurements with that of Belli et al. (2019). To do so, we first derive the quiescent sequence of our sample by conducting a linear regression between the ( U -V ) and ( V -J ) colors. With the best-fit linear relationship between the two colors in hand, we use Equation 3 of Belli et al. (2019) 6 to derive the expected change of stellar age along our quiescent sequence, which is</text>
<figure>
<location><page_10><loc_33><loc_78><loc_51><loc_92></location>
<caption>3800 4000 4200 4400 4600 4800 5000 5200 rest [Å]</caption>
</figure>
<figure>
<location><page_10><loc_52><loc_78><loc_71><loc_92></location>
</figure>
<figure>
<location><page_10><loc_71><loc_76><loc_90><loc_92></location>
<caption>3800 4000 4200 4400 4600 4800 5000 5200 rest [Å]</caption>
</figure>
<figure>
<location><page_11><loc_11><loc_70><loc_49><loc_92></location>
</figure>
<figure>
<location><page_11><loc_51><loc_70><loc_89><loc_92></location>
<caption>Figure 7. Left: UVJ diagram. The quiescent galaxies of this study are shown as individual circles color coded according to stellar ages from our Prospector fitting. The background contours show the distribution of all LEGA-C galaxies with M ∗ > 10 10 . 5 M ⊙ . The black dashed lines mark the UVJ selection criteria of Muzzin et al. (2013). The vector shows the best-fit relation between the stellar age and UVJ colors from Belli et al. (2019). Great qualitative agreement - galaxies with bluer UVJ colors are younger - is seen between the two stellar age inferences. Right: Distribution of the difference in stellar age between our Prospector measurements and those inferred using the best-fit relation of Belli et al. (2019). Our measurements on average return a 0 . 23 ± 0 . 14 dex older stellar age.</caption>
</figure>
<text><location><page_11><loc_8><loc_43><loc_92><loc_57></location>shown as the vector in Figure 7. We run a Pearson correlation test between the stellar ages from Prospector and the stellar ages inferred using the best-fit relation of Belli et al. (2019). We found a strong correlation, with Pearson coefficient r = 0 . 52), which demonstrates good qualitative agreement between our stellar-age measures and those of Belli et al. (2019). Note, however, that the above relationship (Belli et al. 2019) between age and rest-frame colors was derived at z ∼ 1 . 7. As the galaxies age and their colors evolve, we expect the relationship to change as well. While the study of the color evolution of quiescent galaxies is beyond the scope of this work, we note that, as the right panel of Figure 7 shows, a systematic age difference is observed between our measures with Prospector at z ∼ 0 . 8 and those of Belli et al. (2019) at z ∼ 1 . 7 such that the z ∼ 0 . 8 galaxies are older than those at z ∼ 1 . 7 by 0.23 dex, or 2.6 Gyr which is about the time interval between the two redshifts, t z =0 . 8 H -t z =1 . 7 H ≈ 2 . 9 Gyr.</text>
<text><location><page_11><loc_8><loc_35><loc_92><loc_42></location>In summary, by comparing (1) the predicted spectra from Prospector with the observed ones from LEGA-C and (2) the stellar ages from Prospector and the ones inferred using the age-color relationship reported by previous studies, we showed very good agreement among different stellar-age measures. We stress that, because the conclusions of this study only depend on differential stellar-age measures (younger vs. older), rather than absolute stellar-age determinations, we conclude that the results and key conclusions presented in this study are robust.</text>
<section_header_level_1><location><page_11><loc_44><loc_31><loc_56><loc_32></location>REFERENCES</section_header_level_1>
<text><location><page_11><loc_8><loc_25><loc_48><loc_30></location>Balogh, M. L., Morris, S. L., Yee, H. K. C., Carlberg, R. G., & Ellingson, E. 1999, ApJ, 527, 54, doi: 10.1086/308056 Belli, S., Newman, A. B., & Ellis, R. S. 2017, ApJ, 834, 18,</text>
<text><location><page_11><loc_8><loc_12><loc_47><loc_24></location>doi: 10.3847/1538-4357/834/1/18 -. 2019, ApJ, 874, 17, doi: 10.3847/1538-4357/ab07af Beverage, A. G., Kriek, M., Suess, K. A., et al. 2024, ApJ, 966, 234, doi: 10.3847/1538-4357/ad372d Bezanson, R., van der Wel, A., Straatman, C., et al. 2018a, ApJL, 868, L36, doi: 10.3847/2041-8213/aaf16b Bezanson, R., van der Wel, A., Pacifici, C., et al. 2018b,</text>
<text><location><page_11><loc_10><loc_10><loc_40><loc_11></location>ApJ, 858, 60, doi: 10.3847/1538-4357/aabc55</text>
<text><location><page_11><loc_52><loc_27><loc_90><loc_30></location>Byler, N., Dalcanton, J. J., Conroy, C., & Johnson, B. D. 2017, ApJ, 840, 44, doi: 10.3847/1538-4357/aa6c66</text>
<text><location><page_11><loc_52><loc_17><loc_90><loc_26></location>Calzetti, D., Armus, L., Bohlin, R. C., et al. 2000, ApJ, 533, 682, doi: 10.1086/308692 Cappellari, M. 2016, ARA&A, 54, 597, doi: 10.1146/annurev-astro-082214-122432 -. 2017, MNRAS, 466, 798, doi: 10.1093/mnras/stw3020</text>
<text><location><page_11><loc_52><loc_13><loc_92><loc_16></location>Cappellari, M., Emsellem, E., Bacon, R., et al. 2007, MNRAS, 379, 418, doi: 10.1111/j.1365-2966.2007.11963.x</text>
<text><location><page_11><loc_52><loc_10><loc_92><loc_13></location>Cappellari, M., Scott, N., Alatalo, K., et al. 2013, MNRAS, 432, 1709, doi: 10.1093/mnras/stt562</text>
<table>
<location><page_12><loc_8><loc_12><loc_49><loc_92></location>
</table>
<unordered_list>
<list_item><location><page_12><loc_52><loc_89><loc_91><loc_91></location>Laigle, C., McCracken, H. J., Ilbert, O., et al. 2016, ApJS, 224, 24, doi: 10.3847/0067-0049/224/2/24</list_item>
</unordered_list>
<table>
<location><page_12><loc_52><loc_13><loc_92><loc_91></location>
</table>
</document>
[
{
"title": "ABSTRACT",
"content": "We study the evolution of stellar kinematics of a sample of 952 massive quiescent galaxies with M ∗ > 10 10 . 5 M ⊙ at 0 . 6 < z < 1. Utilizing spatially integrated spectroscopy from the LEGA-C survey, we focus on the relationship between the observed integrated stellar velocity dispersion ( σ ' star ) and the morphological axial ratio ( q ), and its variation with the stellar age and mass of quiescent galaxies. For the youngest quiescent galaxies, regardless of stellar mass, σ ' star decreases with increasing q , a trend that is consistent with a system having significant rotation and hence suggests that massive galaxies still retain significant amount of angular momentum in the aftermath of quenching. As they continue to evolve, the variation of the σ ' star -q relationship depends on stellar mass. For quiescent galaxies with M ∗ < 10 11 . 3 M ⊙ , σ ' star decreases with q in all stellar-age bins, suggesting that the quiescent populations of this mass regime retain significant rotation even long time after quenching. In contrast, for more massive quiescent galaxies with M ∗ > 10 11 . 3 M ⊙ , the relationship between σ ' star and q becomes significantly flattened with increasing stellar age. This indicates that, as the very massive galaxy populations continue to evolve after quenching, angular momentum gradually reduces, which eventually transforms them into velocity-dispersion supported systems. We suggest that incoherent, continuous merging and accretion events onto the galaxies are the main drivers of the observed mass-dependent, posting-quenching dynamical evolution, because more massive galaxies are more likely to undergo such interactions. We are witnessing the early formation epoch of fast and slow rotators at z ∼ 0 . 8, when the Universe was only half of its age nowadays. Keywords: Galaxy formation(595); Galaxy evolution(594); Galaxy structure(622); High-redshift galaxies(734)",
"pages": [
1
]
},
{
"title": "Reconstructing the Assembly of Massive Galaxies. III: Quiescent Galaxies Loose Angular Momentum as They Evolve in a Mass-dependent Fashion.",
"content": "Zhiyuan Ji 1 and Mauro Giavalisco 2 1 Steward Observatory, University of Arizona, 933 N. Cherry Avenue, Tucson, AZ 85721, USA 2 University of Massachusetts Amherst, 710 North Pleasant Street, Amherst, MA 01003-9305, USA",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "Early type/quiescent galaxies that no longer actively form stars dominate the cosmic stellar-mass budget in the present-day Universe (Muzzin et al. 2013). At redshift z ∼ 0, integral-field spectroscopic (IFS) observations revealed a bimodal distribution in the stellar kinematics of massive quiescent galaxies (Emsellem et al. 2004, 2007; Cappellari et al. 2007). Two classes - fast and slow rotators - are identified to have distinct V/σ , i.e. the ratio of the ordered ( V ) to random ( σ ) motions in a stellar system. Relative to fast rotators, slow rotators have lower V/σ , and they generally are more mas- zhiyuanji@arizona.edu with stellar masses of M ∗ ≳ 10 11 . 3 M ⊙ and weakly triaxial (Cappellari 2016). Constraining the pathway to establishing the observed kinematical dichotomy at z ∼ 0 is the key for understanding the assembly of massive galaxies, which requires us to push the study of stellar kinematics in quiescent galaxies towards higher redshifts, i.e. closer to the epoch when the dichotomy was emerging. For quiescent galaxies at high redshifts, however, measuring stellar kinematics with spatially resolved spectroscopy is very challenging, owing to their compact morphologies (e.g. van der Wel et al. 2014; Ji et al. 2024), and the lack of strong emission lines. Prior to the launch of James Webb Space Telescope (JWST, Gardner et al. 2023), such measurements only exist for a handful of rare, extremely bright quiescent galaxies whose observed fluxes are highly magnified due to strong grav- itational lensing (Newman et al. 2015; Toft et al. 2017; Newman et al. 2018). The immense gain of JWST in sensitivity and angular resolution at IR wavelengths now enables spatially resolved spectroscopy of more general populations (e.g., unlensed) of high-z quiescent galaxies (D'Eugenio et al. 2023). However, such observations are still time-consuming, requiring ≳ 10 hours on-source exposure with NIRSpec/IFS (Jakobsen et al. 2022) for a single fairly bright ( K < 22 . 5 mag, M ∗ ∼ 10 11 M ⊙ ) quiescent galaxy (Nanayakkara et al. 2022). This makes it not possible - even with JWST - to measure stellar kinematics with spatially resolved spectroscopy in statistically large samples of high-z quiescent galaxies on a rapid timescale. Yet, notwithstanding the very limited sample size of high-z quiescent galaxies with robust measures of stellar kinematics, the findings from existing studies are somewhat surprising. All systems that have been studied show rapid rotation (Newman et al. 2015; Toft et al. 2017; Newman et al. 2018; D'Eugenio et al. 2023), despite that their large stellar mass, i.e. typically ≳ 10 11 . 3 M ⊙ , suggests that they should be the progenitors of z ∼ 0 slow rotators. If those systems are good representative of the underlying population of high-z massive quiescent galaxies, the implication will be profound: Significant dynamical transformations, particularly the loss of angular momentum, must happen after the quenching of massive galaxies. Unfortunately, such an implication can be fraught with systematic errors, considering that the current sample size is rather small and the sample selection function can be complex for observations on target basis. In this work, instead of relying on spatially resolved spectroscopy, we investigate the dynamical transformation of quiescent galaxies using spatially integrated/unresolved stellar kinematics. In such a way we are able to conduct the analysis with a statistically significant sample of ≈ 1000 massive quiescent galaxies at z ∼ 0 . 8, about half the Hubble time of the Universe today. In particular, we focus on the stellar-age dependence of the empirical relationship between σ ' star 1 , i.e. the observed integrated stellar velocity dispersion (after taking into account the instrumental resolution), and q , i.e. the ratio of the semi-minor to semi-major axes of the morphology of galaxies which is a sensitive probe of inclination. The idea is illustrated in Figure 1 and described in detail in what follows. For spatially integrated spectroscopy, both random and ordered (if present) motions contribute to the broadening of intrinsic stellar templates. Therefore, σ ' star equals to the square root of the quadratic sum of the intrinsically random motion ( σ ) and the contribution from projected rotation along the line of sight ( σ rotation ). For a system with significant rotation, because σ rotation decreases with increasing q (from edge-on to face-on), σ ' star decreases with increasing q (black solid line in Figure 1). In contrast, for a system dominated by random motion, because the contribution from σ rotation to σ ' star is negligible compared to σ , a much weaker relationship between σ ' star and q is expected. A very similar idea 2 has been discussed and utilized in an earlier study by Belli et al. (2017) of a much smaller sample of 24 quiescent galaxies at z ∼ 2. With spectral energy distribution (SED) modeling growing in sophistication and accuracy, statistically reconstructing high-fidelity star formation histories (SFHs) is becoming possible for high-z massive galaxies when high-quality, panchromatic data are available. The flexibility of the SFH treatment in SED modeling ensures a much less biased, if at all, inference of physical parameters (Carnall et al. 2019; Leja et al. 2019). Built upon this latest development in SED modeling, in the first two papers of this series (Ji & Giavalisco 2022a,b), we have utilized the fully Bayesian SED fitting code Prospector (Johnson et al. 2021) to reconstruct the nonparametric SFH of massive galaxies at z ∼ 2. Combining together the SFHs and morphological analysis, we were able to reconstruct the timing sequence of the morphological transformation of massive galaxies as they evolve from the main sequence to quiescence. In this third paper, we focus on the dynamical transformation of quiescent galaxies in approximately the last half of the Hubble time. With robust stellar-age estimates from SED fitting, we study the dependence of the relationship between σ ' star and q on the stellar age of galaxies. Any significant change of the σ ' star vs. q relationship with stellar age is an indication of strong evolution in the dynamical state of massive galaxies after they quench. The redshift range considered here is 0 . 6 < z < 1, where statistically significant samples of quiescent galaxies with unresolved stellar kinematics are available. Throughout this paper, we adopt the AB magnitude system and the ΛCDM cosmology with Planck Collaboration et al. 2020 parameters, i.e., Ω m = 0 . 315 and h = H 0 / (100 km s -1 Mpc -1 ) = 0 . 673.",
"pages": [
1,
2,
3
]
},
{
"title": "2. THE SAMPLE",
"content": "The parent sample considered in this study comes from the Large Early Galaxy Astrophysics Census (LEGA-C, van der Wel et al. 2016; Straatman et al. 2018), the latest and final Data Release 3 (van der Wel et al. 2021). The LEGA-C survey is an ESO/Very Large Telescope public survey that observed with deep spectroscopy (median S/N ∼ 15 at 4000 ˚ A) for a sample of ∼ 3500 galaxies at 0 . 6 < z < 1, selected using the K -band flux from the COSMOS/UltraVISTA survey (Muzzin et al. 2013). Here we only focus on the galaxies with M ∗ > 10 10 . 5 M ⊙ , to ensure (1) good stellar mass completeness (see Figure A1 of van der Wel et al. 2021) and (2) that the environmental effects - external to the host halo of a galaxy - on the evolution of galaxies are minor (e.g. Ji et al. 2018). We refine the sample selection using the flags from the LEGA-C data release. We require FLAG MORPH = 0, to ensure that during observations the light through the slit is from a single galaxy with a regular morphology, meaning that mergers and galaxies whose LEGA-C spectra are contaminated by adjacent galaxies are excluded from the sample. We also require FLAG SPEC = 0, to exclude the galaxies with clear AGN presence identified by either IR or X-ray observations. These two constraints together ensure the high-quality spectral measures, and eliminate the cases when the interpretation of the dynamical measures becomes complicated. We cross match the LEGA-C catalog with the photometric catalog of COSMOS2020 (Weaver et al. 2022), and finally select quiescent galaxies using the UVJ criteria of Muzzin et al. (2013). Our final sample contains 952 UVJ-selected quiescent galaxies.",
"pages": [
3
]
},
{
"title": "3. MEASUREMENTS",
"content": "3.1. σ ' star and q The measurements of unresolved stellar kinematics σ ' star and morphological axis ratio q are taken directly from the LEGA-C data release (Bezanson et al. 2018a; van der Wel et al. 2021). We refer readers to those references for technical details. Briefly, q is derived from the HST/ACS I 814 imaging in the COSMOS field, following van der Wel et al. (2012) who used the Galfit package (Peng et al. 2010) to model the 2D light distribution of galaxies assuming a single S'ersic profile. σ ' star were measured with the pPFX package (Cappellari 2017) by fitting the observed spatially integrated spectra with the combination of (1) high-resolution (R = 10000) theoretical single stellar templates and emission lines at the instrumental resolution, (2) a 3rd-order multiplicative polynomial and (3) an additive polynomial. The unresolved stellar velocity dispersion σ ' star and gas velocity dispersion are estimated independently by broadening the templates with Gaussian kernels. 3.2. SED fitting with Prospector The properties of the stellar-populations of the sample galaxies are derived by fitting the multi-band photometry from the COSMOS2020 catalog with the fully Bayesian code Prospector (Johnson et al. 2021). Each one of the sample galaxies has ≈ 40 band photometry that densely samples the rest-frame UV-to-NIR wavelengths. Compared to previous COSMOS catalogs, COSMOS2020 includes the new, significantly deeper optical and NIR imaging from the Subaru/HSC and VISTA/VIRCAM surveys (Weaver et al. 2022). Two catalogs using different aperture photometric methods are available in the COSMOS2020 release, namely the CLASSIC and FARMER catalogs. By default, we use the former where aperture-matched photometry was carried out following Laigle et al. (2016). We note, however, that the difference between the two photometric catalogs is negligible for galaxies in the magnitude range ( K < 21 . 5 mag) considered here (Figure 8 and 9 in Weaver et al. 2022). The basic setups of our Prospector fitting are essentially the same as those in the first two papers of this series (Ji & Giavalisco 2022a,b). We adopt the Flexible Stellar Population Synthesis (FSPS) code (Conroy et al. 2009; Conroy & Gunn 2010) where the stellar isochrone libraries MIST (Choi et al. 2016; Dotter 2016) and the stellar spectral libraries MILES (Falc'onBarroso et al. 2011) are used. We assume the Kroupa (2001) initial mass function and the Byler et al. (2017) nebular emission model. We assume the Calzetti et al. 2000 dust attenuation law and fit the V-band dust op- ical depth with a uniform prior τ V ∈ (0 , 2). We fix the redshift to the spectroscopically-measured values from LEGA-C, and set the stellar metallicity as a free parameter with a uniform prior in the logarithmic space log( Z ∗ /Z ⊙ ) ∈ ( -2 , 0 . 19), where the upper limit of the prior is chosen because it is the highest metallicity that the MILES library covers. We use the nonparametric form of SFH that is critical for unbiased inference of stellar-population properties (e.g. Leja et al. 2019). Specifically, we use a piecewise step function composed of nine lookback time bins, where the star formation rate (SFR) is constant within each bin. We fix the first two bins as 0 -30 and 30 -100 Myr to capture recent episodes of star formation. We also fix the last bin as 0.9 t H -t H where t H is the Hubble Time of observation. The remaining six bins are evenly spaced in the logarithmic lookback time between 100 Myr and 0.9 t H . To ensure the convergence of nonparametric SFH reconstructions and reasonable uncertainty estimations (e.g. Carnall et al. 2019; Leja et al. 2019), we adopt the Dirichlet prior (Leja et al. 2017) during the Prospector SED fitting. This prior has been demonstrated to be able to recover the diverse shape of SFHs (Leja et al. 2019). Moreover, using the synthetic observations of simulated galaxies that have similar data quality like the ones we use here for the LEGA-C galaxies, Ji & Giavalisco (2022a, see their Appendix A) demonstrated that the Dirichlet prior can better recover the stellar age of high-z quiescent galaxies compared to other commonly-used priors, such as the continuity one, which is commonly adopted to measure the SFH of starforming galaxies. In Figure 2, we show the distributions of the sample galaxies in the planes of SFR vs. M ∗ , and of H δ A (Worthey et al. 1994; Worthey & Ottaviani 1997) vs. D N 4000 (4000 ˚ A break, Balogh et al. 1999). From the left panel of the Figure it is immediately clear that the UVJ-selected quiescent galaxies also occupy the parameter space of galaxies below the star-forming main sequence, i.e. with depressed SFR at any given stellar mass. This shows very good consistency among different selection methods of quiescent galaxies. In the right panel of Figure 2, each one of the galaxies is color coded according to mass-weighted stellar age. It has been extensively shown that H δ A and D N 4000 are sensitive diagnostics of galaxy's stellar age (e.g. Kauffmann et al. 2003). As the Figure shows, galaxies with lower SFR (at fixed M ∗ ), larger D N 4000 and larger (negative) H δ A also have larger ages (older stellar populations) from our Prospector fitting, demonstrating the robustness of our stellar-age inference. Because the main conclusion of this study depends on the stellar-age measures, in Appendix A we conduct a number of further tests on the robustness of the age inference. We conclude that the our stellar-age measures are robust.",
"pages": [
3,
4,
5
]
},
{
"title": "4. RESULTS",
"content": "We now present the relationship between σ ' star and q , i.e. the core of this study. We first divide the sample into the low-mass and high-mass subsamples using M ∗ = 10 11 . 3 M ⊙ , i.e. the characteristic mass to separate fast and slow rotators at z ∼ 0 (Cappellari 2016). We then further divide each subsample into three subgroups using the 33th- and 67th- percentiles of the stellar-age distribution of the entire quiescent sample.",
"pages": [
5
]
},
{
"title": "4.1. The median trend",
"content": "We measure the median relationship between σ ' star and q of each one of the subgroups using the Locally Weighted Scatterplot Smoothing (LOWESS 3 ) method that fits a smoothed curve to data points through a nonparametric approach, i.e. the process does not require to assume any specific functional form. We estimate the uncertainty of the median relationship via Monte Carlo simulations. In particular, we use Gaussian distributions to resample the individual σ ' star and q measures with their corresponding uncertainties. We then use LOWESS to measure the median σ ' star vs. q rela- of the resampled data points. We repeat these 1000 times, and use the range between 16th- and 84thpercentiles as 1σ uncertainty. To begin, as the first column of Figure 3 shows, in the youngest age bin, σ ' star decreases with increasing q , which is observed in both mass bins. This suggests that massive galaxies - regardless of their masses - still retain significant rotation in the aftermath of quenching. As they continue evolving and become older (the second and third columns of Figure 3), the relationship between σ ' star and q starts to differ in the two mass bins. For low-mass quiescent galaxies with M ∗ < 10 11 . 3 M ⊙ (the first row of Figure 3), σ ' star decreases with q in all age bins, suggesting quiescent galaxies in this mass regime continue to retain significant rotation even long time after quenching. Overall, the median relationships between σ ' star and q are statistically consistent with each other for all age bins within the uncertainties, although there is some evidence that younger galaxies have a steeper relationship than the older ones. For high-mass quiescent galaxies with M ∗ > 10 11 . 3 M ⊙ (the second row of Figure 3), the decreasing trend of σ ' star with q becomes significantly flattened as they become older. This shows that post-quenching dynamical transformations (1) happen in these very massive systems, which significantly reduce the amount of rotation and (2) these transformations are more profound in more massive quiescent galaxies. In the analysis above we divided the low- and highmass subsamples into only three age bins. We continue to study the variation of the σ ' star -q relationship by dividing the subsamples into more age bins. Specifically, instead of binning the subsamples using arbitrary bins of stellar age, we first sort stellar ages of individual galaxies into an increasing order. And starting from the first 30% of the sorted subsamples, we measure the Spearman's rank correlation coefficient ρ . Then, we keep adding older quiescent galaxies into the correlation test, and study the change of ρ as a function of the maximum age of the galaxies included in the measure. The uncertainty of ρ calculated in this way is estimated by bootstrapping the sample galaxies 1000 times, and during each bootstrapping iteration we also resample the values of σ ' star and q with their corresponding measurement uncertainties using Gaussian distributions. In Figure 4, ρ is plotted against the maximum massweighted age of the quiescent galaxies included to the Spearman's rank correlation test. For galaxies that are freshly quenched, i.e. with relatively young stellar ages, we find a strong negative ( ρ ∼ -0 . 5, i.e. a decreasing trend) correlation between σ ' star and q , regardless of stellar mass. As older quiescent galaxies are added to the correlation test, ρ gradually changes from -0 . 5 to -0 . 3 for the high-mass ( > 10 11 . 3 M ⊙ ) quiescent populations, while it remains approximately unchanged for the lowmass ones. This shows that the variation of the σ ' star vs. q relationship with age is significantly stronger in more massive quiescent galaxies, confirming the conclusion reached above based on Figure 3.",
"pages": [
5,
6
]
},
{
"title": "4.2. A toy model for the σ ' star vs. q relationship",
"content": "We now introduce a simple toy model, in an attempt to quantify the contribution from rotation to the observed relationship between σ ' star and q . As detailed already in Section 1, both random and ordered motions contribute to σ ' star , which can be expressed as In the above equation we have used σ rotation = γV sin i , which describes velocity dispersion observed through a slit (i.e. unresolved spectroscopy) due to a purely rotating disk, where V is the rotational velocity, i is the inclination and γ is the conversion factor. Because to our knowledge there is no direct, statistical estimate of γ at high redshifts, we decide to fix γ = 0 . 7 which is the median value from Cappellari et al. (2013) who measured it using the spatially resolved stellar kinematics of z ∼ 0 early type galaxies from ATLAS 3D . The inclination can be estimated using q as where q z is the thickness of a disk which - following Belli et al. (2017) - we fix to be q z = 0 . 2, namely about the minimum axis ratio observed in large extragalactic imaging surveys. The remaining unknowns in Equation 1 are σ and V (or V/σ ) that we attempt to constrain by fitting Equation 1 to the observed σ ' star vs. q relationship. To estimate the uncertainties of the fitted parameters, we use the same Monte Carlo method mentioned above by resampling the σ ' star and q measures using their uncertainties. The best-fit relationships are plotted as dashed lines in Figure 3. The observations can be reproduced very well with the toy model. The best-fit models are in excellent agreement with the LOWESS median trends. In Figure 5, the inferred V/σ is plotted as a function of stellar age. Regardless of stellar age, the V/σ of the quiescent galaxies with M ∗ < 10 11 . 3 M ⊙ is greater than 1. As the population evolves, the V/σ of these systems decreases from 2 . 3 ± 0 . 3 to 1 . 7 ± 0 . 4, implying that they remain rotationally supported, with V/σ ∼ 1 . 7, even at least 7 Gyr after their formation 4 . In contrast, during the same cosmic time, the V/σ of the highermass quiescent galaxies with M ∗ > 10 11 . 3 M ⊙ monotonically decreases, from 1 . 6 ± 0 . 3 (rotationally supported) to 1 . 0 ± 0 . 3 (velocity dispersion supported), with increasing stellar age. Before moving forward, we note several caveats of the V/σ inferred from our simple model. The misalignment between the slit and the kinematic major axis of galaxies has not been taken into account in our analysis. Neglecting this effect, however - given the large sample size of this study - should lead to an equal/similar systematic error in all subgroups and hence may not cause any substantial impacts on our conclusions, which only rely on a differential comparison. Also, the uncertainty on the value of γ in Equation 1 affects the inferred V/σ , a situation which, unfortunately, cannot be addressed at the moment. However, we note that the γ of gas kinematics has been statistically estimated at high redshifts and found to have a typical value of 0 . 6 -1 (e.g., Weiner et al. 2006). Thus, while the adopted γ = 0 . 7 is bracket by the range determined from high-z gas kinematics, quantitatively the inferred V/σ presented in Figure 5 should be taken with caution. Qualitatively, however, our conclusions about the mass-dependent evolution of V/σ with stellar age should stand despite the over-simplified model. Finally, we also clarify that, when we state, e.g., that the quiescent population of a given stellar age is rotationally supported ( V/σ > 1), we do not mean that each one of the quiescent galaxies of that age bin retains significant rotation or that it has a disk, since the spatially integrated spectroscopy does not allow us to constrain that. Instead, what we really mean is that the quiescent population of that age on average should have significant rotation. With the aforementioned caveats in mind, we now compare the inferred V/σ of this work with previous studies of the stellar kinematics in quiescent galaxies. Bezanson et al. (2018b) pioneered a LEGA-C study of the dynamical transformation of z ∼ 0 . 8 massive quiescent galaxies using a small (relative to this study) sample of ∼ 100 galaxies whose major axes are overall aligned with the slit ( | PA | < 45 · ), which allows spatially resolved analysis of stellar kinematics. They found that the most massive ( > 10 11 . 3 M ⊙ ) quiescent galaxies show much less rotation compared to less massive systems. In broad agreement 5 with Bezanson et al. (2018b), our model suggests that quiescent populations of M ∗ > 10 11 . 3 M ⊙ are significantly less rotationally supported compared to the lower-mass ones, with V/σ = 1 . 2 ± 0 . 1 for the high-mass subsample compared to V/σ = 2 . 0 ± 0 . 2 for the low-mass one. The ≈ 10 × larger in sample size of this study allows us to further group galaxies according to their stellar ages, adding a new piece of information regarding the dynamical transformation of quiescent galaxies as they evolve. We also compare our inferred V/σ with the very limited number of direct V/σ measures at higher redshifts z ∼ 2. In Figure 5, we show the results from Newman et al. (2018) who measured the stellar kinematics in three strongly lensed massive quiescent galaxies with M ∗ ≳ 10 11 . 3 M ⊙ at z ∼ 2 using spatially resolved spectroscopy. On average, those z ∼ 2 quiescent systems have even higher V/σ than that inferred for the youngest z ∼ 0 . 8 quiescent populations of similar masses ( ≳ 10 11 . 3 M ⊙ ). Note that the median stellar age of the youngest bin of our z ∼ 0 . 8 sample is ∼ 3 -4 Gyr which is longer than the Hubble time of z ∼ 2 (i.e. ∼ 3 Gyr). Thus, those z ∼ 2 quiescent galaxies must be - on average - younger (i.e. more freshly quenched) than the youngest quiescent populations considered in this study. If the dynamical state of the three lensed quiescent systems considered by Newman et al. (2018) is representative of the stellar kinematics of the entire quiescent populations at z ∼ 2 of comparable stellar mass, the implication is that the loss of angular momentum continuously and gradually happens after the cessation of star formation (quenching) in very massive ( M ∗ ≳ 10 11 . 3 M ⊙ ) galaxies, transforming them from fast rotators right after quenching to slow rotators in at least ∼ 7 Gyr after their quenching.",
"pages": [
6,
7
]
},
{
"title": "5. DISCUSSION AND SUMMARY",
"content": "To summarize, we studied the relationship between the dynamical transformation and quenching using the unresolved stellar kinematics of a sample of 952 massive ( > 10 10 . 5 M ⊙ ) quiescent galaxies at 0 . 6 < z < 1 from the LEGA-C survey. Using the SED fitting code Prospector , we robustly measured the stellar-population properties of the sample galaxies. We focused on the variation of the relationship between σ ' star and q as a function of stellar age, and of stellar mass. We found a decreasing trend of σ ' star with q for the youngest quiescent galaxies of all masses. The implication is that freshly quenched galaxies, regardless of stellar mass, still have significant rotation. This is strong evidence that the occurrence of quenching in itself at high redshift does not fully transform massive galaxies into dispersion supported systems as it takes place or immediately after. Based on what we have found in our recent study (Ji & Giavalisco 2022b), however, it is very likely that quenching happens in close temporal proximity to whatever mechanism alters the inner structure of galaxies by building dense central stellar cores. We found that the post-quenching dynamical transformation of quiescent galaxies depends on stellar mass, adding an important piece of information regarding the early formation epoch of fast and slow rotators. We remind that M ∗ = 10 11 . 3 M ⊙ is the characteristic mass that separates the fast and slow rotators at z ∼ 0 (Cappellari 2016). For very massive galaxies, with M ∗ > 10 11 . 3 M ⊙ , at z ∼ 0 . 8, we observe that the incidence of rotational support gradually reduces, as the galaxies become older. Using a simple toy model, we infer these very massive systems transform from being rotationally supported in the aftermath of quenching, with V/σ ∼ 1 . 6, to being velocity-dispersion supported ≈ 7 Gyr after their formation, with V/σ ∼ 1 . 0. In contrast, lower-mass quiescent populations with M ∗ < 10 11 . 3 M ⊙ show a much weaker post-quenching dynamical evolution. Even 7 Gyr after their formation, these lowermass quiescent systems still retain significant rotation with V/σ ∼ 1 . 7. Our findings are consistent with the picture that quiescent galaxies at the high-mass end formed in dense environments, presumably in the regions with large overdensities of the primordial density field. After quenching, these very massive galaxies continue changing their dynamical states via continuous gas accretion or inchoerent merging with other adjacent galaxies through dynamic friction. These very massive systems will eventually evolve into slow rotators seen at z ∼ 0, because multiple incoherent merging episodes can cause significant loss of angular momentum (e.g. Emsellem et al. 2011). For lower-mass quiescent galaxies, however, they very likely formed in regions with, comparatively speaking, smaller overdensities, meaning that they have shallower gravitational wells such that the frequency of merging events with other smaller galaxies is much lower than that in more massive halos. Consequently, lower-mass quiescent galaxies can retain significant rotations long time after their formation, i.e. they will evolve into fast rotators. The mass-dependent evolution of massive quiescent galaxies has been revealed in both their morphological (e.g. van der Wel et al. 2014; Ji & Giavalisco 2022a; Ji et al. 2024) and chemical properties (e.g. Kriek et al. 2019; Jafariyazani et al. 2020; Cheng et al. 2024; Beverage et al. 2024) in the high-z Universe. The purely empirical study presented here, which uses unresolved spectroscopy, provides robust evidence that, statistically, the evolution of the dynamical properties (angular momentum in particular) of quiescent galaxies also depends on stellar mass. In other words, we are witnessing the early build-up of the populations of fast and slow rotators at z ∼ 0 . 8, when the Universe was only half of its age nowadays. Undoubtedly, future spatially resolved spectroscopy of individual targets is needed to fully characterize the dispersion and intrinsic distribution of the dynamical state of massive quiescent galaxies and their evolution across cosmic time, including the mechanisms responsible for the dichotomy of stellar kinematics found in z ∼ 0 early type galaxies. Yet, results of the empirical study presented here, and its simplicity, provides robust evidence that the formation of the dichotomy was well underway at half the Hubble time. Software: Prospector (Johnson et al. 2021), FSPS (Conroy et al. 2009; Conroy & Gunn 2010), MIST (Choi et al. 2016; Dotter 2016), MILES (Falc'on-Barroso et al. 2011), GALFIT (Peng et al. 2010)",
"pages": [
7,
8,
9
]
},
{
"title": "A. TESTING THE ROBUSTNESS OF THE STELLAR-AGE INFERENCE",
"content": "Here we present and discuss in detail about our tests on the stellar-age measures from our Prospector fitting. To begin, we compare the best-fit SED models predicted by Prospector with the observed LEGA-C spectra. Specifically, we compare the median observed and predicted spectra of each one of the subgroups presented in the main text (Section 4). We remind that our Prospector fitting only used photometric data ( ≈ 40 bands), i.e. the spectra were not used in the SED modeling. This comparison thus allows us to have a direct and broad view on the quality of our SED fitting. As Figure 6 shows, the median spectra of all subgroups dim at the rest-frame UV wavelengths, and show strong stellar absorption features without any strong emission lines over rest-frame 3800 -5200 ˚ A, which demonstrates, once again, the effectiveness of the UVJ technique in identifying quiescent galaxies. The the best-fit spectra from SED modeling are in excellent agreement with the observed ones, within the uncertainties, for all subgroups. Moreover, as the right-most panel of Figure 6 shows, galaxies with older stellar ages - inferred from Prospector - also have redder observed spectra, even though the spectral information was not included during the SED fitting procedure. This agreement suggests that the stellar-age inference from our SED fitting procedure is robust. We stress, however, that the conclusion above does not at all imply that the spectral information is not needed for SED fitting. In fact, robust measurements of metallicity and elemental abundance are only possible with spectra. What we really mean is that the stellar age of high-z quiescent galaxies can be inferred robustly when densely-sampled, panchromatic photometry is available. Similar conclusions were also reached by Ji & Giavalisco (2022a) using synthetic galaxies from cosmological simulations. rest [Å] Despite the numerous advantages of fitting photometric and spectral data simultaneously (e.g., Tacchella et al. 2022), here we want to highlight one potential, serious systematics: the non-trivial aperture matching that is required when combining photometry and spectroscopy. To tackle this, most studies simply rescale (or perform the fit with the rescaling factor as a free parameter) the observed spectra to the same flux level of photometry. The big assumption behind such a procedure is that there is no strong color/stellar-population variation between the photometric aperture and the spectral slit. This assumption can be problematic given that color gradients have been clearly observed in high-z massive quiescent galaxies (e.g. Suess et al. 2020; Ji et al. 2024). Ideally, in order to simultaneously fitting photometry and spectroscopy, one needs forward modeling the instrumental effects to properly account for e.g. the mismatch of apertures, which however is beyond the scope of this work. The fact that the predicted spectra using photometry alone are in very good agreement with the observed ones provides confidence that the measures of stellar age of the sample galaxies are robust. We continue our tests on the stellar-age measures by comparing the age-color relationship from our Prospector fitting with earlier studies. Whitaker et al. (2013) stacked the 3D-HST grism spectra of massive quiescent galaxies at z ∼ 2, and fit the stacked spectral features with a solar-metallicity, single stellar population model. They found that quiescent galaxies having bluer rest-frame ( U -V ) and ( V -J ) colors are younger than those having the redder colors. We made a similar comparison and found excellent qualitative agreement, as we show in Figure 7. Similarly, Belli et al. (2019) derived and calibrated the relationship between stellar age and rest-frame UVJ colors using a sample of 24 quiescent galaxies at 1 . 5 < z < 2 . 5 with deep rest-optical spectroscopy. The stellar ages of their sample galaxies were estimated by modeling spectra and photometry combined. To assess the consistency between their results and ours, we compare the relationship between stellar age and UVJ colors of our measurements with that of Belli et al. (2019). To do so, we first derive the quiescent sequence of our sample by conducting a linear regression between the ( U -V ) and ( V -J ) colors. With the best-fit linear relationship between the two colors in hand, we use Equation 3 of Belli et al. (2019) 6 to derive the expected change of stellar age along our quiescent sequence, which is shown as the vector in Figure 7. We run a Pearson correlation test between the stellar ages from Prospector and the stellar ages inferred using the best-fit relation of Belli et al. (2019). We found a strong correlation, with Pearson coefficient r = 0 . 52), which demonstrates good qualitative agreement between our stellar-age measures and those of Belli et al. (2019). Note, however, that the above relationship (Belli et al. 2019) between age and rest-frame colors was derived at z ∼ 1 . 7. As the galaxies age and their colors evolve, we expect the relationship to change as well. While the study of the color evolution of quiescent galaxies is beyond the scope of this work, we note that, as the right panel of Figure 7 shows, a systematic age difference is observed between our measures with Prospector at z ∼ 0 . 8 and those of Belli et al. (2019) at z ∼ 1 . 7 such that the z ∼ 0 . 8 galaxies are older than those at z ∼ 1 . 7 by 0.23 dex, or 2.6 Gyr which is about the time interval between the two redshifts, t z =0 . 8 H -t z =1 . 7 H ≈ 2 . 9 Gyr. In summary, by comparing (1) the predicted spectra from Prospector with the observed ones from LEGA-C and (2) the stellar ages from Prospector and the ones inferred using the age-color relationship reported by previous studies, we showed very good agreement among different stellar-age measures. We stress that, because the conclusions of this study only depend on differential stellar-age measures (younger vs. older), rather than absolute stellar-age determinations, we conclude that the results and key conclusions presented in this study are robust.",
"pages": [
9,
10,
11
]
},
{
"title": "REFERENCES",
"content": "Balogh, M. L., Morris, S. L., Yee, H. K. C., Carlberg, R. G., & Ellingson, E. 1999, ApJ, 527, 54, doi: 10.1086/308056 Belli, S., Newman, A. B., & Ellis, R. S. 2017, ApJ, 834, 18, doi: 10.3847/1538-4357/834/1/18 -. 2019, ApJ, 874, 17, doi: 10.3847/1538-4357/ab07af Beverage, A. G., Kriek, M., Suess, K. A., et al. 2024, ApJ, 966, 234, doi: 10.3847/1538-4357/ad372d Bezanson, R., van der Wel, A., Straatman, C., et al. 2018a, ApJL, 868, L36, doi: 10.3847/2041-8213/aaf16b Bezanson, R., van der Wel, A., Pacifici, C., et al. 2018b, ApJ, 858, 60, doi: 10.3847/1538-4357/aabc55 Byler, N., Dalcanton, J. J., Conroy, C., & Johnson, B. D. 2017, ApJ, 840, 44, doi: 10.3847/1538-4357/aa6c66 Calzetti, D., Armus, L., Bohlin, R. C., et al. 2000, ApJ, 533, 682, doi: 10.1086/308692 Cappellari, M. 2016, ARA&A, 54, 597, doi: 10.1146/annurev-astro-082214-122432 -. 2017, MNRAS, 466, 798, doi: 10.1093/mnras/stw3020 Cappellari, M., Emsellem, E., Bacon, R., et al. 2007, MNRAS, 379, 418, doi: 10.1111/j.1365-2966.2007.11963.x Cappellari, M., Scott, N., Alatalo, K., et al. 2013, MNRAS, 432, 1709, doi: 10.1093/mnras/stt562",
"pages": [
11
]
}
]
2024arXiv241003627L
https://arxiv.org/pdf/2410.03627.pdf
<document>
<section_header_level_1><location><page_1><loc_11><loc_82><loc_90><loc_85></location>Cosmological predictions for minor axis stellar density profiles in the inner regions of Milky Way-mass galaxies</section_header_level_1>
<text><location><page_1><loc_10><loc_78><loc_89><loc_81></location>Madeline Lucey , 1 Robyn E. Sanderson , 1 Danny Horta , 2 Aritra Kundu , 1 Philip F. Hopkins , 3 Arpit Arora , 1 Jasjeev Singh , 1 and Nondh Panithanpaisal 4, 3</text>
<text><location><page_1><loc_9><loc_70><loc_90><loc_77></location>1 Department of Physics & Astronomy, University of Pennsylvania, 209 S 33rd St., Philadelphia, PA 19104, USA 2 Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA 3 TAPIR, Mailcode 350-17, California Institute of Technology, Pasadena, CA 91125, USA 4 Observatories of the Carnegie Institution for Science, 813 Santa Barbara St, Pasadena, CA 91101, USA</text>
<section_header_level_1><location><page_1><loc_45><loc_67><loc_55><loc_68></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_38><loc_86><loc_66></location>Λ-CDM cosmology predicts the hierarchical formation of galaxies which build up mass by merger events and accreting smaller systems. The stellar halo of the Milky Way has proven to be useful a tool for tracing this accretion history. However, most of this work has focused on the outer halo where dynamical times are large and the dynamical properties of accreted systems are preserved. In this work, we investigate the inner galaxy regime, where dynamical times are relatively small and systems are generally completely phase-mixed. Using the FIRE-2 and Auriga cosmological zoomin simulation suites of Milky Way-mass galaxies, we find the stellar density profiles along the minor axis (perpendicular to the galactic disk) within the NFW scale radii (R ≈ 15 kpc) are best described as an exponential disk with scale height < 0.3 kpc and a power law component with slope α ≈ -4. The stellar density amplitude and slope for the power law component is not significantly correlated with metrics of the galaxy's accretion history. Instead, we find the stellar profiles strongly correlate with the dark matter profile. Across simulation suites, the galaxies studied in this work have a stellar to dark matter mass ratio that decreases as 1 /r 2 along the minor axis.</text>
<section_header_level_1><location><page_1><loc_19><loc_35><loc_37><loc_36></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_11><loc_48><loc_33></location>Understanding the formation and evolution of galaxies is one of the main goals of astrophysics today. In the inside-out theory of galaxy formation, the innermost regions of galaxies form first and are therefore especially informative for studying the earliest epochs of galaxy formation (Peebles 1969; Larson 1976; Fall & Efstathiou 1980; Mo et al. 1998; Somerville et al. 2008; Dutton et al. 2011). Furthermore, a substantial fraction of a galaxy's stellar mass is within the inner region, making it information-rich. The Milky Way (MW) presents a unique opportu-</text>
<text><location><page_1><loc_52><loc_33><loc_92><loc_36></location>nity to study the inner region of a galaxy in exquisite detail.</text>
<text><location><page_1><loc_52><loc_10><loc_92><loc_32></location>Along with more than a third of massive (M ∗ > 10 10 M ⊙ ) disk galaxies in the local Universe (Sellwood & Wilkinson 1993; Masters et al. 2011; Gavazzi et al. 2015), the MW hosts a galactic bar in its center (Blitz & Spergel 1991; Weiland et al. 1994; Peters 1975; Binney et al. 1991). The bulk of the stellar mass in the inner Galaxy participates in the bar structure (Howard et al. 2009; Shen et al. 2010; Ness et al. 2013a; Debattista et al. 2017). The MW also has an X-shaped structure in its center (Nataf et al. 2010; McWilliam & Zoccali</text>
<text><location><page_2><loc_8><loc_77><loc_48><loc_91></location>2010; Ness et al. 2012; Wegg & Gerhard 2013; Ness & Lang 2016), which is characteristic of a boxy/peanut-shaped (B/P) bulge and consistent with simulations and observations of barred galaxies (Combes et al. 1990; Athanassoula 2005; Martinez-Valpuesta et al. 2006; Bureau et al. 2006; Laurikainen et al. 2014; Debattista et al. 2019).</text>
<text><location><page_2><loc_8><loc_28><loc_48><loc_76></location>It is currently debated whether the MW also hosts a less-massive metal-poor classical bulge component (Babusiaux et al. 2010; Hill et al. 2011; Zoccali et al. 2014). The major evidence for a metal-poor classical bulge is based on the stellar kinematics as a function of metallicity. Specifically, metal-poor stars in the inner Galaxy rotate slower and have a higher velocity dispersion than the metal-rich stars (Ness et al. 2013b; Kunder et al. 2016; Arentsen et al. 2020). However, Debattista et al. (2017) demonstrated that these observations may be the result of the overlapping Galactic halo whose density would peak in the center of the Galaxy. In fact, 25% of the RR Lyrae stars in the inner Galaxy were found on orbits with apocenters > 3.5 kpc from the Galactic center (Kunder et al. 2020). Similarly, Lucey et al. (2020) found that about 50% of metal-poor giants in the inner Galaxy are interlopers with apocenters > 3.5 kpc and that the fraction of interlopers increases with decreasing metallicity. After removing these stars from the sample, Lucey et al. (2020) found that the velocity dispersion decreased and there was no longer evidence for a classical bulge component in the kinematics.</text>
<text><location><page_2><loc_8><loc_11><loc_48><loc_27></location>Whether they comprise a classical bulge or the innermost part of the Galactic halo, the stars which are not part of the bar or B/P bulge in the inner Galaxy give us a unique clue to the MW's formation history and galaxy evolution as a whole (Rix et al. 2022; Horta et al. 2021). Classical bulges and halos are both thought to be made through mergers and galaxy accretion events (Kauffmann et al. 1993;</text>
<text><location><page_2><loc_52><loc_77><loc_92><loc_91></location>Kobayashi & Nakasato 2011; Guedes et al. 2013; Freeman & Bland-Hawthorn 2002; Belokurov 2013; Bland-Hawthorn & Gerhard 2016; Johnston et al. 1995; Helmi 2020). Therefore, it is thought that the total stellar mass and radial profile of the halo/bulge components may reflect the galaxy's cumulative accretion history (Helmi 2020; Han et al. 2022).</text>
<text><location><page_2><loc_52><loc_47><loc_92><loc_76></location>However, there are also suggestions that the inner halo (Galactocentric radius < 10 kpc) may be a distinct Galactic component with in-situ origins. One theory for creating an in-situ inner halo is ELS-like contraction (Eggen, LyndenBell, & Sandage 1962), with early star formation occurring on halo-like random orbits (Carollo et al. 2007, 2010; Belokurov & Kravtsov 2022; El-Badry et al. 2018; Yu et al. 2023). Another proposed theory for the creation of an in-situ inner halo includes disruption of an old thick disk during a major merger (Belokurov et al. 2020). By comparing MW observations to cosmological zoom-in simulations, we can better understand the history of these stars, and the MW's evolution.</text>
<text><location><page_2><loc_52><loc_11><loc_92><loc_46></location>The radial density profile of the MW's halo is typically quantified with a double power law, although single and triple power laws have also been suggested. Studies use a variety of stellar distance tracers, and radial ranges resulting in large spread in the MW estimates. Generally, the breaking radius estimates range from ≈ 1030 kpc (Sesar et al. 2013; Han et al. 2022) with inner slopes ranging from -1 to -3 (Sesar et al. 2010, 2013; Faccioli et al. 2014; Han et al. 2022) and outer slopes from -3 to -6 (Sesar et al. 2010, 2013; Deason et al. 2011). Most of these works are based on stars with Galactocentric radii > 5 kpc. The extension of the stellar density profile within 5 kpc, where Galactic extinction and crowding make observations difficult, is less certain. However, the available estimates are generally consistent with the results at larger radii (Pietrukowicz et al. 2015; P'erez-Villegas et al.</text>
<text><location><page_3><loc_8><loc_81><loc_48><loc_91></location>2017; Yang et al. 2022). The MW's halo density profile is generally consistent with nearby MWmass disk galaxies which have power law slopes of -3 to -5 (Harmsen et al. 2017). Furthermore, our nearest disk galaxy neighbor, M31, has a power law slope of -3.7 (Ibata et al. 2014).</text>
<text><location><page_3><loc_8><loc_37><loc_48><loc_80></location>Cosmological zoom-in simulations provide a crucial tool for interpreting the observational properties of the MW and other galaxies. Using the IllustrisTNG suite of simulations, Pillepich et al. (2018) found that less massive galaxies generally have steeper halo density slopes, and that the profiles flatten towards the galactic center. They also found MW-mass galaxies to have a stellar density power law slope of ≈ -4.3 on average for stars within the half-light radius. Using the Auriga simulations of MW-mass galaxies, Monachesi et al. (2019) found that galaxies with fewer progenitors have more massive halos. Furthermore, they found that galaxies with fewer progenitors have steeper halo stellar density profile slopes, but the correlation is quite weak (see Figure 11 in Monachesi et al. 2019). It has also been suggested that the halo stellar density slope is related to the accretion time of the dominant progenitor (D'Souza & Bell 2018). In general, results between simulation suites differ, and a definitive metric that predicts the stellar halo density slope is yet to be found.</text>
<text><location><page_3><loc_8><loc_11><loc_48><loc_37></location>Similar to MW estimates, results from ΛCDM simulations of Milky Way-mass galaxies show a large range of radial density profile power law slopes. Whether the works use single or broken power laws, the slopes range from ≈ -2 to ≈ -6.5 (Monachesi et al. 2019; Deason et al. 2013; Font et al. 2020; Cooper et al. 2010; Amorisco 2017), which is consistent with the range observed in nearby disk galaxies, including M31 and the MW (Harmsen et al. 2017). However, studies of the simulated stellar halos use either a kinematic or spatial selection to define the halo which can impact the results (Monachesi et al. 2019; Font et al. 2020; Pillepich et al. 2018;</text>
<text><location><page_3><loc_52><loc_71><loc_92><loc_91></location>Cooper et al. 2010). For comparison with observations, a spatial selection is generally preferred since kinematic information is not always available. However, in order to avoid contamination by disk stars many of these works only study the halo profile beyond ∼ 10 kpc from the galactic center (Monachesi et al. 2019; Pillepich et al. 2018; Cooper et al. 2010). As we are interested in the inner most region in this work, we instead study the minor axis profile and simultaneously fit a power-law halo with an exponential disk.</text>
<text><location><page_3><loc_52><loc_43><loc_92><loc_71></location>In this work, we use the FIRE-2 and Auriga MW-mass galaxy simulation suites to study inner minor axis stellar density profiles and explore their relation to the galaxy's mass assembly history. In Section 2 and 2.2, we provide a brief description of the FIRE-2 and Auriga simulation suites, respectively. Our method for parametrizing the density profiles of the simulated galaxies is described in Section 3 and Appendix A in more detail. We compare the stellar density profile parameters with metrics of the galaxy's merger and accretion history in Section 4 and with the dark matter profile parameters in Section 5. Last in Section 6, we discuss the results and summarize the conclusions.</text>
<section_header_level_1><location><page_3><loc_58><loc_38><loc_86><loc_41></location>2. COSMOLOGICAL ZOOM-IN SIMULATIONS</section_header_level_1>
<text><location><page_3><loc_52><loc_17><loc_92><loc_37></location>This work is primarily based on the FIRE-2 cosmological zoom-in simulations of Milky Waymass galaxies (Wetzel et al. 2023). However, we also make use of the publicly available Auriga simulation suite as comparison (Grand et al. 2024). As these two suites are numerically relatively similar with a few key differences in their physical prescriptions, they provide a unique opportunity to learn about galaxy formation physics by studying how the galaxies differ, or stay the same, between suites.</text>
<section_header_level_1><location><page_3><loc_54><loc_13><loc_90><loc_15></location>2.1. FIRE-2 Milky Way-mass Simulations</section_header_level_1>
<text><location><page_3><loc_52><loc_9><loc_92><loc_12></location>From FIRE-2, we use the Latte suite of seven isolated Milky Way-mass galaxies (Wetzel et al.</text>
<figure>
<location><page_4><loc_12><loc_31><loc_89><loc_86></location>
<caption>Figure 1. In the top panels we show the dynamical time as a function of galactocentric radius ( R ) from R =0 kpc to their respecitve NFW scale radii. The dynamical time is calculated from the mean density of dark matter, stars and gas inside radius R . In the lower panels, we show the fraction of accreted stellar mass profile along the minor axis for the same radial range. The left panel shows results for the isolated FIRE-2 simulations while the middle panel shows results for the pairs. Results for these galaxies are colored based on their formation redshift (z 0 . 5 ). The right panel shows results for the Auriga galaxies in random shades of green.</caption>
</figure>
<text><location><page_5><loc_8><loc_71><loc_48><loc_91></location>2016) as well as the ELVIS suite of three Local Group-like pairs of Milky Way-mass galaxies (Garrison-Kimmel et al. 2019a). All of these simulations are run with the FIRE-2 physics model (Hopkins et al. 2018a) using the GIZMO 1 gravity plus hydrodynamics code in meshless finite-mass (MFM) mode (Hopkins 2015). For a complete and detailed description of the simulation implementations, we refer the reader to the above papers. Below, we summarize a few key properties.</text>
<text><location><page_5><loc_8><loc_43><loc_48><loc_71></location>Each simulation assumes flat Λ-CDM cosmology with parameters consistent with Planck Collaboration et al. (2014). Specifically, the Latte suite (excluding m12w) uses Ω m = 0.272, Ω b = 0.0455, σ 8 = 0.807, n s = 0.961, h = 0.702. Of the Elvis suite of galaxy pairs, Thelma & Louise and Romulus & Remus both use the same cosmology as in the original ELVIS darkmatter-only (DMO) suite: Ω m = 0.266, Ω b = 0.0449, σ 8 = 0.801, n s = 0.963, h = 0.71. The third ELVIS galaxy pair, Romeo & Juliet along with one Latte galaxy, m12w, both use the updated parameters from Planck Collaboration et al. (2020), Ω m = 0.31, Ω b = 0.048, σ 8 = 0.82, n s = 0.97, h = 0.68.</text>
<text><location><page_5><loc_8><loc_16><loc_48><loc_42></location>The physical prescription for feedback mechanisms are known to significantly impact the distribution of mass in simulated galaxies (Pontzen & Governato 2014; Lazar et al. 2020). FIRE2 simulations include implementations of stellar feedback from stellar winds, radiation pressure from young stars, Type II and Type Ia supernovae, photoelectric heating, and photoheating from ionizing radiation, which regulates star formation. The gas density threshold for star formation is n SF > 1000 cm -3 . Feedback event rates, luminosities, energies, massloss rates, and other quantities are tabulated directly from stellar evolution models (STAR-</text>
<text><location><page_5><loc_52><loc_75><loc_92><loc_91></location>BURST99; Leitherer et al. 1999). All 13 simulated galaxies have dark matter halo masses at z =0 of M 200 = 1 -2 . 1 × 10 12 M ⊙ (Sanderson et al. 2018). The Latte suite galaxies have initial stellar particle masses of 7070 M ⊙ , while the ELVIS suite galaxies have higher resolution at initial stellar particle masses of 3500 M ⊙ . Star particle softening lengths are ≈ 4 pc and dark matter force softening is ≈ 40 pc.</text>
<text><location><page_5><loc_52><loc_41><loc_92><loc_74></location>These simulated galaxies show agreement with the observed stellar mass-dark matter halo mass relation across cosmic time (Hopkins et al. 2018a). They are also consistent with a number of key observed properties of the Milky Way, including the stellar halo mass fraction (Sanderson et al. 2018), the existence of a metal-rich in-situ stellar halo component (Bonaca et al. 2017), and the radial and vertical structure of the stellar disk (Ma et al. 2017; Sanderson et al. 2020; Bellardini et al. 2021; McCluskey et al. 2024). The simulated satellite populations, including stellar streams, are also consistent with observations of these populations around the Milky Way and M31 (Wetzel et al. 2016; Samuel et al. 2020; Garrison-Kimmel et al. 2019b; Panithanpaisal et al. 2021; Cunningham et al. 2022; Shipp et al. 2023).</text>
<text><location><page_5><loc_52><loc_11><loc_92><loc_40></location>To understand the role of hierarchical formation in building the minor axis stellar profile, we identify and track star particles that formed outside of the main progenitor galaxy across the simulation as a function of time. These star particles are formed in subhalos before they interact with the main branch progenitor. The redshift when the main progenitor reaches a mass that is 3 times larger than the next most massive luminous halo, z MR 3:1 , is when the main progenitor emerges as the dominant host galaxy (Santistevan et al. 2020; Horta et al. 2024). Before this redshift, systems that merge with the main progenitor are labeled as building blocks (Horta et al. 2024). After this redshift, the luminous halos that are within the virial radius of</text>
<text><location><page_6><loc_8><loc_60><loc_48><loc_91></location>the central host galaxy at z = 0 are labeled as accreted. These systems are tracked with the help of the ROCKSTAR halo catalogs and the halo merger trees (Behroozi et al. 2013a,b); along with every star particle that make-up a present-day substructure (satellites, streams, or phase-mixed) within this virial radius. These accreted substructures are classified as phasemixed, if they satisfy the following criteria at the present-day (Panithanpaisal et al. 2021): (1) the total stellar mass is greater than 10 4 . 5 M ⊙ , (2) the maximum separation between any two star particles is greater than 120 kpc, and (3) the median of the local velocity dispersion of the star particles is greater than a stellar-mass dependent threshold value (see Equation 2 of Panithanpaisal et al. (2021)).</text>
<section_header_level_1><location><page_6><loc_18><loc_57><loc_38><loc_58></location>2.2. Auriga Simulations</section_header_level_1>
<text><location><page_6><loc_8><loc_34><loc_48><loc_56></location>The Auriga cosmological zoom-in simulations provide a good comparison to the FIRE-2 simulations because they both have similar resolution and broadly MW-like z = 0 disk galaxies, but different physical prescriptions. In this work, we specifically use the 6 Milky Way-mass galaxies simulated at 'level 3' resolution, which is most similar to the resolution of FIRE-2. For a detailed description of these simulations we refer the reader to Grand et al. (2017) and Grand et al. (2024). Below, we provide a brief overview of a few key details.</text>
<text><location><page_6><loc_8><loc_9><loc_48><loc_33></location>The Auriga simulations assume flat Λ-CDM cosmology with parameters taken from Planck Collaboration et al. (2014). They are run using the gravo-magnetohydrodynamics movingmesh code AREPO (Springel 2010; Pakmor et al. 2016). The halos are selected from the dark-matter-only EAGLE simulations (Schaye et al. 2015) as Milky Way-mass (1 < M 200 / [10 12 M ⊙ ] < 2) halos at z = 0. The dark matter particle mass for these simulations is 50,000 M ⊙ while the stellar particles have mass 6,000 M ⊙ . The softening lengths for the gas, stars, and dark matter are ≈ 188 pc.</text>
<text><location><page_6><loc_52><loc_65><loc_92><loc_91></location>The Auriga simulations implement stellar feedback from Type II supernovae. However, unlike the FIRE-2 supernova rates that evolve in time with the star particles, the Auriga simulations have instantaneous feedback at the time of star formation. Furthermore, the Auriga simulations are missing the early stellar feedback prescriptions included in FIRE-2 which are thought to regulate star formation. The gas density threshold for star formation is n SF > 0 . 13 cm -3 , which is significantly smaller than the FIRE-2 threshold of n SF > 1000 cm -3 . On the other hand, Auriga includes AGN feedback, while FIRE-2 does not.</text>
<text><location><page_6><loc_52><loc_50><loc_92><loc_65></location>For the Auriga galaxies, stellar particles are labeled as accreted if they form in a subhalo and are within the R 200 of the main host galaxy as z = 0. Similar to FIRE-2, the Auriga simulations have been shown to produce galaxies that are disk-dominated with Milky Way-mass stellar masses, sizes, rotation curves, and metallicities (Grand et al. 2017).</text>
<section_header_level_1><location><page_6><loc_55><loc_44><loc_92><loc_49></location>2.3. Inner Galaxy Dynamical Time and Accreted Fraction Profile along the Minor Axis</section_header_level_1>
<text><location><page_6><loc_52><loc_9><loc_92><loc_43></location>To confirm that the inner regions of the simulated galaxies are similarly well phase-mixed, we investigate the dynamical time as a function of galactocentric radius. In the top panels of Figure 1, we show the dynamical time as a function of galactocentric radius out to the NFW scale radius of the main dark matter halo for each simulated galaxy used in this work. Specifically, we show results for the isolated FIRE-2 galaxies in the left panel, the FIRE-2 pair galaxies in the middle panel, and the Auriga galaxies in the right panel. The FIRE-2 galaxies are colored by the redshift where they have reached 50% of their z = 0 mass ( z 0 . 5 ). Since the z 0 . 5 metric is not available to us for the Auriga galaxies, they are shown in random shades of green. The dynamical times are calculated using the following formula:</text>
<formula><location><page_7><loc_13><loc_85><loc_48><loc_89></location>t dyn ( R ) = √ 3 π Gρ ( < R ) (1)</formula>
<text><location><page_7><loc_8><loc_64><loc_48><loc_84></location>where G is the gravitational constant and ρ ( < R ) is the average density inside radius R . As the average density of a galaxy decreases as a function of radius, the dynamical time increases. At the NFW scale radii of these simulated galaxies the dynamical times is on average ≈ 500 Myr with a range of ≈ 250-750 Myr. As the phase-mixing timescale is a few times the dynamical time, we expect mergers with infall times ⪆ 2 Gyr ago to be well phase-mixed inside this radius.</text>
<text><location><page_7><loc_8><loc_32><loc_48><loc_63></location>In the bottom panels of Figure 1, we explore the birth origin of the stars that comprise the minor axis profile. As the halos/bulges of galaxies are thought to be built from accreted systems, we probe the distance along the minor axis from the galactic center where accreted stars begin to outnumber stars that formed insitu. Specifically, we show the fraction of accreted stars within a cylinder along the minor axis which is defined to be perpendicular to the galaxy's disk plane. The cylinder's radius is 1 / √ π so that each cylindrical bin with height of 1 kpc has the volume of 1 kpc 3 . Similar to the top panels, we only calculate the accreted fraction out to the NFW scale radius of each galaxy. The black dashed line indicates the accreted fraction value of 50%.</text>
<text><location><page_7><loc_8><loc_9><loc_48><loc_31></location>The FIRE-2 galaxies generally increase their accreted fraction with increasing distance from the Galactic center. The isolated galaxies (left panel) typically reach an accreted fraction of ∼ 50% just inside of the NFW scale radius, while the accreted fraction is generally < 10% within R < 5 kpc. The paired galaxies (middle panel) show larger spread in their accretedfraction profiles, with the earlier forming galaxies (larger z 0 . 5 ) having lower accreted fractions closer to the galactic center. The Auriga galaxies show significantly different behavior than the</text>
<text><location><page_7><loc_52><loc_65><loc_92><loc_91></location>FIRE-2 galaxies in that their accreted fraction quickly jumps to > 10-40% at ∼ 2 kpc above the galactic center, rather than the gradual increase seen in FIRE-2. In the Auriga galaxies, the accreted fraction only increases slightly with increasing distance from the galactic center so that just within the NFW scale radii (R ∼ 15 kpc) the galaxies generally have an accreted fraction within 10-50%. It is possible this may be due to differences in method of defining accreted stars. Although the methods are philosophically the same, there could be implementation differences which require deeper investigation beyond the scope of this work.</text>
<section_header_level_1><location><page_7><loc_56><loc_60><loc_91><loc_63></location>3. STELLAR AND DARK MATTER PROFILES ALONG THE MINOR AXIS</section_header_level_1>
<text><location><page_7><loc_52><loc_29><loc_92><loc_59></location>The main focus of this work is on the cosmological predictions for minor axis stellar density profiles in the inner regions of Milky Waymass galaxies. We define the inner region as within the NFW profile (Navarro et al. 1997) scale radius of the dark matter halo. Specifically, for the FIRE-2 simulations, we use the value given from the halo finder ROCKSTAR (Behroozi et al. 2013a) for the host dark matter halo, rounded up to an integer. For the Latte galaxies, the NFW scale radii range from 16-24 kpc with an average of 20 kpc. For the ELVIS galaxies, these values range from 13-20 kpc with an average of 16 kpc. The estimate of the NFW scale radii for the Auriga galaxies comes from Callingham et al. (2020).</text>
<text><location><page_7><loc_52><loc_18><loc_92><loc_28></location>We select particles within a cylinder along the minor axis with a radius of 1 / √ π so that each cylindrical bin with height of 1 kpc has the volume of 1 kpc 3 . After multiplying these particles by their mass, we retrieve the mass density profile.</text>
<text><location><page_7><loc_52><loc_14><loc_92><loc_17></location>Wefit a simple power law and exponential disk model to the mass density profile. Explicitly,</text>
<formula><location><page_7><loc_57><loc_8><loc_92><loc_12></location>ρ ∗ ( R ) = ρ 0 ,H 1 + ( R r H ) α + ρ 0 ,D e -R/h D (2)</formula>
<table>
<location><page_8><loc_18><loc_48><loc_92><loc_90></location>
<caption>Table 1. Simulated Galaxy Properties</caption>
</table>
<text><location><page_8><loc_8><loc_38><loc_92><loc_47></location>Note -We provide the following properties for each galaxy at z = 0. M 200c and R 200c are the total mass and spherical radius in which the average density is 200 × the critical density of the universe. M ∗ , 90 is the stellar mass within a spherical radius that encloses 90% of the stellar mass within 20 kpc (Wetzel et al. 2023). N sp is the number of satellites which contribute 90% of the accreted stellar mass (Monachesi et al. 2019). ρ 0 , H and α are the stellar halo central mass density and slope, while ρ 0 , DM and α DM are the corresponding parameters for the dark matter component (see Appendix A).</text>
<text><location><page_8><loc_8><loc_10><loc_48><loc_36></location>where ρ 0 ,H , and ρ 0 ,D are the central halo/bulge and disk mass densities, respectively. The scale height for the disk is h D , the scale radius for the halo is r H and the halo power law slope is given by α . To avoid the impact of binning when fitting to the simulated galaxy profiles, we perform the fit to the reverse cumulative distribution function (CDF) normalized by the total mass within the NFW scale radius and 1 / √ π kpc of the minor axis. This fit provides the disk scale height, halo power law slope, halo scale radius, and the relative strength of the disk mass density compared to the halo. To derive the ρ 0 ,H and ρ 0 ,D we then fit to the unnormalized</text>
<text><location><page_8><loc_52><loc_31><loc_92><loc_36></location>profile. For further details and figures demonstrating the fits, we refer the reader to Appendix A.</text>
<text><location><page_8><loc_52><loc_10><loc_92><loc_30></location>In addition to quantifying the density profile, we are interested in determining the radius at which halo substructures become significant. To do this we measure the density profile along different lines and compare to the profile along the minor (z-)axis. Specifically, we rotate the the simulated galaxy about the x-axis by a random angle, ϕ , between 0 and π/ 4 radians, in order to avoid the galactic disk. We then rotate the galaxy again but this time about the z-axis by a random angle, θ , between 0 and 2 π radians.</text>
<text><location><page_9><loc_8><loc_77><loc_48><loc_91></location>Each time we measure the mass density profile along the new z-axis using the same cylinder as before. We perform 100 unique combinations of rotations to obtain 100 different estimates of the mass density profile. We then compute the fractional difference between this and the original, non-rotated z-axis profile (see Figure 8-4 in Appendix A).</text>
<text><location><page_9><loc_8><loc_62><loc_48><loc_76></location>In order to understand how the stellar component relates to its dark matter counterpart, we also fit the dark matter halo distribution using the same technique. The only difference is that we do not include a exponential disk component in the functional form. Therefore, the functional form of the dark matter mass density distribution is the following power law:</text>
<formula><location><page_9><loc_13><loc_56><loc_48><loc_60></location>ρ DM ( R ) = ρ 0 ,DM 1 + ( R r DM ) α DM (3)</formula>
<text><location><page_9><loc_8><loc_34><loc_48><loc_54></location>where ρ 0 ,DM is the dark matter mass density at R = 0, r DM is the scale radius and α DM is the power law slope. We follow the same procedure as the stellar component for fitting the functional form as well as the angle dependence. We find that the fractional difference between the dark matter density profiles at random angles is significantly smaller at all radii than that of the stellar profiles, indicating that the stellar particles have more significant substructure than the dark matter.</text>
<section_header_level_1><location><page_9><loc_12><loc_25><loc_46><loc_32></location>4. INNER MINOR AXIS STELLAR DENSITY PROFILES ARE WEAKLY CORRELATED WITH ACCRETION HISTORY</section_header_level_1>
<text><location><page_9><loc_8><loc_9><loc_48><loc_24></location>Generally, it is thought that the inner halo stellar density profile traces the cumulative accretion history of a galaxy (Helmi 2020; Han et al. 2022). Similarly, the growth of bulges is thought to occur through merger events (Aguerri et al. 2001; Hopkins et al. 2010). This idea can be directly tested using cosmological zoom-in simulations. In this section, we com-</text>
<text><location><page_9><loc_52><loc_54><loc_92><loc_91></location>pare measured properties of the z = 0 minor axis stellar density profiles of Milky Waymass galaxies (see Section 3) with their accretion histories. In order to quantify and interpret strengths of correlations in this work, we use the Spearman correlation coefficient which measures the monotonicity of the relationship between two variables (Spearman 1904; Dancey & Reidy 2004). Therefore, the Spearman correlation coefficient will be close to 1 if the data have similar rank in the two variables, i.e., the data point with the largest x -value also has the largest y -value and so on. Furthermore, the Spearman correlation coefficient will be close to -1 if the data have almost opposite rank. Generally, a correlation is considered weak for Spearman coeffcients with an absolute value < 0.40. A moderate correlation has an absolute Spearman coefficient between 0.40 and 0.60, while a strong correlation has > 0.60.</text>
<text><location><page_9><loc_52><loc_11><loc_92><loc_54></location>In Figure 2, we show how the amount of accreted stellar mass in each galaxy relates to α , the power law slope (bottom panels), and ρ 0 , H , the central stellar mass density for the powerlaw component (top panels). The left and middle panels show results for the FIRE-2 galaxies while the right panels show results for the Auriga galaxies (green Xs). The FIRE-2 pair galaxies are shown as diamonds, while the isolated galaxies are circles. The FIRE-2 results are colored by the redshift when the proto-Milky Way emerges, z MR 3:1 , defined in Horta et al. (2024) as the redshift when the main galaxy halo becomes 3 times more massive than the next most massive luminous halo. For the Auriga galaxies the horizontal axis is the reported accreted stellar halo mass from Monachesi et al. (2019). For the FIRE-2 galaxies we use two different metrics for the accreted stellar mass. The left panel horizontal axis is the log 10 of the stellar mass of the simulated galaxy's building blocks from Horta et al. (2024). Specifically, the building blocks are defined as luminous ha-</text>
<figure>
<location><page_10><loc_10><loc_49><loc_91><loc_88></location>
<caption>Figure 2. How the minor axis stellar density profile parameters relate to metrics of the accreted mass in the simulated galaxies. Specifically, the top row shows the stellar density amplitude ( ρ 0 ,H ) on the y-axis, while the bottom row shows the stellar density slope ( α ). The left and middle panels show results for the FIRE-2 simulations, colored by z MR 3:1 . The left panel has the stellar mass of the building blocks (i.e., systems accreted before z MR 3:1 ), while the middle panel shows the total stellar mass accreted after z MR 3:1 . The right panels shows results for the Auriga galaxies in green.</caption>
</figure>
<text><location><page_10><loc_8><loc_27><loc_48><loc_34></location>s which merge with the main branch before z MR 3:1 . In the middle panel, the horizontal axis is the log 10 of the phase-mixed stellar mass accreted after z MR 3:1 from Kundu et al. (in prep).</text>
<text><location><page_10><loc_8><loc_10><loc_48><loc_27></location>It is interesting to note that although the Auriga central stellar densities overlap with the FIRE-2 results, the range is smaller: all Auriga halos have ρ 0 , H < 10 9 M ⊙ / kpc 3 while almost a third (4/15) FIRE-2 galaxies have power law components with central densities larger than that. The opposite is true for the power law slopes, α , in that the Auriga galaxies have larger scatter than FIRE-2. However, we note that</text>
<text><location><page_10><loc_52><loc_22><loc_92><loc_34></location>these differences are likely due to fit degeneracies. There is an especially strong degeneracy between ρ 0 , H and r H in that smaller r H indicates a more cuspy density profile with a higher central density. We note that the FIRE-2 galaxies with larger ρ 0 , H all have r H that are smaller than the Auriga galaxies (see Appendix A).</text>
<text><location><page_10><loc_52><loc_10><loc_92><loc_21></location>Both Auriga and FIRE-2 show weak correlations between the various measures of accreted stellar mass and the minor axis stellar density profile. For the FIRE-2 galaxies, the strongest correlation is between the power law slope and the total stellar mass of the building blocks.</text>
<figure>
<location><page_11><loc_10><loc_49><loc_91><loc_88></location>
<caption>Figure 3. Similar to Figure 2, but the horizontal axes now show the number of significant progenitors (left and right panels) or accreted systems (middle panel). As with Figure 2, there are not strong correlations between the minor axis stellar density profiles and the accretion history metrics.</caption>
</figure>
<text><location><page_11><loc_8><loc_23><loc_48><loc_40></location>The Spearman correlation coefficient for these two measures is 0.35, which signifies a weak relationship. Similarly, for the Auriga galaxies the power law slope is most correlated with the accreted stellar mass, with a Spearman correlation coefficient of -0.31. In conclusion, we do not find strong correlations between the accreted stellar mass and the amplitude or slope of the minor axis stellar density profile.</text>
<text><location><page_11><loc_8><loc_10><loc_48><loc_23></location>Figure 3 is the same as Figure 2 except for the horizontal axes. In Figure 3, the horizontal axis for the left and right panels is the number of 'significant progenitors' from Monachesi et al. (2019), defined as the number of satellites that together contribute 90% of the total accreted stellar mass. The middle panel horizontal axis</text>
<text><location><page_11><loc_52><loc_36><loc_92><loc_40></location>is the number of phase- mixed accreted systems from Kundu et al. (in prep).</text>
<text><location><page_11><loc_52><loc_10><loc_92><loc_36></location>The Auriga galaxies have a positive correlation between the number of significant progenitors and the stellar density profile slope with a Spearman correlation coefficient of 0.57. This is classified as a moderate relationship. However, a negative correlation is reported in Monachesi et al. (2019) with the 28 Milky Way-mass Auriga galaxies run at lower resolution, although the correlation is weak. We also see a negative correlation between the stellar density profile slope and number of accreted systems with the FIRE-2 galaxies, but the correlation is weak with a Spearman correlation coefficient of -0.37. In general, we find inconclusive evidence that</text>
<text><location><page_12><loc_8><loc_86><loc_48><loc_91></location>the number of merger events impacts the minor axis stellar density profile for the inner regions of these simulated galaxies.</text>
<section_header_level_1><location><page_12><loc_11><loc_81><loc_45><loc_85></location>5. THE STARS FOLLOW THE DARK MATTER</section_header_level_1>
<text><location><page_12><loc_8><loc_64><loc_48><loc_80></location>It is well accepted that dark matter is a significant component of galactic halos (White & Rees 1978; Bland-Hawthorn & Gerhard 2016; Helmi 2020). In this section, we compare the measured minor axis stellar density profiles with the corresponding dark matter density profiles. In general, we find that the strength and slope of the stellar profile is highly correlated with the dark matter profile.</text>
<text><location><page_12><loc_8><loc_13><loc_48><loc_63></location>In Figure 4, we show a graphical representation of the correlations between the minor axis stellar and dark matter density profile parameters along with parameters describing the accretion history of the galaxy. Specifically the Spearman correlation coefficients are represented as ellipses with ellipticity, orientation and color based on the their value. Strong positive correlations are shown with narrow ellipses that point up to the right in dark blue while strong negative correlations point down to the right in dark red. Weak correlations with coefficients close to zero are shown as white circles. Results for how the amplitude and slope of the FIRE-2 galaxies' minor axis stellar density profiles relate to the stellar mass of the building blocks, accreted stellar mass, number of significant progenitors, and number of accreted systems, along with the minor axis dark matter density profile's amplitude and slope are shown on the left in order from left to right. Results for the correlations of the amplitude and slope of Auriga's minor axis stellar density profiles with the accreted stellar mass, number of significant progenitors, minor axis dark matter density profile amplitude and slope are shown on the right, also in order from left to right.</text>
<text><location><page_12><loc_8><loc_9><loc_48><loc_12></location>We show the correlations for the stellar mass of the building blocks, the accreted stellar mass,</text>
<text><location><page_12><loc_52><loc_82><loc_92><loc_91></location>the number of accreted systems and the number of significant progenitors in Figure 4 in order to compare with the dark matter correlations, although they have already been discussed in Section 4 (see Figures 2 and 3).</text>
<text><location><page_12><loc_52><loc_22><loc_92><loc_82></location>For the FIRE-2 galaxies the strongest correlations are between the corresponding minor-axis stellar and dark matter density profile parameters. Specifically, the stellar and dark matter density amplitudes are strongly positively correlated, along with the slopes. On the other hand the stellar density amplitude is negatively correlated with the dark matter slope and the dark matter density amplitude is negatively correlated with the stellar slope. This is likely due to the combination of the degeneracies in the fit of the density amplitudes and slopes for both components and the strong positive correlations between the components' profiles. For example, using the covariance matrix from the fitting procedure, we find that on average 75% of the variance in the stellar halo central density parameters can be explained by the variance in the stellar halo power-law slope. On the other hand, when comparing across the stars and dark matter, the strength of the positive correlations between the components' amplitudes and slopes are 0.63 and 0.69, respectively, which are considered strong relationships. The negative correlation coefficient between the stellar amplitude and dark matter slope is -0.43, while the dark matter amplitude and stellar slope have a correlation coefficient of -0.46. These are both considered moderate relationships, and are roughly 75% of the strong relationship ( ∼ 0.6) between the corresponding stellar and dark matter components.</text>
<text><location><page_12><loc_52><loc_11><loc_92><loc_22></location>The minor axis stellar density profiles of the Auriga galaxies have weaker correlations with their corresponding dark matter density profiles than the FIRE-2 galaxies. The strength of the correlations between the amplitudes and slopes are both 0.49, which is considered moderate, al-</text>
<figure>
<location><page_13><loc_11><loc_63><loc_92><loc_89></location>
<caption>Figure 4. The Spearman correlation coefficients for the minor axis stellar density profile parameters correlated with metrics of the accretion history and the minor axis dark matter density profile. Specifically, we show an ellipse whose shape, color and orientation demonstrate the coefficient's value. Large positive (negative) correlation coefficients are shown with bluer (redder) ellipses that are more elongated to the top-right (bottom-right). The top row shows the correlations with the minor axis stellar density slope ( α ), while the bottom row shows the correlations with the amplitude of the profile ( ρ 0 ,H ). The left panel shows results for the FIRE-2 simulations with stellar mass of the building blocks ( M ∗ ,BB ), total stellar mass of accreted systems ( M ∗ ,accr ), number of significant progenitors ( N sp ), number of accreted systems ( N accr ), minor axis dark matter density profile amplitude ( ρ 0 ,DM ) and slope ( α DM ), from left to right. The right panel shows the results for the Auriga galaxies with the stellar mass of accreted systems ( M ∗ ,accr ), number of significant progenitors ( N sp ), minor axis dark matter density profile amplitude ( ρ 0 ,DM ) and slope ( α DM ), from left to right.</caption>
</figure>
<text><location><page_13><loc_8><loc_11><loc_48><loc_37></location>though less strong than the FIRE-2 correlations of 0.63 and 0.69. It is difficult to determine why the Auriga galaxies do not show stronger correlations, but it is likely due to worse fit parameters. The functional form was optimized for FIRE-2 galaxies and thus the average log 10 rootmean-squared error (RMSE) is 7.38 for Auriga galaxies while it is only 7.16 for FIRE-2 galaxies. We further discuss the relative goodnessof-fits in Appendix A. It is also possible that the difference in correlation strengths between FIRE-2 and Auriga is due to the difference in physical prescriptions. This is discussed further in Section 6.1.</text>
<text><location><page_13><loc_52><loc_9><loc_92><loc_37></location>In Figure 5, we show the relationship between the minor axis stellar density profile amplitude with the corresponding dark matter density amplitude for the FIRE-2 Milky Way-mass galaxies (top two panels) and the Auriga Milky Waymass galaxies (bottom panel). In the top panels, the FIRE-2 galaxy pairs are shown as diamonds while the isolated galaxies are circles. In the topmost panel, the points are colored by their formation time. Specifically, the color corresponds to the redshift at which the galaxy has reached 50% of its z=0 mass (z 0 . 5 ). The middle panel is the same as the top panel, except here the points are colored by the ratio of the stellar to dark matter scale radii. The bottom panel</text>
<text><location><page_14><loc_5><loc_33><loc_21><loc_35></location>ρ</text>
<figure>
<location><page_14><loc_11><loc_33><loc_46><loc_90></location>
<caption>Figure 5. The relationship between the minor axis stellar density profile amplitude and the dark matter counter-part. The results for FIRE-2 simulations are shown in the top two panels, while results for the Auriga galaxies are in the bottom plot. In the top plot, the points are colored by the formation redshift ( z 0 . 5 ). For the bottom two plots the points are colored by the ratio of the scale radius for the stellar and dark matter profiles. The Auriga galaxies use a different colormap to emphasize the difference in range of the scale radius ratios.</caption>
</figure>
<text><location><page_14><loc_54><loc_80><loc_57><loc_81></location>α</text>
<figure>
<location><page_14><loc_55><loc_52><loc_90><loc_90></location>
<caption>Figure 6. The relationship between the slope of the minor axis stellar density profile with the dark matter counterpart. The FIRE-2 galaxies are colored by the ratio of the scale radii of the stellar and dark matter profiles with isolated galaxies as circles and pairs as diamonds. The results for the Auriga galaxies are shown as Xs also colored by the ratio of the scale radii, but we use a different colormap to emphasize the difference in range compared to the FIRE-2 galaxies.</caption>
</figure>
<text><location><page_14><loc_52><loc_24><loc_92><loc_29></location>shows the same quantities for the Auriga galaxies, but the color bar scale is quite different than the middle panel's.</text>
<text><location><page_14><loc_52><loc_11><loc_92><loc_23></location>Not only do the ratios of the scale radii for the stellar and dark matter density profiles have larger values, but the Auriga halos also have larger dark matter density amplitudes than the FIRE-2 galaxies on average. For example, only 2 out of 15 FIRE-2 galaxies have ρ 0 , DM > 20 × 10 8 M ⊙ / kpc 3 while the least dense Auriga halo</text>
<figure>
<location><page_15><loc_12><loc_61><loc_89><loc_91></location>
<caption>Figure 7. The ratio of stellar density to dark matter density in the simulated galaxies as a function of increasing distance in the direction that is perpendicular to the galaxy's disk plane. The left panel shows results for the isolated FIRE-2 galaxies while the middle panels shows results for the pairs. In both the left and middle panels, the FIRE-2 galaxies are colored by their formation redshift ( z 0 . 5 ). The right panel shows results for the Auriga galaxies in random shades of green. The grey dashed line shows the best fit to the FIRE-2 galaxies which is simply ρ ∗ /ρ DM = 0 . 4 /R 2 .</caption>
</figure>
<text><location><page_15><loc_8><loc_10><loc_48><loc_48></location>has ρ 0 , DM = 24 × 10 8 M ⊙ / kpc 3 . This is consistent with the fact that the FIRE-2 and Auriga galaxies have a similar range of M 200 ≈ 1-2 × 10 12 M ⊙ but FIRE-2 has R 200 m ≈ 320-410 kpc while Auriga has R 200 ≈ 210-260 kpc. The virial radii R 200 are calculated differently in that Auriga uses the radius within which the density is 200 × the critical density for closure, while FIRE-2 uses the radius within which the density is 200 × the mean density of the Universe. Either way, the Auriga galaxies have similar mass inside a smaller radius than the FIRE2 galaxies, and therefore higher mass density. The Auriga dark matter density profiles also have smaller scale radii than the FIRE-2 dark matter profiles, while the stellar profiles have larger scale radii on average. This is likely due to differences in feedback mechanisms between the simulations, which are discussed further in Section 6.1.</text>
<text><location><page_15><loc_52><loc_14><loc_92><loc_47></location>In the top panel of Figure 5, the FIRE-2 galaxy pairs with early formation times are outliers in the relationship between the stellar and dark matter density amplitudes. One possible reason for this is simply that these structures formed earlier when the universe had an overall lower stellar-to-dark-matter mass ratio. In the middle panel, we demonstrate how the degeneracy of the fits impacts the spread in the relationship between the stellar and dark matter density amplitudes for the FIRE-2 galaxies. On the other hand, the Auriga galaxies do not show a strong correlation between the stellar and dark matter density amplitudes. However, given the consistency of the ratio of stellar-to-dark-matter mass seen in Figure 7, the correlation is likely masked by degeneracies in the fit parameters (see Appendix A).</text>
<text><location><page_15><loc_52><loc_10><loc_92><loc_13></location>The relationship between the minor axis stellar density power law slopes and the correspond-</text>
<text><location><page_16><loc_8><loc_77><loc_48><loc_91></location>ing dark matter slopes are shown in Figure 6. The Auriga galaxies are shown as Xs while the FIRE-2 isolated galaxies are shown as circles and the pairs as diamonds. The points are colored by the ratio of their stellar and dark matter scale radii, but as in Figure 5, the Auriga galaxies have a different color scale than the FIRE-2 galaxies.</text>
<text><location><page_16><loc_8><loc_56><loc_48><loc_76></location>The Auriga and FIRE-2 galaxies both show a strong correlation between their stellar and dark matter density slopes. In general, the Auriga galaxies have steeper stellar slope for a given dark matter slope. Similar to the stellar density amplitudes, the largest outliers in the relationship between the stellar and dark matter density slopes for the FIRE-2 galaxies are also outliers in the ratio of the corresponding scale radii. This indicates much of the spread is likely due to degeneracies in the fit parameters.</text>
<text><location><page_16><loc_8><loc_15><loc_48><loc_55></location>Figure 7 shows the ratio of stellar density to dark matter density for the simulated galaxies as a function of distance from the galactic center. Specifically, we calculate these ratios similar to the density profile fits in Section 3 in that we only use particles within a cylinder along the minor axis with radius=1 / √ π . In the left panel we show results for the isolated FIRE-2 galaxies, while the middle panel shows the pairs. The lines in these panels are colored by the redshift at which the galaxy has reached 50% of its z = 0 mass. In the middle panel, we also use different line styles for sets of pairs. Specifically, Romeo and Juliet are shown as solid lines, while Romulus and Remus are in dotted lines and Thelma and Louise in dot-dashed lines. The right panel shows results for the Auriga galaxies colored in random shades of green. Each panel also has a grey dashed line which is simply the function ρ ∗ /ρ DM = 0 . 4 / R 2 which is the best fit to the results for the FIRE-2 pair galaxies (middle panel).</text>
<text><location><page_16><loc_8><loc_11><loc_48><loc_14></location>The ratio of stellar to dark matter density along the minor axis in these simulated galaxies</text>
<text><location><page_16><loc_52><loc_67><loc_92><loc_92></location>follows a r -2 profile. While each simulation suite generally follows this relation, the FIRE-2 and Auriga galaxies show slightly different behavior. The later forming (lower z 0 . 5 ) FIRE-2 pairs have higher ρ ∗ /ρ DM at R > 10 kpc compared to the earlier forming galaxies. The Auriga galaxies generally have higher stellar to dark matter ratios similar to the later forming FIRE-2 galaxy pairs. It is possible that this is because they also form later, when the universe overall has a higher stellar-to-dark-matter density, but we do not have formation redshifts for these galaxies to confirm.</text>
<section_header_level_1><location><page_16><loc_55><loc_64><loc_89><loc_66></location>6. DISCUSSION AND CONCLUSIONS</section_header_level_1>
<text><location><page_16><loc_52><loc_43><loc_92><loc_63></location>In this work, we parameterized the minor axis stellar and dark matter density profiles of Milky Way-mass FIRE-2 and Auriga simulated galaxies. With this parameterization, we compare the stellar profiles to the accretion histories and find weak to no correlation. Instead, we find strong correlations between the distribution of stars and dark matter. Specifically, we find that the ratio of stellar to dark matter density consistently falls off as r -2 along the minor axis across the simulation suites.</text>
<text><location><page_16><loc_52><loc_9><loc_92><loc_43></location>Whether the consistency of the stellar-todark-matter profile is a natural consequence of galaxy formation and Λ-CDM, or if its merely a coincidence in these simulations is unknown. Consistent with the profiles measured in this work, dark matter halos are generally thought to have pseudo-isothermal profiles with r -2 within their NFW scale radii (Navarro et al. 1997). Predictions for the minor axis stellar density profiles in this range are fewer and vary more widely, but generally cluster around r -4 (Monachesi et al. 2019; Deason et al. 2013; Font et al. 2020; Cooper et al. 2010; Amorisco 2017). In this work, we find that degeneracies in the fit parameters can cause more spread in the measured slopes (see Figure 6), but the profile of the ratio of stellar-to-dark-matter is consistent (see Figure 7).</text>
<text><location><page_17><loc_8><loc_41><loc_48><loc_91></location>We have intentionally investigated the minor axis stellar density profiles in a region where galactic component classification is difficult. Specifically, the radial range we study is where the disk, bulge and inner halo overlap. Although it is clear that the exponential component of the stellar density profile is the galactic disk, we do not attempt to classify the powerlaw component. It could be classified as either bulge or inner halo. Classical bulges have long been observed to have r -4 profiles, similar to elliptical galaxies (de Vaucouleurs 1948; Hernquist 1990). However, estimates for the S'ersic indices of the FIRE-2 and Auriga galaxies have n < 2, inconsistent with classical bulges (Sanderson et al. 2018; Gargiulo et al. 2019). Simulations show that that r -4 stellar profiles can be reproduced with dissipationless collapse (van Albada 1982) and also with dissipational mergers (e.g., Hernquist 1992; Barnes 1992). Halo stellar density profiles are less often studied, given their low surface brightness. However, the studies that exist have found profiles with slopes that vary around r -4 for nearby Milky Way-mass disk galaxies (Harmsen et al. 2017). In Section 6.2, we discuss comparisons to the Milky Way.</text>
<section_header_level_1><location><page_17><loc_9><loc_38><loc_47><loc_39></location>6.1. Differences between FIRE-2 and Auriga</section_header_level_1>
<text><location><page_17><loc_8><loc_19><loc_48><loc_37></location>In this work, we note a number of differences between the Auriga and FIRE-2 minor axis stellar and dark matter profiles. While we do not focus on the galactic disks in this work, we find that Auriga's disks have 10-100 × higher stellar mass densities than the FIRE-2 disks (see Appendix A). Relatedly, the Auriga galaxies are known to host stronger bars than the Fire2 galaxies (Ansar et al. 2023; Fragkoudi et al. 2024).</text>
<text><location><page_17><loc_8><loc_9><loc_48><loc_18></location>Another large difference between the FIRE-2 and Auriga galaxies are the dark matter central densities ( ρ 0 , DM ) and scale radii ( r DM ). Although they have never been directly compared, it is known that the Auriga galaxies have sig-</text>
<text><location><page_17><loc_52><loc_77><loc_92><loc_91></location>ificant baryon contraction (Callingham et al. 2020), while the FIRE-2 galaxies form dark matter cores (Lazar et al. 2020). This is likely due to differences in the implementation of feedback, which has been shown to significantly impact the distribution of mass in the center of galaxies (Pontzen & Governato 2014; Lazar et al. 2020).</text>
<text><location><page_17><loc_52><loc_50><loc_92><loc_76></location>The formation of dark matter halo cores in cosmological simulations is dependent on the the star formation prescriptions as they relate to the supernova rates. Specifically, the choice of gas density threshold for star formation has been shown to impact the dark matter distribution in galaxies (Dutton et al. 2019; Ben'ıtezLlambay et al. 2019). Given that the FIRE2 galaxies have a higher gas density threshold for star formation ( n SF > 1000 cm -3 ) than the Auriga simulations ( n SF > 0 . 13 cm -3 ), it is not surprising that they have dark matter cores ( r DM ∼ 1 kpc) and while the Auriga simulations do not ( r DM ∼ 0.3 kpc).</text>
<text><location><page_17><loc_52><loc_26><loc_92><loc_50></location>As discussed in Section 2.2, the supernova prescriptions are also different between the simulations in that FIRE-2 have rates that evolve with the star particles (Hopkins et al. 2018b), while the Auriga simulations have instantaneous supernova feedback at the formation time of the star particle. FIRE-2 also implements radiative feedback from massive stars, which regulates star formation (Hopkins et al. 2014; Orr et al. 2018; Hopkins et al. 2020). Auriga does not include early stellar feedback, but does include AGN feedback, which is missing from the FIRE-2 physics model.</text>
<text><location><page_17><loc_52><loc_11><loc_92><loc_25></location>In total, there are many differences between the implementations of the simulations which could explain the small variations in the stellar and dark matter minor axis profiles. Despite these differences, however, the most interesting result is the remarkable similarity of their stellar-to-dark-matter ratio profiles, which consistently fall off as r -2 for both simulation suites.</text>
<section_header_level_1><location><page_18><loc_15><loc_90><loc_41><loc_91></location>6.2. Comparison to Milky Way</section_header_level_1>
<text><location><page_18><loc_8><loc_65><loc_48><loc_89></location>As discussed in the introduction (Section 1), there are a number of estimates of the MW halo stellar density profile. Each of these measurements have their own selection functions, biases and radial ranges. Therefore, it is difficult to compare our results without a thorough investigation of the impacts of each selection function and method. We plan to perform a detailed comparison with MW data and study the impact of selection functions in future work. Here, we briefly summarize the MW results which are most suitable for comparison with the simulated galaxies.</text>
<text><location><page_18><loc_8><loc_20><loc_48><loc_64></location>Generally, the simulated galaxies have steeper stellar density slopes than what is measured in the MW. Between ≈ 5-20 kpc estimates for the MW's halo stellar density profile slope range from -1 to -3 (Sesar et al. 2013; Faccioli et al. 2014; Sesar et al. 2010; Han et al. 2022). There are few measurements within 5 kpc of the Galactic center given the high levels of dust extinction and crowding. However, using RR Lyrae, the stellar density profile slope was estimated as -3 in the radial range of 0.2-2.8 kpc from the Galactic center (Pietrukowicz et al. 2015). Furthermore, using orbit integration, Yang et al. (2022) found that the halo stellar density profile flattens in the inner regions with the slope increasing to -1.5. Although these works provide estimates in this range, they are based only on a small fraction of the stellar population, with complex selection functions. In order to perform a fair comparison, the MW minor axis profile should model the Galactic disk to minimize the selection bias. We plan to do this in future work using data with a well-modeled selection function.</text>
<text><location><page_18><loc_8><loc_10><loc_48><loc_19></location>The MW's dark matter density profile is not well constrained because methods rely on the poorly constrained baryonic matter distribution. Recent measurements of the Milky Way's circular velocity curve have provided estimates</text>
<text><location><page_18><loc_52><loc_79><loc_92><loc_91></location>of the dark matter profile which give results consistent with the profiles of FIRE-2 galaxies (Ou et al. 2024). However, we note that these results give a larger core and more concentrated dark matter halo than what is typically seen in Milky Way-mass cosmological simulations (Lazar et al. 2020).</text>
<text><location><page_18><loc_52><loc_56><loc_92><loc_78></location>The ratio of stellar-to-dark matter mass profile has not been directly measured in the Milky Way. However, combining results from the individually measured profiles indicate a ratio profile that is roughly constant or linear, given that the stellar profile is less steep than in the simulated galaxies. If this discrepancy is borne out in future work, this could indicate a need for adjustments in the physics models used in FIRE2 and Auriga simulations, likely either in the stellar feedback prescription, or the dark matter model.</text>
<section_header_level_1><location><page_18><loc_60><loc_49><loc_85><loc_51></location>6.3. Summary of Conclusions</section_header_level_1>
<text><location><page_18><loc_52><loc_34><loc_92><loc_48></location>Λ-CDM cosmology predicts that hierarchical growth dominates early galaxy formation and evolution. The minor axis stellar density profile of galaxies is of special interest as it is thought to trace the galaxy's accretion history. Whether classified as halo or bulge, this stellar component is thought to build mass through mergers and accretion events.</text>
<text><location><page_18><loc_52><loc_9><loc_92><loc_33></location>In this work, we have investigated the minor axis stellar and dark matter density profiles of Milky Way-mass galaxies in the cosmological zoom-in simulation suites of FIRE-2 and Auriga. Using only data within their respective NFWscale radiii, we quantify the profiles with a simple power law parameterization, including a exponential disk component for the stars. With the aim of understanding the physical mechanism that shapes the minor axis stellar density profile, we compare the parameters with other properties of the simulated galaxies. In total, we find:</text>
<unordered_list>
<list_item><location><page_19><loc_11><loc_86><loc_48><loc_92></location>· The amplitude and slope of the minor axis power law stellar density profiles do not relate to the galaxy's accretion history.</list_item>
<list_item><location><page_19><loc_11><loc_74><loc_48><loc_85></location>· Instead, the stellar profiles are tightly correlated with the corresponding dark matter profile. Therefore, we find the dark matter potential is the dominate mechanism determining the amplitude and slope of minor axis power law stellar profile.</list_item>
<list_item><location><page_19><loc_11><loc_67><loc_48><loc_73></location>· The ratio of stellar-to-dark-matter mass decreases as 1 /r 2 along the minor axis for all simulated galaxies.</list_item>
</unordered_list>
<text><location><page_19><loc_8><loc_49><loc_48><loc_65></location>As the ratio of light to dark matter is a fundamental observation, it is crucial to understand the interplay of stellar and dark matter mass in galaxies. In future work, we plan to build on this work by investigating the comparison between the MW and simulations more deeply. This includes a new measurement of the minor axis stellar density profile using Gaia data, and an investigation of the Milky Way's ratio of</text>
<text><location><page_19><loc_52><loc_79><loc_92><loc_91></location>stellar-to-dark matter mass profile using stellar kinematics. Finally, we will also look to alternative dark matter models including the FIRE-2 self-interacting dark matter simulations (Sameie et al. 2021; Vargya et al. 2022) to further study the connection between stellar and dark matter mass.</text>
<text><location><page_19><loc_52><loc_65><loc_92><loc_78></location>Software: Astropy (Astropy Collaboration et al. 2013, 2018), Matplotlib (Hunter 2007), IPython (P'erez & Granger 2007), Numpy (Harris et al. 2020), Scipy (Virtanen et al. 2020), GizmoAnalysis (Wetzel & Garrison-Kimmel 2020a) HaloAnalysis (Wetzel & Garrison-Kimmel 2020b; Wetzel et al. 2016)</text>
<section_header_level_1><location><page_19><loc_61><loc_61><loc_83><loc_62></location>ACKNOWLEDGEMENTS</section_header_level_1>
<text><location><page_19><loc_52><loc_49><loc_92><loc_60></location>This material is based upon work supported by the National Science Foundation under Award No. 2303831. We have used simulations from the Auriga Project public data release (Grand et al. 2024) available at https:// wwwmpa.mpa-garching.mpg.de/auriga/data.</text>
<section_header_level_1><location><page_19><loc_45><loc_46><loc_55><loc_47></location>APPENDIX</section_header_level_1>
<section_header_level_1><location><page_19><loc_37><loc_40><loc_63><loc_41></location>A. DENSITY PROFILE FITS</section_header_level_1>
<text><location><page_19><loc_8><loc_28><loc_92><loc_39></location>To parameterize the minor axis stellar and dark matter density profiles, we fit simple power law functions (see Equations 2 and 3) to the distribution of particles inside a cylinder of radius 1 / √ π along the minor axis which is defined to be perpendicular to the galaxy's disk. For the stellar component we also need to account for the exponential disk. To avoid the effect of binning, we first perform the fit to the cumulative distribution function which is normalized by the total stellar mass within the cylinder out to the halo's NFW scale radius (left panels).</text>
<text><location><page_19><loc_8><loc_9><loc_92><loc_28></location>In Figures 8, 9, and 10, we show the fits to the stellar density profiles, while Figures 11, 12, and 13 show the fits to the dark matter profiles. Each row corresponds to one simulated galaxy with the resulting parameterization printed above the corresponding panels. The galaxy's designation is printed in the top right of the leftmost panel. The middle panels show the fit (red dashed line) to the unnormalized density distribution (dark blue solid line). We note that fits to the Auriga stellar density profiles are generally worse then the fits to the FIRE-2 galaxies. For example, the log 10 RMSE is 7.17 for the isolated FIRE-2 galaxies, 7.15 for the pairs and 7.38 for the Auriga galaxies. This is because the functional form (Equation 2) was chosen to optimize the fit to the FIRE-2 galaxies. The Auriga galaxies may be fit better using a different functional form, but in order to facilitate a comparison we keep the functional form the same for both simulations.</text>
<figure>
<location><page_20><loc_11><loc_29><loc_90><loc_90></location>
<caption>Figure 8. The minor axis stellar density profile fits for the FIRE-2 isolated galaxies where each row demonstrates the fit for a different galaxy whose label is in the top right corner of the left panels. The CDF fits are shown in the left panels (red dashed line) with the galaxy's stellar particle CDF (dark blue solid line). In the left and middle panels, we also show 100 different stellar profiles for cylinders that are randomly selected to be up to 45 · away from the minor (z-)axis (dark blue transparent lines). The middle panels shows the unnormalized stellar particle distributions. The final fit distribution is shown as a red dashed line in the middle panels, with the final parameters printed above the panels for each galaxy. In the right panel, we show the fractional difference between particle distribution along the z-axis compared to the 100 random directions (transparent blue lines). The median of these lines is shown as the dark blue solid line with no transparency. The red dashed line indicate the y-axis values of 0, -1 and 1.</caption>
</figure>
<figure>
<location><page_21><loc_11><loc_29><loc_90><loc_90></location>
<caption>Figure 9. The same as Figure 8 but for the FIRE-2 galaxy pairs.</caption>
</figure>
<text><location><page_21><loc_8><loc_10><loc_92><loc_22></location>In the left and middle panels, we also plot in transparent dark blue the stellar particle distributions for cylinders of the same size, but at 100 random angles that are < 45 · from the z-axis. With these distributions, we investigate the level of substructure and asymmetry in the simulated minor axis profiles. The right panels show the difference between the density distributions along the z-axis and the 100 random angles, normalized by the z-axis density at that point (dark blue transparent lines). The median of these 100 lines is shown as the solid dark blue line. We also show three red horizontal dashed lines at y-values of -1, 0, and 1. Generally, we find the distributions do not vary significantly</text>
<figure>
<location><page_22><loc_11><loc_29><loc_90><loc_90></location>
<caption>Figure 10. The same as Figure 8 but for the Auriga galaxies.</caption>
</figure>
<text><location><page_22><loc_8><loc_19><loc_92><loc_22></location>for the FIRE-2 galaxies, while the Auriga galaxies show systematically larger densities at angles closer to the disk, indicating more oblate halos, consistent with results from Monachesi et al. (2019).</text>
<section_header_level_1><location><page_22><loc_43><loc_15><loc_57><loc_16></location>REFERENCES</section_header_level_1>
<text><location><page_22><loc_8><loc_10><loc_46><loc_13></location>Aguerri, J. A. L., Balcells, M., & Peletier, R. F. 2001, A&A, 367, 428,</text>
<text><location><page_22><loc_10><loc_9><loc_36><loc_10></location>doi: 10.1051/0004-6361:20000441</text>
<text><location><page_22><loc_52><loc_12><loc_86><loc_13></location>Amorisco, N. C. 2017, MNRAS, 464, 2882,</text>
<text><location><page_22><loc_54><loc_10><loc_76><loc_11></location>doi: 10.1093/mnras/stw2229</text>
<figure>
<location><page_23><loc_10><loc_29><loc_90><loc_90></location>
<caption>Figure 11. The same as Figure 8 but for the dark matter profiles.</caption>
</figure>
<text><location><page_23><loc_8><loc_18><loc_45><loc_23></location>Ansar, S., Pearson, S., Sanderson, R. E., et al. 2023, arXiv e-prints, arXiv:2309.16811, doi: 10.48550/arXiv.2309.16811</text>
<text><location><page_23><loc_8><loc_10><loc_44><loc_16></location>Arentsen, A., Starkenburg, E., Martin, N. F., et al. 2020, MNRAS, 491, L11, doi: 10.1093/mnrasl/slz156</text>
<text><location><page_23><loc_52><loc_10><loc_92><loc_23></location>Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068 Astropy Collaboration, Price-Whelan, A. M., Sip"ocz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f Athanassoula, E. 2005, MNRAS, 358, 1477, doi: 10.1111/j.1365-2966.2005.08872.x</text>
<figure>
<location><page_24><loc_10><loc_30><loc_89><loc_90></location>
<caption>Figure 12. The same as Figure 11 but for the FIRE-2 galaxy pairs.</caption>
</figure>
<text><location><page_24><loc_8><loc_22><loc_45><loc_23></location>Babusiaux, C., G´omez, A., Hill, V., et al. 2010,</text>
<text><location><page_24><loc_8><loc_10><loc_44><loc_22></location>A&A, 519, A77, doi: 10.1051/0004-6361/201014353 Barnes, J. E. 1992, ApJ, 393, 484, doi: 10.1086/171522 Behroozi, P. S., Wechsler, R. H., & Wu, H.-Y. 2013a, ApJ, 762, 109, doi: 10.1088/0004-637X/762/2/109</text>
<text><location><page_24><loc_52><loc_10><loc_91><loc_23></location>Behroozi, P. S., Wechsler, R. H., Wu, H.-Y., et al. 2013b, ApJ, 763, 18, doi: 10.1088/0004-637X/763/1/18 Bellardini, M. A., Wetzel, A., Loebman, S. R., et al. 2021, MNRAS, 505, 4586, doi: 10.1093/mnras/stab1606 Belokurov, V. 2013, NewAR, 57, 100, doi: 10.1016/j.newar.2013.07.001</text>
<figure>
<location><page_25><loc_10><loc_29><loc_90><loc_90></location>
<caption>Figure 13. The same as Figure 11 but for the Auriga galaxies.</caption>
</figure>
<text><location><page_25><loc_8><loc_10><loc_47><loc_23></location>Belokurov, V., & Kravtsov, A. 2022, MNRAS, 514, 689, doi: 10.1093/mnras/stac1267 Belokurov, V., Sanders, J. L., Fattahi, A., et al. 2020, MNRAS, 494, 3880, doi: 10.1093/mnras/staa876 Ben'ıtez-Llambay, A., Frenk, C. S., Ludlow, A. D., & Navarro, J. F. 2019, MNRAS, 488, 2387, doi: 10.1093/mnras/stz1890</text>
<text><location><page_25><loc_52><loc_22><loc_92><loc_23></location>Binney, J., Gerhard, O. E., Stark, A. A., Bally, J.,</text>
<text><location><page_25><loc_54><loc_20><loc_86><loc_21></location>& Uchida, K. I. 1991, MNRAS, 252, 210,</text>
<text><location><page_25><loc_52><loc_10><loc_90><loc_20></location>doi: 10.1093/mnras/252.2.210 Bland-Hawthorn, J., & Gerhard, O. 2016, ArXiv e-prints. https://arxiv.org/abs/1602.07702 Blitz, L., & Spergel, D. N. 1991, ApJ, 370, 205, doi: 10.1086/169806</text>
<table>
<location><page_26><loc_8><loc_9><loc_48><loc_91></location>
</table>
<table>
<location><page_26><loc_52><loc_10><loc_92><loc_91></location>
</table>
<table>
<location><page_27><loc_8><loc_9><loc_48><loc_92></location>
</table>
<table>
<location><page_27><loc_52><loc_9><loc_92><loc_92></location>
</table>
<table>
<location><page_28><loc_8><loc_9><loc_48><loc_92></location>
</table>
<table>
<location><page_28><loc_52><loc_10><loc_92><loc_92></location>
</table>
<text><location><page_29><loc_52><loc_87><loc_90><loc_91></location>Zoccali, M., Gonzalez, O. A., Vasquez, S., et al. 2014, A&A, 562, A66, doi: 10.1051/0004-6361/201323120</text>
</document>
[
{
"title": "ABSTRACT",
"content": "Λ-CDM cosmology predicts the hierarchical formation of galaxies which build up mass by merger events and accreting smaller systems. The stellar halo of the Milky Way has proven to be useful a tool for tracing this accretion history. However, most of this work has focused on the outer halo where dynamical times are large and the dynamical properties of accreted systems are preserved. In this work, we investigate the inner galaxy regime, where dynamical times are relatively small and systems are generally completely phase-mixed. Using the FIRE-2 and Auriga cosmological zoomin simulation suites of Milky Way-mass galaxies, we find the stellar density profiles along the minor axis (perpendicular to the galactic disk) within the NFW scale radii (R ≈ 15 kpc) are best described as an exponential disk with scale height < 0.3 kpc and a power law component with slope α ≈ -4. The stellar density amplitude and slope for the power law component is not significantly correlated with metrics of the galaxy's accretion history. Instead, we find the stellar profiles strongly correlate with the dark matter profile. Across simulation suites, the galaxies studied in this work have a stellar to dark matter mass ratio that decreases as 1 /r 2 along the minor axis.",
"pages": [
1
]
},
{
"title": "Cosmological predictions for minor axis stellar density profiles in the inner regions of Milky Way-mass galaxies",
"content": "Madeline Lucey , 1 Robyn E. Sanderson , 1 Danny Horta , 2 Aritra Kundu , 1 Philip F. Hopkins , 3 Arpit Arora , 1 Jasjeev Singh , 1 and Nondh Panithanpaisal 4, 3 1 Department of Physics & Astronomy, University of Pennsylvania, 209 S 33rd St., Philadelphia, PA 19104, USA 2 Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA 3 TAPIR, Mailcode 350-17, California Institute of Technology, Pasadena, CA 91125, USA 4 Observatories of the Carnegie Institution for Science, 813 Santa Barbara St, Pasadena, CA 91101, USA",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "Understanding the formation and evolution of galaxies is one of the main goals of astrophysics today. In the inside-out theory of galaxy formation, the innermost regions of galaxies form first and are therefore especially informative for studying the earliest epochs of galaxy formation (Peebles 1969; Larson 1976; Fall & Efstathiou 1980; Mo et al. 1998; Somerville et al. 2008; Dutton et al. 2011). Furthermore, a substantial fraction of a galaxy's stellar mass is within the inner region, making it information-rich. The Milky Way (MW) presents a unique opportu- nity to study the inner region of a galaxy in exquisite detail. Along with more than a third of massive (M ∗ > 10 10 M ⊙ ) disk galaxies in the local Universe (Sellwood & Wilkinson 1993; Masters et al. 2011; Gavazzi et al. 2015), the MW hosts a galactic bar in its center (Blitz & Spergel 1991; Weiland et al. 1994; Peters 1975; Binney et al. 1991). The bulk of the stellar mass in the inner Galaxy participates in the bar structure (Howard et al. 2009; Shen et al. 2010; Ness et al. 2013a; Debattista et al. 2017). The MW also has an X-shaped structure in its center (Nataf et al. 2010; McWilliam & Zoccali 2010; Ness et al. 2012; Wegg & Gerhard 2013; Ness & Lang 2016), which is characteristic of a boxy/peanut-shaped (B/P) bulge and consistent with simulations and observations of barred galaxies (Combes et al. 1990; Athanassoula 2005; Martinez-Valpuesta et al. 2006; Bureau et al. 2006; Laurikainen et al. 2014; Debattista et al. 2019). It is currently debated whether the MW also hosts a less-massive metal-poor classical bulge component (Babusiaux et al. 2010; Hill et al. 2011; Zoccali et al. 2014). The major evidence for a metal-poor classical bulge is based on the stellar kinematics as a function of metallicity. Specifically, metal-poor stars in the inner Galaxy rotate slower and have a higher velocity dispersion than the metal-rich stars (Ness et al. 2013b; Kunder et al. 2016; Arentsen et al. 2020). However, Debattista et al. (2017) demonstrated that these observations may be the result of the overlapping Galactic halo whose density would peak in the center of the Galaxy. In fact, 25% of the RR Lyrae stars in the inner Galaxy were found on orbits with apocenters > 3.5 kpc from the Galactic center (Kunder et al. 2020). Similarly, Lucey et al. (2020) found that about 50% of metal-poor giants in the inner Galaxy are interlopers with apocenters > 3.5 kpc and that the fraction of interlopers increases with decreasing metallicity. After removing these stars from the sample, Lucey et al. (2020) found that the velocity dispersion decreased and there was no longer evidence for a classical bulge component in the kinematics. Whether they comprise a classical bulge or the innermost part of the Galactic halo, the stars which are not part of the bar or B/P bulge in the inner Galaxy give us a unique clue to the MW's formation history and galaxy evolution as a whole (Rix et al. 2022; Horta et al. 2021). Classical bulges and halos are both thought to be made through mergers and galaxy accretion events (Kauffmann et al. 1993; Kobayashi & Nakasato 2011; Guedes et al. 2013; Freeman & Bland-Hawthorn 2002; Belokurov 2013; Bland-Hawthorn & Gerhard 2016; Johnston et al. 1995; Helmi 2020). Therefore, it is thought that the total stellar mass and radial profile of the halo/bulge components may reflect the galaxy's cumulative accretion history (Helmi 2020; Han et al. 2022). However, there are also suggestions that the inner halo (Galactocentric radius < 10 kpc) may be a distinct Galactic component with in-situ origins. One theory for creating an in-situ inner halo is ELS-like contraction (Eggen, LyndenBell, & Sandage 1962), with early star formation occurring on halo-like random orbits (Carollo et al. 2007, 2010; Belokurov & Kravtsov 2022; El-Badry et al. 2018; Yu et al. 2023). Another proposed theory for the creation of an in-situ inner halo includes disruption of an old thick disk during a major merger (Belokurov et al. 2020). By comparing MW observations to cosmological zoom-in simulations, we can better understand the history of these stars, and the MW's evolution. The radial density profile of the MW's halo is typically quantified with a double power law, although single and triple power laws have also been suggested. Studies use a variety of stellar distance tracers, and radial ranges resulting in large spread in the MW estimates. Generally, the breaking radius estimates range from ≈ 1030 kpc (Sesar et al. 2013; Han et al. 2022) with inner slopes ranging from -1 to -3 (Sesar et al. 2010, 2013; Faccioli et al. 2014; Han et al. 2022) and outer slopes from -3 to -6 (Sesar et al. 2010, 2013; Deason et al. 2011). Most of these works are based on stars with Galactocentric radii > 5 kpc. The extension of the stellar density profile within 5 kpc, where Galactic extinction and crowding make observations difficult, is less certain. However, the available estimates are generally consistent with the results at larger radii (Pietrukowicz et al. 2015; P'erez-Villegas et al. 2017; Yang et al. 2022). The MW's halo density profile is generally consistent with nearby MWmass disk galaxies which have power law slopes of -3 to -5 (Harmsen et al. 2017). Furthermore, our nearest disk galaxy neighbor, M31, has a power law slope of -3.7 (Ibata et al. 2014). Cosmological zoom-in simulations provide a crucial tool for interpreting the observational properties of the MW and other galaxies. Using the IllustrisTNG suite of simulations, Pillepich et al. (2018) found that less massive galaxies generally have steeper halo density slopes, and that the profiles flatten towards the galactic center. They also found MW-mass galaxies to have a stellar density power law slope of ≈ -4.3 on average for stars within the half-light radius. Using the Auriga simulations of MW-mass galaxies, Monachesi et al. (2019) found that galaxies with fewer progenitors have more massive halos. Furthermore, they found that galaxies with fewer progenitors have steeper halo stellar density profile slopes, but the correlation is quite weak (see Figure 11 in Monachesi et al. 2019). It has also been suggested that the halo stellar density slope is related to the accretion time of the dominant progenitor (D'Souza & Bell 2018). In general, results between simulation suites differ, and a definitive metric that predicts the stellar halo density slope is yet to be found. Similar to MW estimates, results from ΛCDM simulations of Milky Way-mass galaxies show a large range of radial density profile power law slopes. Whether the works use single or broken power laws, the slopes range from ≈ -2 to ≈ -6.5 (Monachesi et al. 2019; Deason et al. 2013; Font et al. 2020; Cooper et al. 2010; Amorisco 2017), which is consistent with the range observed in nearby disk galaxies, including M31 and the MW (Harmsen et al. 2017). However, studies of the simulated stellar halos use either a kinematic or spatial selection to define the halo which can impact the results (Monachesi et al. 2019; Font et al. 2020; Pillepich et al. 2018; Cooper et al. 2010). For comparison with observations, a spatial selection is generally preferred since kinematic information is not always available. However, in order to avoid contamination by disk stars many of these works only study the halo profile beyond ∼ 10 kpc from the galactic center (Monachesi et al. 2019; Pillepich et al. 2018; Cooper et al. 2010). As we are interested in the inner most region in this work, we instead study the minor axis profile and simultaneously fit a power-law halo with an exponential disk. In this work, we use the FIRE-2 and Auriga MW-mass galaxy simulation suites to study inner minor axis stellar density profiles and explore their relation to the galaxy's mass assembly history. In Section 2 and 2.2, we provide a brief description of the FIRE-2 and Auriga simulation suites, respectively. Our method for parametrizing the density profiles of the simulated galaxies is described in Section 3 and Appendix A in more detail. We compare the stellar density profile parameters with metrics of the galaxy's merger and accretion history in Section 4 and with the dark matter profile parameters in Section 5. Last in Section 6, we discuss the results and summarize the conclusions.",
"pages": [
1,
2,
3
]
},
{
"title": "2. COSMOLOGICAL ZOOM-IN SIMULATIONS",
"content": "This work is primarily based on the FIRE-2 cosmological zoom-in simulations of Milky Waymass galaxies (Wetzel et al. 2023). However, we also make use of the publicly available Auriga simulation suite as comparison (Grand et al. 2024). As these two suites are numerically relatively similar with a few key differences in their physical prescriptions, they provide a unique opportunity to learn about galaxy formation physics by studying how the galaxies differ, or stay the same, between suites.",
"pages": [
3
]
},
{
"title": "2.1. FIRE-2 Milky Way-mass Simulations",
"content": "From FIRE-2, we use the Latte suite of seven isolated Milky Way-mass galaxies (Wetzel et al. 2016) as well as the ELVIS suite of three Local Group-like pairs of Milky Way-mass galaxies (Garrison-Kimmel et al. 2019a). All of these simulations are run with the FIRE-2 physics model (Hopkins et al. 2018a) using the GIZMO 1 gravity plus hydrodynamics code in meshless finite-mass (MFM) mode (Hopkins 2015). For a complete and detailed description of the simulation implementations, we refer the reader to the above papers. Below, we summarize a few key properties. Each simulation assumes flat Λ-CDM cosmology with parameters consistent with Planck Collaboration et al. (2014). Specifically, the Latte suite (excluding m12w) uses Ω m = 0.272, Ω b = 0.0455, σ 8 = 0.807, n s = 0.961, h = 0.702. Of the Elvis suite of galaxy pairs, Thelma & Louise and Romulus & Remus both use the same cosmology as in the original ELVIS darkmatter-only (DMO) suite: Ω m = 0.266, Ω b = 0.0449, σ 8 = 0.801, n s = 0.963, h = 0.71. The third ELVIS galaxy pair, Romeo & Juliet along with one Latte galaxy, m12w, both use the updated parameters from Planck Collaboration et al. (2020), Ω m = 0.31, Ω b = 0.048, σ 8 = 0.82, n s = 0.97, h = 0.68. The physical prescription for feedback mechanisms are known to significantly impact the distribution of mass in simulated galaxies (Pontzen & Governato 2014; Lazar et al. 2020). FIRE2 simulations include implementations of stellar feedback from stellar winds, radiation pressure from young stars, Type II and Type Ia supernovae, photoelectric heating, and photoheating from ionizing radiation, which regulates star formation. The gas density threshold for star formation is n SF > 1000 cm -3 . Feedback event rates, luminosities, energies, massloss rates, and other quantities are tabulated directly from stellar evolution models (STAR- BURST99; Leitherer et al. 1999). All 13 simulated galaxies have dark matter halo masses at z =0 of M 200 = 1 -2 . 1 × 10 12 M ⊙ (Sanderson et al. 2018). The Latte suite galaxies have initial stellar particle masses of 7070 M ⊙ , while the ELVIS suite galaxies have higher resolution at initial stellar particle masses of 3500 M ⊙ . Star particle softening lengths are ≈ 4 pc and dark matter force softening is ≈ 40 pc. These simulated galaxies show agreement with the observed stellar mass-dark matter halo mass relation across cosmic time (Hopkins et al. 2018a). They are also consistent with a number of key observed properties of the Milky Way, including the stellar halo mass fraction (Sanderson et al. 2018), the existence of a metal-rich in-situ stellar halo component (Bonaca et al. 2017), and the radial and vertical structure of the stellar disk (Ma et al. 2017; Sanderson et al. 2020; Bellardini et al. 2021; McCluskey et al. 2024). The simulated satellite populations, including stellar streams, are also consistent with observations of these populations around the Milky Way and M31 (Wetzel et al. 2016; Samuel et al. 2020; Garrison-Kimmel et al. 2019b; Panithanpaisal et al. 2021; Cunningham et al. 2022; Shipp et al. 2023). To understand the role of hierarchical formation in building the minor axis stellar profile, we identify and track star particles that formed outside of the main progenitor galaxy across the simulation as a function of time. These star particles are formed in subhalos before they interact with the main branch progenitor. The redshift when the main progenitor reaches a mass that is 3 times larger than the next most massive luminous halo, z MR 3:1 , is when the main progenitor emerges as the dominant host galaxy (Santistevan et al. 2020; Horta et al. 2024). Before this redshift, systems that merge with the main progenitor are labeled as building blocks (Horta et al. 2024). After this redshift, the luminous halos that are within the virial radius of the central host galaxy at z = 0 are labeled as accreted. These systems are tracked with the help of the ROCKSTAR halo catalogs and the halo merger trees (Behroozi et al. 2013a,b); along with every star particle that make-up a present-day substructure (satellites, streams, or phase-mixed) within this virial radius. These accreted substructures are classified as phasemixed, if they satisfy the following criteria at the present-day (Panithanpaisal et al. 2021): (1) the total stellar mass is greater than 10 4 . 5 M ⊙ , (2) the maximum separation between any two star particles is greater than 120 kpc, and (3) the median of the local velocity dispersion of the star particles is greater than a stellar-mass dependent threshold value (see Equation 2 of Panithanpaisal et al. (2021)).",
"pages": [
3,
5,
6
]
},
{
"title": "2.2. Auriga Simulations",
"content": "The Auriga cosmological zoom-in simulations provide a good comparison to the FIRE-2 simulations because they both have similar resolution and broadly MW-like z = 0 disk galaxies, but different physical prescriptions. In this work, we specifically use the 6 Milky Way-mass galaxies simulated at 'level 3' resolution, which is most similar to the resolution of FIRE-2. For a detailed description of these simulations we refer the reader to Grand et al. (2017) and Grand et al. (2024). Below, we provide a brief overview of a few key details. The Auriga simulations assume flat Λ-CDM cosmology with parameters taken from Planck Collaboration et al. (2014). They are run using the gravo-magnetohydrodynamics movingmesh code AREPO (Springel 2010; Pakmor et al. 2016). The halos are selected from the dark-matter-only EAGLE simulations (Schaye et al. 2015) as Milky Way-mass (1 < M 200 / [10 12 M ⊙ ] < 2) halos at z = 0. The dark matter particle mass for these simulations is 50,000 M ⊙ while the stellar particles have mass 6,000 M ⊙ . The softening lengths for the gas, stars, and dark matter are ≈ 188 pc. The Auriga simulations implement stellar feedback from Type II supernovae. However, unlike the FIRE-2 supernova rates that evolve in time with the star particles, the Auriga simulations have instantaneous feedback at the time of star formation. Furthermore, the Auriga simulations are missing the early stellar feedback prescriptions included in FIRE-2 which are thought to regulate star formation. The gas density threshold for star formation is n SF > 0 . 13 cm -3 , which is significantly smaller than the FIRE-2 threshold of n SF > 1000 cm -3 . On the other hand, Auriga includes AGN feedback, while FIRE-2 does not. For the Auriga galaxies, stellar particles are labeled as accreted if they form in a subhalo and are within the R 200 of the main host galaxy as z = 0. Similar to FIRE-2, the Auriga simulations have been shown to produce galaxies that are disk-dominated with Milky Way-mass stellar masses, sizes, rotation curves, and metallicities (Grand et al. 2017).",
"pages": [
6
]
},
{
"title": "2.3. Inner Galaxy Dynamical Time and Accreted Fraction Profile along the Minor Axis",
"content": "To confirm that the inner regions of the simulated galaxies are similarly well phase-mixed, we investigate the dynamical time as a function of galactocentric radius. In the top panels of Figure 1, we show the dynamical time as a function of galactocentric radius out to the NFW scale radius of the main dark matter halo for each simulated galaxy used in this work. Specifically, we show results for the isolated FIRE-2 galaxies in the left panel, the FIRE-2 pair galaxies in the middle panel, and the Auriga galaxies in the right panel. The FIRE-2 galaxies are colored by the redshift where they have reached 50% of their z = 0 mass ( z 0 . 5 ). Since the z 0 . 5 metric is not available to us for the Auriga galaxies, they are shown in random shades of green. The dynamical times are calculated using the following formula: where G is the gravitational constant and ρ ( < R ) is the average density inside radius R . As the average density of a galaxy decreases as a function of radius, the dynamical time increases. At the NFW scale radii of these simulated galaxies the dynamical times is on average ≈ 500 Myr with a range of ≈ 250-750 Myr. As the phase-mixing timescale is a few times the dynamical time, we expect mergers with infall times ⪆ 2 Gyr ago to be well phase-mixed inside this radius. In the bottom panels of Figure 1, we explore the birth origin of the stars that comprise the minor axis profile. As the halos/bulges of galaxies are thought to be built from accreted systems, we probe the distance along the minor axis from the galactic center where accreted stars begin to outnumber stars that formed insitu. Specifically, we show the fraction of accreted stars within a cylinder along the minor axis which is defined to be perpendicular to the galaxy's disk plane. The cylinder's radius is 1 / √ π so that each cylindrical bin with height of 1 kpc has the volume of 1 kpc 3 . Similar to the top panels, we only calculate the accreted fraction out to the NFW scale radius of each galaxy. The black dashed line indicates the accreted fraction value of 50%. The FIRE-2 galaxies generally increase their accreted fraction with increasing distance from the Galactic center. The isolated galaxies (left panel) typically reach an accreted fraction of ∼ 50% just inside of the NFW scale radius, while the accreted fraction is generally < 10% within R < 5 kpc. The paired galaxies (middle panel) show larger spread in their accretedfraction profiles, with the earlier forming galaxies (larger z 0 . 5 ) having lower accreted fractions closer to the galactic center. The Auriga galaxies show significantly different behavior than the FIRE-2 galaxies in that their accreted fraction quickly jumps to > 10-40% at ∼ 2 kpc above the galactic center, rather than the gradual increase seen in FIRE-2. In the Auriga galaxies, the accreted fraction only increases slightly with increasing distance from the galactic center so that just within the NFW scale radii (R ∼ 15 kpc) the galaxies generally have an accreted fraction within 10-50%. It is possible this may be due to differences in method of defining accreted stars. Although the methods are philosophically the same, there could be implementation differences which require deeper investigation beyond the scope of this work.",
"pages": [
6,
7
]
},
{
"title": "3. STELLAR AND DARK MATTER PROFILES ALONG THE MINOR AXIS",
"content": "The main focus of this work is on the cosmological predictions for minor axis stellar density profiles in the inner regions of Milky Waymass galaxies. We define the inner region as within the NFW profile (Navarro et al. 1997) scale radius of the dark matter halo. Specifically, for the FIRE-2 simulations, we use the value given from the halo finder ROCKSTAR (Behroozi et al. 2013a) for the host dark matter halo, rounded up to an integer. For the Latte galaxies, the NFW scale radii range from 16-24 kpc with an average of 20 kpc. For the ELVIS galaxies, these values range from 13-20 kpc with an average of 16 kpc. The estimate of the NFW scale radii for the Auriga galaxies comes from Callingham et al. (2020). We select particles within a cylinder along the minor axis with a radius of 1 / √ π so that each cylindrical bin with height of 1 kpc has the volume of 1 kpc 3 . After multiplying these particles by their mass, we retrieve the mass density profile. Wefit a simple power law and exponential disk model to the mass density profile. Explicitly, Note -We provide the following properties for each galaxy at z = 0. M 200c and R 200c are the total mass and spherical radius in which the average density is 200 × the critical density of the universe. M ∗ , 90 is the stellar mass within a spherical radius that encloses 90% of the stellar mass within 20 kpc (Wetzel et al. 2023). N sp is the number of satellites which contribute 90% of the accreted stellar mass (Monachesi et al. 2019). ρ 0 , H and α are the stellar halo central mass density and slope, while ρ 0 , DM and α DM are the corresponding parameters for the dark matter component (see Appendix A). where ρ 0 ,H , and ρ 0 ,D are the central halo/bulge and disk mass densities, respectively. The scale height for the disk is h D , the scale radius for the halo is r H and the halo power law slope is given by α . To avoid the impact of binning when fitting to the simulated galaxy profiles, we perform the fit to the reverse cumulative distribution function (CDF) normalized by the total mass within the NFW scale radius and 1 / √ π kpc of the minor axis. This fit provides the disk scale height, halo power law slope, halo scale radius, and the relative strength of the disk mass density compared to the halo. To derive the ρ 0 ,H and ρ 0 ,D we then fit to the unnormalized profile. For further details and figures demonstrating the fits, we refer the reader to Appendix A. In addition to quantifying the density profile, we are interested in determining the radius at which halo substructures become significant. To do this we measure the density profile along different lines and compare to the profile along the minor (z-)axis. Specifically, we rotate the the simulated galaxy about the x-axis by a random angle, ϕ , between 0 and π/ 4 radians, in order to avoid the galactic disk. We then rotate the galaxy again but this time about the z-axis by a random angle, θ , between 0 and 2 π radians. Each time we measure the mass density profile along the new z-axis using the same cylinder as before. We perform 100 unique combinations of rotations to obtain 100 different estimates of the mass density profile. We then compute the fractional difference between this and the original, non-rotated z-axis profile (see Figure 8-4 in Appendix A). In order to understand how the stellar component relates to its dark matter counterpart, we also fit the dark matter halo distribution using the same technique. The only difference is that we do not include a exponential disk component in the functional form. Therefore, the functional form of the dark matter mass density distribution is the following power law: where ρ 0 ,DM is the dark matter mass density at R = 0, r DM is the scale radius and α DM is the power law slope. We follow the same procedure as the stellar component for fitting the functional form as well as the angle dependence. We find that the fractional difference between the dark matter density profiles at random angles is significantly smaller at all radii than that of the stellar profiles, indicating that the stellar particles have more significant substructure than the dark matter.",
"pages": [
7,
8,
9
]
},
{
"title": "4. INNER MINOR AXIS STELLAR DENSITY PROFILES ARE WEAKLY CORRELATED WITH ACCRETION HISTORY",
"content": "Generally, it is thought that the inner halo stellar density profile traces the cumulative accretion history of a galaxy (Helmi 2020; Han et al. 2022). Similarly, the growth of bulges is thought to occur through merger events (Aguerri et al. 2001; Hopkins et al. 2010). This idea can be directly tested using cosmological zoom-in simulations. In this section, we com- pare measured properties of the z = 0 minor axis stellar density profiles of Milky Waymass galaxies (see Section 3) with their accretion histories. In order to quantify and interpret strengths of correlations in this work, we use the Spearman correlation coefficient which measures the monotonicity of the relationship between two variables (Spearman 1904; Dancey & Reidy 2004). Therefore, the Spearman correlation coefficient will be close to 1 if the data have similar rank in the two variables, i.e., the data point with the largest x -value also has the largest y -value and so on. Furthermore, the Spearman correlation coefficient will be close to -1 if the data have almost opposite rank. Generally, a correlation is considered weak for Spearman coeffcients with an absolute value < 0.40. A moderate correlation has an absolute Spearman coefficient between 0.40 and 0.60, while a strong correlation has > 0.60. In Figure 2, we show how the amount of accreted stellar mass in each galaxy relates to α , the power law slope (bottom panels), and ρ 0 , H , the central stellar mass density for the powerlaw component (top panels). The left and middle panels show results for the FIRE-2 galaxies while the right panels show results for the Auriga galaxies (green Xs). The FIRE-2 pair galaxies are shown as diamonds, while the isolated galaxies are circles. The FIRE-2 results are colored by the redshift when the proto-Milky Way emerges, z MR 3:1 , defined in Horta et al. (2024) as the redshift when the main galaxy halo becomes 3 times more massive than the next most massive luminous halo. For the Auriga galaxies the horizontal axis is the reported accreted stellar halo mass from Monachesi et al. (2019). For the FIRE-2 galaxies we use two different metrics for the accreted stellar mass. The left panel horizontal axis is the log 10 of the stellar mass of the simulated galaxy's building blocks from Horta et al. (2024). Specifically, the building blocks are defined as luminous ha- s which merge with the main branch before z MR 3:1 . In the middle panel, the horizontal axis is the log 10 of the phase-mixed stellar mass accreted after z MR 3:1 from Kundu et al. (in prep). It is interesting to note that although the Auriga central stellar densities overlap with the FIRE-2 results, the range is smaller: all Auriga halos have ρ 0 , H < 10 9 M ⊙ / kpc 3 while almost a third (4/15) FIRE-2 galaxies have power law components with central densities larger than that. The opposite is true for the power law slopes, α , in that the Auriga galaxies have larger scatter than FIRE-2. However, we note that these differences are likely due to fit degeneracies. There is an especially strong degeneracy between ρ 0 , H and r H in that smaller r H indicates a more cuspy density profile with a higher central density. We note that the FIRE-2 galaxies with larger ρ 0 , H all have r H that are smaller than the Auriga galaxies (see Appendix A). Both Auriga and FIRE-2 show weak correlations between the various measures of accreted stellar mass and the minor axis stellar density profile. For the FIRE-2 galaxies, the strongest correlation is between the power law slope and the total stellar mass of the building blocks. The Spearman correlation coefficient for these two measures is 0.35, which signifies a weak relationship. Similarly, for the Auriga galaxies the power law slope is most correlated with the accreted stellar mass, with a Spearman correlation coefficient of -0.31. In conclusion, we do not find strong correlations between the accreted stellar mass and the amplitude or slope of the minor axis stellar density profile. Figure 3 is the same as Figure 2 except for the horizontal axes. In Figure 3, the horizontal axis for the left and right panels is the number of 'significant progenitors' from Monachesi et al. (2019), defined as the number of satellites that together contribute 90% of the total accreted stellar mass. The middle panel horizontal axis is the number of phase- mixed accreted systems from Kundu et al. (in prep). The Auriga galaxies have a positive correlation between the number of significant progenitors and the stellar density profile slope with a Spearman correlation coefficient of 0.57. This is classified as a moderate relationship. However, a negative correlation is reported in Monachesi et al. (2019) with the 28 Milky Way-mass Auriga galaxies run at lower resolution, although the correlation is weak. We also see a negative correlation between the stellar density profile slope and number of accreted systems with the FIRE-2 galaxies, but the correlation is weak with a Spearman correlation coefficient of -0.37. In general, we find inconclusive evidence that the number of merger events impacts the minor axis stellar density profile for the inner regions of these simulated galaxies.",
"pages": [
9,
10,
11,
12
]
},
{
"title": "5. THE STARS FOLLOW THE DARK MATTER",
"content": "It is well accepted that dark matter is a significant component of galactic halos (White & Rees 1978; Bland-Hawthorn & Gerhard 2016; Helmi 2020). In this section, we compare the measured minor axis stellar density profiles with the corresponding dark matter density profiles. In general, we find that the strength and slope of the stellar profile is highly correlated with the dark matter profile. In Figure 4, we show a graphical representation of the correlations between the minor axis stellar and dark matter density profile parameters along with parameters describing the accretion history of the galaxy. Specifically the Spearman correlation coefficients are represented as ellipses with ellipticity, orientation and color based on the their value. Strong positive correlations are shown with narrow ellipses that point up to the right in dark blue while strong negative correlations point down to the right in dark red. Weak correlations with coefficients close to zero are shown as white circles. Results for how the amplitude and slope of the FIRE-2 galaxies' minor axis stellar density profiles relate to the stellar mass of the building blocks, accreted stellar mass, number of significant progenitors, and number of accreted systems, along with the minor axis dark matter density profile's amplitude and slope are shown on the left in order from left to right. Results for the correlations of the amplitude and slope of Auriga's minor axis stellar density profiles with the accreted stellar mass, number of significant progenitors, minor axis dark matter density profile amplitude and slope are shown on the right, also in order from left to right. We show the correlations for the stellar mass of the building blocks, the accreted stellar mass, the number of accreted systems and the number of significant progenitors in Figure 4 in order to compare with the dark matter correlations, although they have already been discussed in Section 4 (see Figures 2 and 3). For the FIRE-2 galaxies the strongest correlations are between the corresponding minor-axis stellar and dark matter density profile parameters. Specifically, the stellar and dark matter density amplitudes are strongly positively correlated, along with the slopes. On the other hand the stellar density amplitude is negatively correlated with the dark matter slope and the dark matter density amplitude is negatively correlated with the stellar slope. This is likely due to the combination of the degeneracies in the fit of the density amplitudes and slopes for both components and the strong positive correlations between the components' profiles. For example, using the covariance matrix from the fitting procedure, we find that on average 75% of the variance in the stellar halo central density parameters can be explained by the variance in the stellar halo power-law slope. On the other hand, when comparing across the stars and dark matter, the strength of the positive correlations between the components' amplitudes and slopes are 0.63 and 0.69, respectively, which are considered strong relationships. The negative correlation coefficient between the stellar amplitude and dark matter slope is -0.43, while the dark matter amplitude and stellar slope have a correlation coefficient of -0.46. These are both considered moderate relationships, and are roughly 75% of the strong relationship ( ∼ 0.6) between the corresponding stellar and dark matter components. The minor axis stellar density profiles of the Auriga galaxies have weaker correlations with their corresponding dark matter density profiles than the FIRE-2 galaxies. The strength of the correlations between the amplitudes and slopes are both 0.49, which is considered moderate, al- though less strong than the FIRE-2 correlations of 0.63 and 0.69. It is difficult to determine why the Auriga galaxies do not show stronger correlations, but it is likely due to worse fit parameters. The functional form was optimized for FIRE-2 galaxies and thus the average log 10 rootmean-squared error (RMSE) is 7.38 for Auriga galaxies while it is only 7.16 for FIRE-2 galaxies. We further discuss the relative goodnessof-fits in Appendix A. It is also possible that the difference in correlation strengths between FIRE-2 and Auriga is due to the difference in physical prescriptions. This is discussed further in Section 6.1. In Figure 5, we show the relationship between the minor axis stellar density profile amplitude with the corresponding dark matter density amplitude for the FIRE-2 Milky Way-mass galaxies (top two panels) and the Auriga Milky Waymass galaxies (bottom panel). In the top panels, the FIRE-2 galaxy pairs are shown as diamonds while the isolated galaxies are circles. In the topmost panel, the points are colored by their formation time. Specifically, the color corresponds to the redshift at which the galaxy has reached 50% of its z=0 mass (z 0 . 5 ). The middle panel is the same as the top panel, except here the points are colored by the ratio of the stellar to dark matter scale radii. The bottom panel ρ α shows the same quantities for the Auriga galaxies, but the color bar scale is quite different than the middle panel's. Not only do the ratios of the scale radii for the stellar and dark matter density profiles have larger values, but the Auriga halos also have larger dark matter density amplitudes than the FIRE-2 galaxies on average. For example, only 2 out of 15 FIRE-2 galaxies have ρ 0 , DM > 20 × 10 8 M ⊙ / kpc 3 while the least dense Auriga halo has ρ 0 , DM = 24 × 10 8 M ⊙ / kpc 3 . This is consistent with the fact that the FIRE-2 and Auriga galaxies have a similar range of M 200 ≈ 1-2 × 10 12 M ⊙ but FIRE-2 has R 200 m ≈ 320-410 kpc while Auriga has R 200 ≈ 210-260 kpc. The virial radii R 200 are calculated differently in that Auriga uses the radius within which the density is 200 × the critical density for closure, while FIRE-2 uses the radius within which the density is 200 × the mean density of the Universe. Either way, the Auriga galaxies have similar mass inside a smaller radius than the FIRE2 galaxies, and therefore higher mass density. The Auriga dark matter density profiles also have smaller scale radii than the FIRE-2 dark matter profiles, while the stellar profiles have larger scale radii on average. This is likely due to differences in feedback mechanisms between the simulations, which are discussed further in Section 6.1. In the top panel of Figure 5, the FIRE-2 galaxy pairs with early formation times are outliers in the relationship between the stellar and dark matter density amplitudes. One possible reason for this is simply that these structures formed earlier when the universe had an overall lower stellar-to-dark-matter mass ratio. In the middle panel, we demonstrate how the degeneracy of the fits impacts the spread in the relationship between the stellar and dark matter density amplitudes for the FIRE-2 galaxies. On the other hand, the Auriga galaxies do not show a strong correlation between the stellar and dark matter density amplitudes. However, given the consistency of the ratio of stellar-to-dark-matter mass seen in Figure 7, the correlation is likely masked by degeneracies in the fit parameters (see Appendix A). The relationship between the minor axis stellar density power law slopes and the correspond- ing dark matter slopes are shown in Figure 6. The Auriga galaxies are shown as Xs while the FIRE-2 isolated galaxies are shown as circles and the pairs as diamonds. The points are colored by the ratio of their stellar and dark matter scale radii, but as in Figure 5, the Auriga galaxies have a different color scale than the FIRE-2 galaxies. The Auriga and FIRE-2 galaxies both show a strong correlation between their stellar and dark matter density slopes. In general, the Auriga galaxies have steeper stellar slope for a given dark matter slope. Similar to the stellar density amplitudes, the largest outliers in the relationship between the stellar and dark matter density slopes for the FIRE-2 galaxies are also outliers in the ratio of the corresponding scale radii. This indicates much of the spread is likely due to degeneracies in the fit parameters. Figure 7 shows the ratio of stellar density to dark matter density for the simulated galaxies as a function of distance from the galactic center. Specifically, we calculate these ratios similar to the density profile fits in Section 3 in that we only use particles within a cylinder along the minor axis with radius=1 / √ π . In the left panel we show results for the isolated FIRE-2 galaxies, while the middle panel shows the pairs. The lines in these panels are colored by the redshift at which the galaxy has reached 50% of its z = 0 mass. In the middle panel, we also use different line styles for sets of pairs. Specifically, Romeo and Juliet are shown as solid lines, while Romulus and Remus are in dotted lines and Thelma and Louise in dot-dashed lines. The right panel shows results for the Auriga galaxies colored in random shades of green. Each panel also has a grey dashed line which is simply the function ρ ∗ /ρ DM = 0 . 4 / R 2 which is the best fit to the results for the FIRE-2 pair galaxies (middle panel). The ratio of stellar to dark matter density along the minor axis in these simulated galaxies follows a r -2 profile. While each simulation suite generally follows this relation, the FIRE-2 and Auriga galaxies show slightly different behavior. The later forming (lower z 0 . 5 ) FIRE-2 pairs have higher ρ ∗ /ρ DM at R > 10 kpc compared to the earlier forming galaxies. The Auriga galaxies generally have higher stellar to dark matter ratios similar to the later forming FIRE-2 galaxy pairs. It is possible that this is because they also form later, when the universe overall has a higher stellar-to-dark-matter density, but we do not have formation redshifts for these galaxies to confirm.",
"pages": [
12,
13,
14,
15,
16
]
},
{
"title": "6. DISCUSSION AND CONCLUSIONS",
"content": "In this work, we parameterized the minor axis stellar and dark matter density profiles of Milky Way-mass FIRE-2 and Auriga simulated galaxies. With this parameterization, we compare the stellar profiles to the accretion histories and find weak to no correlation. Instead, we find strong correlations between the distribution of stars and dark matter. Specifically, we find that the ratio of stellar to dark matter density consistently falls off as r -2 along the minor axis across the simulation suites. Whether the consistency of the stellar-todark-matter profile is a natural consequence of galaxy formation and Λ-CDM, or if its merely a coincidence in these simulations is unknown. Consistent with the profiles measured in this work, dark matter halos are generally thought to have pseudo-isothermal profiles with r -2 within their NFW scale radii (Navarro et al. 1997). Predictions for the minor axis stellar density profiles in this range are fewer and vary more widely, but generally cluster around r -4 (Monachesi et al. 2019; Deason et al. 2013; Font et al. 2020; Cooper et al. 2010; Amorisco 2017). In this work, we find that degeneracies in the fit parameters can cause more spread in the measured slopes (see Figure 6), but the profile of the ratio of stellar-to-dark-matter is consistent (see Figure 7). We have intentionally investigated the minor axis stellar density profiles in a region where galactic component classification is difficult. Specifically, the radial range we study is where the disk, bulge and inner halo overlap. Although it is clear that the exponential component of the stellar density profile is the galactic disk, we do not attempt to classify the powerlaw component. It could be classified as either bulge or inner halo. Classical bulges have long been observed to have r -4 profiles, similar to elliptical galaxies (de Vaucouleurs 1948; Hernquist 1990). However, estimates for the S'ersic indices of the FIRE-2 and Auriga galaxies have n < 2, inconsistent with classical bulges (Sanderson et al. 2018; Gargiulo et al. 2019). Simulations show that that r -4 stellar profiles can be reproduced with dissipationless collapse (van Albada 1982) and also with dissipational mergers (e.g., Hernquist 1992; Barnes 1992). Halo stellar density profiles are less often studied, given their low surface brightness. However, the studies that exist have found profiles with slopes that vary around r -4 for nearby Milky Way-mass disk galaxies (Harmsen et al. 2017). In Section 6.2, we discuss comparisons to the Milky Way.",
"pages": [
16,
17
]
},
{
"title": "6.1. Differences between FIRE-2 and Auriga",
"content": "In this work, we note a number of differences between the Auriga and FIRE-2 minor axis stellar and dark matter profiles. While we do not focus on the galactic disks in this work, we find that Auriga's disks have 10-100 × higher stellar mass densities than the FIRE-2 disks (see Appendix A). Relatedly, the Auriga galaxies are known to host stronger bars than the Fire2 galaxies (Ansar et al. 2023; Fragkoudi et al. 2024). Another large difference between the FIRE-2 and Auriga galaxies are the dark matter central densities ( ρ 0 , DM ) and scale radii ( r DM ). Although they have never been directly compared, it is known that the Auriga galaxies have sig- ificant baryon contraction (Callingham et al. 2020), while the FIRE-2 galaxies form dark matter cores (Lazar et al. 2020). This is likely due to differences in the implementation of feedback, which has been shown to significantly impact the distribution of mass in the center of galaxies (Pontzen & Governato 2014; Lazar et al. 2020). The formation of dark matter halo cores in cosmological simulations is dependent on the the star formation prescriptions as they relate to the supernova rates. Specifically, the choice of gas density threshold for star formation has been shown to impact the dark matter distribution in galaxies (Dutton et al. 2019; Ben'ıtezLlambay et al. 2019). Given that the FIRE2 galaxies have a higher gas density threshold for star formation ( n SF > 1000 cm -3 ) than the Auriga simulations ( n SF > 0 . 13 cm -3 ), it is not surprising that they have dark matter cores ( r DM ∼ 1 kpc) and while the Auriga simulations do not ( r DM ∼ 0.3 kpc). As discussed in Section 2.2, the supernova prescriptions are also different between the simulations in that FIRE-2 have rates that evolve with the star particles (Hopkins et al. 2018b), while the Auriga simulations have instantaneous supernova feedback at the formation time of the star particle. FIRE-2 also implements radiative feedback from massive stars, which regulates star formation (Hopkins et al. 2014; Orr et al. 2018; Hopkins et al. 2020). Auriga does not include early stellar feedback, but does include AGN feedback, which is missing from the FIRE-2 physics model. In total, there are many differences between the implementations of the simulations which could explain the small variations in the stellar and dark matter minor axis profiles. Despite these differences, however, the most interesting result is the remarkable similarity of their stellar-to-dark-matter ratio profiles, which consistently fall off as r -2 for both simulation suites.",
"pages": [
17
]
},
{
"title": "6.2. Comparison to Milky Way",
"content": "As discussed in the introduction (Section 1), there are a number of estimates of the MW halo stellar density profile. Each of these measurements have their own selection functions, biases and radial ranges. Therefore, it is difficult to compare our results without a thorough investigation of the impacts of each selection function and method. We plan to perform a detailed comparison with MW data and study the impact of selection functions in future work. Here, we briefly summarize the MW results which are most suitable for comparison with the simulated galaxies. Generally, the simulated galaxies have steeper stellar density slopes than what is measured in the MW. Between ≈ 5-20 kpc estimates for the MW's halo stellar density profile slope range from -1 to -3 (Sesar et al. 2013; Faccioli et al. 2014; Sesar et al. 2010; Han et al. 2022). There are few measurements within 5 kpc of the Galactic center given the high levels of dust extinction and crowding. However, using RR Lyrae, the stellar density profile slope was estimated as -3 in the radial range of 0.2-2.8 kpc from the Galactic center (Pietrukowicz et al. 2015). Furthermore, using orbit integration, Yang et al. (2022) found that the halo stellar density profile flattens in the inner regions with the slope increasing to -1.5. Although these works provide estimates in this range, they are based only on a small fraction of the stellar population, with complex selection functions. In order to perform a fair comparison, the MW minor axis profile should model the Galactic disk to minimize the selection bias. We plan to do this in future work using data with a well-modeled selection function. The MW's dark matter density profile is not well constrained because methods rely on the poorly constrained baryonic matter distribution. Recent measurements of the Milky Way's circular velocity curve have provided estimates of the dark matter profile which give results consistent with the profiles of FIRE-2 galaxies (Ou et al. 2024). However, we note that these results give a larger core and more concentrated dark matter halo than what is typically seen in Milky Way-mass cosmological simulations (Lazar et al. 2020). The ratio of stellar-to-dark matter mass profile has not been directly measured in the Milky Way. However, combining results from the individually measured profiles indicate a ratio profile that is roughly constant or linear, given that the stellar profile is less steep than in the simulated galaxies. If this discrepancy is borne out in future work, this could indicate a need for adjustments in the physics models used in FIRE2 and Auriga simulations, likely either in the stellar feedback prescription, or the dark matter model.",
"pages": [
18
]
},
{
"title": "6.3. Summary of Conclusions",
"content": "Λ-CDM cosmology predicts that hierarchical growth dominates early galaxy formation and evolution. The minor axis stellar density profile of galaxies is of special interest as it is thought to trace the galaxy's accretion history. Whether classified as halo or bulge, this stellar component is thought to build mass through mergers and accretion events. In this work, we have investigated the minor axis stellar and dark matter density profiles of Milky Way-mass galaxies in the cosmological zoom-in simulation suites of FIRE-2 and Auriga. Using only data within their respective NFWscale radiii, we quantify the profiles with a simple power law parameterization, including a exponential disk component for the stars. With the aim of understanding the physical mechanism that shapes the minor axis stellar density profile, we compare the parameters with other properties of the simulated galaxies. In total, we find: As the ratio of light to dark matter is a fundamental observation, it is crucial to understand the interplay of stellar and dark matter mass in galaxies. In future work, we plan to build on this work by investigating the comparison between the MW and simulations more deeply. This includes a new measurement of the minor axis stellar density profile using Gaia data, and an investigation of the Milky Way's ratio of stellar-to-dark matter mass profile using stellar kinematics. Finally, we will also look to alternative dark matter models including the FIRE-2 self-interacting dark matter simulations (Sameie et al. 2021; Vargya et al. 2022) to further study the connection between stellar and dark matter mass. Software: Astropy (Astropy Collaboration et al. 2013, 2018), Matplotlib (Hunter 2007), IPython (P'erez & Granger 2007), Numpy (Harris et al. 2020), Scipy (Virtanen et al. 2020), GizmoAnalysis (Wetzel & Garrison-Kimmel 2020a) HaloAnalysis (Wetzel & Garrison-Kimmel 2020b; Wetzel et al. 2016)",
"pages": [
18,
19
]
},
{
"title": "ACKNOWLEDGEMENTS",
"content": "This material is based upon work supported by the National Science Foundation under Award No. 2303831. We have used simulations from the Auriga Project public data release (Grand et al. 2024) available at https:// wwwmpa.mpa-garching.mpg.de/auriga/data.",
"pages": [
19
]
},
{
"title": "A. DENSITY PROFILE FITS",
"content": "To parameterize the minor axis stellar and dark matter density profiles, we fit simple power law functions (see Equations 2 and 3) to the distribution of particles inside a cylinder of radius 1 / √ π along the minor axis which is defined to be perpendicular to the galaxy's disk. For the stellar component we also need to account for the exponential disk. To avoid the effect of binning, we first perform the fit to the cumulative distribution function which is normalized by the total stellar mass within the cylinder out to the halo's NFW scale radius (left panels). In Figures 8, 9, and 10, we show the fits to the stellar density profiles, while Figures 11, 12, and 13 show the fits to the dark matter profiles. Each row corresponds to one simulated galaxy with the resulting parameterization printed above the corresponding panels. The galaxy's designation is printed in the top right of the leftmost panel. The middle panels show the fit (red dashed line) to the unnormalized density distribution (dark blue solid line). We note that fits to the Auriga stellar density profiles are generally worse then the fits to the FIRE-2 galaxies. For example, the log 10 RMSE is 7.17 for the isolated FIRE-2 galaxies, 7.15 for the pairs and 7.38 for the Auriga galaxies. This is because the functional form (Equation 2) was chosen to optimize the fit to the FIRE-2 galaxies. The Auriga galaxies may be fit better using a different functional form, but in order to facilitate a comparison we keep the functional form the same for both simulations. In the left and middle panels, we also plot in transparent dark blue the stellar particle distributions for cylinders of the same size, but at 100 random angles that are < 45 · from the z-axis. With these distributions, we investigate the level of substructure and asymmetry in the simulated minor axis profiles. The right panels show the difference between the density distributions along the z-axis and the 100 random angles, normalized by the z-axis density at that point (dark blue transparent lines). The median of these 100 lines is shown as the solid dark blue line. We also show three red horizontal dashed lines at y-values of -1, 0, and 1. Generally, we find the distributions do not vary significantly for the FIRE-2 galaxies, while the Auriga galaxies show systematically larger densities at angles closer to the disk, indicating more oblate halos, consistent with results from Monachesi et al. (2019).",
"pages": [
19,
21,
22
]
},
{
"title": "REFERENCES",
"content": "Aguerri, J. A. L., Balcells, M., & Peletier, R. F. 2001, A&A, 367, 428, doi: 10.1051/0004-6361:20000441 Amorisco, N. C. 2017, MNRAS, 464, 2882, doi: 10.1093/mnras/stw2229 Ansar, S., Pearson, S., Sanderson, R. E., et al. 2023, arXiv e-prints, arXiv:2309.16811, doi: 10.48550/arXiv.2309.16811 Arentsen, A., Starkenburg, E., Martin, N. F., et al. 2020, MNRAS, 491, L11, doi: 10.1093/mnrasl/slz156 Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068 Astropy Collaboration, Price-Whelan, A. M., Sip\"ocz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f Athanassoula, E. 2005, MNRAS, 358, 1477, doi: 10.1111/j.1365-2966.2005.08872.x Babusiaux, C., G´omez, A., Hill, V., et al. 2010, A&A, 519, A77, doi: 10.1051/0004-6361/201014353 Barnes, J. E. 1992, ApJ, 393, 484, doi: 10.1086/171522 Behroozi, P. S., Wechsler, R. H., & Wu, H.-Y. 2013a, ApJ, 762, 109, doi: 10.1088/0004-637X/762/2/109 Behroozi, P. S., Wechsler, R. H., Wu, H.-Y., et al. 2013b, ApJ, 763, 18, doi: 10.1088/0004-637X/763/1/18 Bellardini, M. A., Wetzel, A., Loebman, S. R., et al. 2021, MNRAS, 505, 4586, doi: 10.1093/mnras/stab1606 Belokurov, V. 2013, NewAR, 57, 100, doi: 10.1016/j.newar.2013.07.001 Belokurov, V., & Kravtsov, A. 2022, MNRAS, 514, 689, doi: 10.1093/mnras/stac1267 Belokurov, V., Sanders, J. L., Fattahi, A., et al. 2020, MNRAS, 494, 3880, doi: 10.1093/mnras/staa876 Ben'ıtez-Llambay, A., Frenk, C. S., Ludlow, A. D., & Navarro, J. F. 2019, MNRAS, 488, 2387, doi: 10.1093/mnras/stz1890 Binney, J., Gerhard, O. E., Stark, A. A., Bally, J., & Uchida, K. I. 1991, MNRAS, 252, 210, doi: 10.1093/mnras/252.2.210 Bland-Hawthorn, J., & Gerhard, O. 2016, ArXiv e-prints. https://arxiv.org/abs/1602.07702 Blitz, L., & Spergel, D. N. 1991, ApJ, 370, 205, doi: 10.1086/169806 Zoccali, M., Gonzalez, O. A., Vasquez, S., et al. 2014, A&A, 562, A66, doi: 10.1051/0004-6361/201323120",
"pages": [
22,
23,
24,
25,
29
]
}
]
2024arXiv241007084K
https://arxiv.org/pdf/2410.07084.pdf
<document>
<section_header_level_1><location><page_1><loc_10><loc_86><loc_90><loc_87></location>CONSTRAINING THE DISPERSION MEASURE REDSHIFT RELATION WITH SIMULATION-BASED INFERENCE</section_header_level_1>
<section_header_level_1><location><page_1><loc_45><loc_78><loc_55><loc_79></location>K/o.pc/u.pc/s.pc/t.pc/a.pc/v.pc K/o.pc/n.pc/a.pc/r.pc ★</section_header_level_1>
<text><location><page_1><loc_10><loc_76><loc_91><loc_78></location>Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), German Centre for Cosmological Lensing, 44780 Bochum, Germany</text>
<section_header_level_1><location><page_1><loc_44><loc_72><loc_56><loc_74></location>R/o.pc/b.pc/e.pc/r.pc/t.pc R/e.pc/i.pc/s.pc/c.pc/h.pc/k.pc/e.pc †</section_header_level_1>
<text><location><page_1><loc_10><loc_69><loc_91><loc_72></location>Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71, D-53121 Bonn, Germany and Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), German Centre for Cosmological Lensing, 44780 Bochum, Germany</text>
<section_header_level_1><location><page_1><loc_44><loc_66><loc_56><loc_67></location>S/t.pc/e.pc/f.pc/f.pc/e.pc/n.pc H/a.pc/g.pc/s.pc/t.pc/o.pc/t.pc/z.pc</section_header_level_1>
<text><location><page_1><loc_14><loc_63><loc_87><loc_65></location>Universitäts-Sternwarte, Fakultät für Physik, Ludwig-Maximilians Universität München, Scheinerstraße 1, D-81679 München, Germany and Excellence Cluster ORIGINS, Boltzmannstraße 2, D-85748 Garching, Germany</text>
<section_header_level_1><location><page_1><loc_45><loc_60><loc_55><loc_61></location>A/n.pc/d.pc/r.pc/i.pc/n.pc/a.pc N/i.pc/c.pc/o.pc/l.pc/a.pc</section_header_level_1>
<text><location><page_1><loc_25><loc_59><loc_76><loc_60></location>Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71, D-53121 Bonn, Germany</text>
<section_header_level_1><location><page_1><loc_43><loc_55><loc_57><loc_56></location>H/e.pc/n.pc/d.pc/r.pc/i.pc/k.pc H/i.pc/l.pc/d.pc/e.pc/b.pc/r.pc/a.pc/n.pc/d.pc/t.pc</section_header_level_1>
<text><location><page_1><loc_10><loc_54><loc_91><loc_55></location>Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), German Centre for Cosmological Lensing, 44780 Bochum,</text>
<text><location><page_1><loc_48><loc_53><loc_52><loc_54></location>Germany</text>
<text><location><page_1><loc_43><loc_52><loc_57><loc_52></location>Version October 10, 2024</text>
<section_header_level_1><location><page_1><loc_46><loc_49><loc_54><loc_50></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_43><loc_86><loc_48></location>We use the dispersion measure (DM) of localised Fast Radio Bursts (FRBs) to constrain cosmological and host galaxy parameters using simulation-based inference (SBI) for the first time. By simulating the large-scale structure of the electron density with the Generator for Large-Scale Structure (GLASS), we generate log-normal realisations of the free electron density field, accurately capturing the correlations between different FRBs.</text>
<text><location><page_1><loc_14><loc_37><loc_86><loc_43></location>For the host galaxy contribution, we rigorously test various models, including log-normal, truncated Gaussian and Gamma distributions, while modelling the Milky Way component using pulsar data. Through these simulations, we employ the truncated sequential neural posterior estimation method to obtain the posterior. Using current observational data, we successfully recover the amplitude of the DM-redshift relation, consistent with Planck, while also fitting both the mean host contribution and its shape. Notably, we find no clear preference</text>
<text><location><page_1><loc_14><loc_34><loc_86><loc_36></location>for a specific model of the host galaxy contribution. Although SBI may not yet be strictly necessary for FRB inference, this work lays the groundwork for the future,</text>
<text><location><page_1><loc_14><loc_29><loc_86><loc_34></location>as the increasing volume of FRB data will demand precise modelling of both the host and large-scale structure components. Our modular simulation pipeline offers flexibility, allowing for easy integration of improved models as they become available, ensuring scalability and adaptability for upcoming analyses using FRBs. The pipeline is made publicly available under https://github.com/koustav-konar/FastNeuralBurst.</text>
<text><location><page_1><loc_14><loc_27><loc_41><loc_28></location>Keywords: Cosmology, Fast Radio Bursts</text>
<section_header_level_1><location><page_1><loc_22><loc_23><loc_34><loc_25></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_10><loc_48><loc_23></location>Fast Radio Bursts (FRB) have received significant attention over the past decades, both from cosmological and astrophysical perspectives. First discovered in archival data (Lorimer et al. 2007), these broad, millisecond transient pulses in the radio frequency range get dispersed by free electrons along their line of sight. While their origin is still debated (Petroff et al. 2019) and ranges from Magnetars (Thornton et al. 2013; Bochenek et al. 2020) to binary mergers (Liu et al. 2016), it is clear that the majority of them must be of extragalactic origin due to their highly dispersed signal. The proportionality con-</text>
<text><location><page_1><loc_10><loc_7><loc_21><loc_8></location>† reischke@posteo.net</text>
<text><location><page_1><loc_52><loc_21><loc_92><loc_25></location>stant of this dispersion, fittingly called Dispersion Measure (DM), is proportional to the column density of electrons along the line-of-sight.</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_21></location>As a consequence, FRBs have been proposed to be used as a cosmological probe, in particular of the baryon distribution in the Universe. As for all cosmological fields, the electron density can be split into a background and fluctuation component relative to the background. If the host of the FRB is identified, an independent redshift estimate can be obtained. This allows the construction of the DM𝑧 relation, similar to the luminosity distance from supernovae. This relation has been used with current data to measure the baryon density and the Hubble constant (e.g. Zhou et al. 2014; Walters et al. 2018;</text>
<text><location><page_2><loc_8><loc_84><loc_48><loc_92></location>Hagstotz et al. 2022a; Macquart et al. 2020; Wu et al. 2022; James et al. 2022; Reischke & Hagstotz 2023a). Likewise, one can study the statistical properties of the DM fluctuations (e.g. Masui & Sigurdson 2015; Shirasaki et al. 2017; RafieiRavandi et al. 2021; Bhattacharya et al. 2021; Takahashi et al. 2021; Reischke et al. 2021, 2022; Reischke et al. 2023).</text>
<text><location><page_2><loc_8><loc_57><loc_48><loc_84></location>The ever-increasing number of observed FRBs (currently around 600 unique events, see e.g. Newburgh et al. 2016; CHIME/FRB Collaboration et al. 2021; Khrykin et al. 2024a) leads to raising interest in these events. The Square Kilometre Array (SKA /one.sup , Dewdney et al. 2009) should observe > 10 5 FRBs. Also, other surveys like DSA-2000 (Hallinan et al. 2019) are planning to detect > 10 4 FRBs with host identification. This increasing number makes the modelling and inference process prone to systematic effects. Reischke & Hagstotz (2023b) showed that already with ∼ 300 FRBs, it becomes necessary to include their covariance to conduct unbiased parameter inference using the DM𝑧 relation, an effect which has been neglected in all studies so far. With around 10 4 FRBs, additional effects such as magnification can become important as well (Takahashi 2024). Using FRBs as a tool for cosmology and astrophysics therefore requires careful modelling. A lot of these effects can be challenging to model analytically, including the case where systematic effects from the search (or in cosmological terms, survey) strategy will not be tractable.</text>
<text><location><page_2><loc_8><loc_20><loc_48><loc_57></location>In this paper, we want to tackle these issues and present simulation-based inference (SBI) of cosmological and astrophysical models via the DM𝑧 relation of FRBs. SBI, sometimes also referred to as Likelihood-Free Inference or Implicit Likelihood Inference, is a Bayesian inference technique that does not require an explicit expression for the likelihood function of the data given the parameters of interest. Instead, the likelihood is implicitly assessed by evaluating the joint probability of the data and parameters from forward simulations that map the parameters to the corresponding synthetic data vectors. This approach offers several advantages over traditional methods that necessitate an explicit form for the likelihood. Firstly, the likelihood can assume any form, thus allowing one to bypass the common assumption of a Gaussian likelihood or the need to define a complex analytical expression for the likelihood. Secondly, for certain models and measurements, it might be impractical or too resource-intensive to determine an analytical likelihood. On the similar side, for SBI, data compression becomes essential for the high-dimensional data and parameter spaces typical in cosmology (Leclercq 2018; Alsing et al. 2019). The methods available in the SBI framework also vary based on their complexities; from the relatively trivial Approximate Bayesian Computing (ABC, see e.g. Rubin 1984; Pritchard et al. 1999; Beaumont 2019) to the latest development in NN. These procedures have been applied to cosmological data analysis (e.g. Fluri et al. 2022; Lu et al. 2023; Lin et al. 2023; Euclid Collaboration et al. 2023; von Wietersheim-Kramsta et al. 2024; Gatti et al. 2024).</text>
<text><location><page_2><loc_8><loc_10><loc_48><loc_20></location>The NN requires forward simulations to learn the posterior distribution 𝑝 ( 𝜽 | 𝒅 ) where 𝒅 is simulated given 𝜽 . In our case, these forward simulations consist in principle of three components: ( 𝑖 ) The large-scale structure (LSS) and the background component are produced using the Generator for Large-Scale Structure (GLASS, Tessore et al. 2023) using halo model power spectra for the three-dimensional electron</text>
<text><location><page_2><loc_52><loc_71><loc_92><loc_92></location>power spectrum. This will generate log-normal realisations of the electron field with the correct two-point statistics imprinted. ( 𝑖𝑖 ) The host contribution, which is simply sampled from a host model probability density function (PDF). ( 𝑖𝑖𝑖 ) The Milky Way (MW) contribution, for which we will use the standard methods of inferring it from already present electron models (Cordes & Lazio 2002; Yao et al. 2017; Yamasaki & Totani 2020). We will then use those forward simulations to train an NN to learn the posterior and sample from the posterior with traditional MCMC. Here, we will use Truncated Sequential Neural Posterior Estimation (TSNPE, Deistler et al. 2022) to conduct the inference within the SBI framework. We aim to fit the amplitude of the DM𝑧 relation and the median and the width of the log-normal host distribution with the available host identified FRBs, providing a roadmap for future cosmological and astrophysical inference with FRBs.</text>
<text><location><page_2><loc_52><loc_60><loc_92><loc_70></location>The manuscript is structured as follows: In Section 2, we introduce the basics of FRB cosmology and discuss the different components entering the total DM. Section 3 provides an overview of the forward simulation pipeline. The inference techniques are discussed in Section 4. Section 5 introduces the validation techniques we use after the inference process. In Section 6, we present the results and summarise them in Section 7.</text>
<section_header_level_1><location><page_2><loc_59><loc_57><loc_84><loc_58></location>2. DISPERSION MEASURE COMPONENTS</section_header_level_1>
<section_header_level_1><location><page_2><loc_67><loc_56><loc_77><loc_57></location>2.1. FRB basics</section_header_level_1>
<text><location><page_2><loc_52><loc_45><loc_92><loc_55></location>The pulses of FRBs undergo dispersion while travelling through the ionized matter distribution in the Universe, leading to a frequency-dependent, ∝ 𝜈 -2 , offset of the bursts' arrival times. Given this time delay measured as 𝛿𝑡 ( ˆ 𝒙 , 𝑧 ) for an FRB at redshift 𝑧 in direction ˆ 𝒙 , the constant of proportionality is the observed dispersion measure: 𝛿𝑡 ( ˆ 𝒙 , 𝑧 ) = DMtot ( ˆ 𝒙 , 𝑧 ) 𝜈 -2 . This DM can be broken up into different components:</text>
<formula><location><page_2><loc_53><loc_43><loc_92><loc_44></location>DMtot ( ˆ 𝒙 , 𝑧 ) = DMLSS ( ˆ 𝒙 , 𝑧 ) + DMMW ( ˆ 𝒙 ) + DMhost ( 𝑧 ) . (1)</formula>
<text><location><page_2><loc_52><loc_37><loc_92><loc_42></location>The first contribution is DMLSS ( ˆ 𝒙 , 𝑧 ) , caused by free electrons in the LSS. Here, the dependence on the direction is kept explicitly, since the LSS is correlated. In the literature, DMLSS ( ˆ 𝒙 , 𝑧 ) is often split up into an IGM part and a halo part</text>
<formula><location><page_2><loc_57><loc_34><loc_92><loc_36></location>DMLSS ( ˆ 𝒙 , 𝑧 ) = DMIGM ( ˆ 𝒙 , 𝑧 ) + DMhalo ( ˆ 𝒙 , 𝑧 ) . (2)</formula>
<text><location><page_2><loc_52><loc_27><loc_92><loc_33></location>This is equivalent to a halo model prescription (Cooray & Sheth 2002) of the statistical properties of the DM. On the level of the power spectrum, this would amount to the twohalo term (corresponding to DMIGM ( ˆ 𝒙 , 𝑧 ) ) and the one-halo term (corresponding to DMhalo ( ˆ 𝒙 , 𝑧 ) ).</text>
<text><location><page_2><loc_52><loc_10><loc_92><loc_27></location>The MW contribution DMMW ( ˆ 𝒙 ) can itself be split up into a contribution from the ISM and the MW halo. Both will not depend on redshift, as these are local quantities. However, there is a clear directional dependence. Lastly, DMhost ( 𝑧 ) is the contribution of the host galaxy which can, as the MW contribution, be split up into a part originating from the visible galaxy and one of the halo. For this, only a potential redshift dependence is assumed, as the contribution of different hosts should not be correlated, ignoring the unlikely event that two distinct FRBs originate from the same galaxy. Note that the rest-frame DM of the host, DMhost , rf , is observed as DMhost ( 𝑧 ) = ( 1 + 𝑧 ) -1 DMhost , rf .</text>
<text><location><page_3><loc_8><loc_89><loc_48><loc_92></location>First, we will take a more detailed look at the contribution of the LSS. Quite generally this is given by</text>
<formula><location><page_3><loc_12><loc_85><loc_48><loc_89></location>DMLSS ( ˆ 𝒙 , 𝑧 ) = ∫ 𝑧 0 𝑛 e ( ˆ 𝒙 , 𝑧 ' ) 𝑓 IGM ( 𝑧 ' ) 1 + 𝑧 ' 𝐻 ( 𝑧 ' ) d 𝑧 ' , (3)</formula>
<text><location><page_3><loc_8><loc_77><loc_48><loc_85></location>where 𝑛 e ( ˆ 𝒙 , 𝑧 ) is the comoving cosmic free electron density, 𝐻 ( 𝑧 ) = 𝐻 0 𝐸 ( 𝑧 ) is the Hubble function with the expansion function 𝐸 ( 𝑧 ) and the Hubble constant 𝐻 0. 𝑓 IGM ( 𝑧 ) is the fraction of electrons in the IGM and is calculated by subtracting the fraction bound in stars, compact objects and the dense interstellar medium (ISM)</text>
<formula><location><page_3><loc_18><loc_75><loc_48><loc_76></location>𝑓 IGM ( 𝑧 ) = 1 -𝑓 ★ ( 𝑧 ) -𝑓 ISM ( 𝑧 ) . (4)</formula>
<text><location><page_3><loc_8><loc_71><loc_48><loc_74></location>For redshifts 𝑧 < 3 almost all baryons are ionised, and the DM is therefore rewritten as</text>
<formula><location><page_3><loc_13><loc_68><loc_48><loc_71></location>𝑛 e ( ˆ 𝒙 , 𝑧 ) = 𝜒 e 𝜌 b ( ˆ 𝒙 , 𝑧 ) 𝑚 p = 𝜒 e ¯ 𝜌 b 𝑚 p GLYPH<0> 1 + 𝛿 e ( ˆ 𝒙 , 𝑧 )) , (5)</formula>
<text><location><page_3><loc_8><loc_64><loc_48><loc_67></location>with the baryon density 𝜌 b, the proton mass 𝑚 p and the electron fraction</text>
<formula><location><page_3><loc_19><loc_61><loc_48><loc_64></location>𝜒 e = 𝑌 H + 1 2 𝑌 He ≈ 1 -1 2 𝑌 He , (6)</formula>
<text><location><page_3><loc_8><loc_58><loc_48><loc_60></location>calculated from the primordial hydrogen and helium abundances 𝑌 H and 𝑌 He. Altogether, one finds:</text>
<formula><location><page_3><loc_12><loc_54><loc_48><loc_57></location>DMLSS ( ˆ 𝒙 , 𝑧 ) = A ∫ 𝑧 0 1 + 𝑧 ' 𝐸 ( 𝑧 ' ) GLYPH<0> 1 + 𝛿 e ( ˆ 𝒙 , 𝑧 ' ) GLYPH<1> d 𝑧 ' , (7)</formula>
<text><location><page_3><loc_8><loc_37><loc_48><loc_53></location>where we defined A B 3 𝑐 Ω b0 𝐻 0 8 𝜋𝐺𝑚 p 𝜒 e 𝑓 IGM. The LSS contribution is therefore entirely specified by the statistical properties of the electron density field 𝛿 e. Motivated by numerical simulations of the DM which showed that it follows a log-normal distribution (see e.g. Zhang et al. 2021), we model the electron field using GLASS (Tessore et al. 2023) which can, given a three-dimensional power spectrum of a cosmological field, create log-normal realisations. This is done by dividing the LSS into 𝑁 shells non-overlapping and concentric shells. Each shell covers the full sky, spanning the comoving volume between redshift 𝑧 𝑖 and 𝑧 𝑖 + 1. First, we define a matter weight function along the line-of-sight via:</text>
<formula><location><page_3><loc_14><loc_33><loc_48><loc_36></location>𝑊 ( 𝑖 ) ( 𝑧 ) B GLYPH<26> 𝜒 2 ( 𝑧 )/ 𝐸 ( 𝑧 ) if 𝑧 𝑖 ≤ 𝑧 < 𝑧 𝑖 + 1 , 0 else , (8)</formula>
<text><location><page_3><loc_8><loc_29><loc_48><loc_32></location>with the co-moving distance 𝜒 ( 𝑧 ) . The electron density contrast, 𝛿 e ( ˆ 𝒙 , 𝑧 ' ) , can now also be defined in each shell:</text>
<formula><location><page_3><loc_17><loc_26><loc_48><loc_29></location>𝛿 ( 𝑖 ) e ( ˆ 𝒙 ) = ∫ d 𝑧 𝑊 ( 𝑖 ) ( 𝑧 ) 𝛿 e ( ˆ 𝒙 , 𝑧 ) . (9)</formula>
<text><location><page_3><loc_8><loc_22><loc_48><loc_25></location>The statistical properties of this field on the two-point level are calculated in the harmonic space by the angular power spectra:</text>
<formula><location><page_3><loc_10><loc_16><loc_48><loc_21></location>𝐶 ( 𝑖 𝑗 ) 𝛿 e 𝛿 e ( ℓ ) = 2 π ∫ d 𝜒𝑊 ( 𝑖 ) ( 𝑧 ( 𝜒 )) ∫ d 𝜒 ' 𝑊 ( 𝑗 ) ( 𝑧 ( 𝜒 ' )) ∫ d 𝑘 𝑘 2 𝑃 𝛿 e , nl ( 𝑘, 𝑧 ( 𝜒 ) , 𝑧 ( 𝜒 ' )) 𝑗 ℓ ( 𝑘 𝜒 ) 𝑗 ℓ ( 𝑘 𝜒 ' ) , (10)</formula>
<text><location><page_3><loc_8><loc_12><loc_48><loc_14></location>where 𝑗 ℓ ( 𝑥 ) are spherical Bessel functions of order ℓ and ℓ ∈ N . The electron power spectrum is defined as:</text>
<formula><location><page_3><loc_10><loc_9><loc_45><loc_11></location>⟨ ' ' 3 ' '</formula>
<formula><location><page_3><loc_11><loc_7><loc_48><loc_11></location>𝛿 e ( 𝒌 , 𝑧 ) 𝛿 e ( 𝒌 , 𝑧 )⟩ = ( 2 π ) 𝛿 D ( 𝒌 + 𝒌 ) 𝑃 𝛿 e , nl ( 𝑘, 𝑧, 𝑧 ) . (11)</formula>
<text><location><page_3><loc_52><loc_89><loc_92><loc_92></location>Lastly, we approximate the unequal time correlator by its geometric mean:</text>
<formula><location><page_3><loc_55><loc_87><loc_92><loc_89></location>𝑃 𝛿 e , nl ( 𝑘, 𝑧, 𝑧 ' ) ≈ GLYPH<2> 𝑃 𝛿 e , nl ( 𝑘, 𝑧 ) 𝑃 𝛿 e , nl ( 𝑘, 𝑧 ' ) GLYPH<3> 1 / 2 , (12)</formula>
<text><location><page_3><loc_52><loc_78><loc_92><loc_86></location>which has been shown to be an excellent approximation for weak gravitational lensing (Kitching & Heavens 2017; de la Bella et al. 2021). Since the DM of the LSS has a similarly broad kernel as it is an integrated effect, this should also hold for FRBs. Starting from Equation (7), let us define the perturbations to the DM as</text>
<formula><location><page_3><loc_60><loc_74><loc_92><loc_77></location>D( ˆ 𝒙 , 𝑧 ) = A ∫ 𝑧 0 1 + 𝑧 ' 𝐸 ( 𝑧 ' ) 𝛿 e ( ˆ 𝒙 , 𝑧 ' ) d 𝑧 ' . (13)</formula>
<text><location><page_3><loc_52><loc_71><loc_92><loc_73></location>It can be broken down into a discrete sum over the matter shells</text>
<formula><location><page_3><loc_60><loc_67><loc_92><loc_71></location>D( ˆ 𝒙 , 𝑧 𝑁 ) ≈ A 𝑁 shells ∑︁ 𝑖 = 1 1 + ¯ 𝑧 𝑖 𝐸 ( ¯ 𝑧 𝑖 ) 𝑤 𝑖 𝛿 e ,𝑖 ( ˆ 𝒙 ) , (14)</formula>
<text><location><page_3><loc_52><loc_64><loc_92><loc_66></location>where we defined the characteristic redshift of each shell to be its mean:</text>
<formula><location><page_3><loc_64><loc_60><loc_92><loc_64></location>¯ 𝑧 ( 𝑖 ) = ∫ d 𝑧 𝑧 𝑊 ( 𝑖 ) ( 𝑧 ) ∫ d 𝑧 𝑊 ( 𝑖 ) ( 𝑧 ) . (15)</formula>
<text><location><page_3><loc_52><loc_58><loc_90><loc_59></location>Likewise, 𝑤 𝑖 takes into account the weight of the shell via</text>
<formula><location><page_3><loc_62><loc_54><loc_92><loc_57></location>𝑤 𝑖 = 1 𝑊 ( 𝑖 ) ( ¯ 𝑧 𝑖 ) ∫ 𝑊 ( 𝑖 ) ( 𝑧 ) d 𝑧 . (16)</formula>
<section_header_level_1><location><page_3><loc_64><loc_52><loc_79><loc_53></location>2.3. Host Contribution</section_header_level_1>
<text><location><page_3><loc_52><loc_28><loc_92><loc_51></location>The next component in the forward simulation is the host contribution, quantifying the effect the host galaxy on the observed DM. Based on the position of the FRB progenitor in the galaxy, the signal may travel through the whole galaxy or parts of it. As such, the induced DM varies accordingly. The effect of a complete or partial travel path through the local host, which translates to high and low DM, is usually described by a log-normal distribution (Macquart et al. 2020; Wu et al. 2022). Other works, e.g. Hagstotz et al. (2022a), have assumed a Gaussian host contribution. Recently, simulations have also shown that a log-normal distribution can fit the host contribution rather well (Theis et al. 2024). With the current data set of FRBs, this choice does not make a difference if priors on the DM are included, as we will show later. For our fiducial case, however, we will choose a log-normal distribution for DMhost , rf (that is the host contribution in the rest-frame of the host galaxy for which we assume no intrinsic redshift evolution):</text>
<formula><location><page_3><loc_54><loc_23><loc_92><loc_27></location>𝑝 host ( 𝑥 ; 𝜇, 𝜎 LN ) = 1 𝑥𝜎 LN √ 2 𝜋 exp -( ln 𝑥 -𝜇 ) 2 2 𝜎 2 LN ! , (17)</formula>
<text><location><page_3><loc_52><loc_16><loc_92><loc_22></location>whereexp ( 𝜇 ) and exp ( 2 𝜇 + 𝜎 2 LN ) [ exp ( 𝜎 2 LN )-1 ] are the median and variance respectively with 𝑥 = DMhost , rf . The median and the scale ( 𝜎 LN) are free parameters in our study. Note that this samples the rest-frame DM of the host, DMhost , rf .</text>
<section_header_level_1><location><page_3><loc_63><loc_14><loc_81><loc_15></location>2.4. Milky Way contribution</section_header_level_1>
<text><location><page_3><loc_52><loc_7><loc_92><loc_13></location>In our analysis, we assume that the MW contribution is accessible from the models described in Cordes & Lazio (2002); Yao et al. (2017); Yamasaki & Totani (2020) and we simply add the numerical value for DMMW ( ˆ 𝒙 ) in Equation (1). At the current sensitivity level, dictated by the amounts of</text>
<figure>
<location><page_4><loc_10><loc_45><loc_46><loc_92></location>
<caption>Figure 1. Flowchart of the simulation pipeline described in Section 3. Blue boxes indicate the pipeline input and output. Green boxes show intermediate data products, and the labels of the arrows depict the operation applied to the previous data product.</caption>
</figure>
<text><location><page_4><loc_8><loc_28><loc_48><loc_39></location>FRBs available, this addition does not have a sizable influence on the inference of cosmological parameters. However, with more FRBs being observed, it could well be that the addition of the MW contribution leads to a residual correlation in the DMwhich will be falsely picked up as a cosmological signal. This scenario can in principle be tested with our pipeline, as the MW contribution can simply be added to the simulated DM.</text>
<section_header_level_1><location><page_4><loc_21><loc_26><loc_36><loc_27></location>3. MODEL GENERATION</section_header_level_1>
<section_header_level_1><location><page_4><loc_20><loc_24><loc_36><loc_25></location>3.1. Forward simulation</section_header_level_1>
<text><location><page_4><loc_8><loc_13><loc_48><loc_24></location>In this section, we describe how we use the previously described ingredients to construct forward simulations for the individual components of Equation (1), which, by simple summation, yield a forward simulation prediction for a set of DMtot ( ˆ 𝒙 𝑎 , 𝑧 𝑎 ) , with 𝑎 = 1 , . . . , 𝑁 FRB. Here, ˆ 𝒙 𝑎 and 𝑧 𝑎 are the positions and redshifts of the FRBs from the FRB catalogue. Figure 1 summarises the forward simulation pipeline which goes through the following steps:</text>
<unordered_list>
<list_item><location><page_4><loc_10><loc_7><loc_48><loc_12></location>1. Fix the model parameters { 𝜃 𝛼 } and obtain the 3-dimensional non-linear electron power spectrum, 𝑃 𝛿 e 𝛿 e ( 𝑘, 𝑧 ) using CAMB (Lewis et al. 2000; Lewis & Bridle 2002; Howlett et al. 2012) and then HMCODE</list_item>
</unordered_list>
<section_header_level_1><location><page_4><loc_56><loc_91><loc_85><loc_92></location>(Mead et al. 2015, 2020; Tröster et al. 2022).</section_header_level_1>
<unordered_list>
<list_item><location><page_4><loc_54><loc_80><loc_92><loc_89></location>2. Define concentric shells such that there are no discreteness effects, i.e. that a finer resolution along the lineof-sight does not change the results. We found that 𝑁 shells = 17 is enough for our purpose for 𝑧 ∈ [ 0 . 01 , 1 ] . Use Levin (Zieser & Merkel 2016; Leonard et al. 2023) to calculate the angular power spectrum in those shells via Equation (10).</list_item>
<list_item><location><page_4><loc_54><loc_73><loc_92><loc_78></location>3. Run GLASS (Tessore et al. 2023) with the angular power spectra in the shells to generate log-normal or Gaussian realisations of the electron overdensity. Add unity to each shell to arrive at the physical density.</list_item>
<list_item><location><page_4><loc_54><loc_65><loc_92><loc_72></location>4. Place all FRBs from the catalogue in the simulated electron density field and project it along the line of sight. If we label all shells s 𝑖 and 𝑧 < s 𝑖 is interpreted as that 𝑧 is strictly below all redshifts in shell 𝑖 , we can define an auxiliary weight as:</list_item>
</unordered_list>
<formula><location><page_4><loc_57><loc_57><loc_92><loc_64></location>𝑤 DM 𝑖 ( 𝑧 FRB ) B 0 if 𝑧 FRB < s 𝑖 , 𝑧 FRB -𝑧 i , min 𝑧 i , max -𝑧 i , min if 𝑧 FRB ∈ s 𝑖 , 1 if s 𝑖 ≤ 𝑧 FRB . (18)</formula>
<text><location><page_4><loc_67><loc_57><loc_68><loc_57></location></text>
<text><location><page_4><loc_56><loc_55><loc_88><loc_56></location>Including this weight in Equation (14) one finds:</text>
<formula><location><page_4><loc_58><loc_52><loc_92><loc_54></location>D( ˆ 𝒙 𝑎 , 𝑧 𝑎 ) = A " 𝑁 𝑖 = 1 1 + ¯ 𝑧 𝑖 𝐸 ( ¯ 𝑧 𝑖 ) 𝑤 DM 𝑖 ( 𝑧 𝑎 ) 𝑤 𝑖 𝛿 e ,𝑖 ( ˆ 𝒙 𝑎 ) . (19)</formula>
<text><location><page_4><loc_56><loc_50><loc_90><loc_51></location>Lastly, adding the homogeneous contribution gives:</text>
<formula><location><page_4><loc_58><loc_47><loc_92><loc_49></location>DMLSS ( ˆ 𝒙 𝑎 , 𝑧 𝑎 ) = A ∫ 𝑧 𝑎 0 1 + 𝑧 ' 𝐸 ( 𝑧 ' ) d 𝑧 ' + D( ˆ 𝒙 𝑎 , 𝑧 𝑎 ) . (20)</formula>
<unordered_list>
<list_item><location><page_4><loc_54><loc_42><loc_92><loc_46></location>5. Draw samples from the host PDF contribution for each FRB (in our case Equation (17)) and map it to the physical frame by redshifting it.</list_item>
<list_item><location><page_4><loc_54><loc_39><loc_92><loc_41></location>6. Obtain the MW contribution as discussed in Section 2.4.</list_item>
<list_item><location><page_4><loc_54><loc_37><loc_76><loc_38></location>7. Add all contributions together.</list_item>
</unordered_list>
<text><location><page_4><loc_52><loc_22><loc_92><loc_35></location>This procedure provides a pair ({ 𝜃 𝛼 } , { DMtot ( ˆ 𝒙 𝑎 , 𝑧 𝑎 )}) . Rerunning the pipeline, with parameters sampled from a prior distribution, creates a set of forward simulations, which is used to learn the posterior distribution by the NN. The resolution of the simulation is dependent on the parameter 𝑁 side ∈ 2 Z + as GLASS internally uses HEALPix (Górski et al. 2005). From the flowchart in Figure 1 and the list above, it is clear that any component in the pipeline can easily be exchanged for another model. If, for example, there exists a better model for the host contribution with different parameters, it is easy to replace it.</text>
<section_header_level_1><location><page_4><loc_64><loc_19><loc_79><loc_20></location>3.2. Data compression</section_header_level_1>
<text><location><page_4><loc_52><loc_7><loc_92><loc_19></location>The data vector currently is 𝑁 FRB dimensional, thus requiring a compression procedure, essentially translating a 𝑑 -dimensional dataset down to 𝑛 dimensions ( 𝑛 < 𝑑 ). This is an essential step, as training the NN with high-dimensional data is slow and can lead to inaccuracies. Lossless data compression preserves the Fisher information of the original data. Therefore, we use the data reduction scheme prescribed in Alsing et al. (2019) reducing the 𝑑 -dimensional data down to a dimension equal to the number of free parameters. Assuming a</text>
<table>
<location><page_5><loc_16><loc_88><loc_41><loc_92></location>
<caption>Table 1</caption>
</table>
<paragraph><location><page_5><loc_8><loc_84><loc_48><loc_86></location>Parameters fitted in the inference with prior ranges and fiducial values for the data compression.</paragraph>
<text><location><page_5><loc_8><loc_79><loc_48><loc_82></location>Gaussian likelihood, this compressed data ( 𝑡 ) can be obtained as (Tegmark et al. 1997)</text>
<formula><location><page_5><loc_15><loc_73><loc_48><loc_78></location>𝒕 = ∇ 𝝁 𝑻 ∗ C -1 ∗ ( d -𝝁 ∗ ) + 1 2 ( d -𝝁 ∗ ) 𝑻 C -1 ∗ ∇ C ∗ C -1 ∗ ( d -𝝁 ∗ ) , (21)</formula>
<text><location><page_5><loc_8><loc_65><loc_48><loc_73></location>where 𝝁 ∗ , C ∗ are the ensemble mean and covariance of the original data at some fiducial value 𝜽 ∗ and d is the corresponding data vector. ∇ represents the partial derivatives with respect to the free parameters. This fiducial 𝜽 ∗ needs to be optimised to ensure no information loss. Specifically, we use the Fisher scoring method</text>
<formula><location><page_5><loc_22><loc_62><loc_48><loc_64></location>𝜽 𝑘 + 1 = 𝜽 𝑘 + F -1 𝑘 𝒕 𝑘 , (22)</formula>
<text><location><page_5><loc_8><loc_50><loc_48><loc_62></location>where 𝒕 𝑘 is the compressed statistics for 𝜽 𝑘 at the 𝑘 -th step. Depending on the complexity of the parameter space, a larger value of 𝑘 may be required for convergence. In practice, we stop after a finite number of iterations when the increment, the second term on the right-hand side of Equation (22), asymptotically approaches a plateau and further steps are not favoured against the computational time. The components of the Fisher matrix, F 𝑖 𝑗 are evaluated by its full expression for a Gaussian likelihood:</text>
<formula><location><page_5><loc_14><loc_44><loc_48><loc_49></location>F 𝑖 𝑗 = 1 2 Tr h C -1 ∇ 𝑖 C C -1 ∇ 𝑗 C + C -1 (∇ 𝑖 𝝁 ∇ 𝑗 𝝁 T + ∇ 𝑖 𝝁 T ∇ 𝑗 𝝁 ) i . (23)</formula>
<text><location><page_5><loc_8><loc_24><loc_48><loc_43></location>The three free parameters in our model are the amplitude of the DM𝑧 relation, the median and the scale of the lognormal host distribution. We specify the initial 𝜽 ∗ in Table 1 motivated from Reischke et al. (2022), but we quickly converge to 𝜽 optimal = { 0 . 94 , 200 . 74 , 0 . 79 } T . For the current selection of free parameters, all the derivatives have analytic solutions as the derivative of the covariance of the LSS component (Reischke & Hagstotz 2023b) scales with 2 A , with A being the prefactor in Equation (7). Similarly, the derivatives of the log-normal host covariance are trivial. If we are to fit a more complex model in the future, one would have to calculate those derivatives numerically or for more stable results using a differentiable code. This could for example be achieved by emulating the important quantities.</text>
<section_header_level_1><location><page_5><loc_24><loc_22><loc_33><loc_23></location>4. INFERENCE</section_header_level_1>
<text><location><page_5><loc_8><loc_7><loc_48><loc_21></location>Given the simulation pipeline described in the previous section, we are now in the position to infer the posterior of the parameters summarised in Table 1. Inference refers to determining the parameters that best describe the observation. Typically, we have a model based on physical laws, observations from surveys and a likelihood. The latter is then combined with the prior to yield the posterior. The most popular approach to such an inference process has been the Markov Chain Monte Carlo (MCMC) within the Bayesian framework. MCMCmethods create samples from the posterior. However, modelling a likelihood that captures the full complexity of</text>
<text><location><page_5><loc_52><loc_88><loc_92><loc_92></location>the physical processes is one of the most complex parts of any traditional analysis, with cases arising where an analytical likelihood is not accessible.</text>
<section_header_level_1><location><page_5><loc_62><loc_85><loc_82><loc_86></location>4.1. Simulation-based inference</section_header_level_1>
<text><location><page_5><loc_52><loc_69><loc_92><loc_84></location>Approximate Bayesian Calculation (ABC) addresses this issue by comparing data from forward simulation with observation with some arbitrary margin (Rubin 1984; Pritchard et al. 1999). Only the accepted simulations are considered for posterior inference, which leads to critical drawbacks such as a high rejection rate with decreasing margin and the curse of dimensionality (Sisson et al. 2018). Improvements upon this basic rejection sampling include MCMC-ABC (Marjoram et al. 2003) and Sequential Monte Carlo ABC (SMCABC) (Bonassi & West 2015), which only partially solve the rejection rate and do not address the high dimensionality problem.</text>
<text><location><page_5><loc_52><loc_51><loc_92><loc_68></location>Following the development of NNs, a suite of algorithms has been proposed recently that estimate the posterior without access to a likelihood. All of these methods come under the umbrella of SBI or likelihood-free inference (LFI), which introduces a parameterised density estimator that learns from the joint distribution of the data-parameter pair (Cranmer et al. 2020). In this section, we briefly introduce the density estimation mentioned in (Papamakarios & Murray 2016) called sequential neural posterior estimation (SNPE-A) and summarise its subsequent developments in SNPE-B (Lueckmann et al. 2017), SNPE-C or APT (Greenberg et al. 2019) and an improvement on APT called TSNPE (Deistler et al. 2022). Finally, we will be using the TSNPE algorithm in our analysis.</text>
<text><location><page_5><loc_52><loc_30><loc_92><loc_51></location>The fundamental idea behind all of these methods is to approximate the posterior, 𝑝 ( 𝜽 | d ) , with a conditional density estimator 𝑞 𝜙 ( 𝜽 | d ) , where 𝜙 are the trainable parameters. Typically, the parameters, 𝜽 , are sampled from the full prior range, which is inefficient. There exist two schools of SBI. Amortised methods yield a posterior that can be utilised for various observations without the need for retraining, whereas sequential methods concentrate the inference on a specific observation to enhance simulation efficiency. Here we focus on the sequential-(S)NPE methods refer to using a proposal as a subset of the prior, ˜ 𝑝 𝜽 ) ⊆ 𝑝 ( 𝜽 ) . Initially, the proposal is equal to the prior and in subsequent rounds we update the proposal with the approximate posterior, improving the efficiency of the inference process. The predicament is that this procedure does not recover the true posterior, but rather an approximate posterior:</text>
<formula><location><page_5><loc_62><loc_26><loc_92><loc_29></location>˜ 𝑝 ( 𝜽 | d ) = 𝑝 ( 𝜽 | d ) ˜ 𝑝 ( 𝜽 ) 𝑝 ( d ) 𝑝 ( 𝜽 ) ˜ 𝑝 ( d ) , (24)</formula>
<text><location><page_5><loc_52><loc_20><loc_92><loc_25></location>where 𝑝 ( 𝜽 | d ) is the true posterior, ˜ 𝑝 ( 𝜽 | d ) is the approximate posterior, ˜ 𝑝 ( 𝜽 ) is the proposal, 𝑝 ( 𝜽 ) is the prior and ˜ 𝑝 ( d ) is the evidence. The approximation recovers the true posterior only when the proposal is equal to the prior.</text>
<text><location><page_5><loc_52><loc_16><loc_92><loc_20></location>SNPE methods address this issue with a parameterised density estimator, 𝑞 𝜙 ( 𝜽 | d ) , that learns from the joint distribution of the data and parameter</text>
<formula><location><page_5><loc_63><loc_12><loc_92><loc_15></location>˜ 𝑝 ( 𝜽 | d ) ∝ 𝑞 𝜙 ( 𝜽 | d ) ˜ 𝑝 ( 𝜽 ) 𝑝 ( 𝜽 ) . (25)</formula>
<text><location><page_5><loc_52><loc_7><loc_92><loc_11></location>The problem to solve is the extra ratio, ˜ 𝑝 ( 𝜽 )/ 𝑝 ( 𝜽 ) . The way each algorithm addresses this issue is what differentiates the various SNPE methods. For instance, SNPE-A maintains a</text>
<text><location><page_6><loc_8><loc_85><loc_48><loc_92></location>closed-form solution by restricting the choice of the prior, proposal and the density estimator to either a uniform or Gaussian distribution. The training is carried out by minimising the loss function or the negative log-likelihood of the joint probability of the data and parameter defined via</text>
<formula><location><page_6><loc_11><loc_83><loc_48><loc_84></location>L( 𝜙 ) B min E 𝜽 ∼ ˜ 𝑝 ( 𝜽 ) E d ∼ 𝑝 ( d | 𝜽 ) GLYPH<2> -log 𝑞 𝜙 ( 𝜽 | d ) GLYPH<3> . (26)</formula>
<text><location><page_6><loc_8><loc_68><loc_48><loc_82></location>Other methods like SNPE-B offer a solution by including the ratio inside the loss function, whereas SNPE-C incorporates the ratio into the density estimator and finally, APT normalises the density estimator by using uniform subsets of the prior called atoms. This development allows any arbitrary distribution choice for the prior, proposal and the density estimator. For an extensive technical review of the shortcomings and the subsequent developments of the various SNPE methods, the reader is referred to the following articles and the references therein (Durkan et al. 2020; Lueckmann et al. 2021; Xiong et al. 2023).</text>
<text><location><page_6><loc_8><loc_56><loc_48><loc_68></location>The latest development, TSNPE, builds upon the APT method and truncates the prior based on the 1 -𝜖 mass ( 𝜖 being an arbitrary margin) of the highest probability region (HPR) of the approximate posterior. In other words, the subsequent rounds only consider the prior range which is most probable to produce the posterior. This prescription maintains the flexibility of the distribution choice while also accounting for the posterior mass outside the prior boundaries, an issue of the APT method (Deistler et al. 2022).</text>
<text><location><page_6><loc_8><loc_51><loc_48><loc_56></location>As the HPR of the approximate posterior contains the information on the joint probability of the data and parameter in each round, the truncation is data-driven. TSNPE also results in faster overall convergence.</text>
<text><location><page_6><loc_8><loc_36><loc_48><loc_50></location>The various SNPE methods tackle posterior estimation by either changing the loss function or restricting the choice of distribution. Without a general solution, the choice of algorithm affects the performance and accuracy of the analysis. Our tests reveal that TSNPE, due to the truncation, is faster and allows for a more flexible distribution choice for prior and density estimators as compared to SNPE-A. As the field of NN is projected to experience rapid growth, the choice of the best algorithm is expected to change as well; refer to (Cranmer et al. 2020) for a detailed review of the status of SBI at the beginning of this decade.</text>
<section_header_level_1><location><page_6><loc_22><loc_34><loc_35><loc_35></location>4.2. Implementation</section_header_level_1>
<text><location><page_6><loc_8><loc_27><loc_48><loc_33></location>The simulated data in our case are the compressed DM values of FRBs as described in Section 3.2. The priors are defined in Table 1. For the analysis, we use 12 FRBs with host identification as given in Table A.1, where amongst others we list the observed DMs and their references.</text>
<text><location><page_6><loc_8><loc_10><loc_48><loc_27></location>The forward simulations for the fiducial case of log-normal density field and log-normal host contribution are carried out at a resolution of 𝑁 side = 4096. The choice is so that DM contributions from small scales ( ℓ ∼ 10 4 ) are accounted for. This is essential for an unbiased analysis, as we shall discuss. A good consistency check here is the comparison of the numerical covariance we get from the simulations to its analytical counterpart (Reischke & Hagstotz 2023b). The diagonal elements of the analytic covariance matrix show a monotonic behaviour between the DM and the redshift of FRBs. For the numerical covariance, we choose 𝑁 side = 4096 to match this monotonous nature and magnitude of the diagonal elements, i.e. the DM field variance.</text>
<text><location><page_6><loc_8><loc_7><loc_48><loc_9></location>In each round of TSNPE, the density estimator learns the mapping between the simulated data and the corresponding</text>
<text><location><page_6><loc_52><loc_63><loc_92><loc_92></location>parameters. The observed DM is introduced post training, which helps evaluate an approximate posterior for that round. This approximate posterior is then utilised to calculate the HPR region and the prior range for the next round is truncated based on that. As the number of rounds is increased, the approximate posterior converges to the true posterior. Introducing the observed DM is necessary to propel the truncation in the direction of the true posterior. With parameters, simulated data and observation, these are the three components required for the SBI analysis. We implement this using LtU-ILI package for a quick setup with different neural embedding (Ho et al. 2024). We have used the TSNPE algorithm for a total of 1500 simulations in 10 rounds. The initial simulations contain the full prior range and the subsequent truncation is only influenced by the data. As LtU-ILI does not natively support TSNPE yet, we run TSNPE, save the data and then use it. The choice of the number of simulations in each round is dictated by the number of epochs the NN takes for training. We limit the number of epochs to 50 to avoid overfitting and 150 simulations in each round provide ample data for the density estimator 𝑞 𝜙 ( 𝜽 | d ) to approximate the posterior by adjusting the trainable parameter 𝜙 .</text>
<text><location><page_6><loc_52><loc_52><loc_92><loc_62></location>As for the NN architecture, we use two different types; mixture density network (MDN) and masked autoregressive flows (MAF). These architectures aim to produce an arbitrary yet tractable PDF with trainable parameters to be used as the density estimator. The complexity of data mandates arbitrariness, while the backpropagation in the training process requires tractability. MDN, as the name suggests, is the weighted sum of many Gaussian distributions defined as (Bishop 1994)</text>
<formula><location><page_6><loc_60><loc_48><loc_92><loc_51></location>𝑞 MDN 𝜙 ( 𝜽 | d ) B ∑︁ 𝑘 𝛼 𝑘 N( 𝜽 | m 𝑘 , S 𝑘 ) , (27)</formula>
<text><location><page_6><loc_52><loc_21><loc_92><loc_47></location>where m 𝑘 and S 𝑘 are the mean and covariance of the 𝑘 -th Gaussian. This type of density estimator is malleable and can be transformed into any arbitrary distribution by adjusting the weights, means and standard deviations. While flexibility is a property we seek, a value of 𝑘 too large can lead to overfitting. Masked Autoregressive Flows (MAFs, Papamakarios et al. 2017), on the other hand, combine two different architectures called Normalizing Flow (NF, Jimenez Rezende & Mohamed 2015) and Masked Autoencoder for Distribution Estimation (MADE, Germain et al. 2015) into one. In NF, a normal distribution is subjected to 𝑘 invertible transformations, called flow, resulting in an arbitrary complex distribution. These transformations are based on the conditional probability scheme described in MADE. Combining NF and MADE,MAFcanproduce arbitrary distribution optimised for density estimation. For technical details, the reader is referred to the respective articles. The presence of MAF along with MDN(7:3 weight ratio) ensures the model has a better scaling with dimensionality and learns any multimodal, non-Gaussian features of the data while avoiding overfitting.</text>
<text><location><page_6><loc_52><loc_10><loc_92><loc_20></location>For the MDN embedding, we use 50 hidden layers with 10 components in each layer, while the MAF net is constructed using 20 hidden layers with 8 transformations in each layer. When the training is complete, we acquire a mapping (effectively a KDE) from the joint space of parameters to the distribution 𝑝 ( 𝜽 | d ) without the usage of an explicit likelihood function. This posterior can then be sampled by standard MCMCtechniques.</text>
<section_header_level_1><location><page_6><loc_67><loc_7><loc_77><loc_8></location>5. VALIDATION</section_header_level_1>
<figure>
<location><page_7><loc_8><loc_62><loc_46><loc_89></location>
<caption>✶ Figure 2. Marginalised contour plot for the fiducial model in our analysis, i.e. log-normal field and log-normal host. The shaded areas represent the 1 𝜎 error margin around the dotted line or the mean. The best-fit values given as A = 0 . 89 + 0 . 31 -0 . 29 , DM host = 223 + 108 -95 pc cm -3 and 𝜎 LN = 0 . 76 + 0 . 37 -0 . 44 .</caption>
</figure>
<text><location><page_7><loc_8><loc_42><loc_48><loc_54></location>After the completion of the SBI pipeline, we carry out two validation tests, one for the model under consideration and one for its posterior. For the posterior, we use the multivariate coverage test called Tests of Accuracy with Random Points (TARP, Lemos et al. 2023) and for the model, the accuracy is measured via the goodness of fit or 𝜒 2 -test. While TARP validates the SBI, the 𝜒 2 -test is used to investigate if there are preferred models. In the following, we describe the two validation techniques.</text>
<section_header_level_1><location><page_7><loc_22><loc_40><loc_34><loc_41></location>5.1. Coverage Test</section_header_level_1>
<text><location><page_7><loc_8><loc_18><loc_48><loc_39></location>To check the consistency of the posterior distributions, we use the multivariate coverage test called TARP. Coverage probability measures how frequently the estimated posterior contains the true parameter value, and can be used to check the consistency of the posterior (Guo et al. 2017; Hermans et al. 2021). Once we have obtained the posterior, TARP measures the expected coverage probability of random posterior samples within a given credibility level of the learned posterior empirically. In more technical terms, for any instance of the prior 𝜽 ★ and the corresponding simulated data d ★ , samples are drawn from the posterior as 𝑝 ( 𝜽 | d ★ ) . Then, a circle is drawn centred at a random reference point 𝜽 r with radius | 𝜽 ★ -𝜽 r | and the fraction of posterior samples inside the circle is calculated. A higher fraction implies the posterior is more accurate. The fraction is calculated multiple times to get a statistical estimate of the coverage.</text>
<text><location><page_7><loc_8><loc_10><loc_48><loc_18></location>This random sampling is necessary to validate the static HPR region in TSNPE, which is prone to having blind spots. We use the learnt posterior to draw 1000 samples at random parameter points for the TARP test implemented within the LtU-ILI . We also bootstrap the test 100 times at each credibility level, considering the process is random by definition.</text>
<section_header_level_1><location><page_7><loc_21><loc_8><loc_35><loc_9></location>5.2. Goodness-of-Fit</section_header_level_1>
<figure>
<location><page_7><loc_52><loc_70><loc_91><loc_92></location>
<caption>✶ Figure 3. TARP coverage for the fiducial case of log-normal field and lognormal host is shown with 𝑁 side = 4096. Credibility level refers to the fraction of the total PDF of the final posterior in Figure 2 being considered, which intuitively goes from 0 to 1. The expected coverage is the fraction of the posterior samples with a lower posterior probability than the best estimate for that corresponding credibility. The diagonal line is the ideal relation, as increasing the credibility level linearly increases the expected coverage. For example, at a 50% credibility level, we expect at least 50% of the samples to have coverage; anything below 50% implies the samples are not well covered, i.e. the posterior is biased and similarly a value greater than 50% implies that the posterior is conservative. The dark and light-shaded regions are the one and two-sigma error bars from 100 bootstrappings. Our model, the solid blue line, is accurate as the ideal line is within the 1 𝜎 error bar.</caption>
</figure>
<text><location><page_7><loc_52><loc_47><loc_92><loc_52></location>The typical goodness of fit test can not be used in the SBI framework, as it requires an analytical likelihood. Here, we use the implementation of the 𝜒 2 -test (Gelman et al. 1996) described in von Wietersheim-Kramsta et al. (2024).</text>
<text><location><page_7><loc_52><loc_43><loc_92><loc_47></location>After finding the best-fit values for the parameters of interest, 𝚯 ∗ , we sample 𝑛 noise realisations at this set of parameters and define the 𝜒 2 as</text>
<formula><location><page_7><loc_52><loc_39><loc_92><loc_42></location>𝜒 2 ( 𝒕 𝑖 | 𝚯 ) B ( 𝒕 𝑖 -E [ 𝒕 ∗ | 𝚯 ∗ ]) T ( Cov ( 𝒕 ∗ | 𝚯 ∗ )) -1 ( 𝒕 𝑖 -E [ 𝒕 ∗ | 𝚯 ∗ ]) , (28)</formula>
<text><location><page_7><loc_52><loc_26><loc_92><loc_39></location>where E denotes the expectation value and Cov is the covariance at the best-fit cosmology. 𝒕 𝑖 is the score-compressed summarystatistic of the 𝑖 -th realisation. Specifically, we create 1000 new forward simulations at the best-fit values to calculate the mean data vector and the covariance. The 𝜒 2 values from the simulations are plotted as a histogram and compared with the 𝜒 2 value for the observed DM. The accuracy of the model is assessed through the probability mass of the simulated 𝜒 2 that lies beyond this observed 𝜒 2 , which we call the probability to exceed (PTE).</text>
<section_header_level_1><location><page_7><loc_68><loc_23><loc_76><loc_24></location>6. RESULTS</section_header_level_1>
<text><location><page_7><loc_52><loc_7><loc_92><loc_23></location>In this section, we present the major result of our analysis. Weconsider three parameters for the inference. The first one is the amplitude of the DM𝑧 relation in Equation (7) as a scale. This encapsulates the Λ CDMparameters, for which we use the Planck cosmological results from (Planck Collaboration et al. 2020), i.e. A = 1 refers to the input cosmology and this A is varied in different simulations. The motivation behind only varying the amplitude of the DM-redshift relation is that the number of FRBs currently available is not able to discriminate between different expansion histories. The remaining two parameters are the median and the scale of the log-normal host distribution, which we denote as DMhost and 𝜎 LN.</text>
<figure>
<location><page_8><loc_9><loc_70><loc_47><loc_92></location>
<caption>✶ Figure 4. Goodness-of-fit test that shows the distribution of 𝜒 2 values. The solid blue line is obtained by a smoothing kernel applied on the histogram built from 1000 forward simulated DM values at the best-fit value of Figure 2 with 𝑁 side = 4096. The dotted vertical black line is the 𝜒 2 value for the observed DM. The probability mass beyond this line, PTE, is 0.3.</caption>
</figure>
<text><location><page_8><loc_8><loc_41><loc_48><loc_61></location>As for the model, there are two components in our simulations that vary. First, the DMLSS part is modelled using either a log-normal or a Gaussian realisation of the underlying electron density field via GLASS . These are denoted by 'Field: log-normal' and 'Field: Gaussian' respectively. Similarly, we have three host distributions; log-normal, truncated Gaussian and Gamma. With this setup, we have six possible combinations of field and host. That being said, we use a log-normal field and a log-normal host distribution as our fiducial setting. For this, we will discuss and validate the results in detail and use the high resolution setting of 𝑁 side = 4096. For the other combinations, we use a lower resolution 𝑁 side = 512 in order to limit computation time. In the following, we start with the fiducial model and then observe how the changes in field and host affect the results.</text>
<section_header_level_1><location><page_8><loc_17><loc_39><loc_40><loc_40></location>6.1. Fiducial model and validation</section_header_level_1>
<text><location><page_8><loc_8><loc_30><loc_48><loc_38></location>For the fiducial model, we assume the electron density field in GLASS to be log-normal and sample the host DM from a lognormal distribution. The decision is primarily motivated by results from simulations (see e.g. Zhang et al. 2021; Theis et al. 2024). In the subsequent sections, we present the comparisons of the six different models, which support our decision.</text>
<text><location><page_8><loc_8><loc_6><loc_48><loc_30></location>With the simulator set to the desired configuration and priors defined in Table 1, the SBI pipeline is run with the TSNPE algorithm, which returns the posterior distribution for the free parameters. In Figure 2, we show the marginalised contour plots using ChainConsumer (Hinton 2016). The forward simulations have the resolution parameter 𝑁 side = 4096. As mentioned at the beginning of this section, there are three parameters that we constrain; A as a scale for the input cosmology, DMhost as the median and 𝜎 LN as the scale of the log-normal host distribution. The value for the scale parameter A is 0 . 89 + 0 . 31 -0 . 29 at 68% confidence, which is consistent with unity or the input fiducial Planck cosmology. The median and the scale of the log-normal host along with their 68% confidence intervals are given as DMhost = 223 + 108 -95 pc cm -3 and 𝜎 LN = 0 . 76 + 0 . 37 -0 . 44 . This 𝜎 LN is further converted into the standard deviation from the definition of the variance in Equation (17). Then, we write 𝜎 = 263 + 91 -113 . As expected, we find a</text>
<text><location><page_8><loc_52><loc_85><loc_92><loc_92></location>strong anti-correlation between A and DMhost since they both enter the observed DM in an additive fashion. The width of the host contribution is not degenerate with the other parameters, as it rather reduces the error on each measurement than changing the signal.</text>
<text><location><page_8><loc_52><loc_76><loc_92><loc_85></location>Our constraints of 𝜎 LN seem to be slightly prior driven, as can be seen from the posterior hitting the prior boundaries of 𝜎 LB in Figure 2. However, there is clearly some constraining power in the data on the shape of the host distribution. In general, we find excellent agreement with previous work (Macquart et al. 2020; James et al. 2022; Hagstotz et al. 2022b; Reischke & Hagstotz 2023a; Khrykin et al. 2024b).</text>
<text><location><page_8><loc_52><loc_63><loc_92><loc_76></location>Now that the posterior distribution is evaluated, we first assess its accuracy via the TARP coverage in Figure 3 with the solid blue line. The shaded blue regions are the 1 and 2 𝜎 error bars from the 100 bootstraps. The region above the diagonal dotted line is called under-confident or conservative, while the region below suggests overconfidence or bias. As the diagonal dotted line is within the dark-shaded region throughout the whole range of credibility, this means that the learnt posterior is an accurate representation of the true posterior and is not biased.</text>
<text><location><page_8><loc_52><loc_52><loc_92><loc_62></location>Additionally, we check for the goodness of fit of the model and the result is shown in Figure 4. The histograms are the 𝜒 2 values from our simulations at best-fit values of the parameters, and the vertical line corresponds to the 𝜒 2 evaluated at the observed DM. The PTE for the fiducial model is 0.3 (corresponding to a 𝑝 -value), which means that the model is a good fit. We can thus concur that the 'log-normal field and log-normal host' model is an excellent fit for the DM of FRBs.</text>
<text><location><page_8><loc_52><loc_10><loc_92><loc_52></location>With the SBI framework, all the correlations are taken into account without defining any likelihood function. Nonetheless, we can still recover the likelihood from the simulated DMs for the best-fit values. We therefore run our pipeline 1000 times at the best-fit value and plot the corresponding distribution of DM. In Figure 5 we show the DM likelihoods for two FRBs with varying redshift, 𝑧 = 0 . 1178 and 𝑧 = 0 . 66. The first observation is the non-Gaussian nature of the distribution, with a tendency towards higher DMs. This behaviour emerges from the host and LSS DM contribution. If the electron density of the host halo where the FRB progenitor resides is high, the signal experiences higher dispersion. That seems to be especially the case for FRB 20190520B, with an observed DM = 1202 pc cm -3 at a relatively lower redshift of 𝑧 = 0 . 241 as shown in Table A.1. The high DM is due to the local contribution of the dwarf host galaxy, identified as J160204.31-111718.5 (Ocker et al. 2022; Niu et al. 2022; Yan et al. 2024). The implication is that a Gaussian likelihood assumption in the inference can introduce biases, as it does not consider the physical effect of variable travel distances and local environments of the host. Furthermore, we observe that the mean shifts towards higher DMs in keeping with the DM𝑧 relation. We also see that the width of the distribution seems to be getting slightly smaller. This might be counterintuitive at first, as the LSS component should increase with redshift, hence causing a broader distribution. However, the host contribution is scaled down with 1 /( 1 + 𝑧 ) in physical coordinates. At the same time, the density of the electron density field reduces as well with increasing redshift /two.sup . Thus, those two counter-acting effects can lead to a decreasing width of the likelihood as a function of redshift. The strength of this</text>
<text><location><page_9><loc_8><loc_79><loc_48><loc_92></location>effect itself depends on the parameters of the model, in particular on the parameters of the host contribution. To understand this a bit better, let us consider a simple toy model. We call the variance of the host at redshift zero 𝜎 2 host , 0 and label the variance of the LSS at 𝑧 = 1 as 𝜎 2 LSS , 1 . For linear structure growth in a matter-dominated Universe, one has 𝛿 e ∝ 𝑎 . The LSS variance scales roughly linear with redshift in this case (Reischke & Hagstotz 2023b). Therefore, the total variance is given by</text>
<formula><location><page_9><loc_19><loc_75><loc_48><loc_79></location>𝜎 2 ( 𝑧 ) = 𝜎 2 host , 0 ( 1 + 𝑧 ) 2 + 𝜎 2 LSS , 1 𝑧 , (29)</formula>
<text><location><page_9><loc_8><loc_65><loc_48><loc_74></location>which has a minimum at 𝑧 = ( 2 𝜎 2 host , 0 / 𝜎 2 LSS , 1 ) 1 / 3 . Plugging in the values we find in our analysis, the minimum arises around 𝑧 = 1. This simple model shows that the variance of the likelihood can initially decrease and then rise again at larger redshifts. This is exactly what we observe in Figure 5. If FRBs at larger redshifts, 𝑧 ≳ 1, become available, an increase in the width of the likelihood should become visible.</text>
<text><location><page_9><loc_8><loc_34><loc_48><loc_64></location>Finally, we show the model prediction at the global maximum posterior together with the likelihood for all data points in Figure 6. Note that the model prediction includes all components of the DM in Equation (1). Therefore, the relation between DM and redshift is not necessarily monotonous. The medians of the simulated DMs are shown in cyan dots for the redshifts of the FRBs. The colour bars represent the different percentiles. They include the observed data (black cross) within the darker shades, i.e. 25 to 75 percentiles. We would like to remark that those percentiles are in principle correlated due to the LSS contribution. However, this correlation is not important for the number of FRBs considered here (Reischke & Hagstotz 2023b). It is again apparent that the long tails of the likelihood are required to explain the large scatter in DM values. We furthermore investigated the response of the inference to leaving out FRB20190520B, which has a DM > 1000 but is located at a low redshift. Our findings show that the host contribution responds with a lower median and width by roughly 10 and 20 percent respectively. This of course is still fully consistent within the error bars. It shows, however, that this particular FRB increases the values of the inferred host contribution. Lastly, we recover the mean DM𝑧 relation as well by connecting the cyan dots.</text>
<section_header_level_1><location><page_9><loc_16><loc_31><loc_40><loc_32></location>6.2. Variations in the LSS component</section_header_level_1>
<text><location><page_9><loc_8><loc_7><loc_48><loc_31></location>Our fiducial model consisted of a log-normal realisation of the LSS component along with a log-normal host model. Next, we change the LSS component to a Gaussian distribution and observe how the results are affected, which can be done trivially with our pipeline. To that end, the SBI pipeline, as described in Section 4.2, is applied by changing the field in GLASS . For the comparison, we reduce the resolution of the simulations to 𝑁 side = 512. As we have established that the high-resolution run faithfully reproduces the real posterior distribution and is a good fit to the data in the last section, reducing the resolution to 𝑁 side = 512 makes a quantitative comparison while also requiring less computational resources. We rerun the fiducial case at the same resolution as well to make a quantitative comparison. It should be noted that the lower resolution effectively reduces the field variance of the LSScomponent. In appendix A we discuss the effect of reducing 𝑁 side on the coverage tests. It can be seen that the resulting posterior estimates are slightly biased when 𝑁 side is lowered,</text>
<figure>
<location><page_9><loc_52><loc_70><loc_91><loc_92></location>
<caption>✶ Figure 5. Likelihood of the DM for two FRBs at different redshifts as evaluated from the simulations. The blue line is for a lower redshift of 𝑧 = 0 . 1178, while the red line is for 𝑧 = 0 . 66. The non-Gaussian nature is obvious along with the distinct long tail which accounts for the high DM values originating from the disk of the host halo.</caption>
</figure>
<text><location><page_9><loc_52><loc_56><loc_92><loc_61></location>reducing the variance of the LSS component. Since we have established unbiased estimators at high resolution and checked that the constraints and 𝜒 2 -test are not affected, this does not hamper our analysis when comparing different models.</text>
<text><location><page_9><loc_52><loc_34><loc_92><loc_55></location>The contours are shown in Figure A.2 and the numerical values of the means and the 1 𝜎 errors of the parameters are presented in Table 2. As can be observed, there is a significant overlap among the values. Only by observing the contour plots, we cannot distinguish the models. Hence, we rely on the 𝜒 2 -tests, shown in Figure A.3 respectively. Even then, judging from these figures, there is no discernible difference between the Gaussian and log-normal LSS components. All the variations seem to be good fits according to the likelihood in Figure A.4. Consequently, with the current data, one cannot distinguish a Gaussian from a log-normal LSS component. A larger number of host identified FRBs is required for meaningful detection of this difference. In particular, FRBs with higher redshift would be especially important to assess the effect due to the LSS field, as it becomes more dominant as the redshift increases.</text>
<section_header_level_1><location><page_9><loc_59><loc_32><loc_84><loc_33></location>6.3. Variations in the Host component</section_header_level_1>
<text><location><page_9><loc_52><loc_14><loc_92><loc_31></location>Now, we turn our focus on the host model of the DM, changing it from the fiducial log-normal to first a truncated Gaussian (t-Gaussian hereafter) and then to a Gamma distribution. The Gaussian is truncated at zero as the DM is positive by definition, hence, it is not a Gaussian mathematically. For the t-Gaussian and Gamma hosts, the prior on 𝜎 is uniform on [0, 500], which is broad enough to capture the expected long-tail behaviour. It also implies that the Gamma distribution can resemble either a log-normal or a t-Gaussian distribution depending on the data, as all of these distributions come from the exponential family. As before, the contours, 𝜒 2 and likelihood are shown in Figure A.2 Figure A.3 and Figure A.4 respectively, all with 𝑁 side = 512.</text>
<text><location><page_9><loc_52><loc_7><loc_92><loc_14></location>There are three columns for the three host models. The 𝜒 2 -tests and their PTE values suggest that all models are good fits. The likelihoods of the DM for each model in Figure A.4 also agree with the goodness of fit. As can be seen, all observed DMsare within the 25-75 percentile of the simulated DMs. In</text>
<figure>
<location><page_10><loc_9><loc_74><loc_45><loc_92></location>
<caption>Figure 6. Model prediction with likelihoods for the best-fit model from Figure 2. The median (cyan dots) and the observed DM values (black cross) are overlaid on the DM values from the 1000 forward simulations of our fiducial model with log-normal field and log-normal host distribution. The colour bar is representative of the percentiles of the DM values for each FRB, where the darkest shade is at the median and falls off on both sides.</caption>
</figure>
<table>
<location><page_10><loc_8><loc_45><loc_49><loc_64></location>
<caption>Table 2</caption>
</table>
<text><location><page_10><loc_8><loc_36><loc_48><loc_42></location>Best-fit values with 1 𝜎 error bars for all the combinations of density fields and host models in our analysis are presented. The first row is our fiducial model with 𝑁 side = 4096 indicated by the ∗ symbol. All the other values are calculated at a lower resolution of 𝑁 side = 512 for comparison. The corresponding contour plots are shown in Figure A.2. The 𝜎 LN values for the log-normal host are converted to 𝜎 for better comparison.</text>
<text><location><page_10><loc_8><loc_30><loc_48><loc_35></location>summary, the 𝜒 2 -test does not prefer a particular model as the data currently lacks constraining power. The best-fit values in Table 2 for the host model show that we converge on the same behaviour.</text>
<text><location><page_10><loc_8><loc_18><loc_48><loc_29></location>Looking at the best fit values, we can see that the statistical properties of the LSS field are rather sub-dominant with the current FRB sample and if a log-normal distribution is used for the host contribution. If we assume a (truncated) Gaussian host contribution, it cannot explain the high DM of FRBs at low redshifts, Hence the model responds by artificially increasing the variance of the host contribution when the LSS is chosen to be Gaussian as well. In general, however, the constraints are all consistent with each other.</text>
<section_header_level_1><location><page_10><loc_23><loc_15><loc_33><loc_16></location>7. CONCLUSION</section_header_level_1>
<text><location><page_10><loc_8><loc_8><loc_48><loc_15></location>In this paper, we have, for the first time, presented a simulation-based inference (SBI) analysis of the DM𝑧 relation, incorporating the appropriate statistical properties of the electron density field and the host contribution in forward simulations.</text>
<text><location><page_10><loc_10><loc_7><loc_48><loc_8></location>We introduced a novel set of simulations for DM observ-</text>
<text><location><page_10><loc_52><loc_56><loc_92><loc_92></location>ables, which can seamlessly incorporate any contribution to the DM along the line of sight. For the host contribution, we adopted a log-normal distribution as our fiducial setting, as it is widely accepted in the literature. However, we also implemented alternative functional forms of the host contributions, as this merely involves substituting a single function in the simulations. For the large-scale structure component, we utilised GLASS (Tessore et al. 2023), which enabled us to simulate the electron density as either a log-normal or a Gaussian field with the correct correlations up to a given spatial resolution, provided by an input power spectrum of the threedimensional electron field. The power spectrum was calculated using HMCODE (Mead et al. 2020; Tröster et al. 2022), which was fitted to hydrodynamical simulations to jointly fit the matter and gas power spectra using a halo model approach. As output, we obtained concentric shells of the electron overdensity field with a narrow width in redshift. FRBs were then placed in the electron density at their observed redshift and location from the real data (see below). After adding the stochastic host contribution, the line-of-sight integral was performed for each FRB and the Milky Way contribution was added. For the latter, we employed the standard method of using an electron model from prior literature (Cordes & Lazio 2002; Yao et al. 2017), which is, in the spirit of our analysis, also fully flexible. With this approach, we provided realistic simulated realisations of the DM given a cosmological model, which can be easily made more complex.</text>
<text><location><page_10><loc_52><loc_24><loc_92><loc_56></location>In the next step, we performed SBI on the described simulations. This inference method requires no explicit likelihood and works by training an NN to learn either the posterior distribution (non-amortised) or the joint distribution of data and parameters (amortised). In this context, the shape of the likelihood is unconstrained, which is why this approach is often referred to as likelihood-free inference (though it still implicitly requires a likelihood). To demonstrate our pipeline, we applied it to 12 host-identified FRBs as a function of the cosmological amplitude of the DM𝑧 relation, A in Equation (7), and two host contribution parameters. For our fiducial model, we used a log-normal electron density field as well as a log-normal host model. We inferred A = 0 . 89 + 0 . 31 -0 . 29 , consistent with unity or the Planck cosmology. Similarly, the median and scale of the host distribution are DMhost = 223 + 108 -95 pc cm -3 and 𝜎 LN = 0 . 76 + 0 . 37 -0 . 44 . The values for these three parameters are consistent with previous findings, and we indeed found that no parameter (amplitude of the DM𝑧 relation, mean host contribution and its variance) is dominated by its prior, i.e. that there is additional information in the data and no parameter just assumes a flat posterior in the prior range. This is indicating that already 12 FRBs can inform us about the shape of the host contribution to some extent.</text>
<text><location><page_10><loc_52><loc_14><loc_92><loc_23></location>The resulting posterior distributions were also assessed for consistency using standard coverage tests, specifically the TARP test. Our fiducial high-resolution case with a lognormal LSS and log-normal host component demonstrated perfect coverage, indicating that the learned posterior reproduces the coverage expected from random realisations from the simulator.</text>
<text><location><page_10><loc_52><loc_7><loc_92><loc_14></location>Furthermore, we assessed the quality of the fits using a Bayesian goodness-of-fit measure based on Gelman et al. (1996), which was also employed in von Wietersheim-Kramsta et al. (2024). The 𝜒 2 was calculated from a number of data realisations generated from the simulations at the maximum a</text>
<text><location><page_11><loc_8><loc_88><loc_48><loc_92></location>posteriori. This allows us to check whether the actual data is a plausible realisation of the likelihood. With the current data, we found all the models to be a good fit.</text>
<text><location><page_11><loc_8><loc_68><loc_48><loc_88></location>One of the added benefits of the SBI pipeline is that we can test the Gaussian likelihood assumption frequently used in a more traditional Bayesian approach. To that regard, we investigated the shape of the likelihood, finding the expected long-tailed distribution towards high DM values, which is necessary to explain the large DMs observed at low redshifts, such as those seen in FRB 20190520B. Another important check involved examining the evolution of the likelihood with redshift. We found that the mean of the probability distribution function increases with redshift, as expected. However, we also observed a reduction in its scatter. Although this behaviour may seem counterintuitive at first, it is supported by a straightforward analytical calculation, which shows that the width of the total distribution indeed reaches a minimum at redshift 𝑧 < 1 for the parameter values assumed in this analysis.</text>
<text><location><page_11><loc_8><loc_49><loc_48><loc_68></location>Lastly, we explored different modelling choices for both the LSS component and the host contribution. For the LSS component, we considered both Gaussian and log-normal distributions, while for the host contribution, we evaluated log-normal, truncated Gaussian and gamma distributions. We systematically tested all possible combinations of these models and found that they consistently provided similar constraints. To save computational time, we reduced the resolution of these simulations from 𝑁 side = 4048 to 𝑁 side = 512. We found that the TARP test indicates that the learned posterior in these cases is slightly biased. Importantly, none of the models showed any indication of being a poor fit to the data, which is mainly due to the low number of FRBs available with host identification at the moment.</text>
<text><location><page_11><loc_8><loc_32><loc_48><loc_49></location>In conclusion, the simulations and inference pipeline we have developed integrate the precise physical and statistical properties needed to accurately infer the DM𝑧 relation, free from general assumptions about the likelihood or posterior. This approach is highly adaptable and scales effectively with an increasing number of FRBs, thanks to the implemented data compression techniques. Moreover, our simulations provide a robust foundation for investigating systematic effects in cosmological studies involving FRBs, as these can be seamlessly incorporated at the map level. Looking ahead, a key direction will be to include more FRBs without known redshifts, enabling a joint fit of the DM𝑧 properties alongside the statistical properties of the DM.</text>
<text><location><page_11><loc_8><loc_29><loc_48><loc_32></location>The code for this work is publicly available via https://github.com/koustav-konar/FastNeuralBurst.</text>
<section_header_level_1><location><page_11><loc_21><loc_27><loc_35><loc_28></location>ACKNOWLEDGEMENTS</section_header_level_1>
<text><location><page_11><loc_8><loc_22><loc_48><loc_27></location>SH was supported by the Excellence Cluster ORIGINS which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy-EXC-2094-390783311.</text>
<section_header_level_1><location><page_11><loc_23><loc_19><loc_34><loc_20></location>REFERENCES</section_header_level_1>
<text><location><page_11><loc_8><loc_16><loc_47><loc_18></location>Alsing J., Charnock T., Feeney S., Wandelt B., 2019, MNRAS, 488, 4440 Bannister K. W., et al., 2019, Science, 365, 565</text>
<text><location><page_11><loc_8><loc_15><loc_47><loc_16></location>Beaumont M. A., 2019, Annual Review of Statistics and Its Application, 6,</text>
<text><location><page_11><loc_10><loc_14><loc_12><loc_14></location>379</text>
<text><location><page_11><loc_8><loc_12><loc_30><loc_13></location>Bhandari S., et al., 2020, ApJL, 895, L37</text>
<text><location><page_11><loc_8><loc_11><loc_28><loc_12></location>Bhandari S., et al., 2022, ApJ, 163, 69</text>
<text><location><page_11><loc_8><loc_10><loc_47><loc_11></location>Bhattacharya M., Kumar P., Linder E. V., 2021, Phys. Rev. D, 103, 103526</text>
<text><location><page_11><loc_8><loc_9><loc_21><loc_10></location>Bishop C., 1994, NCRG</text>
<text><location><page_11><loc_8><loc_8><loc_48><loc_9></location>Bochenek C. D., Ravi V., Belov K. V., Hallinan G., Kocz J., Kulkarni S. R.,</text>
<text><location><page_11><loc_10><loc_7><loc_30><loc_8></location>McKenna D. L., 2020, Nature, 587, 59</text>
<unordered_list>
<list_item><location><page_11><loc_52><loc_91><loc_79><loc_92></location>Bonassi F. V., West M., 2015, Bayesian Analysis, 10</list_item>
<list_item><location><page_11><loc_52><loc_90><loc_81><loc_90></location>CHIME/FRB Collaboration et al., 2021, ApJS, 257, 59</list_item>
<list_item><location><page_11><loc_52><loc_88><loc_74><loc_89></location>Chatterjee S., et al., 2017, Nature, 541, 58</list_item>
</unordered_list>
<text><location><page_11><loc_52><loc_87><loc_73><loc_88></location>Chittidi J. S., et al., 2021, ApJ, 922, 173</text>
<unordered_list>
<list_item><location><page_11><loc_52><loc_86><loc_76><loc_87></location>Cooray A., Sheth R., 2002, Phys. Rep., 372, 1</list_item>
<list_item><location><page_11><loc_52><loc_85><loc_89><loc_86></location>Cordes J. M., Lazio T. J. W., 2002, arXiv e-prints, pp astro-ph/0207156</list_item>
<list_item><location><page_11><loc_52><loc_83><loc_89><loc_85></location>Cranmer K., Brehmer J., Louppe G., 2020, Proceedings of the National Academy of Science, 117, 30055</list_item>
<list_item><location><page_11><loc_52><loc_81><loc_86><loc_83></location>Deistler M., Goncalves P. J., Macke J. H., 2022, arXiv e-prints, p. arXiv:2210.04815</list_item>
<list_item><location><page_11><loc_52><loc_79><loc_89><loc_81></location>Dewdney P. E., Hall P. J., Schilizzi R. T., Lazio T. J. L. W., 2009, Proc. IEEE, 97, 1482</list_item>
<list_item><location><page_11><loc_52><loc_76><loc_91><loc_78></location>Durkan et al., 2020, in Proceedings of the 37th International Conference on Machine Learning. PMLR, pp 2771-2781,</list_item>
<list_item><location><page_11><loc_52><loc_74><loc_91><loc_76></location>https://proceedings.mlr.press/v119/durkan20a.html Euclid Collaboration et al., 2023, A&A, 675, A120</list_item>
<list_item><location><page_11><loc_52><loc_72><loc_90><loc_74></location>Fluri J., Kacprzak T., Lucchi A., Schneider A., Refregier A., Hofmann T., 2022, Phys. Rev. D, 105, 083518</list_item>
<list_item><location><page_11><loc_52><loc_71><loc_82><loc_72></location>Gatti M., et al., 2024, arXiv e-prints, p. arXiv:2405.10881</list_item>
<list_item><location><page_11><loc_52><loc_67><loc_90><loc_71></location>Gelman A., Meng X.-L., Stern H., 1996, Statistica sinica, pp 733-760 Germain M., Gregor K., Murray I., Larochelle H., 2015, in International conference on machine learning. pp 881-889</list_item>
<list_item><location><page_11><loc_52><loc_65><loc_87><loc_67></location>Górski K. M., Hivon E., Banday A. J., Wandelt B. D., Hansen F. K., Reinecke M., Bartelmann M., 2005, ApJ, 622, 759</list_item>
<list_item><location><page_11><loc_52><loc_62><loc_91><loc_65></location>Greenberg D. S., Nonnenmacher M., Macke J. H., 2019, in Proceedings of the 36th International Conference on Machine Learning. ( arXiv:1905.07488 )</list_item>
<list_item><location><page_11><loc_52><loc_59><loc_92><loc_62></location>Guo C., Pleiss G., Sun Y., Weinberger K. Q., 2017, in Proceedings of the 34th International Conference on Machine Learning. p. arXiv:1706.04599 ( arXiv:1706.04599 ), doi:10.48550/arXiv.1706.04599</list_item>
<list_item><location><page_11><loc_52><loc_58><loc_84><loc_59></location>Hagstotz S., Reischke R., Lilow R., 2022b, MNRAS, 511, 662</list_item>
<list_item><location><page_11><loc_52><loc_56><loc_84><loc_57></location>Hagstotz S., Reischke R., Lilow R., 2022a, MNRAS, 511, 662</list_item>
<list_item><location><page_11><loc_52><loc_55><loc_91><loc_56></location>Hallinan G., et al., 2019, in Bulletin of the American Astronomical Society.</list_item>
<list_item><location><page_11><loc_52><loc_53><loc_87><loc_55></location>p. 255 ( arXiv:1907.07648 ), doi:10.48550/arXiv.1907.07648 Heintz K. E., et al., 2020, ApJ, 903, 152</list_item>
<list_item><location><page_11><loc_52><loc_51><loc_92><loc_53></location>Hermans J., Delaunoy A., Rozet F., Wehenkel A., Begy V., Louppe G., 2021, arXiv e-prints, p. arXiv:2110.06581</list_item>
<list_item><location><page_11><loc_52><loc_50><loc_87><loc_51></location>Hinton S. R., 2016, The Journal of Open Source Software, 1, 00045</list_item>
<list_item><location><page_11><loc_52><loc_49><loc_83><loc_50></location>Ho M., et al., 2024, The Open Journal of Astrophysics, 7, 54</list_item>
<list_item><location><page_11><loc_52><loc_47><loc_91><loc_49></location>Howlett C., Lewis A., Hall A., Challinor A., 2012, J. Cosmology Astropart. Phys., 4, 27</list_item>
<list_item><location><page_11><loc_52><loc_45><loc_76><loc_46></location>James C. W., et al., 2022, MNRAS, 516, 4862</list_item>
</unordered_list>
<text><location><page_11><loc_52><loc_44><loc_83><loc_45></location>Jimenez Rezende D., Mohamed S., 2015, arXiv e-prints, p.</text>
<text><location><page_11><loc_53><loc_43><loc_63><loc_44></location>arXiv:1505.05770</text>
<unordered_list>
<list_item><location><page_11><loc_52><loc_41><loc_91><loc_43></location>Khrykin I. S., et al., 2024a, FLIMFLAM DR1: The First Constraints on the Cosmic Baryon Distribution from 8 FRB sightlines,</list_item>
</unordered_list>
<text><location><page_11><loc_53><loc_40><loc_70><loc_41></location>doi:10.48550/arXiv.2402.00505,</text>
<text><location><page_11><loc_53><loc_39><loc_77><loc_40></location>http://arxiv.org/abs/2402.00505</text>
<unordered_list>
<list_item><location><page_11><loc_52><loc_38><loc_84><loc_39></location>Khrykin I. S., et al., 2024b, arXiv preprint arXiv:2402.00505</list_item>
</unordered_list>
<text><location><page_11><loc_52><loc_37><loc_84><loc_38></location>Kitching T. D., Heavens A. F., 2017, Phys. Rev. D, 95, 063522</text>
<unordered_list>
<list_item><location><page_11><loc_52><loc_36><loc_75><loc_36></location>Leclercq F., 2018, Phys. Rev. D, 98, 063511</list_item>
</unordered_list>
<text><location><page_11><loc_52><loc_30><loc_91><loc_35></location>Lemos P., Coogan A., Hezaveh Y., Perreault-Levasseur L., 2023, in Krause A., Brunskill E., Cho K., Engelhardt B., Sabato S., Scarlett J., eds, Proceedings of Machine Learning Research Vol. 202, Proceedings of the 40th International Conference on Machine Learning. PMLR, pp 19256-19273,</text>
<text><location><page_11><loc_53><loc_29><loc_91><loc_30></location>https://proceedings.mlr.press/v202/lemos23a.html</text>
<text><location><page_11><loc_52><loc_28><loc_87><loc_29></location>Leonard C. D., et al., 2023, The Open Journal of Astrophysics, 6, 8</text>
<text><location><page_11><loc_52><loc_27><loc_79><loc_28></location>Lewis A., Bridle S., 2002, Phys. Rev., D66, 103511</text>
<unordered_list>
<list_item><location><page_11><loc_52><loc_26><loc_86><loc_27></location>Lewis A., Challinor A., Lasenby A., 2000, Astrophys. J., 538, 473</list_item>
<list_item><location><page_11><loc_52><loc_25><loc_87><loc_25></location>Lin K., von wietersheim-Kramsta M., Joachimi B., Feeney S., 2023,</list_item>
</unordered_list>
<text><location><page_11><loc_53><loc_23><loc_64><loc_24></location>MNRAS, 524, 6167</text>
<unordered_list>
<list_item><location><page_11><loc_52><loc_22><loc_83><loc_23></location>Liu T., Romero G. E., Liu M.-L., Li A., 2016, ApJ, 826, 82</list_item>
<list_item><location><page_11><loc_52><loc_20><loc_91><loc_22></location>Lorimer D. R., Bailes M., McLaughlin M. A., Narkevic D. J., Crawford F., 2007, Sci, 318, 777</list_item>
<list_item><location><page_11><loc_52><loc_19><loc_79><loc_20></location>Lu T., Haiman Z., Li X., 2023, MNRAS, 521, 2050</list_item>
<list_item><location><page_11><loc_52><loc_17><loc_92><loc_19></location>Lueckmann J.-M., Goncalves P. J., Bassetto G., Öcal K., Nonnenmacher M., Macke J. H., 2017, arXiv e-prints, p. arXiv:1711.01861</list_item>
<list_item><location><page_11><loc_52><loc_14><loc_91><loc_17></location>Lueckmann J.-M., Boelts J., Greenberg D. S., Gonçalves P. J., Macke J. H., 2021, in Proceedings of The 24th International Conference on Artificial Intelligence and Statistics. p. arXiv:2101.04653</list_item>
</unordered_list>
<text><location><page_11><loc_53><loc_12><loc_54><loc_13></location>(</text>
<text><location><page_11><loc_54><loc_13><loc_66><loc_13></location>arXiv:2101.04653</text>
<text><location><page_11><loc_66><loc_12><loc_84><loc_13></location>), doi:10.48550/arXiv.2101.04653</text>
<text><location><page_11><loc_52><loc_11><loc_75><loc_12></location>Macquart J.-P., et al., 2020, Nature, 581, 391</text>
<unordered_list>
<list_item><location><page_11><loc_52><loc_9><loc_89><loc_11></location>Marjoram P., Molitor J., Plagnol V., Tavaré S., 2003, Proceedings of the National Academy of Sciences, 100, 15324</list_item>
</unordered_list>
<text><location><page_11><loc_52><loc_8><loc_85><loc_9></location>Masui K. W., Sigurdson K., 2015, Phys. Rev. Lett., 115, 121301</text>
<text><location><page_12><loc_8><loc_90><loc_46><loc_92></location>Mead A. J., Peacock J. A., Heymans C., Joudaki S., Heavens A. F., 2015, MNRAS, 454, 1958</text>
<unordered_list>
<list_item><location><page_12><loc_8><loc_87><loc_45><loc_89></location>Mead A. J., Tröster T., Heymans C., Van Waerbeke L., McCarthy I. G., 2020, A&A, 641, A130</list_item>
</unordered_list>
<text><location><page_12><loc_8><loc_86><loc_38><loc_87></location>Newburgh L. B., et al., 2016, arXiv:1607.02059 [astro-ph</text>
<text><location><page_12><loc_10><loc_85><loc_27><loc_86></location>10.1117/12.2234286, p. 99065X</text>
<text><location><page_12><loc_8><loc_84><loc_29><loc_85></location>Niu C. H., et al., 2022, Nature, 611, E10</text>
<text><location><page_12><loc_8><loc_83><loc_28><loc_84></location>Ocker S. K., et al., 2022, ApJ, 931, 87</text>
<text><location><page_12><loc_8><loc_81><loc_43><loc_83></location>Papamakarios G., Murray I., 2016, Advances in neural information processing systems, 29</text>
<text><location><page_12><loc_8><loc_79><loc_43><loc_81></location>Papamakarios G., Pavlakou T., Murray I., 2017, Advances in neural information processing systems, 30</text>
<unordered_list>
<list_item><location><page_12><loc_8><loc_76><loc_47><loc_78></location>Petroff E., Hessels J. W. T., Lorimer D. R., 2019, Astron. Astrophys. Rev., 27, 4</list_item>
</unordered_list>
<text><location><page_12><loc_8><loc_75><loc_34><loc_76></location>Planck Collaboration et al., 2020, A&A, 641, A6</text>
<text><location><page_12><loc_8><loc_74><loc_46><loc_75></location>Pritchard J. K., Seielstad M. T., Perez-Lezaun A., Feldman M. W., 1999,</text>
<text><location><page_12><loc_10><loc_73><loc_32><loc_74></location>Molecular biology and evolution, 16, 1791</text>
<text><location><page_12><loc_8><loc_72><loc_33><loc_73></location>Prochaska J. X., et al., 2019, Science, 366, 231</text>
<text><location><page_12><loc_8><loc_71><loc_32><loc_72></location>Rafiei-Ravandi M., et al., 2021, ApJ, 922, 42</text>
<text><location><page_12><loc_8><loc_70><loc_28><loc_71></location>Ravi V., et al., 2019, Nature, 572, 352</text>
<text><location><page_12><loc_8><loc_69><loc_36><loc_70></location>Reischke R., Hagstotz S., 2023a, MNRAS, 523, 6264</text>
<text><location><page_12><loc_8><loc_67><loc_36><loc_68></location>Reischke R., Hagstotz S., 2023b, MNRAS, 524, 2237</text>
<unordered_list>
<list_item><location><page_12><loc_8><loc_66><loc_44><loc_67></location>Reischke R., Hagstotz S., Lilow R., 2021, Phys. Rev. D, 103, 023517</list_item>
<list_item><location><page_12><loc_8><loc_65><loc_40><loc_66></location>Reischke R., Hagstotz S., Lilow R., 2022, MNRAS, 512, 285</list_item>
</unordered_list>
<text><location><page_12><loc_8><loc_64><loc_46><loc_65></location>Reischke R., Neumann D., Bertmann K. A., Hagstotz S., Hildebrandt H.,</text>
<text><location><page_12><loc_10><loc_63><loc_32><loc_64></location>2023, arXiv e-prints, p. arXiv:2309.09766</text>
<text><location><page_12><loc_8><loc_62><loc_39><loc_63></location>Rubin D. B., 1984, The Annals of Statistics, pp 1151-1172</text>
<text><location><page_12><loc_8><loc_61><loc_46><loc_62></location>Shirasaki M., Kashiyama K., Yoshida N., 2017, Phys. Rev. D, 95, 083012</text>
<text><location><page_12><loc_8><loc_60><loc_44><loc_61></location>Sisson S. A., Fan Y., Beaumont M., 2018, Handbook of approximate</text>
<text><location><page_12><loc_10><loc_59><loc_27><loc_60></location>Bayesian computation. CRC Press</text>
<figure>
<location><page_12><loc_10><loc_38><loc_46><loc_58></location>
<caption>✶ Figure A.1. The multivariate TARP coverages for the fiducial model (using a log-normal distribution for both the field and the host) These are computed from 1000 forward simulations. One can see that and increased resolution of the maps increases the accuracy of the final posterior estimate. This behaviour is found in all cases studied in this work.</caption>
</figure>
<section_header_level_1><location><page_12><loc_25><loc_27><loc_31><loc_28></location>APPENDIX</section_header_level_1>
<section_header_level_1><location><page_12><loc_18><loc_25><loc_38><loc_26></location>A. EFFECT OF REDUCING 𝑁 SIDE</section_header_level_1>
<text><location><page_12><loc_8><loc_7><loc_48><loc_24></location>In Figure A.1, we show the effect of reducing the resolution of the GLASS forward simulation on the TARP coverage. This is necessary to establish the baseline for the comparison of the different models for the host and the LSS contribution, which have been run at a lower resolution. The main effect of decreasing 𝑁 side is that power on small scales gets washed out. In particular, the maximum multipole properly resolved is ℓ max = 3 𝑁 side -1. However, loss of power already occurs earlier. In terms of effects on the inference process, this means that the variance introduced by the LSS component decreases and therefore the scatter in the data will be larger than in the simulations. Therefore, the final constraints might be artificially tight. Figure A.1 demonstrates that the estimated</text>
<unordered_list>
<list_item><location><page_12><loc_52><loc_91><loc_81><loc_92></location>Takahashi R., 2024, arXiv e-prints, p. arXiv:2407.06621</list_item>
<list_item><location><page_12><loc_52><loc_90><loc_90><loc_90></location>Takahashi R., Ioka K., Mori A., Funahashi K., 2021, MNRAS, 502, 2615</list_item>
<list_item><location><page_12><loc_52><loc_88><loc_84><loc_89></location>Tegmark M., Taylor A. N., Heavens A. F., 1997, ApJ, 480, 22</list_item>
<list_item><location><page_12><loc_52><loc_87><loc_91><loc_88></location>Tessore N., Loureiro A., Joachimi B., von Wietersheim-Kramsta M., Jeffrey</list_item>
<list_item><location><page_12><loc_53><loc_86><loc_78><loc_87></location>N., 2023, The Open Journal of Astrophysics, 6,</list_item>
</unordered_list>
<text><location><page_12><loc_53><loc_85><loc_67><loc_86></location>10.21105/astro.2302.01942</text>
<text><location><page_12><loc_52><loc_83><loc_88><loc_85></location>Theis A., Hagstotz S., Reischke R., Weller J., 2024, arXiv e-prints, p. arXiv:2403.08611</text>
<text><location><page_12><loc_52><loc_82><loc_74><loc_83></location>Thornton D., et al., 2013, Science, 341, 53</text>
<text><location><page_12><loc_52><loc_81><loc_72><loc_82></location>Tröster T., et al., 2022, A&A, 660, A27</text>
<text><location><page_12><loc_52><loc_80><loc_90><loc_81></location>Walters A., Weltman A., Gaensler B. M., Ma Y.-Z., Witzemann A., 2018,</text>
<text><location><page_12><loc_53><loc_79><loc_60><loc_79></location>ApJ, 856, 65</text>
<text><location><page_12><loc_52><loc_77><loc_82><loc_78></location>Wu Q., Zhang G.-Q., Wang F.-Y., 2022, MNRAS, 515, L1</text>
<unordered_list>
<list_item><location><page_12><loc_52><loc_76><loc_83><loc_77></location>Xiong Y., Yang X., Zhang S., He Z., 2023, arXiv e-prints, p.</list_item>
</unordered_list>
<text><location><page_12><loc_53><loc_75><loc_63><loc_76></location>arXiv:2311.12530</text>
<text><location><page_12><loc_52><loc_74><loc_75><loc_75></location>Yamasaki S., Totani T., 2020, ApJ, 888, 105</text>
<unordered_list>
<list_item><location><page_12><loc_52><loc_73><loc_81><loc_74></location>Yan Z., et al., 2024, arXiv e-prints, p. arXiv:2402.12084</list_item>
<list_item><location><page_12><loc_52><loc_72><loc_82><loc_73></location>Yao J. M., Manchester R. N., Wang N., 2017, ApJ, 835, 29</list_item>
<list_item><location><page_12><loc_52><loc_71><loc_91><loc_72></location>Zhang Z. J., Yan K., Li C. M., Zhang G. Q., Wang F. Y., 2021, ApJ, 906, 49</list_item>
</unordered_list>
<text><location><page_12><loc_52><loc_70><loc_89><loc_71></location>Zhou B., Li X., Wang T., Fan Y.-Z., Wei D.-M., 2014, Phys. Rev. D, 89,</text>
<text><location><page_12><loc_53><loc_69><loc_57><loc_70></location>107303</text>
<text><location><page_12><loc_52><loc_67><loc_79><loc_68></location>Zieser B., Merkel P. M., 2016, MNRAS, 459, 1586</text>
<unordered_list>
<list_item><location><page_12><loc_52><loc_65><loc_92><loc_67></location>de la Bella L. F., Tessore N., Bridle S., 2021, J. Cosmology Astropart. Phys., 2021, 001</list_item>
</unordered_list>
<text><location><page_12><loc_52><loc_63><loc_92><loc_65></location>von Wietersheim-Kramsta M., Lin K., Tessore N., Joachimi B., Loureiro A., Reischke R., Wright A. H., 2024, arXiv e-prints, p. arXiv:2404.15402</text>
<text><location><page_12><loc_52><loc_52><loc_92><loc_58></location>posterior becomes slightly biased for decreasing 𝑁 side for the fiducial model. Since this will be true for all models and the fact that the high resolution run is unbiased and exact, we can safely do a one-to-one comparison of the different models using a lower resolution.</text>
<section_header_level_1><location><page_12><loc_61><loc_50><loc_83><loc_51></location>B. RESULTS FROM DIFFERENT FITS</section_header_level_1>
<text><location><page_12><loc_52><loc_24><loc_92><loc_49></location>In this section, we present a comprehensive analysis of the different figures resulting from variations in both the host contribution and the large-scale structure (LSS) contribution. Specifically, we display the final contour plots Figure A.2, the goodness-of-fit tests in Figure A.3 and the likelihoods in Figure A.4. Note that we 𝑁 side = 512 in this comparison to conserve computational resources. The reduced resolution underestimates the error from cosmic variance on individual measurements, leading to artificially tighter constraints. This issue has been addressed in our fiducial run, where a higher 𝑁 side was utilised to validate the pipeline, as described in the main text. The key conclusion from these analyses is that all combinations of the model are statistically consistent with one another and provide a satisfactory fit to the data. This is primarily due to the relatively small number of FRBs considered in this study, as well as the conservative error estimates and the expansive posterior volume resulting from the use of a three-parameter model. While these effects limit the current discriminative power, they ensure robustness in our findings.</text>
<text><location><page_12><loc_52><loc_19><loc_92><loc_24></location>However, these tests will become increasingly critical as larger FRB samples become available in the future. The current coverage tests indicate that all posteriors exhibit a slight degree of bias when a Gaussian distribution is involved.</text>
<text><location><page_12><loc_52><loc_16><loc_92><loc_18></location>Finally, for reference, we provide a list of the FRBs used in this analysis in Table A.1.</text>
<text><location><page_12><loc_52><loc_9><loc_92><loc_15></location>This paper was built using the Open Journal of Astrophysics L A T E X template. The OJA is a journal which provides fast and easy peer review for new papers in the astro-ph section of the arXiv, making the reviewing process simpler for authors and referees alike. Learn more at http://astro.theo</text>
<text><location><page_12><loc_52><loc_7><loc_57><loc_8></location>j.org</text>
<text><location><page_12><loc_57><loc_7><loc_57><loc_8></location>.</text>
<figure>
<location><page_13><loc_10><loc_72><loc_90><loc_90></location>
<caption>✶ ✶ ✶ Figure A.2. Marginalised contour plots for different combinations of the density field and host distribution. Each panel have the same host distribution and only the density field is varied, where the blue colour is for the log-normal field and red is for the Gaussian field. The host distribution for the top left, top right and bottom panels are log-normal, t-Gaussian and Gamma respectively. The posteriors for all of these contours are evaluated with 1000 forward simulations in a 10-round TSNPE setup. The resolution of all the contours is 𝑁 side = 512 for comparison. The means with the 1 𝜎 errors are provided in Table 2.</caption>
</figure>
<figure>
<location><page_13><loc_9><loc_20><loc_91><loc_51></location>
<caption>✶ Figure A.3. A comparison of the 𝜒 2 distribution for all the models we tested. The two rows represent the log-normal and Gaussian density fields respectively, while the three columns are for three different host distributions, i.e. log-normal,t-Gaussian and Gamma. The histograms are from 1000 forward simulations at best-fit values from Table 2 with 𝑁 side = 512 for all cases. The corresponding probability mass beyond the observed 𝜒 2 (dotted line) is displayed as the PTE value.</caption>
</figure>
<figure>
<location><page_14><loc_9><loc_57><loc_91><loc_92></location>
<caption>Figure A.4. Likelihoods for all the models in our analysis. The two rows represent the log-normal and Gaussian density fields and the three columns are for the log-normal, t-Gaussian and Gamma host distribution. The simulated DMs for the individual FRBs are shown with a colour bar by mapping it to the percentile values. The cyan dots are the median and the black crosses are the observed DM.</caption>
</figure>
<table>
<location><page_14><loc_18><loc_16><loc_82><loc_47></location>
<caption>Table of localised FRBs used in our analysis with their respective parameters and references. The Milky Way (MW) contributions are calculated from the YMW16 model a (Yao et al. 2017) and have the same unit as DM obs , i.e. pc cm -3 .</caption>
</table>
<section_header_level_1><location><page_14><loc_47><loc_14><loc_53><loc_14></location>Table A.1</section_header_level_1>
<text><location><page_14><loc_9><loc_9><loc_47><loc_10></location>a https://www.atnf.csiro.au/research/pulsar/ymw16/</text>
</document>
[
{
"title": "ABSTRACT",
"content": "We use the dispersion measure (DM) of localised Fast Radio Bursts (FRBs) to constrain cosmological and host galaxy parameters using simulation-based inference (SBI) for the first time. By simulating the large-scale structure of the electron density with the Generator for Large-Scale Structure (GLASS), we generate log-normal realisations of the free electron density field, accurately capturing the correlations between different FRBs. For the host galaxy contribution, we rigorously test various models, including log-normal, truncated Gaussian and Gamma distributions, while modelling the Milky Way component using pulsar data. Through these simulations, we employ the truncated sequential neural posterior estimation method to obtain the posterior. Using current observational data, we successfully recover the amplitude of the DM-redshift relation, consistent with Planck, while also fitting both the mean host contribution and its shape. Notably, we find no clear preference for a specific model of the host galaxy contribution. Although SBI may not yet be strictly necessary for FRB inference, this work lays the groundwork for the future, as the increasing volume of FRB data will demand precise modelling of both the host and large-scale structure components. Our modular simulation pipeline offers flexibility, allowing for easy integration of improved models as they become available, ensuring scalability and adaptability for upcoming analyses using FRBs. The pipeline is made publicly available under https://github.com/koustav-konar/FastNeuralBurst. Keywords: Cosmology, Fast Radio Bursts",
"pages": [
1
]
},
{
"title": "K/o.pc/u.pc/s.pc/t.pc/a.pc/v.pc K/o.pc/n.pc/a.pc/r.pc ★",
"content": "Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), German Centre for Cosmological Lensing, 44780 Bochum, Germany",
"pages": [
1
]
},
{
"title": "R/o.pc/b.pc/e.pc/r.pc/t.pc R/e.pc/i.pc/s.pc/c.pc/h.pc/k.pc/e.pc †",
"content": "Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71, D-53121 Bonn, Germany and Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), German Centre for Cosmological Lensing, 44780 Bochum, Germany",
"pages": [
1
]
},
{
"title": "S/t.pc/e.pc/f.pc/f.pc/e.pc/n.pc H/a.pc/g.pc/s.pc/t.pc/o.pc/t.pc/z.pc",
"content": "Universitäts-Sternwarte, Fakultät für Physik, Ludwig-Maximilians Universität München, Scheinerstraße 1, D-81679 München, Germany and Excellence Cluster ORIGINS, Boltzmannstraße 2, D-85748 Garching, Germany",
"pages": [
1
]
},
{
"title": "A/n.pc/d.pc/r.pc/i.pc/n.pc/a.pc N/i.pc/c.pc/o.pc/l.pc/a.pc",
"content": "Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71, D-53121 Bonn, Germany",
"pages": [
1
]
},
{
"title": "H/e.pc/n.pc/d.pc/r.pc/i.pc/k.pc H/i.pc/l.pc/d.pc/e.pc/b.pc/r.pc/a.pc/n.pc/d.pc/t.pc",
"content": "Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), German Centre for Cosmological Lensing, 44780 Bochum, Germany Version October 10, 2024",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "Fast Radio Bursts (FRB) have received significant attention over the past decades, both from cosmological and astrophysical perspectives. First discovered in archival data (Lorimer et al. 2007), these broad, millisecond transient pulses in the radio frequency range get dispersed by free electrons along their line of sight. While their origin is still debated (Petroff et al. 2019) and ranges from Magnetars (Thornton et al. 2013; Bochenek et al. 2020) to binary mergers (Liu et al. 2016), it is clear that the majority of them must be of extragalactic origin due to their highly dispersed signal. The proportionality con- † reischke@posteo.net stant of this dispersion, fittingly called Dispersion Measure (DM), is proportional to the column density of electrons along the line-of-sight. As a consequence, FRBs have been proposed to be used as a cosmological probe, in particular of the baryon distribution in the Universe. As for all cosmological fields, the electron density can be split into a background and fluctuation component relative to the background. If the host of the FRB is identified, an independent redshift estimate can be obtained. This allows the construction of the DM𝑧 relation, similar to the luminosity distance from supernovae. This relation has been used with current data to measure the baryon density and the Hubble constant (e.g. Zhou et al. 2014; Walters et al. 2018; Hagstotz et al. 2022a; Macquart et al. 2020; Wu et al. 2022; James et al. 2022; Reischke & Hagstotz 2023a). Likewise, one can study the statistical properties of the DM fluctuations (e.g. Masui & Sigurdson 2015; Shirasaki et al. 2017; RafieiRavandi et al. 2021; Bhattacharya et al. 2021; Takahashi et al. 2021; Reischke et al. 2021, 2022; Reischke et al. 2023). The ever-increasing number of observed FRBs (currently around 600 unique events, see e.g. Newburgh et al. 2016; CHIME/FRB Collaboration et al. 2021; Khrykin et al. 2024a) leads to raising interest in these events. The Square Kilometre Array (SKA /one.sup , Dewdney et al. 2009) should observe > 10 5 FRBs. Also, other surveys like DSA-2000 (Hallinan et al. 2019) are planning to detect > 10 4 FRBs with host identification. This increasing number makes the modelling and inference process prone to systematic effects. Reischke & Hagstotz (2023b) showed that already with ∼ 300 FRBs, it becomes necessary to include their covariance to conduct unbiased parameter inference using the DM𝑧 relation, an effect which has been neglected in all studies so far. With around 10 4 FRBs, additional effects such as magnification can become important as well (Takahashi 2024). Using FRBs as a tool for cosmology and astrophysics therefore requires careful modelling. A lot of these effects can be challenging to model analytically, including the case where systematic effects from the search (or in cosmological terms, survey) strategy will not be tractable. In this paper, we want to tackle these issues and present simulation-based inference (SBI) of cosmological and astrophysical models via the DM𝑧 relation of FRBs. SBI, sometimes also referred to as Likelihood-Free Inference or Implicit Likelihood Inference, is a Bayesian inference technique that does not require an explicit expression for the likelihood function of the data given the parameters of interest. Instead, the likelihood is implicitly assessed by evaluating the joint probability of the data and parameters from forward simulations that map the parameters to the corresponding synthetic data vectors. This approach offers several advantages over traditional methods that necessitate an explicit form for the likelihood. Firstly, the likelihood can assume any form, thus allowing one to bypass the common assumption of a Gaussian likelihood or the need to define a complex analytical expression for the likelihood. Secondly, for certain models and measurements, it might be impractical or too resource-intensive to determine an analytical likelihood. On the similar side, for SBI, data compression becomes essential for the high-dimensional data and parameter spaces typical in cosmology (Leclercq 2018; Alsing et al. 2019). The methods available in the SBI framework also vary based on their complexities; from the relatively trivial Approximate Bayesian Computing (ABC, see e.g. Rubin 1984; Pritchard et al. 1999; Beaumont 2019) to the latest development in NN. These procedures have been applied to cosmological data analysis (e.g. Fluri et al. 2022; Lu et al. 2023; Lin et al. 2023; Euclid Collaboration et al. 2023; von Wietersheim-Kramsta et al. 2024; Gatti et al. 2024). The NN requires forward simulations to learn the posterior distribution 𝑝 ( 𝜽 | 𝒅 ) where 𝒅 is simulated given 𝜽 . In our case, these forward simulations consist in principle of three components: ( 𝑖 ) The large-scale structure (LSS) and the background component are produced using the Generator for Large-Scale Structure (GLASS, Tessore et al. 2023) using halo model power spectra for the three-dimensional electron power spectrum. This will generate log-normal realisations of the electron field with the correct two-point statistics imprinted. ( 𝑖𝑖 ) The host contribution, which is simply sampled from a host model probability density function (PDF). ( 𝑖𝑖𝑖 ) The Milky Way (MW) contribution, for which we will use the standard methods of inferring it from already present electron models (Cordes & Lazio 2002; Yao et al. 2017; Yamasaki & Totani 2020). We will then use those forward simulations to train an NN to learn the posterior and sample from the posterior with traditional MCMC. Here, we will use Truncated Sequential Neural Posterior Estimation (TSNPE, Deistler et al. 2022) to conduct the inference within the SBI framework. We aim to fit the amplitude of the DM𝑧 relation and the median and the width of the log-normal host distribution with the available host identified FRBs, providing a roadmap for future cosmological and astrophysical inference with FRBs. The manuscript is structured as follows: In Section 2, we introduce the basics of FRB cosmology and discuss the different components entering the total DM. Section 3 provides an overview of the forward simulation pipeline. The inference techniques are discussed in Section 4. Section 5 introduces the validation techniques we use after the inference process. In Section 6, we present the results and summarise them in Section 7.",
"pages": [
1,
2
]
},
{
"title": "2.1. FRB basics",
"content": "The pulses of FRBs undergo dispersion while travelling through the ionized matter distribution in the Universe, leading to a frequency-dependent, ∝ 𝜈 -2 , offset of the bursts' arrival times. Given this time delay measured as 𝛿𝑡 ( ˆ 𝒙 , 𝑧 ) for an FRB at redshift 𝑧 in direction ˆ 𝒙 , the constant of proportionality is the observed dispersion measure: 𝛿𝑡 ( ˆ 𝒙 , 𝑧 ) = DMtot ( ˆ 𝒙 , 𝑧 ) 𝜈 -2 . This DM can be broken up into different components: The first contribution is DMLSS ( ˆ 𝒙 , 𝑧 ) , caused by free electrons in the LSS. Here, the dependence on the direction is kept explicitly, since the LSS is correlated. In the literature, DMLSS ( ˆ 𝒙 , 𝑧 ) is often split up into an IGM part and a halo part This is equivalent to a halo model prescription (Cooray & Sheth 2002) of the statistical properties of the DM. On the level of the power spectrum, this would amount to the twohalo term (corresponding to DMIGM ( ˆ 𝒙 , 𝑧 ) ) and the one-halo term (corresponding to DMhalo ( ˆ 𝒙 , 𝑧 ) ). The MW contribution DMMW ( ˆ 𝒙 ) can itself be split up into a contribution from the ISM and the MW halo. Both will not depend on redshift, as these are local quantities. However, there is a clear directional dependence. Lastly, DMhost ( 𝑧 ) is the contribution of the host galaxy which can, as the MW contribution, be split up into a part originating from the visible galaxy and one of the halo. For this, only a potential redshift dependence is assumed, as the contribution of different hosts should not be correlated, ignoring the unlikely event that two distinct FRBs originate from the same galaxy. Note that the rest-frame DM of the host, DMhost , rf , is observed as DMhost ( 𝑧 ) = ( 1 + 𝑧 ) -1 DMhost , rf . First, we will take a more detailed look at the contribution of the LSS. Quite generally this is given by where 𝑛 e ( ˆ 𝒙 , 𝑧 ) is the comoving cosmic free electron density, 𝐻 ( 𝑧 ) = 𝐻 0 𝐸 ( 𝑧 ) is the Hubble function with the expansion function 𝐸 ( 𝑧 ) and the Hubble constant 𝐻 0. 𝑓 IGM ( 𝑧 ) is the fraction of electrons in the IGM and is calculated by subtracting the fraction bound in stars, compact objects and the dense interstellar medium (ISM) For redshifts 𝑧 < 3 almost all baryons are ionised, and the DM is therefore rewritten as with the baryon density 𝜌 b, the proton mass 𝑚 p and the electron fraction calculated from the primordial hydrogen and helium abundances 𝑌 H and 𝑌 He. Altogether, one finds: where we defined A B 3 𝑐 Ω b0 𝐻 0 8 𝜋𝐺𝑚 p 𝜒 e 𝑓 IGM. The LSS contribution is therefore entirely specified by the statistical properties of the electron density field 𝛿 e. Motivated by numerical simulations of the DM which showed that it follows a log-normal distribution (see e.g. Zhang et al. 2021), we model the electron field using GLASS (Tessore et al. 2023) which can, given a three-dimensional power spectrum of a cosmological field, create log-normal realisations. This is done by dividing the LSS into 𝑁 shells non-overlapping and concentric shells. Each shell covers the full sky, spanning the comoving volume between redshift 𝑧 𝑖 and 𝑧 𝑖 + 1. First, we define a matter weight function along the line-of-sight via: with the co-moving distance 𝜒 ( 𝑧 ) . The electron density contrast, 𝛿 e ( ˆ 𝒙 , 𝑧 ' ) , can now also be defined in each shell: The statistical properties of this field on the two-point level are calculated in the harmonic space by the angular power spectra: where 𝑗 ℓ ( 𝑥 ) are spherical Bessel functions of order ℓ and ℓ ∈ N . The electron power spectrum is defined as: Lastly, we approximate the unequal time correlator by its geometric mean: which has been shown to be an excellent approximation for weak gravitational lensing (Kitching & Heavens 2017; de la Bella et al. 2021). Since the DM of the LSS has a similarly broad kernel as it is an integrated effect, this should also hold for FRBs. Starting from Equation (7), let us define the perturbations to the DM as It can be broken down into a discrete sum over the matter shells where we defined the characteristic redshift of each shell to be its mean: Likewise, 𝑤 𝑖 takes into account the weight of the shell via",
"pages": [
2,
3
]
},
{
"title": "2.3. Host Contribution",
"content": "The next component in the forward simulation is the host contribution, quantifying the effect the host galaxy on the observed DM. Based on the position of the FRB progenitor in the galaxy, the signal may travel through the whole galaxy or parts of it. As such, the induced DM varies accordingly. The effect of a complete or partial travel path through the local host, which translates to high and low DM, is usually described by a log-normal distribution (Macquart et al. 2020; Wu et al. 2022). Other works, e.g. Hagstotz et al. (2022a), have assumed a Gaussian host contribution. Recently, simulations have also shown that a log-normal distribution can fit the host contribution rather well (Theis et al. 2024). With the current data set of FRBs, this choice does not make a difference if priors on the DM are included, as we will show later. For our fiducial case, however, we will choose a log-normal distribution for DMhost , rf (that is the host contribution in the rest-frame of the host galaxy for which we assume no intrinsic redshift evolution): whereexp ( 𝜇 ) and exp ( 2 𝜇 + 𝜎 2 LN ) [ exp ( 𝜎 2 LN )-1 ] are the median and variance respectively with 𝑥 = DMhost , rf . The median and the scale ( 𝜎 LN) are free parameters in our study. Note that this samples the rest-frame DM of the host, DMhost , rf .",
"pages": [
3
]
},
{
"title": "2.4. Milky Way contribution",
"content": "In our analysis, we assume that the MW contribution is accessible from the models described in Cordes & Lazio (2002); Yao et al. (2017); Yamasaki & Totani (2020) and we simply add the numerical value for DMMW ( ˆ 𝒙 ) in Equation (1). At the current sensitivity level, dictated by the amounts of FRBs available, this addition does not have a sizable influence on the inference of cosmological parameters. However, with more FRBs being observed, it could well be that the addition of the MW contribution leads to a residual correlation in the DMwhich will be falsely picked up as a cosmological signal. This scenario can in principle be tested with our pipeline, as the MW contribution can simply be added to the simulated DM.",
"pages": [
3,
4
]
},
{
"title": "3.1. Forward simulation",
"content": "In this section, we describe how we use the previously described ingredients to construct forward simulations for the individual components of Equation (1), which, by simple summation, yield a forward simulation prediction for a set of DMtot ( ˆ 𝒙 𝑎 , 𝑧 𝑎 ) , with 𝑎 = 1 , . . . , 𝑁 FRB. Here, ˆ 𝒙 𝑎 and 𝑧 𝑎 are the positions and redshifts of the FRBs from the FRB catalogue. Figure 1 summarises the forward simulation pipeline which goes through the following steps:",
"pages": [
4
]
},
{
"title": "(Mead et al. 2015, 2020; Tröster et al. 2022).",
"content": " Including this weight in Equation (14) one finds: Lastly, adding the homogeneous contribution gives: This procedure provides a pair ({ 𝜃 𝛼 } , { DMtot ( ˆ 𝒙 𝑎 , 𝑧 𝑎 )}) . Rerunning the pipeline, with parameters sampled from a prior distribution, creates a set of forward simulations, which is used to learn the posterior distribution by the NN. The resolution of the simulation is dependent on the parameter 𝑁 side ∈ 2 Z + as GLASS internally uses HEALPix (Górski et al. 2005). From the flowchart in Figure 1 and the list above, it is clear that any component in the pipeline can easily be exchanged for another model. If, for example, there exists a better model for the host contribution with different parameters, it is easy to replace it.",
"pages": [
4
]
},
{
"title": "3.2. Data compression",
"content": "The data vector currently is 𝑁 FRB dimensional, thus requiring a compression procedure, essentially translating a 𝑑 -dimensional dataset down to 𝑛 dimensions ( 𝑛 < 𝑑 ). This is an essential step, as training the NN with high-dimensional data is slow and can lead to inaccuracies. Lossless data compression preserves the Fisher information of the original data. Therefore, we use the data reduction scheme prescribed in Alsing et al. (2019) reducing the 𝑑 -dimensional data down to a dimension equal to the number of free parameters. Assuming a Gaussian likelihood, this compressed data ( 𝑡 ) can be obtained as (Tegmark et al. 1997) where 𝝁 ∗ , C ∗ are the ensemble mean and covariance of the original data at some fiducial value 𝜽 ∗ and d is the corresponding data vector. ∇ represents the partial derivatives with respect to the free parameters. This fiducial 𝜽 ∗ needs to be optimised to ensure no information loss. Specifically, we use the Fisher scoring method where 𝒕 𝑘 is the compressed statistics for 𝜽 𝑘 at the 𝑘 -th step. Depending on the complexity of the parameter space, a larger value of 𝑘 may be required for convergence. In practice, we stop after a finite number of iterations when the increment, the second term on the right-hand side of Equation (22), asymptotically approaches a plateau and further steps are not favoured against the computational time. The components of the Fisher matrix, F 𝑖 𝑗 are evaluated by its full expression for a Gaussian likelihood: The three free parameters in our model are the amplitude of the DM𝑧 relation, the median and the scale of the lognormal host distribution. We specify the initial 𝜽 ∗ in Table 1 motivated from Reischke et al. (2022), but we quickly converge to 𝜽 optimal = { 0 . 94 , 200 . 74 , 0 . 79 } T . For the current selection of free parameters, all the derivatives have analytic solutions as the derivative of the covariance of the LSS component (Reischke & Hagstotz 2023b) scales with 2 A , with A being the prefactor in Equation (7). Similarly, the derivatives of the log-normal host covariance are trivial. If we are to fit a more complex model in the future, one would have to calculate those derivatives numerically or for more stable results using a differentiable code. This could for example be achieved by emulating the important quantities.",
"pages": [
4,
5
]
},
{
"title": "4. INFERENCE",
"content": "Given the simulation pipeline described in the previous section, we are now in the position to infer the posterior of the parameters summarised in Table 1. Inference refers to determining the parameters that best describe the observation. Typically, we have a model based on physical laws, observations from surveys and a likelihood. The latter is then combined with the prior to yield the posterior. The most popular approach to such an inference process has been the Markov Chain Monte Carlo (MCMC) within the Bayesian framework. MCMCmethods create samples from the posterior. However, modelling a likelihood that captures the full complexity of the physical processes is one of the most complex parts of any traditional analysis, with cases arising where an analytical likelihood is not accessible.",
"pages": [
5
]
},
{
"title": "4.1. Simulation-based inference",
"content": "Approximate Bayesian Calculation (ABC) addresses this issue by comparing data from forward simulation with observation with some arbitrary margin (Rubin 1984; Pritchard et al. 1999). Only the accepted simulations are considered for posterior inference, which leads to critical drawbacks such as a high rejection rate with decreasing margin and the curse of dimensionality (Sisson et al. 2018). Improvements upon this basic rejection sampling include MCMC-ABC (Marjoram et al. 2003) and Sequential Monte Carlo ABC (SMCABC) (Bonassi & West 2015), which only partially solve the rejection rate and do not address the high dimensionality problem. Following the development of NNs, a suite of algorithms has been proposed recently that estimate the posterior without access to a likelihood. All of these methods come under the umbrella of SBI or likelihood-free inference (LFI), which introduces a parameterised density estimator that learns from the joint distribution of the data-parameter pair (Cranmer et al. 2020). In this section, we briefly introduce the density estimation mentioned in (Papamakarios & Murray 2016) called sequential neural posterior estimation (SNPE-A) and summarise its subsequent developments in SNPE-B (Lueckmann et al. 2017), SNPE-C or APT (Greenberg et al. 2019) and an improvement on APT called TSNPE (Deistler et al. 2022). Finally, we will be using the TSNPE algorithm in our analysis. The fundamental idea behind all of these methods is to approximate the posterior, 𝑝 ( 𝜽 | d ) , with a conditional density estimator 𝑞 𝜙 ( 𝜽 | d ) , where 𝜙 are the trainable parameters. Typically, the parameters, 𝜽 , are sampled from the full prior range, which is inefficient. There exist two schools of SBI. Amortised methods yield a posterior that can be utilised for various observations without the need for retraining, whereas sequential methods concentrate the inference on a specific observation to enhance simulation efficiency. Here we focus on the sequential-(S)NPE methods refer to using a proposal as a subset of the prior, ˜ 𝑝 𝜽 ) ⊆ 𝑝 ( 𝜽 ) . Initially, the proposal is equal to the prior and in subsequent rounds we update the proposal with the approximate posterior, improving the efficiency of the inference process. The predicament is that this procedure does not recover the true posterior, but rather an approximate posterior: where 𝑝 ( 𝜽 | d ) is the true posterior, ˜ 𝑝 ( 𝜽 | d ) is the approximate posterior, ˜ 𝑝 ( 𝜽 ) is the proposal, 𝑝 ( 𝜽 ) is the prior and ˜ 𝑝 ( d ) is the evidence. The approximation recovers the true posterior only when the proposal is equal to the prior. SNPE methods address this issue with a parameterised density estimator, 𝑞 𝜙 ( 𝜽 | d ) , that learns from the joint distribution of the data and parameter The problem to solve is the extra ratio, ˜ 𝑝 ( 𝜽 )/ 𝑝 ( 𝜽 ) . The way each algorithm addresses this issue is what differentiates the various SNPE methods. For instance, SNPE-A maintains a closed-form solution by restricting the choice of the prior, proposal and the density estimator to either a uniform or Gaussian distribution. The training is carried out by minimising the loss function or the negative log-likelihood of the joint probability of the data and parameter defined via Other methods like SNPE-B offer a solution by including the ratio inside the loss function, whereas SNPE-C incorporates the ratio into the density estimator and finally, APT normalises the density estimator by using uniform subsets of the prior called atoms. This development allows any arbitrary distribution choice for the prior, proposal and the density estimator. For an extensive technical review of the shortcomings and the subsequent developments of the various SNPE methods, the reader is referred to the following articles and the references therein (Durkan et al. 2020; Lueckmann et al. 2021; Xiong et al. 2023). The latest development, TSNPE, builds upon the APT method and truncates the prior based on the 1 -𝜖 mass ( 𝜖 being an arbitrary margin) of the highest probability region (HPR) of the approximate posterior. In other words, the subsequent rounds only consider the prior range which is most probable to produce the posterior. This prescription maintains the flexibility of the distribution choice while also accounting for the posterior mass outside the prior boundaries, an issue of the APT method (Deistler et al. 2022). As the HPR of the approximate posterior contains the information on the joint probability of the data and parameter in each round, the truncation is data-driven. TSNPE also results in faster overall convergence. The various SNPE methods tackle posterior estimation by either changing the loss function or restricting the choice of distribution. Without a general solution, the choice of algorithm affects the performance and accuracy of the analysis. Our tests reveal that TSNPE, due to the truncation, is faster and allows for a more flexible distribution choice for prior and density estimators as compared to SNPE-A. As the field of NN is projected to experience rapid growth, the choice of the best algorithm is expected to change as well; refer to (Cranmer et al. 2020) for a detailed review of the status of SBI at the beginning of this decade.",
"pages": [
5,
6
]
},
{
"title": "4.2. Implementation",
"content": "The simulated data in our case are the compressed DM values of FRBs as described in Section 3.2. The priors are defined in Table 1. For the analysis, we use 12 FRBs with host identification as given in Table A.1, where amongst others we list the observed DMs and their references. The forward simulations for the fiducial case of log-normal density field and log-normal host contribution are carried out at a resolution of 𝑁 side = 4096. The choice is so that DM contributions from small scales ( ℓ ∼ 10 4 ) are accounted for. This is essential for an unbiased analysis, as we shall discuss. A good consistency check here is the comparison of the numerical covariance we get from the simulations to its analytical counterpart (Reischke & Hagstotz 2023b). The diagonal elements of the analytic covariance matrix show a monotonic behaviour between the DM and the redshift of FRBs. For the numerical covariance, we choose 𝑁 side = 4096 to match this monotonous nature and magnitude of the diagonal elements, i.e. the DM field variance. In each round of TSNPE, the density estimator learns the mapping between the simulated data and the corresponding parameters. The observed DM is introduced post training, which helps evaluate an approximate posterior for that round. This approximate posterior is then utilised to calculate the HPR region and the prior range for the next round is truncated based on that. As the number of rounds is increased, the approximate posterior converges to the true posterior. Introducing the observed DM is necessary to propel the truncation in the direction of the true posterior. With parameters, simulated data and observation, these are the three components required for the SBI analysis. We implement this using LtU-ILI package for a quick setup with different neural embedding (Ho et al. 2024). We have used the TSNPE algorithm for a total of 1500 simulations in 10 rounds. The initial simulations contain the full prior range and the subsequent truncation is only influenced by the data. As LtU-ILI does not natively support TSNPE yet, we run TSNPE, save the data and then use it. The choice of the number of simulations in each round is dictated by the number of epochs the NN takes for training. We limit the number of epochs to 50 to avoid overfitting and 150 simulations in each round provide ample data for the density estimator 𝑞 𝜙 ( 𝜽 | d ) to approximate the posterior by adjusting the trainable parameter 𝜙 . As for the NN architecture, we use two different types; mixture density network (MDN) and masked autoregressive flows (MAF). These architectures aim to produce an arbitrary yet tractable PDF with trainable parameters to be used as the density estimator. The complexity of data mandates arbitrariness, while the backpropagation in the training process requires tractability. MDN, as the name suggests, is the weighted sum of many Gaussian distributions defined as (Bishop 1994) where m 𝑘 and S 𝑘 are the mean and covariance of the 𝑘 -th Gaussian. This type of density estimator is malleable and can be transformed into any arbitrary distribution by adjusting the weights, means and standard deviations. While flexibility is a property we seek, a value of 𝑘 too large can lead to overfitting. Masked Autoregressive Flows (MAFs, Papamakarios et al. 2017), on the other hand, combine two different architectures called Normalizing Flow (NF, Jimenez Rezende & Mohamed 2015) and Masked Autoencoder for Distribution Estimation (MADE, Germain et al. 2015) into one. In NF, a normal distribution is subjected to 𝑘 invertible transformations, called flow, resulting in an arbitrary complex distribution. These transformations are based on the conditional probability scheme described in MADE. Combining NF and MADE,MAFcanproduce arbitrary distribution optimised for density estimation. For technical details, the reader is referred to the respective articles. The presence of MAF along with MDN(7:3 weight ratio) ensures the model has a better scaling with dimensionality and learns any multimodal, non-Gaussian features of the data while avoiding overfitting. For the MDN embedding, we use 50 hidden layers with 10 components in each layer, while the MAF net is constructed using 20 hidden layers with 8 transformations in each layer. When the training is complete, we acquire a mapping (effectively a KDE) from the joint space of parameters to the distribution 𝑝 ( 𝜽 | d ) without the usage of an explicit likelihood function. This posterior can then be sampled by standard MCMCtechniques.",
"pages": [
6
]
},
{
"title": "5. VALIDATION",
"content": "After the completion of the SBI pipeline, we carry out two validation tests, one for the model under consideration and one for its posterior. For the posterior, we use the multivariate coverage test called Tests of Accuracy with Random Points (TARP, Lemos et al. 2023) and for the model, the accuracy is measured via the goodness of fit or 𝜒 2 -test. While TARP validates the SBI, the 𝜒 2 -test is used to investigate if there are preferred models. In the following, we describe the two validation techniques.",
"pages": [
7
]
},
{
"title": "5.1. Coverage Test",
"content": "To check the consistency of the posterior distributions, we use the multivariate coverage test called TARP. Coverage probability measures how frequently the estimated posterior contains the true parameter value, and can be used to check the consistency of the posterior (Guo et al. 2017; Hermans et al. 2021). Once we have obtained the posterior, TARP measures the expected coverage probability of random posterior samples within a given credibility level of the learned posterior empirically. In more technical terms, for any instance of the prior 𝜽 ★ and the corresponding simulated data d ★ , samples are drawn from the posterior as 𝑝 ( 𝜽 | d ★ ) . Then, a circle is drawn centred at a random reference point 𝜽 r with radius | 𝜽 ★ -𝜽 r | and the fraction of posterior samples inside the circle is calculated. A higher fraction implies the posterior is more accurate. The fraction is calculated multiple times to get a statistical estimate of the coverage. This random sampling is necessary to validate the static HPR region in TSNPE, which is prone to having blind spots. We use the learnt posterior to draw 1000 samples at random parameter points for the TARP test implemented within the LtU-ILI . We also bootstrap the test 100 times at each credibility level, considering the process is random by definition.",
"pages": [
7
]
},
{
"title": "5.2. Goodness-of-Fit",
"content": "The typical goodness of fit test can not be used in the SBI framework, as it requires an analytical likelihood. Here, we use the implementation of the 𝜒 2 -test (Gelman et al. 1996) described in von Wietersheim-Kramsta et al. (2024). After finding the best-fit values for the parameters of interest, 𝚯 ∗ , we sample 𝑛 noise realisations at this set of parameters and define the 𝜒 2 as where E denotes the expectation value and Cov is the covariance at the best-fit cosmology. 𝒕 𝑖 is the score-compressed summarystatistic of the 𝑖 -th realisation. Specifically, we create 1000 new forward simulations at the best-fit values to calculate the mean data vector and the covariance. The 𝜒 2 values from the simulations are plotted as a histogram and compared with the 𝜒 2 value for the observed DM. The accuracy of the model is assessed through the probability mass of the simulated 𝜒 2 that lies beyond this observed 𝜒 2 , which we call the probability to exceed (PTE).",
"pages": [
7
]
},
{
"title": "6. RESULTS",
"content": "In this section, we present the major result of our analysis. Weconsider three parameters for the inference. The first one is the amplitude of the DM𝑧 relation in Equation (7) as a scale. This encapsulates the Λ CDMparameters, for which we use the Planck cosmological results from (Planck Collaboration et al. 2020), i.e. A = 1 refers to the input cosmology and this A is varied in different simulations. The motivation behind only varying the amplitude of the DM-redshift relation is that the number of FRBs currently available is not able to discriminate between different expansion histories. The remaining two parameters are the median and the scale of the log-normal host distribution, which we denote as DMhost and 𝜎 LN. As for the model, there are two components in our simulations that vary. First, the DMLSS part is modelled using either a log-normal or a Gaussian realisation of the underlying electron density field via GLASS . These are denoted by 'Field: log-normal' and 'Field: Gaussian' respectively. Similarly, we have three host distributions; log-normal, truncated Gaussian and Gamma. With this setup, we have six possible combinations of field and host. That being said, we use a log-normal field and a log-normal host distribution as our fiducial setting. For this, we will discuss and validate the results in detail and use the high resolution setting of 𝑁 side = 4096. For the other combinations, we use a lower resolution 𝑁 side = 512 in order to limit computation time. In the following, we start with the fiducial model and then observe how the changes in field and host affect the results.",
"pages": [
7,
8
]
},
{
"title": "6.1. Fiducial model and validation",
"content": "For the fiducial model, we assume the electron density field in GLASS to be log-normal and sample the host DM from a lognormal distribution. The decision is primarily motivated by results from simulations (see e.g. Zhang et al. 2021; Theis et al. 2024). In the subsequent sections, we present the comparisons of the six different models, which support our decision. With the simulator set to the desired configuration and priors defined in Table 1, the SBI pipeline is run with the TSNPE algorithm, which returns the posterior distribution for the free parameters. In Figure 2, we show the marginalised contour plots using ChainConsumer (Hinton 2016). The forward simulations have the resolution parameter 𝑁 side = 4096. As mentioned at the beginning of this section, there are three parameters that we constrain; A as a scale for the input cosmology, DMhost as the median and 𝜎 LN as the scale of the log-normal host distribution. The value for the scale parameter A is 0 . 89 + 0 . 31 -0 . 29 at 68% confidence, which is consistent with unity or the input fiducial Planck cosmology. The median and the scale of the log-normal host along with their 68% confidence intervals are given as DMhost = 223 + 108 -95 pc cm -3 and 𝜎 LN = 0 . 76 + 0 . 37 -0 . 44 . This 𝜎 LN is further converted into the standard deviation from the definition of the variance in Equation (17). Then, we write 𝜎 = 263 + 91 -113 . As expected, we find a strong anti-correlation between A and DMhost since they both enter the observed DM in an additive fashion. The width of the host contribution is not degenerate with the other parameters, as it rather reduces the error on each measurement than changing the signal. Our constraints of 𝜎 LN seem to be slightly prior driven, as can be seen from the posterior hitting the prior boundaries of 𝜎 LB in Figure 2. However, there is clearly some constraining power in the data on the shape of the host distribution. In general, we find excellent agreement with previous work (Macquart et al. 2020; James et al. 2022; Hagstotz et al. 2022b; Reischke & Hagstotz 2023a; Khrykin et al. 2024b). Now that the posterior distribution is evaluated, we first assess its accuracy via the TARP coverage in Figure 3 with the solid blue line. The shaded blue regions are the 1 and 2 𝜎 error bars from the 100 bootstraps. The region above the diagonal dotted line is called under-confident or conservative, while the region below suggests overconfidence or bias. As the diagonal dotted line is within the dark-shaded region throughout the whole range of credibility, this means that the learnt posterior is an accurate representation of the true posterior and is not biased. Additionally, we check for the goodness of fit of the model and the result is shown in Figure 4. The histograms are the 𝜒 2 values from our simulations at best-fit values of the parameters, and the vertical line corresponds to the 𝜒 2 evaluated at the observed DM. The PTE for the fiducial model is 0.3 (corresponding to a 𝑝 -value), which means that the model is a good fit. We can thus concur that the 'log-normal field and log-normal host' model is an excellent fit for the DM of FRBs. With the SBI framework, all the correlations are taken into account without defining any likelihood function. Nonetheless, we can still recover the likelihood from the simulated DMs for the best-fit values. We therefore run our pipeline 1000 times at the best-fit value and plot the corresponding distribution of DM. In Figure 5 we show the DM likelihoods for two FRBs with varying redshift, 𝑧 = 0 . 1178 and 𝑧 = 0 . 66. The first observation is the non-Gaussian nature of the distribution, with a tendency towards higher DMs. This behaviour emerges from the host and LSS DM contribution. If the electron density of the host halo where the FRB progenitor resides is high, the signal experiences higher dispersion. That seems to be especially the case for FRB 20190520B, with an observed DM = 1202 pc cm -3 at a relatively lower redshift of 𝑧 = 0 . 241 as shown in Table A.1. The high DM is due to the local contribution of the dwarf host galaxy, identified as J160204.31-111718.5 (Ocker et al. 2022; Niu et al. 2022; Yan et al. 2024). The implication is that a Gaussian likelihood assumption in the inference can introduce biases, as it does not consider the physical effect of variable travel distances and local environments of the host. Furthermore, we observe that the mean shifts towards higher DMs in keeping with the DM𝑧 relation. We also see that the width of the distribution seems to be getting slightly smaller. This might be counterintuitive at first, as the LSS component should increase with redshift, hence causing a broader distribution. However, the host contribution is scaled down with 1 /( 1 + 𝑧 ) in physical coordinates. At the same time, the density of the electron density field reduces as well with increasing redshift /two.sup . Thus, those two counter-acting effects can lead to a decreasing width of the likelihood as a function of redshift. The strength of this effect itself depends on the parameters of the model, in particular on the parameters of the host contribution. To understand this a bit better, let us consider a simple toy model. We call the variance of the host at redshift zero 𝜎 2 host , 0 and label the variance of the LSS at 𝑧 = 1 as 𝜎 2 LSS , 1 . For linear structure growth in a matter-dominated Universe, one has 𝛿 e ∝ 𝑎 . The LSS variance scales roughly linear with redshift in this case (Reischke & Hagstotz 2023b). Therefore, the total variance is given by which has a minimum at 𝑧 = ( 2 𝜎 2 host , 0 / 𝜎 2 LSS , 1 ) 1 / 3 . Plugging in the values we find in our analysis, the minimum arises around 𝑧 = 1. This simple model shows that the variance of the likelihood can initially decrease and then rise again at larger redshifts. This is exactly what we observe in Figure 5. If FRBs at larger redshifts, 𝑧 ≳ 1, become available, an increase in the width of the likelihood should become visible. Finally, we show the model prediction at the global maximum posterior together with the likelihood for all data points in Figure 6. Note that the model prediction includes all components of the DM in Equation (1). Therefore, the relation between DM and redshift is not necessarily monotonous. The medians of the simulated DMs are shown in cyan dots for the redshifts of the FRBs. The colour bars represent the different percentiles. They include the observed data (black cross) within the darker shades, i.e. 25 to 75 percentiles. We would like to remark that those percentiles are in principle correlated due to the LSS contribution. However, this correlation is not important for the number of FRBs considered here (Reischke & Hagstotz 2023b). It is again apparent that the long tails of the likelihood are required to explain the large scatter in DM values. We furthermore investigated the response of the inference to leaving out FRB20190520B, which has a DM > 1000 but is located at a low redshift. Our findings show that the host contribution responds with a lower median and width by roughly 10 and 20 percent respectively. This of course is still fully consistent within the error bars. It shows, however, that this particular FRB increases the values of the inferred host contribution. Lastly, we recover the mean DM𝑧 relation as well by connecting the cyan dots.",
"pages": [
8,
9
]
},
{
"title": "6.2. Variations in the LSS component",
"content": "Our fiducial model consisted of a log-normal realisation of the LSS component along with a log-normal host model. Next, we change the LSS component to a Gaussian distribution and observe how the results are affected, which can be done trivially with our pipeline. To that end, the SBI pipeline, as described in Section 4.2, is applied by changing the field in GLASS . For the comparison, we reduce the resolution of the simulations to 𝑁 side = 512. As we have established that the high-resolution run faithfully reproduces the real posterior distribution and is a good fit to the data in the last section, reducing the resolution to 𝑁 side = 512 makes a quantitative comparison while also requiring less computational resources. We rerun the fiducial case at the same resolution as well to make a quantitative comparison. It should be noted that the lower resolution effectively reduces the field variance of the LSScomponent. In appendix A we discuss the effect of reducing 𝑁 side on the coverage tests. It can be seen that the resulting posterior estimates are slightly biased when 𝑁 side is lowered, reducing the variance of the LSS component. Since we have established unbiased estimators at high resolution and checked that the constraints and 𝜒 2 -test are not affected, this does not hamper our analysis when comparing different models. The contours are shown in Figure A.2 and the numerical values of the means and the 1 𝜎 errors of the parameters are presented in Table 2. As can be observed, there is a significant overlap among the values. Only by observing the contour plots, we cannot distinguish the models. Hence, we rely on the 𝜒 2 -tests, shown in Figure A.3 respectively. Even then, judging from these figures, there is no discernible difference between the Gaussian and log-normal LSS components. All the variations seem to be good fits according to the likelihood in Figure A.4. Consequently, with the current data, one cannot distinguish a Gaussian from a log-normal LSS component. A larger number of host identified FRBs is required for meaningful detection of this difference. In particular, FRBs with higher redshift would be especially important to assess the effect due to the LSS field, as it becomes more dominant as the redshift increases.",
"pages": [
9
]
},
{
"title": "6.3. Variations in the Host component",
"content": "Now, we turn our focus on the host model of the DM, changing it from the fiducial log-normal to first a truncated Gaussian (t-Gaussian hereafter) and then to a Gamma distribution. The Gaussian is truncated at zero as the DM is positive by definition, hence, it is not a Gaussian mathematically. For the t-Gaussian and Gamma hosts, the prior on 𝜎 is uniform on [0, 500], which is broad enough to capture the expected long-tail behaviour. It also implies that the Gamma distribution can resemble either a log-normal or a t-Gaussian distribution depending on the data, as all of these distributions come from the exponential family. As before, the contours, 𝜒 2 and likelihood are shown in Figure A.2 Figure A.3 and Figure A.4 respectively, all with 𝑁 side = 512. There are three columns for the three host models. The 𝜒 2 -tests and their PTE values suggest that all models are good fits. The likelihoods of the DM for each model in Figure A.4 also agree with the goodness of fit. As can be seen, all observed DMsare within the 25-75 percentile of the simulated DMs. In Best-fit values with 1 𝜎 error bars for all the combinations of density fields and host models in our analysis are presented. The first row is our fiducial model with 𝑁 side = 4096 indicated by the ∗ symbol. All the other values are calculated at a lower resolution of 𝑁 side = 512 for comparison. The corresponding contour plots are shown in Figure A.2. The 𝜎 LN values for the log-normal host are converted to 𝜎 for better comparison. summary, the 𝜒 2 -test does not prefer a particular model as the data currently lacks constraining power. The best-fit values in Table 2 for the host model show that we converge on the same behaviour. Looking at the best fit values, we can see that the statistical properties of the LSS field are rather sub-dominant with the current FRB sample and if a log-normal distribution is used for the host contribution. If we assume a (truncated) Gaussian host contribution, it cannot explain the high DM of FRBs at low redshifts, Hence the model responds by artificially increasing the variance of the host contribution when the LSS is chosen to be Gaussian as well. In general, however, the constraints are all consistent with each other.",
"pages": [
9,
10
]
},
{
"title": "7. CONCLUSION",
"content": "In this paper, we have, for the first time, presented a simulation-based inference (SBI) analysis of the DM𝑧 relation, incorporating the appropriate statistical properties of the electron density field and the host contribution in forward simulations. We introduced a novel set of simulations for DM observ- ables, which can seamlessly incorporate any contribution to the DM along the line of sight. For the host contribution, we adopted a log-normal distribution as our fiducial setting, as it is widely accepted in the literature. However, we also implemented alternative functional forms of the host contributions, as this merely involves substituting a single function in the simulations. For the large-scale structure component, we utilised GLASS (Tessore et al. 2023), which enabled us to simulate the electron density as either a log-normal or a Gaussian field with the correct correlations up to a given spatial resolution, provided by an input power spectrum of the threedimensional electron field. The power spectrum was calculated using HMCODE (Mead et al. 2020; Tröster et al. 2022), which was fitted to hydrodynamical simulations to jointly fit the matter and gas power spectra using a halo model approach. As output, we obtained concentric shells of the electron overdensity field with a narrow width in redshift. FRBs were then placed in the electron density at their observed redshift and location from the real data (see below). After adding the stochastic host contribution, the line-of-sight integral was performed for each FRB and the Milky Way contribution was added. For the latter, we employed the standard method of using an electron model from prior literature (Cordes & Lazio 2002; Yao et al. 2017), which is, in the spirit of our analysis, also fully flexible. With this approach, we provided realistic simulated realisations of the DM given a cosmological model, which can be easily made more complex. In the next step, we performed SBI on the described simulations. This inference method requires no explicit likelihood and works by training an NN to learn either the posterior distribution (non-amortised) or the joint distribution of data and parameters (amortised). In this context, the shape of the likelihood is unconstrained, which is why this approach is often referred to as likelihood-free inference (though it still implicitly requires a likelihood). To demonstrate our pipeline, we applied it to 12 host-identified FRBs as a function of the cosmological amplitude of the DM𝑧 relation, A in Equation (7), and two host contribution parameters. For our fiducial model, we used a log-normal electron density field as well as a log-normal host model. We inferred A = 0 . 89 + 0 . 31 -0 . 29 , consistent with unity or the Planck cosmology. Similarly, the median and scale of the host distribution are DMhost = 223 + 108 -95 pc cm -3 and 𝜎 LN = 0 . 76 + 0 . 37 -0 . 44 . The values for these three parameters are consistent with previous findings, and we indeed found that no parameter (amplitude of the DM𝑧 relation, mean host contribution and its variance) is dominated by its prior, i.e. that there is additional information in the data and no parameter just assumes a flat posterior in the prior range. This is indicating that already 12 FRBs can inform us about the shape of the host contribution to some extent. The resulting posterior distributions were also assessed for consistency using standard coverage tests, specifically the TARP test. Our fiducial high-resolution case with a lognormal LSS and log-normal host component demonstrated perfect coverage, indicating that the learned posterior reproduces the coverage expected from random realisations from the simulator. Furthermore, we assessed the quality of the fits using a Bayesian goodness-of-fit measure based on Gelman et al. (1996), which was also employed in von Wietersheim-Kramsta et al. (2024). The 𝜒 2 was calculated from a number of data realisations generated from the simulations at the maximum a posteriori. This allows us to check whether the actual data is a plausible realisation of the likelihood. With the current data, we found all the models to be a good fit. One of the added benefits of the SBI pipeline is that we can test the Gaussian likelihood assumption frequently used in a more traditional Bayesian approach. To that regard, we investigated the shape of the likelihood, finding the expected long-tailed distribution towards high DM values, which is necessary to explain the large DMs observed at low redshifts, such as those seen in FRB 20190520B. Another important check involved examining the evolution of the likelihood with redshift. We found that the mean of the probability distribution function increases with redshift, as expected. However, we also observed a reduction in its scatter. Although this behaviour may seem counterintuitive at first, it is supported by a straightforward analytical calculation, which shows that the width of the total distribution indeed reaches a minimum at redshift 𝑧 < 1 for the parameter values assumed in this analysis. Lastly, we explored different modelling choices for both the LSS component and the host contribution. For the LSS component, we considered both Gaussian and log-normal distributions, while for the host contribution, we evaluated log-normal, truncated Gaussian and gamma distributions. We systematically tested all possible combinations of these models and found that they consistently provided similar constraints. To save computational time, we reduced the resolution of these simulations from 𝑁 side = 4048 to 𝑁 side = 512. We found that the TARP test indicates that the learned posterior in these cases is slightly biased. Importantly, none of the models showed any indication of being a poor fit to the data, which is mainly due to the low number of FRBs available with host identification at the moment. In conclusion, the simulations and inference pipeline we have developed integrate the precise physical and statistical properties needed to accurately infer the DM𝑧 relation, free from general assumptions about the likelihood or posterior. This approach is highly adaptable and scales effectively with an increasing number of FRBs, thanks to the implemented data compression techniques. Moreover, our simulations provide a robust foundation for investigating systematic effects in cosmological studies involving FRBs, as these can be seamlessly incorporated at the map level. Looking ahead, a key direction will be to include more FRBs without known redshifts, enabling a joint fit of the DM𝑧 properties alongside the statistical properties of the DM. The code for this work is publicly available via https://github.com/koustav-konar/FastNeuralBurst.",
"pages": [
10,
11
]
},
{
"title": "ACKNOWLEDGEMENTS",
"content": "SH was supported by the Excellence Cluster ORIGINS which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy-EXC-2094-390783311.",
"pages": [
11
]
},
{
"title": "REFERENCES",
"content": "Alsing J., Charnock T., Feeney S., Wandelt B., 2019, MNRAS, 488, 4440 Bannister K. W., et al., 2019, Science, 365, 565 Beaumont M. A., 2019, Annual Review of Statistics and Its Application, 6, 379 Bhandari S., et al., 2020, ApJL, 895, L37 Bhandari S., et al., 2022, ApJ, 163, 69 Bhattacharya M., Kumar P., Linder E. V., 2021, Phys. Rev. D, 103, 103526 Bishop C., 1994, NCRG Bochenek C. D., Ravi V., Belov K. V., Hallinan G., Kocz J., Kulkarni S. R., McKenna D. L., 2020, Nature, 587, 59 Chittidi J. S., et al., 2021, ApJ, 922, 173 Jimenez Rezende D., Mohamed S., 2015, arXiv e-prints, p. arXiv:1505.05770 doi:10.48550/arXiv.2402.00505, http://arxiv.org/abs/2402.00505 Kitching T. D., Heavens A. F., 2017, Phys. Rev. D, 95, 063522 Lemos P., Coogan A., Hezaveh Y., Perreault-Levasseur L., 2023, in Krause A., Brunskill E., Cho K., Engelhardt B., Sabato S., Scarlett J., eds, Proceedings of Machine Learning Research Vol. 202, Proceedings of the 40th International Conference on Machine Learning. PMLR, pp 19256-19273, https://proceedings.mlr.press/v202/lemos23a.html Leonard C. D., et al., 2023, The Open Journal of Astrophysics, 6, 8 Lewis A., Bridle S., 2002, Phys. Rev., D66, 103511 MNRAS, 524, 6167 ( arXiv:2101.04653 ), doi:10.48550/arXiv.2101.04653 Macquart J.-P., et al., 2020, Nature, 581, 391 Masui K. W., Sigurdson K., 2015, Phys. Rev. Lett., 115, 121301 Mead A. J., Peacock J. A., Heymans C., Joudaki S., Heavens A. F., 2015, MNRAS, 454, 1958 Newburgh L. B., et al., 2016, arXiv:1607.02059 [astro-ph 10.1117/12.2234286, p. 99065X Niu C. H., et al., 2022, Nature, 611, E10 Ocker S. K., et al., 2022, ApJ, 931, 87 Papamakarios G., Murray I., 2016, Advances in neural information processing systems, 29 Papamakarios G., Pavlakou T., Murray I., 2017, Advances in neural information processing systems, 30 Planck Collaboration et al., 2020, A&A, 641, A6 Pritchard J. K., Seielstad M. T., Perez-Lezaun A., Feldman M. W., 1999, Molecular biology and evolution, 16, 1791 Prochaska J. X., et al., 2019, Science, 366, 231 Rafiei-Ravandi M., et al., 2021, ApJ, 922, 42 Ravi V., et al., 2019, Nature, 572, 352 Reischke R., Hagstotz S., 2023a, MNRAS, 523, 6264 Reischke R., Hagstotz S., 2023b, MNRAS, 524, 2237 Reischke R., Neumann D., Bertmann K. A., Hagstotz S., Hildebrandt H., 2023, arXiv e-prints, p. arXiv:2309.09766 Rubin D. B., 1984, The Annals of Statistics, pp 1151-1172 Shirasaki M., Kashiyama K., Yoshida N., 2017, Phys. Rev. D, 95, 083012 Sisson S. A., Fan Y., Beaumont M., 2018, Handbook of approximate Bayesian computation. CRC Press",
"pages": [
11,
12
]
},
{
"title": "A. EFFECT OF REDUCING 𝑁 SIDE",
"content": "In Figure A.1, we show the effect of reducing the resolution of the GLASS forward simulation on the TARP coverage. This is necessary to establish the baseline for the comparison of the different models for the host and the LSS contribution, which have been run at a lower resolution. The main effect of decreasing 𝑁 side is that power on small scales gets washed out. In particular, the maximum multipole properly resolved is ℓ max = 3 𝑁 side -1. However, loss of power already occurs earlier. In terms of effects on the inference process, this means that the variance introduced by the LSS component decreases and therefore the scatter in the data will be larger than in the simulations. Therefore, the final constraints might be artificially tight. Figure A.1 demonstrates that the estimated 10.21105/astro.2302.01942 Theis A., Hagstotz S., Reischke R., Weller J., 2024, arXiv e-prints, p. arXiv:2403.08611 Thornton D., et al., 2013, Science, 341, 53 Tröster T., et al., 2022, A&A, 660, A27 Walters A., Weltman A., Gaensler B. M., Ma Y.-Z., Witzemann A., 2018, ApJ, 856, 65 Wu Q., Zhang G.-Q., Wang F.-Y., 2022, MNRAS, 515, L1 arXiv:2311.12530 Yamasaki S., Totani T., 2020, ApJ, 888, 105 Zhou B., Li X., Wang T., Fan Y.-Z., Wei D.-M., 2014, Phys. Rev. D, 89, 107303 Zieser B., Merkel P. M., 2016, MNRAS, 459, 1586 von Wietersheim-Kramsta M., Lin K., Tessore N., Joachimi B., Loureiro A., Reischke R., Wright A. H., 2024, arXiv e-prints, p. arXiv:2404.15402 posterior becomes slightly biased for decreasing 𝑁 side for the fiducial model. Since this will be true for all models and the fact that the high resolution run is unbiased and exact, we can safely do a one-to-one comparison of the different models using a lower resolution.",
"pages": [
12
]
},
{
"title": "B. RESULTS FROM DIFFERENT FITS",
"content": "In this section, we present a comprehensive analysis of the different figures resulting from variations in both the host contribution and the large-scale structure (LSS) contribution. Specifically, we display the final contour plots Figure A.2, the goodness-of-fit tests in Figure A.3 and the likelihoods in Figure A.4. Note that we 𝑁 side = 512 in this comparison to conserve computational resources. The reduced resolution underestimates the error from cosmic variance on individual measurements, leading to artificially tighter constraints. This issue has been addressed in our fiducial run, where a higher 𝑁 side was utilised to validate the pipeline, as described in the main text. The key conclusion from these analyses is that all combinations of the model are statistically consistent with one another and provide a satisfactory fit to the data. This is primarily due to the relatively small number of FRBs considered in this study, as well as the conservative error estimates and the expansive posterior volume resulting from the use of a three-parameter model. While these effects limit the current discriminative power, they ensure robustness in our findings. However, these tests will become increasingly critical as larger FRB samples become available in the future. The current coverage tests indicate that all posteriors exhibit a slight degree of bias when a Gaussian distribution is involved. Finally, for reference, we provide a list of the FRBs used in this analysis in Table A.1. This paper was built using the Open Journal of Astrophysics L A T E X template. The OJA is a journal which provides fast and easy peer review for new papers in the astro-ph section of the arXiv, making the reviewing process simpler for authors and referees alike. Learn more at http://astro.theo j.org .",
"pages": [
12
]
},
{
"title": "Table A.1",
"content": "a https://www.atnf.csiro.au/research/pulsar/ymw16/",
"pages": [
14
]
}
]
2024arXiv241007285D
https://arxiv.org/pdf/2410.07285.pdf
<document>
<section_header_level_1><location><page_1><loc_15><loc_82><loc_86><loc_88></location>Impact of Exoplanet Science on Society: Professional Contributions, Citizen Science Engagement and Public Perception</section_header_level_1>
<text><location><page_1><loc_15><loc_73><loc_79><loc_77></location>Hans-Jörg Deeg Instituto de Astrofísica de Canarias, C. Via Lactea S/N, La Laguna, Tenerife, Spain e-mail: hdeeg@iac.es</text>
<section_header_level_1><location><page_1><loc_15><loc_67><loc_24><loc_69></location>Abstract</section_header_level_1>
<text><location><page_1><loc_15><loc_55><loc_86><loc_64></location>The impact of exoplanet science on both the scientific community and on the general public is presented through various indicators and examples. It is estimated that about 3-4% of all refereed astronomy articles focus on exoplanets, and between 15-20% percent of current, and up to 25% of upcoming astronomy space missions are dedicated to exoplanet research. Also, about 15-20% of the science cases for large multi-purpose ground-based astronomical instruments involve exoplanet science.</text>
<text><location><page_1><loc_15><loc_42><loc_86><loc_53></location>Interactions between the scientific community and the public occur on several levels and play a crucial role in shaping the future of exoplanet science. The rise of citizen science platforms and the successes of coordinated observing projects involving amateur astronomers have engaged the public in meaningful scientific contributions, and contribute to some areas of discovery and characterization of exoplanet systems, for which several examples are given. These initiatives not only fuel public interest in the search for extraterrestrial life but also promote STEM education, broadening participation in science.</text>
<text><location><page_1><loc_15><loc_31><loc_86><loc_41></location>Lastly, the changing perception of the informed public about the existence of 'other Earths' and life in the Universe in the light of results from exoplanet science is outlined. Media coverage of results from exoplanet science has furthered the acceptance that extraterrestrial life, be it intelligent of not, is not rare in the Universe. The shift in perception that such life might be detected in a potentially not very distant future has, in turn, promoted public support for the research infrastructure necessary to sustain the growth of exoplanetology.</text>
<section_header_level_1><location><page_1><loc_15><loc_22><loc_28><loc_24></location>Introduction</section_header_level_1>
<text><location><page_1><loc_15><loc_16><loc_86><loc_21></location>Since the discovery of the first exoplanets in the 1990s, the field of exoplanet science, or 'exoplanetology,' has undoubtedly become a central component of astronomy and one of its most rapidly advancing areas. But now, well into the 21st century, how far has the impact of</text>
<text><location><page_2><loc_15><loc_81><loc_86><loc_88></location>exoplanet science reached, both in the professional domain and in the society at large? Which interactions with society has it generated? These questions are significant not only for researchers in the field of exoplanets, who seek to understand the broader impact of their work, but also for policymakers responsible for making high-level decisions regarding support for related research activities.</text>
<text><location><page_2><loc_15><loc_68><loc_86><loc_79></location>The first part of this chapter provides several indicators of exoplanet science's current role within professional astronomy. However, assessing the broader impact on the general public is more complex, with media playing a key role in shaping public perceptions. We present public engagement on multiple levels, beginning with the involvement of interested individuals in citizen science projects, followed by an examination of how media coverage influences the general public's perception of 'other worlds' and the possibility of life in the universe.</text>
<section_header_level_1><location><page_2><loc_15><loc_64><loc_71><loc_66></location>Impact of exoplanet science within professional astronomy .</section_header_level_1>
<text><location><page_2><loc_15><loc_56><loc_86><loc_63></location>Since the discovery of the first widely recognized exoplanet in 1995, it is without doubt that exoplanet research has become a significant field in astronomy. But how big has it become in absolute terms, and in relative terms, against the entire field of astronomy and astrophysics? Here we provide a few indicators that may quantify this issue.</text>
<figure>
<location><page_2><loc_26><loc_51><loc_28><loc_53></location>
</figure>
<text><location><page_2><loc_28><loc_51><loc_37><loc_53></location>refereed</text>
<figure>
<location><page_2><loc_27><loc_18><loc_74><loc_50></location>
</figure>
<text><location><page_3><loc_15><loc_84><loc_85><loc_88></location>Fig.1 Number of publications, in 3-year bins from 1989 to 2024, that contain the terms 'exoplanet' or 'extrasolar AND planet' in abstract or main-text. Data extracted from the astronomy collection of the Astrophysical Data System on 2024 Sept. 4.</text>
<text><location><page_3><loc_15><loc_70><loc_86><loc_81></location>Fig. 1 shows the number of scientific publications that contain the terms 'exoplanet' or 'extrasolar AND planet', be it in their abstract or their main text, from a search in the astronomy collection of the Astrophysical Data System (ADS). Starting from a rather constant level of 10-15 refereed papers per year during the 1980s and a moderate rise in the 1990's, the field started to take off in 1998 or 1999. This was followed by a constant increase in productivity during the first two decades of the 2000's, surpassing 1000 refereed papers per year in 2011 and 2400 papers in 2022.</text>
<text><location><page_3><loc_15><loc_58><loc_86><loc_69></location>To evaluate the relative productivity of exoplanet science within the whole field of astronomy, Fig. 2 shows the yearly fractions of refereed papers found by above search, versus all papers in ADS' astronomy collection. Until 1993, exoplanets were a marginal topic mentioned in less than 0.2% of all papers. This changed after the discovery of the first exoplanets (around pulsars in 1992 and main-sequence stars in 1995). Since the late 1990s, a roughly linear increase has set in, and in the mid 2020s, this fraction is approaching 8%. Exoplanets are therefore truly a science of the 21 st century!</text>
<figure>
<location><page_3><loc_27><loc_21><loc_74><loc_53></location>
<caption>Fig. 2 Fraction of refereed publications that contain the terms 'exoplanet' or 'extrasolar AND planet' in abstract or main-text, relative to all refereed publications in the astronomy collection of ADS.</caption>
</figure>
<text><location><page_4><loc_15><loc_76><loc_86><loc_88></location>Among the exoplanet-papers found in this search, many of them mention exoplanets however only peripherally, and only an estimated 40-50% have a primary focus on exoplanets. This fraction was obtained from a manual revision of the titles (and abstract, if needed) of the last 100 papers from 2023 that were returned by the search, from which only 38 were deemed to focus on exoplanets or on closely related analytical or instrumental techniques. An identical exercise with 100 publications from the year 2016 found 55 papers to focus on exoplanets (Deeg & Belmonte 2018). In conclusion, about 3 - 4% of the current technical works in astronomy are focused onto - or driven by - exoplanet science.</text>
<text><location><page_4><loc_15><loc_62><loc_86><loc_75></location>A further indicator on the impact of exoplanetology within the entire field of astronomy might be the number of conferences related to exoplanets. A review of the titles of all IAU Symposia of the last 10 years (Symposia 306 to 394, covering May 2013 to August 2024) shows that eight of these symposia (about 9%) treated exoplanets as a main topic. Similarly, an analysis of large meetings covering all of astronomy during the 2010's (IAU General Assemblies, American Astronomical Society meetings, European Astronomical Society meetings) showed that about 10% of such meetings were dedicated to exoplanets (Deeg & Belmonte 2018).</text>
<text><location><page_4><loc_15><loc_56><loc_86><loc_61></location>One of the most important consequences of exoplanet science is its influence onto advanced astrophysics instrumentation, both ground and space-based. This is also an indicator of current and future funding that is expended for this science.</text>
<text><location><page_4><loc_15><loc_47><loc_86><loc_54></location>Only relatively small ground based facilities are primarily dedicated to exoplanet science (mainly several transit surveys), whereas exoplanets have been a strong driver in the large expenditures needed for space missions. To date (2024), the following missions with a primary dedication to exoplanets have been put into operation (see chapter -> Space Missions for Extrasolar Planets: Overview and Introduction):</text>
<unordered_list>
<list_item><location><page_4><loc_18><loc_42><loc_85><loc_45></location>-CoRoT (active 2007-2012), led by the French space agency CNES, a dual exoplanetdetection and asteroseismology mission.</list_item>
<list_item><location><page_4><loc_18><loc_39><loc_86><loc_42></location>-Kepler (2009-2012, relabeled as K2 until 2018) by NASA, the most successful exoplanet finder to date.</list_item>
<list_item><location><page_4><loc_18><loc_36><loc_86><loc_39></location>-TESS (currently active, launched 2018) by NASA, an all-sky survey to detect exoplanets.</list_item>
<list_item><location><page_4><loc_18><loc_33><loc_86><loc_36></location>-CHEOPS (currently active, launched 2019) by ESA, primarily dedicated to exoplanet follow-up.</list_item>
</unordered_list>
<text><location><page_4><loc_15><loc_21><loc_86><loc_31></location>Of significant impact in exoplanet science, but not originally or primarily designed for it, are also the Hubble Space Telescope (launched 1990), the Spitzer space telescope (main phase 2003 - 2009, with limited capabilities until 2020), the Gaia mission (launched 2013 and expected to revolutionize astrometric detections of exoplanets), and of course the James Webb Space Telescope (launched 2021), with two of its four principal science cases being strongly related to exoplanets: the birth of Stars & Protoplanetary Systems and Planets & Origins of Life (https://jwst.nasa.gov/science.html).</text>
<text><location><page_4><loc_15><loc_16><loc_86><loc_19></location>In order to place the number of exoplanet-related space missions into context against all operating astrophysics-related missions, NASA's astrophysics program (NASA 2024)</text>
<text><location><page_5><loc_15><loc_81><loc_86><loc_88></location>currently lists 14 missions in active status (as operating or extended missions), while ESA maintains 6 operational astrophysics missions (ESA 2024). Depending on the extent to which multi-purpose missions like JWST are considered to be dedicated to exoplanets, we may estimate that 15 - 20 % of currently active astrophysics space missions are dedicated to exoplanets.</text>
<text><location><page_5><loc_15><loc_59><loc_86><loc_79></location>Regarding missions in development, ESA indicates 10 missions with this status. Most of them are small missions, but it is of note that its two upcoming mid-sized ('M') missions are both dedicated to exoplanets (PLATO for launch in 2026 and Ariel in 2029); exoplanets represent therefore a large fraction of ESA's current efforts. NASA lists 11 astrophysics missions in the implementation phase, of which the Nancy Grace Roman Space Telescope (launch 2026) is partially dedicated to exoplanets; their only fully exoplanet-dedicated mission is however the ESA-led Ariel (which has a NASA contribution). The Chinese Academy of Science advances the Earth 2.0 or ET transit survey with an expected launch in 2028, which will re-observe the Kepler-field (Ge et al. 2022, 2024); little information is however available about China's wider astrophysics-related space programme. Overall, we may expect that exoplanet science is driving a similar or somewhat larger fraction of upcoming (with secured launch) astrophysics space missions than currently in operation, estimated at 20 - 25 %.</text>
<text><location><page_5><loc_15><loc_55><loc_85><loc_58></location>Lastly, exoplanet-related high-level science cases for the largest actual projects for groundbased multi-purpose astronomical instrumentation are indicated:</text>
<unordered_list>
<list_item><location><page_5><loc_18><loc_50><loc_86><loc_53></location>∞ Atacama Large Millimeter Array (ALMA): Planet-forming disks is given as one of six science themes (https://almascience.nrao.edu/alma-science)</list_item>
<list_item><location><page_5><loc_18><loc_47><loc_85><loc_50></location>∞ European Extremely Large Telescope (E-ELT): Exoplanets - Towards other Earths is one of its six principal cases (Kissler-Patig & Lyubenova 2011)</list_item>
<list_item><location><page_5><loc_18><loc_43><loc_86><loc_47></location>∞ Thirty Meter Telescope (TMT): The Birth and Early Lives of Stars and Planets and Exoplanets are two out of nine of its science cases (Skidmore 2015)</list_item>
<list_item><location><page_5><loc_18><loc_39><loc_86><loc_43></location>∞ Square kilometre array (SKA): Seeking the origins of life , with studies on planet and star formation and also of SETI, is one of nine science goals of this radio telescope project (https://www.skao.int/en/explore/science-goals)</list_item>
<list_item><location><page_5><loc_18><loc_32><loc_86><loc_39></location>∞ Vera C. Rubin Observatory (formerly LSST): No specific exoplanet science goal is listed (https://rubinobservatory.org/explore/science-goals), but its large-scale surveys may contribute to the discovery of transiting exoplanets (Lund et al. 2015, Tamburo et al. 2023).</list_item>
</unordered_list>
<text><location><page_5><loc_15><loc_28><loc_86><loc_31></location>From this, a dedication to exoplanet science of about 15% for these major ground facilities can be estimated.</text>
<text><location><page_5><loc_15><loc_17><loc_86><loc_26></location>In summary, exoplanet science currently contributes 3-4% of all publications within the broader field of astrophysics, with this share continuing to grow. It also accounts for approximately 15-20% of ongoing ground and space instrumentation projects, and potentially a higher proportion (20-25%) of upcoming space missions. This significant contribution from a relatively young field reflects the perceived potential of exoplanet science, not only within professional astronomy but also among the informed public and</text>
<text><location><page_6><loc_15><loc_84><loc_86><loc_88></location>policymakers. The focus on the discovery of Earth-like planets and, more broadly, the search for signs of life beyond Earth plays a key role in driving public interest and securing political and financial support.</text>
<section_header_level_1><location><page_6><loc_15><loc_76><loc_48><loc_78></location>Exoplanets and the general public</section_header_level_1>
<text><location><page_6><loc_15><loc_69><loc_86><loc_75></location>Assessing the impact of exoplanet science onto the broader population is more challenging due to the lack of quantitative indicators. In this section, we highlight contexts where exoplanet research reaches and engages the informed public-a group with awareness of natural sciences and capable of understanding fundamental scientific concepts.</text>
<section_header_level_1><location><page_6><loc_15><loc_65><loc_28><loc_66></location>Citizen Science</section_header_level_1>
<text><location><page_6><loc_15><loc_54><loc_86><loc_63></location>Several levels of involvement by non-professional actors may be encompassed as citizen science. Among this, we include also activities oriented towards outreach or within preacademic educational stages, where astronomy-related activities are frequently offered with the aim make young people interested in wider further STEM-related fields (science, technology, engineering and mathematics) and in the related careers (Yüzgeç & Okuşluk 2023, Onuchukwu et al. 2024).</text>
<text><location><page_6><loc_15><loc_49><loc_86><loc_54></location>The involvement by citizen scientists may go from short interactions, like the installation of a screensaver for some distributed computing project, to a very serious level that approaches the work of professionals, such as displayed by some advanced amateur astronomers.</text>
<text><location><page_6><loc_15><loc_28><loc_86><loc_48></location>Distributed computing and analysis projects are one of the most common involvements of citizens in actual science projects. On the most basic level, also known as volunteer computing, a citizen lends private computing power to a science project employing distributed computing. The project seeks to solve a problem which is difficult or infeasible to tackle using other methods, but it does not require further citizen interaction besides the installation of a software running on a networked computer. A historical example of this was SETI@home, a search for signals from extra-terrestrial intelligences in radio data obtained by SETI projects at the Arecibo and Green Bank radio telescopes. SETI@home was released to the public in 1999 and active until 2020, and its iconic screensaver (Fig. 3) was a common sight on many computers in the first years of the 21 st century. Non-interactive volunteer computing projects may have however declined in appeal, partly because advancements in computational power have reduced the need for them, and also because they offer limited educational benefits.</text>
<figure>
<location><page_7><loc_16><loc_63><loc_72><loc_87></location>
<caption>Fig 3 : The SETi@home Classic screensaver and distributed computing project in its original ('classic') version. (Source Wikimedia, SETI@HOME is licensed under the GNU General Public License)</caption>
</figure>
<text><location><page_7><loc_15><loc_23><loc_86><loc_55></location>On the next level of citizen involvement are projects that are computer-based but require active participation. Typically, repetitive tasks that are difficult for computer codes but easy for the human brain are being distributed to the participants. The Zooniverse (www.zooniverse.org; Lintot 2019), which has been a pioneer of this concept, currently (Sept. 2024) hosts 84 projects from a variety of disciplines, mostly involving classification tasks. A fairly large fraction of them, 21, are related to space sciences of with the following being more directly about exoplanet systems: 'Exoasteroids' - a search for asteroids around white dwarf stars, 'Disk detective' - a search and analysis of circumstellar, and potentially protoplanetary disks, and 'Planet Hunters NGTS', which is the latest incarnation of the successful Planet Hunters project. The first version of Planet Hunters was launched in 2010 by a team from Yale University, where participants were employed to spot transit-like features in chunks of light curves from the Kepler space mission. In a second phase, they were also tasked to inspect and confirm the found candidates, with an assessment if further follow-up is warranted (Schwamb et al. 2012). Planet Hunters was later modified to analyze data from the K2 (Kepler's successor) and TESS space missions, while it is currently focused on the ground-based Next Generation Transit Survey (see chapter 'Transit Photometry as an Exoplanet Discovery Method' for an overview over transit detection projects). Planet Hunters has been one of the most successful citizen science project in exoplanetology; it has contributed to the discovery of over 10 exoplanets, of which two are named after it: PH1 b and PH2b, with PH1 b being a relatively rare circumbinary planet (Schwamb et al. 2013). A further noteworthy project is the Exoplanet Explorer (with uncertain status in 2024, listed</text>
<text><location><page_7><loc_15><loc_17><loc_86><loc_23></location>in Zooniverse as 'out of data'), which also analyzed data from the K2 mission. With over 15000 volunteers it aided in the discovery of several planet systems, including one with five transiting planets, named K2-138 (Christiansen et al. 2018). The most impacting result of such projects to date might however have been the discovery of a mysterious object with very</text>
<text><location><page_8><loc_15><loc_76><loc_86><loc_88></location>unusual brightness variations (Fig. 4 ) by members of the Planet Hunters project (Boyajian et al. 2016). A variety of explanations have been brought forward (see Wright & Sigurðsson 2016 for an overview), including very speculative ones involving extra-terrestrial intelligences, which led to significant media attention. The discovery of this object, also known as 'Boyajian's star', has been a wonderful result of the power of citizen science: its light curve had been discarded by the algorithms that previously had sifted through Kepler data, and only through the viewing by real persons was its strange nature recognized, leading to one of the strangest puzzles in current-day astronomy.</text>
<text><location><page_8><loc_15><loc_65><loc_86><loc_76></location>Further resources on space-related citizen-science projects, we also refer to dedicated websites at NASA ( https://exoplanets.nasa.gov/citizen-science/) and ESA ( https://www.esa.int/Enabling_Support/Preparing_for_the_Future/Space_for_Earth/Citizen_ science ). Data from upcoming exoplanet-related missions, like PLATO, the Roman Space Telescope or the Habitable Worlds Observatory will also become available to both the scientific community and to the public, and will provide ample opportunities for future citizen science involvement.</text>
<figure>
<location><page_8><loc_17><loc_43><loc_84><loc_61></location>
<caption>Fig 4 : The unusual light curve of Boyajian's star (KIC 8462852) which was discovered by Citizen Scientists in data of the Kepler space mission. (Figure by JohnPassos - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=46685223 )</caption>
</figure>
<text><location><page_8><loc_15><loc_17><loc_86><loc_31></location>Amateur astronomers in their classical form, with their proper observations of the sky, are also contributing significantly to exoplanet science. In contrary to expectations prior to the discovery of the first transiting planets, exoplanet observations do not exclusively require large professional equipment, instead, small instruments accessible to amateurs may be useful as well. Indeed, several pioneering projects to search for transiting planets were based on amateur equipment - although executed by professionals - and one of the two groups that simultaneously detected the first exoplanet transits, of HD209458 b, used a small 10 cm telescope (Charbonneau et al. 2000). This discovery gave immediately rise to many successful transits observations by amateurs, for which the Exoplanet Transits Database</text>
<text><location><page_9><loc_15><loc_75><loc_86><loc_88></location>(ETD, Poddany et al. 2010, http://var2.astro.cz/ETD) and the database of the American Association of Variable Star observer (AAVSO, https://www.aavso.org/ ) have become the principal repositories. Amateur observations of transits have been aided by the availability of detailed observing instructions (e.g. Gary 2010, Dennis 2016) and by freely available software geared towards differential precision photometry by non-professionals, such as AstroImageJ (https://www.astro.louisville.edu/software/astroimagej, Collins et al. 2017), CMunipack (Motl 2024) or SIRIL (Team-FreeAstro 2024). Some of the citizen science projects have also developed their specific analysis software, and many light curves from amateurs are good matches to data taken by professional observatories (Fig. 5 ).</text>
<text><location><page_9><loc_15><loc_38><loc_86><loc_73></location>The main driver for amateur exoplanet observations is the follow-up of known transiters, with the principal objective being the maintenance or refinement of transit ephemeris. Many exoplanets that were discovered from observations made over shorter time spans (several months or less), like those from CoRoT and most from TESS, have transit-ephemeris with a severely limited precision. Within a few years, transit times will then accumulate uncertainties so big (more than 2-3 hours) that further transit observations cannot be planned well. Prior transit re-observations are hence used to generate improved ephemeris, which in turn permit efficient reobservations by future ground and space-based instrumentations over time-scales of years to decades (Dragomir et al. 2020, Deeg et al. 2020, Zellem et al., 2020). The importance of ephemeris maintenance and refinement is epitomized by the automatic ephemeris fitting in ETD, which is based on quality-weighted sets of the uploaded transit observations. Amateur data have been taken into account in several publications presenting improved transit ephemeris; e.g. for planets found by CoRoT (Klagyivik et al. 2021), TESS (Peluso et al. 2023) or WASP (Noguer et al. 2024). Transit re-observations may also give rise to the detection of transit timing variations (TTVs) - these are deviations from strict periodicity that indicate non-linear ephemeris. TTVs may be due to the presence of further orbiting bodies and/or relevant star-planet interactions; they may also be used to estimate the masses of transiting planets (see chapter Transit-Timing and Duration Variations for the Discovery and Characterization of Exoplanets). An example is CoRoT-11 b, which indicates TTVs from both amateur and professional observations (timings in ETD from 2011- 2024 and Deeg et al. 2020). ETD also tracks variations in transit depths and durations, although conclusive evidence for the presence such variations based on amateur data is more difficult to obtain, given that these parameters are very sensitive to data quality.</text>
<text><location><page_9><loc_15><loc_28><loc_86><loc_37></location>A selection of observational citizen science projects that pursue these and related topics are currently Exoplanet Watch (https://exoplanets.nasa.gov/exoplanet-watch), the Unistellar Network (https://science.unistellar.com/, Peluso et al. 2023) - a private /public partnership supplying also suitably fitted small telescopes (Marchis et al. 2020) - and Exoclock (https://www.exoclock.space/, Kokori et al. 2022, 2023), with the latter one dedicated to the ephemeris-refinement of target planets for the future ARIEL space mission.</text>
<text><location><page_9><loc_15><loc_16><loc_86><loc_27></location>A further driver for amateur involvement is also the identification of false alarms among planet candidates. Such identifications are needed if the photometry from the discoverysurvey (be it ground or space-based) is based on imaging of low spatial resolution. In this case, follow-up photometry from higher resolution imaging is needed, in order to distinguish if a planet-candidate displays a transit on the target star, or if a deeper eclipse appears on a nearby star, which means the presence of a false alarm. It is of note that such follow-up observations are also meaningful for candidates with transits that are too shallow to be</text>
<text><location><page_10><loc_15><loc_72><loc_86><loc_88></location>detected again, as long as the presence of eclipses at all nearby stars can be excluded with certainty. Such work by amateurs was already included in the verification of an exoplanet by the ground-based XO planet survey (McCullough et al. (2007), while more detailed theoretical and observational foundations for these follow-up observations are described by Deeg et al. (2009). For candidates from the TESS mission, the aforementioned Unistellar Network and others are currently pursuing false alarm identifications with the help of citizen observers; see Sgro et al (2024) for a successful example. For the upcoming PLATO transit survey mission (Rauer et al. 2010, 2024 and chapter Space Missions for Exoplanet Science: PLATO), to be launched in 2026, a dedicated program for ground-based photometric followup is currently prepared, with branches for professional and for amateur observers, including outreach organizations (Deeg & Alonso 2024, https://citizen.plato-planets.at/ ).</text>
<text><location><page_10><loc_15><loc_55><loc_86><loc_70></location>Proposed further objectives of amateur exoplanet observations are also the confirmation of long-periodic planets (by re-observation of their transits), the search for transits of planets that are currently only known from radial velocity detections, or the pursuing of microlensing events caused by exoplanets (Peluso et al. 2023). Boyajian's star, the prominent discovery made from Kepler light curves by citizen scientists, has also become a target of intense surveillance by amateur observers: the database of the AAVSO lists presently (2024 Oct 2) 137 345 observations by 137 observers, with new observations arriving most of the days. While photometric observations are clearly the main subject of amateur involvement in exoplanetology, some private initiatives for SETI searches in the optical (e.g. Schuetz 2018) and radio domains (Project Bambi, www.bambi.net/) are also of note.</text>
<figure>
<location><page_10><loc_17><loc_22><loc_83><loc_48></location>
<caption>Fig. 5 . Amateur light curve of a planet transit across the 11.3 mag star XO-1, made with a 14-inch telescope. From Gary (2010), reproduced with permission by the author.</caption>
</figure>
<section_header_level_1><location><page_11><loc_15><loc_82><loc_64><loc_84></location>The Public's View About the Presence of Life in the Universe</section_header_level_1>
<text><location><page_11><loc_15><loc_76><loc_86><loc_81></location>The general public learns about exoplanets through a variety of sources. How has the input from exoplanet science influenced the public's perceptions of Earth's uniqueness in the universe and shaped views on the potential universality of life?</text>
<text><location><page_11><loc_15><loc_53><loc_86><loc_75></location>From a historic perspective, our view about 'other worlds' away from the Sun or about life in the Universe has changed from one of complete speculation to a semi-empirical one. As the best-known tool for this attempted quantification remains Drake's famous equation (Drake 1965, 2011). This equation, whose original form estimates the abundance of detectable intelligent life, is based on the multiplication of several factors, of which only one, the rate of star formation in our Galaxy, could be estimated reasonably well when the equation was originally presented. Other factors, such as the fraction of planets with life that develop technological civilizations, or the length of time over which such civilizations release detectable signals, remain to date entirely speculative. Our increasing knowledge on exoplanets has however raised two of the equation's factors, namely the fraction of stars with planets, and the number of habitable planets around such stars, to estimates that may be reliable to a factor of 'a few'. These estimates constitute one of the major advances of the past 20 years of research on exoplanets, leading for example to an estimate (Petigura et al. 2013) that about 6% of Sun-like stars have Earth-like planets .</text>
<text><location><page_11><loc_15><loc_30><loc_86><loc_51></location>Currently we are able to label planets as 'Earth-like' solely based on their principal physical parameters without crucial information for habitability, such as the presence of water or the type of atmosphere. The quoted 6% is therefore an upper limit for habitable Earth-like planets. The habitability of planets that are not Earth-like is hypothetical but in many cases reasonable (e.g. on planets around low-mass stars or on planet-sized moons); such objects might even provide the vast majority of habitats in the Universe (see also chapters 'Habitability of Planets in Binary Star Systems', and 'Habitability in Brown Dwarf Systems'). In either case is seems safe to assume that habitable planets (in the sense of fulfilling all requirements to develop life) are frequent, and that life - evolved or not - will be frequent as well, unless the unknown factors of Drake's equation are minuscule small. In that sense, Frank & Sullivan (2016) estimate that there is at least one other technological species in the observable Universe, unless the probability that a habitable zone planet develops a technological species is below 10 -24 (see also chapter Where Life May Arise: Habitability).</text>
<text><location><page_11><loc_15><loc_21><loc_86><loc_28></location>These more general results from exoplanet science, together with the numerous individual findings of potentially habitable planets have led to the perception among the informed public that life in the Universe might be frequent and that its discovery is a question of "when" rather than "if". For instance, a survey in 2020 found that nearly 65% of Americans believe there is intelligent life on other planets (Kennedy & Lau 2021).</text>
<text><location><page_11><loc_15><loc_16><loc_86><loc_20></location>At the same time, activities for more direct detections of extraterrestrial life have come within technological reach. Not only classical SETI searches for technological signatures, but also searches for biosignatures on planets but within and outside of our Solar System. Such</text>
<text><location><page_12><loc_15><loc_79><loc_86><loc_88></location>investigations are supported by a wide interest among the public, which is also essential for the support of the many ambitious projects in the coming decades. These projects in turn will advance the entire field of exoplanet science and humanity's understanding of the origin and presence of life in the Universe. And lastly, the public interest and fascination with current findings from exoplanets science may better prepare society for the day when we find out that we are not alone.</text>
<section_header_level_1><location><page_12><loc_15><loc_73><loc_34><loc_75></location>Acknowledgements</section_header_level_1>
<text><location><page_12><loc_15><loc_66><loc_86><loc_72></location>The author acknowledges support from the Spanish Research Agency of the Ministry of Science and Innovation (AEI-MICINN) under grant PID2019-107061GB-C66, DOI: 10.13039/501100011033. This publication has made use of the Astrophysics Data System, funded by NASA under Cooperative Agreement 80NSSC21M00561.</text>
<section_header_level_1><location><page_12><loc_15><loc_60><loc_26><loc_61></location>References</section_header_level_1>
<table>
<location><page_12><loc_15><loc_17><loc_85><loc_58></location>
</table>
<table>
<location><page_13><loc_15><loc_17><loc_85><loc_88></location>
</table>
<table>
<location><page_14><loc_15><loc_38><loc_85><loc_88></location>
</table>
</document>
[
{
"title": "Impact of Exoplanet Science on Society: Professional Contributions, Citizen Science Engagement and Public Perception",
"content": "Hans-Jörg Deeg Instituto de Astrofísica de Canarias, C. Via Lactea S/N, La Laguna, Tenerife, Spain e-mail: hdeeg@iac.es",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "The impact of exoplanet science on both the scientific community and on the general public is presented through various indicators and examples. It is estimated that about 3-4% of all refereed astronomy articles focus on exoplanets, and between 15-20% percent of current, and up to 25% of upcoming astronomy space missions are dedicated to exoplanet research. Also, about 15-20% of the science cases for large multi-purpose ground-based astronomical instruments involve exoplanet science. Interactions between the scientific community and the public occur on several levels and play a crucial role in shaping the future of exoplanet science. The rise of citizen science platforms and the successes of coordinated observing projects involving amateur astronomers have engaged the public in meaningful scientific contributions, and contribute to some areas of discovery and characterization of exoplanet systems, for which several examples are given. These initiatives not only fuel public interest in the search for extraterrestrial life but also promote STEM education, broadening participation in science. Lastly, the changing perception of the informed public about the existence of 'other Earths' and life in the Universe in the light of results from exoplanet science is outlined. Media coverage of results from exoplanet science has furthered the acceptance that extraterrestrial life, be it intelligent of not, is not rare in the Universe. The shift in perception that such life might be detected in a potentially not very distant future has, in turn, promoted public support for the research infrastructure necessary to sustain the growth of exoplanetology.",
"pages": [
1
]
},
{
"title": "Introduction",
"content": "Since the discovery of the first exoplanets in the 1990s, the field of exoplanet science, or 'exoplanetology,' has undoubtedly become a central component of astronomy and one of its most rapidly advancing areas. But now, well into the 21st century, how far has the impact of exoplanet science reached, both in the professional domain and in the society at large? Which interactions with society has it generated? These questions are significant not only for researchers in the field of exoplanets, who seek to understand the broader impact of their work, but also for policymakers responsible for making high-level decisions regarding support for related research activities. The first part of this chapter provides several indicators of exoplanet science's current role within professional astronomy. However, assessing the broader impact on the general public is more complex, with media playing a key role in shaping public perceptions. We present public engagement on multiple levels, beginning with the involvement of interested individuals in citizen science projects, followed by an examination of how media coverage influences the general public's perception of 'other worlds' and the possibility of life in the universe.",
"pages": [
1,
2
]
},
{
"title": "Impact of exoplanet science within professional astronomy .",
"content": "Since the discovery of the first widely recognized exoplanet in 1995, it is without doubt that exoplanet research has become a significant field in astronomy. But how big has it become in absolute terms, and in relative terms, against the entire field of astronomy and astrophysics? Here we provide a few indicators that may quantify this issue. refereed Fig.1 Number of publications, in 3-year bins from 1989 to 2024, that contain the terms 'exoplanet' or 'extrasolar AND planet' in abstract or main-text. Data extracted from the astronomy collection of the Astrophysical Data System on 2024 Sept. 4. Fig. 1 shows the number of scientific publications that contain the terms 'exoplanet' or 'extrasolar AND planet', be it in their abstract or their main text, from a search in the astronomy collection of the Astrophysical Data System (ADS). Starting from a rather constant level of 10-15 refereed papers per year during the 1980s and a moderate rise in the 1990's, the field started to take off in 1998 or 1999. This was followed by a constant increase in productivity during the first two decades of the 2000's, surpassing 1000 refereed papers per year in 2011 and 2400 papers in 2022. To evaluate the relative productivity of exoplanet science within the whole field of astronomy, Fig. 2 shows the yearly fractions of refereed papers found by above search, versus all papers in ADS' astronomy collection. Until 1993, exoplanets were a marginal topic mentioned in less than 0.2% of all papers. This changed after the discovery of the first exoplanets (around pulsars in 1992 and main-sequence stars in 1995). Since the late 1990s, a roughly linear increase has set in, and in the mid 2020s, this fraction is approaching 8%. Exoplanets are therefore truly a science of the 21 st century! Among the exoplanet-papers found in this search, many of them mention exoplanets however only peripherally, and only an estimated 40-50% have a primary focus on exoplanets. This fraction was obtained from a manual revision of the titles (and abstract, if needed) of the last 100 papers from 2023 that were returned by the search, from which only 38 were deemed to focus on exoplanets or on closely related analytical or instrumental techniques. An identical exercise with 100 publications from the year 2016 found 55 papers to focus on exoplanets (Deeg & Belmonte 2018). In conclusion, about 3 - 4% of the current technical works in astronomy are focused onto - or driven by - exoplanet science. A further indicator on the impact of exoplanetology within the entire field of astronomy might be the number of conferences related to exoplanets. A review of the titles of all IAU Symposia of the last 10 years (Symposia 306 to 394, covering May 2013 to August 2024) shows that eight of these symposia (about 9%) treated exoplanets as a main topic. Similarly, an analysis of large meetings covering all of astronomy during the 2010's (IAU General Assemblies, American Astronomical Society meetings, European Astronomical Society meetings) showed that about 10% of such meetings were dedicated to exoplanets (Deeg & Belmonte 2018). One of the most important consequences of exoplanet science is its influence onto advanced astrophysics instrumentation, both ground and space-based. This is also an indicator of current and future funding that is expended for this science. Only relatively small ground based facilities are primarily dedicated to exoplanet science (mainly several transit surveys), whereas exoplanets have been a strong driver in the large expenditures needed for space missions. To date (2024), the following missions with a primary dedication to exoplanets have been put into operation (see chapter -> Space Missions for Extrasolar Planets: Overview and Introduction): Of significant impact in exoplanet science, but not originally or primarily designed for it, are also the Hubble Space Telescope (launched 1990), the Spitzer space telescope (main phase 2003 - 2009, with limited capabilities until 2020), the Gaia mission (launched 2013 and expected to revolutionize astrometric detections of exoplanets), and of course the James Webb Space Telescope (launched 2021), with two of its four principal science cases being strongly related to exoplanets: the birth of Stars & Protoplanetary Systems and Planets & Origins of Life (https://jwst.nasa.gov/science.html). In order to place the number of exoplanet-related space missions into context against all operating astrophysics-related missions, NASA's astrophysics program (NASA 2024) currently lists 14 missions in active status (as operating or extended missions), while ESA maintains 6 operational astrophysics missions (ESA 2024). Depending on the extent to which multi-purpose missions like JWST are considered to be dedicated to exoplanets, we may estimate that 15 - 20 % of currently active astrophysics space missions are dedicated to exoplanets. Regarding missions in development, ESA indicates 10 missions with this status. Most of them are small missions, but it is of note that its two upcoming mid-sized ('M') missions are both dedicated to exoplanets (PLATO for launch in 2026 and Ariel in 2029); exoplanets represent therefore a large fraction of ESA's current efforts. NASA lists 11 astrophysics missions in the implementation phase, of which the Nancy Grace Roman Space Telescope (launch 2026) is partially dedicated to exoplanets; their only fully exoplanet-dedicated mission is however the ESA-led Ariel (which has a NASA contribution). The Chinese Academy of Science advances the Earth 2.0 or ET transit survey with an expected launch in 2028, which will re-observe the Kepler-field (Ge et al. 2022, 2024); little information is however available about China's wider astrophysics-related space programme. Overall, we may expect that exoplanet science is driving a similar or somewhat larger fraction of upcoming (with secured launch) astrophysics space missions than currently in operation, estimated at 20 - 25 %. Lastly, exoplanet-related high-level science cases for the largest actual projects for groundbased multi-purpose astronomical instrumentation are indicated: From this, a dedication to exoplanet science of about 15% for these major ground facilities can be estimated. In summary, exoplanet science currently contributes 3-4% of all publications within the broader field of astrophysics, with this share continuing to grow. It also accounts for approximately 15-20% of ongoing ground and space instrumentation projects, and potentially a higher proportion (20-25%) of upcoming space missions. This significant contribution from a relatively young field reflects the perceived potential of exoplanet science, not only within professional astronomy but also among the informed public and policymakers. The focus on the discovery of Earth-like planets and, more broadly, the search for signs of life beyond Earth plays a key role in driving public interest and securing political and financial support.",
"pages": [
2,
3,
4,
5,
6
]
},
{
"title": "Exoplanets and the general public",
"content": "Assessing the impact of exoplanet science onto the broader population is more challenging due to the lack of quantitative indicators. In this section, we highlight contexts where exoplanet research reaches and engages the informed public-a group with awareness of natural sciences and capable of understanding fundamental scientific concepts.",
"pages": [
6
]
},
{
"title": "Citizen Science",
"content": "Several levels of involvement by non-professional actors may be encompassed as citizen science. Among this, we include also activities oriented towards outreach or within preacademic educational stages, where astronomy-related activities are frequently offered with the aim make young people interested in wider further STEM-related fields (science, technology, engineering and mathematics) and in the related careers (Yüzgeç & Okuşluk 2023, Onuchukwu et al. 2024). The involvement by citizen scientists may go from short interactions, like the installation of a screensaver for some distributed computing project, to a very serious level that approaches the work of professionals, such as displayed by some advanced amateur astronomers. Distributed computing and analysis projects are one of the most common involvements of citizens in actual science projects. On the most basic level, also known as volunteer computing, a citizen lends private computing power to a science project employing distributed computing. The project seeks to solve a problem which is difficult or infeasible to tackle using other methods, but it does not require further citizen interaction besides the installation of a software running on a networked computer. A historical example of this was SETI@home, a search for signals from extra-terrestrial intelligences in radio data obtained by SETI projects at the Arecibo and Green Bank radio telescopes. SETI@home was released to the public in 1999 and active until 2020, and its iconic screensaver (Fig. 3) was a common sight on many computers in the first years of the 21 st century. Non-interactive volunteer computing projects may have however declined in appeal, partly because advancements in computational power have reduced the need for them, and also because they offer limited educational benefits. On the next level of citizen involvement are projects that are computer-based but require active participation. Typically, repetitive tasks that are difficult for computer codes but easy for the human brain are being distributed to the participants. The Zooniverse (www.zooniverse.org; Lintot 2019), which has been a pioneer of this concept, currently (Sept. 2024) hosts 84 projects from a variety of disciplines, mostly involving classification tasks. A fairly large fraction of them, 21, are related to space sciences of with the following being more directly about exoplanet systems: 'Exoasteroids' - a search for asteroids around white dwarf stars, 'Disk detective' - a search and analysis of circumstellar, and potentially protoplanetary disks, and 'Planet Hunters NGTS', which is the latest incarnation of the successful Planet Hunters project. The first version of Planet Hunters was launched in 2010 by a team from Yale University, where participants were employed to spot transit-like features in chunks of light curves from the Kepler space mission. In a second phase, they were also tasked to inspect and confirm the found candidates, with an assessment if further follow-up is warranted (Schwamb et al. 2012). Planet Hunters was later modified to analyze data from the K2 (Kepler's successor) and TESS space missions, while it is currently focused on the ground-based Next Generation Transit Survey (see chapter 'Transit Photometry as an Exoplanet Discovery Method' for an overview over transit detection projects). Planet Hunters has been one of the most successful citizen science project in exoplanetology; it has contributed to the discovery of over 10 exoplanets, of which two are named after it: PH1 b and PH2b, with PH1 b being a relatively rare circumbinary planet (Schwamb et al. 2013). A further noteworthy project is the Exoplanet Explorer (with uncertain status in 2024, listed in Zooniverse as 'out of data'), which also analyzed data from the K2 mission. With over 15000 volunteers it aided in the discovery of several planet systems, including one with five transiting planets, named K2-138 (Christiansen et al. 2018). The most impacting result of such projects to date might however have been the discovery of a mysterious object with very unusual brightness variations (Fig. 4 ) by members of the Planet Hunters project (Boyajian et al. 2016). A variety of explanations have been brought forward (see Wright & Sigurðsson 2016 for an overview), including very speculative ones involving extra-terrestrial intelligences, which led to significant media attention. The discovery of this object, also known as 'Boyajian's star', has been a wonderful result of the power of citizen science: its light curve had been discarded by the algorithms that previously had sifted through Kepler data, and only through the viewing by real persons was its strange nature recognized, leading to one of the strangest puzzles in current-day astronomy. Further resources on space-related citizen-science projects, we also refer to dedicated websites at NASA ( https://exoplanets.nasa.gov/citizen-science/) and ESA ( https://www.esa.int/Enabling_Support/Preparing_for_the_Future/Space_for_Earth/Citizen_ science ). Data from upcoming exoplanet-related missions, like PLATO, the Roman Space Telescope or the Habitable Worlds Observatory will also become available to both the scientific community and to the public, and will provide ample opportunities for future citizen science involvement. Amateur astronomers in their classical form, with their proper observations of the sky, are also contributing significantly to exoplanet science. In contrary to expectations prior to the discovery of the first transiting planets, exoplanet observations do not exclusively require large professional equipment, instead, small instruments accessible to amateurs may be useful as well. Indeed, several pioneering projects to search for transiting planets were based on amateur equipment - although executed by professionals - and one of the two groups that simultaneously detected the first exoplanet transits, of HD209458 b, used a small 10 cm telescope (Charbonneau et al. 2000). This discovery gave immediately rise to many successful transits observations by amateurs, for which the Exoplanet Transits Database (ETD, Poddany et al. 2010, http://var2.astro.cz/ETD) and the database of the American Association of Variable Star observer (AAVSO, https://www.aavso.org/ ) have become the principal repositories. Amateur observations of transits have been aided by the availability of detailed observing instructions (e.g. Gary 2010, Dennis 2016) and by freely available software geared towards differential precision photometry by non-professionals, such as AstroImageJ (https://www.astro.louisville.edu/software/astroimagej, Collins et al. 2017), CMunipack (Motl 2024) or SIRIL (Team-FreeAstro 2024). Some of the citizen science projects have also developed their specific analysis software, and many light curves from amateurs are good matches to data taken by professional observatories (Fig. 5 ). The main driver for amateur exoplanet observations is the follow-up of known transiters, with the principal objective being the maintenance or refinement of transit ephemeris. Many exoplanets that were discovered from observations made over shorter time spans (several months or less), like those from CoRoT and most from TESS, have transit-ephemeris with a severely limited precision. Within a few years, transit times will then accumulate uncertainties so big (more than 2-3 hours) that further transit observations cannot be planned well. Prior transit re-observations are hence used to generate improved ephemeris, which in turn permit efficient reobservations by future ground and space-based instrumentations over time-scales of years to decades (Dragomir et al. 2020, Deeg et al. 2020, Zellem et al., 2020). The importance of ephemeris maintenance and refinement is epitomized by the automatic ephemeris fitting in ETD, which is based on quality-weighted sets of the uploaded transit observations. Amateur data have been taken into account in several publications presenting improved transit ephemeris; e.g. for planets found by CoRoT (Klagyivik et al. 2021), TESS (Peluso et al. 2023) or WASP (Noguer et al. 2024). Transit re-observations may also give rise to the detection of transit timing variations (TTVs) - these are deviations from strict periodicity that indicate non-linear ephemeris. TTVs may be due to the presence of further orbiting bodies and/or relevant star-planet interactions; they may also be used to estimate the masses of transiting planets (see chapter Transit-Timing and Duration Variations for the Discovery and Characterization of Exoplanets). An example is CoRoT-11 b, which indicates TTVs from both amateur and professional observations (timings in ETD from 2011- 2024 and Deeg et al. 2020). ETD also tracks variations in transit depths and durations, although conclusive evidence for the presence such variations based on amateur data is more difficult to obtain, given that these parameters are very sensitive to data quality. A selection of observational citizen science projects that pursue these and related topics are currently Exoplanet Watch (https://exoplanets.nasa.gov/exoplanet-watch), the Unistellar Network (https://science.unistellar.com/, Peluso et al. 2023) - a private /public partnership supplying also suitably fitted small telescopes (Marchis et al. 2020) - and Exoclock (https://www.exoclock.space/, Kokori et al. 2022, 2023), with the latter one dedicated to the ephemeris-refinement of target planets for the future ARIEL space mission. A further driver for amateur involvement is also the identification of false alarms among planet candidates. Such identifications are needed if the photometry from the discoverysurvey (be it ground or space-based) is based on imaging of low spatial resolution. In this case, follow-up photometry from higher resolution imaging is needed, in order to distinguish if a planet-candidate displays a transit on the target star, or if a deeper eclipse appears on a nearby star, which means the presence of a false alarm. It is of note that such follow-up observations are also meaningful for candidates with transits that are too shallow to be detected again, as long as the presence of eclipses at all nearby stars can be excluded with certainty. Such work by amateurs was already included in the verification of an exoplanet by the ground-based XO planet survey (McCullough et al. (2007), while more detailed theoretical and observational foundations for these follow-up observations are described by Deeg et al. (2009). For candidates from the TESS mission, the aforementioned Unistellar Network and others are currently pursuing false alarm identifications with the help of citizen observers; see Sgro et al (2024) for a successful example. For the upcoming PLATO transit survey mission (Rauer et al. 2010, 2024 and chapter Space Missions for Exoplanet Science: PLATO), to be launched in 2026, a dedicated program for ground-based photometric followup is currently prepared, with branches for professional and for amateur observers, including outreach organizations (Deeg & Alonso 2024, https://citizen.plato-planets.at/ ). Proposed further objectives of amateur exoplanet observations are also the confirmation of long-periodic planets (by re-observation of their transits), the search for transits of planets that are currently only known from radial velocity detections, or the pursuing of microlensing events caused by exoplanets (Peluso et al. 2023). Boyajian's star, the prominent discovery made from Kepler light curves by citizen scientists, has also become a target of intense surveillance by amateur observers: the database of the AAVSO lists presently (2024 Oct 2) 137 345 observations by 137 observers, with new observations arriving most of the days. While photometric observations are clearly the main subject of amateur involvement in exoplanetology, some private initiatives for SETI searches in the optical (e.g. Schuetz 2018) and radio domains (Project Bambi, www.bambi.net/) are also of note.",
"pages": [
6,
7,
8,
9,
10
]
},
{
"title": "The Public's View About the Presence of Life in the Universe",
"content": "The general public learns about exoplanets through a variety of sources. How has the input from exoplanet science influenced the public's perceptions of Earth's uniqueness in the universe and shaped views on the potential universality of life? From a historic perspective, our view about 'other worlds' away from the Sun or about life in the Universe has changed from one of complete speculation to a semi-empirical one. As the best-known tool for this attempted quantification remains Drake's famous equation (Drake 1965, 2011). This equation, whose original form estimates the abundance of detectable intelligent life, is based on the multiplication of several factors, of which only one, the rate of star formation in our Galaxy, could be estimated reasonably well when the equation was originally presented. Other factors, such as the fraction of planets with life that develop technological civilizations, or the length of time over which such civilizations release detectable signals, remain to date entirely speculative. Our increasing knowledge on exoplanets has however raised two of the equation's factors, namely the fraction of stars with planets, and the number of habitable planets around such stars, to estimates that may be reliable to a factor of 'a few'. These estimates constitute one of the major advances of the past 20 years of research on exoplanets, leading for example to an estimate (Petigura et al. 2013) that about 6% of Sun-like stars have Earth-like planets . Currently we are able to label planets as 'Earth-like' solely based on their principal physical parameters without crucial information for habitability, such as the presence of water or the type of atmosphere. The quoted 6% is therefore an upper limit for habitable Earth-like planets. The habitability of planets that are not Earth-like is hypothetical but in many cases reasonable (e.g. on planets around low-mass stars or on planet-sized moons); such objects might even provide the vast majority of habitats in the Universe (see also chapters 'Habitability of Planets in Binary Star Systems', and 'Habitability in Brown Dwarf Systems'). In either case is seems safe to assume that habitable planets (in the sense of fulfilling all requirements to develop life) are frequent, and that life - evolved or not - will be frequent as well, unless the unknown factors of Drake's equation are minuscule small. In that sense, Frank & Sullivan (2016) estimate that there is at least one other technological species in the observable Universe, unless the probability that a habitable zone planet develops a technological species is below 10 -24 (see also chapter Where Life May Arise: Habitability). These more general results from exoplanet science, together with the numerous individual findings of potentially habitable planets have led to the perception among the informed public that life in the Universe might be frequent and that its discovery is a question of \"when\" rather than \"if\". For instance, a survey in 2020 found that nearly 65% of Americans believe there is intelligent life on other planets (Kennedy & Lau 2021). At the same time, activities for more direct detections of extraterrestrial life have come within technological reach. Not only classical SETI searches for technological signatures, but also searches for biosignatures on planets but within and outside of our Solar System. Such investigations are supported by a wide interest among the public, which is also essential for the support of the many ambitious projects in the coming decades. These projects in turn will advance the entire field of exoplanet science and humanity's understanding of the origin and presence of life in the Universe. And lastly, the public interest and fascination with current findings from exoplanets science may better prepare society for the day when we find out that we are not alone.",
"pages": [
11,
12
]
},
{
"title": "Acknowledgements",
"content": "The author acknowledges support from the Spanish Research Agency of the Ministry of Science and Innovation (AEI-MICINN) under grant PID2019-107061GB-C66, DOI: 10.13039/501100011033. This publication has made use of the Astrophysics Data System, funded by NASA under Cooperative Agreement 80NSSC21M00561.",
"pages": [
12
]
}
]
2024arXiv241009064T
https://arxiv.org/pdf/2410.09064.pdf
<document>
<text><location><page_1><loc_6><loc_96><loc_11><loc_97></location>REPORT</text>
<section_header_level_1><location><page_1><loc_5><loc_89><loc_87><loc_94></location>Monitoring the daily variation of Sun-Earth magnetic /uniFB01 elds using galactic cosmic rays</section_header_level_1>
<text><location><page_1><loc_5><loc_88><loc_22><loc_89></location>The LHAASO Collaboration 1, *</text>
<text><location><page_1><loc_5><loc_87><loc_53><loc_88></location>*Correspondence: nanyc@ihep.ac.cn (Y.N.); chensz@ihep.ac.cn (S.C.); fengcf@sdu.edu.cn (C.F.)</text>
<text><location><page_1><loc_6><loc_84><loc_95><loc_85></location>2024 The Author(s). Published by Elsevier Inc. on behalf of Youth Innovation Co., Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).</text>
<text><location><page_1><loc_5><loc_84><loc_74><loc_86></location>Received: April 10, 2024; Accepted: September 3, 2024; Published Online: September 6, 2024; https://doi.org/10.1016/j.xinn.2024.100695 ª</text>
<section_header_level_1><location><page_1><loc_5><loc_80><loc_28><loc_82></location>GRAPHICAL ABSTRACT</section_header_level_1>
<figure>
<location><page_1><loc_9><loc_38><loc_91><loc_79></location>
</figure>
<section_header_level_1><location><page_1><loc_5><loc_34><loc_24><loc_35></location>PUBLIC SUMMARY</section_header_level_1>
<unordered_list>
<list_item><location><page_1><loc_5><loc_32><loc_85><loc_33></location>-Daily Sun ' s shadows on very high-energy galactic cosmic rays were observed for the /uniFB01 rst time using LHAASO.</list_item>
<list_item><location><page_1><loc_5><loc_29><loc_64><loc_31></location>-This work therefore proposes a novel method for monitoring the Sun-Earth IMF.</list_item>
<list_item><location><page_1><loc_5><loc_27><loc_86><loc_28></location>-Compared with the near-Earth spacecraft, the Sun ' s shadow can provide 3.3-day earlier predictions for the IMF.</list_item>
<list_item><location><page_1><loc_5><loc_24><loc_68><loc_25></location>-The timing advance signi /uniFB01 cantly deviated from the predictions of current IMF models.</list_item>
<list_item><location><page_1><loc_5><loc_21><loc_84><loc_23></location>-These /uniFB01 ndings may provide valuable insights into the IMF structure, thus improving space weather research.</list_item>
</unordered_list>
<figure>
<location><page_1><loc_92><loc_95><loc_95><loc_97></location>
</figure>
<figure>
<location><page_2><loc_5><loc_95><loc_8><loc_97></location>
</figure>
<section_header_level_1><location><page_2><loc_5><loc_89><loc_88><loc_94></location>Monitoring the daily variation of Sun-Earth magnetic /uniFB01 elds using galactic cosmic rays</section_header_level_1>
<text><location><page_2><loc_5><loc_88><loc_23><loc_89></location>The LHAASO Collaboration 1, *</text>
<text><location><page_2><loc_5><loc_87><loc_37><loc_88></location>1 Further details can be found in the supplemental information</text>
<text><location><page_2><loc_5><loc_85><loc_54><loc_86></location>*Correspondence: nanyc@ihep.ac.cn (Y.N.); chensz@ihep.ac.cn (S.C.); fengcf@sdu.edu.cn (C.F.)</text>
<text><location><page_2><loc_5><loc_84><loc_75><loc_85></location>Received: April 10, 2024; Accepted: September 3, 2024; Published Online: September 6, 2024; https://doi.org/10.1016/j.xinn.2024.100695</text>
<text><location><page_2><loc_5><loc_81><loc_95><loc_84></location>ª 2024 The Author(s). Published by Elsevier Inc. on behalf of Youth Innovation Co., Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Citation: (2024). Monitoring the daily variation of Sun-Earth magnetic /uniFB01 elds using galactic cosmic rays. The Innovation 5(6) , 100695.</text>
<text><location><page_2><loc_5><loc_53><loc_49><loc_80></location>The interplanetary magnetic /uniFB01 eld (IMF) between the Sun and Earth is an extension of the solar magnetic /uniFB01 eld carried by the solar wind into interplanetary space. Monitoring variations in the IMF upstream of the Earth would provide very important information for the prediction of space weather effects, such as effects of solar storms and the solar wind, on human activity. In this study, the IMF between the Sun and Earth was measured daily for the /uniFB01 rst time using a cosmic-ray observatory. Cosmic rays mainly consist of charged particles that are de /uniFB02 ected as they pass through a magnetic /uniFB01 eld. Therefore, the cosmic-ray Sun shadow, caused by high-energy charged cosmic rays blocked by the Sun and de /uniFB02 ected by the magnetic /uniFB01 eld, can be used to explore the transverse IMF between the Sun and Earth. By employing the powerful kilometer-square array at the Large High Altitude Air Shower Observatory, the cosmic-ray Sun shadows were observed daily with high signi /uniFB01 -cance for the /uniFB01 rst time. The displacement of the Sun shadow measured in 2021 correlates well with the transverse IMF component measured in situ by spacecraft near the Earth, with a time lag of 3 : 31 ± 0 : 12 days. The displacement of the Sun shadow was also simulated using Parker ' s classic IMF model, yielding a time lag of 2 : 06 ± 0 : 04 days. This deviation may provide valuable insights into the magnetic /uniFB01 eld structure, which can improve space weather research.</text>
<section_header_level_1><location><page_2><loc_5><loc_48><loc_15><loc_49></location>INTRODUCTION</section_header_level_1>
<text><location><page_2><loc_5><loc_40><loc_49><loc_48></location>The Sun, our nearest star, is the main source of energy for living organisms on Earth, and its activity continually affects our planet ' s environment. As human science and technology advance, along with the increasing use of electronic and space equipment, the impact of solar activity on human activity has steadily grown. Therefore, monitoring solar activity and forecasting space weather are important areas of scienti /uniFB01 c research.</text>
<text><location><page_2><loc_5><loc_23><loc_50><loc_40></location>Solar magnetic /uniFB01 elds play a vital role in understanding diverse solar activities. Since the /uniFB01 rst measurement of the solar magnetic /uniFB01 eldin1908usingtheZeeman effect, 1 the photospheric magnetic /uniFB01 eld at the Sun ' s surface has been continuously monitored by both space- and ground-based observatories. 2 The coronal magnetic /uniFB01 eld (CMF) lies above the photosphere, and direct measurement remains challenging, despite recent attempts using magnetoseismology. 3 The CMF is carried into interplanetary space by the solar wind, forming the interplanetary magnetic /uniFB01 eld (IMF). 4 The IMF provides valuable information for studying the CMF and is critical for understanding space weather and improving forecasting accuracy. 5,6 Since the discovery of the solar wind in 1962, the IMF has been monitored in situ by a series of spacecraft, with considerable monitoring performed from the L 1 Lagrange point of gravitational balance between the Sun and Earth. 7</text>
<text><location><page_2><loc_5><loc_11><loc_50><loc_22></location>Although the ongoing Parker Solar Probe mission can /uniFB02 y from the IMF to the CMF, 8 at locations other than the Sun ' ssurfaceand L 1 point, it remains challenging to continuously monitor the magnetic /uniFB01 eld in the vast space between the Sun and Earth. Currently, the distribution of this magnetic /uniFB01 eld relies on theoretical models that extrapolate the photospheric magnetic /uniFB01 eldtotheCMF(e.g., the classical potential /uniFB01 eld source surface [PFSS] model 9,10 ) and extend the outermost CMF to the IMF (e.g., the classical and widely used Parker model 4 ). The distributions of the CMF and IMF can also be simulated using data-driven models. 11</text>
<text><location><page_2><loc_5><loc_4><loc_49><loc_10></location>Very-high-energy galactic cosmic rays, consisting mainly of positively charged particles moving near the speed of light, can travel from the Sun to the Earth within approximately 8 min. Their trajectories are affected by the magnetic /uniFB01 eld along the Sun-Earth line. Therefore, when the cosmic-ray Sun shadow was detected for the /uniFB01 rst time, it was proposed that the magnetic /uniFB01 eld between the</text>
<text><location><page_2><loc_51><loc_72><loc_95><loc_80></location>Sun and Earth could be studied through measurements of the Sun shadow. 12 It was noted that as observational sensitivity increases, the cosmic-ray de /uniFB01 cit ratio of the Sun shadow was observed or proved to have a relationship with the solar magnetic /uniFB01 eld, including the photospheric magnetic /uniFB01 eld, 13-15 CMF and its annual variations, 16-19 and even coronal mass ejections (CMEs), 20 and it was applied to diagnose different CMF models. 16,17</text>
<text><location><page_2><loc_51><loc_59><loc_95><loc_72></location>In addition, studies have observed 21-25 and proved 23-26 that the displacement of the Sun shadow is also related to the IMF. Based on the annual timescale displacement of the Sun shadow, the ARGO-YBJ collaboration measured the structure and strength of the mean IMF near the sunspot minimum in Solar Cycle 24 for the /uniFB01 rst time. 24 They also proposed the possibility of using Sun shadow measurements for space weather forecasting, given the time advantage this method offers over spacecraft at L 1. However, this approach requires cosmicray arrays with enough sensitivity to measure shadows daily. 24 In addition, the AS g collaboration used annual Sun shadow data to correct the average strength of the Parker model for the IMF near total Solar Cycle 23. 25</text>
<text><location><page_2><loc_51><loc_55><loc_95><loc_58></location>Owing to the limited observational sensitivity of the Sun shadows, researchers have not yet been able to study /uniFB01 ner structures in the magnetic /uniFB01 eld, particularly during short periods.</text>
<section_header_level_1><location><page_2><loc_51><loc_51><loc_68><loc_52></location>RESULTS AND DISCUSSION</section_header_level_1>
<text><location><page_2><loc_51><loc_31><loc_95><loc_51></location>The Large High Altitude Air Shower Observatory (LHAASO) is a composite ground-based cosmic-ray detection facility located at 100.01 /C14 E, 29.35 /C14 N, and an altitude of 4,410 m above sea level in Sichuan, China. 27,28 The 1.3 kilometer square array (KM2A) is one of the main ground-based arrays in LHAASO, which has a detection area of 1 to 2 orders of magnitude larger than those of the ARGOYBJ and AS g experiments. KM2A has been operating with a nearly full-duty cycle since the beginning of 2020. In this study, only the data recorded in 2021 near the sunspot minimum in Solar Cycle 25 were used. Owing to the location of the array near the Tropic of Cancer, the Sun shadow could be observed signi /uniFB01 cantly for 196 days from March 21 to October 2, with /C24 7 h observation on each day. During this period, the pointing accuracy, angular resolution, and energy of the cosmic rays measured by the array were stable according to the observation of the cosmic-ray Moon shadow. 29 For monitoring the IMF, we selected events with 26 -251 /uniFB01 red electromagnetic particle detectors (EDs) in KM2A. The corresponding median energy was /C24 40 TeV, and the angular resolution was 0.5 /C14 .</text>
<text><location><page_2><loc_51><loc_19><loc_95><loc_31></location>The Sun shadow was observed along the longitude and latitude in the geocentric solar ecliptic (GSE) coordinate system. Figure 1A depicts the Sun shadows observed during the 196 days of data collection, with a very high signi /uniFB01 cance exceeding 100 s . Figure 1B displays the Sun shadow obtained from a single day of data, which also demonstrates good signi /uniFB01 cance exceeding 9 s . The daily Sun shadows are clearly de /uniFB02 ected away from the direction of the Sun by the magnetic /uniFB01 eld. The angular distance from the center of the shadow to the Sun is de /uniFB01 ned as the daily displacement of the shadow. Details regarding the Sun shadow analysis are presented in the materials and methods.</text>
<section_header_level_1><location><page_2><loc_51><loc_16><loc_79><loc_17></location>Daily IMFBy measurement using Sun shadow</section_header_level_1>
<text><location><page_2><loc_51><loc_4><loc_95><loc_16></location>Cosmic rays traveling toward the Sun propagate approximately parallel to the Sun-Earth line. In the GSE coordinate system, the x axis points toward the Sun, the z axis is toward the north ecliptic pole, and the y axis is roughly opposite Earth ' s orbital motion. Positively charged cosmic rays are only affected by the y and z components of the IMF in the GSE coordinate system, but not by the x component. By and Bz cause displacement of the Sun shadow along the north-south and west-east directions, respectively, according to the Lorentz force law. However, the strength of Bz is in /uniFB02 uenced not only by the IMF, but also by the geomagnetic /uniFB01 eld, which has a complex effect on the west-east displacement of</text>
<figure>
<location><page_3><loc_5><loc_65><loc_49><loc_94></location>
<caption>Figure 3A presents a comparison of the estimated By derived from D with the By value obtained from OMNI. We performed a c 2 -test to compare these values. The c 2 value is 155.8 at 159 degrees of freedom, and the corresponding probability is 0.56. All By estimates from D and OMNI fall within 3 standard deviations, except for June 1 (3.1 standard deviations) and June 4 (3.3 standard deviations). These results indicate that daily By and its variations can be well estimated by monitoring D of the Sun shadow 3.31 days earlier.</caption>
</figure>
<figure>
<location><page_3><loc_52><loc_65><loc_94><loc_94></location>
<caption>Figure 1. Observed signi /uniFB01 cance map of Sun shadows by LHAASO for two timescales (A) Map for 196 days. The central circle of the contour map indicates a signi /uniFB01 cance of /C0 102.5 s , and the step between the contour lines is 10 s . (B) Map for a single day on May 26, 2021. The central circle of the contour map indicates a signi /uniFB01 cance of /C0 9.2 s , and the step between the contour lines is 1 s . The best/uniFB01 t displacement of the Sun shadow shown as the red plus sign is ( /C0 0 : 16 /C14 ±0 : 08 /C14 ; 0 : 52 /C14 ±0 : 08 /C14 ). This displacement provides a measurement of the transverse magnetic /uniFB01 eld along the Sun-Earth line.</caption>
</figure>
<text><location><page_3><loc_5><loc_55><loc_49><loc_57></location>the Sun shadow. 23,25 Therefore, the dominant displacement of the Sun shadow is in the north-south direction, caused by By . 24,25</text>
<text><location><page_3><loc_5><loc_44><loc_49><loc_54></location>The daily displacement of the Sun shadow along the north-south direction (denoted by D ) and its variation can be monitored by LHAASO, as illustrated in Figure 2A, where only days with a signi /uniFB01 cance exceeding 5 s are shown. The number of effective observation days for D reaches 177 (90% of the total days). The variationindaily D appears to be periodic in each 27-day Carrington rotation (i.e., solar rotation), with the periodicity gradually changing throughout the Carrington rotation. This allows LHAASO to directly test the speci /uniFB01 c correlation between D and IMFBy , potentially enabling the use of D to measure daily By for the /uniFB01 rst time.</text>
<text><location><page_3><loc_5><loc_36><loc_49><loc_44></location>The By values used in our test are observational results at L 1 from OMNI. 30 Figure 2B displays the daily By values, representing the mean value of hourly By measurements within a 24-h period. The number of ef /uniFB01 ciently observed days for By is 181 (92% of the total observation period). As presented in Figures 2A and 2B, D and By exhibit similar trends over each Carrington rotation, with a possible time lag between them.</text>
<text><location><page_3><loc_5><loc_20><loc_49><loc_36></location>To determine the correlation and time lag between D and By ,weusedthe discrete correlation function (DCF) method, 31 which considers the unevenly sampled time series of D and By and their measurement errors. The time lag bin width was set to 1 day, matching the time bins of D and By in Figures 2A and 2B, and we considered time lags of up to 5 days. To achieve higher precision in the time lag determination, the cadence was set to 0.0625 days based on the hourly By measurement and a time lag sliding technique. The DCF coef /uniFB01 cient and its error are shown in Figure 2C. The error of the time lag was estimated using 10 3 random time series of D and By , 32 generated based on a Gaussian probability distribution. The standard deviation of the distribution of time lags between the 10 3 random time series of D and By was taken as the error of the time lag.</text>
<text><location><page_3><loc_5><loc_5><loc_49><loc_20></location>For the entire dataset (labeled as ' All ' ), we tested the correlation between 177 days of D data and 181 days of By data. D is most correlated with By at L 1 when D precedes By by 3 : 31 ± 0.12 days, according to the maximum DCF coef /uniFB01 cient shown with blue markers in Figure 2C. The con /uniFB01 dence level of the maximum DCF coef /uniFB01 cient was estimated using the Monte Carlo method. 33,34 Speci /uniFB01 cally, we generated 10 5 random time series D by randomizing both the phase and amplitude of the Fourier transform of the observed time series D . The DCF was then applied to each random time series D and the observed time series By . The corresponding con /uniFB01 -dence level of the maximum DCF coef /uniFB01 cient exceeded 99.73% (corresponding to 3 s ) with a two-sided p value of 0.0027, as indicated by the</text>
<text><location><page_3><loc_5><loc_4><loc_20><loc_5></location>dashed line in Figure 2C.</text>
<text><location><page_3><loc_51><loc_53><loc_95><loc_57></location>As depicted in Figure 3B, D remains correlated with By after considering a time lag of 3.31 days. The speci /uniFB01 c correlation is /uniFB01 tted using the following linear formula:</text>
<formula><location><page_3><loc_52><loc_50><loc_95><loc_52></location>By ð t Þ = ð 7 : 6 ± 0 : 6 Þ nT = /C14 3 D ð t /C0 3 : 31 Þ + ð 0 : 2 ± 0 : 1 Þ nT : (Equation 1)</formula>
<text><location><page_3><loc_51><loc_42><loc_95><loc_50></location>The corresponding correlation coef /uniFB01 cient is 0.67. The functional form of this formula is essentially the same as the change in position according to the Newton-Lorentz equation. Following Equation 1, the daily By at L 1 can be estimateddirectlybasedon D measured by LHAASO 3.31 days earlier. The By estimated by D re /uniFB02 ects the effective IMF responsible for the cumulative de /uniFB02 ection of cosmic rays along the Sun-Earth line.</text>
<section_header_level_1><location><page_3><loc_51><loc_30><loc_85><loc_31></location>IMF model diagnosis from the time lag between D and By</section_header_level_1>
<text><location><page_3><loc_51><loc_27><loc_95><loc_29></location>Based on the classic Parker model, the IMF in heliocentric spherical coordinates can be expressed as follows:</text>
<formula><location><page_3><loc_53><loc_21><loc_95><loc_26></location>B ð r ; q ; f Þ = Br ð b ; q ; f 0 Þ /C18 b r /C19 2 /C20 b er /C0 u ð r /C0 b Þ sin q n b e f /C21 ; (Equation 2)</formula>
<text><location><page_3><loc_51><loc_14><loc_95><loc_21></location>where Br ð b ; q ; f 0 Þ is the outermost CMF at the boundary radius b and heliolongitude f 0 . Beyond b , the IMF in the model is blown out by the radial solar wind with velocity n . When the Sun rotates with angular velocity u , the streamline of the magnetic /uniFB01 eld with azimuth f 0 at r = b is given by r b /C0 1 /C0 ln /C0 r b /C1 = n b u ð f /C0 f 0 Þ .</text>
<text><location><page_3><loc_51><loc_4><loc_95><loc_14></location>To study the effect of the magnetic /uniFB01 eld predicted by the IMF model, an antiparticle retraction method was adopted to simulate the Sun shadow. The details of the simulation program are presented in the materials and methods.Inthe Parker model calculation, the boundary radius b was set to 2.5 R 1 , and Br ð b ; q ; f 0 Þ was extrapolated from the PFSS model 9 using a 9th-order spherical harmonic expansion. 35 Photospheric magnetograms named ' mrnqs ' from the Global Oscillation Network Group (GONG) 36 served as an input. The calculated IMF was modi /uniFB01 ed by a factor of 5.7 according to the observed average daily</text>
<figure>
<location><page_4><loc_5><loc_40><loc_66><loc_94></location>
<caption>Figure 2. Correlation between N-S displacement D of the LHAASO Sun shadow and By at L 1 from OMNI with 3.31-day time lag. Daily variations of D and By are shown in (A) and (B), respectively. The red open crosses and blue open circles correspond to the periods during interplanetary coronal mass ejections (ICMEs) and stream interaction regions (SIRs) transits, respectively. The error bar shows the statistical error. The vertical dashed lines are the boundaries between Carrington rotations (CRs). The correlation coef /uniFB01 cient of the discrete correlation function (DCF) between D and By as a function of the time lag for the ' All ' data sample (C) and the ' without ICME and SIR ' data sample (D), respectively. The dots are the discrete correlation coef /uniFB01 cients and their error bar are included. The dashed lines are the 3 s con /uniFB01 dence intervals. The vertical lines show the time lag of 3 : 31±0 : 12 days for the ' All ' and 3 : 00±0 : 21 days for the ' without ICME and SIR ' data sample.</caption>
</figure>
<text><location><page_4><loc_5><loc_33><loc_49><loc_38></location>By at L 1 from OMNI. This scaling of magnetogram data used as the input to a solar wind model is a standard practice in the heliophysics community. 37 The Sun ' s rotation period was /C24 25.4 days and the solar wind velocity n was obtained from the daily average value from OMNI.</text>
<text><location><page_4><loc_5><loc_17><loc_49><loc_32></location>The IMF variations are transferred from the Sun to the Earth by the solar wind at velocity n . Cosmic rays can record By between the Sun and Earth through D of the Sun shadow. Hence, the variation in By at L 1 lags behind the variation in D . Because D represents a cumulative effect of By that spreads from the Sun to the Earth, the speci /uniFB01 c time lag value depends on the distribution of By along the Sun-Earth line. Therefore, the time lag between D and By at L 1 provides an opportunity to test the IMF models. Based on the Parker model, the average simulated time lag for the entire dataset is 2 : 06 ± 0.04 days. This simulated result reproduces the phenomenon that D leads By in the observations. However, a deviation between the simulated and observed time lags exists. This suggests a more complicated spiral structure of the IMF than that depicted by the Parker model.</text>
<text><location><page_4><loc_5><loc_3><loc_50><loc_16></location>One possible solution to address this deviation between the simulated and observed time lag is to add a steady, azimuthal IMF component, B f ð b Þ ,atthe CMF boundary b to the Parker model, as proposed by Smith and Bieber. 38 This modi /uniFB01 ed model has been used to explain the deviation of the spiral structure from the Parker model 38 and even to calculate such a deviation to interpret recent Parker Solar Probe results. 39 During the observation time of LHAASO, the additional azimuthal IMF component B f ð b Þ x /C0 0 : 002 Br ð b Þ n n ð b Þ , which corresponds to a ' gardenhose ' angle of the spiral that is /C24 3.6 /C14 larger than that predicted by the Parker model at L 1.Here, n ð b Þ repre-</text>
<text><location><page_4><loc_68><loc_61><loc_95><loc_68></location>velocity at boundary b . Based on this modi /uniFB01 ed model, the simulated time lag becomes 2 : 64 ± 0.04 days with a deviation from the observed lag. This simulated time lag is close to our measured results; however, a deviation still persists.</text>
<text><location><page_4><loc_68><loc_38><loc_95><loc_60></location>The magnetic /uniFB01 eld structures of CMEs can disturb the IMF to form interplanetary CMEs (ICMEs), which may lead to IMF deviations from the Parker model. In addition, solar wind interactions can disturb the IMF by generating stream interaction regions (SIRs), which may also lead to IMF deviations. To check for a possible effect of CMEs and SIRs on the deviation between the observed and simulated time lag, a data sample without ICME and SIR transits (labeled as ' Without ICME+SIR ' ) was segmented, as depicted by the blue dots in Figures 2Aand 2B. We excluded ICMEs based on an updated catalog 40 and excluded SIRs based on coronal hole information 41 and an identi /uniFB01 ed method, 42 as shown in Figures 2A and 2B. The ICME transit of the Sun shadow was counted from the occur-</text>
<text><location><page_4><loc_51><loc_32><loc_95><loc_38></location>rence of the CME until it stops disturbing By at L 1. Similarly, the SIR transit was counted from the time when the coronal hole faces the Earth until the SIR stops disturbing By at L 1. The ICME or SIR transit of By at L 1 was counted from the time when the ICME or SIR starts disturbing By at L 1 until it stops affecting By at L 1.</text>
<text><location><page_4><loc_51><loc_22><loc_95><loc_32></location>For the Without ICME+SIR data sample, we tested the correlation between 59-day D and 130-day By . The observed time lag between D and By at L 1 was 3 : 00 ± 0 : 21 days according to the maximum DCF coef /uniFB01 cient, as indicated by the blue markers in Figure 2D. The corresponding con /uniFB01 dence level of the maximum DCF coef /uniFB01 cient exceeded 3 s as shown by the dashed lines in Figure 2D. There was no signi /uniFB01 cant difference in the observed time lag between the All and Without ICME+SIR data samples. Therefore, the deviation between the observed and simulated time lags was not due to CMEs and SIRs.</text>
<text><location><page_4><loc_51><loc_15><loc_95><loc_22></location>Other possible explanations for the deviation between the observed and simulated time lags include systematic changes in the IMF 43,44 and disturbances with timescales much shorter than the Carrington rotation. 45-47 A more comprehensive discussion about these possibilities is beyond the scope of this study, as it focuses on long-term average time lag.</text>
<section_header_level_1><location><page_4><loc_51><loc_12><loc_69><loc_13></location>CONCLUSION AND OUTLOOK</section_header_level_1>
<text><location><page_4><loc_51><loc_4><loc_95><loc_12></location>Traveling almost at the speed of light, very-high-energy galactic cosmic rays take only approximately 8 min to traverse the distance between the Sun and Earth. This provides us with a useful method for exploring the magnetic /uniFB01 eld between the Sun and Earth. Based on the unprecedented monitoring of the daily cosmic-ray Sun shadow by KM2A in LHAASO, we provided daily measurements of the IMF between the Sun and Earth near the sunspot minimum period. For the</text>
<figure>
<location><page_5><loc_3><loc_68><loc_94><loc_94></location>
<caption>Figure 3. Daily By at L 1 as predicted by LHAASO and measured at L 1 3.31 days later (A) The red full squares and the blue open circles are for LHAASO and OMNI data, respectively. The error bar of LHAASO results is the statistical error, which includes the error from the D and the /uniFB01 tting error from Equation 1. The error bar of OMNI results is the root mean-squared value. The vertical dashed lines are the boundaries between CRs. (B) Scatterplot of D and By at L 1 from OMNI after shifting for the time lag with the ' All ' data sample, and their correlation coef /uniFB01 cient (CC). The error bars of D indicate the statistical errors. The error bars of By is the root mean squared value. The dashed line is the best/uniFB01 t linear formula for the By measurement by LHAASO.</caption>
</figure>
<text><location><page_5><loc_5><loc_54><loc_49><loc_60></location>/uniFB01 rst time, we made it possible for the Sun shadow to achieve an accurate measurement of the transverse magnetic /uniFB01 eld along the entire Sun-Earth line. Using this measurement, the IMF component along the GSE y -direction near the Earth was deduced to be 3 : 31 ± 0 : 12 days ahead of the measurement by spacecraft at L 1.</text>
<text><location><page_5><loc_5><loc_42><loc_49><loc_53></location>With a measured 3 : 31 ± 0 : 12-day time lag, we also tested the classic Parker model of the IMF. We found that the measured lag signi /uniFB01 cantly deviates from the model ' spredictionof2 : 06 ± 0 : 04 days, indicating the need for re /uniFB01 nements to the IMF model. This measurement not only provides a new method for monitoring the IMF and diagnosing existing models but also drives the development of more accurate IMF models. 4,35,38,48 This advancement will contribute to improved understanding of the propagation and forecasting of solar activity events, which can be helpful for research on space weather effects that impact human activities.</text>
<text><location><page_5><loc_5><loc_34><loc_49><loc_41></location>Moreover, the observed time lag implies that the Sun shadow has the potential for predicting the IMF arrival on Earth. Future research will explore the possibility of Sun shadow forecasting of the IMF component along the z axis (north-south direction) when the geomagnetic /uniFB01 eld effect is excluded. Even the possibility of forecasting a speci /uniFB01 c solar activity event, such as CMEs and SIRs, holds promise for further increasing space weather forecasting capabilities.</text>
<section_header_level_1><location><page_5><loc_5><loc_30><loc_22><loc_31></location>MATERIALS AND METHODS</section_header_level_1>
<section_header_level_1><location><page_5><loc_5><loc_29><loc_16><loc_30></location>KM2A experiment</section_header_level_1>
<text><location><page_5><loc_5><loc_19><loc_49><loc_28></location>The KM2A, a component of LHAASO, consists of 5216 EDs with 15 m spacing and 1,188 muondetectors with 30 m spacing, covering an area of 1.3 km 2 . EDs are designed to detect showers from cosmic-ray ions and gamma rays 49 with determining their directions and energies. The 3/4 array has been operating since December 1, 2020, and the full array since July 19, 2021. During this period, the pointing accuracy, angular resolution, and energy of the cosmic rays measured by the array were stable, based on observations of the cosmic-ray Moon shadow. 29</text>
<section_header_level_1><location><page_5><loc_5><loc_16><loc_34><loc_17></location>Signal and displacement analysis of Sun shadow</section_header_level_1>
<text><location><page_5><loc_5><loc_4><loc_49><loc_16></location>TheSunshadow was analyzed on a sky map with 0 : 025 /C14 3 0 : 025 /C14 grid spacing along the longitude and latitude in the GSE coordinate system. The background was estimated using the equal zenith angle method. 50 Speci /uniFB01 cally, there was one on-source window centered around the Sun and 20 off-source windows aligned at the same zenith angle to cover all azimuthal angles. In each grid cell of the on-source window, the background events were estimated according to the number of cosmic rays at the same grid points in the 20 offsource windows. Combined with the data selection described in the results and discussion, wecollected 1.1 billion cosmic-ray events in on- and off-source windows in the data sample. The energy of these events was obtained from the energy distribution of the simulated pri-</text>
<text><location><page_5><loc_51><loc_58><loc_95><loc_60></location>mary cosmic rays. The median energy was /C24 40 TeV, with a 68% interval between 22 and 107 TeV. The angular resolution containing 68% of the events was 0.5 /C14 .</text>
<text><location><page_5><loc_51><loc_48><loc_95><loc_57></location>The signal events were extracted using a smoothing procedure with Gaussian weighting, and the corresponding signi /uniFB01 cance was estimated using the Li and Ma formula (see Equation 2.5 in Nan and Chen 51 ). The signi /uniFB01 cance of the Sun shadow varies with the day, with an average daily signi /uniFB01 cance reaching 8.4 s , ranging from 5 s to 13.8 s . The displacement of the Sun shadow was estimated by a likelihood ratio test between the one-source model and background-only model, 52 with the displacement of the Sun shadow along the west-east and north-south directions as free parameters.</text>
<section_header_level_1><location><page_5><loc_51><loc_45><loc_74><loc_46></location>Monte Carlo simulation of Sun shadow</section_header_level_1>
<text><location><page_5><loc_51><loc_36><loc_95><loc_45></location>For the Monte Carlo simulations of KM2A detection of cosmic-ray showers, the cosmicray primary chemical composition and energy spectrum were speci /uniFB01 ed according to the model of Gaisser et al. 53 The cascade processes induced by the cosmic-ray interactions within the atmosphere and the response of the detector were simulated by the CORSIKA 54 and G4KM2A 52,55 codes, respectively. Then, the primary cosmic-ray properties were reconstructed and selected using the same methods and conditions as in the basic observation 51 and in this work.</text>
<text><location><page_5><loc_51><loc_17><loc_95><loc_36></location>During the Sun shadow simulation, the Sun was tracked in real time. Above the atmosphere, an average of 1 : 27 3 10 7 particles with opposite charge to the primary cosmic rays were isotropically thrown back around the Sun ' s direction within a window of 10 /C14 3 10 /C14 . The simulation included magnetic /uniFB01 elds along the Sun-Earth line and was performed on each day of data collection. The paths of the particles along the Sun-Earth line were tracked according to the changes in momentum and position following the Newton-Lorentz equation. In addition to the IMF, the CMF and geomagnetic /uniFB01 eld were considered in the Sun shadow simulation (the CMF has been described previously). The geomagnetic /uniFB01 eld calculation followed the International Geomagnetic Reference Field-13 (IGRF-13), 56 in which 13 (2) orders of spherical harmonic expansion were used for /uniFB01 elds below (above) 600 km from the Earth ' s surface. When a particle hits the Sun, it is counted as a Sun shadow signal coming from opposite to the initial throwing direction. Finally, the initial throwing directions were smeared using the KM2A angular resolution. 18,57</text>
<section_header_level_1><location><page_5><loc_51><loc_14><loc_59><loc_15></location>REFERENCES</section_header_level_1>
<unordered_list>
<list_item><location><page_5><loc_51><loc_12><loc_95><loc_14></location>1. Hale, G.E. (1908). Solar vortices and the Zeeman effect. Publ. Astron. Soc. Pac. 20 (121): 220 -224. https://doi.org/10.1086/121822.</list_item>
<list_item><location><page_5><loc_51><loc_9><loc_95><loc_12></location>2. Pevtsov, A.A., Bertello, L., Nagovitsyn, Y.A., et al. (2021). Long-term studies of photospheric magnetic /uniFB01 elds on the Sun. J. Space Weather Space Clim. 11 :4.https://doi.org/10.1051/ swsc/2020069.</list_item>
<list_item><location><page_5><loc_51><loc_6><loc_95><loc_8></location>3. Yang, Z., Bethge, C., Tian, H., et al. (2020). Global maps of the magnetic /uniFB01 eld in the solar corona. Science 369 (6504): 694 -697. https://doi.org/10.1126/science.abb4462.</list_item>
<list_item><location><page_5><loc_51><loc_4><loc_95><loc_6></location>4. Parker, E.N. (1958). Dynamics of the interplanetary gas and magnetic /uniFB01 elds. Astrophys. J. 128 (3): 664 -676. https://doi.org/10.1086/146579.</list_item>
<list_item><location><page_6><loc_6><loc_92><loc_49><loc_94></location>5. Owens, M.J., and Forsyth, R.J. (2013). The heliospheric magnetic /uniFB01 eld. Living Rev. Sol. Phys. 10 (1): 5. https://doi.org/10.12942/lrsp-2013-5.</list_item>
<list_item><location><page_6><loc_6><loc_90><loc_49><loc_92></location>6. Temmer, M. (2021). Space weather: the solar perspective. Living Rev. Sol. Phys. 18 (1): 4. https://doi.org/10.1007/s41116-021-00030-3.</list_item>
<list_item><location><page_6><loc_6><loc_88><loc_49><loc_90></location>7. Verscharen, D., Klein, K.G., and Maruca, B.A. (2019). The multi-scale nature of the solar wind. Living Rev. Sol. Phys. 16 (1): 5. https://doi.org/10.1007/s41116-019-0021-0.</list_item>
<list_item><location><page_6><loc_6><loc_84><loc_49><loc_87></location>8. Fox, N.J., Velli, M.C., Bale, S.D., et al. (2016). The solar probe plus mission: humanity ' s /uniFB01 rst visit to our star. Space Sci. Rev. 204 (1): 7 -48. https://doi.org/10.1007/s11214015-0211-6.</list_item>
<list_item><location><page_6><loc_6><loc_82><loc_49><loc_84></location>9. Schatten, K.H., Wilcox, J.M., and Ness, N.F. (1969). A model of interplanetary and coronal magnetic /uniFB01 elds. Sol. Phys. 6 (3): 442 -455. https://doi.org/10.1007/BF00146478.</list_item>
<list_item><location><page_6><loc_5><loc_80><loc_49><loc_82></location>10. Altschuler, M.D., and Newkirk, G. (1969). Magnetic /uniFB01 elds and the structure of the solar corona. Sol. Phys. 9 (1): 131 -149. https://doi.org/10.1007/BF00145734.</list_item>
<list_item><location><page_6><loc_5><loc_76><loc_49><loc_79></location>11. Jiang, C., Feng, X., Guo, Y., et al. (2022). Data-driven modeling of solar coronal magnetic /uniFB01 eld evolution and eruptions. Innovation 3 (3): 100236. https://doi.org/10.1016/j.xinn.2022. 100236.</list_item>
<list_item><location><page_6><loc_5><loc_74><loc_49><loc_76></location>12. Clark, G.W. (1957). Arrival directions of cosmic-ray air showers from the northern sky. Phys. Rev. 108 (2): 450 -457. https://doi.org/10.1103/PhysRev.108.450.</list_item>
<list_item><location><page_6><loc_5><loc_71><loc_49><loc_74></location>13. Amenomori, M., Ayabe, S., Cui, S.W., et al. (2006). Variation of Sun shadow in the Solar Cycle 23 observed with the Tibet air shower array. Adv. Space Res. 38 (5): 936 -941. https://doi. org/10.1016/j.asr.2006.04.023.</list_item>
<list_item><location><page_6><loc_5><loc_67><loc_49><loc_70></location>14. Chen, S.Z., and Nan, Y.C. (2017). Measurement of the solar magnetic /uniFB01 eld effect on cosmic rays using the Sun shadow observed by the ARGO-YBJ experiment. In 35th International Cosmic Ray Conference.</list_item>
<list_item><location><page_6><loc_5><loc_64><loc_49><loc_67></location>15. Alfaro, R., Alvarez, C., Arteaga-Velázquez, J.C., et al. (2024). Exploring the coronal magnetic /uniFB01 eld with Galactic cosmic rays: the Sun shadow observed by HAWC. Astrophys. J. 966 (1): 67. https://doi.org/10.3847/1538-4357/ad3208.</list_item>
<list_item><location><page_6><loc_5><loc_60><loc_49><loc_63></location>16. Amenomori, M., Bi, X.J., Chen, D., et al. (2013). Probe of the solar magnetic /uniFB01 eld using the "cosmic-ray shadow" of the Sun. Phys. Rev. Lett. 111 (1): 011101. https://doi.org/10. 1103/PhysRevLett.111.011101.</list_item>
<list_item><location><page_6><loc_5><loc_56><loc_49><loc_60></location>17. Aartsen, M.G., Abbasi, R., Ackermann, M., et al. (2021). Measurements of the time-dependent cosmic-ray Sun shadow with seven years of IceCube data: comparison with the Solar cycle and magnetic /uniFB01 eld models. Phys. Rev. D 103 (4): 042005. https://doi.org/10.1103/ PhysRevD.103.042005.</list_item>
<list_item><location><page_6><loc_5><loc_53><loc_49><loc_55></location>18. Nan, Y.C., Feng, C.F., Chen, S.Z., et al. (2019). Study of the solar magnetic /uniFB01 eld in /uniFB02 uence on the cosmic ray Sun shadow. In 36th International Cosmic Ray Conference.</list_item>
<list_item><location><page_6><loc_5><loc_50><loc_49><loc_53></location>19. Becker Tjus, J., Desiati, P., Döpper, N., et al. (2020). Cosmic-ray propagation around the Sun: investigating the in /uniFB02 uence of the solar magnetic /uniFB01 eld on the cosmic-ray Sun shadow. Astron. Astrophys. 633 : A83. https://doi.org/10.1051/0004-6361/201936306.</list_item>
<list_item><location><page_6><loc_5><loc_46><loc_49><loc_50></location>20. Amenomori, M., Bi, X.J., Chen, D., et al. (2018). In /uniFB02 uence of Earth-directed coronal mass ejections on the Sun ' s shadow observed by the Tibet-III air shower array. Astrophys. J. 860 (1): 13. https://doi.org/10.3847/1538-4357/aac2e6.</list_item>
<list_item><location><page_6><loc_5><loc_43><loc_49><loc_46></location>21. Amenomori, M., Cao, Z., Ding, L.K., et al. (1993). Direct evidence of the interplanetary magnetic /uniFB01 eld effect on the cosmic-ray shadow by the Sun. Astrophys. J. 415 (2): L147 -L150. https://doi.org/10.1086/187054.</list_item>
<list_item><location><page_6><loc_5><loc_41><loc_49><loc_43></location>22. Amenomori, M., Dai, B.Z., Ding, L.K., et al. (1996). Shadowing of cosmic rays by the Sun near maximum or at the declining phase of solar activity. Astrophys. J. 464 (2): 954 -958.</list_item>
<list_item><location><page_6><loc_5><loc_37><loc_49><loc_40></location>23. Amenomori, M., Ayabe, S., Ding, L.K., et al. (2000). A study of the shadowing of Galactic cosmic rays by the Sun in a quiet phase of solar activity with the Tibet air shower array. Astrophys. J. 541 (2): 1051 -1058. https://doi.org/10.1086/309479.</list_item>
<list_item><location><page_6><loc_5><loc_34><loc_49><loc_37></location>24. Aielli, G., Bacci, C., Bartoli, B., et al. (2011). Mean interplanetary magnetic /uniFB01 eld measurement using the ARGO-YBJ experiment. Astrophys. J. 729 (2): 113. https://doi.org/10.1088/0004637X/729/2/113.</list_item>
<list_item><location><page_6><loc_5><loc_30><loc_49><loc_34></location>25. Amenomori, M., Bi, X.J., Chen, D., et al. (2018). Evaluation of the interplanetary magnetic /uniFB01 eld strength using the cosmic-ray shadow of the Sun. Phys. Rev. Lett. 120 (3): 031101. https:// doi.org/10.1103/PhysRevLett.120.031101.</list_item>
<list_item><location><page_6><loc_5><loc_27><loc_49><loc_30></location>26. Saeed, M., Zha, M., and Cao, Z. (2017). Simulation of the Galactic cosmic ray shadow of the Sun. Chin. Phys. Lett. 34 (12): 129601. https://doi.org/10.1088/0256-307X/34/12/ 129601.</list_item>
<list_item><location><page_6><loc_5><loc_24><loc_49><loc_27></location>27. Cao, Z., Chen, M.J., Chen, S.Z., et al. (2019). Introduction to large high altitude air shower observatory (LHAASO). Chin. Astron. Astrophys. 43 (4): 457 -478. https://doi.org/10.1016/j.chinastron.2019.11.001.</list_item>
<list_item><location><page_6><loc_5><loc_21><loc_49><loc_23></location>28. He, H. (2018). Design of the LHAASO detectors. Radiat. Detect. Technol. Methods 2 (1): 7. https://doi.org/10.1007/s41605-018-0037-3.</list_item>
<list_item><location><page_6><loc_5><loc_18><loc_49><loc_21></location>29. Cao, Z., Aharonian, F., and Zuo, X. (2025). Data quality control system and long-term performance monitor of LHAASO-KM2A. Astropart. Phys. 164 : 103029. https://doi.org/10.1016/j. astropartphys.2024.103029.</list_item>
<list_item><location><page_6><loc_5><loc_14><loc_49><loc_18></location>30. King, J.H., and Papitashvili, N.E. (2005). Solar wind spatial scales in and comparisons of hourly Wind and ACE plasma and magnetic /uniFB01 eld data. J. Geophys. Res. 110 : A02104. https://doi.org/10.1029/2004JA010649.</list_item>
<list_item><location><page_6><loc_5><loc_11><loc_49><loc_14></location>31. Edelson, R.A., and Krolik, J.H. (1988). The discrete correlation function: a new method for analyzing unevenly sampled variability data. Astrophys. J. 333 (2): 646 -659. https://doi. org/10.1086/166773.</list_item>
<list_item><location><page_6><loc_5><loc_8><loc_49><loc_11></location>32. Xiao, S., Xiong, S.L., Zhang, S.N., et al. (2021). Enhanced localization of transients based on a novel cross-correlation method. Astrophys. J. 920 (1): 43. https://doi.org/10.3847/15384357/ac1420.</list_item>
<list_item><location><page_6><loc_5><loc_4><loc_49><loc_7></location>33. Robertson, D.R.S., Gallo, L.C., Zoghbi, A., et al. (2015). Searching for correlations in simultaneous X-ray and UV emission in the narrow-line Seyfert 1 galaxy 1H 0707 -495. Mon. Not. Roy. Astron. Soc. 453 (4): 3456 -3461. https://doi.org/10.1093/mnras/stv1575.</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_6><loc_51><loc_92><loc_95><loc_94></location>34. Timmer, J., and König, M. (1995). On generating power law noise. Astron. Astrophys. 300 : 707 -710.</list_item>
<list_item><location><page_6><loc_51><loc_89><loc_95><loc_92></location>35. Shen, F., Yang, Z., Zhang, J., et al. (2018). Three-dimensional MHD simulation of solar wind using a new boundary treatment: comparison with in situ data at Earth. Astrophys. J. 866 (1): 18. https://doi.org/10.3847/1538-4357/aad806.</list_item>
<list_item><location><page_6><loc_51><loc_85><loc_95><loc_89></location>36. Li, H., Feng, X., and Wei, F. (2021). Comparison of synoptic maps and PFSS solutions for the declining phase of Solar Cycle 24. J. Geophys. Res. 126 (3): e28870. https://doi.org/10. 1029/2020JA028870.</list_item>
<list_item><location><page_6><loc_51><loc_82><loc_95><loc_85></location>37. Ruffolo, D., Matthaeus, W.H., Chhiber, R., et al. (2020). Shear-driven transition to isotropically turbulent solar wind outside the Alfvén critical zone. Astrophys. J. 902 (2): 94. https://doi. org/10.3847/1538-4357/abb594.</list_item>
<list_item><location><page_6><loc_51><loc_80><loc_95><loc_82></location>38. Smith, C.W., and Bieber, J.W. (1991). Solar cycle variation of the interplanetary magnetic /uniFB01 eld spiral. Astrophys. J. 370 (1): 435 -441. https://doi.org/10.1086/169830.</list_item>
<list_item><location><page_6><loc_51><loc_76><loc_95><loc_79></location>39. Mitchell, J.G., Leske, R.A., Nolfo, G.A.D., et al. (2022). First measurements of Jovian electrons by Parker Solar Probe/IS 1 IS within 0.5 au of the Sun. Astrophys. J. 933 (2): 171. https://doi. org/10.3847/1538-4357/ac75ce.</list_item>
<list_item><location><page_6><loc_51><loc_73><loc_95><loc_76></location>40. Richardson, I.G., and Cane, H.V. (2010). Near-Earth interplanetary coronal mass ejections during Solar Cycle 23 (1996 -2009): catalog and summary of properties. Sol. Phys. 264 (1): 189 -237. https://doi.org/10.1007/s11207-010-9568-6.</list_item>
<list_item><location><page_6><loc_51><loc_72><loc_92><loc_72></location>41. Alvestad, J. (2021). Coronal hole history. https://solen.info/solar/coronal_holes.html.</list_item>
<list_item><location><page_6><loc_51><loc_68><loc_95><loc_71></location>42. Hajra, R., Sunny, J.V., Babu, M., et al. (2022). Interplanetary sheaths and corotating interaction regions: a comparative statistical study on their characteristics and geoeffectiveness. Sol. Phys. 297 (7): 97. https://doi.org/10.1007/s11207-022-02020-6.</list_item>
<list_item><location><page_6><loc_51><loc_65><loc_95><loc_68></location>43. Fisk, L.A. (1996). Motion of the footpoints of heliospheric magnetic /uniFB01 eld lines at the Sun: Implications for recurrent energetic particle events at high heliographic latitudes. J. Geophys. Res. 101 (A7): 15547 -15553. https://doi.org/10.1029/96JA01005.</list_item>
<list_item><location><page_6><loc_51><loc_61><loc_95><loc_64></location>44. Forsyth, R.J., Balogh, A., and Smith, E.J. (2002). The underlying direction of the heliospheric magnetic /uniFB01 eld through the Ulysses /uniFB01 rst orbit. J. Geophys. Res. 107 (A11): 1405. https://doi. org/10.1029/2001JA005056.</list_item>
<list_item><location><page_6><loc_51><loc_58><loc_95><loc_61></location>45. Bale, S.D., Badman, S.T., Bonnell, J.W., et al. (2019). Highly structured slow solar wind emerging from an equatorial coronal hole. Nature 576 (7786): 237 -242. https://doi.org/ 10.1038/s41586-019-1818-7.</list_item>
<list_item><location><page_6><loc_51><loc_54><loc_95><loc_58></location>46. Burlaga, L.F., Lepping, R.P., Behannon, K.W., et al. (1982). Large-scale variations of the interplanetary magnetic /uniFB01 eld: Voyager 1 and 2 observations between 1 -5 AU. J. Geophys. Res. 87 (A6): 4345 -4353. https://doi.org/10.1029/JA087iA06p04345.</list_item>
<list_item><location><page_6><loc_51><loc_52><loc_95><loc_54></location>47. Ragot, B.R. (2006). Distributions of magnetic /uniFB01 eld orientations in the turbulent solar wind. Astrophys. J. 651 (2): 1209 -1218. https://doi.org/10.1086/507783.</list_item>
<list_item><location><page_6><loc_51><loc_48><loc_95><loc_52></location>48. Shen, F., Feng, X., Wu, S.T., et al. (2007). Three-dimensional MHD simulation of CMEs in three-dimensional background solar wind with the self-consistent structure on the source surface as input: numerical simulation of the January 1997 Sun-Earth connection event. J. Geophys. Res. 112 (A6): A06109. https://doi.org/10.1029/2006JA012164.</list_item>
<list_item><location><page_6><loc_51><loc_45><loc_95><loc_47></location>49. Yang, R. (2022). LHAASO and Galactic cosmic rays. Innovation 3 (4): 100260. https://doi. org/10.1016/j.xinn.2022.100260.</list_item>
<list_item><location><page_6><loc_51><loc_42><loc_95><loc_45></location>50. Amenomori, M., Cao, Z., Ding, L.K., et al. (1993). Cosmic-ray de /uniFB01 cit from the directions of the Moonandthe Sun detected with the Tibet air-shower array. Phys. Rev. D 47 (7): 2675 -2681. https://doi.org/10.1103/PhysRevD.47.2675.</list_item>
<list_item><location><page_6><loc_51><loc_40><loc_95><loc_42></location>51. Nan, Y.C., and Chen, S.Z. (2017). A study of the methods for signal signi /uniFB01 cance estimation in ground-based gamma-ray detectors. In 35th International Cosmic Ray Conference.</list_item>
<list_item><location><page_6><loc_51><loc_36><loc_95><loc_39></location>52. Aharonian, F., An, Q., and Zuo, X. (2021). Observation of the Crab Nebula with LHAASOKM2A /C0 a performance study. Chin. Phys. C 45 (2): 025002. https://doi.org/10.1088/ 1674-1137/abd01b.</list_item>
<list_item><location><page_6><loc_51><loc_33><loc_95><loc_36></location>53. Gaisser, T.K., Stanev, T., and Tilav, S. (2013). Cosmic ray energy spectrum from measurements of air showers. Front. Physiol. 8 (6): 748 -758. https://doi.org/10.1007/s11467-0130319-7.</list_item>
<list_item><location><page_6><loc_51><loc_30><loc_95><loc_32></location>54. Heck, D., Knapp, J., Capdevielle, J.N., et al. (1998). CORSIKA: a Monte Carlo code to simulate extensive air showers (Forschungszentrum Karlsruhe GmbH).</list_item>
<list_item><location><page_6><loc_51><loc_28><loc_95><loc_30></location>55. Cao, Z., Aharonian, F., An, Q., et al. (2024). LHAASO-KM2A detector simulation using Geant4. Radiat. Detect. Technol. Methods. https://doi.org/10.1007/s41605-024-00467-8.</list_item>
<list_item><location><page_6><loc_51><loc_25><loc_95><loc_28></location>56. Alken, P., Thébault, E., Beggan, C.D., et al. (2021). International geomagnetic reference /uniFB01 eld: the thirteenth generation. Earth Planets Space 73 (1): 49. https://doi.org/10.1186/s40623020-01288-x.</list_item>
<list_item><location><page_6><loc_51><loc_21><loc_95><loc_24></location>57. Nan, Y.C., Chen, S.Z., and Feng, C.F. (2021). Measurement of interplanetary magnetic /uniFB01 eld in short period using the cosmic-ray Sun shadow measured by LHAASO. In 37th International Cosmic Ray Conference.</list_item>
</unordered_list>
<section_header_level_1><location><page_6><loc_51><loc_17><loc_64><loc_18></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_6><loc_51><loc_4><loc_95><loc_17></location>We would like to thank all staff members who worked at the LHAASO site, situated more than 4,400 m above sea level, for their year-round dedication to maintaining the detector and ensuring smooth operation of the water recycling system, electricity power supply, and other experimental components. We are grateful to the Chengdu Management Committee of Tianfu New Area for their constant /uniFB01 nancial support for research using LHAASO data. We appreciate the computing and data service support provided by the National High Energy Physics Data Center for the data analysis presented in this paper. We also thank the OMNI and GONG groups for providing the data, SSW and IAGA for providing the codes, and Shen Zhenning for his guidance on using Smith and Bieber ' smodi /uniFB01 ed model. This research was supported by the following grants: The National Key R&D</text>
<section_header_level_1><location><page_7><loc_6><loc_96><loc_11><loc_97></location>REPORT</section_header_level_1>
<text><location><page_7><loc_5><loc_87><loc_49><loc_94></location>Program of China (no. 2018YFA0404201), National Natural Science Foundation of China (nos. 12393851, 12393854, 12175121, 12205314, 12105301, 12305120, 12261160362, 12105294, U1931201, and 12375107), China Postdoctoral Science Foundation (no. 2022M723150), National Science and Technology Development Agency of Thailand, and National Research Council of Thailand under the High-Potential Research Team Grant Program (N42A650868).</text>
<section_header_level_1><location><page_7><loc_5><loc_84><loc_21><loc_85></location>AUTHOR CONTRIBUTIONS</section_header_level_1>
<text><location><page_7><loc_5><loc_77><loc_49><loc_84></location>Y.N., S.C., and C.F. led the writing of the text for the data analysis and interpretation. S.C. initiated this work. Y.N. analyzed the KM2A data under the guidance of S.C. and C.F., while J.X. crosschecked the results. C.J. and Y.Y. offered expertise on solar magnetic models. Z.C. acted as spokesperson for the LHAASO Collaboration and principal investigator of the LHAASO project. All other authors participated in data analysis (including detector calibra-</text>
<text><location><page_7><loc_51><loc_92><loc_95><loc_94></location>tion, data processing, event reconstruction, data quality checks, and various simulations) and provided comments on the manuscript.</text>
<section_header_level_1><location><page_7><loc_51><loc_89><loc_69><loc_90></location>DECLARATION OF INTERESTS</section_header_level_1>
<text><location><page_7><loc_52><loc_88><loc_74><loc_89></location>The authors declare no competing interests.</text>
<section_header_level_1><location><page_7><loc_51><loc_85><loc_70><loc_86></location>SUPPLEMENTAL INFORMATION</section_header_level_1>
<text><location><page_7><loc_52><loc_84><loc_86><loc_85></location>It can be found online at https://doi.org/10.1016/j.xinn.2024.100695.</text>
<section_header_level_1><location><page_7><loc_51><loc_81><loc_66><loc_82></location>LEAD CONTACT WEBSITE</section_header_level_1>
<text><location><page_7><loc_52><loc_77><loc_80><loc_81></location>Yuncheng Nan: https://orcid.org/0000-0002-7363-0252 Songzhan Chen: https://orcid.org/0000-0003-0703-1275 Cunfeng Feng: https://orcid.org/0000-0001-9138-3200</text>
<section_header_level_1><location><page_8><loc_46><loc_90><loc_54><loc_91></location>Author List</section_header_level_1>
<text><location><page_8><loc_15><loc_31><loc_85><loc_88></location>Zhen Cao 1,2,3 ,F. Aharonian 4,5 ,Axikegu 6 ,Y.X. Bai 1,3 ,Y.W. Bao 7 ,D. Bastieri 8 ,X.J. Bi 1,2,3 ,Y.J. Bi 1,3 ,W. Bian 9 ,A.V. Bukevich 10 ,Q. Cao 11 ,W.Y. Cao 12 ,Zhe Cao 13,12 ,J. Chang 14 ,J.F. Chang 1,3,13 ,A.M. Chen 9 ,E.S. Chen 1,2,3 ,H.X. Chen 15 ,Liang Chen 16 ,Lin Chen 6 ,Long Chen 6 ,M.J. Chen 1,3 ,M.L. Chen 1,3,13 ,Q.H. Chen 6 ,S. Chen 17 ,S.H. Chen 1,2,3 ,S.Z. Chen 1,3 ,T.L. Chen 18 ,Y. Chen 7 ,N. Cheng 1,3 ,Y.D. Cheng 1,2,3 ,M.Y. Cui 14 ,S.W. Cui 11 ,X.H. Cui 19 ,Y.D. Cui 20 ,B.Z. Dai 17 ,H.L. Dai 1,3,13 ,Z.G. Dai 12 ,Danzengluobu 18 ,X.Q. Dong 1,2,3 ,K.K. Duan 14 ,J.H. Fan 8 ,Y.Z. Fan 14 ,J. Fang 17 ,J.H. Fang 15 ,K. Fang 1,3 ,C.F. Feng 21 ,H. Feng 1 ,L. Feng 14 ,S.H. Feng 1,3 ,X.T. Feng 21 ,Y. Feng 15 ,Y.L. Feng 18 ,S. Gabici 22 ,B. Gao 1 , 3 ,C.D. Gao 21 ,Q. Gao 18 ,W. Gao 1,3 ,W.K. Gao 1,2,3 ,M.M. Ge 17 ,L.S. Geng 1,3 ,G. Giacinti 9 ,G.H. Gong 23 ,Q.B. Gou 1,3 ,M.H. Gu 1,3,13 ,F.L. Guo 16 ,X.L. Guo 6 ,Y.Q. Guo 1,3 ,Y.Y. Guo 14 ,Y.A. Han 24 ,M. Hasan 1,2,3 ,H.H. He 1,2,3 ,H.N. He 14 ,J.Y. He 14 ,Y. He 6 ,Y.K. Hor 20 ,B.W. Hou 1,2,3 ,C. Hou 1,3 ,X. Hou 25 ,H.B. Hu 1,2,3 ,Q. Hu 12,14 ,S.C. Hu 1,3,26 ,D.H. Huang 6 ,T.Q. Huang 1,3 ,W.J. Huang 20 ,X.T. Huang 21 ,X.Y. Huang 14 ,Y. Huang 1,2,3 ,X.L. Ji 1,3,13 ,H.Y. Jia 6 ,K. Jia 21 ,K. Jiang 13,12 ,X.W. Jiang 1,3 ,Z.J. Jiang 17 ,M. Jin 6 ,M.M. Kang 27 ,I. Karpikov 10 ,D. Kuleshov 10 ,K. Kurinov 10 ,B.B. Li 11 ,C.M. Li 7 ,Cheng Li 13,12 ,Cong Li 1,3 ,D. Li 1,2,3 ,F. Li 1,3,13 ,H.B. Li 1,3 ,H.C. Li 1,3 ,Jian Li 12 ,Jie Li 1,3,13 ,K. Li 1,3 ,S.D. Li 16,2 ,W.L. Li 21 ,W.L. Li 9 ,X.R. Li 1,3 ,Xin Li 13,12 ,Y.Z. Li 1,2,3 ,Zhe Li 1,3 ,Zhuo Li 28 ,E.W. Liang 29 ,Y.F. Liang 29 ,S.J. Lin 20 ,B. Liu 12 ,C. Liu 1,3 ,D. Liu 21 ,D.B. Liu 9 ,H. Liu 6 ,H.D. Liu 24 ,J. Liu 1,3 ,J.L. Liu 1,3 ,M.Y. Liu 18 ,R.Y. Liu 7 ,S.M. Liu 6 ,W. Liu 1,3 ,Y. Liu 8 ,Y.N. Liu 23 ,Q. Luo 20 ,Y. Luo 9 ,H.K. Lv 1,3 ,B.Q. Ma 28 ,L.L. Ma 1,3 ,X.H. Ma 1,3 ,J.R. Mao 25 ,Z. Min 1,3 ,W. Mitthumsiri 30 ,H.J. Mu 24 ,Y.C. Nan 1,3 ,A. Neronov 22 ,L.J. Ou 8 ,P. Pattarakijwanich 30 ,Z.Y. Pei 8 ,J.C. Qi 1,2,3 ,M.Y. Qi 1,3 ,B.Q. Qiao 1,3 ,J.J. Qin 12 ,A. Raza 1,2,3 ,D. Ruffolo 30 ,A. S'aiz 30 ,M. Saeed 1,2,3 ,D. Semikoz 22 ,L. Shao 11 ,O. Shchegolev 10,31 ,X.D. Sheng 1,3 ,F.W. Shu 32 ,H.C. Song 28 ,Yu.V. Stenkin 10,31 ,V. Stepanov 10 ,Y. Su 14 ,D.X. Sun 12,14 ,Q.N. Sun 6 ,X.N. Sun 29 ,Z.B. Sun 33 ,J. Takata 34 ,P.H.T. Tam 20 ,Q.W. Tang 32 ,R. Tang 9 ,Z.B. Tang 13,12 ,W.W. Tian 2,19 ,C. Wang 33 ,C.B. Wang 6 ,G.W. Wang 12 ,H.G. Wang 8 ,H.H. Wang 20 ,J.C. Wang 25 ,Kai Wang 7 ,Kai Wang 34 ,L.P. Wang 1,2,3 ,L.Y. Wang 1,3 ,P.H. Wang 6 ,R. Wang 21 ,W. Wang 20 ,X.G. Wang 29 ,X.Y. Wang 7 ,Y. Wang 6 ,Y.D. Wang 1,3 ,Y.J. Wang 1,3 ,Z.H. Wang 27 ,Z.X. Wang 17 ,Zhen Wang 9 ,Zheng Wang 1,3,13 ,D.M. Wei 14 ,J.J. Wei 14 ,Y.J. Wei 1,2,3 ,T. Wen 17 ,C.Y. Wu 1,3 ,H.R. Wu 1,3 ,Q.W. Wu 34 ,S. Wu 1,3 ,X.F. Wu 14 ,Y.S. Wu 12 ,S.Q. Xi 1,3 ,J. Xia 12,14 ,G.M. Xiang 16,2 ,D.X. Xiao 11 ,G. Xiao 1,3 ,Y.L. Xin 6 ,Y. Xing 16 ,D.R. Xiong 25 ,Z. Xiong 1,2,3 ,D.L. Xu 9 ,R.F. Xu 1,2,3 ,R.X. Xu 28 ,W.L. Xu 27 ,L. Xue 21 ,D.H. Yan 17 ,J.Z. Yan 14 ,T. Yan 1,3 ,C.W. Yang 27 ,C.Y. Yang 25 ,F. Yang 11 ,F.F. Yang 1,3,13 ,L.L. Yang 20 ,M.J. Yang 1,3 ,R.Z. Yang 12 ,W.X. Yang 8 ,Y.H. Yao 1,3 ,Z.G. Yao 1,3 ,L.Q. Yin 1,3 ,N. Yin 21 ,X.H. You 1,3 ,Z.Y. You 1,3 ,Y.H. Yu 12 ,Q. Yuan 14 ,H. Yue 1,2,3 ,H.D. Zeng 14 ,T.X. Zeng 1,3,13 ,W. Zeng 17 ,M. Zha 1,3 ,B.B. Zhang 7 ,F. Zhang 6 ,H. Zhang 9 ,H.M. Zhang 7 ,H.Y. Zhang 1,3 ,J.L. Zhang 19 ,Li Zhang 17 ,P.F. Zhang 17 ,P.P. Zhang 12,14 ,R. Zhang 12,14 ,S.B. Zhang 2,19 ,S.R. Zhang 11 ,S.S. Zhang 1,3 ,X. Zhang 7 ,X.P. Zhang 1,3 ,Y.F. Zhang 6 ,Yi Zhang 1,14 ,Yong Zhang 1,3 ,B. Zhao 6 ,J. Zhao 1,3 ,L. Zhao 13,12 ,L.Z. Zhao 11 ,S.P. Zhao 14 ,X.H. Zhao 25 ,F. Zheng 33 ,W.J. Zhong 7 ,B. Zhou 1,3 ,H. Zhou 9 ,J.N. Zhou 16 ,M. Zhou 32 ,P. Zhou 7 ,R. Zhou 27 ,X.X. Zhou 1,2,3 ,X.X. Zhou 6 ,B.Y. Zhu 12,14 ,C.G. Zhu 21 ,F.R. Zhu 6 ,H. Zhu 19 ,K.J. Zhu 1,2,3,13 ,Y.C. Zou 34 ,X. Zuo 1,3 ,</text>
<text><location><page_8><loc_41><loc_29><loc_59><loc_30></location>(The LHAASO Collaboration)</text>
<text><location><page_8><loc_42><loc_25><loc_58><loc_26></location>C.W. Jiang 35 and Y. Yang 33 .</text>
<unordered_list>
<list_item><location><page_9><loc_15><loc_90><loc_48><loc_91></location>Jiaotong University, Chengdu 610031, Sichuan, China</list_item>
<list_item><location><page_9><loc_15><loc_88><loc_73><loc_89></location>7 School of Astronomy and Space Science, Nanjing University, Nanjing 210023, Jiangsu, China</list_item>
<list_item><location><page_9><loc_15><loc_86><loc_70><loc_88></location>8 Center for Astrophysics, Guangzhou University, Guangzhou 510006, Guangdong, China</list_item>
<list_item><location><page_9><loc_15><loc_82><loc_85><loc_86></location>9 Tsung-Dao Lee Institute & School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China</list_item>
<list_item><location><page_9><loc_15><loc_81><loc_70><loc_82></location>10 Institute for Nuclear Research of Russian Academy of Sciences, Moscow 117312, Russia</list_item>
<list_item><location><page_9><loc_15><loc_79><loc_54><loc_80></location>11 Hebei Normal University, Shijiazhuang 050024, Hebei, China</list_item>
<list_item><location><page_9><loc_15><loc_77><loc_63><loc_78></location>12 University of Science and Technology of China, Hefei 230026, Anhui, China</list_item>
<list_item><location><page_9><loc_15><loc_75><loc_56><loc_76></location>13 State Key Laboratory of Particle Detection and Electronics, China</list_item>
<list_item><location><page_9><loc_15><loc_71><loc_85><loc_75></location>14 Key Laboratory of Dark Matter and Space Astronomy & Key Laboratory of Radio Astronomy, Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023, Jiangsu, China</list_item>
<list_item><location><page_9><loc_15><loc_69><loc_80><loc_71></location>15 Research Center for Astronomical Computing, Zhejiang Laboratory, Hangzhou 311121, Zhejiang, China</list_item>
<list_item><location><page_9><loc_15><loc_66><loc_85><loc_69></location>16 Key Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, China</list_item>
<list_item><location><page_9><loc_15><loc_64><loc_70><loc_65></location>17 School of Physics and Astronomy, Yunnan University, Kunming 650091, Yunnan, China</list_item>
<list_item><location><page_9><loc_15><loc_62><loc_79><loc_63></location>18 Key Laboratory of Cosmic Rays (Tibet University), Ministry of Education, Lhasa 850000, Tibet, China</list_item>
<list_item><location><page_9><loc_15><loc_58><loc_85><loc_62></location>19 Key Laboratory of Radio Astronomy and Technology, National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China</list_item>
<list_item><location><page_9><loc_15><loc_53><loc_85><loc_58></location>20 School of Physics and Astronomy (Zhuhai) & School of Physics (Guangzhou) & Sino-French Institute of Nuclear Engineering and Technology (Zhuhai), Sun Yat-sen University, Zhuhai 519000& Guangzhou 510275, Guangdong, China</list_item>
<list_item><location><page_9><loc_15><loc_51><loc_82><loc_52></location>21 Institute of Frontier and Interdisciplinary Science, Shandong University, Qingdao 266237, Shandong, China</list_item>
<list_item><location><page_9><loc_15><loc_49><loc_79><loc_50></location>22 APC, Universit'e Paris Cit'e, CNRS/IN2P3, CEA/IRFU, Observatoire de Paris, 119 75205 Paris, France</list_item>
<list_item><location><page_9><loc_15><loc_47><loc_65><loc_49></location>23 Department of Engineering Physics, Tsinghua University, Beijing 100084, China</list_item>
<list_item><location><page_9><loc_15><loc_45><loc_76><loc_47></location>24 School of Physics and Microelectronics, Zhengzhou University, Zhengzhou 450001, Henan, China</list_item>
<list_item><location><page_9><loc_15><loc_43><loc_70><loc_45></location>25 Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216, Yunnan, China</list_item>
<list_item><location><page_9><loc_15><loc_42><loc_62><loc_43></location>26 China Center of Advanced Science and Technology, Beijing 100190, China</list_item>
<list_item><location><page_9><loc_15><loc_40><loc_61><loc_41></location>27 College of Physics, Sichuan University, Chengdu 610065, Sichuan, China</list_item>
<list_item><location><page_9><loc_15><loc_38><loc_53><loc_39></location>28 School of Physics, Peking University, Beijing 100871, China</list_item>
<list_item><location><page_9><loc_15><loc_34><loc_85><loc_38></location>29 Guangxi Key Laboratory for Relativistic Astrophysics, School of Physical Science and Technology, Guangxi University, Nanning 530004, Guangxi, China</list_item>
<list_item><location><page_9><loc_15><loc_32><loc_71><loc_34></location>30 Department of Physics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand</list_item>
<list_item><location><page_9><loc_15><loc_31><loc_59><loc_32></location>31 Moscow Institute of Physics and Technology, Moscow 141700, Russia</list_item>
<list_item><location><page_9><loc_15><loc_27><loc_85><loc_30></location>32 Center for Relativistic Astrophysics and High Energy Physics, School of Physics and Materials Science & Institute of Space Science and Technology, Nanchang University, Nanchang 330031, Jiangxi, China</list_item>
<list_item><location><page_9><loc_15><loc_25><loc_69><loc_26></location>33 National Space Science Center, Chinese Academy of Sciences, Beijing 100190, China</list_item>
<list_item><location><page_9><loc_15><loc_23><loc_76><loc_25></location>34 School of Physics, Huazhong University of Science and Technology, Wuhan 430074, Hubei, China</list_item>
<list_item><location><page_9><loc_15><loc_21><loc_82><loc_23></location>35 Institute of Space Science and Applied Technology, Harbin Institute of Technology, Shenzhen 518055, China</list_item>
</document>
[
{
"title": "ABSTRACT",
"content": "REPORT",
"pages": [
1
]
},
{
"title": "Monitoring the daily variation of Sun-Earth magnetic /uniFB01 elds using galactic cosmic rays",
"content": "The LHAASO Collaboration 1, * 1 Further details can be found in the supplemental information *Correspondence: nanyc@ihep.ac.cn (Y.N.); chensz@ihep.ac.cn (S.C.); fengcf@sdu.edu.cn (C.F.) Received: April 10, 2024; Accepted: September 3, 2024; Published Online: September 6, 2024; https://doi.org/10.1016/j.xinn.2024.100695 ª 2024 The Author(s). Published by Elsevier Inc. on behalf of Youth Innovation Co., Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Citation: (2024). Monitoring the daily variation of Sun-Earth magnetic /uniFB01 elds using galactic cosmic rays. The Innovation 5(6) , 100695. The interplanetary magnetic /uniFB01 eld (IMF) between the Sun and Earth is an extension of the solar magnetic /uniFB01 eld carried by the solar wind into interplanetary space. Monitoring variations in the IMF upstream of the Earth would provide very important information for the prediction of space weather effects, such as effects of solar storms and the solar wind, on human activity. In this study, the IMF between the Sun and Earth was measured daily for the /uniFB01 rst time using a cosmic-ray observatory. Cosmic rays mainly consist of charged particles that are de /uniFB02 ected as they pass through a magnetic /uniFB01 eld. Therefore, the cosmic-ray Sun shadow, caused by high-energy charged cosmic rays blocked by the Sun and de /uniFB02 ected by the magnetic /uniFB01 eld, can be used to explore the transverse IMF between the Sun and Earth. By employing the powerful kilometer-square array at the Large High Altitude Air Shower Observatory, the cosmic-ray Sun shadows were observed daily with high signi /uniFB01 -cance for the /uniFB01 rst time. The displacement of the Sun shadow measured in 2021 correlates well with the transverse IMF component measured in situ by spacecraft near the Earth, with a time lag of 3 : 31 ± 0 : 12 days. The displacement of the Sun shadow was also simulated using Parker ' s classic IMF model, yielding a time lag of 2 : 06 ± 0 : 04 days. This deviation may provide valuable insights into the magnetic /uniFB01 eld structure, which can improve space weather research.",
"pages": [
2
]
},
{
"title": "INTRODUCTION",
"content": "The Sun, our nearest star, is the main source of energy for living organisms on Earth, and its activity continually affects our planet ' s environment. As human science and technology advance, along with the increasing use of electronic and space equipment, the impact of solar activity on human activity has steadily grown. Therefore, monitoring solar activity and forecasting space weather are important areas of scienti /uniFB01 c research. Solar magnetic /uniFB01 elds play a vital role in understanding diverse solar activities. Since the /uniFB01 rst measurement of the solar magnetic /uniFB01 eldin1908usingtheZeeman effect, 1 the photospheric magnetic /uniFB01 eld at the Sun ' s surface has been continuously monitored by both space- and ground-based observatories. 2 The coronal magnetic /uniFB01 eld (CMF) lies above the photosphere, and direct measurement remains challenging, despite recent attempts using magnetoseismology. 3 The CMF is carried into interplanetary space by the solar wind, forming the interplanetary magnetic /uniFB01 eld (IMF). 4 The IMF provides valuable information for studying the CMF and is critical for understanding space weather and improving forecasting accuracy. 5,6 Since the discovery of the solar wind in 1962, the IMF has been monitored in situ by a series of spacecraft, with considerable monitoring performed from the L 1 Lagrange point of gravitational balance between the Sun and Earth. 7 Although the ongoing Parker Solar Probe mission can /uniFB02 y from the IMF to the CMF, 8 at locations other than the Sun ' ssurfaceand L 1 point, it remains challenging to continuously monitor the magnetic /uniFB01 eld in the vast space between the Sun and Earth. Currently, the distribution of this magnetic /uniFB01 eld relies on theoretical models that extrapolate the photospheric magnetic /uniFB01 eldtotheCMF(e.g., the classical potential /uniFB01 eld source surface [PFSS] model 9,10 ) and extend the outermost CMF to the IMF (e.g., the classical and widely used Parker model 4 ). The distributions of the CMF and IMF can also be simulated using data-driven models. 11 Very-high-energy galactic cosmic rays, consisting mainly of positively charged particles moving near the speed of light, can travel from the Sun to the Earth within approximately 8 min. Their trajectories are affected by the magnetic /uniFB01 eld along the Sun-Earth line. Therefore, when the cosmic-ray Sun shadow was detected for the /uniFB01 rst time, it was proposed that the magnetic /uniFB01 eld between the Sun and Earth could be studied through measurements of the Sun shadow. 12 It was noted that as observational sensitivity increases, the cosmic-ray de /uniFB01 cit ratio of the Sun shadow was observed or proved to have a relationship with the solar magnetic /uniFB01 eld, including the photospheric magnetic /uniFB01 eld, 13-15 CMF and its annual variations, 16-19 and even coronal mass ejections (CMEs), 20 and it was applied to diagnose different CMF models. 16,17 In addition, studies have observed 21-25 and proved 23-26 that the displacement of the Sun shadow is also related to the IMF. Based on the annual timescale displacement of the Sun shadow, the ARGO-YBJ collaboration measured the structure and strength of the mean IMF near the sunspot minimum in Solar Cycle 24 for the /uniFB01 rst time. 24 They also proposed the possibility of using Sun shadow measurements for space weather forecasting, given the time advantage this method offers over spacecraft at L 1. However, this approach requires cosmicray arrays with enough sensitivity to measure shadows daily. 24 In addition, the AS g collaboration used annual Sun shadow data to correct the average strength of the Parker model for the IMF near total Solar Cycle 23. 25 Owing to the limited observational sensitivity of the Sun shadows, researchers have not yet been able to study /uniFB01 ner structures in the magnetic /uniFB01 eld, particularly during short periods.",
"pages": [
2
]
},
{
"title": "RESULTS AND DISCUSSION",
"content": "The Large High Altitude Air Shower Observatory (LHAASO) is a composite ground-based cosmic-ray detection facility located at 100.01 /C14 E, 29.35 /C14 N, and an altitude of 4,410 m above sea level in Sichuan, China. 27,28 The 1.3 kilometer square array (KM2A) is one of the main ground-based arrays in LHAASO, which has a detection area of 1 to 2 orders of magnitude larger than those of the ARGOYBJ and AS g experiments. KM2A has been operating with a nearly full-duty cycle since the beginning of 2020. In this study, only the data recorded in 2021 near the sunspot minimum in Solar Cycle 25 were used. Owing to the location of the array near the Tropic of Cancer, the Sun shadow could be observed signi /uniFB01 cantly for 196 days from March 21 to October 2, with /C24 7 h observation on each day. During this period, the pointing accuracy, angular resolution, and energy of the cosmic rays measured by the array were stable according to the observation of the cosmic-ray Moon shadow. 29 For monitoring the IMF, we selected events with 26 -251 /uniFB01 red electromagnetic particle detectors (EDs) in KM2A. The corresponding median energy was /C24 40 TeV, and the angular resolution was 0.5 /C14 . The Sun shadow was observed along the longitude and latitude in the geocentric solar ecliptic (GSE) coordinate system. Figure 1A depicts the Sun shadows observed during the 196 days of data collection, with a very high signi /uniFB01 cance exceeding 100 s . Figure 1B displays the Sun shadow obtained from a single day of data, which also demonstrates good signi /uniFB01 cance exceeding 9 s . The daily Sun shadows are clearly de /uniFB02 ected away from the direction of the Sun by the magnetic /uniFB01 eld. The angular distance from the center of the shadow to the Sun is de /uniFB01 ned as the daily displacement of the shadow. Details regarding the Sun shadow analysis are presented in the materials and methods.",
"pages": [
2
]
},
{
"title": "Daily IMFBy measurement using Sun shadow",
"content": "Cosmic rays traveling toward the Sun propagate approximately parallel to the Sun-Earth line. In the GSE coordinate system, the x axis points toward the Sun, the z axis is toward the north ecliptic pole, and the y axis is roughly opposite Earth ' s orbital motion. Positively charged cosmic rays are only affected by the y and z components of the IMF in the GSE coordinate system, but not by the x component. By and Bz cause displacement of the Sun shadow along the north-south and west-east directions, respectively, according to the Lorentz force law. However, the strength of Bz is in /uniFB02 uenced not only by the IMF, but also by the geomagnetic /uniFB01 eld, which has a complex effect on the west-east displacement of the Sun shadow. 23,25 Therefore, the dominant displacement of the Sun shadow is in the north-south direction, caused by By . 24,25 The daily displacement of the Sun shadow along the north-south direction (denoted by D ) and its variation can be monitored by LHAASO, as illustrated in Figure 2A, where only days with a signi /uniFB01 cance exceeding 5 s are shown. The number of effective observation days for D reaches 177 (90% of the total days). The variationindaily D appears to be periodic in each 27-day Carrington rotation (i.e., solar rotation), with the periodicity gradually changing throughout the Carrington rotation. This allows LHAASO to directly test the speci /uniFB01 c correlation between D and IMFBy , potentially enabling the use of D to measure daily By for the /uniFB01 rst time. The By values used in our test are observational results at L 1 from OMNI. 30 Figure 2B displays the daily By values, representing the mean value of hourly By measurements within a 24-h period. The number of ef /uniFB01 ciently observed days for By is 181 (92% of the total observation period). As presented in Figures 2A and 2B, D and By exhibit similar trends over each Carrington rotation, with a possible time lag between them. To determine the correlation and time lag between D and By ,weusedthe discrete correlation function (DCF) method, 31 which considers the unevenly sampled time series of D and By and their measurement errors. The time lag bin width was set to 1 day, matching the time bins of D and By in Figures 2A and 2B, and we considered time lags of up to 5 days. To achieve higher precision in the time lag determination, the cadence was set to 0.0625 days based on the hourly By measurement and a time lag sliding technique. The DCF coef /uniFB01 cient and its error are shown in Figure 2C. The error of the time lag was estimated using 10 3 random time series of D and By , 32 generated based on a Gaussian probability distribution. The standard deviation of the distribution of time lags between the 10 3 random time series of D and By was taken as the error of the time lag. For the entire dataset (labeled as ' All ' ), we tested the correlation between 177 days of D data and 181 days of By data. D is most correlated with By at L 1 when D precedes By by 3 : 31 ± 0.12 days, according to the maximum DCF coef /uniFB01 cient shown with blue markers in Figure 2C. The con /uniFB01 dence level of the maximum DCF coef /uniFB01 cient was estimated using the Monte Carlo method. 33,34 Speci /uniFB01 cally, we generated 10 5 random time series D by randomizing both the phase and amplitude of the Fourier transform of the observed time series D . The DCF was then applied to each random time series D and the observed time series By . The corresponding con /uniFB01 -dence level of the maximum DCF coef /uniFB01 cient exceeded 99.73% (corresponding to 3 s ) with a two-sided p value of 0.0027, as indicated by the dashed line in Figure 2C. As depicted in Figure 3B, D remains correlated with By after considering a time lag of 3.31 days. The speci /uniFB01 c correlation is /uniFB01 tted using the following linear formula: The corresponding correlation coef /uniFB01 cient is 0.67. The functional form of this formula is essentially the same as the change in position according to the Newton-Lorentz equation. Following Equation 1, the daily By at L 1 can be estimateddirectlybasedon D measured by LHAASO 3.31 days earlier. The By estimated by D re /uniFB02 ects the effective IMF responsible for the cumulative de /uniFB02 ection of cosmic rays along the Sun-Earth line.",
"pages": [
2,
3
]
},
{
"title": "IMF model diagnosis from the time lag between D and By",
"content": "Based on the classic Parker model, the IMF in heliocentric spherical coordinates can be expressed as follows: where Br ð b ; q ; f 0 Þ is the outermost CMF at the boundary radius b and heliolongitude f 0 . Beyond b , the IMF in the model is blown out by the radial solar wind with velocity n . When the Sun rotates with angular velocity u , the streamline of the magnetic /uniFB01 eld with azimuth f 0 at r = b is given by r b /C0 1 /C0 ln /C0 r b /C1 = n b u ð f /C0 f 0 Þ . To study the effect of the magnetic /uniFB01 eld predicted by the IMF model, an antiparticle retraction method was adopted to simulate the Sun shadow. The details of the simulation program are presented in the materials and methods.Inthe Parker model calculation, the boundary radius b was set to 2.5 R 1 , and Br ð b ; q ; f 0 Þ was extrapolated from the PFSS model 9 using a 9th-order spherical harmonic expansion. 35 Photospheric magnetograms named ' mrnqs ' from the Global Oscillation Network Group (GONG) 36 served as an input. The calculated IMF was modi /uniFB01 ed by a factor of 5.7 according to the observed average daily By at L 1 from OMNI. This scaling of magnetogram data used as the input to a solar wind model is a standard practice in the heliophysics community. 37 The Sun ' s rotation period was /C24 25.4 days and the solar wind velocity n was obtained from the daily average value from OMNI. The IMF variations are transferred from the Sun to the Earth by the solar wind at velocity n . Cosmic rays can record By between the Sun and Earth through D of the Sun shadow. Hence, the variation in By at L 1 lags behind the variation in D . Because D represents a cumulative effect of By that spreads from the Sun to the Earth, the speci /uniFB01 c time lag value depends on the distribution of By along the Sun-Earth line. Therefore, the time lag between D and By at L 1 provides an opportunity to test the IMF models. Based on the Parker model, the average simulated time lag for the entire dataset is 2 : 06 ± 0.04 days. This simulated result reproduces the phenomenon that D leads By in the observations. However, a deviation between the simulated and observed time lags exists. This suggests a more complicated spiral structure of the IMF than that depicted by the Parker model. One possible solution to address this deviation between the simulated and observed time lag is to add a steady, azimuthal IMF component, B f ð b Þ ,atthe CMF boundary b to the Parker model, as proposed by Smith and Bieber. 38 This modi /uniFB01 ed model has been used to explain the deviation of the spiral structure from the Parker model 38 and even to calculate such a deviation to interpret recent Parker Solar Probe results. 39 During the observation time of LHAASO, the additional azimuthal IMF component B f ð b Þ x /C0 0 : 002 Br ð b Þ n n ð b Þ , which corresponds to a ' gardenhose ' angle of the spiral that is /C24 3.6 /C14 larger than that predicted by the Parker model at L 1.Here, n ð b Þ repre- velocity at boundary b . Based on this modi /uniFB01 ed model, the simulated time lag becomes 2 : 64 ± 0.04 days with a deviation from the observed lag. This simulated time lag is close to our measured results; however, a deviation still persists. The magnetic /uniFB01 eld structures of CMEs can disturb the IMF to form interplanetary CMEs (ICMEs), which may lead to IMF deviations from the Parker model. In addition, solar wind interactions can disturb the IMF by generating stream interaction regions (SIRs), which may also lead to IMF deviations. To check for a possible effect of CMEs and SIRs on the deviation between the observed and simulated time lag, a data sample without ICME and SIR transits (labeled as ' Without ICME+SIR ' ) was segmented, as depicted by the blue dots in Figures 2Aand 2B. We excluded ICMEs based on an updated catalog 40 and excluded SIRs based on coronal hole information 41 and an identi /uniFB01 ed method, 42 as shown in Figures 2A and 2B. The ICME transit of the Sun shadow was counted from the occur- rence of the CME until it stops disturbing By at L 1. Similarly, the SIR transit was counted from the time when the coronal hole faces the Earth until the SIR stops disturbing By at L 1. The ICME or SIR transit of By at L 1 was counted from the time when the ICME or SIR starts disturbing By at L 1 until it stops affecting By at L 1. For the Without ICME+SIR data sample, we tested the correlation between 59-day D and 130-day By . The observed time lag between D and By at L 1 was 3 : 00 ± 0 : 21 days according to the maximum DCF coef /uniFB01 cient, as indicated by the blue markers in Figure 2D. The corresponding con /uniFB01 dence level of the maximum DCF coef /uniFB01 cient exceeded 3 s as shown by the dashed lines in Figure 2D. There was no signi /uniFB01 cant difference in the observed time lag between the All and Without ICME+SIR data samples. Therefore, the deviation between the observed and simulated time lags was not due to CMEs and SIRs. Other possible explanations for the deviation between the observed and simulated time lags include systematic changes in the IMF 43,44 and disturbances with timescales much shorter than the Carrington rotation. 45-47 A more comprehensive discussion about these possibilities is beyond the scope of this study, as it focuses on long-term average time lag.",
"pages": [
3,
4
]
},
{
"title": "CONCLUSION AND OUTLOOK",
"content": "Traveling almost at the speed of light, very-high-energy galactic cosmic rays take only approximately 8 min to traverse the distance between the Sun and Earth. This provides us with a useful method for exploring the magnetic /uniFB01 eld between the Sun and Earth. Based on the unprecedented monitoring of the daily cosmic-ray Sun shadow by KM2A in LHAASO, we provided daily measurements of the IMF between the Sun and Earth near the sunspot minimum period. For the /uniFB01 rst time, we made it possible for the Sun shadow to achieve an accurate measurement of the transverse magnetic /uniFB01 eld along the entire Sun-Earth line. Using this measurement, the IMF component along the GSE y -direction near the Earth was deduced to be 3 : 31 ± 0 : 12 days ahead of the measurement by spacecraft at L 1. With a measured 3 : 31 ± 0 : 12-day time lag, we also tested the classic Parker model of the IMF. We found that the measured lag signi /uniFB01 cantly deviates from the model ' spredictionof2 : 06 ± 0 : 04 days, indicating the need for re /uniFB01 nements to the IMF model. This measurement not only provides a new method for monitoring the IMF and diagnosing existing models but also drives the development of more accurate IMF models. 4,35,38,48 This advancement will contribute to improved understanding of the propagation and forecasting of solar activity events, which can be helpful for research on space weather effects that impact human activities. Moreover, the observed time lag implies that the Sun shadow has the potential for predicting the IMF arrival on Earth. Future research will explore the possibility of Sun shadow forecasting of the IMF component along the z axis (north-south direction) when the geomagnetic /uniFB01 eld effect is excluded. Even the possibility of forecasting a speci /uniFB01 c solar activity event, such as CMEs and SIRs, holds promise for further increasing space weather forecasting capabilities.",
"pages": [
4,
5
]
},
{
"title": "KM2A experiment",
"content": "The KM2A, a component of LHAASO, consists of 5216 EDs with 15 m spacing and 1,188 muondetectors with 30 m spacing, covering an area of 1.3 km 2 . EDs are designed to detect showers from cosmic-ray ions and gamma rays 49 with determining their directions and energies. The 3/4 array has been operating since December 1, 2020, and the full array since July 19, 2021. During this period, the pointing accuracy, angular resolution, and energy of the cosmic rays measured by the array were stable, based on observations of the cosmic-ray Moon shadow. 29",
"pages": [
5
]
},
{
"title": "Signal and displacement analysis of Sun shadow",
"content": "TheSunshadow was analyzed on a sky map with 0 : 025 /C14 3 0 : 025 /C14 grid spacing along the longitude and latitude in the GSE coordinate system. The background was estimated using the equal zenith angle method. 50 Speci /uniFB01 cally, there was one on-source window centered around the Sun and 20 off-source windows aligned at the same zenith angle to cover all azimuthal angles. In each grid cell of the on-source window, the background events were estimated according to the number of cosmic rays at the same grid points in the 20 offsource windows. Combined with the data selection described in the results and discussion, wecollected 1.1 billion cosmic-ray events in on- and off-source windows in the data sample. The energy of these events was obtained from the energy distribution of the simulated pri- mary cosmic rays. The median energy was /C24 40 TeV, with a 68% interval between 22 and 107 TeV. The angular resolution containing 68% of the events was 0.5 /C14 . The signal events were extracted using a smoothing procedure with Gaussian weighting, and the corresponding signi /uniFB01 cance was estimated using the Li and Ma formula (see Equation 2.5 in Nan and Chen 51 ). The signi /uniFB01 cance of the Sun shadow varies with the day, with an average daily signi /uniFB01 cance reaching 8.4 s , ranging from 5 s to 13.8 s . The displacement of the Sun shadow was estimated by a likelihood ratio test between the one-source model and background-only model, 52 with the displacement of the Sun shadow along the west-east and north-south directions as free parameters.",
"pages": [
5
]
},
{
"title": "Monte Carlo simulation of Sun shadow",
"content": "For the Monte Carlo simulations of KM2A detection of cosmic-ray showers, the cosmicray primary chemical composition and energy spectrum were speci /uniFB01 ed according to the model of Gaisser et al. 53 The cascade processes induced by the cosmic-ray interactions within the atmosphere and the response of the detector were simulated by the CORSIKA 54 and G4KM2A 52,55 codes, respectively. Then, the primary cosmic-ray properties were reconstructed and selected using the same methods and conditions as in the basic observation 51 and in this work. During the Sun shadow simulation, the Sun was tracked in real time. Above the atmosphere, an average of 1 : 27 3 10 7 particles with opposite charge to the primary cosmic rays were isotropically thrown back around the Sun ' s direction within a window of 10 /C14 3 10 /C14 . The simulation included magnetic /uniFB01 elds along the Sun-Earth line and was performed on each day of data collection. The paths of the particles along the Sun-Earth line were tracked according to the changes in momentum and position following the Newton-Lorentz equation. In addition to the IMF, the CMF and geomagnetic /uniFB01 eld were considered in the Sun shadow simulation (the CMF has been described previously). The geomagnetic /uniFB01 eld calculation followed the International Geomagnetic Reference Field-13 (IGRF-13), 56 in which 13 (2) orders of spherical harmonic expansion were used for /uniFB01 elds below (above) 600 km from the Earth ' s surface. When a particle hits the Sun, it is counted as a Sun shadow signal coming from opposite to the initial throwing direction. Finally, the initial throwing directions were smeared using the KM2A angular resolution. 18,57",
"pages": [
5
]
},
{
"title": "ACKNOWLEDGMENTS",
"content": "We would like to thank all staff members who worked at the LHAASO site, situated more than 4,400 m above sea level, for their year-round dedication to maintaining the detector and ensuring smooth operation of the water recycling system, electricity power supply, and other experimental components. We are grateful to the Chengdu Management Committee of Tianfu New Area for their constant /uniFB01 nancial support for research using LHAASO data. We appreciate the computing and data service support provided by the National High Energy Physics Data Center for the data analysis presented in this paper. We also thank the OMNI and GONG groups for providing the data, SSW and IAGA for providing the codes, and Shen Zhenning for his guidance on using Smith and Bieber ' smodi /uniFB01 ed model. This research was supported by the following grants: The National Key R&D",
"pages": [
6
]
},
{
"title": "REPORT",
"content": "Program of China (no. 2018YFA0404201), National Natural Science Foundation of China (nos. 12393851, 12393854, 12175121, 12205314, 12105301, 12305120, 12261160362, 12105294, U1931201, and 12375107), China Postdoctoral Science Foundation (no. 2022M723150), National Science and Technology Development Agency of Thailand, and National Research Council of Thailand under the High-Potential Research Team Grant Program (N42A650868).",
"pages": [
7
]
},
{
"title": "AUTHOR CONTRIBUTIONS",
"content": "Y.N., S.C., and C.F. led the writing of the text for the data analysis and interpretation. S.C. initiated this work. Y.N. analyzed the KM2A data under the guidance of S.C. and C.F., while J.X. crosschecked the results. C.J. and Y.Y. offered expertise on solar magnetic models. Z.C. acted as spokesperson for the LHAASO Collaboration and principal investigator of the LHAASO project. All other authors participated in data analysis (including detector calibra- tion, data processing, event reconstruction, data quality checks, and various simulations) and provided comments on the manuscript.",
"pages": [
7
]
},
{
"title": "DECLARATION OF INTERESTS",
"content": "The authors declare no competing interests.",
"pages": [
7
]
},
{
"title": "SUPPLEMENTAL INFORMATION",
"content": "It can be found online at https://doi.org/10.1016/j.xinn.2024.100695.",
"pages": [
7
]
},
{
"title": "LEAD CONTACT WEBSITE",
"content": "Yuncheng Nan: https://orcid.org/0000-0002-7363-0252 Songzhan Chen: https://orcid.org/0000-0003-0703-1275 Cunfeng Feng: https://orcid.org/0000-0001-9138-3200",
"pages": [
7
]
},
{
"title": "Author List",
"content": "Zhen Cao 1,2,3 ,F. Aharonian 4,5 ,Axikegu 6 ,Y.X. Bai 1,3 ,Y.W. Bao 7 ,D. Bastieri 8 ,X.J. Bi 1,2,3 ,Y.J. Bi 1,3 ,W. Bian 9 ,A.V. Bukevich 10 ,Q. Cao 11 ,W.Y. Cao 12 ,Zhe Cao 13,12 ,J. Chang 14 ,J.F. Chang 1,3,13 ,A.M. Chen 9 ,E.S. Chen 1,2,3 ,H.X. Chen 15 ,Liang Chen 16 ,Lin Chen 6 ,Long Chen 6 ,M.J. Chen 1,3 ,M.L. Chen 1,3,13 ,Q.H. Chen 6 ,S. Chen 17 ,S.H. Chen 1,2,3 ,S.Z. Chen 1,3 ,T.L. Chen 18 ,Y. Chen 7 ,N. Cheng 1,3 ,Y.D. Cheng 1,2,3 ,M.Y. Cui 14 ,S.W. Cui 11 ,X.H. Cui 19 ,Y.D. Cui 20 ,B.Z. Dai 17 ,H.L. Dai 1,3,13 ,Z.G. Dai 12 ,Danzengluobu 18 ,X.Q. Dong 1,2,3 ,K.K. Duan 14 ,J.H. Fan 8 ,Y.Z. Fan 14 ,J. Fang 17 ,J.H. Fang 15 ,K. Fang 1,3 ,C.F. Feng 21 ,H. Feng 1 ,L. Feng 14 ,S.H. Feng 1,3 ,X.T. Feng 21 ,Y. Feng 15 ,Y.L. Feng 18 ,S. Gabici 22 ,B. Gao 1 , 3 ,C.D. Gao 21 ,Q. Gao 18 ,W. Gao 1,3 ,W.K. Gao 1,2,3 ,M.M. Ge 17 ,L.S. Geng 1,3 ,G. Giacinti 9 ,G.H. Gong 23 ,Q.B. Gou 1,3 ,M.H. Gu 1,3,13 ,F.L. Guo 16 ,X.L. Guo 6 ,Y.Q. Guo 1,3 ,Y.Y. Guo 14 ,Y.A. Han 24 ,M. Hasan 1,2,3 ,H.H. He 1,2,3 ,H.N. He 14 ,J.Y. He 14 ,Y. He 6 ,Y.K. Hor 20 ,B.W. Hou 1,2,3 ,C. Hou 1,3 ,X. Hou 25 ,H.B. Hu 1,2,3 ,Q. Hu 12,14 ,S.C. Hu 1,3,26 ,D.H. Huang 6 ,T.Q. Huang 1,3 ,W.J. Huang 20 ,X.T. Huang 21 ,X.Y. Huang 14 ,Y. Huang 1,2,3 ,X.L. Ji 1,3,13 ,H.Y. Jia 6 ,K. Jia 21 ,K. Jiang 13,12 ,X.W. Jiang 1,3 ,Z.J. Jiang 17 ,M. Jin 6 ,M.M. Kang 27 ,I. Karpikov 10 ,D. Kuleshov 10 ,K. Kurinov 10 ,B.B. Li 11 ,C.M. Li 7 ,Cheng Li 13,12 ,Cong Li 1,3 ,D. Li 1,2,3 ,F. Li 1,3,13 ,H.B. Li 1,3 ,H.C. Li 1,3 ,Jian Li 12 ,Jie Li 1,3,13 ,K. Li 1,3 ,S.D. Li 16,2 ,W.L. Li 21 ,W.L. Li 9 ,X.R. Li 1,3 ,Xin Li 13,12 ,Y.Z. Li 1,2,3 ,Zhe Li 1,3 ,Zhuo Li 28 ,E.W. Liang 29 ,Y.F. Liang 29 ,S.J. Lin 20 ,B. Liu 12 ,C. Liu 1,3 ,D. Liu 21 ,D.B. Liu 9 ,H. Liu 6 ,H.D. Liu 24 ,J. Liu 1,3 ,J.L. Liu 1,3 ,M.Y. Liu 18 ,R.Y. Liu 7 ,S.M. Liu 6 ,W. Liu 1,3 ,Y. Liu 8 ,Y.N. Liu 23 ,Q. Luo 20 ,Y. Luo 9 ,H.K. Lv 1,3 ,B.Q. Ma 28 ,L.L. Ma 1,3 ,X.H. Ma 1,3 ,J.R. Mao 25 ,Z. Min 1,3 ,W. Mitthumsiri 30 ,H.J. Mu 24 ,Y.C. Nan 1,3 ,A. Neronov 22 ,L.J. Ou 8 ,P. Pattarakijwanich 30 ,Z.Y. Pei 8 ,J.C. Qi 1,2,3 ,M.Y. Qi 1,3 ,B.Q. Qiao 1,3 ,J.J. Qin 12 ,A. Raza 1,2,3 ,D. Ruffolo 30 ,A. S'aiz 30 ,M. Saeed 1,2,3 ,D. Semikoz 22 ,L. Shao 11 ,O. Shchegolev 10,31 ,X.D. Sheng 1,3 ,F.W. Shu 32 ,H.C. Song 28 ,Yu.V. Stenkin 10,31 ,V. Stepanov 10 ,Y. Su 14 ,D.X. Sun 12,14 ,Q.N. Sun 6 ,X.N. Sun 29 ,Z.B. Sun 33 ,J. Takata 34 ,P.H.T. Tam 20 ,Q.W. Tang 32 ,R. Tang 9 ,Z.B. Tang 13,12 ,W.W. Tian 2,19 ,C. Wang 33 ,C.B. Wang 6 ,G.W. Wang 12 ,H.G. Wang 8 ,H.H. Wang 20 ,J.C. Wang 25 ,Kai Wang 7 ,Kai Wang 34 ,L.P. Wang 1,2,3 ,L.Y. Wang 1,3 ,P.H. Wang 6 ,R. Wang 21 ,W. Wang 20 ,X.G. Wang 29 ,X.Y. Wang 7 ,Y. Wang 6 ,Y.D. Wang 1,3 ,Y.J. Wang 1,3 ,Z.H. Wang 27 ,Z.X. Wang 17 ,Zhen Wang 9 ,Zheng Wang 1,3,13 ,D.M. Wei 14 ,J.J. Wei 14 ,Y.J. Wei 1,2,3 ,T. Wen 17 ,C.Y. Wu 1,3 ,H.R. Wu 1,3 ,Q.W. Wu 34 ,S. Wu 1,3 ,X.F. Wu 14 ,Y.S. Wu 12 ,S.Q. Xi 1,3 ,J. Xia 12,14 ,G.M. Xiang 16,2 ,D.X. Xiao 11 ,G. Xiao 1,3 ,Y.L. Xin 6 ,Y. Xing 16 ,D.R. Xiong 25 ,Z. Xiong 1,2,3 ,D.L. Xu 9 ,R.F. Xu 1,2,3 ,R.X. Xu 28 ,W.L. Xu 27 ,L. Xue 21 ,D.H. Yan 17 ,J.Z. Yan 14 ,T. Yan 1,3 ,C.W. Yang 27 ,C.Y. Yang 25 ,F. Yang 11 ,F.F. Yang 1,3,13 ,L.L. Yang 20 ,M.J. Yang 1,3 ,R.Z. Yang 12 ,W.X. Yang 8 ,Y.H. Yao 1,3 ,Z.G. Yao 1,3 ,L.Q. Yin 1,3 ,N. Yin 21 ,X.H. You 1,3 ,Z.Y. You 1,3 ,Y.H. Yu 12 ,Q. Yuan 14 ,H. Yue 1,2,3 ,H.D. Zeng 14 ,T.X. Zeng 1,3,13 ,W. Zeng 17 ,M. Zha 1,3 ,B.B. Zhang 7 ,F. Zhang 6 ,H. Zhang 9 ,H.M. Zhang 7 ,H.Y. Zhang 1,3 ,J.L. Zhang 19 ,Li Zhang 17 ,P.F. Zhang 17 ,P.P. Zhang 12,14 ,R. Zhang 12,14 ,S.B. Zhang 2,19 ,S.R. Zhang 11 ,S.S. Zhang 1,3 ,X. Zhang 7 ,X.P. Zhang 1,3 ,Y.F. Zhang 6 ,Yi Zhang 1,14 ,Yong Zhang 1,3 ,B. Zhao 6 ,J. Zhao 1,3 ,L. Zhao 13,12 ,L.Z. Zhao 11 ,S.P. Zhao 14 ,X.H. Zhao 25 ,F. Zheng 33 ,W.J. Zhong 7 ,B. Zhou 1,3 ,H. Zhou 9 ,J.N. Zhou 16 ,M. Zhou 32 ,P. Zhou 7 ,R. Zhou 27 ,X.X. Zhou 1,2,3 ,X.X. Zhou 6 ,B.Y. Zhu 12,14 ,C.G. Zhu 21 ,F.R. Zhu 6 ,H. Zhu 19 ,K.J. Zhu 1,2,3,13 ,Y.C. Zou 34 ,X. Zuo 1,3 , (The LHAASO Collaboration) C.W. Jiang 35 and Y. Yang 33 .",
"pages": [
8
]
}
]
2024arXiv241022466T
https://arxiv.org/pdf/2410.22466.pdf
<document>
<section_header_level_1><location><page_1><loc_7><loc_82><loc_93><loc_87></location>Globular clusters as cosmic clocks: new cosmological hints from their integrated light</section_header_level_1>
<text><location><page_1><loc_12><loc_80><loc_87><loc_81></location>Elena Tomasetti 1 , 2, ⋆ , Michele Moresco 1 , 2 , Carmela Lardo 1 , 2 , Andrea Cimatti 1 , 3 , and Raul Jimenez 4 , 5</text>
<unordered_list>
<list_item><location><page_1><loc_10><loc_77><loc_87><loc_78></location>1 Dipartimento di Fisica e Astronomia 'Augusto Righi'-Università di Bologna, via Piero Gobetti 93 / 2, I-40129 Bologna, Italy</list_item>
<list_item><location><page_1><loc_10><loc_76><loc_82><loc_77></location>2 INAF - Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, via Piero Gobetti 93 / 3, I-40129 Bologna, Italy</list_item>
<list_item><location><page_1><loc_10><loc_74><loc_62><loc_75></location>3 INAF - Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125, Firenze, Italy</list_item>
<list_item><location><page_1><loc_10><loc_73><loc_58><loc_74></location>4 ICC, University of Barcelona, Martí i Franquès, 1, E08028, Barcelona, Spain</list_item>
<list_item><location><page_1><loc_10><loc_72><loc_46><loc_73></location>5 ICREA, Pg. Lluis Companys 23, Barcelona, 08010, Spain</list_item>
</unordered_list>
<text><location><page_1><loc_10><loc_70><loc_21><loc_71></location>October 29, 2024</text>
<section_header_level_1><location><page_1><loc_46><loc_67><loc_54><loc_68></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_10><loc_55><loc_90><loc_65></location>Aims. In this work we explore the reliability and robustness in measuring the ages and main physical properties of a sample of old Milky Way globular clusters (GCs) from their integrated light. This approach sets the stage for using GCs as cosmic clocks at high redshift. Additionally, it enables us to establish an independent lower limit to the age of the Universe, and an upper limit to H 0. Methods. We analyse a sample of 77 GCs from the WAGGS project, by first measuring their spectral features (Lick indices and spectroscopic breaks) with PyLick and then performing full-spectral-fitting with BAGPIPES . The analysis of Lick indices o ff ers an initial estimate of the population's age and metallicity, generally aligning well with values reported in the literature. However, it also highlights a subset of old clusters for which we estimate younger ages. This discrepancy is primarily attributed to the presence of horizontal branches with complex morphologies, which are not accounted for in the stellar population models. With full-spectralfitting we measure the GCs' ages, metallicities, and masses, testing how removing the cosmological prior on the ages a ff ects the final</text>
<text><location><page_1><loc_10><loc_54><loc_14><loc_55></location>results.</text>
<text><location><page_1><loc_10><loc_46><loc_90><loc_54></location>Results. Compared to isochrone fitting estimates, ages are best recovered when the cosmological prior is removed, with a 20% increase in the number of GCs showing ages compatible with literature values within ± 1.5 Gyr. The derived metallicity and mass are consistently in good agreement with the reference values, regardless of HB morphology, [Z / H], or the fit settings. The average discrepancies across the entire sample are ⟨ ∆ [Z / H] ⟩ = -0 . 02 ± 0 . 24 dex for metallicity and ⟨ ∆ log(M ⋆/ M ⊙ ) ⟩ = 0 . 04 ± 0 . 28 dex for mass. Metal-rich GCs ([Z / H] ≥ -0.4) showing a red horizontal branch (HBR > 0) are the sub-group in which ages are best recovered. In this group, 70% of the results align with literature values within ± 1.5 Gyr. Identifying the tail of the oldest cosmology-independent ages with a Gaussian Mixture Model, we obtain a sample of 24 objects with ⟨ age ⟩ = 13 . 4 ± 1 . 1 Gyr.</text>
<text><location><page_1><loc_10><loc_41><loc_90><loc_45></location>Conclusions. Being a natural lower limit to the age of the Universe, we use the age of the oldest GCs to constrain the Hubble constant, obtaining H0 = 70 . 5 + 7 . 7 -6 . 3 km s -1 Mpc -1 (stat + syst) when a flat Λ CDMwith Ω m = 0 . 30 ± 0 . 02 (based on low-z measurements) is assumed. Validating the analysis of GCs based on their integrated light lays the foundation to extend this type of study to high redshift, where GCs have begun to appear in lensed fields, thanks to JWST.</text>
<text><location><page_1><loc_10><loc_39><loc_67><loc_40></location>Key words. globular clusters: general - Cosmology:observations - cosmological parameters</text>
<section_header_level_1><location><page_1><loc_6><loc_35><loc_18><loc_36></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_6><loc_17><loc_49><loc_34></location>During the last decades, the flat Λ CDM model has been a fundamental pillar in cosmology thanks to the support of independent cosmological probes, including the cosmic microwave background (CMB; e.g. Bennett et al. 2003; Planck Collaboration et al. 2020), type Ia supernovae (e.g. Riess et al. 1998; Perlmutter et al. 1999), and baryon acoustic oscillations (e.g. Percival et al. 2001; Eisenstein et al. 2005). Nonetheless, further investigation is necessary to unveil the nature of dark matter and dark energy and assess the precise values of cosmological parameters. In fact, due to the higher precision achieved in lateand early-Universe probes, some inconsistencies have emerged regarding the value of the Hubble constant ( H 0), where a tension of 4-5 σ has now been observed (Abdalla et al. 2022).</text>
<text><location><page_1><loc_6><loc_12><loc_49><loc_17></location>In this context, the age of the Universe ( tU ) can play a crucial role, given its sensitivity to H 0. Indeed, in a flat Λ CDM cosmology with Ω M = 0.3 and ΩΛ= 0.7, tU can span a range from ∼ 14.1 Gyr if H 0 = 67 km s -1 Mpc -1 to ∼ 12.9 Gyr if H 0 = 73</text>
<text><location><page_1><loc_51><loc_28><loc_94><loc_36></location>km s -1 Mpc -1 . Thus, measuring the absolute ages of the most long-lived objects at z = 0 can be critical since they naturally place a lower limit on the current age of the Universe ( tU ) and, in turn, an upper limit on H 0. This provides independent constraints on the Hubble constant and valuable information for investigating the origin of the observed tension.</text>
<text><location><page_1><loc_51><loc_10><loc_94><loc_26></location>Globular clusters (GC) are among the oldest objects in the Universe for which we can accurately determine the age (VandenBerg et al. 1996; Soderblom 2010; Brown et al. 2018; Oliveira et al. 2020). Composed of roughly one million stars that formed simultaneously with similar composition (though see reviews by Bastian & Lardo 2018; Gratton et al. 2019; Milone & Marino 2022, for discussions on the multiple population phenomenon in massive clusters), these clusters have remained gravitationally bound for up to a Hubble time. Each GC thus serves as an observable record of the age, metallicity, and kinematics from the time of its formation. Therefore, by measuring their ages, we can use them as "clocks" that began ticking in the early stages of the Universe's evolution (O'Malley et al.</text>
<section_header_level_1><location><page_2><loc_6><loc_91><loc_49><loc_93></location>2017; Jimenez et al. 2019; Valcin et al. 2020, 2021; Cimatti & Moresco 2023).</section_header_level_1>
<text><location><page_2><loc_6><loc_59><loc_49><loc_90></location>The most straightforward method to determine the age of a GC is by exploiting the fact that the position of the mainsequence turn-o ff (MSTO) in the plane of e ff ective temperature (Te ff ) versus luminosity (L) changes with age (or mass). Isochrones, or theoretical tracks of stars with the same chemical composition, are fitted to the MSTO region of colour-magnitude diagrams (CMD) to estimate the age. Even if this is a wellestablished and robust method, it is important to explore new and complementary approaches that can address the case when the CMDis not available. In need of a spatially resolved stellar population, indeed, isochrone fitting can be applied only to nearby systems, while moving further than the Magellanic Clouds becomes either very expensive in terms of exposure time or even impossible. Moreover, recent JWST observations of lensed fields highlighted the presence of GC candidates around lensed galaxies, like the Sparkler (Mowla et al. 2022) at z = 1.38 which, if confirmed with spectroscopy, would extend the study of GCs at high redshift. To do so, we need to explore methods relying on GCs' integrated light and validate them against the traditional methods. In this scenario, one of the best ways to leverage all the integrated light information is to perform full-spectral-fitting (FSF), a technique that enables measuring, alongside the age, all the physical properties of the GC such as metallicity, mass, and dust reddening.</text>
<text><location><page_2><loc_6><loc_23><loc_49><loc_59></location>Previous works have derived physical parameters of GCs like age and metallicity using the integrated light provided by Schiavon et al. (2005) for 41 MW GCs (e.g., Koleva et al. 2008; Cezario et al. 2013; Cabrera-Ziri & Conroy 2022), testing different algorithms, like STECKMAP (Ocvirk et al. 2006), NBURSTS (Chilingarian et al. 2007), ULySS (Koleva et al. 2009) or ALF (Conroy & van Dokkum 2012), and di ff erent simple stellar population (SSP) models (Bruzual & Charlot 2003; Prugniel & Soubiran 2004; Vazdekis et al. 2010, 2015). Others have benefited from the larger spectral coverage and higher resolution of the WiFeS Atlas of Galactic Globular cluster Spectra project (WAGGS, Usher et al. 2017, 2019a), providing integrated spectra for 113 GCs in the MW and its satellite galaxies (Usher et al. 2019b; Gonçalves et al. 2020; Cabrera-Ziri & Conroy 2022). In Gonçalves et al. (2020), for instance, the authors adopted the non-parametric FSF code STARLIGHT (Cid Fernandes et al. 2005), relying on MILES SSP models (Vazdekis et al. 2015), with a focus on how the wavelength range influences the recovery of the stellar parameters compared to the CMD fitting. Cabrera-Ziri & Conroy (2022) extended the Milky Way GCs sample from Schiavon et al. (2005) by including younger objects from the Large and Small Magellanic Clouds (LMC and SMC) with WAGGS spectra, adopting the non-parametric FSF code ALF . While there is broad agreement that the ages of younger GCs can be reliably determined through FSF, these studies highlighted the challenges in dating the oldest GCs from their integrated spectra, often yielding results significantly younger than those from isochrone-fitting methods.</text>
<text><location><page_2><loc_6><loc_10><loc_49><loc_22></location>In this paper we focus on the oldest tail of the WAGGS GCs, analysing 82 GCs in the Milky Way. We take advantage of the high-quality integrated spectra provided by WAGGS, along with the wealth of data available for these objects, the independent age estimates derived with di ff erent techniques, and the fact that GCs are among the simplest stellar systems in the Universe, the closest templates to an SSP we have. In this study, we adopt a parametric FSF method, enabling the reconstruction of GCs' integrated emission within a high-dimensional parameter space. For this purpose, we use the FSF code BAGPIPES (Carnall et al.</text>
<text><location><page_2><loc_51><loc_78><loc_94><loc_93></location>2018) utilizing the 2016 version of the Bruzual & Charlot (2003) SSP models. Previous studies in the literature have typically derived parameters such as age, metallicity, mass, and dust reddening while assuming a cosmological prior on age. In contrast, the novelty of this study lies in removing this prior to explore how the derived ages are a ff ected, as done by Tomasetti et al. (2023) and Jiao et al. (2023), in order to test the potential of the results in a cosmological framework. By testing this approach, we aim to assess its potential in a cosmological context. We also use these cosmology-independent results to place new constraints on H 0 setting the stage for future applications in studying the distant Universe.</text>
<text><location><page_2><loc_51><loc_70><loc_94><loc_77></location>This paper is organised as follows: in Sect. 2 we describe the spectra we used, along with the adjustments needed and the ancillary data; in Sect. 3 the spectroscopic analysis of the sample is presented; in Sect. 4 the FSF method and its result are outlined; in Sect. 5 we report the final cosmological analysis; in Sect. 6 we draw our conclusions.</text>
<section_header_level_1><location><page_2><loc_51><loc_66><loc_57><loc_68></location>2. Data</section_header_level_1>
<text><location><page_2><loc_51><loc_50><loc_94><loc_65></location>The WAGGS project (Usher et al. 2017) is a library of integrated spectra of GCs in the Milky Way (MW) and the Local Group, obtained with the WiFeS integral field spectrograph on the Australian National University 2.3 m telescope. With 112 spectra of GCs in the Local Group, it is one of the largest GCs spectral libraries currently available, with a wide wavelength coverage (3270-9050 Å) and high spectral resolution (R ∼ 6800). The spectra we work with are normalised and consist of four di ff erent gratings, each with its own sampling: 3270-4350 Å (0.27 Å per pixel), 4170-5540 Å (0.37 Å per pixel), 5280-7020 Å (0.44 Å per pixel) and 6800-9050 Å (0.57 Å per pixel). To perform FSF across the entire spectrum, certain adjustments were required.</text>
<text><location><page_2><loc_51><loc_43><loc_94><loc_49></location>First, we had to re-scale each spectrum to match its literature photometry, to retrieve the fluxes in physical units. We used the UBVRI integrated photometry from the 2010 edition of the Harris catalogue (Harris 1996, 2010). The correction factor C , derived via χ 2 minimisation, can be written as:</text>
<formula><location><page_2><loc_51><loc_39><loc_94><loc_42></location>C = P (pJ / eJ) 2 P fJpJ / e 2 J , (1)</formula>
<text><location><page_2><loc_51><loc_33><loc_94><loc_38></location>where p is the photometry in the J -th filter, and f and e are the average flux and corresponding error estimated on the spectrum on a window of 10 Å. We then multiplied the spectrum in each grating by the corresponding factor C .</text>
<text><location><page_2><loc_51><loc_16><loc_94><loc_33></location>Here, we must underline that UBVRI photometry is not available for all the objects in WAGGS, but only for the 82 GCs belonging to the MW. For the younger GCs in the LMC and SMC and in the Fornax dwarf spheroidal, only BV photometry is available, respectively from van den Bergh (1981) and van den Bergh (1969). Anyway, in this work, we want to focus on the oldest tail of the local GCs, so we limit our sample to the MW GCs. Before proceeding with the analysis we performed a visual inspection of the spectra, removing five GCs showing either visibly corrupted or very noisy regions (S / N < 10 in more than 40% of the spectrum), namely NGC6144, NGC6401, NGC6517, NGC6712 and NGC7492. The sample we analyse here is then constituted of 77 GCs.</text>
<text><location><page_2><loc_51><loc_10><loc_94><loc_16></location>To combine the four gratings into a single spectrum, we interpolated all of them onto a common wavelength grid, matching the largest spectral sampling value (0.57 Å per pixel). In the overlap regions, the flux and associated error were estimated by averaging the spectra from the consecutive gratings.</text>
<text><location><page_3><loc_6><loc_83><loc_49><loc_93></location>Throughout the paper, we compare our results to literature values of age, mass, and metallicity, and we also consider additional quantities to complement and expand our analysis, like dust reddening, radial velocities and distances of the GCs. We use as a reference the values listed in Usher et al. (2017) for ages and masses, in Harris (2010) for metallicities ([Fe / H]) and dust reddening (EB -V) and in Baumgardt et al. (2023) for radial velocities, distance from the Sun and associated errors.</text>
<text><location><page_3><loc_6><loc_78><loc_49><loc_83></location>As for the uncertainties on metallicities, we consider the errors found in other spectroscopic investigations based on integrated spectra of Galactic GCs, which are approximately ± 0.15 dex (see Roediger et al. 2014; Colucci et al. 2017).</text>
<text><location><page_3><loc_6><loc_62><loc_49><loc_77></location>Onages, the error budget based on MSTO fitting involves several key contributors. The most significant is distance uncertainty; an error of approximately 0.1-0.15 mag can result in about a 10% uncertainty in age. The error in the initial helium content, known within ∼ 2%, translates to about a 2% uncertainty in age. An error in the global metallicity of ∼ 9-10% and of ∼ 0.15 dex in iron content leads to approximately 4-5% error in age. An uncertainty of ∼ 0.15-0.2 dex in alpha elements translates to about 4% error in age (see a discussion in Cassisi & Salaris 2013). Combining these factors, the overall uncertainty in age can be around 1020% (e.g. O'Malley et al. 2017). For the sake of comparison, we consider a fixed error of ± 1.5 Gyr.</text>
<text><location><page_3><loc_6><loc_54><loc_49><loc_62></location>On mass, a typical uncertainty of ∼ 10% is generally found on single measurements, corresponding to 0.05 dex in log(M ⋆/ M ⊙ ) (Hénault-Brunet et al. 2019). Nevertheless, a di ff erence of ∼ 0.2 dex on average can be observed among di ff erent catalogues (e.g., Usher et al. (2017) with respect to Baumgardt et al. (2023)). For this reason, we adopt a typical error on mass of 0.2 dex.</text>
<section_header_level_1><location><page_3><loc_6><loc_48><loc_27><loc_49></location>3. Spectroscopic analysis</section_header_level_1>
<text><location><page_3><loc_6><loc_36><loc_49><loc_46></location>Before estimating the physical parameters of our GC sample with FSF, we want to derive measurements of the spectroscopic features in our sample and use those to have a preliminary assessment of the age and metal content of our GCs. At high redshift, the study of Lick indices (Burstein et al. 1984; Faber et al. 1985) or spectral breaks is often used to constrain stellar population properties. Here we want to see how GCs, the astrophysical objects that most resemble SSPs, fit inside this framework.</text>
<text><location><page_3><loc_6><loc_22><loc_49><loc_35></location>To do so, we first measured all the absorption features detectable in the spectra using the public code PyLick (Borghi et al. 2022a). We want to compare these data with theoretical models estimated by Thomas et al. (2011) at di ff erent ages, metallicities and alpha-enhancements ([ α / Fe]). Since these models are built with MILES resolution ( ∼ 2.7 Å FWHM), we downgraded the spectral resolution of our sample to match the models. Moreover, if an index lay in the overlap region of two spectral gratings, we ran PyLick on both spectra and then averaged the two values weighting them with their associated errors.</text>
<text><location><page_3><loc_6><loc_10><loc_49><loc_21></location>In particular, we measured indices of the Balmer series, like H β and H γ , indices of the iron group like Fe5270 and Fe5335 or Mgb and broader features like the D4000, a spectral break at 4000 Å. These features are particularly relevant in the study of stellar populations in that they are known to independently correlate with age (like D4000 and H β ) and metallicity (like the iron group), so their analysis can give important insights into the physical properties of the GCs, as we discuss in the following sections.</text>
<section_header_level_1><location><page_3><loc_51><loc_92><loc_67><loc_93></location>3.1. Index-age analysis</section_header_level_1>
<text><location><page_3><loc_51><loc_74><loc_94><loc_91></location>After measuring the spectral features, we can compare their trend in age with the theoretical ones. A variety of stellar libraries are available in the literature, like Bruzual & Charlot (2003), Maraston & Strömbäck (2011) and Conroy & van Dokkum (2012), adopting di ff erent stellar evolutionary models, libraries of stellar spectra and procedures to compute the integrated spectra. In Moresco et al. (2012), however, it is shown how the assumption of di ff erent SPS models has a negligible impact on the slope of the index-age trend, in particular in the case of the D4000. To compare our measurements with theoretical trends, here we are considering the 2016 version of Bruzual & Charlot (2003) models (hereafter BC16), since these are the same SPS models implemented in BAGPIPES .</text>
<text><location><page_3><loc_51><loc_60><loc_94><loc_74></location>In Fig. 1a the measured Dn4000 is shown as a function of the GC literature age for the 75 GCs for which an age estimate is provided in the literature, divided into six metallicity bins. For a qualitative comparison, we report the theoretical trends from BC16 with [Fe / H] varying from -0.33 to -2.25 and alphaenhancement ([ α/ Fe]) fixed to solar value. The trends are almost flat in this age interval, as expected in these ranges of ages and metallicities (see, e.g., Moresco et al. 2022) but show an evident gradient with metallicity, that is in good agreement with the theoretical distribution, with the Dn4000 increasing as metallicity raises.</text>
<text><location><page_3><loc_51><loc_53><loc_94><loc_59></location>Analogous to the Dn4000, in Fig. 1b the trends in age are shown for the Balmer index H β , in comparison with the theoretical trends in the same metallicity bins. The distribution of this feature shows a good agreement with the models as well, this time decreasing with increasing metallicity.</text>
<text><location><page_3><loc_51><loc_40><loc_94><loc_53></location>Testing these observables against the stellar models in the case of GCs, objects for which independent and robust measurements of age and metallicities are available, is of great importance in order to validate the models and their use in cosmological analyses. In the application of cosmic chronometers (Jimenez & Loeb 2002), for instance, especially when the D4000 is directly used to trace the age evolution in redshift (Moresco et al. 2012; Moresco 2015; Moresco et al. 2016) it is fundamental that the D4000 traces correctly the stellar population evolution in time (i.e. the D4000-age slope).</text>
<section_header_level_1><location><page_3><loc_51><loc_37><loc_69><loc_38></location>3.2. Index-index diagrams</section_header_level_1>
<text><location><page_3><loc_51><loc_10><loc_94><loc_35></location>While in the previous section we studied the sensitivity of single features to the age and metallicity of the population, it is also possible to combine them in the analysis, taking advantage of their di ff erent sensitivity to parameters like age, metallicity, and alpha-enhancement. In this case, we are considering a di ff erent set of theoretical models to compare them with, specifically developed to take into account also a variation of the chemical composition (Thomas et al. 2011, TMJ). Historically, this method has been applied in galaxies (e.g., Onodera et al. 2015; Scott et al. 2017; Lonoce et al. 2020; Borghi et al. 2022a) but also in GCs (e.g., Strader & Brodie 2004; Proctor et al. 2004; Mendel et al. 2007; Annibali et al. 2018) to derive constraints on the physical properties of these objects. Applying this method, we select and compare indices that are mostly sensible to age, metallicity, or α -enhancement variations so that we can disentangle their contributions to the feature's equivalent width. The most widely used are indices of the Balmer series, the iron-group ones, like Fe5270 and Fe5335, and Mgb. As for BC16, the models in TMJ have MILES resolution, but di ff erently from BC16, they produce a forecast only for the set of Lick indices and not</text>
<figure>
<location><page_4><loc_9><loc_15><loc_91><loc_94></location>
<caption>Fig. 1: Dn 4000 (top) and H β (bottom) trends with age and metallicity. The indices are shown for a sample of 75 GCs for which a literature value of age is available, divided into six [Fe / H] bins and colour-coded according to it. In the background, the stellar models from BC16 relative to each [Fe / H] bin are shown with di ff erent line styles.</caption>
</figure>
<text><location><page_4><loc_49><loc_14><loc_51><loc_15></location>(b)</text>
<text><location><page_4><loc_6><loc_7><loc_24><loc_8></location>Article number, page 4 of 16</text>
<figure>
<location><page_5><loc_30><loc_88><loc_69><loc_93></location>
</figure>
<figure>
<location><page_5><loc_9><loc_60><loc_48><loc_87></location>
</figure>
<figure>
<location><page_5><loc_52><loc_60><loc_91><loc_87></location>
<caption>Fig. 2: Index-index diagnostic diagrams, on the left H β -Mgb and on the right H γ F-[MgFe]'. The points are colour-coded by their literature value of [Fe / H]. We report TMJ models as a grid with varying metallicity (vertical lines) and varying age (horizontal lines), the first with the same colour code as the data points.</caption>
</figure>
<text><location><page_5><loc_6><loc_44><loc_49><loc_53></location>the entire spectrum. In particular, they model the indices values on a grid of ages, metallicities, and alpha-enhancements. Generally, an age-sensitive and a metallicity-sensitive index are plotted one against the other, and compared to a model grid with varying age and [Fe / H] but fixed [ α / Fe]. In Fig. 2 we present two examples of these diagrams, H β -Mgb and H γ F-[MgFe]', where the latter is defined as:</text>
<formula><location><page_5><loc_6><loc_42><loc_49><loc_43></location>[Mg / Fe] ' = p Mgb(0 . 72 × Fe5270 + 0 . 28 × Fe5335) . (2)</formula>
<text><location><page_5><loc_6><loc_37><loc_49><loc_41></location>Here the [ α / Fe] is fixed at 0.3, which is the closest to the typical value found in the literature for our sample of GCs, around 0.35 (e.g., Pritzl et al. 2005; Mendel et al. 2007).</text>
<text><location><page_5><loc_6><loc_23><loc_49><loc_37></location>This method allows us to have a first estimate of the population's age and metallicity, which aligns well with the literature values, obtained with the independent traditional methods. In terms of age, the sample of GCs populates the area of the oldest objects in the diagrams, with a percentage of data points compatible with an age older than 12 Gyr of 66% and 88%, respectively in the H β -Mgb and H γ F-[MgFe]'. From the diagrams in Fig. 2 we can also see a small percentage of GCs in the area of typically younger objects, above the 8 Gyr grid-line. In particular, in the H β -Mgb this happens for 17 GCs, while in the H γ F-[MgFe]' for 11 GCs.</text>
<text><location><page_5><loc_6><loc_10><loc_49><loc_22></location>This behaviour can be attributed to the possible presence of an extended horizontal branch (HB), which can make the spectrum appear much bluer than expected for an old population and exhibit prominent Balmer lines, resembling a younger object. This is a very well-known e ff ect, that has always made the study of GCs from integrated light challenging, mostly because the parameters determining the presence and the extent of the HB are not fully predictable with the current stellar evolution models. Various works have made progress in developing diagnostics to identify elongated HBs from integrated light, based on the</text>
<text><location><page_5><loc_51><loc_39><loc_94><loc_53></location>Balmer lines (Lee et al. 2000; Schiavon et al. 2004) or on CaII and Mgb (Percival & Salaris 2011). Others have managed to include the HB contribution on top of the SSP models (Jimenez et al. 2004; Koleva et al. 2008; Cabrera-Ziri & Conroy 2022), modelling the emission from the HB hot stars as identified in the GCs' CMD. However, we still lack of a complete modelisation of the HB component, due to the many uncertainties around its origin. Modelling the HB component is beyond the scope of this study, instead, our primary goal is to assess how this unmodeled component may impact studies of integrated populations, using methods commonly employed in galaxy evolution analyses.</text>
<text><location><page_5><loc_51><loc_35><loc_94><loc_38></location>The most common parameter used to quantify the HB extent is the morphology index HBR (Lee 1989; Lee et al. 1994), defined as:</text>
<formula><location><page_5><loc_51><loc_31><loc_94><loc_34></location>HBR = B -R B + R + V (3)</formula>
<text><location><page_5><loc_51><loc_10><loc_94><loc_30></location>where B and R are the number of stars bluer and redder than the RR Lyrae instability strip, and V is the number of RR Lyrae stars. Although this parameter does not fully capture the distribution of stars along the HB, it still provides valuable information about the HB morphology, indicating whether it is predominantly red (HBR ∼ -1) or blue (HBR ∼ 1). In Fig. 3 we report the same H β -Mgb diagram as in Fig. 2a, but this time colour-coded by the HBR value listed in Harris (1996) (2010 edition), which is known for 69 / 82 objects (coloured points). According to this index, among the 14 GCs populating the area above the 8 Gyr grid line and for which the HBR is known, 13 show a blue HB, and for 11 of those HBR is even higher than 0.5, a clue of a very elongated blue HB. The fraction of objects with HBR above zero decreases as we move to the areas belonging to older ages: 72% (18 / 25) between 8 Gyr and 12 Gyr and 33% (10 / 30) over 12 Gyr. The same trend can be observed moving from lower to higher</text>
<text><location><page_6><loc_27><loc_90><loc_31><loc_92></location>HBR</text>
<figure>
<location><page_6><loc_8><loc_62><loc_47><loc_92></location>
<caption>Fig. 3: H β -Mgb diagram, colour-coded by the value of HBR index. Blank points are GCs for which the HBR estimate is not available in Harris (1996) (2010 version). TMJ models are represented as a grid with varying metallicity (vertical lines) and varying age (horizontal lines).</caption>
</figure>
<text><location><page_6><loc_6><loc_47><loc_49><loc_51></location>[Fe / H], with a percentage of GCs showing blue HBs decreasing from 68% (23 / 34) at [Fe / H] < -1 . 35 to 49% (13 / 29) in the range -1.35 ≤ [Fe / H] ≤ -0.33 and dropping to zero at [Fe / H] > -0 . 33.</text>
<text><location><page_6><loc_6><loc_35><loc_49><loc_47></location>As anticipated, stellar evolution models do not currently account for the presence of an extended blue HB, so this must be considered in the FSF analysis, where objects with extended HB morphology might be mistakenly identified as young stellar populations (Schiavon et al. 2004; Jimenez et al. 2004; Koleva et al. 2008; Cabrera-Ziri & Conroy 2022). From our initial qualitative analysis using indices, we expect this issue to be more prevalent in metal-poor objects with prominent H β , which tend to show the highest HBR.</text>
<section_header_level_1><location><page_6><loc_6><loc_32><loc_25><loc_33></location>4. Method and analysis</section_header_level_1>
<text><location><page_6><loc_6><loc_27><loc_49><loc_31></location>In this section, we present the method adopted to estimate the ages and physical properties of the GCs sample, the code used, its settings, and the results obtained.</text>
<section_header_level_1><location><page_6><loc_6><loc_24><loc_33><loc_25></location>4.1. Full-spectral-fitting with BAGPIPES</section_header_level_1>
<text><location><page_6><loc_6><loc_12><loc_49><loc_22></location>We perform FSF using the public code BAGPIPES (Carnall et al. 2018), which allows us to fit spectra and / or photometry adopting a parametric Bayesian approach. A detailed description of all the code's features is presented in Carnall et al. (2019) and Carnall et al. (2022), while an overview of the settings that we adopt is already outlined in Tomasetti et al. (2023). Here we recap the main features of the code, highlighting the aspects that are relevant to this work.</text>
<text><location><page_6><loc_6><loc_10><loc_49><loc_12></location>BAGPIPES is able to model synthetic spectra and photometry, based on a set of instructions, and then fit the so-modelled</text>
<text><location><page_6><loc_51><loc_87><loc_94><loc_93></location>spectro-photometry to the observed one via a Bayesian approach, thus maximising the posterior probability using a nested sampling algorithm, Multinest (Buchner 2016). In this work, we focus on four main model components to construct the synthetic spectra.</text>
<text><location><page_6><loc_51><loc_75><loc_94><loc_86></location>The first component is an SSP model, which is designed to reproduce the continuum emission and the absorption features of a population built up in a single episode of star formation. The SSP models implemented in BAGPIPES are the 2016 version of Bruzual & Charlot (2003) (BC16, see Chevallard & Charlot 2016). They produce di ff erent synthetic spectra based on the wavelength λ range, the age of the stellar population, and its overall metallicity [Z / H], assuming a Kroupa (2001) initial mass function (IMF).</text>
<text><location><page_6><loc_51><loc_67><loc_94><loc_75></location>The second component is the star formation history (SFH). The code allows you to combine di ff erent SFHs, one for each SSP, but dealing with GCs we implement a single SFH, assuming a unique star formation episode. In particular, we adopt the delayed exponentially declining (DED) SFH, which is given by the equation</text>
<formula><location><page_6><loc_51><loc_63><loc_94><loc_66></location>SFR(t) ∝ (t -T0) e -t -T 0 τ , t > T0 0 , t < T0 , (4)</formula>
<text><location><page_6><loc_51><loc_56><loc_94><loc_62></location>where SFR is the star formation rate, τ provides the width of the SFH and T0 sets the age of the Universe at which the star formation begins. Using a single DED is recommendable when dealing with a stellar population whose time scale of formation is much shorter than its age, as we expect for GCs.</text>
<text><location><page_6><loc_51><loc_44><loc_94><loc_56></location>The third component is dust absorption and emission. This is particularly important to model the redder part of the spectrum, which can be largely depressed due to the presence of dust in the system. In the context of MW GCs, this component is necessary to account for the MW dust on the line-of-sight, for which WAGGSspectra are not corrected. The model implemented here is the Salim et al. (2018), represented by a power-law, as in Calzetti et al. (2000), with an additional parameter, δ , representing a slope deviation.</text>
<text><location><page_6><loc_51><loc_39><loc_94><loc_44></location>The last component is a non-physical term representing noise, which can be added to the error spectrum to account for any potential underestimation. This noise is introduced as white noise.</text>
<text><location><page_6><loc_51><loc_27><loc_94><loc_39></location>After running the code, we obtain a best-fit spectrum and the posterior distributions for all the parameters involved, like age, mass formed, overall metallicity, and dust extinction. At the same time, BAGPIPES can provide estimates of derived quantities, like the SFR or the stellar mass formed, parameters that are not directly involved in the fit. In particular, the mass formed (Mformed) and the stellar mass formed (M ⋆ ) are di ff erent in that the first one comprises all the mass formed from t = 0 to the time t at which the GC is observed:</text>
<formula><location><page_6><loc_51><loc_24><loc_94><loc_27></location>Mformed = Z t 0 SFR(t ' )dt ' , (5)</formula>
<text><location><page_6><loc_51><loc_20><loc_94><loc_23></location>including also stellar remnants (Mrem), while the second one includes only the mass of living stars:</text>
<formula><location><page_6><loc_51><loc_17><loc_94><loc_20></location>M ⋆ = Z t 0 SFR(t ' )dt ' -Mrem . (6)</formula>
<text><location><page_6><loc_51><loc_15><loc_89><loc_16></location>From now on, we refer to M ⋆ as the mass of the objects.</text>
<text><location><page_6><loc_51><loc_10><loc_94><loc_15></location>Since we aim to use the resulting ages in a cosmological framework, it is important to avoid any constraint based on cosmology. For this reason, we employ a version of BAGPIPES , described in Jiao et al. (2023), that deviates from the original in</text>
<text><location><page_7><loc_6><loc_74><loc_49><loc_93></location>handling the priors on the stellar population age, allowing them to vary up to a cosmology-independent value (e.g., 15 Gyr, 20 Gyr) at any redshift. This modification has already been tested and validated in VANDELS (Tomasetti et al. 2023) and LEGAC (Jiao et al. 2023). Originally the code assumes a cosmological prior on ages, for which the maximum age resulting from the fit should be smaller than the age of the Universe at the corresponding redshift, given a flat Λ CDM model with parameters Ω M = 0 . 3 , ΩΛ = 0 . 7 and H0 = 70 km s -1 Mpc -1 . Although this e ff ect is of relative interest in stellar population studies and is typically neglected, it can't be ignored in cosmological analyses because the derived ages would be constrained by the cosmological model assumed, leading to results that just recover the assumed prior. Below, we are going to test the e ff ect of di ff erent assumptions of age prior on the sample of GCs.</text>
<section_header_level_1><location><page_7><loc_6><loc_70><loc_30><loc_71></location>4.2. Full-spectral-fitting in WAGGS</section_header_level_1>
<text><location><page_7><loc_6><loc_67><loc_49><loc_69></location>Before inputting the cluster spectra into BAGPIPES, some adjustments were necessary.</text>
<text><location><page_7><loc_6><loc_63><loc_49><loc_66></location>First, we downgraded the spectral resolution to approximately 2.7 Å FWHM, consistent with the BC16 models used in the code.</text>
<text><location><page_7><loc_6><loc_57><loc_49><loc_62></location>Next, we aligned the spectra to the correct frame using distances from Baumgardt et al. (2023) and corrected for radial velocity variations, which could cause minor blueshifts or redshifts in the spectra.</text>
<text><location><page_7><loc_6><loc_47><loc_49><loc_57></location>To prevent underweighting the blue features in the fit - due to the non-uniform error spectrum, with S / N ranging from a few tens to a thousand - we set an upper limit for the S / Nat 100 and adjusted the error spectrum accordingly. We tested various S / N thresholds (e.g., 20, 50), finding that they had minimal impact on the results, except when the S / N ratio between the blue and red ends of the spectrum di ff ered by more than a factor of ten, which resulted in very low weight for the blue features in the fit.</text>
<text><location><page_7><loc_6><loc_26><loc_49><loc_46></location>After these adjustments, we performed multiple tests to optimally use BAGPIPES on GCs spectra and find the best-fit configuration to reproduce their spectral features accurately. In particular, this process involved: adopting di ff erent SFHs (e.g., single burst, delayed exponentially declined) with di ff erent priors on the parameters; fitting di ff erent wavelength ranges, either moving the lower limit to longer wavelengths to reduce the contamination by HB stars or pushing the upper limit to redder features to better constrain dust reddening; testing di ff erent priors on the GC's [Z / H] and mass (e.g., uniform, Gaussian, logarithmic) to assess their impact on the estimation of these parameters, as well as the influence on ages, given the degeneracies at play. It is worth mentioning here that the mass and metallicity estimates proved to be very stable against all the di ff erent changes in the fit setup, while ages were mainly a ff ected by the choice of prior, as we discuss in the following.</text>
<text><location><page_7><loc_6><loc_15><loc_49><loc_25></location>In the end, we converged to a fit configuration in which: (i) as often done in literature (Koleva et al. 2008; Gonçalves et al. 2020) we fit the range 3700-6000 Å to avoid the redder telluric lines and we mask the interval 5870 -5910 Å, where the spectra show a very deep sodium doublet absorption line, since it could be potentially contaminated by interstellar absorption; (ii) we consider a single DED SFH, a dust component and a noise component; (iii) on all the parameters we set uniform, uninformative priors.</text>
<text><location><page_7><loc_6><loc_10><loc_49><loc_15></location>To assess the impact of the cosmological prior on the results, we tested two di ff erent upper limits for the age parameter: 13.47 Gyr, age of the Universe in a flat Λ CDMmodel with Ω M , 0 = 0.3, ΩΛ , 0 = 0.7, H0 = 70 km / s / Mpc; 15 Gyr, a loose limit independent</text>
<text><location><page_7><loc_51><loc_84><loc_94><loc_93></location>of any cosmology. We refer to the first configuration as Config. 13.5 and to the latter as Config. 15 . A summary of the main parameters and relative priors for the two configurations can be found in Tab. 1. We highlight that we assume uniform priors on all the parameters, along with wide ranges so that the results are not constrained by any previous knowledge of the GC's mass, metallicity, dust extinction or age.</text>
<text><location><page_7><loc_51><loc_78><loc_94><loc_84></location>As anticipated, BAGPIPES adopts the overall metallicity [Z / H] as the metallicity parameter. To compare our results with [Fe / H] values from the literature, we need to perform a conversion. We use the conversion formula from Salaris & Cassisi (2005):</text>
<formula><location><page_7><loc_51><loc_71><loc_94><loc_76></location>[Z / H] = log Z Z ⊙ ! = = [Fe / H] + log10 GLYPH<16> 10 [ α/ Fe] 0 . 694 + 0 . 306 GLYPH<17> . (7)</formula>
<text><location><page_7><loc_51><loc_61><loc_94><loc_70></location>For objects with [Fe / H] ≤ -1 we apply this formula with an alpha-enhancement of [ α / Fe] = 0.35, which is typical of the metal-poor MW GCs (e.g., Pritzl et al. 2005; Mendel et al. 2007), while for GCs with [Fe / H] > -1 we use [ α / Fe] = 0.15, average alpha-enhancement at these metallicities (see, e.g., Pagel & Tautvaisiene 1995; Pancino et al. 2017). From now on, we refer to the quantity [Z / H] as the metallicity of the GCs.</text>
<section_header_level_1><location><page_7><loc_51><loc_58><loc_59><loc_59></location>4.3. Results</section_header_level_1>
<text><location><page_7><loc_51><loc_32><loc_94><loc_57></location>We performed a visual inspection to evaluate the quality and convergence of the fits. Specifically, we identified fits that either failed to recover the spectral lines or continuum or exhibited double- or multiple-peaked posterior distributions. As a result, we discarded a significant number of poor fits, totalling 11 objects in both Config. 13.5 and Config. 15 , which represent about 14% of the sample. Among these, 8 GCs had an HBR > 0, and we found that, in these cases, the posterior spectrum underestimated the emission in the wavelength range blueward of 4000 -4500 Å. This issue is likely due to the blue HB emission, which the models cannot fully reproduce. Consequently, the fits converge to younger ages, as observed in these cases where all 8 poor fits have ages younger than 10 Gyr. As discussed in Sect. 3.2, various studies have successfully included a contribution of the HB on top of the SSP models (e.g., Jimenez et al. 2004; Koleva et al. 2008; Cabrera-Ziri & Conroy 2022). However, incorporating this component into BAGPIPES is outside the scope of this work. Removing bad fits, the clean sample counts 66 GCs in both configurations.</text>
<text><location><page_7><loc_51><loc_15><loc_94><loc_32></location>In Fig. 4 two examples of good fits are reported, both converging to ages older than 13 Gyr, one presenting a red HB (NGC6356, HBR = -1) while the other shows a blue HB (NGC6717, HBR = 0.98). The pulls highlight how in the case of NGC6356 the stellar models, plus the dust components, are able to accurately reproduce the GC's spectrum, with residuals compatible with 1σ fluctuations at all wavelengths. In the case of NGC6717, the quality of the fit is still good, but the pulls clearly show a residual at bluer wavelengths, especially concerning the Balmer absorption lines, pointing out the unmodelled hot stars component. This suggests that for GCs characterised by blue HBs may still produce a reliable age estimation, as long as the blue HB stars do not outshine the blue end of the spectrum.</text>
<text><location><page_7><loc_51><loc_10><loc_94><loc_15></location>The quality of both considered setups is highlighted by the median reduced chi-squares, ˜ χ 2 = 1 . 21 in Config. 13.5 and ˜ χ 2 = 1 . 26 in Config. 15 . This is quite noticeable since in this case we adopted the formal spectrum error provided by the analysis,</text>
<table>
<location><page_8><loc_8><loc_75><loc_92><loc_91></location>
<caption>Table 1: Parameters and priors for di ff erent configurations.</caption>
</table>
<text><location><page_8><loc_6><loc_66><loc_49><loc_72></location>with the correction described in Sect. 4.2. These values are further (and as expected) reduced if we take into account the noise parameter, which acts in correcting the error spectrum for potential underestimations, leading to ˜ χ 2 = 0 . 99 in Config. 13.5 and ˜ χ 2 = 0 . 98 in Config. 15 .</text>
<text><location><page_8><loc_6><loc_61><loc_49><loc_66></location>To analyse the derived physical properties, we consider, for each parameter, the median and the 16 th and 84 th percentiles of the posterior distribution respectively as the best-fit value, lower and upper error.</text>
<section_header_level_1><location><page_8><loc_6><loc_57><loc_24><loc_58></location>4.3.1. Configuration 13.5</section_header_level_1>
<text><location><page_8><loc_6><loc_42><loc_49><loc_56></location>In Config. 13.5 we observe that metallicities and GC masses are in good agreement with literature values, with mean deviations of ⟨ ∆ [Z / H] ⟩ = 0 . 09 ± 0 . 21 dex and ⟨ ∆ log(M ⋆/ M ⊙ ) ⟩ = -0 . 09 ± 0 . 24 dex respectively, consistent with the typical errors associated with these quantities (see Sect. 2). In terms of stellar age, instead, a clear bimodality is present. While 17% of the sample (11 GCs) turns out to have ages older than 10 Gyr and only ∼ 0.16 Gyr younger than literature values on average, most of it (55 GCs) shows ages significantly younger than 10 Gyr, ∼ 8.9 Gyr lower on average. We investigate this di ff erence in the following.</text>
<text><location><page_8><loc_6><loc_38><loc_49><loc_42></location>In Fig. 5a and 5b we show the di ff erences in stellar mass and metallicity as a function of this age gap, colour-coded by HBR index. This highlights two important aspects.</text>
<text><location><page_8><loc_6><loc_10><loc_49><loc_38></location>The first is, again, the trend with HBR, showing that when this index is positive (blue HB), the code misinterprets the blue shape of the spectrum and the deep Balmer lines for a young population 87% of the time (27 / 31), resulting in ages on average 8.4 Gyr younger than expected from the literature. This exact behaviour is also observed for most of the red HB population, but in a smaller fraction (73% of the cases, 19 / 26 GCs) and with a less significant age discrepancy of 5.1 Gyr on average. A similar result was already found both in Koleva et al. (2008) and Cabrera-Ziri & Conroy (2022), where the issue was mitigated by adding a fraction of hot stars on top of the SPS models, but, as anticipated, including the HB component goes beyond the purpose of this work. In Sect. 4.6, though, a detailed comparison with the results in Cabrera-Ziri & Conroy (2022) can be found. The second is the existence of a degeneracy among the parameters involved. A lower cluster mass or a higher metallicity can easily mislead the fit to ages much younger than the literature one. This is clear if we compute the median di ff erences in metallicity and mass separately for the GCs resulting older and younger than 10 Gyr: for the former, we find average deviations of ⟨ ∆ [Z / H] ⟩ = -0 . 14 ± 0 . 19 dex and ⟨ ∆ log(M ⋆/ M ⊙ ) ⟩ = 0 . 16 ± 0 . 15 dex, for</text>
<text><location><page_8><loc_51><loc_70><loc_94><loc_72></location>the latter instead ⟨ ∆ [Z / H] ⟩ = 0 . 13 ± 0 . 18 dex and ⟨ ∆ log(M ⋆/ M ⊙ ) ⟩ = -0 . 13 ± 0 . 24 dex.</text>
<section_header_level_1><location><page_8><loc_51><loc_66><loc_67><loc_67></location>4.3.2. Configuration 15</section_header_level_1>
<text><location><page_8><loc_51><loc_25><loc_94><loc_65></location>In Config. 15 the good agreement of metallicity and GC mass estimates with literature values holds, with average di ff erences of ⟨ ∆ [Z / H] ⟩ = -0 . 02 ± 0 . 23 dex and ⟨ ∆ log(M ⋆/ M ⊙ ) ⟩ = 0 . 04 ± 0 . 28. Concerning the stellar ages, even though the only di ff erence with respect to Config. 13.5 is the removal of the cosmological prior, the results improve significantly. Here 36% (24 GCs) of the sample shows ages older than 10 Gyr, more than twice the old population of Config. 13.5 . Among these 24 GCs, the ages result compatible with literature values 92% of the times (22 GCs), on average 0.67 Gyr older, and we find average discrepancies in [Z / H] and mass of ∆ [Z / H] = -0 . 11 ± 0 . 18 and ∆ log(M ⋆/ M ⊙ ) = 0 . 20 ± 0 . 20. In Fig. 5c and Fig. 5d we show the analogous of 5a and Fig. 5b for Config. 15 . We can see that both the trend in HBR and the age-metallicity and age-mass degeneracies are present but with important di ff erences. This time, removing the upper limit on the age parameter has reduced the fraction of blue HB GCs mistaken for young populations to 71% (22 / 31) and the one of the red HB GCs to 54% (14 / 26). This means that 13 GCs previously resulting younger than 10 Gyr are now recognised as old populations, representing a 20% increment. The degeneracies cited above play an important role in this because all of these 13 GCs are here characterised by a lower metallicity ( ∆ [Z / H] ∼ -0 . 18 dex) and a higher mass ( ∆ log(M ⋆/ M ⊙ ) ∼ 0 . 19 dex) than the one found in Config. 13.5 , yet mostly in agreement with literature values within errors. This suggests that for a fraction of GCs an old, more realistic solution does exist beyond the cosmological limit usually set when performing FSF and that it may also be preferred to the younger one if this area of the parameter space is made accessible. For this reason and in light of the subsequent cosmological analysis, we consider Config. 15 our benchmark.</text>
<section_header_level_1><location><page_8><loc_51><loc_21><loc_67><loc_22></location>4.4. Systematic effects</section_header_level_1>
<text><location><page_8><loc_51><loc_10><loc_94><loc_20></location>As anticipated in Sect. 4.2, we performed multiple tests with different settings to find the optimal fit configuration. These analyses have been used to assess the systematic error induced in the age determination with our approach. Specifically, we examined 8 configurations, each di ff ering from our benchmark in one or two key aspects, including variations in SFH type (burst or DED), age prior (15 Gyr or 20 Gyr) metallicity prior (uniform or Gaussian), wavelength range of the fit, and in fitting spec-</text>
<figure>
<location><page_9><loc_7><loc_74><loc_50><loc_93></location>
</figure>
<figure>
<location><page_9><loc_52><loc_74><loc_91><loc_93></location>
<caption>Fig. 4: Examples of good fits. In the top panels, the observed spectra are shown in black and the posterior ones in orange, dashed lines identify the Balmer absorption series, while other main absorption features are highlighted with grey shaded boxes. In the bottom panels, the pulls of each fit ((observed - fit) / error) are shown, with the orange horizontal area representing a 1σ fluctuation.</caption>
</figure>
<text><location><page_9><loc_6><loc_59><loc_49><loc_65></location>tra in physical units or normalised in the window 4500-5000 Å. The latter was applied in just one configuration, the only case in which the mass parameter could not be determined due to the normalisation. All the characteristics of the di ff erent configurations are outlined in Tab. 1, numbered from 1 to 8.</text>
<text><location><page_9><loc_6><loc_46><loc_49><loc_59></location>In this analysis, we discarded all the spurious young solutions with a best-fit age below 10 Gyr for the same reasons discussed in Sect. 4.3, and all the bad fits in Config. 15 and in each of the 8 test configurations. In this way, we end up with a sample of 18 GSs having a good fit in at least 5 out of the 9 runs. For each object, we computed the standard deviation of the age distribution in the 9 runs. Finally, we estimated a global systematic contribution to the age uncertainty as the average of these standard deviations, weighted on the number of good fits for each GC, resulting in 0.71 Gyr 1 .</text>
<section_header_level_1><location><page_9><loc_6><loc_42><loc_24><loc_44></location>4.5. The role of metallicity</section_header_level_1>
<text><location><page_9><loc_6><loc_23><loc_49><loc_41></location>In Sect. 3.2 we observed that not only the presence of an extended blue HB, but also a low metallicity could produce some spectral features that can drag the fit to younger ages due to their degeneracy. Here we want to verify if this trend is present in our results. In Fig. 6a, we plot the metallicity obtained from our fit against the values found in the literature, colour-coded by the di ff erence between the age estimated from the fit and the literature one. We can observe that the best match in age estimation is indeed found at the highest metallicities, while the objects for which the age is most underestimated are also the ones with lower metallicity. To perform a quantitative comparison, we divided our sample into metallicity sub-samples, considering three intervals equally spaced in [Z / H]lit: a metal-rich [ -0 . 7 , 0 . 0], a metal-intermediate [ -1 . 4 , -0 . 7] and a metal-poor [ -2 . 1 , -1 . 4].</text>
<text><location><page_9><loc_6><loc_14><loc_49><loc_23></location>The metal-poor sample counts 15 GCs, among which only the 20% (3 / 15) is recognised as older than 10 Gyr. This can be better understood by examining the top panel of Fig. 7, where the median stacked spectrum of the metal-poor sample is compared to two synthetic spectra, one 13 Gyr old and the other 4 Gyr old, while all the other parameters (e.g., [Z / H], mass, dust) are fixed to the median literature values characterising</text>
<text><location><page_9><loc_51><loc_50><loc_94><loc_65></location>this sub-sample. The spectral shape does indeed resemble the one of a young population, with a stronger emission bluer than ∼ 4400Å with respect to the spectrum of an old population, and stronger Balmer lines. However, in Fig. 6b, where discrepancies in the reproduction of CaII K and H β are shown as a function of literature metallicity, we can see that for most of the metalpoor GCs, the H β line is actually overestimated. This means that the young solution, even if preferred by the fit, does not precisely follow the observed features. In terms of metallicity and mass, the agreement with literature values is very good, with average deviations of ⟨ ∆ [Z / H] ⟩ = -0 . 02 ± 0 . 35 dex and ⟨ ∆ log(M ⋆/ M ⊙ ) ⟩ = -0 . 2 ± 0 . 2 dex respectively.</text>
<text><location><page_9><loc_51><loc_20><loc_94><loc_49></location>The metal-intermediate is the largest sub-group, with 33 GCs, and shows a higher percentage of GCs older than 10 Gyr compared to the metal-poor sample, equal to 33% (11 / 33). Its median stacked spectrum has a better agreement with an old population in terms of the continuum but still fails to reproduce some observed lines of the Balmer series, as we can see in the central panel of Fig. 7. This is mostly evident for the H β line, falling in the region 4500 -5000Å where the spectra have been normalised, which is clearly better reproduced by a young population, as in the metal-poor case. This suggests that there is still non-negligible contamination by hot stars, deepening the Balmer series. We can observe this in more detail in Fig. 6b where to obtain old solutions, the fit has to underestimate the H β feature. In contrast, for the young ones, it is either compatible with observations or overestimated. In this metal-intermediate group, we can also observe the importance of reproducing the CaII K line in recovering ages. In fact, while the old solutions all scatter around ∆ CaIIK ∼ 0, the young ones systematically underestimate this feature. Regarding the metallicity, this subgroup shows a good agreement with literature values, with a ⟨ ∆ [Z / H] ⟩ = 0 . 10 ± 0 . 15 dex, and a discrepancy in mass smaller than before ⟨ ∆ log(M ⋆/ M ⊙ ) ⟩ = -0 . 04 ± 0 . 30 dex.</text>
<text><location><page_9><loc_51><loc_10><loc_94><loc_20></location>In the metal-rich sample instead, we are able to obtain ages older than 10 Gyr for 56% (10 / 18) of the sample. This fraction increases as we move to solar values, reaching 70% for [Z / H]lit ≥ -0 . 4. In the bottom panel of Fig. 7 we can see that, among the three samples, the metal-rich is the most distant from a young population with a much redder continuum. In addition, the metal-rich is the only stacked spectrum in which the old synthetic spectrum accurately reproduces the</text>
<figure>
<location><page_10><loc_8><loc_29><loc_92><loc_94></location>
<caption>Fig. 5: Di ff erences in stellar mass ( ∆ log(M ⋆/ M ⊙ )) and metallicity ( ∆ [Z / H]) as a function of the di ff erence in age as estimated in this work with respect to literature values. The dashed lines correspond to a null di ff erence, the grey shaded areas represent an average representative error on literature values, namely 0.2 dex in mass, 0.15 dex in metallicity and 1.5 Gyr in age. The top two panels refer to Config. 13.5 while the bottom ones to Config. 15 . All the points are colour-coded by their HBR index.</caption>
</figure>
<text><location><page_10><loc_6><loc_13><loc_49><loc_21></location>H β , while the young one leaves a clear residual. Looking at Fig. 6b, the metal-rich sample is the only one for which both H β and CaII K are well reproduced. In terms of metallicity and mass here we find ⟨ ∆ [Z / H] ⟩ = -0 . 08 ± 0 . 15 dex and ⟨ ∆ log(M ⋆/ M ⊙ ) ⟩ = 0 . 09 ± 0 . 16 dex with respect to literature values.</text>
<text><location><page_10><loc_6><loc_11><loc_49><loc_13></location>The fact that we can well recover ages in this metallicity range is a remarkable result also in the context of galaxy evolution stud-</text>
<text><location><page_10><loc_51><loc_12><loc_94><loc_21></location>ies, for which GCs have always been an important test bench, since it shows the reliability of the FSF method in recovering the main physical parameters of the stellar population in a metallicity interval that is typical of high redshift galaxies (see, e.g., Kriek et al. 2019; Lonoce et al. 2020; Carnall et al. 2022; Borghi et al. 2022a). Moreover, we show that spectral features like H β and CaII K prove to be very sensible to possible spurious age</text>
<text><location><page_11><loc_7><loc_77><loc_10><loc_77></location>]</text>
<text><location><page_11><loc_7><loc_76><loc_10><loc_77></location>H</text>
<text><location><page_11><loc_7><loc_75><loc_10><loc_76></location>/</text>
<text><location><page_11><loc_7><loc_75><loc_10><loc_75></location>Z</text>
<text><location><page_11><loc_7><loc_74><loc_10><loc_75></location>[</text>
<figure>
<location><page_11><loc_8><loc_62><loc_48><loc_92></location>
<caption>Fig. 6: Metallicity and di ff erences in CaII K and H β measurements as a function of literature metallicity. Top : [Z / H] obtained in this work against literature [Z / H], colour-coded by the corresponding di ff erence in age. The continuous line represents the one-to-one relation, and the dashed lines the 0.2 dex scatter. Bottom : Di ff erences in CaII K and H β as measured on the posterior spectra with the observed ones, colour-coded by the di ff erence in age.</caption>
</figure>
<text><location><page_11><loc_27><loc_59><loc_28><loc_60></location>(a)</text>
<text><location><page_11><loc_12><loc_57><loc_13><loc_59></location>1</text>
<figure>
<location><page_11><loc_7><loc_31><loc_48><loc_58></location>
</figure>
<text><location><page_11><loc_7><loc_41><loc_9><loc_42></location>H</text>
<text><location><page_11><loc_6><loc_10><loc_49><loc_12></location>determinations and thus can be used as diagnostics to determine the quality of the age estimates.</text>
<text><location><page_11><loc_26><loc_90><loc_30><loc_93></location>agefit</text>
<text><location><page_11><loc_32><loc_90><loc_36><loc_93></location>agelit</text>
<section_header_level_1><location><page_11><loc_51><loc_92><loc_77><loc_93></location>4.6. Comparison with previous works</section_header_level_1>
<text><location><page_11><loc_51><loc_79><loc_94><loc_91></location>As anticipated in Sect. 1, di ff erent works have already been published investigating the potential of analysing GCs' integrated spectra, either relying on di ff erent datasets or adopting di ff erent fitting codes. Here we focus on the work from Cabrera-Ziri & Conroy (2022) (CC22, hereafter), where they use the observations from Schiavon et al. (2005) for a common sub-sample of GCs, and then on the results from Gonçalves et al. (2020) (G20, hereafter), where they analyse the same data but adopt a nonparametric approach.</text>
<section_header_level_1><location><page_11><loc_51><loc_76><loc_88><loc_77></location>4.6.1. Comparison with Cabrera-Ziri & Conroy (2022)</section_header_level_1>
<text><location><page_11><loc_51><loc_44><loc_94><loc_75></location>CC22 used a similar, but non-parametric, FSF approach with the code ALF (Conroy & van Dokkum 2012) to estimate the ages and metallicities of 32 Galactic GCs from Schiavon et al. (2005), fitting normalised spectra in the range ∼ 3300 -6500Å. As in Config. 13.5 , CC22 initially performed the analysis with a standard setting, using a cosmological prior of 14 Gyr. They obtained ages compatible with literature values within 1.5 Gyr for 7 GCs, which constitutes 22% of their sample. To compare the results, we consider the 31 GCs included both in their sample and our good fits. For those, in Config. 13.5 we obtain ages within 1.5 Gyr from literature values for 13% of the sample (4 / 31) while this fraction is more than doubled when we remove the cosmological prior, reaching 32% (10 / 31). Compared to CC22, the performance in recovering old ages with Config. 13.5 is comparable but slightly worse. This is likely due to the additional degrees of freedom in our setting, such as the inclusion of dust and mass parameters, along with the lower age prior. If adopting a multicomponent model can help in reproducing all the spectral features in more detail, this approach is also more prone to possible degeneracies, as we underlined in di ff erent steps of our analysis. Nevertheless, this same choice allows us to obtain better results when removing the cosmological prior, with an increase in the fraction of old objects of 20% with respect to our Config. 13.5 and 10% with respect to the standard setting in CC22.</text>
<text><location><page_11><loc_51><loc_35><loc_94><loc_44></location>In CC22, an additional fit was performed that included a component to account for the hotter fraction of HB stars. This approach allowed them to recover ages older than 10 Gyr for 27 out of 31 GCs, with 24 of these being compatible with literature values within 1.5 Gyr. As already discussed, modelling the HB component is outside the purpose of this work, but represents a promising possibility to be explored in future analyses.</text>
<section_header_level_1><location><page_11><loc_51><loc_31><loc_84><loc_33></location>4.6.2. Comparison with Goncalves et al. (2020)</section_header_level_1>
<text><location><page_11><loc_51><loc_16><loc_94><loc_30></location>In G20 the authors focus on the impact that the wavelength range choice has on the results, adopting the FSF code STARLIGHT (Cid Fernandes et al. 2005). They fit normalised spectra using MILES SSP models (Vazdekis et al. 2015) with ages up to 14 Gyr, [Fe / H] in the range from -2.27 to 0.26 and alpha enhancement either absent or equal to 0.4. Dust reddening is implemented in the code, modelled as in Cardelli et al. (1989). We compare our results to the ones published in Goncalves et al. (2023), obtained by fitting the interval 4828 - 5634 Å, a narrow range where the main features detectable are H β , Mgb triplet, Fe5015, Fe5270 and Fe5335.</text>
<text><location><page_11><loc_51><loc_10><loc_94><loc_16></location>We consider the 58 MW GCs for which we obtain a good fit among the 64 MW GCs published in G20; in this sample, they obtained ages compatible with literature values within 1.5 Gyr for 9 GCs, representing 15% of the total. In this same sub-group, we have 7 GCs compatible with literature ages in Config. 13.5 ,</text>
<figure>
<location><page_12><loc_9><loc_20><loc_91><loc_96></location>
<caption>Fig. 7: Stacked spectra of the metal-poor (top panel, [ -2 . 2 , -1 . 4]), metal-intermediate (central panel, [ -1 . 4 , -0 . 6]) and metal-rich (bottom panel, [ -0 . 6 , 0 . 2]) subsamples, in comparison with a 4 Gyr (in blue) and 13 Gyr old (in red) synthetic spectra. The values of mass, dust absorption and velocity dispersion of the synthetic spectra are set to the median literature values of each subgroup. All the spectra are normalised in the window 4500-5000 Å. At the bottom of each panel the residuals of the stacked spectra with respect to the synthetic ones are shown with corresponding colours. The Balmer series is indicated with dashed lines, other relevant spectral features are highlighted with grey boxes. It can be noticed how the di ff erent sub-groups show di ff erent continuum and spectral features properties, and how the models are able to reproduce more accurately the high-metallicity ones.</caption>
</figure>
<text><location><page_13><loc_6><loc_82><loc_49><loc_93></location>and 15 in Config. 15 , corresponding to 12% and 26% of the sample, respectively. As in the comparison with CC22, also in the case of G20 our results when the cosmological prior is applied are comparable but slightly worse, and as commented above the reason probably resides in the higher number of parameters involved and the lower age prior. Again, when we remove the cosmological prior, we obtain a major improvement in the fraction of ages compatible with literature, both with respect to our Config. 13.5 ( + 14%) and to G20 ( + 11%).</text>
<text><location><page_13><loc_6><loc_62><loc_49><loc_81></location>It is worth mentioning here that we also tested the impact on the results of fitting the wavelength range adopted in G20, first suggested in Walcher et al. (2009), and we considered it when computing the systematic uncertainty on ages in Sect. 4.4. Avoiding all the features bluer than 4828 Å, polluted by the hot HB component, this configuration performs much better for the blue HB, low-metallicity GCs in our sample. In particular, it allows us to recover ages older than 10 Gyr for 63% of metalpoor GCs and 73% of metal-intermediate ones. For the metalrich sample, instead, it yields worse results compared to Config. 15 , with 47% of GCs resulting older than 10 Gyr. Nevertheless, the latter sub-group is the one in which the stellar models should be most e ff ective, thanks to the absence of an extended HB component and low alpha-enhancement in the systems, so Config. 15 was still preferable in terms of the robustness of the results.</text>
<section_header_level_1><location><page_13><loc_6><loc_58><loc_29><loc_60></location>5. Application to cosmology</section_header_level_1>
<text><location><page_13><loc_6><loc_51><loc_49><loc_57></location>In this section, we analyse what impact our results for GCs' ages, interpreted as lower limits to the age of the Universe tU , can have in the determination of the Hubble constant H 0 (Jimenez et al. 2019; Valcin et al. 2020, 2021; Vagnozzi et al. 2022; Cimatti & Moresco 2023).</text>
<section_header_level_1><location><page_13><loc_6><loc_48><loc_15><loc_49></location>5.1. Method</section_header_level_1>
<text><location><page_13><loc_6><loc_43><loc_49><loc_47></location>As anticipated in Sect. 1, and widely described in Cimatti & Moresco (2023), H 0 is very sensitive to the value of tU . In a generic cosmological model, H 0 can be expressed as:</text>
<formula><location><page_13><loc_6><loc_39><loc_49><loc_42></location>H0 = A t Z zF 0 1 (1 + z)E(z) dz , (8)</formula>
<text><location><page_13><loc_6><loc_31><loc_49><loc_39></location>where t is the age of a given object, zF its redshift of formation, E(z) is defined as H(z) / H 0 and A = 977.8 converts the result in units of km s -1 Mpc -1 . The analytical expression of E(z) depends on the cosmological model assumed, in particular, in a flat Λ CDM model it can be expressed as a function of redshift and matter density parameter Ω M , so that H 0 becomes:</text>
<formula><location><page_13><loc_6><loc_27><loc_49><loc_30></location>H0 = A t Z zF 0 1 1 + z [ Ω M(1 + z) 3 + (1 -Ω M)] -1 / 2 dz . (9)</formula>
<text><location><page_13><loc_6><loc_20><loc_49><loc_26></location>Considering the limit of zF = ∞ for which t corresponds to the age of the Universe tU , we can easily understand the sensitivity of the method. Fixing Ω M = 0 . 3 and varying tU from 12.9 Gyr to 14.1 Gyr the resulting value of H 0 spans from 73 km s -1 Mpc -1 to 67 km s -1 Mpc -1 .</text>
<text><location><page_13><loc_6><loc_14><loc_49><loc_20></location>When applying the method to the oldest objects, H 0 can be estimated via a Bayesian approach, in which the likelihood is built on the di ff erence between the measured age and the one predicted by the cosmological model (agem), accounting for the age error ( σ age):</text>
<formula><location><page_13><loc_6><loc_9><loc_49><loc_13></location>L (age | p ) = -0 . 5 X i [agei -agem( p )] 2 σ age , i , (10)</formula>
<figure>
<location><page_13><loc_54><loc_73><loc_92><loc_93></location>
<caption>Fig. 8: Combination of all the 66 GCs ages, drawn as normal distributions peaked on the best-fit value and 1σ equal to the associated error. In black, the three Gaussian components identified with a GMM are shown. In the inner panel, AIC and BIC curves are shown.</caption>
</figure>
<text><location><page_13><loc_51><loc_53><loc_94><loc_62></location>where p = (H0 , zF , Ω M) in a flat Λ CDM cosmology. The posterior distribution then, can be sampled with a Monte Carlo Markov Chain approach like the one implemented in the a ffi neinvariant ensemble sampler emcee (Foreman-Mackey et al. 2013). In the choice of priors, a flat, uninformative one can be adopted on H 0 and zF , while a Gaussian prior is preferable for Ω M in order to break its intrinsic degeneracy with H 0.</text>
<text><location><page_13><loc_51><loc_22><loc_94><loc_53></location>As a final note on the method, it is interesting to highlight that nowadays, with facilities like JWST, this approach is no more limited to the study of local objects but can be extended to higher redshift thanks to the first detections of GCs around lensed galaxies. A great case-study is the one of the Sparkler, a galaxy discovered in Webb's First Deep Field (Mowla et al. 2022) showing a population of compact objects associated with it. Among the 18 compact objects identified (Mowla et al. 2022; Millon et al. 2024), Sparkler shows 5 GC candidates that, if confirmed with spectroscopy, would be the first detection of GCs at z = 1.38. The spectroscopic study of these objects, analogous to the one performed in this work, would allow us to measure their age, potentially with a higher precision than the one obtained here for ∼ 13 Gyr old GCs. In younger stellar populations, indeed, the shape of the spectrum and the width of the absorption lines changes much quicker than it does at very old ages, thus reducing the impact of the parameters' degeneracies on the age uncertainty. In the context of our cosmological analysis, extending the method at higher redshift would just require to replace the lower limit of the integral in Eq. 9 with the redshift of the lensed GC. In the case of the Sparkler at redshift 1.38, for example, the age of the Universe ranges from 4.3 Gyr to 4.7 Gyr adopting the reference values for H 0 given above, 73 km s -1 Mpc -1 and 67 km s -1 Mpc -1 respectively.</text>
<section_header_level_1><location><page_13><loc_51><loc_18><loc_59><loc_19></location>5.2. Results</section_header_level_1>
<text><location><page_13><loc_51><loc_10><loc_94><loc_17></location>As anticipated in Sect. 4.3, we adopted as benchmark Config. 15 , where the cosmological prior is removed. To identify the tail of the oldest objects, we adopt a Gaussian Mixture Model (GMM) on the whole sample, combining normal distributions peaked on the best-fit ages, with 1σ equal to relative uncertainties. We then let the fit decide the optimal number of subsamples in which to</text>
<text><location><page_14><loc_6><loc_84><loc_49><loc_93></location>split our data. Both considering the Bayesian Information Criterion (BIC) or the Akaike Information Criterion (AIC), we find that the optimal number of components is three: a first peak identifying the youngest, blue-HB, spurious solutions; a second one for the intermediate ages; a third one comprising all the 24 oldest GCs, peaking at 13 . 4 ± 1 . 1 Gyr. The latter represents the oldest tail of the GCs' age distribution.</text>
<text><location><page_14><loc_6><loc_70><loc_49><loc_84></location>We applied the method described in Sect. 5.1 for each of these GCs separately, and then on their average, adopting the following priors: uniform on H0 ∈ [0 , 150] km s -1 Mpc -1 and on zF ∈ [11,30], Gaussian on Ω M = 0 . 30 ± 0 . 02. The lower limit on zF is based on the highest redshift at which galaxies have spectroscopic confirmations (Curtis-Lake et al. 2023), the higher limit instead relies on the values found in theoretical models for the redshift of formation of the very first stars (Galli & Palla 2013). As regards Ω M , the value chosen here comes from the combination of di ff erent low-redshift results (Jimenez et al. 2019), thus independent of the CMB.</text>
<text><location><page_14><loc_6><loc_60><loc_49><loc_69></location>Our 24 old GCs span the age range 13.2-13.6 Gyr, with errors around 1 Gyr, thus the resulting H 0 range we find covers the interval 69.5-71.7 km s -1 Mpc -1 , with typical uncertainties of ∼ 5 km s -1 Mpc -1 . To provide a single H 0 measurement, we also ran the MCMC using mean and standard deviation of the old peak found in the GMM fit: 13 . 4 ± 1 . 1 Gyr. This results in a final value for H 0 = 70 . 4 + 6 . 7 -5 7 km s -1 Mpc -1 (stat).</text>
<text><location><page_14><loc_6><loc_55><loc_49><loc_60></location>. To account for the systematic contribution to the error budget computed in Sect. 4.4 we summed it in quadrature to the standard deviation of the distribution and ran again the MCMC. The final result, comprising both statistics and systematic e ff ects is:</text>
<formula><location><page_14><loc_6><loc_52><loc_35><loc_54></location>H 0 = 70 . 5 + 7 . 7 -6 . 3 km s -1 Mpc -1 (stat + syst) .</formula>
<text><location><page_14><loc_6><loc_41><loc_49><loc_51></location>In Fig. 9 the 24 GCs' ages and the corresponding H 0 estimates found after the MCMC run are represented as Gaussian distributions, in the respective domains and combined as ellipses in the H 0-age plane. The corresponding Gaussian curves and ellipses relative to the combined age and H 0 are shown as black solid lines. For comparison, the values from Riess et al. (2022) and Planck Collaboration et al. (2020) are represented respectively with dashed and dotted lines.</text>
<text><location><page_14><loc_6><loc_34><loc_49><loc_40></location>Of course, the results that we obtain here are not able to address the tension but represent a pilot exploration of the use of GCs' ages for cosmological purposes, especially in view of future missions that could potentially discover such objects at higher redshifts.</text>
<section_header_level_1><location><page_14><loc_6><loc_30><loc_19><loc_31></location>6. Conclusions</section_header_level_1>
<text><location><page_14><loc_6><loc_11><loc_49><loc_29></location>In this work, we analysed the integrated spectra of a sample of 77 Milky Way GCs from the WAGGS project (Usher et al. 2017) and measured their physical properties via FSF with the code BAGPIPES (Carnall et al. 2018). In doing this, we aimed to study how well FSF can recover the GCs' ages and physical parameters, and assess, in particular, how the age estimates are a ff ected by the presence or absence of a cosmological prior. This required a modification on the code, already tested and validated in Jiao et al. (2023) and Tomasetti et al. (2023), thanks to which a flat non-cosmological prior can be set at 15 Gyr. At the same time, this allowed us to obtain a cosmology-independent lower limit to the age of the Universe, that we used to derive a new constraint on H 0, performing a pilot study for future potential applications at higher redshift.</text>
<text><location><page_14><loc_9><loc_10><loc_35><loc_11></location>Our results are summarised as follows:</text>
<text><location><page_14><loc_6><loc_7><loc_24><loc_8></location>Article number, page 14 of 16</text>
<figure>
<location><page_14><loc_52><loc_71><loc_93><loc_92></location>
<caption>Fig. 9: H 0 versus age for the sample of 24 old GCs. The ages and corresponding H 0 estimates are shown as Gaussians peaked on the best-fit values and 1σ equal to the relative uncertainties. The 1σ limits are also highlighted in a darker blue. The solid black curves correspond to the average GC's age and relative H 0 estimate. For comparison, the values from Riess et al. (2022) and Planck Collaboration et al. (2020) are represented respectively with dashed and dotted lines in the H 0 domain, and in the age-domain we report the corresponding ages of the Universe as computed in a flat Λ CDMwith Ω M = 0 . 3 and ΩΛ = 0 . 7.</caption>
</figure>
<unordered_list>
<list_item><location><page_14><loc_51><loc_47><loc_94><loc_54></location>1. Measuring age-related spectral features detectable in the spectra, like Dn 4000 and H β , allowed us to build indexage diagrams for these features in di ff erent metallicity bins, showing a distribution that well aligns with the trends from theoretical spectral models.</list_item>
<list_item><location><page_14><loc_51><loc_30><loc_94><loc_47></location>2. Combining age-related and metallicity-related spectral features we built two di ff erent types of index-index diagrams, H β -Mgb and H γ F-[MgFe]', able to disentangle age and metallicity contributions to the spectral features. Based on how the GCs populated these diagrams, we could have a first estimate of their properties, showing an overall good agreement with the corresponding literature values. Ages, however, appeared underestimated for a fraction of the sample characterised by low metallicity ([Fe / H] < -0.33) and blue HB (HBR > 0), that, showing a more prominent H β , populated the area of ages younger than 8 Gyr. This anticipates how the presence of an unmodelled HB component in the spectra can bias the results toward younger solutions.</list_item>
<list_item><location><page_14><loc_51><loc_10><loc_94><loc_30></location>3. Performing FSF with BAGPIPES we could measure simultaneously the age, metallicity, and mass of the GCs. We tested multiple fit configurations, with di ff erent choices of priors, model components, SFHs, and wavelength ranges. Metallicity and mass proved to be very stable against the changes in fit configurations, while age was mostly sensible to the prior limit. In particular, we tested two configurations, one with a cosmological prior set at 13.5 Gyr and another at 15 Gyr ( Config. 15 ), thus independent of cosmology. The percentage of GCs for which ages result compatible with the literature values within ± 1.5 Gyr increases by 20% removing the cosmological prior, demonstrating the relevance of this limit in stellar population studies. In Config. 15 the agreement with literature values is maximum for the subgroup of GCs with [Z / H] > -0.4, the least a ff ected by the presence of blue HBs, reaching 70%. Metallicity and mass al-</list_item>
</unordered_list>
<text><location><page_15><loc_9><loc_87><loc_49><loc_93></location>ays result well compatible with reference values independently of HBR, [Z / H] or fit setting, with average discrepancies on the whole sample of ⟨ ∆ [Z / H] ⟩ = -0 . 02 ± 0 . 24 dex and ⟨ ∆ log(M ⋆/ M ⊙ ) ⟩ = 0 . 04 ± 0 . 28, compatible with the typical uncertainties associated with these quantities.</text>
<unordered_list>
<list_item><location><page_15><loc_7><loc_74><loc_49><loc_86></location>4. Performing a GMM fit on the whole age distribution as derived in Config.15 , we identified a tail of 24 old GCs with ⟨ age ⟩ = 13 . 4 ± 1 . 1 Gyr. In a cosmological framework, this value can be used as a lower limit to the age of the Universe to constrain the Hubble constant. In particular, by fitting the H 0 -tU relation in a flat Λ CDM cosmology and assuming Ω m = 0 . 30 ± 0 . 02, from low-z measurements, we obtained H0 = 70 . 4 + 6 . 7 -5 . 7 km s -1 Mpc -1 (stat). Taking into account a systematic contribution to the age error of 0.71 Gyr, based on the age fluctuations in 8 di ff erent fit configurations, we obtained:</list_item>
</unordered_list>
<formula><location><page_15><loc_9><loc_71><loc_38><loc_73></location>H 0 = 70 . 5 + 7 . 7 -6 . 3 km s -1 Mpc -1 (stat + syst) .</formula>
<text><location><page_15><loc_6><loc_55><loc_49><loc_70></location>While we acknowledge that the method adopted in this paper is not intended to compete with other age estimation techniques (e.g., isochrone fitting) for local and resolved objects, it does offer a viable alternative. This study serves as an initial pilot investigation into the feasibility of using only spectroscopic information to determine GC ages, an approach that may be particularly useful for investigating the properties of lensed GCs at higher redshifts, where isochrone fitting is not feasible. Future developments will include expanding the tests to incorporate models with an extended HB. Nonetheless, even without these enhancements, this work provides valuable diagnostics for identifying the most robust and reliable fits.</text>
<text><location><page_15><loc_6><loc_40><loc_49><loc_54></location>Acknowledgements. We thank Christopher Usher for kindly providing us with WAGGS spectra, Adam Carnall for his help in using Bagpipes , and Frédéric Courbin and Licia Verde for the insightful discussion on the potential of lensed GCs' ages in cosmology. ET acknowledges the support from COST Action CA21136 - 'Addressing observational tensions in cosmology with systematics and fundamental physics (CosmoVerse)', supported by COST (European Cooperation in Science and Technology). Funding for the work of RJ was partially provided by project PID2022-141125NB-I00, and the 'Center of Excellence Maria de Maeztu 2020-2023' award to the ICCUB (CEX2019- 000918M) funded by MCIN / AEI / 10.13039 / 501100011033. MM acknowledges support from MIUR, PRIN 2022 (grant 2022NY2ZRS 001). MM and AC acknowledge support from the grant ASI n. 2024-10-HH.0 'Attività scientifiche per la missione Euclid - fase E'.</text>
<section_header_level_1><location><page_15><loc_6><loc_36><loc_16><loc_37></location>References</section_header_level_1>
<text><location><page_15><loc_6><loc_33><loc_49><loc_35></location>Abdalla, E., Abellán, G. F., Aboubrahim, A., et al. 2022, Journal of High Energy Astrophysics, 34, 49</text>
<unordered_list>
<list_item><location><page_15><loc_6><loc_31><loc_46><loc_33></location>Annibali, F., Morandi, E., Watkins, L. L., et al. 2018, MNRAS, 476, 1942 Bastian, N. & Lardo, C. 2018, ARA&A, 56, 83</list_item>
<list_item><location><page_15><loc_6><loc_29><loc_49><loc_31></location>Baumgardt, H., Hénault-Brunet, V., Dickson, N., & Sollima, A. 2023, MNRAS, 521, 3991</list_item>
<list_item><location><page_15><loc_6><loc_28><loc_42><loc_29></location>Bennett, C. L., Halpern, M., Hinshaw, G., et al. 2003, ApJS, 148, 1</list_item>
<list_item><location><page_15><loc_6><loc_27><loc_41><loc_27></location>Borghi, N., Moresco, M., Cimatti, A., et al. 2022a, ApJ, 927, 164</list_item>
<list_item><location><page_15><loc_6><loc_26><loc_41><loc_26></location>Brown, T. M., Casertano, S., Strader, J., et al. 2018, ApJ, 856, L6</list_item>
<list_item><location><page_15><loc_6><loc_25><loc_34><loc_25></location>Bruzual, G. & Charlot, S. 2003, MNRAS, 344, 1000</list_item>
<list_item><location><page_15><loc_6><loc_23><loc_34><loc_24></location>Buchner, J. 2016, Statistics and Computing, 26, 383</list_item>
<list_item><location><page_15><loc_6><loc_22><loc_48><loc_23></location>Burstein, D., Faber, S. M., Gaskell, C. M., & Krumm, N. 1984, ApJ, 287, 586</list_item>
<list_item><location><page_15><loc_6><loc_21><loc_36><loc_22></location>Cabrera-Ziri, I. & Conroy, C. 2022, MNRAS, 511, 341</list_item>
<list_item><location><page_15><loc_6><loc_20><loc_41><loc_21></location>Calzetti, D., Armus, L., Bohlin, R. C., et al. 2000, ApJ, 533, 682</list_item>
<list_item><location><page_15><loc_6><loc_19><loc_42><loc_20></location>Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345, 245</list_item>
<list_item><location><page_15><loc_6><loc_18><loc_46><loc_19></location>Carnall, A. C., McLure, R. J., Dunlop, J. S., et al. 2019, MNRAS, 490, 417</list_item>
<list_item><location><page_15><loc_6><loc_16><loc_49><loc_18></location>Carnall, A. C., McLure, R. J., Dunlop, J. S., & Davé, R. 2018, MNRAS, 480, 4379</list_item>
<list_item><location><page_15><loc_6><loc_15><loc_44><loc_16></location>Carnall, A. C., McLure, R. J., Dunlop, J. S., et al. 2022, ApJ, 929, 131</list_item>
<list_item><location><page_15><loc_6><loc_13><loc_49><loc_15></location>Cassisi, S. & Salaris, M. 2013, Old Stellar Populations: How to Study the Fossil Record of Galaxy Formation</list_item>
<list_item><location><page_15><loc_6><loc_11><loc_49><loc_13></location>Cezario, E., Coelho, P. R. T., Alves-Brito, A., Forbes, D. A., & Brodie, J. P. 2013, A&A, 549, A60</list_item>
</unordered_list>
<text><location><page_15><loc_6><loc_10><loc_35><loc_11></location>Chevallard, J. & Charlot, S. 2016, MNRAS, 462, 1415</text>
<text><location><page_15><loc_51><loc_90><loc_94><loc_93></location>Chilingarian, I., Prugniel, P., Sil'chenko, O., & Koleva, M. 2007, in IAU Symposium, Vol. 241, Stellar Populations as Building Blocks of Galaxies, ed. A. Vazdekis & R. Peletier, 175-176</text>
<unordered_list>
<list_item><location><page_15><loc_51><loc_89><loc_94><loc_90></location>Cid Fernandes, R., Mateus, A., Sodré, L., Stasi'nska, G., & Gomes, J. M. 2005,</list_item>
</unordered_list>
<text><location><page_15><loc_51><loc_85><loc_89><loc_89></location>MNRAS, 358, 363 Cimatti, A. & Moresco, M. 2023, ApJ, 953, 149 Colucci, J. E., Bernstein, R. A., & McWilliam, A. 2017, ApJ, 834, 105 Conroy, C. & van Dokkum, P. 2012, ApJ, 747, 69</text>
<unordered_list>
<list_item><location><page_15><loc_51><loc_83><loc_94><loc_84></location>Curtis-Lake, E., Carniani, S., Cameron, A., et al. 2023, Nature Astronomy, 7, 622</list_item>
</unordered_list>
<text><location><page_15><loc_51><loc_79><loc_94><loc_82></location>Eisenstein, D. J., Zehavi, I., Hogg, D. W., et al. 2005, ApJ, 633, 560 Faber, S. M., Friel, E. D., Burstein, D., & Gaskell, C. M. 1985, ApJS, 57, 711 Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125,</text>
<text><location><page_15><loc_53><loc_78><loc_55><loc_79></location>306</text>
<unordered_list>
<list_item><location><page_15><loc_51><loc_77><loc_75><loc_78></location>Galli, D. & Palla, F. 2013, ARA&A, 51, 163</list_item>
<list_item><location><page_15><loc_51><loc_74><loc_94><loc_77></location>Gonçalves, G., Coelho, P., Schiavon, R., & Usher, C. 2020, MNRAS, 499, 2327 Goncalves, G., Coelho, P., Schiavon, R., & Usher, C. 2023, VizieR Online Data Catalog, J / MNRAS / 499 / 2327</list_item>
</unordered_list>
<text><location><page_15><loc_51><loc_73><loc_88><loc_74></location>Gratton, R., Bragaglia, A., Carretta, E., et al. 2019, A&A Rev., 27, 8</text>
<unordered_list>
<list_item><location><page_15><loc_51><loc_72><loc_69><loc_73></location>Harris, W. E. 1996, AJ, 112, 1487</list_item>
<list_item><location><page_15><loc_51><loc_71><loc_79><loc_72></location>Harris, W. E. 2010, arXiv e-prints, arXiv:1012.3224</list_item>
<list_item><location><page_15><loc_51><loc_70><loc_92><loc_71></location>Hénault-Brunet, V., Gieles, M., Sollima, A., et al. 2019, MNRAS, 483, 1400</list_item>
</unordered_list>
<text><location><page_15><loc_51><loc_69><loc_89><loc_70></location>Jiao, K., Borghi, N., Moresco, M., & Zhang, T.-J. 2023, ApJS, 265, 48</text>
<unordered_list>
<list_item><location><page_15><loc_51><loc_67><loc_94><loc_69></location>Jimenez, R., Cimatti, A., Verde, L., Moresco, M., & Wandelt, B. 2019, J. Cosmology Astropart. Phys., 2019, 043</list_item>
<list_item><location><page_15><loc_51><loc_66><loc_74><loc_67></location>Jimenez, R. & Loeb, A. 2002, ApJ, 573, 37</list_item>
<list_item><location><page_15><loc_51><loc_64><loc_94><loc_66></location>Jimenez, R., MacDonald, J., Dunlop, J. S., Padoan, P., & Peacock, J. A. 2004, MNRAS, 349, 240</list_item>
</unordered_list>
<text><location><page_15><loc_51><loc_63><loc_90><loc_64></location>Koleva, M., Prugniel, P., Bouchard, A., & Wu, Y. 2009, A&A, 501, 1269</text>
<unordered_list>
<list_item><location><page_15><loc_51><loc_61><loc_94><loc_63></location>Koleva, M., Prugniel, P., Ocvirk, P., Le Borgne, D., & Soubiran, C. 2008, MNRAS, 385, 1998</list_item>
</unordered_list>
<text><location><page_15><loc_51><loc_54><loc_88><loc_60></location>Kriek, M., Price, S. H., Conroy, C., et al. 2019, ApJ, 880, L31 Kroupa, P. 2001, MNRAS, 322, 231 Lee, H.-c., Yoon, S.-J., & Lee, Y.-W. 2000, AJ, 120, 998 Lee, Y.-W. 1989, PhD thesis, Yale University, Connecticut Lee, Y.-W., Demarque, P., & Zinn, R. 1994, ApJ, 423, 248 Lonoce, I., Maraston, C., Thomas, D., et al. 2020, MNRAS, 492, 326</text>
<unordered_list>
<list_item><location><page_15><loc_51><loc_53><loc_82><loc_54></location>Maraston, C. & Strömbäck, G. 2011, MNRAS, 418, 2785</list_item>
<list_item><location><page_15><loc_51><loc_52><loc_90><loc_53></location>Mendel, J. T., Proctor, R. N., & Forbes, D. A. 2007, MNRAS, 379, 1618</list_item>
<list_item><location><page_15><loc_51><loc_50><loc_94><loc_52></location>Millon, M., Michalewicz, K., Dux, F., Courbin, F., & Marshall, P. J. 2024, AJ, 168, 55</list_item>
<list_item><location><page_15><loc_51><loc_49><loc_79><loc_50></location>Milone, A. P. & Marino, A. F. 2022, Universe, 8, 359</list_item>
<list_item><location><page_15><loc_51><loc_48><loc_72><loc_49></location>Moresco, M. 2015, MNRAS, 450, L16</list_item>
<list_item><location><page_15><loc_51><loc_46><loc_94><loc_48></location>Moresco, M., Amati, L., Amendola, L., et al. 2022, Living Reviews in Relativity, 25, 6</list_item>
<list_item><location><page_15><loc_51><loc_44><loc_94><loc_46></location>Moresco, M., Cimatti, A., Jimenez, R., et al. 2012, J. Cosmology Astropart. Phys., 2012, 006</list_item>
<list_item><location><page_15><loc_51><loc_42><loc_94><loc_44></location>Moresco, M., Pozzetti, L., Cimatti, A., et al. 2016, J. Cosmology Astropart. Phys., 2016, 014</list_item>
<list_item><location><page_15><loc_51><loc_41><loc_85><loc_42></location>Mowla, L., Iyer, K. G., Desprez, G., et al. 2022, ApJ, 937, L35</list_item>
<list_item><location><page_15><loc_51><loc_40><loc_91><loc_41></location>Ocvirk, P., Pichon, C., Lançon, A., & Thiébaut, E. 2006, MNRAS, 365, 74</list_item>
</unordered_list>
<text><location><page_15><loc_51><loc_39><loc_89><loc_40></location>Oliveira, R. A. P., Souza, S. O., Kerber, L. O., et al. 2020, ApJ, 891, 37</text>
<text><location><page_15><loc_51><loc_38><loc_87><loc_39></location>O'Malley, E. M., Gilligan, C., & Chaboyer, B. 2017, ApJ, 838, 162</text>
<unordered_list>
<list_item><location><page_15><loc_51><loc_37><loc_87><loc_38></location>Onodera, M., Carollo, C. M., Renzini, A., et al. 2015, ApJ, 808, 161</list_item>
<list_item><location><page_15><loc_51><loc_36><loc_83><loc_37></location>Pagel, B. E. J. & Tautvaisiene, G. 1995, MNRAS, 276, 505</list_item>
<list_item><location><page_15><loc_51><loc_35><loc_86><loc_36></location>Pancino, E., Romano, D., Tang, B., et al. 2017, A&A, 601, A112</list_item>
<list_item><location><page_15><loc_51><loc_34><loc_81><loc_34></location>Percival, S. M. & Salaris, M. 2011, MNRAS, 412, 2445</list_item>
<list_item><location><page_15><loc_51><loc_32><loc_94><loc_33></location>Percival, W. J., Baugh, C. M., Bland-Hawthorn, J., et al. 2001, MNRAS, 327, 1297</list_item>
<list_item><location><page_15><loc_51><loc_30><loc_88><loc_31></location>Perlmutter, S., Aldering, G., Goldhaber, G., et al. 1999, ApJ, 517, 565</list_item>
</unordered_list>
<text><location><page_15><loc_51><loc_29><loc_91><loc_30></location>Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2020, A&A, 641, A6</text>
<unordered_list>
<list_item><location><page_15><loc_51><loc_28><loc_82><loc_29></location>Pritzl, B. J., Venn, K. A., & Irwin, M. 2005, AJ, 130, 2140</list_item>
<list_item><location><page_15><loc_51><loc_27><loc_91><loc_28></location>Proctor, R. N., Forbes, D. A., & Beasley, M. A. 2004, MNRAS, 355, 1327</list_item>
<list_item><location><page_15><loc_51><loc_26><loc_80><loc_27></location>Prugniel, P. & Soubiran, C. 2004, arXiv e-prints, astro</list_item>
<list_item><location><page_15><loc_51><loc_25><loc_88><loc_26></location>Riess, A. G., Filippenko, A. V., Challis, P., et al. 1998, AJ, 116, 1009</list_item>
<list_item><location><page_15><loc_51><loc_24><loc_85><loc_25></location>Riess, A. G., Yuan, W., Macri, L. M., et al. 2022, ApJ, 934, L7</list_item>
<list_item><location><page_15><loc_51><loc_23><loc_94><loc_24></location>Roediger, J. C., Courteau, S., Graves, G., & Schiavon, R. P. 2014, ApJS, 210, 10</list_item>
</unordered_list>
<text><location><page_15><loc_51><loc_22><loc_90><loc_23></location>Salaris, M. & Cassisi, S. 2005, Evolution of Stars and Stellar Populations</text>
<unordered_list>
<list_item><location><page_15><loc_51><loc_21><loc_81><loc_22></location>Salim, S., Boquien, M., & Lee, J. C. 2018, ApJ, 859, 11</list_item>
<list_item><location><page_15><loc_51><loc_19><loc_94><loc_21></location>Schiavon, R. P., Rose, J. A., Courteau, S., & MacArthur, L. A. 2004, ApJ, 608, L33</list_item>
<list_item><location><page_15><loc_51><loc_17><loc_94><loc_19></location>Schiavon, R. P., Rose, J. A., Courteau, S., & MacArthur, L. A. 2005, ApJS, 160, 163</list_item>
<list_item><location><page_15><loc_51><loc_16><loc_88><loc_17></location>Scott, N., Brough, S., Croom, S. M., et al. 2017, MNRAS, 472, 2833</list_item>
<list_item><location><page_15><loc_51><loc_15><loc_74><loc_16></location>Soderblom, D. R. 2010, ARA&A, 48, 581</list_item>
<list_item><location><page_15><loc_51><loc_14><loc_76><loc_15></location>Strader, J. & Brodie, J. P. 2004, AJ, 128, 1671</list_item>
<list_item><location><page_15><loc_51><loc_13><loc_88><loc_14></location>Thomas, D., Maraston, C., & Johansson, J. 2011, MNRAS, 412, 2183</list_item>
<list_item><location><page_15><loc_51><loc_12><loc_87><loc_13></location>Tomasetti, E., Moresco, M., Borghi, N., et al. 2023, A&A, 679, A96</list_item>
<list_item><location><page_15><loc_51><loc_11><loc_89><loc_12></location>Usher, C., Beckwith, T., Bellstedt, S., et al. 2019a, MNRAS, 482, 1275</list_item>
<list_item><location><page_15><loc_51><loc_10><loc_89><loc_11></location>Usher, C., Brodie, J. P., Forbes, D. A., et al. 2019b, MNRAS, 490, 491</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_16><loc_6><loc_92><loc_44><loc_93></location>Usher, C., Pastorello, N., Bellstedt, S., et al. 2017, MNRAS, 468, 3828</list_item>
<list_item><location><page_16><loc_6><loc_90><loc_49><loc_92></location>Vagnozzi, S., Pacucci, F., & Loeb, A. 2022, Journal of High Energy Astrophysics, 36, 27</list_item>
<list_item><location><page_16><loc_6><loc_88><loc_49><loc_90></location>Valcin, D., Bernal, J. L., Jimenez, R., Verde, L., & Wandelt, B. D. 2020, J. Cosmology Astropart. Phys., 2020, 002</list_item>
<list_item><location><page_16><loc_6><loc_86><loc_49><loc_88></location>Valcin, D., Jimenez, R., Verde, L., Bernal, J. L., & Wandelt, B. D. 2021, J. Cosmology Astropart. Phys., 2021, 017</list_item>
<list_item><location><page_16><loc_6><loc_85><loc_27><loc_85></location>van den Bergh, S. 1969, ApJS, 19, 145</list_item>
<list_item><location><page_16><loc_6><loc_83><loc_27><loc_84></location>van den Bergh, S. 1981, A&AS, 46, 79</list_item>
<list_item><location><page_16><loc_6><loc_81><loc_49><loc_83></location>VandenBerg, D. A., Bolte, M., & Stetson, P. B. 1996, Annual Review of Astronomy and Astrophysics, 34, 461</list_item>
<list_item><location><page_16><loc_6><loc_80><loc_43><loc_81></location>Vazdekis, A., Coelho, P., Cassisi, S., et al. 2015, MNRAS, 449, 1177</list_item>
<list_item><location><page_16><loc_6><loc_78><loc_49><loc_80></location>Vazdekis, A., Sánchez-Blázquez, P., Falcón-Barroso, J., et al. 2010, MNRAS, 404, 1639</list_item>
<list_item><location><page_16><loc_6><loc_77><loc_49><loc_78></location>Walcher, C. J., Coelho, P., Gallazzi, A., & Charlot, S. 2009, MNRAS, 398, L44</list_item>
</unordered_list>
</document>
[
{
"title": "ABSTRACT",
"content": "Aims. In this work we explore the reliability and robustness in measuring the ages and main physical properties of a sample of old Milky Way globular clusters (GCs) from their integrated light. This approach sets the stage for using GCs as cosmic clocks at high redshift. Additionally, it enables us to establish an independent lower limit to the age of the Universe, and an upper limit to H 0. Methods. We analyse a sample of 77 GCs from the WAGGS project, by first measuring their spectral features (Lick indices and spectroscopic breaks) with PyLick and then performing full-spectral-fitting with BAGPIPES . The analysis of Lick indices o ff ers an initial estimate of the population's age and metallicity, generally aligning well with values reported in the literature. However, it also highlights a subset of old clusters for which we estimate younger ages. This discrepancy is primarily attributed to the presence of horizontal branches with complex morphologies, which are not accounted for in the stellar population models. With full-spectralfitting we measure the GCs' ages, metallicities, and masses, testing how removing the cosmological prior on the ages a ff ects the final results. Results. Compared to isochrone fitting estimates, ages are best recovered when the cosmological prior is removed, with a 20% increase in the number of GCs showing ages compatible with literature values within ± 1.5 Gyr. The derived metallicity and mass are consistently in good agreement with the reference values, regardless of HB morphology, [Z / H], or the fit settings. The average discrepancies across the entire sample are ⟨ ∆ [Z / H] ⟩ = -0 . 02 ± 0 . 24 dex for metallicity and ⟨ ∆ log(M ⋆/ M ⊙ ) ⟩ = 0 . 04 ± 0 . 28 dex for mass. Metal-rich GCs ([Z / H] ≥ -0.4) showing a red horizontal branch (HBR > 0) are the sub-group in which ages are best recovered. In this group, 70% of the results align with literature values within ± 1.5 Gyr. Identifying the tail of the oldest cosmology-independent ages with a Gaussian Mixture Model, we obtain a sample of 24 objects with ⟨ age ⟩ = 13 . 4 ± 1 . 1 Gyr. Conclusions. Being a natural lower limit to the age of the Universe, we use the age of the oldest GCs to constrain the Hubble constant, obtaining H0 = 70 . 5 + 7 . 7 -6 . 3 km s -1 Mpc -1 (stat + syst) when a flat Λ CDMwith Ω m = 0 . 30 ± 0 . 02 (based on low-z measurements) is assumed. Validating the analysis of GCs based on their integrated light lays the foundation to extend this type of study to high redshift, where GCs have begun to appear in lensed fields, thanks to JWST. Key words. globular clusters: general - Cosmology:observations - cosmological parameters",
"pages": [
1
]
},
{
"title": "Globular clusters as cosmic clocks: new cosmological hints from their integrated light",
"content": "Elena Tomasetti 1 , 2, ⋆ , Michele Moresco 1 , 2 , Carmela Lardo 1 , 2 , Andrea Cimatti 1 , 3 , and Raul Jimenez 4 , 5 October 29, 2024",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "During the last decades, the flat Λ CDM model has been a fundamental pillar in cosmology thanks to the support of independent cosmological probes, including the cosmic microwave background (CMB; e.g. Bennett et al. 2003; Planck Collaboration et al. 2020), type Ia supernovae (e.g. Riess et al. 1998; Perlmutter et al. 1999), and baryon acoustic oscillations (e.g. Percival et al. 2001; Eisenstein et al. 2005). Nonetheless, further investigation is necessary to unveil the nature of dark matter and dark energy and assess the precise values of cosmological parameters. In fact, due to the higher precision achieved in lateand early-Universe probes, some inconsistencies have emerged regarding the value of the Hubble constant ( H 0), where a tension of 4-5 σ has now been observed (Abdalla et al. 2022). In this context, the age of the Universe ( tU ) can play a crucial role, given its sensitivity to H 0. Indeed, in a flat Λ CDM cosmology with Ω M = 0.3 and ΩΛ= 0.7, tU can span a range from ∼ 14.1 Gyr if H 0 = 67 km s -1 Mpc -1 to ∼ 12.9 Gyr if H 0 = 73 km s -1 Mpc -1 . Thus, measuring the absolute ages of the most long-lived objects at z = 0 can be critical since they naturally place a lower limit on the current age of the Universe ( tU ) and, in turn, an upper limit on H 0. This provides independent constraints on the Hubble constant and valuable information for investigating the origin of the observed tension. Globular clusters (GC) are among the oldest objects in the Universe for which we can accurately determine the age (VandenBerg et al. 1996; Soderblom 2010; Brown et al. 2018; Oliveira et al. 2020). Composed of roughly one million stars that formed simultaneously with similar composition (though see reviews by Bastian & Lardo 2018; Gratton et al. 2019; Milone & Marino 2022, for discussions on the multiple population phenomenon in massive clusters), these clusters have remained gravitationally bound for up to a Hubble time. Each GC thus serves as an observable record of the age, metallicity, and kinematics from the time of its formation. Therefore, by measuring their ages, we can use them as \"clocks\" that began ticking in the early stages of the Universe's evolution (O'Malley et al.",
"pages": [
1
]
},
{
"title": "2017; Jimenez et al. 2019; Valcin et al. 2020, 2021; Cimatti & Moresco 2023).",
"content": "The most straightforward method to determine the age of a GC is by exploiting the fact that the position of the mainsequence turn-o ff (MSTO) in the plane of e ff ective temperature (Te ff ) versus luminosity (L) changes with age (or mass). Isochrones, or theoretical tracks of stars with the same chemical composition, are fitted to the MSTO region of colour-magnitude diagrams (CMD) to estimate the age. Even if this is a wellestablished and robust method, it is important to explore new and complementary approaches that can address the case when the CMDis not available. In need of a spatially resolved stellar population, indeed, isochrone fitting can be applied only to nearby systems, while moving further than the Magellanic Clouds becomes either very expensive in terms of exposure time or even impossible. Moreover, recent JWST observations of lensed fields highlighted the presence of GC candidates around lensed galaxies, like the Sparkler (Mowla et al. 2022) at z = 1.38 which, if confirmed with spectroscopy, would extend the study of GCs at high redshift. To do so, we need to explore methods relying on GCs' integrated light and validate them against the traditional methods. In this scenario, one of the best ways to leverage all the integrated light information is to perform full-spectral-fitting (FSF), a technique that enables measuring, alongside the age, all the physical properties of the GC such as metallicity, mass, and dust reddening. Previous works have derived physical parameters of GCs like age and metallicity using the integrated light provided by Schiavon et al. (2005) for 41 MW GCs (e.g., Koleva et al. 2008; Cezario et al. 2013; Cabrera-Ziri & Conroy 2022), testing different algorithms, like STECKMAP (Ocvirk et al. 2006), NBURSTS (Chilingarian et al. 2007), ULySS (Koleva et al. 2009) or ALF (Conroy & van Dokkum 2012), and di ff erent simple stellar population (SSP) models (Bruzual & Charlot 2003; Prugniel & Soubiran 2004; Vazdekis et al. 2010, 2015). Others have benefited from the larger spectral coverage and higher resolution of the WiFeS Atlas of Galactic Globular cluster Spectra project (WAGGS, Usher et al. 2017, 2019a), providing integrated spectra for 113 GCs in the MW and its satellite galaxies (Usher et al. 2019b; Gonçalves et al. 2020; Cabrera-Ziri & Conroy 2022). In Gonçalves et al. (2020), for instance, the authors adopted the non-parametric FSF code STARLIGHT (Cid Fernandes et al. 2005), relying on MILES SSP models (Vazdekis et al. 2015), with a focus on how the wavelength range influences the recovery of the stellar parameters compared to the CMD fitting. Cabrera-Ziri & Conroy (2022) extended the Milky Way GCs sample from Schiavon et al. (2005) by including younger objects from the Large and Small Magellanic Clouds (LMC and SMC) with WAGGS spectra, adopting the non-parametric FSF code ALF . While there is broad agreement that the ages of younger GCs can be reliably determined through FSF, these studies highlighted the challenges in dating the oldest GCs from their integrated spectra, often yielding results significantly younger than those from isochrone-fitting methods. In this paper we focus on the oldest tail of the WAGGS GCs, analysing 82 GCs in the Milky Way. We take advantage of the high-quality integrated spectra provided by WAGGS, along with the wealth of data available for these objects, the independent age estimates derived with di ff erent techniques, and the fact that GCs are among the simplest stellar systems in the Universe, the closest templates to an SSP we have. In this study, we adopt a parametric FSF method, enabling the reconstruction of GCs' integrated emission within a high-dimensional parameter space. For this purpose, we use the FSF code BAGPIPES (Carnall et al. 2018) utilizing the 2016 version of the Bruzual & Charlot (2003) SSP models. Previous studies in the literature have typically derived parameters such as age, metallicity, mass, and dust reddening while assuming a cosmological prior on age. In contrast, the novelty of this study lies in removing this prior to explore how the derived ages are a ff ected, as done by Tomasetti et al. (2023) and Jiao et al. (2023), in order to test the potential of the results in a cosmological framework. By testing this approach, we aim to assess its potential in a cosmological context. We also use these cosmology-independent results to place new constraints on H 0 setting the stage for future applications in studying the distant Universe. This paper is organised as follows: in Sect. 2 we describe the spectra we used, along with the adjustments needed and the ancillary data; in Sect. 3 the spectroscopic analysis of the sample is presented; in Sect. 4 the FSF method and its result are outlined; in Sect. 5 we report the final cosmological analysis; in Sect. 6 we draw our conclusions.",
"pages": [
2
]
},
{
"title": "2. Data",
"content": "The WAGGS project (Usher et al. 2017) is a library of integrated spectra of GCs in the Milky Way (MW) and the Local Group, obtained with the WiFeS integral field spectrograph on the Australian National University 2.3 m telescope. With 112 spectra of GCs in the Local Group, it is one of the largest GCs spectral libraries currently available, with a wide wavelength coverage (3270-9050 Å) and high spectral resolution (R ∼ 6800). The spectra we work with are normalised and consist of four di ff erent gratings, each with its own sampling: 3270-4350 Å (0.27 Å per pixel), 4170-5540 Å (0.37 Å per pixel), 5280-7020 Å (0.44 Å per pixel) and 6800-9050 Å (0.57 Å per pixel). To perform FSF across the entire spectrum, certain adjustments were required. First, we had to re-scale each spectrum to match its literature photometry, to retrieve the fluxes in physical units. We used the UBVRI integrated photometry from the 2010 edition of the Harris catalogue (Harris 1996, 2010). The correction factor C , derived via χ 2 minimisation, can be written as: where p is the photometry in the J -th filter, and f and e are the average flux and corresponding error estimated on the spectrum on a window of 10 Å. We then multiplied the spectrum in each grating by the corresponding factor C . Here, we must underline that UBVRI photometry is not available for all the objects in WAGGS, but only for the 82 GCs belonging to the MW. For the younger GCs in the LMC and SMC and in the Fornax dwarf spheroidal, only BV photometry is available, respectively from van den Bergh (1981) and van den Bergh (1969). Anyway, in this work, we want to focus on the oldest tail of the local GCs, so we limit our sample to the MW GCs. Before proceeding with the analysis we performed a visual inspection of the spectra, removing five GCs showing either visibly corrupted or very noisy regions (S / N < 10 in more than 40% of the spectrum), namely NGC6144, NGC6401, NGC6517, NGC6712 and NGC7492. The sample we analyse here is then constituted of 77 GCs. To combine the four gratings into a single spectrum, we interpolated all of them onto a common wavelength grid, matching the largest spectral sampling value (0.57 Å per pixel). In the overlap regions, the flux and associated error were estimated by averaging the spectra from the consecutive gratings. Throughout the paper, we compare our results to literature values of age, mass, and metallicity, and we also consider additional quantities to complement and expand our analysis, like dust reddening, radial velocities and distances of the GCs. We use as a reference the values listed in Usher et al. (2017) for ages and masses, in Harris (2010) for metallicities ([Fe / H]) and dust reddening (EB -V) and in Baumgardt et al. (2023) for radial velocities, distance from the Sun and associated errors. As for the uncertainties on metallicities, we consider the errors found in other spectroscopic investigations based on integrated spectra of Galactic GCs, which are approximately ± 0.15 dex (see Roediger et al. 2014; Colucci et al. 2017). Onages, the error budget based on MSTO fitting involves several key contributors. The most significant is distance uncertainty; an error of approximately 0.1-0.15 mag can result in about a 10% uncertainty in age. The error in the initial helium content, known within ∼ 2%, translates to about a 2% uncertainty in age. An error in the global metallicity of ∼ 9-10% and of ∼ 0.15 dex in iron content leads to approximately 4-5% error in age. An uncertainty of ∼ 0.15-0.2 dex in alpha elements translates to about 4% error in age (see a discussion in Cassisi & Salaris 2013). Combining these factors, the overall uncertainty in age can be around 1020% (e.g. O'Malley et al. 2017). For the sake of comparison, we consider a fixed error of ± 1.5 Gyr. On mass, a typical uncertainty of ∼ 10% is generally found on single measurements, corresponding to 0.05 dex in log(M ⋆/ M ⊙ ) (Hénault-Brunet et al. 2019). Nevertheless, a di ff erence of ∼ 0.2 dex on average can be observed among di ff erent catalogues (e.g., Usher et al. (2017) with respect to Baumgardt et al. (2023)). For this reason, we adopt a typical error on mass of 0.2 dex.",
"pages": [
2,
3
]
},
{
"title": "3. Spectroscopic analysis",
"content": "Before estimating the physical parameters of our GC sample with FSF, we want to derive measurements of the spectroscopic features in our sample and use those to have a preliminary assessment of the age and metal content of our GCs. At high redshift, the study of Lick indices (Burstein et al. 1984; Faber et al. 1985) or spectral breaks is often used to constrain stellar population properties. Here we want to see how GCs, the astrophysical objects that most resemble SSPs, fit inside this framework. To do so, we first measured all the absorption features detectable in the spectra using the public code PyLick (Borghi et al. 2022a). We want to compare these data with theoretical models estimated by Thomas et al. (2011) at di ff erent ages, metallicities and alpha-enhancements ([ α / Fe]). Since these models are built with MILES resolution ( ∼ 2.7 Å FWHM), we downgraded the spectral resolution of our sample to match the models. Moreover, if an index lay in the overlap region of two spectral gratings, we ran PyLick on both spectra and then averaged the two values weighting them with their associated errors. In particular, we measured indices of the Balmer series, like H β and H γ , indices of the iron group like Fe5270 and Fe5335 or Mgb and broader features like the D4000, a spectral break at 4000 Å. These features are particularly relevant in the study of stellar populations in that they are known to independently correlate with age (like D4000 and H β ) and metallicity (like the iron group), so their analysis can give important insights into the physical properties of the GCs, as we discuss in the following sections.",
"pages": [
3
]
},
{
"title": "3.1. Index-age analysis",
"content": "After measuring the spectral features, we can compare their trend in age with the theoretical ones. A variety of stellar libraries are available in the literature, like Bruzual & Charlot (2003), Maraston & Strömbäck (2011) and Conroy & van Dokkum (2012), adopting di ff erent stellar evolutionary models, libraries of stellar spectra and procedures to compute the integrated spectra. In Moresco et al. (2012), however, it is shown how the assumption of di ff erent SPS models has a negligible impact on the slope of the index-age trend, in particular in the case of the D4000. To compare our measurements with theoretical trends, here we are considering the 2016 version of Bruzual & Charlot (2003) models (hereafter BC16), since these are the same SPS models implemented in BAGPIPES . In Fig. 1a the measured Dn4000 is shown as a function of the GC literature age for the 75 GCs for which an age estimate is provided in the literature, divided into six metallicity bins. For a qualitative comparison, we report the theoretical trends from BC16 with [Fe / H] varying from -0.33 to -2.25 and alphaenhancement ([ α/ Fe]) fixed to solar value. The trends are almost flat in this age interval, as expected in these ranges of ages and metallicities (see, e.g., Moresco et al. 2022) but show an evident gradient with metallicity, that is in good agreement with the theoretical distribution, with the Dn4000 increasing as metallicity raises. Analogous to the Dn4000, in Fig. 1b the trends in age are shown for the Balmer index H β , in comparison with the theoretical trends in the same metallicity bins. The distribution of this feature shows a good agreement with the models as well, this time decreasing with increasing metallicity. Testing these observables against the stellar models in the case of GCs, objects for which independent and robust measurements of age and metallicities are available, is of great importance in order to validate the models and their use in cosmological analyses. In the application of cosmic chronometers (Jimenez & Loeb 2002), for instance, especially when the D4000 is directly used to trace the age evolution in redshift (Moresco et al. 2012; Moresco 2015; Moresco et al. 2016) it is fundamental that the D4000 traces correctly the stellar population evolution in time (i.e. the D4000-age slope).",
"pages": [
3
]
},
{
"title": "3.2. Index-index diagrams",
"content": "While in the previous section we studied the sensitivity of single features to the age and metallicity of the population, it is also possible to combine them in the analysis, taking advantage of their di ff erent sensitivity to parameters like age, metallicity, and alpha-enhancement. In this case, we are considering a di ff erent set of theoretical models to compare them with, specifically developed to take into account also a variation of the chemical composition (Thomas et al. 2011, TMJ). Historically, this method has been applied in galaxies (e.g., Onodera et al. 2015; Scott et al. 2017; Lonoce et al. 2020; Borghi et al. 2022a) but also in GCs (e.g., Strader & Brodie 2004; Proctor et al. 2004; Mendel et al. 2007; Annibali et al. 2018) to derive constraints on the physical properties of these objects. Applying this method, we select and compare indices that are mostly sensible to age, metallicity, or α -enhancement variations so that we can disentangle their contributions to the feature's equivalent width. The most widely used are indices of the Balmer series, the iron-group ones, like Fe5270 and Fe5335, and Mgb. As for BC16, the models in TMJ have MILES resolution, but di ff erently from BC16, they produce a forecast only for the set of Lick indices and not (b) Article number, page 4 of 16 the entire spectrum. In particular, they model the indices values on a grid of ages, metallicities, and alpha-enhancements. Generally, an age-sensitive and a metallicity-sensitive index are plotted one against the other, and compared to a model grid with varying age and [Fe / H] but fixed [ α / Fe]. In Fig. 2 we present two examples of these diagrams, H β -Mgb and H γ F-[MgFe]', where the latter is defined as: Here the [ α / Fe] is fixed at 0.3, which is the closest to the typical value found in the literature for our sample of GCs, around 0.35 (e.g., Pritzl et al. 2005; Mendel et al. 2007). This method allows us to have a first estimate of the population's age and metallicity, which aligns well with the literature values, obtained with the independent traditional methods. In terms of age, the sample of GCs populates the area of the oldest objects in the diagrams, with a percentage of data points compatible with an age older than 12 Gyr of 66% and 88%, respectively in the H β -Mgb and H γ F-[MgFe]'. From the diagrams in Fig. 2 we can also see a small percentage of GCs in the area of typically younger objects, above the 8 Gyr grid-line. In particular, in the H β -Mgb this happens for 17 GCs, while in the H γ F-[MgFe]' for 11 GCs. This behaviour can be attributed to the possible presence of an extended horizontal branch (HB), which can make the spectrum appear much bluer than expected for an old population and exhibit prominent Balmer lines, resembling a younger object. This is a very well-known e ff ect, that has always made the study of GCs from integrated light challenging, mostly because the parameters determining the presence and the extent of the HB are not fully predictable with the current stellar evolution models. Various works have made progress in developing diagnostics to identify elongated HBs from integrated light, based on the Balmer lines (Lee et al. 2000; Schiavon et al. 2004) or on CaII and Mgb (Percival & Salaris 2011). Others have managed to include the HB contribution on top of the SSP models (Jimenez et al. 2004; Koleva et al. 2008; Cabrera-Ziri & Conroy 2022), modelling the emission from the HB hot stars as identified in the GCs' CMD. However, we still lack of a complete modelisation of the HB component, due to the many uncertainties around its origin. Modelling the HB component is beyond the scope of this study, instead, our primary goal is to assess how this unmodeled component may impact studies of integrated populations, using methods commonly employed in galaxy evolution analyses. The most common parameter used to quantify the HB extent is the morphology index HBR (Lee 1989; Lee et al. 1994), defined as: where B and R are the number of stars bluer and redder than the RR Lyrae instability strip, and V is the number of RR Lyrae stars. Although this parameter does not fully capture the distribution of stars along the HB, it still provides valuable information about the HB morphology, indicating whether it is predominantly red (HBR ∼ -1) or blue (HBR ∼ 1). In Fig. 3 we report the same H β -Mgb diagram as in Fig. 2a, but this time colour-coded by the HBR value listed in Harris (1996) (2010 edition), which is known for 69 / 82 objects (coloured points). According to this index, among the 14 GCs populating the area above the 8 Gyr grid line and for which the HBR is known, 13 show a blue HB, and for 11 of those HBR is even higher than 0.5, a clue of a very elongated blue HB. The fraction of objects with HBR above zero decreases as we move to the areas belonging to older ages: 72% (18 / 25) between 8 Gyr and 12 Gyr and 33% (10 / 30) over 12 Gyr. The same trend can be observed moving from lower to higher HBR [Fe / H], with a percentage of GCs showing blue HBs decreasing from 68% (23 / 34) at [Fe / H] < -1 . 35 to 49% (13 / 29) in the range -1.35 ≤ [Fe / H] ≤ -0.33 and dropping to zero at [Fe / H] > -0 . 33. As anticipated, stellar evolution models do not currently account for the presence of an extended blue HB, so this must be considered in the FSF analysis, where objects with extended HB morphology might be mistakenly identified as young stellar populations (Schiavon et al. 2004; Jimenez et al. 2004; Koleva et al. 2008; Cabrera-Ziri & Conroy 2022). From our initial qualitative analysis using indices, we expect this issue to be more prevalent in metal-poor objects with prominent H β , which tend to show the highest HBR.",
"pages": [
3,
4,
5,
6
]
},
{
"title": "4. Method and analysis",
"content": "In this section, we present the method adopted to estimate the ages and physical properties of the GCs sample, the code used, its settings, and the results obtained.",
"pages": [
6
]
},
{
"title": "4.1. Full-spectral-fitting with BAGPIPES",
"content": "We perform FSF using the public code BAGPIPES (Carnall et al. 2018), which allows us to fit spectra and / or photometry adopting a parametric Bayesian approach. A detailed description of all the code's features is presented in Carnall et al. (2019) and Carnall et al. (2022), while an overview of the settings that we adopt is already outlined in Tomasetti et al. (2023). Here we recap the main features of the code, highlighting the aspects that are relevant to this work. BAGPIPES is able to model synthetic spectra and photometry, based on a set of instructions, and then fit the so-modelled spectro-photometry to the observed one via a Bayesian approach, thus maximising the posterior probability using a nested sampling algorithm, Multinest (Buchner 2016). In this work, we focus on four main model components to construct the synthetic spectra. The first component is an SSP model, which is designed to reproduce the continuum emission and the absorption features of a population built up in a single episode of star formation. The SSP models implemented in BAGPIPES are the 2016 version of Bruzual & Charlot (2003) (BC16, see Chevallard & Charlot 2016). They produce di ff erent synthetic spectra based on the wavelength λ range, the age of the stellar population, and its overall metallicity [Z / H], assuming a Kroupa (2001) initial mass function (IMF). The second component is the star formation history (SFH). The code allows you to combine di ff erent SFHs, one for each SSP, but dealing with GCs we implement a single SFH, assuming a unique star formation episode. In particular, we adopt the delayed exponentially declining (DED) SFH, which is given by the equation where SFR is the star formation rate, τ provides the width of the SFH and T0 sets the age of the Universe at which the star formation begins. Using a single DED is recommendable when dealing with a stellar population whose time scale of formation is much shorter than its age, as we expect for GCs. The third component is dust absorption and emission. This is particularly important to model the redder part of the spectrum, which can be largely depressed due to the presence of dust in the system. In the context of MW GCs, this component is necessary to account for the MW dust on the line-of-sight, for which WAGGSspectra are not corrected. The model implemented here is the Salim et al. (2018), represented by a power-law, as in Calzetti et al. (2000), with an additional parameter, δ , representing a slope deviation. The last component is a non-physical term representing noise, which can be added to the error spectrum to account for any potential underestimation. This noise is introduced as white noise. After running the code, we obtain a best-fit spectrum and the posterior distributions for all the parameters involved, like age, mass formed, overall metallicity, and dust extinction. At the same time, BAGPIPES can provide estimates of derived quantities, like the SFR or the stellar mass formed, parameters that are not directly involved in the fit. In particular, the mass formed (Mformed) and the stellar mass formed (M ⋆ ) are di ff erent in that the first one comprises all the mass formed from t = 0 to the time t at which the GC is observed: including also stellar remnants (Mrem), while the second one includes only the mass of living stars: From now on, we refer to M ⋆ as the mass of the objects. Since we aim to use the resulting ages in a cosmological framework, it is important to avoid any constraint based on cosmology. For this reason, we employ a version of BAGPIPES , described in Jiao et al. (2023), that deviates from the original in handling the priors on the stellar population age, allowing them to vary up to a cosmology-independent value (e.g., 15 Gyr, 20 Gyr) at any redshift. This modification has already been tested and validated in VANDELS (Tomasetti et al. 2023) and LEGAC (Jiao et al. 2023). Originally the code assumes a cosmological prior on ages, for which the maximum age resulting from the fit should be smaller than the age of the Universe at the corresponding redshift, given a flat Λ CDM model with parameters Ω M = 0 . 3 , ΩΛ = 0 . 7 and H0 = 70 km s -1 Mpc -1 . Although this e ff ect is of relative interest in stellar population studies and is typically neglected, it can't be ignored in cosmological analyses because the derived ages would be constrained by the cosmological model assumed, leading to results that just recover the assumed prior. Below, we are going to test the e ff ect of di ff erent assumptions of age prior on the sample of GCs.",
"pages": [
6,
7
]
},
{
"title": "4.2. Full-spectral-fitting in WAGGS",
"content": "Before inputting the cluster spectra into BAGPIPES, some adjustments were necessary. First, we downgraded the spectral resolution to approximately 2.7 Å FWHM, consistent with the BC16 models used in the code. Next, we aligned the spectra to the correct frame using distances from Baumgardt et al. (2023) and corrected for radial velocity variations, which could cause minor blueshifts or redshifts in the spectra. To prevent underweighting the blue features in the fit - due to the non-uniform error spectrum, with S / N ranging from a few tens to a thousand - we set an upper limit for the S / Nat 100 and adjusted the error spectrum accordingly. We tested various S / N thresholds (e.g., 20, 50), finding that they had minimal impact on the results, except when the S / N ratio between the blue and red ends of the spectrum di ff ered by more than a factor of ten, which resulted in very low weight for the blue features in the fit. After these adjustments, we performed multiple tests to optimally use BAGPIPES on GCs spectra and find the best-fit configuration to reproduce their spectral features accurately. In particular, this process involved: adopting di ff erent SFHs (e.g., single burst, delayed exponentially declined) with di ff erent priors on the parameters; fitting di ff erent wavelength ranges, either moving the lower limit to longer wavelengths to reduce the contamination by HB stars or pushing the upper limit to redder features to better constrain dust reddening; testing di ff erent priors on the GC's [Z / H] and mass (e.g., uniform, Gaussian, logarithmic) to assess their impact on the estimation of these parameters, as well as the influence on ages, given the degeneracies at play. It is worth mentioning here that the mass and metallicity estimates proved to be very stable against all the di ff erent changes in the fit setup, while ages were mainly a ff ected by the choice of prior, as we discuss in the following. In the end, we converged to a fit configuration in which: (i) as often done in literature (Koleva et al. 2008; Gonçalves et al. 2020) we fit the range 3700-6000 Å to avoid the redder telluric lines and we mask the interval 5870 -5910 Å, where the spectra show a very deep sodium doublet absorption line, since it could be potentially contaminated by interstellar absorption; (ii) we consider a single DED SFH, a dust component and a noise component; (iii) on all the parameters we set uniform, uninformative priors. To assess the impact of the cosmological prior on the results, we tested two di ff erent upper limits for the age parameter: 13.47 Gyr, age of the Universe in a flat Λ CDMmodel with Ω M , 0 = 0.3, ΩΛ , 0 = 0.7, H0 = 70 km / s / Mpc; 15 Gyr, a loose limit independent of any cosmology. We refer to the first configuration as Config. 13.5 and to the latter as Config. 15 . A summary of the main parameters and relative priors for the two configurations can be found in Tab. 1. We highlight that we assume uniform priors on all the parameters, along with wide ranges so that the results are not constrained by any previous knowledge of the GC's mass, metallicity, dust extinction or age. As anticipated, BAGPIPES adopts the overall metallicity [Z / H] as the metallicity parameter. To compare our results with [Fe / H] values from the literature, we need to perform a conversion. We use the conversion formula from Salaris & Cassisi (2005): For objects with [Fe / H] ≤ -1 we apply this formula with an alpha-enhancement of [ α / Fe] = 0.35, which is typical of the metal-poor MW GCs (e.g., Pritzl et al. 2005; Mendel et al. 2007), while for GCs with [Fe / H] > -1 we use [ α / Fe] = 0.15, average alpha-enhancement at these metallicities (see, e.g., Pagel & Tautvaisiene 1995; Pancino et al. 2017). From now on, we refer to the quantity [Z / H] as the metallicity of the GCs.",
"pages": [
7
]
},
{
"title": "4.3. Results",
"content": "We performed a visual inspection to evaluate the quality and convergence of the fits. Specifically, we identified fits that either failed to recover the spectral lines or continuum or exhibited double- or multiple-peaked posterior distributions. As a result, we discarded a significant number of poor fits, totalling 11 objects in both Config. 13.5 and Config. 15 , which represent about 14% of the sample. Among these, 8 GCs had an HBR > 0, and we found that, in these cases, the posterior spectrum underestimated the emission in the wavelength range blueward of 4000 -4500 Å. This issue is likely due to the blue HB emission, which the models cannot fully reproduce. Consequently, the fits converge to younger ages, as observed in these cases where all 8 poor fits have ages younger than 10 Gyr. As discussed in Sect. 3.2, various studies have successfully included a contribution of the HB on top of the SSP models (e.g., Jimenez et al. 2004; Koleva et al. 2008; Cabrera-Ziri & Conroy 2022). However, incorporating this component into BAGPIPES is outside the scope of this work. Removing bad fits, the clean sample counts 66 GCs in both configurations. In Fig. 4 two examples of good fits are reported, both converging to ages older than 13 Gyr, one presenting a red HB (NGC6356, HBR = -1) while the other shows a blue HB (NGC6717, HBR = 0.98). The pulls highlight how in the case of NGC6356 the stellar models, plus the dust components, are able to accurately reproduce the GC's spectrum, with residuals compatible with 1σ fluctuations at all wavelengths. In the case of NGC6717, the quality of the fit is still good, but the pulls clearly show a residual at bluer wavelengths, especially concerning the Balmer absorption lines, pointing out the unmodelled hot stars component. This suggests that for GCs characterised by blue HBs may still produce a reliable age estimation, as long as the blue HB stars do not outshine the blue end of the spectrum. The quality of both considered setups is highlighted by the median reduced chi-squares, ˜ χ 2 = 1 . 21 in Config. 13.5 and ˜ χ 2 = 1 . 26 in Config. 15 . This is quite noticeable since in this case we adopted the formal spectrum error provided by the analysis, with the correction described in Sect. 4.2. These values are further (and as expected) reduced if we take into account the noise parameter, which acts in correcting the error spectrum for potential underestimations, leading to ˜ χ 2 = 0 . 99 in Config. 13.5 and ˜ χ 2 = 0 . 98 in Config. 15 . To analyse the derived physical properties, we consider, for each parameter, the median and the 16 th and 84 th percentiles of the posterior distribution respectively as the best-fit value, lower and upper error.",
"pages": [
7,
8
]
},
{
"title": "4.3.1. Configuration 13.5",
"content": "In Config. 13.5 we observe that metallicities and GC masses are in good agreement with literature values, with mean deviations of ⟨ ∆ [Z / H] ⟩ = 0 . 09 ± 0 . 21 dex and ⟨ ∆ log(M ⋆/ M ⊙ ) ⟩ = -0 . 09 ± 0 . 24 dex respectively, consistent with the typical errors associated with these quantities (see Sect. 2). In terms of stellar age, instead, a clear bimodality is present. While 17% of the sample (11 GCs) turns out to have ages older than 10 Gyr and only ∼ 0.16 Gyr younger than literature values on average, most of it (55 GCs) shows ages significantly younger than 10 Gyr, ∼ 8.9 Gyr lower on average. We investigate this di ff erence in the following. In Fig. 5a and 5b we show the di ff erences in stellar mass and metallicity as a function of this age gap, colour-coded by HBR index. This highlights two important aspects. The first is, again, the trend with HBR, showing that when this index is positive (blue HB), the code misinterprets the blue shape of the spectrum and the deep Balmer lines for a young population 87% of the time (27 / 31), resulting in ages on average 8.4 Gyr younger than expected from the literature. This exact behaviour is also observed for most of the red HB population, but in a smaller fraction (73% of the cases, 19 / 26 GCs) and with a less significant age discrepancy of 5.1 Gyr on average. A similar result was already found both in Koleva et al. (2008) and Cabrera-Ziri & Conroy (2022), where the issue was mitigated by adding a fraction of hot stars on top of the SPS models, but, as anticipated, including the HB component goes beyond the purpose of this work. In Sect. 4.6, though, a detailed comparison with the results in Cabrera-Ziri & Conroy (2022) can be found. The second is the existence of a degeneracy among the parameters involved. A lower cluster mass or a higher metallicity can easily mislead the fit to ages much younger than the literature one. This is clear if we compute the median di ff erences in metallicity and mass separately for the GCs resulting older and younger than 10 Gyr: for the former, we find average deviations of ⟨ ∆ [Z / H] ⟩ = -0 . 14 ± 0 . 19 dex and ⟨ ∆ log(M ⋆/ M ⊙ ) ⟩ = 0 . 16 ± 0 . 15 dex, for the latter instead ⟨ ∆ [Z / H] ⟩ = 0 . 13 ± 0 . 18 dex and ⟨ ∆ log(M ⋆/ M ⊙ ) ⟩ = -0 . 13 ± 0 . 24 dex.",
"pages": [
8
]
},
{
"title": "4.3.2. Configuration 15",
"content": "In Config. 15 the good agreement of metallicity and GC mass estimates with literature values holds, with average di ff erences of ⟨ ∆ [Z / H] ⟩ = -0 . 02 ± 0 . 23 dex and ⟨ ∆ log(M ⋆/ M ⊙ ) ⟩ = 0 . 04 ± 0 . 28. Concerning the stellar ages, even though the only di ff erence with respect to Config. 13.5 is the removal of the cosmological prior, the results improve significantly. Here 36% (24 GCs) of the sample shows ages older than 10 Gyr, more than twice the old population of Config. 13.5 . Among these 24 GCs, the ages result compatible with literature values 92% of the times (22 GCs), on average 0.67 Gyr older, and we find average discrepancies in [Z / H] and mass of ∆ [Z / H] = -0 . 11 ± 0 . 18 and ∆ log(M ⋆/ M ⊙ ) = 0 . 20 ± 0 . 20. In Fig. 5c and Fig. 5d we show the analogous of 5a and Fig. 5b for Config. 15 . We can see that both the trend in HBR and the age-metallicity and age-mass degeneracies are present but with important di ff erences. This time, removing the upper limit on the age parameter has reduced the fraction of blue HB GCs mistaken for young populations to 71% (22 / 31) and the one of the red HB GCs to 54% (14 / 26). This means that 13 GCs previously resulting younger than 10 Gyr are now recognised as old populations, representing a 20% increment. The degeneracies cited above play an important role in this because all of these 13 GCs are here characterised by a lower metallicity ( ∆ [Z / H] ∼ -0 . 18 dex) and a higher mass ( ∆ log(M ⋆/ M ⊙ ) ∼ 0 . 19 dex) than the one found in Config. 13.5 , yet mostly in agreement with literature values within errors. This suggests that for a fraction of GCs an old, more realistic solution does exist beyond the cosmological limit usually set when performing FSF and that it may also be preferred to the younger one if this area of the parameter space is made accessible. For this reason and in light of the subsequent cosmological analysis, we consider Config. 15 our benchmark.",
"pages": [
8
]
},
{
"title": "4.4. Systematic effects",
"content": "As anticipated in Sect. 4.2, we performed multiple tests with different settings to find the optimal fit configuration. These analyses have been used to assess the systematic error induced in the age determination with our approach. Specifically, we examined 8 configurations, each di ff ering from our benchmark in one or two key aspects, including variations in SFH type (burst or DED), age prior (15 Gyr or 20 Gyr) metallicity prior (uniform or Gaussian), wavelength range of the fit, and in fitting spec- tra in physical units or normalised in the window 4500-5000 Å. The latter was applied in just one configuration, the only case in which the mass parameter could not be determined due to the normalisation. All the characteristics of the di ff erent configurations are outlined in Tab. 1, numbered from 1 to 8. In this analysis, we discarded all the spurious young solutions with a best-fit age below 10 Gyr for the same reasons discussed in Sect. 4.3, and all the bad fits in Config. 15 and in each of the 8 test configurations. In this way, we end up with a sample of 18 GSs having a good fit in at least 5 out of the 9 runs. For each object, we computed the standard deviation of the age distribution in the 9 runs. Finally, we estimated a global systematic contribution to the age uncertainty as the average of these standard deviations, weighted on the number of good fits for each GC, resulting in 0.71 Gyr 1 .",
"pages": [
8,
9
]
},
{
"title": "4.5. The role of metallicity",
"content": "In Sect. 3.2 we observed that not only the presence of an extended blue HB, but also a low metallicity could produce some spectral features that can drag the fit to younger ages due to their degeneracy. Here we want to verify if this trend is present in our results. In Fig. 6a, we plot the metallicity obtained from our fit against the values found in the literature, colour-coded by the di ff erence between the age estimated from the fit and the literature one. We can observe that the best match in age estimation is indeed found at the highest metallicities, while the objects for which the age is most underestimated are also the ones with lower metallicity. To perform a quantitative comparison, we divided our sample into metallicity sub-samples, considering three intervals equally spaced in [Z / H]lit: a metal-rich [ -0 . 7 , 0 . 0], a metal-intermediate [ -1 . 4 , -0 . 7] and a metal-poor [ -2 . 1 , -1 . 4]. The metal-poor sample counts 15 GCs, among which only the 20% (3 / 15) is recognised as older than 10 Gyr. This can be better understood by examining the top panel of Fig. 7, where the median stacked spectrum of the metal-poor sample is compared to two synthetic spectra, one 13 Gyr old and the other 4 Gyr old, while all the other parameters (e.g., [Z / H], mass, dust) are fixed to the median literature values characterising this sub-sample. The spectral shape does indeed resemble the one of a young population, with a stronger emission bluer than ∼ 4400Å with respect to the spectrum of an old population, and stronger Balmer lines. However, in Fig. 6b, where discrepancies in the reproduction of CaII K and H β are shown as a function of literature metallicity, we can see that for most of the metalpoor GCs, the H β line is actually overestimated. This means that the young solution, even if preferred by the fit, does not precisely follow the observed features. In terms of metallicity and mass, the agreement with literature values is very good, with average deviations of ⟨ ∆ [Z / H] ⟩ = -0 . 02 ± 0 . 35 dex and ⟨ ∆ log(M ⋆/ M ⊙ ) ⟩ = -0 . 2 ± 0 . 2 dex respectively. The metal-intermediate is the largest sub-group, with 33 GCs, and shows a higher percentage of GCs older than 10 Gyr compared to the metal-poor sample, equal to 33% (11 / 33). Its median stacked spectrum has a better agreement with an old population in terms of the continuum but still fails to reproduce some observed lines of the Balmer series, as we can see in the central panel of Fig. 7. This is mostly evident for the H β line, falling in the region 4500 -5000Å where the spectra have been normalised, which is clearly better reproduced by a young population, as in the metal-poor case. This suggests that there is still non-negligible contamination by hot stars, deepening the Balmer series. We can observe this in more detail in Fig. 6b where to obtain old solutions, the fit has to underestimate the H β feature. In contrast, for the young ones, it is either compatible with observations or overestimated. In this metal-intermediate group, we can also observe the importance of reproducing the CaII K line in recovering ages. In fact, while the old solutions all scatter around ∆ CaIIK ∼ 0, the young ones systematically underestimate this feature. Regarding the metallicity, this subgroup shows a good agreement with literature values, with a ⟨ ∆ [Z / H] ⟩ = 0 . 10 ± 0 . 15 dex, and a discrepancy in mass smaller than before ⟨ ∆ log(M ⋆/ M ⊙ ) ⟩ = -0 . 04 ± 0 . 30 dex. In the metal-rich sample instead, we are able to obtain ages older than 10 Gyr for 56% (10 / 18) of the sample. This fraction increases as we move to solar values, reaching 70% for [Z / H]lit ≥ -0 . 4. In the bottom panel of Fig. 7 we can see that, among the three samples, the metal-rich is the most distant from a young population with a much redder continuum. In addition, the metal-rich is the only stacked spectrum in which the old synthetic spectrum accurately reproduces the H β , while the young one leaves a clear residual. Looking at Fig. 6b, the metal-rich sample is the only one for which both H β and CaII K are well reproduced. In terms of metallicity and mass here we find ⟨ ∆ [Z / H] ⟩ = -0 . 08 ± 0 . 15 dex and ⟨ ∆ log(M ⋆/ M ⊙ ) ⟩ = 0 . 09 ± 0 . 16 dex with respect to literature values. The fact that we can well recover ages in this metallicity range is a remarkable result also in the context of galaxy evolution stud- ies, for which GCs have always been an important test bench, since it shows the reliability of the FSF method in recovering the main physical parameters of the stellar population in a metallicity interval that is typical of high redshift galaxies (see, e.g., Kriek et al. 2019; Lonoce et al. 2020; Carnall et al. 2022; Borghi et al. 2022a). Moreover, we show that spectral features like H β and CaII K prove to be very sensible to possible spurious age ] H / Z [ (a) 1 H determinations and thus can be used as diagnostics to determine the quality of the age estimates. agefit agelit",
"pages": [
9,
10,
11
]
},
{
"title": "4.6. Comparison with previous works",
"content": "As anticipated in Sect. 1, di ff erent works have already been published investigating the potential of analysing GCs' integrated spectra, either relying on di ff erent datasets or adopting di ff erent fitting codes. Here we focus on the work from Cabrera-Ziri & Conroy (2022) (CC22, hereafter), where they use the observations from Schiavon et al. (2005) for a common sub-sample of GCs, and then on the results from Gonçalves et al. (2020) (G20, hereafter), where they analyse the same data but adopt a nonparametric approach.",
"pages": [
11
]
},
{
"title": "4.6.1. Comparison with Cabrera-Ziri & Conroy (2022)",
"content": "CC22 used a similar, but non-parametric, FSF approach with the code ALF (Conroy & van Dokkum 2012) to estimate the ages and metallicities of 32 Galactic GCs from Schiavon et al. (2005), fitting normalised spectra in the range ∼ 3300 -6500Å. As in Config. 13.5 , CC22 initially performed the analysis with a standard setting, using a cosmological prior of 14 Gyr. They obtained ages compatible with literature values within 1.5 Gyr for 7 GCs, which constitutes 22% of their sample. To compare the results, we consider the 31 GCs included both in their sample and our good fits. For those, in Config. 13.5 we obtain ages within 1.5 Gyr from literature values for 13% of the sample (4 / 31) while this fraction is more than doubled when we remove the cosmological prior, reaching 32% (10 / 31). Compared to CC22, the performance in recovering old ages with Config. 13.5 is comparable but slightly worse. This is likely due to the additional degrees of freedom in our setting, such as the inclusion of dust and mass parameters, along with the lower age prior. If adopting a multicomponent model can help in reproducing all the spectral features in more detail, this approach is also more prone to possible degeneracies, as we underlined in di ff erent steps of our analysis. Nevertheless, this same choice allows us to obtain better results when removing the cosmological prior, with an increase in the fraction of old objects of 20% with respect to our Config. 13.5 and 10% with respect to the standard setting in CC22. In CC22, an additional fit was performed that included a component to account for the hotter fraction of HB stars. This approach allowed them to recover ages older than 10 Gyr for 27 out of 31 GCs, with 24 of these being compatible with literature values within 1.5 Gyr. As already discussed, modelling the HB component is outside the purpose of this work, but represents a promising possibility to be explored in future analyses.",
"pages": [
11
]
},
{
"title": "4.6.2. Comparison with Goncalves et al. (2020)",
"content": "In G20 the authors focus on the impact that the wavelength range choice has on the results, adopting the FSF code STARLIGHT (Cid Fernandes et al. 2005). They fit normalised spectra using MILES SSP models (Vazdekis et al. 2015) with ages up to 14 Gyr, [Fe / H] in the range from -2.27 to 0.26 and alpha enhancement either absent or equal to 0.4. Dust reddening is implemented in the code, modelled as in Cardelli et al. (1989). We compare our results to the ones published in Goncalves et al. (2023), obtained by fitting the interval 4828 - 5634 Å, a narrow range where the main features detectable are H β , Mgb triplet, Fe5015, Fe5270 and Fe5335. We consider the 58 MW GCs for which we obtain a good fit among the 64 MW GCs published in G20; in this sample, they obtained ages compatible with literature values within 1.5 Gyr for 9 GCs, representing 15% of the total. In this same sub-group, we have 7 GCs compatible with literature ages in Config. 13.5 , and 15 in Config. 15 , corresponding to 12% and 26% of the sample, respectively. As in the comparison with CC22, also in the case of G20 our results when the cosmological prior is applied are comparable but slightly worse, and as commented above the reason probably resides in the higher number of parameters involved and the lower age prior. Again, when we remove the cosmological prior, we obtain a major improvement in the fraction of ages compatible with literature, both with respect to our Config. 13.5 ( + 14%) and to G20 ( + 11%). It is worth mentioning here that we also tested the impact on the results of fitting the wavelength range adopted in G20, first suggested in Walcher et al. (2009), and we considered it when computing the systematic uncertainty on ages in Sect. 4.4. Avoiding all the features bluer than 4828 Å, polluted by the hot HB component, this configuration performs much better for the blue HB, low-metallicity GCs in our sample. In particular, it allows us to recover ages older than 10 Gyr for 63% of metalpoor GCs and 73% of metal-intermediate ones. For the metalrich sample, instead, it yields worse results compared to Config. 15 , with 47% of GCs resulting older than 10 Gyr. Nevertheless, the latter sub-group is the one in which the stellar models should be most e ff ective, thanks to the absence of an extended HB component and low alpha-enhancement in the systems, so Config. 15 was still preferable in terms of the robustness of the results.",
"pages": [
11,
13
]
},
{
"title": "5. Application to cosmology",
"content": "In this section, we analyse what impact our results for GCs' ages, interpreted as lower limits to the age of the Universe tU , can have in the determination of the Hubble constant H 0 (Jimenez et al. 2019; Valcin et al. 2020, 2021; Vagnozzi et al. 2022; Cimatti & Moresco 2023).",
"pages": [
13
]
},
{
"title": "5.1. Method",
"content": "As anticipated in Sect. 1, and widely described in Cimatti & Moresco (2023), H 0 is very sensitive to the value of tU . In a generic cosmological model, H 0 can be expressed as: where t is the age of a given object, zF its redshift of formation, E(z) is defined as H(z) / H 0 and A = 977.8 converts the result in units of km s -1 Mpc -1 . The analytical expression of E(z) depends on the cosmological model assumed, in particular, in a flat Λ CDM model it can be expressed as a function of redshift and matter density parameter Ω M , so that H 0 becomes: Considering the limit of zF = ∞ for which t corresponds to the age of the Universe tU , we can easily understand the sensitivity of the method. Fixing Ω M = 0 . 3 and varying tU from 12.9 Gyr to 14.1 Gyr the resulting value of H 0 spans from 73 km s -1 Mpc -1 to 67 km s -1 Mpc -1 . When applying the method to the oldest objects, H 0 can be estimated via a Bayesian approach, in which the likelihood is built on the di ff erence between the measured age and the one predicted by the cosmological model (agem), accounting for the age error ( σ age): where p = (H0 , zF , Ω M) in a flat Λ CDM cosmology. The posterior distribution then, can be sampled with a Monte Carlo Markov Chain approach like the one implemented in the a ffi neinvariant ensemble sampler emcee (Foreman-Mackey et al. 2013). In the choice of priors, a flat, uninformative one can be adopted on H 0 and zF , while a Gaussian prior is preferable for Ω M in order to break its intrinsic degeneracy with H 0. As a final note on the method, it is interesting to highlight that nowadays, with facilities like JWST, this approach is no more limited to the study of local objects but can be extended to higher redshift thanks to the first detections of GCs around lensed galaxies. A great case-study is the one of the Sparkler, a galaxy discovered in Webb's First Deep Field (Mowla et al. 2022) showing a population of compact objects associated with it. Among the 18 compact objects identified (Mowla et al. 2022; Millon et al. 2024), Sparkler shows 5 GC candidates that, if confirmed with spectroscopy, would be the first detection of GCs at z = 1.38. The spectroscopic study of these objects, analogous to the one performed in this work, would allow us to measure their age, potentially with a higher precision than the one obtained here for ∼ 13 Gyr old GCs. In younger stellar populations, indeed, the shape of the spectrum and the width of the absorption lines changes much quicker than it does at very old ages, thus reducing the impact of the parameters' degeneracies on the age uncertainty. In the context of our cosmological analysis, extending the method at higher redshift would just require to replace the lower limit of the integral in Eq. 9 with the redshift of the lensed GC. In the case of the Sparkler at redshift 1.38, for example, the age of the Universe ranges from 4.3 Gyr to 4.7 Gyr adopting the reference values for H 0 given above, 73 km s -1 Mpc -1 and 67 km s -1 Mpc -1 respectively.",
"pages": [
13
]
},
{
"title": "5.2. Results",
"content": "As anticipated in Sect. 4.3, we adopted as benchmark Config. 15 , where the cosmological prior is removed. To identify the tail of the oldest objects, we adopt a Gaussian Mixture Model (GMM) on the whole sample, combining normal distributions peaked on the best-fit ages, with 1σ equal to relative uncertainties. We then let the fit decide the optimal number of subsamples in which to split our data. Both considering the Bayesian Information Criterion (BIC) or the Akaike Information Criterion (AIC), we find that the optimal number of components is three: a first peak identifying the youngest, blue-HB, spurious solutions; a second one for the intermediate ages; a third one comprising all the 24 oldest GCs, peaking at 13 . 4 ± 1 . 1 Gyr. The latter represents the oldest tail of the GCs' age distribution. We applied the method described in Sect. 5.1 for each of these GCs separately, and then on their average, adopting the following priors: uniform on H0 ∈ [0 , 150] km s -1 Mpc -1 and on zF ∈ [11,30], Gaussian on Ω M = 0 . 30 ± 0 . 02. The lower limit on zF is based on the highest redshift at which galaxies have spectroscopic confirmations (Curtis-Lake et al. 2023), the higher limit instead relies on the values found in theoretical models for the redshift of formation of the very first stars (Galli & Palla 2013). As regards Ω M , the value chosen here comes from the combination of di ff erent low-redshift results (Jimenez et al. 2019), thus independent of the CMB. Our 24 old GCs span the age range 13.2-13.6 Gyr, with errors around 1 Gyr, thus the resulting H 0 range we find covers the interval 69.5-71.7 km s -1 Mpc -1 , with typical uncertainties of ∼ 5 km s -1 Mpc -1 . To provide a single H 0 measurement, we also ran the MCMC using mean and standard deviation of the old peak found in the GMM fit: 13 . 4 ± 1 . 1 Gyr. This results in a final value for H 0 = 70 . 4 + 6 . 7 -5 7 km s -1 Mpc -1 (stat). . To account for the systematic contribution to the error budget computed in Sect. 4.4 we summed it in quadrature to the standard deviation of the distribution and ran again the MCMC. The final result, comprising both statistics and systematic e ff ects is: In Fig. 9 the 24 GCs' ages and the corresponding H 0 estimates found after the MCMC run are represented as Gaussian distributions, in the respective domains and combined as ellipses in the H 0-age plane. The corresponding Gaussian curves and ellipses relative to the combined age and H 0 are shown as black solid lines. For comparison, the values from Riess et al. (2022) and Planck Collaboration et al. (2020) are represented respectively with dashed and dotted lines. Of course, the results that we obtain here are not able to address the tension but represent a pilot exploration of the use of GCs' ages for cosmological purposes, especially in view of future missions that could potentially discover such objects at higher redshifts.",
"pages": [
13,
14
]
},
{
"title": "6. Conclusions",
"content": "In this work, we analysed the integrated spectra of a sample of 77 Milky Way GCs from the WAGGS project (Usher et al. 2017) and measured their physical properties via FSF with the code BAGPIPES (Carnall et al. 2018). In doing this, we aimed to study how well FSF can recover the GCs' ages and physical parameters, and assess, in particular, how the age estimates are a ff ected by the presence or absence of a cosmological prior. This required a modification on the code, already tested and validated in Jiao et al. (2023) and Tomasetti et al. (2023), thanks to which a flat non-cosmological prior can be set at 15 Gyr. At the same time, this allowed us to obtain a cosmology-independent lower limit to the age of the Universe, that we used to derive a new constraint on H 0, performing a pilot study for future potential applications at higher redshift. Our results are summarised as follows: Article number, page 14 of 16 ays result well compatible with reference values independently of HBR, [Z / H] or fit setting, with average discrepancies on the whole sample of ⟨ ∆ [Z / H] ⟩ = -0 . 02 ± 0 . 24 dex and ⟨ ∆ log(M ⋆/ M ⊙ ) ⟩ = 0 . 04 ± 0 . 28, compatible with the typical uncertainties associated with these quantities. While we acknowledge that the method adopted in this paper is not intended to compete with other age estimation techniques (e.g., isochrone fitting) for local and resolved objects, it does offer a viable alternative. This study serves as an initial pilot investigation into the feasibility of using only spectroscopic information to determine GC ages, an approach that may be particularly useful for investigating the properties of lensed GCs at higher redshifts, where isochrone fitting is not feasible. Future developments will include expanding the tests to incorporate models with an extended HB. Nonetheless, even without these enhancements, this work provides valuable diagnostics for identifying the most robust and reliable fits. Acknowledgements. We thank Christopher Usher for kindly providing us with WAGGS spectra, Adam Carnall for his help in using Bagpipes , and Frédéric Courbin and Licia Verde for the insightful discussion on the potential of lensed GCs' ages in cosmology. ET acknowledges the support from COST Action CA21136 - 'Addressing observational tensions in cosmology with systematics and fundamental physics (CosmoVerse)', supported by COST (European Cooperation in Science and Technology). Funding for the work of RJ was partially provided by project PID2022-141125NB-I00, and the 'Center of Excellence Maria de Maeztu 2020-2023' award to the ICCUB (CEX2019- 000918M) funded by MCIN / AEI / 10.13039 / 501100011033. MM acknowledges support from MIUR, PRIN 2022 (grant 2022NY2ZRS 001). MM and AC acknowledge support from the grant ASI n. 2024-10-HH.0 'Attività scientifiche per la missione Euclid - fase E'.",
"pages": [
14,
15
]
},
{
"title": "References",
"content": "Abdalla, E., Abellán, G. F., Aboubrahim, A., et al. 2022, Journal of High Energy Astrophysics, 34, 49 Chevallard, J. & Charlot, S. 2016, MNRAS, 462, 1415 Chilingarian, I., Prugniel, P., Sil'chenko, O., & Koleva, M. 2007, in IAU Symposium, Vol. 241, Stellar Populations as Building Blocks of Galaxies, ed. A. Vazdekis & R. Peletier, 175-176 MNRAS, 358, 363 Cimatti, A. & Moresco, M. 2023, ApJ, 953, 149 Colucci, J. E., Bernstein, R. A., & McWilliam, A. 2017, ApJ, 834, 105 Conroy, C. & van Dokkum, P. 2012, ApJ, 747, 69 Eisenstein, D. J., Zehavi, I., Hogg, D. W., et al. 2005, ApJ, 633, 560 Faber, S. M., Friel, E. D., Burstein, D., & Gaskell, C. M. 1985, ApJS, 57, 711 Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306 Gratton, R., Bragaglia, A., Carretta, E., et al. 2019, A&A Rev., 27, 8 Jiao, K., Borghi, N., Moresco, M., & Zhang, T.-J. 2023, ApJS, 265, 48 Koleva, M., Prugniel, P., Bouchard, A., & Wu, Y. 2009, A&A, 501, 1269 Kriek, M., Price, S. H., Conroy, C., et al. 2019, ApJ, 880, L31 Kroupa, P. 2001, MNRAS, 322, 231 Lee, H.-c., Yoon, S.-J., & Lee, Y.-W. 2000, AJ, 120, 998 Lee, Y.-W. 1989, PhD thesis, Yale University, Connecticut Lee, Y.-W., Demarque, P., & Zinn, R. 1994, ApJ, 423, 248 Lonoce, I., Maraston, C., Thomas, D., et al. 2020, MNRAS, 492, 326 Oliveira, R. A. P., Souza, S. O., Kerber, L. O., et al. 2020, ApJ, 891, 37 O'Malley, E. M., Gilligan, C., & Chaboyer, B. 2017, ApJ, 838, 162 Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2020, A&A, 641, A6 Salaris, M. & Cassisi, S. 2005, Evolution of Stars and Stellar Populations",
"pages": [
15
]
}
]
2024arXiv241022966C
https://arxiv.org/pdf/2410.22966.pdf
<document>
<section_header_level_1><location><page_1><loc_36><loc_84><loc_64><loc_85></location>C-19 and Hot, Wide, Star Streams</section_header_level_1>
<text><location><page_1><loc_13><loc_80><loc_87><loc_82></location>Raymond G. Carlberg, 1 Rodrigo Ibata, 2 Nicolas F. Martin, 2 Else Starkenburg, 3 David S. Aguado, 4 Khyati Malhan, 5 Kim Venn, 6 and Zhen Yuan ( 袁 珍 ) 7</text>
<text><location><page_1><loc_19><loc_78><loc_80><loc_79></location>1 Department of Astronomy & Astrophysics, University of Toronto, Toronto, ON M5S 3H4, Canada</text>
<text><location><page_1><loc_14><loc_74><loc_85><loc_77></location>2 Universit'e de Strasbourg, CNRS, Observatoire astronomique de Strasbourg, UMR 7550, F-67000 Strasbourg, France 3 Kapteyn Astronomical Institute, University of Groningen, Landleven 12, 9747 AD Groningen, The Netherlands 4</text>
<text><location><page_1><loc_24><loc_73><loc_76><loc_74></location>Instituto de Astrof'ısica de Canarias, V'ıa L'actea, 38205 La Laguna, Tenerife, Spain</text>
<text><location><page_1><loc_20><loc_72><loc_80><loc_73></location>5 DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 Copenhagen, Denmark</text>
<text><location><page_1><loc_20><loc_71><loc_80><loc_72></location>6 Department of Physics and Astronomy, University of Victoria, Victoria, BC, V8W 3P2, Canada</text>
<text><location><page_1><loc_11><loc_68><loc_89><loc_70></location>7 School of Astronomy and Space Science, Nanjing University, Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210093, China</text>
<section_header_level_1><location><page_1><loc_45><loc_65><loc_55><loc_66></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_44><loc_86><loc_64></location>The C-19 star stream has the abundance characteristics of an unusually metal poor globular cluster but kinematically is uncharacteristically hot and wide for a cluster stream, having a line of sight velocity dispersion of 6 km s -1 and a 1-sigma width of 240 pc. We show that the tidal dissolution of an old, lower mass, globular cluster in a CDM galactic halo naturally creates a hot, wide stream currently near orbital apocenter. More generally, simulations show that hot streams, which are all near their orbital apocenter, become thin, cool streams near pericenter. Furthermore, the wide streams from a population of dissolved clusters in the simulations have a mean galactocentric radial velocity dispersion of 7.8 ± 1.0 kms -1 in a CDM cosmology but only 4.1 ± 1.6 kms -1 in a WDM (5.5 keV) simulation. A detailed C-19 model in a simplified Milky Way halo potential with a CDM subhalo population provides a lower bound to stream heating, finding that the stream develops a line of sight velocity dispersion of 4.1 ± 1.1 kms -1 , whereas WDM (5.5 keV) subhalos give 3.1 ± 0.1 km s -1 . Known dwarf galaxies alone provide negligible heating. There are five other currently known streams wider than 200 pc that contain a globular cluster, all near their orbital apocenter.</text>
<section_header_level_1><location><page_1><loc_20><loc_41><loc_36><loc_42></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_15><loc_48><loc_40></location>The C-19 stream of extremely low metallicity stars (Martin et al. 2022) was discovered (Ibata et al. 2021) in the Gaia data (Gaia Collaboration et al. 2016) using the STREAMFINDER algorithm (Malhan & Ibata 2018). High resolution spectroscopy (Martin et al. 2022; Yuan et al. 2022) of photometrically identified low metallicity stream candidates (Starkenburg et al. 2017) established that C-19 has a metal abundance of [Fe/H] = -3.4 with a 95% confidence [Fe/H] spread of less than 0.2 (Martin et al. 2022). Further investigation has found additional stream candidates giving a length of more than 50 degrees (Yuan et al. 2022; Viswanathan et al. 2024). The spread in stellar abundances is close to the detection limit and the abundance patterns are characteristic of globular cluster stars (Yuan et al. 2022). The abundances and extrapolated luminosity of approximately</text>
<text><location><page_1><loc_52><loc_30><loc_92><loc_42></location>10 4 L ⊙ led Martin et al. (2022) to argue that the progenitor was more likely to be a disrupted globular cluster than a disrupted dwarf galaxy, while considering that the 1-sigma width of 240 pc, and line of sight velocity dispersion of 6 km s -1 were more in keeping with a disrupted dwarf. A dynamical model of a disrupting dark matter dominated dwarf galaxy can account for the kinematic properties of the stream (Errani et al. 2022).</text>
<text><location><page_1><loc_52><loc_11><loc_92><loc_29></location>The question addressed here is under what conditions can a globular cluster progenitor be compatible with the C-19 kinematic data. Most of the known streams in the compilation Galstreams (Mateu 2023) are thinner than C-19 with typical FWHM of 100 pc or less. On the other hand, simulations of globular cluster streams in realistic cosmological conditions find that the dark matter subhalos present in the galactic halo cause the velocity dispersion of stream stars to increase with time (Carlberg & Agler 2023). The average velocity distribution of long, thin streams has a core with extended wings (Carlberg 2018; Malhan et al. 2019). The width of the</text>
<text><location><page_2><loc_8><loc_78><loc_48><loc_91></location>wings increases with the numbers of subhalos present in the galactic halo (Carlberg et al. 2024). The velocity at which stars join the stream determines the width of the low velocity core of a stream. Subsequent encounters with subhalos perturb stars into the extended wings of the velocity and width distribution. The growing dataset for the GD-1 stream provides tentative evidence for the existence of a core-wing velocity structure (Ibata et al. 2024; Valluri et al. 2024).</text>
<text><location><page_2><loc_8><loc_64><loc_48><loc_77></location>The C-19 stream is on a nearly polar, inner halo orbit, 8-24 kpc, where the density of subhalos is relatively high, creating the circumstances for significant subhalo heating. The stream is composed of extremely metal poor, [Fe/H] ≃ -3.4, hence likely very old, stars in an inner halo orbit such that the moderate mass progenitor cluster likely dissolved relatively early and the stream has been subject to subhalo heating for much of the lifetime of the Milky Way.</text>
<text><location><page_2><loc_8><loc_43><loc_48><loc_63></location>In § 2 we measure the range of widths and velocity dispersions of the streams that form in both CDM and WDM cosmological simulations from low mass globular clusters. We then use the cosmological simulations to motivate an evolving potential model for the Milky Way in which model star clusters on the orbit of C-19 are integrated and their velocity dispersions are measured to compare to C-19 in § 3. The results are discussed § 4 to assess the accuracy of the model and its ability to constrain the subhalo content of the Milky Way. Other currently known wide streams from globular clusters are briefly discussed along with the likelihood that there are other wide streams yet to be discovered.</text>
<section_header_level_1><location><page_2><loc_14><loc_38><loc_42><loc_41></location>2. LOW MASS STAR CLUSTERS IN COSMOLOGICAL SIMULATIONS</section_header_level_1>
<text><location><page_2><loc_8><loc_22><loc_48><loc_37></location>The total stellar mass of C-19 is estimated to be in the range 3 × 10 3 to 10 4 M ⊙ (Martin et al. 2022; Yuan et al. 2022; Ibata et al. 2024). There is no visible progenitor, as expected for an old star cluster with a half mass relaxation time of 1-2 Gyr orbiting in the Milky Way (Gnedin & Ostriker 1997; Binney & Tremaine 2008; Errani et al. 2022). The C-19 stream has an angular momentum of about 3100 kpc- km s -1 , and is on a moderate eccentricity, e ≃ 0 . 45, nearly polar, orbit within 25 kpc of the galactic center (Martin et al. 2022).</text>
<text><location><page_2><loc_8><loc_9><loc_48><loc_21></location>Two cosmological Milky Way-like simulations are seeded with low mass progenitor star clusters in the mass range 5-20 × 10 3 M ⊙ in a CDM cosmology and a WDM(5.5 keV) cosmology. The cosmological simulations of streams in Carlberg et al. (2024) contained progenitor clusters more massive than 4 × 10 4 M ⊙ , most of which survive to the present epoch. Other than the initial star cluster masses, the simulations here are iden-</text>
<figure>
<location><page_2><loc_53><loc_67><loc_91><loc_91></location>
<caption>Figure 1. The stream radial velocity dispersion (galactocentric) vs stream width as measured perpendicular to the stream track for low mass cluster progenitors in cosmological simulations at the final time. CDM streams are blue points, WDM red. The error bar gives the C-19 values.</caption>
</figure>
<text><location><page_2><loc_52><loc_26><loc_92><loc_55></location>to those in Carlberg et al. (2024) and the same stream analysis procedures are used. We examine the simulation streams above a minimum angular momentum of 2000 kpc- km s -1 , which excludes those that interact strongly with the disk. The streams are selected to be within 60 kpc of the galactic center. The velocity dispersion and width are measured for the central region of the stream above a surface density threshold of 20 M ⊙ deg -2 which approximately defines the readily detected part of streams. Figure 1 shows the stream radial velocity dispersion, after removing the mean, against stream width for the 32 CDM streams and 24 WDM streams longer than 10 degrees, the minimum angle for our search and analysis algorithm. The same plot for the more massive progenitor star clusters in Carlberg et al. (2024) shows a similar correlation of velocity dispersion and width for the streams which are about twice the length of the streams here but show less difference between CDM and WDM.</text>
<text><location><page_2><loc_52><loc_10><loc_92><loc_25></location>The distribution of stream stars projected onto the orbital plane varies from a narrow distribution at pericenter to extended 'feathers' at apocenter, as illustrated in Figure 2 of Carlberg (2015). The orbital variation is a consequence of the galactic tidal field accelerating stars away from the cluster in a narrow radial band near pericenter, the subsequent orbital spreading of the unbound stars at apocenter, then regrouping at pericenter again, somewhat similar to the orbital dependence of velocities of an accretion remnant (Helmi & White 1999).</text>
<figure>
<location><page_3><loc_9><loc_70><loc_36><loc_92></location>
<caption>Figure 5 shows that streams on high angular momentum orbits, those above 8000 kpc- km s -1 , have the velocity dispersion at which stars joined the streams, approximately 2 km s -1 . The medium size squares in Figure 5 identify the streams that are within 30 kpc of the galactic center. The mean of the radial velocity dispersion for streams inside 30 kpc are shown as the large squares, finding 7.8 ± 1 . 0 kms -1 for CDM and 4.1 ± 1 . 6 kms -1 for 5.5 keV WDM, where the error ranges are the spread of the population, not the error in the mean. Lower angular momentum streams orbit closer to the center of the galaxy where the subhalo density is higher than in the outskirts, and there are more subhalos in a CDM galactic halo than in a WDM version. Clearly, a stellar stream from a globular cluster progenitor can easily have a velocity dispersion of 6 km s -1 or more when the stream is near apocenter in a CDM galactic halo.</caption>
</figure>
<figure>
<location><page_3><loc_37><loc_70><loc_64><loc_92></location>
</figure>
<figure>
<location><page_3><loc_65><loc_70><loc_92><loc_92></location>
<caption>Figure 2. The orbital evolution of the stream radial velocity dispersion (left panel) with time for a few of the CDM streams selected to have a high velocity dispersion and L ≃ 3000 kpc- km s -1 at the final time, 14.1 Gyr. The final time is marked with a dot in the two rightmost panels. The middle panel shows tracks the evolution of velocity dispersion with width. The left panel shows the evolution of the radial velocity dispersion with galactocentric distance.</caption>
</figure>
<text><location><page_3><loc_8><loc_36><loc_48><loc_60></location>The variation of the radial velocity dispersion (galactocentric) and the average physical width with time for a few individual inner halo streams are shown in the left hand panel of Figure 2. The streams were selected to have angular momenta around 3000 kpc- km s -1 and velocity dispersion around 8 km s -1 , values comparable to those for C-19. The right hand panel shows the relation between the galactocentric σ r and the galactic radius of the center of the streams. The streams from clusters 154 and 179 are on moderately eccentric orbits with σ r varying from about 2-3 to 7-9 km s -1 as the stream orbits. Stream 86 is on a less eccentric orbit and shows smaller orbital variations in velocity dispersion and width. When near apocenter a stream's structure is often visibly less well organized than at pericenter, with offshoots and bifurcations.</text>
<text><location><page_3><loc_8><loc_12><loc_48><loc_35></location>Figure 3 shows the radial velocity dispersion (left) and width (right) of all the CDM (top) and WDM (bottom) streams at the end of the simulation, over the 13-14.1 Gyr time interval, as a function of the fractional distance to apocenter. The 1.1 Gyr time range is just sufficient to capture the basic radial range of the most distant clusters in these samples, but not the full orbital complexity of the potentials. The point sizes are proportional to the orbital eccentricity. The figure shows that strong correlation of both σ r and σ width with the fractional distance to apocenter depends on orbital eccentricity, with a large scatter. The high eccentricity streams (larger point sizes) have the most extreme variation with apocenter distance and tend to have larger absolute velocity dispersions and widths.</text>
<text><location><page_3><loc_52><loc_40><loc_92><loc_60></location>The velocity distribution within a stream usually has an approximately Gaussian core with extended wings (Carlberg et al. 2024). The kurtosis of these velocity distributions is not very stable, so we characterize the velocity distribution using clipped variances of the distributions. The radial velocity dispersion is calculated along the length of the stream with 3-sigma clipping. The 6-sigma clipped velocity dispersion is typically a factor of two larger for streams with σ r ≤ 3 kms -1 as shown in Figure 4. As the streams pass near pericenter the velocity distribution narrows but leaves behind extended wings which are not present when the velocity distribution widens around apocenter.</text>
<figure>
<location><page_4><loc_9><loc_34><loc_92><loc_92></location>
<caption>Figure 3. The stream radial velocity dispersions and widths with fractional distance to apocenter for the CDM (top) and WDM(5.5 keV) (bottom) streams. The point sizes are proportional to the orbital eccentricity. The colors change with time from 13 to 14.1 Gyr (the final time), in steps of 0.1 Gyr with a cycle of 7 colors.</caption>
</figure>
<text><location><page_4><loc_8><loc_10><loc_48><loc_26></location>Figure 6 shows two streams in the CDM simulation with a final time velocity dispersion above 6 km s -1 that orbit within 30 kpc. The particle plots usefully reveal the entire distribution of the dispersed cluster stars, but only the 20-30 degree segments at a galactic Y coordinate of approximately 20 kpc have sufficiently high sky density to be readily visible. The orbital history of these two clusters is shown in Figure 7. Cluster 154, shown in red, accretes onto the main halo in the first few Gyr of the simulation. Cluster 179, shown in green, is accreted around 7 Gyr, about halfway through the simulation.</text>
<text><location><page_4><loc_52><loc_21><loc_92><loc_26></location>Dynamical friction in the aspherical halo (Binney 1977) draws the subhalo containing the cluster into the plane of the galaxy and reduces the orbital angular momentum of the cluster.</text>
<section_header_level_1><location><page_4><loc_58><loc_18><loc_85><loc_19></location>3. MODELING THE C-19 STREAM</section_header_level_1>
<text><location><page_4><loc_52><loc_9><loc_92><loc_17></location>The globular cluster streams in the cosmological simulations have radial velocity dispersions that range from about 2 to 9 km s -1 , with the highest velocity dispersion at orbital apocenter. The highest density part of C-19 is just past orbital apocenter (Martin et al. 2022;</text>
<figure>
<location><page_5><loc_9><loc_67><loc_48><loc_91></location>
<caption>Figure 4. The ratio of the 6-sigma clipped radial velocity dispersion to the 3-sigma value vs the stream velocity dispersion. The 6-to-3 ratio is a measure of the size of the wings of the velocity distribution function. CDM streams are blue, WDM, red.</caption>
</figure>
<figure>
<location><page_5><loc_9><loc_26><loc_47><loc_51></location>
<caption>Figure 5. The mean galactocentric σ r along the part of the stream above the density threshold with the (dissolved) progenitor angular momentum. All orbital phases are plotted. The CDM simulation streams are marked with blue and the WDM(5.5 keV) with red. The streams inside 30 kpc, with widths greater than 100 pc and angular momentum between 2000 and 4000 kpc- km s -1 are shown as squares with the averages shown with the large squares.</caption>
</figure>
<text><location><page_5><loc_52><loc_79><loc_92><loc_91></location>Viswanathan et al. 2024) and is similar to the high velocity streams in the simulations. Detailed modeling of a star cluster on the C-19 orbit is useful to understand the conditions under which a wide, hot stream develops. In particular the subhalo heating of a stream depends on the number of subhalos present in the primary halo and the length of time that the stream has been orbiting after tidal removal from of its natal subhalo.</text>
<text><location><page_5><loc_52><loc_48><loc_92><loc_79></location>Our C-1 stream model is a simplification of the cosmological simulation to follow a single star cluster in a precomputed version of the galactic potential. The model star cluster has a mass of 1 × 10 4 M ⊙ , composed of 1 M ⊙ star particles using the same star cluster simulation as in the full cosmological model. The initial half-mass radius is set to approximately 4 pc for the models reported here. The star particles have a softening of 1 pc and are integrated with the Gadget4 code (Springel et al. 2021). Two body interactions are included with small random velocities calculated from the relaxation time (Spitzer 1987; Binney & Tremaine 2008) added to star particles within the virial radius, about 5 pc, every 5 Myr (Carlberg & Agler 2023). The tidal acceleration increases with distance from the cluster, pumping the stars into more distant orbits until they are swept away from the cluster (Meiron et al. 2021). The cluster-centric orbits become increasingly complex with distance(Fukushige & Heggie 2000). The cluster-centric radial coordinates of a sample of star particles are shown in Figure 8.</text>
<text><location><page_5><loc_52><loc_10><loc_92><loc_47></location>The potential in which the star cluster orbits is drawn from the cosmological simulation. The primary mass component is a static, triaxial NFW potential, with mass, 9 . 2 × 10 11 M ⊙ , virial radius 200 kpc, scale of 14.4 kpc, and density triaxiality from the AHF halo finder (Gill et al. 2004; Knollmann & Knebe 2009) measurements at the final moment the simulation. The potential triaxiality is set to b/a=0.97, c/a=0.83 using Equation 2.72a of Binney & Tremaine (2008). A Miyamoto-Nagai galactic bulge-disk (Miyamoto & Nagai 1975) grows within the halo using the model of Carlberg et al. (2024). Although not important for the inner halo streams considered here, the LMC is included as a 1 . 8 × 10 11 M ⊙ (Garavito-Camargo et al. 2019) Hernquist sphere (Hernquist 1990), which means that the barycenter of the system is offset from the center of the primary halo. The framework could include a more detailed buildup of the primary halo and include a rotating galactic bar both of which would provide some additional stream heating. The current sparse C-19 data does not require more than the basic model. The positions and velocities of the population of subhalos above 10 6 M ⊙ as found in one of the full cosmological simulations at the chosen starting time is integrated in the</text>
<figure>
<location><page_6><loc_9><loc_57><loc_48><loc_87></location>
<caption>Figure 9 shows the model subhalo population in the inner halo along with the stream at a cosmological time of 13.1 Gyr, 1Gyr before the present time. For a CDM cosmology about 8200 subhalos are followed and 2800 for a WDM (5.5 keV) cosmology, both within the virial radius of 200 kpc. Most of the subhalos orbit well outside the region where C-19 orbits, but they are included for completeness. The dwarf galaxies are shown in green in Figure 9 although interactions are rare and usually do not influence inner halo streams (Bonaca et al. 2019). The XY projection of the stream is shown in grayscale at time 1 Gyr before present when stream particles are still passing through orbital apocenter. Subhalos that are within 2 scale radii of the stream are in blue and the stream particles within that radius are red. The interacting subhalo near the center of the image has a current mass of 2 . 2 × 10 7 M ⊙ and induces a velocity change of about 0.5 km s -1 in the nearest stream particles. The interactions that dominate the stream velocity variations are largely from more massive halos in the first 5-6 Gyr of the simulation, once the tidal tails have acquired appreciable length but before the subhalo interactions diminish as their masses decline.</caption>
</figure>
<figure>
<location><page_6><loc_53><loc_57><loc_92><loc_87></location>
<caption>Figure 6. XY and XZ projections of two low angular momentum streams with radial velocity dispersion of 6.9 (red) and 9.2 kms -1 (green) that develop from two dissolved stars clusters with initial masses of 1.6 and 1.3 × 10 4 M ⊙ in the CDM simulation. The red stream is nearly polar whereas the green stream orbits near the galactic plane. Movies of the entire simulation and individually of these two streams are at CDM Low Mass Clusters.</caption>
</figure>
<text><location><page_6><loc_8><loc_9><loc_48><loc_48></location>combined Milky Way dark halo, disk and LMC potential. The subhalos do not interact with each other. The subhalos' masses decrease in time with an exponential decay in time, with a timescale of 5.5 Gyr, measured from the subhalo numbers with time in the simulations. The decrease in subhalo mass with time means that the subhalo N ( > M ) relation within the main halo declines approximately as measured in the fully dynamical cosmological simulations. Subhalos over the mass range 10 6 M ⊙ to 3 × 10 8 M ⊙ are included. Lower mass halos have little dynamical effect on the stream and higher mass subhalos are expected to contain visible dwarf galaxies. All subhalos are modeled as Hernquist spheres with a scale radius equal to r max from AHF. As the subhalos lose mass the radii are adjusted as M 1 / 3 to keep the characteristic density of each subhalo constant. In addition to the subhalos the 55 dwarf galaxies with kinematics (McConnachie 2012) are included. The dwarf galaxy dark halos are assigned maximal masses, M = 5 × 10 8 ( L/L ⊙ ) 0 . 65 M ⊙ , which roughly match the high mass end of the subhalo mass distribution function. The dwarf galaxy halo radii are set to a = 1 . 0( M/ 10 8 M ⊙ ) 0 . 23 kpc, on the basis of the fit to the r max mass relation of the subhalos in the simulations. All the subhalo orbits are precomputed and define spline</text>
<text><location><page_6><loc_52><loc_45><loc_92><loc_48></location>coefficients for the positions of the masses which the star cluster calculation uses as an external potential.</text>
<figure>
<location><page_7><loc_18><loc_87><loc_43><loc_92></location>
<caption>Figure 7. The orbits of the two clusters in Figure 6 relative to the center of the primary halo. Note that the scale is larger in the lower panels. The coordinates are with respect to the center of the dominant halo. The colors indicate the time within the simulation, from a 1 Gyr start (violet) to the 14.1 Gyr end time (red).</caption>
</figure>
<figure>
<location><page_7><loc_8><loc_31><loc_46><loc_85></location>
</figure>
<text><location><page_7><loc_44><loc_60><loc_47><loc_61></location>100</text>
<text><location><page_7><loc_8><loc_10><loc_48><loc_18></location>The evolving potential model presented here does a reasonable job capturing the interactions of subhalos with the stream but misses the larger scale interactions that scatter the older, more distant ends of the stream over the halo as shown in Figure 6 for the cosmologi-</text>
<figure>
<location><page_7><loc_53><loc_70><loc_93><loc_92></location>
<caption>Figure 8. The distance from the star cluster center of star particles with time. The heating algorithm adds a random velocity, initially about 0.6 km s -1 , every 0.005 Gyr to the star particles between 3 to 4 pc. Tidal pumping raises particles over several orbits to the zero binding energy surface near 100 kpc after which they drift out to join the stream.</caption>
</figure>
<figure>
<location><page_7><loc_53><loc_26><loc_90><loc_55></location>
<caption>Figure 9. The XY projection one Gyr before the final time of the stream (in greyscale), the dwarf galaxies (filled green circles) and the subhalos (grey circles). Subhalo-stream encounters are shown for subhalos within 2 scale radii of stream particles are blue circles and the particles within 2 scale radii are in red. The distances are in kpc.</caption>
</figure>
<text><location><page_7><loc_52><loc_9><loc_92><loc_13></location>cal simulation. Consequently the detailed stream model presented here is a lower bound on the effects of subhalos on the C-19 stream.</text>
<text><location><page_8><loc_8><loc_57><loc_48><loc_91></location>The C-19 stream track is defined as passing through the approximate mean of the stars in the ϕ 1 = [ -10 , 10] region in Yuan et al. (2022). Specifically the progenitor orbit passes through RA=355.0, Dec=25.0 degrees, with a radial velocity of -195 km s -1 , proper motion of 1.25 and -2.8 milli-arcsec per year in RAcos(Dec) and Dec at a heliocentric distance of 20 kpc. The progenitor is then integrated in the potential of the MW plus LMC model. No optimization of the stream track coordinates or potential model was undertaken. The resulting orbit is adequately close to the current data for the purposes of this paper. The stream simulations placed the current epoch progenitor position at -10 degrees along the stream track, then integrated the progenitor backward including the full subhalo distribution to the desired start time. The model star cluster was then put at that position and integrated forward to create the model stream. A consequence is that the model leaves the dissolved progenitor in the ϕ 1 = [ -40 , -10] region of the stream. The precise final time progenitor location depends on the details of encounters with subhalos, so varies from run to run.</text>
<text><location><page_8><loc_8><loc_43><loc_48><loc_57></location>The C-19 model starts the star cluster at a time of 3 Gyr after the Big Bang. Starting the model star cluster at later times means that there are less subhalos to heat the stream and less time for subhalos to encounter the stream, leading to insufficient heating. The model does a reasonable job of accumulating the subhalo interactions of the central region of the stream, but stream ends are more coherent here than they would be in a full cosmological simulation, as seen in Figure 6.</text>
<text><location><page_8><loc_8><loc_13><loc_48><loc_43></location>The subhalos in the mass decade around 10 7 M ⊙ dominate the perturbations to the stream (Carlberg & Agler 2023; Carlberg et al. 2024). Subhalo stream encounters are sufficiently infrequent that there are significant differences of detail in the stream morphology depending on the detailed stream collision history, hence the phases of the orbits of the subhalos. Therefore a set of simulations is done with the subhalo distribution rotated 30 degrees from 0 to 330 degrees between simulations. The 4 streams that are significantly bifurcated around ϕ 1 = 0 in the CDM subhalo distribution are not included in Figure 10 because they lead to artificially wide stream widths which the currently measured C-19 sky distribution does not support. The widths of the model streams, calculated as a standard deviation, are shown in the lower panel of Figure 10. At 20 kpc distance 0.5 degrees corresponds to 170 pc or a FWHM of 410 pc. The narrowing of the streams along their length is an effect of orbital motion.</text>
<text><location><page_8><loc_8><loc_10><loc_48><loc_13></location>The C-19 model streams have considerable structure along their length, but the structure in these model</text>
<text><location><page_8><loc_55><loc_91><loc_56><loc_91></location>40</text>
<text><location><page_8><loc_55><loc_88><loc_56><loc_89></location>30</text>
<text><location><page_8><loc_55><loc_86><loc_56><loc_87></location>20</text>
<text><location><page_8><loc_55><loc_84><loc_56><loc_84></location>10</text>
<text><location><page_8><loc_55><loc_81><loc_56><loc_82></location>0</text>
<text><location><page_8><loc_55><loc_79><loc_56><loc_80></location>10</text>
<text><location><page_8><loc_53><loc_87><loc_54><loc_87></location>]</text>
<text><location><page_8><loc_53><loc_86><loc_54><loc_87></location>e</text>
<text><location><page_8><loc_53><loc_86><loc_54><loc_86></location>e</text>
<text><location><page_8><loc_53><loc_85><loc_54><loc_86></location>r</text>
<text><location><page_8><loc_53><loc_85><loc_54><loc_85></location>g</text>
<text><location><page_8><loc_53><loc_84><loc_54><loc_85></location>e</text>
<text><location><page_8><loc_53><loc_84><loc_54><loc_84></location>d</text>
<text><location><page_8><loc_53><loc_84><loc_54><loc_84></location>[</text>
<text><location><page_8><loc_53><loc_83><loc_54><loc_83></location>2</text>
<text><location><page_8><loc_59><loc_77><loc_60><loc_78></location>40</text>
<text><location><page_8><loc_65><loc_77><loc_66><loc_78></location>20</text>
<text><location><page_8><loc_71><loc_77><loc_72><loc_78></location>0</text>
<text><location><page_8><loc_77><loc_77><loc_78><loc_78></location>20</text>
<text><location><page_8><loc_83><loc_77><loc_84><loc_78></location>40</text>
<text><location><page_8><loc_90><loc_77><loc_90><loc_78></location>60</text>
<figure>
<location><page_8><loc_53><loc_61><loc_93><loc_76></location>
<caption>Figure 10. The model streams in C-19 stream coordinates (top) and the standard deviation of their spread about the mean position along the stream (bottom0. The subhalo distribution is rotated in steps of 30 degrees from 0 to 330 degrees for a dozen variants of the C-19 model stream. The plot does not show the streams which are bifurcated in the region around ϕ 1 = 0.</caption>
</figure>
<text><location><page_8><loc_52><loc_20><loc_92><loc_45></location>streams is significantly less than the two streams drawn from the full cosmological simulation shown in Figure 6. The movies of the two example streams CDM Low Mass Clusters show that large scale merging disperses the 'ends' of the streams until the main halo settles down around 7 Gyr. The primary halo is not subject to major mergers (by design) for the last 7 Gyr of the simulation so at late times is fairly well represented with the halo potential model used here. At the C-19 start time of 3 Gyr the dominant halo of the full cosmological simulation is about 50% less massive, which mainly effects the orbit, not the subhalo density. The C-19 model potential does not capture the larger scale potential flucutations due to accretion and merger buildup between 3 and 7 Gyr, but does include the subhalo effects. Thefore the C-19 model is a lower bound on the heating effects.</text>
<text><location><page_8><loc_52><loc_9><loc_92><loc_20></location>The line of sight velocities and their dispersion along the streams in the CDM subhalo distribution are shown in Figure 11. The velocity dispersion of C-19, 6.2 kms -1 , and its 1-sigma errors from Yuan et al. (2022) is shown in black. The quadrature summed mean of the model streams over ϕ 1 = [ -10 , 10] is 4.1 ± 1.1 kms -1 . The mean velocity dispersion (excluding the bifurcated</text>
<figure>
<location><page_9><loc_9><loc_61><loc_50><loc_92></location>
<caption>Figure 11. The line of sight velocity dispersion (top) and its distribution along the streams (bottom) for the CDM models of the C-19 stream. The four bifurcated streams are not plotted. The dots show the velocities of the observed C-19 stars. The observational estimate of the velocity dispersion and its one-sigma errors is shown in black.</caption>
</figure>
<text><location><page_9><loc_8><loc_28><loc_48><loc_48></location>streams) is within 1.2 sigma (combined) of the observed value. The same modeling procedure for a WDM (5.5 keV) model halo gives velocities shown in Figure 12. Its mean line of sight velocity dispersion of 3.1 ± 0.09 kms -1 , a 2.2 sigma difference from the observed value. The small increase in velocity dispersion in the WDM (5.5 keV) model is consistent with the cosmological simulation results (Carlberg et al. 2024). These simulations disfavor the WDM(5.5 keV) model. However given the significant scatter seen in the full cosmological simulation for clusters in the same orbital range, Figure 5, the result is not very strong and emphasizes that stronger conclusions require more than a single stream.</text>
<text><location><page_9><loc_8><loc_12><loc_48><loc_28></location>A model stream with no subhalos and no dwarf galaxies (designated c84) is shown in Figure 12. It has a velocity dispersion of 3.6 km s -1 in the ϕ 1 = [ -10 , 10] region. Including the known dwarf galaxies (c85) also gives a line of sight velocity dispersion of 3.6 km s -1 . The streams velocity dispersion calculated with 3 σ clipping are 1.41 and 1.75 km s -1 , respectively, indicating unusually strong wings in the velocity profile. The numbers of particles in this section of the two streams are about 20% of those in the models with subhalos which</text>
<text><location><page_9><loc_53><loc_72><loc_54><loc_72></location>1</text>
<figure>
<location><page_9><loc_53><loc_61><loc_94><loc_92></location>
<caption>Figure 12. The line of sight velocity distribution for the WDM (5.5 keV) models of the C-19 stream. The top two lines of points are for models with no subhalos and with no dwarf galaxies (purple, second line) and with known dwarf galaxies (yellow, top line). Streams that bifurcate over the plotted stream latitude range are not included.</caption>
</figure>
<text><location><page_9><loc_52><loc_45><loc_92><loc_48></location>is a consequence of the subalo perturbations gradually changing stream orbital phases.</text>
<section_header_level_1><location><page_9><loc_58><loc_43><loc_87><loc_44></location>4. DISCUSSION AND CONCLUSIONS</section_header_level_1>
<text><location><page_9><loc_52><loc_9><loc_92><loc_42></location>Errani et al. (2022) established, and we confirm, that the dissolution of a globular cluster in a smooth galactic halo cannot explain the the width and line of sight velocity dispersion of the C-19 stream. There are two generic solutions: either the C-19 stream was created from a stellar system hotter and larger than a globular cluster (Errani et al. 2022), or, the velocity dispersion of the stream was increased over the course of its orbital history. We have shown that an old, dissolved, globular cluster stream near apocenter in a galactic halo containing subhalos can explain the kinematics of the C-19 stream, under certain conditions. Detailed modeling of C-19 using a simplified model potential and the subhalos drawn from the cosmological simulations finds that heating C-19 requires the numbers of subhalos found in a CDM cosmology acting on the stream for ≃ 11 Gyr. The model finds that the numbers of WDM (5.5 kev) subhalos are insufficient. The subhalo heating of the stream leads to significant density and velocity variations along the stream, Figures 10 and 11, whereas a disrupted dwarf is smooth (Errani et al. 2022). The</text>
<text><location><page_10><loc_8><loc_89><loc_48><loc_91></location>currently known numbers of stars do not allow a reliable density profile measurement.</text>
<text><location><page_10><loc_8><loc_75><loc_48><loc_88></location>More generally, cosmological Milky Way-like simulations find that dissolved globular cluster streams that are hot and wide near apocenter, like C-19, become thin and cool streams as they orbit through pericenter. The streams discussed here are from progenitor globular clusters with masses below 2 × 10 4 M ⊙ and half mass radii in the 3-5 pc range, which have dissolution times in the galactic tidal field of 1-2 Gyr so few stars are added to the streams in the last 5 Gyr.</text>
<text><location><page_10><loc_8><loc_21><loc_48><loc_74></location>The width of C-19 as a globular cluster stream is not unique. Streams wider than 0.2 kpc containing globular clusters are listed in Table 1 drawing from the Mateu (2023) compendium (see also Bonaca & Price-Whelan (2024)). There are a total of 27 streams wider than 0.2 kpc, with 5 containing globular clusters and the C-19 stream. The widths are all from Ibata et al. (2021) and therefore use a uniform measurement approach. C-19 may be part of an ancient, extremely metal poor group of streams (Malhan et al. 2022). The wide stream of the distant cluster NGC5466 is excluded because it is likely associated with a merger remnant (Malhan et al. 2022) with a complex orbital history. NGC288 is a complex wide stream (Grillmair 2024). All the listed globular clusters are near the apocenters of their orbits (Baumgardt et al. 2019). These clusters orbit within a few kpc of the bulge, whereas C-19 is less eccentric and has a ≃ 8 kpc pericenter. Most of the 22 other wide streams have the abundance spread of a dissolved dwarf galaxy. There should of course be other wide streams from dissolved globular clusters at orbital apocenter but their distance and spread in position and velocity make them harder to find. The distribution of stream widths with current distance in the simulations is shown in Figure 13. About half of the simulated streams are within 30 kpc, whereas about 90% of the Galstreams compendium (Mateu 2023) is within 30 kpc. The radial distribution of streams will have some dependence on the radii at which globular clusters form within pre-galactic subhalos, although that dependence is fairly weak for globular cluster formation within a dwarf galaxy (Carlberg & Keating 2022). On the average the streams are physically wider with galactocentric distance, but approximately the same angular width, around 0.2 degrees.</text>
<text><location><page_10><loc_8><loc_10><loc_48><loc_21></location>The hot, wide streams at apocenter become thin, cool streams at pericenter as a result of stream orbital dynamics. Stars are unbound from their progenitor cluster at pericenter in a narrow range of radii, joining the stream with a spread in velocities. The subhalo perturbations to the stream velocities are more likely to occur near orbital pericenter where the subhalo density</text>
<table>
<location><page_10><loc_56><loc_75><loc_89><loc_88></location>
<caption>Table 1. Wide Globular Cluster Streams</caption>
</table>
<text><location><page_10><loc_52><loc_60><loc_92><loc_71></location>is highest. The velocity differences lead to differences in pericenter angle, orbital tilt, and angular momentum which causes the stream particles to spread apart near apocenter. The streams wider than a σ w of 100 pc in a CDM simulation are on the average about a factor of two hotter than in a WDM (5.5 keV), 7.8 ± 1.0 kms -1 as compared to 4.1 ± 1.6 kms -1 .</text>
<text><location><page_10><loc_52><loc_38><loc_92><loc_60></location>The hot, wide C-19 and thin, cool, GD-1 (Grillmair & Dionatos 2006) streams cover similar radial ranges in their orbits, with the difference in their widths being at least partially explained with C-19 being near apocenter and GD-1 near pericenter. The C-19 model required that it be orbiting in the subhalos of the galaxy for 11 Gyr, which is consistent with the age that being extremely metal poor implies. That is, for C-19 there is very little room for stream age-subhalo abundance degeneracy. GD-1 is less metal poor which opens up the possibility of more age-abundance degeneracy in the stream properties. Finding a common model that explains the internal kinematics of a set of streams is a goal as the data improves.</text>
<text><location><page_10><loc_52><loc_21><loc_92><loc_38></location>There is nearly a factor of two velocity dispersion difference between CDM and WDM(5.5 keV) wide streams in the inner halo. The cosmology dependence of long, thin streams, which are generally near pericenter, is present in the wings of the velocity distribution (Carlberg et al. 2024) with a relatively insensitive core Gaussian velocity spread. On the other hand, the cosmology dependence of wide streams, which are generally near apocenter, is in the velocity dispersion itself, which is easier to measure than the core-wing structure of pericenter streams.</text>
<figure>
<location><page_11><loc_9><loc_68><loc_47><loc_92></location>
<caption>Figure 13. Stream widths with distance. Dissolved clusters are circles, remnant clusters squares, blue CDM, red WDM. Most of the known globular cluster streams are within 30 kpc.</caption>
</figure>
<text><location><page_11><loc_52><loc_47><loc_92><loc_90></location>Adrian Jenkins, Carlos Frenk and Andrew Cooper provided invaluable advice and support for computing. This work used the DiRAC@Durham facility managed by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded by BEIS capital funding via STFC capital grants ST/K00042X/1, ST/P002293/1, ST/R002371/1 and ST/S002502/1, Durham University and STFC operations grant ST/R000832/1. DiRAC is part of the National e-Infrastructure. This work used high-performance computing facilities operated by the Center for Informatics and Computation in Astronomy (CICA) at National Tsing Hua University. This equipment was funded by the Ministry of Education of Taiwan, the National Science and Technology Council of Taiwan, and National Tsing Hua University. Computations were performed on the niagara supercomputer at the SciNet HPC Consortium. SciNet is funded by: the Canada Foundation for Innovation; the Government of Ontario; Ontario Research Fund - Research Excellence; and the University of Toronto. CSF acknowledges support by the European Research Council (ERC) through Advanced Investigator grant, DMIDAS (GA 786910). ARJ and CSF acknowledge support from STFC Consolidated Grant ST/X001075/1. APC acknowledges the support of the Taiwan Ministry of Education Yushan Fellowship and Taiwan National Science and Technology Council grant 112-2112-M-007-017-MY3.</text>
<text><location><page_11><loc_52><loc_40><loc_92><loc_46></location>Software: Gadget4: Springel et al. (2021), Amiga Halo Finder: (Gill et al. 2004; Knollmann & Knebe 2009), ROCKSTAR: (Behroozi et al. 2013), NumPy: (Harris et al. 2020).</text>
<text><location><page_11><loc_52><loc_37><loc_92><loc_40></location>Data Availability: Final snapshots, movies, images, and example scripts are at CDM Low Mass Clusters</text>
<section_header_level_1><location><page_11><loc_44><loc_33><loc_56><loc_34></location>REFERENCES</section_header_level_1>
<text><location><page_11><loc_8><loc_10><loc_47><loc_32></location>Baumgardt, H., Hilker, M., Sollima, A., & Bellini, A. 2019, MNRAS, 482, 5138, doi: 10.1093/mnras/sty2997 Behroozi, P. S., Wechsler, R. H., & Wu, H.-Y. 2013, ApJ, 762, 109, doi: 10.1088/0004-637X/762/2/109 Binney, J. 1977, MNRAS, 181, 735, doi: 10.1093/mnras/181.4.735 Binney, J., & Tremaine, S. 2008, Galactic Dynamics: Second Edition (Princeton University Press) Bonaca, A., Hogg, D. W., Price-Whelan, A. M., & Conroy, C. 2019, ApJ, 880, 38, doi: 10.3847/1538-4357/ab2873 Bonaca, A., & Price-Whelan, A. M. 2024, arXiv e-prints, arXiv:2405.19410, doi: 10.48550/arXiv.2405.19410 Carlberg, R. G. 2015, ApJ, 800, 133, doi: 10.1088/0004-637X/800/2/133</text>
<text><location><page_11><loc_52><loc_31><loc_88><loc_32></location>-. 2018, ApJ, 861, 69, doi: 10.3847/1538-4357/aac88a</text>
<text><location><page_11><loc_52><loc_10><loc_92><loc_30></location>Carlberg, R. G., & Agler, H. 2023, ApJ, 953, 99, doi: 10.3847/1538-4357/ace4be Carlberg, R. G., Jenkins, A., Frenk, C. S., & Cooper, A. P. 2024, arXiv e-prints, arXiv:2405.18522, doi: 10.48550/arXiv.2405.18522 Carlberg, R. G., & Keating, L. C. 2022, ApJ, 924, 77, doi: 10.3847/1538-4357/ac347e Errani, R., Navarro, J. F., Ibata, R., & Pe˜narrubia, J. 2022, MNRAS, 511, 6001, doi: 10.1093/mnras/stac476 Fukushige, T., & Heggie, D. C. 2000, MNRAS, 318, 753, doi: 10.1046/j.1365-8711.2000.03811.x Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016, A&A, 595, A1, doi: 10.1051/0004-6361/201629272</text>
<table>
<location><page_12><loc_8><loc_45><loc_47><loc_92></location>
</table>
<table>
<location><page_12><loc_52><loc_46><loc_92><loc_92></location>
</table>
</document>
[
{
"title": "ABSTRACT",
"content": "The C-19 star stream has the abundance characteristics of an unusually metal poor globular cluster but kinematically is uncharacteristically hot and wide for a cluster stream, having a line of sight velocity dispersion of 6 km s -1 and a 1-sigma width of 240 pc. We show that the tidal dissolution of an old, lower mass, globular cluster in a CDM galactic halo naturally creates a hot, wide stream currently near orbital apocenter. More generally, simulations show that hot streams, which are all near their orbital apocenter, become thin, cool streams near pericenter. Furthermore, the wide streams from a population of dissolved clusters in the simulations have a mean galactocentric radial velocity dispersion of 7.8 ± 1.0 kms -1 in a CDM cosmology but only 4.1 ± 1.6 kms -1 in a WDM (5.5 keV) simulation. A detailed C-19 model in a simplified Milky Way halo potential with a CDM subhalo population provides a lower bound to stream heating, finding that the stream develops a line of sight velocity dispersion of 4.1 ± 1.1 kms -1 , whereas WDM (5.5 keV) subhalos give 3.1 ± 0.1 km s -1 . Known dwarf galaxies alone provide negligible heating. There are five other currently known streams wider than 200 pc that contain a globular cluster, all near their orbital apocenter.",
"pages": [
1
]
},
{
"title": "C-19 and Hot, Wide, Star Streams",
"content": "Raymond G. Carlberg, 1 Rodrigo Ibata, 2 Nicolas F. Martin, 2 Else Starkenburg, 3 David S. Aguado, 4 Khyati Malhan, 5 Kim Venn, 6 and Zhen Yuan ( 袁 珍 ) 7 1 Department of Astronomy & Astrophysics, University of Toronto, Toronto, ON M5S 3H4, Canada 2 Universit'e de Strasbourg, CNRS, Observatoire astronomique de Strasbourg, UMR 7550, F-67000 Strasbourg, France 3 Kapteyn Astronomical Institute, University of Groningen, Landleven 12, 9747 AD Groningen, The Netherlands 4 Instituto de Astrof'ısica de Canarias, V'ıa L'actea, 38205 La Laguna, Tenerife, Spain 5 DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 Copenhagen, Denmark 6 Department of Physics and Astronomy, University of Victoria, Victoria, BC, V8W 3P2, Canada 7 School of Astronomy and Space Science, Nanjing University, Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210093, China",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "The C-19 stream of extremely low metallicity stars (Martin et al. 2022) was discovered (Ibata et al. 2021) in the Gaia data (Gaia Collaboration et al. 2016) using the STREAMFINDER algorithm (Malhan & Ibata 2018). High resolution spectroscopy (Martin et al. 2022; Yuan et al. 2022) of photometrically identified low metallicity stream candidates (Starkenburg et al. 2017) established that C-19 has a metal abundance of [Fe/H] = -3.4 with a 95% confidence [Fe/H] spread of less than 0.2 (Martin et al. 2022). Further investigation has found additional stream candidates giving a length of more than 50 degrees (Yuan et al. 2022; Viswanathan et al. 2024). The spread in stellar abundances is close to the detection limit and the abundance patterns are characteristic of globular cluster stars (Yuan et al. 2022). The abundances and extrapolated luminosity of approximately 10 4 L ⊙ led Martin et al. (2022) to argue that the progenitor was more likely to be a disrupted globular cluster than a disrupted dwarf galaxy, while considering that the 1-sigma width of 240 pc, and line of sight velocity dispersion of 6 km s -1 were more in keeping with a disrupted dwarf. A dynamical model of a disrupting dark matter dominated dwarf galaxy can account for the kinematic properties of the stream (Errani et al. 2022). The question addressed here is under what conditions can a globular cluster progenitor be compatible with the C-19 kinematic data. Most of the known streams in the compilation Galstreams (Mateu 2023) are thinner than C-19 with typical FWHM of 100 pc or less. On the other hand, simulations of globular cluster streams in realistic cosmological conditions find that the dark matter subhalos present in the galactic halo cause the velocity dispersion of stream stars to increase with time (Carlberg & Agler 2023). The average velocity distribution of long, thin streams has a core with extended wings (Carlberg 2018; Malhan et al. 2019). The width of the wings increases with the numbers of subhalos present in the galactic halo (Carlberg et al. 2024). The velocity at which stars join the stream determines the width of the low velocity core of a stream. Subsequent encounters with subhalos perturb stars into the extended wings of the velocity and width distribution. The growing dataset for the GD-1 stream provides tentative evidence for the existence of a core-wing velocity structure (Ibata et al. 2024; Valluri et al. 2024). The C-19 stream is on a nearly polar, inner halo orbit, 8-24 kpc, where the density of subhalos is relatively high, creating the circumstances for significant subhalo heating. The stream is composed of extremely metal poor, [Fe/H] ≃ -3.4, hence likely very old, stars in an inner halo orbit such that the moderate mass progenitor cluster likely dissolved relatively early and the stream has been subject to subhalo heating for much of the lifetime of the Milky Way. In § 2 we measure the range of widths and velocity dispersions of the streams that form in both CDM and WDM cosmological simulations from low mass globular clusters. We then use the cosmological simulations to motivate an evolving potential model for the Milky Way in which model star clusters on the orbit of C-19 are integrated and their velocity dispersions are measured to compare to C-19 in § 3. The results are discussed § 4 to assess the accuracy of the model and its ability to constrain the subhalo content of the Milky Way. Other currently known wide streams from globular clusters are briefly discussed along with the likelihood that there are other wide streams yet to be discovered.",
"pages": [
1,
2
]
},
{
"title": "2. LOW MASS STAR CLUSTERS IN COSMOLOGICAL SIMULATIONS",
"content": "The total stellar mass of C-19 is estimated to be in the range 3 × 10 3 to 10 4 M ⊙ (Martin et al. 2022; Yuan et al. 2022; Ibata et al. 2024). There is no visible progenitor, as expected for an old star cluster with a half mass relaxation time of 1-2 Gyr orbiting in the Milky Way (Gnedin & Ostriker 1997; Binney & Tremaine 2008; Errani et al. 2022). The C-19 stream has an angular momentum of about 3100 kpc- km s -1 , and is on a moderate eccentricity, e ≃ 0 . 45, nearly polar, orbit within 25 kpc of the galactic center (Martin et al. 2022). Two cosmological Milky Way-like simulations are seeded with low mass progenitor star clusters in the mass range 5-20 × 10 3 M ⊙ in a CDM cosmology and a WDM(5.5 keV) cosmology. The cosmological simulations of streams in Carlberg et al. (2024) contained progenitor clusters more massive than 4 × 10 4 M ⊙ , most of which survive to the present epoch. Other than the initial star cluster masses, the simulations here are iden- to those in Carlberg et al. (2024) and the same stream analysis procedures are used. We examine the simulation streams above a minimum angular momentum of 2000 kpc- km s -1 , which excludes those that interact strongly with the disk. The streams are selected to be within 60 kpc of the galactic center. The velocity dispersion and width are measured for the central region of the stream above a surface density threshold of 20 M ⊙ deg -2 which approximately defines the readily detected part of streams. Figure 1 shows the stream radial velocity dispersion, after removing the mean, against stream width for the 32 CDM streams and 24 WDM streams longer than 10 degrees, the minimum angle for our search and analysis algorithm. The same plot for the more massive progenitor star clusters in Carlberg et al. (2024) shows a similar correlation of velocity dispersion and width for the streams which are about twice the length of the streams here but show less difference between CDM and WDM. The distribution of stream stars projected onto the orbital plane varies from a narrow distribution at pericenter to extended 'feathers' at apocenter, as illustrated in Figure 2 of Carlberg (2015). The orbital variation is a consequence of the galactic tidal field accelerating stars away from the cluster in a narrow radial band near pericenter, the subsequent orbital spreading of the unbound stars at apocenter, then regrouping at pericenter again, somewhat similar to the orbital dependence of velocities of an accretion remnant (Helmi & White 1999). The variation of the radial velocity dispersion (galactocentric) and the average physical width with time for a few individual inner halo streams are shown in the left hand panel of Figure 2. The streams were selected to have angular momenta around 3000 kpc- km s -1 and velocity dispersion around 8 km s -1 , values comparable to those for C-19. The right hand panel shows the relation between the galactocentric σ r and the galactic radius of the center of the streams. The streams from clusters 154 and 179 are on moderately eccentric orbits with σ r varying from about 2-3 to 7-9 km s -1 as the stream orbits. Stream 86 is on a less eccentric orbit and shows smaller orbital variations in velocity dispersion and width. When near apocenter a stream's structure is often visibly less well organized than at pericenter, with offshoots and bifurcations. Figure 3 shows the radial velocity dispersion (left) and width (right) of all the CDM (top) and WDM (bottom) streams at the end of the simulation, over the 13-14.1 Gyr time interval, as a function of the fractional distance to apocenter. The 1.1 Gyr time range is just sufficient to capture the basic radial range of the most distant clusters in these samples, but not the full orbital complexity of the potentials. The point sizes are proportional to the orbital eccentricity. The figure shows that strong correlation of both σ r and σ width with the fractional distance to apocenter depends on orbital eccentricity, with a large scatter. The high eccentricity streams (larger point sizes) have the most extreme variation with apocenter distance and tend to have larger absolute velocity dispersions and widths. The velocity distribution within a stream usually has an approximately Gaussian core with extended wings (Carlberg et al. 2024). The kurtosis of these velocity distributions is not very stable, so we characterize the velocity distribution using clipped variances of the distributions. The radial velocity dispersion is calculated along the length of the stream with 3-sigma clipping. The 6-sigma clipped velocity dispersion is typically a factor of two larger for streams with σ r ≤ 3 kms -1 as shown in Figure 4. As the streams pass near pericenter the velocity distribution narrows but leaves behind extended wings which are not present when the velocity distribution widens around apocenter. Figure 6 shows two streams in the CDM simulation with a final time velocity dispersion above 6 km s -1 that orbit within 30 kpc. The particle plots usefully reveal the entire distribution of the dispersed cluster stars, but only the 20-30 degree segments at a galactic Y coordinate of approximately 20 kpc have sufficiently high sky density to be readily visible. The orbital history of these two clusters is shown in Figure 7. Cluster 154, shown in red, accretes onto the main halo in the first few Gyr of the simulation. Cluster 179, shown in green, is accreted around 7 Gyr, about halfway through the simulation. Dynamical friction in the aspherical halo (Binney 1977) draws the subhalo containing the cluster into the plane of the galaxy and reduces the orbital angular momentum of the cluster.",
"pages": [
2,
3,
4
]
},
{
"title": "3. MODELING THE C-19 STREAM",
"content": "The globular cluster streams in the cosmological simulations have radial velocity dispersions that range from about 2 to 9 km s -1 , with the highest velocity dispersion at orbital apocenter. The highest density part of C-19 is just past orbital apocenter (Martin et al. 2022; Viswanathan et al. 2024) and is similar to the high velocity streams in the simulations. Detailed modeling of a star cluster on the C-19 orbit is useful to understand the conditions under which a wide, hot stream develops. In particular the subhalo heating of a stream depends on the number of subhalos present in the primary halo and the length of time that the stream has been orbiting after tidal removal from of its natal subhalo. Our C-1 stream model is a simplification of the cosmological simulation to follow a single star cluster in a precomputed version of the galactic potential. The model star cluster has a mass of 1 × 10 4 M ⊙ , composed of 1 M ⊙ star particles using the same star cluster simulation as in the full cosmological model. The initial half-mass radius is set to approximately 4 pc for the models reported here. The star particles have a softening of 1 pc and are integrated with the Gadget4 code (Springel et al. 2021). Two body interactions are included with small random velocities calculated from the relaxation time (Spitzer 1987; Binney & Tremaine 2008) added to star particles within the virial radius, about 5 pc, every 5 Myr (Carlberg & Agler 2023). The tidal acceleration increases with distance from the cluster, pumping the stars into more distant orbits until they are swept away from the cluster (Meiron et al. 2021). The cluster-centric orbits become increasingly complex with distance(Fukushige & Heggie 2000). The cluster-centric radial coordinates of a sample of star particles are shown in Figure 8. The potential in which the star cluster orbits is drawn from the cosmological simulation. The primary mass component is a static, triaxial NFW potential, with mass, 9 . 2 × 10 11 M ⊙ , virial radius 200 kpc, scale of 14.4 kpc, and density triaxiality from the AHF halo finder (Gill et al. 2004; Knollmann & Knebe 2009) measurements at the final moment the simulation. The potential triaxiality is set to b/a=0.97, c/a=0.83 using Equation 2.72a of Binney & Tremaine (2008). A Miyamoto-Nagai galactic bulge-disk (Miyamoto & Nagai 1975) grows within the halo using the model of Carlberg et al. (2024). Although not important for the inner halo streams considered here, the LMC is included as a 1 . 8 × 10 11 M ⊙ (Garavito-Camargo et al. 2019) Hernquist sphere (Hernquist 1990), which means that the barycenter of the system is offset from the center of the primary halo. The framework could include a more detailed buildup of the primary halo and include a rotating galactic bar both of which would provide some additional stream heating. The current sparse C-19 data does not require more than the basic model. The positions and velocities of the population of subhalos above 10 6 M ⊙ as found in one of the full cosmological simulations at the chosen starting time is integrated in the combined Milky Way dark halo, disk and LMC potential. The subhalos do not interact with each other. The subhalos' masses decrease in time with an exponential decay in time, with a timescale of 5.5 Gyr, measured from the subhalo numbers with time in the simulations. The decrease in subhalo mass with time means that the subhalo N ( > M ) relation within the main halo declines approximately as measured in the fully dynamical cosmological simulations. Subhalos over the mass range 10 6 M ⊙ to 3 × 10 8 M ⊙ are included. Lower mass halos have little dynamical effect on the stream and higher mass subhalos are expected to contain visible dwarf galaxies. All subhalos are modeled as Hernquist spheres with a scale radius equal to r max from AHF. As the subhalos lose mass the radii are adjusted as M 1 / 3 to keep the characteristic density of each subhalo constant. In addition to the subhalos the 55 dwarf galaxies with kinematics (McConnachie 2012) are included. The dwarf galaxy dark halos are assigned maximal masses, M = 5 × 10 8 ( L/L ⊙ ) 0 . 65 M ⊙ , which roughly match the high mass end of the subhalo mass distribution function. The dwarf galaxy halo radii are set to a = 1 . 0( M/ 10 8 M ⊙ ) 0 . 23 kpc, on the basis of the fit to the r max mass relation of the subhalos in the simulations. All the subhalo orbits are precomputed and define spline coefficients for the positions of the masses which the star cluster calculation uses as an external potential. 100 The evolving potential model presented here does a reasonable job capturing the interactions of subhalos with the stream but misses the larger scale interactions that scatter the older, more distant ends of the stream over the halo as shown in Figure 6 for the cosmologi- cal simulation. Consequently the detailed stream model presented here is a lower bound on the effects of subhalos on the C-19 stream. The C-19 stream track is defined as passing through the approximate mean of the stars in the ϕ 1 = [ -10 , 10] region in Yuan et al. (2022). Specifically the progenitor orbit passes through RA=355.0, Dec=25.0 degrees, with a radial velocity of -195 km s -1 , proper motion of 1.25 and -2.8 milli-arcsec per year in RAcos(Dec) and Dec at a heliocentric distance of 20 kpc. The progenitor is then integrated in the potential of the MW plus LMC model. No optimization of the stream track coordinates or potential model was undertaken. The resulting orbit is adequately close to the current data for the purposes of this paper. The stream simulations placed the current epoch progenitor position at -10 degrees along the stream track, then integrated the progenitor backward including the full subhalo distribution to the desired start time. The model star cluster was then put at that position and integrated forward to create the model stream. A consequence is that the model leaves the dissolved progenitor in the ϕ 1 = [ -40 , -10] region of the stream. The precise final time progenitor location depends on the details of encounters with subhalos, so varies from run to run. The C-19 model starts the star cluster at a time of 3 Gyr after the Big Bang. Starting the model star cluster at later times means that there are less subhalos to heat the stream and less time for subhalos to encounter the stream, leading to insufficient heating. The model does a reasonable job of accumulating the subhalo interactions of the central region of the stream, but stream ends are more coherent here than they would be in a full cosmological simulation, as seen in Figure 6. The subhalos in the mass decade around 10 7 M ⊙ dominate the perturbations to the stream (Carlberg & Agler 2023; Carlberg et al. 2024). Subhalo stream encounters are sufficiently infrequent that there are significant differences of detail in the stream morphology depending on the detailed stream collision history, hence the phases of the orbits of the subhalos. Therefore a set of simulations is done with the subhalo distribution rotated 30 degrees from 0 to 330 degrees between simulations. The 4 streams that are significantly bifurcated around ϕ 1 = 0 in the CDM subhalo distribution are not included in Figure 10 because they lead to artificially wide stream widths which the currently measured C-19 sky distribution does not support. The widths of the model streams, calculated as a standard deviation, are shown in the lower panel of Figure 10. At 20 kpc distance 0.5 degrees corresponds to 170 pc or a FWHM of 410 pc. The narrowing of the streams along their length is an effect of orbital motion. The C-19 model streams have considerable structure along their length, but the structure in these model 40 30 20 10 0 10 ] e e r g e d [ 2 40 20 0 20 40 60 streams is significantly less than the two streams drawn from the full cosmological simulation shown in Figure 6. The movies of the two example streams CDM Low Mass Clusters show that large scale merging disperses the 'ends' of the streams until the main halo settles down around 7 Gyr. The primary halo is not subject to major mergers (by design) for the last 7 Gyr of the simulation so at late times is fairly well represented with the halo potential model used here. At the C-19 start time of 3 Gyr the dominant halo of the full cosmological simulation is about 50% less massive, which mainly effects the orbit, not the subhalo density. The C-19 model potential does not capture the larger scale potential flucutations due to accretion and merger buildup between 3 and 7 Gyr, but does include the subhalo effects. Thefore the C-19 model is a lower bound on the heating effects. The line of sight velocities and their dispersion along the streams in the CDM subhalo distribution are shown in Figure 11. The velocity dispersion of C-19, 6.2 kms -1 , and its 1-sigma errors from Yuan et al. (2022) is shown in black. The quadrature summed mean of the model streams over ϕ 1 = [ -10 , 10] is 4.1 ± 1.1 kms -1 . The mean velocity dispersion (excluding the bifurcated streams) is within 1.2 sigma (combined) of the observed value. The same modeling procedure for a WDM (5.5 keV) model halo gives velocities shown in Figure 12. Its mean line of sight velocity dispersion of 3.1 ± 0.09 kms -1 , a 2.2 sigma difference from the observed value. The small increase in velocity dispersion in the WDM (5.5 keV) model is consistent with the cosmological simulation results (Carlberg et al. 2024). These simulations disfavor the WDM(5.5 keV) model. However given the significant scatter seen in the full cosmological simulation for clusters in the same orbital range, Figure 5, the result is not very strong and emphasizes that stronger conclusions require more than a single stream. A model stream with no subhalos and no dwarf galaxies (designated c84) is shown in Figure 12. It has a velocity dispersion of 3.6 km s -1 in the ϕ 1 = [ -10 , 10] region. Including the known dwarf galaxies (c85) also gives a line of sight velocity dispersion of 3.6 km s -1 . The streams velocity dispersion calculated with 3 σ clipping are 1.41 and 1.75 km s -1 , respectively, indicating unusually strong wings in the velocity profile. The numbers of particles in this section of the two streams are about 20% of those in the models with subhalos which 1 is a consequence of the subalo perturbations gradually changing stream orbital phases.",
"pages": [
4,
5,
6,
7,
8,
9
]
},
{
"title": "4. DISCUSSION AND CONCLUSIONS",
"content": "Errani et al. (2022) established, and we confirm, that the dissolution of a globular cluster in a smooth galactic halo cannot explain the the width and line of sight velocity dispersion of the C-19 stream. There are two generic solutions: either the C-19 stream was created from a stellar system hotter and larger than a globular cluster (Errani et al. 2022), or, the velocity dispersion of the stream was increased over the course of its orbital history. We have shown that an old, dissolved, globular cluster stream near apocenter in a galactic halo containing subhalos can explain the kinematics of the C-19 stream, under certain conditions. Detailed modeling of C-19 using a simplified model potential and the subhalos drawn from the cosmological simulations finds that heating C-19 requires the numbers of subhalos found in a CDM cosmology acting on the stream for ≃ 11 Gyr. The model finds that the numbers of WDM (5.5 kev) subhalos are insufficient. The subhalo heating of the stream leads to significant density and velocity variations along the stream, Figures 10 and 11, whereas a disrupted dwarf is smooth (Errani et al. 2022). The currently known numbers of stars do not allow a reliable density profile measurement. More generally, cosmological Milky Way-like simulations find that dissolved globular cluster streams that are hot and wide near apocenter, like C-19, become thin and cool streams as they orbit through pericenter. The streams discussed here are from progenitor globular clusters with masses below 2 × 10 4 M ⊙ and half mass radii in the 3-5 pc range, which have dissolution times in the galactic tidal field of 1-2 Gyr so few stars are added to the streams in the last 5 Gyr. The width of C-19 as a globular cluster stream is not unique. Streams wider than 0.2 kpc containing globular clusters are listed in Table 1 drawing from the Mateu (2023) compendium (see also Bonaca & Price-Whelan (2024)). There are a total of 27 streams wider than 0.2 kpc, with 5 containing globular clusters and the C-19 stream. The widths are all from Ibata et al. (2021) and therefore use a uniform measurement approach. C-19 may be part of an ancient, extremely metal poor group of streams (Malhan et al. 2022). The wide stream of the distant cluster NGC5466 is excluded because it is likely associated with a merger remnant (Malhan et al. 2022) with a complex orbital history. NGC288 is a complex wide stream (Grillmair 2024). All the listed globular clusters are near the apocenters of their orbits (Baumgardt et al. 2019). These clusters orbit within a few kpc of the bulge, whereas C-19 is less eccentric and has a ≃ 8 kpc pericenter. Most of the 22 other wide streams have the abundance spread of a dissolved dwarf galaxy. There should of course be other wide streams from dissolved globular clusters at orbital apocenter but their distance and spread in position and velocity make them harder to find. The distribution of stream widths with current distance in the simulations is shown in Figure 13. About half of the simulated streams are within 30 kpc, whereas about 90% of the Galstreams compendium (Mateu 2023) is within 30 kpc. The radial distribution of streams will have some dependence on the radii at which globular clusters form within pre-galactic subhalos, although that dependence is fairly weak for globular cluster formation within a dwarf galaxy (Carlberg & Keating 2022). On the average the streams are physically wider with galactocentric distance, but approximately the same angular width, around 0.2 degrees. The hot, wide streams at apocenter become thin, cool streams at pericenter as a result of stream orbital dynamics. Stars are unbound from their progenitor cluster at pericenter in a narrow range of radii, joining the stream with a spread in velocities. The subhalo perturbations to the stream velocities are more likely to occur near orbital pericenter where the subhalo density is highest. The velocity differences lead to differences in pericenter angle, orbital tilt, and angular momentum which causes the stream particles to spread apart near apocenter. The streams wider than a σ w of 100 pc in a CDM simulation are on the average about a factor of two hotter than in a WDM (5.5 keV), 7.8 ± 1.0 kms -1 as compared to 4.1 ± 1.6 kms -1 . The hot, wide C-19 and thin, cool, GD-1 (Grillmair & Dionatos 2006) streams cover similar radial ranges in their orbits, with the difference in their widths being at least partially explained with C-19 being near apocenter and GD-1 near pericenter. The C-19 model required that it be orbiting in the subhalos of the galaxy for 11 Gyr, which is consistent with the age that being extremely metal poor implies. That is, for C-19 there is very little room for stream age-subhalo abundance degeneracy. GD-1 is less metal poor which opens up the possibility of more age-abundance degeneracy in the stream properties. Finding a common model that explains the internal kinematics of a set of streams is a goal as the data improves. There is nearly a factor of two velocity dispersion difference between CDM and WDM(5.5 keV) wide streams in the inner halo. The cosmology dependence of long, thin streams, which are generally near pericenter, is present in the wings of the velocity distribution (Carlberg et al. 2024) with a relatively insensitive core Gaussian velocity spread. On the other hand, the cosmology dependence of wide streams, which are generally near apocenter, is in the velocity dispersion itself, which is easier to measure than the core-wing structure of pericenter streams. Adrian Jenkins, Carlos Frenk and Andrew Cooper provided invaluable advice and support for computing. This work used the DiRAC@Durham facility managed by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded by BEIS capital funding via STFC capital grants ST/K00042X/1, ST/P002293/1, ST/R002371/1 and ST/S002502/1, Durham University and STFC operations grant ST/R000832/1. DiRAC is part of the National e-Infrastructure. This work used high-performance computing facilities operated by the Center for Informatics and Computation in Astronomy (CICA) at National Tsing Hua University. This equipment was funded by the Ministry of Education of Taiwan, the National Science and Technology Council of Taiwan, and National Tsing Hua University. Computations were performed on the niagara supercomputer at the SciNet HPC Consortium. SciNet is funded by: the Canada Foundation for Innovation; the Government of Ontario; Ontario Research Fund - Research Excellence; and the University of Toronto. CSF acknowledges support by the European Research Council (ERC) through Advanced Investigator grant, DMIDAS (GA 786910). ARJ and CSF acknowledge support from STFC Consolidated Grant ST/X001075/1. APC acknowledges the support of the Taiwan Ministry of Education Yushan Fellowship and Taiwan National Science and Technology Council grant 112-2112-M-007-017-MY3. Software: Gadget4: Springel et al. (2021), Amiga Halo Finder: (Gill et al. 2004; Knollmann & Knebe 2009), ROCKSTAR: (Behroozi et al. 2013), NumPy: (Harris et al. 2020). Data Availability: Final snapshots, movies, images, and example scripts are at CDM Low Mass Clusters",
"pages": [
9,
10,
11
]
},
{
"title": "REFERENCES",
"content": "Baumgardt, H., Hilker, M., Sollima, A., & Bellini, A. 2019, MNRAS, 482, 5138, doi: 10.1093/mnras/sty2997 Behroozi, P. S., Wechsler, R. H., & Wu, H.-Y. 2013, ApJ, 762, 109, doi: 10.1088/0004-637X/762/2/109 Binney, J. 1977, MNRAS, 181, 735, doi: 10.1093/mnras/181.4.735 Binney, J., & Tremaine, S. 2008, Galactic Dynamics: Second Edition (Princeton University Press) Bonaca, A., Hogg, D. W., Price-Whelan, A. M., & Conroy, C. 2019, ApJ, 880, 38, doi: 10.3847/1538-4357/ab2873 Bonaca, A., & Price-Whelan, A. M. 2024, arXiv e-prints, arXiv:2405.19410, doi: 10.48550/arXiv.2405.19410 Carlberg, R. G. 2015, ApJ, 800, 133, doi: 10.1088/0004-637X/800/2/133 -. 2018, ApJ, 861, 69, doi: 10.3847/1538-4357/aac88a Carlberg, R. G., & Agler, H. 2023, ApJ, 953, 99, doi: 10.3847/1538-4357/ace4be Carlberg, R. G., Jenkins, A., Frenk, C. S., & Cooper, A. P. 2024, arXiv e-prints, arXiv:2405.18522, doi: 10.48550/arXiv.2405.18522 Carlberg, R. G., & Keating, L. C. 2022, ApJ, 924, 77, doi: 10.3847/1538-4357/ac347e Errani, R., Navarro, J. F., Ibata, R., & Pe˜narrubia, J. 2022, MNRAS, 511, 6001, doi: 10.1093/mnras/stac476 Fukushige, T., & Heggie, D. C. 2000, MNRAS, 318, 753, doi: 10.1046/j.1365-8711.2000.03811.x Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016, A&A, 595, A1, doi: 10.1051/0004-6361/201629272",
"pages": [
11
]
}
]
2024arXiv241100100C
https://arxiv.org/pdf/2411.00100.pdf
<document>
<section_header_level_1><location><page_1><loc_6><loc_82><loc_94><loc_87></location>The formation and stability of a cold disc made out of stellar winds in the Galactic Centre</section_header_level_1>
<text><location><page_1><loc_29><loc_80><loc_82><loc_81></location>1,2, 3,4 5 6,7,8</text>
<text><location><page_1><loc_18><loc_78><loc_82><loc_81></location>Diego Calderón ⋆ , Jorge Cuadra , Christopher M. P. Russell , Andreas Burkert , Stephan Rosswog 1,9 , and Mayura Balakrishnan 10</text>
<unordered_list>
<list_item><location><page_1><loc_11><loc_75><loc_69><loc_76></location>1 Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, 21029 Hamburg, Germany</list_item>
<list_item><location><page_1><loc_11><loc_73><loc_69><loc_75></location>2 Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Straße 1, 85748 Garching, Germany e-mail: calderon@mpa-garching.mpg.de</list_item>
<list_item><location><page_1><loc_11><loc_72><loc_90><loc_73></location>3 Departamento de Ciencias, Facultad de Artes Liberales, Universidad Adolfo Ibáñez, Av. Padre Hurtado 750, Viña del Mar, Chile</list_item>
<list_item><location><page_1><loc_11><loc_70><loc_79><loc_72></location>4 Millennium Nucleus on Transversal Research and Technology to Explore Supermassive Black Holes (TITANS)</list_item>
<list_item><location><page_1><loc_11><loc_69><loc_66><loc_70></location>5 Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA</list_item>
<list_item><location><page_1><loc_11><loc_68><loc_78><loc_69></location>6 Universitäts-Sternwarte, Ludwig-Maximilians-Universität München, Scheinerstr. 1, 81679 Munich, Germany</list_item>
<list_item><location><page_1><loc_11><loc_67><loc_71><loc_68></location>7 Max-Planck Institute for Extraterrestrial Physics, Giessenbacherstr. 1, 85748 Garching, Germany</list_item>
<list_item><location><page_1><loc_11><loc_66><loc_59><loc_67></location>8 Excellence Cluster ORIGINS, Boltzmannstrasse 2, 85748 Garching, Germany</list_item>
<list_item><location><page_1><loc_11><loc_65><loc_83><loc_66></location>9 The Oskar Klein Centre, Department of Astronomy, AlbaNova, Stockholm University, SE-106 91 Stockholm, Sweden</list_item>
<list_item><location><page_1><loc_10><loc_63><loc_72><loc_64></location>10 Department of Astronomy, University of Michigan, 1085 S. University, Ann Arbor, MI 48109, USA</list_item>
</unordered_list>
<text><location><page_1><loc_10><loc_61><loc_45><loc_62></location>Received November 4, 2024; accepted November 4, 2024</text>
<section_header_level_1><location><page_1><loc_46><loc_58><loc_54><loc_59></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_10><loc_52><loc_90><loc_57></location>Context. The reported discovery of a cold ( ∼ 10 4 K) disc-like structure around the super-massive black hole at the centre of the Milk Way, Sagittarius A* (Sgr A*), has challenged our understanding of the gas dynamics and thermodynamic state of the plasma in its immediate vicinity. State-of-the-art simulations do not agree on whether or not such a disc can indeed be a product of the multiple stellar wind interactions of the mass-losing stars in the region.</text>
<text><location><page_1><loc_10><loc_49><loc_90><loc_52></location>Aims. This study aims to constrain the conditions for the formation of a cold disc as a natural outcome of the system of the mass-losing stars orbiting around Sgr A*, to investigate if the disc is a transient or long-lasting structure, and to assess the validity of the model through direct comparisons with observations.</text>
<text><location><page_1><loc_10><loc_45><loc_90><loc_49></location>Methods. We conduct a set of hydrodynamic simulations of the observed Wolf-Rayet (WR) stars feeding Sgr A* using the finitevolume adaptive mesh-refinement code Ramses. We focus, for the first time, on the impact of the chemical composition of the plasma emanating from the WR stars.</text>
<text><location><page_1><loc_10><loc_39><loc_90><loc_45></location>Results. The simulations show that the chemical composition of the plasma a ff ects the radiative cooling enough to impact the properties of the medium such as density and temperature and, as a consequence, the rate at which the material inflows onto Sgr A*. We demonstrated that the formation of a cold disc from the stellar winds is possible for certain chemical compositions that are consistent with the current observational constraints. However, even in such a case, it is not possible to reproduce the reported properties of the observed disc-like structure, namely its inclination and hydrogen recombination line fluxes.</text>
<text><location><page_1><loc_10><loc_37><loc_90><loc_39></location>Conclusions. Weconclude that the stellar winds on their own cannot form the cold disc around Sgr A* inferred from the observations. Either relevant ingredients are still missing in the model, or the interpretation of the observed data needs to be revised.</text>
<text><location><page_1><loc_10><loc_35><loc_82><loc_36></location>Key words. accretion, accretion discs - Galaxy: centre - hydrodynamics - Stars: winds, outflows - Stars: Wolf-Rayet</text>
<section_header_level_1><location><page_1><loc_6><loc_31><loc_18><loc_32></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_6><loc_12><loc_49><loc_30></location>The Galactic Centre (GC) hosts the closest super-massive black hole to us, Sagittarius A* (Sgr A*; see Genzel et al. 2010, for a review). Unlike the black holes present in active galactic nuclei, Sgr A* is very underluminous. It does not seem to be currently accreting much material, or to have a standard accretion disc around it (Yuan & Narayan 2014). Murchikova et al. (2019) detected however a disc-like structure around Sgr A*. Using the 1.3-millimetre recombination line H30 α they observed a doublepeaked emission line with full velocity width of ∼ 2200 km s -1 . The centre of the emission coincides with Sgr A* and extends up to 0.11 '' ( ∼ 4.4 × 10 -3 pc) to both the redshifted and blueshifted sides. They interpreted this feature as a rotating disc of mass 10 -5 -10 -4 M ⊙ . Yusef-Zadeh et al. (2020) also reported the presence of broad hydrogen recombination lines, including</text>
<text><location><page_1><loc_51><loc_20><loc_94><loc_32></location>the double-peaked H30 α line, at the position of Sgr A* using the Very Large Array. However, they interpreted the signatures as a jet emanating from the central region. So far, there has not been any detection in the near infrared, despite many observations. Ciurlo et al. (2021) reports an upper limit for Br γ , which is two orders of magnitude below the extrapolation from the reported H30 α flux. Currently, there is no consensus on how such observations can be reconciled or from where the observed cold material comes from.</text>
<text><location><page_1><loc_51><loc_10><loc_94><loc_20></location>The gaseous environment around Sgr A* is dominated by the outflows from around 30 Wolf-Rayet (WR) stars, all located within a fraction of a parsec of the black hole (Paumard et al. 2006; Martins et al. 2007). At such close distances, the winds interact strongly in shocks that thermalise them and create a hot and di ff use X-ray emitting plasma (Quataert 2004). However, given the relatively low velocity of some of the outflows (450600 km s -1 ; Martins et al. 2007), the resulting plasma can be</text>
<text><location><page_2><loc_6><loc_33><loc_49><loc_93></location>prone to radiative cooling and form dense, cold clumps (Cuadra et al. 2005; Calderón et al. 2016), which end up being embedded in the di ff use, hot plasma ( ∼ 10 7 K; Bagano ff et al. 2003). The complex interplay between the stellar winds around Sgr A* has been the subject of hydrodynamic simulations using a variety of numerical techniques. With smoothed particle hydrodynamics (SPH) simulations, Cuadra et al. (2006) showed that, if most of the slow-wind stars are located in a relatively compact stellar disc, many cold clumps form and quickly coalesce in a cold gaseous disc of radius ∼ 1 '' . However, using a stellar distribution closer to the one observed, Cuadra et al. (2008, 2015) showed that no conspicuous disc appears over the 1800 yr time-span of their models. Calderón et al. (2020b) performed finite-volume hydrodynamic simulations on a Cartesian grid of the same system, which are better suited to model shocks, the multi-phase medium, and subsequent cooling, and found that the formation of a cold disc is indeed possible. According to their model, the stellar wind of the star IRS 33E, which is fairly dense ( ∼ 10 -5 M ⊙ yr -1 ) and slow (450 km s -1 ; Martins et al. 2007) interacts with the medium creating a shell that is dense enough to radiate quickly its thermal energy and break into denser and smaller structures. These pieces manage to fall inwards forming a cold ( ∼ 10 4 K) disc with a total mass of ∼ 5 × 10 -3 M ⊙ within a simulation time of 3500 yr. Based on this result, Ciurlo et al. (2021) found that the properties of the modelled disc are consistent with the non-detection in Br γ . Nevertheless, this scenario has not been confirmed by analogous grid-based simulations. Ressler et al. (2020) revisited the same system using the same numerical approach, although a di ff erent code: athena++ (Stone et al. 2020), and found no disc formation even extending their simulation time up to 9000 yr. Building on this model, Solanki et al. (2023) also studied this system through hydrodynamic modelling but encompassing a larger region and evolving it for much longer timescales. They included, for the first time, the presence of the circumnuclear disc (CND), an observed gaseous structure located between 1 . 5-3 . 0 pc (Becklin et al. 1982; Genzel 1989) and with a total mass of 3-4 × 10 4 M ⊙ (Dinh et al. 2021; James et al. 2021). This work showed that the formation of a cold disc is possible on long timescales ( ≳ 300,000 yr) due to the interaction of the innermost boundary of the CND and the WR stellar winds, which results in the transport of CND material to smaller scales. However, the authors warned that the lack of certain physical mechanisms in the model on such timescales (e.g. magnetic fields, supernovae, thermal conduction) might a ff ect the robustness of this result. Thus, the exact formation process of the observed cold disc still remains unknown.</text>
<text><location><page_2><loc_6><loc_10><loc_49><loc_33></location>Akey factor driving the formation of clumps and a disc in the model of Calderón et al. (2020b) is that radiative cooling allows the gas to get rid of its energy e ffi ciently. The strength of this process depends on the chemical composition of the gas, which is highly unusual in the GC given its origin as winds from evolved massive stars. In this work, we explore for the first time the impact of the radiative cooling through studying specific chemical compositions: Solar with 1 Z ⊙ , 3 Z ⊙ , and 5 Z ⊙ . More importantly, we develop a more realistic chemical setup that considers the atmospheric abundances for the di ff erent WR sub-types present in the region, in order to follow the thermodynamic evolution of the gas in a more appropriate manner. Our results show that the formation of the disc is indeed determined by the composition of the gas. However, the properties of such a disc do not agree with the observations when considering realistic wind compositions compatible with current observational constraints. We also improve the numerical setup in order to better assess the disc orientation and more directly compare our results to the work of</text>
<text><location><page_2><loc_51><loc_91><loc_94><loc_93></location>Ressler et al. (2018, 2020) as well as with SPH models (Cuadra et al. 2008, 2015; Russell et al. 2017).</text>
<text><location><page_2><loc_51><loc_71><loc_94><loc_90></location>This article is organised as follows: in Section 2, we present the numerical approach, the setup and the models investigated, Section 3 presents and describes the results of the numerical simulations. In Section 4, we present the synthetic observables obtained through post-processing the models and contrast them with observations. We compare our grid-based models with SPH models in Section 5. In Section 6, we present an analytic analysis of the stability of the observed cold disc. Section 7 discusses our findings and uncertainties in the parameters of our models. Finally, Section 8 presents conclusions and final remarks. Throughout this paper we use the mass and distance to Sgr A* of 4 . 3 × 10 6 M ⊙ and 8 . 33 kpc, respectively (Gillessen et al. 2017; GRAVITY Collaboration et al. 2019, 2021), so that 1 arcsec corresponds to a length of ∼ 0.04 pc ≈ 10 5 RSch, being RSch the Schwarzschild radius.</text>
<section_header_level_1><location><page_2><loc_51><loc_68><loc_72><loc_69></location>2. Numerical simulations</section_header_level_1>
<section_header_level_1><location><page_2><loc_51><loc_66><loc_61><loc_67></location>2.1. Equations</section_header_level_1>
<text><location><page_2><loc_51><loc_58><loc_94><loc_65></location>The simulations were performed using the adaptive-mesh refinement (AMR) hydrodynamic code Ramses (Teyssier 2002). This code uses a second-order Godunov method with a shockcapturing scheme to solve the Euler equations in their conservative form, i.e.</text>
<formula><location><page_2><loc_59><loc_55><loc_94><loc_57></location>∂ρ ∂ t + ∇ · ( ρ u ) = 0 , (1)</formula>
<formula><location><page_2><loc_56><loc_52><loc_94><loc_54></location>∂ ∂ t ( ρ u ) + ∇ · ( ρ uu ) = -∇ p -ρ ∇ ϕ, (2)</formula>
<formula><location><page_2><loc_51><loc_49><loc_94><loc_52></location>∂ ∂ t ( ρ e ) + ∇ · " ρ u e + p ρ !# = -ρ u · ∇ ϕ -Q -tot , (3)</formula>
<formula><location><page_2><loc_55><loc_46><loc_94><loc_48></location>∂ ∂ t ( ρ si ) + ∇ · ( ρ si u ) = 0 (4)</formula>
<text><location><page_2><loc_51><loc_32><loc_94><loc_45></location>where ( ρ, u , P ) are the primitive hydrodynamic variables: density, velocity, and pressure, respectively. The set of quantities si correspond to tracer scalar fields that are advected with the fluid whose usage we introduce in Section 2.3. The sink terms on the right-hand side correspond to the e ff ects of the gravitational potential ϕ = ϕ ( x ), assumed to be time independent, and the total radiative loses due to optically-thin radiative cooling Q -tot = Q -tot ( ρ, T , X i), with x the position vector, T the fluid temperature, and X i the chemical composition of the fluid. The total specific energy density e is given by</text>
<formula><location><page_2><loc_51><loc_29><loc_94><loc_31></location>e = 1 2 u · u + p ( γ -1) ρ , (5)</formula>
<text><location><page_2><loc_51><loc_23><loc_94><loc_28></location>where γ is the adiabatic index that is set to 5 / 3. Additionally, we consider the fluid can be described as an ideal gas so that the temperature can be calculated through P = ( ρ/µ ) k B T , being µ the mean molecular weight and k B the Boltzmann constant.</text>
<section_header_level_1><location><page_2><loc_51><loc_20><loc_66><loc_21></location>2.2. Numerical setup</section_header_level_1>
<text><location><page_2><loc_51><loc_10><loc_94><loc_19></location>The model considers the system of WR stars blowing stellar winds while they move on their observed Keplerian orbits around Sgr A*. The setup is analogous to our previous work (Calderón et al. 2020b). The central black hole gravitational field is modelled as a point mass of 4 . 3 × 10 6 M ⊙ (GRAVITY Collaboration et al. 2019, 2021). The stars are simulated as test particles that only feel the gravitational pull of Sgr A*. Their initial</text>
<table>
<location><page_3><loc_16><loc_80><loc_84><loc_91></location>
<caption>Table 1. Atmospheric mass fractions (in percentage) and mean molecular weights.</caption>
</table>
<text><location><page_3><loc_16><loc_73><loc_84><loc_80></location>Notes. The Solar abundances were taken from Lodders (2003). The abundances for the three WR sub-types correspond to the compilation by Russell et al. (2017) based on previous studies (Herald et al. 2001; Crowther 2007; Onifer et al. 2008). Column 1: chemical mixture. Columns 2-7: hydrogen, helium, carbon, nitrogen, oxygen, and rest of the metals mass fractions, respectively. Column 8: mean molecular weigh assuming full ionisation.</text>
<table>
<location><page_3><loc_6><loc_59><loc_49><loc_68></location>
<caption>Table 2. Simulation runs and parameters.</caption>
</table>
<text><location><page_3><loc_6><loc_54><loc_49><loc_58></location>Notes. Column 1: simulation ID. Column 2: chemical composition of the winds. Column 3: Riemann solver. Column 4: Slope limiter. Column 5: whether or not the final state of the system shows the presence of a cold disc around Sgr A*.</text>
<text><location><page_3><loc_6><loc_29><loc_49><loc_51></location>position and velocity vectors are set so that they move on the orbits constrained by observations (Paumard et al. 2006; Cuadra et al. 2008; Gillessen et al. 2019; von Fellenberg et al. 2022). The stellar winds in the simulation are modelled following the procedure by Lemaster et al. (2007), which has been validated and used in our previous work (Calderón et al. 2020a,b). This consists in defining a 'masked region" around each star that is a spherical volume where the hydrodynamic variables are reset in each timestep, in order to reproduce the free wind expansion solution of a spherical wind with certain mass-loss rate ˙ M w, terminal velocity V w, and temperature T w. For these quantities we used the values constrained through modelling of the infrared spectra (Martins et al. 2007; Cuadra et al. 2008). The wind temperature was set to the lowest temperature allowed in the simulation, T w = 10 4 K which is determined by the strong ultraviolet radiation produced by the hundreds of massive stars in the region.</text>
<text><location><page_3><loc_6><loc_10><loc_49><loc_29></location>The simulations were run in a Cartesian grid in a cubic domain of side 40 '' ( ∼ 1.6 pc) with outflow boundary conditions (zero gradients). We used the exact Riemann solver (e.g. Toro 2009) combined with the MinMod slope limiter, as these choices allow the modelling of hydrodynamic instabilities, which otherwise may be quenched due to numerical di ff usion. We investigated other choices such as the Harten-Lax-van Leer-Contact (HLLC; Toro et al. 1994) and / or the MonCen slope limiter but they produced unwanted numerical artifacts such as spurious oscillations. For a careful analysis of these choices we refer the reader to the Appendix A. Regarding the numerical resolution of our models, the coarse resolution was 64 3 cells, allowing adaptive refinement of four levels, i.e. e ff ectively ∆ x ≈ 0 . 0382 '' . However, the regions around the stars allowed an extra level of refinement ( ∆ x ≈ 0 . 0195 '' ). Additionally, the vicinity around the</text>
<text><location><page_3><loc_51><loc_56><loc_94><loc_70></location>central black hole has a fixed nested grid with eight refinement levels above the coarse resolution ( ∆ x ≈ 2 . 44 mas). At the location of Sgr A*, we defined a spherical region where the hydrodynamic variables are reset to low values of density and pressure at rest, in order to avoid artificial accumulation of material. The refinement strategy is based on density gradients on top of the geometric criteria previously defined. In order to explore the role of the AMR potentially a ff ecting the results we tested di ff erent values of the smoothing parameter that controls the refinement in transitions between refinement levels. We tested quadrupling the smoothing parameter and the results remained unchanged 1 .</text>
<section_header_level_1><location><page_3><loc_51><loc_52><loc_59><loc_53></location>2.3. Models</section_header_level_1>
<text><location><page_3><loc_51><loc_21><loc_94><loc_51></location>The main parameter explored in this work is the optically-thin radiative cooling function. This choice is determined by specifying the chemical composition of the fluid. Unfortunately, the metallicity of the young stars in the GC is still poorly known, yet it has been argued that it should be higher than Solar ( Z = 2 -3 Z ⊙ ; Genzel et al. 2010). Past numerical works have considered that the composition of the gas correspond to Solar abundances but with metallicity three times the Solar value, Z = 3 Z ⊙ (Cuadra et al. 2008; Calderón et al. 2016; Ressler et al. 2018, 2020; Solanki et al. 2023). However, more than the 'bulk' metallicity of the stars, the most appropriate composition choice would be the abundances of the WR stellar atmospheres, which normally di ff er significantly from Solar (Herald et al. 2001; Onifer et al. 2008). Although Ressler et al. (2018, 2020) used chemical compositions lacking hydrogen, the rest of the element abundances remained unchanged relative to Solar with 3 Z ⊙ . As a result, the cooling function varied at low temperature ( < 10 5 K) but at higher temperatures it remains unchanged. Moreover, the composition choice not only determines the cooling function but also the ion mean molecular weight as well as the electron to proton number densities n e / n p. These quantities are taken into account in the sink term in the energy equation (see equation 3), and following Schure et al. (2009) can be expressed as follows</text>
<formula><location><page_3><loc_51><loc_16><loc_94><loc_19></location>Q -i ( ρ, Ti , Xi ) = n 2 p n e n p ! i Λ i ( Ti ) , (6)</formula>
<text><location><page_4><loc_6><loc_84><loc_49><loc_93></location>where the subscript i stands for a given chemical abundance. Bear in mind that di ff erent chemical compositions also correspond to di ff erent temperature Ti , as this is obtained through the ideal gas expression that makes uses of the mean molecular weight of the fluid. In the general case, if we consider a fluid element composed of N mixtures of abundances, the total radiative loses will be given as a summation, i.e.</text>
<formula><location><page_4><loc_6><loc_77><loc_49><loc_81></location>Q -tot ( ρ, Ti , Xi ) = n 2 p P N i = 1 si GLYPH<18> n e n p GLYPH<19> i Λ i ( Ti ) P N i = 1 si , (7)</formula>
<text><location><page_4><loc_6><loc_60><loc_49><loc_74></location>where we have expressed the total radative cooling as a linear combination of the mixtures. Notice that we have introduced the passive scalar fields si , as we used them to quantify the fraction of a given chemical composition. In this work, we introduced three types of compositions that represent the WR subtypes based on their atmosphere abundances. In the region where the winds are generated, the corresponding passive scalar is set to 1 while the rest are set to 0. By doing so, it is possible to identify how much material of a given cell is supplied by which sub-group of WR stars. The mass fractions of each mixture are shown in Table 1.</text>
<text><location><page_4><loc_6><loc_28><loc_49><loc_60></location>In this work, we investigated five models with di ff erent compositions: Solar composition with metallicities Z ⊙ , 3 Z ⊙ , and 5 Z ⊙ , a mixture of three compositions based on the spectroscopic constraints of the sub-types of WR stars, and a variation of the latter one motivated by the uncertainty in the WR stellar atmospheres and, specifically on the H fraction in them that is set to X H = 40% (see Section 7.2, for a discussion). Figure 1 shows the cooling function Λ = Λ ( T ) as a function of temperature for different chemical mixtures. Each curve was calculated as a linear combination of the abundance of a given element and its contribution to the total radiative cooling. The values of the composition per element are shown in Table 1. The contribution of each element to the total cooling was taken from the plasma models developed by Štofanová et al. (2021). The cooling functions corresponding to WR subtypes compositions: WC8-9, WN5-7, and WN8-9 / Ofpe are shown with dashed blue, dotted orange, and dotted-dashed green lines, respectively. The Solar and Solar with 3 Z ⊙ are represented with thin and thick solid black lines, respectively. Furthermore, we added the cooling function used in previous works by Ressler et al. (2018, 2020) as a dotted-dashed red line. Here it can be observed that a Solar composition with 3 Z ⊙ (the typical used value) is between the range of the di ff erent WR compositions but the subtypes WC89 and WN89 / Ofpe are about a factor two higher.</text>
<text><location><page_4><loc_6><loc_10><loc_49><loc_28></location>The list of the runs investigated in this work are shown in Table 2. The initial setup was identical to our previous work (Calderón et al. 2020b), i.e. the medium across the whole domain was set to constant and low-enough values of density ρ = 10 -24 g cm -3 and pressure P = γ -1 ρ c 2 s,f with c s,f = 10 km s -1 , so that the stellar winds do not encounter impediments and fill quickly the domain. All models were run for a total of 3,500 yr but setting the starting state of the system in the past, so that the final state of the simulations corresponds to the current state of the system. This is achieved by integrating the current position and velocity vectors of the stars back in time, assuming that they have followed purely Keplerian orbits within this timescale due to the gravitational field of Sgr A* (Cuadra et al. 2008; Ressler et al. 2018; Calderón et al. 2020b).</text>
<figure>
<location><page_4><loc_51><loc_69><loc_92><loc_91></location>
<caption>Fig. 1. Comparison of cooling functions Λ ( T ) for di ff erent chemical abundances. The thin and thick lines show the radiative cooling for Solar and three times Solar abundances, respectively. The dashed blue, dotted orange, and dot-dashed green lines represent the cooling functions for atmosphere abundances corresponding to WR sub-types: WC8-9, WN5-7, WN8-9 and Ofpe / WN9 based on the compositions compiled shown in Table 1. The contributions of each element to the radiative cooling were taken from the plasma models by Štofanová et al. (2021). Additionally, the cooling function used in Ressler et al. (2018) is shown as a thick black line.</caption>
</figure>
<section_header_level_1><location><page_4><loc_51><loc_52><loc_59><loc_53></location>3. Results</section_header_level_1>
<text><location><page_4><loc_51><loc_38><loc_94><loc_51></location>We proceed to describe the evolution and final state of the simulations at t = 0 that corresponds to the present time. The runs with Solar composition, but varying metallicities, were used for understanding the impact of increasing and decreasing the e ff ect of radiative cooling in general. However, since they do not represent realistic compositions we opted not to discuss them in detail here. Instead, we chose to focus on the more physically motivated models, i.e. WR_f07 and WR_f1 which we also refer as fiducial and enhanced , respectively. For completeness, we give a brief description of the models A1, A2, and A3 in Appendix B.</text>
<section_header_level_1><location><page_4><loc_51><loc_34><loc_65><loc_35></location>3.1. Hydrodynamics</section_header_level_1>
<text><location><page_4><loc_51><loc_10><loc_94><loc_33></location>The simulation evolution of all models is analogous, especially in the initial phases. At t = -3500 yr, the stars begin to orbit around Sgr A* that is located at the centre of the domain, while their winds quickly fill the whole computational domain. After ∼ 500 yr ( t = -3000 yr), the system reaches a quasi-steady state. At this point, the domain is full of di ff use ( ∼ 10 -22 -10 -21 g cm -3 ) and hot plasma ( ∼ 10 7 -10 8 K) due to the shocked stellar winds and their interactions. This is illustrated in Figure 2 that shows density maps across di ff erent simulation times. The maps show projected density fields along the line-of-sight direction ( z axis) weighted by density, i.e. R ρ 2 dz / R ρ dz , in order to highlight the densest regions with the highest resolution. The top panels show two models at t = -700 yr, respectively. In them, it is possible to observe the complex structure developed due to the stellar wind interactions such as bow shocks, instabilities, and dense clumps as a result of condensation through radiative cooling. Left- and right-hand side panels display models with di ff erent chemical compositions WR_f07 and WR_f1, respec-</text>
<formula><location><page_5><loc_25><loc_88><loc_75><loc_94></location>10 -24 10 -22 10 -20 10 -18 ∫ ρ 2 dz/ ∫ ρdz (g cm -3 )</formula>
<figure>
<location><page_5><loc_8><loc_27><loc_50><loc_86></location>
</figure>
<figure>
<location><page_5><loc_50><loc_27><loc_92><loc_86></location>
<caption>Fig. 2. Comparison of the simulations with di ff erent chemical abundances at two di ff erent simulation times. The panels show projected density maps weighed by density along the z axis, i.e. R ρ 2 dz / R ρ dz , which is parallel to the line of sight. Top and bottom panels show the systems at t = -0 . 7 kyr and t = 0 (present), respectively. Left- and right-hand side panels show the runs WR_f07 and WR_f1, respectively. All maps display the full computational domain.</caption>
</figure>
<text><location><page_5><loc_6><loc_10><loc_49><loc_19></location>tively. On the left-hand side, it can be seen how mildy e ffi cient cooling a ff ects mostly one of the stellar winds. This star corresponds to IRS 33E and its wind is the slowest ( ∼ 450 km s -1 ). This fact causes its shocked temperature to be in an e ffi cient region of the radiative cooling functions that allows its material to cool down and become denser. Also, some of its material manages to reach the vicinity of Sgr A*, as the map shows an</text>
<text><location><page_5><loc_51><loc_10><loc_94><loc_19></location>elongated clump being accreted and stretched. The right-hand side also portrays this picture but since the radiative cooling is enhanced more dense clumps can be observed, especially close to IRS 33E and Sgr A*. Finally, the bottom panels of Figure 2 show the models at time t = 0, i.e. the present time. One can clearly see that the amount of dense material that has accumulated around Sgr A* is di ff erent in both maps. Although in the</text>
<figure>
<location><page_6><loc_9><loc_69><loc_46><loc_91></location>
<caption>Fig. 3. Net mass flow rate across a sphere of radius 5 r in = 5 × 10 -4 pc (1 . 25 × 10 -2 '' ). At this radius, the direction of the mass flow is inwards throughout the entire simulation. The models WR_f07 and WR_f1 are shown in solid blue and orange lines, respectively.</caption>
</figure>
<text><location><page_6><loc_6><loc_43><loc_49><loc_60></location>model WR_f07 (see bottom left-hand side panel of Figure 2) some dense material is spiraling towards the black hole overall there are is no clear structure around it. In the case of WR_f1 (see bottom right-hand side panel of Figure 2), the dense material has settled at the centre in a a disc-like structure. This di ff erence is due to the e ffi ciency of the radiative cooling. Since model WR_f1 has enhanced cooling, more dense clumps and filaments form, and some of them manage to fall onto Sgr A*. Overall, the hydrodynamic evolution of the two models is analogous: the winds fill the domain during the first ∼ 500 yr, then the systems reach a quasi-steady state, and in the last 500 yr they diverge as the model WR_f1 creates much more dense, cool material that settles around Sgr A*. A quantitative analysis of the evolution of the properties of system is presented as follows.</text>
<text><location><page_6><loc_6><loc_12><loc_49><loc_42></location>To quantify the accretion rate at di ff erent spatial scales we calculated the mass flux as a function of both radial distance and time ˙ M ( r , t ) = 4 π r 2 ρ ( r , t ) vr ( r , t ) averaged over a spherical shell. Here, positive and negatives signs in the radial velocity refer to outflow and inflow mass fluxes, respectively. First, we analyse the net mass flux at the innermost radius we can resolve properly, which is equivalent to five times the inner boundary radius, i.e. 5 r in = 5 × 10 -4 pc ≈ 1 . 25 × 10 -2 '' . Figure 3 shows | ˙ M ( r = 5 r in , t ) | as a function of time for models WR_f07 and WR_f1 that are displayed as solid blue and orange lines, respectively. In both cases, at t > -3300 yr the net mass flux reaches levels of ∼ 10 -6 M ⊙ yr -1 with short variability episodes that increase the rate by factors of two to four. During this quasi-steady phase, both simulations display similar behaviour qualitatively, likely determined by the identical stellar wind configuration. This stage lasts until t ≈ -500 yr where the mass flow rates start to deviate from each other. The fiducial model continues to display a variability amplitude of the same order of magnitude. However, the WR_f1 model shows a transition to a strongly gas inflow dominated phase, with inflow rates that are enhanced by a factor four to eight. This stage of the evolution is what we refer to as the disc formation phase, analogously to our previous models reported in Calderón et al. (2020b).</text>
<text><location><page_6><loc_6><loc_10><loc_49><loc_12></location>Next, we proceed to analyse the time-averaged behaviour of the simulations over the last 500 yr, as this can give us an idea</text>
<figure>
<location><page_6><loc_51><loc_69><loc_90><loc_91></location>
</figure>
<figure>
<location><page_6><loc_52><loc_45><loc_90><loc_66></location>
<caption>Fig. 4. Radial profiles of time-averaged volume-weighted density (top) and mass-weighted temperature (bottom) over the last 500 yr simulation time. The models WR_f07 and WR_f1 are shown in solid orange and dashed blue lines, respectively.</caption>
</figure>
<text><location><page_6><loc_51><loc_10><loc_94><loc_35></location>of the general state of the system minimising the e ff ects of the stochastic variability. First, we analyse the (volume-weighted) density and (mass-weighted) temperature radial profiles of the simulations, which are shown on the top and bottom panels in Figure 4, respectively. The dashed blue and solid orange lines represent the simulations WR_f07 and WR_f1, respectively. The orange lines in both panels clearly highlight the presence of the cold disc. The density profile decays with r 2 in both cases at large scales ( ≳ 1 '' ). This is the result of most of the material flowing outwards at these scales following a roughly isotropic spherical wind as seen in previous models (Ressler et al. 2018; Calderón et al. 2020b). However, the profiles di ff er at smaller scales ( < 1 '' ) where the fiducial case transitions to ρ ∝ r -1 , while the model WR_f1 shows a density enhancement due to the presence of the disc. This increase in density is more than one order of magnitude. Regarding the temperature profiles, both models match at larger scales ( ≳ 1 '' ) with a constant temperature of the order of 10 7 K, set by the stellar wind collisions. Again, the profiles differ at smaller scales where the fiducial case follows T ∝ r -1 , and the enhanced cooling case displays a temperature profile about</text>
<figure>
<location><page_7><loc_7><loc_69><loc_45><loc_91></location>
<caption>Fig. 5. Radial profiles of the mass inflow ˙ M in (solid lines) and outflow ˙ M out (dashed lines) rates averaged over the last 500 yr of the simulations. The models WR_f07 and WR_f1 are shown in solid orange and blue lines, respectively.</caption>
</figure>
<text><location><page_7><loc_6><loc_57><loc_49><loc_60></location>two orders of magnitudes lower on average. The minimum in temperature corresponds to the region where the disc contributes with most of the mass for a given spherical shell.</text>
<text><location><page_7><loc_6><loc_28><loc_49><loc_56></location>To quantify and characterise the mass inflow and outflow regimes in the simulation domain we calculated them as radial profiles. Figure 5 displays the absolute value of the mass flow rates as a function of distance from Sgr A*. The solid and dashed lines represent the mass inflow and outflow rates, respectively; while the colours follow the same convention as before. Thus, if the solid line is above the dashed line the net mass flow is inwards and vice-versa. Analogous to the model by Ressler et al. (2018), the fiducial model encompasses di ff erent regimes due to the dominant direction of the mass flow at given spatial scales. At r > 3 '' , the outflow component dominates and the inflow is negligible. We find a net mass outflow rate of ∼ 5 × 10 -4 M ⊙ yr -1 . Notice that the enhanced cooling model exhibits exactly the same behaviour at these scales. At smaller scales (1 '' -3 '' ), there is a transition region where | ˙ M in | ∼ | ˙ M out | that corresponds to the location of most mass-losing stars and wind interactions. For model WR_f1, the change is more abrupt due to the presence of the cold disc. In both models, at r < 1 '' the inflow component is larger than the outflow, so the material inflows across these scales and down to the innermost boundary. The net e ff ect makes the mass inflow rate about five times larger which generates the cold disc.</text>
<text><location><page_7><loc_6><loc_10><loc_49><loc_28></location>The origin of the infalling material can be traced in two ways: analysing the angular momentum of the material close to the inner boundary, and using the scalar tracer fields that we introduced to label the chemical abundances of the di ff erent WR sub-types. Figure 6 shows Hammer projections of the angular momentum direction of the gas enclosed in a sphere of radius 0 . 25 '' ( ∼ 0 . 01 pc) for the fiducial and enhanced cooling runs in the top and bottom panels, respectively. Each point represents a di ff erent simulation time over the last 1000 yr, and they are connected with dashed lines. For reference, we added the angular momentum direction of the orbits of the mass-losing stars as black star symbols, and the location of the clockwise disc based on the orbits we used as a grey shaded circle. This analysis shows that the angular momentum direction of the infalling</text>
<figure>
<location><page_7><loc_55><loc_77><loc_91><loc_92></location>
</figure>
<text><location><page_7><loc_90><loc_84><loc_92><loc_85></location>-</text>
<text><location><page_7><loc_90><loc_67><loc_92><loc_67></location>-</text>
<text><location><page_7><loc_72><loc_77><loc_74><loc_78></location>Sun</text>
<figure>
<location><page_7><loc_55><loc_59><loc_91><loc_75></location>
<caption>Fig. 6. Orientation of the angular momentum of the gas enclosed in a sphere of radius 0.01 pc ( ∼ 0.25 '' ), and the WR stars'. The vertical dimension represents inclination i , while the horizontal dimensions stand for the longitude of the ascending node Ω . Thus, a face-on star orbiting clockwise on the sky would be at the north pole of the graph. Top and bottom panels show the runs WR_f07 and WR_f1, respectively. Each dot corresponds to the analysis of a single snapshot. As reference, the grey shaded region corresponds to the average direction of the clockwise disk at (104 · , 126 · ) with a dispersion of 16 · (Yelda et al. 2014).</caption>
</figure>
<figure>
<location><page_7><loc_53><loc_21><loc_90><loc_45></location>
<caption>Fig. 7. Radial profiles of the scalar tracer fields that represent the WR subtypes: WC89 (dashed lines), WN57 (solid lines), and WN89 / Ofpe (dot-dashed) averaged over the last 500 yr of the simulation. The models WR_f07 and WR_f1 are shown in solid orange and blue lines, respectively.</caption>
</figure>
<figure>
<location><page_8><loc_8><loc_69><loc_91><loc_93></location>
<caption>Figure 7 shows radial profiles of the scalar tracers. The dashed, solid, and solid-dashed represent the scalars that correspond to the WR sub-types WC89, WN57, and WN89 / Ofpe, respectively. The blue and orange lines stand for the results for the models WR_f07 and WR_f1. In this analysis, the scalar tracer values represent the fraction of the mass density that comes from a given WR sub-type wind. Thus, in general most of the material supplied into the domain comes from the WN89 / Ofpe stars. At smaller scales ( r < 1 '' ), the dominance is even clearer as the material from these stars makes about 80-90% of the mass budget. This is also the result of these stars having slow winds and colder shocked material. The rest of the stars contribute mostly to the outflow of material, as their contribution is more relevant at larger scales. Notice that both models, fiducial and enhanced, display roughly the same behaviour across the entire domain.</caption>
</figure>
<text><location><page_8><loc_41><loc_68><loc_43><loc_70></location>10</text>
<text><location><page_8><loc_86><loc_68><loc_88><loc_70></location>2.0</text>
<paragraph><location><page_8><loc_6><loc_64><loc_94><loc_67></location>Fig. 8. Density and line-of-sight velocity maps of the central 2.5 '' × 2.5 '' of the model WR_f1 at t = 0. The left- and right-hand side panels show projected density weighed by density and line-of-sight velocity weighed by mass, respectively both integrated along the z-axis, which is parallel to the line of sight.</paragraph>
<text><location><page_8><loc_6><loc_45><loc_49><loc_61></location>material varies stochastically but overall tends to align with the orientation of the orbits in the clockwise disc. However, the degree of variability depends on whether or not the model results in the formation of a disc. The fiducial model shows more variability while the enhanced cooling run displays that the angular momentum aligns more consistently with the orientation of the clockwise disc. The fact that the infalling material at this spatial scales comes from these stars agrees with previous numerical models (Cuadra et al. 2008; Ressler et al. 2018; Calderón et al. 2020b). Furthermore, this is not surprising since these stars orbit closer to Sgr A*, and have winds that are relatively slow ( ∼ 600 kms), which results into shocks more prone to radiative cooling and with smaller angular momentum.</text>
<text><location><page_8><loc_6><loc_10><loc_49><loc_22></location>Overall, the fiducial model displays properties of the medium and gas dynamics consistent with the previous models that do not form a disc (Cuadra et al. 2008; Ressler et al. 2018, 2020; Calderón et al. 2020b, run for 1100 yr). The model with enhanced cooling is consistent with the simulations that show the formation of a cold disc as an outcome of this system (Calderón et al. 2020b). Before discussing if any of the models can be favoured given the current observational constraints we proceed to characterise the cold disc arising in the WR_f1, as its properties will aid us to do so.</text>
<section_header_level_1><location><page_8><loc_51><loc_60><loc_73><loc_61></location>3.2. Properties of the cold disc</section_header_level_1>
<text><location><page_8><loc_51><loc_43><loc_94><loc_59></location>At the present time ( t = 0), the model WR_f1 shows the presence of a cold disc. Figure 8 shows zoomed maps of the central region of 2 . 5 '' × 2 . 5 '' . The left- and right-hand side panels show projected maps of density and line-of-sight velocity (both weighted by density), respectively. Here it is possible to observe that the projected diameter of the disc is roughly 1 '' . We estimated the mass of the disc by isolating the cold material ( T < 10 5 K) and integrating the cell mass content within a sphere of radius 1.0 '' . We found that the total mass of the disc is 0.005 M ⊙ which is consistent with the value in our previous work (Calderón et al. 2020b). However, the mass accumulated in the disc does not converge and keeps increasing at least for the next hundreds of years in the simulation.</text>
<text><location><page_8><loc_51><loc_32><loc_94><loc_42></location>On the right-hand side panel of Figure 8, we can see that the line-of-sight velocity peaks at ∼ 2000 km s -1 along both directions (towards and outwards the observed). This coincides with the maximum extension of the observed H30 α line (Murchikova et al. 2019). Additionally, the disc is observed to be ∼ 45 · tilted in projection. This is not in agreement with the observations since, at least in projection the di ff erence is about ∼ 90 · (see Figure 1 of Murchikova et al. 2019).</text>
<section_header_level_1><location><page_8><loc_51><loc_29><loc_79><loc_30></location>4. Post-processing of observables</section_header_level_1>
<text><location><page_8><loc_51><loc_22><loc_94><loc_28></location>In order to assess the validity of our models for the Galactic centre we proceed to synthesise observational quantities that we could compare easily with observations. We focus on the recombination lines H30 α and Br γ . Afterwards, we calculate the X-ray spectrum from our models.</text>
<section_header_level_1><location><page_8><loc_51><loc_18><loc_79><loc_19></location>4.1. Recombination lines: H30 α and Br γ</section_header_level_1>
<text><location><page_8><loc_51><loc_10><loc_94><loc_17></location>The cold gas at the location of Sgr A* has been observed through the H30 α recombination line (Murchikova et al. 2019; YusefZadeh et al. 2020), and only with an upper limit in the Br γ recombination line (Ciurlo et al. 2021). In order to explore if the models are consistent with these observations we have calculated the expected flux from these emission lines. To compute</text>
<text><location><page_8><loc_81><loc_91><loc_84><loc_93></location>1e3</text>
<figure>
<location><page_9><loc_7><loc_69><loc_49><loc_93></location>
<caption>Figure 9 shows the maps over a region of 2 '' × 2 '' centred in Sgr A* that resulted from this procedure. Although the maps were also calculated for the model WR_f07 we chose not to show them here since the emission is negligible (four orders of magnitude smaller) due to the lack of cold gas that contributes to the recombination line flux. From these maps we estimated the flux as seen from Earth, i.e. at a distance of 8.33 kpc so that they could be compared directly with the observed values. These were calculated by integrating within a circular area of a projected radius p = 0 . 23 '' and p = 1 '' centred at location of Sgr A*. These radii were motivated by the observations that used an aperture of 0.23 '' for extracting the flux observed . However, our models show that the disc actually extends beyond this size, so to get an</caption>
</figure>
<text><location><page_9><loc_41><loc_68><loc_43><loc_70></location>10</text>
<figure>
<location><page_9><loc_51><loc_69><loc_93><loc_93></location>
</figure>
<text><location><page_9><loc_84><loc_68><loc_87><loc_70></location>10</text>
<paragraph><location><page_9><loc_6><loc_65><loc_94><loc_67></location>Fig. 9. Line-of-sight integrated maps of the emission of the H30 α and Br γ recombination lines shown on the left- and right-hand side panels. The maps correspond to the inner 2 '' × 2 '' of the model WR_f1.</paragraph>
<text><location><page_9><loc_6><loc_58><loc_49><loc_63></location>the emission of the H30 α line we used the coe ffi cients given below fitting a piecewise linear function to them as a function of density and considering a decay ∝ T -1 (see Murchikova et al. 2019, supplementary information),</text>
<formula><location><page_9><loc_6><loc_55><loc_49><loc_57></location>ϵ H30 α (10 4 K , 10 4 cm -3 ) = 1 . 05 × 10 -31 erg s -1 cm 3 , (8)</formula>
<formula><location><page_9><loc_6><loc_54><loc_49><loc_55></location>ϵ H30 α (10 4 K , 10 5 cm -3 ) = 1 . 08 × 10 -31 erg s -1 cm 3 , (9)</formula>
<formula><location><page_9><loc_6><loc_52><loc_49><loc_53></location>ϵ H30 α (10 4 K , 10 6 cm -3 ) = 1 . 25 × 10 -31 erg s -1 cm 3 , (10)</formula>
<formula><location><page_9><loc_6><loc_50><loc_49><loc_52></location>ϵ H30 α (10 4 K , 10 7 cm -3 ) = 1 . 36 × 10 -31 erg s -1 cm 3 . (11)</formula>
<text><location><page_9><loc_6><loc_47><loc_49><loc_49></location>In the case of the Br γ line, its emissivity can be calculated following Schartmann et al. (2015) as</text>
<formula><location><page_9><loc_6><loc_43><loc_49><loc_46></location>j Br γ = 3 . 44 × 10 -27 GLYPH<18> T 10 4 K GLYPH<19> -1 . 09 erg s -1 cm 3 . (12)</formula>
<text><location><page_9><loc_6><loc_27><loc_49><loc_42></location>Then, in each cell of the domain we computed the respective emissivity values, multiplied them by their n p n e (assuming full ionisation), and then integrated the data cube along the z coordinate which is perpendicular to the line of sight. It is important to remark that in this calculation it is necessary to take into account the hydrogen nucleus n p and electron n e densities appropriately. Since our models consider di ff erent amounts of hydrogen in the plasma we include only the material that contains some, i.e. the material coming from the WN89 / Ofpe stars. Additionally, we only used the hydrogen fraction of that material which is X H = 11 . 5% and X H = 40% for the models WR_f07 and WR_f1.</text>
<text><location><page_9><loc_51><loc_37><loc_94><loc_63></location>idea of the whole flux of the model we chose a size of p = 1 '' that encloses entirely the disc. The flux values obtained are shown in Table 3 where we also added the upper limit for Br γ from Ciurlo et al. (2021), and the flux of H30 α measured from integrating the spectrum across the velocity channels given by Murchikova et al. (2019). In the case of the Br γ line, we can see that only the model WR_f07 has a flux consistent with the upper limit reported for an aperture of p = 0 . 23 '' . The model WR_f1 has a flux that is ∼ 100 times higher than the upper limit. Also, we see that the disc in our models extends beyond this projected radius, yet its emission is dominated by the inner p = 0 . 23 '' (see Table 3). The situation is more complex for the H30 α line since the observed flux is in both cases higher than the synthetic emission. Although the model that forms a disc is closer it still remains 30% times fainter than the reported value. The model WR_f07 has negligible emission as it is ten order of magnitudes fainter compared to the observation. Thus, it is still not clear how to reconcile the models with both line emission simultaneously as none of them is capable to reproduce the observations. We discuss further the interpretation of these results in Section 7.1.</text>
<section_header_level_1><location><page_9><loc_51><loc_33><loc_71><loc_34></location>4.2. Spectral X-ray emission</section_header_level_1>
<text><location><page_9><loc_51><loc_10><loc_94><loc_32></location>The X-ray emission from the central parsec is also an observable that has allowed validation of previous numerical models (see Russell et al. 2017; Ressler et al. 2018; Wang et al. 2020). Here we also post-processed our models aiming to obtain the Iron Kalpha spectrum of this region at ∼ 6 . 7 keV as this is the strongest X-ray feature from Sgr A* (e.g. Russell et al. 2017; Corrales et al. 2020). Our approach is based on the procedure outlined in Balakrishnan et al. (2024b) but adapted to our finite-volume grid-based hydrodynamic model, which is briefly described as follows. First, we assigned energy dependent X-ray emissivities j k E = n k e n k i Λ ( E , T k ) that were taken from the vvapec model (Smith et al. 2001), which was obtained with xspec (Arnaud 1996). The densities and temperatures were taken from the last output of our simulation, i.e. at t = 0. Additionally, the tracer fields were used to identify the composition fraction of the parcels of gas according to the WR sub-types. Since the optical depth through the simulation domain is optically thin in the X-ray, the radia-</text>
<table>
<location><page_10><loc_21><loc_82><loc_79><loc_91></location>
<caption>Table 3. Radiative flux of the recombination lines from central region.</caption>
</table>
<text><location><page_10><loc_21><loc_75><loc_79><loc_82></location>Notes. All fluxes correspond to quantities seen from Earth, i.e. at a distance of 8.33 kpc, and are given in units erg cm -2 s -1 . The observed H30 α flux was calculated integrating the reported flux density across the velocity channels, and taking into consideration the error propagation of the 0.3 mJy reported (see Figure 1 in Murchikova et al. 2019).</text>
<text><location><page_10><loc_51><loc_72><loc_75><loc_73></location>v th. This function is given as follows</text>
<figure>
<location><page_10><loc_8><loc_43><loc_45><loc_71></location>
<caption>Fig. 10. Synthetic X-ray spectra computed from the numerical models in an annulus of 0.5 '' < p < 1.5 '' . The models WR_f07 and WR_f1 are shown in solid orange and dashed blue lines, respectively. Additionally, the thin solid black line corresponds to the results from the SPH model (Balakrishnan et al. 2024b).</caption>
</figure>
<text><location><page_10><loc_6><loc_25><loc_49><loc_33></location>tive transfer equation to solve is simply given by the integration along the line of sight of the emissivities of the respective cells. Given that we can trace the fraction of the material that comes from a given WR sub-type this actually corresponds to a linear combination of the tracer field si and the abundance dependent emissivity for each cell, i.e.</text>
<formula><location><page_10><loc_6><loc_16><loc_49><loc_23></location>IE ( x , y ) = 3 X i = 1 si Z j k E ( x , y , Xi ) dz (13) = 3 X i = 1 si Z n k ion n k e Λ ( E k , Ti , Xi ) ϕ k ( v los) dz , (14)</formula>
<text><location><page_10><loc_6><loc_10><loc_49><loc_14></location>where ϕ k represents a Gaussian function that it is used to obtain the emission across di ff erent velocity channels, and to take into account the thermal broadening represented with a velocity of</text>
<formula><location><page_10><loc_51><loc_68><loc_94><loc_71></location>ϕ k ( v los) = 1 √ 2 π v th exp -( v k los -v los) 2 2 v 2 th . (15)</formula>
<text><location><page_10><loc_51><loc_41><loc_94><loc_67></location>We performed calculations through velocity channels spanning line-of-sight velocities v los from -3000 km s -1 to 3000 km s -1 . These were later converted into energy space via Doppler broadening. Thus, for each energy bin from the emissivity tables we obtained 150 line profiles using a fine resolution of 6400 per dex. In order to sum them appropriately we interpolate them into a unique energy range so that we can obtain a single spectrum that includes both the continuum and line contributions. The result of this procedure is shown in Figure 10. This contains the X-ray spectrum emitted from a sky-projected annulus of 0 . 5 '' < p < 1 . 5 '' for models WR_f07 (solid blue line) and WR_f1 (dashed orange line). As a reference, we included the estimation from Balakrishnan et al. (2024b) computed from an analogous model that used the SPH approach, which we discuss in the following section. All spectra show qualitative agreement highlighting the di ff erent features of the Fe complex at 6.7 keV and 6.9 keV. This is also reflected in the broadening of the lines due to the velocity across the line of sight, which is expected since the models are based on the same stellar orbits and wind properties.</text>
<text><location><page_10><loc_51><loc_15><loc_94><loc_41></location>It is relevant to highlight the level of agreement between the finite-volume and SPH models. The qualitative agreement is reasonable and the level of the emission is of the same order of magnitude. The exact quantitative di ff erence among the models could be mainly attributed to the exact cooling table (composition) used in each simulation, which is the result of the di ff erent density and temperature distributions that can be observed in Figure 4. This is why the WR_f07 shows the highest flux level as it has the least e ffi cient cooling and, as a result it has hotter material. Analogously, the run WR_f1 with an enhanced cooling specified in the composition of its plasma results in a reduction of ∼ 40% of its flux. In case of the SPH model, the quantitative di ff erences could be a ff ected due to the intrinsic ability of the generic SPH to capture shocks which impacts directly the density, velocity structure, and the temperature of the medium. Notice that the flux level of the spectrum of the SPH model by Balakrishnan et al. (2024b) is within the cases explored in this work. Their cooling model is not the same as the ones explored here but this analysis show that it is equivalent to a cooling e ffi -ciency between both of our models.</text>
<text><location><page_10><loc_51><loc_10><loc_94><loc_15></location>Although not shown here we also compared the X-ray spectra from simulation WR_f1 before and after the disc is formed, and found negligible di ff erences. Finally, we also analysed the slope of the continuum emission within this energy range and</text>
<text><location><page_11><loc_46><loc_76><loc_52><loc_77></location>∫</text>
<text><location><page_11><loc_46><loc_72><loc_52><loc_73></location>∫</text>
<figure>
<location><page_11><loc_12><loc_66><loc_49><loc_90></location>
<caption>Figure 11 shows density and line-of-sight velocity maps projected along the line of sight of the SPH model at t = 0 on the left- and right-hand side panels, respectively. Notice that these panels are analogous to the ones shown in Figure 8 for the Eulerian models presented in this work. First, let us compare the disc density maps between the two approaches. Simply through visual inspection it is possible to see that the structure of the discs is not exactly the same. The Eulerian disc has a more continu-</caption>
</figure>
<figure>
<location><page_11><loc_55><loc_66><loc_93><loc_90></location>
</figure>
<text><location><page_11><loc_86><loc_65><loc_87><loc_66></location>-</text>
<text><location><page_11><loc_87><loc_65><loc_90><loc_66></location>2000</text>
<paragraph><location><page_11><loc_6><loc_60><loc_94><loc_62></location>Fig. 11. Density and line-of-sight velocity maps of the central 2.5 '' × 2.5 '' of the SPH model at t = 0. The left- and right-hand side panels show projected density and line-of-sight velocity, respectively. These panels are analogous to the ones shown in Figure 8.</paragraph>
<text><location><page_11><loc_6><loc_52><loc_49><loc_57></location>found no significant di ff erences among the models. Thus, in this work the presence of the disc does not imprint a feature in the X-ray spectrum that is resolvable by any current or forthcoming X-ray telescope, in agreement with Balakrishnan et al. (2024b).</text>
<section_header_level_1><location><page_11><loc_6><loc_48><loc_39><loc_49></location>5. Comparison with Lagrangian models</section_header_level_1>
<text><location><page_11><loc_6><loc_20><loc_49><loc_47></location>Hydrodynamic simulations of the feeding of Sgr A* by the WR stellar winds have been conducted with di ff erent numerical tools. In this work, we have presented the results of using a finitevolume approach that solves the hydrodynamic equations in the Eulerian form. However, there has been extensive work on this problem using codes that solve the equations in the Lagrangian form. Specifically, the use of SPH has been the main choice for such a task (Cuadra et al. 2005, 2006, 2008, 2015; Russell et al. 2017; Wang et al. 2020). Despite the limitations of the generic SPH technique (e.g. capturing shocks) the models managed to reproduce the observed accretion rate at the Bondi radius (Cuadra et al. 2008) as well as the X-ray emission (Russell et al. 2017). In this context, we have conducted a comparison of the Eulerian models calculated with Ramses, and the Lagrangian models computed with the SPH code Gadget (Springel 2005). The SPH simulation setup is as similar as it can be to our setup. This corresponds to an updated version from the models in Russell et al. (2017). For more details on the setup of the SPH models, we refer the reader to Balakrishnan et al. (2024a,b). We focus the comparison on the analysis on the final state of the system, specifically in the cold discs formed after 3000 yr.</text>
<text><location><page_11><loc_51><loc_35><loc_94><loc_57></location>ous structure towards smaller scales while the Lagrangian disc displays a ring-like structure. This feature can be attributed to the spatial resolution and the inner boundary radius. The Eulerian models indeed manage to resolve smaller scales towards the centre, and the radius of the inner boundary is 5 mas while the Lagrangian models resolve ∼ 1 mas and have an inner boundary of 0 . 1 '' . As a result, the Lagrangian model obtained a disc 50 times lighter ( ∼ 10 -4 M ⊙ ) than in the Eulerian simulation. This di ff erence could be attributed to the radiative cooling employed in each simulation: WR sub-type abundance and Solar with 3 Z ⊙ in the Eulerian and the Lagrangian model, respectively. Regarding the projected orientation of the disc both models seem to be in agreement. This can be analysed more carefully on the lineof-sight velocity maps (see right-hand side panels of Figures 8 and 11). Both maps display that the discs indeed match their projected orientation as well as their most blue- and redshifted velocities are roughly -2000 km s -1 and 2000 km s -1 .</text>
<text><location><page_11><loc_51><loc_27><loc_94><loc_35></location>Overall, the agreement on the outcome a cold disc around Sgr A* whose properties align between two di ff erent approaches indicates that this is a result independent of the numerical technique. Thus, this shows that the determinant factors to obtain this results lies in both the long-enough simulation time ( ≳ 3000 yr ) as well as e ffi cient radiative cooling.</text>
<section_header_level_1><location><page_11><loc_51><loc_23><loc_71><loc_25></location>6. Disc stability analysis</section_header_level_1>
<text><location><page_11><loc_51><loc_10><loc_94><loc_22></location>In this section, we present an analysis of the stability of the observed cold disc in the Galactic Centre. Based on its reported properties, the disc should be depleted by accretion after its viscous timescale t ν ≈ 43 kyr, assuming an α = 0 . 1 thin disc (Shakura & Sunyaev 1973), a process much slower than its generation as modelled in our simulations, although faster than its formation out of CND material (Solanki et al. 2023). Nonetheless, there are several external mechanisms that could be capable of destroying the disc faster than through accretion. Among them are the gravitational and hydrodynamic interactions of the disc</text>
<text><location><page_12><loc_6><loc_85><loc_49><loc_93></location>with the stars, the stellar-wind-disc collisions due to the presence of the S-star cluster in its location, and / or the evaporation due to heat flowing between the hot medium and the cold disc. Although some of these physical scenarios have been studied analytically and numerically, there has not been any direct application to the cold disc discovered in our Galactic Centre.</text>
<section_header_level_1><location><page_12><loc_6><loc_82><loc_34><loc_83></location>6.1. Stellar cluster perturbing a thin disc</section_header_level_1>
<text><location><page_12><loc_6><loc_68><loc_49><loc_81></location>The stability of a cold disc perturbed by stellar passages has been investigated by Ostriker (1983). Their work presented an analytical approach to calculate how the disc loses angular momentum due to the passages, considering the integrated e ff ect of a stellar cluster in stationary state. Besides this work, most e ff orts have been put on following how the stellar dynamics are a ff ected by the disc presence (e.g., Fabj et al. 2020, and references therein) or on the e ff ect of embedded accreting stars and / or black holes on the disc (e.g., Tagawa et al. 2021) rather than on the consequences to the disc properties.</text>
<text><location><page_12><loc_6><loc_50><loc_49><loc_68></location>In this Section, we will summarise the approach of Ostriker (1983) and apply it to the Galactic Centre case. Let us start considering a cold, thin accretion disc around a compact object. Within the thin disc we assume that its height H ( r ) is small compared to the radial coordinate r , i.e. H ( r ) / r ≪ 1. Also, we consider that the sound speed of the material in the disc is small compared to the speed of the stars at the same radius. In this scenario, a star passing through the disc will produce a torque that removes angular momentum from the disc. We consider that the star interacts with the disc gravitationally and hydrodynamically The gravitational interaction of a star crossing the disc can be described using the Bondi-Hoyle approach (Bondi & Hoyle 1944), while the hydrodynamic interaction is just a physical collision of a solid sphere with a gaseous disc.</text>
<text><location><page_12><loc_6><loc_43><loc_49><loc_50></location>Let us consider a system of reference where the x axis is in the direction of rotation and z is in the direction of the rotation pole of the disc. Then, according to Ostriker (1983) the total change of linear momentum along the x direction ∆ px for a single stellar passage is</text>
<formula><location><page_12><loc_6><loc_39><loc_49><loc_42></location>∆ px = -π R 2 ∗ Σ d q + ln Λ D η 4 v 0 v rel ! 4 v cos ϕ -v d sin θ cos ϕ ! GLYPH<18> v rel v GLYPH<19> , (16)</formula>
<text><location><page_12><loc_6><loc_26><loc_49><loc_38></location>where R ∗ is the radius of the star, Σ d = R disc ρ d H ( r ) is the surface density of the disc, ρ d is the volume density of the disc, q is the hydrodynamic drag parameter that is set to q = 2 assuming a large Mach number collision, v 0 is the local stellar velocity, Λ D ≈ H ( r ) v 2 0 / ( R ∗ v 2 ∗ ) is the Coulomb logarithm, v ∗ is the escape velocity from the surface of the star, η = v ∗ / v 0 is the hardness parameter, v d is the disc velocity, v rel is the relative velocity vector, and ( θ, ϕ ) are the spherical coordinate angles along the polar and azimuthal directions.</text>
<text><location><page_12><loc_6><loc_20><loc_49><loc_26></location>In order to consider the e ff ect of the complete stellar cluster interacting with the disc we need to integrate over the cluster phase-space volume. First, we integrate over the local stellar velocity distribution. Let us define dN as the number of stars in the velocity range v → v + dv , i.e</text>
<formula><location><page_12><loc_6><loc_17><loc_49><loc_18></location>dN = 4 π n ∗ v -3 0 f ( v ) v 2 dv , (17)</formula>
<text><location><page_12><loc_6><loc_12><loc_49><loc_16></location>where f ( v ) is the velocity distribution function, n ∗ is the stellar number density. Then, the number of crossings per unit time and area in the ( θ, ϕ ) direction is</text>
<formula><location><page_12><loc_6><loc_9><loc_49><loc_11></location>dN cr = n ∗ v -3 0 f ( v ) v 2 dv sin θ d θ d ϕ sin θ cos ϕ. (18)</formula>
<text><location><page_12><loc_6><loc_7><loc_24><loc_8></location>Article number, page 12 of 19</text>
<text><location><page_12><loc_51><loc_91><loc_94><loc_93></location>Now, integrating over the phase space, the total transfer of momentum to the disc per unit area at r is</text>
<formula><location><page_12><loc_51><loc_82><loc_94><loc_90></location>˙ px , tot( r ) = -2 π 2 R ∗ v -3 0 Σ d( r ) n ∗ ( r ) Z ∞ 0 dvv 2 f ( v ) × Z + 1 -1 d µ ( v µ -v d) v rel h q + η 4 ln Λ D( v / v rel) 4 i , (19)</formula>
<text><location><page_12><loc_51><loc_80><loc_62><loc_81></location>where µ = cos ϕ .</text>
<text><location><page_12><loc_51><loc_76><loc_94><loc_79></location>As the local momentum per unit area in the disc is px , d( r ) = Σ ( r ) /τ d( r ) the inverse of the drag timescale to remove the angular momentum of the disc is τ d = -px , d / ˙ px , tot can be written as</text>
<formula><location><page_12><loc_51><loc_73><loc_94><loc_75></location>τ d = h n ∗ R 2 ∗ v 0 GLYPH<16> qI 0 + η 4 ln Λ D I 1 GLYPH<17>i -1 , (20)</formula>
<text><location><page_12><loc_51><loc_66><loc_94><loc_71></location>where the quantities ( I 0 , I 1) depend solely on the stellar distribution function. For instance, in the case of a Maxwellian distribution the values of these parameters can be obtained numerically and are I 0 = 12 . 553 and I 1 = 0 . 474.</text>
<text><location><page_12><loc_51><loc_61><loc_94><loc_66></location>In order to apply this model to the Galactic Centre we need to calculate the stellar density within the size of the disc. Following Schödel et al. (2007) the stellar mass density at r < 6 arcsec is described by</text>
<formula><location><page_12><loc_51><loc_57><loc_94><loc_60></location>ρ ∗ ( r ) = 2 . 8 ± 1 . 3 × 10 7 M ⊙ pc -3 GLYPH<18> r 6 arcsec GLYPH<19> -1 . 2 . (21)</formula>
<text><location><page_12><loc_51><loc_51><loc_94><loc_56></location>Assuming that the stellar mass density is composed mainly by stars whose radii are R ∗ = 1 R ⊙ and move on their orbits typically at v 0 = 1000 km s -1 we can use Equation 20 to estimate the inverse of the drag timescale,</text>
<formula><location><page_12><loc_51><loc_42><loc_95><loc_50></location>τ d = 185 n ∗ 3 . 8 × 10 8 pc -3 ! -1 R ∗ 1R ⊙ ! -2 GLYPH<18> v 0 10 3 km s -1 GLYPH<19> -1 × " 1 + 0 . 08 GLYPH<18> v ∗ 440 km s -1 GLYPH<19> 4 GLYPH<18> v 0 10 3 km s -1 GLYPH<19> -4 ln Λ D 10 !# -1 Myr . (22)</formula>
<text><location><page_12><loc_51><loc_32><loc_94><loc_41></location>Based on this calculation the angular momentum of the cold disc should be removed after ∼ 185 Myr of interactions of the stars. This is much longer than the formation time-scale found in our simulations. Furthermore, it is even much longer than the age of the Wolf-Rayet stars, which is about 6 Myr (Martins et al. 2007). Then, the gravitational and hydrodynamic interactions of the stars onto the disc should not a ff ect the stability of the disc.</text>
<section_header_level_1><location><page_12><loc_51><loc_29><loc_75><loc_30></location>6.2. Stellar winds perturbing a disc</section_header_level_1>
<text><location><page_12><loc_51><loc_10><loc_94><loc_28></location>In the case of stellar irradiation a ff ecting the state of the accretion flow there have been many studies mainly motivated by the recent pericentre passages of the star S2 around Sgr A*. For instance, Cuadra et al. (2003) could rule out the existence of an optically-thick disc based on the lack of its thermallyreprocessed emission as it gets illuminated by S2. Nayakshin et al. (2004) estimated that a star as luminous as S2 could heat up and ionise an inner disc, which could enhance the accretion rate. Giannios & Sironi (2013); Christie et al. (2016) calculated the bremsstrahlung emission as a result of the stellar wind shocking a Radiatively Ine ffi cient Accretion Flow (RIAF) around Sgr A*, finding an X-ray luminosity comparable to quiescent emission of Sgr A*. In fact, no noticeable increase was detected during 2018 (see Table 1 of Andrés et al. 2022). Hosseini et al. (2020)</text>
<text><location><page_13><loc_6><loc_84><loc_49><loc_93></location>constrained the observed L ' -band variability of S2 to be about 2-3%, based on a model of bow shock of its stellar wind, implying an ambient density < 10 5 cm -3 , which rules out the presence of a standard accretion disc at S2's pericentre, in agreement with the previous studies. It is important to remark however that such a limit marginally allows the existence of both the disc reported by Murchikova et al. (2019), and also the one in our simulation.</text>
<text><location><page_13><loc_6><loc_75><loc_49><loc_84></location>The stars in the S cluster also have relatively powerful stellar winds. As the stars inhabit the black hole in the vicinity of the cold disc, it is expected that the winds interact with it. By either opening a bubble due to the high kinetic energy carried in the winds or depositing thermal energy they a ff ect the disc properties. We proceed to perform analytical estimates to quantify the impact of these processes.</text>
<section_header_level_1><location><page_13><loc_6><loc_72><loc_24><loc_73></location>6.2.1. Bubbles in the disc</section_header_level_1>
<text><location><page_13><loc_6><loc_67><loc_49><loc_71></location>Let us consider a star blowing an isotropic stellar wind with a mass-loss rate ˙ M w at a terminal speed v w. Then, its density ρ w at a distance r from the star is given by</text>
<formula><location><page_13><loc_6><loc_63><loc_49><loc_66></location>ρ w( r ) = ˙ M w 4 π r 2 v w (23)</formula>
<text><location><page_13><loc_6><loc_57><loc_49><loc_62></location>If such a star is immersed in a medium whose number density is n m at rest at a temperature T m, the radius of the bubble r b opened by the wind can be calculated equating the ram pressure of the wind and the thermal energy of the medium, i.e.</text>
<formula><location><page_13><loc_6><loc_49><loc_49><loc_56></location>r b = 1 . 48 × 10 15 ˙ M w 10 -8 M ⊙ yr -1 ! 1 / 2 GLYPH<18> v w 1000 km s -1 GLYPH<19> 1 / 2 × GLYPH<18> n m 10 6 cm -3 GLYPH<19> -1 / 2 GLYPH<18> T m 10 4 K GLYPH<19> -1 / 2 cm , (24)</formula>
<text><location><page_13><loc_6><loc_42><loc_49><loc_48></location>where we have used as fiducial values the disc parameters measured by Murchikova et al. (2019) and typical stellar wind properties for B stars (Oskinova et al. 2011; Krtiˇcka 2014). Then, the radius of the bubble is about 0.012 arcsec, which is approximately one tenth of the disc radius.</text>
<text><location><page_13><loc_6><loc_32><loc_49><loc_42></location>In order to open such a bubble a star would need to be within the disc fort at least the wind crossing time, i.e. r b / v w, which is about ∼ 0 . 5 yr. Let us estimate the duration of a stellar passage through the disc. Along the perpendicular direction a star crossing the disc would take at most 0 . 1 R d / v 0, being H ( r ) / r = 0 . 1. One S star moves typically at v 0 ≈ 1000 km s -1 , then 0 . 1 R d / v 0 ≈ 1 yr. As both timescales are comparable, we conclude that the bubble can be opened.</text>
<text><location><page_13><loc_6><loc_26><loc_49><loc_31></location>For such a perturbation to last we need to check if either shear or sound waves could eliminate it, i.e. close the bubble. In the case of shear, this is given by the orbital speed of the disc and the size of the bubble:</text>
<formula><location><page_13><loc_6><loc_22><loc_49><loc_25></location>t close = r b ∆ v kep = r r GM h r + r b + p r ( r + r b) i . (25)</formula>
<text><location><page_13><loc_6><loc_10><loc_49><loc_21></location>As this expression increases with r , the largest value will be at the outer edge of the disc, then t close < 5 yr. This value should be interpreted as the longest duration of a perturbation of the wind onto the disc. As there are 12 S-stars on orbits whose pericentre distances are shorter than the radius of the disc and their orbital periods are of the order of ∼ 10 yr (Gillessen et al. 2017) we expect a bubble to be opened every year on average. However, as shear would close such bubbles in less than five years there would be at most four bubbles opened on average in a stationary</text>
<text><location><page_13><loc_51><loc_87><loc_94><loc_93></location>state. As the size of the bubbles is very small compared to the size of the disc (two orders of magnitude in area), four of them would not have an impact on its structure. The sound crossing timescale r b / c s ≈ 30 yr is longer than the shear timescale so we do not expect it to have an impact on erasing perturbations.</text>
<section_header_level_1><location><page_13><loc_51><loc_83><loc_77><loc_84></location>6.2.2. Winds injecting thermal energy</section_header_level_1>
<text><location><page_13><loc_51><loc_67><loc_94><loc_82></location>The S stars are of spectral type B with masses of 10 M ⊙ (e.g. Habibi et al. 2017), thus their mass-loss rate must be ∼ 10 -8 M ⊙ yr -1 with terminal velocity of 1000 km s -1 . The supersonic nature of the winds should be enough to compress the shocked material at high temperature. Potentially, this thermal energy could be deposited onto the disc during each stellar passage. If we consider that most of the kinetic energy of the wind is transformed into thermal energy about ∼ 6 . 4 × 10 33 erg would be deposited in the disc during the ∼ 1 yr long passage (see above). In order to check if the wind energy can significantly impact the thermodynamic state of the disc let us estimate its total internal energy U int.</text>
<formula><location><page_13><loc_51><loc_63><loc_94><loc_66></location>U int = n d k B T d γ -1 V d , (26)</formula>
<text><location><page_13><loc_51><loc_49><loc_94><loc_62></location>where k B is the Boltzmann constant and V d is the disc volume. Replacing the observed disc properties we obtain that U int = 1 . 63 × 10 42 erg. From this calculation it is clear that even if all the energy from the wind in a stellar passage is deposited the contribution is still negligible to modify the state of the disc. Even if we consider the integrated e ff ect of all the S stars passages on a timescale comparable to the viscous timescale of the disc ( ∼ 43 kyr) the contribution is still < 0 . 1% of the total thermal energy of the disc. Thus, the energy supplied by the stellar winds into the disc is not enough to a ff ect its state.</text>
<section_header_level_1><location><page_13><loc_51><loc_46><loc_68><loc_47></location>6.3. Thermal conduction</section_header_level_1>
<text><location><page_13><loc_51><loc_32><loc_94><loc_45></location>The vicinity of Sgr A* at the Bondi radius is made out of a diffuse (10 cm -3 ), hot medium (10 7 K; Bagano ff et al. 2003) due to the shocked stellar winds from the Wolf-Rayet stars in the region. At the disc vicinity ( ∼ 10 3 R Sch), the conditions are more extreme as the plasma has a density of about ∼ 5 × 10 3 cm -3 (Gillessen et al. 2019) and temperature of ∼ 10 8 K. As the disc is significantly colder it is expected that heat flows from the medium to the disc. The large temperature gradient should enable thermal conduction to take place and likely evaporate the disc.</text>
<text><location><page_13><loc_51><loc_18><loc_94><loc_32></location>The problem of evaporating a cold structure sitting in a hot environment due to thermal conduction was first studied by Cowie & McKee (1977). In that work, the authors derived an analytical expression for the mass-loss rate and evaporation timescale for a spherically symmetric gas cloud. Unfortunately, these expressions are not valid for more complex geometries. Meyer & Meyer-Hofmeister (1994) studied specifically the evaporation of a cold thin accretion disc in a hot corona. Although the model considered only one radial zone it was able to calculate the full perpendicular structure of the accretion disc and the corona by means of numerical simulations.</text>
<text><location><page_13><loc_51><loc_10><loc_94><loc_17></location>Liu et al. (2004) used the model by Meyer & MeyerHofmeister (1994) to analyse which solutions were consistent with the observational data available at the time in case there were a cold disc in the center of the Galaxy. They found that any transient disc would evaporate quickly. With their obtained evaporation rate of ∼ 10 -4 M ⊙ yr -1 , the life-time of a disc with</text>
<text><location><page_14><loc_6><loc_71><loc_49><loc_93></location>properties such as found by Murchikova et al. (2019) or formed in our models would be ∼ 1 yr. Similar analysis have argued that small clumps ( ∼ M ⊕ ) in the Galactic Centre should also evaporate quickly ( ∼ 1 yr) due to thermal conduction (Burkert et al. 2012; Calderón et al. 2018). According to this, cold gas should not be able to last long if formed, yet we observe many cold gas structures in the central parsec: the many dusty cold clumps detected in the vicinity of IRS 13E (Fritz et al. 2010; Peißker et al. 2023), and a larger gaseous cloud X7 (Ciurlo et al. 2023). Thus, this regime of classical thermal conduction might not be applicable to the Galactic centre environment. Another possibility has been studied by Nayakshin (2004) who focused on the same problem and found solutions in a di ff erent regime, where the electron mean free path is long enough that an accretion disc does not evaporate but rather increases its mass by condensation. A more thorough analysis of the role of thermal conduction is beyond the scope of this work and is deferred to future study.</text>
<text><location><page_14><loc_6><loc_59><loc_49><loc_71></location>Our findings shows that gravitational and hydrodynamic effects require timescales that are too long to have an impact on the disc stability. On the other hand, we find that thermal conduction in the classical case should evaporate the observed disc in about one year. Given the time-scale for disc formation found in our model, the Nayakshin (2004) condensation regime remains to our knowledge as the only viable physical scenario that would allow its continued existence and potential identification with Murchikova et al. (2019)'s reported disc.</text>
<section_header_level_1><location><page_14><loc_6><loc_55><loc_17><loc_56></location>7. Discussion</section_header_level_1>
<text><location><page_14><loc_6><loc_48><loc_49><loc_54></location>In this section, we interpret our findings exploring the uncertainties in the abundances of WR stars in general and of the ones in the central parsec. Additionally, we contrast the properties of the simulated and observed cold discs as well as discuss further implications.</text>
<section_header_level_1><location><page_14><loc_6><loc_45><loc_38><loc_46></location>7.1. Is the simulated disc the observed disc?</section_header_level_1>
<text><location><page_14><loc_6><loc_24><loc_49><loc_43></location>In order to compare the simulated and the observed discs, first we analyse their physical properties. The simulated disc has a total mass of ∼ 5 × 10 -3 M ⊙ , while the structure observed in H30 α was reported to have a mass of 10 -5 -10 -4 M ⊙ . However, comparing these quantities might not be appropriate since the mass of the observed structure is based on three assumptions: a thin disc with uniform density and temperature, and a masing factor of ∼ 100. All of them minimise the value of the mass inferred from the observations. For instance, dropping the maser assumption the inferred mass increases in a factor ten (see Equations 30-31 of Supplementary Material in Murchikova et al. 2019). Bearing this in mind, we could argue that the mass of the simulated and observed disc might be of the same order provided there was no maser. Although removing that assumption would create tension reconciling both the Br γ and H30 α observed fluxes.</text>
<text><location><page_14><loc_6><loc_18><loc_49><loc_24></location>Regarding its extension, the simulated disc has a projected radius of ∼ 0 . 5 '' , while the observed disc has a radius of 0 . 23 '' . However, since most of the recombination line emission of the disc is originated from its central region ( p < 0 . 23 '' , see Table 3) the sizes of the simulated and observed discs are in agreement.</text>
<text><location><page_14><loc_6><loc_10><loc_49><loc_17></location>A more direct comparison can be done through the analysis of the mock observational quantities computed from the simulations. In Section 4.1, we computed the H30 α and Br γ recombination line emission and estimated their fluxes. Here, we have found that both synthetic fluxes are in tension with the observations. First, the simulated Br γ flux is 100 times higher than</text>
<text><location><page_14><loc_51><loc_79><loc_94><loc_93></location>the upper limit. It is relevant to remark that this limit is already extinction corrected. Second, the simulated H30 α flux is 30% lower than the observed one. These results di ff er from the estimates by Ciurlo et al. (2021) from our previous models due to, again, the uniform disc assumptions. The simulated disc is not thin or possess uniform density or temperature. Although the mean density and temperature of the simulated disc is of the order of ∼ 10 -5 cm -3 and ∼ 10 4 K its geometry is more complex with a larger variance in its properties. As a result, both the synthetic Br γ and H30 α fluxes are much higher than the ones inferred by Ciurlo et al. (2021).</text>
<text><location><page_14><loc_51><loc_54><loc_94><loc_79></location>Based on the line-of-sight velocity maps shown in Murchikova et al. (2019), the orientation of the observed disc displays the red- and blueshifted sides to be on the East (lefthand) and West (right-hand) sides, respectively. Although our model WR_f1 also shows this configuration the exact inclination of the discs does not match perfectly. Specifically, the discs are inclined in ∼ 90 · among each other, being the simulated disc tilted in the clockwise direction on the sky. In principle, this argues against the WR stars as the main responsible for the formation of the observed disc. But we cannot rule out completely their role in the process, since a combination of the WR stars with other gaseous structures, such as the minispiral or the CND, acting simultaneously might result in a net e ff ect that manages to reproduce the observed angular momentum. Solanki et al. (2023) showed that material from the circumnuclear disc could fall close to Sgr A* though on much longer timescales. Certainly, this scenario would be much more di ffi cult and challenging to simulate with the appropriate chemical abundances as well as with high-enough spatial resolution.</text>
<text><location><page_14><loc_51><loc_45><loc_94><loc_54></location>It is also relevant to notice that the observed accretion flow orientation at ≲ 10 RSch is consistent with the orientation of the clockwise stellar disc (GRAVITY Collaboration et al. 2020; Event Horizon Telescope Collaboration et al. 2022). Thus, our models reproduce that connection between the dynamics from large ( > 10 5 RSch) to small ( ≲ 10 RSch) scales. However, the observed cold disc does not seem to follow such a connection.</text>
<text><location><page_14><loc_51><loc_36><loc_94><loc_45></location>Finally, an intriguing aspect of our findings is that the WR atmospheres with more H in them would favour the formation of the disc. At the same time, the disc indeed was detected in the H30 α recombination line, which obviously indicates the presence of H. Thus, it is sensible to ask how much H can be in the winds of WR stars but, specifically in the ones of the sub-type WN89 / Ofpe. This point is discussed in the following section.</text>
<section_header_level_1><location><page_14><loc_51><loc_33><loc_74><loc_34></location>7.2. Uncertainties in abundances</section_header_level_1>
<text><location><page_14><loc_51><loc_10><loc_94><loc_32></location>The results of the hydrodynamic models demonstrate that the chemical abundances of the material in the WR winds are the key factor to determining whether or not a cold disc can be formed due to their action. Unfortunately, there are two large uncertainties in this context. First, the metallicity of the young population of the nuclear star cluster is not well constrained (e.g. Genzel et al. 2010). Although there is agreement that it is above Solar its exact value has not been established so far, with most studies quoting the range Z = 2 -3 Z ⊙ . However, even if the metallicity is determined more precisely, the problem would not be entirely solved, since the abundances of the plasma in the region are dominated by the material supplied by the WR star winds whose compositions are much more uncertain. Theoretically, the evolution of these stars is a topic of active research, and improved prescriptions for mass loss are changing the modelled properties of observed WR stars (see e.g., Gormaz-Matamala et al. 2023, and references therein). Simply based on observations, up to now, it</text>
<text><location><page_15><loc_6><loc_66><loc_49><loc_93></location>has been reported that some of these stars might possess from X H = 0 to X H = 50% for Galactic WR stars of the WN subtype (Hamann et al. 2006, 2019), and even as high as X H = 60% for extragalactic ones (Todt et al. 2015). In the vicinity of Sgr A*, Martins et al. (2007) characterised the wind properties of the sample of WR stars through the analysis of their infrared spectra. They reported abundance H / He ratios in the range 2-5 for these stars which does not allow us to rule out the scenario regarding this aspect. A variation on the hydrogen abundance in the chemical mixture in a factor five might change the net radiative cooling rate by a factor 1 . 1 -1 . 4, especially at lower temperatures. In our models, WR_f07 considered a fiducial value of hydrogen mass fraction of 11.5% (Russell et al. 2017) but the model WR_f1 assumed 40%. Such a change was enough to modify the cooling function by about 30%, which ended up affecting significantly the final result and the state of the plasma. Thus, within our current understanding of the abundances in the WR stellar atmospheres both scenario are plausible. Only better quality spectra and more sophisticated models of them could allow us to constrain the wind properties and their composition more precisely.</text>
<section_header_level_1><location><page_15><loc_6><loc_63><loc_19><loc_64></location>8. Conclusions</section_header_level_1>
<text><location><page_15><loc_6><loc_29><loc_49><loc_62></location>We have presented a numerical study of the system of masslosing stars feeding Sgr A* focusing on the impact of the chemical composition of the plasma. Through studying simulations with di ff erent abundances we have found that the system can either form or not form a cold disc around the black hole, provided that the model is run for long-enough timescales ( ≳ 3000 yr) to create a dense enough medium in the inner parsec of the Galaxy. As in our past work, we have confirmed that if formed the cold disc can impact significantly the hydrodynamic and thermodynamic state of the plasma within the central parsec. As a result, the inferred mass in flow rate at 5 × 10 -4 pc can reach up to 8 × 10 -6 M ⊙ yr -1 , which is 4 -8 times higher than the value without the presence of such a disc. Our models still point to the mass-losing stars in the clockwise disc being the main source of the accreted material due to their dense and slow winds as previous models have shown (e.g. Cuadra et al. 2008; Ressler et al. 2018; Calderón et al. 2020b). In this work, we have followed more carefully the origin of the inflowing material through the use of passive scalar fields for di ff erent WR subtypes. This allowed us to confirm the origin of the accreted material but, more importantly led us to conclude that this is mostly provided by the winds of the stars of the WR sub-type WN89 / Ofpe. This provides evidence that the infalling material must have a nonnegligible fraction of hydrogen as these WR stars have not lost entirely their atmospheric hydrogen via winds.</text>
<text><location><page_15><loc_6><loc_14><loc_49><loc_29></location>Our models also showed that the formation of a cold disc depends strictly on the radiative cooling employed, which is determined by the chemical abundances of the plasma. Previous models have only considered Solar abundances with 3 Z ⊙ but had not agreed whether or not a cold disc can be formed. From our results in Appendix B, we speculate that at Z ≈ 3 Z ⊙ there is a transition between the regimes of persistent quasi-steady accretion and disc formation, which would imply that small di ff erences in the cooling function or numerical implementation can result in models reproducing either regime. This may explain the di ff ering results obtained by Calderón et al. (2020b) and Ressler et al. (2020) with similar numerical and physical setups.</text>
<text><location><page_15><loc_6><loc_10><loc_49><loc_13></location>In the case a cold disc is formed around Sgr A* the structure may resemble the extension and the line-of-sight velocity structure of the observed disc. However, direct observational</text>
<text><location><page_15><loc_51><loc_87><loc_94><loc_93></location>quantities could not be reproduced. Specifically, the exact skyprojected inclination of the simulated and observed discs di ff ers in ∼ 90 · . The Br γ emission line upper limit is 100 lower than the synthetic flux. The observed H30 α emission line flux is 30% higher than in our model.</text>
<text><location><page_15><loc_51><loc_62><loc_94><loc_86></location>Furthermore, we have contrasted this work Eulerian (finitevolume grid-based) models with analogous (SPH) Lagrangian models in order to address the potential influence of the chosen approach in the outcome of the simulations. As it has been shown in Balakrishnan et al. (2024b), a cold disc is also formed around Sgr A* if the simulation is run for long timescales. Both cold structures agree on their sizes, their velocity structure, and their sky-projected inclination. Although there are di ff erences in the total mass and specific inner structure of the discs this could be attributed to the resolution employed and the exact assumption to consider the innermost boundary. Despite these di ff erences, the agreement is reasonable among the numerical approaches, especially when analysing the synthetic X-ray emission (see Figure 10). The spectral features agree qualitatively across the models presented in this work and the Lagrangian model. The level of the X-ray emission are all within the same order of magnitude but the exact base level depends on the radiative cooling chosen, which is determined by the chemical abundances in the WR atmospheres.</text>
<text><location><page_15><loc_51><loc_45><loc_94><loc_62></location>We also have explored the stability and long-term evolution of the putative disc under the e ff ect of perturbing agents: the nuclear star cluster, the wind of all the mass-losing stars (including the S stars) impacting the disc through opening bubbles and / or depositing thermal energy via shocks, and the potential e ff ect of thermal conduction. Our analysis showed that gravitational and hydrodynamic e ff ects would impact the disc on longer timescales then the viscous timescale. However, within the classical thermal conduction framework a cold disc in such a region should not exist. Based on this and the disc formation timescale from our simulations, we speculate that the condensation model of Nayakshin et al. (2004) remains a plausible mechanism consistent with the disc existence.</text>
<text><location><page_15><loc_51><loc_24><loc_94><loc_45></location>In conclusion, through this work we have been able to reconcile the discrepancy among the numerical simulations of the system of mass-losing stars feeding Sgr A*. The formation of a cold disc on ∼ 3000 yr timescale is indeed possible for certain chemical abundances which are consistent with the current observational constraints. Nonetheless, it is not possible to favour or disfavour this scenario due to the uncertainties in them. Thus, high-resolution spectra of the WR stars and more sophisticated modelling of them could allow us to discern the validity of such scenarios. But we must warn that even in the case that the cold disc is a result of action of the WR stars it is not possible to reproduce all its observed properties: its inclination and recombination line fluxes. More sophisticated models that include infalling material from other sources might be necessary to produce the exact inclination of the structure which could also impact the emission of the structure.</text>
<text><location><page_15><loc_51><loc_10><loc_94><loc_23></location>Acknowledgements. We thank Prof. Sergei Nayakshin and Prof. Stanley Owocki for very helpful discussions about this project. We also thank Dr. Sean M. Ressler for sharing the radiative cooling function shown in Figure 1. Additionally, we are grateful to Dr. Álex Gormaz-Matamala for helping with references and discussions on the WR wind abundances. DC and JC acknowledge the support of the Kavli Foundation through its summer program at the Max Planck Institute for Astrophysics where fruitful discussions took place. In addition, DC thanks the warm hospitality of the Universidad Adolfo Ibañez in Chile and University of Delaware in the USA where part of this project was developed. We acknowledge the support from ANID in Chile through FONDECYT Regular grant 1211429 and Millennium Science Initiative Program NCN2023_002. The research of DC and SR was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - EXC 2121 -</text>
<text><location><page_16><loc_6><loc_78><loc_49><loc_93></location>'Quantum Universe' - 390833306. Since 01.10.2024, DC has been funded by the Alexander von Humboldt Foundation. SR is also supported by the European Research Council (ERC) Advanced Grant INSPIRATION under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 101053985), the Swedish Research Council (VR) under grant number 202005044, and the research environment grant 'Gravitational Radiation and Electromagnetic Astrophysical Transients' (GREAT) funded by the Swedish Research Council (VR) under Dnr 2016-06012, by the Knut and Alice Wallenberg Foundation under grant Dnr. KAW 2019.0112. The numerical simulations of this work were run on the high-performance computing system cobra of the Max Planck Computing and Data Facility. Most of the analysis of the numerical data was carried out using the python package yt (Turk et al. 2011). Furthermore, this work made use of python libraries numpy (Harris et al. 2020) and matplotlib (Hunter 2007), as well as of the NASA's Astrophysics Data System.</text>
<section_header_level_1><location><page_16><loc_6><loc_74><loc_16><loc_75></location>References</section_header_level_1>
<text><location><page_16><loc_6><loc_69><loc_49><loc_73></location>Andrés, A., van den Eijnden, J., Degenaar, N., et al. 2022, MNRAS, 510, 2851 Arnaud, K. A. 1996, in Astronomical Society of the Pacific Conference Series, Vol. 101, Astronomical Data Analysis Software and Systems V, ed. G. H. Jacoby & J. Barnes, 17</text>
<unordered_list>
<list_item><location><page_16><loc_6><loc_68><loc_41><loc_69></location>Bagano ff , F. K., Maeda, Y., Morris, M., et al. 2003, ApJ, 591, 891</list_item>
<list_item><location><page_16><loc_6><loc_67><loc_44><loc_68></location>Balakrishnan, M., Corrales, L., Marko ff , S., et al. 2024a, ApJ, 974, 98</list_item>
<list_item><location><page_16><loc_6><loc_66><loc_47><loc_66></location>Balakrishnan, M., Russell, C. M. P., Corrales, L., et al. 2024b, ApJ, 974, 99</list_item>
</unordered_list>
<text><location><page_16><loc_6><loc_65><loc_40><loc_65></location>Becklin, E. E., Gatley, I., & Werner, M. W. 1982, ApJ, 258, 135</text>
<unordered_list>
<list_item><location><page_16><loc_6><loc_64><loc_32><loc_64></location>Bondi, H. & Hoyle, F. 1944, MNRAS, 104, 273</list_item>
<list_item><location><page_16><loc_6><loc_63><loc_40><loc_63></location>Burkert, A., Schartmann, M., Alig, C., et al. 2012, ApJ, 750, 58</list_item>
<list_item><location><page_16><loc_6><loc_62><loc_44><loc_62></location>Calderón, D., Ballone, A., Cuadra, J., et al. 2016, MNRAS, 455, 4388</list_item>
<list_item><location><page_16><loc_6><loc_61><loc_46><loc_61></location>Calderón, D., Cuadra, J., Schartmann, M., et al. 2018, MNRAS, 478, 3494</list_item>
<list_item><location><page_16><loc_6><loc_59><loc_46><loc_60></location>Calderón, D., Cuadra, J., Schartmann, M., et al. 2020a, MNRAS, 493, 447</list_item>
<list_item><location><page_16><loc_6><loc_57><loc_49><loc_59></location>Calderón, D., Cuadra, J., Schartmann, M., Burkert, A., & Russell, C. M. P. 2020b, ApJ, 888, L2</list_item>
<list_item><location><page_16><loc_6><loc_55><loc_49><loc_57></location>Christie, I. M., Petropoulou, M., Mimica, P., & Giannios, D. 2016, MNRAS, 459, 2420</list_item>
<list_item><location><page_16><loc_6><loc_54><loc_44><loc_55></location>Ciurlo, A., Campbell, R. D., Morris, M. R., et al. 2023, ApJ, 944, 136</list_item>
<list_item><location><page_16><loc_6><loc_53><loc_44><loc_54></location>Ciurlo, A., Morris, M. R., Campbell, R. D., et al. 2021, ApJ, 910, 143</list_item>
<list_item><location><page_16><loc_6><loc_52><loc_43><loc_53></location>Corrales, L., Bagano ff , F. K., Wang, Q. D., et al. 2020, ApJ, 891, 71</list_item>
<list_item><location><page_16><loc_6><loc_51><loc_33><loc_52></location>Cowie, L. L. & McKee, C. F. 1977, ApJ, 211, 135</list_item>
<list_item><location><page_16><loc_6><loc_50><loc_28><loc_51></location>Crowther, P. A. 2007, ARA&A, 45, 177</list_item>
<list_item><location><page_16><loc_6><loc_49><loc_41><loc_50></location>Cuadra, J., Nayakshin, S., & Martins, F. 2008, MNRAS, 383, 458</list_item>
<list_item><location><page_16><loc_6><loc_47><loc_49><loc_49></location>Cuadra, J., Nayakshin, S., Springel, V., & Di Matteo, T. 2005, MNRAS, 360, L55</list_item>
<list_item><location><page_16><loc_6><loc_46><loc_49><loc_47></location>Cuadra, J., Nayakshin, S., Springel, V., & Di Matteo, T. 2006, MNRAS, 366, 358</list_item>
<list_item><location><page_16><loc_6><loc_45><loc_40><loc_46></location>Cuadra, J., Nayakshin, S., & Sunyaev, R. 2003, A&A, 411, 405</list_item>
<list_item><location><page_16><loc_6><loc_44><loc_42><loc_45></location>Cuadra, J., Nayakshin, S., & Wang, Q. D. 2015, MNRAS, 450, 277</list_item>
<list_item><location><page_16><loc_6><loc_43><loc_44><loc_44></location>Dinh, C. K., Salas, J. M., Morris, M. R., & Naoz, S. 2021, ApJ, 920, 79</list_item>
<list_item><location><page_16><loc_6><loc_41><loc_49><loc_43></location>Event Horizon Telescope Collaboration, Akiyama, K., Alberdi, A., et al. 2022, ApJ, 930, L12</list_item>
<list_item><location><page_16><loc_6><loc_40><loc_41><loc_41></location>Fabj, G., Nasim, S. S., Caban, F., et al. 2020, MNRAS, 499, 2608</list_item>
<list_item><location><page_16><loc_6><loc_39><loc_43><loc_40></location>Fritz, T. K., Gillessen, S., Dodds-Eden, K., et al. 2010, ApJ, 721, 395</list_item>
<list_item><location><page_16><loc_6><loc_38><loc_46><loc_39></location>Genzel, R. 1989, in The Center of the Galaxy, ed. M. Morris, Vol. 136, 393</list_item>
<list_item><location><page_16><loc_6><loc_36><loc_49><loc_38></location>Genzel, R., Eisenhauer, F., & Gillessen, S. 2010, Reviews of Modern Physics, 82, 3121</list_item>
<list_item><location><page_16><loc_6><loc_35><loc_34><loc_35></location>Giannios, D. & Sironi, L. 2013, MNRAS, 433, L25</list_item>
<list_item><location><page_16><loc_6><loc_34><loc_42><loc_34></location>Gillessen, S., Plewa, P. M., Eisenhauer, F., et al. 2017, ApJ, 837, 30</list_item>
<list_item><location><page_16><loc_6><loc_33><loc_42><loc_33></location>Gillessen, S., Plewa, P. M., Widmann, F., et al. 2019, ApJ, 871, 126</list_item>
<list_item><location><page_16><loc_6><loc_30><loc_49><loc_32></location>Gormaz-Matamala, A. C., Cuadra, J., Meynet, G., & Curé, M. 2023, A&A, 673, A109</list_item>
<list_item><location><page_16><loc_6><loc_29><loc_49><loc_30></location>GRAVITY Collaboration, Abuter, R., Amorim, A., et al. 2019, A&A, 625, L10</list_item>
</unordered_list>
<text><location><page_16><loc_6><loc_28><loc_49><loc_29></location>GRAVITY Collaboration, Abuter, R., Amorim, A., et al. 2021, A&A, 647, A59</text>
<text><location><page_16><loc_6><loc_27><loc_49><loc_28></location>GRAVITY Collaboration, Bauböck, M., Dexter, J., et al. 2020, A&A, 635, A143</text>
<unordered_list>
<list_item><location><page_16><loc_6><loc_26><loc_40><loc_27></location>Habibi, M., Gillessen, S., Martins, F., et al. 2017, ApJ, 847, 120</list_item>
<list_item><location><page_16><loc_6><loc_25><loc_44><loc_26></location>Hamann, W. R., Gräfener, G., & Liermann, A. 2006, A&A, 457, 1015</list_item>
<list_item><location><page_16><loc_6><loc_24><loc_45><loc_25></location>Hamann, W. R., Gräfener, G., Liermann, A., et al. 2019, A&A, 625, A57</list_item>
<list_item><location><page_16><loc_6><loc_22><loc_49><loc_24></location>Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357-362</list_item>
<list_item><location><page_16><loc_6><loc_21><loc_46><loc_22></location>Herald, J. E., Hillier, D. J., & Schulte-Ladbeck, R. E. 2001, ApJ, 548, 932</list_item>
</unordered_list>
<text><location><page_16><loc_6><loc_20><loc_49><loc_21></location>Hosseini, S. E., Zajaˇcek, M., Eckart, A., Sabha, N. B., & Labadie, L. 2020, A&A,</text>
<text><location><page_16><loc_8><loc_19><loc_14><loc_20></location>644, A105</text>
<unordered_list>
<list_item><location><page_16><loc_6><loc_18><loc_40><loc_19></location>Hunter, J. D. 2007, Computing in Science & Engineering, 9, 90</list_item>
<list_item><location><page_16><loc_6><loc_16><loc_49><loc_18></location>James, T. A., Viti, S., Yusef-Zadeh, F., Royster, M., & Wardle, M. 2021, ApJ, 916, 69</list_item>
<list_item><location><page_16><loc_6><loc_10><loc_43><loc_16></location>Krtiˇcka, J. 2014, A&A, 564, A70 Lemaster, M. N., Stone, J. M., & Gardiner, T. A. 2007, ApJ, 662, 582 Liu, B. F., Meyer, F., & Meyer-Hofmeister, E. 2004, A&A, 421, 659 Lodders, K. 2003, ApJ, 591, 1220 Martins, F., Genzel, R., Hillier, D. J., et al. 2007, A&A, 468, 233 Meyer, F. & Meyer-Hofmeister, E. 1994, A&A, 288, 175</list_item>
</unordered_list>
<text><location><page_16><loc_6><loc_7><loc_24><loc_8></location>Article number, page 16 of 19</text>
<unordered_list>
<list_item><location><page_16><loc_51><loc_91><loc_94><loc_93></location>Murchikova, E. M., Phinney, E. S., Pancoast, A., & Blandford, R. D. 2019, Nature, 570, 83</list_item>
<list_item><location><page_16><loc_51><loc_90><loc_73><loc_91></location>Nayakshin, S. 2004, arXiv e-prints, astro</list_item>
<list_item><location><page_16><loc_51><loc_89><loc_85><loc_90></location>Nayakshin, S., Cuadra, J., & Sunyaev, R. 2004, A&A, 413, 173</list_item>
<list_item><location><page_16><loc_51><loc_86><loc_94><loc_89></location>Onifer, A., Heger, A., & Abdallah, J. 2008, in Astronomical Society of the Pacific Conference Series, Vol. 391, Hydrogen-Deficient Stars, ed. A. Werner & T. Rauch, 305</list_item>
<list_item><location><page_16><loc_51><loc_83><loc_89><loc_85></location>Oskinova, L. M., Todt, H., Ignace, R., et al. 2011, MNRAS, 416, 1456 Ostriker, J. P. 1983, ApJ, 273, 99</list_item>
<list_item><location><page_16><loc_51><loc_82><loc_85><loc_83></location>Paumard, T., Genzel, R., Martins, F., et al. 2006, ApJ, 643, 1011</list_item>
<list_item><location><page_16><loc_51><loc_81><loc_85><loc_82></location>Peißker, F., Zajaˇcek, M., Thomkins, L., et al. 2023, ApJ, 956, 70</list_item>
<list_item><location><page_16><loc_51><loc_80><loc_69><loc_81></location>Quataert, E. 2004, ApJ, 613, 322</list_item>
<list_item><location><page_16><loc_51><loc_79><loc_89><loc_80></location>Ressler, S. M., Quataert, E., & Stone, J. M. 2018, MNRAS, 478, 3544</list_item>
<list_item><location><page_16><loc_51><loc_78><loc_89><loc_79></location>Ressler, S. M., Quataert, E., & Stone, J. M. 2020, MNRAS, 492, 3272</list_item>
</unordered_list>
<text><location><page_16><loc_51><loc_77><loc_89><loc_78></location>Russell, C. M. P., Wang, Q. D., & Cuadra, J. 2017, MNRAS, 464, 4958</text>
<text><location><page_16><loc_51><loc_76><loc_88><loc_77></location>Schartmann, M., Ballone, A., Burkert, A., et al. 2015, ApJ, 811, 155</text>
<unordered_list>
<list_item><location><page_16><loc_51><loc_75><loc_87><loc_76></location>Schödel, R., Eckart, A., Alexander, T., et al. 2007, A&A, 469, 125</list_item>
<list_item><location><page_16><loc_51><loc_73><loc_94><loc_75></location>Schure, K. M., Kosenko, D., Kaastra, J. S., Keppens, R., & Vink, J. 2009, A&A, 508, 751</list_item>
<list_item><location><page_16><loc_51><loc_72><loc_80><loc_73></location>Shakura, N. I. & Sunyaev, R. A. 1973, A&A, 500, 33</list_item>
<list_item><location><page_16><loc_51><loc_70><loc_94><loc_72></location>Smith, R. K., Brickhouse, N. S., Liedahl, D. A., & Raymond, J. C. 2001, ApJ, 556, L91</list_item>
<list_item><location><page_16><loc_51><loc_67><loc_94><loc_69></location>Solanki, S., Ressler, S. M., Murchikova, L., Stone, J. M., & Morris, M. R. 2023, ApJ, 953, 22</list_item>
<list_item><location><page_16><loc_51><loc_66><loc_72><loc_67></location>Springel, V. 2005, MNRAS, 364, 1105</list_item>
<list_item><location><page_16><loc_51><loc_65><loc_91><loc_66></location>Stone, J. M., Tomida, K., White, C. J., & Felker, K. G. 2020, ApJS, 249, 4</list_item>
<list_item><location><page_16><loc_51><loc_63><loc_94><loc_65></location>Tagawa, H., Kimura, S. S., Haiman, Z., et al. 2021, arXiv e-prints, arXiv:2112.01544</list_item>
<list_item><location><page_16><loc_51><loc_62><loc_69><loc_63></location>Teyssier, R. 2002, A&A, 385, 337</list_item>
<list_item><location><page_16><loc_51><loc_61><loc_84><loc_62></location>Todt, H., Sander, A., Hainich, R., et al. 2015, A&A, 579, A75</list_item>
<list_item><location><page_16><loc_51><loc_60><loc_94><loc_61></location>Toro, E. 2009, Riemann Solvers and Numerical Methods for Fluid Dynamics: A</list_item>
</unordered_list>
<text><location><page_16><loc_53><loc_59><loc_74><loc_60></location>Practical Introduction (Berlin: Springer)</text>
<text><location><page_16><loc_51><loc_52><loc_93><loc_59></location>Toro, E. F., Spruce, M., & Speares, W. 1994, Shock Waves, 4, 25 Turk, M. J., Smith, B. D., Oishi, J. S., et al. 2011, ApJS, 192, 9 von Fellenberg, S. D., Gillessen, S., Stadler, J., et al. 2022, ApJ, 932, L6 Štofanová, L., Kaastra, J., Mehdipour, M., & de Plaa, J. 2021, A&A, 655, A2 Wang, Q. D., Li, J., Russell, C. M. P., & Cuadra, J. 2020, MNRAS, 492, 2481 Yelda, S., Ghez, A. M., Lu, J. R., et al. 2014, ApJ, 783, 131 Yuan, F. & Narayan, R. 2014, ARA&A, 52, 529</text>
<unordered_list>
<list_item><location><page_16><loc_51><loc_50><loc_91><loc_51></location>Yusef-Zadeh, F., Royster, M., Wardle, M., et al. 2020, MNRAS, 499, 3909</list_item>
</unordered_list>
<section_header_level_1><location><page_17><loc_6><loc_92><loc_37><loc_93></location>Appendix A: Numerical setup choices</section_header_level_1>
<text><location><page_17><loc_6><loc_77><loc_49><loc_91></location>Calderón et al. (2020b) simulated the system of WR stars feeding Sgr A* while orbiting around it following an analogous numerical setup compared to the models presented in this work. There, we had found that the resulting cold disc tended to align with the grid after its formation. In order to remedy this numerical artifact we investigated the use of a MinMod slope limiter, instead of MonCen. However, we warned that it was not straightforward to maintain the spherical symmetry of the winds using the MinMod slope limiter. In this work, we investigated more carefully these numerical choices in order to be confident that our results are robust.</text>
<text><location><page_17><loc_6><loc_60><loc_49><loc_77></location>First, we investigated the choice of the MonCen slope limiter. To do this, we made use of the same setup presented in this work but paying attention to the values of the passive scalar fields. In principle, the values of these fields should be between zero and one. However, this experiment showed that the MonCen slope limiter produced non-physical oscillations that in some cases resulted in negative values of the passive scalar fields. This behaviour was not observed when using the MinMod slope limiter. Thus, we selected this choice throughout this work. To ensure the spherical symmetry of the stellar winds we had to impose a less restrictive refinement criterion based on density gradients as well as a smoother transition between refinement levels through the parameter nexpand .</text>
<text><location><page_17><loc_6><loc_27><loc_49><loc_60></location>Second, we tested the choice of the Riemann solver employed but using the MinMod slope limiter. Currently, the HLLC Riemann solver (Toro et al. 1994) is widely used in grid-based hydrodynamic simulations. In this work, we have found that this choice results in a cold disc that tends to align with the Cartesian grid as seen in Calderón et al. (2020b). This could indicate problems in conserving the angular momentum which is a known issue with simulations in Cartesian grids when the relevant regions are not well resolved. With the usage of the exact Riemann solver (Toro 2009), though more computationally expensive, it was possible to avoid this artificial alignment. Thus, the only sensible combination was the use of the exact Riemann solver with a MinMod slope limiter but we had to ensure to count with enough resolution and smoothness in the transition of the refinement levels. As a reference of these experiments, Figure A.1 shows density projection (weighted by density) maps of the final state of the system for two runs with Solar composition with 3 Z ⊙ but with di ff erent Riemann solvers. The left- and righthand side panels display the models using the HLLC and the exact Riemann solvers, respectively. Although overall the density structure looks similar there are subtle di ff erences. The most relevant is related to the disc orientation and structure. On the left-hand side, the disc tries to align with one of the Cartesian axes, while on the right-hand side the disc remains consistently with the shown orientation.</text>
<section_header_level_1><location><page_17><loc_51><loc_92><loc_93><loc_93></location>Appendix B: Solar composition varying metallicity</section_header_level_1>
<text><location><page_17><loc_51><loc_64><loc_94><loc_91></location>Before investigating the models with the new implementation of the di ff erential cooling we ran some experiments simply modifying the contribution of the metals to the total radiative cooling. We simulated three models varying only the composition used: A1 used Solar composition, A3 used Solar composition with 3 Z ⊙ , and A5 used Solar composition with 5 Z ⊙ . The density projected (weighted by density) maps of the final state of the simulations are shown in Figure B.1. The left-hand side, central, and right-hand side panels display the runs A1, A3, and A5, respectively. A higher metallicity in the plasma increases the radiative cooling directly. As a result, more and denser filaments, clumps, and a denser central disc are obtained with increasing the metallicity. On the contrary, if the metallicity is decreased to a Solar value we observe almost no overdensities, and definitively no disc formation. We speculate that at Z ≈ 3 Z ⊙ there is a transition between the regimes of persistent quasi-steady accretion and disc formation, which would imply that small di ff erences in the cooling function or numerical implementation can result in models reproducing either regime. This may explain the di ff ering results obtained by Calderón et al. (2020b) and Ressler et al. (2020) with similar numerical and physical setups.</text>
<text><location><page_17><loc_51><loc_43><loc_94><loc_64></location>In order to perform a more quantitative comparison of these results Figure B.2 presents radial profiles of density (weighted by volume) and temperature (weighted by mass) on the top and bottom panels, respectively. The dashed blue, solid orange, and blue dotted-dashed lines represent the models A1, A3, and A5, respectively. The density profiles clearly shows the impact of cooling in the structure of the medium. More e ffi cient cooling, like in cases A3 and A5, results in an order of magnitude denser medium within 1 '' compared to the case with ine ffi cient cooling, i.e. A1. Notice that even higher metallicity in A5 extends the impact the denser region to slightly larger spatial scales. The e ff ect is similar in the temperature profile. In A1, the temperature profile decays with r within the central 1 '' . But when the metallicity is high enough to make cooling e ffi cient the temperature profile can decrease up to two orders of magnitude depending on the exact value.</text>
<text><location><page_17><loc_51><loc_29><loc_94><loc_43></location>In summary, these experiments allowed us to quantify the impact of radiative cooling by simply decreasing or increasing the radiative cooling influence. However, since the chemical composition in reality is much more complicated and di ff ers from Solar composition significantly these models should be interpreted only as general guides to assess the role of radiative cooling. In the main text of this work, we devoted to explore more physically motivated mixtures that, although depend on more uncertain parameters are more robust when attempting to build a physical model that can be constrasted with observational data.</text>
<formula><location><page_18><loc_25><loc_88><loc_75><loc_94></location>10 -24 10 -22 10 -20 10 -18 ∫ ρ 2 dz/ ∫ ρdz (g cm -3 )</formula>
<figure>
<location><page_18><loc_8><loc_57><loc_50><loc_86></location>
</figure>
<figure>
<location><page_18><loc_50><loc_57><loc_92><loc_86></location>
<caption>Fig. A.1. Comparison of the final state (present time) of the simulations with di ff erent Riemann solvers. Left- and right-hand side panels correspond to runs with Harten-Lax-van Leer-Contact (HLLC) and exact Riemann solvers, respectively. Both models were run using the MinMod slope limiter.</caption>
</figure>
<figure>
<location><page_18><loc_25><loc_46><loc_75><loc_51></location>
</figure>
<figure>
<location><page_18><loc_7><loc_25><loc_36><loc_45></location>
</figure>
<figure>
<location><page_18><loc_36><loc_24><loc_64><loc_45></location>
</figure>
<figure>
<location><page_18><loc_65><loc_25><loc_93><loc_45></location>
<caption>Fig. B.1. Comparison of the simulations with Solar composition varying the metallicity at two di ff erent simulation times. The panels show projected density maps weighed by density along the z axis, i.e. R ρ 2 dz / R ρ dz , which is parallel to the line of sight. Left-hand side, central, and right-hand side panels show the runs A1, A3, and A5, respectively. All maps display the full computational domain.</caption>
</figure>
<figure>
<location><page_19><loc_6><loc_69><loc_45><loc_91></location>
</figure>
<figure>
<location><page_19><loc_8><loc_45><loc_45><loc_66></location>
<caption>Fig. B.2. Radial profiles of time-averaged volume-weighted density (top) and mass-weighted temperature (bottom) over the last 500 yr of simulation time. The models A1, A3, and A5 are shown in solid orange, dashed blue, and dashed-dotted green lines, respectively.</caption>
</figure>
</document>
[
{
"title": "ABSTRACT",
"content": "Context. The reported discovery of a cold ( ∼ 10 4 K) disc-like structure around the super-massive black hole at the centre of the Milk Way, Sagittarius A* (Sgr A*), has challenged our understanding of the gas dynamics and thermodynamic state of the plasma in its immediate vicinity. State-of-the-art simulations do not agree on whether or not such a disc can indeed be a product of the multiple stellar wind interactions of the mass-losing stars in the region. Aims. This study aims to constrain the conditions for the formation of a cold disc as a natural outcome of the system of the mass-losing stars orbiting around Sgr A*, to investigate if the disc is a transient or long-lasting structure, and to assess the validity of the model through direct comparisons with observations. Methods. We conduct a set of hydrodynamic simulations of the observed Wolf-Rayet (WR) stars feeding Sgr A* using the finitevolume adaptive mesh-refinement code Ramses. We focus, for the first time, on the impact of the chemical composition of the plasma emanating from the WR stars. Results. The simulations show that the chemical composition of the plasma a ff ects the radiative cooling enough to impact the properties of the medium such as density and temperature and, as a consequence, the rate at which the material inflows onto Sgr A*. We demonstrated that the formation of a cold disc from the stellar winds is possible for certain chemical compositions that are consistent with the current observational constraints. However, even in such a case, it is not possible to reproduce the reported properties of the observed disc-like structure, namely its inclination and hydrogen recombination line fluxes. Conclusions. Weconclude that the stellar winds on their own cannot form the cold disc around Sgr A* inferred from the observations. Either relevant ingredients are still missing in the model, or the interpretation of the observed data needs to be revised. Key words. accretion, accretion discs - Galaxy: centre - hydrodynamics - Stars: winds, outflows - Stars: Wolf-Rayet",
"pages": [
1
]
},
{
"title": "The formation and stability of a cold disc made out of stellar winds in the Galactic Centre",
"content": "1,2, 3,4 5 6,7,8 Diego Calderón ⋆ , Jorge Cuadra , Christopher M. P. Russell , Andreas Burkert , Stephan Rosswog 1,9 , and Mayura Balakrishnan 10 Received November 4, 2024; accepted November 4, 2024",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "The Galactic Centre (GC) hosts the closest super-massive black hole to us, Sagittarius A* (Sgr A*; see Genzel et al. 2010, for a review). Unlike the black holes present in active galactic nuclei, Sgr A* is very underluminous. It does not seem to be currently accreting much material, or to have a standard accretion disc around it (Yuan & Narayan 2014). Murchikova et al. (2019) detected however a disc-like structure around Sgr A*. Using the 1.3-millimetre recombination line H30 α they observed a doublepeaked emission line with full velocity width of ∼ 2200 km s -1 . The centre of the emission coincides with Sgr A* and extends up to 0.11 '' ( ∼ 4.4 × 10 -3 pc) to both the redshifted and blueshifted sides. They interpreted this feature as a rotating disc of mass 10 -5 -10 -4 M ⊙ . Yusef-Zadeh et al. (2020) also reported the presence of broad hydrogen recombination lines, including the double-peaked H30 α line, at the position of Sgr A* using the Very Large Array. However, they interpreted the signatures as a jet emanating from the central region. So far, there has not been any detection in the near infrared, despite many observations. Ciurlo et al. (2021) reports an upper limit for Br γ , which is two orders of magnitude below the extrapolation from the reported H30 α flux. Currently, there is no consensus on how such observations can be reconciled or from where the observed cold material comes from. The gaseous environment around Sgr A* is dominated by the outflows from around 30 Wolf-Rayet (WR) stars, all located within a fraction of a parsec of the black hole (Paumard et al. 2006; Martins et al. 2007). At such close distances, the winds interact strongly in shocks that thermalise them and create a hot and di ff use X-ray emitting plasma (Quataert 2004). However, given the relatively low velocity of some of the outflows (450600 km s -1 ; Martins et al. 2007), the resulting plasma can be prone to radiative cooling and form dense, cold clumps (Cuadra et al. 2005; Calderón et al. 2016), which end up being embedded in the di ff use, hot plasma ( ∼ 10 7 K; Bagano ff et al. 2003). The complex interplay between the stellar winds around Sgr A* has been the subject of hydrodynamic simulations using a variety of numerical techniques. With smoothed particle hydrodynamics (SPH) simulations, Cuadra et al. (2006) showed that, if most of the slow-wind stars are located in a relatively compact stellar disc, many cold clumps form and quickly coalesce in a cold gaseous disc of radius ∼ 1 '' . However, using a stellar distribution closer to the one observed, Cuadra et al. (2008, 2015) showed that no conspicuous disc appears over the 1800 yr time-span of their models. Calderón et al. (2020b) performed finite-volume hydrodynamic simulations on a Cartesian grid of the same system, which are better suited to model shocks, the multi-phase medium, and subsequent cooling, and found that the formation of a cold disc is indeed possible. According to their model, the stellar wind of the star IRS 33E, which is fairly dense ( ∼ 10 -5 M ⊙ yr -1 ) and slow (450 km s -1 ; Martins et al. 2007) interacts with the medium creating a shell that is dense enough to radiate quickly its thermal energy and break into denser and smaller structures. These pieces manage to fall inwards forming a cold ( ∼ 10 4 K) disc with a total mass of ∼ 5 × 10 -3 M ⊙ within a simulation time of 3500 yr. Based on this result, Ciurlo et al. (2021) found that the properties of the modelled disc are consistent with the non-detection in Br γ . Nevertheless, this scenario has not been confirmed by analogous grid-based simulations. Ressler et al. (2020) revisited the same system using the same numerical approach, although a di ff erent code: athena++ (Stone et al. 2020), and found no disc formation even extending their simulation time up to 9000 yr. Building on this model, Solanki et al. (2023) also studied this system through hydrodynamic modelling but encompassing a larger region and evolving it for much longer timescales. They included, for the first time, the presence of the circumnuclear disc (CND), an observed gaseous structure located between 1 . 5-3 . 0 pc (Becklin et al. 1982; Genzel 1989) and with a total mass of 3-4 × 10 4 M ⊙ (Dinh et al. 2021; James et al. 2021). This work showed that the formation of a cold disc is possible on long timescales ( ≳ 300,000 yr) due to the interaction of the innermost boundary of the CND and the WR stellar winds, which results in the transport of CND material to smaller scales. However, the authors warned that the lack of certain physical mechanisms in the model on such timescales (e.g. magnetic fields, supernovae, thermal conduction) might a ff ect the robustness of this result. Thus, the exact formation process of the observed cold disc still remains unknown. Akey factor driving the formation of clumps and a disc in the model of Calderón et al. (2020b) is that radiative cooling allows the gas to get rid of its energy e ffi ciently. The strength of this process depends on the chemical composition of the gas, which is highly unusual in the GC given its origin as winds from evolved massive stars. In this work, we explore for the first time the impact of the radiative cooling through studying specific chemical compositions: Solar with 1 Z ⊙ , 3 Z ⊙ , and 5 Z ⊙ . More importantly, we develop a more realistic chemical setup that considers the atmospheric abundances for the di ff erent WR sub-types present in the region, in order to follow the thermodynamic evolution of the gas in a more appropriate manner. Our results show that the formation of the disc is indeed determined by the composition of the gas. However, the properties of such a disc do not agree with the observations when considering realistic wind compositions compatible with current observational constraints. We also improve the numerical setup in order to better assess the disc orientation and more directly compare our results to the work of Ressler et al. (2018, 2020) as well as with SPH models (Cuadra et al. 2008, 2015; Russell et al. 2017). This article is organised as follows: in Section 2, we present the numerical approach, the setup and the models investigated, Section 3 presents and describes the results of the numerical simulations. In Section 4, we present the synthetic observables obtained through post-processing the models and contrast them with observations. We compare our grid-based models with SPH models in Section 5. In Section 6, we present an analytic analysis of the stability of the observed cold disc. Section 7 discusses our findings and uncertainties in the parameters of our models. Finally, Section 8 presents conclusions and final remarks. Throughout this paper we use the mass and distance to Sgr A* of 4 . 3 × 10 6 M ⊙ and 8 . 33 kpc, respectively (Gillessen et al. 2017; GRAVITY Collaboration et al. 2019, 2021), so that 1 arcsec corresponds to a length of ∼ 0.04 pc ≈ 10 5 RSch, being RSch the Schwarzschild radius.",
"pages": [
1,
2
]
},
{
"title": "2.1. Equations",
"content": "The simulations were performed using the adaptive-mesh refinement (AMR) hydrodynamic code Ramses (Teyssier 2002). This code uses a second-order Godunov method with a shockcapturing scheme to solve the Euler equations in their conservative form, i.e. where ( ρ, u , P ) are the primitive hydrodynamic variables: density, velocity, and pressure, respectively. The set of quantities si correspond to tracer scalar fields that are advected with the fluid whose usage we introduce in Section 2.3. The sink terms on the right-hand side correspond to the e ff ects of the gravitational potential ϕ = ϕ ( x ), assumed to be time independent, and the total radiative loses due to optically-thin radiative cooling Q -tot = Q -tot ( ρ, T , X i), with x the position vector, T the fluid temperature, and X i the chemical composition of the fluid. The total specific energy density e is given by where γ is the adiabatic index that is set to 5 / 3. Additionally, we consider the fluid can be described as an ideal gas so that the temperature can be calculated through P = ( ρ/µ ) k B T , being µ the mean molecular weight and k B the Boltzmann constant.",
"pages": [
2
]
},
{
"title": "2.2. Numerical setup",
"content": "The model considers the system of WR stars blowing stellar winds while they move on their observed Keplerian orbits around Sgr A*. The setup is analogous to our previous work (Calderón et al. 2020b). The central black hole gravitational field is modelled as a point mass of 4 . 3 × 10 6 M ⊙ (GRAVITY Collaboration et al. 2019, 2021). The stars are simulated as test particles that only feel the gravitational pull of Sgr A*. Their initial Notes. The Solar abundances were taken from Lodders (2003). The abundances for the three WR sub-types correspond to the compilation by Russell et al. (2017) based on previous studies (Herald et al. 2001; Crowther 2007; Onifer et al. 2008). Column 1: chemical mixture. Columns 2-7: hydrogen, helium, carbon, nitrogen, oxygen, and rest of the metals mass fractions, respectively. Column 8: mean molecular weigh assuming full ionisation. Notes. Column 1: simulation ID. Column 2: chemical composition of the winds. Column 3: Riemann solver. Column 4: Slope limiter. Column 5: whether or not the final state of the system shows the presence of a cold disc around Sgr A*. position and velocity vectors are set so that they move on the orbits constrained by observations (Paumard et al. 2006; Cuadra et al. 2008; Gillessen et al. 2019; von Fellenberg et al. 2022). The stellar winds in the simulation are modelled following the procedure by Lemaster et al. (2007), which has been validated and used in our previous work (Calderón et al. 2020a,b). This consists in defining a 'masked region\" around each star that is a spherical volume where the hydrodynamic variables are reset in each timestep, in order to reproduce the free wind expansion solution of a spherical wind with certain mass-loss rate ˙ M w, terminal velocity V w, and temperature T w. For these quantities we used the values constrained through modelling of the infrared spectra (Martins et al. 2007; Cuadra et al. 2008). The wind temperature was set to the lowest temperature allowed in the simulation, T w = 10 4 K which is determined by the strong ultraviolet radiation produced by the hundreds of massive stars in the region. The simulations were run in a Cartesian grid in a cubic domain of side 40 '' ( ∼ 1.6 pc) with outflow boundary conditions (zero gradients). We used the exact Riemann solver (e.g. Toro 2009) combined with the MinMod slope limiter, as these choices allow the modelling of hydrodynamic instabilities, which otherwise may be quenched due to numerical di ff usion. We investigated other choices such as the Harten-Lax-van Leer-Contact (HLLC; Toro et al. 1994) and / or the MonCen slope limiter but they produced unwanted numerical artifacts such as spurious oscillations. For a careful analysis of these choices we refer the reader to the Appendix A. Regarding the numerical resolution of our models, the coarse resolution was 64 3 cells, allowing adaptive refinement of four levels, i.e. e ff ectively ∆ x ≈ 0 . 0382 '' . However, the regions around the stars allowed an extra level of refinement ( ∆ x ≈ 0 . 0195 '' ). Additionally, the vicinity around the central black hole has a fixed nested grid with eight refinement levels above the coarse resolution ( ∆ x ≈ 2 . 44 mas). At the location of Sgr A*, we defined a spherical region where the hydrodynamic variables are reset to low values of density and pressure at rest, in order to avoid artificial accumulation of material. The refinement strategy is based on density gradients on top of the geometric criteria previously defined. In order to explore the role of the AMR potentially a ff ecting the results we tested di ff erent values of the smoothing parameter that controls the refinement in transitions between refinement levels. We tested quadrupling the smoothing parameter and the results remained unchanged 1 .",
"pages": [
2,
3
]
},
{
"title": "2.3. Models",
"content": "The main parameter explored in this work is the optically-thin radiative cooling function. This choice is determined by specifying the chemical composition of the fluid. Unfortunately, the metallicity of the young stars in the GC is still poorly known, yet it has been argued that it should be higher than Solar ( Z = 2 -3 Z ⊙ ; Genzel et al. 2010). Past numerical works have considered that the composition of the gas correspond to Solar abundances but with metallicity three times the Solar value, Z = 3 Z ⊙ (Cuadra et al. 2008; Calderón et al. 2016; Ressler et al. 2018, 2020; Solanki et al. 2023). However, more than the 'bulk' metallicity of the stars, the most appropriate composition choice would be the abundances of the WR stellar atmospheres, which normally di ff er significantly from Solar (Herald et al. 2001; Onifer et al. 2008). Although Ressler et al. (2018, 2020) used chemical compositions lacking hydrogen, the rest of the element abundances remained unchanged relative to Solar with 3 Z ⊙ . As a result, the cooling function varied at low temperature ( < 10 5 K) but at higher temperatures it remains unchanged. Moreover, the composition choice not only determines the cooling function but also the ion mean molecular weight as well as the electron to proton number densities n e / n p. These quantities are taken into account in the sink term in the energy equation (see equation 3), and following Schure et al. (2009) can be expressed as follows where the subscript i stands for a given chemical abundance. Bear in mind that di ff erent chemical compositions also correspond to di ff erent temperature Ti , as this is obtained through the ideal gas expression that makes uses of the mean molecular weight of the fluid. In the general case, if we consider a fluid element composed of N mixtures of abundances, the total radiative loses will be given as a summation, i.e. where we have expressed the total radative cooling as a linear combination of the mixtures. Notice that we have introduced the passive scalar fields si , as we used them to quantify the fraction of a given chemical composition. In this work, we introduced three types of compositions that represent the WR subtypes based on their atmosphere abundances. In the region where the winds are generated, the corresponding passive scalar is set to 1 while the rest are set to 0. By doing so, it is possible to identify how much material of a given cell is supplied by which sub-group of WR stars. The mass fractions of each mixture are shown in Table 1. In this work, we investigated five models with di ff erent compositions: Solar composition with metallicities Z ⊙ , 3 Z ⊙ , and 5 Z ⊙ , a mixture of three compositions based on the spectroscopic constraints of the sub-types of WR stars, and a variation of the latter one motivated by the uncertainty in the WR stellar atmospheres and, specifically on the H fraction in them that is set to X H = 40% (see Section 7.2, for a discussion). Figure 1 shows the cooling function Λ = Λ ( T ) as a function of temperature for different chemical mixtures. Each curve was calculated as a linear combination of the abundance of a given element and its contribution to the total radiative cooling. The values of the composition per element are shown in Table 1. The contribution of each element to the total cooling was taken from the plasma models developed by Štofanová et al. (2021). The cooling functions corresponding to WR subtypes compositions: WC8-9, WN5-7, and WN8-9 / Ofpe are shown with dashed blue, dotted orange, and dotted-dashed green lines, respectively. The Solar and Solar with 3 Z ⊙ are represented with thin and thick solid black lines, respectively. Furthermore, we added the cooling function used in previous works by Ressler et al. (2018, 2020) as a dotted-dashed red line. Here it can be observed that a Solar composition with 3 Z ⊙ (the typical used value) is between the range of the di ff erent WR compositions but the subtypes WC89 and WN89 / Ofpe are about a factor two higher. The list of the runs investigated in this work are shown in Table 2. The initial setup was identical to our previous work (Calderón et al. 2020b), i.e. the medium across the whole domain was set to constant and low-enough values of density ρ = 10 -24 g cm -3 and pressure P = γ -1 ρ c 2 s,f with c s,f = 10 km s -1 , so that the stellar winds do not encounter impediments and fill quickly the domain. All models were run for a total of 3,500 yr but setting the starting state of the system in the past, so that the final state of the simulations corresponds to the current state of the system. This is achieved by integrating the current position and velocity vectors of the stars back in time, assuming that they have followed purely Keplerian orbits within this timescale due to the gravitational field of Sgr A* (Cuadra et al. 2008; Ressler et al. 2018; Calderón et al. 2020b).",
"pages": [
3,
4
]
},
{
"title": "3. Results",
"content": "We proceed to describe the evolution and final state of the simulations at t = 0 that corresponds to the present time. The runs with Solar composition, but varying metallicities, were used for understanding the impact of increasing and decreasing the e ff ect of radiative cooling in general. However, since they do not represent realistic compositions we opted not to discuss them in detail here. Instead, we chose to focus on the more physically motivated models, i.e. WR_f07 and WR_f1 which we also refer as fiducial and enhanced , respectively. For completeness, we give a brief description of the models A1, A2, and A3 in Appendix B.",
"pages": [
4
]
},
{
"title": "3.1. Hydrodynamics",
"content": "The simulation evolution of all models is analogous, especially in the initial phases. At t = -3500 yr, the stars begin to orbit around Sgr A* that is located at the centre of the domain, while their winds quickly fill the whole computational domain. After ∼ 500 yr ( t = -3000 yr), the system reaches a quasi-steady state. At this point, the domain is full of di ff use ( ∼ 10 -22 -10 -21 g cm -3 ) and hot plasma ( ∼ 10 7 -10 8 K) due to the shocked stellar winds and their interactions. This is illustrated in Figure 2 that shows density maps across di ff erent simulation times. The maps show projected density fields along the line-of-sight direction ( z axis) weighted by density, i.e. R ρ 2 dz / R ρ dz , in order to highlight the densest regions with the highest resolution. The top panels show two models at t = -700 yr, respectively. In them, it is possible to observe the complex structure developed due to the stellar wind interactions such as bow shocks, instabilities, and dense clumps as a result of condensation through radiative cooling. Left- and right-hand side panels display models with di ff erent chemical compositions WR_f07 and WR_f1, respec- tively. On the left-hand side, it can be seen how mildy e ffi cient cooling a ff ects mostly one of the stellar winds. This star corresponds to IRS 33E and its wind is the slowest ( ∼ 450 km s -1 ). This fact causes its shocked temperature to be in an e ffi cient region of the radiative cooling functions that allows its material to cool down and become denser. Also, some of its material manages to reach the vicinity of Sgr A*, as the map shows an elongated clump being accreted and stretched. The right-hand side also portrays this picture but since the radiative cooling is enhanced more dense clumps can be observed, especially close to IRS 33E and Sgr A*. Finally, the bottom panels of Figure 2 show the models at time t = 0, i.e. the present time. One can clearly see that the amount of dense material that has accumulated around Sgr A* is di ff erent in both maps. Although in the model WR_f07 (see bottom left-hand side panel of Figure 2) some dense material is spiraling towards the black hole overall there are is no clear structure around it. In the case of WR_f1 (see bottom right-hand side panel of Figure 2), the dense material has settled at the centre in a a disc-like structure. This di ff erence is due to the e ffi ciency of the radiative cooling. Since model WR_f1 has enhanced cooling, more dense clumps and filaments form, and some of them manage to fall onto Sgr A*. Overall, the hydrodynamic evolution of the two models is analogous: the winds fill the domain during the first ∼ 500 yr, then the systems reach a quasi-steady state, and in the last 500 yr they diverge as the model WR_f1 creates much more dense, cool material that settles around Sgr A*. A quantitative analysis of the evolution of the properties of system is presented as follows. To quantify the accretion rate at di ff erent spatial scales we calculated the mass flux as a function of both radial distance and time ˙ M ( r , t ) = 4 π r 2 ρ ( r , t ) vr ( r , t ) averaged over a spherical shell. Here, positive and negatives signs in the radial velocity refer to outflow and inflow mass fluxes, respectively. First, we analyse the net mass flux at the innermost radius we can resolve properly, which is equivalent to five times the inner boundary radius, i.e. 5 r in = 5 × 10 -4 pc ≈ 1 . 25 × 10 -2 '' . Figure 3 shows | ˙ M ( r = 5 r in , t ) | as a function of time for models WR_f07 and WR_f1 that are displayed as solid blue and orange lines, respectively. In both cases, at t > -3300 yr the net mass flux reaches levels of ∼ 10 -6 M ⊙ yr -1 with short variability episodes that increase the rate by factors of two to four. During this quasi-steady phase, both simulations display similar behaviour qualitatively, likely determined by the identical stellar wind configuration. This stage lasts until t ≈ -500 yr where the mass flow rates start to deviate from each other. The fiducial model continues to display a variability amplitude of the same order of magnitude. However, the WR_f1 model shows a transition to a strongly gas inflow dominated phase, with inflow rates that are enhanced by a factor four to eight. This stage of the evolution is what we refer to as the disc formation phase, analogously to our previous models reported in Calderón et al. (2020b). Next, we proceed to analyse the time-averaged behaviour of the simulations over the last 500 yr, as this can give us an idea of the general state of the system minimising the e ff ects of the stochastic variability. First, we analyse the (volume-weighted) density and (mass-weighted) temperature radial profiles of the simulations, which are shown on the top and bottom panels in Figure 4, respectively. The dashed blue and solid orange lines represent the simulations WR_f07 and WR_f1, respectively. The orange lines in both panels clearly highlight the presence of the cold disc. The density profile decays with r 2 in both cases at large scales ( ≳ 1 '' ). This is the result of most of the material flowing outwards at these scales following a roughly isotropic spherical wind as seen in previous models (Ressler et al. 2018; Calderón et al. 2020b). However, the profiles di ff er at smaller scales ( < 1 '' ) where the fiducial case transitions to ρ ∝ r -1 , while the model WR_f1 shows a density enhancement due to the presence of the disc. This increase in density is more than one order of magnitude. Regarding the temperature profiles, both models match at larger scales ( ≳ 1 '' ) with a constant temperature of the order of 10 7 K, set by the stellar wind collisions. Again, the profiles differ at smaller scales where the fiducial case follows T ∝ r -1 , and the enhanced cooling case displays a temperature profile about two orders of magnitudes lower on average. The minimum in temperature corresponds to the region where the disc contributes with most of the mass for a given spherical shell. To quantify and characterise the mass inflow and outflow regimes in the simulation domain we calculated them as radial profiles. Figure 5 displays the absolute value of the mass flow rates as a function of distance from Sgr A*. The solid and dashed lines represent the mass inflow and outflow rates, respectively; while the colours follow the same convention as before. Thus, if the solid line is above the dashed line the net mass flow is inwards and vice-versa. Analogous to the model by Ressler et al. (2018), the fiducial model encompasses di ff erent regimes due to the dominant direction of the mass flow at given spatial scales. At r > 3 '' , the outflow component dominates and the inflow is negligible. We find a net mass outflow rate of ∼ 5 × 10 -4 M ⊙ yr -1 . Notice that the enhanced cooling model exhibits exactly the same behaviour at these scales. At smaller scales (1 '' -3 '' ), there is a transition region where | ˙ M in | ∼ | ˙ M out | that corresponds to the location of most mass-losing stars and wind interactions. For model WR_f1, the change is more abrupt due to the presence of the cold disc. In both models, at r < 1 '' the inflow component is larger than the outflow, so the material inflows across these scales and down to the innermost boundary. The net e ff ect makes the mass inflow rate about five times larger which generates the cold disc. The origin of the infalling material can be traced in two ways: analysing the angular momentum of the material close to the inner boundary, and using the scalar tracer fields that we introduced to label the chemical abundances of the di ff erent WR sub-types. Figure 6 shows Hammer projections of the angular momentum direction of the gas enclosed in a sphere of radius 0 . 25 '' ( ∼ 0 . 01 pc) for the fiducial and enhanced cooling runs in the top and bottom panels, respectively. Each point represents a di ff erent simulation time over the last 1000 yr, and they are connected with dashed lines. For reference, we added the angular momentum direction of the orbits of the mass-losing stars as black star symbols, and the location of the clockwise disc based on the orbits we used as a grey shaded circle. This analysis shows that the angular momentum direction of the infalling - - Sun 10 2.0 material varies stochastically but overall tends to align with the orientation of the orbits in the clockwise disc. However, the degree of variability depends on whether or not the model results in the formation of a disc. The fiducial model shows more variability while the enhanced cooling run displays that the angular momentum aligns more consistently with the orientation of the clockwise disc. The fact that the infalling material at this spatial scales comes from these stars agrees with previous numerical models (Cuadra et al. 2008; Ressler et al. 2018; Calderón et al. 2020b). Furthermore, this is not surprising since these stars orbit closer to Sgr A*, and have winds that are relatively slow ( ∼ 600 kms), which results into shocks more prone to radiative cooling and with smaller angular momentum. Overall, the fiducial model displays properties of the medium and gas dynamics consistent with the previous models that do not form a disc (Cuadra et al. 2008; Ressler et al. 2018, 2020; Calderón et al. 2020b, run for 1100 yr). The model with enhanced cooling is consistent with the simulations that show the formation of a cold disc as an outcome of this system (Calderón et al. 2020b). Before discussing if any of the models can be favoured given the current observational constraints we proceed to characterise the cold disc arising in the WR_f1, as its properties will aid us to do so.",
"pages": [
4,
5,
6,
7,
8
]
},
{
"title": "3.2. Properties of the cold disc",
"content": "At the present time ( t = 0), the model WR_f1 shows the presence of a cold disc. Figure 8 shows zoomed maps of the central region of 2 . 5 '' × 2 . 5 '' . The left- and right-hand side panels show projected maps of density and line-of-sight velocity (both weighted by density), respectively. Here it is possible to observe that the projected diameter of the disc is roughly 1 '' . We estimated the mass of the disc by isolating the cold material ( T < 10 5 K) and integrating the cell mass content within a sphere of radius 1.0 '' . We found that the total mass of the disc is 0.005 M ⊙ which is consistent with the value in our previous work (Calderón et al. 2020b). However, the mass accumulated in the disc does not converge and keeps increasing at least for the next hundreds of years in the simulation. On the right-hand side panel of Figure 8, we can see that the line-of-sight velocity peaks at ∼ 2000 km s -1 along both directions (towards and outwards the observed). This coincides with the maximum extension of the observed H30 α line (Murchikova et al. 2019). Additionally, the disc is observed to be ∼ 45 · tilted in projection. This is not in agreement with the observations since, at least in projection the di ff erence is about ∼ 90 · (see Figure 1 of Murchikova et al. 2019).",
"pages": [
8
]
},
{
"title": "4. Post-processing of observables",
"content": "In order to assess the validity of our models for the Galactic centre we proceed to synthesise observational quantities that we could compare easily with observations. We focus on the recombination lines H30 α and Br γ . Afterwards, we calculate the X-ray spectrum from our models.",
"pages": [
8
]
},
{
"title": "4.1. Recombination lines: H30 α and Br γ",
"content": "The cold gas at the location of Sgr A* has been observed through the H30 α recombination line (Murchikova et al. 2019; YusefZadeh et al. 2020), and only with an upper limit in the Br γ recombination line (Ciurlo et al. 2021). In order to explore if the models are consistent with these observations we have calculated the expected flux from these emission lines. To compute 1e3 10 10 the emission of the H30 α line we used the coe ffi cients given below fitting a piecewise linear function to them as a function of density and considering a decay ∝ T -1 (see Murchikova et al. 2019, supplementary information), In the case of the Br γ line, its emissivity can be calculated following Schartmann et al. (2015) as Then, in each cell of the domain we computed the respective emissivity values, multiplied them by their n p n e (assuming full ionisation), and then integrated the data cube along the z coordinate which is perpendicular to the line of sight. It is important to remark that in this calculation it is necessary to take into account the hydrogen nucleus n p and electron n e densities appropriately. Since our models consider di ff erent amounts of hydrogen in the plasma we include only the material that contains some, i.e. the material coming from the WN89 / Ofpe stars. Additionally, we only used the hydrogen fraction of that material which is X H = 11 . 5% and X H = 40% for the models WR_f07 and WR_f1. idea of the whole flux of the model we chose a size of p = 1 '' that encloses entirely the disc. The flux values obtained are shown in Table 3 where we also added the upper limit for Br γ from Ciurlo et al. (2021), and the flux of H30 α measured from integrating the spectrum across the velocity channels given by Murchikova et al. (2019). In the case of the Br γ line, we can see that only the model WR_f07 has a flux consistent with the upper limit reported for an aperture of p = 0 . 23 '' . The model WR_f1 has a flux that is ∼ 100 times higher than the upper limit. Also, we see that the disc in our models extends beyond this projected radius, yet its emission is dominated by the inner p = 0 . 23 '' (see Table 3). The situation is more complex for the H30 α line since the observed flux is in both cases higher than the synthetic emission. Although the model that forms a disc is closer it still remains 30% times fainter than the reported value. The model WR_f07 has negligible emission as it is ten order of magnitudes fainter compared to the observation. Thus, it is still not clear how to reconcile the models with both line emission simultaneously as none of them is capable to reproduce the observations. We discuss further the interpretation of these results in Section 7.1.",
"pages": [
8,
9
]
},
{
"title": "4.2. Spectral X-ray emission",
"content": "The X-ray emission from the central parsec is also an observable that has allowed validation of previous numerical models (see Russell et al. 2017; Ressler et al. 2018; Wang et al. 2020). Here we also post-processed our models aiming to obtain the Iron Kalpha spectrum of this region at ∼ 6 . 7 keV as this is the strongest X-ray feature from Sgr A* (e.g. Russell et al. 2017; Corrales et al. 2020). Our approach is based on the procedure outlined in Balakrishnan et al. (2024b) but adapted to our finite-volume grid-based hydrodynamic model, which is briefly described as follows. First, we assigned energy dependent X-ray emissivities j k E = n k e n k i Λ ( E , T k ) that were taken from the vvapec model (Smith et al. 2001), which was obtained with xspec (Arnaud 1996). The densities and temperatures were taken from the last output of our simulation, i.e. at t = 0. Additionally, the tracer fields were used to identify the composition fraction of the parcels of gas according to the WR sub-types. Since the optical depth through the simulation domain is optically thin in the X-ray, the radia- Notes. All fluxes correspond to quantities seen from Earth, i.e. at a distance of 8.33 kpc, and are given in units erg cm -2 s -1 . The observed H30 α flux was calculated integrating the reported flux density across the velocity channels, and taking into consideration the error propagation of the 0.3 mJy reported (see Figure 1 in Murchikova et al. 2019). v th. This function is given as follows tive transfer equation to solve is simply given by the integration along the line of sight of the emissivities of the respective cells. Given that we can trace the fraction of the material that comes from a given WR sub-type this actually corresponds to a linear combination of the tracer field si and the abundance dependent emissivity for each cell, i.e. where ϕ k represents a Gaussian function that it is used to obtain the emission across di ff erent velocity channels, and to take into account the thermal broadening represented with a velocity of We performed calculations through velocity channels spanning line-of-sight velocities v los from -3000 km s -1 to 3000 km s -1 . These were later converted into energy space via Doppler broadening. Thus, for each energy bin from the emissivity tables we obtained 150 line profiles using a fine resolution of 6400 per dex. In order to sum them appropriately we interpolate them into a unique energy range so that we can obtain a single spectrum that includes both the continuum and line contributions. The result of this procedure is shown in Figure 10. This contains the X-ray spectrum emitted from a sky-projected annulus of 0 . 5 '' < p < 1 . 5 '' for models WR_f07 (solid blue line) and WR_f1 (dashed orange line). As a reference, we included the estimation from Balakrishnan et al. (2024b) computed from an analogous model that used the SPH approach, which we discuss in the following section. All spectra show qualitative agreement highlighting the di ff erent features of the Fe complex at 6.7 keV and 6.9 keV. This is also reflected in the broadening of the lines due to the velocity across the line of sight, which is expected since the models are based on the same stellar orbits and wind properties. It is relevant to highlight the level of agreement between the finite-volume and SPH models. The qualitative agreement is reasonable and the level of the emission is of the same order of magnitude. The exact quantitative di ff erence among the models could be mainly attributed to the exact cooling table (composition) used in each simulation, which is the result of the di ff erent density and temperature distributions that can be observed in Figure 4. This is why the WR_f07 shows the highest flux level as it has the least e ffi cient cooling and, as a result it has hotter material. Analogously, the run WR_f1 with an enhanced cooling specified in the composition of its plasma results in a reduction of ∼ 40% of its flux. In case of the SPH model, the quantitative di ff erences could be a ff ected due to the intrinsic ability of the generic SPH to capture shocks which impacts directly the density, velocity structure, and the temperature of the medium. Notice that the flux level of the spectrum of the SPH model by Balakrishnan et al. (2024b) is within the cases explored in this work. Their cooling model is not the same as the ones explored here but this analysis show that it is equivalent to a cooling e ffi -ciency between both of our models. Although not shown here we also compared the X-ray spectra from simulation WR_f1 before and after the disc is formed, and found negligible di ff erences. Finally, we also analysed the slope of the continuum emission within this energy range and ∫ ∫ - 2000 found no significant di ff erences among the models. Thus, in this work the presence of the disc does not imprint a feature in the X-ray spectrum that is resolvable by any current or forthcoming X-ray telescope, in agreement with Balakrishnan et al. (2024b).",
"pages": [
9,
10,
11
]
},
{
"title": "5. Comparison with Lagrangian models",
"content": "Hydrodynamic simulations of the feeding of Sgr A* by the WR stellar winds have been conducted with di ff erent numerical tools. In this work, we have presented the results of using a finitevolume approach that solves the hydrodynamic equations in the Eulerian form. However, there has been extensive work on this problem using codes that solve the equations in the Lagrangian form. Specifically, the use of SPH has been the main choice for such a task (Cuadra et al. 2005, 2006, 2008, 2015; Russell et al. 2017; Wang et al. 2020). Despite the limitations of the generic SPH technique (e.g. capturing shocks) the models managed to reproduce the observed accretion rate at the Bondi radius (Cuadra et al. 2008) as well as the X-ray emission (Russell et al. 2017). In this context, we have conducted a comparison of the Eulerian models calculated with Ramses, and the Lagrangian models computed with the SPH code Gadget (Springel 2005). The SPH simulation setup is as similar as it can be to our setup. This corresponds to an updated version from the models in Russell et al. (2017). For more details on the setup of the SPH models, we refer the reader to Balakrishnan et al. (2024a,b). We focus the comparison on the analysis on the final state of the system, specifically in the cold discs formed after 3000 yr. ous structure towards smaller scales while the Lagrangian disc displays a ring-like structure. This feature can be attributed to the spatial resolution and the inner boundary radius. The Eulerian models indeed manage to resolve smaller scales towards the centre, and the radius of the inner boundary is 5 mas while the Lagrangian models resolve ∼ 1 mas and have an inner boundary of 0 . 1 '' . As a result, the Lagrangian model obtained a disc 50 times lighter ( ∼ 10 -4 M ⊙ ) than in the Eulerian simulation. This di ff erence could be attributed to the radiative cooling employed in each simulation: WR sub-type abundance and Solar with 3 Z ⊙ in the Eulerian and the Lagrangian model, respectively. Regarding the projected orientation of the disc both models seem to be in agreement. This can be analysed more carefully on the lineof-sight velocity maps (see right-hand side panels of Figures 8 and 11). Both maps display that the discs indeed match their projected orientation as well as their most blue- and redshifted velocities are roughly -2000 km s -1 and 2000 km s -1 . Overall, the agreement on the outcome a cold disc around Sgr A* whose properties align between two di ff erent approaches indicates that this is a result independent of the numerical technique. Thus, this shows that the determinant factors to obtain this results lies in both the long-enough simulation time ( ≳ 3000 yr ) as well as e ffi cient radiative cooling.",
"pages": [
11
]
},
{
"title": "6. Disc stability analysis",
"content": "In this section, we present an analysis of the stability of the observed cold disc in the Galactic Centre. Based on its reported properties, the disc should be depleted by accretion after its viscous timescale t ν ≈ 43 kyr, assuming an α = 0 . 1 thin disc (Shakura & Sunyaev 1973), a process much slower than its generation as modelled in our simulations, although faster than its formation out of CND material (Solanki et al. 2023). Nonetheless, there are several external mechanisms that could be capable of destroying the disc faster than through accretion. Among them are the gravitational and hydrodynamic interactions of the disc with the stars, the stellar-wind-disc collisions due to the presence of the S-star cluster in its location, and / or the evaporation due to heat flowing between the hot medium and the cold disc. Although some of these physical scenarios have been studied analytically and numerically, there has not been any direct application to the cold disc discovered in our Galactic Centre.",
"pages": [
11,
12
]
},
{
"title": "6.1. Stellar cluster perturbing a thin disc",
"content": "The stability of a cold disc perturbed by stellar passages has been investigated by Ostriker (1983). Their work presented an analytical approach to calculate how the disc loses angular momentum due to the passages, considering the integrated e ff ect of a stellar cluster in stationary state. Besides this work, most e ff orts have been put on following how the stellar dynamics are a ff ected by the disc presence (e.g., Fabj et al. 2020, and references therein) or on the e ff ect of embedded accreting stars and / or black holes on the disc (e.g., Tagawa et al. 2021) rather than on the consequences to the disc properties. In this Section, we will summarise the approach of Ostriker (1983) and apply it to the Galactic Centre case. Let us start considering a cold, thin accretion disc around a compact object. Within the thin disc we assume that its height H ( r ) is small compared to the radial coordinate r , i.e. H ( r ) / r ≪ 1. Also, we consider that the sound speed of the material in the disc is small compared to the speed of the stars at the same radius. In this scenario, a star passing through the disc will produce a torque that removes angular momentum from the disc. We consider that the star interacts with the disc gravitationally and hydrodynamically The gravitational interaction of a star crossing the disc can be described using the Bondi-Hoyle approach (Bondi & Hoyle 1944), while the hydrodynamic interaction is just a physical collision of a solid sphere with a gaseous disc. Let us consider a system of reference where the x axis is in the direction of rotation and z is in the direction of the rotation pole of the disc. Then, according to Ostriker (1983) the total change of linear momentum along the x direction ∆ px for a single stellar passage is where R ∗ is the radius of the star, Σ d = R disc ρ d H ( r ) is the surface density of the disc, ρ d is the volume density of the disc, q is the hydrodynamic drag parameter that is set to q = 2 assuming a large Mach number collision, v 0 is the local stellar velocity, Λ D ≈ H ( r ) v 2 0 / ( R ∗ v 2 ∗ ) is the Coulomb logarithm, v ∗ is the escape velocity from the surface of the star, η = v ∗ / v 0 is the hardness parameter, v d is the disc velocity, v rel is the relative velocity vector, and ( θ, ϕ ) are the spherical coordinate angles along the polar and azimuthal directions. In order to consider the e ff ect of the complete stellar cluster interacting with the disc we need to integrate over the cluster phase-space volume. First, we integrate over the local stellar velocity distribution. Let us define dN as the number of stars in the velocity range v → v + dv , i.e where f ( v ) is the velocity distribution function, n ∗ is the stellar number density. Then, the number of crossings per unit time and area in the ( θ, ϕ ) direction is Article number, page 12 of 19 Now, integrating over the phase space, the total transfer of momentum to the disc per unit area at r is where µ = cos ϕ . As the local momentum per unit area in the disc is px , d( r ) = Σ ( r ) /τ d( r ) the inverse of the drag timescale to remove the angular momentum of the disc is τ d = -px , d / ˙ px , tot can be written as where the quantities ( I 0 , I 1) depend solely on the stellar distribution function. For instance, in the case of a Maxwellian distribution the values of these parameters can be obtained numerically and are I 0 = 12 . 553 and I 1 = 0 . 474. In order to apply this model to the Galactic Centre we need to calculate the stellar density within the size of the disc. Following Schödel et al. (2007) the stellar mass density at r < 6 arcsec is described by Assuming that the stellar mass density is composed mainly by stars whose radii are R ∗ = 1 R ⊙ and move on their orbits typically at v 0 = 1000 km s -1 we can use Equation 20 to estimate the inverse of the drag timescale, Based on this calculation the angular momentum of the cold disc should be removed after ∼ 185 Myr of interactions of the stars. This is much longer than the formation time-scale found in our simulations. Furthermore, it is even much longer than the age of the Wolf-Rayet stars, which is about 6 Myr (Martins et al. 2007). Then, the gravitational and hydrodynamic interactions of the stars onto the disc should not a ff ect the stability of the disc.",
"pages": [
12
]
},
{
"title": "6.2. Stellar winds perturbing a disc",
"content": "In the case of stellar irradiation a ff ecting the state of the accretion flow there have been many studies mainly motivated by the recent pericentre passages of the star S2 around Sgr A*. For instance, Cuadra et al. (2003) could rule out the existence of an optically-thick disc based on the lack of its thermallyreprocessed emission as it gets illuminated by S2. Nayakshin et al. (2004) estimated that a star as luminous as S2 could heat up and ionise an inner disc, which could enhance the accretion rate. Giannios & Sironi (2013); Christie et al. (2016) calculated the bremsstrahlung emission as a result of the stellar wind shocking a Radiatively Ine ffi cient Accretion Flow (RIAF) around Sgr A*, finding an X-ray luminosity comparable to quiescent emission of Sgr A*. In fact, no noticeable increase was detected during 2018 (see Table 1 of Andrés et al. 2022). Hosseini et al. (2020) constrained the observed L ' -band variability of S2 to be about 2-3%, based on a model of bow shock of its stellar wind, implying an ambient density < 10 5 cm -3 , which rules out the presence of a standard accretion disc at S2's pericentre, in agreement with the previous studies. It is important to remark however that such a limit marginally allows the existence of both the disc reported by Murchikova et al. (2019), and also the one in our simulation. The stars in the S cluster also have relatively powerful stellar winds. As the stars inhabit the black hole in the vicinity of the cold disc, it is expected that the winds interact with it. By either opening a bubble due to the high kinetic energy carried in the winds or depositing thermal energy they a ff ect the disc properties. We proceed to perform analytical estimates to quantify the impact of these processes.",
"pages": [
12,
13
]
},
{
"title": "6.2.1. Bubbles in the disc",
"content": "Let us consider a star blowing an isotropic stellar wind with a mass-loss rate ˙ M w at a terminal speed v w. Then, its density ρ w at a distance r from the star is given by If such a star is immersed in a medium whose number density is n m at rest at a temperature T m, the radius of the bubble r b opened by the wind can be calculated equating the ram pressure of the wind and the thermal energy of the medium, i.e. where we have used as fiducial values the disc parameters measured by Murchikova et al. (2019) and typical stellar wind properties for B stars (Oskinova et al. 2011; Krtiˇcka 2014). Then, the radius of the bubble is about 0.012 arcsec, which is approximately one tenth of the disc radius. In order to open such a bubble a star would need to be within the disc fort at least the wind crossing time, i.e. r b / v w, which is about ∼ 0 . 5 yr. Let us estimate the duration of a stellar passage through the disc. Along the perpendicular direction a star crossing the disc would take at most 0 . 1 R d / v 0, being H ( r ) / r = 0 . 1. One S star moves typically at v 0 ≈ 1000 km s -1 , then 0 . 1 R d / v 0 ≈ 1 yr. As both timescales are comparable, we conclude that the bubble can be opened. For such a perturbation to last we need to check if either shear or sound waves could eliminate it, i.e. close the bubble. In the case of shear, this is given by the orbital speed of the disc and the size of the bubble: As this expression increases with r , the largest value will be at the outer edge of the disc, then t close < 5 yr. This value should be interpreted as the longest duration of a perturbation of the wind onto the disc. As there are 12 S-stars on orbits whose pericentre distances are shorter than the radius of the disc and their orbital periods are of the order of ∼ 10 yr (Gillessen et al. 2017) we expect a bubble to be opened every year on average. However, as shear would close such bubbles in less than five years there would be at most four bubbles opened on average in a stationary state. As the size of the bubbles is very small compared to the size of the disc (two orders of magnitude in area), four of them would not have an impact on its structure. The sound crossing timescale r b / c s ≈ 30 yr is longer than the shear timescale so we do not expect it to have an impact on erasing perturbations.",
"pages": [
13
]
},
{
"title": "6.2.2. Winds injecting thermal energy",
"content": "The S stars are of spectral type B with masses of 10 M ⊙ (e.g. Habibi et al. 2017), thus their mass-loss rate must be ∼ 10 -8 M ⊙ yr -1 with terminal velocity of 1000 km s -1 . The supersonic nature of the winds should be enough to compress the shocked material at high temperature. Potentially, this thermal energy could be deposited onto the disc during each stellar passage. If we consider that most of the kinetic energy of the wind is transformed into thermal energy about ∼ 6 . 4 × 10 33 erg would be deposited in the disc during the ∼ 1 yr long passage (see above). In order to check if the wind energy can significantly impact the thermodynamic state of the disc let us estimate its total internal energy U int. where k B is the Boltzmann constant and V d is the disc volume. Replacing the observed disc properties we obtain that U int = 1 . 63 × 10 42 erg. From this calculation it is clear that even if all the energy from the wind in a stellar passage is deposited the contribution is still negligible to modify the state of the disc. Even if we consider the integrated e ff ect of all the S stars passages on a timescale comparable to the viscous timescale of the disc ( ∼ 43 kyr) the contribution is still < 0 . 1% of the total thermal energy of the disc. Thus, the energy supplied by the stellar winds into the disc is not enough to a ff ect its state.",
"pages": [
13
]
},
{
"title": "6.3. Thermal conduction",
"content": "The vicinity of Sgr A* at the Bondi radius is made out of a diffuse (10 cm -3 ), hot medium (10 7 K; Bagano ff et al. 2003) due to the shocked stellar winds from the Wolf-Rayet stars in the region. At the disc vicinity ( ∼ 10 3 R Sch), the conditions are more extreme as the plasma has a density of about ∼ 5 × 10 3 cm -3 (Gillessen et al. 2019) and temperature of ∼ 10 8 K. As the disc is significantly colder it is expected that heat flows from the medium to the disc. The large temperature gradient should enable thermal conduction to take place and likely evaporate the disc. The problem of evaporating a cold structure sitting in a hot environment due to thermal conduction was first studied by Cowie & McKee (1977). In that work, the authors derived an analytical expression for the mass-loss rate and evaporation timescale for a spherically symmetric gas cloud. Unfortunately, these expressions are not valid for more complex geometries. Meyer & Meyer-Hofmeister (1994) studied specifically the evaporation of a cold thin accretion disc in a hot corona. Although the model considered only one radial zone it was able to calculate the full perpendicular structure of the accretion disc and the corona by means of numerical simulations. Liu et al. (2004) used the model by Meyer & MeyerHofmeister (1994) to analyse which solutions were consistent with the observational data available at the time in case there were a cold disc in the center of the Galaxy. They found that any transient disc would evaporate quickly. With their obtained evaporation rate of ∼ 10 -4 M ⊙ yr -1 , the life-time of a disc with properties such as found by Murchikova et al. (2019) or formed in our models would be ∼ 1 yr. Similar analysis have argued that small clumps ( ∼ M ⊕ ) in the Galactic Centre should also evaporate quickly ( ∼ 1 yr) due to thermal conduction (Burkert et al. 2012; Calderón et al. 2018). According to this, cold gas should not be able to last long if formed, yet we observe many cold gas structures in the central parsec: the many dusty cold clumps detected in the vicinity of IRS 13E (Fritz et al. 2010; Peißker et al. 2023), and a larger gaseous cloud X7 (Ciurlo et al. 2023). Thus, this regime of classical thermal conduction might not be applicable to the Galactic centre environment. Another possibility has been studied by Nayakshin (2004) who focused on the same problem and found solutions in a di ff erent regime, where the electron mean free path is long enough that an accretion disc does not evaporate but rather increases its mass by condensation. A more thorough analysis of the role of thermal conduction is beyond the scope of this work and is deferred to future study. Our findings shows that gravitational and hydrodynamic effects require timescales that are too long to have an impact on the disc stability. On the other hand, we find that thermal conduction in the classical case should evaporate the observed disc in about one year. Given the time-scale for disc formation found in our model, the Nayakshin (2004) condensation regime remains to our knowledge as the only viable physical scenario that would allow its continued existence and potential identification with Murchikova et al. (2019)'s reported disc.",
"pages": [
13,
14
]
},
{
"title": "7. Discussion",
"content": "In this section, we interpret our findings exploring the uncertainties in the abundances of WR stars in general and of the ones in the central parsec. Additionally, we contrast the properties of the simulated and observed cold discs as well as discuss further implications.",
"pages": [
14
]
},
{
"title": "7.1. Is the simulated disc the observed disc?",
"content": "In order to compare the simulated and the observed discs, first we analyse their physical properties. The simulated disc has a total mass of ∼ 5 × 10 -3 M ⊙ , while the structure observed in H30 α was reported to have a mass of 10 -5 -10 -4 M ⊙ . However, comparing these quantities might not be appropriate since the mass of the observed structure is based on three assumptions: a thin disc with uniform density and temperature, and a masing factor of ∼ 100. All of them minimise the value of the mass inferred from the observations. For instance, dropping the maser assumption the inferred mass increases in a factor ten (see Equations 30-31 of Supplementary Material in Murchikova et al. 2019). Bearing this in mind, we could argue that the mass of the simulated and observed disc might be of the same order provided there was no maser. Although removing that assumption would create tension reconciling both the Br γ and H30 α observed fluxes. Regarding its extension, the simulated disc has a projected radius of ∼ 0 . 5 '' , while the observed disc has a radius of 0 . 23 '' . However, since most of the recombination line emission of the disc is originated from its central region ( p < 0 . 23 '' , see Table 3) the sizes of the simulated and observed discs are in agreement. A more direct comparison can be done through the analysis of the mock observational quantities computed from the simulations. In Section 4.1, we computed the H30 α and Br γ recombination line emission and estimated their fluxes. Here, we have found that both synthetic fluxes are in tension with the observations. First, the simulated Br γ flux is 100 times higher than the upper limit. It is relevant to remark that this limit is already extinction corrected. Second, the simulated H30 α flux is 30% lower than the observed one. These results di ff er from the estimates by Ciurlo et al. (2021) from our previous models due to, again, the uniform disc assumptions. The simulated disc is not thin or possess uniform density or temperature. Although the mean density and temperature of the simulated disc is of the order of ∼ 10 -5 cm -3 and ∼ 10 4 K its geometry is more complex with a larger variance in its properties. As a result, both the synthetic Br γ and H30 α fluxes are much higher than the ones inferred by Ciurlo et al. (2021). Based on the line-of-sight velocity maps shown in Murchikova et al. (2019), the orientation of the observed disc displays the red- and blueshifted sides to be on the East (lefthand) and West (right-hand) sides, respectively. Although our model WR_f1 also shows this configuration the exact inclination of the discs does not match perfectly. Specifically, the discs are inclined in ∼ 90 · among each other, being the simulated disc tilted in the clockwise direction on the sky. In principle, this argues against the WR stars as the main responsible for the formation of the observed disc. But we cannot rule out completely their role in the process, since a combination of the WR stars with other gaseous structures, such as the minispiral or the CND, acting simultaneously might result in a net e ff ect that manages to reproduce the observed angular momentum. Solanki et al. (2023) showed that material from the circumnuclear disc could fall close to Sgr A* though on much longer timescales. Certainly, this scenario would be much more di ffi cult and challenging to simulate with the appropriate chemical abundances as well as with high-enough spatial resolution. It is also relevant to notice that the observed accretion flow orientation at ≲ 10 RSch is consistent with the orientation of the clockwise stellar disc (GRAVITY Collaboration et al. 2020; Event Horizon Telescope Collaboration et al. 2022). Thus, our models reproduce that connection between the dynamics from large ( > 10 5 RSch) to small ( ≲ 10 RSch) scales. However, the observed cold disc does not seem to follow such a connection. Finally, an intriguing aspect of our findings is that the WR atmospheres with more H in them would favour the formation of the disc. At the same time, the disc indeed was detected in the H30 α recombination line, which obviously indicates the presence of H. Thus, it is sensible to ask how much H can be in the winds of WR stars but, specifically in the ones of the sub-type WN89 / Ofpe. This point is discussed in the following section.",
"pages": [
14
]
},
{
"title": "7.2. Uncertainties in abundances",
"content": "The results of the hydrodynamic models demonstrate that the chemical abundances of the material in the WR winds are the key factor to determining whether or not a cold disc can be formed due to their action. Unfortunately, there are two large uncertainties in this context. First, the metallicity of the young population of the nuclear star cluster is not well constrained (e.g. Genzel et al. 2010). Although there is agreement that it is above Solar its exact value has not been established so far, with most studies quoting the range Z = 2 -3 Z ⊙ . However, even if the metallicity is determined more precisely, the problem would not be entirely solved, since the abundances of the plasma in the region are dominated by the material supplied by the WR star winds whose compositions are much more uncertain. Theoretically, the evolution of these stars is a topic of active research, and improved prescriptions for mass loss are changing the modelled properties of observed WR stars (see e.g., Gormaz-Matamala et al. 2023, and references therein). Simply based on observations, up to now, it has been reported that some of these stars might possess from X H = 0 to X H = 50% for Galactic WR stars of the WN subtype (Hamann et al. 2006, 2019), and even as high as X H = 60% for extragalactic ones (Todt et al. 2015). In the vicinity of Sgr A*, Martins et al. (2007) characterised the wind properties of the sample of WR stars through the analysis of their infrared spectra. They reported abundance H / He ratios in the range 2-5 for these stars which does not allow us to rule out the scenario regarding this aspect. A variation on the hydrogen abundance in the chemical mixture in a factor five might change the net radiative cooling rate by a factor 1 . 1 -1 . 4, especially at lower temperatures. In our models, WR_f07 considered a fiducial value of hydrogen mass fraction of 11.5% (Russell et al. 2017) but the model WR_f1 assumed 40%. Such a change was enough to modify the cooling function by about 30%, which ended up affecting significantly the final result and the state of the plasma. Thus, within our current understanding of the abundances in the WR stellar atmospheres both scenario are plausible. Only better quality spectra and more sophisticated models of them could allow us to constrain the wind properties and their composition more precisely.",
"pages": [
14,
15
]
},
{
"title": "8. Conclusions",
"content": "We have presented a numerical study of the system of masslosing stars feeding Sgr A* focusing on the impact of the chemical composition of the plasma. Through studying simulations with di ff erent abundances we have found that the system can either form or not form a cold disc around the black hole, provided that the model is run for long-enough timescales ( ≳ 3000 yr) to create a dense enough medium in the inner parsec of the Galaxy. As in our past work, we have confirmed that if formed the cold disc can impact significantly the hydrodynamic and thermodynamic state of the plasma within the central parsec. As a result, the inferred mass in flow rate at 5 × 10 -4 pc can reach up to 8 × 10 -6 M ⊙ yr -1 , which is 4 -8 times higher than the value without the presence of such a disc. Our models still point to the mass-losing stars in the clockwise disc being the main source of the accreted material due to their dense and slow winds as previous models have shown (e.g. Cuadra et al. 2008; Ressler et al. 2018; Calderón et al. 2020b). In this work, we have followed more carefully the origin of the inflowing material through the use of passive scalar fields for di ff erent WR subtypes. This allowed us to confirm the origin of the accreted material but, more importantly led us to conclude that this is mostly provided by the winds of the stars of the WR sub-type WN89 / Ofpe. This provides evidence that the infalling material must have a nonnegligible fraction of hydrogen as these WR stars have not lost entirely their atmospheric hydrogen via winds. Our models also showed that the formation of a cold disc depends strictly on the radiative cooling employed, which is determined by the chemical abundances of the plasma. Previous models have only considered Solar abundances with 3 Z ⊙ but had not agreed whether or not a cold disc can be formed. From our results in Appendix B, we speculate that at Z ≈ 3 Z ⊙ there is a transition between the regimes of persistent quasi-steady accretion and disc formation, which would imply that small di ff erences in the cooling function or numerical implementation can result in models reproducing either regime. This may explain the di ff ering results obtained by Calderón et al. (2020b) and Ressler et al. (2020) with similar numerical and physical setups. In the case a cold disc is formed around Sgr A* the structure may resemble the extension and the line-of-sight velocity structure of the observed disc. However, direct observational quantities could not be reproduced. Specifically, the exact skyprojected inclination of the simulated and observed discs di ff ers in ∼ 90 · . The Br γ emission line upper limit is 100 lower than the synthetic flux. The observed H30 α emission line flux is 30% higher than in our model. Furthermore, we have contrasted this work Eulerian (finitevolume grid-based) models with analogous (SPH) Lagrangian models in order to address the potential influence of the chosen approach in the outcome of the simulations. As it has been shown in Balakrishnan et al. (2024b), a cold disc is also formed around Sgr A* if the simulation is run for long timescales. Both cold structures agree on their sizes, their velocity structure, and their sky-projected inclination. Although there are di ff erences in the total mass and specific inner structure of the discs this could be attributed to the resolution employed and the exact assumption to consider the innermost boundary. Despite these di ff erences, the agreement is reasonable among the numerical approaches, especially when analysing the synthetic X-ray emission (see Figure 10). The spectral features agree qualitatively across the models presented in this work and the Lagrangian model. The level of the X-ray emission are all within the same order of magnitude but the exact base level depends on the radiative cooling chosen, which is determined by the chemical abundances in the WR atmospheres. We also have explored the stability and long-term evolution of the putative disc under the e ff ect of perturbing agents: the nuclear star cluster, the wind of all the mass-losing stars (including the S stars) impacting the disc through opening bubbles and / or depositing thermal energy via shocks, and the potential e ff ect of thermal conduction. Our analysis showed that gravitational and hydrodynamic e ff ects would impact the disc on longer timescales then the viscous timescale. However, within the classical thermal conduction framework a cold disc in such a region should not exist. Based on this and the disc formation timescale from our simulations, we speculate that the condensation model of Nayakshin et al. (2004) remains a plausible mechanism consistent with the disc existence. In conclusion, through this work we have been able to reconcile the discrepancy among the numerical simulations of the system of mass-losing stars feeding Sgr A*. The formation of a cold disc on ∼ 3000 yr timescale is indeed possible for certain chemical abundances which are consistent with the current observational constraints. Nonetheless, it is not possible to favour or disfavour this scenario due to the uncertainties in them. Thus, high-resolution spectra of the WR stars and more sophisticated modelling of them could allow us to discern the validity of such scenarios. But we must warn that even in the case that the cold disc is a result of action of the WR stars it is not possible to reproduce all its observed properties: its inclination and recombination line fluxes. More sophisticated models that include infalling material from other sources might be necessary to produce the exact inclination of the structure which could also impact the emission of the structure. Acknowledgements. We thank Prof. Sergei Nayakshin and Prof. Stanley Owocki for very helpful discussions about this project. We also thank Dr. Sean M. Ressler for sharing the radiative cooling function shown in Figure 1. Additionally, we are grateful to Dr. Álex Gormaz-Matamala for helping with references and discussions on the WR wind abundances. DC and JC acknowledge the support of the Kavli Foundation through its summer program at the Max Planck Institute for Astrophysics where fruitful discussions took place. In addition, DC thanks the warm hospitality of the Universidad Adolfo Ibañez in Chile and University of Delaware in the USA where part of this project was developed. We acknowledge the support from ANID in Chile through FONDECYT Regular grant 1211429 and Millennium Science Initiative Program NCN2023_002. The research of DC and SR was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - EXC 2121 - 'Quantum Universe' - 390833306. Since 01.10.2024, DC has been funded by the Alexander von Humboldt Foundation. SR is also supported by the European Research Council (ERC) Advanced Grant INSPIRATION under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 101053985), the Swedish Research Council (VR) under grant number 202005044, and the research environment grant 'Gravitational Radiation and Electromagnetic Astrophysical Transients' (GREAT) funded by the Swedish Research Council (VR) under Dnr 2016-06012, by the Knut and Alice Wallenberg Foundation under grant Dnr. KAW 2019.0112. The numerical simulations of this work were run on the high-performance computing system cobra of the Max Planck Computing and Data Facility. Most of the analysis of the numerical data was carried out using the python package yt (Turk et al. 2011). Furthermore, this work made use of python libraries numpy (Harris et al. 2020) and matplotlib (Hunter 2007), as well as of the NASA's Astrophysics Data System.",
"pages": [
15,
16
]
},
{
"title": "References",
"content": "Andrés, A., van den Eijnden, J., Degenaar, N., et al. 2022, MNRAS, 510, 2851 Arnaud, K. A. 1996, in Astronomical Society of the Pacific Conference Series, Vol. 101, Astronomical Data Analysis Software and Systems V, ed. G. H. Jacoby & J. Barnes, 17 Becklin, E. E., Gatley, I., & Werner, M. W. 1982, ApJ, 258, 135 GRAVITY Collaboration, Abuter, R., Amorim, A., et al. 2021, A&A, 647, A59 GRAVITY Collaboration, Bauböck, M., Dexter, J., et al. 2020, A&A, 635, A143 Hosseini, S. E., Zajaˇcek, M., Eckart, A., Sabha, N. B., & Labadie, L. 2020, A&A, 644, A105 Article number, page 16 of 19 Russell, C. M. P., Wang, Q. D., & Cuadra, J. 2017, MNRAS, 464, 4958 Schartmann, M., Ballone, A., Burkert, A., et al. 2015, ApJ, 811, 155 Practical Introduction (Berlin: Springer) Toro, E. F., Spruce, M., & Speares, W. 1994, Shock Waves, 4, 25 Turk, M. J., Smith, B. D., Oishi, J. S., et al. 2011, ApJS, 192, 9 von Fellenberg, S. D., Gillessen, S., Stadler, J., et al. 2022, ApJ, 932, L6 Štofanová, L., Kaastra, J., Mehdipour, M., & de Plaa, J. 2021, A&A, 655, A2 Wang, Q. D., Li, J., Russell, C. M. P., & Cuadra, J. 2020, MNRAS, 492, 2481 Yelda, S., Ghez, A. M., Lu, J. R., et al. 2014, ApJ, 783, 131 Yuan, F. & Narayan, R. 2014, ARA&A, 52, 529",
"pages": [
16
]
},
{
"title": "Appendix A: Numerical setup choices",
"content": "Calderón et al. (2020b) simulated the system of WR stars feeding Sgr A* while orbiting around it following an analogous numerical setup compared to the models presented in this work. There, we had found that the resulting cold disc tended to align with the grid after its formation. In order to remedy this numerical artifact we investigated the use of a MinMod slope limiter, instead of MonCen. However, we warned that it was not straightforward to maintain the spherical symmetry of the winds using the MinMod slope limiter. In this work, we investigated more carefully these numerical choices in order to be confident that our results are robust. First, we investigated the choice of the MonCen slope limiter. To do this, we made use of the same setup presented in this work but paying attention to the values of the passive scalar fields. In principle, the values of these fields should be between zero and one. However, this experiment showed that the MonCen slope limiter produced non-physical oscillations that in some cases resulted in negative values of the passive scalar fields. This behaviour was not observed when using the MinMod slope limiter. Thus, we selected this choice throughout this work. To ensure the spherical symmetry of the stellar winds we had to impose a less restrictive refinement criterion based on density gradients as well as a smoother transition between refinement levels through the parameter nexpand . Second, we tested the choice of the Riemann solver employed but using the MinMod slope limiter. Currently, the HLLC Riemann solver (Toro et al. 1994) is widely used in grid-based hydrodynamic simulations. In this work, we have found that this choice results in a cold disc that tends to align with the Cartesian grid as seen in Calderón et al. (2020b). This could indicate problems in conserving the angular momentum which is a known issue with simulations in Cartesian grids when the relevant regions are not well resolved. With the usage of the exact Riemann solver (Toro 2009), though more computationally expensive, it was possible to avoid this artificial alignment. Thus, the only sensible combination was the use of the exact Riemann solver with a MinMod slope limiter but we had to ensure to count with enough resolution and smoothness in the transition of the refinement levels. As a reference of these experiments, Figure A.1 shows density projection (weighted by density) maps of the final state of the system for two runs with Solar composition with 3 Z ⊙ but with di ff erent Riemann solvers. The left- and righthand side panels display the models using the HLLC and the exact Riemann solvers, respectively. Although overall the density structure looks similar there are subtle di ff erences. The most relevant is related to the disc orientation and structure. On the left-hand side, the disc tries to align with one of the Cartesian axes, while on the right-hand side the disc remains consistently with the shown orientation.",
"pages": [
17
]
},
{
"title": "Appendix B: Solar composition varying metallicity",
"content": "Before investigating the models with the new implementation of the di ff erential cooling we ran some experiments simply modifying the contribution of the metals to the total radiative cooling. We simulated three models varying only the composition used: A1 used Solar composition, A3 used Solar composition with 3 Z ⊙ , and A5 used Solar composition with 5 Z ⊙ . The density projected (weighted by density) maps of the final state of the simulations are shown in Figure B.1. The left-hand side, central, and right-hand side panels display the runs A1, A3, and A5, respectively. A higher metallicity in the plasma increases the radiative cooling directly. As a result, more and denser filaments, clumps, and a denser central disc are obtained with increasing the metallicity. On the contrary, if the metallicity is decreased to a Solar value we observe almost no overdensities, and definitively no disc formation. We speculate that at Z ≈ 3 Z ⊙ there is a transition between the regimes of persistent quasi-steady accretion and disc formation, which would imply that small di ff erences in the cooling function or numerical implementation can result in models reproducing either regime. This may explain the di ff ering results obtained by Calderón et al. (2020b) and Ressler et al. (2020) with similar numerical and physical setups. In order to perform a more quantitative comparison of these results Figure B.2 presents radial profiles of density (weighted by volume) and temperature (weighted by mass) on the top and bottom panels, respectively. The dashed blue, solid orange, and blue dotted-dashed lines represent the models A1, A3, and A5, respectively. The density profiles clearly shows the impact of cooling in the structure of the medium. More e ffi cient cooling, like in cases A3 and A5, results in an order of magnitude denser medium within 1 '' compared to the case with ine ffi cient cooling, i.e. A1. Notice that even higher metallicity in A5 extends the impact the denser region to slightly larger spatial scales. The e ff ect is similar in the temperature profile. In A1, the temperature profile decays with r within the central 1 '' . But when the metallicity is high enough to make cooling e ffi cient the temperature profile can decrease up to two orders of magnitude depending on the exact value. In summary, these experiments allowed us to quantify the impact of radiative cooling by simply decreasing or increasing the radiative cooling influence. However, since the chemical composition in reality is much more complicated and di ff ers from Solar composition significantly these models should be interpreted only as general guides to assess the role of radiative cooling. In the main text of this work, we devoted to explore more physically motivated mixtures that, although depend on more uncertain parameters are more robust when attempting to build a physical model that can be constrasted with observational data.",
"pages": [
17
]
}
]
2024arXiv241106902L
https://arxiv.org/pdf/2411.06902.pdf
<document>
<section_header_level_1><location><page_1><loc_10><loc_83><loc_90><loc_86></location>A Systematic Search for Candidate Supermassive Black Hole Binaries Using Periodic Mid-Infrared Light Curves of Active Galactic Nuclei</section_header_level_1>
<figure>
<location><page_1><loc_33><loc_81><loc_67><loc_83></location>
</figure>
<text><location><page_1><loc_8><loc_78><loc_92><loc_80></location>1 CAS Key Laboratory for Research in Galaxies and Cosmology, Department of Astronomy, University of Science and Technology of China, Hefei, 230026, China; jnac@ustc.edu.cn</text>
<text><location><page_1><loc_24><loc_76><loc_76><loc_78></location>2 School of Physical Sciences, University of Science and Technology of China, Hefei, 230026, China</text>
<text><location><page_1><loc_20><loc_75><loc_79><loc_76></location>3 School of Astronomy and Space Sciences, University of Science and Technology of China, Hefei, 230026, China</text>
<text><location><page_1><loc_24><loc_74><loc_75><loc_75></location>4 Department of Astronomy, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA</text>
<text><location><page_1><loc_18><loc_73><loc_19><loc_73></location>5</text>
<text><location><page_1><loc_14><loc_71><loc_86><loc_72></location>Center for Artificial Intelligence Innovation, University of Illinois at Urbana-Champaign, 1205 West Clark Street, Urbana, IL 61801, USA</text>
<text><location><page_1><loc_13><loc_71><loc_81><loc_73></location>National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA 6</text>
<text><location><page_1><loc_26><loc_68><loc_74><loc_69></location>(Received 2024 May 24; Revised 2024 November 9; Accepted 2024 November 11)</text>
<text><location><page_1><loc_44><loc_65><loc_56><loc_67></location>Submitted to ApJ</text>
<section_header_level_1><location><page_1><loc_46><loc_62><loc_54><loc_63></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_35><loc_86><loc_62></location>Periodic variability in active galactic nuclei (AGNs) is a promising method for studying sub-parsec supermassive black hole binaries (SMBHBs), which are a challenging detection target. While extensive searches have been made in the optical, X-ray and gamma-ray bands, systematic infrared (IR) studies remain limited. Using data from the Wide-field Infrared Survey Explorer (WISE), which provides unique decade-long mid-IR light curves with a six-month cadence, we have conducted the first systematic search for SMBHB candidates based on IR periodicity. Analyzing a parent sample of 48,932 objects selected from about half a million AGNs, we have identified 28 candidate periodic AGNs with periods ranging from 1,268 to 2,437 days (in the observer frame) by fitting their WISE light curves with sinusoidal functions. However, our mock simulation of the parent sample indicates that stochastic variability can actually produce a similar number of periodic sources, underscoring the difficulty in robustly identifying real periodic signals with WISE light curves, given their current sampling. Notably, we found no overlap between our sample and optical periodic sources, which can be explained by a distinct preference for certain periods due to selection bias. By combining archived data from different surveys, we have identified SDSS J140336.43+174136.1 as a candidate exhibiting periodic behavior in both optical and IR bands, a phenomenon that warrants further validation through observational tests. Our results highlight the potential of IR time-domain surveys, including future missions such as the Nancy GraceRoman Space Telescope, for identifying periodic AGNs, but complementary tests are still needed to determine their physical origins such as SMBHBs.</text>
<text><location><page_1><loc_14><loc_30><loc_86><loc_33></location>Keywords: Active galactic nuclei (16); Infrared astronomy(786); Quasars (1319); Supermassive black holes (1663); Time domain astronomy (2109)</text>
<section_header_level_1><location><page_1><loc_21><loc_27><loc_35><loc_28></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_10><loc_48><loc_26></location>Supermassive black holes (SMBHs) are intriguing celestial objects that reside at the centers of galaxies (Kormendy & Ho 2013). When galaxies merge, it is anticipated that the central black holes will form binary systems known as SMBH binaries (SMBHBs, Begelman et al. 1980). These binary systems offer valuable insights into various astrophysical phenomena, including galaxy formation (Colpi & Dotti 2011), gravitational wave (GW) emission (Hughes 2009), and the evolution of SMBHs (Merritt 2013). More massive binaries are pulsar-timing array (PTA) sources (e.g., Arzoumanian et al. 2018), while less massive binaries are targeted by</text>
<text><location><page_1><loc_52><loc_24><loc_92><loc_28></location>space-based experiments such as LISA (Klein et al. 2016). They provide a laboratory to directly test strong-field general relativity (Hughes 2009; Centrella et al. 2010).</text>
<text><location><page_1><loc_52><loc_10><loc_92><loc_24></location>However, despite their theoretical significance, the direct electromagnetic detection of close SMBHBs below subparsec separation has proven to be a challenging endeavor (see recent reviews such as De Rosa et al. 2019; D'Orazio & Charisi 2023) while numerous SMBH pairs at larger scales have been identified through the presence of dual active galactic nuclei (AGNs) (e.g., Komossa et al. 2003; Zhou et al. 2004; Liu et al. 2010, 2011, 2013; Koss et al. 2011; Liu et al. 2018; Chen et al. 2022). When a binary has exhausted its in-</text>
<text><location><page_2><loc_8><loc_75><loc_48><loc_91></location>teractions with stars but has not approached close enough to emit significant gravitational radiation, its orbit does not have an obvious mechanism for decay. This long-standing challenge presents a major uncertainty in estimating the abundance of SMBHB mergers as GW sources. In theory, the bottleneck may be overcome in gaseous environments (e.g., Gould & Rix 2000; Cuadra et al. 2009; Chapon et al. 2013; del Valle et al. 2015), in triaxial or axisymmetric galaxies (e.g., Khan et al. 2016; Kelley et al. 2017), and/or by interacting with a third BH in hierarchical mergers (e.g., Blaes et al. 2002; Kulkarni & Loeb 2012; Bonetti et al. 2018).</text>
<text><location><page_2><loc_8><loc_21><loc_48><loc_74></location>A recent and promising approach for identifying potential close SMBHB candidates has emerged in the thriving field of time-domain astronomy, which focuses on analyzing the periodic light curves of AGNs. The periodicity observed in these light curves can arise from changes in the accretion rate onto the black holes (Graham et al. 2015b) or from the relativistic Doppler boost resulting from the highly relativistic motion of gas in the mini accretion disk around the smaller black hole in the binary system (D'Orazio et al. 2015). Extensive research has been conducted using optical light curves to search for SMBHBs, and the methods employed in these studies have become well-established (Graham et al. 2015a; Charisi et al. 2016; Zheng et al. 2016; Liu et al. 2019b; Chen et al. 2020; Liao et al. 2021; Chen et al. 2024). However, many of the known candidates have been shown to be subject to false positives due to stochastic quasar variability (e.g., Vaughan et al. 2016). Additionally, recent findings suggest that some AGNs may display periodic variations only at some specific epochs (e.g., Jiang et al. 2022; O'Neill et al. 2022) further enhance the difficulties of periodic searches. Furthermore, previous surveys were only sensitive to the most massive quasars at high redshift ( z ≳ 2) which should have already gone through their major merger process (e.g., Shen 2009). Regardless of the process that imprints periodicity, it is generally accepted that the timescale is primarily determined by the binary orbital period. SMBHBs with masses between 10 6 -10 10 M ⊙ and separations of 0.01 pc possess orbital periods that span from years to several decades, making them possibly detectable by current timedomain surveys. In addition to the optical band, the periodic searches have also been conducted in other wavelength regimes, such as in X-ray (Serafinelli et al. 2020; Liu et al. 2020) and gamma-ray bands (Sandrinelli et al. 2018; Holgado et al. 2018).</text>
<text><location><page_2><loc_8><loc_10><loc_48><loc_21></location>Periodic infrared (IR) light curves are also expected as a result of reverberation mapping of the circumbinary dusty torus in periodic AGNs. However, the search for SMBHBs using IR data has not been extensively explored, despite the detection of an IR time lag in an individual SMBHB candidate (Jun et al. 2015), which highlights the potential of utilizing periodic IR light curves for SMBHB identifi-</text>
<text><location><page_2><loc_52><loc_46><loc_92><loc_91></location>cation (D'Orazio & Haiman 2017). The IR band not only provides an alternative approach to searching for SMBHBs but also offers distinct advantages. Firstly, IR photons are less susceptible to dust extinction, allowing them to penetrate through obscuring material, which is often abundant in galactic nuclear regions and galaxy merger systems. By extending the search to the IR wavelength range, we can uncover a population of SMBHB candidates that may have been previously overlooked, thereby expanding our knowledge of the prevalence and characteristics of these elusive systems. Even for unobscured AGNs, the variability amplitudes in the mid-IR band are typically larger compared to the optical band due to lower background contamination. Additionally, as discussed in D'Orazio & Haiman (2017), the dust-echo model can assist in determining the origin of central periodicity and place constraints on the physical characteristics of SMBHBs and their dust torus. To address this, we perform the first systematic search for SMBHB candidates via periodic IR light curves. The data we use come from the Widefield Infrared Survey Explorer (WISE, Wright et al. 2010) and its successor NEOWISE (Mainzer et al. 2014), which have provided us with a public dataset from February 2010 to December 2023 with a half-year cadence except for a 2.5year gap for each object. In this study, we primarily focus on sinusoidal periodic signals, which have been commonly adopted in modeling the optical light curves of SMBHB candidates (e.g., D'Orazio et al. 2015), and on modeling the IR light curves using the dust-echo model (D'Orazio & Haiman 2017).</text>
<text><location><page_2><loc_52><loc_24><loc_92><loc_46></location>The paper is structured as follows. Section 2 introduces the data and methods employed to search for IR periodic AGNs. Section 3 presents the search and corresponding analysis results, where we show compelling evidence demonstrating the highly improbable nature of these periodic light curves being generated by random processes. Section 4 compares selected candidates with normal AGNs and with previous studies that utilized optical light curves. Additionally, we discuss SDSS J140336.43+174136.1 as a candidate exhibiting periodic behavior in both optical and IR bands. Finally, we summarize the main results and suggest directions for future work in Section 5. We assume a cosmology with H 0 = 66 . 88 km s -1 Mpc -1 , Ω m =0 . 32, and Ω Λ =0 . 68, adopted from Planck Collaboration et al. (2020).</text>
<section_header_level_1><location><page_2><loc_63><loc_21><loc_81><loc_22></location>2. DATA AND METHODS</section_header_level_1>
<section_header_level_1><location><page_2><loc_68><loc_19><loc_76><loc_20></location>2.1. IR data</section_header_level_1>
<text><location><page_2><loc_52><loc_9><loc_92><loc_18></location>The WISE was originally launched to conduct a full-sky survey in four MIR bands centered at 3.4, 4.6, 12, and 22 µ m (labeled W1-W4) from February to August 2010 (Wright et al. 2010). The solid hydrogen cryogen used to cool the W3 and W4 instrumentation was depleted later and the spacecraft was placed in hibernation in 2011 February. However, it was</text>
<text><location><page_3><loc_8><loc_76><loc_48><loc_91></location>reactivated in 2013 October under the new name NEOWISE -R, with only W1 and W2 operational, specifically to search for asteroids that could potentially pose a threat of impact on Earth (Mainzer et al. 2014). The WISE and its new mission NEOWISE operate in the same manner, that is, scanning a specific sky area every six months, with an average of 12 or more individual exposures taken within each epoch, typically spanning a single day. Consequently, each target in the sky has been observed 22-23 times separated by a six-month interval up to December 2023.</text>
<text><location><page_3><loc_8><loc_60><loc_48><loc_76></location>The single-exposure photometry of WISE and NEOWISE are archived in the AllWISE Multiepoch Photometry Table and NEOWISE-R Single Exposure (L1b) Source Table 1 , respectively. The photometry we adopted is measured by pointspread function (PSF) profile fitting, which is suitable for quasars. We first binned the data within each epoch since the intraday IR variability is negligible except for very radioloud AGNs (Jiang et al. 2012) and the anticipated periods we aim to detect are on much longer timescales. The variance of the binned data was recalculated as shown in Equation 1:</text>
<formula><location><page_3><loc_12><loc_54><loc_48><loc_58></location>[ δ mag ( ti )] 2 = 1 ni -1 ni ∑ j =1 [ magj -mag ( ti )] 2 + 1 ni σ 2 s . s . (1)</formula>
<text><location><page_3><loc_8><loc_41><loc_48><loc_53></location>where ni denotes the number of data points within the i th epoch, mag ( ti ) denotes the mean magnitude within the i th epoch, and σ j denotes the photometric uncertainty of the j th data point. The first part represents the contribution from photometric uncertainty and short-term variability, while the second part represents the contribution from the system stability of WISE. We adopt σ s . s . = 0 . 029 mag for WISE and σ s . s . = 0 . 016 mag for NEOWISE from Lyu et al. (2019).</text>
<text><location><page_3><loc_8><loc_27><loc_48><loc_41></location>Prior to binning, the data were filtered based on the quality flags in the catalog following Jiang et al. (2021). Moreover, outliers that fell outside the 3 σ range of the single data point at each epoch were eliminated, using the mean magnitude and mean photometric uncertainty at that epoch as references. The mean and variance of the single data points were then re-computed, resulting in the binned data used for our analysis. This step was crucial to ensure the accuracy and reliability of the data used in our study.</text>
<section_header_level_1><location><page_3><loc_22><loc_25><loc_34><loc_26></location>2.2. AGN Sample</section_header_level_1>
<text><location><page_3><loc_8><loc_15><loc_48><loc_24></location>The AGN sample we chose is the widely-used Million Quasar Catalog (v8, 2 August 2023; Flesch 2023). This is a compendium of 907,144 type-I QSOs and AGNs, largely complete from the literature up to 30 June 2023. 66,026 QSO candidates are also included, calculated via radio/X-ray association (including double radio lobes) as being 99% likely</text>
<text><location><page_3><loc_52><loc_76><loc_92><loc_91></location>to be quasars. Blazars and type-II objects are also included, bringing the total count to 1,021,800. To ensure the purity of our candidates, we only selected objects labeled as types 'Q', 'A', 'B', 'K', 'N' in the catalog (see what the legends of these types refer to in the note of Table 2), as these are confirmed AGNs. As a result, the number of objects amounts to 955,744. To enable a reliable long-term variability analysis, we apply a criterion that requires a minimum of 15 detected epochs (about 2/3 of all detected epochs) for each source. This left us with a subset of 576,260 AGNs.</text>
<text><location><page_3><loc_52><loc_51><loc_92><loc_76></location>In order to rationalize the computational effort in further analysis, we choose to analyze only AGNs with obvious variability. This selection strategy is crucial to ensure that candidates meet the criteria outlined in selection criterion 5 (see Section 2.4). Furthermore, light curves displaying subtle variability tend to pose a challenge in terms of model constraints within the DRW framework (see Section 3.1). To address this issue, we introduce a requirement that, for a given light curve, at least one data point must satisfy mag ( ti ) -2 δ mag ( ti ) > magmean and at least one data point should satisfy mag ( ti ) + 2 δ mag ( ti ) < magmean . This allows us to narrow down our sample to 53,496 AGNs, ensuring that the selected candidates have the necessary characteristics for further analysis. We also require that the light curves be well constrained by the DRW model (see section 3.1), resulting in a parent sample of 48,932 AGNs.</text>
<section_header_level_1><location><page_3><loc_64><loc_48><loc_80><loc_49></location>2.3. Finding Periodicity</section_header_level_1>
<text><location><page_3><loc_52><loc_40><loc_92><loc_47></location>To identify sources exhibiting periodic variability, we employed a sinusoidal function to fit the light curves of both W1 and W2 bands. The fitting process involved minimizing the following equation using the χ 2 statistics as Equation 2 below (D'Orazio & Haiman 2017):</text>
<formula><location><page_3><loc_60><loc_35><loc_92><loc_39></location>χ 2 = n ∑ i =1 [ mag ( ti ) -A sin Ω ( ti -t 0) + B ] 2 [ δ mag ( ti )] 2 (2)</formula>
<text><location><page_3><loc_52><loc_24><loc_92><loc_34></location>The Lomb-Scargle (LS) periodogram (Lomb 1976; Scargle 1982) is widely used for periodicity detection in unevenly sampled data. In our analysis, we utilized the generalized Lomb-Scargle (GLS) periodogram (Zechmeister & Kürster 2009) implemented in the pyastronomy 2 package. In the GLS periodogram, we calculate a series of 'powers' defined as Equation 3:</text>
<formula><location><page_3><loc_68><loc_19><loc_92><loc_22></location>p = χ 2 0 -χ 2 χ 2 0 (3)</formula>
<text><location><page_3><loc_52><loc_14><loc_92><loc_18></location>where χ 2 0 is the residual of a model assuming no variability, and χ 2 is equivalent to Equation 2 in the GLS periodogram (Zechmeister & Kürster 2009). Compared to the</text>
<figure>
<location><page_4><loc_9><loc_42><loc_90><loc_92></location>
<caption>Figure 1. Two special examples of candidates. The top panel specifically highlights the highest redshift source at z = 0 . 684 and the bottom panel highlights the candidate showing the most periods. Left: the grey error bars represent the original data (with outlier data points removed). The blue error bars represent the binned data. The orange sinusoids represent the best-fitting sine curves. Right: the black curves show the periodograms of candidates, while other curves show different significance level calculated from 100,000 simulated light curves.</caption>
</figure>
<text><location><page_4><loc_8><loc_24><loc_48><loc_34></location>LS periodogram, the GLS periodogram provides a more accurate frequency prediction by taking into consideration an offset and weights, which allows the calculation of the maximumpower in the GLS periodogram to be equivalent to minimizing Equation 2. The searched period range of the GLS periodogram is from 700 days to T 0 days, where T 0 is the time span of the light curve.</text>
<section_header_level_1><location><page_4><loc_21><loc_21><loc_35><loc_22></location>2.4. Selection Criteria</section_header_level_1>
<text><location><page_4><loc_8><loc_9><loc_48><loc_20></location>We selected reliable periodic candidates based on five criteria. First, to account for stochastic red noise variability, we adopt a comparative approach by measuring the performance of the candidates against that of simulated light curves, as described in Section 3.1. Rather than applying a single threshold to the normalized periodograms, we require 3 σ significance in both the W1 and W2 bands.</text>
<text><location><page_4><loc_52><loc_28><loc_92><loc_34></location>Secondly, we adopt the signal-to-noise (S/N) ratio defined as Equation 4 (Horne & Baliunas 1986), where A 0 denotes the amplitude and σ 2 r denotes the variance of the residuals. we require that ξ > 2 for both the W1 and W2 bands.</text>
<formula><location><page_4><loc_68><loc_25><loc_92><loc_27></location>ξ = A 2 0 / (2 σ 2 r ) (4)</formula>
<text><location><page_4><loc_52><loc_20><loc_92><loc_24></location>Thirdly, we require that the difference between the frequencies of the W1 and W2 light curves must not exceed 5% of their sum, in accordance with Equation 5:</text>
<formula><location><page_4><loc_67><loc_16><loc_92><loc_19></location>| f 1 -f 2 | f 1 + f 2 < 0 . 05 (5)</formula>
<text><location><page_4><loc_52><loc_9><loc_92><loc_15></location>where f 1 and f 2 represent the frequencies of the W1 and W2 light curves. In addition, the phase difference between them must be less than 1/8 cycles. This criterion reflects the requirements of the dust-echo model and guarantees that the</text>
<text><location><page_5><loc_8><loc_73><loc_48><loc_91></location>two sinusoids have the same frequency and phase. However, we allow for a margin of error of 1/8 cycles to account for potential minor phase differences arising from low photometric accuracy or intrinsic characteristics of the IR light curves. For instance, the W2 band may receive a greater contribution from dust with lower temperatures located further from the BH, leading to a slight delay in the W2 variability compared to the W1 band (Lyu et al. 2019). If the sources are located at high redshifts, i.e. z > 2, the W1 and W2 emission could mainly come from a relatively inner and outer part of the accretion disk, then a time delay can also be expected. We confirm this small delay for our candidates in Section 3.2.</text>
<text><location><page_5><loc_8><loc_31><loc_48><loc_73></location>Fourthly, the available data are required to show more than two cycles in both the W1 and W2 bands. It is important to note that even strictly sinusoidal variations are difficult to distinguish from a simple stochastic process when the number of cycles N cyc is ≲ 5 (Vaughan et al. 2016). Any finite light curve generated by the corresponding variability process represents only a snapshot of that process. The periodogram derived from this light curve will show power distributed around the actual Power Spectral Density (PSD) continuum with some scatter. This scatter can create the appearance of spikes in the periodogram (e.g., see Fig. 1 and Fig. 12), leading the light curve to seemingly exhibit a sinusoidal-like pattern over 2 to 3 cycles, even if the underlying PSD process is purely continuous. Therefore, these patterns can only be reliably distinguished with much longer time series spanning many (e.g., > 4-5) cycles of the putative period. However, most optical searches still adhere to a criterion of > 1.5 cycles due to the limited time span of the data (e.g. Graham et al. 2015a; Charisi et al. 2016; Liu et al. 2019b; Chen et al. 2024). Consequently, false positives would arise in the candidate sample from stochastic red noise variability (Vaughan et al. 2016). Due to the limited number of detected epochs, which amounts to 23, we set the limit to > two cycles. We believe that this is the best approach we can take with the current WISE data, but we note that further improvements should be pursued in future research with more available data.</text>
<text><location><page_5><loc_8><loc_10><loc_48><loc_30></location>Fifthly, we check if the maximum power of the periodogram indicates a 'correct' period, since anomalous data points (e.g., an outlier with a small error) could mislead the periodogram calculation. Additionally, this criterion allows us to check if the light curves show strong variability. We estimate the period of a light curve by examining the data points. If all data points within a time segment satisfy mag ( ti ) -2 δ mag ( ti ) > offset (calculated by the periodogram), we classify it as a 'faint segment'. If all data points within a time segment satisfy mag ( ti ) + 2 δ mag ( ti ) < offset, we classify it as a 'bright segment'. Then we conduct a count in chronological order, recording each change from a faint segment to a bright segment or vice versa, and neglecting normal</text>
<figure>
<location><page_5><loc_53><loc_68><loc_94><loc_91></location>
<caption>Fig. 2 illustrates an AGN that meets all the selection criteria except for the selection Criterion 5 in the W2 band. The periodogram analysis reveals a 3 σ significance in the W2 light curve, indicating a period of T max = 1545 . 6 days within a time span of T 0 = 4939 . 9 days. However, the total count for W2 is 3, so that it does not meet Criterion 5. This failure is attributed to the lack of significant variability in the W2 band, as shown in Fig. 2.</caption>
</figure>
<section_header_level_1><location><page_5><loc_68><loc_90><loc_85><loc_92></location>FBQS J0832+2853</section_header_level_1>
<paragraph><location><page_5><loc_52><loc_61><loc_92><loc_66></location>Figure 2. An example for selection Criterion 5. In this example, the total count for W1 is 4, while W2 has 3 counts. Note that the errorbars show 2 δ mag ( ti ), and the purple lines show the best-fit sine curves.</paragraph>
<text><location><page_5><loc_52><loc_51><loc_92><loc_59></location>data points in this process (see an example shown in Fig. 2). Denoting the total count as n , ideally n / 2 periods would be included in the time span. We require that for both the W1 and W2 bands, the period at the maximum power Tmax and the time span of the light curve T 0 satisfy Equation 6:</text>
<formula><location><page_5><loc_62><loc_47><loc_92><loc_50></location>max { n -2 , 0 } 2 < T 0 Tmax < n + 3 2 (6)</formula>
<section_header_level_1><location><page_5><loc_62><loc_32><loc_83><loc_33></location>3. ANALYSIS AND RESULTS</section_header_level_1>
<section_header_level_1><location><page_5><loc_58><loc_30><loc_86><loc_31></location>3.1. Simulating with Stochastic Variability</section_header_level_1>
<text><location><page_5><loc_52><loc_20><loc_92><loc_29></location>AGN variability can roughly be described by a Gaussian first-order continuous autoregressive model (CAR(1), Kelly et al. 2009), also known as a damped random walk (DRW, MacLeod et al. 2010) or Ornstein-Uhlenbeck process. The likelihood function for the DRW process can be represented as:</text>
<formula><location><page_5><loc_57><loc_14><loc_92><loc_19></location>L ∝ | C | -1 2 exp -1 2 ∑ i , j ( Xi -q )( C -1 ) i j ( Xj -q ) (7)</formula>
<text><location><page_5><loc_52><loc_9><loc_92><loc_13></location>where the Xi represents the flux of the data point i , and q is the long-term mean of the light curve. The covariance matrix C is given by:</text>
<figure>
<location><page_6><loc_9><loc_69><loc_50><loc_92></location>
<caption>Figure 3. Distribution of the ∆ BIC value. BIC DRW( c ) is the BIC value of correlated DRW model, BIC DRW( n ) is the BIC value of the uncorrelated DRW model, while BIC DRW( c ) + sin is the BIC value of the sinusoid +correlated DRW model for the candidates.</caption>
</figure>
<formula><location><page_6><loc_18><loc_56><loc_48><loc_59></location>Cij = δ i j σ 2 i + σ 2 exp ( -| ti -t j | τ ) (8)</formula>
<text><location><page_6><loc_8><loc_49><loc_48><loc_55></location>where δ i j is the Kronecker delta. In our analysis, we assume a strong correlation between the W1 and W2 data. Subsequently, we have devised a modified form of the covariance matrix that accounts for this correlation, expressed as:</text>
<formula><location><page_6><loc_10><loc_44><loc_48><loc_47></location>Cai , bj = δ ab δ i j σ 2 i + [ ρ + (1 -ρ ) δ ab ] σ 2 exp ( -| ti -t j | τ ) (9)</formula>
<text><location><page_6><loc_8><loc_40><loc_48><loc_43></location>where a and b represent the observed band (W1 or W2) of the data points, and ρ is the correlation coefficient.</text>
<text><location><page_6><loc_8><loc_14><loc_48><loc_40></location>Regarding the data quality of WISE, we only use the binned data when estimating DRW parameters. We set a uniform prior for the logarithm of the parameters σ 1, σ 2 and τ , and a uniform prior for ρ W 1 , W 2. We set q as the mean magnitude of the light curve to reduce computational cost and use the Markov Chain Monte Carlo (MCMC) sampler emcee 3 (Foreman-Mackey et al. 2013) to construct the posterior samples of the parameters. However, we find that some light curves lack enough variability but pass the variability criterion (see Section 2.2) due to certain data points with very small errors. Consequently, they are not adequately constrained by the DRW model. To ensure a stable posterior for the MCMC processes, we set a criterion that the 1 σ error of the posterior lg τ sample must be less than 2. This is the final requirement to define our parent sample, resulting in a final number of 48,932 (see Section 2.2). The statistical distributions of σ , τ , and ρ W 1 , W 2 are illustrated in Fig. 4. Upon</text>
<text><location><page_6><loc_52><loc_84><loc_92><loc_91></location>comparison, the selected candidates exhibit similar values for σ 1, σ 2, and τ compared to the parent sample. Notably, they display a stronger correlation, which aids in meeting selection criterion 4 easier. The distribution of ρ W 1 , W 2 serves as validation for our correlation assumption.</text>
<text><location><page_6><loc_52><loc_75><loc_92><loc_84></location>As an alternative test, we perform a model selection to test if the correlation parameter is necessary to explain the light curve on top of a uncorrelated stochastic background. We adopt a maximum likelihood approach for the model comparison and parameter estimation. We use the Bayesian information criterion (BIC), which is defined as:</text>
<formula><location><page_6><loc_65><loc_71><loc_92><loc_73></location>BIC = -2ln L + k ln N (10)</formula>
<text><location><page_6><loc_52><loc_53><loc_92><loc_70></location>where L is the likelihood function, k is the number of free model parameters, and N is the number of data points. A lower BIC value indicates a preferable model. Typically, when ∆ BIC < -10, the model with the lower BIC is strongly favored. For the uncorrelated DRW process, we generate posterior parameter samples independently for each band. Subsequently, we integrate these samples into a covariance matrix to compute L with all correlation components set as zero. Our analysis indicates a strong preference for the correlated model over the uncorrelated one, aligning with our earlier analysis (further details provided in Fig. 3).</text>
<text><location><page_6><loc_52><loc_30><loc_92><loc_53></location>We proceed by randomly selecting a set of parameters to generate two correlated W1 and W2 simulated light curves. These simulated light curves are then compared with the actual observation cadence and measurement errors. This process is repeated 100,000 times to check that the selection criteria are met. In addition, we use the periodogram values of these simulated light curves to determine the significance of our candidates, as described in Criterion 1 in section 2.4. To further validate our simulation methodology, we perform a statistical comparison between the real and simulated light curves. Our analysis shows that they exhibit similar behaviour for each selection criterion (detailed results in Table 1), suggesting that the correlated DRW processes could potentially generate a comparable number of candidates to the real ones.</text>
<text><location><page_6><loc_52><loc_17><loc_92><loc_29></location>Notably, the DRW model is relatively simple and may not fully capture the complexity of actual AGN variability due to degeneracies (Zu et al. 2013). When using the DRW assumption to calculate false alarm probabilities, it might not completely consider the assumptions of the red noise hypothesis. However, using different variability models could lead to some small changes in the results, but usually not significant ones.</text>
<section_header_level_1><location><page_6><loc_60><loc_14><loc_84><loc_16></location>3.2. Candidates of Periodic Sources</section_header_level_1>
<text><location><page_6><loc_52><loc_9><loc_92><loc_13></location>Using the data and criteria outlined in Section 2, we have identified a total of 28 candidates. The catalog of candidates can be found in Table 2. The light curves and the GLS pe-</text>
<figure>
<location><page_7><loc_9><loc_44><loc_91><loc_92></location>
<caption>Figure 4. The DRW parameters distribution for the selection parent sample (blue) and the candidates (orange). Top: the σ distribution in the W1 band (left) and the W2 band (right). Bottom: the τ distribution (left) and the correlation coefficient distribution (right).</caption>
</figure>
<text><location><page_7><loc_8><loc_33><loc_48><loc_39></location>riodograms of two examples are presented in Fig. 1 and the rest can be found in Fig. 12. The W1 and redshift distributions of both the parent sample and the selected candidates are shown in Fig 5.</text>
<text><location><page_7><loc_8><loc_25><loc_48><loc_33></location>It is worth noting that the candidates are predominantly situated in the low redshift range (median z=0.126 , with 27/28 at z < 0.4) and bright region (median W1=12.69, 25/28 satisfy W1 < 14), which can be attributed to a selection effect making it easier for them to fulfill the selection criteria.</text>
<text><location><page_7><loc_8><loc_14><loc_48><loc_25></location>In Section 2.4, we mention the possibility of a slight phase delay between the W2 and W1 light curves. This is confirmed by our candidates, with an average delay of 0.025 periods, as illustrated in Fig. 6. The maximum phase delay is 0.087 period, which remains smaller than the 1/8 period threshold used in our selection process, indicating a reasonable choice.</text>
<text><location><page_7><loc_8><loc_9><loc_48><loc_14></location>As an alternative test, here we use the BIC method (see details in Section 3.1) to assess if an additional periodic signal is needed to explain the light curve on top of a stochastic</text>
<text><location><page_7><loc_52><loc_36><loc_92><loc_39></location>background. The likelihood function for the DRW + sinusoidal model is written as:</text>
<formula><location><page_7><loc_55><loc_29><loc_92><loc_33></location>L ∝ | C | -1 2 exp -1 2 ∑ i , j ( Xi -Si )( C -1 ) i j ( Xj -Sj ) (11)</formula>
<text><location><page_7><loc_52><loc_9><loc_92><loc_26></location>where Si represents the sinusoid signal. Our analysis shows that, for the vast majority of the candidates, BIC DRW + sin -BIC DRW( c ) < -10, indicating robust evidence that the periodic model is significantly favored over the pure stochastic model (refer to Fig. 3 for detailed results). However, this could be explained by the limitation of the DRW model, which might not adequately capture a spiky PSD and may be more suited to smooth continuum-like PSD shapes. Therefore, additional data in the future are necessary to obtain a higher density in frequency sampling for the periodogram and better test the preference for the sinusoid + DRW model.</text>
<table>
<location><page_8><loc_16><loc_69><loc_44><loc_89></location>
<caption>Table 1. Statistical comparison between real light curves and simulated ones.NOTE-Data in the 'Mock' column refer to 1 /100,000 of the total from 4,893,200,000 simulated light curves, allowing direct comparison with the real data. '3 σ significance' refers to the selection criterion 1 in Section 2.4. 'correct period' refers to the selection criterion 5 in Section 2.4.</caption>
</table>
<figure>
<location><page_8><loc_9><loc_36><loc_49><loc_59></location>
<caption>Figure 5. Distribution of the parent sample (blue), and the candidates (red) in the redshift and W1 magnitude space. We use a 40 × 40 grid to visualize the distribution of the parent sample. The color for each grid cell represents the number of AGNs from the parent sample that fall within that cell.</caption>
</figure>
<text><location><page_8><loc_8><loc_13><loc_48><loc_25></location>It's worth noting that the sinusoid +DRW model is a commonly used model due to its simplicity and clear physical picture. However, there exist alternative light curve models, such as the bursty model predicted by circumbinary accretion disk simulations (Farris et al. 2014). Despite the potential of this bursty model, we did not simulate it for our candidates as we could not observe 'bursty' characteristics in the WISE light curves.</text>
<text><location><page_8><loc_8><loc_9><loc_48><loc_12></location>Here we compile the derived quantities for our analysis, which are presented in Table 2. The table includes essen-</text>
<figure>
<location><page_8><loc_53><loc_69><loc_93><loc_92></location>
<caption>Figure 6. The distribution of the phase delay of W2 relative to W1 for the candidates (blue), and the corresponding Gaussian fit of the distribution (orange line).</caption>
</figure>
<text><location><page_8><loc_52><loc_52><loc_92><loc_61></location>tial information such as the Milliquas Name, right ascension (R.A.), declination (Decl.), redshift ( z ), and quasar properties as provided by Wu & Shen (2022) and Liu et al. (2019a). Additionally, we have included periodic properties such as offsets of the W1 and W2 bands, and the period of the candidates.</text>
<section_header_level_1><location><page_8><loc_63><loc_48><loc_82><loc_49></location>3.3. False Alarm Probability</section_header_level_1>
<text><location><page_8><loc_52><loc_37><loc_92><loc_47></location>To estimate the false alarm probability (FAP) for each candidate, we employ the method described in Chen et al. (2020). We estimate the approximate FAP using the effective number of independent frequencies Nef f , which is calculated by dividing the observed frequency window by the expected peak width δ f = 1 / T . The FAP is estimated as (e.g., VanderPlas 2018):</text>
<formula><location><page_8><loc_65><loc_33><loc_92><loc_35></location>FAP ∼ 1 -[ Psingle ] Nef f (12)</formula>
<text><location><page_8><loc_52><loc_24><loc_92><loc_32></location>Where Psingle = 1 -e Z , Z is the periodogram value defined in Zechmeister & Kürster (2009). Since candidates are selected based on the light curves of both the W1 band and W2 band, there are two P single and N eff. To simplify the calculation, we can estimate the periodogram FAP as Equation 13:</text>
<formula><location><page_8><loc_58><loc_20><loc_92><loc_22></location>FAP ∼ 1 -[ Psingle , W 1] Nef f , W 1 [ Psingle , W 2] Nef f , W 2 (13)</formula>
<text><location><page_8><loc_52><loc_10><loc_92><loc_19></location>It is important to acknowledge that this calculation assumes independent FAPs for the W1 and W2 bands, contrary to the strong correlation observed in the DRW simulation (refer to Section 3.1). Nevertheless, this method provides an upper limit estimate for the FAP, as single-band FAP calculations do not consider this correlation.</text>
<table>
<location><page_9><loc_11><loc_31><loc_89><loc_86></location>
<caption>Table 2 . The 28 periodic AGN candidates meeting the selection criteria.</caption>
</table>
<text><location><page_9><loc_9><loc_18><loc_91><loc_30></location>NOTE- (1)-(4): Name, RA, DEC, and redshift of the objects. (5)-(6): Median magnitudes in the W1 and W2 bands, respectively. (7): Period (in the observer-frame) in units of days. (8): Legend of type/class from Flesch (2023): Q = QSO, type-I broad-line core-dominated, 860,100 of these. A = AGN, type-I Seyferts/host-dominated, 47,044 of these. B = BL Lac type object, 2,814 of these. (FSRQs are typed as QSOs here) K = NLQSO, type-II narrow-line core-dominated, 6,048 of these. N = NLAGN, type-II Seyferts/host-dominated, 39,768 of these. Incomplete, and includes an unquantified residue of legacy NELGs/ELGs/LINERs, plus some unclear AGN. This is the catch-all category. S = star classified but showing quasar-like photometry and radio/X-ray association, thus included as a quasar candidate; 124 of these. R = radio association displayed. X = X-ray association displayed. 2 = double radio lobes displayed (declared by data-driven algorithm). (9)-(11): Logarithmic bolometric luminosity in units of erg/s, black hole mass in units of solar mass, and Eddington ratio for sources in the SDSS DR16 quasar catalog (Wu & Shen 2022), the DR7 board-line AGN catalog (Liu et al. 2019a).</text>
<text><location><page_9><loc_8><loc_11><loc_48><loc_15></location>Additionally, we empirically compute the global FAP using the 100,000 simulated light curves for each AGN, following a similar approach as presented by Barth & Stern (2018).</text>
<text><location><page_9><loc_52><loc_11><loc_92><loc_15></location>The global FAP accounts for all false positives that satisfy all selection criteria within the explored frequency range. The global FAP values for candidates range from 4 . 0 × 10 -4 to</text>
<figure>
<location><page_10><loc_9><loc_69><loc_50><loc_92></location>
<caption>Figure 7. The FAP distribution estimated from periodogram calculation (blue), and the global FAP estimated from 100,000 simulated light curves (orange). The Gaussian fit curves for the 2 distributions are also shown.</caption>
</figure>
<text><location><page_10><loc_8><loc_57><loc_48><loc_60></location>4 . 77 × 10 -3 , with a mean of 2 . 07 × 10 -3 by Gaussian fit. Further details are available in Fig. 7.</text>
<text><location><page_10><loc_8><loc_43><loc_48><loc_56></location>It is noteworthy that the global FAPs are significantly higher than the periodogram FAPs. This discrepancy can be attributed to the stronger variability exhibited by the simulated light curves compared to real observations, leading to an increase in the global FAP. Moreover, while most false positives exhibit only a 3 σ significance level, certain candidates display notably stronger significance, such as SDSS J162938.09+362452.2 and 6dF J214907.4-175159 as depicted in Fig. 12.</text>
<section_header_level_1><location><page_10><loc_22><loc_40><loc_34><loc_41></location>4. DISCUSSION</section_header_level_1>
<section_header_level_1><location><page_10><loc_11><loc_38><loc_45><loc_39></location>4.1. Do the Candidates Show Distinctive Properties?</section_header_level_1>
<text><location><page_10><loc_8><loc_17><loc_48><loc_37></location>One might wonder whether our selected periodic sample shows distinct features in other physical properties, such as bolometric luminosity ( L bol), black hole mass ( M BH) and Eddington ratio ( λ edd= L bol / L edd). To investigate this, we try to extract a subsample of candidates with previous measurements in these parameters. We chose to cross-match the candidates with two AGN samples based on Sloan Digital Sky Survey (SDSS), namely the SDSS DR16 quasar catalog (DR16Q) (Wu & Shen 2022) and DR7 broad-line AGN catalog (Liu et al. 2019a). They contained 750,414 and 14,584 sources, concentrated at high and low redshifts, respectively (see Figure 8). We only found a total of 10 candidates that overlapped with the two catalogs.</text>
<text><location><page_10><loc_8><loc_9><loc_48><loc_17></location>Similar to the occupation of the candidates in the whole parent sample (see Figure 5), they are all in the low redshift and low magnitude space (see Figure 8) due to selection effects. Therefore, we need to construct a control sample for a fair comparison, at least in redshift and IR magnitudes.</text>
<figure>
<location><page_10><loc_53><loc_68><loc_93><loc_92></location>
<caption>Figure 8. Distributions of the selection parent sample (blue), quasars present in the DR16Q catalog that overlap with the parent sample (black), AGNs present in the DR7 broad-line AGN catalog that overlap with the parent sample (green), and the overlapping candidates (red) in the redshift and W1 magnitude space. Enclosed percentiles are marked for each contour level.</caption>
</figure>
<text><location><page_10><loc_52><loc_33><loc_92><loc_56></location>Specifically, we selected the 10 closest AGNs with a redshift ratio less than 1.01 and with differences in W1 and W2 magnitudes less than 0.1. In cases when no AGN meet the specified criteria, we choose the closest AGN in the catalog as the control sample. After controlling, we then compare the M BH, L bol and λ edd of the periodic sources with normal AGNs. Additionally, we have also considered another parameter dust covering factor ( fd ), defined as the ratio of the W1monochromatic luminosity ( LW 1 = ν W 1 F ν W 1) and the L bol ( fd = L W1 / L bol, e.g., Ma & Wang 2013), to indicate the extent of dust obscuration. However, no significant differences in the distributions of these parameters can be seen visually (see Fig. 9). We have also checked the mean and standard deviation (SD) of their distributions (see Table 3) and confirmed it.</text>
<text><location><page_10><loc_52><loc_14><loc_92><loc_32></location>We further performed the K-S statistics for the L bol, M BH, λ edd, and fd distributions between the candidates and the control sample, yielding values of 0.375, 0.400, 0.225, and 0.400, respectively. The K-S tests suggest significant differences in these properties with a critical value of 0.480 for a 95% confidence level, so none of them show significant differences. It is worth emphasizing that the systematic uncertainties in the estimates of AGN properties (e.g., singleepoch BH mass, see discussions in Shen 2013) are nonnegligible and would dilute the difference, if there were any. This may be particularly important for the small sample size for comparison here.</text>
<text><location><page_10><loc_54><loc_10><loc_90><loc_13></location>4.2. Comparison with SMBHB Candidates Selected by Optical Periodicity</text>
<figure>
<location><page_11><loc_9><loc_69><loc_49><loc_92></location>
<caption>Figure 10. Redshift and period distributions of SMBHB candidates selected by this work (red), and those reported in previous works by optical search (other colors).</caption>
</figure>
<figure>
<location><page_11><loc_51><loc_69><loc_91><loc_91></location>
<caption>Figure 9. Left: distribution of the control samples (blue), which control for redshift and W1, W2, and the candidates (red) in the log ( M BH) and log ( L bol) space. Right: distribution of the control samples (blue), which control for redshift and L bol, and the candidates (red) in the log ( L bol) and log ( fd ) space (right).The right panel also tests the anti-correlation between L bol and fd .</caption>
</figure>
<table>
<location><page_11><loc_13><loc_46><loc_47><loc_59></location>
<caption>Table 3. Statistical comparison of quasar properties between candidates and control samples.NOTE- 'control 1' refers to the control sample that controls z, W1 and W2 magnitude. 'control 2' refers to the control sample that controls z and L bol.</caption>
</table>
<text><location><page_11><loc_8><loc_19><loc_48><loc_39></location>As mentioned in the introduction, previous studies have identified samples of SMBHB candidates based on optical periodic light curves. It is intriguing to investigate whether there are any common sources between our IR-selected sample and these optically-selected samples. Such overlap would provide compelling evidence for the periodicity of the candidates. Additionally, if the periods derived from optical light curves align with those obtained from IR light curves, we can employ the dust-echo model to further explore the properties of the candidates. For this purpose, we compile a collection of optical SMBHB candidates reported in various literature, including Graham et al. (2015a); Charisi et al. (2016); Zheng et al. (2016); Liu et al. (2019b); Chen et al. (2020, 2024).</text>
<text><location><page_11><loc_8><loc_11><loc_48><loc_18></location>Unfortunately, we do not find any overlap between our sample and those samples. The lack of matches is not entirely unexpected, as the two searches prioritize candidates with different periods. Specifically, we find that the periods of the optical candidates predominantly fell below 1700 days,</text>
<figure>
<location><page_11><loc_53><loc_39><loc_93><loc_62></location>
</figure>
<text><location><page_11><loc_53><loc_52><loc_55><loc_52></location>z</text>
<text><location><page_11><loc_52><loc_24><loc_92><loc_31></location>whereas the periods of our selected candidates were mostly longer than 1700 days (see details shown in Fig. 10). Additionally, most of our candidates have low redshifts (z < 0.4) due to the detection ability of WISE, while the redshifts of candidates from optical searches span from 0 to ∼ 3.</text>
<section_header_level_1><location><page_11><loc_53><loc_21><loc_91><loc_22></location>4.3. Optical Light Curves and A Possible Periodic Source</section_header_level_1>
<text><location><page_11><loc_52><loc_9><loc_92><loc_20></location>Given the abundance of optical photometric archival data, we then examined the optical light curves of the 28 candidates using data from various surveys, such as the Catalina Real Time Transient Survey (CRTS, Drake et al. 2009), the Palomar Transient Factory (PTF, Rau et al. 2009), the Zwicky Transient Facility (ZTF, Masci et al. 2019), and the All-Sky Automated Survey for Supernovae (ASAS-SN, Kochanek</text>
<text><location><page_12><loc_9><loc_77><loc_11><loc_80></location>mag</text>
<figure>
<location><page_12><loc_9><loc_62><loc_50><loc_92></location>
<caption>Figure 11. The W1, W2 and optical light curves of SDSS J140336.43+174136.1. The orange, purple and black lines represent the corresponding best-fitting sinusoids. We assume that the periods of IR light curves are the same as that of the optical light curves.</caption>
</figure>
<text><location><page_12><loc_8><loc_38><loc_48><loc_52></location>et al. 2017). Their combined dataset provides optical light curves spanning about two decades. We have normalized the data from different surveys and binned them at 20-day intervals. However, they generally show a small amplitude of optical variability, probably due to either intrinsically weak variability in the optical band or too shallow surveys. Only one candidate, SDSS J140336.43+174136.1, shows significant variability and periodicity in its optical light curve (see Fig. 11).</text>
<text><location><page_12><loc_8><loc_24><loc_48><loc_37></location>The periodogram analysis of the optical light curve of SDSS J140336.43+174136.1 reveals a period of 2891 ± 22 days, that is slightly longer than the period fitted from the IR light curves, which is 2346 ± 87 days. The errors are given by periodogram peak statistics based on 100,000 perturbed light curves for each real light curve. The discrepancy could be caused by the poor cadence of the WISE surveys, and so in the following analysis we choose the optical period for this particular source.</text>
<text><location><page_12><loc_8><loc_11><loc_48><loc_23></location>We present the corresponding best-fitting sinusoids for the W1, W2, and optical light curves in Figure 11. The time delay between the W1 (W2) and optical light curves is 630 (727) days (in the rest frame), respectively. Assuming the simplest scenario of a hollow spherical shell of dust with the source located at its center, we can easily estimate that the inner radius of the shell is 0.53 pc. These values are roughly consistent with previous statistical correlations between the</text>
<text><location><page_12><loc_52><loc_89><loc_92><loc_91></location>time delay and bolometric luminosity of AGNs. For example, using the relations obtained by Lyu et al. (2019), i.e.</text>
<formula><location><page_12><loc_56><loc_84><loc_92><loc_86></location>∆ tW 1 / day = 10 2 . 10 ± 0 . 06 ( L bol / 10 11 L ⊙ ) 0 . 47 ± 0 . 06 (14)</formula>
<text><location><page_12><loc_54><loc_81><loc_67><loc_82></location>for the W1 band and</text>
<formula><location><page_12><loc_56><loc_77><loc_92><loc_78></location>∆ tW 2 / day = 10 2 . 20 ± 0 . 06 ( L bol / 10 11 L ⊙ ) 0 . 45 ± 0 . 05 (15)</formula>
<text><location><page_12><loc_52><loc_68><loc_92><loc_75></location>for the W2 band, and a given bolometric luminosity of 10 45 . 81 erg s -1 for SDSS J140336.43+174136.1 (see Table 2), the W1 and W2 time delays are estimated to be 476 ± 147 days and 566 ± 158 days, respectively, which is in good agreement with our measurements.</text>
<section_header_level_1><location><page_12><loc_66><loc_65><loc_78><loc_66></location>5. CONCLUSION</section_header_level_1>
<text><location><page_12><loc_52><loc_9><loc_92><loc_64></location>In this work, we have conducted the first systematic search for SMBHBs in the IR band and identified 28 AGNs exhibiting IR periodic variability using the decade-long WISE and NEOWISE light curves. We performed extensive simulations to evaluate the potential influence of stochastic variability on candidate selection. By counting all the false positives in the simulated light curves, the probability of these periodic light curves being generated by random processes is an average of 0.207% by Gaussian fit. However, the number can be reproduced by our mock simulations with the DRW process of the parent sample. This indicates the challenge of identifying reliable SMBHBs with periodic emission using only WISE light curves, which are constrained by their time span and visit cadence, necessitating a low threshold on the minimum number of cycles ( N cyc > 2). Consequently, the nature of these periodic sources, whether driven by SMBHBs or not, should be carefully tested in future observations. We subsequently investigated whether these IR periodic sources exhibit distinct properties compared to normal AGNs. However, no significant biases were found, and the small sample size prevents us from statistically identifying potential weak differences. On the other hand, we found that there is no overlap between our sample (IRselected) and the SMBHB candidates reported in the literature (optically selected), likely due to differences in their preferred period ranges. Interestingly, we have identified SDSS J140336.43+174136.1 as a candidate displaying periodicity in both optical and IR bands. Further extended and more sensitive observations of these candidates are essential to confirm whether or not they are real periodic sources in both optical and IR bands. As suggested by D'Orazio & Haiman (2017), the periodic sources can also help test the physical mechanisms that produce the observed periodicity, due to relativistic Doppler modulation or accretion rate fluctuations.</text>
<text><location><page_13><loc_8><loc_67><loc_48><loc_91></location>This study highlights the promising potential of identifying SMBHB candidates through IR time-domain observations, which can significantly complement searches in other bands, although there are challenges in distinguishing genuine sources. It opens up a new avenue for SMBHB searches and can be applied to future surveys. Although NEOWISE unfortunately ended on July 31, 2024, the upcoming NEO Surveyor (Mainzer et al. 2023), the successor to NEOWISE , and the Nancy Grace Roman Space Telescope (Spergel et al. 2015) will ensure a bright future for IR time-domain astronomy, particularly for SMBHB searches (Haiman et al. 2023). These different surveys somewhat form a relay in time and provide us with IR light curves with a long time baseline, which is essential for verifying the periodicity of the candidates with N cyc < 4 and for detailed comparisons against physical models.</text>
<section_header_level_1><location><page_13><loc_19><loc_62><loc_37><loc_63></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_13><loc_8><loc_56><loc_48><loc_59></location>We sincerely thank the expert referee for very constructive and insightful comments, which helped improve our</text>
<text><location><page_13><loc_52><loc_57><loc_92><loc_91></location>manuscript greatly. This work is supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB0550200), the National Natural Science Foundation of China (grants 12073025, 12192221, the science research grants from the China Manned Space Project and the Cyrus Chun Ying Tang Foundations. D.L. acknowledges the support from the National Undergraduate Training Program for Innovation and Entrepreneurship. X.L. acknowledges support by NSF grant AST-2206499. This research makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. This research also makes use of data products from NEOWISE-R, which is a project of the Jet Propulsion Laboratory/California Institute of Technology, funded by the Planetary Science Division of the National Aeronautics and Space Administration. This research has made use of the NASA/IPAC Infrared Science Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration.</text>
<section_header_level_1><location><page_13><loc_45><loc_52><loc_55><loc_53></location>REFERENCES</section_header_level_1>
<code><location><page_13><loc_8><loc_9><loc_48><loc_51></location>Arzoumanian Z., et al., 2018, ApJ, 859, 47 Barth A. J., Stern D., 2018, ApJ, 859, 10 Begelman M. C., Blandford R. D., Rees M. J., 1980, Nature, 287, 307 Blaes O., Lee M. H., Socrates A., 2002, ApJ, 578, 775 Bonetti M., Sesana A., Barausse E., Haardt F., 2018, MNRAS, 477, 2599 Centrella J., Baker J. G., Kelly B. J., van Meter J. R., 2010, Reviews of Modern Physics, 82, 3069 Chapon D., Mayer L., Teyssier R., 2013, MNRAS, 429, 3114 Charisi M., Bartos I., Haiman Z., Price-Whelan A. M., Graham M. J., Bellm E. C., Laher R. R., Márka S., 2016, MNRAS, 463, 2145 Chen Y.-C., et al., 2020, MNRAS, 499, 2245 Chen Y.-C., Hwang H.-C., Shen Y., Liu X., Zakamska N. L., Yang Q., Li J. I., 2022, ApJ, 925, 162 Chen Y.-J., et al., 2024, MNRAS, 527, 12154 Colpi M., Dotti M., 2011, Advanced Science Letters, 4, 181 Cuadra J., Armitage P. J., Alexander R. D., Begelman M. C., 2009, MNRAS, 393, 1423 D'Orazio D. J., Charisi M., 2023, arXiv e-prints, p. arXiv:2310.16896 D'Orazio D. J., Haiman Z., 2017, MNRAS, 470, 1198 D'Orazio D. J., Haiman Z., Schiminovich D., 2015, Nature, 525, 351</code>
<code><location><page_13><loc_52><loc_11><loc_91><loc_51></location>De Rosa A., et al., 2019, NewAR, 86, 101525 Drake A. J., et al., 2009, ApJ, 696, 870 Farris B. D., Duffell P., MacFadyen A. I., Haiman Z., 2014, ApJ, 783, 134 Flesch E. W., 2023, The Open Journal of Astrophysics, 6, 49 Foreman-Mackey D., Hogg D. W., Lang D., Goodman J., 2013, PASP, 125, 306 Gould A., Rix H.-W., 2000, ApJL, 532, L29 Graham M. J., et al., 2015a, MNRAS, 453, 1562 Graham M. J., et al., 2015b, Nature, 518, 74 Haiman Z., et al., 2023, arXiv e-prints, p. arXiv:2306.14990 Holgado A. M., Sesana A., Sandrinelli A., Covino S., Treves A., Liu X., Ricker P., 2018, MNRAS, 481, L74 Horne J. H., Baliunas S. L., 1986, ApJ, 302, 757 Hughes S. A., 2009, ARA&A, 47, 107 Jiang N., et al., 2012, ApJL, 759, L31 Jiang N., et al., 2021, ApJS, 252, 32 Jiang N., et al., 2022, arXiv e-prints, p. arXiv:2201.11633 Jun H. D., Stern D., Graham M. J., Djorgovski S. G., Mainzer A., Cutri R. M., Drake A. J., Mahabal A. A., 2015, ApJL, 814, L12 Kelley L. Z., Blecha L., Hernquist L., 2017, MNRAS, 464, 3131 Kelly B. C., Bechtold J., Siemiginowska A., 2009, ApJ, 698, 895 Khan F. M., Fiacconi D., Mayer L., Berczik P., Just A., 2016, ApJ, 828, 73</code>
<text><location><page_13><loc_52><loc_9><loc_77><loc_10></location>Klein A., et al., 2016, PhRvD, 93, 024003</text>
<table>
<location><page_14><loc_8><loc_49><loc_48><loc_91></location>
</table>
<table>
<location><page_14><loc_52><loc_50><loc_91><loc_91></location>
</table>
<section_header_level_1><location><page_15><loc_46><loc_89><loc_54><loc_90></location>APPENDIX</section_header_level_1>
<section_header_level_1><location><page_15><loc_9><loc_83><loc_47><loc_85></location>A. THE IR LIGHT CURVES OF THE PERIODIC AGN SAMPLE</section_header_level_1>
<text><location><page_15><loc_8><loc_77><loc_48><loc_82></location>We have shown the light curves of 2 special candidates in Fig. 1. Here we show the light curves of the rest 26 candidates in Fig. 12.</text>
<figure>
<location><page_16><loc_9><loc_15><loc_91><loc_90></location>
<caption>Figure 12. The light curves and periodograms of the rest 26 candidates. The legends correspond to those in Fig. 1.</caption>
</figure>
<figure>
<location><page_17><loc_9><loc_15><loc_91><loc_90></location>
<caption>Figure 12. (Continued).</caption>
</figure>
<figure>
<location><page_18><loc_9><loc_14><loc_91><loc_90></location>
<caption>Figure 12. (Continued).</caption>
</figure>
<figure>
<location><page_19><loc_9><loc_14><loc_91><loc_89></location>
<caption>Figure 12. (Continued).</caption>
</figure>
<figure>
<location><page_20><loc_9><loc_13><loc_91><loc_89></location>
<caption>Figure 12. (Continued).</caption>
</figure>
<section_header_level_1><location><page_20><loc_61><loc_88><loc_86><loc_90></location>DESI 39632956414755490</section_header_level_1>
<figure>
<location><page_21><loc_9><loc_15><loc_91><loc_90></location>
<caption>Figure 12. (Continued).</caption>
</figure>
<figure>
<location><page_22><loc_9><loc_14><loc_91><loc_90></location>
<caption>Figure 12. (Continued).</caption>
</figure>
<figure>
<location><page_23><loc_9><loc_15><loc_91><loc_90></location>
<caption>Figure 12. (Continued).</caption>
</figure>
<figure>
<location><page_24><loc_9><loc_28><loc_91><loc_77></location>
<caption>Figure 12. (Continued).</caption>
</figure>
</document>
[
{
"title": "ABSTRACT",
"content": "Periodic variability in active galactic nuclei (AGNs) is a promising method for studying sub-parsec supermassive black hole binaries (SMBHBs), which are a challenging detection target. While extensive searches have been made in the optical, X-ray and gamma-ray bands, systematic infrared (IR) studies remain limited. Using data from the Wide-field Infrared Survey Explorer (WISE), which provides unique decade-long mid-IR light curves with a six-month cadence, we have conducted the first systematic search for SMBHB candidates based on IR periodicity. Analyzing a parent sample of 48,932 objects selected from about half a million AGNs, we have identified 28 candidate periodic AGNs with periods ranging from 1,268 to 2,437 days (in the observer frame) by fitting their WISE light curves with sinusoidal functions. However, our mock simulation of the parent sample indicates that stochastic variability can actually produce a similar number of periodic sources, underscoring the difficulty in robustly identifying real periodic signals with WISE light curves, given their current sampling. Notably, we found no overlap between our sample and optical periodic sources, which can be explained by a distinct preference for certain periods due to selection bias. By combining archived data from different surveys, we have identified SDSS J140336.43+174136.1 as a candidate exhibiting periodic behavior in both optical and IR bands, a phenomenon that warrants further validation through observational tests. Our results highlight the potential of IR time-domain surveys, including future missions such as the Nancy GraceRoman Space Telescope, for identifying periodic AGNs, but complementary tests are still needed to determine their physical origins such as SMBHBs. Keywords: Active galactic nuclei (16); Infrared astronomy(786); Quasars (1319); Supermassive black holes (1663); Time domain astronomy (2109)",
"pages": [
1
]
},
{
"title": "A Systematic Search for Candidate Supermassive Black Hole Binaries Using Periodic Mid-Infrared Light Curves of Active Galactic Nuclei",
"content": "1 CAS Key Laboratory for Research in Galaxies and Cosmology, Department of Astronomy, University of Science and Technology of China, Hefei, 230026, China; jnac@ustc.edu.cn 2 School of Physical Sciences, University of Science and Technology of China, Hefei, 230026, China 3 School of Astronomy and Space Sciences, University of Science and Technology of China, Hefei, 230026, China 4 Department of Astronomy, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA 5 Center for Artificial Intelligence Innovation, University of Illinois at Urbana-Champaign, 1205 West Clark Street, Urbana, IL 61801, USA National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA 6 (Received 2024 May 24; Revised 2024 November 9; Accepted 2024 November 11) Submitted to ApJ",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "Supermassive black holes (SMBHs) are intriguing celestial objects that reside at the centers of galaxies (Kormendy & Ho 2013). When galaxies merge, it is anticipated that the central black holes will form binary systems known as SMBH binaries (SMBHBs, Begelman et al. 1980). These binary systems offer valuable insights into various astrophysical phenomena, including galaxy formation (Colpi & Dotti 2011), gravitational wave (GW) emission (Hughes 2009), and the evolution of SMBHs (Merritt 2013). More massive binaries are pulsar-timing array (PTA) sources (e.g., Arzoumanian et al. 2018), while less massive binaries are targeted by space-based experiments such as LISA (Klein et al. 2016). They provide a laboratory to directly test strong-field general relativity (Hughes 2009; Centrella et al. 2010). However, despite their theoretical significance, the direct electromagnetic detection of close SMBHBs below subparsec separation has proven to be a challenging endeavor (see recent reviews such as De Rosa et al. 2019; D'Orazio & Charisi 2023) while numerous SMBH pairs at larger scales have been identified through the presence of dual active galactic nuclei (AGNs) (e.g., Komossa et al. 2003; Zhou et al. 2004; Liu et al. 2010, 2011, 2013; Koss et al. 2011; Liu et al. 2018; Chen et al. 2022). When a binary has exhausted its in- teractions with stars but has not approached close enough to emit significant gravitational radiation, its orbit does not have an obvious mechanism for decay. This long-standing challenge presents a major uncertainty in estimating the abundance of SMBHB mergers as GW sources. In theory, the bottleneck may be overcome in gaseous environments (e.g., Gould & Rix 2000; Cuadra et al. 2009; Chapon et al. 2013; del Valle et al. 2015), in triaxial or axisymmetric galaxies (e.g., Khan et al. 2016; Kelley et al. 2017), and/or by interacting with a third BH in hierarchical mergers (e.g., Blaes et al. 2002; Kulkarni & Loeb 2012; Bonetti et al. 2018). A recent and promising approach for identifying potential close SMBHB candidates has emerged in the thriving field of time-domain astronomy, which focuses on analyzing the periodic light curves of AGNs. The periodicity observed in these light curves can arise from changes in the accretion rate onto the black holes (Graham et al. 2015b) or from the relativistic Doppler boost resulting from the highly relativistic motion of gas in the mini accretion disk around the smaller black hole in the binary system (D'Orazio et al. 2015). Extensive research has been conducted using optical light curves to search for SMBHBs, and the methods employed in these studies have become well-established (Graham et al. 2015a; Charisi et al. 2016; Zheng et al. 2016; Liu et al. 2019b; Chen et al. 2020; Liao et al. 2021; Chen et al. 2024). However, many of the known candidates have been shown to be subject to false positives due to stochastic quasar variability (e.g., Vaughan et al. 2016). Additionally, recent findings suggest that some AGNs may display periodic variations only at some specific epochs (e.g., Jiang et al. 2022; O'Neill et al. 2022) further enhance the difficulties of periodic searches. Furthermore, previous surveys were only sensitive to the most massive quasars at high redshift ( z ≳ 2) which should have already gone through their major merger process (e.g., Shen 2009). Regardless of the process that imprints periodicity, it is generally accepted that the timescale is primarily determined by the binary orbital period. SMBHBs with masses between 10 6 -10 10 M ⊙ and separations of 0.01 pc possess orbital periods that span from years to several decades, making them possibly detectable by current timedomain surveys. In addition to the optical band, the periodic searches have also been conducted in other wavelength regimes, such as in X-ray (Serafinelli et al. 2020; Liu et al. 2020) and gamma-ray bands (Sandrinelli et al. 2018; Holgado et al. 2018). Periodic infrared (IR) light curves are also expected as a result of reverberation mapping of the circumbinary dusty torus in periodic AGNs. However, the search for SMBHBs using IR data has not been extensively explored, despite the detection of an IR time lag in an individual SMBHB candidate (Jun et al. 2015), which highlights the potential of utilizing periodic IR light curves for SMBHB identifi- cation (D'Orazio & Haiman 2017). The IR band not only provides an alternative approach to searching for SMBHBs but also offers distinct advantages. Firstly, IR photons are less susceptible to dust extinction, allowing them to penetrate through obscuring material, which is often abundant in galactic nuclear regions and galaxy merger systems. By extending the search to the IR wavelength range, we can uncover a population of SMBHB candidates that may have been previously overlooked, thereby expanding our knowledge of the prevalence and characteristics of these elusive systems. Even for unobscured AGNs, the variability amplitudes in the mid-IR band are typically larger compared to the optical band due to lower background contamination. Additionally, as discussed in D'Orazio & Haiman (2017), the dust-echo model can assist in determining the origin of central periodicity and place constraints on the physical characteristics of SMBHBs and their dust torus. To address this, we perform the first systematic search for SMBHB candidates via periodic IR light curves. The data we use come from the Widefield Infrared Survey Explorer (WISE, Wright et al. 2010) and its successor NEOWISE (Mainzer et al. 2014), which have provided us with a public dataset from February 2010 to December 2023 with a half-year cadence except for a 2.5year gap for each object. In this study, we primarily focus on sinusoidal periodic signals, which have been commonly adopted in modeling the optical light curves of SMBHB candidates (e.g., D'Orazio et al. 2015), and on modeling the IR light curves using the dust-echo model (D'Orazio & Haiman 2017). The paper is structured as follows. Section 2 introduces the data and methods employed to search for IR periodic AGNs. Section 3 presents the search and corresponding analysis results, where we show compelling evidence demonstrating the highly improbable nature of these periodic light curves being generated by random processes. Section 4 compares selected candidates with normal AGNs and with previous studies that utilized optical light curves. Additionally, we discuss SDSS J140336.43+174136.1 as a candidate exhibiting periodic behavior in both optical and IR bands. Finally, we summarize the main results and suggest directions for future work in Section 5. We assume a cosmology with H 0 = 66 . 88 km s -1 Mpc -1 , Ω m =0 . 32, and Ω Λ =0 . 68, adopted from Planck Collaboration et al. (2020).",
"pages": [
1,
2
]
},
{
"title": "2.1. IR data",
"content": "The WISE was originally launched to conduct a full-sky survey in four MIR bands centered at 3.4, 4.6, 12, and 22 µ m (labeled W1-W4) from February to August 2010 (Wright et al. 2010). The solid hydrogen cryogen used to cool the W3 and W4 instrumentation was depleted later and the spacecraft was placed in hibernation in 2011 February. However, it was reactivated in 2013 October under the new name NEOWISE -R, with only W1 and W2 operational, specifically to search for asteroids that could potentially pose a threat of impact on Earth (Mainzer et al. 2014). The WISE and its new mission NEOWISE operate in the same manner, that is, scanning a specific sky area every six months, with an average of 12 or more individual exposures taken within each epoch, typically spanning a single day. Consequently, each target in the sky has been observed 22-23 times separated by a six-month interval up to December 2023. The single-exposure photometry of WISE and NEOWISE are archived in the AllWISE Multiepoch Photometry Table and NEOWISE-R Single Exposure (L1b) Source Table 1 , respectively. The photometry we adopted is measured by pointspread function (PSF) profile fitting, which is suitable for quasars. We first binned the data within each epoch since the intraday IR variability is negligible except for very radioloud AGNs (Jiang et al. 2012) and the anticipated periods we aim to detect are on much longer timescales. The variance of the binned data was recalculated as shown in Equation 1: where ni denotes the number of data points within the i th epoch, mag ( ti ) denotes the mean magnitude within the i th epoch, and σ j denotes the photometric uncertainty of the j th data point. The first part represents the contribution from photometric uncertainty and short-term variability, while the second part represents the contribution from the system stability of WISE. We adopt σ s . s . = 0 . 029 mag for WISE and σ s . s . = 0 . 016 mag for NEOWISE from Lyu et al. (2019). Prior to binning, the data were filtered based on the quality flags in the catalog following Jiang et al. (2021). Moreover, outliers that fell outside the 3 σ range of the single data point at each epoch were eliminated, using the mean magnitude and mean photometric uncertainty at that epoch as references. The mean and variance of the single data points were then re-computed, resulting in the binned data used for our analysis. This step was crucial to ensure the accuracy and reliability of the data used in our study.",
"pages": [
2,
3
]
},
{
"title": "2.2. AGN Sample",
"content": "The AGN sample we chose is the widely-used Million Quasar Catalog (v8, 2 August 2023; Flesch 2023). This is a compendium of 907,144 type-I QSOs and AGNs, largely complete from the literature up to 30 June 2023. 66,026 QSO candidates are also included, calculated via radio/X-ray association (including double radio lobes) as being 99% likely to be quasars. Blazars and type-II objects are also included, bringing the total count to 1,021,800. To ensure the purity of our candidates, we only selected objects labeled as types 'Q', 'A', 'B', 'K', 'N' in the catalog (see what the legends of these types refer to in the note of Table 2), as these are confirmed AGNs. As a result, the number of objects amounts to 955,744. To enable a reliable long-term variability analysis, we apply a criterion that requires a minimum of 15 detected epochs (about 2/3 of all detected epochs) for each source. This left us with a subset of 576,260 AGNs. In order to rationalize the computational effort in further analysis, we choose to analyze only AGNs with obvious variability. This selection strategy is crucial to ensure that candidates meet the criteria outlined in selection criterion 5 (see Section 2.4). Furthermore, light curves displaying subtle variability tend to pose a challenge in terms of model constraints within the DRW framework (see Section 3.1). To address this issue, we introduce a requirement that, for a given light curve, at least one data point must satisfy mag ( ti ) -2 δ mag ( ti ) > magmean and at least one data point should satisfy mag ( ti ) + 2 δ mag ( ti ) < magmean . This allows us to narrow down our sample to 53,496 AGNs, ensuring that the selected candidates have the necessary characteristics for further analysis. We also require that the light curves be well constrained by the DRW model (see section 3.1), resulting in a parent sample of 48,932 AGNs.",
"pages": [
3
]
},
{
"title": "2.3. Finding Periodicity",
"content": "To identify sources exhibiting periodic variability, we employed a sinusoidal function to fit the light curves of both W1 and W2 bands. The fitting process involved minimizing the following equation using the χ 2 statistics as Equation 2 below (D'Orazio & Haiman 2017): The Lomb-Scargle (LS) periodogram (Lomb 1976; Scargle 1982) is widely used for periodicity detection in unevenly sampled data. In our analysis, we utilized the generalized Lomb-Scargle (GLS) periodogram (Zechmeister & Kürster 2009) implemented in the pyastronomy 2 package. In the GLS periodogram, we calculate a series of 'powers' defined as Equation 3: where χ 2 0 is the residual of a model assuming no variability, and χ 2 is equivalent to Equation 2 in the GLS periodogram (Zechmeister & Kürster 2009). Compared to the LS periodogram, the GLS periodogram provides a more accurate frequency prediction by taking into consideration an offset and weights, which allows the calculation of the maximumpower in the GLS periodogram to be equivalent to minimizing Equation 2. The searched period range of the GLS periodogram is from 700 days to T 0 days, where T 0 is the time span of the light curve.",
"pages": [
3,
4
]
},
{
"title": "2.4. Selection Criteria",
"content": "We selected reliable periodic candidates based on five criteria. First, to account for stochastic red noise variability, we adopt a comparative approach by measuring the performance of the candidates against that of simulated light curves, as described in Section 3.1. Rather than applying a single threshold to the normalized periodograms, we require 3 σ significance in both the W1 and W2 bands. Secondly, we adopt the signal-to-noise (S/N) ratio defined as Equation 4 (Horne & Baliunas 1986), where A 0 denotes the amplitude and σ 2 r denotes the variance of the residuals. we require that ξ > 2 for both the W1 and W2 bands. Thirdly, we require that the difference between the frequencies of the W1 and W2 light curves must not exceed 5% of their sum, in accordance with Equation 5: where f 1 and f 2 represent the frequencies of the W1 and W2 light curves. In addition, the phase difference between them must be less than 1/8 cycles. This criterion reflects the requirements of the dust-echo model and guarantees that the two sinusoids have the same frequency and phase. However, we allow for a margin of error of 1/8 cycles to account for potential minor phase differences arising from low photometric accuracy or intrinsic characteristics of the IR light curves. For instance, the W2 band may receive a greater contribution from dust with lower temperatures located further from the BH, leading to a slight delay in the W2 variability compared to the W1 band (Lyu et al. 2019). If the sources are located at high redshifts, i.e. z > 2, the W1 and W2 emission could mainly come from a relatively inner and outer part of the accretion disk, then a time delay can also be expected. We confirm this small delay for our candidates in Section 3.2. Fourthly, the available data are required to show more than two cycles in both the W1 and W2 bands. It is important to note that even strictly sinusoidal variations are difficult to distinguish from a simple stochastic process when the number of cycles N cyc is ≲ 5 (Vaughan et al. 2016). Any finite light curve generated by the corresponding variability process represents only a snapshot of that process. The periodogram derived from this light curve will show power distributed around the actual Power Spectral Density (PSD) continuum with some scatter. This scatter can create the appearance of spikes in the periodogram (e.g., see Fig. 1 and Fig. 12), leading the light curve to seemingly exhibit a sinusoidal-like pattern over 2 to 3 cycles, even if the underlying PSD process is purely continuous. Therefore, these patterns can only be reliably distinguished with much longer time series spanning many (e.g., > 4-5) cycles of the putative period. However, most optical searches still adhere to a criterion of > 1.5 cycles due to the limited time span of the data (e.g. Graham et al. 2015a; Charisi et al. 2016; Liu et al. 2019b; Chen et al. 2024). Consequently, false positives would arise in the candidate sample from stochastic red noise variability (Vaughan et al. 2016). Due to the limited number of detected epochs, which amounts to 23, we set the limit to > two cycles. We believe that this is the best approach we can take with the current WISE data, but we note that further improvements should be pursued in future research with more available data. Fifthly, we check if the maximum power of the periodogram indicates a 'correct' period, since anomalous data points (e.g., an outlier with a small error) could mislead the periodogram calculation. Additionally, this criterion allows us to check if the light curves show strong variability. We estimate the period of a light curve by examining the data points. If all data points within a time segment satisfy mag ( ti ) -2 δ mag ( ti ) > offset (calculated by the periodogram), we classify it as a 'faint segment'. If all data points within a time segment satisfy mag ( ti ) + 2 δ mag ( ti ) < offset, we classify it as a 'bright segment'. Then we conduct a count in chronological order, recording each change from a faint segment to a bright segment or vice versa, and neglecting normal",
"pages": [
4,
5
]
},
{
"title": "FBQS J0832+2853",
"content": "data points in this process (see an example shown in Fig. 2). Denoting the total count as n , ideally n / 2 periods would be included in the time span. We require that for both the W1 and W2 bands, the period at the maximum power Tmax and the time span of the light curve T 0 satisfy Equation 6:",
"pages": [
5
]
},
{
"title": "3.1. Simulating with Stochastic Variability",
"content": "AGN variability can roughly be described by a Gaussian first-order continuous autoregressive model (CAR(1), Kelly et al. 2009), also known as a damped random walk (DRW, MacLeod et al. 2010) or Ornstein-Uhlenbeck process. The likelihood function for the DRW process can be represented as: where the Xi represents the flux of the data point i , and q is the long-term mean of the light curve. The covariance matrix C is given by: where δ i j is the Kronecker delta. In our analysis, we assume a strong correlation between the W1 and W2 data. Subsequently, we have devised a modified form of the covariance matrix that accounts for this correlation, expressed as: where a and b represent the observed band (W1 or W2) of the data points, and ρ is the correlation coefficient. Regarding the data quality of WISE, we only use the binned data when estimating DRW parameters. We set a uniform prior for the logarithm of the parameters σ 1, σ 2 and τ , and a uniform prior for ρ W 1 , W 2. We set q as the mean magnitude of the light curve to reduce computational cost and use the Markov Chain Monte Carlo (MCMC) sampler emcee 3 (Foreman-Mackey et al. 2013) to construct the posterior samples of the parameters. However, we find that some light curves lack enough variability but pass the variability criterion (see Section 2.2) due to certain data points with very small errors. Consequently, they are not adequately constrained by the DRW model. To ensure a stable posterior for the MCMC processes, we set a criterion that the 1 σ error of the posterior lg τ sample must be less than 2. This is the final requirement to define our parent sample, resulting in a final number of 48,932 (see Section 2.2). The statistical distributions of σ , τ , and ρ W 1 , W 2 are illustrated in Fig. 4. Upon comparison, the selected candidates exhibit similar values for σ 1, σ 2, and τ compared to the parent sample. Notably, they display a stronger correlation, which aids in meeting selection criterion 4 easier. The distribution of ρ W 1 , W 2 serves as validation for our correlation assumption. As an alternative test, we perform a model selection to test if the correlation parameter is necessary to explain the light curve on top of a uncorrelated stochastic background. We adopt a maximum likelihood approach for the model comparison and parameter estimation. We use the Bayesian information criterion (BIC), which is defined as: where L is the likelihood function, k is the number of free model parameters, and N is the number of data points. A lower BIC value indicates a preferable model. Typically, when ∆ BIC < -10, the model with the lower BIC is strongly favored. For the uncorrelated DRW process, we generate posterior parameter samples independently for each band. Subsequently, we integrate these samples into a covariance matrix to compute L with all correlation components set as zero. Our analysis indicates a strong preference for the correlated model over the uncorrelated one, aligning with our earlier analysis (further details provided in Fig. 3). We proceed by randomly selecting a set of parameters to generate two correlated W1 and W2 simulated light curves. These simulated light curves are then compared with the actual observation cadence and measurement errors. This process is repeated 100,000 times to check that the selection criteria are met. In addition, we use the periodogram values of these simulated light curves to determine the significance of our candidates, as described in Criterion 1 in section 2.4. To further validate our simulation methodology, we perform a statistical comparison between the real and simulated light curves. Our analysis shows that they exhibit similar behaviour for each selection criterion (detailed results in Table 1), suggesting that the correlated DRW processes could potentially generate a comparable number of candidates to the real ones. Notably, the DRW model is relatively simple and may not fully capture the complexity of actual AGN variability due to degeneracies (Zu et al. 2013). When using the DRW assumption to calculate false alarm probabilities, it might not completely consider the assumptions of the red noise hypothesis. However, using different variability models could lead to some small changes in the results, but usually not significant ones.",
"pages": [
5,
6
]
},
{
"title": "3.2. Candidates of Periodic Sources",
"content": "Using the data and criteria outlined in Section 2, we have identified a total of 28 candidates. The catalog of candidates can be found in Table 2. The light curves and the GLS pe- riodograms of two examples are presented in Fig. 1 and the rest can be found in Fig. 12. The W1 and redshift distributions of both the parent sample and the selected candidates are shown in Fig 5. It is worth noting that the candidates are predominantly situated in the low redshift range (median z=0.126 , with 27/28 at z < 0.4) and bright region (median W1=12.69, 25/28 satisfy W1 < 14), which can be attributed to a selection effect making it easier for them to fulfill the selection criteria. In Section 2.4, we mention the possibility of a slight phase delay between the W2 and W1 light curves. This is confirmed by our candidates, with an average delay of 0.025 periods, as illustrated in Fig. 6. The maximum phase delay is 0.087 period, which remains smaller than the 1/8 period threshold used in our selection process, indicating a reasonable choice. As an alternative test, here we use the BIC method (see details in Section 3.1) to assess if an additional periodic signal is needed to explain the light curve on top of a stochastic background. The likelihood function for the DRW + sinusoidal model is written as: where Si represents the sinusoid signal. Our analysis shows that, for the vast majority of the candidates, BIC DRW + sin -BIC DRW( c ) < -10, indicating robust evidence that the periodic model is significantly favored over the pure stochastic model (refer to Fig. 3 for detailed results). However, this could be explained by the limitation of the DRW model, which might not adequately capture a spiky PSD and may be more suited to smooth continuum-like PSD shapes. Therefore, additional data in the future are necessary to obtain a higher density in frequency sampling for the periodogram and better test the preference for the sinusoid + DRW model. It's worth noting that the sinusoid +DRW model is a commonly used model due to its simplicity and clear physical picture. However, there exist alternative light curve models, such as the bursty model predicted by circumbinary accretion disk simulations (Farris et al. 2014). Despite the potential of this bursty model, we did not simulate it for our candidates as we could not observe 'bursty' characteristics in the WISE light curves. Here we compile the derived quantities for our analysis, which are presented in Table 2. The table includes essen- tial information such as the Milliquas Name, right ascension (R.A.), declination (Decl.), redshift ( z ), and quasar properties as provided by Wu & Shen (2022) and Liu et al. (2019a). Additionally, we have included periodic properties such as offsets of the W1 and W2 bands, and the period of the candidates.",
"pages": [
6,
7,
8
]
},
{
"title": "3.3. False Alarm Probability",
"content": "To estimate the false alarm probability (FAP) for each candidate, we employ the method described in Chen et al. (2020). We estimate the approximate FAP using the effective number of independent frequencies Nef f , which is calculated by dividing the observed frequency window by the expected peak width δ f = 1 / T . The FAP is estimated as (e.g., VanderPlas 2018): Where Psingle = 1 -e Z , Z is the periodogram value defined in Zechmeister & Kürster (2009). Since candidates are selected based on the light curves of both the W1 band and W2 band, there are two P single and N eff. To simplify the calculation, we can estimate the periodogram FAP as Equation 13: It is important to acknowledge that this calculation assumes independent FAPs for the W1 and W2 bands, contrary to the strong correlation observed in the DRW simulation (refer to Section 3.1). Nevertheless, this method provides an upper limit estimate for the FAP, as single-band FAP calculations do not consider this correlation. NOTE- (1)-(4): Name, RA, DEC, and redshift of the objects. (5)-(6): Median magnitudes in the W1 and W2 bands, respectively. (7): Period (in the observer-frame) in units of days. (8): Legend of type/class from Flesch (2023): Q = QSO, type-I broad-line core-dominated, 860,100 of these. A = AGN, type-I Seyferts/host-dominated, 47,044 of these. B = BL Lac type object, 2,814 of these. (FSRQs are typed as QSOs here) K = NLQSO, type-II narrow-line core-dominated, 6,048 of these. N = NLAGN, type-II Seyferts/host-dominated, 39,768 of these. Incomplete, and includes an unquantified residue of legacy NELGs/ELGs/LINERs, plus some unclear AGN. This is the catch-all category. S = star classified but showing quasar-like photometry and radio/X-ray association, thus included as a quasar candidate; 124 of these. R = radio association displayed. X = X-ray association displayed. 2 = double radio lobes displayed (declared by data-driven algorithm). (9)-(11): Logarithmic bolometric luminosity in units of erg/s, black hole mass in units of solar mass, and Eddington ratio for sources in the SDSS DR16 quasar catalog (Wu & Shen 2022), the DR7 board-line AGN catalog (Liu et al. 2019a). Additionally, we empirically compute the global FAP using the 100,000 simulated light curves for each AGN, following a similar approach as presented by Barth & Stern (2018). The global FAP accounts for all false positives that satisfy all selection criteria within the explored frequency range. The global FAP values for candidates range from 4 . 0 × 10 -4 to 4 . 77 × 10 -3 , with a mean of 2 . 07 × 10 -3 by Gaussian fit. Further details are available in Fig. 7. It is noteworthy that the global FAPs are significantly higher than the periodogram FAPs. This discrepancy can be attributed to the stronger variability exhibited by the simulated light curves compared to real observations, leading to an increase in the global FAP. Moreover, while most false positives exhibit only a 3 σ significance level, certain candidates display notably stronger significance, such as SDSS J162938.09+362452.2 and 6dF J214907.4-175159 as depicted in Fig. 12.",
"pages": [
8,
9,
10
]
},
{
"title": "4.1. Do the Candidates Show Distinctive Properties?",
"content": "One might wonder whether our selected periodic sample shows distinct features in other physical properties, such as bolometric luminosity ( L bol), black hole mass ( M BH) and Eddington ratio ( λ edd= L bol / L edd). To investigate this, we try to extract a subsample of candidates with previous measurements in these parameters. We chose to cross-match the candidates with two AGN samples based on Sloan Digital Sky Survey (SDSS), namely the SDSS DR16 quasar catalog (DR16Q) (Wu & Shen 2022) and DR7 broad-line AGN catalog (Liu et al. 2019a). They contained 750,414 and 14,584 sources, concentrated at high and low redshifts, respectively (see Figure 8). We only found a total of 10 candidates that overlapped with the two catalogs. Similar to the occupation of the candidates in the whole parent sample (see Figure 5), they are all in the low redshift and low magnitude space (see Figure 8) due to selection effects. Therefore, we need to construct a control sample for a fair comparison, at least in redshift and IR magnitudes. Specifically, we selected the 10 closest AGNs with a redshift ratio less than 1.01 and with differences in W1 and W2 magnitudes less than 0.1. In cases when no AGN meet the specified criteria, we choose the closest AGN in the catalog as the control sample. After controlling, we then compare the M BH, L bol and λ edd of the periodic sources with normal AGNs. Additionally, we have also considered another parameter dust covering factor ( fd ), defined as the ratio of the W1monochromatic luminosity ( LW 1 = ν W 1 F ν W 1) and the L bol ( fd = L W1 / L bol, e.g., Ma & Wang 2013), to indicate the extent of dust obscuration. However, no significant differences in the distributions of these parameters can be seen visually (see Fig. 9). We have also checked the mean and standard deviation (SD) of their distributions (see Table 3) and confirmed it. We further performed the K-S statistics for the L bol, M BH, λ edd, and fd distributions between the candidates and the control sample, yielding values of 0.375, 0.400, 0.225, and 0.400, respectively. The K-S tests suggest significant differences in these properties with a critical value of 0.480 for a 95% confidence level, so none of them show significant differences. It is worth emphasizing that the systematic uncertainties in the estimates of AGN properties (e.g., singleepoch BH mass, see discussions in Shen 2013) are nonnegligible and would dilute the difference, if there were any. This may be particularly important for the small sample size for comparison here. 4.2. Comparison with SMBHB Candidates Selected by Optical Periodicity As mentioned in the introduction, previous studies have identified samples of SMBHB candidates based on optical periodic light curves. It is intriguing to investigate whether there are any common sources between our IR-selected sample and these optically-selected samples. Such overlap would provide compelling evidence for the periodicity of the candidates. Additionally, if the periods derived from optical light curves align with those obtained from IR light curves, we can employ the dust-echo model to further explore the properties of the candidates. For this purpose, we compile a collection of optical SMBHB candidates reported in various literature, including Graham et al. (2015a); Charisi et al. (2016); Zheng et al. (2016); Liu et al. (2019b); Chen et al. (2020, 2024). Unfortunately, we do not find any overlap between our sample and those samples. The lack of matches is not entirely unexpected, as the two searches prioritize candidates with different periods. Specifically, we find that the periods of the optical candidates predominantly fell below 1700 days, z whereas the periods of our selected candidates were mostly longer than 1700 days (see details shown in Fig. 10). Additionally, most of our candidates have low redshifts (z < 0.4) due to the detection ability of WISE, while the redshifts of candidates from optical searches span from 0 to ∼ 3.",
"pages": [
10,
11
]
},
{
"title": "4.3. Optical Light Curves and A Possible Periodic Source",
"content": "Given the abundance of optical photometric archival data, we then examined the optical light curves of the 28 candidates using data from various surveys, such as the Catalina Real Time Transient Survey (CRTS, Drake et al. 2009), the Palomar Transient Factory (PTF, Rau et al. 2009), the Zwicky Transient Facility (ZTF, Masci et al. 2019), and the All-Sky Automated Survey for Supernovae (ASAS-SN, Kochanek mag et al. 2017). Their combined dataset provides optical light curves spanning about two decades. We have normalized the data from different surveys and binned them at 20-day intervals. However, they generally show a small amplitude of optical variability, probably due to either intrinsically weak variability in the optical band or too shallow surveys. Only one candidate, SDSS J140336.43+174136.1, shows significant variability and periodicity in its optical light curve (see Fig. 11). The periodogram analysis of the optical light curve of SDSS J140336.43+174136.1 reveals a period of 2891 ± 22 days, that is slightly longer than the period fitted from the IR light curves, which is 2346 ± 87 days. The errors are given by periodogram peak statistics based on 100,000 perturbed light curves for each real light curve. The discrepancy could be caused by the poor cadence of the WISE surveys, and so in the following analysis we choose the optical period for this particular source. We present the corresponding best-fitting sinusoids for the W1, W2, and optical light curves in Figure 11. The time delay between the W1 (W2) and optical light curves is 630 (727) days (in the rest frame), respectively. Assuming the simplest scenario of a hollow spherical shell of dust with the source located at its center, we can easily estimate that the inner radius of the shell is 0.53 pc. These values are roughly consistent with previous statistical correlations between the time delay and bolometric luminosity of AGNs. For example, using the relations obtained by Lyu et al. (2019), i.e. for the W1 band and for the W2 band, and a given bolometric luminosity of 10 45 . 81 erg s -1 for SDSS J140336.43+174136.1 (see Table 2), the W1 and W2 time delays are estimated to be 476 ± 147 days and 566 ± 158 days, respectively, which is in good agreement with our measurements.",
"pages": [
11,
12
]
},
{
"title": "5. CONCLUSION",
"content": "In this work, we have conducted the first systematic search for SMBHBs in the IR band and identified 28 AGNs exhibiting IR periodic variability using the decade-long WISE and NEOWISE light curves. We performed extensive simulations to evaluate the potential influence of stochastic variability on candidate selection. By counting all the false positives in the simulated light curves, the probability of these periodic light curves being generated by random processes is an average of 0.207% by Gaussian fit. However, the number can be reproduced by our mock simulations with the DRW process of the parent sample. This indicates the challenge of identifying reliable SMBHBs with periodic emission using only WISE light curves, which are constrained by their time span and visit cadence, necessitating a low threshold on the minimum number of cycles ( N cyc > 2). Consequently, the nature of these periodic sources, whether driven by SMBHBs or not, should be carefully tested in future observations. We subsequently investigated whether these IR periodic sources exhibit distinct properties compared to normal AGNs. However, no significant biases were found, and the small sample size prevents us from statistically identifying potential weak differences. On the other hand, we found that there is no overlap between our sample (IRselected) and the SMBHB candidates reported in the literature (optically selected), likely due to differences in their preferred period ranges. Interestingly, we have identified SDSS J140336.43+174136.1 as a candidate displaying periodicity in both optical and IR bands. Further extended and more sensitive observations of these candidates are essential to confirm whether or not they are real periodic sources in both optical and IR bands. As suggested by D'Orazio & Haiman (2017), the periodic sources can also help test the physical mechanisms that produce the observed periodicity, due to relativistic Doppler modulation or accretion rate fluctuations. This study highlights the promising potential of identifying SMBHB candidates through IR time-domain observations, which can significantly complement searches in other bands, although there are challenges in distinguishing genuine sources. It opens up a new avenue for SMBHB searches and can be applied to future surveys. Although NEOWISE unfortunately ended on July 31, 2024, the upcoming NEO Surveyor (Mainzer et al. 2023), the successor to NEOWISE , and the Nancy Grace Roman Space Telescope (Spergel et al. 2015) will ensure a bright future for IR time-domain astronomy, particularly for SMBHB searches (Haiman et al. 2023). These different surveys somewhat form a relay in time and provide us with IR light curves with a long time baseline, which is essential for verifying the periodicity of the candidates with N cyc < 4 and for detailed comparisons against physical models.",
"pages": [
12,
13
]
},
{
"title": "ACKNOWLEDGMENTS",
"content": "We sincerely thank the expert referee for very constructive and insightful comments, which helped improve our manuscript greatly. This work is supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB0550200), the National Natural Science Foundation of China (grants 12073025, 12192221, the science research grants from the China Manned Space Project and the Cyrus Chun Ying Tang Foundations. D.L. acknowledges the support from the National Undergraduate Training Program for Innovation and Entrepreneurship. X.L. acknowledges support by NSF grant AST-2206499. This research makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. This research also makes use of data products from NEOWISE-R, which is a project of the Jet Propulsion Laboratory/California Institute of Technology, funded by the Planetary Science Division of the National Aeronautics and Space Administration. This research has made use of the NASA/IPAC Infrared Science Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration.",
"pages": [
13
]
},
{
"title": "REFERENCES",
"content": "Klein A., et al., 2016, PhRvD, 93, 024003",
"pages": [
13
]
},
{
"title": "A. THE IR LIGHT CURVES OF THE PERIODIC AGN SAMPLE",
"content": "We have shown the light curves of 2 special candidates in Fig. 1. Here we show the light curves of the rest 26 candidates in Fig. 12.",
"pages": [
15
]
}
]
2024arXiv241107305R
https://arxiv.org/pdf/2411.07305.pdf
<document>
<section_header_level_1><location><page_1><loc_17><loc_85><loc_83><loc_87></location>PICZL : Image-based photometric redshifts for AGN</section_header_level_1>
<text><location><page_1><loc_8><loc_81><loc_92><loc_84></location>William Roster , ⋆ 1 , M. Salvato 1 , 2 , S. Krippendorf 3 , 4 , A. Saxena 5 , R. Shirley 1 , J. Buchner 1 , J. Wolf 2 , 6 , T. Dwelly 1 , F. E. Bauer 7 , 8 , 9 , 10 , J. Aird 11 , C. Ricci 12 , 13 , R. J. Assef 12 , S.F. Anderson 14 , X. Liu 15 , 16 , 17 , A. Merloni 1 , J. Weller 1 , 4</text>
<text><location><page_1><loc_46><loc_79><loc_54><loc_81></location>K. Nandra 1</text>
<text><location><page_1><loc_36><loc_77><loc_64><loc_78></location>(A ffi liations can be found after the references)</text>
<text><location><page_1><loc_44><loc_75><loc_56><loc_76></location>November 14, 2024</text>
<section_header_level_1><location><page_1><loc_46><loc_72><loc_54><loc_73></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_6><loc_57><loc_94><loc_71></location>Context. Computing reliable photometric redshifts (photo-z) for active galactic nuclei (AGN) is a challenging task, primarily due to the complex interplay between the unresolved relative emissions associated with the supermassive black hole and its host galaxy. Spectral energy distribution (SED) fitting methods, while e ff ective for galaxies and AGN in pencil-beam surveys, face limitations in wide or all-sky surveys with fewer bands available, lacking the ability to accurately capture the AGN contribution to the SED, hindering reliable redshift estimation. This limitation is a ff ecting the many 10s of millions of AGN detected in existing datasets, e.g., those AGN clearly singled out and identified by SRG / eROSITA. Aims. Our goal is to enhance photometric redshift performance for AGN in all-sky surveys while simultaneously simplifying the approach by avoiding the need to merge multiple data sets. Instead, we employ readily available data products from the 10th Data Release of the Imaging Legacy Survey for the Dark Energy Spectroscopic Instrument, which covers > 20,000 deg 2 of extragalactic sky with deep imaging and catalogbased photometry in the grizW1-W4 bands. We fully utilize the spatial flux distribution in the vicinity of each source to produce reliable photo-z. Methods. We introduce PICZL, a machine-learning algorithm leveraging an ensemble of convolutional neural networks. Utilizing a cross-channel approach, the algorithm integrates distinct SED features from images with those obtained from catalog-level data. Full probability distributions are achieved via the integration of Gaussian mixture models.</text>
<text><location><page_1><loc_6><loc_52><loc_94><loc_56></location>Results. On a validation sample of 8098 AGN, PICZL achieves an accuracy σ NMAD of 4.5% with an outlier fraction η of 5.6%. These results significantly outperform previous attempts to compute accurate photo-z for AGN using machine learning. We highlight that the model's performance depends on many variables, predominantly the depth of the data and associated photometric error. A thorough evaluation of these dependencies is presented in the paper.</text>
<text><location><page_1><loc_6><loc_48><loc_94><loc_52></location>Conclusions. Our streamlined methodology maintains consistent performance across the entire survey area, when accounting for di ff ering data quality. The same approach can be adopted for future deep photometric surveys such as LSST and Euclid, showcasing its potential for wide-scale realization. With this paper, we release updated photo-z (including errors) for the XMM-SERVS W-CDF-S, ELAIS-S1 and LSS fields.</text>
<text><location><page_1><loc_6><loc_47><loc_49><loc_47></location>Key words. Photo-z, AGN, Extragalactic Surveys, Machine Learning</text>
<section_header_level_1><location><page_1><loc_6><loc_42><loc_18><loc_43></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_6><loc_13><loc_49><loc_41></location>In recent decades, our understanding of Active Galactic Nuclei (AGN) and their role in galaxy and cosmic evolution has significantly advanced. These luminous celestial powerhouses are thought to be fueled by the accretion of matter onto supermassive black holes (SMBHs) located at the centers of galaxies, exerting intense energetic radiation across the entire electromagnetic spectrum, ranging from radio to γ -rays (Padovani et al. 2017). The close correlation observed between the mass of the central SMBH, whether active or inactive and the properties of its host galaxy's bulge - such as the galaxy's mass and velocity dispersion (e.g., Gebhardt et al. 2000; Ferrarese & Merritt 2000) - suggests a co-evolutionary relationship between galaxies and their central engines (Kormendy & Ho 2013; Heckman & Best 2014). Ongoing research focuses on understanding scaling relations, the evolution of SMBHs within galaxies, and the interconnected rates of star formation (SFR) and black hole accretion (BHAR) over cosmic time (e.g., Madau & Dickinson 2014). To further explore and address these unresolved topics requires diverse AGN samples with reliable redshifts to determine BH demographics and constrain models of galaxy evolution. For all these studies, redshift is an indispensable quantity, with spectroscopic redshifts (spec-z) remaining the preferred es-</text>
<text><location><page_1><loc_51><loc_15><loc_94><loc_43></location>imates for determining precise cosmic distances (Hoyle 2016). However, while multi-objects spectrographs, such as the Sloan Digital Sky Survey (SDSS-V; York et al. 2000; Kollmeier et al. 2019), the Dark Energy Spectroscopic Instrument (DESI; DESICollaboration et al. 2016), the Subaru Prime Focus Spectrograph (PFS; Tamura et al. 2016) or the 4-metre Multi-Object Spectroscopic Telescope (4MOST; De Jong et al. 2019), are set to provide a drastic rise in the number of observed sources over the next several years, we are currently in the situation in which millions of AGN have been detected all-sky by various surveys (e.g., by the Wide-field Infrared Survey Explorer mission (WISE; Wright et al. 2010), and the extended Roentgen Survey with an Imaging Telescope Array (eROSITA; Merloni et al. 2012; Predehl et al. 2021), with only the brightest sources having been observed spectroscopically (Dahlen et al. 2013). The growing disparity between photometric and spectroscopic observations will only widen with upcoming surveys such as the Legacy Survey of Space and Time (LSST; Ivezic et al. 2019) and Euclid (Collaboration et al. 2024), covering unprecedented areas and depths (Newman & Gruen 2022). Thus for the bulk of AGN, we must make use of multiband photometry and rely on photometric redshifts (photo-z).</text>
<text><location><page_1><loc_51><loc_10><loc_94><loc_15></location>First implemented by Baum (1957) for inactive galaxies, these low-precision redshift estimates utilize photometric observations to e ff ectively obtain a sparsely sampled spectral energy distribution (SED), trading precision for scalability. They en-</text>
<text><location><page_2><loc_6><loc_63><loc_49><loc_93></location>compass an array of techniques assuming color-redshift evolution (Connolly et al. 1995; Steidel et al. 1996; Illingworth 1999; Bell et al. 2004), including template-based approaches (e.g., Bolzonella et al. 2000; Ilbert et al. 2006; Salvato et al. 2008; Beck et al. 2017), where redshifted models built on theoretical or empirical SEDs are fitted to observed multi-band photometry. Although a limited number of available bands can introduce uncertainties (see review by Salvato et al. 2018), photo-z methods o ff er an e ffi cient way to estimate distances for all sources in an imaging survey, yielding highly accurate estimates with as few as three bands for passive galaxies (Benitez 2000; Abdalla et al. 2011; Arnouts & Ilbert 2011; Brescia et al. 2014; Desprez et al. 2020). By contrast, reliable photo-z for AGN have historically required highly homogenized photometry across > 20 filters, which was only achievable in pencil-beam surveys (Salvato et al. 2011). As such, this level of detail continues to be unfeasible for wide-area surveys. However, with the 10th data release of the DESI Legacy Imaging Surveys (LS10, Dey et al. 2019), we now have a broad-sky survey that, while lacking NIR coverage, includes a few optical bands supplemented by mid-IR WISE data. This allows us to explore the possibility of generating reliable photo-z for AGN over the full sky, despite having fewer filters compared to the densely sampled pencil-beam surveys.</text>
<text><location><page_2><loc_6><loc_31><loc_49><loc_63></location>SED fitting applied to a broad population of AGN remains particularly challenging due to the uncertainty and di ffi culty of disentangling the relative contributions of the nucleus and respective host to a given band (e.g., Luo et al. 2010; Salvato et al. 2011; Brescia et al. 2019). Since the accretion properties of SMBHs, often characterized as the bolometric luminosity divided by the Eddington limit, or the Eddington ratio, significantly influence the SED of AGN, the intense power-law continuum radiation can either partly (host-dominated) or entirely (quasar-dominated), outshine the respective host, hiding key spectral features that lead to redshift degeneracies (Pierce et al. 2010; Povi'c et al. 2012; Bettoni et al. 2015). Consequently, selecting a limited number of templates can be insufficient for correct redshift determination, while increasing the number of templates raises the degeneracy (see discussion in Salvato et al. 2011; Ananna et al. 2017). In this regime of accounting for AGN contributions to galaxy photo-z, one potential approach involves modeling objects as a combination of quasar and galaxy templates (eg., Cardamone et al. 2010), performed with EAZY (Brammer et al. 2008). In addition, surveys typically estimate fluxes with models that do not account for a mixed contribution from AGN and host galaxy. Ultimately, AGN are also intrinsically variable sources on the timescales explored by the previously mentioned surveys leading to incongruent photometry acquired across di ff erent epochs.</text>
<text><location><page_2><loc_6><loc_10><loc_49><loc_30></location>In contrast to template-fitting methods, more recent approaches have shifted towards the use of empirical Machine Learning (ML) models, performing regression or classification, to tackle photo-z applied predominantly to inactive galaxies (Collister & Lahav 2004; Laurino et al. 2011; Zhang et al. 2013; Hoyle 2016; D'Isanto & Polsterer 2018; Brescia et al. 2019; Eriksen et al. 2020; Li et al. 2021). Provided with a very large and complete spec-z sample, ML architectures manipulate photometric input features to minimize the divergence between spectroscopic and ML-derived redshifts. Over the years, a plethora of ML architectures, including decision trees (Breiman 2001; Carliles et al. 2010; Li et al. 2022), Gaussian processes (Almosallam et al. 2016) and K-nearest neighbours (Zhang et al. 2013; Luken et al. 2019) have been employed, yielding accurate point predictions and, more interestingly, full probability density functions (PDFs) (Kind & Brunner 2013; Cavuoti et al.</text>
<text><location><page_2><loc_51><loc_80><loc_94><loc_93></location>2016; Rau et al. 2015; Sadeh et al. 2016). The latter grants access to the prediction uncertainty, as otherwise naturally provided by template-fitting approaches, relevant for studies dealing with, e.g. luminosity functions (Aird et al. 2010; Buchner et al. 2015; Georgakakis et al. 2015). However, the limited availability of a sizable training sample of AGN has resulted in only a few attempts to compute photo-z for mostly nucleus-dominated objects with ML-based methods (Mountrichas et al. 2017; Fotopoulou & Paltani 2018; Ruiz et al. 2018; Meshcheryakov et al. 2018; Brescia et al. 2019; Nishizawa et al. 2020).</text>
<text><location><page_2><loc_51><loc_54><loc_94><loc_80></location>More recently, the conventional approach of manually selecting photometric features for ML has been replaced by bright, well-resolved galaxies at low redshift (Hoyle 2016; Pasquet et al. 2018; Campagne 2020; Hayat et al. 2021). In this regime, integrating galaxy images into deep neural networks inherently captures essential details like flux, morphology, and other features that would typically be extracted from catalogs based on predefined assumptions, leading to a more comprehensive redshift estimation process. This approach is particularly advantageous for addressing current limitations faced by photo-z methods for AGN, as it leverages model-independent fluxes and redshift indicative features, including surface brightness profiles (Stabenau et al. 2008; Jones & Singal 2017; Gomes et al. 2017; Zhou et al. 2021, 2023). Unlike creating a single SED from total flux measurements, projects employing images with independent pixelby-pixel SEDs at identical redshift have demonstrated increased photo-z constraining power, alleviating previous empirical approaches by decreasing the fraction of outliers (Henghes et al. 2022; Schuldt et al. 2021; Lin et al. 2022; Dey et al. 2022a; Newman & Gruen 2022).</text>
<text><location><page_2><loc_51><loc_35><loc_94><loc_54></location>Here, we introduce PICZL (Photometrically Inferred CNN redshift(Z) Likelihoods), an enhanced approach to photoz estimation that builds upon (C ircle Z by Saxena et al. 2024). While the authors demonstrated that redshift degeneracies encountered for AGN, typical in cases of limited photometry, can be broken by integrating aperture photometry alongside traditional total / model fluxes and colors, PICZL instead computes photo-z PDFs for AGN directly from homogenized flux band cutouts by leveraging the more detailed spatial light profile. All inputs are obtained utilizing LS10 exclusively. Similar to Saxena et al. (2024), PICZL can produce reliable photo-z PDFs for all Legacy-detected sources associated with an AGN. However, the model can, in principle, be applied to other extragalactic sources (e.g, inactive galaxies, Götzenberger et al. in prep.) granted that a dedicated training sample is used.</text>
<text><location><page_2><loc_51><loc_23><loc_94><loc_34></location>We employ an ensemble of the same ML algorithm, notably convolutional neural networks (CNNs), known for their proficiency in learning intricate patterns, as outlined by (Lecun et al. 1998a). Specifically designed for image analysis, CNNs excel at identifying and extracting relevant predictive features directly from images, thereby reducing computational overhead compared to fully connected architectures. Harnessing this more extensive pool of information, these models surpass alternative models based on condensed feature-based input sets.</text>
<text><location><page_2><loc_51><loc_10><loc_94><loc_22></location>The paper is structured as follows: Sect. 2 introduces the AGN training sample down-selection. Sect. 3 focuses on the photometric data products available within LS10. Sect. 4 details the photometric data preprocessing, followed by Sect. 5, which outlines the model pipeline. Sect. 6 presents and quantifies the redshift results, while Sect. 7 evaluates the photo-z released for the XMM-SERVS (Chen et al. 2018; Ni et al. 2021) fields. Sect. 8 outlines current limitations and discusses how we can achieve further improvements. Sect. 9 explores implications for future surveys, concluding with a summary.</text>
<text><location><page_3><loc_6><loc_89><loc_49><loc_93></location>In this paper, unless stated di ff erently, we express magnitudes in the AB system and adopt a Λ CDM cosmology with H 0 = 69 . 8 km s -1 Mpc -1 , Ω m = 0 . 28 and Λ = 0 . 72.</text>
<section_header_level_1><location><page_3><loc_6><loc_85><loc_33><loc_86></location>2. AGN training sample selection</section_header_level_1>
<text><location><page_3><loc_6><loc_69><loc_49><loc_84></location>In X-ray surveys, the identification of AGN has two distinct advantages - i) the reduced impact of moderate obscuration and ii) the lack of host dilution. Due to the inherent brightness of accreting SMBHs compared to their host galaxies, this results in a significantly higher nuclei-to-host emission ratio compared to observations in some of the neighbouring wavelength windows, such as UV-optical-NIR (Padovani et al. 2017). This naturally leads to a larger diversity of AGN observed by an X-ray telescope. That being said, surveys in the more accessible optical and NIR regime can increase the likelihood of detecting higherz , and in the case of MIR more heavily obscured AGN, compared to the soft X-ray bands.</text>
<text><location><page_3><loc_6><loc_54><loc_49><loc_69></location>MLapproaches for photoz estimation in large surveys (e.g., Fotopoulou & Paltani 2018; Duncan 2022) typically classify objects into three broad categories: galaxies, quasars (QSOs), and stars, before computing photo-z. However, this classification is usually based on the optical properties and hence fails for obscured and / or lower-luminosity AGN. Since our goal is to improve on the quality of photo-z estimates for X-ray detected extragalactic sources, including type 2 AGN and low-redshift Seyfert 1 galaxies, generally, our training sample has to replicate this diversity. We achieve this by combining AGN selected across multiple wavelength bands.</text>
<text><location><page_3><loc_6><loc_39><loc_49><loc_54></location>As a starting point, we include the same X-ray samples used in Saxena et al. (2024), namely the latest version of the XMM catalog, 4XMM, which spans 19 years of observations made with XMM-Newton (Webb et al. 2020) and data from the eROSITA CalPV-phase Final Equatorial-Depth Survey (eFEDS; Brunner et al. 2022), as they provide a reasonably representative and complete set of diverse AGN spanning 5 dex in X-ray flux out to redshift z ≲ 4. However, with just these, some portions of AGNparameter space remain imbalanced, such as highly luminious and / or high-z AGN. Thus, we expand the dataset by adding bright, optical and MIR, selections. We describe each of these samples in more detail below.</text>
<text><location><page_3><loc_6><loc_27><loc_49><loc_38></location>This approach enhances the completeness of our training sample, which is essential to mitigate covariate shift, i.e., the shift in parameter space between the training and validation samples, so that the model generalizes e ff ectively to new data (Norris et al. 2019). Subsequently, a non-representative training sample may lead to systematically biased outcomes (Newman & Gruen 2022). Accordingly, algorithms will be strongly weighted towards the most densely populated regions of the training space (Duncan 2022).</text>
<section_header_level_1><location><page_3><loc_6><loc_24><loc_28><loc_25></location>2.1. Beyond eFEDS and 4XMM</section_header_level_1>
<text><location><page_3><loc_6><loc_10><loc_49><loc_22></location>We can enhance the redshift distribution within our sample, particularly towards high ( z ≥ 3) redshift, in this otherwise underrepresented parameter space due to observational selection e ff ects. We recognize the subsequent incorporation of unavoidable selection biases in each survey while restricting the inclusion of sources at low redshift to a minimum. While the balance between dataset quality and size is critical, deep learning algorithms that operate on pixel-level inputs tend to perform optimally only when training datasets contain ≥ 400 000 galaxy images (Schuldt et al. 2021; Dey et al. 2022a; Newman & Gruen</text>
<figure>
<location><page_3><loc_52><loc_40><loc_93><loc_93></location>
<caption>Fig. 1: Flowchart depicting the training sample down-selection pipeline. This includes the sample preprocessing (grey box) and the sample refinement, including redshift extension and duplicate removal below the dashed red line.</caption>
</figure>
<text><location><page_3><loc_51><loc_18><loc_94><loc_30></location>2022). Since our method would benefit from a larger sample (see Table 1), we chose not to apply stringent quality criteria by only considering high-quality data, significantly reducing the number of sources available training. Such a reduction would also prevent the model from learning to handle lower-quality data, limiting its application to only high-quality validation data. By not making an initial down-selection, we retain the flexibility to apply quality cuts to future blind samples by using LS10 flags later.</text>
<section_header_level_1><location><page_3><loc_51><loc_15><loc_77><loc_16></location>2.1.1. Samples from optical selection</section_header_level_1>
<text><location><page_3><loc_51><loc_10><loc_94><loc_13></location>We include the 2dF QSO Redshift Survey (2QZ, Croom et al. 2004) with ∼ 23k color selected QSOs in the magnitude range 18.25 ≤ bJ ≤ 20.85 at redshifts lower than z ∼ 3 and the QUasars</text>
<table>
<location><page_4><loc_7><loc_70><loc_48><loc_90></location>
<caption>Table 1: Overview of the relative fractions of source catalogs used in compiling our AGN training sample.</caption>
</table>
<text><location><page_4><loc_6><loc_63><loc_49><loc_67></location>as BRIght beacons for Cosmology in the Southern hemisphere survey (QUBRICS, Boutsia et al. 2020) with 224 bright (i ≤ 18) QSOs at redshifts of z ≥ 2 . 5.</text>
<section_header_level_1><location><page_4><loc_6><loc_59><loc_44><loc_60></location>2.1.2. Samples from optical follow-up of X-ray sources</section_header_level_1>
<text><location><page_4><loc_6><loc_41><loc_49><loc_58></location>Additionally, we incorporate the SDSS-IV (Blanton et al. 2017) quasar catalog from Data Release 16 (DR16Q, Lyke et al. 2020) with ∼ 150k quasars collected from various subprogrammes including optical / IR selection down to g ≤ 22 in the LS10 footprint after subselecting high quality specz , as well as follow-up of Xray sources from ROSAT (Voges et al. 1999; Boller et al. 2016; Salvato et al. 2018) and XMM (e.g. LaMassa et al. 2019). As successor science programme, we also consider the Black Hole Mapper (BHM) SPectroscopic IDentfication of ERosita Sources (SPIDERS, Anderson et al., in prep, Aydar et al., in prep) from SDSS-V (Kollmeier et al., in prep) Data Release 18 (Dwelly et al. 2017; Co ff ey et al. 2019; Comparat et al. 2020; Almeida et al. 2023).</text>
<section_header_level_1><location><page_4><loc_6><loc_38><loc_29><loc_39></location>2.1.3. Samples of high-z sources</section_header_level_1>
<text><location><page_4><loc_6><loc_30><loc_49><loc_36></location>Given the strong imbalance above z ∼ 3 . 5, we also include 400 optically / IR selected quasars at redshifts 4.8 ≤ z ≤ 6 . 6 down to g ≤ 24 from the high-redshift quasar survey in the DESI Early Data Release (EDR, Yang et al. 2023) and a compilation of highz quasars at z ≥ 5 . 3 published in literature (Fan et al. 2022).</text>
<section_header_level_1><location><page_4><loc_6><loc_26><loc_32><loc_27></location>2.2. Spectroscopic cross-referencing</section_header_level_1>
<text><location><page_4><loc_6><loc_10><loc_49><loc_25></location>The parent sample of AGN is annotated with specz , where available. According to Figure 1, we also consider sources, including those from eFEDS and 4XMM, with spatial counterparts from a compilation of public redshifts (Kluge et al. 2024). The procedure by which we match optical counterparts in our combined sample to a compilation of quality criteria down-selected specz , is outlined in Sect. 3.1 of Saxena et al. (2024). Due to overlaps between surveys, we remove duplicates when combining samples. The final sample of sources with specz comprises 40 489 objects, with a breakdown in Table 1. Correspondingly, the (cumulative) histograms illustrating the n( z ) distributions that collectively constitute the PICZL sample are presented in Figure 2.</text>
<figure>
<location><page_4><loc_52><loc_67><loc_93><loc_93></location>
</figure>
<text><location><page_4><loc_55><loc_66><loc_56><loc_67></location>0</text>
<figure>
<location><page_4><loc_52><loc_52><loc_93><loc_65></location>
<caption>Fig. 2: Binned redshift histogram (top panel) and cumulative distribution (bottom panel) of the sources utilized in the PICZL AGN sample. Note that these samples are not necessarily rank ordered by importance but for improved readability.</caption>
</figure>
<section_header_level_1><location><page_4><loc_51><loc_42><loc_62><loc_43></location>3. The survey</section_header_level_1>
<text><location><page_4><loc_51><loc_33><loc_94><loc_41></location>To streamline and simplify our methodology, we have chosen to employ data from LS10 exclusively to mitigate potential complications arising from the heterogeneity of multiple datasets. Crucially, the survey area now extends over 20 000 deg 2 of optical griz and WISE W 1 -W 4 forced photometry, by incorporating the following datasets:</text>
<unordered_list>
<list_item><location><page_4><loc_52><loc_25><loc_94><loc_32></location>-DECam Legacy Survey observations (DECaLS, Flaugher et al. 2015; Dey et al. 2019), including data from the Dark Energy Survey (DES, Collaboration: et al. 2016), which covers a 5000 deg 2 contiguous area in the South Galactic Cap. In the DES area, the depth reached is higher than elsewhere in the footprint.</list_item>
<list_item><location><page_4><loc_52><loc_16><loc_94><loc_25></location>-DECam observations from a range of non-DECaLS surveys, including the full six years of the Dark Energy Survey, publicly released DECam imaging (NOIRLab Archive) from other projects, including the DECam Local Volume Exploration Survey (DELVE, Drlica-Wagner et al. 2021) and the DECam eROSITA survey (DeROSITAs, PI: A. Zenteno, Zenteno et al. in prep).</list_item>
</unordered_list>
<text><location><page_4><loc_51><loc_10><loc_94><loc_15></location>In the north ( δ > 32.375 deg), LS10 uses the Beijing-Arizona Sky Survey (BASS, Zou et al. 2017) for g - and r -band coverage, and the Mayall z -band Legacy Survey (MzLS, Silva et al. 2016) for z -band coverage (Kluge et al. 2024).</text>
<text><location><page_4><loc_51><loc_78><loc_53><loc_81></location>Count</text>
<figure>
<location><page_5><loc_7><loc_63><loc_93><loc_93></location>
<caption>Fig. 3: Grid of subplots showing various model inputs for PICZL. In the upper row: the LS10 g-band image (a), along with its 2-D model flux (b), residual (c), and aperture flux map (d). In the bottom row: the original g-r color is shown in (a). The presence of a saturated pixel in the top right corner is visible, indicating the need for pre-processing. The result of the preprocessing is shown in b). The bottom panels c) and d) show the g / r-band and w1 / w2-band aperture flux maps, respectively.</caption>
</figure>
<section_header_level_1><location><page_5><loc_6><loc_52><loc_21><loc_54></location>3.1. Photometric data</section_header_level_1>
<text><location><page_5><loc_6><loc_22><loc_49><loc_51></location>LS10 o ff ers registered, background-subtracted, and photometrically calibrated point spread function (PSF)-forced photometry, including corresponding errors. To extend their wavelength coverage, DR10 catalogs incorporate mid-infrared (mid-IR) forced photometry at wavelengths of 3.4, 4.6, 12, and 22 µ m(referred to as W1, W2, W3, and W4, respectively) for all optically detected sources in the LS10 via the Near-Earth Object Wide-field Infrared Survey Explorer (NEOWISE) (Mainzer et al. 2011; Lang 2014; Meisner et al. 2017). Sources are modeled simultaneously across all optical bands, ensuring consistency in shape and size measurements by fitting a set of light profiles, even for spatially extended sources. Consequently, alongside reliable total and multi-aperture (8 annuli ≤ 7 arcseconds, five annuli ≤ 11 arcseconds for the optical and mid-infrared bands, respectively) flux measurements, the LS10 catalog o ff ers seeing-convolved PSF, de Vaucouleurs, exponential disk, or composite de Vaucouleurs + exponential disk models obtained with the Tractor algorithm (Lang et al. 2016). Additionally, providing fluxes rather than magnitudes, enables considering sources with very low signalto-noise ratios without introducing biases at faint levels. This characteristic also facilitates flux stacking at the catalog level, enhancing the overall versatility and utility of the classification and fitting process within LS10 1 .</text>
<section_header_level_1><location><page_5><loc_6><loc_18><loc_19><loc_19></location>3.2. Imaging data</section_header_level_1>
<text><location><page_5><loc_6><loc_13><loc_49><loc_17></location>In addition to catalog data, LS10 provides a rich set of imaging products. These include observations, flux model images, and residual maps for all available bands. For instance, the top pan-</text>
<text><location><page_5><loc_51><loc_51><loc_94><loc_54></location>els (a, b, c) of Figure 3 display all image products for a g -band observation, respectively.</text>
<section_header_level_1><location><page_5><loc_51><loc_47><loc_65><loc_48></location>4. Preprocessing</section_header_level_1>
<text><location><page_5><loc_51><loc_42><loc_94><loc_45></location>Here we detail the preprocessing steps taken to prepare our dataset, ensuring that it is clean, normalized, and structured appropriately.</text>
<section_header_level_1><location><page_5><loc_51><loc_38><loc_69><loc_39></location>4.1. Image preprocessing</section_header_level_1>
<text><location><page_5><loc_51><loc_23><loc_94><loc_37></location>Building on the approach of Saxena et al. (2024), which demonstrated significant improvements by shifting from total to aperture flux utilizing information on the 2D light distribution, we aim to further refine the spatial characterization of sources. This is achieved by incorporating pixel-level flux resolution through imaging as base input. Images in individual bands or in combination (i.e., colors) reflect the surface brightness, angular size, and sub-component structures of the sources, indirectly providing redshift information (Stabenau et al. 2008). To obtain reliable photo-z directly from images, we utilize flux-calibrated optical cutouts across as many filters as available.</text>
<text><location><page_5><loc_51><loc_10><loc_94><loc_22></location>With an average seeing FWHM of 1.3 arcseconds under nominal conditions, LS10 provides a pixel resolution of 0.262 arcseconds per pixel for the optical bands, reaching depths between 23 and 24.7 AB, depending on the specific band and region of the sky (see Dey et al. (2019) and Figure 1 in Saxena et al. (2024)). To enhance computational e ffi ciency and mitigate contamination from nearby sources, we restrict our cutout dimensions to 23 × 23 pixel, centered on the AGN coordinates in the four griz LS10 bands. Our cutouts correspond to a field of view (FOV) of approximately 6 arcseconds × 6 arcseconds. We</text>
<figure>
<location><page_6><loc_7><loc_70><loc_48><loc_93></location>
<caption>Fig. 4: Angular size of sources with a fixed physical size of 2040 kpc, as a function of redshift. The orange dashed line depicts a fixed image dimension of 23 pixels, assuming the LS10 spatial resolution of 0.262 arc seconds per pixel, which su ffi ces to cover objects of 30 kpc diameters in size out to redshifts of ∼ 0 . 3 ≤ z ≤ 7 . 8.</caption>
</figure>
<text><location><page_6><loc_6><loc_56><loc_49><loc_58></location>base our choice of FOV on the angular size-redshift relation by computing the angular diameter distance d Λ via:</text>
<formula><location><page_6><loc_6><loc_51><loc_49><loc_54></location>c H 0 1 (1 + z ) Z z 0 d z ' p Ω M(1 + z ' ) 3 + ΩΛ . (1)</formula>
<text><location><page_6><loc_6><loc_42><loc_49><loc_50></location>Equation 1 and Figure 4 elucidate the connection between an object's physical size, its angular size, and redshift. Notably, we can e ff ectively map galaxies with a diameter of 30 kpc - representative of main sequence galaxies (Wuyts et al. 2011) within the confines of a 23 × 23 pixel cutout, covering the range of 0 . 5 ≤ z ≤ 7 . 7.</text>
<text><location><page_6><loc_6><loc_33><loc_49><loc_42></location>Given that the FWHM of the W1, W2, and W3 images is 6 arcseconds, and of 12 arcseconds for W4, WISE band cutouts do not provide meaningful spatial information at this scale (see Figure 5). Therefore, we have opted to use images solely from the optical bands. Problematic sources, exhibiting signs of defects in various ways are flagged in the Legacy Survey by specific bitmasks (Dey et al. 2019).</text>
<text><location><page_6><loc_6><loc_10><loc_49><loc_33></location>We acknowledge that LS10 images exhibit varying quality due to di ff erences in seeing conditions during observations taken over many years, which impact in particular the measurements of color within source apertures. Despite this, we rely on the model's ability to adapt to these intrinsic variations given the size and diversity of our training sample, as the sources withheld from training represent a shu ffl ed subsample of the main dataset, ensuring robust evaluation. We have verified that the distribution in seeing quality (expressed as the weighted average PSF FWHMofthe images) for the training and validation are comparable (refer to Figure B.1). However, preliminary tests indicate that adding PSF size and PSF depth of the observations-both available in LS10-as additional features enhances model performance (Götzenberger et al., in prep.). While it is not unreasonable to expect the model to implicitly infer the PSF or a related abstract representation thereof from the images themselves, these features will be included by default in future PICZL versions. With ongoing developments, PSF cutouts are expected to</text>
<figure>
<location><page_6><loc_52><loc_79><loc_72><loc_93></location>
</figure>
<figure>
<location><page_6><loc_73><loc_79><loc_93><loc_93></location>
<caption>Fig. 5: Example AGN in our sample, as seen by LS10, in the optical RGB image (left) and in the W1 image from NEOWISE7 (right). The spatial resolution is 0.262 arcseconds and 2.75 arcseconds per pixel, respectively. The size of the cutouts is 27.5 arc seconds × 27.5 arc seconds.</caption>
</figure>
<text><location><page_6><loc_51><loc_62><loc_94><loc_69></location>become more accessible for integration into the image stack (see Table 2). In the longer term, we anticipate that upcoming surveys like LSST, with their improved consistency in image quality, will further reduce these limitations and boost the precision of pixelbased analyses such as ours.</text>
<section_header_level_1><location><page_6><loc_51><loc_59><loc_64><loc_60></location>4.2. Color images</section_header_level_1>
<text><location><page_6><loc_51><loc_27><loc_94><loc_58></location>It proves advantageous to provide the network with color images (ratio between images from di ff erent bands) as an additional input. This approach avoids the necessity for the model to learn the significance of colors solely from the flux images, which is inherently a more di ffi cult task. Likewise, rather than processing numerical features separately and merging them with the information extracted from the image cube at a later stage, we find it beneficial to integrate them directly at the pixel level. As a result, whenever possible, we transform catalog features into 2D arrays to align them with the original images in the same thread, enabling smoother integration and more coherent analysis (see e.g. Hayat et al. 2021). We improve the depth of our data cube by converting catalog-based quantities, e.g., flux measurements from apertures across di ff erent bands, into synthetic images, with a respective image size depending on the cutout size (see Figure 3). We expand this approach by generating images for all viable color combinations of aperture fluxes, constrained to those with matching aperture sizes. Additionally, we produce color images for flux cutouts where pixel resolutions are consistent (see lower panels b) and c) of Figure 3). Since the WISEand optical bands di ff er in both aperture size and pixel resolution, cross-wavelength color images are not feasible; instead, color combinations are restricted to within the optical or within the WISE bands (refer to Table 2).</text>
<text><location><page_6><loc_51><loc_18><loc_94><loc_26></location>To maintain FOV consistency, we integrate WISE data for only the two innermost apertures (see Figure 5), preserving the 23x23 pixel data cube format. Although no additional spatial details are expected at this scale (see Sect. 4.1), the WISE data still captures aperture flux in a format that enables direct crosschannel connections between optical and mid-infrared data at the image level.</text>
<text><location><page_6><loc_51><loc_10><loc_94><loc_17></location>Nevertheless, defected images can introduce challenges when generating color images (see bottom panel a) in Figure 3). Given that colors are derived from the ratio of images in di ff erent bands, the occurrence of unphysical negative or abnormally high / low values poses a significant concern. To address this issue, we examine whether the median value of neighbouring pix-</text>
<table>
<location><page_7><loc_11><loc_75><loc_89><loc_91></location>
<caption>Table 2: Overview of the various feature formats and dimensions utilized in PICZL's multi-channel approach.</caption>
</table>
<text><location><page_7><loc_6><loc_72><loc_94><loc_74></location>Notes. Each column represents a distinct processing channel, detailing its input data type and the respective dimensions. All flux-related inputs have been corrected for reddening e ff ects.</text>
<text><location><page_7><loc_6><loc_53><loc_49><loc_69></location>els is more than three times lower than the value of noisy pixels per band. If so, the central pixel's value is substituted with the median value of the surrounding pixels to smooth out fluctuations. In cases of non-detections or severely corrupted images where the largest value among all pixels in an image is < 0.001, the image is treated as a non-detection and all pixels are set to zero as default. After undergoing preprocessing, the images are utilized to create color images by exploring the six possible color combinations: gr , gi , gz , ri , rz and iz . If either of the two flux images involved in creating a color image is identified as a nondetection, the resulting color image is set to a default value of -99.</text>
<text><location><page_7><loc_6><loc_43><loc_49><loc_53></location>At the catalog level, spatially invariant features, such as bestfit model classifications and signal-to-noise ratios (S / N), are processed separately in a dedicated channel, as they cannot be converted into image data. Notably, regarding normalization, we adopt a uniform min-max scaling approach to handle columns with cross-dependencies, such as flux bands. This strategy aims to preserve crucial information, such as the original shape of the SED.</text>
<text><location><page_7><loc_6><loc_21><loc_49><loc_43></location>Contrary to the conventional approach of stacking all available images into a single input (Hoyle 2016; D'Isanto & Polsterer 2018; Pasquet et al. 2018; Dey et al. 2022a; Treyer et al. 2023), we find it advantageous to separate the color image data cube (23,23,24) from the flux band cutouts (23,23,32). This distinction is necessary because the color images often have di ff ering pixel value scales, including negative values, which require specialized processing in our machine learning application, such as tailored loss functions in the CNN. Non-spatial attributes are then combined with image-based data at a subsequent stage of the model, allowing the model to capture both spatial and nonspatial aspects. By processing data types in parallel channels, we leverage their complementary information by merging the data at a later stage, enhancing the extraction of inter-band correlations and ultimately improving redshift precision (Ma et al. 2015; Ait-Ouahmed et al. 2023). A detailed breakdown of the features integrated into each channel is provided in Table 2.</text>
<section_header_level_1><location><page_7><loc_6><loc_17><loc_21><loc_18></location>5. Neural network</section_header_level_1>
<text><location><page_7><loc_6><loc_10><loc_49><loc_16></location>ML embodies an artificial intelligence paradigm where computers learn patterns and relationships from data, enabling them to make predictions or perform tasks without explicit programming. Multilayer perceptrons (MLPs), a feature-based feedforward neural network, draw inspiration from their biological</text>
<text><location><page_7><loc_51><loc_52><loc_94><loc_69></location>counterparts, namely excitable cells responsible for processing and transmitting information (Rosenblatt 1958; I. Goodfellow 2016; M. Deru 2019). Likewise, each assigned a distinct weight, computational input vectors can be organized into layers, relayed to one or more hidden layers, to compute a scalar output value (Géron 2019). During training, these models learn data mappings by adjusting the weights and biases associated with their connections. The margin of change to the model after every training epoch is dictated by the choice of optimizer and loss function, which e ff ectively calculates the Euclidean distance between the prediction and so-called ground truth in multi-dimensional feature space, thereby significantly impacting model convergence and performance.</text>
<text><location><page_7><loc_51><loc_48><loc_94><loc_52></location>The current state-of-the-art deep learning (DL) networks, characterized by their many hidden layers, have shown exceptional capabilities in handling complex non-linear tasks.</text>
<section_header_level_1><location><page_7><loc_51><loc_45><loc_73><loc_46></location>5.1. Convolutional model layers</section_header_level_1>
<text><location><page_7><loc_51><loc_25><loc_94><loc_43></location>Among such architectures, Convolutional Neural Networks (CNNs; Fukushima 1980; LeCun et al. 1989; Lecun et al. 1998b) distinguish themselves by their remarkable e ff ectiveness in handling grid-like data, a prevalent form of which is represented by images. CNNs leverage a model architecture that is particularly e ff ective in tasks like image recognition, object detection, and image segmentation (O'Shea & Nash 2015; Liu et al. 2022). In convolutional layers, neurons establish connections exclusively with pixels within their receptive field, ensuring successive layers are linked only to specific regions of the previous layer. Subsequently, the model extracts low, image-level features in early layers and progressively complex, higher-level features in later layers. This allows the CNN to learn representations of the input data at multiple levels of abstraction.</text>
<text><location><page_7><loc_51><loc_10><loc_94><loc_25></location>The convolutional operation involves sliding several kernels K , here of sizes 3 × 3 or 5 × 5 pixel, across the images with a fixed stride of size s = 1, compressing each mosaic element into a single scalar. This process entails element-wise multiplication, generating a set of K feature maps. Filters, the learnable parameters of convolutional layers, enable the network to detect and highlight di ff erent aspects of the input data, such as edges or corners. Each convolution is followed by a pooling layer that reduces spatial dimensions and computational load. Pooling involves sliding a kernel, in our case of size 2 × 2 and s = 1, across the feature maps, selecting the maximum value (max pooling) within each kernel window. After flattening the data cube into a single array,</text>
<text><location><page_8><loc_6><loc_87><loc_49><loc_93></location>it is passed through (several) fully connected (FC) layers. Each FC layer contains a layer-specific number of neurons n , tailored to the model's specific needs. The final FC layer's neuron count varies based on the task, with n = 1 referring to regression tasks and n ≥ 2 to other endeavors such as multi-label classification.</text>
<section_header_level_1><location><page_8><loc_6><loc_83><loc_27><loc_84></location>5.2. Gaussian mixture models</section_header_level_1>
<text><location><page_8><loc_6><loc_51><loc_49><loc_82></location>By default, single output regression models (Schuldt et al. 2021) provide point estimates without quantifying uncertainty. This severely limits their area of application, particularly in scenarios where quantifying the uncertainty is crucial, for example, when performing precision cosmology with Euclid (Bordoloi et al. 2010; Scaramella et al. 2022; Newman & Gruen 2022). By instead employing an architecture that provides PDFs, we can not only retrieve point estimates but also encapsulate the uncertainty associated with the predictions in a concise format (D'Isanto & Polsterer 2018). Given the inherent complexity in determining redshifts from only a few broadband cutouts, our results are expected to often exhibit degeneracy with multi-modal posteriors, making a single Gaussian insu ffi cient for representing the photoz PDFs. Therefore, estimates are computed using Bayesian Gaussian Mixture Models (GMMs, Duda et al. 1973; Viroli & McLachlan 2017; Hatfield et al. 2020). These networks provide a unique probabilistic modeling approach that di ff erentiates them from traditional MLPs. One of the key distinctions lies in the output nature, where GMMs provide a set of variables to compute weighted multi-Gaussian distributions as opposed to single-point estimates. Subsequently, each component is characterized by its mean µ , standard deviation σ , and weight w , allowing GMMs to produce a full PDF for a given set of inputs x . The PDF is expressed as</text>
<formula><location><page_8><loc_6><loc_47><loc_49><loc_50></location>P ( x ) = K X k = 1 wk · N ( x | µ k , σ 2 k ) , (2)</formula>
<text><location><page_8><loc_6><loc_26><loc_49><loc_46></location>with, wk representing the weight and N ( x | µ k , σ 2 k ) denoting the Gaussian distribution of the k -th component. As such, we extend our CNN approach by a GMM backend to output PDFs based on the information-rich feature maps produced during the front-end phase of the network. However, we encounter a limiting challenge with inputs of such small dimensions, i.e. 23 × 23, as they are not well-suited for established image-based Deep Learning architectures, such as "ResNet" (He et al. 2015), which typically require larger dimension scales, typically exceeding 200 × 200 pixels, to accommodate the large number of pooling layers they employ. Therefore, we developed a custom architecture to fit our data dimensionality. The resulting model architecture features roughly 490,000 trainable parameters, far fewer than found in comparable studies (see Pasquet et al. 2018; Treyer et al. 2023), and is displayed in Figure A.1 of Appendix A.</text>
<section_header_level_1><location><page_8><loc_6><loc_23><loc_22><loc_24></location>5.3. Model refinement</section_header_level_1>
<text><location><page_8><loc_6><loc_13><loc_49><loc_22></location>Numerous hyper-parameters are crucial in shaping the network architecture while influencing training and convergence. Extensive optimization has been conducted across various parameters utilizing the Optuna framework (Akiba et al. 2019). The current configuration accounts for the vast array of potential combinations. The key parameters with the most significant impact are outlined below:</text>
<unordered_list>
<list_item><location><page_8><loc_7><loc_10><loc_49><loc_12></location>-Batch size [8-2048]: The number of objects utilized in a single training iteration per epoch. Larger batch sizes o ff er com-</list_item>
</unordered_list>
<text><location><page_8><loc_53><loc_91><loc_94><loc_93></location>putational e ffi ciency, while smaller batches may help generalize better.</text>
<unordered_list>
<list_item><location><page_8><loc_52><loc_87><loc_94><loc_90></location>-Learning rate [0.1-0.00001]: Controls the size of the model adjustment step during optimization. It influences the convergence speed and the risk of overshooting optimal settings.</list_item>
<list_item><location><page_8><loc_52><loc_83><loc_94><loc_87></location>-Number of Gaussians [1-50]: Determines the complexity and flexibility of the GMM, influencing how well the model captures the underlying data distribution.</list_item>
<list_item><location><page_8><loc_52><loc_79><loc_94><loc_83></location>-Convolutional & pooling layers [2-10]: the dimension of the kernel influences the size of the receptive field and the learnable features.</list_item>
<list_item><location><page_8><loc_52><loc_74><loc_94><loc_79></location>-Number of neurons and layers [10-300]: Determines the depth and complexity of the neural network, impacting its capacity to capture hierarchical features and relationships in the input data.</list_item>
<list_item><location><page_8><loc_52><loc_69><loc_94><loc_74></location>-Activation function: Introduces non-linearity to the model, enabling it to learn complex mappings between inputs and outputs, commonly a sigmoid, hyperbolic tangent (tanh), rectified linear unit (ReLu) or versions thereof.</list_item>
<list_item><location><page_8><loc_52><loc_64><loc_94><loc_69></location>-Dropout layer [0-0.9]: Refers to the fraction of randomly selected neurons that are temporarily dropped or ignored during training, helping to prevent overfitting by enhancing network robustness and generalization.</list_item>
<list_item><location><page_8><loc_52><loc_58><loc_94><loc_63></location>-Max and average pooling: Pooling layers, such as Max Pooling and Average Pooling, are used to downsample the spatial dimensions of the input feature maps, reducing the computational load.</list_item>
<list_item><location><page_8><loc_52><loc_53><loc_94><loc_58></location>-Batch normalization: Batch Normalization can improve the training stability and speed of convergence in neural networks by subtracting the batch mean and dividing by the batch standard deviation.</list_item>
</unordered_list>
<section_header_level_1><location><page_8><loc_51><loc_50><loc_64><loc_51></location>5.4. Loss functions</section_header_level_1>
<text><location><page_8><loc_51><loc_37><loc_94><loc_49></location>When utilizing PDFs instead of point estimates, it is crucial to quantify whether a predicted PDF e ff ectively reflects the ground truth, in our supervised ML case, specz (Sánchez et al. 2014; Malz et al. 2018; Schmidt et al. 2020; Treyer et al. 2023). The PDF predicted by a model should be concentrated near the true value. A straight-forward and meaningful proper scoring rule (i.e. one which is lowest when the prediction is at the truth), is the product of probabilities at the true redshifts. The negative log-likelihood (NLL) 2 of which passed to a minimizer</text>
<formula><location><page_8><loc_51><loc_32><loc_94><loc_35></location>E ( w ) = -N X n = 1 ln GLYPH<18> k X k = 1 wk ( xn ) N ( y | µ k ( xn ) , σ k ( xn ) 2 ) GLYPH<19> , (3)</formula>
<text><location><page_8><loc_51><loc_26><loc_94><loc_31></location>yields the likelihood of observing a data distribution given a specific set of model parameters. This expression coincides with the average Kullback-Leibler divergence (KL, Kullback & Leibler 1951) when going from a delta PDF at specz to the photo-z PDF.</text>
<text><location><page_8><loc_51><loc_15><loc_94><loc_26></location>An alternative growing in popularity is given by the Continuous Ranked Probability Score (CRPS), initially applied in weather forecasting (Grimit et al. 2006), which serves as a valuable metric for photo-z estimation via PDF quantification (D'Isanto & Polsterer 2018). Computationally, CRPS calculates the integral of the squared di ff erence between the predicted cumulative probability distribution function (CDF) and a Heaviside step-function H ( x ) at the value of the specz ( xz ) as</text>
<formula><location><page_9><loc_6><loc_88><loc_49><loc_91></location>CRPS( F , xz ) = Z ∞ -∞ GLYPH<20> Z x -∞ f ( t ) dt -H ( x -xz ) GLYPH<21> 2 d x . (4)</formula>
<text><location><page_9><loc_6><loc_85><loc_49><loc_88></location>For a finite mixture of normal distributions M , the CRPS can be expressed in closed form:</text>
<formula><location><page_9><loc_6><loc_80><loc_49><loc_83></location>CRPS = M X i = 1 wiA ( δ i , σ 2 i ) -1 2 M X i = 1 M X j = 1 wiwjA ( µ i -µ j , σ 2 i + σ 2 j ) , (5)</formula>
<text><location><page_9><loc_6><loc_76><loc_49><loc_79></location>with δ i = y -µ i , the uncertainty σ i and weight wi of the i -th component respectively, as well as</text>
<formula><location><page_9><loc_6><loc_65><loc_49><loc_74></location>A ( µ, σ 2 ) = µ GLYPH<20> 2 Φ GLYPH<18> µ σ GLYPH<19> -1 GLYPH<21> + 2 σϕ GLYPH<18> µ σ GLYPH<19> , ϕ ( x ) = 1 √ 2 π exp GLYPH<18> -x 2 2 GLYPH<19> , Φ ( x ) = Z x -∞ ϕ ( t )d t . (6)</formula>
<text><location><page_9><loc_6><loc_53><loc_49><loc_64></location>By extending the CRPS loss via normalizing it by (1 + z), we adjust the penalty for prediction errors based on redshift. In doing so, we prioritize accuracy for low and intermediate redshift sources by imposing higher constraints while allowing for more leniency in less critical high-redshift areas. Integrating over the entire range of possible redshift values, the CRPS takes into account both location and spread of the predicted PDF. It, therefore, provides a comprehensive evaluation metric, generating globally well-calibrated PDFs (Dey et al. 2022b).</text>
<section_header_level_1><location><page_9><loc_6><loc_50><loc_30><loc_51></location>5.5. Training & data augmentation</section_header_level_1>
<text><location><page_9><loc_6><loc_27><loc_49><loc_48></location>Our dataset is divided into training and validation sets in an 80:20 ratio. The model is trained over 1000 training epochs utilizing the Adam algorithm (Kingma & Ba 2017) along with a learning rate scheduler to adjust the learning rates during training dynamically. The inclusion of both is a consequence of the Optuna hyperparameter optimization. Typically, the model achieves its lowest validation loss around the 600th epoch when considering both loss functions (see Figure 6). After this point, although the training loss continues to decrease, the validation loss begins to increase, indicating overfitting. This overfitting likely arises due to the limited size of our training sample, allowing the model to memorize specific patterns rather than learning generalizable features. We implement a checkpoint system to mitigate this, saving the model's architecture and parameters when it reaches its lowest validation loss. This approach ensures that we capture the best-performing model configuration while preventing overfitting.</text>
<text><location><page_9><loc_6><loc_10><loc_49><loc_26></location>To further exploit the advantages of deep learning, particularly its capacity to perform well with extensive datasets, we have tested the incorporation of data augmentation techniques into our methodology. Recognizing the substantial impact of training data amount on model performance, we employ common augmentation data products, such as rotated and mirrored images. Although these transformations significantly increase the apparent size of the dataset, we do not observe an increase in performance when including augmentation techniques (Dey et al. 2022a; Treyer et al. 2023). This is potentially not surprising as the influence of augmentation techniques depends on several factors. While CNNs are inherently equivariant to translations within images, enabling them to recognize objects even if</text>
<text><location><page_9><loc_51><loc_76><loc_94><loc_93></location>they are shifted or o ff -centered, they and most other machinelearning approaches are not naturally equivariant to rotations and reflections. This lack of rotational equivariance means that the model output e ff ectively depends on the rotation of the input images. However, as most of our inputs are approximately point sources with radial symmetry, we do not explore this aspect further in this study. More recent publications have focused on developing rotationally equivariant CNN architectures, which have shown improved performance despite increased computational costs (Cohen & Welling 2016; Weiler & Cesa 2021; Lines et al. 2024). Incorporating such architectures could be particularly beneficial for future works dealing with large amounts of imaging data.</text>
<section_header_level_1><location><page_9><loc_51><loc_73><loc_66><loc_74></location>5.6. Model ensemble</section_header_level_1>
<text><location><page_9><loc_51><loc_50><loc_94><loc_72></location>Ensemble models capitalize on the diversity of multiple models to improve prediction accuracy. Our approach involves diversifying models by adjusting parameters such as the number of Gaussians per GMM, learning rates, batch sizes, and the choice of loss function. We employ both NLL and CRPS as complementary loss functions to achieve this. While CRPS excels in achieving lower outlier fractions, NLL enhances overall variance. Each loss function trains a set of 144 models, resulting in a diverse pool of 288. In line with Treyer et al. (2023), we achieve superior performance by randomly combining equally weighted CRPS and NLL models from the pool, compared to the best-performing individual model (Dahlen et al. 2013; Newman & Gruen 2022). This finding is also aided by the result that roughly 30% of outliers are model-specific as opposed to 20% of outliers appearing in all models, therefore considered to be genuine. Put di ff erently, 80% of outliers were found to not be an outlier in at least one other model.</text>
<text><location><page_9><loc_51><loc_41><loc_94><loc_50></location>We additionally recognize that the initial configuration of each model has a minor influence on the performance. Although training additional models could further enhance our results, the computational resources required to generate three to five sets of models would make this approach impractical relative to the potential performance gains. To create an e ff ective ensemble, we evaluate the outlier fraction of all individual models on an un-</text>
<figure>
<location><page_9><loc_51><loc_19><loc_93><loc_38></location>
<caption>Fig. 6: Training (dashed) and Validation (solid) loss curves for a single PICZL model. The solid or dashed lines represent the mean loss values computed over a window size of 20, while the shaded areas denote the 1 σ error range around the mean. The NLL loss is depicted in blue, while the normalized CRPS loss is shown in red.</caption>
</figure>
<figure>
<location><page_10><loc_7><loc_54><loc_96><loc_93></location>
<caption>Fig. 7: Binned Scatter plots of photo-z obtained from our model ZPICZL versus spectroscopic redshift Zspec, for sources classified as type point-like (PSF, left) and extended (EXT, right). Each plot includes identity and two lines denoting the outlier boundary of | z phot -z spec | (1 + z spec) > 0 . 15. Normalized residuals and trend lines and errors are shown in the bottom panels, aiding in the visualization of systematic deviations across the redshift range.</caption>
</figure>
<text><location><page_10><loc_6><loc_33><loc_49><loc_45></location>seen test sample. From this evaluation, we select models based on their relative influence on the model performance. We then observe how the ensemble's performance evolves as more models are continuously incorporated. Our ensemble ultimately comprises 10 models or 84 Gaussians, evenly incorporating models optimized using both NLL and CRPS approaches. To refine the model weights within the ensemble, we employ Optuna for finetuning. Subsequently, the ensemble posterior likelihood distribution P ensemble(x) is given by:</text>
<formula><location><page_10><loc_6><loc_28><loc_49><loc_31></location>P ensemble( x ) = N X i = 1 wiP ( i )( x ) , (7)</formula>
<text><location><page_10><loc_6><loc_26><loc_38><loc_27></location>where the weights wi are normalized such that:</text>
<formula><location><page_10><loc_6><loc_21><loc_49><loc_24></location>N X i = 1 wi = 1 , (8)</formula>
<text><location><page_10><loc_6><loc_10><loc_49><loc_20></location>to assure that the integral of the ensemble PDF is equal to 1. We obtain a point estimate for our photoz prediction by identifying the dominant mode of the resulting PDF, recognizing that the mean or median could be misleading for highly non-Gaussian and bi- or multi-modal distributions. Additionally, we provide asymmetric 1 and 3 sigma upper and lower errors by evaluating the PDF within the 16th / 84th and 0.15 / 99.85th percentiles, respectively, along with the entire PDF.</text>
<section_header_level_1><location><page_10><loc_51><loc_44><loc_66><loc_45></location>6. Photo-z results</section_header_level_1>
<text><location><page_10><loc_51><loc_34><loc_94><loc_43></location>We evaluate the performance of PICLZ on both point estimates and PDFs. This comprehensive assessment includes testing the statistical reliability and accuracy of our predictions. We adopt several commonly employed statistical metrics, providing insights into the accuracy, precision, and reliability of the photo-z estimates. In line with the definitions from the literature, we use the following metrics:</text>
<unordered_list>
<list_item><location><page_10><loc_52><loc_29><loc_94><loc_32></location>-Prediction bias: the mean of the normalised residuals, ⟨ ∆ z ⟩ = ( z spec -z phot) (1 + z spec) .</list_item>
<list_item><location><page_10><loc_52><loc_25><loc_94><loc_29></location>-Variance: following Ilbert et al. (2006), we adopt the standard deviation from the normalised median absolute deviation σ NMAD = 1 . 4826 × Median | z spec -z phot | (1 + z spec) .</list_item>
<list_item><location><page_10><loc_52><loc_20><loc_94><loc_25></location>-Fraction of outliers η is determined by the proportion of photo-z estimates with absolute normalised residuals exceeding η = | z spec -z phot | (1 + z spec) > 0 . 15.</list_item>
<list_item><location><page_10><loc_52><loc_15><loc_94><loc_20></location>-PIT score: a diagnostic tool used to evaluate the calibration of probabilistic forecasts by evaluating the cumulative distribution function (CDF) given the PDF at the (true) specz z corr as CDF( z corr) = R z corr 0 PDF( z ) dz .</list_item>
</unordered_list>
<text><location><page_10><loc_51><loc_10><loc_94><loc_13></location>We quantify PICZL's performances on a validation sample of 8098 sources and find an outlier fraction η , of 5.6% with a variance σ NMAD, of 0.045. Since the validation sample is used</text>
<text><location><page_11><loc_6><loc_84><loc_49><loc_93></location>for feedback in training, it should be mentioned that the results mentioned above are likely overestimating the performance and therefore may not representative of what a future user could achieve with say, a test set previously unknown to the model, leave-one-out, or k-fold cross-validation. To this end, we have estimated our performance on independent, blind, X-rayselected samples (see Sect. 6.2 and 7).</text>
<text><location><page_11><loc_6><loc_66><loc_49><loc_84></location>Figure 7 compares photometric versus spectroscopic redshifts, split by point-like (type = PSF) and extended (type = EXT) morphology. With respect to comparable work (e.g., Figure 13 from Salvato et al. 2022), we observe enhanced performance with a much-reduced fraction of outliers, especially for PSF sources. These objects lack morphological information at redshifts z ≳ 1, hence we attribute this improvement to the model's ability to recognise how the radial extension of the azimuth profile of sources changes with redshift (refer to Figure 4). The scatter, for both PSF and EXT distributions, is tight and symmetrically distributed around the z PICZL = z spec identity line, with outliers appearing randomly scattered, suggesting stable performance across the redshift range and minimal systematic errors (see Figure 8 and Dey et al. (2019) for reference).</text>
<figure>
<location><page_11><loc_8><loc_41><loc_47><loc_64></location>
<caption>Fig. 8: Histogram distribution of the normalised residuals, overlaid with a Gaussian fit (orange). The parameters of the Gaussian distribution are determined by the prediction bias and σ NMAD values obtained from Table 5 for the validation sample.</caption>
</figure>
<text><location><page_11><loc_6><loc_15><loc_49><loc_33></location>Given that our point estimates are derived from PDFs, we must ensure their global calibration and accuracy. To achieve this, we employ the probability integral transform (PIT) statistic (Dawid 1984; Gneiting et al. 2005), a widely accepted method in the field for assessing the quality of redshift PDFs (Pasquet et al. 2018; Schuldt et al. 2021; Newman & Gruen 2022). An ideal scenario is represented by a uniformly distributed histogram of PIT values. Any deviation from uniformity can indicate issues in PDF calibration. Under-dispersion may suggest overly narrow PDFs, while over-dispersion often results from excessively wide PDFs (D'Isanto & Polsterer 2018). Peaks close to zero or one can be explained by catastrophic outliers, where the true redshift lies so far in the wing of the PDF that it essentially falls outside of it.</text>
<text><location><page_11><loc_6><loc_10><loc_49><loc_15></location>We employ the PIT histogram (top panel Figure 9) alongside quantile-quantile (QQ) plots (bottom panel Figure 9), to visually assess the statistical properties of our model predictions. For reference, these can be compared to Figure 2 of in Schmidt et al.</text>
<figure>
<location><page_11><loc_51><loc_52><loc_93><loc_93></location>
<caption>Fig. 9: The top panel displays the QQ plot, comparing identity (flat histogram) against the PIT values derived from our redshift PDFs of the validation sample. The bottom panel shows the differences between the QQ plot and identity, highlighting systematic biases or trends for the two morphological classes (PSF and EXT). For all panels, blue refers to results achieved with a single model (see Figure A.1) while orange reflects the results obtained when utilizing ensemble results (refer to Sect. 5.6)</caption>
</figure>
<text><location><page_11><loc_72><loc_52><loc_78><loc_53></location>PIT Score</text>
<text><location><page_11><loc_72><loc_39><loc_73><loc_40></location>.</text>
<text><location><page_11><loc_51><loc_30><loc_94><loc_36></location>(2020); however, those code performances are completely dominated by inactive galaxies, making a direct comparison impractical. The QQ plot compares the CDFs of observed PIT values and identity U (0 , 1), to visualize QQ di ff erences split by morphological type.</text>
<text><location><page_11><loc_51><loc_10><loc_94><loc_29></location>A well-calibrated model will exhibit a QQ plot closely following identity with a flat PIT histogram, indicating accurate and reliable redshift predictions. Our analysis shows that, while asymmetry in the PIT distribution could suggest systematic bias, the QQ plot reveals small residuals overall. Notably, for both the single- (refer to Sect. 5.2 and Figure A.1) and ensemble model (refer to Sect. 5.6), the curves do not deviate significantly from zero, lesser so for PSF-type than EXT-type objects, indicating well-calibrated models. A distinct observation is the shift towards central PIT values for the ensemble model. This shift is not unexpected, given that the ensemble approach integrates multiple redshift solutions, each potentially contributing di ff erent peaks to the resulting ensemble PDF. Consequently, the ensemble PDF exhibits a broader bulk of probability across redshift, with PIT scores accumulating more area under the curve</text>
<table>
<location><page_12><loc_14><loc_80><loc_86><loc_90></location>
<caption>Table 3: Overview of wall-clock time τ , ∆ z , σ NMAD and η for sources classified as being extended (EXT) or point-like (PSF) from a single model (as opposed to an ensemble utilized for the results depicted in Figure 7) trained with four input configurations.</caption>
</table>
<text><location><page_12><loc_6><loc_77><loc_94><loc_79></location>Notes. The last two setups are an improvement over the previous performances, with the last setup providing comparatively good results, despite ignoring the features provided by catalogs, particularly relevant for potential implementation in e.g., LSST.</text>
<unordered_list>
<list_item><location><page_12><loc_6><loc_75><loc_94><loc_77></location>( a ) Catalog-based model fluxes only, as traditionally done; ( b ) Catalog-based aperture fluxes and respective colors as done in Saxena et al. (2024);</list_item>
</unordered_list>
<text><location><page_12><loc_6><loc_63><loc_49><loc_72></location>when integrated to the main mode. As a result, rather than yielding a flat PIT distribution, the histogram shifts towards having a concentration of values around 0.5. While this phenomenon aligns with our expectations, further enhancements may be realized by incorporating metrics tailored instead to local calibration accounting for population-specific subgroups (see e.g. Zhao et al. 2021; Dey et al. 2022b).</text>
<section_header_level_1><location><page_12><loc_6><loc_59><loc_36><loc_60></location>6.1. Model performance for different inputs</section_header_level_1>
<text><location><page_12><loc_6><loc_36><loc_49><loc_58></location>Images provide a more comprehensive view of astronomical sources by capturing their full spatial structure and light distribution, o ff ering finer details on morphology, apparent size and extended features that are often lost in the averaging process of aperture photometry. This is pivotal, as SED features crucial for determining redshift solutions and resolving degeneracies mostly reside within the host galaxy (Soo et al. 2017; Wilson et al. 2020). In particular for AGN, images help to spatially separate the pixels corresponding to AGN-dominated emission and those corresponding mostly to the host galaxies. This approach enhances our ability to isolate and analyze individual pixel strings across multiple filters, to e ff ectively construct SEDs for the host galaxies independently. We thereby capture subtle features, neighbors, and patterns that may not be discernible from total flux measures. Critically, the independent photo-z from the AGN and host must agree, thus narrowing the overall source PDF.</text>
<text><location><page_12><loc_6><loc_19><loc_49><loc_36></location>Table 3 presents the fraction of outliers for both EXT and PSF sources using di ff erent configurations of the PICZL algorithm. The first setup replicates the feature-only method, relying solely on total fluxes from the catalog, which, as seen in other studies, results in the fraction of outliers for PSF sources being nearly three times higher than for EXT sources (Salvato et al. 2022). In the second configuration, we adopt the approach of Saxena et al. (2024), incorporating the 2D light distribution and color gradients within annuli. The third configuration, i.e., PICZL as outlined above, enhances this by incorporating images, supplemented with catalog values transformed into image format, which leads to an additional improvement over the method in Saxena et al. (2024).</text>
<text><location><page_12><loc_6><loc_10><loc_49><loc_19></location>Since images inherently capture all features typically extracted numerically and presented in catalogs, we show in the fourth case that PICZL achieves excellent results without feature / catalog information, using solely optical images supplemented by WISE aperture flux maps, even without explicit hyperparameter tuning. This approach is particularly promising for future surveys such as LSST and Euclid, where relying solely on</text>
<text><location><page_12><loc_51><loc_68><loc_94><loc_72></location>multi-band imaging, including NIR / IR, will eliminate the need for complex source modeling or aperture photometry, regardless of the sources being galaxies or AGN.</text>
<section_header_level_1><location><page_12><loc_51><loc_65><loc_72><loc_66></location>6.2. Blind sample comparison</section_header_level_1>
<text><location><page_12><loc_51><loc_51><loc_94><loc_64></location>To evaluate the robustness of our approach, we tested PICZL on an independent blind sample from the Chandra Source Catalog 2 (CSC2 3 ). This sample, selected to ensure no overlap with the training data, but yield a comparable X-ray to MIR distribution, is the same one used in the C ircle Z analysis (Saxena et al. 2024), allowing for a direct comparison with their results. After filtering for sources with spectroscopic redshift and removing duplicates with the training dataset, the sample comprises 416 sources within the LS10-South area. For comparison, Saxena et al. (2024) achieve an outlier fraction, η , of 12.3% and a nor-</text>
<figure>
<location><page_12><loc_52><loc_20><loc_94><loc_46></location>
<caption>Fig. 10: PICZL point estimates as a function of PDF width, represented by the di ff erence between upper and lower 1 σ values. Each data point is color-coded according to its r -band magnitude from the LS10. The tilt of the distribution traces the increased uncertainties for PDFs of fainter sources. Horizontal distributions represent areas of large uncertainty in the photo-z, indicating that PICZL errors are realistic.</caption>
</figure>
<text><location><page_13><loc_45><loc_65><loc_46><loc_67></location>7</text>
<figure>
<location><page_13><loc_7><loc_64><loc_46><loc_93></location>
<caption>Figure 10 displays PICZL point estimates as a function of PDF width, represented by the upper 1 σ minus lower 1 σ , colorcoded by the Legacy Survey r -band magnitude, as it o ff ers the most extensive coverage compared to the g , i , or z bands, containing the largest number of sources. We observe wider PDFs correlating with higher redshifts and fainter sources characterized by higher photometric errors or lower S / N, as discussed</caption>
</figure>
<figure>
<location><page_13><loc_47><loc_64><loc_96><loc_93></location>
<caption>Fig. 11: Left: Examples of an inlier (orange) and an outlier (purple) PDF. By definition, the point estimate derived for the two sources from the main mode falls within or outside the grey shaded region. Both sources show a secondary peak in the PDF. We can define the probability of being an inlier (colored area under the PDF which coincides with the grey shaded area). Right: Normalized | ∆ Z | (1 + z spec) as a function of inlier probability, color-coded by number density. The majority of the inliers have a high inlier probability, indicating the stability of our results (see main text for details).</caption>
</figure>
<text><location><page_13><loc_6><loc_45><loc_49><loc_53></location>malized median absolute deviation, σ NMAD, of 0.055, while our algorithm yields a superior σ NMAD of 0.046 and η of 8.3% (see Table 5 and Figure C.1). This comparison, therefore, demonstrates how utilizing image cubes further improves the already good results achieved using 2D-information from a catalog. For more details on the sample, see Saxena et al. (2024).</text>
<section_header_level_1><location><page_13><loc_6><loc_42><loc_35><loc_43></location>6.3. Prediction uncertainty quantification</section_header_level_1>
<text><location><page_13><loc_6><loc_19><loc_49><loc_41></location>Accurate error estimation is crucial for assessing the reliability of any prediction, particularly in those astrophysical contexts where uncertainties can significantly impact the interpretation of data. To this end, we provide asymmetrical 1 σ and 3 σ errors for every source, together with the photoz PDF, as detailed in Sect. 5.6. These error estimates o ff er a comprehensive understanding of the potential variance in our predictions, accurately reflecting the inherent uncertainties in our data and model. We include asymmetrical errors, as the true distribution of uncertainties is often not symmetrical. This asymmetry arises from factors such as photometric noise and degeneracies in the highly nonlinear color-redshift space, where certain higher or lower redshift values may be more probable than their counterparts, leading to skewed PDFs. Although symmetrical redshift errors are observed for the majority of sources, approximately one-third of the sources exhibit asymmetrical 1 σ errors with deviations in redshift up to | ∆ z asym | ≃ 0 . 25.</text>
<text><location><page_13><loc_51><loc_36><loc_94><loc_53></location>in Sect. 8.4. Additionally, the widths of the PDFs exhibit noticeable horizontal structures, largely influenced by the nature of the LS10 filters. We find that PDFs with wider modes, indicating less certain redshift estimates, often correspond to redshift ranges where key spectral features, such as the Ca break, fall outside the filter coverage (e.g. g & r at z ≃ 0 . 4, r & i at z ≃ 0 . 8 and z ≃ 1.5 or z ≃ 2.2). At redshifts approaching z ≈ 3 and extending below z ≈ 5, the Lymanα break starts to fall within the filter ranges, which contributes to narrower PDFs with increased PDF width only observed for extremely faint sources at even higher redshifts. Lastly, we find that the number of sources with wider PDFs increases as a function of | ∆ Z | (1 + Z spec) , indicating that narrower PDFs correspond to more accurate point predictions.</text>
<section_header_level_1><location><page_13><loc_51><loc_32><loc_76><loc_33></location>6.4. Insights from multimodal PDFs</section_header_level_1>
<text><location><page_13><loc_51><loc_17><loc_94><loc_30></location>To investigate whether predicted outliers are genuine anomalies or instead stem from machine learning processes, we inquire whether secondary peaks in PDFs are physically meaningful or training artifacts. For each source, we compute the fraction of its PDF, that satisfies the condition | ∆ Z | (1 + z spec) ≥ 0 . 15, to provide the probability of being considered an inlier. The left panel of Figure 11 illustrates this with examples of two PDFs (one for an inlier and one for an outlier), where we highlight the corresponding inlier probabilities. In the right panel of Figure 11, we plot | ∆ Z | (1 + z spec) re-normalised to scale [0,1], such that</text>
<formula><location><page_13><loc_51><loc_9><loc_89><loc_13></location>| ∆ z | (1 + z spec) norm = | ∆ z | (1 + z spec) · 0 . 5 0 . 15 if | ∆ z | (1 + z spec) < 0 . 15 | ∆ z | (1 + z spec) · 0 . 5 0 . 15 + 0 . 5 if | ∆ z | (1 + z spec) ≥ 0 . 15</formula>
<text><location><page_14><loc_6><loc_80><loc_49><loc_93></location>as a function of inlier probability for all sources. The horizontal dashed black line separates the inliers ( < 0.5) from the outliers ( ≥ 0.5). The majority of inliers are concentrated at high inlier probabilities. This suggests that most inliers provide confident, unimodal PDFs with low errors. Likewise, most outliers have low inlier probability, with only a few outliers having more than 50% inlier probability. While this quantity cannot be computed for sources lacking spectroscopic redshift, it can be used to evaluate the reliability of our point estimates considering their associated PDFs.</text>
<text><location><page_14><loc_6><loc_62><loc_49><loc_80></location>To evaluate whether ensemble solutions with broad or complex distributions e ff ectively capture the true redshift, we identify sources with multiple peaks and measure the proportion of PDFs exhibiting more than one mode. For these sources, we define a secondary peak as significant if it accounts for at least 10% of the height of the primary peak, which is the minimum prominence threshold used in our analysis. The top panel of Figure 12 shows that only 3% of inliers exhibit secondary peaks as opposed to the roughly 30% of outliers, where occurrences of strong secondary modes decrease with increasing prominence for both cases. While a single mode typically indicates a secure estimate for inliers, the presence of a unique mode alone can subsequently not be used to determine whether a source is an outlier.</text>
<text><location><page_14><loc_6><loc_51><loc_49><loc_62></location>Conversely, in the bottom panel of Figure 12, we present the recovery fraction, which denotes how often a secondary peak in PDFs exhibiting multi-modal distributions corresponds to the true redshift. Among the outliers with 30% multi-modal PDF, more than 70% have one of their secondary peaks corresponding to | ∆ Z | (1 + Z spec) ≤ 0 . 15, indicating that there is significant probability that the redshift could consequently be considered as an inlier if the peak heights were reversed. Given the PIT histograms shown</text>
<figure>
<location><page_14><loc_7><loc_18><loc_49><loc_49></location>
<caption>Fig. 12: Top: Fraction of sources having a PDFs presenting secondary peaks for outliers (burgundy) and inliers (blue). Bottom: Corresponding recovery fraction for outliers. Both metrics are plotted as a function of the relative prominence threshold of the secondary peak.</caption>
</figure>
<text><location><page_14><loc_51><loc_80><loc_94><loc_93></location>in Figure 9, it appears likely that the the PDFs are accurately capturing the relative frequency of multi-& bimodal PDFs, as if they were not, there would likely be bias evident in the overall PIT distribution. In other words, using this particular dataset, despite incorporating full multiwavelength image cutouts and cataloglevel information, there persist areas of parameter space that are legitimately degenerate in redshift, and the PDF parameterization looks to be capturing that degeneracy accurately. Consequently, we recommend using the entire PDF rather than point estimates, when possible.</text>
<section_header_level_1><location><page_14><loc_51><loc_76><loc_79><loc_77></location>7. PICZL applied to other surveys</section_header_level_1>
<text><location><page_14><loc_51><loc_65><loc_94><loc_75></location>We want to investigate how PICZL, utilizing LS10 photometry and imaging, performs in determining photo-z for AGN in a more generic setting. Our focus is set on the LSST deep drilling fields (DDFs), selected to study SMBH growth across the full range of cosmic environments. These fields o ff er deeper, more comprehensive spectroscopic redshift coverage, along with a broader range of high-sensitivity bands, providing an enhanced dataset for photo-z estimation.</text>
<section_header_level_1><location><page_14><loc_51><loc_62><loc_64><loc_63></location>7.1. XMM-SERVS</section_header_level_1>
<text><location><page_14><loc_51><loc_45><loc_94><loc_61></location>The XMM-Spitzer Extragalactic Representative Volume Survey (XMM-SERVS, Mauduit et al. 2012) encompasses three key fields: the XMM-Large Scale Structure (LSS, Chen et al. 2018), spannning 5.3 deg 2 with a flux limit of 6 . 5 × 10 -15 erg cm -2 s -1 over 90% of the survey area in the 0.5-10keV band (Savi'c et al. 2023); the Wide Chandra Deep Field-South (W-CDF-S, Ni et al. 2021) and the European Large-Area ISO Survey-South 1 (ELAIS-S1, Ni et al. 2021), covering approximately 4.6 deg 2 and 3.2 deg 2 (Brandt et al. 2018; Scolnic et al. 2018), limited to 1 . 3 × 10 -14 erg cm -2 s -1 , respectively, each selected for their exceptional multiwavelength coverage and strategic alignment with the DDFs.</text>
<section_header_level_1><location><page_14><loc_51><loc_42><loc_69><loc_43></location>7.2. Photo-z computation</section_header_level_1>
<text><location><page_14><loc_51><loc_10><loc_94><loc_41></location>Considering that the original XMM-SERVS photo-z benefit from high S / N photometry and broader wavelength coverage, including u -band and NIR coverage, we would expect significantly better results compared to those achievable by PICZL using LS10 data alone. However, PICZL obtains comparable if not better results when limiting the sample to the depth at which it was trained on (see XMM-SERVS magnitude distribution in Figure E.1). Like any ML model, it struggles to extrapolate to faint sources outside its training distribution. Therefore, a fair comparison requires limiting the analysis to a similar feature space, while the results in Table 4 reflect performance across the entire, diverse XMM-SERVS samples. To facilitate a comparison with PICZL, our analysis begins with the counterpart associations of X-ray emissions as presented in Ni et al. (2021) and Chen et al. (2018). Whenever possible, we limit our samples to sources flagged as AGN (as defined in Sect. 6 of Ni et al. 2021). Matching to objects detected in LS10, due to its more limited depth, we identify approximately one-third of the original sources (see Table 4). This limitation is particularly pronounced in the XMM-LSS survey, where, in addition to deeper X-ray data corresponding to fainter counterparts, Chen et al. (2018) restrict their sample of AGN for which they calculate photo-z, to strictly non-broad-line X-ray sources. We select all sources with spectroscopic redshift and exclude those previously included in the</text>
<table>
<location><page_15><loc_6><loc_83><loc_95><loc_91></location>
<caption>Table 4: Summary of the XMM-SERVS samples and their respective photoz metrics.</caption>
</table>
<text><location><page_15><loc_6><loc_77><loc_94><loc_82></location>Notes. From left to right, the columns list: Sample field; the original number of X-ray sources; the number of sources in each field that were detected in the LS10 survey; have spectroscopic redshift z > 0 . 002; and were not already present in the training sample of PICZL (i.e. previously unseen); the fraction of failed photo-z in XMM-SERVS zphot = -99; and the bias ⟨ ∆ z ⟩ , variance σ NMAD and outlier fraction η obtained with XMMSERVS (S, excluding zphot = -99) and PICZL (P) photo-z.</text>
<figure>
<location><page_15><loc_7><loc_50><loc_49><loc_74></location>
<caption>Fig. 13: Top: Outlier fraction as a function of LS10 i-band magnitude (left) and LS10 i-band S / N (right), comparing the original photo-z from XMM-SERVS to the photo-z computed with PICZL. The XMM-SERVS photo-z results are shown both with and without the inclusion of sources for which XMM-SERVS photo-z failed (photo-z = -99). The shaded regions represent the range of XMM-SERVS outlier fractions across the ELAIS-S1, W-CDF-S, and LSS fields. Bottom: the range of XMM-SERVS variance in the three fields as a function of LS10 i-band magnitude and S / N. PICZL demonstrates a lower outlier fraction for brighter sources while maintaining comparable accuracy to the original photo-z estimates. Additionally, PICZL successfully computes photo-z for sources that are failures in the original methods.</caption>
</figure>
<text><location><page_15><loc_6><loc_10><loc_49><loc_28></location>PICZL training sample (see Table 4). Unlike the original works, PICZL also successfully computes photo-z for the significant fraction of sources for which the SED fitting adopted in XMMSERVS failed (refer to w / photoz S in Table 4). Figure 13 visualizes the outlier fractions (top) and variances (bottom) for the aggregate XMM-SERVS samples. Blue and burgundy represent the distributions including / excluding catastrophic failures in the XMM-SERVScatalogs. The plot is limited to outlier fractions of 30% and variances of 10%, as XMM-SERVS curves including zphot = -99 solutions continue to rise for fainter magnitudes or lower S / N. This suggests that, with increasing magnitude, a decreasing number of accurate estimates are available, making the subset excluding such solutions increasingly non-representative in terms of outlier fraction.</text>
<figure>
<location><page_15><loc_52><loc_52><loc_94><loc_74></location>
<caption>Fig. 14: Normalized residuals for PICZL photo-z compared to XMM-SERVS photo-z, using data from XMM-SERVS ELAISS1. PICZL outliers are shown as triangles, XMM-SERVS outliers as stars, and sources that are outliers in both methods are depicted as octagons. All sources are color-coded based on their LS10 i-band magnitude.</caption>
</figure>
<text><location><page_15><loc_51><loc_26><loc_94><loc_40></location>However, by limiting our comparison to sources with photoz obtained from XMM-SERVS, we find that the accuracy of these photo-z remains comparable up to magnitudes around 23.5 AB, despite the more limited data being used. Additionally, we observe enhanced performance in PICZL photo-z for sources with S / N values ≳ 60, depending on the specific XMM-SERVS sample. Importantly, sources with S / N ≳ 10-20 mark a critical threshold range in LS10 where lower S / N values lead to an exponential rise in photometric errors (see Figure D.1). This trend is particularly evident for magnitudes fainter than 21.5 in the iband (Saxena et al. 2024) and visible in all panels of Figure 13.</text>
<text><location><page_15><loc_51><loc_10><loc_94><loc_25></location>Figure 14 presents a visual comparison of normalized residuals for outliers identified by either PICZL or the original photo-z from Ni et al. (2021), exemplified via the ELAIS-S1 field. The spread of biases, excluding failures from XMM-SERVS, is similar for both methods, falling mostly within the range of [-0.5, 0.5]. The outliers in common appear to be modestly brighter than those that are outliers for a single method only and are evenly distributed between overestimation and underestimation. Notably, the outliers from Ni et al. (2021) are over a narrower range, while PICZL has a few brighter and several more fainter, the majority of which with magnitudes beyond the range covered during training. The observed di ff erence in outlier rates, is</text>
<figure>
<location><page_16><loc_8><loc_46><loc_92><loc_93></location>
<caption>Fig. 15: Prediction bias (top row), fraction of outliers (second row), variance (third row), and number count (fourth row) as a function of magnitude ( r -band from LS10), 0.2-2.3 keV X-ray flux, and specz , categorized by type (extended (EXT) / point-like (PSF)).</caption>
</figure>
<text><location><page_16><loc_49><loc_46><loc_49><loc_47></location>-</text>
<text><location><page_16><loc_6><loc_28><loc_49><loc_39></location>not primarily due to the use of template-fitting versus machinelearning methods but rather a reflection of the available photometry. XMM-SERVS, with its deeper photometry, is wellsuited for faint objects, though saturation in brighter sources may contribute to some outliers. PICZL, which leverages the LS10 dataset, is optimized for brighter sources due to the shallower photometry. While it performs well in this regime, fainter objects that fall outside the known parameter space are consequently less constrained.</text>
<section_header_level_1><location><page_16><loc_6><loc_25><loc_17><loc_26></location>8. Discussion</section_header_level_1>
<text><location><page_16><loc_6><loc_10><loc_49><loc_24></location>While Saxena et al. (2024) made significant strides in overcoming the limitations of relying solely on total or model fluxes for photoz estimation of AGN by utilizing all redshift-correlated features in the LS10 catalog, we have further advanced this approach. By integrating data from both optical and MIR wavelengths, our study highlights the transformative potential of combining imaging with catalog-level data. Despite the state-of-theart advancements in photoz estimation for AGN in this work, further improvements will be possible only by solving issues related to data quality. By overcoming these issues, we can significantly reduce the fraction of catastrophic outliers and achieve</text>
<text><location><page_16><loc_51><loc_36><loc_94><loc_39></location>even greater accuracy in our predictions. In the following, we discuss the limitations that need to addressed in order to apply PICZL to upcoming imaging surveys.</text>
<section_header_level_1><location><page_16><loc_51><loc_32><loc_65><loc_33></location>8.1. Incorrect spec-z</section_header_level_1>
<text><location><page_16><loc_51><loc_10><loc_94><loc_30></location>First and foremost, we need to address the reliability of the labels used in our, and generally any supervised machine learning algorithm. While spectroscopic redshift are typically trusted as accurate representations of true redshifts, this assumption does not always hold. As surveys continue to expand in scale, automated pipelines become indispensable for assigning spectroscopic redshift and identifying artifacts / problems for each source, as manual verification by visual inspection is impractical and will remain so for future surveys (Bolton et al. 2012). Importantly, most pipelines are trained on determining the redshift of the bulk of objects (Alexander et al. 2023), which are inactive galaxies and not AGN. This implies that a non-negligible fraction of AGN have wrong redshift estimates, including many labelled with "no warnings" (Hewett & Wild 2010; Wu & Shen 2022). Consequently, our performance is constrained by the unknown amount of incorrectly identified spectroscopic redshift</text>
<text><location><page_17><loc_9><loc_68><loc_28><loc_70></location>Absolute LS10 r band [mag]</text>
<text><location><page_17><loc_29><loc_68><loc_48><loc_70></location>Absolute LS10 r band [mag]</text>
<figure>
<location><page_17><loc_7><loc_69><loc_48><loc_93></location>
<caption>Fig. 16: Cumulative fraction of outliers as function of absolute magnitudes, to validate their classification based on TYPE. The left panel shows outliers classified as PSF, where most sources have absolute magnitudes consistent with AGN or QSO characteristics. In the right panel, 16% of EXT-classified outliers exhibit absolute magnitudes more typical of QSOs.</caption>
</figure>
<text><location><page_17><loc_6><loc_42><loc_49><loc_53></location>per survey. While this fraction may be smaller for galaxies without nuclear activity, AGN present a range of challenges where pipelines fail. Examples of failures include cases where e.g. MgII is misidentified as Lymanα , leading to ambiguous redshifts around z = 0 . 7 -0 . 9 ⇔ 1 . 5 -2 . 5 or sources with strong featureless continuum typical of Blazars or, finally, sources with intense broad emission lines compared to the background continuum (Chen et al. 2018; Ni et al. 2021), as well as those with weak emission lines and a flat continuum background.</text>
<section_header_level_1><location><page_17><loc_6><loc_37><loc_28><loc_38></location>8.2. Incorrect type classification</section_header_level_1>
<text><location><page_17><loc_6><loc_10><loc_49><loc_34></location>A subset of sources in the training sample, although having redshifts of z ≥ 1, are classified as extended (i.e. non-PSF). This classification is likely incorrect, given the pixel resolution of LS10. The apparent extension is more plausibly a result of poor seeing conditions that were not properly accounted for, or the presence of nearby neighboring sources (see Sect. 5.3 of Hsu et al. 2014). Since the morphological type a ff ects predictions, akin to a prior in template fitting methods, this misclassification explains the increase in outliers within the EXT sample as redshift increases (see middle row in right panel of Figure 15). To verify this, we computed the absolute g magnitudes for outliers to assess whether their values fall within the expected range for galaxies and AGN (Véron-Cetty & Véron 2010). Figure 16 shows that 16% of sources classified as EXT (and thus galaxy dominated) have an absolute magnitude which exceeds the range typical of galaxies MEXT [-16,-24], confirming that their TYPE is wrongly assigned. Interestingly, approximately 96% of PSF sources meet the absolute magnitude requirement of MPSF [-20,31], indicating that their TYPE is generally accurate.</text>
<section_header_level_1><location><page_17><loc_51><loc_92><loc_67><loc_93></location>8.3. Faulty photometry</section_header_level_1>
<text><location><page_17><loc_51><loc_68><loc_94><loc_91></location>In addition to issues connected to the reliability of labels, predictions also face challenges due to data inconsistencies arising from faulty photometry. Surveys often flag issues of this kind directly, including problems such as hot pixels, saturation, cosmic rays, bleed trails, and other uncategorized anomalies. Detecting and addressing such observational defects is crucial because the accuracy and reliability of photoz predictions cannot be guaranteed for compromised photometric inputs. We experimented with removing various groups of sources a ff ected by observational artefacts but found no improvement when excluding any single group. We therefore assume that, while removing some problematic sources could potentially improve performance, the reduction in training data size o ff sets any gains. Adding this kind of noise to the current training sample could be an approach to decouple these two e ff ects. Consequently, we treat the inclusion of all problematic sources as natural noise, as opposed to artificially degrading images employing methods such as Gaussian noise (Hayat et al. 2021).</text>
<section_header_level_1><location><page_17><loc_51><loc_64><loc_64><loc_65></location>8.4. Survey depth</section_header_level_1>
<text><location><page_17><loc_51><loc_38><loc_94><loc_63></location>When splitting the validation sample by S / N across all bands, we find a higher outlier fraction in sources with low S / N (see Table 5). Consequently, to evaluate the reliability of our photoz, we need to consider the photometric depth of LS10 and how increased photometric errors for fainter sources impact redshift estimate uncertainties. Figure 15 presents the prediction bias ⟨ z ⟩ , outlier fraction η , and dispersion σ NMAD as functions of r -band magnitude in LS10, X-ray flux (when available), and specz . As the r -band magnitude increases, as expected, the outlier fraction and scatter rise. This pattern is consistent with the exponential increase in photometric errors for, e.g. i -band magnitudes ≳ 21.5, beyond which the outlier fraction, which otherwise remains well below 10%, and scatter also appears to rise, especially for sources of type EXT (refer to Figure D.1). While faint LS10 sources have unreliable photo-z, future surveys such as LSST and Euclid will probe deeper with improved S / N. As such, our methodology can be adapted to these next-generation surveys, ensuring high accuracy across a broader range of magnitudes and providing robust redshift estimates for the extensive AGN populations these and other surveys will detect.</text>
<section_header_level_1><location><page_17><loc_51><loc_34><loc_64><loc_35></location>8.5. Missing i-band</section_header_level_1>
<text><location><page_17><loc_51><loc_26><loc_94><loc_33></location>For DR10, the Legacy Survey footprint was extended by incorporating all available DECam data from various contributing surveys (refer to Sect. 3), including coverage in the i -band, though limited to a single pass. As a result, ∼ 10% of sources lack i -band observations. Unsurprisingly, and as shown in Table 5, the accuracy of photo-z increases when this band is also available.</text>
<section_header_level_1><location><page_17><loc_51><loc_22><loc_59><loc_23></location>8.6. Biases</section_header_level_1>
<text><location><page_17><loc_51><loc_10><loc_94><loc_21></location>While the photo-z residuals exhibit a symmetric distribution around zero with minimal scatter, they do not consistently center at zero, suggesting a potential systematic bias in the photo-z estimates. Ideally, we would like to achieve normalized residuals that remain consistent irrespective of redshift or selection. Currently, this is not the case, as biases are not corrected for and are mostly influenced by the distribution of our training samples, which are specific to their respective survey and inherently biased.</text>
<table>
<location><page_18><loc_18><loc_63><loc_82><loc_91></location>
<caption>Table 5: Comparison of bias, fraction of outliers, and variance between photo-z computed with PICZL for various subsamples.</caption>
</table>
<text><location><page_18><loc_6><loc_57><loc_94><loc_62></location>Notes. ( 1 ) all sources; ( 2 ) split by type (PSF vs. EXT); ( 3 ) split by signal-to-noise ratio (S / N) of all available bands; ( 4 ) split by availability of the i -band; ( 5 ) split by selection criteria; ( 6 ) split by the presence of a faint [(mAGN - mNeigh . ) within -3 ≤ ∆ mag < -1 for all available bands] or bright [(mAGN - mNeigh . ) within ∆ mag > -1 for all available bands] neighbor within a 5 arcsecond radius. Additionally, results derived for the CSC2 blind sample for both PICZL and C ircle Z are provided for comparison.</text>
<section_header_level_1><location><page_18><loc_6><loc_54><loc_22><loc_55></location>8.6.1. Selection effects</section_header_level_1>
<text><location><page_18><loc_6><loc_43><loc_49><loc_53></location>We investigate whether the combination of various ways of selecting AGN entering the training sample a ff ects the quality of the photo-z for specific subsamples. Using the classification outlined in Table 1, we split the validation sample based on observational criteria, creating a binary division between sources selected in optical or via X-ray. While there is only a minor difference in σ NMAD, we observe a higher outlier fraction for Xray-detected sources.</text>
<text><location><page_18><loc_6><loc_31><loc_49><loc_42></location>While strong X-ray emission from an AGN implies strong optical and mid-IR from an AGN, which should overpower the galaxy emission in the latter two bands, complicating the process of determining accurate photo-z, local Seyfert 1 galaxies with high X-ray flux have their host remain distinctly visible. As shown in the central panel of Figure 15, however, the outlier fractions and accuracy remain consistent across the full range of X-ray fluxes, aside from small-number statistics for the wings of the distribution.</text>
<section_header_level_1><location><page_18><loc_6><loc_27><loc_27><loc_28></location>8.6.2. Sample characteristics</section_header_level_1>
<text><location><page_18><loc_6><loc_20><loc_49><loc_26></location>Unlike for galaxies, where the redshift distribution n( z ) is wellestablished and can serve as a reliable prior, the AGN n( z ) is not su ffi ciently characterised. Therefore uncertainty surrounds the ML algorithm subliminally adopting a distribution similar to the n( z ) of the training sample.</text>
<text><location><page_18><loc_6><loc_10><loc_49><loc_20></location>At very low redshifts, AGN have a low surface density, resulting in only a few rare objects that can be considered for training. This phenomenon is intensified by the fact that deep surveys, as opposed to wide-field surveys, usually bypass such bright nearby objects in search of intermediate and high redshift sources. This leads to areas of scarcity in the training sample's specz distribution, with biased predictions towards redshift values where more data points exist. To mitigate this, we nor-</text>
<text><location><page_18><loc_51><loc_46><loc_94><loc_55></location>malize the CRPS score by (1 + z ), emphasizing the accuracy of low-redshift predictions. Additionally, at extremely low redshifts where the 4000 Å break is barely covered by the g filter, accurate redshift predictions are more di ffi cult to obtain. At very low redshift, the 6 arcsecond × 6 arcsecond cutout may be entirely filled by the galaxy, potentially misleading the algorithm into interpreting it as excess noise.</text>
<section_header_level_1><location><page_18><loc_51><loc_42><loc_81><loc_43></location>8.6.3. Non-representative training samples</section_header_level_1>
<text><location><page_18><loc_51><loc_24><loc_94><loc_41></location>One method to tackle covariate shift by the imbalance of the bright-end dominated spectroscopic sample, is given by subdividing both the training and validation samples into subsets of n -dimensional feature space of distinguishable properties (Newman & Gruen 2022). Specifically, the prediction accuracy improves if the model used to generate a posterior for a blind source was trained exclusively on training sources residing within the corresponding feature space (Rosenbaum & Rubin 1984; Revsbech et al. 2017; Autenrieth et al. 2023). However, this approach might significantly reduce the training sample size per model making it more applicable to photoz codes dealing with large datasets, such as those for inactive galaxies (Newman & Gruen 2022).</text>
<text><location><page_18><loc_51><loc_10><loc_94><loc_24></location>In this work we have demonstrated that, in contrast to just a few years ago, AGN-targeted training samples are now su ffi -ciently large to provide reliable photo-z. Overcoming inherent biases still requires either gathering more representative samples that uniformly cover the full redshift range-particularly underrepresented regions-or applying statistical corrections. Thus, the focus has shifted from simply increasing the number of spectra to strategically obtaining spectra that cover specific regions of parameter space (Masters et al. 2015). However, given the substantial gains our model has shown over existing methods, a detailed examination of how these biases a ff ect prac-</text>
<text><location><page_19><loc_6><loc_78><loc_49><loc_93></location>tical applications is beyond the scope of this study. Looking ahead, spectroscopic follow-up campaigns such as DESI, SDSSV / BHM (Kollmeier et al., in prep), and 4MOST (De Jong et al. 2019) will further enhance our capabilities. Nevertheless, predicting photo-z for AGN in deeper Euclid and LSST datasets remains challenging due to the limited availability of spectroscopic data for faint sources. The upcoming Subaru Prime Focus Spectrograph (PFS, Tamura et al. 2016) and Multi Object Optical and Near-infrared Spectrograph for the Very Large Telescope (VLT / MOONS; Cirasuolo et al. 2020) are expected to help address this shortfall by providing much-needed spectra for faint (& obscured / red) objects.</text>
<section_header_level_1><location><page_19><loc_6><loc_74><loc_40><loc_75></location>8.7. Observational constraints: source crowding</section_header_level_1>
<text><location><page_19><loc_6><loc_44><loc_49><loc_73></location>One factor beyond our control is the local environment or conditions along the line of sight, which can significantly influence the emission characteristics of AGN. Nearby objects can add extra flux, which can a ff ect the accuracy of photometric measurements. We perform a positional cross-match between the validation sample and all LS10 sources within a 5 arcsecond radius of their optical counterparts. For each neighbour meeting this proximity criterion, we calculate apparent magnitudes and flag all sources where the brightest neighbour is no less than 1 magnitude dimmer. Table 5 shows that isolated sources exhibit better overall performance, while those with bright neighbours show drastically reduced quality compared to those with fainter neighbours. The presence of bright neighbours influences the observed flux in two ways: firstly, it complicates the derivation of sourcespecific colors due to convolved fluxes, and secondly, it a ff ects the observed spectra, potentially leading to the detection of two sets of emission lines and incorrect specz identification (Newman & Gruen 2022). A potential solution to this issue could be the implementation of segmentation maps, similar to SExtractor (Source Extractor, Bertin, E. & Arnouts, S. 1996) at image level, as LS10 currently only masks neighbours during their model flux fitting procedure.</text>
<text><location><page_19><loc_6><loc_17><loc_49><loc_44></location>A persistent physical issue that cannot be entirely mitigated is blending. Unlike bright neighbours, which can be masked in principle, blending a ff ects both observed photometry and spectroscopy. While particularly faint neighbours do not substantially a ff ect the predictions negatively, sources identified as blends should be excluded from samples where accurate photoz estimates are required, until we can partially mask sources at the pixel level. The Rubin Observatory expects overlapping (inactive) galaxies to contribute at least 1% of the total flux within their pixels (Sanchez et al. 2021; Newman & Gruen 2022). Blends are also expected to occur in the spectroscopic sample, increasing systematic uncertainty and subsequently the fraction of outliers by up to 5% (Masters et al. 2019). This may be mitigated in future by higher-quality imaging from spacebased data from missions such as Euclid or better tools for deblending (e.g., SCARLET / blendz ; Melchior et al. 2018; Jones & Heavens 2018). Alternatively, data augmentation or standard computer vision techniques could be implemented, such as artificially introducing blending e ff ects in the training sample. However, this approach is not trivial, as LS10 currently lacks a corresponding blending flag.</text>
<section_header_level_1><location><page_19><loc_6><loc_13><loc_16><loc_14></location>8.8. Variability</section_header_level_1>
<text><location><page_19><loc_6><loc_10><loc_49><loc_12></location>AGN are inherently variable sources, meaning that their observed flux can change significantly across di ff erent epochs and</text>
<text><location><page_19><loc_51><loc_80><loc_94><loc_93></location>wavelengths, especially when observations are separated by substantial time gaps. This extends to images that are created by stacking multiple observations taken over extended periods (as in LS10), where the resulting flux represents an average value. Such variability complicates the accurate prediction of AGN redshifts (Simm et al. 2015). However, future time-resolved imaging from LSST will allow us to better account for these variations and potentially use the correlation between AGN variability and their physical properties as a feature to improve photo-z predictions.</text>
<section_header_level_1><location><page_19><loc_51><loc_77><loc_78><loc_78></location>8.9. Scalability in image-based photo-z</section_header_level_1>
<text><location><page_19><loc_51><loc_68><loc_94><loc_76></location>Open issues remain regarding the feasibility of handling the vast data volumes and computational requirements associated with photo-z estimated from images. Exemplified by the calculation of new estimates for XMM-SERVS concerning storage, processing time, and computational resources, we give an overview in Appendix G.</text>
<text><location><page_19><loc_51><loc_60><loc_94><loc_68></location>A crucial future direction involves exploring ways to utilize imaging data e ffi ciently without necessitating extensive local downloads and computations for all image-based analyses of such data sets, potentially leveraging online platforms for realtime analysis, bringing the compute to the data (Zhang & Zhao 2015).</text>
<section_header_level_1><location><page_19><loc_51><loc_56><loc_71><loc_57></location>9. Summary and outlook</section_header_level_1>
<text><location><page_19><loc_51><loc_44><loc_94><loc_55></location>This work has been driven by the goal of determining reliable photo-z for X-ray detected AGN in wide-field surveys, such as eROSITA (Merloni et al. 2024). For that reason, we have concentrated on utilizing data from LS10, which provides su ffi ciently homogeneous coverage and depth in 4 optical bands, enriched by the flux measured on NEOWISE7 data for all identified sources (see Sect. 3). LS10 overlaps almost entirely with the eROSITA footprint, simplifying the cross calibration of data, a processing step typically necessary when merging di ff erent surveys.</text>
<text><location><page_19><loc_51><loc_19><loc_94><loc_43></location>We introduce PICZL, a CNN-based machine learning model designed primarily for AGN redshift estimation, but which holds the potential to reliably measure photoz s for a broad range of extragalactic sources, provided an appropriately constructed training sample is available. In this study, the training sample comprises both X-ray and optically selected AGN. Across our validation set, PICZL demonstrates consistently robust performance, with comparable accuracies and lower outlier fractions particularly for PSF sources, as opposed to the results obtained by Salvato et al. (2022) for similar objects using SED-fitting (see Figure 7 and Table 5). The results show significant improvements for both point-like and extended sources, indicating that the model's performance is primarily driven by the depth and hence photometric error of the training data. Additionally, when tested on a blind sample of X-ray-selected AGN, PICZL maintained comparable results with respect to σ NMAD and η . Notably, on this test set it outperformed (C irclez ; Saxena et al. 2024) by achieving a 20% improvement in accuracy and a 30% reduction in the fraction of outliers (refer to Sect. 6).</text>
<text><location><page_19><loc_51><loc_10><loc_94><loc_19></location>We also applied PICZL to estimate photo-z for the approximately 30% of XMM-SERVS sources (Chen et al. 2018; Ni et al. 2021), detected in LS10 (see Table 4), demonstrating comparable variance and a substantially improved outlier fraction up to a limiting magnitude of 23.5 AB, using much fewer bands and significantly less sensitive imaging. However, it is important to note that the spectroscopic sample at this faint limit is small</text>
<text><location><page_20><loc_6><loc_85><loc_49><loc_93></location>and likely biased toward sources with higher spectroscopic success rates. Consequently, the most reliable photo-z results are expected at brighter magnitudes (as discussed in Saxena et al. 2024). In response to these findings, we are releasing a new catalog of photo-z, complete with 1 and 3 σ error margins for the XMM-SERVS fields (see Sect. 7).</text>
<text><location><page_20><loc_6><loc_57><loc_49><loc_85></location>PICZL will be used in the next generation of photo-z for sources detected by eROSITA, possibly switching from LS10 to Euclid and, most importantly, LSST, as they are poised to deliver more homogeneous and deeper data with broader wavelength coverage. In particular the availability of NIR data at image level, will improve our ability to determine accurate redshift for faint and high-z sources. Our study has shown that the performance of PICZL, when relying predominantly on images, is already robust (see Table 3). Crucially, we have shown that wellcalibrated images alone can su ffi ce for accurate photo-z estimation, eliminating the need for catalog creation, which is often based on predefined models. This suggests that future surveys could reduce their dependence on catalog-based data for photo-z computation (refer to Sect. 6). Although this study has focused on calculating photo-z for AGN, the approach can be generalised and adapted to other source types, e.g., normal galaxies, with an appropriately constructed training sample (Götzenberger et al. in prep.). Subsequently our findings point to a bright future for all-sky surveys, also thanks to the continue expansion of training sample sizes, supported by initiatives like SDSS-V / BHM, 4MOST and VLT / MOONS, which will enhance the reliability and completeness of photoz estimates (refer to Sect. 8).</text>
<text><location><page_20><loc_6><loc_40><loc_49><loc_56></location>Looking forward, the next step in advancing photo-z estimation for AGN lies in exploring more sophisticated machine learning architectures beyond CNNs. For example, transformers with shared latent space embeddings o ff er a promising avenue. These models have shown success in integrating information from various data sources, such as images and entire spectra, potentially reducing uncertainties associated with traditional specz pipelines by leveraging multi-modal data fusion (DonosoOliva et al. 2023; Parker et al. 2024). Additionally, incorporating other informative features like X-ray flux, when available, holds promise. Integrating these diverse data sources within a unified framework has the potential to refine redshift estimates even further.</text>
<section_header_level_1><location><page_20><loc_6><loc_36><loc_22><loc_37></location>10. Data availability</section_header_level_1>
<text><location><page_20><loc_6><loc_22><loc_49><loc_35></location>With this paper, we present a new catalog of photo-z derived using PICZL for the ∼ 30% of sources within the XMM-SERVS (ELAIS-S1, W-CDF-S, and LSS) X-ray source catalogs (Chen et al. 2018; Ni et al. 2021) that are detected in the LS10 survey. Hence, it includes updated photo-z for sources with catastrophic failures in the original works. A detailed description of the catalog columns can be found in Appendix F. The full catalog is only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http:// cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/ .</text>
<text><location><page_20><loc_6><loc_10><loc_49><loc_21></location>Acknowledgements. WRand MS are grateful for the constant support of Dustin Lang in handling Legacy Survey-related issues. Part of this work was supported by the German Deutsche Forschungsgemeinschaft, DFG , under Germany's Excellence Strategy - EXC 2094 - 390783311. We gratefully acknowledge funding from FONDECYT Regular - 1231718 (RJA), 1230345 (CR), and 1241005 (FEB), CATA-BASAL - FB210003 (RJA, CR, FEB), and ANID - Millennium Science Initiative - AIM23-0001 (FEB). JA acknowledges support from a UKRI Future Leaders Fellowship (grant code: MR / T020989 / 1) This work is based on data from eROSITA, the soft X-ray instrument aboard SRG, a joint RussianGerman science mission supported by the Russian Space Agency (Roskosmos), in the interests of the Russian Academy of Sciences represented by its Space</text>
<text><location><page_20><loc_51><loc_10><loc_94><loc_93></location>Research Institute (IKI), and the Deutsches Zentrum für Luft- und Raumfahrt (DLR). The SRG spacecraft was built by Lavochkin Association (NPOL) and its subcontractors and is operated by NPOL with support from the Max Planck Institute for Extraterrestrial Physics (MPE). The development and construction of the eROSITA X-ray instrument were led by MPE, with contributions from the Dr. Karl Remeis Observatory Bamberg & ECAP (FAU Erlangen-Nuernberg), the University of Hamburg Observatory, the Leibniz Institute for Astrophysics Potsdam (AIP), and the Institute for Astronomy and Astrophysics of the University of Tübingen, with the support of DLR and the Max Planck Society. The Argelander Institute for Astronomy of the University of Bonn and the Ludwig Maximilians Universität Munich also participated in the science preparation for eROSITA. The Legacy Surveys consist of three individual and complementary projects: the Dark Energy Camera Legacy Survey (DECaLS; Proposal ID 2014B-0404; PIs: David Schlegel and Arjun Dey), the Beijing-Arizona Sky Survey (BASS; NOAO Prop. ID 2015A-0801; PIs: Zhou Xu and Xiaohui Fan), and the Mayall z-band Legacy Survey (MzLS; Prop. ID 2016A-0453; PI: Arjun Dey). DECaLS, BASS, and MzLS together include data obtained, respectively, at the Blanco telescope, Cerro Tololo Inter-American Observatory, NSF's NOIRLab; the Bok telescope, Steward Observatory, University of Arizona; and the Mayall telescope, Kitt Peak National Observatory, NOIRLab. Pipeline processing and analyses of the data were supported by NOIRLab and the Lawrence Berkeley National Laboratory (LBNL). The Legacy Surveys project is honored to be permitted to conduct astronomical research on Iolkam Du'ag (Kitt Peak), a mountain with particular significance to the Tohono O'odham Nation. NOIRLab is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. LBNL is managed by the Regents of the University of California under contract to the U.S. Department of Energy. This project used data obtained with the Dark Energy Camera (DECam), which was constructed by the Dark Energy Survey (DES) collaboration. Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A & M University, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho de Amparo, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Cientifico e Tecnologico and the Ministerio da Ciencia, Tecnologia e Inovacao, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey. The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Energeticas, Medioambientales y Tecnologicas-Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgenossische Technische Hochschule (ETH) Zurich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ciencies de l'Espai (IEEC / CSIC), the Institut de Fisica d'Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig Maximilians Universitat Munchen and the associated Excellence Cluster Universe, the University of Michigan, NSF's NOIRLab, the University of Nottingham, the Ohio State University, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, and Texas A&M University. BASS is a key project of the Telescope Access Program (TAP), which has been funded by the National Astronomical Observatories of China, the Chinese Academy of Sciences (the Strategic Priority Research Program 'The Emergence of Cosmological Structures' Grant # XDB09000000), and the Special Fund for Astronomy from the Ministry of Finance. The BASS is also supported by the External Cooperation Program of Chinese Academy of Sciences (Grant # 114A11KYSB20160057), and Chinese National Natural Science Foundation (Grant # 12120101003, # 11433005). The Legacy Survey team uses data products from the Near-Earth Object Wide-field Infrared Survey Explorer (NEOWISE), a project of the Jet Propulsion Laboratory / California Institute of Technology. NEOWISE is funded by the National Aeronautics and Space Administration. The Legacy Surveys imaging of the DESI footprint is supported by the Director, O ffi ce of Science, O ffi ce of High Energy Physics of the U.S. Department of Energy under Contract No. DE-AC02-05CH1123, by the National Energy Research Scientific Computing Center, a DOE O ffi ce of Science User Facility under the same contract, and by the U.S. National Science Foundation, Division of Astronomical Sciences under Contract No. AST-0950945 to NOAO. Funding for the Sloan Digital Sky Survey V has been provided by the Alfred P. Sloan Foundation, the Heising-Simons Foundation, the National Science Foundation, and the Participating Institutions. SDSS acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. SDSS telescopes are located at Apache Point Observatory, funded by the Astrophysical Research Consortium and operated by New Mexico State University, and at Las Campanas Observatory, operated by the Carnegie Institution for</text>
<text><location><page_21><loc_6><loc_76><loc_49><loc_93></location>Science. The SDSS web site is www.sdss.org . SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration, including Caltech, The Carnegie Institution for Science, Chilean National Time Allocation Committee (CNTAC) ratified researchers, The Flatiron Institute, the Gotham Participation Group, Harvard University, Heidelberg University, The Johns Hopkins University, L'Ecole polytechnique fédérale de Lausanne (EPFL), Leibniz-Institut für Astrophysik Potsdam (AIP), Max-PlanckInstitut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Extraterrestrische Physik (MPE), Nanjing University, National Astronomical Observatories of China (NAOC), New Mexico State University, The Ohio State University, Pennsylvania State University, Smithsonian Astrophysical Observatory, Space Telescope Science Institute (STScI), the Stellar Astrophysics Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Illinois at Urbana-Champaign, University of Toronto, University of Utah, University of Virginia, Yale University, and Yunnan University.</text>
<section_header_level_1><location><page_21><loc_6><loc_72><loc_16><loc_73></location>References</section_header_level_1>
<text><location><page_21><loc_6><loc_69><loc_49><loc_71></location>Abdalla, F. B., Banerji, M., Lahav, O., & Rashkov, V. 2011, MNRAS, 417, 1891 Aird, J., Nandra, K., Laird, E. S., et al. 2010, MNRAS, 401, 2531-2551</text>
<unordered_list>
<list_item><location><page_21><loc_6><loc_67><loc_49><loc_69></location>Ait-Ouahmed, R., Arnouts, S., Pasquet, J., Treyer, M., & Bertin, E. 2023, Multimodality for improved CNN photometric redshifts</list_item>
<list_item><location><page_21><loc_6><loc_64><loc_49><loc_66></location>Akiba, T., Sano, S., Yanase, T., Ohta, T., & Koyama, M. 2019, Optuna: A Nextgeneration Hyperparameter Optimization Framework</list_item>
</unordered_list>
<text><location><page_21><loc_6><loc_63><loc_46><loc_64></location>Alexander, D. M., Davis, T. M., Chaussidon, E., et al. 2023, ApJ, 165, 124</text>
<text><location><page_21><loc_6><loc_62><loc_49><loc_63></location>Almeida, A., Anderson, S. F., Argudo-Fernández, M., et al. 2023, ApJ Supple-</text>
<text><location><page_21><loc_8><loc_61><loc_19><loc_62></location>ment Series, 267, 44</text>
<text><location><page_21><loc_6><loc_60><loc_46><loc_61></location>Almosallam, I. A., Jarvis, M. J., & Roberts, S. J. 2016, MNRAS, 462, 726</text>
<text><location><page_21><loc_6><loc_59><loc_41><loc_60></location>Ananna, T. T., Salvato, M., LaMassa, S., et al. 2017, ApJ, 850, 66</text>
<unordered_list>
<list_item><location><page_21><loc_6><loc_58><loc_45><loc_59></location>Arnouts, S. & Ilbert, O. 2011, Astrophysics Source Code Library, 08009</list_item>
<list_item><location><page_21><loc_6><loc_55><loc_49><loc_58></location>Autenrieth, M., van Dyk, D. A., Trotta, R., & Stenning, D. C. 2023, Stratified Learning: A General-Purpose Statistical Method for Improved Learning under Covariate Shift</list_item>
</unordered_list>
<text><location><page_21><loc_6><loc_54><loc_21><loc_55></location>Baum, W. 1957, ApJ, 62, 6</text>
<unordered_list>
<list_item><location><page_21><loc_6><loc_52><loc_49><loc_54></location>Beck, R., Dobos, L., Budavári, T., Szalay, A., & Csabai, I. 2017, Astronomy and Computing, 19, 34</list_item>
</unordered_list>
<text><location><page_21><loc_6><loc_50><loc_44><loc_52></location>Bell, E. F., Wolf, C., Meisenheimer, K., et al. 2004, ApJ, 608, 752-767 Benitez, N. 2000, ApJ, 536, 571</text>
<text><location><page_21><loc_6><loc_49><loc_44><loc_50></location>Bertin, E. & Arnouts, S. 1996, Astron. Astrophys. Suppl. Ser., 117, 393</text>
<text><location><page_21><loc_6><loc_48><loc_49><loc_49></location>Bettoni, D., Falomo, R., Kotilainen, J. K., Karhunen, K., & Uslenghi, M. 2015,</text>
<text><location><page_21><loc_8><loc_47><loc_19><loc_47></location>MNRAS, 454, 4103</text>
<text><location><page_21><loc_6><loc_46><loc_45><loc_46></location>Blanton, M. R., Bershady, M. A., Abolfathi, B., et al. 2017, ApJ, 154, 28</text>
<text><location><page_21><loc_6><loc_45><loc_43><loc_45></location>Boller, T., Freyberg, M. J., Trümper, J., et al. 2016, A&A, 588, A103</text>
<text><location><page_21><loc_6><loc_43><loc_43><loc_44></location>Bolton, A. S., Schlegel, D. J., Aubourg, E., et al. 2012, ApJ, 144, 144</text>
<unordered_list>
<list_item><location><page_21><loc_6><loc_41><loc_49><loc_43></location>Bolzonella, M., Miralles, J.-M., & Pello', R. 2000, Photometric Redshifts based on standard SED fitting procedures</list_item>
</unordered_list>
<text><location><page_21><loc_6><loc_40><loc_37><loc_41></location>Bordoloi, R., Lilly, S. J., & Amara, A. 2010, MNRAS, no</text>
<text><location><page_21><loc_6><loc_38><loc_49><loc_40></location>Boutsia, K., Grazian, A., Calderone, G., et al. 2020, ApJ Supplement Series, 250, 26</text>
<unordered_list>
<list_item><location><page_21><loc_6><loc_34><loc_49><loc_38></location>Brammer, G. B., van Dokkum, P. G., & Coppi, P. 2008, ApJ, 686, 1503-1513 Brandt, W. N., Ni, Q., Yang, G., et al. 2018, Active Galaxy Science in the LSST Deep-Drilling Fields: Footprints, Cadence Requirements, and Total-Depth Requirements</list_item>
</unordered_list>
<text><location><page_21><loc_6><loc_33><loc_30><loc_34></location>Breiman, L. 2001, Machine Learning, 545, 5</text>
<text><location><page_21><loc_6><loc_32><loc_48><loc_33></location>Brescia, M., Cavuoti, S., Longo, G., & Stefano, V. D. 2014, A&A, 568, A126</text>
<unordered_list>
<list_item><location><page_21><loc_6><loc_31><loc_43><loc_32></location>Brescia, M., Salvato, M., Cavuoti, S., et al. 2019, MNRAS, 489, 663</list_item>
<list_item><location><page_21><loc_6><loc_30><loc_38><loc_31></location>Brunner, H., Liu, T., Lamer, G., et al. 2022, A&A, 661, A1</list_item>
<list_item><location><page_21><loc_6><loc_29><loc_42><loc_30></location>Buchner, J., Georgakakis, A., Nandra, K., et al. 2015, ApJ, 802, 89</list_item>
<list_item><location><page_21><loc_6><loc_27><loc_49><loc_29></location>Campagne, J.-E. 2020, Adversarial training applied to Convolutional Neural Network for photometric redshift predictions</list_item>
<list_item><location><page_21><loc_6><loc_23><loc_49><loc_26></location>Cardamone, C. N., van Dokkum, P. G., Urry, C. M., et al. 2010, ApJS, 189, 270 Carliles, S., Budavári, T., Heinis, S., Priebe, C., & Szalay, A. S. 2010, ApJ, 712, 511</list_item>
<list_item><location><page_21><loc_6><loc_22><loc_43><loc_23></location>Cavuoti, S., Amaro, V., Brescia, M., et al. 2016, MNRAS, 465, 1959</list_item>
<list_item><location><page_21><loc_6><loc_21><loc_48><loc_22></location>Chen, C.-T. J., Brandt, W. N., Luo, B., et al. 2018, MNRAS, 478, 2132-2163</list_item>
<list_item><location><page_21><loc_6><loc_19><loc_49><loc_21></location>Cirasuolo, M., Fairley, A., Rees, P., et al. 2020, Published in The Messenger vol. 180, pp. 10-17, June 2020.</list_item>
<list_item><location><page_21><loc_6><loc_18><loc_42><loc_19></location>Co ff ey, D., Salvato, M., Merloni, A., et al. 2019, A&A, 625, A123</list_item>
<list_item><location><page_21><loc_6><loc_14><loc_49><loc_18></location>Cohen, T. & Welling, M. 2016, in Proceedings of Machine Learning Research, Vol. 48, Proceedings of The 33rd International Conference on Machine Learning, ed. M. F. Balcan & K. Q. Weinberger (New York, New York, USA: PMLR), 2990-2999</list_item>
<list_item><location><page_21><loc_6><loc_12><loc_49><loc_14></location>Collaboration:, D. E. S., Abbott, T., Abdalla, F. B., et al. 2016, MNRAS, 460, 1270</list_item>
<list_item><location><page_21><loc_6><loc_10><loc_49><loc_12></location>Collaboration, E., Mellier, Y., Abdurro'uf, et al. 2024, Euclid. I. Overview of the Euclid mission</list_item>
<list_item><location><page_21><loc_51><loc_91><loc_94><loc_93></location>Collister, A. A. & Lahav, O. 2004, Publications of the Astronomical Society of the Pacific, 116, 345</list_item>
</unordered_list>
<text><location><page_21><loc_51><loc_86><loc_94><loc_91></location>Comparat, J., Merloni, A., Dwelly, T., et al. 2020, A&A, 636, A97 Connolly, A. J., Csabai, I., Szalay, A. S., et al. 1995, ApJ, 110, 2655 Croom, S. M., Smith, R. J., Boyle, B. J., et al. 2004, MNRAS, 349, 1397-1418 Dahlen, T., Mobasher, B., Faber, S. M., et al. 2013, ApJ, 775, 93 Dawid, A. P. 1984, Journal of the Royal Statistical Society. Series A (General),</text>
<unordered_list>
<list_item><location><page_21><loc_53><loc_85><loc_57><loc_85></location>147, 278</list_item>
<list_item><location><page_21><loc_51><loc_82><loc_94><loc_84></location>DeJong, R. S., Agertz, O., Berbel, A. A., et al. 2019, Published in The Messenger vol. 175, pp. 3-11, March 2019.</list_item>
<list_item><location><page_21><loc_51><loc_80><loc_94><loc_82></location>DESI-Collaboration, Aghamousa, A., Aguilar, J., et al. 2016, The DESI Experiment Part I: Science,Targeting, and Survey Design</list_item>
<list_item><location><page_21><loc_51><loc_79><loc_85><loc_80></location>Desprez, G., Paltani, S., Coupon, J., et al. 2020, A&A, 644, A31</list_item>
<list_item><location><page_21><loc_51><loc_78><loc_83><loc_79></location>Dey, A., Schlegel, D. J., Lang, D., et al. 2019, ApJ, 157, 168</list_item>
<list_item><location><page_21><loc_51><loc_77><loc_94><loc_78></location>Dey, B., Andrews, B. H., Newman, J. A., et al. 2022a, MNRAS, 515, 5285-5305</list_item>
</unordered_list>
<text><location><page_21><loc_51><loc_76><loc_94><loc_77></location>Dey, B., Newman, J. A., Andrews, B. H., et al. 2022b, Re-calibrating Photometric</text>
<text><location><page_21><loc_53><loc_75><loc_88><loc_76></location>Redshift Probability Distributions Using Feature-space Regression</text>
<text><location><page_21><loc_51><loc_74><loc_80><loc_75></location>D'Isanto, A. & Polsterer, K. L. 2018, A&A, 609, A111</text>
<unordered_list>
<list_item><location><page_21><loc_51><loc_73><loc_90><loc_74></location>Donoso-Oliva, C., Becker, I., Protopapas, P., et al. 2023, A&A, 670, A54</list_item>
<list_item><location><page_21><loc_51><loc_71><loc_94><loc_73></location>Drlica-Wagner, A., Carlin, J. L., Nidever, D. L., Ferguson, P. S., & Kuropatkin, N. 2021, ApJ Supplement Series, 256, 2</list_item>
<list_item><location><page_21><loc_51><loc_69><loc_94><loc_71></location>Duda, R. O., Hart, P. E., & Stork, D. G. 1973, Pattern Classification, 2nd edn. (New York: Wiley)</list_item>
<list_item><location><page_21><loc_51><loc_68><loc_76><loc_69></location>Duncan, K. J. 2022, MNRAS, 512, 3662-3683</list_item>
<list_item><location><page_21><loc_51><loc_67><loc_88><loc_68></location>Dwelly, T., Salvato, M., Merloni, A., et al. 2017, MNRAS, 469, 1065</list_item>
</unordered_list>
<text><location><page_21><loc_51><loc_66><loc_92><loc_66></location>Eriksen, M., Alarcon, A., Cabayol, L., et al. 2020, MNRAS, 497, 4565-4579</text>
<unordered_list>
<list_item><location><page_21><loc_51><loc_64><loc_94><loc_65></location>Fan, X., Banados, E., & Simcoe, R. A. 2022, Quasars and the Intergalactic Medium at Cosmic Dawn</list_item>
<list_item><location><page_21><loc_51><loc_62><loc_79><loc_63></location>Ferrarese, L. & Merritt, D. 2000, ApJ, 539, L9-L12</list_item>
<list_item><location><page_21><loc_51><loc_61><loc_87><loc_62></location>Flaugher, B., Diehl, H. T., Honscheid, K., et al. 2015, ApJ, 150, 150</list_item>
<list_item><location><page_21><loc_51><loc_60><loc_78><loc_61></location>Fotopoulou, S. & Paltani, S. 2018, A&A, 619, A14</list_item>
<list_item><location><page_21><loc_51><loc_59><loc_80><loc_60></location>Fukushima, K. 1980, Biological Cybernetics, 36, 193</list_item>
<list_item><location><page_21><loc_51><loc_58><loc_88><loc_59></location>Gebhardt, K., Bender, R., Bower, G., et al. 2000, ApJ, 539, L13-L16</list_item>
<list_item><location><page_21><loc_51><loc_57><loc_92><loc_58></location>Georgakakis, A., Aird, J., Buchner, J., et al. 2015, MNRAS, 453, 1946-1964</list_item>
<list_item><location><page_21><loc_51><loc_55><loc_94><loc_57></location>Gneiting, T., Raftery, A., Westveld, A., & Goldman, A. 2005, Monthly Weather Review - MON WEATHER REV, 133</list_item>
<list_item><location><page_21><loc_51><loc_53><loc_94><loc_55></location>Gomes, Z., Jarvis, M. J., Almosallam, I. A., & Roberts, S. J. 2017, MNRAS, 475, 331</list_item>
<list_item><location><page_21><loc_51><loc_51><loc_94><loc_53></location>Grimit, E., Gneiting, T., Berrocal, V., & Johnson, N. 2006, Quarterly Journal of the Royal Meteorological Society, 132, 2925</list_item>
<list_item><location><page_21><loc_51><loc_49><loc_94><loc_51></location>Géron, A. 2019, Hands-On Machine Learning with Scikit-Learn, Keras & TensorFlow (O'Reilly, Kiwisoft S.A.S.)</list_item>
<list_item><location><page_21><loc_51><loc_46><loc_94><loc_49></location>Hatfield, P. W., Almosallam, I. A., Jarvis, M. J., et al. 2020, MNRAS, 498, 5498 Hayat, M. A., Stein, G., Harrington, P., Luki'c , Z., & Mustafa, M. 2021, ApJ Letters, 911, L33</list_item>
<list_item><location><page_21><loc_51><loc_43><loc_94><loc_45></location>He, K., Zhang, X., Ren, S., & Sun, J. 2015, Deep Residual Learning for Image Recognition</list_item>
</unordered_list>
<text><location><page_21><loc_51><loc_42><loc_88><loc_43></location>Heckman, T. M. & Best, P. N. 2014, Annual Review of A&A, 52, 589</text>
<unordered_list>
<list_item><location><page_21><loc_51><loc_40><loc_94><loc_42></location>Henghes, B., Thiyagalingam, J., Pettitt, C., Hey, T., & Lahav, O. 2022, MNRAS, 512, 1696</list_item>
<list_item><location><page_21><loc_51><loc_39><loc_75><loc_40></location>Hewett, P. C. & Wild, V. 2010, MNRAS, no</list_item>
<list_item><location><page_21><loc_51><loc_37><loc_94><loc_39></location>Hoyle, B. 2016, Measuring photometric redshifts using galaxy images and Deep Neural Networks</list_item>
<list_item><location><page_21><loc_51><loc_36><loc_84><loc_37></location>Hsu, L.-T., Salvato, M., Nandra, K., et al. 2014, ApJ, 796, 60</list_item>
<list_item><location><page_21><loc_51><loc_34><loc_94><loc_36></location>I. Goodfellow, Y. Bengio, A. C. 2016, Deep Learning (Massachusetts Institute of Technology)</list_item>
<list_item><location><page_21><loc_51><loc_32><loc_91><loc_34></location>Ilbert, O., Arnouts, S., McCracken, H. J., et al. 2006, A&A, 457, 841-856 Illingworth, G. D. 1999, Astrophysics, 269, 165</list_item>
<list_item><location><page_21><loc_51><loc_31><loc_85><loc_32></location>Ivezic, Z., Kahn, S. M., Tyson, J. A., et al. 2019, ApJ, 873, 111</list_item>
<list_item><location><page_21><loc_51><loc_30><loc_85><loc_31></location>Jones, D. M. & Heavens, A. F. 2018, MNRAS, 483, 2487-2505</list_item>
<list_item><location><page_21><loc_51><loc_29><loc_75><loc_30></location>Jones, E. & Singal, J. 2017, A&A, 600, A113</list_item>
<list_item><location><page_21><loc_51><loc_28><loc_81><loc_29></location>Kind, M. C. & Brunner, R. J. 2013, MNRAS, 432, 1483</list_item>
<list_item><location><page_21><loc_51><loc_27><loc_91><loc_27></location>Kingma, D. P. & Ba, J. 2017, Adam: A Method for Stochastic Optimization</list_item>
<list_item><location><page_21><loc_51><loc_23><loc_94><loc_26></location>Kluge, M., Comparat, J., Liu, A., et al. 2024, The First SRG / eROSITA All-Sky Survey: Optical Identification and Properties of Galaxy Clusters and Groups in the Western Galactic Hemisphere</list_item>
<list_item><location><page_21><loc_51><loc_21><loc_94><loc_23></location>Kollmeier, J., Anderson, S. F., Blanc, G. A., et al. 2019, Bulletin of the AAS, 51, https: // baas.aas.org / pub / 2020n7i274</list_item>
<list_item><location><page_21><loc_51><loc_20><loc_86><loc_21></location>Kormendy, J. & Ho, L. C. 2013, Annual Review of A&A, 51, 511</list_item>
<list_item><location><page_21><loc_51><loc_18><loc_94><loc_20></location>Kullback, S. & Leibler, R. A. 1951, The Annals of Mathematical Statistics, 22, 79</list_item>
<list_item><location><page_21><loc_51><loc_16><loc_89><loc_18></location>LaMassa, S. M., Georgakakis, A., Vivek, M., et al. 2019, ApJ, 876, 50 Lang, D. 2014, ApJ, 147, 108</list_item>
<list_item><location><page_21><loc_51><loc_14><loc_94><loc_16></location>Lang, D., Hogg, D. W., & Mykytyn, D. 2016, The Tractor: Probabilistic astronomical source detection and measurement</list_item>
<list_item><location><page_21><loc_51><loc_10><loc_94><loc_14></location>Laurino, O., D'Abrusco, R., Longo, G., & Riccio, G. 2011, MNRAS, 418, 2165 LeCun, Y., Boser, B., Denker, J. S., et al. 1989, Neural Computation, 1, 541 Lecun, Y., Bottou, L., Bengio, Y., & Ha ff ner, P. 1998a, Proceedings of the IEEE, 86, 2278</list_item>
</unordered_list>
<text><location><page_22><loc_6><loc_92><loc_49><loc_93></location>Lecun, Y., Bottou, L., Bengio, Y., & Ha ff ner, P. 1998b, Proceedings of the IEEE,</text>
<text><location><page_22><loc_8><loc_91><loc_13><loc_92></location>86, 2278</text>
<text><location><page_22><loc_6><loc_88><loc_39><loc_91></location>Li, C., Zhang, Y., Cui, C., et al. 2021, MNRAS, 509, 2289 Li, C., Zhang, Y., Cui, C., et al. 2022, MNRAS, 518, 513 Lin, Q., Fouchez, D., Pasquet, J., et al. 2022, A&A, 662, A36</text>
<text><location><page_22><loc_6><loc_85><loc_49><loc_88></location>Lines, N. E. P., Roset, J. F.-Q., & Scaife, A. M. M. 2024, E(2)-Equivariant Features in Machine Learning for Morphological Classification of Radio Galaxies</text>
<text><location><page_22><loc_6><loc_84><loc_43><loc_84></location>Liu, Z., Mao, H., Wu, C.-Y., et al. 2022, in Proceedings of the IEEE</text>
<text><location><page_22><loc_43><loc_84><loc_43><loc_84></location>/</text>
<text><location><page_22><loc_43><loc_84><loc_49><loc_84></location>CVF Con-</text>
<text><location><page_22><loc_8><loc_83><loc_49><loc_83></location>ference on Computer Vision and Pattern Recognition (CVPR), 11976-11986</text>
<text><location><page_22><loc_6><loc_82><loc_49><loc_82></location>Luken, K. J., Norris, R. P., & Park, L. A. F. 2019, Publications of the Astronom-</text>
<text><location><page_22><loc_6><loc_77><loc_49><loc_81></location>ical Society of the Pacific, 131, 108003 Luo, B., Brandt, W. N., Xue, Y. Q., et al. 2010, ApJ Supplement Series, 187, 560 Lyke, B. W., Higley, A. N., McLane, J. N., et al. 2020, ApJ Supplement Series, 250, 8</text>
<unordered_list>
<list_item><location><page_22><loc_6><loc_75><loc_49><loc_77></location>M. Deru, A. N. 2019, Deep Learning mit TensorFlow, Keras und TensorFlow.js. (Rheinwerk Verlag)</list_item>
</unordered_list>
<text><location><page_22><loc_6><loc_73><loc_49><loc_75></location>Ma, L., Lu, Z., Shang, L., & Li, H. 2015, Multimodal Convolutional Neural Networks for Matching Image and Sentence</text>
<text><location><page_22><loc_6><loc_72><loc_45><loc_73></location>Madau, P. & Dickinson, M. 2014, Annual Review of A&A, 52, 415-486</text>
<text><location><page_22><loc_6><loc_71><loc_37><loc_72></location>Mainzer, A., Bauer, J., Grav, T., et al. 2011, ApJ, 731, 53</text>
<unordered_list>
<list_item><location><page_22><loc_6><loc_70><loc_40><loc_71></location>Malz, A. I., Marshall, P. J., DeRose, J., et al. 2018, ApJ, 156, 35</list_item>
</unordered_list>
<text><location><page_22><loc_6><loc_69><loc_37><loc_70></location>Masters, D., Capak, P., Stern, D., et al. 2015, ApJ, 813, 53</text>
<unordered_list>
<list_item><location><page_22><loc_6><loc_68><loc_42><loc_69></location>Masters, D. C., Stern, D. K., Cohen, J. G., et al. 2019, ApJ, 877, 81</list_item>
</unordered_list>
<text><location><page_22><loc_6><loc_66><loc_49><loc_68></location>Mauduit, J.-C., Lacy, M., Farrah, D., et al. 2012, Publications of the Astronomical Society of the Pacific, 124, 714-736</text>
<text><location><page_22><loc_6><loc_65><loc_40><loc_66></location>Meisner, A. M., Lang, D., & Schlegel, D. J. 2017, ApJ, 153, 38</text>
<unordered_list>
<list_item><location><page_22><loc_6><loc_63><loc_49><loc_65></location>Melchior, P., Moolekamp, F., Jerdee, M., et al. 2018, Astronomy and Computing, 24, 129-142</list_item>
<list_item><location><page_22><loc_6><loc_62><loc_38><loc_63></location>Merloni, A., Lamer, G., Liu, T., et al. 2024, A&A, 682, A34</list_item>
</unordered_list>
<text><location><page_22><loc_6><loc_60><loc_49><loc_62></location>Merloni, A., Predehl, P., Becker, W., et al. 2012, eROSITA Science Book: Mapping the Structure of the Energetic Universe</text>
<text><location><page_22><loc_6><loc_58><loc_49><loc_60></location>Meshcheryakov, A., Glazkova, V., Gerasimov, S., & Mashechkin, I. 2018, Astronomy Letters, 44, 735</text>
<text><location><page_22><loc_6><loc_56><loc_45><loc_57></location>Mountrichas, G., Corral, A., Masoura, V. A., et al. 2017, A&A, 608, A39 Newman, J. A. & Gruen, D. 2022, Annual Review of A&A, 60</text>
<text><location><page_22><loc_6><loc_51><loc_49><loc_55></location>Ni, Q., Brandt, W. N., Chen, C.-T., et al. 2021, ApJ Supplement Series, 256, 21 Nishizawa, A. J., Hsieh, B.-C., Tanaka, M., & Takata, T. 2020, Photometric Redshifts for the Hyper Suprime-Cam Subaru Strategic Program Data Release 2</text>
<unordered_list>
<list_item><location><page_22><loc_6><loc_49><loc_49><loc_51></location>Norris, R. P., Salvato, M., Longo, G., et al. 2019, Publications of the Astronomical Society of the Pacific, 131, 108004</list_item>
</unordered_list>
<text><location><page_22><loc_6><loc_46><loc_49><loc_49></location>O'Shea, K. & Nash, R. 2015, An Introduction to Convolutional Neural Networks Padovani, P., Alexander, D. M., Assef, R. J., et al. 2017, The A&A Review, 25 Parker, L., Lanusse, F., Golkar, S., et al. 2024, MNRAS, 531, 4990-5011</text>
<unordered_list>
<list_item><location><page_22><loc_6><loc_44><loc_49><loc_46></location>Pasquet, J., Bertin, E., Treyer, M., Arnouts, S., & Fouchez, D. 2018, A&A, 621, A26</list_item>
</unordered_list>
<text><location><page_22><loc_6><loc_43><loc_40><loc_44></location>Pierce, C. M., Lotz, J. M., Primack, J. R., et al. 2010, MNRAS</text>
<text><location><page_22><loc_6><loc_41><loc_48><loc_43></location>Povi'c , M., Sánchez-Portal, M., García, A. M. P., et al. 2012, A&A, 541, A118 Predehl, P., Andritschke, R., Arefiev, V., et al. 2021, A&A, 647, A1</text>
<unordered_list>
<list_item><location><page_22><loc_6><loc_39><loc_49><loc_41></location>Rau, M. M., Seitz, S., Brimioulle, F., et al. 2015, Accurate photometric redshift probability density estimation - method comparison and application</list_item>
</unordered_list>
<text><location><page_22><loc_6><loc_38><loc_49><loc_39></location>Revsbech, E. A., Trotta, R., & van Dyk, D. A. 2017, MNRAS, 473, 3969-3986</text>
<text><location><page_22><loc_6><loc_37><loc_49><loc_38></location>Rosenbaum, P. & Rubin, D. 1984, Journal of the American Statistical Associa-</text>
<text><location><page_22><loc_8><loc_36><loc_12><loc_37></location>tion, 79</text>
<unordered_list>
<list_item><location><page_22><loc_6><loc_34><loc_49><loc_36></location>Rosenblatt, F. 1958, The perceptron: A probabilistic model for information storage and organization in the brain</list_item>
<list_item><location><page_22><loc_6><loc_32><loc_49><loc_34></location>Ruiz, A., Corral, A., Mountrichas, G., & Georgantopoulos, I. 2018, A&A, 618, A52</list_item>
<list_item><location><page_22><loc_6><loc_30><loc_49><loc_31></location>Sadeh, I., Abdalla, F. B., & Lahav, O. 2016, Publications of the Astronomical Society of the Pacific, 128, 104502</list_item>
<list_item><location><page_22><loc_6><loc_28><loc_40><loc_29></location>Salvato, M., Hasinger, G., Ilbert, O., et al. 2008, ApJ, 690, 1250</list_item>
<list_item><location><page_22><loc_6><loc_27><loc_39><loc_28></location>Salvato, M., Ilbert, O., Hasinger, G., et al. 2011, ApJ, 742, 61</list_item>
<list_item><location><page_22><loc_6><loc_25><loc_49><loc_27></location>Salvato, M., Ilbert, O., & Hoyle, B. 2018, The many flavours of photometric redshifts</list_item>
<list_item><location><page_22><loc_6><loc_24><loc_38><loc_25></location>Salvato, M., Wolf, J., Dwelly, T., et al. 2022, A&A, 661, A3</list_item>
<list_item><location><page_22><loc_6><loc_23><loc_42><loc_24></location>Sánchez, C., Kind, M. C., Lin, H., et al. 2014, MNRAS, 445, 1482</list_item>
<list_item><location><page_22><loc_6><loc_21><loc_49><loc_23></location>Sanchez, J., Mendoza, I., Kirkby, D. P., & Burchat, P. R. 2021, Journal of Cosmology and Astroparticle Physics, 2021, 043</list_item>
<list_item><location><page_22><loc_6><loc_19><loc_49><loc_21></location>Savi'c, D. V., Jankov, I., Yu, W., et al. 2023, The LSST AGN Data Challenge: Selection methods</list_item>
</unordered_list>
<text><location><page_22><loc_6><loc_16><loc_49><loc_19></location>Saxena, A., Salvato, M., Roster, W., et al. 2024, CircleZ: Reliable Photometric redshifts for AGN computed using only photometry from Legacy Survey Imaging for DESI</text>
<text><location><page_22><loc_6><loc_15><loc_43><loc_16></location>Scaramella, R., Amiaux, J., Mellier, Y., et al. 2022, A&A, 662, A112</text>
<text><location><page_22><loc_6><loc_14><loc_45><loc_15></location>Schmidt, S. J., Malz, A. I., Soo, J. Y. H., et al. 2020, MNRAS, 499, 1587</text>
<text><location><page_22><loc_6><loc_13><loc_43><loc_14></location>Schuldt, S., Suyu, S. H., Cañ ameras, R., et al. 2021, A&A, 651, A55</text>
<unordered_list>
<list_item><location><page_22><loc_6><loc_10><loc_49><loc_13></location>Scolnic, D. M., Lochner, M., Gris, P., et al. 2018, Optimizing the LSST Observing Strategy for Dark Energy Science: DESC Recommendations for the Deep Drilling Fields and other Special Programs</list_item>
<list_item><location><page_22><loc_51><loc_90><loc_94><loc_93></location>Silva, D. R., Blum, R. D., Allen, L., et al. 2016, in American Astronomical Society Meeting Abstracts, Vol. 228, American Astronomical Society Meeting Abstracts #228, 317.02</list_item>
</unordered_list>
<text><location><page_22><loc_51><loc_87><loc_93><loc_90></location>Simm, T., Saglia, R., Salvato, M., et al. 2015, A&A, 584, A106 Soo, J. Y. H., Moraes, B., Joachimi, B., et al. 2017, MNRAS, 475, 3613-3632 Stabenau, H. F., Connolly, A., & Jain, B. 2008, MNRAS, 387, 1215</text>
<unordered_list>
<list_item><location><page_22><loc_51><loc_85><loc_94><loc_86></location>Steidel, C. C., Giavalisco, M., Dickinson, M., & Adelberger, K. L. 1996, ApJ, 112, 352</list_item>
<list_item><location><page_22><loc_51><loc_81><loc_94><loc_84></location>Tamura, N., Takato, N., Shimono, A., et al. 2016, in Ground-based and Airborne Instrumentation for Astronomy VI, ed. C. J. Evans, L. Simard, & H. Takami (SPIE)</list_item>
<list_item><location><page_22><loc_51><loc_79><loc_94><loc_81></location>Treyer, M., Ait-Ouahmed, R., Pasquet, J., et al. 2023, CNN photometric redshifts in the SDSS at r ≤ 20</list_item>
<list_item><location><page_22><loc_51><loc_78><loc_80><loc_79></location>Véron-Cetty, M. P. & Véron, P. 2010, A&A, 518, A10</list_item>
<list_item><location><page_22><loc_51><loc_77><loc_88><loc_78></location>Viroli, C. & McLachlan, G. J. 2017, Deep Gaussian Mixture Models</list_item>
<list_item><location><page_22><loc_51><loc_75><loc_94><loc_77></location>Voges, W., Aschenbach, B., Boller, T., et al. 1999, The ROSAT All-Sky Survey Bright Source Catalogue</list_item>
</unordered_list>
<text><location><page_22><loc_51><loc_74><loc_87><loc_75></location>Webb, N. A., Coriat, M., Traulsen, I., et al. 2020, A&A, 641, A136</text>
<text><location><page_22><loc_51><loc_73><loc_71><loc_74></location>Weiler, M. & Cesa, G. 2021, General</text>
<text><location><page_22><loc_71><loc_73><loc_72><loc_74></location>E</text>
<text><location><page_22><loc_72><loc_73><loc_89><loc_74></location>(2)-Equivariant Steerable CNNs</text>
<text><location><page_22><loc_51><loc_72><loc_89><loc_73></location>Wilson, D., Nayyeri, H., Cooray, A., & Häußler, B. 2020, ApJ, 888, 83</text>
<unordered_list>
<list_item><location><page_22><loc_51><loc_70><loc_94><loc_72></location>Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, ApJ, 140, 1868-1881</list_item>
</unordered_list>
<text><location><page_22><loc_51><loc_69><loc_82><loc_69></location>Wu, Q. & Shen, Y. 2022, ApJ Supplement Series, 263, 42</text>
<unordered_list>
<list_item><location><page_22><loc_51><loc_67><loc_93><loc_68></location>Wuyts, S., Förster Schreiber, N. M., van der Wel, A., et al. 2011, ApJ, 742, 96</list_item>
</unordered_list>
<text><location><page_22><loc_51><loc_66><loc_90><loc_67></location>Yang, J., Fan, X., Gupta, A., et al. 2023, ApJ Supplement Series, 269, 27</text>
<text><location><page_22><loc_51><loc_65><loc_91><loc_66></location>York, D. G., Adelman, J., John E. Anderson, J., et al. 2000, ApJ, 120, 1579</text>
<unordered_list>
<list_item><location><page_22><loc_51><loc_63><loc_90><loc_65></location>Zhang, Y., Ma, H., Peng, N., Zhao, Y., & bing Wu, X. 2013, ApJ, 146, 22 Zhang, Y. & Zhao, Y. 2015, Data Science Journal</list_item>
<list_item><location><page_22><loc_51><loc_61><loc_94><loc_63></location>Zhao, D., Dalmasso, N., Izbicki, R., & Lee, A. B. 2021, Diagnostics for Conditional Density Models and Bayesian Inference Algorithms</list_item>
<list_item><location><page_22><loc_51><loc_59><loc_94><loc_61></location>Zhou, R., Ferraro, S., White, M., et al. 2023, J. Cosmology Astropart. Phys., 2023, 097</list_item>
<list_item><location><page_22><loc_51><loc_57><loc_89><loc_59></location>Zhou, R., Newman, J. A., Mao, Y.-Y., et al. 2021, MNRAS, 501, 3309 Zou, H., Zhang, T., Zhou, Z., et al. 2017, ApJ, 153</list_item>
<list_item><location><page_22><loc_52><loc_52><loc_94><loc_54></location>1 Max-Planck-Institut für extraterrestrische Physik, Giessenbachstr. 1, 85748 Garching, Germany</list_item>
<list_item><location><page_22><loc_52><loc_50><loc_94><loc_52></location>2 Exzellenzcluster ORIGINS, Boltzmannstr. 2, D-85748 Garching, Germany</list_item>
<list_item><location><page_22><loc_52><loc_47><loc_94><loc_49></location>3 LMU Munich, Arnold Sommerfeld Center for Theoretical Physics, Theresienstr. 37 80333 München Germany</list_item>
<list_item><location><page_22><loc_52><loc_45><loc_94><loc_47></location>4 LMU Munich, Universitäts-Sternwarte, Scheinerstr. 1, 81679 München, Germany</list_item>
<list_item><location><page_22><loc_52><loc_42><loc_94><loc_45></location>5 Technical University of Munich Department of Computer Science I26 Boltzmannstr. 3 85748 Garching b. München Germany</list_item>
<list_item><location><page_22><loc_52><loc_40><loc_94><loc_42></location>6 Max Planck Institut für Astronomie, Königstuhl 17, D-69117, Heidelberg, Germany</list_item>
<list_item><location><page_22><loc_52><loc_37><loc_94><loc_40></location>7 Instituto de Astrofísica, Facultad de Física, Pontificia Universidad Católica de Chile, Campus San Joaquín, Av. Vicuña Mackenna 4860, Macul Santiago, Chile, 7820436</list_item>
<list_item><location><page_22><loc_52><loc_33><loc_94><loc_36></location>8 Centro de Astroingeniería, Facultad de Física, Pontificia Universidad Católica de Chile, Campus San Joaquín, Av. Vicuña Mackenna 4860, Macul Santiago, Chile, 7820436</list_item>
<list_item><location><page_22><loc_52><loc_31><loc_94><loc_33></location>9 Millennium Institute of Astrophysics, Nuncio Monseñor Sótero Sanz 100, Of 104, Providencia, Santiago, Chile</list_item>
<list_item><location><page_22><loc_51><loc_28><loc_94><loc_31></location>10 Space Science Institute, 4750 Walnut Street, Suite 205, Boulder, Colorado 80301</list_item>
<list_item><location><page_22><loc_51><loc_26><loc_94><loc_28></location>11 Institute for Astronomy, University of Edinburgh, Royal Observatory, Edinburgh EH9 3HJ, UK</list_item>
<list_item><location><page_22><loc_51><loc_24><loc_94><loc_26></location>12 Instituto de Estudios Astrofísicos, Universidad Diego Portales, Av. Ejército Libertador 441, Santiago 8370191, Chile</list_item>
<list_item><location><page_22><loc_51><loc_21><loc_94><loc_23></location>13 Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China</list_item>
<list_item><location><page_22><loc_51><loc_19><loc_94><loc_21></location>14 Department of Astronomy, University of Washington, Box 351580, Seattle, WA, 98195, USA</list_item>
<list_item><location><page_22><loc_51><loc_16><loc_94><loc_19></location>15 Department of Astronomy, University of Illinois at UrbanaChampaign, Urbana, IL 61801, USA</list_item>
<list_item><location><page_22><loc_51><loc_14><loc_94><loc_16></location>16 National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA</list_item>
<list_item><location><page_22><loc_51><loc_10><loc_94><loc_14></location>17 Center for Artificial Intelligence Innovation, University of Illinois at Urbana-Champaign, 1205 West Clark Street, Urbana, IL 61801, USA</list_item>
</unordered_list>
<text><location><page_23><loc_59><loc_94><loc_65><loc_95></location>02.09.24, 21</text>
<text><location><page_23><loc_65><loc_94><loc_65><loc_95></location>:</text>
<text><location><page_23><loc_65><loc_94><loc_67><loc_95></location>39</text>
<section_header_level_1><location><page_23><loc_51><loc_92><loc_90><loc_93></location>Appendix B: LS10 seeing for different samples</section_header_level_1>
<figure>
<location><page_23><loc_51><loc_67><loc_93><loc_89></location>
<caption>Fig. B.1: Histograms of the weighted average FWHM (seeing) values for the PICZL training, validation, and CSC2 blind samples. The close alignment of these distributions indicates no significant mismatch between the datasets, as they represent all-sky sampling. This similarity supports the applicability of the training sample to blind samples of comparable seeing.</caption>
</figure>
<section_header_level_1><location><page_23><loc_51><loc_52><loc_84><loc_53></location>Appendix C: CSC2 blind sample results</section_header_level_1>
<figure>
<location><page_23><loc_51><loc_22><loc_93><loc_50></location>
<caption>Page 1 of 2 Fig. C.1: Scatter plot of the CSC2 sample color-coded by point density, illustrating the performance of PICZL on previously unseen data, as the majority of sources lie within the defined outlier boundary, demonstrating the prediction robustness. Sources identified as outliers generally scatter close to the boundary, with only a few instances of significant deviations or catastrophic failures.</caption>
</figure>
<section_header_level_1><location><page_23><loc_6><loc_92><loc_37><loc_93></location>Appendix A: CNN model architecture</section_header_level_1>
<figure>
<location><page_23><loc_19><loc_23><loc_36><loc_89></location>
<caption>Fig. A.1: PICZL architecture, showcasing its three-threaded design. Two threads are dedicated to processing image inputs, while the third handles numerical data. The model's output layer generates distinct vectors corresponding to the means, standard deviations, and weights of multiple Gaussian distributions. This design enables the production of full PDFs, allowing the network to capture uncertainties. Question marks as the first dimension per thread input correspond to the (variable) sample size.</caption>
</figure>
<section_header_level_1><location><page_24><loc_6><loc_92><loc_39><loc_93></location>Appendix D: Photometric errors in LS10</section_header_level_1>
<section_header_level_1><location><page_24><loc_51><loc_90><loc_88><loc_93></location>Appendix F: Column description of released photo-z catalog</section_header_level_1>
<figure>
<location><page_24><loc_6><loc_67><loc_49><loc_89></location>
<caption>Fig. D.1: Relationship between S / N and the associated photometric error for the LS10 i -band, considering all sources from Table 1. Each point is color-coded according to the i -band magnitude. A noticeable turning point is observed for sources with S / N ≲ 15, indicating a significant increase in magnitude error for sources fainter than ∼ 22 magnitude.</caption>
</figure>
<section_header_level_1><location><page_24><loc_6><loc_51><loc_41><loc_52></location>Appendix E: XMM-SERVS sources in LS10</section_header_level_1>
<figure>
<location><page_24><loc_7><loc_26><loc_48><loc_48></location>
<caption>Fig. E.1: Histograms of LS10 i -band magnitudes for the three XMM-SERVS subsamples: ELAIS-S1, W-CDF-S, and LSS. The magnitudes are displayed as normalized densities to allow for direct comparison across the fields. These distributions extend to fainter magnitudes than those shown in Figure 15, accounting for the increased η P denoted in Table 4. As the depth reached in PICZL's LS10 training set is limited toward brighter magnitudes, a direct comparison between subsamples is only valid when conducted within a controlled parameter space, as shown in Figure 13.</caption>
</figure>
<unordered_list>
<list_item><location><page_24><loc_51><loc_87><loc_94><loc_89></location>1. XID: Unique source ID assigned to each X-ray source in the original papers (see Chen et al. 2018; Ni et al. 2021)</list_item>
<list_item><location><page_24><loc_51><loc_86><loc_92><loc_87></location>2. SURVEY: Original survey where the source was detected</list_item>
<list_item><location><page_24><loc_51><loc_82><loc_94><loc_85></location>3. LS10_FULLID: Unique LS source ID assigned to optical counterpart. It is created by concatenating the LS coloumns RELEASE, BRICKID and OBJID.</list_item>
<list_item><location><page_24><loc_51><loc_79><loc_94><loc_82></location>4. CTP_LS10_RA: Right Ascension in degrees of the LS10 optical counterpart</list_item>
<list_item><location><page_24><loc_51><loc_77><loc_94><loc_79></location>5. CTP_LS10_DEC: Declination in degrees of the LS10 optical counterpart</list_item>
<list_item><location><page_24><loc_51><loc_75><loc_78><loc_76></location>6. SPECZ: specz from original catalog</list_item>
<list_item><location><page_24><loc_51><loc_74><loc_77><loc_75></location>7. PHZ_PICZL: Photo-z from PICZL</list_item>
<list_item><location><page_24><loc_51><loc_73><loc_86><loc_74></location>8. PHZ_PICZL_l68: PICZL photo-z min at 1 sigma</list_item>
<list_item><location><page_24><loc_51><loc_71><loc_87><loc_72></location>9. PHZ_PICZL_u68: PICZL photo-z max at 1 sigma</list_item>
<list_item><location><page_24><loc_51><loc_70><loc_86><loc_71></location>10. PHZ_PICZL_l99: PICZL photo-z min at 3 sigma</list_item>
<list_item><location><page_24><loc_51><loc_69><loc_87><loc_70></location>11. PHZ_PICZL_u99: PICZL photo-z max at 3 sigma</list_item>
</unordered_list>
<section_header_level_1><location><page_24><loc_51><loc_65><loc_74><loc_66></location>Appendix G: PICZL run time</section_header_level_1>
<text><location><page_24><loc_51><loc_16><loc_94><loc_64></location>The computational demands for training and deploying PICZL depend mostly on the hardware configuration and dataset size. For our experiments, we leveraged a set of two Tesla V100PCIE graphics processing units (GPUs) each equipped with 32 GB of ready access memory (RAM), accompanied by a 48 core multi-thread CPU to accelerate the computational workload. This parallel processing capability significantly expedited the model training process, compared to running solely on a central CPU, allowing for faster convergence. Training a single model (refer to Figure A.1) on a dataset of 32 391 sources, corresponding to the 80:20 train-test split (refer to Table 1), each with 108 features (56 of which are images), takes approximately 25 minutes for 600 epochs. The use of an ensemble further scales the processing time by the number of models trained, with ensemble optimization depending on user preferences, making it di ffi -cult to provide a fixed time estimate. After finalizing the model, computing photo-z for e.g. 1393 sources in the XMM-SERVS W-CDF-S field, as displayed via Table 4, takes roughly 17 seconds. The storage requirements for the data used in this sample corresponds to approximately 250 MB (refer to Table 2). Looking ahead, upcoming surveys such as Euclid and LSST will produce higher-resolution images, demanding increased storage and computational resources due to larger data volumes and pixel counts. If the input dimensions were to increase, for instance, from 23x23 pixels to 64x64 pixels, the computational burden would rise significantly. While our current architecture of 32 GB RAM per GPU is e ffi cient for the current input sizes, processing larger cutouts may necessitate either more GPUs or GPUs with higher memory capacity to maintain feasible training times and performance. In scenarios where larger input sizes are anticipated, a thoughtful approach to model architecture and resource allocation will be crucial. Therefore, adapting to the capabilities of the hardware will be key to successfully utilizing the methods in the context of future data from LSST and Euclid, particularly given the much larger sample sizes that we expect from these surveys. Finally, performance will also improve by having PICZL revised by an expert software developer.</text>
</document>
[
{
"title": "ABSTRACT",
"content": "Context. Computing reliable photometric redshifts (photo-z) for active galactic nuclei (AGN) is a challenging task, primarily due to the complex interplay between the unresolved relative emissions associated with the supermassive black hole and its host galaxy. Spectral energy distribution (SED) fitting methods, while e ff ective for galaxies and AGN in pencil-beam surveys, face limitations in wide or all-sky surveys with fewer bands available, lacking the ability to accurately capture the AGN contribution to the SED, hindering reliable redshift estimation. This limitation is a ff ecting the many 10s of millions of AGN detected in existing datasets, e.g., those AGN clearly singled out and identified by SRG / eROSITA. Aims. Our goal is to enhance photometric redshift performance for AGN in all-sky surveys while simultaneously simplifying the approach by avoiding the need to merge multiple data sets. Instead, we employ readily available data products from the 10th Data Release of the Imaging Legacy Survey for the Dark Energy Spectroscopic Instrument, which covers > 20,000 deg 2 of extragalactic sky with deep imaging and catalogbased photometry in the grizW1-W4 bands. We fully utilize the spatial flux distribution in the vicinity of each source to produce reliable photo-z. Methods. We introduce PICZL, a machine-learning algorithm leveraging an ensemble of convolutional neural networks. Utilizing a cross-channel approach, the algorithm integrates distinct SED features from images with those obtained from catalog-level data. Full probability distributions are achieved via the integration of Gaussian mixture models. Results. On a validation sample of 8098 AGN, PICZL achieves an accuracy σ NMAD of 4.5% with an outlier fraction η of 5.6%. These results significantly outperform previous attempts to compute accurate photo-z for AGN using machine learning. We highlight that the model's performance depends on many variables, predominantly the depth of the data and associated photometric error. A thorough evaluation of these dependencies is presented in the paper. Conclusions. Our streamlined methodology maintains consistent performance across the entire survey area, when accounting for di ff ering data quality. The same approach can be adopted for future deep photometric surveys such as LSST and Euclid, showcasing its potential for wide-scale realization. With this paper, we release updated photo-z (including errors) for the XMM-SERVS W-CDF-S, ELAIS-S1 and LSS fields. Key words. Photo-z, AGN, Extragalactic Surveys, Machine Learning",
"pages": [
1
]
},
{
"title": "PICZL : Image-based photometric redshifts for AGN",
"content": "William Roster , ⋆ 1 , M. Salvato 1 , 2 , S. Krippendorf 3 , 4 , A. Saxena 5 , R. Shirley 1 , J. Buchner 1 , J. Wolf 2 , 6 , T. Dwelly 1 , F. E. Bauer 7 , 8 , 9 , 10 , J. Aird 11 , C. Ricci 12 , 13 , R. J. Assef 12 , S.F. Anderson 14 , X. Liu 15 , 16 , 17 , A. Merloni 1 , J. Weller 1 , 4 K. Nandra 1 (A ffi liations can be found after the references) November 14, 2024",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "In recent decades, our understanding of Active Galactic Nuclei (AGN) and their role in galaxy and cosmic evolution has significantly advanced. These luminous celestial powerhouses are thought to be fueled by the accretion of matter onto supermassive black holes (SMBHs) located at the centers of galaxies, exerting intense energetic radiation across the entire electromagnetic spectrum, ranging from radio to γ -rays (Padovani et al. 2017). The close correlation observed between the mass of the central SMBH, whether active or inactive and the properties of its host galaxy's bulge - such as the galaxy's mass and velocity dispersion (e.g., Gebhardt et al. 2000; Ferrarese & Merritt 2000) - suggests a co-evolutionary relationship between galaxies and their central engines (Kormendy & Ho 2013; Heckman & Best 2014). Ongoing research focuses on understanding scaling relations, the evolution of SMBHs within galaxies, and the interconnected rates of star formation (SFR) and black hole accretion (BHAR) over cosmic time (e.g., Madau & Dickinson 2014). To further explore and address these unresolved topics requires diverse AGN samples with reliable redshifts to determine BH demographics and constrain models of galaxy evolution. For all these studies, redshift is an indispensable quantity, with spectroscopic redshifts (spec-z) remaining the preferred es- imates for determining precise cosmic distances (Hoyle 2016). However, while multi-objects spectrographs, such as the Sloan Digital Sky Survey (SDSS-V; York et al. 2000; Kollmeier et al. 2019), the Dark Energy Spectroscopic Instrument (DESI; DESICollaboration et al. 2016), the Subaru Prime Focus Spectrograph (PFS; Tamura et al. 2016) or the 4-metre Multi-Object Spectroscopic Telescope (4MOST; De Jong et al. 2019), are set to provide a drastic rise in the number of observed sources over the next several years, we are currently in the situation in which millions of AGN have been detected all-sky by various surveys (e.g., by the Wide-field Infrared Survey Explorer mission (WISE; Wright et al. 2010), and the extended Roentgen Survey with an Imaging Telescope Array (eROSITA; Merloni et al. 2012; Predehl et al. 2021), with only the brightest sources having been observed spectroscopically (Dahlen et al. 2013). The growing disparity between photometric and spectroscopic observations will only widen with upcoming surveys such as the Legacy Survey of Space and Time (LSST; Ivezic et al. 2019) and Euclid (Collaboration et al. 2024), covering unprecedented areas and depths (Newman & Gruen 2022). Thus for the bulk of AGN, we must make use of multiband photometry and rely on photometric redshifts (photo-z). First implemented by Baum (1957) for inactive galaxies, these low-precision redshift estimates utilize photometric observations to e ff ectively obtain a sparsely sampled spectral energy distribution (SED), trading precision for scalability. They en- compass an array of techniques assuming color-redshift evolution (Connolly et al. 1995; Steidel et al. 1996; Illingworth 1999; Bell et al. 2004), including template-based approaches (e.g., Bolzonella et al. 2000; Ilbert et al. 2006; Salvato et al. 2008; Beck et al. 2017), where redshifted models built on theoretical or empirical SEDs are fitted to observed multi-band photometry. Although a limited number of available bands can introduce uncertainties (see review by Salvato et al. 2018), photo-z methods o ff er an e ffi cient way to estimate distances for all sources in an imaging survey, yielding highly accurate estimates with as few as three bands for passive galaxies (Benitez 2000; Abdalla et al. 2011; Arnouts & Ilbert 2011; Brescia et al. 2014; Desprez et al. 2020). By contrast, reliable photo-z for AGN have historically required highly homogenized photometry across > 20 filters, which was only achievable in pencil-beam surveys (Salvato et al. 2011). As such, this level of detail continues to be unfeasible for wide-area surveys. However, with the 10th data release of the DESI Legacy Imaging Surveys (LS10, Dey et al. 2019), we now have a broad-sky survey that, while lacking NIR coverage, includes a few optical bands supplemented by mid-IR WISE data. This allows us to explore the possibility of generating reliable photo-z for AGN over the full sky, despite having fewer filters compared to the densely sampled pencil-beam surveys. SED fitting applied to a broad population of AGN remains particularly challenging due to the uncertainty and di ffi culty of disentangling the relative contributions of the nucleus and respective host to a given band (e.g., Luo et al. 2010; Salvato et al. 2011; Brescia et al. 2019). Since the accretion properties of SMBHs, often characterized as the bolometric luminosity divided by the Eddington limit, or the Eddington ratio, significantly influence the SED of AGN, the intense power-law continuum radiation can either partly (host-dominated) or entirely (quasar-dominated), outshine the respective host, hiding key spectral features that lead to redshift degeneracies (Pierce et al. 2010; Povi'c et al. 2012; Bettoni et al. 2015). Consequently, selecting a limited number of templates can be insufficient for correct redshift determination, while increasing the number of templates raises the degeneracy (see discussion in Salvato et al. 2011; Ananna et al. 2017). In this regime of accounting for AGN contributions to galaxy photo-z, one potential approach involves modeling objects as a combination of quasar and galaxy templates (eg., Cardamone et al. 2010), performed with EAZY (Brammer et al. 2008). In addition, surveys typically estimate fluxes with models that do not account for a mixed contribution from AGN and host galaxy. Ultimately, AGN are also intrinsically variable sources on the timescales explored by the previously mentioned surveys leading to incongruent photometry acquired across di ff erent epochs. In contrast to template-fitting methods, more recent approaches have shifted towards the use of empirical Machine Learning (ML) models, performing regression or classification, to tackle photo-z applied predominantly to inactive galaxies (Collister & Lahav 2004; Laurino et al. 2011; Zhang et al. 2013; Hoyle 2016; D'Isanto & Polsterer 2018; Brescia et al. 2019; Eriksen et al. 2020; Li et al. 2021). Provided with a very large and complete spec-z sample, ML architectures manipulate photometric input features to minimize the divergence between spectroscopic and ML-derived redshifts. Over the years, a plethora of ML architectures, including decision trees (Breiman 2001; Carliles et al. 2010; Li et al. 2022), Gaussian processes (Almosallam et al. 2016) and K-nearest neighbours (Zhang et al. 2013; Luken et al. 2019) have been employed, yielding accurate point predictions and, more interestingly, full probability density functions (PDFs) (Kind & Brunner 2013; Cavuoti et al. 2016; Rau et al. 2015; Sadeh et al. 2016). The latter grants access to the prediction uncertainty, as otherwise naturally provided by template-fitting approaches, relevant for studies dealing with, e.g. luminosity functions (Aird et al. 2010; Buchner et al. 2015; Georgakakis et al. 2015). However, the limited availability of a sizable training sample of AGN has resulted in only a few attempts to compute photo-z for mostly nucleus-dominated objects with ML-based methods (Mountrichas et al. 2017; Fotopoulou & Paltani 2018; Ruiz et al. 2018; Meshcheryakov et al. 2018; Brescia et al. 2019; Nishizawa et al. 2020). More recently, the conventional approach of manually selecting photometric features for ML has been replaced by bright, well-resolved galaxies at low redshift (Hoyle 2016; Pasquet et al. 2018; Campagne 2020; Hayat et al. 2021). In this regime, integrating galaxy images into deep neural networks inherently captures essential details like flux, morphology, and other features that would typically be extracted from catalogs based on predefined assumptions, leading to a more comprehensive redshift estimation process. This approach is particularly advantageous for addressing current limitations faced by photo-z methods for AGN, as it leverages model-independent fluxes and redshift indicative features, including surface brightness profiles (Stabenau et al. 2008; Jones & Singal 2017; Gomes et al. 2017; Zhou et al. 2021, 2023). Unlike creating a single SED from total flux measurements, projects employing images with independent pixelby-pixel SEDs at identical redshift have demonstrated increased photo-z constraining power, alleviating previous empirical approaches by decreasing the fraction of outliers (Henghes et al. 2022; Schuldt et al. 2021; Lin et al. 2022; Dey et al. 2022a; Newman & Gruen 2022). Here, we introduce PICZL (Photometrically Inferred CNN redshift(Z) Likelihoods), an enhanced approach to photoz estimation that builds upon (C ircle Z by Saxena et al. 2024). While the authors demonstrated that redshift degeneracies encountered for AGN, typical in cases of limited photometry, can be broken by integrating aperture photometry alongside traditional total / model fluxes and colors, PICZL instead computes photo-z PDFs for AGN directly from homogenized flux band cutouts by leveraging the more detailed spatial light profile. All inputs are obtained utilizing LS10 exclusively. Similar to Saxena et al. (2024), PICZL can produce reliable photo-z PDFs for all Legacy-detected sources associated with an AGN. However, the model can, in principle, be applied to other extragalactic sources (e.g, inactive galaxies, Götzenberger et al. in prep.) granted that a dedicated training sample is used. We employ an ensemble of the same ML algorithm, notably convolutional neural networks (CNNs), known for their proficiency in learning intricate patterns, as outlined by (Lecun et al. 1998a). Specifically designed for image analysis, CNNs excel at identifying and extracting relevant predictive features directly from images, thereby reducing computational overhead compared to fully connected architectures. Harnessing this more extensive pool of information, these models surpass alternative models based on condensed feature-based input sets. The paper is structured as follows: Sect. 2 introduces the AGN training sample down-selection. Sect. 3 focuses on the photometric data products available within LS10. Sect. 4 details the photometric data preprocessing, followed by Sect. 5, which outlines the model pipeline. Sect. 6 presents and quantifies the redshift results, while Sect. 7 evaluates the photo-z released for the XMM-SERVS (Chen et al. 2018; Ni et al. 2021) fields. Sect. 8 outlines current limitations and discusses how we can achieve further improvements. Sect. 9 explores implications for future surveys, concluding with a summary. In this paper, unless stated di ff erently, we express magnitudes in the AB system and adopt a Λ CDM cosmology with H 0 = 69 . 8 km s -1 Mpc -1 , Ω m = 0 . 28 and Λ = 0 . 72.",
"pages": [
1,
2,
3
]
},
{
"title": "2. AGN training sample selection",
"content": "In X-ray surveys, the identification of AGN has two distinct advantages - i) the reduced impact of moderate obscuration and ii) the lack of host dilution. Due to the inherent brightness of accreting SMBHs compared to their host galaxies, this results in a significantly higher nuclei-to-host emission ratio compared to observations in some of the neighbouring wavelength windows, such as UV-optical-NIR (Padovani et al. 2017). This naturally leads to a larger diversity of AGN observed by an X-ray telescope. That being said, surveys in the more accessible optical and NIR regime can increase the likelihood of detecting higherz , and in the case of MIR more heavily obscured AGN, compared to the soft X-ray bands. MLapproaches for photoz estimation in large surveys (e.g., Fotopoulou & Paltani 2018; Duncan 2022) typically classify objects into three broad categories: galaxies, quasars (QSOs), and stars, before computing photo-z. However, this classification is usually based on the optical properties and hence fails for obscured and / or lower-luminosity AGN. Since our goal is to improve on the quality of photo-z estimates for X-ray detected extragalactic sources, including type 2 AGN and low-redshift Seyfert 1 galaxies, generally, our training sample has to replicate this diversity. We achieve this by combining AGN selected across multiple wavelength bands. As a starting point, we include the same X-ray samples used in Saxena et al. (2024), namely the latest version of the XMM catalog, 4XMM, which spans 19 years of observations made with XMM-Newton (Webb et al. 2020) and data from the eROSITA CalPV-phase Final Equatorial-Depth Survey (eFEDS; Brunner et al. 2022), as they provide a reasonably representative and complete set of diverse AGN spanning 5 dex in X-ray flux out to redshift z ≲ 4. However, with just these, some portions of AGNparameter space remain imbalanced, such as highly luminious and / or high-z AGN. Thus, we expand the dataset by adding bright, optical and MIR, selections. We describe each of these samples in more detail below. This approach enhances the completeness of our training sample, which is essential to mitigate covariate shift, i.e., the shift in parameter space between the training and validation samples, so that the model generalizes e ff ectively to new data (Norris et al. 2019). Subsequently, a non-representative training sample may lead to systematically biased outcomes (Newman & Gruen 2022). Accordingly, algorithms will be strongly weighted towards the most densely populated regions of the training space (Duncan 2022).",
"pages": [
3
]
},
{
"title": "2.1. Beyond eFEDS and 4XMM",
"content": "We can enhance the redshift distribution within our sample, particularly towards high ( z ≥ 3) redshift, in this otherwise underrepresented parameter space due to observational selection e ff ects. We recognize the subsequent incorporation of unavoidable selection biases in each survey while restricting the inclusion of sources at low redshift to a minimum. While the balance between dataset quality and size is critical, deep learning algorithms that operate on pixel-level inputs tend to perform optimally only when training datasets contain ≥ 400 000 galaxy images (Schuldt et al. 2021; Dey et al. 2022a; Newman & Gruen 2022). Since our method would benefit from a larger sample (see Table 1), we chose not to apply stringent quality criteria by only considering high-quality data, significantly reducing the number of sources available training. Such a reduction would also prevent the model from learning to handle lower-quality data, limiting its application to only high-quality validation data. By not making an initial down-selection, we retain the flexibility to apply quality cuts to future blind samples by using LS10 flags later.",
"pages": [
3
]
},
{
"title": "2.1.1. Samples from optical selection",
"content": "We include the 2dF QSO Redshift Survey (2QZ, Croom et al. 2004) with ∼ 23k color selected QSOs in the magnitude range 18.25 ≤ bJ ≤ 20.85 at redshifts lower than z ∼ 3 and the QUasars as BRIght beacons for Cosmology in the Southern hemisphere survey (QUBRICS, Boutsia et al. 2020) with 224 bright (i ≤ 18) QSOs at redshifts of z ≥ 2 . 5.",
"pages": [
3,
4
]
},
{
"title": "2.1.2. Samples from optical follow-up of X-ray sources",
"content": "Additionally, we incorporate the SDSS-IV (Blanton et al. 2017) quasar catalog from Data Release 16 (DR16Q, Lyke et al. 2020) with ∼ 150k quasars collected from various subprogrammes including optical / IR selection down to g ≤ 22 in the LS10 footprint after subselecting high quality specz , as well as follow-up of Xray sources from ROSAT (Voges et al. 1999; Boller et al. 2016; Salvato et al. 2018) and XMM (e.g. LaMassa et al. 2019). As successor science programme, we also consider the Black Hole Mapper (BHM) SPectroscopic IDentfication of ERosita Sources (SPIDERS, Anderson et al., in prep, Aydar et al., in prep) from SDSS-V (Kollmeier et al., in prep) Data Release 18 (Dwelly et al. 2017; Co ff ey et al. 2019; Comparat et al. 2020; Almeida et al. 2023).",
"pages": [
4
]
},
{
"title": "2.1.3. Samples of high-z sources",
"content": "Given the strong imbalance above z ∼ 3 . 5, we also include 400 optically / IR selected quasars at redshifts 4.8 ≤ z ≤ 6 . 6 down to g ≤ 24 from the high-redshift quasar survey in the DESI Early Data Release (EDR, Yang et al. 2023) and a compilation of highz quasars at z ≥ 5 . 3 published in literature (Fan et al. 2022).",
"pages": [
4
]
},
{
"title": "2.2. Spectroscopic cross-referencing",
"content": "The parent sample of AGN is annotated with specz , where available. According to Figure 1, we also consider sources, including those from eFEDS and 4XMM, with spatial counterparts from a compilation of public redshifts (Kluge et al. 2024). The procedure by which we match optical counterparts in our combined sample to a compilation of quality criteria down-selected specz , is outlined in Sect. 3.1 of Saxena et al. (2024). Due to overlaps between surveys, we remove duplicates when combining samples. The final sample of sources with specz comprises 40 489 objects, with a breakdown in Table 1. Correspondingly, the (cumulative) histograms illustrating the n( z ) distributions that collectively constitute the PICZL sample are presented in Figure 2. 0",
"pages": [
4
]
},
{
"title": "3. The survey",
"content": "To streamline and simplify our methodology, we have chosen to employ data from LS10 exclusively to mitigate potential complications arising from the heterogeneity of multiple datasets. Crucially, the survey area now extends over 20 000 deg 2 of optical griz and WISE W 1 -W 4 forced photometry, by incorporating the following datasets: In the north ( δ > 32.375 deg), LS10 uses the Beijing-Arizona Sky Survey (BASS, Zou et al. 2017) for g - and r -band coverage, and the Mayall z -band Legacy Survey (MzLS, Silva et al. 2016) for z -band coverage (Kluge et al. 2024). Count",
"pages": [
4
]
},
{
"title": "3.1. Photometric data",
"content": "LS10 o ff ers registered, background-subtracted, and photometrically calibrated point spread function (PSF)-forced photometry, including corresponding errors. To extend their wavelength coverage, DR10 catalogs incorporate mid-infrared (mid-IR) forced photometry at wavelengths of 3.4, 4.6, 12, and 22 µ m(referred to as W1, W2, W3, and W4, respectively) for all optically detected sources in the LS10 via the Near-Earth Object Wide-field Infrared Survey Explorer (NEOWISE) (Mainzer et al. 2011; Lang 2014; Meisner et al. 2017). Sources are modeled simultaneously across all optical bands, ensuring consistency in shape and size measurements by fitting a set of light profiles, even for spatially extended sources. Consequently, alongside reliable total and multi-aperture (8 annuli ≤ 7 arcseconds, five annuli ≤ 11 arcseconds for the optical and mid-infrared bands, respectively) flux measurements, the LS10 catalog o ff ers seeing-convolved PSF, de Vaucouleurs, exponential disk, or composite de Vaucouleurs + exponential disk models obtained with the Tractor algorithm (Lang et al. 2016). Additionally, providing fluxes rather than magnitudes, enables considering sources with very low signalto-noise ratios without introducing biases at faint levels. This characteristic also facilitates flux stacking at the catalog level, enhancing the overall versatility and utility of the classification and fitting process within LS10 1 .",
"pages": [
5
]
},
{
"title": "3.2. Imaging data",
"content": "In addition to catalog data, LS10 provides a rich set of imaging products. These include observations, flux model images, and residual maps for all available bands. For instance, the top pan- els (a, b, c) of Figure 3 display all image products for a g -band observation, respectively.",
"pages": [
5
]
},
{
"title": "4. Preprocessing",
"content": "Here we detail the preprocessing steps taken to prepare our dataset, ensuring that it is clean, normalized, and structured appropriately.",
"pages": [
5
]
},
{
"title": "4.1. Image preprocessing",
"content": "Building on the approach of Saxena et al. (2024), which demonstrated significant improvements by shifting from total to aperture flux utilizing information on the 2D light distribution, we aim to further refine the spatial characterization of sources. This is achieved by incorporating pixel-level flux resolution through imaging as base input. Images in individual bands or in combination (i.e., colors) reflect the surface brightness, angular size, and sub-component structures of the sources, indirectly providing redshift information (Stabenau et al. 2008). To obtain reliable photo-z directly from images, we utilize flux-calibrated optical cutouts across as many filters as available. With an average seeing FWHM of 1.3 arcseconds under nominal conditions, LS10 provides a pixel resolution of 0.262 arcseconds per pixel for the optical bands, reaching depths between 23 and 24.7 AB, depending on the specific band and region of the sky (see Dey et al. (2019) and Figure 1 in Saxena et al. (2024)). To enhance computational e ffi ciency and mitigate contamination from nearby sources, we restrict our cutout dimensions to 23 × 23 pixel, centered on the AGN coordinates in the four griz LS10 bands. Our cutouts correspond to a field of view (FOV) of approximately 6 arcseconds × 6 arcseconds. We base our choice of FOV on the angular size-redshift relation by computing the angular diameter distance d Λ via: Equation 1 and Figure 4 elucidate the connection between an object's physical size, its angular size, and redshift. Notably, we can e ff ectively map galaxies with a diameter of 30 kpc - representative of main sequence galaxies (Wuyts et al. 2011) within the confines of a 23 × 23 pixel cutout, covering the range of 0 . 5 ≤ z ≤ 7 . 7. Given that the FWHM of the W1, W2, and W3 images is 6 arcseconds, and of 12 arcseconds for W4, WISE band cutouts do not provide meaningful spatial information at this scale (see Figure 5). Therefore, we have opted to use images solely from the optical bands. Problematic sources, exhibiting signs of defects in various ways are flagged in the Legacy Survey by specific bitmasks (Dey et al. 2019). We acknowledge that LS10 images exhibit varying quality due to di ff erences in seeing conditions during observations taken over many years, which impact in particular the measurements of color within source apertures. Despite this, we rely on the model's ability to adapt to these intrinsic variations given the size and diversity of our training sample, as the sources withheld from training represent a shu ffl ed subsample of the main dataset, ensuring robust evaluation. We have verified that the distribution in seeing quality (expressed as the weighted average PSF FWHMofthe images) for the training and validation are comparable (refer to Figure B.1). However, preliminary tests indicate that adding PSF size and PSF depth of the observations-both available in LS10-as additional features enhances model performance (Götzenberger et al., in prep.). While it is not unreasonable to expect the model to implicitly infer the PSF or a related abstract representation thereof from the images themselves, these features will be included by default in future PICZL versions. With ongoing developments, PSF cutouts are expected to become more accessible for integration into the image stack (see Table 2). In the longer term, we anticipate that upcoming surveys like LSST, with their improved consistency in image quality, will further reduce these limitations and boost the precision of pixelbased analyses such as ours.",
"pages": [
5,
6
]
},
{
"title": "4.2. Color images",
"content": "It proves advantageous to provide the network with color images (ratio between images from di ff erent bands) as an additional input. This approach avoids the necessity for the model to learn the significance of colors solely from the flux images, which is inherently a more di ffi cult task. Likewise, rather than processing numerical features separately and merging them with the information extracted from the image cube at a later stage, we find it beneficial to integrate them directly at the pixel level. As a result, whenever possible, we transform catalog features into 2D arrays to align them with the original images in the same thread, enabling smoother integration and more coherent analysis (see e.g. Hayat et al. 2021). We improve the depth of our data cube by converting catalog-based quantities, e.g., flux measurements from apertures across di ff erent bands, into synthetic images, with a respective image size depending on the cutout size (see Figure 3). We expand this approach by generating images for all viable color combinations of aperture fluxes, constrained to those with matching aperture sizes. Additionally, we produce color images for flux cutouts where pixel resolutions are consistent (see lower panels b) and c) of Figure 3). Since the WISEand optical bands di ff er in both aperture size and pixel resolution, cross-wavelength color images are not feasible; instead, color combinations are restricted to within the optical or within the WISE bands (refer to Table 2). To maintain FOV consistency, we integrate WISE data for only the two innermost apertures (see Figure 5), preserving the 23x23 pixel data cube format. Although no additional spatial details are expected at this scale (see Sect. 4.1), the WISE data still captures aperture flux in a format that enables direct crosschannel connections between optical and mid-infrared data at the image level. Nevertheless, defected images can introduce challenges when generating color images (see bottom panel a) in Figure 3). Given that colors are derived from the ratio of images in di ff erent bands, the occurrence of unphysical negative or abnormally high / low values poses a significant concern. To address this issue, we examine whether the median value of neighbouring pix- Notes. Each column represents a distinct processing channel, detailing its input data type and the respective dimensions. All flux-related inputs have been corrected for reddening e ff ects. els is more than three times lower than the value of noisy pixels per band. If so, the central pixel's value is substituted with the median value of the surrounding pixels to smooth out fluctuations. In cases of non-detections or severely corrupted images where the largest value among all pixels in an image is < 0.001, the image is treated as a non-detection and all pixels are set to zero as default. After undergoing preprocessing, the images are utilized to create color images by exploring the six possible color combinations: gr , gi , gz , ri , rz and iz . If either of the two flux images involved in creating a color image is identified as a nondetection, the resulting color image is set to a default value of -99. At the catalog level, spatially invariant features, such as bestfit model classifications and signal-to-noise ratios (S / N), are processed separately in a dedicated channel, as they cannot be converted into image data. Notably, regarding normalization, we adopt a uniform min-max scaling approach to handle columns with cross-dependencies, such as flux bands. This strategy aims to preserve crucial information, such as the original shape of the SED. Contrary to the conventional approach of stacking all available images into a single input (Hoyle 2016; D'Isanto & Polsterer 2018; Pasquet et al. 2018; Dey et al. 2022a; Treyer et al. 2023), we find it advantageous to separate the color image data cube (23,23,24) from the flux band cutouts (23,23,32). This distinction is necessary because the color images often have di ff ering pixel value scales, including negative values, which require specialized processing in our machine learning application, such as tailored loss functions in the CNN. Non-spatial attributes are then combined with image-based data at a subsequent stage of the model, allowing the model to capture both spatial and nonspatial aspects. By processing data types in parallel channels, we leverage their complementary information by merging the data at a later stage, enhancing the extraction of inter-band correlations and ultimately improving redshift precision (Ma et al. 2015; Ait-Ouahmed et al. 2023). A detailed breakdown of the features integrated into each channel is provided in Table 2.",
"pages": [
6,
7
]
},
{
"title": "5. Neural network",
"content": "ML embodies an artificial intelligence paradigm where computers learn patterns and relationships from data, enabling them to make predictions or perform tasks without explicit programming. Multilayer perceptrons (MLPs), a feature-based feedforward neural network, draw inspiration from their biological counterparts, namely excitable cells responsible for processing and transmitting information (Rosenblatt 1958; I. Goodfellow 2016; M. Deru 2019). Likewise, each assigned a distinct weight, computational input vectors can be organized into layers, relayed to one or more hidden layers, to compute a scalar output value (Géron 2019). During training, these models learn data mappings by adjusting the weights and biases associated with their connections. The margin of change to the model after every training epoch is dictated by the choice of optimizer and loss function, which e ff ectively calculates the Euclidean distance between the prediction and so-called ground truth in multi-dimensional feature space, thereby significantly impacting model convergence and performance. The current state-of-the-art deep learning (DL) networks, characterized by their many hidden layers, have shown exceptional capabilities in handling complex non-linear tasks.",
"pages": [
7
]
},
{
"title": "5.1. Convolutional model layers",
"content": "Among such architectures, Convolutional Neural Networks (CNNs; Fukushima 1980; LeCun et al. 1989; Lecun et al. 1998b) distinguish themselves by their remarkable e ff ectiveness in handling grid-like data, a prevalent form of which is represented by images. CNNs leverage a model architecture that is particularly e ff ective in tasks like image recognition, object detection, and image segmentation (O'Shea & Nash 2015; Liu et al. 2022). In convolutional layers, neurons establish connections exclusively with pixels within their receptive field, ensuring successive layers are linked only to specific regions of the previous layer. Subsequently, the model extracts low, image-level features in early layers and progressively complex, higher-level features in later layers. This allows the CNN to learn representations of the input data at multiple levels of abstraction. The convolutional operation involves sliding several kernels K , here of sizes 3 × 3 or 5 × 5 pixel, across the images with a fixed stride of size s = 1, compressing each mosaic element into a single scalar. This process entails element-wise multiplication, generating a set of K feature maps. Filters, the learnable parameters of convolutional layers, enable the network to detect and highlight di ff erent aspects of the input data, such as edges or corners. Each convolution is followed by a pooling layer that reduces spatial dimensions and computational load. Pooling involves sliding a kernel, in our case of size 2 × 2 and s = 1, across the feature maps, selecting the maximum value (max pooling) within each kernel window. After flattening the data cube into a single array, it is passed through (several) fully connected (FC) layers. Each FC layer contains a layer-specific number of neurons n , tailored to the model's specific needs. The final FC layer's neuron count varies based on the task, with n = 1 referring to regression tasks and n ≥ 2 to other endeavors such as multi-label classification.",
"pages": [
7,
8
]
},
{
"title": "5.2. Gaussian mixture models",
"content": "By default, single output regression models (Schuldt et al. 2021) provide point estimates without quantifying uncertainty. This severely limits their area of application, particularly in scenarios where quantifying the uncertainty is crucial, for example, when performing precision cosmology with Euclid (Bordoloi et al. 2010; Scaramella et al. 2022; Newman & Gruen 2022). By instead employing an architecture that provides PDFs, we can not only retrieve point estimates but also encapsulate the uncertainty associated with the predictions in a concise format (D'Isanto & Polsterer 2018). Given the inherent complexity in determining redshifts from only a few broadband cutouts, our results are expected to often exhibit degeneracy with multi-modal posteriors, making a single Gaussian insu ffi cient for representing the photoz PDFs. Therefore, estimates are computed using Bayesian Gaussian Mixture Models (GMMs, Duda et al. 1973; Viroli & McLachlan 2017; Hatfield et al. 2020). These networks provide a unique probabilistic modeling approach that di ff erentiates them from traditional MLPs. One of the key distinctions lies in the output nature, where GMMs provide a set of variables to compute weighted multi-Gaussian distributions as opposed to single-point estimates. Subsequently, each component is characterized by its mean µ , standard deviation σ , and weight w , allowing GMMs to produce a full PDF for a given set of inputs x . The PDF is expressed as with, wk representing the weight and N ( x | µ k , σ 2 k ) denoting the Gaussian distribution of the k -th component. As such, we extend our CNN approach by a GMM backend to output PDFs based on the information-rich feature maps produced during the front-end phase of the network. However, we encounter a limiting challenge with inputs of such small dimensions, i.e. 23 × 23, as they are not well-suited for established image-based Deep Learning architectures, such as \"ResNet\" (He et al. 2015), which typically require larger dimension scales, typically exceeding 200 × 200 pixels, to accommodate the large number of pooling layers they employ. Therefore, we developed a custom architecture to fit our data dimensionality. The resulting model architecture features roughly 490,000 trainable parameters, far fewer than found in comparable studies (see Pasquet et al. 2018; Treyer et al. 2023), and is displayed in Figure A.1 of Appendix A.",
"pages": [
8
]
},
{
"title": "5.3. Model refinement",
"content": "Numerous hyper-parameters are crucial in shaping the network architecture while influencing training and convergence. Extensive optimization has been conducted across various parameters utilizing the Optuna framework (Akiba et al. 2019). The current configuration accounts for the vast array of potential combinations. The key parameters with the most significant impact are outlined below: putational e ffi ciency, while smaller batches may help generalize better.",
"pages": [
8
]
},
{
"title": "5.4. Loss functions",
"content": "When utilizing PDFs instead of point estimates, it is crucial to quantify whether a predicted PDF e ff ectively reflects the ground truth, in our supervised ML case, specz (Sánchez et al. 2014; Malz et al. 2018; Schmidt et al. 2020; Treyer et al. 2023). The PDF predicted by a model should be concentrated near the true value. A straight-forward and meaningful proper scoring rule (i.e. one which is lowest when the prediction is at the truth), is the product of probabilities at the true redshifts. The negative log-likelihood (NLL) 2 of which passed to a minimizer yields the likelihood of observing a data distribution given a specific set of model parameters. This expression coincides with the average Kullback-Leibler divergence (KL, Kullback & Leibler 1951) when going from a delta PDF at specz to the photo-z PDF. An alternative growing in popularity is given by the Continuous Ranked Probability Score (CRPS), initially applied in weather forecasting (Grimit et al. 2006), which serves as a valuable metric for photo-z estimation via PDF quantification (D'Isanto & Polsterer 2018). Computationally, CRPS calculates the integral of the squared di ff erence between the predicted cumulative probability distribution function (CDF) and a Heaviside step-function H ( x ) at the value of the specz ( xz ) as For a finite mixture of normal distributions M , the CRPS can be expressed in closed form: with δ i = y -µ i , the uncertainty σ i and weight wi of the i -th component respectively, as well as By extending the CRPS loss via normalizing it by (1 + z), we adjust the penalty for prediction errors based on redshift. In doing so, we prioritize accuracy for low and intermediate redshift sources by imposing higher constraints while allowing for more leniency in less critical high-redshift areas. Integrating over the entire range of possible redshift values, the CRPS takes into account both location and spread of the predicted PDF. It, therefore, provides a comprehensive evaluation metric, generating globally well-calibrated PDFs (Dey et al. 2022b).",
"pages": [
8,
9
]
},
{
"title": "5.5. Training & data augmentation",
"content": "Our dataset is divided into training and validation sets in an 80:20 ratio. The model is trained over 1000 training epochs utilizing the Adam algorithm (Kingma & Ba 2017) along with a learning rate scheduler to adjust the learning rates during training dynamically. The inclusion of both is a consequence of the Optuna hyperparameter optimization. Typically, the model achieves its lowest validation loss around the 600th epoch when considering both loss functions (see Figure 6). After this point, although the training loss continues to decrease, the validation loss begins to increase, indicating overfitting. This overfitting likely arises due to the limited size of our training sample, allowing the model to memorize specific patterns rather than learning generalizable features. We implement a checkpoint system to mitigate this, saving the model's architecture and parameters when it reaches its lowest validation loss. This approach ensures that we capture the best-performing model configuration while preventing overfitting. To further exploit the advantages of deep learning, particularly its capacity to perform well with extensive datasets, we have tested the incorporation of data augmentation techniques into our methodology. Recognizing the substantial impact of training data amount on model performance, we employ common augmentation data products, such as rotated and mirrored images. Although these transformations significantly increase the apparent size of the dataset, we do not observe an increase in performance when including augmentation techniques (Dey et al. 2022a; Treyer et al. 2023). This is potentially not surprising as the influence of augmentation techniques depends on several factors. While CNNs are inherently equivariant to translations within images, enabling them to recognize objects even if they are shifted or o ff -centered, they and most other machinelearning approaches are not naturally equivariant to rotations and reflections. This lack of rotational equivariance means that the model output e ff ectively depends on the rotation of the input images. However, as most of our inputs are approximately point sources with radial symmetry, we do not explore this aspect further in this study. More recent publications have focused on developing rotationally equivariant CNN architectures, which have shown improved performance despite increased computational costs (Cohen & Welling 2016; Weiler & Cesa 2021; Lines et al. 2024). Incorporating such architectures could be particularly beneficial for future works dealing with large amounts of imaging data.",
"pages": [
9
]
},
{
"title": "5.6. Model ensemble",
"content": "Ensemble models capitalize on the diversity of multiple models to improve prediction accuracy. Our approach involves diversifying models by adjusting parameters such as the number of Gaussians per GMM, learning rates, batch sizes, and the choice of loss function. We employ both NLL and CRPS as complementary loss functions to achieve this. While CRPS excels in achieving lower outlier fractions, NLL enhances overall variance. Each loss function trains a set of 144 models, resulting in a diverse pool of 288. In line with Treyer et al. (2023), we achieve superior performance by randomly combining equally weighted CRPS and NLL models from the pool, compared to the best-performing individual model (Dahlen et al. 2013; Newman & Gruen 2022). This finding is also aided by the result that roughly 30% of outliers are model-specific as opposed to 20% of outliers appearing in all models, therefore considered to be genuine. Put di ff erently, 80% of outliers were found to not be an outlier in at least one other model. We additionally recognize that the initial configuration of each model has a minor influence on the performance. Although training additional models could further enhance our results, the computational resources required to generate three to five sets of models would make this approach impractical relative to the potential performance gains. To create an e ff ective ensemble, we evaluate the outlier fraction of all individual models on an un- seen test sample. From this evaluation, we select models based on their relative influence on the model performance. We then observe how the ensemble's performance evolves as more models are continuously incorporated. Our ensemble ultimately comprises 10 models or 84 Gaussians, evenly incorporating models optimized using both NLL and CRPS approaches. To refine the model weights within the ensemble, we employ Optuna for finetuning. Subsequently, the ensemble posterior likelihood distribution P ensemble(x) is given by: where the weights wi are normalized such that: to assure that the integral of the ensemble PDF is equal to 1. We obtain a point estimate for our photoz prediction by identifying the dominant mode of the resulting PDF, recognizing that the mean or median could be misleading for highly non-Gaussian and bi- or multi-modal distributions. Additionally, we provide asymmetric 1 and 3 sigma upper and lower errors by evaluating the PDF within the 16th / 84th and 0.15 / 99.85th percentiles, respectively, along with the entire PDF.",
"pages": [
9,
10
]
},
{
"title": "6. Photo-z results",
"content": "We evaluate the performance of PICLZ on both point estimates and PDFs. This comprehensive assessment includes testing the statistical reliability and accuracy of our predictions. We adopt several commonly employed statistical metrics, providing insights into the accuracy, precision, and reliability of the photo-z estimates. In line with the definitions from the literature, we use the following metrics: We quantify PICZL's performances on a validation sample of 8098 sources and find an outlier fraction η , of 5.6% with a variance σ NMAD, of 0.045. Since the validation sample is used for feedback in training, it should be mentioned that the results mentioned above are likely overestimating the performance and therefore may not representative of what a future user could achieve with say, a test set previously unknown to the model, leave-one-out, or k-fold cross-validation. To this end, we have estimated our performance on independent, blind, X-rayselected samples (see Sect. 6.2 and 7). Figure 7 compares photometric versus spectroscopic redshifts, split by point-like (type = PSF) and extended (type = EXT) morphology. With respect to comparable work (e.g., Figure 13 from Salvato et al. 2022), we observe enhanced performance with a much-reduced fraction of outliers, especially for PSF sources. These objects lack morphological information at redshifts z ≳ 1, hence we attribute this improvement to the model's ability to recognise how the radial extension of the azimuth profile of sources changes with redshift (refer to Figure 4). The scatter, for both PSF and EXT distributions, is tight and symmetrically distributed around the z PICZL = z spec identity line, with outliers appearing randomly scattered, suggesting stable performance across the redshift range and minimal systematic errors (see Figure 8 and Dey et al. (2019) for reference). Given that our point estimates are derived from PDFs, we must ensure their global calibration and accuracy. To achieve this, we employ the probability integral transform (PIT) statistic (Dawid 1984; Gneiting et al. 2005), a widely accepted method in the field for assessing the quality of redshift PDFs (Pasquet et al. 2018; Schuldt et al. 2021; Newman & Gruen 2022). An ideal scenario is represented by a uniformly distributed histogram of PIT values. Any deviation from uniformity can indicate issues in PDF calibration. Under-dispersion may suggest overly narrow PDFs, while over-dispersion often results from excessively wide PDFs (D'Isanto & Polsterer 2018). Peaks close to zero or one can be explained by catastrophic outliers, where the true redshift lies so far in the wing of the PDF that it essentially falls outside of it. We employ the PIT histogram (top panel Figure 9) alongside quantile-quantile (QQ) plots (bottom panel Figure 9), to visually assess the statistical properties of our model predictions. For reference, these can be compared to Figure 2 of in Schmidt et al. PIT Score . (2020); however, those code performances are completely dominated by inactive galaxies, making a direct comparison impractical. The QQ plot compares the CDFs of observed PIT values and identity U (0 , 1), to visualize QQ di ff erences split by morphological type. A well-calibrated model will exhibit a QQ plot closely following identity with a flat PIT histogram, indicating accurate and reliable redshift predictions. Our analysis shows that, while asymmetry in the PIT distribution could suggest systematic bias, the QQ plot reveals small residuals overall. Notably, for both the single- (refer to Sect. 5.2 and Figure A.1) and ensemble model (refer to Sect. 5.6), the curves do not deviate significantly from zero, lesser so for PSF-type than EXT-type objects, indicating well-calibrated models. A distinct observation is the shift towards central PIT values for the ensemble model. This shift is not unexpected, given that the ensemble approach integrates multiple redshift solutions, each potentially contributing di ff erent peaks to the resulting ensemble PDF. Consequently, the ensemble PDF exhibits a broader bulk of probability across redshift, with PIT scores accumulating more area under the curve Notes. The last two setups are an improvement over the previous performances, with the last setup providing comparatively good results, despite ignoring the features provided by catalogs, particularly relevant for potential implementation in e.g., LSST. when integrated to the main mode. As a result, rather than yielding a flat PIT distribution, the histogram shifts towards having a concentration of values around 0.5. While this phenomenon aligns with our expectations, further enhancements may be realized by incorporating metrics tailored instead to local calibration accounting for population-specific subgroups (see e.g. Zhao et al. 2021; Dey et al. 2022b).",
"pages": [
10,
11,
12
]
},
{
"title": "6.1. Model performance for different inputs",
"content": "Images provide a more comprehensive view of astronomical sources by capturing their full spatial structure and light distribution, o ff ering finer details on morphology, apparent size and extended features that are often lost in the averaging process of aperture photometry. This is pivotal, as SED features crucial for determining redshift solutions and resolving degeneracies mostly reside within the host galaxy (Soo et al. 2017; Wilson et al. 2020). In particular for AGN, images help to spatially separate the pixels corresponding to AGN-dominated emission and those corresponding mostly to the host galaxies. This approach enhances our ability to isolate and analyze individual pixel strings across multiple filters, to e ff ectively construct SEDs for the host galaxies independently. We thereby capture subtle features, neighbors, and patterns that may not be discernible from total flux measures. Critically, the independent photo-z from the AGN and host must agree, thus narrowing the overall source PDF. Table 3 presents the fraction of outliers for both EXT and PSF sources using di ff erent configurations of the PICZL algorithm. The first setup replicates the feature-only method, relying solely on total fluxes from the catalog, which, as seen in other studies, results in the fraction of outliers for PSF sources being nearly three times higher than for EXT sources (Salvato et al. 2022). In the second configuration, we adopt the approach of Saxena et al. (2024), incorporating the 2D light distribution and color gradients within annuli. The third configuration, i.e., PICZL as outlined above, enhances this by incorporating images, supplemented with catalog values transformed into image format, which leads to an additional improvement over the method in Saxena et al. (2024). Since images inherently capture all features typically extracted numerically and presented in catalogs, we show in the fourth case that PICZL achieves excellent results without feature / catalog information, using solely optical images supplemented by WISE aperture flux maps, even without explicit hyperparameter tuning. This approach is particularly promising for future surveys such as LSST and Euclid, where relying solely on multi-band imaging, including NIR / IR, will eliminate the need for complex source modeling or aperture photometry, regardless of the sources being galaxies or AGN.",
"pages": [
12
]
},
{
"title": "6.2. Blind sample comparison",
"content": "To evaluate the robustness of our approach, we tested PICZL on an independent blind sample from the Chandra Source Catalog 2 (CSC2 3 ). This sample, selected to ensure no overlap with the training data, but yield a comparable X-ray to MIR distribution, is the same one used in the C ircle Z analysis (Saxena et al. 2024), allowing for a direct comparison with their results. After filtering for sources with spectroscopic redshift and removing duplicates with the training dataset, the sample comprises 416 sources within the LS10-South area. For comparison, Saxena et al. (2024) achieve an outlier fraction, η , of 12.3% and a nor- 7 malized median absolute deviation, σ NMAD, of 0.055, while our algorithm yields a superior σ NMAD of 0.046 and η of 8.3% (see Table 5 and Figure C.1). This comparison, therefore, demonstrates how utilizing image cubes further improves the already good results achieved using 2D-information from a catalog. For more details on the sample, see Saxena et al. (2024).",
"pages": [
12,
13
]
},
{
"title": "6.3. Prediction uncertainty quantification",
"content": "Accurate error estimation is crucial for assessing the reliability of any prediction, particularly in those astrophysical contexts where uncertainties can significantly impact the interpretation of data. To this end, we provide asymmetrical 1 σ and 3 σ errors for every source, together with the photoz PDF, as detailed in Sect. 5.6. These error estimates o ff er a comprehensive understanding of the potential variance in our predictions, accurately reflecting the inherent uncertainties in our data and model. We include asymmetrical errors, as the true distribution of uncertainties is often not symmetrical. This asymmetry arises from factors such as photometric noise and degeneracies in the highly nonlinear color-redshift space, where certain higher or lower redshift values may be more probable than their counterparts, leading to skewed PDFs. Although symmetrical redshift errors are observed for the majority of sources, approximately one-third of the sources exhibit asymmetrical 1 σ errors with deviations in redshift up to | ∆ z asym | ≃ 0 . 25. in Sect. 8.4. Additionally, the widths of the PDFs exhibit noticeable horizontal structures, largely influenced by the nature of the LS10 filters. We find that PDFs with wider modes, indicating less certain redshift estimates, often correspond to redshift ranges where key spectral features, such as the Ca break, fall outside the filter coverage (e.g. g & r at z ≃ 0 . 4, r & i at z ≃ 0 . 8 and z ≃ 1.5 or z ≃ 2.2). At redshifts approaching z ≈ 3 and extending below z ≈ 5, the Lymanα break starts to fall within the filter ranges, which contributes to narrower PDFs with increased PDF width only observed for extremely faint sources at even higher redshifts. Lastly, we find that the number of sources with wider PDFs increases as a function of | ∆ Z | (1 + Z spec) , indicating that narrower PDFs correspond to more accurate point predictions.",
"pages": [
13
]
},
{
"title": "6.4. Insights from multimodal PDFs",
"content": "To investigate whether predicted outliers are genuine anomalies or instead stem from machine learning processes, we inquire whether secondary peaks in PDFs are physically meaningful or training artifacts. For each source, we compute the fraction of its PDF, that satisfies the condition | ∆ Z | (1 + z spec) ≥ 0 . 15, to provide the probability of being considered an inlier. The left panel of Figure 11 illustrates this with examples of two PDFs (one for an inlier and one for an outlier), where we highlight the corresponding inlier probabilities. In the right panel of Figure 11, we plot | ∆ Z | (1 + z spec) re-normalised to scale [0,1], such that as a function of inlier probability for all sources. The horizontal dashed black line separates the inliers ( < 0.5) from the outliers ( ≥ 0.5). The majority of inliers are concentrated at high inlier probabilities. This suggests that most inliers provide confident, unimodal PDFs with low errors. Likewise, most outliers have low inlier probability, with only a few outliers having more than 50% inlier probability. While this quantity cannot be computed for sources lacking spectroscopic redshift, it can be used to evaluate the reliability of our point estimates considering their associated PDFs. To evaluate whether ensemble solutions with broad or complex distributions e ff ectively capture the true redshift, we identify sources with multiple peaks and measure the proportion of PDFs exhibiting more than one mode. For these sources, we define a secondary peak as significant if it accounts for at least 10% of the height of the primary peak, which is the minimum prominence threshold used in our analysis. The top panel of Figure 12 shows that only 3% of inliers exhibit secondary peaks as opposed to the roughly 30% of outliers, where occurrences of strong secondary modes decrease with increasing prominence for both cases. While a single mode typically indicates a secure estimate for inliers, the presence of a unique mode alone can subsequently not be used to determine whether a source is an outlier. Conversely, in the bottom panel of Figure 12, we present the recovery fraction, which denotes how often a secondary peak in PDFs exhibiting multi-modal distributions corresponds to the true redshift. Among the outliers with 30% multi-modal PDF, more than 70% have one of their secondary peaks corresponding to | ∆ Z | (1 + Z spec) ≤ 0 . 15, indicating that there is significant probability that the redshift could consequently be considered as an inlier if the peak heights were reversed. Given the PIT histograms shown in Figure 9, it appears likely that the the PDFs are accurately capturing the relative frequency of multi-& bimodal PDFs, as if they were not, there would likely be bias evident in the overall PIT distribution. In other words, using this particular dataset, despite incorporating full multiwavelength image cutouts and cataloglevel information, there persist areas of parameter space that are legitimately degenerate in redshift, and the PDF parameterization looks to be capturing that degeneracy accurately. Consequently, we recommend using the entire PDF rather than point estimates, when possible.",
"pages": [
13,
14
]
},
{
"title": "7. PICZL applied to other surveys",
"content": "We want to investigate how PICZL, utilizing LS10 photometry and imaging, performs in determining photo-z for AGN in a more generic setting. Our focus is set on the LSST deep drilling fields (DDFs), selected to study SMBH growth across the full range of cosmic environments. These fields o ff er deeper, more comprehensive spectroscopic redshift coverage, along with a broader range of high-sensitivity bands, providing an enhanced dataset for photo-z estimation.",
"pages": [
14
]
},
{
"title": "7.1. XMM-SERVS",
"content": "The XMM-Spitzer Extragalactic Representative Volume Survey (XMM-SERVS, Mauduit et al. 2012) encompasses three key fields: the XMM-Large Scale Structure (LSS, Chen et al. 2018), spannning 5.3 deg 2 with a flux limit of 6 . 5 × 10 -15 erg cm -2 s -1 over 90% of the survey area in the 0.5-10keV band (Savi'c et al. 2023); the Wide Chandra Deep Field-South (W-CDF-S, Ni et al. 2021) and the European Large-Area ISO Survey-South 1 (ELAIS-S1, Ni et al. 2021), covering approximately 4.6 deg 2 and 3.2 deg 2 (Brandt et al. 2018; Scolnic et al. 2018), limited to 1 . 3 × 10 -14 erg cm -2 s -1 , respectively, each selected for their exceptional multiwavelength coverage and strategic alignment with the DDFs.",
"pages": [
14
]
},
{
"title": "7.2. Photo-z computation",
"content": "Considering that the original XMM-SERVS photo-z benefit from high S / N photometry and broader wavelength coverage, including u -band and NIR coverage, we would expect significantly better results compared to those achievable by PICZL using LS10 data alone. However, PICZL obtains comparable if not better results when limiting the sample to the depth at which it was trained on (see XMM-SERVS magnitude distribution in Figure E.1). Like any ML model, it struggles to extrapolate to faint sources outside its training distribution. Therefore, a fair comparison requires limiting the analysis to a similar feature space, while the results in Table 4 reflect performance across the entire, diverse XMM-SERVS samples. To facilitate a comparison with PICZL, our analysis begins with the counterpart associations of X-ray emissions as presented in Ni et al. (2021) and Chen et al. (2018). Whenever possible, we limit our samples to sources flagged as AGN (as defined in Sect. 6 of Ni et al. 2021). Matching to objects detected in LS10, due to its more limited depth, we identify approximately one-third of the original sources (see Table 4). This limitation is particularly pronounced in the XMM-LSS survey, where, in addition to deeper X-ray data corresponding to fainter counterparts, Chen et al. (2018) restrict their sample of AGN for which they calculate photo-z, to strictly non-broad-line X-ray sources. We select all sources with spectroscopic redshift and exclude those previously included in the Notes. From left to right, the columns list: Sample field; the original number of X-ray sources; the number of sources in each field that were detected in the LS10 survey; have spectroscopic redshift z > 0 . 002; and were not already present in the training sample of PICZL (i.e. previously unseen); the fraction of failed photo-z in XMM-SERVS zphot = -99; and the bias ⟨ ∆ z ⟩ , variance σ NMAD and outlier fraction η obtained with XMMSERVS (S, excluding zphot = -99) and PICZL (P) photo-z. PICZL training sample (see Table 4). Unlike the original works, PICZL also successfully computes photo-z for the significant fraction of sources for which the SED fitting adopted in XMMSERVS failed (refer to w / photoz S in Table 4). Figure 13 visualizes the outlier fractions (top) and variances (bottom) for the aggregate XMM-SERVS samples. Blue and burgundy represent the distributions including / excluding catastrophic failures in the XMM-SERVScatalogs. The plot is limited to outlier fractions of 30% and variances of 10%, as XMM-SERVS curves including zphot = -99 solutions continue to rise for fainter magnitudes or lower S / N. This suggests that, with increasing magnitude, a decreasing number of accurate estimates are available, making the subset excluding such solutions increasingly non-representative in terms of outlier fraction. However, by limiting our comparison to sources with photoz obtained from XMM-SERVS, we find that the accuracy of these photo-z remains comparable up to magnitudes around 23.5 AB, despite the more limited data being used. Additionally, we observe enhanced performance in PICZL photo-z for sources with S / N values ≳ 60, depending on the specific XMM-SERVS sample. Importantly, sources with S / N ≳ 10-20 mark a critical threshold range in LS10 where lower S / N values lead to an exponential rise in photometric errors (see Figure D.1). This trend is particularly evident for magnitudes fainter than 21.5 in the iband (Saxena et al. 2024) and visible in all panels of Figure 13. Figure 14 presents a visual comparison of normalized residuals for outliers identified by either PICZL or the original photo-z from Ni et al. (2021), exemplified via the ELAIS-S1 field. The spread of biases, excluding failures from XMM-SERVS, is similar for both methods, falling mostly within the range of [-0.5, 0.5]. The outliers in common appear to be modestly brighter than those that are outliers for a single method only and are evenly distributed between overestimation and underestimation. Notably, the outliers from Ni et al. (2021) are over a narrower range, while PICZL has a few brighter and several more fainter, the majority of which with magnitudes beyond the range covered during training. The observed di ff erence in outlier rates, is - not primarily due to the use of template-fitting versus machinelearning methods but rather a reflection of the available photometry. XMM-SERVS, with its deeper photometry, is wellsuited for faint objects, though saturation in brighter sources may contribute to some outliers. PICZL, which leverages the LS10 dataset, is optimized for brighter sources due to the shallower photometry. While it performs well in this regime, fainter objects that fall outside the known parameter space are consequently less constrained.",
"pages": [
14,
15,
16
]
},
{
"title": "8. Discussion",
"content": "While Saxena et al. (2024) made significant strides in overcoming the limitations of relying solely on total or model fluxes for photoz estimation of AGN by utilizing all redshift-correlated features in the LS10 catalog, we have further advanced this approach. By integrating data from both optical and MIR wavelengths, our study highlights the transformative potential of combining imaging with catalog-level data. Despite the state-of-theart advancements in photoz estimation for AGN in this work, further improvements will be possible only by solving issues related to data quality. By overcoming these issues, we can significantly reduce the fraction of catastrophic outliers and achieve even greater accuracy in our predictions. In the following, we discuss the limitations that need to addressed in order to apply PICZL to upcoming imaging surveys.",
"pages": [
16
]
},
{
"title": "8.1. Incorrect spec-z",
"content": "First and foremost, we need to address the reliability of the labels used in our, and generally any supervised machine learning algorithm. While spectroscopic redshift are typically trusted as accurate representations of true redshifts, this assumption does not always hold. As surveys continue to expand in scale, automated pipelines become indispensable for assigning spectroscopic redshift and identifying artifacts / problems for each source, as manual verification by visual inspection is impractical and will remain so for future surveys (Bolton et al. 2012). Importantly, most pipelines are trained on determining the redshift of the bulk of objects (Alexander et al. 2023), which are inactive galaxies and not AGN. This implies that a non-negligible fraction of AGN have wrong redshift estimates, including many labelled with \"no warnings\" (Hewett & Wild 2010; Wu & Shen 2022). Consequently, our performance is constrained by the unknown amount of incorrectly identified spectroscopic redshift Absolute LS10 r band [mag] Absolute LS10 r band [mag] per survey. While this fraction may be smaller for galaxies without nuclear activity, AGN present a range of challenges where pipelines fail. Examples of failures include cases where e.g. MgII is misidentified as Lymanα , leading to ambiguous redshifts around z = 0 . 7 -0 . 9 ⇔ 1 . 5 -2 . 5 or sources with strong featureless continuum typical of Blazars or, finally, sources with intense broad emission lines compared to the background continuum (Chen et al. 2018; Ni et al. 2021), as well as those with weak emission lines and a flat continuum background.",
"pages": [
16,
17
]
},
{
"title": "8.2. Incorrect type classification",
"content": "A subset of sources in the training sample, although having redshifts of z ≥ 1, are classified as extended (i.e. non-PSF). This classification is likely incorrect, given the pixel resolution of LS10. The apparent extension is more plausibly a result of poor seeing conditions that were not properly accounted for, or the presence of nearby neighboring sources (see Sect. 5.3 of Hsu et al. 2014). Since the morphological type a ff ects predictions, akin to a prior in template fitting methods, this misclassification explains the increase in outliers within the EXT sample as redshift increases (see middle row in right panel of Figure 15). To verify this, we computed the absolute g magnitudes for outliers to assess whether their values fall within the expected range for galaxies and AGN (Véron-Cetty & Véron 2010). Figure 16 shows that 16% of sources classified as EXT (and thus galaxy dominated) have an absolute magnitude which exceeds the range typical of galaxies MEXT [-16,-24], confirming that their TYPE is wrongly assigned. Interestingly, approximately 96% of PSF sources meet the absolute magnitude requirement of MPSF [-20,31], indicating that their TYPE is generally accurate.",
"pages": [
17
]
},
{
"title": "8.3. Faulty photometry",
"content": "In addition to issues connected to the reliability of labels, predictions also face challenges due to data inconsistencies arising from faulty photometry. Surveys often flag issues of this kind directly, including problems such as hot pixels, saturation, cosmic rays, bleed trails, and other uncategorized anomalies. Detecting and addressing such observational defects is crucial because the accuracy and reliability of photoz predictions cannot be guaranteed for compromised photometric inputs. We experimented with removing various groups of sources a ff ected by observational artefacts but found no improvement when excluding any single group. We therefore assume that, while removing some problematic sources could potentially improve performance, the reduction in training data size o ff sets any gains. Adding this kind of noise to the current training sample could be an approach to decouple these two e ff ects. Consequently, we treat the inclusion of all problematic sources as natural noise, as opposed to artificially degrading images employing methods such as Gaussian noise (Hayat et al. 2021).",
"pages": [
17
]
},
{
"title": "8.4. Survey depth",
"content": "When splitting the validation sample by S / N across all bands, we find a higher outlier fraction in sources with low S / N (see Table 5). Consequently, to evaluate the reliability of our photoz, we need to consider the photometric depth of LS10 and how increased photometric errors for fainter sources impact redshift estimate uncertainties. Figure 15 presents the prediction bias ⟨ z ⟩ , outlier fraction η , and dispersion σ NMAD as functions of r -band magnitude in LS10, X-ray flux (when available), and specz . As the r -band magnitude increases, as expected, the outlier fraction and scatter rise. This pattern is consistent with the exponential increase in photometric errors for, e.g. i -band magnitudes ≳ 21.5, beyond which the outlier fraction, which otherwise remains well below 10%, and scatter also appears to rise, especially for sources of type EXT (refer to Figure D.1). While faint LS10 sources have unreliable photo-z, future surveys such as LSST and Euclid will probe deeper with improved S / N. As such, our methodology can be adapted to these next-generation surveys, ensuring high accuracy across a broader range of magnitudes and providing robust redshift estimates for the extensive AGN populations these and other surveys will detect.",
"pages": [
17
]
},
{
"title": "8.5. Missing i-band",
"content": "For DR10, the Legacy Survey footprint was extended by incorporating all available DECam data from various contributing surveys (refer to Sect. 3), including coverage in the i -band, though limited to a single pass. As a result, ∼ 10% of sources lack i -band observations. Unsurprisingly, and as shown in Table 5, the accuracy of photo-z increases when this band is also available.",
"pages": [
17
]
},
{
"title": "8.6. Biases",
"content": "While the photo-z residuals exhibit a symmetric distribution around zero with minimal scatter, they do not consistently center at zero, suggesting a potential systematic bias in the photo-z estimates. Ideally, we would like to achieve normalized residuals that remain consistent irrespective of redshift or selection. Currently, this is not the case, as biases are not corrected for and are mostly influenced by the distribution of our training samples, which are specific to their respective survey and inherently biased. Notes. ( 1 ) all sources; ( 2 ) split by type (PSF vs. EXT); ( 3 ) split by signal-to-noise ratio (S / N) of all available bands; ( 4 ) split by availability of the i -band; ( 5 ) split by selection criteria; ( 6 ) split by the presence of a faint [(mAGN - mNeigh . ) within -3 ≤ ∆ mag < -1 for all available bands] or bright [(mAGN - mNeigh . ) within ∆ mag > -1 for all available bands] neighbor within a 5 arcsecond radius. Additionally, results derived for the CSC2 blind sample for both PICZL and C ircle Z are provided for comparison.",
"pages": [
17,
18
]
},
{
"title": "8.6.1. Selection effects",
"content": "We investigate whether the combination of various ways of selecting AGN entering the training sample a ff ects the quality of the photo-z for specific subsamples. Using the classification outlined in Table 1, we split the validation sample based on observational criteria, creating a binary division between sources selected in optical or via X-ray. While there is only a minor difference in σ NMAD, we observe a higher outlier fraction for Xray-detected sources. While strong X-ray emission from an AGN implies strong optical and mid-IR from an AGN, which should overpower the galaxy emission in the latter two bands, complicating the process of determining accurate photo-z, local Seyfert 1 galaxies with high X-ray flux have their host remain distinctly visible. As shown in the central panel of Figure 15, however, the outlier fractions and accuracy remain consistent across the full range of X-ray fluxes, aside from small-number statistics for the wings of the distribution.",
"pages": [
18
]
},
{
"title": "8.6.2. Sample characteristics",
"content": "Unlike for galaxies, where the redshift distribution n( z ) is wellestablished and can serve as a reliable prior, the AGN n( z ) is not su ffi ciently characterised. Therefore uncertainty surrounds the ML algorithm subliminally adopting a distribution similar to the n( z ) of the training sample. At very low redshifts, AGN have a low surface density, resulting in only a few rare objects that can be considered for training. This phenomenon is intensified by the fact that deep surveys, as opposed to wide-field surveys, usually bypass such bright nearby objects in search of intermediate and high redshift sources. This leads to areas of scarcity in the training sample's specz distribution, with biased predictions towards redshift values where more data points exist. To mitigate this, we nor- malize the CRPS score by (1 + z ), emphasizing the accuracy of low-redshift predictions. Additionally, at extremely low redshifts where the 4000 Å break is barely covered by the g filter, accurate redshift predictions are more di ffi cult to obtain. At very low redshift, the 6 arcsecond × 6 arcsecond cutout may be entirely filled by the galaxy, potentially misleading the algorithm into interpreting it as excess noise.",
"pages": [
18
]
},
{
"title": "8.6.3. Non-representative training samples",
"content": "One method to tackle covariate shift by the imbalance of the bright-end dominated spectroscopic sample, is given by subdividing both the training and validation samples into subsets of n -dimensional feature space of distinguishable properties (Newman & Gruen 2022). Specifically, the prediction accuracy improves if the model used to generate a posterior for a blind source was trained exclusively on training sources residing within the corresponding feature space (Rosenbaum & Rubin 1984; Revsbech et al. 2017; Autenrieth et al. 2023). However, this approach might significantly reduce the training sample size per model making it more applicable to photoz codes dealing with large datasets, such as those for inactive galaxies (Newman & Gruen 2022). In this work we have demonstrated that, in contrast to just a few years ago, AGN-targeted training samples are now su ffi -ciently large to provide reliable photo-z. Overcoming inherent biases still requires either gathering more representative samples that uniformly cover the full redshift range-particularly underrepresented regions-or applying statistical corrections. Thus, the focus has shifted from simply increasing the number of spectra to strategically obtaining spectra that cover specific regions of parameter space (Masters et al. 2015). However, given the substantial gains our model has shown over existing methods, a detailed examination of how these biases a ff ect prac- tical applications is beyond the scope of this study. Looking ahead, spectroscopic follow-up campaigns such as DESI, SDSSV / BHM (Kollmeier et al., in prep), and 4MOST (De Jong et al. 2019) will further enhance our capabilities. Nevertheless, predicting photo-z for AGN in deeper Euclid and LSST datasets remains challenging due to the limited availability of spectroscopic data for faint sources. The upcoming Subaru Prime Focus Spectrograph (PFS, Tamura et al. 2016) and Multi Object Optical and Near-infrared Spectrograph for the Very Large Telescope (VLT / MOONS; Cirasuolo et al. 2020) are expected to help address this shortfall by providing much-needed spectra for faint (& obscured / red) objects.",
"pages": [
18,
19
]
},
{
"title": "8.7. Observational constraints: source crowding",
"content": "One factor beyond our control is the local environment or conditions along the line of sight, which can significantly influence the emission characteristics of AGN. Nearby objects can add extra flux, which can a ff ect the accuracy of photometric measurements. We perform a positional cross-match between the validation sample and all LS10 sources within a 5 arcsecond radius of their optical counterparts. For each neighbour meeting this proximity criterion, we calculate apparent magnitudes and flag all sources where the brightest neighbour is no less than 1 magnitude dimmer. Table 5 shows that isolated sources exhibit better overall performance, while those with bright neighbours show drastically reduced quality compared to those with fainter neighbours. The presence of bright neighbours influences the observed flux in two ways: firstly, it complicates the derivation of sourcespecific colors due to convolved fluxes, and secondly, it a ff ects the observed spectra, potentially leading to the detection of two sets of emission lines and incorrect specz identification (Newman & Gruen 2022). A potential solution to this issue could be the implementation of segmentation maps, similar to SExtractor (Source Extractor, Bertin, E. & Arnouts, S. 1996) at image level, as LS10 currently only masks neighbours during their model flux fitting procedure. A persistent physical issue that cannot be entirely mitigated is blending. Unlike bright neighbours, which can be masked in principle, blending a ff ects both observed photometry and spectroscopy. While particularly faint neighbours do not substantially a ff ect the predictions negatively, sources identified as blends should be excluded from samples where accurate photoz estimates are required, until we can partially mask sources at the pixel level. The Rubin Observatory expects overlapping (inactive) galaxies to contribute at least 1% of the total flux within their pixels (Sanchez et al. 2021; Newman & Gruen 2022). Blends are also expected to occur in the spectroscopic sample, increasing systematic uncertainty and subsequently the fraction of outliers by up to 5% (Masters et al. 2019). This may be mitigated in future by higher-quality imaging from spacebased data from missions such as Euclid or better tools for deblending (e.g., SCARLET / blendz ; Melchior et al. 2018; Jones & Heavens 2018). Alternatively, data augmentation or standard computer vision techniques could be implemented, such as artificially introducing blending e ff ects in the training sample. However, this approach is not trivial, as LS10 currently lacks a corresponding blending flag.",
"pages": [
19
]
},
{
"title": "8.8. Variability",
"content": "AGN are inherently variable sources, meaning that their observed flux can change significantly across di ff erent epochs and wavelengths, especially when observations are separated by substantial time gaps. This extends to images that are created by stacking multiple observations taken over extended periods (as in LS10), where the resulting flux represents an average value. Such variability complicates the accurate prediction of AGN redshifts (Simm et al. 2015). However, future time-resolved imaging from LSST will allow us to better account for these variations and potentially use the correlation between AGN variability and their physical properties as a feature to improve photo-z predictions.",
"pages": [
19
]
},
{
"title": "8.9. Scalability in image-based photo-z",
"content": "Open issues remain regarding the feasibility of handling the vast data volumes and computational requirements associated with photo-z estimated from images. Exemplified by the calculation of new estimates for XMM-SERVS concerning storage, processing time, and computational resources, we give an overview in Appendix G. A crucial future direction involves exploring ways to utilize imaging data e ffi ciently without necessitating extensive local downloads and computations for all image-based analyses of such data sets, potentially leveraging online platforms for realtime analysis, bringing the compute to the data (Zhang & Zhao 2015).",
"pages": [
19
]
},
{
"title": "9. Summary and outlook",
"content": "This work has been driven by the goal of determining reliable photo-z for X-ray detected AGN in wide-field surveys, such as eROSITA (Merloni et al. 2024). For that reason, we have concentrated on utilizing data from LS10, which provides su ffi ciently homogeneous coverage and depth in 4 optical bands, enriched by the flux measured on NEOWISE7 data for all identified sources (see Sect. 3). LS10 overlaps almost entirely with the eROSITA footprint, simplifying the cross calibration of data, a processing step typically necessary when merging di ff erent surveys. We introduce PICZL, a CNN-based machine learning model designed primarily for AGN redshift estimation, but which holds the potential to reliably measure photoz s for a broad range of extragalactic sources, provided an appropriately constructed training sample is available. In this study, the training sample comprises both X-ray and optically selected AGN. Across our validation set, PICZL demonstrates consistently robust performance, with comparable accuracies and lower outlier fractions particularly for PSF sources, as opposed to the results obtained by Salvato et al. (2022) for similar objects using SED-fitting (see Figure 7 and Table 5). The results show significant improvements for both point-like and extended sources, indicating that the model's performance is primarily driven by the depth and hence photometric error of the training data. Additionally, when tested on a blind sample of X-ray-selected AGN, PICZL maintained comparable results with respect to σ NMAD and η . Notably, on this test set it outperformed (C irclez ; Saxena et al. 2024) by achieving a 20% improvement in accuracy and a 30% reduction in the fraction of outliers (refer to Sect. 6). We also applied PICZL to estimate photo-z for the approximately 30% of XMM-SERVS sources (Chen et al. 2018; Ni et al. 2021), detected in LS10 (see Table 4), demonstrating comparable variance and a substantially improved outlier fraction up to a limiting magnitude of 23.5 AB, using much fewer bands and significantly less sensitive imaging. However, it is important to note that the spectroscopic sample at this faint limit is small and likely biased toward sources with higher spectroscopic success rates. Consequently, the most reliable photo-z results are expected at brighter magnitudes (as discussed in Saxena et al. 2024). In response to these findings, we are releasing a new catalog of photo-z, complete with 1 and 3 σ error margins for the XMM-SERVS fields (see Sect. 7). PICZL will be used in the next generation of photo-z for sources detected by eROSITA, possibly switching from LS10 to Euclid and, most importantly, LSST, as they are poised to deliver more homogeneous and deeper data with broader wavelength coverage. In particular the availability of NIR data at image level, will improve our ability to determine accurate redshift for faint and high-z sources. Our study has shown that the performance of PICZL, when relying predominantly on images, is already robust (see Table 3). Crucially, we have shown that wellcalibrated images alone can su ffi ce for accurate photo-z estimation, eliminating the need for catalog creation, which is often based on predefined models. This suggests that future surveys could reduce their dependence on catalog-based data for photo-z computation (refer to Sect. 6). Although this study has focused on calculating photo-z for AGN, the approach can be generalised and adapted to other source types, e.g., normal galaxies, with an appropriately constructed training sample (Götzenberger et al. in prep.). Subsequently our findings point to a bright future for all-sky surveys, also thanks to the continue expansion of training sample sizes, supported by initiatives like SDSS-V / BHM, 4MOST and VLT / MOONS, which will enhance the reliability and completeness of photoz estimates (refer to Sect. 8). Looking forward, the next step in advancing photo-z estimation for AGN lies in exploring more sophisticated machine learning architectures beyond CNNs. For example, transformers with shared latent space embeddings o ff er a promising avenue. These models have shown success in integrating information from various data sources, such as images and entire spectra, potentially reducing uncertainties associated with traditional specz pipelines by leveraging multi-modal data fusion (DonosoOliva et al. 2023; Parker et al. 2024). Additionally, incorporating other informative features like X-ray flux, when available, holds promise. Integrating these diverse data sources within a unified framework has the potential to refine redshift estimates even further.",
"pages": [
19,
20
]
},
{
"title": "10. Data availability",
"content": "With this paper, we present a new catalog of photo-z derived using PICZL for the ∼ 30% of sources within the XMM-SERVS (ELAIS-S1, W-CDF-S, and LSS) X-ray source catalogs (Chen et al. 2018; Ni et al. 2021) that are detected in the LS10 survey. Hence, it includes updated photo-z for sources with catastrophic failures in the original works. A detailed description of the catalog columns can be found in Appendix F. The full catalog is only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http:// cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/ . Acknowledgements. WRand MS are grateful for the constant support of Dustin Lang in handling Legacy Survey-related issues. Part of this work was supported by the German Deutsche Forschungsgemeinschaft, DFG , under Germany's Excellence Strategy - EXC 2094 - 390783311. We gratefully acknowledge funding from FONDECYT Regular - 1231718 (RJA), 1230345 (CR), and 1241005 (FEB), CATA-BASAL - FB210003 (RJA, CR, FEB), and ANID - Millennium Science Initiative - AIM23-0001 (FEB). JA acknowledges support from a UKRI Future Leaders Fellowship (grant code: MR / T020989 / 1) This work is based on data from eROSITA, the soft X-ray instrument aboard SRG, a joint RussianGerman science mission supported by the Russian Space Agency (Roskosmos), in the interests of the Russian Academy of Sciences represented by its Space Research Institute (IKI), and the Deutsches Zentrum für Luft- und Raumfahrt (DLR). The SRG spacecraft was built by Lavochkin Association (NPOL) and its subcontractors and is operated by NPOL with support from the Max Planck Institute for Extraterrestrial Physics (MPE). The development and construction of the eROSITA X-ray instrument were led by MPE, with contributions from the Dr. Karl Remeis Observatory Bamberg & ECAP (FAU Erlangen-Nuernberg), the University of Hamburg Observatory, the Leibniz Institute for Astrophysics Potsdam (AIP), and the Institute for Astronomy and Astrophysics of the University of Tübingen, with the support of DLR and the Max Planck Society. The Argelander Institute for Astronomy of the University of Bonn and the Ludwig Maximilians Universität Munich also participated in the science preparation for eROSITA. The Legacy Surveys consist of three individual and complementary projects: the Dark Energy Camera Legacy Survey (DECaLS; Proposal ID 2014B-0404; PIs: David Schlegel and Arjun Dey), the Beijing-Arizona Sky Survey (BASS; NOAO Prop. ID 2015A-0801; PIs: Zhou Xu and Xiaohui Fan), and the Mayall z-band Legacy Survey (MzLS; Prop. ID 2016A-0453; PI: Arjun Dey). DECaLS, BASS, and MzLS together include data obtained, respectively, at the Blanco telescope, Cerro Tololo Inter-American Observatory, NSF's NOIRLab; the Bok telescope, Steward Observatory, University of Arizona; and the Mayall telescope, Kitt Peak National Observatory, NOIRLab. Pipeline processing and analyses of the data were supported by NOIRLab and the Lawrence Berkeley National Laboratory (LBNL). The Legacy Surveys project is honored to be permitted to conduct astronomical research on Iolkam Du'ag (Kitt Peak), a mountain with particular significance to the Tohono O'odham Nation. NOIRLab is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. LBNL is managed by the Regents of the University of California under contract to the U.S. Department of Energy. This project used data obtained with the Dark Energy Camera (DECam), which was constructed by the Dark Energy Survey (DES) collaboration. Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A & M University, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho de Amparo, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Cientifico e Tecnologico and the Ministerio da Ciencia, Tecnologia e Inovacao, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey. The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Energeticas, Medioambientales y Tecnologicas-Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgenossische Technische Hochschule (ETH) Zurich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ciencies de l'Espai (IEEC / CSIC), the Institut de Fisica d'Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig Maximilians Universitat Munchen and the associated Excellence Cluster Universe, the University of Michigan, NSF's NOIRLab, the University of Nottingham, the Ohio State University, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, and Texas A&M University. BASS is a key project of the Telescope Access Program (TAP), which has been funded by the National Astronomical Observatories of China, the Chinese Academy of Sciences (the Strategic Priority Research Program 'The Emergence of Cosmological Structures' Grant # XDB09000000), and the Special Fund for Astronomy from the Ministry of Finance. The BASS is also supported by the External Cooperation Program of Chinese Academy of Sciences (Grant # 114A11KYSB20160057), and Chinese National Natural Science Foundation (Grant # 12120101003, # 11433005). The Legacy Survey team uses data products from the Near-Earth Object Wide-field Infrared Survey Explorer (NEOWISE), a project of the Jet Propulsion Laboratory / California Institute of Technology. NEOWISE is funded by the National Aeronautics and Space Administration. The Legacy Surveys imaging of the DESI footprint is supported by the Director, O ffi ce of Science, O ffi ce of High Energy Physics of the U.S. Department of Energy under Contract No. DE-AC02-05CH1123, by the National Energy Research Scientific Computing Center, a DOE O ffi ce of Science User Facility under the same contract, and by the U.S. National Science Foundation, Division of Astronomical Sciences under Contract No. AST-0950945 to NOAO. Funding for the Sloan Digital Sky Survey V has been provided by the Alfred P. Sloan Foundation, the Heising-Simons Foundation, the National Science Foundation, and the Participating Institutions. SDSS acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. SDSS telescopes are located at Apache Point Observatory, funded by the Astrophysical Research Consortium and operated by New Mexico State University, and at Las Campanas Observatory, operated by the Carnegie Institution for Science. The SDSS web site is www.sdss.org . SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration, including Caltech, The Carnegie Institution for Science, Chilean National Time Allocation Committee (CNTAC) ratified researchers, The Flatiron Institute, the Gotham Participation Group, Harvard University, Heidelberg University, The Johns Hopkins University, L'Ecole polytechnique fédérale de Lausanne (EPFL), Leibniz-Institut für Astrophysik Potsdam (AIP), Max-PlanckInstitut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Extraterrestrische Physik (MPE), Nanjing University, National Astronomical Observatories of China (NAOC), New Mexico State University, The Ohio State University, Pennsylvania State University, Smithsonian Astrophysical Observatory, Space Telescope Science Institute (STScI), the Stellar Astrophysics Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Illinois at Urbana-Champaign, University of Toronto, University of Utah, University of Virginia, Yale University, and Yunnan University.",
"pages": [
20,
21
]
},
{
"title": "References",
"content": "Abdalla, F. B., Banerji, M., Lahav, O., & Rashkov, V. 2011, MNRAS, 417, 1891 Aird, J., Nandra, K., Laird, E. S., et al. 2010, MNRAS, 401, 2531-2551 Alexander, D. M., Davis, T. M., Chaussidon, E., et al. 2023, ApJ, 165, 124 Almeida, A., Anderson, S. F., Argudo-Fernández, M., et al. 2023, ApJ Supple- ment Series, 267, 44 Almosallam, I. A., Jarvis, M. J., & Roberts, S. J. 2016, MNRAS, 462, 726 Ananna, T. T., Salvato, M., LaMassa, S., et al. 2017, ApJ, 850, 66 Baum, W. 1957, ApJ, 62, 6 Bell, E. F., Wolf, C., Meisenheimer, K., et al. 2004, ApJ, 608, 752-767 Benitez, N. 2000, ApJ, 536, 571 Bertin, E. & Arnouts, S. 1996, Astron. Astrophys. Suppl. Ser., 117, 393 Bettoni, D., Falomo, R., Kotilainen, J. K., Karhunen, K., & Uslenghi, M. 2015, MNRAS, 454, 4103 Blanton, M. R., Bershady, M. A., Abolfathi, B., et al. 2017, ApJ, 154, 28 Boller, T., Freyberg, M. J., Trümper, J., et al. 2016, A&A, 588, A103 Bolton, A. S., Schlegel, D. J., Aubourg, E., et al. 2012, ApJ, 144, 144 Bordoloi, R., Lilly, S. J., & Amara, A. 2010, MNRAS, no Boutsia, K., Grazian, A., Calderone, G., et al. 2020, ApJ Supplement Series, 250, 26 Breiman, L. 2001, Machine Learning, 545, 5 Brescia, M., Cavuoti, S., Longo, G., & Stefano, V. D. 2014, A&A, 568, A126 Comparat, J., Merloni, A., Dwelly, T., et al. 2020, A&A, 636, A97 Connolly, A. J., Csabai, I., Szalay, A. S., et al. 1995, ApJ, 110, 2655 Croom, S. M., Smith, R. J., Boyle, B. J., et al. 2004, MNRAS, 349, 1397-1418 Dahlen, T., Mobasher, B., Faber, S. M., et al. 2013, ApJ, 775, 93 Dawid, A. P. 1984, Journal of the Royal Statistical Society. Series A (General), Dey, B., Newman, J. A., Andrews, B. H., et al. 2022b, Re-calibrating Photometric Redshift Probability Distributions Using Feature-space Regression D'Isanto, A. & Polsterer, K. L. 2018, A&A, 609, A111 Eriksen, M., Alarcon, A., Cabayol, L., et al. 2020, MNRAS, 497, 4565-4579 Heckman, T. M. & Best, P. N. 2014, Annual Review of A&A, 52, 589 Lecun, Y., Bottou, L., Bengio, Y., & Ha ff ner, P. 1998b, Proceedings of the IEEE, 86, 2278 Li, C., Zhang, Y., Cui, C., et al. 2021, MNRAS, 509, 2289 Li, C., Zhang, Y., Cui, C., et al. 2022, MNRAS, 518, 513 Lin, Q., Fouchez, D., Pasquet, J., et al. 2022, A&A, 662, A36 Lines, N. E. P., Roset, J. F.-Q., & Scaife, A. M. M. 2024, E(2)-Equivariant Features in Machine Learning for Morphological Classification of Radio Galaxies Liu, Z., Mao, H., Wu, C.-Y., et al. 2022, in Proceedings of the IEEE / CVF Con- ference on Computer Vision and Pattern Recognition (CVPR), 11976-11986 Luken, K. J., Norris, R. P., & Park, L. A. F. 2019, Publications of the Astronom- ical Society of the Pacific, 131, 108003 Luo, B., Brandt, W. N., Xue, Y. Q., et al. 2010, ApJ Supplement Series, 187, 560 Lyke, B. W., Higley, A. N., McLane, J. N., et al. 2020, ApJ Supplement Series, 250, 8 Ma, L., Lu, Z., Shang, L., & Li, H. 2015, Multimodal Convolutional Neural Networks for Matching Image and Sentence Madau, P. & Dickinson, M. 2014, Annual Review of A&A, 52, 415-486 Mainzer, A., Bauer, J., Grav, T., et al. 2011, ApJ, 731, 53 Masters, D., Capak, P., Stern, D., et al. 2015, ApJ, 813, 53 Mauduit, J.-C., Lacy, M., Farrah, D., et al. 2012, Publications of the Astronomical Society of the Pacific, 124, 714-736 Meisner, A. M., Lang, D., & Schlegel, D. J. 2017, ApJ, 153, 38 Merloni, A., Predehl, P., Becker, W., et al. 2012, eROSITA Science Book: Mapping the Structure of the Energetic Universe Meshcheryakov, A., Glazkova, V., Gerasimov, S., & Mashechkin, I. 2018, Astronomy Letters, 44, 735 Mountrichas, G., Corral, A., Masoura, V. A., et al. 2017, A&A, 608, A39 Newman, J. A. & Gruen, D. 2022, Annual Review of A&A, 60 Ni, Q., Brandt, W. N., Chen, C.-T., et al. 2021, ApJ Supplement Series, 256, 21 Nishizawa, A. J., Hsieh, B.-C., Tanaka, M., & Takata, T. 2020, Photometric Redshifts for the Hyper Suprime-Cam Subaru Strategic Program Data Release 2 O'Shea, K. & Nash, R. 2015, An Introduction to Convolutional Neural Networks Padovani, P., Alexander, D. M., Assef, R. J., et al. 2017, The A&A Review, 25 Parker, L., Lanusse, F., Golkar, S., et al. 2024, MNRAS, 531, 4990-5011 Pierce, C. M., Lotz, J. M., Primack, J. R., et al. 2010, MNRAS Povi'c , M., Sánchez-Portal, M., García, A. M. P., et al. 2012, A&A, 541, A118 Predehl, P., Andritschke, R., Arefiev, V., et al. 2021, A&A, 647, A1 Revsbech, E. A., Trotta, R., & van Dyk, D. A. 2017, MNRAS, 473, 3969-3986 Rosenbaum, P. & Rubin, D. 1984, Journal of the American Statistical Associa- tion, 79 Saxena, A., Salvato, M., Roster, W., et al. 2024, CircleZ: Reliable Photometric redshifts for AGN computed using only photometry from Legacy Survey Imaging for DESI Scaramella, R., Amiaux, J., Mellier, Y., et al. 2022, A&A, 662, A112 Schmidt, S. J., Malz, A. I., Soo, J. Y. H., et al. 2020, MNRAS, 499, 1587 Schuldt, S., Suyu, S. H., Cañ ameras, R., et al. 2021, A&A, 651, A55 Simm, T., Saglia, R., Salvato, M., et al. 2015, A&A, 584, A106 Soo, J. Y. H., Moraes, B., Joachimi, B., et al. 2017, MNRAS, 475, 3613-3632 Stabenau, H. F., Connolly, A., & Jain, B. 2008, MNRAS, 387, 1215 Webb, N. A., Coriat, M., Traulsen, I., et al. 2020, A&A, 641, A136 Weiler, M. & Cesa, G. 2021, General E (2)-Equivariant Steerable CNNs Wilson, D., Nayyeri, H., Cooray, A., & Häußler, B. 2020, ApJ, 888, 83 Wu, Q. & Shen, Y. 2022, ApJ Supplement Series, 263, 42 Yang, J., Fan, X., Gupta, A., et al. 2023, ApJ Supplement Series, 269, 27 York, D. G., Adelman, J., John E. Anderson, J., et al. 2000, ApJ, 120, 1579 02.09.24, 21 : 39",
"pages": [
21,
22,
23
]
},
{
"title": "Appendix G: PICZL run time",
"content": "The computational demands for training and deploying PICZL depend mostly on the hardware configuration and dataset size. For our experiments, we leveraged a set of two Tesla V100PCIE graphics processing units (GPUs) each equipped with 32 GB of ready access memory (RAM), accompanied by a 48 core multi-thread CPU to accelerate the computational workload. This parallel processing capability significantly expedited the model training process, compared to running solely on a central CPU, allowing for faster convergence. Training a single model (refer to Figure A.1) on a dataset of 32 391 sources, corresponding to the 80:20 train-test split (refer to Table 1), each with 108 features (56 of which are images), takes approximately 25 minutes for 600 epochs. The use of an ensemble further scales the processing time by the number of models trained, with ensemble optimization depending on user preferences, making it di ffi -cult to provide a fixed time estimate. After finalizing the model, computing photo-z for e.g. 1393 sources in the XMM-SERVS W-CDF-S field, as displayed via Table 4, takes roughly 17 seconds. The storage requirements for the data used in this sample corresponds to approximately 250 MB (refer to Table 2). Looking ahead, upcoming surveys such as Euclid and LSST will produce higher-resolution images, demanding increased storage and computational resources due to larger data volumes and pixel counts. If the input dimensions were to increase, for instance, from 23x23 pixels to 64x64 pixels, the computational burden would rise significantly. While our current architecture of 32 GB RAM per GPU is e ffi cient for the current input sizes, processing larger cutouts may necessitate either more GPUs or GPUs with higher memory capacity to maintain feasible training times and performance. In scenarios where larger input sizes are anticipated, a thoughtful approach to model architecture and resource allocation will be crucial. Therefore, adapting to the capabilities of the hardware will be key to successfully utilizing the methods in the context of future data from LSST and Euclid, particularly given the much larger sample sizes that we expect from these surveys. Finally, performance will also improve by having PICZL revised by an expert software developer.",
"pages": [
24
]
}
]
2024arXiv241108310G
https://arxiv.org/pdf/2411.08310.pdf
<document>
<text><location><page_1><loc_68><loc_88><loc_76><loc_89></location>R esearch in</text>
<text><location><page_1><loc_68><loc_87><loc_70><loc_88></location>A</text>
<text><location><page_1><loc_70><loc_87><loc_78><loc_88></location>stronomy and</text>
<text><location><page_1><loc_68><loc_86><loc_70><loc_87></location>A</text>
<text><location><page_1><loc_70><loc_86><loc_76><loc_87></location>strophysics</text>
<section_header_level_1><location><page_1><loc_12><loc_76><loc_78><loc_80></location>Measurements of the solar coronal magnetic field based on coronal seismology with propagating Alfv'enic waves: forward modeling</section_header_level_1>
<text><location><page_1><loc_12><loc_71><loc_75><loc_74></location>Yuhang Gao 1 , 2 , Hui Tian 1 , Tom Van Doorsselaere 2 , Zihao Yang 1 , 3 , Mingzhe Guo 2 , 4 and Konstantinos Karampelas 2</text>
<unordered_list>
<list_item><location><page_1><loc_12><loc_68><loc_75><loc_71></location>1 School of Earth and Space Sciences, Peking University, Beijing, 100871, People's Republic of China; huitian@pku.edu.cn</list_item>
<list_item><location><page_1><loc_12><loc_65><loc_71><loc_68></location>2 Centre for mathematical Plasma Astrophysics, Department of Mathematics, KU Leuven, Celestijnenlaan 200B bus 2400, B-3001 Leuven, Belgium</list_item>
<list_item><location><page_1><loc_12><loc_64><loc_77><loc_65></location>3 High Altitude Observatory, National Center for Atmospheric Research, Boulder, CO 80307, USA</list_item>
<list_item><location><page_1><loc_12><loc_61><loc_75><loc_64></location>4 Shandong Provincial Key Laboratory of Optical Astronomy and Solar-Terrestrial Environment, Institute of Space Sciences, Shandong University, Weihai 264209, China</list_item>
</unordered_list>
<text><location><page_1><loc_12><loc_59><loc_43><loc_60></location>Received 20xx month day; accepted 20xx month day</text>
<text><location><page_1><loc_16><loc_39><loc_74><loc_56></location>Abstract Recent observations have demonstrated the capability of mapping the solar coronal magnetic field using the technique of coronal seismology based on the ubiquitous propagating Alfv'enic/kink waves through imaging spectroscopy. We established a magnetohydrodynamic (MHD) model of a gravitationally stratified open magnetic flux tube, exciting kink waves propagating upwards along the tube. Forward modeling was performed to synthesize the Fe XIII 1074.7 and 1079.8 nm spectral line profiles, which were then used to determine the wave phase speed, plasma density, and magnetic field with seismology method. A comparison between the seismologically inferred results and the corresponding input values verifies the reliability of the seismology method. In addition, we also identified some factors that could lead to errors during magnetic field measurements. Our results may serve as a valuable reference for current and future coronal magnetic field measurements based on observations of propagating kink waves.</text>
<text><location><page_1><loc_16><loc_36><loc_67><loc_37></location>Key words: Sun: corona - Sun: magnetic fields - magnetohydrodynamics</text>
<section_header_level_1><location><page_1><loc_12><loc_32><loc_27><loc_33></location>1 INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_12><loc_16><loc_79><loc_30></location>The magnetic field plays a crucial role in various physical processes in the solar and stellar coronae. The dissipation of magnetic energy is believed to drive solar eruptive events (e.g., flares and coronal mass ejections) and cause heating of the corona. While the magnetic field in the lower solar atmosphere can be reliably measured through spectro-polarimetric observations, direct measurements of the coronal magnetic field have remained challenging for decades. Several methods, including the spectro-polarimetry of coronal infrared lines (Lin et al. 2000, 2004; Schad et al. 2024), coronal radio observations (e.g., Fleishman et al. 2020; Chen et al. 2020; Tan 2022), and magnetic-field-induced transitions of extreme ultraviolet emission lines (e.g., Li et al. 2015, 2016; Landi et al. 2020; Chen et al. 2021, 2023), have been proposed and attempts of measurements have been made. However, these approaches all face limitations and none of them could be used for routine measurements of the global coronal magnetic field.</text>
<text><location><page_1><loc_12><loc_12><loc_79><loc_16></location>Another technique that could be used to measure the coronal magnetic field is coronal seismology. This technique combines the magneto-hydrodynamic (MHD) wave theory with observed wave parameters (e.g., period, amplitude, propagation speed, damping time) to diagnose various physical properties,</text>
<text><location><page_2><loc_12><loc_60><loc_79><loc_87></location>particularly the magnetic field. Different wave phenomena in the corona have been used for coronal seismology, including standing kink waves in magnetic loops (e.g. Nakariakov & Ofman 2001; Aschwanden et al. 2002; Van Doorsselaere et al. 2007; Tian et al. 2012; Zhang et al. 2022; Gao et al. 2022, 2024a; Zhong et al. 2023; Li & Long 2023, to name but a few), propagating slow magneto-acoustic waves (e.g., Jess et al. 2016), sausage waves (Chen et al. 2015; Guo et al. 2016; see Li et al. 2020 for a review), propagating kink waves in streamers (Chen et al. 2011; Guo et al. 2022), and torsional oscillations in solar surges (Kohutova et al. 2020). However, these studies provided only one-dimensional (1D) distributions or single values of the magnetic field in specific coronal structures, such as oscillating loops or streamers. To create global 2D magnetic field maps, we need to utilize ubiquitous and continuous wave phenomena. The pervasive propagating disturbances in Dopplergrams (Tomczyk et al. 2007; Tomczyk & McIntosh 2009; Liu et al. 2015; Morton et al. 2015, 2019) observed by the Coronal Multi-channel Polarimeter (CoMP; Tomczyk et al. 2008) are ideal for this purpose. These propagating disturbances are interpreted as kink or Alfv'enic waves (Van Doorsselaere et al. 2008), and their propagation speeds are naturally linked to the local magnetic field. Based on these CoMP observations, Yang et al. (2020b,a) have successfully measured the global distribution of the coronal magnetic field for the first time. Following these successful attempts, a routine (continuous) measurement of the global coronal magnetic field based on similar observations from the Upgraded CoMP (UCoMP; Landi et al. 2016) has been recently achieved, which allows for the construction of coronal synoptic magnetograms (Carrington maps) (Yang et al. 2024).</text>
<text><location><page_2><loc_12><loc_51><loc_79><loc_59></location>The CoMP and UCoMP instruments can conduct spectroscopic observations of the Fe XIII lines at 1074.7 and 1079.8 nm in the coronal region above the solar limb. From the Dopplergrams of Fe XIII 1074.7 nm, propagating kink waves can be identified throughout the corona. The propagating or phase speed c k of these waves (also named the kink speed) could be obtained by constructing a timedistance map of Doppler velocity, while the coronal density can be inferred from the observed Fe XIII 1079.8-nm/1074.7-nm intensity ratio.</text>
<text><location><page_2><loc_15><loc_50><loc_31><loc_51></location>For kink waves, we have</text>
<formula><location><page_2><loc_39><loc_46><loc_79><loc_50></location>c 2 k = B 2 i + B 2 e µ 0 ( ρ i + ρ e ) , (1)</formula>
<text><location><page_2><loc_12><loc_37><loc_79><loc_46></location>where µ 0 is the magnetic permeability, B and ρ are the magnetic field and mass density, respectively. The subscripts i and e refer to parameters inside and outside the magnetic flux tubes (the waveguides). Since CoMP and UCoMP likely cannot resolve individual flux tubes, we can only work with an average density ⟨ ρ ⟩ . Furthermore, in the lowβ coronal environment, the internal and external magnetic fields are often assumed to be approximately equal (see e.g., Tomczyk & McIntosh 2009; Morton et al. 2015; Zhong et al. 2023). This leads to the simplified expression for the kink speed:</text>
<formula><location><page_2><loc_41><loc_33><loc_79><loc_36></location>c 2 k = B 2 µ 0 ⟨ ρ ⟩ , (2)</formula>
<text><location><page_2><loc_12><loc_29><loc_79><loc_32></location>which is widely used in estimations of coronal magnetic fields (Long et al. 2017; Yang et al. 2020a,b; Yang et al. 2024).</text>
<text><location><page_2><loc_12><loc_16><loc_79><loc_29></location>Given the potential of these measurements to provide routine coronal magnetograms on a daily basis, which could play a crucial role in future solar physics research, it is essential to thoroughly assess the reliability and robustness of the methodology used in Yang et al. (2020a,b) and Yang et al. (2024). Magyar & Van Doorsselaere (2018) performed 3D MHD simulations of propagating kink waves under various conditions and conducted forward modeling to evaluate the reliability of this method in deriving the magnetic field strength. Their findings indicated that the magnetic field strengths inferred through seismology closely match the input values, typically with an error less than ∼ 20%. However, there is a limitation in their simulation. They utilized a non-stratified setup that excluded the effects of gravity, resulting in uniform initial density and propagation speed in the vertical direction.</text>
<text><location><page_2><loc_12><loc_12><loc_79><loc_16></location>The gravitational stratification can play a significant role as it causes the Alfv'en speed and kink speed c k to vary with height. This variation can affect the wave tracking method which typically relies on a linear fit of velocity signals (e.g., see Figure 6 in Tomczyk & McIntosh 2009). Therefore, it</text>
<text><location><page_3><loc_12><loc_76><loc_79><loc_87></location>is important to assess how the gravitational stratification affects the seismological results and estimate the possible error range. So we conducted 3D MHD simulations of propagating kink waves in stratified coronal open flux tubes. Following Magyar & Van Doorsselaere (2018), we employed forward modeling to compare the seismologically derived results with the actual values of physical parameters from our simulation. This paper is organized as follows: Section 2 describes our simulation setup and methodology, Section 3 presents the simulation and forward-modeling results, along with detailed comparisons between seismology results and input values, and Section 4 provides a discussion and summary of our findings.</text>
<section_header_level_1><location><page_3><loc_12><loc_69><loc_21><loc_71></location>2 METHOD</section_header_level_1>
<text><location><page_3><loc_12><loc_55><loc_79><loc_67></location>The model used in this study is a gravitationally stratified, open magnetic flux tube with a radius of 1 Mm, similar to that in Gao et al. (2024b) (hereafter referred to as Paper I). The main difference is that the initial magnetic field in this study is set at around 4 G, instead of 10 G, to better match the seismologically inferred results in Yang et al. (2020b). As in Paper I, the magnetic field is oriented in the z direction and remains nearly uniform across the simulation domain, with only small spatial gradients to maintain total pressure balance. As a result, the Alfv'en and kink speeds increase with height as the density decreases due to stratification. A relaxation process of 2400 s was conducted to achieve a quasimagnetohydrostatic state, as shown in Figure 1. The figure shows that the post-relaxation magnetic field has some spatial variation but mostly within 0.7 G.</text>
<text><location><page_3><loc_12><loc_51><loc_79><loc_54></location>The kink wave driver is also similar to that in Paper I, with a velocity amplitude of 8 km s -1 and a period of 300 s. The primary velocity perturbation is along the x direction.</text>
<text><location><page_3><loc_12><loc_36><loc_79><loc_50></location>We ran the 3D MHD simulation in Cartesian coordinates with the PLUTO code (Mignone et al. 2007). The simulation domain spans [-4, 4] Mm × [-4, 4] Mm × [0, 150] Mm, with a uniform grid of 128 × 128 × 1024 cells, providing spatial resolutions of 62.5 km in the horizontal ( x and y ) directions and 146.5 km in the vertical ( z ) direction. We chose a second-order parabolic spatial scheme and a Roe Riemann solver. The boundary conditions were set to be outflow, except for the lower boundary, where the kink wave driver was introduced by adjusting v x and v y , while other parameters were fixed. As in Paper I, the upper 50 Mm ( z > 100 Mm) was set as a velocity absorption region (VAR) to minimize numerical reflections from the upper boundary (see also Pelouze et al. 2023; Guo et al. 2023; Gao et al. 2023). For the subsequent analysis, we only considered the region below z = 100 Mmwhich can give us physical results.</text>
<text><location><page_3><loc_12><loc_25><loc_79><loc_35></location>Once the kink wave driver was applied, the propagating waves were rapidly excited, with their properties thoroughly analyzed in Paper I. Here, we focus on assessing the reliability of the seismology method described in Section 1 using the simulation outputs. To do so, we performed forward calculations to synthesize spectroscopic observables of CoMP and UCoMP, namely, the Fe XIII 1074.7 and 1079.8 nm spectral line profiles. Specifically, we used synthesized observation of the 1074.7 nm line to perform the wave tracking and determine the propagation speed, while synthesized observation of the 1079.8 nm line was only employed for the density diagnostic using the intensity ratio method (Yang et al. 2020a,b).</text>
<text><location><page_3><loc_12><loc_12><loc_79><loc_25></location>We applied the FoMo code (Van Doorsselaere et al. 2016) to synthesize the spectral profiles at all pixels in the yz plane. The photo-excitation (see Young et al. 2003; Yang et al. 2020a) was not considered for simplification, as their effects on the spectral lines are not significant at the lower corona. The line of sight (LOS) was chosen as the x axis. In this way, we can reconstruct 2D maps (in the yz plane) of the intensity and Doppler velocity by fitting a single Gaussian to each spectral profile. The resulting maps consist of 128 pixels in the y direction from -4 Mm to 4 Mm, and 500 pixels in the z direction from 0 Mmto 100 Mm. Since the primary velocity perturbation is along the x direction (or in the LOS plane), the Doppler velocity maps capture the wave propagation signals, while the intensity maps do not show any transverse displacement.</text>
<figure>
<location><page_4><loc_14><loc_48><loc_76><loc_87></location>
<caption>Fig. 1: Model of the open magnetic flux tube. (A) Variation at the x = 0 plane after relaxation. (B)-(C) Horizontal profiles of density and magnetic field along the x axis at three different heights (indicated by different line styles). (D) Vertical profile of density demonstrating gravitational stratification, with solid and dashed lines indicating density inside and outside the flux tube, respectively.</caption>
</figure>
<text><location><page_4><loc_14><loc_67><loc_15><loc_69></location>z (Mm)</text>
<section_header_level_1><location><page_4><loc_12><loc_36><loc_21><loc_37></location>3 RESULTS</section_header_level_1>
<section_header_level_1><location><page_4><loc_12><loc_33><loc_27><loc_35></location>3.1 Before Degrading</section_header_level_1>
<text><location><page_4><loc_12><loc_12><loc_79><loc_32></location>We first generated time-distance (TD) maps of the Doppler velocity along the z direction, which can be produced at each y position. In Figure 2(A), the TD map along the flux tube's axis (i.e., y = 0 ) is presented, clearly illustrating the propagation of Doppler velocity disturbances. The increasing slope reflects the growing propagation speed with height. However, after 400 s, unusual patterns appear due to wave reflections. Despite efforts to suppress numerical reflections from the upper boundary at z = 150 Mm using a VAR (see Section 2), it is still difficult to eliminate all reflections. Some reflections may come from the layer of z = 100 Mm, which is the lower boundary of the VAR; while others could be attributed to the vertical inhomogeneities of density and phase speed. As noted in previous studies, even smooth phase speed gradients can cause partial wave reflection (e.g., Verdini & Velli 2007; Hahn et al. 2018; Pascoe et al. 2022; Bose et al. 2024). In fact, real CoMP observations of coronal kink waves often reveal both upward and downward propagating components (Tomczyk & McIntosh 2009; Morton et al. 2015). However, when calculating the propagation speed, these reflected waves can introduce significant errors. To reduce this, we applied the Fourier filtering method to separate the upward and downward propagating wave components (see e.g., Tomczyk & McIntosh 2009; Threlfall et al. 2013; Liu et al.</text>
<figure>
<location><page_5><loc_12><loc_71><loc_78><loc_87></location>
<caption>Fig. 2: Time distance maps of the Doppler velocity along the z axis at y = 0 . (A) corresponds to the original forward modeling output, while (B) corresponds to the upward propagating component obtained from Fourier filtering.</caption>
</figure>
<figure>
<location><page_5><loc_12><loc_29><loc_79><loc_64></location>
<caption>Fig. 3: Seismology results for the case with original resolution (before degrading). (A) Spatial distribution of the phase speed. (B) Spatial distribution of the magnetic field ( B seis). (C) Spatial distribution of the relative error of B seis compared with the input magnetic field B los. Blue and red contours correspond to the relative error of -30% and 30%, respectively.</caption>
</figure>
<text><location><page_5><loc_12><loc_18><loc_79><loc_20></location>2014; Tiwari et al. 2019; Yang et al. 2020b). Figure 2(B) shows the TD map for the upward-propagating component, which is used to determine the phase speed.</text>
<text><location><page_5><loc_12><loc_12><loc_79><loc_17></location>By applying the wave-tracking method to all pixels (for more details, see e.g., Yang et al. 2020b), we obtained a 2D phase speed distribution, as shown in Figure 3(A). For most pixels, the phase speed falls in the range of 200-800 km s -1 (see also Figure 4(B)), with a general trend of increasing with altitude, though some fluctuations are present (see relevant discussions in Section 4).</text>
<figure>
<location><page_6><loc_13><loc_37><loc_77><loc_88></location>
<caption>Fig. 4: (A) 2D histogram comparing the density derived from the Fe XIII intensity ratio and the input density ρ los. (B) 2D histogram comparing the propagation speed obtained from wave tracking and the Alfv'en speed in the model. (C)-(D) Histograms of B seis and the relative error. The Gaussian fit results are overplotted, with the mean value ( µ ) and standard deviation ( σ ) indicated at the top.</caption>
</figure>
<text><location><page_6><loc_12><loc_20><loc_79><loc_27></location>Next, we derived the density by calculating the intensity ratio of the synthesized 1074.7 nm and 1079.8 nm intensity maps. The theoretical relationship between intensity ratio and density can be obtained from the CHIANTI database (Dere et al. 2023). The obtained density values were compared with the input values, as shown in Figure 4(A). The input density corresponds to the emissivity-weighted density along the LOS (Yang et al. 2024):</text>
<formula><location><page_6><loc_38><loc_15><loc_79><loc_19></location>ρ los = ∫ x ρ ( x ) ϵ ( x )d x ∫ x ϵ ( x )d x , (3)</formula>
<text><location><page_6><loc_12><loc_12><loc_79><loc_14></location>where ϵ is the Fe XIII 1074.7 nm line emissivity at the corresponding pixel, calculated using the IDL routine emiss calc.pro from the CHIANTI software package. The comparison shows that the density</text>
<figure>
<location><page_7><loc_13><loc_70><loc_31><loc_86></location>
</figure>
<figure>
<location><page_7><loc_59><loc_70><loc_76><loc_86></location>
</figure>
<figure>
<location><page_7><loc_36><loc_70><loc_54><loc_86></location>
<caption>Fig. 5: Phase speed, density, and magnetic field as a function of height ( z ). In panel (A), the red lines correspond to input values, including the internal Alfv'en speed (dot-dashed line), the external Alfv'en speed (dashed line), and the kink speed (solid line). The solid black line represents the wave propagation speed inferred from the wave tracking. In panel (B), the solid black line corresponds to the density derived from the Fe XIII intensity ratio ( ρ Fe XIII), and the dotted black line corresponds to the input value ( ρ los). The red lines represent the density in the model (similar to Figure 1(D)), including the internal density ρ i (dashed line), the external density ρ e (dotted line), and the average density ⟨ ρ ⟩ = ( ρ i + ρ e ) / 2 (solid line). In panel (C), the solid black line depicts the magnetic field inferred from coronal seismology ( B seis). The dashed and dotted red lines correspond to the internal and external magnetic fields ( B i and B e) in the model, respectively. The solid red line represents the input magnetic field ( B los).</caption>
</figure>
<text><location><page_7><loc_12><loc_48><loc_79><loc_52></location>derived from the intensity ratio is reliable for most pixels, though about 20% pixels show an overestimation ( ≳ 15%). However, this overestimation would not lead to large errors in the seismologically inferred magnetic field, as only the square root of density is needed during the calculation.</text>
<text><location><page_7><loc_12><loc_36><loc_79><loc_48></location>With the derived phase speed and density, we calculated the magnetic field B seis. Here we chose to treat the phase speed as the local Alfv'en speed (i.e., c ph = B seis / √ µ 0 ρ ), rather than employ Equation(1). Because in our magnetic field configuration, Equation (1) can only provide the phase speed as a function of z ; however, here with a high spatial resolution, we are also interested in the horizontal distribution of phase speed and magnetic field. We note that such a choice may lead to some errors, especially within the flux tube region, which will be further discussed in Section 4. In Figure 4(B), we compared the Alfv'en speed in the model and the wave propagation speed obtained from wave tracking. Despite some noticeable discrepancies, the overall correspondence between the two is reasonably good.</text>
<text><location><page_7><loc_12><loc_26><loc_79><loc_36></location>The calculated magnetic field is shown in Figure 3(B), and its relative error (compared to the input value, B los, which was calculated in the same manner as ρ los) is presented in Figure 3(C). The contours that correspond to the values of ± 30% indicate that the relative error is less than 30% for most pixels. Figure 4(C) and (D) display histograms of B seis and its relative error. Statistically, there is a slight overestimation of the magnetic field by ∼ 5% (with a standard deviation of 17%). Given that the 5% bias is quite small, the results support the reliability of the coronal seismology technique. Further discussion on the error distributions in Figure 3(C) and the cause of the overestimation can be found in Section 4.</text>
<section_header_level_1><location><page_7><loc_12><loc_23><loc_26><loc_24></location>3.2 After Degrading</section_header_level_1>
<text><location><page_7><loc_12><loc_15><loc_79><loc_22></location>We then degraded the spatial and temporal resolutions to match those of CoMP observations, with a pixel size of approximately 3.3 Mm and a cadence of 36 s. We focused on the central pixels (from y = -1 . 65 Mm to y = 1 . 65 Mm), which fully encompass the flux tube (diameter ∼ 2 Mm). At this resolution, the tube boundary and finer structures are unresolved, similar to the case in observational studies (Yang et al. 2020a,b).</text>
<text><location><page_7><loc_12><loc_12><loc_79><loc_14></location>We can now track the physical parameters along the tube axis. Figure 5 shows the phase speed, density, and magnetic field as a function of height z . In panel (A), the phase speed derived using the</text>
<text><location><page_8><loc_12><loc_83><loc_79><loc_87></location>wave tracking method ( c seis) is compared to the characteristic speeds in the model (input values). The c seis closely matches the input kink speed ( c k, input), which lies between the internal and external Alfv'en speeds.</text>
<text><location><page_8><loc_12><loc_70><loc_79><loc_82></location>Panel (B) shows the density derived from the Fe XIII intensity ratio ( ρ Fe XIII), which also falls between the internal density ( ρ i) and external density ( ρ e) in the model. The ρ Fe XIII is slightly higher than the emissivity-weighted density along the LOS ( ρ los), similar to the pre-degradation results shown in Figure 4(A). When calculating the magnetic field with Equation (2), the density value should ideally be the average of internal and external density (i.e., ⟨ ρ ⟩ = ( ρ i + ρ e ) / 2 ), because Equation (2) is derived from Equation (1) when assuming B i = B e. However, in practice, ρ Fe XIII is used, which is slightly lower than ⟨ ρ ⟩ . It means that using ρ Fe XIII as a representative of ⟨ ρ ⟩ can lead to a slight underestimation. The density underestimation is about 12-20% based on Figure 5(B), and the resulting error in B seis would be minimal (less than 5%) since the density is taken the square root of.</text>
<text><location><page_8><loc_12><loc_62><loc_79><loc_69></location>In fact, the deviation of ρ Fe XIII from ⟨ ρ ⟩ can vary with the filling factor, which represents the fraction of the pixel occupied by the flux tube. Given the current pixel size (3.3 Mm), LOS integration length (8 Mm), and flux tube radius (1 Mm), the filling factor is approximately 12%. If the filling factor is lower, the ρ Fe XIII will be closer to ρ e, increasing the density underestimation. The maximum underestimation depends on the density contrast ζ = ρ i /ρ e, with the deviation factor given by:</text>
<formula><location><page_8><loc_33><loc_58><loc_79><loc_61></location>α = ρ Fe XIII ⟨ ρ ⟩ ∼ ρ e ( ρ e + ρ i ) / 2 = 2 1 + ζ . (4)</formula>
<text><location><page_8><loc_12><loc_50><loc_79><loc_57></location>In our simulation, ζ was initially set to be 3, but after relaxation, it decreased to around 2 (see Figure 1(D)). Such a value is comparable with previous observational estimates (Tian et al. 2012; Verwichte et al. 2013; Morton et al. 2021). We tested the case with a reduced filling factor ( 5%), where a density contrast of 2 led to an underestimation of ∼ 30%. Therefore, parameters like the filling factor and density contrast can be crucial when assessing the accuracy of coronal magnetic field measurements.</text>
<text><location><page_8><loc_12><loc_44><loc_79><loc_49></location>Figure 5(C) presents the magnetic field inferred from coronal seismology ( B seis), compared to the internal ( B i), external ( B e), and emissivity-weighted ( B los) magnetic fields. The B seis ranges from 3.9 to 5.1 G, with errors below 15%, indicating that the technique of coronal seismolgy can be used for reliable measurements of the coronal magnetic field strengths.</text>
<section_header_level_1><location><page_8><loc_12><loc_41><loc_39><loc_42></location>4 DISCUSSION AND CONCLUSION</section_header_level_1>
<text><location><page_8><loc_12><loc_35><loc_79><loc_39></location>In this study, we tested the accuracy of the previously developed seismological techniques based on observations of propagating kink waves for deriving coronal magnetic fields. The results indicate that seismology-based measurements of the magnetic field are accurate and reliable to a large extent.</text>
<text><location><page_8><loc_12><loc_12><loc_79><loc_35></location>In the high-resolution case, we obtained a 2D distribution of the magnetic field ( B seis) and its relative error. For most regions, the relative error is less than 30%, as shown in Figure 3(C). Statistically, the average magnetic field derived from coronal seismology is about 5% larger than the input value. Although this case has a much higher resolution than the CoMP and UCoMP observations, the pixel size (62.5 km in the z direction and 200 km in the y direction) and cadence (12 s) can be comparable to those of the Cryogenic Near-Infrared Spectro-Polarimeter (Cryo-NIRSP; Fehlmann et al. 2023) and the Diffraction-Limited Near-Infrared Spectropolarimeter (DL-NIRSP; Jaeggli et al. 2022) of the Daniel K. Inouye Solar Telescope (DKIST; Rimmele et al. 2020). The intruments offer high-resolution coronal spectroscopic observations with the Fe XIII lines (Schad et al. 2023), and recently Schad et al. (2024) successfully obtained a coronal LOS magnetogram with Cryo-NIRSP observation based on the Zeeman effect. With rapid repeated raster scans, it is possible that DKIST could also detect propagating transverse waves via Doppler velocity measurements, enabling seismological diagnostics of the plane-of-sky (POS) coronal magnetic field. In this way, DKIST observation can also provide us with maps of the POS magnetic field, but with a much higher spatial resolution compared to CoMP and UCoMP. Our results, particularly Figure 3, can serve as a reference for such diagnostics. Combined with the Stokes-V measurements, DKIST may then be able to achieve measurements of the full magnetic field vector.</text>
<text><location><page_9><loc_12><loc_71><loc_79><loc_87></location>Nevertheless, we note that some errors appear in the seismologically inferred magnetic field. First, large errors appeared near the upper and lower boundaries (around z = 0 and z = 100 Mm). This is due to the Fourier filtering method used to subtract downward propagating wave components. As shown in Figure 2(B), the upward propagating components of the Doppler velocity show some artificial velocity amplification near the upper and lower boundaries, especially after 250 s. This could affect the phase speed determination through wave tracking, leading to errors in B seis near the z boundaries. Thus, in observations, it may be useful to exclude boundary signals before calculating phase speeds. Another contributing factor comes from the wave-tracking process itself. When calculating the phase speed, a certain number of data points along the propagating direction is required. Near the lower and upper boundaries, fewer data points will be available to perform cross-correlation, leading to larger uncertainties in the calculated phase speed and consequently in the inferred magnetic field.</text>
<text><location><page_9><loc_12><loc_57><loc_79><loc_71></location>Second, overestimations were observed inside or along the lateral boundaries of the flux tube. This is likely because we treat the phase velocity as the Alfv'en speed, which applies well to the regions away from the flux tube or cases with spatial averaging (see Yang et al. 2020b and Section 3.2). However, in this case, we modeled kink waves, and the flux tube can be well resolved. Thus, at the flux tube region, particularly the tube boundary, it would be more appropriate to apply Equation (1) since the waves have a dominant kink wave characteristic (e.g., Goossens et al. 2009). Calculating the magnetic field with B seis = c ph √ µ 0 ρ leads to overestimation at high-density regions, explaining the positive errors within the flux tube in Figure 3(C). In future DKIST observations, we would suggest first applying Equation (2) to diagnose the magnetic field outside the flux tube ( B e), then calculate the magnetic field inside the flux tube ( B i) with Equation (1).</text>
<text><location><page_9><loc_12><loc_51><loc_79><loc_56></location>In addition, there are some other confusing patterns in Figure 3(B) and (C). For instance, B seis and the relative error manifest fluctuations along the z direction. It might be related to longitudinal oscillations excited by kink waves due to some non-linear effects (e.g., Goldstein 1978; Del Zanna et al. 2001; Terradas & Ofman 2004). Further investigations are needed to understand these patterns.</text>
<text><location><page_9><loc_12><loc_38><loc_79><loc_51></location>When we degraded the spatial and temporal resolutions to approximately match those of CoMP, we found that the distributions of phase speed, density, and B seis along z all show remarkable similarities to the input values (Figure 5). Again we can notice a deviation near the upper and lower boundaries, particularly the lower one. However, within the height range of 20-80 Mm, the magnetic field error is generally less than 10%. Additionally, the CoMP instrument has recently been upgraded to UCoMP, which has a slightly higher spatial resolution and a larger field of view. Our main conclusions should also apply to the case with the UCoMP observations (Yang et al. 2024). We also tested the case when degrading to UCoMP's pixel size ( ∼ 2.2 Mm), and the results are largely similar to those shown in Figure 5, with a slightly smaller error.</text>
<text><location><page_9><loc_12><loc_22><loc_79><loc_38></location>Another factor that may impact the phase speed and magnetic field measurements is the magnitude of the input magnetic field and phase speed. A stronger magnetic field and higher phase speed result in steeper slopes in the time-distance maps of Doppler velocity, which can reduce the accuracy of the wave-tracking method, particularly when the spatial and temporal resolutions are degraded. We ran a separate simulation with a background magnetic field of ∼ 10 G, which gives phase speeds of 1-2 Mm s -1 . We found a systematic underestimation of the magnetic field by approximately 30% in this case. Nevertheless, the observed coronal magnetic field around 1 . 05 -1 . 5 R ⊙ is 1-4 G (Lin et al. 2004; Gopalswamy et al. 2012; Kumari et al. 2019; Yang et al. 2020b; Zhong et al. 2023), which is more comparable with the case discussed in Section 3. In conclusion, the coronal seismology technique based on propagating kink waves can provide reliable magnetic field measurements according to our forward modeling.</text>
<text><location><page_9><loc_12><loc_12><loc_79><loc_22></location>We note that this study only focuses on open coronal structures, including plumes in coronal holes and fan loops at the boundaries of active regions (Morton et al. 2015; Banerjee et al. 2021). Specifically, in our simulation, the magnetic flux tube is perpendicular to the solar surface, with both gravity and magnetic field aligned along the tube axis. However, the propagating kink waves can be detected not only in these structures but also in closed-field regions (e.g., Tomczyk et al. 2007; Yang et al. 2020a,b). Our magnetic configuration can roughly describe one leg of a large-scale closed coronal loop, where the curvature can be neglected. For smaller loops where curvature cannot be ignored, our model is no</text>
<text><location><page_10><loc_12><loc_80><loc_79><loc_87></location>longer applicable due to differences in gravitational stratification. Nevertheless, for such loops, phase speed along the axis often shows minimal variation McIntosh et al. (2011); Threlfall et al. (2013); Zhong et al. (2023), thus similar to the case in Magyar & Van Doorsselaere (2018), where a model without any vertical gradient in phase speed was used. Therefore, this study complements previous research by highlighting the importance of phase speed variation along the flux tube in coronal seismology.</text>
<text><location><page_10><loc_12><loc_65><loc_79><loc_80></location>Finally, we would like to mention that our model has some limitations. The vertical gradient of the magnetic field and the magnetic expansion are not included. Additionally, we didn't consider the effect of internal flows along the flux tube, which are frequently reported and can affect the apparent wave propagation speed (e.g., Soler et al. 2011; Morton et al. 2015). Moreover, in real observations, there are often multiple flux tubes overlapping along the LOS, introducing further complexities that may affect wave-tracking accuracy and magnetic field measurements. In Figure 5 of Yang et al. (2024), comparisons between seismological results and global coronal MHD models revealed some discrepancies, particularly at higher latitudes where open field lines may dominate. Future work that incorporates more realistic models could offer a more sophisticated evaluation of the magnetic field measurements through coronal seismology and help clarify the discrepancies in Yang et al. (2024).</text>
<text><location><page_10><loc_12><loc_44><loc_79><loc_64></location>Acknowledgements This work is supported by the National Natural Science Foundation of China grant 12425301 and the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB0560000). M.G. acknowledges support from the National Natural Science Foundation of China (NSFC, 12203030). T.V.D was supported by the C1 grant TRACEspace of Internal Funds KU Leuven and a Senior Research Project (G088021N) of the FWO Vlaanderen. Furthermore, TVD received financial support from the Flemish Government under the long-term structural Methusalem funding program, project SOUL: Stellar evolution in full glory, grant METH/24/012 at KU Leuven. The research that led to these results was subsidised by the Belgian Federal Science Policy Office through the contract B2/223/P1/CLOSE-UP. It is also part of the DynaSun project and has thus received funding under the Horizon Europe programme of the European Union under grant agreement (no. 101131534). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union and therefore the European Union cannot be held responsible for them. K.K. acknowledges support by an FWO (Fonds voor Wetenschappelijk Onderzoek - Vlaanderen) postdoctoral fellowship (1273221N).</text>
<section_header_level_1><location><page_10><loc_12><loc_40><loc_19><loc_42></location>References</section_header_level_1>
<text><location><page_10><loc_12><loc_12><loc_74><loc_39></location>Aschwanden, M. J., De Pontieu, B., Schrijver, C. J., & Title, A. M. 2002, Sol. Phys., 206, 99 2 Banerjee, D., Krishna Prasad, S., Pant, V., et al. 2021, Space Sci. Rev., 217, 76 9 Bose, S., TenBarge, J. M., Carter, T., et al. 2024, ApJ, 971, 72 4 Chen, B., Shen, C., Gary, D. E., et al. 2020, Nature Astronomy, 4, 1140 1 Chen, S.-X., Li, B., Xiong, M., Yu, H., & Guo, M.-Z. 2015, ApJ, 812, 22 2 Chen, Y., Feng, S. W., Li, B., et al. 2011, ApJ, 728, 147 2 Chen, Y., Li, W., Tian, H., et al. 2023, Research in Astronomy and Astrophysics, 23, 022001 1 Chen, Y., Li, W., Tian, H., et al. 2021, ApJ, 920, 116 1 Del Zanna, L., Velli, M., & Londrillo, P. 2001, A&A, 367, 705 9 Dere, K. P., Del Zanna, G., Young, P. R., & Landi, E. 2023, ApJS, 268, 52 6 Fehlmann, A., Kuhn, J. R., Schad, T. A., et al. 2023, Sol. Phys., 298, 5 8 Fleishman, G. D., Gary, D. E., Chen, B., et al. 2020, Science, 367, 278 1 Gao, Y., Guo, M., Van Doorsselaere, T., Tian, H., & Skirvin, S. J. 2023, ApJ, 955, 73 3 Gao, Y., Hou, Z., Van Doorsselaere, T., & Guo, M. 2024a, A&A, 681, L4 2 Gao, Y., Tian, H., Van Doorsselaere, T., & Chen, Y. 2022, ApJ, 930, 55 2 Gao, Y., Van Doorsselaere, T., Tian, H., Guo, M., & Karampelas, K. 2024b, A&A, 689, A195 3 Goldstein, M. L. 1978, ApJ, 219, 700 9 Goossens, M., Terradas, J., Andries, J., Arregui, I., & Ballester, J. L. 2009, A&A, 503, 213 9 Gopalswamy, N., Nitta, N., Akiyama, S., M¨akel¨a, P., & Yashiro, S. 2012, ApJ, 744, 72 9</text>
<table>
<location><page_11><loc_12><loc_12><loc_80><loc_87></location>
</table>
<unordered_list>
<list_item><location><page_12><loc_12><loc_86><loc_57><loc_87></location>Young, P. R., Del Zanna, G., Landi, E., et al. 2003, ApJS, 144, 135 3</list_item>
<list_item><location><page_12><loc_12><loc_84><loc_64><loc_85></location>Zhang, Q. M., Chen, J. L., Li, S. T., Lu, L., & Li, D. 2022, Sol. Phys., 297, 18 2</list_item>
<list_item><location><page_12><loc_12><loc_83><loc_79><loc_84></location>Zhong, S., Nakariakov, V. M., Miao, Y., Fu, L., & Yuan, D. 2023, Scientific Reports, 13, 12963 2, 9, 10</list_item>
</document>
[
{
"title": "ABSTRACT",
"content": "R esearch in A stronomy and A strophysics",
"pages": [
1
]
},
{
"title": "Measurements of the solar coronal magnetic field based on coronal seismology with propagating Alfv'enic waves: forward modeling",
"content": "Yuhang Gao 1 , 2 , Hui Tian 1 , Tom Van Doorsselaere 2 , Zihao Yang 1 , 3 , Mingzhe Guo 2 , 4 and Konstantinos Karampelas 2 Received 20xx month day; accepted 20xx month day Abstract Recent observations have demonstrated the capability of mapping the solar coronal magnetic field using the technique of coronal seismology based on the ubiquitous propagating Alfv'enic/kink waves through imaging spectroscopy. We established a magnetohydrodynamic (MHD) model of a gravitationally stratified open magnetic flux tube, exciting kink waves propagating upwards along the tube. Forward modeling was performed to synthesize the Fe XIII 1074.7 and 1079.8 nm spectral line profiles, which were then used to determine the wave phase speed, plasma density, and magnetic field with seismology method. A comparison between the seismologically inferred results and the corresponding input values verifies the reliability of the seismology method. In addition, we also identified some factors that could lead to errors during magnetic field measurements. Our results may serve as a valuable reference for current and future coronal magnetic field measurements based on observations of propagating kink waves. Key words: Sun: corona - Sun: magnetic fields - magnetohydrodynamics",
"pages": [
1
]
},
{
"title": "1 INTRODUCTION",
"content": "The magnetic field plays a crucial role in various physical processes in the solar and stellar coronae. The dissipation of magnetic energy is believed to drive solar eruptive events (e.g., flares and coronal mass ejections) and cause heating of the corona. While the magnetic field in the lower solar atmosphere can be reliably measured through spectro-polarimetric observations, direct measurements of the coronal magnetic field have remained challenging for decades. Several methods, including the spectro-polarimetry of coronal infrared lines (Lin et al. 2000, 2004; Schad et al. 2024), coronal radio observations (e.g., Fleishman et al. 2020; Chen et al. 2020; Tan 2022), and magnetic-field-induced transitions of extreme ultraviolet emission lines (e.g., Li et al. 2015, 2016; Landi et al. 2020; Chen et al. 2021, 2023), have been proposed and attempts of measurements have been made. However, these approaches all face limitations and none of them could be used for routine measurements of the global coronal magnetic field. Another technique that could be used to measure the coronal magnetic field is coronal seismology. This technique combines the magneto-hydrodynamic (MHD) wave theory with observed wave parameters (e.g., period, amplitude, propagation speed, damping time) to diagnose various physical properties, particularly the magnetic field. Different wave phenomena in the corona have been used for coronal seismology, including standing kink waves in magnetic loops (e.g. Nakariakov & Ofman 2001; Aschwanden et al. 2002; Van Doorsselaere et al. 2007; Tian et al. 2012; Zhang et al. 2022; Gao et al. 2022, 2024a; Zhong et al. 2023; Li & Long 2023, to name but a few), propagating slow magneto-acoustic waves (e.g., Jess et al. 2016), sausage waves (Chen et al. 2015; Guo et al. 2016; see Li et al. 2020 for a review), propagating kink waves in streamers (Chen et al. 2011; Guo et al. 2022), and torsional oscillations in solar surges (Kohutova et al. 2020). However, these studies provided only one-dimensional (1D) distributions or single values of the magnetic field in specific coronal structures, such as oscillating loops or streamers. To create global 2D magnetic field maps, we need to utilize ubiquitous and continuous wave phenomena. The pervasive propagating disturbances in Dopplergrams (Tomczyk et al. 2007; Tomczyk & McIntosh 2009; Liu et al. 2015; Morton et al. 2015, 2019) observed by the Coronal Multi-channel Polarimeter (CoMP; Tomczyk et al. 2008) are ideal for this purpose. These propagating disturbances are interpreted as kink or Alfv'enic waves (Van Doorsselaere et al. 2008), and their propagation speeds are naturally linked to the local magnetic field. Based on these CoMP observations, Yang et al. (2020b,a) have successfully measured the global distribution of the coronal magnetic field for the first time. Following these successful attempts, a routine (continuous) measurement of the global coronal magnetic field based on similar observations from the Upgraded CoMP (UCoMP; Landi et al. 2016) has been recently achieved, which allows for the construction of coronal synoptic magnetograms (Carrington maps) (Yang et al. 2024). The CoMP and UCoMP instruments can conduct spectroscopic observations of the Fe XIII lines at 1074.7 and 1079.8 nm in the coronal region above the solar limb. From the Dopplergrams of Fe XIII 1074.7 nm, propagating kink waves can be identified throughout the corona. The propagating or phase speed c k of these waves (also named the kink speed) could be obtained by constructing a timedistance map of Doppler velocity, while the coronal density can be inferred from the observed Fe XIII 1079.8-nm/1074.7-nm intensity ratio. For kink waves, we have where µ 0 is the magnetic permeability, B and ρ are the magnetic field and mass density, respectively. The subscripts i and e refer to parameters inside and outside the magnetic flux tubes (the waveguides). Since CoMP and UCoMP likely cannot resolve individual flux tubes, we can only work with an average density ⟨ ρ ⟩ . Furthermore, in the lowβ coronal environment, the internal and external magnetic fields are often assumed to be approximately equal (see e.g., Tomczyk & McIntosh 2009; Morton et al. 2015; Zhong et al. 2023). This leads to the simplified expression for the kink speed: which is widely used in estimations of coronal magnetic fields (Long et al. 2017; Yang et al. 2020a,b; Yang et al. 2024). Given the potential of these measurements to provide routine coronal magnetograms on a daily basis, which could play a crucial role in future solar physics research, it is essential to thoroughly assess the reliability and robustness of the methodology used in Yang et al. (2020a,b) and Yang et al. (2024). Magyar & Van Doorsselaere (2018) performed 3D MHD simulations of propagating kink waves under various conditions and conducted forward modeling to evaluate the reliability of this method in deriving the magnetic field strength. Their findings indicated that the magnetic field strengths inferred through seismology closely match the input values, typically with an error less than ∼ 20%. However, there is a limitation in their simulation. They utilized a non-stratified setup that excluded the effects of gravity, resulting in uniform initial density and propagation speed in the vertical direction. The gravitational stratification can play a significant role as it causes the Alfv'en speed and kink speed c k to vary with height. This variation can affect the wave tracking method which typically relies on a linear fit of velocity signals (e.g., see Figure 6 in Tomczyk & McIntosh 2009). Therefore, it is important to assess how the gravitational stratification affects the seismological results and estimate the possible error range. So we conducted 3D MHD simulations of propagating kink waves in stratified coronal open flux tubes. Following Magyar & Van Doorsselaere (2018), we employed forward modeling to compare the seismologically derived results with the actual values of physical parameters from our simulation. This paper is organized as follows: Section 2 describes our simulation setup and methodology, Section 3 presents the simulation and forward-modeling results, along with detailed comparisons between seismology results and input values, and Section 4 provides a discussion and summary of our findings.",
"pages": [
1,
2,
3
]
},
{
"title": "2 METHOD",
"content": "The model used in this study is a gravitationally stratified, open magnetic flux tube with a radius of 1 Mm, similar to that in Gao et al. (2024b) (hereafter referred to as Paper I). The main difference is that the initial magnetic field in this study is set at around 4 G, instead of 10 G, to better match the seismologically inferred results in Yang et al. (2020b). As in Paper I, the magnetic field is oriented in the z direction and remains nearly uniform across the simulation domain, with only small spatial gradients to maintain total pressure balance. As a result, the Alfv'en and kink speeds increase with height as the density decreases due to stratification. A relaxation process of 2400 s was conducted to achieve a quasimagnetohydrostatic state, as shown in Figure 1. The figure shows that the post-relaxation magnetic field has some spatial variation but mostly within 0.7 G. The kink wave driver is also similar to that in Paper I, with a velocity amplitude of 8 km s -1 and a period of 300 s. The primary velocity perturbation is along the x direction. We ran the 3D MHD simulation in Cartesian coordinates with the PLUTO code (Mignone et al. 2007). The simulation domain spans [-4, 4] Mm × [-4, 4] Mm × [0, 150] Mm, with a uniform grid of 128 × 128 × 1024 cells, providing spatial resolutions of 62.5 km in the horizontal ( x and y ) directions and 146.5 km in the vertical ( z ) direction. We chose a second-order parabolic spatial scheme and a Roe Riemann solver. The boundary conditions were set to be outflow, except for the lower boundary, where the kink wave driver was introduced by adjusting v x and v y , while other parameters were fixed. As in Paper I, the upper 50 Mm ( z > 100 Mm) was set as a velocity absorption region (VAR) to minimize numerical reflections from the upper boundary (see also Pelouze et al. 2023; Guo et al. 2023; Gao et al. 2023). For the subsequent analysis, we only considered the region below z = 100 Mmwhich can give us physical results. Once the kink wave driver was applied, the propagating waves were rapidly excited, with their properties thoroughly analyzed in Paper I. Here, we focus on assessing the reliability of the seismology method described in Section 1 using the simulation outputs. To do so, we performed forward calculations to synthesize spectroscopic observables of CoMP and UCoMP, namely, the Fe XIII 1074.7 and 1079.8 nm spectral line profiles. Specifically, we used synthesized observation of the 1074.7 nm line to perform the wave tracking and determine the propagation speed, while synthesized observation of the 1079.8 nm line was only employed for the density diagnostic using the intensity ratio method (Yang et al. 2020a,b). We applied the FoMo code (Van Doorsselaere et al. 2016) to synthesize the spectral profiles at all pixels in the yz plane. The photo-excitation (see Young et al. 2003; Yang et al. 2020a) was not considered for simplification, as their effects on the spectral lines are not significant at the lower corona. The line of sight (LOS) was chosen as the x axis. In this way, we can reconstruct 2D maps (in the yz plane) of the intensity and Doppler velocity by fitting a single Gaussian to each spectral profile. The resulting maps consist of 128 pixels in the y direction from -4 Mm to 4 Mm, and 500 pixels in the z direction from 0 Mmto 100 Mm. Since the primary velocity perturbation is along the x direction (or in the LOS plane), the Doppler velocity maps capture the wave propagation signals, while the intensity maps do not show any transverse displacement. z (Mm)",
"pages": [
3,
4
]
},
{
"title": "3.1 Before Degrading",
"content": "We first generated time-distance (TD) maps of the Doppler velocity along the z direction, which can be produced at each y position. In Figure 2(A), the TD map along the flux tube's axis (i.e., y = 0 ) is presented, clearly illustrating the propagation of Doppler velocity disturbances. The increasing slope reflects the growing propagation speed with height. However, after 400 s, unusual patterns appear due to wave reflections. Despite efforts to suppress numerical reflections from the upper boundary at z = 150 Mm using a VAR (see Section 2), it is still difficult to eliminate all reflections. Some reflections may come from the layer of z = 100 Mm, which is the lower boundary of the VAR; while others could be attributed to the vertical inhomogeneities of density and phase speed. As noted in previous studies, even smooth phase speed gradients can cause partial wave reflection (e.g., Verdini & Velli 2007; Hahn et al. 2018; Pascoe et al. 2022; Bose et al. 2024). In fact, real CoMP observations of coronal kink waves often reveal both upward and downward propagating components (Tomczyk & McIntosh 2009; Morton et al. 2015). However, when calculating the propagation speed, these reflected waves can introduce significant errors. To reduce this, we applied the Fourier filtering method to separate the upward and downward propagating wave components (see e.g., Tomczyk & McIntosh 2009; Threlfall et al. 2013; Liu et al. 2014; Tiwari et al. 2019; Yang et al. 2020b). Figure 2(B) shows the TD map for the upward-propagating component, which is used to determine the phase speed. By applying the wave-tracking method to all pixels (for more details, see e.g., Yang et al. 2020b), we obtained a 2D phase speed distribution, as shown in Figure 3(A). For most pixels, the phase speed falls in the range of 200-800 km s -1 (see also Figure 4(B)), with a general trend of increasing with altitude, though some fluctuations are present (see relevant discussions in Section 4). Next, we derived the density by calculating the intensity ratio of the synthesized 1074.7 nm and 1079.8 nm intensity maps. The theoretical relationship between intensity ratio and density can be obtained from the CHIANTI database (Dere et al. 2023). The obtained density values were compared with the input values, as shown in Figure 4(A). The input density corresponds to the emissivity-weighted density along the LOS (Yang et al. 2024): where ϵ is the Fe XIII 1074.7 nm line emissivity at the corresponding pixel, calculated using the IDL routine emiss calc.pro from the CHIANTI software package. The comparison shows that the density derived from the intensity ratio is reliable for most pixels, though about 20% pixels show an overestimation ( ≳ 15%). However, this overestimation would not lead to large errors in the seismologically inferred magnetic field, as only the square root of density is needed during the calculation. With the derived phase speed and density, we calculated the magnetic field B seis. Here we chose to treat the phase speed as the local Alfv'en speed (i.e., c ph = B seis / √ µ 0 ρ ), rather than employ Equation(1). Because in our magnetic field configuration, Equation (1) can only provide the phase speed as a function of z ; however, here with a high spatial resolution, we are also interested in the horizontal distribution of phase speed and magnetic field. We note that such a choice may lead to some errors, especially within the flux tube region, which will be further discussed in Section 4. In Figure 4(B), we compared the Alfv'en speed in the model and the wave propagation speed obtained from wave tracking. Despite some noticeable discrepancies, the overall correspondence between the two is reasonably good. The calculated magnetic field is shown in Figure 3(B), and its relative error (compared to the input value, B los, which was calculated in the same manner as ρ los) is presented in Figure 3(C). The contours that correspond to the values of ± 30% indicate that the relative error is less than 30% for most pixels. Figure 4(C) and (D) display histograms of B seis and its relative error. Statistically, there is a slight overestimation of the magnetic field by ∼ 5% (with a standard deviation of 17%). Given that the 5% bias is quite small, the results support the reliability of the coronal seismology technique. Further discussion on the error distributions in Figure 3(C) and the cause of the overestimation can be found in Section 4.",
"pages": [
4,
5,
6,
7
]
},
{
"title": "3.2 After Degrading",
"content": "We then degraded the spatial and temporal resolutions to match those of CoMP observations, with a pixel size of approximately 3.3 Mm and a cadence of 36 s. We focused on the central pixels (from y = -1 . 65 Mm to y = 1 . 65 Mm), which fully encompass the flux tube (diameter ∼ 2 Mm). At this resolution, the tube boundary and finer structures are unresolved, similar to the case in observational studies (Yang et al. 2020a,b). We can now track the physical parameters along the tube axis. Figure 5 shows the phase speed, density, and magnetic field as a function of height z . In panel (A), the phase speed derived using the wave tracking method ( c seis) is compared to the characteristic speeds in the model (input values). The c seis closely matches the input kink speed ( c k, input), which lies between the internal and external Alfv'en speeds. Panel (B) shows the density derived from the Fe XIII intensity ratio ( ρ Fe XIII), which also falls between the internal density ( ρ i) and external density ( ρ e) in the model. The ρ Fe XIII is slightly higher than the emissivity-weighted density along the LOS ( ρ los), similar to the pre-degradation results shown in Figure 4(A). When calculating the magnetic field with Equation (2), the density value should ideally be the average of internal and external density (i.e., ⟨ ρ ⟩ = ( ρ i + ρ e ) / 2 ), because Equation (2) is derived from Equation (1) when assuming B i = B e. However, in practice, ρ Fe XIII is used, which is slightly lower than ⟨ ρ ⟩ . It means that using ρ Fe XIII as a representative of ⟨ ρ ⟩ can lead to a slight underestimation. The density underestimation is about 12-20% based on Figure 5(B), and the resulting error in B seis would be minimal (less than 5%) since the density is taken the square root of. In fact, the deviation of ρ Fe XIII from ⟨ ρ ⟩ can vary with the filling factor, which represents the fraction of the pixel occupied by the flux tube. Given the current pixel size (3.3 Mm), LOS integration length (8 Mm), and flux tube radius (1 Mm), the filling factor is approximately 12%. If the filling factor is lower, the ρ Fe XIII will be closer to ρ e, increasing the density underestimation. The maximum underestimation depends on the density contrast ζ = ρ i /ρ e, with the deviation factor given by: In our simulation, ζ was initially set to be 3, but after relaxation, it decreased to around 2 (see Figure 1(D)). Such a value is comparable with previous observational estimates (Tian et al. 2012; Verwichte et al. 2013; Morton et al. 2021). We tested the case with a reduced filling factor ( 5%), where a density contrast of 2 led to an underestimation of ∼ 30%. Therefore, parameters like the filling factor and density contrast can be crucial when assessing the accuracy of coronal magnetic field measurements. Figure 5(C) presents the magnetic field inferred from coronal seismology ( B seis), compared to the internal ( B i), external ( B e), and emissivity-weighted ( B los) magnetic fields. The B seis ranges from 3.9 to 5.1 G, with errors below 15%, indicating that the technique of coronal seismolgy can be used for reliable measurements of the coronal magnetic field strengths.",
"pages": [
7,
8
]
},
{
"title": "4 DISCUSSION AND CONCLUSION",
"content": "In this study, we tested the accuracy of the previously developed seismological techniques based on observations of propagating kink waves for deriving coronal magnetic fields. The results indicate that seismology-based measurements of the magnetic field are accurate and reliable to a large extent. In the high-resolution case, we obtained a 2D distribution of the magnetic field ( B seis) and its relative error. For most regions, the relative error is less than 30%, as shown in Figure 3(C). Statistically, the average magnetic field derived from coronal seismology is about 5% larger than the input value. Although this case has a much higher resolution than the CoMP and UCoMP observations, the pixel size (62.5 km in the z direction and 200 km in the y direction) and cadence (12 s) can be comparable to those of the Cryogenic Near-Infrared Spectro-Polarimeter (Cryo-NIRSP; Fehlmann et al. 2023) and the Diffraction-Limited Near-Infrared Spectropolarimeter (DL-NIRSP; Jaeggli et al. 2022) of the Daniel K. Inouye Solar Telescope (DKIST; Rimmele et al. 2020). The intruments offer high-resolution coronal spectroscopic observations with the Fe XIII lines (Schad et al. 2023), and recently Schad et al. (2024) successfully obtained a coronal LOS magnetogram with Cryo-NIRSP observation based on the Zeeman effect. With rapid repeated raster scans, it is possible that DKIST could also detect propagating transverse waves via Doppler velocity measurements, enabling seismological diagnostics of the plane-of-sky (POS) coronal magnetic field. In this way, DKIST observation can also provide us with maps of the POS magnetic field, but with a much higher spatial resolution compared to CoMP and UCoMP. Our results, particularly Figure 3, can serve as a reference for such diagnostics. Combined with the Stokes-V measurements, DKIST may then be able to achieve measurements of the full magnetic field vector. Nevertheless, we note that some errors appear in the seismologically inferred magnetic field. First, large errors appeared near the upper and lower boundaries (around z = 0 and z = 100 Mm). This is due to the Fourier filtering method used to subtract downward propagating wave components. As shown in Figure 2(B), the upward propagating components of the Doppler velocity show some artificial velocity amplification near the upper and lower boundaries, especially after 250 s. This could affect the phase speed determination through wave tracking, leading to errors in B seis near the z boundaries. Thus, in observations, it may be useful to exclude boundary signals before calculating phase speeds. Another contributing factor comes from the wave-tracking process itself. When calculating the phase speed, a certain number of data points along the propagating direction is required. Near the lower and upper boundaries, fewer data points will be available to perform cross-correlation, leading to larger uncertainties in the calculated phase speed and consequently in the inferred magnetic field. Second, overestimations were observed inside or along the lateral boundaries of the flux tube. This is likely because we treat the phase velocity as the Alfv'en speed, which applies well to the regions away from the flux tube or cases with spatial averaging (see Yang et al. 2020b and Section 3.2). However, in this case, we modeled kink waves, and the flux tube can be well resolved. Thus, at the flux tube region, particularly the tube boundary, it would be more appropriate to apply Equation (1) since the waves have a dominant kink wave characteristic (e.g., Goossens et al. 2009). Calculating the magnetic field with B seis = c ph √ µ 0 ρ leads to overestimation at high-density regions, explaining the positive errors within the flux tube in Figure 3(C). In future DKIST observations, we would suggest first applying Equation (2) to diagnose the magnetic field outside the flux tube ( B e), then calculate the magnetic field inside the flux tube ( B i) with Equation (1). In addition, there are some other confusing patterns in Figure 3(B) and (C). For instance, B seis and the relative error manifest fluctuations along the z direction. It might be related to longitudinal oscillations excited by kink waves due to some non-linear effects (e.g., Goldstein 1978; Del Zanna et al. 2001; Terradas & Ofman 2004). Further investigations are needed to understand these patterns. When we degraded the spatial and temporal resolutions to approximately match those of CoMP, we found that the distributions of phase speed, density, and B seis along z all show remarkable similarities to the input values (Figure 5). Again we can notice a deviation near the upper and lower boundaries, particularly the lower one. However, within the height range of 20-80 Mm, the magnetic field error is generally less than 10%. Additionally, the CoMP instrument has recently been upgraded to UCoMP, which has a slightly higher spatial resolution and a larger field of view. Our main conclusions should also apply to the case with the UCoMP observations (Yang et al. 2024). We also tested the case when degrading to UCoMP's pixel size ( ∼ 2.2 Mm), and the results are largely similar to those shown in Figure 5, with a slightly smaller error. Another factor that may impact the phase speed and magnetic field measurements is the magnitude of the input magnetic field and phase speed. A stronger magnetic field and higher phase speed result in steeper slopes in the time-distance maps of Doppler velocity, which can reduce the accuracy of the wave-tracking method, particularly when the spatial and temporal resolutions are degraded. We ran a separate simulation with a background magnetic field of ∼ 10 G, which gives phase speeds of 1-2 Mm s -1 . We found a systematic underestimation of the magnetic field by approximately 30% in this case. Nevertheless, the observed coronal magnetic field around 1 . 05 -1 . 5 R ⊙ is 1-4 G (Lin et al. 2004; Gopalswamy et al. 2012; Kumari et al. 2019; Yang et al. 2020b; Zhong et al. 2023), which is more comparable with the case discussed in Section 3. In conclusion, the coronal seismology technique based on propagating kink waves can provide reliable magnetic field measurements according to our forward modeling. We note that this study only focuses on open coronal structures, including plumes in coronal holes and fan loops at the boundaries of active regions (Morton et al. 2015; Banerjee et al. 2021). Specifically, in our simulation, the magnetic flux tube is perpendicular to the solar surface, with both gravity and magnetic field aligned along the tube axis. However, the propagating kink waves can be detected not only in these structures but also in closed-field regions (e.g., Tomczyk et al. 2007; Yang et al. 2020a,b). Our magnetic configuration can roughly describe one leg of a large-scale closed coronal loop, where the curvature can be neglected. For smaller loops where curvature cannot be ignored, our model is no longer applicable due to differences in gravitational stratification. Nevertheless, for such loops, phase speed along the axis often shows minimal variation McIntosh et al. (2011); Threlfall et al. (2013); Zhong et al. (2023), thus similar to the case in Magyar & Van Doorsselaere (2018), where a model without any vertical gradient in phase speed was used. Therefore, this study complements previous research by highlighting the importance of phase speed variation along the flux tube in coronal seismology. Finally, we would like to mention that our model has some limitations. The vertical gradient of the magnetic field and the magnetic expansion are not included. Additionally, we didn't consider the effect of internal flows along the flux tube, which are frequently reported and can affect the apparent wave propagation speed (e.g., Soler et al. 2011; Morton et al. 2015). Moreover, in real observations, there are often multiple flux tubes overlapping along the LOS, introducing further complexities that may affect wave-tracking accuracy and magnetic field measurements. In Figure 5 of Yang et al. (2024), comparisons between seismological results and global coronal MHD models revealed some discrepancies, particularly at higher latitudes where open field lines may dominate. Future work that incorporates more realistic models could offer a more sophisticated evaluation of the magnetic field measurements through coronal seismology and help clarify the discrepancies in Yang et al. (2024). Acknowledgements This work is supported by the National Natural Science Foundation of China grant 12425301 and the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB0560000). M.G. acknowledges support from the National Natural Science Foundation of China (NSFC, 12203030). T.V.D was supported by the C1 grant TRACEspace of Internal Funds KU Leuven and a Senior Research Project (G088021N) of the FWO Vlaanderen. Furthermore, TVD received financial support from the Flemish Government under the long-term structural Methusalem funding program, project SOUL: Stellar evolution in full glory, grant METH/24/012 at KU Leuven. The research that led to these results was subsidised by the Belgian Federal Science Policy Office through the contract B2/223/P1/CLOSE-UP. It is also part of the DynaSun project and has thus received funding under the Horizon Europe programme of the European Union under grant agreement (no. 101131534). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union and therefore the European Union cannot be held responsible for them. K.K. acknowledges support by an FWO (Fonds voor Wetenschappelijk Onderzoek - Vlaanderen) postdoctoral fellowship (1273221N).",
"pages": [
8,
9,
10
]
},
{
"title": "References",
"content": "Aschwanden, M. J., De Pontieu, B., Schrijver, C. J., & Title, A. M. 2002, Sol. Phys., 206, 99 2 Banerjee, D., Krishna Prasad, S., Pant, V., et al. 2021, Space Sci. Rev., 217, 76 9 Bose, S., TenBarge, J. M., Carter, T., et al. 2024, ApJ, 971, 72 4 Chen, B., Shen, C., Gary, D. E., et al. 2020, Nature Astronomy, 4, 1140 1 Chen, S.-X., Li, B., Xiong, M., Yu, H., & Guo, M.-Z. 2015, ApJ, 812, 22 2 Chen, Y., Feng, S. W., Li, B., et al. 2011, ApJ, 728, 147 2 Chen, Y., Li, W., Tian, H., et al. 2023, Research in Astronomy and Astrophysics, 23, 022001 1 Chen, Y., Li, W., Tian, H., et al. 2021, ApJ, 920, 116 1 Del Zanna, L., Velli, M., & Londrillo, P. 2001, A&A, 367, 705 9 Dere, K. P., Del Zanna, G., Young, P. R., & Landi, E. 2023, ApJS, 268, 52 6 Fehlmann, A., Kuhn, J. R., Schad, T. A., et al. 2023, Sol. Phys., 298, 5 8 Fleishman, G. D., Gary, D. E., Chen, B., et al. 2020, Science, 367, 278 1 Gao, Y., Guo, M., Van Doorsselaere, T., Tian, H., & Skirvin, S. J. 2023, ApJ, 955, 73 3 Gao, Y., Hou, Z., Van Doorsselaere, T., & Guo, M. 2024a, A&A, 681, L4 2 Gao, Y., Tian, H., Van Doorsselaere, T., & Chen, Y. 2022, ApJ, 930, 55 2 Gao, Y., Van Doorsselaere, T., Tian, H., Guo, M., & Karampelas, K. 2024b, A&A, 689, A195 3 Goldstein, M. L. 1978, ApJ, 219, 700 9 Goossens, M., Terradas, J., Andries, J., Arregui, I., & Ballester, J. L. 2009, A&A, 503, 213 9 Gopalswamy, N., Nitta, N., Akiyama, S., M¨akel¨a, P., & Yashiro, S. 2012, ApJ, 744, 72 9",
"pages": [
10
]
}
]
2024arXiv241110270B
https://arxiv.org/pdf/2411.10270.pdf
<document>
<section_header_level_1><location><page_1><loc_21><loc_82><loc_79><loc_87></location>Strong magnetic fields of old white dwarfs are symmetric about the stellar rotation axes</section_header_level_1>
<text><location><page_1><loc_37><loc_80><loc_63><loc_81></location>S. Bagnulo 1 and J.D. Landstreet 1 , 2</text>
<unordered_list>
<list_item><location><page_1><loc_10><loc_77><loc_69><loc_78></location>1 Armagh Observatory and Planetarium, College Hill, Armagh BT61 9DG, Northern Ireland, UK</list_item>
<list_item><location><page_1><loc_10><loc_76><loc_71><loc_77></location>2 Dept. of Physics & Astronomy, University of Western Ontario, London, Ontario N6A 3K7, Canada</list_item>
</unordered_list>
<text><location><page_1><loc_10><loc_73><loc_40><loc_74></location>Received July 5, 2024, accepted October 22, 2024</text>
<section_header_level_1><location><page_1><loc_46><loc_70><loc_54><loc_71></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_10><loc_51><loc_90><loc_69></location>Many magnetic white dwarfs exhibit a polarised spectrum that periodically varies as the star rotates because the magnetic field is not symmetric about the rotation axis. In this work, we report the discovery that while weakly magnetic white dwarfs of all ages with M ≤ 1 M ⊙ show polarimetric variability with a period between hours and several days, the large majority of magnetic white dwarfs in the same mass range with cooling ages older than 2 Gyr and field strengths ≥ 10 MG show little or no polarimetric variability. This could be interpreted as extremely slow rotation, but a lack of known white dwarfs with measured periods longer than two weeks means that we do not see white dwarfs slowing their rotation. We therefore suggest a di ff erent interpretation: old strongly magnetic white dwarfs do not vary because their fields are roughly symmetric about the rotation axes. Symmetry may either be a consequence of field evolution or a physical characteristic intrinsic to the way strong fields are generated in older stars. Specifically, a strong magnetic field could distort the shape of a star, forcing the principal axis of maximum inertia away from the spin axis. Eventually, as a result of energy dissipation, the magnetic axis will align with the angular momentum axis. Alternatively, symmetry could be the hallmark of a dynamo that operates after the beginning of core crystallisation. We also find that the higher-mass strongly magnetised white dwarfs, which are likely the products of the merging of two white dwarfs, may appear as either polarimetrically variable or constant. This may be the symptom of two di ff erent formation channels or the consequence of the fact that a dynamo operating during a merger may produce diverse magnetic configurations. Alternatively, the massive white dwarfs with constant polarisation may be rotating with periods much shorter than the typical exposure times of the observations.</text>
<text><location><page_1><loc_10><loc_50><loc_53><loc_51></location>Key words. polarisation - stars: white dwarfs - stars: magnetic fields</text>
<section_header_level_1><location><page_1><loc_6><loc_45><loc_18><loc_46></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_6><loc_38><loc_49><loc_44></location>Most white dwarfs do not show any variability, and more than 97% of them can be safely considered as suitable flux standard stars (Hermes et al. 2017). Because of flux stability, it is generally impossible to measure the rotation period of white dwarfs except when they pulsate or when they have a magnetic field.</text>
<text><location><page_1><loc_6><loc_10><loc_49><loc_37></location>It is well known that magnetic white dwarfs often show periodically variable signals of polarisation and sometimes also periodic photometric variability. In the local 20 pc volume, more than 20% of white dwarfs have a magnetic field (Bagnulo & Landstreet 2021), and about 20% of the magnetic white dwarfs are photometrically variable (Farihi et al. 2024). Polarimetric variability is explained in terms of a magnetic field not symmetric about the rotation axis so that the magnetic configuration seen by the observer (as encoded in the polarisation signal) varies as the star rotates. The causes of photometric variability are not well understood (see, e.g., Bagnulo et al. 2024b), yet photometric variability of non-pulsating white dwarfs is so clearly correlated with the presence of the magnetic field variations that one may hypothesise that all non-pulsating white dwarfs with light variations are magnetic. If a star belongs to the small class of white dwarfs with a well-detected light curve, then the rotational period may usually be deduced from photometry (see Brinkworth et al. 2013; Hernandez et al. 2024; Oliveira da Rosa et al. 2024), but, in general, the time series of polarisation measurements may allow one to recover the rotational period of a larger sample of magnetic white dwarfs.</text>
<text><location><page_1><loc_51><loc_23><loc_94><loc_46></location>Investigations of magnetic fields and of rotational periods are closely related not only because the presence of a magnetic field enables period measurements but also because one could expect that the occurrence of a magnetic field and its strength is physically related to stellar rotation. For example, should the field be supported by a dynamo, one could predict fields to be stronger in more rapidly rotating stars than in those that are more slowly rotating. Correlations between the presence of a magnetic field and the rotational period of a star exist even when the field has a fossil origin. Most of the chemically peculiar stars of the main sequence (Ap and Bp stars) have a fossil magnetic field, and they all rotate more slowly than the non-magnetic non-chemically peculiar stars in the same region of the Hertzsprung-Russel diagram (e.g. Donati & Landstreet 2009). Some of the magnetic Ap and Bp stars have well measured rotational periods as long as several months, years, or even decades (e.g. γ Equ, Leroy et al. 1994). There is no clear explanation for this well-established correlation between magnetic fields and slow stellar rotation.</text>
<text><location><page_1><loc_51><loc_10><loc_94><loc_22></location>In the case of white dwarfs, various samples of data have gradually become available in the past few years to constrain possible scenarios for the evolution of their magnetic fields. It now appears that among the normal-mass white dwarfs ( M ≃ 0 . 6 M ⊙ ) that largely descend from single-star evolution, magnetic fields are very rare and weak during the first 1-2 Gyr of cooling but then gradually become much more common and often very strong after about 2-3 Gyr of cooling (Bagnulo & Landstreet 2021). This evolution may occur as a result of gradual relaxation to the surface of fields present in stellar cores during earlier evo-</text>
<text><location><page_2><loc_6><loc_72><loc_49><loc_93></location>on and / or as a result of the operation of a dynamo during core crystallisation (Isern et al. 2017; Schreiber et al. 2021; Bagnulo & Landstreet 2021, 2022; Ginzburg et al. 2022). Among normal-mass stars, those with a magnetic field seem to rotate faster than non-magnetic white dwarfs (Hernandez et al. 2024), supporting the idea that crystallisation-driven dynamo may play a role in the formation of their magnetic fields. Many highly massive white dwarfs ( M ≥ 1 . 0 M ⊙ ), which are probably the product of mergers of two white dwarfs, appear magnetic very early in their cooling age, and the origin of the magnetic field may be due to a dynamo operating during the merging (Tout et al. 2008; García-Berro et al. 2012; Briggs et al. 2015; Bagnulo & Landstreet 2022). This idea is supported also by the evidence that some white dwarfs have very short rotation periods (e.g. Feige 7: 131 min, Liebert et al. 1977; Cl Oct: 12.1 min, Barstow et al. 1995; WD2209 + 113: 70 s, Kilic et al. 2021).</text>
<text><location><page_2><loc_6><loc_49><loc_49><loc_72></location>At the same time, some polarimetric studies have hinted that a few old magnetic white dwarfs with a field strength of the order of tens of to a hundred megagauss have extremely long rotational periods, showing at most quite small variations even on timescales of decades (e.g. Berdyugin & Piirola 1999; Bagnulo &Landstreet 2019b). Traditionally, this lack of obvious variability is explained by the assumption that such non-varying magnetic white dwarfs have very long rotation periods of the order of centuries (Schmidt & Norsworthy 1991). In the absence of clear observational support or contradiction, this assumption has been widely accepted (e.g. Ferrario et al. 2015). The conclusion is basically that the known periods, P , fall into two very di ff erent families: one with P < ∼ 2 weeks, characterising most of the sample, and a few percent with strong fields that have periods of centuries (i.e. no clearly detected rotation). There is no explanation as to why some old white dwarfs would have extremely long rotation periods. Magnetic braking has generally been invoked but without the support of numerical calculations.</text>
<text><location><page_2><loc_6><loc_42><loc_49><loc_49></location>In this work, we report new polarimetric observations and combine them with data collected from the literature. We analyse the magnetic variability and thus the rotation of a sample of 74 white dwarfs, and we discuss the constraints that these data place on possible evolution paths for magnetic fields in white dwarfs.</text>
<section_header_level_1><location><page_2><loc_6><loc_37><loc_45><loc_40></location>2. Polarimetric observations of magnetic white dwarfs</section_header_level_1>
<section_header_level_1><location><page_2><loc_6><loc_35><loc_36><loc_36></location>2.1. Observations not previously published</section_header_level_1>
<text><location><page_2><loc_6><loc_10><loc_49><loc_34></location>We present here 49 unpublished spectropolarimetric observations of 13 magnetic white dwarfs. Fourteen spectra of six white dwarfs were recently obtained with the FORS2 instrument (Appenzeller et al. 1998) of the ESO VLT in the course of a spectropolarimetric survey of the solar neighbourhood. Seven polarised spectra of two other white dwarfs were retrieved from the ESO archive: one of the spectra was obtained with FORS1 and the remaining ones with FORS2 (we note that FORS1 and FORS2 are virtually identical instruments). Fourteen new spectra of four white dwarfs were obtained with the ISIS instrument of the William Herschel Telescope (WHT). For all of these data, the observing strategy and data reduction are identical to the procedures described by Bagnulo & Landstreet (2018) for circular polarisation data and by Bagnulo & Landstreet (2019b) for linear polarisation data. For two of the four stars recently observed with ISIS, we also present here three previously unpublished spectra obtained in the 1970s that we used to study the long-term variability of magnetic white dwarfs. These data consist of lowresolution polarised spectra obtained by Angel and Landstreet</text>
<text><location><page_2><loc_51><loc_84><loc_94><loc_93></location>using the multi-channel spectrophotometer (hereafter MCSP) on the 5-m Palomar telescope. The instrument set-up and data reduction of these observations are described in detail by Angel & Landstreet (1974). Finally, eleven spectra of WD 1105-340 were obtained with the ESPaDOnS instrument of the CanadaFrance-Hawaii Telescope (CFHT). These data were reduced by the automatic CFHT pipeline LibreEsprit.</text>
<text><location><page_2><loc_51><loc_71><loc_94><loc_84></location>The log of these unpublished observations is given in Table A.1, and results are described in the sections dedicated to individual stars of Appendix B. Apart from their general purpose of assessing polarimetric variability, our new observations confirm that star WD 1116 -470 is magnetic (it was previously considered as suspected to be magnetic by Bagnulo & Landstreet 2021) and bring the number of magnetic white dwarfs in the local 20 pc volume to 34 (see Bagnulo & Landstreet 2021). We have also discovered two new massive magnetic white dwarfs: WD1619 + 054 and WD1754 -550 (both rapidly variable).</text>
<section_header_level_1><location><page_2><loc_51><loc_68><loc_64><loc_69></location>2.2. Literature data</section_header_level_1>
<text><location><page_2><loc_51><loc_59><loc_94><loc_67></location>We searched the literature and collected data for all magnetic white dwarfs that were observed in polarimetric mode at least twice as well as for all magnetic white dwarfs for which multiple spectroscopic intensity observations revealed magnetic variability. The way literature data and new observations have been used is described in the next section.</text>
<section_header_level_1><location><page_2><loc_51><loc_54><loc_92><loc_56></location>3. Determination of the variability of the magnetic white dwarfs</section_header_level_1>
<text><location><page_2><loc_51><loc_27><loc_94><loc_52></location>We are interested in establishing whether the magnetic field of a white dwarf appears constant or variable with time to the observer. Magnetic variability is ascribed to changes in the apparent field geometry as a star rotates. It can be detected mainly via circular spectro- or broadband polarimetry, which are both sensitive to the longitudinal component of the magnetic field. Linear polarisation, which is sensitive to the transverse component of the magnetic field, is usually much weaker than circular polarisation, and it is rarely detected in white dwarfs. Intensity spectra (if the star is not featureless) are sensitive to the mean field modulus and may also be used to detect magnetic variability, but they are less sensitive to changes of the apparent magnetic configuration than polarimetry. Magnetic white dwarfs that have been repeatedly observed in spectroscopic mode and that do not show sign of variability have not been considered here because having observed a constant intensity spectrum is not a su ffi ciently strong indication that a star is not magnetically variable. An example is WD2047 + 372, as it shows an almost constant Zeeman triplet in multi-epoch intensity spectra and a variable, sign-reversing circular polarisation spectrum (Landstreet et al. 2017).</text>
<text><location><page_2><loc_51><loc_11><loc_94><loc_26></location>Photometric variability is observed in magnetic white dwarfs, but in the absence of convincing observational evidence or theoretical arguments, we have chosen not to consider it alone as a proxy for magnetism nor magnetic variability. For example, Brinkworth et al. (2013, see in particular their Tables 1 and 3) detected light variability for a number of magnetic stars. For many of them, however, the light amplitude is quite low, and there are no multi-epoch polarimetric data to confirm magnetic variability. Furthermore, recent analysis of TESS data made by Hernandez et al. (2024) and Oliveira da Rosa et al. (2024) failed to detect a periodic light curve for many of these targets. These stars are not included in our sample. In summary:</text>
<text><location><page_2><loc_51><loc_10><loc_94><loc_11></location>1) Two or more intensity spectra, or polarisation spectra, or</text>
<text><location><page_3><loc_6><loc_91><loc_49><loc_93></location>broadband polarisation measurements that clearly di ff er from each other are interpreted as being due to magnetic variability.</text>
<unordered_list>
<list_item><location><page_3><loc_6><loc_84><loc_49><loc_90></location>2) Repeated observations of circular polarisation (either spectropolarimetry or broadband polarimetry) that are consistent among themselves within uncertainties are interpreted as evidence of a lack of magnetic variability (that is, the observer always sees the same magnetic configuration).</list_item>
<list_item><location><page_3><loc_6><loc_80><loc_49><loc_84></location>3) No conclusion will be drawn from a series of intensity spectra obtained in di ff erent epochs that appear similar to each other within uncertainties.</list_item>
<list_item><location><page_3><loc_6><loc_76><loc_49><loc_80></location>4) Photometric variability that is not associated with observed spectroscopic or polarimetric variability will not be used to decide that a star is magnetically variable.</list_item>
</unordered_list>
<text><location><page_3><loc_6><loc_65><loc_49><loc_76></location>Wegathered reasonable evidence for polarimetric variability, or non-variability, for the sample of 74 magnetic white dwarfs listed in Table 2. Admittedly, for some stars, we found that establishing whether or not two or more observations are consistent among themselves was somewhat subjective. In Sect. 5.3, we argue that our results are not a ff ected by these uncertain cases. Appendix B discusses in detail all individual stars. In this section, we summarise the results and flag special cases.</text>
<text><location><page_3><loc_6><loc_58><loc_49><loc_65></location>Among the 74 white dwarfs, 44 are magnetically variable (Sect. 3.1), while for 27 there is so far no evidence of variability (Sect. 3.2). The remaining three white dwarfs, which have strong fields and have been observed over several decades, may be further tested for subtle or possibly very slow variability (Sect. 3.3).</text>
<section_header_level_1><location><page_3><loc_6><loc_54><loc_43><loc_56></location>3.1. Stars that show evidence of magnetic variability</section_header_level_1>
<text><location><page_3><loc_6><loc_38><loc_49><loc_53></location>For 30 white dwarfs among the 44 stars that show polarimetric variability, a rotational period, or at least a good candidate for it, has been established in the literature. This period is given (in days) in the last column of Table 2 and is followed by ':' when only tentative or approximated values are known. Rotational periods are generally of the order of hours; about 25% are of the order of one day or longer, and only two stars have a rotation period longer than one week (the slowest magnetic white dwarf for which the period is known, WD 2316 + 123, has P ≃ 17 . 4 d, Schmidt & Norsworthy 1991). Most of these stars were listed in Table C.1 of Hernandez et al. (2024), and they are shown with blue symbols in Fig. 1 of Sect. 4.</text>
<text><location><page_3><loc_6><loc_10><loc_49><loc_37></location>Among the other 14 variable stars, we do not have enough data to estimate the rotation period. For ten of them, the evidence of polarimetric variability has been securely established by two or more observations. These ten variable stars are marked in last column of Table 2 with 'var.' and are also represented with a blue symbol in Fig. 1. The remaining four white dwarfs show subtle signs of variability among the observations obtained with the same instrument. They are marked in Table 2 with 'var.:' and are shown with light blue symbols in Fig. 1. For WD 0446-789 and WD1009-184, our conclusion is that the field variations are real but small due to a configuration probably nearly symmetric about the rotation axis. WD 1105-048 was repeatedly observed with FORS1, FORS2, and ESPaDOnS, and a field was detected on only two occasions out of 12 observing epochs. If the star is magnetic, then it is certainly variable with a timescale of the order of days or weeks, but, admittedly, one could suspect that the two detections are in fact spurious. We decided to define the star as a probable variable star. WD 2049-222 shows a signal of polarisation that is marginally higher than instrumental polarisation, but for the reasons explained in Sect. B.68, we decided to consider its subtle variations as real.</text>
<section_header_level_1><location><page_3><loc_51><loc_91><loc_88><loc_93></location>3.2. Stars for which there is no evidence of magnetic variability</section_header_level_1>
<text><location><page_3><loc_51><loc_71><loc_94><loc_89></location>Nineteen stars have been observed three or more times in polarimetric mode, and the observations are always consistent among themselves within uncertainties. We consider these stars as established non-variable white dwarfs. They are marked with 'n.v.' in Table 2 and represented with red symbols in Fig. 1. Among these 19 stars, four were observed over a time interval of decades and with di ff erent instruments: WD 0548 -001 = G9937, WD0553 + 053 = G99-47, WD1036 -204 = LP790 -29, and WD1829 + 547 = G227-35. Obviously, it is possible that the other members of this group of 'non-variable stars', for which the observations span a time interval of up to a few months or years, are actually variable on a timescale of decades and that as such they could be stars with (so far undetected) long-term variability.</text>
<text><location><page_3><loc_51><loc_47><loc_94><loc_71></location>Eight stars should be considered simply as candidate nonvariable white dwarfs ('n.v.:' in Table 2 and magenta symbols in Fig. 1). For seven of them, the reason for considering them only as candidate non-variable stars is that they have only been observed twice. In particular we mention that for normal-mass WD0236-269 and for the massive WD0330 -000, the nonvariability is inferred based on comments in the original discovery paper by Schmidt et al. (2001), which represents especially weak evidence. Another candidate non-variable star, WD1750 -311, was observed only once (with FORS2). The comparison of spectra obtained within that single observing series shows some change of circular polarisation and certainly a rapidly variable intensity spectrum (timescale of minutes) in the regions of the Balmer lines, particularly around H β . The star also shows photometric variability with a 35-min period (Macfarlane et al. 2017). Our interpretation is that WD 1750-311 is probably a magnetic white dwarf with a field symmetric, or nearly symmetric about the rotation axis, showing strong abundance patches of hydrogen.</text>
<section_header_level_1><location><page_3><loc_51><loc_43><loc_81><loc_44></location>3.3. Stars possibly subtly or slowly variable</section_header_level_1>
<text><location><page_3><loc_51><loc_10><loc_94><loc_42></location>There are three stars that have been monitored for decades, WD1748 + 708 = G240 + 708, WD1900 + 310 = Grw + 70 · 8247, and WD2010 + 310 = GD229, for which our conclusions are uncertain. These stars, marked with 's.v.:' in Table 2 and shown with black symbols in Fig. 1, may indeed show some low amplitude and possible long-term variability, but there might be room for a di ff erent interpretation. The recent detailed comparison of old and new polarisation spectra of Grw + 70 · 8247 by Bagnulo & Landstreet (2019b) identifies small changes between spectra taken decades apart (see in particular their Fig. 5), but for the two other stars, it is not entirely clear which changes are real and which are due to di ff erences between observational equipment and techniques. Bagnulo & Landstreet (2019b) assumed that timescales of the observed small changes were those corresponding to the large time gaps between isolated data sets. However, because the changes are generally small, one could argue that the subtle changes have timescales as short as weeks and have not have been noticed because the stars were not adequately monitored. For example, as previously discussed, WD 1105 -048 appears constant (and non-magnetic) in ten out of 12 observations, and a magnetic field was detected in only two epochs. In case of WD 1748 + 708, photometry leads to ambiguous results. The star was found photometrically variable with a period between 5 and 48 h by Antonyuk et al. (2016). Brinkworth et al. (2013) did not confirm its short-term photometric variability (nor</text>
<text><location><page_4><loc_6><loc_91><loc_49><loc_93></location>did Hernandez et al. 2024) but instead claimed detection of light changes over a period of ten months.</text>
<section_header_level_1><location><page_4><loc_6><loc_87><loc_33><loc_88></location>4. Clustering in parameter space</section_header_level_1>
<text><location><page_4><loc_6><loc_56><loc_49><loc_86></location>In this section, we correlate the variability or non-variability of the stars with their mass, field strength, and stellar temperature. The left panels of Fig. 1 show the field strength versus e ff ective temperature and cooling age for the stars of our sample, with the cooling age on both a linear scale and a logarithmic scale. In these plots, stars are represented with symbols that increase in size with a star's mass, while di ff erent colours are used to mark the characteristics in terms of variability. The right panels of Fig. 1 show the cooling age-mass diagrams for the same sample, with cooling age reported again both with linear and logarithmic scales. The size of the symbols is proportional to the field strength. With the help of these figures and Table 2, we studied the relationships between field strength, mass, age, and variability of the magnetic white dwarfs. We recall that our data do not come from a volume-limited sample of stars and do not reflect the relative density of magnetic white dwarfs in di ff erent regions of the diagrams of Fig. 1. In fact, both young magnetic normalmass white dwarfs and ultra-massive white dwarfs are quite rare objects in space, and they are over-represented in our sample because of biases of the surveys (see Bagnulo & Landstreet 2022). The following analysis is about the relative frequency of variable and non-variable stars in di ff erent regions of the diagrams where we noticed the existence of statistically remarkable di ff erences.</text>
<section_header_level_1><location><page_4><loc_6><loc_53><loc_43><loc_54></location>4.1. Normal-mass and weakly magnetic white dwarfs</section_header_level_1>
<text><location><page_4><loc_6><loc_44><loc_49><loc_52></location>There are 25 white dwarfs with a field strength of ≤ 1 MG, and all of them but one (WD 2051-208) have M ≤ 1 . 0 M ⊙ . These 'normal-mass' 'weak-field' white dwarfs span an age range up to τ ≈ 6 Gyr. Twenty-one of these 24 stars are variable. Among the three that are non-variable, one is younger than 1 Gyr: WD 1105 -340 ( τ = 0 . 34 Gyr).</text>
<section_header_level_1><location><page_4><loc_6><loc_41><loc_44><loc_42></location>4.2. Normal-mass and strongly magnetic white dwarfs</section_header_level_1>
<text><location><page_4><loc_6><loc_32><loc_49><loc_40></location>There are 25 magnetic white dwarfs with M ≤ 1 . 0 M ⊙ and field strength of ≥ 10 MG. Among those that are younger than 2 Gyr, six out of eight white dwarfs are variable; however, we recall that young strongly magnetic white dwarfs are rare in space (Bagnulo & Landstreet 2022). Among the magnetic white dwarfs older than 2 Gyr, only one out of the 17 shows polarimetric variability.</text>
<section_header_level_1><location><page_4><loc_6><loc_29><loc_31><loc_30></location>4.3. Massive magnetic white dwarfs</section_header_level_1>
<text><location><page_4><loc_6><loc_19><loc_49><loc_28></location>Fifteen stars of our sample have M > 1 . 0 M ⊙ , and all but one are younger than about 1 Gyr. Eight of them are magnetically variable, including the only old ( τ ≃ 4 Gyr) star WD 0756 + 437. Nearly all of these massive magnetically variable white dwarfs have a strong field (tens to hundreds megagauss), except for WD2051-208, which is the only ultra-massive star ( M = 1 . 2 M ⊙ ) with a sub-MG field strength (0.25 MG) in our sample.</text>
<text><location><page_4><loc_6><loc_10><loc_49><loc_19></location>In contrast, seven massive magnetic white dwarfs show no obvious signs of variability, although none of them may be safely considered as non-variable, either because they were observed only twice (or even just once, i.e. WD 1750-311; see Appendix B.60) or because some subtle sign of variability may have been detected (in WD 1900 + 705 and WD2010 + 310). WD1658 + 440 and WD1750-311 are the only massive non-</text>
<table>
<location><page_4><loc_51><loc_82><loc_95><loc_89></location>
<caption>Table 1. Frequency of polarimetrically non-variable stars among normal-mass white dwarfs ( M ≤ 1 M ⊙ ).</caption>
</table>
<text><location><page_4><loc_51><loc_75><loc_94><loc_80></location>Notes. The full list includes 59 stars, 23 of which are non-variable. Between brackets, the table also gives the number of non-variable stars N K divided by the number of stars N tot in the subset as defined by the age and field values.</text>
<text><location><page_4><loc_51><loc_69><loc_94><loc_73></location>variable white dwarfs in our sample with a field as low as ≈ 2 MG. The remaining five non-variable stars have magnetic fields with strengths of the order of hundreds of megagauss.</text>
<section_header_level_1><location><page_4><loc_51><loc_66><loc_88><loc_67></location>4.4. Statistical analysis of normal-mass white dwarfs</section_header_level_1>
<text><location><page_4><loc_51><loc_50><loc_94><loc_65></location>Because we have only one example of an old massive magnetic white dwarf, we do not know how variability in massive magnetic white dwarfs evolves with time. Vice versa, an evolutionary path is clearly seen in normal-mass ( M < ∼ 1 M ⊙ ) white dwarfs. In our sample, field variability is nearly ubiquitous among weakly magnetic white dwarfs of all ages and in young white dwarfs regardless of the field strength, it is but very rare in old ( τ ≥ 2 Gyr) strongly magnetic ( B ≥ 10 MG) white dwarfs. Next, we assess the statistical significance of this pattern, specifically whether it can be attributed to small number statistics or if it reflects a genuine correlation between cooling age, field strength, and field variability.</text>
<text><location><page_4><loc_51><loc_35><loc_94><loc_49></location>We first split the sample of stars with M ≤ 1 M ⊙ into two groups: those younger than 2 Gyr and those older than 2 Gyr. Each of these two groups was divided into two subsets: stars with a field strength ⟨| B |⟩ ≤ 1 MG and stars with ⟨| B |⟩ > 10 MG. For each of these four subsets, we considered the ratio between the number of non-variable magnetic stars, N K, and the total number of magnetic stars, N tot. From these numbers, we estimated the probability, Pr , that the sample frequency of non-variable stars is between r and r + d r , normalised to one and obtained assuming that all numbers between zero and one are a priori equally probable, using</text>
<formula><location><page_4><loc_51><loc_31><loc_94><loc_34></location>Pr = ( N tot + 1)! N K! ( N tot -N K)! r N K (1 -r ) ( N tot -N K) . (1)</formula>
<text><location><page_4><loc_51><loc_23><loc_94><loc_30></location>Figure 2 shows the probability density functions for the stars belonging to these sets. It clearly appears that there is little to no overlap between the density functions of the symmetric field in old strongly magnetic white dwarfs and of young white dwarfs. Table 1 reports the points of maximum for these distributions, f = N K / N tot, and their uncertainties</text>
<formula><location><page_4><loc_51><loc_18><loc_94><loc_21></location>r f (1 -f ) N tot . (2)</formula>
<text><location><page_4><loc_51><loc_12><loc_94><loc_17></location>Table 1 also includes the results for the smaller sets of stars with intermediate strength, 1 ≤ ⟨| B |⟩ ≤ 10 MG (with endpoints overlapping with the other two sets), which could possibly be considered a 'transitioning' field strength range.</text>
<text><location><page_4><loc_51><loc_10><loc_94><loc_12></location>Figure 3 shows the ratio between the number of non-variable magnetic white dwarfs with a field strength lower than a given</text>
<figure>
<location><page_5><loc_7><loc_25><loc_92><loc_94></location>
<caption>Fig. 1. Correlations between field strength, magnetic variability, and other stellar parameters. Left panels: Field strength versus e ff ective temperature and versus cooling age for variable and non-variable stars of Table 2. The size of the symbols is related to the mass of the star as shown in the legend. Red symbols refer to stars that were observed at least three times with no evidence for variability; magenta symbols are for stars that were observed only twice and are tentatively assumed non-variable. Blue symbols refer to stars that show variability. Light blue symbols indicate stars that show marginal but probably real signs of variability, maybe due to a field nearly aligned with the rotation axis. Black symbols are used for stars that show signs of long-term variability that we were not able to interpret. Right panels: Temperature-mass and cooling age-mass diagrams for the same sample of stars. The size of the symbols is related to the field strength as shown in the legend. The meaning of the colour is the same as for the left panels. Black lines represent the onset of crystallisation (solid line for H-thick envelop and dashed line of H-thin envelop) obtained via interpolation of the cooling tables by Bédard et al. (2020). In all panels, the yellow vertical and horizontal lines highlight the position of the box tick marks. We recall that these diagrams do not refer to a volume-limited sample and that both young normal-mass magnetic white dwarfs and highly massive magnetic white dwarfs, which are rare objects in space, are over-represented.</caption>
</figure>
<figure>
<location><page_6><loc_7><loc_65><loc_48><loc_94></location>
<caption>Fig. 3. Cumulative fractions of variable and non-variable magnetic white dwarfs with M ≤ 1 M ⊙ . The blue solid line represents the number of non-variable stars younger than 2 Gyr divided by the total number of stars younger than 2 Gyr with an average field strength lower than the value in the x -axis. The red solid line refers to the same quantity for magnetic white dwarfs older than 2 Gyr. Dotted lines show the estimate of the uncertainties.</caption>
</figure>
<figure>
<location><page_6><loc_52><loc_66><loc_93><loc_93></location>
<caption>Fig. 2. Probability density functions of the non-variable fields in normal-mass white dwarfs with the age and field strength specified in the legend.</caption>
</figure>
<text><location><page_6><loc_6><loc_43><loc_49><loc_55></location>value ¯ ⟨| B |⟩ and the total number of magnetic white dwarfs with a field strength lower than that value as a function of ¯ ⟨| B |⟩ for stars with cooling ages τ ≤ 2 Gyr (blue lines) and stars with τ > 2 Gyr (red lines). This plot supports our claim that nonvariable fields are much more common in old, strongly magnetic white dwarfs than in young strongly magnetic white dwarfs, and than in weakly magnetic white dwarfs of all ages. It suggests also that the minimum field strength required for a field to become non-variable is in the range of 5 to 10 MG.</text>
<section_header_level_1><location><page_6><loc_6><loc_39><loc_47><loc_41></location>5. Explanation for the lack of observed variability</section_header_level_1>
<text><location><page_6><loc_6><loc_35><loc_49><loc_38></location>There are three possible reasons for the observed non-variability of so many magnetic white dwarfs. We examine them in the following sections.</text>
<section_header_level_1><location><page_6><loc_6><loc_31><loc_26><loc_32></location>5.1. Extremely slow rotation</section_header_level_1>
<text><location><page_6><loc_6><loc_10><loc_49><loc_30></location>We first consider the possibility that the rotation of magnetic white dwarfs slows with age and / or magnetic field strength. Perhaps, as the surface field becomes stronger, the white dwarf loses angular momentum to its environment by electromagnetic dipole radiation (García-Berro et al. 2012), by coupling with gas clouds in the ISM, or by a very weak magnetically coupled wind. Each of these possibilities would shed stellar angular momentum faster from a magnetic white dwarf with a strong field than from one with a weak field, preferentially slowing the magnetic white dwarfs with strong fields. However, all of these mechanisms are expected to exert at most a very weak influence on the rotation of a white dwarf, and it seems quite unlikely that any of them could slow the rotation of a magnetic white dwarf so much that repeated observations even a year apart would not show any rotation. Nevertheless, the idea of extremely slowly rotating white dwarfs has been generally accepted. This hypothesis likely origi-</text>
<text><location><page_6><loc_51><loc_41><loc_94><loc_51></location>ates from Sect. 3 of Schmidt & Norsworthy (1991), which says: 'We therefore assign long periods to these stars, with the recognition that other explanations for their polarimetric constancy are possible.' This statement was accompanied by the first plots of field strength versus rotational period, a plot that in updated form has reappeared in numerous reviews (e.g. Ferrario et al. 2015; Kawka 2020; Ferrario et al. 2020) but has never been critically revisited.</text>
<text><location><page_6><loc_51><loc_16><loc_94><loc_41></location>We note that there is a complete lack of white dwarfs that are known to vary on a timescale between weeks and decades. In this respect, there is a profound di ff erence between candidate longterm variable magnetic white dwarfs and long period magnetic Ap and Bp stars. While we do not know of any white dwarf with a firmly established polarimetric variability with a period longer than approximately two weeks, we are sure that a number of Ap and Bp stars are very slowly rotating stars because they definitely show clearly periodic field variation, even ⟨ Bz ⟩ sign reversals, on a very long but measured timescale that may be months, years, or decades. Furthermore, the distribution of periods is roughly continuous between periods of less than one day and periods of several years (Mathys 2008). If we wanted to interpret the lack of polarimetric variability in white dwarfs as the e ff ect of extremely long rotation periods, we would need to accept that the rotation periods of white dwarfs show an extremely bimodal distribution that peaks around hours and days and around centuries with nothing in between. This appears to be a very unlikely scenario.</text>
<section_header_level_1><location><page_6><loc_51><loc_13><loc_70><loc_14></location>5.2. Extremely fast rotation</section_header_level_1>
<text><location><page_6><loc_51><loc_10><loc_94><loc_12></location>Asecond possibility is that some or all of the non-variable white dwarfs actually are very rapidly rotating stars. If the star has a</text>
<text><location><page_7><loc_6><loc_89><loc_49><loc_93></location>rotation period much shorter than the individual frame exposure time, then the polarimetric variability would be smeared out, and the star would appear constant.</text>
<text><location><page_7><loc_6><loc_56><loc_49><loc_89></location>In fact, it has been possible to probe variability on the timescale of the exposure time of each individual frame (typically 10 min or less). In very general terms, our FORS2 and ISIS polarimetric observations allowed us to identify a star as variable provided that its rotation period is longer than ≈ 10 min and its magnetic configuration is clearly not symmetric about the rotation axis (for example, WD 1712-590 in Sects. B.57). Isolated white dwarfs that are the product of single-star evolution can hardly have rotation periods shorter than that (Kawaler 2015). This is confirmed by the results of Hernandez et al. (2024), who have shown that rotation periods of young magnetic white dwarfs in the normal mass range are only marginally shorter than the rotation periods of non-magnetic white dwarfs. Merger products, in contrast, can have rotation periods of the order of 1 min (Schwab 2021; Kilic et al. 2021) if most of the binary angular momentum is retained by the merger product. It is therefore possible that the non-variable massive white dwarfs have nonaxisymmetric fields but are rotating very rapidly. Periods down to 1 min or less can be probed via rapid cadence photometry. None of the massive, magnetically non-variable white dwarfs show TESS light variability (Hernandez, priv. comm.; Ramsay, priv. comm.), although of course one cannot rule out that more accurate photometry could reveal some extremely rapid rotators. Provisionally, we rule out very rapid rotation for the massive magnetic white dwarfs that show a constant polarisation.</text>
<section_header_level_1><location><page_7><loc_6><loc_51><loc_48><loc_53></location>5.3. Magnetic fields are symmetric about the stellar rotation axes</section_header_level_1>
<text><location><page_7><loc_6><loc_42><loc_49><loc_49></location>After ruling out both extremely slow and extremely fast rotation, we are left with the hypothesis that a non-variable field is approximately axisymmetric about the rotation axis. This means that as the star rotates with a normal period between a fraction of an hour and several days, the observer does not see any significant change in the signal of circular polarisation.</text>
<text><location><page_7><loc_6><loc_35><loc_49><loc_41></location>This interpretation naturally accounts for some outliers, such as the non-variable weakly magnetic star WD 1105-340. It is reasonable to hypothesise that this one star does not appear variable simply because its rotation axis is tilted at a small angle with respect to the line of sight.</text>
<text><location><page_7><loc_6><loc_25><loc_49><loc_34></location>In Sect. 3, we highlighted that it may be di ffi cult to firmly assess whether a star is magnetically variable because changes could be too subtle to be detected. However, the risk of classifying as 'non-variable' a star that in fact exhibits subtle but real changes does not weaken our analysis because small changes of the circular polarisation spectrum are still symptoms of a field nearly symmetric about the rotation axis.</text>
<section_header_level_1><location><page_7><loc_6><loc_19><loc_17><loc_20></location>6. Discussion</section_header_level_1>
<text><location><page_7><loc_6><loc_10><loc_49><loc_17></location>Here, we first consider the case of normal-mass white dwarfs (Sect. 6.1) and then that of massive stars (Sect. 6.2). We note that our analysis applies only to isolated white dwarfs; symmetric fields seem uncommon among magnetic white dwarfs accreting material from a companion (Cropper 1988; Reimers et al. 1999; Schmidt et al. 1995).</text>
<section_header_level_1><location><page_7><loc_51><loc_92><loc_73><loc_93></location>6.1. Normal-mass white dwarfs</section_header_level_1>
<text><location><page_7><loc_51><loc_83><loc_94><loc_91></location>It is clear that most of the weak fields of normal-mass stars of all ages are not symmetric about the stellar rotation axis. Most of the strong fields of normal-mass, young white dwarfs are also not axisymmetric. Most of the strong field of stars older than 2 Gyr are symmetric about the star's rotation axis. We now investigate what are the relationships between these weak and strong fields.</text>
<text><location><page_7><loc_51><loc_60><loc_94><loc_83></location>Our hypothesis is that the weak and non-symmetric magnetic fields of the young normal-mass white dwarfs are due to a relaxation process of a field that was produced in a previous evolutionary stage of the star, a field that was buried below the surface of the newly formed white dwarf, and that it starts to reveal itself with time as the star cools down. Such seed fields could be those found in red-giant stars, in the vicinity of the hydrogenburning shell (e.g. Li et al. 2022, 2023). The question is whether the older, generally stronger fields share essentially the same origin as the weak, younger ones, having just evolved for longer time, or if the strong fields in older stars formed by a totally different mechanism than the weaker, younger ones, for instance by a crystallisation dynamo. Whatever its origin is, we consider first the possibility that the small distortion of the shape of a magnetic white dwarf produced by magnetic forces in outer layers might actually drive the evolution of the global field towards axisymmetry. In Sect. 6.1.1, we discuss one example of a physical e ff ect that might drive such evolution.</text>
<section_header_level_1><location><page_7><loc_51><loc_55><loc_93><loc_57></location>6.1.1. The magnetic field structure of white dwarfs may evolve towards rotational axisymmetry during cooling</section_header_level_1>
<text><location><page_7><loc_51><loc_18><loc_94><loc_54></location>In the absence of a magnetic field, the rotation of the white dwarf will lead to an equatorial bulge, increasing the component of the principal moment of inertia that is parallel to the rotation axis. In this case, the rotation about the spin axis, which coincides with the largest moment of inertia, is stable. Next, we introduce a dipolar magnetic field inclined to the axis of rotation by, say, 45 · . We suppose that the visible surface field is maintained by a fossil current system deep in the star, which is gradually decaying due to Ohmic losses. As the interior field strength decreases, an azimuthal electric field will be produced by Faraday induction outside the white dwarf's initial current loop. This field will generate a current around the magnetic axis that will in turn interact with the local meridional magnetic field (Landstreet 1987) to produce an outward-directed force, distorting the stellar outer layers into a slightly oblate shape, with the largest radius around the magnetic equator. Because of this e ff ect, the principal moment of inertia of the star will be rotated towards the symmetry axis of the field, which is not aligned with the rotation axis or the angular momentum axis of the star. The white dwarf e ff ectively becomes an unstable free asymmetric rotator (an asymmetric top). If such objects can dissipate some of the energy supporting the asymmetric shape, the field axis will gradually shift to bring the principal moment of inertia closer to the angular momentum axis, which is a stable minimum energy state of an oblate system. Energy dissipation will gradually lead to the alignment of the magnetic axis with the rotation axis, which is the e ff ect we observe in old normal-mass strongly magnetic white dwarfs.</text>
<text><location><page_7><loc_51><loc_10><loc_94><loc_17></location>This basic physical e ff ect was first identified in the 1970s as possibly operating in main-sequence magnetic Ap stars, which, similar to magnetic white dwarfs, possess global fossil magnetic fields that are usually roughly dipolar and are generally oblique to the rotation axis (Stibbs 1950; Preston 1971). A number of efforts to estimate the timescale of possible evolution of an oblique</text>
<text><location><page_8><loc_6><loc_65><loc_49><loc_93></location>magnetic field to a state of small or vanishing obliquity have been published, for example, by Mestel & Takhar (1972). Much of this work is summarised in Chapter 9 of Mestel (1999). However, this line of investigation did not lead to a convergence of clear results about possible timescales or about dependence on basic input physics or parameters, such as interior field strength, or details of induced forces in outer layers and their consequences. As the increasing sample of magnetic Ap star models failed to reveal an obliquity distribution suggestive of this e ff ect, possibly because of the relatively short (of order 10 8 yr) mainsequence lifetimes of magnetic Ap stars, theoretical studies of stellar fossil fields moved on to other e ff ects. However, a similar e ff ect may be at work aligning the magnetic fields to the rotation axis in white dwarfs, which have a very di ff erent structure and much longer evolutionary times than Ap stars. This hypothesis could be explored with the aid of numerical modelling of the global (interior and surface) structure and evolution of a rotating white dwarf with an oblique magnetic field. Modelling of comparably complex white dwarf states (e.g. polars, mergers) that include magnetic fields (e.g. Franzon & Schramm 2015; Bisikalo et al. 2021; Zhong et al. 2024) suggest that numerical methods for exploring this problem are already available.</text>
<section_header_level_1><location><page_8><loc_6><loc_61><loc_49><loc_62></location>6.1.2. Crystallisation-driven dynamo and axisymmetric fields</section_header_level_1>
<text><location><page_8><loc_6><loc_44><loc_49><loc_60></location>The line that marks the beginning of core crystallisation in the age-mass diagram separates variable from non-variable magnetic white dwarfs perhaps in a cleaner way compared to a massindependent age threshold (see Fig. 1). Before core crystallisation begins, magnetic fields are almost always non-symmetric about the rotation axis. From volume-limited surveys, we know that in normal-mass white dwarfs, before crystallisation, strong fields are rare (Bagnulo & Landstreet 2021, 2022), but those that have been discovered and monitored show polarimetric variability. After the beginning of core crystallisation, many strong fields appear, and most of them are symmetric about the rotation axis. White dwarfs with weak and non-symmetric fields continue to appear also after the beginning of core-crystallisation.</text>
<text><location><page_8><loc_6><loc_18><loc_49><loc_43></location>The strong magnetic fields of old normal-mass white dwarfs could be generated by the crystallisation convective dynamo mechanism (Isern et al. 2017; Schreiber et al. 2021), as almost all have cooling ages longer than the cooling age required for the onset of this dynamo. In this case, the conclusion would be that the crystallisation dynamo is responsible for approximate symmetry about the rotation axis. This situation could have similarities with what has been observed in fully convective late type stars with no di ff erential rotation, which are known to be able to generate strong and simple large-scale, mostly axisymmetric, poloidal fields (Donati et al. 2006; Morin et al. 2008b,a; Kochukhov 2021, Fig. 14). However, how the crystallisation dynamo alone can produce fields stronger than ∼ 1 MG is still unclear (Isern et al. 2017; Castro-Tapia et al. 2024). In fact, Montgomery & Dunlap (2024) have argued that fluid mixing by phase separation is not a viable mechanism to produce the strong fields observed in old white dwarfs. Perhaps the crystallisation dynamo could be more powerful if it were to amplify a pre-existing internal field, such as the same seed field that is seen in some of the white dwarfs prior to the beginning of crystallisation.</text>
<text><location><page_8><loc_6><loc_10><loc_49><loc_17></location>Montgomery & Dunlap (2024) have proposed that corecrystallisation could trigger temporary di ff erential rotation. Therefore, one could suspect that rapid rotation in normal-mass white dwarfs might generate a magnetic field in the stellar core using rotational shear on a seed field left from an earlier point in the star's evolutionary history. In this situation, we might also</text>
<text><location><page_8><loc_51><loc_84><loc_94><loc_93></location>expect that the field would be roughly axisymmetric, although it is not clear why such fields would always be strong or how they might be related to the weaker oblique fields of stars of similar mass. Furthermore, Spruit (1999) has shown that di ff erential rotation would suppress the non-axisymmetric field component of weak fields, but not in the stronger fields. This is the opposite of what we observed.</text>
<text><location><page_8><loc_51><loc_80><loc_94><loc_84></location>The origin of strong non-axisymmetric fields of normal-mass white dwarfs that appear before crystallisation could possibly be similar to that of higher-mass white dwarfs (see Sect. 6.2 below).</text>
<section_header_level_1><location><page_8><loc_51><loc_77><loc_88><loc_78></location>6.2. The variability of massive magnetic white dwarfs</section_header_level_1>
<text><location><page_8><loc_51><loc_58><loc_94><loc_76></location>Compared to normal-mass white dwarfs, the magnetic fields of massive white dwarfs present a di ff erent behaviour. Magnetic fields appear when the stars are still very young, and fields may be either symmetric or non-symmetric about the rotation axis. In massive white dwarfs, strong fields (tens or hundreds of megagauss) seem much more common than weaker sub-megagauss fields. It is widely thought that many of these high-mass objects are the result of WD-WD mergers that cause rapid generation of a strong magnetic field (García-Berro et al. 2012; Bagnulo & Landstreet 2022). However, Camisassa et al. (2022) and Blatman & Ginzburg (2024) have shown that at least some of them, depending on their core composition, may have started the process of core crystallisation. Hence, the origin of their field could be linked to the crystallisation dynamo, at least in some cases.</text>
<text><location><page_8><loc_51><loc_43><loc_94><loc_57></location>The bimodal distribution of the morphologies of the fields of the massive magnetic white dwarfs could indeed reflect two di ff erent channels of formation, one from WD-WD merger (or by a di ff erent binary evolution path), and one by massive singlestar evolution, such as that of normal-mass white dwarfs. Alternatively, the dynamo stimulated by WD-WD merging may be capable of generating both axisymmetric and non-axisymmetric global fields, perhaps depending on the initial angular momentum vectors of the individual merging white dwarfs relative to the orbital angular momentum vector or the mass ratio of the two merging white dwarfs.</text>
<text><location><page_8><loc_51><loc_31><loc_94><loc_43></location>A viable alternative explanation is that the massive white dwarfs that do not show variability are actually rotating with a period much shorter than the typical exposure time of individual polarimetric measurements. In any case, the large mass of stars in this sample and the occurrence of some rotation periods as short as minutes make it very reasonable to suppose that the evolution of the magnetic fields and rotation periods follows a di ff erent course from the evolution of fields in normal-mass magnetic white dwarfs.</text>
<text><location><page_8><loc_51><loc_20><loc_94><loc_31></location>Because almost all of the massive magnetic white dwarfs of our sample, except one, are very young, we cannot test whether the morphology of older stars is generally axisymmetric. Remarkably, however, the only example we know of a massive star older than 2 Gyr shows a non-axisymmetric field. With an age of more than 4 Gyr, high mass, and a very strong and variable field, the very unusual WD 0756 + 437 may be considered further evidence that the origin of the fields in massive stars is di ff erent than in normal-mass stars.</text>
<section_header_level_1><location><page_8><loc_51><loc_16><loc_63><loc_17></location>7. Conclusions</section_header_level_1>
<text><location><page_8><loc_51><loc_10><loc_94><loc_15></location>Since the earliest discoveries of magnetic white dwarfs (Kemp et al. 1970; Angel & Landstreet 1970, 1971b), two quite di ff erent categories of them have been known to exist. Some magnetic white dwarfs show periodic variations of circular polarisation</text>
<text><location><page_9><loc_6><loc_88><loc_49><loc_93></location>(which probes the longitudinal magnetic field) with timescales ranging from minutes to days. Classically, a signal of circular polarisation constant with time has been interpreted as indicating either a very long rotation period or a lack of rotation.</text>
<text><location><page_9><loc_6><loc_66><loc_49><loc_88></location>Using both literature data and new observations, we have studied the polarimetric variability of a sample of 74 magnetic white dwarfs. We find that among white dwarfs with M ≤ 1 . 0 M ⊙ ('normal-mass white dwarfs'), nearly all stars with fields weaker than about 1 MG show circular polarisation varying with time. Furthermore, the rare normal-mass white dwarfs younger than ≈ 2 Gyr with strong magnetic fields also show polarimetric variability. In striking contrast, 16 out of the 17 normal-mass stars older than 2 Gyr with fields stronger than about 10 MG in our sample show constant polarisation. Magnetic white dwarfs with M ≥ 1 . 0 M ⊙ ('massive white dwarfs'), many of which are the product of WD-WD merging, show a mixed behaviour. Many of them have a strong magnetic and variable field, but some have a strong and constant magnetic field. In our sample, nearly all the massive magnetic white dwarfs are younger than ≈ 1 Gyr, with the exception of WD 0756 + 437, an old strongly magnetic variable star.</text>
<text><location><page_9><loc_6><loc_58><loc_49><loc_66></location>The lack of evidence for major variations of circular polarisation on any timescale longer than about two weeks suggests that the interpretation of non-variability arising from extremely long rotational periods is incorrect. We have argued that the nonvariability is due to magnetic field structures roughly symmetric around the star's rotation axis.</text>
<text><location><page_9><loc_6><loc_34><loc_49><loc_58></location>A possible explanation for the observed symmetry could be that most or all of the magnetic fields evolved from pre-existing fields that formed during pre-white dwarf evolution stages (Li et al. 2022, 2023) and gradually relaxed to the stellar surface over a relation time of 1 or 2 Gyr; in fact, very recently, Camisassa et al. (2024) have shown that the magnetic fields of old white dwarfs with M ≥ 0 . 65 M ⊙ may have been generated by a coreconvection dynamo when the star was in the main sequence, and emerged at the stellar surface during the white dwarf cooling phase. As the surface field increases in strength, at some point the Lorentz forces in the outer layers, together with the Coriolis forces acting on rotating convective flows in what are asymmetric rotators, could lead to a gradual relaxation, through energy dissipation, of the global field structure to a form that is symmetric about the rotation axis. The timescale would depend on the field strength, and for a field weaker than several megagauss, it could require a timescale too long to be observed in white dwarfs that still show spectral lines.</text>
<text><location><page_9><loc_6><loc_10><loc_49><loc_34></location>Alternatively, because these normal-mass large-field magnetic white dwarfs mostly occur after the start of crystallisation (Bagnulo & Landstreet 2021, 2022), one could speculate that the crystallisation dynamo (Isern et al. 2017; Schreiber et al. 2021; Ginzburg et al. 2022) may generate some fraction of the observed fields and further that this dynamo produces essentially axisymmetric surface magnetic field structures with extremely strong fields. This kind of origin would have similarities with the axisymmetric fields that are commonly found in fully convective strongly magnetic M-dwarfs and, much weaker, in the planets Earth and Jupiter. However, it has been argued that the instability of the mantle surrounding the core that starts to crystallise cannot produce fields much stronger than 1 MG (Isern et al. 2017; Montgomery & Dunlap 2024; Castro-Tapia et al. 2024). Furthermore, Camisassa et al. (2024) find that, even if the crystallisation-driven dynamo could generate a strong magnetic field, this field would take too long to emerge at the stellar surface. On the other hand, Montgomery & Dunlap (2024) have suggested that crystallisation may still play a role by trig-</text>
<text><location><page_9><loc_51><loc_89><loc_94><loc_93></location>gering a temporary phenomenon of di ff erential rotation, which in turn would generate a magnetic field - the di ff usion timescale of which has not been discussed.</text>
<text><location><page_9><loc_51><loc_76><loc_94><loc_89></location>The situation for ultra-massive white dwarfs (with M > ∼ 1 . 0 M ⊙ ) is di ff erent, as both strongly magnetic variable and nonvariable white dwarfs are found among them. Some of these massive white dwarfs are the product of WD-WD merging, during which a dynamo could create a strong magnetic field (García-Berro et al. 2012), though it would not necessarily be symmetric about the rotation axis. In addition, some massive magnetic white dwarfs could be the result of single-star evolution and have acquired an axisymmetric field following the same mechanism acting in normal-mass white dwarfs.</text>
<text><location><page_9><loc_51><loc_70><loc_94><loc_76></location>Further investigation into the variability of magnetic white dwarfs is necessary. Nevertheless, current observations have already imposed significant constraints that any theory explaining the origin and evolution of magnetic fields in degenerate stars must take into account.</text>
<table>
<location><page_10><loc_10><loc_10><loc_90><loc_90></location>
<caption>Table 2. Stars used in this work, their main physical parameters, and a note on their magnetic variability.</caption>
</table>
<paragraph><location><page_11><loc_25><loc_95><loc_75><loc_96></location>S. Bagnulo and J.D. Landstreet : Strong fields of old white dwarfs are axisymmetric</paragraph>
<table>
<location><page_11><loc_10><loc_78><loc_90><loc_91></location>
<caption>Table 2. Continued.</caption>
</table>
<text><location><page_11><loc_6><loc_67><loc_94><loc_76></location>Notes. A number in the last column represents the established rotation period (in d) of a star that shows polarimetric variability; tentative periods are followed by the symbol ':'. Other symbols have the following meaning: 'var.' means that the star is certainly polarimetric variable but the period is still unknown; 'var.:' means that hints of subtle variability have been detected over a short timescale; 'n.v.' means that the observed polarisation was constant (within uncertainties) and the star was observed at least three times; 'n.v.:' means that observed polarisation was constant (within uncertainties) but the star was observed only twice; 's.v.' means that polarisation shows some sign of subtle variability over a timescale of a decade or longer. For stars within the local 40 pc volume, the parameters of stellar magnitude, distance, spectral type, atmospheric composition, temperature, mass, and age are from O'Brien et al. (2024); for the remaining stars, we used the catalogue from Gentile Fusillo et al. (2021) with ages interpolated from the tables by Bédard et al. (2020).</text>
<text><location><page_12><loc_6><loc_74><loc_49><loc_93></location>Acknowledgements. The new observations presented in this work were made with the FORS2 instrument at the ESO Telescopes at the La Silla Paranal Observatory under program ID 110.243J.001, 110.23XV.001, 110.23XV.002, and 112.25C9.001, and with the ISIS instrument at the William Herschel Telescope (operated on the island of La Palma by the Isaac Newton Group), under programmes P10 in 19A and P8 in 19B. This research has made use also of additional FORS1 and FORS2 data obtained from the ESO Science Archive Facility: data for WD 1036-204 were obtained under programmes IDs 70.D-0259(A), 087.D-0714(A), 089.D-0612(A), 090.D-0269(A). Data for WD 1105-340 was acquired with ESPaDOnS on the Canada-France-Hawaii Telescope (CFHT) (operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii), under programmes 17AC01, 19AC04, 19BC02, and 21BC02. We thank the anonymous referee for their very constructive criticism. We thank Matthias Schreiber and Antonino Lanza for very useful comments. JDL acknowledges the financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference number 6377-2016.</text>
<section_header_level_1><location><page_12><loc_6><loc_69><loc_16><loc_71></location>References</section_header_level_1>
<text><location><page_12><loc_6><loc_68><loc_39><loc_69></location>Achilleos, N. & Wickramasinghe, D. T. 1989, ApJ, 346, 444</text>
<unordered_list>
<list_item><location><page_12><loc_6><loc_66><loc_49><loc_68></location>Achilleos, N., Wickramasinghe, D. T., Liebert, J., Sa ff er, R. A., & Grauer, A. D. 1992, ApJ, 396, 273</list_item>
<list_item><location><page_12><loc_6><loc_65><loc_27><loc_65></location>Angel, J. R. P. 1978, ARA&A, 16, 487</list_item>
<list_item><location><page_12><loc_6><loc_64><loc_43><loc_64></location>Angel, J. R. P., Borra, E. F., & Landstreet, J. D. 1981, ApJS, 45, 457</list_item>
<list_item><location><page_12><loc_6><loc_63><loc_43><loc_63></location>Angel, J. R. P., Hintzen, P., & Landstreet, J. D. 1975, ApJ, 196, L27</list_item>
<list_item><location><page_12><loc_6><loc_60><loc_49><loc_62></location>Angel, J. R. P., Hintzen, P., Strittmatter, P. A., & Martin, P. G. 1974, ApJ, 190, L71</list_item>
</unordered_list>
<text><location><page_12><loc_6><loc_59><loc_45><loc_60></location>Angel, J. R. P., Illing, R. M. E., & Landstreet, J. D. 1972a, ApJ, 175, L85</text>
<unordered_list>
<list_item><location><page_12><loc_6><loc_58><loc_46><loc_59></location>Angel, J. R. P., Illing, R. M. E., & Landstreet, J. D. 1972b, ApJ, 175, L85</list_item>
<list_item><location><page_12><loc_6><loc_57><loc_36><loc_58></location>Angel, J. R. P. & Landstreet, J. D. 1970, ApJ, 162, L61</list_item>
<list_item><location><page_12><loc_6><loc_56><loc_36><loc_57></location>Angel, J. R. P. & Landstreet, J. D. 1971a, ApJ, 164, L15</list_item>
<list_item><location><page_12><loc_6><loc_55><loc_36><loc_56></location>Angel, J. R. P. & Landstreet, J. D. 1971b, ApJ, 165, L71</list_item>
<list_item><location><page_12><loc_6><loc_54><loc_36><loc_55></location>Angel, J. R. P. & Landstreet, J. D. 1972, ApJ, 178, L21</list_item>
<list_item><location><page_12><loc_6><loc_53><loc_36><loc_54></location>Angel, J. R. P. & Landstreet, J. D. 1974, ApJ, 191, 457</list_item>
<list_item><location><page_12><loc_6><loc_51><loc_49><loc_53></location>Antonyuk, K. A., Kolesnikov, S. V., Pit, N. V., et al. 2016, Astrophysical Bulletin, 71, 475</list_item>
<list_item><location><page_12><loc_6><loc_50><loc_45><loc_51></location>Appenzeller, I., Fricke, K., Fürtig, W., et al. 1998, The Messenger, 94, 1</list_item>
</unordered_list>
<text><location><page_12><loc_6><loc_49><loc_48><loc_50></location>Aznar Cuadrado, R., Jordan, S., Napiwotzki, R., et al. 2004, A&A, 423, 1081</text>
<unordered_list>
<list_item><location><page_12><loc_6><loc_48><loc_48><loc_49></location>Bagnulo, S., Farihi, J., Landstreet, J. D., & Folsom, C. P. 2024a, ApJ, 963, L22</list_item>
<list_item><location><page_12><loc_6><loc_47><loc_36><loc_48></location>Bagnulo, S. & Landstreet, J. D. 2018, A&A, 618, A113</list_item>
<list_item><location><page_12><loc_6><loc_46><loc_36><loc_47></location>Bagnulo, S. & Landstreet, J. D. 2019a, A&A, 630, A65</list_item>
<list_item><location><page_12><loc_6><loc_45><loc_38><loc_46></location>Bagnulo, S. & Landstreet, J. D. 2019b, MNRAS, 486, 4655</list_item>
<list_item><location><page_12><loc_6><loc_44><loc_36><loc_45></location>Bagnulo, S. & Landstreet, J. D. 2020, A&A, 643, A134</list_item>
<list_item><location><page_12><loc_6><loc_43><loc_37><loc_44></location>Bagnulo, S. & Landstreet, J. D. 2021, MNRAS, 507, 5902</list_item>
<list_item><location><page_12><loc_6><loc_42><loc_34><loc_43></location>Bagnulo, S. & Landstreet, J. D. 2022, ApJ, 935, L12</list_item>
<list_item><location><page_12><loc_6><loc_41><loc_43><loc_42></location>Bagnulo, S., Landstreet, J. D., Farihi, J., et al. 2024b, A&A, 688, L14</list_item>
<list_item><location><page_12><loc_6><loc_40><loc_47><loc_41></location>Barstow, M. A., Jordan, S., O'Donoghue, D., et al. 1995, MNRAS, 277, 971</list_item>
<list_item><location><page_12><loc_6><loc_39><loc_45><loc_40></location>Bédard, A., Bergeron, P., Brassard, P., & Fontaine, G. 2020, ApJ, 901, 93</list_item>
<list_item><location><page_12><loc_6><loc_37><loc_49><loc_39></location>Berdyugin, A., Landstreet, J. D., Bagnulo, S., Piirola, V., & Berdyugina, S. V. 2024, A&A, 690, A10</list_item>
<list_item><location><page_12><loc_6><loc_36><loc_34><loc_37></location>Berdyugin, A. V. 1995, Astronomy Letters, 21, 695</list_item>
<list_item><location><page_12><loc_6><loc_35><loc_34><loc_36></location>Berdyugin, A. V. & Piirola, V. 1999, A&A, 352, 619</list_item>
<list_item><location><page_12><loc_6><loc_33><loc_49><loc_35></location>Berdyugin, A. V., Piirola, V., Bagnulo, S., Landstreet, J. D., & Berdyugina, S. V. 2022, A&A, 657, A105</list_item>
<list_item><location><page_12><loc_6><loc_31><loc_49><loc_32></location>Berdyugin, A. V., Piirola, V., Bagnulo, S., Landstreet, J. D., & Berdyugina, S. V. 2023, A&A, 670, A2</list_item>
<list_item><location><page_12><loc_6><loc_28><loc_49><loc_30></location>Berdyugina, S. V., Berdyugin, A. V., & Piirola, V. 2007, Phys. Rev. Lett., 99, 091101</list_item>
<list_item><location><page_12><loc_6><loc_27><loc_40><loc_28></location>Bergeron, P., Ruiz, M.-T., & Leggett, S. K. 1992, ApJ, 400, 315</list_item>
<list_item><location><page_12><loc_6><loc_26><loc_34><loc_27></location>Beuermann, K. & Reinsch, K. 2002, A&A, 381, 487</list_item>
<list_item><location><page_12><loc_6><loc_25><loc_40><loc_26></location>Bisikalo, D., Sobolev, A., & Zhilkin, A. 2021, Galaxies, 9, 110</list_item>
<list_item><location><page_12><loc_6><loc_24><loc_35><loc_25></location>Blatman, D. & Ginzburg, S. 2024, MNRAS, 533, L13</list_item>
<list_item><location><page_12><loc_6><loc_22><loc_49><loc_24></location>Briggs, G. P., Ferrario, L., Tout, C. A., Wickramasinghe, D. T., & Hurley, J. R. 2015, MNRAS, 447, 1713</list_item>
<list_item><location><page_12><loc_6><loc_20><loc_49><loc_22></location>Brinkworth, C. S., Burleigh, M. R., Lawrie, K., Marsh, T. R., & Knigge, C. 2013, ApJ, 773, 47</list_item>
<list_item><location><page_12><loc_6><loc_19><loc_42><loc_20></location>Burleigh, M. R., Jordan, S., & Schweizer, W. 1999, ApJ, 510, L37</list_item>
<list_item><location><page_12><loc_6><loc_17><loc_49><loc_19></location>Bychkov, V. D., Fabrika, S. N., & Shtol', V. G. 1991, Pisma v Astronomicheskii Zhurnal, 17, 43</list_item>
<list_item><location><page_12><loc_6><loc_15><loc_49><loc_17></location>Camisassa, M., Fuentes, J. R., Schreiber, M. R., et al. 2024, arXiv e-prints, arXiv:2411.02296</list_item>
<list_item><location><page_12><loc_6><loc_14><loc_46><loc_15></location>Camisassa, M. E., Raddi, R., Althaus, L. G., et al. 2022, MNRAS, 516, L1</list_item>
<list_item><location><page_12><loc_6><loc_12><loc_49><loc_14></location>Castro-Tapia, M., Zhang, S., & Cumming, A. 2024, arXiv e-prints, arXiv:2406.01807</list_item>
<list_item><location><page_12><loc_6><loc_11><loc_42><loc_12></location>Cohen, M. H., Putney, A., & Goodrich, R. W. 1993, ApJ, 405, L67</list_item>
<list_item><location><page_12><loc_6><loc_10><loc_27><loc_11></location>Cropper, M. 1988, MNRAS, 231, 597</list_item>
<list_item><location><page_12><loc_51><loc_92><loc_91><loc_93></location>Donati, J. F., Howarth, I. D., Jardine, M. M., et al. 2006, MNRAS, 370, 629</list_item>
<list_item><location><page_12><loc_51><loc_91><loc_81><loc_92></location>Donati, J.-F. & Landstreet, J. D. 2009, ARA&A, 47, 333</list_item>
<list_item><location><page_12><loc_51><loc_90><loc_78><loc_91></location>Downes, R. A. & Margon, B. 1983, PASP, 95, 358</list_item>
<list_item><location><page_12><loc_51><loc_89><loc_88><loc_90></location>Dufour, P., Bergeron, P., Schmidt, G. D., et al. 2006, ApJ, 651, 1112</list_item>
<list_item><location><page_12><loc_51><loc_87><loc_94><loc_89></location>Efimov, I. S. 1981, Izvestiya Ordena Trudovogo Krasnogo Znameni Krymskoj Astrofizicheskoj Observatorii, 63, 118</list_item>
<list_item><location><page_12><loc_51><loc_85><loc_94><loc_87></location>Euchner, F., Jordan, S., Beuermann, K., Reinsch, K., & Gänsicke, B. T. 2006, A&A, 451, 671</list_item>
<list_item><location><page_12><loc_51><loc_83><loc_94><loc_85></location>Euchner, F., Reinsch, K., Jordan, S., Beuermann, K., & Gänsicke, B. T. 2005, A&A, 442, 651</list_item>
<list_item><location><page_12><loc_51><loc_81><loc_94><loc_82></location>Fabrika, S. N., Shtol', V. G., Valyavin, G. G., & Bychkov, V. D. 1997, Astronomy Letters, 23, 43</list_item>
<list_item><location><page_12><loc_51><loc_79><loc_88><loc_80></location>Farihi, J., Fossati, L., Wheatley, P. J., et al. 2018, MNRAS, 474, 947</list_item>
<list_item><location><page_12><loc_51><loc_78><loc_85><loc_79></location>Farihi, J., Robert, A., & Walters, N. 2024, MNRAS, 529, L164</list_item>
<list_item><location><page_12><loc_51><loc_77><loc_93><loc_78></location>Ferrario, L., de Martino, D., & Gänsicke, B. T. 2015, Space Sci. Rev., 191, 111</list_item>
<list_item><location><page_12><loc_51><loc_75><loc_94><loc_77></location>Ferrario, L., Vennes, S., Wickramasinghe, D. T., Bailey, J. A., & Christian, D. J. 1997, MNRAS, 292, 205</list_item>
<list_item><location><page_12><loc_51><loc_73><loc_94><loc_75></location>Ferrario, L., Wickramasinghe, D., & Kawka, A. 2020, Advances in Space Research, 66, 1025</list_item>
<list_item><location><page_12><loc_51><loc_72><loc_83><loc_73></location>Franzon, B. & Schramm, S. 2015, Phys. Rev. D, 92, 083006</list_item>
<list_item><location><page_12><loc_51><loc_71><loc_94><loc_72></location>Friedrich, S., Östreicher, R., & Ruder, H. 1993, in Astronomische Gesellschaft</list_item>
</unordered_list>
<text><location><page_12><loc_6><loc_7><loc_24><loc_8></location>Article number, page 12 of 29</text>
<text><location><page_12><loc_53><loc_70><loc_91><loc_71></location>Abstract Series, Vol. 9, Astronomische Gesellschaft Abstract Series, 145</text>
<unordered_list>
<list_item><location><page_12><loc_51><loc_68><loc_94><loc_70></location>Gänsicke, B. T., Rodríguez-Gil, P., Gentile Fusillo, N. P., et al. 2020, MNRAS, 499, 2564</list_item>
<list_item><location><page_12><loc_51><loc_65><loc_94><loc_68></location>García-Berro, E., Lorén-Aguilar, P., Aznar-Siguán, G., et al. 2012, ApJ, 749, 25 Gentile Fusillo, N. P., Tremblay, P. E., Cukanovaite, E., et al. 2021, MNRAS, 508, 3877</list_item>
<list_item><location><page_12><loc_51><loc_64><loc_92><loc_65></location>Ginzburg, S., Fuller, J., Kawka, A., & Caiazzo, I. 2022, MNRAS, 514, 4111</list_item>
<list_item><location><page_12><loc_51><loc_63><loc_90><loc_64></location>Hermes, J. J., Gänsicke, B. T., Kawaler, S. D., et al. 2017, ApJS, 232, 23</list_item>
<list_item><location><page_12><loc_51><loc_61><loc_94><loc_63></location>Hernandez, M. S., Schreiber, M. R., Landstreet, J. D., et al. 2024, MNRAS, 528, 6056</list_item>
<list_item><location><page_12><loc_51><loc_58><loc_94><loc_61></location>Isern, J., García-Berro, E., Külebi, B., & Lorén-Aguilar, P. 2017, ApJ, 836, L28 Jordan, S. 2003, in NATO Advanced Study Institute (ASI) Series B, Vol. 105, White Dwarfs, 175</list_item>
<list_item><location><page_12><loc_51><loc_57><loc_77><loc_57></location>Jordan, S. & Friedrich, S. 2002, A&A, 383, 519</list_item>
<list_item><location><page_12><loc_51><loc_56><loc_85><loc_56></location>Jordan, S., Schmelcher, P., & Becken, W. 2001, A&A, 376, 614</list_item>
<list_item><location><page_12><loc_51><loc_51><loc_94><loc_55></location>Jordan, S., Schmelcher, P., Becken, W., & Schweizer, W. 1998, A&A, 336, L33 Kawaler, S. D. 2015, in Astronomical Society of the Pacific Conference Series, Vol. 493, 19th European Workshop on White Dwarfs, ed. P. Dufour, P. Bergeron, & G. Fontaine, 65</list_item>
<list_item><location><page_12><loc_51><loc_48><loc_94><loc_51></location>Kawka, A. 2020, in White Dwarfs as Probes of Fundamental Physics: Tracers of Planetary, Stellar and Galactic Evolution, ed. M. A. Barstow, S. J. Kleinman, J. L. Provencal, & L. Ferrario, Vol. 357, 60-74</list_item>
<list_item><location><page_12><loc_51><loc_47><loc_88><loc_48></location>Kawka, A., Briggs, G. P., Vennes, S., et al. 2017, MNRAS, 466, 1127</list_item>
<list_item><location><page_12><loc_51><loc_46><loc_78><loc_47></location>Kawka, A. & Vennes, S. 2012, MNRAS, 425, 1394</list_item>
<list_item><location><page_12><loc_51><loc_45><loc_86><loc_46></location>Kawka, A., Vennes, S., & Thorstensen, J. R. 2004, AJ, 127, 1702</list_item>
<list_item><location><page_12><loc_51><loc_43><loc_94><loc_45></location>Kemp, J. C., Coyne, G. V., Swedlund, S. J. J. B., & Wolstencroft, R. D. 1974, ApJ, 189, L79</list_item>
<list_item><location><page_12><loc_51><loc_41><loc_94><loc_43></location>Kemp, J. C., Swedlund, J. B., Landstreet, J. D., & Angel, J. R. P. 1970, ApJ, 161, L77</list_item>
<list_item><location><page_12><loc_51><loc_39><loc_94><loc_41></location>Kilic, M., Kosakowski, A., Moss, A. G., Bergeron, P., & Conly, A. A. 2021, ApJ, 923, L6</list_item>
<list_item><location><page_12><loc_51><loc_37><loc_88><loc_39></location>Kilic, M., Rolland, B., Bergeron, P., et al. 2019, MNRAS, 489, 3648 Kochukhov, O. 2021, A&A Rev., 29, 1</list_item>
<list_item><location><page_12><loc_51><loc_35><loc_93><loc_37></location>Koester, D., Dreizler, S., Weidemann, V., & Allard, N. F. 1998, A&A, 338, 612 Koester, D., Voss, B., Napiwotzki, R., et al. 2009, A&A, 505, 441</list_item>
<list_item><location><page_12><loc_51><loc_33><loc_94><loc_35></location>Landi Degl'Innocenti, E. & Landolfi, M., eds. 2004, Astrophysics and Space Science Library, Vol. 307, Polarization in Spectral Lines</list_item>
<list_item><location><page_12><loc_51><loc_32><loc_74><loc_32></location>Landstreet, J. D. 1987, MNRAS, 225, 437</list_item>
<list_item><location><page_12><loc_51><loc_31><loc_80><loc_31></location>Landstreet, J. D. & Angel, J. R. P. 1971, ApJ, 165, L67</list_item>
<list_item><location><page_12><loc_51><loc_30><loc_80><loc_30></location>Landstreet, J. D. & Angel, J. R. P. 1974, ApJ, 190, L25</list_item>
<list_item><location><page_12><loc_51><loc_28><loc_80><loc_29></location>Landstreet, J. D. & Bagnulo, S. 2019, A&A, 623, A46</list_item>
<list_item><location><page_12><loc_51><loc_27><loc_80><loc_28></location>Landstreet, J. D. & Bagnulo, S. 2020, A&A, 634, L10</list_item>
<list_item><location><page_12><loc_51><loc_26><loc_94><loc_27></location>Landstreet, J. D., Bagnulo, S., Martin, A., & Valyavin, G. 2016, A&A, 591, A80</list_item>
<list_item><location><page_12><loc_51><loc_24><loc_94><loc_26></location>Landstreet, J. D., Bagnulo, S., Valyavin, G., & Valeev, A. F. 2017, A&A, 607, A92</list_item>
<list_item><location><page_12><loc_51><loc_23><loc_91><loc_24></location>Landstreet, J. D., Bagnulo, S., Valyavin, G. G., et al. 2012, A&A, 545, A30</list_item>
<list_item><location><page_12><loc_51><loc_22><loc_86><loc_23></location>Landstreet, J. D., Villaver, E., & Bagnulo, S. 2023, ApJ, 952, 129</list_item>
<list_item><location><page_12><loc_51><loc_18><loc_94><loc_22></location>Lawrie, K. A., Burleigh, M. R., Brinkworth, C. S., et al. 2013, in Astronomical Society of the Pacific Conference Series, Vol. 469, 18th European White Dwarf Workshop., ed. J. Krzesi'nski, G. Stachowski, P. Moskalik, & K. Bajan, 429</list_item>
<list_item><location><page_12><loc_51><loc_16><loc_94><loc_18></location>Leroy, J. L., Bagnulo, S., Landolfi, M., & Landi Degl'Innocenti, E. 1994, A&A, 284, 174</list_item>
<list_item><location><page_12><loc_51><loc_15><loc_88><loc_16></location>Li, G., Deheuvels, S., Ballot, J., & Lignières, F. 2022, Nature, 610, 43</list_item>
<list_item><location><page_12><loc_51><loc_14><loc_92><loc_15></location>Li, G., Deheuvels, S., Li, T., Ballot, J., & Lignières, F. 2023, A&A, 680, A26</list_item>
<list_item><location><page_12><loc_51><loc_13><loc_88><loc_14></location>Liebert, J., Angel, J. R. P., & Landstreet, J. D. 1975, ApJ, 202, L139</list_item>
<list_item><location><page_12><loc_51><loc_12><loc_94><loc_13></location>Liebert, J., Angel, J. R. P., Stockman, H. S., & Beaver, E. A. 1978, ApJ, 225, 181</list_item>
<list_item><location><page_12><loc_51><loc_10><loc_94><loc_12></location>Liebert, J., Angel, J. R. P., Stockman, H. S., Spinrad, H., & Beaver, E. A. 1977, ApJ, 214, 457</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_13><loc_6><loc_91><loc_49><loc_93></location>Liebert, J., Schmidt, G. D., Green, R. F., Stockman, H. S., & McGraw, J. T. 1983, ApJ, 264, 262</list_item>
</unordered_list>
<text><location><page_13><loc_6><loc_90><loc_42><loc_91></location>Liebert, J., Schmidt, G. D., Sion, E. M., et al. 1985, PASP, 97, 158</text>
<text><location><page_13><loc_6><loc_89><loc_46><loc_90></location>Macfarlane, S. A., Woudt, P. A., Dufour, P., et al. 2017, MNRAS, 470, 732</text>
<text><location><page_13><loc_6><loc_88><loc_46><loc_89></location>Manser, C. J., Gänsicke, B. T., Inight, K., et al. 2023, MNRAS, 521, 4976</text>
<text><location><page_13><loc_6><loc_87><loc_40><loc_88></location>Martin, B. & Wickramasinghe, D. T. 1978, MNRAS, 183, 533</text>
<text><location><page_13><loc_6><loc_85><loc_49><loc_86></location>Mathys, G. 2008, Contributions of the Astronomical Observatory Skalnate Pleso, 38, 217</text>
<text><location><page_13><loc_6><loc_82><loc_49><loc_84></location>Maxted, P. F. L., Ferrario, L., Marsh, T. R., & Wickramasinghe, D. T. 2000, MNRAS, 315, L41</text>
<text><location><page_13><loc_6><loc_80><loc_49><loc_82></location>McCook, G. P. & Sion, E. M. 1977, A Catalogue of spectroscopically identified white dwarfs</text>
<text><location><page_13><loc_6><loc_79><loc_33><loc_80></location>McCook, G. P. & Sion, E. M. 1999, ApJS, 121, 1</text>
<text><location><page_13><loc_6><loc_78><loc_25><loc_79></location>Mestel, L. 1999, Stellar magnetism</text>
<text><location><page_13><loc_6><loc_77><loc_34><loc_78></location>Mestel, L. & Takhar, H. S. 1972, MNRAS, 156, 419</text>
<text><location><page_13><loc_6><loc_76><loc_37><loc_77></location>Montgomery, M. H. & Dunlap, B. H. 2024, ApJ, 961, 197</text>
<text><location><page_13><loc_6><loc_75><loc_42><loc_76></location>Morin, J., Donati, J. F., Forveille, T., et al. 2008a, MNRAS, 384, 77</text>
<text><location><page_13><loc_6><loc_74><loc_41><loc_75></location>Morin, J., Donati, J. F., Petit, P., et al. 2008b, MNRAS, 390, 567</text>
<text><location><page_13><loc_6><loc_73><loc_42><loc_74></location>Moss, A., Bergeron, P., Kilic, M., et al. 2024, MNRAS, 527, 10111</text>
<text><location><page_13><loc_6><loc_71><loc_49><loc_73></location>O'Brien, M. W., Tremblay, P. E., Gentile Fusillo, N. P., et al. 2023, MNRAS, 518, 3055</text>
<text><location><page_13><loc_6><loc_67><loc_49><loc_70></location>O'Brien, M. W., Tremblay, P. E., Klein, B. L., et al. 2024, MNRAS, 527, 8687 Oliveira da Rosa, G., Kepler, S. O., Soethe, L. T. T., Romero, A. D., & Bell, K. J. 2024, arXiv e-prints, arXiv:2407.05214</text>
<text><location><page_13><loc_6><loc_66><loc_30><loc_67></location>Piirola, V. & Reiz, A. 1992, A&A, 259, 143</text>
<text><location><page_13><loc_6><loc_65><loc_25><loc_66></location>Preston, G. W. 1971, PASP, 83, 571</text>
<text><location><page_13><loc_6><loc_64><loc_23><loc_65></location>Putney, A. 1995, ApJ, 451, L67</text>
<text><location><page_13><loc_6><loc_63><loc_24><loc_64></location>Putney, A. 1997, ApJS, 112, 527</text>
<text><location><page_13><loc_6><loc_62><loc_30><loc_63></location>Putney, A. & Jordan, S. 1995, ApJ, 449, 863</text>
<text><location><page_13><loc_6><loc_61><loc_44><loc_62></location>Reding, J. S., Hermes, J. J., Vanderbosch, Z., et al. 2020, ApJ, 894, 19</text>
<text><location><page_13><loc_6><loc_60><loc_39><loc_61></location>Reimers, D., Hagen, H. J., & Hopp, U. 1999, A&A, 343, 157</text>
<unordered_list>
<list_item><location><page_13><loc_6><loc_59><loc_41><loc_60></location>Reimers, D., Jordan, S., Koester, D., et al. 1996, A&A, 311, 572</list_item>
</unordered_list>
<text><location><page_13><loc_6><loc_58><loc_40><loc_59></location>Schmidt, G. D., Bergeron, P., & Fegley, B. 1995, ApJ, 443, 274</text>
<unordered_list>
<list_item><location><page_13><loc_6><loc_56><loc_49><loc_58></location>Schmidt, G. D., Liebert, J., Harris, H. C., Dahn, C. C., & Leggett, S. K. 1999, ApJ, 512, 916</list_item>
</unordered_list>
<text><location><page_13><loc_6><loc_55><loc_37><loc_56></location>Schmidt, G. D. & Norsworthy, J. E. 1991, ApJ, 366, 270</text>
<text><location><page_13><loc_6><loc_54><loc_34><loc_54></location>Schmidt, G. D. & Smith, P. S. 1994, ApJ, 423, L63</text>
<text><location><page_13><loc_6><loc_53><loc_33><loc_53></location>Schmidt, G. D. & Smith, P. S. 1995, ApJ, 448, 305</text>
<unordered_list>
<list_item><location><page_13><loc_6><loc_50><loc_49><loc_52></location>Schmidt, G. D., Vennes, S., Wickramasinghe, D. T., & Ferrario, L. 2001, MNRAS, 328, 203</list_item>
<list_item><location><page_13><loc_6><loc_48><loc_49><loc_50></location>Schmidt, G. D., West, S. C., Liebert, J., Green, R. F., & Stockman, H. S. 1986, ApJ, 309, 218</list_item>
</unordered_list>
<text><location><page_13><loc_6><loc_46><loc_49><loc_48></location>Schreiber, M. R., Belloni, D., Gänsicke, B. T., Parsons, S. G., & Zorotovic, M. 2021, Nature Astronomy [ arXiv:2104.14607 ]</text>
<text><location><page_13><loc_6><loc_45><loc_22><loc_46></location>Schwab, J. 2021, ApJ, 906, 53</text>
<text><location><page_13><loc_6><loc_44><loc_49><loc_45></location>Siebenmorgen, R., Voshchinnikov, N. V., & Bagnulo, S. 2014, A&A, 561, A82</text>
<text><location><page_13><loc_6><loc_43><loc_49><loc_44></location>Sion, E. M., Liebert, J., Schmidt, G., & Starrfield, S. G. 1984, in Bulletin of the</text>
<text><location><page_13><loc_8><loc_42><loc_32><loc_43></location>American Astronomical Society, Vol. 16, 725</text>
<text><location><page_13><loc_6><loc_41><loc_25><loc_42></location>Spruit, H. C. 1999, A&A, 349, 189</text>
<text><location><page_13><loc_6><loc_40><loc_29><loc_41></location>Stibbs, D. W. N. 1950, MNRAS, 110, 395</text>
<text><location><page_13><loc_6><loc_38><loc_49><loc_40></location>Swedlund, J. B., Wolstencroft, R. D., Michalsky, Jr., J. J., & Kemp, J. C. 1974, ApJ, 187, L121</text>
<unordered_list>
<list_item><location><page_13><loc_6><loc_36><loc_49><loc_37></location>Tout, C. A., Wickramasinghe, D. T., Liebert, J., Ferrario, L., & Pringle, J. E. 2008, MNRAS, 387, 897</list_item>
</unordered_list>
<text><location><page_13><loc_6><loc_34><loc_42><loc_35></location>Valyavin, G., Bagnulo, S., Monin, D., et al. 2005, A&A, 439, 1099</text>
<text><location><page_13><loc_6><loc_33><loc_42><loc_34></location>Valyavin, G., Wade, G. A., Bagnulo, S., et al. 2008, ApJ, 683, 466</text>
<unordered_list>
<list_item><location><page_13><loc_6><loc_31><loc_49><loc_33></location>Vennes, S., Kawka, A., Ferrario, L., & Paunzen, E. 2018, Contributions of the Astronomical Observatory Skalnate Pleso, 48, 307</list_item>
</unordered_list>
<text><location><page_13><loc_6><loc_30><loc_42><loc_31></location>Vennes, S., Schmidt, G. D., Ferrario, L., et al. 2003, ApJ, 593, 1040</text>
<text><location><page_13><loc_6><loc_29><loc_49><loc_30></location>Vornanen, T., Berdyugina, S. V., Berdyugin, A. V., & Piirola, V. 2010, ApJ, 720,</text>
<text><location><page_13><loc_8><loc_28><loc_10><loc_29></location>L52</text>
<text><location><page_13><loc_6><loc_27><loc_43><loc_28></location>Walters, N., Farihi, J., Marsh, T. R., et al. 2021, MNRAS, 503, 3743</text>
<text><location><page_13><loc_6><loc_26><loc_34><loc_27></location>Wegner, G. 1977, Mem. Soc. Astron. Italiana, 48, 27</text>
<text><location><page_13><loc_6><loc_25><loc_43><loc_26></location>Wesemael, F., Liebert, J., Schmidt, G. D., et al. 2001, ApJ, 554, 1118</text>
<text><location><page_13><loc_6><loc_24><loc_24><loc_25></location>West, S. C. 1989a, ApJ, 345, 511</text>
<text><location><page_13><loc_6><loc_23><loc_24><loc_24></location>West, S. C. 1989b, ApJ, 345, 511</text>
<text><location><page_13><loc_6><loc_22><loc_39><loc_23></location>Wickramasinghe, D. T. & Bessell, M. S. 1976, ApJ, 203, L39</text>
<text><location><page_13><loc_6><loc_21><loc_41><loc_21></location>Wickramasinghe, D. T. & Cropper, M. 1988, MNRAS, 235, 1451</text>
<unordered_list>
<list_item><location><page_13><loc_6><loc_18><loc_49><loc_20></location>Zhong, Y., Kashiyama, K., Takasao, S., Shigeyama, T., & Fujisawa, K. 2024, ApJ, 963, 26</list_item>
</unordered_list>
<section_header_level_1><location><page_14><loc_6><loc_92><loc_32><loc_93></location>Appendix A: New observations</section_header_level_1>
<table>
<location><page_14><loc_6><loc_10><loc_95><loc_88></location>
<caption>Table A.1. Newly presented spectropolarimetric observations</caption>
</table>
<text><location><page_14><loc_6><loc_7><loc_24><loc_8></location>Article number, page 14 of 29</text>
<paragraph><location><page_15><loc_25><loc_95><loc_75><loc_96></location>S. Bagnulo and J.D. Landstreet : Strong fields of old white dwarfs are axisymmetric</paragraph>
<table>
<location><page_15><loc_6><loc_72><loc_95><loc_91></location>
<caption>Table A.1. Continued.</caption>
</table>
<section_header_level_1><location><page_16><loc_6><loc_90><loc_47><loc_93></location>Appendix B: Variability of magnetic white dwarfs: Comments on individual stars</section_header_level_1>
<text><location><page_16><loc_6><loc_77><loc_49><loc_88></location>We preliminary note that to compare observations taken by different authors, we need to know how circular polarisation was defined in di ff erent works. This problem has been highlighted in detail, for example in Sect. 2 of Bagnulo & Landstreet (2020). Here, we have adopted the definition of positive handedness of circular polarisation given by Landi Degl'Innocenti & Landolfi (2004), and converted some of the literature data to it by changing the sign of circular polarisation with respect to the originally published data.</text>
<text><location><page_16><loc_6><loc_59><loc_49><loc_76></location>Numerous examples in the literature (e.g. Bagnulo & Landstreet 2020) and in this section show that many magnetic white dwarfs with very strong fields have a circular polarisation spectrum that varies rapidly with wavelength. The consequence is that the same source observed with even slightly di ff erent broadband filters may result in di ff erences of the measured broadband polarisation signal that are larger than the formal uncertainties. This specific issue is discussed in Sect. 7 of Bagnulo & Landstreet (2019b), where some numerical examples are presented. As a consequence, small di ff erences between broadband polarimetric measurements obtained with di ff erent instruments cannot reliably provide strong evidence that a star is magnetically variable.</text>
<text><location><page_16><loc_6><loc_48><loc_49><loc_58></location>Spectropolarimetry should allow comparisons that are less instrument dependent, provided the observations overlap in wavelength, and that the resolving power of di ff erent instrument is similar. There is no doubt, however, that the safest way to establish variability or constancy of polarisation is to repeatedly observe the star with the same instrument and instrument setting, and to always perform the data reduction using exactly the same technique.</text>
<text><location><page_16><loc_6><loc_35><loc_49><loc_47></location>In the following, we discuss all stars of our sample (including two stars that did not make it into the final list of Table 2). The titles of the individual subsections contain the star's name in the Villanova system (McCook & Sion 1977, 1999), the main SIMBADidentifier, and our conclusions about the magnetic variability (using the same designation as in Table 2, followed by the rotation period P , if this is known). When the rotation period is obtained only from photometric data, we use the designation P phot.</text>
<section_header_level_1><location><page_16><loc_6><loc_31><loc_46><loc_32></location>Appendix B.1: WD 0004+122 = LP 464-57 (non-variable)</section_header_level_1>
<text><location><page_16><loc_6><loc_10><loc_49><loc_29></location>This star was discovered to be magnetic, using spectropolarimetry, by Bagnulo & Landstreet (2020), and later observed four times by Berdyugin et al. (2024) in broadband circular polarisation. These measurements are shown in the top left panel of Fig. B.1. None of them deviated more than ≃ 1 σ from the mean value of the measurements obtained in the same filter, except for a ∼ 2 . 5 σ di ff erence between on measurement and the other three in the B filter, which might reflect a real small variability. We reanalysed the FORS2 spectropolarimetry published by Bagnulo & Landstreet (2020) to check for di ff erences between the V / I profiles obtained from the first and the second pair of exposures, sampling an interval in time of about 15 m, without finding any di ff erence (see Fig. B.2). We conclude that the star does not show convincing sign of variability. We estimate a field strength of ≃ 100 MG.</text>
<section_header_level_1><location><page_16><loc_51><loc_91><loc_88><loc_93></location>Appendix B.2: WD 0009+501 = EGGR 381 (variable, P ≃ 8 h)</section_header_level_1>
<text><location><page_16><loc_51><loc_83><loc_94><loc_90></location>This white dwarf was well monitored and modelled as a magnetic variable with ⟨| B |⟩ between 150 and 250 kG by Valyavin et al. (2005), who presented a model obtained by adopting a rotational period of 0 . 3337 ± 0 . 0031 d. Recent unpublished ESPaDOnS data obtained by us confirm the rotational period.</text>
<section_header_level_1><location><page_16><loc_51><loc_78><loc_87><loc_81></location>Appendix B.3: WD 0011-134 = G 158-45 (variable, P phot ≃ 0 . 73 h)</section_header_level_1>
<text><location><page_16><loc_51><loc_62><loc_94><loc_77></location>Discovered magnetic (with a field of about 9 MG) by Bergeron et al. (1992), and as a magnetic variable by Putney (1997). From photometry, Lawrie et al. (2013) found a rotational period of 44 ± 0 . 43 min. While there is no guarantee that the photometric period reflects the rotational period of the star, the star is definitely a magnetic variable, and in fact, this very short photometric period is completely consistent with several of the many acceptable periods found from the six widely spaced ⟨ Bz ⟩ data values published by Putney (1997) from spectropolarimetric observations made between 1994 Sep 29 and 1994 Dec 31. This star is one of the magnetic white dwarfs that clearly shows longitudinal field reversal.</text>
<section_header_level_1><location><page_16><loc_51><loc_59><loc_92><loc_60></location>Appendix B.4: WD 0041-102 = Feige 7 (variable, P ≃ 2 h)</section_header_level_1>
<text><location><page_16><loc_51><loc_42><loc_94><loc_57></location>Liebert et al. (1977) showed that the spectrum is rich in faint lines of H and He, and identified a magnetic field of about 18-20 MG from the spectrum. They found, on the basis of broadband circular polarisation measurements, that the field varies with a period of 131.6 min. (Note that the broadband circular polarisation changes sign as the star rotates.) A decentred dipole model of the field structure based on a series of flux spectra was derived by Achilleos et al. (1992); both H and He apparently vary substantially in abundance over the surface. Achilleos et al. (1992) also discovered that the star varies photometrically (in a double wave) with the same period as the spectrum and broad-band polarisation.</text>
<section_header_level_1><location><page_16><loc_51><loc_39><loc_94><loc_40></location>Appendix B.5: WD 0051+117 = PHL 886 (variable, P ≃ 4 . 7 d)</section_header_level_1>
<text><location><page_16><loc_51><loc_31><loc_94><loc_38></location>From unpublished spectropolarimetric data (Landstreet & Bagnulo, in prep., hereafter LB25) it is found that the star is periodically variable ( P = 4 . 703 d) with ⟨| B |⟩ ≃ 270 kG. The field ⟨ Bz ⟩ reverses sign as the star rotates. Modelling will be presented in a forthcoming paper.</text>
<section_header_level_1><location><page_16><loc_51><loc_28><loc_91><loc_29></location>Appendix B.6: WD 0058-044 = GD 9 (variable, P ≃ 2 d?)</section_header_level_1>
<text><location><page_16><loc_51><loc_19><loc_94><loc_27></location>From our unpublished spectropolarimetric data we found that the star is periodically variable (the period is not determined uniquely from available data, but the most probable period is P = 1 . 96 d), with ⟨| B |⟩ ≃ 300 kG (LB25). The field ⟨ Bz ⟩ varies sinusoidally but does not (quite) reverse sign. Modelling will be presented in a forthcoming paper.</text>
<section_header_level_1><location><page_16><loc_51><loc_16><loc_88><loc_17></location>Appendix B.7: WD 0232+525 = EGGR 314 (variable)</section_header_level_1>
<text><location><page_16><loc_51><loc_10><loc_94><loc_15></location>This is a well-known star that was repeatedly included in studies of bright white dwarfs and been searched for a magnetic field (Bychkov et al. 1991; Schmidt & Smith 1995; Fabrika et al. 1997) without any detection being reported. Recent particularly</text>
<figure>
<location><page_17><loc_9><loc_51><loc_93><loc_93></location>
<caption>Fig. B.1. Variation in the broadband circular polarisation measurements observed in three filters B ' V ' R ' of the DiPol-UF instrument at the NOT (Berdyugin et al. 2022, 2023, 2024). In each panel, the three dotted lines represent the average of all data obtained in a particular filter: from top to bottom, the R ' filter (red dotted line), the V ' filter (green dotted line) and the B ' filter (blue dotted line). These lines are separated by ∆ V / I = 0 . 3 %. The x -axis represents the time of the observations in days from the first measurement. The solid circles represent the di ff erence between each measurement and the average of all measurements in a given filter (red circles for filter R ' , green circles for filter V ' , and blue circles for filter B ' .</caption>
</figure>
<figure>
<location><page_17><loc_7><loc_23><loc_49><loc_40></location>
<caption>Fig. B.2. WD0004 + 122: Stokes V / I from two pairs of observations obtained within 15 m on 2019-10-07.</caption>
</figure>
<text><location><page_17><loc_6><loc_10><loc_49><loc_15></location>sensitive observations (Bagnulo & Landstreet 2022) succeeded in detecting a weak field. The three published measurements (two with ISIS, one with ESPaDOnS) reveal a very weak, and probably variable longitudinal field (the measurement obtained</text>
<text><location><page_17><loc_51><loc_38><loc_94><loc_40></location>with ESPaDOnS has the opposite sign as those obtained with ISIS).</text>
<section_header_level_1><location><page_17><loc_51><loc_33><loc_92><loc_35></location>Appendix B.8: WD 0233-242A = LP 830-14 = NLTT 8435A (variable, P phot ≃ 1 . 5 h)</section_header_level_1>
<text><location><page_17><loc_51><loc_26><loc_94><loc_32></location>The star is discussed by Bagnulo & Landstreet (2021), who suggested the presence of a variable field of the order of 3.8 MG. In fact, ⟨| B |⟩ does not vary between two FORS observations, while ⟨ Bz ⟩ clearly changes sign. According to Vennes et al. (2018) it has a photometric variation period of 95 min.</text>
<section_header_level_1><location><page_17><loc_51><loc_21><loc_90><loc_23></location>Appendix B.9: WD 0236-269 = PHL 4227 (n.v.: - only 2 meas.)</section_header_level_1>
<text><location><page_17><loc_51><loc_10><loc_94><loc_20></location>Schmidt et al. (2001) report two spectropolarimetric detections of circular polarisation at about the 1.2% level. The two observations were obtained three days apart, with no changes detected. Furthermore no significant change in the spectrum-average circular polarisation was observed. We tentatively assume that is not a magnetic variable, but the evidence for non-variability is not compelling (the original paper just reports: "consistent results were obtained on the two occasions").</text>
<section_header_level_1><location><page_18><loc_6><loc_91><loc_48><loc_93></location>Appendix B.10: WD 0253+508 = KPD 0253+5052 (variable, P ≃ 4 h)</section_header_level_1>
<text><location><page_18><loc_6><loc_81><loc_49><loc_89></location>The field was discovered by Downes & Margon (1983), who estimated ⟨| B |⟩ ≈ 13 MG, and modelled by Achilleos & Wickramasinghe (1989) using only flux spectra. The star shows continuum polarisation up to 0.4% that is periodically variable according to Schmidt & Norsworthy (1991) with P = 0 . 170 d. The sign of the broadband continuum circular polarisation briefly reverses during each cycle.</text>
<section_header_level_1><location><page_18><loc_6><loc_76><loc_40><loc_78></location>Appendix B.11: WD 0301+059 = SDSS J030350.63+060748.9 (n.v.: - only 2 meas.)</section_header_level_1>
<text><location><page_18><loc_6><loc_66><loc_49><loc_74></location>The star was discovered to be magnetic by Landstreet & Bagnulo (2020), who did not find any obvious sign of variability between two observations obtained 1 day apart, nor between individual exposures of the same observing series (exposure times of single frames were 225 s). We tentatively assume that it is not magnetically variable. The star therefore appears as a strongly magnetic, massive, non-variable star.</text>
<section_header_level_1><location><page_18><loc_6><loc_61><loc_48><loc_63></location>Appendix B.12: WD 0313-084 = GALEX J031613.8-081637 (non-variable)</section_header_level_1>
<text><location><page_18><loc_6><loc_55><loc_49><loc_59></location>The star was discovered to be magnetic by O'Brien et al. (2023). It was observed five times with FORS2 (see Sect. 2.1), and no variability was found within timescales of 1 h, 1 d, and 8 months (see Fig. B.3). The spectrum indicates that ⟨| B |⟩ ≃ 800 kG.</text>
<section_header_level_1><location><page_18><loc_6><loc_50><loc_49><loc_52></location>Appendix B.13: WD 0316-849 = EUVE J0317-855 (variable, P ≃ 0 . 2 h)</section_header_level_1>
<text><location><page_18><loc_6><loc_33><loc_49><loc_48></location>Studied by Ferrario et al. (1997) Burleigh et al. (1999), and Vennes et al. (2003), the star has a rotation period P = 725 s. A detailed model based on UV spectra was obtained by Burleigh et al. (1999), who found that the field varies in strength from about 180 to 800 MG over the surface. Vennes et al. (2003) shown that the star has a light curve with a minimum that corresponds to the maximum of the polarisation. This is one of the very few white dwarfs that show magnetic and photometric variability, for which we have a firmly established phase relationship between photometry and circular polarisation. This star is erroneously missing in the list of such stars compiled by Bagnulo et al. (2024b).</text>
<section_header_level_1><location><page_18><loc_6><loc_30><loc_44><loc_31></location>Appendix B.14: WD 0322-019 = EGGR 566 (variable)</section_header_level_1>
<text><location><page_18><loc_6><loc_22><loc_49><loc_28></location>The star was discussed by Bagnulo & Landstreet (2021). Two FORS2 measurements by Farihi et al. (2018) obtained during two consecutive nights ( ⟨ Bz ⟩ = -5 . 4 ± 3 . 0 kG and -16 . 5 ± 2 . 3 kG) suggest that the longitudinal field may change on a timescale of a few days.</text>
<section_header_level_1><location><page_18><loc_6><loc_17><loc_45><loc_20></location>Appendix B.15: WD 0330-000 = HE 0330-0002 (only 2 meas., and contradicting data)</section_header_level_1>
<text><location><page_18><loc_6><loc_10><loc_49><loc_16></location>Schmidt et al. (2001) report two spectropolarimetric observations obtained one day apart, and concluded that no significant di ff erences had been detected in the overall spectrum. However, the mean circular polarisation averaged over the observed wavelength range reported in their Table 1 di ff ers by 0.5%, contra-</text>
<figure>
<location><page_18><loc_52><loc_47><loc_93><loc_93></location>
<caption>Fig. B.3. Top and middle panels: WD 0313 -084 observed with FORS2 in five di ff erent epochs. Bottom panels: Stokes V / I from four pairs of observations obtained within 1 h on 2023-09-19.</caption>
</figure>
<text><location><page_18><loc_51><loc_35><loc_94><loc_38></location>dicting the assessment of non-variability. Because of this, this star will not be considered in this paper.</text>
<section_header_level_1><location><page_18><loc_51><loc_30><loc_91><loc_33></location>Appendix B.16: WD 0410-114 = G 160-51 = NLTT 12758 (variable, P ≃ 0 . 3 h)</section_header_level_1>
<text><location><page_18><loc_51><loc_20><loc_94><loc_29></location>A magnetic field of about 1.7 MG was discovered by Kawka & Vennes (2012). Kawka et al. (2017) found that circular polarisation in the H α sigma components varies with P = 22 . 6 min. This is another example of a star for which the phase relationship between light curve and magnetic field are known (Kawka et al. 2017) and it is also missing from the list compiled by Bagnulo et al. (2024b).</text>
<section_header_level_1><location><page_18><loc_51><loc_16><loc_84><loc_17></location>Appendix B.17: WD 0446-789 = WG 47 (var.:)</section_header_level_1>
<text><location><page_18><loc_51><loc_10><loc_94><loc_15></location>Bagnulo & Landstreet (2018) show that this is a weak-field star with ⟨ Bz ⟩ ≃ -5 kG; it probably has a dipolar field with axis nearly parallel to the rotation axis, because only very mild variations in ⟨ Bz ⟩ occur on a timescale of days.</text>
<figure>
<location><page_19><loc_7><loc_64><loc_49><loc_93></location>
<caption>Fig. B.4. Upper panel: WD 0548 -001 = G99-37 flux spectra obtained with FORS1 and FORS2 compared. The Stokes I FORS spectra have not been calibrated, and the di ff erences in slopes are almost certainly explained by instrument and atmospheric e ff ects. Lower pane: FORS V / I spectra compared to the low-resolution MCSP spectrum of Angel &Landstreet (1974).</caption>
</figure>
<section_header_level_1><location><page_19><loc_6><loc_50><loc_45><loc_52></location>Appendix B.18: WD 0548-001 = EGGR 248 = G 99-37 (non-variable)</section_header_level_1>
<text><location><page_19><loc_6><loc_41><loc_49><loc_49></location>The magnetic field was discovered from continuum circular polarisation by Landstreet & Angel (1971). Observations of broadband CP on six nights showed no variations. During one hour, broad-band data were printed every 6 s. These data were Fourier analysed without finding any evidence of short-period variability.</text>
<text><location><page_19><loc_6><loc_32><loc_49><loc_41></location>Angel & Landstreet (1974) obtained an MCSP circular polarisation spectrum in 1972, and the star was re-observed by Vornanen et al. (2010), who remarked that similarity with the previous spectra, reporting also "no unambiguous variations" between several polarisation spectra taken in 2003, 2005, 2008. Using FORS1 data obtained in 2005, they also reported no detection of linear polarisation.</text>
<text><location><page_19><loc_6><loc_19><loc_49><loc_32></location>We retrieved from the ESO FORS1 and FORS2 archives all circular polarisation spectra obtained in 2005, 2008 and 2012. Figure B.4 shows all these spectra compared to the polarisation spectrum of Angel & Landstreet (1974) . We note that there are three spectra obtained in three consecutive nights in November 2012. The spectrum obtained during the last night (2012-1119) shows zero circular polarisation (see green solid line), but a Stokes I flux consistent with that of previous observations. We ascribe this to a non-detected instrument failure (perhaps the retarder waveplate did not move) rather than to stellar variability.</text>
<text><location><page_19><loc_6><loc_14><loc_49><loc_19></location>Observations obtained by Berdyugin et al. (2023) and Berdyugin et al. (2024) show no variability in broadband polarimetry between two measurements obtained 8 months apart (see the top panel of Fig. B.1).</text>
<text><location><page_19><loc_6><loc_10><loc_49><loc_13></location>This is a nearly unique DQp white dwarf which shows not only the Swan bands in the flux spectrum, but also the CH G band. The magnetic field was modelled by Angel & Landstreet</text>
<text><location><page_19><loc_51><loc_88><loc_94><loc_93></location>(1974) who deduced ⟨ Bz ⟩ ≈ 3 . 6 MG from the strong polarisation signature of the CH G-band. Berdyugina et al. (2007) and Vornanen et al. (2010) modelled FORS1 spectra and found ⟨ Bz ⟩ ≃ 2 . 5 MG.</text>
<section_header_level_1><location><page_19><loc_51><loc_83><loc_90><loc_85></location>Appendix B.19: WD 0553+053 = EGGR 290 = G 99-47 (non-variable)</section_header_level_1>
<text><location><page_19><loc_51><loc_61><loc_94><loc_81></location>The magnetic field of this star was discovered by Angel & Landstreet (1972), who also made 14 separate BBCP measurements during 10 months, without detecting any significant variation. Low resolution MCSP spectropolarimetry was obtained by Liebert et al. (1975), and 20 years later, a higher resolution polarised spectrum was obtained also by Putney & Jordan (1995). No significant variation has been detected on any timescale up to decades (for details, see Bagnulo & Landstreet 2021). Brinkworth et al. (2013) observed a light curve with 26.8 m period and semi-amplitude = 0.3 %, which suggest that polarimetric observations could have missed short-term variability. However, Angel & Landstreet (1972) had carried out Fourier-analysis of a polarimetric run, and found no significant variability for any period between 11 seconds and 2 h. They claimed that periodic variation with an amplitude of 0.17 % would certainly have been detected. Our conclusion is therefore that the star is non-variable.</text>
<section_header_level_1><location><page_19><loc_51><loc_57><loc_93><loc_58></location>Appendix B.20: WD 0637+478 = GD 77 (variable, P ≃ 1 . 4 d)</section_header_level_1>
<text><location><page_19><loc_51><loc_49><loc_94><loc_55></location>Magnetic variability was reported by Schmidt & Smith (1995). Eight unpublished ISIS polarisation spectra and one ESPaDOnS spectrum show that ⟨ Bz ⟩ varies with P ≃ 1 . 362 d. The value of ⟨| B |⟩ ≈ 1 . 0 MG hardly changes with rotation, but ⟨ Bz ⟩ varies sinusoidally between about + 400 and -250 kG (LB25).</text>
<section_header_level_1><location><page_19><loc_51><loc_44><loc_80><loc_46></location>Appendix B.21: WD 0654+059 = 2MASS J06572938+0550479 (non-variable)</section_header_level_1>
<text><location><page_19><loc_51><loc_36><loc_94><loc_42></location>Three BBCP observations obtained 8 months apart by Berdyugin et al. (2024) show no variability of the polarisation in the three B ' V ' R ' filters (see Fig. B.1). In all three filters, all three observations have V / I ≃ -0 . 3 %. We assume that the star is not variable.</text>
<section_header_level_1><location><page_19><loc_51><loc_31><loc_92><loc_33></location>Appendix B.22: WD 0708-670 = SCR J0708-6706 (n.v.: only 2 meas.)</section_header_level_1>
<text><location><page_19><loc_51><loc_23><loc_94><loc_29></location>Two spectra published by Bagnulo & Landstreet (2020) show no variability over a 2 month interval. The strength and wavelength dependence of the circular polarisation spectrum suggests that the underlying field could be of the order of 60 to 200 MG. We tentatively assume that the star is magnetically non-variable.</text>
<section_header_level_1><location><page_19><loc_51><loc_18><loc_94><loc_20></location>Appendix B.23: WD0745 + 115 = GALEX J074842.4+112502 (non-variable)</section_header_level_1>
<text><location><page_19><loc_51><loc_10><loc_94><loc_16></location>Strong broadband circular polarisation was detected by Berdyugin et al. (2024). The polarisation changes sign between the B ' and the other bands, and its absolute value is about 1% in all bands. Observed four times, the star does not show variability (see the top right panel of Fig. B.1).</text>
<section_header_level_1><location><page_20><loc_6><loc_91><loc_44><loc_93></location>Appendix B.24: WD 0756+437 = EGGR 428 (variable, P phot ≃ 6 . 5 h )</section_header_level_1>
<text><location><page_20><loc_6><loc_69><loc_49><loc_89></location>The star was discovered to be magnetic and discussed in detail by Putney (1995). She estimated its magnetic field to be ≃ 200 MG. Recent BBCP observations at NOT (Berdyugin et al. 2024) showed that it is rapidly variable. The first two observations, about 3 h apart, report the largest and smallest polarisation seen in the full data set of five observations; this suggests a rotation period of the order of 6 h. In fact, the photometric study of Brinkworth et al. (2013) found a unique, very large amplitude ( ± 4 %) light variation with a period of P = 6 . 68 h. The similarity of this period with the period range deduced from polarisation measurements strongly confirms that 6.68 h is the rotation period of this white dwarf. A further remarkable fact is the combination of high field (200 MG), high mass (1 . 04 M ⊙ ) with advanced age (4.45 Gyr): this star seems to be a unique example of a very old, still strongly magnetic and rapidly rotating WD-WD merger.</text>
<section_header_level_1><location><page_20><loc_6><loc_64><loc_42><loc_66></location>Appendix B.25: WD 0810-353 = UPM J0812-3529 (non-variable)</section_header_level_1>
<text><location><page_20><loc_6><loc_55><loc_49><loc_62></location>No variability detected among six polarised spectra obtained over a four year period, as described in a detailed study by Landstreet et al. (2023). According to their modelling, the star shows two regions of di ff erent field strength: one with magnetic field of predominantly 30 MG strength, outward, and one showing a field strength of 45 MG, inward.</text>
<section_header_level_1><location><page_20><loc_6><loc_49><loc_49><loc_51></location>Appendix B.26: WD 0816-310 = SCR J0818-3110 (variable, P ≃ 10 d)</section_header_level_1>
<text><location><page_20><loc_6><loc_40><loc_49><loc_47></location>This is a DZ white dwarf with strong flux, spectrum and ⟨ Bz ⟩ variations observed in five FORS polarised spectra in 2023 obtained with grism 1200B. It is clear that the surface abundances vary over the surface, and appear to be locked to the surface magnetic field. The rotational period is of the order of 10 d, and ⟨| B |⟩ ≃ 100 kG (Bagnulo et al. 2024a).</text>
<section_header_level_1><location><page_20><loc_6><loc_34><loc_42><loc_36></location>Appendix B.27: WD 0850+192 = LB 8915 (variable, P ∼ hours?)</section_header_level_1>
<text><location><page_20><loc_6><loc_25><loc_49><loc_32></location>This DBA white dwarf shows very weak, variable H lines, slightly variable He, and variable ⟨ Bz ⟩ (Wesemael et al. 2001). It is found that ⟨| B |⟩ ≃ 850 kG, and the rotation period is relatively short, probably some hours. We note that the correct identification of this star is LB 8915, and not LB 8827 as given in title of the paper by Wesemael et al. (2001).</text>
<text><location><page_20><loc_6><loc_19><loc_49><loc_21></location>Appendix B.28: WD0907+213 = GALEX J091016.5+210555 (variable, P ≃ 10 h)</text>
<text><location><page_20><loc_6><loc_10><loc_49><loc_17></location>Moss et al. (2024) discovered that this is a spectroscopically and magnetically variable DBA star with a rotation period of either 7.7 or 11.3 h (the ambiguity is due to aliasing). They have modelled the field and abundance geometry with a simple model like that used for Feige 7 = WD0041-102. The line splitting in the flux spectra suggests a field of ⟨| B |⟩ ≃ 0 . 5 MG.</text>
<figure>
<location><page_20><loc_52><loc_60><loc_93><loc_93></location>
<caption>Fig. B.5. Circular polarisation spectra of WD 1008 -242 obtained with FORS2 and the 300V grism on 2022-02-10 (top panel) and with the 600B grism on 2022-01-01.</caption>
</figure>
<section_header_level_1><location><page_20><loc_51><loc_47><loc_90><loc_49></location>Appendix B.29: WD 0912+536 = EGGR 250 = G 195-19 (variable, P ≃ 1 . 3 d)</section_header_level_1>
<text><location><page_20><loc_51><loc_27><loc_94><loc_44></location>This is the second magnetic white dwarf discovered (Angel & Landstreet 1971a), and the first magnetic white dwarf to be discovered to be rotationally variable (Angel & Landstreet 1971b). An improved ephemeris was provided by Angel et al. (1972a). The star shows a large variation of its circular polarisation with a period of 1.33 d (Angel et al. 1972b). Hernandez et al. (2024) measured the period of the photometric variability from TESS data as 1 . 3304 ± 0 . 0054 d (note that high accuracy of period relies on two widely separated TESS observational data sets). Six MCSP V / I spectra roughly uniformly distributed in phase were obtained by Landstreet & Angel, (unpublished). Between 4000 and 5000 Å, V / I reverses sign during rotation; redwards of this, the strong variations retain one sign.</text>
<text><location><page_20><loc_51><loc_21><loc_94><loc_23></location>Appendix B.30: WD 1008-242 = UCAC4 328-061594 (n.v.: only 2 meas.)</text>
<text><location><page_20><loc_51><loc_10><loc_94><loc_19></location>Observed twice with FORS2 in spectropolarimetric mode by Bagnulo & Landstreet (2022), this white dwarf did not show any variation between two spectra obtained 40 d apart, nor within the same observing series (see Fig. B.5). Probably the field is of order 100 MG or more. We consider that it is probably not variable. The star therefore appears a young, strongly magnetic, ultra-massive, non-variable star.</text>
<section_header_level_1><location><page_21><loc_6><loc_91><loc_46><loc_93></location>Appendix B.31: WD 1008+290 = LP 315-42 = LHS 2229 (non-variable)</section_header_level_1>
<text><location><page_21><loc_6><loc_82><loc_49><loc_89></location>This white dwarf is a cool peculiar DQ star discovered to be magnetic by Schmidt et al. (1999), who tentatively suggested that the field strength is of the order of 100 MG or more. Four observations obtained by Berdyugin et al. (2024) show little to no variability of broadband circular polarisation (see Fig. B.1). We assume that it is not magnetically variable.</text>
<section_header_level_1><location><page_21><loc_6><loc_78><loc_40><loc_80></location>Appendix B.32: WD 1009-184 = WT 1759 (var.:)</section_header_level_1>
<text><location><page_21><loc_6><loc_71><loc_49><loc_77></location>The star was discovered to be a magnetic white dwarf by Bagnulo & Landstreet (2019a), who published one measurement obtained with FORS2 and one obtained with ISIS, showing that the star has a weak and variable field ( ⟨ Bz ⟩ ≃ 50 kG). Additional unpublished data suggest also weak variability.</text>
<section_header_level_1><location><page_21><loc_6><loc_66><loc_46><loc_69></location>Appendix B.33: WD 1015+014 = PG 1015+014 (variable, P = 98 . 75 m)</section_header_level_1>
<text><location><page_21><loc_6><loc_47><loc_49><loc_65></location>This is a very strongly polarised white dwarf with V / I ≃ 1 . 5 %. The flux and polarisation spectra are strongly variable with P = 98 . 75 m and a polar field strength of about 120 MG (Angel 1978; Wickramasinghe & Cropper 1988). Schmidt & Norsworthy (1991) report that BBCP also varies with P = 98.75 min, approximately sinusoidally, with extrema of + 1 and -1 % polarisation. Brinkworth et al. (2013) found that the star's light curve has a period consistent, within uncertainties, with that obtained from polarimetry. Euchner et al. (2006) obtained a series of polarised spectra with FORS using the 300V grism. They describe strong I and V spectrum variations, and model the field structure using a multipole field expansion. This white dwarf displays spectral features originating from regions with typical field strengths between about 50 and 90 MG.</text>
<section_header_level_1><location><page_21><loc_6><loc_42><loc_47><loc_45></location>Appendix B.34: WD 1031+234 = Ton 527 = PG 1031+234 (variable, P ≃ 3 . 5 h)</section_header_level_1>
<text><location><page_21><loc_6><loc_30><loc_49><loc_41></location>Schmidt et al. (1986) report strongly variable intensity, circular and linear polarisation spectra with a 3 h 24 min period, and proposed a simple magnetic model with field strength in the range of 200 - 500 MG. Further broadband polarisation observations through the rotation period were obtained by Piirola & Reiz (1992), who measured also a light variation in anti-phase with circular polarisation (that is, the star appears darker when the absolute value of the polarisation is maximum). Brinkworth et al. (2013) found P phot ≃ 3 . 5 h.</text>
<section_header_level_1><location><page_21><loc_6><loc_26><loc_47><loc_27></location>Appendix B.35: WD 1036-204 = LP 790-29 (non-variable)</section_header_level_1>
<text><location><page_21><loc_6><loc_10><loc_49><loc_25></location>The star was discovered to be magnetic via spectropolarimetry by Liebert et al. (1978), and repeatedly studied (West 1989b; Schmidt et al. 1995, 1999; Beuermann & Reinsch 2002; Jordan & Friedrich 2002). Beuermann & Reinsch (2002) have monitored the star with EFOSC in spectropolarimetric mode to search for short-term variability, without finding any significant variations. Beuermann & Reinsch (2002), however, pointed out that Schmidt et al. (1995) measured a polarisation signal of ≃ -6 % around 6500 Å, a measurement at odds with broadband polarimetric measurements obtained in 1977, 1986, 1994 and 2000, which were all about -9 % (see their Table 1; we recall here that we use the opposite definition for the sign of circular polarisa-</text>
<figure>
<location><page_21><loc_52><loc_64><loc_93><loc_93></location>
</figure>
<figure>
<location><page_21><loc_52><loc_47><loc_93><loc_63></location>
<caption>Fig. B.6. Top and middle panels: Intensity and circular polarisation spectra of WD 1036 -204 obtained with FORS1 and FORS2 at four di ff erent epochs; circular polarisation from Schmidt et al. (1995) is also shown with small circles. Bottom panel: Stokes V / I from four pairs of observations obtained within 10 m from each other on 2013-02-21.</caption>
</figure>
<text><location><page_21><loc_51><loc_10><loc_94><loc_37></location>n). Schmidt et al. (1995) therefore suggested the possibility of a very long rotation period of the star. Jordan & Friedrich (2002) carried out a similar study, with similar results regarding short term variation. They also proposed a rotation period in the range of about 24 - 29 yr. We have reduced FORS1 and FORS2 archival polarisation spectra from 2003, 2011, 2012, and 2013, finding absolutely no hint of variability among them. Figure B.6 shows a comparison of archive observations obtained with grism 600B. The flux is not corrected for atmospheric and instrument transmission, and the discrepancies in the slope of the flux measured in 2003 can be explained by the use of a different CCD. A comparison between FORS spectra with those obtained by Schmidt et al. (1995) on May 7 and 8, 1994, (wavelength range 4160-7460 Å) does not show obvious long-term variability, except in the range 5700 -6200 Å. In that range, our data are instead consistent with most of the literature and point to a value of ≃ -9 %. In addition to the polarisation spectra, FORS1 and FORS2 archive contains another two broadband circular polarisation (BBCP) observations in the R filter: a FORS1 BBCP measurement obtained in April 2006, and one obtained with FORS2 in March 2024, using a similar (but not per-fec</text>
<text><location><page_22><loc_6><loc_68><loc_49><loc_93></location>identical) R filter (program ID 112.25C9.001). We found both measurements consistent with a polarisation signal of about -9 . 4 %. These measurements rule out any significant change in the region around 6500 Å over an interval of 18 years. Finally, Berdyugin et al. (2024) have published the series of BBCP observations obtained in November 2022, November 2023 and February 2024, all consistent among themselves. In the R ' filter they report a polarisation signal of ≃ -9 . 1 %. For the reasons explained at the beginning of this section, it is not possible to accurately compare BBCP measurements obtained with di ff erent instruments, but it is clear that the only deviant point of a series of polarimetric observations obtained in the course of almost half a century is a small portion of a spectrum obtained in 1994. Remarkably, data obtained with the same instrument (FORS) over nearly two decades are fully consistent among themselves, and point strongly to a constant circular polarisation spectrum. Our conclusion is that the star is actually not variable. From the measured signal of circular polarisation of ≃ 10 % we estimate a longitudinal field of the order of 100 MG.</text>
<section_header_level_1><location><page_22><loc_6><loc_64><loc_48><loc_66></location>Appendix B.36: WD 1043-050 = HE 1043-0502 (n.v.: - only 2 meas.)</section_header_level_1>
<text><location><page_22><loc_6><loc_52><loc_49><loc_62></location>This DBA star was discovered magnetic by Schmidt et al. (2001), who proposed that the field is ≃ 800 MG (although its continuum is not highly polarised). Two observations were taken a few days apart, and Schmidt et al. (2001) state that they did not show variability; the reported wavelength integrated circular polarisation of about 1.5% is also essentially unchanged between the two spectra. So we consider it as candidate nonvariable magnetic white dwarf.</text>
<section_header_level_1><location><page_22><loc_6><loc_47><loc_47><loc_50></location>Appendix B.37: WD 1045-091 = HE 1045-0908 (variable, P ≃ 3 h)</section_header_level_1>
<text><location><page_22><loc_6><loc_37><loc_49><loc_46></location>This white dwarf was discovered to be magnetic by Reimers et al. (1996) from a flux spectrum. Circular polarisation was confirmed, and shown to be variable by Schmidt et al. (2001). Euchner et al. (2005) obtained a series of I and V spectra and, assuming a rotational period of about 2.7 h, derived a detailed surface field model with a dominant field strength of 16 MG, but local field strength ranging between about 10 and 75 MG.</text>
<section_header_level_1><location><page_22><loc_6><loc_32><loc_43><loc_35></location>Appendix B.38: WD 1105-340 = SCR J1107-3420A (non-variable)</section_header_level_1>
<text><location><page_22><loc_6><loc_19><loc_49><loc_31></location>Eleven ESPaDOnS spectra taken between 2018 and 2022, with ten of them during 1 week in 2019 (see Table A.1), show that the star has a weak, non-variable field with ⟨ Bz ⟩ ≈ -22 ± 4 . 5 ˙ kG and ⟨| B |⟩ ≈ 125 ± 5 kG (see Fig. B.7). We also have two FORS spectra, one taken with 1200B and one with 1200R, with lower resolving power but higher signal-to-noise ratio (S / N). Owing to their di ff erence resolution, these spectra were not used in this work. This star is perhaps the best studied non-variable weakfield magnetic white dwarf.</text>
<section_header_level_1><location><page_22><loc_6><loc_15><loc_48><loc_17></location>Appendix B.39: WD 1105-048 = EGGR 76 (ultra-weak field, var.:)</section_header_level_1>
<text><location><page_22><loc_6><loc_10><loc_49><loc_13></location>The star was repeatedly observed, and a longitudinal field of the order of 1 kG was detected only in two measurements (Bagnulo & Landstreet 2018). The star seems to have a very weak and</text>
<figure>
<location><page_22><loc_52><loc_65><loc_93><loc_93></location>
<caption>Fig. B.7. WD 1105 -340: Stokes I and V / I profiles of H α observed with ESPaDONs at four di ff erent epochs: 2019-03-21 (two observations), 2019-05-31, and 2022-02-22. Only negligible changes are seen between spectra.</caption>
</figure>
<text><location><page_22><loc_51><loc_52><loc_94><loc_54></location>variable field, but additional measurements should be obtained to confirm the existence of a magnetic field.</text>
<section_header_level_1><location><page_22><loc_51><loc_47><loc_87><loc_49></location>Appendix B.40: WD 1116-470 = SCR J1118-4721 (non-variable)</section_header_level_1>
<text><location><page_22><loc_51><loc_33><loc_94><loc_45></location>This white dwarf was observed twice with FORS2 by Bagnulo & Landstreet (2021), who flagged it as suspected magnetic star. Both observations show a similar signal of circular polarisation at -0 . 2 %, close to the FORS2 instrumental detection limit. A third observations was obtained in January 2023, and the V / I spectrum is again consistent with that measured previously. The star is definitely magnetic, and most likely not variable. This confirmation (see Table A.1) brings the number of magnetic white dwarfs in the 20 pc volume to 34.</text>
<section_header_level_1><location><page_22><loc_51><loc_28><loc_92><loc_30></location>Appendix B.41: WD 1211-171 = HE 1211-1707 (variable, P ≃ 2 h)</section_header_level_1>
<text><location><page_22><loc_51><loc_10><loc_94><loc_26></location>A magnetic field was suspected in this white dwarf by Reimers et al. (1996), which was confirmed with polarimetry by Schmidt et al. (2001). Both papers show varying flux spectra. Schmidt et al. (2001) estimates P ≃ 100 -120 min and ⟨| B |⟩ ≃ 50 MG. Brinkworth et al. (2013) measured a photometric period of 1.79 h, consistent with the previous estimates from polarimetric data. The star is polarised at a level that varies between 0 and 3% during the rotation cycle. Modelling by Schmidt et al. (2001) strongly suggests a He dominated atmosphere with T e ff ≃ 12000 K, but Reimers et al. (1996), using IUE data, estimates 23000 K. Gentile Fusillo et al. (2021) gives 30000 K and 1 . 20 M ⊙ . We tentatively assume T e ff = 23000 K , M = 1 . 2 M ⊙ , and an He-rich atmosphere.</text>
<figure>
<location><page_23><loc_7><loc_60><loc_49><loc_93></location>
<caption>Fig. B.8. Top panel: WD 1116 -470 observed with FORS2 in three different epochs. Bottom panel: Stokes V / I (re-binned at 410 Å) from individual pairs of exposures as shown in the legend.</caption>
</figure>
<section_header_level_1><location><page_23><loc_6><loc_49><loc_47><loc_51></location>Appendix B.42: WD 1217+475 = SDSS J121929.45+471522.8 (DAHe variable, P phot = 15 . 26 h)</section_header_level_1>
<text><location><page_23><loc_6><loc_40><loc_49><loc_48></location>DAHe with 18.5 MG field strength and a photometric period of about 15.25 h (Gänsicke et al. 2020). Spectroscopy reveals Zeeman components of the Balmer lines varying in strength but with constant splitting. Published data do not demonstrate that the star is magnetically variable but cannot rule out this possibility either. Therefore we have decided not to include this star in our sample.</text>
<section_header_level_1><location><page_23><loc_6><loc_35><loc_48><loc_38></location>Appendix B.43: WD 1249-022 = GALEX J125230.9-023417 (DAHe variable, P phot = 0 . 09 h)</section_header_level_1>
<text><location><page_23><loc_6><loc_29><loc_49><loc_34></location>This is a DAHe white dwarf that shows variable H β and H α lines that appear sometimes in emission and sometimes in absorption. Field strength has been estimated 5 MG and rotational period = 0.09 h (Reding et al. 2020).</text>
<section_header_level_1><location><page_23><loc_6><loc_24><loc_48><loc_27></location>Appendix B.44: WD 1312+098 = PG 1312+099 - (variable, P ≃ 5 . 4 h)</section_header_level_1>
<text><location><page_23><loc_6><loc_18><loc_49><loc_23></location>Variable continuum circular polarisation was detected in this hot DAH by Schmidt & Norsworthy (1991), who present over 100 measures, and find P = 5 . 43 h. Circular polarisation varies approximately between + 1% and -1%.</text>
<section_header_level_1><location><page_23><loc_6><loc_13><loc_46><loc_16></location>Appendix B.45: WD 1315-781 = LAWD 45 (n.v.: - only 2 meas.)</section_header_level_1>
<text><location><page_23><loc_6><loc_10><loc_49><loc_12></location>No change between two ⟨| B |⟩ ≃ 5 . 5 MG and ⟨ Bz ⟩ ≃ 0 MG measurements from FORS 300V spectra taken five nights apart</text>
<text><location><page_23><loc_51><loc_91><loc_94><loc_93></location>by Bagnulo & Landstreet (2020). We tentatively assume that it is magnetically non-variable.</text>
<section_header_level_1><location><page_23><loc_51><loc_87><loc_93><loc_88></location>Appendix B.46: WD 1315+222 = LP 378-956 (non-variable)</section_header_level_1>
<text><location><page_23><loc_51><loc_81><loc_94><loc_85></location>Two observations by Berdyugin et al. (2023) and one by Berdyugin et al. (2024) show no obvious sign of variability (see Fig B.1).</text>
<section_header_level_1><location><page_23><loc_51><loc_78><loc_87><loc_79></location>Appendix B.47: WD 1328+307 = G 165-7 (variable)</section_header_level_1>
<text><location><page_23><loc_51><loc_65><loc_94><loc_76></location>This star is found to host a magnetic field of ⟨| B |⟩ ≃ 650 kG based on line splitting observed in a good S / NSDSSspectrum (Dufour et al. 2006). These authors have also obtained low-resolution polarised spectra. They state that three 600 sec polarisation spectra were taken on 2005-12-30 at Steward Obs, all yielding essentially the same ⟨ Bz ⟩ ≈ 150 kG, but that the polarisation amplitude in similar Steward polarised spectra from 2006-05-03 (apparently not measured) is at least two times weaker than in 2005 spectrum. We conclude that the star is magnetically variable.</text>
<section_header_level_1><location><page_23><loc_51><loc_61><loc_92><loc_62></location>Appendix B.48: WD 1346+121 = LP 498-66 (non-variable)</section_header_level_1>
<text><location><page_23><loc_51><loc_53><loc_94><loc_59></location>Observed three times by Berdyugin et al. (2023) and Berdyugin et al. (2024), we assume the star, which shows circular polarisation ranging between -1 and 0 % in the three DIPol-UF bands, is magnetically non-variable (see Fig. B.1). The field strength of this white dwarf is probably in the tens of MG.</text>
<section_header_level_1><location><page_23><loc_51><loc_48><loc_92><loc_50></location>Appendix B.49: WD 1350-090 = PG 1350-090 = GJ 3814 (non-variable)</section_header_level_1>
<text><location><page_23><loc_51><loc_40><loc_94><loc_46></location>Discovered by Schmidt & Smith (1994), who measured ⟨ Bz ⟩ = 85 ± 9 kG. We have obtained four polarised spectra with ESPaDOnS that show ⟨| B |⟩ ≃ 450 -465 kG. There is no strong evidence of field variability, see also Schmidt & Smith (1994). Data will be published in a forthcoming paper (LB25).</text>
<section_header_level_1><location><page_23><loc_51><loc_36><loc_88><loc_37></location>Appendix B.50: WD 1532+129 = G 137-24 (variable)</section_header_level_1>
<text><location><page_23><loc_51><loc_25><loc_94><loc_34></location>Originally classified as DZ white dwarf by Kawka et al. (2004), the star was discovered to be a magnetic white dwarf by Bagnulo &Landstreet (2019a), who published two FORS2 measurements and one ISIS measurement. The star is variable, with ⟨ Bz ⟩ values from FORS2 measurements of -21 ± 1 kG and -4 ± 1 kG, while ⟨| B |⟩ is not strong enough to split spectral lines, leading to ⟨| B |⟩ < ∼ 300 kG.</text>
<section_header_level_1><location><page_23><loc_51><loc_20><loc_87><loc_22></location>Appendix B.51: WD 1556+044 = PM J15589+0417 (non-variable)</section_header_level_1>
<text><location><page_23><loc_51><loc_10><loc_94><loc_19></location>Discovered to be magnetic by Berdyugin et al. (2022), the star was re-observed three more times by Berdyugin et al. (2024). The observed circular polarisation, which is detected at the 10 σ level, ranges between -0 . 3 and + 0 . 4% in the three filter bands of DIPol-UF, so the order of magnitude of the field strength is probably some tens of MG. The polarisation does not show any variability (see Fig. B.1).</text>
<figure>
<location><page_24><loc_7><loc_56><loc_49><loc_94></location>
<caption>Fig. B.9. Spectra of WD 1619 + 046 around H β obtained with FORS2 on 2023-06-16.</caption>
</figure>
<section_header_level_1><location><page_24><loc_6><loc_46><loc_49><loc_49></location>Appendix B.52: WD 1615+542 = GALEX J161634.4+541011 (DAHe magnetically variable, P phot = 1 . 59 h)</section_header_level_1>
<text><location><page_24><loc_6><loc_41><loc_49><loc_45></location>DAHe with a variable magnetic field (from 3.5 to 6.5 MG) and a rotation period P = 95 . 29 m estimate via photometry (Manser et al. 2023).</text>
<section_header_level_1><location><page_24><loc_6><loc_37><loc_49><loc_39></location>Appendix B.53: WD 1619+046 = GALEX J162157.7+043219 (variable, P ≃ 40 m?)</section_header_level_1>
<text><location><page_24><loc_6><loc_24><loc_49><loc_36></location>This white dwarf was discovered to be magnetic and rapidly variable in this work (see Sect. 2.1 and Fig. B.9). From the H β regions, it is clear that the star is a DAH with a rather non-uniform field of ⟨| B |⟩ ≃ 15 MG. We observe clear changes in intensity between spectra obtained a few minutes apart, and also polarisation (which is measured by the combination of two spectra) shows some variation, but without sign reversal. The close similarity between the first pair and the last pair of intensity spectra suggest that the star's rotation period is approximately 40 min.</text>
<section_header_level_1><location><page_24><loc_6><loc_20><loc_48><loc_22></location>Appendix B.54: WD 1639+537 = GD356 (non-variable, but with a light curve and P phot = 1.93 h)</section_header_level_1>
<text><location><page_24><loc_6><loc_10><loc_49><loc_19></location>This star is the prototype of DAHe white dwarfs. It shows a light curve with low amplitude and P phot ≃ 1 . 93 h (Walters et al. 2021). A time series of circular polarisation spectra obtained during an entire rotational cycle shows now variability while subtle sinusoidal variability is seen in the position of the σ components of H β and H α lines (Walters et al. 2021). Walters et al. (2021) compared spectropolarimetry obtained in 2019, with that</text>
<figure>
<location><page_24><loc_53><loc_64><loc_93><loc_93></location>
</figure>
<figure>
<location><page_24><loc_52><loc_32><loc_93><loc_61></location>
<caption>Fig. B.10. Top panels: Comparison of spectra of WD 1658 + 440 obtained in 1980 (Liebert et al. 1983) and in 2019 (this work). Bottom panels: Overplot of eight intensity spectra obtained in sequence with 450s exposure time, and of four circular polarisation spectra obtained from pairs of frames obtained in sequence.</caption>
</figure>
<text><location><page_24><loc_51><loc_10><loc_94><loc_15></location>published by Ferrario et al. (1997), finding no changes. Its field strength is of the order of 10 MG (Walters et al. 2021). We classify the star as non-variable, but showing very small changes of its apparent magnetic field with a ∼ 2 h rotation period.</text>
<section_header_level_1><location><page_25><loc_6><loc_91><loc_48><loc_93></location>Appendix B.55: WD 1658+440 = PG 1658+441 (n.v.: - only 2 meas., one very old)</section_header_level_1>
<text><location><page_25><loc_6><loc_49><loc_49><loc_89></location>The star was discovered to be magnetic by Liebert et al. (1983), who measured ⟨ Bz ⟩ ≃ 0 . 7 MG, and ⟨| B |⟩ ≃ 2 . 3 MG from spectropolarimetry of H α , H β and H γ (see their Fig. 5). Their observations were obtained on July 20 1980, with a spectral resolution of 10 Å. The same authors measured a signal of broadband circular polarisation = -0 . 016 ± 0 . 033% and -0 . 044 ± 0 . 019 % in the range 3300-8600 Å and concluded that a 3 σ upper limit of 0.10% semi-amplitude could be set for any presumed sinusoidal variation with a period between 0.5 and 5 h. For possible periods between 4 minutes and 0.5 hours, a less stringent limit of 0.30% polarisation semi-amplitude may be deduced. A comparison with our ISIS spectra obtained on April 21, 2019 (Sect. 2.1) is shown in the top panels of Fig. B.10. We observe a wavelength shift possibly due to bad calibration; most remarkable is the different strength of H α , but perhaps this is instrumental, because the ISIS spectrum was obtained with a setting that puts H α at the edge of the CCD, and no good flat-fielding correction could be applied. Although our comparison between 1980 and 2019 data is not conclusive, we certainly do not see any convincing change of the spectral features that demonstrate long-term variability. A comparison between the four pairs of Stokes V / I profiles obtained with ISIS in 2019 show no di ff erences within the error bars (see the bottom panels of Fig. B.10); therefore we rule out variability on a timescale of 10-15 minutes. This star seems to be a young, strongly magnetic, massive, tentatively classified as non-variable star. Photometric studies lead to inconsistent results: Brinkworth et al. (2013) found that the star is photometrically variable with a period between 6 h and 4 d, while, using TESS photometry, Oliveira da Rosa et al. (2024) derived a period shorter than 1 h. Hernandez et al. (2024), instead, did not detect periodicity in TESS data.</text>
<section_header_level_1><location><page_25><loc_6><loc_44><loc_44><loc_47></location>Appendix B.56: WD 1703-266 = UCAC4 317-104829 (variable)</section_header_level_1>
<text><location><page_25><loc_6><loc_37><loc_49><loc_43></location>This DA white dwarf was discovered to have a magnetic field by Bagnulo & Landstreet (2020), with ⟨| B |⟩ ≈ 8 MG. They also found that two FORS2 polarised spectra four days apart are significantly di ff erent and yield di ff erent values of ⟨ Bz ⟩ , so the star is variable on a timescale of some days or less.</text>
<section_header_level_1><location><page_25><loc_6><loc_32><loc_36><loc_34></location>Appendix B.57: WD 1712-590 = Gaia DR3 5915797694789556096 (variable)</section_header_level_1>
<text><location><page_25><loc_6><loc_19><loc_49><loc_31></location>Discovered to be magnetic by O'Brien et al. (2023). We observed this white dwarf in polarimetric mode with grism 1200B three times, twice during the same night. Intensity spectra show very similar splitting of Balmer lines, but the flux distribution within the line cores changes quite significantly, rather similarly to WD2359-434 (Landstreet et al. 2017). The deduced field strength is ⟨| B |⟩ < ∼ 0 . 8 MG. The Stokes V spectra also show that the star is strongly variable within 1-2 h (see Fig. B.11), and that the longitudinal field reverses its polarity during rotation.</text>
<section_header_level_1><location><page_25><loc_6><loc_15><loc_46><loc_17></location>Appendix B.58: WD 1743-521 = L 270-31 = BPM25114 (variable, P = 2 . 84 d)</section_header_level_1>
<text><location><page_25><loc_6><loc_10><loc_49><loc_13></location>Wickramasinghe & Bessell (1976) reported a magnetic field of about 35 MG in this southern DA star from close examination of the peculiar flux spectrum. Wegner (1977) found vari-</text>
<figure>
<location><page_25><loc_52><loc_64><loc_93><loc_93></location>
<caption>Fig. B.11. WD1712 -590 observed with FORS2 in the dates shown in the bottom panel. H β detail.</caption>
</figure>
<text><location><page_25><loc_51><loc_49><loc_94><loc_57></location>ability of light and of the flux and polarisation spectrum with P ≈ 2 . 84 d, and modelled the spectrum variations, finding results consistent with a magnetic dipole field. Field modelling was carried out in more detail by Martin & Wickramasinghe (1978), who deduced a dipole field strength of 36 MG, corresponding to ⟨| B |⟩ ≃ 18 MG.</text>
<section_header_level_1><location><page_25><loc_51><loc_44><loc_90><loc_47></location>Appendix B.59: WD 1748+708 = EGGR 372 = G 240-72 (s.v.:)</section_header_level_1>
<text><location><page_25><loc_51><loc_10><loc_94><loc_43></location>Intrinsic broadband circular and linear polarisation of this magnetic white dwarf were discovered by Angel et al. (1974), who reported no variations in repeated observations of broad-band circular polarisation over a month, and linear polarisation over two nights. The star was later observed in broadband circular and linear polarimetry by West (1989a), who generally found similar polarisation levels and position angles to earlier work, and in broadband linear polarimetry by Berdyugin & Piirola (1999). Berdyugin & Piirola (1999) found evidence of clear change of the position angle of linear polarisation (mainly rotation of the polarisation angle) on a timescale of 20 years. We have observed the star with ISIS both in circular (twice) and in linear polarisation, and compared these observations with MCSP lowresolution spectra of I , V and P obtained by Angel & Landstreet on September 6 and 7, 1974, that were never published until now. Our ISIS spectra are consistent among themselves. When compared with spectropolarimetry obtained in 1974 and the 1990s, our new ISIS linear polarisation data find small changes in the V and P polarisation amplitude in the blue and visual, whereas the overall position angle behaviour is almost identical to that observed observed in 1974, thus we cannot confirm the changes in the polarisation position angle seen in 1997 by Berdyugin & Piirola (1999). On the other side, Antonyuk et al. (2016) found that the star is photometrically variable with a period between 5 hours and two days. Brinkworth et al. (2013) detected no short period variability, but claim that photometry varied over a ten</text>
<figure>
<location><page_26><loc_6><loc_55><loc_49><loc_94></location>
<caption>Fig. B.12. Polarisation spectra of WD 1748 + 708 = G240-272.</caption>
</figure>
<text><location><page_26><loc_6><loc_46><loc_49><loc_48></location>month interval. We conclude that the star show some signs of subtle long term variability that should be further investigated.</text>
<text><location><page_26><loc_6><loc_39><loc_45><loc_43></location>Appendix B.60: WD1750 -311 = UCAC4 295-140552 = [MTR2015] OW J175358.85-310728.9 (n.v.: - but P phot ≃ 0 . 5 h)</text>
<text><location><page_26><loc_6><loc_22><loc_49><loc_37></location>Amagnetic field of ⟨| B |⟩ ≈ 2 . 1 MG was identified by Macfarlane et al. (2017) in this hot DQ white dwarf, which has strong lines of neutral C as well as fairly strong Balmer lines. They observed clear light variability of the star with an amplitude of about ± 2 % and a period of 35 min. These authors discuss the origin of the light variations: they argue that the variability is not due to pulsation, as no subsidiary frequencies are observed, and conclude that the variation could be due to rotation. They do not seem to have considered testing this with the spectra that they have collected of the object by looking for spectral variations, although they do test the spectra for radial velocity variations, and find none.</text>
<text><location><page_26><loc_6><loc_10><loc_49><loc_21></location>Our four FORS2 300V spectra, taken with 13 m spacing through the light variation cycle, show virtually no changes in either flux or polarisation except for H β , which has a strongly variable I profile depth but almost constant V / I ( λ ) (see Fig. B.13). Possibly this is due to a magnetic field distribution that is nearly axisymmetric about the rotation axis but a distribution of H that varies strongly around that axis. This might be related to the light variability that has been detected. We have decided to classify the star as candidate magnetically non-variable.</text>
<figure>
<location><page_26><loc_51><loc_64><loc_93><loc_93></location>
<caption>Fig. B.13. WD 1750 -311 observed with FORS2 on 2023-06-12 with grism 300V.</caption>
</figure>
<figure>
<location><page_26><loc_52><loc_41><loc_93><loc_57></location>
<caption>Fig. B.14. FORS2 V / I spectra of WD 1754-550, rebinned at 45 Å steps, from four exposure pairs obtained on 2023-06-12 at the times shown in the legend. The intensity spectrum is featureless.</caption>
</figure>
<section_header_level_1><location><page_26><loc_51><loc_30><loc_93><loc_33></location>Appendix B.61: WD1754 -550 = GALEX J175845.9-550117 (variable)</section_header_level_1>
<text><location><page_26><loc_51><loc_20><loc_94><loc_29></location>This star was discovered to be magnetic in this work (Sect. 2.1). There is a strong signal of circular polarisation that seems variable on a short timescale between 0 and 4%, suggesting a rotation period of the order of 15 min: see Fig B.14. The inferred field strength is presumably of the order of 100 MG or more. The intensity spectrum appears featureless, so the T e ff ≃ 35 000 K star could be defined a hot DC.</text>
<section_header_level_1><location><page_26><loc_51><loc_17><loc_88><loc_18></location>Appendix B.62: WD 1814+248 = G 183-35 (variable)</section_header_level_1>
<text><location><page_26><loc_51><loc_10><loc_94><loc_16></location>Putney (1997) showed that this white dwarf, formerly classified as a DC, is in fact a cool DAH that shows weak H α and H β , and that both lines are split by a field of about 6.8 MG, roughly the same in two observations. Kilic et al. (2019) observed that the white dwarf shows spectral variations. A series of spectra taken</text>
<figure>
<location><page_27><loc_6><loc_54><loc_49><loc_93></location>
<caption>Fig. B.15. Polarisation spectra of WD 1829 + 547 = G227-35. Top panel: Comparison between MP spectra obtained in 1974 and our ISIS spectra obtained in 2019, after degrading their resolution. Bottom panel: Comparison between ISIS spectra obtained in 2019 and a spectrum obtained by Putney & Jordan (1995) in August 1992.</caption>
</figure>
<text><location><page_27><loc_6><loc_27><loc_49><loc_42></location>over several hours show that the separation of the σ components of H α from the central π component changes rather abruptly between about 90 Å and about 120 Å, equivalent to fairly sudden jumps between ⟨| B |⟩ values of about 4.6 and 6 MG, repeating with a period of about 4 h. The authors suggest that this unusual form of variability may be due to a patchy distribution of H over an He-rich envelope. A plausible model that might explain the observations could be a dipole oblique to the rotation axis, decentred in the direction of one pole so that the polar strengths at the two magnetic poles are unequal, with H-rich patches around both poles, but little or no H in a belt around the magnetic equator.</text>
<section_header_level_1><location><page_27><loc_6><loc_23><loc_46><loc_24></location>Appendix B.63: WD 1829+547 = G 227-35 (non-variable)</section_header_level_1>
<text><location><page_27><loc_6><loc_10><loc_49><loc_21></location>This white dwarf was discovered to be strongly magnetic by Angel et al. (1975) using both broadband polarimetry and lowresolution spectropolarimetry (on September 9, 1974). Limited tests of broadband variability were negative (Angel et al. 1975, 1981). It was observed again in spectropolarimetric mode by Cohen et al. (1993) and by Putney & Jordan (1995) who estimated a dipolar field strength of 170-180 MG. We observed the same star with ISIS twice in circular polarisation and once in linear polarisation. There are strong circular polarisation features at 6960,</text>
<figure>
<location><page_27><loc_51><loc_54><loc_94><loc_93></location>
<caption>Fig. B.16. MCSP linear and circular polarisation spectra of WD2010 + 310 = GD229.</caption>
</figure>
<text><location><page_27><loc_51><loc_43><loc_94><loc_45></location>7450 and 7930 Å. Figure B.15 shows a comparison between all these spectra. We do not see any obvious sign of variation.</text>
<section_header_level_1><location><page_27><loc_51><loc_37><loc_94><loc_40></location>Appendix B.64: WD 1900+705 = LAWD 73 = Grw + 70 · 8247 (s.v.:)</section_header_level_1>
<text><location><page_27><loc_51><loc_25><loc_94><loc_35></location>The star has a magnetic field of the order of 180 MG (Jordan 2003), and shows hints of small variability. We refer to Bagnulo & Landstreet (2019b), who present a review and analysis of its polarimetric characteristics, and in particular, a comparison with observations obtained 50 years apart show some discrepancies in both linear and circular polarisation. More recent broadband circular polarisation obtained at NOT (Berdyugin et al. 2022, 2023) show no variability on a timescale of 2 years (see Fig. B.1).</text>
<section_header_level_1><location><page_27><loc_51><loc_21><loc_94><loc_22></location>Appendix B.65: WD 1953-011 = GJ 772 (variable, P ≃ 1 . 5 d)</section_header_level_1>
<text><location><page_27><loc_51><loc_10><loc_94><loc_19></location>Maxted et al. (2000) modelled a series of Stokes I spectra in terms of a global dipolar field with polar field strength of order 100 kG, together with a spot with a field of order 500 kG. This basic modelling was confirmed by the analysis of a series of polarised spectra obtained with FORS1 and on the Russian 6 m telescope, which also revealed a rotation period of 1.448 d (Valyavin et al. 2008).</text>
<section_header_level_1><location><page_28><loc_6><loc_92><loc_39><loc_93></location>Appendix B.66: WD 2010+310 = GD 229 (s.v.:)</section_header_level_1>
<text><location><page_28><loc_6><loc_80><loc_49><loc_91></location>This star was the fifth circularly polarised white dwarf to be discovered. Swedlund et al. (1974) reported a series of measurements indicating the presence of elevated levels of both circular polarisation and linear polarisation, and strongly suggesting polarisation variability. The claim of variability was then questioned by Kemp et al. (1974), who argued that the initial measurements had been badly contaminated by polarised foreground zodiacal light contamination.</text>
<text><location><page_28><loc_6><loc_75><loc_49><loc_80></location>Linear and / or circular polarisation of GD 229 has been measured by Landstreet & Angel (1974), Efimov (1981), Angel et al. (1981), West (1989a), Berdyugin (1995), and Berdyugin & Piirola (1999).</text>
<text><location><page_28><loc_6><loc_70><loc_49><loc_75></location>Angel & Landstreet (1974) obtained three low-resolution I and V spectra of the star on the nights of 1973 Nov 6-8 using the MCSP. Synthetic broad-band polarimetry derived from these low-resolution spectra revealed no variations over three nights.</text>
<text><location><page_28><loc_6><loc_65><loc_49><loc_69></location>West (1989b) observed linear polarisation in GD 229 again in 1986 using broadband filters, but found no strong evidence of variation, not even of the position angle of linear polarisation.</text>
<text><location><page_28><loc_6><loc_42><loc_49><loc_65></location>Observations of GD 229 using broadband polarimetry was also carried out for linear polarisation by Berdyugin (1995) and for both circular and linear polarisation, with higher precision, by Berdyugin & Piirola (1999), who compared their results to earlier work. Their results appear to show some quite significant changes in both circular and linear polarimetry compared to earlier observations by other groups. A major di ffi culty of comparing the various measurements is that each group has been made with di ff erent instrumentation. The consequences of using di ff erent filter passbands, di ff erent detector sensitivities as functions of wavelength, and even di ff erent (and possibly incorrect) calibration of instrumental polarimetric e ffi ciency, make it very hard to compare these data. However, Berdyugin & Piirola (1999) do o ff er strong evidence for significant rotation of the angle of linear polarisation, by about 30 · over 20 years, which is di ffi cult to explain by instrumental e ff ects. This is probably the most robust e ff ect that emerges from comparison of the many kinds of polarimetric observations</text>
<text><location><page_28><loc_6><loc_26><loc_49><loc_41></location>Linear and circular polarisation spectra of GD 229 were obtained in 2018 and 2019 using the ISIS spectropolarimeter on the William Herschel Telescope (Sect. 2.1). These new data are shown in Fig. B.16, and compared to the circular polarised spectra by Angel & Landstreet (1974), and to previously unpublished linear spectropolarimetry of Angel and Landstreet (see Table A.1). Remarkably, the angle of linear polarisation appears to have returned to its value during the 1970s. Other di ff erences compared to the spectra of Angel and Landstreet are observed, but some of these are undoubtedly due to greatly di ff erent resolving power, and some may be due to uncertainties in the calibration of the 1970s data, particularly in the UV.</text>
<text><location><page_28><loc_6><loc_14><loc_49><loc_25></location>It is thus di ffi cult to establish clearly how much variability has occurred in GD 229. There is evidence for variations, but these appear to have relatively small amplitude compared to the overall scale of polarisation, except possibly for position angle rotation of linear polarisation (which however has not been confirmed by our most recent linear spectropolarimetry). Furthermore, because of very limited sampling with a variety of instruments, it is not possible to establish clearly any particular timescale for variations.</text>
<text><location><page_28><loc_6><loc_10><loc_49><loc_13></location>We note that the spectrum of GD 229 has been modelled as due to He in a field of hundreds of MG by Jordan et al. (1998, 2001) and Jordan (2003).</text>
<figure>
<location><page_28><loc_52><loc_78><loc_93><loc_93></location>
<caption>Fig. B.17. Circular polarisation spectra of WD 2049 -222, rebinned at ≃ 410 Å, and obtained at the time specified in the legend.</caption>
</figure>
<section_header_level_1><location><page_28><loc_51><loc_67><loc_89><loc_70></location>Appendix B.67: WD 2047+372 = EGGR 261 (variable, P ≃ 6 h)</section_header_level_1>
<text><location><page_28><loc_51><loc_61><loc_94><loc_66></location>Originally, the star was observed polarimetrically by Schmidt & Smith (1995), who did not detect its weak and sign reversing field (their measurements were ⟨ Bz ⟩ = -42 ± 59 kG and -2 . 5 ± 4 . 6 kG).</text>
<text><location><page_28><loc_51><loc_51><loc_94><loc_60></location>This star was discovered to be magnetic by Landstreet et al. (2016), and monitored and modelled by Landstreet et al. (2017). It is currently the weakest white dwarf field ( ⟨| B |⟩ ≈ 60 kG) that has been modelled in detail, mainly on the basis of a series of 18 ESPaDOnS spectra. The rotation period, determined from the variation of ⟨ Bz ⟩ , is 0.243 d. The observed variations of ⟨ Bz ⟩ and ⟨| B |⟩ are well modelled using a simple dipole model.</text>
<text><location><page_28><loc_51><loc_48><loc_94><loc_51></location>No rotational variation is detected in TESS photometry (Hernandez et al. 2024).</text>
<section_header_level_1><location><page_28><loc_51><loc_45><loc_86><loc_46></location>Appendix B.68: WD 2049-222 = LP 872-48 (var.:)</section_header_level_1>
<text><location><page_28><loc_51><loc_24><loc_94><loc_43></location>Discovered to be magnetic by Berdyugin et al. (2022) with BBCP measurements. The star has V / I ≃ + 0 . 1 %, and is one of the weakest polarisation levels securely detected. The inferred ⟨ Bz ⟩ field strength is only of the order of a few MG. Broad-band measurements were repeated in July 2022 (Berdyugin et al. 2024) to check for variability, which was not detected (see Fig. B.1). We also observed the white dwarf three times with FORS2 with grism 300V (Sect. 2.1). The data are barely above the threshold of instrumental polarisation ( ≃ 0 . 1%; see Siebenmorgen et al. 2014), and therefore it is hard to establish whether the hints of variability seen in the spectra are real or not (see Fig. B.17). However, star WD 1116-440 shows a similar level of polarisation, and constant over about 4 year (see Fig. B.8), suggesting that the tiny variability observed in WD 2049-222 may be real and not an instrumental artefact.</text>
<section_header_level_1><location><page_28><loc_51><loc_19><loc_89><loc_21></location>Appendix B.69: WD 2049-253 = UCAC4 325-215293 (non-variable)</section_header_level_1>
<text><location><page_28><loc_51><loc_10><loc_94><loc_17></location>Discovered to be magnetic by Bagnulo & Landstreet (2020), who observed continuum circular polarisation of order 0.4% and deduced a field strength of order 20 MG. This white dwarf was re-observed in broadband circular polarisation once by Berdyugin et al. (2022) and two more times by Berdyugin et al. (2024), but shows no sign of variability (see Fig. B.1).</text>
<section_header_level_1><location><page_29><loc_6><loc_91><loc_44><loc_93></location>Appendix B.70: WD 2051-208 = BPS CS 22880-0134 (variable, P ≃ 1 . 5 h)</section_header_level_1>
<text><location><page_29><loc_6><loc_86><loc_49><loc_89></location>The magnetic field of this white dwarf was discovered from the shape of the H α line by Koester et al. (2009).</text>
<text><location><page_29><loc_6><loc_74><loc_49><loc_85></location>We have two series of five polarised spectra each, one using FORS grism 1200B and one with grism 1200R (LB25). These unpublished data clearly reveal very rapid rotation of this massive magnetic white dwarf. The stellar rotation period is 0.0594 d = 1.425 h. The value of ⟨ Bz ⟩ ranges approximately sinusoidally between about + 50 kG and -30 kG, while the corresponding values of ⟨| B |⟩ increase from about 200 kG to nearly 300 kG, in good agreement with the two values of <| B |> (220 and 290 kG) obtained by Koester et al. (2009) from UVES SPY spectra.</text>
<section_header_level_1><location><page_29><loc_6><loc_70><loc_43><loc_71></location>Appendix B.71: WD 2105-820 = LAWD 83 (variable)</section_header_level_1>
<text><location><page_29><loc_6><loc_55><loc_49><loc_68></location>This star was suspected to have a weak magnetic field by Koester et al. (1998), who observed that the core of H α was abnormally broad, but they could not decide whether this was due to rapid rotation of v sin i ≈ 65 km / s or to a magnetic field of about 43 kG. Five FORS1 polarised spectra of the star by Landstreet et al. (2012) detected an apparently nearly constant magnetic field of ⟨ Bz ⟩ ≈ 10 kG. Later, FORS2 polarised spectra by Bagnulo & Landstreet (2018) and Farihi et al. (2018) reveal that ⟨ Bz ⟩ sometimes decreases to ⟨ Bz ⟩ ≈ 4 kG, so the star is apparently variable (see Bagnulo & Landstreet 2021).</text>
<section_header_level_1><location><page_29><loc_6><loc_50><loc_43><loc_52></location>Appendix B.72: WD 2138-332 = L 570-26 (variable, P ≃ 6 . 19 h)</section_header_level_1>
<text><location><page_29><loc_6><loc_38><loc_49><loc_48></location>The DZA star was discovered to be magnetic, with a variable ⟨ Bz ⟩ of the order of 10 kG (Bagnulo & Landstreet 2019a) and a rotational period of P = 6 . 19 h (Hernandez et al. 2024; Farihi et al. 2024). The same period was found from the analysis of the equivalent width and ⟨ Bz ⟩ curves by Bagnulo et al. (2024b), who proposed a magnetic model with a dipolar field with ⟨| B |⟩ ≃ 50 kG. The star shows photometric and ⟨ Bz ⟩ curves with light minimum corresponding to ⟨ Bz ⟩ maximum.</text>
<section_header_level_1><location><page_29><loc_6><loc_33><loc_44><loc_35></location>Appendix B.73: WD 2150+591 = UCAC4 747-070768 (variable, P ≃ 2 . 4 d)</section_header_level_1>
<text><location><page_29><loc_6><loc_22><loc_49><loc_31></location>The star was discovered to be magnetic by Landstreet & Bagnulo (2019), who reported two ISIS measurements that showed clearly that the field is variable with a period of hours or days. We subsequently monitored the star with one ESPaDOnS observation and several more ISIS spectra, and confirmed variability. These observations and a model of the star's magnetic field will be presented in a forthcoming paper (LB25).</text>
<section_header_level_1><location><page_29><loc_6><loc_18><loc_47><loc_19></location>Appendix B.74: WD 2211+372 = LP 287-35 (non-variable)</section_header_level_1>
<text><location><page_29><loc_6><loc_10><loc_49><loc_16></location>This DC white dwarf was discovered to be magnetic by Berdyugin et al. (2023), who found circular polarisation in excess of 1% in the blue. It was re-observed by Berdyugin et al. (2024). It does not seem to be variable (see Fig. B.1). The field strength ⟨| B |⟩ is probably of the order of 60 MG or more.</text>
<section_header_level_1><location><page_29><loc_51><loc_91><loc_94><loc_93></location>Appendix B.75: WD 2316+123 = KUV 23162+1220 (variable, P ≃ 18 d)</section_header_level_1>
<text><location><page_29><loc_51><loc_78><loc_94><loc_90></location>Discovered magnetic by Sion et al. (1984). Schmidt & Norsworthy (1991) reported BBCP of amplitude up to nearly 1% that varies sinusoidally with P = 17.86 d. Both the flux spectrum and the linear and circular polarisation spectra have been modelled repeatedly (Liebert et al. 1985; Achilleos & Wickramasinghe 1989; Friedrich et al. 1993; Putney & Jordan 1995); all agree that the global field strength is of the order of 30 MG. This star has the longest rotational period firmly established for a white dwarf.</text>
<section_header_level_1><location><page_29><loc_51><loc_74><loc_88><loc_76></location>Appendix B.76: WD2359 -434 = LAWD 96 (variable, P ≃ 2 . 7 h)</section_header_level_1>
<text><location><page_29><loc_51><loc_57><loc_94><loc_73></location>WD2359 -434 was suggested to be a magnetic star by Koester et al. (1998) on the basis of the peculiar profile of the H α core, and ⟨ Bz ⟩ was found to be non-zero by Aznar Cuadrado et al. (2004). A series of polarised ESPaDoNS spectra revealed a rotation period of 0.1123 d (Landstreet et al. 2017). The star is also a photometric variable with the same period. The field structure has been modelled approximately, and found to be distinctly more complex than a simple dipole, with ⟨| B |⟩ varying approximately between 50 and 100 kG. This white dwarf o ff ers one of the clearest examples known of a field structure that is substantially more complex than a simple co-linear multipole expansion (Landstreet et al. 2017).</text>
</document>
[
{
"title": "ABSTRACT",
"content": "Many magnetic white dwarfs exhibit a polarised spectrum that periodically varies as the star rotates because the magnetic field is not symmetric about the rotation axis. In this work, we report the discovery that while weakly magnetic white dwarfs of all ages with M ≤ 1 M ⊙ show polarimetric variability with a period between hours and several days, the large majority of magnetic white dwarfs in the same mass range with cooling ages older than 2 Gyr and field strengths ≥ 10 MG show little or no polarimetric variability. This could be interpreted as extremely slow rotation, but a lack of known white dwarfs with measured periods longer than two weeks means that we do not see white dwarfs slowing their rotation. We therefore suggest a di ff erent interpretation: old strongly magnetic white dwarfs do not vary because their fields are roughly symmetric about the rotation axes. Symmetry may either be a consequence of field evolution or a physical characteristic intrinsic to the way strong fields are generated in older stars. Specifically, a strong magnetic field could distort the shape of a star, forcing the principal axis of maximum inertia away from the spin axis. Eventually, as a result of energy dissipation, the magnetic axis will align with the angular momentum axis. Alternatively, symmetry could be the hallmark of a dynamo that operates after the beginning of core crystallisation. We also find that the higher-mass strongly magnetised white dwarfs, which are likely the products of the merging of two white dwarfs, may appear as either polarimetrically variable or constant. This may be the symptom of two di ff erent formation channels or the consequence of the fact that a dynamo operating during a merger may produce diverse magnetic configurations. Alternatively, the massive white dwarfs with constant polarisation may be rotating with periods much shorter than the typical exposure times of the observations. Key words. polarisation - stars: white dwarfs - stars: magnetic fields",
"pages": [
1
]
},
{
"title": "Strong magnetic fields of old white dwarfs are symmetric about the stellar rotation axes",
"content": "S. Bagnulo 1 and J.D. Landstreet 1 , 2 Received July 5, 2024, accepted October 22, 2024",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "Most white dwarfs do not show any variability, and more than 97% of them can be safely considered as suitable flux standard stars (Hermes et al. 2017). Because of flux stability, it is generally impossible to measure the rotation period of white dwarfs except when they pulsate or when they have a magnetic field. It is well known that magnetic white dwarfs often show periodically variable signals of polarisation and sometimes also periodic photometric variability. In the local 20 pc volume, more than 20% of white dwarfs have a magnetic field (Bagnulo & Landstreet 2021), and about 20% of the magnetic white dwarfs are photometrically variable (Farihi et al. 2024). Polarimetric variability is explained in terms of a magnetic field not symmetric about the rotation axis so that the magnetic configuration seen by the observer (as encoded in the polarisation signal) varies as the star rotates. The causes of photometric variability are not well understood (see, e.g., Bagnulo et al. 2024b), yet photometric variability of non-pulsating white dwarfs is so clearly correlated with the presence of the magnetic field variations that one may hypothesise that all non-pulsating white dwarfs with light variations are magnetic. If a star belongs to the small class of white dwarfs with a well-detected light curve, then the rotational period may usually be deduced from photometry (see Brinkworth et al. 2013; Hernandez et al. 2024; Oliveira da Rosa et al. 2024), but, in general, the time series of polarisation measurements may allow one to recover the rotational period of a larger sample of magnetic white dwarfs. Investigations of magnetic fields and of rotational periods are closely related not only because the presence of a magnetic field enables period measurements but also because one could expect that the occurrence of a magnetic field and its strength is physically related to stellar rotation. For example, should the field be supported by a dynamo, one could predict fields to be stronger in more rapidly rotating stars than in those that are more slowly rotating. Correlations between the presence of a magnetic field and the rotational period of a star exist even when the field has a fossil origin. Most of the chemically peculiar stars of the main sequence (Ap and Bp stars) have a fossil magnetic field, and they all rotate more slowly than the non-magnetic non-chemically peculiar stars in the same region of the Hertzsprung-Russel diagram (e.g. Donati & Landstreet 2009). Some of the magnetic Ap and Bp stars have well measured rotational periods as long as several months, years, or even decades (e.g. γ Equ, Leroy et al. 1994). There is no clear explanation for this well-established correlation between magnetic fields and slow stellar rotation. In the case of white dwarfs, various samples of data have gradually become available in the past few years to constrain possible scenarios for the evolution of their magnetic fields. It now appears that among the normal-mass white dwarfs ( M ≃ 0 . 6 M ⊙ ) that largely descend from single-star evolution, magnetic fields are very rare and weak during the first 1-2 Gyr of cooling but then gradually become much more common and often very strong after about 2-3 Gyr of cooling (Bagnulo & Landstreet 2021). This evolution may occur as a result of gradual relaxation to the surface of fields present in stellar cores during earlier evo- on and / or as a result of the operation of a dynamo during core crystallisation (Isern et al. 2017; Schreiber et al. 2021; Bagnulo & Landstreet 2021, 2022; Ginzburg et al. 2022). Among normal-mass stars, those with a magnetic field seem to rotate faster than non-magnetic white dwarfs (Hernandez et al. 2024), supporting the idea that crystallisation-driven dynamo may play a role in the formation of their magnetic fields. Many highly massive white dwarfs ( M ≥ 1 . 0 M ⊙ ), which are probably the product of mergers of two white dwarfs, appear magnetic very early in their cooling age, and the origin of the magnetic field may be due to a dynamo operating during the merging (Tout et al. 2008; García-Berro et al. 2012; Briggs et al. 2015; Bagnulo & Landstreet 2022). This idea is supported also by the evidence that some white dwarfs have very short rotation periods (e.g. Feige 7: 131 min, Liebert et al. 1977; Cl Oct: 12.1 min, Barstow et al. 1995; WD2209 + 113: 70 s, Kilic et al. 2021). At the same time, some polarimetric studies have hinted that a few old magnetic white dwarfs with a field strength of the order of tens of to a hundred megagauss have extremely long rotational periods, showing at most quite small variations even on timescales of decades (e.g. Berdyugin & Piirola 1999; Bagnulo &Landstreet 2019b). Traditionally, this lack of obvious variability is explained by the assumption that such non-varying magnetic white dwarfs have very long rotation periods of the order of centuries (Schmidt & Norsworthy 1991). In the absence of clear observational support or contradiction, this assumption has been widely accepted (e.g. Ferrario et al. 2015). The conclusion is basically that the known periods, P , fall into two very di ff erent families: one with P < ∼ 2 weeks, characterising most of the sample, and a few percent with strong fields that have periods of centuries (i.e. no clearly detected rotation). There is no explanation as to why some old white dwarfs would have extremely long rotation periods. Magnetic braking has generally been invoked but without the support of numerical calculations. In this work, we report new polarimetric observations and combine them with data collected from the literature. We analyse the magnetic variability and thus the rotation of a sample of 74 white dwarfs, and we discuss the constraints that these data place on possible evolution paths for magnetic fields in white dwarfs.",
"pages": [
1,
2
]
},
{
"title": "2.1. Observations not previously published",
"content": "We present here 49 unpublished spectropolarimetric observations of 13 magnetic white dwarfs. Fourteen spectra of six white dwarfs were recently obtained with the FORS2 instrument (Appenzeller et al. 1998) of the ESO VLT in the course of a spectropolarimetric survey of the solar neighbourhood. Seven polarised spectra of two other white dwarfs were retrieved from the ESO archive: one of the spectra was obtained with FORS1 and the remaining ones with FORS2 (we note that FORS1 and FORS2 are virtually identical instruments). Fourteen new spectra of four white dwarfs were obtained with the ISIS instrument of the William Herschel Telescope (WHT). For all of these data, the observing strategy and data reduction are identical to the procedures described by Bagnulo & Landstreet (2018) for circular polarisation data and by Bagnulo & Landstreet (2019b) for linear polarisation data. For two of the four stars recently observed with ISIS, we also present here three previously unpublished spectra obtained in the 1970s that we used to study the long-term variability of magnetic white dwarfs. These data consist of lowresolution polarised spectra obtained by Angel and Landstreet using the multi-channel spectrophotometer (hereafter MCSP) on the 5-m Palomar telescope. The instrument set-up and data reduction of these observations are described in detail by Angel & Landstreet (1974). Finally, eleven spectra of WD 1105-340 were obtained with the ESPaDOnS instrument of the CanadaFrance-Hawaii Telescope (CFHT). These data were reduced by the automatic CFHT pipeline LibreEsprit. The log of these unpublished observations is given in Table A.1, and results are described in the sections dedicated to individual stars of Appendix B. Apart from their general purpose of assessing polarimetric variability, our new observations confirm that star WD 1116 -470 is magnetic (it was previously considered as suspected to be magnetic by Bagnulo & Landstreet 2021) and bring the number of magnetic white dwarfs in the local 20 pc volume to 34 (see Bagnulo & Landstreet 2021). We have also discovered two new massive magnetic white dwarfs: WD1619 + 054 and WD1754 -550 (both rapidly variable).",
"pages": [
2
]
},
{
"title": "2.2. Literature data",
"content": "We searched the literature and collected data for all magnetic white dwarfs that were observed in polarimetric mode at least twice as well as for all magnetic white dwarfs for which multiple spectroscopic intensity observations revealed magnetic variability. The way literature data and new observations have been used is described in the next section.",
"pages": [
2
]
},
{
"title": "3. Determination of the variability of the magnetic white dwarfs",
"content": "We are interested in establishing whether the magnetic field of a white dwarf appears constant or variable with time to the observer. Magnetic variability is ascribed to changes in the apparent field geometry as a star rotates. It can be detected mainly via circular spectro- or broadband polarimetry, which are both sensitive to the longitudinal component of the magnetic field. Linear polarisation, which is sensitive to the transverse component of the magnetic field, is usually much weaker than circular polarisation, and it is rarely detected in white dwarfs. Intensity spectra (if the star is not featureless) are sensitive to the mean field modulus and may also be used to detect magnetic variability, but they are less sensitive to changes of the apparent magnetic configuration than polarimetry. Magnetic white dwarfs that have been repeatedly observed in spectroscopic mode and that do not show sign of variability have not been considered here because having observed a constant intensity spectrum is not a su ffi ciently strong indication that a star is not magnetically variable. An example is WD2047 + 372, as it shows an almost constant Zeeman triplet in multi-epoch intensity spectra and a variable, sign-reversing circular polarisation spectrum (Landstreet et al. 2017). Photometric variability is observed in magnetic white dwarfs, but in the absence of convincing observational evidence or theoretical arguments, we have chosen not to consider it alone as a proxy for magnetism nor magnetic variability. For example, Brinkworth et al. (2013, see in particular their Tables 1 and 3) detected light variability for a number of magnetic stars. For many of them, however, the light amplitude is quite low, and there are no multi-epoch polarimetric data to confirm magnetic variability. Furthermore, recent analysis of TESS data made by Hernandez et al. (2024) and Oliveira da Rosa et al. (2024) failed to detect a periodic light curve for many of these targets. These stars are not included in our sample. In summary: 1) Two or more intensity spectra, or polarisation spectra, or broadband polarisation measurements that clearly di ff er from each other are interpreted as being due to magnetic variability. Wegathered reasonable evidence for polarimetric variability, or non-variability, for the sample of 74 magnetic white dwarfs listed in Table 2. Admittedly, for some stars, we found that establishing whether or not two or more observations are consistent among themselves was somewhat subjective. In Sect. 5.3, we argue that our results are not a ff ected by these uncertain cases. Appendix B discusses in detail all individual stars. In this section, we summarise the results and flag special cases. Among the 74 white dwarfs, 44 are magnetically variable (Sect. 3.1), while for 27 there is so far no evidence of variability (Sect. 3.2). The remaining three white dwarfs, which have strong fields and have been observed over several decades, may be further tested for subtle or possibly very slow variability (Sect. 3.3).",
"pages": [
2,
3
]
},
{
"title": "3.1. Stars that show evidence of magnetic variability",
"content": "For 30 white dwarfs among the 44 stars that show polarimetric variability, a rotational period, or at least a good candidate for it, has been established in the literature. This period is given (in days) in the last column of Table 2 and is followed by ':' when only tentative or approximated values are known. Rotational periods are generally of the order of hours; about 25% are of the order of one day or longer, and only two stars have a rotation period longer than one week (the slowest magnetic white dwarf for which the period is known, WD 2316 + 123, has P ≃ 17 . 4 d, Schmidt & Norsworthy 1991). Most of these stars were listed in Table C.1 of Hernandez et al. (2024), and they are shown with blue symbols in Fig. 1 of Sect. 4. Among the other 14 variable stars, we do not have enough data to estimate the rotation period. For ten of them, the evidence of polarimetric variability has been securely established by two or more observations. These ten variable stars are marked in last column of Table 2 with 'var.' and are also represented with a blue symbol in Fig. 1. The remaining four white dwarfs show subtle signs of variability among the observations obtained with the same instrument. They are marked in Table 2 with 'var.:' and are shown with light blue symbols in Fig. 1. For WD 0446-789 and WD1009-184, our conclusion is that the field variations are real but small due to a configuration probably nearly symmetric about the rotation axis. WD 1105-048 was repeatedly observed with FORS1, FORS2, and ESPaDOnS, and a field was detected on only two occasions out of 12 observing epochs. If the star is magnetic, then it is certainly variable with a timescale of the order of days or weeks, but, admittedly, one could suspect that the two detections are in fact spurious. We decided to define the star as a probable variable star. WD 2049-222 shows a signal of polarisation that is marginally higher than instrumental polarisation, but for the reasons explained in Sect. B.68, we decided to consider its subtle variations as real.",
"pages": [
3
]
},
{
"title": "3.2. Stars for which there is no evidence of magnetic variability",
"content": "Nineteen stars have been observed three or more times in polarimetric mode, and the observations are always consistent among themselves within uncertainties. We consider these stars as established non-variable white dwarfs. They are marked with 'n.v.' in Table 2 and represented with red symbols in Fig. 1. Among these 19 stars, four were observed over a time interval of decades and with di ff erent instruments: WD 0548 -001 = G9937, WD0553 + 053 = G99-47, WD1036 -204 = LP790 -29, and WD1829 + 547 = G227-35. Obviously, it is possible that the other members of this group of 'non-variable stars', for which the observations span a time interval of up to a few months or years, are actually variable on a timescale of decades and that as such they could be stars with (so far undetected) long-term variability. Eight stars should be considered simply as candidate nonvariable white dwarfs ('n.v.:' in Table 2 and magenta symbols in Fig. 1). For seven of them, the reason for considering them only as candidate non-variable stars is that they have only been observed twice. In particular we mention that for normal-mass WD0236-269 and for the massive WD0330 -000, the nonvariability is inferred based on comments in the original discovery paper by Schmidt et al. (2001), which represents especially weak evidence. Another candidate non-variable star, WD1750 -311, was observed only once (with FORS2). The comparison of spectra obtained within that single observing series shows some change of circular polarisation and certainly a rapidly variable intensity spectrum (timescale of minutes) in the regions of the Balmer lines, particularly around H β . The star also shows photometric variability with a 35-min period (Macfarlane et al. 2017). Our interpretation is that WD 1750-311 is probably a magnetic white dwarf with a field symmetric, or nearly symmetric about the rotation axis, showing strong abundance patches of hydrogen.",
"pages": [
3
]
},
{
"title": "3.3. Stars possibly subtly or slowly variable",
"content": "There are three stars that have been monitored for decades, WD1748 + 708 = G240 + 708, WD1900 + 310 = Grw + 70 · 8247, and WD2010 + 310 = GD229, for which our conclusions are uncertain. These stars, marked with 's.v.:' in Table 2 and shown with black symbols in Fig. 1, may indeed show some low amplitude and possible long-term variability, but there might be room for a di ff erent interpretation. The recent detailed comparison of old and new polarisation spectra of Grw + 70 · 8247 by Bagnulo & Landstreet (2019b) identifies small changes between spectra taken decades apart (see in particular their Fig. 5), but for the two other stars, it is not entirely clear which changes are real and which are due to di ff erences between observational equipment and techniques. Bagnulo & Landstreet (2019b) assumed that timescales of the observed small changes were those corresponding to the large time gaps between isolated data sets. However, because the changes are generally small, one could argue that the subtle changes have timescales as short as weeks and have not have been noticed because the stars were not adequately monitored. For example, as previously discussed, WD 1105 -048 appears constant (and non-magnetic) in ten out of 12 observations, and a magnetic field was detected in only two epochs. In case of WD 1748 + 708, photometry leads to ambiguous results. The star was found photometrically variable with a period between 5 and 48 h by Antonyuk et al. (2016). Brinkworth et al. (2013) did not confirm its short-term photometric variability (nor did Hernandez et al. 2024) but instead claimed detection of light changes over a period of ten months.",
"pages": [
3,
4
]
},
{
"title": "4. Clustering in parameter space",
"content": "In this section, we correlate the variability or non-variability of the stars with their mass, field strength, and stellar temperature. The left panels of Fig. 1 show the field strength versus e ff ective temperature and cooling age for the stars of our sample, with the cooling age on both a linear scale and a logarithmic scale. In these plots, stars are represented with symbols that increase in size with a star's mass, while di ff erent colours are used to mark the characteristics in terms of variability. The right panels of Fig. 1 show the cooling age-mass diagrams for the same sample, with cooling age reported again both with linear and logarithmic scales. The size of the symbols is proportional to the field strength. With the help of these figures and Table 2, we studied the relationships between field strength, mass, age, and variability of the magnetic white dwarfs. We recall that our data do not come from a volume-limited sample of stars and do not reflect the relative density of magnetic white dwarfs in di ff erent regions of the diagrams of Fig. 1. In fact, both young magnetic normalmass white dwarfs and ultra-massive white dwarfs are quite rare objects in space, and they are over-represented in our sample because of biases of the surveys (see Bagnulo & Landstreet 2022). The following analysis is about the relative frequency of variable and non-variable stars in di ff erent regions of the diagrams where we noticed the existence of statistically remarkable di ff erences.",
"pages": [
4
]
},
{
"title": "4.1. Normal-mass and weakly magnetic white dwarfs",
"content": "There are 25 white dwarfs with a field strength of ≤ 1 MG, and all of them but one (WD 2051-208) have M ≤ 1 . 0 M ⊙ . These 'normal-mass' 'weak-field' white dwarfs span an age range up to τ ≈ 6 Gyr. Twenty-one of these 24 stars are variable. Among the three that are non-variable, one is younger than 1 Gyr: WD 1105 -340 ( τ = 0 . 34 Gyr).",
"pages": [
4
]
},
{
"title": "4.2. Normal-mass and strongly magnetic white dwarfs",
"content": "There are 25 magnetic white dwarfs with M ≤ 1 . 0 M ⊙ and field strength of ≥ 10 MG. Among those that are younger than 2 Gyr, six out of eight white dwarfs are variable; however, we recall that young strongly magnetic white dwarfs are rare in space (Bagnulo & Landstreet 2022). Among the magnetic white dwarfs older than 2 Gyr, only one out of the 17 shows polarimetric variability.",
"pages": [
4
]
},
{
"title": "4.3. Massive magnetic white dwarfs",
"content": "Fifteen stars of our sample have M > 1 . 0 M ⊙ , and all but one are younger than about 1 Gyr. Eight of them are magnetically variable, including the only old ( τ ≃ 4 Gyr) star WD 0756 + 437. Nearly all of these massive magnetically variable white dwarfs have a strong field (tens to hundreds megagauss), except for WD2051-208, which is the only ultra-massive star ( M = 1 . 2 M ⊙ ) with a sub-MG field strength (0.25 MG) in our sample. In contrast, seven massive magnetic white dwarfs show no obvious signs of variability, although none of them may be safely considered as non-variable, either because they were observed only twice (or even just once, i.e. WD 1750-311; see Appendix B.60) or because some subtle sign of variability may have been detected (in WD 1900 + 705 and WD2010 + 310). WD1658 + 440 and WD1750-311 are the only massive non- Notes. The full list includes 59 stars, 23 of which are non-variable. Between brackets, the table also gives the number of non-variable stars N K divided by the number of stars N tot in the subset as defined by the age and field values. variable white dwarfs in our sample with a field as low as ≈ 2 MG. The remaining five non-variable stars have magnetic fields with strengths of the order of hundreds of megagauss.",
"pages": [
4
]
},
{
"title": "4.4. Statistical analysis of normal-mass white dwarfs",
"content": "Because we have only one example of an old massive magnetic white dwarf, we do not know how variability in massive magnetic white dwarfs evolves with time. Vice versa, an evolutionary path is clearly seen in normal-mass ( M < ∼ 1 M ⊙ ) white dwarfs. In our sample, field variability is nearly ubiquitous among weakly magnetic white dwarfs of all ages and in young white dwarfs regardless of the field strength, it is but very rare in old ( τ ≥ 2 Gyr) strongly magnetic ( B ≥ 10 MG) white dwarfs. Next, we assess the statistical significance of this pattern, specifically whether it can be attributed to small number statistics or if it reflects a genuine correlation between cooling age, field strength, and field variability. We first split the sample of stars with M ≤ 1 M ⊙ into two groups: those younger than 2 Gyr and those older than 2 Gyr. Each of these two groups was divided into two subsets: stars with a field strength ⟨| B |⟩ ≤ 1 MG and stars with ⟨| B |⟩ > 10 MG. For each of these four subsets, we considered the ratio between the number of non-variable magnetic stars, N K, and the total number of magnetic stars, N tot. From these numbers, we estimated the probability, Pr , that the sample frequency of non-variable stars is between r and r + d r , normalised to one and obtained assuming that all numbers between zero and one are a priori equally probable, using Figure 2 shows the probability density functions for the stars belonging to these sets. It clearly appears that there is little to no overlap between the density functions of the symmetric field in old strongly magnetic white dwarfs and of young white dwarfs. Table 1 reports the points of maximum for these distributions, f = N K / N tot, and their uncertainties Table 1 also includes the results for the smaller sets of stars with intermediate strength, 1 ≤ ⟨| B |⟩ ≤ 10 MG (with endpoints overlapping with the other two sets), which could possibly be considered a 'transitioning' field strength range. Figure 3 shows the ratio between the number of non-variable magnetic white dwarfs with a field strength lower than a given value ¯ ⟨| B |⟩ and the total number of magnetic white dwarfs with a field strength lower than that value as a function of ¯ ⟨| B |⟩ for stars with cooling ages τ ≤ 2 Gyr (blue lines) and stars with τ > 2 Gyr (red lines). This plot supports our claim that nonvariable fields are much more common in old, strongly magnetic white dwarfs than in young strongly magnetic white dwarfs, and than in weakly magnetic white dwarfs of all ages. It suggests also that the minimum field strength required for a field to become non-variable is in the range of 5 to 10 MG.",
"pages": [
4,
6
]
},
{
"title": "5. Explanation for the lack of observed variability",
"content": "There are three possible reasons for the observed non-variability of so many magnetic white dwarfs. We examine them in the following sections.",
"pages": [
6
]
},
{
"title": "5.1. Extremely slow rotation",
"content": "We first consider the possibility that the rotation of magnetic white dwarfs slows with age and / or magnetic field strength. Perhaps, as the surface field becomes stronger, the white dwarf loses angular momentum to its environment by electromagnetic dipole radiation (García-Berro et al. 2012), by coupling with gas clouds in the ISM, or by a very weak magnetically coupled wind. Each of these possibilities would shed stellar angular momentum faster from a magnetic white dwarf with a strong field than from one with a weak field, preferentially slowing the magnetic white dwarfs with strong fields. However, all of these mechanisms are expected to exert at most a very weak influence on the rotation of a white dwarf, and it seems quite unlikely that any of them could slow the rotation of a magnetic white dwarf so much that repeated observations even a year apart would not show any rotation. Nevertheless, the idea of extremely slowly rotating white dwarfs has been generally accepted. This hypothesis likely origi- ates from Sect. 3 of Schmidt & Norsworthy (1991), which says: 'We therefore assign long periods to these stars, with the recognition that other explanations for their polarimetric constancy are possible.' This statement was accompanied by the first plots of field strength versus rotational period, a plot that in updated form has reappeared in numerous reviews (e.g. Ferrario et al. 2015; Kawka 2020; Ferrario et al. 2020) but has never been critically revisited. We note that there is a complete lack of white dwarfs that are known to vary on a timescale between weeks and decades. In this respect, there is a profound di ff erence between candidate longterm variable magnetic white dwarfs and long period magnetic Ap and Bp stars. While we do not know of any white dwarf with a firmly established polarimetric variability with a period longer than approximately two weeks, we are sure that a number of Ap and Bp stars are very slowly rotating stars because they definitely show clearly periodic field variation, even ⟨ Bz ⟩ sign reversals, on a very long but measured timescale that may be months, years, or decades. Furthermore, the distribution of periods is roughly continuous between periods of less than one day and periods of several years (Mathys 2008). If we wanted to interpret the lack of polarimetric variability in white dwarfs as the e ff ect of extremely long rotation periods, we would need to accept that the rotation periods of white dwarfs show an extremely bimodal distribution that peaks around hours and days and around centuries with nothing in between. This appears to be a very unlikely scenario.",
"pages": [
6
]
},
{
"title": "5.2. Extremely fast rotation",
"content": "Asecond possibility is that some or all of the non-variable white dwarfs actually are very rapidly rotating stars. If the star has a rotation period much shorter than the individual frame exposure time, then the polarimetric variability would be smeared out, and the star would appear constant. In fact, it has been possible to probe variability on the timescale of the exposure time of each individual frame (typically 10 min or less). In very general terms, our FORS2 and ISIS polarimetric observations allowed us to identify a star as variable provided that its rotation period is longer than ≈ 10 min and its magnetic configuration is clearly not symmetric about the rotation axis (for example, WD 1712-590 in Sects. B.57). Isolated white dwarfs that are the product of single-star evolution can hardly have rotation periods shorter than that (Kawaler 2015). This is confirmed by the results of Hernandez et al. (2024), who have shown that rotation periods of young magnetic white dwarfs in the normal mass range are only marginally shorter than the rotation periods of non-magnetic white dwarfs. Merger products, in contrast, can have rotation periods of the order of 1 min (Schwab 2021; Kilic et al. 2021) if most of the binary angular momentum is retained by the merger product. It is therefore possible that the non-variable massive white dwarfs have nonaxisymmetric fields but are rotating very rapidly. Periods down to 1 min or less can be probed via rapid cadence photometry. None of the massive, magnetically non-variable white dwarfs show TESS light variability (Hernandez, priv. comm.; Ramsay, priv. comm.), although of course one cannot rule out that more accurate photometry could reveal some extremely rapid rotators. Provisionally, we rule out very rapid rotation for the massive magnetic white dwarfs that show a constant polarisation.",
"pages": [
6,
7
]
},
{
"title": "5.3. Magnetic fields are symmetric about the stellar rotation axes",
"content": "After ruling out both extremely slow and extremely fast rotation, we are left with the hypothesis that a non-variable field is approximately axisymmetric about the rotation axis. This means that as the star rotates with a normal period between a fraction of an hour and several days, the observer does not see any significant change in the signal of circular polarisation. This interpretation naturally accounts for some outliers, such as the non-variable weakly magnetic star WD 1105-340. It is reasonable to hypothesise that this one star does not appear variable simply because its rotation axis is tilted at a small angle with respect to the line of sight. In Sect. 3, we highlighted that it may be di ffi cult to firmly assess whether a star is magnetically variable because changes could be too subtle to be detected. However, the risk of classifying as 'non-variable' a star that in fact exhibits subtle but real changes does not weaken our analysis because small changes of the circular polarisation spectrum are still symptoms of a field nearly symmetric about the rotation axis.",
"pages": [
7
]
},
{
"title": "6. Discussion",
"content": "Here, we first consider the case of normal-mass white dwarfs (Sect. 6.1) and then that of massive stars (Sect. 6.2). We note that our analysis applies only to isolated white dwarfs; symmetric fields seem uncommon among magnetic white dwarfs accreting material from a companion (Cropper 1988; Reimers et al. 1999; Schmidt et al. 1995).",
"pages": [
7
]
},
{
"title": "6.1. Normal-mass white dwarfs",
"content": "It is clear that most of the weak fields of normal-mass stars of all ages are not symmetric about the stellar rotation axis. Most of the strong fields of normal-mass, young white dwarfs are also not axisymmetric. Most of the strong field of stars older than 2 Gyr are symmetric about the star's rotation axis. We now investigate what are the relationships between these weak and strong fields. Our hypothesis is that the weak and non-symmetric magnetic fields of the young normal-mass white dwarfs are due to a relaxation process of a field that was produced in a previous evolutionary stage of the star, a field that was buried below the surface of the newly formed white dwarf, and that it starts to reveal itself with time as the star cools down. Such seed fields could be those found in red-giant stars, in the vicinity of the hydrogenburning shell (e.g. Li et al. 2022, 2023). The question is whether the older, generally stronger fields share essentially the same origin as the weak, younger ones, having just evolved for longer time, or if the strong fields in older stars formed by a totally different mechanism than the weaker, younger ones, for instance by a crystallisation dynamo. Whatever its origin is, we consider first the possibility that the small distortion of the shape of a magnetic white dwarf produced by magnetic forces in outer layers might actually drive the evolution of the global field towards axisymmetry. In Sect. 6.1.1, we discuss one example of a physical e ff ect that might drive such evolution.",
"pages": [
7
]
},
{
"title": "6.1.1. The magnetic field structure of white dwarfs may evolve towards rotational axisymmetry during cooling",
"content": "In the absence of a magnetic field, the rotation of the white dwarf will lead to an equatorial bulge, increasing the component of the principal moment of inertia that is parallel to the rotation axis. In this case, the rotation about the spin axis, which coincides with the largest moment of inertia, is stable. Next, we introduce a dipolar magnetic field inclined to the axis of rotation by, say, 45 · . We suppose that the visible surface field is maintained by a fossil current system deep in the star, which is gradually decaying due to Ohmic losses. As the interior field strength decreases, an azimuthal electric field will be produced by Faraday induction outside the white dwarf's initial current loop. This field will generate a current around the magnetic axis that will in turn interact with the local meridional magnetic field (Landstreet 1987) to produce an outward-directed force, distorting the stellar outer layers into a slightly oblate shape, with the largest radius around the magnetic equator. Because of this e ff ect, the principal moment of inertia of the star will be rotated towards the symmetry axis of the field, which is not aligned with the rotation axis or the angular momentum axis of the star. The white dwarf e ff ectively becomes an unstable free asymmetric rotator (an asymmetric top). If such objects can dissipate some of the energy supporting the asymmetric shape, the field axis will gradually shift to bring the principal moment of inertia closer to the angular momentum axis, which is a stable minimum energy state of an oblate system. Energy dissipation will gradually lead to the alignment of the magnetic axis with the rotation axis, which is the e ff ect we observe in old normal-mass strongly magnetic white dwarfs. This basic physical e ff ect was first identified in the 1970s as possibly operating in main-sequence magnetic Ap stars, which, similar to magnetic white dwarfs, possess global fossil magnetic fields that are usually roughly dipolar and are generally oblique to the rotation axis (Stibbs 1950; Preston 1971). A number of efforts to estimate the timescale of possible evolution of an oblique magnetic field to a state of small or vanishing obliquity have been published, for example, by Mestel & Takhar (1972). Much of this work is summarised in Chapter 9 of Mestel (1999). However, this line of investigation did not lead to a convergence of clear results about possible timescales or about dependence on basic input physics or parameters, such as interior field strength, or details of induced forces in outer layers and their consequences. As the increasing sample of magnetic Ap star models failed to reveal an obliquity distribution suggestive of this e ff ect, possibly because of the relatively short (of order 10 8 yr) mainsequence lifetimes of magnetic Ap stars, theoretical studies of stellar fossil fields moved on to other e ff ects. However, a similar e ff ect may be at work aligning the magnetic fields to the rotation axis in white dwarfs, which have a very di ff erent structure and much longer evolutionary times than Ap stars. This hypothesis could be explored with the aid of numerical modelling of the global (interior and surface) structure and evolution of a rotating white dwarf with an oblique magnetic field. Modelling of comparably complex white dwarf states (e.g. polars, mergers) that include magnetic fields (e.g. Franzon & Schramm 2015; Bisikalo et al. 2021; Zhong et al. 2024) suggest that numerical methods for exploring this problem are already available.",
"pages": [
7,
8
]
},
{
"title": "6.1.2. Crystallisation-driven dynamo and axisymmetric fields",
"content": "The line that marks the beginning of core crystallisation in the age-mass diagram separates variable from non-variable magnetic white dwarfs perhaps in a cleaner way compared to a massindependent age threshold (see Fig. 1). Before core crystallisation begins, magnetic fields are almost always non-symmetric about the rotation axis. From volume-limited surveys, we know that in normal-mass white dwarfs, before crystallisation, strong fields are rare (Bagnulo & Landstreet 2021, 2022), but those that have been discovered and monitored show polarimetric variability. After the beginning of core crystallisation, many strong fields appear, and most of them are symmetric about the rotation axis. White dwarfs with weak and non-symmetric fields continue to appear also after the beginning of core-crystallisation. The strong magnetic fields of old normal-mass white dwarfs could be generated by the crystallisation convective dynamo mechanism (Isern et al. 2017; Schreiber et al. 2021), as almost all have cooling ages longer than the cooling age required for the onset of this dynamo. In this case, the conclusion would be that the crystallisation dynamo is responsible for approximate symmetry about the rotation axis. This situation could have similarities with what has been observed in fully convective late type stars with no di ff erential rotation, which are known to be able to generate strong and simple large-scale, mostly axisymmetric, poloidal fields (Donati et al. 2006; Morin et al. 2008b,a; Kochukhov 2021, Fig. 14). However, how the crystallisation dynamo alone can produce fields stronger than ∼ 1 MG is still unclear (Isern et al. 2017; Castro-Tapia et al. 2024). In fact, Montgomery & Dunlap (2024) have argued that fluid mixing by phase separation is not a viable mechanism to produce the strong fields observed in old white dwarfs. Perhaps the crystallisation dynamo could be more powerful if it were to amplify a pre-existing internal field, such as the same seed field that is seen in some of the white dwarfs prior to the beginning of crystallisation. Montgomery & Dunlap (2024) have proposed that corecrystallisation could trigger temporary di ff erential rotation. Therefore, one could suspect that rapid rotation in normal-mass white dwarfs might generate a magnetic field in the stellar core using rotational shear on a seed field left from an earlier point in the star's evolutionary history. In this situation, we might also expect that the field would be roughly axisymmetric, although it is not clear why such fields would always be strong or how they might be related to the weaker oblique fields of stars of similar mass. Furthermore, Spruit (1999) has shown that di ff erential rotation would suppress the non-axisymmetric field component of weak fields, but not in the stronger fields. This is the opposite of what we observed. The origin of strong non-axisymmetric fields of normal-mass white dwarfs that appear before crystallisation could possibly be similar to that of higher-mass white dwarfs (see Sect. 6.2 below).",
"pages": [
8
]
},
{
"title": "6.2. The variability of massive magnetic white dwarfs",
"content": "Compared to normal-mass white dwarfs, the magnetic fields of massive white dwarfs present a di ff erent behaviour. Magnetic fields appear when the stars are still very young, and fields may be either symmetric or non-symmetric about the rotation axis. In massive white dwarfs, strong fields (tens or hundreds of megagauss) seem much more common than weaker sub-megagauss fields. It is widely thought that many of these high-mass objects are the result of WD-WD mergers that cause rapid generation of a strong magnetic field (García-Berro et al. 2012; Bagnulo & Landstreet 2022). However, Camisassa et al. (2022) and Blatman & Ginzburg (2024) have shown that at least some of them, depending on their core composition, may have started the process of core crystallisation. Hence, the origin of their field could be linked to the crystallisation dynamo, at least in some cases. The bimodal distribution of the morphologies of the fields of the massive magnetic white dwarfs could indeed reflect two di ff erent channels of formation, one from WD-WD merger (or by a di ff erent binary evolution path), and one by massive singlestar evolution, such as that of normal-mass white dwarfs. Alternatively, the dynamo stimulated by WD-WD merging may be capable of generating both axisymmetric and non-axisymmetric global fields, perhaps depending on the initial angular momentum vectors of the individual merging white dwarfs relative to the orbital angular momentum vector or the mass ratio of the two merging white dwarfs. A viable alternative explanation is that the massive white dwarfs that do not show variability are actually rotating with a period much shorter than the typical exposure time of individual polarimetric measurements. In any case, the large mass of stars in this sample and the occurrence of some rotation periods as short as minutes make it very reasonable to suppose that the evolution of the magnetic fields and rotation periods follows a di ff erent course from the evolution of fields in normal-mass magnetic white dwarfs. Because almost all of the massive magnetic white dwarfs of our sample, except one, are very young, we cannot test whether the morphology of older stars is generally axisymmetric. Remarkably, however, the only example we know of a massive star older than 2 Gyr shows a non-axisymmetric field. With an age of more than 4 Gyr, high mass, and a very strong and variable field, the very unusual WD 0756 + 437 may be considered further evidence that the origin of the fields in massive stars is di ff erent than in normal-mass stars.",
"pages": [
8
]
},
{
"title": "7. Conclusions",
"content": "Since the earliest discoveries of magnetic white dwarfs (Kemp et al. 1970; Angel & Landstreet 1970, 1971b), two quite di ff erent categories of them have been known to exist. Some magnetic white dwarfs show periodic variations of circular polarisation (which probes the longitudinal magnetic field) with timescales ranging from minutes to days. Classically, a signal of circular polarisation constant with time has been interpreted as indicating either a very long rotation period or a lack of rotation. Using both literature data and new observations, we have studied the polarimetric variability of a sample of 74 magnetic white dwarfs. We find that among white dwarfs with M ≤ 1 . 0 M ⊙ ('normal-mass white dwarfs'), nearly all stars with fields weaker than about 1 MG show circular polarisation varying with time. Furthermore, the rare normal-mass white dwarfs younger than ≈ 2 Gyr with strong magnetic fields also show polarimetric variability. In striking contrast, 16 out of the 17 normal-mass stars older than 2 Gyr with fields stronger than about 10 MG in our sample show constant polarisation. Magnetic white dwarfs with M ≥ 1 . 0 M ⊙ ('massive white dwarfs'), many of which are the product of WD-WD merging, show a mixed behaviour. Many of them have a strong magnetic and variable field, but some have a strong and constant magnetic field. In our sample, nearly all the massive magnetic white dwarfs are younger than ≈ 1 Gyr, with the exception of WD 0756 + 437, an old strongly magnetic variable star. The lack of evidence for major variations of circular polarisation on any timescale longer than about two weeks suggests that the interpretation of non-variability arising from extremely long rotational periods is incorrect. We have argued that the nonvariability is due to magnetic field structures roughly symmetric around the star's rotation axis. A possible explanation for the observed symmetry could be that most or all of the magnetic fields evolved from pre-existing fields that formed during pre-white dwarf evolution stages (Li et al. 2022, 2023) and gradually relaxed to the stellar surface over a relation time of 1 or 2 Gyr; in fact, very recently, Camisassa et al. (2024) have shown that the magnetic fields of old white dwarfs with M ≥ 0 . 65 M ⊙ may have been generated by a coreconvection dynamo when the star was in the main sequence, and emerged at the stellar surface during the white dwarf cooling phase. As the surface field increases in strength, at some point the Lorentz forces in the outer layers, together with the Coriolis forces acting on rotating convective flows in what are asymmetric rotators, could lead to a gradual relaxation, through energy dissipation, of the global field structure to a form that is symmetric about the rotation axis. The timescale would depend on the field strength, and for a field weaker than several megagauss, it could require a timescale too long to be observed in white dwarfs that still show spectral lines. Alternatively, because these normal-mass large-field magnetic white dwarfs mostly occur after the start of crystallisation (Bagnulo & Landstreet 2021, 2022), one could speculate that the crystallisation dynamo (Isern et al. 2017; Schreiber et al. 2021; Ginzburg et al. 2022) may generate some fraction of the observed fields and further that this dynamo produces essentially axisymmetric surface magnetic field structures with extremely strong fields. This kind of origin would have similarities with the axisymmetric fields that are commonly found in fully convective strongly magnetic M-dwarfs and, much weaker, in the planets Earth and Jupiter. However, it has been argued that the instability of the mantle surrounding the core that starts to crystallise cannot produce fields much stronger than 1 MG (Isern et al. 2017; Montgomery & Dunlap 2024; Castro-Tapia et al. 2024). Furthermore, Camisassa et al. (2024) find that, even if the crystallisation-driven dynamo could generate a strong magnetic field, this field would take too long to emerge at the stellar surface. On the other hand, Montgomery & Dunlap (2024) have suggested that crystallisation may still play a role by trig- gering a temporary phenomenon of di ff erential rotation, which in turn would generate a magnetic field - the di ff usion timescale of which has not been discussed. The situation for ultra-massive white dwarfs (with M > ∼ 1 . 0 M ⊙ ) is di ff erent, as both strongly magnetic variable and nonvariable white dwarfs are found among them. Some of these massive white dwarfs are the product of WD-WD merging, during which a dynamo could create a strong magnetic field (García-Berro et al. 2012), though it would not necessarily be symmetric about the rotation axis. In addition, some massive magnetic white dwarfs could be the result of single-star evolution and have acquired an axisymmetric field following the same mechanism acting in normal-mass white dwarfs. Further investigation into the variability of magnetic white dwarfs is necessary. Nevertheless, current observations have already imposed significant constraints that any theory explaining the origin and evolution of magnetic fields in degenerate stars must take into account. Notes. A number in the last column represents the established rotation period (in d) of a star that shows polarimetric variability; tentative periods are followed by the symbol ':'. Other symbols have the following meaning: 'var.' means that the star is certainly polarimetric variable but the period is still unknown; 'var.:' means that hints of subtle variability have been detected over a short timescale; 'n.v.' means that the observed polarisation was constant (within uncertainties) and the star was observed at least three times; 'n.v.:' means that observed polarisation was constant (within uncertainties) but the star was observed only twice; 's.v.' means that polarisation shows some sign of subtle variability over a timescale of a decade or longer. For stars within the local 40 pc volume, the parameters of stellar magnitude, distance, spectral type, atmospheric composition, temperature, mass, and age are from O'Brien et al. (2024); for the remaining stars, we used the catalogue from Gentile Fusillo et al. (2021) with ages interpolated from the tables by Bédard et al. (2020). Acknowledgements. The new observations presented in this work were made with the FORS2 instrument at the ESO Telescopes at the La Silla Paranal Observatory under program ID 110.243J.001, 110.23XV.001, 110.23XV.002, and 112.25C9.001, and with the ISIS instrument at the William Herschel Telescope (operated on the island of La Palma by the Isaac Newton Group), under programmes P10 in 19A and P8 in 19B. This research has made use also of additional FORS1 and FORS2 data obtained from the ESO Science Archive Facility: data for WD 1036-204 were obtained under programmes IDs 70.D-0259(A), 087.D-0714(A), 089.D-0612(A), 090.D-0269(A). Data for WD 1105-340 was acquired with ESPaDOnS on the Canada-France-Hawaii Telescope (CFHT) (operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii), under programmes 17AC01, 19AC04, 19BC02, and 21BC02. We thank the anonymous referee for their very constructive criticism. We thank Matthias Schreiber and Antonino Lanza for very useful comments. JDL acknowledges the financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference number 6377-2016.",
"pages": [
8,
9,
11,
12
]
},
{
"title": "References",
"content": "Achilleos, N. & Wickramasinghe, D. T. 1989, ApJ, 346, 444 Angel, J. R. P., Illing, R. M. E., & Landstreet, J. D. 1972a, ApJ, 175, L85 Aznar Cuadrado, R., Jordan, S., Napiwotzki, R., et al. 2004, A&A, 423, 1081 Article number, page 12 of 29 Abstract Series, Vol. 9, Astronomische Gesellschaft Abstract Series, 145 Liebert, J., Schmidt, G. D., Sion, E. M., et al. 1985, PASP, 97, 158 Macfarlane, S. A., Woudt, P. A., Dufour, P., et al. 2017, MNRAS, 470, 732 Manser, C. J., Gänsicke, B. T., Inight, K., et al. 2023, MNRAS, 521, 4976 Martin, B. & Wickramasinghe, D. T. 1978, MNRAS, 183, 533 Mathys, G. 2008, Contributions of the Astronomical Observatory Skalnate Pleso, 38, 217 Maxted, P. F. L., Ferrario, L., Marsh, T. R., & Wickramasinghe, D. T. 2000, MNRAS, 315, L41 McCook, G. P. & Sion, E. M. 1977, A Catalogue of spectroscopically identified white dwarfs McCook, G. P. & Sion, E. M. 1999, ApJS, 121, 1 Mestel, L. 1999, Stellar magnetism Mestel, L. & Takhar, H. S. 1972, MNRAS, 156, 419 Montgomery, M. H. & Dunlap, B. H. 2024, ApJ, 961, 197 Morin, J., Donati, J. F., Forveille, T., et al. 2008a, MNRAS, 384, 77 Morin, J., Donati, J. F., Petit, P., et al. 2008b, MNRAS, 390, 567 Moss, A., Bergeron, P., Kilic, M., et al. 2024, MNRAS, 527, 10111 O'Brien, M. W., Tremblay, P. E., Gentile Fusillo, N. P., et al. 2023, MNRAS, 518, 3055 O'Brien, M. W., Tremblay, P. E., Klein, B. L., et al. 2024, MNRAS, 527, 8687 Oliveira da Rosa, G., Kepler, S. O., Soethe, L. T. T., Romero, A. D., & Bell, K. J. 2024, arXiv e-prints, arXiv:2407.05214 Piirola, V. & Reiz, A. 1992, A&A, 259, 143 Preston, G. W. 1971, PASP, 83, 571 Putney, A. 1995, ApJ, 451, L67 Putney, A. 1997, ApJS, 112, 527 Putney, A. & Jordan, S. 1995, ApJ, 449, 863 Reding, J. S., Hermes, J. J., Vanderbosch, Z., et al. 2020, ApJ, 894, 19 Reimers, D., Hagen, H. J., & Hopp, U. 1999, A&A, 343, 157 Schmidt, G. D., Bergeron, P., & Fegley, B. 1995, ApJ, 443, 274 Schmidt, G. D. & Norsworthy, J. E. 1991, ApJ, 366, 270 Schmidt, G. D. & Smith, P. S. 1994, ApJ, 423, L63 Schmidt, G. D. & Smith, P. S. 1995, ApJ, 448, 305 Schreiber, M. R., Belloni, D., Gänsicke, B. T., Parsons, S. G., & Zorotovic, M. 2021, Nature Astronomy [ arXiv:2104.14607 ] Schwab, J. 2021, ApJ, 906, 53 Siebenmorgen, R., Voshchinnikov, N. V., & Bagnulo, S. 2014, A&A, 561, A82 Sion, E. M., Liebert, J., Schmidt, G., & Starrfield, S. G. 1984, in Bulletin of the American Astronomical Society, Vol. 16, 725 Spruit, H. C. 1999, A&A, 349, 189 Stibbs, D. W. N. 1950, MNRAS, 110, 395 Swedlund, J. B., Wolstencroft, R. D., Michalsky, Jr., J. J., & Kemp, J. C. 1974, ApJ, 187, L121 Valyavin, G., Bagnulo, S., Monin, D., et al. 2005, A&A, 439, 1099 Valyavin, G., Wade, G. A., Bagnulo, S., et al. 2008, ApJ, 683, 466 Vennes, S., Schmidt, G. D., Ferrario, L., et al. 2003, ApJ, 593, 1040 Vornanen, T., Berdyugina, S. V., Berdyugin, A. V., & Piirola, V. 2010, ApJ, 720, L52 Walters, N., Farihi, J., Marsh, T. R., et al. 2021, MNRAS, 503, 3743 Wegner, G. 1977, Mem. Soc. Astron. Italiana, 48, 27 Wesemael, F., Liebert, J., Schmidt, G. D., et al. 2001, ApJ, 554, 1118 West, S. C. 1989a, ApJ, 345, 511 West, S. C. 1989b, ApJ, 345, 511 Wickramasinghe, D. T. & Bessell, M. S. 1976, ApJ, 203, L39 Wickramasinghe, D. T. & Cropper, M. 1988, MNRAS, 235, 1451",
"pages": [
12,
13
]
},
{
"title": "Appendix A: New observations",
"content": "Article number, page 14 of 29",
"pages": [
14
]
},
{
"title": "Appendix B: Variability of magnetic white dwarfs: Comments on individual stars",
"content": "We preliminary note that to compare observations taken by different authors, we need to know how circular polarisation was defined in di ff erent works. This problem has been highlighted in detail, for example in Sect. 2 of Bagnulo & Landstreet (2020). Here, we have adopted the definition of positive handedness of circular polarisation given by Landi Degl'Innocenti & Landolfi (2004), and converted some of the literature data to it by changing the sign of circular polarisation with respect to the originally published data. Numerous examples in the literature (e.g. Bagnulo & Landstreet 2020) and in this section show that many magnetic white dwarfs with very strong fields have a circular polarisation spectrum that varies rapidly with wavelength. The consequence is that the same source observed with even slightly di ff erent broadband filters may result in di ff erences of the measured broadband polarisation signal that are larger than the formal uncertainties. This specific issue is discussed in Sect. 7 of Bagnulo & Landstreet (2019b), where some numerical examples are presented. As a consequence, small di ff erences between broadband polarimetric measurements obtained with di ff erent instruments cannot reliably provide strong evidence that a star is magnetically variable. Spectropolarimetry should allow comparisons that are less instrument dependent, provided the observations overlap in wavelength, and that the resolving power of di ff erent instrument is similar. There is no doubt, however, that the safest way to establish variability or constancy of polarisation is to repeatedly observe the star with the same instrument and instrument setting, and to always perform the data reduction using exactly the same technique. In the following, we discuss all stars of our sample (including two stars that did not make it into the final list of Table 2). The titles of the individual subsections contain the star's name in the Villanova system (McCook & Sion 1977, 1999), the main SIMBADidentifier, and our conclusions about the magnetic variability (using the same designation as in Table 2, followed by the rotation period P , if this is known). When the rotation period is obtained only from photometric data, we use the designation P phot.",
"pages": [
16
]
},
{
"title": "Appendix B.1: WD 0004+122 = LP 464-57 (non-variable)",
"content": "This star was discovered to be magnetic, using spectropolarimetry, by Bagnulo & Landstreet (2020), and later observed four times by Berdyugin et al. (2024) in broadband circular polarisation. These measurements are shown in the top left panel of Fig. B.1. None of them deviated more than ≃ 1 σ from the mean value of the measurements obtained in the same filter, except for a ∼ 2 . 5 σ di ff erence between on measurement and the other three in the B filter, which might reflect a real small variability. We reanalysed the FORS2 spectropolarimetry published by Bagnulo & Landstreet (2020) to check for di ff erences between the V / I profiles obtained from the first and the second pair of exposures, sampling an interval in time of about 15 m, without finding any di ff erence (see Fig. B.2). We conclude that the star does not show convincing sign of variability. We estimate a field strength of ≃ 100 MG.",
"pages": [
16
]
},
{
"title": "Appendix B.2: WD 0009+501 = EGGR 381 (variable, P ≃ 8 h)",
"content": "This white dwarf was well monitored and modelled as a magnetic variable with ⟨| B |⟩ between 150 and 250 kG by Valyavin et al. (2005), who presented a model obtained by adopting a rotational period of 0 . 3337 ± 0 . 0031 d. Recent unpublished ESPaDOnS data obtained by us confirm the rotational period.",
"pages": [
16
]
},
{
"title": "Appendix B.3: WD 0011-134 = G 158-45 (variable, P phot ≃ 0 . 73 h)",
"content": "Discovered magnetic (with a field of about 9 MG) by Bergeron et al. (1992), and as a magnetic variable by Putney (1997). From photometry, Lawrie et al. (2013) found a rotational period of 44 ± 0 . 43 min. While there is no guarantee that the photometric period reflects the rotational period of the star, the star is definitely a magnetic variable, and in fact, this very short photometric period is completely consistent with several of the many acceptable periods found from the six widely spaced ⟨ Bz ⟩ data values published by Putney (1997) from spectropolarimetric observations made between 1994 Sep 29 and 1994 Dec 31. This star is one of the magnetic white dwarfs that clearly shows longitudinal field reversal.",
"pages": [
16
]
},
{
"title": "Appendix B.4: WD 0041-102 = Feige 7 (variable, P ≃ 2 h)",
"content": "Liebert et al. (1977) showed that the spectrum is rich in faint lines of H and He, and identified a magnetic field of about 18-20 MG from the spectrum. They found, on the basis of broadband circular polarisation measurements, that the field varies with a period of 131.6 min. (Note that the broadband circular polarisation changes sign as the star rotates.) A decentred dipole model of the field structure based on a series of flux spectra was derived by Achilleos et al. (1992); both H and He apparently vary substantially in abundance over the surface. Achilleos et al. (1992) also discovered that the star varies photometrically (in a double wave) with the same period as the spectrum and broad-band polarisation.",
"pages": [
16
]
},
{
"title": "Appendix B.5: WD 0051+117 = PHL 886 (variable, P ≃ 4 . 7 d)",
"content": "From unpublished spectropolarimetric data (Landstreet & Bagnulo, in prep., hereafter LB25) it is found that the star is periodically variable ( P = 4 . 703 d) with ⟨| B |⟩ ≃ 270 kG. The field ⟨ Bz ⟩ reverses sign as the star rotates. Modelling will be presented in a forthcoming paper.",
"pages": [
16
]
},
{
"title": "Appendix B.6: WD 0058-044 = GD 9 (variable, P ≃ 2 d?)",
"content": "From our unpublished spectropolarimetric data we found that the star is periodically variable (the period is not determined uniquely from available data, but the most probable period is P = 1 . 96 d), with ⟨| B |⟩ ≃ 300 kG (LB25). The field ⟨ Bz ⟩ varies sinusoidally but does not (quite) reverse sign. Modelling will be presented in a forthcoming paper.",
"pages": [
16
]
},
{
"title": "Appendix B.7: WD 0232+525 = EGGR 314 (variable)",
"content": "This is a well-known star that was repeatedly included in studies of bright white dwarfs and been searched for a magnetic field (Bychkov et al. 1991; Schmidt & Smith 1995; Fabrika et al. 1997) without any detection being reported. Recent particularly sensitive observations (Bagnulo & Landstreet 2022) succeeded in detecting a weak field. The three published measurements (two with ISIS, one with ESPaDOnS) reveal a very weak, and probably variable longitudinal field (the measurement obtained with ESPaDOnS has the opposite sign as those obtained with ISIS).",
"pages": [
16,
17
]
},
{
"title": "Appendix B.8: WD 0233-242A = LP 830-14 = NLTT 8435A (variable, P phot ≃ 1 . 5 h)",
"content": "The star is discussed by Bagnulo & Landstreet (2021), who suggested the presence of a variable field of the order of 3.8 MG. In fact, ⟨| B |⟩ does not vary between two FORS observations, while ⟨ Bz ⟩ clearly changes sign. According to Vennes et al. (2018) it has a photometric variation period of 95 min.",
"pages": [
17
]
},
{
"title": "Appendix B.9: WD 0236-269 = PHL 4227 (n.v.: - only 2 meas.)",
"content": "Schmidt et al. (2001) report two spectropolarimetric detections of circular polarisation at about the 1.2% level. The two observations were obtained three days apart, with no changes detected. Furthermore no significant change in the spectrum-average circular polarisation was observed. We tentatively assume that is not a magnetic variable, but the evidence for non-variability is not compelling (the original paper just reports: \"consistent results were obtained on the two occasions\").",
"pages": [
17
]
},
{
"title": "Appendix B.10: WD 0253+508 = KPD 0253+5052 (variable, P ≃ 4 h)",
"content": "The field was discovered by Downes & Margon (1983), who estimated ⟨| B |⟩ ≈ 13 MG, and modelled by Achilleos & Wickramasinghe (1989) using only flux spectra. The star shows continuum polarisation up to 0.4% that is periodically variable according to Schmidt & Norsworthy (1991) with P = 0 . 170 d. The sign of the broadband continuum circular polarisation briefly reverses during each cycle.",
"pages": [
18
]
},
{
"title": "Appendix B.11: WD 0301+059 = SDSS J030350.63+060748.9 (n.v.: - only 2 meas.)",
"content": "The star was discovered to be magnetic by Landstreet & Bagnulo (2020), who did not find any obvious sign of variability between two observations obtained 1 day apart, nor between individual exposures of the same observing series (exposure times of single frames were 225 s). We tentatively assume that it is not magnetically variable. The star therefore appears as a strongly magnetic, massive, non-variable star.",
"pages": [
18
]
},
{
"title": "Appendix B.12: WD 0313-084 = GALEX J031613.8-081637 (non-variable)",
"content": "The star was discovered to be magnetic by O'Brien et al. (2023). It was observed five times with FORS2 (see Sect. 2.1), and no variability was found within timescales of 1 h, 1 d, and 8 months (see Fig. B.3). The spectrum indicates that ⟨| B |⟩ ≃ 800 kG.",
"pages": [
18
]
},
{
"title": "Appendix B.13: WD 0316-849 = EUVE J0317-855 (variable, P ≃ 0 . 2 h)",
"content": "Studied by Ferrario et al. (1997) Burleigh et al. (1999), and Vennes et al. (2003), the star has a rotation period P = 725 s. A detailed model based on UV spectra was obtained by Burleigh et al. (1999), who found that the field varies in strength from about 180 to 800 MG over the surface. Vennes et al. (2003) shown that the star has a light curve with a minimum that corresponds to the maximum of the polarisation. This is one of the very few white dwarfs that show magnetic and photometric variability, for which we have a firmly established phase relationship between photometry and circular polarisation. This star is erroneously missing in the list of such stars compiled by Bagnulo et al. (2024b).",
"pages": [
18
]
},
{
"title": "Appendix B.14: WD 0322-019 = EGGR 566 (variable)",
"content": "The star was discussed by Bagnulo & Landstreet (2021). Two FORS2 measurements by Farihi et al. (2018) obtained during two consecutive nights ( ⟨ Bz ⟩ = -5 . 4 ± 3 . 0 kG and -16 . 5 ± 2 . 3 kG) suggest that the longitudinal field may change on a timescale of a few days.",
"pages": [
18
]
},
{
"title": "Appendix B.15: WD 0330-000 = HE 0330-0002 (only 2 meas., and contradicting data)",
"content": "Schmidt et al. (2001) report two spectropolarimetric observations obtained one day apart, and concluded that no significant di ff erences had been detected in the overall spectrum. However, the mean circular polarisation averaged over the observed wavelength range reported in their Table 1 di ff ers by 0.5%, contra- dicting the assessment of non-variability. Because of this, this star will not be considered in this paper.",
"pages": [
18
]
},
{
"title": "Appendix B.16: WD 0410-114 = G 160-51 = NLTT 12758 (variable, P ≃ 0 . 3 h)",
"content": "A magnetic field of about 1.7 MG was discovered by Kawka & Vennes (2012). Kawka et al. (2017) found that circular polarisation in the H α sigma components varies with P = 22 . 6 min. This is another example of a star for which the phase relationship between light curve and magnetic field are known (Kawka et al. 2017) and it is also missing from the list compiled by Bagnulo et al. (2024b).",
"pages": [
18
]
},
{
"title": "Appendix B.17: WD 0446-789 = WG 47 (var.:)",
"content": "Bagnulo & Landstreet (2018) show that this is a weak-field star with ⟨ Bz ⟩ ≃ -5 kG; it probably has a dipolar field with axis nearly parallel to the rotation axis, because only very mild variations in ⟨ Bz ⟩ occur on a timescale of days.",
"pages": [
18
]
},
{
"title": "Appendix B.18: WD 0548-001 = EGGR 248 = G 99-37 (non-variable)",
"content": "The magnetic field was discovered from continuum circular polarisation by Landstreet & Angel (1971). Observations of broadband CP on six nights showed no variations. During one hour, broad-band data were printed every 6 s. These data were Fourier analysed without finding any evidence of short-period variability. Angel & Landstreet (1974) obtained an MCSP circular polarisation spectrum in 1972, and the star was re-observed by Vornanen et al. (2010), who remarked that similarity with the previous spectra, reporting also \"no unambiguous variations\" between several polarisation spectra taken in 2003, 2005, 2008. Using FORS1 data obtained in 2005, they also reported no detection of linear polarisation. We retrieved from the ESO FORS1 and FORS2 archives all circular polarisation spectra obtained in 2005, 2008 and 2012. Figure B.4 shows all these spectra compared to the polarisation spectrum of Angel & Landstreet (1974) . We note that there are three spectra obtained in three consecutive nights in November 2012. The spectrum obtained during the last night (2012-1119) shows zero circular polarisation (see green solid line), but a Stokes I flux consistent with that of previous observations. We ascribe this to a non-detected instrument failure (perhaps the retarder waveplate did not move) rather than to stellar variability. Observations obtained by Berdyugin et al. (2023) and Berdyugin et al. (2024) show no variability in broadband polarimetry between two measurements obtained 8 months apart (see the top panel of Fig. B.1). This is a nearly unique DQp white dwarf which shows not only the Swan bands in the flux spectrum, but also the CH G band. The magnetic field was modelled by Angel & Landstreet (1974) who deduced ⟨ Bz ⟩ ≈ 3 . 6 MG from the strong polarisation signature of the CH G-band. Berdyugina et al. (2007) and Vornanen et al. (2010) modelled FORS1 spectra and found ⟨ Bz ⟩ ≃ 2 . 5 MG.",
"pages": [
19
]
},
{
"title": "Appendix B.19: WD 0553+053 = EGGR 290 = G 99-47 (non-variable)",
"content": "The magnetic field of this star was discovered by Angel & Landstreet (1972), who also made 14 separate BBCP measurements during 10 months, without detecting any significant variation. Low resolution MCSP spectropolarimetry was obtained by Liebert et al. (1975), and 20 years later, a higher resolution polarised spectrum was obtained also by Putney & Jordan (1995). No significant variation has been detected on any timescale up to decades (for details, see Bagnulo & Landstreet 2021). Brinkworth et al. (2013) observed a light curve with 26.8 m period and semi-amplitude = 0.3 %, which suggest that polarimetric observations could have missed short-term variability. However, Angel & Landstreet (1972) had carried out Fourier-analysis of a polarimetric run, and found no significant variability for any period between 11 seconds and 2 h. They claimed that periodic variation with an amplitude of 0.17 % would certainly have been detected. Our conclusion is therefore that the star is non-variable.",
"pages": [
19
]
},
{
"title": "Appendix B.20: WD 0637+478 = GD 77 (variable, P ≃ 1 . 4 d)",
"content": "Magnetic variability was reported by Schmidt & Smith (1995). Eight unpublished ISIS polarisation spectra and one ESPaDOnS spectrum show that ⟨ Bz ⟩ varies with P ≃ 1 . 362 d. The value of ⟨| B |⟩ ≈ 1 . 0 MG hardly changes with rotation, but ⟨ Bz ⟩ varies sinusoidally between about + 400 and -250 kG (LB25).",
"pages": [
19
]
},
{
"title": "Appendix B.21: WD 0654+059 = 2MASS J06572938+0550479 (non-variable)",
"content": "Three BBCP observations obtained 8 months apart by Berdyugin et al. (2024) show no variability of the polarisation in the three B ' V ' R ' filters (see Fig. B.1). In all three filters, all three observations have V / I ≃ -0 . 3 %. We assume that the star is not variable.",
"pages": [
19
]
},
{
"title": "Appendix B.22: WD 0708-670 = SCR J0708-6706 (n.v.: only 2 meas.)",
"content": "Two spectra published by Bagnulo & Landstreet (2020) show no variability over a 2 month interval. The strength and wavelength dependence of the circular polarisation spectrum suggests that the underlying field could be of the order of 60 to 200 MG. We tentatively assume that the star is magnetically non-variable.",
"pages": [
19
]
},
{
"title": "Appendix B.23: WD0745 + 115 = GALEX J074842.4+112502 (non-variable)",
"content": "Strong broadband circular polarisation was detected by Berdyugin et al. (2024). The polarisation changes sign between the B ' and the other bands, and its absolute value is about 1% in all bands. Observed four times, the star does not show variability (see the top right panel of Fig. B.1).",
"pages": [
19
]
},
{
"title": "Appendix B.24: WD 0756+437 = EGGR 428 (variable, P phot ≃ 6 . 5 h )",
"content": "The star was discovered to be magnetic and discussed in detail by Putney (1995). She estimated its magnetic field to be ≃ 200 MG. Recent BBCP observations at NOT (Berdyugin et al. 2024) showed that it is rapidly variable. The first two observations, about 3 h apart, report the largest and smallest polarisation seen in the full data set of five observations; this suggests a rotation period of the order of 6 h. In fact, the photometric study of Brinkworth et al. (2013) found a unique, very large amplitude ( ± 4 %) light variation with a period of P = 6 . 68 h. The similarity of this period with the period range deduced from polarisation measurements strongly confirms that 6.68 h is the rotation period of this white dwarf. A further remarkable fact is the combination of high field (200 MG), high mass (1 . 04 M ⊙ ) with advanced age (4.45 Gyr): this star seems to be a unique example of a very old, still strongly magnetic and rapidly rotating WD-WD merger.",
"pages": [
20
]
},
{
"title": "Appendix B.25: WD 0810-353 = UPM J0812-3529 (non-variable)",
"content": "No variability detected among six polarised spectra obtained over a four year period, as described in a detailed study by Landstreet et al. (2023). According to their modelling, the star shows two regions of di ff erent field strength: one with magnetic field of predominantly 30 MG strength, outward, and one showing a field strength of 45 MG, inward.",
"pages": [
20
]
},
{
"title": "Appendix B.26: WD 0816-310 = SCR J0818-3110 (variable, P ≃ 10 d)",
"content": "This is a DZ white dwarf with strong flux, spectrum and ⟨ Bz ⟩ variations observed in five FORS polarised spectra in 2023 obtained with grism 1200B. It is clear that the surface abundances vary over the surface, and appear to be locked to the surface magnetic field. The rotational period is of the order of 10 d, and ⟨| B |⟩ ≃ 100 kG (Bagnulo et al. 2024a).",
"pages": [
20
]
},
{
"title": "Appendix B.27: WD 0850+192 = LB 8915 (variable, P ∼ hours?)",
"content": "This DBA white dwarf shows very weak, variable H lines, slightly variable He, and variable ⟨ Bz ⟩ (Wesemael et al. 2001). It is found that ⟨| B |⟩ ≃ 850 kG, and the rotation period is relatively short, probably some hours. We note that the correct identification of this star is LB 8915, and not LB 8827 as given in title of the paper by Wesemael et al. (2001). Appendix B.28: WD0907+213 = GALEX J091016.5+210555 (variable, P ≃ 10 h) Moss et al. (2024) discovered that this is a spectroscopically and magnetically variable DBA star with a rotation period of either 7.7 or 11.3 h (the ambiguity is due to aliasing). They have modelled the field and abundance geometry with a simple model like that used for Feige 7 = WD0041-102. The line splitting in the flux spectra suggests a field of ⟨| B |⟩ ≃ 0 . 5 MG.",
"pages": [
20
]
},
{
"title": "Appendix B.29: WD 0912+536 = EGGR 250 = G 195-19 (variable, P ≃ 1 . 3 d)",
"content": "This is the second magnetic white dwarf discovered (Angel & Landstreet 1971a), and the first magnetic white dwarf to be discovered to be rotationally variable (Angel & Landstreet 1971b). An improved ephemeris was provided by Angel et al. (1972a). The star shows a large variation of its circular polarisation with a period of 1.33 d (Angel et al. 1972b). Hernandez et al. (2024) measured the period of the photometric variability from TESS data as 1 . 3304 ± 0 . 0054 d (note that high accuracy of period relies on two widely separated TESS observational data sets). Six MCSP V / I spectra roughly uniformly distributed in phase were obtained by Landstreet & Angel, (unpublished). Between 4000 and 5000 Å, V / I reverses sign during rotation; redwards of this, the strong variations retain one sign. Appendix B.30: WD 1008-242 = UCAC4 328-061594 (n.v.: only 2 meas.) Observed twice with FORS2 in spectropolarimetric mode by Bagnulo & Landstreet (2022), this white dwarf did not show any variation between two spectra obtained 40 d apart, nor within the same observing series (see Fig. B.5). Probably the field is of order 100 MG or more. We consider that it is probably not variable. The star therefore appears a young, strongly magnetic, ultra-massive, non-variable star.",
"pages": [
20
]
},
{
"title": "Appendix B.31: WD 1008+290 = LP 315-42 = LHS 2229 (non-variable)",
"content": "This white dwarf is a cool peculiar DQ star discovered to be magnetic by Schmidt et al. (1999), who tentatively suggested that the field strength is of the order of 100 MG or more. Four observations obtained by Berdyugin et al. (2024) show little to no variability of broadband circular polarisation (see Fig. B.1). We assume that it is not magnetically variable.",
"pages": [
21
]
},
{
"title": "Appendix B.32: WD 1009-184 = WT 1759 (var.:)",
"content": "The star was discovered to be a magnetic white dwarf by Bagnulo & Landstreet (2019a), who published one measurement obtained with FORS2 and one obtained with ISIS, showing that the star has a weak and variable field ( ⟨ Bz ⟩ ≃ 50 kG). Additional unpublished data suggest also weak variability.",
"pages": [
21
]
},
{
"title": "Appendix B.33: WD 1015+014 = PG 1015+014 (variable, P = 98 . 75 m)",
"content": "This is a very strongly polarised white dwarf with V / I ≃ 1 . 5 %. The flux and polarisation spectra are strongly variable with P = 98 . 75 m and a polar field strength of about 120 MG (Angel 1978; Wickramasinghe & Cropper 1988). Schmidt & Norsworthy (1991) report that BBCP also varies with P = 98.75 min, approximately sinusoidally, with extrema of + 1 and -1 % polarisation. Brinkworth et al. (2013) found that the star's light curve has a period consistent, within uncertainties, with that obtained from polarimetry. Euchner et al. (2006) obtained a series of polarised spectra with FORS using the 300V grism. They describe strong I and V spectrum variations, and model the field structure using a multipole field expansion. This white dwarf displays spectral features originating from regions with typical field strengths between about 50 and 90 MG.",
"pages": [
21
]
},
{
"title": "Appendix B.34: WD 1031+234 = Ton 527 = PG 1031+234 (variable, P ≃ 3 . 5 h)",
"content": "Schmidt et al. (1986) report strongly variable intensity, circular and linear polarisation spectra with a 3 h 24 min period, and proposed a simple magnetic model with field strength in the range of 200 - 500 MG. Further broadband polarisation observations through the rotation period were obtained by Piirola & Reiz (1992), who measured also a light variation in anti-phase with circular polarisation (that is, the star appears darker when the absolute value of the polarisation is maximum). Brinkworth et al. (2013) found P phot ≃ 3 . 5 h.",
"pages": [
21
]
},
{
"title": "Appendix B.35: WD 1036-204 = LP 790-29 (non-variable)",
"content": "The star was discovered to be magnetic via spectropolarimetry by Liebert et al. (1978), and repeatedly studied (West 1989b; Schmidt et al. 1995, 1999; Beuermann & Reinsch 2002; Jordan & Friedrich 2002). Beuermann & Reinsch (2002) have monitored the star with EFOSC in spectropolarimetric mode to search for short-term variability, without finding any significant variations. Beuermann & Reinsch (2002), however, pointed out that Schmidt et al. (1995) measured a polarisation signal of ≃ -6 % around 6500 Å, a measurement at odds with broadband polarimetric measurements obtained in 1977, 1986, 1994 and 2000, which were all about -9 % (see their Table 1; we recall here that we use the opposite definition for the sign of circular polarisa- n). Schmidt et al. (1995) therefore suggested the possibility of a very long rotation period of the star. Jordan & Friedrich (2002) carried out a similar study, with similar results regarding short term variation. They also proposed a rotation period in the range of about 24 - 29 yr. We have reduced FORS1 and FORS2 archival polarisation spectra from 2003, 2011, 2012, and 2013, finding absolutely no hint of variability among them. Figure B.6 shows a comparison of archive observations obtained with grism 600B. The flux is not corrected for atmospheric and instrument transmission, and the discrepancies in the slope of the flux measured in 2003 can be explained by the use of a different CCD. A comparison between FORS spectra with those obtained by Schmidt et al. (1995) on May 7 and 8, 1994, (wavelength range 4160-7460 Å) does not show obvious long-term variability, except in the range 5700 -6200 Å. In that range, our data are instead consistent with most of the literature and point to a value of ≃ -9 %. In addition to the polarisation spectra, FORS1 and FORS2 archive contains another two broadband circular polarisation (BBCP) observations in the R filter: a FORS1 BBCP measurement obtained in April 2006, and one obtained with FORS2 in March 2024, using a similar (but not per-fec identical) R filter (program ID 112.25C9.001). We found both measurements consistent with a polarisation signal of about -9 . 4 %. These measurements rule out any significant change in the region around 6500 Å over an interval of 18 years. Finally, Berdyugin et al. (2024) have published the series of BBCP observations obtained in November 2022, November 2023 and February 2024, all consistent among themselves. In the R ' filter they report a polarisation signal of ≃ -9 . 1 %. For the reasons explained at the beginning of this section, it is not possible to accurately compare BBCP measurements obtained with di ff erent instruments, but it is clear that the only deviant point of a series of polarimetric observations obtained in the course of almost half a century is a small portion of a spectrum obtained in 1994. Remarkably, data obtained with the same instrument (FORS) over nearly two decades are fully consistent among themselves, and point strongly to a constant circular polarisation spectrum. Our conclusion is that the star is actually not variable. From the measured signal of circular polarisation of ≃ 10 % we estimate a longitudinal field of the order of 100 MG.",
"pages": [
21,
22
]
},
{
"title": "Appendix B.36: WD 1043-050 = HE 1043-0502 (n.v.: - only 2 meas.)",
"content": "This DBA star was discovered magnetic by Schmidt et al. (2001), who proposed that the field is ≃ 800 MG (although its continuum is not highly polarised). Two observations were taken a few days apart, and Schmidt et al. (2001) state that they did not show variability; the reported wavelength integrated circular polarisation of about 1.5% is also essentially unchanged between the two spectra. So we consider it as candidate nonvariable magnetic white dwarf.",
"pages": [
22
]
},
{
"title": "Appendix B.37: WD 1045-091 = HE 1045-0908 (variable, P ≃ 3 h)",
"content": "This white dwarf was discovered to be magnetic by Reimers et al. (1996) from a flux spectrum. Circular polarisation was confirmed, and shown to be variable by Schmidt et al. (2001). Euchner et al. (2005) obtained a series of I and V spectra and, assuming a rotational period of about 2.7 h, derived a detailed surface field model with a dominant field strength of 16 MG, but local field strength ranging between about 10 and 75 MG.",
"pages": [
22
]
},
{
"title": "Appendix B.38: WD 1105-340 = SCR J1107-3420A (non-variable)",
"content": "Eleven ESPaDOnS spectra taken between 2018 and 2022, with ten of them during 1 week in 2019 (see Table A.1), show that the star has a weak, non-variable field with ⟨ Bz ⟩ ≈ -22 ± 4 . 5 ˙ kG and ⟨| B |⟩ ≈ 125 ± 5 kG (see Fig. B.7). We also have two FORS spectra, one taken with 1200B and one with 1200R, with lower resolving power but higher signal-to-noise ratio (S / N). Owing to their di ff erence resolution, these spectra were not used in this work. This star is perhaps the best studied non-variable weakfield magnetic white dwarf.",
"pages": [
22
]
},
{
"title": "Appendix B.39: WD 1105-048 = EGGR 76 (ultra-weak field, var.:)",
"content": "The star was repeatedly observed, and a longitudinal field of the order of 1 kG was detected only in two measurements (Bagnulo & Landstreet 2018). The star seems to have a very weak and variable field, but additional measurements should be obtained to confirm the existence of a magnetic field.",
"pages": [
22
]
},
{
"title": "Appendix B.40: WD 1116-470 = SCR J1118-4721 (non-variable)",
"content": "This white dwarf was observed twice with FORS2 by Bagnulo & Landstreet (2021), who flagged it as suspected magnetic star. Both observations show a similar signal of circular polarisation at -0 . 2 %, close to the FORS2 instrumental detection limit. A third observations was obtained in January 2023, and the V / I spectrum is again consistent with that measured previously. The star is definitely magnetic, and most likely not variable. This confirmation (see Table A.1) brings the number of magnetic white dwarfs in the 20 pc volume to 34.",
"pages": [
22
]
},
{
"title": "Appendix B.41: WD 1211-171 = HE 1211-1707 (variable, P ≃ 2 h)",
"content": "A magnetic field was suspected in this white dwarf by Reimers et al. (1996), which was confirmed with polarimetry by Schmidt et al. (2001). Both papers show varying flux spectra. Schmidt et al. (2001) estimates P ≃ 100 -120 min and ⟨| B |⟩ ≃ 50 MG. Brinkworth et al. (2013) measured a photometric period of 1.79 h, consistent with the previous estimates from polarimetric data. The star is polarised at a level that varies between 0 and 3% during the rotation cycle. Modelling by Schmidt et al. (2001) strongly suggests a He dominated atmosphere with T e ff ≃ 12000 K, but Reimers et al. (1996), using IUE data, estimates 23000 K. Gentile Fusillo et al. (2021) gives 30000 K and 1 . 20 M ⊙ . We tentatively assume T e ff = 23000 K , M = 1 . 2 M ⊙ , and an He-rich atmosphere.",
"pages": [
22
]
},
{
"title": "Appendix B.42: WD 1217+475 = SDSS J121929.45+471522.8 (DAHe variable, P phot = 15 . 26 h)",
"content": "DAHe with 18.5 MG field strength and a photometric period of about 15.25 h (Gänsicke et al. 2020). Spectroscopy reveals Zeeman components of the Balmer lines varying in strength but with constant splitting. Published data do not demonstrate that the star is magnetically variable but cannot rule out this possibility either. Therefore we have decided not to include this star in our sample.",
"pages": [
23
]
},
{
"title": "Appendix B.43: WD 1249-022 = GALEX J125230.9-023417 (DAHe variable, P phot = 0 . 09 h)",
"content": "This is a DAHe white dwarf that shows variable H β and H α lines that appear sometimes in emission and sometimes in absorption. Field strength has been estimated 5 MG and rotational period = 0.09 h (Reding et al. 2020).",
"pages": [
23
]
},
{
"title": "Appendix B.44: WD 1312+098 = PG 1312+099 - (variable, P ≃ 5 . 4 h)",
"content": "Variable continuum circular polarisation was detected in this hot DAH by Schmidt & Norsworthy (1991), who present over 100 measures, and find P = 5 . 43 h. Circular polarisation varies approximately between + 1% and -1%.",
"pages": [
23
]
},
{
"title": "Appendix B.45: WD 1315-781 = LAWD 45 (n.v.: - only 2 meas.)",
"content": "No change between two ⟨| B |⟩ ≃ 5 . 5 MG and ⟨ Bz ⟩ ≃ 0 MG measurements from FORS 300V spectra taken five nights apart by Bagnulo & Landstreet (2020). We tentatively assume that it is magnetically non-variable.",
"pages": [
23
]
},
{
"title": "Appendix B.46: WD 1315+222 = LP 378-956 (non-variable)",
"content": "Two observations by Berdyugin et al. (2023) and one by Berdyugin et al. (2024) show no obvious sign of variability (see Fig B.1).",
"pages": [
23
]
},
{
"title": "Appendix B.47: WD 1328+307 = G 165-7 (variable)",
"content": "This star is found to host a magnetic field of ⟨| B |⟩ ≃ 650 kG based on line splitting observed in a good S / NSDSSspectrum (Dufour et al. 2006). These authors have also obtained low-resolution polarised spectra. They state that three 600 sec polarisation spectra were taken on 2005-12-30 at Steward Obs, all yielding essentially the same ⟨ Bz ⟩ ≈ 150 kG, but that the polarisation amplitude in similar Steward polarised spectra from 2006-05-03 (apparently not measured) is at least two times weaker than in 2005 spectrum. We conclude that the star is magnetically variable.",
"pages": [
23
]
},
{
"title": "Appendix B.48: WD 1346+121 = LP 498-66 (non-variable)",
"content": "Observed three times by Berdyugin et al. (2023) and Berdyugin et al. (2024), we assume the star, which shows circular polarisation ranging between -1 and 0 % in the three DIPol-UF bands, is magnetically non-variable (see Fig. B.1). The field strength of this white dwarf is probably in the tens of MG.",
"pages": [
23
]
},
{
"title": "Appendix B.49: WD 1350-090 = PG 1350-090 = GJ 3814 (non-variable)",
"content": "Discovered by Schmidt & Smith (1994), who measured ⟨ Bz ⟩ = 85 ± 9 kG. We have obtained four polarised spectra with ESPaDOnS that show ⟨| B |⟩ ≃ 450 -465 kG. There is no strong evidence of field variability, see also Schmidt & Smith (1994). Data will be published in a forthcoming paper (LB25).",
"pages": [
23
]
},
{
"title": "Appendix B.50: WD 1532+129 = G 137-24 (variable)",
"content": "Originally classified as DZ white dwarf by Kawka et al. (2004), the star was discovered to be a magnetic white dwarf by Bagnulo &Landstreet (2019a), who published two FORS2 measurements and one ISIS measurement. The star is variable, with ⟨ Bz ⟩ values from FORS2 measurements of -21 ± 1 kG and -4 ± 1 kG, while ⟨| B |⟩ is not strong enough to split spectral lines, leading to ⟨| B |⟩ < ∼ 300 kG.",
"pages": [
23
]
},
{
"title": "Appendix B.51: WD 1556+044 = PM J15589+0417 (non-variable)",
"content": "Discovered to be magnetic by Berdyugin et al. (2022), the star was re-observed three more times by Berdyugin et al. (2024). The observed circular polarisation, which is detected at the 10 σ level, ranges between -0 . 3 and + 0 . 4% in the three filter bands of DIPol-UF, so the order of magnitude of the field strength is probably some tens of MG. The polarisation does not show any variability (see Fig. B.1).",
"pages": [
23
]
},
{
"title": "Appendix B.52: WD 1615+542 = GALEX J161634.4+541011 (DAHe magnetically variable, P phot = 1 . 59 h)",
"content": "DAHe with a variable magnetic field (from 3.5 to 6.5 MG) and a rotation period P = 95 . 29 m estimate via photometry (Manser et al. 2023).",
"pages": [
24
]
},
{
"title": "Appendix B.53: WD 1619+046 = GALEX J162157.7+043219 (variable, P ≃ 40 m?)",
"content": "This white dwarf was discovered to be magnetic and rapidly variable in this work (see Sect. 2.1 and Fig. B.9). From the H β regions, it is clear that the star is a DAH with a rather non-uniform field of ⟨| B |⟩ ≃ 15 MG. We observe clear changes in intensity between spectra obtained a few minutes apart, and also polarisation (which is measured by the combination of two spectra) shows some variation, but without sign reversal. The close similarity between the first pair and the last pair of intensity spectra suggest that the star's rotation period is approximately 40 min.",
"pages": [
24
]
},
{
"title": "Appendix B.54: WD 1639+537 = GD356 (non-variable, but with a light curve and P phot = 1.93 h)",
"content": "This star is the prototype of DAHe white dwarfs. It shows a light curve with low amplitude and P phot ≃ 1 . 93 h (Walters et al. 2021). A time series of circular polarisation spectra obtained during an entire rotational cycle shows now variability while subtle sinusoidal variability is seen in the position of the σ components of H β and H α lines (Walters et al. 2021). Walters et al. (2021) compared spectropolarimetry obtained in 2019, with that published by Ferrario et al. (1997), finding no changes. Its field strength is of the order of 10 MG (Walters et al. 2021). We classify the star as non-variable, but showing very small changes of its apparent magnetic field with a ∼ 2 h rotation period.",
"pages": [
24
]
},
{
"title": "Appendix B.55: WD 1658+440 = PG 1658+441 (n.v.: - only 2 meas., one very old)",
"content": "The star was discovered to be magnetic by Liebert et al. (1983), who measured ⟨ Bz ⟩ ≃ 0 . 7 MG, and ⟨| B |⟩ ≃ 2 . 3 MG from spectropolarimetry of H α , H β and H γ (see their Fig. 5). Their observations were obtained on July 20 1980, with a spectral resolution of 10 Å. The same authors measured a signal of broadband circular polarisation = -0 . 016 ± 0 . 033% and -0 . 044 ± 0 . 019 % in the range 3300-8600 Å and concluded that a 3 σ upper limit of 0.10% semi-amplitude could be set for any presumed sinusoidal variation with a period between 0.5 and 5 h. For possible periods between 4 minutes and 0.5 hours, a less stringent limit of 0.30% polarisation semi-amplitude may be deduced. A comparison with our ISIS spectra obtained on April 21, 2019 (Sect. 2.1) is shown in the top panels of Fig. B.10. We observe a wavelength shift possibly due to bad calibration; most remarkable is the different strength of H α , but perhaps this is instrumental, because the ISIS spectrum was obtained with a setting that puts H α at the edge of the CCD, and no good flat-fielding correction could be applied. Although our comparison between 1980 and 2019 data is not conclusive, we certainly do not see any convincing change of the spectral features that demonstrate long-term variability. A comparison between the four pairs of Stokes V / I profiles obtained with ISIS in 2019 show no di ff erences within the error bars (see the bottom panels of Fig. B.10); therefore we rule out variability on a timescale of 10-15 minutes. This star seems to be a young, strongly magnetic, massive, tentatively classified as non-variable star. Photometric studies lead to inconsistent results: Brinkworth et al. (2013) found that the star is photometrically variable with a period between 6 h and 4 d, while, using TESS photometry, Oliveira da Rosa et al. (2024) derived a period shorter than 1 h. Hernandez et al. (2024), instead, did not detect periodicity in TESS data.",
"pages": [
25
]
},
{
"title": "Appendix B.56: WD 1703-266 = UCAC4 317-104829 (variable)",
"content": "This DA white dwarf was discovered to have a magnetic field by Bagnulo & Landstreet (2020), with ⟨| B |⟩ ≈ 8 MG. They also found that two FORS2 polarised spectra four days apart are significantly di ff erent and yield di ff erent values of ⟨ Bz ⟩ , so the star is variable on a timescale of some days or less.",
"pages": [
25
]
},
{
"title": "Appendix B.57: WD 1712-590 = Gaia DR3 5915797694789556096 (variable)",
"content": "Discovered to be magnetic by O'Brien et al. (2023). We observed this white dwarf in polarimetric mode with grism 1200B three times, twice during the same night. Intensity spectra show very similar splitting of Balmer lines, but the flux distribution within the line cores changes quite significantly, rather similarly to WD2359-434 (Landstreet et al. 2017). The deduced field strength is ⟨| B |⟩ < ∼ 0 . 8 MG. The Stokes V spectra also show that the star is strongly variable within 1-2 h (see Fig. B.11), and that the longitudinal field reverses its polarity during rotation.",
"pages": [
25
]
},
{
"title": "Appendix B.58: WD 1743-521 = L 270-31 = BPM25114 (variable, P = 2 . 84 d)",
"content": "Wickramasinghe & Bessell (1976) reported a magnetic field of about 35 MG in this southern DA star from close examination of the peculiar flux spectrum. Wegner (1977) found vari- ability of light and of the flux and polarisation spectrum with P ≈ 2 . 84 d, and modelled the spectrum variations, finding results consistent with a magnetic dipole field. Field modelling was carried out in more detail by Martin & Wickramasinghe (1978), who deduced a dipole field strength of 36 MG, corresponding to ⟨| B |⟩ ≃ 18 MG.",
"pages": [
25
]
},
{
"title": "Appendix B.59: WD 1748+708 = EGGR 372 = G 240-72 (s.v.:)",
"content": "Intrinsic broadband circular and linear polarisation of this magnetic white dwarf were discovered by Angel et al. (1974), who reported no variations in repeated observations of broad-band circular polarisation over a month, and linear polarisation over two nights. The star was later observed in broadband circular and linear polarimetry by West (1989a), who generally found similar polarisation levels and position angles to earlier work, and in broadband linear polarimetry by Berdyugin & Piirola (1999). Berdyugin & Piirola (1999) found evidence of clear change of the position angle of linear polarisation (mainly rotation of the polarisation angle) on a timescale of 20 years. We have observed the star with ISIS both in circular (twice) and in linear polarisation, and compared these observations with MCSP lowresolution spectra of I , V and P obtained by Angel & Landstreet on September 6 and 7, 1974, that were never published until now. Our ISIS spectra are consistent among themselves. When compared with spectropolarimetry obtained in 1974 and the 1990s, our new ISIS linear polarisation data find small changes in the V and P polarisation amplitude in the blue and visual, whereas the overall position angle behaviour is almost identical to that observed observed in 1974, thus we cannot confirm the changes in the polarisation position angle seen in 1997 by Berdyugin & Piirola (1999). On the other side, Antonyuk et al. (2016) found that the star is photometrically variable with a period between 5 hours and two days. Brinkworth et al. (2013) detected no short period variability, but claim that photometry varied over a ten month interval. We conclude that the star show some signs of subtle long term variability that should be further investigated. Appendix B.60: WD1750 -311 = UCAC4 295-140552 = [MTR2015] OW J175358.85-310728.9 (n.v.: - but P phot ≃ 0 . 5 h) Amagnetic field of ⟨| B |⟩ ≈ 2 . 1 MG was identified by Macfarlane et al. (2017) in this hot DQ white dwarf, which has strong lines of neutral C as well as fairly strong Balmer lines. They observed clear light variability of the star with an amplitude of about ± 2 % and a period of 35 min. These authors discuss the origin of the light variations: they argue that the variability is not due to pulsation, as no subsidiary frequencies are observed, and conclude that the variation could be due to rotation. They do not seem to have considered testing this with the spectra that they have collected of the object by looking for spectral variations, although they do test the spectra for radial velocity variations, and find none. Our four FORS2 300V spectra, taken with 13 m spacing through the light variation cycle, show virtually no changes in either flux or polarisation except for H β , which has a strongly variable I profile depth but almost constant V / I ( λ ) (see Fig. B.13). Possibly this is due to a magnetic field distribution that is nearly axisymmetric about the rotation axis but a distribution of H that varies strongly around that axis. This might be related to the light variability that has been detected. We have decided to classify the star as candidate magnetically non-variable.",
"pages": [
25,
26
]
},
{
"title": "Appendix B.61: WD1754 -550 = GALEX J175845.9-550117 (variable)",
"content": "This star was discovered to be magnetic in this work (Sect. 2.1). There is a strong signal of circular polarisation that seems variable on a short timescale between 0 and 4%, suggesting a rotation period of the order of 15 min: see Fig B.14. The inferred field strength is presumably of the order of 100 MG or more. The intensity spectrum appears featureless, so the T e ff ≃ 35 000 K star could be defined a hot DC.",
"pages": [
26
]
},
{
"title": "Appendix B.62: WD 1814+248 = G 183-35 (variable)",
"content": "Putney (1997) showed that this white dwarf, formerly classified as a DC, is in fact a cool DAH that shows weak H α and H β , and that both lines are split by a field of about 6.8 MG, roughly the same in two observations. Kilic et al. (2019) observed that the white dwarf shows spectral variations. A series of spectra taken over several hours show that the separation of the σ components of H α from the central π component changes rather abruptly between about 90 Å and about 120 Å, equivalent to fairly sudden jumps between ⟨| B |⟩ values of about 4.6 and 6 MG, repeating with a period of about 4 h. The authors suggest that this unusual form of variability may be due to a patchy distribution of H over an He-rich envelope. A plausible model that might explain the observations could be a dipole oblique to the rotation axis, decentred in the direction of one pole so that the polar strengths at the two magnetic poles are unequal, with H-rich patches around both poles, but little or no H in a belt around the magnetic equator.",
"pages": [
26,
27
]
},
{
"title": "Appendix B.63: WD 1829+547 = G 227-35 (non-variable)",
"content": "This white dwarf was discovered to be strongly magnetic by Angel et al. (1975) using both broadband polarimetry and lowresolution spectropolarimetry (on September 9, 1974). Limited tests of broadband variability were negative (Angel et al. 1975, 1981). It was observed again in spectropolarimetric mode by Cohen et al. (1993) and by Putney & Jordan (1995) who estimated a dipolar field strength of 170-180 MG. We observed the same star with ISIS twice in circular polarisation and once in linear polarisation. There are strong circular polarisation features at 6960, 7450 and 7930 Å. Figure B.15 shows a comparison between all these spectra. We do not see any obvious sign of variation.",
"pages": [
27
]
},
{
"title": "Appendix B.64: WD 1900+705 = LAWD 73 = Grw + 70 · 8247 (s.v.:)",
"content": "The star has a magnetic field of the order of 180 MG (Jordan 2003), and shows hints of small variability. We refer to Bagnulo & Landstreet (2019b), who present a review and analysis of its polarimetric characteristics, and in particular, a comparison with observations obtained 50 years apart show some discrepancies in both linear and circular polarisation. More recent broadband circular polarisation obtained at NOT (Berdyugin et al. 2022, 2023) show no variability on a timescale of 2 years (see Fig. B.1).",
"pages": [
27
]
},
{
"title": "Appendix B.65: WD 1953-011 = GJ 772 (variable, P ≃ 1 . 5 d)",
"content": "Maxted et al. (2000) modelled a series of Stokes I spectra in terms of a global dipolar field with polar field strength of order 100 kG, together with a spot with a field of order 500 kG. This basic modelling was confirmed by the analysis of a series of polarised spectra obtained with FORS1 and on the Russian 6 m telescope, which also revealed a rotation period of 1.448 d (Valyavin et al. 2008).",
"pages": [
27
]
},
{
"title": "Appendix B.66: WD 2010+310 = GD 229 (s.v.:)",
"content": "This star was the fifth circularly polarised white dwarf to be discovered. Swedlund et al. (1974) reported a series of measurements indicating the presence of elevated levels of both circular polarisation and linear polarisation, and strongly suggesting polarisation variability. The claim of variability was then questioned by Kemp et al. (1974), who argued that the initial measurements had been badly contaminated by polarised foreground zodiacal light contamination. Linear and / or circular polarisation of GD 229 has been measured by Landstreet & Angel (1974), Efimov (1981), Angel et al. (1981), West (1989a), Berdyugin (1995), and Berdyugin & Piirola (1999). Angel & Landstreet (1974) obtained three low-resolution I and V spectra of the star on the nights of 1973 Nov 6-8 using the MCSP. Synthetic broad-band polarimetry derived from these low-resolution spectra revealed no variations over three nights. West (1989b) observed linear polarisation in GD 229 again in 1986 using broadband filters, but found no strong evidence of variation, not even of the position angle of linear polarisation. Observations of GD 229 using broadband polarimetry was also carried out for linear polarisation by Berdyugin (1995) and for both circular and linear polarisation, with higher precision, by Berdyugin & Piirola (1999), who compared their results to earlier work. Their results appear to show some quite significant changes in both circular and linear polarimetry compared to earlier observations by other groups. A major di ffi culty of comparing the various measurements is that each group has been made with di ff erent instrumentation. The consequences of using di ff erent filter passbands, di ff erent detector sensitivities as functions of wavelength, and even di ff erent (and possibly incorrect) calibration of instrumental polarimetric e ffi ciency, make it very hard to compare these data. However, Berdyugin & Piirola (1999) do o ff er strong evidence for significant rotation of the angle of linear polarisation, by about 30 · over 20 years, which is di ffi cult to explain by instrumental e ff ects. This is probably the most robust e ff ect that emerges from comparison of the many kinds of polarimetric observations Linear and circular polarisation spectra of GD 229 were obtained in 2018 and 2019 using the ISIS spectropolarimeter on the William Herschel Telescope (Sect. 2.1). These new data are shown in Fig. B.16, and compared to the circular polarised spectra by Angel & Landstreet (1974), and to previously unpublished linear spectropolarimetry of Angel and Landstreet (see Table A.1). Remarkably, the angle of linear polarisation appears to have returned to its value during the 1970s. Other di ff erences compared to the spectra of Angel and Landstreet are observed, but some of these are undoubtedly due to greatly di ff erent resolving power, and some may be due to uncertainties in the calibration of the 1970s data, particularly in the UV. It is thus di ffi cult to establish clearly how much variability has occurred in GD 229. There is evidence for variations, but these appear to have relatively small amplitude compared to the overall scale of polarisation, except possibly for position angle rotation of linear polarisation (which however has not been confirmed by our most recent linear spectropolarimetry). Furthermore, because of very limited sampling with a variety of instruments, it is not possible to establish clearly any particular timescale for variations. We note that the spectrum of GD 229 has been modelled as due to He in a field of hundreds of MG by Jordan et al. (1998, 2001) and Jordan (2003).",
"pages": [
28
]
},
{
"title": "Appendix B.67: WD 2047+372 = EGGR 261 (variable, P ≃ 6 h)",
"content": "Originally, the star was observed polarimetrically by Schmidt & Smith (1995), who did not detect its weak and sign reversing field (their measurements were ⟨ Bz ⟩ = -42 ± 59 kG and -2 . 5 ± 4 . 6 kG). This star was discovered to be magnetic by Landstreet et al. (2016), and monitored and modelled by Landstreet et al. (2017). It is currently the weakest white dwarf field ( ⟨| B |⟩ ≈ 60 kG) that has been modelled in detail, mainly on the basis of a series of 18 ESPaDOnS spectra. The rotation period, determined from the variation of ⟨ Bz ⟩ , is 0.243 d. The observed variations of ⟨ Bz ⟩ and ⟨| B |⟩ are well modelled using a simple dipole model. No rotational variation is detected in TESS photometry (Hernandez et al. 2024).",
"pages": [
28
]
},
{
"title": "Appendix B.68: WD 2049-222 = LP 872-48 (var.:)",
"content": "Discovered to be magnetic by Berdyugin et al. (2022) with BBCP measurements. The star has V / I ≃ + 0 . 1 %, and is one of the weakest polarisation levels securely detected. The inferred ⟨ Bz ⟩ field strength is only of the order of a few MG. Broad-band measurements were repeated in July 2022 (Berdyugin et al. 2024) to check for variability, which was not detected (see Fig. B.1). We also observed the white dwarf three times with FORS2 with grism 300V (Sect. 2.1). The data are barely above the threshold of instrumental polarisation ( ≃ 0 . 1%; see Siebenmorgen et al. 2014), and therefore it is hard to establish whether the hints of variability seen in the spectra are real or not (see Fig. B.17). However, star WD 1116-440 shows a similar level of polarisation, and constant over about 4 year (see Fig. B.8), suggesting that the tiny variability observed in WD 2049-222 may be real and not an instrumental artefact.",
"pages": [
28
]
},
{
"title": "Appendix B.69: WD 2049-253 = UCAC4 325-215293 (non-variable)",
"content": "Discovered to be magnetic by Bagnulo & Landstreet (2020), who observed continuum circular polarisation of order 0.4% and deduced a field strength of order 20 MG. This white dwarf was re-observed in broadband circular polarisation once by Berdyugin et al. (2022) and two more times by Berdyugin et al. (2024), but shows no sign of variability (see Fig. B.1).",
"pages": [
28
]
},
{
"title": "Appendix B.70: WD 2051-208 = BPS CS 22880-0134 (variable, P ≃ 1 . 5 h)",
"content": "The magnetic field of this white dwarf was discovered from the shape of the H α line by Koester et al. (2009). We have two series of five polarised spectra each, one using FORS grism 1200B and one with grism 1200R (LB25). These unpublished data clearly reveal very rapid rotation of this massive magnetic white dwarf. The stellar rotation period is 0.0594 d = 1.425 h. The value of ⟨ Bz ⟩ ranges approximately sinusoidally between about + 50 kG and -30 kG, while the corresponding values of ⟨| B |⟩ increase from about 200 kG to nearly 300 kG, in good agreement with the two values of <| B |> (220 and 290 kG) obtained by Koester et al. (2009) from UVES SPY spectra.",
"pages": [
29
]
},
{
"title": "Appendix B.71: WD 2105-820 = LAWD 83 (variable)",
"content": "This star was suspected to have a weak magnetic field by Koester et al. (1998), who observed that the core of H α was abnormally broad, but they could not decide whether this was due to rapid rotation of v sin i ≈ 65 km / s or to a magnetic field of about 43 kG. Five FORS1 polarised spectra of the star by Landstreet et al. (2012) detected an apparently nearly constant magnetic field of ⟨ Bz ⟩ ≈ 10 kG. Later, FORS2 polarised spectra by Bagnulo & Landstreet (2018) and Farihi et al. (2018) reveal that ⟨ Bz ⟩ sometimes decreases to ⟨ Bz ⟩ ≈ 4 kG, so the star is apparently variable (see Bagnulo & Landstreet 2021).",
"pages": [
29
]
},
{
"title": "Appendix B.72: WD 2138-332 = L 570-26 (variable, P ≃ 6 . 19 h)",
"content": "The DZA star was discovered to be magnetic, with a variable ⟨ Bz ⟩ of the order of 10 kG (Bagnulo & Landstreet 2019a) and a rotational period of P = 6 . 19 h (Hernandez et al. 2024; Farihi et al. 2024). The same period was found from the analysis of the equivalent width and ⟨ Bz ⟩ curves by Bagnulo et al. (2024b), who proposed a magnetic model with a dipolar field with ⟨| B |⟩ ≃ 50 kG. The star shows photometric and ⟨ Bz ⟩ curves with light minimum corresponding to ⟨ Bz ⟩ maximum.",
"pages": [
29
]
},
{
"title": "Appendix B.73: WD 2150+591 = UCAC4 747-070768 (variable, P ≃ 2 . 4 d)",
"content": "The star was discovered to be magnetic by Landstreet & Bagnulo (2019), who reported two ISIS measurements that showed clearly that the field is variable with a period of hours or days. We subsequently monitored the star with one ESPaDOnS observation and several more ISIS spectra, and confirmed variability. These observations and a model of the star's magnetic field will be presented in a forthcoming paper (LB25).",
"pages": [
29
]
},
{
"title": "Appendix B.74: WD 2211+372 = LP 287-35 (non-variable)",
"content": "This DC white dwarf was discovered to be magnetic by Berdyugin et al. (2023), who found circular polarisation in excess of 1% in the blue. It was re-observed by Berdyugin et al. (2024). It does not seem to be variable (see Fig. B.1). The field strength ⟨| B |⟩ is probably of the order of 60 MG or more.",
"pages": [
29
]
},
{
"title": "Appendix B.75: WD 2316+123 = KUV 23162+1220 (variable, P ≃ 18 d)",
"content": "Discovered magnetic by Sion et al. (1984). Schmidt & Norsworthy (1991) reported BBCP of amplitude up to nearly 1% that varies sinusoidally with P = 17.86 d. Both the flux spectrum and the linear and circular polarisation spectra have been modelled repeatedly (Liebert et al. 1985; Achilleos & Wickramasinghe 1989; Friedrich et al. 1993; Putney & Jordan 1995); all agree that the global field strength is of the order of 30 MG. This star has the longest rotational period firmly established for a white dwarf.",
"pages": [
29
]
},
{
"title": "Appendix B.76: WD2359 -434 = LAWD 96 (variable, P ≃ 2 . 7 h)",
"content": "WD2359 -434 was suggested to be a magnetic star by Koester et al. (1998) on the basis of the peculiar profile of the H α core, and ⟨ Bz ⟩ was found to be non-zero by Aznar Cuadrado et al. (2004). A series of polarised ESPaDoNS spectra revealed a rotation period of 0.1123 d (Landstreet et al. 2017). The star is also a photometric variable with the same period. The field structure has been modelled approximately, and found to be distinctly more complex than a simple dipole, with ⟨| B |⟩ varying approximately between 50 and 100 kG. This white dwarf o ff ers one of the clearest examples known of a field structure that is substantially more complex than a simple co-linear multipole expansion (Landstreet et al. 2017).",
"pages": [
29
]
}
]
2024arXiv241114911B
https://arxiv.org/pdf/2411.14911.pdf
<document>
<section_header_level_1><location><page_1><loc_9><loc_82><loc_91><loc_87></location>Radii, masses, and transit-timing variations of the three-planet system orbiting the naked-eye star TOI-396 ⋆</section_header_level_1>
<text><location><page_1><loc_6><loc_69><loc_94><loc_81></location>A. Bonfanti 1 , I. Amateis 2 , 1 , 3 , D. Gandolfi 2 , L. Borsato 4 , J. A. Egger 5 , P. E. Cubillos 1 , 6 , D. Armstrong 7 , 8 , I. C. Leão 9 , M. Fridlund 10 , 11 , B. L. Canto Martins 9 , 12 , S. G. Sousa 13 , J. R. De Medeiros 9 , L. Fossati 1 , V. Adibekyan 13 , A. Collier Cameron 14 , S. Grziwa 15 , K. W. F. Lam 16 , E. Go ff o 17 , L. D. Nielsen 18 , F. Rodler 19 , J. Alarcon 20 , J. Lillo-Box 21 , W. D. Cochran 22 , 23 , R. Luque 24 , S. Redfield 25 , N. C. Santos 13 , 26 , S. C. C. Barros 13 , 26 , D. Bayliss 7 , 8 , X. Dumusque 27 , M. A. F. Keniger 7 , 8 , J. Livingston 28 , 29 , 30 , F. Murgas 31 , 32 , G. Nowak 33 , A. Osborn 34 , H. P. Osborn 5 , 35 , E. Pallé 31 , 32 , C. M. Persson 11 , L. M. Serrano 2 , P. A. Strøm 7 , 8 , S. Udry 27 , and P. J. Wheatley 7 , 8</text>
<text><location><page_1><loc_36><loc_67><loc_64><loc_68></location>(A ffi liations can be found after the references)</text>
<section_header_level_1><location><page_1><loc_46><loc_63><loc_54><loc_64></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_6><loc_58><loc_94><loc_61></location>Context. TOI-396 is an F6 V bright naked-eye star ( V ≈ 6.4) orbited by three small ( Rp ≈ 2 R ⊕ ) transiting planets discovered thanks to space-based photometry from two TESS sectors. The orbital periods of the two innermost planets, namely TOI-396 b and c, are close to the 5:3 commensurability ( Pb ∼ 3.6 d and Pc ∼ 6.0 d), suggesting that the planets might be trapped in a mean motion resonance (MMR).</text>
<text><location><page_1><loc_6><loc_56><loc_94><loc_58></location>Aims. To measure the masses of the three planets, refine their radii, and investigate whether planets b and c are in MMR, we carried out HARPS radial velocity (RV) observations of TOI-396 and retrieved archival high-precision transit photometry from four TESS sectors.</text>
<text><location><page_1><loc_6><loc_51><loc_94><loc_55></location>Methods. We extracted the RVs via a skew-normal fit onto the HARPS cross-correlation functions and performed a Markov chain Monte Carlo joint analysis of the Doppler measurements and transit photometry while employing the breakpoint method to remove stellar activity from the RV time series. We also performed a transit timing variation (TTV) dynamical analysis of the system and simulated the temporal evolution of the TTV amplitudes of the three planets following an N-body numerical integration.</text>
<text><location><page_1><loc_6><loc_39><loc_94><loc_51></location>Results. Our analysis confirms that the three planets have similar sizes ( Rb = 2 . 004 + 0 . 045 -0 . 047 R ⊕ ; Rc = 1 . 979 + 0 . 054 -0 . 051 R ⊕ ; Rd = 2 . 001 + 0 . 063 -0 . 064 R ⊕ ) and is thus in agreement with previous findings. However, our measurements are ∼ 1.4 times more precise thanks to the use of two additional TESS sectors. For the first time, we have determined the RV masses for TOI-396 b and d, finding them to be Mb = 3 . 55 + 0 . 94 -0 . 96 M ⊕ and Md = 7 . 1 ± 1 . 6 M ⊕ , which implies bulk densities of ρ b = 2 . 44 + 0 . 69 -0 . 68 g cm -3 and ρ d = 4 . 9 + 1 . 2 -1 . 1 g cm -3 , respectively. Our results suggest a quite unusual system architecture, with the outermost planet being the densest. Based on a frequency analysis of the HARPS activity indicators and TESS light curves, we find the rotation period of the star to be P rot ,⋆ = 6 . 7 ± 1 . 3 d, in agreement with the value predicted from log R ' HK -based empirical relations. The Doppler reflex motion induced by TOI-396 c remains undetected in our RV time series, likely due to the proximity of the planet's orbital period to the star's rotation period. We also discovered that TOI-396 b and c display significant TTVs. While the TTV dynamical analysis returns a formally precise mass for TOI-396 c of Mc , dyn = 2 . 24 + 0 . 13 -0 . 67 M ⊕ , the result might not be accurate, owing to the poor sampling of the TTV phase. We also conclude that TOI-396 b and c are close to but out of the 5:3 MMR.</text>
<text><location><page_1><loc_6><loc_35><loc_94><loc_39></location>Conclusions. A TTV dynamical analysis of additional transit photometry evenly covering the TTV phase and super-period is likely the most e ff ective approach for precisely and accurately determining the mass of TOI-396 c. Our numerical simulation suggests TTV semi-amplitudes of up to 5 hours over a temporal baseline of ∼ 5.2 years, which should be duly taken into account when scheduling future observations of TOI-396.</text>
<text><location><page_1><loc_6><loc_32><loc_94><loc_34></location>Key words. planets and satellites: fundamental parameters - stars: fundamental parameters - techniques: photometric - techniques: radial velocities</text>
<section_header_level_1><location><page_1><loc_6><loc_28><loc_18><loc_29></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_6><loc_14><loc_49><loc_27></location>Multi-planet systems enable us to place significantly stronger constraints on formation and evolution mechanisms compared to single-planet systems (e.g. Lissauer et al. 2011; Fabrycky et al. 2014; Winn & Fabrycky 2015; Mishra et al. 2023). As a matter of fact, the temporal evolution of the gas content in the proto-planetary disc influences planet migration (e.g. Lin & Papaloizou 1979; Goldreich & Tremaine 1980; Tanaka et al. 2002; D'Angelo & Lubow 2008; Alexander & Armitage 2009), which shapes the orbital architecture of a planetary system. The latter is further expected to correlate with the planet composition (e.g.</text>
<text><location><page_1><loc_51><loc_24><loc_94><loc_29></location>Thiabaud et al. 2014, 2015; Walsh et al. 2015; Bergner et al. 2020; Li et al. 2020) that can be inferred once the physical parameters of the planets are known (e.g. Dorn et al. 2017; Otegi et al. 2020b; Leleu et al. 2021; Haldemann et al. 2024).</text>
<text><location><page_1><loc_51><loc_10><loc_94><loc_22></location>If planets are found in mean motion resonance (MMR; e.g. Lee & Peale 2002; Correia et al. 2018), systems can also shed light on the migration mechanisms during formation, as well as on the impact of tidal e ff ects occurring later on (e.g. Delisle et al. 2012; Izidoro et al. 2017). In addition, planets in, or close to, MMR likely exhibit transit timing variations (TTVs; see e.g. Agol et al. 2005; Agol & Fabrycky 2018; Leleu et al. 2021) that enable one to infer planetary masses without necessarily relying on radial velocity (RV) measurements, which are not always possible (e.g. Hatzes 2016).</text>
<text><location><page_2><loc_6><loc_80><loc_49><loc_93></location>Multi-planet systems also give insights into correlations between physical and orbital parameters of exoplanets. For example, Weiss et al. (2018) noticed that planets in adjacent orbits usually show similar radii, hence the 'peas in a pod' label to describe this scenario. Weiss et al. (2018) further noticed that the outermost planet is the largest in the majority of cases, which agrees with the observed bulk planet density ( ρ p ) trend highlighted by Mishra et al. (2023), where ρ p decreases with the distance from the stellar host as outer planets are expected to be richer in volatiles (thus bigger and less dense).</text>
<text><location><page_2><loc_6><loc_61><loc_49><loc_80></location>The object TOI-396 represents an interesting laboratory to test these theories as it is the brightest star known so far to host three transiting planets (Vanderburg et al. 2019) after ν 2 Lupi (Delrez et al. 2021). After analysing two sectors from the Transiting Exoplanet Survey Satellite ( TESS ; Ricker et al. 2015), Vanderburg et al. (2019) found that the three planets have radii of ∼ 2 R ⊕ and orbital periods of ∼ 3.6, ∼ 6.0, and ∼ 11.2 d, with TOI-396 c and b showing a period commensurability close to the 5:3 ratio. Following the notation introduced in Mishra et al. (2023), the three planets are 'similar' in terms of radii, and one may wonder whether this architecture class is kept also in the mass-period diagram. Mishra et al. (2023) found a positive and strong correlation of the coe ffi cients of similarity between radii and masses, though exceptions are possible (e.g. Weiss & Marcy 2014; Otegi et al. 2020a, 2022).</text>
<text><location><page_2><loc_6><loc_39><loc_49><loc_60></location>In this work, we complement the photometric analysis of new TESS light curves (LCs) with RV observations taken with the High Accuracy Radial Velocity Planet Searcher ( HARPS ; Mayor et al. 2003) spectrograph to refine the planet radii and constrain for the first time the planetary masses. Considering the possible 5:3 MMR between TOI-396 c and b, we also simulate the evolution in time of the TTV amplitudes. This paper is organised as follows: Section 2 presents the stellar properties, and Sect. 3 describes the photometric and RV data. After outlining the method to jointly analyse the TESS LCs and the HARPS RV time series in Sect. 4, we present the corresponding results in Sect. 5. We attempt to dynamically model TTV and RV data simultaneously and track the temporal evolution of the TTV signals in Sect. 6, and we study the planets' internal structure in Sect. 7 and explore the prospects for characterising the system with the James Webb Space Telescope (JWST; Gardner et al. 2006) in Sect. 8. Finally Sect. 9 gathers the conclusions.</text>
<section_header_level_1><location><page_2><loc_6><loc_35><loc_30><loc_36></location>2. Host star characterisation</section_header_level_1>
<text><location><page_2><loc_6><loc_26><loc_49><loc_34></location>TOI-396 is an F6 V (Gray et al. 2006) bright naked-eye star with an apparent visual magnitude of V ≈ 6.4 (Perryman et al. 1997). It is located ∼ 31.7 pc away and is visible in the constellation of Fornax in the southern hemisphere. It is member of a visual binary system and its companion HR 858 B is a faint M-dwarf ( G ∼ 16 mag), about 8.4 '' away from the main component.</text>
<text><location><page_2><loc_6><loc_12><loc_49><loc_26></location>We co-added 78 HARPS spectra (see Sect. 3.2 for further details) and then modelled it with Spectroscopy Made Easy 1 ( SME ; Piskunov & Valenti 2017) version 5.2.2. SME computes synthetic spectra from a grid of well established stellar atmosphere models and adjusts a chosen free parameter based on comparison with the observed spectrum. Here we used the stellar atmosphere grid A tlas 12 (Kurucz 2013) together with atomic line lists from the V ald database (Piskunov et al. 1995) in order to produce the synthetic spectra. We modelled one parameter at a time utilising spectral features sensitive to di ff erent photospheric parameters iterating until convergence of all free parameters. Throughout</text>
<text><location><page_2><loc_51><loc_85><loc_94><loc_93></location>the modelling, we held the macro- and micro-turbulent velocities, v mac and v mic, fixed to 6 km s -1 (Doyle et al. 2014) and 1.34 km s -1 (Bruntt et al. 2010), respectively. A description of the modelling procedure is detailed in Persson et al. (2018). Finally, we obtained T e ff = 6354 ± 70 K, [Fe / H] = 0 . 025 ± 0 . 050, log g = 4 . 30 ± 0 . 06, and v sin i ⋆ = 7 . 5 ± 0 . 2 kms -1 .</text>
<text><location><page_2><loc_51><loc_63><loc_94><loc_85></location>To double-check the derived spectroscopy parameters we performed an additional analysis employing ARES + MOOG (Sousa et al. 2021; Sousa 2014; Santos et al. 2013). In detail, we used the latest version of ARES 2 (Sousa et al. 2007, 2015) to consistently measure the equivalent widths (EW) for the list of iron lines presented in Sousa et al. (2008). Following a minimisation process, we then find the ionisation and excitation equilibria to converge for the best set of spectroscopic parameters. This process uses a grid of Kurucz model atmospheres (Kurucz 1993) and the radiative transfer code MOOG (Sneden 1973). We obtained T e ff = 6389 ± 67 K, [Fe / H] = -0 . 014 ± 0 . 045 dex, log g = 4 . 58 ± 0 . 11, and v mic = 1 . 54 ± 0 . 04 kms -1 . In this process we also derived a more accurate trigonometric surface gravity (log g trig = 4 . 34 ± 0 . 02) using recent GAIA data following the same procedure as described in Sousa et al. (2021). In the end ARES + MOOG provides consistent values when compared with the ones derived with SME .</text>
<text><location><page_2><loc_51><loc_53><loc_94><loc_63></location>Using the SME stellar atmospheric parameters, we determined the abundances of Mg and Si following the classical curve-of-growth analysis method described in Adibekyan et al. (2012, 2015). Similar to the stellar parameter determination, we used ARES to measure the EWs of the spectral lines of these elements and a grid of Kurucz model atmospheres (Kurucz 1993) along with the radiative transfer code MOOG to convert the EWs into abundances, assuming local thermodynamic equilibrium.</text>
<text><location><page_2><loc_51><loc_28><loc_94><loc_52></location>The stellar radius R ⋆ , mass M ⋆ , and age t ⋆ were derived homogeneously using the isochrone placement algorithm (Bonfanti et al. 2015, 2016) and its capability of interpolating a flexible set of input parameters within pre-computed grids of PARSEC 3 v1.2S (Marigo et al. 2017) isochrones and tracks. For each magnitude listed in Table 1, we performed an isochrone placement run by inserting the spectroscopic parameters derived above, the Gaia parallax π (Gaia Collaboration et al. 2023, o ff set-corrected following Lindegren et al. 2021), and the magnitude value to obtain an estimate for the stellar radius, mass, and age along with their uncertainties. From these results, we built the corresponding Gaussian probability density functions (PDFs) and then we merged (i.e. we summed) the PDFs derived from the di ff erent runs to obtain robust estimates for R ⋆ , M ⋆ , and t ⋆ . The radius R ⋆ derives essentially from T e ff , π , and the stellar magnitude, while M ⋆ and t ⋆ are much more model-dependent; therefore we conservatively inflated their internal uncertainties by 4% and 20%, respectively, following Bonfanti et al. (2021). Our adopted stellar parameters are listed in Table 1.</text>
<section_header_level_1><location><page_2><loc_51><loc_24><loc_69><loc_25></location>3. Observational data</section_header_level_1>
<section_header_level_1><location><page_2><loc_51><loc_22><loc_67><loc_23></location>3.1. TESS photometry</section_header_level_1>
<text><location><page_2><loc_51><loc_15><loc_94><loc_21></location>As presented in Vanderburg et al. (2019), TOI-396 was photometrically monitored by TESS during the first year of its nominal mission in Sector 3 from 20 September to 17 October 2018 (UT) in CCD 2 of Camera 2, and in Sector 4 from 19 October to 14 November 2018 (UT) in CCD 2 of Camera 1. TOI-396 was later</text>
<figure>
<location><page_3><loc_19><loc_95><loc_21><loc_96></location>
</figure>
<text><location><page_3><loc_15><loc_92><loc_41><loc_93></location>Table 1: Stellar properties of TOI-396.</text>
<text><location><page_3><loc_7><loc_85><loc_15><loc_86></location>Star names</text>
<text><location><page_3><loc_20><loc_89><loc_26><loc_90></location>TOI-396</text>
<text><location><page_3><loc_20><loc_87><loc_31><loc_88></location>TIC 178155732</text>
<text><location><page_3><loc_20><loc_86><loc_26><loc_87></location>HR 858</text>
<text><location><page_3><loc_20><loc_84><loc_27><loc_85></location>HD 17926</text>
<text><location><page_3><loc_20><loc_82><loc_28><loc_83></location>HIP 13363</text>
<text><location><page_3><loc_20><loc_81><loc_43><loc_82></location>Gaia DR3 5064574724769475968</text>
<table>
<location><page_3><loc_6><loc_34><loc_49><loc_80></location>
</table>
<text><location><page_3><loc_6><loc_29><loc_49><loc_33></location>Notes. RA & DEC are reported as in J2000 reference frame. Values in the bottom half of the table have been derived as part of this paper. ( a ) Zero-point correction from Lindegren et al. (2021) applied.</text>
<text><location><page_3><loc_6><loc_14><loc_49><loc_21></location>re-observed by TESS during the first year of its extended mission in Sector 30 from 23 September to 20 October 2020 (UT) in CCD 2 of Camera 2, and in Sector 31 from 22 October to 16 November 2020 (UT) in CCD 2 of Camera 1. All data were collected at a 2-minute cadence, except for Sector 3 for which TESS only sent down data at 30-minute cadence.</text>
<text><location><page_3><loc_6><loc_10><loc_49><loc_13></location>We analyse all available TESS time series, including the Sector 3 and 4 data already presented in (Vanderburg et al. 2019). For Sector 3 we used the TESS Asteroseismic Science Opera-</text>
<text><location><page_3><loc_51><loc_70><loc_94><loc_93></location>tion Center (TASOC) photometry (Handberg et al. 2021; Lund et al. 2021), while for the other sectors we analysed the Presearch Data Conditioning Simple Aperture Photometry (PDCSAP) LCs generated by the TESS Science Processing Operation Center (SPOC) pipeline Jenkins et al. (2016), which removes the majority of instrumental artefacts and systematic trends (Smith et al. 2012; Stumpe et al. 2012, 2014). We rejected those data marked with a bad quality flag and we performed a 5 medianabsolute-deviation (MAD) clipping of flux data points. After that, we built our custom LCs by splitting the TESS sectors in temporal windows centred around the transit events keeping ∼ 4 h of out-of-transit data points both before and after the transit for de-trending purposes. We ended up with 41 TESS LCs reporting the epoch of observation ( t ), the flux and its error, and further ancillary vectors available from the TESS science data product, that is mom _ centr 1, mom _ centr 2 (hereafter denoted with x and y , respectively), pos _ corr 1, and pos _ corr 2 (hereafter denoted with pc1 and pc2 , respectively) 4 .</text>
<section_header_level_1><location><page_3><loc_51><loc_66><loc_80><loc_67></location>3.2. HARPS high-resolution spectroscopy</section_header_level_1>
<text><location><page_3><loc_51><loc_40><loc_94><loc_65></location>We performed the radial velocity (RV) follow-up of TOI-396 with the HARPS spectrograph mounted at the ESO-3.6 m telescope at La Silla Observatory (Chile). We acquired 77 high resolution ( R ≈ 115 000) spectra between 31 January and 27 July 2019 (UT), covering a baseline of ∼ 177 days, as part of the follow-up programs of TESS systems carried out with the HARPS spectrograph (IDs: 1102.C-0923, 1102.C-0249, 0102.C0584; PIs: Gandolfi, Armstrong, De Medeiros). One additional spectrum was acquired during a technical night (ID: 60.A-9700) in February 2019. Following Dumusque et al. (2011b), we averaged out p-mode stellar pulsations by setting the exposure time to 900 s, which led to a median signal-to-noise (S / N) ratio of ∼ 320 per pixel at 550 nm. We used the second fibre of the HARPS spectrograph to simultaneously observe a Fabry-Perot lamp and trace possible instrumental drift down to the sub-metre per second level (Wildi et al. 2010). We reduced the data using the dedicated HARPS data reduction software ( DRS ; Pepe et al. 2002; Lovis, C. & Pepe, F. 2007) and computed the crosscorrelation function (CCF) for each spectrum using a G2 numerical mask (Baranne et al. 1996).</text>
<text><location><page_3><loc_51><loc_24><loc_94><loc_39></location>We then performed a skew-normal (SN) fit on each CCF (Simola et al. 2019) in orderto extract the stellar radial velocity along with its error, the full width at half maximum (FWHMSN), the contrast ( A ), and the skewness parameter ( γ ) of the CCF. The advantages of using an SN-fit rather than a Normal fit are thoroughly discussed in Simola et al. (2019), while the SN-fitting details are outlined in, for example, Bonfanti et al. (2023); Luque et al. (2023); Fridlund et al. (2024). After the SN-based extraction, we ended up with an RV time series, whose ancillary vectors (FWHMSN, A , γ ) are the activity indicators used to constrain the polynomial basis to model and de-trend the RV component of the stellar activity (see Table A.1).</text>
<text><location><page_3><loc_51><loc_15><loc_94><loc_23></location>As stellar activity is not stationary, the correlations between the RV observations and the activity indicators are expected to change over time as discussed in Simola et al. (2022), who proposed to apply the breakpoint ( bp ) technique (Bai & Perron 2003) to check whether these correlation changes are statistically significant. If so, the bp algorithm finds the optimal locations of correlation changes, which defines the segmenta-</text>
<text><location><page_4><loc_6><loc_83><loc_49><loc_93></location>tion characterised by the lowest Bayesian Information Criterion (BIC; Schwarz 1978). Indeed, we found one breakpoint at observation 48 (BJD = 2458667.940719; ∆ BIC ≈ -14 with respect to the zero-breakpoint solution). Thus, we split our RV time series in two piecewise stationary segments and de-trended it on a chunk-wise base, rather than performing a global de-trending over the whole time series, similarly to what has already been done in Bonfanti et al. (2023); Luque et al. (2023).</text>
<section_header_level_1><location><page_4><loc_6><loc_79><loc_15><loc_80></location>4. Methods</section_header_level_1>
<text><location><page_4><loc_6><loc_43><loc_49><loc_78></location>We jointly analysed the TESS LCs and the RV time series within a Markov chain Monte Carlo (MCMC) framework using the MCMCI code (Bonfanti & Gillon 2020). When fitting the TESS LCs extracted from Sector 3 that has a cadence of 30 min, we generated the transit model using a cadence of 2 min (the same as the other TESS sectors) and then rebinned it to 30 min following Kipping (2010), who warns that long-cadence photometry may lead to retrieve erroneous system parameters. We imposed Normal priors on the input stellar parameters (i.e. T e ff , [Fe / H], R ⋆ , and M ⋆ ), as derived in Sect. 2 with a twofold aim: (i) the induced prior on the mean stellar density ρ⋆ (via M ⋆ and R ⋆ ) helps the convergence of the transit model; (ii) limb darkening (LD) coe ffi cients for the TESS filter may be retrieved following interpolation within A tlas 9-based 5 grids that were precomputed using the get_lds.py code 6 by Espinoza & Jordán (2015). We then set Normal priors on the quadratic LD coe ffi -cients using the values coming from the grid interpolation (i.e. u 1 , TESS = 0 . 2318 ± 0 . 0065 and u 2 , TESS = 0 . 3085 ± 0 . 0028) after re-parameterising them following Holman et al. (2006). On the planetary side, we adopted unbounded (except for the physical limits) uniform priors on the transit depth d F , the impact parameter b , the orbital period P , the transit timing T 0, and the RV semi-amplitude K , while we set the eccentricity e = 0 and the argument of periastron ω = 90 · for all planets. We come back to the assumption on the eccentricity below. The specific parameterisations of the jump parameters (aka step parameters) are outlined in Bonfanti & Gillon (2020).</text>
<text><location><page_4><loc_6><loc_21><loc_49><loc_43></location>The LCs and the RV time series were de-trended simultaneously during the MCMC analysis using polynomials. To assess the polynomial orders to be associated with the di ff erent de-trending parameters for each time series, we first launch several MCMC preliminary runs made of 10 000 steps where we varied only one polynomial order at a time. We then selected the best de-trending baseline (see Table A.2) as the one having the lowest BIC. After that, we performed a preliminary MCMCI run comprising 200 000 steps to evaluate the contribution of both the white and red noise in the LCs following Pont et al. (2006); Bonfanti & Gillon (2020), so to properly rescale the photometric errors and get reliable uncertainties on the output parameters. Finally, three independent MCMCI runs comprising 200 000 steps each (burn-in length equal to 20%) were performed to build the posterior distributions of the output parameters after checking their convergence via the Gelman-Rubin statistic ( ˆ R ; Gelman & Rubin 1992).</text>
<text><location><page_4><loc_6><loc_13><loc_49><loc_21></location>We also tested the possibility of eccentric orbits by imposing uniform priors on ( √ e cos ω , √ e cos ω ) either bounded to imply e ≲ 0 . 3 or completely unbounded (except for the physical limits). The wider the eccentricity range to be explored by the MCMC scheme, the poorer the parameter convergence, which suggests that the available data are not enough to con-</text>
<text><location><page_4><loc_51><loc_82><loc_94><loc_93></location>he planetary eccentricities well. Moreover, the MCMCI runs with e , 0 are disfavoured by the ∆ BIC criterion (e.g. Kass & Raftery 1995; Trotta 2007) as well, in fact we obtained ∆ BIC = BIC e , 0 -BIC e = 0 ≳ + 100. This is also in agreement with the simulations performed by Vanderburg et al. (2019), who suggested that the periods' commensurability state of planets b and c is more likely maintained if the system is characterised by low eccentricities. Therefore, we adopted the circular solution as the reference one.</text>
<section_header_level_1><location><page_4><loc_51><loc_78><loc_79><loc_79></location>5. LC and RV data analysis results</section_header_level_1>
<section_header_level_1><location><page_4><loc_51><loc_76><loc_89><loc_77></location>5.1. Joint LC and RV analysis with linear ephemerides</section_header_level_1>
<text><location><page_4><loc_51><loc_65><loc_94><loc_75></location>As mentioned in Sect. 4, we set P and T 0 as free parameters under the control of a uniform prior, which implies assuming linear ephemerides. With this setup, we improved the transit depth precision of all three planets by a factor ∼ 1.4 if compared with the results of Vanderburg et al. (2019). This improvement level is consistent with having twice the number of data points with respect to the LC analysis performed by Vanderburg et al. (2019), as well as with low TTV amplitudes (see Sect. 5.2).</text>
<text><location><page_4><loc_51><loc_54><loc_94><loc_65></location>By combining the SN-fit onto the HARPS CCFs along with the bp method, we were able to estimate the masses of TOI-396 b and TOI-396 d to Mb = 3 . 56 + 0 . 92 -0 . 94 M ⊕ and Md = 7 . 2 ± 1 . 6 M ⊕ (detections at the 3.8 and 4.5 σ -level, respectively). Instead, we did not detect any significant Keplerian signal at the orbital period of TOI-396 c within the RV time series. In detail, we obtained a median Kc = 0 . 28 + 0 . 29 -0 . 20 ms -1 (3 σ upper limit K up c = 1 . 2 ms -1 ), which implies Mc = 0 . 92 + 0 . 94 -0 63 M ⊕ ( M up c = 4 . 0 M ⊕ ).</text>
<text><location><page_4><loc_51><loc_25><loc_94><loc_54></location>. When combining the mass and radius values of the three planets, we obtain the following median estimates for the bulk planetary densities: ρ b = 2 . 56 + 0 . 71 -0 . 70 , ρ c = 0 . 67 + 0 . 69 -0 . 46 ( ρ up c = 3 . 1), and ρ d = 5 . 1 + 1 . 3 -1 . 2 g cm -3 . We note that the RV-undetected TOI396 c would be the least dense planet, while the densest planet is the outermost one (i.e. TOI-396 d), which constitutes a quite atypical architecture within the observed exoplanet population (e.g. Ciardi et al. 2013; Weiss et al. 2018; Mishra et al. 2023). However, this conclusion is just tentative, given the uncertainties on the mean planetary densities. Moreover, the detection level of the RV-inferred parameters of TOI-396 c is not statistically significant. We further tested whether including the MINERVAAustralis (Addison et al. 2019) RV data (30 measurements as taken from Vanderburg et al. 2019) can help detect the elusive planet. However, it turned out that their precision level ( ∼ 6 m s -1 ) is not high enough to improve the characterisation of the system. In other words, the RV semi-amplitudes we obtained are consistent and indistinguishable within the statistical fluctuation with what was derived from the more precise HARPS data set. This led us to further check that TOI-396 c indeed belongs to this system (Sect. 5.2) and to investigate the reason for its RV non-detection (Sect. 5.3).</text>
<section_header_level_1><location><page_4><loc_51><loc_22><loc_86><loc_23></location>5.2. Joint LC and RV analysis accounting for TTVs</section_header_level_1>
<text><location><page_4><loc_51><loc_14><loc_94><loc_21></location>As planet c is undetected in the RV time series, one may wonder whether TOI-396 c is a false positive. However, by using the VESPA tool (Morton 2012, 2015) that accounts for the constraints from the TESS LCs, imaging, and spectroscopy, Vanderburg et al. (2019) already computed that the false-positive probabilities (FPPs) are lower than 10 -3 for all three planets.</text>
<text><location><page_4><loc_51><loc_10><loc_94><loc_13></location>In line with Vanderburg et al. (2019), we also confirmed that the Pc / Pb period ratio is commensurable and di ff ers from the 5:3 ratio by less than 0.027%. As planets with orbits in, or close to,</text>
<figure>
<location><page_5><loc_19><loc_95><loc_21><loc_96></location>
<caption>Fig. 3 shows the three planets of the system (star symbol) along with other exoplanets whose density is more significant than the 3 σ level 7 (circular marker) in the mass-radius (MR) diagram. The superimposed theoretical MR models as taken from Aguichine et al. (2021, A21) and Haldemann et al. (2024) help guide the eye; however, we warn the reader of possible degeneracies occurring in the MR plane. A thorough internal structure analysis of the well characterised TOI-396 b and d is presented in Sect. 7.</caption>
</figure>
<text><location><page_5><loc_6><loc_76><loc_49><loc_93></location>resonances are likely to show TTVs, we decided to repeat the same MCMC analysis outlined above, but enabling the transit timings of each transit event to vary to then compute the TTV amplitude with respect to the linear ephemerides model derived in Sect. 5.1. All jump parameters converged ( ˆ R ≲ 1 . 01). The medians of the posterior distributions of the most relevant system parameters along with the 68.3% confidence intervals are listed in Table 2. The phase-folded LCs of the three planets are shown in Figure 1, while the phase-folded RV time series are displayed in Figure 2. In particular, the middle panel of Fig. 2 shows that, after subtracting the RV signals of both planets b and d, the RV time series looks flat consistently with the non-detection of TOI396 c.</text>
<text><location><page_5><loc_6><loc_68><loc_49><loc_76></location>This analysis gives Rb = 2 . 004 + 0 . 045 -0 . 047 R ⊕ , Rc = 1 . 979 + 0 . 054 -0 . 051 R ⊕ , and Rd = 2 . 001 + 0 . 063 -0 . 064 R ⊕ and confirms the RV detection of TOI396 b and TOI-396 d (at the 3.8 and 4.5 σ -level, respectively) with Mb = 3 . 55 + 0 . 94 -0 . 96 M ⊕ and Md = 7 . 1 ± 1 . 6 M ⊕ . These outcomes are consistent with the results obtained from the analysis that assumes linear ephemerides (Sect. 5.1).</text>
<text><location><page_5><loc_6><loc_46><loc_49><loc_56></location>We found significant TTV signals for planet b and c (see Figure 4, Top and Middle panels) as expected from the commensurability of their periods. The TTV statistical significance of TOI-396 b and c is evident even by eye when comparing the data points location with the shaded regions that represent the 1 σ uncertainties of the linear ephemerides. For each planet, we further quantified the reducedχ 2 ( ˆ χ 2 ) characterising the TTV amplitudes via</text>
<formula><location><page_5><loc_6><loc_41><loc_49><loc_44></location>ˆ χ 2 j = 1 N tr , j -2 N tr , j X i = 1 TTV j , i σ j , i ! 2 (1)</formula>
<text><location><page_5><loc_6><loc_34><loc_49><loc_39></location>where N tr , j is the number of transit event of the j -th planet. We obtained ˆ χ 2 b = 4 . 4, ˆ χ 2 c = 4 . 1, and ˆ χ 2 d = 0 . 7 for planets b, c, and d, respectively, which confirms the significance of the TTV amplitudes for TOI-396 b and c.</text>
<text><location><page_5><loc_6><loc_24><loc_49><loc_34></location>Furthermore, the TTV amplitudes of TOI-396 b and c exhibit a clear anti-correlation pattern that we highlight by superimposing the TTV measurements in Figure A.1. This is a typical signature of gravitational interaction between the two planets, which confirms that TOI-396 c belongs to the system despite its elusiveness in the RV time series. For each planet, the timing of each transit event and the corresponding TTV amplitude computed with respect to the linear ephemerides are listed in Table A.3.</text>
<section_header_level_1><location><page_5><loc_6><loc_19><loc_47><loc_22></location>5.3. Discussion regarding why TOI-396 c is not detected in the RV time series</section_header_level_1>
<text><location><page_5><loc_6><loc_14><loc_49><loc_18></location>Magnetic activity combined with stellar rotation induces RV variations that can hide, a ff ect, or even mimic planetary signals (e.g. Queloz et al. 2001; Hatzes et al. 2010; Dumusque et al.</text>
<figure>
<location><page_5><loc_52><loc_25><loc_92><loc_92></location>
<caption>Fig. 1: TESS detrended and phase-folded LCs (blue dots) of TOI396 b ( Top panel ), TOI-396 c ( Middle panel ), and TOI-396 d ( Bottom panel ) with the transit model superimposed in red. The black markers are the binned data points (binning 20 min).</caption>
</figure>
<text><location><page_5><loc_51><loc_10><loc_94><loc_16></location>2011a; Haywood et al. 2014; Suárez Mascareño et al. 2017; Gandolfi et al. 2017). A possible explanation for the non-detection of the Doppler reflex motion induced by TOI-396 c is that stellar activity destructively interferes with the Keplerian signal of the planet. This may happen if the star has a rotation period</text>
<table>
<location><page_6><loc_13><loc_52><loc_87><loc_91></location>
<caption>Table 2: System parameters as derived from the joint LC and RV analysis accounting for TTVs.</caption>
</table>
<text><location><page_6><loc_6><loc_49><loc_94><loc_51></location>Notes. All jump parameters, but the LD coe ffi cients, were subject to unbounded uniform priors following the parameterisations specified in Bonfanti & Gillon (2020); see text for further details.</text>
<text><location><page_6><loc_6><loc_46><loc_94><loc_49></location>( a ) Uncertainties from the run assuming linear ephemerides (no TTVs). T 0 values are shifted by -2 450 000. ( b ) Assuming zero albedo and full recirculation. ( c ) The 3 σ upper limits on the RV semi-amplitude, the planet mass, and the mean planet density of TOI-396 c are K up c = 1 . 2 ms -1 , M up c = 3 . 8 M ⊕ , and ρ up c = 2 . 9 g cm -3 , respectively.</text>
<text><location><page_6><loc_6><loc_37><loc_49><loc_43></location>comparable to the orbital period of the planet (e.g. Vanderburg et al. 2016). Disentangling the planetary signal from stellar activity and retrieving the Doppler motion induced by the orbiting planet would then be challenging (e.g. Dragomir et al. 2012; Kossakowski et al. 2022).</text>
<text><location><page_6><loc_6><loc_10><loc_49><loc_35></location>In order to investigate this hypothesis, we performed a frequency analysis of the line profile variation diagnostics (FWHM, contrast, and skewness) and activity indicators (log R ' HK and H α indexes). Figure 5 displays the time series (left column), along with the respective generalised Lomb-Scargle (GLS, Zechmeister & Kürster 2009) periodograms (right column). We assessed the significance of the peaks detected in the power spectra by estimating their false alarm probability (FAP), that is the probability that noise could produce a peak with power higher than the one we found in the time series. To account for the possible presence of non-Gaussian noise in the data, we estimated the FAP using the bootstrap randomisation method (see, e.g., Murdoch et al. 1993; Kuerster et al. 1997; Hatzes 2019). Briefly, we computed the GLS periodogram of 10 5 mock time series obtained by randomly shu ffl ing the data points along with their error bars, while keeping the timestamps fixed. We defined the FAP as the fraction of those mock periodograms whose highest power exceeds the power of the real data at any frequency. In the present work, we considered a peak to be significant if its false alarm probability is FAP < 0.1 %.</text>
<text><location><page_6><loc_51><loc_20><loc_94><loc_43></location>We found that the time series of the log R ' HK and H α indexes display long-term trends likely due to the magnetic cycles of the star (Fig. 5, left column, first and third panels). In the Fourier domain, these trends translate into a significant (FAP < 0.1 %) excess of power at frequencies lower than the spectral resolution 8 of our HARPS observations (Fig. 5, right column, first and third panels). We modelled these long-term signals as quadratic trends and subtracted the best-fitting parabolas from the respective time-series. The GLS periodograms of the residuals of the activity indicators display significant peaks between ∼ 6 and 8 d (i.e, between ∼ 0.0125 and 0.167 d -1 ; Fig. 5, yellow area). The peaks are equally spaced by ∼ 0.0068 d -1 , which coincides with a peak found in the periodogram of the window function. Given the current data at our disposal, we are not able to distinguish between true frequencies and aliases. Although not significant, the power spectra of the contrast and skewness show peaks in the same frequency range, suggesting that the rotation period of the star might be ∼ 6-8 d.</text>
<text><location><page_6><loc_51><loc_15><loc_94><loc_20></location>We note that the periodogram of the FWHM also displays an excess of power at low frequencies. However, the corresponding peak remains below our 0.1 %-FAP significance threshold (see Fig. 5, second to last panel). Yet, if we apply the same procedure</text>
<figure>
<location><page_7><loc_19><loc_95><loc_21><loc_96></location>
</figure>
<figure>
<location><page_7><loc_6><loc_25><loc_47><loc_93></location>
<caption>Fig. 2: HARPS detrended and phase-folded RV time series of TOI-396 b ( Top panel ), TOI-396 c ( Middle panel ), and TOI396 d ( Bottom panel ) with the Keplerian model superimposed in red. For each planet, the time series were obtained after subtracting the RV contribution of the other planets. The error bars also account for the jitter contribution (displayed in grey).</caption>
</figure>
<text><location><page_7><loc_6><loc_10><loc_49><loc_13></location>described above and remove this signal by fitting a quadratic trend to the FWHM time series, we find no significant peak in the residuals.</text>
<figure>
<location><page_7><loc_54><loc_70><loc_94><loc_92></location>
<caption>Fig. 3: Mass-radius diagram showing the three planets orbiting TOI-396 (star symbol) along with the exoplanets whose density is more significant than the 3 σ level (circle). All the markers are colour-coded according to the equilibrium temperature ( T eq) of the planets. The thick lines are the theoretical MR BICEPS models as detailed in the legend, except for the light-blue line denoted with A21, which is taken from Aguichine et al. (2021). Earth-like means 32.5% iron + 67.5% silicates, while Mercurylike means 70% iron + 30% silicates. Models were computed for T eq = T eq , b = 1552 K (dashed-dotted lines), T eq = T eq , c = 1309 K (solid lines), and T eq = T eq , d = 1061 K (dashed lines). The di ff erence between the A21 model and its BICEPS counterpart is due to di ff erent assumptions for e.g. pressure-temperature profiles and opacities. The dotted black lines are the iso-density loci of points corresponding to 0.5, 1, 3, 5, 10 g cm -3 (going from top to bottom). We recall that the mass estimate of TOI-396 c (the leftmost star symbol) is not statistically significant and its 3 σ upper limit is M up c ∼ 4 M ⊕ .</caption>
</figure>
<text><location><page_7><loc_51><loc_25><loc_94><loc_41></location>The projected equatorial velocity of the star ( v sin i ⋆ = 7 . 5 ± 0 . 2 km s -1 ), along with its radius ( R ⋆ = 1 . 258 ± 0 . 019 R ⊙ ), yields an upper limit for the rotation period of P up rot = 8 . 5 ± 0 . 3 d. Using the mean value 9 of log R ' HK = -4 . 926 ± 0 . 014, we inferred a stellar rotation period of 6.7 ± 1.3 d and 6.9 ± 1.3 d from the empirical equations of Noyes et al. (1984) and Mamajek & Hillenbrand (2008), respectively. In addition, by inputting the isochronal age into the gyrochronological relation from Barnes (2010), we computed a stellar rotation period of 7 . 1 + 1 . 0 -1 . 1 d. These results corroborate our interpretations that the peaks between 6 and 8 d significantly detected in the power spectra of the activity indicators originate from stellar rotation.</text>
<text><location><page_7><loc_51><loc_15><loc_94><loc_25></location>To check whether a quasi-periodic signal compatible with ∼ 7 d is also present in the photometric data, for each TESS sector we extracted custom LCs from pixel data using lightkurve (Lightkurve Collaboration et al. 2018). In detail, we adopted the default quality bitmask and set the aperture to 'all', which corresponds to an aperture larger than the one used by the o ffi cial SAP pipeline. In fact, larger apertures mitigate the e ff ect of slow image drifts that could interfere with slow flux changes, such</text>
<text><location><page_8><loc_7><loc_82><loc_8><loc_85></location>TV [min]</text>
<text><location><page_8><loc_7><loc_82><loc_8><loc_82></location>T</text>
<text><location><page_8><loc_7><loc_60><loc_8><loc_63></location>TV [min]</text>
<text><location><page_8><loc_7><loc_60><loc_8><loc_60></location>T</text>
<text><location><page_8><loc_9><loc_91><loc_11><loc_92></location>40</text>
<text><location><page_8><loc_9><loc_88><loc_11><loc_89></location>30</text>
<text><location><page_8><loc_9><loc_86><loc_11><loc_87></location>20</text>
<text><location><page_8><loc_9><loc_83><loc_11><loc_84></location>10</text>
<text><location><page_8><loc_10><loc_81><loc_11><loc_81></location>0</text>
<text><location><page_8><loc_9><loc_78><loc_11><loc_79></location>10</text>
<text><location><page_8><loc_9><loc_76><loc_11><loc_76></location>20</text>
<text><location><page_8><loc_9><loc_70><loc_11><loc_71></location>60</text>
<text><location><page_8><loc_9><loc_68><loc_11><loc_69></location>50</text>
<text><location><page_8><loc_9><loc_66><loc_11><loc_67></location>40</text>
<text><location><page_8><loc_9><loc_64><loc_11><loc_65></location>30</text>
<text><location><page_8><loc_9><loc_62><loc_11><loc_62></location>20</text>
<text><location><page_8><loc_9><loc_59><loc_11><loc_60></location>10</text>
<text><location><page_8><loc_10><loc_57><loc_11><loc_58></location>0</text>
<text><location><page_8><loc_9><loc_55><loc_11><loc_56></location>10</text>
<text><location><page_8><loc_9><loc_53><loc_11><loc_54></location>20</text>
<text><location><page_8><loc_12><loc_73><loc_27><loc_74></location>8360 8380 8400 8420 8440</text>
<text><location><page_8><loc_29><loc_73><loc_48><loc_74></location>9100 9120 9140 9160 9180 9200</text>
<text><location><page_8><loc_25><loc_72><loc_34><loc_73></location>BJD - 2450000 [d]</text>
<text><location><page_8><loc_12><loc_51><loc_27><loc_52></location>8360 8380 8400 8420 8440</text>
<text><location><page_8><loc_29><loc_51><loc_48><loc_52></location>9100 9120 9140 9160 9180 9200</text>
<text><location><page_8><loc_25><loc_50><loc_34><loc_51></location>BJD - 2450000 [d]</text>
<figure>
<location><page_8><loc_7><loc_28><loc_49><loc_49></location>
<caption>Fig. 4: TTV amplitudes obtained for TOI-396 b ( Top panel ), TOI-396 c ( Middle panel ), and TOI-396 d ( Bottom panel ). The grey shaded region highlights the 1 σ uncertainty region as derived from error propagation of the linear ephemerides.</caption>
</figure>
<text><location><page_8><loc_6><loc_10><loc_49><loc_19></location>as the 6-8 d rotation period signals we aim to detect. After removing the temporal windows containing the transit events, we computed the GLS periodograms of these lightkurve -based LCs. The FAP was computed following the same bootstrap technique outlined above for the RV activity indicators. The four periodograms (Fig. A.2, first column) exhibit very significant peaks at ∼ 7.7, 7.9, 7.5, and 6.8 days for TESS Sectors 3, 4, 30, and 31,</text>
<table>
<location><page_8><loc_52><loc_77><loc_93><loc_87></location>
<caption>Table 3: Radial velocity semi-amplitudes K out as retrieved from MCMCIanalyses of the RV time series obtained by adding an artificial Keplerian signal with period P = Pc and semi-amplitudes K in to the original HARPS time series.Notes. Columns ρ p and Mp translate the injected synthetic signals into the corresponding physical parameters of the planet, while the last column quantifies the K out detection level in terms of σ .</caption>
</table>
<text><location><page_8><loc_51><loc_51><loc_94><loc_70></location>respectively. Except for Sector 30, they are not the most prominent peaks; however, they persist even after removing the most significant signals (Fig. A.2, second column). Whereas we acknowledge that the likely rotation period of the star is close to the first harmonic of the orbital period of TESS around the Earth ( P rot ,⋆ ∼ 1 2 P TESS ∼ 14 days), the photometric signal at 6-8 d is significant in all the four TESS sectors and persists after prewhitening the data. This signal is consistent with the rotation period detected in the HARPS activity indicators and inferred from log R ' HK , v sin i ⋆ , and gyrochronology, suggesting it is astrophysical in nature and due to the presence of active regions carried around by stellar rotation. Assuming the log R ' HK -based P rot ,⋆ = 6 . 7 ± 1 . 3 d as our reference estimate, the orbital period of TOI-396 c ( Pc ∼ 6 d) is close to P rot ,⋆ , which may explain the non-detection of planet c within the RV time series.</text>
<text><location><page_8><loc_51><loc_43><loc_94><loc_50></location>If some kind of destructive interference between the Keplerian signal of planet c and the stellar activity has occurred, any artificial Keplerian signals with period P = Pc added to the observed RV time series should in principle be retrieved. To test this hypothesis, we considered Keplerian signals of the following form</text>
<formula><location><page_8><loc_51><loc_39><loc_94><loc_42></location>RV art = -K art sin " 2 π P ( t -T 0) # (2)</formula>
<text><location><page_8><loc_51><loc_29><loc_94><loc_38></location>and generated four di ff erent RV time series by separately adding to the HARPS time series synthetic RV signals following Equation (2) with P = Pc , T 0 = T 0 , c , and K art = K in, where K in are the four di ff erent amplitude values listed in the first column of Table 3. For each RV time series, we then performed an MCMCI analysis to retrieve the RV semi-amplitude of the artificial signal ( K out; see Table 3).</text>
<text><location><page_8><loc_51><loc_20><loc_94><loc_29></location>We note that the resulting K out ≈ K in + Kc , where Kc = 0 . 28 + 0 . 29 -0 . 19 ms -1 is the RV semi-amplitude of planet c as derived from the analysis on the original RV time series. As we essentially retrieved what we inserted in the HARPS time series, we may conclude that the destructive interference between the RV signals induced by the star and by planet c has already occurred and any further RV signal added to the RV time series is detected.</text>
<text><location><page_8><loc_51><loc_10><loc_94><loc_20></location>As P rot ,⋆ is not exactly equal to Pc , we then repeated the test outlined above, but this time we injected into the original HARPS time series artificial Keplerian signals with P = P rot ,⋆ . The K out values obtained by the MCMC analyses depending on the di ff erent K in values are reported in Table 4. The K out values are systematically and significantly smaller than the corresponding K in values, which a posteriori supports the conclusion that the stellar rotation period is around 6-8 d. Furthermore, a planet with this</text>
<figure>
<location><page_9><loc_19><loc_95><loc_21><loc_96></location>
</figure>
<figure>
<location><page_9><loc_15><loc_34><loc_85><loc_93></location>
<caption>Fig. 5: Time series (left panels) and GLS periodograms (right panels) of the line profile variation diagnostics and activity indicators extracted from TOI-396's HARPS spectra. In the left panels, the red curves in the first four panels mark the quadratic trends and sine functions as obtained from the best fit to the most significant peaks (false alarm probability FAP < 0.1%) identified in the corresponding GLS periodograms. In the right panels, the vertical dashed blue lines mark the orbital frequencies of the tree transiting planets. The yellow area encompasses the peaks likely due to stellar rotation. The horizontal dashed red lines mark the 0.1% false alarm probability.</caption>
</figure>
<text><location><page_9><loc_6><loc_12><loc_49><loc_22></location>orbital period would be firmly detected if its RV semi-amplitude were greater than Kd , which is the largest RV semi-amplitude detected for this system. Instead, by injecting a Keplerian signal with K in = Kd and P = P rot ,⋆ , the MCMCI analysis is able to barely detect (at ∼ 2 σ ) a planetary signal whose amplitude is half of that expected. The detection level increases when K in increases; however, we still underestimate K out. These tests prove that it is di ffi cult to retrieve planetary signals with P ∼ P rot ,⋆ .</text>
<text><location><page_9><loc_51><loc_12><loc_94><loc_22></location>In summary, we conclude that stellar activity is responsible for generating spurious RV signals whose harmonics also include the stellar rotation period. As a consequence, it is hard to reliably detect planets with orbital periods comparable to P rot ,⋆ via the RV technique and Table 4 quantifies the magnitude of this e ff ect. We note that the Kc we obtained from the MCMCI analysis in Sect. 5.2 is comparable to the K out retrieved when inserting an artificial signal having K in = 1 . 0 ms -1 , which let us</text>
<table>
<location><page_10><loc_7><loc_82><loc_48><loc_90></location>
<caption>Table 4: Same as Table 3, but this time the Keplerian signals with K = K in have period P = P rot ,⋆ .</caption>
</table>
<text><location><page_10><loc_6><loc_76><loc_49><loc_81></location>Notes. Column ∆ K K gives the relative di ff erence (in percentage) between the obtained RV semi-amplitude ( K out) and the expected one ( K in), while the last column ∆ K ≡ K out -K in lists the semi-amplitude di ff erence in terms of the 1 σ uncertainty of K out.</text>
<text><location><page_10><loc_6><loc_71><loc_49><loc_74></location>to speculate that TOI-396 c might have Mc ∼ 3.0 M ⊕ and ρ c ∼ 2.0 g cm -3 .</text>
<section_header_level_1><location><page_10><loc_6><loc_67><loc_39><loc_68></location>6. Joint RV and TTV dynamical analysis</section_header_level_1>
<text><location><page_10><loc_6><loc_51><loc_49><loc_66></location>As mentioned, the anti-correlation pattern of the observed TTVs (see Fig. A.1) is a typical signal of the dynamical interaction between TOI-396 b and c. However, the data in our hands are not enough to currently derive meaningful planetary masses from the TTVs. Indeed, the photometric observations are clustered in two groups (about two years apart), which results in a partial coverage of the curvature of the TTVs. This prevents us from accurately mapping the full phases and super-periods of the TTV signals. Hence a dynamical fit onto the transit times would lead to a high fraction of low-amplitude TTVs with a short superperiod or to a low fraction of high-amplitude long-period TTV signals.</text>
<text><location><page_10><loc_6><loc_26><loc_49><loc_50></location>Despite this, we attempted to run a dynamical joint fit of the TTV and RV data set with TRADES 10 (Borsato et al. 2014, 2019, 2021), a Fortran-python code developed to model TTVs and RVs simultaneously along with N-body integration. We have taken and fixed the stellar mass and radius values from Table 1. We further fixed the orbital inclinations i to the values in Table 2 and the longitude of the ascending nodes to Ω = 180 · 11 for all the planets. We fitted the planetary masses scaled by the stellar mass ( M b , c , d / M ⋆ ), the orbital periods ( P ), the eccentricities ( e ) and the arguments of the pericentre ( ω ) in the form ( √ e cos ω , √ e sin ω ), and the mean longitudes ( λ 12 ). We also fitted for an RV o ff set ( γ RV) and for an RV jitter term ( σ jitter) by adopting log 2 σ jitter as step parameter, although we used the de-trended RV data set as derived from Sect. 5.2, where a jitter term was already included in the RV errors. All fitting parameters were subject to uniform priors that account for their respective physical boundaries (see Table 5); for the eccentricities we applied a log-penalty (log p e ) based on the half-Gaussian ( e = 0 , σ e = 0 . 083) from Van Eylen et al. (2019).</text>
<text><location><page_10><loc_6><loc_19><loc_49><loc_26></location>The reference time for the dynamical integration of the orbital parameters was set at T ref = 2 458 379 BJDTDB, that is before all available observations. We combined the quasiglobal di ff erential evolution (Storn & Price 1997) optimisation algorithm implemented in P y DE (Parviainen et al. 2016)</text>
<text><location><page_10><loc_51><loc_47><loc_94><loc_93></location>with the A ffi ne Invariant MCMC Ensemble sampler (Goodman & Weare 2010) emcee implemented by Foreman-Mackey et al. (2019, 2013). We first ran P y DE and evolved 68 di ff erent initial configurations of parameter sets for 50 000 generations (number of steps for which each parameter is evolved). To perform the dynamical analysis in an MCMC fashion, we then ran emcee , assuming as starting point the outcome obtained with P y DE. We set up 68 chains for 600 000 steps each. To sample the parameter space e ffi ciently, we mixed the DEMove() and DESnookerMove() di ff erential evolution moves 13 in the proportion 80%-20% (Nelson et al. 2014; ter Braak & Vrugt 2008). We repeated the sequence P y DE + emcee twice, with di ff erent seeds for the random number generator. The chains reached convergence according to visual inspection and statistical indicators, such as the Gelman-Rubin ˆ R , the Geweke's statistic (Geweke 1991), and the auto-correlation function 14 . For both runs, we applied a conservative thinning factor of 100 and discarded the first 50% of the chains (burn-in). For each run, we derived the reference outcome (hereinafter also referred to as best-fit), as the maximum-a-posteriori (MAP) parameters' set, that is the set of parameters that maximises the log-probability 15 . The uncertainty of each parameter is quantified by the high-density interval (HDI) at 68.27% 16 of its posterior distribution. For each parameter we computed a Z-score defined as Z-score = | MAP1 -MAP2 | / p max | ERR1 | 2 + max | ERR2 | 2 , where the subscripts denote the two di ff erent runs and ERR = HDI -MAP. It turned out that Z-score < 1 for each parameter, which allows us to merge the posterior distributions deriving from the two runs to finally compute the MAP and the respective HDI from the merged posterior distributions. We further checked the Hill stability of the system (Sundman 1913) when assuming the entire merged posterior distributions by calculating the angular momentum deficit (AMD, Laskar 1997, 2000; Laskar & Petit 2017) criterion (Eq. 26 from Petit et al. 2018). Table 5 lists the parameters returned by TRADES (MAP and HDI) along with their respective priors.</text>
<text><location><page_10><loc_51><loc_23><loc_94><loc_47></location>We note that the dynamical integration with TRADES allowed for a significant detection ( > 3 σ ) of the mass of TOI-396 c, that is Mc , dyn = 2 . 24 + 0 . 13 -0 . 67 M ⊕ . However, as explained above, TESS data do not allow us to fully map the TTV pattern. Therefore, the present-day estimate of Mc , dyn only provides an indication of the possible mass of the planet that might not be accurate, despite its formal precision. Indeed, to accurately and reliably determine planetary masses via TTVs, it is necessary to monitor the TTV signal by benefiting of a full sampling coverage, as demonstrated for example by the cases of Kepler-9 (Holman et al. 2010) and K2-24 (Petigura et al. 2016), whose orbital parameters and masses were comprehensively revisited by Borsato et al. (2014) and Petigura et al. (2018); Nascimbeni et al. (2024), respectively. In detail, Holman et al. (2010) had reported the masses of Kepler-9 b and c with a precision better than 8%, by performing a TTV analysis on the first three quarters of the Kepler data. Later on, by benefiting of twelve sectors of Kepler data that enabled the full mapping of the TTV phase, Borsato et al. (2014) obtained TTV-based masses di ff ering by a</text>
<figure>
<location><page_11><loc_19><loc_95><loc_21><loc_96></location>
<caption>Figures 10 and 11 show the resulting posteriors of the most important interior structure parameters for TOI-396 b and d, respectively, in comparison with the chosen priors (dotted lines). Tables A.4 and A.5 in the Appendix summarise the median and one sigma error intervals for the full set of internal structure parameters. For both planets, the posterior distributions for the core and mantle mass fractions largely agree with the chosen priors for each of the six models that we ran. Indeed, the only planetary structure parameters for which the observational data contributed to their characterisation were the envelope mass fractions. If we assume that the planets formed outside the iceline, we find envelope mass fractions of 28 ± 10% (A1), 32 + 9 -11 %(A2) and 30 + 11 -13 %(A3) for planet b and 23 + 12 -10 %(A1), 28 ± 11% (A2) and 26 + 12 -13 % (A3) for planet d, with water mass fractions in the envelope of almost 100%. Conversely, if the planets were to have formed inside the iceline, we infer envelope mass fractions of the order of 10 -5 for planet b and 10 -4 for planet d, almost entirely made up of H / He.</caption>
</figure>
<text><location><page_11><loc_6><loc_87><loc_49><loc_93></location>factor ∼ 2 from the estimate of Holman et al. (2010). Similarly, by accounting on more photometric data, Petigura et al. (2018); Nascimbeni et al. (2024) found that the mass of K2-24 c is lower by almost 2 σ than the estimate of Petigura et al. (2016) who had claimed a detection at the ∼ 4 σ level.</text>
<text><location><page_11><loc_6><loc_79><loc_49><loc_86></location>After extracting all the synthetic transit timings T tr (i.e. the 'observed': O) from TRADES' analysis, we computed the 'calculated' (C) counterpart according to the linear ephemerides model based on what listed in Table 2. We plotted the O -C as a function of time as well as the RV best-fit model in Figures 6 and 7.</text>
<text><location><page_11><loc_6><loc_70><loc_49><loc_79></location>Additionally, we investigated if the best-fit configuration is in or close to an MMR. We integrated the MAP parameters with the N-body code rebound (Rein & Liu 2012) and the symplectic Wisdom-Holman integrator whfast (Rein & Tamayo 2015; Wisdom & Holman 1991) for 10 000 years. We computed the evolution of the critical resonance angles of TOI-396 b and TOI396 c</text>
<formula><location><page_11><loc_6><loc_66><loc_49><loc_68></location>ϕ b = p λ b -( p + q ) λ c + q ϖ b ϕ c = p λ b -( p + q ) λ c + q ϖ c , (3)</formula>
<text><location><page_11><loc_6><loc_42><loc_49><loc_64></location>where p = 3 and q = 2 (for a second order 5:3 MMR), while ϖ ≡ ω + Ω is the longitude of the pericentre. We also computed the evolution of ∆ ϖ ≡ ϖ b -ϖ c = ( ϕ b -ϕ c) / q . In case of MMR, we expect that both ϕ b and ϕ c librate (i.e. oscillate) around a fixed value for the entire orbital integration. Instead, if these angles circulate, that is they span the full 0 · -360 · range (or equivalently the -180 · -180 · range), then the planet pair is not in resonance. We found that both ϕ b and ϕ c circulate (see the two upper panels of Fig. 8), which indicates that the system is not in an exact 5:3 MMR. Even if ∆ ϖ seems to oscillate around 0 · , it circulates every ∼ 2 000 years (bottom panel of Fig. 8), which further confirms that the system is not trapped in an MMR state. We also found the same behaviour for the resonant angles of 200 random samples drawn from the posterior distribution (not shown here). Our conclusions are consistent with the simulations performed by Vanderburg et al. (2019), who found that most realisations of the system are not in resonance.</text>
<text><location><page_11><loc_6><loc_10><loc_49><loc_42></location>As shown in Fig. 6, the MAP parameter set predicts that the TTV super-period is longer than the time spanning the two clustered TESS observations. We decided to track the potential evolution of the TTV signals over a temporal baseline of ∼ 5.2 years. To this end, we ran forward numerical N-body simulations with TRADES, setting the initial conditions of the orbital parameters to be integrated at t = T ref. We ran a first simulation taking all the parameters from the MAP solution. Then, we further ran 200 simulations, where the sets of the system's parameters were randomly drawn from the merged posterior distributions. We computed the synthetic (i.e. observed: O ) transit times T tr and created a simulated O -C plot against time, where the C counterpart of T tr was computed assuming the linear ephemerides of Table 2. The results are displayed in Fig. 9 that emphasises a progressive drift of the O -C values inferred from the MAP parameters (black line) with respect to the zero value. As a consequence, the TTV amplitudes of planets b and c increase with time. In particular, the semi-amplitude of the O -C black curves are about two and five hours for b and c, respectively. The TTV super-period seems to be roughly equal to or larger than the integration time span, that is ∼ 5 years. The variance of the TTV amplitude (shaded area) is inferred from the results of the additional 200 simulations and reflects the widths of the posterior distributions from which the system's parameters were drawn. The remarkable O -C drifts (i.e. the poorly constrained linear</text>
<text><location><page_11><loc_51><loc_85><loc_94><loc_93></location>ephemerides) combined with the uncertainty on the TTV amplitudes make challenging to plan future observations of planets b and c, as the actual transit timings might di ff er from the linear ephemerides predictions by ∼ 5 and ∼ 10 hours for TOI-396 b and TOI-396 c, respectively. TESS will not observe the target in the foreseeable future. 17</text>
<section_header_level_1><location><page_11><loc_51><loc_76><loc_67><loc_77></location>7. Internal structure</section_header_level_1>
<text><location><page_11><loc_51><loc_59><loc_94><loc_73></location>Using the masses and transit depths reported in Table 2, we ran the neural network based internal structure modelling framework plaNETic 18 (Egger et al. 2024) to infer the internal structure of TOI-396 b and d. plaNETic uses a full grid accept-reject sampling method in combination with a deep neural network (DNN) that was trained on the forward model of BICEPS (Haldemann et al. 2024) to infer the internal structure of observed planets. Each planet is modelled as a three-layered structure: an inner iron-dominated core, a silicate mantle and a fully mixed envelope made up of water and H / He. In the case of multi-planet systems, all planets are modelled simultaneously.</text>
<text><location><page_11><loc_51><loc_35><loc_94><loc_58></location>As modelling the internal structure of exoplanets is a highly degenerate problem, the resulting inferred structure is, at least to a certain extent, dependent on the chosen priors. To mitigate this e ff ect, we ran a total of six models assuming six di ff erent combinations of priors. Most importantly, we use two di ff erent priors for the water content of the modelled planet, one motivated by a formation scenario outside the iceline (case A, water-rich) and one compatible with a formation inside the iceline (case B, water-poor). For both of these water priors, we choose three different options for the planetary Si / Mg / Fe ratios. In a first case, we assume that these match the stellar Si / Mg / Fe ratios exactly Thiabaud et al. (2015). Second, we assume that the planet is enriched in iron compared to its host star by using the fit of Adibekyan et al. (2021). For option 3, we model the planet independent of the stellar Si / Mg / Fe ratios, but just sampling the planetary ratios uniformly from the simplex where the molar Si, Mgand Fe ratios add up to 1, with an upper bound of 0.75 for Fe. These priors are described in more detail in Egger et al. (2024).</text>
<table>
<location><page_12><loc_7><loc_63><loc_93><loc_90></location>
<caption>Table 5: Best-fit parameters (MAP and HDI at 68.27%) along with their respective priors as inferred from the dynamical joint modelling of RVs and TTVs with TRADES.Notes. All parameters have been defined at the reference time T ref = BJDTDB -2 450 000 = 8379. U ( X , Y ) means uniform distribution between X and Y values; HG ( µ, σ ) means half-normal distribution with mean µ and standard deviation σ .</caption>
</table>
<figure>
<location><page_12><loc_7><loc_29><loc_48><loc_58></location>
</figure>
<figure>
<location><page_12><loc_51><loc_29><loc_91><loc_58></location>
<caption>Fig. 6: Observed minus calculated synthetic diagrams derived from the joint RV and TTV dynamical analysis with TRADES for planet b ( left panel ) and c ( right panel ). The O -C for the best-fit (MAP) model is plotted with a black line, while the observed data points are the orange circles. The shaded grey regions displays the confidence intervals at 1, 2, and 3 σ , as inferred from the 200 samples randomly drawn from the merged posterior distributions. Residuals are shown in the lower panels.</caption>
</figure>
<section_header_level_1><location><page_12><loc_6><loc_18><loc_36><loc_19></location>8. JWST characterisation prospects</section_header_level_1>
<text><location><page_12><loc_6><loc_15><loc_49><loc_17></location>All three planets in the TOI-396 system share similar radii ( ∼ 2 R ⊕ ), but span a wide range of masses (0.9-7.1 M ⊕ ), which</text>
<text><location><page_12><loc_51><loc_13><loc_94><loc_19></location>leaves open the question of whether they have primary or secondary atmospheres. Furthermore, the progression of bulk densities with distance from the host star varies in ways that cannot be described by simple formation and evolution models (e.g. Weiss et al. 2018; Mishra et al. 2023).</text>
<text><location><page_12><loc_51><loc_10><loc_94><loc_12></location>Given the bright host star and combination of planetary masses, radii, and equilibrium temperatures, the three planets</text>
<figure>
<location><page_13><loc_19><loc_95><loc_21><loc_96></location>
</figure>
<figure>
<location><page_13><loc_7><loc_64><loc_46><loc_91></location>
</figure>
<figure>
<location><page_13><loc_48><loc_64><loc_86><loc_91></location>
<caption>Fig. 7: Left panel : Same as Fig. 6, but for TOI-396 d. Right panel : Combined RV model of the three planets (black line) superimposed to the de-trended HARPS observations (purple circles). The grey shaded area is determined from the 200 sets of system parameters randomly drawn from the merged posterior distributions as obtained from the joint dynamical analysis with TRADES.</caption>
</figure>
<figure>
<location><page_13><loc_7><loc_25><loc_46><loc_52></location>
<caption>Fig. 8: Temporal evolution of the critical resonance angles ϕ b ( top panel ) and ϕ c ( middle panel ) as well as of ∆ ϖ = ( ϕ b -ϕ c) / q ( bottom panel ) as inferred when assuming the MAP parameters derived by the TRADES dynamical analysis and integrated with rebound+whfast for 10 000 years.</caption>
</figure>
<text><location><page_13><loc_6><loc_10><loc_49><loc_15></location>have favourable metrics for atmospheric characterisation in both transmission and emission among sub-Neptunes (Kempton et al. 2018, see Fig. 12). This makes the TOI-396 system a highly valuable laboratory to study the formation and evolution of plan-</text>
<text><location><page_13><loc_51><loc_49><loc_94><loc_55></location>etary systems. Thus, we explored the prospects for characterisation with JWST. We focused these simulations on emission observations, but we note that transmission and emission have their own advantages and disadvantages in terms of achievable science goals and challenges.</text>
<text><location><page_13><loc_51><loc_16><loc_94><loc_49></location>We employed the open-source P yrat B ay modelling framework (Cubillos & Blecic 2021) to compute synthetic spectra of the TOI-396 planets. These models consist of 1D cloud-free atmospheres in radiative, thermochemical, and hydrostatic equilibrium (Cubillos et al., in prep.). We varied the models' atmospheric elemental content to explore the wide range of compositions that the planets span. For this comparison we settled on two models to represent a primary- and a secondary-atmosphere scenario: the first is a gas giant with a 5 × solar metallicity, the second is a water world with a 80% H 2 O plus 20% CO 2 composition (based on the C / O ratios seen in the solar system minor bodies, see, e.g., Mumma & Charnley 2011; McKay et al. 2019). For the thermochemical-equilibrium calculations we considered a set of 45 neutral and ionic species, which are the main actors determining the thermal structure. For the radiative-transfer calculation we considered opacities from molecular species for CO, CO 2 , CH 4 , H 2 O, HCN, NH 3 , and C 2 H 2 from hitemp and E xo M ol (Rothman et al. 2010; Tennyson et al. 2016); Na and K resonant lines (Burrows et al. 2000); H, H2, and He Rayleigh (Kurucz 1970); and H2-H2 and H2-He collision-induced absorption (Borysow et al. 2001; Borysow 2002; Richard et al. 2012). We preprocessed the large E xo M ol line lists with the R epack algorithm (Cubillos 2017) to extract the dominant transitions. Figure 13 (top panels) shows the resulting thermal and composition structure for TOI-396 b (planets c and d follow a similar trend).</text>
<text><location><page_13><loc_51><loc_10><loc_94><loc_16></location>The infrared synthetic emission spectra (Fig. 13, bottom panels) are mainly shaped by H 2 O, CO 2 , and CO features. At most wavelengths the primary- and secondary-atmosphere scenarios roughly di ff er by an o ff set, which would be hard to distinguish unless the energy budget of the planets are known. In contrast,</text>
<figure>
<location><page_14><loc_8><loc_63><loc_45><loc_92></location>
<caption>Fig. 9: Synthetic O -C diagrams obtained after performing forward numerical N-body simulations with TRADES (integration of 5.2 years). The C represents the timings calculated from the linear ephemerides in Table 2. The MAP model is plotted with a black line, while the confidence intervals at 1, 2, and 3 σ are marked as shaded grey regions and come from 200 random samples drawn from the merged posterior distributions derived from the joint RV and TTV dynamical analysis.</caption>
</figure>
<text><location><page_14><loc_6><loc_44><loc_49><loc_49></location>the 4-5 µ mwindow shows the most distinctive spectral features; here the strong CO 2 absorption band at 4.4 µ m mainly allows one to distinguish primary from secondary atmospheres. Thus, in the following we focus on this region of the spectrum.</text>
<text><location><page_14><loc_6><loc_24><loc_49><loc_43></location>We simulated JWST observations using the P andeia exposure time calculator Pontoppidan et al. (2016). The brightness of TOI-396 limits the instrument selection to NIRCam (F444W filter) to avoid saturation. We selected the fastest readout and subarray modes, 5 groups per integration, to optimise the S / N. We generated a distribution of (noised up) realisations for each model to estimate how many eclipses are required to distinguish between primary and secondary atmospheres at the 3 σ level. We found that 2, 4, and 8 eclipses (for planets b, c, and d, respectively) would be su ffi cient to di ff erentiate between these two models. Figure 13 shows one of those random realisations when including the required number of eclipses. The decreasing equilibrium temperature of the planets as they are located further away from TOI-396 plays the major role in the decreasing S / N for planets c and d.</text>
<section_header_level_1><location><page_14><loc_6><loc_20><loc_19><loc_21></location>9. Conclusions</section_header_level_1>
<text><location><page_14><loc_6><loc_9><loc_49><loc_19></location>The object TOI-396 is an F6 V bright naked-eye star orbited by three planets of almost equal size, and the two inner planets are close to but out of a 5:3 MMR. A photometric analysis of the system was already performed by Vanderburg et al. (2019), but by benefiting from two additional TESS sectors, we improved the precision on the planet radii by a factor of ∼ 1.4, obtaining Rb = 2 . 004 + 0 . 045 -0 . 047 R ⊕ , Rc = 1 . 979 + 0 . 054 -0 . 051 R ⊕ , and Rd = 2 . 001 + 0 . 063 -0 . 064 R ⊕ .</text>
<text><location><page_14><loc_51><loc_79><loc_94><loc_93></location>We determined the masses of the planets by extracting the RV time series from HARPS CCFs using an SN fit followed by a joint LC and RV MCMC analysis, where the RV de-trending uses the breakpoint method. We obtained a firm detection of the RV signals of planets b and d, deriving Mb = 3 . 55 + 0 . 94 -0 . 96 M ⊕ and Md = 7 . 1 ± 1 . 6 M ⊕ , but we can provide only a 3 σ upper limit for the mass of TOI-396 c of M up c = 3 . 8 M ⊕ . This yields the following mean planet densities: ρ b = 2 . 44 + 0 . 69 -0 . 68 , ρ up c = 2 . 9, and ρ d = 4 . 9 + 1 . 2 -1 . 1 g cm -3 , implying a quite unusual system architecture (Mishra et al. 2023) where the mid planet is the least dense and the outermost planet is the densest.</text>
<text><location><page_14><loc_51><loc_55><loc_94><loc_78></location>The reason for the RV non-detection of any Keplerian signal at P = Pc ∼ 6 d is likely to be ascribed to the vicinity of Pc to the stellar rotation period. As a matter of fact, from the GLS periodograms of both the RV-related activity indices and the TESS raw LCs and from log R ' HK -based empirical relations, we consistently inferred P rot ,⋆ = 6 . 7 ± 1 . 3 d. After injecting synthetic Keplerian signals at P = P rot ,⋆ and di ff erent semi-amplitudes ( K in) into the RV time series, we empirically find that the RV semi-amplitudes output by the MCMC analyses ( K out) are systematically lower than the input ones by almost 3 σ , and they are statistically non-significant as far as K in ≲ Kd . In addition, we find that K out ≈ Kc when considering a planet with Mp ∼ 3 M ⊕ (i.e. ρ p ∼ 2 g cm -3 ), which might correspond to the properties of TOI-396 c. On a more general perspective, these simulations confirm that stellar activity destructively interferes with Keplerian signals having P ∼ P rot ,⋆ (e.g. Vanderburg et al. 2016), and furthermore, they indicate that - even in the case of firm detection - values of K out are significantly underestimated.</text>
<text><location><page_14><loc_51><loc_24><loc_94><loc_54></location>Longer-baseline RV observations may help disentangle coherent signals originated by Keplerian motions from noncoherent signals due to stellar activity, even if degeneracy issues still hold when the planet orbital period is close to the stellar rotation period (Kossakowski et al. 2022). Alternatively, a possible constraint on Mc may come from TTVs, as planets b and c are close to an MMR of the second order. Indeed, the TTV amplitudes of the two planets show a characteristic anti-correlation pattern, as expected; however, the phase coverage given by the available observations is too poor to perform a conclusive TTV dynamical analysis based on the observed transit timings of the planets. We also attempted to fit the TTV and RV simultaneously while integrating the orbits of the system. We found that the masses and densities of planets b and d are consistent with the results from the joint LC and RV analysis. TOI-396 c shows a dynamical mass of Mc , dyn = 2 . 24 + 0 . 13 -0 . 67 M ⊕ , which is greater than that inferred from the joint LC and RV analysis, but it is consistent (Z-score = 1 . 2 σ ); the density is consistent at the 1 . 1 σ level. However, we emphasise that, although formally precise, the Mc , dyn estimate might not be accurate, as the full coverage of the TTV phase is needed to reliably compute TTV-based masses. Therefore, to fully confirm the system architecture, a reliable estimate of the mass of TOI-396 c is still missing.</text>
<text><location><page_14><loc_51><loc_10><loc_94><loc_24></location>We also checked the evolution of the system over 10 000 years, and the critical resonance angles showed that planets b and c are close to but not in a 5:3 MMR. We further performed forward N-body simulations over a temporal baseline of ∼ 5.2 years in order to track the transit epochs and evaluate the expected TTV amplitudes during time. It turns out that TOI-396 b and TOI-396 c may exhibit TTVs with a super-period of about 5 years and semi-amplitudes of ∼ 2 and ∼ 5 hours, respectively. This translates into a temporal drift of the transit timings that can rise up to ∼ 5 and ∼ 10 hours with respect to the linear ephemerides computed from TESS data.</text>
<figure>
<location><page_15><loc_19><loc_95><loc_21><loc_96></location>
</figure>
<figure>
<location><page_15><loc_7><loc_70><loc_94><loc_93></location>
<caption>Fig. 10: Inferred posteriors for the most important internal structure parameters of TOI-396 b. The depicted parameters are the mass fractions of the inner core (wcore), mantle (wmantle) and envelope layers (wenvelope) with respect to the total planet mass, and the mass fraction of water in the envelope (Zenvelope). The top row shows the results when assuming a water prior motivated by a formation of the planet outside the iceline (case A), while the bottom row uses a water prior compatible with a formation inside the iceline (case B). At the same time, we run models with three di ff erent compositional priors for the planetary Si / Mg / Fe ratios: stellar (purple, option 1), iron-enriched compared to the star (pink, option 2) and sampled using a uniform prior (blue, option 3). The dotted lines show the prior distributions, while the dashed vertical lines show the median values of the posteriors.</caption>
</figure>
<figure>
<location><page_15><loc_7><loc_35><loc_94><loc_57></location>
<caption>Fig. 11: Same as Figure 10 but for TOI-396 d.</caption>
</figure>
<text><location><page_15><loc_6><loc_19><loc_49><loc_29></location>Studying the planetary atmospheres with JWST would take advantage of the favourable spectroscopy metrics of the system (Kempton et al. 2018). Therefore, we set up 1D cloud-free atmospheric models, generated the synthetic emission spectra of the three planets, and simulated eclipse observations with JWST. It turns out that 2, 4, and 8 eclipses (for TOI-396 b, c, and d, respectively) would be su ffi cient to distinguish between primary and secondary atmosphere scenarios at the 3 σ level.</text>
<text><location><page_15><loc_6><loc_10><loc_49><loc_17></location>Characterising the nature of the planetary atmosphere is also key to correctly assessing the planetary bulk densities (in particular for planet c). The potentially high TTVs inferred from our simulations should be duly taken into account when scheduling future observations of the target. This holds not only for JWST, but also for CHEOPS (Benz et al. 2021), which appears espe-</text>
<text><location><page_15><loc_51><loc_27><loc_94><loc_29></location>cially suitable for collecting exquisite photometric data to enable the full characterisation of the system.</text>
<text><location><page_15><loc_51><loc_10><loc_94><loc_25></location>Acknowledgements. We thank the anonymous referee for all the valuable comments that significantly improved the quality of the manuscript. This work has made use of data from the European Space Agency (ESA) mission Gaia ( https://www.cosmos.esa.int/gaia ), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/ gaia/dpac/consortium ). This research made use of Lightkurve, a Python package for Kepler and TESS data analysis (Lightkurve Collaboration et al. 2018). We thank contributors to NumPy (Harris et al. 2020), SciPy (Virtanen et al. 2020), matplotlib (Hunter 2007), astropy (Astropy Collaboration et al. 2013, 2018, 2022), astroquery (Ginsburg et al. 2019), and tesscut (Brasseur et al. 2019). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. We acknowledge financial support from the Agencia Estatal de Investigación of the Ministerio de Ciencia e Innovación MCIN / AEI / 10.13039 / 501100011033 and the ERDF 'A way of making Europe' through project PID2021-125627OB-C32,</text>
<figure>
<location><page_16><loc_6><loc_75><loc_49><loc_93></location>
</figure>
<figure>
<location><page_16><loc_7><loc_54><loc_49><loc_73></location>
<caption>Fig. 12: Transit ( top panel ) and eclipse ( bottom panel ) spectroscopic metrics for the TOI-396 planets (see legend). The metrics were calculated using the 2MASS Ks-band magnitude. The grey markers show the metrics for the known sample of transiting exoplanets to date. The blue markers show the metrics for targets with approved JWST programmes.</caption>
</figure>
<text><location><page_16><loc_6><loc_10><loc_49><loc_41></location>and from the Centre of Excellence 'Severo Ochoa' award to the Instituto de Astrofisica de Canarias. This research was funded in part by the UKRI, (Grants ST / X001121 / 1, EP / X027562 / 1). This work was supported by FCT - Fundação para a Ciência e a Tecnologia through national funds and by FEDER through COMPETE2020 - Programa Operacional Competitividade e Internacionalização by these grants: UIDB / 04434 / 2020; UIDP / 04434 / 2020. D.G., A.B., L.F., and L.M.S. gratefully acknowledge the financial support from the grant for internationalization (GAND_GFI_23_01) provided by the University of Turin (Italy). S.G.S acknowledges the support from FCT through Investigador FCT contract nr. CEECIND / 00826 / 2018 and POPH / FSE (EC). P.J.W. acknowledges support from the UK Science and Technology Facilities Council (STFC) through consolidated grants ST / P000495 / 1, ST / T000406 / 1 and ST / X001121 / 1. N.C.S. is funded by the European Union (ERC, FIERCE, 101052347). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. J.L.-B. is funded by the MICIU / AEI / 10.13039 / 501100011033 and NextGenerationEU / PRTR grant PID2019-107061GB-C61 and CNS2023-144309. X.D. acknowledges the support from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement SCORE No 851555) and from the Swiss National Science Foundation under the grant SPECTRE (No 200021_215200). This work has been carried out within the framework of the NCCR PlanetS supported by the Swiss National Science Foundation under grants 51NF40_182901 and 51NF40_205606. G.N. thanks for the research funding from the Ministry of Science and Higher Education programme the "Excellence Initiative - Research University" conducted at the Centre of Excellence in Astrophysics and Astrochemistry of the Nicolaus Copernicus University in Toru'n, Poland. Research activities of the Board of Observational and Instrumental Astronomy at the Federal University of Rio Grande do Norte are supported by continuous grants from the Brazilian funding</text>
<figure>
<location><page_16><loc_51><loc_74><loc_94><loc_93></location>
</figure>
<figure>
<location><page_16><loc_51><loc_43><loc_94><loc_73></location>
<caption>Fig. 13: Simulations of the atmospheric pressure profiles, eclipse spectra, and JWST observations. Top-left panel : Radiativeequilibrium thermal profile of TOI-396 b assuming a gas-giant atmosphere (5 × solar metallicity) and a secondary atmosphere (80% H 2 O plus 20% CO 2 ). Top-right panels : Volume mixing ratio of TOI-396 b for the most relevant species shaping the infrared spectrum. Bottom panels : Synthetic secondary eclipse spectra of the three TOI-396 planets (solid curves). The round markers with error bars show a realisation of JWST observation with NIRCam / F444W (and their expected uncertainties) when accumulating 2, 4, and 8 observations for planets b, c, and d, respectively. The vertical dashed lines mark the spectral window covered by NIRCam / F444W.</caption>
</figure>
<text><location><page_16><loc_51><loc_11><loc_94><loc_19></location>agencies CNPq. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-Brasil (CAPES)-Finance Code 001 and CAPES-Print program. B.L.C.M., I.C.L., and J.R.M. acknowledge CNPq research fellowships. K.W.F.L. was supported by Deutsche Forschungsgemeinschaft grants RA714 / 14-1, RA714 / 14-2 within the DFG Schwerpunkt SPP 1992, Exploring the Diversity of Extrasolar Planets. L.B. acknowledges support from CHEOPS ASI-INAF agreement n. 2019-29-HH.0. D.G. sincerely thanks Stefano Camera for the inspiring and valuable discussions on the properties of TOI-396.</text>
<figure>
<location><page_17><loc_19><loc_95><loc_21><loc_96></location>
</figure>
<section_header_level_1><location><page_17><loc_6><loc_92><loc_16><loc_93></location>References</section_header_level_1>
<text><location><page_17><loc_6><loc_86><loc_49><loc_91></location>Addison, B., Wright, D. J., Wittenmyer, R. A., et al. 2019, PASP, 131, 115003 Adibekyan, V., Dorn, C., Sousa, S. G., et al. 2021, Science, 374, 330 Adibekyan, V., Figueira, P., Santos, N. C., et al. 2015, A&A, 583, A94 Adibekyan, V. Z., Sousa, S. G., Santos, N. C., et al. 2012, A&A, 545, A32 Agol, E. & Fabrycky, D. C. 2018, in Handbook of Exoplanets, ed. H. J. Deeg &</text>
<unordered_list>
<list_item><location><page_17><loc_6><loc_82><loc_45><loc_86></location>J. A. Belmonte (Cambridge University Press), 7 Agol, E., Ste ff en, J., Sari, R., & Clarkson, W. 2005, MNRAS, 359, 567 Aguichine, A., Mousis, O., Deleuil, M., & Marcq, E. 2021, ApJ, 914, 84 Alexander, R. D. & Armitage, P. J. 2009, ApJ, 704, 989</list_item>
<list_item><location><page_17><loc_6><loc_80><loc_49><loc_81></location>Astropy Collaboration, Price-Whelan, A. M., Lim, P. L., et al. 2022, ApJ, 935, 167</list_item>
<list_item><location><page_17><loc_6><loc_77><loc_49><loc_79></location>Astropy Collaboration, Price-Whelan, A. M., Sip"ocz, B. M., et al. 2018, AJ, 156, 123</list_item>
<list_item><location><page_17><loc_6><loc_75><loc_49><loc_77></location>Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33</list_item>
<list_item><location><page_17><loc_6><loc_74><loc_41><loc_75></location>Bai, J. & Perron, P. 2003, Journal of Applied Econometrics, 18, 1</list_item>
</unordered_list>
<text><location><page_17><loc_6><loc_73><loc_42><loc_74></location>Baranne, A., Queloz, D., Mayor, M., et al. 1996, A&AS, 119, 373</text>
<text><location><page_17><loc_6><loc_72><loc_24><loc_73></location>Barnes, S. A. 2010, ApJ, 722, 222</text>
<unordered_list>
<list_item><location><page_17><loc_6><loc_71><loc_48><loc_72></location>Benz, W., Broeg, C., Fortier, A., et al. 2021, Experimental Astronomy, 51, 109</list_item>
<list_item><location><page_17><loc_6><loc_70><loc_42><loc_71></location>Bergner, J. B., Öberg, K. I., Bergin, E. A., et al. 2020, ApJ, 898, 97</list_item>
<list_item><location><page_17><loc_6><loc_69><loc_43><loc_70></location>Bonfanti, A., Delrez, L., Hooton, M. J., et al. 2021, A&A, 646, A157</list_item>
<list_item><location><page_17><loc_6><loc_68><loc_42><loc_69></location>Bonfanti, A., Gandolfi, D., Egger, J. A., et al. 2023, A&A, 671, L8</list_item>
<list_item><location><page_17><loc_6><loc_67><loc_32><loc_68></location>Bonfanti, A. & Gillon, M. 2020, A&A, 635, A6</list_item>
<list_item><location><page_17><loc_6><loc_66><loc_42><loc_67></location>Bonfanti, A., Ortolani, S., & Nascimbeni, V. 2016, A&A, 585, A5</list_item>
<list_item><location><page_17><loc_6><loc_65><loc_48><loc_66></location>Bonfanti, A., Ortolani, S., Piotto, G., & Nascimbeni, V. 2015, A&A, 575, A18</list_item>
<list_item><location><page_17><loc_6><loc_64><loc_40><loc_65></location>Borsato, L., Degen, D., Leleu, A., et al. 2024, A&A, 689, A52</list_item>
<list_item><location><page_17><loc_6><loc_63><loc_44><loc_64></location>Borsato, L., Malavolta, L., Piotto, G., et al. 2019, MNRAS, 484, 3233</list_item>
<list_item><location><page_17><loc_6><loc_62><loc_43><loc_62></location>Borsato, L., Marzari, F., Nascimbeni, V., et al. 2014, A&A, 571, A38</list_item>
<list_item><location><page_17><loc_6><loc_60><loc_43><loc_61></location>Borsato, L., Piotto, G., Gandolfi, D., et al. 2021, MNRAS, 506, 3810 Borysow, A. 2002, A&A, 390, 779</list_item>
<list_item><location><page_17><loc_6><loc_57><loc_49><loc_59></location>Borysow, A., Jorgensen, U. G., & Fu, Y. 2001, J. Quant. Spectr. Rad. Transf., 68, 235</list_item>
<list_item><location><page_17><loc_6><loc_54><loc_49><loc_57></location>Brasseur, C. E., Phillip, C., Fleming, S. W., Mullally, S. E., & White, R. L. 2019, Astrocut: Tools for creating cutouts of TESS images, Astrophysics Source Code Library, record ascl:1905.007</list_item>
</unordered_list>
<text><location><page_17><loc_6><loc_53><loc_41><loc_54></location>Bruntt, H., Deleuil, M., Fridlund, M., et al. 2010, A&A, 519, A51</text>
<text><location><page_17><loc_6><loc_52><loc_41><loc_53></location>Burrows, A., Marley, M. S., & Sharp, C. M. 2000, ApJ, 531, 438</text>
<unordered_list>
<list_item><location><page_17><loc_6><loc_51><loc_43><loc_52></location>Ciardi, D. R., Fabrycky, D. C., Ford, E. B., et al. 2013, ApJ, 763, 41</list_item>
<list_item><location><page_17><loc_6><loc_50><loc_49><loc_51></location>Correia, A. C. M., Delisle, J.-B., & Laskar, J. 2018, in Handbook of Exoplanets,</list_item>
<list_item><location><page_17><loc_8><loc_48><loc_49><loc_50></location>ed. H. J. Deeg & J. A. Belmonte (Springer International Publishing AG, part of Springer Nature), 12</list_item>
<list_item><location><page_17><loc_6><loc_47><loc_24><loc_48></location>Cubillos, P. E. 2017, ApJ, 850, 32</list_item>
<list_item><location><page_17><loc_6><loc_46><loc_35><loc_47></location>Cubillos, P. E. & Blecic, J. 2021, MNRAS, 505, 2675</list_item>
<list_item><location><page_17><loc_6><loc_43><loc_49><loc_46></location>Cutri, R. M., Skrutskie, M. F., van Dyk, S., et al. 2003, 2MASS All Sky Catalog of point sources. (University of Massachusetts and Infrared Processing and Analysis Center (IPAC / California Institute of Technology))</list_item>
</unordered_list>
<text><location><page_17><loc_6><loc_41><loc_34><loc_42></location>D'Angelo, G. & Lubow, S. H. 2008, ApJ, 685, 560</text>
<text><location><page_17><loc_6><loc_40><loc_48><loc_41></location>Delisle, J. B., Laskar, J., Correia, A. C. M., & Boué, G. 2012, A&A, 546, A71</text>
<text><location><page_17><loc_6><loc_39><loc_47><loc_40></location>Delrez, L., Ehrenreich, D., Alibert, Y., et al. 2021, Nature Astronomy, 5, 775</text>
<text><location><page_17><loc_6><loc_38><loc_39><loc_39></location>Dorn, C., Venturini, J., Khan, A., et al. 2017, A&A, 597, A37</text>
<unordered_list>
<list_item><location><page_17><loc_6><loc_36><loc_49><loc_38></location>Doyle, A. P., Davies, G. R., Smalley, B., Chaplin, W. J., & Elsworth, Y. 2014, MNRAS, 444, 3592</list_item>
<list_item><location><page_17><loc_6><loc_35><loc_42><loc_36></location>Dragomir, D., Kane, S. R., Henry, G. W., et al. 2012, ApJ, 754, 37</list_item>
<list_item><location><page_17><loc_6><loc_33><loc_49><loc_35></location>Dumusque, X., Santos, N. C., Udry, S., Lovis, C., & Bonfils, X. 2011a, A&A, 527, A82</list_item>
<list_item><location><page_17><loc_6><loc_31><loc_49><loc_33></location>Dumusque, X., Udry, S., Lovis, C., Santos, N. C., & Monteiro, M. J. P. F. G. 2011b, A&A, 525, A140</list_item>
</unordered_list>
<text><location><page_17><loc_6><loc_30><loc_46><loc_31></location>Egger, J. A., Osborn, H. P., Kubyshkina, D., et al. 2024, A&A, 688, A223</text>
<unordered_list>
<list_item><location><page_17><loc_6><loc_29><loc_35><loc_30></location>Espinoza, N. & Jordán, A. 2015, MNRAS, 450, 1879</list_item>
<list_item><location><page_17><loc_6><loc_28><loc_45><loc_29></location>Fabrycky, D. C., Lissauer, J. J., Ragozzine, D., et al. 2014, ApJ, 790, 146</list_item>
<list_item><location><page_17><loc_6><loc_26><loc_49><loc_28></location>Foreman-Mackey, D., Farr, W., Sinha, M., et al. 2019, The Journal of Open Source Software, 4, 1864</list_item>
<list_item><location><page_17><loc_6><loc_23><loc_49><loc_25></location>Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306</list_item>
<list_item><location><page_17><loc_6><loc_22><loc_45><loc_23></location>Fridlund, M., Georgieva, I. Y., Bonfanti, A., et al. 2024, A&A, 684, A12</list_item>
<list_item><location><page_17><loc_6><loc_21><loc_48><loc_22></location>Gaia Collaboration, Vallenari, A., Brown, A. G. A., et al. 2023, A&A, 674, A1</list_item>
<list_item><location><page_17><loc_6><loc_20><loc_42><loc_21></location>Gandolfi, D., Barragán, O., Hatzes, A. P., et al. 2017, AJ, 154, 123</list_item>
<list_item><location><page_17><loc_6><loc_18><loc_49><loc_20></location>Gardner, J. P., Mather, J. C., Clampin, M., et al. 2006, Space Sci. Rev., 123, 485 Gelman, A. & Rubin, D. B. 1992, Statistical Science, 7, 457</list_item>
<list_item><location><page_17><loc_6><loc_15><loc_49><loc_18></location>Geweke, J. F. 1991, Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments, Sta ff Report 148, Federal Reserve Bank of Minneapolis</list_item>
<list_item><location><page_17><loc_6><loc_14><loc_43><loc_15></location>Ginsburg, A., Sip"ocz, B. M., Brasseur, C. E., et al. 2019, AJ, 157, 98</list_item>
<list_item><location><page_17><loc_6><loc_13><loc_33><loc_14></location>Goldreich, P. & Tremaine, S. 1980, ApJ, 241, 425</list_item>
<list_item><location><page_17><loc_6><loc_11><loc_49><loc_13></location>Goodman, J. & Weare, J. 2010, Communications in Applied Mathematics and Computational Science, 5, 65</list_item>
<list_item><location><page_17><loc_6><loc_10><loc_43><loc_11></location>Gray, R. O., Corbally, C. J., Garrison, R. F., et al. 2006, AJ, 132, 161</list_item>
<list_item><location><page_17><loc_51><loc_91><loc_94><loc_93></location>Haldemann, J., Dorn, C., Venturini, J., Alibert, Y., & Benz, W. 2024, A&A, 681, A96</list_item>
<list_item><location><page_17><loc_51><loc_90><loc_86><loc_91></location>Handberg, R., Lund, M. N., White, T. R., et al. 2021, AJ, 162, 170</list_item>
<list_item><location><page_17><loc_51><loc_89><loc_93><loc_90></location>Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357</list_item>
</unordered_list>
<text><location><page_17><loc_51><loc_88><loc_94><loc_89></location>Hatzes, A. P. 2016, in Astrophysics and Space Science Library, Vol. 428, Meth-</text>
<unordered_list>
<list_item><location><page_17><loc_53><loc_86><loc_94><loc_87></location>ods of Detecting Exoplanets: 1st Advanced School on Exoplanetary Science, ed. V. Bozza, L. Mancini, & A. Sozzetti, 3</list_item>
<list_item><location><page_17><loc_51><loc_83><loc_94><loc_85></location>Hatzes, A. P. 2019, The Doppler Method for the Detection of Exoplanets (Institute of Physics Publishing)</list_item>
<list_item><location><page_17><loc_51><loc_80><loc_94><loc_83></location>Hatzes, A. P., Dvorak, R., Wuchterl, G., et al. 2010, A&A, 520, A93 Haywood, R. D., Collier Cameron, A., Queloz, D., et al. 2014, MNRAS, 443, 2517</list_item>
<list_item><location><page_17><loc_51><loc_79><loc_87><loc_80></location>Høg, E., Fabricius, C., Makarov, V. V., et al. 2000, A&A, 355, L27</list_item>
</unordered_list>
<text><location><page_17><loc_51><loc_78><loc_92><loc_79></location>Holman, M. J., Fabrycky, D. C., Ragozzine, D., et al. 2010, Science, 330, 51</text>
<text><location><page_17><loc_51><loc_77><loc_89><loc_78></location>Holman, M. J., Winn, J. N., Latham, D. W., et al. 2006, ApJ, 652, 1715</text>
<unordered_list>
<list_item><location><page_17><loc_51><loc_76><loc_85><loc_77></location>Hunter, J. D. 2007, Computing in Science & Engineering, 9, 90</list_item>
</unordered_list>
<text><location><page_17><loc_51><loc_75><loc_91><loc_76></location>Izidoro, A., Ogihara, M., Raymond, S. N., et al. 2017, MNRAS, 470, 1750</text>
<unordered_list>
<list_item><location><page_17><loc_51><loc_70><loc_94><loc_74></location>Jenkins, J. M., Twicken, J. D., McCauli ff , S., et al. 2016, in Society of PhotoOptical Instrumentation Engineers (SPIE) Conference Series, Vol. 9913, Software and Cyberinfrastructure for Astronomy IV, ed. G. Chiozzi & J. C. Guzman, 99133E</list_item>
<list_item><location><page_17><loc_51><loc_68><loc_94><loc_70></location>Kass, R. E. & Raftery, A. E. 1995, Journal of the American Statistical Association, 90, 773</list_item>
<list_item><location><page_17><loc_51><loc_66><loc_92><loc_68></location>Kempton, E. M. R., Bean, J. L., Louie, D. R., et al. 2018, PASP, 130, 114401 Kipping, D. M. 2010, MNRAS, 408, 1758</list_item>
<list_item><location><page_17><loc_51><loc_65><loc_90><loc_66></location>Kossakowski, D., Kürster, M., Henning, T., et al. 2022, A&A, 666, A143</list_item>
<list_item><location><page_17><loc_51><loc_63><loc_94><loc_65></location>Kuerster, M., Schmitt, J. H. M. M., Cutispoto, G., & Dennerl, K. 1997, A&A, 320, 831</list_item>
<list_item><location><page_17><loc_51><loc_62><loc_75><loc_63></location>Kurucz, R. L. 1970, SAO Special Report, 309</list_item>
<list_item><location><page_17><loc_51><loc_60><loc_94><loc_62></location>Kurucz, R. L. 1993, SYNTHE spectrum synthesis programs and line data (Astrophysics Source Code Library)</list_item>
<list_item><location><page_17><loc_51><loc_59><loc_93><loc_59></location>Kurucz, R. L. 2013, ATLAS12: Opacity sampling model atmosphere program</list_item>
<list_item><location><page_17><loc_51><loc_57><loc_68><loc_58></location>Laskar, J. 1997, A&A, 317, L75</list_item>
<list_item><location><page_17><loc_51><loc_56><loc_74><loc_57></location>Laskar, J. 2000, Phys. Rev. Lett., 84, 3240</list_item>
<list_item><location><page_17><loc_51><loc_55><loc_76><loc_56></location>Laskar, J. & Petit, A. C. 2017, A&A, 605, A72</list_item>
<list_item><location><page_17><loc_51><loc_54><loc_79><loc_55></location>Lee, M. H. & Peale, S. J. 2002, arXiv e-prints, astro</list_item>
<list_item><location><page_17><loc_51><loc_53><loc_85><loc_54></location>Leleu, A., Alibert, Y., Hara, N. C., et al. 2021, A&A, 649, A26</list_item>
<list_item><location><page_17><loc_51><loc_51><loc_94><loc_53></location>Li, M., Huang, S., Petaev, M. I., Zhu, Z., & Ste ff en, J. H. 2020, MNRAS, 495, 2543</list_item>
<list_item><location><page_17><loc_51><loc_48><loc_94><loc_51></location>Lightkurve Collaboration, Cardoso, J. V. d. M., Hedges, C., et al. 2018, Lightkurve: Kepler and TESS time series analysis in Python, Astrophysics Source Code Library</list_item>
<list_item><location><page_17><loc_51><loc_47><loc_81><loc_47></location>Lin, D. N. C. & Papaloizou, J. 1979, MNRAS, 186, 799</list_item>
<list_item><location><page_17><loc_51><loc_46><loc_88><loc_46></location>Lindegren, L., Bastian, U., Biermann, M., et al. 2021, A&A, 649, A4</list_item>
</unordered_list>
<text><location><page_17><loc_51><loc_44><loc_90><loc_45></location>Lissauer, J. J., Ragozzine, D., Fabrycky, D. C., et al. 2011, ApJS, 197, 8</text>
<text><location><page_17><loc_51><loc_43><loc_75><loc_44></location>Lovis, C. & Pepe, F. 2007, A&A, 468, 1115</text>
<unordered_list>
<list_item><location><page_17><loc_51><loc_42><loc_88><loc_43></location>Lund, M. N., Handberg, R., Buzasi, D. L., et al. 2021, ApJS, 257, 53</list_item>
<list_item><location><page_17><loc_51><loc_41><loc_86><loc_42></location>Luque, R., Osborn, H. P., Leleu, A., et al. 2023, Nature, 623, 932</list_item>
</unordered_list>
<text><location><page_17><loc_51><loc_40><loc_83><loc_41></location>Mamajek, E. E. & Hillenbrand, L. A. 2008, ApJ, 687, 1264</text>
<unordered_list>
<list_item><location><page_17><loc_51><loc_39><loc_84><loc_40></location>Marigo, P., Girardi, L., Bressan, A., et al. 2017, ApJ, 835, 77</list_item>
<list_item><location><page_17><loc_51><loc_38><loc_88><loc_39></location>Mayor, M., Pepe, F., Queloz, D., et al. 2003, The Messenger, 114, 20</list_item>
</unordered_list>
<text><location><page_17><loc_51><loc_37><loc_90><loc_38></location>McKay, A. J., DiSanti, M. A., Kelley, M. S. P., et al. 2019, AJ, 158, 128</text>
<unordered_list>
<list_item><location><page_17><loc_51><loc_35><loc_90><loc_37></location>Mishra, L., Alibert, Y., Udry, S., & Mordasini, C. 2023, A&A, 670, A68 Morton, T. D. 2012, ApJ, 761, 6</list_item>
<list_item><location><page_17><loc_51><loc_33><loc_94><loc_34></location>Morton, T. D. 2015, VESPA: False positive probabilities calculator, Astrophysics Source Code Library, record ascl:1503.011</list_item>
<list_item><location><page_17><loc_51><loc_31><loc_82><loc_32></location>Mumma, M. J. & Charnley, S. B. 2011, ARA&A, 49, 471</list_item>
<list_item><location><page_17><loc_51><loc_30><loc_88><loc_31></location>Murdoch, K. A., Hearnshaw, J. B., & Clark, M. 1993, ApJ, 413, 349</list_item>
<list_item><location><page_17><loc_51><loc_29><loc_89><loc_30></location>Nascimbeni, V., Borsato, L., Leonardi, P., et al. 2024, A&A, 690, A349</list_item>
<list_item><location><page_17><loc_51><loc_28><loc_88><loc_29></location>Nascimbeni, V., Borsato, L., Zingales, T., et al. 2023, A&A, 673, A42</list_item>
<list_item><location><page_17><loc_51><loc_27><loc_83><loc_28></location>Nelson, B., Ford, E. B., & Payne, M. J. 2014, ApJS, 210, 11</list_item>
<list_item><location><page_17><loc_51><loc_25><loc_94><loc_27></location>Noyes, R. W., Hartmann, L. W., Baliunas, S. L., Duncan, D. K., & Vaughan, A. H. 1984, ApJ, 279, 763</list_item>
<list_item><location><page_17><loc_51><loc_24><loc_84><loc_25></location>Otegi, J. F., Bouchy, F., & Helled, R. 2020a, A&A, 634, A43</list_item>
<list_item><location><page_17><loc_51><loc_23><loc_85><loc_24></location>Otegi, J. F., Dorn, C., Helled, R., et al. 2020b, A&A, 640, A135</list_item>
<list_item><location><page_17><loc_51><loc_22><loc_84><loc_23></location>Otegi, J. F., Helled, R., & Bouchy, F. 2022, A&A, 658, A107</list_item>
<list_item><location><page_17><loc_51><loc_21><loc_88><loc_21></location>Parviainen, H., Pallé, E., Nortmann, L., et al. 2016, A&A, 585, A114</list_item>
<list_item><location><page_17><loc_51><loc_19><loc_83><loc_20></location>Pepe, F., Mayor, M., Galland, F., et al. 2002, A&A, 388, 632</list_item>
<list_item><location><page_17><loc_51><loc_18><loc_93><loc_19></location>Perryman, M. A. C., Lindegren, L., Kovalevsky, J., et al. 1997, A&A, 323, L49</list_item>
<list_item><location><page_17><loc_51><loc_17><loc_89><loc_18></location>Persson, C. M., Fridlund, M., Barragán, O., et al. 2018, A&A, 618, A33</list_item>
<list_item><location><page_17><loc_51><loc_16><loc_86><loc_17></location>Petigura, E. A., Benneke, B., Batygin, K., et al. 2018, AJ, 156, 89</list_item>
<list_item><location><page_17><loc_51><loc_15><loc_89><loc_16></location>Petigura, E. A., Howard, A. W., Lopez, E. D., et al. 2016, ApJ, 818, 36</list_item>
<list_item><location><page_17><loc_51><loc_14><loc_82><loc_15></location>Petit, A. C., Laskar, J., & Boué, G. 2018, A&A, 617, A93</list_item>
<list_item><location><page_17><loc_51><loc_13><loc_79><loc_14></location>Piskunov, N. & Valenti, J. A. 2017, A&A, 597, A16</list_item>
<list_item><location><page_17><loc_51><loc_11><loc_94><loc_13></location>Piskunov, N. E., Kupka, F., Ryabchikova, T. A., Weiss, W. W., & Je ff ery, C. S. 1995, A&AS, 112, 525</list_item>
<list_item><location><page_17><loc_51><loc_10><loc_83><loc_11></location>Pont, F., Zucker, S., & Queloz, D. 2006, MNRAS, 373, 231</list_item>
</unordered_list>
<text><location><page_18><loc_6><loc_90><loc_49><loc_93></location>Pontoppidan, K. M., Pickering, T. E., Laidler, V. G., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9910, Observatory Operations: Strategies, Processes, and Systems VI, ed.</text>
<unordered_list>
<list_item><location><page_18><loc_8><loc_89><loc_34><loc_90></location>A. B. Peck, R. L. Seaman, & C. R. Benn, 991016</list_item>
</unordered_list>
<text><location><page_18><loc_6><loc_88><loc_42><loc_89></location>Queloz, D., Henry, G. W., Sivan, J. P., et al. 2001, A&A, 379, 279</text>
<text><location><page_18><loc_6><loc_87><loc_30><loc_88></location>Rein, H. & Liu, S. F. 2012, A&A, 537, A128</text>
<unordered_list>
<list_item><location><page_18><loc_6><loc_86><loc_32><loc_87></location>Rein, H. & Tamayo, D. 2015, MNRAS, 452, 376</list_item>
</unordered_list>
<text><location><page_18><loc_6><loc_85><loc_11><loc_85></location>Richard,</text>
<text><location><page_18><loc_12><loc_85><loc_14><loc_85></location>C.,</text>
<text><location><page_18><loc_16><loc_85><loc_20><loc_85></location>Gordon,</text>
<text><location><page_18><loc_22><loc_85><loc_23><loc_85></location>I.</text>
<text><location><page_18><loc_24><loc_85><loc_26><loc_85></location>E.,</text>
<text><location><page_18><loc_27><loc_85><loc_33><loc_85></location>Rothman,</text>
<text><location><page_18><loc_34><loc_85><loc_36><loc_85></location>L.</text>
<text><location><page_18><loc_37><loc_85><loc_39><loc_85></location>S.,</text>
<text><location><page_18><loc_40><loc_85><loc_41><loc_85></location>et</text>
<text><location><page_18><loc_43><loc_85><loc_44><loc_85></location>al.</text>
<text><location><page_18><loc_46><loc_85><loc_49><loc_85></location>2012,</text>
<text><location><page_18><loc_8><loc_84><loc_30><loc_84></location>J. Quant. Spectr. Rad. Transf., 113, 1276</text>
<unordered_list>
<list_item><location><page_18><loc_6><loc_81><loc_49><loc_83></location>Ricker, G. R., Winn, J. N., Vanderspek, R., et al. 2015, Journal of Astronomical Telescopes, Instruments, and Systems, 1, 014003</list_item>
<list_item><location><page_18><loc_6><loc_79><loc_49><loc_81></location>Rothman, L. S., Gordon, I. E., Barber, R. J., et al. 2010, J. Quant. Spectr. Rad. Transf., 111, 2139</list_item>
<list_item><location><page_18><loc_6><loc_77><loc_44><loc_79></location>Santos, N. C., Sousa, S. G., Mortier, A., et al. 2013, A&A, 556, A150 Schwarz, G. 1978, Annals of Statistics, 6, 461</list_item>
<list_item><location><page_18><loc_6><loc_76><loc_44><loc_77></location>Simola, U., Bonfanti, A., Dumusque, X., et al. 2022, A&A, 664, A127</list_item>
<list_item><location><page_18><loc_6><loc_75><loc_45><loc_76></location>Simola, U., Dumusque, X., & Cisewski-Kehe, J. 2019, A&A, 622, A131</list_item>
<list_item><location><page_18><loc_6><loc_74><loc_46><loc_75></location>Smith, J. C., Stumpe, M. C., Van Cleve, J. E., et al. 2012, PASP, 124, 1000</list_item>
</unordered_list>
<text><location><page_18><loc_6><loc_73><loc_49><loc_74></location>Sneden, C. A. 1973, PhD thesis, THE UNIVERSITY OF TEXAS AT AUSTIN.</text>
<unordered_list>
<list_item><location><page_18><loc_6><loc_70><loc_49><loc_73></location>Sousa, S. G. 2014, in Determination of Atmospheric Parameters of B-, A-, Fand G-Type Stars., ed. E. Niemczura, B. Smalley, & W. Pych (Springer International Publishing (Cham)), 297-310</list_item>
<list_item><location><page_18><loc_6><loc_67><loc_49><loc_70></location>Sousa, S. G., Adibekyan, V., Delgado-Mena, E., et al. 2021, A&A, 656, A53 Sousa, S. G., Santos, N. C., Adibekyan, V., Delgado-Mena, E., & Israelian, G. 2015, A&A, 577, A67</list_item>
<list_item><location><page_18><loc_6><loc_65><loc_49><loc_67></location>Sousa, S. G., Santos, N. C., Israelian, G., Mayor, M., & Monteiro, M. J. P. F. G. 2007, A&A, 469, 783</list_item>
<list_item><location><page_18><loc_6><loc_64><loc_42><loc_64></location>Sousa, S. G., Santos, N. C., Mayor, M., et al. 2008, A&A, 487, 373</list_item>
<list_item><location><page_18><loc_6><loc_63><loc_43><loc_63></location>Storn, R. & Price, K. 1997, Journal of Global Optimization, 11, 341</list_item>
<list_item><location><page_18><loc_6><loc_62><loc_46><loc_62></location>Stumpe, M. C., Smith, J. C., Catanzarite, J. H., et al. 2014, PASP, 126, 100</list_item>
<list_item><location><page_18><loc_6><loc_61><loc_45><loc_61></location>Stumpe, M. C., Smith, J. C., Van Cleve, J. E., et al. 2012, PASP, 124, 985</list_item>
<list_item><location><page_18><loc_6><loc_58><loc_49><loc_60></location>Suárez Mascareño, A., Rebolo, R., González Hernández, J. I., & Esposito, M. 2017, MNRAS, 468, 4772</list_item>
<list_item><location><page_18><loc_6><loc_57><loc_33><loc_58></location>Sundman, K. F. 1913, Acta Mathematica, 36, 105</list_item>
<list_item><location><page_18><loc_6><loc_56><loc_40><loc_57></location>Tanaka, H., Takeuchi, T., & Ward, W. R. 2002, ApJ, 565, 1257</list_item>
<list_item><location><page_18><loc_6><loc_54><loc_49><loc_56></location>Tennyson, J., Yurchenko, S. N., Al-Refaie, A. F., et al. 2016, Journal of Molecular Spectroscopy, 327, 73</list_item>
</unordered_list>
<text><location><page_18><loc_6><loc_53><loc_45><loc_54></location>ter Braak, C. J. F. & Vrugt, J. A. 2008, Statistics and Computing, 18, 435</text>
<text><location><page_18><loc_6><loc_52><loc_43><loc_53></location>Thiabaud, A., Marboeuf, U., Alibert, Y., et al. 2014, A&A, 562, A27</text>
<unordered_list>
<list_item><location><page_18><loc_6><loc_50><loc_49><loc_52></location>Thiabaud, A., Marboeuf, U., Alibert, Y., Leya, I., & Mezger, K. 2015, A&A, 574, A138</list_item>
<list_item><location><page_18><loc_6><loc_49><loc_24><loc_50></location>Trotta, R. 2007, MNRAS, 378, 72</list_item>
<list_item><location><page_18><loc_6><loc_48><loc_40><loc_49></location>Van Eylen, V., Albrecht, S., Huang, X., et al. 2019, AJ, 157, 61</list_item>
<list_item><location><page_18><loc_6><loc_47><loc_46><loc_48></location>Vanderburg, A., Huang, C. X., Rodriguez, J. E., et al. 2019, ApJ, 881, L19</list_item>
<list_item><location><page_18><loc_6><loc_46><loc_47><loc_47></location>Vanderburg, A., Plavchan, P., Johnson, J. A., et al. 2016, MNRAS, 459, 3565</list_item>
<list_item><location><page_18><loc_6><loc_44><loc_49><loc_46></location>Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261 Walsh, C., Nomura, H., & van Dishoeck, E. 2015, A&A, 582, A88</list_item>
<list_item><location><page_18><loc_6><loc_43><loc_33><loc_44></location>Weiss, L. M. & Marcy, G. W. 2014, ApJ, 783, L6</list_item>
<list_item><location><page_18><loc_6><loc_42><loc_43><loc_43></location>Weiss, L. M., Marcy, G. W., Petigura, E. A., et al. 2018, AJ, 155, 48</list_item>
<list_item><location><page_18><loc_6><loc_37><loc_49><loc_42></location>Wildi, F., Pepe, F., Chazelas, B., Lo Curto, G., & Lovis, C. 2010, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7735, Ground-based and Airborne Instrumentation for Astronomy III, ed. I. S. McLean, S. K. Ramsay, & H. Takami, 77354X</list_item>
<list_item><location><page_18><loc_6><loc_35><loc_49><loc_37></location>Winn, J. N. 2010, in Exoplanets, ed. S. Seager (University of Arizona Press, Tucson, AZ), 55-77</list_item>
<list_item><location><page_18><loc_6><loc_34><loc_36><loc_35></location>Winn, J. N. & Fabrycky, D. C. 2015, ARA&A, 53, 409</list_item>
</unordered_list>
<text><location><page_18><loc_6><loc_33><loc_32><loc_34></location>Wisdom, J. & Holman, M. 1991, AJ, 102, 1528</text>
<unordered_list>
<list_item><location><page_18><loc_6><loc_32><loc_35><loc_33></location>Zechmeister, M. & Kürster, M. 2009, A&A, 496, 577</list_item>
<list_item><location><page_18><loc_7><loc_27><loc_49><loc_30></location>1 Austrian Academy of Sciences, Schmiedlstrasse 6, A-8042 Graz, Austria</list_item>
<list_item><location><page_18><loc_8><loc_26><loc_33><loc_27></location>e-mail: andrea.bonfanti@oeaw.ac.at</list_item>
<list_item><location><page_18><loc_7><loc_24><loc_49><loc_26></location>2 Dipartimento di Fisica, Università degli Studi di Torino, via Pietro Giuria 1, I-10125, Torino, Italy</list_item>
<list_item><location><page_18><loc_7><loc_22><loc_49><loc_24></location>3 Department of Physics and Astronomy, Uppsala University, Box 516, Uppsala SE-75120, Sweden</list_item>
<list_item><location><page_18><loc_7><loc_19><loc_49><loc_21></location>4 INAF, Osservatorio Astronomico di Padova, Vicolo dell'Osservatorio 5, 35122 Padova, Italy</list_item>
<list_item><location><page_18><loc_7><loc_17><loc_49><loc_19></location>5 Weltraumforschung und Planetologie, Physikalisches Institut, University of Bern, Gesellschaftsstrasse 6, 3012 Bern, Switzerland</list_item>
<list_item><location><page_18><loc_7><loc_14><loc_49><loc_17></location>6 INAF, Osservatorio Astrofisico di Torino, Via Osservatorio, 20, I10025 Pino Torinese To, Italy</list_item>
<list_item><location><page_18><loc_7><loc_12><loc_49><loc_14></location>7 Department of Physics, University of Warwick, Coventry CV4 7AL, UK</list_item>
<list_item><location><page_18><loc_7><loc_10><loc_49><loc_12></location>8 Centre for Exoplanets and Habitability, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_18><loc_52><loc_90><loc_94><loc_93></location>9 Departamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, Campus Universitário, Natal, RN, 59072-970, Brazil</list_item>
<list_item><location><page_18><loc_51><loc_87><loc_94><loc_90></location>10 Leiden Observatory, University of Leiden, PO Box 9513, 2300 RA Leiden, The Netherlands</list_item>
<list_item><location><page_18><loc_51><loc_84><loc_94><loc_87></location>11 Chalmers University of Technology, Department of Space, Earth and Environment, Onsala Space Observatory, SE-439 92 Onsala, Sweden</list_item>
<list_item><location><page_18><loc_51><loc_81><loc_94><loc_84></location>12 INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5 Florence, Italy</list_item>
<list_item><location><page_18><loc_51><loc_79><loc_94><loc_81></location>13 Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal</list_item>
<list_item><location><page_18><loc_51><loc_77><loc_94><loc_79></location>14 School of Physics and Astronomy, Physical Science Building, North Haugh, St Andrews, United Kingdom</list_item>
<list_item><location><page_18><loc_51><loc_73><loc_94><loc_76></location>15 Rhenish Institute for Environmental Research, Dep. of Planetary Research, University of Cologne, Aachener Str. 209, 50931 Cologne, Germany</list_item>
<list_item><location><page_18><loc_51><loc_71><loc_94><loc_73></location>16 Institute of Planetary Research, German Aerospace Center (DLR), Rutherfordstrasse 2, D-12489 Berlin, Germany</list_item>
<list_item><location><page_18><loc_51><loc_68><loc_94><loc_71></location>17 Thüringer Landessternwarte Tautenburg, Sternwarte 5, D-07778 Tautenburg, Germany</list_item>
<list_item><location><page_18><loc_51><loc_66><loc_94><loc_68></location>18 University Observatory Munich, Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679 Munich, Germany</list_item>
<list_item><location><page_18><loc_51><loc_64><loc_94><loc_66></location>19 European Southern Observatory, Alonso de Cordova 3107, Vitacura, Casilla, 19001, Santiago, Chile</list_item>
<list_item><location><page_18><loc_51><loc_61><loc_94><loc_63></location>20 European Southern Observatory, Alonso de Cordova 3107, Vitacura, Santiago de Chile, Chile</list_item>
<list_item><location><page_18><loc_51><loc_59><loc_94><loc_61></location>21 Centro de Astrobiología (CAB), CSIC-INTA, Camino Bajo del Castillo s / n, 28692, Villanueva de la Cañada (Madrid), Spain</list_item>
<list_item><location><page_18><loc_51><loc_58><loc_94><loc_59></location>22 McDonald Observatory, The University of Texas, Austin Texas USA</list_item>
<list_item><location><page_18><loc_51><loc_55><loc_94><loc_58></location>23 Center for Planetary Systems Habitability, The University of Texas, Austin Texas</list_item>
<list_item><location><page_18><loc_51><loc_53><loc_94><loc_55></location>24 Department of Astronomy & Astrophysics, University of Chicago, Chicago, IL 60637, USA</list_item>
<list_item><location><page_18><loc_51><loc_51><loc_94><loc_53></location>25 Astronomy Department and Van Vleck Observatory, Wesleyan University, Middletown, CT 06459, USA</list_item>
<list_item><location><page_18><loc_51><loc_48><loc_94><loc_50></location>26 Departamento de Física e Astronomia, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre, 4169-007 Porto, Portugal</list_item>
<list_item><location><page_18><loc_51><loc_46><loc_94><loc_48></location>27 Department of Astronomy of the University of Geneva, chemin Pegasi 51, 1290 Versoix, Switzerland</list_item>
<list_item><location><page_18><loc_51><loc_43><loc_94><loc_46></location>28 Astrobiology Center, NINS, 2-21-1 Osawa, Mitaka, Tokyo 1818588, Japan</list_item>
<list_item><location><page_18><loc_51><loc_41><loc_94><loc_43></location>29 National Astronomical Observatory of Japan, NINS, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan</list_item>
<list_item><location><page_18><loc_51><loc_38><loc_94><loc_41></location>30 Astronomical Science Program, Graduate University for Advanced Studies, SOKENDAI, 2-21-1, Osawa, Mitaka, Tokyo, 181-8588, Japan</list_item>
<list_item><location><page_18><loc_51><loc_35><loc_94><loc_37></location>31 Instituto de Astrofísica de Canarias (IAC), calle Vía Láctea s / n, 38205 La Laguna, Tenerife, Spain</list_item>
<list_item><location><page_18><loc_51><loc_33><loc_94><loc_35></location>32 Departamento de Astrofísica, Universidad de La Laguna (ULL), 38206 La Laguna, Tenerife, Spain</list_item>
<list_item><location><page_18><loc_51><loc_29><loc_94><loc_33></location>33 Institute of Astronomy, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudzia¸dzka 5, 87-100 Toru'n, Poland</list_item>
<list_item><location><page_18><loc_51><loc_27><loc_94><loc_29></location>34 Department of Physics and Astronomy, McMaster University, 1280 Main St W, Hamilton, ON, L8S 4L8, Canada</list_item>
<list_item><location><page_18><loc_51><loc_25><loc_94><loc_27></location>35 Department of Physics, ETH Zurich, Wolfgang-Pauli-Strasse 2, CH8093 Zurich, Switzerland</list_item>
</unordered_list>
<text><location><page_19><loc_11><loc_95><loc_18><loc_96></location>A. Bonfanti</text>
<figure>
<location><page_19><loc_7><loc_72><loc_48><loc_93></location>
<caption>Fig. A.1: TTV amplitudes obtained for TOI-396 b (black markers) and TOI-396 c (red markers). This is a superposition of the first two panels of Figure 4 emphasising the anti-correlation pattern.</caption>
</figure>
<section_header_level_1><location><page_19><loc_6><loc_62><loc_37><loc_63></location>Appendix A: Supplementary material</section_header_level_1>
<figure>
<location><page_20><loc_16><loc_16><loc_89><loc_93></location>
<caption>Fig. A.2: Left column : Generalised Lomb-Scargle periodograms of TESS Sector 3 ( first row ), 4 ( second row ), 30 ( third row ), and 31 ( fourth row ) raw photometry. Right column : Generalised Lomb-Scargle periodograms of the residuals after removing the best fitting sinusoidal signals at the most significant frequency identified in the periodograms of the corresponding TESS time series (left column). The long dashed red line marks the FAP at 0.1%. Significant peaks are detected in the 6-8 day range, which we attribute to the presence of active regions co-rotating with the star.</caption>
</figure>
<text><location><page_21><loc_11><loc_95><loc_18><loc_96></location>A. Bonfanti</text>
<text><location><page_21><loc_21><loc_95><loc_89><loc_96></location>et al.: Radii, masses, and transit-timing variations of the three-planet system orbiting the naked-eye star TOI-396</text>
<figure>
<location><page_21><loc_19><loc_95><loc_21><loc_96></location>
<caption>Table A.1: Radial velocities RV as extracted from the centred CCFs with their errors σ RV. They are followed by the hyperparameters inferred from the SN fit onto the CCFs (i.e. FWHMSN, A , and γ ) and by the bp -detrended RV values ( RV bp) with their errors which also account for the jitter ( σ RV(bp + jitter)).</caption>
</figure>
<table>
<location><page_21><loc_17><loc_11><loc_83><loc_88></location>
</table>
<paragraph><location><page_22><loc_39><loc_95><loc_61><loc_96></location>A & A proofs: manuscript no. TOI-396</paragraph>
<table>
<location><page_22><loc_17><loc_33><loc_83><loc_91></location>
<caption>Table A.1: continued.</caption>
</table>
<figure>
<location><page_23><loc_19><loc_95><loc_21><loc_96></location>
</figure>
<table>
<location><page_23><loc_11><loc_21><loc_43><loc_90></location>
<caption>Table A.2: Polynomial detrending baseline applied to the TESS LCs within the MCMC scheme.</caption>
</table>
<text><location><page_23><loc_6><loc_14><loc_49><loc_20></location>Notes. TESS (TE) LCs are identified by a counter based on the chronological order of observation. In particular, LCs from 1 to 11, from 12 to 21, from 22 to 32, and from 33 to 41 were extracted from Sector 3, 4, 30, and 31, respectively. c indicates a normalisation scalar; see text for further details.</text>
<table>
<location><page_24><loc_7><loc_45><loc_38><loc_90></location>
<caption>Table A.3: Timing of each transit event T tr and the corresponding TTV amplitude as computed with respect to linear ephemerides. The last column specifies the TESS sector where the transit occurs.</caption>
</table>
<text><location><page_24><loc_51><loc_86><loc_52><loc_87></location>T</text>
<text><location><page_24><loc_52><loc_86><loc_58><loc_87></location>tr [BJD]</text>
<text><location><page_24><loc_67><loc_86><loc_74><loc_87></location>TTV [min]</text>
<text><location><page_24><loc_76><loc_86><loc_80><loc_87></location>Sector</text>
<text><location><page_24><loc_51><loc_84><loc_55><loc_85></location>8385</text>
<text><location><page_24><loc_55><loc_84><loc_55><loc_84></location>.</text>
<text><location><page_24><loc_55><loc_84><loc_57><loc_85></location>794</text>
<text><location><page_24><loc_55><loc_83><loc_55><loc_83></location>.</text>
<text><location><page_24><loc_51><loc_83><loc_55><loc_84></location>8391</text>
<text><location><page_24><loc_55><loc_81><loc_55><loc_81></location>.</text>
<text><location><page_24><loc_51><loc_81><loc_55><loc_82></location>8397</text>
<text><location><page_24><loc_57><loc_85><loc_58><loc_85></location>+</text>
<text><location><page_24><loc_58><loc_85><loc_59><loc_85></location>0</text>
<text><location><page_24><loc_57><loc_84><loc_58><loc_85></location>-</text>
<text><location><page_24><loc_58><loc_84><loc_59><loc_85></location>0</text>
<text><location><page_24><loc_59><loc_85><loc_59><loc_85></location>.</text>
<text><location><page_24><loc_59><loc_85><loc_61><loc_85></location>010</text>
<text><location><page_24><loc_59><loc_84><loc_59><loc_84></location>.</text>
<text><location><page_24><loc_58><loc_83><loc_59><loc_84></location>+</text>
<text><location><page_24><loc_59><loc_83><loc_60><loc_84></location>0</text>
<text><location><page_24><loc_58><loc_82><loc_59><loc_83></location>-</text>
<text><location><page_24><loc_59><loc_82><loc_60><loc_83></location>0</text>
<text><location><page_24><loc_55><loc_83><loc_58><loc_84></location>7447</text>
<text><location><page_24><loc_58><loc_81><loc_59><loc_82></location>+</text>
<text><location><page_24><loc_59><loc_81><loc_60><loc_82></location>0</text>
<text><location><page_24><loc_58><loc_81><loc_59><loc_81></location>-</text>
<text><location><page_24><loc_59><loc_81><loc_60><loc_81></location>0</text>
<text><location><page_24><loc_55><loc_81><loc_58><loc_82></location>7195</text>
<text><location><page_24><loc_55><loc_79><loc_55><loc_80></location>.</text>
<text><location><page_24><loc_51><loc_79><loc_55><loc_80></location>8403</text>
<text><location><page_24><loc_55><loc_78><loc_55><loc_78></location>.</text>
<text><location><page_24><loc_51><loc_78><loc_55><loc_79></location>8415</text>
<text><location><page_24><loc_55><loc_76><loc_55><loc_76></location>.</text>
<text><location><page_24><loc_51><loc_76><loc_55><loc_77></location>8421</text>
<text><location><page_24><loc_55><loc_74><loc_55><loc_75></location>.</text>
<text><location><page_24><loc_51><loc_74><loc_55><loc_75></location>8427</text>
<text><location><page_24><loc_55><loc_73><loc_55><loc_73></location>.</text>
<text><location><page_24><loc_51><loc_73><loc_55><loc_74></location>8433</text>
<text><location><page_24><loc_55><loc_71><loc_55><loc_71></location>.</text>
<text><location><page_24><loc_51><loc_71><loc_55><loc_72></location>9120</text>
<text><location><page_24><loc_55><loc_79><loc_58><loc_80></location>6857</text>
<text><location><page_24><loc_59><loc_79><loc_60><loc_80></location>±</text>
<text><location><page_24><loc_60><loc_79><loc_61><loc_80></location>0</text>
<text><location><page_24><loc_58><loc_78><loc_59><loc_79></location>+</text>
<text><location><page_24><loc_59><loc_78><loc_60><loc_79></location>0</text>
<text><location><page_24><loc_58><loc_77><loc_59><loc_78></location>-</text>
<text><location><page_24><loc_59><loc_77><loc_60><loc_78></location>0</text>
<text><location><page_24><loc_55><loc_78><loc_58><loc_79></location>6337</text>
<text><location><page_24><loc_58><loc_77><loc_59><loc_77></location>+</text>
<text><location><page_24><loc_59><loc_77><loc_60><loc_77></location>0</text>
<text><location><page_24><loc_58><loc_76><loc_59><loc_76></location>-</text>
<text><location><page_24><loc_59><loc_76><loc_60><loc_76></location>0</text>
<text><location><page_24><loc_55><loc_76><loc_58><loc_77></location>6062</text>
<text><location><page_24><loc_58><loc_75><loc_59><loc_76></location>+</text>
<text><location><page_24><loc_59><loc_75><loc_60><loc_76></location>0</text>
<text><location><page_24><loc_58><loc_74><loc_59><loc_75></location>-</text>
<text><location><page_24><loc_59><loc_74><loc_60><loc_75></location>0</text>
<text><location><page_24><loc_55><loc_74><loc_58><loc_75></location>5817</text>
<text><location><page_24><loc_58><loc_73><loc_59><loc_74></location>+</text>
<text><location><page_24><loc_59><loc_73><loc_60><loc_74></location>0</text>
<text><location><page_24><loc_58><loc_73><loc_59><loc_73></location>-</text>
<text><location><page_24><loc_59><loc_72><loc_60><loc_73></location>0</text>
<text><location><page_24><loc_55><loc_73><loc_58><loc_74></location>5482</text>
<text><location><page_24><loc_58><loc_72><loc_59><loc_72></location>+</text>
<text><location><page_24><loc_59><loc_72><loc_60><loc_72></location>0</text>
<text><location><page_24><loc_58><loc_71><loc_59><loc_72></location>-</text>
<text><location><page_24><loc_59><loc_71><loc_60><loc_72></location>0</text>
<text><location><page_24><loc_60><loc_78><loc_60><loc_78></location>.</text>
<text><location><page_24><loc_60><loc_77><loc_60><loc_78></location>.</text>
<text><location><page_24><loc_60><loc_77><loc_60><loc_77></location>.</text>
<text><location><page_24><loc_60><loc_76><loc_60><loc_76></location>.</text>
<text><location><page_24><loc_60><loc_75><loc_60><loc_75></location>.</text>
<text><location><page_24><loc_60><loc_74><loc_60><loc_74></location>.</text>
<text><location><page_24><loc_60><loc_73><loc_60><loc_73></location>.</text>
<text><location><page_24><loc_60><loc_72><loc_60><loc_73></location>.</text>
<text><location><page_24><loc_60><loc_72><loc_60><loc_72></location>.</text>
<text><location><page_24><loc_60><loc_71><loc_60><loc_71></location>.</text>
<text><location><page_24><loc_55><loc_71><loc_58><loc_72></location>5407</text>
<text><location><page_24><loc_55><loc_70><loc_55><loc_70></location>.</text>
<text><location><page_24><loc_51><loc_70><loc_55><loc_71></location>9126</text>
<text><location><page_24><loc_55><loc_70><loc_57><loc_71></location>518</text>
<text><location><page_24><loc_58><loc_70><loc_59><loc_71></location>±</text>
<text><location><page_24><loc_59><loc_70><loc_60><loc_71></location>0</text>
<text><location><page_24><loc_55><loc_68><loc_55><loc_68></location>.</text>
<text><location><page_24><loc_51><loc_68><loc_55><loc_69></location>9132</text>
<text><location><page_24><loc_55><loc_66><loc_55><loc_67></location>.</text>
<text><location><page_24><loc_51><loc_66><loc_55><loc_67></location>9138</text>
<text><location><page_24><loc_55><loc_65><loc_55><loc_65></location>.</text>
<text><location><page_24><loc_51><loc_65><loc_55><loc_66></location>9150</text>
<text><location><page_24><loc_55><loc_63><loc_55><loc_63></location>.</text>
<text><location><page_24><loc_51><loc_63><loc_55><loc_64></location>9156</text>
<text><location><page_24><loc_55><loc_61><loc_55><loc_62></location>.</text>
<text><location><page_24><loc_51><loc_61><loc_55><loc_62></location>9162</text>
<text><location><page_24><loc_55><loc_60><loc_55><loc_60></location>.</text>
<text><location><page_24><loc_51><loc_60><loc_55><loc_61></location>9168</text>
<text><location><page_24><loc_55><loc_56><loc_55><loc_56></location>.</text>
<text><location><page_24><loc_51><loc_56><loc_55><loc_57></location>8387</text>
<text><location><page_24><loc_55><loc_54><loc_55><loc_54></location>.</text>
<text><location><page_24><loc_51><loc_54><loc_55><loc_55></location>8398</text>
<text><location><page_24><loc_55><loc_53><loc_55><loc_53></location>.</text>
<text><location><page_24><loc_51><loc_52><loc_55><loc_54></location>8432</text>
<text><location><page_24><loc_55><loc_51><loc_55><loc_51></location>.</text>
<text><location><page_24><loc_51><loc_51><loc_55><loc_52></location>9117</text>
<text><location><page_24><loc_55><loc_49><loc_55><loc_49></location>.</text>
<text><location><page_24><loc_51><loc_49><loc_55><loc_50></location>9139</text>
<text><location><page_24><loc_55><loc_48><loc_55><loc_48></location>.</text>
<text><location><page_24><loc_51><loc_48><loc_55><loc_49></location>9150</text>
<text><location><page_24><loc_55><loc_46><loc_55><loc_46></location>.</text>
<text><location><page_24><loc_51><loc_46><loc_55><loc_47></location>9162</text>
<text><location><page_24><loc_55><loc_68><loc_58><loc_69></location>4916</text>
<text><location><page_24><loc_58><loc_68><loc_59><loc_69></location>+</text>
<text><location><page_24><loc_59><loc_68><loc_60><loc_69></location>0</text>
<text><location><page_24><loc_58><loc_68><loc_59><loc_68></location>-</text>
<text><location><page_24><loc_59><loc_68><loc_60><loc_68></location>0</text>
<text><location><page_24><loc_58><loc_67><loc_59><loc_68></location>+</text>
<text><location><page_24><loc_59><loc_67><loc_60><loc_68></location>0</text>
<text><location><page_24><loc_58><loc_66><loc_59><loc_67></location>-</text>
<text><location><page_24><loc_59><loc_66><loc_60><loc_67></location>0</text>
<text><location><page_24><loc_55><loc_66><loc_58><loc_67></location>4731</text>
<text><location><page_24><loc_58><loc_65><loc_59><loc_66></location>+</text>
<text><location><page_24><loc_59><loc_65><loc_60><loc_66></location>0</text>
<text><location><page_24><loc_58><loc_64><loc_59><loc_65></location>-</text>
<text><location><page_24><loc_59><loc_64><loc_60><loc_65></location>0</text>
<text><location><page_24><loc_55><loc_65><loc_58><loc_66></location>4169</text>
<text><location><page_24><loc_58><loc_64><loc_59><loc_64></location>+</text>
<text><location><page_24><loc_59><loc_64><loc_60><loc_64></location>0</text>
<text><location><page_24><loc_58><loc_63><loc_59><loc_63></location>-</text>
<text><location><page_24><loc_59><loc_63><loc_60><loc_63></location>0</text>
<text><location><page_24><loc_55><loc_63><loc_58><loc_64></location>4050</text>
<text><location><page_24><loc_58><loc_62><loc_59><loc_63></location>+</text>
<text><location><page_24><loc_59><loc_62><loc_60><loc_63></location>0</text>
<text><location><page_24><loc_58><loc_61><loc_59><loc_62></location>-</text>
<text><location><page_24><loc_59><loc_61><loc_60><loc_62></location>0</text>
<text><location><page_24><loc_55><loc_61><loc_58><loc_62></location>3798</text>
<text><location><page_24><loc_58><loc_60><loc_59><loc_61></location>+</text>
<text><location><page_24><loc_59><loc_60><loc_60><loc_61></location>0</text>
<text><location><page_24><loc_58><loc_59><loc_59><loc_60></location>-</text>
<text><location><page_24><loc_59><loc_59><loc_60><loc_60></location>0</text>
<text><location><page_24><loc_55><loc_60><loc_58><loc_61></location>3562</text>
<text><location><page_24><loc_58><loc_56><loc_59><loc_57></location>+</text>
<text><location><page_24><loc_59><loc_56><loc_60><loc_57></location>0</text>
<text><location><page_24><loc_58><loc_55><loc_59><loc_56></location>-</text>
<text><location><page_24><loc_59><loc_55><loc_60><loc_56></location>0</text>
<text><location><page_24><loc_55><loc_56><loc_58><loc_57></location>2711</text>
<text><location><page_24><loc_58><loc_55><loc_59><loc_55></location>+</text>
<text><location><page_24><loc_59><loc_55><loc_60><loc_55></location>0</text>
<text><location><page_24><loc_58><loc_54><loc_59><loc_55></location>-</text>
<text><location><page_24><loc_59><loc_54><loc_60><loc_55></location>0</text>
<text><location><page_24><loc_55><loc_54><loc_58><loc_55></location>5047</text>
<text><location><page_24><loc_58><loc_53><loc_59><loc_54></location>+</text>
<text><location><page_24><loc_59><loc_53><loc_60><loc_54></location>0</text>
<text><location><page_24><loc_58><loc_52><loc_59><loc_53></location>-</text>
<text><location><page_24><loc_59><loc_52><loc_60><loc_53></location>0</text>
<text><location><page_24><loc_55><loc_52><loc_58><loc_54></location>1919</text>
<text><location><page_24><loc_58><loc_51><loc_59><loc_52></location>+</text>
<text><location><page_24><loc_59><loc_51><loc_60><loc_52></location>0</text>
<text><location><page_24><loc_58><loc_51><loc_59><loc_51></location>-</text>
<text><location><page_24><loc_59><loc_51><loc_60><loc_51></location>0</text>
<text><location><page_24><loc_55><loc_51><loc_58><loc_52></location>2589</text>
<text><location><page_24><loc_58><loc_50><loc_59><loc_50></location>+</text>
<text><location><page_24><loc_59><loc_50><loc_60><loc_50></location>0</text>
<text><location><page_24><loc_58><loc_49><loc_59><loc_50></location>-</text>
<text><location><page_24><loc_59><loc_49><loc_60><loc_50></location>0</text>
<text><location><page_24><loc_55><loc_49><loc_58><loc_50></location>7171</text>
<text><location><page_24><loc_58><loc_48><loc_59><loc_49></location>+</text>
<text><location><page_24><loc_59><loc_48><loc_60><loc_49></location>0</text>
<text><location><page_24><loc_58><loc_47><loc_59><loc_48></location>-</text>
<text><location><page_24><loc_59><loc_47><loc_60><loc_48></location>0</text>
<text><location><page_24><loc_55><loc_48><loc_58><loc_49></location>9371</text>
<text><location><page_24><loc_58><loc_46><loc_59><loc_47></location>+</text>
<text><location><page_24><loc_59><loc_46><loc_60><loc_47></location>0</text>
<text><location><page_24><loc_58><loc_46><loc_59><loc_46></location>-</text>
<text><location><page_24><loc_59><loc_46><loc_60><loc_46></location>0</text>
<text><location><page_24><loc_60><loc_68><loc_60><loc_69></location>.</text>
<text><location><page_24><loc_60><loc_68><loc_60><loc_68></location>.</text>
<text><location><page_24><loc_60><loc_67><loc_60><loc_67></location>.</text>
<text><location><page_24><loc_60><loc_66><loc_60><loc_66></location>.</text>
<text><location><page_24><loc_60><loc_65><loc_60><loc_65></location>.</text>
<text><location><page_24><loc_60><loc_64><loc_60><loc_64></location>.</text>
<text><location><page_24><loc_60><loc_64><loc_60><loc_64></location>.</text>
<text><location><page_24><loc_60><loc_63><loc_60><loc_63></location>.</text>
<text><location><page_24><loc_60><loc_62><loc_60><loc_62></location>.</text>
<text><location><page_24><loc_60><loc_61><loc_60><loc_61></location>.</text>
<text><location><page_24><loc_60><loc_60><loc_60><loc_60></location>.</text>
<text><location><page_24><loc_60><loc_59><loc_60><loc_60></location>.</text>
<text><location><page_24><loc_60><loc_56><loc_60><loc_56></location>.</text>
<text><location><page_24><loc_60><loc_55><loc_60><loc_56></location>.</text>
<text><location><page_24><loc_60><loc_55><loc_60><loc_55></location>.</text>
<text><location><page_24><loc_60><loc_54><loc_60><loc_54></location>.</text>
<text><location><page_24><loc_60><loc_53><loc_60><loc_53></location>.</text>
<text><location><page_24><loc_60><loc_52><loc_60><loc_52></location>.</text>
<text><location><page_24><loc_60><loc_51><loc_60><loc_52></location>.</text>
<text><location><page_24><loc_60><loc_51><loc_60><loc_51></location>.</text>
<text><location><page_24><loc_60><loc_50><loc_60><loc_50></location>.</text>
<text><location><page_24><loc_60><loc_49><loc_60><loc_49></location>.</text>
<text><location><page_24><loc_60><loc_48><loc_60><loc_48></location>.</text>
<text><location><page_24><loc_60><loc_47><loc_60><loc_47></location>.</text>
<text><location><page_24><loc_60><loc_47><loc_60><loc_47></location>.</text>
<text><location><page_24><loc_60><loc_46><loc_60><loc_46></location>.</text>
<text><location><page_24><loc_59><loc_84><loc_61><loc_85></location>016</text>
<text><location><page_24><loc_60><loc_83><loc_60><loc_83></location>.</text>
<text><location><page_24><loc_60><loc_82><loc_60><loc_82></location>.</text>
<text><location><page_24><loc_60><loc_81><loc_60><loc_82></location>.</text>
<text><location><page_24><loc_60><loc_81><loc_60><loc_81></location>.</text>
<text><location><page_24><loc_60><loc_83><loc_62><loc_84></location>0068</text>
<text><location><page_24><loc_60><loc_82><loc_62><loc_83></location>0063</text>
<text><location><page_24><loc_60><loc_81><loc_62><loc_82></location>0046</text>
<text><location><page_24><loc_60><loc_81><loc_62><loc_81></location>0047</text>
<text><location><page_24><loc_61><loc_79><loc_61><loc_80></location>.</text>
<text><location><page_24><loc_61><loc_79><loc_65><loc_80></location>0033</text>
<text><location><page_24><loc_68><loc_79><loc_69><loc_80></location>0</text>
<text><location><page_24><loc_60><loc_78><loc_62><loc_79></location>0015</text>
<text><location><page_24><loc_60><loc_77><loc_62><loc_78></location>0013</text>
<text><location><page_24><loc_60><loc_77><loc_62><loc_77></location>0017</text>
<text><location><page_24><loc_60><loc_76><loc_62><loc_76></location>0016</text>
<text><location><page_24><loc_60><loc_75><loc_62><loc_76></location>0065</text>
<text><location><page_24><loc_60><loc_74><loc_62><loc_75></location>0033</text>
<text><location><page_24><loc_60><loc_73><loc_62><loc_74></location>0042</text>
<text><location><page_24><loc_60><loc_72><loc_62><loc_73></location>0077</text>
<text><location><page_24><loc_60><loc_72><loc_62><loc_72></location>0052</text>
<text><location><page_24><loc_60><loc_71><loc_62><loc_72></location>0043</text>
<text><location><page_24><loc_60><loc_70><loc_61><loc_70></location>.</text>
<text><location><page_24><loc_61><loc_70><loc_63><loc_71></location>010</text>
<text><location><page_24><loc_68><loc_70><loc_70><loc_71></location>-</text>
<text><location><page_24><loc_70><loc_70><loc_70><loc_71></location>8</text>
<text><location><page_24><loc_60><loc_68><loc_62><loc_69></location>0032</text>
<text><location><page_24><loc_60><loc_68><loc_62><loc_68></location>0020</text>
<text><location><page_24><loc_60><loc_67><loc_62><loc_68></location>0057</text>
<text><location><page_24><loc_60><loc_66><loc_62><loc_67></location>0040</text>
<text><location><page_24><loc_60><loc_65><loc_62><loc_66></location>0085</text>
<text><location><page_24><loc_60><loc_64><loc_62><loc_65></location>0090</text>
<text><location><page_24><loc_60><loc_64><loc_62><loc_64></location>0029</text>
<text><location><page_24><loc_60><loc_63><loc_62><loc_63></location>0044</text>
<text><location><page_24><loc_60><loc_62><loc_62><loc_63></location>0027</text>
<text><location><page_24><loc_60><loc_61><loc_62><loc_62></location>0030</text>
<text><location><page_24><loc_60><loc_60><loc_62><loc_61></location>0068</text>
<text><location><page_24><loc_60><loc_59><loc_62><loc_60></location>0084</text>
<text><location><page_24><loc_60><loc_56><loc_62><loc_57></location>0036</text>
<text><location><page_24><loc_60><loc_55><loc_62><loc_56></location>0034</text>
<text><location><page_24><loc_60><loc_55><loc_62><loc_55></location>0056</text>
<text><location><page_24><loc_60><loc_54><loc_62><loc_55></location>0051</text>
<text><location><page_24><loc_60><loc_53><loc_62><loc_54></location>0050</text>
<text><location><page_24><loc_60><loc_52><loc_62><loc_53></location>0034</text>
<text><location><page_24><loc_60><loc_51><loc_62><loc_52></location>0076</text>
<text><location><page_24><loc_60><loc_51><loc_62><loc_51></location>0068</text>
<text><location><page_24><loc_60><loc_50><loc_62><loc_50></location>0047</text>
<text><location><page_24><loc_60><loc_49><loc_62><loc_50></location>0025</text>
<text><location><page_24><loc_60><loc_48><loc_62><loc_49></location>0072</text>
<text><location><page_24><loc_60><loc_47><loc_62><loc_48></location>0037</text>
<text><location><page_24><loc_60><loc_46><loc_62><loc_47></location>0038</text>
<text><location><page_24><loc_60><loc_46><loc_62><loc_46></location>0031</text>
<text><location><page_24><loc_67><loc_63><loc_68><loc_64></location>+</text>
<text><location><page_24><loc_68><loc_63><loc_70><loc_64></location>17</text>
<text><location><page_24><loc_67><loc_61><loc_68><loc_62></location>+</text>
<text><location><page_24><loc_68><loc_61><loc_70><loc_62></location>19</text>
<text><location><page_24><loc_68><loc_60><loc_69><loc_61></location>+</text>
<text><location><page_24><loc_69><loc_60><loc_71><loc_61></location>23</text>
<text><location><page_24><loc_68><loc_68><loc_69><loc_69></location>-</text>
<text><location><page_24><loc_69><loc_68><loc_70><loc_69></location>8</text>
<text><location><page_24><loc_68><loc_66><loc_69><loc_67></location>+</text>
<text><location><page_24><loc_69><loc_66><loc_70><loc_67></location>2</text>
<text><location><page_24><loc_68><loc_65><loc_70><loc_66></location>-</text>
<text><location><page_24><loc_70><loc_65><loc_70><loc_66></location>3</text>
<text><location><page_24><loc_70><loc_68><loc_70><loc_68></location>.</text>
<text><location><page_24><loc_70><loc_66><loc_70><loc_67></location>.</text>
<text><location><page_24><loc_68><loc_78><loc_69><loc_79></location>+</text>
<text><location><page_24><loc_69><loc_78><loc_70><loc_79></location>0</text>
<text><location><page_24><loc_68><loc_76><loc_69><loc_77></location>-</text>
<text><location><page_24><loc_69><loc_76><loc_70><loc_77></location>1</text>
<text><location><page_24><loc_68><loc_74><loc_69><loc_75></location>+</text>
<text><location><page_24><loc_69><loc_74><loc_70><loc_75></location>0</text>
<text><location><page_24><loc_68><loc_73><loc_69><loc_74></location>-</text>
<text><location><page_24><loc_69><loc_73><loc_71><loc_74></location>10</text>
<text><location><page_24><loc_67><loc_71><loc_68><loc_72></location>-</text>
<text><location><page_24><loc_68><loc_71><loc_70><loc_72></location>12</text>
<text><location><page_24><loc_70><loc_71><loc_70><loc_71></location>.</text>
<text><location><page_24><loc_68><loc_84><loc_69><loc_85></location>+</text>
<text><location><page_24><loc_69><loc_84><loc_71><loc_85></location>43</text>
<text><location><page_24><loc_68><loc_83><loc_69><loc_84></location>+</text>
<text><location><page_24><loc_69><loc_83><loc_70><loc_84></location>9</text>
<text><location><page_24><loc_67><loc_81><loc_68><loc_82></location>+</text>
<text><location><page_24><loc_68><loc_81><loc_70><loc_82></location>11</text>
<text><location><page_24><loc_69><loc_79><loc_70><loc_80></location>.</text>
<text><location><page_24><loc_70><loc_83><loc_70><loc_83></location>.</text>
<text><location><page_24><loc_71><loc_85><loc_72><loc_85></location>+</text>
<text><location><page_24><loc_72><loc_85><loc_73><loc_85></location>15</text>
<text><location><page_24><loc_71><loc_84><loc_72><loc_85></location>-</text>
<text><location><page_24><loc_72><loc_84><loc_73><loc_85></location>22</text>
<text><location><page_24><loc_71><loc_83><loc_72><loc_84></location>+</text>
<text><location><page_24><loc_72><loc_83><loc_72><loc_84></location>9</text>
<text><location><page_24><loc_71><loc_82><loc_72><loc_83></location>-</text>
<text><location><page_24><loc_72><loc_82><loc_72><loc_83></location>9</text>
<text><location><page_24><loc_71><loc_81><loc_72><loc_82></location>+</text>
<text><location><page_24><loc_72><loc_81><loc_73><loc_82></location>6</text>
<text><location><page_24><loc_71><loc_81><loc_72><loc_81></location>-</text>
<text><location><page_24><loc_72><loc_81><loc_73><loc_81></location>6</text>
<text><location><page_24><loc_72><loc_80><loc_72><loc_80></location>.</text>
<text><location><page_24><loc_70><loc_83><loc_71><loc_84></location>7</text>
<text><location><page_24><loc_70><loc_81><loc_70><loc_81></location>.</text>
<text><location><page_24><loc_70><loc_81><loc_71><loc_82></location>0</text>
<text><location><page_24><loc_70><loc_80><loc_71><loc_81></location>+</text>
<text><location><page_24><loc_71><loc_80><loc_72><loc_81></location>4</text>
<text><location><page_24><loc_70><loc_79><loc_71><loc_80></location>-</text>
<text><location><page_24><loc_71><loc_79><loc_72><loc_80></location>4</text>
<text><location><page_24><loc_71><loc_78><loc_72><loc_79></location>+</text>
<text><location><page_24><loc_72><loc_78><loc_72><loc_79></location>2</text>
<text><location><page_24><loc_71><loc_77><loc_72><loc_78></location>-</text>
<text><location><page_24><loc_72><loc_77><loc_72><loc_78></location>1</text>
<text><location><page_24><loc_70><loc_79><loc_70><loc_80></location>0</text>
<text><location><page_24><loc_70><loc_78><loc_70><loc_78></location>.</text>
<text><location><page_24><loc_70><loc_76><loc_70><loc_76></location>.</text>
<text><location><page_24><loc_70><loc_74><loc_70><loc_75></location>.</text>
<text><location><page_24><loc_70><loc_78><loc_71><loc_79></location>4</text>
<text><location><page_24><loc_71><loc_77><loc_72><loc_77></location>+</text>
<text><location><page_24><loc_72><loc_77><loc_72><loc_77></location>2</text>
<text><location><page_24><loc_71><loc_76><loc_72><loc_76></location>-</text>
<text><location><page_24><loc_72><loc_76><loc_72><loc_76></location>2</text>
<text><location><page_24><loc_70><loc_76><loc_71><loc_77></location>7</text>
<text><location><page_24><loc_71><loc_75><loc_72><loc_76></location>+</text>
<text><location><page_24><loc_72><loc_75><loc_72><loc_76></location>9</text>
<text><location><page_24><loc_71><loc_74><loc_72><loc_75></location>-</text>
<text><location><page_24><loc_72><loc_74><loc_72><loc_75></location>4</text>
<text><location><page_24><loc_70><loc_74><loc_71><loc_75></location>7</text>
<text><location><page_24><loc_71><loc_73><loc_72><loc_74></location>+</text>
<text><location><page_24><loc_72><loc_73><loc_72><loc_74></location>6</text>
<text><location><page_24><loc_71><loc_73><loc_72><loc_73></location>-</text>
<text><location><page_24><loc_72><loc_72><loc_73><loc_73></location>11</text>
<text><location><page_24><loc_71><loc_72><loc_72><loc_72></location>+</text>
<text><location><page_24><loc_72><loc_72><loc_73><loc_72></location>7</text>
<text><location><page_24><loc_71><loc_71><loc_72><loc_72></location>-</text>
<text><location><page_24><loc_72><loc_71><loc_73><loc_72></location>6</text>
<text><location><page_24><loc_73><loc_72><loc_73><loc_72></location>.</text>
<text><location><page_24><loc_73><loc_71><loc_73><loc_71></location>.</text>
<text><location><page_24><loc_73><loc_72><loc_74><loc_72></location>5</text>
<text><location><page_24><loc_73><loc_71><loc_74><loc_72></location>3</text>
<text><location><page_24><loc_72><loc_68><loc_73><loc_69></location>5</text>
<text><location><page_24><loc_70><loc_71><loc_71><loc_72></location>8</text>
<text><location><page_24><loc_70><loc_70><loc_71><loc_71></location>+</text>
<text><location><page_24><loc_71><loc_70><loc_72><loc_71></location>14</text>
<text><location><page_24><loc_70><loc_69><loc_71><loc_70></location>-</text>
<text><location><page_24><loc_71><loc_69><loc_72><loc_70></location>15</text>
<text><location><page_24><loc_71><loc_68><loc_72><loc_69></location>+</text>
<text><location><page_24><loc_72><loc_68><loc_72><loc_69></location>4</text>
<text><location><page_24><loc_71><loc_68><loc_72><loc_68></location>-</text>
<text><location><page_24><loc_72><loc_68><loc_72><loc_68></location>2</text>
<text><location><page_24><loc_70><loc_68><loc_71><loc_69></location>2</text>
<text><location><page_24><loc_71><loc_67><loc_72><loc_68></location>+</text>
<text><location><page_24><loc_72><loc_67><loc_72><loc_68></location>8</text>
<text><location><page_24><loc_71><loc_66><loc_72><loc_67></location>-</text>
<text><location><page_24><loc_72><loc_66><loc_72><loc_67></location>5</text>
<text><location><page_24><loc_70><loc_66><loc_71><loc_67></location>8</text>
<text><location><page_24><loc_70><loc_65><loc_71><loc_66></location>+</text>
<text><location><page_24><loc_71><loc_65><loc_72><loc_66></location>12</text>
<text><location><page_24><loc_70><loc_64><loc_71><loc_65></location>-</text>
<text><location><page_24><loc_71><loc_64><loc_72><loc_65></location>13</text>
<text><location><page_24><loc_70><loc_63><loc_70><loc_63></location>.</text>
<text><location><page_24><loc_70><loc_61><loc_70><loc_62></location>.</text>
<text><location><page_24><loc_70><loc_63><loc_71><loc_64></location>6</text>
<text><location><page_24><loc_71><loc_64><loc_72><loc_64></location>+</text>
<text><location><page_24><loc_72><loc_64><loc_73><loc_64></location>4</text>
<text><location><page_24><loc_71><loc_63><loc_72><loc_63></location>-</text>
<text><location><page_24><loc_72><loc_63><loc_73><loc_63></location>6</text>
<text><location><page_24><loc_71><loc_62><loc_72><loc_63></location>+</text>
<text><location><page_24><loc_72><loc_62><loc_73><loc_63></location>3</text>
<text><location><page_24><loc_71><loc_61><loc_72><loc_62></location>-</text>
<text><location><page_24><loc_72><loc_61><loc_73><loc_62></location>4</text>
<text><location><page_24><loc_70><loc_61><loc_71><loc_62></location>0</text>
<text><location><page_24><loc_71><loc_60><loc_72><loc_61></location>+</text>
<text><location><page_24><loc_72><loc_60><loc_73><loc_61></location>10</text>
<text><location><page_24><loc_71><loc_59><loc_72><loc_60></location>-</text>
<text><location><page_24><loc_72><loc_59><loc_73><loc_60></location>12</text>
<text><location><page_24><loc_62><loc_57><loc_69><loc_58></location>TOI-396 d</text>
<text><location><page_24><loc_68><loc_56><loc_69><loc_57></location>-</text>
<text><location><page_24><loc_69><loc_56><loc_70><loc_57></location>0</text>
<text><location><page_24><loc_69><loc_54><loc_70><loc_54></location>.</text>
<text><location><page_24><loc_68><loc_54><loc_69><loc_55></location>4</text>
<text><location><page_24><loc_68><loc_53><loc_69><loc_54></location>-</text>
<text><location><page_24><loc_69><loc_52><loc_70><loc_54></location>2</text>
<text><location><page_24><loc_68><loc_51><loc_70><loc_52></location>+</text>
<text><location><page_24><loc_70><loc_51><loc_70><loc_52></location>6</text>
<text><location><page_24><loc_68><loc_49><loc_69><loc_50></location>+</text>
<text><location><page_24><loc_69><loc_49><loc_70><loc_50></location>2</text>
<text><location><page_24><loc_68><loc_48><loc_69><loc_49></location>-</text>
<text><location><page_24><loc_69><loc_48><loc_71><loc_49></location>13</text>
<text><location><page_24><loc_68><loc_46><loc_69><loc_47></location>+</text>
<text><location><page_24><loc_69><loc_46><loc_70><loc_47></location>0</text>
<text><location><page_24><loc_70><loc_46><loc_70><loc_46></location>.</text>
<text><location><page_24><loc_70><loc_56><loc_70><loc_56></location>.</text>
<text><location><page_24><loc_70><loc_56><loc_71><loc_57></location>4</text>
<text><location><page_24><loc_71><loc_56><loc_72><loc_57></location>+</text>
<text><location><page_24><loc_72><loc_56><loc_72><loc_57></location>5</text>
<text><location><page_24><loc_71><loc_55><loc_72><loc_56></location>-</text>
<text><location><page_24><loc_72><loc_55><loc_72><loc_56></location>4</text>
<text><location><page_24><loc_70><loc_55><loc_71><loc_55></location>+</text>
<text><location><page_24><loc_71><loc_55><loc_72><loc_55></location>8</text>
<text><location><page_24><loc_70><loc_54><loc_71><loc_55></location>-</text>
<text><location><page_24><loc_71><loc_54><loc_72><loc_55></location>7</text>
<text><location><page_24><loc_71><loc_53><loc_72><loc_54></location>+</text>
<text><location><page_24><loc_72><loc_53><loc_72><loc_54></location>7</text>
<text><location><page_24><loc_71><loc_52><loc_72><loc_53></location>-</text>
<text><location><page_24><loc_72><loc_52><loc_72><loc_53></location>4</text>
<text><location><page_24><loc_70><loc_54><loc_70><loc_55></location>0</text>
<text><location><page_24><loc_70><loc_53><loc_70><loc_53></location>.</text>
<text><location><page_24><loc_70><loc_52><loc_71><loc_54></location>2</text>
<text><location><page_24><loc_70><loc_51><loc_71><loc_52></location>+</text>
<text><location><page_24><loc_71><loc_51><loc_72><loc_52></location>11</text>
<text><location><page_24><loc_70><loc_51><loc_71><loc_51></location>-</text>
<text><location><page_24><loc_71><loc_51><loc_72><loc_51></location>10</text>
<text><location><page_24><loc_71><loc_50><loc_72><loc_50></location>+</text>
<text><location><page_24><loc_72><loc_50><loc_72><loc_50></location>6</text>
<text><location><page_24><loc_71><loc_49><loc_72><loc_50></location>-</text>
<text><location><page_24><loc_72><loc_49><loc_72><loc_50></location>3</text>
<text><location><page_24><loc_70><loc_49><loc_71><loc_50></location>1</text>
<text><location><page_24><loc_71><loc_48><loc_72><loc_49></location>+</text>
<text><location><page_24><loc_72><loc_48><loc_73><loc_49></location>10</text>
<text><location><page_24><loc_71><loc_47><loc_72><loc_48></location>-</text>
<text><location><page_24><loc_72><loc_47><loc_72><loc_48></location>5</text>
<text><location><page_24><loc_71><loc_46><loc_72><loc_47></location>+</text>
<text><location><page_24><loc_72><loc_46><loc_72><loc_47></location>5</text>
<text><location><page_24><loc_71><loc_46><loc_72><loc_46></location>-</text>
<text><location><page_24><loc_72><loc_46><loc_72><loc_46></location>4</text>
<text><location><page_24><loc_72><loc_47><loc_72><loc_47></location>.</text>
<text><location><page_24><loc_72><loc_46><loc_73><loc_47></location>5</text>
<text><location><page_24><loc_72><loc_46><loc_72><loc_46></location>.</text>
<text><location><page_24><loc_72><loc_46><loc_73><loc_46></location>5</text>
<table>
<location><page_24><loc_9><loc_17><loc_91><loc_39></location>
<caption>Table A.4: Results of the internal structure modelling for TOI-396 b. The 'w · ' symbol represents the mass fraction with respect to the total planet mass, Zenvelope is the water mass fraction in the planet envelope, while the 'x · ' symbol represents the molar fraction of a given chemical element either in the planet core (x · , core) or in the mantle (x · , mantle).</caption>
</table>
<text><location><page_24><loc_70><loc_46><loc_71><loc_47></location>3</text>
<text><location><page_24><loc_72><loc_50><loc_72><loc_50></location>.</text>
<text><location><page_24><loc_72><loc_56><loc_72><loc_56></location>.</text>
<text><location><page_24><loc_72><loc_56><loc_73><loc_57></location>2</text>
<text><location><page_24><loc_72><loc_55><loc_72><loc_56></location>.</text>
<text><location><page_24><loc_72><loc_55><loc_73><loc_56></location>9</text>
<text><location><page_24><loc_72><loc_55><loc_73><loc_55></location>1</text>
<text><location><page_24><loc_72><loc_54><loc_73><loc_55></location>4</text>
<text><location><page_24><loc_72><loc_53><loc_72><loc_53></location>.</text>
<text><location><page_24><loc_72><loc_53><loc_73><loc_54></location>2</text>
<text><location><page_24><loc_72><loc_52><loc_72><loc_52></location>.</text>
<text><location><page_24><loc_72><loc_52><loc_73><loc_53></location>9</text>
<text><location><page_24><loc_72><loc_50><loc_73><loc_50></location>8</text>
<text><location><page_24><loc_72><loc_49><loc_72><loc_49></location>.</text>
<text><location><page_24><loc_72><loc_49><loc_73><loc_50></location>6</text>
<text><location><page_24><loc_72><loc_55><loc_72><loc_55></location>.</text>
<text><location><page_24><loc_72><loc_54><loc_72><loc_54></location>.</text>
<text><location><page_24><loc_70><loc_49><loc_70><loc_49></location>.</text>
<text><location><page_24><loc_72><loc_68><loc_72><loc_69></location>.</text>
<text><location><page_24><loc_72><loc_68><loc_72><loc_68></location>.</text>
<text><location><page_24><loc_72><loc_68><loc_73><loc_68></location>9</text>
<text><location><page_24><loc_72><loc_67><loc_72><loc_67></location>.</text>
<text><location><page_24><loc_72><loc_67><loc_73><loc_68></location>2</text>
<text><location><page_24><loc_72><loc_66><loc_72><loc_66></location>.</text>
<text><location><page_24><loc_72><loc_66><loc_73><loc_67></location>7</text>
<text><location><page_24><loc_73><loc_64><loc_73><loc_64></location>.</text>
<text><location><page_24><loc_73><loc_63><loc_73><loc_63></location>.</text>
<text><location><page_24><loc_73><loc_62><loc_73><loc_62></location>.</text>
<text><location><page_24><loc_73><loc_61><loc_73><loc_61></location>.</text>
<text><location><page_24><loc_73><loc_64><loc_74><loc_64></location>2</text>
<text><location><page_24><loc_73><loc_63><loc_74><loc_63></location>4</text>
<text><location><page_24><loc_73><loc_62><loc_74><loc_63></location>8</text>
<text><location><page_24><loc_73><loc_61><loc_74><loc_62></location>4</text>
<text><location><page_24><loc_72><loc_83><loc_72><loc_83></location>.</text>
<text><location><page_24><loc_72><loc_83><loc_73><loc_84></location>8</text>
<text><location><page_24><loc_72><loc_82><loc_72><loc_82></location>.</text>
<text><location><page_24><loc_72><loc_82><loc_73><loc_83></location>1</text>
<text><location><page_24><loc_73><loc_81><loc_73><loc_82></location>.</text>
<text><location><page_24><loc_73><loc_81><loc_73><loc_81></location>.</text>
<text><location><page_24><loc_72><loc_80><loc_73><loc_81></location>7</text>
<text><location><page_24><loc_72><loc_79><loc_73><loc_80></location>8</text>
<text><location><page_24><loc_72><loc_78><loc_72><loc_78></location>.</text>
<text><location><page_24><loc_72><loc_78><loc_73><loc_79></location>2</text>
<text><location><page_24><loc_72><loc_77><loc_72><loc_78></location>.</text>
<text><location><page_24><loc_72><loc_77><loc_73><loc_78></location>9</text>
<text><location><page_24><loc_72><loc_77><loc_72><loc_77></location>.</text>
<text><location><page_24><loc_72><loc_77><loc_73><loc_77></location>5</text>
<text><location><page_24><loc_72><loc_76><loc_72><loc_76></location>.</text>
<text><location><page_24><loc_72><loc_76><loc_73><loc_76></location>3</text>
<text><location><page_24><loc_72><loc_75><loc_72><loc_75></location>.</text>
<text><location><page_24><loc_72><loc_75><loc_73><loc_76></location>4</text>
<text><location><page_24><loc_72><loc_74><loc_72><loc_74></location>.</text>
<text><location><page_24><loc_72><loc_74><loc_73><loc_75></location>7</text>
<text><location><page_24><loc_73><loc_81><loc_74><loc_82></location>6</text>
<text><location><page_24><loc_73><loc_81><loc_74><loc_81></location>8</text>
<text><location><page_24><loc_72><loc_79><loc_72><loc_79></location>.</text>
<text><location><page_24><loc_55><loc_46><loc_58><loc_47></location>1769</text>
<text><location><page_24><loc_78><loc_84><loc_79><loc_85></location>3</text>
<text><location><page_24><loc_78><loc_83><loc_79><loc_84></location>3</text>
<text><location><page_24><loc_78><loc_81><loc_79><loc_82></location>3</text>
<text><location><page_24><loc_78><loc_79><loc_79><loc_80></location>3</text>
<text><location><page_24><loc_78><loc_78><loc_79><loc_79></location>4</text>
<text><location><page_24><loc_78><loc_76><loc_79><loc_77></location>4</text>
<text><location><page_24><loc_78><loc_74><loc_79><loc_75></location>4</text>
<text><location><page_24><loc_78><loc_73><loc_79><loc_74></location>4</text>
<text><location><page_24><loc_77><loc_71><loc_79><loc_72></location>30</text>
<text><location><page_24><loc_77><loc_70><loc_79><loc_71></location>30</text>
<text><location><page_24><loc_77><loc_68><loc_79><loc_69></location>30</text>
<text><location><page_24><loc_77><loc_66><loc_79><loc_67></location>30</text>
<text><location><page_24><loc_77><loc_65><loc_79><loc_66></location>31</text>
<text><location><page_24><loc_77><loc_63><loc_79><loc_64></location>31</text>
<text><location><page_24><loc_77><loc_61><loc_79><loc_62></location>31</text>
<text><location><page_24><loc_77><loc_60><loc_79><loc_61></location>31</text>
<text><location><page_24><loc_78><loc_56><loc_79><loc_57></location>3</text>
<text><location><page_24><loc_78><loc_54><loc_79><loc_55></location>3</text>
<text><location><page_24><loc_78><loc_52><loc_79><loc_54></location>4</text>
<text><location><page_24><loc_77><loc_51><loc_79><loc_52></location>30</text>
<text><location><page_24><loc_77><loc_49><loc_79><loc_50></location>30</text>
<text><location><page_24><loc_77><loc_48><loc_79><loc_49></location>31</text>
<text><location><page_24><loc_77><loc_46><loc_79><loc_47></location>31</text>
<text><location><page_24><loc_62><loc_87><loc_69><loc_89></location>TOI-396 c</text>
<figure>
<location><page_25><loc_11><loc_95><loc_24><loc_96></location>
</figure>
<table>
<location><page_25><loc_9><loc_69><loc_91><loc_91></location>
<caption>Table A.5: Same as Tab. A.4, but for TOI-396 d.</caption>
</table>
</document>
[
{
"title": "ABSTRACT",
"content": "Context. TOI-396 is an F6 V bright naked-eye star ( V ≈ 6.4) orbited by three small ( Rp ≈ 2 R ⊕ ) transiting planets discovered thanks to space-based photometry from two TESS sectors. The orbital periods of the two innermost planets, namely TOI-396 b and c, are close to the 5:3 commensurability ( Pb ∼ 3.6 d and Pc ∼ 6.0 d), suggesting that the planets might be trapped in a mean motion resonance (MMR). Aims. To measure the masses of the three planets, refine their radii, and investigate whether planets b and c are in MMR, we carried out HARPS radial velocity (RV) observations of TOI-396 and retrieved archival high-precision transit photometry from four TESS sectors. Methods. We extracted the RVs via a skew-normal fit onto the HARPS cross-correlation functions and performed a Markov chain Monte Carlo joint analysis of the Doppler measurements and transit photometry while employing the breakpoint method to remove stellar activity from the RV time series. We also performed a transit timing variation (TTV) dynamical analysis of the system and simulated the temporal evolution of the TTV amplitudes of the three planets following an N-body numerical integration. Results. Our analysis confirms that the three planets have similar sizes ( Rb = 2 . 004 + 0 . 045 -0 . 047 R ⊕ ; Rc = 1 . 979 + 0 . 054 -0 . 051 R ⊕ ; Rd = 2 . 001 + 0 . 063 -0 . 064 R ⊕ ) and is thus in agreement with previous findings. However, our measurements are ∼ 1.4 times more precise thanks to the use of two additional TESS sectors. For the first time, we have determined the RV masses for TOI-396 b and d, finding them to be Mb = 3 . 55 + 0 . 94 -0 . 96 M ⊕ and Md = 7 . 1 ± 1 . 6 M ⊕ , which implies bulk densities of ρ b = 2 . 44 + 0 . 69 -0 . 68 g cm -3 and ρ d = 4 . 9 + 1 . 2 -1 . 1 g cm -3 , respectively. Our results suggest a quite unusual system architecture, with the outermost planet being the densest. Based on a frequency analysis of the HARPS activity indicators and TESS light curves, we find the rotation period of the star to be P rot ,⋆ = 6 . 7 ± 1 . 3 d, in agreement with the value predicted from log R ' HK -based empirical relations. The Doppler reflex motion induced by TOI-396 c remains undetected in our RV time series, likely due to the proximity of the planet's orbital period to the star's rotation period. We also discovered that TOI-396 b and c display significant TTVs. While the TTV dynamical analysis returns a formally precise mass for TOI-396 c of Mc , dyn = 2 . 24 + 0 . 13 -0 . 67 M ⊕ , the result might not be accurate, owing to the poor sampling of the TTV phase. We also conclude that TOI-396 b and c are close to but out of the 5:3 MMR. Conclusions. A TTV dynamical analysis of additional transit photometry evenly covering the TTV phase and super-period is likely the most e ff ective approach for precisely and accurately determining the mass of TOI-396 c. Our numerical simulation suggests TTV semi-amplitudes of up to 5 hours over a temporal baseline of ∼ 5.2 years, which should be duly taken into account when scheduling future observations of TOI-396. Key words. planets and satellites: fundamental parameters - stars: fundamental parameters - techniques: photometric - techniques: radial velocities",
"pages": [
1
]
},
{
"title": "Radii, masses, and transit-timing variations of the three-planet system orbiting the naked-eye star TOI-396 ⋆",
"content": "A. Bonfanti 1 , I. Amateis 2 , 1 , 3 , D. Gandolfi 2 , L. Borsato 4 , J. A. Egger 5 , P. E. Cubillos 1 , 6 , D. Armstrong 7 , 8 , I. C. Leão 9 , M. Fridlund 10 , 11 , B. L. Canto Martins 9 , 12 , S. G. Sousa 13 , J. R. De Medeiros 9 , L. Fossati 1 , V. Adibekyan 13 , A. Collier Cameron 14 , S. Grziwa 15 , K. W. F. Lam 16 , E. Go ff o 17 , L. D. Nielsen 18 , F. Rodler 19 , J. Alarcon 20 , J. Lillo-Box 21 , W. D. Cochran 22 , 23 , R. Luque 24 , S. Redfield 25 , N. C. Santos 13 , 26 , S. C. C. Barros 13 , 26 , D. Bayliss 7 , 8 , X. Dumusque 27 , M. A. F. Keniger 7 , 8 , J. Livingston 28 , 29 , 30 , F. Murgas 31 , 32 , G. Nowak 33 , A. Osborn 34 , H. P. Osborn 5 , 35 , E. Pallé 31 , 32 , C. M. Persson 11 , L. M. Serrano 2 , P. A. Strøm 7 , 8 , S. Udry 27 , and P. J. Wheatley 7 , 8 (A ffi liations can be found after the references)",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "Multi-planet systems enable us to place significantly stronger constraints on formation and evolution mechanisms compared to single-planet systems (e.g. Lissauer et al. 2011; Fabrycky et al. 2014; Winn & Fabrycky 2015; Mishra et al. 2023). As a matter of fact, the temporal evolution of the gas content in the proto-planetary disc influences planet migration (e.g. Lin & Papaloizou 1979; Goldreich & Tremaine 1980; Tanaka et al. 2002; D'Angelo & Lubow 2008; Alexander & Armitage 2009), which shapes the orbital architecture of a planetary system. The latter is further expected to correlate with the planet composition (e.g. Thiabaud et al. 2014, 2015; Walsh et al. 2015; Bergner et al. 2020; Li et al. 2020) that can be inferred once the physical parameters of the planets are known (e.g. Dorn et al. 2017; Otegi et al. 2020b; Leleu et al. 2021; Haldemann et al. 2024). If planets are found in mean motion resonance (MMR; e.g. Lee & Peale 2002; Correia et al. 2018), systems can also shed light on the migration mechanisms during formation, as well as on the impact of tidal e ff ects occurring later on (e.g. Delisle et al. 2012; Izidoro et al. 2017). In addition, planets in, or close to, MMR likely exhibit transit timing variations (TTVs; see e.g. Agol et al. 2005; Agol & Fabrycky 2018; Leleu et al. 2021) that enable one to infer planetary masses without necessarily relying on radial velocity (RV) measurements, which are not always possible (e.g. Hatzes 2016). Multi-planet systems also give insights into correlations between physical and orbital parameters of exoplanets. For example, Weiss et al. (2018) noticed that planets in adjacent orbits usually show similar radii, hence the 'peas in a pod' label to describe this scenario. Weiss et al. (2018) further noticed that the outermost planet is the largest in the majority of cases, which agrees with the observed bulk planet density ( ρ p ) trend highlighted by Mishra et al. (2023), where ρ p decreases with the distance from the stellar host as outer planets are expected to be richer in volatiles (thus bigger and less dense). The object TOI-396 represents an interesting laboratory to test these theories as it is the brightest star known so far to host three transiting planets (Vanderburg et al. 2019) after ν 2 Lupi (Delrez et al. 2021). After analysing two sectors from the Transiting Exoplanet Survey Satellite ( TESS ; Ricker et al. 2015), Vanderburg et al. (2019) found that the three planets have radii of ∼ 2 R ⊕ and orbital periods of ∼ 3.6, ∼ 6.0, and ∼ 11.2 d, with TOI-396 c and b showing a period commensurability close to the 5:3 ratio. Following the notation introduced in Mishra et al. (2023), the three planets are 'similar' in terms of radii, and one may wonder whether this architecture class is kept also in the mass-period diagram. Mishra et al. (2023) found a positive and strong correlation of the coe ffi cients of similarity between radii and masses, though exceptions are possible (e.g. Weiss & Marcy 2014; Otegi et al. 2020a, 2022). In this work, we complement the photometric analysis of new TESS light curves (LCs) with RV observations taken with the High Accuracy Radial Velocity Planet Searcher ( HARPS ; Mayor et al. 2003) spectrograph to refine the planet radii and constrain for the first time the planetary masses. Considering the possible 5:3 MMR between TOI-396 c and b, we also simulate the evolution in time of the TTV amplitudes. This paper is organised as follows: Section 2 presents the stellar properties, and Sect. 3 describes the photometric and RV data. After outlining the method to jointly analyse the TESS LCs and the HARPS RV time series in Sect. 4, we present the corresponding results in Sect. 5. We attempt to dynamically model TTV and RV data simultaneously and track the temporal evolution of the TTV signals in Sect. 6, and we study the planets' internal structure in Sect. 7 and explore the prospects for characterising the system with the James Webb Space Telescope (JWST; Gardner et al. 2006) in Sect. 8. Finally Sect. 9 gathers the conclusions.",
"pages": [
1,
2
]
},
{
"title": "2. Host star characterisation",
"content": "TOI-396 is an F6 V (Gray et al. 2006) bright naked-eye star with an apparent visual magnitude of V ≈ 6.4 (Perryman et al. 1997). It is located ∼ 31.7 pc away and is visible in the constellation of Fornax in the southern hemisphere. It is member of a visual binary system and its companion HR 858 B is a faint M-dwarf ( G ∼ 16 mag), about 8.4 '' away from the main component. We co-added 78 HARPS spectra (see Sect. 3.2 for further details) and then modelled it with Spectroscopy Made Easy 1 ( SME ; Piskunov & Valenti 2017) version 5.2.2. SME computes synthetic spectra from a grid of well established stellar atmosphere models and adjusts a chosen free parameter based on comparison with the observed spectrum. Here we used the stellar atmosphere grid A tlas 12 (Kurucz 2013) together with atomic line lists from the V ald database (Piskunov et al. 1995) in order to produce the synthetic spectra. We modelled one parameter at a time utilising spectral features sensitive to di ff erent photospheric parameters iterating until convergence of all free parameters. Throughout the modelling, we held the macro- and micro-turbulent velocities, v mac and v mic, fixed to 6 km s -1 (Doyle et al. 2014) and 1.34 km s -1 (Bruntt et al. 2010), respectively. A description of the modelling procedure is detailed in Persson et al. (2018). Finally, we obtained T e ff = 6354 ± 70 K, [Fe / H] = 0 . 025 ± 0 . 050, log g = 4 . 30 ± 0 . 06, and v sin i ⋆ = 7 . 5 ± 0 . 2 kms -1 . To double-check the derived spectroscopy parameters we performed an additional analysis employing ARES + MOOG (Sousa et al. 2021; Sousa 2014; Santos et al. 2013). In detail, we used the latest version of ARES 2 (Sousa et al. 2007, 2015) to consistently measure the equivalent widths (EW) for the list of iron lines presented in Sousa et al. (2008). Following a minimisation process, we then find the ionisation and excitation equilibria to converge for the best set of spectroscopic parameters. This process uses a grid of Kurucz model atmospheres (Kurucz 1993) and the radiative transfer code MOOG (Sneden 1973). We obtained T e ff = 6389 ± 67 K, [Fe / H] = -0 . 014 ± 0 . 045 dex, log g = 4 . 58 ± 0 . 11, and v mic = 1 . 54 ± 0 . 04 kms -1 . In this process we also derived a more accurate trigonometric surface gravity (log g trig = 4 . 34 ± 0 . 02) using recent GAIA data following the same procedure as described in Sousa et al. (2021). In the end ARES + MOOG provides consistent values when compared with the ones derived with SME . Using the SME stellar atmospheric parameters, we determined the abundances of Mg and Si following the classical curve-of-growth analysis method described in Adibekyan et al. (2012, 2015). Similar to the stellar parameter determination, we used ARES to measure the EWs of the spectral lines of these elements and a grid of Kurucz model atmospheres (Kurucz 1993) along with the radiative transfer code MOOG to convert the EWs into abundances, assuming local thermodynamic equilibrium. The stellar radius R ⋆ , mass M ⋆ , and age t ⋆ were derived homogeneously using the isochrone placement algorithm (Bonfanti et al. 2015, 2016) and its capability of interpolating a flexible set of input parameters within pre-computed grids of PARSEC 3 v1.2S (Marigo et al. 2017) isochrones and tracks. For each magnitude listed in Table 1, we performed an isochrone placement run by inserting the spectroscopic parameters derived above, the Gaia parallax π (Gaia Collaboration et al. 2023, o ff set-corrected following Lindegren et al. 2021), and the magnitude value to obtain an estimate for the stellar radius, mass, and age along with their uncertainties. From these results, we built the corresponding Gaussian probability density functions (PDFs) and then we merged (i.e. we summed) the PDFs derived from the di ff erent runs to obtain robust estimates for R ⋆ , M ⋆ , and t ⋆ . The radius R ⋆ derives essentially from T e ff , π , and the stellar magnitude, while M ⋆ and t ⋆ are much more model-dependent; therefore we conservatively inflated their internal uncertainties by 4% and 20%, respectively, following Bonfanti et al. (2021). Our adopted stellar parameters are listed in Table 1.",
"pages": [
2
]
},
{
"title": "3.1. TESS photometry",
"content": "As presented in Vanderburg et al. (2019), TOI-396 was photometrically monitored by TESS during the first year of its nominal mission in Sector 3 from 20 September to 17 October 2018 (UT) in CCD 2 of Camera 2, and in Sector 4 from 19 October to 14 November 2018 (UT) in CCD 2 of Camera 1. TOI-396 was later Table 1: Stellar properties of TOI-396. Star names TOI-396 TIC 178155732 HR 858 HD 17926 HIP 13363 Gaia DR3 5064574724769475968 Notes. RA & DEC are reported as in J2000 reference frame. Values in the bottom half of the table have been derived as part of this paper. ( a ) Zero-point correction from Lindegren et al. (2021) applied. re-observed by TESS during the first year of its extended mission in Sector 30 from 23 September to 20 October 2020 (UT) in CCD 2 of Camera 2, and in Sector 31 from 22 October to 16 November 2020 (UT) in CCD 2 of Camera 1. All data were collected at a 2-minute cadence, except for Sector 3 for which TESS only sent down data at 30-minute cadence. We analyse all available TESS time series, including the Sector 3 and 4 data already presented in (Vanderburg et al. 2019). For Sector 3 we used the TESS Asteroseismic Science Opera- tion Center (TASOC) photometry (Handberg et al. 2021; Lund et al. 2021), while for the other sectors we analysed the Presearch Data Conditioning Simple Aperture Photometry (PDCSAP) LCs generated by the TESS Science Processing Operation Center (SPOC) pipeline Jenkins et al. (2016), which removes the majority of instrumental artefacts and systematic trends (Smith et al. 2012; Stumpe et al. 2012, 2014). We rejected those data marked with a bad quality flag and we performed a 5 medianabsolute-deviation (MAD) clipping of flux data points. After that, we built our custom LCs by splitting the TESS sectors in temporal windows centred around the transit events keeping ∼ 4 h of out-of-transit data points both before and after the transit for de-trending purposes. We ended up with 41 TESS LCs reporting the epoch of observation ( t ), the flux and its error, and further ancillary vectors available from the TESS science data product, that is mom _ centr 1, mom _ centr 2 (hereafter denoted with x and y , respectively), pos _ corr 1, and pos _ corr 2 (hereafter denoted with pc1 and pc2 , respectively) 4 .",
"pages": [
2,
3
]
},
{
"title": "3.2. HARPS high-resolution spectroscopy",
"content": "We performed the radial velocity (RV) follow-up of TOI-396 with the HARPS spectrograph mounted at the ESO-3.6 m telescope at La Silla Observatory (Chile). We acquired 77 high resolution ( R ≈ 115 000) spectra between 31 January and 27 July 2019 (UT), covering a baseline of ∼ 177 days, as part of the follow-up programs of TESS systems carried out with the HARPS spectrograph (IDs: 1102.C-0923, 1102.C-0249, 0102.C0584; PIs: Gandolfi, Armstrong, De Medeiros). One additional spectrum was acquired during a technical night (ID: 60.A-9700) in February 2019. Following Dumusque et al. (2011b), we averaged out p-mode stellar pulsations by setting the exposure time to 900 s, which led to a median signal-to-noise (S / N) ratio of ∼ 320 per pixel at 550 nm. We used the second fibre of the HARPS spectrograph to simultaneously observe a Fabry-Perot lamp and trace possible instrumental drift down to the sub-metre per second level (Wildi et al. 2010). We reduced the data using the dedicated HARPS data reduction software ( DRS ; Pepe et al. 2002; Lovis, C. & Pepe, F. 2007) and computed the crosscorrelation function (CCF) for each spectrum using a G2 numerical mask (Baranne et al. 1996). We then performed a skew-normal (SN) fit on each CCF (Simola et al. 2019) in orderto extract the stellar radial velocity along with its error, the full width at half maximum (FWHMSN), the contrast ( A ), and the skewness parameter ( γ ) of the CCF. The advantages of using an SN-fit rather than a Normal fit are thoroughly discussed in Simola et al. (2019), while the SN-fitting details are outlined in, for example, Bonfanti et al. (2023); Luque et al. (2023); Fridlund et al. (2024). After the SN-based extraction, we ended up with an RV time series, whose ancillary vectors (FWHMSN, A , γ ) are the activity indicators used to constrain the polynomial basis to model and de-trend the RV component of the stellar activity (see Table A.1). As stellar activity is not stationary, the correlations between the RV observations and the activity indicators are expected to change over time as discussed in Simola et al. (2022), who proposed to apply the breakpoint ( bp ) technique (Bai & Perron 2003) to check whether these correlation changes are statistically significant. If so, the bp algorithm finds the optimal locations of correlation changes, which defines the segmenta- tion characterised by the lowest Bayesian Information Criterion (BIC; Schwarz 1978). Indeed, we found one breakpoint at observation 48 (BJD = 2458667.940719; ∆ BIC ≈ -14 with respect to the zero-breakpoint solution). Thus, we split our RV time series in two piecewise stationary segments and de-trended it on a chunk-wise base, rather than performing a global de-trending over the whole time series, similarly to what has already been done in Bonfanti et al. (2023); Luque et al. (2023).",
"pages": [
3,
4
]
},
{
"title": "4. Methods",
"content": "We jointly analysed the TESS LCs and the RV time series within a Markov chain Monte Carlo (MCMC) framework using the MCMCI code (Bonfanti & Gillon 2020). When fitting the TESS LCs extracted from Sector 3 that has a cadence of 30 min, we generated the transit model using a cadence of 2 min (the same as the other TESS sectors) and then rebinned it to 30 min following Kipping (2010), who warns that long-cadence photometry may lead to retrieve erroneous system parameters. We imposed Normal priors on the input stellar parameters (i.e. T e ff , [Fe / H], R ⋆ , and M ⋆ ), as derived in Sect. 2 with a twofold aim: (i) the induced prior on the mean stellar density ρ⋆ (via M ⋆ and R ⋆ ) helps the convergence of the transit model; (ii) limb darkening (LD) coe ffi cients for the TESS filter may be retrieved following interpolation within A tlas 9-based 5 grids that were precomputed using the get_lds.py code 6 by Espinoza & Jordán (2015). We then set Normal priors on the quadratic LD coe ffi -cients using the values coming from the grid interpolation (i.e. u 1 , TESS = 0 . 2318 ± 0 . 0065 and u 2 , TESS = 0 . 3085 ± 0 . 0028) after re-parameterising them following Holman et al. (2006). On the planetary side, we adopted unbounded (except for the physical limits) uniform priors on the transit depth d F , the impact parameter b , the orbital period P , the transit timing T 0, and the RV semi-amplitude K , while we set the eccentricity e = 0 and the argument of periastron ω = 90 · for all planets. We come back to the assumption on the eccentricity below. The specific parameterisations of the jump parameters (aka step parameters) are outlined in Bonfanti & Gillon (2020). The LCs and the RV time series were de-trended simultaneously during the MCMC analysis using polynomials. To assess the polynomial orders to be associated with the di ff erent de-trending parameters for each time series, we first launch several MCMC preliminary runs made of 10 000 steps where we varied only one polynomial order at a time. We then selected the best de-trending baseline (see Table A.2) as the one having the lowest BIC. After that, we performed a preliminary MCMCI run comprising 200 000 steps to evaluate the contribution of both the white and red noise in the LCs following Pont et al. (2006); Bonfanti & Gillon (2020), so to properly rescale the photometric errors and get reliable uncertainties on the output parameters. Finally, three independent MCMCI runs comprising 200 000 steps each (burn-in length equal to 20%) were performed to build the posterior distributions of the output parameters after checking their convergence via the Gelman-Rubin statistic ( ˆ R ; Gelman & Rubin 1992). We also tested the possibility of eccentric orbits by imposing uniform priors on ( √ e cos ω , √ e cos ω ) either bounded to imply e ≲ 0 . 3 or completely unbounded (except for the physical limits). The wider the eccentricity range to be explored by the MCMC scheme, the poorer the parameter convergence, which suggests that the available data are not enough to con- he planetary eccentricities well. Moreover, the MCMCI runs with e , 0 are disfavoured by the ∆ BIC criterion (e.g. Kass & Raftery 1995; Trotta 2007) as well, in fact we obtained ∆ BIC = BIC e , 0 -BIC e = 0 ≳ + 100. This is also in agreement with the simulations performed by Vanderburg et al. (2019), who suggested that the periods' commensurability state of planets b and c is more likely maintained if the system is characterised by low eccentricities. Therefore, we adopted the circular solution as the reference one.",
"pages": [
4
]
},
{
"title": "5.1. Joint LC and RV analysis with linear ephemerides",
"content": "As mentioned in Sect. 4, we set P and T 0 as free parameters under the control of a uniform prior, which implies assuming linear ephemerides. With this setup, we improved the transit depth precision of all three planets by a factor ∼ 1.4 if compared with the results of Vanderburg et al. (2019). This improvement level is consistent with having twice the number of data points with respect to the LC analysis performed by Vanderburg et al. (2019), as well as with low TTV amplitudes (see Sect. 5.2). By combining the SN-fit onto the HARPS CCFs along with the bp method, we were able to estimate the masses of TOI-396 b and TOI-396 d to Mb = 3 . 56 + 0 . 92 -0 . 94 M ⊕ and Md = 7 . 2 ± 1 . 6 M ⊕ (detections at the 3.8 and 4.5 σ -level, respectively). Instead, we did not detect any significant Keplerian signal at the orbital period of TOI-396 c within the RV time series. In detail, we obtained a median Kc = 0 . 28 + 0 . 29 -0 . 20 ms -1 (3 σ upper limit K up c = 1 . 2 ms -1 ), which implies Mc = 0 . 92 + 0 . 94 -0 63 M ⊕ ( M up c = 4 . 0 M ⊕ ). . When combining the mass and radius values of the three planets, we obtain the following median estimates for the bulk planetary densities: ρ b = 2 . 56 + 0 . 71 -0 . 70 , ρ c = 0 . 67 + 0 . 69 -0 . 46 ( ρ up c = 3 . 1), and ρ d = 5 . 1 + 1 . 3 -1 . 2 g cm -3 . We note that the RV-undetected TOI396 c would be the least dense planet, while the densest planet is the outermost one (i.e. TOI-396 d), which constitutes a quite atypical architecture within the observed exoplanet population (e.g. Ciardi et al. 2013; Weiss et al. 2018; Mishra et al. 2023). However, this conclusion is just tentative, given the uncertainties on the mean planetary densities. Moreover, the detection level of the RV-inferred parameters of TOI-396 c is not statistically significant. We further tested whether including the MINERVAAustralis (Addison et al. 2019) RV data (30 measurements as taken from Vanderburg et al. 2019) can help detect the elusive planet. However, it turned out that their precision level ( ∼ 6 m s -1 ) is not high enough to improve the characterisation of the system. In other words, the RV semi-amplitudes we obtained are consistent and indistinguishable within the statistical fluctuation with what was derived from the more precise HARPS data set. This led us to further check that TOI-396 c indeed belongs to this system (Sect. 5.2) and to investigate the reason for its RV non-detection (Sect. 5.3).",
"pages": [
4
]
},
{
"title": "5.2. Joint LC and RV analysis accounting for TTVs",
"content": "As planet c is undetected in the RV time series, one may wonder whether TOI-396 c is a false positive. However, by using the VESPA tool (Morton 2012, 2015) that accounts for the constraints from the TESS LCs, imaging, and spectroscopy, Vanderburg et al. (2019) already computed that the false-positive probabilities (FPPs) are lower than 10 -3 for all three planets. In line with Vanderburg et al. (2019), we also confirmed that the Pc / Pb period ratio is commensurable and di ff ers from the 5:3 ratio by less than 0.027%. As planets with orbits in, or close to, resonances are likely to show TTVs, we decided to repeat the same MCMC analysis outlined above, but enabling the transit timings of each transit event to vary to then compute the TTV amplitude with respect to the linear ephemerides model derived in Sect. 5.1. All jump parameters converged ( ˆ R ≲ 1 . 01). The medians of the posterior distributions of the most relevant system parameters along with the 68.3% confidence intervals are listed in Table 2. The phase-folded LCs of the three planets are shown in Figure 1, while the phase-folded RV time series are displayed in Figure 2. In particular, the middle panel of Fig. 2 shows that, after subtracting the RV signals of both planets b and d, the RV time series looks flat consistently with the non-detection of TOI396 c. This analysis gives Rb = 2 . 004 + 0 . 045 -0 . 047 R ⊕ , Rc = 1 . 979 + 0 . 054 -0 . 051 R ⊕ , and Rd = 2 . 001 + 0 . 063 -0 . 064 R ⊕ and confirms the RV detection of TOI396 b and TOI-396 d (at the 3.8 and 4.5 σ -level, respectively) with Mb = 3 . 55 + 0 . 94 -0 . 96 M ⊕ and Md = 7 . 1 ± 1 . 6 M ⊕ . These outcomes are consistent with the results obtained from the analysis that assumes linear ephemerides (Sect. 5.1). We found significant TTV signals for planet b and c (see Figure 4, Top and Middle panels) as expected from the commensurability of their periods. The TTV statistical significance of TOI-396 b and c is evident even by eye when comparing the data points location with the shaded regions that represent the 1 σ uncertainties of the linear ephemerides. For each planet, we further quantified the reducedχ 2 ( ˆ χ 2 ) characterising the TTV amplitudes via where N tr , j is the number of transit event of the j -th planet. We obtained ˆ χ 2 b = 4 . 4, ˆ χ 2 c = 4 . 1, and ˆ χ 2 d = 0 . 7 for planets b, c, and d, respectively, which confirms the significance of the TTV amplitudes for TOI-396 b and c. Furthermore, the TTV amplitudes of TOI-396 b and c exhibit a clear anti-correlation pattern that we highlight by superimposing the TTV measurements in Figure A.1. This is a typical signature of gravitational interaction between the two planets, which confirms that TOI-396 c belongs to the system despite its elusiveness in the RV time series. For each planet, the timing of each transit event and the corresponding TTV amplitude computed with respect to the linear ephemerides are listed in Table A.3.",
"pages": [
4,
5
]
},
{
"title": "5.3. Discussion regarding why TOI-396 c is not detected in the RV time series",
"content": "Magnetic activity combined with stellar rotation induces RV variations that can hide, a ff ect, or even mimic planetary signals (e.g. Queloz et al. 2001; Hatzes et al. 2010; Dumusque et al. 2011a; Haywood et al. 2014; Suárez Mascareño et al. 2017; Gandolfi et al. 2017). A possible explanation for the non-detection of the Doppler reflex motion induced by TOI-396 c is that stellar activity destructively interferes with the Keplerian signal of the planet. This may happen if the star has a rotation period Notes. All jump parameters, but the LD coe ffi cients, were subject to unbounded uniform priors following the parameterisations specified in Bonfanti & Gillon (2020); see text for further details. ( a ) Uncertainties from the run assuming linear ephemerides (no TTVs). T 0 values are shifted by -2 450 000. ( b ) Assuming zero albedo and full recirculation. ( c ) The 3 σ upper limits on the RV semi-amplitude, the planet mass, and the mean planet density of TOI-396 c are K up c = 1 . 2 ms -1 , M up c = 3 . 8 M ⊕ , and ρ up c = 2 . 9 g cm -3 , respectively. comparable to the orbital period of the planet (e.g. Vanderburg et al. 2016). Disentangling the planetary signal from stellar activity and retrieving the Doppler motion induced by the orbiting planet would then be challenging (e.g. Dragomir et al. 2012; Kossakowski et al. 2022). In order to investigate this hypothesis, we performed a frequency analysis of the line profile variation diagnostics (FWHM, contrast, and skewness) and activity indicators (log R ' HK and H α indexes). Figure 5 displays the time series (left column), along with the respective generalised Lomb-Scargle (GLS, Zechmeister & Kürster 2009) periodograms (right column). We assessed the significance of the peaks detected in the power spectra by estimating their false alarm probability (FAP), that is the probability that noise could produce a peak with power higher than the one we found in the time series. To account for the possible presence of non-Gaussian noise in the data, we estimated the FAP using the bootstrap randomisation method (see, e.g., Murdoch et al. 1993; Kuerster et al. 1997; Hatzes 2019). Briefly, we computed the GLS periodogram of 10 5 mock time series obtained by randomly shu ffl ing the data points along with their error bars, while keeping the timestamps fixed. We defined the FAP as the fraction of those mock periodograms whose highest power exceeds the power of the real data at any frequency. In the present work, we considered a peak to be significant if its false alarm probability is FAP < 0.1 %. We found that the time series of the log R ' HK and H α indexes display long-term trends likely due to the magnetic cycles of the star (Fig. 5, left column, first and third panels). In the Fourier domain, these trends translate into a significant (FAP < 0.1 %) excess of power at frequencies lower than the spectral resolution 8 of our HARPS observations (Fig. 5, right column, first and third panels). We modelled these long-term signals as quadratic trends and subtracted the best-fitting parabolas from the respective time-series. The GLS periodograms of the residuals of the activity indicators display significant peaks between ∼ 6 and 8 d (i.e, between ∼ 0.0125 and 0.167 d -1 ; Fig. 5, yellow area). The peaks are equally spaced by ∼ 0.0068 d -1 , which coincides with a peak found in the periodogram of the window function. Given the current data at our disposal, we are not able to distinguish between true frequencies and aliases. Although not significant, the power spectra of the contrast and skewness show peaks in the same frequency range, suggesting that the rotation period of the star might be ∼ 6-8 d. We note that the periodogram of the FWHM also displays an excess of power at low frequencies. However, the corresponding peak remains below our 0.1 %-FAP significance threshold (see Fig. 5, second to last panel). Yet, if we apply the same procedure described above and remove this signal by fitting a quadratic trend to the FWHM time series, we find no significant peak in the residuals. The projected equatorial velocity of the star ( v sin i ⋆ = 7 . 5 ± 0 . 2 km s -1 ), along with its radius ( R ⋆ = 1 . 258 ± 0 . 019 R ⊙ ), yields an upper limit for the rotation period of P up rot = 8 . 5 ± 0 . 3 d. Using the mean value 9 of log R ' HK = -4 . 926 ± 0 . 014, we inferred a stellar rotation period of 6.7 ± 1.3 d and 6.9 ± 1.3 d from the empirical equations of Noyes et al. (1984) and Mamajek & Hillenbrand (2008), respectively. In addition, by inputting the isochronal age into the gyrochronological relation from Barnes (2010), we computed a stellar rotation period of 7 . 1 + 1 . 0 -1 . 1 d. These results corroborate our interpretations that the peaks between 6 and 8 d significantly detected in the power spectra of the activity indicators originate from stellar rotation. To check whether a quasi-periodic signal compatible with ∼ 7 d is also present in the photometric data, for each TESS sector we extracted custom LCs from pixel data using lightkurve (Lightkurve Collaboration et al. 2018). In detail, we adopted the default quality bitmask and set the aperture to 'all', which corresponds to an aperture larger than the one used by the o ffi cial SAP pipeline. In fact, larger apertures mitigate the e ff ect of slow image drifts that could interfere with slow flux changes, such TV [min] T TV [min] T 40 30 20 10 0 10 20 60 50 40 30 20 10 0 10 20 8360 8380 8400 8420 8440 9100 9120 9140 9160 9180 9200 BJD - 2450000 [d] 8360 8380 8400 8420 8440 9100 9120 9140 9160 9180 9200 BJD - 2450000 [d] as the 6-8 d rotation period signals we aim to detect. After removing the temporal windows containing the transit events, we computed the GLS periodograms of these lightkurve -based LCs. The FAP was computed following the same bootstrap technique outlined above for the RV activity indicators. The four periodograms (Fig. A.2, first column) exhibit very significant peaks at ∼ 7.7, 7.9, 7.5, and 6.8 days for TESS Sectors 3, 4, 30, and 31, respectively. Except for Sector 30, they are not the most prominent peaks; however, they persist even after removing the most significant signals (Fig. A.2, second column). Whereas we acknowledge that the likely rotation period of the star is close to the first harmonic of the orbital period of TESS around the Earth ( P rot ,⋆ ∼ 1 2 P TESS ∼ 14 days), the photometric signal at 6-8 d is significant in all the four TESS sectors and persists after prewhitening the data. This signal is consistent with the rotation period detected in the HARPS activity indicators and inferred from log R ' HK , v sin i ⋆ , and gyrochronology, suggesting it is astrophysical in nature and due to the presence of active regions carried around by stellar rotation. Assuming the log R ' HK -based P rot ,⋆ = 6 . 7 ± 1 . 3 d as our reference estimate, the orbital period of TOI-396 c ( Pc ∼ 6 d) is close to P rot ,⋆ , which may explain the non-detection of planet c within the RV time series. If some kind of destructive interference between the Keplerian signal of planet c and the stellar activity has occurred, any artificial Keplerian signals with period P = Pc added to the observed RV time series should in principle be retrieved. To test this hypothesis, we considered Keplerian signals of the following form and generated four di ff erent RV time series by separately adding to the HARPS time series synthetic RV signals following Equation (2) with P = Pc , T 0 = T 0 , c , and K art = K in, where K in are the four di ff erent amplitude values listed in the first column of Table 3. For each RV time series, we then performed an MCMCI analysis to retrieve the RV semi-amplitude of the artificial signal ( K out; see Table 3). We note that the resulting K out ≈ K in + Kc , where Kc = 0 . 28 + 0 . 29 -0 . 19 ms -1 is the RV semi-amplitude of planet c as derived from the analysis on the original RV time series. As we essentially retrieved what we inserted in the HARPS time series, we may conclude that the destructive interference between the RV signals induced by the star and by planet c has already occurred and any further RV signal added to the RV time series is detected. As P rot ,⋆ is not exactly equal to Pc , we then repeated the test outlined above, but this time we injected into the original HARPS time series artificial Keplerian signals with P = P rot ,⋆ . The K out values obtained by the MCMC analyses depending on the di ff erent K in values are reported in Table 4. The K out values are systematically and significantly smaller than the corresponding K in values, which a posteriori supports the conclusion that the stellar rotation period is around 6-8 d. Furthermore, a planet with this orbital period would be firmly detected if its RV semi-amplitude were greater than Kd , which is the largest RV semi-amplitude detected for this system. Instead, by injecting a Keplerian signal with K in = Kd and P = P rot ,⋆ , the MCMCI analysis is able to barely detect (at ∼ 2 σ ) a planetary signal whose amplitude is half of that expected. The detection level increases when K in increases; however, we still underestimate K out. These tests prove that it is di ffi cult to retrieve planetary signals with P ∼ P rot ,⋆ . In summary, we conclude that stellar activity is responsible for generating spurious RV signals whose harmonics also include the stellar rotation period. As a consequence, it is hard to reliably detect planets with orbital periods comparable to P rot ,⋆ via the RV technique and Table 4 quantifies the magnitude of this e ff ect. We note that the Kc we obtained from the MCMCI analysis in Sect. 5.2 is comparable to the K out retrieved when inserting an artificial signal having K in = 1 . 0 ms -1 , which let us Notes. Column ∆ K K gives the relative di ff erence (in percentage) between the obtained RV semi-amplitude ( K out) and the expected one ( K in), while the last column ∆ K ≡ K out -K in lists the semi-amplitude di ff erence in terms of the 1 σ uncertainty of K out. to speculate that TOI-396 c might have Mc ∼ 3.0 M ⊕ and ρ c ∼ 2.0 g cm -3 .",
"pages": [
5,
6,
7,
8,
9,
10
]
},
{
"title": "6. Joint RV and TTV dynamical analysis",
"content": "As mentioned, the anti-correlation pattern of the observed TTVs (see Fig. A.1) is a typical signal of the dynamical interaction between TOI-396 b and c. However, the data in our hands are not enough to currently derive meaningful planetary masses from the TTVs. Indeed, the photometric observations are clustered in two groups (about two years apart), which results in a partial coverage of the curvature of the TTVs. This prevents us from accurately mapping the full phases and super-periods of the TTV signals. Hence a dynamical fit onto the transit times would lead to a high fraction of low-amplitude TTVs with a short superperiod or to a low fraction of high-amplitude long-period TTV signals. Despite this, we attempted to run a dynamical joint fit of the TTV and RV data set with TRADES 10 (Borsato et al. 2014, 2019, 2021), a Fortran-python code developed to model TTVs and RVs simultaneously along with N-body integration. We have taken and fixed the stellar mass and radius values from Table 1. We further fixed the orbital inclinations i to the values in Table 2 and the longitude of the ascending nodes to Ω = 180 · 11 for all the planets. We fitted the planetary masses scaled by the stellar mass ( M b , c , d / M ⋆ ), the orbital periods ( P ), the eccentricities ( e ) and the arguments of the pericentre ( ω ) in the form ( √ e cos ω , √ e sin ω ), and the mean longitudes ( λ 12 ). We also fitted for an RV o ff set ( γ RV) and for an RV jitter term ( σ jitter) by adopting log 2 σ jitter as step parameter, although we used the de-trended RV data set as derived from Sect. 5.2, where a jitter term was already included in the RV errors. All fitting parameters were subject to uniform priors that account for their respective physical boundaries (see Table 5); for the eccentricities we applied a log-penalty (log p e ) based on the half-Gaussian ( e = 0 , σ e = 0 . 083) from Van Eylen et al. (2019). The reference time for the dynamical integration of the orbital parameters was set at T ref = 2 458 379 BJDTDB, that is before all available observations. We combined the quasiglobal di ff erential evolution (Storn & Price 1997) optimisation algorithm implemented in P y DE (Parviainen et al. 2016) with the A ffi ne Invariant MCMC Ensemble sampler (Goodman & Weare 2010) emcee implemented by Foreman-Mackey et al. (2019, 2013). We first ran P y DE and evolved 68 di ff erent initial configurations of parameter sets for 50 000 generations (number of steps for which each parameter is evolved). To perform the dynamical analysis in an MCMC fashion, we then ran emcee , assuming as starting point the outcome obtained with P y DE. We set up 68 chains for 600 000 steps each. To sample the parameter space e ffi ciently, we mixed the DEMove() and DESnookerMove() di ff erential evolution moves 13 in the proportion 80%-20% (Nelson et al. 2014; ter Braak & Vrugt 2008). We repeated the sequence P y DE + emcee twice, with di ff erent seeds for the random number generator. The chains reached convergence according to visual inspection and statistical indicators, such as the Gelman-Rubin ˆ R , the Geweke's statistic (Geweke 1991), and the auto-correlation function 14 . For both runs, we applied a conservative thinning factor of 100 and discarded the first 50% of the chains (burn-in). For each run, we derived the reference outcome (hereinafter also referred to as best-fit), as the maximum-a-posteriori (MAP) parameters' set, that is the set of parameters that maximises the log-probability 15 . The uncertainty of each parameter is quantified by the high-density interval (HDI) at 68.27% 16 of its posterior distribution. For each parameter we computed a Z-score defined as Z-score = | MAP1 -MAP2 | / p max | ERR1 | 2 + max | ERR2 | 2 , where the subscripts denote the two di ff erent runs and ERR = HDI -MAP. It turned out that Z-score < 1 for each parameter, which allows us to merge the posterior distributions deriving from the two runs to finally compute the MAP and the respective HDI from the merged posterior distributions. We further checked the Hill stability of the system (Sundman 1913) when assuming the entire merged posterior distributions by calculating the angular momentum deficit (AMD, Laskar 1997, 2000; Laskar & Petit 2017) criterion (Eq. 26 from Petit et al. 2018). Table 5 lists the parameters returned by TRADES (MAP and HDI) along with their respective priors. We note that the dynamical integration with TRADES allowed for a significant detection ( > 3 σ ) of the mass of TOI-396 c, that is Mc , dyn = 2 . 24 + 0 . 13 -0 . 67 M ⊕ . However, as explained above, TESS data do not allow us to fully map the TTV pattern. Therefore, the present-day estimate of Mc , dyn only provides an indication of the possible mass of the planet that might not be accurate, despite its formal precision. Indeed, to accurately and reliably determine planetary masses via TTVs, it is necessary to monitor the TTV signal by benefiting of a full sampling coverage, as demonstrated for example by the cases of Kepler-9 (Holman et al. 2010) and K2-24 (Petigura et al. 2016), whose orbital parameters and masses were comprehensively revisited by Borsato et al. (2014) and Petigura et al. (2018); Nascimbeni et al. (2024), respectively. In detail, Holman et al. (2010) had reported the masses of Kepler-9 b and c with a precision better than 8%, by performing a TTV analysis on the first three quarters of the Kepler data. Later on, by benefiting of twelve sectors of Kepler data that enabled the full mapping of the TTV phase, Borsato et al. (2014) obtained TTV-based masses di ff ering by a factor ∼ 2 from the estimate of Holman et al. (2010). Similarly, by accounting on more photometric data, Petigura et al. (2018); Nascimbeni et al. (2024) found that the mass of K2-24 c is lower by almost 2 σ than the estimate of Petigura et al. (2016) who had claimed a detection at the ∼ 4 σ level. After extracting all the synthetic transit timings T tr (i.e. the 'observed': O) from TRADES' analysis, we computed the 'calculated' (C) counterpart according to the linear ephemerides model based on what listed in Table 2. We plotted the O -C as a function of time as well as the RV best-fit model in Figures 6 and 7. Additionally, we investigated if the best-fit configuration is in or close to an MMR. We integrated the MAP parameters with the N-body code rebound (Rein & Liu 2012) and the symplectic Wisdom-Holman integrator whfast (Rein & Tamayo 2015; Wisdom & Holman 1991) for 10 000 years. We computed the evolution of the critical resonance angles of TOI-396 b and TOI396 c where p = 3 and q = 2 (for a second order 5:3 MMR), while ϖ ≡ ω + Ω is the longitude of the pericentre. We also computed the evolution of ∆ ϖ ≡ ϖ b -ϖ c = ( ϕ b -ϕ c) / q . In case of MMR, we expect that both ϕ b and ϕ c librate (i.e. oscillate) around a fixed value for the entire orbital integration. Instead, if these angles circulate, that is they span the full 0 · -360 · range (or equivalently the -180 · -180 · range), then the planet pair is not in resonance. We found that both ϕ b and ϕ c circulate (see the two upper panels of Fig. 8), which indicates that the system is not in an exact 5:3 MMR. Even if ∆ ϖ seems to oscillate around 0 · , it circulates every ∼ 2 000 years (bottom panel of Fig. 8), which further confirms that the system is not trapped in an MMR state. We also found the same behaviour for the resonant angles of 200 random samples drawn from the posterior distribution (not shown here). Our conclusions are consistent with the simulations performed by Vanderburg et al. (2019), who found that most realisations of the system are not in resonance. As shown in Fig. 6, the MAP parameter set predicts that the TTV super-period is longer than the time spanning the two clustered TESS observations. We decided to track the potential evolution of the TTV signals over a temporal baseline of ∼ 5.2 years. To this end, we ran forward numerical N-body simulations with TRADES, setting the initial conditions of the orbital parameters to be integrated at t = T ref. We ran a first simulation taking all the parameters from the MAP solution. Then, we further ran 200 simulations, where the sets of the system's parameters were randomly drawn from the merged posterior distributions. We computed the synthetic (i.e. observed: O ) transit times T tr and created a simulated O -C plot against time, where the C counterpart of T tr was computed assuming the linear ephemerides of Table 2. The results are displayed in Fig. 9 that emphasises a progressive drift of the O -C values inferred from the MAP parameters (black line) with respect to the zero value. As a consequence, the TTV amplitudes of planets b and c increase with time. In particular, the semi-amplitude of the O -C black curves are about two and five hours for b and c, respectively. The TTV super-period seems to be roughly equal to or larger than the integration time span, that is ∼ 5 years. The variance of the TTV amplitude (shaded area) is inferred from the results of the additional 200 simulations and reflects the widths of the posterior distributions from which the system's parameters were drawn. The remarkable O -C drifts (i.e. the poorly constrained linear ephemerides) combined with the uncertainty on the TTV amplitudes make challenging to plan future observations of planets b and c, as the actual transit timings might di ff er from the linear ephemerides predictions by ∼ 5 and ∼ 10 hours for TOI-396 b and TOI-396 c, respectively. TESS will not observe the target in the foreseeable future. 17",
"pages": [
10,
11
]
},
{
"title": "7. Internal structure",
"content": "Using the masses and transit depths reported in Table 2, we ran the neural network based internal structure modelling framework plaNETic 18 (Egger et al. 2024) to infer the internal structure of TOI-396 b and d. plaNETic uses a full grid accept-reject sampling method in combination with a deep neural network (DNN) that was trained on the forward model of BICEPS (Haldemann et al. 2024) to infer the internal structure of observed planets. Each planet is modelled as a three-layered structure: an inner iron-dominated core, a silicate mantle and a fully mixed envelope made up of water and H / He. In the case of multi-planet systems, all planets are modelled simultaneously. As modelling the internal structure of exoplanets is a highly degenerate problem, the resulting inferred structure is, at least to a certain extent, dependent on the chosen priors. To mitigate this e ff ect, we ran a total of six models assuming six di ff erent combinations of priors. Most importantly, we use two di ff erent priors for the water content of the modelled planet, one motivated by a formation scenario outside the iceline (case A, water-rich) and one compatible with a formation inside the iceline (case B, water-poor). For both of these water priors, we choose three different options for the planetary Si / Mg / Fe ratios. In a first case, we assume that these match the stellar Si / Mg / Fe ratios exactly Thiabaud et al. (2015). Second, we assume that the planet is enriched in iron compared to its host star by using the fit of Adibekyan et al. (2021). For option 3, we model the planet independent of the stellar Si / Mg / Fe ratios, but just sampling the planetary ratios uniformly from the simplex where the molar Si, Mgand Fe ratios add up to 1, with an upper bound of 0.75 for Fe. These priors are described in more detail in Egger et al. (2024).",
"pages": [
11
]
},
{
"title": "8. JWST characterisation prospects",
"content": "All three planets in the TOI-396 system share similar radii ( ∼ 2 R ⊕ ), but span a wide range of masses (0.9-7.1 M ⊕ ), which leaves open the question of whether they have primary or secondary atmospheres. Furthermore, the progression of bulk densities with distance from the host star varies in ways that cannot be described by simple formation and evolution models (e.g. Weiss et al. 2018; Mishra et al. 2023). Given the bright host star and combination of planetary masses, radii, and equilibrium temperatures, the three planets have favourable metrics for atmospheric characterisation in both transmission and emission among sub-Neptunes (Kempton et al. 2018, see Fig. 12). This makes the TOI-396 system a highly valuable laboratory to study the formation and evolution of plan- etary systems. Thus, we explored the prospects for characterisation with JWST. We focused these simulations on emission observations, but we note that transmission and emission have their own advantages and disadvantages in terms of achievable science goals and challenges. We employed the open-source P yrat B ay modelling framework (Cubillos & Blecic 2021) to compute synthetic spectra of the TOI-396 planets. These models consist of 1D cloud-free atmospheres in radiative, thermochemical, and hydrostatic equilibrium (Cubillos et al., in prep.). We varied the models' atmospheric elemental content to explore the wide range of compositions that the planets span. For this comparison we settled on two models to represent a primary- and a secondary-atmosphere scenario: the first is a gas giant with a 5 × solar metallicity, the second is a water world with a 80% H 2 O plus 20% CO 2 composition (based on the C / O ratios seen in the solar system minor bodies, see, e.g., Mumma & Charnley 2011; McKay et al. 2019). For the thermochemical-equilibrium calculations we considered a set of 45 neutral and ionic species, which are the main actors determining the thermal structure. For the radiative-transfer calculation we considered opacities from molecular species for CO, CO 2 , CH 4 , H 2 O, HCN, NH 3 , and C 2 H 2 from hitemp and E xo M ol (Rothman et al. 2010; Tennyson et al. 2016); Na and K resonant lines (Burrows et al. 2000); H, H2, and He Rayleigh (Kurucz 1970); and H2-H2 and H2-He collision-induced absorption (Borysow et al. 2001; Borysow 2002; Richard et al. 2012). We preprocessed the large E xo M ol line lists with the R epack algorithm (Cubillos 2017) to extract the dominant transitions. Figure 13 (top panels) shows the resulting thermal and composition structure for TOI-396 b (planets c and d follow a similar trend). The infrared synthetic emission spectra (Fig. 13, bottom panels) are mainly shaped by H 2 O, CO 2 , and CO features. At most wavelengths the primary- and secondary-atmosphere scenarios roughly di ff er by an o ff set, which would be hard to distinguish unless the energy budget of the planets are known. In contrast, the 4-5 µ mwindow shows the most distinctive spectral features; here the strong CO 2 absorption band at 4.4 µ m mainly allows one to distinguish primary from secondary atmospheres. Thus, in the following we focus on this region of the spectrum. We simulated JWST observations using the P andeia exposure time calculator Pontoppidan et al. (2016). The brightness of TOI-396 limits the instrument selection to NIRCam (F444W filter) to avoid saturation. We selected the fastest readout and subarray modes, 5 groups per integration, to optimise the S / N. We generated a distribution of (noised up) realisations for each model to estimate how many eclipses are required to distinguish between primary and secondary atmospheres at the 3 σ level. We found that 2, 4, and 8 eclipses (for planets b, c, and d, respectively) would be su ffi cient to di ff erentiate between these two models. Figure 13 shows one of those random realisations when including the required number of eclipses. The decreasing equilibrium temperature of the planets as they are located further away from TOI-396 plays the major role in the decreasing S / N for planets c and d.",
"pages": [
12,
13,
14
]
},
{
"title": "9. Conclusions",
"content": "The object TOI-396 is an F6 V bright naked-eye star orbited by three planets of almost equal size, and the two inner planets are close to but out of a 5:3 MMR. A photometric analysis of the system was already performed by Vanderburg et al. (2019), but by benefiting from two additional TESS sectors, we improved the precision on the planet radii by a factor of ∼ 1.4, obtaining Rb = 2 . 004 + 0 . 045 -0 . 047 R ⊕ , Rc = 1 . 979 + 0 . 054 -0 . 051 R ⊕ , and Rd = 2 . 001 + 0 . 063 -0 . 064 R ⊕ . We determined the masses of the planets by extracting the RV time series from HARPS CCFs using an SN fit followed by a joint LC and RV MCMC analysis, where the RV de-trending uses the breakpoint method. We obtained a firm detection of the RV signals of planets b and d, deriving Mb = 3 . 55 + 0 . 94 -0 . 96 M ⊕ and Md = 7 . 1 ± 1 . 6 M ⊕ , but we can provide only a 3 σ upper limit for the mass of TOI-396 c of M up c = 3 . 8 M ⊕ . This yields the following mean planet densities: ρ b = 2 . 44 + 0 . 69 -0 . 68 , ρ up c = 2 . 9, and ρ d = 4 . 9 + 1 . 2 -1 . 1 g cm -3 , implying a quite unusual system architecture (Mishra et al. 2023) where the mid planet is the least dense and the outermost planet is the densest. The reason for the RV non-detection of any Keplerian signal at P = Pc ∼ 6 d is likely to be ascribed to the vicinity of Pc to the stellar rotation period. As a matter of fact, from the GLS periodograms of both the RV-related activity indices and the TESS raw LCs and from log R ' HK -based empirical relations, we consistently inferred P rot ,⋆ = 6 . 7 ± 1 . 3 d. After injecting synthetic Keplerian signals at P = P rot ,⋆ and di ff erent semi-amplitudes ( K in) into the RV time series, we empirically find that the RV semi-amplitudes output by the MCMC analyses ( K out) are systematically lower than the input ones by almost 3 σ , and they are statistically non-significant as far as K in ≲ Kd . In addition, we find that K out ≈ Kc when considering a planet with Mp ∼ 3 M ⊕ (i.e. ρ p ∼ 2 g cm -3 ), which might correspond to the properties of TOI-396 c. On a more general perspective, these simulations confirm that stellar activity destructively interferes with Keplerian signals having P ∼ P rot ,⋆ (e.g. Vanderburg et al. 2016), and furthermore, they indicate that - even in the case of firm detection - values of K out are significantly underestimated. Longer-baseline RV observations may help disentangle coherent signals originated by Keplerian motions from noncoherent signals due to stellar activity, even if degeneracy issues still hold when the planet orbital period is close to the stellar rotation period (Kossakowski et al. 2022). Alternatively, a possible constraint on Mc may come from TTVs, as planets b and c are close to an MMR of the second order. Indeed, the TTV amplitudes of the two planets show a characteristic anti-correlation pattern, as expected; however, the phase coverage given by the available observations is too poor to perform a conclusive TTV dynamical analysis based on the observed transit timings of the planets. We also attempted to fit the TTV and RV simultaneously while integrating the orbits of the system. We found that the masses and densities of planets b and d are consistent with the results from the joint LC and RV analysis. TOI-396 c shows a dynamical mass of Mc , dyn = 2 . 24 + 0 . 13 -0 . 67 M ⊕ , which is greater than that inferred from the joint LC and RV analysis, but it is consistent (Z-score = 1 . 2 σ ); the density is consistent at the 1 . 1 σ level. However, we emphasise that, although formally precise, the Mc , dyn estimate might not be accurate, as the full coverage of the TTV phase is needed to reliably compute TTV-based masses. Therefore, to fully confirm the system architecture, a reliable estimate of the mass of TOI-396 c is still missing. We also checked the evolution of the system over 10 000 years, and the critical resonance angles showed that planets b and c are close to but not in a 5:3 MMR. We further performed forward N-body simulations over a temporal baseline of ∼ 5.2 years in order to track the transit epochs and evaluate the expected TTV amplitudes during time. It turns out that TOI-396 b and TOI-396 c may exhibit TTVs with a super-period of about 5 years and semi-amplitudes of ∼ 2 and ∼ 5 hours, respectively. This translates into a temporal drift of the transit timings that can rise up to ∼ 5 and ∼ 10 hours with respect to the linear ephemerides computed from TESS data. Studying the planetary atmospheres with JWST would take advantage of the favourable spectroscopy metrics of the system (Kempton et al. 2018). Therefore, we set up 1D cloud-free atmospheric models, generated the synthetic emission spectra of the three planets, and simulated eclipse observations with JWST. It turns out that 2, 4, and 8 eclipses (for TOI-396 b, c, and d, respectively) would be su ffi cient to distinguish between primary and secondary atmosphere scenarios at the 3 σ level. Characterising the nature of the planetary atmosphere is also key to correctly assessing the planetary bulk densities (in particular for planet c). The potentially high TTVs inferred from our simulations should be duly taken into account when scheduling future observations of the target. This holds not only for JWST, but also for CHEOPS (Benz et al. 2021), which appears espe- cially suitable for collecting exquisite photometric data to enable the full characterisation of the system. Acknowledgements. We thank the anonymous referee for all the valuable comments that significantly improved the quality of the manuscript. This work has made use of data from the European Space Agency (ESA) mission Gaia ( https://www.cosmos.esa.int/gaia ), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/ gaia/dpac/consortium ). This research made use of Lightkurve, a Python package for Kepler and TESS data analysis (Lightkurve Collaboration et al. 2018). We thank contributors to NumPy (Harris et al. 2020), SciPy (Virtanen et al. 2020), matplotlib (Hunter 2007), astropy (Astropy Collaboration et al. 2013, 2018, 2022), astroquery (Ginsburg et al. 2019), and tesscut (Brasseur et al. 2019). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. We acknowledge financial support from the Agencia Estatal de Investigación of the Ministerio de Ciencia e Innovación MCIN / AEI / 10.13039 / 501100011033 and the ERDF 'A way of making Europe' through project PID2021-125627OB-C32, and from the Centre of Excellence 'Severo Ochoa' award to the Instituto de Astrofisica de Canarias. This research was funded in part by the UKRI, (Grants ST / X001121 / 1, EP / X027562 / 1). This work was supported by FCT - Fundação para a Ciência e a Tecnologia through national funds and by FEDER through COMPETE2020 - Programa Operacional Competitividade e Internacionalização by these grants: UIDB / 04434 / 2020; UIDP / 04434 / 2020. D.G., A.B., L.F., and L.M.S. gratefully acknowledge the financial support from the grant for internationalization (GAND_GFI_23_01) provided by the University of Turin (Italy). S.G.S acknowledges the support from FCT through Investigador FCT contract nr. CEECIND / 00826 / 2018 and POPH / FSE (EC). P.J.W. acknowledges support from the UK Science and Technology Facilities Council (STFC) through consolidated grants ST / P000495 / 1, ST / T000406 / 1 and ST / X001121 / 1. N.C.S. is funded by the European Union (ERC, FIERCE, 101052347). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. J.L.-B. is funded by the MICIU / AEI / 10.13039 / 501100011033 and NextGenerationEU / PRTR grant PID2019-107061GB-C61 and CNS2023-144309. X.D. acknowledges the support from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement SCORE No 851555) and from the Swiss National Science Foundation under the grant SPECTRE (No 200021_215200). This work has been carried out within the framework of the NCCR PlanetS supported by the Swiss National Science Foundation under grants 51NF40_182901 and 51NF40_205606. G.N. thanks for the research funding from the Ministry of Science and Higher Education programme the \"Excellence Initiative - Research University\" conducted at the Centre of Excellence in Astrophysics and Astrochemistry of the Nicolaus Copernicus University in Toru'n, Poland. Research activities of the Board of Observational and Instrumental Astronomy at the Federal University of Rio Grande do Norte are supported by continuous grants from the Brazilian funding agencies CNPq. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-Brasil (CAPES)-Finance Code 001 and CAPES-Print program. B.L.C.M., I.C.L., and J.R.M. acknowledge CNPq research fellowships. K.W.F.L. was supported by Deutsche Forschungsgemeinschaft grants RA714 / 14-1, RA714 / 14-2 within the DFG Schwerpunkt SPP 1992, Exploring the Diversity of Extrasolar Planets. L.B. acknowledges support from CHEOPS ASI-INAF agreement n. 2019-29-HH.0. D.G. sincerely thanks Stefano Camera for the inspiring and valuable discussions on the properties of TOI-396.",
"pages": [
14,
15,
16
]
},
{
"title": "References",
"content": "Addison, B., Wright, D. J., Wittenmyer, R. A., et al. 2019, PASP, 131, 115003 Adibekyan, V., Dorn, C., Sousa, S. G., et al. 2021, Science, 374, 330 Adibekyan, V., Figueira, P., Santos, N. C., et al. 2015, A&A, 583, A94 Adibekyan, V. Z., Sousa, S. G., Santos, N. C., et al. 2012, A&A, 545, A32 Agol, E. & Fabrycky, D. C. 2018, in Handbook of Exoplanets, ed. H. J. Deeg & Baranne, A., Queloz, D., Mayor, M., et al. 1996, A&AS, 119, 373 Barnes, S. A. 2010, ApJ, 722, 222 Bruntt, H., Deleuil, M., Fridlund, M., et al. 2010, A&A, 519, A51 Burrows, A., Marley, M. S., & Sharp, C. M. 2000, ApJ, 531, 438 D'Angelo, G. & Lubow, S. H. 2008, ApJ, 685, 560 Delisle, J. B., Laskar, J., Correia, A. C. M., & Boué, G. 2012, A&A, 546, A71 Delrez, L., Ehrenreich, D., Alibert, Y., et al. 2021, Nature Astronomy, 5, 775 Dorn, C., Venturini, J., Khan, A., et al. 2017, A&A, 597, A37 Egger, J. A., Osborn, H. P., Kubyshkina, D., et al. 2024, A&A, 688, A223 Hatzes, A. P. 2016, in Astrophysics and Space Science Library, Vol. 428, Meth- Holman, M. J., Fabrycky, D. C., Ragozzine, D., et al. 2010, Science, 330, 51 Holman, M. J., Winn, J. N., Latham, D. W., et al. 2006, ApJ, 652, 1715 Izidoro, A., Ogihara, M., Raymond, S. N., et al. 2017, MNRAS, 470, 1750 Lissauer, J. J., Ragozzine, D., Fabrycky, D. C., et al. 2011, ApJS, 197, 8 Lovis, C. & Pepe, F. 2007, A&A, 468, 1115 Mamajek, E. E. & Hillenbrand, L. A. 2008, ApJ, 687, 1264 McKay, A. J., DiSanti, M. A., Kelley, M. S. P., et al. 2019, AJ, 158, 128 Pontoppidan, K. M., Pickering, T. E., Laidler, V. G., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9910, Observatory Operations: Strategies, Processes, and Systems VI, ed. Queloz, D., Henry, G. W., Sivan, J. P., et al. 2001, A&A, 379, 279 Rein, H. & Liu, S. F. 2012, A&A, 537, A128 Richard, C., Gordon, I. E., Rothman, L. S., et al. 2012, J. Quant. Spectr. Rad. Transf., 113, 1276 Sneden, C. A. 1973, PhD thesis, THE UNIVERSITY OF TEXAS AT AUSTIN. ter Braak, C. J. F. & Vrugt, J. A. 2008, Statistics and Computing, 18, 435 Thiabaud, A., Marboeuf, U., Alibert, Y., et al. 2014, A&A, 562, A27 Wisdom, J. & Holman, M. 1991, AJ, 102, 1528 A. Bonfanti",
"pages": [
17,
18,
19
]
},
{
"title": "Appendix A: Supplementary material",
"content": "A. Bonfanti et al.: Radii, masses, and transit-timing variations of the three-planet system orbiting the naked-eye star TOI-396 Notes. TESS (TE) LCs are identified by a counter based on the chronological order of observation. In particular, LCs from 1 to 11, from 12 to 21, from 22 to 32, and from 33 to 41 were extracted from Sector 3, 4, 30, and 31, respectively. c indicates a normalisation scalar; see text for further details. T tr [BJD] TTV [min] Sector 8385 . 794 . 8391 . 8397 + 0 - 0 . 010 . + 0 - 0 7447 + 0 - 0 7195 . 8403 . 8415 . 8421 . 8427 . 8433 . 9120 6857 ± 0 + 0 - 0 6337 + 0 - 0 6062 + 0 - 0 5817 + 0 - 0 5482 + 0 - 0 . . . . . . . . . . 5407 . 9126 518 ± 0 . 9132 . 9138 . 9150 . 9156 . 9162 . 9168 . 8387 . 8398 . 8432 . 9117 . 9139 . 9150 . 9162 4916 + 0 - 0 + 0 - 0 4731 + 0 - 0 4169 + 0 - 0 4050 + 0 - 0 3798 + 0 - 0 3562 + 0 - 0 2711 + 0 - 0 5047 + 0 - 0 1919 + 0 - 0 2589 + 0 - 0 7171 + 0 - 0 9371 + 0 - 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 016 . . . . 0068 0063 0046 0047 . 0033 0 0015 0013 0017 0016 0065 0033 0042 0077 0052 0043 . 010 - 8 0032 0020 0057 0040 0085 0090 0029 0044 0027 0030 0068 0084 0036 0034 0056 0051 0050 0034 0076 0068 0047 0025 0072 0037 0038 0031 + 17 + 19 + 23 - 8 + 2 - 3 . . + 0 - 1 + 0 - 10 - 12 . + 43 + 9 + 11 . . + 15 - 22 + 9 - 9 + 6 - 6 . 7 . 0 + 4 - 4 + 2 - 1 0 . . . 4 + 2 - 2 7 + 9 - 4 7 + 6 - 11 + 7 - 6 . . 5 3 5 8 + 14 - 15 + 4 - 2 2 + 8 - 5 8 + 12 - 13 . . 6 + 4 - 6 + 3 - 4 0 + 10 - 12 TOI-396 d - 0 . 4 - 2 + 6 + 2 - 13 + 0 . . 4 + 5 - 4 + 8 - 7 + 7 - 4 0 . 2 + 11 - 10 + 6 - 3 1 + 10 - 5 + 5 - 4 . 5 . 5 3 . . 2 . 9 1 4 . 2 . 9 8 . 6 . . . . . 9 . 2 . 7 . . . . 2 4 8 4 . 8 . 1 . . 7 8 . 2 . 9 . 5 . 3 . 4 . 7 6 8 . 1769 3 3 3 3 4 4 4 4 30 30 30 30 31 31 31 31 3 3 4 30 30 31 31 TOI-396 c",
"pages": [
21,
23,
24
]
}
]
2024arXiv241118738F
https://arxiv.org/pdf/2411.18738.pdf
<document>
<section_header_level_1><location><page_1><loc_10><loc_84><loc_91><loc_86></location>The Galaxy Activity, Torus, and Outflow Survey (GATOS). VII. The 20-214 𝜇 mimaging atlas of active galactic nuclei using SOFIA</section_header_level_1>
<text><location><page_1><loc_9><loc_76><loc_90><loc_83></location>L/i.pc/n.pc/d.pc/s.pc/a.pc/y.pc F/u.pc/l.pc/l.pc/e.pc/r.pc, 1 E/n.pc/r.pc/i.pc/q.pc/u.pc/e.pc L/o.pc/p.pc/e.pc/z.pc-R/o.pc/d.pc/r.pc/i.pc/g.pc/u.pc/e.pc/z.pc, 2, 3 I/s.pc/m.pc/a.pc/e.pc/l.pc G/a.pc/r.pc/c.pc'/dotlessi.pc/a.pc-B/e.pc/r.pc/n.pc/e.pc/t.pc/e.pc, 4 C/r.pc/i.pc/s.pc/t.pc/i.pc/n.pc/a.pc R/a.pc/m.pc/o.pc/s.pc A/l.pc/m.pc/e.pc/i.pc/d.pc/a.pc, 5, 6 A/l.pc/m.pc/u.pc/d.pc/e.pc/n.pc/a.pc A/l.pc/o.pc/n.pc/s.pc/o.pc-H/e.pc/r.pc/r.pc/e.pc/r.pc/o.pc, 7 C/h.pc/r.pc/i.pc/s.pc P/a.pc/c.pc/k.pc/h.pc/a.pc/m.pc, 1, 8 L/u.pc/l.pc/u.pc Z/h.pc/a.pc/n.pc/g.pc, 1 M/a.pc/s.pc/o.pc/n.pc L/e.pc/i.pc/s.pc/t.pc, 1 N/a.pc/n.pc/c.pc/y.pc L/e.pc/v.pc/e.pc/n.pc/s.pc/o.pc/n.pc, 9 M/a.pc/s.pc/a.pc I/m.pc/a.pc/n.pc/i.pc/s.pc/h.pc/i.pc, 8 S/e.pc/b.pc/a.pc/s.pc/t.pc/i.pc/a.pc/n.pc H/o.pc/e.pc/n.pc/i.pc/g.pc, 10 M/a.pc/r.pc/k.pc/o.pc S/t.pc/a.pc/l.pc/e.pc/v.pc/s.pc/k.pc/i.pc, 11, 12 C/l.pc/a.pc/u.pc/d.pc/i.pc/o.pc R/i.pc/c.pc/c.pc/i.pc, 13, 14 E/r.pc/i.pc/n.pc H/i.pc/c.pc/k.pc/s.pc, 15 E/n.pc/r.pc/i.pc/c.pc/a.pc B/e.pc/l.pc/l.pc/o.pc/c.pc/c.pc/h.pc/i.pc, 16, 17 F/r.pc/a.pc/n.pc/c.pc/o.pc/i.pc/s.pc/e.pc C/o.pc/m.pc/b.pc/e.pc/s.pc, 18 R/i.pc/c.pc D/a.pc/v.pc/i.pc/e.pc/s.pc, 19 S/a.pc/n.pc/t.pc/i.pc/a.pc/g.pc/o.pc G/a.pc/r.pc/c.pc'/dotlessi.pc/a.pc B/u.pc/r.pc/i.pc/l.pc/l.pc/o.pc, 20 O/m.pc/a.pc/i.pc/r.pc/a.pc G/o.pc/n.pc/z.pc '/a.pc/l.pc/e.pc/z.pc M/a.pc/r.pc/t.pc'/dotlessi.pc/n.pc, 21 T/a.pc/k.pc/u.pc/m.pc/a.pc I/z.pc/u.pc/m.pc/i.pc, 8 A/l.pc/v.pc/a.pc/r.pc/o.pc L/a.pc/b.pc/i.pc/a.pc/n.pc/o.pc, 22 M/i.pc/g.pc/u.pc/e.pc/l.pc P/e.pc/r.pc/e.pc/i.pc/r.pc/a.pc S/a.pc/n.pc/t.pc/a.pc/e.pc/l.pc/l.pc/a.pc, 23 D/i.pc/m.pc/i.pc/t.pc/r.pc/a.pc R/i.pc/g.pc/o.pc/p.pc/o.pc/u.pc/l.pc/o.pc/u.pc, 4, 24 D/a.pc/v.pc/i.pc/d.pc R/o.pc/s.pc/a.pc/r.pc/i.pc/o.pc, 25 D/a.pc/n.pc/i.pc/e.pc/l.pc R/o.pc/u.pc/a.pc/n.pc, 26 T/a.pc/r.pc/o.pc S/h.pc/i.pc/m.pc/i.pc/z.pc/u.pc, 19 /a.pc/n.pc/d.pc M/a.pc/r.pc/t.pc/i.pc/n.pc W/a.pc/r.pc/d.pc 27</text>
<text><location><page_1><loc_27><loc_74><loc_72><loc_75></location>1 University of Texas at San Antonio, One UTSA Circle, San Antonio, TX, 78249, USA</text>
<text><location><page_1><loc_25><loc_73><loc_75><loc_74></location>2 Department of Physics & Astronomy, University of South Carolina, Columbia, SC 29208, USA</text>
<text><location><page_1><loc_21><loc_71><loc_79><loc_72></location>3 Kavli Institute for Particle Astrophysics & Cosmology (KIPAC), Stanford University, Stanford, CA 94305, USA</text>
<text><location><page_1><loc_28><loc_70><loc_71><loc_71></location>4 Department of Physics, University of Oxford, Keble Road, Oxford OX1 3RH, UK</text>
<text><location><page_1><loc_25><loc_68><loc_75><loc_69></location>5 Instituto de Astrof'ısica de Canarias, Calle V'ıa L'actea, s/n, E-38205 La Laguna, Tenerife, Spain</text>
<text><location><page_1><loc_25><loc_67><loc_75><loc_68></location>6 Departamento de Astrof'ısica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain</text>
<text><location><page_1><loc_17><loc_65><loc_83><loc_67></location>7 Centro de Astrobiolog'ıa (CAB), CSIC-INTA, Camino Bajo del Castillo s/n, E-28692, Villanueva de la Ca˜nada, Madrid, Spain</text>
<text><location><page_1><loc_13><loc_63><loc_87><loc_65></location>8 National Astronomical Observatory of Japan, National Institutes of Natural Sciences (NINS), 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 9</text>
<text><location><page_1><loc_27><loc_63><loc_73><loc_63></location>Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA</text>
<text><location><page_1><loc_26><loc_61><loc_73><loc_62></location>10 School of Physics & Astronomy, University of Southampton, Southampton SO17 1BJ, UK</text>
<text><location><page_1><loc_33><loc_60><loc_67><loc_61></location>11 Astronomical Observatory, Volgina 7, 11060 Belgrade, Serbia</text>
<text><location><page_1><loc_25><loc_58><loc_74><loc_59></location>12 Sterrenkundig Observatorium, Universiteit Gent, Krijgslaan 281-S9, Gent B-9000, Belgium</text>
<text><location><page_1><loc_16><loc_57><loc_83><loc_58></location>13 N'ucleo de Astronom'ıa de la Facultad de Ingenier'ıa, Universidad Diego Portales, Av. Ej'ercito Libertador 441, Santiago, Chile</text>
<text><location><page_1><loc_20><loc_55><loc_79><loc_56></location>14 Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, People's Republic of China</text>
<text><location><page_1><loc_22><loc_54><loc_78><loc_55></location>15 Department of Physics and Astronomy, University of Alaska Anchorage, Anchorage, AK 99508-4664, USA</text>
<text><location><page_1><loc_15><loc_52><loc_85><loc_54></location>16 Departmento de F'ısica de la Tierra y Astrof'ısica, Fac. de CC F'ısicas, Universidad Complutense de Madrid, E-28040 Madrid, Spain</text>
<text><location><page_1><loc_15><loc_51><loc_86><loc_52></location>Instituto de F'ısica de Part'ıculas y del Cosmos IPARCOS, Fac. CC F'ısicas, Universidad Complutense de Madrid, E-28040 Madrid, Spain</text>
<text><location><page_1><loc_20><loc_50><loc_79><loc_51></location>18 LERMA, Observatoire de Paris, Coll'ege de France, PSL University, CNRS, Sorbonne University, Paris, France</text>
<text><location><page_1><loc_19><loc_48><loc_80><loc_49></location>19 Max Planck Institut fur Extraterrestrische Physik, Giessenbachstrasse 1, D-85748 Garching bei Munchen, Germany</text>
<text><location><page_1><loc_29><loc_47><loc_70><loc_48></location>20 Observatorio de Madrid, OAN-IGN, Alfonso XII, 3, E-28014 Madrid, Spain</text>
<text><location><page_1><loc_8><loc_44><loc_91><loc_46></location>21 Instituto de Radioastronom'ıa y Astrof'ısica (IRyA), Universidad Nacional Aut'onoma de M'exico, Antigua Carretera a P'tzcuaro #8701, ExHda. San Jos'e de la Huerta, Morelia, Michoac'an, C.P. 58089, Mexico</text>
<text><location><page_1><loc_16><loc_43><loc_83><loc_44></location>22 Telespazio UK for the European Space Agency, ESAC, Camino Bajo del Castillo s/n, E-28692 Villanueva de la Ca˜nada, Spain</text>
<text><location><page_1><loc_28><loc_41><loc_72><loc_42></location>23 Instituto de F'ısica Fundamental, CSIC, Calle Serrano 123, E-28006 Madrid, Spain</text>
<text><location><page_1><loc_24><loc_40><loc_76><loc_41></location>24 School of Sciences, European University Cyprus, Diogenes street, Engomi, 1516 Nicosia, Cyprus</text>
<text><location><page_1><loc_22><loc_38><loc_78><loc_39></location>25 School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne NE1 7RU, UK</text>
<text><location><page_1><loc_11><loc_37><loc_89><loc_38></location>26 LESIA, Observatoire de Paris, Universit'e PSL, CNRS, Sorbonne Universit'e, Sorbonne Paris Cite'e, 5 place Jules Janssen, F-92195 Meudon, France</text>
<text><location><page_1><loc_19><loc_35><loc_81><loc_36></location>27 Centre for Extragalactic Astronomy, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK</text>
<section_header_level_1><location><page_1><loc_46><loc_32><loc_54><loc_33></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_13><loc_86><loc_32></location>Wepresent a 19 . 7 -214 𝜇 mimaging atlas of local (4 -181 Mpc; median 43 Mpc) active galactic nuclei (AGN) observed with FORCAST and HAWC+ on board the SOFIA telescope with angular resolutions ∼ 3 '' -20 '' . This atlas comprises 22 Seyferts (17 Type 2 and 5 Type 1) with a total of 69 images, 41 of which have not been previously published. The AGN span a range of luminosities of log 10 ( 𝐿 bol [ erg / s ]) = [ 42 , 46 ] with a median of log 10 ( 𝐿 bol [ erg / s ]) = 44 . 1 ± 1 . 0. We provide total fluxes of our sample using aperture photometry for point source objects and a 2-D Gaussian fitting for objects with extended host galaxy emission, which was used to estimate the unresolved nuclear component. Most galaxies in our sample are point-like sources, however, four sources (Centaurus A, Circinus, NGC 1068, and NGC 4388) show extended emission in all wavelengths. The 30 -40 𝜇 m extended emission in NGC 4388 is coincident with the narrow line region at PA ∼ 50 · , while the dusty extension at longer wavelengths arises from the host galaxy at PA ∼ 90 · . Our new observations allow us to construct the best sampled spectral energy distributions (SEDs) available between 30 - 500 𝜇 m for a sample of nearby AGN. We estimate that the average peak wavelength of the nuclear SEDs is ∼ 40 𝜇 m in 𝜈𝐹 𝜈 , which we associate with an unresolved extended dusty region heated by the AGN.</text>
<text><location><page_1><loc_13><loc_51><loc_14><loc_52></location>17</text>
<text><location><page_2><loc_52><loc_77><loc_92><loc_87></location>CO(2-1) emission in NGC 5643 as a nuclear molecular gas component of the torus that is likely collimating the ionization cone. Conditions favorable for launching a cold and molecular wind likely depend on Eddington ratio and nuclear hydrogencolumndensities (e.g., Venanzi et al. 2020; Garc'ıa-Burillo et al. 2021; Alonso-Herrero et al. 2021; Garc'ıa-Bernete et al. 2022a).</text>
<text><location><page_2><loc_52><loc_44><loc_92><loc_77></location>This pc-scale dusty component is possibly associated with larger scale MIR emission detected out to 100s pc scales. In the case of Circinus, high angular resolution MIR imaging, optical polarimetry and integral field spectra, coupled with state-of-the-art radiative transfer simulations, provide evidence that extended dust emission from pc to tens of pc scales in this object is a result of a hollow dusty cone illuminated by a tilted accretion disk (Stalevski et al. 2017, 2019, 2023; Kakkad et al. 2023). MIR extended emission out to 1' ( ∼ 75 pc) was clearly detected in NGC 1068 by Bock et al. (2000). Later 10 . 8 and 18 . 2 𝜇 m emission extending 3 . '' 5 ( ∼ 200 pc) across NGC 4151 was also attributed to dust in the NLR heated by the central engine (Radomski et al. 2003). Likewise, at similar wavelengths, extended emission in 18 AGN at distances out to hundreds of parsecs was detected (Asmus et al. 2016; Garc'ıa-Bernete et al. 2016; Asmus 2019). Using the 37 . 1 𝜇 m filter on SOFIA/FORCAST and thanks to the increase in angular resolution compared with 𝑆𝑝𝑖𝑡𝑧𝑒𝑟 , extended dust emission in Mrk 3, NGC 4151, and NGC4388 was found on ∼ 100s pc-scales (Fuller et al. 2019) coincident with the NLR and radio axis. This emission may be due to dust along the wall of ionization cones (Mason et al. 2009) or a dusty NLR (Mor et al. 2009; Mor & Netzer 2012).</text>
<text><location><page_2><loc_52><loc_24><loc_92><loc_44></location>In this manuscript we present an imaging atlas of 22 local (D = 4 -181 Mpc; median 42.8 Mpc) AGN obtained using the FORCAST and HAWC+ instruments on the 2.7-m SOFIA telescope in the wavelength range 20 -214 𝜇 m. Most of these datasets are unpublished or dispersed throughout the literature. We provide a mid- to far-IR imaging atlas at angular scales of ∼ 3 -20 '' . At these scales, contribution from several dust sources is expected and we expect to disentangle the emission sources in a future study. Instead, here we aim to determine whether these objects are extended or not, and at what wavelengths within the resolution of the SOFIA telescope. We also explore the wavelength of turnover in the SED. This atlas is complementary to JWST observations up to ∼ 25 𝜇 mand archival Herschel data (70 -500 𝜇 m).</text>
<text><location><page_2><loc_52><loc_17><loc_92><loc_24></location>The manuscript is organzed as follows. Section 2 describes the observations and AGN sample definition; Section 3 shows the new IR images; Section 4 contains details of the imaging analysis and Section 5 shows the resulting SEDS; we present results about the data in Section 6.</text>
<section_header_level_1><location><page_2><loc_56><loc_15><loc_88><loc_16></location>2. AGN SAMPLE AND OBSERVATION DATA</section_header_level_1>
<text><location><page_2><loc_65><loc_13><loc_79><loc_14></location>2.1. Sample Selection</text>
<text><location><page_2><loc_52><loc_9><loc_92><loc_12></location>This imaging atlas was drawn from the ongoing AGN survey performed by the Galactic Activity, Torus, and Out-</text>
<text><location><page_2><loc_14><loc_89><loc_44><loc_90></location>Keywords: galaxies -- active galaxies -- agn</text>
<section_header_level_1><location><page_2><loc_21><loc_86><loc_35><loc_87></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_2><loc_8><loc_51><loc_48><loc_85></location>There is clear evidence that a considerable amount of dust in the vicinity of supermassive black holes (SMBHs) in active galaxies obscures the central engine (i.e., accretion disk and SMBH) in some lines of sight. Through spectropolarimetric observations of NGC 1068, Antonucci & Miller (1985) showed that its optical polarized spectrum contained broad optical polarized emission lines not originally observed by direct total intensity observations. It was subsequently presumed that an optically and geometrically thick dusty structure ('torus') blocked the central engine in some lines of sight (Antonucci 1993; Urry & Padovani 1995). Under this unified scheme a Type 1 AGN is seen face-on and shows broadened optical lines, while in Type 2 AGN the broadened lines are obscured. This model also predicts that broad silicate features at 10 and 18 𝜇 m will be seen in emission in Type 1 and in absorption in Type 2. However, silicate emission can be seen in emission in some Type 2 AGN, while absorption can be seen in some Type 1 (e.g., Hatziminaoglou et al. 2015). This and other observational features are explained by the inhomogeneous nature of the torus. Clumpy torus models (Nenkova et al. 2008a,b) predict shallower silicate features, more similar infrared spectral energy distributions (SEDs) between Type 1 and Type 2 AGN, etc. (see Ramos Almeida & Ricci 2017, for a review).</text>
<text><location><page_2><loc_8><loc_25><loc_48><loc_51></location>A region of narrow forbidden line emission extends above and below the midplane of the dusty torus structure out to several kpc scales. Recent sub-arcsecond interferometric imaging observations have shown a dust component at pc-scales co-spatial with the base of the narrow line region (NLR; Honig et al. 2012; Tristram et al. 2014; L'opez-Gonzaga et al. 2014, 2016; Burtscher et al. 2013; G'amez Rosas et al. 2022; Isbell et al. 2022). This dusty structure is interpreted as part of a dusty wind launched from the inner hot part of the torus driven by radiation pressure at pc-scales (Honig 2019), but generated by a magnetohydrodynamical wind at sub-pc scales (e.g., Emmering et al. 1992; Lopez-Rodriguez et al. 2015; Takasao et al. 2022; Lopez-Rodriguez et al. 2023). This extended dusty structure has been resolved in a nearby galaxy, ESO 418-G14, using mid-infrared (MIR) images with JWST/MIRI finding that the dust is primarily heated by the AGN and/or radiative jet-induced shocks in the NLR rather than a wind (Haidar et al. 2024).</text>
<text><location><page_2><loc_8><loc_10><loc_48><loc_24></location>ALMA observations provide observational support for a dusty torus+outflow scenario. Emission from the nucleus of NGC1068 was mapped with a resolution of ∼ 4 pc, resolving a 7 -10 pc diameter disk interpreted as the sub-mm counterpart of the torus (Garc'ıa-Burillo et al. 2016). Rotation of the compact emission was detected in HCN J=3-2 and HCO+ J=3-2 (Imanishi et al. 2018, 2020). A molecular outflowing wind co-spatial with the dusty and molecular torus was also observed (Garc'ıa-Burillo et al. 2019). Alonso-Herrero et al. (2018) interpreted the measured nuclear (10 -20 pc)</text>
<text><location><page_3><loc_8><loc_82><loc_48><loc_92></location>flow Survey (GATOS; Garc'ıa-Burillo et al. 2021; AlonsoHerrero et al. 2021; Garc'ıa-Bernete et al. 2024). GATOS /one.sup aims to characterize the dynamics and composition of the dusty and molecular torus and multi-phase outflows in AGN. The GATOS parent sample is selected from the 70 Month Swift /BAT AGN catalog, which is flux limited in the ultrahard 14 -195 keV X-rays band (Baumgartner et al. 2013).</text>
<text><location><page_3><loc_8><loc_46><loc_48><loc_81></location>In the initial study of AGN using SOFIA observations (Fuller et al. 2016), sources from the GATOS survey were selected based on the criteria that the galaxies had been previously studied using C/l.pc/u.pc/m.pc/p.pc/y.pc (Nenkova et al. 2008a,b) torus models and were well-sampled in the 1 -18 𝜇 m regime (Ramos Almeida et al. 2009, 2011; Alonso-Herrero et al. 2011). The study of the 11 objects included the 31 . 5 𝜇 m photometry in the SEDs and found that including the 31.5 𝜇 m photometry reduces the number of C/l.pc/u.pc/m.pc/p.pc/y.pc torus models that are compatible with the data and modifies the model output for the torus outer radius. Fuller et al. (2019) further extended the wavelength range of a subset of 7 AGN SEDs to 37.1 𝜇 m. They subsequently found extended emission in the PSF-subtracted images of Mrk 3, NGC 4151, and NGC 4388 that is coincident with the radio axis and NLR. In a separate study, Lopez-Rodriguez et al. (2018) modeled the torus of NGC 1068 using ∼ 20 -53 𝜇 m FORCAST and HAWC+ observations. They showed that the peak wavelength range of emission from the torus is ∼ 30 - 40 𝜇 m with a characteristic temperature 70 - 100 K. The use of observations ¿ 30 𝜇 m in that study from SOFIA and ALMA highlights the importance of longer wavelength observations to put constraints on MIR emission sources. Based on these results, we extend the wavelength range in objects previously observed, and also expand the number of AGN observed.</text>
<text><location><page_3><loc_8><loc_18><loc_48><loc_45></location>We present the complete imaging atlas of 22 Seyferts observed by SOFIA in the wavelength range 19 . 7 -214 𝜇 m using FORCAST and HAWC+. The final set of observations presented here was part of a multi-year AGN survey over several observing SOFIA cycles (Proposal IDs: 02 0035, 04 0048, 06 0066, 08 0014; PI: Lopez-Rodriguez; 70 0400 PI: Herter). The SOFIA atlas of AGN in the far-IR (FIR) is a flux-limited sample of nearby, bright, and well-studied AGN. All objects have a point source flux of > 200 mJy at 31 . 5 𝜇 m, which ensures that each band can be observed within 1 hr of on-source time with a signal to noise ratio > 10 using FORCAST/SOFIA. Although the original AGN sample is larger than that presented here, only 22 AGN were observed in total by SOFIA before end of operations in 2022. Note that there are gaps in the 20 -214 𝜇 m wavelength range due to the fact that SOFIA only flies with a single instrument per night. For each SOFIA cycle, we prioritized the objects with observations acquired in a single instrument from the previous cycle.</text>
<text><location><page_3><loc_8><loc_14><loc_48><loc_18></location>The sample properties are given in Table 1. For most objects, we retrieved redshift data from the NASA Extragalactic Database (NED). However, for nearby objects Centaurus A</text>
<table>
<location><page_3><loc_54><loc_55><loc_91><loc_89></location>
<caption>Table 1. SOFIA AGN Sample</caption>
</table>
<text><location><page_3><loc_53><loc_48><loc_92><loc_55></location>Redshifts and spectral type were taken from NED. Distances to most sources were obtained using H0 = 70 km s -1 Mpc -1 . Distances to nearby sources Centaurus A and Circinus were taken from Harris et al. (2010) and Tully et al. (2009), respectively. References for log 𝐿 𝑏𝑜𝑙 : a) Borkar et al. (2021) b) Marinucci et al. (2012) c) Alonso-Herrero et al. (2011) d) Ichikawa et al. (2017), e) Leighly et al. (2014) f) Marconi et al. (2004), g) Baumgartner et al. (2013), h) Garc'ıa-Bernete et al. (2015), i) Ramos Almeida et al. (2011), j) Yuan et al. (2002), k) Duras et al. (2020),</text>
<text><location><page_3><loc_52><loc_34><loc_92><loc_46></location>and Circinus, distances were obtained individually (Harris et al. 2010; Tully et al. 2009). The 22 objects in this atlas cover the luminosity range of log 10 ( 𝐿 bol [ erg / s ]) = [ 42 , 46 ] with a median of log 10 ( 𝐿 bol [ erg / s ]) = 44 . 1 ± 1 . 0, and a distance of 4 -181 Mpc with a median of 42.8 Mpc. Figure 1 shows bolometric luminosity plotted against distance, where Seyfert 1 objects are shown as red triangles and Seyfert 2 objects are shown as purple stars.</text>
<section_header_level_1><location><page_3><loc_57><loc_32><loc_87><loc_33></location>2.2. SOFIA Observations and Data Reduction</section_header_level_1>
<section_header_level_1><location><page_3><loc_67><loc_30><loc_77><loc_31></location>2.2.1. FORCAST</section_header_level_1>
<text><location><page_3><loc_52><loc_9><loc_92><loc_29></location>FORCAST is an IR camera and spectrograph sensitive in the wavelength range 5 -40 𝜇 m with a field of view (FOV) of 3.4 ' × 3.2 ' and pixel scale 0.768 '' /pixel. With one exception (NGC 1068 in the 19.7 𝜇 m filter), we only used the Long Wavelength Camera (LWC; 25 -40 𝜇 m) due to the abundance of ground-based images at shorter wavelengths for the objects in our sample. FORCAST observations were made in dual channel mode using the two-position chop-nod (C2N) method with symmetric nod-match-chop (NMC) to remove telescope thermal emission and time variable sky background, and to reduce the effect of 1/ 𝑓 noise from the array. Data were reduced by the SOFIA Science Center using the /f.pc/o.pc/r.pc/c.pc/a.pc/s.pc/t.pc /r.pc/e.pc/d.pc/u.pc/x.pc pipeline following the methods described by Herter et al. (2012). Most of the pipeline changes over the</text>
<figure>
<location><page_4><loc_9><loc_69><loc_44><loc_90></location>
<caption>Figure 1. Luminosity plotted against distance of the 22 AGN in the atlas. Red triangles represent Sy1 and purple stars represent Sy2.</caption>
</figure>
<text><location><page_4><loc_8><loc_61><loc_48><loc_63></location>cycles were to refine the spectroscopic mode of FORCAST with little or no change to the image mode presented here.</text>
<text><location><page_4><loc_8><loc_42><loc_48><loc_60></location>Observations were flux-calibrated using the set of standard stars of the observing run, which provides flux uncertainties of ∼ 10 %. The point spread function (PSF) of the 31.5 𝜇 m observations from Cycle 2 (see Fuller et al. 2016) was the co-average of a set of standard stars from that cycle. Its FWHMwas 3.40 '' , in agreement with the SOFIA Observer's Handbook v3.0.0. The PSFs for Cycle 4 in the 30 - 40 𝜇 mwavelength range were determined by using standard star observations from the individual flights (see Fuller et al. 2019) and averaged at FWHM ∼ 4.33 '' and 4.58 '' in the 31.5 and 37.1 𝜇 m filters, respectively. The FORCAST PSFs for the observations of NGC1068 are detailed in Lopez-Rodriguez et al. (2018).</text>
<section_header_level_1><location><page_4><loc_23><loc_38><loc_33><loc_39></location>2.2.2. HAWC+</section_header_level_1>
<text><location><page_4><loc_8><loc_31><loc_48><loc_36></location>HAWC+isaFIRimaging polarimeter designed to allow total and polarized intensity imaging observations in four broad bands centered at 53, 89, 155, and 214 𝜇 m, corresponding to Bands A, C, D, and E respectively (see Table 2). On-</text>
<text><location><page_4><loc_52><loc_70><loc_92><loc_92></location>-fly mapping (OTFMAP) observing modes were used for both imaging polarimetry and total intensity imaging. Observing modes for the individual observations are given in Table 3. Data taken in polarization mode was reduced using the /h.pc/a.pc/w.pc/c.pc /d.pc/p.pc/r.pc /p.pc/i.pc/p.pc/e.pc/l.pc/i.pc/n.pc/e.pc and the reduction steps presented in Lopez-Rodriguez et al. (2022b). The Comprehensive Reduction Utility for SHARC II (CRUSH; Kov'acs et al. 2006; Kov'acs 2008) was used to obtain the total intensity observations. HAWC+ observations were reduced following the same reduction steps. We quote the pipeline versions or CRUSH versions in Table 3 to differentiate between the observing modes. There are no differences between CRUSH versions to obtain the total intensity images as all the changes in the pipeline were done for the polarimetric mode. Table 3 also shows the pixel scale of each image.</text>
<table>
<location><page_4><loc_57><loc_59><loc_88><loc_69></location>
<caption>Table 2. HAWC+ filter suite</caption>
</table>
<text><location><page_4><loc_52><loc_41><loc_92><loc_57></location>As in FORCAST observations, the source of uncertainty in the photometry for HAWC+ results from calibration factors of the standard stars associated with the observation, giving an uncertainty of ∼ 10 % (Lopez-Rodriguez et al. 2022b). HAWC+ PSFs were estimated using standard star observations in 2017. Pallas was observed in Bands A and C on 7 November 2017, while Neptune was observed in Bands D and E on 19 October, 2017. The FWHM of these standards are 5.25 '' , 8.26 '' , 14.74 '' , and 19.65 '' in Bands A, C, D, and E, respectively. The FWHMs from the Observer's Handbook /two.sup are given in Table 2.</text>
<section_header_level_1><location><page_4><loc_65><loc_39><loc_79><loc_40></location>2.3. Observing Data</section_header_level_1>
<text><location><page_4><loc_52><loc_32><loc_92><loc_38></location>Table 3 provides the final AGN sample with information about the wavelength, observing mode, observation and mission details, versions of the separate pipelines, and also the field-of-view (FOV) of the individual images.</text>
<paragraph><location><page_5><loc_12><loc_49><loc_14><loc_53></location>T able 3 .</paragraph>
<table>
<location><page_5><loc_16><loc_10><loc_88><loc_92></location>
<caption>T able 3 continued</caption>
</table>
<table>
<location><page_6><loc_10><loc_10><loc_87><loc_92></location>
<caption>T able 3 (continued)T able 3 continued</caption>
</table>
<paragraph><location><page_7><loc_8><loc_46><loc_10><loc_55></location>T able 3 (continued)</paragraph>
<section_header_level_1><location><page_7><loc_45><loc_19><loc_51><loc_80></location>N /o.pc/t.pc/e.pc -Obser v ation Data -Column 1: Object; Column 2: W a v elength; Column 3: Ins tr ument; Column 4: Obser v ation date; Column 5: Obser ving mode; Column 6: Pipeline/CR USH v ershion; Column 7: On-source time; Column 8: Aircraft s tar ting altitude; Column 9: Mission ID; Column 10: Prog ram ID; Column 11: FO V of imag es in Section 3 ; Column 12: Pix el scale</section_header_level_1>
<table>
<location><page_7><loc_11><loc_10><loc_46><loc_92></location>
</table>
<section_header_level_1><location><page_8><loc_24><loc_90><loc_32><loc_92></location>3. IMAGES</section_header_level_1>
<text><location><page_8><loc_8><loc_77><loc_48><loc_90></location>Images of the 22 AGN in the 19 . 7 -214 𝜇 m wavelength range are presented in Figures 2, 3, 4, and 5. The orange scale on the bottom left of the images indicates a scale of 500 pc. The beam size is depicted in white in the top right of the images. In all images, north is up and east is to the left. Complementary Herschel 70 -500 𝜇 m images are shown in Appendix A (Fig. 10, 11, 12, 13). These fully reduced images were obtained through the Herschel Science Archive /three.sup . All objects are presented and analyzed individually.</text>
<text><location><page_8><loc_8><loc_58><loc_48><loc_77></location>Centaurus A . The host galaxy of Centaurus A is clearly visible in the 53 𝜇 m image with a bright compact center. However, the host galaxy becomes more dominant in the 89 𝜇 mimage(seeadetailed analysis of host galaxy dust emission at 89 𝜇 m in Lopez-Rodriguez 2021). The kpc-scale warped dust and gas lane was first observed with Spitzer imaging using IRAC and MIPS (Quillen et al. 2006). On subarcsecond scales, Radomski et al. (2008) observed the nucleus of Centaurus A using the 8.8, 10.4, and 18.3 𝜇 m filters on T-ReCS at Gemini South. They concluded that the mostly likely sources of nuclear MIR emission are an unresolved clumpy dusty torus in the core, and a dusty NLR for the arcsecondscale extended emission (see also Garc'ıa-Bernete et al. 2016).</text>
<text><location><page_8><loc_8><loc_36><loc_48><loc_56></location>Circinus Galaxy . HAWC+ 53 and 89 𝜇 m images of the Circinus Galaxy show a very bright FIR core with extended emission at a PA ∼ 30 · , whereas the 215 𝜇 m image shows a slightly different PA ∼ 55 · . We estimate the FWHM of the extended FIR nuclear emission to be ∼ 6 '' × 6 '' , 13.5 '' × 11.5 '' , and 22.3 '' × 25.6 '' at 53, 89, and 214 𝜇 m, respectively. These are larger than the PSF FWHMs given in Section 2.2.2, which indicates extended emission along the axis of the inner bar of the galaxy. MIR emission was resolved at 8.7 and 18.3 𝜇 m out to 2 '' in an approximate east-west direction, coincident with the ionization cones at PA ∼ 100 · (Packham et al. 2005; Stalevski et al. 2017). However, the elongation seen in the SOFIAimages(andin Herschel imagesinAppendixA)seems to be arising from dust in the host galaxy.</text>
<text><location><page_8><loc_8><loc_19><loc_48><loc_36></location>MCG-5-23-16 . MCG-5-23-16 appears as a point-like source in the 31 . 5 -155 𝜇 mwavelength range whose brightness decreases with increasing wavelength. However, this galaxy appears as an extended source in the MIR using high angular resolution data from VLT (Garc'ıa-Bernete et al. 2016). Likewise, Ferruit et al. (2000) found that this nucleus has an extended optical NLR at a PA of 40 · . They found a dust lane extending 2 '' on either side of the nucleus, parallel to the axis of the galaxy. Their extended dusty emission at 40 · was detected at a 3 𝜎 level up to ∼ 4 '' from the core. This extended structure has no thermal emission counterpart within the 31 . 5 -155 𝜇 mwavelength range.</text>
<text><location><page_8><loc_8><loc_15><loc_48><loc_19></location>Mrk3 . Although the FORCAST and HAWC+ images generally appear to be point-like sources, Fuller et al. (2019) found extended emission in the PSF-subtracted 37.1 𝜇 m im-</text>
<text><location><page_8><loc_52><loc_77><loc_92><loc_92></location>of Mrk 3 in the direction of the radio axis (84 · ; Kukula et al. 1993) and the NLR ( ∼ 70 · ; Capetti et al. 1995). Asmus et al. (2013) found an elongated nucleus out to ∼ 170 pc with PA ∼ 70 · in the Si-5 (11.6 𝜇 m) filter using Gemini/MICHELLE. However, the Si-2 (8.7 𝜇 m) image from GTC/Canaricam appears point-like (Alonso-Herrero et al. 2016). The large scale east-west structure in the 53 𝜇 mimage here is a background artifact produced by the data reduction due to the small spatial coverage and short integration time of the observation.</text>
<text><location><page_8><loc_52><loc_56><loc_92><loc_77></location>Mrk 231 . The 89 𝜇 mimage shown here is point-like with a FWHM of ∼ 8.5', similar to the standard FWHM of 8.26' (see Section 2.2.2). Mrk 231 is a Type 1 Ultra Luminous Infrared Galaxy (ULIRG) and is the nearest known quasar at a distance of 181 Mpc. It is known for its multi-phase and multi-scale outflows (see Rupke & Veilleux 2011), with a neutral outflow up to 3 kpc in radius (Rupke et al. 2005). Mrk 573 . Although the SNR is very low (3 -4 𝜎 ), both 31.5 and 37.1 𝜇 mimages of Mrk 573 show marginally resolved ∼ 4 . 5 '' elongation in the east-west direction at PA ∼ 110 · . Mrk 573 was previously shown to have a biconical NLR coincident with radio emission 3-4' from the nucleus at a PA ∼ 125 · (Ulvestad & Wilson 1984; Pogge & De Robertis 1995). The marginal detection here may be cold extended dust in the outer layers of the NLR.</text>
<text><location><page_8><loc_52><loc_35><loc_92><loc_55></location>NGC 1068 . The 19 - 53 𝜇 m images of NGC 1068 were published previously in Lopez-Rodriguez et al. (2018), where it was shown that the peak emission from the torus occurs between 30 - 40 𝜇 m with a corresponding temperature of 70 - 100 K. The 89 𝜇 m image was published as a polarimetric observation (Lopez-Rodriguez et al. 2020). Within a scale of about 1 kpc, NGC 1068 shows extended emission in the NE to SW direction at a PA ∼ 45 · , similar to MIR observations using VISIR/VLT (Asmus et al. 2014). Their observations revealed a nuclear structure in the north-south direction and extended structures to the NE and SW. From the N-band spectrum, Mason et al. (2006) concluded that while torus emission dominates NIR wavelengths, large-scale MIR emission is dominated by diffuse dust within the ionization cones.</text>
<text><location><page_8><loc_52><loc_17><loc_92><loc_35></location>NGC 1275 . All 30 - 53 𝜇 m images are dominated by a point-like source. However, the 31.5 𝜇 m image (Fuller et al. 2019) shows 3 𝜎 extended emission along the PA ∼ 140 · . This AGN is known to have a network of H 𝛼 filaments extending out to ∼ 100' (see Conselice et al. 2001) and is possibly the result of a merger (Holtzman et al. 1992). The MIR core shows silicate dust emission in both 10 and 18 𝜇 mbands (see Fuller et al. 2019). Hence, both dust and gas are extended covering several kpc around the core. In the HAWC+ 89 𝜇 mfilter, Lopez-Rodriguez et al. (2023) found extended dust emission at a PA ∼ 125 · out to a 12 kpc radius potentially associated with a magnetized dusty filament along the NW direction (Fabian et al. 2008).</text>
<text><location><page_8><loc_52><loc_9><loc_92><loc_16></location>NGC 2110 . The 30 - 215 𝜇 mimages of NGC 2110 are all point-like. The north-south pattern in the 53 𝜇 mimage is due to background noise and does not represent extended dust. NGC2110is a Type 2 AGN that shows silicate emission at 10 and18 𝜇 mthatis interpreted as a result of a clumpy torus, or as</text>
<figure>
<location><page_9><loc_12><loc_38><loc_89><loc_87></location>
<caption>Figure 2. FORCAST 31.5, and 37.1 𝜇 m images, and HAWC+ 53, 89, 155, and 215 𝜇 m images. Each image has a differing FOV, which can be found in Table 3. For bright objects (Centaurus A, Circinus, and Mrk 231) contours start at 3 𝜎 , then follow log(maximum) from [-1.2 to 0.8], [-1.8 to 0.8] [-1.4,0.8] in steps of 0.2. For MCG-5-23-16 and Mrk 3, the lowest contours are 3 𝜎 and increase in steps of 5 𝜎 . The white transparent circle on the top right indicates the telescope beam size. The orange bar on the bottom left of the images is scaled to 500 pc. For all images, north is up and east is to the left.</caption>
</figure>
<text><location><page_9><loc_8><loc_16><loc_48><loc_25></location>dust within the ionization cones (PA ∼ 160 · ; Mulchaey et al. 1994) in the inner 32 pc of the AGN (Mason et al. 2009). This galaxy appears as an extended source in Gemini/MICHELLE high resolution N-band observations (Garc'ıa-Bernete et al. 2016). However, any structure within the NLR or ionization cones is not resolved by our observations.</text>
<text><location><page_9><loc_8><loc_11><loc_48><loc_16></location>NGC 2273 . The full set of 30 - 215 𝜇 m images of NGC 2273 show a point-like source. The north/south pattern in the 53 𝜇 m image is due to background noise and does not represent extended dust. Within the FWHM of these images</text>
<text><location><page_9><loc_52><loc_15><loc_92><loc_25></location>(see Sections 2.2.1, 2.2.2), there is a known star-forming ring within ∼ 2' of the nucleus (Ferruit et al. 2000; Martini et al. 2003; Sani et al. 2012). GTC/Canaricam observations (Alonso-Herrero et al. 2014, 2016) at 8.7 𝜇 mshowelongation from the north-east to the south-west, likely with contribution from PAH. This structure is consistent with extension seen in the PSF-subtracted 37.1 𝜇 mSOFIAimage(Fulleretal. 2019).</text>
<text><location><page_9><loc_52><loc_11><loc_92><loc_15></location>NGC 2992 . The image of NGC 2992 in the 31.5 𝜇 mfilter is published in Fuller et al. (2016) and appears as a point-like source. Subarcsecond N-band imaging (Garc'ıa-Bernete et al.</text>
<figure>
<location><page_10><loc_12><loc_39><loc_89><loc_88></location>
<caption>Figure 3. FORCAST 31.5 and 37.1 𝜇 m images, and HAWC+ 53, 89, 155, and 215 𝜇 m images. Note that the wavelength range for NGC 1068 starts 19.7 𝜇 m, so its range is shifted. Each image has a differing FOV, which can be found in Table 3. For NGC 1068 FORCAST images, contours begin at 3 𝜎 and follow log(maximum) from [-2.0,0.8] in steps of 0.2, while the HAWC+ images follow the same steps but log (max) ranges [-1.6,0.8]. For all other images, the lowest contours are 3 𝜎 and increase in steps of 5 𝜎 . The white transparent circle on the top right indicates the telescope beam size. The orange bar on the bottom left of the images is scaled to 500 pc. For all images, north is up and east is to the left.</caption>
</figure>
<text><location><page_10><loc_8><loc_12><loc_48><loc_24></location>2015) reveals extended emission along PA ∼ 30 · out to ∼ 3 kpc which is attributed to dust heated by star formation based on corresponding N-band spectroscopy. The FWHM of the SOFIA image is ∼ 3.5' × 3.5' (560 × 560 pc 2 ) so the extension should be resolvable within the SOFIA image. Since we do not see the extension in the image here, we conclude that either the extended dust emission tapers at wavelengths ¿ 20 𝜇 mor SOFIA does not have enough sensitivity to detect it.</text>
<text><location><page_10><loc_8><loc_9><loc_48><loc_12></location>NGC 3081 . The 31.5 𝜇 m image of NGC 3081 was published in Fuller et al. (2016) while the 37.1 𝜇 m image was</text>
<text><location><page_10><loc_52><loc_9><loc_92><loc_23></location>published in Fuller et al. (2019). The nucleus is known to harbor a region of strong optical emission ∼ 1' from the AGN (Ferruit et al. 2000) likely due to dust or gas heated by the AGN. Fuller et al. (2019) estimated that ∼ 35% of the MIRemission within the central few arcseconds (few hundred parsecs) of the AGN originates in the NLR. High angular resolution N- and Q- band observations show extension towards the north, extending out to ∼ 450 pc from the south-east to the north-west (PA ∼ 160 · ; Garc'ıa-Bernete et al. 2016). On larger scales, optical and NIR observations reveal a series of star</text>
<figure>
<location><page_11><loc_10><loc_34><loc_85><loc_91></location>
<caption>Figure 4. FORCAST 31.5 and 37.1 𝜇 mimages, and HAWC+ 53, 89, 155, and 215 𝜇 mimages. Each image has a differing FOV, which can be found in Table 3. The lowest contours are 3 𝜎 and increase in steps of 5 𝜎 . The white transparent circle on the top right indicates the telescope beam size. The orange bar on the bottom left of the images is scaled to 500 pc. For all images, north is up and east is to the left.</caption>
</figure>
<text><location><page_11><loc_8><loc_21><loc_48><loc_28></location>forming resonance rings at distances of 2.3, 11.0, 26.9 kpc and 33.1 kpc (Buta 1990; Buta et al. 1998, 2004). At longer wavelengths (¿200 𝜇 m), Ramos Almeida et al. (2011) concluded that FIR emission is contaminated by the star-forming ring 2.3 kpc in diameter.</text>
<text><location><page_11><loc_8><loc_9><loc_48><loc_21></location>NGC 3227 . The 31.5 𝜇 m image was published in Fuller et al. (2016) while the 37.1 𝜇 mimage was published in Fuller et al. (2019). These images show a point-like source, although NGC 3227 is known to harbor a nuclear star-forming region (Schinnerer et al. 2001; Davies et al. 2006) with a nuclear cluster within ∼ 70 pc ( ∼ 1') from the core. The 8.7 𝜇 mimage from Alonso-Herrero et al. (2016) shows a slight north/south elongation and the corresponding spectrum shows clear PAH</text>
<text><location><page_11><loc_52><loc_24><loc_92><loc_28></location>in the nucleus (see also Garc'ıa-Bernete et al. 2016). These star forming regions likely contaminate the nuclear MIR emission within the FWHM of our images.</text>
<text><location><page_11><loc_52><loc_12><loc_92><loc_23></location>NGC 3281 . While the 31.5 𝜇 m FORCAST image is published in Fuller et al. (2016), the HAWC+ images at 53, 89, 154, and 214 𝜇 m are presented here for the first time and appear point-like in all filters. The images taken at 53 and 89 𝜇 mappear to have significant noise in their backgrounds. The subarcsecond (0.35') N-band spectrum in Gonz'alez-Mart'ın et al. (2013) shows a deep 10 𝜇 m silicate absorption feature which originates in the inner ∼ 80 pc of the AGN.</text>
<text><location><page_11><loc_52><loc_9><loc_92><loc_12></location>NGC 4151 . The SOFIA images of NGC 4151 appear as a point-like source, with a potential detection of extended</text>
<figure>
<location><page_12><loc_11><loc_83><loc_24><loc_91></location>
<caption>Figure 5. FORCAST 31.5 and 37.1 𝜇 m images, and HAWC+ 53, 89, 155, and 215 𝜇 m images. Each image has a differing FOV, which can be found in Table 3. The white transparent circle on the top right indicates the telescope beam size. The orange bar on the bottom left of the images is scaled to 500 pc. For all images, north is up and east is to the left.</caption>
</figure>
<text><location><page_12><loc_17><loc_52><loc_21><loc_52></location>NGC 7469</text>
<text><location><page_12><loc_19><loc_45><loc_20><loc_45></location>15 5</text>
<text><location><page_12><loc_17><loc_42><loc_21><loc_43></location>NGC 7674</text>
<text><location><page_12><loc_11><loc_38><loc_12><loc_40></location>1</text>
<text><location><page_12><loc_8><loc_18><loc_48><loc_28></location>emission at PA ∼ 120 · at a 3 𝜎 level at 37 . 1 𝜇 m. Fuller et al. (2019) confirmed this elongation in the PSF-subtracted 37.1 𝜇 mimagecoincident with the NLR and radio axes. Radomski et al. (2003) show extended emission in 10.8 and 18.2 𝜇 m images that coincides with the NLR axis at PA ∼ -60 · . For ≥ 37 . 1 𝜇 m, we conclude that any extended emission due to NLR dust is within the FWHM of the SOFIA instruments.</text>
<text><location><page_12><loc_8><loc_11><loc_48><loc_18></location>NGC4258 . NGC 4258 was not detected but we include the data here since it is part of the sample. It has been observed and analyzed in the N-band with Gemini/Michelle by Mason et al. (2012). These authors found a compact nucleus that is marginally resolved at 10 𝜇 m(FWHM ∼ 0 . 5 '' ).</text>
<text><location><page_12><loc_52><loc_11><loc_92><loc_28></location>NGC 4388 . NGC 4388 is an edge-on spiral that shows the most interesting mid- to far-IR morphology in this study. Notably, in the 30 - 40 𝜇 mFORCAST images of NGC 4388, extension can be seen in the NE to SW direction at PA ∼ 40 · (see also Fuller et al. 2019), coincident with the NLR. This emission is seen on smaller scales at shorter wavelengths (Asmus et al. 2016; Garc'ıa-Bernete et al. 2016). The 53 𝜇 m image decreases in intensity and does not show a strong central core of emission as in the 31 . 5 -37 . 1 wavelength range. However, at longer wavelengths (89 -214 𝜇 m), host galaxy emission clearly dominates the images in the east-west direction at PA ∼ 90 · .</text>
<text><location><page_13><loc_8><loc_83><loc_48><loc_92></location>NGC 4941 . NGC 4941 is a low-luminosity AGN that appears here as a faint point-like source in the 31 . 5 𝜇 m and 37 . 1 𝜇 m images, but brighter at 53 and 89 𝜇 m. Subarcsecond resolution N-band imaging on VLT/VISIR (Asmus et al. 2011) showed no significant extended MIR sources outside of the nucleus.</text>
<text><location><page_13><loc_8><loc_67><loc_48><loc_83></location>NGC 5506 NGC 5506 appears as a bright point source in both the 31.5 and 37.1 𝜇 m filters. While the nucleus is unresolved, extended MIR emission has been detected up to a few arcseconds to the northeast at 11.9 𝜇 m (Raban et al. 2008). Extended emission in the north-south direction was detected in the N-band out to ∼ 560 pc, while faint extended emission towards the east in the Q-band was also detected (Garc'ıa-Bernete et al. 2016). However, the PSF-subtracted 12.27 𝜇 m2' × 2' VLT/VISIR image of Alonso-Herrero et al. (2021) shows that the PA of extended emission varies from 30 · in the central ∼ 0.5' to nearly 90 · in the outer regions.</text>
<text><location><page_13><loc_8><loc_59><loc_48><loc_67></location>NGC 7465 . The 31.5 𝜇 m FORCAST image appears faint with a 3 𝜎 upper-limit in the 37.1 𝜇 m image. The 53 and 89 𝜇 m HAWC+ images here appear increasingly brighter, albeit as point-like sources. Cold molecular gas observations (Young et al. 2021) reveal that NGC 7465 is quite gas-rich, possibly from a recent merger.</text>
<text><location><page_13><loc_8><loc_44><loc_48><loc_58></location>NGC 7469 . NGC 7469 appears as a very bright source in the 31.5 𝜇 m image with FWHM ∼ 4.3'. After PSF subtraction, Fuller et al. (2016) found extended emission in the north-south direction. This AGN is known to have a circumnuclear ring of star formation at a radius of ∼ 480 pc ( ∼ 1.4'; Ramos Almeida et al. 2011) in 8.7 and 18.3 𝜇 mimages taken on Gemini/T-ReCS. Recent JWST observations reveal prominent PAH emission, indicative of star formation, in the circumnuclear ring (Garc'ıa-Bernete et al. 2022b; Zhang & Ho 2023).</text>
<text><location><page_13><loc_8><loc_37><loc_48><loc_44></location>NGC 7674 . The previously published (Fuller et al. 2016) FORCAST 31.5 𝜇 m image appears as a point-like source. Asmus et al. (2013) found that the nucleus of NGC 7674 is extended at PA ∼ 125 · at subarcsecond scale resolution, where the extension roughly aligns with the ionization cone.</text>
<section_header_level_1><location><page_13><loc_16><loc_33><loc_40><loc_34></location>4. NUCLEAR FLUX EXTRACTION</section_header_level_1>
<text><location><page_13><loc_8><loc_9><loc_48><loc_32></location>We aim to construct well-sampled mid- to far-IR SEDs of the nuclear emission of AGN at scales of several arcseconds, depending on the PSF of the observation and possible extended emission. On these scales, we expect multiple dust sources (i.e. torus, star forming regions, dusty outflow), however disentangling these sources is beyond the scope of this imaging atlas. Because the images span a range of observing cycles, instruments, and observing modes, we analyzed each image individually. Of our sample, 17 objects appear visually as point sources. For these sources, we performed aperture photometry where the aperture size was set to be 2 × the FWHM at a given band. For objects that show host galaxy emission, we extract the central PSF to construct the SEDs as described below. We complement our SOFIA data with Herschel imaging data (see Appendix A) and use a similar analysis method to construct the full IR SEDs.</text>
<section_header_level_1><location><page_13><loc_57><loc_90><loc_87><loc_92></location>4.1. Extended Sources: 2D Gaussian Fitting</section_header_level_1>
<text><location><page_13><loc_52><loc_81><loc_92><loc_90></location>For sources with extended dust emission, we performed a two-component simultaneous fit to accurately model both the central source based on the PSF, and the host galaxy whose fit assumes an elongated 2D Gaussian profile. In order to supplement the SOFIA data for the full mid- to far-IR SEDs, we used a similar methodology with Herschel images.</text>
<text><location><page_13><loc_52><loc_55><loc_92><loc_81></location>For the SOFIA images, the PSF used was based on the standard stars of the observing runs for each cycle as explained in Section 2.2. However, the same analysis could not be performed on Herschel images due to the threefold lobes associated with the instrument PSFs. To accommodate this, we compared three different PSF models to reproduce and fit the central source. Two of the PSFs were point source images while the third was an approximation of the theoretical instrumental PSF using a Gaussian profile. The fitting routine used four free parameters for the Gaussian profile: (1,2) the position in x and y of the PSF center according to the image center, (3) the amplitude of the PSF, (4) the fourth parameter was dependent on the PSF type used. For archival PSFs, this parameter represents the rotation angle that needs to be applied to the PSF to match the orientation of the image. For the Gaussian PSFs, this fourth parameter represents a scaling factor to the width of the Gaussian compared to its ideal value for a perfect instrument (1 . 22 × 𝜆 / 𝐷 ).</text>
<text><location><page_13><loc_52><loc_36><loc_92><loc_55></location>The galaxy background is defined by 7 parameters: (1,2) the 2D Gaussian's center position (x0 and y0), (3,4) its width ( 𝜎 𝑥 and 𝜎 𝑦 ) in both directions, (5) its amplitude, and (6) its orientation on the image ( 𝜃 ). To these 6 parameters we added a constant background as a 7th free parameter. We combined these components and fit this simulated intensity map to the observed map using the 11 total free parameters (4 from the PSF and 7 from the 2D gaussian). We thus derived the parameters describing the best central source for our intensity maps, and then studied the properties extracted for the central source and removed it from the initial map to study the host galaxy itself. An example of this procedure is given in the Appendix in Figure 14.</text>
<section_header_level_1><location><page_13><loc_59><loc_34><loc_85><loc_35></location>4.2. AGN and host galaxy contribution</section_header_level_1>
<text><location><page_13><loc_52><loc_16><loc_92><loc_33></location>While most SOFIA images were treated as point sources, Centaurus A, Circinus, NGC 1068, and NGC 4388 all had significant host galaxy contamination that needed to be subtracted from at least some of the images. Figure 6 shows the PSF subtracted images of these sources and Table 4 gives the percentage of the contribution of the PSF to the total flux of the object. In several other objects, the shorter wavelength ( ∼ 30 - 100 𝜇 m) images did not show host galaxy contamination, but longer wavelength Herschel images show the colder extended dust. Because this is an atlas of SOFIA images, we include objects with host galaxy contamination only in Herschel images in Appendix C for completeness.</text>
<text><location><page_13><loc_52><loc_9><loc_92><loc_16></location>The53 𝜇 mimageofCentaurusAcontains ( < 5%)emission from extended sources, so we performed aperture photometry to account for the central emission. At wavelengths ≳ 70 𝜇 m, the host galaxy substantially ( ∼ 56 -70%) contributes to the central AGN emission, so the extended emission was</text>
<text><location><page_14><loc_8><loc_86><loc_48><loc_92></location>subtracted. The PSF of Centaurus A at these wavelengths contributes ∼ 35 %. This can be interpreted as the nucleus having a relatively constant IR contribution, so the brightness of the nucleus coincides with IR brightness of the host galaxy.</text>
<text><location><page_14><loc_8><loc_77><loc_48><loc_86></location>The PSF contribution of the Circinus Galaxy decreases between 53 and 160 𝜇 m from 58 % to 35 %. At longer FIR wavelengths, the contribution of the PSF appears to be from the host galaxy and the fitting no longer provides information about the AGN. We interpret this as a decreasing IR contribution from the nucleus compared to the extended emission.</text>
<text><location><page_14><loc_8><loc_59><loc_48><loc_77></location>For the completeness of the SOFIA Atlas presented here, we used the 19-53 𝜇 m images of NGC 1068 from LopezRodriguez et al. (2018). These datasets were analyzed as described in that study and here we only present the results and images in that wavelength range. The study showed that the fractional contribution from star formation increases from 20 - 50 𝜇 m, while extended emission from 200 K dust decreases. Emission from the torus peaks in this range, a result which is in agreement with Fuller et al. (2016) who found that the turnover in torus emission occurs at wavelengths ¿ 31.5 𝜇 m. Extended emission is observed here at all wavelengths ≳ 70 𝜇 m arising from dust in the host galaxy and star formation regions.</text>
<text><location><page_14><loc_8><loc_25><loc_48><loc_58></location>For NGC 4388, we show the results of PSF subtraction at all wavelengths, but only use the results in wavelengths ¿ 40 𝜇 m for the SED. In the 30 - 40 𝜇 m range, the NE to SW extension is clear in the PSF-subtracted images. However, almost all of the extended emission lies within the FWHM of the observation; the FWHM of these images are only ∼ 10% greater than the FWHM of the PSF. Thus, while we show the PSF subtracted images of NGC 4388 here, for the SED we use the total 30 - 40 𝜇 mfluxes which encompass the apparent extended emission due to the NLR. The change in the extended emission source and morphology between 40 and 70 𝜇 m is clear in the PSF-subtracted images (Figure 6). The host galaxy clearly dominates the extended emission in the FIR while the NLR region dominates the extended emission in MIR wavelengths. The 53 𝜇 m HAWC+ image appears to show the transition between dominant extended sources. The contribution of the PSF in the images of NGC 4388 is ∼ 60 % in the 30 - 40 𝜇 m range, where the extended emission is in the NE to SW direction. The contribution then decreases drastically to ∼ 20 %. This reflects the turnover in extended emission seen in the images in Figure 5. The contribution of the PSF returns to ∼ 70 % between 70 - 100 𝜇 m, which suggests two separate but significant IR emission sources.</text>
<section_header_level_1><location><page_14><loc_13><loc_23><loc_43><loc_24></location>5. SPECTRAL ENERGY DISTRIBUTIONS</section_header_level_1>
<text><location><page_14><loc_8><loc_14><loc_48><loc_22></location>Tables 5 (SOFIA) and 6 ( Herschel ) give the nuclear fluxes of the AGN in our sample along with their associated errors in units of Jy. The sources of uncertainty here are the instrument calibration, sky background, and the 2D gaussian fitting, where applicable. We estimate FORCAST and HAWC+ errors at ∼ 10%. We use PACS instrument errors at 5% /four.sup and</text>
<text><location><page_14><loc_52><loc_85><loc_92><loc_92></location>SPIRE instrument errors as 5.5% /five.sup . The uncertainty due to sky background is determined on an individual basis, but averages ∼ 5%. The average uncertainty due to the 2-D gaussian fitting is ∼ 1.5%. We add these uncertainties in quadrature for the error bar estimation.</text>
<text><location><page_14><loc_52><loc_64><loc_92><loc_84></location>The nuclear SEDs are shown in Figure 7 in 𝜈 F 𝜈 . The pink diamonds represent SOFIA observations while the black circles represent the complementary Herschel data. We obtained Spitzer /IRS spectra from the Sptizer /CASSIS database (Lebouteiller et al. 2011) for 21 of the 22 objects in our sample (solid black line). There was no spectrum available for NGC 7465. Low-resolution spectra (R ∼ 100) were obtained for 18 of the objects, while moderate resolution spectra (R ∼ 600) were available for Circinus, NGC 1068, and NGC 7674. This dataset provides the most completed SED coverage available between 30 - 500 𝜇 m. Decomposing the SEDs in this sample is outside the scope of this manuscript, as we are presenting an imaging atlas. Here, we provide the main results and features of the SEDs of these objects.</text>
<text><location><page_14><loc_52><loc_54><loc_92><loc_64></location>The morphological changes seen in the extended emission source in Figure 5 for NGC 4388 are reflected in the SED at 53 𝜇 m, where there is a marked decline in the SED. The drastic decrease seems to be due the change of dominant emitting sources. The extended emission at wavelengths ≲ 40 𝜇 m is due to dust in the direction of the radio axis, and the extended emission at wavelengths ≳ 50 𝜇 mis due to the host galaxy.</text>
<text><location><page_14><loc_52><loc_33><loc_92><loc_54></location>The wavelength of peak emission can give insight to the primary processes that drive MIR emission. The peak wavelength, determined by the highest flux from photometry and spectroscopy, ranges from 18 to 100 𝜇 m in 𝜈 F 𝜈 with an average of ∼ 40 𝜇 m. This average only includes the peak in continuum values and does not take into account fine structure lines. Most (73%; 11 out of 15) Seyfert 2 have a peak emission at wavelengths ≲ 40 𝜇 m. The SEDs of MCG-5-2316, Mrk3, Mrk 573, NGC 1068, NGC 3081, and NGC 4151 peak at ∼ 18 - 20 𝜇 m. This is in agreement with the correlation peak between the hard-X-rays and the mid-IR for Type 1 AGN in Garc'ıa-Bernete et al. (2017). The peak wavelengths in 𝐹 𝜈 (Jy) range ∼ 20 - 160 𝜇 m, with an average ∼ 93 𝜇 m. NGC 1068 is the only AGN to peak at the same wavelength in both sets of units.</text>
<text><location><page_14><loc_52><loc_22><loc_92><loc_32></location>The 𝑆𝑝𝑖𝑡𝑧𝑒𝑟 spectrum for NGC 1068 does not align with the SOFIA photometry because of the extensive PSF subtraction that we performed in the photometry that was not accounted for in the spectroscopy. This is the only object that not only has overlapping 20 - 40 𝜇 m 𝑆𝑝𝑖𝑡𝑧𝑒𝑟 and SOFIA data, but also that has had the background emission subtracted at these wavelengths.</text>
<section_header_level_1><location><page_14><loc_60><loc_20><loc_85><loc_21></location>5.1. Luminosity and Peak Wavelength</section_header_level_1>
<text><location><page_14><loc_52><loc_14><loc_92><loc_19></location>To test whether the peak wavelength is a function of luminosity, we plot L 𝑏𝑜𝑙 vs 𝜆 𝑝𝑒𝑎𝑘 . Figure 8 shows the bolometric luminosities of the AGN plotted against the peak wavelength in the SEDs for both Sy1s, shown as red triangles, and Sy2s,</text>
<figure>
<location><page_15><loc_10><loc_61><loc_87><loc_89></location>
<caption>Figure 6. PSF-subtracted images of host galaxy backgroundsTable 4. Contribution of the PSF to the host galaxy extended emission</caption>
</figure>
<table>
<location><page_15><loc_20><loc_46><loc_90><loc_56></location>
</table>
<text><location><page_15><loc_8><loc_32><loc_48><loc_45></location>shown as purple stars. The correlation coefficient between the luminosity and peak wavelength is | 𝑅 | ∼ 0.63 with statistical significance 𝑝 = 0 . 0015. While it is argued that | 𝑅 | of 0.6 - 0.7 may show moderate to strong correlation (see Section 3.2 in Messenger et al. 2013), a 𝑝 -value ≤ 0.05 is generally accepted as statistically significant. The data suggests that higher luminosity objects have SEDs that peak at shorter wavelengths, which indicates the presence of a hot dust component in the vicinity of the AGN.</text>
<section_header_level_1><location><page_15><loc_19><loc_27><loc_37><loc_28></location>5.2. Mid- to Far-IR Colors</section_header_level_1>
<text><location><page_15><loc_8><loc_9><loc_48><loc_26></location>The ratio of F 𝜈 (70)/F 𝜈 (160) has been used as a proxy for dust temperature (Mel'endez et al. 2014; Garc'ıa-Gonz'alez et al. 2016), where the ratio is higher for dust heated by the AGN and lower for dust heated by star formation. Here we perform this analysis using the ratio F 𝜈 (31)/F 𝜈 (70) by using the 31.5 𝜇 mSOFIA data in our atlas. For objects that do not have data in the 31.5 𝜇 mfilter, we supplement that with data from the Spitzer /IRS continuum. NGC 7465 did not have 31.5 𝜇 m flux data, nor did it have Spitzer data so we leave that object out of this analysis. Using the fluxes in Table 5, we plot a color-color diagram in F 𝜈 in Figure 9. This figure also visually shows the peak wavelength from the SEDs (in 𝜈𝐹 𝜈 ).</text>
<text><location><page_15><loc_52><loc_42><loc_92><loc_45></location>Longer peak wavelengths tend to cluster at F 𝜈 (70)/F 𝜈 (160) ∼ 1 and F 𝜈 (31)/F 𝜈 (70) between 0.25 - 0.5.</text>
<text><location><page_15><loc_52><loc_34><loc_92><loc_42></location>In this sample we find an average F 𝜈 (70)/F 𝜈 (160) ratio of 1 . 4 ± 0 . 7. Previous studies (Mel'endez et al. 2014; Garc'ıaGonz'alez et al. 2016) with larger sample sizes (313 and 33, respectively) have found an average ratio of ∼ 0.8, albeit the data was analyzed using independent methods. This suggests a higher amount of AGN heated dust in our sample.</text>
<text><location><page_15><loc_52><loc_21><loc_92><loc_33></location>We find that 18 objects have F 𝜈 (31)/F 𝜈 (70) ¡1, with an average F 𝜈 (31)/F 𝜈 (70) of 0 . 6 ± 0 . 3. The only object with both F 𝜈 (70)/F 𝜈 (160) and F 𝜈 (31)/F 𝜈 (70) ¿1 is MCG-5-23-16, and an SED that peaks ∼ 20 𝜇 m. This object may be the most AGN dominated source in our sample. The other objects that show a peak at ∼ 20 𝜇 min 𝜈 F 𝜈 still show F 𝜈 (31)/F 𝜈 (70) ¡1. Only one object, NGC 4388, shows F 𝜈 (31)/F 𝜈 (70) ¿1 while F 𝜈 (70)/F 𝜈 (160) ¡1. This reflects the change in extended emission seen in Figure 5.</text>
<text><location><page_15><loc_52><loc_11><loc_92><loc_20></location>Half (11) of the objects in the sample show ratios F 𝜈 (31)/F 𝜈 (70) ¡1 while F 𝜈 (70)/F 𝜈 (160) ¿1. These objects (Circinus, Mrk 231, Mrk 573, NGC 1275, NGC 2110, NGC 3081, NGC 3227, NGC 3281, NGC 4151, NGC 4941, NGC 5506, NGC 7469) are likely AGN dominated. Six objects show F 𝜈 (31)/F 𝜈 (70) ¡1 and F 𝜈 (70)/F 𝜈 (160) ¡1, meaning that their SEDs peak at longer wavelengths. The emission from</text>
<table>
<location><page_16><loc_20><loc_46><loc_88><loc_90></location>
<caption>Table 5. Nuclear fluxes for SOFIA/FORCAST and HAWC+ images.</caption>
</table>
<text><location><page_16><loc_10><loc_42><loc_92><loc_45></location>R/e.pc/f.pc/e.pc/r.pc/e.pc/n.pc/c.pc/e.pc/s.pc: a) Fuller et al. (2016), b) Fuller et al. (2019), c) Lopez-Rodriguez et al. (2018), d) Lopez-Rodriguez et al. (2020), e) LopezRodriguez et al. (2022b). *Lopez-Rodriguez et al. (2018) measured the flux of NGC 1068 at 19.7 𝜇 mto be 22.0 ± 1.4.</text>
<text><location><page_16><loc_8><loc_37><loc_48><loc_41></location>these objects (Centaurus A, NGC 1068, NGC 2273, NGC 2992, NGC 4258, NGC 7674) are likely dominated by star formation.</text>
<section_header_level_1><location><page_16><loc_21><loc_34><loc_35><loc_35></location>6. CONCLUSIONS</section_header_level_1>
<text><location><page_16><loc_8><loc_15><loc_48><loc_33></location>We have presented a SOFIA atlas of nearby AGN in the 20 - 215 𝜇 m wavelength range using FORCAST and HAWC+. We have released 69 observations of which 41 are newly published and 28 have been previously published (Fuller et al. 2016, 2019; Lopez-Rodriguez et al. 2018, 2022a). From these observations, NGC 4388 shows the most dramatic visual change in emission morphology. The 30 - 40 𝜇 m images show a NE to SW dusty extension associated with the NLR, while the ¿ 50 𝜇 mimages show a East to West dusty emission associated with the plane of the host galaxy. Our observations show that < 10 '' resolution 30 - 70 𝜇 m observations are crucial to disentangle the emitting contribution from AGN and host galaxy.</text>
<text><location><page_16><loc_8><loc_9><loc_48><loc_15></location>We measured arcsecond scale unresolved nuclear fluxes in order to construct SEDs of the objects in our sample. We included complementary Herschel data to cover up to 500 𝜇 m. For point sources we used aperture photometry to determine</text>
<text><location><page_16><loc_52><loc_33><loc_92><loc_41></location>the flux. For extended sources we used a 2D gaussian fitting method to extract the central unresolved source(s) of emission from the galaxy background. For this method, the PSF is scaled to represent the central emission while a 2D gaussian represents host galaxy or background emission. Based on the SEDs, we make the following conclusions:</text>
<unordered_list>
<list_item><location><page_16><loc_52><loc_28><loc_92><loc_32></location>- There is a sharp drop in the SED of NGC 4388 that corresponds to the wavelength where the angle of extended emission transitions from NE/SW (NLR) to E/W (host galaxy).</list_item>
<list_item><location><page_16><loc_52><loc_25><loc_92><loc_28></location>- The average peak of the SEDs is 40 𝜇 m in 𝜈 F 𝜈 , spanning a range of [20,100] 𝜇 m.</list_item>
<list_item><location><page_16><loc_52><loc_21><loc_92><loc_25></location>- The peak wavelength of the SED appears to be a function of AGN luminosity, where higher luminosity objects peak at shorter wavelengths.</list_item>
<list_item><location><page_16><loc_52><loc_17><loc_92><loc_21></location>- MCG-5-23-16 is the only object whose color diagram shows both F 𝜈 (31)/F 𝜈 (70) and F 𝜈 (70)/F 𝜈 (160) ¿1, which may indicate an AGN dominated source.</list_item>
<list_item><location><page_16><loc_52><loc_12><loc_92><loc_16></location>- Half of the objects in the sample have flux ratios which suggest that the SED is dominated by AGN heated dust, while six objects show ratios consistent with heating by SF.</list_item>
</unordered_list>
<table>
<location><page_17><loc_21><loc_46><loc_85><loc_90></location>
<caption>Table 6. Nuclear fluxes for Herschel /PACS and SPIRE images.</caption>
</table>
<text><location><page_17><loc_8><loc_34><loc_48><loc_42></location>In future studies, we will combine data from this atlas with incoming data from JWST to update our IR datasets with the latest and highest resolution data available. Newly obtained JWST/MIRI observations will provide new higher angular resolution data for some of the sources in the wavelength range 5 - 28 𝜇 m.</text>
<section_header_level_1><location><page_17><loc_20><loc_30><loc_36><loc_31></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_17><loc_8><loc_24><loc_48><loc_29></location>We acknowledge Dr. Lucas Grosset for his effort in subtracting the image backgrounds. E.L.-R. is supported by the NASA/DLR Stratospheric Observatory for Infrared Astronomy (SOFIA) under the 08 0012 Program. SOFIA is</text>
<text><location><page_17><loc_52><loc_27><loc_92><loc_42></location>jointly operated by the Universities Space Research Association,Inc.(USRA), under NASA contract NNA17BF53C, and the Deutsches SOFIA Institut (DSI) under DLR contract 50OK0901 to the University of Stuttgart. E.L.-R. is supported by the NASA Astrophysics Decadal Survey Precursor Science (ADSPS) Program (NNH22ZDA001NADSPS) with ID 22-ADSPS22-0009 and agreement number 80NSSC23K1585. I.G.B. acknowledges support from STFC through grants ST/S000488/1 and ST/W000903/1. C.R. acknowledges support from Fondecyt Regular grant 1230345 and ANID BASAL project FB210003.</text>
<text><location><page_17><loc_52><loc_24><loc_92><loc_27></location>Software: /a.pc/s.pc/t.pc/r.pc/o.pc/p.pc/y.pc (Astropy Collaboration et al. 2022, 2018, 2013); S/c.pc/i.pcP/y.pc Virtanen et al. (2020)</text>
<section_header_level_1><location><page_17><loc_45><loc_20><loc_55><loc_21></location>REFERENCES</section_header_level_1>
<text><location><page_17><loc_8><loc_18><loc_46><loc_19></location>Alonso-Herrero, A., Ramos Almeida, C., Mason, R., et al. 2011,</text>
<text><location><page_17><loc_10><loc_16><loc_38><loc_17></location>ApJ, 736, 82, doi: 10.1088/0004-637X/736/2/82</text>
<text><location><page_17><loc_8><loc_15><loc_46><loc_16></location>Alonso-Herrero, A., Ramos Almeida, C., Esquej, P., et al. 2014,</text>
<text><location><page_17><loc_10><loc_13><loc_39><loc_14></location>MNRAS, 443, 2766, doi: 10.1093/mnras/stu1293</text>
<text><location><page_17><loc_8><loc_11><loc_47><loc_12></location>Alonso-Herrero, A., Esquej, P., Roche, P. F., et al. 2016, MNRAS,</text>
<text><location><page_17><loc_10><loc_9><loc_32><loc_10></location>455, 563, doi: 10.1093/mnras/stv2342</text>
<text><location><page_17><loc_52><loc_16><loc_92><loc_19></location>Alonso-Herrero, A., Pereira-Santaella, M., Garc´ıa-Burillo, S., et al. 2018, ApJ, 859, 144, doi: 10.3847/1538-4357/aabe30</text>
<text><location><page_17><loc_52><loc_15><loc_90><loc_16></location>Alonso-Herrero, A., Garc´ıa-Burillo, S., H¨onig, S. F., et al. 2021,</text>
<text><location><page_17><loc_54><loc_13><loc_85><loc_14></location>A&A, 652, A99, doi: 10.1051/0004-6361/202141219</text>
<text><location><page_17><loc_52><loc_11><loc_75><loc_12></location>Antonucci, R. 1993, ARA&A, 31, 473,</text>
<text><location><page_17><loc_54><loc_9><loc_79><loc_10></location>doi: 10.1146/annurev.aa.31.090193.002353</text>
<figure>
<location><page_18><loc_10><loc_47><loc_89><loc_91></location>
<caption>Figure 7. Mid- to far-IR SEDs of the several-arcsecond-scale nuclear fluxes in our sample of AGN. Pink diamonds represent SOFIA observations while black dots represent complementary Herschel observations. The solid black lines correspond to Spitzer spectra.</caption>
</figure>
<figure>
<location><page_18><loc_9><loc_19><loc_44><loc_39></location>
<caption>Figure 8. Bolometric luminosity vs the peak wavelength of the SED for both Sy1 (red triangles) and Sy2 (purple stars).</caption>
</figure>
<text><location><page_18><loc_8><loc_12><loc_41><loc_13></location>Antonucci, R. R. J., & Miller, J. S. 1985, ApJ, 297, 621,</text>
<text><location><page_18><loc_10><loc_10><loc_22><loc_11></location>doi: 10.1086/163559</text>
<figure>
<location><page_18><loc_53><loc_20><loc_89><loc_41></location>
<caption>Figure 9. Color diagram of 21 of the 22 AGN in our sample. Sy 1 are represented by triangles while Sy 2 are stars. The scale on the right shows peak wavelength by color.</caption>
</figure>
<text><location><page_18><loc_52><loc_11><loc_92><loc_12></location>Asmus, D. 2019, MNRAS, 489, 2177, doi: 10.1093/mnras/stz2289</text>
<table>
<location><page_19><loc_8><loc_9><loc_48><loc_92></location>
</table>
<table>
<location><page_19><loc_52><loc_9><loc_92><loc_92></location>
</table>
<unordered_list>
<list_item><location><page_20><loc_8><loc_89><loc_47><loc_91></location>Kov'acs, A., Chapman, S. C., Dowell, C. D., et al. 2006, ApJ, 650, 592, doi: 10.1086/506341</list_item>
<list_item><location><page_20><loc_8><loc_85><loc_45><loc_88></location>Kukula, M. J., Ghosh, T., Pedlar, A., et al. 1993, MNRAS, 264, 893, doi: 10.1093/mnras/264.4.893</list_item>
<list_item><location><page_20><loc_8><loc_82><loc_47><loc_85></location>Lebouteiller, V., Barry, D. J., Spoon, H. W. W., et al. 2011, ApJS, 196, 8, doi: 10.1088/0067-0049/196/1/8</list_item>
<list_item><location><page_20><loc_8><loc_79><loc_47><loc_81></location>Leighly, K. M., Terndrup, D. M., Baron, E., et al. 2014, ApJ, 788, 123, doi: 10.1088/0004-637X/788/2/123</list_item>
<list_item><location><page_20><loc_8><loc_74><loc_46><loc_78></location>L'opez-Gonzaga, N., Burtscher, L., Tristram, K. R. W., Meisenheimer, K., & Schartmann, M. 2016, A&A, 591, A47, doi: 10.1051/0004-6361/201527590</list_item>
<list_item><location><page_20><loc_8><loc_69><loc_47><loc_73></location>L'opez-Gonzaga, N., Jaffe, W., Burtscher, L., Tristram, K. R. W., & Meisenheimer, K. 2014, A&A, 565, A71, doi: 10.1051/0004-6361/201323002</list_item>
<list_item><location><page_20><loc_8><loc_65><loc_40><loc_68></location>Lopez-Rodriguez, E. 2021, Nature Astronomy, 5, 604, doi: 10.1038/s41550-021-01329-9</list_item>
<list_item><location><page_20><loc_8><loc_62><loc_47><loc_64></location>Lopez-Rodriguez, E., Kishimoto, M., Antonucci, R., et al. 2022a, arXiv e-prints, arXiv:2207.09466.</list_item>
</unordered_list>
<text><location><page_20><loc_10><loc_60><loc_29><loc_61></location>https://arxiv.org/abs/2207.09466</text>
<unordered_list>
<list_item><location><page_20><loc_8><loc_58><loc_41><loc_59></location>-. 2023, ApJ, 951, 31, doi: 10.3847/1538-4357/accb96</list_item>
<list_item><location><page_20><loc_8><loc_55><loc_43><loc_58></location>Lopez-Rodriguez, E., Packham, C., Jones, T. J., et al. 2015, MNRAS, 452, 1902, doi: 10.1093/mnras/stv1410</list_item>
<list_item><location><page_20><loc_8><loc_52><loc_46><loc_54></location>Lopez-Rodriguez, E., Fuller, L., Alonso-Herrero, A., et al. 2018, ArXiv e-prints. https://arxiv.org/abs/1804.04134</list_item>
<list_item><location><page_20><loc_8><loc_48><loc_47><loc_51></location>Lopez-Rodriguez, E., Dowell, C. D., Jones, T. J., et al. 2020, ApJ, 888, 66, doi: 10.3847/1538-4357/ab5849</list_item>
<list_item><location><page_20><loc_8><loc_45><loc_46><loc_48></location>Lopez-Rodriguez, E., Clarke, M., Shenoy, S., et al. 2022b, ApJ, 936, 65, doi: 10.3847/1538-4357/ac83ac</list_item>
<list_item><location><page_20><loc_8><loc_42><loc_47><loc_44></location>Marconi, A., Risaliti, G., Gilli, R., et al. 2004, MNRAS, 351, 169, doi: 10.1111/j.1365-2966.2004.07765.x</list_item>
<list_item><location><page_20><loc_8><loc_35><loc_47><loc_41></location>Marinucci, A., Bianchi, S., Nicastro, F., Matt, G., & Goulding, A. D. 2012, ApJ, 748, 130, doi: 10.1088/0004-637X/748/2/130 Martini, P., Regan, M. W., Mulchaey, J. S., & Pogge, R. W. 2003, ApJS, 146, 353, doi: 10.1086/367817</list_item>
<list_item><location><page_20><loc_8><loc_31><loc_46><loc_34></location>Mason, R. E., Geballe, T. R., Packham, C., et al. 2006, ApJ, 640, 612, doi: 10.1086/500299</list_item>
<list_item><location><page_20><loc_8><loc_28><loc_45><loc_31></location>Mason, R. E., Levenson, N. A., Shi, Y., et al. 2009, ApJL, 693, L136, doi: 10.1088/0004-637X/693/2/L136</list_item>
<list_item><location><page_20><loc_8><loc_25><loc_47><loc_27></location>Mason, R. E., Lopez-Rodriguez, E., Packham, C., et al. 2012, AJ, 144, 11, doi: 10.1088/0004-6256/144/1/11</list_item>
<list_item><location><page_20><loc_8><loc_21><loc_46><loc_24></location>Mel'endez, M., Mushotzky, R. F., Shimizu, T. T., Barger, A. J., & Cowie, L. L. 2014, ApJ, 794, 152,</list_item>
</unordered_list>
<text><location><page_20><loc_10><loc_20><loc_31><loc_21></location>doi: 10.1088/0004-637X/794/2/152</text>
<unordered_list>
<list_item><location><page_20><loc_8><loc_16><loc_44><loc_19></location>Messenger, S. J., Speck, A., & Volk, K. 2013, ApJ, 764, 142, doi: 10.1088/0004-637X/764/2/142</list_item>
<list_item><location><page_20><loc_8><loc_13><loc_37><loc_16></location>Mor, R., & Netzer, H. 2012, MNRAS, 420, 526, doi: 10.1111/j.1365-2966.2011.20060.x</list_item>
</unordered_list>
<text><location><page_20><loc_8><loc_11><loc_41><loc_12></location>Mor, R., Netzer, H., & Elitzur, M. 2009, ApJ, 705, 298,</text>
<unordered_list>
<list_item><location><page_20><loc_10><loc_9><loc_31><loc_10></location>doi: 10.1088/0004-637X/705/1/298</list_item>
</unordered_list>
<table>
<location><page_20><loc_52><loc_9><loc_92><loc_92></location>
</table>
<text><location><page_21><loc_8><loc_90><loc_45><loc_91></location>Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature</text>
<text><location><page_21><loc_10><loc_89><loc_40><loc_90></location>Methods, 17, 261, doi: 10.1038/s41592-019-0686-2</text>
<text><location><page_21><loc_8><loc_85><loc_47><loc_88></location>Young, L. M., Meier, D. S., Bureau, M., et al. 2021, ApJ, 909, 98, doi: 10.3847/1538-4357/abe126</text>
<text><location><page_21><loc_52><loc_86><loc_90><loc_91></location>Yuan, F., Markoff, S., Falcke, H., & Biermann, P. L. 2002, A&A, 391, 139, doi: 10.1051/0004-6361:20020817 Zhang, L., & Ho, L. C. 2023, ApJL, 953, L9, doi: 10.3847/2041-8213/acea73</text>
<section_header_level_1><location><page_22><loc_46><loc_93><loc_55><loc_94></location>F/u.pc/l.pc/l.pc/e.pc/r.pc /e.pc/t.pc /a.pc/l.pc.</section_header_level_1>
<figure>
<location><page_22><loc_9><loc_32><loc_90><loc_91></location>
<caption>Figure 10. Herschel PACS and SPIRE FIR images. The white circle in the top right indicates the beam size of the observation, while the pink line in the bottom left indicates a distance of 1 kpc.</caption>
</figure>
<section_header_level_1><location><page_22><loc_46><loc_23><loc_54><loc_24></location>APPENDIX</section_header_level_1>
<section_header_level_1><location><page_22><loc_42><loc_17><loc_58><loc_18></location>A. HERSCHEL IMAGES</section_header_level_1>
<text><location><page_22><loc_8><loc_9><loc_92><loc_16></location>We used images from the Herschel Archive to supplement SOFIA data in FIR wavelengths. Images from the PACS and SPIRE instruments covering the wavelength range 70 - 500 𝜇 m are shown in Figures 10, 11, 12, and 13. The white circle in the upper right indicates the beam size while the pink scale in the bottom left indicates a distance of 1 kpc. To extract nuclear fluxes on scales of several arcseconds, we use the methods described in Section 4. For point sources, we performed aperture photometry. For extended sources, we used the 2D gaussian routine outlined in Section 4.1.</text>
<figure>
<location><page_23><loc_9><loc_30><loc_93><loc_91></location>
<caption>Figure 11. Herschel PACS and SPIRE FIR images. The white circle in the top right indicates the beam size of the observation, while the pink line in the bottom left indicates a distance of 1 kpc.</caption>
</figure>
<section_header_level_1><location><page_23><loc_40><loc_22><loc_60><loc_24></location>B. BACKGROUND FITTING</section_header_level_1>
<text><location><page_23><loc_8><loc_15><loc_92><loc_22></location>Figure 14 shows an example of the 2-D gaussian fitting we used to extract the central flux from the background of the host galaxy in wavelengths 30 - 500 𝜇 m. The object used here is NGC 4388. In the first column, the original observation image is shown. The second column shows a model of the image using a PSF (Column 3) combined with the model background. Column 4 shows a PSF-subtracted image (Column 1 - Column 3) used to make the model background (Column 5). Column 6 shows the PSF- and background- subtracted image of the residuals.</text>
<section_header_level_1><location><page_23><loc_32><loc_12><loc_68><loc_13></location>C. HOST GALAXY BACKGROUNDS HERSCHEL</section_header_level_1>
<text><location><page_23><loc_10><loc_10><loc_92><loc_11></location>Here we show the results of PSF-subtracted images for objects in which only Herschel data showed host galaxy contamination.</text>
<section_header_level_1><location><page_24><loc_46><loc_93><loc_55><loc_94></location>F/u.pc/l.pc/l.pc/e.pc/r.pc /e.pc/t.pc /a.pc/l.pc.</section_header_level_1>
<figure>
<location><page_24><loc_9><loc_81><loc_26><loc_91></location>
</figure>
<figure>
<location><page_24><loc_26><loc_81><loc_39><loc_91></location>
</figure>
<figure>
<location><page_24><loc_52><loc_81><loc_66><loc_91></location>
</figure>
<text><location><page_24><loc_38><loc_89><loc_39><loc_90></location>19'00</text>
<figure>
<location><page_24><loc_39><loc_81><loc_53><loc_91></location>
</figure>
<figure>
<location><page_24><loc_65><loc_81><loc_79><loc_91></location>
</figure>
<text><location><page_24><loc_38><loc_71><loc_39><loc_72></location>50'00</text>
<figure>
<location><page_24><loc_26><loc_68><loc_39><loc_79></location>
</figure>
<text><location><page_24><loc_38><loc_64><loc_40><loc_64></location>52'30"</text>
<text><location><page_24><loc_38><loc_60><loc_40><loc_61></location>51'30"</text>
<figure>
<location><page_24><loc_9><loc_68><loc_26><loc_79></location>
</figure>
<figure>
<location><page_24><loc_38><loc_68><loc_53><loc_79></location>
</figure>
<figure>
<location><page_24><loc_66><loc_68><loc_79><loc_79></location>
</figure>
<figure>
<location><page_24><loc_79><loc_68><loc_91><loc_79></location>
</figure>
<figure>
<location><page_24><loc_25><loc_56><loc_39><loc_66></location>
</figure>
<figure>
<location><page_24><loc_39><loc_56><loc_52><loc_66></location>
</figure>
<figure>
<location><page_24><loc_52><loc_56><loc_66><loc_66></location>
</figure>
<figure>
<location><page_24><loc_66><loc_56><loc_79><loc_66></location>
</figure>
<figure>
<location><page_24><loc_79><loc_56><loc_91><loc_66></location>
</figure>
<figure>
<location><page_24><loc_10><loc_56><loc_26><loc_66></location>
</figure>
<figure>
<location><page_24><loc_9><loc_44><loc_26><loc_54></location>
</figure>
<figure>
<location><page_24><loc_25><loc_44><loc_39><loc_54></location>
</figure>
<figure>
<location><page_24><loc_39><loc_44><loc_52><loc_54></location>
</figure>
<figure>
<location><page_24><loc_52><loc_44><loc_65><loc_54></location>
</figure>
<text><location><page_24><loc_38><loc_52><loc_40><loc_53></location>50'40"</text>
<text><location><page_24><loc_38><loc_51><loc_40><loc_51></location>51'00"</text>
<text><location><page_24><loc_38><loc_45><loc_40><loc_46></location>52'00"</text>
<text><location><page_24><loc_38><loc_38><loc_40><loc_38></location>24 30"</text>
<text><location><page_24><loc_38><loc_34><loc_40><loc_34></location>73'30"</text>
<figure>
<location><page_24><loc_26><loc_32><loc_39><loc_42></location>
</figure>
<text><location><page_24><loc_18><loc_44><loc_21><loc_45></location>RA (J2000}</text>
<text><location><page_24><loc_45><loc_44><loc_47><loc_45></location>(2000)</text>
<figure>
<location><page_24><loc_10><loc_32><loc_26><loc_42></location>
</figure>
<figure>
<location><page_24><loc_26><loc_20><loc_39><loc_30></location>
</figure>
<figure>
<location><page_24><loc_10><loc_20><loc_26><loc_30></location>
</figure>
<figure>
<location><page_24><loc_39><loc_32><loc_51><loc_42></location>
</figure>
<figure>
<location><page_24><loc_52><loc_32><loc_65><loc_42></location>
</figure>
<figure>
<location><page_24><loc_79><loc_33><loc_91><loc_42></location>
</figure>
<text><location><page_24><loc_38><loc_28><loc_39><loc_28></location>19'00</text>
<text><location><page_24><loc_38><loc_26><loc_39><loc_26></location>18'30</text>
<figure>
<location><page_24><loc_39><loc_20><loc_52><loc_30></location>
</figure>
<figure>
<location><page_24><loc_52><loc_20><loc_64><loc_30></location>
</figure>
<figure>
<location><page_24><loc_65><loc_20><loc_78><loc_30></location>
</figure>
<text><location><page_24><loc_64><loc_53><loc_66><loc_53></location>34*50'</text>
<figure>
<location><page_24><loc_65><loc_44><loc_79><loc_54></location>
</figure>
<figure>
<location><page_24><loc_78><loc_44><loc_91><loc_54></location>
</figure>
<figure>
<location><page_24><loc_65><loc_32><loc_79><loc_42></location>
</figure>
<figure>
<location><page_24><loc_78><loc_20><loc_91><loc_30></location>
<caption>Figure 12. Herschel PACS and SPIRE FIR images. The white circle in the top right indicates the beam size of the observation, while the pink line in the bottom left indicates a distance of 1 kpc.</caption>
</figure>
<text><location><page_24><loc_70><loc_32><loc_73><loc_32></location>RA U2000)</text>
<figure>
<location><page_24><loc_52><loc_68><loc_66><loc_79></location>
</figure>
<text><location><page_24><loc_51><loc_64><loc_53><loc_65></location>19*53'</text>
<figure>
<location><page_24><loc_79><loc_81><loc_91><loc_91></location>
</figure>
<text><location><page_24><loc_78><loc_78><loc_79><loc_78></location>2*48'</text>
<figure>
<location><page_25><loc_9><loc_81><loc_25><loc_91></location>
<caption>Figure 13. Herschel PACS and SPIRE FIR images. The white circle in the top right indicates the beam size of the observation, while the pink line in the bottom left indicates a distance of 1 kpc.</caption>
</figure>
<figure>
<location><page_25><loc_37><loc_81><loc_53><loc_91></location>
</figure>
<figure>
<location><page_25><loc_53><loc_81><loc_66><loc_91></location>
</figure>
<figure>
<location><page_25><loc_66><loc_81><loc_78><loc_91></location>
</figure>
<figure>
<location><page_25><loc_79><loc_81><loc_92><loc_91></location>
</figure>
<figure>
<location><page_25><loc_9><loc_68><loc_25><loc_79></location>
</figure>
<figure>
<location><page_25><loc_37><loc_68><loc_52><loc_79></location>
</figure>
<figure>
<location><page_25><loc_52><loc_68><loc_66><loc_79></location>
</figure>
<figure>
<location><page_25><loc_65><loc_68><loc_78><loc_79></location>
</figure>
<figure>
<location><page_25><loc_79><loc_68><loc_92><loc_79></location>
</figure>
<figure>
<location><page_25><loc_9><loc_56><loc_26><loc_67></location>
</figure>
<figure>
<location><page_25><loc_26><loc_56><loc_39><loc_67></location>
</figure>
<text><location><page_25><loc_17><loc_56><loc_20><loc_57></location>RA (J2000)</text>
<text><location><page_25><loc_57><loc_56><loc_61><loc_57></location>RA (J2000)</text>
<text><location><page_25><loc_71><loc_56><loc_74><loc_57></location>RA (J2000)</text>
<figure>
<location><page_25><loc_9><loc_44><loc_26><loc_54></location>
</figure>
<text><location><page_25><loc_51><loc_76><loc_53><loc_77></location>5*32</text>
<figure>
<location><page_25><loc_38><loc_56><loc_53><loc_67></location>
</figure>
<figure>
<location><page_25><loc_53><loc_56><loc_66><loc_67></location>
</figure>
<figure>
<location><page_25><loc_66><loc_56><loc_78><loc_67></location>
</figure>
<text><location><page_25><loc_38><loc_64><loc_40><loc_65></location>12'00"</text>
<text><location><page_25><loc_38><loc_59><loc_40><loc_60></location>13'00"</text>
<figure>
<location><page_25><loc_26><loc_44><loc_39><loc_54></location>
</figure>
<text><location><page_25><loc_17><loc_44><loc_21><loc_45></location>RA (2000)</text>
<text><location><page_25><loc_31><loc_44><loc_34><loc_45></location>RA (J2000)</text>
<text><location><page_25><loc_44><loc_44><loc_47><loc_45></location>RA (J2000)</text>
<text><location><page_25><loc_57><loc_44><loc_61><loc_45></location>RA (J2000)</text>
<text><location><page_25><loc_71><loc_44><loc_74><loc_45></location>RA (J2000)</text>
<text><location><page_25><loc_84><loc_44><loc_87><loc_45></location>RA (2000)</text>
<figure>
<location><page_25><loc_10><loc_32><loc_26><loc_42></location>
</figure>
<figure>
<location><page_25><loc_39><loc_44><loc_53><loc_54></location>
</figure>
<figure>
<location><page_25><loc_25><loc_32><loc_39><loc_42></location>
</figure>
<figure>
<location><page_25><loc_40><loc_32><loc_52><loc_42></location>
</figure>
<figure>
<location><page_25><loc_9><loc_19><loc_26><loc_29></location>
</figure>
<figure>
<location><page_25><loc_26><loc_19><loc_39><loc_29></location>
</figure>
<text><location><page_25><loc_38><loc_40><loc_40><loc_40></location>53'00</text>
<text><location><page_25><loc_38><loc_38><loc_40><loc_39></location>52'40"</text>
<text><location><page_25><loc_38><loc_33><loc_40><loc_34></location>51'40"</text>
<figure>
<location><page_25><loc_38><loc_19><loc_52><loc_29></location>
</figure>
<figure>
<location><page_25><loc_52><loc_19><loc_65><loc_29></location>
</figure>
<figure>
<location><page_25><loc_65><loc_19><loc_78><loc_29></location>
</figure>
<figure>
<location><page_25><loc_78><loc_19><loc_91><loc_29></location>
</figure>
<figure>
<location><page_25><loc_52><loc_44><loc_66><loc_54></location>
</figure>
<text><location><page_25><loc_51><loc_41><loc_53><loc_41></location>8*54'</text>
<figure>
<location><page_25><loc_65><loc_44><loc_78><loc_54></location>
</figure>
<figure>
<location><page_25><loc_79><loc_44><loc_92><loc_54></location>
</figure>
<text><location><page_25><loc_65><loc_41><loc_67><loc_41></location>8*54'</text>
<figure>
<location><page_25><loc_65><loc_32><loc_79><loc_42></location>
</figure>
<figure>
<location><page_25><loc_52><loc_32><loc_66><loc_42></location>
</figure>
<text><location><page_25><loc_51><loc_28><loc_53><loc_28></location>8*48'</text>
<text><location><page_25><loc_65><loc_28><loc_66><loc_28></location>8*48'</text>
<text><location><page_25><loc_78><loc_28><loc_80><loc_28></location>8*48</text>
<text><location><page_25><loc_78><loc_41><loc_80><loc_41></location>8*54'</text>
<figure>
<location><page_25><loc_79><loc_32><loc_92><loc_42></location>
</figure>
<figure>
<location><page_25><loc_79><loc_56><loc_92><loc_67></location>
</figure>
<section_header_level_1><location><page_26><loc_46><loc_93><loc_55><loc_94></location>F/u.pc/l.pc/l.pc/e.pc/r.pc /e.pc/t.pc /a.pc/l.pc.</section_header_level_1>
<section_header_level_1><location><page_26><loc_45><loc_90><loc_60><loc_91></location>Fitting result - NGC4388 - best</section_header_level_1>
<figure>
<location><page_26><loc_18><loc_42><loc_91><loc_88></location>
<caption>Figure 14. Example of the 2D gaussian fitting routine explained in Section 4.1</caption>
</figure>
<section_header_level_1><location><page_27><loc_45><loc_93><loc_54><loc_94></location>SOFIA A/t.pc/l.pc/a.pc/s.pc</section_header_level_1>
<figure>
<location><page_27><loc_22><loc_45><loc_76><loc_91></location>
<caption>Figure 15. PSF-subtracted host galaxy images from archive Herschel data</caption>
</figure>
</document>
[
{
"title": "ABSTRACT",
"content": "Wepresent a 19 . 7 -214 𝜇 mimaging atlas of local (4 -181 Mpc; median 43 Mpc) active galactic nuclei (AGN) observed with FORCAST and HAWC+ on board the SOFIA telescope with angular resolutions ∼ 3 '' -20 '' . This atlas comprises 22 Seyferts (17 Type 2 and 5 Type 1) with a total of 69 images, 41 of which have not been previously published. The AGN span a range of luminosities of log 10 ( 𝐿 bol [ erg / s ]) = [ 42 , 46 ] with a median of log 10 ( 𝐿 bol [ erg / s ]) = 44 . 1 ± 1 . 0. We provide total fluxes of our sample using aperture photometry for point source objects and a 2-D Gaussian fitting for objects with extended host galaxy emission, which was used to estimate the unresolved nuclear component. Most galaxies in our sample are point-like sources, however, four sources (Centaurus A, Circinus, NGC 1068, and NGC 4388) show extended emission in all wavelengths. The 30 -40 𝜇 m extended emission in NGC 4388 is coincident with the narrow line region at PA ∼ 50 · , while the dusty extension at longer wavelengths arises from the host galaxy at PA ∼ 90 · . Our new observations allow us to construct the best sampled spectral energy distributions (SEDs) available between 30 - 500 𝜇 m for a sample of nearby AGN. We estimate that the average peak wavelength of the nuclear SEDs is ∼ 40 𝜇 m in 𝜈𝐹 𝜈 , which we associate with an unresolved extended dusty region heated by the AGN. 17 CO(2-1) emission in NGC 5643 as a nuclear molecular gas component of the torus that is likely collimating the ionization cone. Conditions favorable for launching a cold and molecular wind likely depend on Eddington ratio and nuclear hydrogencolumndensities (e.g., Venanzi et al. 2020; Garc'ıa-Burillo et al. 2021; Alonso-Herrero et al. 2021; Garc'ıa-Bernete et al. 2022a). This pc-scale dusty component is possibly associated with larger scale MIR emission detected out to 100s pc scales. In the case of Circinus, high angular resolution MIR imaging, optical polarimetry and integral field spectra, coupled with state-of-the-art radiative transfer simulations, provide evidence that extended dust emission from pc to tens of pc scales in this object is a result of a hollow dusty cone illuminated by a tilted accretion disk (Stalevski et al. 2017, 2019, 2023; Kakkad et al. 2023). MIR extended emission out to 1' ( ∼ 75 pc) was clearly detected in NGC 1068 by Bock et al. (2000). Later 10 . 8 and 18 . 2 𝜇 m emission extending 3 . '' 5 ( ∼ 200 pc) across NGC 4151 was also attributed to dust in the NLR heated by the central engine (Radomski et al. 2003). Likewise, at similar wavelengths, extended emission in 18 AGN at distances out to hundreds of parsecs was detected (Asmus et al. 2016; Garc'ıa-Bernete et al. 2016; Asmus 2019). Using the 37 . 1 𝜇 m filter on SOFIA/FORCAST and thanks to the increase in angular resolution compared with 𝑆𝑝𝑖𝑡𝑧𝑒𝑟 , extended dust emission in Mrk 3, NGC 4151, and NGC4388 was found on ∼ 100s pc-scales (Fuller et al. 2019) coincident with the NLR and radio axis. This emission may be due to dust along the wall of ionization cones (Mason et al. 2009) or a dusty NLR (Mor et al. 2009; Mor & Netzer 2012). In this manuscript we present an imaging atlas of 22 local (D = 4 -181 Mpc; median 42.8 Mpc) AGN obtained using the FORCAST and HAWC+ instruments on the 2.7-m SOFIA telescope in the wavelength range 20 -214 𝜇 m. Most of these datasets are unpublished or dispersed throughout the literature. We provide a mid- to far-IR imaging atlas at angular scales of ∼ 3 -20 '' . At these scales, contribution from several dust sources is expected and we expect to disentangle the emission sources in a future study. Instead, here we aim to determine whether these objects are extended or not, and at what wavelengths within the resolution of the SOFIA telescope. We also explore the wavelength of turnover in the SED. This atlas is complementary to JWST observations up to ∼ 25 𝜇 mand archival Herschel data (70 -500 𝜇 m). The manuscript is organzed as follows. Section 2 describes the observations and AGN sample definition; Section 3 shows the new IR images; Section 4 contains details of the imaging analysis and Section 5 shows the resulting SEDS; we present results about the data in Section 6.",
"pages": [
1,
2
]
},
{
"title": "The Galaxy Activity, Torus, and Outflow Survey (GATOS). VII. The 20-214 𝜇 mimaging atlas of active galactic nuclei using SOFIA",
"content": "L/i.pc/n.pc/d.pc/s.pc/a.pc/y.pc F/u.pc/l.pc/l.pc/e.pc/r.pc, 1 E/n.pc/r.pc/i.pc/q.pc/u.pc/e.pc L/o.pc/p.pc/e.pc/z.pc-R/o.pc/d.pc/r.pc/i.pc/g.pc/u.pc/e.pc/z.pc, 2, 3 I/s.pc/m.pc/a.pc/e.pc/l.pc G/a.pc/r.pc/c.pc'/dotlessi.pc/a.pc-B/e.pc/r.pc/n.pc/e.pc/t.pc/e.pc, 4 C/r.pc/i.pc/s.pc/t.pc/i.pc/n.pc/a.pc R/a.pc/m.pc/o.pc/s.pc A/l.pc/m.pc/e.pc/i.pc/d.pc/a.pc, 5, 6 A/l.pc/m.pc/u.pc/d.pc/e.pc/n.pc/a.pc A/l.pc/o.pc/n.pc/s.pc/o.pc-H/e.pc/r.pc/r.pc/e.pc/r.pc/o.pc, 7 C/h.pc/r.pc/i.pc/s.pc P/a.pc/c.pc/k.pc/h.pc/a.pc/m.pc, 1, 8 L/u.pc/l.pc/u.pc Z/h.pc/a.pc/n.pc/g.pc, 1 M/a.pc/s.pc/o.pc/n.pc L/e.pc/i.pc/s.pc/t.pc, 1 N/a.pc/n.pc/c.pc/y.pc L/e.pc/v.pc/e.pc/n.pc/s.pc/o.pc/n.pc, 9 M/a.pc/s.pc/a.pc I/m.pc/a.pc/n.pc/i.pc/s.pc/h.pc/i.pc, 8 S/e.pc/b.pc/a.pc/s.pc/t.pc/i.pc/a.pc/n.pc H/o.pc/e.pc/n.pc/i.pc/g.pc, 10 M/a.pc/r.pc/k.pc/o.pc S/t.pc/a.pc/l.pc/e.pc/v.pc/s.pc/k.pc/i.pc, 11, 12 C/l.pc/a.pc/u.pc/d.pc/i.pc/o.pc R/i.pc/c.pc/c.pc/i.pc, 13, 14 E/r.pc/i.pc/n.pc H/i.pc/c.pc/k.pc/s.pc, 15 E/n.pc/r.pc/i.pc/c.pc/a.pc B/e.pc/l.pc/l.pc/o.pc/c.pc/c.pc/h.pc/i.pc, 16, 17 F/r.pc/a.pc/n.pc/c.pc/o.pc/i.pc/s.pc/e.pc C/o.pc/m.pc/b.pc/e.pc/s.pc, 18 R/i.pc/c.pc D/a.pc/v.pc/i.pc/e.pc/s.pc, 19 S/a.pc/n.pc/t.pc/i.pc/a.pc/g.pc/o.pc G/a.pc/r.pc/c.pc'/dotlessi.pc/a.pc B/u.pc/r.pc/i.pc/l.pc/l.pc/o.pc, 20 O/m.pc/a.pc/i.pc/r.pc/a.pc G/o.pc/n.pc/z.pc '/a.pc/l.pc/e.pc/z.pc M/a.pc/r.pc/t.pc'/dotlessi.pc/n.pc, 21 T/a.pc/k.pc/u.pc/m.pc/a.pc I/z.pc/u.pc/m.pc/i.pc, 8 A/l.pc/v.pc/a.pc/r.pc/o.pc L/a.pc/b.pc/i.pc/a.pc/n.pc/o.pc, 22 M/i.pc/g.pc/u.pc/e.pc/l.pc P/e.pc/r.pc/e.pc/i.pc/r.pc/a.pc S/a.pc/n.pc/t.pc/a.pc/e.pc/l.pc/l.pc/a.pc, 23 D/i.pc/m.pc/i.pc/t.pc/r.pc/a.pc R/i.pc/g.pc/o.pc/p.pc/o.pc/u.pc/l.pc/o.pc/u.pc, 4, 24 D/a.pc/v.pc/i.pc/d.pc R/o.pc/s.pc/a.pc/r.pc/i.pc/o.pc, 25 D/a.pc/n.pc/i.pc/e.pc/l.pc R/o.pc/u.pc/a.pc/n.pc, 26 T/a.pc/r.pc/o.pc S/h.pc/i.pc/m.pc/i.pc/z.pc/u.pc, 19 /a.pc/n.pc/d.pc M/a.pc/r.pc/t.pc/i.pc/n.pc W/a.pc/r.pc/d.pc 27 1 University of Texas at San Antonio, One UTSA Circle, San Antonio, TX, 78249, USA 2 Department of Physics & Astronomy, University of South Carolina, Columbia, SC 29208, USA 3 Kavli Institute for Particle Astrophysics & Cosmology (KIPAC), Stanford University, Stanford, CA 94305, USA 4 Department of Physics, University of Oxford, Keble Road, Oxford OX1 3RH, UK 5 Instituto de Astrof'ısica de Canarias, Calle V'ıa L'actea, s/n, E-38205 La Laguna, Tenerife, Spain 6 Departamento de Astrof'ısica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain 7 Centro de Astrobiolog'ıa (CAB), CSIC-INTA, Camino Bajo del Castillo s/n, E-28692, Villanueva de la Ca˜nada, Madrid, Spain 8 National Astronomical Observatory of Japan, National Institutes of Natural Sciences (NINS), 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 9 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA 10 School of Physics & Astronomy, University of Southampton, Southampton SO17 1BJ, UK 11 Astronomical Observatory, Volgina 7, 11060 Belgrade, Serbia 12 Sterrenkundig Observatorium, Universiteit Gent, Krijgslaan 281-S9, Gent B-9000, Belgium 13 N'ucleo de Astronom'ıa de la Facultad de Ingenier'ıa, Universidad Diego Portales, Av. Ej'ercito Libertador 441, Santiago, Chile 14 Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, People's Republic of China 15 Department of Physics and Astronomy, University of Alaska Anchorage, Anchorage, AK 99508-4664, USA 16 Departmento de F'ısica de la Tierra y Astrof'ısica, Fac. de CC F'ısicas, Universidad Complutense de Madrid, E-28040 Madrid, Spain Instituto de F'ısica de Part'ıculas y del Cosmos IPARCOS, Fac. CC F'ısicas, Universidad Complutense de Madrid, E-28040 Madrid, Spain 18 LERMA, Observatoire de Paris, Coll'ege de France, PSL University, CNRS, Sorbonne University, Paris, France 19 Max Planck Institut fur Extraterrestrische Physik, Giessenbachstrasse 1, D-85748 Garching bei Munchen, Germany 20 Observatorio de Madrid, OAN-IGN, Alfonso XII, 3, E-28014 Madrid, Spain 21 Instituto de Radioastronom'ıa y Astrof'ısica (IRyA), Universidad Nacional Aut'onoma de M'exico, Antigua Carretera a P'tzcuaro #8701, ExHda. San Jos'e de la Huerta, Morelia, Michoac'an, C.P. 58089, Mexico 22 Telespazio UK for the European Space Agency, ESAC, Camino Bajo del Castillo s/n, E-28692 Villanueva de la Ca˜nada, Spain 23 Instituto de F'ısica Fundamental, CSIC, Calle Serrano 123, E-28006 Madrid, Spain 24 School of Sciences, European University Cyprus, Diogenes street, Engomi, 1516 Nicosia, Cyprus 25 School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne NE1 7RU, UK 26 LESIA, Observatoire de Paris, Universit'e PSL, CNRS, Sorbonne Universit'e, Sorbonne Paris Cite'e, 5 place Jules Janssen, F-92195 Meudon, France 27 Centre for Extragalactic Astronomy, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK",
"pages": [
1
]
},
{
"title": "2. AGN SAMPLE AND OBSERVATION DATA",
"content": "2.1. Sample Selection This imaging atlas was drawn from the ongoing AGN survey performed by the Galactic Activity, Torus, and Out- Keywords: galaxies -- active galaxies -- agn",
"pages": [
2
]
},
{
"title": "1. INTRODUCTION",
"content": "There is clear evidence that a considerable amount of dust in the vicinity of supermassive black holes (SMBHs) in active galaxies obscures the central engine (i.e., accretion disk and SMBH) in some lines of sight. Through spectropolarimetric observations of NGC 1068, Antonucci & Miller (1985) showed that its optical polarized spectrum contained broad optical polarized emission lines not originally observed by direct total intensity observations. It was subsequently presumed that an optically and geometrically thick dusty structure ('torus') blocked the central engine in some lines of sight (Antonucci 1993; Urry & Padovani 1995). Under this unified scheme a Type 1 AGN is seen face-on and shows broadened optical lines, while in Type 2 AGN the broadened lines are obscured. This model also predicts that broad silicate features at 10 and 18 𝜇 m will be seen in emission in Type 1 and in absorption in Type 2. However, silicate emission can be seen in emission in some Type 2 AGN, while absorption can be seen in some Type 1 (e.g., Hatziminaoglou et al. 2015). This and other observational features are explained by the inhomogeneous nature of the torus. Clumpy torus models (Nenkova et al. 2008a,b) predict shallower silicate features, more similar infrared spectral energy distributions (SEDs) between Type 1 and Type 2 AGN, etc. (see Ramos Almeida & Ricci 2017, for a review). A region of narrow forbidden line emission extends above and below the midplane of the dusty torus structure out to several kpc scales. Recent sub-arcsecond interferometric imaging observations have shown a dust component at pc-scales co-spatial with the base of the narrow line region (NLR; Honig et al. 2012; Tristram et al. 2014; L'opez-Gonzaga et al. 2014, 2016; Burtscher et al. 2013; G'amez Rosas et al. 2022; Isbell et al. 2022). This dusty structure is interpreted as part of a dusty wind launched from the inner hot part of the torus driven by radiation pressure at pc-scales (Honig 2019), but generated by a magnetohydrodynamical wind at sub-pc scales (e.g., Emmering et al. 1992; Lopez-Rodriguez et al. 2015; Takasao et al. 2022; Lopez-Rodriguez et al. 2023). This extended dusty structure has been resolved in a nearby galaxy, ESO 418-G14, using mid-infrared (MIR) images with JWST/MIRI finding that the dust is primarily heated by the AGN and/or radiative jet-induced shocks in the NLR rather than a wind (Haidar et al. 2024). ALMA observations provide observational support for a dusty torus+outflow scenario. Emission from the nucleus of NGC1068 was mapped with a resolution of ∼ 4 pc, resolving a 7 -10 pc diameter disk interpreted as the sub-mm counterpart of the torus (Garc'ıa-Burillo et al. 2016). Rotation of the compact emission was detected in HCN J=3-2 and HCO+ J=3-2 (Imanishi et al. 2018, 2020). A molecular outflowing wind co-spatial with the dusty and molecular torus was also observed (Garc'ıa-Burillo et al. 2019). Alonso-Herrero et al. (2018) interpreted the measured nuclear (10 -20 pc) flow Survey (GATOS; Garc'ıa-Burillo et al. 2021; AlonsoHerrero et al. 2021; Garc'ıa-Bernete et al. 2024). GATOS /one.sup aims to characterize the dynamics and composition of the dusty and molecular torus and multi-phase outflows in AGN. The GATOS parent sample is selected from the 70 Month Swift /BAT AGN catalog, which is flux limited in the ultrahard 14 -195 keV X-rays band (Baumgartner et al. 2013). In the initial study of AGN using SOFIA observations (Fuller et al. 2016), sources from the GATOS survey were selected based on the criteria that the galaxies had been previously studied using C/l.pc/u.pc/m.pc/p.pc/y.pc (Nenkova et al. 2008a,b) torus models and were well-sampled in the 1 -18 𝜇 m regime (Ramos Almeida et al. 2009, 2011; Alonso-Herrero et al. 2011). The study of the 11 objects included the 31 . 5 𝜇 m photometry in the SEDs and found that including the 31.5 𝜇 m photometry reduces the number of C/l.pc/u.pc/m.pc/p.pc/y.pc torus models that are compatible with the data and modifies the model output for the torus outer radius. Fuller et al. (2019) further extended the wavelength range of a subset of 7 AGN SEDs to 37.1 𝜇 m. They subsequently found extended emission in the PSF-subtracted images of Mrk 3, NGC 4151, and NGC 4388 that is coincident with the radio axis and NLR. In a separate study, Lopez-Rodriguez et al. (2018) modeled the torus of NGC 1068 using ∼ 20 -53 𝜇 m FORCAST and HAWC+ observations. They showed that the peak wavelength range of emission from the torus is ∼ 30 - 40 𝜇 m with a characteristic temperature 70 - 100 K. The use of observations ¿ 30 𝜇 m in that study from SOFIA and ALMA highlights the importance of longer wavelength observations to put constraints on MIR emission sources. Based on these results, we extend the wavelength range in objects previously observed, and also expand the number of AGN observed. We present the complete imaging atlas of 22 Seyferts observed by SOFIA in the wavelength range 19 . 7 -214 𝜇 m using FORCAST and HAWC+. The final set of observations presented here was part of a multi-year AGN survey over several observing SOFIA cycles (Proposal IDs: 02 0035, 04 0048, 06 0066, 08 0014; PI: Lopez-Rodriguez; 70 0400 PI: Herter). The SOFIA atlas of AGN in the far-IR (FIR) is a flux-limited sample of nearby, bright, and well-studied AGN. All objects have a point source flux of > 200 mJy at 31 . 5 𝜇 m, which ensures that each band can be observed within 1 hr of on-source time with a signal to noise ratio > 10 using FORCAST/SOFIA. Although the original AGN sample is larger than that presented here, only 22 AGN were observed in total by SOFIA before end of operations in 2022. Note that there are gaps in the 20 -214 𝜇 m wavelength range due to the fact that SOFIA only flies with a single instrument per night. For each SOFIA cycle, we prioritized the objects with observations acquired in a single instrument from the previous cycle. The sample properties are given in Table 1. For most objects, we retrieved redshift data from the NASA Extragalactic Database (NED). However, for nearby objects Centaurus A Redshifts and spectral type were taken from NED. Distances to most sources were obtained using H0 = 70 km s -1 Mpc -1 . Distances to nearby sources Centaurus A and Circinus were taken from Harris et al. (2010) and Tully et al. (2009), respectively. References for log 𝐿 𝑏𝑜𝑙 : a) Borkar et al. (2021) b) Marinucci et al. (2012) c) Alonso-Herrero et al. (2011) d) Ichikawa et al. (2017), e) Leighly et al. (2014) f) Marconi et al. (2004), g) Baumgartner et al. (2013), h) Garc'ıa-Bernete et al. (2015), i) Ramos Almeida et al. (2011), j) Yuan et al. (2002), k) Duras et al. (2020), and Circinus, distances were obtained individually (Harris et al. 2010; Tully et al. 2009). The 22 objects in this atlas cover the luminosity range of log 10 ( 𝐿 bol [ erg / s ]) = [ 42 , 46 ] with a median of log 10 ( 𝐿 bol [ erg / s ]) = 44 . 1 ± 1 . 0, and a distance of 4 -181 Mpc with a median of 42.8 Mpc. Figure 1 shows bolometric luminosity plotted against distance, where Seyfert 1 objects are shown as red triangles and Seyfert 2 objects are shown as purple stars.",
"pages": [
2,
3
]
},
{
"title": "2.2.1. FORCAST",
"content": "FORCAST is an IR camera and spectrograph sensitive in the wavelength range 5 -40 𝜇 m with a field of view (FOV) of 3.4 ' × 3.2 ' and pixel scale 0.768 '' /pixel. With one exception (NGC 1068 in the 19.7 𝜇 m filter), we only used the Long Wavelength Camera (LWC; 25 -40 𝜇 m) due to the abundance of ground-based images at shorter wavelengths for the objects in our sample. FORCAST observations were made in dual channel mode using the two-position chop-nod (C2N) method with symmetric nod-match-chop (NMC) to remove telescope thermal emission and time variable sky background, and to reduce the effect of 1/ 𝑓 noise from the array. Data were reduced by the SOFIA Science Center using the /f.pc/o.pc/r.pc/c.pc/a.pc/s.pc/t.pc /r.pc/e.pc/d.pc/u.pc/x.pc pipeline following the methods described by Herter et al. (2012). Most of the pipeline changes over the cycles were to refine the spectroscopic mode of FORCAST with little or no change to the image mode presented here. Observations were flux-calibrated using the set of standard stars of the observing run, which provides flux uncertainties of ∼ 10 %. The point spread function (PSF) of the 31.5 𝜇 m observations from Cycle 2 (see Fuller et al. 2016) was the co-average of a set of standard stars from that cycle. Its FWHMwas 3.40 '' , in agreement with the SOFIA Observer's Handbook v3.0.0. The PSFs for Cycle 4 in the 30 - 40 𝜇 mwavelength range were determined by using standard star observations from the individual flights (see Fuller et al. 2019) and averaged at FWHM ∼ 4.33 '' and 4.58 '' in the 31.5 and 37.1 𝜇 m filters, respectively. The FORCAST PSFs for the observations of NGC1068 are detailed in Lopez-Rodriguez et al. (2018).",
"pages": [
3,
4
]
},
{
"title": "2.2.2. HAWC+",
"content": "HAWC+isaFIRimaging polarimeter designed to allow total and polarized intensity imaging observations in four broad bands centered at 53, 89, 155, and 214 𝜇 m, corresponding to Bands A, C, D, and E respectively (see Table 2). On- -fly mapping (OTFMAP) observing modes were used for both imaging polarimetry and total intensity imaging. Observing modes for the individual observations are given in Table 3. Data taken in polarization mode was reduced using the /h.pc/a.pc/w.pc/c.pc /d.pc/p.pc/r.pc /p.pc/i.pc/p.pc/e.pc/l.pc/i.pc/n.pc/e.pc and the reduction steps presented in Lopez-Rodriguez et al. (2022b). The Comprehensive Reduction Utility for SHARC II (CRUSH; Kov'acs et al. 2006; Kov'acs 2008) was used to obtain the total intensity observations. HAWC+ observations were reduced following the same reduction steps. We quote the pipeline versions or CRUSH versions in Table 3 to differentiate between the observing modes. There are no differences between CRUSH versions to obtain the total intensity images as all the changes in the pipeline were done for the polarimetric mode. Table 3 also shows the pixel scale of each image. As in FORCAST observations, the source of uncertainty in the photometry for HAWC+ results from calibration factors of the standard stars associated with the observation, giving an uncertainty of ∼ 10 % (Lopez-Rodriguez et al. 2022b). HAWC+ PSFs were estimated using standard star observations in 2017. Pallas was observed in Bands A and C on 7 November 2017, while Neptune was observed in Bands D and E on 19 October, 2017. The FWHM of these standards are 5.25 '' , 8.26 '' , 14.74 '' , and 19.65 '' in Bands A, C, D, and E, respectively. The FWHMs from the Observer's Handbook /two.sup are given in Table 2.",
"pages": [
4
]
},
{
"title": "2.3. Observing Data",
"content": "Table 3 provides the final AGN sample with information about the wavelength, observing mode, observation and mission details, versions of the separate pipelines, and also the field-of-view (FOV) of the individual images.",
"pages": [
4
]
},
{
"title": "3. IMAGES",
"content": "Images of the 22 AGN in the 19 . 7 -214 𝜇 m wavelength range are presented in Figures 2, 3, 4, and 5. The orange scale on the bottom left of the images indicates a scale of 500 pc. The beam size is depicted in white in the top right of the images. In all images, north is up and east is to the left. Complementary Herschel 70 -500 𝜇 m images are shown in Appendix A (Fig. 10, 11, 12, 13). These fully reduced images were obtained through the Herschel Science Archive /three.sup . All objects are presented and analyzed individually. Centaurus A . The host galaxy of Centaurus A is clearly visible in the 53 𝜇 m image with a bright compact center. However, the host galaxy becomes more dominant in the 89 𝜇 mimage(seeadetailed analysis of host galaxy dust emission at 89 𝜇 m in Lopez-Rodriguez 2021). The kpc-scale warped dust and gas lane was first observed with Spitzer imaging using IRAC and MIPS (Quillen et al. 2006). On subarcsecond scales, Radomski et al. (2008) observed the nucleus of Centaurus A using the 8.8, 10.4, and 18.3 𝜇 m filters on T-ReCS at Gemini South. They concluded that the mostly likely sources of nuclear MIR emission are an unresolved clumpy dusty torus in the core, and a dusty NLR for the arcsecondscale extended emission (see also Garc'ıa-Bernete et al. 2016). Circinus Galaxy . HAWC+ 53 and 89 𝜇 m images of the Circinus Galaxy show a very bright FIR core with extended emission at a PA ∼ 30 · , whereas the 215 𝜇 m image shows a slightly different PA ∼ 55 · . We estimate the FWHM of the extended FIR nuclear emission to be ∼ 6 '' × 6 '' , 13.5 '' × 11.5 '' , and 22.3 '' × 25.6 '' at 53, 89, and 214 𝜇 m, respectively. These are larger than the PSF FWHMs given in Section 2.2.2, which indicates extended emission along the axis of the inner bar of the galaxy. MIR emission was resolved at 8.7 and 18.3 𝜇 m out to 2 '' in an approximate east-west direction, coincident with the ionization cones at PA ∼ 100 · (Packham et al. 2005; Stalevski et al. 2017). However, the elongation seen in the SOFIAimages(andin Herschel imagesinAppendixA)seems to be arising from dust in the host galaxy. MCG-5-23-16 . MCG-5-23-16 appears as a point-like source in the 31 . 5 -155 𝜇 mwavelength range whose brightness decreases with increasing wavelength. However, this galaxy appears as an extended source in the MIR using high angular resolution data from VLT (Garc'ıa-Bernete et al. 2016). Likewise, Ferruit et al. (2000) found that this nucleus has an extended optical NLR at a PA of 40 · . They found a dust lane extending 2 '' on either side of the nucleus, parallel to the axis of the galaxy. Their extended dusty emission at 40 · was detected at a 3 𝜎 level up to ∼ 4 '' from the core. This extended structure has no thermal emission counterpart within the 31 . 5 -155 𝜇 mwavelength range. Mrk3 . Although the FORCAST and HAWC+ images generally appear to be point-like sources, Fuller et al. (2019) found extended emission in the PSF-subtracted 37.1 𝜇 m im- of Mrk 3 in the direction of the radio axis (84 · ; Kukula et al. 1993) and the NLR ( ∼ 70 · ; Capetti et al. 1995). Asmus et al. (2013) found an elongated nucleus out to ∼ 170 pc with PA ∼ 70 · in the Si-5 (11.6 𝜇 m) filter using Gemini/MICHELLE. However, the Si-2 (8.7 𝜇 m) image from GTC/Canaricam appears point-like (Alonso-Herrero et al. 2016). The large scale east-west structure in the 53 𝜇 mimage here is a background artifact produced by the data reduction due to the small spatial coverage and short integration time of the observation. Mrk 231 . The 89 𝜇 mimage shown here is point-like with a FWHM of ∼ 8.5', similar to the standard FWHM of 8.26' (see Section 2.2.2). Mrk 231 is a Type 1 Ultra Luminous Infrared Galaxy (ULIRG) and is the nearest known quasar at a distance of 181 Mpc. It is known for its multi-phase and multi-scale outflows (see Rupke & Veilleux 2011), with a neutral outflow up to 3 kpc in radius (Rupke et al. 2005). Mrk 573 . Although the SNR is very low (3 -4 𝜎 ), both 31.5 and 37.1 𝜇 mimages of Mrk 573 show marginally resolved ∼ 4 . 5 '' elongation in the east-west direction at PA ∼ 110 · . Mrk 573 was previously shown to have a biconical NLR coincident with radio emission 3-4' from the nucleus at a PA ∼ 125 · (Ulvestad & Wilson 1984; Pogge & De Robertis 1995). The marginal detection here may be cold extended dust in the outer layers of the NLR. NGC 1068 . The 19 - 53 𝜇 m images of NGC 1068 were published previously in Lopez-Rodriguez et al. (2018), where it was shown that the peak emission from the torus occurs between 30 - 40 𝜇 m with a corresponding temperature of 70 - 100 K. The 89 𝜇 m image was published as a polarimetric observation (Lopez-Rodriguez et al. 2020). Within a scale of about 1 kpc, NGC 1068 shows extended emission in the NE to SW direction at a PA ∼ 45 · , similar to MIR observations using VISIR/VLT (Asmus et al. 2014). Their observations revealed a nuclear structure in the north-south direction and extended structures to the NE and SW. From the N-band spectrum, Mason et al. (2006) concluded that while torus emission dominates NIR wavelengths, large-scale MIR emission is dominated by diffuse dust within the ionization cones. NGC 1275 . All 30 - 53 𝜇 m images are dominated by a point-like source. However, the 31.5 𝜇 m image (Fuller et al. 2019) shows 3 𝜎 extended emission along the PA ∼ 140 · . This AGN is known to have a network of H 𝛼 filaments extending out to ∼ 100' (see Conselice et al. 2001) and is possibly the result of a merger (Holtzman et al. 1992). The MIR core shows silicate dust emission in both 10 and 18 𝜇 mbands (see Fuller et al. 2019). Hence, both dust and gas are extended covering several kpc around the core. In the HAWC+ 89 𝜇 mfilter, Lopez-Rodriguez et al. (2023) found extended dust emission at a PA ∼ 125 · out to a 12 kpc radius potentially associated with a magnetized dusty filament along the NW direction (Fabian et al. 2008). NGC 2110 . The 30 - 215 𝜇 mimages of NGC 2110 are all point-like. The north-south pattern in the 53 𝜇 mimage is due to background noise and does not represent extended dust. NGC2110is a Type 2 AGN that shows silicate emission at 10 and18 𝜇 mthatis interpreted as a result of a clumpy torus, or as dust within the ionization cones (PA ∼ 160 · ; Mulchaey et al. 1994) in the inner 32 pc of the AGN (Mason et al. 2009). This galaxy appears as an extended source in Gemini/MICHELLE high resolution N-band observations (Garc'ıa-Bernete et al. 2016). However, any structure within the NLR or ionization cones is not resolved by our observations. NGC 2273 . The full set of 30 - 215 𝜇 m images of NGC 2273 show a point-like source. The north/south pattern in the 53 𝜇 m image is due to background noise and does not represent extended dust. Within the FWHM of these images (see Sections 2.2.1, 2.2.2), there is a known star-forming ring within ∼ 2' of the nucleus (Ferruit et al. 2000; Martini et al. 2003; Sani et al. 2012). GTC/Canaricam observations (Alonso-Herrero et al. 2014, 2016) at 8.7 𝜇 mshowelongation from the north-east to the south-west, likely with contribution from PAH. This structure is consistent with extension seen in the PSF-subtracted 37.1 𝜇 mSOFIAimage(Fulleretal. 2019). NGC 2992 . The image of NGC 2992 in the 31.5 𝜇 mfilter is published in Fuller et al. (2016) and appears as a point-like source. Subarcsecond N-band imaging (Garc'ıa-Bernete et al. 2015) reveals extended emission along PA ∼ 30 · out to ∼ 3 kpc which is attributed to dust heated by star formation based on corresponding N-band spectroscopy. The FWHM of the SOFIA image is ∼ 3.5' × 3.5' (560 × 560 pc 2 ) so the extension should be resolvable within the SOFIA image. Since we do not see the extension in the image here, we conclude that either the extended dust emission tapers at wavelengths ¿ 20 𝜇 mor SOFIA does not have enough sensitivity to detect it. NGC 3081 . The 31.5 𝜇 m image of NGC 3081 was published in Fuller et al. (2016) while the 37.1 𝜇 m image was published in Fuller et al. (2019). The nucleus is known to harbor a region of strong optical emission ∼ 1' from the AGN (Ferruit et al. 2000) likely due to dust or gas heated by the AGN. Fuller et al. (2019) estimated that ∼ 35% of the MIRemission within the central few arcseconds (few hundred parsecs) of the AGN originates in the NLR. High angular resolution N- and Q- band observations show extension towards the north, extending out to ∼ 450 pc from the south-east to the north-west (PA ∼ 160 · ; Garc'ıa-Bernete et al. 2016). On larger scales, optical and NIR observations reveal a series of star forming resonance rings at distances of 2.3, 11.0, 26.9 kpc and 33.1 kpc (Buta 1990; Buta et al. 1998, 2004). At longer wavelengths (¿200 𝜇 m), Ramos Almeida et al. (2011) concluded that FIR emission is contaminated by the star-forming ring 2.3 kpc in diameter. NGC 3227 . The 31.5 𝜇 m image was published in Fuller et al. (2016) while the 37.1 𝜇 mimage was published in Fuller et al. (2019). These images show a point-like source, although NGC 3227 is known to harbor a nuclear star-forming region (Schinnerer et al. 2001; Davies et al. 2006) with a nuclear cluster within ∼ 70 pc ( ∼ 1') from the core. The 8.7 𝜇 mimage from Alonso-Herrero et al. (2016) shows a slight north/south elongation and the corresponding spectrum shows clear PAH in the nucleus (see also Garc'ıa-Bernete et al. 2016). These star forming regions likely contaminate the nuclear MIR emission within the FWHM of our images. NGC 3281 . While the 31.5 𝜇 m FORCAST image is published in Fuller et al. (2016), the HAWC+ images at 53, 89, 154, and 214 𝜇 m are presented here for the first time and appear point-like in all filters. The images taken at 53 and 89 𝜇 mappear to have significant noise in their backgrounds. The subarcsecond (0.35') N-band spectrum in Gonz'alez-Mart'ın et al. (2013) shows a deep 10 𝜇 m silicate absorption feature which originates in the inner ∼ 80 pc of the AGN. NGC 4151 . The SOFIA images of NGC 4151 appear as a point-like source, with a potential detection of extended NGC 7469 15 5 NGC 7674 1 emission at PA ∼ 120 · at a 3 𝜎 level at 37 . 1 𝜇 m. Fuller et al. (2019) confirmed this elongation in the PSF-subtracted 37.1 𝜇 mimagecoincident with the NLR and radio axes. Radomski et al. (2003) show extended emission in 10.8 and 18.2 𝜇 m images that coincides with the NLR axis at PA ∼ -60 · . For ≥ 37 . 1 𝜇 m, we conclude that any extended emission due to NLR dust is within the FWHM of the SOFIA instruments. NGC4258 . NGC 4258 was not detected but we include the data here since it is part of the sample. It has been observed and analyzed in the N-band with Gemini/Michelle by Mason et al. (2012). These authors found a compact nucleus that is marginally resolved at 10 𝜇 m(FWHM ∼ 0 . 5 '' ). NGC 4388 . NGC 4388 is an edge-on spiral that shows the most interesting mid- to far-IR morphology in this study. Notably, in the 30 - 40 𝜇 mFORCAST images of NGC 4388, extension can be seen in the NE to SW direction at PA ∼ 40 · (see also Fuller et al. 2019), coincident with the NLR. This emission is seen on smaller scales at shorter wavelengths (Asmus et al. 2016; Garc'ıa-Bernete et al. 2016). The 53 𝜇 m image decreases in intensity and does not show a strong central core of emission as in the 31 . 5 -37 . 1 wavelength range. However, at longer wavelengths (89 -214 𝜇 m), host galaxy emission clearly dominates the images in the east-west direction at PA ∼ 90 · . NGC 4941 . NGC 4941 is a low-luminosity AGN that appears here as a faint point-like source in the 31 . 5 𝜇 m and 37 . 1 𝜇 m images, but brighter at 53 and 89 𝜇 m. Subarcsecond resolution N-band imaging on VLT/VISIR (Asmus et al. 2011) showed no significant extended MIR sources outside of the nucleus. NGC 5506 NGC 5506 appears as a bright point source in both the 31.5 and 37.1 𝜇 m filters. While the nucleus is unresolved, extended MIR emission has been detected up to a few arcseconds to the northeast at 11.9 𝜇 m (Raban et al. 2008). Extended emission in the north-south direction was detected in the N-band out to ∼ 560 pc, while faint extended emission towards the east in the Q-band was also detected (Garc'ıa-Bernete et al. 2016). However, the PSF-subtracted 12.27 𝜇 m2' × 2' VLT/VISIR image of Alonso-Herrero et al. (2021) shows that the PA of extended emission varies from 30 · in the central ∼ 0.5' to nearly 90 · in the outer regions. NGC 7465 . The 31.5 𝜇 m FORCAST image appears faint with a 3 𝜎 upper-limit in the 37.1 𝜇 m image. The 53 and 89 𝜇 m HAWC+ images here appear increasingly brighter, albeit as point-like sources. Cold molecular gas observations (Young et al. 2021) reveal that NGC 7465 is quite gas-rich, possibly from a recent merger. NGC 7469 . NGC 7469 appears as a very bright source in the 31.5 𝜇 m image with FWHM ∼ 4.3'. After PSF subtraction, Fuller et al. (2016) found extended emission in the north-south direction. This AGN is known to have a circumnuclear ring of star formation at a radius of ∼ 480 pc ( ∼ 1.4'; Ramos Almeida et al. 2011) in 8.7 and 18.3 𝜇 mimages taken on Gemini/T-ReCS. Recent JWST observations reveal prominent PAH emission, indicative of star formation, in the circumnuclear ring (Garc'ıa-Bernete et al. 2022b; Zhang & Ho 2023). NGC 7674 . The previously published (Fuller et al. 2016) FORCAST 31.5 𝜇 m image appears as a point-like source. Asmus et al. (2013) found that the nucleus of NGC 7674 is extended at PA ∼ 125 · at subarcsecond scale resolution, where the extension roughly aligns with the ionization cone.",
"pages": [
8,
9,
10,
11,
12,
13
]
},
{
"title": "4. NUCLEAR FLUX EXTRACTION",
"content": "We aim to construct well-sampled mid- to far-IR SEDs of the nuclear emission of AGN at scales of several arcseconds, depending on the PSF of the observation and possible extended emission. On these scales, we expect multiple dust sources (i.e. torus, star forming regions, dusty outflow), however disentangling these sources is beyond the scope of this imaging atlas. Because the images span a range of observing cycles, instruments, and observing modes, we analyzed each image individually. Of our sample, 17 objects appear visually as point sources. For these sources, we performed aperture photometry where the aperture size was set to be 2 × the FWHM at a given band. For objects that show host galaxy emission, we extract the central PSF to construct the SEDs as described below. We complement our SOFIA data with Herschel imaging data (see Appendix A) and use a similar analysis method to construct the full IR SEDs.",
"pages": [
13
]
},
{
"title": "4.1. Extended Sources: 2D Gaussian Fitting",
"content": "For sources with extended dust emission, we performed a two-component simultaneous fit to accurately model both the central source based on the PSF, and the host galaxy whose fit assumes an elongated 2D Gaussian profile. In order to supplement the SOFIA data for the full mid- to far-IR SEDs, we used a similar methodology with Herschel images. For the SOFIA images, the PSF used was based on the standard stars of the observing runs for each cycle as explained in Section 2.2. However, the same analysis could not be performed on Herschel images due to the threefold lobes associated with the instrument PSFs. To accommodate this, we compared three different PSF models to reproduce and fit the central source. Two of the PSFs were point source images while the third was an approximation of the theoretical instrumental PSF using a Gaussian profile. The fitting routine used four free parameters for the Gaussian profile: (1,2) the position in x and y of the PSF center according to the image center, (3) the amplitude of the PSF, (4) the fourth parameter was dependent on the PSF type used. For archival PSFs, this parameter represents the rotation angle that needs to be applied to the PSF to match the orientation of the image. For the Gaussian PSFs, this fourth parameter represents a scaling factor to the width of the Gaussian compared to its ideal value for a perfect instrument (1 . 22 × 𝜆 / 𝐷 ). The galaxy background is defined by 7 parameters: (1,2) the 2D Gaussian's center position (x0 and y0), (3,4) its width ( 𝜎 𝑥 and 𝜎 𝑦 ) in both directions, (5) its amplitude, and (6) its orientation on the image ( 𝜃 ). To these 6 parameters we added a constant background as a 7th free parameter. We combined these components and fit this simulated intensity map to the observed map using the 11 total free parameters (4 from the PSF and 7 from the 2D gaussian). We thus derived the parameters describing the best central source for our intensity maps, and then studied the properties extracted for the central source and removed it from the initial map to study the host galaxy itself. An example of this procedure is given in the Appendix in Figure 14.",
"pages": [
13
]
},
{
"title": "4.2. AGN and host galaxy contribution",
"content": "While most SOFIA images were treated as point sources, Centaurus A, Circinus, NGC 1068, and NGC 4388 all had significant host galaxy contamination that needed to be subtracted from at least some of the images. Figure 6 shows the PSF subtracted images of these sources and Table 4 gives the percentage of the contribution of the PSF to the total flux of the object. In several other objects, the shorter wavelength ( ∼ 30 - 100 𝜇 m) images did not show host galaxy contamination, but longer wavelength Herschel images show the colder extended dust. Because this is an atlas of SOFIA images, we include objects with host galaxy contamination only in Herschel images in Appendix C for completeness. The53 𝜇 mimageofCentaurusAcontains ( < 5%)emission from extended sources, so we performed aperture photometry to account for the central emission. At wavelengths ≳ 70 𝜇 m, the host galaxy substantially ( ∼ 56 -70%) contributes to the central AGN emission, so the extended emission was subtracted. The PSF of Centaurus A at these wavelengths contributes ∼ 35 %. This can be interpreted as the nucleus having a relatively constant IR contribution, so the brightness of the nucleus coincides with IR brightness of the host galaxy. The PSF contribution of the Circinus Galaxy decreases between 53 and 160 𝜇 m from 58 % to 35 %. At longer FIR wavelengths, the contribution of the PSF appears to be from the host galaxy and the fitting no longer provides information about the AGN. We interpret this as a decreasing IR contribution from the nucleus compared to the extended emission. For the completeness of the SOFIA Atlas presented here, we used the 19-53 𝜇 m images of NGC 1068 from LopezRodriguez et al. (2018). These datasets were analyzed as described in that study and here we only present the results and images in that wavelength range. The study showed that the fractional contribution from star formation increases from 20 - 50 𝜇 m, while extended emission from 200 K dust decreases. Emission from the torus peaks in this range, a result which is in agreement with Fuller et al. (2016) who found that the turnover in torus emission occurs at wavelengths ¿ 31.5 𝜇 m. Extended emission is observed here at all wavelengths ≳ 70 𝜇 m arising from dust in the host galaxy and star formation regions. For NGC 4388, we show the results of PSF subtraction at all wavelengths, but only use the results in wavelengths ¿ 40 𝜇 m for the SED. In the 30 - 40 𝜇 m range, the NE to SW extension is clear in the PSF-subtracted images. However, almost all of the extended emission lies within the FWHM of the observation; the FWHM of these images are only ∼ 10% greater than the FWHM of the PSF. Thus, while we show the PSF subtracted images of NGC 4388 here, for the SED we use the total 30 - 40 𝜇 mfluxes which encompass the apparent extended emission due to the NLR. The change in the extended emission source and morphology between 40 and 70 𝜇 m is clear in the PSF-subtracted images (Figure 6). The host galaxy clearly dominates the extended emission in the FIR while the NLR region dominates the extended emission in MIR wavelengths. The 53 𝜇 m HAWC+ image appears to show the transition between dominant extended sources. The contribution of the PSF in the images of NGC 4388 is ∼ 60 % in the 30 - 40 𝜇 m range, where the extended emission is in the NE to SW direction. The contribution then decreases drastically to ∼ 20 %. This reflects the turnover in extended emission seen in the images in Figure 5. The contribution of the PSF returns to ∼ 70 % between 70 - 100 𝜇 m, which suggests two separate but significant IR emission sources.",
"pages": [
13,
14
]
},
{
"title": "5. SPECTRAL ENERGY DISTRIBUTIONS",
"content": "Tables 5 (SOFIA) and 6 ( Herschel ) give the nuclear fluxes of the AGN in our sample along with their associated errors in units of Jy. The sources of uncertainty here are the instrument calibration, sky background, and the 2D gaussian fitting, where applicable. We estimate FORCAST and HAWC+ errors at ∼ 10%. We use PACS instrument errors at 5% /four.sup and SPIRE instrument errors as 5.5% /five.sup . The uncertainty due to sky background is determined on an individual basis, but averages ∼ 5%. The average uncertainty due to the 2-D gaussian fitting is ∼ 1.5%. We add these uncertainties in quadrature for the error bar estimation. The nuclear SEDs are shown in Figure 7 in 𝜈 F 𝜈 . The pink diamonds represent SOFIA observations while the black circles represent the complementary Herschel data. We obtained Spitzer /IRS spectra from the Sptizer /CASSIS database (Lebouteiller et al. 2011) for 21 of the 22 objects in our sample (solid black line). There was no spectrum available for NGC 7465. Low-resolution spectra (R ∼ 100) were obtained for 18 of the objects, while moderate resolution spectra (R ∼ 600) were available for Circinus, NGC 1068, and NGC 7674. This dataset provides the most completed SED coverage available between 30 - 500 𝜇 m. Decomposing the SEDs in this sample is outside the scope of this manuscript, as we are presenting an imaging atlas. Here, we provide the main results and features of the SEDs of these objects. The morphological changes seen in the extended emission source in Figure 5 for NGC 4388 are reflected in the SED at 53 𝜇 m, where there is a marked decline in the SED. The drastic decrease seems to be due the change of dominant emitting sources. The extended emission at wavelengths ≲ 40 𝜇 m is due to dust in the direction of the radio axis, and the extended emission at wavelengths ≳ 50 𝜇 mis due to the host galaxy. The wavelength of peak emission can give insight to the primary processes that drive MIR emission. The peak wavelength, determined by the highest flux from photometry and spectroscopy, ranges from 18 to 100 𝜇 m in 𝜈 F 𝜈 with an average of ∼ 40 𝜇 m. This average only includes the peak in continuum values and does not take into account fine structure lines. Most (73%; 11 out of 15) Seyfert 2 have a peak emission at wavelengths ≲ 40 𝜇 m. The SEDs of MCG-5-2316, Mrk3, Mrk 573, NGC 1068, NGC 3081, and NGC 4151 peak at ∼ 18 - 20 𝜇 m. This is in agreement with the correlation peak between the hard-X-rays and the mid-IR for Type 1 AGN in Garc'ıa-Bernete et al. (2017). The peak wavelengths in 𝐹 𝜈 (Jy) range ∼ 20 - 160 𝜇 m, with an average ∼ 93 𝜇 m. NGC 1068 is the only AGN to peak at the same wavelength in both sets of units. The 𝑆𝑝𝑖𝑡𝑧𝑒𝑟 spectrum for NGC 1068 does not align with the SOFIA photometry because of the extensive PSF subtraction that we performed in the photometry that was not accounted for in the spectroscopy. This is the only object that not only has overlapping 20 - 40 𝜇 m 𝑆𝑝𝑖𝑡𝑧𝑒𝑟 and SOFIA data, but also that has had the background emission subtracted at these wavelengths.",
"pages": [
14
]
},
{
"title": "5.1. Luminosity and Peak Wavelength",
"content": "To test whether the peak wavelength is a function of luminosity, we plot L 𝑏𝑜𝑙 vs 𝜆 𝑝𝑒𝑎𝑘 . Figure 8 shows the bolometric luminosities of the AGN plotted against the peak wavelength in the SEDs for both Sy1s, shown as red triangles, and Sy2s, shown as purple stars. The correlation coefficient between the luminosity and peak wavelength is | 𝑅 | ∼ 0.63 with statistical significance 𝑝 = 0 . 0015. While it is argued that | 𝑅 | of 0.6 - 0.7 may show moderate to strong correlation (see Section 3.2 in Messenger et al. 2013), a 𝑝 -value ≤ 0.05 is generally accepted as statistically significant. The data suggests that higher luminosity objects have SEDs that peak at shorter wavelengths, which indicates the presence of a hot dust component in the vicinity of the AGN.",
"pages": [
14,
15
]
},
{
"title": "5.2. Mid- to Far-IR Colors",
"content": "The ratio of F 𝜈 (70)/F 𝜈 (160) has been used as a proxy for dust temperature (Mel'endez et al. 2014; Garc'ıa-Gonz'alez et al. 2016), where the ratio is higher for dust heated by the AGN and lower for dust heated by star formation. Here we perform this analysis using the ratio F 𝜈 (31)/F 𝜈 (70) by using the 31.5 𝜇 mSOFIA data in our atlas. For objects that do not have data in the 31.5 𝜇 mfilter, we supplement that with data from the Spitzer /IRS continuum. NGC 7465 did not have 31.5 𝜇 m flux data, nor did it have Spitzer data so we leave that object out of this analysis. Using the fluxes in Table 5, we plot a color-color diagram in F 𝜈 in Figure 9. This figure also visually shows the peak wavelength from the SEDs (in 𝜈𝐹 𝜈 ). Longer peak wavelengths tend to cluster at F 𝜈 (70)/F 𝜈 (160) ∼ 1 and F 𝜈 (31)/F 𝜈 (70) between 0.25 - 0.5. In this sample we find an average F 𝜈 (70)/F 𝜈 (160) ratio of 1 . 4 ± 0 . 7. Previous studies (Mel'endez et al. 2014; Garc'ıaGonz'alez et al. 2016) with larger sample sizes (313 and 33, respectively) have found an average ratio of ∼ 0.8, albeit the data was analyzed using independent methods. This suggests a higher amount of AGN heated dust in our sample. We find that 18 objects have F 𝜈 (31)/F 𝜈 (70) ¡1, with an average F 𝜈 (31)/F 𝜈 (70) of 0 . 6 ± 0 . 3. The only object with both F 𝜈 (70)/F 𝜈 (160) and F 𝜈 (31)/F 𝜈 (70) ¿1 is MCG-5-23-16, and an SED that peaks ∼ 20 𝜇 m. This object may be the most AGN dominated source in our sample. The other objects that show a peak at ∼ 20 𝜇 min 𝜈 F 𝜈 still show F 𝜈 (31)/F 𝜈 (70) ¡1. Only one object, NGC 4388, shows F 𝜈 (31)/F 𝜈 (70) ¿1 while F 𝜈 (70)/F 𝜈 (160) ¡1. This reflects the change in extended emission seen in Figure 5. Half (11) of the objects in the sample show ratios F 𝜈 (31)/F 𝜈 (70) ¡1 while F 𝜈 (70)/F 𝜈 (160) ¿1. These objects (Circinus, Mrk 231, Mrk 573, NGC 1275, NGC 2110, NGC 3081, NGC 3227, NGC 3281, NGC 4151, NGC 4941, NGC 5506, NGC 7469) are likely AGN dominated. Six objects show F 𝜈 (31)/F 𝜈 (70) ¡1 and F 𝜈 (70)/F 𝜈 (160) ¡1, meaning that their SEDs peak at longer wavelengths. The emission from R/e.pc/f.pc/e.pc/r.pc/e.pc/n.pc/c.pc/e.pc/s.pc: a) Fuller et al. (2016), b) Fuller et al. (2019), c) Lopez-Rodriguez et al. (2018), d) Lopez-Rodriguez et al. (2020), e) LopezRodriguez et al. (2022b). *Lopez-Rodriguez et al. (2018) measured the flux of NGC 1068 at 19.7 𝜇 mto be 22.0 ± 1.4. these objects (Centaurus A, NGC 1068, NGC 2273, NGC 2992, NGC 4258, NGC 7674) are likely dominated by star formation.",
"pages": [
15,
16
]
},
{
"title": "6. CONCLUSIONS",
"content": "We have presented a SOFIA atlas of nearby AGN in the 20 - 215 𝜇 m wavelength range using FORCAST and HAWC+. We have released 69 observations of which 41 are newly published and 28 have been previously published (Fuller et al. 2016, 2019; Lopez-Rodriguez et al. 2018, 2022a). From these observations, NGC 4388 shows the most dramatic visual change in emission morphology. The 30 - 40 𝜇 m images show a NE to SW dusty extension associated with the NLR, while the ¿ 50 𝜇 mimages show a East to West dusty emission associated with the plane of the host galaxy. Our observations show that < 10 '' resolution 30 - 70 𝜇 m observations are crucial to disentangle the emitting contribution from AGN and host galaxy. We measured arcsecond scale unresolved nuclear fluxes in order to construct SEDs of the objects in our sample. We included complementary Herschel data to cover up to 500 𝜇 m. For point sources we used aperture photometry to determine the flux. For extended sources we used a 2D gaussian fitting method to extract the central unresolved source(s) of emission from the galaxy background. For this method, the PSF is scaled to represent the central emission while a 2D gaussian represents host galaxy or background emission. Based on the SEDs, we make the following conclusions: In future studies, we will combine data from this atlas with incoming data from JWST to update our IR datasets with the latest and highest resolution data available. Newly obtained JWST/MIRI observations will provide new higher angular resolution data for some of the sources in the wavelength range 5 - 28 𝜇 m.",
"pages": [
16,
17
]
},
{
"title": "ACKNOWLEDGMENTS",
"content": "We acknowledge Dr. Lucas Grosset for his effort in subtracting the image backgrounds. E.L.-R. is supported by the NASA/DLR Stratospheric Observatory for Infrared Astronomy (SOFIA) under the 08 0012 Program. SOFIA is jointly operated by the Universities Space Research Association,Inc.(USRA), under NASA contract NNA17BF53C, and the Deutsches SOFIA Institut (DSI) under DLR contract 50OK0901 to the University of Stuttgart. E.L.-R. is supported by the NASA Astrophysics Decadal Survey Precursor Science (ADSPS) Program (NNH22ZDA001NADSPS) with ID 22-ADSPS22-0009 and agreement number 80NSSC23K1585. I.G.B. acknowledges support from STFC through grants ST/S000488/1 and ST/W000903/1. C.R. acknowledges support from Fondecyt Regular grant 1230345 and ANID BASAL project FB210003. Software: /a.pc/s.pc/t.pc/r.pc/o.pc/p.pc/y.pc (Astropy Collaboration et al. 2022, 2018, 2013); S/c.pc/i.pcP/y.pc Virtanen et al. (2020)",
"pages": [
17
]
},
{
"title": "REFERENCES",
"content": "Alonso-Herrero, A., Ramos Almeida, C., Mason, R., et al. 2011, ApJ, 736, 82, doi: 10.1088/0004-637X/736/2/82 Alonso-Herrero, A., Ramos Almeida, C., Esquej, P., et al. 2014, MNRAS, 443, 2766, doi: 10.1093/mnras/stu1293 Alonso-Herrero, A., Esquej, P., Roche, P. F., et al. 2016, MNRAS, 455, 563, doi: 10.1093/mnras/stv2342 Alonso-Herrero, A., Pereira-Santaella, M., Garc´ıa-Burillo, S., et al. 2018, ApJ, 859, 144, doi: 10.3847/1538-4357/aabe30 Alonso-Herrero, A., Garc´ıa-Burillo, S., H¨onig, S. F., et al. 2021, A&A, 652, A99, doi: 10.1051/0004-6361/202141219 Antonucci, R. 1993, ARA&A, 31, 473, doi: 10.1146/annurev.aa.31.090193.002353 Antonucci, R. R. J., & Miller, J. S. 1985, ApJ, 297, 621, doi: 10.1086/163559 Asmus, D. 2019, MNRAS, 489, 2177, doi: 10.1093/mnras/stz2289 https://arxiv.org/abs/2207.09466 doi: 10.1088/0004-637X/794/2/152 Mor, R., Netzer, H., & Elitzur, M. 2009, ApJ, 705, 298, Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261, doi: 10.1038/s41592-019-0686-2 Young, L. M., Meier, D. S., Bureau, M., et al. 2021, ApJ, 909, 98, doi: 10.3847/1538-4357/abe126 Yuan, F., Markoff, S., Falcke, H., & Biermann, P. L. 2002, A&A, 391, 139, doi: 10.1051/0004-6361:20020817 Zhang, L., & Ho, L. C. 2023, ApJL, 953, L9, doi: 10.3847/2041-8213/acea73",
"pages": [
17,
18,
20,
21
]
},
{
"title": "A. HERSCHEL IMAGES",
"content": "We used images from the Herschel Archive to supplement SOFIA data in FIR wavelengths. Images from the PACS and SPIRE instruments covering the wavelength range 70 - 500 𝜇 m are shown in Figures 10, 11, 12, and 13. The white circle in the upper right indicates the beam size while the pink scale in the bottom left indicates a distance of 1 kpc. To extract nuclear fluxes on scales of several arcseconds, we use the methods described in Section 4. For point sources, we performed aperture photometry. For extended sources, we used the 2D gaussian routine outlined in Section 4.1.",
"pages": [
22
]
},
{
"title": "B. BACKGROUND FITTING",
"content": "Figure 14 shows an example of the 2-D gaussian fitting we used to extract the central flux from the background of the host galaxy in wavelengths 30 - 500 𝜇 m. The object used here is NGC 4388. In the first column, the original observation image is shown. The second column shows a model of the image using a PSF (Column 3) combined with the model background. Column 4 shows a PSF-subtracted image (Column 1 - Column 3) used to make the model background (Column 5). Column 6 shows the PSF- and background- subtracted image of the residuals.",
"pages": [
23
]
},
{
"title": "C. HOST GALAXY BACKGROUNDS HERSCHEL",
"content": "Here we show the results of PSF-subtracted images for objects in which only Herschel data showed host galaxy contamination.",
"pages": [
23
]
},
{
"title": "F/u.pc/l.pc/l.pc/e.pc/r.pc /e.pc/t.pc /a.pc/l.pc.",
"content": "19'00 50'00 52'30\" 51'30\" 50'40\" 51'00\" 52'00\" 24 30\" 73'30\" RA (J2000} (2000) 19'00 18'30 34*50' RA U2000) 19*53' 2*48' RA (J2000) RA (J2000) RA (J2000) 5*32 12'00\" 13'00\" RA (2000) RA (J2000) RA (J2000) RA (J2000) RA (J2000) RA (2000) 53'00 52'40\" 51'40\" 8*54' 8*54' 8*48' 8*48' 8*48 8*54'",
"pages": [
24,
25
]
}
]
2024arXiv241119306P
https://arxiv.org/pdf/2411.19306.pdf
<document>
<section_header_level_1><location><page_1><loc_22><loc_84><loc_74><loc_88></location>Methods to Characterise Exoplanet Host Stars from Spectroscopy</section_header_level_1>
<text><location><page_1><loc_22><loc_80><loc_34><loc_81></location>Carina M. Persson</text>
<text><location><page_1><loc_22><loc_55><loc_76><loc_65></location>Abstract A key to understand exoplanets is characterisation of their host stars. One of the most powerful tools to characterise stellar properties like effective temperature, surface gravity and metallicity, is spectroscopy based on observations of stellar atmospheres. This chapter describes the stellar parameters that can be derived from a spectrum with examples of well established methods and theoretical model atmospheres. Combined with photometry and parallax measurements, the outcome of the spectroscopic modelling can be used to derive stellar radii and masses.</text>
<section_header_level_1><location><page_1><loc_22><loc_50><loc_33><loc_51></location>Introduction</section_header_level_1>
<text><location><page_1><loc_22><loc_37><loc_76><loc_47></location>Exoplanets are intimately connected to their host stars through formation and evolution. In addition, detection and characterisation of exoplanets depend on detailed knowledge of their host stars since the current major detection techniques, transit photometry and the radial velocity (RV) method, detect planet sizes and masses relative to their host star (see Chapters by Deeg & Alonso and Wright in Volume 1 of the Handbook of Exoplanets). Uncertainties in a host star's parameters propagate directly to the planets.</text>
<text><location><page_1><loc_22><loc_33><loc_76><loc_37></location>Stellar modelling is normally based on two major techniques - photometry and spectroscopy. These methods are model-dependent in contrast to the direct measurements by interferometry, eclipsing binaries, and asteroseismology.</text>
<text><location><page_1><loc_22><loc_25><loc_76><loc_32></location>For eclipsing binaries (e.g. Andersen 1991; Torres et al. 2010; Serenelli et al. 2021), and for the few large and nearby stars that enable interferometric measurements (e.g. Quirrenbach 2001; Eisenhauer et al. 2023), it is possible to derive the stellar radius with an accuracy of a few per cent. The timing of the duration of the eclipses of eclipsing binaries and their orbital velocities allows accurate estimates</text>
<text><location><page_1><loc_22><loc_20><loc_76><loc_22></location>Chalmers University of Technology, department of Space, Earth, and Environment, Onsala space observatory, 439 92 ONSALA, Sweden, e-mail: carina.persson@chalmers.se</text>
<text><location><page_2><loc_22><loc_76><loc_76><loc_87></location>of their sizes. Similarly, stellar masses can be accurately determined for visual binaries from observed separations from the common centre-of-mass with Kepler's third law that relates their masses with observed orbital period and separation. Asteroseismology can also be used to derive stellar mass, radius, and age with a high precision (see Chapter by Lundkvist, Huber, Aguirre & Chaplin in this volume of the Handbook of Exoplanets). Recently, the seismic surface gravity of the star has also been used to obtain the effective temperature and metallicity, in particular with APOGEE (Ahumada et al. 2020, 2022).</text>
<text><location><page_2><loc_22><loc_65><loc_76><loc_75></location>These methods are, however, currently only possible to apply to a small subset of all stars. Spectroscopic measurements open a window to derive stellar parameters from a larger pool of stars than from direct methods. High-resolution spectroscopy is a powerful tool that provides a wealth of information; the effective temperature, surface gravity, chemical composition, and velocities. The stellar radius, and the luminosity, can then readily be derived from its spectral energy distribution (SED) via the spectroscopic parameters combined with photometry and parallax.</text>
<text><location><page_2><loc_22><loc_53><loc_76><loc_65></location>The spectroscopic parameters also serve as a base to model characteristics that normally cannot be directly measured, like mass and age, with stellar evolution models and the complementary tool isochrones (e.g. Dotter 2016; Hidalgo et al. 2018). An isochrone is an evolutionary track for a population of stars with different masses on the Hertzsprung-Russell diagram with the same ( iso ) age ( chrone ). The mass and radius can also be obtained from the spectroscopic parameters and empirical calibration equations (e.g. Torres et al. 2010; Enoch et al. 2010; Southworth 2011) albeit often with a higher uncertainty.</text>
<text><location><page_2><loc_22><loc_37><loc_76><loc_53></location>The downside of high-resolution spectroscopy is that it is expensive; the observations are time-consuming and also requires bright stars or a large collecting area. Modelling of stellar parameters has therefore traditionally been performed by photometry because a much larger number of stars can quickly be observed and analysed. The effective temperature of a star can for instance be derived from color-color diagrams which have been scaled to stars measured with direct methods (e.g. Bell and Gustafsson 1989; Alonso et al. 1996; Nordstrom et al. 2004). However, such models have in general higher uncertainties than based on spectroscopic measurements. In that respect they may only be reliable in a statistical sense and not for individual stars. Thus for exoplanet host stars (for which direct measurements are not applicable), high-resolution spectroscopy is preferred.</text>
<text><location><page_2><loc_22><loc_24><loc_76><loc_36></location>This chapter begins with a short summary of a few basic requirements in order to obtain spectroscopic measurements of a star. It continues with an overview of the parameters that can be extracted from a high-resolution spectrum and a few examples of well established methods. Some advantages and caveats are highlighted. The chapter ends with a brief summary of how to combine spectroscopic parameters, photometric measurements, and stellar evolution models to obtain stellar radii and masses. Modelling of stellar ages are described by Christensen-Dalsgaard & Aguirre in a Chapter in this volume of the Handbook of Exoplanets.</text>
<section_header_level_1><location><page_3><loc_22><loc_86><loc_52><loc_88></location>Instruments and spectral resolution</section_header_level_1>
<text><location><page_3><loc_22><loc_69><loc_76><loc_84></location>Important properties of a spectrum is high-resolution, high signal-to-noise (S/N), and wavelength coverage. There are several different types of spectrometers and many textbooks have been written about this topic including advantages and challenges for different types of instruments (e.g. Gray 2008). The most successfull type in observations of exoplanets is echelle spectrographs. The main advantages is the high resolution combined with a wide wavelength coverage obtained in a single exposure. A description of high-precision cross-dispersed echelle spectrographs for exoplanet research (CORAVEL, ELODIE, CORALIE, SOPHIE and HARPS) is found in a Chapter by Fransesco Pepe in this volume of the Handbook of Exoplanets.</text>
<text><location><page_3><loc_22><loc_66><loc_76><loc_69></location>For the purpose of spectroscopic modelling of host stars, we want high-resolution in order to resolve the spectral lines. The spectral resolution ∆λ at wavelength λ is</text>
<formula><location><page_3><loc_46><loc_62><loc_76><loc_65></location>∆λ λ = 1 R , (1)</formula>
<text><location><page_3><loc_22><loc_58><loc_76><loc_61></location>where R is the resolving power of the spectrograph. It can be translated into a velocity resolution according to</text>
<formula><location><page_3><loc_43><loc_54><loc_76><loc_57></location>∆ V = ∆λ λ c = c R . (2)</formula>
<text><location><page_3><loc_22><loc_46><loc_76><loc_53></location>Doppler broadening of spectral lines due to thermal and turbulent motions of absorbing species in the atmospheres produce line widths of ∼ 6 kms -1 for latetype stars. This corresponds to a spectral resolution of 50 000. Low-mass, slowly rotating stars can have line widths of only ≈ 1 -2 km s -1 which require R ≳ 300000 to resolve the spectral lines and disentangle blended lines.</text>
<text><location><page_3><loc_22><loc_21><loc_76><loc_45></location>In terms of high resolution, ultra-high precision, and long-term stability, the High Accuracy Radial Velocity Planet Searcher (HARPS; Mayor et al. 2003) and its decade younger sibling HARPS-North (Cosentino et al. 2012) mounted on the ESO 3.6 m telescope (La Silla observatory, Chile) and the Telescopio Nazionale Galileo (TNG) of Roque de los Muchachos Observatory (La Palma, Spain), respectively, have been the leading instruments in detecting exoplanets over the last two decades. Both HARPS instruments are fiber-fed cross-dispersed high-precision echelle spectrographs covering 380 nm to 690 nm. The spectroscopic resolution is approximately 115000 at visual wavelengths corresponding to a velocity resolution of 2.6 km s -1 . The design is based on experience with the groundbreaking ELODIE and CORALIE instruments where the former was used to detect the first exoplanet 51 Peg b (Mayor and Queloz 1995). Both HARPS spectrographs can be considered as the 'gold standard' when searching for exoplanets in RV data and are also used for characterisation of exoplanet host stars. When searching for exoplanets with the RV method, many measurements are collected, sometimes over a period over many years. The individual spectra can be co-added (after correcting for the peri-</text>
<text><location><page_4><loc_22><loc_85><loc_76><loc_87></location>odic changes in radial velocity) in order to increase the S/N enabling spectroscopic characterisation of the host star.</text>
<text><location><page_4><loc_22><loc_77><loc_76><loc_84></location>In addition to the HARPS spectrographs, there are many other instruments used for exoplanet detection that can also be used to characterise host stars e.g. ESPRESSO (Pepe et al. 2021), FIES (Telting et al. 2014), HIRES (Vogt et al. 1994), CHIRON (Tokovinin et al. 2013), TULL (Tull et al. 1995) with different resolutions operating at different wavelengths.</text>
<section_header_level_1><location><page_4><loc_22><loc_72><loc_52><loc_73></location>Stellar properties from spectroscopy</section_header_level_1>
<text><location><page_4><loc_22><loc_65><loc_76><loc_69></location>Only a few parameters characterise a stellar atmosphere: the effective temperature ( T eff ), the surface gravity (log g ⋆ ), the overall metal abundance ([M/H]), and atmospheric and rotational velocities.</text>
<text><location><page_4><loc_22><loc_43><loc_76><loc_65></location>The surface of a star is defined as the location where photons escape from the star which occurs at a characteristic optical depth of 2/3. This occurs within the photosphere, the innermost ≈ 500 km of a star's atmosphere which overlies the opaque interior. The photospheric temperature and density varies with depth and depends on the surface gravity as well as the abundances and opacity of the gases. The Sun's photosphere has a temperature that varies between 4 400 K and 6 600 K with an effective temperature of 5 772 K, while the density is approximately 3 × 10 -4 kg m -3 , increasing with depth into the Sun. Other stars may have hotter or cooler photospheres. Spectral absorption lines originate from different depths and opacities within the photosphere. Weak and optically thin spectral lines, and the line wings of optically thick lines, originate essentially in the same layer as the continuum. In contrast, the cores of optically thick saturated lines develop in higher layers. An example is the core of the hydrogen α line at 6562.81 ˚ A which originates in the hotter and less dense chromosphere which lies on top of the photosphere, where the assumption of local thermal equilibrium (LTE) is no longer valid.</text>
<section_header_level_1><location><page_4><loc_22><loc_38><loc_37><loc_39></location>Effective temperature</section_header_level_1>
<text><location><page_4><loc_22><loc_31><loc_76><loc_36></location>Stars are classified according to effective temperature from the hot O-stars, also called early-type stars, to the cool M-stars (late-type stars). If a star is on the main sequence, the effective temperature immediately signals which type of star it is along with typical mass and radius.</text>
<text><location><page_4><loc_22><loc_24><loc_76><loc_30></location>Instead of choosing a particular depth to define a star's surface temperature, the effective temperature is defined in terms of flux. The effective temperature is defined via the Stefan-Bolztmann law in terms of the total power per unit area, radiated by the star (e.g. Gray 2008)</text>
<formula><location><page_4><loc_43><loc_22><loc_76><loc_24></location>∫ ∞ 0 F ν d ν = σ T 4 eff . (3)</formula>
<text><location><page_5><loc_22><loc_82><loc_76><loc_88></location>Here F ν is the total flux passing through the star's surface and σ is the Boltzmann constant. The effective temperature is thus the temperature of a black body having the same power output per unit area as a star. It is related to the flux we measure at Earth, f ν , via (Gray 2008)</text>
<formula><location><page_5><loc_33><loc_77><loc_76><loc_80></location>∫ ∞ 0 f ν d ν = ( R ⋆ d ) 2 ∫ ∞ 0 F ν d ν = ( θ ⋆ 2 ) 2 σ T 4 eff , (4)</formula>
<text><location><page_5><loc_22><loc_51><loc_76><loc_76></location>where the left integral is the total radiative flux from the star received at the top of the Earth's atmosphere (bolometric flux), R ⋆ is the stellar radius, d is the distance, and θ ⋆ is the angular diameter of the star. The effective temperature can hence be derived by measuring the angular size of the star from interferometry and the received flux at Earth over a wide spectral range. By combining the angular size with distance from parallax measurements, the linear radius of the star can be inferred for a wide range of spectral types nearly model-independent except for the dependence on the adopted limb-darkening coefficients and bolometric correction. Alternatively, if the radius of the star is known from e.g. eclipsing binaries and the distance from parallax measurments, this will give the angular size which then can be used to compute T eff . Since the target stars have to be nearby to measure their angular sizes, interstellar absorption can be neglected. The infrared flux method (IRFM Blackwell and Shallis 1977; Blackwell et al. 1980; Bell and Gustafsson 1989; Blackwell et al. 1990; Casagrande et al. 2006, 2010) is based on observations of the angular size of the star and the measured infrared flux at the top of the Earth's atmosphere. The bolometric flux is derived taking into account the bolometric correction which gives the effective temperature.</text>
<text><location><page_5><loc_22><loc_43><loc_76><loc_50></location>In addition to the above methods, T eff can also be derived from spectroscopy as described in the following section. However, it is worth highlighting that the temperature derived from spectroscopy is a microscopic value which is close to, but not exactly the same as the effective temperature due to its definition being a macroscopic description.</text>
<section_header_level_1><location><page_5><loc_22><loc_38><loc_33><loc_40></location>Surface gravity</section_header_level_1>
<text><location><page_5><loc_22><loc_31><loc_76><loc_37></location>The surface gravity is an indication of the luminosity class of a star where V is the main sequence and I - IV is different types of giants. Thus surface gravity contains information of the size and age of a star. A low surface gravity immediately indicates that the star has left the main sequence.</text>
<text><location><page_5><loc_24><loc_29><loc_62><loc_30></location>The surface gravity of a star is defined by (e.g. Gray 2008)</text>
<formula><location><page_5><loc_45><loc_25><loc_76><loc_28></location>g ⋆ = g ⊙ M ⋆ R 2 ⋆ , (5)</formula>
<text><location><page_5><loc_22><loc_19><loc_76><loc_24></location>where g ⊙ is the surface gravity of the Sun (2 . 740 × 10 4 cm s -2 ) and the radius, R ⋆ , and mass, M ⋆ , of the star are in solar units. The surface gravity is commonly measured in a logarithmic scale, log g ⋆ , where the solar value is 4.44.</text>
<text><location><page_6><loc_22><loc_80><loc_76><loc_87></location>The surface gravity determines the gas density in the photosphere and is the spectroscopic parameter that has the highest impact on the stellar radius. Unfortunately, log g ⋆ is often poorly constrained by spectral analysis. Since the uncertainties of T eff and metallicity can be strongly correlated with surface gravity this can lead to a significant source of systematic error in some analysis techniques.</text>
<section_header_level_1><location><page_6><loc_22><loc_76><loc_30><loc_77></location>Metallicity</section_header_level_1>
<text><location><page_6><loc_22><loc_65><loc_76><loc_74></location>A stars chemical composition is an outcome of the nucleosynthesis by previous generations of stars. This is important when reconstructing star and planet formation history in our Galaxy. There is a large variation of metallicity, i.e. the abundance of all elements heavier than helium denoted with [M/H], in the Milky Way up to about twice the solar value to hundreds of thousands of times lower than the solar value (e.g. Christlieb et al. 2004; Li et al. 2022; Nepal et al. 2024).</text>
<text><location><page_6><loc_22><loc_50><loc_76><loc_65></location>The chemical composition of a star is commonly fixed to the overall metallicity of a star relative to the Sun. The Grevesse et al. (2007), Asplund et al. (2009) and Lodders (2003) abundance scales for the Sun are currently the most adopted. However, individual abundances of a star may not follow solar composition and may require modelling of individual elemental abundances. Abundances are generally measured on a logarithmic scale normalised to the Sun where zero equals the Sun's metallicity. In the case of iron we have [Fe/H] ⋆ = log(Fe/H) ⋆ - log(Fe/H) ⊙ . For example, an iron abundance of [Fe/H] = -0 . 5 means that the abundance is 10 -0 . 5 relative to the Sun. Since iron is by far the most abundant species in a stellar atmosphere after hydrogen and helium, measurements of iron has become a proxy for the metallicity.</text>
<text><location><page_6><loc_22><loc_38><loc_76><loc_50></location>Stellar abundances is also important when modelling exoplanet interiors in particular rocky super-Earths without significant gaseous envelopes. The degeneracy of interior composition inferred from radius and mass measurements can for this type of planet be reduced assuming an interior structure with a differentiated iron core and a rocky mantle. In these cases, the host star abundances is often used as a proxy of the primary planet-building elements Fe, Mg, and Si, which are expected to be reflected in the planet composition, planet interior, and core mass fraction (Dorn et al. 2015; Acu˜na et al. 2023).</text>
<section_header_level_1><location><page_6><loc_22><loc_34><loc_29><loc_35></location>Velocities</section_header_level_1>
<text><location><page_6><loc_22><loc_20><loc_76><loc_32></location>Thermal widths of spectral lines are only a fraction of the observed line widths for dwarf stars hotter than spectral type K0. The line widths are instead mainly governed by Doppler shifts produced by motions of the star's photospheric gases. The radial velocity of the star is only shifting the wavelengths of all spectral lines in the observed spectrum compared to the observations and can easily be corrected for. The velocity that dominates the line shape and width for hot stars is the projected equatorial rotational velocity of the star, V sin i ⋆ , where i ⋆ is the inclination of the stellar rotation axis relative to the line of sight. It can be measured via the full width</text>
<text><location><page_7><loc_22><loc_62><loc_76><loc_87></location>at half maximum (FWHM) of a large number of optically thin and unblended lines not sensitive to pressure broadening. The line shapes are, however, also affected by turbulence from convective motion, granulation, high-order pulsations, stellar activity, and other types of local flows in the photosphere. Turbulence is represented in the models by the macro-turbulent velocity ( V mac) that describes motions on scales larger than the mean free path within the photosphere that induce a change in the line shape; and the micro-turbulent velocity ( V mic ) that describes motions on scales smaller than the mean free path leading to increased line opacity (Gray 2008; Bruntt et al. 2010; Doyle et al. 2014). The latter velocity is a 'fudge' factor originally introduced to reconcile observed and predicted equivalent widths (Mihalas 1978). It includes all remaining types of broadening mechanisms and is at present standard to include in analyses of solar-type stars. Both turbulent velocities depend on temperature and to a lesser extent on surface gravity. The micro-turbulent velocity has a width of the order of 1 km s -1 for dwarfs and several km s -1 for giants. For lowmass stars, V mac and V sin i ⋆ have comparable widths of the order of a few km s -1 . As a reference, the Sun's V sin i ⋆ is 2 km s -1 at the equator while hotter stars have much higher rotational velocities (tens to hundreds of km s -1 ).</text>
<section_header_level_1><location><page_7><loc_22><loc_57><loc_53><loc_58></location>Methods for spectroscopic modelling</section_header_level_1>
<text><location><page_7><loc_22><loc_43><loc_76><loc_54></location>There are several ways to model a spectrum which can be divided into two main groups. The first is based on spectral synthesis. Here observations are fitted to a synthetic spectrum of stellar atmosphere models by comparison of line profiles. The second is a line-by-line analysis based on measured strengths of observed spectral lines, their equivalent widths (EWs). Detailed description of the physics can be found in many textbooks e.g. Gray (2008). Spectroscopic observations can also be compared to a library of spectra of well-characterised stars via for example interferometric measurements or spectroscopic binaries.</text>
<text><location><page_7><loc_22><loc_20><loc_76><loc_42></location>One major problem in spectroscopic analysis is to accurately determine the continuum which can introduce large errors. This is particularly difficult for poor spectra with low spectral resolution or low S/N. It also depends on the spectral type of the star and the wavelength region. The number of spectral lines increases towards shorter wavelengths for all types of stars. In addition, late-type stars have a much higher density of spectral lines than early-type stars arising from both atoms and molecules leading to blending and confusion of the continuum location. The higher temperatures of early-type stars ionize a large fraction of their atoms, leading to significantly fewer spectral features than low-mass stars. Differences in rotational velocities also affect the spectral line density. In contrast to late-type stars, the early types have very high rotational velocities which leads to very broad spectral lines that smear out spectral features. Thus solar-type stars (FGK) are the easiest stars to model, while high- and low-mass stars often entails a significantly higher degree of difficulty in the modelling. M-dwarfs have in addition generally a much longer period of high stellar activity than FGK stars, exacerbating the problems (e.g. Mignon</text>
<text><location><page_8><loc_22><loc_85><loc_76><loc_87></location>et al. 2023). This is unfortunate since M-dwarfs are popular exoplanet host stars due to their small masses and sizes which increase the exoplanet signals.</text>
<text><location><page_8><loc_22><loc_71><loc_76><loc_84></location>Care must be taken when selecting which spectral lines to model. A large set of narrow, non-blended spectral lines are preferred (unless modelling pressure broadened line wings, see below). If a spectral line becomes optically thick, the abundance of a species stops growing linearly with absorption depth. The characteristics of an optically thick line is a saturated line centre which flattens the bottom and broaden the line wings. Not all optically thin spectral lines may, however, be useful since a large number comes with poorly determined atomic parameters which are needed to compute synthetic spectra. This can be circumvented for solar-type stars if adopting new atomic parameters after comparing the lines from observations of the Sun.</text>
<section_header_level_1><location><page_8><loc_22><loc_66><loc_54><loc_67></location>Fitting observations to synthetic spectra</section_header_level_1>
<text><location><page_8><loc_22><loc_47><loc_76><loc_63></location>Computations of a synthetic spectrum requires a model atmosphere based on solutions to the stellar structure equations to synthesize a spectrum. Most stellar atmosphere models are pre-calculated and tabulated on grids describing the profiles of the temperature, surface gravity and abundances as functions of atmospheric depth. Each layer in the model atmosphere is contributing to the formation of absorption line profiles in the final spectrum. Some widely used atmospheric model atmospheres are Atlas12 (Kurucz 2013), Atlas9 (Kurucz 1993a; Heiter et al. 2002), MARCS (Gustafsson et al. 2008) for cool and giant stars, and LL models (Shulyak et al. 2004) for hot main sequence stars. Line lists of atomic and molecular data needed in the computation can be provided by the Vienna Atomic Line Database (VALD3; Piskunov et al. 1995; Ryabchikova et al. 2015).</text>
<text><location><page_8><loc_22><loc_38><loc_76><loc_47></location>A radiative transfer code then computes a synthetic model spectrum of the star for the chosen set of stellar parameters ( T eff , log g ⋆ , [Fe/H], V sin i ⋆ , V mic , V mac) which are matched against the observed spectra based on the spectral line shapes and strengths. The parameters generally affect either the line strength ( T eff , log g ⋆ , abundances, and V mic ) or the line shape ( V mac, V sin i ⋆ , and the instrumental resolution). Degeneracies are stronger within the subsets.</text>
<text><location><page_8><loc_22><loc_20><loc_76><loc_38></location>The dependence of the line profile on the V mac parameter is broadened line wings and a cusp-shaped core. Unfortunately, disentangling the effect on the line profile from V sin i ⋆ and V mac is difficult, leading to a degeneracy between the two. Prior information of V sin i ⋆ may be obtained from time-resolved photometry, available for the known transiting planets, or from asteroseismology (cf. Doyle et al. 2014). If no prior information is available, calibration equations of both turbulent velocities are often used (e.g. Bruntt et al. 2010; Doyle et al. 2014) which allows modelling of V sin i ⋆ . Note that in order to properly model the different velocities above, it is imperative to take into account how the spectrograph itself broadens the lines. Spectral lines can also be pressure broadened through various mechanisms which further broaden the lines, e.g. the Balmer lines. Details of the various broadening mechanisms can be found in many textbooks such as Gray (2008).</text>
<text><location><page_9><loc_22><loc_61><loc_76><loc_87></location>There are many softwares that computes synthetic spectra to be used as a model constraint to interpret the observed spectrum. The popular open-source spectroscopic tool iSpec (Blanco-Cuaresma et al. 2014; Blanco-Cuaresma 2019) support several of the most well-known radiative transfer codes such as Spectrocsopy Made Easy ( SME ; Valenti and Piskunov 1996; Piskunov and Valenti 2017; Wehrhahn et al. 2023), SPECTRUM (Gray and Corbally 1994), Turbospectrum (Plez 2012; de Laverny et al. 2012; Gerber et al. 2023), Synthe/WIDTH9 (Kurucz 1993b), and MOOG (Sneden 1973) where the latter is based on the equivalent width method. The codes often adopt LTE and a plane-parallel geometry as default motivated by the fact that for main-sequence stars, the photosphere constitutes ≪ 1 % of the stellar radius. Aspherically symmetric geometry is on the other hand required for giant stars where the atmosphere makes up a substantial portion of its radius (Heiter and Eriksson 2006). However, since complex interaction of gas particles and the non-local radiation fields leads to deviations from LTE in the atmospheres of FGKM-type stars, some of the softwares have also the option of non-local thermodynamic equilibrium (NLTE). For instance, SME includes NLTE departure coefficients for the MARCS and LL models atmospheres (Piskunov and Valenti 2017), and Turbospectrum also have the option of NLTE (Gerber et al. 2023).</text>
<text><location><page_9><loc_22><loc_49><loc_76><loc_60></location>Fitting can in several of the softwares be made for one or several parameters at the same time using a χ 2 -minimization algorithm. However, care must be taken when solving for several parameters simultaneously (Torres et al. 2012) due to degeneracies and difficulties in the modelling. Different initial assumptions of one free parameter at a time can therefore be made in the fitting process to iterate to the final solution, thereby mitigating degeneracies. Also, since the surface gravity is often difficult to constrain it is thus advantageous to have external information about the stellar density that can facilitate modelling.</text>
<text><location><page_9><loc_22><loc_27><loc_76><loc_48></location>Spectroscopic modelling can take advantage of the sensitivity of certain spectral lines to specific parameters. For instance, all Balmer lines exhibit pressure (collisionally) broadened line wings which makes their profiles strongly sensitive to temperature (e.g. Fuhrmann et al. 1993, 1994; Barklem et al. 2000, 2002) as shown in Fig. 1. The Balmer line wings are formed in the deepest photospheric layers likely close to the LTE. The metal-line blending increase from H α at 6562.81 ˚ A to H δ at 4101.75 ˚ A which perturb the higher transitions of the Balmer line profiles making H α the best choice. For late F, G, and early K-stars, H α is very insensitive to log g ⋆ and abundance making its line wings an excellent T eff indicator. Two examples of H α line profiles towards a K5V and a G0V host star are shown in Fig. 1. It is obvious that the line wings have almost completely disappeared in the K5V spectrum, while prominent and very broad towards the G-star. For M-dwarfs where the line wings of H α are absent, TiO lines can also be used as an T eff indicator (Valenti et al. 1998).</text>
<text><location><page_9><loc_22><loc_20><loc_76><loc_27></location>For late F- and G-dwarfs there are several pressure-sensitive lines that can be used to constrain log g ⋆ : the Mg I triplet at 5167.33 ˚ A, 5172.70 ˚ A and 5183.62 ˚ A (e.g. Fuhrmann et al. 1997; Valenti and Fischer 2005) as shown in Fig. 2, and the Ca I lines at 6122.23 ˚ A and 6162.18 ˚ A (e.g. Gray 2008). For lower gravity or higher temperature, the line wings disappear due to lower photospheric density or ionisa-</text>
<figure>
<location><page_10><loc_26><loc_54><loc_72><loc_88></location>
<caption>Fig. 1 Examples of SME modelling of H α using the MARCS (LTE) stellar atmosphere model towards a K5V (HD85512) and a G0V (HD164509) host star observed with HARPS. The observations are plotted in black and the models in blue. The line wings of H α are very sensitive to the effective temperature. Towards the K5V star they have almost disappeared, while they are very broad towards the G0V star.</caption>
</figure>
<text><location><page_10><loc_22><loc_26><loc_76><loc_44></location>he Mg I lines are, however, very wide which makes normalization difficult. Two of the lines lie very close without continuum between them, and there are numerous overlying narrow metal lines in particular for late-type stars. In these cases, only the 5183.62 ˚ A line can be used reliably. Figure 2 shows the Mg I triplet line profiles towards the same host stars as in Fig. 1. As already seen in Fig. 1, the spectral line density is much higher towards the K5V star than the G0V star. However, the effect is even more pronounciated at shorter wavelengths. In addition to surface gravity, the Ca and Mg pressure broadened line wings are, however, to some degree also sensitive to temperature and abundance. The procedure is therefore to model the temperature first, e.g. via H α , and the abundance via narrow, unblended Ca (e.g. 6156.02 ˚ A, 6166.439 ˚ A, 6169.042 ˚ A, 6455.985 ˚ A) and Mg (e.g. 5711.09 ˚ A) lines and iterate to a solution.</text>
<text><location><page_10><loc_22><loc_22><loc_76><loc_26></location>The broad line wings of Na I D-lines (5889.97 ˚ A and 5895.94 ˚ A) are sensitive to both T eff , log g ⋆ (and the Na abundance). Hence these lines can be used to check the final model for consistency.</text>
<figure>
<location><page_11><loc_26><loc_54><loc_72><loc_88></location>
<caption>Fig. 2 Examples of SME modelling of the Mg I triplet (5167.33 ˚ A, 5172.70 ˚ A, and 5183.62 ˚ A) towards the same stars as in Fig. 1. The line wings are sensitive to log g ⋆ . Note the difference in spectral line density and line blanketing in HD85512 which introduce severe problems in the spectral modelling.</caption>
</figure>
<text><location><page_11><loc_22><loc_42><loc_76><loc_45></location>More details and exampels of spectral modelling can for instance be found in Valenti and Fischer (2005), Jofr'e et al. (2014), and Brewer et al. (2016).</text>
<section_header_level_1><location><page_11><loc_22><loc_36><loc_42><loc_38></location>Fitting equivalent widths</section_header_level_1>
<text><location><page_11><loc_22><loc_27><loc_76><loc_34></location>In contrast to synthetic spectral synthesis methods, the equivalent width (EW) method begins with the observed spectrum by measuring the strengths of selected absorption lines which are translated into individual line abundances. The method is based on theoretical atmosphere models and excitation and ionisation equilibrium which determines the population in a certain level of an atom or an ion.</text>
<text><location><page_11><loc_22><loc_19><loc_76><loc_27></location>The EW of an absorption line is a convenient measurement of its strength. It is defined as the width of a rectangle reaching up to the continuum (with length one) having the same area as the spectral line. Measuring EWs of well-defined weak neutral iron (Fe I) and ionised iron (Fe II) lines is a traditional method to model stellar spectroscopic parameters since there are numerous iron lines in stellar spectra, many</text>
<text><location><page_12><loc_22><loc_82><loc_76><loc_87></location>with accurate atomic parameters. Metal lines can be very sensitive to temperature although large variations between lines exist. Depending on spectral type, neutral metals are often used as a temperature indicator in solar-type stars, while ionised metal lines are better tracers in early-type stars.</text>
<text><location><page_12><loc_22><loc_77><loc_76><loc_81></location>Assuming LTE, the ratio of population in two levels n and m of an atom (or ion) of a species can be computed with the Bolzmann equation (e.g. Gray 2008; Carroll and Ostlie 2017)</text>
<formula><location><page_12><loc_41><loc_74><loc_76><loc_77></location>Nn Nm = gn gm e -( χ n -χ m ) / kT , (6)</formula>
<text><location><page_12><loc_22><loc_69><loc_76><loc_74></location>where gn and gm are the statistical weights of the two levels, χ n and χ m are the corresponding excitation potentials, k is the Boltzmann constant, and T is the temperature. Different line ratios have different sensitivity to temperature.</text>
<text><location><page_12><loc_22><loc_63><loc_76><loc_69></location>As the temperature increases so will ionisation which occurs quite abruptly once the threshold temperature is reached. In a star's photosphere, elements exist mainly in just two ionisation stages which can be computed with the Saha equation (e.g. Gray 2008; Carroll and Ostlie 2017) between ionisation states i and i + 1 as</text>
<formula><location><page_12><loc_34><loc_59><loc_76><loc_62></location>N i + 1 Ni Pe = ( 2 π me ) 3 / 2 ( kT ) 5 / 2 h 3 2 u i + 1 ( T ) ui ( T ) e -I / kT , (7)</formula>
<text><location><page_12><loc_22><loc_53><loc_76><loc_57></location>where the electron pressure is Pe = NekT (indicator of the surface gravity), me is the electron mass, h is Planck's constant, u ( T ) = ∑ gi e -χ i / kT is the partition function, and I is the ionisation energy.</text>
<text><location><page_12><loc_22><loc_44><loc_76><loc_53></location>The EW method requires measurements of a large number of equivalent widths either measured by direct integration over the entire line or by fitting a Gaussian profile (or a Lorentzian profile that may fit optically thick lines better). Measuring EWs manually is, however, very time-consuming and therefore several softwares have been developed to automate the measurements for example ARES (Sousa et al. 2007) and DAOSPEC (Stetson and Pancino 2008).</text>
<text><location><page_12><loc_22><loc_32><loc_76><loc_44></location>Weak and optically thin iron lines depend mainly on T eff and the iron abundance and less on log g ⋆ and V mic . In order to avoid abundance dependence with the EW method, it is best to choose two lines from the same element. However, since continuum normalisation is often a large source of error it may sometimes be necessary to use pairs of lines at nearby wavelengths. In these cases, pairs of similar elements like Fe, V, and Ti that normally have similar abundances, are often used. Different sets of lines are chosen for different spectral types since they are useful in different temperature ranges.</text>
<text><location><page_12><loc_22><loc_23><loc_76><loc_32></location>The effective temperature is constrained in the following modelling by the correlation between the excitation potential and the iron abundance of each individual line, the microturbulence is constrained by the correlation between the abundance of each line and the reduced equivalent width, while the surface gravity is constrained by the ionisation balance. The parameters are adjusted until there are no correlations left and all individual abundances are the same (Sousa 2014).</text>
<text><location><page_12><loc_22><loc_20><loc_76><loc_23></location>An example of a widely used open radiative transfer code based on the EW method is MOOG (Sneden 1973). This software uses a grid of the ATLAS9 (Ku-</text>
<text><location><page_13><loc_22><loc_71><loc_76><loc_87></location>rucz 1993a) plane-parallel model atmospheres to produce model spectra which is compared to the measured EWs of individual Fe I and Fe II lines. The equilibrium conditions are solved simultaneously to derive T eff , log g ⋆ , [Fe/H], and V mic . A good description of the process is found in Sousa (2014). It has been used by for instance Sousa et al. (2021) together with ARES for a homogeneous spectroscopic characterisation of almost one thousand exoplanet host stars (the SWEETCat online catalogue). Another example of a software based on the EW method is ODUSSEAS (Observing Dwarfs Using Stellar Spectroscopic Energy-Absorption Shapes; Antoniadis-Karnavas et al. 2020). This code uses the machine learning Python package scikit-learn to offer a quick and automatic derivation of T eff and [Fe/H] for M dwarfs from optical spectroscopy.</text>
<text><location><page_13><loc_22><loc_50><loc_76><loc_71></location>When comparing the outcome of the EW and models based on synthetic spectra, T eff and log g ⋆ normally agree within the uncertainties and also with results from asteroseismology within 100 K and 0.1 dex, respectively. The methods, however, do not perform equally well for all spectral types. Since the EW method is based on differential analysis with respect to the Sun, it can be applied to FGK stars with T eff ≈ 4500 -6400 K (Sousa et al. 2011). Modelling based on synthetic spectra and the line shapes are also sensitive to spectral type since some of the fundamental traits, e.g. the broad H α line wings, can only be made for solar-type stars since the line wings disappear for colder and hotter stars. In these cases it may still, however, be possible to use other spectral lines (e.g. TiO for low-mass stars). Furthermore, the EWmethod cannot be used on fast-rotator stars because of the severe line blending. For these stars, it may still be possible to use the synthetic method. Another difference is that the EW method is normally much faster than the synthetic method, while synthetic models provide a more complete description of the star.</text>
<section_header_level_1><location><page_13><loc_22><loc_45><loc_37><loc_46></location>Empirical methods</section_header_level_1>
<text><location><page_13><loc_22><loc_24><loc_76><loc_42></location>Instead of the above methods, it is also possible to compare an observed highresolution spectrum to spectra of well-characterised stars and in this way derive stellar parameters. An example of such software is Spechmatch-emp (Yee et al. 2017) that compares an observed optical spectrum with a dense empirical spectral library thereby deriving R ⋆ (or log g ⋆ ), T eff , and [Fe/H]. The library contains 404 stars observed with the HIRES instrument with R ≈ 60000 at the Keck telescope. The properties of all the library stars have been derived from asteroseismoloy, interferometry, spectrophotometry and LTE spectral synthesis and represents spectral types from approximately M5 to F1. This method performs very well for the difficult late type stars (K4 stars and later) which are challenging for other methods reaching accuracies of 10 % in stellar radius, 70 K in effective temperature, and 0.09 dex in metallicity ([Fe/H]). The software and the library are publicly available.</text>
<section_header_level_1><location><page_14><loc_22><loc_86><loc_42><loc_88></location>Stellar radius and mass</section_header_level_1>
<text><location><page_14><loc_22><loc_81><loc_76><loc_84></location>Once we have obtained the stellar spectroscopic parameters for our host star we now want to find out the radius and the mass of the star.</text>
<section_header_level_1><location><page_14><loc_22><loc_76><loc_69><loc_77></location>Radius from spectroscopy and spectral energy distribution</section_header_level_1>
<text><location><page_14><loc_22><loc_50><loc_76><loc_73></location>When spectroscopic parameters and photometric measurements are available it is straightforward to obtain the stellar radius from a fit of the observed magnitudes in different bands to its spectral energy distribution (SED). In addition to magnitudes, the SED also depends on the distance (most accurately computed from the observed parallax), the spectroscopic parameters (derived from spectroscopic modelling), the extinction along the line-of-sight, and the stellar radius which is a free parameter in the fit. Parallax measurements have been performed by the European space missions Hipparchos (van Leeuwen 1997) and Gaia (Gaia Collaboration et al. 2016) launched in 1989 and 2013, respectively. The latter mission have provided the community with astrometric and photometric measurements of almost two billion stars in the Milky Way. Gaia has also delivered excellent measurements of the magnitudes in the Gaia optical band. Observations will end in early 2025. Figure 3 shows an example of a SED fit with the Phoenix (Husser et al. 2013) atmospheric model grid. An example of a publicly available software that automatically fits broadband photometry to six different stellar atmosphere models using Nested Sampling algorithms is the ARIADNE (Vines and Jenkins 2022) software.</text>
<section_header_level_1><location><page_14><loc_22><loc_44><loc_32><loc_45></location>Stellar mass</section_header_level_1>
<text><location><page_14><loc_22><loc_25><loc_76><loc_42></location>Unless the star is a binary, the mass cannot usually be determined directly but can be modeled via stellar evolution models. Some of the most popular models are BaSTI (Hidalgo et al. 2018), Padova (Bertelli et al. 2008, 2009), DESP (the Dartmouth Stellar Evolution Database; Dotter et al. 2008), MIST (MESA Isochrones and Stellar Tracks; Choi et al. 2016), PARSEC (the PAdova and TRieste Stellar Evolution Code; Bressan et al. 2012), and Y 2 (Yonsei-Yale; Yi et al. 2003; Demarque et al. 2004). A quick way to obtain a Bayesian estimation of both mass and radius based on PARSEC and MESA (Rodrigues et al. 2017) models are the web interfaces Param1.3 and Param1.5 (da Silva et al. 2006). Another publicly available software that provides a simple interface for MCMC fitting of MIST stellar model grids is isochrones (Morton 2015).</text>
<text><location><page_14><loc_22><loc_21><loc_76><loc_25></location>A complementing method is to use a set of stars with well-known masses and radii, e.g. through interferometry and eclipsing binaries, to derive empirical calibration equations. Such equations can give the mass and radius of a star given a set</text>
<figure>
<location><page_15><loc_25><loc_63><loc_73><loc_88></location>
<caption>Fig. 3 Example of a spectral energy distribution (SED) fit of the host star TOI-2196 discovered by the Transiting Exoplanet Survey Satellite (Persson et al. 2022). The observed photometric measurements are plotted with blue circles and the effective width of the passband are represented with horizontal bars. The Phoenix (Husser et al. 2013) atmosphere model is outlined in black and the magenta diamond symbols are the corresponding model fluxes.</caption>
</figure>
<text><location><page_15><loc_22><loc_50><loc_76><loc_53></location>of stellar atmosphere values (e.g. Torres et al. 2010; Enoch et al. 2010; Southworth 2011).</text>
<section_header_level_1><location><page_15><loc_22><loc_44><loc_32><loc_46></location>Final checks</section_header_level_1>
<text><location><page_15><loc_22><loc_36><loc_76><loc_42></location>If a planet is transiting a final important check can be made. The stellar density obtained from the above radius and mass can be checked against the density obtained from transit photometry and Kepler's third law. Assuming a circular orbit we have (Seager and Mall'en-Ornelas 2003)</text>
<formula><location><page_15><loc_42><loc_31><loc_76><loc_35></location>ρ ⋆ = 3 π GP 2 rot ( a R ⋆ ) 3 , (8)</formula>
<text><location><page_15><loc_22><loc_25><loc_76><loc_30></location>where G is the gravitational constant, P rot is the orbital period, a is the semi-major axis, and R ⋆ is the stellar radius. If this density does not agree with the above derived value, you need to check your modelling again. Equation 8 can be modified to include the eccentricity if known (e.g. Tingley et al. 2011).</text>
<text><location><page_15><loc_22><loc_20><loc_76><loc_24></location>Another checkpoint is to compare the stellar mass computed from the spectroscopic log g ⋆ combined with R ⋆ which should be consistent with the final mass from other methods.</text>
<text><location><page_16><loc_22><loc_85><loc_76><loc_87></location>A final remark: if possible, it is best to use several methods to make sure that the derived stellar parameters are robust and consistent.</text>
<section_header_level_1><location><page_16><loc_22><loc_79><loc_31><loc_81></location>References</section_header_level_1>
<table>
<location><page_16><loc_22><loc_19><loc_76><loc_78></location>
</table>
<unordered_list>
<list_item><location><page_17><loc_22><loc_85><loc_76><loc_87></location>Casagrande L, Ram'ırez I, Mel'endez J, Bessell M Asplund M (2010) An absolutely calibrated T e f f scale from the infrared flux method. Dwarfs and subgiants. A&A512:A54</list_item>
<list_item><location><page_17><loc_22><loc_83><loc_76><loc_85></location>Choi J, Dotter A, Conroy C et al. (2016) Mesa Isochrones and Stellar Tracks (MIST). I. Solarscaled Models. ApJ823(2):102</list_item>
<list_item><location><page_17><loc_22><loc_80><loc_76><loc_82></location>Christlieb N, Gustafsson B, Korn AJ et al. (2004) HE 0107-5240, a Chemically Ancient Star. I. A Detailed Abundance Analysis. ApJ603(2):708-728</list_item>
<list_item><location><page_17><loc_22><loc_76><loc_76><loc_80></location>Cosentino R, Lovis C, Pepe F et al. (2012) Harps-N: the new planet hunter at TNG. In: Groundbased and Airborne Instrumentation for Astronomy IV, Proc SPIE, vol 8446, p 84461V, DOI 10.1117/12.925738</list_item>
<list_item><location><page_17><loc_22><loc_74><loc_76><loc_76></location>da Silva L, Girardi L, Pasquini L et al. (2006) Basic physical parameters of a selected sample of evolved stars. A&A458:609-623</list_item>
<list_item><location><page_17><loc_22><loc_71><loc_76><loc_74></location>de Laverny P, Recio-Blanco A, Worley CC Plez B (2012) The AMBRE project: A new synthetic grid of high-resolution FGKM stellar spectra. A&A544:A126</list_item>
<list_item><location><page_17><loc_22><loc_69><loc_76><loc_71></location>Demarque P, Woo JH, Kim YC Yi SK (2004) Y 2 Isochrones with an Improved Core Overshoot Treatment. ApJS155(2):667-674</list_item>
</unordered_list>
<text><location><page_17><loc_22><loc_66><loc_76><loc_68></location>Dorn C, Khan A, Heng K et al. (2015) Can we constrain the interior structure of rocky exoplanets from mass and radius measurements? A&A577:A83</text>
<text><location><page_17><loc_22><loc_64><loc_76><loc_66></location>Dotter A (2016) MESA Isochrones and Stellar Tracks (MIST) 0: Methods for the Construction of Stellar Isochrones. ApJS222(1):8</text>
<text><location><page_17><loc_22><loc_63><loc_76><loc_63></location>Dotter A, Chaboyer B, Jevremovi'c D et al. (2008) The Dartmouth Stellar Evolution Database.</text>
<text><location><page_17><loc_24><loc_61><loc_33><loc_62></location>ApJS178:89-101</text>
<text><location><page_17><loc_22><loc_59><loc_76><loc_61></location>Doyle AP, Davies GR, Smalley B, Chaplin WJ Elsworth Y (2014) Determining stellar macroturbulence using asteroseismic rotational velocities from Kepler. MNRAS444:3592-3602</text>
<unordered_list>
<list_item><location><page_17><loc_22><loc_56><loc_76><loc_58></location>Eisenhauer F, Monnier JD Pfuhl O (2023) Advances in Optical/Infrared Interferometry. ARA&A61:237-285</list_item>
</unordered_list>
<text><location><page_17><loc_22><loc_55><loc_76><loc_56></location>Enoch B, Collier Cameron A, Parley NR Hebb L (2010) An improved method for estimating the</text>
<text><location><page_17><loc_24><loc_54><loc_54><loc_55></location>masses of stars with transiting planets. A&A516:A33</text>
<unordered_list>
<list_item><location><page_17><loc_22><loc_51><loc_76><loc_53></location>Fuhrmann K, Axer M Gehren T (1993) Balmer lines in cool dwarf stars. 1. Basic influence of atmospheric models. A&A271:451</list_item>
<list_item><location><page_17><loc_22><loc_49><loc_76><loc_51></location>Fuhrmann K, Axer M Gehren T (1994) Balmer lines in cool dwarf stars II. Effective temperatures and calibration of colour indices. A&A285:585-594</list_item>
<list_item><location><page_17><loc_22><loc_46><loc_76><loc_48></location>Fuhrmann K, Pfeiffer M, Frank C, Reetz J Gehren T (1997) The surface gravities of cool dwarf stars revisited. A&A323:909-922</list_item>
<list_item><location><page_17><loc_22><loc_45><loc_72><loc_46></location>Gaia Collaboration, Prusti T, de Bruijne JHJ et al. (2016) The Gaia mission. A&A595:A1</list_item>
<list_item><location><page_17><loc_22><loc_42><loc_76><loc_45></location>Gerber JM, Magg E, Plez B et al. (2023) Non-LTE radiative transfer with Turbospectrum. A&A669:A43</list_item>
<list_item><location><page_17><loc_22><loc_40><loc_76><loc_42></location>Gray DF (2008) The Observation and Analysis of Stellar Photospheres. Cambridge University Press</list_item>
<list_item><location><page_17><loc_22><loc_37><loc_76><loc_40></location>Gray RO Corbally CJ (1994) The Calibration of MK Spectral Classes Using Spectral Synthesis. I. The Effective Temperature Calibration of Dwarf Stars. AJ107:742</list_item>
<list_item><location><page_17><loc_22><loc_35><loc_76><loc_37></location>Grevesse N, Asplund M Sauval AJ (2007) The Solar Chemical Composition. Space Sci Rev130(14):105-114</list_item>
<list_item><location><page_17><loc_22><loc_32><loc_76><loc_35></location>Gustafsson B, Edvardsson B, Eriksson K et al. (2008) A grid of MARCS model atmospheres for late-type stars. I. Methods and general properties. A&A486:951-970</list_item>
<list_item><location><page_17><loc_22><loc_30><loc_76><loc_32></location>Heiter U Eriksson K (2006) Geometry of giant star model atmospheres: a consistency test. A&A452(3):1039-1048</list_item>
<list_item><location><page_17><loc_22><loc_26><loc_76><loc_29></location>Heiter U, Kupka F, van't Veer-Menneret C et al. (2002) New grids of ATLAS9 atmospheres I: Influence of convection treatments on model structure and on observable quantities. A&A392:619636</list_item>
<list_item><location><page_17><loc_22><loc_24><loc_76><loc_26></location>Hidalgo SL, Pietrinferni A, Cassisi S et al. (2018) The Updated BaSTI Stellar Evolution Models and Isochrones. I. Solar-scaled Calculations. ApJ856(2):125</list_item>
<list_item><location><page_17><loc_22><loc_21><loc_76><loc_23></location>Husser TO, Wende-von Berg S, Dreizler S et al. (2013) A new extensive library of PHOENIX stellar atmospheres and synthetic spectra. A&A553:A6</list_item>
<list_item><location><page_17><loc_22><loc_20><loc_76><loc_21></location>Jofr'e P, Heiter U, Soubiran C et al. (2014) Gaia FGK benchmark stars: Metallicity. A&A564:A133</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_18><loc_22><loc_84><loc_76><loc_87></location>Kurucz R (1993a) ATLAS9 Stellar Atmosphere Programs and 2 km/s grid. ATLAS9 Stellar Atmosphere Programs and 2 km/s grid Kurucz CD-ROM No 13 Cambridge, Mass: Smithsonian Astrophysical Observatory, 1993 13</list_item>
<list_item><location><page_18><loc_22><loc_80><loc_76><loc_84></location>Kurucz R (1993b) SYNTHE Spectrum Synthesis Programs and Line Data. SYNTHE Spectrum Synthesis Programs and Line Data Kurucz CD-ROM No 18 Cambridge, Mass: Smithsonian Astrophysical Observatory, 1993 18</list_item>
<list_item><location><page_18><loc_22><loc_78><loc_76><loc_80></location>Kurucz RL (2013) ATLAS12: Opacity sampling model atmosphere program. Astrophysics Source Code Library</list_item>
<list_item><location><page_18><loc_22><loc_75><loc_76><loc_77></location>Li H, Aoki W, Matsuno T et al. (2022) Four-hundred Very Metal-poor Stars Studied with LAMOST and Subaru. II. Elemental Abundances. ApJ931(2):147</list_item>
<list_item><location><page_18><loc_22><loc_73><loc_76><loc_75></location>Lodders K (2003) Solar System Abundances and Condensation Temperatures of the Elements. ApJ591(2):1220-1247</list_item>
<list_item><location><page_18><loc_22><loc_69><loc_76><loc_72></location>Mayor M Queloz D (1995) A Jupiter-mass companion to a solar-type star. Nature378:355-359 Mayor M, Pepe F, Queloz D et al. (2003) Setting New Standards with HARPS. The Messenger 114:20-24</list_item>
</unordered_list>
<text><location><page_18><loc_22><loc_65><loc_76><loc_68></location>Mignon L, Meunier N, Delfosse X et al. (2023) Characterisation of stellar activity of M dwarfs. I. Long-timescale variability in a large sample and detection of new cycles. A&A675:A168 Mihalas D (1978) Stellar atmospheres</text>
<unordered_list>
<list_item><location><page_18><loc_22><loc_64><loc_54><loc_65></location>Morton TD (2015) isochrones: Stellar model grid package</list_item>
<list_item><location><page_18><loc_22><loc_61><loc_76><loc_63></location>Nepal S, Chiappini C, Guiglion G et al. (2024) Insights from super-metal-rich stars: Is the Milky Way bar young? A&A681:L8</list_item>
</unordered_list>
<text><location><page_18><loc_22><loc_57><loc_76><loc_61></location>Nordstrom B, Mayor M, Andersen J et al. (2004) The Geneva-Copenhagen survey of the Solar neighbourhood. Ages, metallicities, and kinematic properties of ∼ 14 000 F and G dwarfs. A&A418:989-1019</text>
<text><location><page_18><loc_22><loc_55><loc_76><loc_57></location>Pepe F, Cristiani S, Rebolo R et al. (2021) ESPRESSO at VLT. On-sky performance and first results. A&A645:A96</text>
<text><location><page_18><loc_22><loc_52><loc_76><loc_55></location>Persson CM, Georgieva IY, Gandolfi D et al. (2022) TOI-2196 b: Rare planet in the hot Neptune desert transiting a G-type star. A&A666:A184</text>
<text><location><page_18><loc_22><loc_51><loc_68><loc_52></location>Piskunov N Valenti JA (2017) Spectroscopy Made Easy: Evolution. A&A597:A16</text>
<unordered_list>
<list_item><location><page_18><loc_22><loc_49><loc_76><loc_51></location>Piskunov NE, Kupka F, Ryabchikova TA, Weiss WW Jeffery CS (1995) VALD: The Vienna Atomic Line Data Base. A&AS112:525</list_item>
</unordered_list>
<text><location><page_18><loc_22><loc_46><loc_76><loc_48></location>Plez B (2012) Turbospectrum: Code for spectral synthesis. Astrophysics Source Code Library, record ascl:1205.004</text>
<text><location><page_18><loc_22><loc_45><loc_59><loc_46></location>Quirrenbach A (2001) Optical Interferometry. ARA&A39:353-401</text>
<text><location><page_18><loc_22><loc_42><loc_76><loc_45></location>Rodrigues TS, Bossini D, Miglio A et al. (2017) Determining stellar parameters of asteroseismic targets: going beyond the use of scaling relations. MNRAS467(2):1433-1448</text>
<text><location><page_18><loc_22><loc_40><loc_76><loc_42></location>Ryabchikova T, Piskunov N, Kurucz RL et al. (2015) A major upgrade of the VALD database. Phys Scr90(5):054005</text>
<text><location><page_18><loc_22><loc_37><loc_76><loc_40></location>Seager S Mall'en-Ornelas G (2003) A Unique Solution of Planet and Star Parameters from an Extrasolar Planet Transit Light Curve. ApJ585:1038-1055</text>
<unordered_list>
<list_item><location><page_18><loc_22><loc_35><loc_76><loc_37></location>Serenelli A, Weiss A, Aerts C et al. (2021) Weighing stars from birth to death: mass determination methods across the HRD. A&A Rev29(1):4</list_item>
<list_item><location><page_18><loc_22><loc_32><loc_76><loc_35></location>Shulyak D, Tsymbal V, Ryabchikova T, Stutz C Weiss WW (2004) Line-by-line opacity stellar model atmospheres. A&A428:993-1000</list_item>
<list_item><location><page_18><loc_22><loc_30><loc_76><loc_32></location>Sneden CA (1973) Carbon and Nitrogen Abundances in Metal-Poor Stars. PhD thesis, University of Texas, Austin</list_item>
<list_item><location><page_18><loc_22><loc_26><loc_76><loc_29></location>Sousa SG (2014) ARES + MOOG: A Practical Overview of an Equivalent Width (EW) Method to Derive Stellar Parameters. In: Determination of Atmospheric Parameters of B, pp 297-310, DOI 10.1007/978-3-319-06956-2 26</list_item>
</unordered_list>
<text><location><page_18><loc_22><loc_22><loc_76><loc_26></location>Sousa SG, Santos NC, Israelian G, Mayor M Monteiro MJPFG (2007) A new code for automatic determination of equivalent widths: Automatic Routine for line Equivalent widths in stellar Spectra (ARES). A&A469:783-791</text>
<text><location><page_19><loc_22><loc_84><loc_76><loc_87></location>Sousa SG, Santos NC, Israelian G, Mayor M Udry S (2011) Spectroscopic stellar parameters for 582 FGK stars in the HARPS volume-limited sample. Revising the metallicity-planet correlation. A&A533:A141</text>
<text><location><page_19><loc_22><loc_81><loc_76><loc_84></location>Sousa SG, Adibekyan V, Delgado-Mena E et al. (2021) SWEET-Cat 2.0: The Cat just got SWEETer. Higher quality spectra and precise parallaxes from Gaia eDR3. A&A656:A53</text>
<text><location><page_19><loc_22><loc_79><loc_76><loc_81></location>Southworth J (2011) Homogeneous studies of transiting extrasolar planets - IV. Thirty systems with space-based light curves. MNRAS417:2166-2196</text>
<text><location><page_19><loc_22><loc_76><loc_76><loc_79></location>Stetson PB Pancino E (2008) DAOSPEC: An Automatic Code for Measuring Equivalent Widths in High-Resolution Stellar Spectra. PASP120(874):1332</text>
<text><location><page_19><loc_22><loc_74><loc_76><loc_76></location>Telting JH, Avila G, Buchhave L et al. (2014) FIES: The high-resolution Fiber-fed Echelle Spectrograph at the Nordic Optical Telescope. Astronomische Nachrichten 335:41</text>
<text><location><page_19><loc_22><loc_71><loc_76><loc_74></location>Tingley B, Bonomo AS Deeg HJ (2011) Using Stellar Densities to Evaluate Transiting Exoplanetary Candidates. ApJ726(2):112</text>
<text><location><page_19><loc_22><loc_69><loc_76><loc_71></location>Tokovinin A, Fischer DA, Bonati M et al. (2013) CHIRON-A Fiber Fed Spectrometer for Precise Radial Velocities. PASP125(933):1336</text>
<text><location><page_19><loc_22><loc_66><loc_76><loc_68></location>Torres G, Andersen J Gim'enez A (2010) Accurate masses and radii of normal stars: modern results and applications. A&A Rev18(1-2):67-126</text>
<text><location><page_19><loc_22><loc_64><loc_76><loc_66></location>Torres G, Fischer DA, Sozzetti A et al. (2012) Improved Spectroscopic Parameters for Transiting Planet Hosts. ApJ757(2):161</text>
<text><location><page_19><loc_22><loc_61><loc_76><loc_63></location>Tull RG, MacQueen PJ, Sneden C Lambert DL (1995) The high-resolution cross-dispersed echelle white-pupil spectrometer of the McDonald Observatory 2.7-m telescope. PASP107:251-264</text>
<text><location><page_19><loc_22><loc_59><loc_76><loc_61></location>Valenti JA Fischer DA (2005) Spectroscopic Properties of Cool Stars (SPOCS). I. 1040 F, G, and K Dwarfs from Keck, Lick, and AAT Planet Search Programs. ApJS159:141-166</text>
<text><location><page_19><loc_22><loc_56><loc_76><loc_58></location>Valenti JA Piskunov N (1996) Spectroscopy made easy: A new tool for fitting observations with synthetic spectra. A&AS118:595-603</text>
<text><location><page_19><loc_22><loc_54><loc_76><loc_56></location>Valenti JA, Piskunov N Johns-Krull CM (1998) Spectral Synthesis of TiO Lines. ApJ498(2):851862</text>
<text><location><page_19><loc_22><loc_52><loc_65><loc_53></location>van Leeuwen F (1997) The HIPPARCOS Mission. Space Sci Rev81:201-409</text>
<text><location><page_19><loc_22><loc_50><loc_76><loc_52></location>Vines JI Jenkins JS (2022) ARIADNE: measuring accurate and precise stellar parameters through SED fitting. MNRAS513(2):2719-2731</text>
<text><location><page_19><loc_22><loc_45><loc_76><loc_50></location>Vogt SS, Allen SL, Bigelow BC et al. (1994) HIRES: the high-resolution echelle spectrometer on the Keck 10-m Telescope. In: Crawford DL Craine ER (eds) Instrumentation in Astronomy VIII, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol 2198, p 362, DOI 10.1117/12.176725</text>
<text><location><page_19><loc_22><loc_42><loc_76><loc_45></location>Wehrhahn A, Piskunov N Ryabchikova T (2023) PySME. Spectroscopy Made Easier. A&A671:A171</text>
<text><location><page_19><loc_22><loc_40><loc_76><loc_42></location>Yee SW, Petigura EA von Braun K (2017) Precision Stellar Characterization of FGKM Stars using an Empirical Spectral Library. ApJ836:77</text>
<unordered_list>
<list_item><location><page_19><loc_22><loc_39><loc_75><loc_40></location>Yi SK, Kim YC Demarque P (2003) The Y 2 Stellar Evolutionary Tracks. ApJS144(2):259-261</list_item>
</document>
[
{
"title": "Methods to Characterise Exoplanet Host Stars from Spectroscopy",
"content": "Carina M. Persson Abstract A key to understand exoplanets is characterisation of their host stars. One of the most powerful tools to characterise stellar properties like effective temperature, surface gravity and metallicity, is spectroscopy based on observations of stellar atmospheres. This chapter describes the stellar parameters that can be derived from a spectrum with examples of well established methods and theoretical model atmospheres. Combined with photometry and parallax measurements, the outcome of the spectroscopic modelling can be used to derive stellar radii and masses.",
"pages": [
1
]
},
{
"title": "Introduction",
"content": "Exoplanets are intimately connected to their host stars through formation and evolution. In addition, detection and characterisation of exoplanets depend on detailed knowledge of their host stars since the current major detection techniques, transit photometry and the radial velocity (RV) method, detect planet sizes and masses relative to their host star (see Chapters by Deeg & Alonso and Wright in Volume 1 of the Handbook of Exoplanets). Uncertainties in a host star's parameters propagate directly to the planets. Stellar modelling is normally based on two major techniques - photometry and spectroscopy. These methods are model-dependent in contrast to the direct measurements by interferometry, eclipsing binaries, and asteroseismology. For eclipsing binaries (e.g. Andersen 1991; Torres et al. 2010; Serenelli et al. 2021), and for the few large and nearby stars that enable interferometric measurements (e.g. Quirrenbach 2001; Eisenhauer et al. 2023), it is possible to derive the stellar radius with an accuracy of a few per cent. The timing of the duration of the eclipses of eclipsing binaries and their orbital velocities allows accurate estimates Chalmers University of Technology, department of Space, Earth, and Environment, Onsala space observatory, 439 92 ONSALA, Sweden, e-mail: carina.persson@chalmers.se of their sizes. Similarly, stellar masses can be accurately determined for visual binaries from observed separations from the common centre-of-mass with Kepler's third law that relates their masses with observed orbital period and separation. Asteroseismology can also be used to derive stellar mass, radius, and age with a high precision (see Chapter by Lundkvist, Huber, Aguirre & Chaplin in this volume of the Handbook of Exoplanets). Recently, the seismic surface gravity of the star has also been used to obtain the effective temperature and metallicity, in particular with APOGEE (Ahumada et al. 2020, 2022). These methods are, however, currently only possible to apply to a small subset of all stars. Spectroscopic measurements open a window to derive stellar parameters from a larger pool of stars than from direct methods. High-resolution spectroscopy is a powerful tool that provides a wealth of information; the effective temperature, surface gravity, chemical composition, and velocities. The stellar radius, and the luminosity, can then readily be derived from its spectral energy distribution (SED) via the spectroscopic parameters combined with photometry and parallax. The spectroscopic parameters also serve as a base to model characteristics that normally cannot be directly measured, like mass and age, with stellar evolution models and the complementary tool isochrones (e.g. Dotter 2016; Hidalgo et al. 2018). An isochrone is an evolutionary track for a population of stars with different masses on the Hertzsprung-Russell diagram with the same ( iso ) age ( chrone ). The mass and radius can also be obtained from the spectroscopic parameters and empirical calibration equations (e.g. Torres et al. 2010; Enoch et al. 2010; Southworth 2011) albeit often with a higher uncertainty. The downside of high-resolution spectroscopy is that it is expensive; the observations are time-consuming and also requires bright stars or a large collecting area. Modelling of stellar parameters has therefore traditionally been performed by photometry because a much larger number of stars can quickly be observed and analysed. The effective temperature of a star can for instance be derived from color-color diagrams which have been scaled to stars measured with direct methods (e.g. Bell and Gustafsson 1989; Alonso et al. 1996; Nordstrom et al. 2004). However, such models have in general higher uncertainties than based on spectroscopic measurements. In that respect they may only be reliable in a statistical sense and not for individual stars. Thus for exoplanet host stars (for which direct measurements are not applicable), high-resolution spectroscopy is preferred. This chapter begins with a short summary of a few basic requirements in order to obtain spectroscopic measurements of a star. It continues with an overview of the parameters that can be extracted from a high-resolution spectrum and a few examples of well established methods. Some advantages and caveats are highlighted. The chapter ends with a brief summary of how to combine spectroscopic parameters, photometric measurements, and stellar evolution models to obtain stellar radii and masses. Modelling of stellar ages are described by Christensen-Dalsgaard & Aguirre in a Chapter in this volume of the Handbook of Exoplanets.",
"pages": [
1,
2
]
},
{
"title": "Instruments and spectral resolution",
"content": "Important properties of a spectrum is high-resolution, high signal-to-noise (S/N), and wavelength coverage. There are several different types of spectrometers and many textbooks have been written about this topic including advantages and challenges for different types of instruments (e.g. Gray 2008). The most successfull type in observations of exoplanets is echelle spectrographs. The main advantages is the high resolution combined with a wide wavelength coverage obtained in a single exposure. A description of high-precision cross-dispersed echelle spectrographs for exoplanet research (CORAVEL, ELODIE, CORALIE, SOPHIE and HARPS) is found in a Chapter by Fransesco Pepe in this volume of the Handbook of Exoplanets. For the purpose of spectroscopic modelling of host stars, we want high-resolution in order to resolve the spectral lines. The spectral resolution ∆λ at wavelength λ is where R is the resolving power of the spectrograph. It can be translated into a velocity resolution according to Doppler broadening of spectral lines due to thermal and turbulent motions of absorbing species in the atmospheres produce line widths of ∼ 6 kms -1 for latetype stars. This corresponds to a spectral resolution of 50 000. Low-mass, slowly rotating stars can have line widths of only ≈ 1 -2 km s -1 which require R ≳ 300000 to resolve the spectral lines and disentangle blended lines. In terms of high resolution, ultra-high precision, and long-term stability, the High Accuracy Radial Velocity Planet Searcher (HARPS; Mayor et al. 2003) and its decade younger sibling HARPS-North (Cosentino et al. 2012) mounted on the ESO 3.6 m telescope (La Silla observatory, Chile) and the Telescopio Nazionale Galileo (TNG) of Roque de los Muchachos Observatory (La Palma, Spain), respectively, have been the leading instruments in detecting exoplanets over the last two decades. Both HARPS instruments are fiber-fed cross-dispersed high-precision echelle spectrographs covering 380 nm to 690 nm. The spectroscopic resolution is approximately 115000 at visual wavelengths corresponding to a velocity resolution of 2.6 km s -1 . The design is based on experience with the groundbreaking ELODIE and CORALIE instruments where the former was used to detect the first exoplanet 51 Peg b (Mayor and Queloz 1995). Both HARPS spectrographs can be considered as the 'gold standard' when searching for exoplanets in RV data and are also used for characterisation of exoplanet host stars. When searching for exoplanets with the RV method, many measurements are collected, sometimes over a period over many years. The individual spectra can be co-added (after correcting for the peri- odic changes in radial velocity) in order to increase the S/N enabling spectroscopic characterisation of the host star. In addition to the HARPS spectrographs, there are many other instruments used for exoplanet detection that can also be used to characterise host stars e.g. ESPRESSO (Pepe et al. 2021), FIES (Telting et al. 2014), HIRES (Vogt et al. 1994), CHIRON (Tokovinin et al. 2013), TULL (Tull et al. 1995) with different resolutions operating at different wavelengths.",
"pages": [
3,
4
]
},
{
"title": "Stellar properties from spectroscopy",
"content": "Only a few parameters characterise a stellar atmosphere: the effective temperature ( T eff ), the surface gravity (log g ⋆ ), the overall metal abundance ([M/H]), and atmospheric and rotational velocities. The surface of a star is defined as the location where photons escape from the star which occurs at a characteristic optical depth of 2/3. This occurs within the photosphere, the innermost ≈ 500 km of a star's atmosphere which overlies the opaque interior. The photospheric temperature and density varies with depth and depends on the surface gravity as well as the abundances and opacity of the gases. The Sun's photosphere has a temperature that varies between 4 400 K and 6 600 K with an effective temperature of 5 772 K, while the density is approximately 3 × 10 -4 kg m -3 , increasing with depth into the Sun. Other stars may have hotter or cooler photospheres. Spectral absorption lines originate from different depths and opacities within the photosphere. Weak and optically thin spectral lines, and the line wings of optically thick lines, originate essentially in the same layer as the continuum. In contrast, the cores of optically thick saturated lines develop in higher layers. An example is the core of the hydrogen α line at 6562.81 ˚ A which originates in the hotter and less dense chromosphere which lies on top of the photosphere, where the assumption of local thermal equilibrium (LTE) is no longer valid.",
"pages": [
4
]
},
{
"title": "Effective temperature",
"content": "Stars are classified according to effective temperature from the hot O-stars, also called early-type stars, to the cool M-stars (late-type stars). If a star is on the main sequence, the effective temperature immediately signals which type of star it is along with typical mass and radius. Instead of choosing a particular depth to define a star's surface temperature, the effective temperature is defined in terms of flux. The effective temperature is defined via the Stefan-Bolztmann law in terms of the total power per unit area, radiated by the star (e.g. Gray 2008) Here F ν is the total flux passing through the star's surface and σ is the Boltzmann constant. The effective temperature is thus the temperature of a black body having the same power output per unit area as a star. It is related to the flux we measure at Earth, f ν , via (Gray 2008) where the left integral is the total radiative flux from the star received at the top of the Earth's atmosphere (bolometric flux), R ⋆ is the stellar radius, d is the distance, and θ ⋆ is the angular diameter of the star. The effective temperature can hence be derived by measuring the angular size of the star from interferometry and the received flux at Earth over a wide spectral range. By combining the angular size with distance from parallax measurements, the linear radius of the star can be inferred for a wide range of spectral types nearly model-independent except for the dependence on the adopted limb-darkening coefficients and bolometric correction. Alternatively, if the radius of the star is known from e.g. eclipsing binaries and the distance from parallax measurments, this will give the angular size which then can be used to compute T eff . Since the target stars have to be nearby to measure their angular sizes, interstellar absorption can be neglected. The infrared flux method (IRFM Blackwell and Shallis 1977; Blackwell et al. 1980; Bell and Gustafsson 1989; Blackwell et al. 1990; Casagrande et al. 2006, 2010) is based on observations of the angular size of the star and the measured infrared flux at the top of the Earth's atmosphere. The bolometric flux is derived taking into account the bolometric correction which gives the effective temperature. In addition to the above methods, T eff can also be derived from spectroscopy as described in the following section. However, it is worth highlighting that the temperature derived from spectroscopy is a microscopic value which is close to, but not exactly the same as the effective temperature due to its definition being a macroscopic description.",
"pages": [
4,
5
]
},
{
"title": "Surface gravity",
"content": "The surface gravity is an indication of the luminosity class of a star where V is the main sequence and I - IV is different types of giants. Thus surface gravity contains information of the size and age of a star. A low surface gravity immediately indicates that the star has left the main sequence. The surface gravity of a star is defined by (e.g. Gray 2008) where g ⊙ is the surface gravity of the Sun (2 . 740 × 10 4 cm s -2 ) and the radius, R ⋆ , and mass, M ⋆ , of the star are in solar units. The surface gravity is commonly measured in a logarithmic scale, log g ⋆ , where the solar value is 4.44. The surface gravity determines the gas density in the photosphere and is the spectroscopic parameter that has the highest impact on the stellar radius. Unfortunately, log g ⋆ is often poorly constrained by spectral analysis. Since the uncertainties of T eff and metallicity can be strongly correlated with surface gravity this can lead to a significant source of systematic error in some analysis techniques.",
"pages": [
5,
6
]
},
{
"title": "Metallicity",
"content": "A stars chemical composition is an outcome of the nucleosynthesis by previous generations of stars. This is important when reconstructing star and planet formation history in our Galaxy. There is a large variation of metallicity, i.e. the abundance of all elements heavier than helium denoted with [M/H], in the Milky Way up to about twice the solar value to hundreds of thousands of times lower than the solar value (e.g. Christlieb et al. 2004; Li et al. 2022; Nepal et al. 2024). The chemical composition of a star is commonly fixed to the overall metallicity of a star relative to the Sun. The Grevesse et al. (2007), Asplund et al. (2009) and Lodders (2003) abundance scales for the Sun are currently the most adopted. However, individual abundances of a star may not follow solar composition and may require modelling of individual elemental abundances. Abundances are generally measured on a logarithmic scale normalised to the Sun where zero equals the Sun's metallicity. In the case of iron we have [Fe/H] ⋆ = log(Fe/H) ⋆ - log(Fe/H) ⊙ . For example, an iron abundance of [Fe/H] = -0 . 5 means that the abundance is 10 -0 . 5 relative to the Sun. Since iron is by far the most abundant species in a stellar atmosphere after hydrogen and helium, measurements of iron has become a proxy for the metallicity. Stellar abundances is also important when modelling exoplanet interiors in particular rocky super-Earths without significant gaseous envelopes. The degeneracy of interior composition inferred from radius and mass measurements can for this type of planet be reduced assuming an interior structure with a differentiated iron core and a rocky mantle. In these cases, the host star abundances is often used as a proxy of the primary planet-building elements Fe, Mg, and Si, which are expected to be reflected in the planet composition, planet interior, and core mass fraction (Dorn et al. 2015; Acu˜na et al. 2023).",
"pages": [
6
]
},
{
"title": "Velocities",
"content": "Thermal widths of spectral lines are only a fraction of the observed line widths for dwarf stars hotter than spectral type K0. The line widths are instead mainly governed by Doppler shifts produced by motions of the star's photospheric gases. The radial velocity of the star is only shifting the wavelengths of all spectral lines in the observed spectrum compared to the observations and can easily be corrected for. The velocity that dominates the line shape and width for hot stars is the projected equatorial rotational velocity of the star, V sin i ⋆ , where i ⋆ is the inclination of the stellar rotation axis relative to the line of sight. It can be measured via the full width at half maximum (FWHM) of a large number of optically thin and unblended lines not sensitive to pressure broadening. The line shapes are, however, also affected by turbulence from convective motion, granulation, high-order pulsations, stellar activity, and other types of local flows in the photosphere. Turbulence is represented in the models by the macro-turbulent velocity ( V mac) that describes motions on scales larger than the mean free path within the photosphere that induce a change in the line shape; and the micro-turbulent velocity ( V mic ) that describes motions on scales smaller than the mean free path leading to increased line opacity (Gray 2008; Bruntt et al. 2010; Doyle et al. 2014). The latter velocity is a 'fudge' factor originally introduced to reconcile observed and predicted equivalent widths (Mihalas 1978). It includes all remaining types of broadening mechanisms and is at present standard to include in analyses of solar-type stars. Both turbulent velocities depend on temperature and to a lesser extent on surface gravity. The micro-turbulent velocity has a width of the order of 1 km s -1 for dwarfs and several km s -1 for giants. For lowmass stars, V mac and V sin i ⋆ have comparable widths of the order of a few km s -1 . As a reference, the Sun's V sin i ⋆ is 2 km s -1 at the equator while hotter stars have much higher rotational velocities (tens to hundreds of km s -1 ).",
"pages": [
6,
7
]
},
{
"title": "Methods for spectroscopic modelling",
"content": "There are several ways to model a spectrum which can be divided into two main groups. The first is based on spectral synthesis. Here observations are fitted to a synthetic spectrum of stellar atmosphere models by comparison of line profiles. The second is a line-by-line analysis based on measured strengths of observed spectral lines, their equivalent widths (EWs). Detailed description of the physics can be found in many textbooks e.g. Gray (2008). Spectroscopic observations can also be compared to a library of spectra of well-characterised stars via for example interferometric measurements or spectroscopic binaries. One major problem in spectroscopic analysis is to accurately determine the continuum which can introduce large errors. This is particularly difficult for poor spectra with low spectral resolution or low S/N. It also depends on the spectral type of the star and the wavelength region. The number of spectral lines increases towards shorter wavelengths for all types of stars. In addition, late-type stars have a much higher density of spectral lines than early-type stars arising from both atoms and molecules leading to blending and confusion of the continuum location. The higher temperatures of early-type stars ionize a large fraction of their atoms, leading to significantly fewer spectral features than low-mass stars. Differences in rotational velocities also affect the spectral line density. In contrast to late-type stars, the early types have very high rotational velocities which leads to very broad spectral lines that smear out spectral features. Thus solar-type stars (FGK) are the easiest stars to model, while high- and low-mass stars often entails a significantly higher degree of difficulty in the modelling. M-dwarfs have in addition generally a much longer period of high stellar activity than FGK stars, exacerbating the problems (e.g. Mignon et al. 2023). This is unfortunate since M-dwarfs are popular exoplanet host stars due to their small masses and sizes which increase the exoplanet signals. Care must be taken when selecting which spectral lines to model. A large set of narrow, non-blended spectral lines are preferred (unless modelling pressure broadened line wings, see below). If a spectral line becomes optically thick, the abundance of a species stops growing linearly with absorption depth. The characteristics of an optically thick line is a saturated line centre which flattens the bottom and broaden the line wings. Not all optically thin spectral lines may, however, be useful since a large number comes with poorly determined atomic parameters which are needed to compute synthetic spectra. This can be circumvented for solar-type stars if adopting new atomic parameters after comparing the lines from observations of the Sun.",
"pages": [
7,
8
]
},
{
"title": "Fitting observations to synthetic spectra",
"content": "Computations of a synthetic spectrum requires a model atmosphere based on solutions to the stellar structure equations to synthesize a spectrum. Most stellar atmosphere models are pre-calculated and tabulated on grids describing the profiles of the temperature, surface gravity and abundances as functions of atmospheric depth. Each layer in the model atmosphere is contributing to the formation of absorption line profiles in the final spectrum. Some widely used atmospheric model atmospheres are Atlas12 (Kurucz 2013), Atlas9 (Kurucz 1993a; Heiter et al. 2002), MARCS (Gustafsson et al. 2008) for cool and giant stars, and LL models (Shulyak et al. 2004) for hot main sequence stars. Line lists of atomic and molecular data needed in the computation can be provided by the Vienna Atomic Line Database (VALD3; Piskunov et al. 1995; Ryabchikova et al. 2015). A radiative transfer code then computes a synthetic model spectrum of the star for the chosen set of stellar parameters ( T eff , log g ⋆ , [Fe/H], V sin i ⋆ , V mic , V mac) which are matched against the observed spectra based on the spectral line shapes and strengths. The parameters generally affect either the line strength ( T eff , log g ⋆ , abundances, and V mic ) or the line shape ( V mac, V sin i ⋆ , and the instrumental resolution). Degeneracies are stronger within the subsets. The dependence of the line profile on the V mac parameter is broadened line wings and a cusp-shaped core. Unfortunately, disentangling the effect on the line profile from V sin i ⋆ and V mac is difficult, leading to a degeneracy between the two. Prior information of V sin i ⋆ may be obtained from time-resolved photometry, available for the known transiting planets, or from asteroseismology (cf. Doyle et al. 2014). If no prior information is available, calibration equations of both turbulent velocities are often used (e.g. Bruntt et al. 2010; Doyle et al. 2014) which allows modelling of V sin i ⋆ . Note that in order to properly model the different velocities above, it is imperative to take into account how the spectrograph itself broadens the lines. Spectral lines can also be pressure broadened through various mechanisms which further broaden the lines, e.g. the Balmer lines. Details of the various broadening mechanisms can be found in many textbooks such as Gray (2008). There are many softwares that computes synthetic spectra to be used as a model constraint to interpret the observed spectrum. The popular open-source spectroscopic tool iSpec (Blanco-Cuaresma et al. 2014; Blanco-Cuaresma 2019) support several of the most well-known radiative transfer codes such as Spectrocsopy Made Easy ( SME ; Valenti and Piskunov 1996; Piskunov and Valenti 2017; Wehrhahn et al. 2023), SPECTRUM (Gray and Corbally 1994), Turbospectrum (Plez 2012; de Laverny et al. 2012; Gerber et al. 2023), Synthe/WIDTH9 (Kurucz 1993b), and MOOG (Sneden 1973) where the latter is based on the equivalent width method. The codes often adopt LTE and a plane-parallel geometry as default motivated by the fact that for main-sequence stars, the photosphere constitutes ≪ 1 % of the stellar radius. Aspherically symmetric geometry is on the other hand required for giant stars where the atmosphere makes up a substantial portion of its radius (Heiter and Eriksson 2006). However, since complex interaction of gas particles and the non-local radiation fields leads to deviations from LTE in the atmospheres of FGKM-type stars, some of the softwares have also the option of non-local thermodynamic equilibrium (NLTE). For instance, SME includes NLTE departure coefficients for the MARCS and LL models atmospheres (Piskunov and Valenti 2017), and Turbospectrum also have the option of NLTE (Gerber et al. 2023). Fitting can in several of the softwares be made for one or several parameters at the same time using a χ 2 -minimization algorithm. However, care must be taken when solving for several parameters simultaneously (Torres et al. 2012) due to degeneracies and difficulties in the modelling. Different initial assumptions of one free parameter at a time can therefore be made in the fitting process to iterate to the final solution, thereby mitigating degeneracies. Also, since the surface gravity is often difficult to constrain it is thus advantageous to have external information about the stellar density that can facilitate modelling. Spectroscopic modelling can take advantage of the sensitivity of certain spectral lines to specific parameters. For instance, all Balmer lines exhibit pressure (collisionally) broadened line wings which makes their profiles strongly sensitive to temperature (e.g. Fuhrmann et al. 1993, 1994; Barklem et al. 2000, 2002) as shown in Fig. 1. The Balmer line wings are formed in the deepest photospheric layers likely close to the LTE. The metal-line blending increase from H α at 6562.81 ˚ A to H δ at 4101.75 ˚ A which perturb the higher transitions of the Balmer line profiles making H α the best choice. For late F, G, and early K-stars, H α is very insensitive to log g ⋆ and abundance making its line wings an excellent T eff indicator. Two examples of H α line profiles towards a K5V and a G0V host star are shown in Fig. 1. It is obvious that the line wings have almost completely disappeared in the K5V spectrum, while prominent and very broad towards the G-star. For M-dwarfs where the line wings of H α are absent, TiO lines can also be used as an T eff indicator (Valenti et al. 1998). For late F- and G-dwarfs there are several pressure-sensitive lines that can be used to constrain log g ⋆ : the Mg I triplet at 5167.33 ˚ A, 5172.70 ˚ A and 5183.62 ˚ A (e.g. Fuhrmann et al. 1997; Valenti and Fischer 2005) as shown in Fig. 2, and the Ca I lines at 6122.23 ˚ A and 6162.18 ˚ A (e.g. Gray 2008). For lower gravity or higher temperature, the line wings disappear due to lower photospheric density or ionisa- he Mg I lines are, however, very wide which makes normalization difficult. Two of the lines lie very close without continuum between them, and there are numerous overlying narrow metal lines in particular for late-type stars. In these cases, only the 5183.62 ˚ A line can be used reliably. Figure 2 shows the Mg I triplet line profiles towards the same host stars as in Fig. 1. As already seen in Fig. 1, the spectral line density is much higher towards the K5V star than the G0V star. However, the effect is even more pronounciated at shorter wavelengths. In addition to surface gravity, the Ca and Mg pressure broadened line wings are, however, to some degree also sensitive to temperature and abundance. The procedure is therefore to model the temperature first, e.g. via H α , and the abundance via narrow, unblended Ca (e.g. 6156.02 ˚ A, 6166.439 ˚ A, 6169.042 ˚ A, 6455.985 ˚ A) and Mg (e.g. 5711.09 ˚ A) lines and iterate to a solution. The broad line wings of Na I D-lines (5889.97 ˚ A and 5895.94 ˚ A) are sensitive to both T eff , log g ⋆ (and the Na abundance). Hence these lines can be used to check the final model for consistency. More details and exampels of spectral modelling can for instance be found in Valenti and Fischer (2005), Jofr'e et al. (2014), and Brewer et al. (2016).",
"pages": [
8,
9,
10,
11
]
},
{
"title": "Fitting equivalent widths",
"content": "In contrast to synthetic spectral synthesis methods, the equivalent width (EW) method begins with the observed spectrum by measuring the strengths of selected absorption lines which are translated into individual line abundances. The method is based on theoretical atmosphere models and excitation and ionisation equilibrium which determines the population in a certain level of an atom or an ion. The EW of an absorption line is a convenient measurement of its strength. It is defined as the width of a rectangle reaching up to the continuum (with length one) having the same area as the spectral line. Measuring EWs of well-defined weak neutral iron (Fe I) and ionised iron (Fe II) lines is a traditional method to model stellar spectroscopic parameters since there are numerous iron lines in stellar spectra, many with accurate atomic parameters. Metal lines can be very sensitive to temperature although large variations between lines exist. Depending on spectral type, neutral metals are often used as a temperature indicator in solar-type stars, while ionised metal lines are better tracers in early-type stars. Assuming LTE, the ratio of population in two levels n and m of an atom (or ion) of a species can be computed with the Bolzmann equation (e.g. Gray 2008; Carroll and Ostlie 2017) where gn and gm are the statistical weights of the two levels, χ n and χ m are the corresponding excitation potentials, k is the Boltzmann constant, and T is the temperature. Different line ratios have different sensitivity to temperature. As the temperature increases so will ionisation which occurs quite abruptly once the threshold temperature is reached. In a star's photosphere, elements exist mainly in just two ionisation stages which can be computed with the Saha equation (e.g. Gray 2008; Carroll and Ostlie 2017) between ionisation states i and i + 1 as where the electron pressure is Pe = NekT (indicator of the surface gravity), me is the electron mass, h is Planck's constant, u ( T ) = ∑ gi e -χ i / kT is the partition function, and I is the ionisation energy. The EW method requires measurements of a large number of equivalent widths either measured by direct integration over the entire line or by fitting a Gaussian profile (or a Lorentzian profile that may fit optically thick lines better). Measuring EWs manually is, however, very time-consuming and therefore several softwares have been developed to automate the measurements for example ARES (Sousa et al. 2007) and DAOSPEC (Stetson and Pancino 2008). Weak and optically thin iron lines depend mainly on T eff and the iron abundance and less on log g ⋆ and V mic . In order to avoid abundance dependence with the EW method, it is best to choose two lines from the same element. However, since continuum normalisation is often a large source of error it may sometimes be necessary to use pairs of lines at nearby wavelengths. In these cases, pairs of similar elements like Fe, V, and Ti that normally have similar abundances, are often used. Different sets of lines are chosen for different spectral types since they are useful in different temperature ranges. The effective temperature is constrained in the following modelling by the correlation between the excitation potential and the iron abundance of each individual line, the microturbulence is constrained by the correlation between the abundance of each line and the reduced equivalent width, while the surface gravity is constrained by the ionisation balance. The parameters are adjusted until there are no correlations left and all individual abundances are the same (Sousa 2014). An example of a widely used open radiative transfer code based on the EW method is MOOG (Sneden 1973). This software uses a grid of the ATLAS9 (Ku- rucz 1993a) plane-parallel model atmospheres to produce model spectra which is compared to the measured EWs of individual Fe I and Fe II lines. The equilibrium conditions are solved simultaneously to derive T eff , log g ⋆ , [Fe/H], and V mic . A good description of the process is found in Sousa (2014). It has been used by for instance Sousa et al. (2021) together with ARES for a homogeneous spectroscopic characterisation of almost one thousand exoplanet host stars (the SWEETCat online catalogue). Another example of a software based on the EW method is ODUSSEAS (Observing Dwarfs Using Stellar Spectroscopic Energy-Absorption Shapes; Antoniadis-Karnavas et al. 2020). This code uses the machine learning Python package scikit-learn to offer a quick and automatic derivation of T eff and [Fe/H] for M dwarfs from optical spectroscopy. When comparing the outcome of the EW and models based on synthetic spectra, T eff and log g ⋆ normally agree within the uncertainties and also with results from asteroseismology within 100 K and 0.1 dex, respectively. The methods, however, do not perform equally well for all spectral types. Since the EW method is based on differential analysis with respect to the Sun, it can be applied to FGK stars with T eff ≈ 4500 -6400 K (Sousa et al. 2011). Modelling based on synthetic spectra and the line shapes are also sensitive to spectral type since some of the fundamental traits, e.g. the broad H α line wings, can only be made for solar-type stars since the line wings disappear for colder and hotter stars. In these cases it may still, however, be possible to use other spectral lines (e.g. TiO for low-mass stars). Furthermore, the EWmethod cannot be used on fast-rotator stars because of the severe line blending. For these stars, it may still be possible to use the synthetic method. Another difference is that the EW method is normally much faster than the synthetic method, while synthetic models provide a more complete description of the star.",
"pages": [
11,
12,
13
]
},
{
"title": "Empirical methods",
"content": "Instead of the above methods, it is also possible to compare an observed highresolution spectrum to spectra of well-characterised stars and in this way derive stellar parameters. An example of such software is Spechmatch-emp (Yee et al. 2017) that compares an observed optical spectrum with a dense empirical spectral library thereby deriving R ⋆ (or log g ⋆ ), T eff , and [Fe/H]. The library contains 404 stars observed with the HIRES instrument with R ≈ 60000 at the Keck telescope. The properties of all the library stars have been derived from asteroseismoloy, interferometry, spectrophotometry and LTE spectral synthesis and represents spectral types from approximately M5 to F1. This method performs very well for the difficult late type stars (K4 stars and later) which are challenging for other methods reaching accuracies of 10 % in stellar radius, 70 K in effective temperature, and 0.09 dex in metallicity ([Fe/H]). The software and the library are publicly available.",
"pages": [
13
]
},
{
"title": "Stellar radius and mass",
"content": "Once we have obtained the stellar spectroscopic parameters for our host star we now want to find out the radius and the mass of the star.",
"pages": [
14
]
},
{
"title": "Radius from spectroscopy and spectral energy distribution",
"content": "When spectroscopic parameters and photometric measurements are available it is straightforward to obtain the stellar radius from a fit of the observed magnitudes in different bands to its spectral energy distribution (SED). In addition to magnitudes, the SED also depends on the distance (most accurately computed from the observed parallax), the spectroscopic parameters (derived from spectroscopic modelling), the extinction along the line-of-sight, and the stellar radius which is a free parameter in the fit. Parallax measurements have been performed by the European space missions Hipparchos (van Leeuwen 1997) and Gaia (Gaia Collaboration et al. 2016) launched in 1989 and 2013, respectively. The latter mission have provided the community with astrometric and photometric measurements of almost two billion stars in the Milky Way. Gaia has also delivered excellent measurements of the magnitudes in the Gaia optical band. Observations will end in early 2025. Figure 3 shows an example of a SED fit with the Phoenix (Husser et al. 2013) atmospheric model grid. An example of a publicly available software that automatically fits broadband photometry to six different stellar atmosphere models using Nested Sampling algorithms is the ARIADNE (Vines and Jenkins 2022) software.",
"pages": [
14
]
},
{
"title": "Stellar mass",
"content": "Unless the star is a binary, the mass cannot usually be determined directly but can be modeled via stellar evolution models. Some of the most popular models are BaSTI (Hidalgo et al. 2018), Padova (Bertelli et al. 2008, 2009), DESP (the Dartmouth Stellar Evolution Database; Dotter et al. 2008), MIST (MESA Isochrones and Stellar Tracks; Choi et al. 2016), PARSEC (the PAdova and TRieste Stellar Evolution Code; Bressan et al. 2012), and Y 2 (Yonsei-Yale; Yi et al. 2003; Demarque et al. 2004). A quick way to obtain a Bayesian estimation of both mass and radius based on PARSEC and MESA (Rodrigues et al. 2017) models are the web interfaces Param1.3 and Param1.5 (da Silva et al. 2006). Another publicly available software that provides a simple interface for MCMC fitting of MIST stellar model grids is isochrones (Morton 2015). A complementing method is to use a set of stars with well-known masses and radii, e.g. through interferometry and eclipsing binaries, to derive empirical calibration equations. Such equations can give the mass and radius of a star given a set of stellar atmosphere values (e.g. Torres et al. 2010; Enoch et al. 2010; Southworth 2011).",
"pages": [
14,
15
]
},
{
"title": "Final checks",
"content": "If a planet is transiting a final important check can be made. The stellar density obtained from the above radius and mass can be checked against the density obtained from transit photometry and Kepler's third law. Assuming a circular orbit we have (Seager and Mall'en-Ornelas 2003) where G is the gravitational constant, P rot is the orbital period, a is the semi-major axis, and R ⋆ is the stellar radius. If this density does not agree with the above derived value, you need to check your modelling again. Equation 8 can be modified to include the eccentricity if known (e.g. Tingley et al. 2011). Another checkpoint is to compare the stellar mass computed from the spectroscopic log g ⋆ combined with R ⋆ which should be consistent with the final mass from other methods. A final remark: if possible, it is best to use several methods to make sure that the derived stellar parameters are robust and consistent.",
"pages": [
15,
16
]
},
{
"title": "References",
"content": "Dorn C, Khan A, Heng K et al. (2015) Can we constrain the interior structure of rocky exoplanets from mass and radius measurements? A&A577:A83 Dotter A (2016) MESA Isochrones and Stellar Tracks (MIST) 0: Methods for the Construction of Stellar Isochrones. ApJS222(1):8 Dotter A, Chaboyer B, Jevremovi'c D et al. (2008) The Dartmouth Stellar Evolution Database. ApJS178:89-101 Doyle AP, Davies GR, Smalley B, Chaplin WJ Elsworth Y (2014) Determining stellar macroturbulence using asteroseismic rotational velocities from Kepler. MNRAS444:3592-3602 Enoch B, Collier Cameron A, Parley NR Hebb L (2010) An improved method for estimating the masses of stars with transiting planets. A&A516:A33 Mignon L, Meunier N, Delfosse X et al. (2023) Characterisation of stellar activity of M dwarfs. I. Long-timescale variability in a large sample and detection of new cycles. A&A675:A168 Mihalas D (1978) Stellar atmospheres Nordstrom B, Mayor M, Andersen J et al. (2004) The Geneva-Copenhagen survey of the Solar neighbourhood. Ages, metallicities, and kinematic properties of ∼ 14 000 F and G dwarfs. A&A418:989-1019 Pepe F, Cristiani S, Rebolo R et al. (2021) ESPRESSO at VLT. On-sky performance and first results. A&A645:A96 Persson CM, Georgieva IY, Gandolfi D et al. (2022) TOI-2196 b: Rare planet in the hot Neptune desert transiting a G-type star. A&A666:A184 Piskunov N Valenti JA (2017) Spectroscopy Made Easy: Evolution. A&A597:A16 Plez B (2012) Turbospectrum: Code for spectral synthesis. Astrophysics Source Code Library, record ascl:1205.004 Quirrenbach A (2001) Optical Interferometry. ARA&A39:353-401 Rodrigues TS, Bossini D, Miglio A et al. (2017) Determining stellar parameters of asteroseismic targets: going beyond the use of scaling relations. MNRAS467(2):1433-1448 Ryabchikova T, Piskunov N, Kurucz RL et al. (2015) A major upgrade of the VALD database. Phys Scr90(5):054005 Seager S Mall'en-Ornelas G (2003) A Unique Solution of Planet and Star Parameters from an Extrasolar Planet Transit Light Curve. ApJ585:1038-1055 Sousa SG, Santos NC, Israelian G, Mayor M Monteiro MJPFG (2007) A new code for automatic determination of equivalent widths: Automatic Routine for line Equivalent widths in stellar Spectra (ARES). A&A469:783-791 Sousa SG, Santos NC, Israelian G, Mayor M Udry S (2011) Spectroscopic stellar parameters for 582 FGK stars in the HARPS volume-limited sample. Revising the metallicity-planet correlation. A&A533:A141 Sousa SG, Adibekyan V, Delgado-Mena E et al. (2021) SWEET-Cat 2.0: The Cat just got SWEETer. Higher quality spectra and precise parallaxes from Gaia eDR3. A&A656:A53 Southworth J (2011) Homogeneous studies of transiting extrasolar planets - IV. Thirty systems with space-based light curves. MNRAS417:2166-2196 Stetson PB Pancino E (2008) DAOSPEC: An Automatic Code for Measuring Equivalent Widths in High-Resolution Stellar Spectra. PASP120(874):1332 Telting JH, Avila G, Buchhave L et al. (2014) FIES: The high-resolution Fiber-fed Echelle Spectrograph at the Nordic Optical Telescope. Astronomische Nachrichten 335:41 Tingley B, Bonomo AS Deeg HJ (2011) Using Stellar Densities to Evaluate Transiting Exoplanetary Candidates. ApJ726(2):112 Tokovinin A, Fischer DA, Bonati M et al. (2013) CHIRON-A Fiber Fed Spectrometer for Precise Radial Velocities. PASP125(933):1336 Torres G, Andersen J Gim'enez A (2010) Accurate masses and radii of normal stars: modern results and applications. A&A Rev18(1-2):67-126 Torres G, Fischer DA, Sozzetti A et al. (2012) Improved Spectroscopic Parameters for Transiting Planet Hosts. ApJ757(2):161 Tull RG, MacQueen PJ, Sneden C Lambert DL (1995) The high-resolution cross-dispersed echelle white-pupil spectrometer of the McDonald Observatory 2.7-m telescope. PASP107:251-264 Valenti JA Fischer DA (2005) Spectroscopic Properties of Cool Stars (SPOCS). I. 1040 F, G, and K Dwarfs from Keck, Lick, and AAT Planet Search Programs. ApJS159:141-166 Valenti JA Piskunov N (1996) Spectroscopy made easy: A new tool for fitting observations with synthetic spectra. A&AS118:595-603 Valenti JA, Piskunov N Johns-Krull CM (1998) Spectral Synthesis of TiO Lines. ApJ498(2):851862 van Leeuwen F (1997) The HIPPARCOS Mission. Space Sci Rev81:201-409 Vines JI Jenkins JS (2022) ARIADNE: measuring accurate and precise stellar parameters through SED fitting. MNRAS513(2):2719-2731 Vogt SS, Allen SL, Bigelow BC et al. (1994) HIRES: the high-resolution echelle spectrometer on the Keck 10-m Telescope. In: Crawford DL Craine ER (eds) Instrumentation in Astronomy VIII, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol 2198, p 362, DOI 10.1117/12.176725 Wehrhahn A, Piskunov N Ryabchikova T (2023) PySME. Spectroscopy Made Easier. A&A671:A171 Yee SW, Petigura EA von Braun K (2017) Precision Stellar Characterization of FGKM Stars using an Empirical Spectral Library. ApJ836:77",
"pages": [
17,
18,
19
]
}
]
2024arXiv241119893Y
https://arxiv.org/pdf/2411.19893.pdf
<document>
<section_header_level_1><location><page_1><loc_10><loc_84><loc_90><loc_86></location>A Galaxy with an Extremely Blue UV Slope β = -3 at z = 9 . 25 Identified by JWST Spectroscopy: Evidence for a Weak Nebular Continuum and Efficient Ionizing Photon Escape?</section_header_level_1>
<text><location><page_1><loc_12><loc_78><loc_88><loc_82></location>Hiroto Yanagisawa , 1, 2 Masami Ouchi , 3, 1, 4, 5 Kimihiko Nakajima , 3 Yuichi Harikane , 1 Seiji Fujimoto , 6, 7, 8 Yoshiaki Ono , 1 Hiroya Umeda , 1, 2 Minami Nakane , 1, 2 Hidenobu Yajima , 9 Hajime Fukushima , 9 and Yi Xu 1, 10</text>
<text><location><page_1><loc_9><loc_70><loc_91><loc_77></location>1 Institute for Cosmic Ray Research, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan 2 Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan 3 National Astronomical Observatory of Japan, National Institutes of Natural Sciences, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 4 Department of Astronomical Science, SOKENDAI (The Graduate University for Advanced Studies), 2-21-1 Osawa, Mitaka, Tokyo, 181-8588, Japan</text>
<text><location><page_1><loc_11><loc_66><loc_89><loc_70></location>5 Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, Chiba 277-8583, Japan 6 Department of Astronomy, The University of Texas at Austin, Austin, TX 78712, USA 7 Cosmic Dawn Center (DAWN), Denmark</text>
<text><location><page_1><loc_20><loc_65><loc_79><loc_66></location>8 Niels Bohr Institute, University of Copenhagen, Lyngbyvej 2, DK2100 Copenhagen Ø, Denmark</text>
<text><location><page_1><loc_10><loc_62><loc_11><loc_63></location>10</text>
<text><location><page_1><loc_11><loc_62><loc_90><loc_64></location>9 Center for Computational Sciences, University of Tsukuba, Ten-nodai, 1-1-1 Tsukuba, Ibaraki 305-8577, Japan Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan</text>
<section_header_level_1><location><page_1><loc_45><loc_59><loc_55><loc_60></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_32><loc_86><loc_58></location>We investigate UV continuum slopes β of 974 galaxies at z = 4 -14 using archival JWST/NIRSpec PRISM spectra obtained from major JWST GTO, ERS, and GO programs, including JADES, CEERS, and UNCOVER. Among these galaxies, we identify a remarkable galaxy at z = 9 . 25, dubbed EBG-1, with a significantly blue UV slope β = -2 . 99 ± 0 . 15, unlike the rest of the galaxies that exhibit red continua or ambiguous blue continua hindered by large uncertainties. We confirm that the β value negligibly changes by the data reduction and fitting wavelength ranges for UV emission/absorption line masking. The extreme blue slope, β = -3 . 0, rules out significant contributions from dust extinction or AGN activity. Comparing with stellar and nebular emission models, we find that such a blue UV slope cannot be reproduced solely by stellar models even with very young, metal-poor, or top-heavy contiguous star formation associated with strong nebular continua making the UV slopes red, but with a high ionizing photon escape fraction, f ion esc ≳ 0 . 5, for a weak nebular continuum. While the H β emission line is not detected, likely due to the limited sensitivity of the spectrum, we find moderately weak [O iii ] λ 4959,5007 emission lines for the given star-formation rate (3 M ⊙ yr -1 ) and stellar mass (10 8 . 0 M ⊙ ) that are about three times weaker than the average emission lines, again suggestive of the high ionizing photon escape fraction, f ion esc ∼ 0 . 7 or more. EBG-1 would provide crucial insights into stellar and nebular continuum emission in high-redshift galaxies, serving as an example of the ionizing photon escaping site at the epoch of reionization.</text>
<text><location><page_1><loc_14><loc_27><loc_86><loc_28></location>Keywords: Ultraviolet color; Reionization; Galaxy evolution; Galaxy formation; High-redshift galaxies</text>
<section_header_level_1><location><page_1><loc_20><loc_22><loc_36><loc_23></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_11><loc_48><loc_21></location>The first galaxies initiate star formation in the early universe, with young massive stars exhibiting blue UV continua and producing substantial amounts of ionizing photons. Although the ionizing photons are mainly used to ionize the interstellar medium, a fraction of the ionizing photons escape from the galaxy and ionize the intergalactic medium, driving the cosmic reionization.</text>
<text><location><page_1><loc_52><loc_17><loc_92><loc_23></location>The escape of the hydrogen ionizing photons is characterized by an escape fraction, f ion esc , which is the key quantity to understand how the ionizing photons of the galaxies contribute to the cosmic reionization.</text>
<text><location><page_1><loc_52><loc_9><loc_92><loc_17></location>The UV continuum slope β ( f ( λ ) ∝ λ β ) is an important indicator of the production and escape of ionizing photons. Young stellar populations produce β < -2, while dust extinction and active galactic nucleus (AGN) provide a red ( β ≳ -2) UV continuum (e.g., Bouwens</text>
<text><location><page_2><loc_8><loc_82><loc_48><loc_91></location>et al. 2012; Finkelstein et al. 2012). Typically the β values are larger than ∼ -2 . 6 (Chisholm et al. 2022), because an intrinsically blue UV continuum intensely ionizes the nebula, leading to a significant contribution from a nebular continuum, which has β ≳ -2 (Katz et al. 2024; Cameron et al. 2024).</text>
<text><location><page_2><loc_8><loc_68><loc_48><loc_82></location>However, without the nebular continuum, the β values can be as low as -3 . 4 (e.g., Bouwens et al. 2010). This situation is achieved if the escape fraction of the ionizing photon f ion esc is large, because in that case the nebula is less ionized and the nebular continuum has less contribution to the UV spectrum (Zackrisson et al. 2017) . It is thus important to search for galaxies with β < -2 . 6 because such blue galaxies may have extremely high f ion esc , which contribute to the cosmic reionization.</text>
<text><location><page_2><loc_8><loc_56><loc_48><loc_68></location>Because galaxies with such high f ion esc are probably rare (Leitet et al. 2013; Matthee et al. 2017), it is necessary to search large sample of galaxies. Photometric studies are conducted with JWST data by Topping et al. (2022, 2024); Morales et al. (2024); Cullen et al. (2023, 2024), although deriving β using photometry suffers from contamination by emission lines. The accurate measurement of β requires the high quality spectroscopic data.</text>
<text><location><page_2><loc_8><loc_42><loc_48><loc_55></location>In this work, we search a large spectroscopic sample of galaxies provided by the DAWN JWST Archive for a galaxy with a blue UV slope, and report a galaxy at z = 9 . 25 that have an extremely blue UV slope. In Section 2, we describe our sample and method of UV slope measurements. In Section 3, we present the results the extremely blue object. We discuss physical origins of the extremely blue UV continuum in Section 4. Section 5 summarizes our results.</text>
<section_header_level_1><location><page_2><loc_10><loc_39><loc_46><loc_40></location>2. SEARCH FOR BLUE UV SLOPE OBJECTS</section_header_level_1>
<section_header_level_1><location><page_2><loc_16><loc_36><loc_40><loc_38></location>2.1. Lower Limit of β for f ion esc = 0</section_header_level_1>
<text><location><page_2><loc_8><loc_9><loc_48><loc_36></location>We first define a quantitative criterion for a blue UV continuum. We calculate β assuming f ion esc = 0 using Cloudy version 23.01 (Gunasekera et al. 2023) with incident radiations of Kroupa IMF (Kroupa 2001) with a mass range of 0 . 1 -100 M ⊙ provided by BPASS v2.2.1 (Stanway & Eldridge 2018), top-heavy IMF with a mass range of 50 -500 M ⊙ taken from Yggdrasil Pop III.1 model (Zackrisson et al. 2011), and blackbody. We assume a hydrogen density of n e = 10 2 cm -3 , nebular metallicity of log Z neb / Z ⊙ = -2, number of ionizing photon Q (H) = 10 50 s -1 , and inner radius of 10 14 cm. We assume stellar metallicity of log Z star / Z ⊙ = -2 for the Kroupa IMF model, while Z star = 0 is used for top-heavy IMF. Figure 1 shows time evolution of β values. Although the incident radiation of the top-heavy IMF is bluer than that of the Kroupa IMF, the topheavy IMF+nebular continuum model is redder than the</text>
<figure>
<location><page_2><loc_53><loc_70><loc_91><loc_92></location>
<caption>Figure 1. UV slope β as a function of age calculated with Cloudy. The blue, green, and gray lines present the incident radiations of top-heavy IMF (Zackrisson et al. 2011), Kroupa IMF (Kroupa 2001), and blackbody, respectively. The solid lines denote the models with f ion esc = 0 (i.e., the ionizing photons are completely consumed to ionize the nebula), while the dashed lines represent the models with f ion esc = 1 (i.e., the ionizing photons are not used to ionize the nebula). For the blackbody models, the blackbody temperatures are converted into the age using the typical lifetimes of stars having the same temperature.</caption>
</figure>
<text><location><page_2><loc_52><loc_39><loc_92><loc_51></location>Kroupa+nebular continuum model. This is because the intrinsically blue incident radiation in top-heavy IMF model intensely ionize the nebula, which leads to a significant contribution from the red nebular continuum. One can also see that the lower limit of β value for f ion esc = 0 is -2 . 6, while the β value reach as low as -3 . 4 if nebular continuum is not included. We thus define a criterion for a blue UV continuum as β = -2 . 6.</text>
<section_header_level_1><location><page_2><loc_66><loc_36><loc_78><loc_37></location>2.2. Our Sample</section_header_level_1>
<text><location><page_2><loc_52><loc_20><loc_92><loc_35></location>We use 974 PRISM/CLEAR spectra of galaxies provided by the DAWN JWST Archive (DJA) 1 , which compile the major JWST GTO, ERS, and GO programs. 2 We select the galaxies within the redshift range of 4 < z < 14, whose UV continua are covered by NIRSpec. The spectra in the DJA are reduced with msaexp (Brammer 2023; Heintz et al. 2024). Hereafter, these 974 galaxies are referred to as our sample. The redshift distribution of galaxies in our sample is shown in the top panel of Figure 2.</text>
<section_header_level_1><location><page_2><loc_62><loc_17><loc_83><loc_18></location>2.3. UV Slope Measurements</section_header_level_1>
<figure>
<location><page_3><loc_9><loc_60><loc_47><loc_92></location>
<caption>Figure 2. (Top) Redshift distribution of our sample. (Bottom) Fitted β value as a function of redshift. The blue points represent the galaxies in our sample. The β values with unreliably large error ( > 0 . 5) are omitted in this figure. The red point indicates EBG-1. The dotted line denote the lower limit of β if f ion esc = 0 is assumed.</caption>
</figure>
<text><location><page_3><loc_10><loc_47><loc_47><loc_48></location>To derive β , we fit a galaxy spectra with a function</text>
<formula><location><page_3><loc_23><loc_44><loc_48><loc_45></location>f ( λ ) = Aλ β , (1)</formula>
<text><location><page_3><loc_8><loc_37><loc_48><loc_43></location>where A is a constant for normalization and λ is wavelengths. We employ a Markov Chain Monte Carlo (MCMC) method for the fitting using emcee (ForemanMackey et al. 2013). We minimize a likelihood function</text>
<formula><location><page_3><loc_8><loc_30><loc_48><loc_35></location>log( L ) = -1 2 ∑ λ [ ( f mod ( λ ) -f obs ( λ ) σ ( λ ) ) 2 +log( σ ( λ ) 2 ) ] , (2)</formula>
<text><location><page_3><loc_8><loc_9><loc_48><loc_29></location>where f mod and f obs are the model and observed flux, respectively, and σ ( λ ) is the 1 σ error of the flux. We use a rest-frame wavelength range of 1268 -2580 ˚ A for the fitting, following the fitting range presented by Calzetti et al. (1994). The measured β values are shown in the bottom panel of Figure 2. Most of the galaxies show β > -2 . 6, which is larger than the lower limit for f ion esc = 0 (Figure 1). Among our sample, we find one galaxy at z = 9 . 25, showing β = -2 . 99 ± 0 . 15, which is smaller than -2 . 6 beyond the 2 σ level. Hereafter, we refer to this object as extremely blue galaxy 1 (EBG-1). EBG-1 is originally identified as MACS0647-z9-20158 at z phot = 9 . 5 by the program GO 1433 (PI: Coe; McLeod et al.</text>
<text><location><page_3><loc_52><loc_86><loc_92><loc_91></location>2024), and then spectroscopically observed by the same program GO 1433 as a filler target for MACS0647-JD. The NIRCam images and spectra of EBG-1 are shown in Figure 3.</text>
<text><location><page_3><loc_52><loc_60><loc_92><loc_85></location>Recently, Saxena et al. (2024) have found six galaxies at 5 . 5 < z < 8 showing β ∼ -3 from the spectroscopic sample obtained from the JADES survey. One out of six galaxies, JADES-GS-210003 at z = 5 . 779, is also included in our sample. Using the same fitting range as Saxena et al. (2024), we obtain β = -2 . 70 ± 0 . 13 for JADES-GS-210003 (Figure 4). There is a ∼ 2 σ difference between our and Saxena et al. (2024) of the β values. This is probably because Saxena et al. (2024) conduct a sigma-clipping method to exclude outlying pixels. JADES-GS-210003 is not selected in our study because we cannot distinguish from β = -2 . 6 at the 2 σ level. On the other hand, EBG-1 is not selected by Saxena et al. (2024), as EBG-1 falls in the MACS0647 lensing field that is not covered in the sample of Saxena et al. (2024).</text>
<text><location><page_3><loc_52><loc_49><loc_92><loc_60></location>We also compare our β values with those in previous studies in Figure 4. We plot photometric β measurements of Cullen et al. (2024), who conduct β measurements for the sample galaxies taken from NGDEEP, JADES DR1, UNCOVER, and those from McLeod et al. (2024). Cullen et al. (2024) also derive β for EBG-1, which show good agreement with our measurement.</text>
<section_header_level_1><location><page_3><loc_58><loc_47><loc_86><loc_48></location>2.4. Observations and Data of EBG-1</section_header_level_1>
<text><location><page_3><loc_52><loc_32><loc_92><loc_46></location>In the previous section, we identify EBG-1 out of 974 objects from the spectra reduced by DJA. In this section, we first describe how EBG-1 was observed in Section 2.4.1. We next conduct the photometry for EBG-1 to confirm whether the photometry is consistent with spectrum in Section 2.4.2. We then independently performed data reduction to verify whether the β value changes with different data reduction procedures, as described in Section 2.4.3.</text>
<section_header_level_1><location><page_3><loc_65><loc_30><loc_79><loc_31></location>2.4.1. Observations</section_header_level_1>
<text><location><page_3><loc_52><loc_19><loc_92><loc_29></location>MACS0647 lensing field was observed with JWST/NIRCam in January 8th 2023 in GO 1433 (PI: Coe) targeting MACS0647-JD (Coe et al. 2013; Hsiao et al. 2023). EBG-1 is falling on the footprints of this NIRCam observations, which is photometrically identified at R.A.=06:47:36.95 and Decl.=+70:14:34.69 by McLeod et al. (2024).</text>
<text><location><page_3><loc_52><loc_9><loc_92><loc_18></location>EBG-1 was then spectroscopically observed with JWST/NIRSpec as a filler target for MACS0647-JD in GO 1433 in February 20th, 2023. The observations were performed with PRISM/CLEAR ( R ∼ 100) for a total exposure time of 2200 s. The data were reduced with msaexp by DJA. For details of the reduction, see Heintz</text>
<figure>
<location><page_4><loc_9><loc_60><loc_91><loc_92></location>
<caption>Figure 3. Top and bottom panels show the 2D and 1D NIRSpec spectra of EBG-1 reduced by DJA, respectively. In the bottom panel, the black histogram and gray shaded regions represent the spectrum and its 1 σ uncertainty, respectively. The red line presents the best-fit UV slope derived with the Calzetti et al. (1994) fitting windows. The dark blue shaded regions are the regions that are not used for the UV slope fitting. The light blue shaded regions denote the masks presented by Calzetti et al. (1994) and the mask for the artifact at 2500 ˚ A, which are not used for fitting. The dotted lines present the positions of emission lines. The F150W image and slit position are presented in the inset.</caption>
</figure>
<figure>
<location><page_4><loc_9><loc_28><loc_48><loc_50></location>
<caption>Figure 4. Comparison of β measurements in this work and previous work. The red circle represents the values for EBG-1, whose β values in the previous works are taken from the photometric measurement of Cullen et al. (2024). The crosses denote the β values taken from the photometric measurements of Cullen et al. (2024), while the blue diamond represents those taken from the spectroscopic value of Saxena et al. (2024). The black solid line present the lower limit of β for no ionizing photon escape. The gray dashed line denotes the line of equality.</caption>
</figure>
<text><location><page_4><loc_52><loc_47><loc_92><loc_50></location>et al. (2024). The spectrum reduced by DJA is shown in Figure 3.</text>
<section_header_level_1><location><page_4><loc_66><loc_45><loc_78><loc_46></location>2.4.2. Photometry</section_header_level_1>
<text><location><page_4><loc_52><loc_33><loc_92><loc_43></location>Calibrated NIRCam data are collected from DJA. We conduct aperture photometry with 0 . '' 35 aperture size, which are shown in Table 1. The errors are estimated by conducting photometry within a 0 . '' 80 annulus and taking the standard deviations. An apparent magnitude of F200W is derived as 27.7, which is consistent with that derived by McLeod et al. (2024).</text>
<section_header_level_1><location><page_4><loc_60><loc_30><loc_84><loc_31></location>2.4.3. Reanalysis of EBG-1 spectra</section_header_level_1>
<text><location><page_4><loc_52><loc_9><loc_92><loc_29></location>We performed the data reduction in this work following the method described in Nakajima et al. (2023). Starting from the level 1 products provided by MAST, we executed Spec2 and Spec3 pipelines using Python library jwst (ver. 1.16.1; Bushouse et al. 2024). The reference files stored in the latest pmap file of jwst 1299.pmap were used. The pathloss corrections were conducted by comparing the position of the source and MSA shutter, where we assumed that the source was point-like. We then combined 2D spectra by medianstacking to reduce the effect of hot pixels, with extractions of 3 pixels in the spatial direction. For more details, see Nakajima et al. (2023). In Figure 5 we present</text>
<text><location><page_5><loc_8><loc_87><loc_48><loc_91></location>the spectrum reduced in this work, which is consistent with both of the spectrum reduced by DJA and the NIRCam photometry.</text>
<text><location><page_5><loc_8><loc_75><loc_48><loc_87></location>The spectrum of EBG-1 shows [O iii ] λλ 4959, 5007 and tentative C iv λλ 1548, 1550 emission lines. We measure flux values and 3 σ upper limits of emission lines by integrating the flux and error in each wavelength bin (Table 2). The C iv emission might be associated with an AGN, although the extremely blue UV slope disfavors the contribution from an AGN, whose dust content reddens the spectrum (Francis et al. 1991).</text>
<section_header_level_1><location><page_5><loc_23><loc_72><loc_33><loc_73></location>3. RESULTS</section_header_level_1>
<section_header_level_1><location><page_5><loc_14><loc_70><loc_42><loc_71></location>3.1. Confirmation of the Blue UV Slope</section_header_level_1>
<text><location><page_5><loc_8><loc_52><loc_48><loc_69></location>To check whether the extremely blue UV continuum of EBG-1 is caused by systematics or not, we adopt three fitting methods: 1) conventional Calzetti et al. (1994) windows, 2) using the whole wavelength range without masking, and 3) simply avoiding the possible C iv emission line (note that we mask out at 2480 -2520 ˚ A contaminated by the artifact). For the spectrum reduced by DJA, each fitting method gives β = -3 . 03 ± 0 . 22 , -2 . 99 ± 0 . 15 , and -3 . 08 ± 0 . 15, respectively, all of which are smaller than β = -2 . 6 at the ∼ 2 σ level.</text>
<text><location><page_5><loc_8><loc_38><loc_48><loc_51></location>We derive β also for the spectrum reduced in this work in the same manner as described above. The fitting methods of 1), 2), and 3) yield β = -2 . 96 ± 0 . 19 , -2 . 92 ± 0 . 13 , and -2 . 79 ± 0 . 14, respectively. Although the significance levels are slightly lower than the β values from the DJA spectrum, the results of the extremely blue UV slope of EBG-1 does not significantly change. We thus conclude that the extremely blue UV slope of EBG-1 is not attributed to observational systematics.</text>
<section_header_level_1><location><page_5><loc_22><loc_32><loc_34><loc_33></location>3.2. SED Fitting</section_header_level_1>
<text><location><page_5><loc_8><loc_9><loc_48><loc_31></location>To measure stellar properties, we conduct SED fitting to the NIRCam data presented in Table 1 using Prospector (Johnson et al. 2021a). We use the photometry values that are corrected for magnification factor. The Binary Population and Spectral Synthesis (BPASS; Eldridge et al. 2017) model is used for an isochrone library. We apply Calzetti et al. (2000) dust extinction law. We assume a non-parametric star-formation history (SFH) with five bins that are evenly spaced in logarithmic times between 0 Myr and a look-back time corresponding to z = 30. The redshift of the source is fixed at z spec = 9 . 25. We vary the stellar mass M ∗ , metallicity Z , optical depth of dust attenuation at 5500 ˚ A τ dust (5500 ˚ A), and ionization parameter U . The best-fit</text>
<table>
<location><page_5><loc_61><loc_59><loc_83><loc_90></location>
<caption>Table 1. Properties of EBG-1</caption>
</table>
<text><location><page_5><loc_61><loc_56><loc_82><loc_58></location>Note -a The value is corrected for magnification factor.</text>
<table>
<location><page_5><loc_56><loc_37><loc_88><loc_48></location>
<caption>Table 2. Line fluxes and 3 σ upper limits</caption>
</table>
<text><location><page_5><loc_56><loc_35><loc_61><loc_36></location>Note -</text>
<text><location><page_5><loc_52><loc_18><loc_92><loc_29></location>parameters and SFH are shown in Table 1 and Figure 6, respectively. The results imply the dust-free condition and the recent starburst, which agree with the extremely blue UV continuum of EBG-1. Because f ion esc = 0 is assumed in our SED fitting, the weak emission line feature of EBG-1 indicates a low metallicity, which compensates for the effect of nonzero f ion esc .</text>
<section_header_level_1><location><page_5><loc_60><loc_16><loc_85><loc_17></location>3.3. Weak Nebular Emission Lines</section_header_level_1>
<text><location><page_5><loc_52><loc_9><loc_92><loc_15></location>The extremely blue UV continuum in EBG-1 may be explained by high f esc . However, the detection of [O iii ] λλ 4959,5007 emission indicates a certain amount of the ionizing photon is used to ionize the nebulae. To</text>
<figure>
<location><page_6><loc_9><loc_66><loc_91><loc_92></location>
<caption>Figure 5. Comparison of NIRSpec spectrum of EBG-1 reduced in this work and DJA. The yellow histogram and shaded region represent the spectrum reduced in this work and its 1 σ error, respectively. The blue points denote the NIRCam photometry. The other symbols are the same as Figure 3.</caption>
</figure>
<figure>
<location><page_6><loc_9><loc_37><loc_47><loc_59></location>
<caption>Figure 6. Best-fit SFH for EBG-1. The black line and gray shaded region represent the best-fit and 1 σ error of SFH, respectively.</caption>
</figure>
<text><location><page_6><loc_8><loc_10><loc_48><loc_30></location>quantify the escape of ionizing photons in EBG-1, we plot ratios of luminosity of [O iii ] λ 5007 line, L [OIII] , to star-formation rate (SFR) as a function of stellar mass in Figure 7. For comparison, we also plot average values calculated from 126 galaxies at 4 < z < 9 compiled by Nakajima et al. (2023). The SFRs are estimated from UV luminosities of galaxies by using Equation (1) of Kennicutt (1998). The L [OIII] /SFR value of EBG-1 is ∼ 0 . 5 dex smaller than the average value at the same stellar mass, suggesting that [O iii ] emission in EBG-1 is ∼ 3 times weaker than the average. This can be interpreted as a result of the escape of ionizing photon without ionizing the nebulae. If we assume f ion esc = 0 for the</text>
<text><location><page_6><loc_52><loc_41><loc_92><loc_59></location>galaxies in Nakajima et al. (2023), the [O iii ] emission three times weaker than the average suggests f ion esc ∼ 0 . 7 for EBG-1. However, the f ion esc values of the galaxies in Nakajima et al. (2023) can be larger than zero, in which case our estimate of f ion esc becomes larger. Furthermore, if we assume density-bounded nebulae, weak [O iii ] emission means an excess of O ++ ionizing photons (ionization energy of 35 eV) compared to the amount of the gas, with H + ionizing photon (ionization energy of 13 . 6 eV) being even more abundant. In such a situation, the escape fraction of H + ionizing photon can be larger than that of O ++ ionizing photon.</text>
<text><location><page_6><loc_52><loc_12><loc_92><loc_40></location>However, the estimate of f ion esc from the [O iii ] emission is susceptible to metallicity. To estimate f ion esc independently of the assumption of metallicity, we use the β and EW(H β ) values of EBG-1. We derive the upper limit of the H β flux by summing up the error spectrum in quadrature. We then divide the upper limit of H β by the continuum flux density, obtaining log (EW(H β ) / ˚ A) < 2 . 8. In Figure 8, we compare the β and EW(H β ) values with the Cloudy models, which are the same as those in Figure 1 except for varying f ion esc . In all of the models, the very small β values of EBG1 imply f esc ≳ 0 . 5 (although models with extremely metal-poor or metal-free stellar populations reproduce β ∼ -3 even with f ion esc = 0 see Figure 4 of (Bouwens et al. 2010), the detection of [O iii ] emission in EBG-1 disfavors this scenario). However, due to the weak upper limit of EW(H β ), it is difficult to constrain the ionizing radiation and f esc with current data.</text>
<section_header_level_1><location><page_6><loc_66><loc_10><loc_78><loc_11></location>4. DISCUSSION</section_header_level_1>
<figure>
<location><page_7><loc_8><loc_66><loc_48><loc_92></location>
<caption>Figure 7. L [OIII] /SFR as a function of stellar mass. The red circle represents EBG-1. The blue squares denote the average values of galaxies at 4 < z < 9 taken from Nakajima et al. (2023). The ticks at the top of the figure denote metallicity, which is converted from stellar mass using the mass-metallicity relation at z = 4 -10 derived by Nakajima et al. (2023).</caption>
</figure>
<text><location><page_7><loc_8><loc_16><loc_48><loc_53></location>There are two scenarios that explain the high escape fraction: density-bounded nebulae, or ionizationbounded nebulae with holes (Zackrisson et al. 2013). In density-bounded nebulae, ionizing photons exceeding the gas supply escape the nebula, while in ionizationbounded nebulae with holes, ionizing photons leak through these holes. Both of the scenarios are consistent with our results, and it is difficult to test these two scenarios with the current data. However, the morphology of EBG-1 (Figure 3) may provide a clue to the origin of the high escape fraction. EBG-1 has a small tail extending towards north-west direction from the main component. The slit covers around the tail of the EBG1. The tail may be stripped from EBG-1 by galaxy interactions, where the gas could be lacking. We may be looking at this gas-deficient region, where the nebulae are density-bounded or with many holes. In such a case, the center of EBG-1 can be redder than the tail (see also Schombert et al. (1990), who claim that tidal tails tend to be bluer than primary galaxies from observations). To examine this scenario, it is necessary to conduct spectroscopy by placing a slit along the northwest direction that covers from the center to the tail of the EBG-1 with spatial extent.</text>
<text><location><page_7><loc_8><loc_9><loc_48><loc_15></location>The fraction of the extremely blue galaxy in this work is low (one out of 974 galaxies) compared to the other studies such as Saxena et al. (2024). This is probably because of the low signal-to-noise ratio of most of the</text>
<text><location><page_7><loc_52><loc_82><loc_92><loc_91></location>galaxies. Although some galaxies show the best-fit β values smaller than -2 . 6 in Figure 2, the large error bars hinder us from confirming that these galaxies truly have blue UV continua. There may be more galaxies like EBG-1, while deeper spectroscopic observations are required to confirm.</text>
<text><location><page_7><loc_52><loc_70><loc_92><loc_82></location>The non-detection of H β line and the large uncertainty in [O iii ] emission lines make it difficult to constrain the f ion esc value and type of ionizing source of EBG-1. It is thus necessary to conduct deeper spectroscopy on EBG1 to definitively confirm the high f ion esc and the ionizing source, by which one can utilize EBG-1 as an example to study the process of the significant ionizing photon escape during the epoch of reionization.</text>
<section_header_level_1><location><page_7><loc_67><loc_67><loc_77><loc_68></location>5. SUMMARY</section_header_level_1>
<text><location><page_7><loc_52><loc_55><loc_92><loc_66></location>In this work, we search the large JWST/NIRSpec spectroscopic sample for a galaxy with an extremely blue UV continuum. Among our sample consisting of the 974 galaxies at 4 < z < 14 taken from the major JWST GTO, ERS, and GO programs, we identify EBG-1, a galaxy at z = 9 . 25 showing β ∼ -3. Our major findings are summarized below:</text>
<unordered_list>
<list_item><location><page_7><loc_55><loc_42><loc_92><loc_54></location>· By fitting f ( λ ) ∝ λ β to the UV continua for our sample, we find EBG-1 showing β = -2 . 99 ± 0 . 15, which is below the lower limit for no ionizing photon escape, β = -2 . 6, beyond the 2 σ level. This small β value does not change significantly by changing the fitting method and data reduction procedure, which suggest that the extremely blue UV continuum is not caused by systematics.</list_item>
<list_item><location><page_7><loc_55><loc_26><loc_92><loc_40></location>· The NIRSpec PRISM spectrum of EBG-1 shows [O iii ] λλ 4959, 5007 emission lines. By estimating the SFR from the UV luminosity, we calculate the L [OIII] / SFR ratio for EBG-1. The comparison with the galaxies at 4 < z < 9 compiled by Nakajima et al. (2023) suggests that the [O iii ] emission is slightly weaker than expected from the UV luminosity if f ion esc = 0 is assumed, indicating f ion esc ∼ 0 . 7 for EBG-1.</list_item>
<list_item><location><page_7><loc_55><loc_13><loc_92><loc_25></location>· We compare the observed β and EW(H β ) with our Cloudy modeling. The extremely blue β value imply f ion esc ≳ 0 . 5, which is consistent with the f ion esc value inferred from the [O iii ] emission. However, the weak upper limit of H β emission line prevents us from break the degeneracy between f ion esc and the shape of the ionizing source. It is thus important to conduct deeper spectroscopy on EBG-1.</list_item>
</unordered_list>
<section_header_level_1><location><page_7><loc_63><loc_10><loc_81><loc_11></location>ACKNOWLEDGMENTS</section_header_level_1>
<figure>
<location><page_8><loc_17><loc_60><loc_84><loc_92></location>
<caption>Figure 8. Relation of β and EW(H β ). The blue, green, and gray grids denote the same models in Figure 1. The numbers shown beside the grids represent f ion esc . The red circle presents the current constraint of β = -2 . 99 ± 0 . 15.</caption>
</figure>
<text><location><page_8><loc_8><loc_19><loc_48><loc_55></location>We are grateful to Steven L. Finkelstein, Moka Nishigaki, and Kuria Watanabe for the valuable discussions. This work is based on observations made with the NASA/ESA/CSA James Webb Space Telescope. The NIRSpec data of EBG-1 were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. The data products presented herein were retrieved from DJA. DJA is an initiative of the Cosmic Dawn Center (DAWN), which is funded by the Danish National Research Foundation under grant DNRF140. We thank DJA for providing the reduced NIRSpec data. The observational data collected from DJA are associated with programs ERS 1345 (CEERS; PI: Finkelstein), DDT 2756 (PI: Chen), DDT 2750 (CEERS; PI: Arrabal Haro), GTO 1180 (PI: Eisenstein), GTO 1181 (PI: Eisenstein), GTO 1210 (PI: Luetzgendorf), GTO 1211 (PI: Isaak), GTO 1286 (PI: Luetzgendorf), DDT 6541 (PI: Egami), GO 1433 (PI: Coe), GO 1747 (PI: Roberts-Borsani), GO 1810 (PI: Belli), GO 1871 (PI: Chisholm), GO 2110 (PI: Kriek), GO 2198 (PI: Barrufet), GO 2561 (UN-</text>
<text><location><page_8><loc_52><loc_34><loc_92><loc_55></location>COVER; PI: Labbe), GO 2565 (PI: Glazebrook), DDT 2767 (PI: Kelly), ERO 2736 (PI: Pontoppidan), GO 3215 (PI: Eisenstein), GO 4233 (PI: de Graaff) GO 4246 (PI: Abdurro'uf), DDT 4446(PI: Frye), and DDT 4557 (PI: Yan). The authors acknowledge the teams conducting these observations for publicly releasing the data. This publication is based on work supported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, KAKENHI (20H00180, 21H04467, 21H04489, 24K07102) through the Japan Society for the Promotion of Science, and JST FOREST Program (JP-MJFR202Z). This work was supported by the joint research program of the Institute for Cosmic Ray Research (ICRR), University of Tokyo.</text>
<section_header_level_1><location><page_8><loc_54><loc_29><loc_62><loc_31></location>Facilities:</section_header_level_1>
<text><location><page_8><loc_52><loc_19><loc_92><loc_28></location>Software: astropy (Astropy Collaboration et al. 2013, 2018, 2022), NumPy (Harris et al. 2020), matplotlib (Hunter 2007), SciPy (Virtanen et al. 2020), corner (Foreman-Mackey 2016) Cloudy (Ferland et al. 2013), emcee (Foreman-Mackey et al. 2013), msaexp (Brammer 2023) Prospector (Johnson et al. 2021b)</text>
<section_header_level_1><location><page_8><loc_44><loc_16><loc_56><loc_17></location>REFERENCES</section_header_level_1>
<text><location><page_8><loc_8><loc_10><loc_45><loc_14></location>Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068</text>
<text><location><page_8><loc_52><loc_13><loc_92><loc_14></location>Astropy Collaboration, Price-Whelan, A. M., Sip"ocz, B. M.,</text>
<text><location><page_8><loc_54><loc_11><loc_91><loc_12></location>et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f</text>
<table>
<location><page_9><loc_8><loc_16><loc_48><loc_92></location>
</table>
<text><location><page_9><loc_52><loc_89><loc_92><loc_91></location>Hunter, J. D. 2007, Computing in Science and Engineering, 9, 90, doi: 10.1109/MCSE.2007.55</text>
<table>
<location><page_9><loc_52><loc_17><loc_92><loc_89></location>
</table>
</document>
[
{
"title": "ABSTRACT",
"content": "We investigate UV continuum slopes β of 974 galaxies at z = 4 -14 using archival JWST/NIRSpec PRISM spectra obtained from major JWST GTO, ERS, and GO programs, including JADES, CEERS, and UNCOVER. Among these galaxies, we identify a remarkable galaxy at z = 9 . 25, dubbed EBG-1, with a significantly blue UV slope β = -2 . 99 ± 0 . 15, unlike the rest of the galaxies that exhibit red continua or ambiguous blue continua hindered by large uncertainties. We confirm that the β value negligibly changes by the data reduction and fitting wavelength ranges for UV emission/absorption line masking. The extreme blue slope, β = -3 . 0, rules out significant contributions from dust extinction or AGN activity. Comparing with stellar and nebular emission models, we find that such a blue UV slope cannot be reproduced solely by stellar models even with very young, metal-poor, or top-heavy contiguous star formation associated with strong nebular continua making the UV slopes red, but with a high ionizing photon escape fraction, f ion esc ≳ 0 . 5, for a weak nebular continuum. While the H β emission line is not detected, likely due to the limited sensitivity of the spectrum, we find moderately weak [O iii ] λ 4959,5007 emission lines for the given star-formation rate (3 M ⊙ yr -1 ) and stellar mass (10 8 . 0 M ⊙ ) that are about three times weaker than the average emission lines, again suggestive of the high ionizing photon escape fraction, f ion esc ∼ 0 . 7 or more. EBG-1 would provide crucial insights into stellar and nebular continuum emission in high-redshift galaxies, serving as an example of the ionizing photon escaping site at the epoch of reionization. Keywords: Ultraviolet color; Reionization; Galaxy evolution; Galaxy formation; High-redshift galaxies",
"pages": [
1
]
},
{
"title": "A Galaxy with an Extremely Blue UV Slope β = -3 at z = 9 . 25 Identified by JWST Spectroscopy: Evidence for a Weak Nebular Continuum and Efficient Ionizing Photon Escape?",
"content": "Hiroto Yanagisawa , 1, 2 Masami Ouchi , 3, 1, 4, 5 Kimihiko Nakajima , 3 Yuichi Harikane , 1 Seiji Fujimoto , 6, 7, 8 Yoshiaki Ono , 1 Hiroya Umeda , 1, 2 Minami Nakane , 1, 2 Hidenobu Yajima , 9 Hajime Fukushima , 9 and Yi Xu 1, 10 1 Institute for Cosmic Ray Research, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan 2 Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan 3 National Astronomical Observatory of Japan, National Institutes of Natural Sciences, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 4 Department of Astronomical Science, SOKENDAI (The Graduate University for Advanced Studies), 2-21-1 Osawa, Mitaka, Tokyo, 181-8588, Japan 5 Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, Chiba 277-8583, Japan 6 Department of Astronomy, The University of Texas at Austin, Austin, TX 78712, USA 7 Cosmic Dawn Center (DAWN), Denmark 8 Niels Bohr Institute, University of Copenhagen, Lyngbyvej 2, DK2100 Copenhagen Ø, Denmark 10 9 Center for Computational Sciences, University of Tsukuba, Ten-nodai, 1-1-1 Tsukuba, Ibaraki 305-8577, Japan Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "The first galaxies initiate star formation in the early universe, with young massive stars exhibiting blue UV continua and producing substantial amounts of ionizing photons. Although the ionizing photons are mainly used to ionize the interstellar medium, a fraction of the ionizing photons escape from the galaxy and ionize the intergalactic medium, driving the cosmic reionization. The escape of the hydrogen ionizing photons is characterized by an escape fraction, f ion esc , which is the key quantity to understand how the ionizing photons of the galaxies contribute to the cosmic reionization. The UV continuum slope β ( f ( λ ) ∝ λ β ) is an important indicator of the production and escape of ionizing photons. Young stellar populations produce β < -2, while dust extinction and active galactic nucleus (AGN) provide a red ( β ≳ -2) UV continuum (e.g., Bouwens et al. 2012; Finkelstein et al. 2012). Typically the β values are larger than ∼ -2 . 6 (Chisholm et al. 2022), because an intrinsically blue UV continuum intensely ionizes the nebula, leading to a significant contribution from a nebular continuum, which has β ≳ -2 (Katz et al. 2024; Cameron et al. 2024). However, without the nebular continuum, the β values can be as low as -3 . 4 (e.g., Bouwens et al. 2010). This situation is achieved if the escape fraction of the ionizing photon f ion esc is large, because in that case the nebula is less ionized and the nebular continuum has less contribution to the UV spectrum (Zackrisson et al. 2017) . It is thus important to search for galaxies with β < -2 . 6 because such blue galaxies may have extremely high f ion esc , which contribute to the cosmic reionization. Because galaxies with such high f ion esc are probably rare (Leitet et al. 2013; Matthee et al. 2017), it is necessary to search large sample of galaxies. Photometric studies are conducted with JWST data by Topping et al. (2022, 2024); Morales et al. (2024); Cullen et al. (2023, 2024), although deriving β using photometry suffers from contamination by emission lines. The accurate measurement of β requires the high quality spectroscopic data. In this work, we search a large spectroscopic sample of galaxies provided by the DAWN JWST Archive for a galaxy with a blue UV slope, and report a galaxy at z = 9 . 25 that have an extremely blue UV slope. In Section 2, we describe our sample and method of UV slope measurements. In Section 3, we present the results the extremely blue object. We discuss physical origins of the extremely blue UV continuum in Section 4. Section 5 summarizes our results.",
"pages": [
1,
2
]
},
{
"title": "2.1. Lower Limit of β for f ion esc = 0",
"content": "We first define a quantitative criterion for a blue UV continuum. We calculate β assuming f ion esc = 0 using Cloudy version 23.01 (Gunasekera et al. 2023) with incident radiations of Kroupa IMF (Kroupa 2001) with a mass range of 0 . 1 -100 M ⊙ provided by BPASS v2.2.1 (Stanway & Eldridge 2018), top-heavy IMF with a mass range of 50 -500 M ⊙ taken from Yggdrasil Pop III.1 model (Zackrisson et al. 2011), and blackbody. We assume a hydrogen density of n e = 10 2 cm -3 , nebular metallicity of log Z neb / Z ⊙ = -2, number of ionizing photon Q (H) = 10 50 s -1 , and inner radius of 10 14 cm. We assume stellar metallicity of log Z star / Z ⊙ = -2 for the Kroupa IMF model, while Z star = 0 is used for top-heavy IMF. Figure 1 shows time evolution of β values. Although the incident radiation of the top-heavy IMF is bluer than that of the Kroupa IMF, the topheavy IMF+nebular continuum model is redder than the Kroupa+nebular continuum model. This is because the intrinsically blue incident radiation in top-heavy IMF model intensely ionize the nebula, which leads to a significant contribution from the red nebular continuum. One can also see that the lower limit of β value for f ion esc = 0 is -2 . 6, while the β value reach as low as -3 . 4 if nebular continuum is not included. We thus define a criterion for a blue UV continuum as β = -2 . 6.",
"pages": [
2
]
},
{
"title": "2.2. Our Sample",
"content": "We use 974 PRISM/CLEAR spectra of galaxies provided by the DAWN JWST Archive (DJA) 1 , which compile the major JWST GTO, ERS, and GO programs. 2 We select the galaxies within the redshift range of 4 < z < 14, whose UV continua are covered by NIRSpec. The spectra in the DJA are reduced with msaexp (Brammer 2023; Heintz et al. 2024). Hereafter, these 974 galaxies are referred to as our sample. The redshift distribution of galaxies in our sample is shown in the top panel of Figure 2.",
"pages": [
2
]
},
{
"title": "2.3. UV Slope Measurements",
"content": "To derive β , we fit a galaxy spectra with a function where A is a constant for normalization and λ is wavelengths. We employ a Markov Chain Monte Carlo (MCMC) method for the fitting using emcee (ForemanMackey et al. 2013). We minimize a likelihood function where f mod and f obs are the model and observed flux, respectively, and σ ( λ ) is the 1 σ error of the flux. We use a rest-frame wavelength range of 1268 -2580 ˚ A for the fitting, following the fitting range presented by Calzetti et al. (1994). The measured β values are shown in the bottom panel of Figure 2. Most of the galaxies show β > -2 . 6, which is larger than the lower limit for f ion esc = 0 (Figure 1). Among our sample, we find one galaxy at z = 9 . 25, showing β = -2 . 99 ± 0 . 15, which is smaller than -2 . 6 beyond the 2 σ level. Hereafter, we refer to this object as extremely blue galaxy 1 (EBG-1). EBG-1 is originally identified as MACS0647-z9-20158 at z phot = 9 . 5 by the program GO 1433 (PI: Coe; McLeod et al. 2024), and then spectroscopically observed by the same program GO 1433 as a filler target for MACS0647-JD. The NIRCam images and spectra of EBG-1 are shown in Figure 3. Recently, Saxena et al. (2024) have found six galaxies at 5 . 5 < z < 8 showing β ∼ -3 from the spectroscopic sample obtained from the JADES survey. One out of six galaxies, JADES-GS-210003 at z = 5 . 779, is also included in our sample. Using the same fitting range as Saxena et al. (2024), we obtain β = -2 . 70 ± 0 . 13 for JADES-GS-210003 (Figure 4). There is a ∼ 2 σ difference between our and Saxena et al. (2024) of the β values. This is probably because Saxena et al. (2024) conduct a sigma-clipping method to exclude outlying pixels. JADES-GS-210003 is not selected in our study because we cannot distinguish from β = -2 . 6 at the 2 σ level. On the other hand, EBG-1 is not selected by Saxena et al. (2024), as EBG-1 falls in the MACS0647 lensing field that is not covered in the sample of Saxena et al. (2024). We also compare our β values with those in previous studies in Figure 4. We plot photometric β measurements of Cullen et al. (2024), who conduct β measurements for the sample galaxies taken from NGDEEP, JADES DR1, UNCOVER, and those from McLeod et al. (2024). Cullen et al. (2024) also derive β for EBG-1, which show good agreement with our measurement.",
"pages": [
3
]
},
{
"title": "2.4. Observations and Data of EBG-1",
"content": "In the previous section, we identify EBG-1 out of 974 objects from the spectra reduced by DJA. In this section, we first describe how EBG-1 was observed in Section 2.4.1. We next conduct the photometry for EBG-1 to confirm whether the photometry is consistent with spectrum in Section 2.4.2. We then independently performed data reduction to verify whether the β value changes with different data reduction procedures, as described in Section 2.4.3.",
"pages": [
3
]
},
{
"title": "2.4.1. Observations",
"content": "MACS0647 lensing field was observed with JWST/NIRCam in January 8th 2023 in GO 1433 (PI: Coe) targeting MACS0647-JD (Coe et al. 2013; Hsiao et al. 2023). EBG-1 is falling on the footprints of this NIRCam observations, which is photometrically identified at R.A.=06:47:36.95 and Decl.=+70:14:34.69 by McLeod et al. (2024). EBG-1 was then spectroscopically observed with JWST/NIRSpec as a filler target for MACS0647-JD in GO 1433 in February 20th, 2023. The observations were performed with PRISM/CLEAR ( R ∼ 100) for a total exposure time of 2200 s. The data were reduced with msaexp by DJA. For details of the reduction, see Heintz et al. (2024). The spectrum reduced by DJA is shown in Figure 3.",
"pages": [
3,
4
]
},
{
"title": "2.4.2. Photometry",
"content": "Calibrated NIRCam data are collected from DJA. We conduct aperture photometry with 0 . '' 35 aperture size, which are shown in Table 1. The errors are estimated by conducting photometry within a 0 . '' 80 annulus and taking the standard deviations. An apparent magnitude of F200W is derived as 27.7, which is consistent with that derived by McLeod et al. (2024).",
"pages": [
4
]
},
{
"title": "2.4.3. Reanalysis of EBG-1 spectra",
"content": "We performed the data reduction in this work following the method described in Nakajima et al. (2023). Starting from the level 1 products provided by MAST, we executed Spec2 and Spec3 pipelines using Python library jwst (ver. 1.16.1; Bushouse et al. 2024). The reference files stored in the latest pmap file of jwst 1299.pmap were used. The pathloss corrections were conducted by comparing the position of the source and MSA shutter, where we assumed that the source was point-like. We then combined 2D spectra by medianstacking to reduce the effect of hot pixels, with extractions of 3 pixels in the spatial direction. For more details, see Nakajima et al. (2023). In Figure 5 we present the spectrum reduced in this work, which is consistent with both of the spectrum reduced by DJA and the NIRCam photometry. The spectrum of EBG-1 shows [O iii ] λλ 4959, 5007 and tentative C iv λλ 1548, 1550 emission lines. We measure flux values and 3 σ upper limits of emission lines by integrating the flux and error in each wavelength bin (Table 2). The C iv emission might be associated with an AGN, although the extremely blue UV slope disfavors the contribution from an AGN, whose dust content reddens the spectrum (Francis et al. 1991).",
"pages": [
4,
5
]
},
{
"title": "3.1. Confirmation of the Blue UV Slope",
"content": "To check whether the extremely blue UV continuum of EBG-1 is caused by systematics or not, we adopt three fitting methods: 1) conventional Calzetti et al. (1994) windows, 2) using the whole wavelength range without masking, and 3) simply avoiding the possible C iv emission line (note that we mask out at 2480 -2520 ˚ A contaminated by the artifact). For the spectrum reduced by DJA, each fitting method gives β = -3 . 03 ± 0 . 22 , -2 . 99 ± 0 . 15 , and -3 . 08 ± 0 . 15, respectively, all of which are smaller than β = -2 . 6 at the ∼ 2 σ level. We derive β also for the spectrum reduced in this work in the same manner as described above. The fitting methods of 1), 2), and 3) yield β = -2 . 96 ± 0 . 19 , -2 . 92 ± 0 . 13 , and -2 . 79 ± 0 . 14, respectively. Although the significance levels are slightly lower than the β values from the DJA spectrum, the results of the extremely blue UV slope of EBG-1 does not significantly change. We thus conclude that the extremely blue UV slope of EBG-1 is not attributed to observational systematics.",
"pages": [
5
]
},
{
"title": "3.2. SED Fitting",
"content": "To measure stellar properties, we conduct SED fitting to the NIRCam data presented in Table 1 using Prospector (Johnson et al. 2021a). We use the photometry values that are corrected for magnification factor. The Binary Population and Spectral Synthesis (BPASS; Eldridge et al. 2017) model is used for an isochrone library. We apply Calzetti et al. (2000) dust extinction law. We assume a non-parametric star-formation history (SFH) with five bins that are evenly spaced in logarithmic times between 0 Myr and a look-back time corresponding to z = 30. The redshift of the source is fixed at z spec = 9 . 25. We vary the stellar mass M ∗ , metallicity Z , optical depth of dust attenuation at 5500 ˚ A τ dust (5500 ˚ A), and ionization parameter U . The best-fit Note -a The value is corrected for magnification factor. Note - parameters and SFH are shown in Table 1 and Figure 6, respectively. The results imply the dust-free condition and the recent starburst, which agree with the extremely blue UV continuum of EBG-1. Because f ion esc = 0 is assumed in our SED fitting, the weak emission line feature of EBG-1 indicates a low metallicity, which compensates for the effect of nonzero f ion esc .",
"pages": [
5
]
},
{
"title": "3.3. Weak Nebular Emission Lines",
"content": "The extremely blue UV continuum in EBG-1 may be explained by high f esc . However, the detection of [O iii ] λλ 4959,5007 emission indicates a certain amount of the ionizing photon is used to ionize the nebulae. To quantify the escape of ionizing photons in EBG-1, we plot ratios of luminosity of [O iii ] λ 5007 line, L [OIII] , to star-formation rate (SFR) as a function of stellar mass in Figure 7. For comparison, we also plot average values calculated from 126 galaxies at 4 < z < 9 compiled by Nakajima et al. (2023). The SFRs are estimated from UV luminosities of galaxies by using Equation (1) of Kennicutt (1998). The L [OIII] /SFR value of EBG-1 is ∼ 0 . 5 dex smaller than the average value at the same stellar mass, suggesting that [O iii ] emission in EBG-1 is ∼ 3 times weaker than the average. This can be interpreted as a result of the escape of ionizing photon without ionizing the nebulae. If we assume f ion esc = 0 for the galaxies in Nakajima et al. (2023), the [O iii ] emission three times weaker than the average suggests f ion esc ∼ 0 . 7 for EBG-1. However, the f ion esc values of the galaxies in Nakajima et al. (2023) can be larger than zero, in which case our estimate of f ion esc becomes larger. Furthermore, if we assume density-bounded nebulae, weak [O iii ] emission means an excess of O ++ ionizing photons (ionization energy of 35 eV) compared to the amount of the gas, with H + ionizing photon (ionization energy of 13 . 6 eV) being even more abundant. In such a situation, the escape fraction of H + ionizing photon can be larger than that of O ++ ionizing photon. However, the estimate of f ion esc from the [O iii ] emission is susceptible to metallicity. To estimate f ion esc independently of the assumption of metallicity, we use the β and EW(H β ) values of EBG-1. We derive the upper limit of the H β flux by summing up the error spectrum in quadrature. We then divide the upper limit of H β by the continuum flux density, obtaining log (EW(H β ) / ˚ A) < 2 . 8. In Figure 8, we compare the β and EW(H β ) values with the Cloudy models, which are the same as those in Figure 1 except for varying f ion esc . In all of the models, the very small β values of EBG1 imply f esc ≳ 0 . 5 (although models with extremely metal-poor or metal-free stellar populations reproduce β ∼ -3 even with f ion esc = 0 see Figure 4 of (Bouwens et al. 2010), the detection of [O iii ] emission in EBG-1 disfavors this scenario). However, due to the weak upper limit of EW(H β ), it is difficult to constrain the ionizing radiation and f esc with current data.",
"pages": [
5,
6
]
},
{
"title": "4. DISCUSSION",
"content": "There are two scenarios that explain the high escape fraction: density-bounded nebulae, or ionizationbounded nebulae with holes (Zackrisson et al. 2013). In density-bounded nebulae, ionizing photons exceeding the gas supply escape the nebula, while in ionizationbounded nebulae with holes, ionizing photons leak through these holes. Both of the scenarios are consistent with our results, and it is difficult to test these two scenarios with the current data. However, the morphology of EBG-1 (Figure 3) may provide a clue to the origin of the high escape fraction. EBG-1 has a small tail extending towards north-west direction from the main component. The slit covers around the tail of the EBG1. The tail may be stripped from EBG-1 by galaxy interactions, where the gas could be lacking. We may be looking at this gas-deficient region, where the nebulae are density-bounded or with many holes. In such a case, the center of EBG-1 can be redder than the tail (see also Schombert et al. (1990), who claim that tidal tails tend to be bluer than primary galaxies from observations). To examine this scenario, it is necessary to conduct spectroscopy by placing a slit along the northwest direction that covers from the center to the tail of the EBG-1 with spatial extent. The fraction of the extremely blue galaxy in this work is low (one out of 974 galaxies) compared to the other studies such as Saxena et al. (2024). This is probably because of the low signal-to-noise ratio of most of the galaxies. Although some galaxies show the best-fit β values smaller than -2 . 6 in Figure 2, the large error bars hinder us from confirming that these galaxies truly have blue UV continua. There may be more galaxies like EBG-1, while deeper spectroscopic observations are required to confirm. The non-detection of H β line and the large uncertainty in [O iii ] emission lines make it difficult to constrain the f ion esc value and type of ionizing source of EBG-1. It is thus necessary to conduct deeper spectroscopy on EBG1 to definitively confirm the high f ion esc and the ionizing source, by which one can utilize EBG-1 as an example to study the process of the significant ionizing photon escape during the epoch of reionization.",
"pages": [
7
]
},
{
"title": "5. SUMMARY",
"content": "In this work, we search the large JWST/NIRSpec spectroscopic sample for a galaxy with an extremely blue UV continuum. Among our sample consisting of the 974 galaxies at 4 < z < 14 taken from the major JWST GTO, ERS, and GO programs, we identify EBG-1, a galaxy at z = 9 . 25 showing β ∼ -3. Our major findings are summarized below:",
"pages": [
7
]
},
{
"title": "ACKNOWLEDGMENTS",
"content": "We are grateful to Steven L. Finkelstein, Moka Nishigaki, and Kuria Watanabe for the valuable discussions. This work is based on observations made with the NASA/ESA/CSA James Webb Space Telescope. The NIRSpec data of EBG-1 were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. The data products presented herein were retrieved from DJA. DJA is an initiative of the Cosmic Dawn Center (DAWN), which is funded by the Danish National Research Foundation under grant DNRF140. We thank DJA for providing the reduced NIRSpec data. The observational data collected from DJA are associated with programs ERS 1345 (CEERS; PI: Finkelstein), DDT 2756 (PI: Chen), DDT 2750 (CEERS; PI: Arrabal Haro), GTO 1180 (PI: Eisenstein), GTO 1181 (PI: Eisenstein), GTO 1210 (PI: Luetzgendorf), GTO 1211 (PI: Isaak), GTO 1286 (PI: Luetzgendorf), DDT 6541 (PI: Egami), GO 1433 (PI: Coe), GO 1747 (PI: Roberts-Borsani), GO 1810 (PI: Belli), GO 1871 (PI: Chisholm), GO 2110 (PI: Kriek), GO 2198 (PI: Barrufet), GO 2561 (UN- COVER; PI: Labbe), GO 2565 (PI: Glazebrook), DDT 2767 (PI: Kelly), ERO 2736 (PI: Pontoppidan), GO 3215 (PI: Eisenstein), GO 4233 (PI: de Graaff) GO 4246 (PI: Abdurro'uf), DDT 4446(PI: Frye), and DDT 4557 (PI: Yan). The authors acknowledge the teams conducting these observations for publicly releasing the data. This publication is based on work supported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, KAKENHI (20H00180, 21H04467, 21H04489, 24K07102) through the Japan Society for the Promotion of Science, and JST FOREST Program (JP-MJFR202Z). This work was supported by the joint research program of the Institute for Cosmic Ray Research (ICRR), University of Tokyo.",
"pages": [
8
]
},
{
"title": "Facilities:",
"content": "Software: astropy (Astropy Collaboration et al. 2013, 2018, 2022), NumPy (Harris et al. 2020), matplotlib (Hunter 2007), SciPy (Virtanen et al. 2020), corner (Foreman-Mackey 2016) Cloudy (Ferland et al. 2013), emcee (Foreman-Mackey et al. 2013), msaexp (Brammer 2023) Prospector (Johnson et al. 2021b)",
"pages": [
8
]
},
{
"title": "REFERENCES",
"content": "Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068 Astropy Collaboration, Price-Whelan, A. M., Sip\"ocz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f Hunter, J. D. 2007, Computing in Science and Engineering, 9, 90, doi: 10.1109/MCSE.2007.55",
"pages": [
8,
9
]
}
]
2024arXiv241200191E
https://arxiv.org/pdf/2412.00191.pdf
<document>
<section_header_level_1><location><page_1><loc_38><loc_84><loc_62><loc_86></location>Euclid preparation</section_header_level_1>
<section_header_level_1><location><page_1><loc_6><loc_77><loc_94><loc_82></location>The impact of line-of-sight projections on the covariance between galaxy cluster multi-wavelength observable properties - insights from hydrodynamic simulations</section_header_level_1>
<text><location><page_1><loc_3><loc_22><loc_94><loc_79></location>Euclid Collaboration: A. Ragagnin ⋆ 1 , 2 , 3 , 4 , A. Saro 5 , 2 , 6 , 7 , 4 , S. Andreon 8 , A. Biviano 6 , 2 , K. Dolag 9 , S. Ettori 1 , 10 , C. Giocoli 1 , 11 , A. M. C. Le Brun 12 , G. A. Mamon 13 , 14 , B. J. Maughan 15 , M. Meneghetti 1 , 16 , L. Moscardini 3 , 1 , 16 , F. Pacaud 17 , G. W. Pratt 18 , M. Sereno 1 , 16 , S. Borgani 5 , 2 , 6 , 7 , F. Calura 1 , G. Castignani 1 , M. De Petris 19 , D. Eckert 20 , G. F. Lesci 3 , 1 , J. Macias-Perez 21 , M. Maturi 22 , 23 , A. Amara 24 , N. Auricchio 1 , C. Baccigalupi 2 , 6 , 7 , 25 , M. Baldi 26 , 1 , 16 , S. Bardelli 1 , D. Bonino 27 , E. Branchini 28 , 29 , 8 , M. Brescia 30 , 31 , 32 , J. Brinchmann 33 , S. Camera 34 , 35 , 27 , V. Capobianco 27 , C. Carbone 36 , J. Carretero 37 , 38 , S. Casas 39 , M. Castellano 40 , S. Cavuoti 31 , 32 , A. Cimatti 41 , C. Colodro-Conde 42 , G. Congedo 43 , C. J. Conselice 44 , L. Conversi 45 , 46 , Y. Copin 47 , F. Courbin 48 , H. M. Courtois 49 , A. Da Silva 50 , 51 , H. Degaudenzi 20 , G. De Lucia 6 , J. Dinis 50 , 51 , F. Dubath 20 , X. Dupac 46 , M. Farina 52 , S. Farrens 18 , S. Ferriol 47 , M. Frailis 6 , E. Franceschi 1 , M. Fumana 36 , K. George 9 , B. Gillis 43 , A. Grazian 53 , F. Grupp 54 , 9 , S. V. H. Haugan 55 , W. Holmes 56 , I. Hook 57 , F. Hormuth 58 , A. Hornstrup 59 , 60 , K. Jahnke 61 , E. Keihänen 62 , S. Kermiche 63 , A. Kiessling 56 , M. Kilbinger 18 , B. Kubik 47 , M. Kümmel 9 , M. Kunz 64 , H. Kurki-Suonio 65 , 66 , S. Ligori 27 , P. B. Lilje 55 , V. Lindholm 65 , 66 , I. Lloro 67 , D. Maino 68 , 36 , 69 , E. Maiorano 1 , O. Mansutti 6 , O. Marggraf 17 , K. Markovic 56 , M. Martinelli 40 , 70 , N. Martinet 71 , F. Marulli 3 , 1 , 16 , R. Massey 72 , S. Maurogordato 73 , E. Medinaceli 1 , S. Mei 74 , Y. Mellier 13 , 14 , G. Meylan 48 , M. Moresco 3 , 1 , E. Munari 6 , 2 , C. Neissner 75 , 38 , S.-M. Niemi 76 , J. W. Nightingale 77 , 72 , C. Padilla 75 , S. Paltani 20 , F. Pasian 6 , K. Pedersen 78 , V. Pettorino 76 , G. Polenta 79 , M. Poncet 80 , L. A. Popa 81 , L. Pozzetti 1 , F. Raison 54 , A. Renzi 82 , 83 , J. Rhodes 56 , G. Riccio 31 , E. Romelli 6 , M. Roncarelli 1 , E. Rossetti 26 , R. Saglia 9 , 54 , Z. Sakr 22 , 84 , 85 , A. G. Sánchez 54 , D. Sapone 86 , B. Sartoris 9 , 6 , R. Scaramella 40 , 70 , P. Schneider 17 , T. Schrabback 87 , A. Secroun 63 , E. Sefusatti 6 , 2 , 7 , G. Seidel 61 , S. Serrano 88 , 89 , 90 , C. Sirignano 82 , 83 , G. Sirri 16 , L. Stanco 83 , J. Steinwagner 54 , P. Tallada-Crespí 37 , 38 , I. Tereno 50 , 91 , R. Toledo-Moreo 92 , F. Torradeflot 38 , 37 , I. Tutusaus 84 , L. Valenziano 1 , 10 , T. Vassallo 9 , 6 , G. Verdoes Kleijn 93 , A. Veropalumbo 8 , 29 , Y. Wang 94 , J. Weller 9 , 54 , G. Zamorani 1 , E. Zucca 1 , M. Bolzonella 1 , A. Boucaud 74 , E. Bozzo 20 , C. Burigana 95 , 10 , M. Calabrese 96 , 36 , D. Di Ferdinando 16 , J. A. Escartin Vigo 54 , R. Farinelli 1 , J. Gracia-Carpio 54 , N. Mauri 41 , 16 , V. Scottez 13 , 97 , M. Tenti 16 , M. Viel 2 , 6 , 25 , 7 , 4 , M. Wiesmann 55 , Y. Akrami 98 , 99 , V. Allevato 31 , S. Anselmi 83 , 82 , 12 , M. Ballardini 100 , 1 , 101 , P. Bergamini 68 , 1 , A. Blanchard 84 , L. Blot 102 , 12 , S. Bruton 103 , R. Cabanac 84 , A. Calabro 40 , G. Canas-Herrera 76 , 104 , A. Cappi 1 , 73 , C. S. Carvalho 91 , T. Castro 6 , 7 , 2 , 4 , K. C. Chambers 105 , S. Contarini 54 , 3 , A. R. Cooray 106 , M. Costanzi 5 , 6 , 2 , B. De Caro 83 , 82 , S. de la Torre 71 , G. Desprez 107 , A. Díaz-Sánchez 108 , S. Di Domizio 28 , 29 , H. Dole 109 , S. Esco ffi er 63 , A. G. Ferrari 41 , 16 , P. G. Ferreira 110 , I. Ferrero 55 , F. Finelli 1 , 10 , F. Fornari 10 , L. Gabarra 110 , K. Ganga 74 , J. García-Bellido 98 , E. Gaztanaga 89 , 88 , 111 , F. Giacomini 16 , G. Gozaliasl 112 , 65 , A. Hall 43 , H. Hildebrandt 113 , J. Hjorth 114 , A. Jimenez Muñoz 21 , J. J. E. Kajava 115 , 116 , V. Kansal 117 , 118 , D. Karagiannis 119 , 120 , C. C. Kirkpatrick 62 , L. Legrand 121 , G. Libet 80 , A. Loureiro 122 , 123 , G. Maggio 6 , M. Magliocchetti 52 , F. Mannucci 124 , R. Maoli 19 , 40 , C. J. A. P. Martins 125 , 33 , S. Matthew 43 , L. Maurin 109 , R. B. Metcalf 3 , 1 , P. Monaco 5 , 6 , 7 , 2 , C. Moretti 25 , 4 , 6 , 2 , 7 , G. Morgante 1 , Nicholas A. Walton 126 , L. Patrizii 16 , A. Pezzotta 54 , M. Pöntinen 65 , V. Popa 81 , C. Porciani 17 , D. Potter 127 , I. Risso 128 , P.-F. Rocci 109 , M. Sahlén 129 , A. Schneider 127 , M. Schultheis 73 , P. Simon 17 , A. Spurio Mancini 130 , 131 , C. Tao 63 , G. Testera 29 , R. Teyssier 132 , S. Toft 60 , 133 , 134 , S. Tosi 28 , 29 , 8 , A. Troja 82 , 83 , M. Tucci 20 , C. Valieri 16 , J. Valiviita 65 , 66 , D. Vergani 1 , and G. Verza 135 , 136 arXiv:2412.00191v1 [astro-ph.CO] 29 Nov 2024</text>
<text><location><page_1><loc_36><loc_20><loc_64><loc_21></location>(A ffi liations can be found after the references)</text>
<text><location><page_1><loc_46><loc_17><loc_54><loc_18></location>June 16, 2023</text>
<section_header_level_1><location><page_1><loc_46><loc_14><loc_54><loc_14></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_6><loc_10><loc_94><loc_12></location>Context. Cluster cosmology can benefit from combining multi-wavelength studies, which themselves can benefit from a characterisation of the correlation coe ffi cients between di ff erent mass-observable relations.</text>
<text><location><page_2><loc_6><loc_90><loc_94><loc_93></location>Aims. In this work, we aim to provide information on the scatter, the skewness, and the covariance of various mass-observable relations in galaxy clusters in cosmological hydrodynamic simulations. This information will help future analyses to better tackle accretion histories and projection e ff ects and model mass observable relations for cosmology studies.</text>
<text><location><page_2><loc_6><loc_85><loc_94><loc_90></location>Methods. Weidentify galaxy clusters in Magneticum Box2b simulations with mass M 200c > 10 14 M ⊙ at redshift z = 0 . 24 and z = 0 . 90. Our analysis includes Euclid -derived properties such as richness, stellar mass, lensing mass, and concentration. Additionally, we investigate complementary multi-wavelength data, including X-ray luminosity, integrated Comptony parameter, gas mass, and temperature. The impact of projection e ff ects on mass-observable residuals and correlations is then examined.</text>
<text><location><page_2><loc_6><loc_78><loc_94><loc_85></location>Results. We find that at intermediate redshift ( z = 0 . 24) projection e ff ects impact lensing concentration, richness, and gas mass the most in terms of scatter and skewness of log-residuals of scaling relations. The contribution of projection e ff ects can be significant enough to boost a spurious hot- vs. cold-baryons correlation and consequently hide underlying correlations due to halo accretion histories. At high redshift ( z = 0 . 9), the richness has a much lower scatter (of log-residuals), and the quantity that is most impacted by projection e ff ects is the lensing mass. Lensing concentration reconstruction, in particular, is a ff ected by deviations of the reduced-shear profile shape from the one derived by an NFW profile rather than interlopers in the line of sight.</text>
<paragraph><location><page_2><loc_6><loc_76><loc_82><loc_77></location>Key words. Galaxies: clusters: general - Cosmology: cosmological parameters - galaxies: abundances - methods: numerical</paragraph>
<section_header_level_1><location><page_2><loc_6><loc_72><loc_18><loc_73></location>1. Introduction</section_header_level_1>
<text><location><page_2><loc_6><loc_58><loc_49><loc_71></location>Galaxy clusters are the largest gravitationally bound, collapsed, and virialised structures in our Universe and represent unique laboratories for testing cosmological models, galaxy evolution, and thermodynamics of the intracluster medium (ICM, see Kravtsov & Borgani 2012, for a review on galaxy clusters). Regarding galaxy cluster cosmology studies (see, e.g., Rozo et al. 2010; Bocquet et al. 2019), an accurate characterisation of the selection function and of the mass-observable scaling relations represent the dominant systematic uncertainties (see the review on the cluster mass scale in Pratt et al. 2019).</text>
<text><location><page_2><loc_6><loc_37><loc_49><loc_57></location>Cluster masses cannot be observed directly, and their reconstruction requires both a number of assumptions and highquality data (see, e.g., Meneghetti et al. 2010). This means that precise estimates are rare (Okabe et al. 2010; Hoekstra et al. 2012; Melchior et al. 2015; van Uitert et al. 2016; Stern et al. 2019; Sugiyama et al. 2023; Bocquet et al. 2024). Once a set of highly accurate mass determinations are available, together with other mass proxies recovered from multi-band observations, well-calibrated mass-observable relations (for instance, the mass-richness relation or the mass-temperature relation) can be established and used to estimate galaxy cluster masses for larger samples with known observable properties. For this purpose, it is important to calibrate accurately the mass-observable relations (Giodini et al. 2013; Allen et al. 2011; Schrabback et al. 2021), including proper modelling of their associated scatter (Lima & Hu 2005; Bocquet et al. 2019).</text>
<text><location><page_2><loc_6><loc_12><loc_49><loc_37></location>This process is complicated by the fact that studies at di ff erent wavelengths are biased by various astrophysical processes and projection e ff ects to various degrees. For instance, X-ray surveys tend to favour the selection of clusters with centrally peaked gas distributions (Pacaud et al. 2007; Hudson et al. 2010; Andreon & Moretti 2011; Andreon et al. 2016; Xu et al. 2018) and su ff er from AGN contamination (see, e.g., Bhargava et al. 2023), while projection e ff ects are known to strongly impact weak lensing mass reconstructions (Meneghetti et al. 2014; Euclid Collaboration: Giocoli et al. 2024) and richness evaluations (e.g., Castignani & Benoist 2016). This is particularly relevant for cluster cosmology studies, where the aim is to reduce uncertainty by combining constraints on di ff erent massobservable relations. For Euclid (Euclid Collaboration: Mellier et al. 2024), this will include quantities such as richness, stellar mass, and properties of stacked weak lensing signals (Pires et al. 2020), of the cluster samples detected using tools such as AMICO (Bellagamba et al. 2018a; Maturi et al. 2019) or PZWav (Euclid Collaboration: Adam et al. 2019), possibly to-</text>
<text><location><page_2><loc_51><loc_57><loc_94><loc_73></location>gether with other multi-wavelength observations Allen et al. (2011). These properties are known to be biased by projection e ff ects (Meneghetti et al. 2014), accretion histories (Ragagnin et al. 2022), mis-centring (Sommer et al. 2022, 2023), and the fit procedure (Sereno et al. 2016). Projection e ff ects, especially, are expected to generate some covariance between the richness and weak lensing signal, the uncertainty of which may significantly a ff ect the performance of the mission for cluster population analyses. This e ff ect is one the major sources of systematics for current optical cluster surveys (Costanzi et al. 2019; Abbott et al. 2020), and thus is expected to play an even more critical role for the Euclid cluster sample.</text>
<text><location><page_2><loc_51><loc_42><loc_94><loc_56></location>Numerical simulations are thus a critical tool to mitigate the impact of the aforementioned biases on cosmological cluster studies. Indeed, the power of observations to constrain them is limited, thus increasing the final uncertainty budget. However, scatter and covariance parameters are also prime sources of uncertainty when aiming to combine information originating from di ff erent wavelengths. For instance, various observational works hint towards di ff erent directions for the hot- vs. cold-baryon covariance (Farahi et al. 2019; Puddu & Andreon 2022; Ragagnin et al. 2022), as di ff erent formation times are related with satellite accretion history (Giocoli et al. 2008).</text>
<text><location><page_2><loc_51><loc_21><loc_94><loc_41></location>In this context, numerical simulations have proven to be a very powerful tool for helping observational studies in modelling mass-observable relations, which are strongly a ff ected by galaxy cluster accretion histories (Ludlow et al. 2012; Bose et al. 2019; Davies et al. 2020; Anbajagane et al. 2020; Ragagnin et al. 2022), projection e ff ects (Meneghetti et al. 2014), and deviations (see, e.g., Ragagnin et al. 2021) from the Navarro-FrenkWhite density profile (NFW, Navarro et al. 1997), which is often adopted in weak lensing studies. Thus, simulations can suggest the most suitable functional forms of scaling relations for cosmological studies (as in the works of Costanzi et al. 2019; Bocquet et al. 2016, 2019; Ghirardini et al. 2024). They can provide informative priors on their correlation coe ffi cients, which are among the most di ffi cult parameters to be constrained directly from observed quantities, guiding the forward modelling setup of cluster cosmology studies.</text>
<text><location><page_2><loc_51><loc_10><loc_94><loc_20></location>There are various works in the literature that study how simulations can help disentangle physical models (see e.g., Cui et al. 2022; Angelinelli et al. 2023a) cosmological models (see e.g., Bocquet et al. 2020; Angulo et al. 2021; Villaescusa-Navarro et al. 2022) or dark matter types (see e.g., Ragagnin et al. 2024; Fischer et al. 2024; Contreras-Santos et al. 2024), and study observable cross-correlations (see e.g., Stanek et al. 2010; Anbajagane et al. 2020).</text>
<text><location><page_3><loc_6><loc_74><loc_49><loc_93></location>In this work, besides focusing on correlations between observable properties of interests for multi-wave length studies, we also study the impact of projection e ff ects. The impact of uncorrelated large-scale structure on the covariance between observable properties can be modelled analytically (Hoekstra 2003; McClintock et al. 2019; Costanzi et al. 2019), but the covariance of di ff erent observable properties below a few tens of Mpc requires dedicated simulations. At these scales, numerical hydrodynamic simulations, with their self-consistent depiction of the ICM, emerge as an ideal tool for exploring multiwavelength observable properties since they incorporate the effects of large-scale structures within which clusters are situated. Indeed, baryon feedback influences the ICM not only within cluster virial radii but also beyond (see, e.g., Angelinelli et al. 2022, 2023b).</text>
<text><location><page_3><loc_6><loc_54><loc_49><loc_73></location>While it is true that cosmological simulations are influenced by the underlying sub-grid prescriptions, and while it is true that these simulations may diverge on small scales, they generally exhibit agreement on quantities integrated up to the sizes of galaxy groups and clusters (see, e.g., Anbajagane et al. 2020). At the same time, di ff erent cosmological parameters can a ff ect galaxy cluster properties, such as their masses (Ragagnin et al. 2021), satellite abundance (van den Bosch et al. 2005), and massobservable relations (Singh et al. 2020). On the other hand, the qualitative significance of covariances and projection e ff ects on observable properties is not expected to significantly hinge on cosmological parameters (Bocquet et al. 2019), and possible deviations from this expectation could be estimated using emulators (see, e.g., Bocquet et al. 2020; Ragagnin et al. 2021; Angulo et al. 2021; Ragagnin et al. 2023).</text>
<text><location><page_3><loc_6><loc_48><loc_49><loc_54></location>We will study the impact of projection e ff ects using hydrodynamic simulations in order to gain insight into which fraction of the scatter and skewness of scaling relations originates from projection e ff ects (i.e., alignment with filaments and objects) or di ff erent accretion histories.</text>
<text><location><page_3><loc_6><loc_33><loc_49><loc_47></location>In Sect. 2, we present how we set up our Euclid -like observable properties and the others coming from the other wavelengths. In Sect. 3, we study how projection e ff ects impact the scatter and skewness of log-residuals of scaling relations and discuss the impact of projection e ff ects on observable covariance. In Sect. 4, we focus on the mass-concentration relation and how it is a ff ected by projection e ff ects and deviations from the functional form of profiles and the radial ranges of the fits. In Sect. 5, we focus on the covariance between observable properties and study how di ff erent accretion histories and projection e ff ects impact them. Finally, we draw our conclusions in Sect. 6.</text>
<section_header_level_1><location><page_3><loc_6><loc_30><loc_22><loc_31></location>2. Numerical Setup</section_header_level_1>
<text><location><page_3><loc_6><loc_12><loc_49><loc_29></location>We will conduct our study by analysing clusters obtained from the Magneticum 1 hydrodynamic cosmological simulations (Bi ffi et al. 2013; Saro et al. 2014; Steinborn et al. 2015; Dolag et al. 2016, 2015; Teklu et al. 2015; Steinborn et al. 2016; Bocquet et al. 2016; Ragagnin et al. 2019). They are based on the N -body code Gadget3 , which is built upon Gadget2 (Springel et al. 2005b; Springel 2005; Boylan-Kolchin et al. 2009) with an improved smoothed particle hydrodynamics (SPH) solver from Beck et al. (2016). Magneticum initial conditions are generated using a standard Λ CDM cosmology with Wilkinson Microwave Anisotropy Probe 7 (Komatsu et al. 2011) cosmological parameters. The large-scale structure evolution in Magneticum simulations includes a treatment of radiative cooling, heating</text>
<text><location><page_3><loc_51><loc_74><loc_94><loc_93></location>from a uniform redshift-dependent ultraviolet (UV) background, star formation, and stellar feedback processes as in Springel et al. (2005a). The stellar feedback is then connected to a detailed chemical evolution and enrichment model as in Tornatore et al. (2007), which follows 11 chemical elements (H, He, C, N, O, Ne, Mg, Si, S, Ca, and Fe, with cooling tables from Wiersma et al. 2009) which are produced with the CLOUDY photo-ionisation code (Ferland et al. 1998). Fabjan et al. (2010) and Hirschmann et al. (2014) described prescriptions for black hole growth and feedback from AGNs. Haloes that host galaxy clusters and groups are identified using the friends-of-friends halo finder (Davis et al. 1985), and subhaloes together with their associated galaxies are identified with an improved version of SUBFIND (Springel et al. 2001), which takes into account the presence of baryons (Dolag et al. 2009).</text>
<text><location><page_3><loc_51><loc_70><loc_94><loc_73></location>We define r ∆ c as the radius that encloses an average density of ∆ c ρ cr , where ρ cr is the critical density of the Universe at a given redshift,</text>
<formula><location><page_3><loc_51><loc_66><loc_94><loc_69></location>M ∆ c = 4 3 π r 3 ∆ c ∆ c ρ cr . (1)</formula>
<text><location><page_3><loc_51><loc_63><loc_94><loc_65></location>Throughout this paper, when we omit ∆ c from masses and radii we imply the usage of ∆ c = 200 (i.e., M = M 200c) .</text>
<text><location><page_3><loc_51><loc_41><loc_94><loc_62></location>To disentangle the scatter of the mass-observable relation from projection e ff ects, we compute quantities within a sphere of radius r 200c and as integrated into a cylinder. Projected quantities will be denoted with the superscript 2D (for instance, the total mass inside the cylinder is denoted as M 2D ). We opted to employ a random projection plane for each cluster. Additionally, we set an integration depth of 20 comoving h -1 Mpc, corresponding to approximately 23 Mpc at z = 0 . 24 and 15 Mpc at z = 0 . 9 (with h = 0 . 704). This cylinder depth is smaller than Euclid 's galaxy cluster photoz equivalent uncertainty (Euclid Collaboration: Desprez et al. 2020) and, while we exclude some uncorrelated projection e ff ects, they are known not to play an important role (Sunayama et al. 2020; Wu et al. 2022). Thus, we ensure that we do not overestimate any projection e ff ect that could be mitigated using photoz . Consequently, all projection e ff ects examined in this paper hold relevance for interpreting forthcoming Euclid -based catalogues.</text>
<text><location><page_3><loc_51><loc_12><loc_94><loc_40></location>This study is based on the results from Box2b / hr (Hirschmann et al. 2014) Magneticum simulation, which covers a length of 900 comoving Mpc, with dark matter particle masses m DM = 9 . 8 × 10 8 M ⊙ , gas initial particle masses of m gas = 2 × 10 8 M ⊙ , and a gravitational softening of both gas and dark matter of ϵ = 3 . 75 comoving kpc. Euclid is expected to detect clusters with masses M > 10 14 M ⊙ up to a redshift of approximately z ≈ 2 (Sartoris et al. 2016; Euclid Collaboration: Adam et al. 2019), where the bulk of the cluster population which will be used for mass-calibration will lie below redshift z ≈ 1 . Furthermore, the number of haloes contained in this Magneticum simulation drops significantly beyond the same redshift value. Therefore, we decided to extract haloes at two representative redshift slices: at an intermediate redshift of approximately z ≈ 0 . 24, yielding 4300 objects, and at a higher redshift of approximately z ≈ 0 . 9, yielding 1300 objects. These extractions were performed using the web portal 2 introduced in Ragagnin et al. (2017). We focus most of the analyses on the qualitative e ff ect of projection e ff ects at our intermediate redshift slice because of the larger statistics of clusters to help us determine projection e ff ects. We stress that this mass threshold is high enough so that all of our galaxy clusters have at least</text>
<text><location><page_4><loc_6><loc_91><loc_49><loc_93></location>10 4 particles and, therefore, can be considered well resolved in terms of density profile fitting (Navarro et al. 1997).</text>
<section_header_level_1><location><page_4><loc_6><loc_87><loc_25><loc_88></location>2.1. Observable properties</section_header_level_1>
<text><location><page_4><loc_6><loc_56><loc_49><loc_86></location>We now discuss the properties that we compute for each cluster and report a summary in Table 1. We compute the total stellar masses M ⋆ and M 2D ⋆ as the sum of all stellar particles within the respective volumes. We compute the richness n with a cut of satellites of log 10 ( M ⋆/ M ⊙ ) > 10 . 65 , and the projected version n 2D that includes Euclid -like corrections for projection effects where similarly to Andreon et al. (2016). In particular, we compute the average projected richness between 3 . 5 and 8 Mpc radii from the cluster centre, divided the annulus in 8 slices of equal angles, excluded the two least dense and most dense slices and removed the average projected number density of the 4 remaining octants from the projected richness within r 200c; 3 We compute the X-ray luminosities LX , and L 2D X , in the [0 . 5 , 2] keV energy band computed using the APEC model (Smith et al. 2001), using SPH particle temperatures together with the XSPEC package 4 (Arnaud 1996), which considers the emission of a collisionally-ionized, chemically-enriched plasma implemented with metallicity values taken from the simulated particles 5 . We compute the temperature T and T 2D as weighted by the X-ray emissivity of gas particles. We compute the hot gas mass M g and M 3D g , computed as the sum of the mass of SPH particles with cold gas fraction greater than 0 . 1 and T > 3 × 10 4 K in order to filter out cold gas.</text>
<text><location><page_4><loc_6><loc_43><loc_49><loc_56></location>Note that the projected gas mass is not to be confused with the one inferred from X-ray observations, as X-ray observational works typically perform a de-projection of the surface brightness ∝ n 2 e , which provides a gas-mass estimate that is closer to the spherical M g , with the addition of some possible alignment e ff ects coming from the central region of clusters. Moreover, observational works have the capability to mask possible bright substructures, thus minimising the presence of interlopers. Consequently, we can conceptualise the observed projected gas mass as an intermediate value between our M g and M 2D g .</text>
<text><location><page_4><loc_6><loc_39><loc_49><loc_43></location>We estimate the integrated Comptony parameter produced by thermal Sunyaev-Zeldovich (SZ, Sunyaev & Zeldovich 1972). The Comptony parameter is defined as</text>
<formula><location><page_4><loc_6><loc_35><loc_49><loc_38></location>y = k B σ T m e c 2 Z T n e d l , (2)</formula>
<text><location><page_4><loc_6><loc_26><loc_49><loc_34></location>where T is the temperature, n e the number density of the electrons, k B the Boltzmann constant, σ T the Thomson cross-section, c the speed of light, and m e the electron rest mass. We compute the integrated Comptony parameter Y = R y d Ω , both within the volume of sphere of ( Y ) and a cylinder ( Y 2D ). We estimate the integral in Eq. (2) as</text>
<formula><location><page_4><loc_6><loc_22><loc_49><loc_25></location>Z T n e d l ≈ 1 π R 2 X i Ti f e , i mi m p , (3)</formula>
<text><location><page_4><loc_51><loc_88><loc_94><loc_93></location>where the sum runs over all SPH particles, mi is the i -th SPH particle mass, Ti its temperature and f e , i is its electron fraction, expressed as local electron number density normalised to the hydrogen number density, and m p is the proton mass.</text>
<text><location><page_4><loc_51><loc_85><loc_94><loc_88></location>For each halo, we also perform fits of the NFW profile ρ NFW, defined as</text>
<formula><location><page_4><loc_51><loc_82><loc_94><loc_85></location>ρ NFW ( r ) = ρ 0 r / r s (1 + r / r s) 2 , (4)</formula>
<text><location><page_4><loc_51><loc_68><loc_94><loc_81></location>where the scaling density ρ 0 and the scale radius r s (that is the radius where the density log-slope equals -2) are free parameters. We perform this fit on the total matter density profile on 100 log-spaced radial bins between 75 ckpc (which corresponds to 60 kpc at z = 0 . 24 , and to 40 kpc at z = 0 . 9; as it is enough to exclude the deep central potential of baryons) and r 200c , and define the corresponding NFW masses and concentration parameters as M NFW and c NFW respectively, and the concentration as c NFW = r NFW / r s , where r NFW is obtained from M NFW via the Eq. (1).</text>
<text><location><page_4><loc_51><loc_55><loc_94><loc_68></location>The projected version of the mass and concentrations are obtained by mimicking a lensing reconstruction procedure by fitting the corresponding reduced shear. We define the derived masses and concentrations as M 2D NFW and c 2D NFW , where c 2D NFW = R 2D NFW / r s , where R 2D NFW is obtained from M 2D NFW via the Eq. (1). Note that the "2D" here, as for the other quantities, means that the quantity is computed in projection, however, a correct fit of the mass from reduced shear NFW profile (namely, our M 2D NFW ) should provide an estimate of the same NFW halo mass M NFW that would be recovered from a 3D fit.</text>
<text><location><page_4><loc_51><loc_41><loc_94><loc_55></location>The fit is computed within the cylinder described above, with a projected radial range of [300 , 3000] kpc at z = 0 . 2 . We performed the analyses at z = 0 . 9 by rescaling that range with H -2 / 3 ( z ) , where H ( z ) is the Hubble parameter, in order to retain the same fractional distances from the virial radius (at fixed mass), which resulted in a range of [234 , 2300] kpc . Note that in this work we are not interested in estimating the contribution of the uncorrelated large-scale structure in the reduced shear reconstruction, therefore we limit our density projection to a cylinder of the depth of 20 cMpc , (see, e.g., Euclid Collaboration: Giocoli et al. 2024; Becker & Kravtsov 2011).</text>
<text><location><page_4><loc_51><loc_37><loc_94><loc_41></location>The signal from the source-averaged excess surface mass density ∆Σ gt , averaged over circular radii R and a population of sources distributed in redshift, can be written as</text>
<formula><location><page_4><loc_51><loc_33><loc_94><loc_36></location>D ∆Σ gt E ( R ) ≃ ⟨ ∆Σ t ⟩ ( R ) 1 -D Σ -1 cr E ⟨ Σ ⟩ ( R ) . (5)</formula>
<text><location><page_4><loc_51><loc_24><loc_94><loc_32></location>Here Σ denotes the surface mass density. The symbol ⟨ ... ⟩ denotes an average over radial bins and redshift lens sources, where we used a redshift distribution as proposed in Euclid Collaboration: Ajani et al. (2023), and Euclid Collaboration: Giocoli et al. (2024). The quantity ⟨ ∆Σ t ⟩ is the excess of surface mass density, averaged over polar coordinates and defined as</text>
<formula><location><page_4><loc_51><loc_20><loc_94><loc_23></location>⟨ ∆Σ t ⟩ ( R ) = 1 π R 2 Z R 0 2 π r ⟨ Σ ⟩ ( r ) d r - ⟨ Σ ⟩ ( R ) . (6)</formula>
<text><location><page_4><loc_51><loc_17><loc_94><loc_19></location>The symbol Σ cr in Eq. (5) is the critical surface mass density, that for a given redshift source equals to</text>
<formula><location><page_4><loc_51><loc_13><loc_94><loc_16></location>Σ cr = c 2 D s 4 π GD ds D d , (7)</formula>
<text><location><page_4><loc_51><loc_10><loc_94><loc_12></location>where G the universal gravity constant, D s the angular diameter distance to the source, D d the angular distance to the lens, and</text>
<table>
<location><page_5><loc_6><loc_75><loc_95><loc_91></location>
<caption>Table 1: List of observable properties used in this work and presented in Sect. 2.1</caption>
</table>
<text><location><page_5><loc_6><loc_69><loc_49><loc_72></location>D ds the angular distance between the source and the lens. Similarly to Euclid Collaboration: Giocoli et al. (2024), we define the error associated with each radial bin of the profile in Eq. (5) as</text>
<formula><location><page_5><loc_6><loc_64><loc_49><loc_67></location>δ D ∆Σ gt E = ⟨ Σ cr ⟩ σϵ q π n g GLYPH<16> R 2 2 -R 2 1 GLYPH<17> , (8)</formula>
<text><location><page_5><loc_6><loc_52><loc_49><loc_63></location>where σϵ = 0 . 3 (Hoekstra et al. 2012; Euclid Collaboration: Blanchard et al. 2020) is the dispersion of the total intrinsic ellipticity ϵ = (1 -q ) / (1 + q ) , where q is the axis ratio, R 1 and R 2 represent the inner and outer radius of a bin, and n g is the number density of galaxies. For our redshift source distribution (we assume the same as in Euclid Collaboration: Giocoli et al. 2024), we find that n g ≈ 28 arcmin -2 for lenses at redshift z = 0 . 24 and n g ≈ 14 arcmin -2 for lenses at redshift z = 0 . 9 .</text>
<section_header_level_1><location><page_5><loc_6><loc_49><loc_21><loc_50></location>2.2. Scaling relations</section_header_level_1>
<text><location><page_5><loc_6><loc_35><loc_49><loc_48></location>In Fig. 1 we show the observable properties vs. true mass M of clusters, derived from Magneticum Box2b / hr simulation for properties that could be derived by Euclid -like catalogues, such as lensing concentration (first row from top), lensing mass (second row), projected richness (third row), and projected stellar mass (last row), as presented in Sect. 2.1. For each property, we fit a scaling relation performed using a linear regression in the log-log space. We utilise a log-log linear regression because a single power law proves to be e ff ective in modelling our scaling relations.</text>
<text><location><page_5><loc_6><loc_25><loc_49><loc_35></location>In the right panel of Fig. 1, we show the log-residual distribution for both low-mass haloes ( M < 2 × 10 14 M ⊙ ), highmass haloes ( M > 2 × 10 14 M ⊙ ), and for the complete sample of the log-residuals σ ln , i , defined as the logarithmic ratio between the i -th cluster property and the corresponding scaling relation value at its mass. In the second column, we also report the logscatter σ ln defined here as the corresponding standard deviation of the log-residual, namely</text>
<formula><location><page_5><loc_6><loc_21><loc_49><loc_23></location>σ ln = E h GLYPH<0> σ ln , i -E GLYPH<2> σ ln , i GLYPH<3>GLYPH<1> 2 i 1 / 2 , (9)</formula>
<text><location><page_5><loc_6><loc_12><loc_49><loc_20></location>where E is the expectation operator that averages over our catalogue data. We note that the concentration has a scatter of 0 . 45 which is higher than theoretical expectations (see, e.g., Child et al. 2018). Throughout this paper, we will show that this is due to projection e ff ects; in fact, the 3D concentration has a scatter of ≈ 0 . 33 .</text>
<text><location><page_5><loc_6><loc_10><loc_49><loc_12></location>Note that our scatter in temperature exceeds that reported in the theoretical work by Truong et al. (2018). We verified that, if</text>
<figure>
<location><page_5><loc_52><loc_38><loc_93><loc_72></location>
<caption>Fig. 1: Magneticum mass-observable relation for Euclid -like derived quantities. The left column shows scaling relations, relative fit (solid grey line), and a corridor corresponding to one standard deviation (dashed grey line). The right column shows the residual PDF and scatter of log-residuals σ ln . We report the following properties: lensing concentration c 2D NFW (first row), lensing mass M 2D NFW (second row), projected richness n 2D (third row), and projected stellar mass M 2D ⋆ (fourth row). The histogram of residuals for haloes with M < 2 × 10 14 M ⊙ is in blue dotted lines, for haloes with M > 2 × 10 14 M ⊙ is in orange dashed lines, and for the complete sample is in solid grey lines. Note that the three histograms almost overlap. Each distribution panel reports the value of the natural log scatter σ ln for the complete sample.</caption>
</figure>
<text><location><page_5><loc_51><loc_10><loc_94><loc_15></location>we compute mass-weighted temperature, which is known to behave very well in a power law scaling relation, reveals a log scatter of 0 . 07 , in agreement with their work. The additional scatter that we see may be due to di ff erent X-ray temperature computa-</text>
<figure>
<location><page_6><loc_7><loc_59><loc_48><loc_93></location>
<caption>Fig. 2: As Fig. 1, but for the following multi-wavelength observable properties: projected integrated Comptony parameter Y 2D (first row), the 3D gas mass M g , 500c (second row), the projected X-ray luminosity in the soft band (in range [0 . 5 , 2] keV) L 2D X , 500c (third row), and the projected temperature T 2D 500c (fourth row). The values of X-ray luminosity, gas mass, and temperatures are reported within an overdensity of r 500c as this definition is a typical choice in X-ray-based observations.</caption>
</figure>
<text><location><page_6><loc_6><loc_40><loc_49><loc_42></location>tions (they use core-excised temperature while we take the contribution of the core into account).</text>
<text><location><page_6><loc_6><loc_28><loc_49><loc_40></location>In Fig. 2 we show the mass-observable relations of quantities that could potentially be obtained from various multi-wavelength observations that could enrich studies based on Euclid -like data products: the integrated Comptony parameter, the gas mass M g , 500c, the X-ray luminosity L 2D X , 500c converted in the soft band [0 . 5 , 2] keV , and the temperature T 500c . Wedecided to plot the Xray luminosity, gas mass, and temperature within r 500c because this radius is typically used in various X-ray observations (see, e.g., Vikhlinin et al. 2006; Sun et al. 2009).</text>
<text><location><page_6><loc_6><loc_10><loc_49><loc_28></location>The typical Euclid cluster cosmology analysis will therefore rely on mass-observable relations calibrated within r 200 c (e.g., richness, weak-lensing mass), and follow-up observations calibrated within r 500 c (e.g., X-ray and SZ mass-proxies). We will thus need to take into account the covariance between observable properties extracted at di ff erent radii. . Finally, we note that generally, X-ray observations are expected to align more closely with the 3D mass rather than the projected one (Ettori et al. 2013), although several details adopted to analyse the Xray observations (e.g., including masking of substructures, deprojection procedures, etc.) can significantly impact the final result. These choices critically depend on the quality of the observations themselves. Dedicated mocks will thus be required to properly take into account all these e ff ects and are, therefore,</text>
<text><location><page_6><loc_51><loc_89><loc_94><loc_93></location>beyond the purpose of this work. For simplicity, in this work, we consider an X-ray-derived gas mass as close to M g , while an SZ-derived gas mass closer to M 2D g .</text>
<section_header_level_1><location><page_6><loc_51><loc_85><loc_68><loc_86></location>3. Projection effects</section_header_level_1>
<text><location><page_6><loc_51><loc_69><loc_94><loc_84></location>The main objective of this work is to disentangle the amount of scatter and skewness in scaling relations that is purely due to projection e ff ects. Note that observational data are also a ff ected by measurement errors that we do not tackle in this work (as, for instance, the Poisson error of the limited number of galaxies used to infer the richness). In this Section, we discuss the scatter and skewness of mass-observable relation qualitatively and limit the discussion for the data at redshift z = 0 . 24 , as we have a larger sample of galaxy clusters, and the results are qualitatively similar to the ones at z = 0 . 9. We stress that we leave the quantitative discussion of the scatter and correlation coe ffi cients for both redshifts on Sect. 5.</text>
<text><location><page_6><loc_51><loc_59><loc_94><loc_68></location>To assess the role of projection e ff ects, Fig. 3 reports the 3D and 2D scatter of our cluster properties. In the left panel of Fig. 3 we show the value of the scatter (see Eq. 9) of log-residuals σ ln for all our mass-observable relations. In the shaded region, we also report the values computed within r 500c for the X-ray luminosity, gas mass, and temperature values because this is the characteristic overdensity used in X-ray analyses.</text>
<text><location><page_6><loc_51><loc_50><loc_94><loc_59></location>For each observable, in Fig. 3, we report (with di ff erent symbols) both the scatter of the complete sample as well as the one of two separate mass range of 10 14 < M < 2 × 10 14 M ⊙ and M > 2 × 10 14 M ⊙ respectively. We note that the lower-mass bin ( M < 2 × 10 14 M ⊙ ) is the one with the largest scatter because, for a given external object in the line-of-sight (LoS), the profile of a small cluster will be more perturbed with respect to a cluster.</text>
<text><location><page_6><loc_51><loc_32><loc_94><loc_50></location>In the left panel of Fig. 3, we see that some quantities have a low scatter in the 3D space and do gain a large amount of scatter once they are seen in projection. To better quantify what is the actual impact of projection e ff ects, in the right panel of Fig. 3 we present the metric q σ 2 ln , 2D -σ 2 ln , 3D . This metric shows that the quantities that are most a ff ected by projection e ff ects are the weak lensing concentration, integrated Comptony parameter, gas mass, and NFW profile lensing mass. This is expected as all these observable properties (except for weak lensing concentration and X-ray luminosity) scale linearly with the respective observable mass. Further, we note that the scatter in the richness agrees to the theoretical predictions from Castignani & Benoist (2016).</text>
<text><location><page_6><loc_51><loc_16><loc_94><loc_32></location>We observe that X-ray luminosity and temperature are the least a ff ected by projection e ff ects. This is attributed to the fact that X-ray luminosity is contingent upon the square of gas density, thereby being primarily influenced by the most bright regions within an image. Similarly, the temperature is predominantly influenced by the innermost regions of a cluster. Note that we lack the value of projection e ff ects for L X , 500c because the 2D scatter is slightly smaller than the 3D one. This happened because projection e ff ects impact under-luminous haloes more strongly than overly luminous haloes (at a fixed mass bin), with the consequence of the projected X-ray luminosity having a higher normalisation and a lower scatter (see Fig. B.1).</text>
<text><location><page_6><loc_51><loc_9><loc_94><loc_16></location>Some mass-observable relations have a large skewness, to aid observational works in modelling these relations, we will estimate their skewness. Therefore, we also quantify deviations of residuals from a symmetrical distribution by means of the Fisher-Pearson coe ffi cient of skewness m 3 / m 3 / 2 2 , where mk is the</text>
<figure>
<location><page_7><loc_7><loc_72><loc_93><loc_93></location>
<caption>Fig. 3: Scatter of our mass-observable relations, paired with their projected version: concentration, integrated Comptony parameter, gas mass, NFW mass, richness, stellar mass, X-ray luminosity, and temperature, within a radius of r 200c; the grey band reports the gas mass, X-ray luminosity, and temperature within r 500c . The left panel reports the fractional scatter of both 3D and 2D quantities. Each dotted vertical line separates the regions that report a given quantity and its 2D version. The right panel reports the contribution of projection e ff ects. Points are coloured by their mass range as in Fig. 1, blue down-triangles represent the low mass bin, orange up-triangles represent the high mass bin, and grey crosses represent the complete sample. Note that we lack the value of projection e ff ects for L X , 500c (see discussion). Points are ordered according to the value of the second panel. Error bars are computed using the jackknife method.</caption>
</figure>
<figure>
<location><page_7><loc_7><loc_38><loc_48><loc_56></location>
<caption>Fig. 4: Skewness parameters for our cluster properties. Data, line styles and colours are as in Fig. 3. Error bars are computed using the jackknife method.</caption>
</figure>
<text><location><page_7><loc_6><loc_16><loc_49><loc_29></location>sample k th central moment. 6 We report its dependency on projection e ff ects in Fig. 4, showing that the properties whose skewness is most impacted by projection e ff ects are the integrated Comptony parameter, the gas mass, and the lensing NFW mass. We also notice that some scaling relation residuals move from having a negative skewness (for the NFW concentration, for instance, due to un-relaxed and merging clusters) to a positive one once projection e ff ects are taken into account (namely, from an asymmetry towards negative residuals towards an asymmetry towards positive residuals).</text>
<figure>
<location><page_7><loc_52><loc_42><loc_89><loc_56></location>
<caption>Fig. 5: Probability density distribution of M 2D / M at di ff erent mass bins: for haloes with M < 2 × 10 14 M ⊙ as a blue dotted line, for haloes with M > 2 × 10 14 M ⊙ as a dashed orange line and for the complete sample as a grey solid line. For each mass bin, we report a vertical line with the median values (note that the three lines are very close together).</caption>
</figure>
<text><location><page_7><loc_51><loc_10><loc_94><loc_28></location>To assess the impact of projection e ff ects, we introduce a variable to quantify the amount of additional matter in the line of sight that can skew our observable properties. We define it as the ratio of the mass within the cylinder and the mass within a sphere M 2D / M , both of radius r 200c , where the length of the cylinder is described in Sect. 2. We present the distribution of M 2D / M in Fig. 5, where we can see that this quantity is strongly skewed and its median value is M 2D / M ≈ 1 . 26 (note that for a NFW profile with c = 4 , the corresponding analytical cylinder vs. spherical mass is 1 . 25). Although this quantity is not directly observable, we will use it to assess the contribution of LoS objects in the scatter of scaling relations. Note that besides objects in the LoS, di ff erent fitting procedures may impact the scatter of projection e ff ects, as we will see in Sect. 4.</text>
<text><location><page_8><loc_7><loc_93><loc_8><loc_93></location>2D</text>
<figure>
<location><page_8><loc_6><loc_75><loc_94><loc_93></location>
<caption>Fig. 6: Projected maps along a cylinder of length 23 Mpc , with radius r 200c and centred on a random sample of our galaxy clusters, ordered by their M 2D / M (over-plotted above each map) values decreasing from left to right. The pixel red, green, and blue channels are used as follows: the red channel maps the gas projected mass, the green channel maps the dark matter projected mass, and the blue channel maps the stellar projected mass. Columns widths are proportional to the cluster radii.</caption>
</figure>
<figure>
<location><page_8><loc_7><loc_21><loc_93><loc_66></location>
<caption>Fig. 7: Impact of LoS contamination in scaling relations. We show halo properties as a function of halo mass M in the left column, and colour-coded by the fractional amount of mass in a cylinder ( M 2D / M ), and the residuals PDFs in the right column (grey shaded histogram). Rows correspond to richness, the integrated Comptony parameter, lensing mass, and lensing concentration. We also show the residuals of a subset of haloes with low LoS contamination (in particular M 2D / M < 1 . 26 , dashed line histogram). Each panel reports the scatter of the residuals σ and the mean µ of the low LoS residuals.</caption>
</figure>
<figure>
<location><page_9><loc_7><loc_68><loc_48><loc_93></location>
<caption>Fig. 8: Integrated Comptony parameter vs halo mass, colourcoded by a stellar mass fraction. The top panel shows quantities computed within a sphere of radius r 200c, while on the bottom panel, they are computed within a cylinder (of radius r 200c and length 23 Mpc as already presented in Sect. 2). We limit the plot in the mass range M ∈ [1 , 2] 10 14 M ⊙ .</caption>
</figure>
<text><location><page_9><loc_6><loc_48><loc_49><loc_56></location>In Fig. 6, we show a random selection of clusters, ordered by decreasing M 2D / M from left to right. Objects with high M 2D / M (the objects in the left-most panels) include clusters that are merging, elongated, or in the LoS. In the rest of this paper, we will refer to the objects having M 2D / M greater than the median of the distribution 1 . 26 as having LoS excess.</text>
<text><location><page_9><loc_6><loc_35><loc_49><loc_48></location>To study the impact of projection e ff ects in scaling relations, in Fig. 7 we show the scaling relations of the following projected quantities: richness, integrated Comptony parameter, lensing mass, and concentrations, that are the ones that are most a ff ected by projection e ff ects. We colour-code these points by M 2D / M and focus on a narrow mass range of M ∈ [1 , 2] × 10 14 M ⊙ in order to visualise better how LoS excess impacts these scaling relations. On the left column, we can visually see that, except for concentration, they strongly correlate with M 2D / M , as the upper points of the scatter plot tend to have higher values of M 2D / M .</text>
<text><location><page_9><loc_6><loc_15><loc_49><loc_34></location>We quantify this finding in the right column by comparing the residual distributions (of the power-law fit over the complete mass range presented in Sect. 2.2) of the complete sample with the distribution of objects with low LoS contamination only (we adopt the criteria of M 2D / M < 1 . 26 being the median of the M 2D / M distribution, as shown in Fig. 5), and report the respective scatter σ and average value µ of the residual distributions (for the complete sample we have that µ = 0). Except for the concentration, residuals of haloes with low M 2D / M (see dashed histogram) significantly shift towards negative values of µ and σ. For instance, when we consider only objects with a low LoS excess, the scatter of Y 2D decreases from 0 . 35 down to 0 . 11 , and the M 2D NFW scatter goes from 0 . 23 to 0 . 9. The lensing mass is, in fact, well known to be a ff ected by projection e ff ects (Meneghetti et al. 2014; Euclid Collaboration: Giocoli et al. 2024).</text>
<text><location><page_9><loc_6><loc_10><loc_49><loc_15></location>In this paper, we refer to projection e ff ects as all e ff ects that take place when going from 3D to projection; they include both LoS e ff ects and model uncertainties. This definition becomes relevant when dealing with concentration, which is not impacted</text>
<figure>
<location><page_9><loc_51><loc_73><loc_93><loc_93></location>
<caption>Fig. 9: Probability density distribution of the parameters γ (inner slope, upper panel) and β (outer slope, right panel) of Eq. (10) of the successful gNFW profile fits. The central panel shows the scatter plot between the two parameters colour-coded by M . The dotted lines show the NFW parameters γ = 1 and β = 3 .</caption>
</figure>
<text><location><page_9><loc_51><loc_52><loc_94><loc_62></location>only by LoS objects but also by the NFW profile fitting procedure. We stress that in Fig. 3 we proved that our projected concentration is actually highly impacted by projection e ff ects, yet only weakly a ff ected by LoS e ff ects. We show in the next Section the reason for concentration being strongly a ff ected by projection e ff ects is that their reduced shear profile deviates strongly from the one produced by NFW profile (see, e.g., Ragagnin et al. 2021), which is used to reconstruct the reduced shear profile.</text>
<text><location><page_9><loc_51><loc_40><loc_94><loc_51></location>To conclude this section, we now study how correlations between cluster observable properties can be a ff ected by projection e ff ects. To this end, we take the case of a possible hot- vs. coldbaryon correlation by studying the stellar mass vs. integrated Comptony parameter (as the latter should strongly correlate with the gas mass). In Fig. 8, we show the integrated Comptony parameter scaling relation, colour-coded by stellar mass for the 3D quantities (top panel) and projected quantities (bottom panel).</text>
<text><location><page_9><loc_51><loc_24><loc_94><loc_40></location>Examining the correlations at a constant halo mass among the computed quantities within spheres (as depicted in the top panel of Fig. 8), we find no discernible weak anti-correlation between stellar mass and the integrated Comptony parameter (which is defined in Sect. 5). Conversely, when investigating the properties in the projected space, a more pronounced correlation becomes evident. This implies that projection e ff ects can strongly impact the correlation between observable properties. While this analysis is purely qualitative, we will quantify the impact of these projection e ff ects in Sect. 5, where we will compute the correlation coe ffi cients for both 3D quantities and 2D quantities.</text>
<section_header_level_1><location><page_9><loc_51><loc_21><loc_89><loc_22></location>4. Projection effects on lensing concentration</section_header_level_1>
<text><location><page_9><loc_51><loc_12><loc_94><loc_20></location>As we found in the previous section, projection e ff ects significantly increase the scatter and skewness in the scaling of lensing concentration with mass. However, this scatter increase is not related to external objects along the LoS. Now, we assess if the high scatter of lensing concentration is due to deviations of the reduced shear profile from the one induced by an NFW profile.</text>
<text><location><page_9><loc_51><loc_10><loc_94><loc_12></location>In this work, we will not delve into the origin of this deviation as it falls beyond the scope of this paper. Such deviation</text>
<figure>
<location><page_10><loc_7><loc_55><loc_46><loc_93></location>
<caption>Fig. 10: Ratio between 2D NFW profile fit parameters and 3D parameters for haloes with a successful 3D gNFW profile fit. Upper panel: the ratio between concentrations; lower panel: the ratio between halo masses. Points are colour-coded by the external log-slope β of the 3D fit of gNFW.</caption>
</figure>
<text><location><page_10><loc_6><loc_32><loc_49><loc_43></location>may arise due to halo elongations, suggesting that alternative profiles such as truncated NFW profiles may better suit galaxy clusters (Oguri & Hamana 2011). Alternatively, it could stem from the expectation that the NFW profile is intended to describe stacked haloes rather than individual objects. Our focus in this paper is to understand the impact of assuming an NFW profile for each of our haloes. We emphasize that these NFW deviations only a ff ect weak lensing signal reconstruction, as the NFW profile is highly e ff ective in recovering halo mass in 3D.</text>
<text><location><page_10><loc_6><loc_25><loc_49><loc_31></location>To study deviations from the NFW profile of haloes we fit a generalised NFW profile (Nagai et al. 2007), hereafter gNFW, in spherical coordinates over the same radial range as our previous NFW profile (described in Sect. 2.1), where the density profile ρ gNFW ( r ) is defined as</text>
<formula><location><page_10><loc_6><loc_21><loc_49><loc_24></location>ρ gNFW ( r ) = ρ 0 ( r / r s) γ (1 + r / r s) β -γ , (10)</formula>
<text><location><page_10><loc_6><loc_12><loc_49><loc_20></location>where γ and β are respectively the internal and external logslopes of the total matter density profiles. The case γ = 1 and β = 3 produces the NFW profile as in Eq. (4). Note that the Nagai et al. (2007) gNFW profile also depends on the parameter α that we fix to α = 1 in this work in order to explore internal and external log-slope variations only.</text>
<text><location><page_10><loc_6><loc_10><loc_49><loc_12></location>We present the PDF for the gNFW profile parameters γ and β in Fig. 9, where the fit was performed in 3D with a flat priors</text>
<figure>
<location><page_10><loc_52><loc_68><loc_89><loc_93></location>
<caption>Fig. 11: Residuals of lensing concentrations with respect to the power-law fit. As in Fig. 7, the dashed line histogram indicates the residuals for objects with low LoS e ff ects (low value of M 2D / M ). The solid line histogram contains the additional constraints of haloes with β and γ parameters close to the ones of an NFW profile (2 . 8 < β < 3 . 2 and 0 . 8 < γ < 1 . 2). Each histogram label reports the scatter σ of the residuals.</caption>
</figure>
<figure>
<location><page_10><loc_52><loc_32><loc_89><loc_54></location>
<caption>Fig. 12: Scatter plot of ratio between 2D concentration and 3D concentration against β, namely the outer slope of Eq. (10), of well behaving 3D gNFW profile fits in a narrow mass range of M ∈ [1 , 2] × 10 14 M ⊙ . We also report the correlation coe ffi cient.</caption>
</figure>
<text><location><page_10><loc_51><loc_10><loc_94><loc_22></location>for γ ∈ [0 , 3] and β ∈ [0 , 6]. The data points are colour-coded according to the variable M , revealing no discernible strong trend with respect to the fitted parameters. For 19% of the objects, the resulting best-fit parameters hit the boundaries of hard-cut priors. Upon visual inspection, these objects are characterised by a very steep matter density profile at large cluster-centric distances, possibly suggesting that a truncated NFW profile might be a better model choice. As our objective is to examine the effects of deviations from the generalised NFW profile, we omit these objects from the subsequent analysis in this Section. Given</text>
<text><location><page_11><loc_6><loc_87><loc_49><loc_93></location>the substantial deviation of these objects from NFW profile, they could potentially o ff er additional insights for our analyses. However, incorporating them would necessitate the use of a profile more general than Eq. 10. Therefore, we excluded them in order to make our analysis clearer.</text>
<text><location><page_11><loc_6><loc_76><loc_49><loc_86></location>We observe that the external logslope of Magneticum profiles appears to be slightly flatter than -3 . While we emphasize that this discrepancy does not a ff ect the accurate recovery of mass and concentration parameters in 3D NFW fits (such fits can still yield precise estimates of halo mass and concentrations). However, these deviations in the NFW profiles may a ff ect the reduced shear fit, particularly when observed over large radii (remember that in this work, we use 3 Mpc).</text>
<text><location><page_11><loc_6><loc_67><loc_49><loc_76></location>Furthermore, we observe a degeneracy between the β and γ parameters, indicating that our profiles deviating from NFW profile tend to exhibit a flatter profile compared to NFW profile (as illustrated in Fig. A.1). However, investigating this discrepancy is beyond the scope of this paper, as the internal log slope of clusters is not currently captured by existing weak lensing studies.</text>
<text><location><page_11><loc_6><loc_51><loc_49><loc_67></location>In Fig. 10, we plot the values of concentration and mass obtained from reduced shear fit, divided by the corresponding 3D quantities and colour-coded by the external 3D gNFW slope β for our intermediate redshift haloes. As we can see, haloes with large values of β have a projected concentration that is significantly higher than the 3D one (see upper panel). In Appendix A we report the example of a simulated halo with low LoS excess (see Fig. A.1) and an analytical one (see Fig. A.2), both with a flat external log-slope, and we show how the under-estimation of the concentration is caused by the fact that the NFW profile fit on the reduced shear is weighting too much the external part of the profile, that deviates from an NFW profile.</text>
<text><location><page_11><loc_6><loc_37><loc_49><loc_51></location>In Fig. 11, we show the concentration residual distribution and report their scatter. We note that the projected concentration scatter is not a ff ected by external material along the LoS (dashed line and shaded histograms match). However, if one restricts our sample to objects having NFW-like profile log-slopes (we used criteria of 2 . 8 < β < 3 . 2 and 0 . 8 < γ < 1 . 2), then the scatter distribution changes drastically. The concentration residuals decrease from 0 . 43 to 0 . 38, and the residuals shift towards higher values, suggesting that these objects are more relaxed. Such e ff ect is well known, as studied for instance in Macciò et al. (2007).</text>
<text><location><page_11><loc_6><loc_28><loc_49><loc_37></location>We also show how the external log-slope of the halo profile a ff ects the lensing reconstruction by plotting the ratio between the projected and 3D concentration (i.e., c 2D NFW / c NFW) value versus the 3D log-slope β in the narrow mass bin of M ∈ [1 , 2] × 10 14 M ⊙ in Fig. 12, where we find a positive correlation coe ffi cient of ≈ 0 . 28 , in agreement with a shift of residuals we showed in Fig. 11.</text>
<section_header_level_1><location><page_11><loc_6><loc_24><loc_35><loc_25></location>5. Correlations between properties</section_header_level_1>
<text><location><page_11><loc_6><loc_10><loc_49><loc_23></location>While in the last sections, we investigated the origin of the impact of projection e ff ects in the scatter and skewness of observable properties, we will now quantify how projection e ff ects impact the correlation between observable properties. To this end, we quantify the Pearson correlation coe ffi cients between their log-residuals (as defined in Sect. 2.2). We adopt the standard error associated with the Pearson coe ffi cient ρ as derived from two normal distributions, given by σρ = p 1 -ρ 2 / √ N -2 (see Eq. 12-93 in Pugh & Winslow 1966), where N represents the number of objects. This corresponds to a maximum error of 0 . 015</text>
<text><location><page_11><loc_51><loc_83><loc_94><loc_93></location>(for ρ = 0) for the sample size at z = 0 . 24 and a maximum error of 0 . 028 for the sample size at z = 0 . 90. It is worth noting that in the correlation coe ffi cient matrices generated in subsequent analyses, we will only colour values with correlation coe ffi cients | ρ | > 0 . 3, aiming to highlight strongly correlating properties. We define mild correlation as 0 . 2 < | ρ | < 0 . 3, as we choose to exercise caution. Correlation coe ffi cients with | ρ | < 0 . 1 are disregarded.</text>
<section_header_level_1><location><page_11><loc_51><loc_79><loc_68><loc_80></location>5.1. Analysis at z = 0 . 24</section_header_level_1>
<text><location><page_11><loc_51><loc_68><loc_94><loc_78></location>In this Section we focus on the haloes at intermediate redshift z = 0 . 24 . In Fig. 13 we show the correlation coe ffi cient matrix between log-residuals at fixed halo mass of our projected observable both from Euclid -like data (lensing concentration, lensing mass, richness, and stellar mass, respectively) and possible outcomes from multi-wavelength observations (integrated Comptony parameter, gas mass, X-ray luminosity, and temperature) for intermediate-redshift objects.</text>
<text><location><page_11><loc_51><loc_47><loc_94><loc_67></location>In the lower triangle, we present scatter plots alongside the slope derived from the correlation coe ffi cient. This visualization allows for the identification of instances where the correlation coe ffi cient slope accurately captures the trend of the residuals. Typically, this alignment occurs for quantities that exhibit strong correlation coe ffi cients. For example, our data points show a robust correlation between certain hot baryon tracers ( M 2D g and Y 2D ), some cold baryon components ( n 2D and M 2D ⋆ ), and weak lensing mass M 2D NFW . The underlying reason for these strong correlations lies in projection e ff ects: the greater the amount of matter along the line of sight, the higher the observed values. We will further demonstrate this in the subsequent section by presenting the correlation coe ffi cient matrix for 3D quantities, where many of these correlations diminish. This can be anticipated by observing that L 2D x and c 2D NFW do not exhibit this positive trend of correlations.</text>
<text><location><page_11><loc_51><loc_31><loc_94><loc_46></location>Notably, we observe that the correlation between richness and stellar mass ( ρ = 0 . 48) is not exceptionally high. Moreover, the stellar mass appears to be more influenced by projection e ff ects compared to richness (evident in their correlations with M 2D NFW , where they exhibit ρ = 0 . 69 and ρ = 0 . 42, respectively). We can speculate on two potential causes: firstly, unlike stellar mass, our richness computation incorporates some observationally-motivated background subtraction; alternatively, since stellar mass encompasses all stellar particles (while richness involves a luminosity-motivated galaxy stellar-mass cut), it is plausible that small subhaloes are influencing the projected stellar mass.</text>
<text><location><page_11><loc_51><loc_10><loc_94><loc_30></location>We note that concentration anti-correlates with gas-mass (and integrated Comptony parameter). This is in agreement with recent analyses of simulations. In fact, richness at fixed mass anti-correlates with concentration (Bose et al. 2019); low concentration is an index of the system being perturbed (Ludlow et al. 2012); and un-relaxed systems tend to be gas-rich (Davies et al. 2020). We refer to Ragagnin et al. (2022) for a more comprehensive study on low luminous groups. Moreover, at fixed halo mass, the lensing mass correlates strongly with total projected stellar mass ( ρ = 0 . 69) and projected gas mass ( ρ = 0 . 59), which may be due to the fact that both correlate strongly with LoS contamination. The same holds for the correlation among richness, gas mass, and stellar mass. This is due to projection e ff ects, where LoS excess amplifies all these quantities, as discussed in Section 2.2. We note that the 2D lensing mass and projected X-ray luminosity have a slight positive ( ρ = 0 . 20) cor-</text>
<text><location><page_12><loc_6><loc_91><loc_49><loc_93></location>ation, in agreement with the observational work of Sereno et al. (2020).</text>
<text><location><page_12><loc_6><loc_65><loc_49><loc_90></location>In Fig. 14 we show the covariance matrix of non-projected quantities for intermediate redshift objects. We see that as opposed to Fig. 13, the 3D covariance matrix shows a mild yet negative covariance between gas mass and stellar mass ( ρ = -0 . 24), and a positive correlation between richness and gas mass ( ρ = 0 . 23) because most of their correlations in the projection are due to Line of sights excess, which significantly increases the values of the gas mass, the richness, and the stellar mass. In Fig. 15 we report the correlation matrix as in Fig. 13 where we present X-ray luminosity, gas mass, and temperature, as computed within r 500c , which shows an anti-correlation between the gas mass and the concentration residuals ( ρ = -0 . 14) that is significantly lower than the one found in Fig. 13 and Fig. 14 ( ρ equals to -0 . 26 and -0 . 34 respectively). One possibility is that this change in sign of the correlation is caused by the fact that mixing overdensities (concentration is within ∆ c = 200 and gasmass is within ∆ c = 500) does introduce an additional correlation with the sparsity (Balmès et al. 2014; Corasaniti et al. 2022) that itself correlates with the concentration (see Appendix B).</text>
<text><location><page_12><loc_6><loc_52><loc_49><loc_65></location>For completeness, we report the correlation coe ffi cient matrix and the scatter of log-residuals of all quantities in Table B.1. There we also added the core-excised projected X-ray luminosity L 2D X , ce500c , as it is typically used in X-ray-based observational studies, where we can see that the scatter and most of the correlation coe ffi cients are smaller than L 2D ce500c , while the correlations with the concentration and gas mass increase. Note that we do not report the 3D NFW mass ( M NFW) because it has an extremely low intrinsic scatter σ ln ( M NFW) ≈ 0 . 01 and its correlation coefficients are not meaningful.</text>
<section_header_level_1><location><page_12><loc_6><loc_48><loc_22><loc_49></location>5.2. Analysis at z = 0 . 9</section_header_level_1>
<text><location><page_12><loc_6><loc_33><loc_49><loc_47></location>In this Section, we focus on observational property covariance matrixes of our haloes z = 0 . 9. At this redshift, we computed projected quantities within a cylinder depth of 35 Mpc in order to retain the same relative ratio as the photoz uncertainty of the low-redshift analysis (it scales with 1 + z ). For what concerns the cylinder used to integrate ∆Σ gt, we rescaled so as to keep it constant in comoving units with the low-redshift analyses. We rescaled the 3D NFW profile minimum radius to 40 kpc while we kept the maximum radius at r 200c . For what concerns the radial range of the lensing fit, we rescaled it with H -2 / 3 ( z ), therefore performing it in the range of [234 , 2300] kpc .</text>
<text><location><page_12><loc_6><loc_23><loc_49><loc_32></location>We stress that we do not model observational uncertainty. Therefore, the decrease in background source count with redshift does not impact our best fits. However, it still impacts the fact that we weigh external radial bins more than internal ones. We report the values of the scatter and the projection contribution at z = 0 . 9 in Fig. 16, while we report the log-residuals and the skewness for each property in Fig. 17.</text>
<text><location><page_12><loc_6><loc_10><loc_49><loc_22></location>In particular, the quantities most a ff ected by projection effects are the lensing mass and concentration, whereas the temperature is the lowest. These results are qualitatively similar to the low redshift analyses, with the ComptonY parameter and gas mass being slightly less a ff ected by projection e ff ects. Note that since the virial radius is smaller at higher redshift values, our radial range of the reduced shear is closer to the NFW scale radius; therefore, the weak lensing reconstruction is more e ff ective in capturing the scale radius and more sensitive to deviations from an NFW profile. As a consequence, we found that the in-</text>
<text><location><page_12><loc_51><loc_91><loc_94><loc_93></location>crease of scatter going from c NFW to c 2D NFW compared to the low redshift analyses.</text>
<text><location><page_12><loc_51><loc_83><loc_94><loc_90></location>We report the correlation coe ffi cient matrix and the scatter log-residuals of the quantities at z = 0 . 9 in Table B.2. As for the case at z = 0 . 24 , note that we do not report the 3D NFW mass ( M NFW) because it has an extremely small intrinsic scatter of 0 . 01 , and thus its correlation coe ffi cients have no impact in our study.</text>
<section_header_level_1><location><page_12><loc_51><loc_79><loc_63><loc_80></location>6. Conclusions</section_header_level_1>
<text><location><page_12><loc_51><loc_55><loc_94><loc_78></location>In this work, we analysed a number of galaxy clusters from Magneticum hydrodynamic simulation Box2b / hr. We did so in a mass range, tailored for Euclid -like data products (see Sartoris et al. 2016; Euclid Collaboration: Adam et al. 2019), namely with a mass of M 200c > 10 14 M ⊙ . To this end, we computed properties that could come from Euclid catalogues of galaxy clusters, such as richness, stellar mass, and lensing masses and concentration, and possible properties coming from multi-wavelength studies such as X-ray luminosity, integrated Comptony parameter, gas mass, and temperature. All these properties were computed both within a sphere and within a cylinder (both with radius r 200c) to account for projection e ff ects. Our study considers the remarkable capabilities of Euclid photoz measurements in identifying interlopers. However, their importance decreases significantly at scales as small as a few tens Mpc. This depth is still long enough to contain multiple haloes along the LoS. Hence, we studied the projection e ff ects on a scale that is significantly smaller than the Euclid photoz uncertainty.</text>
<text><location><page_12><loc_51><loc_51><loc_94><loc_55></location>We then studied how the scatter and skewness change when one measures quantities in 3D space or in projection. Below, we summarise our findings:</text>
<unordered_list>
<list_item><location><page_12><loc_52><loc_43><loc_94><loc_50></location>-The properties that are most a ff ected by projection e ff ects are the mass and concentration from lensing, the integrated Comptony parameter, and the gas mass. In contrast, temperature and X-ray luminosity are the quantities least a ff ected by projection e ff ects.</list_item>
<list_item><location><page_12><loc_52><loc_33><loc_94><loc_43></location>-In both redshift slices ( z = 0 . 24 and z = 0 . 9), the influence of LoS e ff ects is substantial and potentially leads to a spurious correlation between gas and stellar masses. These projection e ff ects have the capacity to markedly enhance correlations between gas and stellar mass (they go from a negative value of -0 . 24 to a significantly high value of 0 . 57), e ff ectively masking the intrinsic correlation (for instance, driven by distinct accretion histories) beneath.</list_item>
<list_item><location><page_12><loc_52><loc_24><loc_94><loc_33></location>-The lensing concentration, on the other hand, is mainly affected by the fact that the profile outskirt of reduced shear deviates from the one coming from an NFW profile (which is the profile typically used in WL analyses). We found that deviations from an ideal NFW profile increase the skewness from 0 . 6 to 2 . 5 and increase the scatter of log-residuals from 0 . 33 (in agreement with theoretical works) up to 0 . 46 .</list_item>
</unordered_list>
<text><location><page_12><loc_51><loc_16><loc_94><loc_22></location>The analysis presented here has been carried out using a single suite of hydrodynamic simulations. Regarding weak lensing masses and concentration, since in this work, we did not consider the profile noise due to the finite number of background galaxies, future studies are needed to improve our estimations.</text>
<text><location><page_12><loc_51><loc_10><loc_94><loc_16></location>Some works show that both scatter and correlation coefficients vary between cosmological simulations with di ff erent cosmologies (Ragagnin et al. 2023), the presence of feedback schemes (Stanek et al. 2010), and di ff erent cosmological simulation suite in the market (see Fig. 7 in Anbajagane et al. 2020).</text>
<text><location><page_13><loc_6><loc_88><loc_49><loc_93></location>So, while simulations can provide directions on how to model correlation coe ffi cients, it is possible that when using this kind of data, one needs to allow for variation due to the di ff erent baryon physics.</text>
<text><location><page_13><loc_6><loc_71><loc_49><loc_88></location>Furthermore, when striving for even more precise results, it is important to acknowledge that mass-observable relations are not exact power laws. Therefore, employing more generic fitting techniques, such as a running median, could yield improvements. Additionally, there is room for enhancement in how we compute correlation coe ffi cients in future studies. One potential approach could involve simultaneously fitting both the massobservable relation scatter and the correlation coe ffi cients by maximizing multivariate likelihoods. We anticipate that future studies combining Euclid data with multi-wavelength observations may encounter challenges in shedding light on currently puzzling residual correlations, primarily dominated by projection e ff ects.</text>
<text><location><page_13><loc_6><loc_34><loc_49><loc_70></location>Acknowledgements. We thanks the anonymous referee for the useful comments. The Magneticum Pathfinder simulations were partially performed at the LeibnizRechenzentrum with CPU time assigned to the Project 'pr86re'. AR and LM acknowledge support from the grant PRIN-MIUR 2017 WSCC32 and acknowledges the usage of the INAF-OATs IT framework (Ta ff oni et al. 2020; Bertocco et al. 2020), and the space filling curve improvement on Gadget3 (Ragagnin et al. 2016). Antonio Ragagnin thanks Veronica Bi ffi and Elena Rasia for the X-ray computation routines and tables. KD acknowledges support by the COMPLEX project from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program grant agreement ERC-2019-AdG 882679. LM acknowledges the financial contribution from the grant PRIN-MUR 2022 20227RNLY3 'The concordance cosmological model: stress-tests with galaxy clusters' supported by Next Generation EU. CG and LM acknowledge support from the grant ASI n.2018-23-HH.0. AR and CG acknowledge funding from INAF theory Grant 2022: Illuminating Dark Matter using Weak Lensing by Cluster Satellites, PI: Carlo Giocoli. SB acknowledges partial financial support from the INFN InDark grant. AMCLB was supported by a fellowship of PSL University hosted by the Paris Observatory. We used the package colossus (see Diemer 2018) for computing Σ and ∆Σ as expected from NFW profiles. AR and FC acknowledge co-funding by the European Union - NextGenerationEU within PRIN 2022 project n.20229YBSAN - Globular clusters in cosmological simulations and in lensed fields: from their birth to the present epoch. The Euclid Consortium acknowledges the European Space Agency and a number of agencies and institutes that have supported the development of Euclid , in particular the Academy of Finland, the Agenzia Spaziale Italiana, the Belgian Science Policy, the Canadian Euclid Consortium, the French Centre National d'Etudes Spatiales, the Deutsches Zentrum für Luft- und Raumfahrt, the Danish Space Research Institute, the Fundação para a Ciência e a Tecnologia, the Ministerio de Ciencia e Innovación, the National Aeronautics and Space Administration, the National Astronomical Observatory of Japan, the Netherlandse Onderzoekschool Voor Astronomie, the Norwegian Space Agency, the Romanian Space Agency, the State Secretariat for Education, Research and Innovation (SERI) at the Swiss Space O ffi ce (SSO), and the United Kingdom Space Agency. A complete and detailed list is available on the Euclid web site ( http://www.euclid-ec.org ).</text>
<section_header_level_1><location><page_13><loc_6><loc_29><loc_20><loc_30></location>Data Availability</section_header_level_1>
<text><location><page_13><loc_6><loc_23><loc_49><loc_28></location>Raw simulation data were generated at C 2 PAP / LRZ cosmology simulation web portal https://c2papcosmosim.uc. lrz.de/ . Derived data supporting the findings of this study are available from the corresponding author AR on request.</text>
<section_header_level_1><location><page_13><loc_6><loc_19><loc_16><loc_20></location>References</section_header_level_1>
<unordered_list>
<list_item><location><page_13><loc_6><loc_16><loc_49><loc_18></location>Abbott, T. M. C., Aguena, M., Alarcon, A., et al. 2020, Phys. Rev. D, 102, 023509</list_item>
<list_item><location><page_13><loc_6><loc_15><loc_43><loc_16></location>Allen, S. W., Evrard, A. E., & Mantz, A. B. 2011, ARA&A, 49, 409</list_item>
<list_item><location><page_13><loc_6><loc_14><loc_45><loc_15></location>Anbajagane, D., Evrard, A. E., Farahi, A., et al. 2020, MNRAS, 495, 686</list_item>
<list_item><location><page_13><loc_6><loc_13><loc_39><loc_14></location>Andreon, S., Dong, H., & Raichoor, A. 2016, A&A, 593, A2</list_item>
<list_item><location><page_13><loc_6><loc_12><loc_33><loc_13></location>Andreon, S. & Moretti, A. 2011, A&A, 536, A37</list_item>
<list_item><location><page_13><loc_6><loc_10><loc_49><loc_12></location>Angelinelli, M., Ettori, S., Dolag, K., Vazza, F., & Ragagnin, A. 2022, A&A, 663, L6</list_item>
<list_item><location><page_13><loc_51><loc_91><loc_94><loc_93></location>Angelinelli, M., Ettori, S., Dolag, K., Vazza, F., & Ragagnin, A. 2023a, A&A, 675, A188</list_item>
<list_item><location><page_13><loc_51><loc_89><loc_94><loc_91></location>Angelinelli, M., Ettori, S., Dolag, K., Vazza, F., & Ragagnin, A. 2023b, A&A, 675, A188</list_item>
<list_item><location><page_13><loc_51><loc_88><loc_91><loc_89></location>Angulo, R. E., Zennaro, M., Contreras, S., et al. 2021, MNRAS, 507, 5869</list_item>
<list_item><location><page_13><loc_51><loc_85><loc_94><loc_88></location>Arnaud, K. A. 1996, in Astronomical Society of the Pacific Conference Series, Vol. 101, Astronomical Data Analysis Software and Systems V, ed. G. H. Jacoby & J. Barnes, 17</list_item>
<list_item><location><page_13><loc_51><loc_83><loc_94><loc_84></location>Balmès, I., Rasera, Y., Corasaniti, P. S., & Alimi, J. M. 2014, MNRAS, 437, 2328</list_item>
<list_item><location><page_13><loc_51><loc_81><loc_88><loc_82></location>Beck, A. M., Murante, G., Arth, A., et al. 2016, MNRAS, 455, 2110</list_item>
<list_item><location><page_13><loc_51><loc_80><loc_79><loc_81></location>Becker, M. R. & Kravtsov, A. V. 2011, ApJ, 740, 25</list_item>
<list_item><location><page_13><loc_51><loc_78><loc_94><loc_80></location>Bellagamba, F., Roncarelli, M., Maturi, M., & Moscardini, L. 2018a, MNRAS, 473, 5221</list_item>
<list_item><location><page_13><loc_51><loc_76><loc_94><loc_78></location>Bellagamba, F., Roncarelli, M., Maturi, M., & Moscardini, L. 2018b, MNRAS, 473, 5221</list_item>
<list_item><location><page_13><loc_51><loc_72><loc_94><loc_76></location>Bertocco, S., Goz, D., Tornatore, L., et al. 2020, in Astronomical Society of the Pacific Conference Series, Vol. 527, Astronomical Society of the Pacific Conference Series, ed. R. Pizzo, E. R. Deul, J. D. Mol, J. de Plaa, & H. Verkouter, 303</list_item>
<list_item><location><page_13><loc_51><loc_63><loc_94><loc_72></location>Bhargava, S., Garrel, C., Koulouridis, E., et al. 2023, A&A, 673, A92 Bi ffi , V., Dolag, K., & Böhringer, H. 2013, MNRAS, 428, 1395 Bi ffi , V., Planelles, S., Borgani, S., et al. 2017, MNRAS, 468, 531 Bocquet, S., Dietrich, J. P., Schrabback, T., et al. 2019, ApJ, 878, 55 Bocquet, S., Grandis, S., Bleem, L. E., et al. 2024, Phys. Rev. D, 110, 083510 Bocquet, S., Heitmann, K., Habib, S., et al. 2020, ApJ, 901, 5 Bocquet, S., Saro, A., Dolag, K., & Mohr, J. J. 2016, MNRAS, 456, 2361 Bose, S., Eisenstein, D. J., Hernquist, L., et al. 2019, MNRAS, 490, 5693 Boylan-Kolchin, M., Springel, V., White, S. D. M., Jenkins, A., & Lemson, G.</list_item>
<list_item><location><page_13><loc_53><loc_62><loc_67><loc_63></location>2009, MNRAS, 398, 1150</list_item>
<list_item><location><page_13><loc_51><loc_61><loc_79><loc_62></location>Castignani, G. & Benoist, C. 2016, A&A, 595, A111</list_item>
<list_item><location><page_13><loc_51><loc_60><loc_85><loc_60></location>Child, H. L., Habib, S., Heitmann, K., et al. 2018, ApJ, 859, 55</list_item>
<list_item><location><page_13><loc_51><loc_59><loc_92><loc_59></location>Contreras-Santos, A., Buitrago, F., Knebe, A., et al. 2024, A&A, 690, A109</list_item>
<list_item><location><page_13><loc_51><loc_57><loc_94><loc_58></location>Corasaniti, P. S., Le Brun, A. M. C., Richardson, T. R. G., et al. 2022, MNRAS, 516, 437</list_item>
<list_item><location><page_13><loc_51><loc_55><loc_87><loc_56></location>Costanzi, M., Rozo, E., Simet, M., et al. 2019, MNRAS, 488, 4779</list_item>
<list_item><location><page_13><loc_51><loc_54><loc_84><loc_55></location>Cui, W., Dave, R., Knebe, A., et al. 2022, MNRAS, 514, 977</list_item>
<list_item><location><page_13><loc_51><loc_52><loc_94><loc_54></location>Davies, J. J., Crain, R. A., Oppenheimer, B. D., & Schaye, J. 2020, MNRAS, 491, 4462</list_item>
<list_item><location><page_13><loc_51><loc_50><loc_93><loc_52></location>Davis, M., Efstathiou, G., Frenk, C. S., & White, S. D. M. 1985, ApJ, 292, 371 Diemer, B. 2018, ApJS, 239, 35</list_item>
<list_item><location><page_13><loc_51><loc_49><loc_92><loc_50></location>Dolag, K., Borgani, S., Murante, G., & Springel, V. 2009, MNRAS, 399, 497</list_item>
<list_item><location><page_13><loc_51><loc_47><loc_94><loc_49></location>Dolag, K., Gaensler, B. M., Beck, A. M., & Beck, M. C. 2015, MNRAS, 451, 4277</list_item>
<list_item><location><page_13><loc_51><loc_46><loc_86><loc_47></location>Dolag, K., Komatsu, E., & Sunyaev, R. 2016, MNRAS, 463, 1797</list_item>
<list_item><location><page_13><loc_51><loc_44><loc_94><loc_46></location>Ettori, S., Donnarumma, A., Pointecouteau, E., et al. 2013, Space Sci. Rev., 177, 119</list_item>
<list_item><location><page_13><loc_51><loc_42><loc_94><loc_44></location>Euclid Collaboration: Adam, R., Vannier, M., Maurogordato, S., et al. 2019, A&A, 627, A23</list_item>
<list_item><location><page_13><loc_51><loc_40><loc_94><loc_42></location>Euclid Collaboration: Ajani, V., Baldi, M., Barthelemy, A., et al. 2023, A&A, 675, A120</list_item>
<list_item><location><page_13><loc_51><loc_38><loc_94><loc_40></location>Euclid Collaboration: Blanchard, A., Camera, S., Carbone, C., et al. 2020, A&A, 642, A191</list_item>
<list_item><location><page_13><loc_51><loc_36><loc_94><loc_38></location>Euclid Collaboration: Desprez, G., Paltani, S., Coupon, J., et al. 2020, A&A, 644, A31</list_item>
<list_item><location><page_13><loc_51><loc_34><loc_94><loc_35></location>Euclid Collaboration: Giocoli, C., Meneghetti, M., Rasia, E., et al. 2024, A&A, 681, A67</list_item>
<list_item><location><page_13><loc_51><loc_31><loc_94><loc_33></location>Euclid Collaboration: Mellier, Y., Abdurro'uf, Acevedo Barroso, J., Achúcarro, A., et al. 2024, A&A, submitted, arXiv:2405.13491</list_item>
<list_item><location><page_13><loc_51><loc_30><loc_89><loc_31></location>Fabjan, D., Borgani, S., Tornatore, L., et al. 2010, MNRAS, 401, 1670</list_item>
<list_item><location><page_13><loc_51><loc_28><loc_94><loc_30></location>Farahi, A., Mulroy, S. L., Evrard, A. E., et al. 2019, Nature Communications, 10, 2504</list_item>
<list_item><location><page_13><loc_51><loc_27><loc_89><loc_28></location>Ferland, G. J., Korista, K. T., Verner, D. A., et al. 1998, PASP, 110, 761</list_item>
<list_item><location><page_13><loc_51><loc_21><loc_94><loc_27></location>Fischer, M. S., Kasselmann, L., Brüggen, M., et al. 2024, MNRAS, 529, 2327 Ghirardini, V., Bulbul, E., Artis, E., et al. 2024, arXiv e-prints, arXiv:2402.08458 Giocoli, C., Tormen, G., & van den Bosch, F. C. 2008, MNRAS, 386, 2135 Giodini, S., Lovisari, L., Pointecouteau, E., et al. 2013, Space Sci. Rev., 177, 247 Hirschmann, M., Dolag, K., Saro, A., et al. 2014, MNRAS, 442, 2304 Hoekstra, H. 2003, MNRAS, 339, 1155</list_item>
<list_item><location><page_13><loc_51><loc_19><loc_93><loc_21></location>Hoekstra, H., Mahdavi, A., Babul, A., & Bildfell, C. 2012, MNRAS, 427, 1298 Hudson, D. S., Mittal, R., Reiprich, T. H., et al. 2010, A&A, 513, A37</list_item>
<list_item><location><page_13><loc_51><loc_18><loc_87><loc_19></location>Komatsu, E., Smith, K. M., Dunkley, J., et al. 2011, ApJS, 192, 18</list_item>
</unordered_list>
<text><location><page_13><loc_51><loc_17><loc_80><loc_18></location>Kravtsov, A. V. & Borgani, S. 2012, ARA&A, 50, 353</text>
<unordered_list>
<list_item><location><page_13><loc_51><loc_16><loc_79><loc_17></location>Lima, M. & Hu, W. 2005, Phys. Rev. D, 72, 043006</list_item>
<list_item><location><page_13><loc_51><loc_15><loc_88><loc_16></location>Ludlow, A. D., Navarro, J. F., Li, M., et al. 2012, MNRAS, 427, 1322</list_item>
<list_item><location><page_13><loc_51><loc_13><loc_94><loc_15></location>Macciò, A. V., Dutton, A. A., van den Bosch, F. C., et al. 2007, MNRAS, 378, 55</list_item>
<list_item><location><page_13><loc_51><loc_10><loc_91><loc_13></location>Maturi, M., Bellagamba, F., Radovich, M., et al. 2019, MNRAS, 485, 498 McClintock, T., Varga, T. N., Gruen, D., et al. 2019, MNRAS, 482, 1352 Melchior, P., Suchyta, E., Hu ff , E., et al. 2015, MNRAS, 449, 2219</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_14><loc_6><loc_78><loc_49><loc_93></location>Meneghetti, M., Rasia, E., Merten, J., et al. 2010, A&A, 514, A93 Meneghetti, M., Rasia, E., Vega, J., et al. 2014, ApJ, 797, 34 Nagai, D., Kravtsov, A. V., & Vikhlinin, A. 2007, ApJ, 668, 1 Navarro, J. F., Frenk, C. S., & White, S. D. M. 1997, ApJ, 490, 493 Oguri, M. & Hamana, T. 2011, MNRAS, 414, 1851 Okabe, N., Zhang, Y. Y., Finoguenov, A., et al. 2010, ApJ, 721, 875 Pacaud, F., Pierre, M., Adami, C., et al. 2007, MNRAS, 382, 1289 Pires, S., Vandenbussche, V., Kansal, V., et al. 2020, A&A, 638, A141 Pratt, G. W., Arnaud, M., Biviano, A., et al. 2019, Space Sci. Rev., 215, 25 Puddu, E. & Andreon, S. 2022, MNRAS, 511, 2968 Pugh, E. & Winslow, G. 1966, The Analysis of Physical Measurements (By > Emerson M. Pugh (And > George H. Winslow, Addison-Wesley Series in Physics</list_item>
</unordered_list>
<text><location><page_14><loc_6><loc_77><loc_40><loc_78></location>Ragagnin, A., Andreon, S., & Puddu, E. 2022, A&A, 666, A22</text>
<unordered_list>
<list_item><location><page_14><loc_6><loc_75><loc_49><loc_77></location>Ragagnin, A., Dolag, K., Bi ffi , V., et al. 2017, Astronomy and Computing, 20, 52</list_item>
<list_item><location><page_14><loc_6><loc_72><loc_49><loc_74></location>Ragagnin, A., Dolag, K., Moscardini, L., Biviano, A., & D'Onofrio, M. 2019, MNRAS, 486, 4001</list_item>
</unordered_list>
<text><location><page_14><loc_6><loc_71><loc_43><loc_72></location>Ragagnin, A., Fumagalli, A., Castro, T., et al. 2023, A&A, 675, A77</text>
<unordered_list>
<list_item><location><page_14><loc_6><loc_70><loc_44><loc_71></location>Ragagnin, A., Meneghetti, M., Calura, F., et al. 2024, A&A, 687, A270</list_item>
<list_item><location><page_14><loc_6><loc_69><loc_46><loc_70></location>Ragagnin, A., Saro, A., Singh, P., & Dolag, K. 2021, MNRAS, 500, 5056</list_item>
<list_item><location><page_14><loc_6><loc_63><loc_49><loc_69></location>Ragagnin, A., Tchipev, N., Bader, M., Dolag, K., & Hammer, N. J. 2016, in Advances in Parallel Computing, Volume 27: Parallel Computing: On the Road to Exascale, Edited by Gerhard R. Joubert, Hugh Leather, Mark Parsons, Frans Peters, Mark Sawyer. IOP Ebook, ISBN: 978-1-61499-621-7, pages 411-420</list_item>
<list_item><location><page_14><loc_6><loc_57><loc_49><loc_63></location>Rozo, E., Wechsler, R. H., Ryko ff , E. S., et al. 2010, ApJ, 708, 645 Saro, A., Liu, J., Mohr, J. J., et al. 2014, MNRAS, 440, 2610 Sartoris, B., Biviano, A., Fedeli, C., et al. 2016, MNRAS, 459, 1764 Schrabback, T., Bocquet, S., Sommer, M., et al. 2021, MNRAS, 505, 3923 Sereno, M., Fedeli, C., & Moscardini, L. 2016, J. Cosmology Astropart. Phys., 2016, 042</list_item>
</unordered_list>
<text><location><page_14><loc_6><loc_55><loc_42><loc_56></location>Sereno, M., Umetsu, K., Ettori, S., et al. 2020, MNRAS, 492, 4528</text>
<unordered_list>
<list_item><location><page_14><loc_6><loc_54><loc_45><loc_55></location>Singh, P., Saro, A., Costanzi, M., & Dolag, K. 2020, MNRAS, 494, 3728</list_item>
<list_item><location><page_14><loc_6><loc_52><loc_49><loc_54></location>Smith, R. K., Brickhouse, N. S., Liedahl, D. A., & Raymond, J. C. 2001, ApJ, 556, L91</list_item>
<list_item><location><page_14><loc_6><loc_50><loc_49><loc_52></location>Sommer, M. W., Schrabback, T., Applegate, D. E., et al. 2022, MNRAS, 509, 1127</list_item>
<list_item><location><page_14><loc_6><loc_47><loc_49><loc_49></location>Sommer, M. W., Schrabback, T., Ragagnin, A., & Rockenfeller, R. 2023, arXiv, arXiv:2306.13187</list_item>
<list_item><location><page_14><loc_6><loc_46><loc_27><loc_47></location>Springel, V. 2005, MNRAS, 364, 1105</list_item>
<list_item><location><page_14><loc_6><loc_45><loc_44><loc_46></location>Springel, V., Di Matteo, T., & Hernquist, L. 2005a, MNRAS, 361, 776</list_item>
<list_item><location><page_14><loc_6><loc_44><loc_45><loc_45></location>Springel, V., White, S. D. M., Jenkins, A., et al. 2005b, Nature, 435, 629</list_item>
<list_item><location><page_14><loc_6><loc_42><loc_49><loc_44></location>Springel, V., White, S. D. M., Tormen, G., & Kau ff mann, G. 2001, MNRAS, 328, 726</list_item>
<list_item><location><page_14><loc_6><loc_39><loc_49><loc_41></location>Stanek, R., Rasia, E., Evrard, A. E., Pearce, F., & Gazzola, L. 2010, ApJ, 715, 1508</list_item>
<list_item><location><page_14><loc_6><loc_36><loc_49><loc_39></location>Steinborn, L. K., Dolag, K., Comerford, J. M., et al. 2016, MNRAS, 458, 1013 Steinborn, L. K., Dolag, K., Hirschmann, M., Prieto, M. A., & Remus, R.-S. 2015, MNRAS, 448, 1504</list_item>
<list_item><location><page_14><loc_6><loc_35><loc_42><loc_36></location>Stern, C., Dietrich, J. P., Bocquet, S., et al. 2019, MNRAS, 485, 69</list_item>
</unordered_list>
<text><location><page_14><loc_6><loc_34><loc_48><loc_35></location>Sugiyama, S., Miyatake, H., More, S., et al. 2023, Phys. Rev. D, 108, 123521</text>
<text><location><page_14><loc_6><loc_32><loc_40><loc_33></location>Sun, M., Voit, G. M., Donahue, M., et al. 2009, ApJ, 693, 1142</text>
<unordered_list>
<list_item><location><page_14><loc_6><loc_31><loc_43><loc_32></location>Sunayama, T., Park, Y., Takada, M., et al. 2020, MNRAS, 496, 4468</list_item>
<list_item><location><page_14><loc_6><loc_29><loc_49><loc_31></location>Sunyaev, R. A. & Zeldovich, Y. B. 1972, Comments on Astrophysics and Space Physics, 4, 173</list_item>
<list_item><location><page_14><loc_6><loc_25><loc_49><loc_29></location>Ta ff oni, G., Becciani, U., Garilli, B., et al. 2020, in Astronomical Society of the Pacific Conference Series, Vol. 527, Astronomical Society of the Pacific Conference Series, ed. R. Pizzo, E. R. Deul, J. D. Mol, J. de Plaa, & H. Verkouter, 307</list_item>
</unordered_list>
<text><location><page_14><loc_6><loc_24><loc_40><loc_24></location>Teklu, A. F., Remus, R.-S., Dolag, K., et al. 2015, ApJ, 812, 29</text>
<text><location><page_14><loc_6><loc_21><loc_49><loc_23></location>Tornatore, L., Borgani, S., Dolag, K., & Matteucci, F. 2007, MNRAS, 382, 1050 Truong, N., Rasia, E., Mazzotta, P., et al. 2018, MNRAS, 474, 4089</text>
<unordered_list>
<list_item><location><page_14><loc_6><loc_19><loc_49><loc_21></location>van den Bosch, F. C., Yang, X., Mo, H. J., & Norberg, P. 2005, MNRAS, 356, 1233</list_item>
</unordered_list>
<text><location><page_14><loc_6><loc_18><loc_46><loc_19></location>van Uitert, E., Cacciato, M., Hoekstra, H., et al. 2016, MNRAS, 459, 3251</text>
<unordered_list>
<list_item><location><page_14><loc_6><loc_17><loc_42><loc_17></location>Vikhlinin, A., Kravtsov, A., Forman, W., et al. 2006, ApJ, 640, 691</list_item>
<list_item><location><page_14><loc_6><loc_15><loc_44><loc_16></location>Villaescusa-Navarro, F., Ding, J., Genel, S., et al. 2022, ApJ, 929, 132</list_item>
<list_item><location><page_14><loc_6><loc_13><loc_49><loc_15></location>Wiersma, R. P. C., Schaye, J., Theuns, T., Dalla Vecchia, C., & Tornatore, L. 2009, MNRAS, 399, 574</list_item>
<list_item><location><page_14><loc_6><loc_12><loc_43><loc_13></location>Wu, H.-Y., Costanzi, M., To, C.-H., et al. 2022, MNRAS, 515, 4471</list_item>
<list_item><location><page_14><loc_6><loc_10><loc_49><loc_12></location>Xu, W., Ramos-Ceja, M. E., Pacaud, F., Reiprich, T. H., & Erben, T. 2018, A&A, 619, A162</list_item>
<list_item><location><page_14><loc_52><loc_89><loc_94><loc_91></location>1 INAF-Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via Piero Gobetti 93 / 3, 40129 Bologna, Italy</list_item>
<list_item><location><page_14><loc_52><loc_87><loc_94><loc_89></location>2 IFPU, Institute for Fundamental Physics of the Universe, via Beirut 2, 34151 Trieste, Italy</list_item>
<list_item><location><page_14><loc_52><loc_83><loc_94><loc_87></location>3 Dipartimento di Fisica e Astronomia "Augusto Righi" - Alma Mater Studiorum Università di Bologna, via Piero Gobetti 93 / 2, 40129 Bologna, Italy</list_item>
<list_item><location><page_14><loc_52><loc_80><loc_94><loc_83></location>4 ICSC - Centro Nazionale di Ricerca in High Performance Computing, Big Data e Quantum Computing, Via Magnanelli 2, Bologna, Italy</list_item>
<list_item><location><page_14><loc_52><loc_77><loc_94><loc_80></location>5 Dipartimento di Fisica - Sezione di Astronomia, Università di Trieste, Via Tiepolo 11, 34131 Trieste, Italy</list_item>
<list_item><location><page_14><loc_52><loc_75><loc_94><loc_77></location>6 INAF-Osservatorio Astronomico di Trieste, Via G. B. Tiepolo 11, 34143 Trieste, Italy</list_item>
<list_item><location><page_14><loc_52><loc_74><loc_92><loc_75></location>7 INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste TS, Italy</list_item>
<list_item><location><page_14><loc_52><loc_71><loc_94><loc_74></location>8 INAF-Osservatorio Astronomico di Brera, Via Brera 28, 20122 Milano, Italy</list_item>
<list_item><location><page_14><loc_52><loc_68><loc_94><loc_71></location>9 Universitäts-Sternwarte München, Fakultät für Physik, LudwigMaximilians-Universität München, Scheinerstrasse 1, 81679 München, Germany</list_item>
<list_item><location><page_14><loc_52><loc_67><loc_85><loc_68></location>10 INFN-Bologna, Via Irnerio 46, 40126 Bologna, Italy</list_item>
<list_item><location><page_14><loc_52><loc_64><loc_94><loc_67></location>11 Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Via Irnerio 46, 40126 Bologna, Italy</list_item>
<list_item><location><page_14><loc_52><loc_62><loc_94><loc_64></location>12 Laboratoire Univers et Théorie, Observatoire de Paris, Université PSL, Université Paris Cité, CNRS, 92190 Meudon, France</list_item>
<list_item><location><page_14><loc_52><loc_60><loc_94><loc_62></location>13 Institut d'Astrophysique de Paris, 98bis Boulevard Arago, 75014, Paris, France</list_item>
<list_item><location><page_14><loc_52><loc_57><loc_94><loc_59></location>14 Institut d'Astrophysique de Paris, UMR 7095, CNRS, and Sorbonne Université, 98 bis boulevard Arago, 75014 Paris, France</list_item>
<list_item><location><page_14><loc_52><loc_55><loc_94><loc_57></location>15 School of Physics, HH Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, UK</list_item>
<list_item><location><page_14><loc_52><loc_52><loc_94><loc_55></location>16 INFN-Sezione di Bologna, Viale Berti Pichat 6 / 2, 40127 Bologna, Italy</list_item>
<list_item><location><page_14><loc_52><loc_50><loc_94><loc_52></location>17 Universität Bonn, Argelander-Institut für Astronomie, Auf dem Hügel 71, 53121 Bonn, Germany</list_item>
</unordered_list>
<text><location><page_14><loc_6><loc_7><loc_24><loc_8></location>Article number, page 14 of 26</text>
<text><location><page_14><loc_52><loc_49><loc_53><loc_50></location>18</text>
<text><location><page_14><loc_54><loc_49><loc_94><loc_50></location>Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM,</text>
<text><location><page_14><loc_54><loc_48><loc_72><loc_49></location>91191, Gif-sur-Yvette, France</text>
<unordered_list>
<list_item><location><page_14><loc_52><loc_45><loc_94><loc_48></location>19 Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy</list_item>
<list_item><location><page_14><loc_52><loc_43><loc_94><loc_45></location>20 Department of Astronomy, University of Geneva, ch. d'Ecogia 16, 1290 Versoix, Switzerland</list_item>
<list_item><location><page_14><loc_52><loc_41><loc_94><loc_43></location>21 Univ. Grenoble Alpes, CNRS, Grenoble INP, LPSC-IN2P3, 53, Avenue des Martyrs, 38000, Grenoble, France</list_item>
<list_item><location><page_14><loc_52><loc_38><loc_94><loc_40></location>22 Institut für Theoretische Physik, University of Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany</list_item>
</unordered_list>
<text><location><page_14><loc_52><loc_37><loc_53><loc_38></location>23</text>
<text><location><page_14><loc_54><loc_37><loc_94><loc_38></location>Zentrum für Astronomie, Universität Heidelberg, Philosophenweg</text>
<text><location><page_14><loc_54><loc_36><loc_73><loc_37></location>12, 69120 Heidelberg, Germany</text>
<unordered_list>
<list_item><location><page_14><loc_52><loc_33><loc_94><loc_36></location>24 School of Mathematics and Physics, University of Surrey, Guildford, Surrey, GU2 7XH, UK</list_item>
<list_item><location><page_14><loc_52><loc_31><loc_94><loc_33></location>25 SISSA, International School for Advanced Studies, Via Bonomea 265, 34136 Trieste TS, Italy</list_item>
<list_item><location><page_14><loc_52><loc_29><loc_94><loc_31></location>26 Dipartimento di Fisica e Astronomia, Università di Bologna, Via Gobetti 93 / 2, 40129 Bologna, Italy</list_item>
<list_item><location><page_14><loc_52><loc_26><loc_94><loc_29></location>27 INAF-Osservatorio Astrofisico di Torino, Via Osservatorio 20, 10025 Pino Torinese (TO), Italy</list_item>
<list_item><location><page_14><loc_52><loc_24><loc_94><loc_26></location>28 Dipartimento di Fisica, Università di Genova, Via Dodecaneso 33, 16146, Genova, Italy</list_item>
<list_item><location><page_14><loc_52><loc_22><loc_94><loc_24></location>29 INFN-Sezione di Genova, Via Dodecaneso 33, 16146, Genova, Italy</list_item>
<list_item><location><page_14><loc_52><loc_19><loc_94><loc_21></location>30 Department of Physics "E. Pancini", University Federico II, Via Cinthia 6, 80126, Napoli, Italy</list_item>
<list_item><location><page_14><loc_52><loc_17><loc_94><loc_19></location>31 INAF-Osservatorio Astronomico di Capodimonte, Via Moiariello 16, 80131 Napoli, Italy</list_item>
<list_item><location><page_14><loc_52><loc_16><loc_90><loc_17></location>32 INFN section of Naples, Via Cinthia 6, 80126, Napoli, Italy</list_item>
<list_item><location><page_14><loc_52><loc_13><loc_94><loc_16></location>33 Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, PT4150-762 Porto, Portugal</list_item>
<list_item><location><page_14><loc_52><loc_11><loc_94><loc_13></location>34 Dipartimento di Fisica, Università degli Studi di Torino, Via P. Giuria 1, 10125 Torino, Italy</list_item>
<list_item><location><page_14><loc_52><loc_10><loc_90><loc_11></location>35 INFN-Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_15><loc_7><loc_92><loc_47><loc_93></location>36 INAF-IASF Milano, Via Alfonso Corti 12, 20133 Milano, Italy</list_item>
<list_item><location><page_15><loc_7><loc_88><loc_49><loc_92></location>37 Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Avenida Complutense 40, 28040 Madrid, Spain</list_item>
<list_item><location><page_15><loc_7><loc_86><loc_49><loc_88></location>38 Port d'Informació Científica, Campus UAB, C. Albareda s / n, 08193 Bellaterra (Barcelona), Spain</list_item>
<list_item><location><page_15><loc_7><loc_84><loc_49><loc_86></location>39 Institute for Theoretical Particle Physics and Cosmology (TTK), RWTH Aachen University, 52056 Aachen, Germany</list_item>
<list_item><location><page_15><loc_7><loc_81><loc_49><loc_84></location>40 INAF-Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monteporzio Catone, Italy</list_item>
<list_item><location><page_15><loc_7><loc_78><loc_49><loc_81></location>41 Dipartimento di Fisica e Astronomia "Augusto Righi" - Alma Mater Studiorum Università di Bologna, Viale Berti Pichat 6 / 2, 40127 Bologna, Italy</list_item>
<list_item><location><page_15><loc_7><loc_75><loc_49><loc_78></location>42 Instituto de Astrofísica de Canarias, Calle Vía Láctea s / n, 38204, San Cristóbal de La Laguna, Tenerife, Spain</list_item>
<list_item><location><page_15><loc_7><loc_73><loc_49><loc_75></location>43 Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK</list_item>
<list_item><location><page_15><loc_7><loc_69><loc_49><loc_73></location>44 Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, UK</list_item>
<list_item><location><page_15><loc_7><loc_67><loc_49><loc_69></location>45 European Space Agency / ESRIN, Largo Galileo Galilei 1, 00044 Frascati, Roma, Italy</list_item>
<list_item><location><page_15><loc_7><loc_65><loc_49><loc_67></location>46 ESAC / ESA, Camino Bajo del Castillo, s / n., Urb. Villafranca del Castillo, 28692 Villanueva de la Cañada, Madrid, Spain</list_item>
<list_item><location><page_15><loc_7><loc_62><loc_49><loc_64></location>47 Université Claude Bernard Lyon 1, CNRS / IN2P3, IP2I Lyon, UMR 5822, Villeurbanne, F-69100, France</list_item>
<list_item><location><page_15><loc_7><loc_59><loc_49><loc_62></location>48 Institute of Physics, Laboratory of Astrophysics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland</list_item>
<list_item><location><page_15><loc_7><loc_56><loc_49><loc_59></location>49 UCB Lyon 1, CNRS / IN2P3, IUF, IP2I Lyon, 4 rue Enrico Fermi, 69622 Villeurbanne, France</list_item>
<list_item><location><page_15><loc_7><loc_54><loc_49><loc_56></location>50 Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Edifício C8, Campo Grande, PT1749-016 Lisboa, Portugal</list_item>
<list_item><location><page_15><loc_7><loc_50><loc_49><loc_54></location>51 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal</list_item>
<list_item><location><page_15><loc_7><loc_48><loc_49><loc_50></location>52 INAF-Istituto di Astrofisica e Planetologia Spaziali, via del Fosso del Cavaliere, 100, 00100 Roma, Italy</list_item>
<list_item><location><page_15><loc_7><loc_45><loc_49><loc_48></location>53 INAF-Osservatorio Astronomico di Padova, Via dell'Osservatorio 5, 35122 Padova, Italy</list_item>
<list_item><location><page_15><loc_7><loc_43><loc_49><loc_45></location>54 Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1, 85748 Garching, Germany</list_item>
<list_item><location><page_15><loc_7><loc_41><loc_49><loc_43></location>55 Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, 0315 Oslo, Norway</list_item>
<list_item><location><page_15><loc_7><loc_38><loc_49><loc_41></location>56 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA, 91109, USA</list_item>
<list_item><location><page_15><loc_7><loc_36><loc_49><loc_38></location>57 Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK</list_item>
<list_item><location><page_15><loc_7><loc_34><loc_49><loc_36></location>58 Felix Hormuth Engineering, Goethestr. 17, 69181 Leimen, Germany</list_item>
<list_item><location><page_15><loc_7><loc_31><loc_49><loc_33></location>59 Technical University of Denmark, Elektrovej 327, 2800 Kgs. Lyngby, Denmark</list_item>
<list_item><location><page_15><loc_7><loc_30><loc_34><loc_31></location>60 Cosmic Dawn Center (DAWN), Denmark</list_item>
<list_item><location><page_15><loc_7><loc_28><loc_49><loc_30></location>61 Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany</list_item>
<list_item><location><page_15><loc_7><loc_25><loc_49><loc_28></location>62 Department of Physics and Helsinki Institute of Physics, Gustaf Hällströmin katu 2, 00014 University of Helsinki, Finland</list_item>
<list_item><location><page_15><loc_7><loc_24><loc_49><loc_25></location>63 Aix-Marseille Université, CNRS / IN2P3, CPPM, Marseille, France</list_item>
<list_item><location><page_15><loc_7><loc_20><loc_49><loc_24></location>64 Université de Genève, Département de Physique Théorique and Centre for Astroparticle Physics, 24 quai Ernest-Ansermet, CH1211 Genève 4, Switzerland</list_item>
<list_item><location><page_15><loc_7><loc_18><loc_49><loc_20></location>65 Department of Physics, P.O. Box 64, 00014 University of Helsinki, Finland</list_item>
<list_item><location><page_15><loc_7><loc_16><loc_49><loc_18></location>66 Helsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, Finland</list_item>
<list_item><location><page_15><loc_7><loc_13><loc_49><loc_16></location>67 NOVA optical infrared instrumentation group at ASTRON, Oude Hoogeveensedijk 4, 7991PD, Dwingeloo, The Netherlands</list_item>
<list_item><location><page_15><loc_7><loc_11><loc_49><loc_13></location>68 Dipartimento di Fisica "Aldo Pontremoli", Università degli Studi di Milano, Via Celoria 16, 20133 Milano, Italy</list_item>
<list_item><location><page_15><loc_7><loc_10><loc_46><loc_11></location>69 INFN-Sezione di Milano, Via Celoria 16, 20133 Milano, Italy</list_item>
<list_item><location><page_15><loc_52><loc_91><loc_94><loc_93></location>70 INFN-Sezione di Roma, Piazzale Aldo Moro, 2 - c / o Dipartimento di Fisica, Edificio G. Marconi, 00185 Roma, Italy</list_item>
<list_item><location><page_15><loc_52><loc_90><loc_93><loc_91></location>71 Aix-Marseille Université, CNRS, CNES, LAM, Marseille, France</list_item>
<list_item><location><page_15><loc_52><loc_87><loc_94><loc_89></location>72 Department of Physics, Institute for Computational Cosmology, Durham University, South Road, Durham, DH1 3LE, UK</list_item>
<list_item><location><page_15><loc_52><loc_84><loc_94><loc_87></location>73 Université Côte d'Azur, Observatoire de la Côte d'Azur, CNRS, Laboratoire Lagrange, Bd de l'Observatoire, CS 34229, 06304 Nice cedex 4, France</list_item>
<list_item><location><page_15><loc_52><loc_81><loc_94><loc_83></location>74 Université Paris Cité, CNRS, Astroparticule et Cosmologie, 75013 Paris, France</list_item>
<list_item><location><page_15><loc_52><loc_77><loc_94><loc_81></location>75 Institut de Física d'Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona), Spain</list_item>
<list_item><location><page_15><loc_52><loc_75><loc_94><loc_77></location>76 European Space Agency / ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands</list_item>
<list_item><location><page_15><loc_52><loc_73><loc_94><loc_75></location>77 School of Mathematics, Statistics and Physics, Newcastle University, Herschel Building, Newcastle-upon-Tyne, NE1 7RU, UK</list_item>
<list_item><location><page_15><loc_52><loc_70><loc_94><loc_72></location>78 Department of Physics and Astronomy, University of Aarhus, Ny Munkegade 120, DK-8000 Aarhus C, Denmark</list_item>
<list_item><location><page_15><loc_52><loc_68><loc_94><loc_70></location>79 Space Science Data Center, Italian Space Agency, via del Politecnico snc, 00133 Roma, Italy</list_item>
<list_item><location><page_15><loc_52><loc_65><loc_94><loc_68></location>80 Centre National d'Etudes Spatiales - Centre spatial de Toulouse, 18 avenue Edouard Belin, 31401 Toulouse Cedex 9, France</list_item>
<list_item><location><page_15><loc_52><loc_63><loc_94><loc_65></location>81 Institute of Space Science, Str. Atomistilor, nr. 409 M˘agurele, Ilfov, 077125, Romania</list_item>
<list_item><location><page_15><loc_52><loc_61><loc_94><loc_63></location>82 Dipartimento di Fisica e Astronomia "G. Galilei", Università di Padova, Via Marzolo 8, 35131 Padova, Italy</list_item>
<list_item><location><page_15><loc_52><loc_59><loc_84><loc_60></location>83 INFN-Padova, Via Marzolo 8, 35131 Padova, Italy</list_item>
<list_item><location><page_15><loc_52><loc_56><loc_94><loc_59></location>84 Institut de Recherche en Astrophysique et Planétologie (IRAP), Université de Toulouse, CNRS, UPS, CNES, 14 Av. Edouard Belin, 31400 Toulouse, France</list_item>
<list_item><location><page_15><loc_52><loc_54><loc_89><loc_56></location>85 Université St Joseph; Faculty of Sciences, Beirut, Lebanon</list_item>
<list_item><location><page_15><loc_52><loc_52><loc_94><loc_54></location>86 Departamento de Física, FCFM, Universidad de Chile, Blanco Encalada 2008, Santiago, Chile</list_item>
<list_item><location><page_15><loc_52><loc_50><loc_94><loc_52></location>87 Universität Innsbruck, Institut für Astro- und Teilchenphysik, Technikerstr. 25 / 8, 6020 Innsbruck, Austria</list_item>
<list_item><location><page_15><loc_52><loc_47><loc_94><loc_49></location>88 Institut d'Estudis Espacials de Catalunya (IEEC), Edifici RDIT, Campus UPC, 08860 Castelldefels, Barcelona, Spain</list_item>
<list_item><location><page_15><loc_52><loc_45><loc_94><loc_47></location>89 Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, s / n, 08193 Barcelona, Spain</list_item>
<list_item><location><page_15><loc_52><loc_42><loc_94><loc_45></location>90 Satlantis, University Science Park, Sede Bld 48940, Leioa-Bilbao, Spain</list_item>
<list_item><location><page_15><loc_52><loc_39><loc_94><loc_42></location>91 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, Tapada da Ajuda, 1349-018 Lisboa, Portugal</list_item>
<list_item><location><page_15><loc_52><loc_35><loc_94><loc_39></location>92 Universidad Politécnica de Cartagena, Departamento de Electrónica y Tecnología de Computadoras, Plaza del Hospital 1, 30202 Cartagena, Spain</list_item>
<list_item><location><page_15><loc_52><loc_33><loc_94><loc_35></location>93 Kapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands</list_item>
<list_item><location><page_15><loc_52><loc_30><loc_94><loc_33></location>94 Infrared Processing and Analysis Center, California Institute of Technology, Pasadena, CA 91125, USA</list_item>
<list_item><location><page_15><loc_52><loc_28><loc_94><loc_30></location>95 INAF, Istituto di Radioastronomia, Via Piero Gobetti 101, 40129 Bologna, Italy</list_item>
<list_item><location><page_15><loc_52><loc_24><loc_94><loc_28></location>96 Astronomical Observatory of the Autonomous Region of the Aosta Valley (OAVdA), Loc. Lignan 39, I-11020, Nus (Aosta Valley), Italy</list_item>
<list_item><location><page_15><loc_52><loc_22><loc_94><loc_24></location>97 ICL, Junia, Université Catholique de Lille, LITL, 59000 Lille, France</list_item>
<list_item><location><page_15><loc_52><loc_19><loc_94><loc_22></location>98 Instituto de Física Teórica UAM-CSIC, Campus de Cantoblanco, 28049 Madrid, Spain</list_item>
<list_item><location><page_15><loc_52><loc_17><loc_94><loc_19></location>99 CERCA / ISO, Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106, USA</list_item>
<list_item><location><page_15><loc_51><loc_15><loc_94><loc_17></location>100 Dipartimento di Fisica e Scienze della Terra, Università degli Studi di Ferrara, Via Giuseppe Saragat 1, 44122 Ferrara, Italy</list_item>
<list_item><location><page_15><loc_51><loc_12><loc_94><loc_14></location>101 Istituto Nazionale di Fisica Nucleare, Sezione di Ferrara, Via Giuseppe Saragat 1, 44122 Ferrara, Italy</list_item>
<list_item><location><page_15><loc_51><loc_10><loc_94><loc_12></location>102 Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, Chiba 277-8583, Japan</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_16><loc_7><loc_91><loc_49><loc_93></location>103 Minnesota Institute for Astrophysics, University of Minnesota, 116 Church St SE, Minneapolis, MN 55455, USA</list_item>
<list_item><location><page_16><loc_7><loc_88><loc_49><loc_91></location>104 Institute Lorentz, Leiden University, Niels Bohrweg 2, 2333 CA Leiden, The Netherlands</list_item>
<list_item><location><page_16><loc_7><loc_86><loc_49><loc_88></location>105 Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, USA</list_item>
<list_item><location><page_16><loc_7><loc_84><loc_49><loc_86></location>106 Department of Physics & Astronomy, University of California Irvine, Irvine CA 92697, USA</list_item>
<list_item><location><page_16><loc_7><loc_80><loc_49><loc_84></location>107 Department of Astronomy & Physics and Institute for Computational Astrophysics, Saint Mary's University, 923 Robie Street, Halifax, Nova Scotia, B3H 3C3, Canada</list_item>
<list_item><location><page_16><loc_7><loc_78><loc_49><loc_80></location>108 Departamento Física Aplicada, Universidad Politécnica de Cartagena, Campus Muralla del Mar, 30202 Cartagena, Murcia, Spain</list_item>
<list_item><location><page_16><loc_7><loc_75><loc_49><loc_78></location>109 Université Paris-Saclay, CNRS, Institut d'astrophysique spatiale, 91405, Orsay, France</list_item>
<list_item><location><page_16><loc_7><loc_73><loc_49><loc_75></location>110 Department of Physics, Oxford University, Keble Road, Oxford OX1 3RH, UK</list_item>
<list_item><location><page_16><loc_7><loc_71><loc_49><loc_73></location>111 Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth PO1 3FX, UK</list_item>
<list_item><location><page_16><loc_7><loc_68><loc_49><loc_70></location>112 Department of Computer Science, Aalto University, PO Box 15400, Espoo, FI-00 076, Finland</list_item>
<list_item><location><page_16><loc_7><loc_65><loc_49><loc_68></location>113 Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), German Centre for Cosmological Lensing (GCCL), 44780 Bochum, Germany</list_item>
<list_item><location><page_16><loc_7><loc_62><loc_49><loc_64></location>114 DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 155, 2200 Copenhagen, Denmark</list_item>
<list_item><location><page_16><loc_7><loc_60><loc_49><loc_62></location>115 Department of Physics and Astronomy, Vesilinnantie 5, 20014 University of Turku, Finland</list_item>
<list_item><location><page_16><loc_7><loc_56><loc_49><loc_60></location>116 Serco for European Space Agency (ESA), Camino bajo del Castillo, s / n, Urbanizacion Villafranca del Castillo, Villanueva de la Cañada, 28692 Madrid, Spain</list_item>
<list_item><location><page_16><loc_7><loc_54><loc_49><loc_56></location>117 ARC Centre of Excellence for Dark Matter Particle Physics, Melbourne, Australia</list_item>
<list_item><location><page_16><loc_7><loc_51><loc_49><loc_54></location>118 Centre for Astrophysics & Supercomputing, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia</list_item>
<list_item><location><page_16><loc_7><loc_49><loc_49><loc_51></location>119 School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London E1 4NS, UK</list_item>
<list_item><location><page_16><loc_7><loc_47><loc_49><loc_49></location>120 Department of Physics and Astronomy, University of the Western Cape, Bellville, Cape Town, 7535, South Africa</list_item>
<list_item><location><page_16><loc_7><loc_43><loc_49><loc_47></location>121 ICTP South American Institute for Fundamental Research, Instituto de Física Teórica, Universidade Estadual Paulista, São Paulo, Brazil</list_item>
<list_item><location><page_16><loc_7><loc_41><loc_49><loc_43></location>122 Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, Stockholm, SE-106 91, Sweden</list_item>
<list_item><location><page_16><loc_7><loc_38><loc_49><loc_41></location>123 Astrophysics Group, Blackett Laboratory, Imperial College London, London SW7 2AZ, UK</list_item>
<list_item><location><page_16><loc_7><loc_36><loc_49><loc_38></location>124 INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125, Firenze, Italy</list_item>
<list_item><location><page_16><loc_7><loc_34><loc_49><loc_36></location>125 Centro de Astrofísica da Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal</list_item>
<list_item><location><page_16><loc_7><loc_31><loc_49><loc_33></location>126 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK</list_item>
<list_item><location><page_16><loc_7><loc_29><loc_49><loc_31></location>127 Department of Astrophysics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland</list_item>
<list_item><location><page_16><loc_7><loc_25><loc_49><loc_29></location>128 Dipartimento di Fisica, Università degli studi di Genova, and INFN-Sezione di Genova, via Dodecaneso 33, 16146, Genova, Italy</list_item>
<list_item><location><page_16><loc_7><loc_23><loc_49><loc_25></location>129 Theoretical astrophysics, Department of Physics and Astronomy, Uppsala University, Box 515, 751 20 Uppsala, Sweden</list_item>
<list_item><location><page_16><loc_7><loc_20><loc_49><loc_23></location>130 Department of Physics, Royal Holloway, University of London, TW20 0EX, UK</list_item>
<list_item><location><page_16><loc_7><loc_18><loc_49><loc_20></location>131 Mullard Space Science Laboratory, University College London, Holmbury St Mary, Dorking, Surrey RH5 6NT, UK</list_item>
<list_item><location><page_16><loc_7><loc_16><loc_49><loc_18></location>132 Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA</list_item>
<list_item><location><page_16><loc_7><loc_14><loc_28><loc_16></location>133 Cosmic Dawn Center (DAWN)</list_item>
<list_item><location><page_16><loc_7><loc_12><loc_49><loc_14></location>134 Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 Copenhagen, Denmark</list_item>
<list_item><location><page_16><loc_7><loc_10><loc_49><loc_12></location>135 Center for Cosmology and Particle Physics, Department of Physics, New York University, New York, NY 10003, USA</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_16><loc_51><loc_91><loc_94><loc_93></location>136 Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, 10010, New York, NY, USA</list_item>
</unordered_list>
<figure>
<location><page_17><loc_8><loc_26><loc_88><loc_93></location>
<caption>Fig. 13: Correlation coe ffi cients matrix (upper-right triangle) and scatter plot (bottom-left triangle) of power-law log-residuals of Euclid -data (lensing concentration, lensing mass, richness, and stellar mass, respectively) and possible outcomes from multiwavelength observations (integrated Comptony parameter, gas mass, X-ray luminosity, and temperature, respectively). Cell colouring goes from blue (negative correlation coe ffi cients) to red (positive correlation coe ffi cients) and is white in the interval [ -0 . 35 , 0 . 35] in order to enhance the visibility of the most significant coe ffi cients.</caption>
</figure>
<figure>
<location><page_18><loc_7><loc_26><loc_88><loc_93></location>
<caption>Fig. 14: As Fig. 13, here we show the quantities computed in the 3D space.</caption>
</figure>
<figure>
<location><page_19><loc_7><loc_22><loc_93><loc_93></location>
<caption>Fig. 15: As Fig. 13, but we show the projected X-ray luminosity, the projected gas mass, and the projected temperature computed within the overdensity of ∆ c = 500 instead of the respective quantities within ∆ c = 200 .</caption>
</figure>
<figure>
<location><page_20><loc_7><loc_72><loc_93><loc_93></location>
<caption>Fig. 16: Same as Fig. 3 for the data at z = 0 . 9 .</caption>
</figure>
<figure>
<location><page_21><loc_7><loc_74><loc_48><loc_93></location>
<caption>Fig. 17: Same as Fig. 4 for the data at z = 0 . 9 . We do not report the value for the concentration because in our fit radial range the lensing one does not correlate with the 3D one.</caption>
</figure>
<section_header_level_1><location><page_22><loc_6><loc_92><loc_27><loc_93></location>Appendix A: Fit of gNFW</section_header_level_1>
<text><location><page_22><loc_6><loc_76><loc_49><loc_90></location>In Fig. A.1, we present the density profile of a halo that deviates from NFW profile and has no LoS contamination. In particular, it has β ≈ 1 . 8 and γ = 1 . 5 . We show its NFW profile fit profile on the 3D density in the upper panel of Fig. A.1, where we can see that 3D NFW profile (performed on radial bins in a sphere) is capable of capturing the shape of the halo and to estimate its mass with high accuracy (within ≈ 5%). In the central panel, we show the reduced shear profile and best fits, where we can see that the fit performed on the reduced shear underestimated the concentration and is not able to capture the more internal part of the shear profile, as it is done by the profile that was fit in 3D.</text>
<text><location><page_22><loc_6><loc_68><loc_49><loc_76></location>We first exclude this mismatch as being due to projection effects by showing that both the reduced shear from the particle data (orange line) matches the one recovered by performing an analytical projection of the 3D profile (blue solid line). In particular we project the density profile ρ ( r ) and derive the surface mass density Σ conv ., as follows</text>
<formula><location><page_22><loc_6><loc_61><loc_49><loc_65></location>Σ conv . ( R ) = Z + ∞ -∞ ρ GLYPH<18> p R 2 + z 2 GLYPH<19> d z = 2 Z ∞ R ρ ( r ) r √ r 2 -R 2 d r . (A.1)</formula>
<text><location><page_22><loc_6><loc_53><loc_49><loc_58></location>What we find is that the shear obtained with the aid of an analytical projection Σ conv . matches very well the real one (i.e., orange and blue lines do match). This hints that for this cluster there are no strong LoS e ff ects.</text>
<text><location><page_22><loc_6><loc_39><loc_49><loc_52></location>To understand why the fit on the shear is not able to capture the concentration of the original halo, we zoom our fit in the bottom panel of Fig. A.1, where it looks like the fit is very good in capturing the final part of the profile and not able to capture the internal. It is crucial to emphasise that we did not include observational uncertainties in these analyses. Therefore, the uncertainty outlined in Eq. (8) a ff ects the fit by assigning more weight to external radial bins compared to internal ones. It is worth noting that the proportionality factors in Eq. (8) will not a ff ect our best fit.</text>
<text><location><page_22><loc_6><loc_18><loc_49><loc_39></location>To validate this point in Fig. A.2 we study the bias on fitting an NFW profile on a mock gNFW profile that has β = 1 . 8 and γ = 1 . 5 , a mass of 3 × 10 14 M ⊙ and a concentration c = 2 . 4 (as the halo presented in Fig. A.1). We see that the 3D NFW profile is capable of estimating both its mass and its gNFW concentration with high accuracy (see top panel match between blue and dashed black lines). On the other hand the fit of the shear (we report in the bottom panel of Fig. A.2) has the same problems as the one on the cluster in Fig. A.1: it recovers a low concentration (with a value of 1 . 5). This may be because, at outer radii, the model fits the data. It is possible that the under-estimation of concentration at low radii is caused by the combination of two factors: the fit under-estimates the shear at lower radii (with the result of under-estimating the lensing concentration), or the fact that γ is di ff erent than 3 induces an NFW profile fit with a low concentration.</text>
<text><location><page_22><loc_6><loc_10><loc_49><loc_17></location>We then performed the experiment of fitting the analytical profile with constant (yet unrealistic) error bars. While the fit was able to capture the shape of the profile, it recovered a concentration of 1 . 6 , implying that there is indeed a degeneracy between the shear of low-concentrated NFW profiles and steeper-NFW profiles.</text>
<figure>
<location><page_22><loc_53><loc_78><loc_91><loc_93></location>
</figure>
<figure>
<location><page_22><loc_53><loc_61><loc_91><loc_76></location>
</figure>
<figure>
<location><page_22><loc_52><loc_45><loc_93><loc_60></location>
<caption>Fig. A.1: Density profiles of a simulated halo and the corresponding NFW profile fit. Upper panel: total matter density profile (blue solid line) and the respective NFW profile fit profile (black dashed line). Central panel: reduced shear from simulated particles (orange solid line), and from the analytical projection of the density profile Σ conv . presented in Eq. (A.1) and performed in the radial range [60 , 3000] kpc, in the blue solid line. The dashed vertical line indicates the minimum radius of the shear fit, and the fit profiles (black lines) are extrapolated down to 10 kpc to enhance the central densities predicted by the two fits. The bottom panel shows the same as the central panel but focuses on the radial range of the fit. The error bars indicate the uncertainty for each radial bin, as defined in Eq. (8).</caption>
</figure>
<section_header_level_1><location><page_22><loc_51><loc_21><loc_83><loc_24></location>Appendix B: Correlations with different overdensities</section_header_level_1>
<text><location><page_22><loc_51><loc_12><loc_94><loc_20></location>In this appendix, we discuss the di ff erences between scaling relation scatters and covariance values at di ff erent overdensities. First of all, we tackle the fact that when we compute X-ray luminosity within r 500c (instead of r 200c), we find that the scatter of the scaling relation of the projected quantity is larger than the 3D one.</text>
<text><location><page_22><loc_51><loc_10><loc_94><loc_12></location>To investigate this feature, we will focus on the bolometric X-ray luminosity. We report the 3D and projected bolometric X-</text>
<figure>
<location><page_23><loc_8><loc_78><loc_48><loc_93></location>
</figure>
<figure>
<location><page_23><loc_7><loc_62><loc_47><loc_77></location>
<caption>Fig. A.2: Density profiles of a mock halo that deviates from NFW and the corresponding NFW profile fit. The mock halo mass, concentration parameter and gNFW log-slopes are chosen to match the ones of the simulated halo presented in Fig. A.1. The upper panel reports the total matter density profile of the mock halo (solid blue line) and the profile from the corresponding NFW profile fit (dashed black line). The bottom panel shows the reduced shear and the profile from the corresponding NFW profile fit (dotted black line). The error bars indicate the uncertainty for each radial bin, as defined in Eq. (8).</caption>
</figure>
<text><location><page_23><loc_6><loc_25><loc_49><loc_44></location>ray luminosity in Fig. B.1 (top panel), where it is visually clear that the projected X-ray luminosity is (as expected) always larger than the 3D one. One can also notice that the increase in X-ray luminosity depends on the fact that a halo is over-luminous or not: the increase of luminosity growing from the 3D to 2D is larger for under-luminous haloes than for over-luminous haloes. We prove this point in the bottom panel of Fig. B.1 where we show the ratio between the 2D and 3D luminosity as a function of their residual of the 3D scaling relation (the higher the value of the x axis, the more over-luminous is the object for its mass bin), where we can see a strong anti-correlation: overly luminous objects (for a given mass bin) are not going to be a ff ected much by the fact that their luminosity is computed in 3D or 2D. The possible cause is that an interloper in the LoS will not a ff ect much an overly luminous object.</text>
<text><location><page_23><loc_6><loc_16><loc_49><loc_24></location>For completeness, in Fig. B.2 we show the correlation coefficients between the gas mass and stellar mass computed within both r 500c and r 200c and the concentration. Here we can see a change of sign between M ⋆, 500cMg , 500c correlations and M ⋆, 500cM g correlations and a change in the sign between c NFWM g correlations and c NFWMg , 500c correlations.</text>
<text><location><page_23><loc_6><loc_10><loc_49><loc_16></location>Finally, in Fig. B.3 we report the scatter of observable properties at fixed mass for both Euclid -like quantities (lensing mass, richness, and projected stellar mass), and possible multiwavelength properties (integrated Comptony parameter, X-ray luminosity, and temperature), where we compute X-ray lumi-</text>
<figure>
<location><page_23><loc_52><loc_69><loc_91><loc_93></location>
</figure>
<figure>
<location><page_23><loc_52><loc_45><loc_93><loc_69></location>
<caption>Fig. B.1: Comparison between 3D and projected X-ray luminosities. Top panel shows a scatter plot of the two mass-observable relations, while the bottom panel shows their ratio as a function of the 3D X-ray scaling relation residuals.</caption>
</figure>
<text><location><page_23><loc_51><loc_26><loc_94><loc_36></location>nosity and temperature within r 500c as they are typically derived within this overdensity. The upper panel of Fig. B.3 shows the residuals of the log-log linear regression where we see that in terms of 2D scatter, the properties with the lowest scatter are the stellar mass and the temperature. The bottom panel shows the data points used to perform the fit (in black) where we used a visually-inspected cut on the halo mass values in order to ensure that mass values are complete for a given observable value.</text>
<table>
<location><page_24><loc_6><loc_62><loc_95><loc_91></location>
<caption>Table B.1: Scatter and correlation coe ffi cient matrix between z = 0 . 24 log-residuals of scaling relations.</caption>
</table>
<text><location><page_24><loc_6><loc_56><loc_94><loc_61></location>Notes. Diagonal terms report the scatter of the log-residuals of each quantity, namely σ ln of Eq. (9), while the o ff -diagonal terms report the correlation coe ffi cient between the log-residuals. We do not report values of the correlation coe ffi cient below 0 . 20 because they are not significant. We do not report the values for the 3D NFW mass M 200c because it has a very low scatter of log-residuals ( ≈ 0 . 01) and its correlation coe ffi cients are not meaningful. Note that in this table we also added the core-excised X-ray luminosity.</text>
<table>
<location><page_24><loc_6><loc_23><loc_96><loc_52></location>
<caption>Table B.2: Scatter and correlation coe ffi cient matrix between z = 0 . 9 log-residuals of scaling relations.</caption>
</table>
<text><location><page_24><loc_6><loc_21><loc_34><loc_22></location>Notes. Rows and columns are as in Table B.1.</text>
<figure>
<location><page_25><loc_7><loc_59><loc_48><loc_93></location>
<caption>Fig. B.2: We report the correlation coe ffi cient between concentration, gas mass and stellar mass computed at both r 500c and r 200c .</caption>
</figure>
<figure>
<location><page_26><loc_7><loc_69><loc_93><loc_93></location>
<caption>Fig. B.3: Halo masses at fixed observable properties. We report the lensing mass, lensing richness, projected stellar mass, projected integrated Comptony , projected X-ray luminosity, and projected temperature in each column, respectively. The top panel shows residuals of the observable-mass relations and respective scatter of log-residuals σ ln , and its axes are on the upper part of the plot. The bottom panel shows the scaling relation fit (blue solid line); the data used to perform the fit (black data points) over-plotted on top of the total sample (grey data points) of the mass M as a function of the observable properties.</caption>
</figure>
</document>
[
{
"title": "ABSTRACT",
"content": "Context. Cluster cosmology can benefit from combining multi-wavelength studies, which themselves can benefit from a characterisation of the correlation coe ffi cients between di ff erent mass-observable relations. Aims. In this work, we aim to provide information on the scatter, the skewness, and the covariance of various mass-observable relations in galaxy clusters in cosmological hydrodynamic simulations. This information will help future analyses to better tackle accretion histories and projection e ff ects and model mass observable relations for cosmology studies. Methods. Weidentify galaxy clusters in Magneticum Box2b simulations with mass M 200c > 10 14 M ⊙ at redshift z = 0 . 24 and z = 0 . 90. Our analysis includes Euclid -derived properties such as richness, stellar mass, lensing mass, and concentration. Additionally, we investigate complementary multi-wavelength data, including X-ray luminosity, integrated Comptony parameter, gas mass, and temperature. The impact of projection e ff ects on mass-observable residuals and correlations is then examined. Results. We find that at intermediate redshift ( z = 0 . 24) projection e ff ects impact lensing concentration, richness, and gas mass the most in terms of scatter and skewness of log-residuals of scaling relations. The contribution of projection e ff ects can be significant enough to boost a spurious hot- vs. cold-baryons correlation and consequently hide underlying correlations due to halo accretion histories. At high redshift ( z = 0 . 9), the richness has a much lower scatter (of log-residuals), and the quantity that is most impacted by projection e ff ects is the lensing mass. Lensing concentration reconstruction, in particular, is a ff ected by deviations of the reduced-shear profile shape from the one derived by an NFW profile rather than interlopers in the line of sight.",
"pages": [
1,
2
]
},
{
"title": "The impact of line-of-sight projections on the covariance between galaxy cluster multi-wavelength observable properties - insights from hydrodynamic simulations",
"content": "Euclid Collaboration: A. Ragagnin ⋆ 1 , 2 , 3 , 4 , A. Saro 5 , 2 , 6 , 7 , 4 , S. Andreon 8 , A. Biviano 6 , 2 , K. Dolag 9 , S. Ettori 1 , 10 , C. Giocoli 1 , 11 , A. M. C. Le Brun 12 , G. A. Mamon 13 , 14 , B. J. Maughan 15 , M. Meneghetti 1 , 16 , L. Moscardini 3 , 1 , 16 , F. Pacaud 17 , G. W. Pratt 18 , M. Sereno 1 , 16 , S. Borgani 5 , 2 , 6 , 7 , F. Calura 1 , G. Castignani 1 , M. De Petris 19 , D. Eckert 20 , G. F. Lesci 3 , 1 , J. Macias-Perez 21 , M. Maturi 22 , 23 , A. Amara 24 , N. Auricchio 1 , C. Baccigalupi 2 , 6 , 7 , 25 , M. Baldi 26 , 1 , 16 , S. Bardelli 1 , D. Bonino 27 , E. Branchini 28 , 29 , 8 , M. Brescia 30 , 31 , 32 , J. Brinchmann 33 , S. Camera 34 , 35 , 27 , V. Capobianco 27 , C. Carbone 36 , J. Carretero 37 , 38 , S. Casas 39 , M. Castellano 40 , S. Cavuoti 31 , 32 , A. Cimatti 41 , C. Colodro-Conde 42 , G. Congedo 43 , C. J. Conselice 44 , L. Conversi 45 , 46 , Y. Copin 47 , F. Courbin 48 , H. M. Courtois 49 , A. Da Silva 50 , 51 , H. Degaudenzi 20 , G. De Lucia 6 , J. Dinis 50 , 51 , F. Dubath 20 , X. Dupac 46 , M. Farina 52 , S. Farrens 18 , S. Ferriol 47 , M. Frailis 6 , E. Franceschi 1 , M. Fumana 36 , K. George 9 , B. Gillis 43 , A. Grazian 53 , F. Grupp 54 , 9 , S. V. H. Haugan 55 , W. Holmes 56 , I. Hook 57 , F. Hormuth 58 , A. Hornstrup 59 , 60 , K. Jahnke 61 , E. Keihänen 62 , S. Kermiche 63 , A. Kiessling 56 , M. Kilbinger 18 , B. Kubik 47 , M. Kümmel 9 , M. Kunz 64 , H. Kurki-Suonio 65 , 66 , S. Ligori 27 , P. B. Lilje 55 , V. Lindholm 65 , 66 , I. Lloro 67 , D. Maino 68 , 36 , 69 , E. Maiorano 1 , O. Mansutti 6 , O. Marggraf 17 , K. Markovic 56 , M. Martinelli 40 , 70 , N. Martinet 71 , F. Marulli 3 , 1 , 16 , R. Massey 72 , S. Maurogordato 73 , E. Medinaceli 1 , S. Mei 74 , Y. Mellier 13 , 14 , G. Meylan 48 , M. Moresco 3 , 1 , E. Munari 6 , 2 , C. Neissner 75 , 38 , S.-M. Niemi 76 , J. W. Nightingale 77 , 72 , C. Padilla 75 , S. Paltani 20 , F. Pasian 6 , K. Pedersen 78 , V. Pettorino 76 , G. Polenta 79 , M. Poncet 80 , L. A. Popa 81 , L. Pozzetti 1 , F. Raison 54 , A. Renzi 82 , 83 , J. Rhodes 56 , G. Riccio 31 , E. Romelli 6 , M. Roncarelli 1 , E. Rossetti 26 , R. Saglia 9 , 54 , Z. Sakr 22 , 84 , 85 , A. G. Sánchez 54 , D. Sapone 86 , B. Sartoris 9 , 6 , R. Scaramella 40 , 70 , P. Schneider 17 , T. Schrabback 87 , A. Secroun 63 , E. Sefusatti 6 , 2 , 7 , G. Seidel 61 , S. Serrano 88 , 89 , 90 , C. Sirignano 82 , 83 , G. Sirri 16 , L. Stanco 83 , J. Steinwagner 54 , P. Tallada-Crespí 37 , 38 , I. Tereno 50 , 91 , R. Toledo-Moreo 92 , F. Torradeflot 38 , 37 , I. Tutusaus 84 , L. Valenziano 1 , 10 , T. Vassallo 9 , 6 , G. Verdoes Kleijn 93 , A. Veropalumbo 8 , 29 , Y. Wang 94 , J. Weller 9 , 54 , G. Zamorani 1 , E. Zucca 1 , M. Bolzonella 1 , A. Boucaud 74 , E. Bozzo 20 , C. Burigana 95 , 10 , M. Calabrese 96 , 36 , D. Di Ferdinando 16 , J. A. Escartin Vigo 54 , R. Farinelli 1 , J. Gracia-Carpio 54 , N. Mauri 41 , 16 , V. Scottez 13 , 97 , M. Tenti 16 , M. Viel 2 , 6 , 25 , 7 , 4 , M. Wiesmann 55 , Y. Akrami 98 , 99 , V. Allevato 31 , S. Anselmi 83 , 82 , 12 , M. Ballardini 100 , 1 , 101 , P. Bergamini 68 , 1 , A. Blanchard 84 , L. Blot 102 , 12 , S. Bruton 103 , R. Cabanac 84 , A. Calabro 40 , G. Canas-Herrera 76 , 104 , A. Cappi 1 , 73 , C. S. Carvalho 91 , T. Castro 6 , 7 , 2 , 4 , K. C. Chambers 105 , S. Contarini 54 , 3 , A. R. Cooray 106 , M. Costanzi 5 , 6 , 2 , B. De Caro 83 , 82 , S. de la Torre 71 , G. Desprez 107 , A. Díaz-Sánchez 108 , S. Di Domizio 28 , 29 , H. Dole 109 , S. Esco ffi er 63 , A. G. Ferrari 41 , 16 , P. G. Ferreira 110 , I. Ferrero 55 , F. Finelli 1 , 10 , F. Fornari 10 , L. Gabarra 110 , K. Ganga 74 , J. García-Bellido 98 , E. Gaztanaga 89 , 88 , 111 , F. Giacomini 16 , G. Gozaliasl 112 , 65 , A. Hall 43 , H. Hildebrandt 113 , J. Hjorth 114 , A. Jimenez Muñoz 21 , J. J. E. Kajava 115 , 116 , V. Kansal 117 , 118 , D. Karagiannis 119 , 120 , C. C. Kirkpatrick 62 , L. Legrand 121 , G. Libet 80 , A. Loureiro 122 , 123 , G. Maggio 6 , M. Magliocchetti 52 , F. Mannucci 124 , R. Maoli 19 , 40 , C. J. A. P. Martins 125 , 33 , S. Matthew 43 , L. Maurin 109 , R. B. Metcalf 3 , 1 , P. Monaco 5 , 6 , 7 , 2 , C. Moretti 25 , 4 , 6 , 2 , 7 , G. Morgante 1 , Nicholas A. Walton 126 , L. Patrizii 16 , A. Pezzotta 54 , M. Pöntinen 65 , V. Popa 81 , C. Porciani 17 , D. Potter 127 , I. Risso 128 , P.-F. Rocci 109 , M. Sahlén 129 , A. Schneider 127 , M. Schultheis 73 , P. Simon 17 , A. Spurio Mancini 130 , 131 , C. Tao 63 , G. Testera 29 , R. Teyssier 132 , S. Toft 60 , 133 , 134 , S. Tosi 28 , 29 , 8 , A. Troja 82 , 83 , M. Tucci 20 , C. Valieri 16 , J. Valiviita 65 , 66 , D. Vergani 1 , and G. Verza 135 , 136 arXiv:2412.00191v1 [astro-ph.CO] 29 Nov 2024 (A ffi liations can be found after the references) June 16, 2023",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "Galaxy clusters are the largest gravitationally bound, collapsed, and virialised structures in our Universe and represent unique laboratories for testing cosmological models, galaxy evolution, and thermodynamics of the intracluster medium (ICM, see Kravtsov & Borgani 2012, for a review on galaxy clusters). Regarding galaxy cluster cosmology studies (see, e.g., Rozo et al. 2010; Bocquet et al. 2019), an accurate characterisation of the selection function and of the mass-observable scaling relations represent the dominant systematic uncertainties (see the review on the cluster mass scale in Pratt et al. 2019). Cluster masses cannot be observed directly, and their reconstruction requires both a number of assumptions and highquality data (see, e.g., Meneghetti et al. 2010). This means that precise estimates are rare (Okabe et al. 2010; Hoekstra et al. 2012; Melchior et al. 2015; van Uitert et al. 2016; Stern et al. 2019; Sugiyama et al. 2023; Bocquet et al. 2024). Once a set of highly accurate mass determinations are available, together with other mass proxies recovered from multi-band observations, well-calibrated mass-observable relations (for instance, the mass-richness relation or the mass-temperature relation) can be established and used to estimate galaxy cluster masses for larger samples with known observable properties. For this purpose, it is important to calibrate accurately the mass-observable relations (Giodini et al. 2013; Allen et al. 2011; Schrabback et al. 2021), including proper modelling of their associated scatter (Lima & Hu 2005; Bocquet et al. 2019). This process is complicated by the fact that studies at di ff erent wavelengths are biased by various astrophysical processes and projection e ff ects to various degrees. For instance, X-ray surveys tend to favour the selection of clusters with centrally peaked gas distributions (Pacaud et al. 2007; Hudson et al. 2010; Andreon & Moretti 2011; Andreon et al. 2016; Xu et al. 2018) and su ff er from AGN contamination (see, e.g., Bhargava et al. 2023), while projection e ff ects are known to strongly impact weak lensing mass reconstructions (Meneghetti et al. 2014; Euclid Collaboration: Giocoli et al. 2024) and richness evaluations (e.g., Castignani & Benoist 2016). This is particularly relevant for cluster cosmology studies, where the aim is to reduce uncertainty by combining constraints on di ff erent massobservable relations. For Euclid (Euclid Collaboration: Mellier et al. 2024), this will include quantities such as richness, stellar mass, and properties of stacked weak lensing signals (Pires et al. 2020), of the cluster samples detected using tools such as AMICO (Bellagamba et al. 2018a; Maturi et al. 2019) or PZWav (Euclid Collaboration: Adam et al. 2019), possibly to- gether with other multi-wavelength observations Allen et al. (2011). These properties are known to be biased by projection e ff ects (Meneghetti et al. 2014), accretion histories (Ragagnin et al. 2022), mis-centring (Sommer et al. 2022, 2023), and the fit procedure (Sereno et al. 2016). Projection e ff ects, especially, are expected to generate some covariance between the richness and weak lensing signal, the uncertainty of which may significantly a ff ect the performance of the mission for cluster population analyses. This e ff ect is one the major sources of systematics for current optical cluster surveys (Costanzi et al. 2019; Abbott et al. 2020), and thus is expected to play an even more critical role for the Euclid cluster sample. Numerical simulations are thus a critical tool to mitigate the impact of the aforementioned biases on cosmological cluster studies. Indeed, the power of observations to constrain them is limited, thus increasing the final uncertainty budget. However, scatter and covariance parameters are also prime sources of uncertainty when aiming to combine information originating from di ff erent wavelengths. For instance, various observational works hint towards di ff erent directions for the hot- vs. cold-baryon covariance (Farahi et al. 2019; Puddu & Andreon 2022; Ragagnin et al. 2022), as di ff erent formation times are related with satellite accretion history (Giocoli et al. 2008). In this context, numerical simulations have proven to be a very powerful tool for helping observational studies in modelling mass-observable relations, which are strongly a ff ected by galaxy cluster accretion histories (Ludlow et al. 2012; Bose et al. 2019; Davies et al. 2020; Anbajagane et al. 2020; Ragagnin et al. 2022), projection e ff ects (Meneghetti et al. 2014), and deviations (see, e.g., Ragagnin et al. 2021) from the Navarro-FrenkWhite density profile (NFW, Navarro et al. 1997), which is often adopted in weak lensing studies. Thus, simulations can suggest the most suitable functional forms of scaling relations for cosmological studies (as in the works of Costanzi et al. 2019; Bocquet et al. 2016, 2019; Ghirardini et al. 2024). They can provide informative priors on their correlation coe ffi cients, which are among the most di ffi cult parameters to be constrained directly from observed quantities, guiding the forward modelling setup of cluster cosmology studies. There are various works in the literature that study how simulations can help disentangle physical models (see e.g., Cui et al. 2022; Angelinelli et al. 2023a) cosmological models (see e.g., Bocquet et al. 2020; Angulo et al. 2021; Villaescusa-Navarro et al. 2022) or dark matter types (see e.g., Ragagnin et al. 2024; Fischer et al. 2024; Contreras-Santos et al. 2024), and study observable cross-correlations (see e.g., Stanek et al. 2010; Anbajagane et al. 2020). In this work, besides focusing on correlations between observable properties of interests for multi-wave length studies, we also study the impact of projection e ff ects. The impact of uncorrelated large-scale structure on the covariance between observable properties can be modelled analytically (Hoekstra 2003; McClintock et al. 2019; Costanzi et al. 2019), but the covariance of di ff erent observable properties below a few tens of Mpc requires dedicated simulations. At these scales, numerical hydrodynamic simulations, with their self-consistent depiction of the ICM, emerge as an ideal tool for exploring multiwavelength observable properties since they incorporate the effects of large-scale structures within which clusters are situated. Indeed, baryon feedback influences the ICM not only within cluster virial radii but also beyond (see, e.g., Angelinelli et al. 2022, 2023b). While it is true that cosmological simulations are influenced by the underlying sub-grid prescriptions, and while it is true that these simulations may diverge on small scales, they generally exhibit agreement on quantities integrated up to the sizes of galaxy groups and clusters (see, e.g., Anbajagane et al. 2020). At the same time, di ff erent cosmological parameters can a ff ect galaxy cluster properties, such as their masses (Ragagnin et al. 2021), satellite abundance (van den Bosch et al. 2005), and massobservable relations (Singh et al. 2020). On the other hand, the qualitative significance of covariances and projection e ff ects on observable properties is not expected to significantly hinge on cosmological parameters (Bocquet et al. 2019), and possible deviations from this expectation could be estimated using emulators (see, e.g., Bocquet et al. 2020; Ragagnin et al. 2021; Angulo et al. 2021; Ragagnin et al. 2023). We will study the impact of projection e ff ects using hydrodynamic simulations in order to gain insight into which fraction of the scatter and skewness of scaling relations originates from projection e ff ects (i.e., alignment with filaments and objects) or di ff erent accretion histories. In Sect. 2, we present how we set up our Euclid -like observable properties and the others coming from the other wavelengths. In Sect. 3, we study how projection e ff ects impact the scatter and skewness of log-residuals of scaling relations and discuss the impact of projection e ff ects on observable covariance. In Sect. 4, we focus on the mass-concentration relation and how it is a ff ected by projection e ff ects and deviations from the functional form of profiles and the radial ranges of the fits. In Sect. 5, we focus on the covariance between observable properties and study how di ff erent accretion histories and projection e ff ects impact them. Finally, we draw our conclusions in Sect. 6.",
"pages": [
2,
3
]
},
{
"title": "2. Numerical Setup",
"content": "We will conduct our study by analysing clusters obtained from the Magneticum 1 hydrodynamic cosmological simulations (Bi ffi et al. 2013; Saro et al. 2014; Steinborn et al. 2015; Dolag et al. 2016, 2015; Teklu et al. 2015; Steinborn et al. 2016; Bocquet et al. 2016; Ragagnin et al. 2019). They are based on the N -body code Gadget3 , which is built upon Gadget2 (Springel et al. 2005b; Springel 2005; Boylan-Kolchin et al. 2009) with an improved smoothed particle hydrodynamics (SPH) solver from Beck et al. (2016). Magneticum initial conditions are generated using a standard Λ CDM cosmology with Wilkinson Microwave Anisotropy Probe 7 (Komatsu et al. 2011) cosmological parameters. The large-scale structure evolution in Magneticum simulations includes a treatment of radiative cooling, heating from a uniform redshift-dependent ultraviolet (UV) background, star formation, and stellar feedback processes as in Springel et al. (2005a). The stellar feedback is then connected to a detailed chemical evolution and enrichment model as in Tornatore et al. (2007), which follows 11 chemical elements (H, He, C, N, O, Ne, Mg, Si, S, Ca, and Fe, with cooling tables from Wiersma et al. 2009) which are produced with the CLOUDY photo-ionisation code (Ferland et al. 1998). Fabjan et al. (2010) and Hirschmann et al. (2014) described prescriptions for black hole growth and feedback from AGNs. Haloes that host galaxy clusters and groups are identified using the friends-of-friends halo finder (Davis et al. 1985), and subhaloes together with their associated galaxies are identified with an improved version of SUBFIND (Springel et al. 2001), which takes into account the presence of baryons (Dolag et al. 2009). We define r ∆ c as the radius that encloses an average density of ∆ c ρ cr , where ρ cr is the critical density of the Universe at a given redshift, Throughout this paper, when we omit ∆ c from masses and radii we imply the usage of ∆ c = 200 (i.e., M = M 200c) . To disentangle the scatter of the mass-observable relation from projection e ff ects, we compute quantities within a sphere of radius r 200c and as integrated into a cylinder. Projected quantities will be denoted with the superscript 2D (for instance, the total mass inside the cylinder is denoted as M 2D ). We opted to employ a random projection plane for each cluster. Additionally, we set an integration depth of 20 comoving h -1 Mpc, corresponding to approximately 23 Mpc at z = 0 . 24 and 15 Mpc at z = 0 . 9 (with h = 0 . 704). This cylinder depth is smaller than Euclid 's galaxy cluster photoz equivalent uncertainty (Euclid Collaboration: Desprez et al. 2020) and, while we exclude some uncorrelated projection e ff ects, they are known not to play an important role (Sunayama et al. 2020; Wu et al. 2022). Thus, we ensure that we do not overestimate any projection e ff ect that could be mitigated using photoz . Consequently, all projection e ff ects examined in this paper hold relevance for interpreting forthcoming Euclid -based catalogues. This study is based on the results from Box2b / hr (Hirschmann et al. 2014) Magneticum simulation, which covers a length of 900 comoving Mpc, with dark matter particle masses m DM = 9 . 8 × 10 8 M ⊙ , gas initial particle masses of m gas = 2 × 10 8 M ⊙ , and a gravitational softening of both gas and dark matter of ϵ = 3 . 75 comoving kpc. Euclid is expected to detect clusters with masses M > 10 14 M ⊙ up to a redshift of approximately z ≈ 2 (Sartoris et al. 2016; Euclid Collaboration: Adam et al. 2019), where the bulk of the cluster population which will be used for mass-calibration will lie below redshift z ≈ 1 . Furthermore, the number of haloes contained in this Magneticum simulation drops significantly beyond the same redshift value. Therefore, we decided to extract haloes at two representative redshift slices: at an intermediate redshift of approximately z ≈ 0 . 24, yielding 4300 objects, and at a higher redshift of approximately z ≈ 0 . 9, yielding 1300 objects. These extractions were performed using the web portal 2 introduced in Ragagnin et al. (2017). We focus most of the analyses on the qualitative e ff ect of projection e ff ects at our intermediate redshift slice because of the larger statistics of clusters to help us determine projection e ff ects. We stress that this mass threshold is high enough so that all of our galaxy clusters have at least 10 4 particles and, therefore, can be considered well resolved in terms of density profile fitting (Navarro et al. 1997).",
"pages": [
3,
4
]
},
{
"title": "2.1. Observable properties",
"content": "We now discuss the properties that we compute for each cluster and report a summary in Table 1. We compute the total stellar masses M ⋆ and M 2D ⋆ as the sum of all stellar particles within the respective volumes. We compute the richness n with a cut of satellites of log 10 ( M ⋆/ M ⊙ ) > 10 . 65 , and the projected version n 2D that includes Euclid -like corrections for projection effects where similarly to Andreon et al. (2016). In particular, we compute the average projected richness between 3 . 5 and 8 Mpc radii from the cluster centre, divided the annulus in 8 slices of equal angles, excluded the two least dense and most dense slices and removed the average projected number density of the 4 remaining octants from the projected richness within r 200c; 3 We compute the X-ray luminosities LX , and L 2D X , in the [0 . 5 , 2] keV energy band computed using the APEC model (Smith et al. 2001), using SPH particle temperatures together with the XSPEC package 4 (Arnaud 1996), which considers the emission of a collisionally-ionized, chemically-enriched plasma implemented with metallicity values taken from the simulated particles 5 . We compute the temperature T and T 2D as weighted by the X-ray emissivity of gas particles. We compute the hot gas mass M g and M 3D g , computed as the sum of the mass of SPH particles with cold gas fraction greater than 0 . 1 and T > 3 × 10 4 K in order to filter out cold gas. Note that the projected gas mass is not to be confused with the one inferred from X-ray observations, as X-ray observational works typically perform a de-projection of the surface brightness ∝ n 2 e , which provides a gas-mass estimate that is closer to the spherical M g , with the addition of some possible alignment e ff ects coming from the central region of clusters. Moreover, observational works have the capability to mask possible bright substructures, thus minimising the presence of interlopers. Consequently, we can conceptualise the observed projected gas mass as an intermediate value between our M g and M 2D g . We estimate the integrated Comptony parameter produced by thermal Sunyaev-Zeldovich (SZ, Sunyaev & Zeldovich 1972). The Comptony parameter is defined as where T is the temperature, n e the number density of the electrons, k B the Boltzmann constant, σ T the Thomson cross-section, c the speed of light, and m e the electron rest mass. We compute the integrated Comptony parameter Y = R y d Ω , both within the volume of sphere of ( Y ) and a cylinder ( Y 2D ). We estimate the integral in Eq. (2) as where the sum runs over all SPH particles, mi is the i -th SPH particle mass, Ti its temperature and f e , i is its electron fraction, expressed as local electron number density normalised to the hydrogen number density, and m p is the proton mass. For each halo, we also perform fits of the NFW profile ρ NFW, defined as where the scaling density ρ 0 and the scale radius r s (that is the radius where the density log-slope equals -2) are free parameters. We perform this fit on the total matter density profile on 100 log-spaced radial bins between 75 ckpc (which corresponds to 60 kpc at z = 0 . 24 , and to 40 kpc at z = 0 . 9; as it is enough to exclude the deep central potential of baryons) and r 200c , and define the corresponding NFW masses and concentration parameters as M NFW and c NFW respectively, and the concentration as c NFW = r NFW / r s , where r NFW is obtained from M NFW via the Eq. (1). The projected version of the mass and concentrations are obtained by mimicking a lensing reconstruction procedure by fitting the corresponding reduced shear. We define the derived masses and concentrations as M 2D NFW and c 2D NFW , where c 2D NFW = R 2D NFW / r s , where R 2D NFW is obtained from M 2D NFW via the Eq. (1). Note that the \"2D\" here, as for the other quantities, means that the quantity is computed in projection, however, a correct fit of the mass from reduced shear NFW profile (namely, our M 2D NFW ) should provide an estimate of the same NFW halo mass M NFW that would be recovered from a 3D fit. The fit is computed within the cylinder described above, with a projected radial range of [300 , 3000] kpc at z = 0 . 2 . We performed the analyses at z = 0 . 9 by rescaling that range with H -2 / 3 ( z ) , where H ( z ) is the Hubble parameter, in order to retain the same fractional distances from the virial radius (at fixed mass), which resulted in a range of [234 , 2300] kpc . Note that in this work we are not interested in estimating the contribution of the uncorrelated large-scale structure in the reduced shear reconstruction, therefore we limit our density projection to a cylinder of the depth of 20 cMpc , (see, e.g., Euclid Collaboration: Giocoli et al. 2024; Becker & Kravtsov 2011). The signal from the source-averaged excess surface mass density ∆Σ gt , averaged over circular radii R and a population of sources distributed in redshift, can be written as Here Σ denotes the surface mass density. The symbol ⟨ ... ⟩ denotes an average over radial bins and redshift lens sources, where we used a redshift distribution as proposed in Euclid Collaboration: Ajani et al. (2023), and Euclid Collaboration: Giocoli et al. (2024). The quantity ⟨ ∆Σ t ⟩ is the excess of surface mass density, averaged over polar coordinates and defined as The symbol Σ cr in Eq. (5) is the critical surface mass density, that for a given redshift source equals to where G the universal gravity constant, D s the angular diameter distance to the source, D d the angular distance to the lens, and D ds the angular distance between the source and the lens. Similarly to Euclid Collaboration: Giocoli et al. (2024), we define the error associated with each radial bin of the profile in Eq. (5) as where σϵ = 0 . 3 (Hoekstra et al. 2012; Euclid Collaboration: Blanchard et al. 2020) is the dispersion of the total intrinsic ellipticity ϵ = (1 -q ) / (1 + q ) , where q is the axis ratio, R 1 and R 2 represent the inner and outer radius of a bin, and n g is the number density of galaxies. For our redshift source distribution (we assume the same as in Euclid Collaboration: Giocoli et al. 2024), we find that n g ≈ 28 arcmin -2 for lenses at redshift z = 0 . 24 and n g ≈ 14 arcmin -2 for lenses at redshift z = 0 . 9 .",
"pages": [
4,
5
]
},
{
"title": "2.2. Scaling relations",
"content": "In Fig. 1 we show the observable properties vs. true mass M of clusters, derived from Magneticum Box2b / hr simulation for properties that could be derived by Euclid -like catalogues, such as lensing concentration (first row from top), lensing mass (second row), projected richness (third row), and projected stellar mass (last row), as presented in Sect. 2.1. For each property, we fit a scaling relation performed using a linear regression in the log-log space. We utilise a log-log linear regression because a single power law proves to be e ff ective in modelling our scaling relations. In the right panel of Fig. 1, we show the log-residual distribution for both low-mass haloes ( M < 2 × 10 14 M ⊙ ), highmass haloes ( M > 2 × 10 14 M ⊙ ), and for the complete sample of the log-residuals σ ln , i , defined as the logarithmic ratio between the i -th cluster property and the corresponding scaling relation value at its mass. In the second column, we also report the logscatter σ ln defined here as the corresponding standard deviation of the log-residual, namely where E is the expectation operator that averages over our catalogue data. We note that the concentration has a scatter of 0 . 45 which is higher than theoretical expectations (see, e.g., Child et al. 2018). Throughout this paper, we will show that this is due to projection e ff ects; in fact, the 3D concentration has a scatter of ≈ 0 . 33 . Note that our scatter in temperature exceeds that reported in the theoretical work by Truong et al. (2018). We verified that, if we compute mass-weighted temperature, which is known to behave very well in a power law scaling relation, reveals a log scatter of 0 . 07 , in agreement with their work. The additional scatter that we see may be due to di ff erent X-ray temperature computa- tions (they use core-excised temperature while we take the contribution of the core into account). In Fig. 2 we show the mass-observable relations of quantities that could potentially be obtained from various multi-wavelength observations that could enrich studies based on Euclid -like data products: the integrated Comptony parameter, the gas mass M g , 500c, the X-ray luminosity L 2D X , 500c converted in the soft band [0 . 5 , 2] keV , and the temperature T 500c . Wedecided to plot the Xray luminosity, gas mass, and temperature within r 500c because this radius is typically used in various X-ray observations (see, e.g., Vikhlinin et al. 2006; Sun et al. 2009). The typical Euclid cluster cosmology analysis will therefore rely on mass-observable relations calibrated within r 200 c (e.g., richness, weak-lensing mass), and follow-up observations calibrated within r 500 c (e.g., X-ray and SZ mass-proxies). We will thus need to take into account the covariance between observable properties extracted at di ff erent radii. . Finally, we note that generally, X-ray observations are expected to align more closely with the 3D mass rather than the projected one (Ettori et al. 2013), although several details adopted to analyse the Xray observations (e.g., including masking of substructures, deprojection procedures, etc.) can significantly impact the final result. These choices critically depend on the quality of the observations themselves. Dedicated mocks will thus be required to properly take into account all these e ff ects and are, therefore, beyond the purpose of this work. For simplicity, in this work, we consider an X-ray-derived gas mass as close to M g , while an SZ-derived gas mass closer to M 2D g .",
"pages": [
5,
6
]
},
{
"title": "3. Projection effects",
"content": "The main objective of this work is to disentangle the amount of scatter and skewness in scaling relations that is purely due to projection e ff ects. Note that observational data are also a ff ected by measurement errors that we do not tackle in this work (as, for instance, the Poisson error of the limited number of galaxies used to infer the richness). In this Section, we discuss the scatter and skewness of mass-observable relation qualitatively and limit the discussion for the data at redshift z = 0 . 24 , as we have a larger sample of galaxy clusters, and the results are qualitatively similar to the ones at z = 0 . 9. We stress that we leave the quantitative discussion of the scatter and correlation coe ffi cients for both redshifts on Sect. 5. To assess the role of projection e ff ects, Fig. 3 reports the 3D and 2D scatter of our cluster properties. In the left panel of Fig. 3 we show the value of the scatter (see Eq. 9) of log-residuals σ ln for all our mass-observable relations. In the shaded region, we also report the values computed within r 500c for the X-ray luminosity, gas mass, and temperature values because this is the characteristic overdensity used in X-ray analyses. For each observable, in Fig. 3, we report (with di ff erent symbols) both the scatter of the complete sample as well as the one of two separate mass range of 10 14 < M < 2 × 10 14 M ⊙ and M > 2 × 10 14 M ⊙ respectively. We note that the lower-mass bin ( M < 2 × 10 14 M ⊙ ) is the one with the largest scatter because, for a given external object in the line-of-sight (LoS), the profile of a small cluster will be more perturbed with respect to a cluster. In the left panel of Fig. 3, we see that some quantities have a low scatter in the 3D space and do gain a large amount of scatter once they are seen in projection. To better quantify what is the actual impact of projection e ff ects, in the right panel of Fig. 3 we present the metric q σ 2 ln , 2D -σ 2 ln , 3D . This metric shows that the quantities that are most a ff ected by projection e ff ects are the weak lensing concentration, integrated Comptony parameter, gas mass, and NFW profile lensing mass. This is expected as all these observable properties (except for weak lensing concentration and X-ray luminosity) scale linearly with the respective observable mass. Further, we note that the scatter in the richness agrees to the theoretical predictions from Castignani & Benoist (2016). We observe that X-ray luminosity and temperature are the least a ff ected by projection e ff ects. This is attributed to the fact that X-ray luminosity is contingent upon the square of gas density, thereby being primarily influenced by the most bright regions within an image. Similarly, the temperature is predominantly influenced by the innermost regions of a cluster. Note that we lack the value of projection e ff ects for L X , 500c because the 2D scatter is slightly smaller than the 3D one. This happened because projection e ff ects impact under-luminous haloes more strongly than overly luminous haloes (at a fixed mass bin), with the consequence of the projected X-ray luminosity having a higher normalisation and a lower scatter (see Fig. B.1). Some mass-observable relations have a large skewness, to aid observational works in modelling these relations, we will estimate their skewness. Therefore, we also quantify deviations of residuals from a symmetrical distribution by means of the Fisher-Pearson coe ffi cient of skewness m 3 / m 3 / 2 2 , where mk is the sample k th central moment. 6 We report its dependency on projection e ff ects in Fig. 4, showing that the properties whose skewness is most impacted by projection e ff ects are the integrated Comptony parameter, the gas mass, and the lensing NFW mass. We also notice that some scaling relation residuals move from having a negative skewness (for the NFW concentration, for instance, due to un-relaxed and merging clusters) to a positive one once projection e ff ects are taken into account (namely, from an asymmetry towards negative residuals towards an asymmetry towards positive residuals). To assess the impact of projection e ff ects, we introduce a variable to quantify the amount of additional matter in the line of sight that can skew our observable properties. We define it as the ratio of the mass within the cylinder and the mass within a sphere M 2D / M , both of radius r 200c , where the length of the cylinder is described in Sect. 2. We present the distribution of M 2D / M in Fig. 5, where we can see that this quantity is strongly skewed and its median value is M 2D / M ≈ 1 . 26 (note that for a NFW profile with c = 4 , the corresponding analytical cylinder vs. spherical mass is 1 . 25). Although this quantity is not directly observable, we will use it to assess the contribution of LoS objects in the scatter of scaling relations. Note that besides objects in the LoS, di ff erent fitting procedures may impact the scatter of projection e ff ects, as we will see in Sect. 4. 2D In Fig. 6, we show a random selection of clusters, ordered by decreasing M 2D / M from left to right. Objects with high M 2D / M (the objects in the left-most panels) include clusters that are merging, elongated, or in the LoS. In the rest of this paper, we will refer to the objects having M 2D / M greater than the median of the distribution 1 . 26 as having LoS excess. To study the impact of projection e ff ects in scaling relations, in Fig. 7 we show the scaling relations of the following projected quantities: richness, integrated Comptony parameter, lensing mass, and concentrations, that are the ones that are most a ff ected by projection e ff ects. We colour-code these points by M 2D / M and focus on a narrow mass range of M ∈ [1 , 2] × 10 14 M ⊙ in order to visualise better how LoS excess impacts these scaling relations. On the left column, we can visually see that, except for concentration, they strongly correlate with M 2D / M , as the upper points of the scatter plot tend to have higher values of M 2D / M . We quantify this finding in the right column by comparing the residual distributions (of the power-law fit over the complete mass range presented in Sect. 2.2) of the complete sample with the distribution of objects with low LoS contamination only (we adopt the criteria of M 2D / M < 1 . 26 being the median of the M 2D / M distribution, as shown in Fig. 5), and report the respective scatter σ and average value µ of the residual distributions (for the complete sample we have that µ = 0). Except for the concentration, residuals of haloes with low M 2D / M (see dashed histogram) significantly shift towards negative values of µ and σ. For instance, when we consider only objects with a low LoS excess, the scatter of Y 2D decreases from 0 . 35 down to 0 . 11 , and the M 2D NFW scatter goes from 0 . 23 to 0 . 9. The lensing mass is, in fact, well known to be a ff ected by projection e ff ects (Meneghetti et al. 2014; Euclid Collaboration: Giocoli et al. 2024). In this paper, we refer to projection e ff ects as all e ff ects that take place when going from 3D to projection; they include both LoS e ff ects and model uncertainties. This definition becomes relevant when dealing with concentration, which is not impacted only by LoS objects but also by the NFW profile fitting procedure. We stress that in Fig. 3 we proved that our projected concentration is actually highly impacted by projection e ff ects, yet only weakly a ff ected by LoS e ff ects. We show in the next Section the reason for concentration being strongly a ff ected by projection e ff ects is that their reduced shear profile deviates strongly from the one produced by NFW profile (see, e.g., Ragagnin et al. 2021), which is used to reconstruct the reduced shear profile. To conclude this section, we now study how correlations between cluster observable properties can be a ff ected by projection e ff ects. To this end, we take the case of a possible hot- vs. coldbaryon correlation by studying the stellar mass vs. integrated Comptony parameter (as the latter should strongly correlate with the gas mass). In Fig. 8, we show the integrated Comptony parameter scaling relation, colour-coded by stellar mass for the 3D quantities (top panel) and projected quantities (bottom panel). Examining the correlations at a constant halo mass among the computed quantities within spheres (as depicted in the top panel of Fig. 8), we find no discernible weak anti-correlation between stellar mass and the integrated Comptony parameter (which is defined in Sect. 5). Conversely, when investigating the properties in the projected space, a more pronounced correlation becomes evident. This implies that projection e ff ects can strongly impact the correlation between observable properties. While this analysis is purely qualitative, we will quantify the impact of these projection e ff ects in Sect. 5, where we will compute the correlation coe ffi cients for both 3D quantities and 2D quantities.",
"pages": [
6,
7,
8,
9
]
},
{
"title": "4. Projection effects on lensing concentration",
"content": "As we found in the previous section, projection e ff ects significantly increase the scatter and skewness in the scaling of lensing concentration with mass. However, this scatter increase is not related to external objects along the LoS. Now, we assess if the high scatter of lensing concentration is due to deviations of the reduced shear profile from the one induced by an NFW profile. In this work, we will not delve into the origin of this deviation as it falls beyond the scope of this paper. Such deviation may arise due to halo elongations, suggesting that alternative profiles such as truncated NFW profiles may better suit galaxy clusters (Oguri & Hamana 2011). Alternatively, it could stem from the expectation that the NFW profile is intended to describe stacked haloes rather than individual objects. Our focus in this paper is to understand the impact of assuming an NFW profile for each of our haloes. We emphasize that these NFW deviations only a ff ect weak lensing signal reconstruction, as the NFW profile is highly e ff ective in recovering halo mass in 3D. To study deviations from the NFW profile of haloes we fit a generalised NFW profile (Nagai et al. 2007), hereafter gNFW, in spherical coordinates over the same radial range as our previous NFW profile (described in Sect. 2.1), where the density profile ρ gNFW ( r ) is defined as where γ and β are respectively the internal and external logslopes of the total matter density profiles. The case γ = 1 and β = 3 produces the NFW profile as in Eq. (4). Note that the Nagai et al. (2007) gNFW profile also depends on the parameter α that we fix to α = 1 in this work in order to explore internal and external log-slope variations only. We present the PDF for the gNFW profile parameters γ and β in Fig. 9, where the fit was performed in 3D with a flat priors for γ ∈ [0 , 3] and β ∈ [0 , 6]. The data points are colour-coded according to the variable M , revealing no discernible strong trend with respect to the fitted parameters. For 19% of the objects, the resulting best-fit parameters hit the boundaries of hard-cut priors. Upon visual inspection, these objects are characterised by a very steep matter density profile at large cluster-centric distances, possibly suggesting that a truncated NFW profile might be a better model choice. As our objective is to examine the effects of deviations from the generalised NFW profile, we omit these objects from the subsequent analysis in this Section. Given the substantial deviation of these objects from NFW profile, they could potentially o ff er additional insights for our analyses. However, incorporating them would necessitate the use of a profile more general than Eq. 10. Therefore, we excluded them in order to make our analysis clearer. We observe that the external logslope of Magneticum profiles appears to be slightly flatter than -3 . While we emphasize that this discrepancy does not a ff ect the accurate recovery of mass and concentration parameters in 3D NFW fits (such fits can still yield precise estimates of halo mass and concentrations). However, these deviations in the NFW profiles may a ff ect the reduced shear fit, particularly when observed over large radii (remember that in this work, we use 3 Mpc). Furthermore, we observe a degeneracy between the β and γ parameters, indicating that our profiles deviating from NFW profile tend to exhibit a flatter profile compared to NFW profile (as illustrated in Fig. A.1). However, investigating this discrepancy is beyond the scope of this paper, as the internal log slope of clusters is not currently captured by existing weak lensing studies. In Fig. 10, we plot the values of concentration and mass obtained from reduced shear fit, divided by the corresponding 3D quantities and colour-coded by the external 3D gNFW slope β for our intermediate redshift haloes. As we can see, haloes with large values of β have a projected concentration that is significantly higher than the 3D one (see upper panel). In Appendix A we report the example of a simulated halo with low LoS excess (see Fig. A.1) and an analytical one (see Fig. A.2), both with a flat external log-slope, and we show how the under-estimation of the concentration is caused by the fact that the NFW profile fit on the reduced shear is weighting too much the external part of the profile, that deviates from an NFW profile. In Fig. 11, we show the concentration residual distribution and report their scatter. We note that the projected concentration scatter is not a ff ected by external material along the LoS (dashed line and shaded histograms match). However, if one restricts our sample to objects having NFW-like profile log-slopes (we used criteria of 2 . 8 < β < 3 . 2 and 0 . 8 < γ < 1 . 2), then the scatter distribution changes drastically. The concentration residuals decrease from 0 . 43 to 0 . 38, and the residuals shift towards higher values, suggesting that these objects are more relaxed. Such e ff ect is well known, as studied for instance in Macciò et al. (2007). We also show how the external log-slope of the halo profile a ff ects the lensing reconstruction by plotting the ratio between the projected and 3D concentration (i.e., c 2D NFW / c NFW) value versus the 3D log-slope β in the narrow mass bin of M ∈ [1 , 2] × 10 14 M ⊙ in Fig. 12, where we find a positive correlation coe ffi cient of ≈ 0 . 28 , in agreement with a shift of residuals we showed in Fig. 11.",
"pages": [
9,
10,
11
]
},
{
"title": "5. Correlations between properties",
"content": "While in the last sections, we investigated the origin of the impact of projection e ff ects in the scatter and skewness of observable properties, we will now quantify how projection e ff ects impact the correlation between observable properties. To this end, we quantify the Pearson correlation coe ffi cients between their log-residuals (as defined in Sect. 2.2). We adopt the standard error associated with the Pearson coe ffi cient ρ as derived from two normal distributions, given by σρ = p 1 -ρ 2 / √ N -2 (see Eq. 12-93 in Pugh & Winslow 1966), where N represents the number of objects. This corresponds to a maximum error of 0 . 015 (for ρ = 0) for the sample size at z = 0 . 24 and a maximum error of 0 . 028 for the sample size at z = 0 . 90. It is worth noting that in the correlation coe ffi cient matrices generated in subsequent analyses, we will only colour values with correlation coe ffi cients | ρ | > 0 . 3, aiming to highlight strongly correlating properties. We define mild correlation as 0 . 2 < | ρ | < 0 . 3, as we choose to exercise caution. Correlation coe ffi cients with | ρ | < 0 . 1 are disregarded.",
"pages": [
11
]
},
{
"title": "5.1. Analysis at z = 0 . 24",
"content": "In this Section we focus on the haloes at intermediate redshift z = 0 . 24 . In Fig. 13 we show the correlation coe ffi cient matrix between log-residuals at fixed halo mass of our projected observable both from Euclid -like data (lensing concentration, lensing mass, richness, and stellar mass, respectively) and possible outcomes from multi-wavelength observations (integrated Comptony parameter, gas mass, X-ray luminosity, and temperature) for intermediate-redshift objects. In the lower triangle, we present scatter plots alongside the slope derived from the correlation coe ffi cient. This visualization allows for the identification of instances where the correlation coe ffi cient slope accurately captures the trend of the residuals. Typically, this alignment occurs for quantities that exhibit strong correlation coe ffi cients. For example, our data points show a robust correlation between certain hot baryon tracers ( M 2D g and Y 2D ), some cold baryon components ( n 2D and M 2D ⋆ ), and weak lensing mass M 2D NFW . The underlying reason for these strong correlations lies in projection e ff ects: the greater the amount of matter along the line of sight, the higher the observed values. We will further demonstrate this in the subsequent section by presenting the correlation coe ffi cient matrix for 3D quantities, where many of these correlations diminish. This can be anticipated by observing that L 2D x and c 2D NFW do not exhibit this positive trend of correlations. Notably, we observe that the correlation between richness and stellar mass ( ρ = 0 . 48) is not exceptionally high. Moreover, the stellar mass appears to be more influenced by projection e ff ects compared to richness (evident in their correlations with M 2D NFW , where they exhibit ρ = 0 . 69 and ρ = 0 . 42, respectively). We can speculate on two potential causes: firstly, unlike stellar mass, our richness computation incorporates some observationally-motivated background subtraction; alternatively, since stellar mass encompasses all stellar particles (while richness involves a luminosity-motivated galaxy stellar-mass cut), it is plausible that small subhaloes are influencing the projected stellar mass. We note that concentration anti-correlates with gas-mass (and integrated Comptony parameter). This is in agreement with recent analyses of simulations. In fact, richness at fixed mass anti-correlates with concentration (Bose et al. 2019); low concentration is an index of the system being perturbed (Ludlow et al. 2012); and un-relaxed systems tend to be gas-rich (Davies et al. 2020). We refer to Ragagnin et al. (2022) for a more comprehensive study on low luminous groups. Moreover, at fixed halo mass, the lensing mass correlates strongly with total projected stellar mass ( ρ = 0 . 69) and projected gas mass ( ρ = 0 . 59), which may be due to the fact that both correlate strongly with LoS contamination. The same holds for the correlation among richness, gas mass, and stellar mass. This is due to projection e ff ects, where LoS excess amplifies all these quantities, as discussed in Section 2.2. We note that the 2D lensing mass and projected X-ray luminosity have a slight positive ( ρ = 0 . 20) cor- ation, in agreement with the observational work of Sereno et al. (2020). In Fig. 14 we show the covariance matrix of non-projected quantities for intermediate redshift objects. We see that as opposed to Fig. 13, the 3D covariance matrix shows a mild yet negative covariance between gas mass and stellar mass ( ρ = -0 . 24), and a positive correlation between richness and gas mass ( ρ = 0 . 23) because most of their correlations in the projection are due to Line of sights excess, which significantly increases the values of the gas mass, the richness, and the stellar mass. In Fig. 15 we report the correlation matrix as in Fig. 13 where we present X-ray luminosity, gas mass, and temperature, as computed within r 500c , which shows an anti-correlation between the gas mass and the concentration residuals ( ρ = -0 . 14) that is significantly lower than the one found in Fig. 13 and Fig. 14 ( ρ equals to -0 . 26 and -0 . 34 respectively). One possibility is that this change in sign of the correlation is caused by the fact that mixing overdensities (concentration is within ∆ c = 200 and gasmass is within ∆ c = 500) does introduce an additional correlation with the sparsity (Balmès et al. 2014; Corasaniti et al. 2022) that itself correlates with the concentration (see Appendix B). For completeness, we report the correlation coe ffi cient matrix and the scatter of log-residuals of all quantities in Table B.1. There we also added the core-excised projected X-ray luminosity L 2D X , ce500c , as it is typically used in X-ray-based observational studies, where we can see that the scatter and most of the correlation coe ffi cients are smaller than L 2D ce500c , while the correlations with the concentration and gas mass increase. Note that we do not report the 3D NFW mass ( M NFW) because it has an extremely low intrinsic scatter σ ln ( M NFW) ≈ 0 . 01 and its correlation coefficients are not meaningful.",
"pages": [
11,
12
]
},
{
"title": "5.2. Analysis at z = 0 . 9",
"content": "In this Section, we focus on observational property covariance matrixes of our haloes z = 0 . 9. At this redshift, we computed projected quantities within a cylinder depth of 35 Mpc in order to retain the same relative ratio as the photoz uncertainty of the low-redshift analysis (it scales with 1 + z ). For what concerns the cylinder used to integrate ∆Σ gt, we rescaled so as to keep it constant in comoving units with the low-redshift analyses. We rescaled the 3D NFW profile minimum radius to 40 kpc while we kept the maximum radius at r 200c . For what concerns the radial range of the lensing fit, we rescaled it with H -2 / 3 ( z ), therefore performing it in the range of [234 , 2300] kpc . We stress that we do not model observational uncertainty. Therefore, the decrease in background source count with redshift does not impact our best fits. However, it still impacts the fact that we weigh external radial bins more than internal ones. We report the values of the scatter and the projection contribution at z = 0 . 9 in Fig. 16, while we report the log-residuals and the skewness for each property in Fig. 17. In particular, the quantities most a ff ected by projection effects are the lensing mass and concentration, whereas the temperature is the lowest. These results are qualitatively similar to the low redshift analyses, with the ComptonY parameter and gas mass being slightly less a ff ected by projection e ff ects. Note that since the virial radius is smaller at higher redshift values, our radial range of the reduced shear is closer to the NFW scale radius; therefore, the weak lensing reconstruction is more e ff ective in capturing the scale radius and more sensitive to deviations from an NFW profile. As a consequence, we found that the in- crease of scatter going from c NFW to c 2D NFW compared to the low redshift analyses. We report the correlation coe ffi cient matrix and the scatter log-residuals of the quantities at z = 0 . 9 in Table B.2. As for the case at z = 0 . 24 , note that we do not report the 3D NFW mass ( M NFW) because it has an extremely small intrinsic scatter of 0 . 01 , and thus its correlation coe ffi cients have no impact in our study.",
"pages": [
12
]
},
{
"title": "6. Conclusions",
"content": "In this work, we analysed a number of galaxy clusters from Magneticum hydrodynamic simulation Box2b / hr. We did so in a mass range, tailored for Euclid -like data products (see Sartoris et al. 2016; Euclid Collaboration: Adam et al. 2019), namely with a mass of M 200c > 10 14 M ⊙ . To this end, we computed properties that could come from Euclid catalogues of galaxy clusters, such as richness, stellar mass, and lensing masses and concentration, and possible properties coming from multi-wavelength studies such as X-ray luminosity, integrated Comptony parameter, gas mass, and temperature. All these properties were computed both within a sphere and within a cylinder (both with radius r 200c) to account for projection e ff ects. Our study considers the remarkable capabilities of Euclid photoz measurements in identifying interlopers. However, their importance decreases significantly at scales as small as a few tens Mpc. This depth is still long enough to contain multiple haloes along the LoS. Hence, we studied the projection e ff ects on a scale that is significantly smaller than the Euclid photoz uncertainty. We then studied how the scatter and skewness change when one measures quantities in 3D space or in projection. Below, we summarise our findings: The analysis presented here has been carried out using a single suite of hydrodynamic simulations. Regarding weak lensing masses and concentration, since in this work, we did not consider the profile noise due to the finite number of background galaxies, future studies are needed to improve our estimations. Some works show that both scatter and correlation coefficients vary between cosmological simulations with di ff erent cosmologies (Ragagnin et al. 2023), the presence of feedback schemes (Stanek et al. 2010), and di ff erent cosmological simulation suite in the market (see Fig. 7 in Anbajagane et al. 2020). So, while simulations can provide directions on how to model correlation coe ffi cients, it is possible that when using this kind of data, one needs to allow for variation due to the di ff erent baryon physics. Furthermore, when striving for even more precise results, it is important to acknowledge that mass-observable relations are not exact power laws. Therefore, employing more generic fitting techniques, such as a running median, could yield improvements. Additionally, there is room for enhancement in how we compute correlation coe ffi cients in future studies. One potential approach could involve simultaneously fitting both the massobservable relation scatter and the correlation coe ffi cients by maximizing multivariate likelihoods. We anticipate that future studies combining Euclid data with multi-wavelength observations may encounter challenges in shedding light on currently puzzling residual correlations, primarily dominated by projection e ff ects. Acknowledgements. We thanks the anonymous referee for the useful comments. The Magneticum Pathfinder simulations were partially performed at the LeibnizRechenzentrum with CPU time assigned to the Project 'pr86re'. AR and LM acknowledge support from the grant PRIN-MIUR 2017 WSCC32 and acknowledges the usage of the INAF-OATs IT framework (Ta ff oni et al. 2020; Bertocco et al. 2020), and the space filling curve improvement on Gadget3 (Ragagnin et al. 2016). Antonio Ragagnin thanks Veronica Bi ffi and Elena Rasia for the X-ray computation routines and tables. KD acknowledges support by the COMPLEX project from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program grant agreement ERC-2019-AdG 882679. LM acknowledges the financial contribution from the grant PRIN-MUR 2022 20227RNLY3 'The concordance cosmological model: stress-tests with galaxy clusters' supported by Next Generation EU. CG and LM acknowledge support from the grant ASI n.2018-23-HH.0. AR and CG acknowledge funding from INAF theory Grant 2022: Illuminating Dark Matter using Weak Lensing by Cluster Satellites, PI: Carlo Giocoli. SB acknowledges partial financial support from the INFN InDark grant. AMCLB was supported by a fellowship of PSL University hosted by the Paris Observatory. We used the package colossus (see Diemer 2018) for computing Σ and ∆Σ as expected from NFW profiles. AR and FC acknowledge co-funding by the European Union - NextGenerationEU within PRIN 2022 project n.20229YBSAN - Globular clusters in cosmological simulations and in lensed fields: from their birth to the present epoch. The Euclid Consortium acknowledges the European Space Agency and a number of agencies and institutes that have supported the development of Euclid , in particular the Academy of Finland, the Agenzia Spaziale Italiana, the Belgian Science Policy, the Canadian Euclid Consortium, the French Centre National d'Etudes Spatiales, the Deutsches Zentrum für Luft- und Raumfahrt, the Danish Space Research Institute, the Fundação para a Ciência e a Tecnologia, the Ministerio de Ciencia e Innovación, the National Aeronautics and Space Administration, the National Astronomical Observatory of Japan, the Netherlandse Onderzoekschool Voor Astronomie, the Norwegian Space Agency, the Romanian Space Agency, the State Secretariat for Education, Research and Innovation (SERI) at the Swiss Space O ffi ce (SSO), and the United Kingdom Space Agency. A complete and detailed list is available on the Euclid web site ( http://www.euclid-ec.org ).",
"pages": [
12,
13
]
},
{
"title": "Data Availability",
"content": "Raw simulation data were generated at C 2 PAP / LRZ cosmology simulation web portal https://c2papcosmosim.uc. lrz.de/ . Derived data supporting the findings of this study are available from the corresponding author AR on request.",
"pages": [
13
]
},
{
"title": "References",
"content": "Kravtsov, A. V. & Borgani, S. 2012, ARA&A, 50, 353 Ragagnin, A., Andreon, S., & Puddu, E. 2022, A&A, 666, A22 Ragagnin, A., Fumagalli, A., Castro, T., et al. 2023, A&A, 675, A77 Sereno, M., Umetsu, K., Ettori, S., et al. 2020, MNRAS, 492, 4528 Sugiyama, S., Miyatake, H., More, S., et al. 2023, Phys. Rev. D, 108, 123521 Sun, M., Voit, G. M., Donahue, M., et al. 2009, ApJ, 693, 1142 Teklu, A. F., Remus, R.-S., Dolag, K., et al. 2015, ApJ, 812, 29 Tornatore, L., Borgani, S., Dolag, K., & Matteucci, F. 2007, MNRAS, 382, 1050 Truong, N., Rasia, E., Mazzotta, P., et al. 2018, MNRAS, 474, 4089 van Uitert, E., Cacciato, M., Hoekstra, H., et al. 2016, MNRAS, 459, 3251 Article number, page 14 of 26 18 Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France 23 Zentrum für Astronomie, Universität Heidelberg, Philosophenweg 12, 69120 Heidelberg, Germany",
"pages": [
13,
14
]
},
{
"title": "Appendix A: Fit of gNFW",
"content": "In Fig. A.1, we present the density profile of a halo that deviates from NFW profile and has no LoS contamination. In particular, it has β ≈ 1 . 8 and γ = 1 . 5 . We show its NFW profile fit profile on the 3D density in the upper panel of Fig. A.1, where we can see that 3D NFW profile (performed on radial bins in a sphere) is capable of capturing the shape of the halo and to estimate its mass with high accuracy (within ≈ 5%). In the central panel, we show the reduced shear profile and best fits, where we can see that the fit performed on the reduced shear underestimated the concentration and is not able to capture the more internal part of the shear profile, as it is done by the profile that was fit in 3D. We first exclude this mismatch as being due to projection effects by showing that both the reduced shear from the particle data (orange line) matches the one recovered by performing an analytical projection of the 3D profile (blue solid line). In particular we project the density profile ρ ( r ) and derive the surface mass density Σ conv ., as follows What we find is that the shear obtained with the aid of an analytical projection Σ conv . matches very well the real one (i.e., orange and blue lines do match). This hints that for this cluster there are no strong LoS e ff ects. To understand why the fit on the shear is not able to capture the concentration of the original halo, we zoom our fit in the bottom panel of Fig. A.1, where it looks like the fit is very good in capturing the final part of the profile and not able to capture the internal. It is crucial to emphasise that we did not include observational uncertainties in these analyses. Therefore, the uncertainty outlined in Eq. (8) a ff ects the fit by assigning more weight to external radial bins compared to internal ones. It is worth noting that the proportionality factors in Eq. (8) will not a ff ect our best fit. To validate this point in Fig. A.2 we study the bias on fitting an NFW profile on a mock gNFW profile that has β = 1 . 8 and γ = 1 . 5 , a mass of 3 × 10 14 M ⊙ and a concentration c = 2 . 4 (as the halo presented in Fig. A.1). We see that the 3D NFW profile is capable of estimating both its mass and its gNFW concentration with high accuracy (see top panel match between blue and dashed black lines). On the other hand the fit of the shear (we report in the bottom panel of Fig. A.2) has the same problems as the one on the cluster in Fig. A.1: it recovers a low concentration (with a value of 1 . 5). This may be because, at outer radii, the model fits the data. It is possible that the under-estimation of concentration at low radii is caused by the combination of two factors: the fit under-estimates the shear at lower radii (with the result of under-estimating the lensing concentration), or the fact that γ is di ff erent than 3 induces an NFW profile fit with a low concentration. We then performed the experiment of fitting the analytical profile with constant (yet unrealistic) error bars. While the fit was able to capture the shape of the profile, it recovered a concentration of 1 . 6 , implying that there is indeed a degeneracy between the shear of low-concentrated NFW profiles and steeper-NFW profiles.",
"pages": [
22
]
},
{
"title": "Appendix B: Correlations with different overdensities",
"content": "In this appendix, we discuss the di ff erences between scaling relation scatters and covariance values at di ff erent overdensities. First of all, we tackle the fact that when we compute X-ray luminosity within r 500c (instead of r 200c), we find that the scatter of the scaling relation of the projected quantity is larger than the 3D one. To investigate this feature, we will focus on the bolometric X-ray luminosity. We report the 3D and projected bolometric X- ray luminosity in Fig. B.1 (top panel), where it is visually clear that the projected X-ray luminosity is (as expected) always larger than the 3D one. One can also notice that the increase in X-ray luminosity depends on the fact that a halo is over-luminous or not: the increase of luminosity growing from the 3D to 2D is larger for under-luminous haloes than for over-luminous haloes. We prove this point in the bottom panel of Fig. B.1 where we show the ratio between the 2D and 3D luminosity as a function of their residual of the 3D scaling relation (the higher the value of the x axis, the more over-luminous is the object for its mass bin), where we can see a strong anti-correlation: overly luminous objects (for a given mass bin) are not going to be a ff ected much by the fact that their luminosity is computed in 3D or 2D. The possible cause is that an interloper in the LoS will not a ff ect much an overly luminous object. For completeness, in Fig. B.2 we show the correlation coefficients between the gas mass and stellar mass computed within both r 500c and r 200c and the concentration. Here we can see a change of sign between M ⋆, 500cMg , 500c correlations and M ⋆, 500cM g correlations and a change in the sign between c NFWM g correlations and c NFWMg , 500c correlations. Finally, in Fig. B.3 we report the scatter of observable properties at fixed mass for both Euclid -like quantities (lensing mass, richness, and projected stellar mass), and possible multiwavelength properties (integrated Comptony parameter, X-ray luminosity, and temperature), where we compute X-ray lumi- nosity and temperature within r 500c as they are typically derived within this overdensity. The upper panel of Fig. B.3 shows the residuals of the log-log linear regression where we see that in terms of 2D scatter, the properties with the lowest scatter are the stellar mass and the temperature. The bottom panel shows the data points used to perform the fit (in black) where we used a visually-inspected cut on the halo mass values in order to ensure that mass values are complete for a given observable value. Notes. Diagonal terms report the scatter of the log-residuals of each quantity, namely σ ln of Eq. (9), while the o ff -diagonal terms report the correlation coe ffi cient between the log-residuals. We do not report values of the correlation coe ffi cient below 0 . 20 because they are not significant. We do not report the values for the 3D NFW mass M 200c because it has a very low scatter of log-residuals ( ≈ 0 . 01) and its correlation coe ffi cients are not meaningful. Note that in this table we also added the core-excised X-ray luminosity. Notes. Rows and columns are as in Table B.1.",
"pages": [
22,
23,
24
]
}
]
2024arXiv241200975R
https://arxiv.org/pdf/2412.00975.pdf
<document>
<section_header_level_1><location><page_1><loc_25><loc_86><loc_75><loc_89></location>On the Orbital Effects of Stellar Collisions in Galactic Nuclei: Tidal Disruption Events and Ejected Stars</section_header_level_1>
<figure>
<location><page_1><loc_33><loc_84><loc_66><loc_85></location>
</figure>
<text><location><page_1><loc_11><loc_81><loc_89><loc_83></location>1 Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA), Northwestern University, 1800 Sherman Ave, Evanston, IL 60201, USA</text>
<text><location><page_1><loc_23><loc_79><loc_76><loc_80></location>2 The Observatories of the Carnegie Institution for Science, Pasadena, CA 91101, USA</text>
<section_header_level_1><location><page_1><loc_45><loc_76><loc_55><loc_77></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_51><loc_86><loc_75></location>Dense stellar clusters surround the supermassive black holes (SMBH) in galactic nuclei. Interactions within the cluster can alter the stellar orbits, occasionally driving a star into the SMBH's tidal radius where it becomes ruptured. This proof-of-concept study examines the orbital effects of stellar collisions using a semianalytic model. Both low and high speed collisions occur in the SMBH's sphere of influence. Our model treats stars in low speed collisions as sticky spheres. For high-speed collisions, we develop a simple prescription based on the limiting case of a hyperbolic encounter. We test a range of collision treatments and cluster conditions. We find that collisions can place stars on nearly radial orbits. Depositing stars within the tidal radius, collisions may drive the disruption of stars with unusual masses and structures: depending on the nature of the collision, the star could be the product of a recent merger, or it could have lost its outer layers in a high speed impact, appearing as a stripped star. We also find that high speed collisions near the periapsis of an eccentric orbit can unbind stars from the SMBH. However, dissipation during these high-speed collisions can substantially reduce the number of unbound stars achieved in our simulations. We conclude that TDEs and ejected stars, even in the hypervelocity regime, are plausible outcomes of stellar collisions, though their frequency in a three-dimensional nuclear star cluster are uncertain. Future work will address the rates and properties of these events.</text>
<text><location><page_1><loc_14><loc_46><loc_67><loc_47></location>Keywords: Stellar dynamics; Galactic center; Star clusters; Stellar mergers</text>
<section_header_level_1><location><page_1><loc_20><loc_43><loc_36><loc_44></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_17><loc_48><loc_42></location>Asupermassive black hole resides at the center of most galaxies, where it is surrounded by a dense stellar cluster (e.g. Ferrarese & Ford 2005; Kormendy & Ho 2013; Schodel et al. 2003; Ghez et al. 2005, 2008; Gillessen et al. 2009, 2017; Neumayer et al. 2020). A tidal disruption event (TDE) occurs when a star from the cluster passes within a critical distance from the SMBH and becomes ruptured by tidal forces (e.g., Hills 1975; Rees 1988; Alexander 1999; Magorrian & Tremaine 1999; Wang & Merritt 2004; MacLeod et al. 2012). The SMBH then accretes the stellar material, producing an electromagnetic signature (e.g., Guillochon & Ramirez-Ruiz 2013). Spectra of these events encode valuable information about the mass, structure, and composition of the ruptured star (e.g., Kochanek 2016a,b; Yang et al. 2017; Mockler et al. 2022; Miller et al. 2023). Observations of</text>
<text><location><page_1><loc_8><loc_13><loc_32><loc_14></location>Corresponding author: Sanaea C. Rose</text>
<text><location><page_1><loc_8><loc_12><loc_26><loc_13></location>sanaea.rose@northwestern.edu</text>
<text><location><page_1><loc_52><loc_40><loc_92><loc_44></location>TDEs represent a powerful way to probe the stellar populations in galactic nuclei and the processes that shape them.</text>
<text><location><page_1><loc_52><loc_10><loc_92><loc_39></location>Amongst these physical processes are direct collisions. Collisions can occur within the sphere of influence of the SMBH due to the high densities and velocity dispersion (e.g., Freitag & Benz 2002; Dale et al. 2009; Dale & Davies 2006; Rubin & Loeb 2011; Balberg et al. 2013; Balberg 2024; Mastrobuono-Battisti et al. 2014; Rose & MacLeod 2024). These events may produce electromagnetic signatures, both from the collisions themselves and from interactions between liberated material and the SMBH (e.g., Rosswog et al. 2009; Lee et al. 2010; Balberg et al. 2013; Dessart et al. 2024; Ryu et al. 2024b,a; Brutman et al. 2024), though it is difficult to ignite a main-sequence star with compression (Guillochon et al. 2009). They can also shape the stellar population. Both low and high speed collisions can alter the mass of a star - the latter can destroy stars completely and the former can give rise to blue stragglers in dense stellar systems (e.g., Lai et al. 1993; Rauch 1999; Sills et al. 1997, 2001; Lombardi et al. 2002; MacLeod et al. 2013; Leigh et al.</text>
<text><location><page_2><loc_8><loc_86><loc_48><loc_91></location>2016; Rose et al. 2023). Recently, Gibson et al. (2024) have shown that high speed collisions can also produce stripped stars similar to what might be seen through binary evolution.</text>
<text><location><page_2><loc_8><loc_45><loc_48><loc_85></location>Collisions that produce stellar mergers or stripped stars may explain recent unexpected TDE observations. For example, detections of high nitrogen-tocarbon (N/C) ratios in TDEs point to the disruption of more stars that burn on the CNO cycle (as first proposed by Kochanek 2016a,b), and that are ≳ 1 -2 M ⊙ (Kochanek 2016a; Yang et al. 2017; Gallegos-Garcia et al. 2018; Mockler et al. 2023) than is predicted by the host galaxies' stellar populations (Mockler et al. 2023). One particular TDE has such an extremely high N/C abundance ratio that it is difficult to explain with single stellar evolution alone (Miller et al. 2023), but could be the result of the disruption of a stripped star that has lost its nitrogen-poor envelope (Mockler et al. 2024). Additionally, a recently discovered population of extremely bright nuclear transients has also been suggested to originate from the disruption of high mass stars (e.g. ≳ 10 M ⊙ , Subrayan et al. 2023; Hinkle et al. 2024). Because stars that end in disruptions are expected to be drawn approximately at random from the stellar mass function (with small adjustments for stellar type, MacLeod et al. 2012), this preference for higher mass stars may imply that the mass function in galactic nuclei is more top-heavy (and/or bottom-light) than in the rest of the galaxy (see e.g., Lu et al. 2013; Hosek et al. 2019).</text>
<text><location><page_2><loc_8><loc_15><loc_48><loc_44></location>The potential link between TDEs and collisions is intriguing: collisions can simultaneously affect both the properties of the stars and their orbits about the SMBH. MacLeod et al. (2012) first considered collisions, particularly destructive ones, in the context of the TDE rate. Changes to a star's trajectory from collisions have also been considered in general dynamical models of dense stellar systems (e.g., Sanders 1970; Rauch 1999; Freitag & Benz 2002; Kremer et al. 2020; Gonz'alez et al. 2021; Rodriguez et al. 2022). In this proof-of-concept study, we study the orbital effects of stellar collisions in galactic nuclei. We assess whether the collisions that produce unusual stars could plausibly deposit those same stars onto TDE-producing orbits. Our models leverage simple, intuitive treatments for collision outcomes and fitting formulae from previous studies (e.g. Lai et al. 1993; Rauch 1999). We test a range of initial conditions and nuclear star cluster properties. Our paper is organized as follows:</text>
<text><location><page_2><loc_8><loc_10><loc_48><loc_14></location>In Section 2, we discuss our general approach to modeling the Milky Way's nuclear star cluster. Section 3 describes the treatment of various physics in our code, with</text>
<text><location><page_2><loc_52><loc_73><loc_92><loc_91></location>Section 3.2 in particular outlining our methodology for updating the stellar orbits post collision. These orbital changes represent the most major addition to the code as compared to previous iterations (Rose et al. 2022, 2023). Section 4 presents simulated results for direct collisions. Sections 4.1, 4.2, and 4.3 discuss the implications for TDEs, orbital properties of collision-affected stars, and unbound and hypervelocity stars. We then incorporate relaxation into our simulations in addition to stellar collisions and present results in Section 5. Lastly, we summarize the scope and findings of our study in Section 6.</text>
<section_header_level_1><location><page_2><loc_57><loc_70><loc_87><loc_71></location>2. MODEL NUCLEAR STAR CLUSTER</section_header_level_1>
<text><location><page_2><loc_52><loc_52><loc_92><loc_69></location>We leverage semi-analytic models to study the effects of collisions on the nuclear star cluster. Our fiducial model uses the conditions and properties of the Milky Way's GN, whose proximity makes it the best studied galactic center. We follow a sample of stars embedded in a fixed, unevolving cluster. For simplicity, both the evolving sample and the surrounding cluster are composed of 1 M ⊙ stars. The cluster can be understood in two key properties, density and velocity dispersion, which govern the dynamical processes unfolding within it.</text>
<text><location><page_2><loc_52><loc_47><loc_92><loc_52></location>The stellar density sets the the frequency with which stars interact. We describe the stellar density as a function of distance from the SMBH using a power law:</text>
<formula><location><page_2><loc_64><loc_43><loc_92><loc_46></location>ρ ( r · ) = ρ 0 ( r · r 0 ) -α , (1)</formula>
<text><location><page_2><loc_52><loc_18><loc_92><loc_41></location>where α is the slope and r · , distance from the SMBH. Based on observations of the cluster within the sphere of influence, this equation is normalized using ρ 0 = 1 . 35 × 10 6 M ⊙ / pc 3 at r 0 = 0 . 25 pc (Genzel et al. 2010). Our fiducial model uses a slope of 1 . 75, the expectation for a single-mass population (Bahcall & Wolf 1976), consistent with the fact that our simple model cluster has only solar mass stars. However, in order to capture the range of theoretical predictions and observational constraints on the stellar cusp (e.g., Bahcall & Wolf 1976; Gallego-Cano et al. 2018a; Linial & Sari 2022), we also test a few simulations with α = 1 . 25, shown in Appendix A. We assume that the slope of the stellar cusp is roughly constant, or varying slowly, over the timescales of interest in our simulations.</text>
<text><location><page_2><loc_52><loc_13><loc_92><loc_17></location>The velocity dispersion within the cluster also influences the frequency and nature of stellar interactions. It decreases with distance from the SMBH:</text>
<formula><location><page_2><loc_64><loc_8><loc_92><loc_12></location>σ ( r · ) = √ GM · r · (1 + α ) , (2)</formula>
<text><location><page_3><loc_8><loc_82><loc_48><loc_91></location>where α is the slope of the density profile and M · is the mass of the SMBH (Alexander 1999; Alexander & Pfuhl 2014). We take M · to be 4 × 10 6 M ⊙ , like the Milky Way's SMBH (e.g., Ghez et al. 2003). For a uniform mass cluster of 1 M ⊙ stars, the number density n is simply ρ ( r · ) 1 M ⊙ .</text>
<section_header_level_1><location><page_3><loc_17><loc_80><loc_39><loc_81></location>3. SEMIANALYTIC MODEL</section_header_level_1>
<text><location><page_3><loc_8><loc_67><loc_48><loc_79></location>We follow a sample of 1 M ⊙ tracer stars embedded in our model cluster. We draw their orbital eccentricities from a thermal distribution. We select their semimajor axes so that they lie on a cusp with slope α , matching the background cluster. These stars are allowed to evolve under the influence of two main dynamical processes, direct collisions and two-body relaxation, using a model first developed by Rose et al. (2022, 2023).</text>
<section_header_level_1><location><page_3><loc_23><loc_64><loc_33><loc_65></location>3.1. Collisions</section_header_level_1>
<text><location><page_3><loc_8><loc_53><loc_48><loc_63></location>Direct collisions occur over a characteristic timescale t -1 coll = nσA , where A is the cross-section of interaction, n is the number density, and σ is the velocity dispersion. For an impact to occur, A is the physical crosssection enhanced by gravitational focusing. The collision timescale also depends weakly on the star's orbital eccentricity and can be written as:</text>
<formula><location><page_3><loc_12><loc_46><loc_48><loc_51></location>t -1 coll = πn ( a · ) σ ( a · ) × ( f 1 ( e · ) r 2 c + f 2 ( e · ) r c 2 G ( M ⊙ + M ) σ ( a · ) 2 ) . (3)</formula>
<text><location><page_3><loc_8><loc_25><loc_48><loc_45></location>where f 1 ( e · ) and f 2 ( e · ) are equations 20 and 21 from Rose et al. (2020), G is the gravitational constant, and a · is the star's semimajor axis. r c is the sum of the radii of the colliding stars, or 2 R ⊙ for a uniform population of solar mass stars. We plot this timescale in red in the upper panel of Figure 1. We consider a range of slopes for the stellar density profile, spanning α = 1 . 25 (dashed line) to α = 1 . 75 (solid line). The horizontal grey line shows the total simulation time, included to guide the eye. Where the collision timescale is less than the simulation time, within 0 . 1 pc of the SMBH, collisions become important to understanding the evolution of the cluster (e.g., Rose & MacLeod 2024).</text>
<text><location><page_3><loc_8><loc_9><loc_48><loc_24></location>We treat stellar collisions using a statistical approach. We begin by computing the probability that a star in our sample will experience a collision. Over a timestep ∆ t , this probability equals ∆ t/t coll . ∆ t is taken to be 10 6 years so that the probability ∆ t/t coll is always less than one; t coll ≳ 10 7 years for the parameter space we consider. The code then draws a random number between 0 and 1, which, if less than or equal to the collision probability ∆ t/t coll , means a collision has occurred. We repeat this prescription until the desired simulation</text>
<text><location><page_3><loc_52><loc_89><loc_92><loc_91></location>runtime or the star's main-sequence lifetime has been reached, whichever is shorter.</text>
<text><location><page_3><loc_52><loc_54><loc_92><loc_88></location>If a collision has occurred over a given timestep, we update the mass and age of the star using a prescription detailed in Rose et al. (2023). A full discussion of our approach can be found in previous papers (Rose et al. 2023; Rose & MacLeod 2024). In brief, we combine fitting formulae from hydrodynamics simulations of stellar collisions (Rauch 1999; Lai et al. 1993) with heuristic arguments to (1) determine if a given collision will result in a merger and (2) determine the amount of mass lost from either the merger product or the unbound individual stars. Since speeds in galactic nuclei often exceed hundreds of kilometers per second, collisions can eject anywhere from a percent to all of the star's mass, effectively blowing it up (Spitzer & Saslaw 1966; Lai et al. 1993; Balberg et al. 2013; Balberg 2024; Brutman et al. 2024). Mergers with minimal mass loss are most likely to occur outside of about 0 . 01 pc, where the relative speeds tend to be less than the escape speed from the stars. Within 0 . 01 pc, however, velocities exceed the escape speed from the stars, and collisions can result in peculiar, low-mass 'stripped stars' (Rose et al. 2023; Rose & MacLeod 2024; Gibson et al. 2024).</text>
<text><location><page_3><loc_52><loc_35><loc_92><loc_54></location>In practice, the outcome of a collision is more complex to determine, depending on properties such as the impact parameter and stellar structure (e.g., Freitag & Benz 2005; Gibson et al. 2024, Rose et al. in prep.). For the purposes of this study, which focuses on the general implications of collisions for stellar orbits in galactic nuclei, the specific mass loss and merger prescriptions should not qualitatively change our results. Unless otherwise specified, all simulations shown herein use fitting formulae from Rauch (1999) for the mass loss and escape speed arguments to determine whether or not a collision results in a merger (Rose et al. 2023).</text>
<text><location><page_3><loc_52><loc_24><loc_92><loc_35></location>In previous iterations of this code, the collision speed was taken to be simply the velocity dispersion at the star's distance from the supermassive black hole, given by Eq. (2) (Rose et al. 2022, 2023). However, this approach is insufficient for studying the effects of collisions on stellar orbits. In this study, we draw the orbit of the second colliding star, using a procedure detailed below.</text>
<section_header_level_1><location><page_3><loc_64><loc_21><loc_80><loc_22></location>3.2. Orbital Dynamics</section_header_level_1>
<text><location><page_3><loc_52><loc_9><loc_92><loc_20></location>If a star in our sample with semimajor axis a · and eccentricity e · collides over timestep ∆ t , we find a plausible orbit for the second colliding star. As we operate under the assumption that the cluster is composed of a uniform stellar population, we always take the second star to have m collider = 1 M ⊙ . We then draw semimajor axes and eccentricities from the cluster's property distri-</text>
<figure>
<location><page_4><loc_8><loc_72><loc_47><loc_92></location>
<caption>Figure 1. Upper Panel: We plot the relevant timescales as a function of distance from the SMBH for a range of stellar density profiles, α = 1 . 25 (solid line) to α = 1 . 75 (dashed line). The collision and relaxation timescales are in red and green, respectively, while the grey line marks the total simulation time of 10 Gyr. The vertical red line emphasizes the radius at which the collision timescale equals the simulation time; within this radius, collisions are common and shape the evolution of the stars and cluster. We also mark the radius at which the velocity dispersion is approximately the escape speed from the surface of a Sun-like star using the green vertical line. External to this radius, we expect collisions to result in mergers. Lower Panel: We consider a hyperbolic encounter between two stars in the cluster. This plot shows the impact parameter in solar radii for a 90 · deflection as a function of the relative speed between the stars. Note that the x-axis is inverted to reflect the dependence of velocity dispersion on distance from the SMBH; it parallels the x-axis of the plot above, with the largest velocities occurring near the SMBH and decreasing further out. We have marked the speeds that correspond to the inner and outer allowed initial semimajor axes of the stars in our sample, 0 . 001 and 1 pc, respectively. Additionally, we mark where b 90 becomes equal to the star's radius. This point coincides well with the speed at which we would expect the collision outcome to transition from mergers to the sorts of high-speed encounters that might strip or destroy stars, but leave them unbound from each other.</caption>
</figure>
<figure>
<location><page_4><loc_9><loc_51><loc_47><loc_71></location>
</figure>
<text><location><page_4><loc_52><loc_78><loc_92><loc_91></location>as described in Section 2, until we find an orbit that intersects with that of our tracer star. Our code assumes that the cluster is spherically symmetric and only tracks the semimajor axis and eccentricity of each tracer star. In this set up, our tracer star always has argument of periapsis equal to zero, however the second colliding star's periapsis need not be aligned with it. The angle between them is drawn from a uniform distribution. The orbits are always assumed to be coplanar.</text>
<text><location><page_4><loc_52><loc_49><loc_92><loc_77></location>Observations suggest that a subset of the stars in the Milky Way's galactic center reside in a disk, while others have an isotropic distribution (e.g., Levin & Beloborodov 2003; Ghez et al. 2003, 2005; Gillessen et al. 2009; Paumard et al. 2006; Lu et al. 2009; Bartko et al. 2009; Yelda et al. 2014). In our co-planar physical picture, we have two options: either both of the colliding stars can orbit in the same direction about the supermassive black hole, or they can be equally likely to be prograde or retrograde. We test two extremes to capture the effects of different orbital orientations. Our first case, 'disk-like', has the two colliding stars orbiting the SMBH in the same direction. In the second case, 'isotropic-like', we assume that fifty percent of the time the two stars orbit the SMBH in the same direction and fifty percent of the time in opposite directions. The latter orientation means that the orbital angular momentum vectors are anti-parallel to one another.</text>
<text><location><page_4><loc_52><loc_37><loc_92><loc_49></location>Once we have found an intersecting orbit for our colliding tracer star, we determine the intersection point and compute the velocity vectors of the two stars at that point. The relative speed tells us whether or not a merger occurs and the degree of fractional mass loss from the system (see above section). From here, determining the final orbit(s) will depend on the type of outcome and therefore the relative speed.</text>
<section_header_level_1><location><page_4><loc_61><loc_34><loc_84><loc_35></location>3.2.1. Final Orbit in Merger Case</section_header_level_1>
<text><location><page_4><loc_52><loc_9><loc_92><loc_32></location>A stellar merger is the natural outcome of a low-speed collision. We calculate the final mass, age, and now trajectory of the product assuming the stars act as 'sticky spheres' (e.g., Rodriguez et al. 2022; Rose et al. 2023). The two stars approach each other with velocity vectors v · and v collider as determined by the intersection point of their orbit. Momentum conservation demands that the final velocity of the merger product, v final , equals ( M v · + M ⊙ v collider ) / ( M + M ⊙ ). Based on hydrodynamics studies of stellar mergers, the mass loss in these collisions should be low (e.g.., Lai et al. 1993; Rauch 1999, Rose et al. in prep.). With the final velocity, mass, and location of the collision - the intersection point of the original orbits - we can calculate the new orbit of the merger product about the SMBH.</text>
<section_header_level_1><location><page_5><loc_10><loc_90><loc_46><loc_91></location>3.2.2. Final Orbits Following a High-Speed Collision</section_header_level_1>
<text><location><page_5><loc_8><loc_74><loc_48><loc_89></location>Orbital changes from high-speed collisions are much more difficult to determine in the absence of hydrodynamics simulations. However, we present a framework for treating these interactions. In order to understand these collisions, we start at the limit where the two stars barely graze. This interaction should unfold as a hyperbolic encounter. In the center of mass frame of the two stars, their speeds remain constant, but their velocity vectors are deflected by angle θ hyp . θ hyp can be found analytically:</text>
<formula><location><page_5><loc_19><loc_70><loc_48><loc_71></location>θ hyp = 2arctan( b 90 /b ) , (4)</formula>
<text><location><page_5><loc_8><loc_61><loc_48><loc_67></location>where b is the impact parameter and b 90 is defined as the impact parameter needed for a 90 · deflection. b 90 equals 2 G/v 2 rel , where v rel is the relative speed between the two stars (e.g., Binney & Tremaine 2008).</text>
<text><location><page_5><loc_8><loc_34><loc_48><loc_61></location>We plot b 90 as a function of the relative speed in the bottom panel of Figure 1. In the nuclear star cluster, the velocity dispersion can be understood as the characteristic relative speed between stars at a given distance from the SMBH (see Eq. 2). We have therefore inverted the x-axis of the bottom plot to parallel that of the upper plot, distance from the SMBH, and marked the velocity dispersion at key distances using vertical dashed lines. In both the upper and lower plots, we also indicate the regions in which we expect mergers versus high-speed collisions, which leave the stars unbound from each other. Interestingly, for most of the parameter space where these high-speed, non-sticky sphere collisions occur, b 90 is less than the star's radius. Another way of interpreting this statement is that at high speeds, physical collisions are required for a strong-angle deflection.</text>
<text><location><page_5><loc_8><loc_9><loc_48><loc_34></location>If the stars were point particles, the impact parameter could be arbitrarily small and the interaction would still unfold as a hyperbolic encounter. Two Sun-like stars begin to touch when b = 2 R ⊙ . With b < 2 R ⊙ , the stars physically impede each other as they interact, leading to a smaller deflection angle than the one given in Eq. (4). Furthermore, if the stars approach each other perfectly head-on with b = 0, the center of mass velocity is 0 and there is no angular momentum. In this case, there would be no deflection. Heuristically, then, we expect the deflection angle to be given by Eq. (4) for grazing encounters, and some fraction of this angle for encounters with b < 2 R . That fraction should decrease with impact parameter until they are both zero. The precise dependence is impossible to determine in the absence of hydro simulations, but we define a collision deflection</text>
<text><location><page_5><loc_52><loc_90><loc_83><loc_91></location>angle that meets the two limiting criterion:</text>
<formula><location><page_5><loc_63><loc_86><loc_92><loc_89></location>θ coll = 2 b r c arctan( b 90 /b ) , (5)</formula>
<text><location><page_5><loc_52><loc_84><loc_92><loc_85></location>where r c is the sum of the radii of the two colliding stars.</text>
<text><location><page_5><loc_52><loc_71><loc_92><loc_83></location>It is reasonable to expect that the collision affects the stars' speeds as well. Even more so than the deflection angle, changes in speed can only be understood through hydrodynamic simulations. In our models, we simply test three cases: one in which the speed is not affected at all, one in which the speed in the center of mass frame is always reduced by 10%, and one in which it is always reduced by a factor of 2.</text>
<text><location><page_5><loc_52><loc_48><loc_92><loc_71></location>We treat these collisions as follows: at the intersection point of the orbits of the two colliding stars, we calculate their center of mass velocity. We transform to the center of mass frame. Only then do we rotate and scale their velocity vectors. While we only track semimajor axes and eccentricities, we do allow the deflections to be three dimensional, with components out of the plane. The final velocity can be any vector, chosen randomly, along a cone defined by angle θ coll with respect to the original velocity. Isotropizing the deflection is important because otherwise the colliding stars would always be scattered exactly towards or away from the SMBH, leading to an gross overestimate of TDE rates. We then transform back to the frame of the SMBH and calculate the new orbits given the velocity and position vectors.</text>
<text><location><page_5><loc_52><loc_40><loc_92><loc_47></location>We note that our calculation assumes that the center of mass is not moving in the frame of the supermassive black hole. In actuality, the center of mass of the two stars would orbit the supermassive black hole, but the effects will be negligable due to the high collision speeds.</text>
<section_header_level_1><location><page_5><loc_63><loc_37><loc_82><loc_38></location>3.3. Two-Body Relaxation</section_header_level_1>
<text><location><page_5><loc_52><loc_27><loc_92><loc_36></location>In addition to collisions, stars in the cluster experience the weak gravitational effects of nearby neighbors. The effects of these interactions can accumulate, eventually changing the star's orbital energy and angular momentum by order of itself. The original orbit is 'erased' over a characteristic timescale:</text>
<formula><location><page_5><loc_62><loc_23><loc_92><loc_26></location>t rlx = 0 . 34 σ 3 G 2 ρ ⟨ M ∗ ⟩ ln Λ rlx , (6)</formula>
<text><location><page_5><loc_52><loc_14><loc_92><loc_21></location>where ⟨ M ∗ ⟩ is the average star's mass, here taken to be 1 M ⊙ , and ln Λ rlx is the coulomb logarithm (e.g., Binney & Tremaine 2008; Merritt 2013). Figure 1 shows the relaxation timescale as a function of distance from the SMBH in blue for a range of stellar density profiles.</text>
<text><location><page_5><loc_52><loc_9><loc_92><loc_13></location>We account for relaxation by allowing the orbital eccentricity and semimajor axis of each of our tracer stars to slowly evolve. Once per orbit, we apply a small in-</text>
<text><location><page_6><loc_8><loc_75><loc_48><loc_91></location>stantaneous change in velocity to each star (e.g., Bradnick et al. 2017; Lu & Naoz 2019; Rose et al. 2022, 2023; Naoz et al. 2022, see the latter for the full set of equations). The kick is calibrated so that ∆ v/v ∼ √ ∆ t/t rlx , and if ∆ t = t rlx , ∆ v is of order of the velocity. This prescription simulates the diffusion of the orbital orbital parameters over time from interactions with other stars in the cluster. Previously, it has been used in studies of TDEs and extreme mass ratio inspirals of stellar mass black holes into the SMBH (Naoz et al. 2022; Melchor et al. 2024).</text>
<section_header_level_1><location><page_6><loc_16><loc_71><loc_40><loc_72></location>3.4. Orbital Stopping Conditions</section_header_level_1>
<text><location><page_6><loc_8><loc_49><loc_48><loc_70></location>As noted in Section 3.1, we terminate the simulation when the desired runtime of 10 billion years is reached or when the time elapsed has exceeded the star's mainsequence lifetime, whichever comes first. However, orbital changes from collisions or relaxation can also send stars into the tidal radius, where they will be destroyed. We trigger a stopping condition if the star's periapsis a · (1 -e · ) becomes less than twice the tidal radius from the SMBH, R star ( M · /M star ) 1 / 3 (e.g., Guillochon & Ramirez-Ruiz 2013; Mockler et al. 2023). Tidal disruption events can be characterized by impact parameter β = R tidal / ( a · (1 -e · )). Our stopping condition corresponds to β = 0 . 5, allowing us to capture partial as well as full disruptions.</text>
<text><location><page_6><loc_8><loc_30><loc_48><loc_48></location>Because both direct collisions and relaxation processes can place a star onto a nearly radial orbit, we log whether the critical orbit was reached through a direct collision or our relaxation prescription. A star that becomes a TDE due to relaxation processes can still have experienced one or more collision previously in its life. However, a star that collides and becomes a TDE due to the collision itself may still be inflated from the impact when it reaches its periapsis. In this case, the stopping condition noted above may be conservative; the R star is simply the radius expected using a mass-radius relation for a main-sequence star of mass m star .</text>
<text><location><page_6><loc_8><loc_9><loc_48><loc_29></location>High-speed collisions, unlike those that result in mergers, can also cause stars to be ejected from the nuclear star cluster. Consider two stars that collide on elliptical orbits that intersect near periapsis. As the eccentricity approaches unity, the speed at periapsis approaches the escape speed, just shy of becoming an unbound, parabolic orbit. In the limit of a hyperbolic encounter, the speeds of the stars are unchanged in the center of mass frame, but they undergo a deflection. In the frame of the SMBH, this change can boost one star's speed enough to unbind it from the SMBH, while the other star ends up on a more tightly bound orbit. Close encounters have previously been shown to eject stars from</text>
<text><location><page_6><loc_52><loc_79><loc_92><loc_91></location>dense stellar systems (e.g., Henon 1969; Lin & Tremaine 1980). Additionally, high-speed collisions often lead to mass loss, which can unbind the orbit not unlike a supernova (e.g., Lu & Naoz 2019). Therefore, we include a stopping condition for orbital energy ≥ 0 and eccentricity ≥ 1. We do not consider stars that become unbound after losing mass in a tidal disruption, as proposed by Manukian et al. (2013).</text>
<section_header_level_1><location><page_6><loc_57><loc_74><loc_87><loc_77></location>4. NUMERICAL RESULTS WITHOUT RELAXATION</section_header_level_1>
<text><location><page_6><loc_52><loc_56><loc_92><loc_73></location>We run large sets of 10000 tracer stars and let them evolve over 10 billion years. We begin by presenting simulated results without our relaxation prescription. These simulations allows us to isolate the orbital effects of collisions unobscured by relaxation. As in a nuclear star cluster, most of our tracer stars reside near ∼ 1 pc, the edge of the sphere of influence. About 500 of our tracer stars, or 5% of the sample, lie within 0 . 1 pc, where collisions are most common (see Figure 1). Sampling in this way gives a more complete picture of the relative rates of each outcome.</text>
<text><location><page_6><loc_52><loc_53><loc_92><loc_56></location>Figure 2 juxtaposes results for slightly different treatments of the collision physics, labeled as follows:</text>
<text><location><page_6><loc_52><loc_42><loc_92><loc_53></location>Type A: In this simulation, two colliding stars have a 50% chance of orbiting the SMBH in the same direction and a 50% chance of orbiting in opposite directions. The final velocities after a high-speed collision are calculated using Eq. 5. We assume that these high speed collisions do not impact the speed of the stars in their center of mass frame.</text>
<text><location><page_6><loc_52><loc_38><loc_92><loc_42></location>Type B: Simulations with type B conditions differ from type A by assuming that all stars orbit the SMBH in the same direction.</text>
<text><location><page_6><loc_52><loc_30><loc_92><loc_37></location>Type C: These simulations are identical to type A, except high speed collisions decrease the speeds in the stars' center of mass frame by 10%. In other words, we scale the velocity vectors of the stars by 0 . 9 before tranforming back to the frame of the SMBH.</text>
<text><location><page_6><loc_52><loc_25><loc_92><loc_29></location>Type D: These conditions are identical to types A and C except high speed collisions drecrease the speed in the center of mass frame by 50%.</text>
<text><location><page_6><loc_52><loc_9><loc_92><loc_24></location>The left column of Figure 2 shows the final masses and semimajor axes of the tracer stars. Red open circles represent stars at the end of the simulation or their mainsequence lifetime, whichever came first. Grey points represent the initial conditions. Blue diamonds indicate stars that were placed on unbound orbits by high speed collisions. Note that unbound orbits have negative semimajor axes, so we plot the absolute value. We provide the total number of stars that escape on unbound orbits in the upper right corner of each plot. In addition to</text>
<figure>
<location><page_7><loc_14><loc_32><loc_91><loc_92></location>
<caption>Figure 2. Left Column: We show the final masses of our tracer stars versus their final distances from the SMBH for four select simulations without relaxation. Turning off relaxation allows us to isolate the effect of collisions. The final mass and distance are either the stars properties when it evolved off the main-sequence or at the end of the 10 Gyr integration time, whichever is shorter. The grey points are the initial conditions, while the red points are the final tracer stars that remained in the cluster. Blue diamonds denote stars which escaped from the cluster, and black and lime stars represent TDEs from different types of collisions, mergers and high-speed collision deflections, respectively. The crosses represents stars which were placed onto such radial orbits that they plunged right into the Schwarschild radius of the SMBH. In the upper left hand corner, we indicate the number of stars that were placed onto unbound orbits. Since unbound orbits are defined to have negative semimajor axis, here we are plotting the absolute value for their final orbits. Right Column: These plots show the orbital period of the stars when they become TDEs versus β , the ratio of the tidal radius to the orbit's periapsis. The grey dashed horizontal line marks 30 yr, approximately the threshold at which we would expect repeating TDEs because the orbital period is short enough to observe multiple passages. Rows: Each row corresponds to slightly different initial conditions or collision treatment. A uses Eq. 5 to calculate the post-high speed collision deflection and assumes our so-called 'isotropic' conditions for the relative velocities between the colliding stars. B is identical to the above except the collision conditions are 'disky', meaning stars are always assumed to orbit the SMBH in the same direction. C uses isotropic conditions and reduces the post-collision speed in the center of mass frame by 10% for high-speed collisions. D scales the post-collision speed by 50% in center of mass frame following high-speed, destructive collision cases.</caption>
</figure>
<text><location><page_8><loc_8><loc_89><loc_48><loc_91></location>these unbound orbits, collisions can also place stars on orbits with semimajor axes outside the inner pc.</text>
<text><location><page_8><loc_8><loc_76><loc_48><loc_88></location>TDEs from direct collisions are represented by the star shaped symbols. The color indicates the type of collision responsible for the radial orbit: low-speed collisions, which we refer to as merger TDEs, are shown in black, while lime represents high speed collisions. Some of these orbits are so extreme that the periapsis distance lies within the Schwarzschild radius. We mark these plunging stars using an 'x' symbol.</text>
<text><location><page_8><loc_8><loc_60><loc_48><loc_76></location>The right column shows the properties of the TDEs and plunges for each simulation. The y-axis shows the orbital period of the final orbit, which carries them into the tidal radius. On the x-axis, we plot the parameter β , which quantifies how deeply the orbit penetrates the tidal radius. 0 . 5 < β < 1 indicate a partial disruption. To guide the eye, grey dashed line marks where the orbital period equals 30 yr. TDEs with periods less than this could conceivably be observed as a repeating TDE. We discuss the implications for TDEs below.</text>
<section_header_level_1><location><page_8><loc_18><loc_57><loc_38><loc_58></location>4.1. Tidal Disruption Events</section_header_level_1>
<text><location><page_8><loc_8><loc_42><loc_48><loc_56></location>Collision-induced mergers can result in TDEs under conditions A, C, and D, but not B. These merged stars are placed on nearly radial orbits when the colliding stars are orbiting in opposite directions. Their angular momenta vector are anti-aligned, giving a low angular momentum to the final orbit of the merger product. Under type B collision conditions, in contrast, there are no merger TDEs. The orbital angular momenta are never in a position to cancel each other out.</text>
<text><location><page_8><loc_8><loc_20><loc_48><loc_42></location>Fewer TDEs result from high speed collisions. The relative abundances reflect the relative frequency of each type of collision. High speed collisions tend to occur near the SMBH where speeds are sufficiently high, but fewer stars reside. These high speed TDEs are relatively rarer because this type of collision is relatively rarer. Collisions in general tend to become common within the inner 0 . 1 pc of the cluster. However, only about 500 stars from our sample lie in this region. We find that when we sample this region in greater detail, both high and low speed collision TDEs become more common. One high speed collision even produced a repeating TDE (see additional simulations in Section A.2). Others were close to our repeating TDE threshold with 30-50 yr periods.</text>
<text><location><page_8><loc_8><loc_9><loc_48><loc_20></location>There are some initial conditions that consistently fail to produce high speed collision TDEs, notably Type D conditions. A shallower density profile will also lead to a lower collision rate overall because the timescale is longer (see Figure 1), in turn reducing the rate of collision TDEs (see Appendix A.2). Additionally, we note that our prescription for determining whether or</text>
<text><location><page_8><loc_52><loc_67><loc_92><loc_91></location>not a merger results from a collision may over-predict the number of mergers. We use fitting formulae from Rauch (1999) to determine the mass loss in a collision and heuristic arguments, similar to the aforementioned study, to determine if a merger occurs. Previous studies with this code (Rose et al. 2023; Rose & MacLeod 2024) have also tested simulations using fitting formulae from Lai et al. (1993), who include a fitting formula for the merger capture radius. Their formula leads to fewer mergers compared to our first approach. However, collision-induced mergers would still be viable as a mechanism to create TDEs; they would simply occur at a much lower rate. As with other additional simulations, we show examples using the Lai et al. (1993) fitting formulae in the Appendix (Figure 8) to avoid overcrowding the main text.</text>
<section_header_level_1><location><page_8><loc_59><loc_62><loc_85><loc_63></location>4.2. Orbital Changes from Collisions</section_header_level_1>
<text><location><page_8><loc_52><loc_27><loc_92><loc_61></location>We consider general trends in the effects of collisions on the orbital properties of the stars. We do this by examining sub-populations of the tracer stars based on their final mass. We stress that this classification does not necessarily give the full picture of a star's collision history. While all stars with M final > 1 M ⊙ must have undergone a merger - recall that all stars in are cluster are initially 1 M ⊙ - and stars with M final < 1 M ⊙ must have lost mass in a high speed collision, these two populations are not mutually exclusive. A single star can experience both types of collisions over its lifetime depending on where it is in its orbit and the orbit of the second colliding star. However, final mass still represents the best 'observable' probe of a star's collision history. We compare the semimajor axis versus 1 -e of the stars in Figure 3, the latter being proportional to the periapsis. Grey dots show the initial conditions for all the stars. Green circles represent stars with M final > 1 M ⊙ , while orange dots represent stars with M final < 1 M ⊙ . TDEs and unbound stars are represented using the same symbols as in previous figures. We confirm that the TDEs in our simulation do indeed come from nearly radial orbits.</text>
<text><location><page_8><loc_52><loc_9><loc_92><loc_26></location>Mergers and high speed collisions differ in terms of their orbital outcomes. Conservation of energy demands that mergers always shrink the semimajor axis, while high speed collisions can place stars on both wider or smaller orbits by facilitating an energy exchange between the two stars. These trends are visible in the Figure. The green open circles are mostly confined to the inner cluster, while orange dots have an extended distribution. A few merged stars lie outside our initial cluster. These merged stars experienced a high speed collision that deflected them onto wider orbits.</text>
<figure>
<location><page_9><loc_8><loc_75><loc_47><loc_92></location>
</figure>
<figure>
<location><page_9><loc_8><loc_57><loc_47><loc_74></location>
</figure>
<figure>
<location><page_9><loc_8><loc_38><loc_47><loc_56></location>
<caption>Figure 3. This figure shows orbital parameters of our tracer stars from the simulations in the first three rows of Figure 2. We plot semimajor axis versus 1 -e , a quantity proportional to the periapsis distance. Systems with e f > 1, that is, stars placed on unbound orbits, are ommitted from this plot. Grey dots represent the initial population. Stars that experienced a post-collision merger over their lifetime are represented by green open circles. These stars are defined as having M > 1 M ⊙ . Orange dots, or stripped stars, are defined as having M < 1 M ⊙ , the result of a high speed collision. TDEs and plunges are marked with stars and crosses, respectively. As can be seen in the figure, these stars were placed on nearly radial orbits with extreme eccentricity values. We omit the final properties of stars that have not experienced a collision. These simulations have no relaxation, so un-collided stars' parameters remain unchanged.</caption>
</figure>
<text><location><page_9><loc_52><loc_59><loc_92><loc_91></location>In simulations A and B, a roughly equal number of stripped stars moved to more (less) tightly bound orbits. We count unbound stars in the less tightly bound category because their final orbital energies are larger than the initial values. Additionally, about 10 stars are placed on orbits with semimajor axes outside of 1 pc. This result suggests that there may be collision-affected stripped stars masquerading outside of the sphere of influence. Including a treatment for dissipation during a high speed collision, simulation C presents a different story. The orbits of stripped stars are more likely to shrink and become more circular. About 75% become more tightly bound. Additionally, a population of short period stripped stars emerges. These stars likely experienced multiple high speed collisions, giving them multiple opportunities to shrink their semimajor axes. The effects are even more pronounced for simulation D (not shown). Under type D conditions, stripped stars are overwhelmingly on less eccentric orbits, explaining why the high speed collision TDE rate for these conditions are so low.</text>
<text><location><page_9><loc_52><loc_26><loc_92><loc_58></location>Eccentricity trends can also be gleaned from Figure 3. For type B conditions, mergers always act to make the orbit more circular. In addition to energy, angular momentum must be conserved. For stars orbiting in the same direction, their angular momentum vectors are always aligned. Since the semimajor axis shrinks after a collision, the eccentricity must decrease to conserve angular momentum. This effect can be seen in vertical extent of the green circles in B compared to A. The most eccentric orbits for merged stars in B have been largely removed. We reiterate that the results are somewhat muddied by the occurrence of high speed collisions in a subset of this population. Under type A conditions, low speed collisions both decrease and increase the orbital eccentricity. The most eccentric final orbits become TDEs or plunges, removing the stars from the population. We note that in actuality, the orbital properties of the stars can be modified by resonant and non-resonant relaxation processes, not accounted for in these results (e.g., Rauch & Tremaine 1996; Hopman & Alexander 2006; Kocsis & Tremaine 2011).</text>
<section_header_level_1><location><page_9><loc_65><loc_22><loc_79><loc_23></location>4.3. Unbound Stars</section_header_level_1>
<text><location><page_9><loc_52><loc_9><loc_92><loc_21></location>High speed collisions can unbind stars from the SMBH. These interactions, treated similarly to a hyperbolic encounter, can place one star on a more tightly bound orbit while giving the other star a positive orbital energy. We show properties of the unbound stars from simulations A and B in Figure 4. The initial orbits tend to be eccentric. We also confirmed by inspection of specific cases that the collisions tend to occur near periapsis.</text>
<text><location><page_10><loc_8><loc_73><loc_48><loc_91></location>For the most part, the final orbits have eccentricity just above unity. We calculate the speed at infinity, v inf , for these stars based on their final orbital energy and color code the points in Figure 4 based on its value. Typical speeds range from ∼ 100 to 600 km/s, but occasionally one star will have a v inf above 1000 km/s, as was the case in simulation B. For simulation C, which includes some form of dissipation during high speed collisions, there were fewer unbound stars overall, but their distribution was similar. The maximum v inf was 642 km/s. The single unbound star from the fourth row of Figure 2 (simulation D) had v inf equal to 377 km/s.</text>
<text><location><page_10><loc_8><loc_27><loc_48><loc_73></location>These maximum values suggest that high speed collisions may represent another mechanism to launch hypervelocity stars from galactic nuclei. The most famous of these mechanisms is the Hills Mechanism, in which a binary is disrupted by the SMBH such that one star is ejected at high speed while the other is retained on a tightly bound orbit (e.g., Hills 1988; Generozov & Madigan 2020), though other mechanisms exist (e.g., Yu & Tremaine 2003; Perets 2009a). Close encounters between single stars are known to eject stars from dense stellar clusters (Henon 1969; Lin & Tremaine 1980), and in fact Yu & Tremaine (2003) consider such interactions as a means of producing hypervelocity stars. Omitting collisions, they find a low ejection rate at speeds ≳ 300 km/s. We consider collisions exclusively and treat them as modified close encounters. While our largest speed is consistent with those generated by the Hill's Mechanism ( ≳ 1000 km/s, Hills 1988), about 30% have v inf > 300 km/s. Observations of hypervelocity stars with origins pointing to the Galactic center exhibit a range of speeds, from the hundreds to thousands of km/s (e.g., Brown et al. 2005, 2018; Koposov et al. 2020; Generozov & Perets 2022, to quote from the latter, stars need ejection speed ≳ 750 km/s from the Galactic center to have 275 km/s at 20 kpc). The v inf of our stars shown in Figure 4 do not account for any additional terms in the potential beyond the SMBH. We reserve a detailed comparison of our results to observations for future work.</text>
<text><location><page_10><loc_8><loc_10><loc_48><loc_27></location>The mass loss fitting formulae from Rauch (1999) give very low mass loss for larger impact parameter collisions ( b ≳ 0 . 6 r c ) at moderately high speeds ( ∼ 1000 km/s); generally, the fractional mass loss in these cases is less than a percent, and unbound stars in Figure 2 appear as if they have not lost any mass at all. However, fitting formulae from high speed collisions can be unreliable (Freitag & Benz 2005). We therefore refrain from commenting on the masses of the unbound stars in detail except to state that some of them may look like stripped stars (Gibson et al. 2024).</text>
<text><location><page_10><loc_52><loc_78><loc_92><loc_92></location>As can be seen in Figure 2, a maximum of ∼ 1% of the stars in the inner parsec region may become unbound from the SMBH. This number far exceeds those that become TDEs through high speed collisions. Furthermore, this outcome represents a surprisingly high fraction of stars that experience such collisions: about a fifth of stars that experience high speed collisions become unbound. This high number may owe to a few conspiring conditions:</text>
<text><location><page_10><loc_52><loc_51><loc_92><loc_77></location>For reasons discussed in Section 3.4 and as supported by Figure 4, unbound stars tend come from high speed collisions near the periapsis of eccentric orbits. Near periapsis, a small boost in the frame of the SMBH can tip the star's speed over the escape speed. However, the merger-to-high speed collision boundary also lies around 800 km/s. The velocity dispersion only exceeds this value within about 0 . 02 pc, and the vast majority of stars reside outside this distance. In consequence, most high speed collisions can only occur near periapsis, where speeds are high enough to no longer be in the merger regime and also where unbinding the star from the SMBH becomes a more favorable outcome. Our thermal initial eccentricity distribution ensures that plenty of stars begin on very eccentric orbits (see Appendix A.1 for the impact of the eccentricity distribution).</text>
<text><location><page_10><loc_52><loc_34><loc_92><loc_51></location>Dissipation during the collision can reduce the number of unbound stars. As discussed in Section 3.2.2, it is possible that high speed direct collisions reduce the speeds of the stars in their center of mass frame. The physical impact of the collision may reduce the number of stars that get enough of a boost to escape the inner parsec. Type C and D collision conditions test the limits of this mechanism in producing unbound stars. Notably, if high speed collisions reduce the post-collision speed by 50% in the center of mass frame, only a few tenths of a percent of the cluster stars escape.</text>
<section_header_level_1><location><page_10><loc_54><loc_30><loc_91><loc_31></location>5. NUMERICAL RESULTS WITH RELAXATION</section_header_level_1>
<text><location><page_10><loc_52><loc_9><loc_92><loc_29></location>Now that we have built a physical picture of how collisions shape the stellar cluster, we turn on our prescription for two-body relaxation in the code. These simulations take longer to run, so the sample size of the tracer stars presented herein number at 4000 instead of 10000. We present results in Figure 5. The symbols are all the same as in previous figures, with one addition: stars that became TDEs through two body relaxation are shown in dark green. The orbital evolution of these stars where still affected by collisions, as can be seen by their masses. Interestingly, the stripped star that became a TDE via relaxation had experienced a collision within the same timestep. The collision had placed it on</text>
<section_header_level_1><location><page_11><loc_17><loc_91><loc_38><loc_92></location>Unbound Orbits Properties</section_header_level_1>
<figure>
<location><page_11><loc_9><loc_55><loc_47><loc_91></location>
<caption>Figure 4. We plot the final versus initial eccentricities for the stars on unbound orbits from simulations A and B (first and second row in Figure 2). Both the initial and final eccentricities are clustered near 1, the initial eccentricities slightly under this value and the final eccentricities above. The diamonds are colorcoded by the speed at infinity of the unbound stars. We note that this calculation does not take into account any additional contribution to the potential, such as the stellar cluster, beyond the mass of the SMBH. Typical v inf are order 100 km/s, though some exceed 1000 km/s.</caption>
</figure>
<text><location><page_11><loc_8><loc_27><loc_48><loc_37></location>an eccentric orbit just shy of the tidal radius, periapsis of 1 . 135 AU versus 0 . 956 AU needed for partial disruption. Had the star been inflated at all from the collision, material would have been siphoned off by the SMBH. As the star's parameters did not trigger the stopping condition, our relaxation prescription tipped the star into the tidal radius.</text>
<text><location><page_11><loc_8><loc_10><loc_48><loc_26></location>We place a caveat on the TDE inside 0 . 001 pc. The closest known star to the Milky Way's SMBH has a periapsis just under 0 . 001 pc (e.g., Gillessen et al. 2017). It is questionable whether our two-body relaxation prescription applies within this radius. However, this star too experienced a collision in the previous timestep which, had the star been inflated, would have led to a disruption. Altogether, our results suggest that collisions represent a viable mechanism to preferentially deposit stars that are (1) more massive than the surrounding population due to a merger or (2) stripped stars into the</text>
<text><location><page_11><loc_52><loc_84><loc_92><loc_91></location>tidal radius. Some of these TDEs could be observed as repeating TDEs. Additionally, we may be missing disruptions from stars that are inflated post collision, leading to qualitatively different electromagnetic signatures (e.g., MacLeod et al. 2013; Gibson et al. 2024).</text>
<text><location><page_11><loc_52><loc_54><loc_92><loc_84></location>The unbound stars present something of a puzzle. Even accounting for the difference in sample size, we find much fewer compared to Figure 2 (no relaxation). There are a few possible explanations. First, the relaxation prescription changes the orbital energy by roughly order of itself over the relaxation timescale, which is less than 10 Gyr in the model cluster. Stars at larger radii can therefore wander towards the SMBH, where destructive collisions become possible. Once in this 'danger zone', they can experience enough high-speed collisions to become destroyed. Roughly 500 stars were destroyed over this simulation. While this process is slow (see e.g., Rose & MacLeod 2024), it does drain stars from the sample population. In the simulation without relaxation, the majority of stars began and remained outside of this region. Only ∼ 50 stars were destroyed, a much smaller fraction of the sample, and most unbound stars had initial semimajor axes outside of 0 . 1 pc (see Appendix A.4).</text>
<text><location><page_11><loc_52><loc_27><loc_92><loc_54></location>The reduction of unbound stars may also be related to the interplay between the two dynamical processes. We speculate on why this second option might be true. Collisions are discrete events. Relaxation, in contrast, is a diffusion process that slowly alters a star's orbital parameters. This timescale is less than the collision timescale for most of the stars in our sample (see Figure 1), particularly where the unbound stars originate in our earlier simulations. The implication is that when a collision does occur, it may be less likely to catch the orbit in a state favorable for unbinding the star. The last possibility is that there is something in our relaxation prescription which we have yet to detect that is artificially suppressing the unbound stars. 1 Regardless, the interplay of relaxation and collisions merit a more thorough examination. We reserve a detailed study for future work.</text>
<section_header_level_1><location><page_11><loc_65><loc_24><loc_79><loc_25></location>6. CONCLUSIONS</section_header_level_1>
<text><location><page_11><loc_52><loc_16><loc_92><loc_23></location>Collisions between main-sequence stars occur in the nuclear star cluster and become common within the inner 0 . 1 pc (e.g., Rose & MacLeod 2024). In this proof-ofconcept study, we examine the orbital changes that result from these collisions. Specifically, we assess whether</text>
<figure>
<location><page_12><loc_10><loc_72><loc_91><loc_92></location>
<caption>Figure 5. We show a simulation with type A collision conditions that includes relaxation. The symbols are consistent with those in our other figures, with the addition of the dark green stars to represent TDEs from our relaxation prescription. We place a caveat on the dark green star (relaxation TDE) with final semimajor axis within 0 . 001 pc, also the TDE with the shortest orbital period. The applicability of our relaxation prescription within this region is uncertain.</caption>
</figure>
<text><location><page_12><loc_8><loc_60><loc_48><loc_65></location>physical collisions can contribute to the production of TDEs and ejected stars. Collisions in galactic nuclei can be understood in two types.</text>
<text><location><page_12><loc_8><loc_30><loc_48><loc_60></location>In the first type, the relative speed between the stars is low and the collision results in a merger. Little mass loss occurs (e.g., Lai et al. 1993) and we can treat the stars as sticky spheres (e.g., Kremer et al. 2020; Gonz'alez et al. 2021). Conservation of momentum allows us to calculate the final orbit of the merged star. The second type of collision occurs at speeds that exceed the escape velocity from the stars. While the high speeds ensure that the stars remain unbound from each other, they can also drive mass loss from the stars, in some cases producing a stripped star (e.g., Lai et al. 1993; Rauch 1999; Freitag & Benz 2005; Gibson et al. 2024). We use fitting formulae from Rauch (1999) to calculate the mass loss. However, we refrain from commenting in detail on the final masses of these stars because fitting formulae are not always accurate for collisions in galactic nuclei (Freitag & Benz 2005). We are examining these collisions further using sph simulations in forthcoming work (Rose et al. in prep.).</text>
<text><location><page_12><loc_8><loc_13><loc_48><loc_30></location>We use limiting cases and heuristic arguments to determine the final orbits from high speed collisions. If the stars were point masses, these interactions would unfold as a hyperbolic encounter in the center of mass frame of the stars. As an upper limit, we could calculate the deflection angle using Eq. 4. However, while a grazing collision approaches the limit given by Eq. 4, a head-on collision should not result in any deflection at all. We therefore adopt Eq. 5, which meets both criteria. We also test the role of dissipation by reducing the speeds of the stars post-collision in their center of mass frame.</text>
<text><location><page_12><loc_52><loc_60><loc_92><loc_65></location>Our simulations follow a sample of tracer stars embedded in a fixed, uniform cluster. We test a variety of cluster conditions and find the following:</text>
<unordered_list>
<list_item><location><page_12><loc_55><loc_26><loc_92><loc_59></location>1. Tidal disruption events: We find that both high and low speed collisions can produce TDEs by placing stars on highly eccentric orbits. Certain conditions, however, can be prohibitive to this mechanism. If stars all orbit the SMBH in the same direction, for example, collision-induced mergers cannot produce TDEs because the orbital angular momentum vectors of the colliding stars are aligned. If high speed collisions are highly dissipative, then TDEs from this class of collisions are similarly suppressed. The overall rates of collision TDEs remain to be seen. Generally, however, our results mean that collision-affected stars can be delivered to the tidal radius by the collisions themselves, in addition to standard two-body relaxation that affects the population at large. The TDEs immediately post-collision versus after the star has relaxed may look quite difficult (see discussion of imminent versus eventual TDEs in Gibson et al. 2024), adding to the richness of the observations that this dynamical process can yield.</list_item>
<list_item><location><page_12><loc_55><loc_9><loc_92><loc_24></location>2. Unbound stars: High speed collisions can place stars on unbound orbits, especially if the collisions occur near the periapsis of an eccentric orbit. As evidenced by the low mass loss in Figure 2, these collisions tend to have larger impact parameter. These grazing collisions approach the limit given by Eq. 4. Of course, in a physical collision, stars impede each other's motion. We test simple cases in which the star is only deflected a fraction of the angle and its speed is reduced post-collision.</list_item>
</unordered_list>
<text><location><page_13><loc_12><loc_59><loc_48><loc_91></location>Despite these changes, the unbound stars persist, albeit in much lower numbers. We reserve a precise examination of their rates and properties for future work, though some may look like stripped stars (Gibson et al. 2024). In the optimistic case, stellar collisions represent a mechanism to launch hypervelocity stars from galactic nuclei, joining the list of existing mechanisms (e.g., Hills 1988; Yu & Tremaine 2003; Ginsburg & Loeb 2007; Perets 2009a,b; Generozov & Madigan 2020). The Hills Mechanism elegantly explains both the origins of the S-star cluster, young-seeming massive stars in the vicinity of the SMBH, and hypervelocity stars using the same dynamical process (e.g., Hills 1988; Ghez et al. 2003; Ginsburg & Loeb 2007; Perets et al. 2007; Madigan et al. 2009; Lockmann et al. 2009; Generozov & Madigan 2020; Generozov 2021). Collisions present another possibility, where S-stars are the high-mass tail of the merger products created by successive low-speed collisions (see e.g., Rose et al. 2023), and hypervelocity stars</text>
<text><location><page_13><loc_56><loc_82><loc_92><loc_91></location>trace to high speed collisions near periapsis. However, the viability of this ejection mechanism still faces tests in the form of dissipation during collisions, a stellar mass function, interplay with other dynamical processes such as relaxation, and overall rates in a three-dimensional cluster.</text>
<text><location><page_13><loc_52><loc_58><loc_92><loc_80></location>We thank Fred Rasio, Enrico Ramirez-Ruiz, Jamie Lombardi, Fulya Kiroglu, Charles Gibson, and Claire Ye for invaluable discussion and input. SR thanks the CIERA Lindheimer Fellowship for support. BM thanks the Carnegie CTAC postdoctoral fellowship for support. This project began while SCR and BM were at the Aspen Center for Physics, which is supported by NSF grant PHY-2210452. This research was supported in part through the computational resources and staff contributions provided for the Quest high performance computing facility at Northwestern University which is jointly supported by the Office of the Provost, the Office for Research, and Northwestern University Information Technology.</text>
<section_header_level_1><location><page_13><loc_46><loc_54><loc_54><loc_55></location>APPENDIX</section_header_level_1>
<section_header_level_1><location><page_13><loc_38><loc_49><loc_62><loc_50></location>A. SUPPLEMENTAL FIGURES</section_header_level_1>
<section_header_level_1><location><page_13><loc_33><loc_47><loc_67><loc_48></location>A.1. Role of the Initial Eccentricity Distribution</section_header_level_1>
<text><location><page_13><loc_8><loc_36><loc_92><loc_46></location>Less eccentric orbits on average should result in fewer unbound stars. We test this hypothesis by drawing the eccentricities of tracer stars from a uniform distribution (average e = 0 . 5) instead of a thermal one (average e = 0 . 67). We compare the results in the first row of Figure 2, which uses a thermal initial eccentricity distribution, to that shown in Figure 6. Despite both having type A initial conditions, changing the eccentricity distribution to uniform reduces the number of unbound stars by about a factor of 2. There were also fewer high speed collisions overall, a ∼ 35% reduction. These results come from reducing the number of stars with speeds at periapsis that place them in the high speed collision regime.</text>
<section_header_level_1><location><page_13><loc_31><loc_33><loc_69><loc_34></location>A.2. Sampling the Inner Collision-dominated Region</section_header_level_1>
<text><location><page_13><loc_8><loc_17><loc_92><loc_32></location>Most collisions occur in the inner 0 . 1 parsec of the nuclear star cluster, where the collision timescale becomes shorter than 10 Gyr. In order to probe this region in greater detail, we run a similar set of simulations to those shown in Figure 2, but with the stars sampled from 0 . 001 to 0 . 1 pc. We note that a cusp with slope α = 1 . 75 results in a very high density in this region. The actual cusp in the Galactic center may be closer to 1 . 4 (Gallego-Cano et al. 2018b). Additionally, collisions themselves can deplete the cusp in the inner 0 . 1 pc. A shallower cusp would lead to a lower collision rate. We therefore test α = 1 . 25 with the idea that reality may lie somewhere in the middle. These simulations are marked with an asterisk in Figure 7. The shallower profile means that the collision timescale is longer and that fewer collisions occur within the cluster over the integration time. However, as can be seen in the Figure, the collisions that do occur can stil produce TDEs. Additionally, type B conditions, which prohibit merger TDEs, can still have collision-driven TDEs of a different kind.</text>
<section_header_level_1><location><page_13><loc_38><loc_14><loc_62><loc_16></location>A.3. Alternative Fitting Formulae</section_header_level_1>
<text><location><page_13><loc_8><loc_9><loc_92><loc_13></location>As mentioned in Section 3.1, we leverage fitting formulae and intuitive toy models to calculate the mass loss from a collision and determine whether a merger occurs. Simulations shown in the main text and in Figure 7 use fitting formulae from Rauch (1999) and heuristic arguments to determine if a merger occurs. In this section, we present</text>
<figure>
<location><page_14><loc_8><loc_72><loc_88><loc_92></location>
<caption>Figure 6. We show a simulation with type A collision conditions that uses a uniform eccentricity distribution for the stars instead of a thermal one. The presentation of the figure and symbols are the same as those of Figure 2. Having less eccentric orbits on average reduces the number of unbound stars compared to the thermal case.</caption>
</figure>
<text><location><page_14><loc_8><loc_52><loc_92><loc_64></location>simulations with type A collision conditions that use fitting formulae from Lai et al. (1993). Lai et al. (1993) include fitting formula for the capture radius, a function of the relative speed between the stars and their impact parameter. This formula leads to less mergers than the simple prescription that we use in the main text of this paper, which is also similar to the prescription used in Rauch (1999) (see for a full discussion Rose et al. 2023). We present the results of two simulations using the Lai et al. (1993) formulae in Figure 8. The upper row samples tracer stars on a cusp from 0 . 001 to 1 pc, while the second row focuses on the inner 0 . 1 pc. As expected, with the overall reduction in mergers - they instead are treated as high speed collisions - there are fewer merger TDEs. However, it is still possible to get TDEs of all types and unbound stars.</text>
<section_header_level_1><location><page_14><loc_41><loc_49><loc_59><loc_51></location>A.4. Orbital Parameters</section_header_level_1>
<text><location><page_14><loc_8><loc_43><loc_92><loc_48></location>We expand upon Section 4.2 in this appendix. We compare the final semimajor axes versus the initial semimajor axes of the stars in Figure 9. As is our convention in the main text, green circles represent stars with M final > 1 M ⊙ , while orange dots have M final < 1 M ⊙ . TDEs and unbound stars are represented using stars and diamonds. The black dashed line shows where the initial and final semimajor axes are equal.</text>
<text><location><page_14><loc_8><loc_36><loc_92><loc_42></location>Mergers work to shrink the semimajor axis, while high speed collisions can increase or decrease it, as is visible in the Figure. For the most part, green circles lie below the black dashed lines. A few lie above the black line, mostly within 0 . 1 pc, because they also experienced a high speed collision. They initially resided within ∼ 0 . 1 pc, where high speed collisions become more likely.</text>
<text><location><page_14><loc_8><loc_32><loc_92><loc_36></location>We examine the final eccentricities for merger-affected stars in Figure 10. We compare the final eccentricity distribution to the initial thermal distribution. For reasons outlined in Section 4.2, type B mergers circularize orbits. This effect can be seen in Figure 10, where there is a dearth of eccentric orbits.</text>
<text><location><page_14><loc_8><loc_22><loc_92><loc_31></location>The effects on eccentricity in type A conditions are harder to tease out. 50% of the time the orbits are aligned and the eccentricity decreases. The other 50% of the time, the angular momenta vector are anti-aligned, so even though the semimajor axis still shrinks, the eccentricity can increase. The most eccentric merged stars are removed from the population because they become TDEs or plunges. The result is an eccentricity distribution that flattens at high ( e > 0 . 5) eccentricities. However, the actual number of merged star orbits that became more (less) eccentric are similar.</text>
<section_header_level_1><location><page_14><loc_44><loc_18><loc_56><loc_19></location>REFERENCES</section_header_level_1>
<text><location><page_14><loc_8><loc_16><loc_45><loc_17></location>Alexander, T. 1999, ApJ, 527, 835, doi: 10.1086/308129</text>
<text><location><page_14><loc_8><loc_13><loc_40><loc_14></location>Alexander, T., & Pfuhl, O. 2014, ApJ, 780, 148,</text>
<text><location><page_14><loc_10><loc_10><loc_33><loc_11></location>doi: 10.1088/0004-637X/780/2/148</text>
<text><location><page_14><loc_52><loc_16><loc_86><loc_17></location>Bahcall, J. N., & Wolf, R. A. 1976, ApJ, 209, 214,</text>
<text><location><page_14><loc_54><loc_14><loc_67><loc_15></location>doi: 10.1086/154711</text>
<text><location><page_14><loc_52><loc_12><loc_74><loc_13></location>Balberg, S. 2024, ApJ, 962, 150,</text>
<text><location><page_14><loc_54><loc_10><loc_74><loc_11></location>doi: 10.3847/1538-4357/ad1690</text>
<section_header_level_1><location><page_15><loc_32><loc_85><loc_61><loc_87></location>Sampling Tracer Stars from the Inner 0.1 pc</section_header_level_1>
<figure>
<location><page_15><loc_14><loc_24><loc_91><loc_85></location>
<caption>Figure 7. This Figure has the same form as Figure 2, except the 10000 tracer stars have been sampled on a cusp from 0 . 001 to 0 . 1 pc. In this region, the collision timescale becomes comparable to the lifetime of a solar mass star, and collision become common. The simulations conditions are denoted by the letter on the left and correspond to those with the same letter in Figure 2. Simulations with an asterisk use a shallower slope for the stellar density profile, α = 1 . 25 instead of 1 . 75.</caption>
</figure>
<text><location><page_15><loc_57><loc_22><loc_78><loc_23></location>*uses a = 1.25 instead of 1.75</text>
<figure>
<location><page_16><loc_14><loc_55><loc_91><loc_90></location>
<caption>Figure 8. Again with the same form as Figure 2, here we present results using fitting formulae from Lai et al. (1993). The upper row has tracer stars sampled on a cusp from 0 . 001 to 1 pc, while the outer limit in the bottom row for the initial semimajor axes is 0 . 1.</caption>
</figure>
<figure>
<location><page_16><loc_9><loc_27><loc_91><loc_45></location>
<caption>Figure 9. We compare the final semimajor axes of interesting tracer stars to their initial values. We note that since there was no relaxation in any of these simulations, all changes are due to direct collisions. The black dashed line shows where the initial semimajor axes equals the final. Stars that experienced a post-collision merger over their lifetime are represented by green open circles. These stars are defined as having M > 1 M ⊙ . Orange dots, or stripped stars, are defined as having M < 1 M ⊙ , the result of a high speed collision. Blue diamonds represent unbound stars. We plot the absolute value of their semimajor axes. Black (lime) stars represent TDEs from merger (high speed) collisions. Each initial condition case are qualitatively similar, with mergers producing more tightly bound orbits and high speed collisions moving stars to both tighter and wider orbits. The exception is the simulation with type C initial conditions. This prescription slows stars down by 10% following a high-speed collision in the stars' center of mass frame. As a result, a population ot stripped stars that have sunk towards the SMBH emerges.</caption>
</figure>
<figure>
<location><page_17><loc_14><loc_89><loc_34><loc_92></location>
</figure>
<figure>
<location><page_17><loc_49><loc_89><loc_69><loc_92></location>
</figure>
<figure>
<location><page_17><loc_8><loc_65><loc_83><loc_87></location>
<caption>Figure 10. We compare the final eccentricity distribution of stars that have experienced a merger with the initial (thermal) eccentricity distribution. The first column uses type A inital conditions ('isotropic'), in which the colliding stars can orbit the SMBH in opposite direction, while the second column uses type B initial conditions ('disky'), in which stars always orbit the SMBH in the same direction. Mergers seem to circularize the stellar orbits, completely erasing the peak at high eccentricities of a thermal distribution. We note that these simulations do not include relaxation; all orbital changes are due to direct collisions.</caption>
</figure>
<text><location><page_17><loc_8><loc_11><loc_48><loc_56></location>Balberg, S., Sari, R., & Loeb, A. 2013, MNRAS, 434, L26, doi: 10.1093/mnrasl/slt071 Bartko, H., Martins, F., Fritz, T. K., et al. 2009, ApJ, 697, 1741, doi: 10.1088/0004-637X/697/2/1741 Binney, J., & Tremaine, S. 2008, Galactic Dynamics: Second Edition Bradnick, B., Mandel, I., & Levin, Y. 2017, MNRAS, 469, 2042, doi: 10.1093/mnras/stx1007 Brown, W. R., Geller, M. J., Kenyon, S. J., & Kurtz, M. J. 2005, ApJL, 622, L33, doi: 10.1086/429378 Brown, W. R., Lattanzi, M. G., Kenyon, S. J., & Geller, M. J. 2018, ApJ, 866, 39, doi: 10.3847/1538-4357/aadb8e Brutman, Y., Steinberg, E., & Balberg, S. 2024, arXiv e-prints, arXiv:2408.16383, doi: 10.48550/arXiv.2408.16383 Dale, J. E., & Davies, M. B. 2006, MNRAS, 366, 1424, doi: 10.1111/j.1365-2966.2005.09937.x Dale, J. E., Davies, M. B., Church, R. P., & Freitag, M. 2009, MNRAS, 393, 1016, doi: 10.1111/j.1365-2966.2008.14254.x Dessart, L., Ryu, T., Amaro Seoane, P., & Taylor, A. M. 2024, A&A, 682, A58, doi: 10.1051/0004-6361/202348228 Ferrarese, L., & Ford, H. 2005, SSRv, 116, 523, doi: 10.1007/s11214-005-3947-6 Freitag, M., & Benz, W. 2002, A&A, 394, 345, doi: 10.1051/0004-6361:20021142 -. 2005, MNRAS, 358, 1133,</text>
<text><location><page_17><loc_10><loc_10><loc_35><loc_11></location>doi: 10.1111/j.1365-2966.2005.08770.x</text>
<text><location><page_17><loc_52><loc_13><loc_92><loc_56></location>Gallego-Cano, E., Sch¨odel, R., Dong, H., et al. 2018a, A&A, 609, A26, doi: 10.1051/0004-6361/201730451 -. 2018b, A&A, 609, A26, doi: 10.1051/0004-6361/201730451 Gallegos-Garcia, M., Law-Smith, J., & Ramirez-Ruiz, E. 2018, ApJ, 857, 109, doi: 10.3847/1538-4357/aab5b8 Generozov, A. 2021, MNRAS, 501, 3088, doi: 10.1093/mnras/staa3851 Generozov, A., & Madigan, A.-M. 2020, ApJ, 896, 137, doi: 10.3847/1538-4357/ab94bc Generozov, A., & Perets, H. B. 2022, MNRAS, 513, 4257, doi: 10.1093/mnras/stac1108 Genzel, R., Eisenhauer, F., & Gillessen, S. 2010, Reviews of Modern Physics, 82, 3121, doi: 10.1103/RevModPhys.82.3121 Ghez, A. M., Salim, S., Hornstein, S. D., et al. 2005, ApJ, 620, 744, doi: 10.1086/427175 Ghez, A. M., Duchˆene, G., Matthews, K., et al. 2003, ApJL, 586, L127, doi: 10.1086/374804 Ghez, A. M., Salim, S., Weinberg, N. N., et al. 2008, ApJ, 689, 1044, doi: 10.1086/592738 Gibson, C., Kıro˘glu, F., Lombardi, J. C., J., et al. 2024, arXiv e-prints, arXiv:2410.02146, doi: 10.48550/arXiv.2410.02146 Gillessen, S., Eisenhauer, F., Trippe, S., et al. 2009, ApJ, 692, 1075, doi: 10.1088/0004-637X/692/2/1075</text>
<text><location><page_17><loc_52><loc_10><loc_92><loc_12></location>Gillessen, S., Plewa, P. M., Eisenhauer, F., et al. 2017, ApJ, 837, 30, doi: 10.3847/1538-4357/aa5c41</text>
<table>
<location><page_18><loc_8><loc_9><loc_48><loc_92></location>
</table>
<table>
<location><page_18><loc_52><loc_9><loc_92><loc_92></location>
</table>
<table>
<location><page_19><loc_8><loc_55><loc_48><loc_92></location>
</table>
<table>
<location><page_19><loc_52><loc_56><loc_92><loc_92></location>
</table>
</document>
[
{
"title": "ABSTRACT",
"content": "Dense stellar clusters surround the supermassive black holes (SMBH) in galactic nuclei. Interactions within the cluster can alter the stellar orbits, occasionally driving a star into the SMBH's tidal radius where it becomes ruptured. This proof-of-concept study examines the orbital effects of stellar collisions using a semianalytic model. Both low and high speed collisions occur in the SMBH's sphere of influence. Our model treats stars in low speed collisions as sticky spheres. For high-speed collisions, we develop a simple prescription based on the limiting case of a hyperbolic encounter. We test a range of collision treatments and cluster conditions. We find that collisions can place stars on nearly radial orbits. Depositing stars within the tidal radius, collisions may drive the disruption of stars with unusual masses and structures: depending on the nature of the collision, the star could be the product of a recent merger, or it could have lost its outer layers in a high speed impact, appearing as a stripped star. We also find that high speed collisions near the periapsis of an eccentric orbit can unbind stars from the SMBH. However, dissipation during these high-speed collisions can substantially reduce the number of unbound stars achieved in our simulations. We conclude that TDEs and ejected stars, even in the hypervelocity regime, are plausible outcomes of stellar collisions, though their frequency in a three-dimensional nuclear star cluster are uncertain. Future work will address the rates and properties of these events. Keywords: Stellar dynamics; Galactic center; Star clusters; Stellar mergers",
"pages": [
1
]
},
{
"title": "On the Orbital Effects of Stellar Collisions in Galactic Nuclei: Tidal Disruption Events and Ejected Stars",
"content": "1 Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA), Northwestern University, 1800 Sherman Ave, Evanston, IL 60201, USA 2 The Observatories of the Carnegie Institution for Science, Pasadena, CA 91101, USA",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "Asupermassive black hole resides at the center of most galaxies, where it is surrounded by a dense stellar cluster (e.g. Ferrarese & Ford 2005; Kormendy & Ho 2013; Schodel et al. 2003; Ghez et al. 2005, 2008; Gillessen et al. 2009, 2017; Neumayer et al. 2020). A tidal disruption event (TDE) occurs when a star from the cluster passes within a critical distance from the SMBH and becomes ruptured by tidal forces (e.g., Hills 1975; Rees 1988; Alexander 1999; Magorrian & Tremaine 1999; Wang & Merritt 2004; MacLeod et al. 2012). The SMBH then accretes the stellar material, producing an electromagnetic signature (e.g., Guillochon & Ramirez-Ruiz 2013). Spectra of these events encode valuable information about the mass, structure, and composition of the ruptured star (e.g., Kochanek 2016a,b; Yang et al. 2017; Mockler et al. 2022; Miller et al. 2023). Observations of Corresponding author: Sanaea C. Rose sanaea.rose@northwestern.edu TDEs represent a powerful way to probe the stellar populations in galactic nuclei and the processes that shape them. Amongst these physical processes are direct collisions. Collisions can occur within the sphere of influence of the SMBH due to the high densities and velocity dispersion (e.g., Freitag & Benz 2002; Dale et al. 2009; Dale & Davies 2006; Rubin & Loeb 2011; Balberg et al. 2013; Balberg 2024; Mastrobuono-Battisti et al. 2014; Rose & MacLeod 2024). These events may produce electromagnetic signatures, both from the collisions themselves and from interactions between liberated material and the SMBH (e.g., Rosswog et al. 2009; Lee et al. 2010; Balberg et al. 2013; Dessart et al. 2024; Ryu et al. 2024b,a; Brutman et al. 2024), though it is difficult to ignite a main-sequence star with compression (Guillochon et al. 2009). They can also shape the stellar population. Both low and high speed collisions can alter the mass of a star - the latter can destroy stars completely and the former can give rise to blue stragglers in dense stellar systems (e.g., Lai et al. 1993; Rauch 1999; Sills et al. 1997, 2001; Lombardi et al. 2002; MacLeod et al. 2013; Leigh et al. 2016; Rose et al. 2023). Recently, Gibson et al. (2024) have shown that high speed collisions can also produce stripped stars similar to what might be seen through binary evolution. Collisions that produce stellar mergers or stripped stars may explain recent unexpected TDE observations. For example, detections of high nitrogen-tocarbon (N/C) ratios in TDEs point to the disruption of more stars that burn on the CNO cycle (as first proposed by Kochanek 2016a,b), and that are ≳ 1 -2 M ⊙ (Kochanek 2016a; Yang et al. 2017; Gallegos-Garcia et al. 2018; Mockler et al. 2023) than is predicted by the host galaxies' stellar populations (Mockler et al. 2023). One particular TDE has such an extremely high N/C abundance ratio that it is difficult to explain with single stellar evolution alone (Miller et al. 2023), but could be the result of the disruption of a stripped star that has lost its nitrogen-poor envelope (Mockler et al. 2024). Additionally, a recently discovered population of extremely bright nuclear transients has also been suggested to originate from the disruption of high mass stars (e.g. ≳ 10 M ⊙ , Subrayan et al. 2023; Hinkle et al. 2024). Because stars that end in disruptions are expected to be drawn approximately at random from the stellar mass function (with small adjustments for stellar type, MacLeod et al. 2012), this preference for higher mass stars may imply that the mass function in galactic nuclei is more top-heavy (and/or bottom-light) than in the rest of the galaxy (see e.g., Lu et al. 2013; Hosek et al. 2019). The potential link between TDEs and collisions is intriguing: collisions can simultaneously affect both the properties of the stars and their orbits about the SMBH. MacLeod et al. (2012) first considered collisions, particularly destructive ones, in the context of the TDE rate. Changes to a star's trajectory from collisions have also been considered in general dynamical models of dense stellar systems (e.g., Sanders 1970; Rauch 1999; Freitag & Benz 2002; Kremer et al. 2020; Gonz'alez et al. 2021; Rodriguez et al. 2022). In this proof-of-concept study, we study the orbital effects of stellar collisions in galactic nuclei. We assess whether the collisions that produce unusual stars could plausibly deposit those same stars onto TDE-producing orbits. Our models leverage simple, intuitive treatments for collision outcomes and fitting formulae from previous studies (e.g. Lai et al. 1993; Rauch 1999). We test a range of initial conditions and nuclear star cluster properties. Our paper is organized as follows: In Section 2, we discuss our general approach to modeling the Milky Way's nuclear star cluster. Section 3 describes the treatment of various physics in our code, with Section 3.2 in particular outlining our methodology for updating the stellar orbits post collision. These orbital changes represent the most major addition to the code as compared to previous iterations (Rose et al. 2022, 2023). Section 4 presents simulated results for direct collisions. Sections 4.1, 4.2, and 4.3 discuss the implications for TDEs, orbital properties of collision-affected stars, and unbound and hypervelocity stars. We then incorporate relaxation into our simulations in addition to stellar collisions and present results in Section 5. Lastly, we summarize the scope and findings of our study in Section 6.",
"pages": [
1,
2
]
},
{
"title": "2. MODEL NUCLEAR STAR CLUSTER",
"content": "We leverage semi-analytic models to study the effects of collisions on the nuclear star cluster. Our fiducial model uses the conditions and properties of the Milky Way's GN, whose proximity makes it the best studied galactic center. We follow a sample of stars embedded in a fixed, unevolving cluster. For simplicity, both the evolving sample and the surrounding cluster are composed of 1 M ⊙ stars. The cluster can be understood in two key properties, density and velocity dispersion, which govern the dynamical processes unfolding within it. The stellar density sets the the frequency with which stars interact. We describe the stellar density as a function of distance from the SMBH using a power law: where α is the slope and r · , distance from the SMBH. Based on observations of the cluster within the sphere of influence, this equation is normalized using ρ 0 = 1 . 35 × 10 6 M ⊙ / pc 3 at r 0 = 0 . 25 pc (Genzel et al. 2010). Our fiducial model uses a slope of 1 . 75, the expectation for a single-mass population (Bahcall & Wolf 1976), consistent with the fact that our simple model cluster has only solar mass stars. However, in order to capture the range of theoretical predictions and observational constraints on the stellar cusp (e.g., Bahcall & Wolf 1976; Gallego-Cano et al. 2018a; Linial & Sari 2022), we also test a few simulations with α = 1 . 25, shown in Appendix A. We assume that the slope of the stellar cusp is roughly constant, or varying slowly, over the timescales of interest in our simulations. The velocity dispersion within the cluster also influences the frequency and nature of stellar interactions. It decreases with distance from the SMBH: where α is the slope of the density profile and M · is the mass of the SMBH (Alexander 1999; Alexander & Pfuhl 2014). We take M · to be 4 × 10 6 M ⊙ , like the Milky Way's SMBH (e.g., Ghez et al. 2003). For a uniform mass cluster of 1 M ⊙ stars, the number density n is simply ρ ( r · ) 1 M ⊙ .",
"pages": [
2,
3
]
},
{
"title": "3. SEMIANALYTIC MODEL",
"content": "We follow a sample of 1 M ⊙ tracer stars embedded in our model cluster. We draw their orbital eccentricities from a thermal distribution. We select their semimajor axes so that they lie on a cusp with slope α , matching the background cluster. These stars are allowed to evolve under the influence of two main dynamical processes, direct collisions and two-body relaxation, using a model first developed by Rose et al. (2022, 2023).",
"pages": [
3
]
},
{
"title": "3.1. Collisions",
"content": "Direct collisions occur over a characteristic timescale t -1 coll = nσA , where A is the cross-section of interaction, n is the number density, and σ is the velocity dispersion. For an impact to occur, A is the physical crosssection enhanced by gravitational focusing. The collision timescale also depends weakly on the star's orbital eccentricity and can be written as: where f 1 ( e · ) and f 2 ( e · ) are equations 20 and 21 from Rose et al. (2020), G is the gravitational constant, and a · is the star's semimajor axis. r c is the sum of the radii of the colliding stars, or 2 R ⊙ for a uniform population of solar mass stars. We plot this timescale in red in the upper panel of Figure 1. We consider a range of slopes for the stellar density profile, spanning α = 1 . 25 (dashed line) to α = 1 . 75 (solid line). The horizontal grey line shows the total simulation time, included to guide the eye. Where the collision timescale is less than the simulation time, within 0 . 1 pc of the SMBH, collisions become important to understanding the evolution of the cluster (e.g., Rose & MacLeod 2024). We treat stellar collisions using a statistical approach. We begin by computing the probability that a star in our sample will experience a collision. Over a timestep ∆ t , this probability equals ∆ t/t coll . ∆ t is taken to be 10 6 years so that the probability ∆ t/t coll is always less than one; t coll ≳ 10 7 years for the parameter space we consider. The code then draws a random number between 0 and 1, which, if less than or equal to the collision probability ∆ t/t coll , means a collision has occurred. We repeat this prescription until the desired simulation runtime or the star's main-sequence lifetime has been reached, whichever is shorter. If a collision has occurred over a given timestep, we update the mass and age of the star using a prescription detailed in Rose et al. (2023). A full discussion of our approach can be found in previous papers (Rose et al. 2023; Rose & MacLeod 2024). In brief, we combine fitting formulae from hydrodynamics simulations of stellar collisions (Rauch 1999; Lai et al. 1993) with heuristic arguments to (1) determine if a given collision will result in a merger and (2) determine the amount of mass lost from either the merger product or the unbound individual stars. Since speeds in galactic nuclei often exceed hundreds of kilometers per second, collisions can eject anywhere from a percent to all of the star's mass, effectively blowing it up (Spitzer & Saslaw 1966; Lai et al. 1993; Balberg et al. 2013; Balberg 2024; Brutman et al. 2024). Mergers with minimal mass loss are most likely to occur outside of about 0 . 01 pc, where the relative speeds tend to be less than the escape speed from the stars. Within 0 . 01 pc, however, velocities exceed the escape speed from the stars, and collisions can result in peculiar, low-mass 'stripped stars' (Rose et al. 2023; Rose & MacLeod 2024; Gibson et al. 2024). In practice, the outcome of a collision is more complex to determine, depending on properties such as the impact parameter and stellar structure (e.g., Freitag & Benz 2005; Gibson et al. 2024, Rose et al. in prep.). For the purposes of this study, which focuses on the general implications of collisions for stellar orbits in galactic nuclei, the specific mass loss and merger prescriptions should not qualitatively change our results. Unless otherwise specified, all simulations shown herein use fitting formulae from Rauch (1999) for the mass loss and escape speed arguments to determine whether or not a collision results in a merger (Rose et al. 2023). In previous iterations of this code, the collision speed was taken to be simply the velocity dispersion at the star's distance from the supermassive black hole, given by Eq. (2) (Rose et al. 2022, 2023). However, this approach is insufficient for studying the effects of collisions on stellar orbits. In this study, we draw the orbit of the second colliding star, using a procedure detailed below.",
"pages": [
3
]
},
{
"title": "3.2. Orbital Dynamics",
"content": "If a star in our sample with semimajor axis a · and eccentricity e · collides over timestep ∆ t , we find a plausible orbit for the second colliding star. As we operate under the assumption that the cluster is composed of a uniform stellar population, we always take the second star to have m collider = 1 M ⊙ . We then draw semimajor axes and eccentricities from the cluster's property distri- as described in Section 2, until we find an orbit that intersects with that of our tracer star. Our code assumes that the cluster is spherically symmetric and only tracks the semimajor axis and eccentricity of each tracer star. In this set up, our tracer star always has argument of periapsis equal to zero, however the second colliding star's periapsis need not be aligned with it. The angle between them is drawn from a uniform distribution. The orbits are always assumed to be coplanar. Observations suggest that a subset of the stars in the Milky Way's galactic center reside in a disk, while others have an isotropic distribution (e.g., Levin & Beloborodov 2003; Ghez et al. 2003, 2005; Gillessen et al. 2009; Paumard et al. 2006; Lu et al. 2009; Bartko et al. 2009; Yelda et al. 2014). In our co-planar physical picture, we have two options: either both of the colliding stars can orbit in the same direction about the supermassive black hole, or they can be equally likely to be prograde or retrograde. We test two extremes to capture the effects of different orbital orientations. Our first case, 'disk-like', has the two colliding stars orbiting the SMBH in the same direction. In the second case, 'isotropic-like', we assume that fifty percent of the time the two stars orbit the SMBH in the same direction and fifty percent of the time in opposite directions. The latter orientation means that the orbital angular momentum vectors are anti-parallel to one another. Once we have found an intersecting orbit for our colliding tracer star, we determine the intersection point and compute the velocity vectors of the two stars at that point. The relative speed tells us whether or not a merger occurs and the degree of fractional mass loss from the system (see above section). From here, determining the final orbit(s) will depend on the type of outcome and therefore the relative speed.",
"pages": [
3,
4
]
},
{
"title": "3.2.1. Final Orbit in Merger Case",
"content": "A stellar merger is the natural outcome of a low-speed collision. We calculate the final mass, age, and now trajectory of the product assuming the stars act as 'sticky spheres' (e.g., Rodriguez et al. 2022; Rose et al. 2023). The two stars approach each other with velocity vectors v · and v collider as determined by the intersection point of their orbit. Momentum conservation demands that the final velocity of the merger product, v final , equals ( M v · + M ⊙ v collider ) / ( M + M ⊙ ). Based on hydrodynamics studies of stellar mergers, the mass loss in these collisions should be low (e.g.., Lai et al. 1993; Rauch 1999, Rose et al. in prep.). With the final velocity, mass, and location of the collision - the intersection point of the original orbits - we can calculate the new orbit of the merger product about the SMBH.",
"pages": [
4
]
},
{
"title": "3.2.2. Final Orbits Following a High-Speed Collision",
"content": "Orbital changes from high-speed collisions are much more difficult to determine in the absence of hydrodynamics simulations. However, we present a framework for treating these interactions. In order to understand these collisions, we start at the limit where the two stars barely graze. This interaction should unfold as a hyperbolic encounter. In the center of mass frame of the two stars, their speeds remain constant, but their velocity vectors are deflected by angle θ hyp . θ hyp can be found analytically: where b is the impact parameter and b 90 is defined as the impact parameter needed for a 90 · deflection. b 90 equals 2 G/v 2 rel , where v rel is the relative speed between the two stars (e.g., Binney & Tremaine 2008). We plot b 90 as a function of the relative speed in the bottom panel of Figure 1. In the nuclear star cluster, the velocity dispersion can be understood as the characteristic relative speed between stars at a given distance from the SMBH (see Eq. 2). We have therefore inverted the x-axis of the bottom plot to parallel that of the upper plot, distance from the SMBH, and marked the velocity dispersion at key distances using vertical dashed lines. In both the upper and lower plots, we also indicate the regions in which we expect mergers versus high-speed collisions, which leave the stars unbound from each other. Interestingly, for most of the parameter space where these high-speed, non-sticky sphere collisions occur, b 90 is less than the star's radius. Another way of interpreting this statement is that at high speeds, physical collisions are required for a strong-angle deflection. If the stars were point particles, the impact parameter could be arbitrarily small and the interaction would still unfold as a hyperbolic encounter. Two Sun-like stars begin to touch when b = 2 R ⊙ . With b < 2 R ⊙ , the stars physically impede each other as they interact, leading to a smaller deflection angle than the one given in Eq. (4). Furthermore, if the stars approach each other perfectly head-on with b = 0, the center of mass velocity is 0 and there is no angular momentum. In this case, there would be no deflection. Heuristically, then, we expect the deflection angle to be given by Eq. (4) for grazing encounters, and some fraction of this angle for encounters with b < 2 R . That fraction should decrease with impact parameter until they are both zero. The precise dependence is impossible to determine in the absence of hydro simulations, but we define a collision deflection angle that meets the two limiting criterion: where r c is the sum of the radii of the two colliding stars. It is reasonable to expect that the collision affects the stars' speeds as well. Even more so than the deflection angle, changes in speed can only be understood through hydrodynamic simulations. In our models, we simply test three cases: one in which the speed is not affected at all, one in which the speed in the center of mass frame is always reduced by 10%, and one in which it is always reduced by a factor of 2. We treat these collisions as follows: at the intersection point of the orbits of the two colliding stars, we calculate their center of mass velocity. We transform to the center of mass frame. Only then do we rotate and scale their velocity vectors. While we only track semimajor axes and eccentricities, we do allow the deflections to be three dimensional, with components out of the plane. The final velocity can be any vector, chosen randomly, along a cone defined by angle θ coll with respect to the original velocity. Isotropizing the deflection is important because otherwise the colliding stars would always be scattered exactly towards or away from the SMBH, leading to an gross overestimate of TDE rates. We then transform back to the frame of the SMBH and calculate the new orbits given the velocity and position vectors. We note that our calculation assumes that the center of mass is not moving in the frame of the supermassive black hole. In actuality, the center of mass of the two stars would orbit the supermassive black hole, but the effects will be negligable due to the high collision speeds.",
"pages": [
5
]
},
{
"title": "3.3. Two-Body Relaxation",
"content": "In addition to collisions, stars in the cluster experience the weak gravitational effects of nearby neighbors. The effects of these interactions can accumulate, eventually changing the star's orbital energy and angular momentum by order of itself. The original orbit is 'erased' over a characteristic timescale: where ⟨ M ∗ ⟩ is the average star's mass, here taken to be 1 M ⊙ , and ln Λ rlx is the coulomb logarithm (e.g., Binney & Tremaine 2008; Merritt 2013). Figure 1 shows the relaxation timescale as a function of distance from the SMBH in blue for a range of stellar density profiles. We account for relaxation by allowing the orbital eccentricity and semimajor axis of each of our tracer stars to slowly evolve. Once per orbit, we apply a small in- stantaneous change in velocity to each star (e.g., Bradnick et al. 2017; Lu & Naoz 2019; Rose et al. 2022, 2023; Naoz et al. 2022, see the latter for the full set of equations). The kick is calibrated so that ∆ v/v ∼ √ ∆ t/t rlx , and if ∆ t = t rlx , ∆ v is of order of the velocity. This prescription simulates the diffusion of the orbital orbital parameters over time from interactions with other stars in the cluster. Previously, it has been used in studies of TDEs and extreme mass ratio inspirals of stellar mass black holes into the SMBH (Naoz et al. 2022; Melchor et al. 2024).",
"pages": [
5,
6
]
},
{
"title": "3.4. Orbital Stopping Conditions",
"content": "As noted in Section 3.1, we terminate the simulation when the desired runtime of 10 billion years is reached or when the time elapsed has exceeded the star's mainsequence lifetime, whichever comes first. However, orbital changes from collisions or relaxation can also send stars into the tidal radius, where they will be destroyed. We trigger a stopping condition if the star's periapsis a · (1 -e · ) becomes less than twice the tidal radius from the SMBH, R star ( M · /M star ) 1 / 3 (e.g., Guillochon & Ramirez-Ruiz 2013; Mockler et al. 2023). Tidal disruption events can be characterized by impact parameter β = R tidal / ( a · (1 -e · )). Our stopping condition corresponds to β = 0 . 5, allowing us to capture partial as well as full disruptions. Because both direct collisions and relaxation processes can place a star onto a nearly radial orbit, we log whether the critical orbit was reached through a direct collision or our relaxation prescription. A star that becomes a TDE due to relaxation processes can still have experienced one or more collision previously in its life. However, a star that collides and becomes a TDE due to the collision itself may still be inflated from the impact when it reaches its periapsis. In this case, the stopping condition noted above may be conservative; the R star is simply the radius expected using a mass-radius relation for a main-sequence star of mass m star . High-speed collisions, unlike those that result in mergers, can also cause stars to be ejected from the nuclear star cluster. Consider two stars that collide on elliptical orbits that intersect near periapsis. As the eccentricity approaches unity, the speed at periapsis approaches the escape speed, just shy of becoming an unbound, parabolic orbit. In the limit of a hyperbolic encounter, the speeds of the stars are unchanged in the center of mass frame, but they undergo a deflection. In the frame of the SMBH, this change can boost one star's speed enough to unbind it from the SMBH, while the other star ends up on a more tightly bound orbit. Close encounters have previously been shown to eject stars from dense stellar systems (e.g., Henon 1969; Lin & Tremaine 1980). Additionally, high-speed collisions often lead to mass loss, which can unbind the orbit not unlike a supernova (e.g., Lu & Naoz 2019). Therefore, we include a stopping condition for orbital energy ≥ 0 and eccentricity ≥ 1. We do not consider stars that become unbound after losing mass in a tidal disruption, as proposed by Manukian et al. (2013).",
"pages": [
6
]
},
{
"title": "4. NUMERICAL RESULTS WITHOUT RELAXATION",
"content": "We run large sets of 10000 tracer stars and let them evolve over 10 billion years. We begin by presenting simulated results without our relaxation prescription. These simulations allows us to isolate the orbital effects of collisions unobscured by relaxation. As in a nuclear star cluster, most of our tracer stars reside near ∼ 1 pc, the edge of the sphere of influence. About 500 of our tracer stars, or 5% of the sample, lie within 0 . 1 pc, where collisions are most common (see Figure 1). Sampling in this way gives a more complete picture of the relative rates of each outcome. Figure 2 juxtaposes results for slightly different treatments of the collision physics, labeled as follows: Type A: In this simulation, two colliding stars have a 50% chance of orbiting the SMBH in the same direction and a 50% chance of orbiting in opposite directions. The final velocities after a high-speed collision are calculated using Eq. 5. We assume that these high speed collisions do not impact the speed of the stars in their center of mass frame. Type B: Simulations with type B conditions differ from type A by assuming that all stars orbit the SMBH in the same direction. Type C: These simulations are identical to type A, except high speed collisions decrease the speeds in the stars' center of mass frame by 10%. In other words, we scale the velocity vectors of the stars by 0 . 9 before tranforming back to the frame of the SMBH. Type D: These conditions are identical to types A and C except high speed collisions drecrease the speed in the center of mass frame by 50%. The left column of Figure 2 shows the final masses and semimajor axes of the tracer stars. Red open circles represent stars at the end of the simulation or their mainsequence lifetime, whichever came first. Grey points represent the initial conditions. Blue diamonds indicate stars that were placed on unbound orbits by high speed collisions. Note that unbound orbits have negative semimajor axes, so we plot the absolute value. We provide the total number of stars that escape on unbound orbits in the upper right corner of each plot. In addition to these unbound orbits, collisions can also place stars on orbits with semimajor axes outside the inner pc. TDEs from direct collisions are represented by the star shaped symbols. The color indicates the type of collision responsible for the radial orbit: low-speed collisions, which we refer to as merger TDEs, are shown in black, while lime represents high speed collisions. Some of these orbits are so extreme that the periapsis distance lies within the Schwarzschild radius. We mark these plunging stars using an 'x' symbol. The right column shows the properties of the TDEs and plunges for each simulation. The y-axis shows the orbital period of the final orbit, which carries them into the tidal radius. On the x-axis, we plot the parameter β , which quantifies how deeply the orbit penetrates the tidal radius. 0 . 5 < β < 1 indicate a partial disruption. To guide the eye, grey dashed line marks where the orbital period equals 30 yr. TDEs with periods less than this could conceivably be observed as a repeating TDE. We discuss the implications for TDEs below.",
"pages": [
6,
8
]
},
{
"title": "4.1. Tidal Disruption Events",
"content": "Collision-induced mergers can result in TDEs under conditions A, C, and D, but not B. These merged stars are placed on nearly radial orbits when the colliding stars are orbiting in opposite directions. Their angular momenta vector are anti-aligned, giving a low angular momentum to the final orbit of the merger product. Under type B collision conditions, in contrast, there are no merger TDEs. The orbital angular momenta are never in a position to cancel each other out. Fewer TDEs result from high speed collisions. The relative abundances reflect the relative frequency of each type of collision. High speed collisions tend to occur near the SMBH where speeds are sufficiently high, but fewer stars reside. These high speed TDEs are relatively rarer because this type of collision is relatively rarer. Collisions in general tend to become common within the inner 0 . 1 pc of the cluster. However, only about 500 stars from our sample lie in this region. We find that when we sample this region in greater detail, both high and low speed collision TDEs become more common. One high speed collision even produced a repeating TDE (see additional simulations in Section A.2). Others were close to our repeating TDE threshold with 30-50 yr periods. There are some initial conditions that consistently fail to produce high speed collision TDEs, notably Type D conditions. A shallower density profile will also lead to a lower collision rate overall because the timescale is longer (see Figure 1), in turn reducing the rate of collision TDEs (see Appendix A.2). Additionally, we note that our prescription for determining whether or not a merger results from a collision may over-predict the number of mergers. We use fitting formulae from Rauch (1999) to determine the mass loss in a collision and heuristic arguments, similar to the aforementioned study, to determine if a merger occurs. Previous studies with this code (Rose et al. 2023; Rose & MacLeod 2024) have also tested simulations using fitting formulae from Lai et al. (1993), who include a fitting formula for the merger capture radius. Their formula leads to fewer mergers compared to our first approach. However, collision-induced mergers would still be viable as a mechanism to create TDEs; they would simply occur at a much lower rate. As with other additional simulations, we show examples using the Lai et al. (1993) fitting formulae in the Appendix (Figure 8) to avoid overcrowding the main text.",
"pages": [
8
]
},
{
"title": "4.2. Orbital Changes from Collisions",
"content": "We consider general trends in the effects of collisions on the orbital properties of the stars. We do this by examining sub-populations of the tracer stars based on their final mass. We stress that this classification does not necessarily give the full picture of a star's collision history. While all stars with M final > 1 M ⊙ must have undergone a merger - recall that all stars in are cluster are initially 1 M ⊙ - and stars with M final < 1 M ⊙ must have lost mass in a high speed collision, these two populations are not mutually exclusive. A single star can experience both types of collisions over its lifetime depending on where it is in its orbit and the orbit of the second colliding star. However, final mass still represents the best 'observable' probe of a star's collision history. We compare the semimajor axis versus 1 -e of the stars in Figure 3, the latter being proportional to the periapsis. Grey dots show the initial conditions for all the stars. Green circles represent stars with M final > 1 M ⊙ , while orange dots represent stars with M final < 1 M ⊙ . TDEs and unbound stars are represented using the same symbols as in previous figures. We confirm that the TDEs in our simulation do indeed come from nearly radial orbits. Mergers and high speed collisions differ in terms of their orbital outcomes. Conservation of energy demands that mergers always shrink the semimajor axis, while high speed collisions can place stars on both wider or smaller orbits by facilitating an energy exchange between the two stars. These trends are visible in the Figure. The green open circles are mostly confined to the inner cluster, while orange dots have an extended distribution. A few merged stars lie outside our initial cluster. These merged stars experienced a high speed collision that deflected them onto wider orbits. In simulations A and B, a roughly equal number of stripped stars moved to more (less) tightly bound orbits. We count unbound stars in the less tightly bound category because their final orbital energies are larger than the initial values. Additionally, about 10 stars are placed on orbits with semimajor axes outside of 1 pc. This result suggests that there may be collision-affected stripped stars masquerading outside of the sphere of influence. Including a treatment for dissipation during a high speed collision, simulation C presents a different story. The orbits of stripped stars are more likely to shrink and become more circular. About 75% become more tightly bound. Additionally, a population of short period stripped stars emerges. These stars likely experienced multiple high speed collisions, giving them multiple opportunities to shrink their semimajor axes. The effects are even more pronounced for simulation D (not shown). Under type D conditions, stripped stars are overwhelmingly on less eccentric orbits, explaining why the high speed collision TDE rate for these conditions are so low. Eccentricity trends can also be gleaned from Figure 3. For type B conditions, mergers always act to make the orbit more circular. In addition to energy, angular momentum must be conserved. For stars orbiting in the same direction, their angular momentum vectors are always aligned. Since the semimajor axis shrinks after a collision, the eccentricity must decrease to conserve angular momentum. This effect can be seen in vertical extent of the green circles in B compared to A. The most eccentric orbits for merged stars in B have been largely removed. We reiterate that the results are somewhat muddied by the occurrence of high speed collisions in a subset of this population. Under type A conditions, low speed collisions both decrease and increase the orbital eccentricity. The most eccentric final orbits become TDEs or plunges, removing the stars from the population. We note that in actuality, the orbital properties of the stars can be modified by resonant and non-resonant relaxation processes, not accounted for in these results (e.g., Rauch & Tremaine 1996; Hopman & Alexander 2006; Kocsis & Tremaine 2011).",
"pages": [
8,
9
]
},
{
"title": "4.3. Unbound Stars",
"content": "High speed collisions can unbind stars from the SMBH. These interactions, treated similarly to a hyperbolic encounter, can place one star on a more tightly bound orbit while giving the other star a positive orbital energy. We show properties of the unbound stars from simulations A and B in Figure 4. The initial orbits tend to be eccentric. We also confirmed by inspection of specific cases that the collisions tend to occur near periapsis. For the most part, the final orbits have eccentricity just above unity. We calculate the speed at infinity, v inf , for these stars based on their final orbital energy and color code the points in Figure 4 based on its value. Typical speeds range from ∼ 100 to 600 km/s, but occasionally one star will have a v inf above 1000 km/s, as was the case in simulation B. For simulation C, which includes some form of dissipation during high speed collisions, there were fewer unbound stars overall, but their distribution was similar. The maximum v inf was 642 km/s. The single unbound star from the fourth row of Figure 2 (simulation D) had v inf equal to 377 km/s. These maximum values suggest that high speed collisions may represent another mechanism to launch hypervelocity stars from galactic nuclei. The most famous of these mechanisms is the Hills Mechanism, in which a binary is disrupted by the SMBH such that one star is ejected at high speed while the other is retained on a tightly bound orbit (e.g., Hills 1988; Generozov & Madigan 2020), though other mechanisms exist (e.g., Yu & Tremaine 2003; Perets 2009a). Close encounters between single stars are known to eject stars from dense stellar clusters (Henon 1969; Lin & Tremaine 1980), and in fact Yu & Tremaine (2003) consider such interactions as a means of producing hypervelocity stars. Omitting collisions, they find a low ejection rate at speeds ≳ 300 km/s. We consider collisions exclusively and treat them as modified close encounters. While our largest speed is consistent with those generated by the Hill's Mechanism ( ≳ 1000 km/s, Hills 1988), about 30% have v inf > 300 km/s. Observations of hypervelocity stars with origins pointing to the Galactic center exhibit a range of speeds, from the hundreds to thousands of km/s (e.g., Brown et al. 2005, 2018; Koposov et al. 2020; Generozov & Perets 2022, to quote from the latter, stars need ejection speed ≳ 750 km/s from the Galactic center to have 275 km/s at 20 kpc). The v inf of our stars shown in Figure 4 do not account for any additional terms in the potential beyond the SMBH. We reserve a detailed comparison of our results to observations for future work. The mass loss fitting formulae from Rauch (1999) give very low mass loss for larger impact parameter collisions ( b ≳ 0 . 6 r c ) at moderately high speeds ( ∼ 1000 km/s); generally, the fractional mass loss in these cases is less than a percent, and unbound stars in Figure 2 appear as if they have not lost any mass at all. However, fitting formulae from high speed collisions can be unreliable (Freitag & Benz 2005). We therefore refrain from commenting on the masses of the unbound stars in detail except to state that some of them may look like stripped stars (Gibson et al. 2024). As can be seen in Figure 2, a maximum of ∼ 1% of the stars in the inner parsec region may become unbound from the SMBH. This number far exceeds those that become TDEs through high speed collisions. Furthermore, this outcome represents a surprisingly high fraction of stars that experience such collisions: about a fifth of stars that experience high speed collisions become unbound. This high number may owe to a few conspiring conditions: For reasons discussed in Section 3.4 and as supported by Figure 4, unbound stars tend come from high speed collisions near the periapsis of eccentric orbits. Near periapsis, a small boost in the frame of the SMBH can tip the star's speed over the escape speed. However, the merger-to-high speed collision boundary also lies around 800 km/s. The velocity dispersion only exceeds this value within about 0 . 02 pc, and the vast majority of stars reside outside this distance. In consequence, most high speed collisions can only occur near periapsis, where speeds are high enough to no longer be in the merger regime and also where unbinding the star from the SMBH becomes a more favorable outcome. Our thermal initial eccentricity distribution ensures that plenty of stars begin on very eccentric orbits (see Appendix A.1 for the impact of the eccentricity distribution). Dissipation during the collision can reduce the number of unbound stars. As discussed in Section 3.2.2, it is possible that high speed direct collisions reduce the speeds of the stars in their center of mass frame. The physical impact of the collision may reduce the number of stars that get enough of a boost to escape the inner parsec. Type C and D collision conditions test the limits of this mechanism in producing unbound stars. Notably, if high speed collisions reduce the post-collision speed by 50% in the center of mass frame, only a few tenths of a percent of the cluster stars escape.",
"pages": [
9,
10
]
},
{
"title": "5. NUMERICAL RESULTS WITH RELAXATION",
"content": "Now that we have built a physical picture of how collisions shape the stellar cluster, we turn on our prescription for two-body relaxation in the code. These simulations take longer to run, so the sample size of the tracer stars presented herein number at 4000 instead of 10000. We present results in Figure 5. The symbols are all the same as in previous figures, with one addition: stars that became TDEs through two body relaxation are shown in dark green. The orbital evolution of these stars where still affected by collisions, as can be seen by their masses. Interestingly, the stripped star that became a TDE via relaxation had experienced a collision within the same timestep. The collision had placed it on",
"pages": [
10
]
},
{
"title": "Unbound Orbits Properties",
"content": "an eccentric orbit just shy of the tidal radius, periapsis of 1 . 135 AU versus 0 . 956 AU needed for partial disruption. Had the star been inflated at all from the collision, material would have been siphoned off by the SMBH. As the star's parameters did not trigger the stopping condition, our relaxation prescription tipped the star into the tidal radius. We place a caveat on the TDE inside 0 . 001 pc. The closest known star to the Milky Way's SMBH has a periapsis just under 0 . 001 pc (e.g., Gillessen et al. 2017). It is questionable whether our two-body relaxation prescription applies within this radius. However, this star too experienced a collision in the previous timestep which, had the star been inflated, would have led to a disruption. Altogether, our results suggest that collisions represent a viable mechanism to preferentially deposit stars that are (1) more massive than the surrounding population due to a merger or (2) stripped stars into the tidal radius. Some of these TDEs could be observed as repeating TDEs. Additionally, we may be missing disruptions from stars that are inflated post collision, leading to qualitatively different electromagnetic signatures (e.g., MacLeod et al. 2013; Gibson et al. 2024). The unbound stars present something of a puzzle. Even accounting for the difference in sample size, we find much fewer compared to Figure 2 (no relaxation). There are a few possible explanations. First, the relaxation prescription changes the orbital energy by roughly order of itself over the relaxation timescale, which is less than 10 Gyr in the model cluster. Stars at larger radii can therefore wander towards the SMBH, where destructive collisions become possible. Once in this 'danger zone', they can experience enough high-speed collisions to become destroyed. Roughly 500 stars were destroyed over this simulation. While this process is slow (see e.g., Rose & MacLeod 2024), it does drain stars from the sample population. In the simulation without relaxation, the majority of stars began and remained outside of this region. Only ∼ 50 stars were destroyed, a much smaller fraction of the sample, and most unbound stars had initial semimajor axes outside of 0 . 1 pc (see Appendix A.4). The reduction of unbound stars may also be related to the interplay between the two dynamical processes. We speculate on why this second option might be true. Collisions are discrete events. Relaxation, in contrast, is a diffusion process that slowly alters a star's orbital parameters. This timescale is less than the collision timescale for most of the stars in our sample (see Figure 1), particularly where the unbound stars originate in our earlier simulations. The implication is that when a collision does occur, it may be less likely to catch the orbit in a state favorable for unbinding the star. The last possibility is that there is something in our relaxation prescription which we have yet to detect that is artificially suppressing the unbound stars. 1 Regardless, the interplay of relaxation and collisions merit a more thorough examination. We reserve a detailed study for future work.",
"pages": [
11
]
},
{
"title": "6. CONCLUSIONS",
"content": "Collisions between main-sequence stars occur in the nuclear star cluster and become common within the inner 0 . 1 pc (e.g., Rose & MacLeod 2024). In this proof-ofconcept study, we examine the orbital changes that result from these collisions. Specifically, we assess whether physical collisions can contribute to the production of TDEs and ejected stars. Collisions in galactic nuclei can be understood in two types. In the first type, the relative speed between the stars is low and the collision results in a merger. Little mass loss occurs (e.g., Lai et al. 1993) and we can treat the stars as sticky spheres (e.g., Kremer et al. 2020; Gonz'alez et al. 2021). Conservation of momentum allows us to calculate the final orbit of the merged star. The second type of collision occurs at speeds that exceed the escape velocity from the stars. While the high speeds ensure that the stars remain unbound from each other, they can also drive mass loss from the stars, in some cases producing a stripped star (e.g., Lai et al. 1993; Rauch 1999; Freitag & Benz 2005; Gibson et al. 2024). We use fitting formulae from Rauch (1999) to calculate the mass loss. However, we refrain from commenting in detail on the final masses of these stars because fitting formulae are not always accurate for collisions in galactic nuclei (Freitag & Benz 2005). We are examining these collisions further using sph simulations in forthcoming work (Rose et al. in prep.). We use limiting cases and heuristic arguments to determine the final orbits from high speed collisions. If the stars were point masses, these interactions would unfold as a hyperbolic encounter in the center of mass frame of the stars. As an upper limit, we could calculate the deflection angle using Eq. 4. However, while a grazing collision approaches the limit given by Eq. 4, a head-on collision should not result in any deflection at all. We therefore adopt Eq. 5, which meets both criteria. We also test the role of dissipation by reducing the speeds of the stars post-collision in their center of mass frame. Our simulations follow a sample of tracer stars embedded in a fixed, uniform cluster. We test a variety of cluster conditions and find the following: Despite these changes, the unbound stars persist, albeit in much lower numbers. We reserve a precise examination of their rates and properties for future work, though some may look like stripped stars (Gibson et al. 2024). In the optimistic case, stellar collisions represent a mechanism to launch hypervelocity stars from galactic nuclei, joining the list of existing mechanisms (e.g., Hills 1988; Yu & Tremaine 2003; Ginsburg & Loeb 2007; Perets 2009a,b; Generozov & Madigan 2020). The Hills Mechanism elegantly explains both the origins of the S-star cluster, young-seeming massive stars in the vicinity of the SMBH, and hypervelocity stars using the same dynamical process (e.g., Hills 1988; Ghez et al. 2003; Ginsburg & Loeb 2007; Perets et al. 2007; Madigan et al. 2009; Lockmann et al. 2009; Generozov & Madigan 2020; Generozov 2021). Collisions present another possibility, where S-stars are the high-mass tail of the merger products created by successive low-speed collisions (see e.g., Rose et al. 2023), and hypervelocity stars trace to high speed collisions near periapsis. However, the viability of this ejection mechanism still faces tests in the form of dissipation during collisions, a stellar mass function, interplay with other dynamical processes such as relaxation, and overall rates in a three-dimensional cluster. We thank Fred Rasio, Enrico Ramirez-Ruiz, Jamie Lombardi, Fulya Kiroglu, Charles Gibson, and Claire Ye for invaluable discussion and input. SR thanks the CIERA Lindheimer Fellowship for support. BM thanks the Carnegie CTAC postdoctoral fellowship for support. This project began while SCR and BM were at the Aspen Center for Physics, which is supported by NSF grant PHY-2210452. This research was supported in part through the computational resources and staff contributions provided for the Quest high performance computing facility at Northwestern University which is jointly supported by the Office of the Provost, the Office for Research, and Northwestern University Information Technology.",
"pages": [
11,
12,
13
]
},
{
"title": "A.1. Role of the Initial Eccentricity Distribution",
"content": "Less eccentric orbits on average should result in fewer unbound stars. We test this hypothesis by drawing the eccentricities of tracer stars from a uniform distribution (average e = 0 . 5) instead of a thermal one (average e = 0 . 67). We compare the results in the first row of Figure 2, which uses a thermal initial eccentricity distribution, to that shown in Figure 6. Despite both having type A initial conditions, changing the eccentricity distribution to uniform reduces the number of unbound stars by about a factor of 2. There were also fewer high speed collisions overall, a ∼ 35% reduction. These results come from reducing the number of stars with speeds at periapsis that place them in the high speed collision regime.",
"pages": [
13
]
},
{
"title": "A.2. Sampling the Inner Collision-dominated Region",
"content": "Most collisions occur in the inner 0 . 1 parsec of the nuclear star cluster, where the collision timescale becomes shorter than 10 Gyr. In order to probe this region in greater detail, we run a similar set of simulations to those shown in Figure 2, but with the stars sampled from 0 . 001 to 0 . 1 pc. We note that a cusp with slope α = 1 . 75 results in a very high density in this region. The actual cusp in the Galactic center may be closer to 1 . 4 (Gallego-Cano et al. 2018b). Additionally, collisions themselves can deplete the cusp in the inner 0 . 1 pc. A shallower cusp would lead to a lower collision rate. We therefore test α = 1 . 25 with the idea that reality may lie somewhere in the middle. These simulations are marked with an asterisk in Figure 7. The shallower profile means that the collision timescale is longer and that fewer collisions occur within the cluster over the integration time. However, as can be seen in the Figure, the collisions that do occur can stil produce TDEs. Additionally, type B conditions, which prohibit merger TDEs, can still have collision-driven TDEs of a different kind.",
"pages": [
13
]
},
{
"title": "A.3. Alternative Fitting Formulae",
"content": "As mentioned in Section 3.1, we leverage fitting formulae and intuitive toy models to calculate the mass loss from a collision and determine whether a merger occurs. Simulations shown in the main text and in Figure 7 use fitting formulae from Rauch (1999) and heuristic arguments to determine if a merger occurs. In this section, we present simulations with type A collision conditions that use fitting formulae from Lai et al. (1993). Lai et al. (1993) include fitting formula for the capture radius, a function of the relative speed between the stars and their impact parameter. This formula leads to less mergers than the simple prescription that we use in the main text of this paper, which is also similar to the prescription used in Rauch (1999) (see for a full discussion Rose et al. 2023). We present the results of two simulations using the Lai et al. (1993) formulae in Figure 8. The upper row samples tracer stars on a cusp from 0 . 001 to 1 pc, while the second row focuses on the inner 0 . 1 pc. As expected, with the overall reduction in mergers - they instead are treated as high speed collisions - there are fewer merger TDEs. However, it is still possible to get TDEs of all types and unbound stars.",
"pages": [
13,
14
]
},
{
"title": "A.4. Orbital Parameters",
"content": "We expand upon Section 4.2 in this appendix. We compare the final semimajor axes versus the initial semimajor axes of the stars in Figure 9. As is our convention in the main text, green circles represent stars with M final > 1 M ⊙ , while orange dots have M final < 1 M ⊙ . TDEs and unbound stars are represented using stars and diamonds. The black dashed line shows where the initial and final semimajor axes are equal. Mergers work to shrink the semimajor axis, while high speed collisions can increase or decrease it, as is visible in the Figure. For the most part, green circles lie below the black dashed lines. A few lie above the black line, mostly within 0 . 1 pc, because they also experienced a high speed collision. They initially resided within ∼ 0 . 1 pc, where high speed collisions become more likely. We examine the final eccentricities for merger-affected stars in Figure 10. We compare the final eccentricity distribution to the initial thermal distribution. For reasons outlined in Section 4.2, type B mergers circularize orbits. This effect can be seen in Figure 10, where there is a dearth of eccentric orbits. The effects on eccentricity in type A conditions are harder to tease out. 50% of the time the orbits are aligned and the eccentricity decreases. The other 50% of the time, the angular momenta vector are anti-aligned, so even though the semimajor axis still shrinks, the eccentricity can increase. The most eccentric merged stars are removed from the population because they become TDEs or plunges. The result is an eccentricity distribution that flattens at high ( e > 0 . 5) eccentricities. However, the actual number of merged star orbits that became more (less) eccentric are similar.",
"pages": [
14
]
},
{
"title": "REFERENCES",
"content": "Alexander, T. 1999, ApJ, 527, 835, doi: 10.1086/308129 Alexander, T., & Pfuhl, O. 2014, ApJ, 780, 148, doi: 10.1088/0004-637X/780/2/148 Bahcall, J. N., & Wolf, R. A. 1976, ApJ, 209, 214, doi: 10.1086/154711 Balberg, S. 2024, ApJ, 962, 150, doi: 10.3847/1538-4357/ad1690",
"pages": [
14
]
},
{
"title": "Sampling Tracer Stars from the Inner 0.1 pc",
"content": "*uses a = 1.25 instead of 1.75 Balberg, S., Sari, R., & Loeb, A. 2013, MNRAS, 434, L26, doi: 10.1093/mnrasl/slt071 Bartko, H., Martins, F., Fritz, T. K., et al. 2009, ApJ, 697, 1741, doi: 10.1088/0004-637X/697/2/1741 Binney, J., & Tremaine, S. 2008, Galactic Dynamics: Second Edition Bradnick, B., Mandel, I., & Levin, Y. 2017, MNRAS, 469, 2042, doi: 10.1093/mnras/stx1007 Brown, W. R., Geller, M. J., Kenyon, S. J., & Kurtz, M. J. 2005, ApJL, 622, L33, doi: 10.1086/429378 Brown, W. R., Lattanzi, M. G., Kenyon, S. J., & Geller, M. J. 2018, ApJ, 866, 39, doi: 10.3847/1538-4357/aadb8e Brutman, Y., Steinberg, E., & Balberg, S. 2024, arXiv e-prints, arXiv:2408.16383, doi: 10.48550/arXiv.2408.16383 Dale, J. E., & Davies, M. B. 2006, MNRAS, 366, 1424, doi: 10.1111/j.1365-2966.2005.09937.x Dale, J. E., Davies, M. B., Church, R. P., & Freitag, M. 2009, MNRAS, 393, 1016, doi: 10.1111/j.1365-2966.2008.14254.x Dessart, L., Ryu, T., Amaro Seoane, P., & Taylor, A. M. 2024, A&A, 682, A58, doi: 10.1051/0004-6361/202348228 Ferrarese, L., & Ford, H. 2005, SSRv, 116, 523, doi: 10.1007/s11214-005-3947-6 Freitag, M., & Benz, W. 2002, A&A, 394, 345, doi: 10.1051/0004-6361:20021142 -. 2005, MNRAS, 358, 1133, doi: 10.1111/j.1365-2966.2005.08770.x Gallego-Cano, E., Sch¨odel, R., Dong, H., et al. 2018a, A&A, 609, A26, doi: 10.1051/0004-6361/201730451 -. 2018b, A&A, 609, A26, doi: 10.1051/0004-6361/201730451 Gallegos-Garcia, M., Law-Smith, J., & Ramirez-Ruiz, E. 2018, ApJ, 857, 109, doi: 10.3847/1538-4357/aab5b8 Generozov, A. 2021, MNRAS, 501, 3088, doi: 10.1093/mnras/staa3851 Generozov, A., & Madigan, A.-M. 2020, ApJ, 896, 137, doi: 10.3847/1538-4357/ab94bc Generozov, A., & Perets, H. B. 2022, MNRAS, 513, 4257, doi: 10.1093/mnras/stac1108 Genzel, R., Eisenhauer, F., & Gillessen, S. 2010, Reviews of Modern Physics, 82, 3121, doi: 10.1103/RevModPhys.82.3121 Ghez, A. M., Salim, S., Hornstein, S. D., et al. 2005, ApJ, 620, 744, doi: 10.1086/427175 Ghez, A. M., Duchˆene, G., Matthews, K., et al. 2003, ApJL, 586, L127, doi: 10.1086/374804 Ghez, A. M., Salim, S., Weinberg, N. N., et al. 2008, ApJ, 689, 1044, doi: 10.1086/592738 Gibson, C., Kıro˘glu, F., Lombardi, J. C., J., et al. 2024, arXiv e-prints, arXiv:2410.02146, doi: 10.48550/arXiv.2410.02146 Gillessen, S., Eisenhauer, F., Trippe, S., et al. 2009, ApJ, 692, 1075, doi: 10.1088/0004-637X/692/2/1075 Gillessen, S., Plewa, P. M., Eisenhauer, F., et al. 2017, ApJ, 837, 30, doi: 10.3847/1538-4357/aa5c41",
"pages": [
15,
17
]
}
]
2024arXiv241201150D
https://arxiv.org/pdf/2412.01150.pdf
<document>
<text><location><page_1><loc_14><loc_95><loc_21><loc_96></location>, 1-23 (2024)</text>
<section_header_level_1><location><page_1><loc_7><loc_86><loc_88><loc_90></location>Representation Learning for Time-Domain High-Energy Astrophysics: Discovery of Extragalactic Fast X-ray Transient XRT 200515</section_header_level_1>
<text><location><page_1><loc_7><loc_81><loc_78><loc_84></location>Steven Dillmann 1 , 2 ★ † , Rafael Martínez-Galarza 3 , Roberto Soria 4 , 5 , Rosanne Di Stefano 3 and Vinay L. Kashyap 3</text>
<unordered_list>
<list_item><location><page_1><loc_7><loc_79><loc_65><loc_80></location>1 Stanford University, Institute for Computational and Mathematical Engineering, Stanford, CA 94305, USA</list_item>
<list_item><location><page_1><loc_7><loc_78><loc_62><loc_79></location>2 University of Cambridge, Department of Physics, Cavendish Laboratory, Cambridge, CB3 0HE, UK</list_item>
<list_item><location><page_1><loc_7><loc_77><loc_51><loc_78></location>3 Center for Astrophysics | Harvard & Smithsonian, Cambridge, MA 02138, USA</list_item>
<list_item><location><page_1><loc_7><loc_75><loc_59><loc_76></location>4 INAF-Osservatorio Astrofisico di Torino, Strada Osservatorio 20, I-10025 Pino Torinese, Italy</list_item>
<list_item><location><page_1><loc_7><loc_74><loc_67><loc_75></location>5 Sydney Institute for Astronomy, School of Physics A28, The University of Sydney, Sydney, NSW 2006, Australia</list_item>
</unordered_list>
<text><location><page_1><loc_7><loc_70><loc_36><loc_71></location>Accepted XXX. Received YYY; in original form ZZZ</text>
<section_header_level_1><location><page_1><loc_7><loc_66><loc_15><loc_67></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_7><loc_43><loc_92><loc_65></location>We present a novel representation learning method for downstream tasks such as anomaly detection and unsupervised transient classification in high-energy datasets. This approach enabled the discovery of a new fast X-ray transient (FXT) in the Chandra archive, XRT 200515, a needle-in-the-haystack event and the first Chandra FXT of its kind. Recent serendipitous breakthroughs in X-ray astronomy, including FXTs from binary neutron star mergers and an extragalactic planetary transit candidate, highlight the need for systematic transient searches in X-ray archives. We introduce new event file representations, 𝐸 -𝑡 Maps and 𝐸 -𝑡 -𝑑𝑡 Cubes, designed to capture both temporal and spectral information, effectively addressing the challenges posed by variable-length event file time series in machine learning applications. Our pipeline extracts low-dimensional, informative features from these representations using principal component analysis or sparse autoencoders, followed by clustering in the embedding space with DBSCAN. New transients are identified within transient-dominant clusters or through nearest-neighbor searches around known transients, producing a catalog of 3,539 candidates (3,427 flares and 112 dips). XRT 200515 exhibits unique temporal and spectral variability, including an intense, hard < 10 s initial burst followed by spectral softening in an ∼ 800 s oscillating tail. We interpret XRT 200515 as either the first giant magnetar flare observed at low X-ray energies or the first extragalactic Type I X-ray burst from a faint LMXB in the LMC. Our method extends to datasets from other observatories such as XMM-Newton , Swift-XRT , eROSITA , Einstein Probe , and upcoming missions like AXIS .</text>
<text><location><page_1><loc_7><loc_40><loc_92><loc_42></location>Key words: software: machine learning, methods: data analysis, X-rays: bursts, stars: magnetars, transients: gamma-ray bursts, stars: peculiar</text>
<section_header_level_1><location><page_1><loc_7><loc_33><loc_21><loc_34></location>1 INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_7><loc_12><loc_48><loc_32></location>Recent serendipitous discoveries, such as extragalactic fast X-ray transients (FXTs) linked to neutron star merger candidates as electromagnetic counterparts to gravitational wave events (Lin et al. 2022) and an X-ray dip associated with the first extragalactic planet candidate (Di Stefano et al. 2021), underscore the challenges of identifying such rare events within large X-ray catalogs. Beyond magnetarpowered FXTs as the aftermath of binary neutron star mergers (Dai et al. 2006; Metzger et al. 2008; Zhang 2013; Sun et al. 2017; Bauer et al. 2017; Xue et al. 2019), other interesting origins of extragalactic FXTs include supernova shock breakouts (SBOs) (Soderberg et al. 2008; Modjaz et al. 2009; Alp & Larsson 2020; Novara et al. 2020), tidal disruption events (TDEs) (Jonker et al. 2013) including quasiperiodic eruptions (QPEs) (Arcodia et al. 2021; Chakraborty et al. 2021), thermonuclear (Type I) X-ray bursts from accreting neutron stars (in't Zand et al. 2013), or binary self-lensing events (D'Orazio</text>
<unordered_list>
<list_item><location><page_1><loc_7><loc_8><loc_25><loc_8></location>★ E-mail: stevendi@stanford.edu</list_item>
<list_item><location><page_1><loc_7><loc_6><loc_44><loc_7></location>† Present address: 450 Jane Stanford Way, Stanford, CA 94305, USA</list_item>
</unordered_list>
<text><location><page_1><loc_51><loc_7><loc_92><loc_34></location>&DiStefano 2018, 2020; Hu et al. 2020). Both because of their very stochastic nature, and because narrow field X-ray missions such as the Chandra X-ray Observatory ( Chandra ) (Weisskopf et al. 2000), XMM-Newton (Jansen et al. 2001) and Swift-XRT (Burrows et al. 2005) are not designed as wide time-domain surveys, X-ray transient discoveries are often serendipitous. They can be found in observations that were originally proposed for a completely unrelated science objective and are rarely the target of the observation. In many cases serendipitously found X-ray sources do not get characterized or classified, since their transient nature is not immediately obvious. Instead, observations with X-ray transients often get stored in large data archives and remain unnoticed. This raises the need for a systematic search for short-duration phenomena in high-energy catalogs. New missions such as eROSITA (Predehl et al. 2021), Einstein Probe (Yuan et al. 2022) and the upcoming AXIS Observatory (Reynolds et al. 2024) target X-ray transients more directly, thus the development of novel transient detection methods is becoming even more relevant. The temporary, unpredictable and 'unusual' nature of X-ray transients distinguishes them from 'normal' X-ray source emissions. From a data science perspective, they can be understood</text>
<text><location><page_2><loc_7><loc_64><loc_48><loc_93></location>as 'anomalies' within a large dataset. Existing methods for identifying X-ray transients primarily rely on statistical tests of variability (Yang et al. 2019; Pastor-Marazuela et al. 2020; Quirola-Vásquez et al. 2022; Quirola-Vásquez et al. 2023). While effective within specific constraints, these approaches are inherently limited by their underlying assumptions, which may not capture the diverse nature of transient phenomena. In contrast, machine learning offers a more flexible, expressive, and scalable framework, making it particularly well-suited for anomaly detection in large, high-dimensional datasets with diverse transient types. While optical time-domain surveys are at the forefront of leveraging extensive observational programs, like ZTF (Bellm et al. 2019) or the upcoming LSST survey (Ivezić et al. 2019), and neural network-based anomaly detection tools to identify rare sources among countless ordinary objects (Villar et al. 2021; Muthukrishna et al. 2022), the X-ray astronomy community has only recently begun exploring the potential of machine learning to classify sources (Yang et al. 2022; Pérez-Díaz et al. 2024) or to search for needle-in-a-haystack events in large X-ray datasets and archives (Kovačević et al. 2022; Dillmann & Martínez-Galarza 2023). The effectiveness of machine learning methods largely depends on the algorithm's ability to learn useful representations from the data.</text>
<text><location><page_2><loc_7><loc_50><loc_48><loc_63></location>Representation learning (Bengio et al. 2013) is an increasingly popular technique in astronomy used in supervised, semi-supervised, self-supervised and unsupervised frameworks (Naul et al. 2018; Hayat et al. 2021; Walmsley et al. 2022; Slijepcevic et al. 2024; Mohale & Lochner 2024). It involves creating or learning meaningful representations for specific modalities of scientific data, which can then be used for downstream tasks such as regression, classification, or, as in this work, anomaly detection. The compressed representations live in a low-dimensional embedding space, in which anomalous data samples are well-separated from more ordinary ones.</text>
<text><location><page_2><loc_7><loc_20><loc_48><loc_49></location>We propose a new unsupervised representation learning method to perform a large-scale search for X-ray transients in the Chandra archive. High-energy catalogs include individual X-ray source observations in the form of event files. The variable length of these time series poses a challenge in creating consistent representations suitable for transient searches with machine learning. Most deep learning algorithms take a fixed-length input for all data samples. In order to effectively represent event files over a broad range of lengths, we introduce novel fixed-length event file representations, which take into account both their time-domain and energy-domain information. Applying feature extraction and dimensionality reduction techniques, for example with sparse autoencoders, we create a representation space that encodes scientifically meaningful information, such as the spectral and variability properties of the astrophysical sources. Previously identified X-ray transients occupy distinct, well-isolated clusters in the embedding space. Using clustering techniques and nearest neighbor searches allows us to effectively explore these transient-dominant clusters to discover new X-ray transients. We collect the identified X-ray flare and dip candidates in a publicly available catalog, serving as a fertile ground for new discoveries in time-domain high-energy astrophysics.</text>
<text><location><page_2><loc_7><loc_6><loc_48><loc_19></location>Among these candidates, we identify an intriguing extragalactic FXT, XRT 200515, which exhibits unique temporal and spectral characteristics distinct from any previously reported Chandra FXTs. The transient's initial hard < 10 s burst shows a sharp rise exceeding 4 orders of magnitude, followed by spectral softening in an ∼ 800 s oscillating tail. This transient is likely related to either a giant magnetar flare (GMF) from a distant soft gamma repeater (SGR) behind the Large Magellanic Cloud (LMC) or an extragalactic Type I X-ray burst from a faint LMXB in the LMC. Each of these interpretations presents its own set of challenges. Alternatively, XRT 200515 could</text>
<text><location><page_2><loc_51><loc_91><loc_92><loc_93></location>be a new type of astronomical phenomenon found by our anomaly detection method using machine learning.</text>
<text><location><page_2><loc_51><loc_72><loc_92><loc_90></location>Our method is the first representation learning approach for anomaly detection in high-energy astrophysics. It is applicable to datasets from high-energy catalogs like Chandra , XMM-Newton , Swift-XRT , eROSITA , and Einstein Probe . We created semantically meaningful representations that can be aligned with other data modalities, such as optical images or infrared spectra to design multi-modal models (Parker et al. 2024; Mishra-Sharma et al. 2024; Zhang et al. 2024; Rizhko & Bloom 2024) using contrastive learning (Radford et al. 2021), that can improve on current state-of-the-art algorithms used to characterize the physics of the associated objects. Ultimately, this work and other representation and contrastive learning approaches lay the groundwork for developing generalized foundation model in astronomy.</text>
<text><location><page_2><loc_51><loc_58><loc_92><loc_72></location>The paper is organized as follows: In § 2, we provide information on the dataset of Chandra event files used in this analysis. In § 3, we describe in detail the implementation of our novel transient detection approach leveraging representation learning. In § 4, we present and discuss the results in form of the semantically meaningful representation space of the event files, the catalog of X-ray transient candidates and the discovery of the new Chandra transient XRT 200515. Finally, we highlight our contributions to time-domain high-energy astrophysics and outline potential directions for extending this work in the future in § 5.</text>
<text><location><page_2><loc_51><loc_53><loc_92><loc_58></location>The relevant code, a demonstration of the pipeline, and an interactive embedding selection, transient search and lightcurve plotting tool are available online at the GitHub repository https://github.com/StevenDillmann/ml-xraytransients-mnras.</text>
<section_header_level_1><location><page_2><loc_51><loc_48><loc_60><loc_49></location>2 DATASET</section_header_level_1>
<text><location><page_2><loc_51><loc_19><loc_92><loc_47></location>We use data from the Chandra Source Catalog (CSC) version 2.1 (Evans et al. 2024), which includes all publicly available X-ray sources detected by Chandra as of December 2021. For this study, we focus specifically on observations from the Advanced CCD Imaging Spectrometer (ACIS). CSC 2.1 had not been fully released at the time our analysis was performed, but catalog data was available for sources that had completed processing in the Current Database View 1 , a snapshot of which we took on 11 April 2023. CSC 2.1 performs source detection on stacked observations, and catalog properties are provided both for these stack-level detections, and for each of observation-level detection that contribute to a stack detection. Because we are interested in short-time variability that happens within a single observation of a source, we use the catalog products for the observation-level detections in our analysis. For a given X-ray detection, two types of products are provided in the CSC: (i) database tables with source properties, such as fluxes in the different X-ray energy bands, hardness ratios, variability indices, etc., and (ii) filebased data products for each detection of a source, such as the detect regions, the Chandra PSF at that location, etc. The following observation-level catalog properties are relevant for our analysis:</text>
<unordered_list>
<list_item><location><page_2><loc_51><loc_13><loc_92><loc_18></location>· var_prob_b : The probability that a source detection is variable in time for the broad energy band (0.5-7 keV), as estimated using the Gregory-Loredo algorithm (Gregory & Loredo 1992). In this paper we call this quantity 𝑝 𝑏 var .</list_item>
<list_item><location><page_2><loc_51><loc_10><loc_92><loc_12></location>· var_index_b : The variability index in the broad band, which indicates the level of confidence for time variability. A variability</list_item>
</unordered_list>
<text><location><page_3><loc_7><loc_90><loc_48><loc_93></location>index of 6 or larger indicates variability at a confidence of at least 2 𝜎 . In this paper we call this quantity 𝐼 𝑏 var .</text>
<unordered_list>
<list_item><location><page_3><loc_7><loc_79><loc_48><loc_90></location>· hard_<hs/ms/hm> : The hardness ratios, which quantify the relative fraction of photons detected in two given bands chosen between the soft (0.5-1.2 keV), medium (1.2-2 keV), and hard (27 keV) bands for a source detection. For example, a value of the hard-to-soft hardness ratio close to 1 indicates that most of the photons detected are in the hard energy band, whereas a value close to -1 indicates that most photons are detected in the soft band. In this paper we call these quantities 𝐻𝑅 hs , 𝐻𝑅 ms , and 𝐻𝑅 hm .</list_item>
</unordered_list>
<text><location><page_3><loc_7><loc_36><loc_48><loc_78></location>From the catalog data products available for observation-level Xray detections, we are interested in the region event file. This event file consists of a list of all individual photon events detected in a small bounding box around a source detection, listing their energies, arrival times, and detector coordinates. These event files are the basis for the characterization of an X-ray source: lightcurves, spectra, images, coordinates, and other properties are derived from the distribution of the listed quantities. In this analysis, we directly use these event files as our primary data products. The values of the catalog properties listed above serve as summary statistics for the detection associated with a given region event file. We only include event files with more than 5 events and a signal-to-noise ratio above 5 to minimize spurious signals from low number statistics in faint sources. We also exclude detections that are flagged for pile-up 2 , i.e., those with a pileup fraction larger than 5%, which corresponds to a maximum pileup warning of 0.1 in CSC 2.1. For the resulting detections, we filter the event files to include only events contained within the detection region for each source. These detection regions are also provided as data products in CSC 2.1, and consist of the ellipse that includes the 90% encircled counts fraction of the PSF at the source location. Due to the low background level in Chandra observations, the majority of events selected after this spatial filtering are expected to be events associated with the X-ray source, not the background. In the selected event files, we only include photon events within good time intervals (GTIs), which are time periods of valid, high-quality data. No other pre-processing is required. The final dataset consists of 95,473 filtered event files from 58,932 sources, resulting in an average of 1 . 62 observations per source. This includes 9,003 new sources that have been added as part of the CSC 2.1 release, in addition to the sources from the previous release.</text>
<section_header_level_1><location><page_3><loc_7><loc_31><loc_16><loc_32></location>3 METHODS</section_header_level_1>
<text><location><page_3><loc_7><loc_15><loc_48><loc_29></location>In this work, we introduce a novel representation learning based anomaly detection method to systematically search for X-ray transients in high-energy archives. We begin with an overview of the method here and provide detailed explanations of each step in individual subsections. The full pipeline is illustrated in Figure 1. Starting with the event files described in § 2, we (i) build two novel and uniform event file representations by binning their arrival times and energies into 𝐸 -𝑡 Maps (Event File Representation I) or 𝐸 -𝑡 -𝑑𝑡 Cubes (Event File Representation II); (ii) use principal component analysis (Feature Extraction I) or sparse autoencoders (Feature Extraction II) to extract informative features from the event</text>
<table>
<location><page_3><loc_51><loc_81><loc_92><loc_89></location>
<caption>Table 1. Naming of the different embedding result cases based on the event file representation and feature extraction method.</caption>
</table>
<text><location><page_3><loc_51><loc_52><loc_92><loc_78></location>file representations; (iii) apply dimensionality reduction to the extracted features to create a low-dimensional embedding space; (iv) use density-based clustering to create embedding clusters that group event files with similar characteristics, for example transient behavior or certain spectral features. Previously identified transients like the extragalactic magnetar-powered flare candidate reported by Lin et al. (2022) and the extragalactic planet candidate dip reported by Di Stefano et al. (2021), shown in Figure 2, occupy well-isolated clusters in the embedding space. Exploring these clusters and conducting nearest-neighbor searches enables us to effectively find analogs to bona-fide time-domain anomalies, while at the same time grouping them according to their spectral properties. We compile the identified transient candidates in a catalog. While our approach is designed and tested using Chandra data, it is applicable to any dataset consisting of event lists, like those from other high-energy telescopes. The described transient detection approach is applied to both types of event file representations with both feature extraction methods, resulting in four different embeddings. We denote the different cases as described in Table 1.</text>
<section_header_level_1><location><page_3><loc_51><loc_47><loc_71><loc_48></location>3.1 Event File Representation</section_header_level_1>
<text><location><page_3><loc_51><loc_29><loc_92><loc_46></location>The different event files in the dataset are variable in length 𝑁 and duration 𝑇 , as shown in Appendix A. The large variation in the number of events and duration highlights the challenge in producing uniform data representations that preserve relevant information on time variability and spectral properties. While there exist machine learning architectures that take variable length inputs, the significant differences in the number of events from object to object make standardization of the inputs challenging, even when these architectures are used (Martínez-Galarza & Makinen 2022). As a first step in our analysis, we introduce 2-dimensional and 3-dimensional fixed-length representations based on an informed binning strategy for the event files, similar to the DMDT maps for optical lightcurves introduced by Mahabal et al. (2017).</text>
<section_header_level_1><location><page_3><loc_51><loc_24><loc_81><loc_25></location>3.1.1 2D Histogram Representation ( 𝐸 -𝑡 Maps)</section_header_level_1>
<text><location><page_3><loc_51><loc_6><loc_92><loc_23></location>Assume an event file with 𝑁 photons and a photon arrival time column 𝒕 with entries { 𝑡 𝑘 } 𝑁 𝑘 = 1 and energy column 𝑬 with entries { 𝐸 𝑘 } 𝑁 𝑘 = 1 . The event file duration is given by 𝑇 = 𝑡 𝑁 -𝑡 1 . The energy column entries take values in the broad energy band of Chandra 's ACIS instrument, i.e. 𝐸 𝑘 ∈ [ 𝐸 𝑚𝑖𝑛 , 𝐸 𝑚𝑎𝑥 ] , where 𝐸 𝑚𝑖𝑛 = 0 . 5 keV and 𝐸 𝑚𝑎𝑥 = 7 keV comes from considering appropriate boundaries for the energy response of Chandra 's ACIS instrument. Beyond these boundaries, the telescope's aperture effective area is low for the majority of detected sources. First, we obtain the normalized time column, given by 𝝉 = 𝒕 -𝑡 1 𝑇 , and the logarithm of the energy column, given by 𝝐 = log 𝑬 . The resulting boundaries for normalized time column are 𝝉 ∈ [ 𝜏 𝑚𝑖𝑛 , 𝜏 𝑚𝑎𝑥 ] , where 𝜏 𝑚𝑖𝑛 = 0 and 𝜏 𝑚𝑎𝑥 = 1.</text>
<figure>
<location><page_4><loc_8><loc_44><loc_91><loc_93></location>
<caption>Figure 1. Flowchart of the proposed representation learning approach for anomaly detection in time-domain high-energy astrophysics, enabling the systematic detection of transients in high-energy archives. The first step is to create uniform event file representations by binning photon arrival times and energies in the event files into into 𝐸 -𝑡 Maps (Event File Representation I) or 𝐸 -𝑡 -𝑑𝑡 Cubes (Event File Representation II). The second step involves extracting informative features from these event file representations via principal component analysis (Feature Extraction I) or sparse autoencoders (Feature Extraction II). The third step is to apply dimensionality reduction to the extracted features and to create a low-dimensional embedding space, which is clustered in the fourth step using density-based clustering. Previously identified transients occupy well-isolated clusters on the edges of the embedding space, thus new transients can be identified by exploring these edge clusters and performing nearest-neighbor searches around known bona-fide flares and dips. Finally, we compile these search results in a publicly available catalog of transient candidates, serving as a fertile ground for the discovery of new X-ray transients.</caption>
</figure>
<figure>
<location><page_4><loc_11><loc_11><loc_88><loc_29></location>
<caption>Figure 2. Left panel: Lightcurve of the first extragalactic planet candidate dip reported by Di Stefano et al. (2021) detected in the Chandra observation ObsID 13814. Right panel: Lightcurve of the magnetar-powered X-ray flare candidate reported by Lin et al. (2022) detected in the Chandra observation ObsID 4062.</caption>
</figure>
<figure>
<location><page_5><loc_7><loc_81><loc_92><loc_93></location>
<caption>Figure 3. The distribution of the optimal number of bins for the energy dimension 𝑛 𝜖 (left), time dimension 𝑛 𝜏 (middle), time difference dimension 𝑛 𝛿𝜏 (right). The distribution of 𝑛 𝜏 only includes event files for which 𝑝 𝑏 var > 0 . 9. The vertical lines indicate the number of bins chosen for the event file representations.</caption>
</figure>
<text><location><page_5><loc_7><loc_70><loc_48><loc_72></location>The range for the log-energy column is 𝝐 ∈ [ 𝜖 𝑚𝑖𝑛 , 𝜖 𝑚𝑎𝑥 ] , where 𝜖 𝑚𝑖𝑛 = log 0 . 5 keV and 𝜖 𝑚𝑎𝑥 = log 7 keV.</text>
<text><location><page_5><loc_7><loc_60><loc_48><loc_69></location>Next, we determine the dimensionality of our representations. For a each event file, we determine the optimal number of bins in the energy dimension, 𝑛 𝜖 , with the Freedman-Diaconis rule (Freedman & Diaconis 1981), a widely used method that balances the trade-off between too noisy histograms (too many bins) and not informative enough histograms (too few bins). The optimal bin width 𝑏 𝜖 according to this rule is calculated in the following way:</text>
<formula><location><page_5><loc_9><loc_56><loc_48><loc_59></location>𝑏 𝜖 = 2 𝐼𝑄𝑅 ( 𝝐 ) 𝑁 1 3 , (1)</formula>
<text><location><page_5><loc_7><loc_52><loc_48><loc_55></location>where 𝐼𝑄𝑅 ( 𝜖 ) represents the interquartile range of the 𝜖 values for a given event file of length 𝑁 . Subsequently, we obtain the optimal number of energy bins 𝑛 𝜖 with:</text>
<formula><location><page_5><loc_7><loc_48><loc_48><loc_51></location>𝑛 𝜖 = 𝜖 𝑚𝑎𝑥 -𝜖 𝑚𝑖𝑛 𝑏 𝜖 . (2)</formula>
<text><location><page_5><loc_7><loc_28><loc_48><loc_47></location>For each event file, we determine the optimal number of bins in the time dimension, 𝑛 𝜏 , with the help of the Bayesian Blocks algorithm, which was specifically developed for time series analysis in astronomy (Scargle et al. 2013). This algorithm partitions the time series into adaptive width bins or blocks that are statistically distinct from neighboring blocks; that is, within a given time-ordered Bayesian block, events grouped in that block are consistent with having a similar event arrival rate. We use the default Astropy implementation of Bayesian blocks, and set the false alarm probability parameter to 𝑝 0 = 0 . 01 (Astropy Collaboration et al. 2013), which implies a 1% probability of declaring a change of rate when there is none. For each event file, we define the optimal uniform bin width 𝑏 𝜏 as the minimum bin width calculated by the Bayesian Blocks algorithm, and then find the optimal number of time bins 𝑛 𝜏 with:</text>
<formula><location><page_5><loc_7><loc_25><loc_48><loc_27></location>𝑛 𝜏 = 𝜏 𝑚𝑎𝑥 -𝜏 𝑚𝑖𝑛 𝑏 𝜏 . (3)</formula>
<text><location><page_5><loc_7><loc_15><loc_48><loc_24></location>The optimal number of bins is different for each event file, due to their different lengths 𝑁 and durations 𝑇 . To select a bin size that can be applied to all event files, we consider the distributions of these optimal bin sizes, which are shown in Figure 3. For the distribution of 𝑛 𝜏 values we only use those event files for which 𝑝 𝑏 var > 0 . 9. The intent of this is to effectively capture variability timescales that are associated with short time-domain events, such as flares and dips.</text>
<text><location><page_5><loc_7><loc_6><loc_48><loc_14></location>We choose the 90th percentile value of each distribution to set the final number of bins in each dimension. That is, only 10% of the event files will have an optimal number of bins that is larger than the chosen values 𝑛 𝜖 = 16 and 𝑛 𝜏 = 24. The choice of the 90th percentile, rather than the mean or mode, is motivated by the need to capture sufficient statistical detail even for long event files, while</text>
<text><location><page_5><loc_51><loc_65><loc_92><loc_72></location>keeping the size of the resulting representations computationally tractable. Choosing a lower resolution would risk losing significant details in the representation, particularly short-duration events such as flares and dips within longer event files. The 𝐸 -𝑡 Maps are the 2D histogram representations with size ( 𝑛 𝜏 , 𝑛 𝜖 ) = ( 24 , 16 ) that result from binning the events according to the optimized number of bins.</text>
<text><location><page_5><loc_51><loc_60><loc_92><loc_64></location>Figure 4 shows the 𝐸 -𝑡 Maps for the known extragalactic dip reported by Di Stefano et al. (2021) and known extragalactic flare reported by Lin et al. (2022).</text>
<section_header_level_1><location><page_5><loc_51><loc_57><loc_85><loc_58></location>3.1.2 3D Histogram Representation ( 𝐸 -𝑡 -𝑑𝑡 Cubes)</section_header_level_1>
<text><location><page_5><loc_51><loc_45><loc_92><loc_55></location>Wenowintroduce the 𝐸 -𝑡 -𝑑𝑡 Cubes, which extend the 𝐸 -𝑡 Maps by a third dimension that serves as a proxy for the photon arrival rate. For an event file of length 𝑁 , consider the array of time differences between consecutive photon arrivals 𝚫 𝒕 with entries Δ 𝑡 𝑘 = 𝑡 𝑘 + 1 -𝑡 𝑘 for 𝑘 = 1 , 2 , . . . , 𝑁 -1. We again scale and normalize the obtained values, so that they adopt values between 0 and 1, using in each case the minimum value Δ 𝑡 𝑚𝑖𝑛 and maximum value Δ 𝑡 𝑚𝑎𝑥 . This provides the third dimension 𝜹𝝉 :</text>
<formula><location><page_5><loc_51><loc_41><loc_92><loc_44></location>𝜹𝝉 = 𝚫 𝒕 -Δ 𝑡 𝑚𝑖𝑛 Δ 𝑡 𝑚𝑎𝑥 -Δ 𝑡 𝑚𝑖𝑛 . (4)</formula>
<text><location><page_5><loc_51><loc_30><loc_92><loc_40></location>The additional dimension is intended to better isolate short-duration features in time variability by capturing high photon arrival rates, which are typical of flares, as well as very low photon arrival rates, which are typical of dips. The boundaries of our histogram representations in this dimension are 𝜹𝝉 ∈ [ 𝛿𝜏 𝑚𝑖𝑛 , 𝛿𝜏 𝑚𝑎𝑥 ] , where 𝛿𝜏 𝑚𝑖𝑛 = 0 and 𝛿𝜏 𝑚𝑎𝑥 = 1. We determine the optimal number of bins in the 𝜹𝝉 dimension, 𝑛 𝛿𝜏 , again by computing the optimal bin width 𝑏 𝛿𝜏 with the Freedman-Diaconis rule and dividing the range for 𝜹𝝉 by 𝑏 𝛿𝜏 :</text>
<formula><location><page_5><loc_51><loc_26><loc_92><loc_29></location>𝑏 𝛿𝜏 = 2 𝐼𝑄𝑅 ( 𝜹𝝉 ) 𝑁 1 3 , (5)</formula>
<formula><location><page_5><loc_51><loc_22><loc_92><loc_25></location>𝑛 𝛿𝜏 = 𝛿𝜏 𝑚𝑎𝑥 -𝛿𝜏 𝑚𝑖𝑛 𝑏 𝛿𝜏 . (6)</formula>
<text><location><page_5><loc_51><loc_15><loc_92><loc_21></location>The distribution of 𝑛 𝛿𝜏 across the event files is shown in Figure 3. Most of the relevant time-domain information is already captured by 𝝉 , but adding 𝜹𝝉 provides an additional marker for dips and flares that can be shorter than the timescales probed by our chosen binning of 𝝉 .</text>
<text><location><page_5><loc_51><loc_6><loc_92><loc_14></location>Unlike in the other two dimensions, we choose the 75th percentile value of the distribution as our final choice of common binning, which results in 𝑛 𝛿𝜏 = 16. This is because in order to identify short transients, we need to capture strong deviations in 𝜹𝝉 only. Choosing a lower value for 𝑛 𝛿𝜏 reduces noise an improves computational tractability. Having both 𝝉 and 𝜹𝝉 represented also breaks</text>
<figure>
<location><page_6><loc_7><loc_79><loc_27><loc_93></location>
<caption>Figure 4 shows the 𝐸 -𝑡 -𝑑𝑡 Cubes for the known extragalactic dip reported by Di Stefano et al. (2021) and known extragalactic flare reported by Lin et al. (2022).</caption>
</figure>
<figure>
<location><page_6><loc_28><loc_79><loc_48><loc_93></location>
</figure>
<figure>
<location><page_6><loc_7><loc_64><loc_26><loc_77></location>
</figure>
<figure>
<location><page_6><loc_28><loc_64><loc_47><loc_77></location>
<caption>Figure 4. Upper panel: 2D histogram representations ( 𝐸 -𝑡 Maps) for the extragalactic dip in (Di Stefano et al. 2021) (left) and extragalactic flare in (Lin et al. 2022) (right). Lower panel: 3D histogram representations ( 𝐸 -𝑡 -𝑑𝑡 Cubes) for the same events. Darker bins indicate higher counts, while lighter bins indicate lower counts.</caption>
</figure>
<text><location><page_6><loc_7><loc_44><loc_48><loc_52></location>any assumption of stationarity, in that we can be sensitive to transient events happening at any time during the observation of the source, and break degeneracies between periodic and non-periodic features in the representations presented by Martínez-Galarza & Makinen (2022). The 𝐸 -𝑡 -𝑑𝑡 Cubes are the resulting 3D histogram event file representations with size ( 𝑛 𝜏 , 𝑛 𝜖 , 𝑛 𝛿𝜏 ) = ( 24 , 16 , 16 ) .</text>
<section_header_level_1><location><page_6><loc_7><loc_36><loc_21><loc_37></location>3.1.3 Feature Notation</section_header_level_1>
<text><location><page_6><loc_7><loc_25><loc_48><loc_34></location>The event file representations can now be used as inputs for various statistical learning and machine learning algorithms. For the 𝑖 𝑡 ℎ event file in the dataset of length 𝑚 = 95,473, we denote the corresponding feature vector as fi 𝑥 𝑖 = [ 𝑥 1 , 𝑥 2 , . . . , 𝑥 𝑛 ] 𝑖 , where 𝑛 = 𝑛 𝜏 · 𝑛 𝜖 = 384 for the 𝐸 -𝑡 Maps and 𝑛 = 𝑛 𝜏 · 𝑛 𝜖 · 𝑛 𝛿𝜏 = 6 , 144 for the 𝐸 -𝑡 -𝑑𝑡 Cubes. The set of all feature vectors is denoted as X = [fi 𝑥 1 , fi 𝑥 2 , . . . , fi 𝑥 𝑚 ] ⊤ with size ( 𝑚, 𝑛 ) .</text>
<section_header_level_1><location><page_6><loc_7><loc_21><loc_43><loc_22></location>3.2 Feature Extraction I: Principal Component Analysis</section_header_level_1>
<text><location><page_6><loc_7><loc_6><loc_48><loc_20></location>We use Principal Component Analysis (PCA) (Pearson 1901) provided by scikit-learn (Pedregosa et al. 2011) as our first feature extraction method. The extracted principal components should encode relevant time-domain and spectral information of the event file they represent. PCA involves transforming a dataset into a new coordinate system by finding the principal components of the data that capture most of the variance in the data. By projecting the dataset onto principal components, PCA reduces the dimensionality of the data while retaining the most important information, which increases the interpretability of high-dimensional data.</text>
<figure>
<location><page_6><loc_52><loc_59><loc_91><loc_93></location>
<caption>Figure 5. Scree plot for the principal components of the 𝐸 -𝑡 Maps (top) and 𝐸 -𝑡 -𝑑𝑡 Cubes (bottom). The scree plots show the amount of variance explained by each individual principal component including the knee point in the cumulative variance.</caption>
</figure>
<section_header_level_1><location><page_6><loc_51><loc_48><loc_64><loc_49></location>3.2.1 PCA Algorithm</section_header_level_1>
<text><location><page_6><loc_51><loc_39><loc_92><loc_47></location>We start with the feature vector set X of size ( 𝑚, 𝑛 ) representing our dataset with 𝑚 samples and 𝑛 dimensions. PCA aims to find a new coordinate system defined by a set of orthogonal axes, i.e. the principal components, that captures the maximum amount of variance in the data. The PCA result is a transformed dataset Xpc obtained by projecting X onto the principal components:</text>
<formula><location><page_6><loc_51><loc_37><loc_92><loc_38></location>Xpc = XW , (7)</formula>
<text><location><page_6><loc_51><loc_31><loc_92><loc_36></location>where W is matrix of size ( 𝑛, 𝑛 𝑝𝑐 ) containing the first 𝑛 𝑝𝑐 principal components to be retained as its columns and Xpc is of size ( 𝑚 , 𝑛 𝑝𝑐 ) with a reduced dimensionality of 𝑛 𝑝𝑐 . For a more detailed explanation of the algorithm, we refer the reader to Jolliffe (2002).</text>
<section_header_level_1><location><page_6><loc_51><loc_27><loc_74><loc_28></location>3.2.2 Principal Components Retained</section_header_level_1>
<text><location><page_6><loc_51><loc_6><loc_92><loc_25></location>The main PCA hyperparameter is the number of principal components 𝑛 𝑝𝑐 to retain. Figure 5 shows two scree plots illustrating the amount of variance explained by each principal component in descending order and the cumulative proportion of variance explained by the principal components for both 𝐸 -𝑡 Mapsand 𝐸 -𝑡 -𝑑𝑡 Cubes. Acommon approach to determine the optimal value of 𝑛 𝑝𝑐 is to find the knee point in the cumulative scree plot of the principal components. This balances the objective of minimizing the dimensionality while retaining as much information as possible. Defining the knee point as the point beyond which adding additional principal components increases the amount of variance by less than 0 . 1% gives 𝑛 𝑝𝑐 = 15 for 𝐸 -𝑡 Maps and 𝑛 𝑝𝑐 = 22 for 𝐸 -𝑡 -𝑑𝑡 Cubes as indicated in Figure 5. These capture 94 . 1% and 89 . 9% of the variance respectively.</text>
<figure>
<location><page_7><loc_8><loc_80><loc_48><loc_93></location>
</figure>
<figure>
<location><page_7><loc_8><loc_61><loc_48><loc_78></location>
<caption>Figure 6. Encoder architecture of the convolutional autoencoder used for the 𝐸 -𝑡 Maps (top) and of the fully connected autoencoder used for the 𝐸 -𝑡 -𝑑𝑡 Cubes (bottom). The decoder architecture is simply a mirror image of the encoder.</caption>
</figure>
<section_header_level_1><location><page_7><loc_7><loc_46><loc_48><loc_47></location>3.3 Feature Extraction II: Sparse Autoencoder Neural Network</section_header_level_1>
<text><location><page_7><loc_7><loc_8><loc_48><loc_45></location>As an alternative to PCA, we now build Autoencoder (Hinton & Salakhutdinov 2006) models with TensorFlow (Abadi et al. 2015) to learn a set of latent features from the 𝐸 -𝑡 Maps and 𝐸 -𝑡 -𝑑𝑡 Cubes that can be used to isolate transients and encode specific spectral properties. An autoencoder is composed of two neural networks, an encoder and a decoder, which work together to learn a compressed representation of the input data. The encoder network takes the input data and maps it to a lower-dimensional representation, often called 'latent space' or 'bottleneck'. The number of neurons in the bottleneck determines the dimensionality of the learned representation. The decoder network then aims to reconstruct the original input from this compressed representation. The decoder is typically a mirrored version of the encoder gradually upsampling the latent space until the output matches the dimensions of the original input. By minimizing the reconstruction error between input and output during training, the model learns a low-dimensional representation of the input. The bottleneck forces the encoder to capture the most important information necessary for accurate reconstruction, effectively compressing the input and learning to extract informative features in an unsupervised manner. Once the autoencoder is trained, the encoder network can be used as a standalone feature extractor to obtain a compressed representation of the input data, which can be used for downstream tasks such as clustering or anomaly detection. As opposed to PCA, which is a linear technique that works well for linearly correlated data but fails to capture complex non-linear relationships, an autoencoder is able to learn complex non-linear relationships. We design two different autoencoders to process the 𝐸 -𝑡 Maps and 𝐸 -𝑡 -𝑑𝑡 Cubes.</text>
<table>
<location><page_7><loc_52><loc_72><loc_90><loc_87></location>
<caption>Table 2. Summary of the encoder architecture of the convolutional autoencoder used to extract informative features from the 𝐸 -𝑡 Maps. Note that each layer has a Leaky ReLU activation function and that each standard convolutional layer is followed by batch normalization with momentum 0.9.</caption>
</table>
<section_header_level_1><location><page_7><loc_51><loc_68><loc_72><loc_69></location>3.3.1 Convolutional Autoencoder</section_header_level_1>
<text><location><page_7><loc_51><loc_33><loc_92><loc_67></location>In a convolutional autoencoder (Masci et al. 2011), both the encoder and decoder network consist of convolutional layers (LeCun et al. 1998), which perform convolutions over the input using a filter. These filters are small matrix kernels with learnable weights that slide across the input, allowing the network to capture high-level features while preserving important spatial hierarchies and relationships, which is why they are often used for image-like data. This makes this architecture particularly well-suited to recognize spatial patterns such as dips or flares in our 𝐸 -𝑡 Maps. To gradually reduce the dimension of the input while it is being passed through the encoder network, we use stride convolution layers (Simonyan & Zisserman 2014) with a stride value of 2 for downsampling. This means that the learnable filter jumps two pixels at a time as it slides over the input. The output of the convolutional layers is a feature map, which is then flattened to a feature vector and passed through a series of fully connected layers, where every neuron in the previous layer is connected to every neuron in the next layer. These fully connected layers are responsible for mapping the learned features to a lower-dimensional latent representation in the bottleneck and perform non-linear transformations while downsampling through the use of non-linear activation functions. The final latent space has 𝑛 𝑎𝑒 = 12 elements, representing the most essential features of the input data, which can now be used for further downstream tasks. Figure 6 shows a diagram of the encoder part of the model and Table 2 summarizes its architecture.</text>
<section_header_level_1><location><page_7><loc_51><loc_29><loc_73><loc_30></location>3.3.2 Fully Connected Autoencoder</section_header_level_1>
<text><location><page_7><loc_51><loc_14><loc_92><loc_27></location>Our 𝐸 -𝑡 -𝑑𝑡 Cubes introduce an additional dimension resulting in sparse 3D input data. Convolutional layers assume regular grid-like data, making them less effective for handling sparse data. Moreover, very expensive 3D convolutional operations would substantially increase complexity of the model. Therefore, we use a simple fully connected autoencoder for the 𝐸 -𝑡 -𝑑𝑡 Cubes. Its encoder network consists of a series of fully connected layers, which gradually map the original input data to a latent space with 𝑛 𝑎𝑒 = 24 elements. Figure 6 shows a diagram of the encoder part of the model and Table 3 summarizes its architecture.</text>
<section_header_level_1><location><page_7><loc_51><loc_10><loc_67><loc_11></location>3.3.3 Activation Functions</section_header_level_1>
<text><location><page_7><loc_51><loc_6><loc_92><loc_8></location>Neural networks are able to learn and represent complex non-linear relationships due to the introduction of non-linear activation func-</text>
<table>
<location><page_8><loc_16><loc_76><loc_38><loc_86></location>
<caption>Table 3. Summary of the encoder architecture of the fully connected autoencoder used to extract informative features from the 𝐸 -𝑡 -𝑑𝑡 Cubes. Note that each layer has a Leaky ReLU activation function and that each standard fully-connected layer is followed by batch normalization with momentum 0.9.</caption>
</table>
<text><location><page_8><loc_7><loc_59><loc_48><loc_73></location>tions within their layers. An activation function is a mathematical function used in a neural network to determine whether a neuron should be activated or not, based on its input. It essentially decides how much of the input signal should pass through the neuron, producing an output that can either be passed to the next layer or used to make predictions. The popular Rectified Linear Unit (ReLU) activation function 𝑅𝑒𝐿𝑈 ( 𝑥 ) = max ( 0 , 𝑥 ) (Nair & Hinton 2010) is simple and computationally efficient. To mitigate any potential encounters of the 'dying the ReLU problem', where neurons become non-responsive during training, we choose an extended version called Leaky ReLU (Maas et al. 2013):</text>
<formula><location><page_8><loc_7><loc_56><loc_48><loc_57></location>𝐿𝑒𝑎𝑘𝑦𝑅𝑒𝐿𝑈 ( 𝑥 ) = max ( 𝛼𝑥, 𝑥 ) , (8)</formula>
<text><location><page_8><loc_7><loc_46><loc_48><loc_55></location>where 𝛼 = 0 . 1 is a hyperparameter that defines the slope of the function for negative input values. ReLU sets all negative values in the input to zero, while Leaky ReLU allows a small negative slope for negative inputs, which can help prevent neurons from dying. As for the output layer, we want any values to be mapped to a range between 0 and 1, which is achieved by using the sigmoid activation function:</text>
<formula><location><page_8><loc_7><loc_42><loc_48><loc_45></location>𝑠𝑖𝑔𝑚𝑜𝑖𝑑 ( 𝑥 ) = 1 1 + 𝑒 -𝑥 . (9)</formula>
<section_header_level_1><location><page_8><loc_7><loc_39><loc_37><loc_40></location>3.3.4 Loss Function and Sparsity Regularization</section_header_level_1>
<text><location><page_8><loc_7><loc_34><loc_48><loc_37></location>In order to encourage the autoencoder to generate reconstructions close to the original inputs, we use the mean squared error ( 𝑀𝑆𝐸 ) as as a measure of the reconstruction quality given by:</text>
<formula><location><page_8><loc_7><loc_29><loc_48><loc_33></location>𝑀𝑆𝐸 = 1 𝑚 𝑚 ∑︁ 𝑖 = 1 ( 𝑥 𝑖 -ˆ 𝑥 𝑖 ) 2 , (10)</formula>
<text><location><page_8><loc_7><loc_22><loc_48><loc_28></location>where 𝑥 𝑖 is the 𝑖 𝑡 ℎ element of the input vector and ˆ 𝑥 𝑖 is the corresponding is reconstructed output. The 𝑀𝑆𝐸 is a straightforward measure of reconstruction error, and its differentiability allows efficient gradient computation for updating model weights via gradient-based optimization.</text>
<text><location><page_8><loc_7><loc_11><loc_48><loc_21></location>Our neural networks are so called sparse autoencoders (Ng et al. 2011), which promote sparsity in the learned representation, meaning only a small subset of the neurons in the network are active at any given time. Sparse representations are valuable for our work because they help extract highly informative features from the input, while disregarding irrelevant or noisy information. To encourage sparsity in the latent space, we introduce a L1 regularization term in the objective, resulting in the following loss function:</text>
<formula><location><page_8><loc_7><loc_6><loc_48><loc_9></location>𝐿 = 𝑀𝑆𝐸 + 𝜆 · 𝑛 𝑤 ∑︁ 𝑗 = 1 | 𝑤 𝑗 | = 1 𝑚 𝑚 ∑︁ 𝑖 = 1 ( 𝑥 𝑖 -ˆ 𝑥 𝑖 ) 2 + 𝜆 · 𝑛 𝑤 ∑︁ 𝑗 = 1 | 𝑤 𝑗 | , (11)</formula>
<text><location><page_8><loc_51><loc_86><loc_92><loc_93></location>where 𝜆 = 0 . 1 is the regularization strength and 𝑤 𝑗 are the individual bottleneck weight values of which there are 𝑛 𝑤 in total. L1 regularization pushes small weights to zero and thus helps the model prioritize the most significant features of the input data, leading to a semantically meaningful latent space.</text>
<section_header_level_1><location><page_8><loc_51><loc_82><loc_60><loc_83></location>3.3.5 Training</section_header_level_1>
<text><location><page_8><loc_51><loc_48><loc_92><loc_81></location>Starting with the original dataset with a 𝑚 = 95,473 samples and using a test split of 0 . 1 gives us a training and validation set of length 85,925 and a test set of length 9,548. Further using a validation split of 0 . 2, gives 68,740 samples for training and 17,185 for validation. We run the training process for a maximum of 200 epochs with a batch size of 1,024 samples. The initial learning rate was set to 0 . 01 along with an on plateau learning rate scheduler, which dynamically reduces the learning rate by a factor of 0 . 1 if the validation loss plateaus for longer than 10 epochs. Reducing the learning rate when a plateau is detected can help escape local minima in the loss surface and converge to a more optimal solution in the parameter space. This scheduler is used in combination with the Adaptive Moment Estimation (Adam) optimizer (Kingma & Ba 2014), which is a stochastic gradient descent algorithm combining the benefits of both adaptive learning rates (Duchi et al. 2011) and momentum-based optimization techniques (Sutskever et al. 2013). Finally, we use an early stopping callback to monitor the validation loss. It automatically interrupts the training process if the validation loss does not improve for 25 epochs and restores the weights of the model to the best observed weights during training. The training process for both autoencoder models is shown in Appendix B. Once the autoencoder is trained, we can use the encoder to transform the original dataset X to the feature vector space Xae of size ( 𝑚 , 𝑛 𝑎𝑒 ) with a reduced dimensionality of 𝑛 𝑎𝑒 features.</text>
<section_header_level_1><location><page_8><loc_51><loc_43><loc_71><loc_44></location>3.4 Dimensionality Reduction</section_header_level_1>
<text><location><page_8><loc_51><loc_24><loc_92><loc_42></location>Using t-SNE (Maaten & Hinton 2008), short for t-Distributed Stochastic Neighbor Embedding, we create two-dimensional embeddings of the informative features previously extracted from the event file representations using PCA or sparse autoencoders. The t-SNE algorithm is a method used to map the input data onto a low-dimensional embedding space, and is particularly useful for the visualization of clusters and patterns in high-dimensional datasets. Each high-dimensional sample is transformed into a low-dimensional embedding in such a way that similar object are nearby points, while dissimilar objects are distant points in the embedding space. Essentially, it aims to capture the local structure of the data by preserving the pairwise similarities between objects while mapping them to a lower-dimensional embedding space.</text>
<section_header_level_1><location><page_8><loc_51><loc_20><loc_61><loc_21></location>3.4.1 Algorithm</section_header_level_1>
<text><location><page_8><loc_51><loc_6><loc_92><loc_18></location>We use our informative features, X if = Xpc or X if = Xae , as input to the t-SNE algorithm to reduce the data to a two-dimensional embedding, denoted as Z . First, t-SNE creates a probability distribution 𝑃 for pairs of high-dimensional data points in X if , assigning higher probabilities to similar pairs and lower probabilities to dissimilar ones. This is done by modeling pairwise similarities using a Gaussian kernel with a specific perplexity parameter, which controls the effective number of neighbors considered for each point. Next, t-SNE defines a similar probability distribution 𝑄 for the pairwise</text>
<table>
<location><page_9><loc_9><loc_82><loc_45><loc_90></location>
<caption>Table 4. Chosen t-SNE hyperparameters for different embedding cases.</caption>
</table>
<text><location><page_9><loc_7><loc_75><loc_48><loc_80></location>similarities in the low-dimensional space Z , modeled using a Student's t-distribution. The goal of t-SNE is to minimize the difference between 𝑃 and 𝑄 using gradient descent, with the Kullback-Leibler (KL) divergence (Kullback & Leibler 1951) as the cost function:</text>
<formula><location><page_9><loc_7><loc_70><loc_48><loc_73></location>𝐷 𝐾𝐿 ( 𝑃 | 𝑄 ) = ∑︁ 𝑖 ≠ 𝑗 𝑃 𝑖 𝑗 log 𝑃 𝑖 𝑗 𝑄 𝑖 𝑗 , (12)</formula>
<text><location><page_9><loc_7><loc_60><loc_48><loc_69></location>where 𝑃 𝑖 𝑗 and 𝑄 𝑖 𝑗 represent pairwise similarities in the high- and low-dimensional spaces, respectively. The algorithm iteratively adjusts the low-dimensional embedding Z to minimize the KL divergence, often requiring hundreds to thousands of iterations for convergence. The result of this optimization is a two-dimensional representation Z of size ( 𝑚, 2 ) , where similar points in the high-dimensional space are clustered closely together.</text>
<section_header_level_1><location><page_9><loc_7><loc_56><loc_29><loc_57></location>3.4.2 Hyperparameter Optimization</section_header_level_1>
<text><location><page_9><loc_7><loc_41><loc_48><loc_54></location>The t-SNE algorithm has a number of important hyperparameters to be tuned. The two most important parameters are the perplexity and the learning_rate . The perplexity parameter controls the balance between capturing the local versus global structure in the data, while the learning_rate controls the step size at each iteration of the optimization process. The n_iter parameter is the number of iterations. To ensure reproducibility, we set a fixed random_state . Our t-SNE hyperparameter optimization approach is detailed in Appendix C. A summary of the final t-SNE hyperparameters is provided in Table 4.</text>
<section_header_level_1><location><page_9><loc_7><loc_37><loc_16><loc_38></location>3.5 Clustering</section_header_level_1>
<text><location><page_9><loc_7><loc_28><loc_48><loc_35></location>The next step is the identification of individual clusters in the embedding space using DBSCAN (Hartigan & Wong 1979), short for Density-Based Spatial Clustering of Applications with Noise. Unlike traditional clustering algorithms such as k-means, DBSCAN does not require the number of clusters to be specified, as it identifies dense regions in the data space based on a density criterion.</text>
<section_header_level_1><location><page_9><loc_7><loc_24><loc_17><loc_25></location>3.5.1 Algorithm</section_header_level_1>
<text><location><page_9><loc_7><loc_6><loc_48><loc_23></location>We use our t-SNE embedding space Z as input to the DBSCAN algorithm, which segments the embedding space into multiple clusters. The DBSCAN algorithm has two main hyperparameters. The eps parameter defines the radius of the neighborhood surrounding each point in the dataset, while the minPts parameter specifies the minimum number of points required within this neighborhood for a data point to be classified as a core point. A border point is defined as a point that is in the vicinity of at least one core point but has fewer than minPts within its neighborhood. All other points are considered to be noise points. Clusters are then created from the aggregation of core points and their associated border points, with noise points being categorized as outliers. Figure 7 visualizes the clustering method.</text>
<figure>
<location><page_9><loc_58><loc_71><loc_85><loc_93></location>
<caption>Figure 7. Illustration of the DBSCAN clustering algorithm, showing core points as densely connected regions, border points along cluster edges, and noise points as outliers. Adapted from Slipski et al. (2024) with permission.</caption>
</figure>
<table>
<location><page_9><loc_54><loc_54><loc_89><loc_60></location>
<caption>Table 5. Chosen DBSCAN hyperparameters for different embedding cases.</caption>
</table>
<section_header_level_1><location><page_9><loc_51><loc_51><loc_73><loc_52></location>3.5.2 Hyperparameter Optimization</section_header_level_1>
<text><location><page_9><loc_51><loc_46><loc_92><loc_49></location>Our DBSCAN hyperparameter optimization approach is detailed in Appendix C. A summary of the final t-SNE hyperparameters is provided in Table 5.</text>
<section_header_level_1><location><page_9><loc_51><loc_42><loc_74><loc_43></location>3.6 Previously Reported Transients</section_header_level_1>
<text><location><page_9><loc_51><loc_22><loc_92><loc_41></location>We highlight the embeddings of previously reported bona-fide transients, listed in Table 6, in our low-dimensional representation space to identify transient-dominant clusters. The flares include extragalactic FXTs reported by Jonker et al. (2013), Glennie et al. (2015), Yang et al. (2019), Lin et al. (2021), Lin et al. (2022), Quirola-Vásquez et al. (2022) and a set of stellar flares found in the dataset by manual inspection. The dips include the extragalactic planet candidate in M51 reported by Di Stefano et al. (2021), the ultraluminous X-ray source (ULX) 2E 1402 . 4+5440 in NGC 5457 (Colbert & Ptak 2002; Swartz et al. 2004) and the well-studied eclipsing and bursting lowmass X-ray binary (LMXB) EXO 0748 -676 (Parmar et al. 1986; D'Aì et al. 2014). These transients occupy well-isolated clusters. Exploring transient-dominant clusters and performing nearest-neighbor searches around known transients allows us to find new transients.</text>
<section_header_level_1><location><page_9><loc_51><loc_18><loc_67><loc_19></location>3.7 Candidate Selection</section_header_level_1>
<text><location><page_9><loc_51><loc_6><loc_92><loc_17></location>Newtransients are identified in embedding clusters containing previously reported transients. For well-isolated clusters containing known discovered transients, we use the entire cluster to define new transient candidates. The well-isolated transient-dominant clusters used for candidate selection are listed in Appendix E. However, in a few cases known discovered transients reside within larger poorly separated clusters. Selecting the entire cluster would result in a high number of false positives. To address this, we instead use the k-nearest</text>
<table>
<location><page_10><loc_11><loc_33><loc_88><loc_90></location>
<caption>Table 6. Previously reported flares and dips used to identify transient-dominant clusters.</caption>
</table>
<text><location><page_10><loc_7><loc_27><loc_48><loc_30></location>neighbors ( kNN ) algorithm (Cover & Hart 1967), identifying the 50 nearest neighbors for each known transient residing in a poorly separated cluster to define additional transient candidates.</text>
<section_header_level_1><location><page_10><loc_7><loc_22><loc_20><loc_23></location>3.8 Cross Matching</section_header_level_1>
<text><location><page_10><loc_7><loc_6><loc_48><loc_21></location>We use an existing cross-match table (Green et al. 2023) between CSC2.1 and five other catalogs - Gaia DR3 (Gaia Collaboration et al. 2021), DESI Legacy Survey DR10 (Dey et al. 2019), PanSTARRS-1 (Chamberset al. 2016), 2MASS (Skrutskie et al. 2006), and the SDSS DR17 catalog - to complement the X-ray properties derived from the CSC with additional multi-wavelength observations. This includes catalog identifiers, positions, magnitudes, source type classifications and other columns. We cross-matched our transient candidates with the SIMBAD database (Wenger et al. 2000) by associating each candidate with the nearest SIMBAD object, provided the object is located within a 5 arcsec radius of the candidate's coordinates listed</text>
<text><location><page_10><loc_51><loc_27><loc_92><loc_30></location>in the CSC. The multi-wavelength observations of the transient candidates provide valuable information for their characterization and classification.</text>
<section_header_level_1><location><page_10><loc_51><loc_22><loc_73><loc_23></location>4 RESULTS AND DISCUSSION</section_header_level_1>
<text><location><page_10><loc_51><loc_18><loc_92><loc_20></location>We now present the results of applying the methods in § 3 to the set of representations of X-ray event files in the dataset from § 2.</text>
<section_header_level_1><location><page_10><loc_51><loc_14><loc_84><loc_15></location>4.1 Representation Embedding Space and Clusters</section_header_level_1>
<text><location><page_10><loc_51><loc_6><loc_92><loc_13></location>Figure 8 shows the t-SNE embedding space for the 3D-PCA and 3DAE cases color-coded by the hardness ratio 𝐻𝑅 hs . The embedding space for the other two cases, 2D-PCA and 2D-AE , are shown in Appendix D. The observed hardness ratio gradients in all embedding spaces indicate that the learned representations effectively encode</text>
<figure>
<location><page_11><loc_9><loc_70><loc_90><loc_93></location>
<caption>Figure 8. Embedding representations color-coded by 𝐻𝑅 hs for the 3D-PCA case (left) and 3D-AE case (right).</caption>
</figure>
<figure>
<location><page_11><loc_9><loc_39><loc_90><loc_65></location>
<caption>Figure 9. Embedding representations color-coded by 𝐼 𝑏 var for 3D-PCA (left) and 3D-AE (right). Known transients and XRT 200515 are highlighted.</caption>
</figure>
<figure>
<location><page_11><loc_9><loc_10><loc_90><loc_34></location>
<caption>Figure 10. Embedding clusters for the 3D-PCA case (left) and 3D-AE case (right).</caption>
</figure>
<table>
<location><page_12><loc_8><loc_81><loc_91><loc_90></location>
<caption>Table 7. The first 5 samples of our transient candidates catalog showing a subset of selected columns.</caption>
</table>
<figure>
<location><page_12><loc_7><loc_11><loc_92><loc_79></location>
<caption>Figure 11. Lightcurves in the 0.5-7 kev energy range for different examples of dips (blue), flares (red) and pulsating or quasi-periodic sources (green) in the transient candidates catalog. The shown pulsating or quasi-periodic sources are part of the flare candidates.</caption>
</figure>
<text><location><page_13><loc_7><loc_81><loc_48><loc_93></location>spectral information, in particular at the level of individual clusters, allowing for the identification of X-ray sources with specific spectral signatures. For the 2D-PCA and 2D-AE cases, these gradients are more uniform across the embedding space, because the temporal and spectral information of event files are captured by one axis each in the 𝐸 -𝑡 Maps. Moreover, some clusters consist exclusively of soft or hard sources, demonstrating that our representations can be leveraged not only to identify transients but also to find analogs to sources with specific spectral characteristics.</text>
<text><location><page_13><loc_7><loc_60><loc_48><loc_80></location>Figure 9 shows the 3D-PCA and 3D-AE embedding spaces, now color-coded by the variability index 𝐼 𝑏 index with the other two cases shown in Appendix D. The learned embeddings also encode the temporal behavior of the sources, with some clusters being dominated by X-ray detections with significant variability, including transient behavior. To demonstrate this, we also highlight the embeddings of the bona-fide flares and dips listed in Table 6. Note that these occupy very well-defined clusters on the edges of the representation space, allowing for queries of analog transient behavior. In the 2D-PCA and 2D-AE cases, transient sources are distributed across multiple small clusters on the edges of the embedding spaces. In contrast, the 3D-PCA and 3D-AE embedding spaces achieve a significantly more compact clustering of bona-fide transients because temporal features in the event files are given a higher importance by the introduction of an additional time-related axis in the 𝐸 -𝑡 -𝑑𝑡 Cubes.</text>
<text><location><page_13><loc_7><loc_28><loc_48><loc_59></location>Figure 10 shows the clusters identified by the DBSCAN algorithm in the 3D-PCA and 3D-AE cases. The clusters for the other two cases, 2D-PCA and 2D-AE , are shown in Appendix D. The largest cluster in all cases (Cluster 1) corresponds to observations that are not 'anomalous', for example non-variable sources or noisy detections in the low-count regime. We also see multiple smaller clusters on the edges of the embedding space clearly separated from this main cluster. Of special interest are clusters that contain known discovered transients, as these likely host other interesting transients that have not yet been discovered. Some of the edge clusters group observations with similar temporal and spectral behavior. For example, Cluster 4 in the 3D-PCA case only contains flares with high hardness ratios. Other clusters instead group observations primarily by similar temporal behavior, but then show a within-cluster grouping of similar spectral behaviors. For example, Cluster 4 in the 3D-AE case contains many dipping sources, but show a hardness ratio gradient within the cluster. When comparing the results of different feature extraction methods, we observe that in the 3D-AE embedding space, nearly all previously identified extragalactic FXTs live within a single, wellisolated cluster (Cluster 8). In contrast, the 3D-PCA embedding space distributes these extragalactic FXTs across multiple clusters. All of these points underline the effectiveness of our method and that the created representation space is highly informative.</text>
<section_header_level_1><location><page_13><loc_7><loc_24><loc_37><loc_25></location>4.2 Catalog of X-ray Flare and Dip Candidates</section_header_level_1>
<text><location><page_13><loc_7><loc_6><loc_48><loc_22></location>We identify new transient candidates within clusters that are occupied by previously reported transients and by conducting nearestneighbor searches around these known transients. We compile these in a catalog of X-ray transient candidates, which includes both flares and dips. Table E1 lists the selected clusters used to define the new flare and dip candidates in addition to the 50 nearest neighbors of each bona-fide transient. From each selected cluster, we only include X-ray detections for which the variability index 𝐼 𝑏 var ≥ 5. This threshold corresponds to detections for which the Gregory-Loredo algorithm yields a confidence in the variability of at least 90%, and allows us to discard flare and dip-like behaviors that are not statistically significant. We also manually exclude a small fraction of</text>
<figure>
<location><page_13><loc_51><loc_58><loc_92><loc_93></location>
<caption>Figure 12. Distribution SIMBAD object types in the dip candidates catalog (blue) and flare candidates catalog (red). There are 11 dip candidates and 897 flare candidates, for which no SIMBAD match is found.</caption>
</figure>
<text><location><page_13><loc_51><loc_6><loc_92><loc_48></location>false positives identified by visual inspection of the lightcurves for both flare and dip candidates. The resulting catalog contains a total of 3539 detections, and the columns included are described in Appendix F. Table 7 shows the first 5 samples in our catalog for a subset of selected columns. Figure 11 shows a number of example lightcurves of the dips and flares in our catalog. The dip selection shows dips from LMXBs, a low-mass X-ray binary (HMXB), an ULX, an eclipsing binary, a cataclysmic binary, and a quasar. The flare selection shows flares from an eruptive variable, a pulsar, an AGN, a HMXB, a cataclysmic variables and young stars. We also find a number of pulsating or quasi-periodic lightcurves from pulsars, magnetic cataclysmic variables and SGRs. Figure 12 shows the distribution of SIMBAD object types in our transient catalog. About 25% of the transient candidates do not have a SIMBAD match, making them particularly interesting sources for new transient discoveries. Our dip candidates include 6 Chandra observations with prominent dips from the known source CXOGlb J002400.9 -720453 in the globular cluster NGC 104 (47 Tuc). The catalog identifiers for these are CATALOG_ID: 2737_139, 16527_79, 15747_79, 16529_79, 15748_79, 16528_14 . Our flare candidates include a newly discovered extragalactic FXT, which is characterized and discussed in detail in § 4.3. Its catalog identifier is CATALOG_ID: 23022_122 . We recommend using our catalog to identify a diverse range of flares and dips. While this work is primarily motivated by the discovery of new extragalactic transients, we intentionally did not exclude galactic stellar flares to enable systematic follow-up studies to study flare incidence rates and the rotational evolution of stars. Users interested exclusively in extragalactic transients can filter out galactic sources using metadata from the CSC and the cross-match columns in the catalog.</text>
<section_header_level_1><location><page_14><loc_7><loc_92><loc_45><loc_93></location>4.3 XRT 200515: A New Extragalactic Fast X-ray Transient</section_header_level_1>
<text><location><page_14><loc_7><loc_69><loc_48><loc_90></location>Among the flare candidates in our catalog, we discovered an intriguing new extragalactic Chandra FXT in an observation of the supernova remnant SNR 0509 -67.5 in the LMC on May 15, 2020 (Guest et al. 2022). What made this transient stand out from thousands of other flares discovered in this work is the unique temporal variability in its lightcurve, which exhibits no detectable pre-flare X-ray emission, a sharp rise of at least 4 orders of magnitude in the count rate to peak intensity followed by a sharp fall, all in a matter of a < 10 s, down to ∼ 800 s long oscillating tail. There is also notable spectral variability during the flare, characterized by an initially hard spectrum at the peak, followed by spectral softening in the tail. The combination of these temporal and spectral properties establishes this transient as the first of its kind within the sample of discovered Chandra FXTs. We designate this newly discovered FXT as XRT 200515 and present a detailed study and discussion of its potential origins.</text>
<section_header_level_1><location><page_14><loc_7><loc_63><loc_28><loc_64></location>4.3.1 X-Ray Detection by Chandra</section_header_level_1>
<text><location><page_14><loc_7><loc_28><loc_48><loc_61></location>The transient XRT 200515 was detected in Chandra ObsID 23022. The target of the observation was the supernova remnant SNR 0509 -67.5 in the LMC, which is shown in Figure 13 alongside the newly discovered FXT event. Table 8 summarizes the properties of XRT 200515 and its associated Chandra source 2CXO J051117.2 -672556 in ObdID 23022. The transient was captured by the ACIS camera in the S4 chip, and is located significantly off-axis in this observation, at an angular distance of 11.75 arcmin from the aimpoint in the S3 chip. This leads to an elongated and relatively large PSF, which, in this case, is advantageous as it substantially reduces photon pile-up in the initial spike, by spreading the counts over many pixels. We processed the data of Chandra observation ObsID 23022 with the Chandra Interactive Analysis of Observations (/c.pc/i.pc/a.pc/o.pc) version 4.15 (Fruscione et al. 2006), with calibration data base version 4.9.8. In particular, we created a new level-2 event file with the /c.pc/i.pc/a.pc/o.pc task chandra_repro and filter it in energy and time with dmcopy . We obtained the sky position in Table 8 using the /c.pc/i.pc/a.pc/o.pc tool wavdetect . To reduce background noise and improve the determination of the source centroid, we applied wavdetect on an image filtered to include only the time interval from the beginning of the flare ( 𝑡 0 ) until a time 𝑡 0 + 920 s. The 90% uncertainty radius of 2.0 arcsec is the combination of the uncertainty in the source centroid position reported by wavdetect , and the absolute astrometry uncertainty in a typical ACIS observation for off-axis sources 3 .</text>
<text><location><page_14><loc_7><loc_13><loc_48><loc_28></location>The field was previously covered by four other Chandra observations (ObsIDs 776, 7635, 8554, and 23023) with no source detections at the location of 2CXO J051117.2 -672556. We estimated modelindependent upper limits to the source flux and luminosity with /c.pc/i.pc/a.pc/o.pc tool srcflux . In the pre-flare part of ObsID 23022, we obtained a 90% confidence limit of 𝐿 X < 1 . 0 × 10 34 erg / s in the 0.3-7 keV band at the LMC distance of 50 kpc. Stacking the data from all the ObsIDs with non-detections, including the pre-flare part of ObsID 23022, results in a total observed exposure of approximately ∼ 150 ks, and yields a 90% confidence upper limit on the X-ray luminosity is 𝐿 X < 3 × 10 33 erg / s.</text>
<table>
<location><page_14><loc_52><loc_74><loc_91><loc_89></location>
<caption>Table 8. Properties of the Chandra observation ObsID 23022 and source 2CXO J051117.2 -672556 associated with XRT 200515.</caption>
</table>
<figure>
<location><page_14><loc_51><loc_55><loc_92><loc_72></location>
<caption>Figure 13. ACIS-S image for Chandra observation ObsID 23022 showing the target, SNR 0509 -67.5, on the bottom right and the transient event, XRT 200515, on the top left. For this image, the event file was filtered to include only the time interval 𝑡 0 + 920 s around the flare. Red counts correspond to photons in the 0.3-1.2 keV band, yellow counts correspond to photons in the 1.2-2.4 keV band, and blue photons correspond to photons in the 2.4-7 keV band. The inset image is a 1.0 arcmin × 1.0 arcmin zoomed-in view. The dashed ellipse has semi-minor and semi-major axes of 15 arcsec × 20 arcsec, and is the source region used for spectral extraction.</caption>
</figure>
<section_header_level_1><location><page_14><loc_51><loc_37><loc_70><loc_38></location>4.3.2 X-ray Temporal Analysis</section_header_level_1>
<text><location><page_14><loc_51><loc_27><loc_92><loc_36></location>We used the /c.pc/i.pc/a.pc/o.pc tool dmextract to extract background-subtracted lightcurves in several energy bands, from the reprocessed event file of Chandra ObsID 23022. We defined an elliptical source extraction region, with semi-minor and semi-major axes of 15 arcsec and 20 arcsec (matching the point-source PSF at the source location); the local background region was chosen in the same ACIS chip, with an area approximately eight times larger.</text>
<text><location><page_14><loc_51><loc_11><loc_92><loc_26></location>Figure 14 shows the 0.3-7 keV background-subtracted lightcurve of XRT 200515 with a time resolution of 20 s. The lightcurve is consistent with no source detection at the location of the transient, before the start of the flare at around 23.5 ks into the observation. The few pre-flare counts are consistent with background noise. The lightcurve exhibits a strong initial spike with a sharp rise of at least 4 orders of magnitude in < 10 s, containing 44 out of all ∼ 180 flare counts. This initial burst is followed by a sudden drop to a ∼ 800 s long pulsating and decaying tail. We estimate a 𝑇 90 ∼ 580-740 s for the photons observed in the 0.3-7 keV band 4 , depending on the definition of total flare counts.</text>
<figure>
<location><page_15><loc_7><loc_74><loc_47><loc_93></location>
<caption>Figure 14. Background-subtracted lightcurve of XRT 200515 in the 0.37 keV energy range with a bin size of 20 s. The zero start time is taken as the start of the Chandra observation ObsID 23022. The inset shows the last ∼ 2 ks of the observations, and captures the initial burst and tail of the transient event. The bin size is chosen to better visualize the oscillatory decay of the tail. The actual duration of the initial burst peak is < 10 s. The presence of a few negative counts arises from the process of background subtraction.</caption>
</figure>
<text><location><page_15><loc_7><loc_44><loc_48><loc_61></location>Figure 15 shows the lightcurve of XRT 200515 at a resolution matching the ACIS frame time of 3.2 s, the hardness ratio, and the energy evolution for the time interval 𝑡 0 + 920 s. The lightcurve exhibits a spike in the count rate across only 3 bins (with a total of 4, 31 and 9 counts, respectively), hence the burst duration of < 10 s. The rise and fall times of the burst are both between 3.2 s and 6.4 s. The maximum count rate at the Chandra frame time resolution is ∼ 9.7 counts/s, acting as the lower bound for the peak count rate of the burst. Those counts are spatially spread over a PSF area of ∼ 3000 pixels; therefore, pile-up is not an issue. We evaluated the hardness ratio evolution during the flare with the Bayesian estimation method BEHR (Park et al. 2006). Here, the hardness ratio is defined as:</text>
<formula><location><page_15><loc_7><loc_41><loc_48><loc_44></location>𝐻𝑅 = ℎ -𝑚 -𝑠 ℎ + 𝑚 + 𝑠 , (13)</formula>
<text><location><page_15><loc_7><loc_30><loc_48><loc_40></location>where 𝑠 is the number of soft photons (0.3-1.2 keV), 𝑚 is the number of medium photons (1.2-2 keV), and ℎ is the number of hard photons (2-7 keV) in each bin. We also track the running average of the photon energies during the flare with a moving window of ± 10 counts. The hardness ratio and energy evolution indicate spectral softening during the flare, with the hardness ratio starting at 1 during the hard burst peak and decreasing to a range of 0.4 to 0.6 in the tail, highlighting the notable spectral variability of XRT 200515.</text>
<section_header_level_1><location><page_15><loc_7><loc_26><loc_25><loc_27></location>4.3.3 X-ray Spectral Analysis</section_header_level_1>
<text><location><page_15><loc_7><loc_10><loc_48><loc_25></location>We used the /c.pc/i.pc/a.pc/o.pc tool specextract to extract the spectrum and the associated response and ancillary response files from the reprocessed event file of Chandra ObsID 23022. We used the same source and background extraction regions defined for the lightcurve extraction. To improve the signal-to-noise ratio of the source, we extracted the spectrum only from the time interval 𝑡 0 + 920 s. We binned the spectrum to a minimum of 1 count per bin with the grppha task within the /f.pc/t.pc/o.pc/o.pc/l.pc/s.pc package suite (Blackburn 1995) from NASA's High Energy Astrophysics Science Archive Research Center (HEASARC) 5 . For all spectral modelling and flux estimates, we used the /x.pc/s.pc/p.pc/e.pc/c.pc software version 12.13.0 (Arnaud 1996). With only 179 net counts, we</text>
<figure>
<location><page_15><loc_51><loc_67><loc_92><loc_93></location>
<caption>Figure 15. Upperpanel: Background-subtracted count rate lightcurve of XRT 200515 in the 0.3-7 keV energy range at the full Chandra detector resolution of 3.2 s and hardness ratio evolution during the flare obtained for a minimum of 20 counts per bin. The zero start time is taken as the flare start time 𝑡 0 of XRT 200515. The initial peak is < 10 s long and is very hard, while the ∼ 800 s long oscillatory tail is significantly softer. Bottom panel: The energy evolution during the flare obtained from the running average energy with a moving window of ± 10 counts, showing significant spectral variability and softening during the flare. The scatter points represent the time and energy of individual photons from XRT 200515 in the event file associated with Chandra observation ObsID 23022.</caption>
</figure>
<text><location><page_15><loc_51><loc_37><loc_92><loc_46></location>are unable to fit complex spectral models; thus, we limit our analysis to the simplest one-component models representative of opposite scenarios: a power law ( powerlaw ) and a blackbody model ( bbody ), both modified by photo-electric absorption ( tbabs ). In both cases, we adopted the Tuebingen-Boulder absorption model with Wilms abundances (Wilms et al. 2000). We minimized the Cash statistic (Cash 1979), as we do not have enough counts for 𝜒 2 fitting.</text>
<text><location><page_15><loc_51><loc_22><loc_92><loc_37></location>The best-fitting power-law model (Table 9 and Figure 16) has a photon index of Γ = 0 . 5 ± 0 . 3. The fit statistics yield a null hypothesis probability of 3 . 5 × 10 -3 , with a Cstat value of 132.7 for 137 degrees of freedom. For the blackbody model, the best-fitting temperature is 𝑘𝑇 bb = 1 . 8 ± 0 . 3 keV (Table 9). The fit statistics yield a null hypothesis probability of 1 . 2 × 10 -2 , with a Cstat value of 129.6 for 137 degrees of freedom. The reason this blackbody spectrum may appear hard in the Chandra band, resembling a Γ ∼ 0 . 5 power law, is that at a temperature of 𝑘𝑇 bb ∼ 2 keV, the ACIS detector samples only the peak and the Rayleigh-Jeans (rising) portion of the blackbody emission.</text>
<text><location><page_15><loc_51><loc_6><loc_92><loc_21></location>We can use either model to determine an average conversion between the count rate and luminosity. This will then enable us to estimate the peak luminosity in the initial spike, for which we have previously estimated a peak count rate of ≳ 10 counts/s. The best-fitting power law model implies a peak flux of 𝐹 p ≳ 5 . 6 × 10 -10 erg/s/cm 2 , a total flare fluence of 𝐸 f ≳ 1 . 1 × 10 -8 erg/cm 2 , and a peak unabsorbed 0.3-10 keV luminosity of 𝐿 X ≳ 1 . 7 × 10 38 erg/s at the LMC distance of 50 kpc. For the best-fitting blackbody model, the peak flux and flare fluence would be 𝐹 p ≳ 4 . 0 × 10 -10 erg/s/cm 2 and 𝐸 f ≳ 0 . 8 × 10 -8 erg/cm 2 respectively. The peak unabsorbed 0.3-10 keV luminosity would be 𝐿 X ≳ 1 . 2 × 10 38 erg/s and the peak</text>
<table>
<location><page_16><loc_10><loc_60><loc_44><loc_84></location>
<caption>Table 9. Best-fitting parameters of the Chandra /ACIS-S spectrum of XRT 200515, fitted with the Cash statistics, for an absorbed power law and an absorbed blackbody model. Because of the relatively low number of counts, parameter uncertainties are reported at the confidence interval Δ 𝐶 = ± 1 . 0: this is asymptotically equivalent to the 68% confidence interval (1 𝜎 ) in the 𝜒 2 statistics.</caption>
</table>
<figure>
<location><page_16><loc_7><loc_33><loc_48><loc_56></location>
<caption>Figure 16. Upper panel: Observed X-ray spectral energy distribution from ∼ 920 s around the flare XRT 200515 and the best-fit absorbed power law model. The data have been rebinned to a minimum of 10 counts per bin for plotting purposes only; a binning of 1 count per bin was instead used for the fitting (Cash statistics). Lower panel: Residuals between the data and the best-fit model.</caption>
</figure>
<text><location><page_16><loc_7><loc_6><loc_48><loc_17></location>bolometric luminosity would be 𝐿 bol ≳ 1 . 5 × 10 38 erg/s. These values should be considered conservative lower limits for two reasons: (i) the peak count rate provides only a lower bound estimate, as it is constrained by the Chandra frame time resolution of the observations, potentially underestimating the true peak count rate; and (ii) the conversion factor applied is derived from the average spectrum over the entire flare, even though the spectrum of the initial spike is significantly harder compared to the tail, as shown in Figure 15.</text>
<text><location><page_16><loc_51><loc_77><loc_92><loc_90></location>We searched for potential detections of XRT 200515 by other highenergy facilities. However, no significant X-ray or 𝛾 -ray events in the field around the X-ray source coordinates and flare start time 𝑡 0 reported in Table 8 were detected by the Fermi Gamma-ray Space Telescope ( Fermi ), the Burst Alert Telescope ( BAT ) on the Neil Gehrels Swift Observatory ( Swift ), the International Gamma-Ray Astrophysics Laboratory ( INTEGRAL ), or the Monitor of All-sky X-ray Image MAXI . LIGO was not operational during the time of the FXT, hence no gravitational wave signal could have been detected if the origin of XRT 200515 was a compact object merger.</text>
<section_header_level_1><location><page_16><loc_51><loc_73><loc_72><loc_74></location>4.3.5 Optical Counterpart Search</section_header_level_1>
<text><location><page_16><loc_51><loc_27><loc_92><loc_71></location>We used the X-ray source coordinates reported in Table 8 to search for optical and infrared counterparts to XRT 200515. The field of XRT 200515 was covered by the Survey of Magellanic Stellar History ( SMASH ) (Nidever et al. 2017), a deep optical survey in the ugriz bands with the Dark Energy Camera (DECam) mounted on the Víctor M. Blanco Telescope at the Cerro Tololo Inter-American Observatory (CTIO) in Chile. We used the Astro Data Lab Jupyter Notebook server (Nikutta et al. 2020; Juneau et al. 2021) to access and visualize the SMASH catalog 6 . Figure 17 shows a color image of the field created from the deepest available stacked images in the u , g and i bands; the 5 𝜎 detection limits in these bands are 23.9 mag, 24.8 mag and 24.2 mag, respectively. The images were taken on December 7, 2015 with exposure times of 1,179 s, 981 s and 1,179 s respectively. The astrometry of the SMASH images is calibrated on the Gaia DR3 reference frame, thus their positional uncertainty is negligible compared to the X-ray source position uncertainty. Within the Chandra position error circle in Figure 17, there is no obvious optical counterpart that stands out in brightness or color from the surrounding stellar population. We performed relative photometry on the sources inside the error circle, comparing them to several nearby sources with known positions and brightnesses listed in the Gaia DR3 catalog. We used the SMASH g band as the closest approximation to Gaia's G band. We estimate the brightest optical source within the error circle to have a Vega magnitude of 𝑔 = 22 . 7 ± 0 . 1 mag, corresponding to an absolute magnitude of 𝑀 𝑔 ≈ 4 . 2, assuming it is in the LMC. Additionally, three other point-like sources are detected with 𝑔 band magnitudes in the range of 23-24 mag. All four sources appear pointlike, consistent with the seeing conditions of the SMASH survey, with no evidence of any spatially extended background galaxies. The three brightest stars visible in Figure 17 within ∼ 12 arcsec of the Chandra source are solar-mass stars on the red giant branch, indicative of an old stellar population.</text>
<text><location><page_16><loc_51><loc_11><loc_92><loc_27></location>The lack of bright optical counterparts and the short burst duration of < 10 s rules out a stellar flare from a foreground Galactic low-mass star (Güdel 2004; Reale 2007; Reale & Landi 2012; Pye et al. 2015; Kuznetsov & Kolotkov 2021). A flare from a Be/X-ray binary or any other HMXB in the LMC is also excluded by the lack of a bright optical counterpart (Ducci et al. 2019, 2022). The temporal and spectral properties of XRT 200515, combined with the absence of an optical counterpart, suggests three possibilities: (i) a relativistic jet phenomenon, such as a 𝛾 -ray burst (GRB); (ii) a rapid, high-energy process linked to extreme magnetic fields, such as a giant magnetar flare (GMF); or (iii) a thermonuclear Type I X-ray burst caused by surface nuclear burning on a neutron star.</text>
<figure>
<location><page_17><loc_7><loc_68><loc_48><loc_93></location>
<caption>Figure 17. SMASH survey color image of the field of XRT 200515 created from the deepest available stacked images in the u , g an i bands. Red corresponds to the i band, green to the g band, and blue to the u band. The dashed circle has a radius of 2 arcsec, and is the 90% position uncertainty of the Chandra source.</caption>
</figure>
<section_header_level_1><location><page_17><loc_7><loc_50><loc_42><loc_51></location>4.3.6 Gamma Ray Burst from a Compact Object Merger?</section_header_level_1>
<text><location><page_17><loc_7><loc_6><loc_48><loc_49></location>Evidence in favor or against the association of at least some Chandra FXTs with low-luminosity long-GRBs or off-axis short-GRBs (see Berger 2014 for a review), at moderate or high redshifts, is extensively discussed in Quirola-Vásquez et al. (2022), Quirola-Vásquez et al. (2023), and Wichern et al. (2024). A detailed re-investigation of this issue is beyond the scope of this work. Here, we simply point out that XRT 200515, like the other Chandra FXTs in the literature, does not have any 𝛾 -ray detection. On the other hand, XRT 200515 has a significantly harder spectrum ( Γ = 0 . 5 ± 0 . 3) in the Chandra band than the rest of the FXT sample, all of which have photon indices of Γ > 1 (Jonker et al. 2013; Glennie et al. 2015; Bauer et al. 2017; Xue et al. 2019; Lin et al. 2022; Quirola-Vásquez et al. 2022; Quirola-Vásquez et al. 2023; Eappachen et al. 2023). A photon index of Γ ∼ 0 . 5 below 10 keV is indeed expected and observed from both core-collapse GRBs and compact-merger GRBs (Ghirlanda et al. 2009; Bromberg et al. 2013; Oganesyan et al. 2018; Ravasio et al. 2019; Toffano et al. 2021). This might support the association of the initial spike of XRT 200515 with a GRB. However, the presence and properties of the ∼ 800 s tail (candidate GRB afterglow) is puzzling. The 𝑇 90 ∼ 580-740 s value for XRT 200515 is significantly shorter than in most other Chandra FXTs (Quirola-Vásquez et al. 2022; Lin et al. 2022; Quirola-Vásquez et al. 2023), which have 𝑇 90 values on the order of several ks and are already pushing the limit for a GRB afterglow detection (Wichern et al. 2024). Moreover, XRT 200515's initial burst duration ( < 10 s), its short rise and fall times (3.2-6.4 s), and the lack of a peak plateau are inconsistent with the lightcurves of Chandra FXTs interpreted as magnetar-powered GRBs as the aftermath of a binary neutron star merger, such as CDF-S XT1 (Bauer et al. 2017), CDF-S XT2 (Xue et al. 2019) and the sample in Lin et al. (2022). Finally, the lack of any optical evidence for a host galaxy is another element disfavoring the high-redshift GRB interpretation.</text>
<text><location><page_17><loc_51><loc_92><loc_87><loc_93></location>4.3.7 Giant Magnetar Flare from a Soft Gamma Repeater?</text>
<text><location><page_17><loc_51><loc_18><loc_92><loc_90></location>Based on its temporal and spectral variability, it is tempting to interpret XRT 200515 as a rare GMF from a SGR (Mereghetti 2008; Turolla et al. 2015) in the LMC or behind it, which can easily explain the burst's strong increase of at least 4 orders of magnitude in < 10 s (Coti Zelati et al. 2018). Similar to XRT 200515, GMFs are characterized by a short and hard initial spike and a longer and softer, pulsating tail. GMFs are extremely rare, with only a select few ever discovered. Well-studied examples are SGR 0526 -66 in the LMC (Mazets et al. 1979), and the Galactic sources SGR 1900 + 14 (Hurley et al. 1999) and SGR 1806 -20 (Hurley et al. 2005; Palmer et al. 2005; Israel et al. 2005). More recently, GMFs have been identified in M 31 (Mazets et al. 2008a), NGC 253 (Fermi-LAT Collaboration et al. 2021; Svinkin et al. 2021; Roberts et al. 2021; Trigg et al. 2024) and M82 (Mereghetti et al. 2024). All of these have been observed by high time resolution instruments in the hard X-rays and soft 𝛾 -rays with luminosities above 10 46 erg/s for a fraction of a second in the initial spike. The tails of GMFs are often modulated by magnetar spin periods of 2-12 s, leading to quasi-periodic oscillations (QPOs). For XRT 200515, there is no hard X-ray or 𝛾 -ray detection, despite the LMC direction being in good visibility for most of the previously mentioned high-energy facilities. We were unable to identify any significant periodicities in the tail of XRT 200515 through periodogram analysis, which is unsurprising given the low time resolution of Chandra observations. No X-ray activity has been observed by Chandra or other X-ray telescopes in the years before or after XRT 200515, which may be because SGRs are very faint when they are not bursting. The strongest argument against a magnetar in the LMC as the origin of XRT 200515 is that magnetars are short-lived objects ( ≲ 10 5 yr) associated to young stellar populations (Olausen & Kaspi 2014; Nakano et al. 2015; Mondal 2021). Even allowing for the persistence of magnetar-like activity in ordinary radio pulsars as old as ∼ 10 7 yr (Rea et al. 2010), this scenario is still inconsistent with the old stellar population (several Gyr) in the LMC field shown in Figure 17. The nearest star-forming regions in the LMC are ∼ 10 arcmin ( ∼ 150 pc) away. If (in a very contrived scenario), we assume that XRT 200515 is powered by a young neutron star ejected from one of those regions, we estimate a characteristic time of 1 Myr to travel that distance at a speed of 150 km/s. Therefore, if XRT 200515 is a GMF, it must be located behind the LMC, in a low-redshift galaxy (Hurley et al. 2005; Tanvir et al. 2005). Since GMFs have been observed only a few times and never at soft X-ray energies, their properties in the soft X-ray band detectable by Chandra remain largely unexplored. XRT 200515 could indeed be the first GMF detected at soft X-ray energies. Distinguishing distant short GRBs from GMFs has historically been difficult and there are multiple studies suggesting that a subset of short GRBs are actually extragalactic GMFs (Hurley et al. 2005; Palmer et al. 2005; Tanvir et al. 2005; Ofek et al. 2006; Mazets et al. 2008b; Hurley 2011; Yang et al. 2020; Svinkin et al. 2021; Negro & Burns 2023). Just as for the distant GRB interpretation, the non-detection of any optical counterpart remains puzzling for a distant GMF scenario, unless we are dealing with a very distant and exceptionally luminous GMF.</text>
<section_header_level_1><location><page_17><loc_51><loc_14><loc_92><loc_15></location>4.3.8 Thermonuclear X-ray Burst from a quiet LMXB in the LMC?</section_header_level_1>
<text><location><page_17><loc_51><loc_6><loc_92><loc_12></location>If XRT 200515 is in the LMC, a peak luminosity near the Eddington luminosity 𝐿 Edd ∼ 10 38 erg/s and sharp rise time of the flare suggests a Type I X-ray burst interpretation, which is a thermonuclear explosion on the surface of a weakly magnetized, accreting neutron star (Lewin et al. 1993; Strohmayer & Bildsten 2003; Gal-</text>
<section_header_level_1><location><page_18><loc_7><loc_92><loc_48><loc_93></location>loway et al. 2008, 2020; Galloway & Keek 2021; Alizai et al. 2023).</section_header_level_1>
<text><location><page_18><loc_7><loc_22><loc_48><loc_91></location>The old stellar population in the field of XRT 200515 is consistent with the presence of neutron star LMXBs. Following the definition of burst timescale 𝜏 = 𝐸 f / 𝐹 p in Galloway et al. (2008), we estimate 𝜏 ∼ 20 s for XRT 200515, which is consistent with Type I X-ray bursts (Galloway & Keek 2021; Alizai et al. 2023). The fitted temperature 𝑘𝑇 bb ∼ 2 keV when the average spectrum is fitted with a simple blackbody, and the softening of the spectrum (temperature decrease) in the tail is also typical of Type I X-ray bursts (Galloway et al. 2008, 2020; Güver et al. 2012). On the other hand, several observed properties of XRT 200515 are unusual for Type I X-ray bursts. In particular, most Type I X-ray bursts occur when the persistent luminosity (proportional to the accretion rate) of a LMXB is 𝐿 X > 10 -4 𝐿 Edd (and, in most cases, 𝐿 X > 10 -3 𝐿 Edd ) (Galloway et al. 2008). Instead, in the initial part of ObsID 23022, the upper limit on the X-ray luminosity at the position of XRT 200515 is 𝐿 X < 10 -4 𝐿 Edd , so that the X-ray flux increased by at least 4 orders of magnitudes. On another note, the sharp decline after the initial burst of XRT 200515 would be unusual for Type I X-ray bursts, which typically exhibit a gradual and exponential decay. However, note that most Type I X-ray bursters were observed by the Rossi X-Ray Timing Explorer ( RXTE ) (Jahoda et al. 1996), which has a high time resolution. The low time resolution of Chandra may have obscured such a decay for XRT 200515. Moreover, most Type I bursts tend to repeat every few hours (Galloway et al. 2008); instead, XRT 200515 is the only event detected at that location over a total observed time of ∼ 150 ks. No LMXB has ever been noted at that position before or after the event. The time interval between bursts is related to an index 𝛼 defined as the ratio between the integrated persistent fluence between subsequent bursts and the burst fluence; from a comparison of the energy released by accretion (contributing to the persistent fluence) and by thermonuclear burning (burst fluence), we expect 𝛼 ≳ 40, in agreement with the observations of Type I bursts (Galloway et al. 2008). If we apply the same criterion ( 𝛼 ≳ 40) to the persistent and flare fluences of XRT 200515, we would have to wait > 10 7 s (4 months) to observe another similar event, assuming the persistent flux level upper limit in ObsID 23022 before the transient event. This waiting time extends to at least one year if we assume the persistent flux upper limit derived from the stacked ∼ 150 ks Chandra observations. Only a few one-off bursts from Galactic neutron stars at a very low persistent luminosity ( 𝐿 X ∼ 10 32 -10 33 erg/s) were found by Cornelisse et al. (2002a,b) with estimated recurrence times of tens of years. The vast majority of Type I X-ray bursts are Galactic, due to their lower flux at large distances. Only a handful of extragalactic Type I X-ray bursts are documented, for example in M 31 (Pastor-Marazuela et al. 2020) and the Magellanic Bridge (Haberl et al. 2023). If XRT 200515 is a Type I X-ray burst, it is the first extragalactic Type I X-ray burster in the LMC and represents the tip of the iceberg for a vast population of faint LMXBs in nearby galaxies, too dim to be detected by Chandra or XMM-Newton , but which may occasionally reveal themselves via thermonuclear bursts with a long duty cycle.</text>
<section_header_level_1><location><page_18><loc_7><loc_18><loc_41><loc_19></location>4.3.9 Concluding Remarks and Outlook for XRT 200515</section_header_level_1>
<text><location><page_18><loc_7><loc_6><loc_48><loc_17></location>XRT 200515 is a unique and intriguing extragalactic Chandra FXT. The combination of its temporal and spectral properties is unlike any of the other Chandra FXT samples. Based on our analysis, the two most likely scenarios for XRT 200515 are: (i) a distant GMF from a SGR behind the LMC; the first observed in the low X-ray energy band, missed by any other high-energy facilities, or (ii) an unusual Type I X-ray burst from a previously unknown faint LMXB; the first extragalactic X-ray burster in the LMC. Nevertheless, both</text>
<figure>
<location><page_18><loc_51><loc_79><loc_92><loc_93></location>
<caption>Figure 18. 𝐸 -𝑡 Mapevent file representation (left) and 𝐸 -𝑡 -𝑑𝑡 Cube event file representation (right) of XRT 200515 in Chandra observation ObsID 23022. The catalog identifier for XRT 200515 is CATALOG_ID: 23022_122 .</caption>
</figure>
<text><location><page_18><loc_51><loc_48><loc_92><loc_69></location>of these interpretations come with their own unique challenges. XRT 200515 could, in fact, represent an entirely new type of astronomical phenomenon. After all, the primary objective of our work was to use machine learning to find rare, needle-in-the-haystack anomalies hidden within vast astronomical datasets. We invite further detailed studies of XRT 200515 to evaluate our interpretations and explore alternative scenarios, such as potential associations with a fast radio burst (FRB) or a SBO. We highly recommend follow-up multi-band observations at the source coordinates of XRT 200515 to better constrain its nature. Lastly, we note that XRT 200515 and the second transient discovered by Glennie et al. (2015), XRT 120830, have remarkably similar temporal evolutions in their lightcurves (J. Irwin, personal communication, November 2024), however with very different spectral properties ( Γ ∼ 2 . 5 for XRT 120830 versus Γ ∼ 0 . 5 for XRT 200515). We leave a detailed comparative analysis of these transients for future work.</text>
<text><location><page_18><loc_51><loc_42><loc_92><loc_47></location>Figure 18 shows the 𝐸 -𝑡 Map and 𝐸 -𝑡 -𝑑𝑡 Cube event file representations for XRT 200515. These exhibit high counts at high energies in a narrow time window, which is in line with the hard spectrum and transient nature of XRT 200515.</text>
<section_header_level_1><location><page_18><loc_51><loc_38><loc_65><loc_39></location>4.4 Technical Caveats</section_header_level_1>
<text><location><page_18><loc_51><loc_6><loc_92><loc_36></location>The main technical caveat of our approach is related to the representation of event files. While our new event file representations enable a simple, yet powerful representation learning approach to find new and rare X-ray transients, any simplification of raw event files, like the fixed number of time bins we use across all event files, is associated with a loss of information. This could lead to us missing a small amount of transients. To minimize this, we have implemented a rigorous approach to justify the resolution of the event file representations in § 3.1. Moreover, flares, in particular known extragalactic FXTs, cluster notably well in our representation spaces. This is because their distinctive features are less dependent on the temporal binning resolution in the 𝐸 -𝑡 Maps and 𝐸 -𝑡 -𝑑𝑡 Cubes. To improve the effectiveness of dip searches with our proposed method, we suggest using higher resolution event file representations. Nevertheless, our comprehensive transient candidate catalog includes numerous newly identified transients that were previously overlooked by other X-ray transient searches in the Chandra archive. Among these is the remarkable needle-in-the-haystack event XRT 200515 discovered in this work, underscoring the effectiveness of our method. A followup representation learning algorithm will learn informative features from raw and unbinned event files while accounting for the Poisson nature of X-ray observations (Song et al., 2025, in preparation).</text>
<section_header_level_1><location><page_19><loc_7><loc_92><loc_19><loc_93></location>5 CONCLUSION</section_header_level_1>
<text><location><page_19><loc_7><loc_74><loc_48><loc_90></location>We have introduced a novel representation learning method, the first of its kind applied to X-ray event files, enabling downstream tasks such as unsupervised classification and anomaly detection in highenergy astrophysics. We have used the learned representation to investigate time-domain properties of sources in the Chandra archive, with a particular emphasis on the discovery of X-ray transients. As a result, we have compiled the identified X-ray flares and dips in a comprehensive catalog of transient candidates. Notably, our method led to the discovery of XRT 200515; a previously unidentified extragalactic FXT with unique temporal and spectral properties, representing a genuine needle-in-the-haystack discovery. Our key results are as follows:</text>
<unordered_list>
<list_item><location><page_19><loc_7><loc_70><loc_48><loc_72></location>(i) Weintroduce novel event file representations, the E-t Mapsand E-t-dt Cubes, which capture both temporal and spectral information.</list_item>
<list_item><location><page_19><loc_7><loc_65><loc_48><loc_69></location>(ii) We apply two feature extraction methods to the event file representations, PCA and sparse autoencoder neural networks, to extract or learn informative features that can be utilized for downstream tasks, such as unsupervised classification or anomaly detection.</list_item>
<list_item><location><page_19><loc_7><loc_60><loc_48><loc_64></location>(iii) We project the learned features to two-dimensional embedding spaces, enabling interpretable queries of analogs to objects of interest based on their temporal and spectral properties.</list_item>
<list_item><location><page_19><loc_7><loc_55><loc_48><loc_60></location>(iv) We cluster the embedding spaces with DBSCAN, successfully isolating previously identified X-ray transients. We identify new transient candidates within specific transient-dominant clusters or through nearest-neighbor searches using kNN.</list_item>
<list_item><location><page_19><loc_7><loc_51><loc_48><loc_54></location>(v) We compile a catalog of the X-ray transient candidates, including 3,427 flares and 112 dips, and make it openly accessible to the community and the broader scientific audience.</list_item>
<list_item><location><page_19><loc_7><loc_45><loc_48><loc_50></location>(vi) We report the discovery of XRT 200515, a rare extragalactic FXT characterized by unique temporal and spectral features. We explore its potential origins and suggest that it may be associated with one of the following scenarios, presented in no particular order:</list_item>
<list_item><location><page_19><loc_9><loc_38><loc_48><loc_43></location>· A rare GMF from an SGR behind the LMC, marking the first GMF detected in the low X-ray energy range covered by telescopes like Chandra , XMM-Newton , Swift-XRT , eROSITA , or Einstein Probe .</list_item>
<list_item><location><page_19><loc_9><loc_35><loc_48><loc_38></location>· A rare extragalactic Type I X-ray burst from a faint LMXB in the LMC, representing the first such detection in the LMC.</list_item>
<list_item><location><page_19><loc_9><loc_32><loc_48><loc_35></location>· A new type of astronomical phenomenon and a genuine anomaly, previously hidden in the vast Chandra archive.</list_item>
</unordered_list>
<text><location><page_19><loc_7><loc_26><loc_48><loc_31></location>XRT200515wasonlydetectedby Chandra , with no identified optical counterparts. We strongly encourage a multi-wavelength search for additional signals from the source associated with XRT 200515 to better understand its origin and nature.</text>
<text><location><page_19><loc_7><loc_13><loc_48><loc_24></location>Ourworkadvancestime-domainhigh-energy astrophysics by making the Chandra transient candidates catalog publicly available and open-sourcing the representation learning based transient search pipeline 7 . The catalog enables queries to identify new Chandra transients. Future work involves applying the detection pipeline to additional high-energy archives and adapting it to a variety of other scientific datasets, paving the way for further machine learning driven discoveries of rare transients and other scientific anomalies.</text>
<section_header_level_1><location><page_19><loc_51><loc_92><loc_69><loc_93></location>ACKNOWLEDGEMENTS</section_header_level_1>
<text><location><page_19><loc_51><loc_87><loc_92><loc_90></location>This research has made use of data obtained from the Chandra Source Catalog, and software provided by the Chandra X-ray Center (CXC) in the /c.pc/i.pc/a.pc/o.pc application package.</text>
<text><location><page_19><loc_51><loc_72><loc_92><loc_86></location>SD's work was partially funded by the UK government's Turing Scheme and mainly carried out at the Center for Astrophysics | Harvard & Smithsonian as part of the SAO Predoctoral Program, with the support of AstroAI. SD acknowledges hospitality at the Institute of Astronomy at the University of Cambridge and at the Stanford Center for Decoding the Universe (Stanford Data Science) during the later parts of this project. RS's work was performed in part at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-2210452. RS acknowledges support and hospitality at the National Astronomical Observatories of China (NAOC) in Beijing, during part of this project.</text>
<text><location><page_19><loc_51><loc_65><loc_92><loc_71></location>We thank Edo Berger, Massimiliano De Pasquale, Ken Ebisawa, Duncan Galloway, Jimmy Irwin, Peter Jonker, Daniel Kocevski, Amy Lien, Sandro Meregetthi, Daniel Muthukrishna, Nicola Omodei, Jonathan Quirola-Vásquez, Shivam Raval, Ashley Villar, and Silvia Zane for their fruitful discussions.</text>
<section_header_level_1><location><page_19><loc_51><loc_60><loc_67><loc_61></location>DATA AVAILABILITY</section_header_level_1>
<text><location><page_19><loc_51><loc_45><loc_92><loc_58></location>The data used in this paper, composed of X-ray event files and source detection regions, was obtained from the publicly available CSC, using their public interfaces (https://cxc.cfa.harvard.edu/csc/). The catalog of transient candidates and the clustered embedding spaces generated using our unsupervised representation learning method can be accessed in the supplementary material. All intermediate data products, i.e. as 𝐸 -𝑡 Maps, 𝐸 -𝑡 -𝑑𝑡 Cubes, principal components and latent features, feature embeddings and embedding clusters can be produced using the code provided in the GitHub repository https://github.com/StevenDillmann/ml-xraytransients-mnras.</text>
<section_header_level_1><location><page_19><loc_51><loc_40><loc_61><loc_41></location>REFERENCES</section_header_level_1>
<unordered_list>
<list_item><location><page_19><loc_51><loc_8><loc_92><loc_39></location>Abadi M., et al., 2015, TensorFlow: Large-Scale Machine Learning on Heterogeneous Systems, https://www.tensorflow.org/ Alizai K., et al., 2023, Monthly Notices of the RAS, 521, 3608 Alp D., Larsson J., 2020, The Astrophysical Journal, 896, 39 Arcodia R., et al., 2021, Nature, 592, 704 Arnaud K., 1996, in ASP Conf.. Astropy Collaboration et al., 2013, Astronomy and Astrophysics, 558, A33 Bauer F. E., et al., 2017, Monthly Notices of the RAS, 467, 4841 Bellm E. C., et al., 2019, PASP, 131, 018002 Bengio Y., Courville A., Vincent P., 2013, IEEE transactions on pattern analysis and machine intelligence, 35, 1798 Berger E., 2014, Annual Review of Astronomy and Astrophysics, 52, 43 Blackburn J. K., 1995, in Shaw R. A., Payne H. E., Hayes J. J. E., eds, Astronomical Society of the Pacific Conference Series Vol. 77, Astronomical Data Analysis Software and Systems IV. p. 367 Bromberg O., Nakar E., Piran T., Sari R., 2013, The Astrophysical Journal, 764, 179 Burrows D. N., et al., 2005, SSR, 120, 165 Caliński T., Harabasz J., 1974, Communications in Statistics-theory and Methods, 3, 1 Cash W., 1979, The Astrophysical Journal, 228, 939 Chakraborty J., Kara E., Masterson M., Giustini M., Miniutti G., Saxton R., 2021, The Astrophysical Journal, Letters, 921, L40 Chambers K. C., et al., 2016, arXiv e-prints, p. arXiv:1612.05560 Colbert E. J. M., Ptak A. F., 2002, The Astrophysical Journal, Supplement,</list_item>
</unordered_list>
<table>
<location><page_20><loc_7><loc_6><loc_48><loc_93></location>
</table>
<table>
<location><page_20><loc_51><loc_6><loc_92><loc_93></location>
</table>
<table>
<location><page_21><loc_7><loc_6><loc_48><loc_93></location>
</table>
<text><location><page_21><loc_51><loc_92><loc_87><loc_93></location>Walmsley M., et al., 2022, Monthly Notices of the RAS, 513, 1581</text>
<text><location><page_21><loc_51><loc_86><loc_92><loc_91></location>Weisskopf M. C., Tananbaum H. D., Van Speybroeck L. P., O'Dell S. L., 2000, in Truemper J. E., Aschenbach B., eds, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Vol. 4012, X-Ray Optics, Instruments, and Missions III. pp 2-16 ( arXiv:astro-ph/0004127 ), doi:10.1117/12.391545</text>
<unordered_list>
<list_item><location><page_21><loc_51><loc_84><loc_91><loc_85></location>Wenger M., et al., 2000, Astronomy and Astrophysics, Supplement, 143, 9</list_item>
</unordered_list>
<text><location><page_21><loc_51><loc_83><loc_92><loc_84></location>Wichern H. C. I., Ravasio M. E., Jonker P. G., Quirola-Vásquez J. A., Levan</text>
<unordered_list>
<list_item><location><page_21><loc_54><loc_81><loc_92><loc_82></location>A. J., Bauer F. E., Kann D. A., 2024, Astronomy and Astrophysics, 690, A101</list_item>
<list_item><location><page_21><loc_51><loc_78><loc_91><loc_80></location>Wilms J., Allen A., McCray R., 2000, The Astrophysical Journal, 542, 914 Xue Y. Q., et al., 2019, Nature, 568, 198</list_item>
<list_item><location><page_21><loc_51><loc_77><loc_92><loc_77></location>Yang G., Brandt W. N., Zhu S. F., Bauer F. E., Luo B., Xue Y. Q., Zheng</list_item>
</unordered_list>
<text><location><page_21><loc_54><loc_76><loc_82><loc_76></location>X. C., 2019, Monthly Notices of the RAS, 487, 4721</text>
<unordered_list>
<list_item><location><page_21><loc_51><loc_74><loc_82><loc_75></location>Yang J., et al., 2020, The Astrophysical Journal, 899, 106</list_item>
<list_item><location><page_21><loc_51><loc_72><loc_92><loc_73></location>Yang H., Hare J., Kargaltsev O., Volkov I., Chen S., Rangelov B., 2022, The Astrophysical Journal, 941, 104</list_item>
<list_item><location><page_21><loc_51><loc_68><loc_92><loc_71></location>Yuan W., Zhang C., Chen Y., Ling Z., 2022, in Bambi C., Sangangelo A., eds, , Handbook of X-ray and Gamma-ray Astrophysics. p. 86, doi:10.1007/978-981-16-4544-0_151-1</list_item>
<list_item><location><page_21><loc_51><loc_67><loc_84><loc_67></location>Zhang B., 2013, The Astrophysical Journal, Letters, 763, L22</list_item>
<list_item><location><page_21><loc_51><loc_64><loc_92><loc_66></location>Zhang G., Helfer T., Gagliano A. T., Mishra-Sharma S., Villar V. A., 2024, arXiv e-prints, p. arXiv:2408.16829</list_item>
<list_item><location><page_21><loc_51><loc_63><loc_89><loc_63></location>in't Zand J. J. M., et al., 2013, Astronomy and Astrophysics, 553, A83</list_item>
</unordered_list>
<figure>
<location><page_22><loc_10><loc_60><loc_45><loc_93></location>
<caption>Figure B1. Training process for the convolutional autoencoder applied to the 𝐸 -𝑡 Maps (top) and fully connected autoencoder applied to the 𝐸 -𝑡 -𝑑𝑡 Cubes (bottom). The plots show the evolution of the training and validation loss with the number of epochs including the early stopping point and epoch of the restored weights.</caption>
</figure>
<figure>
<location><page_22><loc_54><loc_58><loc_89><loc_93></location>
<caption>Figure A1. Distribution of Chandra event file lengths 𝑁 (top) and durations 𝑇 (bottom) in the dataset used in this work.</caption>
</figure>
<section_header_level_1><location><page_22><loc_7><loc_51><loc_41><loc_54></location>APPENDIX A: DISTRIBUTION OF EVENT FILE LENGTHS AND DURATIONS</section_header_level_1>
<text><location><page_22><loc_7><loc_48><loc_48><loc_50></location>Figure A1 shows the distribution of the length 𝑁 and duration 𝑇 of event files in the dataset used in this work.</text>
<section_header_level_1><location><page_22><loc_7><loc_43><loc_45><loc_44></location>APPENDIX B: AUTOENCODER TRAINING PROCESS</section_header_level_1>
<text><location><page_22><loc_7><loc_39><loc_48><loc_41></location>Figure B1 shows the training process of the autoencoders used in this work.</text>
<section_header_level_1><location><page_22><loc_7><loc_34><loc_44><loc_35></location>APPENDIX C: HYPERPARAMETER OPTIMIZATION</section_header_level_1>
<text><location><page_22><loc_7><loc_29><loc_48><loc_33></location>Below, we summarize the optimization strategy for the t-SNE and DBSCAN hyperparameters. For even more details on this approach, please refer to Dillmann & Martínez-Galarza (2023).</text>
<section_header_level_1><location><page_22><loc_7><loc_25><loc_25><loc_26></location>C1 t-SNE Hyperparameters</section_header_level_1>
<text><location><page_22><loc_7><loc_6><loc_48><loc_24></location>The choice of the perplexity and learning_rate can have a large impact on the resulting t-SNE embedding space. Ideally, we want the two-dimensional embedding space to effectively capture both energy information (in form of the hardness ratio 𝐻𝑅 ) and variability information (in form of the variability probability 𝑝 var ). That means that event files with similar values for 𝐻𝑅 and 𝑝 var should live close to each other in the final embedding space. We can use this information to define a performance metric for different t-SNE hyperparameter inputs. First, we compute the pairwise distance matrix D Z of size ( 𝑚, 𝑚 ) , where the distance 𝐷 𝑍 𝑖 𝑗 between points 𝑖 and 𝑗 is computed using a Euclidean distance metric. Next, we define the property vector Y , which includes 7 CSC properties (hardness ratios 𝐻𝑅 hm , 𝐻𝑅 hs , 𝐻𝑅 ms and variability probabilities 𝑝 b var , 𝑝 h var , 𝑝 m var , 𝑝 s var ) for</text>
<text><location><page_22><loc_51><loc_6><loc_92><loc_45></location>each event file and thus each t-SNE point. As a measure of similarity between the labels of different points, we can again compute a pairwise similarity matrix D Y of size ( 𝑚, 𝑚 ) . To compute the similarity distance 𝐷 𝑌 𝑖 𝑗 between sample 𝑖 and 𝑗 , we use the Mahalanobis distance metric (Mahalanobis 1936). Unlike the Euclidean distance metric, the Mahalanobis distance metric accounts for the correlation between different labels by taking into account the covariance structure of the data. Note that our hardness ratios are correlated with each other, and that the same holds for the variability probabilities. Accounting for these correlations provides a more accurate measure of the similarity distance between different samples. Having computed D Z and D Y , we can define a performance metric that allows us to compare the performance of different t-SNE hyperparameters. The smaller the distance 𝐷 𝑍 𝑖 𝑗 between two points 𝑖 and 𝑗 in the t-SNE embedding, the smaller should be difference in their associated labels as measured by the distance 𝐷 𝑌 𝑖 𝑗 . We can thus define a performance metric based on the statistical correlation of D Z and D Y using the Spearman's rank correlation coefficient 𝜌 𝑍𝑌 (Spearman 1904). The higher 𝜌 𝑍𝑌 , the higher is the positive correlation between D Z and D Y and the better the performance of the t-SNE embedding. The hyperparameter space is given by the ranges learning_rate ∈ ( 20 , 200 ) with a step size of 20 and perplexity ∈ ( 10 , 100 ) with a step size of 10. This optimization process is performed using a reduced dataset of 15,353 samples for 2,000 iterations per hyperparameter combination due to computational constraints. While subsampling, the overall structure of the data was preserved by selecting the same distributions between any combinations of hard, medium, soft, variable and non-variable samples. This ensures that the sample set is</text>
<text><location><page_23><loc_7><loc_90><loc_48><loc_93></location>representative of the original data. We choose the hyperparameter combination that produces the highest value of 𝜌 𝑍𝑌 .</text>
<section_header_level_1><location><page_23><loc_7><loc_87><loc_28><loc_88></location>C2 DBSCAN Hyperparameters</section_header_level_1>
<text><location><page_23><loc_7><loc_73><loc_48><loc_85></location>Different hyperparameter combinations of eps and minPts can have a large impact on the resulting DBSCAN clusters. We use a combination of the Davies-Bouldin index 𝐷𝐵 (Davies & Bouldin 1979) and Calinski-Harabasz index 𝐶𝐻 (Caliński & Harabasz 1974) as a performance metric to find the optimal DBSCAN hyperparameter inputs. The 𝐷𝐵 index is a measure of the average similarity between each cluster and its most similar cluster, relative to the average distance between points within each cluster. The 𝐷𝐵 index is given by the following formula:</text>
<formula><location><page_23><loc_7><loc_69><loc_48><loc_72></location>𝐷𝐵 = 1 𝑛 𝑐 𝑛 𝑐 ∑︁ 𝑖 = 1 max 𝑗 ≠ 𝑖 GLYPH<18> 𝑊 𝑖 + 𝑊 𝑗 𝑑 ( 𝑐 𝑖 , 𝑐 𝑗 ) GLYPH<19> , (C1)</formula>
<text><location><page_23><loc_7><loc_55><loc_48><loc_68></location>where 𝑛 𝑐 is the number of clusters, 𝑊 𝑖 and 𝑊 𝑗 are the within-cluster sum of squares for cluster 𝑖 and 𝑗 , and 𝑑 ( 𝑐 𝑖 , 𝑐 𝑗 ) is the distance between the centroids of clusters 𝑖 and 𝑗 . On the other hand, the 𝐶𝐻 index is based on the concept that good clusters should have high intra-cluster similarity (cohesion) measured by the between-cluster dispersion 𝐵 and low inter-cluster similarity (separation) measured by the within-cluster dispersion 𝑊 . 𝐵 is the sum of the pairwise distances between cluster centroids, and 𝑊 is the sum of the pairwise distances between points within each cluster. The 𝐶𝐻 index is given by the following formula:</text>
<formula><location><page_23><loc_7><loc_51><loc_48><loc_53></location>𝐶𝐻 = 𝐵 𝑊 × 𝑚 -𝑛 𝑐 𝑛 𝑐 -1 , (C2)</formula>
<text><location><page_23><loc_7><loc_37><loc_48><loc_50></location>where the scaling factor 𝑚 -𝑛 𝑐 𝑛 𝑐 -1 accounts for the total number of data points 𝑚 and the number of clusters 𝑛 𝑐 . A lower 𝐷𝐵 index and higher 𝐶𝐻 index indicate that the clustering algorithm is more effective in grouping similar data points together and separating different data points into distinct clusters. We thus define the performance metric 𝜌 𝐷𝐶 as the ratio of the normalized indices 𝐷𝐵 𝑛 = 𝐷𝐵 max ( 𝐷𝐵 ) and 𝐶𝐻 𝑛 = 𝐶𝐻 max ( 𝐶𝐻 ) in the hyperparameter space given by eps ∈ ( 1 . 0 , 3 . 0 ) with a step size of 0 . 1 and minPts ∈ ( 10 , 30 ) with a step size of 1:</text>
<formula><location><page_23><loc_7><loc_34><loc_48><loc_36></location>𝜌 𝐷𝐵𝑆𝐶𝐴𝑁 = 𝐶𝐻 𝑛 𝐷𝐵 𝑛 . (C3)</formula>
<text><location><page_23><loc_7><loc_31><loc_48><loc_33></location>We choose the hyperparameter combination that produces the highest value of 𝜌 𝐷𝐵𝑆𝐶𝐴𝑁 .</text>
<section_header_level_1><location><page_23><loc_7><loc_26><loc_28><loc_27></location>APPENDIX D: EMBEDDINGS</section_header_level_1>
<text><location><page_23><loc_7><loc_24><loc_48><loc_25></location>Figures D1, D2 and D3 show the 2D-PCA and 2D-AE embeddedings.</text>
<section_header_level_1><location><page_23><loc_7><loc_18><loc_45><loc_20></location>APPENDIX E: TRANSIENT-DOMINANT EMBEDDING CLUSTERS</section_header_level_1>
<text><location><page_23><loc_7><loc_15><loc_48><loc_16></location>Table E1 lists the transient-dominant clusters in the different embedding spaces used for the selection of transient candidates.</text>
<section_header_level_1><location><page_23><loc_7><loc_10><loc_33><loc_11></location>APPENDIX F: CATALOG COLUMNS</section_header_level_1>
<text><location><page_23><loc_7><loc_6><loc_48><loc_8></location>Table F1 shows the of X-ray transient candidate catalog column descriptions.</text>
<table>
<location><page_23><loc_52><loc_82><loc_91><loc_91></location>
<caption>Table E1. Transient-dominant clusters used to find new transient candidates.</caption>
</table>
<text><location><page_23><loc_51><loc_79><loc_91><loc_80></location>This paper has been typeset from a T E X/L A T E X file prepared by the author.</text>
<figure>
<location><page_24><loc_9><loc_70><loc_90><loc_93></location>
<caption>Figure D1. Embedding representations color-coded by 𝐻𝑅 hs for the 2D-PCA case (left) and 2D-AE case (right).</caption>
</figure>
<figure>
<location><page_24><loc_8><loc_39><loc_90><loc_65></location>
<caption>Figure D2. Embedding representations color-coded by 𝐼 𝑏 var for 2D-PCA (left) and 2D-AE (right). The bona-fide transients and XRT 200515 are highlighted.</caption>
</figure>
<figure>
<location><page_24><loc_9><loc_13><loc_90><loc_34></location>
<caption>Figure D3. Embedding clusters for the 2D-PCA case (left) and 2D-AE case (right).</caption>
</figure>
<table>
<location><page_25><loc_20><loc_15><loc_78><loc_90></location>
<caption>Table F1. Column descriptions of the catalog of X-ray transient candidates found in this work.</caption>
</table>
</document>
[
{
"title": "ABSTRACT",
"content": "We present a novel representation learning method for downstream tasks such as anomaly detection and unsupervised transient classification in high-energy datasets. This approach enabled the discovery of a new fast X-ray transient (FXT) in the Chandra archive, XRT 200515, a needle-in-the-haystack event and the first Chandra FXT of its kind. Recent serendipitous breakthroughs in X-ray astronomy, including FXTs from binary neutron star mergers and an extragalactic planetary transit candidate, highlight the need for systematic transient searches in X-ray archives. We introduce new event file representations, 𝐸 -𝑡 Maps and 𝐸 -𝑡 -𝑑𝑡 Cubes, designed to capture both temporal and spectral information, effectively addressing the challenges posed by variable-length event file time series in machine learning applications. Our pipeline extracts low-dimensional, informative features from these representations using principal component analysis or sparse autoencoders, followed by clustering in the embedding space with DBSCAN. New transients are identified within transient-dominant clusters or through nearest-neighbor searches around known transients, producing a catalog of 3,539 candidates (3,427 flares and 112 dips). XRT 200515 exhibits unique temporal and spectral variability, including an intense, hard < 10 s initial burst followed by spectral softening in an ∼ 800 s oscillating tail. We interpret XRT 200515 as either the first giant magnetar flare observed at low X-ray energies or the first extragalactic Type I X-ray burst from a faint LMXB in the LMC. Our method extends to datasets from other observatories such as XMM-Newton , Swift-XRT , eROSITA , Einstein Probe , and upcoming missions like AXIS . Key words: software: machine learning, methods: data analysis, X-rays: bursts, stars: magnetars, transients: gamma-ray bursts, stars: peculiar",
"pages": [
1
]
},
{
"title": "Representation Learning for Time-Domain High-Energy Astrophysics: Discovery of Extragalactic Fast X-ray Transient XRT 200515",
"content": "Steven Dillmann 1 , 2 ★ † , Rafael Martínez-Galarza 3 , Roberto Soria 4 , 5 , Rosanne Di Stefano 3 and Vinay L. Kashyap 3 Accepted XXX. Received YYY; in original form ZZZ",
"pages": [
1
]
},
{
"title": "1 INTRODUCTION",
"content": "Recent serendipitous discoveries, such as extragalactic fast X-ray transients (FXTs) linked to neutron star merger candidates as electromagnetic counterparts to gravitational wave events (Lin et al. 2022) and an X-ray dip associated with the first extragalactic planet candidate (Di Stefano et al. 2021), underscore the challenges of identifying such rare events within large X-ray catalogs. Beyond magnetarpowered FXTs as the aftermath of binary neutron star mergers (Dai et al. 2006; Metzger et al. 2008; Zhang 2013; Sun et al. 2017; Bauer et al. 2017; Xue et al. 2019), other interesting origins of extragalactic FXTs include supernova shock breakouts (SBOs) (Soderberg et al. 2008; Modjaz et al. 2009; Alp & Larsson 2020; Novara et al. 2020), tidal disruption events (TDEs) (Jonker et al. 2013) including quasiperiodic eruptions (QPEs) (Arcodia et al. 2021; Chakraborty et al. 2021), thermonuclear (Type I) X-ray bursts from accreting neutron stars (in't Zand et al. 2013), or binary self-lensing events (D'Orazio &DiStefano 2018, 2020; Hu et al. 2020). Both because of their very stochastic nature, and because narrow field X-ray missions such as the Chandra X-ray Observatory ( Chandra ) (Weisskopf et al. 2000), XMM-Newton (Jansen et al. 2001) and Swift-XRT (Burrows et al. 2005) are not designed as wide time-domain surveys, X-ray transient discoveries are often serendipitous. They can be found in observations that were originally proposed for a completely unrelated science objective and are rarely the target of the observation. In many cases serendipitously found X-ray sources do not get characterized or classified, since their transient nature is not immediately obvious. Instead, observations with X-ray transients often get stored in large data archives and remain unnoticed. This raises the need for a systematic search for short-duration phenomena in high-energy catalogs. New missions such as eROSITA (Predehl et al. 2021), Einstein Probe (Yuan et al. 2022) and the upcoming AXIS Observatory (Reynolds et al. 2024) target X-ray transients more directly, thus the development of novel transient detection methods is becoming even more relevant. The temporary, unpredictable and 'unusual' nature of X-ray transients distinguishes them from 'normal' X-ray source emissions. From a data science perspective, they can be understood as 'anomalies' within a large dataset. Existing methods for identifying X-ray transients primarily rely on statistical tests of variability (Yang et al. 2019; Pastor-Marazuela et al. 2020; Quirola-Vásquez et al. 2022; Quirola-Vásquez et al. 2023). While effective within specific constraints, these approaches are inherently limited by their underlying assumptions, which may not capture the diverse nature of transient phenomena. In contrast, machine learning offers a more flexible, expressive, and scalable framework, making it particularly well-suited for anomaly detection in large, high-dimensional datasets with diverse transient types. While optical time-domain surveys are at the forefront of leveraging extensive observational programs, like ZTF (Bellm et al. 2019) or the upcoming LSST survey (Ivezić et al. 2019), and neural network-based anomaly detection tools to identify rare sources among countless ordinary objects (Villar et al. 2021; Muthukrishna et al. 2022), the X-ray astronomy community has only recently begun exploring the potential of machine learning to classify sources (Yang et al. 2022; Pérez-Díaz et al. 2024) or to search for needle-in-a-haystack events in large X-ray datasets and archives (Kovačević et al. 2022; Dillmann & Martínez-Galarza 2023). The effectiveness of machine learning methods largely depends on the algorithm's ability to learn useful representations from the data. Representation learning (Bengio et al. 2013) is an increasingly popular technique in astronomy used in supervised, semi-supervised, self-supervised and unsupervised frameworks (Naul et al. 2018; Hayat et al. 2021; Walmsley et al. 2022; Slijepcevic et al. 2024; Mohale & Lochner 2024). It involves creating or learning meaningful representations for specific modalities of scientific data, which can then be used for downstream tasks such as regression, classification, or, as in this work, anomaly detection. The compressed representations live in a low-dimensional embedding space, in which anomalous data samples are well-separated from more ordinary ones. We propose a new unsupervised representation learning method to perform a large-scale search for X-ray transients in the Chandra archive. High-energy catalogs include individual X-ray source observations in the form of event files. The variable length of these time series poses a challenge in creating consistent representations suitable for transient searches with machine learning. Most deep learning algorithms take a fixed-length input for all data samples. In order to effectively represent event files over a broad range of lengths, we introduce novel fixed-length event file representations, which take into account both their time-domain and energy-domain information. Applying feature extraction and dimensionality reduction techniques, for example with sparse autoencoders, we create a representation space that encodes scientifically meaningful information, such as the spectral and variability properties of the astrophysical sources. Previously identified X-ray transients occupy distinct, well-isolated clusters in the embedding space. Using clustering techniques and nearest neighbor searches allows us to effectively explore these transient-dominant clusters to discover new X-ray transients. We collect the identified X-ray flare and dip candidates in a publicly available catalog, serving as a fertile ground for new discoveries in time-domain high-energy astrophysics. Among these candidates, we identify an intriguing extragalactic FXT, XRT 200515, which exhibits unique temporal and spectral characteristics distinct from any previously reported Chandra FXTs. The transient's initial hard < 10 s burst shows a sharp rise exceeding 4 orders of magnitude, followed by spectral softening in an ∼ 800 s oscillating tail. This transient is likely related to either a giant magnetar flare (GMF) from a distant soft gamma repeater (SGR) behind the Large Magellanic Cloud (LMC) or an extragalactic Type I X-ray burst from a faint LMXB in the LMC. Each of these interpretations presents its own set of challenges. Alternatively, XRT 200515 could be a new type of astronomical phenomenon found by our anomaly detection method using machine learning. Our method is the first representation learning approach for anomaly detection in high-energy astrophysics. It is applicable to datasets from high-energy catalogs like Chandra , XMM-Newton , Swift-XRT , eROSITA , and Einstein Probe . We created semantically meaningful representations that can be aligned with other data modalities, such as optical images or infrared spectra to design multi-modal models (Parker et al. 2024; Mishra-Sharma et al. 2024; Zhang et al. 2024; Rizhko & Bloom 2024) using contrastive learning (Radford et al. 2021), that can improve on current state-of-the-art algorithms used to characterize the physics of the associated objects. Ultimately, this work and other representation and contrastive learning approaches lay the groundwork for developing generalized foundation model in astronomy. The paper is organized as follows: In § 2, we provide information on the dataset of Chandra event files used in this analysis. In § 3, we describe in detail the implementation of our novel transient detection approach leveraging representation learning. In § 4, we present and discuss the results in form of the semantically meaningful representation space of the event files, the catalog of X-ray transient candidates and the discovery of the new Chandra transient XRT 200515. Finally, we highlight our contributions to time-domain high-energy astrophysics and outline potential directions for extending this work in the future in § 5. The relevant code, a demonstration of the pipeline, and an interactive embedding selection, transient search and lightcurve plotting tool are available online at the GitHub repository https://github.com/StevenDillmann/ml-xraytransients-mnras.",
"pages": [
1,
2
]
},
{
"title": "2 DATASET",
"content": "We use data from the Chandra Source Catalog (CSC) version 2.1 (Evans et al. 2024), which includes all publicly available X-ray sources detected by Chandra as of December 2021. For this study, we focus specifically on observations from the Advanced CCD Imaging Spectrometer (ACIS). CSC 2.1 had not been fully released at the time our analysis was performed, but catalog data was available for sources that had completed processing in the Current Database View 1 , a snapshot of which we took on 11 April 2023. CSC 2.1 performs source detection on stacked observations, and catalog properties are provided both for these stack-level detections, and for each of observation-level detection that contribute to a stack detection. Because we are interested in short-time variability that happens within a single observation of a source, we use the catalog products for the observation-level detections in our analysis. For a given X-ray detection, two types of products are provided in the CSC: (i) database tables with source properties, such as fluxes in the different X-ray energy bands, hardness ratios, variability indices, etc., and (ii) filebased data products for each detection of a source, such as the detect regions, the Chandra PSF at that location, etc. The following observation-level catalog properties are relevant for our analysis: index of 6 or larger indicates variability at a confidence of at least 2 𝜎 . In this paper we call this quantity 𝐼 𝑏 var . From the catalog data products available for observation-level Xray detections, we are interested in the region event file. This event file consists of a list of all individual photon events detected in a small bounding box around a source detection, listing their energies, arrival times, and detector coordinates. These event files are the basis for the characterization of an X-ray source: lightcurves, spectra, images, coordinates, and other properties are derived from the distribution of the listed quantities. In this analysis, we directly use these event files as our primary data products. The values of the catalog properties listed above serve as summary statistics for the detection associated with a given region event file. We only include event files with more than 5 events and a signal-to-noise ratio above 5 to minimize spurious signals from low number statistics in faint sources. We also exclude detections that are flagged for pile-up 2 , i.e., those with a pileup fraction larger than 5%, which corresponds to a maximum pileup warning of 0.1 in CSC 2.1. For the resulting detections, we filter the event files to include only events contained within the detection region for each source. These detection regions are also provided as data products in CSC 2.1, and consist of the ellipse that includes the 90% encircled counts fraction of the PSF at the source location. Due to the low background level in Chandra observations, the majority of events selected after this spatial filtering are expected to be events associated with the X-ray source, not the background. In the selected event files, we only include photon events within good time intervals (GTIs), which are time periods of valid, high-quality data. No other pre-processing is required. The final dataset consists of 95,473 filtered event files from 58,932 sources, resulting in an average of 1 . 62 observations per source. This includes 9,003 new sources that have been added as part of the CSC 2.1 release, in addition to the sources from the previous release.",
"pages": [
2,
3
]
},
{
"title": "3 METHODS",
"content": "In this work, we introduce a novel representation learning based anomaly detection method to systematically search for X-ray transients in high-energy archives. We begin with an overview of the method here and provide detailed explanations of each step in individual subsections. The full pipeline is illustrated in Figure 1. Starting with the event files described in § 2, we (i) build two novel and uniform event file representations by binning their arrival times and energies into 𝐸 -𝑡 Maps (Event File Representation I) or 𝐸 -𝑡 -𝑑𝑡 Cubes (Event File Representation II); (ii) use principal component analysis (Feature Extraction I) or sparse autoencoders (Feature Extraction II) to extract informative features from the event file representations; (iii) apply dimensionality reduction to the extracted features to create a low-dimensional embedding space; (iv) use density-based clustering to create embedding clusters that group event files with similar characteristics, for example transient behavior or certain spectral features. Previously identified transients like the extragalactic magnetar-powered flare candidate reported by Lin et al. (2022) and the extragalactic planet candidate dip reported by Di Stefano et al. (2021), shown in Figure 2, occupy well-isolated clusters in the embedding space. Exploring these clusters and conducting nearest-neighbor searches enables us to effectively find analogs to bona-fide time-domain anomalies, while at the same time grouping them according to their spectral properties. We compile the identified transient candidates in a catalog. While our approach is designed and tested using Chandra data, it is applicable to any dataset consisting of event lists, like those from other high-energy telescopes. The described transient detection approach is applied to both types of event file representations with both feature extraction methods, resulting in four different embeddings. We denote the different cases as described in Table 1.",
"pages": [
3
]
},
{
"title": "3.1 Event File Representation",
"content": "The different event files in the dataset are variable in length 𝑁 and duration 𝑇 , as shown in Appendix A. The large variation in the number of events and duration highlights the challenge in producing uniform data representations that preserve relevant information on time variability and spectral properties. While there exist machine learning architectures that take variable length inputs, the significant differences in the number of events from object to object make standardization of the inputs challenging, even when these architectures are used (Martínez-Galarza & Makinen 2022). As a first step in our analysis, we introduce 2-dimensional and 3-dimensional fixed-length representations based on an informed binning strategy for the event files, similar to the DMDT maps for optical lightcurves introduced by Mahabal et al. (2017).",
"pages": [
3
]
},
{
"title": "3.1.1 2D Histogram Representation ( 𝐸 -𝑡 Maps)",
"content": "Assume an event file with 𝑁 photons and a photon arrival time column 𝒕 with entries { 𝑡 𝑘 } 𝑁 𝑘 = 1 and energy column 𝑬 with entries { 𝐸 𝑘 } 𝑁 𝑘 = 1 . The event file duration is given by 𝑇 = 𝑡 𝑁 -𝑡 1 . The energy column entries take values in the broad energy band of Chandra 's ACIS instrument, i.e. 𝐸 𝑘 ∈ [ 𝐸 𝑚𝑖𝑛 , 𝐸 𝑚𝑎𝑥 ] , where 𝐸 𝑚𝑖𝑛 = 0 . 5 keV and 𝐸 𝑚𝑎𝑥 = 7 keV comes from considering appropriate boundaries for the energy response of Chandra 's ACIS instrument. Beyond these boundaries, the telescope's aperture effective area is low for the majority of detected sources. First, we obtain the normalized time column, given by 𝝉 = 𝒕 -𝑡 1 𝑇 , and the logarithm of the energy column, given by 𝝐 = log 𝑬 . The resulting boundaries for normalized time column are 𝝉 ∈ [ 𝜏 𝑚𝑖𝑛 , 𝜏 𝑚𝑎𝑥 ] , where 𝜏 𝑚𝑖𝑛 = 0 and 𝜏 𝑚𝑎𝑥 = 1. The range for the log-energy column is 𝝐 ∈ [ 𝜖 𝑚𝑖𝑛 , 𝜖 𝑚𝑎𝑥 ] , where 𝜖 𝑚𝑖𝑛 = log 0 . 5 keV and 𝜖 𝑚𝑎𝑥 = log 7 keV. Next, we determine the dimensionality of our representations. For a each event file, we determine the optimal number of bins in the energy dimension, 𝑛 𝜖 , with the Freedman-Diaconis rule (Freedman & Diaconis 1981), a widely used method that balances the trade-off between too noisy histograms (too many bins) and not informative enough histograms (too few bins). The optimal bin width 𝑏 𝜖 according to this rule is calculated in the following way: where 𝐼𝑄𝑅 ( 𝜖 ) represents the interquartile range of the 𝜖 values for a given event file of length 𝑁 . Subsequently, we obtain the optimal number of energy bins 𝑛 𝜖 with: For each event file, we determine the optimal number of bins in the time dimension, 𝑛 𝜏 , with the help of the Bayesian Blocks algorithm, which was specifically developed for time series analysis in astronomy (Scargle et al. 2013). This algorithm partitions the time series into adaptive width bins or blocks that are statistically distinct from neighboring blocks; that is, within a given time-ordered Bayesian block, events grouped in that block are consistent with having a similar event arrival rate. We use the default Astropy implementation of Bayesian blocks, and set the false alarm probability parameter to 𝑝 0 = 0 . 01 (Astropy Collaboration et al. 2013), which implies a 1% probability of declaring a change of rate when there is none. For each event file, we define the optimal uniform bin width 𝑏 𝜏 as the minimum bin width calculated by the Bayesian Blocks algorithm, and then find the optimal number of time bins 𝑛 𝜏 with: The optimal number of bins is different for each event file, due to their different lengths 𝑁 and durations 𝑇 . To select a bin size that can be applied to all event files, we consider the distributions of these optimal bin sizes, which are shown in Figure 3. For the distribution of 𝑛 𝜏 values we only use those event files for which 𝑝 𝑏 var > 0 . 9. The intent of this is to effectively capture variability timescales that are associated with short time-domain events, such as flares and dips. We choose the 90th percentile value of each distribution to set the final number of bins in each dimension. That is, only 10% of the event files will have an optimal number of bins that is larger than the chosen values 𝑛 𝜖 = 16 and 𝑛 𝜏 = 24. The choice of the 90th percentile, rather than the mean or mode, is motivated by the need to capture sufficient statistical detail even for long event files, while keeping the size of the resulting representations computationally tractable. Choosing a lower resolution would risk losing significant details in the representation, particularly short-duration events such as flares and dips within longer event files. The 𝐸 -𝑡 Maps are the 2D histogram representations with size ( 𝑛 𝜏 , 𝑛 𝜖 ) = ( 24 , 16 ) that result from binning the events according to the optimized number of bins. Figure 4 shows the 𝐸 -𝑡 Maps for the known extragalactic dip reported by Di Stefano et al. (2021) and known extragalactic flare reported by Lin et al. (2022).",
"pages": [
3,
5
]
},
{
"title": "3.1.2 3D Histogram Representation ( 𝐸 -𝑡 -𝑑𝑡 Cubes)",
"content": "Wenowintroduce the 𝐸 -𝑡 -𝑑𝑡 Cubes, which extend the 𝐸 -𝑡 Maps by a third dimension that serves as a proxy for the photon arrival rate. For an event file of length 𝑁 , consider the array of time differences between consecutive photon arrivals 𝚫 𝒕 with entries Δ 𝑡 𝑘 = 𝑡 𝑘 + 1 -𝑡 𝑘 for 𝑘 = 1 , 2 , . . . , 𝑁 -1. We again scale and normalize the obtained values, so that they adopt values between 0 and 1, using in each case the minimum value Δ 𝑡 𝑚𝑖𝑛 and maximum value Δ 𝑡 𝑚𝑎𝑥 . This provides the third dimension 𝜹𝝉 : The additional dimension is intended to better isolate short-duration features in time variability by capturing high photon arrival rates, which are typical of flares, as well as very low photon arrival rates, which are typical of dips. The boundaries of our histogram representations in this dimension are 𝜹𝝉 ∈ [ 𝛿𝜏 𝑚𝑖𝑛 , 𝛿𝜏 𝑚𝑎𝑥 ] , where 𝛿𝜏 𝑚𝑖𝑛 = 0 and 𝛿𝜏 𝑚𝑎𝑥 = 1. We determine the optimal number of bins in the 𝜹𝝉 dimension, 𝑛 𝛿𝜏 , again by computing the optimal bin width 𝑏 𝛿𝜏 with the Freedman-Diaconis rule and dividing the range for 𝜹𝝉 by 𝑏 𝛿𝜏 : The distribution of 𝑛 𝛿𝜏 across the event files is shown in Figure 3. Most of the relevant time-domain information is already captured by 𝝉 , but adding 𝜹𝝉 provides an additional marker for dips and flares that can be shorter than the timescales probed by our chosen binning of 𝝉 . Unlike in the other two dimensions, we choose the 75th percentile value of the distribution as our final choice of common binning, which results in 𝑛 𝛿𝜏 = 16. This is because in order to identify short transients, we need to capture strong deviations in 𝜹𝝉 only. Choosing a lower value for 𝑛 𝛿𝜏 reduces noise an improves computational tractability. Having both 𝝉 and 𝜹𝝉 represented also breaks any assumption of stationarity, in that we can be sensitive to transient events happening at any time during the observation of the source, and break degeneracies between periodic and non-periodic features in the representations presented by Martínez-Galarza & Makinen (2022). The 𝐸 -𝑡 -𝑑𝑡 Cubes are the resulting 3D histogram event file representations with size ( 𝑛 𝜏 , 𝑛 𝜖 , 𝑛 𝛿𝜏 ) = ( 24 , 16 , 16 ) .",
"pages": [
5,
6
]
},
{
"title": "3.1.3 Feature Notation",
"content": "The event file representations can now be used as inputs for various statistical learning and machine learning algorithms. For the 𝑖 𝑡 ℎ event file in the dataset of length 𝑚 = 95,473, we denote the corresponding feature vector as fi 𝑥 𝑖 = [ 𝑥 1 , 𝑥 2 , . . . , 𝑥 𝑛 ] 𝑖 , where 𝑛 = 𝑛 𝜏 · 𝑛 𝜖 = 384 for the 𝐸 -𝑡 Maps and 𝑛 = 𝑛 𝜏 · 𝑛 𝜖 · 𝑛 𝛿𝜏 = 6 , 144 for the 𝐸 -𝑡 -𝑑𝑡 Cubes. The set of all feature vectors is denoted as X = [fi 𝑥 1 , fi 𝑥 2 , . . . , fi 𝑥 𝑚 ] ⊤ with size ( 𝑚, 𝑛 ) .",
"pages": [
6
]
},
{
"title": "3.2 Feature Extraction I: Principal Component Analysis",
"content": "We use Principal Component Analysis (PCA) (Pearson 1901) provided by scikit-learn (Pedregosa et al. 2011) as our first feature extraction method. The extracted principal components should encode relevant time-domain and spectral information of the event file they represent. PCA involves transforming a dataset into a new coordinate system by finding the principal components of the data that capture most of the variance in the data. By projecting the dataset onto principal components, PCA reduces the dimensionality of the data while retaining the most important information, which increases the interpretability of high-dimensional data.",
"pages": [
6
]
},
{
"title": "3.2.1 PCA Algorithm",
"content": "We start with the feature vector set X of size ( 𝑚, 𝑛 ) representing our dataset with 𝑚 samples and 𝑛 dimensions. PCA aims to find a new coordinate system defined by a set of orthogonal axes, i.e. the principal components, that captures the maximum amount of variance in the data. The PCA result is a transformed dataset Xpc obtained by projecting X onto the principal components: where W is matrix of size ( 𝑛, 𝑛 𝑝𝑐 ) containing the first 𝑛 𝑝𝑐 principal components to be retained as its columns and Xpc is of size ( 𝑚 , 𝑛 𝑝𝑐 ) with a reduced dimensionality of 𝑛 𝑝𝑐 . For a more detailed explanation of the algorithm, we refer the reader to Jolliffe (2002).",
"pages": [
6
]
},
{
"title": "3.2.2 Principal Components Retained",
"content": "The main PCA hyperparameter is the number of principal components 𝑛 𝑝𝑐 to retain. Figure 5 shows two scree plots illustrating the amount of variance explained by each principal component in descending order and the cumulative proportion of variance explained by the principal components for both 𝐸 -𝑡 Mapsand 𝐸 -𝑡 -𝑑𝑡 Cubes. Acommon approach to determine the optimal value of 𝑛 𝑝𝑐 is to find the knee point in the cumulative scree plot of the principal components. This balances the objective of minimizing the dimensionality while retaining as much information as possible. Defining the knee point as the point beyond which adding additional principal components increases the amount of variance by less than 0 . 1% gives 𝑛 𝑝𝑐 = 15 for 𝐸 -𝑡 Maps and 𝑛 𝑝𝑐 = 22 for 𝐸 -𝑡 -𝑑𝑡 Cubes as indicated in Figure 5. These capture 94 . 1% and 89 . 9% of the variance respectively.",
"pages": [
6
]
},
{
"title": "3.3 Feature Extraction II: Sparse Autoencoder Neural Network",
"content": "As an alternative to PCA, we now build Autoencoder (Hinton & Salakhutdinov 2006) models with TensorFlow (Abadi et al. 2015) to learn a set of latent features from the 𝐸 -𝑡 Maps and 𝐸 -𝑡 -𝑑𝑡 Cubes that can be used to isolate transients and encode specific spectral properties. An autoencoder is composed of two neural networks, an encoder and a decoder, which work together to learn a compressed representation of the input data. The encoder network takes the input data and maps it to a lower-dimensional representation, often called 'latent space' or 'bottleneck'. The number of neurons in the bottleneck determines the dimensionality of the learned representation. The decoder network then aims to reconstruct the original input from this compressed representation. The decoder is typically a mirrored version of the encoder gradually upsampling the latent space until the output matches the dimensions of the original input. By minimizing the reconstruction error between input and output during training, the model learns a low-dimensional representation of the input. The bottleneck forces the encoder to capture the most important information necessary for accurate reconstruction, effectively compressing the input and learning to extract informative features in an unsupervised manner. Once the autoencoder is trained, the encoder network can be used as a standalone feature extractor to obtain a compressed representation of the input data, which can be used for downstream tasks such as clustering or anomaly detection. As opposed to PCA, which is a linear technique that works well for linearly correlated data but fails to capture complex non-linear relationships, an autoencoder is able to learn complex non-linear relationships. We design two different autoencoders to process the 𝐸 -𝑡 Maps and 𝐸 -𝑡 -𝑑𝑡 Cubes.",
"pages": [
7
]
},
{
"title": "3.3.1 Convolutional Autoencoder",
"content": "In a convolutional autoencoder (Masci et al. 2011), both the encoder and decoder network consist of convolutional layers (LeCun et al. 1998), which perform convolutions over the input using a filter. These filters are small matrix kernels with learnable weights that slide across the input, allowing the network to capture high-level features while preserving important spatial hierarchies and relationships, which is why they are often used for image-like data. This makes this architecture particularly well-suited to recognize spatial patterns such as dips or flares in our 𝐸 -𝑡 Maps. To gradually reduce the dimension of the input while it is being passed through the encoder network, we use stride convolution layers (Simonyan & Zisserman 2014) with a stride value of 2 for downsampling. This means that the learnable filter jumps two pixels at a time as it slides over the input. The output of the convolutional layers is a feature map, which is then flattened to a feature vector and passed through a series of fully connected layers, where every neuron in the previous layer is connected to every neuron in the next layer. These fully connected layers are responsible for mapping the learned features to a lower-dimensional latent representation in the bottleneck and perform non-linear transformations while downsampling through the use of non-linear activation functions. The final latent space has 𝑛 𝑎𝑒 = 12 elements, representing the most essential features of the input data, which can now be used for further downstream tasks. Figure 6 shows a diagram of the encoder part of the model and Table 2 summarizes its architecture.",
"pages": [
7
]
},
{
"title": "3.3.2 Fully Connected Autoencoder",
"content": "Our 𝐸 -𝑡 -𝑑𝑡 Cubes introduce an additional dimension resulting in sparse 3D input data. Convolutional layers assume regular grid-like data, making them less effective for handling sparse data. Moreover, very expensive 3D convolutional operations would substantially increase complexity of the model. Therefore, we use a simple fully connected autoencoder for the 𝐸 -𝑡 -𝑑𝑡 Cubes. Its encoder network consists of a series of fully connected layers, which gradually map the original input data to a latent space with 𝑛 𝑎𝑒 = 24 elements. Figure 6 shows a diagram of the encoder part of the model and Table 3 summarizes its architecture.",
"pages": [
7
]
},
{
"title": "3.3.3 Activation Functions",
"content": "Neural networks are able to learn and represent complex non-linear relationships due to the introduction of non-linear activation func- tions within their layers. An activation function is a mathematical function used in a neural network to determine whether a neuron should be activated or not, based on its input. It essentially decides how much of the input signal should pass through the neuron, producing an output that can either be passed to the next layer or used to make predictions. The popular Rectified Linear Unit (ReLU) activation function 𝑅𝑒𝐿𝑈 ( 𝑥 ) = max ( 0 , 𝑥 ) (Nair & Hinton 2010) is simple and computationally efficient. To mitigate any potential encounters of the 'dying the ReLU problem', where neurons become non-responsive during training, we choose an extended version called Leaky ReLU (Maas et al. 2013): where 𝛼 = 0 . 1 is a hyperparameter that defines the slope of the function for negative input values. ReLU sets all negative values in the input to zero, while Leaky ReLU allows a small negative slope for negative inputs, which can help prevent neurons from dying. As for the output layer, we want any values to be mapped to a range between 0 and 1, which is achieved by using the sigmoid activation function:",
"pages": [
7,
8
]
},
{
"title": "3.3.4 Loss Function and Sparsity Regularization",
"content": "In order to encourage the autoencoder to generate reconstructions close to the original inputs, we use the mean squared error ( 𝑀𝑆𝐸 ) as as a measure of the reconstruction quality given by: where 𝑥 𝑖 is the 𝑖 𝑡 ℎ element of the input vector and ˆ 𝑥 𝑖 is the corresponding is reconstructed output. The 𝑀𝑆𝐸 is a straightforward measure of reconstruction error, and its differentiability allows efficient gradient computation for updating model weights via gradient-based optimization. Our neural networks are so called sparse autoencoders (Ng et al. 2011), which promote sparsity in the learned representation, meaning only a small subset of the neurons in the network are active at any given time. Sparse representations are valuable for our work because they help extract highly informative features from the input, while disregarding irrelevant or noisy information. To encourage sparsity in the latent space, we introduce a L1 regularization term in the objective, resulting in the following loss function: where 𝜆 = 0 . 1 is the regularization strength and 𝑤 𝑗 are the individual bottleneck weight values of which there are 𝑛 𝑤 in total. L1 regularization pushes small weights to zero and thus helps the model prioritize the most significant features of the input data, leading to a semantically meaningful latent space.",
"pages": [
8
]
},
{
"title": "3.3.5 Training",
"content": "Starting with the original dataset with a 𝑚 = 95,473 samples and using a test split of 0 . 1 gives us a training and validation set of length 85,925 and a test set of length 9,548. Further using a validation split of 0 . 2, gives 68,740 samples for training and 17,185 for validation. We run the training process for a maximum of 200 epochs with a batch size of 1,024 samples. The initial learning rate was set to 0 . 01 along with an on plateau learning rate scheduler, which dynamically reduces the learning rate by a factor of 0 . 1 if the validation loss plateaus for longer than 10 epochs. Reducing the learning rate when a plateau is detected can help escape local minima in the loss surface and converge to a more optimal solution in the parameter space. This scheduler is used in combination with the Adaptive Moment Estimation (Adam) optimizer (Kingma & Ba 2014), which is a stochastic gradient descent algorithm combining the benefits of both adaptive learning rates (Duchi et al. 2011) and momentum-based optimization techniques (Sutskever et al. 2013). Finally, we use an early stopping callback to monitor the validation loss. It automatically interrupts the training process if the validation loss does not improve for 25 epochs and restores the weights of the model to the best observed weights during training. The training process for both autoencoder models is shown in Appendix B. Once the autoencoder is trained, we can use the encoder to transform the original dataset X to the feature vector space Xae of size ( 𝑚 , 𝑛 𝑎𝑒 ) with a reduced dimensionality of 𝑛 𝑎𝑒 features.",
"pages": [
8
]
},
{
"title": "3.4 Dimensionality Reduction",
"content": "Using t-SNE (Maaten & Hinton 2008), short for t-Distributed Stochastic Neighbor Embedding, we create two-dimensional embeddings of the informative features previously extracted from the event file representations using PCA or sparse autoencoders. The t-SNE algorithm is a method used to map the input data onto a low-dimensional embedding space, and is particularly useful for the visualization of clusters and patterns in high-dimensional datasets. Each high-dimensional sample is transformed into a low-dimensional embedding in such a way that similar object are nearby points, while dissimilar objects are distant points in the embedding space. Essentially, it aims to capture the local structure of the data by preserving the pairwise similarities between objects while mapping them to a lower-dimensional embedding space.",
"pages": [
8
]
},
{
"title": "3.4.1 Algorithm",
"content": "We use our informative features, X if = Xpc or X if = Xae , as input to the t-SNE algorithm to reduce the data to a two-dimensional embedding, denoted as Z . First, t-SNE creates a probability distribution 𝑃 for pairs of high-dimensional data points in X if , assigning higher probabilities to similar pairs and lower probabilities to dissimilar ones. This is done by modeling pairwise similarities using a Gaussian kernel with a specific perplexity parameter, which controls the effective number of neighbors considered for each point. Next, t-SNE defines a similar probability distribution 𝑄 for the pairwise similarities in the low-dimensional space Z , modeled using a Student's t-distribution. The goal of t-SNE is to minimize the difference between 𝑃 and 𝑄 using gradient descent, with the Kullback-Leibler (KL) divergence (Kullback & Leibler 1951) as the cost function: where 𝑃 𝑖 𝑗 and 𝑄 𝑖 𝑗 represent pairwise similarities in the high- and low-dimensional spaces, respectively. The algorithm iteratively adjusts the low-dimensional embedding Z to minimize the KL divergence, often requiring hundreds to thousands of iterations for convergence. The result of this optimization is a two-dimensional representation Z of size ( 𝑚, 2 ) , where similar points in the high-dimensional space are clustered closely together.",
"pages": [
8,
9
]
},
{
"title": "3.4.2 Hyperparameter Optimization",
"content": "The t-SNE algorithm has a number of important hyperparameters to be tuned. The two most important parameters are the perplexity and the learning_rate . The perplexity parameter controls the balance between capturing the local versus global structure in the data, while the learning_rate controls the step size at each iteration of the optimization process. The n_iter parameter is the number of iterations. To ensure reproducibility, we set a fixed random_state . Our t-SNE hyperparameter optimization approach is detailed in Appendix C. A summary of the final t-SNE hyperparameters is provided in Table 4.",
"pages": [
9
]
},
{
"title": "3.5 Clustering",
"content": "The next step is the identification of individual clusters in the embedding space using DBSCAN (Hartigan & Wong 1979), short for Density-Based Spatial Clustering of Applications with Noise. Unlike traditional clustering algorithms such as k-means, DBSCAN does not require the number of clusters to be specified, as it identifies dense regions in the data space based on a density criterion.",
"pages": [
9
]
},
{
"title": "3.5.1 Algorithm",
"content": "We use our t-SNE embedding space Z as input to the DBSCAN algorithm, which segments the embedding space into multiple clusters. The DBSCAN algorithm has two main hyperparameters. The eps parameter defines the radius of the neighborhood surrounding each point in the dataset, while the minPts parameter specifies the minimum number of points required within this neighborhood for a data point to be classified as a core point. A border point is defined as a point that is in the vicinity of at least one core point but has fewer than minPts within its neighborhood. All other points are considered to be noise points. Clusters are then created from the aggregation of core points and their associated border points, with noise points being categorized as outliers. Figure 7 visualizes the clustering method.",
"pages": [
9
]
},
{
"title": "3.5.2 Hyperparameter Optimization",
"content": "Our DBSCAN hyperparameter optimization approach is detailed in Appendix C. A summary of the final t-SNE hyperparameters is provided in Table 5.",
"pages": [
9
]
},
{
"title": "3.6 Previously Reported Transients",
"content": "We highlight the embeddings of previously reported bona-fide transients, listed in Table 6, in our low-dimensional representation space to identify transient-dominant clusters. The flares include extragalactic FXTs reported by Jonker et al. (2013), Glennie et al. (2015), Yang et al. (2019), Lin et al. (2021), Lin et al. (2022), Quirola-Vásquez et al. (2022) and a set of stellar flares found in the dataset by manual inspection. The dips include the extragalactic planet candidate in M51 reported by Di Stefano et al. (2021), the ultraluminous X-ray source (ULX) 2E 1402 . 4+5440 in NGC 5457 (Colbert & Ptak 2002; Swartz et al. 2004) and the well-studied eclipsing and bursting lowmass X-ray binary (LMXB) EXO 0748 -676 (Parmar et al. 1986; D'Aì et al. 2014). These transients occupy well-isolated clusters. Exploring transient-dominant clusters and performing nearest-neighbor searches around known transients allows us to find new transients.",
"pages": [
9
]
},
{
"title": "3.7 Candidate Selection",
"content": "Newtransients are identified in embedding clusters containing previously reported transients. For well-isolated clusters containing known discovered transients, we use the entire cluster to define new transient candidates. The well-isolated transient-dominant clusters used for candidate selection are listed in Appendix E. However, in a few cases known discovered transients reside within larger poorly separated clusters. Selecting the entire cluster would result in a high number of false positives. To address this, we instead use the k-nearest neighbors ( kNN ) algorithm (Cover & Hart 1967), identifying the 50 nearest neighbors for each known transient residing in a poorly separated cluster to define additional transient candidates.",
"pages": [
9,
10
]
},
{
"title": "3.8 Cross Matching",
"content": "We use an existing cross-match table (Green et al. 2023) between CSC2.1 and five other catalogs - Gaia DR3 (Gaia Collaboration et al. 2021), DESI Legacy Survey DR10 (Dey et al. 2019), PanSTARRS-1 (Chamberset al. 2016), 2MASS (Skrutskie et al. 2006), and the SDSS DR17 catalog - to complement the X-ray properties derived from the CSC with additional multi-wavelength observations. This includes catalog identifiers, positions, magnitudes, source type classifications and other columns. We cross-matched our transient candidates with the SIMBAD database (Wenger et al. 2000) by associating each candidate with the nearest SIMBAD object, provided the object is located within a 5 arcsec radius of the candidate's coordinates listed in the CSC. The multi-wavelength observations of the transient candidates provide valuable information for their characterization and classification.",
"pages": [
10
]
},
{
"title": "4 RESULTS AND DISCUSSION",
"content": "We now present the results of applying the methods in § 3 to the set of representations of X-ray event files in the dataset from § 2.",
"pages": [
10
]
},
{
"title": "4.1 Representation Embedding Space and Clusters",
"content": "Figure 8 shows the t-SNE embedding space for the 3D-PCA and 3DAE cases color-coded by the hardness ratio 𝐻𝑅 hs . The embedding space for the other two cases, 2D-PCA and 2D-AE , are shown in Appendix D. The observed hardness ratio gradients in all embedding spaces indicate that the learned representations effectively encode spectral information, in particular at the level of individual clusters, allowing for the identification of X-ray sources with specific spectral signatures. For the 2D-PCA and 2D-AE cases, these gradients are more uniform across the embedding space, because the temporal and spectral information of event files are captured by one axis each in the 𝐸 -𝑡 Maps. Moreover, some clusters consist exclusively of soft or hard sources, demonstrating that our representations can be leveraged not only to identify transients but also to find analogs to sources with specific spectral characteristics. Figure 9 shows the 3D-PCA and 3D-AE embedding spaces, now color-coded by the variability index 𝐼 𝑏 index with the other two cases shown in Appendix D. The learned embeddings also encode the temporal behavior of the sources, with some clusters being dominated by X-ray detections with significant variability, including transient behavior. To demonstrate this, we also highlight the embeddings of the bona-fide flares and dips listed in Table 6. Note that these occupy very well-defined clusters on the edges of the representation space, allowing for queries of analog transient behavior. In the 2D-PCA and 2D-AE cases, transient sources are distributed across multiple small clusters on the edges of the embedding spaces. In contrast, the 3D-PCA and 3D-AE embedding spaces achieve a significantly more compact clustering of bona-fide transients because temporal features in the event files are given a higher importance by the introduction of an additional time-related axis in the 𝐸 -𝑡 -𝑑𝑡 Cubes. Figure 10 shows the clusters identified by the DBSCAN algorithm in the 3D-PCA and 3D-AE cases. The clusters for the other two cases, 2D-PCA and 2D-AE , are shown in Appendix D. The largest cluster in all cases (Cluster 1) corresponds to observations that are not 'anomalous', for example non-variable sources or noisy detections in the low-count regime. We also see multiple smaller clusters on the edges of the embedding space clearly separated from this main cluster. Of special interest are clusters that contain known discovered transients, as these likely host other interesting transients that have not yet been discovered. Some of the edge clusters group observations with similar temporal and spectral behavior. For example, Cluster 4 in the 3D-PCA case only contains flares with high hardness ratios. Other clusters instead group observations primarily by similar temporal behavior, but then show a within-cluster grouping of similar spectral behaviors. For example, Cluster 4 in the 3D-AE case contains many dipping sources, but show a hardness ratio gradient within the cluster. When comparing the results of different feature extraction methods, we observe that in the 3D-AE embedding space, nearly all previously identified extragalactic FXTs live within a single, wellisolated cluster (Cluster 8). In contrast, the 3D-PCA embedding space distributes these extragalactic FXTs across multiple clusters. All of these points underline the effectiveness of our method and that the created representation space is highly informative.",
"pages": [
10,
13
]
},
{
"title": "4.2 Catalog of X-ray Flare and Dip Candidates",
"content": "We identify new transient candidates within clusters that are occupied by previously reported transients and by conducting nearestneighbor searches around these known transients. We compile these in a catalog of X-ray transient candidates, which includes both flares and dips. Table E1 lists the selected clusters used to define the new flare and dip candidates in addition to the 50 nearest neighbors of each bona-fide transient. From each selected cluster, we only include X-ray detections for which the variability index 𝐼 𝑏 var ≥ 5. This threshold corresponds to detections for which the Gregory-Loredo algorithm yields a confidence in the variability of at least 90%, and allows us to discard flare and dip-like behaviors that are not statistically significant. We also manually exclude a small fraction of false positives identified by visual inspection of the lightcurves for both flare and dip candidates. The resulting catalog contains a total of 3539 detections, and the columns included are described in Appendix F. Table 7 shows the first 5 samples in our catalog for a subset of selected columns. Figure 11 shows a number of example lightcurves of the dips and flares in our catalog. The dip selection shows dips from LMXBs, a low-mass X-ray binary (HMXB), an ULX, an eclipsing binary, a cataclysmic binary, and a quasar. The flare selection shows flares from an eruptive variable, a pulsar, an AGN, a HMXB, a cataclysmic variables and young stars. We also find a number of pulsating or quasi-periodic lightcurves from pulsars, magnetic cataclysmic variables and SGRs. Figure 12 shows the distribution of SIMBAD object types in our transient catalog. About 25% of the transient candidates do not have a SIMBAD match, making them particularly interesting sources for new transient discoveries. Our dip candidates include 6 Chandra observations with prominent dips from the known source CXOGlb J002400.9 -720453 in the globular cluster NGC 104 (47 Tuc). The catalog identifiers for these are CATALOG_ID: 2737_139, 16527_79, 15747_79, 16529_79, 15748_79, 16528_14 . Our flare candidates include a newly discovered extragalactic FXT, which is characterized and discussed in detail in § 4.3. Its catalog identifier is CATALOG_ID: 23022_122 . We recommend using our catalog to identify a diverse range of flares and dips. While this work is primarily motivated by the discovery of new extragalactic transients, we intentionally did not exclude galactic stellar flares to enable systematic follow-up studies to study flare incidence rates and the rotational evolution of stars. Users interested exclusively in extragalactic transients can filter out galactic sources using metadata from the CSC and the cross-match columns in the catalog.",
"pages": [
13
]
},
{
"title": "4.3 XRT 200515: A New Extragalactic Fast X-ray Transient",
"content": "Among the flare candidates in our catalog, we discovered an intriguing new extragalactic Chandra FXT in an observation of the supernova remnant SNR 0509 -67.5 in the LMC on May 15, 2020 (Guest et al. 2022). What made this transient stand out from thousands of other flares discovered in this work is the unique temporal variability in its lightcurve, which exhibits no detectable pre-flare X-ray emission, a sharp rise of at least 4 orders of magnitude in the count rate to peak intensity followed by a sharp fall, all in a matter of a < 10 s, down to ∼ 800 s long oscillating tail. There is also notable spectral variability during the flare, characterized by an initially hard spectrum at the peak, followed by spectral softening in the tail. The combination of these temporal and spectral properties establishes this transient as the first of its kind within the sample of discovered Chandra FXTs. We designate this newly discovered FXT as XRT 200515 and present a detailed study and discussion of its potential origins.",
"pages": [
14
]
},
{
"title": "4.3.1 X-Ray Detection by Chandra",
"content": "The transient XRT 200515 was detected in Chandra ObsID 23022. The target of the observation was the supernova remnant SNR 0509 -67.5 in the LMC, which is shown in Figure 13 alongside the newly discovered FXT event. Table 8 summarizes the properties of XRT 200515 and its associated Chandra source 2CXO J051117.2 -672556 in ObdID 23022. The transient was captured by the ACIS camera in the S4 chip, and is located significantly off-axis in this observation, at an angular distance of 11.75 arcmin from the aimpoint in the S3 chip. This leads to an elongated and relatively large PSF, which, in this case, is advantageous as it substantially reduces photon pile-up in the initial spike, by spreading the counts over many pixels. We processed the data of Chandra observation ObsID 23022 with the Chandra Interactive Analysis of Observations (/c.pc/i.pc/a.pc/o.pc) version 4.15 (Fruscione et al. 2006), with calibration data base version 4.9.8. In particular, we created a new level-2 event file with the /c.pc/i.pc/a.pc/o.pc task chandra_repro and filter it in energy and time with dmcopy . We obtained the sky position in Table 8 using the /c.pc/i.pc/a.pc/o.pc tool wavdetect . To reduce background noise and improve the determination of the source centroid, we applied wavdetect on an image filtered to include only the time interval from the beginning of the flare ( 𝑡 0 ) until a time 𝑡 0 + 920 s. The 90% uncertainty radius of 2.0 arcsec is the combination of the uncertainty in the source centroid position reported by wavdetect , and the absolute astrometry uncertainty in a typical ACIS observation for off-axis sources 3 . The field was previously covered by four other Chandra observations (ObsIDs 776, 7635, 8554, and 23023) with no source detections at the location of 2CXO J051117.2 -672556. We estimated modelindependent upper limits to the source flux and luminosity with /c.pc/i.pc/a.pc/o.pc tool srcflux . In the pre-flare part of ObsID 23022, we obtained a 90% confidence limit of 𝐿 X < 1 . 0 × 10 34 erg / s in the 0.3-7 keV band at the LMC distance of 50 kpc. Stacking the data from all the ObsIDs with non-detections, including the pre-flare part of ObsID 23022, results in a total observed exposure of approximately ∼ 150 ks, and yields a 90% confidence upper limit on the X-ray luminosity is 𝐿 X < 3 × 10 33 erg / s.",
"pages": [
14
]
},
{
"title": "4.3.2 X-ray Temporal Analysis",
"content": "We used the /c.pc/i.pc/a.pc/o.pc tool dmextract to extract background-subtracted lightcurves in several energy bands, from the reprocessed event file of Chandra ObsID 23022. We defined an elliptical source extraction region, with semi-minor and semi-major axes of 15 arcsec and 20 arcsec (matching the point-source PSF at the source location); the local background region was chosen in the same ACIS chip, with an area approximately eight times larger. Figure 14 shows the 0.3-7 keV background-subtracted lightcurve of XRT 200515 with a time resolution of 20 s. The lightcurve is consistent with no source detection at the location of the transient, before the start of the flare at around 23.5 ks into the observation. The few pre-flare counts are consistent with background noise. The lightcurve exhibits a strong initial spike with a sharp rise of at least 4 orders of magnitude in < 10 s, containing 44 out of all ∼ 180 flare counts. This initial burst is followed by a sudden drop to a ∼ 800 s long pulsating and decaying tail. We estimate a 𝑇 90 ∼ 580-740 s for the photons observed in the 0.3-7 keV band 4 , depending on the definition of total flare counts. Figure 15 shows the lightcurve of XRT 200515 at a resolution matching the ACIS frame time of 3.2 s, the hardness ratio, and the energy evolution for the time interval 𝑡 0 + 920 s. The lightcurve exhibits a spike in the count rate across only 3 bins (with a total of 4, 31 and 9 counts, respectively), hence the burst duration of < 10 s. The rise and fall times of the burst are both between 3.2 s and 6.4 s. The maximum count rate at the Chandra frame time resolution is ∼ 9.7 counts/s, acting as the lower bound for the peak count rate of the burst. Those counts are spatially spread over a PSF area of ∼ 3000 pixels; therefore, pile-up is not an issue. We evaluated the hardness ratio evolution during the flare with the Bayesian estimation method BEHR (Park et al. 2006). Here, the hardness ratio is defined as: where 𝑠 is the number of soft photons (0.3-1.2 keV), 𝑚 is the number of medium photons (1.2-2 keV), and ℎ is the number of hard photons (2-7 keV) in each bin. We also track the running average of the photon energies during the flare with a moving window of ± 10 counts. The hardness ratio and energy evolution indicate spectral softening during the flare, with the hardness ratio starting at 1 during the hard burst peak and decreasing to a range of 0.4 to 0.6 in the tail, highlighting the notable spectral variability of XRT 200515.",
"pages": [
14,
15
]
},
{
"title": "4.3.3 X-ray Spectral Analysis",
"content": "We used the /c.pc/i.pc/a.pc/o.pc tool specextract to extract the spectrum and the associated response and ancillary response files from the reprocessed event file of Chandra ObsID 23022. We used the same source and background extraction regions defined for the lightcurve extraction. To improve the signal-to-noise ratio of the source, we extracted the spectrum only from the time interval 𝑡 0 + 920 s. We binned the spectrum to a minimum of 1 count per bin with the grppha task within the /f.pc/t.pc/o.pc/o.pc/l.pc/s.pc package suite (Blackburn 1995) from NASA's High Energy Astrophysics Science Archive Research Center (HEASARC) 5 . For all spectral modelling and flux estimates, we used the /x.pc/s.pc/p.pc/e.pc/c.pc software version 12.13.0 (Arnaud 1996). With only 179 net counts, we are unable to fit complex spectral models; thus, we limit our analysis to the simplest one-component models representative of opposite scenarios: a power law ( powerlaw ) and a blackbody model ( bbody ), both modified by photo-electric absorption ( tbabs ). In both cases, we adopted the Tuebingen-Boulder absorption model with Wilms abundances (Wilms et al. 2000). We minimized the Cash statistic (Cash 1979), as we do not have enough counts for 𝜒 2 fitting. The best-fitting power-law model (Table 9 and Figure 16) has a photon index of Γ = 0 . 5 ± 0 . 3. The fit statistics yield a null hypothesis probability of 3 . 5 × 10 -3 , with a Cstat value of 132.7 for 137 degrees of freedom. For the blackbody model, the best-fitting temperature is 𝑘𝑇 bb = 1 . 8 ± 0 . 3 keV (Table 9). The fit statistics yield a null hypothesis probability of 1 . 2 × 10 -2 , with a Cstat value of 129.6 for 137 degrees of freedom. The reason this blackbody spectrum may appear hard in the Chandra band, resembling a Γ ∼ 0 . 5 power law, is that at a temperature of 𝑘𝑇 bb ∼ 2 keV, the ACIS detector samples only the peak and the Rayleigh-Jeans (rising) portion of the blackbody emission. We can use either model to determine an average conversion between the count rate and luminosity. This will then enable us to estimate the peak luminosity in the initial spike, for which we have previously estimated a peak count rate of ≳ 10 counts/s. The best-fitting power law model implies a peak flux of 𝐹 p ≳ 5 . 6 × 10 -10 erg/s/cm 2 , a total flare fluence of 𝐸 f ≳ 1 . 1 × 10 -8 erg/cm 2 , and a peak unabsorbed 0.3-10 keV luminosity of 𝐿 X ≳ 1 . 7 × 10 38 erg/s at the LMC distance of 50 kpc. For the best-fitting blackbody model, the peak flux and flare fluence would be 𝐹 p ≳ 4 . 0 × 10 -10 erg/s/cm 2 and 𝐸 f ≳ 0 . 8 × 10 -8 erg/cm 2 respectively. The peak unabsorbed 0.3-10 keV luminosity would be 𝐿 X ≳ 1 . 2 × 10 38 erg/s and the peak bolometric luminosity would be 𝐿 bol ≳ 1 . 5 × 10 38 erg/s. These values should be considered conservative lower limits for two reasons: (i) the peak count rate provides only a lower bound estimate, as it is constrained by the Chandra frame time resolution of the observations, potentially underestimating the true peak count rate; and (ii) the conversion factor applied is derived from the average spectrum over the entire flare, even though the spectrum of the initial spike is significantly harder compared to the tail, as shown in Figure 15. We searched for potential detections of XRT 200515 by other highenergy facilities. However, no significant X-ray or 𝛾 -ray events in the field around the X-ray source coordinates and flare start time 𝑡 0 reported in Table 8 were detected by the Fermi Gamma-ray Space Telescope ( Fermi ), the Burst Alert Telescope ( BAT ) on the Neil Gehrels Swift Observatory ( Swift ), the International Gamma-Ray Astrophysics Laboratory ( INTEGRAL ), or the Monitor of All-sky X-ray Image MAXI . LIGO was not operational during the time of the FXT, hence no gravitational wave signal could have been detected if the origin of XRT 200515 was a compact object merger.",
"pages": [
15,
16
]
},
{
"title": "4.3.5 Optical Counterpart Search",
"content": "We used the X-ray source coordinates reported in Table 8 to search for optical and infrared counterparts to XRT 200515. The field of XRT 200515 was covered by the Survey of Magellanic Stellar History ( SMASH ) (Nidever et al. 2017), a deep optical survey in the ugriz bands with the Dark Energy Camera (DECam) mounted on the Víctor M. Blanco Telescope at the Cerro Tololo Inter-American Observatory (CTIO) in Chile. We used the Astro Data Lab Jupyter Notebook server (Nikutta et al. 2020; Juneau et al. 2021) to access and visualize the SMASH catalog 6 . Figure 17 shows a color image of the field created from the deepest available stacked images in the u , g and i bands; the 5 𝜎 detection limits in these bands are 23.9 mag, 24.8 mag and 24.2 mag, respectively. The images were taken on December 7, 2015 with exposure times of 1,179 s, 981 s and 1,179 s respectively. The astrometry of the SMASH images is calibrated on the Gaia DR3 reference frame, thus their positional uncertainty is negligible compared to the X-ray source position uncertainty. Within the Chandra position error circle in Figure 17, there is no obvious optical counterpart that stands out in brightness or color from the surrounding stellar population. We performed relative photometry on the sources inside the error circle, comparing them to several nearby sources with known positions and brightnesses listed in the Gaia DR3 catalog. We used the SMASH g band as the closest approximation to Gaia's G band. We estimate the brightest optical source within the error circle to have a Vega magnitude of 𝑔 = 22 . 7 ± 0 . 1 mag, corresponding to an absolute magnitude of 𝑀 𝑔 ≈ 4 . 2, assuming it is in the LMC. Additionally, three other point-like sources are detected with 𝑔 band magnitudes in the range of 23-24 mag. All four sources appear pointlike, consistent with the seeing conditions of the SMASH survey, with no evidence of any spatially extended background galaxies. The three brightest stars visible in Figure 17 within ∼ 12 arcsec of the Chandra source are solar-mass stars on the red giant branch, indicative of an old stellar population. The lack of bright optical counterparts and the short burst duration of < 10 s rules out a stellar flare from a foreground Galactic low-mass star (Güdel 2004; Reale 2007; Reale & Landi 2012; Pye et al. 2015; Kuznetsov & Kolotkov 2021). A flare from a Be/X-ray binary or any other HMXB in the LMC is also excluded by the lack of a bright optical counterpart (Ducci et al. 2019, 2022). The temporal and spectral properties of XRT 200515, combined with the absence of an optical counterpart, suggests three possibilities: (i) a relativistic jet phenomenon, such as a 𝛾 -ray burst (GRB); (ii) a rapid, high-energy process linked to extreme magnetic fields, such as a giant magnetar flare (GMF); or (iii) a thermonuclear Type I X-ray burst caused by surface nuclear burning on a neutron star.",
"pages": [
16
]
},
{
"title": "4.3.6 Gamma Ray Burst from a Compact Object Merger?",
"content": "Evidence in favor or against the association of at least some Chandra FXTs with low-luminosity long-GRBs or off-axis short-GRBs (see Berger 2014 for a review), at moderate or high redshifts, is extensively discussed in Quirola-Vásquez et al. (2022), Quirola-Vásquez et al. (2023), and Wichern et al. (2024). A detailed re-investigation of this issue is beyond the scope of this work. Here, we simply point out that XRT 200515, like the other Chandra FXTs in the literature, does not have any 𝛾 -ray detection. On the other hand, XRT 200515 has a significantly harder spectrum ( Γ = 0 . 5 ± 0 . 3) in the Chandra band than the rest of the FXT sample, all of which have photon indices of Γ > 1 (Jonker et al. 2013; Glennie et al. 2015; Bauer et al. 2017; Xue et al. 2019; Lin et al. 2022; Quirola-Vásquez et al. 2022; Quirola-Vásquez et al. 2023; Eappachen et al. 2023). A photon index of Γ ∼ 0 . 5 below 10 keV is indeed expected and observed from both core-collapse GRBs and compact-merger GRBs (Ghirlanda et al. 2009; Bromberg et al. 2013; Oganesyan et al. 2018; Ravasio et al. 2019; Toffano et al. 2021). This might support the association of the initial spike of XRT 200515 with a GRB. However, the presence and properties of the ∼ 800 s tail (candidate GRB afterglow) is puzzling. The 𝑇 90 ∼ 580-740 s value for XRT 200515 is significantly shorter than in most other Chandra FXTs (Quirola-Vásquez et al. 2022; Lin et al. 2022; Quirola-Vásquez et al. 2023), which have 𝑇 90 values on the order of several ks and are already pushing the limit for a GRB afterglow detection (Wichern et al. 2024). Moreover, XRT 200515's initial burst duration ( < 10 s), its short rise and fall times (3.2-6.4 s), and the lack of a peak plateau are inconsistent with the lightcurves of Chandra FXTs interpreted as magnetar-powered GRBs as the aftermath of a binary neutron star merger, such as CDF-S XT1 (Bauer et al. 2017), CDF-S XT2 (Xue et al. 2019) and the sample in Lin et al. (2022). Finally, the lack of any optical evidence for a host galaxy is another element disfavoring the high-redshift GRB interpretation. 4.3.7 Giant Magnetar Flare from a Soft Gamma Repeater? Based on its temporal and spectral variability, it is tempting to interpret XRT 200515 as a rare GMF from a SGR (Mereghetti 2008; Turolla et al. 2015) in the LMC or behind it, which can easily explain the burst's strong increase of at least 4 orders of magnitude in < 10 s (Coti Zelati et al. 2018). Similar to XRT 200515, GMFs are characterized by a short and hard initial spike and a longer and softer, pulsating tail. GMFs are extremely rare, with only a select few ever discovered. Well-studied examples are SGR 0526 -66 in the LMC (Mazets et al. 1979), and the Galactic sources SGR 1900 + 14 (Hurley et al. 1999) and SGR 1806 -20 (Hurley et al. 2005; Palmer et al. 2005; Israel et al. 2005). More recently, GMFs have been identified in M 31 (Mazets et al. 2008a), NGC 253 (Fermi-LAT Collaboration et al. 2021; Svinkin et al. 2021; Roberts et al. 2021; Trigg et al. 2024) and M82 (Mereghetti et al. 2024). All of these have been observed by high time resolution instruments in the hard X-rays and soft 𝛾 -rays with luminosities above 10 46 erg/s for a fraction of a second in the initial spike. The tails of GMFs are often modulated by magnetar spin periods of 2-12 s, leading to quasi-periodic oscillations (QPOs). For XRT 200515, there is no hard X-ray or 𝛾 -ray detection, despite the LMC direction being in good visibility for most of the previously mentioned high-energy facilities. We were unable to identify any significant periodicities in the tail of XRT 200515 through periodogram analysis, which is unsurprising given the low time resolution of Chandra observations. No X-ray activity has been observed by Chandra or other X-ray telescopes in the years before or after XRT 200515, which may be because SGRs are very faint when they are not bursting. The strongest argument against a magnetar in the LMC as the origin of XRT 200515 is that magnetars are short-lived objects ( ≲ 10 5 yr) associated to young stellar populations (Olausen & Kaspi 2014; Nakano et al. 2015; Mondal 2021). Even allowing for the persistence of magnetar-like activity in ordinary radio pulsars as old as ∼ 10 7 yr (Rea et al. 2010), this scenario is still inconsistent with the old stellar population (several Gyr) in the LMC field shown in Figure 17. The nearest star-forming regions in the LMC are ∼ 10 arcmin ( ∼ 150 pc) away. If (in a very contrived scenario), we assume that XRT 200515 is powered by a young neutron star ejected from one of those regions, we estimate a characteristic time of 1 Myr to travel that distance at a speed of 150 km/s. Therefore, if XRT 200515 is a GMF, it must be located behind the LMC, in a low-redshift galaxy (Hurley et al. 2005; Tanvir et al. 2005). Since GMFs have been observed only a few times and never at soft X-ray energies, their properties in the soft X-ray band detectable by Chandra remain largely unexplored. XRT 200515 could indeed be the first GMF detected at soft X-ray energies. Distinguishing distant short GRBs from GMFs has historically been difficult and there are multiple studies suggesting that a subset of short GRBs are actually extragalactic GMFs (Hurley et al. 2005; Palmer et al. 2005; Tanvir et al. 2005; Ofek et al. 2006; Mazets et al. 2008b; Hurley 2011; Yang et al. 2020; Svinkin et al. 2021; Negro & Burns 2023). Just as for the distant GRB interpretation, the non-detection of any optical counterpart remains puzzling for a distant GMF scenario, unless we are dealing with a very distant and exceptionally luminous GMF.",
"pages": [
17
]
},
{
"title": "4.3.8 Thermonuclear X-ray Burst from a quiet LMXB in the LMC?",
"content": "If XRT 200515 is in the LMC, a peak luminosity near the Eddington luminosity 𝐿 Edd ∼ 10 38 erg/s and sharp rise time of the flare suggests a Type I X-ray burst interpretation, which is a thermonuclear explosion on the surface of a weakly magnetized, accreting neutron star (Lewin et al. 1993; Strohmayer & Bildsten 2003; Gal-",
"pages": [
17
]
},
{
"title": "loway et al. 2008, 2020; Galloway & Keek 2021; Alizai et al. 2023).",
"content": "The old stellar population in the field of XRT 200515 is consistent with the presence of neutron star LMXBs. Following the definition of burst timescale 𝜏 = 𝐸 f / 𝐹 p in Galloway et al. (2008), we estimate 𝜏 ∼ 20 s for XRT 200515, which is consistent with Type I X-ray bursts (Galloway & Keek 2021; Alizai et al. 2023). The fitted temperature 𝑘𝑇 bb ∼ 2 keV when the average spectrum is fitted with a simple blackbody, and the softening of the spectrum (temperature decrease) in the tail is also typical of Type I X-ray bursts (Galloway et al. 2008, 2020; Güver et al. 2012). On the other hand, several observed properties of XRT 200515 are unusual for Type I X-ray bursts. In particular, most Type I X-ray bursts occur when the persistent luminosity (proportional to the accretion rate) of a LMXB is 𝐿 X > 10 -4 𝐿 Edd (and, in most cases, 𝐿 X > 10 -3 𝐿 Edd ) (Galloway et al. 2008). Instead, in the initial part of ObsID 23022, the upper limit on the X-ray luminosity at the position of XRT 200515 is 𝐿 X < 10 -4 𝐿 Edd , so that the X-ray flux increased by at least 4 orders of magnitudes. On another note, the sharp decline after the initial burst of XRT 200515 would be unusual for Type I X-ray bursts, which typically exhibit a gradual and exponential decay. However, note that most Type I X-ray bursters were observed by the Rossi X-Ray Timing Explorer ( RXTE ) (Jahoda et al. 1996), which has a high time resolution. The low time resolution of Chandra may have obscured such a decay for XRT 200515. Moreover, most Type I bursts tend to repeat every few hours (Galloway et al. 2008); instead, XRT 200515 is the only event detected at that location over a total observed time of ∼ 150 ks. No LMXB has ever been noted at that position before or after the event. The time interval between bursts is related to an index 𝛼 defined as the ratio between the integrated persistent fluence between subsequent bursts and the burst fluence; from a comparison of the energy released by accretion (contributing to the persistent fluence) and by thermonuclear burning (burst fluence), we expect 𝛼 ≳ 40, in agreement with the observations of Type I bursts (Galloway et al. 2008). If we apply the same criterion ( 𝛼 ≳ 40) to the persistent and flare fluences of XRT 200515, we would have to wait > 10 7 s (4 months) to observe another similar event, assuming the persistent flux level upper limit in ObsID 23022 before the transient event. This waiting time extends to at least one year if we assume the persistent flux upper limit derived from the stacked ∼ 150 ks Chandra observations. Only a few one-off bursts from Galactic neutron stars at a very low persistent luminosity ( 𝐿 X ∼ 10 32 -10 33 erg/s) were found by Cornelisse et al. (2002a,b) with estimated recurrence times of tens of years. The vast majority of Type I X-ray bursts are Galactic, due to their lower flux at large distances. Only a handful of extragalactic Type I X-ray bursts are documented, for example in M 31 (Pastor-Marazuela et al. 2020) and the Magellanic Bridge (Haberl et al. 2023). If XRT 200515 is a Type I X-ray burst, it is the first extragalactic Type I X-ray burster in the LMC and represents the tip of the iceberg for a vast population of faint LMXBs in nearby galaxies, too dim to be detected by Chandra or XMM-Newton , but which may occasionally reveal themselves via thermonuclear bursts with a long duty cycle.",
"pages": [
18
]
},
{
"title": "4.3.9 Concluding Remarks and Outlook for XRT 200515",
"content": "XRT 200515 is a unique and intriguing extragalactic Chandra FXT. The combination of its temporal and spectral properties is unlike any of the other Chandra FXT samples. Based on our analysis, the two most likely scenarios for XRT 200515 are: (i) a distant GMF from a SGR behind the LMC; the first observed in the low X-ray energy band, missed by any other high-energy facilities, or (ii) an unusual Type I X-ray burst from a previously unknown faint LMXB; the first extragalactic X-ray burster in the LMC. Nevertheless, both of these interpretations come with their own unique challenges. XRT 200515 could, in fact, represent an entirely new type of astronomical phenomenon. After all, the primary objective of our work was to use machine learning to find rare, needle-in-the-haystack anomalies hidden within vast astronomical datasets. We invite further detailed studies of XRT 200515 to evaluate our interpretations and explore alternative scenarios, such as potential associations with a fast radio burst (FRB) or a SBO. We highly recommend follow-up multi-band observations at the source coordinates of XRT 200515 to better constrain its nature. Lastly, we note that XRT 200515 and the second transient discovered by Glennie et al. (2015), XRT 120830, have remarkably similar temporal evolutions in their lightcurves (J. Irwin, personal communication, November 2024), however with very different spectral properties ( Γ ∼ 2 . 5 for XRT 120830 versus Γ ∼ 0 . 5 for XRT 200515). We leave a detailed comparative analysis of these transients for future work. Figure 18 shows the 𝐸 -𝑡 Map and 𝐸 -𝑡 -𝑑𝑡 Cube event file representations for XRT 200515. These exhibit high counts at high energies in a narrow time window, which is in line with the hard spectrum and transient nature of XRT 200515.",
"pages": [
18
]
},
{
"title": "4.4 Technical Caveats",
"content": "The main technical caveat of our approach is related to the representation of event files. While our new event file representations enable a simple, yet powerful representation learning approach to find new and rare X-ray transients, any simplification of raw event files, like the fixed number of time bins we use across all event files, is associated with a loss of information. This could lead to us missing a small amount of transients. To minimize this, we have implemented a rigorous approach to justify the resolution of the event file representations in § 3.1. Moreover, flares, in particular known extragalactic FXTs, cluster notably well in our representation spaces. This is because their distinctive features are less dependent on the temporal binning resolution in the 𝐸 -𝑡 Maps and 𝐸 -𝑡 -𝑑𝑡 Cubes. To improve the effectiveness of dip searches with our proposed method, we suggest using higher resolution event file representations. Nevertheless, our comprehensive transient candidate catalog includes numerous newly identified transients that were previously overlooked by other X-ray transient searches in the Chandra archive. Among these is the remarkable needle-in-the-haystack event XRT 200515 discovered in this work, underscoring the effectiveness of our method. A followup representation learning algorithm will learn informative features from raw and unbinned event files while accounting for the Poisson nature of X-ray observations (Song et al., 2025, in preparation).",
"pages": [
18
]
},
{
"title": "5 CONCLUSION",
"content": "We have introduced a novel representation learning method, the first of its kind applied to X-ray event files, enabling downstream tasks such as unsupervised classification and anomaly detection in highenergy astrophysics. We have used the learned representation to investigate time-domain properties of sources in the Chandra archive, with a particular emphasis on the discovery of X-ray transients. As a result, we have compiled the identified X-ray flares and dips in a comprehensive catalog of transient candidates. Notably, our method led to the discovery of XRT 200515; a previously unidentified extragalactic FXT with unique temporal and spectral properties, representing a genuine needle-in-the-haystack discovery. Our key results are as follows: XRT200515wasonlydetectedby Chandra , with no identified optical counterparts. We strongly encourage a multi-wavelength search for additional signals from the source associated with XRT 200515 to better understand its origin and nature. Ourworkadvancestime-domainhigh-energy astrophysics by making the Chandra transient candidates catalog publicly available and open-sourcing the representation learning based transient search pipeline 7 . The catalog enables queries to identify new Chandra transients. Future work involves applying the detection pipeline to additional high-energy archives and adapting it to a variety of other scientific datasets, paving the way for further machine learning driven discoveries of rare transients and other scientific anomalies.",
"pages": [
19
]
},
{
"title": "ACKNOWLEDGEMENTS",
"content": "This research has made use of data obtained from the Chandra Source Catalog, and software provided by the Chandra X-ray Center (CXC) in the /c.pc/i.pc/a.pc/o.pc application package. SD's work was partially funded by the UK government's Turing Scheme and mainly carried out at the Center for Astrophysics | Harvard & Smithsonian as part of the SAO Predoctoral Program, with the support of AstroAI. SD acknowledges hospitality at the Institute of Astronomy at the University of Cambridge and at the Stanford Center for Decoding the Universe (Stanford Data Science) during the later parts of this project. RS's work was performed in part at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-2210452. RS acknowledges support and hospitality at the National Astronomical Observatories of China (NAOC) in Beijing, during part of this project. We thank Edo Berger, Massimiliano De Pasquale, Ken Ebisawa, Duncan Galloway, Jimmy Irwin, Peter Jonker, Daniel Kocevski, Amy Lien, Sandro Meregetthi, Daniel Muthukrishna, Nicola Omodei, Jonathan Quirola-Vásquez, Shivam Raval, Ashley Villar, and Silvia Zane for their fruitful discussions.",
"pages": [
19
]
},
{
"title": "DATA AVAILABILITY",
"content": "The data used in this paper, composed of X-ray event files and source detection regions, was obtained from the publicly available CSC, using their public interfaces (https://cxc.cfa.harvard.edu/csc/). The catalog of transient candidates and the clustered embedding spaces generated using our unsupervised representation learning method can be accessed in the supplementary material. All intermediate data products, i.e. as 𝐸 -𝑡 Maps, 𝐸 -𝑡 -𝑑𝑡 Cubes, principal components and latent features, feature embeddings and embedding clusters can be produced using the code provided in the GitHub repository https://github.com/StevenDillmann/ml-xraytransients-mnras.",
"pages": [
19
]
},
{
"title": "REFERENCES",
"content": "Walmsley M., et al., 2022, Monthly Notices of the RAS, 513, 1581 Weisskopf M. C., Tananbaum H. D., Van Speybroeck L. P., O'Dell S. L., 2000, in Truemper J. E., Aschenbach B., eds, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Vol. 4012, X-Ray Optics, Instruments, and Missions III. pp 2-16 ( arXiv:astro-ph/0004127 ), doi:10.1117/12.391545 Wichern H. C. I., Ravasio M. E., Jonker P. G., Quirola-Vásquez J. A., Levan X. C., 2019, Monthly Notices of the RAS, 487, 4721",
"pages": [
21
]
},
{
"title": "APPENDIX A: DISTRIBUTION OF EVENT FILE LENGTHS AND DURATIONS",
"content": "Figure A1 shows the distribution of the length 𝑁 and duration 𝑇 of event files in the dataset used in this work.",
"pages": [
22
]
},
{
"title": "APPENDIX B: AUTOENCODER TRAINING PROCESS",
"content": "Figure B1 shows the training process of the autoencoders used in this work.",
"pages": [
22
]
},
{
"title": "APPENDIX C: HYPERPARAMETER OPTIMIZATION",
"content": "Below, we summarize the optimization strategy for the t-SNE and DBSCAN hyperparameters. For even more details on this approach, please refer to Dillmann & Martínez-Galarza (2023).",
"pages": [
22
]
},
{
"title": "C1 t-SNE Hyperparameters",
"content": "The choice of the perplexity and learning_rate can have a large impact on the resulting t-SNE embedding space. Ideally, we want the two-dimensional embedding space to effectively capture both energy information (in form of the hardness ratio 𝐻𝑅 ) and variability information (in form of the variability probability 𝑝 var ). That means that event files with similar values for 𝐻𝑅 and 𝑝 var should live close to each other in the final embedding space. We can use this information to define a performance metric for different t-SNE hyperparameter inputs. First, we compute the pairwise distance matrix D Z of size ( 𝑚, 𝑚 ) , where the distance 𝐷 𝑍 𝑖 𝑗 between points 𝑖 and 𝑗 is computed using a Euclidean distance metric. Next, we define the property vector Y , which includes 7 CSC properties (hardness ratios 𝐻𝑅 hm , 𝐻𝑅 hs , 𝐻𝑅 ms and variability probabilities 𝑝 b var , 𝑝 h var , 𝑝 m var , 𝑝 s var ) for each event file and thus each t-SNE point. As a measure of similarity between the labels of different points, we can again compute a pairwise similarity matrix D Y of size ( 𝑚, 𝑚 ) . To compute the similarity distance 𝐷 𝑌 𝑖 𝑗 between sample 𝑖 and 𝑗 , we use the Mahalanobis distance metric (Mahalanobis 1936). Unlike the Euclidean distance metric, the Mahalanobis distance metric accounts for the correlation between different labels by taking into account the covariance structure of the data. Note that our hardness ratios are correlated with each other, and that the same holds for the variability probabilities. Accounting for these correlations provides a more accurate measure of the similarity distance between different samples. Having computed D Z and D Y , we can define a performance metric that allows us to compare the performance of different t-SNE hyperparameters. The smaller the distance 𝐷 𝑍 𝑖 𝑗 between two points 𝑖 and 𝑗 in the t-SNE embedding, the smaller should be difference in their associated labels as measured by the distance 𝐷 𝑌 𝑖 𝑗 . We can thus define a performance metric based on the statistical correlation of D Z and D Y using the Spearman's rank correlation coefficient 𝜌 𝑍𝑌 (Spearman 1904). The higher 𝜌 𝑍𝑌 , the higher is the positive correlation between D Z and D Y and the better the performance of the t-SNE embedding. The hyperparameter space is given by the ranges learning_rate ∈ ( 20 , 200 ) with a step size of 20 and perplexity ∈ ( 10 , 100 ) with a step size of 10. This optimization process is performed using a reduced dataset of 15,353 samples for 2,000 iterations per hyperparameter combination due to computational constraints. While subsampling, the overall structure of the data was preserved by selecting the same distributions between any combinations of hard, medium, soft, variable and non-variable samples. This ensures that the sample set is representative of the original data. We choose the hyperparameter combination that produces the highest value of 𝜌 𝑍𝑌 .",
"pages": [
22,
23
]
},
{
"title": "C2 DBSCAN Hyperparameters",
"content": "Different hyperparameter combinations of eps and minPts can have a large impact on the resulting DBSCAN clusters. We use a combination of the Davies-Bouldin index 𝐷𝐵 (Davies & Bouldin 1979) and Calinski-Harabasz index 𝐶𝐻 (Caliński & Harabasz 1974) as a performance metric to find the optimal DBSCAN hyperparameter inputs. The 𝐷𝐵 index is a measure of the average similarity between each cluster and its most similar cluster, relative to the average distance between points within each cluster. The 𝐷𝐵 index is given by the following formula: where 𝑛 𝑐 is the number of clusters, 𝑊 𝑖 and 𝑊 𝑗 are the within-cluster sum of squares for cluster 𝑖 and 𝑗 , and 𝑑 ( 𝑐 𝑖 , 𝑐 𝑗 ) is the distance between the centroids of clusters 𝑖 and 𝑗 . On the other hand, the 𝐶𝐻 index is based on the concept that good clusters should have high intra-cluster similarity (cohesion) measured by the between-cluster dispersion 𝐵 and low inter-cluster similarity (separation) measured by the within-cluster dispersion 𝑊 . 𝐵 is the sum of the pairwise distances between cluster centroids, and 𝑊 is the sum of the pairwise distances between points within each cluster. The 𝐶𝐻 index is given by the following formula: where the scaling factor 𝑚 -𝑛 𝑐 𝑛 𝑐 -1 accounts for the total number of data points 𝑚 and the number of clusters 𝑛 𝑐 . A lower 𝐷𝐵 index and higher 𝐶𝐻 index indicate that the clustering algorithm is more effective in grouping similar data points together and separating different data points into distinct clusters. We thus define the performance metric 𝜌 𝐷𝐶 as the ratio of the normalized indices 𝐷𝐵 𝑛 = 𝐷𝐵 max ( 𝐷𝐵 ) and 𝐶𝐻 𝑛 = 𝐶𝐻 max ( 𝐶𝐻 ) in the hyperparameter space given by eps ∈ ( 1 . 0 , 3 . 0 ) with a step size of 0 . 1 and minPts ∈ ( 10 , 30 ) with a step size of 1: We choose the hyperparameter combination that produces the highest value of 𝜌 𝐷𝐵𝑆𝐶𝐴𝑁 .",
"pages": [
23
]
},
{
"title": "APPENDIX D: EMBEDDINGS",
"content": "Figures D1, D2 and D3 show the 2D-PCA and 2D-AE embeddedings.",
"pages": [
23
]
},
{
"title": "APPENDIX E: TRANSIENT-DOMINANT EMBEDDING CLUSTERS",
"content": "Table E1 lists the transient-dominant clusters in the different embedding spaces used for the selection of transient candidates.",
"pages": [
23
]
},
{
"title": "APPENDIX F: CATALOG COLUMNS",
"content": "Table F1 shows the of X-ray transient candidate catalog column descriptions. This paper has been typeset from a T E X/L A T E X file prepared by the author.",
"pages": [
23
]
}
]
2024arXiv241201157B
https://arxiv.org/pdf/2412.01157.pdf
<document>
<section_header_level_1><location><page_1><loc_10><loc_85><loc_90><loc_87></location>LOCAL VARIATIONS OF THE RADIAL METALLICITY GRADIENT IN A SIMULATED NIHAO-UHD MILKY WAY ANALOGUE AND THEIR IMPLICATIONS FOR (EXTRA-)GALACTIC STUDIES</section_header_level_1>
<text><location><page_1><loc_20><loc_82><loc_79><loc_83></location>S/v.pc/e.pc/n.pc B/u.pc/d.pc/e.pc/r.pc 1 , 2 , ∗ , T/o.pc/b.pc/i.pc/a.pc/s.pc B/u.pc/c.pc/k.pc 3 , 4 , Q/i.pc/a.pc/n.pc-H/u.pc/i.pc C/h.pc/e.pc/n.pc ( 陈 千 惠 ) 1 , 2 , /a.pc/n.pc/d.pc K/a.pc/t.pc/h.pc/r.pc/y.pc/n.pc G/r.pc/a.pc/s.pc/h.pc/a.pc 1 , 2 , ∗</text>
<text><location><page_1><loc_21><loc_81><loc_80><loc_82></location>1 Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT 2611, Australia</text>
<text><location><page_1><loc_26><loc_79><loc_75><loc_81></location>2 ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Australia</text>
<text><location><page_1><loc_12><loc_78><loc_89><loc_79></location>3 Universität Heidelberg, Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, Im Neuenheimer Feld 205, D-69120 Heidelberg, Germany and</text>
<text><location><page_1><loc_13><loc_76><loc_88><loc_78></location>4 Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische Astrophysik, Albert-Ueberle-Strae 2, D-69120 Heidelberg, Germany Version December 3, 2024</text>
<section_header_level_1><location><page_1><loc_46><loc_73><loc_54><loc_74></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_53><loc_86><loc_73></location>Radial metallicity gradients are fundamental to understanding galaxy formation and evolution. In our high-resolution simulation of a NIHAO-UHD Milky Way analogue, we analyze the linearity, scatter, spatial coherence, and age-related variations of metallicity gradients using young stars and gas. While a global linear model generally captures the gradient, it ever so slightly overestimates metallicity in the inner galaxy and underestimates it in the outer regions of our simulated galaxy. Both a quadratic model, showing an initially steeper gradient that smoothly flattens outward, and a piecewise linear model with a break radius at 10 kpc (2.5 effective radii) fit the data equally better. The spread of [Fe/H] of young stars in the simulation increases by tenfold from the innermost to the outer galaxy at a radius of 20 kpc. We find that stars born at similar times along radial spirals drive this spread in the outer galaxy, with a chemical under- and over-enhancement of up to 0.1 dex at leading and trailing regions of such spirals, respectively. This localised chemical variance highlights the need to examine radial and azimuthal selection effects for both Galactic and extragalactic observational studies. The arguably idealised but volume-complete simulations suggest that future studies should not only test linear and piecewise linear gradients, but also non-linear functions such as quadratic ones to test for a smooth gradient rather than one with a break radius. Either finding would help to determine the importance of different enrichment or mixing pathways and thus our understanding of galaxy formation and evolution scenarios.</text>
<text><location><page_1><loc_14><loc_51><loc_82><loc_52></location>Subject headings: Galaxy: structure - Galaxy: abundances - galaxies: structure - galaxies: abundances</text>
<section_header_level_1><location><page_1><loc_21><loc_48><loc_35><loc_49></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_15><loc_48><loc_47></location>Understanding the radial metallicity gradient, defined as the change in heavy element abundance with galactocentric radius, in galaxies provides critical insights into their formation and evolutionary processes, such as inside-out formation, gas accretion, outflows, and radial migration (e.g. Quirk & Tinsley 1973; Tinsley 1980; Lacey & Fall 1985; Wyse & Silk 1989; Kauffmann 1996; Chiappini et al. 1997; Schönrich & Binney 2009; Moran et al. 2012; Bird et al. 2013). The decrease in metallicity with increasing distance from the Galactic centre is well-established both theoretically (Larson 1976; Tinsley 1980; Chiosi 1980) and observationally in the Milky Way (Searle 1971; Janes 1979; Twarog et al. 1997) and other massive spiral galaxies (e.g. Tinsley 1980; Zaritsky et al. 1994; Bresolin et al. 2012). The Milky Way, being the only galaxy where we can resolve millions of stars, provides a unique opportunity to study these gradients and deviations from them in detail. Early evidence by Janes (1979) suggested a linear gradient on the order of d [ Fe / H ]/ d 𝑅 = -0 . 05 ± 0 . 01 dex kpc -1 for the Milky Way which aligns very well with more recent measurements (Anders et al. 2017; Hayden et al. 2015). However, these gradients are accompanied by a significant spread in [Fe/H] of 0 . 1 -0 . 15 dex, as noted by Twarog (1980), which may imply a fine structure of the metallicity gradient (see Genovali et al. 2014).</text>
<text><location><page_1><loc_8><loc_11><loc_48><loc_15></location>With increasing sample size and measurement precision, the specific shape and characteristics of this gradient remain somewhat unclear (Chiappini 2002). Previous studies have</text>
<text><location><page_1><loc_10><loc_7><loc_34><loc_8></location>∗ Australian Research Council DECRA Fellow</text>
<text><location><page_1><loc_52><loc_38><loc_92><loc_49></location>for example claimed more intricate non-linear trends, bends, and flattening in the gradient of the Milky Way (e.g. Donor et al. 2020) and other galaxies (e.g. Pilyugin 2003; Sánchez et al. 2014) or even sequences of shapes (Pilyugin et al. 2017; Pilyugin & Tautvaišien˙e 2024), which were fitted with different models (Rosales-Ortega et al. 2011; Bresolin et al. 2012), such as piecewise linear ones (e.g. Sánchez-Menguiano et al. 2016) or non-linear ones (e.g. Scarano & Lépine 2013).</text>
<text><location><page_1><loc_52><loc_22><loc_92><loc_38></location>Variations in the metal distribution, including breaks of the gradient at specific radii, give rise to a plethora of possible physical explanations, such as star formation efficiency variations and localised star formation bursts (Sánchez et al. 2014; Sánchez-Blázquez et al. 2014; Ho et al. 2015), gas accretion and dilution at different rates (Bresolin et al. 2012; Sánchez et al. 2013; Belfiore et al. 2016; Sánchez-Menguiano et al. 2016), gas outflows and feedback (Lilly et al. 2013; Ma et al. 2017a), as well as disk instabilities or local overdensities (Grand et al. 2016; Ho et al. 2017). In particular, Scarano & Lépine (2013) suggested that gradient break radii coincided with the corotation radii of spiral arms.</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_22></location>Recent advancements in both computations and observations have significantly expanded our capabilities. For example, in terms of observational data in the Milky Way, the Gaia mission (Gaia Collaboration et al. 2016) enables more detailed studies of these gradients. New suites of large-scale simulations now allow us to gain insights into radial metallicity gradients across a range of simulated galaxies, including Milky Way analogues. This presents opportunities to revisit outstanding challenges of the detailed shape of the radial metallicity gradient. For instance, Hogg et al. (2019) created an extensive metallicity map of the Milky Way using APOGEE and Gaia</text>
<text><location><page_2><loc_8><loc_83><loc_48><loc_92></location>data, while Poggio et al. (2022) mapped young stars and found metallicity variations around spiral arms (see also Zari et al. 2018, 2021; Poggio et al. 2021; Hackshaw et al. 2024). Similarly, Imig et al. (2023, among others) traced gradients across different stellar populations and ages, emphasizing the importance of considering radial migration effects (Binney & Tremaine 2008; Frankel et al. 2018, 2020).</text>
<text><location><page_2><loc_8><loc_69><loc_48><loc_82></location>Historically, radial metallicity gradients have been measured using various stellar populations and gas tracers. Estimated gradients seem to be broadly consistent across different stellar tracers, such as young open clusters (e.g. Yong et al. 2012; Cunha et al. 2016; Magrini et al. 2017; Casamiquela et al. 2019; Donor et al. 2020; Spina et al. 2021; Myers et al. 2022), young hot (OB-type) stars (Zari et al. 2018, 2021; Poggio et al. 2021, 2022), field stars close to the Galactic plane (e.g. Bergemann et al. 2014) or Cepheids (Andrievsky et al. 2002a,b; Lemasle et al. 2007, 2013).</text>
<text><location><page_2><loc_8><loc_45><loc_48><loc_69></location>Despite extensive observational efforts, several challenges persist for studies in the Milky Way. The completeness (or patchiness) of observed datasets remains a fundamental issue (Bergemann et al. 2014). The robustness of fits to the incomplete data is still contentious, including the need to actually fit two linear gradients with a break radius at corotation radius (Bresolin et al. 2012, and references therein) or further out (Yong et al. 2012; Donor et al. 2020) - or even more complicated functions (see e.g. Chiappini et al. 2001; Kubryk et al. 2015). Furthermore, methodologies for fitting linear models to scattered data need re-evaluation (Metha et al. 2021). Different samples yield varying gradients, potentially due to biases in data or the inclusion of older stars (e.g. Allende Prieto et al. 2006; Hayden et al. 2014; Anders et al. 2014; Vickers et al. 2021; Willett et al. 2023). The impact of spiral arm structures (Poggio et al. 2021), the Galactic warp (Lemasle et al. 2022) or bar-driven mixing (Di Matteo et al. 2013) on metallicity gradients is not fully understood.</text>
<text><location><page_2><loc_8><loc_25><loc_48><loc_45></location>Understanding these gradients in the Milky Way is also crucial for extragalactic studies, where spatial resolution limits observations in different ways. In extragalactic systems, metals are mainly traced via gas, because it provides a more direct measure of the ongoing enrichment processes, unlike stars, which primarily reflect the integrated chemical history of the past. Observationally, gas emission lines are typically brighter and more accessible across large distances than stellar absorption lines, allowing for broader spatial coverage, especially in distant galaxies. Consequently, extragalactic studies often focus on gas-phase metallicity as traced by oxygen, A ( O ) = 12 + log ( O / H ) , while Galactic studies typically use stellar iron abundance [ Fe / H ] = A ( Fe ) -A ( Fe ) ⊙ as a metallicity tracer (e.g. Nicholls et al. 2017; Fraser-McKelvie et al. 2022).</text>
<text><location><page_2><loc_8><loc_7><loc_48><loc_25></location>New instruments like the MUSE integral field spectrograph have enabled a plethora of extragalactic studies to contrast the Milky Way and techniques like the spectroscopy of H /i.pc/i.pc regions and planetary nebulae have helped to infer gas metallicity distributions in external galaxies (Shaver et al. 1983; Vilchez & Esteban 1996; Rolleston et al. 2000; Bresolin et al. 2012). Recent examples include Sánchez et al. (2014) with CALIFA galaxy observations as well as Mun et al. (2024) and Chen et al. (2024a) who use MAGPI observations to probe for example the effects of spiral arms. Notable is also the scatter that Chen et al. (2023) found for the gas metallicity across galactic radii with TYPHOON observations (see their. Figs. 4-6). Grasha et al. (2022) found that the gas metallicity gradient plateaus at a lower limit in their TYPHOON galaxies at the outermost</text>
<text><location><page_2><loc_52><loc_89><loc_92><loc_92></location>radii - an observation replicated by IllustrisTNG simulations (Hemler et al. 2021; Garcia et al. 2023).</text>
<text><location><page_2><loc_52><loc_75><loc_92><loc_89></location>From a modelling perspective, galactic chemical evolution models can both test understanding of radial metallicity gradients and make predictions beyond the limited volumes and tracers tested by Milky Way and extragalactic studies. Such galactic chemical evolution models include Chiappini et al. (2001); Matteucci & Recchi (2001); Minchev et al. (2014); Kubryk et al. (2015); Stanghellini et al. (2015); Rybizki et al. (2017); Spitoni et al. (2023); Johnson et al. (2024). Sharda et al. (2021) even presented a model for gas phase metallicity gradients in galaxies and their evolution from first principles (see also Krumholz & Ting 2018).</text>
<text><location><page_2><loc_52><loc_32><loc_92><loc_74></location>In exploring radial metallicity gradients through simulations, we have better understood how different processes influence these gradients across galactic models and temporal scales. Studies such as Pilkington et al. (2012) in RaDES simulations reveal that gradients are typically established via inside-out galaxy formation. Khoperskov et al. (2023) quantified the scatter of gas metallicity to ≈ 0 . 04 -0 . 06 dex at a given galactocentric distance in their simulations. Meanwhile, the EAGLE simulations used by Tissera et al. (2019) provide insights into how these gradients vary with galaxy characteristics like stellar mass and merger history, emphasizing the dynamic nature of metallicity distributions. The plethora of simulations such as AURIGA (Grand et al. 2016), FIRE (Ma et al. 2017b, see their Fig. 6) or VINTERGATAN (see their Fig. 9; Agertz et al. 2021) also allow us to explore the gradient evolution of galactic timescales. Buck et al. (2023), for example, found a link of major accretion events with periods of unexpected steepening in the metallicity gradient within NIHAO-UHD simulations - closely resembling findings for the Milky Way by Lu et al. (2022) and Ratcliffe et al. (2023). FIRE simulations, examined by Bellardini et al. (2021), Bellardini et al. (2022), and Graf et al. (2024), extend these findings by comparing radial metallicity gradients and their azimuthal scatter across gas and stellar components and illustrate the complex interplay between galactic structure and metal enrichment processes. Similarly, Grand et al. (2015, 2016) highlight temporal changes in metallicity gradients already within 120 Myr, or roughly one galactic rotation. Such rapid changes underscore the impact of transient galactic events on the metal distribution, linking them to star formation patterns along spiral arms and the broader evolutionary history of the galaxy.</text>
<text><location><page_2><loc_52><loc_21><loc_92><loc_32></location>In this study, we analyze a high-resolution NIHAO-UHD simulation of a Milky Way analogue to bridge the observational gap between detailed studies of our galaxy and broader extragalactic surveys. We aim to reveal subtle features of the radial metallicity gradient, which may be obscured by observational constraints in both the Milky Way and distant galaxies, by testing the following properties within the observationally probed inner 𝑅 Gal . ≤ 20 kpc:</text>
<unordered_list>
<list_item><location><page_2><loc_54><loc_18><loc_92><loc_20></location>1. Linearity of the gradient: Assess the extent to which the radial metallicity gradient of young stars is linear.</list_item>
<list_item><location><page_2><loc_54><loc_14><loc_92><loc_17></location>2. Scatter in the gradient: Quantify the expected scatter in the radial metallicity gradient of young stars.</list_item>
<list_item><location><page_2><loc_54><loc_11><loc_92><loc_13></location>3. Coherence of the gradient with position: Investigate the gradient's variation with radial coverage and azimuth.</list_item>
<list_item><location><page_2><loc_54><loc_7><loc_92><loc_9></location>4. Coherence of the gradient with age: Test the reliability of stars as tracers of the gas disk over different ages.</list_item>
</unordered_list>
<figure>
<location><page_3><loc_13><loc_58><loc_87><loc_92></location>
</figure>
<text><location><page_3><loc_23><loc_57><loc_27><loc_60></location>XGal.</text>
<figure>
<location><page_3><loc_13><loc_10><loc_87><loc_53></location>
<caption>F/i.pc/g.pc. 1.- Logarithmic spatial density distribution of stars (upper panels) and gas (lower panels) within 𝑅 < 20 kpc ∼ 5 Re of the NIHAO-UHD Milky Way analogue g8.26e11 in galactocentric cartesian and cylindrical coordinates. Panel c) shows the influence of selecting only young stars with ages below 0 . 5 Gyr.F/i.pc/g.pc. 2.- Face-on view of average simulated metallicity (left panels), a linear radial fit to it (middle panels, see Section 3.1 and Eq. 1) and the fit residuals (right panels) in bins of 0 . 5 kpc. Shown are metallicity as traced by young star iron abundances (top panels) and gas phase metallicity (bottom) panels.</caption>
</figure>
<text><location><page_4><loc_8><loc_73><loc_48><loc_92></location>The paper is structured as follows: Section 2 describes the data of our Milky Way analogue NIHAO-UHD simulation. Section 3 analyses the linearity of the radial metallicity gradient of the simulation, the first of our four objectives. Section 4 then analyses both the scatter and local deviations from the gradient as well as the coherence of the gradient with vertical and azimuthal position as well as age (the remaining three objectives). Section 5 discusses them individually. We note that in this research we are mainly interested in the specific shape of the radial metallicity gradient for radii relevant to Galactic observations. However, we also discuss our results in the context of extragalactic results that probe beyond the inner 20 kpc of a galaxy. Section 6 bundles our results into an overarching conclusion.</text>
<section_header_level_1><location><page_4><loc_10><loc_70><loc_47><loc_72></location>2. DATA: A NIHAO-UHD MILKY WAY ANALOGUE SIMULATION</section_header_level_1>
<text><location><page_4><loc_8><loc_55><loc_48><loc_69></location>For this project, we use a cosmological zoom-in simulation of a Milky Way analogue ( g8.26e11 ) from the Numerical Investigation of a Hundred Astronomical Objects (NIHAO, Wang et al. 2015) suite. This model galaxy was calculated as part of the NIHAO-UHD project (Buck et al. 2020) and has previously been used in various works studying Milky Way satellites (Buck et al. 2019), Milky Way's dark halo spin (Obreja et al. 2022), inferring birth properties of stars with abundance clustering (Ratcliffe et al. 2022), as well as the evolution of the interstellar medium's radial metallicity gradient since redshift three (Ratcliffe et al. 2024).</text>
<text><location><page_4><loc_8><loc_39><loc_48><loc_55></location>Simulations were carried out with the smoothed particle hydrodynamics code Gasoline2 (Wadsley et al. 2017), including sub-grid turbulent mixing, using cosmological parameters from Planck Collaboration et al. (2014) with initial conditions and energetic feedback descriptions from the NIHAO project (Wang et al. 2015). Zoom-in simulations were then performed as described in detail by Buck et al. (2021) with star formation following Stinson et al. (2006) and energetic feedback following Stinson et al. (2013). We note that this is a slightly different rerun of the same simulation than the one studied by Buder et al. (2024); in this work, we use a higher resolution version and updated chemical yields.</text>
<text><location><page_4><loc_8><loc_11><loc_48><loc_39></location>Because computational resources still limit the mass resolution of simulations, we are relying on tracer particles that represent simple stellar populations (SSPs) with the same age, overall metallicity and discrete initial mass function (IMF). Buck et al. (2021) have implemented the flexible chemical evolution code /c.pc/h.pc/e.pc/m.pc/p.pc/y.pc (Rybizki et al. 2017) to calculate the chemical yields for the SSPs. In particular, we use the alternative ( alt ) setup of /c.pc/h.pc/e.pc/m.pc/p.pc/y.pc that assumes a Chabrier (2003) IMF with high-mass slope of 𝛼 IMF = -2 . 3 over a mass range of 0 . 1 -100 M ⊙ for SSPs across a metallicity range of 𝑍 / 𝑍 ⊙ ∈ [ 10 -5 , 2 ] . The code calculates the contribution from asymptotic giant branch (AGB) stars, CCSN across a mass range of 8 -40 M ⊙ , and SNIa with an exponential function with exponent -1 . 12, a delay time of 40 Myr, and a normalization of the SNIa rate of log 10 ( NIa ) = -2 . 9. For each of these nucleosynthetic channels, yields from the following studies are used: Chieffi & Limongi (2004) for CCSN, Seitenzahl et al. (2013) for SNIa, and Karakas & Lugaro (2016) for AGB stars ( new_fit model in Buck et al. 2021). Contrary to a previous study by Buder et al. (2024), we take the elemental abundances at face value and do not apply any shifts.</text>
<text><location><page_4><loc_8><loc_7><loc_48><loc_11></location>We limit the simulation data to the main halo by applying /p.pc/y.pc/n.pc/b.pc/o.pc/d.pc/y.pc's implementation of the Amiga Halo Finder (Knollmann & Knebe 2009) and then reposition and rotate this</text>
<text><location><page_4><loc_52><loc_84><loc_92><loc_92></location>main halo to be face-on based on the angular momentum with /p.pc/y.pc/n.pc/b.pc/o.pc/d.pc/y.pc's /a.pc/n.pc/a.pc/l.pc/y.pc/s.pc/i.pc/s.pc./a.pc/n.pc/g.pc/m.pc/o.pc/m.pc./f.pc/a.pc/c.pc/e.pc/o.pc/n.pc module (Pontzen et al. 2013). We then further transform the resulting galactocentric Cartesian coordinate ( 𝑋,𝑌, 𝑍 ) and velocities ( 𝑉 𝑋 , 𝑉 𝑌 , 𝑉 𝑍 ) to Cylindrical ones as done in a previous study of this main halo by Buder et al. (2024).</text>
<text><location><page_4><loc_52><loc_72><loc_92><loc_84></location>The model galaxy has a virial radius of 𝑅 vir = 𝑅 200 = 206 kpc and a total mass (gas, stars and dark matter) inside 𝑅 vir of 9 . 1 · 10 11 M ⊙ . At redshift zero, it contains 8 . 2 · 10 11 M ⊙ dark matter, 6 . 4 · 10 10 M ⊙ gas mass and 2 . 3 · 10 10 M ⊙ stellar mass with a stellar mass resolution of around 7 . 5 · 10 3 M ⊙ . When using a fifth of the virial radius as a reference to calculate total luminosity /one.sup and mass, we estimate a half-light radius, that is, effective radius of 𝑅 𝑒 = 3 . 79 kpc and a half-stellar-mass radius of 2 . 97 kpc.</text>
<text><location><page_4><loc_52><loc_62><loc_92><loc_71></location>To achieve a roughly similar selection as the observational data of the Milky Way (Genovali et al. 2014) and other galaxies (e.g. Chen et al. 2023), we restrict the simulation data to a galactocentric radius of 𝑅 Gal ≤ 20 kpc and | 𝑧 | ≤ 10 kpc, as shown in Fig. 1. Similar to the Milky Way (Poggio et al. 2018; Lemasle et al. 2022), we note a warp of the stellar and gaseous disk (see Figs. 1b and 1e, respectively).</text>
<text><location><page_4><loc_52><loc_45><loc_92><loc_62></location>To avoid too strong effects of radial migration (Binney & Tremaine 2008; Frankel et al. 2018; Grand et al. 2016; Minchev et al. 2018) while maintaining a sufficiently large sample size we further enforce stars to be younger than 0 . 5 Gyr, corresponding to roughly the time of four galactic rotations, and being half the value found by Minchev et al. (2018) for very limited migration in the Milky Way. This selection defacto limits the vertical range of 99% of stars to | 𝑧 | = 1 . 4 kpc. The strong influence of this age cut on the vertical distribution of stars in the Milky Way analogue can best be appreciated from the difference of vertical density distributions of stars in Figs. 1b and 1c. We are applying these cuts for all following analyses of the radial metallicity gradient in Section 3.</text>
<section_header_level_1><location><page_4><loc_53><loc_41><loc_91><loc_44></location>3. THE LINEARITY OF THE RADIAL METALLICITY GRADIENT IN NIHAO-UHD</section_header_level_1>
<text><location><page_4><loc_52><loc_30><loc_92><loc_40></location>In this section, we analyse the functional shape of the radial metallicity gradient. To get a first impression of possible shapes, we show the face-on view of the decreasing radial metallicity gradient of the simulation in Figs. 2a (for young stars) and Fig. 2d (for gas). Foreshadowing the later parts of this work on local variations, we also show a linear radial fit to either distribution in Fig. 2b and 2e and show the fit residuals in Figs. 2c and 2f.</text>
<text><location><page_4><loc_52><loc_14><loc_92><loc_30></location>At the moment, however, we focus on the linearity and thus the logarithmic density distribution of star particle iron abundances [Fe/H] across different galactocentric radii 𝑅 Gal . . This distribution is shown in Fig. 3a and strongly suggests that the gradient is predominantly linear, similar to findings for the Milky Way. More complex shapes, such as piecewise linear ones have been suggested based on incomplete and limited data in the literature. We are thus also analysing these shapes with the complete and better-sampled data points of the NIHAO-UHD simulation. We firstly test different global fits in Section 3.1, before testing the influence of binning and coverage in Sections 3.2 and 3.3, respectively.</text>
<table>
<location><page_5><loc_8><loc_81><loc_48><loc_87></location>
<caption>TABLE 1 G/l.pc/o.pc/b.pc/a.pc/l.pc /l.pc/i.pc/n.pc/e.pc/a.pc/r.pc /g.pc/r.pc/a.pc/d.pc/i.pc/e.pc/n.pc/t.pc /f.pc/i.pc/t.pc /r.pc/e.pc/s.pc/u.pc/l.pc/t.pc/s.pc /w.pc/i.pc/t.pc/h.pc /d.pc/i.pc/f.pc/f.pc/e.pc/r.pc/e.pc/n.pc/t.pc /m.pc/e.pc/t.pc/h.pc/o.pc/d.pc/s.pc. L/i.pc/n.pc/e.pc/a.pc/r.pcR/e.pc/g.pc/r.pc/e.pc/s.pc/s.pc/i.pc/o.pc/n.pc /i.pc/s.pc /p.pc/a.pc/r.pc/t.pc /o.pc/f.pc /t.pc/h.pc/e.pc /s.pc/k.pc/l.pc/e.pc/a.pc/r.pc/n.pc /p.pc/a.pc/c.pc/k.pc/a.pc/g.pc/e.pc.</caption>
</table>
<section_header_level_1><location><page_5><loc_20><loc_78><loc_36><loc_79></location>3.1. Global gradient fits</section_header_level_1>
<text><location><page_5><loc_8><loc_75><loc_48><loc_77></location>We fit three different functional forms to the global data: a linear function (used for Fig. 2b)</text>
<formula><location><page_5><loc_20><loc_72><loc_48><loc_74></location>𝑓 lin ( 𝑅 Gal . ) = 𝑐 1 · 𝑅 Gal . + 𝑐 2 , (1)</formula>
<text><location><page_5><loc_8><loc_70><loc_37><loc_72></location>a piecewise linear with a break radius 𝑅 break</text>
<formula><location><page_5><loc_11><loc_66><loc_48><loc_69></location>𝑓 piece ( 𝑅 Gal . ) = GLYPH<26> 𝑐 1 · 𝑅 Gal . + 𝑐 2 if 𝑅 Gal . ≤ 𝑅 break 𝑐 3 · 𝑅 Gal . + 𝑐 4 if 𝑅 Gal . > 𝑅 break , (2)</formula>
<text><location><page_5><loc_8><loc_64><loc_24><loc_65></location>and a quadratic function</text>
<formula><location><page_5><loc_15><loc_62><loc_48><loc_63></location>𝑓 quad ( 𝑅 Gal . ) = 𝑐 1 · 𝑅 2 Gal . + 𝑐 2 · 𝑅 Gal . + 𝑐 3 . (3)</formula>
<text><location><page_5><loc_8><loc_49><loc_48><loc_61></location>The coefficients of the functions are fitted with the /s.pc/c.pc/i.pc/p.pc/y.pc./o.pc/p.pc/t.pc/i.pc/m.pc/i.pc/z.pc/e.pc function /c.pc/u.pc/r.pc/v.pc/e.pc_/f.pc/i.pc/t.pc (Virtanen et al. 2020) and listed in Table 2. To estimate the uncertainty of the break radius 𝑅 break, we use the profile likelihood method to identify the radii at which the residual sum of squares (RSS) values are increased by 1 𝜎 from the best RSS radius in steps of Δ 𝑅 break = 0 . 1 kpc and 0 . 5 kpc for the full and binned data set, respectively. We compute the RSS for each model 𝑓 𝑖 (see Eqs. 1-3) based on the 𝑁 data points as</text>
<formula><location><page_5><loc_16><loc_44><loc_48><loc_48></location>RSS 𝑖 = 𝑁 ∑︁ 𝑛 = 1 GLYPH<0> [ Fe / H ] 𝑛 -𝑓 𝑖 ( 𝑅 Gal .,𝑛 ) GLYPH<1> 2 . (4)</formula>
<text><location><page_5><loc_8><loc_38><loc_48><loc_43></location>We have confirmed the robustness of our fits by applying other fitting routines as outlined in Table 1. After fitting three different functional forms, we use a combination of parameters to determine which model provides the best fit.</text>
<text><location><page_5><loc_8><loc_34><loc_48><loc_38></location>In Table 2, the RSS value is the smallest (although only by a small margin) for the quadratic function. When assuming 𝜎 2 = 𝑅𝑆𝑆 / 𝑁 , we can also define a logarithmic likelihood</text>
<formula><location><page_5><loc_16><loc_30><loc_48><loc_33></location>ln 𝐿 = -𝑁 2 ln ( 2 𝜋 ) -𝑁 2 ln 𝑅𝑆𝑆 𝑁 -𝑁 2 (5)</formula>
<text><location><page_5><loc_8><loc_26><loc_48><loc_30></location>for the 𝑁 data points. For 𝑘 free parameters, we then calculate the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) as</text>
<formula><location><page_5><loc_13><loc_24><loc_48><loc_25></location>𝐴𝐼𝐶 = 2 𝑘 -2 ln 𝐿 𝐵𝐼𝐶 = 𝑘 ln 𝑁 -2 ln 𝐿. (6)</formula>
<text><location><page_5><loc_8><loc_20><loc_48><loc_23></location>For these criteria, the quadratic function performs slightly better than the linear or piecewise linear functions (see Table 2).</text>
<text><location><page_5><loc_8><loc_7><loc_48><loc_20></location>We show the fit residuals in Fig. 3b as density distribution as well as in Fig. 3c as percentile distributions in radial bins of Δ 𝑅 Gal = 0 . 5 kpc. While the density distribution shows substructure, which we investigate later in Section 4, we note an increase in the median residuals of the linear fit in Fig. 3c towards the inner and outer radii, especially for 𝑅 Gal . > 17 kpc. A quadratic fit (see orange lines in Fig. 3) results in a slightly steeper linear component of the gradient (from -0 . 0411 to -0 . 0497 dex kpc -1 ), which is counteracted by the quadratic flattening term of + 0 . 0005 dex kpc -2 . The latter leads to an</text>
<figure>
<location><page_5><loc_52><loc_59><loc_91><loc_92></location>
</figure>
<text><location><page_5><loc_70><loc_59><loc_73><loc_61></location>RGal.</text>
<paragraph><location><page_5><loc_52><loc_51><loc_92><loc_58></location>F/i.pc/g.pc. 3.- Global fits and deviation to the radial metallicity gradient 𝑅 -[ Fe / H ] . Functional forms of the linear (red) and quadratic (orange) lines are shown in the legend. Panel a) shows the underlying data of all data points as logarithmic density and the global fit to them as red dashed line. Panel b) shows the deviation of data from a linear gradient as a logarithmic density plot, whereas panel c) shows the 16th and 84th percentile around the median deviation as error bars in Δ 𝑅 Gal = 0 . 5 kpc bins.</paragraph>
<text><location><page_5><loc_52><loc_39><loc_92><loc_49></location>effective flattening of -0 . 172 + 0 . 200 = 0 . 028 dex (linear vs. quadratic terms) at 𝑅 Gal . = 20 dex. While seemingly only a nuisance correction across the large extent of [Fe/H] and 𝑅 Gal . , this quadratic function outperforms the linear fit. This is most apparent in Fig. 3c, where the orange line better traces the median residuals from the linear function across all radii. This suggests that non-linear functions, such as piecewise linear or quadratic ones describe the gradient better.</text>
<text><location><page_5><loc_52><loc_31><loc_92><loc_38></location>Distinguishing between the two latter functional forms is challenging. The quantitative performance indicators-RSS, AIC, and BIC-show very similar values for both forms, and a closer examination of the residuals in Fig. 4 reveals no clear visual advantage for either the piecewise linear or quadratic model.</text>
<text><location><page_5><loc_52><loc_23><loc_92><loc_29></location>TAKE-AWAY: Both piecewise linear and quadratic functions provide a better fit to the radial metallicity relation than a simple linear model. However, based on our assessments, there is no clear preference between the piecewise and quadratic functions.</text>
<section_header_level_1><location><page_5><loc_63><loc_20><loc_82><loc_22></location>3.2. The influence of binning</section_header_level_1>
<text><location><page_5><loc_52><loc_11><loc_92><loc_20></location>In this section, we test the influence of fitting a function to all points of the distribution or binned data in steps of Δ 𝑅 Gal . = 0 . 5 kpc, using median values as data points and standard deviations /two.sup as uncertainty (see also Hemler et al. 2021, who fitted functions to radially binned IllustrisTNG data). The results are shown in Fig. 4. Given that more than half of the young star particles of the galaxy are within</text>
<text><location><page_5><loc_52><loc_7><loc_92><loc_9></location>/two.sup Wenote that this 𝜎 is not equivalent to observational uncertainty and can thus not be directly applied onto observational analyses.</text>
<paragraph><location><page_6><loc_48><loc_90><loc_53><loc_91></location>TABLE 2</paragraph>
<table>
<location><page_6><loc_8><loc_76><loc_92><loc_86></location>
<caption>F/i.pc/t.pc E/v.pc/a.pc/l.pc/u.pc/a.pc/t.pc/i.pc/o.pc/n.pc /o.pc/f.pc /l.pc/i.pc/n.pc/e.pc/a.pc/r.pc, /q.pc/u.pc/a.pc/d.pc/r.pc/a.pc/t.pc/i.pc/c.pc, /a.pc/n.pc/d.pc /p.pc/i.pc/e.pc/c.pc/e.pc/w.pc/i.pc/s.pc/e.pc /l.pc/i.pc/n.pc/e.pc/a.pc/r.pc /f.pc/i.pc/t.pc/s.pc. E/x.pc/t.pc/r.pc/a.pc /p.pc/a.pc/r.pc/a.pc/m.pc/e.pc/t.pc/e.pc/r.pc/s.pc /a.pc/r.pc/e.pc /q.pc/u.pc/a.pc/d.pc/r.pc/a.pc/t.pc/i.pc/c.pc /t.pc/e.pc/r.pc/m.pc /a.pc/n.pc/d.pc /b.pc/r.pc/e.pc/a.pc/k.pc /r.pc/a.pc/d.pc/i.pc/u.pc/s.pc /f.pc/o.pc/r.pc /t.pc/h.pc/e.pc /q.pc/u.pc/a.pc/d.pc/r.pc/a.pc/t.pc/i.pc/c.pc /a.pc/n.pc/d.pc /p.pc/i.pc/e.pc/c.pc/e.pc/w.pc/i.pc/s.pc/e.pc /f.pc/i.pc/t.pc. RSS /s.pc/t.pc/a.pc/n.pc/d.pc/s.pc /f.pc/o.pc/r.pc R/e.pc/s.pc/i.pc/d.pc/u.pc/a.pc/l.pc S/u.pc/m.pc /o.pc/f.pc S/q.pc/u.pc/a.pc/r.pc/e.pc/s.pc (E/q.pc. 4). AIC /s.pc/t.pc/a.pc/n.pc/d.pc/s.pc /f.pc/o.pc/r.pc A/k.pc/a.pc/i.pc/k.pc/e.pc I/n.pc/f.pc/o.pc/r.pc/m.pc/a.pc/t.pc/i.pc/o.pc/n.pc C/r.pc/i.pc/t.pc/e.pc/r.pc/i.pc/o.pc/n.pc /a.pc/n.pc/d.pc BIC /s.pc/t.pc/a.pc/n.pc/d.pc/s.pc /f.pc/o.pc/r.pc B/a.pc/y.pc/e.pc/s.pc/i.pc/a.pc/n.pc I/n.pc/f.pc/o.pc/r.pc/m.pc/a.pc/t.pc/i.pc/o.pc/n.pc C/r.pc/i.pc/t.pc/e.pc/r.pc/i.pc/o.pc/n.pc (/s.pc/e.pc/e.pc E/q.pc. 6).</caption>
</table>
<figure>
<location><page_6><loc_9><loc_55><loc_48><loc_75></location>
<caption>F/i.pc/g.pc. 4.- Deviation of different radial metallicity gradient functions from the global linear fit. Shown are the different functions (linear, quadratic, and piecewise linear) estimated from the full distribution (solid lines) or medians and standard deviations (error bars) in Δ 𝑅 Gal . = 0 . 5 kpc bins (dashed lines).</caption>
</figure>
<text><location><page_6><loc_8><loc_37><loc_48><loc_48></location>𝑅 Gal . < 4 kpc, this binning - although counteracted by the smaller spread of [Fe/H] in the inner galaxy - weighs the distribution of the inner galaxy significantly less than when weighing all particles equally (20 vs. 34 000 data points). The parameters fitted to the binned data exhibit a larger uncertainty due to our use of the spread of [Fe/H] per bin as absolute uncertainty 𝜎 , but the fitted parameters agree well within the fitting uncertainties.</text>
<text><location><page_6><loc_8><loc_33><loc_48><loc_36></location>TAKE-AWAY: While the specific slopes differ when fitting all points or binned data, they agree within the small fitting uncertainties.</text>
<section_header_level_1><location><page_6><loc_12><loc_30><loc_45><loc_31></location>3.3. The influence of radial coverage on linear fits</section_header_level_1>
<text><location><page_6><loc_8><loc_10><loc_48><loc_29></location>Although we have gained useful insight into the global function, observational data will rarely cover the full extent of the stellar disk. Milky Way studies have previously been limited to the range of around 5 -15 kpc. There are often similar limitations and even gaps in extragalactic data. Using smaller ranges, observational studies have found hints of piece-wise linear gradients with a break radius in them based on limited radial coverage (e.g. Andrievsky et al. 2002a; Yong et al. 2012; Boeche et al. 2013; Hayden et al. 2014; Anders et al. 2017; Donor et al. 2020; Chen et al. 2023). These results are intriguing, since a quadratic function can, to first order, be approximated by two linear functions with a break radius. We therefore want to use our simulation to test if the radial coverage may indeed delude us into identifying broken linear gradients.</text>
<text><location><page_6><loc_8><loc_7><loc_48><loc_9></location>Wetest how smaller coverage and piecewise linear fits could mimic a complex global gradient by fitting in piecewise lin-</text>
<figure>
<location><page_6><loc_52><loc_41><loc_91><loc_75></location>
<caption>F/i.pc/g.pc. 5.- Impact of different coverage in galactocentric radius when fitting a linear radial metallicity gradient to young stars. Each horizontal segment uses a different running radial fitting range between 0.25 and 15 kpc as outlined on the right. For better contrast, the global linear fit is subtracted from the local gradient estimates and each line is colored by the gradient slope with a color scale centered around the global fit slope. Additionally, the slope of each line segment highlights the difference between global and local slopes. Thus, if the local fit exactly matches the global fit, we display it as a flat line (zero slope difference) on top of the gray dashed line (zero offset deviation) and give it a gray color signaling the same gradient value as the global fit.</caption>
</figure>
<text><location><page_6><loc_52><loc_7><loc_92><loc_27></location>ear radial ranges of 0.25, 0.5, 1, 2, 5, 8, 10, 15, and 20 kpc. We show their difference with respect to a global linear fit in Fig. 5, with color-coding indicating the slope of the local gradient. A horizontal dashed line indicates the same slope as the global fit, whereas the offset of a line from the said horizontal dashed line indicates the local deviation from the global gradient intercept. Differences in line slopes are visualising the difference in gradient slopes between the global and local fits. We see that all ranges suggest more or less significant deviations from a global linear fit. The innermost fit suggests a significantly different gradient than the outermost fit. We also note increasing slope differences towards the smallest scales, hinting at local deviations from a global pattern. We follow these up in Section 4, but for now, focus on the larger-scale trends.</text>
<text><location><page_7><loc_8><loc_76><loc_48><loc_92></location>When directly comparing an inner and outer radius fit, such as between 𝑅 Gal . = 5 -10 kpc (thick grey line in Fig. 5) and 𝑅 Gal . = 10 -15 kpc (thick black line in Fig. 5), we note a significant change, similar to previous estimates of the Milky Way (e.g. Yong et al. 2012; Lemasle et al. 2008). In our case, the gradient estimate changes from [ Fe / H ] ( 𝑅 Gal . ) = 0 . 471 -0 . 044 · 𝑅 Gal . to [ Fe / H ] ( 𝑅 Gal . ) = 0 . 375 -0 . 034 · 𝑅 Gal . . When looking at linear gradient fits across the radial coverage of Δ 𝑅 Gal . = 5 -15 kpc in Fig. 5, the gradient is steeper (bluer color) for smaller radii and flatter (redder) for larger radii. Indeed, a piecewise linear function can well mimic a complex global gradient.</text>
<text><location><page_7><loc_8><loc_68><loc_48><loc_76></location>We note that in the simulated data, we see local deviations that become traceable below Δ 𝑅 Gal . ≤ 2 kpc ∼ 0 . 5 Re (bottom part of Fig. 5). This might indicate the spatial resolution required to see local effects, such as spiral arms, for extragalactic studies (see also Krumholz & Ting 2018; Li et al. 2024). We pursue this observation in the following Section 4.</text>
<text><location><page_7><loc_8><loc_61><loc_48><loc_67></location>TAKE-AWAY: We find that a piecewise linear function can well mimicaquadraticfunction across the scales used in Milky Way and extragalactic studies. Local deviations become traceable below are spatial resolution of Δ 𝑅 Gal . ≤ 2 kpc (or Δ 𝑅 Gal . ≤ 0 . 5 Re).</text>
<section_header_level_1><location><page_7><loc_10><loc_57><loc_47><loc_59></location>4. SCATTER AND LOCAL DEVIATIONS FROM THE GRADIENT</section_header_level_1>
<text><location><page_7><loc_8><loc_44><loc_48><loc_56></location>Now that we are sufficiently satisfied that our flattening gradient function reproduces the overall shape of the radial metallicity gradient, we are concerned with both the scatter and local slope deviations across the galactocentric radii in this section. In detail, we analyse the scatter (Section 4.2), vertical variations (Section 4.2), azimuthal variations (Section 4.3, particularly motivated by the localised, spiral-shaped fit residuals of Figs. 2c and 2f) and deviations across different ages (Section 4.4).</text>
<section_header_level_1><location><page_7><loc_25><loc_42><loc_32><loc_43></location>4.1. Scatter</section_header_level_1>
<text><location><page_7><loc_8><loc_38><loc_48><loc_41></location>When investigating the change in scatter from the innermost radii to the outermost (see Fig. 3c), we see a steady increase in 1 -𝜎 spread. This spread increases from</text>
<text><location><page_7><loc_8><loc_36><loc_41><loc_37></location>𝜎 [ Fe / H ] = 0 . 01 dex at 𝑅 Gal . = 0 . 25 ± 0 . 25 kpc to</text>
<text><location><page_7><loc_8><loc_35><loc_48><loc_36></location>𝜎 [ Fe / H ] = 0 . 06 dex at 𝑅 Gal . = 8 . 25 ± 0 . 25 kpc and reaches</text>
<text><location><page_7><loc_8><loc_33><loc_40><loc_35></location>𝜎 [ Fe / H ] = 0 . 10 dex at 𝑅 Gal . = 19 . 75 ± 0 . 25 kpc.</text>
<text><location><page_7><loc_8><loc_22><loc_48><loc_33></location>When we recall the observed significant spread in metallicities of young open clusters at the solar radius beyond observational uncertainty (e.g. Donor et al. 2020; Spina et al. 2021) and our selection of only young ( < 0 . 5 Gyr) stars from the simulation, a strong impact of this scatter by radial migration should be excluded. At this point, we can imagine that this chemical diversity might be caused by less well-mixed gas or non-radial effects (such as vertical or azimuthal ones), which we investigate subsequently.</text>
<section_header_level_1><location><page_7><loc_21><loc_19><loc_36><loc_20></location>4.2. Vertical deviations</section_header_level_1>
<text><location><page_7><loc_8><loc_7><loc_48><loc_19></location>In this section, we now look at deviations with respect to the vertical dimension, that is, 𝑅 -𝑧 . In Fig. 6 we show the previously identified local gradient deviations (lines following the left axis label) on top of the vertical density distribution ( 𝑅 -𝑧 ) of young stars (Fig. 6a) and gas (Fig. 6b) between -3 < 𝑧 < 3 kpc. Although the quickly decreasing number of young stars (Fig. 6a) at outer radii does not show substructure in the density plots for reasonable bin sizes, we see more substructure for the gaseous component in Fig. 6b). In particular,</text>
<figure>
<location><page_7><loc_52><loc_77><loc_91><loc_92></location>
</figure>
<figure>
<location><page_7><loc_52><loc_62><loc_91><loc_76></location>
<caption>F/i.pc/g.pc. 6.- Local gradient deviations (red-blue lines) similar to the second lowest row of Fig. 5 for radial gradients in 0 . 5 kpc steps (but compared to a global quadratic function) overlapped on top of the logarithmic density distribution in 𝑅 -𝑧 for | 𝑧 | < 3 kpc of gas (panel a) and stars (panel b). We see no strong correlation between local gradient slopes (red-blue lines) and star or gas density in this projection.</caption>
</figure>
<text><location><page_7><loc_52><loc_41><loc_92><loc_53></location>we see rather minor deviations at small radii (where most stars and gas are close to the plane). At increasing radii, we notice an increase in both the vertical distribution of stars, and increasing local gradient deviations. In particular, we note a significant deviation of the slope around 𝑅 Gal . ∼ 15 kpc, where the gradient deviation line is steep and blue (indicating a much steeper gradient at this radius), and we notice a significant overdensity of gas around 𝑧 ∼ 1 kpc. Overall, however, we do not see strong correlations in this particular plane.</text>
<text><location><page_7><loc_52><loc_23><loc_92><loc_41></location>This could, however, be caused by a super-position effect of the up- and downturn at larger radii due to the galactic warp (see Figs. 1b and 1e). Although the warp of the stellar disk in Fig. 1b is not as clear, we confirm that both the gas disk and the youngest stars below 0 . 5 Gyr are tracing each other across the simulation in both galactocentric radius 𝑅 Gal . and height 𝑧 Gal . for different sectors in the azimuthal direction. We note that the superposition in Fig. 6 could smear out local correlations of slope changes with gas overdensities, for example, by spiral arms. Although such an edge-on view of the galaxy may indeed be the only observable one for extragalactic targets, for example, of the GECKOS survey of edge-on galaxies (van de Sande et al. 2023), we have the luxury of being able to analyse the azimuthal direction of our simulated galaxy, too.</text>
<text><location><page_7><loc_52><loc_15><loc_92><loc_22></location>TAKE-AWAY: We see no strong correlations of deviations in the vertical direction throughout the simulation. Such correlations could, however, be blurred by azimuthal effects, like the galactic warp, which needs to be disentangled in the azimuthal dimension.</text>
<section_header_level_1><location><page_7><loc_64><loc_13><loc_81><loc_14></location>4.3. Azimuthal deviations</section_header_level_1>
<text><location><page_7><loc_52><loc_7><loc_92><loc_12></location>To analyse the deviations from a global gradient across different azimuthal viewing angles, we divide the galaxy into 8 sectors with Δ 𝜛 Gal . = 45 · (see Fig. 7a). This allows us to study the positions around the upturn and downturn of the</text>
<figure>
<location><page_8><loc_13><loc_53><loc_88><loc_91></location>
<caption>F/i.pc/g.pc. 7.- Stellar density variation across 8 different sectors (with color-code visualised in panel a) of the radial metallicity gradient 𝑅 - [ Fe / H ] across 8 different azimuth ranges (panels b-i). A rotating lighthouse-like GIF animation of the median age and median density of the 𝑅 - [ Fe / H ] -relation across different azimuths is freely available on a repository.</caption>
</figure>
<figure>
<location><page_8><loc_10><loc_11><loc_90><loc_46></location>
<caption>F/i.pc/g.pc. 8.- Same as Fig. 7, but colored by median age per bin. We identify 3 groups with boxes in panels e, f, and h. A rotating lighthouse-like GIF animation of the median age and median density of the 𝑅 - [ Fe / H ] -relation across different azimuths is freely available a repository.</caption>
</figure>
<text><location><page_9><loc_8><loc_87><loc_48><loc_92></location>galactic warp with the median azimuth of young star particles below and above the plane being 𝜑 Gal . ∼ 183 · and 𝜑 Gal . ∼ 4 · , respectively (see Fig. 1e), while maintaining a reasonable sample size.</text>
<text><location><page_9><loc_8><loc_60><loc_48><loc_86></location>At face value, the distribution of 𝑅 Gal . - [ Fe / H ] for each sector follows a similar, rather linear shape with most stars being born in the inner 5 kpc of the galaxy. However, we find significant deviations in different sectors of the galaxy (Fig. 7). On the one hand, we find non-linear deviations as bumps with slightly increased or decreased iron abundance - up to 0 . 1 -0 . 2 dex - in Figs. 7c at 𝑅 Gal ∼ 18 kpc, 7d at 𝑅 Gal ∼ 10 kpc, 7f at 𝑅 Gal ∼ 14 kpc, and 7g at 𝑅 Gal ∼ 17 kpc. On the other hand, we find significant gaps in the distribution at similar [Fe/H], most strikingly at the upturn of the galactic warp in Fig. 7e ( 𝜛 Gal . = 135 -180 · ) at [ Fe / H ] ∼ 0 dex and 𝑅 Gal . ∼ 8 -14 kpc. We note that the sector e) with the gap is surrounded by two sectors (d and f) with significant overabundance at the same radius. This could be indicative of stars having formed as a result of gas moving from sector e towards either azimuthal direction, causing a gas overdensity which could in turn lead to higher star formation activity. To establish this observation, we take a closer look at the timedomain, that is, stellar age as well as the spatial domain of 𝑅 Gal . -𝜑 Gal . in the next section.</text>
<text><location><page_9><loc_8><loc_50><loc_48><loc_59></location>TAKE-AWAY: We find various deviations from the global trend in the azimuthal direction, including gaps and isolated streaks of stars with similar [Fe/H] throughout Δ 𝑅 Gal . = 2 -6 kpc. These can introduce local over- and under-enhancement of up to ± 0 . 2 dex in [Fe/H] at a given radius. In the next section, we analyse whether the stars of these streaks have been born at the same or different time.</text>
<section_header_level_1><location><page_9><loc_17><loc_47><loc_40><loc_48></location>4.4. Deviations with time and age</section_header_level_1>
<text><location><page_9><loc_8><loc_19><loc_48><loc_47></location>In this section, we examine the radial metallicity gradient in a small age range less than 0 . 5 Gyr. To do so, we color Fig. 7 by the median stellar age rather than the logarithmic density in Fig. 8. We find an overall significant scatter across time, suggesting a good mix of star formation across all sectors for stars born within less than 0 . 5 Gyr. For stars within this restricted age range, we do not see a strong correlation with radius, such as older stars being born further inside, but a larger amount of stars being born closer to the galactic centre. Wenote that stars with similar [Fe/H] in each sector tend to be formed at similar times (within 50 Myr), that is, as flat lines with the same color (age) in Figs. 8b-i. To guide the eye, we have identified Group 1 in Fig. 8e (around 𝑅 Gal . ∼ 14 kpc at 𝜛 Gal . = 180 -225 · ). We further note that the enriched bumps identified earlier are born at similar times, see, for example, Group 2 in Fig. 8f. The coloring by age also reveals that stars with lower [Fe/H] than expected (see Group 3 in Fig. 8h) are born at similar times. In some cases, these extend to Δ 𝑅 Gal . = 2 -6 kpc, see Groups 1, 2, and 3 in Fig. 8. At a given radius 𝑅 Gal . , these streaks cause a significant spread in local [Fe/H] of up to ± 0 . 2 dex (see Fig. 8).</text>
<text><location><page_9><loc_8><loc_7><loc_48><loc_19></location>From the analysis of azimuthal sectors, the impression arose that the star formation in this simulated Milky Way analogue is - as expected for a spiral galaxy - rather patchy and localised on the smallest timescales. This is confirmed by looking at the spatial distribution of azimuth 𝜑 Gal . and radius 𝑅 Gal . in Fig. 9. Already when looking at the density distribution of all stars born within less than 0 . 5 Gyr in Fig. 9a, multiple streams are visible, stars on spiral patterns (see also Kreckel et al. 2019; Chen et al. 2024b). When following up the previously identi-</text>
<figure>
<location><page_9><loc_52><loc_77><loc_91><loc_92></location>
</figure>
<figure>
<location><page_9><loc_52><loc_61><loc_91><loc_77></location>
<caption>F/i.pc/g.pc. 9.- Density distribution (panel a) and age distribution (panel b) of young stars in the azimuthal and radial direction 𝜑 Gal . -𝑅 Gal . . In panel a), we also show the groups previously identified in Fig. 8.</caption>
</figure>
<text><location><page_9><loc_52><loc_39><loc_92><loc_55></location>roups 1, 2, and 3, we recover them on said spiral patterns (Groups 1 and 3) or a local overdensity (Group 2). Although one could imagine that radial migration might induce such a spiral-like shape for the stars of groups 1 and 3, their low age of less than 250 Myr would require a significant migration effect of several kpc, while having no influence on the older stars of group 2. When tracing the position of significant overdensities from Fig. 9a in the same projection colored by age in Fig. 9b, we note that for radii above 𝑅 Gal . > 5 kpc these overdensities are colored in red, that is, containing indeed young stars with ages below 200 Myr and being consistent with the most recent star formation along these spiral patterns in the outer galaxy.</text>
<text><location><page_9><loc_52><loc_30><loc_92><loc_38></location>TAKE-AWAY: We find significant scatter across the radial metallicity distribution caused by streaks of stars born with similar [Fe/H] at similar times (within 50 Myr) across either very local or radially extended spiral-shaped regions of the galaxy, suggesting local enhancement patterns in small overdensities or along spiral arms.</text>
<section_header_level_1><location><page_9><loc_66><loc_28><loc_78><loc_29></location>5. DISCUSSION</section_header_level_1>
<text><location><page_9><loc_52><loc_17><loc_92><loc_27></location>Having presented the analysis, we now put our results into the context of other work in terms of our initial aims: to analyse the shape (Section 5.1), scatter (Section 5.2), local deviations (Section 5.3), and time-dependence (Section 5.4) of the radial metallicity gradient. These initial discussions inform our thoughts on the implications of this work for Milky Way studies in Section 5.5 and the studies of other galaxies in Section 5.6.</text>
<section_header_level_1><location><page_9><loc_57><loc_14><loc_87><loc_15></location>5.1. Linearity of the radial metallicity gradient</section_header_level_1>
<text><location><page_9><loc_52><loc_7><loc_92><loc_13></location>The radial metallicity gradient of our simulated NIHAOUHDMilky Way analogue showed an overall decreasing, predominantly linear shape, as established in Section 3. Motivated by previous works by Sánchez-Menguiano et al. (2016), among others, we also fitted piecewise linear and quadratic</text>
<figure>
<location><page_10><loc_9><loc_71><loc_91><loc_92></location>
<caption>F/i.pc/g.pc. 10.- Comparison of the Milky Way's radial metallicity trend as traced by Cepheids (black triangles, compiled from literature by Genovali et al. 2014, G+14) as well as young ( < 0 . 5 Gyr) open cluster of the Milky Way as traced by the literature compilation from Genovali et al. (2014, G+14 as squares), APOGEE DR17 from Myers et al. (2022, M+22 as crosses), and GALAH DR3 from Spina et al. (2021, S+21 as circles). The latter two are compiled based on the membership and age catalogue by Cantat-Gaudin & Anders (2020, CG+20).</caption>
</figure>
<text><location><page_10><loc_8><loc_42><loc_48><loc_65></location>functions to the data in Section 3.1. Both forms perform better than a linear trend. The very similar fitting performances indicate no significant preference between either piecewise linear or quadratic function. Due to both functions' rather good overall fit, we have not tried more exotic non-linear functions as done by Scarano & Lépine (2013). Increasing the flexibility of the gradient function could, however, improve the fit at the innermost kpc, where a flattening is predicted by our simulation, but chemical enrichment is also harder to simulate (see also Minchev et al. 2013; Sun et al. 2024). We have found no significant influence of binning for our gradient estimates (Section 3.2), but have found that a limited radial coverage - as is the case for the Milky Way - could mimic a truly quadratic function with two piecewise linear fits (Section 3.3). This is important, as it has significant implications for the conclusions we draw from the incomplete data of our Milky Way, as we will discuss in more detail in Section 5.5.</text>
<text><location><page_10><loc_8><loc_8><loc_48><loc_42></location>The balance between a quadratic and piecewise linear radial metallicity gradient teeters at the breaking radius. If present, our analysis of the Milky Way analogue would place it at 𝑅 break ∼ 10 ± 0 . 5 kpc. This radius is strikingly close to the radius of 9 kpc found by Hemler et al. (2021) for a TNG50 galaxy simulation with a stellar mass of log ( 𝑀 ★ / M ⊙ ) = 10 . 72, that is, close to the Milky Way's (see their Fig. 2). In terms of physical reasons for a breaking radius at this location, a direct and secular influence of a stellar bar with non-symmetric effects around the corotation radius (Di Matteo et al. 2013; Scarano & Lépine 2013) should be minor for our specific scenario due to the low ages of the stars considered in our analysis. In particular, our identified break radius is significantly larger than the corotation radius of the Milky Way bar at 4 . 5 -7 . 0 kpc (Bland-Hawthorn & Gerhard 2016, and references therein) anyway. We are intrigued, however, by the proposition by Garcia et al. (2023) of galactic discs consisting of a star-forming inner disc with a steep gradient and a mixing-dominated outer disc with a flat gradient, with the break radius marking the region of transition between them. In Illustris TNG50-1 data, they found such a transition and break radius to be situated much further out at 30 kpc for Milky Way mass galaxies (10 . 1 ≤ log ( 𝑀 ★ / M ⊙ ) ≤ 10 . 6). While our bestfitting break radius - if present - is inconsistent with theirs, we will follow this up in more detail in Section 5.6, where we also discuss the implications for extragalactic studies in general.</text>
<figure>
<location><page_10><loc_52><loc_32><loc_91><loc_65></location>
<caption>F/i.pc/g.pc. 11.- Same as Fig. 3, but for gas.</caption>
</figure>
<section_header_level_1><location><page_10><loc_57><loc_27><loc_87><loc_28></location>5.2. Scatter of the radial metallicity gradient</section_header_level_1>
<text><location><page_10><loc_52><loc_8><loc_92><loc_27></location>In Section 4.1 we found an increasing scatter from 𝜎 [ Fe / H ] = 0 . 01 dex in the inner galaxy to 𝜎 [ Fe / H ] = 0 . 10 dex around 𝑅 Gal . ∼ 20 kpc. Comparing these values with simulations other than TNG50 with a similar metallicity spread (see Fig. 2 by Hemler et al. 2021) is rather difficult, as the literature focuses on the shape and density distribution (see e.g. Minchev et al. 2014, their Fig. 10). When comparing with Milky Way studies (e.g. Anders et al. 2017), the scatter in the simulation is smaller than the observed spread of [Fe/H]. This can be visually appreciated by comparing the combinations of different measurements in the Milky Way (Genovali et al. 2014; Spina et al. 2021; Myers et al. 2022) in Fig. 10a and our simulation in Fig. 10b and c. We discuss the implications of this on studies of the Milky Way's gradient in Section 5.5.</text>
<text><location><page_10><loc_53><loc_7><loc_92><loc_8></location>When assuming that young star and gas phase abundances</text>
<figure>
<location><page_11><loc_19><loc_68><loc_82><loc_92></location>
<caption>F/i.pc/g.pc. 12.- Tracing young stars and gas across galactocentric radii 𝑅 Gal . and height 𝑧 Gal . across the whole galaxy (panel a) and different azimuthal ranges/sectors (panels b-i). Small rectangles with cool-warm colors along the horizontal axis indicate the local gradient slopes as in Fig. 6.</caption>
</figure>
<text><location><page_11><loc_8><loc_39><loc_48><loc_63></location>are similar, we find comparable scatter of abundances for example with respect to TYPHOON observations by Chen et al. (2023). To test this assumption, we also show the gas phase metallicity in Fig. 11, for which we find a similar shape and scatter of the gradient, but systematically less scatter or spread than observed gas phase abundance, thus urging us to treat the absolute values of abundances and abundance scatters as well as spreads with caution. We furthermore note that the spread of abundance in observations does only increase for some but not all of the observed (and thus observationally limited) galaxies by Chen et al. (2023). This potentially limits the range of galaxies to which our conclusions may apply. While the simulated abundance scatter is consistent with the predictions by the theoretical forced-diffusion model by Krumholz & Ting (2018), that is, a scatter of ∼ 0 . 1 dex over timescales of ∼ 100 -300 Myr, our simulations suggest that the scatter is driven by the radial structure and large-scale spiral arms, which were not included in their model.</text>
<section_header_level_1><location><page_11><loc_9><loc_35><loc_47><loc_38></location>5.3. Localised vertical and azimuthal deviations and their correlation with gas</section_header_level_1>
<text><location><page_11><loc_8><loc_25><loc_48><loc_35></location>In Sections 4.2 and 4.3 we established that local deviations contribute significantly to the spread of the global metallicity gradient above 𝑅 Gal . > 8 kpc ∼ 2 Re. We noted in particular a void of stars where we found an upturning warp of the galaxy around 𝜑 Gal . ∼ 180 · spatially close to regions of the galaxy (Groups 1 and 2) that deviated most significantly from the overall trend in Fig. 8 and Fig. 9.</text>
<text><location><page_11><loc_8><loc_7><loc_48><loc_25></location>These stellar voids pose the question if they are also void of gas thus suggesting the gas has shifted to the more enhanced regions. In Fig. 12 we are thus tracing both the spatial distribution of gas as colored density distribution and stars as grey contour lines and gas. We find that while stars and gas trace each other in the vertical direction, we do not always see a match between the two tracers in the radial direction. In particular, we do find a significant amount of gas around the stellar void of 𝜑 Gal . ∼ 180 · and 𝑅 Gal . ∼ 8 -11 kpc in Figs. 12e and 12f. This gas seems to be more tightly concentrated though for example in the tight wave around 𝑅 Gal . ∼ 8 -11 kpc in Fig. 12e. We also note that significant gas overdensities, for example around 𝜑 Gal . ∼ 0 -45 · and 𝑅 Gal . ∼ 7 kpc in Fig. 12a do not seem to correlate with significant overenhancement in</text>
<text><location><page_11><loc_52><loc_57><loc_92><loc_63></location>iron abundance (compare to Fig. 7a). While we see a hint of a coinciding deviation of Δ [ Fe / H ] for larger deviations from the galactic plane Δ 𝑧 in the upturning outer region of Fig. 12f, this does not seem to be the case for the downturning outer region of Figs. 12b and 12i.</text>
<text><location><page_11><loc_52><loc_42><loc_92><loc_56></location>As the edge-on projection is not providing conclusive insights, we are now looking into the phase-on projection in Fig. 13. We have chosen a region of the simulated galaxy whose gas density at solar radius (Fig. 13c) matches with the recently measured distribution of young stars in the Milky Way at face value (Fig. 13a) by Poggio et al. (2021). Comparing simulated gas and observed young stars is preferable in this case, as the density of simulated stars is too low to easily identify overdensities (Fig. 13b). The region and its gas spiral structures appear to be representative, as these structures exist throughout the whole galaxy (see Fig. 1d).</text>
<text><location><page_11><loc_52><loc_12><loc_92><loc_42></location>In the different panels of Fig. 14, we thus show this representative region of the galaxy, but color each spatial bin by stellar metallicity (Fig. 14a), the deviation from the global linear trend (Fig. 14b) as well as the gas metallicity (Fig. 14c) and its deviation from the global linear trend (Fig. 14d). In all cases, we also overlay the density contours of the significant gas overdensities (red regions in Fig. 13c). As expected, we see that the metallicity color map of the stars in Fig. 14a shows a decreasing trend from right to left (inner to outer galaxy) and an increasing scatter (more blue and red points towards the left) in the residual plot of Fig. 14b. We cannot identify a strong correlation between gas overdensities and stellar metallicity or residuals in either plot - possibly caused by the low number density. In Figs. 14c and 14d, however, the radial gas metallicity gradient shows significant local variations, that is, a trend from left to right that is not very smooth. In particular, we find significant deviations of up to + 0 . 15 dex in [Fe/H] behind the outer gas spiral (lower left of Fig. 14d) and -0 . 1 dex in [Fe/H] in front of the inner gas spiral (upper center of Fig. 14d) with a steep edge consistent with the gas spiral edge. We have identified the same patterns in both [Fe/H] and A(O) as both elements trace each other rather well in the simulation of young stars (see Fig. A1).</text>
<text><location><page_11><loc_52><loc_8><loc_92><loc_11></location>Tentatively, we even see a slight enhancement of A(O) at the trailing edge of the inner spiral arm (top of Fig. 14d). We convince ourselves of the step-like behaviour by selecting a</text>
<figure>
<location><page_12><loc_12><loc_71><loc_89><loc_92></location>
<caption>F/i.pc/g.pc. 13.- Comparison of the density distribution of young stars and gas in the Milky Way and the NIHAO Milky Way analogue simulation. Panel a) shows the measurements of the Solar vicinity within 5 kpc by Poggio et al. (2021). Panels b) and c) show young stars and gas NIHAO, respectively, for a selected region similar to panel a). Black and white contour lines in panel b) trace overdensities in the gas distribution of panel c).</caption>
</figure>
<figure>
<location><page_12><loc_9><loc_45><loc_91><loc_65></location>
<caption>F/i.pc/g.pc. 14.- Comparison of density distribution of young stars and gas in the NIHAO-UHD Milky Way analogue simulation for the same regions as Figs. 13b and 13c. Panels a) and c) trace median young star Fe and gas O abundances, respectively. Panels b) and d) plot the metallicity residuals of stars and gas, respectively, when correcting with a radial metallicity gradient fit. Black and white contour lines in each panel trace overdensities in the gas distribution of Fig. 13c).</caption>
</figure>
<text><location><page_12><loc_8><loc_28><loc_48><loc_40></location>small slit-like region of 𝜑 Gal . ∼ 0 · and -2 < 𝑌 Gal . / kpc < -1 and tracing the gas metallicity and gas density as a function of radius in Fig. 15. We indeed find steps and confirm that they coincide with the location of significant gas overdensities. These step-patterns have also been found by Grand et al. (2015) in another simulation and observationally by Ho et al. (2017). In Fig. 15, we note an extended flat region just beyond 𝑅 Gal . > 10 kpc, the best fitting 𝑅 break of an assumed piecewise linear fit.</text>
<text><location><page_12><loc_8><loc_9><loc_48><loc_28></location>Our analyses suggest that the correlation of void and overdensities with chemical enrichment of gas and young stars is more complicated and should better be followed up by tracing these structures over simulation look-back time in a dedicated follow-up analysis to unravel the physical mechanisms of star formation feedback cycles . This could also involve the tracing of star formation bursts and disk instabilities (Sánchez et al. 2014; Sánchez-Blázquez et al. 2014; Ho et al. 2015) as well as tracing how much self-enrichment as well as mixing and dilution takes place around the gas spirals (Ho et al. 2017). Rather than going back in simulation time, the present simulation data of a single snapshot in time already allows us to look back in terms of stellar lifetime - similar to Milky Way studies, as we discuss subsequently.</text>
<figure>
<location><page_12><loc_52><loc_16><loc_91><loc_40></location>
<caption>F/i.pc/g.pc. 15.- Radial gas metallicity gradient of a slit-like region ( -2 < 𝑌 Gal . / kpc < -1) from Fig. 14. The plot extends towards larger and smaller radii and shows the step-like distribution of individual gas particle metallicities colored by their deviation from a global fit. A running median along 1000 particles is shown as a black line. The gray histogram indicates the gas density along the radius with prominent overdensities coinciding with step edges.</caption>
</figure>
<figure>
<location><page_13><loc_9><loc_51><loc_48><loc_92></location>
<caption>F/i.pc/g.pc. 16.- Stellar density distribution and spread of [Fe/H] across different galactocentric radii with respect to a global linear radial metallicity gradient across different age ranges. Panels a-j) show young stars and exhibit a rather similar trend, whereas the scatter increases significantly for stars above 0 . 5 Gyr in panels k) and l).</caption>
</figure>
<section_header_level_1><location><page_13><loc_10><loc_42><loc_47><loc_43></location>5.4. The impact of time and age: mixing and migration</section_header_level_1>
<text><location><page_13><loc_8><loc_27><loc_48><loc_41></location>Although we have chosen a rather small stellar age window of 0 . 5 Gyr to trace the radial metallicity gradient without the expected significant impact of radial mixing and migration, we are testing and discussing this particular choice in this section in two ways. Firstly, we test the deviation of the radial metallicity gradient from the same global shape as well as the abundance spread across smaller age bins of 100 Myr between 0 -1000 Myr in Fig. 16. Secondly, we trace the distribution of stellar metallicity across galactic radii for increasing age bins from 50 Myr up to the maximum stellar age of 13 . 8 Gyr in Fig. 17.</text>
<text><location><page_13><loc_8><loc_19><loc_48><loc_27></location>Our first test in Fig. 16 shows that the deviation from a global trend remains similar in functional form. We find that the spread of iron abundance does indeed scatter significantly, but the distributions stay within the same overall shape across the ten age bins. We note though, that the smallest age bin of 0 -100 Myr shows the least abundance scatter.</text>
<text><location><page_13><loc_8><loc_7><loc_48><loc_19></location>This is consistent with the picture from our second test of increasing age ranges in Fig. 17. Here we find the first significant deviation from a tighter and already slightly quadratic relation for an age of 100 -150 Myr in Fig. 17c - our previously identified Group 3. As expected from previous simulations and observations, we see an increase in the scatter as we include more and more older stars. We note a still similar albeit more scattered shape for stars below 4 Gyr in Fig. 17h, before we start to see a more metal-poor population of stars in the inner</text>
<figure>
<location><page_13><loc_53><loc_52><loc_90><loc_92></location>
<caption>F/i.pc/g.pc. 17.- Radial metallicity gradients and quadratic fits for different maximum age ranges. The quadratic fit to stars below 0 . 5 Gyr is shown as a dashed red line for reference and the quadratic fit to each shown distribution is overlaid as a solid red line with the functional form given as inset text. At 𝑅 Gal . = 8 . 21 kpc, spread increases from 𝜎 [ Fe / H ] = 0 . 05 for youngest stars to 0.09 and 0.11 for stars below 4 and 8 Gyr, respectively.</caption>
</figure>
<text><location><page_13><loc_52><loc_29><loc_92><loc_43></location>galaxy appear between 4 -8 Gyr in Fig. 17i. These also begin to significantly impact the quadratic fit to the radial metallicity distribution, shown as a solid red line, in contrast to our reference fit, represented by a dashed red line. The significant amount of metal-poor stars in the inner galaxy then completely tilts the distribution when also including stars between 8 -13 . 8 Gyr in Fig. 17j (see also Johnson et al. 2024). Similar to the Milky Way (Bland-Hawthorn & Gerhard 2016), these oldest stars are those of the relatively more metal-poor thick disk that are confined to the inner disk with a shorter scale length.</text>
<text><location><page_13><loc_52><loc_19><loc_92><loc_28></location>While we cannot exclude radial migration playing a role for change of radius for the youngest stars of the simulation, since Frankel et al. (2018) predicted significant shifts even for ages below 0 . 5 Gyr (see their Fig. 10), the larger scatter for older stars is certainly suggesting a larger (re-)distribution of stars along the radial axis, as found in previous simulations (Minchev & Famaey 2010; Grand et al. 2015).</text>
<section_header_level_1><location><page_13><loc_59><loc_17><loc_85><loc_18></location>5.5. Implications for Milky Way studies</section_header_level_1>
<text><location><page_13><loc_52><loc_12><loc_92><loc_16></location>Our analysis of the radial metallicity gradient in a simulated NIHAO-UHD galaxy offers several insights that are directly applicable to understanding the Milky Way's gradient.</text>
<text><location><page_13><loc_52><loc_7><loc_92><loc_12></location>First, the nature of the gradient - whether it is linear or better described by more complex functional forms - remains a critical question. Previous studies, such as those by Lépine et al. (2011) and Donor et al. (2020), have suggested the po-</text>
<text><location><page_14><loc_8><loc_72><loc_48><loc_92></location>ential for a break radius, possibly at the corotation radius or further out (Scarano & Lépine 2013), which could indicate two distinct linear regimes. In our analysis, we find evidence that the gradient is at least is not purely linear, but could also be smoothly flattening. Applying a smooth quadratic function on observational data (Yong et al. 2012; Andrievsky et al. 2004; Genovali et al. 2014), might provide a better or at least consistent fit for the Milky Way data without the need for a break radius. However, even this may not fully capture the nuances observed in our simulations. Chemical evolution models propose a more sophisticated behaviour (e.g. Chiappini et al. 2001; Kubryk et al. 2015; Palla et al. 2024), reflecting varying influences of galactic processes at different radii. Understanding this structure in the simulated galaxy provides a framework for interpreting similar complexities in the Milky Way.</text>
<text><location><page_14><loc_8><loc_49><loc_48><loc_72></location>Given these complexities, it is also essential to consider how local sampling biases might affect our understanding of the Milky Way's metallicity gradient. For instance, incomplete samples that omit low [Fe/H] clusters or stars could skew gradient estimates, as suggested by our comparisons in Figures 10a and 10b. Our results indicate that young clusters with lower (or higher) [Fe/H] than expected at a given radius could indicate the previous presence of a spiral arm (see our identified Groups in Figs. 8 and 9). Furthermore, we caution that localised effects - both intrinsic and in terms of selection function - could also mimic non-linear shapes and more spatial coverage is needed in the Milky Way. Our results also indicate that older clusters, which have been found more frequently at larger distances than young clusters - are likely influenced by radial migration - and thus complicate the interpretation of these radial metallicity gradients (Magrini et al. 2009; Lépine et al. 2011).</text>
<text><location><page_14><loc_8><loc_33><loc_48><loc_49></location>Cosmological zoom-in simulations like NIHAO-UHD are approaching the resolution needed to examine regions analogous to the solar vicinity, though the star particle numbers and mass resolution remain a limiting factor. Nonetheless, we observe distinct patterns in the distribution of young stars and gas, including lower [Fe/H] and A(O) in the leading edges of gas overdensities and higher [Fe/H] and A(O) in the trailing edges, consistent with findings by Grand et al. (2016), Ho et al. (2017), and Kreckel et al. (2019). These trends suggest that local metallicity variations, driven by gas dynamics, may also play a significant role in shaping the observed gradients in the Milky Way.</text>
<text><location><page_14><loc_8><loc_14><loc_48><loc_33></location>Additionally, our study hints at the potential for more nuanced variations in [Fe/H] across different regions of the galaxy. In particular, the gas shows a step-like behavior of A(O)and[Fe/H] changes around the edges of gas overdensities (Fig. 15), with significant deviations from the global gradient in specific regions. We have also found a larger stellar void around -12 < 𝑅 Gal < -10kpc. Although further investigation is needed, these findings could have important implications for understanding localized star formation events and their impact on the overall metallicity distribution in the Milky Way (Sánchez et al. 2014; Sánchez-Blázquez et al. 2014; Ho et al. 2015). It will certainly be exciting to see how much more insights (Poggio et al. 2021; Hackshaw et al. 2024) we will get from the more extended data of future data releases of Gaia and spectroscopic surveys.</text>
<text><location><page_14><loc_8><loc_10><loc_48><loc_13></location>Wecannot directly link spiral arms to bar resonances or bardriven mixing in our simulation, because of a negligible bar strength in our galaxy /three.sup (but see Minchev & Famaey 2010; Di</text>
<text><location><page_14><loc_52><loc_79><loc_92><loc_92></location>Matteo et al. 2013). However, the influence of a galactic bar on the spiral arms and, by extension, on the radial metallicity gradient, remains a possibility (see again Chen et al. 2023). Disk instabilities and warps might further complicate the interpretation of these gradients and progress will likely rely on the detailed disentangling of these effects from both cosmological simulations as well as idealised simulations and models (Minchev et al. 2013; Grand et al. 2015, 2016; Krumholz et al. 2018; Sharda et al. 2021; Bland-Hawthorn et al. 2024; Tepper-Garcia et al. 2024).</text>
<section_header_level_1><location><page_14><loc_58><loc_76><loc_86><loc_77></location>5.6. Implications for extragalactic studies</section_header_level_1>
<text><location><page_14><loc_52><loc_70><loc_92><loc_75></location>The insights gained from our analysis of the radial metallicity gradient in a simulated NIHAO-UHD galaxy extend beyond the Milky Way, offering valuable implications for the study of extragalactic systems.</text>
<text><location><page_14><loc_52><loc_49><loc_92><loc_70></location>One key observation is that deviations from a purely linear metallicity gradient, as seen in our Milky Way analogue, are common in other galaxies as well. When fitting a piecewise linear fit to our data, we found a break radius at 𝑅 Gal . = 10 . 0 ± 0 . 5 kpc. Converted to effective radii Re or radii R25 covering the 25 mag arcsec -2 isophote /four.sup , this corresponds to 𝑅 break ∼ 2 . 5 Re ≡ 0 . 7 R25 for our simulation. This would be consistent with the observational results by Sánchez et al. (2014) who found that breaks in metallicity gradients are common in both spiral and barred galaxies, with flattening of the abundance being evident beyond ∼ 2 Re (compare also to Belfiore et al. 2017). Similar to our suggestion for Milky Way studies, we suggest to also test a smooth function, such as a quadratic one, on extragalactic observational data (e.g. Bresolin et al. 2012; Chen et al. 2023) to test the preference of a distinct break radius.</text>
<text><location><page_14><loc_52><loc_28><loc_92><loc_49></location>Although the focus of this research lies on the observable region of the Milky Way ( 𝑅 Gal . < 20 kpc) and most other galaxies ( 𝑅 Gal . < 2 . 5 Re), the finding of significant gradient changes in the outskirts of galaxies by Garcia et al. (2023), suggests to also test this region of our Milky Way analogue. Garcia et al. (2023, see their Fig. 4) found a metallicity floor in IllustrisTNG galaxies. When using their sample to identify a metallicity floor radius for a Milky Way mass galaxy with log ( 𝑀 ★ / M ⊙ ) = 10 . 7 (Bland-Hawthorn & Gerhard 2016) at redshift 𝑧 ∼ 0, we would expect to find it around 25 -30 kpc. We therefore extend the analysed radius to 𝑅 Gal . ≤ 100 kpc (see Fig. 18) and indeed find a similar abundance floor of [ Fe / H ] ≥ -0 . 64 for young stars and A ( O ) ≥ 8 . 12 for the majority of gas (see Fig. 19 at a similar radius. We note that another galaxy without gas in this figure is a sufficiently large distance of 92 kpc, that is, ( 𝑌,𝑌, 𝑍 ) = (-50 , -75 , 20 ) kpc.</text>
<text><location><page_14><loc_52><loc_10><loc_92><loc_27></location>These lowest abundances remind us of two observational results. Firstly, the iron abundance floor is consistent with the lower end of the Milky Way thin - and coincidentally outer disk of [ Fe / H ] ∼ -0 . 7 (Bensby et al. 2014; Buder et al. 2019). Secondly this oxygen abundance floor is consistent with the results by Grasha et al. (2022) from TYPHOON galaxy observations. Grasha et al. (2022) suggested this could be caused by changes in the ratio of supernovae II and AGB reflected by a changing ratio of nitrogen to oxygen abundance N/O which also flattens towards a lower plateau below metallicities of A ( O ) ∼ 8 . 0 (Nicholls et al. 2017). While we cannot follow this observation up with the present simulation, a similar simulation used by Buder et al. (2024) has traced the relative</text>
<figure>
<location><page_15><loc_9><loc_73><loc_36><loc_92></location>
</figure>
<text><location><page_15><loc_21><loc_73><loc_24><loc_75></location>XGal.</text>
<figure>
<location><page_15><loc_9><loc_55><loc_36><loc_73></location>
</figure>
<figure>
<location><page_15><loc_37><loc_55><loc_64><loc_72></location>
</figure>
<text><location><page_15><loc_21><loc_54><loc_25><loc_56></location>XGal.</text>
<text><location><page_15><loc_48><loc_54><loc_52><loc_56></location>XGal.</text>
<figure>
<location><page_15><loc_10><loc_13><loc_47><loc_50></location>
<caption>F/i.pc/g.pc. 18.- Same as Fig. 1, but for an extended 𝑅 Gal . ≤ 100 kpc and | 𝑧 Gal . | ≤ 50 kpc.F/i.pc/g.pc. 19.- Radial metallicity functions for all stars (panel a), young stars (panel b), and gas (panels c and d for iron and oxygen as metallicity tracers) out to 𝑅 Gal . ≤ 100 kpc. Panels b and c are comparable to Figs. 3a and 11a for a smaller radial coverage.</caption>
</figure>
<text><location><page_15><loc_52><loc_48><loc_92><loc_50></location>contribution of both supernovae II and AGB and should be used to test this hypothesis in the future.</text>
<text><location><page_15><loc_52><loc_27><loc_92><loc_48></location>It is important to note that the chemical evolution model in the NIHAO-UHD simulations is constrained by the current, incomplete understanding of evolutionary pathways and yields (Buck et al. 2021), as well as by limitations in resolution and the imperfect physics inherent to cosmological zoom-in simulations (Buck 2020). Both could contribute to the identified differences in absolute and relative abundances across different scales - including a different scatter of abundances for example of the gas phase metallicity between NIHAO-UHD of up to 0 . 1 dex and the low scatter of 0 . 03 -0 . 05 dex (and even lower on local scales) found by PHANGS-MUSE faceon observations (Kreckel et al. 2020). Extending our analysis to other simulations and further improving the resolution and physics of the simulations will be key in uniting the observational and theoretical insights into galactic chemical evolution on small and large scales.</text>
<text><location><page_15><loc_52><loc_16><loc_92><loc_27></location>Similar to more resolved and higher quality observations in the Milky Way, we also expect more, better, and diverse faceon and edge-on observations and analyses across a range of wavelengths by the PHANGS and GECKOS teams (Kreckel et al. 2019, 2020; van de Sande et al. 2023) as well as the SDSS-V and MAGPI collaborations (Kollmeier et al. 2017; Foster et al. 2021; Mun et al. 2024; Chen et al. 2024a), among many other ongoing efforts.</text>
<section_header_level_1><location><page_15><loc_65><loc_14><loc_78><loc_15></location>6. CONCLUSIONS</section_header_level_1>
<text><location><page_15><loc_52><loc_9><loc_92><loc_13></location>To conclude our study, we first iterate the main take-away of our research in Section 6.1 before giving suggestions for future research in Section 6.2.</text>
<figure>
<location><page_15><loc_37><loc_73><loc_64><loc_91></location>
</figure>
<figure>
<location><page_15><loc_65><loc_73><loc_91><loc_91></location>
</figure>
<figure>
<location><page_15><loc_65><loc_55><loc_91><loc_72></location>
</figure>
<section_header_level_1><location><page_16><loc_23><loc_91><loc_34><loc_92></location>6.1. Take-Away</section_header_level_1>
<text><location><page_16><loc_8><loc_78><loc_48><loc_90></location>Wehaveanalysed the radial metallicity distribution of young stars and gas in the inner 20 kpc of a NIHAO-UHD Milky Way analogue (Fig. 1), finding a predominantly linear decrease (Figs. 2 and 3). Although our analysis of a single spiral galaxy simulation has limited applicability to the entire population of diverse spiral galaxies, it reveals several intriguing findings about the shape and local metallicity variations. The results we find in this work hold relevance for both the Milky Way and extragalactic research communities:</text>
<unordered_list>
<list_item><location><page_16><loc_10><loc_68><loc_48><loc_77></location>· Looking into the shape in detail, we find that piecewise linear and quadratic functions both perform better than a linear fit to the radial metallicity relation. However, we see no significant preference between piecewise and quadratic functions based on our assessments (Fig. 4). While the specific slopes differ when fitting all points or binned data, they agree within the still rather small fitting uncertainties.</list_item>
<list_item><location><page_16><loc_10><loc_60><loc_48><loc_67></location>· We find that a piecewise linear function can effectively approximate a quadratic function across scales commonly applied in Milky Way and extragalactic studies. Local deviations become traceable below a spatial resolution of Δ 𝑅 Gal . ≤ 2 kpc (Fig. 5).</list_item>
<list_item><location><page_16><loc_10><loc_52><loc_48><loc_59></location>· We see no strong correlations of deviations in the vertical direction across the whole simulation (Fig. 6). Such correlations could, however, be blurred by azimuthal effects, like the galactic warp, which needs to be disentangled in the azimuthal dimension.</list_item>
<list_item><location><page_16><loc_10><loc_45><loc_48><loc_52></location>· We find various deviations from the global trend in azimuthal direction, including gaps as well as isolated streaks of stars with similar [Fe/H] across Δ 𝑅 Gal . = 2 -6 kpc (Fig. 7). These can introduce significant local over-/underenhancement of up to ± 0 . 2 dex in [Fe/H] at a given radius.</list_item>
<list_item><location><page_16><loc_10><loc_39><loc_48><loc_44></location>· We find significant scatter across the radial metallicity distribution caused by streaks of stars born with similar [Fe/H] at similar times and similar but slightly extended regions of the galaxy (Figs. 8 and 9).</list_item>
<list_item><location><page_16><loc_10><loc_32><loc_48><loc_38></location>· Our results imply the need for more careful consideration of local intrinsic effects and selection effects on radial metallicity gradient and scatter studies in the Milky Way (Fig. 10).</list_item>
<list_item><location><page_16><loc_10><loc_22><loc_48><loc_32></location>· Expanding our work to the gas phase metallicity gradient (Fig. 11), we perform a preliminary comparison of observed and simulated young stars as well as simulated gas distribution and chemistry (Figs. 12 and 13), finding significant step-like changes in the gas chemistry at the leading and trailing edges of gas spirals, with lower and higher enhancement respectively (Figs. 14 and 15).</list_item>
<list_item><location><page_16><loc_10><loc_13><loc_48><loc_21></location>· We have further identified that the abundance scatter, which increases towards larger radii, is as large as 0 . 1 dex and already present at the youngest ages of 100 Myr (Fig. 16). While not the focus of our analysis, we have also confirmed that the scatter significantly increases towards larger ages (Fig. 17).</list_item>
<list_item><location><page_16><loc_10><loc_7><loc_48><loc_12></location>· We have discussed the implications of our findings for studies of the Milky Way (Section 5.5) as well as external galaxies (Section. 5.6). Here, we firstly suggest to explore the spread of abundances across different radii in</list_item>
</unordered_list>
<text><location><page_16><loc_55><loc_85><loc_92><loc_92></location>more detail. Secondly, we suggest approaching the fitting of gradients in external galaxies in a more agnostic way to the shape. This will be particularly interesting when we can observe the outermost regions of galaxies, where simulations predict an abundance floor (Figs. 18 and 19).</text>
<section_header_level_1><location><page_16><loc_65><loc_83><loc_79><loc_84></location>6.2. Future Research</section_header_level_1>
<text><location><page_16><loc_52><loc_60><loc_92><loc_82></location>In our study, we have focused on the present-day snapshot of the NIHAO-UHD Milky Way analogue simulation - similar to present-day observations that are possible in our local Universe. Given that the simulation is tracing particles and gas over time, a detailed follow-up study should trace the evolution and coherence of spatial and chemical over- and underdensities over time and different elements (see also Zhang et al. 2024). This would in particular allow us to quantify the change of abundance in the leading and trailing edges of spiral arms and subsequently track the mixing and blurring of these over time. Certainly, more studies are needed to establish a link to a physical mechanism and further quantify its importance. More results are expected as we extend the reach, number, and quality of stellar measurements in our Galaxy (e.g. Barbillon et al. 2024) and beyond. These improvements will allow us to move beyond a one-dimensional analysis of gradients and better incorporate and model local variations.</text>
<text><location><page_16><loc_52><loc_8><loc_92><loc_59></location>While previous works are showing that relative trends for several elemental abundances do not strongly disagree from observations (Buck et al. 2021; Buder et al. 2024), we are still missing several details on the origin of elements, such as a complete picture of the synthesis sites, environments, and yields for elements. Not least because of these imperfections of absolute chemical enrichment predictions, we are refraining from quantitatively comparing the shape of our Milky Way analogue with the actual Milky Way. We have previously also mentioned the limitations in mass resolution of stars and gas, which may introduce unrealistic effects and could for example drive deviations from actual chemical enrichment at the smallest scales. We also note that the results of our simulation may not apply to the actual Milky Way due to different galaxy properties. These could be different due to different galaxy formation pathways, such as the amount and importance of mergers (Buck et al. 2023; Buder et al. 2024). In our discussion, we have already eluded to the weak bar in this simulation. Motivated by the analysis by Tuntipong et al. (2024), we have also investigated the bulge to total stellar mass ratio 𝐵 / 𝑇 . We find a strong bulge with 𝐵 / 𝑇 = 0 . 48 when selecting bulge stars with orbit circularity 𝑗 𝑧 / 𝑗 𝑐 < 0 . 5 and disk stars with 𝑗 𝑧 / 𝑗 𝑐 > 0 . 7 based on actions 𝑗 , consistent with values found by Obreja et al. (2019) of a lower resolution simulation of 8.26e11 . Our simulated galaxy has both a very weak bar and a smoothly changing radial metallicity gradient. This is in line with the findings by Chen et al. (2023) for five strongly and weakly barred spiral galaxies, where bars seem to drive the dominance of break radii in radial metallicity gradients. Future work should certainly look at a variety of galaxies to establish the causality of bar strength with the smoothness or abrupt change at a break radius for the radial metallicity gradient. Expanding our study to more and other simulations such as the VINTERGATAN suite (Renaud et al. 2024) and subsequently comparing to observations of galaxies with varying parameters like mass, formation history, bar strength or environment would allow us to quantify their effect on metallicity gradients and further disentangle the influence of different enrichment mechanisms on the chemical evolution of galaxies.</text>
<section_header_level_1><location><page_17><loc_24><loc_91><loc_33><loc_92></location>SOFTWARE</section_header_level_1>
<text><location><page_17><loc_8><loc_78><loc_48><loc_90></location>The research for this publication was coded in /p.pc/y.pc/t.pc/h.pc/o.pc/n.pc (version 3.7.4) and included its packages /a.pc/s.pc/t.pc/r.pc/o.pc/p.pc/y.pc (v. 3.2.2; Astropy Collaboration et al. 2013, 2018), IP/y.pc/t.pc/h.pc/o.pc/n.pc (v. 7.8.0; Pérez & Granger 2007), /m.pc/a.pc/t.pc/p.pc/l.pc/o.pc/t.pc/l.pc/i.pc/b.pc (v. 3.1.3; Hunter 2007), N/u.pc/m.pcP/y.pc (v. 1.17.2; Walt et al. 2011), /p.pc/y.pc/n.pc/b.pc/o.pc/d.pc/y.pc (v. 1.1.0; Pontzen et al. 2013), /s.pc/c.pc/i.pc/p.pc/y.pc (v. 1.3.1; Virtanen et al. 2020), /s.pc/k.pc/l.pc/e.pc/a.pc/r.pc/n.pc (v. 1.5.1 Pedregosa et al. 2011) /s.pc/t.pc/a.pc/t.pc/s.pc/m.pc/o.pc/d.pc/e.pc/l.pc/s.pc (v. 0.14.2 Perktold et al. 2024) We further made use of /t.pc/o.pc/p.pc/c.pc/a.pc/t.pc (version 4.7; Taylor 2005);</text>
<section_header_level_1><location><page_17><loc_20><loc_76><loc_36><loc_77></location>DATA AVAILABILITY</section_header_level_1>
<text><location><page_17><loc_8><loc_63><loc_48><loc_75></location>All code to reproduce the analysis and figures can be publicly accessed via https://github.com/svenbuder/ nihao_radial_metallicity_gradients . The used simulationsnapshot can be accessed as FITS file via https: //github.com/svenbuder/preparing_NIHAO . Original data, more snapshots and other galaxies can be found at https://tobias-buck.de/#sim_data . We encourage interested readers to get in contact with the authors for full data access and advice for use and cite Buck et al. (2020, 2021).</text>
<section_header_level_1><location><page_17><loc_63><loc_91><loc_80><loc_92></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_17><loc_52><loc_83><loc_92><loc_90></location>Weacknowledge the traditional owners of the land on which the ANU stands, the Ngunnawal and Ngambri people. We pay our respects to elders past, and present and are proud to continue their tradition of surveying the night sky and its mysteries to better understand our Universe.</text>
<text><location><page_17><loc_52><loc_65><loc_92><loc_83></location>This work was supported by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013. SB and KGacknowledge support from the Australian Research Council under grant numbers DE240100150 and DE220100766, respectively. TB acknowledges funding from the Carl Zeiss Stiftung and support from the European Research Council under ERC-CoG grant CRAGSMAN-646955. We gratefully acknowledge the Gauss Centre for Supercomputing e.V. ( www.gauss-centre.eu ) for funding this project by providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre ( www.lrz.de ). Simulations were partially computed with High Performance Computing resources at New York University, Abu Dhabi.</text>
<section_header_level_1><location><page_17><loc_46><loc_61><loc_54><loc_62></location>REFERENCES</section_header_level_1>
<unordered_list>
<list_item><location><page_17><loc_8><loc_59><loc_31><loc_60></location>Agertz O., et al., 2021, MNRAS, 503, 5826</list_item>
<list_item><location><page_17><loc_52><loc_59><loc_84><loc_60></location>Cunha K., et al., 2016, Astronomische Nachrichten, 337, 922</list_item>
<list_item><location><page_17><loc_8><loc_58><loc_47><loc_59></location>Allende Prieto C., Beers T. C., Wilhelm R., Newberg H. J., Rockosi C. M.,</list_item>
</unordered_list>
<text><location><page_17><loc_10><loc_57><loc_31><loc_58></location>Yanny B., Lee Y. S., 2006, ApJ, 636, 804</text>
<unordered_list>
<list_item><location><page_17><loc_8><loc_56><loc_29><loc_57></location>Anders F., et al., 2014, A&A, 564, A115</list_item>
<list_item><location><page_17><loc_8><loc_55><loc_29><loc_56></location>Anders F., et al., 2017, A&A, 600, A70</list_item>
<list_item><location><page_17><loc_8><loc_54><loc_33><loc_55></location>Andrievsky S. M., et al., 2002a, A&A, 381, 32</list_item>
<list_item><location><page_17><loc_8><loc_53><loc_46><loc_54></location>Andrievsky S. M., Bersier D., Kovtyukh V. V., Luck R. E., Maciel W. J.,</list_item>
<list_item><location><page_17><loc_10><loc_52><loc_38><loc_53></location>Lépine J. R. D., Beletsky Y. V., 2002b, A&A, 384, 140</list_item>
<list_item><location><page_17><loc_8><loc_50><loc_48><loc_52></location>Andrievsky S. M., Luck R. E., Martin P., Lépine J. R. D., 2004, A&A, 413, 159</list_item>
<list_item><location><page_17><loc_8><loc_49><loc_35><loc_50></location>Astropy Collaboration et al., 2013, A&A, 558, A33</list_item>
<list_item><location><page_17><loc_8><loc_48><loc_33><loc_49></location>Astropy Collaboration et al., 2018, AJ, 156, 123</list_item>
<list_item><location><page_17><loc_8><loc_47><loc_45><loc_48></location>Barbillon M., Recio-Blanco A., Poggio E., Palicio P. A., Spitoni E., de</list_item>
<list_item><location><page_17><loc_10><loc_46><loc_44><loc_47></location>Laverny P., Cescutti G., 2024, arXiv e-prints, p. arXiv:2411.10007</list_item>
<list_item><location><page_17><loc_8><loc_45><loc_31><loc_46></location>Belfiore F., et al., 2016, MNRAS, 461, 3111</list_item>
<list_item><location><page_17><loc_8><loc_44><loc_31><loc_45></location>Belfiore F., et al., 2017, MNRAS, 469, 151</list_item>
<list_item><location><page_17><loc_8><loc_43><loc_48><loc_44></location>Bellardini M. A., Wetzel A., Loebman S. R., Faucher-Giguère C.-A., Ma X., Feldmann R., 2021, MNRAS, 505, 4586</list_item>
<list_item><location><page_17><loc_8><loc_41><loc_48><loc_42></location>Bellardini M. A., Wetzel A., Loebman S. R., Bailin J., 2022, MNRAS, 514, 4270</list_item>
<list_item><location><page_17><loc_8><loc_40><loc_38><loc_41></location>Bensby T., Feltzing S., Oey M. S., 2014, A&A, 562, A71</list_item>
<list_item><location><page_17><loc_8><loc_39><loc_32><loc_40></location>Bergemann M., et al., 2014, A&A, 565, A89</list_item>
<list_item><location><page_17><loc_8><loc_37><loc_48><loc_39></location>Binney J., Tremaine S., 2008, Galactic Dynamics: Second Edition. Princeton University Press</list_item>
<list_item><location><page_17><loc_8><loc_35><loc_47><loc_37></location>Bird J. C., Kazantzidis S., Weinberg D. H., Guedes J., Callegari S., Mayer L., Madau P., 2013, ApJ, 773, 43</list_item>
<list_item><location><page_17><loc_8><loc_34><loc_38><loc_35></location>Bland-Hawthorn J., Gerhard O., 2016, ARA&A, 54, 529</list_item>
<list_item><location><page_17><loc_8><loc_32><loc_47><loc_34></location>Bland-Hawthorn J., Tepper-Garcia T., Agertz O., Federrath C., 2024, ApJ, 968, 86</list_item>
<list_item><location><page_17><loc_8><loc_31><loc_29><loc_32></location>Boeche C., et al., 2013, A&A, 559, A59</list_item>
<list_item><location><page_17><loc_8><loc_30><loc_43><loc_31></location>Bresolin F., Kennicutt R. C., Ryan-Weber E., 2012, ApJ, 750, 122</list_item>
<list_item><location><page_17><loc_8><loc_29><loc_27><loc_30></location>Buck T., 2020, MNRAS, 491, 5435</list_item>
<list_item><location><page_17><loc_8><loc_27><loc_47><loc_29></location>Buck T., Macciò A. V., Dutton A. A., Obreja A., Frings J., 2019, MNRAS, 483, 1314</list_item>
<list_item><location><page_17><loc_8><loc_25><loc_47><loc_27></location>Buck T., Obreja A., Macciò A. V., Minchev I., Dutton A. A., Ostriker J. P., 2020, MNRAS, 491, 3461</list_item>
<list_item><location><page_17><loc_8><loc_23><loc_45><loc_25></location>Buck T., Rybizki J., Buder S., Obreja A., Macciò A. V., Pfrommer C., Steinmetz M., Ness M., 2021, MNRAS, 508, 3365</list_item>
<list_item><location><page_17><loc_8><loc_22><loc_46><loc_23></location>Buck T., Obreja A., Ratcliffe B., Lu Y., Minchev I., Macciò A. V., 2023,</list_item>
<list_item><location><page_17><loc_10><loc_21><loc_20><loc_22></location>MNRAS, 523, 1565</list_item>
<list_item><location><page_17><loc_8><loc_20><loc_28><loc_21></location>Buder S., et al., 2019, A&A, 624, A19</list_item>
<list_item><location><page_17><loc_8><loc_19><loc_40><loc_20></location>Buder S., Mijnarends L., Buck T., 2024, MNRAS, 532, 1010</list_item>
<list_item><location><page_17><loc_8><loc_18><loc_35><loc_19></location>Cantat-Gaudin T., Anders F., 2020, A&A, 633, A99</list_item>
<list_item><location><page_17><loc_8><loc_17><loc_34><loc_18></location>Casamiquela L., et al., 2019, MNRAS, 490, 1821</list_item>
<list_item><location><page_17><loc_8><loc_16><loc_27><loc_17></location>Chabrier G., 2003, PASP, 115, 763</list_item>
<list_item><location><page_17><loc_8><loc_14><loc_46><loc_16></location>Chen Q.-H., Grasha K., Battisti A. J., Kewley L. J., Madore B. F., Seibert M., Rich J. A., Beaton R. L., 2023, MNRAS, 519, 4801</list_item>
<list_item><location><page_17><loc_8><loc_14><loc_32><loc_14></location>Chen Q.-H., et al., 2024a, MNRAS, 527, 2991</list_item>
<list_item><location><page_17><loc_8><loc_13><loc_32><loc_13></location>Chen Q.-H., et al., 2024b, MNRAS, 534, 883</list_item>
<list_item><location><page_17><loc_8><loc_12><loc_28><loc_13></location>Chiappini C., 2002, Ap&SS, 281, 253</list_item>
<list_item><location><page_17><loc_8><loc_11><loc_39><loc_12></location>Chiappini C., Matteucci F., Gratton R., 1997, ApJ, 477, 765</list_item>
<list_item><location><page_17><loc_8><loc_10><loc_41><loc_11></location>Chiappini C., Matteucci F., Romano D., 2001, ApJ, 554, 1044</list_item>
<list_item><location><page_17><loc_8><loc_9><loc_32><loc_10></location>Chieffi A., Limongi M., 2004, ApJ, 608, 405</list_item>
<list_item><location><page_17><loc_8><loc_8><loc_25><loc_9></location>Chiosi C., 1980, A&A, 83, 206</list_item>
<list_item><location><page_17><loc_52><loc_57><loc_89><loc_59></location>Di Matteo P., Haywood M., Combes F., Semelin B., Snaith O. N., 2013, A&A, 553, A102</list_item>
<list_item><location><page_17><loc_52><loc_56><loc_70><loc_57></location>Donor J., et al., 2020, AJ, 159, 199</list_item>
<list_item><location><page_17><loc_52><loc_55><loc_72><loc_56></location>Foster C., et al., 2021, PASA, 38, e031</list_item>
<list_item><location><page_17><loc_52><loc_54><loc_92><loc_55></location>Frankel N., Rix H.-W., Ting Y.-S., Ness M., Hogg D. W., 2018, ApJ, 865, 96</list_item>
<list_item><location><page_17><loc_52><loc_53><loc_86><loc_54></location>Frankel N., Sanders J., Ting Y.-S., Rix H.-W., 2020, ApJ, 896, 15</list_item>
<list_item><location><page_17><loc_52><loc_52><loc_79><loc_53></location>Fraser-McKelvie A., et al., 2022, MNRAS, 510, 320</list_item>
<list_item><location><page_17><loc_52><loc_51><loc_76><loc_52></location>Gaia Collaboration et al., 2016, A&A, 595, A1</list_item>
<list_item><location><page_17><loc_52><loc_50><loc_76><loc_51></location>Garcia A. M., et al., 2023, MNRAS, 519, 4716</list_item>
<list_item><location><page_17><loc_52><loc_49><loc_74><loc_50></location>Genovali K., et al., 2014, A&A, 566, A37</list_item>
<list_item><location><page_17><loc_52><loc_47><loc_90><loc_49></location>Graf R. L., Wetzel A., Bellardini M. A., Bailin J., 2024, arXiv e-prints, p. arXiv:2402.15614</list_item>
<list_item><location><page_17><loc_52><loc_46><loc_86><loc_47></location>Grand R. J. J., Kawata D., Cropper M., 2015, MNRAS, 447, 4018</list_item>
<list_item><location><page_17><loc_52><loc_45><loc_76><loc_46></location>Grand R. J. J., et al., 2016, MNRAS, 460, L94</list_item>
<list_item><location><page_17><loc_52><loc_44><loc_72><loc_45></location>Grasha K., et al., 2022, ApJ, 929, 118</list_item>
<list_item><location><page_17><loc_52><loc_43><loc_90><loc_44></location>Hackshaw Z., Hawkins K., Filion C., Horta D., Laporte C. F. P., Carr C.,</list_item>
<list_item><location><page_17><loc_53><loc_43><loc_86><loc_43></location>Price-Whelan A. M., 2024, arXiv e-prints, p. arXiv:2405.18120</list_item>
<list_item><location><page_17><loc_52><loc_42><loc_73><loc_42></location>Hayden M. R., et al., 2014, AJ, 147, 116</list_item>
<list_item><location><page_17><loc_52><loc_41><loc_74><loc_41></location>Hayden M. R., et al., 2015, ApJ, 808, 132</list_item>
<list_item><location><page_17><loc_52><loc_40><loc_76><loc_41></location>Hemler Z. S., et al., 2021, MNRAS, 506, 3024</list_item>
<list_item><location><page_17><loc_52><loc_39><loc_91><loc_40></location>Ho I. T., Kudritzki R.-P., Kewley L. J., Zahid H. J., Dopita M. A., Bresolin</list_item>
<list_item><location><page_17><loc_53><loc_38><loc_77><loc_39></location>F., Rupke D. S. N., 2015, MNRAS, 448, 2030</list_item>
<list_item><location><page_17><loc_52><loc_37><loc_70><loc_38></location>Ho I. T., et al., 2017, ApJ, 846, 39</list_item>
<list_item><location><page_17><loc_52><loc_36><loc_82><loc_37></location>Hogg D. W., Eilers A.-C., Rix H.-W., 2019, AJ, 158, 147</list_item>
<list_item><location><page_17><loc_52><loc_35><loc_74><loc_36></location>Hunter J. D., 2007, Comput Sci Eng, 9, 90</list_item>
<list_item><location><page_17><loc_52><loc_34><loc_70><loc_35></location>Imig J., et al., 2023, ApJ, 954, 124</list_item>
<list_item><location><page_17><loc_52><loc_33><loc_69><loc_34></location>Janes K. A., 1979, ApJS, 39, 135</list_item>
<list_item><location><page_17><loc_52><loc_32><loc_85><loc_33></location>Johnson J. W., et al., 2024, arXiv e-prints, p. arXiv:2410.13256</list_item>
<list_item><location><page_17><loc_52><loc_31><loc_76><loc_32></location>Karakas A. I., Lugaro M., 2016, ApJ, 825, 26</list_item>
<list_item><location><page_17><loc_52><loc_30><loc_73><loc_31></location>Kauffmann G., 1996, MNRAS, 281, 475</list_item>
<list_item><location><page_17><loc_52><loc_28><loc_91><loc_30></location>Khoperskov S., Sivkova E., Saburova A., Vasiliev E., Shustov B., Minchev I., Walcher C. J., 2023, A&A, 671, A56</list_item>
<list_item><location><page_17><loc_52><loc_27><loc_78><loc_28></location>Knollmann S. R., Knebe A., 2009, ApJS, 182, 608</list_item>
<list_item><location><page_17><loc_52><loc_26><loc_86><loc_27></location>Kollmeier J. A., et al., 2017, arXiv e-prints, p. arXiv:1711.03234</list_item>
<list_item><location><page_17><loc_52><loc_25><loc_71><loc_26></location>Kreckel K., et al., 2019, ApJ, 887, 80</list_item>
<list_item><location><page_17><loc_52><loc_24><loc_74><loc_25></location>Kreckel K., et al., 2020, MNRAS, 499, 193</list_item>
<list_item><location><page_17><loc_52><loc_23><loc_81><loc_24></location>Krumholz M. R., Ting Y.-S., 2018, MNRAS, 475, 2236</list_item>
<list_item><location><page_17><loc_52><loc_21><loc_91><loc_23></location>Krumholz M. R., Burkhart B., Forbes J. C., Crocker R. M., 2018, MNRAS, 477, 2716</list_item>
<list_item><location><page_17><loc_52><loc_20><loc_86><loc_21></location>Kubryk M., Prantzos N., Athanassoula E., 2015, A&A, 580, A126</list_item>
<list_item><location><page_17><loc_52><loc_19><loc_75><loc_20></location>Lacey C. G., Fall S. M., 1985, ApJ, 290, 154</list_item>
<list_item><location><page_17><loc_52><loc_18><loc_72><loc_19></location>Larson R. B., 1976, MNRAS, 176, 31</list_item>
<list_item><location><page_17><loc_52><loc_16><loc_90><loc_18></location>Lemasle B., François P., Bono G., Mottini M., Primas F., Romaniello M., 2007, A&A, 467, 283</list_item>
<list_item><location><page_17><loc_52><loc_14><loc_91><loc_16></location>Lemasle B., François P., Piersimoni A., Pedicelli S., Bono G., Laney C. D., Primas F., Romaniello M., 2008, A&A, 490, 613</list_item>
<list_item><location><page_17><loc_52><loc_14><loc_73><loc_14></location>Lemasle B., et al., 2013, A&A, 558, A31</list_item>
<list_item><location><page_17><loc_52><loc_13><loc_73><loc_13></location>Lemasle B., et al., 2022, A&A, 668, A40</list_item>
<list_item><location><page_17><loc_52><loc_12><loc_77><loc_13></location>Lépine J. R. D., et al., 2011, MNRAS, 417, 698</list_item>
<list_item><location><page_17><loc_52><loc_11><loc_72><loc_12></location>Li Z., et al., 2024, MNRAS, 528, 7103</list_item>
<list_item><location><page_17><loc_52><loc_9><loc_90><loc_11></location>Lilly S. J., Carollo C. M., Pipino A., Renzini A., Peng Y., 2013, ApJ, 772, 119</list_item>
<list_item><location><page_17><loc_52><loc_8><loc_86><loc_9></location>Lu Y., Buck T., Minchev I., Ness M. K., 2022, MNRAS, 515, L34</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_18><loc_8><loc_90><loc_48><loc_92></location>Ma X., Hopkins P. F., Feldmann R., Torrey P., Faucher-Giguère C.-A., Kereš D., 2017a, MNRAS, 466, 4780</list_item>
<list_item><location><page_18><loc_8><loc_89><loc_44><loc_90></location>Ma X., Hopkins P. F., Wetzel A. R., Kirby E. N., Anglés-Alcázar D.,</list_item>
<list_item><location><page_18><loc_10><loc_88><loc_48><loc_89></location>Faucher-Giguère C.-A., Kereš D., Quataert E., 2017b, MNRAS, 467, 2430</list_item>
<list_item><location><page_18><loc_8><loc_87><loc_42><loc_88></location>Magrini L., Sestito P., Randich S., Galli D., 2009, A&A, 494, 95</list_item>
<list_item><location><page_18><loc_8><loc_86><loc_29><loc_87></location>Magrini L., et al., 2017, A&A, 603, A2</list_item>
<list_item><location><page_18><loc_8><loc_85><loc_32><loc_86></location>Matteucci F., Recchi S., 2001, ApJ, 558, 351</list_item>
<list_item><location><page_18><loc_8><loc_84><loc_37><loc_85></location>Metha B., Trenti M., Chu T., 2021, MNRAS, 508, 489</list_item>
<list_item><location><page_18><loc_8><loc_83><loc_31><loc_84></location>Minchev I., Famaey B., 2010, ApJ, 722, 112</list_item>
<list_item><location><page_18><loc_8><loc_82><loc_39><loc_83></location>Minchev I., Chiappini C., Martig M., 2013, A&A, 558, A9</list_item>
<list_item><location><page_18><loc_8><loc_81><loc_40><loc_82></location>Minchev I., Chiappini C., Martig M., 2014, A&A, 572, A92</list_item>
<list_item><location><page_18><loc_8><loc_80><loc_32><loc_81></location>Minchev I., et al., 2018, MNRAS, 481, 1645</list_item>
<list_item><location><page_18><loc_8><loc_79><loc_29><loc_80></location>Moran S. M., et al., 2012, ApJ, 745, 66</list_item>
<list_item><location><page_18><loc_8><loc_78><loc_30><loc_79></location>Mun M., et al., 2024, MNRAS, 530, 5072</list_item>
<list_item><location><page_18><loc_8><loc_77><loc_26><loc_78></location>Myers N., et al., 2022, AJ, 164, 85</list_item>
<list_item><location><page_18><loc_8><loc_75><loc_47><loc_77></location>Nicholls D. C., Sutherland R. S., Dopita M. A., Kewley L. J., Groves B. A., 2017, MNRAS, 466, 4403</list_item>
<list_item><location><page_18><loc_8><loc_74><loc_31><loc_75></location>Obreja A., et al., 2019, MNRAS, 487, 4424</list_item>
<list_item><location><page_18><loc_8><loc_73><loc_38><loc_74></location>Obreja A., Buck T., Macciò A. V., 2022, A&A, 657, A15</list_item>
<list_item><location><page_18><loc_8><loc_72><loc_45><loc_73></location>Palla M., Magrini L., Spitoni E., Matteucci F., Viscasillas Vázquez C.,</list_item>
</unordered_list>
<text><location><page_18><loc_10><loc_71><loc_42><loc_72></location>Franchini M., Molero M., Randich S., 2024, A&A, 690, A334</text>
<text><location><page_18><loc_8><loc_70><loc_37><loc_71></location>Pedregosa F., et al., 2011, J Mach Learn Res, 12, 2825</text>
<unordered_list>
<list_item><location><page_18><loc_8><loc_69><loc_36><loc_70></location>Pérez F., Granger B. E., 2007, Comput Sci Eng, 9, 21</list_item>
</unordered_list>
<text><location><page_18><loc_8><loc_68><loc_42><loc_69></location>Perktold J., et al., 2024, statsmodels/statsmodels: Release 0.14.2,</text>
<text><location><page_18><loc_10><loc_67><loc_24><loc_68></location>doi:10.5281/zenodo.593847</text>
<unordered_list>
<list_item><location><page_18><loc_8><loc_66><loc_31><loc_67></location>Pilkington K., et al., 2012, A&A, 540, A56</list_item>
<list_item><location><page_18><loc_8><loc_65><loc_28><loc_66></location>Pilyugin L. S., 2003, A&A, 397, 109</list_item>
<list_item><location><page_18><loc_8><loc_65><loc_37><loc_65></location>Pilyugin L. S., Tautvaišien˙e G., 2024, A&A, 682, A41</list_item>
<list_item><location><page_18><loc_8><loc_63><loc_48><loc_64></location>Pilyugin L. S., Grebel E. K., Zinchenko I. A., Nefedyev Y. A., Vílchez J. M., 2017, A&A, 608, A127</list_item>
<list_item><location><page_18><loc_8><loc_62><loc_34><loc_63></location>Planck Collaboration et al., 2014, A&A, 571, A16</list_item>
<list_item><location><page_18><loc_8><loc_61><loc_30><loc_62></location>Poggio E., et al., 2018, MNRAS, 481, L21</list_item>
<list_item><location><page_18><loc_8><loc_60><loc_30><loc_61></location>Poggio E., et al., 2021, A&A, 651, A104</list_item>
<list_item><location><page_18><loc_8><loc_59><loc_28><loc_60></location>Poggio E., et al., 2022, A&A, 666, L4</list_item>
<list_item><location><page_18><loc_8><loc_58><loc_42><loc_59></location>Pontzen A., Roškar R., Stinson G. S., Woods R., 2013, pynbody:</list_item>
<list_item><location><page_18><loc_10><loc_56><loc_47><loc_58></location>Astrophysics Simulation Analysis for Python, Astrophysics Source Code Library, record ascl:1305.002</list_item>
<list_item><location><page_18><loc_8><loc_55><loc_33><loc_56></location>Quirk W. J., Tinsley B. M., 1973, ApJ, 179, 69</list_item>
</unordered_list>
<text><location><page_18><loc_8><loc_54><loc_48><loc_55></location>Ratcliffe B. L., Ness M. K., Buck T., Johnston K. V., Sen B., Beraldo e Silva</text>
<unordered_list>
<list_item><location><page_18><loc_10><loc_53><loc_30><loc_54></location>L., Debattista V. P., 2022, ApJ, 924, 60</list_item>
<list_item><location><page_18><loc_8><loc_52><loc_32><loc_53></location>Ratcliffe B., et al., 2023, MNRAS, 525, 2208</list_item>
<list_item><location><page_18><loc_8><loc_50><loc_48><loc_52></location>Ratcliffe B., Khoperskov S., Minchev I., Lee N. D., Buck T., Marques L., Lu L., Steinmetz M., 2024, arXiv e-prints, p. arXiv:2410.17326</list_item>
<list_item><location><page_18><loc_8><loc_48><loc_47><loc_50></location>Renaud F., Ratcliffe B., Minchev I., Haywood M., Di Matteo P., Agertz O., Romeo A. B., 2024, arXiv e-prints, p. arXiv:2409.10598</list_item>
<list_item><location><page_18><loc_8><loc_46><loc_46><loc_48></location>Rolleston W. R. J., Smartt S. J., Dufton P. L., Ryans R. S. I., 2000, A&A, 363, 537</list_item>
<list_item><location><page_18><loc_8><loc_44><loc_45><loc_46></location>Rosales-Ortega F. F., Díaz A. I., Kennicutt R. C., Sánchez S. F., 2011, MNRAS, 415, 2439</list_item>
</unordered_list>
<text><location><page_18><loc_8><loc_43><loc_36><loc_44></location>Rybizki J., Just A., Rix H.-W., 2017, A&A, 605, A59</text>
<figure>
<location><page_18><loc_11><loc_21><loc_46><loc_42></location>
<caption>F/i.pc/g.pc. A1.- Comparison of gas abundances for oxygen and iron. Shown are absolute oxygen abundances A(O) and the comparison of relative iron and oxygen abundances [Fe/H] - [O/H]. The top panel shows values at face values, whereas the bottom panel shows the comparison for a linear approximation of [Fe/H] from [O/H].</caption>
</figure>
<unordered_list>
<list_item><location><page_18><loc_52><loc_91><loc_78><loc_92></location>Sánchez-Blázquez P., et al., 2014, A&A, 570, A6</list_item>
<list_item><location><page_18><loc_52><loc_90><loc_80><loc_91></location>Sánchez-Menguiano L., et al., 2016, A&A, 587, A70</list_item>
<list_item><location><page_18><loc_52><loc_89><loc_74><loc_90></location>Sánchez S. F., et al., 2013, A&A, 554, A58</list_item>
<list_item><location><page_18><loc_52><loc_88><loc_74><loc_89></location>Sánchez S. F., et al., 2014, A&A, 563, A49</list_item>
<list_item><location><page_18><loc_52><loc_87><loc_80><loc_88></location>Scarano S., Lépine J. R. D., 2013, MNRAS, 428, 625</list_item>
</unordered_list>
<text><location><page_18><loc_52><loc_86><loc_78><loc_87></location>Schönrich R., Binney J., 2009, MNRAS, 396, 203</text>
<unordered_list>
<list_item><location><page_18><loc_52><loc_85><loc_68><loc_86></location>Searle L., 1971, ApJ, 168, 327</list_item>
<list_item><location><page_18><loc_52><loc_84><loc_78><loc_85></location>Seitenzahl I. R., et al., 2013, MNRAS, 429, 1156</list_item>
</unordered_list>
<text><location><page_18><loc_52><loc_83><loc_88><loc_84></location>Sharda P., Krumholz M. R., Wisnioski E., Forbes J. C., Federrath C.,</text>
<text><location><page_18><loc_53><loc_82><loc_74><loc_83></location>Acharyya A., 2021, MNRAS, 502, 5935</text>
<unordered_list>
<list_item><location><page_18><loc_52><loc_80><loc_89><loc_82></location>Shaver P. A., McGee R. X., Newton L. M., Danks A. C., Pottasch S. R., 1983, MNRAS, 204, 53</list_item>
<list_item><location><page_18><loc_52><loc_79><loc_74><loc_80></location>Spina L., et al., 2021, MNRAS, 503, 3279</list_item>
<list_item><location><page_18><loc_52><loc_78><loc_73><loc_79></location>Spitoni E., et al., 2023, A&A, 680, A85</list_item>
<list_item><location><page_18><loc_52><loc_77><loc_83><loc_78></location>Stanghellini L., Magrini L., Casasola V., 2015, ApJ, 812, 39</list_item>
<list_item><location><page_18><loc_52><loc_75><loc_89><loc_77></location>Stinson G., Seth A., Katz N., Wadsley J., Governato F., Quinn T., 2006, MNRAS, 373, 1074</list_item>
<list_item><location><page_18><loc_52><loc_73><loc_91><loc_75></location>Stinson G. S., Brook C., Macciò A. V., Wadsley J., Quinn T. R., Couchman H. M. P., 2013, MNRAS, 428, 129</list_item>
<list_item><location><page_18><loc_52><loc_72><loc_81><loc_73></location>Sun X., et al., 2024, arXiv e-prints, p. arXiv:2409.09290</list_item>
</unordered_list>
<text><location><page_18><loc_52><loc_71><loc_70><loc_72></location>Taylor M. B., 2005, ASPC, 347, 29</text>
<unordered_list>
<list_item><location><page_18><loc_52><loc_70><loc_90><loc_71></location>Tepper-Garcia T., Bland-Hawthorn J., Vasiliev E., Agertz O., Teyssier R.,</list_item>
</unordered_list>
<text><location><page_18><loc_53><loc_69><loc_82><loc_70></location>Federrath C., 2024, arXiv e-prints, p. arXiv:2406.00342</text>
<unordered_list>
<list_item><location><page_18><loc_52><loc_68><loc_77><loc_69></location>Tinsley B. M., 1980, Fund. Cosmic Phys., 5, 287</list_item>
<list_item><location><page_18><loc_52><loc_67><loc_91><loc_68></location>Tissera P. B., Rosas-Guevara Y., Bower R. G., Crain R. A., del P Lagos C.,</list_item>
</unordered_list>
<text><location><page_18><loc_53><loc_66><loc_85><loc_67></location>Schaller M., Schaye J., Theuns T., 2019, MNRAS, 482, 2208</text>
<unordered_list>
<list_item><location><page_18><loc_52><loc_65><loc_84><loc_66></location>Tuntipong S., et al., 2024, arXiv e-prints, p. arXiv:2408.12223</list_item>
<list_item><location><page_18><loc_52><loc_65><loc_70><loc_65></location>Twarog B. A., 1980, ApJ, 242, 242</list_item>
<list_item><location><page_18><loc_52><loc_64><loc_91><loc_64></location>Twarog B. A., Ashman K. M., Anthony-Twarog B. J., 1997, AJ, 114, 2556</list_item>
<list_item><location><page_18><loc_52><loc_63><loc_78><loc_63></location>Vickers J. J., Shen J., Li Z.-Y., 2021, ApJ, 922, 189</list_item>
<list_item><location><page_18><loc_52><loc_62><loc_79><loc_63></location>Vilchez J. M., Esteban C., 1996, MNRAS, 280, 720</list_item>
<list_item><location><page_18><loc_52><loc_61><loc_78><loc_62></location>Virtanen P., et al., 2020, Nature Methods, 17, 261</list_item>
<list_item><location><page_18><loc_52><loc_60><loc_88><loc_61></location>Wadsley J. W., Keller B. W., Quinn T. R., 2017, MNRAS, 471, 2357</list_item>
</unordered_list>
<text><location><page_18><loc_52><loc_59><loc_90><loc_60></location>Walt S. v. d., Colbert S. C., Varoquaux G., 2011, Comput Sci Eng, 13, 22</text>
<text><location><page_18><loc_52><loc_58><loc_90><loc_59></location>Wang L., Dutton A. A., Stinson G. S., Macciò A. V., Penzo C., Kang X.,</text>
<text><location><page_18><loc_53><loc_57><loc_79><loc_58></location>Keller B. W., Wadsley J., 2015, MNRAS, 454, 83</text>
<text><location><page_18><loc_52><loc_56><loc_75><loc_57></location>Willett E., et al., 2023, MNRAS, 526, 2141</text>
<text><location><page_18><loc_52><loc_54><loc_88><loc_55></location>Williams M. J., Bureau M., Cappellari M., 2009, MNRAS, 400, 1665</text>
<unordered_list>
<list_item><location><page_18><loc_52><loc_54><loc_74><loc_54></location>Wyse R. F. G., Silk J., 1989, ApJ, 339, 700</list_item>
<list_item><location><page_18><loc_52><loc_53><loc_80><loc_53></location>Yong D., Carney B. W., Friel E. D., 2012, AJ, 144, 95</list_item>
<list_item><location><page_18><loc_52><loc_51><loc_89><loc_53></location>Zari E., Hashemi H., Brown A. G. A., Jardine K., de Zeeuw P. T., 2018, A&A, 620, A172</list_item>
<list_item><location><page_18><loc_52><loc_49><loc_86><loc_51></location>Zari E., Rix H. W., Frankel N., Xiang M., Poggio E., Drimmel R., Tkachenko A., 2021, A&A, 650, A112</list_item>
</unordered_list>
<text><location><page_18><loc_52><loc_48><loc_88><loc_49></location>Zaritsky D., Kennicutt Robert C. J., Huchra J. P., 1994, ApJ, 420, 87</text>
<text><location><page_18><loc_52><loc_47><loc_86><loc_48></location>Zhang C., Li Z., Hu Z., Krumholz M. R., 2024, arXiv e-prints, p.</text>
<text><location><page_18><loc_53><loc_46><loc_63><loc_47></location>arXiv:2411.01518</text>
<text><location><page_18><loc_52><loc_43><loc_89><loc_46></location>van de Sande J., Fraser-McKelvie A., Fisher D. B., Martig M., Hayden M. R., the GECKOS Survey collaboration 2023, arXiv e-prints, p. arXiv:2306.00059</text>
<section_header_level_1><location><page_18><loc_68><loc_38><loc_76><loc_39></location>APPENDIX</section_header_level_1>
<section_header_level_1><location><page_18><loc_62><loc_36><loc_82><loc_37></location>A. ADDITIONAL FIGURES</section_header_level_1>
<text><location><page_18><loc_52><loc_32><loc_92><loc_35></location>Fig. A1 demonstrates the tight correlation of gas phase iron and oxygen abundance and how to approximate them linearly.</text>
<text><location><page_18><loc_52><loc_15><loc_92><loc_21></location>This paper was built using the Open Journal of Astrophysics L A T E X template. The OJA is a journal which provides fast and easy peer review for new papers in the astro-ph section of the arXiv, making the reviewing process simpler for authors and referees alike. Learn more at http://astro.theoj.org .</text>
</document>
[
{
"title": "ABSTRACT",
"content": "Radial metallicity gradients are fundamental to understanding galaxy formation and evolution. In our high-resolution simulation of a NIHAO-UHD Milky Way analogue, we analyze the linearity, scatter, spatial coherence, and age-related variations of metallicity gradients using young stars and gas. While a global linear model generally captures the gradient, it ever so slightly overestimates metallicity in the inner galaxy and underestimates it in the outer regions of our simulated galaxy. Both a quadratic model, showing an initially steeper gradient that smoothly flattens outward, and a piecewise linear model with a break radius at 10 kpc (2.5 effective radii) fit the data equally better. The spread of [Fe/H] of young stars in the simulation increases by tenfold from the innermost to the outer galaxy at a radius of 20 kpc. We find that stars born at similar times along radial spirals drive this spread in the outer galaxy, with a chemical under- and over-enhancement of up to 0.1 dex at leading and trailing regions of such spirals, respectively. This localised chemical variance highlights the need to examine radial and azimuthal selection effects for both Galactic and extragalactic observational studies. The arguably idealised but volume-complete simulations suggest that future studies should not only test linear and piecewise linear gradients, but also non-linear functions such as quadratic ones to test for a smooth gradient rather than one with a break radius. Either finding would help to determine the importance of different enrichment or mixing pathways and thus our understanding of galaxy formation and evolution scenarios. Subject headings: Galaxy: structure - Galaxy: abundances - galaxies: structure - galaxies: abundances",
"pages": [
1
]
},
{
"title": "LOCAL VARIATIONS OF THE RADIAL METALLICITY GRADIENT IN A SIMULATED NIHAO-UHD MILKY WAY ANALOGUE AND THEIR IMPLICATIONS FOR (EXTRA-)GALACTIC STUDIES",
"content": "S/v.pc/e.pc/n.pc B/u.pc/d.pc/e.pc/r.pc 1 , 2 , ∗ , T/o.pc/b.pc/i.pc/a.pc/s.pc B/u.pc/c.pc/k.pc 3 , 4 , Q/i.pc/a.pc/n.pc-H/u.pc/i.pc C/h.pc/e.pc/n.pc ( 陈 千 惠 ) 1 , 2 , /a.pc/n.pc/d.pc K/a.pc/t.pc/h.pc/r.pc/y.pc/n.pc G/r.pc/a.pc/s.pc/h.pc/a.pc 1 , 2 , ∗ 1 Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT 2611, Australia 2 ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Australia 3 Universität Heidelberg, Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, Im Neuenheimer Feld 205, D-69120 Heidelberg, Germany and 4 Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische Astrophysik, Albert-Ueberle-Strae 2, D-69120 Heidelberg, Germany Version December 3, 2024",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "Understanding the radial metallicity gradient, defined as the change in heavy element abundance with galactocentric radius, in galaxies provides critical insights into their formation and evolutionary processes, such as inside-out formation, gas accretion, outflows, and radial migration (e.g. Quirk & Tinsley 1973; Tinsley 1980; Lacey & Fall 1985; Wyse & Silk 1989; Kauffmann 1996; Chiappini et al. 1997; Schönrich & Binney 2009; Moran et al. 2012; Bird et al. 2013). The decrease in metallicity with increasing distance from the Galactic centre is well-established both theoretically (Larson 1976; Tinsley 1980; Chiosi 1980) and observationally in the Milky Way (Searle 1971; Janes 1979; Twarog et al. 1997) and other massive spiral galaxies (e.g. Tinsley 1980; Zaritsky et al. 1994; Bresolin et al. 2012). The Milky Way, being the only galaxy where we can resolve millions of stars, provides a unique opportunity to study these gradients and deviations from them in detail. Early evidence by Janes (1979) suggested a linear gradient on the order of d [ Fe / H ]/ d 𝑅 = -0 . 05 ± 0 . 01 dex kpc -1 for the Milky Way which aligns very well with more recent measurements (Anders et al. 2017; Hayden et al. 2015). However, these gradients are accompanied by a significant spread in [Fe/H] of 0 . 1 -0 . 15 dex, as noted by Twarog (1980), which may imply a fine structure of the metallicity gradient (see Genovali et al. 2014). With increasing sample size and measurement precision, the specific shape and characteristics of this gradient remain somewhat unclear (Chiappini 2002). Previous studies have ∗ Australian Research Council DECRA Fellow for example claimed more intricate non-linear trends, bends, and flattening in the gradient of the Milky Way (e.g. Donor et al. 2020) and other galaxies (e.g. Pilyugin 2003; Sánchez et al. 2014) or even sequences of shapes (Pilyugin et al. 2017; Pilyugin & Tautvaišien˙e 2024), which were fitted with different models (Rosales-Ortega et al. 2011; Bresolin et al. 2012), such as piecewise linear ones (e.g. Sánchez-Menguiano et al. 2016) or non-linear ones (e.g. Scarano & Lépine 2013). Variations in the metal distribution, including breaks of the gradient at specific radii, give rise to a plethora of possible physical explanations, such as star formation efficiency variations and localised star formation bursts (Sánchez et al. 2014; Sánchez-Blázquez et al. 2014; Ho et al. 2015), gas accretion and dilution at different rates (Bresolin et al. 2012; Sánchez et al. 2013; Belfiore et al. 2016; Sánchez-Menguiano et al. 2016), gas outflows and feedback (Lilly et al. 2013; Ma et al. 2017a), as well as disk instabilities or local overdensities (Grand et al. 2016; Ho et al. 2017). In particular, Scarano & Lépine (2013) suggested that gradient break radii coincided with the corotation radii of spiral arms. Recent advancements in both computations and observations have significantly expanded our capabilities. For example, in terms of observational data in the Milky Way, the Gaia mission (Gaia Collaboration et al. 2016) enables more detailed studies of these gradients. New suites of large-scale simulations now allow us to gain insights into radial metallicity gradients across a range of simulated galaxies, including Milky Way analogues. This presents opportunities to revisit outstanding challenges of the detailed shape of the radial metallicity gradient. For instance, Hogg et al. (2019) created an extensive metallicity map of the Milky Way using APOGEE and Gaia data, while Poggio et al. (2022) mapped young stars and found metallicity variations around spiral arms (see also Zari et al. 2018, 2021; Poggio et al. 2021; Hackshaw et al. 2024). Similarly, Imig et al. (2023, among others) traced gradients across different stellar populations and ages, emphasizing the importance of considering radial migration effects (Binney & Tremaine 2008; Frankel et al. 2018, 2020). Historically, radial metallicity gradients have been measured using various stellar populations and gas tracers. Estimated gradients seem to be broadly consistent across different stellar tracers, such as young open clusters (e.g. Yong et al. 2012; Cunha et al. 2016; Magrini et al. 2017; Casamiquela et al. 2019; Donor et al. 2020; Spina et al. 2021; Myers et al. 2022), young hot (OB-type) stars (Zari et al. 2018, 2021; Poggio et al. 2021, 2022), field stars close to the Galactic plane (e.g. Bergemann et al. 2014) or Cepheids (Andrievsky et al. 2002a,b; Lemasle et al. 2007, 2013). Despite extensive observational efforts, several challenges persist for studies in the Milky Way. The completeness (or patchiness) of observed datasets remains a fundamental issue (Bergemann et al. 2014). The robustness of fits to the incomplete data is still contentious, including the need to actually fit two linear gradients with a break radius at corotation radius (Bresolin et al. 2012, and references therein) or further out (Yong et al. 2012; Donor et al. 2020) - or even more complicated functions (see e.g. Chiappini et al. 2001; Kubryk et al. 2015). Furthermore, methodologies for fitting linear models to scattered data need re-evaluation (Metha et al. 2021). Different samples yield varying gradients, potentially due to biases in data or the inclusion of older stars (e.g. Allende Prieto et al. 2006; Hayden et al. 2014; Anders et al. 2014; Vickers et al. 2021; Willett et al. 2023). The impact of spiral arm structures (Poggio et al. 2021), the Galactic warp (Lemasle et al. 2022) or bar-driven mixing (Di Matteo et al. 2013) on metallicity gradients is not fully understood. Understanding these gradients in the Milky Way is also crucial for extragalactic studies, where spatial resolution limits observations in different ways. In extragalactic systems, metals are mainly traced via gas, because it provides a more direct measure of the ongoing enrichment processes, unlike stars, which primarily reflect the integrated chemical history of the past. Observationally, gas emission lines are typically brighter and more accessible across large distances than stellar absorption lines, allowing for broader spatial coverage, especially in distant galaxies. Consequently, extragalactic studies often focus on gas-phase metallicity as traced by oxygen, A ( O ) = 12 + log ( O / H ) , while Galactic studies typically use stellar iron abundance [ Fe / H ] = A ( Fe ) -A ( Fe ) ⊙ as a metallicity tracer (e.g. Nicholls et al. 2017; Fraser-McKelvie et al. 2022). New instruments like the MUSE integral field spectrograph have enabled a plethora of extragalactic studies to contrast the Milky Way and techniques like the spectroscopy of H /i.pc/i.pc regions and planetary nebulae have helped to infer gas metallicity distributions in external galaxies (Shaver et al. 1983; Vilchez & Esteban 1996; Rolleston et al. 2000; Bresolin et al. 2012). Recent examples include Sánchez et al. (2014) with CALIFA galaxy observations as well as Mun et al. (2024) and Chen et al. (2024a) who use MAGPI observations to probe for example the effects of spiral arms. Notable is also the scatter that Chen et al. (2023) found for the gas metallicity across galactic radii with TYPHOON observations (see their. Figs. 4-6). Grasha et al. (2022) found that the gas metallicity gradient plateaus at a lower limit in their TYPHOON galaxies at the outermost radii - an observation replicated by IllustrisTNG simulations (Hemler et al. 2021; Garcia et al. 2023). From a modelling perspective, galactic chemical evolution models can both test understanding of radial metallicity gradients and make predictions beyond the limited volumes and tracers tested by Milky Way and extragalactic studies. Such galactic chemical evolution models include Chiappini et al. (2001); Matteucci & Recchi (2001); Minchev et al. (2014); Kubryk et al. (2015); Stanghellini et al. (2015); Rybizki et al. (2017); Spitoni et al. (2023); Johnson et al. (2024). Sharda et al. (2021) even presented a model for gas phase metallicity gradients in galaxies and their evolution from first principles (see also Krumholz & Ting 2018). In exploring radial metallicity gradients through simulations, we have better understood how different processes influence these gradients across galactic models and temporal scales. Studies such as Pilkington et al. (2012) in RaDES simulations reveal that gradients are typically established via inside-out galaxy formation. Khoperskov et al. (2023) quantified the scatter of gas metallicity to ≈ 0 . 04 -0 . 06 dex at a given galactocentric distance in their simulations. Meanwhile, the EAGLE simulations used by Tissera et al. (2019) provide insights into how these gradients vary with galaxy characteristics like stellar mass and merger history, emphasizing the dynamic nature of metallicity distributions. The plethora of simulations such as AURIGA (Grand et al. 2016), FIRE (Ma et al. 2017b, see their Fig. 6) or VINTERGATAN (see their Fig. 9; Agertz et al. 2021) also allow us to explore the gradient evolution of galactic timescales. Buck et al. (2023), for example, found a link of major accretion events with periods of unexpected steepening in the metallicity gradient within NIHAO-UHD simulations - closely resembling findings for the Milky Way by Lu et al. (2022) and Ratcliffe et al. (2023). FIRE simulations, examined by Bellardini et al. (2021), Bellardini et al. (2022), and Graf et al. (2024), extend these findings by comparing radial metallicity gradients and their azimuthal scatter across gas and stellar components and illustrate the complex interplay between galactic structure and metal enrichment processes. Similarly, Grand et al. (2015, 2016) highlight temporal changes in metallicity gradients already within 120 Myr, or roughly one galactic rotation. Such rapid changes underscore the impact of transient galactic events on the metal distribution, linking them to star formation patterns along spiral arms and the broader evolutionary history of the galaxy. In this study, we analyze a high-resolution NIHAO-UHD simulation of a Milky Way analogue to bridge the observational gap between detailed studies of our galaxy and broader extragalactic surveys. We aim to reveal subtle features of the radial metallicity gradient, which may be obscured by observational constraints in both the Milky Way and distant galaxies, by testing the following properties within the observationally probed inner 𝑅 Gal . ≤ 20 kpc: XGal. The paper is structured as follows: Section 2 describes the data of our Milky Way analogue NIHAO-UHD simulation. Section 3 analyses the linearity of the radial metallicity gradient of the simulation, the first of our four objectives. Section 4 then analyses both the scatter and local deviations from the gradient as well as the coherence of the gradient with vertical and azimuthal position as well as age (the remaining three objectives). Section 5 discusses them individually. We note that in this research we are mainly interested in the specific shape of the radial metallicity gradient for radii relevant to Galactic observations. However, we also discuss our results in the context of extragalactic results that probe beyond the inner 20 kpc of a galaxy. Section 6 bundles our results into an overarching conclusion.",
"pages": [
1,
2,
3,
4
]
},
{
"title": "2. DATA: A NIHAO-UHD MILKY WAY ANALOGUE SIMULATION",
"content": "For this project, we use a cosmological zoom-in simulation of a Milky Way analogue ( g8.26e11 ) from the Numerical Investigation of a Hundred Astronomical Objects (NIHAO, Wang et al. 2015) suite. This model galaxy was calculated as part of the NIHAO-UHD project (Buck et al. 2020) and has previously been used in various works studying Milky Way satellites (Buck et al. 2019), Milky Way's dark halo spin (Obreja et al. 2022), inferring birth properties of stars with abundance clustering (Ratcliffe et al. 2022), as well as the evolution of the interstellar medium's radial metallicity gradient since redshift three (Ratcliffe et al. 2024). Simulations were carried out with the smoothed particle hydrodynamics code Gasoline2 (Wadsley et al. 2017), including sub-grid turbulent mixing, using cosmological parameters from Planck Collaboration et al. (2014) with initial conditions and energetic feedback descriptions from the NIHAO project (Wang et al. 2015). Zoom-in simulations were then performed as described in detail by Buck et al. (2021) with star formation following Stinson et al. (2006) and energetic feedback following Stinson et al. (2013). We note that this is a slightly different rerun of the same simulation than the one studied by Buder et al. (2024); in this work, we use a higher resolution version and updated chemical yields. Because computational resources still limit the mass resolution of simulations, we are relying on tracer particles that represent simple stellar populations (SSPs) with the same age, overall metallicity and discrete initial mass function (IMF). Buck et al. (2021) have implemented the flexible chemical evolution code /c.pc/h.pc/e.pc/m.pc/p.pc/y.pc (Rybizki et al. 2017) to calculate the chemical yields for the SSPs. In particular, we use the alternative ( alt ) setup of /c.pc/h.pc/e.pc/m.pc/p.pc/y.pc that assumes a Chabrier (2003) IMF with high-mass slope of 𝛼 IMF = -2 . 3 over a mass range of 0 . 1 -100 M ⊙ for SSPs across a metallicity range of 𝑍 / 𝑍 ⊙ ∈ [ 10 -5 , 2 ] . The code calculates the contribution from asymptotic giant branch (AGB) stars, CCSN across a mass range of 8 -40 M ⊙ , and SNIa with an exponential function with exponent -1 . 12, a delay time of 40 Myr, and a normalization of the SNIa rate of log 10 ( NIa ) = -2 . 9. For each of these nucleosynthetic channels, yields from the following studies are used: Chieffi & Limongi (2004) for CCSN, Seitenzahl et al. (2013) for SNIa, and Karakas & Lugaro (2016) for AGB stars ( new_fit model in Buck et al. 2021). Contrary to a previous study by Buder et al. (2024), we take the elemental abundances at face value and do not apply any shifts. We limit the simulation data to the main halo by applying /p.pc/y.pc/n.pc/b.pc/o.pc/d.pc/y.pc's implementation of the Amiga Halo Finder (Knollmann & Knebe 2009) and then reposition and rotate this main halo to be face-on based on the angular momentum with /p.pc/y.pc/n.pc/b.pc/o.pc/d.pc/y.pc's /a.pc/n.pc/a.pc/l.pc/y.pc/s.pc/i.pc/s.pc./a.pc/n.pc/g.pc/m.pc/o.pc/m.pc./f.pc/a.pc/c.pc/e.pc/o.pc/n.pc module (Pontzen et al. 2013). We then further transform the resulting galactocentric Cartesian coordinate ( 𝑋,𝑌, 𝑍 ) and velocities ( 𝑉 𝑋 , 𝑉 𝑌 , 𝑉 𝑍 ) to Cylindrical ones as done in a previous study of this main halo by Buder et al. (2024). The model galaxy has a virial radius of 𝑅 vir = 𝑅 200 = 206 kpc and a total mass (gas, stars and dark matter) inside 𝑅 vir of 9 . 1 · 10 11 M ⊙ . At redshift zero, it contains 8 . 2 · 10 11 M ⊙ dark matter, 6 . 4 · 10 10 M ⊙ gas mass and 2 . 3 · 10 10 M ⊙ stellar mass with a stellar mass resolution of around 7 . 5 · 10 3 M ⊙ . When using a fifth of the virial radius as a reference to calculate total luminosity /one.sup and mass, we estimate a half-light radius, that is, effective radius of 𝑅 𝑒 = 3 . 79 kpc and a half-stellar-mass radius of 2 . 97 kpc. To achieve a roughly similar selection as the observational data of the Milky Way (Genovali et al. 2014) and other galaxies (e.g. Chen et al. 2023), we restrict the simulation data to a galactocentric radius of 𝑅 Gal ≤ 20 kpc and | 𝑧 | ≤ 10 kpc, as shown in Fig. 1. Similar to the Milky Way (Poggio et al. 2018; Lemasle et al. 2022), we note a warp of the stellar and gaseous disk (see Figs. 1b and 1e, respectively). To avoid too strong effects of radial migration (Binney & Tremaine 2008; Frankel et al. 2018; Grand et al. 2016; Minchev et al. 2018) while maintaining a sufficiently large sample size we further enforce stars to be younger than 0 . 5 Gyr, corresponding to roughly the time of four galactic rotations, and being half the value found by Minchev et al. (2018) for very limited migration in the Milky Way. This selection defacto limits the vertical range of 99% of stars to | 𝑧 | = 1 . 4 kpc. The strong influence of this age cut on the vertical distribution of stars in the Milky Way analogue can best be appreciated from the difference of vertical density distributions of stars in Figs. 1b and 1c. We are applying these cuts for all following analyses of the radial metallicity gradient in Section 3.",
"pages": [
4
]
},
{
"title": "3. THE LINEARITY OF THE RADIAL METALLICITY GRADIENT IN NIHAO-UHD",
"content": "In this section, we analyse the functional shape of the radial metallicity gradient. To get a first impression of possible shapes, we show the face-on view of the decreasing radial metallicity gradient of the simulation in Figs. 2a (for young stars) and Fig. 2d (for gas). Foreshadowing the later parts of this work on local variations, we also show a linear radial fit to either distribution in Fig. 2b and 2e and show the fit residuals in Figs. 2c and 2f. At the moment, however, we focus on the linearity and thus the logarithmic density distribution of star particle iron abundances [Fe/H] across different galactocentric radii 𝑅 Gal . . This distribution is shown in Fig. 3a and strongly suggests that the gradient is predominantly linear, similar to findings for the Milky Way. More complex shapes, such as piecewise linear ones have been suggested based on incomplete and limited data in the literature. We are thus also analysing these shapes with the complete and better-sampled data points of the NIHAO-UHD simulation. We firstly test different global fits in Section 3.1, before testing the influence of binning and coverage in Sections 3.2 and 3.3, respectively.",
"pages": [
4
]
},
{
"title": "3.1. Global gradient fits",
"content": "We fit three different functional forms to the global data: a linear function (used for Fig. 2b) a piecewise linear with a break radius 𝑅 break and a quadratic function The coefficients of the functions are fitted with the /s.pc/c.pc/i.pc/p.pc/y.pc./o.pc/p.pc/t.pc/i.pc/m.pc/i.pc/z.pc/e.pc function /c.pc/u.pc/r.pc/v.pc/e.pc_/f.pc/i.pc/t.pc (Virtanen et al. 2020) and listed in Table 2. To estimate the uncertainty of the break radius 𝑅 break, we use the profile likelihood method to identify the radii at which the residual sum of squares (RSS) values are increased by 1 𝜎 from the best RSS radius in steps of Δ 𝑅 break = 0 . 1 kpc and 0 . 5 kpc for the full and binned data set, respectively. We compute the RSS for each model 𝑓 𝑖 (see Eqs. 1-3) based on the 𝑁 data points as We have confirmed the robustness of our fits by applying other fitting routines as outlined in Table 1. After fitting three different functional forms, we use a combination of parameters to determine which model provides the best fit. In Table 2, the RSS value is the smallest (although only by a small margin) for the quadratic function. When assuming 𝜎 2 = 𝑅𝑆𝑆 / 𝑁 , we can also define a logarithmic likelihood for the 𝑁 data points. For 𝑘 free parameters, we then calculate the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) as For these criteria, the quadratic function performs slightly better than the linear or piecewise linear functions (see Table 2). We show the fit residuals in Fig. 3b as density distribution as well as in Fig. 3c as percentile distributions in radial bins of Δ 𝑅 Gal = 0 . 5 kpc. While the density distribution shows substructure, which we investigate later in Section 4, we note an increase in the median residuals of the linear fit in Fig. 3c towards the inner and outer radii, especially for 𝑅 Gal . > 17 kpc. A quadratic fit (see orange lines in Fig. 3) results in a slightly steeper linear component of the gradient (from -0 . 0411 to -0 . 0497 dex kpc -1 ), which is counteracted by the quadratic flattening term of + 0 . 0005 dex kpc -2 . The latter leads to an RGal. effective flattening of -0 . 172 + 0 . 200 = 0 . 028 dex (linear vs. quadratic terms) at 𝑅 Gal . = 20 dex. While seemingly only a nuisance correction across the large extent of [Fe/H] and 𝑅 Gal . , this quadratic function outperforms the linear fit. This is most apparent in Fig. 3c, where the orange line better traces the median residuals from the linear function across all radii. This suggests that non-linear functions, such as piecewise linear or quadratic ones describe the gradient better. Distinguishing between the two latter functional forms is challenging. The quantitative performance indicators-RSS, AIC, and BIC-show very similar values for both forms, and a closer examination of the residuals in Fig. 4 reveals no clear visual advantage for either the piecewise linear or quadratic model. TAKE-AWAY: Both piecewise linear and quadratic functions provide a better fit to the radial metallicity relation than a simple linear model. However, based on our assessments, there is no clear preference between the piecewise and quadratic functions.",
"pages": [
5
]
},
{
"title": "3.2. The influence of binning",
"content": "In this section, we test the influence of fitting a function to all points of the distribution or binned data in steps of Δ 𝑅 Gal . = 0 . 5 kpc, using median values as data points and standard deviations /two.sup as uncertainty (see also Hemler et al. 2021, who fitted functions to radially binned IllustrisTNG data). The results are shown in Fig. 4. Given that more than half of the young star particles of the galaxy are within /two.sup Wenote that this 𝜎 is not equivalent to observational uncertainty and can thus not be directly applied onto observational analyses. 𝑅 Gal . < 4 kpc, this binning - although counteracted by the smaller spread of [Fe/H] in the inner galaxy - weighs the distribution of the inner galaxy significantly less than when weighing all particles equally (20 vs. 34 000 data points). The parameters fitted to the binned data exhibit a larger uncertainty due to our use of the spread of [Fe/H] per bin as absolute uncertainty 𝜎 , but the fitted parameters agree well within the fitting uncertainties. TAKE-AWAY: While the specific slopes differ when fitting all points or binned data, they agree within the small fitting uncertainties.",
"pages": [
5,
6
]
},
{
"title": "3.3. The influence of radial coverage on linear fits",
"content": "Although we have gained useful insight into the global function, observational data will rarely cover the full extent of the stellar disk. Milky Way studies have previously been limited to the range of around 5 -15 kpc. There are often similar limitations and even gaps in extragalactic data. Using smaller ranges, observational studies have found hints of piece-wise linear gradients with a break radius in them based on limited radial coverage (e.g. Andrievsky et al. 2002a; Yong et al. 2012; Boeche et al. 2013; Hayden et al. 2014; Anders et al. 2017; Donor et al. 2020; Chen et al. 2023). These results are intriguing, since a quadratic function can, to first order, be approximated by two linear functions with a break radius. We therefore want to use our simulation to test if the radial coverage may indeed delude us into identifying broken linear gradients. Wetest how smaller coverage and piecewise linear fits could mimic a complex global gradient by fitting in piecewise lin- ear radial ranges of 0.25, 0.5, 1, 2, 5, 8, 10, 15, and 20 kpc. We show their difference with respect to a global linear fit in Fig. 5, with color-coding indicating the slope of the local gradient. A horizontal dashed line indicates the same slope as the global fit, whereas the offset of a line from the said horizontal dashed line indicates the local deviation from the global gradient intercept. Differences in line slopes are visualising the difference in gradient slopes between the global and local fits. We see that all ranges suggest more or less significant deviations from a global linear fit. The innermost fit suggests a significantly different gradient than the outermost fit. We also note increasing slope differences towards the smallest scales, hinting at local deviations from a global pattern. We follow these up in Section 4, but for now, focus on the larger-scale trends. When directly comparing an inner and outer radius fit, such as between 𝑅 Gal . = 5 -10 kpc (thick grey line in Fig. 5) and 𝑅 Gal . = 10 -15 kpc (thick black line in Fig. 5), we note a significant change, similar to previous estimates of the Milky Way (e.g. Yong et al. 2012; Lemasle et al. 2008). In our case, the gradient estimate changes from [ Fe / H ] ( 𝑅 Gal . ) = 0 . 471 -0 . 044 · 𝑅 Gal . to [ Fe / H ] ( 𝑅 Gal . ) = 0 . 375 -0 . 034 · 𝑅 Gal . . When looking at linear gradient fits across the radial coverage of Δ 𝑅 Gal . = 5 -15 kpc in Fig. 5, the gradient is steeper (bluer color) for smaller radii and flatter (redder) for larger radii. Indeed, a piecewise linear function can well mimic a complex global gradient. We note that in the simulated data, we see local deviations that become traceable below Δ 𝑅 Gal . ≤ 2 kpc ∼ 0 . 5 Re (bottom part of Fig. 5). This might indicate the spatial resolution required to see local effects, such as spiral arms, for extragalactic studies (see also Krumholz & Ting 2018; Li et al. 2024). We pursue this observation in the following Section 4. TAKE-AWAY: We find that a piecewise linear function can well mimicaquadraticfunction across the scales used in Milky Way and extragalactic studies. Local deviations become traceable below are spatial resolution of Δ 𝑅 Gal . ≤ 2 kpc (or Δ 𝑅 Gal . ≤ 0 . 5 Re).",
"pages": [
6,
7
]
},
{
"title": "4. SCATTER AND LOCAL DEVIATIONS FROM THE GRADIENT",
"content": "Now that we are sufficiently satisfied that our flattening gradient function reproduces the overall shape of the radial metallicity gradient, we are concerned with both the scatter and local slope deviations across the galactocentric radii in this section. In detail, we analyse the scatter (Section 4.2), vertical variations (Section 4.2), azimuthal variations (Section 4.3, particularly motivated by the localised, spiral-shaped fit residuals of Figs. 2c and 2f) and deviations across different ages (Section 4.4).",
"pages": [
7
]
},
{
"title": "4.1. Scatter",
"content": "When investigating the change in scatter from the innermost radii to the outermost (see Fig. 3c), we see a steady increase in 1 -𝜎 spread. This spread increases from 𝜎 [ Fe / H ] = 0 . 01 dex at 𝑅 Gal . = 0 . 25 ± 0 . 25 kpc to 𝜎 [ Fe / H ] = 0 . 06 dex at 𝑅 Gal . = 8 . 25 ± 0 . 25 kpc and reaches 𝜎 [ Fe / H ] = 0 . 10 dex at 𝑅 Gal . = 19 . 75 ± 0 . 25 kpc. When we recall the observed significant spread in metallicities of young open clusters at the solar radius beyond observational uncertainty (e.g. Donor et al. 2020; Spina et al. 2021) and our selection of only young ( < 0 . 5 Gyr) stars from the simulation, a strong impact of this scatter by radial migration should be excluded. At this point, we can imagine that this chemical diversity might be caused by less well-mixed gas or non-radial effects (such as vertical or azimuthal ones), which we investigate subsequently.",
"pages": [
7
]
},
{
"title": "4.2. Vertical deviations",
"content": "In this section, we now look at deviations with respect to the vertical dimension, that is, 𝑅 -𝑧 . In Fig. 6 we show the previously identified local gradient deviations (lines following the left axis label) on top of the vertical density distribution ( 𝑅 -𝑧 ) of young stars (Fig. 6a) and gas (Fig. 6b) between -3 < 𝑧 < 3 kpc. Although the quickly decreasing number of young stars (Fig. 6a) at outer radii does not show substructure in the density plots for reasonable bin sizes, we see more substructure for the gaseous component in Fig. 6b). In particular, we see rather minor deviations at small radii (where most stars and gas are close to the plane). At increasing radii, we notice an increase in both the vertical distribution of stars, and increasing local gradient deviations. In particular, we note a significant deviation of the slope around 𝑅 Gal . ∼ 15 kpc, where the gradient deviation line is steep and blue (indicating a much steeper gradient at this radius), and we notice a significant overdensity of gas around 𝑧 ∼ 1 kpc. Overall, however, we do not see strong correlations in this particular plane. This could, however, be caused by a super-position effect of the up- and downturn at larger radii due to the galactic warp (see Figs. 1b and 1e). Although the warp of the stellar disk in Fig. 1b is not as clear, we confirm that both the gas disk and the youngest stars below 0 . 5 Gyr are tracing each other across the simulation in both galactocentric radius 𝑅 Gal . and height 𝑧 Gal . for different sectors in the azimuthal direction. We note that the superposition in Fig. 6 could smear out local correlations of slope changes with gas overdensities, for example, by spiral arms. Although such an edge-on view of the galaxy may indeed be the only observable one for extragalactic targets, for example, of the GECKOS survey of edge-on galaxies (van de Sande et al. 2023), we have the luxury of being able to analyse the azimuthal direction of our simulated galaxy, too. TAKE-AWAY: We see no strong correlations of deviations in the vertical direction throughout the simulation. Such correlations could, however, be blurred by azimuthal effects, like the galactic warp, which needs to be disentangled in the azimuthal dimension.",
"pages": [
7
]
},
{
"title": "4.3. Azimuthal deviations",
"content": "To analyse the deviations from a global gradient across different azimuthal viewing angles, we divide the galaxy into 8 sectors with Δ 𝜛 Gal . = 45 · (see Fig. 7a). This allows us to study the positions around the upturn and downturn of the galactic warp with the median azimuth of young star particles below and above the plane being 𝜑 Gal . ∼ 183 · and 𝜑 Gal . ∼ 4 · , respectively (see Fig. 1e), while maintaining a reasonable sample size. At face value, the distribution of 𝑅 Gal . - [ Fe / H ] for each sector follows a similar, rather linear shape with most stars being born in the inner 5 kpc of the galaxy. However, we find significant deviations in different sectors of the galaxy (Fig. 7). On the one hand, we find non-linear deviations as bumps with slightly increased or decreased iron abundance - up to 0 . 1 -0 . 2 dex - in Figs. 7c at 𝑅 Gal ∼ 18 kpc, 7d at 𝑅 Gal ∼ 10 kpc, 7f at 𝑅 Gal ∼ 14 kpc, and 7g at 𝑅 Gal ∼ 17 kpc. On the other hand, we find significant gaps in the distribution at similar [Fe/H], most strikingly at the upturn of the galactic warp in Fig. 7e ( 𝜛 Gal . = 135 -180 · ) at [ Fe / H ] ∼ 0 dex and 𝑅 Gal . ∼ 8 -14 kpc. We note that the sector e) with the gap is surrounded by two sectors (d and f) with significant overabundance at the same radius. This could be indicative of stars having formed as a result of gas moving from sector e towards either azimuthal direction, causing a gas overdensity which could in turn lead to higher star formation activity. To establish this observation, we take a closer look at the timedomain, that is, stellar age as well as the spatial domain of 𝑅 Gal . -𝜑 Gal . in the next section. TAKE-AWAY: We find various deviations from the global trend in the azimuthal direction, including gaps and isolated streaks of stars with similar [Fe/H] throughout Δ 𝑅 Gal . = 2 -6 kpc. These can introduce local over- and under-enhancement of up to ± 0 . 2 dex in [Fe/H] at a given radius. In the next section, we analyse whether the stars of these streaks have been born at the same or different time.",
"pages": [
7,
9
]
},
{
"title": "4.4. Deviations with time and age",
"content": "In this section, we examine the radial metallicity gradient in a small age range less than 0 . 5 Gyr. To do so, we color Fig. 7 by the median stellar age rather than the logarithmic density in Fig. 8. We find an overall significant scatter across time, suggesting a good mix of star formation across all sectors for stars born within less than 0 . 5 Gyr. For stars within this restricted age range, we do not see a strong correlation with radius, such as older stars being born further inside, but a larger amount of stars being born closer to the galactic centre. Wenote that stars with similar [Fe/H] in each sector tend to be formed at similar times (within 50 Myr), that is, as flat lines with the same color (age) in Figs. 8b-i. To guide the eye, we have identified Group 1 in Fig. 8e (around 𝑅 Gal . ∼ 14 kpc at 𝜛 Gal . = 180 -225 · ). We further note that the enriched bumps identified earlier are born at similar times, see, for example, Group 2 in Fig. 8f. The coloring by age also reveals that stars with lower [Fe/H] than expected (see Group 3 in Fig. 8h) are born at similar times. In some cases, these extend to Δ 𝑅 Gal . = 2 -6 kpc, see Groups 1, 2, and 3 in Fig. 8. At a given radius 𝑅 Gal . , these streaks cause a significant spread in local [Fe/H] of up to ± 0 . 2 dex (see Fig. 8). From the analysis of azimuthal sectors, the impression arose that the star formation in this simulated Milky Way analogue is - as expected for a spiral galaxy - rather patchy and localised on the smallest timescales. This is confirmed by looking at the spatial distribution of azimuth 𝜑 Gal . and radius 𝑅 Gal . in Fig. 9. Already when looking at the density distribution of all stars born within less than 0 . 5 Gyr in Fig. 9a, multiple streams are visible, stars on spiral patterns (see also Kreckel et al. 2019; Chen et al. 2024b). When following up the previously identi- roups 1, 2, and 3, we recover them on said spiral patterns (Groups 1 and 3) or a local overdensity (Group 2). Although one could imagine that radial migration might induce such a spiral-like shape for the stars of groups 1 and 3, their low age of less than 250 Myr would require a significant migration effect of several kpc, while having no influence on the older stars of group 2. When tracing the position of significant overdensities from Fig. 9a in the same projection colored by age in Fig. 9b, we note that for radii above 𝑅 Gal . > 5 kpc these overdensities are colored in red, that is, containing indeed young stars with ages below 200 Myr and being consistent with the most recent star formation along these spiral patterns in the outer galaxy. TAKE-AWAY: We find significant scatter across the radial metallicity distribution caused by streaks of stars born with similar [Fe/H] at similar times (within 50 Myr) across either very local or radially extended spiral-shaped regions of the galaxy, suggesting local enhancement patterns in small overdensities or along spiral arms.",
"pages": [
9
]
},
{
"title": "5. DISCUSSION",
"content": "Having presented the analysis, we now put our results into the context of other work in terms of our initial aims: to analyse the shape (Section 5.1), scatter (Section 5.2), local deviations (Section 5.3), and time-dependence (Section 5.4) of the radial metallicity gradient. These initial discussions inform our thoughts on the implications of this work for Milky Way studies in Section 5.5 and the studies of other galaxies in Section 5.6.",
"pages": [
9
]
},
{
"title": "5.1. Linearity of the radial metallicity gradient",
"content": "The radial metallicity gradient of our simulated NIHAOUHDMilky Way analogue showed an overall decreasing, predominantly linear shape, as established in Section 3. Motivated by previous works by Sánchez-Menguiano et al. (2016), among others, we also fitted piecewise linear and quadratic functions to the data in Section 3.1. Both forms perform better than a linear trend. The very similar fitting performances indicate no significant preference between either piecewise linear or quadratic function. Due to both functions' rather good overall fit, we have not tried more exotic non-linear functions as done by Scarano & Lépine (2013). Increasing the flexibility of the gradient function could, however, improve the fit at the innermost kpc, where a flattening is predicted by our simulation, but chemical enrichment is also harder to simulate (see also Minchev et al. 2013; Sun et al. 2024). We have found no significant influence of binning for our gradient estimates (Section 3.2), but have found that a limited radial coverage - as is the case for the Milky Way - could mimic a truly quadratic function with two piecewise linear fits (Section 3.3). This is important, as it has significant implications for the conclusions we draw from the incomplete data of our Milky Way, as we will discuss in more detail in Section 5.5. The balance between a quadratic and piecewise linear radial metallicity gradient teeters at the breaking radius. If present, our analysis of the Milky Way analogue would place it at 𝑅 break ∼ 10 ± 0 . 5 kpc. This radius is strikingly close to the radius of 9 kpc found by Hemler et al. (2021) for a TNG50 galaxy simulation with a stellar mass of log ( 𝑀 ★ / M ⊙ ) = 10 . 72, that is, close to the Milky Way's (see their Fig. 2). In terms of physical reasons for a breaking radius at this location, a direct and secular influence of a stellar bar with non-symmetric effects around the corotation radius (Di Matteo et al. 2013; Scarano & Lépine 2013) should be minor for our specific scenario due to the low ages of the stars considered in our analysis. In particular, our identified break radius is significantly larger than the corotation radius of the Milky Way bar at 4 . 5 -7 . 0 kpc (Bland-Hawthorn & Gerhard 2016, and references therein) anyway. We are intrigued, however, by the proposition by Garcia et al. (2023) of galactic discs consisting of a star-forming inner disc with a steep gradient and a mixing-dominated outer disc with a flat gradient, with the break radius marking the region of transition between them. In Illustris TNG50-1 data, they found such a transition and break radius to be situated much further out at 30 kpc for Milky Way mass galaxies (10 . 1 ≤ log ( 𝑀 ★ / M ⊙ ) ≤ 10 . 6). While our bestfitting break radius - if present - is inconsistent with theirs, we will follow this up in more detail in Section 5.6, where we also discuss the implications for extragalactic studies in general.",
"pages": [
9,
10
]
},
{
"title": "5.2. Scatter of the radial metallicity gradient",
"content": "In Section 4.1 we found an increasing scatter from 𝜎 [ Fe / H ] = 0 . 01 dex in the inner galaxy to 𝜎 [ Fe / H ] = 0 . 10 dex around 𝑅 Gal . ∼ 20 kpc. Comparing these values with simulations other than TNG50 with a similar metallicity spread (see Fig. 2 by Hemler et al. 2021) is rather difficult, as the literature focuses on the shape and density distribution (see e.g. Minchev et al. 2014, their Fig. 10). When comparing with Milky Way studies (e.g. Anders et al. 2017), the scatter in the simulation is smaller than the observed spread of [Fe/H]. This can be visually appreciated by comparing the combinations of different measurements in the Milky Way (Genovali et al. 2014; Spina et al. 2021; Myers et al. 2022) in Fig. 10a and our simulation in Fig. 10b and c. We discuss the implications of this on studies of the Milky Way's gradient in Section 5.5. When assuming that young star and gas phase abundances are similar, we find comparable scatter of abundances for example with respect to TYPHOON observations by Chen et al. (2023). To test this assumption, we also show the gas phase metallicity in Fig. 11, for which we find a similar shape and scatter of the gradient, but systematically less scatter or spread than observed gas phase abundance, thus urging us to treat the absolute values of abundances and abundance scatters as well as spreads with caution. We furthermore note that the spread of abundance in observations does only increase for some but not all of the observed (and thus observationally limited) galaxies by Chen et al. (2023). This potentially limits the range of galaxies to which our conclusions may apply. While the simulated abundance scatter is consistent with the predictions by the theoretical forced-diffusion model by Krumholz & Ting (2018), that is, a scatter of ∼ 0 . 1 dex over timescales of ∼ 100 -300 Myr, our simulations suggest that the scatter is driven by the radial structure and large-scale spiral arms, which were not included in their model.",
"pages": [
10,
11
]
},
{
"title": "5.3. Localised vertical and azimuthal deviations and their correlation with gas",
"content": "In Sections 4.2 and 4.3 we established that local deviations contribute significantly to the spread of the global metallicity gradient above 𝑅 Gal . > 8 kpc ∼ 2 Re. We noted in particular a void of stars where we found an upturning warp of the galaxy around 𝜑 Gal . ∼ 180 · spatially close to regions of the galaxy (Groups 1 and 2) that deviated most significantly from the overall trend in Fig. 8 and Fig. 9. These stellar voids pose the question if they are also void of gas thus suggesting the gas has shifted to the more enhanced regions. In Fig. 12 we are thus tracing both the spatial distribution of gas as colored density distribution and stars as grey contour lines and gas. We find that while stars and gas trace each other in the vertical direction, we do not always see a match between the two tracers in the radial direction. In particular, we do find a significant amount of gas around the stellar void of 𝜑 Gal . ∼ 180 · and 𝑅 Gal . ∼ 8 -11 kpc in Figs. 12e and 12f. This gas seems to be more tightly concentrated though for example in the tight wave around 𝑅 Gal . ∼ 8 -11 kpc in Fig. 12e. We also note that significant gas overdensities, for example around 𝜑 Gal . ∼ 0 -45 · and 𝑅 Gal . ∼ 7 kpc in Fig. 12a do not seem to correlate with significant overenhancement in iron abundance (compare to Fig. 7a). While we see a hint of a coinciding deviation of Δ [ Fe / H ] for larger deviations from the galactic plane Δ 𝑧 in the upturning outer region of Fig. 12f, this does not seem to be the case for the downturning outer region of Figs. 12b and 12i. As the edge-on projection is not providing conclusive insights, we are now looking into the phase-on projection in Fig. 13. We have chosen a region of the simulated galaxy whose gas density at solar radius (Fig. 13c) matches with the recently measured distribution of young stars in the Milky Way at face value (Fig. 13a) by Poggio et al. (2021). Comparing simulated gas and observed young stars is preferable in this case, as the density of simulated stars is too low to easily identify overdensities (Fig. 13b). The region and its gas spiral structures appear to be representative, as these structures exist throughout the whole galaxy (see Fig. 1d). In the different panels of Fig. 14, we thus show this representative region of the galaxy, but color each spatial bin by stellar metallicity (Fig. 14a), the deviation from the global linear trend (Fig. 14b) as well as the gas metallicity (Fig. 14c) and its deviation from the global linear trend (Fig. 14d). In all cases, we also overlay the density contours of the significant gas overdensities (red regions in Fig. 13c). As expected, we see that the metallicity color map of the stars in Fig. 14a shows a decreasing trend from right to left (inner to outer galaxy) and an increasing scatter (more blue and red points towards the left) in the residual plot of Fig. 14b. We cannot identify a strong correlation between gas overdensities and stellar metallicity or residuals in either plot - possibly caused by the low number density. In Figs. 14c and 14d, however, the radial gas metallicity gradient shows significant local variations, that is, a trend from left to right that is not very smooth. In particular, we find significant deviations of up to + 0 . 15 dex in [Fe/H] behind the outer gas spiral (lower left of Fig. 14d) and -0 . 1 dex in [Fe/H] in front of the inner gas spiral (upper center of Fig. 14d) with a steep edge consistent with the gas spiral edge. We have identified the same patterns in both [Fe/H] and A(O) as both elements trace each other rather well in the simulation of young stars (see Fig. A1). Tentatively, we even see a slight enhancement of A(O) at the trailing edge of the inner spiral arm (top of Fig. 14d). We convince ourselves of the step-like behaviour by selecting a small slit-like region of 𝜑 Gal . ∼ 0 · and -2 < 𝑌 Gal . / kpc < -1 and tracing the gas metallicity and gas density as a function of radius in Fig. 15. We indeed find steps and confirm that they coincide with the location of significant gas overdensities. These step-patterns have also been found by Grand et al. (2015) in another simulation and observationally by Ho et al. (2017). In Fig. 15, we note an extended flat region just beyond 𝑅 Gal . > 10 kpc, the best fitting 𝑅 break of an assumed piecewise linear fit. Our analyses suggest that the correlation of void and overdensities with chemical enrichment of gas and young stars is more complicated and should better be followed up by tracing these structures over simulation look-back time in a dedicated follow-up analysis to unravel the physical mechanisms of star formation feedback cycles . This could also involve the tracing of star formation bursts and disk instabilities (Sánchez et al. 2014; Sánchez-Blázquez et al. 2014; Ho et al. 2015) as well as tracing how much self-enrichment as well as mixing and dilution takes place around the gas spirals (Ho et al. 2017). Rather than going back in simulation time, the present simulation data of a single snapshot in time already allows us to look back in terms of stellar lifetime - similar to Milky Way studies, as we discuss subsequently.",
"pages": [
11,
12
]
},
{
"title": "5.4. The impact of time and age: mixing and migration",
"content": "Although we have chosen a rather small stellar age window of 0 . 5 Gyr to trace the radial metallicity gradient without the expected significant impact of radial mixing and migration, we are testing and discussing this particular choice in this section in two ways. Firstly, we test the deviation of the radial metallicity gradient from the same global shape as well as the abundance spread across smaller age bins of 100 Myr between 0 -1000 Myr in Fig. 16. Secondly, we trace the distribution of stellar metallicity across galactic radii for increasing age bins from 50 Myr up to the maximum stellar age of 13 . 8 Gyr in Fig. 17. Our first test in Fig. 16 shows that the deviation from a global trend remains similar in functional form. We find that the spread of iron abundance does indeed scatter significantly, but the distributions stay within the same overall shape across the ten age bins. We note though, that the smallest age bin of 0 -100 Myr shows the least abundance scatter. This is consistent with the picture from our second test of increasing age ranges in Fig. 17. Here we find the first significant deviation from a tighter and already slightly quadratic relation for an age of 100 -150 Myr in Fig. 17c - our previously identified Group 3. As expected from previous simulations and observations, we see an increase in the scatter as we include more and more older stars. We note a still similar albeit more scattered shape for stars below 4 Gyr in Fig. 17h, before we start to see a more metal-poor population of stars in the inner galaxy appear between 4 -8 Gyr in Fig. 17i. These also begin to significantly impact the quadratic fit to the radial metallicity distribution, shown as a solid red line, in contrast to our reference fit, represented by a dashed red line. The significant amount of metal-poor stars in the inner galaxy then completely tilts the distribution when also including stars between 8 -13 . 8 Gyr in Fig. 17j (see also Johnson et al. 2024). Similar to the Milky Way (Bland-Hawthorn & Gerhard 2016), these oldest stars are those of the relatively more metal-poor thick disk that are confined to the inner disk with a shorter scale length. While we cannot exclude radial migration playing a role for change of radius for the youngest stars of the simulation, since Frankel et al. (2018) predicted significant shifts even for ages below 0 . 5 Gyr (see their Fig. 10), the larger scatter for older stars is certainly suggesting a larger (re-)distribution of stars along the radial axis, as found in previous simulations (Minchev & Famaey 2010; Grand et al. 2015).",
"pages": [
13
]
},
{
"title": "5.5. Implications for Milky Way studies",
"content": "Our analysis of the radial metallicity gradient in a simulated NIHAO-UHD galaxy offers several insights that are directly applicable to understanding the Milky Way's gradient. First, the nature of the gradient - whether it is linear or better described by more complex functional forms - remains a critical question. Previous studies, such as those by Lépine et al. (2011) and Donor et al. (2020), have suggested the po- ential for a break radius, possibly at the corotation radius or further out (Scarano & Lépine 2013), which could indicate two distinct linear regimes. In our analysis, we find evidence that the gradient is at least is not purely linear, but could also be smoothly flattening. Applying a smooth quadratic function on observational data (Yong et al. 2012; Andrievsky et al. 2004; Genovali et al. 2014), might provide a better or at least consistent fit for the Milky Way data without the need for a break radius. However, even this may not fully capture the nuances observed in our simulations. Chemical evolution models propose a more sophisticated behaviour (e.g. Chiappini et al. 2001; Kubryk et al. 2015; Palla et al. 2024), reflecting varying influences of galactic processes at different radii. Understanding this structure in the simulated galaxy provides a framework for interpreting similar complexities in the Milky Way. Given these complexities, it is also essential to consider how local sampling biases might affect our understanding of the Milky Way's metallicity gradient. For instance, incomplete samples that omit low [Fe/H] clusters or stars could skew gradient estimates, as suggested by our comparisons in Figures 10a and 10b. Our results indicate that young clusters with lower (or higher) [Fe/H] than expected at a given radius could indicate the previous presence of a spiral arm (see our identified Groups in Figs. 8 and 9). Furthermore, we caution that localised effects - both intrinsic and in terms of selection function - could also mimic non-linear shapes and more spatial coverage is needed in the Milky Way. Our results also indicate that older clusters, which have been found more frequently at larger distances than young clusters - are likely influenced by radial migration - and thus complicate the interpretation of these radial metallicity gradients (Magrini et al. 2009; Lépine et al. 2011). Cosmological zoom-in simulations like NIHAO-UHD are approaching the resolution needed to examine regions analogous to the solar vicinity, though the star particle numbers and mass resolution remain a limiting factor. Nonetheless, we observe distinct patterns in the distribution of young stars and gas, including lower [Fe/H] and A(O) in the leading edges of gas overdensities and higher [Fe/H] and A(O) in the trailing edges, consistent with findings by Grand et al. (2016), Ho et al. (2017), and Kreckel et al. (2019). These trends suggest that local metallicity variations, driven by gas dynamics, may also play a significant role in shaping the observed gradients in the Milky Way. Additionally, our study hints at the potential for more nuanced variations in [Fe/H] across different regions of the galaxy. In particular, the gas shows a step-like behavior of A(O)and[Fe/H] changes around the edges of gas overdensities (Fig. 15), with significant deviations from the global gradient in specific regions. We have also found a larger stellar void around -12 < 𝑅 Gal < -10kpc. Although further investigation is needed, these findings could have important implications for understanding localized star formation events and their impact on the overall metallicity distribution in the Milky Way (Sánchez et al. 2014; Sánchez-Blázquez et al. 2014; Ho et al. 2015). It will certainly be exciting to see how much more insights (Poggio et al. 2021; Hackshaw et al. 2024) we will get from the more extended data of future data releases of Gaia and spectroscopic surveys. Wecannot directly link spiral arms to bar resonances or bardriven mixing in our simulation, because of a negligible bar strength in our galaxy /three.sup (but see Minchev & Famaey 2010; Di Matteo et al. 2013). However, the influence of a galactic bar on the spiral arms and, by extension, on the radial metallicity gradient, remains a possibility (see again Chen et al. 2023). Disk instabilities and warps might further complicate the interpretation of these gradients and progress will likely rely on the detailed disentangling of these effects from both cosmological simulations as well as idealised simulations and models (Minchev et al. 2013; Grand et al. 2015, 2016; Krumholz et al. 2018; Sharda et al. 2021; Bland-Hawthorn et al. 2024; Tepper-Garcia et al. 2024).",
"pages": [
13,
14
]
},
{
"title": "5.6. Implications for extragalactic studies",
"content": "The insights gained from our analysis of the radial metallicity gradient in a simulated NIHAO-UHD galaxy extend beyond the Milky Way, offering valuable implications for the study of extragalactic systems. One key observation is that deviations from a purely linear metallicity gradient, as seen in our Milky Way analogue, are common in other galaxies as well. When fitting a piecewise linear fit to our data, we found a break radius at 𝑅 Gal . = 10 . 0 ± 0 . 5 kpc. Converted to effective radii Re or radii R25 covering the 25 mag arcsec -2 isophote /four.sup , this corresponds to 𝑅 break ∼ 2 . 5 Re ≡ 0 . 7 R25 for our simulation. This would be consistent with the observational results by Sánchez et al. (2014) who found that breaks in metallicity gradients are common in both spiral and barred galaxies, with flattening of the abundance being evident beyond ∼ 2 Re (compare also to Belfiore et al. 2017). Similar to our suggestion for Milky Way studies, we suggest to also test a smooth function, such as a quadratic one, on extragalactic observational data (e.g. Bresolin et al. 2012; Chen et al. 2023) to test the preference of a distinct break radius. Although the focus of this research lies on the observable region of the Milky Way ( 𝑅 Gal . < 20 kpc) and most other galaxies ( 𝑅 Gal . < 2 . 5 Re), the finding of significant gradient changes in the outskirts of galaxies by Garcia et al. (2023), suggests to also test this region of our Milky Way analogue. Garcia et al. (2023, see their Fig. 4) found a metallicity floor in IllustrisTNG galaxies. When using their sample to identify a metallicity floor radius for a Milky Way mass galaxy with log ( 𝑀 ★ / M ⊙ ) = 10 . 7 (Bland-Hawthorn & Gerhard 2016) at redshift 𝑧 ∼ 0, we would expect to find it around 25 -30 kpc. We therefore extend the analysed radius to 𝑅 Gal . ≤ 100 kpc (see Fig. 18) and indeed find a similar abundance floor of [ Fe / H ] ≥ -0 . 64 for young stars and A ( O ) ≥ 8 . 12 for the majority of gas (see Fig. 19 at a similar radius. We note that another galaxy without gas in this figure is a sufficiently large distance of 92 kpc, that is, ( 𝑌,𝑌, 𝑍 ) = (-50 , -75 , 20 ) kpc. These lowest abundances remind us of two observational results. Firstly, the iron abundance floor is consistent with the lower end of the Milky Way thin - and coincidentally outer disk of [ Fe / H ] ∼ -0 . 7 (Bensby et al. 2014; Buder et al. 2019). Secondly this oxygen abundance floor is consistent with the results by Grasha et al. (2022) from TYPHOON galaxy observations. Grasha et al. (2022) suggested this could be caused by changes in the ratio of supernovae II and AGB reflected by a changing ratio of nitrogen to oxygen abundance N/O which also flattens towards a lower plateau below metallicities of A ( O ) ∼ 8 . 0 (Nicholls et al. 2017). While we cannot follow this observation up with the present simulation, a similar simulation used by Buder et al. (2024) has traced the relative XGal. XGal. XGal. contribution of both supernovae II and AGB and should be used to test this hypothesis in the future. It is important to note that the chemical evolution model in the NIHAO-UHD simulations is constrained by the current, incomplete understanding of evolutionary pathways and yields (Buck et al. 2021), as well as by limitations in resolution and the imperfect physics inherent to cosmological zoom-in simulations (Buck 2020). Both could contribute to the identified differences in absolute and relative abundances across different scales - including a different scatter of abundances for example of the gas phase metallicity between NIHAO-UHD of up to 0 . 1 dex and the low scatter of 0 . 03 -0 . 05 dex (and even lower on local scales) found by PHANGS-MUSE faceon observations (Kreckel et al. 2020). Extending our analysis to other simulations and further improving the resolution and physics of the simulations will be key in uniting the observational and theoretical insights into galactic chemical evolution on small and large scales. Similar to more resolved and higher quality observations in the Milky Way, we also expect more, better, and diverse faceon and edge-on observations and analyses across a range of wavelengths by the PHANGS and GECKOS teams (Kreckel et al. 2019, 2020; van de Sande et al. 2023) as well as the SDSS-V and MAGPI collaborations (Kollmeier et al. 2017; Foster et al. 2021; Mun et al. 2024; Chen et al. 2024a), among many other ongoing efforts.",
"pages": [
14,
15
]
},
{
"title": "6. CONCLUSIONS",
"content": "To conclude our study, we first iterate the main take-away of our research in Section 6.1 before giving suggestions for future research in Section 6.2.",
"pages": [
15
]
},
{
"title": "6.1. Take-Away",
"content": "Wehaveanalysed the radial metallicity distribution of young stars and gas in the inner 20 kpc of a NIHAO-UHD Milky Way analogue (Fig. 1), finding a predominantly linear decrease (Figs. 2 and 3). Although our analysis of a single spiral galaxy simulation has limited applicability to the entire population of diverse spiral galaxies, it reveals several intriguing findings about the shape and local metallicity variations. The results we find in this work hold relevance for both the Milky Way and extragalactic research communities: more detail. Secondly, we suggest approaching the fitting of gradients in external galaxies in a more agnostic way to the shape. This will be particularly interesting when we can observe the outermost regions of galaxies, where simulations predict an abundance floor (Figs. 18 and 19).",
"pages": [
16
]
},
{
"title": "6.2. Future Research",
"content": "In our study, we have focused on the present-day snapshot of the NIHAO-UHD Milky Way analogue simulation - similar to present-day observations that are possible in our local Universe. Given that the simulation is tracing particles and gas over time, a detailed follow-up study should trace the evolution and coherence of spatial and chemical over- and underdensities over time and different elements (see also Zhang et al. 2024). This would in particular allow us to quantify the change of abundance in the leading and trailing edges of spiral arms and subsequently track the mixing and blurring of these over time. Certainly, more studies are needed to establish a link to a physical mechanism and further quantify its importance. More results are expected as we extend the reach, number, and quality of stellar measurements in our Galaxy (e.g. Barbillon et al. 2024) and beyond. These improvements will allow us to move beyond a one-dimensional analysis of gradients and better incorporate and model local variations. While previous works are showing that relative trends for several elemental abundances do not strongly disagree from observations (Buck et al. 2021; Buder et al. 2024), we are still missing several details on the origin of elements, such as a complete picture of the synthesis sites, environments, and yields for elements. Not least because of these imperfections of absolute chemical enrichment predictions, we are refraining from quantitatively comparing the shape of our Milky Way analogue with the actual Milky Way. We have previously also mentioned the limitations in mass resolution of stars and gas, which may introduce unrealistic effects and could for example drive deviations from actual chemical enrichment at the smallest scales. We also note that the results of our simulation may not apply to the actual Milky Way due to different galaxy properties. These could be different due to different galaxy formation pathways, such as the amount and importance of mergers (Buck et al. 2023; Buder et al. 2024). In our discussion, we have already eluded to the weak bar in this simulation. Motivated by the analysis by Tuntipong et al. (2024), we have also investigated the bulge to total stellar mass ratio 𝐵 / 𝑇 . We find a strong bulge with 𝐵 / 𝑇 = 0 . 48 when selecting bulge stars with orbit circularity 𝑗 𝑧 / 𝑗 𝑐 < 0 . 5 and disk stars with 𝑗 𝑧 / 𝑗 𝑐 > 0 . 7 based on actions 𝑗 , consistent with values found by Obreja et al. (2019) of a lower resolution simulation of 8.26e11 . Our simulated galaxy has both a very weak bar and a smoothly changing radial metallicity gradient. This is in line with the findings by Chen et al. (2023) for five strongly and weakly barred spiral galaxies, where bars seem to drive the dominance of break radii in radial metallicity gradients. Future work should certainly look at a variety of galaxies to establish the causality of bar strength with the smoothness or abrupt change at a break radius for the radial metallicity gradient. Expanding our study to more and other simulations such as the VINTERGATAN suite (Renaud et al. 2024) and subsequently comparing to observations of galaxies with varying parameters like mass, formation history, bar strength or environment would allow us to quantify their effect on metallicity gradients and further disentangle the influence of different enrichment mechanisms on the chemical evolution of galaxies.",
"pages": [
16
]
},
{
"title": "SOFTWARE",
"content": "The research for this publication was coded in /p.pc/y.pc/t.pc/h.pc/o.pc/n.pc (version 3.7.4) and included its packages /a.pc/s.pc/t.pc/r.pc/o.pc/p.pc/y.pc (v. 3.2.2; Astropy Collaboration et al. 2013, 2018), IP/y.pc/t.pc/h.pc/o.pc/n.pc (v. 7.8.0; Pérez & Granger 2007), /m.pc/a.pc/t.pc/p.pc/l.pc/o.pc/t.pc/l.pc/i.pc/b.pc (v. 3.1.3; Hunter 2007), N/u.pc/m.pcP/y.pc (v. 1.17.2; Walt et al. 2011), /p.pc/y.pc/n.pc/b.pc/o.pc/d.pc/y.pc (v. 1.1.0; Pontzen et al. 2013), /s.pc/c.pc/i.pc/p.pc/y.pc (v. 1.3.1; Virtanen et al. 2020), /s.pc/k.pc/l.pc/e.pc/a.pc/r.pc/n.pc (v. 1.5.1 Pedregosa et al. 2011) /s.pc/t.pc/a.pc/t.pc/s.pc/m.pc/o.pc/d.pc/e.pc/l.pc/s.pc (v. 0.14.2 Perktold et al. 2024) We further made use of /t.pc/o.pc/p.pc/c.pc/a.pc/t.pc (version 4.7; Taylor 2005);",
"pages": [
17
]
},
{
"title": "DATA AVAILABILITY",
"content": "All code to reproduce the analysis and figures can be publicly accessed via https://github.com/svenbuder/ nihao_radial_metallicity_gradients . The used simulationsnapshot can be accessed as FITS file via https: //github.com/svenbuder/preparing_NIHAO . Original data, more snapshots and other galaxies can be found at https://tobias-buck.de/#sim_data . We encourage interested readers to get in contact with the authors for full data access and advice for use and cite Buck et al. (2020, 2021).",
"pages": [
17
]
},
{
"title": "ACKNOWLEDGMENTS",
"content": "Weacknowledge the traditional owners of the land on which the ANU stands, the Ngunnawal and Ngambri people. We pay our respects to elders past, and present and are proud to continue their tradition of surveying the night sky and its mysteries to better understand our Universe. This work was supported by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013. SB and KGacknowledge support from the Australian Research Council under grant numbers DE240100150 and DE220100766, respectively. TB acknowledges funding from the Carl Zeiss Stiftung and support from the European Research Council under ERC-CoG grant CRAGSMAN-646955. We gratefully acknowledge the Gauss Centre for Supercomputing e.V. ( www.gauss-centre.eu ) for funding this project by providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre ( www.lrz.de ). Simulations were partially computed with High Performance Computing resources at New York University, Abu Dhabi.",
"pages": [
17
]
},
{
"title": "REFERENCES",
"content": "Yanny B., Lee Y. S., 2006, ApJ, 636, 804 Franchini M., Molero M., Randich S., 2024, A&A, 690, A334 Pedregosa F., et al., 2011, J Mach Learn Res, 12, 2825 Perktold J., et al., 2024, statsmodels/statsmodels: Release 0.14.2, doi:10.5281/zenodo.593847 Ratcliffe B. L., Ness M. K., Buck T., Johnston K. V., Sen B., Beraldo e Silva Rybizki J., Just A., Rix H.-W., 2017, A&A, 605, A59 Schönrich R., Binney J., 2009, MNRAS, 396, 203 Sharda P., Krumholz M. R., Wisnioski E., Forbes J. C., Federrath C., Acharyya A., 2021, MNRAS, 502, 5935 Taylor M. B., 2005, ASPC, 347, 29 Federrath C., 2024, arXiv e-prints, p. arXiv:2406.00342 Schaller M., Schaye J., Theuns T., 2019, MNRAS, 482, 2208 Walt S. v. d., Colbert S. C., Varoquaux G., 2011, Comput Sci Eng, 13, 22 Wang L., Dutton A. A., Stinson G. S., Macciò A. V., Penzo C., Kang X., Keller B. W., Wadsley J., 2015, MNRAS, 454, 83 Willett E., et al., 2023, MNRAS, 526, 2141 Williams M. J., Bureau M., Cappellari M., 2009, MNRAS, 400, 1665 Zaritsky D., Kennicutt Robert C. J., Huchra J. P., 1994, ApJ, 420, 87 Zhang C., Li Z., Hu Z., Krumholz M. R., 2024, arXiv e-prints, p. arXiv:2411.01518 van de Sande J., Fraser-McKelvie A., Fisher D. B., Martig M., Hayden M. R., the GECKOS Survey collaboration 2023, arXiv e-prints, p. arXiv:2306.00059",
"pages": [
17,
18
]
},
{
"title": "A. ADDITIONAL FIGURES",
"content": "Fig. A1 demonstrates the tight correlation of gas phase iron and oxygen abundance and how to approximate them linearly. This paper was built using the Open Journal of Astrophysics L A T E X template. The OJA is a journal which provides fast and easy peer review for new papers in the astro-ph section of the arXiv, making the reviewing process simpler for authors and referees alike. Learn more at http://astro.theoj.org .",
"pages": [
18
]
}
]
2024arXiv241201746M
https://arxiv.org/pdf/2412.01746.pdf
<document>
<section_header_level_1><location><page_1><loc_7><loc_88><loc_84><loc_90></location>Strong gravitational lensing with upcoming wide-field radio surveys</section_header_level_1>
<text><location><page_1><loc_8><loc_84><loc_37><loc_86></location>Samuel McCarty 1 , 2 , ★ Liam Connor 3</text>
<text><location><page_1><loc_7><loc_83><loc_53><loc_84></location>1 Department of Astronomy, University of Washington, Seattle, WA 98195-1580, USA</text>
<text><location><page_1><loc_7><loc_82><loc_73><loc_83></location>2 Cahill Center for Astronomy and Astrophysics, MC 249-17, California Institute of Technology, Pasadena CA 91125, USA</text>
<text><location><page_1><loc_7><loc_81><loc_54><loc_82></location>3 Center for Astrophysics | Harvard & Smithsonian, Cambridge, MA 02138-1516, USA</text>
<text><location><page_1><loc_7><loc_77><loc_36><loc_77></location>Accepted XXX. Received YYY; in original form ZZZ</text>
<section_header_level_1><location><page_1><loc_7><loc_72><loc_15><loc_73></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_7><loc_55><loc_92><loc_72></location>The number of strong lensing systems will soon increase by orders of magnitude thanks to sensitive, wide-field optical and infrared imaging surveys such as Euclid, Rubin-LSST, and Roman. A dramatic increase in strong lenses will also occur at radio wavelengths. The 2000-antenna Deep Synoptic Array (DSA-2000) will detect over 10 9 continuum sources in the Northern Hemisphere with a high mean redshift ( ⟨ 𝑧 𝑠 ⟩ ≈ 2) and the Square Kilometer Array (SKA) will observe a large sample of extragalactic sources in the South with sub-arcsecond resolution. We forecast lensing rates, finding that the DSA-2000 will discover O( 10 5 ) strongly lensed systems, many of which will be galaxy group and cluster lenses. We propose strategies for strong lensing discovery in the limit where the Einstein radii are comparable to the PSF angular scale, taking advantage of modern computer vision techniques and multi-survey data. We also forecast synergies with optical and infrared surveys, which will provide redshifts as well as multiwavelength information about the lens systems. Finally, we describe applications of radio strong lensing systems, including time-delay cosmography with transient and variable sources. We find that ∼ 100 time-variable flat-spectrum AGN discovered by the DSA-2000 could be used to constrain 𝐻 0 at the percent level with the appropriate follow-up.</text>
<text><location><page_1><loc_7><loc_53><loc_53><loc_54></location>Key words: gravitational lensing: strong - radio continuum: general</text>
<section_header_level_1><location><page_1><loc_7><loc_46><loc_21><loc_47></location>1 INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_7><loc_29><loc_48><loc_45></location>Strong gravitational lensing has a multitude of applications in astrophysics and cosmology (Treu 2010). Previously theoretical ideas have been put into practice in recent decades as the number of known lensed systems has increased and observational data have improved. For example, strongly lensed time-variable and transient sources can be used to constrain the Hubble constant, 𝐻 0 , because the time delay of a multiply imaged source depends on the geometry of the Universe (Refsdal 1964). The technique is known as time-delay cosmography. With just six lensed quasar systems, the The 𝐻 0 Lenses in COSMOGRAIL's Wellspring (H0LiCOW) collaboration has reported 2.4 % precision on their 𝐻 0 measurement, which is independent of the distance ladder and the CMB (Wong et al. 2019).</text>
<text><location><page_1><loc_7><loc_15><loc_48><loc_28></location>In addition to the Universe's large-scale geometry, lensing observables are sensitive to the total mass of the deflector galaxy or cluster, allowing one to measure the spatial distribution of matter and test different dark matter models (Massey et al. 2010; Vegetti et al. 2024). Lensing magnification allows astronomers to observe objects in the distant Universe, as was pointed out at the field's inception (Zwicky 1937). Dramatic examples have come from the James Webb Space Telescope (JWST), including a red supergiant star at 𝑧 ≈ 2 . 2 that appears to be magnified by a factor of several thousand due to its proximity to caustics in a cluster lens (Diego et al. 2023).</text>
<text><location><page_1><loc_7><loc_11><loc_48><loc_14></location>Nearly all of these applications would benefit from a larger sample of strong lensing systems. To date, roughly 10 3 strong lensing systems have been discovered, most of which were identified at optical</text>
<text><location><page_1><loc_51><loc_32><loc_92><loc_47></location>and infrared wavelengths (O/IR). Fortunately, upcoming wide-field imaging surveys such as Euclid and The Vera C. Rubin Observatory's Legacy Survey of Space and Time (Rubin-LSST) are each expected to detect as many as ∼ 10 5 strong lenses (Collett 2015). An early release of a 0.7 deg 2 field from Euclid has recently affirmed these forecasts (Barroso et al. 2024). The Nancy Grace Roman Space Telescope's 2000 square degree survey could find of order 20,000 strong lenses (Weiner et al. 2020), and many more if the proposed multi-epoch 4 𝜋 sr survey is carried out (Han et al. 2023). An increase in the total number of lenses by two-orders of magnitude will usher in a new era of strong lensing science.</text>
<text><location><page_1><loc_51><loc_19><loc_92><loc_32></location>The first strongly lensed system ever discovered was co-detected at radio wavelengths (Walsh et al. 1979). The first survey for lenses, the Mit-Green Bank survey, was also in the radio (Bennett et al. 1986). Despite this, fewer than ∼ 100 radio lensing systems have been detected to date. This is due to the relatively small total number of known radio sources ( ∼ 10 7 ) and the lack of wide-field imaging surveys with ∼ arcsecond resolution. Both of those limitations will soon be overcome with the advent of next-generation radio survey telescopes.</text>
<text><location><page_1><loc_51><loc_6><loc_92><loc_18></location>The 2000-antenna Deep Synoptic Array (DSA-2000) will detect over one billion radio sources with a deep redshift distribution, most of which will be star-forming radio galaxies (SFRG) or active galactic nuclei (AGN) (Hallinan et al. 2019). The DSA-2000 is expected to see first light in 2027 with key surveys running between 2028 and 2033. Its point-spread function (PSF) will be roughly 2' at the top of the 0.7-2 GHz radio band. A fifty-fold increase in the total radio source catalog is made possible by the DSA-2000's high survey speed, driven by a large field-of-view ( ∼ 10 deg 2 ) and high sensitivity (the expected</text>
<text><location><page_2><loc_7><loc_85><loc_48><loc_93></location>system-equivalent flux density or 'SEFD' is just 2.5 Jy). Its cadenced all-sky survey will map out the 3 𝜋 sr above declination -30 · down to 500 nJy/beam root-mean square noise (Hallinan et al. 2019). While the nominal cadence of the continuum survey is four months, certain fields may be visited more regularly, presenting an opportunity to measure lensing time delays with better temporal sampling.</text>
<text><location><page_2><loc_7><loc_64><loc_48><loc_84></location>The mid-frequency telescope for the square kilometre array (SKAMid) will consist of 197 fully steerable 13.5 m dishes (including the existing MeerKAT radio telescope), operating between 350 MHz and 15.4 GHz with a field-of-view of roughly 1 deg 2 at 1400 MHz. The end of its contruction is anticipated to be July 2029 1 . Although the SKA's lower survey speed will result in fewer sources than the DSA-2000, the wide frequency range and long maximum baseline (150 km) will enable high-resolution imaging, an asset to strong lensing studies (McKean et al. 2015). The Next Generation Very Large Array (ngVLA) is a planned interferometer with extraordinary sensitivity covering a wide range of frequencies (1.2-116 GHz) (Selina et al. 2018a). It will be able to resolve features at milliarcseconds scales. While its broad science goals did not require optimizing the instrument for mapping speed (Selina et al. 2018b), the ngVLA will be a world-class instrument for strong lensing science.</text>
<text><location><page_2><loc_7><loc_33><loc_48><loc_63></location>Radio strong lensing offers distinct advantages to studies at shorter wavelengths and will complement O/IR imaging surveys. Starforming radio galaxies and AGN can be detected to great distances, boosting the mean optical depth of radio continuum sources. Obscuration by dust in lensing galaxies is not an issue at radio wavelengths, nor is the variable 'seeing' that impacts ground-based O/IR telescopes. Relatedly, the point-spread function (PSF) of a radio interferometer is directly determined by observing frequency and array configuration, allowing us to accurately forward model the instrument's response. In the limit of a large number of antennas and a filled aperture (a 'radio camera'), the deterministic PSF allows us to be more ambitious in image-plane deconvolution, enabling techniques such as superresolution (Connor et al. 2022). Radio telescopes also measure full polarization information, which is conserved under gravitational lensing (Greenfield et al. 1985; Dyer & Shaver 1992). The rotation measure (RM) of a lensed source allows one to measure the magnetic field and ionized gas properties of intervening halos (Mao et al. 2017). Finally, the larger emission regions of radio AGN render them less susceptible to microlensing by substructure in the lens, and may provide cleaner modelling of the deflector mass distribution (Birrer et al. 2024). This is critical for measuring 𝐻 0 via time-delay cosmography.</text>
<text><location><page_2><loc_7><loc_18><loc_48><loc_33></location>There are of course several drawbacks to strong lensing studies in radio surveys, often precluding the use of standalone radio observations. For example, continuum sources will not contain redshift information, and multi-wavelength datasets or follow-up will be necessary for the majority of lensed systems at radio frequencies. Secondly, the morphology of radio lobes can mimic lensing arcs, exacerbating the already-challenging lens identification problem (consider, for example, head-tail galaxies 3C 465 or 3C 129 at redshift 2). In the case of the DSA-2000, the angular resolution is such that 75-90% of galaxy-galaxy lensing systems will be unresolved in the absence of superresolution.</text>
<text><location><page_2><loc_7><loc_11><loc_48><loc_17></location>In this work we seek to study radio strong lensing in upcoming interferometric surveys, and develop strategies that alleviate the drawbacks of radio lensing. We focus on the DSA-2000, first by forecasting its strong lensing rates for different source classes. We compare this with a forecast for strong lensing discovery on the SKA. Next, due to</text>
<figure>
<location><page_2><loc_52><loc_62><loc_87><loc_93></location>
<caption>Figure 1. The theoretical distribution of strongly lensed image separations (2 𝜃 𝐸 ) from our model for galaxies (solid), galaxy groups (dashed), and galaxy clusters (dotted), assuming a DSA-2000 source redshift distribution. The portion that will be discoverable by the DSA-2000 is shown in the blue-shaded region, while the portion that could be discoverable with superresolution is shown in the green-shaded region. The galaxy group distribution is flat at the top because the density profile is constant over the entire mass range in our model.</caption>
</figure>
<text><location><page_2><loc_51><loc_39><loc_92><loc_47></location>the 2-3' PSF of the DSA-2000, we discuss methods for superresolving lensing candidates with modern computer vision techniques in order to increase the yield of strongly lensed systems. We then consider lensed time-variable and transient sources that the DSA-2000 will find and forecast constraints on 𝐻 0 that can be achieved with the appropriate follow-up, as well as other applications of strong lensing.</text>
<section_header_level_1><location><page_2><loc_51><loc_34><loc_92><loc_35></location>2 OPTICAL DEPTH CALCULATION & EXPECTED RATES</section_header_level_1>
<text><location><page_2><loc_51><loc_28><loc_92><loc_33></location>First, we offer a simple empirical forecast based on observational data and cosmological simulations, and then offer a more detailed forward model that accounts for different source classes and the deflector population.</text>
<section_header_level_1><location><page_2><loc_51><loc_24><loc_66><loc_25></location>2.1 Empirical forecast</section_header_level_1>
<text><location><page_2><loc_51><loc_6><loc_92><loc_23></location>The Cosmic Lens All Sky Survey (CLASS) (Myers et al. 2003) sought to study a statistical sample of radio-loud gravitationally lensed systems. Despite significant advances in both radio instruments and lens-finding algorithms in the two decades since CLASS, the number of known radio lenses has not increased dramatically. We can use CLASS for a rough estimate of the number of lenses the DSA-2000 will find. CLASS found that a complete sample of 8,958 flat-spectrum radio point sources brighter than 30 mJy had a mean lensing optical depth of 1 . 5 + 0 . 5 -0 . 3 × 10 -3 (Browne et al. 2003). By targeting compact sources with flat spectra, many of their objects are quasars at high redshifts, comparable to typical redshifts of DSA-discovered sources, despite the difference in flux scales. Of the</text>
<text><location><page_3><loc_7><loc_82><loc_48><loc_93></location>confirmed radio lenses, approximately 20% had angular separations larger than 2 arcseconds, which could be detected by the DSA-2000. Thus, a rough estimate indicates that for every 10,000 sources detected by the DSA-2000, several could be identified as strong lenses. If we assume a detection threshold of 10 𝜎 , the number of extragalactic DSA-2000 sources becomes roughly 5 × 10 8 (Section 2.3). This would result in O( 10 5 ) new galaxy-scale radio lenses, depending on the practical signal-to-noise threshold for candidate systems.</text>
<text><location><page_3><loc_7><loc_62><loc_48><loc_81></location>Galaxy group and cluster scale lenses have been neglected in previous investigations of lensing statistics for upcoming surveys because they make up a smaller, but still significant, fraction of the total lenses. Additionally, lens modeling is more complicated in this regime. The dark matter halos of groups, in particular, are complex and not well-understood structures. From Oguri (2006), the number of group and cluster lenses are around 11% and 3% of the total galaxy lenses respectively (excluding sub-halo lensing). However, these systems make up a considerable portion of lens systems with angular separation of order 1', and almost all of the lenses at ≥ 10'. Because the PSF of the DSA-2000 is relatively large (2' at 2 GHz and 3.3' at 1.4 GHz), these lenses will make up a large fraction of the discoverable lensing systems and so must be accounted for in our investigation.</text>
<text><location><page_3><loc_7><loc_26><loc_48><loc_61></location>A radio group/cluster lens survey as complete as CLASS does not exist, but we can make a simple empirical estimate from the literature. The total number of galaxy scale lenses should be O( 10 6 ) assuming 10 9 sources and the CLASS lensing optical depth. Taking the percentages of group/cluster lenses from Oguri (2006), we should see O( 10 5 ) groupscale lenses and O( 10 4 ) cluster scale lenses, almost all of which will have angular separation large enough to be detected in the DSA-2000. The first and largest survey for group-scale lenses was the Strong Lensing Legacy Survey (SL2S) (Cabanac, R. A. et al. 2007). SL2S found 13 strong lens systems in the mass range of groups in ∼ 100 deg 2 of the Canada France Hawaii Telescope Legacy Survey (CFHTLS), giving a rate of ∼ 0.13 deg -2 (Limousin, M. et al. 2009). A later publication identified at least 54 promising group lenses in ∼ 150 deg 2 of the CFHTLS (More et al. 2012). However, the mean redshift of the CFHTLS sources used in SL2S is ⟨ 𝑧 𝑠 ⟩ < 1, and we expect ⟨ 𝑧 𝑠 ⟩ ≈ 2 for the DSA-2000 (see section 2.3, figure 3 below) (Coupon, J. et al. 2009). At these redshifts, we expect the lensing optical depth to be roughly ∝ 𝑧 2 (see equation 31 of Oguri (2019); figure 2 below). This means that we would expect to see ∼ 10 + 6 -6 times more group lenses in the DSA-2000, for a total of ∼ 1 . 1 + 0 . 6 -0 . 9 × 10 5 . Similarly, the Red Sequence Cluster Survey found 8 cluster lenses in ∼ 90 deg 2 with shallower CFHT imaging, indicating ∼ 2 . 6 + 1 . 6 -1 . 6 × 10 4 cluster lenses in the DSA-2000 (Gladders et al. 2003). These estimates are of the same order of magnitude as expected from Oguri (2006).</text>
<text><location><page_3><loc_7><loc_8><loc_48><loc_26></location>Alternatively, if we take the cross section for giant arcs from the simulations of Puchwein & Hilbert (2009), Fedeli, C. et al. (2010), and Mahdi et al. (2014), assume typical values of 𝑀 = 1 × 10 14 M ⊙ , 𝑧 𝑑 = 0 . 5, and 𝑧 𝑠 = 2, we can calculate a rough estimate of the cluster lensing optical depth at 𝑧 𝑠 = 2. The cross sections at these values are all on the order of 10 arcsec 2 . From Böhringer, Hans et al. (2017), the number density at 𝑀 = 1 × 10 14 M ⊙ is approximately ≳ 1 × 10 -6 Mpc -3 , and after multiplying by the differential coming volumeat 𝑧 𝑠 = 2andskycoverageoftheDSA-2000weget ≈ 5 × 10 -5 for all three simulations. When taking into account that this is the cross-section for giant arcs only, this means we should see several tens of thousands of cluster lenses in the DSA-2000, in good agreement with the previous estimates.</text>
<text><location><page_3><loc_9><loc_6><loc_48><loc_7></location>Wealso make an empirical estimate of the rate of lensed transients</text>
<text><location><page_3><loc_51><loc_61><loc_92><loc_93></location>in the DSA-2000. These objects are of particular interest for 𝐻 0 measurements but are much harder to forward model because of the lack of observational data. We can use data from the Very Large Array Sky Survey (VLASS), which is currently the most comparable survey to the DSA-2000, for a simple estimate (Lacy et al. 2020). The determined log rates of supernovae (SNe) and tidal disruption events (TDEs) in deg -2 yr -1 are -1.91 + 0 . 15 -0 . 16 and -2.85 + 0 . 28 -0 . 38 respectively (Dong et al. 2024 in prep). If we assume that the total number of observable sources scales as 𝑆 -1 . 5 min , the value for a Euclidean universe, and that the flux limit of VLASS is 0.7 mJy (10 𝜎 from Lacy et al. (2020)), the DSA-2000 should find ∼ 200 more sources of each type above 20 𝜇 Jy, which is a 10 𝜎 detection in a single epoch of the DSA-2000. We can also assume a lensing optical depth of 1 × 10 -4 and a magnification bias of 2, which are typical for the low redshifts at which we expect to detect transients. Because typical radio rise times of TDEs are on the order of 10 3 days the cadence of the DSA-2000, ∼ 1 / 3 yr, is fast enough to catch most TDEs (Cendes et al. 2023). Scaling for the sky coverage of the DSA-2000, and the cadence for SNe, gives a total yield of 5 + 2 -1 yr -1 and 0.6 + 0 . 5 -0 . 3 yr -1 lensed sources per year for SNe and TDEs respectively. Applying a similar estimate of the lensed rate of GRB afterglows using the predictions of Ghirlanda et al. (2013, 2014) gives a rate much less than 1 per year, and so we ignore them in the rest of this investigation.</text>
<text><location><page_3><loc_51><loc_34><loc_92><loc_61></location>Further, Yao et al. (2023) find a volumetric rate of optically selected TDEs of 290 + 60 -130 Gpc -3 yr -1 . Cendes et al. (2023) find that ≈ 50% of optically selected TDEs emit in the radio on longer timescales. Becausethere are likely TDEs that emit in radio but not the optical, we take 50% of the optically selected rate to be a conservative estimate. The luminosities of the TDEs in the sample from Cendes et al. (2023) are ∼ 10 37 -10 39 ergs/s, but they note that many of these are likely a lower limit because most of the TDEs still had rising emission at the time of observation. Given this, the DSA-2000 should be able to detect this rate of TDEs out to 𝑧 ≈ 0 . 5, which, when combined with the same optical depth and magnification bias as above, gives 0 . 8 + 0 . 2 -0 . 4 yr -1 lensed TDEs. This number is in good agreement with the previous estimate, so in general, we expect to see about 1 lensed TDE per year in the DSA-2000. However, as noted before, this is a conservative estimate, and there is reason to believe that the actual rate of total and lensed TDEs in the DSA-2000 might be significantly higher. Additionally, because many of the TDEs will be bright in multiple epochs, the DSA-2000 could detect them well below the 20 𝜇 Jy limit, which would increase the rate by a large factor.</text>
<text><location><page_3><loc_51><loc_15><loc_92><loc_34></location>The DSA-2000 will also discover tens of thousands of distant fast radio bursts (FRBs) (Petroff et al. 2019; Cordes & Chatterjee 2019), some of which will be strongly lensed (Connor & Ravi 2023). The key advantage to using FRBs for time-delay lensing is that their short duration and coherence allows for exceptional precision on the gravitational lensing time delay (Wucknitz et al. 2021). Radio telescopes can preserve phase information about the electromagnetic waveform at nanosecond sampling, which means microlensing signals can be searched for at ultrashort timescales (Leung et al. 2022; Kader et al. 2022). However, for cosmological lensing time delays longer than a pointing (i.e. deflectors more massive than ∼ 10 8 M ⊙ ), one needs to catch the lensed images by pointing at the same patch of sky when it arrives. We do not forecast lensed FRB rates here and point the reader to previous estimates (Connor & Ravi 2023).</text>
<section_header_level_1><location><page_3><loc_51><loc_10><loc_61><loc_11></location>2.2 Lens Model</section_header_level_1>
<text><location><page_3><loc_51><loc_6><loc_92><loc_9></location>Next, we build a forward model based on the lens and source distributions. Following Yue et al. (2022b), the lensing optical depth for a</text>
<table>
<location><page_4><loc_21><loc_82><loc_77><loc_93></location>
<caption>Table 1. Total sources above 10 𝜎 𝑛 and expected number of discoverable lensing events by deflector type in the DSA-2000, organized by source classes. Blazars are both FSRQs and BLLacs and transients here include ccSNe and TDEs.</caption>
</table>
<text><location><page_4><loc_7><loc_75><loc_28><loc_76></location>singular isothermal sphere (SIS) is</text>
<formula><location><page_4><loc_7><loc_71><loc_48><loc_74></location>𝜏 ( 𝑧 ) = ∫ 𝑧 𝑠 0 𝑑𝑧 𝑑 ∫ 𝑑𝜎 Φ ( 𝜎, 𝑧 𝑑 ) 𝑑 2 𝑉 𝑐 𝑑 Ω 𝑑𝑧 𝑑 𝜋𝜃 𝐸 ( 𝜎, 𝑧 𝑑 , 𝑧 𝑠 ) 2 𝐷 2 𝑑 , (1)</formula>
<text><location><page_4><loc_7><loc_53><loc_48><loc_70></location>where 𝑧 𝑑 is the redshift of the deflector, 𝑧 𝑠 is the redshift of the source, 𝜎 is the 1D velocity dispersion of the deflector, Φ ( 𝜎, 𝑧 𝑑 ) is the velocity dispersion function (VDF) of the deflectors, 𝑑 2 𝑉 𝑐 𝑑 Ω 𝑑𝑧 𝑑 = ( 1 + 𝑧 𝑑 ) 3 𝑐 𝑑𝑡 𝑑𝑧 𝑑 is the differential comoving volume, 𝜃 𝐸 ( 𝜎, 𝑧 𝑑 , 𝑧 𝑠 ) is the Einstein radius, and 𝐷 𝑑 is the angular diameter distance at the deflector redshift. The SIS model (or its elliptical generalization) has been shown to replicate the properties of early-type galaxies, which make up the majority of galaxy lenses (e.g. Gavazzi et al. (2007); Koopmans et al. (2009); Li et al. (2018)), and is a widely used model for galaxy strong lensing statistics (e.g. Oguri & Marshall (2010); Collett (2015); Yue et al. (2022a). For a SIS, the Eintein radius becomes</text>
<formula><location><page_4><loc_7><loc_48><loc_48><loc_50></location>𝜃 𝐸 = 4 𝜋 GLYPH<16> 𝜎 𝑐 GLYPH<17> 2 𝐷 𝑑𝑠 𝐷 𝑠 , (2)</formula>
<text><location><page_4><loc_7><loc_37><loc_48><loc_47></location>where 𝐷 𝑑𝑠 is the angular diameter distance from the deflector to the source and 𝐷 𝑠 is the angular diameter distance from the observer to the source. We use the analytical VDF from Yue et al. (2022b) to model the galaxy deflector population, which they show to be in good agreement with observations. We include a sharp exponential cutoff on the VDF at 300 kms -1 which corresponds roughly to the transition between galaxies and galaxy groups.</text>
<text><location><page_4><loc_7><loc_32><loc_48><loc_37></location>An important ingredient in the calculation is the magnification bias, which increases the rate of lensed sources by about a factor of two at low redshifts, or by orders of magnitude at high redshifts. The general magnification bias is</text>
<formula><location><page_4><loc_7><loc_26><loc_48><loc_29></location>𝐵 = ∫ +∞ 𝜇 min 𝑑𝜇𝑝 ( 𝜇 ) 𝑁 ( > 𝐿 min / 𝜇 ) 𝑁 ( > 𝐿 min ) , (3)</formula>
<text><location><page_4><loc_7><loc_11><loc_48><loc_25></location>where 𝜇 is the magnification of a lensed source, 𝑝 ( 𝜇 ) is the probability distribution of the magnification, 𝐿 min is the smallest observable luminosity, and 𝑁 ( > 𝐿 min ) is the number of sources brighter than 𝐿 min . For SIS, 𝜇 min = 2 for the total magnification of the multiple images. A SIS will produce only two multiple images; if the separation of the multiple lens images is large enough to be resolved then 𝑝 ( 𝜇 ) = 2 ( 𝜇 ± 1 ) 3 describes the magnification of the fainter (+) and brighter (-) image (Wyithe et al. 2001). If the lens is not resolved then 𝑝 ( 𝜇 ) = 8 𝜇 3 is the total magnification of both images. Combining 𝐵 and 𝜏 for the total fraction of lensed sources:</text>
<formula><location><page_4><loc_7><loc_6><loc_48><loc_8></location>𝐹 lensed = 𝐵𝜏 𝐵𝜏 + 𝐵 ' ( 1 -𝜏 ) , (4)</formula>
<text><location><page_4><loc_51><loc_73><loc_92><loc_76></location>where 𝐵 ' is the magnification bias of sources that are not lensed, which is assumed to be unity.</text>
<text><location><page_4><loc_51><loc_44><loc_92><loc_73></location>While the SIS model describes deflector galaxies well, the mass distributions of galaxy groups and clusters behave differently. It has been traditionally thought that in the limit of large mass and large image separation, i.e. clusters, the distribution will be dominated by the dark matter halo, generally following the Navarro-Frenk-White (NFW) profile (Navarro et al. 1997). Recent works suggest that NFW profiles may not be the most accurate representation of large dark matter halos (e.g. Klypin et al. (2016)), but we use them here because they are a decent approximation and their lensing properties are well known. Regardless, for the mass range of groups, it is clear that some intermediate model between SIS and NFW is necessary (Williams et al. 1999; Oguri 2006; More et al. 2012). In a SIS the density 𝜌 ∝ 𝑟 -2 , while the NFW has 𝜌 ∝ 𝑟 -1 on small scales and 𝜌 ∝ 𝑟 -3 on larger scales. The shallower NFW profile has a significantly smaller cross-section than SIS. Oguri (2006) include the effects of baryon cooling and the large elliptical galaxies found at the center of most halos in their model, which will steepen the central density profile and increase the cross-section of halos (especially for groups). Without introducing these complexities, we can steepen the inner profile of halos by modeling them as Generalized NFW (GNFW) profiles. The GNFW is:</text>
<formula><location><page_4><loc_51><loc_39><loc_92><loc_42></location>𝜌 ( 𝑟 ) = 𝜌 𝑠 𝑟 3 𝑠 𝑟 𝛼 ( 𝑟 + 𝑟 𝑠 ) 3 -𝛼 , (5)</formula>
<text><location><page_4><loc_51><loc_6><loc_92><loc_38></location>where 𝑟 𝑠 is the scale radius and 𝜌 𝑠 is the scale density (Li & Ostriker 2002). When 𝛼 = 1 we have the standard NFW. When 𝛼 = 2 the profile resembles an SIS below the scale radius and when 1 < 𝛼 < 2 the inner profile is somewhere in between. Using weak lensing and stellar kinematics, Wang et al. (2023) find the inner dark matter density profiles of group (10 13 𝑀 ⊙ ≤ 𝑀 < 10 14 𝑀 ⊙ ) and cluster ( 𝑀 ≥ 10 14 𝑀 ⊙ ) halos are 1 . 82 + 0 . 15 -0 . 25 and 1 . 48 + 0 . 2 -0 . 41 respectively. Similarly, Mandelbaum et al. (2006) find that the total inner profiles of groups with mass ∼ 2 . 5 × 10 13 𝑀 ⊙ are decently described by a power-law of slope 𝛾 tot = 1 . 85. However, at ∼ 7 × 10 13 𝑀 ⊙ they are less steep than 1.85 and closer to the traditional NFW. Newman et al. (2013) find 𝛾 tot = 1 . 16 + 0 . 10 -0 . 12 for the inner profile of clusters with 𝑀 = 0 . 4 -2 × 10 15 𝑀 ⊙ , and Newman et al. (2015) find the total inner profile slope 𝛾 tot ≈ 1 . 7 while the dark matter slope 𝛼 ≈ 1 . 35 for galaxy groups with ⟨ 𝑀 ⟩ ≈ 10 14 𝑀 ⊙ . In this work, we use the GNFW profile with 𝛼 = 1 . 6 to model groups and 𝛼 = 1 . 2 to model clusters. The resulting GNFW profiles will capture the inner profile well, which is crucial for lensing, and also converge to the standard NFW profile at large radii where we expect the density to be dominated by dark matter (e.g. figures 3 and 4 of Wang et al. (2023)). The GNFW is completely parameterized by 𝛼 , its mass 𝑀 , and the concentration parameter, 𝑐 , from which 𝑟 𝑠 and 𝜌 𝑠 can be determined as in Li & Ostriker (2002). The mass and 𝑐 are correlated with some scatter,</text>
<text><location><page_5><loc_7><loc_82><loc_48><loc_93></location>we use the relationship presented by Dutton & Macciò (2014) for 𝑐 200 . The scatter in 𝑐 is lognormal with 𝜎 log 𝑐 = 0 . 11. We include an additional factor of ( 2 -𝛼 ) as in Oguri et al. (2001) to correct for the generalized form of the NFW which ensures that the radius at which the logarithmic density slope becomes -2 is the same as in the standard NFW for all 𝛼 . The lensing power of the GNFW halo is very sensitive to 𝑐 and its scatter, so having an accurate relationship is important; early investigations tend to overestimate these values.</text>
<text><location><page_5><loc_7><loc_79><loc_48><loc_82></location>The lens equation relates a position on the lens plane, 𝑥 , to a position on the source plane, 𝑦 , which for a GNFW profile is</text>
<formula><location><page_5><loc_7><loc_75><loc_48><loc_77></location>𝑦 = 𝑥 -𝜇 𝑠 𝑔 ( 𝑥, 𝛼 ) 𝑥 , (6)</formula>
<text><location><page_5><loc_7><loc_73><loc_11><loc_74></location>where</text>
<formula><location><page_5><loc_7><loc_68><loc_48><loc_71></location>𝜇 𝑠 = 4 𝑝 𝑠 𝑟 𝑠 Σ 𝑐𝑟 , Σ 𝑐𝑟 = 𝑐 2 4 𝜋𝐺 𝐷 𝑠 𝐷 𝑑 𝐷 𝑑𝑠 , (7)</formula>
<text><location><page_5><loc_7><loc_59><loc_48><loc_67></location>and 𝑔 ( 𝑥, 𝛼 ) is the same as in Li & Ostriker (2002). The lens equation has three solutions inside the radial caustic, 𝑦 𝑐𝑟 , which are the multiple images produced by the GNFW profile. The cross-section for lensing is the area in the source plane in which multiple images will be produced, i.e. the area inside the radial caustic. This is approximated as:</text>
<formula><location><page_5><loc_7><loc_56><loc_48><loc_57></location>𝜎 ( 𝑀, 𝑧 ) ≈ 𝜋𝑦 2 𝑐𝑟 𝑟 2 𝑠 (8)</formula>
<text><location><page_5><loc_7><loc_49><loc_48><loc_55></location>where 𝑦 𝑐𝑟 = -𝑦 ( 𝑥 𝑐𝑟 ) and 𝑥 𝑐𝑟 is the location of the minimum of equation 6 (Li & Ostriker 2002). The image separation is given by the separation between the outer two images, which can be approximated as</text>
<formula><location><page_5><loc_7><loc_45><loc_48><loc_47></location>Δ 𝜃 ≈ 2 𝑥 0 𝑟 𝑠 𝐷 𝑑 (9)</formula>
<text><location><page_5><loc_7><loc_40><loc_48><loc_44></location>where 𝑥 0 is the position of the tangential critical curve and can be found as the root of equation 6 (Li & Ostriker 2002). The Einstein radius is just half of the image separation, 𝜃 𝐸 = Δ 𝜃 / 2.</text>
<text><location><page_5><loc_7><loc_29><loc_48><loc_40></location>In general, because our GNFW profiles are shallower than a SIS, they will be worse at producing multiple images but significantly better at magnifying. This is particularly sensitive to 𝑐 , which defines the shallowness of the GNFW profile. For our concentration parameters, especially at high redshift, we expect the magnifications from the GNFWs to be several times larger than from SIS (Wyithe et al. 2001). To calculate the magnification bias for a GNFW profile, we first determine the minimum magnification as (Li & Ostriker 2002):</text>
<formula><location><page_5><loc_7><loc_25><loc_48><loc_27></location>𝜇 min ≈ 2 𝑥 0 𝑦 𝑐𝑟 . (10)</formula>
<text><location><page_5><loc_7><loc_22><loc_48><loc_24></location>With 𝜇 min wecalculate B according to equation 3 with (Li & Ostriker 2002):</text>
<formula><location><page_5><loc_7><loc_17><loc_48><loc_20></location>𝑝 nfw ( 𝜇 ) = 2 𝜇 2 min 𝜇 3 . (11)</formula>
<text><location><page_5><loc_7><loc_10><loc_48><loc_16></location>The minimum magnification for both the first and second brightest image can be approximated as half of the total minimum magnification (Oguri et al. 2002). Substituting 𝜇 min = 2 into equation 11 recovers the total magnification distribution for SIS.</text>
<text><location><page_5><loc_7><loc_6><loc_48><loc_10></location>To model the populations of galaxy groups and clusters we determine Mass Functions (MF) from the CosmoDC2 synthetic catalogue Korytov et al. (2019). For simplicity, we define a galaxy group as a</text>
<text><location><page_5><loc_51><loc_81><loc_92><loc_93></location>system with mass 10 13 𝑀 ⊙ ≤ 𝑀 < 10 14 𝑀 ⊙ and a galaxy cluster as 𝑀 ≥ 10 14 𝑀 ⊙ . We find all dark matter halos in a redshift interval where the mass of the halo and its constituent galaxies fall in these ranges. We fit Schechter functions to the binned results. The fitted functions match the results of Böhringer, Hans et al. (2017) well at 𝑧 = 0 and show a realistic decline in the mass function towards higher redshift. To calculate the optical depth, we replace 𝜋𝜃 𝐸 ( 𝜎, 𝑧 𝑑 , 𝑧 𝑠 ) 2 𝐷 2 𝑑 in equation 1 with equation 8, Φ ( 𝜎, 𝑧 𝑑 ) with the CosmoDC2 MF, and integrate over 𝑀 instead of 𝜎 .</text>
<text><location><page_5><loc_51><loc_54><loc_92><loc_80></location>While equation 4 gives the total fraction of lensed sightlines, we are more interested in the number of discoverable lenses. We define a lens as discoverable if its image separation Δ 𝜃 = 2 𝜃 𝐸 is larger than 2/3rds of the FWHM of the PSF of the survey instrument (as in Oguri & Marshall (2010); Yue et al. (2022a)) so that multiple images can be resolved. Further, for point-like sources (i.e. flat-spectrum sources and transients), we require the second faintest image to be detectable at 10 𝜎 confidence so that the lens can be identified. To account for this we integrate equation 1 from 𝜎 min (or equivalently 𝑀 min for groups and clusters) corresponding to 𝜃 𝐸 min and use the magnification bias of the fainter image. The second image from a SIS will often be demagnified, so the magnification bias of the fainter image is typically a bit less than 1, while the magnification bias of the fainter image from an NFW will be about half of the total. For SFRGs and SS-AGN, which in general have extended emission regions, the lensed images will be stretched into arcs and rings, and so for this case we require the brightest image to be detectable at 10 𝜎 confidence (using the magnification bias of the brightest image) because only one arc is needed to identify a lens.</text>
<text><location><page_5><loc_51><loc_23><loc_92><loc_53></location>The image separation distribution predicted by our model and the lensing optical depths for galaxies, groups, and clusters are shown in Figures 1 and 2 respectively. We see that the optical depth increases rapidly at small redshift as expected, and approximately matches the estimates given in Section 2.1. The galaxy lens optical depth is smaller than the approximation from Oguri (2019). The discoverable optical depth, limited by the PSF of the DSA-2000, is significantly smaller than the total optical depth for galaxies but similar to the total for groups and clusters. The image separation distribution matches the results of Oguri (2006) well despite the simpler model. As expected, galaxies lie mostly at Δ 𝜃 ≈ 1', while groups and clusters correspond to Δ 𝜃 ≈ 10' and 10' ≲ Δ 𝜃 ≲ 100' respectively. The largest uncertainty is in the model of groups, as our simple model does not capture many of the complexities of halo profiles and is constant over the whole mass range. Further, varying 𝛼 between 1 and 2 creates a significant change in the calculated group optical depth. On the other hand, there is inherent uncertainty in all three models from the assumption of spherical symmetry. Using SIS instead of Singular Isothermal Ellipsoids (SIE) for galaxies will overestimate the galaxy lens population by ∼ 10%, which is corrected for in our final numbers (Yue et al. 2022b; Ferrami & Wyithe 2024). Introducing ellipticity will similarly reduce the cross-section of NFW profiles.</text>
<text><location><page_5><loc_51><loc_6><loc_92><loc_23></location>We do not account for effects such as sub-halo lensing or line-ofsight structures and lens environments. It is shown in Oguri (2006) that a significant fraction of lenses, roughly half for groups and clusters and 10% for galaxies, come from lensing by sub-halos. The environment around these sub-halos boosts their lens capabilities and image separations. At least one of the CLASS lenses is caused by a galaxy within a galaxy group (Auger et al. 2007). In the context of cluster lensing, this is termed galaxy-galaxy strong lensing (GGSL) and it is reported that observational numbers of GGSL exceed expectations of the Λ CDM model (Meneghetti, Massimo et al. 2022; Tokayer et al. 2024). This means that we are likely underestimating the number of lenses in the image separation range of galaxy scale</text>
<figure>
<location><page_6><loc_8><loc_71><loc_48><loc_93></location>
<caption>Figure 2. Lensing optical depth vs. redshift without magnification bias. Solid lines are the total optical depth while dashed lines are the optical depth for lenses discoverable by the DSA-2000.</caption>
</figure>
<text><location><page_6><loc_7><loc_50><loc_48><loc_61></location>lenses. However, most of these will be below the discoverable limit of the DSA-2000. Further, it is well known that intervening masses along the line of sight between the deflector and the observer could increase the probability of multiple imaging by a significant fraction, especially at high source redshifts, which we also neglect in this study (Fleury et al. 2021). Because these higher-order aspects of the model are likely to increase the number of lenses, for the purpose of this paper it is safe to ignore them.</text>
<section_header_level_1><location><page_6><loc_7><loc_46><loc_22><loc_47></location>2.3 Source Populations</section_header_level_1>
<text><location><page_6><loc_7><loc_27><loc_48><loc_45></location>The other important piece of the model is the population of the sources. The DSA-2000 will map ∼ 30,000 deg 2 of the sky to a combined 𝜎 𝑛 = 500 nJy/beam rms noise (Hallinan et al. 2019). Matthews et al. (2021) presents detailed radio source counts from the MeerKat DEEP2 image, including direct source counts above 10 𝜇 Jy and statistical counts extrapolating below 10 𝜇 Jy. They find that for sources <10 𝜇 Jy, the differential source count is significantly flatter than Euclidean. Scaling for the 30,000 deg 2 footprint of DSA2000's continuum survey and integrating down to a minimum flux density 𝑆 = 2 . 5 𝜇 Jy for a 5 𝜎 𝑛 combined detection gives an expected total source count of 1 . 45 + 0 . 25 -0 . 10 × 10 9 , while for 𝑆 ≥ 5 𝜇 Jy = 10 𝜎 𝑛 that number is 8 . 6 + 1 . 1 -0 . 4 × 10 8 .</text>
<text><location><page_6><loc_7><loc_13><loc_48><loc_28></location>To get redshift distributions for different source classes, we model the Star-Forming Radio Galaxy (SFRG) and Active Galactic Nucleus (AGN) populations using the luminosity functions (LF) from the Tiered Radio Extra-galactic Continuum Simulation (T-RECS) (Bonaldi et al. 2018). These have been shown to match observations out to high redshift. Following Bonaldi et al. (2018), we divide the AGN into three sub populations: Flat Spectrum Radio Quasars (FSRQ), BL-Laceratae (BLLac), and Steep Spectrum AGN (SSAGN). We expect SFRGs to make up roughly 95% of all of the sources in any synoptic radio survey, and BLLacs to be the rarest source.</text>
<text><location><page_6><loc_7><loc_10><loc_48><loc_12></location>From the LF we calculate the number of sources in a redshift interval as</text>
<formula><location><page_6><loc_7><loc_6><loc_48><loc_9></location>𝑛 ( 𝑧 ) 𝑑𝑧 = ∫ 𝐿 min ( 𝑧 ) Φ ( 𝐿, 𝑧 ) 𝑑 log 𝐿 × 𝑑 2 𝑉 𝑐 𝑑 Ω 𝑑𝑧 𝑑 × 𝑑𝑧 × Ω survey (12)</formula>
<text><location><page_6><loc_51><loc_80><loc_92><loc_93></location>where Ω survey is the sky coverage of the survey in steradians ( ≈ 3 𝜋 for the DSA-2000) and Φ ( 𝐿, 𝑧 ) is the LF of the source population. 𝐿 min ( 𝑧 ) = 4 𝜋𝐷 2 𝐿 ( 1 + 𝑧 ) 1 + 𝛼 𝑆 min is the minimum observable luminosity at a redshift 𝑧 , where 𝐷 𝐿 is the luminosity distance, 𝛼 is the spectral index of the source population, 𝑆 min is the 10 𝜎 𝑛 flux sensitivity of the DSA-2000, and the factor ( 1 + 𝑧 ) -( 1 + 𝛼 ) is the standard cosmological radio K-correction. We scaled the total number of sources in the LF model to match the observational prediction from the DEEP2 image given above.</text>
<text><location><page_6><loc_51><loc_34><loc_92><loc_80></location>In addition to SFRGs and AGN, we build a model for radio SNe. To date, only ∼ 100 radio core-collapse supernovae (ccSNe) have been detected and the first Type Ia radio supernova was detected last year (Bietenholz et al. 2021; Kool et al. 2023). To model the expected rate of radio ccSNe we follow Lien et al. (2011). Because ccSNe are shortlived, the rate of ccSNe is closely related to the cosmic Star Formation Rate (SFR), which has been observational measured to high redshift and accuracy. We use the ccSNe radio luminosity distribution of Bietenholz et al. (2021), which is more complete than the one in Lien et al. (2011). The best-fit distribution is log-normal and incorporates all known radio ccSNe as well as radio non-detections of supernovae. Bietenholz et al. (2021) note that their data is likely biased towards radio detections, finding that 30% of all ccSNe are detected in the radio. We divide the total number of radio ccSNe in the model by 3 to get a more conservative 10% as in Lien et al. (2011). From the luminosity distribution, we determine the number of radio ccSNe at each redshift above the flux limit of the DSA-2000. Because the SNe observations used in Bietenholz et al. (2021) have frequencies ranging from 2-10 GHz, there is uncertainty when converting to 1.4 GHz. We correct 𝐿 min 6 GHz ( 𝑧 ) = 𝐿 min 1.4 GHz GLYPH<16> 6 1 . 4 GLYPH<17> 𝛼 , where we assume 𝛼 = -0 . 7 for the synchrotron emission of radio ccSNe. Further, because radio ccSNe have a mean rise time of log 10 ( 𝑡 rise ) = 1 . 7 days (Bietenholz et al. 2021), we use the single epoch flux limit of the DSA-2000 and account for the DSA-2000's cadence. The rms noise for a single epoch is 𝜎 𝑛 = 2 𝜇 Jy/beam, so a 10 𝜎 𝑛 detection is 20 𝜇 Jy. The cadence will be ∼ 4 months, so conservatively we will detect 1/3 of the radio ccSNe per year. None of the known radio ccSNe have luminosities above 10 29 erg s -1 Hz -1 , but given the wide spread we expect that the highest luminosities could be several times larger. Because of this we set an exponential cutoff at 10 30 erg s -1 Hz -1 , which we can expect to see out to about 𝑧 ≈ 1 . 5 in a single epoch of the DSA-2000.</text>
<text><location><page_6><loc_51><loc_16><loc_92><loc_34></location>The predicted redshift distributions of SFRGs, AGN, and ccSNe from our model are shown in Figure 3 and compared to the expected distribution of Rubin-LSST sources. As expected, the total source count is dominated by SFRGs until high redshift. Blazars, meaning both FSRQs and BLLacs, are the AGN that will be most useful for time delay measurements, which is explored in Section 5.1. These are the most common sources at very high redshift. We expect the DSA2000 to probe much higher redshifts than the Rubin-LSST because radio emission is much less susceptible to intervening gas or seeing. This is one of the reasons that the DSA-2000 will be effective at lens finding. The total number of expected sources of each type in a DSA-2000 all-sky survey with a 10 𝜎 𝑛 detection are listed in the first row of Table 1.</text>
<section_header_level_1><location><page_6><loc_51><loc_13><loc_63><loc_14></location>2.4 Expected rates</section_header_level_1>
<text><location><page_6><loc_51><loc_6><loc_92><loc_11></location>The final results of our model are shown in Table 1. The results agree quite well with the empirical estimates given in Section 2.1. As expected from CLASS there are roughly 10 6 total galaxy scale lenses contained in the DSA-2000 all-sky survey, less than 10% of</text>
<figure>
<location><page_7><loc_7><loc_70><loc_44><loc_91></location>
<caption>Figure 3. Expected source redshift distribution for the DSA-2000 from the model of Section 2.3. Blazars refers to both FSRQs and BLLacs. The distribution for Rubin-LSST is taken from Alonso & Ferreira (2015).</caption>
</figure>
<text><location><page_7><loc_7><loc_26><loc_48><loc_60></location>which will be discoverable. The total number of group and cluster lenses are about 10% and 2% of the galaxy number respectively, which is consistent with Oguri (2006). Roughly half of the group and cluster lenses will be discoverable, leading to them making up about a third of the total discoverable lenses, a significant fraction. Yue et al. (2022a) find that the Rubin-LSST will discover about 2000 lensed QSOs; our number is similar. This is a result of the competing mechanisms of the DSA-2000 detecting many more at higher redshifts and simultaneously being able to discover a smaller fraction because of the PSF size. The main purpose of Yue et al. (2022b) is to investigate the discrepancy between the fraction of lensed high redshift ( 𝑧 𝑠 ≳ 5) quasars in observations ( ∼ 0 . 2%) and previous predictions ( ∼ 4%).Yueetal.(2022b)find ∼ 0 . 4 -0 . 8%with their model, and we have ∼ 0 . 3%, which is consistent. The number of discoverable lensed ccSNe per year in the DSA-2000 from the full model is about 2, which is of the same order of magnitude as the empirical VLASS estimate. Oguri & Marshall (2010) conclude that Rubin-LSST will find roughly 8 SNe per year. We expect there to be less lensed SNe at radio wavelengths because not all SNe emit in the radio and the radio emission is weaker than in other wavelengths. The redshift distribution of the deflectors and sources is shown in Figure 4. Almost all deflectors are at 𝑧 𝑑 < 3 and most are at 𝑧 𝑑 ≈ 1, consistent with Oguri & Marshall (2010); Collett (2015); Yue et al. (2022a,b). This justifies the use of the CosmoDC2 catalog, which is limited to 𝑧 < 3, to determine the MF of groups and clusters.</text>
<text><location><page_7><loc_7><loc_6><loc_48><loc_25></location>We also run the simulation for the survey specs of the SKA-mid and VLASS. The main differences are the sensitivity and resolution. The expected rms noise of the SKA-mid at 1.4 GHz will be 𝜎 𝑛 = 2 𝜇 Jy/beam, and the PSF size will be about 0.4' (Braun et al. 2019). From the DEEP2 source counts, the SKA-mid should find about 3 × 10 8 sources above 10 𝜎 𝑛 . With this, our model predicts that the SKA-mid should see 1.8 ± 0.7 × 10 5 , 1.8 ± 1.2 × 10 4 , and 2.7 ± 1.1 × 10 3 galaxy, group, and cluster lenses, given the same discoverability limit defined in Section 2.2. Despite less depth leading to around a third of the total galaxy lenses, the PSF of the SKA-mid is such that it will resolve over half of them, resulting in about twice as many discoverable galaxy lenses as in the DSA-2000. The SKA-mid will find fewer group and cluster scale lenses than the DSA-2000 because sensitivity is more important for discovering these systems</text>
<figure>
<location><page_7><loc_52><loc_70><loc_89><loc_91></location>
<caption>Figure 4. Redshift distribution of the deflectors (blue) and sources (orange) for galaxies (solid), groups (dashed), and clusters (dotted) expected in the DSA-2000 with magnification bias.</caption>
</figure>
<text><location><page_7><loc_51><loc_43><loc_92><loc_61></location>than angular resolution. In addition, we expect about 14 lensed radio ccSNe per year in an SKA-mid all-sky survey. McKean et al. (2015) estimate ∼ 3 × 10 5 lenses in an SKA-mid all-sky survey with 𝜎 𝑛 = 3 𝜇 Jy/beam and a lens detection limit of 15 𝜎 𝑛 and >0.3'. This is slightly more optimistic than our forecast, but broadly consistent. VLASS has 𝜎 𝑛 = 70 𝜇 Jy/beam combined and an angular resolution of 2.5' (Lacy et al. 2020). The DEEP2 source counts indicate that we should expect ∼ 5 × 10 6 sources above 10 𝜎 𝑛 in VLASS (Lacy et al. (2020) estimate 5.3 × 10 6 above 5 𝜎 𝑛 ). With these numbers, we estimate that there should be about 300 discoverable lenses in the full sky survey. As VLASS enters its epoch 3, many of these lenses are likely already in the data (several have recently been identified (Martinez et al. 2024)).</text>
<section_header_level_1><location><page_7><loc_51><loc_39><loc_66><loc_40></location>3 LENS DISCOVERY</section_header_level_1>
<section_header_level_1><location><page_7><loc_51><loc_36><loc_64><loc_37></location>3.1 Superresolution</section_header_level_1>
<text><location><page_7><loc_51><loc_24><loc_92><loc_35></location>In the past decade, tremendous progress has been made by the computer vision community with respect to the classical ill-posed inverse problems. These include deblurring (Zhang et al. 2022), deconvolution and superresolution (Alzubaidi et al. 2021), and image inpainting (Yu et al. 2018b). Nearly all of these advances were borne out of the deep learning revolution, as efficient neural network architectures and training strategies have enabled powerful learning-based tools for machine vision.</text>
<text><location><page_7><loc_51><loc_6><loc_92><loc_24></location>Astronomy naturally lends itself to these tools because sparse sampling and ill-posedness arise in many astronomical imaging and reconstruction contexts, especially in interferometry. Several groups have begun developing machine learning methods for interferometric image reconstruction (Connor et al. 2022; Aghabiglou et al. 2024; Mars et al. 2024). One reason this approach is suitable for radio astronomy is that the PSF of an array is given deterministically by the spatial distribution of antennas and the observing frequency: The on-sky response of an interferometer is the 2D Fourier Transform of its sampling in UV-space (the aperture plane). Therefore, prior physical knowledge of the PSF can be incorporated in the model either via training data (Connor et al. 2022) or fed directly to the network itself (Mars et al. 2024). This is not the case in ground-</text>
<figure>
<location><page_8><loc_8><loc_65><loc_92><loc_93></location>
<caption>Figure 5. An example of superresolution image reconstruction on radio strong lenses. The simulated sky model is a mix of star-forming radio galaxies and AGN point sources. We randomly select 5% of sources to be strongly lensed. The artificially high value ensures that we have several lensing examples to reconstruct. The left panel shows a 3.3'x3.3' region observed (i.e. the 'dirty image') with the DSA-2000 full-band PSF (size shown with mauve circle) without any deconvolution. The middle panel is a reconstruction of that field with the POLISH algorithm. The rightmost panel is the true sky. The dirty images have been gamma encoded with 𝐹𝑙𝑢𝑥 0 . 75 with a value range of the inset figures chosen to highlight structure. In this toy example, strong lensing systems with Einstein radii below the PSF scale can be identified.</caption>
</figure>
<text><location><page_8><loc_7><loc_50><loc_48><loc_52></location>based optical astronomy where 'seeing' and complex optics mean the PSF is not known a priori.</text>
<text><location><page_8><loc_7><loc_36><loc_48><loc_49></location>Asasimple demonstration, we use the super-resolution and imageplane deconvolution method POLISH to show how strong lenses with image separations below the PSF scale can be recovered with machine learning. POLISH is a supervised machine learning model that learns the mapping between the true sky and observed images (the 'dirty image' in radio parlance) (Connor et al. 2022). It uses the Wide Activation for Efficient and Accurate Image Super-Resolution (WDSR) architecture (Yu et al. 2018a), but any super-resolution neural network could be swapped in (e.g., a standard U-Net or the Efficient Super-Resolution Transformer (Lim et al. 2017)).</text>
<text><location><page_8><loc_7><loc_17><loc_48><loc_35></location>Wehave trained a POLISH network on forward-modelled synthetic data using a DSA-2000 PSF averaged over the whole band, resulting in an angular resolution of ∼ 3.3'. This is worse resolution than the top of the band, where many strong lenses could be found, but we offer this as a toy example. The sky model is described in Connor et al. (2022), with the addition of strongly lensed galaxies in both the training and validation set. In Figure 5 we show an example validation image that contains multiple lenses. Lensed systems that are undetected in dirty image, including both arcs (left inset panels) and Einstein rings (right inset panels), are identifiable in the POLISH reconstruction. In the future, we plan to develop super-resolution methods explicitly for lens finding, but we take this as a promising sign that the DSA-2000 will be able to find systems with 𝜃 𝐸 ≳ 0 . 5 '' .</text>
<section_header_level_1><location><page_8><loc_7><loc_13><loc_18><loc_14></location>3.2 Identification</section_header_level_1>
<text><location><page_8><loc_7><loc_6><loc_48><loc_11></location>The first major hurdle for strong lensing science with DSA-2000 and other surveys is identification. The large number of sources produced by these surveys makes visual inspection by experts impossible. Despite much work on automated lens discovery (see Lemon et al.</text>
<text><location><page_8><loc_51><loc_41><loc_92><loc_52></location>(2023) for a review), little effort has been spent on identification in the radio. On the one hand, radio lens images are made cleaner by the lack of radio emission from the massive quiescent galaxies that make up the deflector population. However, the large, extended lobes of radio galaxies can easily be misidentified as lensing features. Further, color and photometric redshift information from UV and optical surveys have traditionally played a role in lens identification, despite being a foreground to the morphology of the lensed source.</text>
<text><location><page_8><loc_51><loc_6><loc_92><loc_39></location>Catalog-level searches, where large cuts are made based on certain features before a smaller subset is visually inspected, have been successful in the past. CLASS selected for flat spectrum radio sources ( 𝛼 > -0 . 5) (Myers et al. 2003). This had the advantage of picking out only blazars, whose radio emission is dominated by a small compact core, thus eliminating any confusion with intrinsic structure. A similar strategy would likely be effective in the DSA-2000. However, dedicated follow up of all of these sources, as in the ∼ 11,000 blazars in the initial CLASS sample, will be impossible. Similarly, a visual inspection of even a small subset of the O( 10 7 ) blazars expected in the DSA-2000 is ambitious (hence the need for intermediate automated steps in the identification pipeline). Still, targeting flat spectrum sources could increase the yield of any search, and spectral information from the whole DSA-2000 band will be useful for determining whether components are different sources or multiple images. Jackson & Browne (2006) show that combining astrometric data from radio and optical surveys can significantly improve the efficiency of lens searches. If the optical emission is dominated by the deflector galaxy then there will exist an offset between the centroid position of the optical and radio sources in a lens system. This offset was exploited to find lenses with separation down to ∼ 1' even with the poor 5' resolution of the Faint Images of the Radio Sky at Twenty-one centimeters (FIRST) survey. They predict that this effect will become more efficient at lower radio fluxes because the optical</text>
<text><location><page_9><loc_7><loc_85><loc_48><loc_93></location>flux is more likely to be dominated by the defelctor galaxy. Given the sensitivity of the DSA-2000 and its significant overlap with current and planned large surveys in other wavelengths, this could prove to be a very effective way of overcoming the limitations of its PSF. Regardless, exploiting the extra information from overlap with other surveys will be crucial for any lens search in the radio.</text>
<text><location><page_9><loc_7><loc_60><loc_48><loc_84></location>Machine learning models have been successfully used to discover many new candidate lens systems. Convolutional Neural Networks (CNNs) in particular are effective at classifying astronomical images. The main disadvantage with CNNs and other supervised learners is the need for large realistic training sets. Using CNNs on simulated data for the International LOFAR Telescope (ILT) at 150 MHz, Rezaei et al. (2022) are able to recover over 90% of galaxy-size lenses with a false-positive rate of only 0.008%. They find that a strong 20 𝜎 detection and large separation 𝜃 𝐸 ≥ 3 / 2 beam size (stricter than the discoverable limit imposed in Section 2.2) are necessary for reliable identification with the CNN. Incorporating superresolution models into the lens-finding routine would significantly increase the yield of such a CNN, especially at high S/N. Resolving lenses down to 𝜃 𝐸 ≈ 0 . 5' with would enable the same level of accuracy as in Rezaei et al. (2022). Simulating accurate DSA-2000 lenses and training models for identifying them is a goal of our future work. The SKA-mid will already have sufficient resolution for these purposes, making CNNs an attractive option for lens identification in its all-sky survey.</text>
<text><location><page_9><loc_7><loc_46><loc_48><loc_59></location>The main limitation of any ML-based approach to group/cluster lens finding is the difficulty of creating realistic training sets. The irregularity of group/cluster lensing potentials may require building a forward-modelled training set from ray-tracing in large cosmological magnetohydrodynamical simulations such as TNG-Cluster, which should have a heterogenous population of massive clusters (Nelson et al. 2024). More generally, a realistic training set of strong lenses across scales could be produced with radio source samples from T-RECS and a deflector population drawn from cosmological simulations.</text>
<section_header_level_1><location><page_9><loc_7><loc_41><loc_33><loc_42></location>4 MULTIWAVELENGTH SYNERGIES</section_header_level_1>
<text><location><page_9><loc_7><loc_33><loc_48><loc_40></location>Acquiring redshifts for lensed galaxies and deflectors is critical for the application of strong lensing to cosmology and astrophysics. We consider here how well we could do in the absence of targeted followup observations, capitalizing on the suite of large survey telescopes that will be operating in the mid- to late-2020s.</text>
<text><location><page_9><loc_7><loc_8><loc_48><loc_33></location>DSA-2000 spectroscopic HI survey In addition to the cadenced allsky survey, the DSA-2000 will undertake a large spectroscopic HI survey over the full visible sky (Dec > -30 · ) (Hallinan et al. 2019). This is expected to produce ∼ 10 6 HI galaxies at 𝑧 < 1. Most deflectors will be massive elliptical galaxies with limited 21 cm emission, but a sub-set of early-type galaxies with 𝑛 𝐻 ≥ 10 20 cm -2 𝑧 -2 𝑑 will be detected. Some late-type deflector galaxies will have enough cold gas to obtain a redshift. Additionally, 21 cm absorption in the inner CGM of deflector galaxies may be present in the lensed source's continuum spectrum, providing a measurement of cold gas along different sightlines of the same CGM (see Rudie et al. (2019) for a similar application in lensed quasar systems). Thus, some deflector galaxies will have a spectroscopic redshift without any external O/IR data. Of the ∼ 10 6 HI galaxies detected by the DSA-2000 in emission, we find that only a handful will be strongly lensed, due to their shallow redshift distribution. Spectroscopic radio observations will not provide a significant portion of redshifts required for our purposes, but may be a minority of interesting deflectors.</text>
<text><location><page_9><loc_7><loc_6><loc_48><loc_7></location>The Dark Energy Spectroscopic Instrument (DESI) will produce</text>
<text><location><page_9><loc_51><loc_69><loc_92><loc_93></location>redshifts of ∼ 40 million galaxies and quasars (DESI Collaboration et al. 2016; Dey et al. 2019). Its 3 yr public data release is expected before the DSA-2000's first light. DESI's Bright Galaxy Survey (BGS) targets galaxies with 𝑟 -band magnitudes brighter than 19.5. For the luminous red galaxies (LRGs), emission line galaxies (ELGs), and quasars, the depth will be 𝑟 ≈ 23. Each class of target will have a different redshift distribution, but together they will provide a rich catalog of spectroscopic redshifts for deflectors in the lensing systems discovered by DSA-2000. The BGS and LRG samples will be over-represented as deflector galaxies, whereas the ELG and QSO samples may make up a small fraction of lensed DSA-2000 sources. ELGs, by definition, have strong emission lines indicative of star formation, which produce synchrotron radiation detectable by radio surveys. Some of these will be strongly lensed star-forming radio galaxies with 𝑆 𝜈 above 5 𝜇 Jy. Still, we do not expect more than a small minority of the radio continuum lenses to be detected in DESI due to the relative number of total sources.</text>
<text><location><page_9><loc_51><loc_59><loc_92><loc_69></location>SPHEREx is a near-infrared space mission expected to launch before Spring 2025. It will map the full 4 𝜋 sr sky with 6 arcsecond pixels and 96 color bands, producing a catalog of hundreds of millions of galaxies with 'spectro-photometric redshifts' (Doré et al. 2018). Although its PSF it too large to discover strong lenses directly, the galaxy catalog will be valuable for obtaining reliable redshifts of both deflectors and lensed galaxies.</text>
<text><location><page_9><loc_51><loc_48><loc_92><loc_58></location>The Rubin Observatory will find billions of galaxies in its Legacy Survey of Space and Time (LSST) (Ivezić et al. 2019). Most of these galaxies will be at 𝑧 𝑠 ≲ 1 . 5 (Collaboration et al. 2021), providing an excellent deflector catalog for the DSA-2000 lens candidates. In the DSA-2000/LSST overlapping footprint between -30 and +30 Declination, we find that a significant fraction of deflectors and several thousand lensed sources in the DSA-2000 sample will have photometric redshifts from the Rubin Observatory.</text>
<text><location><page_9><loc_51><loc_33><loc_92><loc_47></location>Roman The High Latitude Spectroscopic Survey (HLSS) is expected to detect 10 million galaxies in 𝐻𝛼 between 1 < 𝑧 < 2and ∼ 2 million [OIII] galaxies at 𝑧 = 2 -3 using the near-IR grism and wide-field camera (Wang et al. 2022). Roman's 2,000 deg 2 footprint will be fully mapped by DSA-2000. With a dust-attenuated 𝐻𝛼 flux higher than 10 -16 erg s -1 cm -2 , several million of these galaxies will be detectable at GHz frequencies at > 5 𝜇 Jy (Murphy et al. 2011). Assuming a mean lensing optical depth of 5 × 10 -4 , this suggests that O( 10 3 ) will be strongly lensed and detected by both the DSA-2000 and Roman , the latter providing 0.1' resolution and a source redshift.</text>
<text><location><page_9><loc_51><loc_23><loc_92><loc_32></location>Euclid is a visible and near-infrared space telescope that launched in July 2023 (Collaboration et al. 2024). Its large photometric survey will find over one billion galaxies, with a spectroscopic survey that will obtain accurate redshifts for ∼ 30 million galaxies mostly at 0 . 9 < 𝑧 < 1 . 8. Euclid and the DSA-2000 share over 10,000 deg 2 of sky, providing an excellent sample of photometric and spectroscopic redshifts of lens systems.</text>
<section_header_level_1><location><page_9><loc_51><loc_14><loc_78><loc_15></location>5 STRONG LENSING APPLICATIONS</section_header_level_1>
<text><location><page_9><loc_51><loc_6><loc_92><loc_13></location>A key limitation of strong lensing science is the small number of known systems, especially at radio wavelengths. We discuss the impact that the large expected number of radio lenses and their multiwavelength counterparts will have on several important applications of strong lensing.</text>
<section_header_level_1><location><page_10><loc_7><loc_92><loc_30><loc_93></location>5.1 Time-delay cosmography & 𝐻 0</section_header_level_1>
<text><location><page_10><loc_7><loc_71><loc_48><loc_90></location>Light from multiple images in a gravitational lens will reach the observer at different times. For continuum sources, we do not observe the difference in arrival times, but transients or time-variable sources allow us to measure the time delay. This time delay is due to a difference in path lengths and gravitational time dilation near the deflector. As such, the time delay encodes information about the geometry of the universe, and is inversely proportional to 𝐻 0 (Oguri 2019). Much work has been dedicated to measuring these time delays and using them to constrain 𝐻 0 (Treu & Marshall 2016; Birrer et al. 2024). The H0LiCOW project has measured 𝐻 0 to 2.4%, which is independent of and competitive with state-of-the-art 𝐻 0 constraints (Wong et al. 2019). A large increase in the number of measurable time-delay systems will allow even tighter constraints and could help settle the current Hubble tension.</text>
<text><location><page_10><loc_7><loc_29><loc_48><loc_70></location>We estimate the number of lensed variable AGN that will be useful for measuring 𝐻 0 in the DSA-2000. To first order, all blazars (FSRQs and BLLacs) are inherently variable, ∼ 10 3 of which will be discoverable as lenses in DSA-2000 data (Table 1). However, several factors complicate our ability to use them for time delay measurements; we would like to know how many will be in a "gold sample" for which the time-delay can be reliably determined. To date, the most extensive study of radio AGN variability is the OVRO 40 m blazar monitoring campaign. Since 2007 this program has monitored around 1500 blazars with a cadence of two weeks (Richards et al. 2011). Richards et al. (2011) provides distributions of the intrinsic modulation index, a measure of the relative intrinsic variability of the AGN, for FSRQs and BLLacs. The DSA-2000 will measure flux variability at the level of a few percent, but to be conservative we consider candidates with ≥ 10% variability. We also want sources with a strong detection, so we chose sources with flux ≥ 20 𝜎 𝑛 = 10 𝜇 Jy. The OVRO blazars are monitored at 15 GHz but we expect lower variability at 1.4 GHz. Fan, J. H. et al. (2007) find that blazars are about 20% less variable at 4.8 GHz than 14.5 GHz. Sotnikova et al. (2024) find that radio AGN at 22.3 GHz are about 35% more variable than at 2.3 GHz, although there is little difference between 11.2 GHz and 2.3 GHz. We assume that the mean modulation index drops by 40% from the OVRO 40 m data to 1.4 GHz for the DSA-2000. Further, for multiple images, we need at least two of the images to be bright enough to measure the time delay so we use the magnification bias of the fainter image. With these restrictions, the OVRO data, and our lens model, we estimate that there will be 68 ± 30 lensed FSRQs and 23 ± 10 lensed BLLacs with sufficient brightness, variability, and image separation to measure time delays. The same analysis for the SKA-mid gives 120 ± 50 and 43 ± 16 lensed FSRQs and BLLacs.</text>
<text><location><page_10><loc_7><loc_6><loc_48><loc_28></location>Of the lensed AGN, the most useful systems for time delay analysis will be those that have four or more images because they allow for multiple constraints on the time delay. A SIS is only capable of creating two images and a spherical NFW can make three, but factors such as ellipticity, irregularities in the lens potential, and external environments of real halos will allow a number of these systems to make four or more images. The fraction of quad galaxy-QSO lenses predicted in Rubin-LSST is ∼ 10-15% (Oguri & Marshall 2010; Yue et al. 2022a). However, of the statistically complete CLASS sample, 5 out of 13 lensed blazars were quadruply imaged and one had six images (B1933+503 has two sets of quadruply imaged sources Sykes et al. (1998)). Therefore we can expect 100-400 quadruply-lensed blazars in the DSA-2000 all-sky survey and about twice as many in the SKA-mid, roughly 10% of which may be used in practice for measuring 𝐻 0 . We note that quad systems often have higher external shear due to their preponderance of group or cluster environments,</text>
<text><location><page_10><loc_51><loc_90><loc_92><loc_93></location>which results in an added systematic in lens modeling (Holder & Schechter 2003).</text>
<text><location><page_10><loc_51><loc_51><loc_92><loc_90></location>Wong et al. (2019) reach a 2.4% measurement of 𝐻 0 with a sample of 6 lensed quasars, and predict that around 40 lenses are needed to constrain 𝐻 0 to within 1%. Jee et al. (2016) argue that with a well-defined sample of quadruply imaged quasars, and combining time-delay distances and velocity dispersion measurements to estimate angular diameter distances within 5%, 𝐻 0 can be measured to the same precision as Planck with as few as 10 systems. Napier et al. (2023) reach a 10% measurement with three galaxy clusters, and estimate that roughly 50 will be needed for a sub 1% constraint, assuming that modeling uncertainties can be reduced in coming years. While we expect uncertainty to decrease significantly with a larger sample of lensed quasars, there are several systematics that must be kept under control for an accurate measurement. Namely, an accurate redshift measurement is necessary, ideally of both the source and deflector, as well as precise measurements of the time delay, complete sampling over many cycles of the time delay to negate contamination from micro-lensing, accurate lens models, and constraints on any line of sight structures. If we assume that all of the lenses in our gold sample can be modeled to the accuracy of the H0LiCOW sample - a roughly 6% combined uncertainty on the time delay, mass model, and line of sight contribution for each lens - then a sample of ∼ 100 lensed quasars would give about a 0.6% constraint on 𝐻 0 . This is of course not possible with the DSA-2000 or SKA-mid alone but will require dedicated follow-up at multiple wavelengths. As discussed in Section 4, we expect that a significant minority of lenses should have redshifts available in other concurrent public surveys. Interesting subsamples of lenses will require targeted follow-up for both mass modeling and spectroscopic redshifts.</text>
<text><location><page_10><loc_51><loc_43><loc_92><loc_51></location>The DSA-2000's cadenced all-sky survey will image each field above -30 Dec once every 4 months. Typical time delays for galaxy, group, and cluster lenses in our model are a month or less, several months, and a year or more respectively (Oguri et al. 2002). Special systems could therefore be visited with a higher cadence, similar to how pulsar fields will be visited regularly for timing experiments.</text>
<text><location><page_10><loc_51><loc_27><loc_92><loc_42></location>So far we have only considered lensed blazars for time delay measurements, which are desirable because of their characteristic variability and compact emission regions. Lensed transients will also be useful for time delay measurements, provided that they are detected early and monitored closely. The variability of radio emission from TDEs is too slow to make them useful for this application (on-axis jetted TDEs can have shorter rise times, but they are significantly rarer such that we do not expect any lensed jetted TDEs in upcoming radio surveys), but SNe have faster rise times. We expect O( 1 ) lensed radio ccSNe per year in the DSA-2000 and O( 10 ) in the SKA-mid that can be used for 𝐻 0 measurements.</text>
<section_header_level_1><location><page_10><loc_51><loc_24><loc_69><loc_25></location>5.2 Dark Matter Structure</section_header_level_1>
<text><location><page_10><loc_51><loc_6><loc_92><loc_23></location>Gravitational lensing has been very useful for determining the structure of galaxies, groups, and clusters at cosmological distances and distinguishing between dark matter models (see Vegetti et al. (2024); Natarajan et al. (2024) for a review). The best of these studies use high-resolution HST images, but radio observations also played a pivotal early role. Cohn et al. (2001) used the 10-image CLASS lens B1933+503, which provides many more constraints than usual, to determine the density profile of the deflecting galaxy, finding it close to isothermal. Wucknitz et al. (2004) find the same for B0218+357. Dalal & Kochanek (2002) used radio observations of flux ratio anomalies between multiple images to constrain the mass fraction of dark matter substructure to 2%. Radio observations are especially</text>
<text><location><page_11><loc_7><loc_76><loc_48><loc_93></location>suited for this type of study because they are less susceptible to micro-lensing (Koopmans et al. 2003; Kochanek & Dalal 2004). Despite this, dark matter studies with radio lenses have been limited for many years by the small sample size (Vegetti et al. 2024). Because resolution is a key factor in the ability to accurately model mass distributions or detect substructure, the DSA-2000 alone will not be able to use lens systems to study dark matter. Its main utility will come from identifying huge numbers of systems for detailed followup. The extraordinary resolution of the ngVLA on a large sample of lenses selected by the DSA-2000 will revolutionize this field. In the more immediate future, the SKA will also provide enough resolution for these studies in their overlapping sky coverage.</text>
<text><location><page_11><loc_7><loc_44><loc_48><loc_76></location>Another exciting application of lensing is using a central image to study small scales near the centers of galaxies (Treu 2010; McKean et al. 2015; Shajib et al. 2024). A SIS lens only forms two images, but for realistic galaxy profiles a highly demagnified image is expected to form at the center of the image configuration. This image will be sensitive to the mass contained within very small distances of the deflector's center, and so can be used to study supermassive black holes at cosmological distances. This is especially suited for radio wavelengths because in many cases the deflector will be radio-quiet allowing the center image to be detected. Because the center image is highly demagnified, only sensitive radio surveys such as the DSA2000 will be able to reliably identify them. At least one such central image has been detected in the radio with brightness ∼ 0 . 8 mJy at 8.46 GHz (Winn et al. 2004), so we can likely expect O( 10 2 ) or more with the DSA-2000. A main challenge will be disentangling the faint emission of the central image from the other images. For groups and clusters, whose inner density profiles are shallower than isothermal, central images are more common and often less demagnified. The very inner profiles of groups/clusters can thus be studied, which are hard to constrain with images near the Einstein radius alone. These central images will be easier to identify because the other images will be farther away (although many clusters have large central elliptical galaxies emitting in the radio that will obscure the image).</text>
<section_header_level_1><location><page_11><loc_7><loc_38><loc_18><loc_39></location>5.3 Polarization</section_header_level_1>
<text><location><page_11><loc_7><loc_22><loc_48><loc_37></location>Measuring full Stokes information of gravitationally lensed signals offers insight into the source plane because polarization is not affected by gravitational potentials. This is an advantage of radio surveys, where polarization information is often stored and Faraday rotation measure (RM) can be measured. This enables us to study propagation effects along different sightlines for lensed objects. For example, Mao et al. (2017) used the VLA to measure the polarization properties of two lensed images of CLASS B1152+199, finding a large difference in RM ( + 9 . 7 ± 0 . 5 rad m -2 for image ' 𝐴 ' and + 517 ± 3 rad m -2 for image ' 𝐵 '), which is due to the magnetized plasma in the lens galaxy's interstellar medium.</text>
<text><location><page_11><loc_7><loc_6><loc_48><loc_21></location>The DSA-2000 will have the ability to measure full polarization information for any source it detects, with a maximum |RM| of roughly 10 4 rad m -2 . Compact sources such as Blazars (and especially BL Lac objects) can show significant polarization fractions at 1 GHz. Assuming a 50 𝜎 detection threshold in total power, we estimate that the DSA-2000 could detect Stokes Q and U for 5-10 million AGN in its 5 year continuum survey. Using the empirical CLASS lensing optical depth and image separation cut, we take 𝜏 𝐴𝐺𝑁 ≈ 5 × 10 -4 . We estimate that the DSA-2000 could find O( 10 3 ) strong lenses for which polarization properties could be used to model the lens distribution and study magnetic fields at cosmological distances.</text>
<section_header_level_1><location><page_11><loc_51><loc_92><loc_66><loc_93></location>5.4 Other applications</section_header_level_1>
<text><location><page_11><loc_51><loc_59><loc_92><loc_91></location>In addition to constraining 𝐻 0 , time-delay measurements are weakly sensitive to other cosmological parameters, such as Ω 𝑚 , Ω Λ , and the dark energy equation of state parameter 𝑤 (Natarajan et al. 2024). Cluster lenses are used for studying dark energy as well (Macciò 2005; Gilmore & Natarajan 2009; Jullo et al. 2010). A large influx in measured time delays and observed cluster lenses with the DSA2000 will allow better constraints on these cosmological parameters. Lenses are also commonly used as 'nature's telescopes'. The magnification of background sources allows very distant galaxies to be studied even by smaller telescopes (Jackson 2011). This is especially true for group/cluster lenses, where we expect the magnifications to be larger (Robertson et al. 2020). We expect to see O( 10 4 ) lenses with 𝑧 𝑠 > 5 in the DSA-2000 (figure 4), many of which will be group and cluster lenses due to their high magnifications. These high redshift sources will provide insight into the population of radio sources in the early universe. Lensing statistics can also be used to test cosmological models. The abundance of giant arcs in cluster lenses was a topic of much debate in the past few decades (Meneghetti et al. 2013). Similarly, the rate of GGSL events in clusters may be in tension with the Λ CDM(Meneghetti et al. 2020; Meneghetti, Massimo et al. 2022; Tokayer et al. 2024). These studies require a large sample of lenses for accurate statistics, which is only now becoming possible with large surveys such as the DSA-2000.</text>
<section_header_level_1><location><page_11><loc_51><loc_51><loc_64><loc_53></location>6 CONCLUSIONS</section_header_level_1>
<text><location><page_11><loc_51><loc_21><loc_92><loc_50></location>In this paper, we forecast expected strong lensing rates in the upcoming DSA-2000 and SKA-mid wide-field radio surveys. We first provide empirical estimates based on previous surveys and simulations, and then develop a detailed forward model that accounts for the source and deflector populations, finding them to be in good agreement. Notably, we model the expected number of galaxy group and cluster scale lenses because these systems will be easily discovered due to their wide angular separations. We find that both the DSA-2000 and the SKA-mid will discover roughly 10 5 strong lens systems. We discuss strategies for identifying these lenses in the data, which will all benefit from emerging superresolution techniques. Finally, we discuss the scientific application of the huge numbers of lenses that will be discovered by these surveys. One of the most exciting applications is 𝐻 0 cosmography with variable and transient sources. The DSA-2000 and SKA-mid will discover about 100 and 200 lensed flat spectrum AGN with >10% variability respectively, as well as about O( 1 ) and O( 10 ) lensed transients per year. With dedicated multi-wavelength follow up these systems could be used to constrain 𝐻 0 to within 1%. The new lens systems will also be useful for studying the distribution of dark matter at cosmological distances, among other applications.</text>
<section_header_level_1><location><page_11><loc_51><loc_14><loc_69><loc_15></location>ACKNOWLEDGEMENTS</section_header_level_1>
<text><location><page_11><loc_51><loc_6><loc_92><loc_13></location>Weare grateful to Schmidt Sciences for supporting Samuel McCarty as a Summer Undergraduate Research Fellow at Caltech. We thank Kim-Vy Tran, Tony Readhead, and Tommaso Treu for helpful conversations on strong lensing, as well as Paul Schechter for insights into quad systems.</text>
<section_header_level_1><location><page_12><loc_7><loc_92><loc_22><loc_93></location>DATA AVAILABILITY</section_header_level_1>
<text><location><page_12><loc_7><loc_87><loc_48><loc_91></location>We have placed a reproduction package on the public GitHub repository available at https://github.com/smmccrty/ radiolensing .</text>
<section_header_level_1><location><page_12><loc_7><loc_82><loc_17><loc_83></location>REFERENCES</section_header_level_1>
<table>
<location><page_12><loc_7><loc_6><loc_48><loc_82></location>
</table>
<table>
<location><page_12><loc_51><loc_6><loc_92><loc_93></location>
</table>
<text><location><page_12><loc_54><loc_6><loc_83><loc_7></location>R., Motta V., 2012, The Astrophysical Journal, 749, 38</text>
<table>
<location><page_13><loc_7><loc_6><loc_48><loc_93></location>
</table>
<unordered_list>
<list_item><location><page_13><loc_51><loc_92><loc_85><loc_93></location>Yao Y., et al., 2023, The Astrophysical Journal Letters, 955, L6</list_item>
<list_item><location><page_13><loc_51><loc_88><loc_92><loc_91></location>Yu J., Fan Y., Yang J., Xu N., Wang Z., Wang X., Huang T., 2018a, Wide Activation for Efficient and Accurate Image Super-Resolution ( arXiv:1808.08718 ), https://arxiv.org/abs/1808.08718</list_item>
<list_item><location><page_13><loc_51><loc_84><loc_92><loc_88></location>Yu J., Lin Z., Yang J., Shen X., Lu X., Huang T. S., 2018b, in Proceedings of the IEEE conference on computer vision and pattern recognition. pp 5505-5514</list_item>
<list_item><location><page_13><loc_51><loc_82><loc_92><loc_84></location>Yue M., Fan X., Yang J., Wang F., 2022a, The Astronomical Journal, 163, 139</list_item>
<list_item><location><page_13><loc_51><loc_79><loc_92><loc_81></location>Yue M., Fan X., Yang J., Wang F., 2022b, The Astrophysical Journal, 925, 169</list_item>
<list_item><location><page_13><loc_51><loc_77><loc_92><loc_79></location>Zhang K., Ren W., Luo W., Lai W.-S., Stenger B., Yang M.-H., Li H., 2022, International Journal of Computer Vision, 130, 2103</list_item>
<list_item><location><page_13><loc_51><loc_75><loc_68><loc_76></location>Zwicky F., 1937, ApJ, 86, 217</list_item>
</unordered_list>
<text><location><page_13><loc_51><loc_72><loc_91><loc_73></location>This paper has been typeset from a T E X/L A T E X file prepared by the author.</text>
</document>
[
{
"title": "ABSTRACT",
"content": "The number of strong lensing systems will soon increase by orders of magnitude thanks to sensitive, wide-field optical and infrared imaging surveys such as Euclid, Rubin-LSST, and Roman. A dramatic increase in strong lenses will also occur at radio wavelengths. The 2000-antenna Deep Synoptic Array (DSA-2000) will detect over 10 9 continuum sources in the Northern Hemisphere with a high mean redshift ( ⟨ 𝑧 𝑠 ⟩ ≈ 2) and the Square Kilometer Array (SKA) will observe a large sample of extragalactic sources in the South with sub-arcsecond resolution. We forecast lensing rates, finding that the DSA-2000 will discover O( 10 5 ) strongly lensed systems, many of which will be galaxy group and cluster lenses. We propose strategies for strong lensing discovery in the limit where the Einstein radii are comparable to the PSF angular scale, taking advantage of modern computer vision techniques and multi-survey data. We also forecast synergies with optical and infrared surveys, which will provide redshifts as well as multiwavelength information about the lens systems. Finally, we describe applications of radio strong lensing systems, including time-delay cosmography with transient and variable sources. We find that ∼ 100 time-variable flat-spectrum AGN discovered by the DSA-2000 could be used to constrain 𝐻 0 at the percent level with the appropriate follow-up. Key words: gravitational lensing: strong - radio continuum: general",
"pages": [
1
]
},
{
"title": "Strong gravitational lensing with upcoming wide-field radio surveys",
"content": "Samuel McCarty 1 , 2 , ★ Liam Connor 3 1 Department of Astronomy, University of Washington, Seattle, WA 98195-1580, USA 2 Cahill Center for Astronomy and Astrophysics, MC 249-17, California Institute of Technology, Pasadena CA 91125, USA 3 Center for Astrophysics | Harvard & Smithsonian, Cambridge, MA 02138-1516, USA Accepted XXX. Received YYY; in original form ZZZ",
"pages": [
1
]
},
{
"title": "1 INTRODUCTION",
"content": "Strong gravitational lensing has a multitude of applications in astrophysics and cosmology (Treu 2010). Previously theoretical ideas have been put into practice in recent decades as the number of known lensed systems has increased and observational data have improved. For example, strongly lensed time-variable and transient sources can be used to constrain the Hubble constant, 𝐻 0 , because the time delay of a multiply imaged source depends on the geometry of the Universe (Refsdal 1964). The technique is known as time-delay cosmography. With just six lensed quasar systems, the The 𝐻 0 Lenses in COSMOGRAIL's Wellspring (H0LiCOW) collaboration has reported 2.4 % precision on their 𝐻 0 measurement, which is independent of the distance ladder and the CMB (Wong et al. 2019). In addition to the Universe's large-scale geometry, lensing observables are sensitive to the total mass of the deflector galaxy or cluster, allowing one to measure the spatial distribution of matter and test different dark matter models (Massey et al. 2010; Vegetti et al. 2024). Lensing magnification allows astronomers to observe objects in the distant Universe, as was pointed out at the field's inception (Zwicky 1937). Dramatic examples have come from the James Webb Space Telescope (JWST), including a red supergiant star at 𝑧 ≈ 2 . 2 that appears to be magnified by a factor of several thousand due to its proximity to caustics in a cluster lens (Diego et al. 2023). Nearly all of these applications would benefit from a larger sample of strong lensing systems. To date, roughly 10 3 strong lensing systems have been discovered, most of which were identified at optical and infrared wavelengths (O/IR). Fortunately, upcoming wide-field imaging surveys such as Euclid and The Vera C. Rubin Observatory's Legacy Survey of Space and Time (Rubin-LSST) are each expected to detect as many as ∼ 10 5 strong lenses (Collett 2015). An early release of a 0.7 deg 2 field from Euclid has recently affirmed these forecasts (Barroso et al. 2024). The Nancy Grace Roman Space Telescope's 2000 square degree survey could find of order 20,000 strong lenses (Weiner et al. 2020), and many more if the proposed multi-epoch 4 𝜋 sr survey is carried out (Han et al. 2023). An increase in the total number of lenses by two-orders of magnitude will usher in a new era of strong lensing science. The first strongly lensed system ever discovered was co-detected at radio wavelengths (Walsh et al. 1979). The first survey for lenses, the Mit-Green Bank survey, was also in the radio (Bennett et al. 1986). Despite this, fewer than ∼ 100 radio lensing systems have been detected to date. This is due to the relatively small total number of known radio sources ( ∼ 10 7 ) and the lack of wide-field imaging surveys with ∼ arcsecond resolution. Both of those limitations will soon be overcome with the advent of next-generation radio survey telescopes. The 2000-antenna Deep Synoptic Array (DSA-2000) will detect over one billion radio sources with a deep redshift distribution, most of which will be star-forming radio galaxies (SFRG) or active galactic nuclei (AGN) (Hallinan et al. 2019). The DSA-2000 is expected to see first light in 2027 with key surveys running between 2028 and 2033. Its point-spread function (PSF) will be roughly 2' at the top of the 0.7-2 GHz radio band. A fifty-fold increase in the total radio source catalog is made possible by the DSA-2000's high survey speed, driven by a large field-of-view ( ∼ 10 deg 2 ) and high sensitivity (the expected system-equivalent flux density or 'SEFD' is just 2.5 Jy). Its cadenced all-sky survey will map out the 3 𝜋 sr above declination -30 · down to 500 nJy/beam root-mean square noise (Hallinan et al. 2019). While the nominal cadence of the continuum survey is four months, certain fields may be visited more regularly, presenting an opportunity to measure lensing time delays with better temporal sampling. The mid-frequency telescope for the square kilometre array (SKAMid) will consist of 197 fully steerable 13.5 m dishes (including the existing MeerKAT radio telescope), operating between 350 MHz and 15.4 GHz with a field-of-view of roughly 1 deg 2 at 1400 MHz. The end of its contruction is anticipated to be July 2029 1 . Although the SKA's lower survey speed will result in fewer sources than the DSA-2000, the wide frequency range and long maximum baseline (150 km) will enable high-resolution imaging, an asset to strong lensing studies (McKean et al. 2015). The Next Generation Very Large Array (ngVLA) is a planned interferometer with extraordinary sensitivity covering a wide range of frequencies (1.2-116 GHz) (Selina et al. 2018a). It will be able to resolve features at milliarcseconds scales. While its broad science goals did not require optimizing the instrument for mapping speed (Selina et al. 2018b), the ngVLA will be a world-class instrument for strong lensing science. Radio strong lensing offers distinct advantages to studies at shorter wavelengths and will complement O/IR imaging surveys. Starforming radio galaxies and AGN can be detected to great distances, boosting the mean optical depth of radio continuum sources. Obscuration by dust in lensing galaxies is not an issue at radio wavelengths, nor is the variable 'seeing' that impacts ground-based O/IR telescopes. Relatedly, the point-spread function (PSF) of a radio interferometer is directly determined by observing frequency and array configuration, allowing us to accurately forward model the instrument's response. In the limit of a large number of antennas and a filled aperture (a 'radio camera'), the deterministic PSF allows us to be more ambitious in image-plane deconvolution, enabling techniques such as superresolution (Connor et al. 2022). Radio telescopes also measure full polarization information, which is conserved under gravitational lensing (Greenfield et al. 1985; Dyer & Shaver 1992). The rotation measure (RM) of a lensed source allows one to measure the magnetic field and ionized gas properties of intervening halos (Mao et al. 2017). Finally, the larger emission regions of radio AGN render them less susceptible to microlensing by substructure in the lens, and may provide cleaner modelling of the deflector mass distribution (Birrer et al. 2024). This is critical for measuring 𝐻 0 via time-delay cosmography. There are of course several drawbacks to strong lensing studies in radio surveys, often precluding the use of standalone radio observations. For example, continuum sources will not contain redshift information, and multi-wavelength datasets or follow-up will be necessary for the majority of lensed systems at radio frequencies. Secondly, the morphology of radio lobes can mimic lensing arcs, exacerbating the already-challenging lens identification problem (consider, for example, head-tail galaxies 3C 465 or 3C 129 at redshift 2). In the case of the DSA-2000, the angular resolution is such that 75-90% of galaxy-galaxy lensing systems will be unresolved in the absence of superresolution. In this work we seek to study radio strong lensing in upcoming interferometric surveys, and develop strategies that alleviate the drawbacks of radio lensing. We focus on the DSA-2000, first by forecasting its strong lensing rates for different source classes. We compare this with a forecast for strong lensing discovery on the SKA. Next, due to the 2-3' PSF of the DSA-2000, we discuss methods for superresolving lensing candidates with modern computer vision techniques in order to increase the yield of strongly lensed systems. We then consider lensed time-variable and transient sources that the DSA-2000 will find and forecast constraints on 𝐻 0 that can be achieved with the appropriate follow-up, as well as other applications of strong lensing.",
"pages": [
1,
2
]
},
{
"title": "2 OPTICAL DEPTH CALCULATION & EXPECTED RATES",
"content": "First, we offer a simple empirical forecast based on observational data and cosmological simulations, and then offer a more detailed forward model that accounts for different source classes and the deflector population.",
"pages": [
2
]
},
{
"title": "2.1 Empirical forecast",
"content": "The Cosmic Lens All Sky Survey (CLASS) (Myers et al. 2003) sought to study a statistical sample of radio-loud gravitationally lensed systems. Despite significant advances in both radio instruments and lens-finding algorithms in the two decades since CLASS, the number of known radio lenses has not increased dramatically. We can use CLASS for a rough estimate of the number of lenses the DSA-2000 will find. CLASS found that a complete sample of 8,958 flat-spectrum radio point sources brighter than 30 mJy had a mean lensing optical depth of 1 . 5 + 0 . 5 -0 . 3 × 10 -3 (Browne et al. 2003). By targeting compact sources with flat spectra, many of their objects are quasars at high redshifts, comparable to typical redshifts of DSA-discovered sources, despite the difference in flux scales. Of the confirmed radio lenses, approximately 20% had angular separations larger than 2 arcseconds, which could be detected by the DSA-2000. Thus, a rough estimate indicates that for every 10,000 sources detected by the DSA-2000, several could be identified as strong lenses. If we assume a detection threshold of 10 𝜎 , the number of extragalactic DSA-2000 sources becomes roughly 5 × 10 8 (Section 2.3). This would result in O( 10 5 ) new galaxy-scale radio lenses, depending on the practical signal-to-noise threshold for candidate systems. Galaxy group and cluster scale lenses have been neglected in previous investigations of lensing statistics for upcoming surveys because they make up a smaller, but still significant, fraction of the total lenses. Additionally, lens modeling is more complicated in this regime. The dark matter halos of groups, in particular, are complex and not well-understood structures. From Oguri (2006), the number of group and cluster lenses are around 11% and 3% of the total galaxy lenses respectively (excluding sub-halo lensing). However, these systems make up a considerable portion of lens systems with angular separation of order 1', and almost all of the lenses at ≥ 10'. Because the PSF of the DSA-2000 is relatively large (2' at 2 GHz and 3.3' at 1.4 GHz), these lenses will make up a large fraction of the discoverable lensing systems and so must be accounted for in our investigation. A radio group/cluster lens survey as complete as CLASS does not exist, but we can make a simple empirical estimate from the literature. The total number of galaxy scale lenses should be O( 10 6 ) assuming 10 9 sources and the CLASS lensing optical depth. Taking the percentages of group/cluster lenses from Oguri (2006), we should see O( 10 5 ) groupscale lenses and O( 10 4 ) cluster scale lenses, almost all of which will have angular separation large enough to be detected in the DSA-2000. The first and largest survey for group-scale lenses was the Strong Lensing Legacy Survey (SL2S) (Cabanac, R. A. et al. 2007). SL2S found 13 strong lens systems in the mass range of groups in ∼ 100 deg 2 of the Canada France Hawaii Telescope Legacy Survey (CFHTLS), giving a rate of ∼ 0.13 deg -2 (Limousin, M. et al. 2009). A later publication identified at least 54 promising group lenses in ∼ 150 deg 2 of the CFHTLS (More et al. 2012). However, the mean redshift of the CFHTLS sources used in SL2S is ⟨ 𝑧 𝑠 ⟩ < 1, and we expect ⟨ 𝑧 𝑠 ⟩ ≈ 2 for the DSA-2000 (see section 2.3, figure 3 below) (Coupon, J. et al. 2009). At these redshifts, we expect the lensing optical depth to be roughly ∝ 𝑧 2 (see equation 31 of Oguri (2019); figure 2 below). This means that we would expect to see ∼ 10 + 6 -6 times more group lenses in the DSA-2000, for a total of ∼ 1 . 1 + 0 . 6 -0 . 9 × 10 5 . Similarly, the Red Sequence Cluster Survey found 8 cluster lenses in ∼ 90 deg 2 with shallower CFHT imaging, indicating ∼ 2 . 6 + 1 . 6 -1 . 6 × 10 4 cluster lenses in the DSA-2000 (Gladders et al. 2003). These estimates are of the same order of magnitude as expected from Oguri (2006). Alternatively, if we take the cross section for giant arcs from the simulations of Puchwein & Hilbert (2009), Fedeli, C. et al. (2010), and Mahdi et al. (2014), assume typical values of 𝑀 = 1 × 10 14 M ⊙ , 𝑧 𝑑 = 0 . 5, and 𝑧 𝑠 = 2, we can calculate a rough estimate of the cluster lensing optical depth at 𝑧 𝑠 = 2. The cross sections at these values are all on the order of 10 arcsec 2 . From Böhringer, Hans et al. (2017), the number density at 𝑀 = 1 × 10 14 M ⊙ is approximately ≳ 1 × 10 -6 Mpc -3 , and after multiplying by the differential coming volumeat 𝑧 𝑠 = 2andskycoverageoftheDSA-2000weget ≈ 5 × 10 -5 for all three simulations. When taking into account that this is the cross-section for giant arcs only, this means we should see several tens of thousands of cluster lenses in the DSA-2000, in good agreement with the previous estimates. Wealso make an empirical estimate of the rate of lensed transients in the DSA-2000. These objects are of particular interest for 𝐻 0 measurements but are much harder to forward model because of the lack of observational data. We can use data from the Very Large Array Sky Survey (VLASS), which is currently the most comparable survey to the DSA-2000, for a simple estimate (Lacy et al. 2020). The determined log rates of supernovae (SNe) and tidal disruption events (TDEs) in deg -2 yr -1 are -1.91 + 0 . 15 -0 . 16 and -2.85 + 0 . 28 -0 . 38 respectively (Dong et al. 2024 in prep). If we assume that the total number of observable sources scales as 𝑆 -1 . 5 min , the value for a Euclidean universe, and that the flux limit of VLASS is 0.7 mJy (10 𝜎 from Lacy et al. (2020)), the DSA-2000 should find ∼ 200 more sources of each type above 20 𝜇 Jy, which is a 10 𝜎 detection in a single epoch of the DSA-2000. We can also assume a lensing optical depth of 1 × 10 -4 and a magnification bias of 2, which are typical for the low redshifts at which we expect to detect transients. Because typical radio rise times of TDEs are on the order of 10 3 days the cadence of the DSA-2000, ∼ 1 / 3 yr, is fast enough to catch most TDEs (Cendes et al. 2023). Scaling for the sky coverage of the DSA-2000, and the cadence for SNe, gives a total yield of 5 + 2 -1 yr -1 and 0.6 + 0 . 5 -0 . 3 yr -1 lensed sources per year for SNe and TDEs respectively. Applying a similar estimate of the lensed rate of GRB afterglows using the predictions of Ghirlanda et al. (2013, 2014) gives a rate much less than 1 per year, and so we ignore them in the rest of this investigation. Further, Yao et al. (2023) find a volumetric rate of optically selected TDEs of 290 + 60 -130 Gpc -3 yr -1 . Cendes et al. (2023) find that ≈ 50% of optically selected TDEs emit in the radio on longer timescales. Becausethere are likely TDEs that emit in radio but not the optical, we take 50% of the optically selected rate to be a conservative estimate. The luminosities of the TDEs in the sample from Cendes et al. (2023) are ∼ 10 37 -10 39 ergs/s, but they note that many of these are likely a lower limit because most of the TDEs still had rising emission at the time of observation. Given this, the DSA-2000 should be able to detect this rate of TDEs out to 𝑧 ≈ 0 . 5, which, when combined with the same optical depth and magnification bias as above, gives 0 . 8 + 0 . 2 -0 . 4 yr -1 lensed TDEs. This number is in good agreement with the previous estimate, so in general, we expect to see about 1 lensed TDE per year in the DSA-2000. However, as noted before, this is a conservative estimate, and there is reason to believe that the actual rate of total and lensed TDEs in the DSA-2000 might be significantly higher. Additionally, because many of the TDEs will be bright in multiple epochs, the DSA-2000 could detect them well below the 20 𝜇 Jy limit, which would increase the rate by a large factor. The DSA-2000 will also discover tens of thousands of distant fast radio bursts (FRBs) (Petroff et al. 2019; Cordes & Chatterjee 2019), some of which will be strongly lensed (Connor & Ravi 2023). The key advantage to using FRBs for time-delay lensing is that their short duration and coherence allows for exceptional precision on the gravitational lensing time delay (Wucknitz et al. 2021). Radio telescopes can preserve phase information about the electromagnetic waveform at nanosecond sampling, which means microlensing signals can be searched for at ultrashort timescales (Leung et al. 2022; Kader et al. 2022). However, for cosmological lensing time delays longer than a pointing (i.e. deflectors more massive than ∼ 10 8 M ⊙ ), one needs to catch the lensed images by pointing at the same patch of sky when it arrives. We do not forecast lensed FRB rates here and point the reader to previous estimates (Connor & Ravi 2023).",
"pages": [
2,
3
]
},
{
"title": "2.2 Lens Model",
"content": "Next, we build a forward model based on the lens and source distributions. Following Yue et al. (2022b), the lensing optical depth for a singular isothermal sphere (SIS) is where 𝑧 𝑑 is the redshift of the deflector, 𝑧 𝑠 is the redshift of the source, 𝜎 is the 1D velocity dispersion of the deflector, Φ ( 𝜎, 𝑧 𝑑 ) is the velocity dispersion function (VDF) of the deflectors, 𝑑 2 𝑉 𝑐 𝑑 Ω 𝑑𝑧 𝑑 = ( 1 + 𝑧 𝑑 ) 3 𝑐 𝑑𝑡 𝑑𝑧 𝑑 is the differential comoving volume, 𝜃 𝐸 ( 𝜎, 𝑧 𝑑 , 𝑧 𝑠 ) is the Einstein radius, and 𝐷 𝑑 is the angular diameter distance at the deflector redshift. The SIS model (or its elliptical generalization) has been shown to replicate the properties of early-type galaxies, which make up the majority of galaxy lenses (e.g. Gavazzi et al. (2007); Koopmans et al. (2009); Li et al. (2018)), and is a widely used model for galaxy strong lensing statistics (e.g. Oguri & Marshall (2010); Collett (2015); Yue et al. (2022a). For a SIS, the Eintein radius becomes where 𝐷 𝑑𝑠 is the angular diameter distance from the deflector to the source and 𝐷 𝑠 is the angular diameter distance from the observer to the source. We use the analytical VDF from Yue et al. (2022b) to model the galaxy deflector population, which they show to be in good agreement with observations. We include a sharp exponential cutoff on the VDF at 300 kms -1 which corresponds roughly to the transition between galaxies and galaxy groups. An important ingredient in the calculation is the magnification bias, which increases the rate of lensed sources by about a factor of two at low redshifts, or by orders of magnitude at high redshifts. The general magnification bias is where 𝜇 is the magnification of a lensed source, 𝑝 ( 𝜇 ) is the probability distribution of the magnification, 𝐿 min is the smallest observable luminosity, and 𝑁 ( > 𝐿 min ) is the number of sources brighter than 𝐿 min . For SIS, 𝜇 min = 2 for the total magnification of the multiple images. A SIS will produce only two multiple images; if the separation of the multiple lens images is large enough to be resolved then 𝑝 ( 𝜇 ) = 2 ( 𝜇 ± 1 ) 3 describes the magnification of the fainter (+) and brighter (-) image (Wyithe et al. 2001). If the lens is not resolved then 𝑝 ( 𝜇 ) = 8 𝜇 3 is the total magnification of both images. Combining 𝐵 and 𝜏 for the total fraction of lensed sources: where 𝐵 ' is the magnification bias of sources that are not lensed, which is assumed to be unity. While the SIS model describes deflector galaxies well, the mass distributions of galaxy groups and clusters behave differently. It has been traditionally thought that in the limit of large mass and large image separation, i.e. clusters, the distribution will be dominated by the dark matter halo, generally following the Navarro-Frenk-White (NFW) profile (Navarro et al. 1997). Recent works suggest that NFW profiles may not be the most accurate representation of large dark matter halos (e.g. Klypin et al. (2016)), but we use them here because they are a decent approximation and their lensing properties are well known. Regardless, for the mass range of groups, it is clear that some intermediate model between SIS and NFW is necessary (Williams et al. 1999; Oguri 2006; More et al. 2012). In a SIS the density 𝜌 ∝ 𝑟 -2 , while the NFW has 𝜌 ∝ 𝑟 -1 on small scales and 𝜌 ∝ 𝑟 -3 on larger scales. The shallower NFW profile has a significantly smaller cross-section than SIS. Oguri (2006) include the effects of baryon cooling and the large elliptical galaxies found at the center of most halos in their model, which will steepen the central density profile and increase the cross-section of halos (especially for groups). Without introducing these complexities, we can steepen the inner profile of halos by modeling them as Generalized NFW (GNFW) profiles. The GNFW is: where 𝑟 𝑠 is the scale radius and 𝜌 𝑠 is the scale density (Li & Ostriker 2002). When 𝛼 = 1 we have the standard NFW. When 𝛼 = 2 the profile resembles an SIS below the scale radius and when 1 < 𝛼 < 2 the inner profile is somewhere in between. Using weak lensing and stellar kinematics, Wang et al. (2023) find the inner dark matter density profiles of group (10 13 𝑀 ⊙ ≤ 𝑀 < 10 14 𝑀 ⊙ ) and cluster ( 𝑀 ≥ 10 14 𝑀 ⊙ ) halos are 1 . 82 + 0 . 15 -0 . 25 and 1 . 48 + 0 . 2 -0 . 41 respectively. Similarly, Mandelbaum et al. (2006) find that the total inner profiles of groups with mass ∼ 2 . 5 × 10 13 𝑀 ⊙ are decently described by a power-law of slope 𝛾 tot = 1 . 85. However, at ∼ 7 × 10 13 𝑀 ⊙ they are less steep than 1.85 and closer to the traditional NFW. Newman et al. (2013) find 𝛾 tot = 1 . 16 + 0 . 10 -0 . 12 for the inner profile of clusters with 𝑀 = 0 . 4 -2 × 10 15 𝑀 ⊙ , and Newman et al. (2015) find the total inner profile slope 𝛾 tot ≈ 1 . 7 while the dark matter slope 𝛼 ≈ 1 . 35 for galaxy groups with ⟨ 𝑀 ⟩ ≈ 10 14 𝑀 ⊙ . In this work, we use the GNFW profile with 𝛼 = 1 . 6 to model groups and 𝛼 = 1 . 2 to model clusters. The resulting GNFW profiles will capture the inner profile well, which is crucial for lensing, and also converge to the standard NFW profile at large radii where we expect the density to be dominated by dark matter (e.g. figures 3 and 4 of Wang et al. (2023)). The GNFW is completely parameterized by 𝛼 , its mass 𝑀 , and the concentration parameter, 𝑐 , from which 𝑟 𝑠 and 𝜌 𝑠 can be determined as in Li & Ostriker (2002). The mass and 𝑐 are correlated with some scatter, we use the relationship presented by Dutton & Macciò (2014) for 𝑐 200 . The scatter in 𝑐 is lognormal with 𝜎 log 𝑐 = 0 . 11. We include an additional factor of ( 2 -𝛼 ) as in Oguri et al. (2001) to correct for the generalized form of the NFW which ensures that the radius at which the logarithmic density slope becomes -2 is the same as in the standard NFW for all 𝛼 . The lensing power of the GNFW halo is very sensitive to 𝑐 and its scatter, so having an accurate relationship is important; early investigations tend to overestimate these values. The lens equation relates a position on the lens plane, 𝑥 , to a position on the source plane, 𝑦 , which for a GNFW profile is where and 𝑔 ( 𝑥, 𝛼 ) is the same as in Li & Ostriker (2002). The lens equation has three solutions inside the radial caustic, 𝑦 𝑐𝑟 , which are the multiple images produced by the GNFW profile. The cross-section for lensing is the area in the source plane in which multiple images will be produced, i.e. the area inside the radial caustic. This is approximated as: where 𝑦 𝑐𝑟 = -𝑦 ( 𝑥 𝑐𝑟 ) and 𝑥 𝑐𝑟 is the location of the minimum of equation 6 (Li & Ostriker 2002). The image separation is given by the separation between the outer two images, which can be approximated as where 𝑥 0 is the position of the tangential critical curve and can be found as the root of equation 6 (Li & Ostriker 2002). The Einstein radius is just half of the image separation, 𝜃 𝐸 = Δ 𝜃 / 2. In general, because our GNFW profiles are shallower than a SIS, they will be worse at producing multiple images but significantly better at magnifying. This is particularly sensitive to 𝑐 , which defines the shallowness of the GNFW profile. For our concentration parameters, especially at high redshift, we expect the magnifications from the GNFWs to be several times larger than from SIS (Wyithe et al. 2001). To calculate the magnification bias for a GNFW profile, we first determine the minimum magnification as (Li & Ostriker 2002): With 𝜇 min wecalculate B according to equation 3 with (Li & Ostriker 2002): The minimum magnification for both the first and second brightest image can be approximated as half of the total minimum magnification (Oguri et al. 2002). Substituting 𝜇 min = 2 into equation 11 recovers the total magnification distribution for SIS. To model the populations of galaxy groups and clusters we determine Mass Functions (MF) from the CosmoDC2 synthetic catalogue Korytov et al. (2019). For simplicity, we define a galaxy group as a system with mass 10 13 𝑀 ⊙ ≤ 𝑀 < 10 14 𝑀 ⊙ and a galaxy cluster as 𝑀 ≥ 10 14 𝑀 ⊙ . We find all dark matter halos in a redshift interval where the mass of the halo and its constituent galaxies fall in these ranges. We fit Schechter functions to the binned results. The fitted functions match the results of Böhringer, Hans et al. (2017) well at 𝑧 = 0 and show a realistic decline in the mass function towards higher redshift. To calculate the optical depth, we replace 𝜋𝜃 𝐸 ( 𝜎, 𝑧 𝑑 , 𝑧 𝑠 ) 2 𝐷 2 𝑑 in equation 1 with equation 8, Φ ( 𝜎, 𝑧 𝑑 ) with the CosmoDC2 MF, and integrate over 𝑀 instead of 𝜎 . While equation 4 gives the total fraction of lensed sightlines, we are more interested in the number of discoverable lenses. We define a lens as discoverable if its image separation Δ 𝜃 = 2 𝜃 𝐸 is larger than 2/3rds of the FWHM of the PSF of the survey instrument (as in Oguri & Marshall (2010); Yue et al. (2022a)) so that multiple images can be resolved. Further, for point-like sources (i.e. flat-spectrum sources and transients), we require the second faintest image to be detectable at 10 𝜎 confidence so that the lens can be identified. To account for this we integrate equation 1 from 𝜎 min (or equivalently 𝑀 min for groups and clusters) corresponding to 𝜃 𝐸 min and use the magnification bias of the fainter image. The second image from a SIS will often be demagnified, so the magnification bias of the fainter image is typically a bit less than 1, while the magnification bias of the fainter image from an NFW will be about half of the total. For SFRGs and SS-AGN, which in general have extended emission regions, the lensed images will be stretched into arcs and rings, and so for this case we require the brightest image to be detectable at 10 𝜎 confidence (using the magnification bias of the brightest image) because only one arc is needed to identify a lens. The image separation distribution predicted by our model and the lensing optical depths for galaxies, groups, and clusters are shown in Figures 1 and 2 respectively. We see that the optical depth increases rapidly at small redshift as expected, and approximately matches the estimates given in Section 2.1. The galaxy lens optical depth is smaller than the approximation from Oguri (2019). The discoverable optical depth, limited by the PSF of the DSA-2000, is significantly smaller than the total optical depth for galaxies but similar to the total for groups and clusters. The image separation distribution matches the results of Oguri (2006) well despite the simpler model. As expected, galaxies lie mostly at Δ 𝜃 ≈ 1', while groups and clusters correspond to Δ 𝜃 ≈ 10' and 10' ≲ Δ 𝜃 ≲ 100' respectively. The largest uncertainty is in the model of groups, as our simple model does not capture many of the complexities of halo profiles and is constant over the whole mass range. Further, varying 𝛼 between 1 and 2 creates a significant change in the calculated group optical depth. On the other hand, there is inherent uncertainty in all three models from the assumption of spherical symmetry. Using SIS instead of Singular Isothermal Ellipsoids (SIE) for galaxies will overestimate the galaxy lens population by ∼ 10%, which is corrected for in our final numbers (Yue et al. 2022b; Ferrami & Wyithe 2024). Introducing ellipticity will similarly reduce the cross-section of NFW profiles. We do not account for effects such as sub-halo lensing or line-ofsight structures and lens environments. It is shown in Oguri (2006) that a significant fraction of lenses, roughly half for groups and clusters and 10% for galaxies, come from lensing by sub-halos. The environment around these sub-halos boosts their lens capabilities and image separations. At least one of the CLASS lenses is caused by a galaxy within a galaxy group (Auger et al. 2007). In the context of cluster lensing, this is termed galaxy-galaxy strong lensing (GGSL) and it is reported that observational numbers of GGSL exceed expectations of the Λ CDM model (Meneghetti, Massimo et al. 2022; Tokayer et al. 2024). This means that we are likely underestimating the number of lenses in the image separation range of galaxy scale lenses. However, most of these will be below the discoverable limit of the DSA-2000. Further, it is well known that intervening masses along the line of sight between the deflector and the observer could increase the probability of multiple imaging by a significant fraction, especially at high source redshifts, which we also neglect in this study (Fleury et al. 2021). Because these higher-order aspects of the model are likely to increase the number of lenses, for the purpose of this paper it is safe to ignore them.",
"pages": [
3,
4,
5,
6
]
},
{
"title": "2.3 Source Populations",
"content": "The other important piece of the model is the population of the sources. The DSA-2000 will map ∼ 30,000 deg 2 of the sky to a combined 𝜎 𝑛 = 500 nJy/beam rms noise (Hallinan et al. 2019). Matthews et al. (2021) presents detailed radio source counts from the MeerKat DEEP2 image, including direct source counts above 10 𝜇 Jy and statistical counts extrapolating below 10 𝜇 Jy. They find that for sources <10 𝜇 Jy, the differential source count is significantly flatter than Euclidean. Scaling for the 30,000 deg 2 footprint of DSA2000's continuum survey and integrating down to a minimum flux density 𝑆 = 2 . 5 𝜇 Jy for a 5 𝜎 𝑛 combined detection gives an expected total source count of 1 . 45 + 0 . 25 -0 . 10 × 10 9 , while for 𝑆 ≥ 5 𝜇 Jy = 10 𝜎 𝑛 that number is 8 . 6 + 1 . 1 -0 . 4 × 10 8 . To get redshift distributions for different source classes, we model the Star-Forming Radio Galaxy (SFRG) and Active Galactic Nucleus (AGN) populations using the luminosity functions (LF) from the Tiered Radio Extra-galactic Continuum Simulation (T-RECS) (Bonaldi et al. 2018). These have been shown to match observations out to high redshift. Following Bonaldi et al. (2018), we divide the AGN into three sub populations: Flat Spectrum Radio Quasars (FSRQ), BL-Laceratae (BLLac), and Steep Spectrum AGN (SSAGN). We expect SFRGs to make up roughly 95% of all of the sources in any synoptic radio survey, and BLLacs to be the rarest source. From the LF we calculate the number of sources in a redshift interval as where Ω survey is the sky coverage of the survey in steradians ( ≈ 3 𝜋 for the DSA-2000) and Φ ( 𝐿, 𝑧 ) is the LF of the source population. 𝐿 min ( 𝑧 ) = 4 𝜋𝐷 2 𝐿 ( 1 + 𝑧 ) 1 + 𝛼 𝑆 min is the minimum observable luminosity at a redshift 𝑧 , where 𝐷 𝐿 is the luminosity distance, 𝛼 is the spectral index of the source population, 𝑆 min is the 10 𝜎 𝑛 flux sensitivity of the DSA-2000, and the factor ( 1 + 𝑧 ) -( 1 + 𝛼 ) is the standard cosmological radio K-correction. We scaled the total number of sources in the LF model to match the observational prediction from the DEEP2 image given above. In addition to SFRGs and AGN, we build a model for radio SNe. To date, only ∼ 100 radio core-collapse supernovae (ccSNe) have been detected and the first Type Ia radio supernova was detected last year (Bietenholz et al. 2021; Kool et al. 2023). To model the expected rate of radio ccSNe we follow Lien et al. (2011). Because ccSNe are shortlived, the rate of ccSNe is closely related to the cosmic Star Formation Rate (SFR), which has been observational measured to high redshift and accuracy. We use the ccSNe radio luminosity distribution of Bietenholz et al. (2021), which is more complete than the one in Lien et al. (2011). The best-fit distribution is log-normal and incorporates all known radio ccSNe as well as radio non-detections of supernovae. Bietenholz et al. (2021) note that their data is likely biased towards radio detections, finding that 30% of all ccSNe are detected in the radio. We divide the total number of radio ccSNe in the model by 3 to get a more conservative 10% as in Lien et al. (2011). From the luminosity distribution, we determine the number of radio ccSNe at each redshift above the flux limit of the DSA-2000. Because the SNe observations used in Bietenholz et al. (2021) have frequencies ranging from 2-10 GHz, there is uncertainty when converting to 1.4 GHz. We correct 𝐿 min 6 GHz ( 𝑧 ) = 𝐿 min 1.4 GHz GLYPH<16> 6 1 . 4 GLYPH<17> 𝛼 , where we assume 𝛼 = -0 . 7 for the synchrotron emission of radio ccSNe. Further, because radio ccSNe have a mean rise time of log 10 ( 𝑡 rise ) = 1 . 7 days (Bietenholz et al. 2021), we use the single epoch flux limit of the DSA-2000 and account for the DSA-2000's cadence. The rms noise for a single epoch is 𝜎 𝑛 = 2 𝜇 Jy/beam, so a 10 𝜎 𝑛 detection is 20 𝜇 Jy. The cadence will be ∼ 4 months, so conservatively we will detect 1/3 of the radio ccSNe per year. None of the known radio ccSNe have luminosities above 10 29 erg s -1 Hz -1 , but given the wide spread we expect that the highest luminosities could be several times larger. Because of this we set an exponential cutoff at 10 30 erg s -1 Hz -1 , which we can expect to see out to about 𝑧 ≈ 1 . 5 in a single epoch of the DSA-2000. The predicted redshift distributions of SFRGs, AGN, and ccSNe from our model are shown in Figure 3 and compared to the expected distribution of Rubin-LSST sources. As expected, the total source count is dominated by SFRGs until high redshift. Blazars, meaning both FSRQs and BLLacs, are the AGN that will be most useful for time delay measurements, which is explored in Section 5.1. These are the most common sources at very high redshift. We expect the DSA2000 to probe much higher redshifts than the Rubin-LSST because radio emission is much less susceptible to intervening gas or seeing. This is one of the reasons that the DSA-2000 will be effective at lens finding. The total number of expected sources of each type in a DSA-2000 all-sky survey with a 10 𝜎 𝑛 detection are listed in the first row of Table 1.",
"pages": [
6
]
},
{
"title": "2.4 Expected rates",
"content": "The final results of our model are shown in Table 1. The results agree quite well with the empirical estimates given in Section 2.1. As expected from CLASS there are roughly 10 6 total galaxy scale lenses contained in the DSA-2000 all-sky survey, less than 10% of which will be discoverable. The total number of group and cluster lenses are about 10% and 2% of the galaxy number respectively, which is consistent with Oguri (2006). Roughly half of the group and cluster lenses will be discoverable, leading to them making up about a third of the total discoverable lenses, a significant fraction. Yue et al. (2022a) find that the Rubin-LSST will discover about 2000 lensed QSOs; our number is similar. This is a result of the competing mechanisms of the DSA-2000 detecting many more at higher redshifts and simultaneously being able to discover a smaller fraction because of the PSF size. The main purpose of Yue et al. (2022b) is to investigate the discrepancy between the fraction of lensed high redshift ( 𝑧 𝑠 ≳ 5) quasars in observations ( ∼ 0 . 2%) and previous predictions ( ∼ 4%).Yueetal.(2022b)find ∼ 0 . 4 -0 . 8%with their model, and we have ∼ 0 . 3%, which is consistent. The number of discoverable lensed ccSNe per year in the DSA-2000 from the full model is about 2, which is of the same order of magnitude as the empirical VLASS estimate. Oguri & Marshall (2010) conclude that Rubin-LSST will find roughly 8 SNe per year. We expect there to be less lensed SNe at radio wavelengths because not all SNe emit in the radio and the radio emission is weaker than in other wavelengths. The redshift distribution of the deflectors and sources is shown in Figure 4. Almost all deflectors are at 𝑧 𝑑 < 3 and most are at 𝑧 𝑑 ≈ 1, consistent with Oguri & Marshall (2010); Collett (2015); Yue et al. (2022a,b). This justifies the use of the CosmoDC2 catalog, which is limited to 𝑧 < 3, to determine the MF of groups and clusters. We also run the simulation for the survey specs of the SKA-mid and VLASS. The main differences are the sensitivity and resolution. The expected rms noise of the SKA-mid at 1.4 GHz will be 𝜎 𝑛 = 2 𝜇 Jy/beam, and the PSF size will be about 0.4' (Braun et al. 2019). From the DEEP2 source counts, the SKA-mid should find about 3 × 10 8 sources above 10 𝜎 𝑛 . With this, our model predicts that the SKA-mid should see 1.8 ± 0.7 × 10 5 , 1.8 ± 1.2 × 10 4 , and 2.7 ± 1.1 × 10 3 galaxy, group, and cluster lenses, given the same discoverability limit defined in Section 2.2. Despite less depth leading to around a third of the total galaxy lenses, the PSF of the SKA-mid is such that it will resolve over half of them, resulting in about twice as many discoverable galaxy lenses as in the DSA-2000. The SKA-mid will find fewer group and cluster scale lenses than the DSA-2000 because sensitivity is more important for discovering these systems than angular resolution. In addition, we expect about 14 lensed radio ccSNe per year in an SKA-mid all-sky survey. McKean et al. (2015) estimate ∼ 3 × 10 5 lenses in an SKA-mid all-sky survey with 𝜎 𝑛 = 3 𝜇 Jy/beam and a lens detection limit of 15 𝜎 𝑛 and >0.3'. This is slightly more optimistic than our forecast, but broadly consistent. VLASS has 𝜎 𝑛 = 70 𝜇 Jy/beam combined and an angular resolution of 2.5' (Lacy et al. 2020). The DEEP2 source counts indicate that we should expect ∼ 5 × 10 6 sources above 10 𝜎 𝑛 in VLASS (Lacy et al. (2020) estimate 5.3 × 10 6 above 5 𝜎 𝑛 ). With these numbers, we estimate that there should be about 300 discoverable lenses in the full sky survey. As VLASS enters its epoch 3, many of these lenses are likely already in the data (several have recently been identified (Martinez et al. 2024)).",
"pages": [
6,
7
]
},
{
"title": "3.1 Superresolution",
"content": "In the past decade, tremendous progress has been made by the computer vision community with respect to the classical ill-posed inverse problems. These include deblurring (Zhang et al. 2022), deconvolution and superresolution (Alzubaidi et al. 2021), and image inpainting (Yu et al. 2018b). Nearly all of these advances were borne out of the deep learning revolution, as efficient neural network architectures and training strategies have enabled powerful learning-based tools for machine vision. Astronomy naturally lends itself to these tools because sparse sampling and ill-posedness arise in many astronomical imaging and reconstruction contexts, especially in interferometry. Several groups have begun developing machine learning methods for interferometric image reconstruction (Connor et al. 2022; Aghabiglou et al. 2024; Mars et al. 2024). One reason this approach is suitable for radio astronomy is that the PSF of an array is given deterministically by the spatial distribution of antennas and the observing frequency: The on-sky response of an interferometer is the 2D Fourier Transform of its sampling in UV-space (the aperture plane). Therefore, prior physical knowledge of the PSF can be incorporated in the model either via training data (Connor et al. 2022) or fed directly to the network itself (Mars et al. 2024). This is not the case in ground- based optical astronomy where 'seeing' and complex optics mean the PSF is not known a priori. Asasimple demonstration, we use the super-resolution and imageplane deconvolution method POLISH to show how strong lenses with image separations below the PSF scale can be recovered with machine learning. POLISH is a supervised machine learning model that learns the mapping between the true sky and observed images (the 'dirty image' in radio parlance) (Connor et al. 2022). It uses the Wide Activation for Efficient and Accurate Image Super-Resolution (WDSR) architecture (Yu et al. 2018a), but any super-resolution neural network could be swapped in (e.g., a standard U-Net or the Efficient Super-Resolution Transformer (Lim et al. 2017)). Wehave trained a POLISH network on forward-modelled synthetic data using a DSA-2000 PSF averaged over the whole band, resulting in an angular resolution of ∼ 3.3'. This is worse resolution than the top of the band, where many strong lenses could be found, but we offer this as a toy example. The sky model is described in Connor et al. (2022), with the addition of strongly lensed galaxies in both the training and validation set. In Figure 5 we show an example validation image that contains multiple lenses. Lensed systems that are undetected in dirty image, including both arcs (left inset panels) and Einstein rings (right inset panels), are identifiable in the POLISH reconstruction. In the future, we plan to develop super-resolution methods explicitly for lens finding, but we take this as a promising sign that the DSA-2000 will be able to find systems with 𝜃 𝐸 ≳ 0 . 5 '' .",
"pages": [
7,
8
]
},
{
"title": "3.2 Identification",
"content": "The first major hurdle for strong lensing science with DSA-2000 and other surveys is identification. The large number of sources produced by these surveys makes visual inspection by experts impossible. Despite much work on automated lens discovery (see Lemon et al. (2023) for a review), little effort has been spent on identification in the radio. On the one hand, radio lens images are made cleaner by the lack of radio emission from the massive quiescent galaxies that make up the deflector population. However, the large, extended lobes of radio galaxies can easily be misidentified as lensing features. Further, color and photometric redshift information from UV and optical surveys have traditionally played a role in lens identification, despite being a foreground to the morphology of the lensed source. Catalog-level searches, where large cuts are made based on certain features before a smaller subset is visually inspected, have been successful in the past. CLASS selected for flat spectrum radio sources ( 𝛼 > -0 . 5) (Myers et al. 2003). This had the advantage of picking out only blazars, whose radio emission is dominated by a small compact core, thus eliminating any confusion with intrinsic structure. A similar strategy would likely be effective in the DSA-2000. However, dedicated follow up of all of these sources, as in the ∼ 11,000 blazars in the initial CLASS sample, will be impossible. Similarly, a visual inspection of even a small subset of the O( 10 7 ) blazars expected in the DSA-2000 is ambitious (hence the need for intermediate automated steps in the identification pipeline). Still, targeting flat spectrum sources could increase the yield of any search, and spectral information from the whole DSA-2000 band will be useful for determining whether components are different sources or multiple images. Jackson & Browne (2006) show that combining astrometric data from radio and optical surveys can significantly improve the efficiency of lens searches. If the optical emission is dominated by the deflector galaxy then there will exist an offset between the centroid position of the optical and radio sources in a lens system. This offset was exploited to find lenses with separation down to ∼ 1' even with the poor 5' resolution of the Faint Images of the Radio Sky at Twenty-one centimeters (FIRST) survey. They predict that this effect will become more efficient at lower radio fluxes because the optical flux is more likely to be dominated by the defelctor galaxy. Given the sensitivity of the DSA-2000 and its significant overlap with current and planned large surveys in other wavelengths, this could prove to be a very effective way of overcoming the limitations of its PSF. Regardless, exploiting the extra information from overlap with other surveys will be crucial for any lens search in the radio. Machine learning models have been successfully used to discover many new candidate lens systems. Convolutional Neural Networks (CNNs) in particular are effective at classifying astronomical images. The main disadvantage with CNNs and other supervised learners is the need for large realistic training sets. Using CNNs on simulated data for the International LOFAR Telescope (ILT) at 150 MHz, Rezaei et al. (2022) are able to recover over 90% of galaxy-size lenses with a false-positive rate of only 0.008%. They find that a strong 20 𝜎 detection and large separation 𝜃 𝐸 ≥ 3 / 2 beam size (stricter than the discoverable limit imposed in Section 2.2) are necessary for reliable identification with the CNN. Incorporating superresolution models into the lens-finding routine would significantly increase the yield of such a CNN, especially at high S/N. Resolving lenses down to 𝜃 𝐸 ≈ 0 . 5' with would enable the same level of accuracy as in Rezaei et al. (2022). Simulating accurate DSA-2000 lenses and training models for identifying them is a goal of our future work. The SKA-mid will already have sufficient resolution for these purposes, making CNNs an attractive option for lens identification in its all-sky survey. The main limitation of any ML-based approach to group/cluster lens finding is the difficulty of creating realistic training sets. The irregularity of group/cluster lensing potentials may require building a forward-modelled training set from ray-tracing in large cosmological magnetohydrodynamical simulations such as TNG-Cluster, which should have a heterogenous population of massive clusters (Nelson et al. 2024). More generally, a realistic training set of strong lenses across scales could be produced with radio source samples from T-RECS and a deflector population drawn from cosmological simulations.",
"pages": [
8,
9
]
},
{
"title": "4 MULTIWAVELENGTH SYNERGIES",
"content": "Acquiring redshifts for lensed galaxies and deflectors is critical for the application of strong lensing to cosmology and astrophysics. We consider here how well we could do in the absence of targeted followup observations, capitalizing on the suite of large survey telescopes that will be operating in the mid- to late-2020s. DSA-2000 spectroscopic HI survey In addition to the cadenced allsky survey, the DSA-2000 will undertake a large spectroscopic HI survey over the full visible sky (Dec > -30 · ) (Hallinan et al. 2019). This is expected to produce ∼ 10 6 HI galaxies at 𝑧 < 1. Most deflectors will be massive elliptical galaxies with limited 21 cm emission, but a sub-set of early-type galaxies with 𝑛 𝐻 ≥ 10 20 cm -2 𝑧 -2 𝑑 will be detected. Some late-type deflector galaxies will have enough cold gas to obtain a redshift. Additionally, 21 cm absorption in the inner CGM of deflector galaxies may be present in the lensed source's continuum spectrum, providing a measurement of cold gas along different sightlines of the same CGM (see Rudie et al. (2019) for a similar application in lensed quasar systems). Thus, some deflector galaxies will have a spectroscopic redshift without any external O/IR data. Of the ∼ 10 6 HI galaxies detected by the DSA-2000 in emission, we find that only a handful will be strongly lensed, due to their shallow redshift distribution. Spectroscopic radio observations will not provide a significant portion of redshifts required for our purposes, but may be a minority of interesting deflectors. The Dark Energy Spectroscopic Instrument (DESI) will produce redshifts of ∼ 40 million galaxies and quasars (DESI Collaboration et al. 2016; Dey et al. 2019). Its 3 yr public data release is expected before the DSA-2000's first light. DESI's Bright Galaxy Survey (BGS) targets galaxies with 𝑟 -band magnitudes brighter than 19.5. For the luminous red galaxies (LRGs), emission line galaxies (ELGs), and quasars, the depth will be 𝑟 ≈ 23. Each class of target will have a different redshift distribution, but together they will provide a rich catalog of spectroscopic redshifts for deflectors in the lensing systems discovered by DSA-2000. The BGS and LRG samples will be over-represented as deflector galaxies, whereas the ELG and QSO samples may make up a small fraction of lensed DSA-2000 sources. ELGs, by definition, have strong emission lines indicative of star formation, which produce synchrotron radiation detectable by radio surveys. Some of these will be strongly lensed star-forming radio galaxies with 𝑆 𝜈 above 5 𝜇 Jy. Still, we do not expect more than a small minority of the radio continuum lenses to be detected in DESI due to the relative number of total sources. SPHEREx is a near-infrared space mission expected to launch before Spring 2025. It will map the full 4 𝜋 sr sky with 6 arcsecond pixels and 96 color bands, producing a catalog of hundreds of millions of galaxies with 'spectro-photometric redshifts' (Doré et al. 2018). Although its PSF it too large to discover strong lenses directly, the galaxy catalog will be valuable for obtaining reliable redshifts of both deflectors and lensed galaxies. The Rubin Observatory will find billions of galaxies in its Legacy Survey of Space and Time (LSST) (Ivezić et al. 2019). Most of these galaxies will be at 𝑧 𝑠 ≲ 1 . 5 (Collaboration et al. 2021), providing an excellent deflector catalog for the DSA-2000 lens candidates. In the DSA-2000/LSST overlapping footprint between -30 and +30 Declination, we find that a significant fraction of deflectors and several thousand lensed sources in the DSA-2000 sample will have photometric redshifts from the Rubin Observatory. Roman The High Latitude Spectroscopic Survey (HLSS) is expected to detect 10 million galaxies in 𝐻𝛼 between 1 < 𝑧 < 2and ∼ 2 million [OIII] galaxies at 𝑧 = 2 -3 using the near-IR grism and wide-field camera (Wang et al. 2022). Roman's 2,000 deg 2 footprint will be fully mapped by DSA-2000. With a dust-attenuated 𝐻𝛼 flux higher than 10 -16 erg s -1 cm -2 , several million of these galaxies will be detectable at GHz frequencies at > 5 𝜇 Jy (Murphy et al. 2011). Assuming a mean lensing optical depth of 5 × 10 -4 , this suggests that O( 10 3 ) will be strongly lensed and detected by both the DSA-2000 and Roman , the latter providing 0.1' resolution and a source redshift. Euclid is a visible and near-infrared space telescope that launched in July 2023 (Collaboration et al. 2024). Its large photometric survey will find over one billion galaxies, with a spectroscopic survey that will obtain accurate redshifts for ∼ 30 million galaxies mostly at 0 . 9 < 𝑧 < 1 . 8. Euclid and the DSA-2000 share over 10,000 deg 2 of sky, providing an excellent sample of photometric and spectroscopic redshifts of lens systems.",
"pages": [
9
]
},
{
"title": "5 STRONG LENSING APPLICATIONS",
"content": "A key limitation of strong lensing science is the small number of known systems, especially at radio wavelengths. We discuss the impact that the large expected number of radio lenses and their multiwavelength counterparts will have on several important applications of strong lensing.",
"pages": [
9
]
},
{
"title": "5.1 Time-delay cosmography & 𝐻 0",
"content": "Light from multiple images in a gravitational lens will reach the observer at different times. For continuum sources, we do not observe the difference in arrival times, but transients or time-variable sources allow us to measure the time delay. This time delay is due to a difference in path lengths and gravitational time dilation near the deflector. As such, the time delay encodes information about the geometry of the universe, and is inversely proportional to 𝐻 0 (Oguri 2019). Much work has been dedicated to measuring these time delays and using them to constrain 𝐻 0 (Treu & Marshall 2016; Birrer et al. 2024). The H0LiCOW project has measured 𝐻 0 to 2.4%, which is independent of and competitive with state-of-the-art 𝐻 0 constraints (Wong et al. 2019). A large increase in the number of measurable time-delay systems will allow even tighter constraints and could help settle the current Hubble tension. We estimate the number of lensed variable AGN that will be useful for measuring 𝐻 0 in the DSA-2000. To first order, all blazars (FSRQs and BLLacs) are inherently variable, ∼ 10 3 of which will be discoverable as lenses in DSA-2000 data (Table 1). However, several factors complicate our ability to use them for time delay measurements; we would like to know how many will be in a \"gold sample\" for which the time-delay can be reliably determined. To date, the most extensive study of radio AGN variability is the OVRO 40 m blazar monitoring campaign. Since 2007 this program has monitored around 1500 blazars with a cadence of two weeks (Richards et al. 2011). Richards et al. (2011) provides distributions of the intrinsic modulation index, a measure of the relative intrinsic variability of the AGN, for FSRQs and BLLacs. The DSA-2000 will measure flux variability at the level of a few percent, but to be conservative we consider candidates with ≥ 10% variability. We also want sources with a strong detection, so we chose sources with flux ≥ 20 𝜎 𝑛 = 10 𝜇 Jy. The OVRO blazars are monitored at 15 GHz but we expect lower variability at 1.4 GHz. Fan, J. H. et al. (2007) find that blazars are about 20% less variable at 4.8 GHz than 14.5 GHz. Sotnikova et al. (2024) find that radio AGN at 22.3 GHz are about 35% more variable than at 2.3 GHz, although there is little difference between 11.2 GHz and 2.3 GHz. We assume that the mean modulation index drops by 40% from the OVRO 40 m data to 1.4 GHz for the DSA-2000. Further, for multiple images, we need at least two of the images to be bright enough to measure the time delay so we use the magnification bias of the fainter image. With these restrictions, the OVRO data, and our lens model, we estimate that there will be 68 ± 30 lensed FSRQs and 23 ± 10 lensed BLLacs with sufficient brightness, variability, and image separation to measure time delays. The same analysis for the SKA-mid gives 120 ± 50 and 43 ± 16 lensed FSRQs and BLLacs. Of the lensed AGN, the most useful systems for time delay analysis will be those that have four or more images because they allow for multiple constraints on the time delay. A SIS is only capable of creating two images and a spherical NFW can make three, but factors such as ellipticity, irregularities in the lens potential, and external environments of real halos will allow a number of these systems to make four or more images. The fraction of quad galaxy-QSO lenses predicted in Rubin-LSST is ∼ 10-15% (Oguri & Marshall 2010; Yue et al. 2022a). However, of the statistically complete CLASS sample, 5 out of 13 lensed blazars were quadruply imaged and one had six images (B1933+503 has two sets of quadruply imaged sources Sykes et al. (1998)). Therefore we can expect 100-400 quadruply-lensed blazars in the DSA-2000 all-sky survey and about twice as many in the SKA-mid, roughly 10% of which may be used in practice for measuring 𝐻 0 . We note that quad systems often have higher external shear due to their preponderance of group or cluster environments, which results in an added systematic in lens modeling (Holder & Schechter 2003). Wong et al. (2019) reach a 2.4% measurement of 𝐻 0 with a sample of 6 lensed quasars, and predict that around 40 lenses are needed to constrain 𝐻 0 to within 1%. Jee et al. (2016) argue that with a well-defined sample of quadruply imaged quasars, and combining time-delay distances and velocity dispersion measurements to estimate angular diameter distances within 5%, 𝐻 0 can be measured to the same precision as Planck with as few as 10 systems. Napier et al. (2023) reach a 10% measurement with three galaxy clusters, and estimate that roughly 50 will be needed for a sub 1% constraint, assuming that modeling uncertainties can be reduced in coming years. While we expect uncertainty to decrease significantly with a larger sample of lensed quasars, there are several systematics that must be kept under control for an accurate measurement. Namely, an accurate redshift measurement is necessary, ideally of both the source and deflector, as well as precise measurements of the time delay, complete sampling over many cycles of the time delay to negate contamination from micro-lensing, accurate lens models, and constraints on any line of sight structures. If we assume that all of the lenses in our gold sample can be modeled to the accuracy of the H0LiCOW sample - a roughly 6% combined uncertainty on the time delay, mass model, and line of sight contribution for each lens - then a sample of ∼ 100 lensed quasars would give about a 0.6% constraint on 𝐻 0 . This is of course not possible with the DSA-2000 or SKA-mid alone but will require dedicated follow-up at multiple wavelengths. As discussed in Section 4, we expect that a significant minority of lenses should have redshifts available in other concurrent public surveys. Interesting subsamples of lenses will require targeted follow-up for both mass modeling and spectroscopic redshifts. The DSA-2000's cadenced all-sky survey will image each field above -30 Dec once every 4 months. Typical time delays for galaxy, group, and cluster lenses in our model are a month or less, several months, and a year or more respectively (Oguri et al. 2002). Special systems could therefore be visited with a higher cadence, similar to how pulsar fields will be visited regularly for timing experiments. So far we have only considered lensed blazars for time delay measurements, which are desirable because of their characteristic variability and compact emission regions. Lensed transients will also be useful for time delay measurements, provided that they are detected early and monitored closely. The variability of radio emission from TDEs is too slow to make them useful for this application (on-axis jetted TDEs can have shorter rise times, but they are significantly rarer such that we do not expect any lensed jetted TDEs in upcoming radio surveys), but SNe have faster rise times. We expect O( 1 ) lensed radio ccSNe per year in the DSA-2000 and O( 10 ) in the SKA-mid that can be used for 𝐻 0 measurements.",
"pages": [
10
]
},
{
"title": "5.2 Dark Matter Structure",
"content": "Gravitational lensing has been very useful for determining the structure of galaxies, groups, and clusters at cosmological distances and distinguishing between dark matter models (see Vegetti et al. (2024); Natarajan et al. (2024) for a review). The best of these studies use high-resolution HST images, but radio observations also played a pivotal early role. Cohn et al. (2001) used the 10-image CLASS lens B1933+503, which provides many more constraints than usual, to determine the density profile of the deflecting galaxy, finding it close to isothermal. Wucknitz et al. (2004) find the same for B0218+357. Dalal & Kochanek (2002) used radio observations of flux ratio anomalies between multiple images to constrain the mass fraction of dark matter substructure to 2%. Radio observations are especially suited for this type of study because they are less susceptible to micro-lensing (Koopmans et al. 2003; Kochanek & Dalal 2004). Despite this, dark matter studies with radio lenses have been limited for many years by the small sample size (Vegetti et al. 2024). Because resolution is a key factor in the ability to accurately model mass distributions or detect substructure, the DSA-2000 alone will not be able to use lens systems to study dark matter. Its main utility will come from identifying huge numbers of systems for detailed followup. The extraordinary resolution of the ngVLA on a large sample of lenses selected by the DSA-2000 will revolutionize this field. In the more immediate future, the SKA will also provide enough resolution for these studies in their overlapping sky coverage. Another exciting application of lensing is using a central image to study small scales near the centers of galaxies (Treu 2010; McKean et al. 2015; Shajib et al. 2024). A SIS lens only forms two images, but for realistic galaxy profiles a highly demagnified image is expected to form at the center of the image configuration. This image will be sensitive to the mass contained within very small distances of the deflector's center, and so can be used to study supermassive black holes at cosmological distances. This is especially suited for radio wavelengths because in many cases the deflector will be radio-quiet allowing the center image to be detected. Because the center image is highly demagnified, only sensitive radio surveys such as the DSA2000 will be able to reliably identify them. At least one such central image has been detected in the radio with brightness ∼ 0 . 8 mJy at 8.46 GHz (Winn et al. 2004), so we can likely expect O( 10 2 ) or more with the DSA-2000. A main challenge will be disentangling the faint emission of the central image from the other images. For groups and clusters, whose inner density profiles are shallower than isothermal, central images are more common and often less demagnified. The very inner profiles of groups/clusters can thus be studied, which are hard to constrain with images near the Einstein radius alone. These central images will be easier to identify because the other images will be farther away (although many clusters have large central elliptical galaxies emitting in the radio that will obscure the image).",
"pages": [
10,
11
]
},
{
"title": "5.3 Polarization",
"content": "Measuring full Stokes information of gravitationally lensed signals offers insight into the source plane because polarization is not affected by gravitational potentials. This is an advantage of radio surveys, where polarization information is often stored and Faraday rotation measure (RM) can be measured. This enables us to study propagation effects along different sightlines for lensed objects. For example, Mao et al. (2017) used the VLA to measure the polarization properties of two lensed images of CLASS B1152+199, finding a large difference in RM ( + 9 . 7 ± 0 . 5 rad m -2 for image ' 𝐴 ' and + 517 ± 3 rad m -2 for image ' 𝐵 '), which is due to the magnetized plasma in the lens galaxy's interstellar medium. The DSA-2000 will have the ability to measure full polarization information for any source it detects, with a maximum |RM| of roughly 10 4 rad m -2 . Compact sources such as Blazars (and especially BL Lac objects) can show significant polarization fractions at 1 GHz. Assuming a 50 𝜎 detection threshold in total power, we estimate that the DSA-2000 could detect Stokes Q and U for 5-10 million AGN in its 5 year continuum survey. Using the empirical CLASS lensing optical depth and image separation cut, we take 𝜏 𝐴𝐺𝑁 ≈ 5 × 10 -4 . We estimate that the DSA-2000 could find O( 10 3 ) strong lenses for which polarization properties could be used to model the lens distribution and study magnetic fields at cosmological distances.",
"pages": [
11
]
},
{
"title": "5.4 Other applications",
"content": "In addition to constraining 𝐻 0 , time-delay measurements are weakly sensitive to other cosmological parameters, such as Ω 𝑚 , Ω Λ , and the dark energy equation of state parameter 𝑤 (Natarajan et al. 2024). Cluster lenses are used for studying dark energy as well (Macciò 2005; Gilmore & Natarajan 2009; Jullo et al. 2010). A large influx in measured time delays and observed cluster lenses with the DSA2000 will allow better constraints on these cosmological parameters. Lenses are also commonly used as 'nature's telescopes'. The magnification of background sources allows very distant galaxies to be studied even by smaller telescopes (Jackson 2011). This is especially true for group/cluster lenses, where we expect the magnifications to be larger (Robertson et al. 2020). We expect to see O( 10 4 ) lenses with 𝑧 𝑠 > 5 in the DSA-2000 (figure 4), many of which will be group and cluster lenses due to their high magnifications. These high redshift sources will provide insight into the population of radio sources in the early universe. Lensing statistics can also be used to test cosmological models. The abundance of giant arcs in cluster lenses was a topic of much debate in the past few decades (Meneghetti et al. 2013). Similarly, the rate of GGSL events in clusters may be in tension with the Λ CDM(Meneghetti et al. 2020; Meneghetti, Massimo et al. 2022; Tokayer et al. 2024). These studies require a large sample of lenses for accurate statistics, which is only now becoming possible with large surveys such as the DSA-2000.",
"pages": [
11
]
},
{
"title": "6 CONCLUSIONS",
"content": "In this paper, we forecast expected strong lensing rates in the upcoming DSA-2000 and SKA-mid wide-field radio surveys. We first provide empirical estimates based on previous surveys and simulations, and then develop a detailed forward model that accounts for the source and deflector populations, finding them to be in good agreement. Notably, we model the expected number of galaxy group and cluster scale lenses because these systems will be easily discovered due to their wide angular separations. We find that both the DSA-2000 and the SKA-mid will discover roughly 10 5 strong lens systems. We discuss strategies for identifying these lenses in the data, which will all benefit from emerging superresolution techniques. Finally, we discuss the scientific application of the huge numbers of lenses that will be discovered by these surveys. One of the most exciting applications is 𝐻 0 cosmography with variable and transient sources. The DSA-2000 and SKA-mid will discover about 100 and 200 lensed flat spectrum AGN with >10% variability respectively, as well as about O( 1 ) and O( 10 ) lensed transients per year. With dedicated multi-wavelength follow up these systems could be used to constrain 𝐻 0 to within 1%. The new lens systems will also be useful for studying the distribution of dark matter at cosmological distances, among other applications.",
"pages": [
11
]
},
{
"title": "ACKNOWLEDGEMENTS",
"content": "Weare grateful to Schmidt Sciences for supporting Samuel McCarty as a Summer Undergraduate Research Fellow at Caltech. We thank Kim-Vy Tran, Tony Readhead, and Tommaso Treu for helpful conversations on strong lensing, as well as Paul Schechter for insights into quad systems.",
"pages": [
11
]
},
{
"title": "DATA AVAILABILITY",
"content": "We have placed a reproduction package on the public GitHub repository available at https://github.com/smmccrty/ radiolensing .",
"pages": [
12
]
},
{
"title": "REFERENCES",
"content": "R., Motta V., 2012, The Astrophysical Journal, 749, 38 This paper has been typeset from a T E X/L A T E X file prepared by the author.",
"pages": [
12,
13
]
}
]
2024arXiv241204059S
https://arxiv.org/pdf/2412.04059.pdf
<document>
<section_header_level_1><location><page_1><loc_24><loc_85><loc_76><loc_86></location>The puzzling long GRB 191019A: Evidence for Kilonova Light</section_header_level_1>
<text><location><page_1><loc_11><loc_81><loc_89><loc_84></location>G. Stratta , 1, 2, 3, 4 A. M. Nicuesa Guelbenzu , 5 S. Klose , 5 A. Rossi , 6 P. Singh, 1, 6 E. Palazzi , 6 C. Guidorzi, 7, 8, 6 A. Camisasca, 9 S. Bernuzzi, 10 A. Rau, 11 M. Bulla, 7, 8, 12 F. Ragosta, 13, 14 E. Maiorano, 6 and 15</text>
<text><location><page_1><loc_46><loc_80><loc_52><loc_81></location>D. Paris</text>
<text><location><page_1><loc_14><loc_56><loc_85><loc_79></location>1 Institut fur Theoretische Physik, Goethe Universitat, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany 2 INAF - Osservatorio di Astrofisica e Scienza dello Spazio, via Piero Gobetti 93/3, 40129 Bologna, Italy 3 Istituto di Astrofisica e Planetologia Spaziali, via Fosso del Cavaliere 100, I-00133 Roma, Italy 4 Istituto Nazionale di Fisica Nucleare-Roma 1, Piazzale Aldo Moro 2, I-00185 Roma, Italy 5 Thuringer Landessternwarte Tautenburg, Sternwarte 5, 07778 Tautenburg, Germany 6 INAF - Osservatorio di Astrofisica e Scienza dello Spazio, via Piero Gobetti 93/3, 40129 Bologna, Italy 7 Department of Physics and Earth Science, University of Ferrara, Via Saragat 1, 44122 Ferrara, Italy 8 INFN - Sezione di Ferrara, Via Saragat 1, 44122 Ferrara, Italy 9 Astronomical Observatory of the Autonomous Region of the Aosta Valley (OAVdA), Loc. Lignan 39, I-11020, Nus (Aosta Valley), Italy 10 Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universitat Jena, 07743, Jena, Germany 11 Max-Planck-Institut fur Extraterrestrische Physik, Giessenbachstraße 1, 85748 Garching, Germany 12 INAF, Osservatorio Astronomico d'Abruzzo, via Mentore Maggini snc, 64100 Teramo, Italy 13 Dipartimento di Fisica 'Ettore Pancini', Universit'a di Napoli Federico II, Via Cinthia 9, 80126 Naples, Italy 14 INAF - Osservatorio Astronomico di Capodimonte, Via Moiariello 16, I-80131 Naples, Italy 15 INAF - Osservatorio Astronomico di Roma, via Frascati 33, 00040 Monte Porzio Catone, Italy</text>
<section_header_level_1><location><page_1><loc_45><loc_53><loc_55><loc_54></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_27><loc_86><loc_52></location>GRB 191019A was a long Gamma-ray burst (GRB) lasting ∼ 65 s and, as such, originally thought to be linked to a core-collapse supernova. However, even though follow-up observations identified the optical counterpart close to the bright nucleus of a nearby ancient galaxy ( z =0.248), no associated supernova was found. This led to the suggestion that the burst was caused by the merger of two compact stellar objects, likely in a dense circumnuclear environment. By using a recently developed diagnostic tool based on prompt emission temporal properties, we noticed that GRB 191019A falls among those long GRBs which are associated with compact mergers and with evidence of kilonova light. We thus re-analyzed unpublished GROND multi-color ( g ' r ' i ' z ' JHK s ) data obtained between 0.4 and 15 days post trigger. Image subtraction confirmed the optical counterpart in all four optical bands, with GROND tracking its fading until 1.5 days post-burst. Incorporating publicly available Swift -XRT data, a joint fit of an afterglow plus a kilonova model revealed a better match than an afterglow-only scenario. The resulting kilonova properties resemble those of AT2017gfo associated with the binary neutron star merger GW170817, with a total ejected mass of ∼ 0 . 06 M ⊙ . Contrary to previous findings inferring a high-density circumburst environment ( n 0 ∼ 10 7 -8 cm -3 ), our analysis finds standard conditions ( n 0 ∼ 1 cm -3 ), suggesting the long duration of GRB 191019A was intrinsic rather than due to jet interaction with a dense external medium.</text>
<text><location><page_1><loc_14><loc_23><loc_50><loc_24></location>Keywords: Gamma-ray bursts - Compact objects</text>
<section_header_level_1><location><page_1><loc_42><loc_19><loc_58><loc_20></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_9><loc_92><loc_18></location>The empirical classification of gamma-ray bursts (GRBs) into two classes according to the bimodal burst duration and hardness distribution (see, e.g., Kouveliotou et al. 1993), has been interpreted as the evidence of two different progenitor channels. Indeed, it was found that the vast majority of sufficiently nearby long GRBs are found to be spatially and temporally coincident with core-collapse (CC) supernovae (SNe) and hosted in star-forming galaxies. On the other hand, short GRBs lack any associated SN and are typically located in the outer regions of early-type galaxies, suggesting a progenitor belonging to an old stellar population, formally consistent with compact binary mergers. This</text>
<text><location><page_2><loc_8><loc_79><loc_92><loc_92></location>picture was confirmed by the association of the short GRB 170817A with the gravitational wave event GW170817 that was compatible with a binary neutron star (NS-NS) merger (e.g., Abbott et al. 2017). From the very same event and thanks to its spatial proximity (40 Mpc), the first robust evidence of a kilonova (AT2017gfo) was obtained (e.g., Coulter et al. 2017), confirming the predicted thermal emission from matter released and heated during a NS-NS merger (e.g., Li & Paczy'nski 1998). In the last decade, kilonova signatures were observed in a number of past short GRBs that were sufficiently nearby, as for instance GRB 130603B ( z =0.3565, Tanvir et al. 2013), GRB 150101B ( z =0.134, Troja et al. 2018), GRB 160821B ( z =0.1613, Troja et al. 2019), though with much lower significance than AT2017gfo due to the larger distances of these events (see, e.g., Troja 2023 for a review).</text>
<text><location><page_2><loc_8><loc_65><loc_92><loc_79></location>In recent years, the standard long and short-GRB progenitor paradigm has been blurred by mounting evidence of the existence of long GRBs with no associated CC-SN but with evidence of a kilonova component, more consistent with a compact binary merger progenitor. Two striking examples are the long GRB 211211A at redshift z = 0 . 076 (350 Mpc; e.g., Troja et al. 2018; Mei et al. 2022; Rastinejad et al. 2022; Yang et al. 2022; Gompertz et al. 2023) and GRB 230307A at z = 0 . 0646 (300 Mpc; e.g., Levan et al. 2023), for which a kilonova was clearly identified in the optical afterglow. In addition, evidence of short GRBs showing massive star collapse properties were also found (e.g., Ahumada et al. 2021; Rossi et al. 2022) further contributed to dismantle the standard GRB classification scheme. These results suggest that some of the past GRB progenitor classifications based on the burst duration and hardness only, have to be revisited. This holds in particular for GRB 191019A.</text>
<text><location><page_2><loc_8><loc_40><loc_92><loc_65></location>GRB 191019A was discovered (Simpson et al. 2019) with the Burst Alert Telescope (BAT; Barthelmy et al. 2005) on board the Neil Gehrels Swift Observatory (Gehrels et al. 2004). It showed a complex burst light curve with multiple peaks with a duration T 90 = 64 . 6 ± 4 . 5 s. Its 15-150 keV spectrum could be well modeled with a power law with a photon index of 2 . 25 ± 0 . 05 and a fluence of 10 -5 erg cm -2 (Krimm et al. 2019) putting this source among the soft/long GRBs. Follow-up observations with the Swift X-ray telescope (XRT; Burrows et al. 2005) robustly identified an uncatalogued X-ray source (Evans et al. 2019). Rapid follow-up observations with the Zeiss-1000 1-m telescope of Tien Shan Astronomical Observatory revealed only one optical source within the XRT error circle with R = 18 . 37 ± 0 . 03 mag at T 0 +14 min (Reva et al. 2019). The position of this source was coincident with a catalogued object quoted to be fainter than the observed one, leading to the interpretation this is the host galaxy of GRB 191019A with the afterglow on top of it (Reva et al. 2019). The Zadko telescope (Coward et al. 2017) observed GRB 191019A field at similar epochs and confirmed the object reported by Reva et al. (2019) with R =18.92 mag, with no significant flux variation within 1.5 hours (Gendre et al. 2019). Similar claims of a lack of flux variations were provided by other teams performing early-epoch observations (Fynbo et al. 2019, Nicuesa Guelbenzu 2019, Perley et al. 2019a, Zhu et al. 2019). However, late observations performed with the Nordic Optical Telescope (NOT) at T 0 +3.25 days were analyzed with image subtraction methods and revealed that the source was fading, confirming the optical transient plus host identification (Perley et al. 2019b).</text>
<text><location><page_2><loc_8><loc_24><loc_92><loc_40></location>Spectroscopic analysis of the host galaxy showed several absorption lines at redshift z = 0 . 248 and a spectral energy distribution consistent with a galaxy dominated by an old stellar population ( > 1 Gyr), with a star formation rate (SFR) of 0 . 06 ± 0 . 03 M ⊙ yr -1 , a stellar mass of 3 × 10 10 M ⊙ , and small dust extinction of A V = 0 . 19 ± 0 . 08 mag (Levan et al. 2023). The measured SFR is at odds with the typical high values inferred in long GRB hosts (but see Rossi et al. 2014). Further hints of anomalies for the long GRB 191019A came from the absence of an associated SN, despite the low redshift. Deep limits in g, r and z obtained between 2 and 73 days with the NOT and optical imaging with the Hubble Space Telescope at 30 and 184 days, put strong constraints on any SN emission up to > 20 times less luminous than SN1998bw (Levan et al. 2023). All these properties are at odds with a massive-star origin of GRB 191019A, while formally consistent with a compact object binary merger progenitor. Though, the deep limits obtained with the NOT Telescope at > 2 days could not confirm the presence of an early kilonova emission (Levan et al. 2023).</text>
<text><location><page_2><loc_8><loc_10><loc_92><loc_24></location>An interesting feature characterizing GRB 191019A is its projected distance from the host galaxy center of ≲ 100 pc, the closest distance among all known short GRBs to their corresponding galactic nucleus (Levan et al. 2023). Indeed, compact binary progenitors formed in stellar binary systems are thought to have large kick velocities acquired during the CC-SN phase of the binary components, and for this reason short GRBs are typically found in the outskirts of their host galaxies, with large offsets from the center (e.g., Berger 2014; Fong et al. 2022; O'Connor et al. 2022). As noted by Levan et al. (2023), the proximity of GRB 191019A to the host galaxy center suggests that the possible progenitor compact binary system could have formed through dynamical encounters, which are thought to be favored in the dense gaseous environment of supermassive black hole surrounding disks through kinetic energy dissipation ('gas-capture' binary formation channel, Tagawa et al. 2020). An exciting consequence of this scenario is that dense environments</text>
<text><location><page_3><loc_8><loc_78><loc_92><loc_92></location>could also alter the prompt emission properties of a short GRB, making it longer and softer (Lazzati et al. 2023). This possibility was explored for GRB 191019A and it was found compatible with an environmental density of the order of 10 7 to 10 8 cm -3 (Lazzati et al. 2023). Starting from Lazzati et al.'s conclusions, Wang et al. (2024) investigated the interaction of a possible kilonova ejecta from GRB 191019A with a dense circumstellar medium (CSM) that could be detected years after the merger. Their model predicts that in a very dense environment, the smaller the kilonova ejected mass, the fainter is the radioactively-powered luminosity but the higher is the contribution from kilonova-CSM interaction at late times. No evidence of kilonova ejecta-CSM interaction was found for GRB 191019A so far, and a possible detection in the future would require an ejected mass less than 2 × 10 -5 M ⊙ (Wang et al. 2024).</text>
<text><location><page_3><loc_8><loc_65><loc_92><loc_79></location>Here we further investigate the properties of GRB 1901910A and the nature of the optical transient that followed the burst, with the goal to find arguments in favour or against its possible compact merger origin. We use the recently developed GRB prompt emission minimum variability (MVT) criterion (Camisasca et al. 2023) to explore the properties of the burst, and perform a complete reanalysis of the light curve of the optical/X-ray transient with particular emphasis on so far not analyzed GROND (Greiner et al. 2008) observations starting 10.3 hours post burst (Nicuesa Guelbenzu 2019). We use the sophisticated Nuclear - Multi-Messenger Astronomy (NMMA, Dietrich et al. 2020; Pang et al. 2023) software package which allows for afterglow and kilonova joint Bayesian inference, to explore if the multi-color light curve of the optical transient can be understood as powered by afterglow emission only or if a kilonova component was present too.</text>
<text><location><page_3><loc_8><loc_53><loc_92><loc_65></location>Throughout this paper, we adopt a flat cosmological model with H 0 =67.4 km s -1 Mpc -1 , Ω M =0.315, and Ω Λ =0.685 (Planck Collaboration et al. 2020). For these parameters a redshift of z =0.248 (Fynbo et al. 2019) corresponds to a luminosity distance of d L = 1.29 Gpc, 1 arcsec corresponds to 4.03 kpc projected distance, and distance modulus is m -M = 40 . 55 mag. The Milky Way reddening E ( B -V ) along the line of sight towards the source is between 0.03 and 0.04 mag (Schlegel et al. 1998; Schlafly & Finkbeiner 2011). We present the observations and data reduction in Sect. 2, and our results in Sect. 3. In our data analysis we corrected the apparent magnitudes for the Galactic reddening and adopted a host-galaxy visual extinction along the line of sight of A host V = 0.06 mag (Levan et al. 2023). Discussion and conclusions are presented in Sect. 4.</text>
<section_header_level_1><location><page_3><loc_33><loc_50><loc_68><loc_51></location>2. OBSERVATIONS AND DATA REDUCTION</section_header_level_1>
<section_header_level_1><location><page_3><loc_37><loc_48><loc_64><loc_49></location>2.1. GROND early-time Observations</section_header_level_1>
<text><location><page_3><loc_8><loc_38><loc_92><loc_47></location>With a duration of about 65 seconds, GRB 191019A was a classical long GRB. At a redshift of z = 0 . 248, a rising SN 1998bw component was thus expected to be detectable with GROND within about two weeks after the GRB trigger. Therefore, following earlier efforts with GROND to explore GRB-SN light curves (Olivares E. et al. 2012a, 2015; Greiner et al. 2015; Kann et al. 2019a; Klose et al. 2019) we performed follow-up observations of the field during several epochs between 0.4 (Nicuesa Guelbenzu 2019) and 15.4 days post GRB trigger (Table 1). Observations were then terminated since no evidence for SN light was found.</text>
<text><location><page_3><loc_8><loc_35><loc_92><loc_37></location>GROND data were reduced in a standard fashion (bias subtraction, flat fielding, co-adding; Kruhler et al. 2008; Yolda¸s et al. 2008), based on numerical routines in IRAF (Image Reduction and Analysis Facility; Tody 1993).</text>
<section_header_level_1><location><page_3><loc_38><loc_29><loc_62><loc_30></location>2.2. LBT late-time Observations</section_header_level_1>
<text><location><page_3><loc_8><loc_22><loc_92><loc_28></location>Late-time observations of the field were performed with the Large Binocular Telescope (LBT) using the two twin LBC instruments equipped with the Sloan filters g ' r ' i ' z ' on October 20, 2023. These data were reduced using the data reduction pipeline developed at INAF - Osservatorio Astronomico di Roma (Fontana et al. 2014) which includes bias subtraction and flat-fielding, bad pixel and cosmic ray masking, astrometric calibration, and coaddition.</text>
<text><location><page_3><loc_8><loc_16><loc_92><loc_22></location>On the LBT images, we measure the AB magnitudes for the host given in Table 4 (see Appendix A). Within the errors, these values are consistent with those measured with GROND in epoch 4 ( T -T 0 = 7 . 4756 days). Based on these data we conclude that 7 days post burst the transient did not contribute anymore to the observed flux in g ' r ' i ' z ' in a measurable quantity.</text>
<section_header_level_1><location><page_3><loc_36><loc_13><loc_64><loc_14></location>2.3. Image subtraction and photometry</section_header_level_1>
<text><location><page_3><loc_8><loc_9><loc_92><loc_12></location>In order to search for a transient hidden by the relatively bright host galaxy, we performed image subtraction on the GROND images using HOTPANTS (Becker 2015). HOTPANTS convolves the template and the source images to the same</text>
<section_header_level_1><location><page_4><loc_44><loc_93><loc_57><loc_94></location>Stratta et al.</section_header_level_1>
<table>
<location><page_4><loc_17><loc_68><loc_82><loc_90></location>
<caption>Table 1. Measured AB magnitudes and upper limits of the transient.</caption>
</table>
<text><location><page_4><loc_18><loc_62><loc_83><loc_67></location>Note -Column (2) provides the midtime of the observation, in units of days from burst trigger. Magnitudes are not corrected for Galactic foreground extinction. The photometry for epochs 1a to 3 was obtained after image subtraction against epoch 4 (see § 2.3 for details). On the contrary, the 3 σ upper limits for epoch 4 to 6 have been measured directly, without image subtraction.</text>
<text><location><page_4><loc_8><loc_52><loc_92><loc_56></location>point spread function (PSF) and photometric scale. As a template image we used the GROND 4th-epoch observations (Table 1) since these images have the best seeing ( ∼ 1 . '' 0). We aligned the template and the source images by means of wcsremap 1 .</text>
<text><location><page_4><loc_8><loc_37><loc_92><loc_52></location>The Gaussian input parameters in HOTPANTS were calculated following Becker (2015) considering the case in which the template FWHM is smaller than the source FWHM. In this way, the template images were smoothed to the seeing of the corresponding source images. In addition, we fixed several other parameters for HOTPANTS following Hu et al. (2022), but let free to vary the number of each region's stamps, the convolution kernel half width, and the size of Gaussians which compose the kernel. In doing so, we selected those input parameters that minimized the background noise on the residual image in a star-free region close to the position of the host galaxy. All resulting residual images were then photometrically calibrated with respect to the source images (see below). In the case of a non-detection of the transient the upper limit we used was measured locally on the subtracted image (Table 1). We have confirmed our results using also the 5th GROND epoch as template (seeing ∼ 1 . '' 3), although we obtained a lower SNR.</text>
<text><location><page_4><loc_8><loc_24><loc_92><loc_38></location>We double checked the detections using the late LBT images (about 4 years or 1460 days after the burst trigger, see Sect. 2.2 and Table 4 in Appendix A). The g ' i ' z ' detections are confirmed for epochs 1a and 2, but with lower SNR likely due to the additional plate scaling, and a general worse PSF shape due to not perfect collimation. We have not checked the r ' -band since the above mentioned problem was worse in this filter, although it did not compromise the aperture photometry in Table 4. For this filter we have instead used (as template) a stack of the best Gemini-S images 2 obtained at 58-63 days after the burst (see Levan et al. 2023), and we obtained consistent results. In other words, the results obtained with GROND could not be improved. Finally, to better constrain the late transient decay, we performed image subtraction on the Gemini r -band image at 11.4 days, using the later Gemini images as template (see above), and obtained for the optical transient an upper limit of r > 25 . 8 AB mag.</text>
<text><location><page_4><loc_8><loc_16><loc_92><loc_23></location>Apparent magnitudes were obtained by performing aperture photometry using DAOPHOT and APPHOT under PyRAF/IRAF with increasing apertures up to 4 times the FWHM and by PSF photometry with the DAOPHOT and ALLSTAR tasks of IRAF. PSF-fitting was used to measure the magnitudes of the transient after image subtraction. In the optical bands the data were calibrated using Pan-STARRS (Chambers et al. 2016), in the NIR bands using 2MASS (Skrutskie et al. 2006). Central wavelengths of the GROND filter bands are listed in Rossi et al. (2011).</text>
<section_header_level_1><location><page_5><loc_45><loc_90><loc_55><loc_92></location>3. RESULTS</section_header_level_1>
<section_header_level_1><location><page_5><loc_25><loc_88><loc_75><loc_89></location>3.1. Application of the prompt emission minimum variability criterion</section_header_level_1>
<text><location><page_5><loc_8><loc_78><loc_92><loc_87></location>According to Camisasca et al. (2023), GRB prompt emission minimum variability timescale (MVT) represents a diagnostic to infer the progenitor nature of a GRB. By estimating MVT from the full width at half maximum of the shortest, statistically significant peak in the prompt emission light curve (FWHM min ) of hundreds of GRBs, it was found that high variability (low FWHM min values) typically belongs to compact binary mergers. Interestingly, this result was found to be independent of the burst duration and for this reason it is an important probe for the nature of peculiar long GRBs with properties more similar to those of short GRBs.</text>
<text><location><page_5><loc_8><loc_63><loc_92><loc_78></location>By analyzing the results from the Swift GRB sample analyzed by Camisasca et al. (2023), we checked which long GRBs sufficiently nearby for a possible kilonova detection, i.e. at z < 0 . 5, and without any associated SN, had an FWHM min compatible with compact binary merger values, and we found that GRB 191019A was satisfying our criteria. Specifically, for this GRB an MVT of FWHM min = 0 . 196 +0 . 068 -0 . 05 s has been measured, which is compatible with the values typically found for GRBs associated with compact binary mergers (Fig. 1). In particular, given the long burst duration of GRB 191019A ( T 90 = 64 . 35 ± 4 . 35 s), its position in the FWHM min vs T 90 diagram lies among the short GRBs showing a soft 'Extended Emission' (SEE, e.g., Norris & Bonnell 2006; Minaev et al. 2010). Interestingly, two famous long GRBs for which a kilonova component was found in the optical afterglow, namely GRB 211211A (Rastinejad et al. 2022; Troja et al. 2022; Yang et al. 2022; Gompertz et al. 2023) and GRB 230307A (e.g., Levan et al. 2023), lie in the same region of the diagram of GRB 191019A (Fig. 1).</text>
<text><location><page_5><loc_8><loc_52><loc_92><loc_62></location>The SEE distribution in the T 90 -FWHM min plane clearly overlaps with the distribution of long GRBs associated with SNe, but at the same time the centroids of the two distributions are well separated: interestingly, GRB 191019A lies in the middle of the SEE region and in the outskirts of the SN-associated cases. This property alone cannot be taken as compelling evidence that GRB 191019A behaves like a SEE, but suggests a probable SEE-like nature than a collapsar one. These findings further support past interpretation that also GRB 191019A originated from a compact binary merger (e.g., Levan et al. 2023), and address the possible presence of a kilonova component, similarly to GRB 211211A and GRB 230307A.</text>
<section_header_level_1><location><page_5><loc_24><loc_49><loc_76><loc_50></location>3.2. Re-analysis of the light curve of the Transient that followed the burst</section_header_level_1>
<text><location><page_5><loc_45><loc_47><loc_55><loc_48></location>3.2.1. X-rays</text>
<text><location><page_5><loc_8><loc_33><loc_92><loc_46></location>Adopting a power-law spectral energy distribution (SED), the best fit of the Swift -XRT 0.3-10.0 keV spectrum, computed by the automatic algorithm of the UK Swift Science Data Center (UKSSDC) Swift -XRT GRB Spectrum Repository 3 in the 3 . 2 -32 . 4 ks temporal window, provides a photon index of Γ = 2 . 0 +0 . 4 -0 . 3 and an intrinsic equivalent hydrogen column density of N H , z = 1 . 2 +1 . 6 -1 . 2 × 10 21 cm -2 in addition to a Galactic N H = 3 . 28 × 10 20 cm -2 . We obtained from the UKSSDC Burst Analyzer tool (Evans et al. 2010), the absorption-corrected X-ray fluxes in the energy range 0.3-10.0 keV, and we converted them into 1 keV flux densities by assuming a power law spectrum and the corresponding photon index provided by the Burst Analyzer in each temporal bin of the light curve. The flux density evolution in the X-ray band is compatible with a power-law decay 4 with α X = 1 . 44 ± 0 . 14.</text>
<section_header_level_1><location><page_5><loc_43><loc_31><loc_57><loc_33></location>3.2.2. Optical bands</section_header_level_1>
<text><location><page_5><loc_8><loc_25><loc_92><loc_31></location>We detected the optical transient during the first three GROND observations of the field, up to 1.5 days post burst (Table 1, Fig. 2). The transient has coordinates R.A., decl. (J2000) = 22:40:05.861, -17:19:42.77 ( ± 0.3 arcsec), measured on the combined g ' r ' i ' residual images of the 2nd-epoch observations (Fig. 3). Within the error, these coordinates agree with what has been reported by Perley et al. (2019b).</text>
<text><location><page_5><loc_8><loc_15><loc_92><loc_24></location>Based on a joint fit of the griz -band magnitudes provided by Levan et al. (2023) and GROND's first visit of the field (epoch 1a in Table 1), we find that between both observing runs the optical transient was fading with a decay slope α opt = 1 . 43 ± 0 . 07. Within the errors, α X ∼ α opt up to T 0 +0 . 4 days (epoch 1a in Table 1), suggesting a common cooling regime of the electrons radiating optical and X-ray synchrotron light (see Fig. 4). This finding is compatible with results by (Levan et al. 2023) who showed that a single power-law SED from X-rays to the optical bands can describe the observations at T 0 +0 . 21 days.</text>
<figure>
<location><page_6><loc_9><loc_32><loc_91><loc_91></location>
<caption>Figure 1. Burst duration versus prompt emission minimum temporal variability of 1291 GRBs detected with Swift -BAT (from January 2005 to July 2022) and the corresponding marginal distributions. The sample includes 78 Short GRBs (blue) and 24 short GRBs with 'Extended Emission' (SEE, green). Gold points are Long GRBs with an associated SN (adapted from Camisasca et al. 2023). The location of GRB 191019A is indicated by a green star in the top-left side of the diagram where SEE are located, together with two long GRBs better identified with being originated by compact binary coalescence progenitors (GRB 211211A and GRB 230307A).</caption>
</figure>
<text><location><page_6><loc_8><loc_12><loc_92><loc_19></location>However, GROND's following observing runs 1b and 2 (Table 1) clearly show a significant flattening of the light curve, with a new decay slope of α opt ∼ 0 . 5 ± 0 . 1 (Fig. 4). The rescaled X-ray power-law model underpredicts the flux in the optical bands, with a > 3 σ deviation observed at T 0 +1 . 5 days. Even by assuming a shallower optical decay index (i.e., α opt = α X -0 . 25), as for the case where the synchrotron cooling frequency lies between the X-ray and the optical regime (e.g., Sari et al. 1998), the discrepancy at late times ( > 1 day) persists (Fig. 4).</text>
<figure>
<location><page_7><loc_14><loc_61><loc_87><loc_89></location>
<caption>Figure 2. Multi-band light curve of the transient that followed GRB 191019A. Optical and NIR data are from Levan et al. (2023) (from their table 4 of the supplementary material, where flux densities corrected for dust extinction and host galaxy substracted are quoted: we consider here only those with relative error ≤ 30%) and from this work (larger markers) where GROND fluxes have been computed after image subtraction of the host galaxy observed at epoch 4 with GROND, and corrected for both Milky Way ( A V =0.10 mag) and host-galaxy dust extinction ( A V =0.06 mag). The vertical dashed lines mark the first three observation epochs quoted in Table 1. X-ray data were taken from the Swift -XRT GRB light curve Repository (Evans et al. 2007, 2009) as unabsorbed 0.3-10 keV fluxes computed with the Burst Analyzer (see Evans et al. 2010), and converted to 1 keV flux densities by assuming a power law spectrum and the corresponding photon index provided in each temporal bin of the light curve.</caption>
</figure>
<figure>
<location><page_7><loc_8><loc_24><loc_92><loc_46></location>
<caption>Figure 3. From left to right, GROND r ' -band observations obtained during epochs #2 at 1.5 days, #4 at 7.5 days and the residual of the image subtraction (epoch #2 minus epoch #4) using HOTPANTS . The optical transient is indicated by the circle.</caption>
</figure>
<text><location><page_7><loc_8><loc_9><loc_92><loc_18></location>To quantify the null hypothesis that a power law model cannot fit the data and test for chromaticity of flux evolution, we considered the early data by Levan et al. (2023) and our GROND detections (i.e. from ∼ 0.2 to 1.5 days) for the two filters with best temporal sampling (i.e. r ' and g ' ): a simple power law model does not provide an acceptable fit, with probability of having the observed data set, given the null hypothesis (p-values), of 1 . 7 × 10 -5 and 4 × 10 -4 , respectively. By assuming a broken power law, with a steep-to-shallow behavior and a temporal break at 0.44 days, we find a significant improvement of the fit, with p-values of 0 . 79 and 0 . 73 in the r ' and g ' bands, respectively. The best</text>
<text><location><page_8><loc_8><loc_89><loc_92><loc_92></location>fit broken power laws show that the flux evolution after the break is slightly shallower in the redder filter, suggesting a chromatic behaviour.</text>
<text><location><page_8><loc_8><loc_82><loc_92><loc_88></location>We note that this peculiar steep-to-shallow optical light curve morphology is not unique, it was already observed in a small subset of GRBs in the early years of the Swift era (see, e.g., Kann et al. 2010), although with light curve breaks on average much earlier than what we observe for GRB 191019A. For instance, the optical light curve of GRB 090102 (Gendre et al. 2010) is similar to GRB 191019A but the break time was ∼ 0 . 1 hr (rest frame) after the burst.</text>
<text><location><page_8><loc_8><loc_68><loc_92><loc_82></location>In the standard fireball model, which fairly accounts for late afterglow phenomenology as forward-shock radiation, a steep-to-shallow light curve is not envisioned (e.g., Sari et al. 1998). Such morphology could be produced by the presence of a reverse shock component dominating the forward shock at early times. However, reverse shock is expected on timescales of the order of minutes (e.g., Nakar & Piran 2004), in contrast with what we observe for GRB 191019A. Alternatively, energy injection into the forward shock could originate a light curve flattening. The spin-down radiation from a newly born magnetar (e.g., Zhang & M'esz'aros 2001) or a slightly off-axis structured jet (e.g., Beniamini et al. 2020) are plausible energy sources that are often invoked to explain the X-ray afterglow 'plateaus' observed in GRBs, and that might have an optical counterpart. In the case of GRB 191019A the lack of any evidence of a shallow phase in X-rays cast some doubts against the energy injection scenario, however.</text>
<section_header_level_1><location><page_8><loc_37><loc_66><loc_63><loc_67></location>3.2.3. Constraints on Supernova Light</section_header_level_1>
<text><location><page_8><loc_8><loc_59><loc_92><loc_65></location>If GRB 191019A had been a classical long burst, then for a redshift of z =0.248 a supernova (SN) component was expected to be detectable with GROND. Observational constraints on SN light were first reported by Levan et al. (2023) based on HST observations 30 and 184 days post GRB trigger. No evidence for transient emission was found ( g > 24 , r > 23 . 5, and z > 22 mag).</text>
<text><location><page_8><loc_8><loc_50><loc_92><loc_59></location>Figure 5 shows the upper limits we can set on any SN that followed GRB 191019A in comparison to the r ' -band light curves of 13 GRB-SNe that have been observed with GROND between 2007 and 2014 (see Table 2 in Klose et al. 2019). These events cover the whole GRB-SN luminosity distribution of well sampled GRB-SNe, from GRB 100316D/SN 2010bh (Olivares E. et al. 2012a), one of the faintest GRB-SN ever detected (e.g., Melandri et al. 2014; Cano et al. 2017; Dainotti et al. 2022), to GRB 111209A / SN 2011kl (Kann et al. 2019a), the most luminous GRB-SN ever observed.</text>
<text><location><page_8><loc_8><loc_40><loc_92><loc_49></location>A visual inspection of Figure 5 shows that all light curves fall into a strip with a width of about 2-3 mag, which is limited by the very faint GRB 100316D/SN 2010bh and the very bright GRB 111209/SN 2011kl. The strongest limit we can set stamps from the time span between 7 and 12 days post burst and is well below the identified GRB-SN magnitude strip: if a CC-SN followed GRB 191019A, in r ' it was at least 3 mag less luminous than SN 1998bw and 2 mag fainter than SN 2010bh. According to the archived Gemini data (Sect. 2.3), the constraint on the luminosity is even stronger, > 4 mag in r at 11.4 days post burst.</text>
<text><location><page_8><loc_8><loc_36><loc_92><loc_40></location>In conclusion, even if we take into account that the time evolution of GRB-SNe light curves show a certain parameter range (characterized by stretch factor), the SN limit we can set provides a constraint on the entire SN light curves. In this respect, any SN related to GRB 191019A must have been less luminous than each GRB-SN (see Appendix B).</text>
<section_header_level_1><location><page_8><loc_34><loc_33><loc_66><loc_34></location>3.3. Joint Afterglow and Kilonova modelling</section_header_level_1>
<text><location><page_8><loc_8><loc_28><loc_92><loc_32></location>Since the flattening observed in the optical light curve at late times (epochs 1b and 2 in Table 1) cannot be explained by the presence of a slowly rising SN (Sect. 3.2.3), and probably not by a reverse shock or energy injection either (Sect. 3.2.2), we explore now the possibility that the source of this additional radiation component was kilonova light.</text>
<text><location><page_8><loc_8><loc_14><loc_92><loc_27></location>Given the relatively small redshift of GRB 191019A ( z = 0 . 248), a kilonova with a luminosity similar to AT2017gfo is expected to lie within the discovery space of a 2-m telescope, if not hidden by an intrinsic low luminosity. Therefore, following the procedure described in Rossi et al. (2020), we compared our data at T 0 +1 . 5 days in each filter, with the flux of the kilonova AT2017gfo (Coulter et al. 2017) associated with the NS-NS merger GW170817 (Abbott et al. 2017), shifted to the redshift of GRB 191019A. We find that an emission component similar to AT2017gfo but ∼ 4 times more luminous could reproduce the observations in the i ' and r ' bands, while our g ' -band flux seems to be brighter (Fig. 4). Apparently, the presence of a kilonova associated with GRB 191019A is overall compatible with the general properties of the observed optical-NIR emission at 0.4 and 1.5 days, although with slightly different brightness than AT2017gfo 5 .</text>
<figure>
<location><page_9><loc_10><loc_37><loc_84><loc_86></location>
<caption>Figure 4. X-ray versus optical light curve, where larger circles indicate GROND data from this work and the other data is taken from the literature (as in Figure 2). The blue dashed line shows the X-ray flux best fit power-law model with a decay index α X = 1 . 44 ± 0 . 14. This model is then scaled to match the optical data (dashed lines) together with a shallower power-law decay with α X -0 . 25 (dotted lines). The two power-law models are expected if the optical emission is produced by electrons in the same cooling regime as those emitting in X-rays (dashed line) or in a different cooling regime (dotted line). The solid lines indicate the flux of a AT2017gfo-like kilonova if it were at the redshift of GRB 191019A and ∼ 4 times more luminous. For visual purposes only, the r ' , i ' , and z ' -band data have been rescaled by a factor of 3, 5, and 7 respectively.</caption>
</figure>
<text><location><page_9><loc_8><loc_15><loc_92><loc_23></location>Motivated by the results obtained with our previous simplistic approach, we then explored the possible presence of a kilonova by comparing our multi-band dataset ( g ' , r ' , i ' , z ' , and X-ray) with a much more sophisticated model that takes into account simultaneously the presence of an afterglow and a kilonova component through a joint fit. For this purpose, we exploited the Nuclear - Multi-Messenger Astronomy (NMMA v0.2.0) framework that allows to estimate best-fit parameters with a bayesian inference method (Dietrich et al. 2020; Pang et al. 2023).</text>
<text><location><page_9><loc_8><loc_11><loc_92><loc_15></location>For the afterglow modelling, NMMA uses the Afterglowpy python module (Ryan et al. 2020). Afterglowpy allows to model GRB afterglow light curves and spectra by taking into account the possible effects due to a complex jet structure and an off-axis observer. We here assumed a Gaussian profile for the jet structure, and that the whole</text>
<figure>
<location><page_10><loc_9><loc_39><loc_91><loc_92></location>
<caption>Figure 5. The r ' -band light curves of all 13 GRB-SNe observed by GROND between 2007 and 2014 which are listed in Table 2 in Klose et al. (2019). All light curves have been calculated based on the luminosity and stretch factors found for these SNe and have been corrected for Galactic and host galaxy extinction along the line of sight. They have been red-shifted to z=0.248 taking into account the appropriate cosmological corrections following the procedure described in Zeh et al. (2004). Also shown is the r ' -band light curve of SN1998bw (in blue) shifted to z=0.248 taking into account the appropriate cosmological corrections. Overplotted are the upper limits we can set based on the GROND data (Table 1; r ' -band) and the late Gemini observations (Sect. 2.3; dark green).</caption>
</figure>
<text><location><page_10><loc_8><loc_23><loc_92><loc_26></location>( ξ N = 1) electron population is shock-accelerated. With the exception of the viewing angle ι , which has a sine function, to all parameters we assigned a uniform function to model the prior probability (for details see Table 2).</text>
<text><location><page_10><loc_8><loc_10><loc_92><loc_22></location>For the kilonova emission, NMMA allows to fit and simulate data using several numerical and analytical models. Assuming a binary neutron star merger as a progenitor system, we adopted here the kilonova modelling resulting from the time dependent 3D Monte Carlo code POSSIS for modelling radiation transport (Bulla 2019, 2023). We used the kilonova model grid published in Dietrich et al. (2020) based on the first version of POSSIS (Bulla 2019) where the kilonova ejecta is represented with two components: 1) a high-velocity (0 . 08 < v dyn /c < 0 . 3) dynamical ejecta of mass M dyn rj with a lanthanide-rich composition distributed about the equatorial plane with half-opening angle Φ, and lanthanide-poor composition at higher latitudes, and 2) a slower (0 . 025 < v wind /c < 0 . 08) wind (or 'post-merger') component, which is a spherical ejecta released from the merger remnant and debris disk, with an intermediate</text>
<text><location><page_11><loc_8><loc_81><loc_92><loc_92></location>lanthanide content and mass M wind ej (see Dietrich et al. 2020). In the fit we fixed the GRB 191019A luminosity distance at 1289.3 Mpc, as obtained from the measured redshift ( z = 0 . 248) and assuming a flat cosmological model (see Sect. 1). We also considered an additional error budget (in magnitudes), which takes into account the uncertainties on the model predictions as well as possible systematic errors ( em syserr in the corner plot), to which we assigned a uniform prior with range from 0 to 2 mag. The resulting best-fit light curves computed in each band are plotted in Figure 6, the fit corner plot is in Figure 7, while the 90% confidence interval and median values of each parameter , as well as the adopted prior functions, are quoted in Table 2.</text>
<text><location><page_11><loc_8><loc_66><loc_92><loc_81></location>By simply inspecting the resulting light curves, it is evident that the afterglow component is constrained mostly by the X-ray data and it is dominant during the early epochs in the optical bands. The afterglow best fit jet core semiaperture angle is about ∼ 10 deg (9 +4 -3 deg), and the observer viewing angle is ∼ 4 deg (4 +3 -2 deg), thus within the jet core. The fireball isotropic kinetic energy E 0 (1 . 4 +1 . 8 -1 . 0 × 10 52 erg) is nicely compatible with the observed prompt emission equivalent isotropic energy E iso , by assuming a reasonable efficiency of about η ∼ 10%, where η = E iso E iso + E 0 . Indeed, the 15-150 keV fluence measured by Swift -BAT is (1 . 00 ± 0 . 03) × 10 -5 erg cm -2 (Krimm et al. 2019), corresponding to a radiated energy E iso = (1 . 70 ± 0 . 05) × 10 51 erg. The latter was computed by assuming a power-law spectrum in the BAT bandpass: this assumption is reasonable given the softness of the spectrum, which is best fit with a photon index Γ = 2 . 25 ± 0 . 05 (Krimm et al. 2019), suggesting that the prompt emission peak energy lies below the BAT bandpass.</text>
<text><location><page_11><loc_8><loc_57><loc_92><loc_66></location>Figure 6 also clearly shows that at later epochs ( > 1 day), the observed flux lies above the predicted levels of the afterglow component, and the presence of a kilonova provides a better match. Indeed, by assuming only the afterglow model, i.e. by removing the kilonova component from our initial model, we obtained a worse fit, with Bayesian evidence ln( Z ) = -21 . 1. By considering the ratio with the Bayesian evidences of the joint afterglow and kilonova model, for which we obtain ln( Z 0 ) = -14 . 1, the resulting Bayes factor (ln( B ) =ln( Z/Z 0 ) = -7 . 0) indicates a strong preference for the model which includes the kilonova component (see, e.g., Kunert et al. 2024 and references therein).</text>
<text><location><page_11><loc_8><loc_49><loc_92><loc_57></location>The kilonova properties we find from our fit are compatible with a dynamical ejecta mass of M dyn ej ∼ 0 . 02 M ⊙ and a wind mass M wind ej ∼ 0 . 04 M ⊙ , though with large uncertainties (relative errors ≥ 60%). These values are slightly higher (yet consistent within the uncertainties) with those found for AT2017gfo by assuming the same kilonova modelling (Dietrich et al. 2020). This is in line with the need of a brighter kilonova (of about a factor 4, see Sect. 3.3 and Fig. 4) by simply superposing an AT2017gfo-like light curve on the data.</text>
<text><location><page_11><loc_8><loc_30><loc_92><loc_49></location>We stress here that the real picture is likely much more complicated than a two-component scenario, and the dynamical/wind masses inferred should rather be interpreted as belonging to some high/low velocity components of a multi-component scenario with multiple ejecta episodes (e.g., Bernuzzi 2020; Nedora et al. 2021). At the same time, the kilonova model we assumed is among the most sophisticated ones publicly available, and the assumption of a kilonova from a NS-NS merger progenitor is the most reasonable choice given that, so far, the only GRB with kilonova emission for which we were able to infer the progenitor nature was GRB170817/AT2017gfo, wich we know from gravitational wave data analysis being originated from a binary neutron star merger. Nevertheless, we explored also other kilonova models available within the NMMA framework, with different levels of sophistication, and which have different assumptions on the progenitor nature and on the kilonova ejecta properties (see Appendix C). We find that the dataset for GRB 191019A did not allow us to confidently distinguish among different kilonova models, apart disfavouring the most simplistic ones. However, in all cases, we find that a joint afterglow plus kilonova fit is preferred with respect to an afterglow-only model.</text>
<text><location><page_11><loc_8><loc_23><loc_92><loc_30></location>Interestingly, both the afterglow-only and afterglow plus kilonova model, provide as a best-fit for the circumburst environment particle density a value n 0 < 1 cm -3 , which is typically deduced for short-GRB environments (e.g., Berger 2014; Fong et al. 2015). This result is in stark contrast with the interpretation by Lazzati et al. (2023) that invokes the presence of a very high density environment with n 0 ∼ 10 7 -10 8 cm -3 (see also Levan et al. 2023).</text>
<text><location><page_12><loc_13><loc_7><loc_21><loc_89></location>T able 2 . The 90% confidence in terv al and median v alues of eac h parameter inferred from the sim ultaneous afterglo w ( Ry an et al. 202 0 ) and kilono v a ( Dietric h et al. 2020 ) mo dell ing p erformed b y using the co de NMMA. F or the afterglo w comp onen t, w e assumed a Gaussian jet profi le and fiducial v alue s for the microph ysical parameters (see Sect. 3.3 an d Fig. 6 and Figure 7 ). The b ottom part of the table quotes the assumed p r i ors in the Ba y esian inference analysis, where 'U' stands for Uniform function, with minim um and maxim um v alue s quoted in the brac k ets.</text>
<table>
<location><page_12><loc_24><loc_10><loc_42><loc_87></location>
</table>
<text><location><page_12><loc_42><loc_19><loc_51><loc_83></location>Note -E 0 = kinetic fireball energy; n = particle n um b er densit y in the circum burst en vironmen t; θ c =half-op ening angle of the jet core; θ w =half-op ening a ngle of the jet truncated-wings; ι =viewing angle with resp ect to jet axis; p = electron energy distribution p o w er-la w index; ϵ e = sho c k e nergy fraction that go es in to th e electrons; ϵ B = sho c k energy fraction that go es in to the magne tic energy densit y; M dyn ej = dynamical ejecta mass; M wej = wind ejecta mass; Φ = half-op ening angle of lan thanide-ric h equatorial ejecta.</text>
<table>
<location><page_13><loc_25><loc_75><loc_75><loc_90></location>
<caption>Table 3. Expected and observed SN AB magnitudes.</caption>
</table>
<text><location><page_13><loc_25><loc_69><loc_75><loc_74></location>Note -All observed (obs) AB magnitudes in the r ' , i ' and z ' filters are 3 σ upper limits (Fig. 5). Columns (2), (5), and (8) quote the expected (exp) magnitudes. All the corresponding differences (observed minus expected; columns 4, 7, 10) are lower limits.</text>
<section_header_level_1><location><page_13><loc_36><loc_57><loc_65><loc_58></location>4. DISCUSSION AND CONCLUSIONS</section_header_level_1>
<text><location><page_13><loc_8><loc_34><loc_92><loc_56></location>Recent analysis of an increasing number of GRBs which were initially classified as long bursts, and therefore thought to be originating from collapsing massive stars, suggest a better compatibility of the observational data with compact binary merger progenitors. By ignoring the standard burst duration classification, typical features of GRBs associated with compact binary mergers are (1) the absence of an associated CC-SN, (2) the presence of an optical/NIR rebrightening compatible with a kilonova, (3) an early-type host galaxy, and (4) a distant GRB explosion site with respect to the corresponding galaxy center. GRB 191019A was suggested to belong to this sample of misclassified long GRBs, given the non detection of any SN component (Levan et al. 2023) despite its close cosmological distance ( z =0.248), and the evidence of a host galaxy dominated by an old stellar population. However, the proximity of the GRB 191019A explosion site to the photometric center of its host galaxy ( ≲ 100 pc) is at odds with typical short-GRB galactocentric distances, and could suggest a compact binary formed in the dense circumnuclear disk of an AGN. In this scenario (Lazzati et al. 2022) a very high-density environment ( n 0 > 10 7 cm -3 ) could be the origin of the long duration of the prompt emission (see Lazzati et al. 2023). Given the low distance of this burst, a kilonova ligh is expected to be seen. However, the temporal coverage and limiting magnitudes of previous data sets, did not allow to address this question yet.</text>
<text><location><page_13><loc_8><loc_19><loc_92><loc_34></location>In this work, we further investigate the burst properties of GRB 1901910A and the nature of the optical transient that followed the burst, with the goal to find arguments in favour or against its possible compact merger origin. We found an independent support on the compact binary merger progenitor nature from the high-energy prompt emission temporal variability, which has been recently found to be an additional potential diagnostic to infer the progenitor nature, where high variability values suggest a compact binary merger progenitor (Camisasca et al. 2023). We noticed that, for GRB 191019A, the obtained MVT value of ∼ 20 ms locates this burst close to the centroid of the distribution of those GRBs associated with compact binary merger progenitors (Camisasca et al. 2023 and Fig. 1). The same variability properties were found also for two other long GRBs with evidence of kilonova (GRB 111210A and GRB 230307A). These results independently support past interpretations on the progenitor nature of GRB 191019A and address to the possible presence of a kilonova component in its optical transient.</text>
<text><location><page_13><loc_8><loc_9><loc_92><loc_18></location>An optical transient following GRB 191019A was at first reported by Perley et al. (2019b) but its nature could not be clearly pinned down. Using GROND multi-color data obtained between 0.4 and 15 days post burst, we argued here that the temporal evolution of the transient's brightness disfavours a pure afterglow emission (Fig. 4). While the GROND data confirm the absence of a SN component (Fig. 5), we found that the luminosity evolution of the optical transient is in agreement with an afterglow plus a kilonova signal compatible with AT2017gfo redshifted to the distance of the GRB host galaxy, though slightly brighter by a factor of a few.</text>
<figure>
<location><page_14><loc_28><loc_16><loc_72><loc_92></location>
<caption>Figure 6. Joint fit of an afterglow plus a kilonova model, performed with NMMA (see Table 2). The dashed lines show the median light curve, while the shaded areas show the 95% interval. Red circles and black triangles mark the detections and upper limits, respectively, in AB mag.</caption>
</figure>
<figure>
<location><page_15><loc_8><loc_27><loc_91><loc_92></location>
<caption>Figure 7. Corner plot obtained with NMMA from a joint fit assuming a model that incorporates both an afterglow and a kilonova component (see Table 2 and Fig. 6). Different shadings mark the 68%, 95%, and 99% confidence intervals. For the 1D posterior probability distributions, the 90% confidence interval (dashed lines) and median values above each panel are indicated.</caption>
</figure>
<text><location><page_15><loc_8><loc_11><loc_92><loc_20></location>By modelling the afterglow and the kilonova component, we were able to estimate a kilonova ejecta mass which is slightly higher but still consistent, within the large uncertainties, with the one measured for AT2017gfo with similar assumptions on the progenitor system (a binary neutron star) and on the kilonova modelling (e.g., Dietrich et al. 2020). By assuming other kilonova models with different levels of complexity and different progenitor assumptions, the data did not allow to discriminate among them, apart disfavouring the most simplistic one. However, in all cases, results strongly favour the presence of a kilonova with respect to an afterglow-only model (Table 5).</text>
<text><location><page_16><loc_8><loc_67><loc_92><loc_92></location>Our findings strongly suggest that GRB 191019A might belong to the increasing list of long GRBs with an associated kilonova, beside GRB 211211A and GRB 230307A, and other cases with less robust evidence, as for instance GRB 060614 (e.g., Jin et al. 2015). Another interesting findings from our analysis, is the evidence of a low circumburst density (Table 2). This result is in stark constrast with the one obtained by Lazzati et al. (2023) from prompt emission light curve modelling, where extreme values of n 0 > 10 7 cm -3 were found. These high density values were considered consistent with the apparent position of the GRB, very close to the center of an AGN-like host galaxy (though no direct proofs on the existence of an AGN have been provided yet, see Levan et al. 2023). The interaction between a jet and an AGN dense circumnuclear environment, however, may choke or strongly suppress GRB relativistic jets, unless particularly bright and long, as also supported by recent studies (e.g., Zhang et al. 2024). GRB 191019A is classified as a normal burst, with a well detected optical counterpart (see, e.g., Zhang et al. 2024), and with low dust extinction along the line-of-sight within the host ( A host V in the range from 0.06 to 0.10 mag, Levan et al. 2023), further supported by the UV-band detections (LaPorte et al. 2019). These properties are more consistent with a low density environment, as the one we obtained from a joint fitting of an afterglow and a kilonova component model on the X-ray and optical data of GRB 191019A. More in general, if GRB 191019A is a disguised short GRB with a compact binary merger origin, as both past studies and this work suggest, a low density environment is in agreement with the typical density values found in short GRBs and with the environments expected for compact binary mergers.</text>
<text><location><page_16><loc_8><loc_56><loc_92><loc_66></location>In a low density scenario, the very small offset from the host galaxy of GRB 191019A, can be explained with a negligible transverse projection of a GRB located far off centre. In this scenario, the long duration of the prompt emission, which Lazzati et al. (2023) explained as caused by the extremely dense circumburst environment, may have instead an intrinsic origin. An interesting hypothesis that was proposed in the past to explain short GRBs with soft extended emission (e.g., Barkov & Pozanenko 2011; Kisaka & Ioka 2015) and which has been recently studied in much greater details (e.g., Musolino et al. 2024), invokes fall-back accretion onto the remnant compact object. Whether this is the case for GRB 191019A goes beyond the scope of this work and will be addressed in another study.</text>
<text><location><page_16><loc_8><loc_32><loc_92><loc_54></location>We thank the anonymous referee for useful comments and suggestions which improved our work. We are grateful to P. Pang for his precious support on NMMA and useful discussions. ANG acknowledges logistic support by the Thuringer Landessternwarte Tautenburg, Germany. G.S. and P.S acknowledge the support by the State of Hesse within the Research Cluster ELEMENTS (Project ID 500/10.006). A.R. and E.P. acknowledge support by PRIN-MIUR 2017 (grant 20179ZF5KS). Part of the funding for GROND (both hardware and personnel) was generously granted by the Leibniz-Prize to G. Hasinger (DFG grant HA 1850/28-1) and by the Thuringer Landessternwarte Tautenburg. S.B. aknowledges funding from the EU Horizon under ERC Consolidator Grant, no. InspiReM-101043372. A.E.C. is partially supported by the 2023/24 'Research and Education' grant from Fondazione CRT. The OAVdA is managed by the Fondazione Cl'ement Fillietroz-ONLUS, which is supported by the Regional Government of the Aosta Valley, the Town Municipality of Nus and the Unit'e des Communes valdotaines Mont-Emilius. The LBT is an international collaboration of the University of Arizona, Italy (INAF: Istituto Nazionale di Astrofisica), Germany (LBTB: LBT Beteiligungsgesellschaft), The Ohio State University, representing also the University of Minnesota, the University of Virginia, and the University of Notre Dame. This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester.</text>
<section_header_level_1><location><page_16><loc_10><loc_27><loc_52><loc_28></location>Facilities: GROND, Swift (BAT, XRT and UVOT), LBT</section_header_level_1>
<text><location><page_16><loc_8><loc_22><loc_92><loc_26></location>Software: Afterglowpy (Ryan et al. 2020), NMMA (Pang et al. 2023), astropy (Astropy Collaboration et al. 2013, 2018), HOTPANTS (Becker 2015), PyRAF (Science Software Branch at STScI 2012), DRAGONS (Labrie et al. 2019), WCSTools (Mink 2019).</text>
<section_header_level_1><location><page_16><loc_46><loc_18><loc_54><loc_19></location>APPENDIX</section_header_level_1>
<section_header_level_1><location><page_16><loc_43><loc_13><loc_57><loc_14></location>A. HOST GALAXY</section_header_level_1>
<table>
<location><page_17><loc_32><loc_66><loc_67><loc_90></location>
<caption>Table 4. Measured AB magnitudes of the host.</caption>
</table>
<text><location><page_17><loc_8><loc_63><loc_92><loc_65></location>Note -Magnitudes are measured within an aperture with radius of 4 × FWHM and are not corrected for Galactic foreground extinction.</text>
<text><location><page_17><loc_8><loc_54><loc_92><loc_57></location>Table 4 provides the magnitude of the GRB host galaxy based on GROND's 4th-epoch observations and LBT's visit of the field four years after the burst (see Sect. 2.2).</text>
<section_header_level_1><location><page_17><loc_34><loc_51><loc_66><loc_52></location>B. GRB-SN PEAK LUMINOSITY RANGE</section_header_level_1>
<text><location><page_17><loc_8><loc_21><loc_92><loc_50></location>We further investigated on possible biases towards the faint end of our GRB-SN sample by comparing with SN Ic peak luminosity properties observed so far. The r-band absolute peak-magnitude ( M r,peak ) distribution of broad line (BL) Type Ic SNe spans at most 2.5 mags (between -17 . 7 and -20 . 8 mag, with an average of M r,peak = -18 . 7 ± 0 . 7 mag, Taddia et al. 2019; Barbarino et al. 2021; Gomez et al. 2022). GRB-SNe observed to date with good data sampling, are at least ∼ 1 mag brighter and trace the high-luminosity end of the above Type Ic BL distribution (e.g., Hjorth & Bloom 2012; Hjorth 2013; Cano et al. 2017; Klose et al. 2019), but the width of their peak luminosity distribution (Hjorth & Bloom 2012; Cano et al. 2017; Kann et al. 2019b; Klose et al. 2019) is not substantially different to the width of the corresponding peak luminosity distribution of type Ic and BL-SNe (see also Soderberg et al. 2006). By considering the GRB-SNe with the best follow-up in multiple bands (including spectroscopy), on the high end site of the peak luminosity distribution remains GRB 111209A/SN 2011kl (e.g., Nakauchi et al. 2013; Kann et al. 2019b) while on the low-end site there are GRB 100316D/SN 2010bh (e.g., Starling et al. 2011; Olivares E. et al. 2012b) and GRB 060218/SN 2006aj (e.g., Pian et al. 2006; Ferrero et al. 2006): the r -band peak magnitudes of these extreme GRB-SNe have a difference that lies between 2 and 3 mags, similar to the broad line (BL) Type Ic SNe. In principle, the long bursts GRB 990712 (Bjornsson et al. 2001), GRB 021211 (Della Valle et al. 2003; Zeh et al. 2004), GRB 040924 (Soderberg et al. 2006; Wiersema et al. 2008), and GRB 060904B (Cano et al. 2017) could have been followed by even slightly fainter SNe than SN 2010bh, but in these cases the data base is comparably poor and no strong conclusions could be made. While, e.g., any SN that was associated with the long bursts GRB 060605 and GRB 060614 must have had a luminosity < 1% of the luminosity of the prototypical SN 1998bw (Fynbo et al. 2006), here and in similar cases (e.g., GRB 111005A; Michaglyph[suppress]lowskI et al. 2018; Tanga et al. 2018) there is no evidence for SN light.</text>
<text><location><page_17><loc_8><loc_16><loc_92><loc_21></location>In conclusion, if substantially less luminous GRB-SNe do exist formally remains an open question, at least from the obervational point of view, and the peak luminosity range of GRB-SNe sample we have used to infer the luminosity of any SN associated with GRB 191019A is the most robust so far.</text>
<section_header_level_1><location><page_17><loc_26><loc_13><loc_74><loc_14></location>C. AFTERGLOW AND KILONOVA JOINT FIT COMPARISON</section_header_level_1>
<text><location><page_17><loc_8><loc_9><loc_92><loc_12></location>In Table 5 we present the results obtained for different kilonova models within the NMMA framework, in addition to the kilonova model presented in Sect. 3.3.</text>
<text><location><page_18><loc_8><loc_74><loc_92><loc_92></location>At first, we have tested a model from Kasen et al. (2017) (Kasen17-Jet in Table 5) which assumes only one ejecta component and has three parameters: the ejecta mass, the ejecta velocity and the lanthanide mass fraction ( χ lan ). Then we made a different assumption on the progenitor by considering a kilonova emission from a neutron star black hole binary system as predicted with POSSIS (Bu19-NSBH-Jet in Table 5), which has three parameters: the dynamical ejecta mass, the wind mass and the orbital plane inclination, which is linked to the viewing angle of the jet assumed in the afterglow modelling. These models were finally compared with the results obtained by assuming the simple analytical model described in Metzger (2017) (Metzger17-Jet in Table 5), which assumes one component and has four parameters: the ejecta mass, the ejecta velocity, the power-law index β of the ejecta mass distribution expressed as a function of its velocity (the faster ejecta matter lies ahead of slower matter and the distribution of mass with velocity greater than a value v 0 can be approximated with a power-law M ( > v 0 ) = M ( v/v 0 ) -β , and the opacity k r .</text>
<text><location><page_18><loc_8><loc_68><loc_92><loc_74></location>The afterglow emission was modelled within the Afterglowpy framework (Afterglow in Table 5), as described in Sect. 3.3. However, contrary to our previous analysis, the microphyiscal parameters ϵ e and ϵ B are now fixed to 0 . 5 and 0 . 01, respectively. In doing so, we aim at reproduce similar assumptions to Lazzati et al. (2023) and further investigate on the circumburst density estimates.</text>
<text><location><page_18><loc_8><loc_60><loc_92><loc_68></location>The highest Bayesian evidence ( Z ) is obtained for the Bu19-NSBH-Jet model. Following Kunert et al. (2024) and references therein, by considering Bu19-NSBH-Jet as the reference model, we find -1 . 10 < ln( Z/Z Bu 19 -NSBH -Jet ) < 0 for the Kasen17-Jet and Bu19-BNS-Jet models, while ln( Z/Z Bu 19 -NSBH -Jet ) < -4 . 5 for Metzger17-Jet and Afterglow models, indicating a strong evidence against the latter two models, while no preference could be set among Bu19-BNSJet, Bu19-NSBH-Jet and Kasen17-Jet models.</text>
<section_header_level_1><location><page_18><loc_44><loc_53><loc_56><loc_54></location>REFERENCES</section_header_level_1>
<text><location><page_18><loc_8><loc_51><loc_36><loc_52></location>Abbott, B. P., et al. 2017, ApJ, 848, L13,</text>
<text><location><page_18><loc_10><loc_49><loc_30><loc_50></location>doi: 10.3847/2041-8213/aa920c</text>
<text><location><page_18><loc_8><loc_42><loc_47><loc_48></location>Ahumada, T., Singer, L. P., Anand, S., et al. 2021, Nature Astronomy, 5, 917, doi: 10.1038/s41550-021-01428-7 Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33,</text>
<text><location><page_18><loc_10><loc_41><loc_33><loc_42></location>doi: 10.1051/0004-6361/201322068</text>
<text><location><page_18><loc_8><loc_35><loc_48><loc_40></location>Astropy Collaboration, Price-Whelan, A. M., Sip"ocz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f Barbarino, C., Sollerman, J., Taddia, F., et al. 2021, A&A,</text>
<text><location><page_18><loc_10><loc_34><loc_39><loc_35></location>651, A81, doi: 10.1051/0004-6361/202038890</text>
<text><location><page_18><loc_8><loc_30><loc_45><loc_33></location>Barkov, M. V., & Pozanenko, A. S. 2011, MNRAS, 417, 2161, doi: 10.1111/j.1365-2966.2011.19398.x</text>
<text><location><page_18><loc_8><loc_27><loc_46><loc_30></location>Barthelmy, S. D., Barbier, L. M., Cummings, J. R., et al. 2005, SSRv, 120, 143, doi: 10.1007/s11214-005-5096-3</text>
<text><location><page_18><loc_8><loc_22><loc_45><loc_26></location>Becker, A. 2015, HOTPANTS: High Order Transform of PSF ANd Template Subtraction, Astrophysics Source Code Library, record ascl:1504.004.</text>
<text><location><page_18><loc_10><loc_20><loc_26><loc_21></location>http://ascl.net/1504.004</text>
<text><location><page_18><loc_8><loc_15><loc_46><loc_19></location>Beniamini, P., Duque, R., Daigne, F., & Mochkovitch, R. 2020, MNRAS, 492, 2847, doi: 10.1093/mnras/staa070 Berger, E. 2014, ARA&A, 52, 43,</text>
<text><location><page_18><loc_10><loc_13><loc_38><loc_14></location>doi: 10.1146/annurev-astro-081913-035926</text>
<text><location><page_18><loc_8><loc_10><loc_47><loc_12></location>Bernuzzi, S. 2020, General Relativity and Gravitation, 52, 108, doi: 10.1007/s10714-020-02752-5</text>
<text><location><page_18><loc_52><loc_47><loc_92><loc_52></location>Bjornsson, G., Hjorth, J., Jakobsson, P., Christensen, L., & Holland, S. 2001, ApJL, 552, L121, doi: 10.1086/320328 Bulla, M. 2019, MNRAS, 489, 5037,</text>
<text><location><page_18><loc_54><loc_46><loc_72><loc_47></location>doi: 10.1093/mnras/stz2495</text>
<text><location><page_18><loc_52><loc_41><loc_91><loc_45></location>-. 2023, MNRAS, 520, 2558, doi: 10.1093/mnras/stad232 Burrows, D. N., Hill, J. E., Nousek, J. A., et al. 2005, SSRv, 120, 165, doi: 10.1007/s11214-005-5097-2</text>
<text><location><page_18><loc_52><loc_37><loc_88><loc_40></location>Camisasca, A. E., Guidorzi, C., Amati, L., et al. 2023, A&A, 671, A112, doi: 10.1051/0004-6361/202245657</text>
<text><location><page_18><loc_52><loc_32><loc_88><loc_36></location>Cano, Z., Wang, S.-Q., Dai, Z.-G., & Wu, X.-F. 2017, Advances in Astronomy, 2017, 8929054, doi: 10.1155/2017/8929054</text>
<text><location><page_18><loc_52><loc_29><loc_91><loc_31></location>Chambers, K. C., Magnier, E. A., Metcalfe, N., et al. 2016, arXiv e-prints, arXiv:1612.05560,</text>
<text><location><page_18><loc_54><loc_27><loc_75><loc_28></location>doi: 10.48550/arXiv.1612.05560</text>
<text><location><page_18><loc_52><loc_23><loc_91><loc_26></location>Coulter, D. A., Foley, R. J., Kilpatrick, C. D., et al. 2017, Science, 358, 1556, doi: 10.1126/science.aap9811</text>
<text><location><page_18><loc_52><loc_20><loc_91><loc_23></location>Coward, D. M., Gendre, B., Tanga, P., et al. 2017, PASA, 34, e005, doi: 10.1017/pasa.2016.61</text>
<text><location><page_18><loc_52><loc_18><loc_91><loc_19></location>Dainotti, M. G., De Simone, B., Islam, K. M., et al. 2022,</text>
<text><location><page_18><loc_54><loc_16><loc_84><loc_18></location>ApJ, 938, 41, doi: 10.3847/1538-4357/ac8b77</text>
<text><location><page_18><loc_52><loc_13><loc_92><loc_16></location>Della Valle, M., Malesani, D., Benetti, S., et al. 2003, A&A, 406, L33, doi: 10.1051/0004-6361:20030855</text>
<text><location><page_18><loc_52><loc_10><loc_91><loc_12></location>Dietrich, T., Coughlin, M. W., Pang, P. T. H., et al. 2020, Science, 370, 1450, doi: 10.1126/science.abb4317</text>
<table>
<location><page_19><loc_9><loc_55><loc_90><loc_88></location>
<caption>Table 5. The 90% confidence interval and median values for each parameter inferred from joint afterglow and kilonova and afterglow only jet modelling performed by using the code NMMA.</caption>
</table>
<text><location><page_19><loc_8><loc_45><loc_92><loc_54></location>Note -E 0 = kinetic fireball energy; n = particle number density in the circumburst environment; θ c =half-opening angle of the jet core; ι =viewing angle with respect to jet axis; p = electron energy distribution power-law index; M dyn ej = dynamical ejecta mass; M wind ej = wind ejecta mass; M ej = total ejecta mass; v ej = ejecta velocity; Φ = half-opening angle of lanthanide-rich equatorial ejecta; ϵ e = shock energy fraction that goes into the electrons; ϵ B = shock energy fraction that goes into the magnetic energy density; χ lan = lanthanide mass fraction (Kasen et al. 2017) ; κ r = opacity; β = power-law index of the ejecta mass distribution as a function of its velocity (Metzger 2017); ln( Z ) = natural logarithm of the bayes evidence. We note that the Bu19-BNS-Jet model is the same as the one presented in Sect. 3.3 where now ϵ e and ϵ B are fixed.</text>
<text><location><page_19><loc_8><loc_39><loc_46><loc_40></location>Evans, P. A., Osborne, J. P., Burrows, D. N., et al. 2019,</text>
<text><location><page_19><loc_8><loc_11><loc_48><loc_39></location>GRB Coordinates Network, 26034, 1 Evans, P. A., Beardmore, A. P., Page, K. L., et al. 2007, A&A, 469, 379, doi: 10.1051/0004-6361:20077530 -. 2009, MNRAS, 397, 1177, doi: 10.1111/j.1365-2966.2009.14913.x Evans, P. A., Willingale, R., Osborne, J. P., et al. 2010, A&A, 519, A102, doi: 10.1051/0004-6361/201014819 Ferrero, P., Kann, D. A., Zeh, A., et al. 2006, A&A, 457, 857, doi: 10.1051/0004-6361:20065530 Fong, W., Berger, E., Margutti, R., & Zauderer, B. A. 2015, ApJ, 815, 102, doi: 10.1088/0004-637X/815/2/102 Fong, W.-f., Nugent, A. E., Dong, Y., et al. 2022, ApJ, 940, 56, doi: 10.3847/1538-4357/ac91d0 Fontana, A., Dunlop, J. S., Paris, D., et al. 2014, A&A, 570, A11, doi: 10.1051/0004-6361/201423543 Fynbo, J. P. U., Perley, D. A., de Ugarte Postigo, A., et al.</text>
<text><location><page_19><loc_10><loc_10><loc_38><loc_11></location>2019, GRB Coordinates Network, 26041, 1</text>
<text><location><page_19><loc_52><loc_11><loc_90><loc_40></location>Fynbo, J. P. U., Watson, D., Thone, C. C., et al. 2006, Nature, 444, 1047, doi: 10.1038/nature05375 Gehrels, N., Chincarini, G., Giommi, P., et al. 2004, ApJ, 611, 1005, doi: 10.1086/422091 Gendre, B., Crisp, H., Coward, D., et al. 2019, GRB Coordinates Network, 26098, 1 Gomez, S., Berger, E., Nicholl, M., Blanchard, P. K., & Hosseinzadeh, G. 2022, ApJ, 941, 107, doi: 10.3847/1538-4357/ac9842 Gompertz, B. P., Ravasio, M. E., Nicholl, M., et al. 2023, Nature Astronomy, 7, 67, doi: 10.1038/s41550-022-01819-4 Greiner, J., Bornemann, W., Clemens, C., et al. 2008, PASP, 120, 405, doi: 10.1086/587032 Greiner, J., Mazzali, P. A., Kann, D. A., et al. 2015, Nature, 523, 189, doi: 10.1038/nature14579 Hjorth, J. 2013, Philosophical Transactions of the Royal Society of London Series A, 371, 20120275,</text>
<text><location><page_19><loc_54><loc_10><loc_72><loc_11></location>doi: 10.1098/rsta.2012.0275</text>
<table>
<location><page_20><loc_8><loc_9><loc_48><loc_92></location>
</table>
<table>
<location><page_20><loc_52><loc_9><loc_92><loc_92></location>
</table>
<table>
<location><page_21><loc_8><loc_43><loc_48><loc_92></location>
</table>
<unordered_list>
<list_item><location><page_21><loc_52><loc_90><loc_78><loc_91></location>Tody, D. 1993, ASP Conf. Ser., 52, 173</list_item>
</unordered_list>
<text><location><page_21><loc_52><loc_89><loc_74><loc_90></location>Troja, E. 2023, Universe, 9, 245,</text>
<text><location><page_21><loc_54><loc_87><loc_73><loc_88></location>doi: 10.3390/universe9060245</text>
<unordered_list>
<list_item><location><page_21><loc_52><loc_82><loc_84><loc_86></location>Troja, E., Ryan, G., Piro, L., et al. 2018, Nature Communications, 9, 4089, doi: 10.1038/s41467-018-06558-7</list_item>
<list_item><location><page_21><loc_52><loc_79><loc_91><loc_81></location>Troja, E., Castro-Tirado, A. J., Becerra Gonz'alez, J., et al. 2019, MNRAS, 489, 2104, doi: 10.1093/mnras/stz2255</list_item>
<list_item><location><page_21><loc_52><loc_75><loc_90><loc_78></location>Troja, E., Fryer, C. L., O'Connor, B., et al. 2022, Nature, 612, 228, doi: 10.1038/s41586-022-05327-3</list_item>
<list_item><location><page_21><loc_52><loc_72><loc_89><loc_75></location>Wang, S.-N., Lu, H.-J., Yuan, Y., et al. 2024, ApJ, 963, 156, doi: 10.3847/1538-4357/ad2205</list_item>
<list_item><location><page_21><loc_52><loc_69><loc_90><loc_71></location>Wiersema, K., van der Horst, A. J., Kann, D. A., et al. 2008, A&A, 481, 319, doi: 10.1051/0004-6361:20078050</list_item>
<list_item><location><page_21><loc_52><loc_65><loc_92><loc_68></location>Yang, J., Ai, S., Zhang, B.-B., et al. 2022, Nature, 612, 232, doi: 10.1038/s41586-022-05403-8</list_item>
</unordered_list>
<text><location><page_21><loc_52><loc_57><loc_90><loc_65></location>Yolda¸s, A. K., Kruhler, T., Greiner, J., et al. 2008, in American Institute of Physics Conference Series, Vol. 1000, American Institute of Physics Conference Series, ed. M. Galassi, D. Palmer, & E. Fenimore, 227-231, doi: 10.1063/1.2943450</text>
<unordered_list>
<list_item><location><page_21><loc_52><loc_54><loc_92><loc_56></location>Zeh, A., Klose, S., & Hartmann, D. H. 2004, ApJ, 609, 952, doi: 10.1086/421100</list_item>
<list_item><location><page_21><loc_52><loc_50><loc_85><loc_53></location>Zhang, B., & M'esz'aros, P. 2001, ApJL, 552, L35, doi: 10.1086/320255</list_item>
<list_item><location><page_21><loc_52><loc_47><loc_92><loc_50></location>Zhang, H.-H., Zhu, J.-P., & Yu, Y.-W. 2024, arXiv e-prints, arXiv:2406.10904, doi: 10.48550/arXiv.2406.10904</list_item>
<list_item><location><page_21><loc_52><loc_44><loc_87><loc_46></location>Zhu, Z. P., Yu, B. Y., Xu, D., & Gao, X. 2019, GRB Coordinates Network, 26059, 1</list_item>
</unordered_list>
</document>
[
{
"title": "ABSTRACT",
"content": "GRB 191019A was a long Gamma-ray burst (GRB) lasting ∼ 65 s and, as such, originally thought to be linked to a core-collapse supernova. However, even though follow-up observations identified the optical counterpart close to the bright nucleus of a nearby ancient galaxy ( z =0.248), no associated supernova was found. This led to the suggestion that the burst was caused by the merger of two compact stellar objects, likely in a dense circumnuclear environment. By using a recently developed diagnostic tool based on prompt emission temporal properties, we noticed that GRB 191019A falls among those long GRBs which are associated with compact mergers and with evidence of kilonova light. We thus re-analyzed unpublished GROND multi-color ( g ' r ' i ' z ' JHK s ) data obtained between 0.4 and 15 days post trigger. Image subtraction confirmed the optical counterpart in all four optical bands, with GROND tracking its fading until 1.5 days post-burst. Incorporating publicly available Swift -XRT data, a joint fit of an afterglow plus a kilonova model revealed a better match than an afterglow-only scenario. The resulting kilonova properties resemble those of AT2017gfo associated with the binary neutron star merger GW170817, with a total ejected mass of ∼ 0 . 06 M ⊙ . Contrary to previous findings inferring a high-density circumburst environment ( n 0 ∼ 10 7 -8 cm -3 ), our analysis finds standard conditions ( n 0 ∼ 1 cm -3 ), suggesting the long duration of GRB 191019A was intrinsic rather than due to jet interaction with a dense external medium. Keywords: Gamma-ray bursts - Compact objects",
"pages": [
1
]
},
{
"title": "The puzzling long GRB 191019A: Evidence for Kilonova Light",
"content": "G. Stratta , 1, 2, 3, 4 A. M. Nicuesa Guelbenzu , 5 S. Klose , 5 A. Rossi , 6 P. Singh, 1, 6 E. Palazzi , 6 C. Guidorzi, 7, 8, 6 A. Camisasca, 9 S. Bernuzzi, 10 A. Rau, 11 M. Bulla, 7, 8, 12 F. Ragosta, 13, 14 E. Maiorano, 6 and 15 D. Paris 1 Institut fur Theoretische Physik, Goethe Universitat, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany 2 INAF - Osservatorio di Astrofisica e Scienza dello Spazio, via Piero Gobetti 93/3, 40129 Bologna, Italy 3 Istituto di Astrofisica e Planetologia Spaziali, via Fosso del Cavaliere 100, I-00133 Roma, Italy 4 Istituto Nazionale di Fisica Nucleare-Roma 1, Piazzale Aldo Moro 2, I-00185 Roma, Italy 5 Thuringer Landessternwarte Tautenburg, Sternwarte 5, 07778 Tautenburg, Germany 6 INAF - Osservatorio di Astrofisica e Scienza dello Spazio, via Piero Gobetti 93/3, 40129 Bologna, Italy 7 Department of Physics and Earth Science, University of Ferrara, Via Saragat 1, 44122 Ferrara, Italy 8 INFN - Sezione di Ferrara, Via Saragat 1, 44122 Ferrara, Italy 9 Astronomical Observatory of the Autonomous Region of the Aosta Valley (OAVdA), Loc. Lignan 39, I-11020, Nus (Aosta Valley), Italy 10 Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universitat Jena, 07743, Jena, Germany 11 Max-Planck-Institut fur Extraterrestrische Physik, Giessenbachstraße 1, 85748 Garching, Germany 12 INAF, Osservatorio Astronomico d'Abruzzo, via Mentore Maggini snc, 64100 Teramo, Italy 13 Dipartimento di Fisica 'Ettore Pancini', Universit'a di Napoli Federico II, Via Cinthia 9, 80126 Naples, Italy 14 INAF - Osservatorio Astronomico di Capodimonte, Via Moiariello 16, I-80131 Naples, Italy 15 INAF - Osservatorio Astronomico di Roma, via Frascati 33, 00040 Monte Porzio Catone, Italy",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "The empirical classification of gamma-ray bursts (GRBs) into two classes according to the bimodal burst duration and hardness distribution (see, e.g., Kouveliotou et al. 1993), has been interpreted as the evidence of two different progenitor channels. Indeed, it was found that the vast majority of sufficiently nearby long GRBs are found to be spatially and temporally coincident with core-collapse (CC) supernovae (SNe) and hosted in star-forming galaxies. On the other hand, short GRBs lack any associated SN and are typically located in the outer regions of early-type galaxies, suggesting a progenitor belonging to an old stellar population, formally consistent with compact binary mergers. This picture was confirmed by the association of the short GRB 170817A with the gravitational wave event GW170817 that was compatible with a binary neutron star (NS-NS) merger (e.g., Abbott et al. 2017). From the very same event and thanks to its spatial proximity (40 Mpc), the first robust evidence of a kilonova (AT2017gfo) was obtained (e.g., Coulter et al. 2017), confirming the predicted thermal emission from matter released and heated during a NS-NS merger (e.g., Li & Paczy'nski 1998). In the last decade, kilonova signatures were observed in a number of past short GRBs that were sufficiently nearby, as for instance GRB 130603B ( z =0.3565, Tanvir et al. 2013), GRB 150101B ( z =0.134, Troja et al. 2018), GRB 160821B ( z =0.1613, Troja et al. 2019), though with much lower significance than AT2017gfo due to the larger distances of these events (see, e.g., Troja 2023 for a review). In recent years, the standard long and short-GRB progenitor paradigm has been blurred by mounting evidence of the existence of long GRBs with no associated CC-SN but with evidence of a kilonova component, more consistent with a compact binary merger progenitor. Two striking examples are the long GRB 211211A at redshift z = 0 . 076 (350 Mpc; e.g., Troja et al. 2018; Mei et al. 2022; Rastinejad et al. 2022; Yang et al. 2022; Gompertz et al. 2023) and GRB 230307A at z = 0 . 0646 (300 Mpc; e.g., Levan et al. 2023), for which a kilonova was clearly identified in the optical afterglow. In addition, evidence of short GRBs showing massive star collapse properties were also found (e.g., Ahumada et al. 2021; Rossi et al. 2022) further contributed to dismantle the standard GRB classification scheme. These results suggest that some of the past GRB progenitor classifications based on the burst duration and hardness only, have to be revisited. This holds in particular for GRB 191019A. GRB 191019A was discovered (Simpson et al. 2019) with the Burst Alert Telescope (BAT; Barthelmy et al. 2005) on board the Neil Gehrels Swift Observatory (Gehrels et al. 2004). It showed a complex burst light curve with multiple peaks with a duration T 90 = 64 . 6 ± 4 . 5 s. Its 15-150 keV spectrum could be well modeled with a power law with a photon index of 2 . 25 ± 0 . 05 and a fluence of 10 -5 erg cm -2 (Krimm et al. 2019) putting this source among the soft/long GRBs. Follow-up observations with the Swift X-ray telescope (XRT; Burrows et al. 2005) robustly identified an uncatalogued X-ray source (Evans et al. 2019). Rapid follow-up observations with the Zeiss-1000 1-m telescope of Tien Shan Astronomical Observatory revealed only one optical source within the XRT error circle with R = 18 . 37 ± 0 . 03 mag at T 0 +14 min (Reva et al. 2019). The position of this source was coincident with a catalogued object quoted to be fainter than the observed one, leading to the interpretation this is the host galaxy of GRB 191019A with the afterglow on top of it (Reva et al. 2019). The Zadko telescope (Coward et al. 2017) observed GRB 191019A field at similar epochs and confirmed the object reported by Reva et al. (2019) with R =18.92 mag, with no significant flux variation within 1.5 hours (Gendre et al. 2019). Similar claims of a lack of flux variations were provided by other teams performing early-epoch observations (Fynbo et al. 2019, Nicuesa Guelbenzu 2019, Perley et al. 2019a, Zhu et al. 2019). However, late observations performed with the Nordic Optical Telescope (NOT) at T 0 +3.25 days were analyzed with image subtraction methods and revealed that the source was fading, confirming the optical transient plus host identification (Perley et al. 2019b). Spectroscopic analysis of the host galaxy showed several absorption lines at redshift z = 0 . 248 and a spectral energy distribution consistent with a galaxy dominated by an old stellar population ( > 1 Gyr), with a star formation rate (SFR) of 0 . 06 ± 0 . 03 M ⊙ yr -1 , a stellar mass of 3 × 10 10 M ⊙ , and small dust extinction of A V = 0 . 19 ± 0 . 08 mag (Levan et al. 2023). The measured SFR is at odds with the typical high values inferred in long GRB hosts (but see Rossi et al. 2014). Further hints of anomalies for the long GRB 191019A came from the absence of an associated SN, despite the low redshift. Deep limits in g, r and z obtained between 2 and 73 days with the NOT and optical imaging with the Hubble Space Telescope at 30 and 184 days, put strong constraints on any SN emission up to > 20 times less luminous than SN1998bw (Levan et al. 2023). All these properties are at odds with a massive-star origin of GRB 191019A, while formally consistent with a compact object binary merger progenitor. Though, the deep limits obtained with the NOT Telescope at > 2 days could not confirm the presence of an early kilonova emission (Levan et al. 2023). An interesting feature characterizing GRB 191019A is its projected distance from the host galaxy center of ≲ 100 pc, the closest distance among all known short GRBs to their corresponding galactic nucleus (Levan et al. 2023). Indeed, compact binary progenitors formed in stellar binary systems are thought to have large kick velocities acquired during the CC-SN phase of the binary components, and for this reason short GRBs are typically found in the outskirts of their host galaxies, with large offsets from the center (e.g., Berger 2014; Fong et al. 2022; O'Connor et al. 2022). As noted by Levan et al. (2023), the proximity of GRB 191019A to the host galaxy center suggests that the possible progenitor compact binary system could have formed through dynamical encounters, which are thought to be favored in the dense gaseous environment of supermassive black hole surrounding disks through kinetic energy dissipation ('gas-capture' binary formation channel, Tagawa et al. 2020). An exciting consequence of this scenario is that dense environments could also alter the prompt emission properties of a short GRB, making it longer and softer (Lazzati et al. 2023). This possibility was explored for GRB 191019A and it was found compatible with an environmental density of the order of 10 7 to 10 8 cm -3 (Lazzati et al. 2023). Starting from Lazzati et al.'s conclusions, Wang et al. (2024) investigated the interaction of a possible kilonova ejecta from GRB 191019A with a dense circumstellar medium (CSM) that could be detected years after the merger. Their model predicts that in a very dense environment, the smaller the kilonova ejected mass, the fainter is the radioactively-powered luminosity but the higher is the contribution from kilonova-CSM interaction at late times. No evidence of kilonova ejecta-CSM interaction was found for GRB 191019A so far, and a possible detection in the future would require an ejected mass less than 2 × 10 -5 M ⊙ (Wang et al. 2024). Here we further investigate the properties of GRB 1901910A and the nature of the optical transient that followed the burst, with the goal to find arguments in favour or against its possible compact merger origin. We use the recently developed GRB prompt emission minimum variability (MVT) criterion (Camisasca et al. 2023) to explore the properties of the burst, and perform a complete reanalysis of the light curve of the optical/X-ray transient with particular emphasis on so far not analyzed GROND (Greiner et al. 2008) observations starting 10.3 hours post burst (Nicuesa Guelbenzu 2019). We use the sophisticated Nuclear - Multi-Messenger Astronomy (NMMA, Dietrich et al. 2020; Pang et al. 2023) software package which allows for afterglow and kilonova joint Bayesian inference, to explore if the multi-color light curve of the optical transient can be understood as powered by afterglow emission only or if a kilonova component was present too. Throughout this paper, we adopt a flat cosmological model with H 0 =67.4 km s -1 Mpc -1 , Ω M =0.315, and Ω Λ =0.685 (Planck Collaboration et al. 2020). For these parameters a redshift of z =0.248 (Fynbo et al. 2019) corresponds to a luminosity distance of d L = 1.29 Gpc, 1 arcsec corresponds to 4.03 kpc projected distance, and distance modulus is m -M = 40 . 55 mag. The Milky Way reddening E ( B -V ) along the line of sight towards the source is between 0.03 and 0.04 mag (Schlegel et al. 1998; Schlafly & Finkbeiner 2011). We present the observations and data reduction in Sect. 2, and our results in Sect. 3. In our data analysis we corrected the apparent magnitudes for the Galactic reddening and adopted a host-galaxy visual extinction along the line of sight of A host V = 0.06 mag (Levan et al. 2023). Discussion and conclusions are presented in Sect. 4.",
"pages": [
1,
2,
3
]
},
{
"title": "2.1. GROND early-time Observations",
"content": "With a duration of about 65 seconds, GRB 191019A was a classical long GRB. At a redshift of z = 0 . 248, a rising SN 1998bw component was thus expected to be detectable with GROND within about two weeks after the GRB trigger. Therefore, following earlier efforts with GROND to explore GRB-SN light curves (Olivares E. et al. 2012a, 2015; Greiner et al. 2015; Kann et al. 2019a; Klose et al. 2019) we performed follow-up observations of the field during several epochs between 0.4 (Nicuesa Guelbenzu 2019) and 15.4 days post GRB trigger (Table 1). Observations were then terminated since no evidence for SN light was found. GROND data were reduced in a standard fashion (bias subtraction, flat fielding, co-adding; Kruhler et al. 2008; Yolda¸s et al. 2008), based on numerical routines in IRAF (Image Reduction and Analysis Facility; Tody 1993).",
"pages": [
3
]
},
{
"title": "2.2. LBT late-time Observations",
"content": "Late-time observations of the field were performed with the Large Binocular Telescope (LBT) using the two twin LBC instruments equipped with the Sloan filters g ' r ' i ' z ' on October 20, 2023. These data were reduced using the data reduction pipeline developed at INAF - Osservatorio Astronomico di Roma (Fontana et al. 2014) which includes bias subtraction and flat-fielding, bad pixel and cosmic ray masking, astrometric calibration, and coaddition. On the LBT images, we measure the AB magnitudes for the host given in Table 4 (see Appendix A). Within the errors, these values are consistent with those measured with GROND in epoch 4 ( T -T 0 = 7 . 4756 days). Based on these data we conclude that 7 days post burst the transient did not contribute anymore to the observed flux in g ' r ' i ' z ' in a measurable quantity.",
"pages": [
3
]
},
{
"title": "2.3. Image subtraction and photometry",
"content": "In order to search for a transient hidden by the relatively bright host galaxy, we performed image subtraction on the GROND images using HOTPANTS (Becker 2015). HOTPANTS convolves the template and the source images to the same",
"pages": [
3
]
},
{
"title": "Stratta et al.",
"content": "Note -Column (2) provides the midtime of the observation, in units of days from burst trigger. Magnitudes are not corrected for Galactic foreground extinction. The photometry for epochs 1a to 3 was obtained after image subtraction against epoch 4 (see § 2.3 for details). On the contrary, the 3 σ upper limits for epoch 4 to 6 have been measured directly, without image subtraction. point spread function (PSF) and photometric scale. As a template image we used the GROND 4th-epoch observations (Table 1) since these images have the best seeing ( ∼ 1 . '' 0). We aligned the template and the source images by means of wcsremap 1 . The Gaussian input parameters in HOTPANTS were calculated following Becker (2015) considering the case in which the template FWHM is smaller than the source FWHM. In this way, the template images were smoothed to the seeing of the corresponding source images. In addition, we fixed several other parameters for HOTPANTS following Hu et al. (2022), but let free to vary the number of each region's stamps, the convolution kernel half width, and the size of Gaussians which compose the kernel. In doing so, we selected those input parameters that minimized the background noise on the residual image in a star-free region close to the position of the host galaxy. All resulting residual images were then photometrically calibrated with respect to the source images (see below). In the case of a non-detection of the transient the upper limit we used was measured locally on the subtracted image (Table 1). We have confirmed our results using also the 5th GROND epoch as template (seeing ∼ 1 . '' 3), although we obtained a lower SNR. We double checked the detections using the late LBT images (about 4 years or 1460 days after the burst trigger, see Sect. 2.2 and Table 4 in Appendix A). The g ' i ' z ' detections are confirmed for epochs 1a and 2, but with lower SNR likely due to the additional plate scaling, and a general worse PSF shape due to not perfect collimation. We have not checked the r ' -band since the above mentioned problem was worse in this filter, although it did not compromise the aperture photometry in Table 4. For this filter we have instead used (as template) a stack of the best Gemini-S images 2 obtained at 58-63 days after the burst (see Levan et al. 2023), and we obtained consistent results. In other words, the results obtained with GROND could not be improved. Finally, to better constrain the late transient decay, we performed image subtraction on the Gemini r -band image at 11.4 days, using the later Gemini images as template (see above), and obtained for the optical transient an upper limit of r > 25 . 8 AB mag. Apparent magnitudes were obtained by performing aperture photometry using DAOPHOT and APPHOT under PyRAF/IRAF with increasing apertures up to 4 times the FWHM and by PSF photometry with the DAOPHOT and ALLSTAR tasks of IRAF. PSF-fitting was used to measure the magnitudes of the transient after image subtraction. In the optical bands the data were calibrated using Pan-STARRS (Chambers et al. 2016), in the NIR bands using 2MASS (Skrutskie et al. 2006). Central wavelengths of the GROND filter bands are listed in Rossi et al. (2011).",
"pages": [
4
]
},
{
"title": "3.1. Application of the prompt emission minimum variability criterion",
"content": "According to Camisasca et al. (2023), GRB prompt emission minimum variability timescale (MVT) represents a diagnostic to infer the progenitor nature of a GRB. By estimating MVT from the full width at half maximum of the shortest, statistically significant peak in the prompt emission light curve (FWHM min ) of hundreds of GRBs, it was found that high variability (low FWHM min values) typically belongs to compact binary mergers. Interestingly, this result was found to be independent of the burst duration and for this reason it is an important probe for the nature of peculiar long GRBs with properties more similar to those of short GRBs. By analyzing the results from the Swift GRB sample analyzed by Camisasca et al. (2023), we checked which long GRBs sufficiently nearby for a possible kilonova detection, i.e. at z < 0 . 5, and without any associated SN, had an FWHM min compatible with compact binary merger values, and we found that GRB 191019A was satisfying our criteria. Specifically, for this GRB an MVT of FWHM min = 0 . 196 +0 . 068 -0 . 05 s has been measured, which is compatible with the values typically found for GRBs associated with compact binary mergers (Fig. 1). In particular, given the long burst duration of GRB 191019A ( T 90 = 64 . 35 ± 4 . 35 s), its position in the FWHM min vs T 90 diagram lies among the short GRBs showing a soft 'Extended Emission' (SEE, e.g., Norris & Bonnell 2006; Minaev et al. 2010). Interestingly, two famous long GRBs for which a kilonova component was found in the optical afterglow, namely GRB 211211A (Rastinejad et al. 2022; Troja et al. 2022; Yang et al. 2022; Gompertz et al. 2023) and GRB 230307A (e.g., Levan et al. 2023), lie in the same region of the diagram of GRB 191019A (Fig. 1). The SEE distribution in the T 90 -FWHM min plane clearly overlaps with the distribution of long GRBs associated with SNe, but at the same time the centroids of the two distributions are well separated: interestingly, GRB 191019A lies in the middle of the SEE region and in the outskirts of the SN-associated cases. This property alone cannot be taken as compelling evidence that GRB 191019A behaves like a SEE, but suggests a probable SEE-like nature than a collapsar one. These findings further support past interpretation that also GRB 191019A originated from a compact binary merger (e.g., Levan et al. 2023), and address the possible presence of a kilonova component, similarly to GRB 211211A and GRB 230307A.",
"pages": [
5
]
},
{
"title": "3.2. Re-analysis of the light curve of the Transient that followed the burst",
"content": "3.2.1. X-rays Adopting a power-law spectral energy distribution (SED), the best fit of the Swift -XRT 0.3-10.0 keV spectrum, computed by the automatic algorithm of the UK Swift Science Data Center (UKSSDC) Swift -XRT GRB Spectrum Repository 3 in the 3 . 2 -32 . 4 ks temporal window, provides a photon index of Γ = 2 . 0 +0 . 4 -0 . 3 and an intrinsic equivalent hydrogen column density of N H , z = 1 . 2 +1 . 6 -1 . 2 × 10 21 cm -2 in addition to a Galactic N H = 3 . 28 × 10 20 cm -2 . We obtained from the UKSSDC Burst Analyzer tool (Evans et al. 2010), the absorption-corrected X-ray fluxes in the energy range 0.3-10.0 keV, and we converted them into 1 keV flux densities by assuming a power law spectrum and the corresponding photon index provided by the Burst Analyzer in each temporal bin of the light curve. The flux density evolution in the X-ray band is compatible with a power-law decay 4 with α X = 1 . 44 ± 0 . 14.",
"pages": [
5
]
},
{
"title": "3.2.2. Optical bands",
"content": "We detected the optical transient during the first three GROND observations of the field, up to 1.5 days post burst (Table 1, Fig. 2). The transient has coordinates R.A., decl. (J2000) = 22:40:05.861, -17:19:42.77 ( ± 0.3 arcsec), measured on the combined g ' r ' i ' residual images of the 2nd-epoch observations (Fig. 3). Within the error, these coordinates agree with what has been reported by Perley et al. (2019b). Based on a joint fit of the griz -band magnitudes provided by Levan et al. (2023) and GROND's first visit of the field (epoch 1a in Table 1), we find that between both observing runs the optical transient was fading with a decay slope α opt = 1 . 43 ± 0 . 07. Within the errors, α X ∼ α opt up to T 0 +0 . 4 days (epoch 1a in Table 1), suggesting a common cooling regime of the electrons radiating optical and X-ray synchrotron light (see Fig. 4). This finding is compatible with results by (Levan et al. 2023) who showed that a single power-law SED from X-rays to the optical bands can describe the observations at T 0 +0 . 21 days. However, GROND's following observing runs 1b and 2 (Table 1) clearly show a significant flattening of the light curve, with a new decay slope of α opt ∼ 0 . 5 ± 0 . 1 (Fig. 4). The rescaled X-ray power-law model underpredicts the flux in the optical bands, with a > 3 σ deviation observed at T 0 +1 . 5 days. Even by assuming a shallower optical decay index (i.e., α opt = α X -0 . 25), as for the case where the synchrotron cooling frequency lies between the X-ray and the optical regime (e.g., Sari et al. 1998), the discrepancy at late times ( > 1 day) persists (Fig. 4). To quantify the null hypothesis that a power law model cannot fit the data and test for chromaticity of flux evolution, we considered the early data by Levan et al. (2023) and our GROND detections (i.e. from ∼ 0.2 to 1.5 days) for the two filters with best temporal sampling (i.e. r ' and g ' ): a simple power law model does not provide an acceptable fit, with probability of having the observed data set, given the null hypothesis (p-values), of 1 . 7 × 10 -5 and 4 × 10 -4 , respectively. By assuming a broken power law, with a steep-to-shallow behavior and a temporal break at 0.44 days, we find a significant improvement of the fit, with p-values of 0 . 79 and 0 . 73 in the r ' and g ' bands, respectively. The best fit broken power laws show that the flux evolution after the break is slightly shallower in the redder filter, suggesting a chromatic behaviour. We note that this peculiar steep-to-shallow optical light curve morphology is not unique, it was already observed in a small subset of GRBs in the early years of the Swift era (see, e.g., Kann et al. 2010), although with light curve breaks on average much earlier than what we observe for GRB 191019A. For instance, the optical light curve of GRB 090102 (Gendre et al. 2010) is similar to GRB 191019A but the break time was ∼ 0 . 1 hr (rest frame) after the burst. In the standard fireball model, which fairly accounts for late afterglow phenomenology as forward-shock radiation, a steep-to-shallow light curve is not envisioned (e.g., Sari et al. 1998). Such morphology could be produced by the presence of a reverse shock component dominating the forward shock at early times. However, reverse shock is expected on timescales of the order of minutes (e.g., Nakar & Piran 2004), in contrast with what we observe for GRB 191019A. Alternatively, energy injection into the forward shock could originate a light curve flattening. The spin-down radiation from a newly born magnetar (e.g., Zhang & M'esz'aros 2001) or a slightly off-axis structured jet (e.g., Beniamini et al. 2020) are plausible energy sources that are often invoked to explain the X-ray afterglow 'plateaus' observed in GRBs, and that might have an optical counterpart. In the case of GRB 191019A the lack of any evidence of a shallow phase in X-rays cast some doubts against the energy injection scenario, however.",
"pages": [
5,
6,
7,
8
]
},
{
"title": "3.2.3. Constraints on Supernova Light",
"content": "If GRB 191019A had been a classical long burst, then for a redshift of z =0.248 a supernova (SN) component was expected to be detectable with GROND. Observational constraints on SN light were first reported by Levan et al. (2023) based on HST observations 30 and 184 days post GRB trigger. No evidence for transient emission was found ( g > 24 , r > 23 . 5, and z > 22 mag). Figure 5 shows the upper limits we can set on any SN that followed GRB 191019A in comparison to the r ' -band light curves of 13 GRB-SNe that have been observed with GROND between 2007 and 2014 (see Table 2 in Klose et al. 2019). These events cover the whole GRB-SN luminosity distribution of well sampled GRB-SNe, from GRB 100316D/SN 2010bh (Olivares E. et al. 2012a), one of the faintest GRB-SN ever detected (e.g., Melandri et al. 2014; Cano et al. 2017; Dainotti et al. 2022), to GRB 111209A / SN 2011kl (Kann et al. 2019a), the most luminous GRB-SN ever observed. A visual inspection of Figure 5 shows that all light curves fall into a strip with a width of about 2-3 mag, which is limited by the very faint GRB 100316D/SN 2010bh and the very bright GRB 111209/SN 2011kl. The strongest limit we can set stamps from the time span between 7 and 12 days post burst and is well below the identified GRB-SN magnitude strip: if a CC-SN followed GRB 191019A, in r ' it was at least 3 mag less luminous than SN 1998bw and 2 mag fainter than SN 2010bh. According to the archived Gemini data (Sect. 2.3), the constraint on the luminosity is even stronger, > 4 mag in r at 11.4 days post burst. In conclusion, even if we take into account that the time evolution of GRB-SNe light curves show a certain parameter range (characterized by stretch factor), the SN limit we can set provides a constraint on the entire SN light curves. In this respect, any SN related to GRB 191019A must have been less luminous than each GRB-SN (see Appendix B).",
"pages": [
8
]
},
{
"title": "3.3. Joint Afterglow and Kilonova modelling",
"content": "Since the flattening observed in the optical light curve at late times (epochs 1b and 2 in Table 1) cannot be explained by the presence of a slowly rising SN (Sect. 3.2.3), and probably not by a reverse shock or energy injection either (Sect. 3.2.2), we explore now the possibility that the source of this additional radiation component was kilonova light. Given the relatively small redshift of GRB 191019A ( z = 0 . 248), a kilonova with a luminosity similar to AT2017gfo is expected to lie within the discovery space of a 2-m telescope, if not hidden by an intrinsic low luminosity. Therefore, following the procedure described in Rossi et al. (2020), we compared our data at T 0 +1 . 5 days in each filter, with the flux of the kilonova AT2017gfo (Coulter et al. 2017) associated with the NS-NS merger GW170817 (Abbott et al. 2017), shifted to the redshift of GRB 191019A. We find that an emission component similar to AT2017gfo but ∼ 4 times more luminous could reproduce the observations in the i ' and r ' bands, while our g ' -band flux seems to be brighter (Fig. 4). Apparently, the presence of a kilonova associated with GRB 191019A is overall compatible with the general properties of the observed optical-NIR emission at 0.4 and 1.5 days, although with slightly different brightness than AT2017gfo 5 . Motivated by the results obtained with our previous simplistic approach, we then explored the possible presence of a kilonova by comparing our multi-band dataset ( g ' , r ' , i ' , z ' , and X-ray) with a much more sophisticated model that takes into account simultaneously the presence of an afterglow and a kilonova component through a joint fit. For this purpose, we exploited the Nuclear - Multi-Messenger Astronomy (NMMA v0.2.0) framework that allows to estimate best-fit parameters with a bayesian inference method (Dietrich et al. 2020; Pang et al. 2023). For the afterglow modelling, NMMA uses the Afterglowpy python module (Ryan et al. 2020). Afterglowpy allows to model GRB afterglow light curves and spectra by taking into account the possible effects due to a complex jet structure and an off-axis observer. We here assumed a Gaussian profile for the jet structure, and that the whole ( ξ N = 1) electron population is shock-accelerated. With the exception of the viewing angle ι , which has a sine function, to all parameters we assigned a uniform function to model the prior probability (for details see Table 2). For the kilonova emission, NMMA allows to fit and simulate data using several numerical and analytical models. Assuming a binary neutron star merger as a progenitor system, we adopted here the kilonova modelling resulting from the time dependent 3D Monte Carlo code POSSIS for modelling radiation transport (Bulla 2019, 2023). We used the kilonova model grid published in Dietrich et al. (2020) based on the first version of POSSIS (Bulla 2019) where the kilonova ejecta is represented with two components: 1) a high-velocity (0 . 08 < v dyn /c < 0 . 3) dynamical ejecta of mass M dyn rj with a lanthanide-rich composition distributed about the equatorial plane with half-opening angle Φ, and lanthanide-poor composition at higher latitudes, and 2) a slower (0 . 025 < v wind /c < 0 . 08) wind (or 'post-merger') component, which is a spherical ejecta released from the merger remnant and debris disk, with an intermediate lanthanide content and mass M wind ej (see Dietrich et al. 2020). In the fit we fixed the GRB 191019A luminosity distance at 1289.3 Mpc, as obtained from the measured redshift ( z = 0 . 248) and assuming a flat cosmological model (see Sect. 1). We also considered an additional error budget (in magnitudes), which takes into account the uncertainties on the model predictions as well as possible systematic errors ( em syserr in the corner plot), to which we assigned a uniform prior with range from 0 to 2 mag. The resulting best-fit light curves computed in each band are plotted in Figure 6, the fit corner plot is in Figure 7, while the 90% confidence interval and median values of each parameter , as well as the adopted prior functions, are quoted in Table 2. By simply inspecting the resulting light curves, it is evident that the afterglow component is constrained mostly by the X-ray data and it is dominant during the early epochs in the optical bands. The afterglow best fit jet core semiaperture angle is about ∼ 10 deg (9 +4 -3 deg), and the observer viewing angle is ∼ 4 deg (4 +3 -2 deg), thus within the jet core. The fireball isotropic kinetic energy E 0 (1 . 4 +1 . 8 -1 . 0 × 10 52 erg) is nicely compatible with the observed prompt emission equivalent isotropic energy E iso , by assuming a reasonable efficiency of about η ∼ 10%, where η = E iso E iso + E 0 . Indeed, the 15-150 keV fluence measured by Swift -BAT is (1 . 00 ± 0 . 03) × 10 -5 erg cm -2 (Krimm et al. 2019), corresponding to a radiated energy E iso = (1 . 70 ± 0 . 05) × 10 51 erg. The latter was computed by assuming a power-law spectrum in the BAT bandpass: this assumption is reasonable given the softness of the spectrum, which is best fit with a photon index Γ = 2 . 25 ± 0 . 05 (Krimm et al. 2019), suggesting that the prompt emission peak energy lies below the BAT bandpass. Figure 6 also clearly shows that at later epochs ( > 1 day), the observed flux lies above the predicted levels of the afterglow component, and the presence of a kilonova provides a better match. Indeed, by assuming only the afterglow model, i.e. by removing the kilonova component from our initial model, we obtained a worse fit, with Bayesian evidence ln( Z ) = -21 . 1. By considering the ratio with the Bayesian evidences of the joint afterglow and kilonova model, for which we obtain ln( Z 0 ) = -14 . 1, the resulting Bayes factor (ln( B ) =ln( Z/Z 0 ) = -7 . 0) indicates a strong preference for the model which includes the kilonova component (see, e.g., Kunert et al. 2024 and references therein). The kilonova properties we find from our fit are compatible with a dynamical ejecta mass of M dyn ej ∼ 0 . 02 M ⊙ and a wind mass M wind ej ∼ 0 . 04 M ⊙ , though with large uncertainties (relative errors ≥ 60%). These values are slightly higher (yet consistent within the uncertainties) with those found for AT2017gfo by assuming the same kilonova modelling (Dietrich et al. 2020). This is in line with the need of a brighter kilonova (of about a factor 4, see Sect. 3.3 and Fig. 4) by simply superposing an AT2017gfo-like light curve on the data. We stress here that the real picture is likely much more complicated than a two-component scenario, and the dynamical/wind masses inferred should rather be interpreted as belonging to some high/low velocity components of a multi-component scenario with multiple ejecta episodes (e.g., Bernuzzi 2020; Nedora et al. 2021). At the same time, the kilonova model we assumed is among the most sophisticated ones publicly available, and the assumption of a kilonova from a NS-NS merger progenitor is the most reasonable choice given that, so far, the only GRB with kilonova emission for which we were able to infer the progenitor nature was GRB170817/AT2017gfo, wich we know from gravitational wave data analysis being originated from a binary neutron star merger. Nevertheless, we explored also other kilonova models available within the NMMA framework, with different levels of sophistication, and which have different assumptions on the progenitor nature and on the kilonova ejecta properties (see Appendix C). We find that the dataset for GRB 191019A did not allow us to confidently distinguish among different kilonova models, apart disfavouring the most simplistic ones. However, in all cases, we find that a joint afterglow plus kilonova fit is preferred with respect to an afterglow-only model. Interestingly, both the afterglow-only and afterglow plus kilonova model, provide as a best-fit for the circumburst environment particle density a value n 0 < 1 cm -3 , which is typically deduced for short-GRB environments (e.g., Berger 2014; Fong et al. 2015). This result is in stark contrast with the interpretation by Lazzati et al. (2023) that invokes the presence of a very high density environment with n 0 ∼ 10 7 -10 8 cm -3 (see also Levan et al. 2023). T able 2 . The 90% confidence in terv al and median v alues of eac h parameter inferred from the sim ultaneous afterglo w ( Ry an et al. 202 0 ) and kilono v a ( Dietric h et al. 2020 ) mo dell ing p erformed b y using the co de NMMA. F or the afterglo w comp onen t, w e assumed a Gaussian jet profi le and fiducial v alue s for the microph ysical parameters (see Sect. 3.3 an d Fig. 6 and Figure 7 ). The b ottom part of the table quotes the assumed p r i ors in the Ba y esian inference analysis, where 'U' stands for Uniform function, with minim um and maxim um v alue s quoted in the brac k ets. Note -E 0 = kinetic fireball energy; n = particle n um b er densit y in the circum burst en vironmen t; θ c =half-op ening angle of the jet core; θ w =half-op ening a ngle of the jet truncated-wings; ι =viewing angle with resp ect to jet axis; p = electron energy distribution p o w er-la w index; ϵ e = sho c k e nergy fraction that go es in to th e electrons; ϵ B = sho c k energy fraction that go es in to the magne tic energy densit y; M dyn ej = dynamical ejecta mass; M wej = wind ejecta mass; Φ = half-op ening angle of lan thanide-ric h equatorial ejecta. Note -All observed (obs) AB magnitudes in the r ' , i ' and z ' filters are 3 σ upper limits (Fig. 5). Columns (2), (5), and (8) quote the expected (exp) magnitudes. All the corresponding differences (observed minus expected; columns 4, 7, 10) are lower limits.",
"pages": [
8,
9,
10,
11,
12,
13
]
},
{
"title": "4. DISCUSSION AND CONCLUSIONS",
"content": "Recent analysis of an increasing number of GRBs which were initially classified as long bursts, and therefore thought to be originating from collapsing massive stars, suggest a better compatibility of the observational data with compact binary merger progenitors. By ignoring the standard burst duration classification, typical features of GRBs associated with compact binary mergers are (1) the absence of an associated CC-SN, (2) the presence of an optical/NIR rebrightening compatible with a kilonova, (3) an early-type host galaxy, and (4) a distant GRB explosion site with respect to the corresponding galaxy center. GRB 191019A was suggested to belong to this sample of misclassified long GRBs, given the non detection of any SN component (Levan et al. 2023) despite its close cosmological distance ( z =0.248), and the evidence of a host galaxy dominated by an old stellar population. However, the proximity of the GRB 191019A explosion site to the photometric center of its host galaxy ( ≲ 100 pc) is at odds with typical short-GRB galactocentric distances, and could suggest a compact binary formed in the dense circumnuclear disk of an AGN. In this scenario (Lazzati et al. 2022) a very high-density environment ( n 0 > 10 7 cm -3 ) could be the origin of the long duration of the prompt emission (see Lazzati et al. 2023). Given the low distance of this burst, a kilonova ligh is expected to be seen. However, the temporal coverage and limiting magnitudes of previous data sets, did not allow to address this question yet. In this work, we further investigate the burst properties of GRB 1901910A and the nature of the optical transient that followed the burst, with the goal to find arguments in favour or against its possible compact merger origin. We found an independent support on the compact binary merger progenitor nature from the high-energy prompt emission temporal variability, which has been recently found to be an additional potential diagnostic to infer the progenitor nature, where high variability values suggest a compact binary merger progenitor (Camisasca et al. 2023). We noticed that, for GRB 191019A, the obtained MVT value of ∼ 20 ms locates this burst close to the centroid of the distribution of those GRBs associated with compact binary merger progenitors (Camisasca et al. 2023 and Fig. 1). The same variability properties were found also for two other long GRBs with evidence of kilonova (GRB 111210A and GRB 230307A). These results independently support past interpretations on the progenitor nature of GRB 191019A and address to the possible presence of a kilonova component in its optical transient. An optical transient following GRB 191019A was at first reported by Perley et al. (2019b) but its nature could not be clearly pinned down. Using GROND multi-color data obtained between 0.4 and 15 days post burst, we argued here that the temporal evolution of the transient's brightness disfavours a pure afterglow emission (Fig. 4). While the GROND data confirm the absence of a SN component (Fig. 5), we found that the luminosity evolution of the optical transient is in agreement with an afterglow plus a kilonova signal compatible with AT2017gfo redshifted to the distance of the GRB host galaxy, though slightly brighter by a factor of a few. By modelling the afterglow and the kilonova component, we were able to estimate a kilonova ejecta mass which is slightly higher but still consistent, within the large uncertainties, with the one measured for AT2017gfo with similar assumptions on the progenitor system (a binary neutron star) and on the kilonova modelling (e.g., Dietrich et al. 2020). By assuming other kilonova models with different levels of complexity and different progenitor assumptions, the data did not allow to discriminate among them, apart disfavouring the most simplistic one. However, in all cases, results strongly favour the presence of a kilonova with respect to an afterglow-only model (Table 5). Our findings strongly suggest that GRB 191019A might belong to the increasing list of long GRBs with an associated kilonova, beside GRB 211211A and GRB 230307A, and other cases with less robust evidence, as for instance GRB 060614 (e.g., Jin et al. 2015). Another interesting findings from our analysis, is the evidence of a low circumburst density (Table 2). This result is in stark constrast with the one obtained by Lazzati et al. (2023) from prompt emission light curve modelling, where extreme values of n 0 > 10 7 cm -3 were found. These high density values were considered consistent with the apparent position of the GRB, very close to the center of an AGN-like host galaxy (though no direct proofs on the existence of an AGN have been provided yet, see Levan et al. 2023). The interaction between a jet and an AGN dense circumnuclear environment, however, may choke or strongly suppress GRB relativistic jets, unless particularly bright and long, as also supported by recent studies (e.g., Zhang et al. 2024). GRB 191019A is classified as a normal burst, with a well detected optical counterpart (see, e.g., Zhang et al. 2024), and with low dust extinction along the line-of-sight within the host ( A host V in the range from 0.06 to 0.10 mag, Levan et al. 2023), further supported by the UV-band detections (LaPorte et al. 2019). These properties are more consistent with a low density environment, as the one we obtained from a joint fitting of an afterglow and a kilonova component model on the X-ray and optical data of GRB 191019A. More in general, if GRB 191019A is a disguised short GRB with a compact binary merger origin, as both past studies and this work suggest, a low density environment is in agreement with the typical density values found in short GRBs and with the environments expected for compact binary mergers. In a low density scenario, the very small offset from the host galaxy of GRB 191019A, can be explained with a negligible transverse projection of a GRB located far off centre. In this scenario, the long duration of the prompt emission, which Lazzati et al. (2023) explained as caused by the extremely dense circumburst environment, may have instead an intrinsic origin. An interesting hypothesis that was proposed in the past to explain short GRBs with soft extended emission (e.g., Barkov & Pozanenko 2011; Kisaka & Ioka 2015) and which has been recently studied in much greater details (e.g., Musolino et al. 2024), invokes fall-back accretion onto the remnant compact object. Whether this is the case for GRB 191019A goes beyond the scope of this work and will be addressed in another study. We thank the anonymous referee for useful comments and suggestions which improved our work. We are grateful to P. Pang for his precious support on NMMA and useful discussions. ANG acknowledges logistic support by the Thuringer Landessternwarte Tautenburg, Germany. G.S. and P.S acknowledge the support by the State of Hesse within the Research Cluster ELEMENTS (Project ID 500/10.006). A.R. and E.P. acknowledge support by PRIN-MIUR 2017 (grant 20179ZF5KS). Part of the funding for GROND (both hardware and personnel) was generously granted by the Leibniz-Prize to G. Hasinger (DFG grant HA 1850/28-1) and by the Thuringer Landessternwarte Tautenburg. S.B. aknowledges funding from the EU Horizon under ERC Consolidator Grant, no. InspiReM-101043372. A.E.C. is partially supported by the 2023/24 'Research and Education' grant from Fondazione CRT. The OAVdA is managed by the Fondazione Cl'ement Fillietroz-ONLUS, which is supported by the Regional Government of the Aosta Valley, the Town Municipality of Nus and the Unit'e des Communes valdotaines Mont-Emilius. The LBT is an international collaboration of the University of Arizona, Italy (INAF: Istituto Nazionale di Astrofisica), Germany (LBTB: LBT Beteiligungsgesellschaft), The Ohio State University, representing also the University of Minnesota, the University of Virginia, and the University of Notre Dame. This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester.",
"pages": [
13,
15,
16
]
},
{
"title": "Facilities: GROND, Swift (BAT, XRT and UVOT), LBT",
"content": "Software: Afterglowpy (Ryan et al. 2020), NMMA (Pang et al. 2023), astropy (Astropy Collaboration et al. 2013, 2018), HOTPANTS (Becker 2015), PyRAF (Science Software Branch at STScI 2012), DRAGONS (Labrie et al. 2019), WCSTools (Mink 2019).",
"pages": [
16
]
},
{
"title": "A. HOST GALAXY",
"content": "Note -Magnitudes are measured within an aperture with radius of 4 × FWHM and are not corrected for Galactic foreground extinction. Table 4 provides the magnitude of the GRB host galaxy based on GROND's 4th-epoch observations and LBT's visit of the field four years after the burst (see Sect. 2.2).",
"pages": [
17
]
},
{
"title": "B. GRB-SN PEAK LUMINOSITY RANGE",
"content": "We further investigated on possible biases towards the faint end of our GRB-SN sample by comparing with SN Ic peak luminosity properties observed so far. The r-band absolute peak-magnitude ( M r,peak ) distribution of broad line (BL) Type Ic SNe spans at most 2.5 mags (between -17 . 7 and -20 . 8 mag, with an average of M r,peak = -18 . 7 ± 0 . 7 mag, Taddia et al. 2019; Barbarino et al. 2021; Gomez et al. 2022). GRB-SNe observed to date with good data sampling, are at least ∼ 1 mag brighter and trace the high-luminosity end of the above Type Ic BL distribution (e.g., Hjorth & Bloom 2012; Hjorth 2013; Cano et al. 2017; Klose et al. 2019), but the width of their peak luminosity distribution (Hjorth & Bloom 2012; Cano et al. 2017; Kann et al. 2019b; Klose et al. 2019) is not substantially different to the width of the corresponding peak luminosity distribution of type Ic and BL-SNe (see also Soderberg et al. 2006). By considering the GRB-SNe with the best follow-up in multiple bands (including spectroscopy), on the high end site of the peak luminosity distribution remains GRB 111209A/SN 2011kl (e.g., Nakauchi et al. 2013; Kann et al. 2019b) while on the low-end site there are GRB 100316D/SN 2010bh (e.g., Starling et al. 2011; Olivares E. et al. 2012b) and GRB 060218/SN 2006aj (e.g., Pian et al. 2006; Ferrero et al. 2006): the r -band peak magnitudes of these extreme GRB-SNe have a difference that lies between 2 and 3 mags, similar to the broad line (BL) Type Ic SNe. In principle, the long bursts GRB 990712 (Bjornsson et al. 2001), GRB 021211 (Della Valle et al. 2003; Zeh et al. 2004), GRB 040924 (Soderberg et al. 2006; Wiersema et al. 2008), and GRB 060904B (Cano et al. 2017) could have been followed by even slightly fainter SNe than SN 2010bh, but in these cases the data base is comparably poor and no strong conclusions could be made. While, e.g., any SN that was associated with the long bursts GRB 060605 and GRB 060614 must have had a luminosity < 1% of the luminosity of the prototypical SN 1998bw (Fynbo et al. 2006), here and in similar cases (e.g., GRB 111005A; Michaglyph[suppress]lowskI et al. 2018; Tanga et al. 2018) there is no evidence for SN light. In conclusion, if substantially less luminous GRB-SNe do exist formally remains an open question, at least from the obervational point of view, and the peak luminosity range of GRB-SNe sample we have used to infer the luminosity of any SN associated with GRB 191019A is the most robust so far.",
"pages": [
17
]
},
{
"title": "C. AFTERGLOW AND KILONOVA JOINT FIT COMPARISON",
"content": "In Table 5 we present the results obtained for different kilonova models within the NMMA framework, in addition to the kilonova model presented in Sect. 3.3. At first, we have tested a model from Kasen et al. (2017) (Kasen17-Jet in Table 5) which assumes only one ejecta component and has three parameters: the ejecta mass, the ejecta velocity and the lanthanide mass fraction ( χ lan ). Then we made a different assumption on the progenitor by considering a kilonova emission from a neutron star black hole binary system as predicted with POSSIS (Bu19-NSBH-Jet in Table 5), which has three parameters: the dynamical ejecta mass, the wind mass and the orbital plane inclination, which is linked to the viewing angle of the jet assumed in the afterglow modelling. These models were finally compared with the results obtained by assuming the simple analytical model described in Metzger (2017) (Metzger17-Jet in Table 5), which assumes one component and has four parameters: the ejecta mass, the ejecta velocity, the power-law index β of the ejecta mass distribution expressed as a function of its velocity (the faster ejecta matter lies ahead of slower matter and the distribution of mass with velocity greater than a value v 0 can be approximated with a power-law M ( > v 0 ) = M ( v/v 0 ) -β , and the opacity k r . The afterglow emission was modelled within the Afterglowpy framework (Afterglow in Table 5), as described in Sect. 3.3. However, contrary to our previous analysis, the microphyiscal parameters ϵ e and ϵ B are now fixed to 0 . 5 and 0 . 01, respectively. In doing so, we aim at reproduce similar assumptions to Lazzati et al. (2023) and further investigate on the circumburst density estimates. The highest Bayesian evidence ( Z ) is obtained for the Bu19-NSBH-Jet model. Following Kunert et al. (2024) and references therein, by considering Bu19-NSBH-Jet as the reference model, we find -1 . 10 < ln( Z/Z Bu 19 -NSBH -Jet ) < 0 for the Kasen17-Jet and Bu19-BNS-Jet models, while ln( Z/Z Bu 19 -NSBH -Jet ) < -4 . 5 for Metzger17-Jet and Afterglow models, indicating a strong evidence against the latter two models, while no preference could be set among Bu19-BNSJet, Bu19-NSBH-Jet and Kasen17-Jet models.",
"pages": [
17,
18
]
},
{
"title": "REFERENCES",
"content": "Abbott, B. P., et al. 2017, ApJ, 848, L13, doi: 10.3847/2041-8213/aa920c Ahumada, T., Singer, L. P., Anand, S., et al. 2021, Nature Astronomy, 5, 917, doi: 10.1038/s41550-021-01428-7 Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068 Astropy Collaboration, Price-Whelan, A. M., Sip\"ocz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f Barbarino, C., Sollerman, J., Taddia, F., et al. 2021, A&A, 651, A81, doi: 10.1051/0004-6361/202038890 Barkov, M. V., & Pozanenko, A. S. 2011, MNRAS, 417, 2161, doi: 10.1111/j.1365-2966.2011.19398.x Barthelmy, S. D., Barbier, L. M., Cummings, J. R., et al. 2005, SSRv, 120, 143, doi: 10.1007/s11214-005-5096-3 Becker, A. 2015, HOTPANTS: High Order Transform of PSF ANd Template Subtraction, Astrophysics Source Code Library, record ascl:1504.004. http://ascl.net/1504.004 Beniamini, P., Duque, R., Daigne, F., & Mochkovitch, R. 2020, MNRAS, 492, 2847, doi: 10.1093/mnras/staa070 Berger, E. 2014, ARA&A, 52, 43, doi: 10.1146/annurev-astro-081913-035926 Bernuzzi, S. 2020, General Relativity and Gravitation, 52, 108, doi: 10.1007/s10714-020-02752-5 Bjornsson, G., Hjorth, J., Jakobsson, P., Christensen, L., & Holland, S. 2001, ApJL, 552, L121, doi: 10.1086/320328 Bulla, M. 2019, MNRAS, 489, 5037, doi: 10.1093/mnras/stz2495 -. 2023, MNRAS, 520, 2558, doi: 10.1093/mnras/stad232 Burrows, D. N., Hill, J. E., Nousek, J. A., et al. 2005, SSRv, 120, 165, doi: 10.1007/s11214-005-5097-2 Camisasca, A. E., Guidorzi, C., Amati, L., et al. 2023, A&A, 671, A112, doi: 10.1051/0004-6361/202245657 Cano, Z., Wang, S.-Q., Dai, Z.-G., & Wu, X.-F. 2017, Advances in Astronomy, 2017, 8929054, doi: 10.1155/2017/8929054 Chambers, K. C., Magnier, E. A., Metcalfe, N., et al. 2016, arXiv e-prints, arXiv:1612.05560, doi: 10.48550/arXiv.1612.05560 Coulter, D. A., Foley, R. J., Kilpatrick, C. D., et al. 2017, Science, 358, 1556, doi: 10.1126/science.aap9811 Coward, D. M., Gendre, B., Tanga, P., et al. 2017, PASA, 34, e005, doi: 10.1017/pasa.2016.61 Dainotti, M. G., De Simone, B., Islam, K. M., et al. 2022, ApJ, 938, 41, doi: 10.3847/1538-4357/ac8b77 Della Valle, M., Malesani, D., Benetti, S., et al. 2003, A&A, 406, L33, doi: 10.1051/0004-6361:20030855 Dietrich, T., Coughlin, M. W., Pang, P. T. H., et al. 2020, Science, 370, 1450, doi: 10.1126/science.abb4317 Note -E 0 = kinetic fireball energy; n = particle number density in the circumburst environment; θ c =half-opening angle of the jet core; ι =viewing angle with respect to jet axis; p = electron energy distribution power-law index; M dyn ej = dynamical ejecta mass; M wind ej = wind ejecta mass; M ej = total ejecta mass; v ej = ejecta velocity; Φ = half-opening angle of lanthanide-rich equatorial ejecta; ϵ e = shock energy fraction that goes into the electrons; ϵ B = shock energy fraction that goes into the magnetic energy density; χ lan = lanthanide mass fraction (Kasen et al. 2017) ; κ r = opacity; β = power-law index of the ejecta mass distribution as a function of its velocity (Metzger 2017); ln( Z ) = natural logarithm of the bayes evidence. We note that the Bu19-BNS-Jet model is the same as the one presented in Sect. 3.3 where now ϵ e and ϵ B are fixed. Evans, P. A., Osborne, J. P., Burrows, D. N., et al. 2019, GRB Coordinates Network, 26034, 1 Evans, P. A., Beardmore, A. P., Page, K. L., et al. 2007, A&A, 469, 379, doi: 10.1051/0004-6361:20077530 -. 2009, MNRAS, 397, 1177, doi: 10.1111/j.1365-2966.2009.14913.x Evans, P. A., Willingale, R., Osborne, J. P., et al. 2010, A&A, 519, A102, doi: 10.1051/0004-6361/201014819 Ferrero, P., Kann, D. A., Zeh, A., et al. 2006, A&A, 457, 857, doi: 10.1051/0004-6361:20065530 Fong, W., Berger, E., Margutti, R., & Zauderer, B. A. 2015, ApJ, 815, 102, doi: 10.1088/0004-637X/815/2/102 Fong, W.-f., Nugent, A. E., Dong, Y., et al. 2022, ApJ, 940, 56, doi: 10.3847/1538-4357/ac91d0 Fontana, A., Dunlop, J. S., Paris, D., et al. 2014, A&A, 570, A11, doi: 10.1051/0004-6361/201423543 Fynbo, J. P. U., Perley, D. A., de Ugarte Postigo, A., et al. 2019, GRB Coordinates Network, 26041, 1 Fynbo, J. P. U., Watson, D., Thone, C. C., et al. 2006, Nature, 444, 1047, doi: 10.1038/nature05375 Gehrels, N., Chincarini, G., Giommi, P., et al. 2004, ApJ, 611, 1005, doi: 10.1086/422091 Gendre, B., Crisp, H., Coward, D., et al. 2019, GRB Coordinates Network, 26098, 1 Gomez, S., Berger, E., Nicholl, M., Blanchard, P. K., & Hosseinzadeh, G. 2022, ApJ, 941, 107, doi: 10.3847/1538-4357/ac9842 Gompertz, B. P., Ravasio, M. E., Nicholl, M., et al. 2023, Nature Astronomy, 7, 67, doi: 10.1038/s41550-022-01819-4 Greiner, J., Bornemann, W., Clemens, C., et al. 2008, PASP, 120, 405, doi: 10.1086/587032 Greiner, J., Mazzali, P. A., Kann, D. A., et al. 2015, Nature, 523, 189, doi: 10.1038/nature14579 Hjorth, J. 2013, Philosophical Transactions of the Royal Society of London Series A, 371, 20120275, doi: 10.1098/rsta.2012.0275 Troja, E. 2023, Universe, 9, 245, doi: 10.3390/universe9060245 Yolda¸s, A. K., Kruhler, T., Greiner, J., et al. 2008, in American Institute of Physics Conference Series, Vol. 1000, American Institute of Physics Conference Series, ed. M. Galassi, D. Palmer, & E. Fenimore, 227-231, doi: 10.1063/1.2943450",
"pages": [
18,
19,
21
]
}
]
2024arXiv241205099M
https://arxiv.org/pdf/2412.05099.pdf
<document>
<figure>
<location><page_1><loc_81><loc_94><loc_93><loc_96></location>
</figure>
<section_header_level_1><location><page_1><loc_7><loc_91><loc_17><loc_92></location>ARTICLE TYPE</section_header_level_1>
<section_header_level_1><location><page_1><loc_7><loc_87><loc_93><loc_89></location>C/O ratios in self-gravitating protoplanetary discs with dust evolution</section_header_level_1>
<text><location><page_1><loc_7><loc_84><loc_50><loc_85></location>Tamara Molyarova, 1,2 Eduard Vorobyov, 1,2 and Vitaly Akimkin 1</text>
<text><location><page_1><loc_7><loc_80><loc_67><loc_83></location>1 Institute of Astronomy, Russian Academy of Sciences, 48 Pyatnitskaya St., Moscow, 119017, Russia 2 Research Institute of Physics, Southern Federal University, Stachki Ave. 194, Rostov-on-Don 344090, Russia Author for correspondence: Tamara Molyarova, Email: moliarova@sfedu.ru.</text>
<section_header_level_1><location><page_1><loc_9><loc_75><loc_14><loc_76></location>Abstract</section_header_level_1>
<text><location><page_1><loc_9><loc_54><loc_92><loc_74></location>Elemental abundances, particularly the C/O ratio, are seen as a way to connect the composition of planetary atmospheres with planet formation scenario and the disc chemical environment. We model the chemical composition of gas and ices in a self-gravitating disc on timescales of 0.5 Myr since its formation to study the evolution of C/O ratio due to dust dynamics and growth, and phase transitions of the volatile species. We use the thin-disc hydrodynamic code FEOSAD, which includes disc self-gravity, thermal balance, dust evolution and turbulent diffusion, and treats dust as a dynamically different and evolving component interacting with the gas. It also describes freeze-out, sublimation and advection of four most abundant volatile species: H2O, CO2, CH4 and CO. We demonstrate the effect of gas and dust substructures such as spirals and rings on the distribution of volatiles and C/O ratios, including the formation of multiple snowlines of one species, and point out the anticorrelation between dust-to-gas ratio and total C/O ratio emerging due to the contribution of oxygen-rich ice mantles. We identify time and spatial locations where two distinct trigger mechanisms for planet formation are operating and differentiate them by C/O ratio range: wide range of the C/O ratios of 0 - 1.4 for streaming instability, and a much narrower range 0.3 - 0.6 for gravitational instability (with the initial value of 0.34). This conclusion is corroborated by observations, showing that transiting exoplanets, which possibly experienced migration through a variety of disc conditions, have significantly larger spread of C/O in comparison with directly imaged exoplanets likely formed in gravitationally unstable outer disk regions. We show that the ice-phase C/O ≈ 0.2 - 0.3 between the CO, CO2 and CH4 snowlines corresponds to the composition of the Solar system comets, that represent primordial planetesimals.</text>
<text><location><page_1><loc_7><loc_50><loc_38><loc_52></location>Keywords: protoplanetary disc, volatiles, dust evolution</text>
<section_header_level_1><location><page_1><loc_7><loc_46><loc_19><loc_47></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_7><loc_24><loc_49><loc_45></location>The protoplanetary disc matter can be roughly divided into three component: gaseous chemical species, solid dust particles, and icy mantles covering the surface of dust grains. Gas and solid particles become dynamically decoupled, as evolving dust grows and acquires relative velocities leading to the redistribution of elements in the disc and between the phases, and creating the premises for different chemical environments. When planets start to form, their properties, including chemical composition of the atmosphere, are inevitably affected by the location and the mechanism of their formation. This suggests that the origin of (exo)planets might be investigated using their observed chemical composition, and makes understanding the disc chemical evolution vital for creating a consistent planet formation theory.</text>
<text><location><page_1><loc_7><loc_6><loc_49><loc_24></location>One of the key parameters that govern the chemical setup of a planetary atmosphere is the relation between the abundances of carbon and oxygen, often referred to as carbon-tooxygen ratio (hereafter C/O ratio). The variations of C/O ratio in the ice and gas phases at the snowlines of main disc volatiles (CO, CO 2 , and H 2 O) and the prospects of connecting them to planet formation were discussed in Öberg, Murray-Clay, and Bergin (2011) within a qualitative freeze-out model. Since then, C/O ratio received a lot of attention in this context. It was thoroughly investigated in modelling (see, e.g., Booth et al. 2017; Eistrup, Walsh, and van Dishoeck 2018; Cridland, Eistrup, and van Dishoeck 2019; Cridland et al. 2019; Crid-</text>
<text><location><page_1><loc_52><loc_34><loc_94><loc_48></location>and, Bosman, and van Dishoeck 2020; Cridland et al. 2020; Krijt et al. 2020; Turrini et al. 2021; Schneider and Bitsch 2021). The connection of disc chemical composition with C/O in exoplanetary atmospheres was modelled using core accretion model (Thiabaud et al. 2015) and 'chain' planet population synthesis model (Mordasini et al. 2016). Paul Mollière et al. (2022) considered a simple formation retrieval pipeline and found that this task requires careful consideration of the model assumptions.</text>
<text><location><page_1><loc_52><loc_6><loc_94><loc_33></location>The measurements of molecular abundances in the atmospheres of giant exoplanets obtained by a variety of modern facilities, such as HST, Spitzer, VLTI, JWST, Gemini, indicate a diversity of C/O ratios: from low C/O ratio values ( ≈ 0.4, below the solar value of 0.54; Benneke et al. 2019; GRAVITY Collaboration et al. 2020; Worthen et al. 2024; Xue et al. 2024) to stellar ( ≈ 0.5, close to solar; P. Mollière et al. 2020; Zhang et al. 2021; Smith et al. 2024) and close to or above unity (Swain et al. 2009; Madhusudhan et al. 2011). A variety of solar and super-solar C/O ratios is observed in four planets within the HR8799 system (Nasedkin et al. 2024). Chemical composition of the atmospheres of many hot Jupiters indicates high C/O>1 of the forming material (Moses et al. 2013). There is an observational evidence of young planets in PDS 70 disc accreting the material with C/O>1 (Facchini et al. 2021). The number of exoplanets with constrained atmospheric C/O ratios grows with large studies of multiple planets such as Changeat et al. (2022), which allows us to make some statistical conclusions. The</text>
<text><location><page_2><loc_7><loc_81><loc_48><loc_92></location>population study of C/O ratios in exoplanetary atmospheres reveals that there are two populations with different elemental ratio, which are likely formed in different mechanisms (Hoch et al. 2023). Khorshid, Min, and Désert (2023) were able to restrict the formation scenario for WASP-77b based on the measured C/O ratio of the planet (Line et al. 2021) and the modelling of planet formation and migration.</text>
<text><location><page_2><loc_7><loc_52><loc_48><loc_81></location>Elemental abundances in protoplanetary discs can be constrained from observations (Fedele and Favre 2020), and C/O ratio in the gas can be estimated. Spatially resolved observations can help distinguish between different C/O ratios spectroscopically (Matter, Pignatale, and Lopez 2020). Cleeves et al. (2018) report C/O ≈ 0.8 in the molecular layer of IM Lup disc. ALMA observations of hydrocarbons and sulphur-bearing species indicate C/O > 1 in the upper disc layers and in the outer disc in TW Hya and DM Tau (Dutrey et al. 2011; Bergin et al. 2016; Semenov et al. 2018) and for a population of discs in Lupus (Miotello et al. 2019). For the nearby discs the solar elemental composition with C/O ≈ 0.54 is usually expected, thus the observed higher values confirm redistribution of carbon and oxygen in discs. In addition to high C/O in disc atmospheres, there is evidence of both carbon and oxygen depletion from gas (Kama et al. 2016; Miotello et al. 2019). However, some of heavy oxygen carriers might not be observable, leading to overestimated C/O in disc observations (Walsh, Nomura, and van Dishoeck 2015).</text>
<text><location><page_2><loc_7><loc_27><loc_48><loc_53></location>The volatile composition is also used to constrain the origin of bodies in the Solar System. Fraction of CO and CO 2 ices relative to water in cometary comae indicate their formation between the CO and CO 2 snowlines or exterior to the CO snowline (A'Hearn et al. 2012; Seligman et al. 2022). Abundances of CO and N 2 ices were used to analyse the original location of Pluto and Triton (Mousis, Anderson, et al. 2024). Observed elemental abundances were used to constrain the Jupiter formation scenario (Lodders 2004), relying also on abundances of nitrogen (Öberg and Wordsworth 2019; Bosman, Cridland, and Miguel 2019) and chemically inactive species like Ar. However, the model assumptions can lead to different interpretation of the observations: while Öberg and Wordsworth (2019) and Bosman, Cridland, and Miguel (2019) suggest that Jupiter formed outside N 2 snowline (at > 30 au), Ohno and Ueda (2021) consider the concept disc shadow, which allows Jupiter to form near its current location.</text>
<text><location><page_2><loc_7><loc_6><loc_48><loc_27></location>Chemical processes other than freeze-out and sublimation at the snowlines can alter the composition of ice and gas as well. Due to gas-phase and surface reactions, snowlines can become important for the redistribution of elements. More detailed chemical modelling shows that the C/O ratio in the gas and in the ice depends also on the initial chemical setup and ionisation by cosmic rays and radioactive nuclei (Eistrup, Walsh, and van Dishoeck 2016, 2018). It directly affects the interpretation of observations. Another essential chemical process is CO depletion from the gas, resulting in its transformation to CO 2 ice (Bosman, Tielens, and van Dishoeck 2018). For example, stellar C/O ratio in the atmosphere of HR 8799e indicates that the planet accreted its material beyond CO snowline ( ≈ 45 au), but chemical modelling suggests that due to CO depletion,</text>
<text><location><page_2><loc_51><loc_87><loc_93><loc_92></location>the C/O in the ice already approaches the stellar ratio beyond CO 2 snowline ( ≈ 20 au), which is closer to the star (P. Mollière et al. 2020).</text>
<text><location><page_2><loc_51><loc_64><loc_93><loc_87></location>Another key process affecting the elemental ratio is dust drift, which leads to spatial segregation between the chemical constituents of the gas and the grains covered with ice. The distribution of CO in the gas and ice phases was studied within dynamical models of dust evolution (Stammler et al. 2017; Krijt et al. 2018). Even without chemical processes, dust evolution and dynamics can substantially alter C/O ratio in the atmospheres of forming planets (Booth et al. 2017). Some models combine chemical reactions treatment with dust evolution and transport, usually within 1D viscous models. Dust transport can have a strong effect on the abundances of volatiles in the inner disc regions (Bosman, Tielens, and van Dishoeck 2018). However, for discs with low turbulence and high cosmic ray ionisation rate, C/O ratio is rather defined by chemical evolution (Booth and Ilee 2019).</text>
<text><location><page_2><loc_51><loc_40><loc_93><loc_64></location>In our previous work (Molyarova et al. 2021) we showed that the volatile species tend to concentrate around their snowlines both in the gas and more notably on the dust surface. This accumulation was found to be caused by effective transport of volatiles through the snowlines by azimuthal variations in the gas and dust radial and angular velocity, an effect that cannot be captured in 1D viscous disc models. Such accumulation should immediately affect the local C/O ratio, which suggests the connection between the snowlines of various volatiles and the formation of planets with altered C/O in their atmospheres. In this work, we follow the distribution of the main volatile species in the disc to investigate the distribution of C/O ratio in gas and ice in a 2D thin-disc hydrodynamic model. We study the effect of dust growth and dynamics on the elemental ratios and consider the role of the initial mass of the collapsing core on the distribution of volatiles.</text>
<text><location><page_2><loc_51><loc_27><loc_93><loc_40></location>The paper is organised as follows. The main features of the used FEOSAD model are described in Section 2, with the details of the treatment of the volatiles given in Section 2.3. In Section 3 we describe the results of the simulations, focusing on distribution of the volatiles in Section 3.2, the C/O ratios in Section 3.3, and their evolution in Section 3.4. In Section 4, we discuss the implications of our results in the context of planet formation via different mechanisms. The main conclusions are listed in Section 5.</text>
<section_header_level_1><location><page_2><loc_51><loc_23><loc_59><loc_24></location>2. Model</section_header_level_1>
<text><location><page_2><loc_51><loc_6><loc_93><loc_22></location>We use the global model of protoplanetary disc formation FEOSAD (Vorobyov et al. 2018), which includes disc selfgravity, dust evolution and interaction with gas (including backreaction of dust on gas), turbulent viscosity, adiabatic and radiative cooling and heating. It describes the formation of a protostar and a protoplanetary disc from a collapsing cloud in a 2D thin-disc approach. The model also includes freeze-out of main volatile species as in Molyarova et al. (2021), with the feedback from ice mantles on dust evolution via fragmentation velocity. Here we summarise the key characteristics of the model, more details can be found in the previous works (Vorobyov</text>
<text><location><page_3><loc_7><loc_84><loc_49><loc_92></location>et al. 2018; Molyarova et al. 2021; Kadam, Vorobyov, and Basu 2022). The main difference from our previous study in Molyarova et al. (2021) is that here we consider the formation of dead zones via variable α -parameter of Shakura and Sunyaev and also include turbulent diffusion.</text>
<section_header_level_1><location><page_3><loc_7><loc_81><loc_20><loc_82></location>2.1 Gas evolution</section_header_level_1>
<text><location><page_3><loc_7><loc_76><loc_49><loc_80></location>For the gas component, the hydrodynamic equations for mass, momentum, and internal energy conservation are the following</text>
<formula><location><page_3><loc_21><loc_72><loc_49><loc_75></location>∂Σ g ∂ t + ∇· ( Σ g v ) = 0, (1)</formula>
<formula><location><page_3><loc_12><loc_65><loc_49><loc_70></location>∂ ∂ t ( Σ g v ) + [ ∇· ( Σ g v ⊗ v ) ] = -∇P + Σ g g + + ∇· Π -Σ d,gr f , (2)</formula>
<formula><location><page_3><loc_14><loc_60><loc_49><loc_63></location>∂ e ∂ t + ∇· ( e v ) = -P ( ∇· v ) -Λ + Γ + ∇ v : Π , (3)</formula>
<text><location><page_3><loc_7><loc_41><loc_49><loc_60></location>where subscripts p and p ' denote the planar components ( r , ϕ ) in polar coordinates, Σ g is the gas mass surface density, e is the internal energy per surface area, P is the vertically integrated gas pressure calculated via the ideal equation of state as P = ( γ - 1) e with γ = 7/5, f is the friction force between gas and dust, v = vr ˆ r + v ϕ ˆ ϕ is the gas velocity in the disc plane, and ∇ = ˆ r ∂ / ∂ r + ˆ ϕ r -1 ∂ / ∂ϕ is the gradient along the planar coordinates of the disc. The gravitational acceleration in the disc plane, g = gr ˆ r + g ϕ ˆ ϕ , includes the gravity of the central protostar when formed and takes into account disc self-gravity of both gas and dust found by solving the Poisson integral (Binney and Tremaine 1987).</text>
<text><location><page_3><loc_7><loc_29><loc_50><loc_41></location>Theconsideration of time-dependent energy balance (Eq. (3)) allows us to accurately calculate the midplane temperature T mp and is particularly important to describe the phase state of the volatiles and the level of turbulent viscosity. The terms Λ and Γ describe the rates of dust cooling and heating, respectively, by stellar and background irradiation. They are calculated based on the analytical solution of the radiation transfer equations in the vertical direction (Dong et al. 2016; Vorobyov et al. 2018)</text>
<formula><location><page_3><loc_13><loc_23><loc_49><loc_28></location>Λ = 8 τ P σ T 4 mp 1 + 2 τ P + 3 2 τ P τ R , Γ = 8 τ P σ T 4 irr 1 + 2 τ P + 3 2 τ P τ R . (4)</formula>
<text><location><page_3><loc_7><loc_6><loc_49><loc_22></location>Here, σ is the Stefan-Boltzmann constant, τ P and τ R are the Planck and Rosseland mean optical depths to the disc midplane, calculated as τ = κΣ dust from Planck and Rosseland mean opacities κ P and κ R (Semenov et al. 2003) and total dust surface density Σ dust . Gas and dust temperatures are assumed to be equal, and the midplane temperature is linked with gas pressure as T mp = P µ / R Σ g, where µ = 2.3 is the mean molecular weight of the gas, and R is the universal gas constant. The irradiation temperature at the disc surface T irr is determined by both stellar and background irradiation. Stellar irradiation includes the luminosity from the photosphere of the protostar</text>
<text><location><page_3><loc_52><loc_87><loc_94><loc_92></location>and accretion luminosity. The background radiation is assumed as a black body with the temperature of 15 K. For more details on the irradiation we refer to Vorobyov et al. (2018).</text>
<text><location><page_3><loc_52><loc_24><loc_94><loc_87></location>Turbulent viscosity is described using the common α -parameter approach of Shakura and Sunyaev (1973). It is taken into account via the viscous stress tensor Π (see Vorobyov and Basu 2010, for explicit expressions for the components of the terms with Π ). The magnitude of kinematic viscosity is ν = α c s H , where c s is the sound speed and H is the vertical scale height of the gas disc calculated using an assumption of local hydrostatic equilibrium of a self-gravitating disc (see Vorobyov and Basu 2010, Appendix A). Here, we use the adaptive α approach implying accretion through a layered disc (Gammie 1996; Armitage, Livio, and Pringle 2001; Kadam, Vorobyov, and Basu 2022). Turbulence is assumed to be generated by magneto-rotational instability (MRI) which only develops in layers of the disc where the ionisation level is high enough. The MRI-active layer is characterised by its surface density Σ MRI and relatively high value of turbulent viscosity α MRI = 10 -3 . As thermal and photo-ionisation are not efficient enough for the relatively cold and dense matter in the disc at >0.5 au, the main process determining the thickness of the MRI-active layer is ionisation by cosmic rays. It is assumed to be constant Σ MRI = 100g cm -2 , which is the typical depth of Galactic cosmic rays penetration in the ISM (Umebayashi and Nakano 1981) and in protoplanetary discs (Padovani, Galli, and Glassgold 2009). The dead zone is characterised by surface density from the midplane Σ dz = Σ g/2Σ MRI with residual turbulence α dz . The turbulence in this layer is only hydrodynamic turbulence driven by the Maxwell stress in the active layer, and small value α dz = 10 -5 is adopted (Okuzumi and Hirose 2011). However, if local temperature exceeds the critical value of 1300 K, thermal ionisation becomes possible, the MRI develops and the dead zone is no longer dead, in which case α dz = 10 -1 (Zhu et al. 2010; Kadam et al. 2019). This value is higher than α MRI in the outer disc due to the different ionisation processes and local conditions. In the outer disc, the MRI can be suppressed by non-ideal magneto-hydrodynamic (MHD) effects such as ambipolar diffusion and Ohmic resistivity (Bai and Stone 2013; Gressel et al. 2015). In the dead zone in the inner disc, α can reach higher values when the MRI is triggered by thermal ionisation, as shown by 3D MHD simulations (Zhu, Jiang, and Stone 2020). The adopted parameterization makes use of an effective parameter α eff , which at any given location is calculated as</text>
<formula><location><page_3><loc_52><loc_17><loc_94><loc_23></location>α eff = Σ MRI α MRI + Σ dz α dz Σ MRI + Σ dz , α dz = { 10 -5 , if T mp < 1300K; 10 -1 , if T mp ≥ 1300K. (5)</formula>
<section_header_level_1><location><page_3><loc_52><loc_14><loc_66><loc_15></location>2.2 Dust evolution</section_header_level_1>
<text><location><page_3><loc_52><loc_6><loc_94><loc_14></location>Dust is described as consisting of two components: small grains that are dynamically coupled to the gas, with the mass surface density Σ d,sm , and grown grains with the mass surface density Σ d,gr that can move relative to the gas and change in size. The total dust surface density necessary for the calculation</text>
<text><location><page_4><loc_7><loc_78><loc_48><loc_92></location>of the optical depths for Eq. (4) is Σ dust = ( Σ d,gr + Σ d,sm )/2. The factor 1/2 appears as the optical depths is calculated to the midplane. Each dust population has a power-law size distribution f ( a ) = dN / da = Ca -p with a normalisation constant C and a fixed exponent p = 3.5. Small dust has sizes between a min = 5 × 10 -7 cm and a ∗ = 10 -4 cm, grown dust has sizes between a ∗ and a max, which can vary due to dust coagulation and fragmentation. Dynamics of these dust components follows the continuity and momentum equations</text>
<formula><location><page_4><loc_7><loc_72><loc_48><loc_77></location>∂Σ d,sm ∂ t + ∇· ( Σ d,sm v ) = -S ( a max) + ∇· ( D Σ g ∇ ( Σ d,sm Σ g )) , (6)</formula>
<formula><location><page_4><loc_8><loc_68><loc_48><loc_72></location>∂Σ d,gr ∂ t + ∇· ( Σ d,gr u ) = S ( a max)+ ∇· ( D Σ g ∇ ( Σ d,gr Σ g )) , (7)</formula>
<formula><location><page_4><loc_11><loc_61><loc_48><loc_66></location>∂ ∂ t ( Σ d,gr u ) + [ ∇· ( Σ d,gr u ⊗ u )] = Σ d,gr g + + Σ d,gr f + S ( a max) v , (8)</formula>
<text><location><page_4><loc_7><loc_43><loc_48><loc_60></location>where u is the grown dust velocity. The term S ( a max) is responsible for the exchange of matter between the dust populations, as dust is converted from the small to grown component due to coagulation and back due to fragmentation. The details of the dust evolution model are presented in Vorobyov et al. (2018). The last term in Eqs. (6) and (7) is responsible for dust turbulent diffusion, similar to Vorobyov, Elbakyan, et al. (2020). The coefficient of turbulent diffusion D is related to the kinematic viscosity ν as D = ν /(1 + St 2 ) (Birnstiel 2023). Diffusion affects dust grains along with their ice mantles, as well as the gas-phase species (see Section 2.3).</text>
<text><location><page_4><loc_7><loc_28><loc_48><loc_43></location>The innermost regions of the disc are challenging to simulate explicitly due to the Courant criterion: in the highly dynamic inner regions (fraction of au) the timescales are so short that the code demands very small time step in order to preserve stability. Therefore, the inner regions are represented by a sink cell, with a carefully chosen inflow-outflow boundary condition at the sink cell and a parametric description of the accretion onto the star (see Vorobyov et al. 2018, for details). In the simulations presented below, the radius of the sink cell is 0.62 au.</text>
<text><location><page_4><loc_7><loc_6><loc_48><loc_28></location>We consider two disc models with different initial mass of the collapsing cloud, 0.66 and 1 M ⊙ . We note that about 10% of the gas mass that crosses the sink cell is assumed to be evacuated by jets and outflows, and the other 90% lands on to the star. A small amount of mass remains in the envelope by the end of simulations. In both models, the initial gas temperature T init = 15 K and the ratio of rotational to gravitational energy β = 0.28%. The simulations start from the collapse of a molecular cloud, with only small dust grains. The simulation continues until the age of the system becomes equal to 0.5 Myr. Masses of the central protostar and the disc by the end of simulation are M ⋆ = 0.4 M ⊙ and M disc = 0.22 M ⊙ in model M1 and M ⋆ = 0.58 M ⊙ and M disc = 0.35 M ⊙ in model M2. The disc masses are around 0.5 stellar masses, which makes them essentially self-gravitating.</text>
<text><location><page_4><loc_51><loc_52><loc_93><loc_92></location>A number of recent studies develop the idea that the mass infall from the ambient ISM continues during the lifetime of the disc, including the Class II stage (Padoan et al. 2024; Pelkonen et al. 2024; Winter, Benisty, and Andrews 2024). Such models describe a Bondi-Hoyle accretion regime and are in good agreement with the observed properties of the disc population, such as accretion rates, masses and sizes. This input of matter can play an important role in disc evolution and planet formation process (Vorobyov, Lin, and Guedel 2015; Vorobyov et al. 2016). In the FEOSAD model, the mass infall to the disc can be accounted for (Vorobyov, Lin, and Guedel 2015; Vorobyov et al. 2016), but in the present simulation this effect is not included. The mass infall from the envelope continues until the cloud is depleted of matter. Because of the thin-disc geometry, the gravitational contraction of the cloud proceeds in the plane of the disc. The matter is landing on the disc outer edge and is transported towards the star by the combined action of gravitational and viscous torques. The infall on the disc inner regions is therefore neglected. This is a reasonable approximation, considering that most of the matter and angular momentum in a three-dimensional cloud is located at relatively large polar angles and a flared outer edge of the disc intercepts most of them (see, e.g., Visser et al. 2009, Figure 1). In our modelling, we do not consider the continuous Bondi-Hoyle accretion and concentrate on the internal disc processes.</text>
<section_header_level_1><location><page_4><loc_51><loc_48><loc_70><loc_49></location>2.3 Evolution of volatiles</section_header_level_1>
<text><location><page_4><loc_51><loc_21><loc_93><loc_47></location>We follow the evolution of four main volatile species: H 2 O, CO 2 , CO and CH 4 . These are the most abundant carbon- and oxygen-bearing ices observed in protostellar cores (Karin I. Öberg et al. 2011). In the model, each of these species can be present in three states: in the gas, in the ice on the surface of small dust, and in the ice on the surface of grown dust. Each species s is described by its surface density in the gas Σ gas s , on small dust Σ sm s , and on grown dust Σ gr s . Their distributions in the disc can change through three main processes: advection together with the corresponding component (gas or small/grown dust); exchange of mantles between dust populations due to grain collisions; phase transitions, including adsorption from gas to dust, and thermal and photo-desorption. Initially, all ices are on small grains and no volatiles present in the gas. The treatment of volatiles is adopted from Molyarova et al. (2021), who describe the models in more details. Here we recap main features of the chemical model.</text>
<text><location><page_4><loc_51><loc_6><loc_93><loc_21></location>Our chemical model only describes phase transitions, i.e. adsorption and desorption, which includes thermal desorption and photo-desorption by interstellar UV radiation. These reactions were shown to be he most important for gas-phase abundances of most species (Ilee et al. 2011). Due to high computational costs, no other chemical processes, either gasphase or surface reactions are included, although they also may have significant effect on the composition of both ice and gas (Semenov and Wiebe 2011). The chemical evolution of the surface densities of volatile species is calculated from the</text>
<text><location><page_5><loc_7><loc_90><loc_20><loc_92></location>system of equations</text>
<formula><location><page_5><loc_10><loc_88><loc_12><loc_89></location>∂Σ</formula>
<formula><location><page_5><loc_11><loc_71><loc_49><loc_90></location>gas s ∂ t + ∇· ( Σ gas s v ) -∇· ( D Σ g ∇ ( Σ gas s Σ g )) = -λ s Σ gas s + η sm s + η gr s , (9) ∂Σ sm s ∂ t + ∇· ( Σ sm s v ) -∇· ( D Σ g ∇ ( Σ sm s Σ g )) = λ sm s Σ gas s -η sm s , (10) ∂Σ gr s ∂ t + ∇· ( Σ gr s u ) -∇· ( D Σ g ∇ ( Σ gr s Σ g )) = λ gr s Σ gas s -η gr s , (11)</formula>
<text><location><page_5><loc_7><loc_45><loc_49><loc_71></location>where the mass rate coefficients per disc unit area of adsorption λ s and desorption η s for the species s are calculated for local conditions at every hydrodynamic step, separately for small and grown dust populations. Eqs. (9)-(11) are solved in two steps: first, the right-hand side is considered without the advection term. The case of pure adsorption/desorption represented by the right-hand side of the equation can be solved analytically (see Appendix A in Molyarova et al. 2021). This is done at every hydrodynamic step, before the dust growth step, when ices on small and grown grains are redistributed proportionally to mass exchange between the dust populations. This is followed by a transport step, when the surface densities of the volatiles are changed according to the fluxes of their respective gas or dust components between the cells. Restricting chemical processes to only adsorption and desorption allows the chemical step to be calculated fast, which is very important for computationally demanding hydrodynamic simulations.</text>
<text><location><page_5><loc_7><loc_25><loc_49><loc_45></location>For each dust population, the desorption rate is a sum of thermal desorption and photo-desorption η = η td + η pd (the indices 'sm' and 'gr' are omitted for convenience). We operate under the assumption of zeroth-order desorption, i.e. the desorption rate does not depend on the present amount of ice ( Σ sm s or Σ gr s ). It implies that only the upper layers of the ice mantle are able to sublimate, which is a more appropriate approach for thick mantles. This assumption better describes desorption of CO and H 2 Oin temperature programmed desorption (TPD) experiments (Fraser et al. 2001; K. I. Öberg et al. 2005; Bisschop et al. 2006) than first-order desorption, which is more suitable for thin mantles of several monolayers. The rate of thermal desorption is calculated as</text>
<formula><location><page_5><loc_11><loc_20><loc_49><loc_24></location>η td = ˜ σ tot N ss µ s m p √ 2 N ss E b k B π 2 µ s m p exp ( -E b T mp ) , (12)</formula>
<text><location><page_5><loc_7><loc_6><loc_49><loc_20></location>where N ss = 10 15 cm -2 is the surface density of binding sites (Cuppen et al. 2017), E b (K) is the binding energy of the species to the surface, µ s is the species molecular mass, m p is atomic mass unit, k B is the Boltzmann constant. We follow Hasegawa and Herbst (1993) and use binding energy to calculate the pre-exponential factor in Eq. (12), the same way it was done in Molyarova et al. (2021). However, this approach was recently demonstrated by Minissale et al. (2022) to underestimate the pre-exponential factor by a few orders of magnitude,</text>
<text><location><page_5><loc_52><loc_84><loc_93><loc_92></location>as it does not account for the rotational part of the partition functions of desorbing molecules. Total surface area of dust grains (small or grown) per disc unit area ˜ σ tot (cm 2 cm -2 ) is calculated for the adopted power-law size distribution with p = 3.5 as</text>
<formula><location><page_5><loc_65><loc_79><loc_93><loc_83></location>˜ σ sm tot = 3 Σ d,sm ρ s √ a min a ∗ , (13)</formula>
<formula><location><page_5><loc_65><loc_76><loc_93><loc_79></location>˜ σ gr tot = 3 Σ d,gr ρ s √ a ∗ a max . (14)</formula>
<text><location><page_5><loc_52><loc_71><loc_93><loc_75></location>Here, ρ s = 3 g cm -3 is the material density of silicate cores of the dust grains.</text>
<text><location><page_5><loc_52><loc_61><loc_94><loc_71></location>The model includes photodesorption of volatiles by interstellar irradiation, which is mostly relevant in the outer disc regions with lower optical depth. We do not consider the UV radiation field produced by the star and the accretion region as a source of photodesorption, assuming that they do not reach disc midplane due to high optical depth. The photo-desorption rate is calculated as</text>
<formula><location><page_5><loc_65><loc_58><loc_93><loc_59></location>η pd = ˜ σ tot µ sp m p YF UV . (15)</formula>
<text><location><page_5><loc_52><loc_39><loc_94><loc_56></location>where F UV = F UV 0 G UV (photons cm -2 s -1 ) is the UV photon flux expressed in the units of standard UV field and Y = 3.5 × 10 -3 + 0.13 exp ( -336K/ T mp ) (mol photon -1 ) is the photodesorption yield adopted from Westley et al. (1995) for water ice. The intensity of the interstellar UV radiation field with G 0 = 1 is F UV 0 = 4.63 × 10 7 photon cm -2 s -1 (Draine 1978). We assume that the disc situated in a star-forming region is illuminated by a slightly elevated unattenuated UV field with G env = 5.5 G 0 . For the disc midplane, which is illuminated from above and below, this field scales with the UV optical depth τ UV towards the disc midplane as</text>
<formula><location><page_5><loc_65><loc_37><loc_93><loc_39></location>G UV = 0.5 G env e -τ UV . (16)</formula>
<text><location><page_5><loc_52><loc_29><loc_94><loc_35></location>The optical depth can be calculated as τ UV = 0.5( κ sm Σ d,sm + κ gr Σ d,gr ), where κ sm = 10 4 cm 2 g -1 , κ gr = 2 × 10 2 cm 2 g -1 are typical values of absorption coefficients in the UV for small and grown grains (Pavlyuchenkov et al. 2019, Fig. 1).</text>
<text><location><page_5><loc_52><loc_17><loc_94><loc_29></location>We calculate the adsorption rate λ following Brown and Charnley (1990). It is proportional to the total cross-section of dust grains per unit volume, which can be obtained from the total surface area of dust grains per unit disc surface ˜ σ tot. To change the normalisation to the 2D case, ˜ σ tot needs to be multiplied by 1/2 H . Another factor of 1/4 follows from the difference between cross-section and surface area of a sphere. As a result, the adsorption rate is calculated as</text>
<formula><location><page_5><loc_66><loc_12><loc_93><loc_15></location>λ = ˜ σ tot 8 H √ 8 k B T mp πµ sp m p . (17)</formula>
<text><location><page_5><loc_52><loc_6><loc_93><loc_10></location>A more detailed derivation of rate coefficients for adsorption and desorption is presented in Section 2.3 of Molyarova et al. (2021).</text>
<table>
<location><page_6><loc_12><loc_75><loc_43><loc_85></location>
<caption>Table 1. Binding energies, molecular weights, and initial abundances for the considered volatiles adopted in the modelling. Initial abundances of the species f s are shown relative to number density of water molecules in ice phase, and Σ sm s / Σ init g is the corresponding initial mass fraction of the ices relative to gas surface density.</caption>
</table>
<text><location><page_6><loc_7><loc_39><loc_48><loc_73></location>Table 1 summarises the molecular parameters used in both models in this work. Binding energies for H 2 O, CO 2 , and CO are based on the experimental data from Cuppen et al. (2017) for desorption from crystalline water ice. The binding energy for methane is taken from Aikawa et al. (1996). The values of the initial abundances relative to water f s are based on the observations of ices in protostellar cores around the low-mass protostars (Karin I. Öberg et al. 2011). They are transformed to the mass fraction relative to gas assuming the water abundance of 5 × 10 -5 relative to gas number density. Total initial mass of ices comprises ≈ 8.5% of the total mass of refractory grain cores. This is relatively low compared to the estimates suggesting comparable masses of ices and refractories in the discs (Pontoppidan et al. 2014). However, the lower fraction of ices is more suitable in our approach, that suggests that ice mantles do not change mass and radius of dust grains (Molyarova et al. 2021). As we are mostly interested in the elemental ratios and relative abundances of the considered ices, lower ice fraction is an appropriate simplification. However, we note that dust dynamics can lead to accumulation of ices and ice mantles exceeding masses of silicate cores in some disc regions, as was shown previously in Molyarova et al. (2021).</text>
<text><location><page_6><loc_7><loc_12><loc_48><loc_39></location>Ice mantles also provide feedback on dust evolution. The model includes the effect of ices on fragmentation velocity v frag , which is the the maximum collision velocity leading to sticking instead of fragmentation. According to laboratory experiments, icy grains have higher v frag than bare silicate grains by an order of magnitude (Wada et al. 2009; Gundlach and Blum 2015). In Molyarova et al. (2021) we used the values of fragmentation velocity v frag = 1.5 and 15 m s -1 for bare and icy grains, correspondingly. Here we follow Okuzumi and Tazaki (2019) and adopt lower values of v frag = 0.5 and 5 m s -1 , which are more relevant for grains consisting of µ m-sized monomers. These lower values of v frag will lead to higher importance of fragmentation compared to Molyarova et al. (2021). To determine if a dust grain should be considered icy or bare, we compare the local total surface density of all ices on grown dust divided by Σ d,gr with the threshold value K , which is calculated as</text>
<formula><location><page_6><loc_22><loc_9><loc_48><loc_13></location>K = 3 a ml ρ ice √ a ∗ a max ρ s , (18)</formula>
<text><location><page_6><loc_7><loc_5><loc_48><loc_9></location>i.e., an icy grain must have at least one monolayer of ice. Here, a ml is the thickness of the ice monolayer estimated as the size</text>
<text><location><page_6><loc_51><loc_86><loc_93><loc_92></location>of a water molecule 3 × 10 -8 cm. The material density of ice ρ ice = 1 g cm -3 and the mean radius of a grown grain is calculated as √ a ∗ a max for the power-law distribution with p = 3.5.</text>
<section_header_level_1><location><page_6><loc_51><loc_82><loc_60><loc_83></location>3. Results</section_header_level_1>
<text><location><page_6><loc_51><loc_68><loc_93><loc_82></location>To understand the distribution of the species, we first need to consider the global evolution of the disc and its structure. The distribution of volatiles and the C/O ratio is very sensitive to gas and dust substructures appearing in the disc. Variations in temperature and pressure lead to the complex shape and temporal evolution of the snowlines. The dependence of dust fragmentation velocity on the presence of ice mantles implies the feedback from the volatiles on dust and (through backreaction) on gas.</text>
<text><location><page_6><loc_51><loc_50><loc_93><loc_68></location>Our modelling starts with the gravitational collapse of a flattened, slowly rotating molecular cloud. The protoplanetary disc is formed after the formation of the protostar, when the in-spiraling layers of the contracting cloud hit the centrifugal barrier near the inner edge of the sink cell, at a time instance depending on the initial core mass. The disc and the protostar are formed ≈ 53 kyr after the beginning of the cloud collapse in model M1 and at ≈ 78kyr in model M2. If not stated otherwise, times are specified counting from the beginning of the simulation, e.g. the 100 kyr time instance for model M1 refers to a ≈ 50kyr old disc, as the first stage includes cloud collapse as well.</text>
<text><location><page_6><loc_51><loc_27><loc_93><loc_50></location>An important characteristic of young stellar objects is their variable accretion rate. Our modelling produces accretion bursts with the magnitude of ∼ 100 L ⊙ occurring every ∼ 10 4 years during the first hundred thousands years of disc evolution. These burst parameters are in line with the episodic accretion scenario (Hartmann and Kenyon 1985) and resemble the observed phenomenon of FU Ori type stars (see Audard et al. 2014, for a review). The luminosity outbursts heat up the disc and typically shift the snowlines further away from the star. Although this effect is temporary, it can be reflected in the distributions of the volatiles, and leave long-term imprints in the observed dust properties (Vorobyov et al. 2022). The detailed analysis of the effect of such outbursts on the distribution of the elements is worthy of a separate study and lies beyond the scope of this paper.</text>
<section_header_level_1><location><page_6><loc_51><loc_24><loc_72><loc_25></location>3.1 Dust and gas structures</section_header_level_1>
<text><location><page_6><loc_51><loc_6><loc_93><loc_24></location>During the first hundred thousands years of evolution, protoplanetary discs change from highly asymmetric and dynamic objects to nearly axisymmetric and slowly evolving structures. Figure 1 shows the examples of different gas and dust substructures that are present in the disc at different stages. The snapshots are shown for model M1, they include a young disc (160 kyr), an intermediate state (350 kyr), and a more evolved and axisymmetric disc (490 kyr). Each of the selected time instances represent some characteristic morphological features addressed below. In model M2, similar structures appear, although sometimes at different evolution times. In this subsection, we consider model M1 as an example and discuss</text>
<figure>
<location><page_7><loc_7><loc_55><loc_93><loc_92></location>
<caption>Figure 1. Surface density of gas and grown dust, Toomre Q -parameter, maximum dust radius, temperature and viscous α -parameter in model M1 at selected time moments: 160 kyr, 80 × 80 au; 350 kyr, 35 × 35 au; 490 kyr, 9 × 9 au. The contours indicate the position of the water snowline. Note that at the panels with multiple water snowlines, water is frozen outside the outer line and inside an inner dust ring at 1 - 2 au.</caption>
</figure>
<text><location><page_7><loc_7><loc_43><loc_49><loc_47></location>these features, highlighting the properties of dust and gas substructures that are most relevant for the distribution of the volatiles.</text>
<text><location><page_7><loc_7><loc_17><loc_49><loc_42></location>The earliest phases of disc evolution are characterised by a large-scale spiral structure in both dust and gas, as well as episodic appearance of clumps. They are the result of gravitational instability (GI) in a massive disc, as in our modelling, the disc mass comprises more than 0.1 of the stellar mass, which is roughly a threshold of the disc stability against GI (Vorobyov 2013; Kratter and Lodato 2016). This is illustrated by the Toomre Q -parameter (Toomre 1964) in the third column of Figure 1: inside the spirals and the clump Q < 1, which indicates the dominance of self-gravity over Keplerian sheer and gas density. The spiral structures become less prominent with time as the disc looses mass and the Q -parameter increases. However, spirals persist in the model throughout the disc evolution up to 0.5 Myr. For example, at 350 kyr, a very tight spiral is present in the gas, starting at the gas and dust ring at ≈ 10au. At 490 kyr, the spiral pattern is weak and exists at > 10 au distances, which are not displayed in Figure 1.</text>
<text><location><page_7><loc_7><loc_6><loc_49><loc_16></location>One- or two-armed grand design spirals are common 100 - 200 kyr after the disc formation in the models. The analogues of such spirals in the observed young protoplanetary discs around low-mass stars are found, for example, in Elias 227 or WaOph 6 (Pérez et al. 2016; Huang, Andrews, Pérez, et al. 2018). It is not yet clear if the observed spirals are the result of GI or caused by a perturbation from a companion</text>
<text><location><page_7><loc_52><loc_41><loc_93><loc_47></location>planet or a (sub)stellar object (Meru et al. 2017; Brown-Sevilla et al. 2021). A spiral with multiple clumps produced by GI was recently observed in the disc around a FUor V960 Mon (Weber et al. 2023).</text>
<text><location><page_7><loc_52><loc_18><loc_94><loc_41></location>Another important feature of gas and dust spatial distributions is ring-like structures at various scales. The system of rings starts to form as early as 100 kyr, and develops to the high-contrast multiple ring structure (1 - 3 orders of magnitude difference between surface densities in rings and gaps), which is evident in the middle and bottom rows of Figure 1. While gas rings are also common, the annual structures are more prominent in the dust surface density, as well as in dust size. Some of the dust rings correspond spatially to the gas rings, while others are barely reflected in the gas distribution. Overall, the difference between the gas and dust structures develops with time, as the result of dust growth and drift (Testi et al. 2014). Only some particular substructures, such as the ring between 1 - 2 au, are present in the distributions of both gas and dust components.</text>
<text><location><page_7><loc_52><loc_6><loc_94><loc_18></location>Another location where prominent rings form in both dust and gas is the water snowline. Here, the water snowline is defined as the location in the disc midplane where the amount of water in the gas equals to its amount in the ice (on both dust populations). There are multiple location in the disc where this happens. Water is frozen in most of the disc beyond 5 - 10 au, and we will refer to the furthest snowline dividing these outer frozen region from the inner one with the gas-phase water</text>
<text><location><page_8><loc_7><loc_74><loc_48><loc_92></location>as a primary snowline. Generally, the primary snowline is roughly circular, but it can have asymmetries due to the spiral structure and an additional snowline may appear, e.g., around a gravitationally bound clump (upper rows in Figure 1). Besides, at ≈ 260kyr, another region rich in water ice appears in the inner disc, creating a pattern of double or triple water snowline at later times (middle and lower rows in Figure 1). We will refer to these inner additional snowlines as the secondary snowlines. They circumcise a cold and dense gas-dust ring that forms at 1 - 2 au. As the disc cools down with time, the primary snowline position moves from ≈ 10au distance at 160 kyr, to ≈ 6au at 350 kyr and ≈ 4au at 490 kyr.</text>
<text><location><page_8><loc_7><loc_37><loc_48><loc_74></location>Snowlines are known to be associated with the enhancement of dust and volatiles (Stevenson and Lunine 1988; Cuzzi and Zahnle 2004; Drążkowska and Alibert 2017; Molyarova et al. 2021). Dust enhancement was also shown to affect the distribution of gas and its accretion rate through the disc (Gárate et al. 2020) by means of dust back reaction, which is also accounted for in our modelling. In our models, an about 2 au wide dust ring is formed at the inner edge of the water snowline (at ≈ 10au) as soon as 50 kyr after the disc formation. Grown dust grains drifting towards the star through the snowline lose their mantles, their fragmentation velocity drops, rendering them more vulnerable to fragmentation. Consequently, the grain maximum size a max decreases, their drift towards the star slows down, which leads to the accumulation of grown dust, as well as small dust as a product of fragmentation. Increase of total dust density also leads to less efficient cooling and results in higher temperature inside the snowline (see the right panels in Figure 1). Later, at times > 200 kyr, several additional rings form outside the water snowline at distances up to 100 au, the most notable one being 1 - 2 au exterior to the primary snowline. Dust is trapped inside the gas pressure maxima in these rings, which is a self-supporting phenomenon as the temperature also increases inside the dust ring due to high optical depth.</text>
<text><location><page_8><loc_7><loc_12><loc_48><loc_37></location>These rings are worthy of attention in the context of possible planet formation. Dust surface density and size are higher in the rings, with a max reaching centimetres, dust-to-gas ratio up to 0.1 - 0.2, and Stokes number up to 0.05 - 0.1. This could ease the development of the streaming instability, which typically requires pebble-size grains with St ≳ 0.01 and dustto-gas ratio ≳ 0.02 (Carrera, Johansen, and Davies 2015). Multiple ring-like structures are commonly observed in protoplanetary discs at a range of ages and display a variety of examples, with different widths, contrasts and numbers of rings (Huang, Andrews, Dullemond, et al. 2018, and many others). However, to directly compare the ring structures in the simulated dust surface density with the observed dust emission, require radiative transfer modelling is needed. Some of the observed ring structures could be produced by radial variation in dust size even in the absence of gaps in dust surface density (Akimkin et al. 2020).</text>
<text><location><page_8><loc_7><loc_6><loc_48><loc_12></location>The most prominent dust rings are located in the vicinity of the water snowline: the ring outside the primary snowline at 5 - 8 au (depending on the time) and the ring at 1 - 2 au, inside the primary snowline, which at later times also contains water</text>
<text><location><page_8><loc_51><loc_65><loc_93><loc_92></location>ice and additional snowlines. The main effect of the snowlines is the change in fragmentation velocity between mantled and non-mantled grains: dust size sharply decreases by ≈ 2 orders of magnitude for the latter. Immediately outside the water snowline, the midplane temperature is lower, due to lower dust opacity and thus more efficient cooling. Turbulent α is on the contrary, higher, providing more efficient radial transport of matter. It leads to lowering the gas surface density in this gap, which in turn increases α (see Eq. (5)), creating the positive feedback and further deepening the gap. One of the possible mechanisms to create the initial decrease in the gas surface density is dust back-reaction, which can affect the inward flow of gas at the snowline. This effect was investigated by Gárate et al. (2020) for different initial dust-to-gas ratios. The radial variation of α itself lead to the appearance of gas substructures (Tong, Alexander, and Rosotti 2024). The combined effect of lower temperature and density creates a pressure minimum, which dust tends to avoid.</text>
<text><location><page_8><loc_51><loc_42><loc_93><loc_64></location>The dead zone, where α -parameter values are lower than 10 -3 , includes the ice-free inner disc and has an approximately two times larger radial span than the iceless region. The distribution of α -parameter is shown in the right column of Figure 1. Inside the primary water snowline, the values of α are the lowest due to high surface density of gas. The dead zone is not axisymmetric and reflects the spiral structures of the gas distribution, as it is sensitive to Σ g (see Eq. (5)). The spiral arms of the dead zone span to 15 - 25 au at 160 kyr, to 10 - 18 au at 350 kyr and to 10 - 14 au at 490 kyr. The comparison with the temperature distribution in Figure 1 indicates that α and T mp are anticorrelated, as higher viscosity provides faster accretion, hence lower surface densities and more efficient cooling. Similarly, a lot of dust accumulates in the dead zone, increasing opacity, which hampers cooling.</text>
<text><location><page_8><loc_51><loc_26><loc_93><loc_41></location>The icy dust ring at 1 - 2 au is especially interesting in the context of planet formation. In this ring, dust-to-gas ratio exceeds 0.1, and surface densities of both gas and dust are increased by more than an order of magnitude compared to the adjacent regions. Due to the ice mantles that protect dust from fragmentation, the dust size reaches several cm, close to the values behind the water snowline. When the ring is established, it is self-supporting, in a sense that without external perturbation (e.g. sharp change in accretion flow from the outer disc), it can be stable for a long time, over tens of kyr.</text>
<text><location><page_8><loc_51><loc_9><loc_93><loc_26></location>The presence of water ice inside the dead zone was investigated by Vallet et al. (2023). They showed that in the discs around lower-mass stars, the turbulent heating in the inner disc can be low enough to allow the existence of ices. In our modelling the cooling of the inner region is assisted by the inner dust ring. The freeze-out has a positive feedback on dust growth due to higher v frag , which helps dust grow larger and further accumulate toward the pressure maximum in the ring. So in our modelling ices coexist with a lot of grown dust in the dead zone. Such an icy inner region appears to be a promising place for the formation of the volatile rich planets.</text>
<figure>
<location><page_9><loc_7><loc_55><loc_94><loc_89></location>
<caption>Figure 2. Radial distribution of azimuthally averaged surface densities of the volatiles in the gas and in the ice at various time instances in M1 ( M core = 0.66 M ⊙ ). Pale lines indicate the total surface density of species. The upper panels show surface densities of gas, small dust and grown dust, and the midplane temperature.</caption>
</figure>
<figure>
<location><page_9><loc_7><loc_10><loc_94><loc_45></location>
<caption>Figure 3. Same as Figure 3 but for model M2 ( M core = 1 M ⊙ ).</caption>
</figure>
<section_header_level_1><location><page_10><loc_7><loc_90><loc_27><loc_91></location>3.2 Distribution of volatiles</section_header_level_1>
<text><location><page_10><loc_7><loc_71><loc_48><loc_90></location>Even in the absence of chemical reactions, over the years of disc evolution the distribution of volatiles significantly changes compared to the initial one. This is the result of both phase transitions and dust growth and advection. Dust drift brings ices from the outer disc, enriching the inner disc with volatiles. Snowlines provide the conditions favouring accumulation of ices and gas-phase species. The formation of the established disc structures, such as dust and gas rings and spirals (see Section 3.1) leads to a complex pattern of intermittent snowlines. In this Section, we describe the main features of the molecular distributions and their implications for the composition of dust and gas at early stages of protoplanetary disc evolution.</text>
<text><location><page_10><loc_7><loc_50><loc_48><loc_71></location>Figures 2 and 3 show the azimuthally averaged radial profiles of the volatile species in models M1 and M2, respectively. As mentioned earlier, we define the snowline position as the location in the disc midplane where the amount of species in the gas equals to its amount in the ice (on both dust populations). This definition allows the snowline to have complex shape, characterised by different positions for each azimuthal direction. In the azimuthally averaged distributions shown in Figures 2 and 3, the position would only be approximate. The slope of the species distribution in the vicinity of the snowline reflects the degree of the axial asymmetry in the disc. Flatter distributions appear as the contributions sum up from the snowlines, the radial position of which vary at different azimuthal angles ϕ .</text>
<text><location><page_10><loc_7><loc_29><loc_48><loc_50></location>In our model setup, the volatiles can either have no snowline in the disc, or have two or more snowlines depending on the local conditions. Snowlines are absent for more volatile species (CO and CH 4 ) at earlier times or during bright luminosity outbursts, when the disc is too hot for them to freeze out. When there are two snowlines, the inner snowline in the disc is the one determined by thermal desorption. We refer to it as the primary snowline. The outer snowline is determined by photo-desorption, it lies in the embedding envelope outside of the body of the disc where optical depth is low. It must be noted that the gas-phase species outside this photo-snowline are vulnerable to photo-dissociation by the UV radiation. This process is not explicitly included in our chemical model, but can be assessed as in Molyarova et al. (2021).</text>
<text><location><page_10><loc_7><loc_6><loc_48><loc_28></location>More than two snowlines appear when the disc physical structure becomes more complex, mainly due to the presence of the ring-like structures. Species can freeze inside a cold dense dust ring, creating additional secondary snowlines, also governed by thermal desorption. The concepts of primary and secondary snowline is necessary for H 2 O and CO 2 , which have multiple snowlines at the later stages of disc evolution. For water, the inner icy region appears at ≈ 1au at 490 kyr (see right column of Figure 3) inside a dense dust ring. For CO 2 , the ring that appears at 7 au, outside the primary water snowline at 4 au, creates the inverted thermal structure in the region with T mp close to the typical CO2 sublimation temperature of 70 - 90 K. Increase in T mp in these ring is caused by higher optical depth and consequently lower cooling on the viscously heated midplane.</text>
<text><location><page_10><loc_51><loc_61><loc_93><loc_92></location>Apart from the snowlines, the distributions of volatiles in Figures 2 and 3 present local radial variations in all of the species components, including total abundance of the species. The initial total abundance is kept only in the envelope. As the matter is being redistributed, abundances of all volatiles in the inner disc grow. Particularly, the process responsible for this is dust drift. It brings the grown ice-covered grains to the inner disc regions, where their ice mantles are sublimated and no longer move with the drifting silicate dust cores. The effect is stronger for less volatile species H 2 Oand CO 2 : their abundance in the gas grows by a factor of a few inside their primary snowlines. For CO and CH 4 the effect is weaker, because there is less grown dust outside their snowlines, and those snowlines are not very much established at earlier times. Thus their abundances inside the snowline only grow by a factor of unity, except for the immediate vicinity of the snowline. Moreover, both CO and CH 4 have a bump in the gas-phase distribution just outside the dust ring at 1 au, that is absent in H 2 Oand CO 2 . At the inner disc edge the abundances of CO and CH 4 in the gas are lower than the initial value.</text>
<text><location><page_10><loc_51><loc_37><loc_93><loc_61></location>The abundance enhancements appear most notably at the snowlines, with local bumps in all three phases at later times (after ≈ 350kyr). They are produced by the combination of the dust radial drift and the azimuthal oscillations of dust and gas radial velocity, described in more detail in Molyarova et al. (2021) and Molyarova et al., in prep. By 500 kyr, the surface density of gas-phase H 2 Oexceeds the initial value by a factor of 5, of CO 2 - by a factor of 7, of CH 4 - by 3.5 and of CO by 2.5. Similar accumulation powered by turbulent diffusion was previously studied for CO molecule in axially symmetric model setup (Stammler et al. 2017; Krijt et al. 2018). Their results indicated similar enhancement in the gas phase by a factor of a few. In our non-axisymmetric approach, diffusion is not a necessary requirement, and the necessary transport is rather provided by azimuthal variations of radial velocity induced by disc self-gravity (Molyarova et al., in prep).</text>
<text><location><page_10><loc_51><loc_10><loc_93><loc_36></location>Distributions of ices on small and grown dust are different as they are affected by dust growth and drift. In general, there is more grown dust than small by mass, and in most of the disc there is more ices on grown dust, particularly at later times. However, in the outer disc and at the earlier times, ices on small grains dominate. As initially all ices are on small dust, it seems inevitable that they will gradually move to the grown grains as small dust coagulates and turns into grown grains. However, near all the snowlines, amount of ices on small dust increases due to the effect of the spiral pattern. Ice abundances are determined by episodic sublimation and freeze-out in the warm spiral arms of the complex density and temperature pattern (see Section 3.3.2. in Molyarova et al. 2021). As adsorption preferably happens to the smaller grains due to their larger total surface area, there is much more ices on small dust in the wake of the spiral arms. This concerns the photo-desorption snowlines as well.</text>
<figure>
<location><page_11><loc_8><loc_68><loc_49><loc_91></location>
<caption>Figure 4 shows radial profiles of the C/O ratios averaged over the azimuthal angle ϕ by the end of the simulations, at 490kyr. The key values of C/O ratio relevant in the context of planet formation are the initial value of 0.34, motivated by the comparison with the initial abundances, and 1.0, which is a boarder between carbon- and oxygen-dominated chemical regimes in the ISM and (exo)planetary atmospheres (see, e.g., Tielens 2005; Seager et al. 2005; Madhusudhan 2012).</caption>
</figure>
<figure>
<location><page_11><loc_51><loc_68><loc_92><loc_91></location>
<caption>Figure 4. Radial profiles of the C/O ratio at 490 kyr in models M1 (le/f_t) and M2 (right). The plots show C/O in total (black), in the gas (red), and in the ice (blue). The C/O ratio in the ice (gas) is only shown for radial distances where the mass of the volatiles in the ice (gas) is larger than 0.1% of the total mass of the volatiles in the gas (ice). The grey horizontal line indicates the baseline C/O = 0.34 . Positions of the snowlines are highlighted by vertical dashed lines. The regions where water (thus all other species) is in the gas are shaded with purple.</caption>
</figure>
<section_header_level_1><location><page_11><loc_7><loc_58><loc_18><loc_59></location>3.3 C/O ratios</section_header_level_1>
<text><location><page_11><loc_7><loc_38><loc_49><loc_57></location>Here we consider the relative amount of carbon and oxygen in the gas, on the surface of small and grown grains, and in total for all for all phases and disc locations. C/O ratio is seen as a perspective tracer of planet formation mechanisms (see Öberg, Murray-Clay, and Bergin 2011; Thiabaud et al. 2015). Therefore, we are especially interested in the regions of the disc and the volatile phase component where the C/O ratio declines from the total initial value (see below). Particularly, weare interested in C/O ratio noticeably above the initial value, as it was observed in atmospheres of exoplanets and suggests the formation of these planets in similarly carbon-enriched environments (Swain et al. 2009; Madhusudhan et al. 2011; Moses et al. 2013; Facchini et al. 2021).</text>
<text><location><page_11><loc_7><loc_15><loc_49><loc_38></location>The C/O ratio is calculated as the total number of carbon atoms contained in the molecules, divided by the total number of oxygen atoms in the molecules. This calculation can be done separately for the species contained in the gas phase, species on dust surface, or for the species in all phases. Thus, we calculate the C/O ratio in the gas, in the ice, and total C/O, respectively. For the ice species, we include both ices on grown and small dust grains.In some disc regions, the amount of carbon and oxygen in in a particular phase is very low, e.g. in the ice phase inside the water snowline, where all the molecules are in the gas. For such cases, C/O ratio would not be representative of the chemical composition, so we exclude the corresponding computational cells from the consideration. We only display the C/O ratio if the mass of the volatiles in the phase is higher than 10 -3 of the total mass of volatiles at a given location.</text>
<text><location><page_11><loc_7><loc_6><loc_49><loc_15></location>For the adopted molecular abundances based on Karin I. Öberg et al. (2011) and also used by Eistrup, Walsh, and van Dishoeck (2018), the baseline value is C/O = 0.34, which is in line with the gas-phase abundances in the ISM ( ≈ 0.36, Przybilla, Nieva, and Butler 2008). This value is different from the typical solar value of ≈ 0.5 (Przybilla, Nieva, and Butler</text>
<text><location><page_11><loc_52><loc_28><loc_94><loc_59></location>2008), which is commonly used as a reference (e.g., Öberg, Murray-Clay, and Bergin 2011; Semenov et al. 2018). First, local galactic abundances changed since the Solar system formation 4.6 billion years ago (see, e.g., Appendix A in Bergin et al. 2024, for the comparison). Second, the difference also stems from inclusion or non-inclusion of the dust component. The stellar atmospheric abundances comprise all the elements present in the medium where the star formed, while the elements available for the volatiles are only a fraction of that. The values of the elemental abundances can be determined separately for the volatile (gas and ices) and the refractory (rocky dust grain cores) components of the ISM (Hensley and Draine 2021). Here, we do not take into account carbon and oxygen from the rocky cores of dust grains. Due to the model assumptions, the initial ice-to-rock mass ratio in our model is quite low (0.08 compared to 2 - 4 in Pontoppidan et al. 2014). Therefore, adding the elements from solid dust grains to those contained in ices would be misleading, as the former would dominate in the resulting C/O ratio. In this work, we concentrate on considering the C/O ratios of the volatile component (ice and gas), and compare them with the initial value of 0.34.</text>
<text><location><page_11><loc_52><loc_6><loc_93><loc_15></location>By the age of 490 yr, the C/O ratio in both models is significantly different from the initial value. This concerns the total C/O ratio, as well as the C/O ratio in the gas and in the ice. Let us analyse the distributions of C/O ratios that we can see from Figure 4, moving inwards from the envelope to the centre. The main features of the C/O distributions are the</text>
<text><location><page_12><loc_7><loc_90><loc_14><loc_92></location>following:</text>
<unordered_list>
<list_item><location><page_12><loc_8><loc_84><loc_48><loc_89></location>· the C/O ratio in the envelope is close to the initial value 0.34 in the gas and in total, and it grows up to unity from the envelope to the CO snowline in the disc;</list_item>
<list_item><location><page_12><loc_8><loc_83><loc_45><loc_84></location>· around the CO snowline, the C/O approaches unity;</list_item>
<list_item><location><page_12><loc_8><loc_80><loc_48><loc_83></location>· between CO 2 and CH 4 snowlines the C/O in the gas is > 1, and the C/O in the ice is below the initial;</list_item>
<list_item><location><page_12><loc_8><loc_75><loc_48><loc_80></location>· at the distances of tens of au, there are variations in the total C/O ratio that are not connected with any snowlines, or variations in gas- or ice-phase C/O;</list_item>
<list_item><location><page_12><loc_8><loc_72><loc_48><loc_75></location>· ice-phase C/O ratios peak around all snowlines except water;</list_item>
<list_item><location><page_12><loc_8><loc_68><loc_48><loc_72></location>· inside the primary water snowline is the region rich in volatiles, with both gas- and ice-phase C/O are the lowest in the disc.</list_item>
</unordered_list>
<text><location><page_12><loc_7><loc_50><loc_48><loc_66></location>As expected, the key changes in the C/O ratio distribution are associated with the positions of the snowlines. There are two main processes. First, the freeze-out and desorption at the snowlines transfer the elements between phases, altering the C/O in the gas and in the ice. Second, the snowlines favour the accumulation of the respective volatiles both in the gas and in the ice, as was discussed in Section 3.2, pumping up the amount of both components and altering their proportion. Besides the snowlines, radial drift of grown grains (Weidenschilling 1977) transports the volatiles inwards. We discuss below which processes are responsible for the formation of the listed features.</text>
<text><location><page_12><loc_7><loc_6><loc_48><loc_49></location>C/O in the outer disc. In the surrounding envelope, outside the disc, all volatiles are in the gas due to photo-desorption, and C/O in the gas is close to the initial value of 0.34. Around the COsnowline and beyond, the C/O in the gas is close to unity, and in the gas C/O is higher that the initial, which requires explanation. The region beyond the CO snowline is usually described as the place where all species are frozen, thus having a stellar C/O ratio in the solid phase and practically no carbon or oxygen in the gas (e.g., Öberg, Murray-Clay, and Bergin 2011; Öberg and Wordsworth 2019; P. Mollière et al. 2020). In our simulations, only in model M2 there is a region where less than 10 -3 of C and O is in the gas, and in model M1 such region is absent. Due to the asymmetric spiral structure that persists even at 490 kyr, even though most of CO is frozen beyond the snowline, there is a significant (> 10 -3 ) fraction of it in the gas. Additionally, the position of CO snowline itself is significantly affected by the dust drift, as it declines from the equilibrium between adsorption and desorption.The indistinctness of the CO snowline also helps CO to persist in the outer regions: all other species are ultimately frozen, so they are efficiently carried away to the inner disc via radial drift, while this works worse for only partially frozen CO. It creates relative overabundance of CO in the outer disc, elevating the total C/O ratio and later the ice-phase C/O ratio, when the preserved CO freezes out. Between the CO ice line and the envelope, there is a gradient of C/O in the gas due to photodesorption of the ices. The last molecule to be photo-desorbed is water, which returns oxygen to the gas phase at the farthest radial distance.</text>
<text><location><page_12><loc_51><loc_77><loc_93><loc_92></location>C / O ≈ 1 around the CO snowline. Beyond the CH 4 snowline, CO dominates the composition of volatiles in the gas phase, leading to the gas-phase C/O close to unity, the value characteristic of the CO molecule. At the CO snowline, the CO dominates the ice-phase composition as well. While dust drift substantially lowered the abundances of CO 2 and H 2 Oby 490 kyr in these regions, CO ice accumulated at the snowline. This leads to the ice-phase C/O closer to 1, too, making the vicinity of the CO snowline a region where total amounts of C and O are similar.</text>
<text><location><page_12><loc_51><loc_47><loc_93><loc_76></location>High C/O in the gas, low C/O in the ice. In the disc regions beyond the primary CO 2 snowline, only CO and CH 4 are in the gas, meaning the dominance of carbon and C/O > 1. Consequently, the C/O in the ice is generally lower, around 0.2, as the ices are mainly H 2 O and CO 2 rich in oxygen. This is consistent with the classical step-like picture of Öberg, Murray-Clay, and Bergin (2011), with the addition of carbonrich methane allowing the C/O > 1. The midplane C/O ratios we simulate are difficult to directly compare with observations, which mostly trace the molecular layer above the disc midplane. High C/O ratios in the gas are indeed observed in many protoplanetary discs (e.g. Miotello et al. 2019), but they are considered to be a natural consequence of dust settling (Krijt et al. 2018; Krijt et al. 2020). Dust inward drift could also enhance this effect in the outer disc regions, creating the radial gradient of total C/O ratio in the disc. Observations of CS and SO emission coming from close to the midplane layers potentially indicate the presence of such radial gradient of the gas-phase C/O in the PDS 70 disc (Rampinelli et al. 2024).</text>
<text><location><page_12><loc_51><loc_21><loc_93><loc_47></location>Variations of total C/O. Throughout the disc, there are sharp changes in total C/O ratio not connected with any snowline. Most noticeable are the variations between CO 2 and CH 4 snowlines, where gas-phase and ice-phase C/O ratios are stable. These variations are associated with the disc substructures, particularly with the dense dust rings described in Section 3.1. The total C/O changes due to radial variations in dust-to-gas ratio: when it is higher, the total C/O is closer to the ice-phase C/O, and vice versa. Inside the dust-rich rings, H 2 Oand CO 2 ices are abundant, due to high surface density of dust relative to gas. At the same time, CO and CH 4 in the gas phase have similar surface densities inside dust rings and between them. Thus, in the dust rings, CO 2 and water are overabundant, leading to lower total C/O ratio. We consider this effect in more detail in Section 3.5. Variations of the total C/O ratio due to dust substructures are also present in the cold ring at ≈ 1au.</text>
<text><location><page_12><loc_51><loc_6><loc_93><loc_21></location>C/O peaks at the snowlines. There are peaks of the C/O ratio in the ice right outside the snowlines of CO 2 , CH 4 and CO, produced by the accumulation of the respective ices (see Figures 2 and 3). In M2 model, the amount of CH 4 and CO ices at their respective snowlines becomes comparable with or even larger than those of CO 2 and H 2 O(compare the right panels of Figure 3), leading to C/O in the ice ≈ 0.6 - 0.9. In model M1, the accumulation is more prominent, so the C/O ratio in the ice approaches unity at CO snowline and > 1 at the methane snowline. These peaks distort the pattern of generally</text>
<text><location><page_13><loc_7><loc_84><loc_49><loc_92></location>low C/O ratio in the ice and preset additional regions where carbon-rich planetesimals could be formed, and carbon-rich pebbles could be accreted onto forming protoplanets. At the snowlines of H 2 Oand CO 2 , the C/O in the ice approaches the C/O ratios of these molecules, 0 and 0.5, respectively.</text>
<text><location><page_13><loc_7><loc_63><loc_49><loc_84></location>Lower C/O inside the water snowline. Inside the primary water snowline, the C/O ratio is generally lower than outside of it, close to the initial 0.34. Contrary to the outer regions depleted of ices due to the radial drift, the disc parts inside and around the water snowline are enriched in volatiles, and particularly of oxygen-rich water and CO 2 . Total and gas-phase C/O ratios vary from ≈ 0.2 to 0.6, as the secondary water snowlines add more substructure to the C/O distribution. The regions where the only ice is water and the ice-phase C/O = 0 are the inner cold dust ring at 1 au and the narrow annuli between water and CO 2 snowlines. The enrichment of the inner disc regions with oxygen as a result of dust radial drift is suggested by the resolved observations of molecules in protoplanetary discs (Banzatti et al. 2020).</text>
<text><location><page_13><loc_7><loc_45><loc_49><loc_63></location>These characteristic features of the C/O distributions are similar in the two presented models. The main difference is the radial distances where the borders between the zones are located; they are closer to the star in the less massive and thus colder model M1. This is mainly due to slightly different masses of the central star accumulated throughout 490 kyr of non-identical protostellar accretion history, which lead to different luminosity and thermal structure (see upper panels of Figure 5). Particularly, the stellar masses and luminosities at this time instance are: 1.07 L ⊙ and 0.34 M ⊙ (for M1), 1.89 L ⊙ and 0.58 M ⊙ (for M2). The less massive model M1 demonstrates overall higher C/O ratio in both phases.</text>
<section_header_level_1><location><page_13><loc_7><loc_41><loc_42><loc_42></location>3.4 Evolution of the snowlines and the C/O ratios</section_header_level_1>
<text><location><page_13><loc_7><loc_27><loc_49><loc_41></location>The positions of the snowlines are crucial for the values of C/O ratios, both because of the direct change through the phase transitions and the associated accumulation of volatiles. They depend on the local gas and dust properties, particularly on temperature. They can also be shifted inward due to dust drift (Piso et al. 2015). Snowlines evolve as the disc structure changes with time. In this Section, we consider the co-evolution of the snowlines and the C/O ratios and discuss the mechanisms of species redistribution over the disc.</text>
<text><location><page_13><loc_7><loc_18><loc_49><loc_27></location>The temporal evolution of the azimuthally averaged C/O ratios and the equilibrium positions of the snowlines is shown in Figure 5. It is evident from Figure 5 that the C/O ratio indeed follows the snowlines, particularly inside ≈ 10 au. The C/O structure changes throughout the disc evolution, and some key features only appear at later times.</text>
<text><location><page_13><loc_7><loc_6><loc_49><loc_18></location>One of the key factors affecting the disc thermal structure is the luminosity of the central source. In the upper panels of Figure 5, we show the evolution of total luminosity, which directly affects the positions of the snowlines. The luminosity is the sum of stellar and accretion components, coming from the protorstar itself and gravitational potential energy of the accreted matter. The stellar luminosity gradually decreases as the protostar becomes more compact. Accretion luminosity</text>
<text><location><page_13><loc_52><loc_75><loc_94><loc_92></location>depends on the accretion rate, which is highly variable as a result of magnetorotational and gravitational instabilities in the disc (Kadam et al. 2019, 2020; Vorobyov, Khaibrakhmanov, et al. 2020). The simulated episodic luminosity outbursts are similar to those occurring in the observed YSOs (Connelley and Reipurth 2018), with their amplitudes of tens and hundreds L ⊙ . In the more massive model M2, the outbursts are more frequent, brighter and occur until later times. The reasons behind this difference is the massive disc being more prone to both MRI and GI, which needs to be investigated in more detail in a separate study.</text>
<text><location><page_13><loc_52><loc_55><loc_94><loc_75></location>Snowlines of the least volatile of the considered species, H 2 Oand CO 2 , exist in the model since the earliest phases of the disc formation. Even during bright luminosity outbursts ( ∼ 100 L ⊙ ) they do not disappear, but move farther away from the star. During the first ≈ 50kyr the disc is spreading out, it is highly asymmetric and dynamic, so the snowline positions oscillate. Later on, the disc generally cools down, and the snowlines gradually move toward the star (except during the outbursts). In model M2, between 130 and 500 kyr, the primary water snowline moves from 12 to 4.5 au; the primary CO 2 snowline moves from 23 to 7 au. In model M1, the water snowline moves from 9 to 4 au, and the CO 2 snowline moves from 17 to 4.5 au.</text>
<text><location><page_13><loc_52><loc_27><loc_94><loc_56></location>Despite a factor of two different binding energies of H 2 O and CO 2 (5770 and 2360 K, respectively), the locations of their primary snowlines do not differ much due to the steep radial temperature gradient around these distances (see upper panels of Figures 3 and 2). Inside 10 - 20 au, T mp is determined by heating mechanisms other than external irradiation: viscous heating, gas work (PdV heating), heating by shocks and energy transport with advection. This also means that the water snowline is less sensitive to the level of irradiative heating, thus only slightly affected by the luminosity outbursts. The temperature change is particularly sharp at the water snowline(s), with the absolute value of the approximated power-law slope of 1.5 - 6. High surface density of dust in water-ice-free regions makes cooling less efficient and leads to locking up the produced heat and consequently higher temperatures. Besides, both these species have multiple snowlines due to the formation of ring-like substructures with the conditions close to the borderline between their frozen and gaseous state. These additional snowlines also affect the C/O distributions.</text>
<text><location><page_13><loc_52><loc_6><loc_94><loc_27></location>Methane and CO are the more volatile species in our model. They either have zero or two snowlines in the disc. The snowlines are absent during the outbursts brighter than ≈ 100 L ⊙ for methane and ≈ 200 L ⊙ for CO. Until approximately 200 - 250 kyr, there is no established CO snowline inside the disc. For example, at 160 kyr (see lower left panel in Figures 3 and 2) there is CO ice on both small and grown dust, but their amount is an order of magnitude lower than that of the gas. Additionally, most of this ice is located at the outer disc edge, where gas and dust surface densities sharply drop. Similar distribution appears for CO and CH 4 during bright outbursts: their ices are present in some disc regions, but due to non-axisymmetric disc structure, they do not dominate in the averaged profiles, and there is no common snowline for</text>
<figure>
<location><page_14><loc_8><loc_18><loc_92><loc_88></location>
<caption>Figure 5. Evolution of central source luminosity and C/O ratio in models M1 ( M core = 0.66 M ⊙ , le/f_t) and M2 ( M core = 1 M ⊙ , right). The upper panels show stellar and accretion luminosity depending on time. Below are, successively, total C/O ratio, C/O in the gas, and C/O in the ice, depending on time. The C/O values above and below the initial value of 0.34 are coloured in shades of red and blue, respectively. The regions with low abundances of both carbon and oxygen, either in the gas or ice phases, are shown in white. Coloured contours correspond to the positions of the snowlines. Photo-dissociation snowlines are not shown.</caption>
</figure>
<text><location><page_15><loc_7><loc_87><loc_49><loc_92></location>the whole disc. There are no secondary snowlines of CO or CH 4 , because there is no prominent gas and dust structures in the outer disc where these species are frozen.</text>
<text><location><page_15><loc_7><loc_43><loc_49><loc_87></location>Snowlines divide the disc into several zones with different characteristic C/O ratios. However, the chemical composition and C/O ratios in these zones change with time. One of the distinct zones is the region where water is not frozen, shaded with purple in Figure 4 and circumscribed by the dark purple dotted line in Figure 5. In this zone, all the species are in the gas phase, so the C/O ratio is initially close to 0.34. However, as the disc evolves, dust brings more volatiles from the outer disc. Particularly, the abundance of water grows most significantly, making C/O in the gas decrease with time. This happens because the main mechanism of the transport of the volatiles is dust radial drift, which works best in the regions where grown grains can sustain their mantles. Banzatti et al. (2020) suggest that dust growth and drift can be responsible for the observed anticorrelation between disc radius and H 2 O emission, implying that the inner regions of small discs with are enriched in water brought by efficiently drifting grains. Water is the least volatile species in our model, it is frozen in the largest part of the disc, thus its distribution is most strongly affected by the dust drift. At later times, this zone is divided into two, when a cold dense dust ring forms at 1 - 2 au, which happens at ≈ 460 kyr in model M2 and at ≈ 270 kyr in model M1. Interior to the ring, water abundance in the gas is around an order of magnitude lower than outside of it in both models. This happens because the inward flow of gasphase H 2 Ois 'blocked' by the cold ring where it freezes and accumulates with the grown grains in the pressure maximum. So the C/O ratio in the gas at r ⪅ 2au is determined by CO 2 and thus close to 0.5.</text>
<text><location><page_15><loc_7><loc_6><loc_49><loc_43></location>The C/O ratio in the ice is not defined in the envelope, where all ices are photo-desorbed, and in the warmest inner disc, where even water is in the gas. In disc regions where only water is frozen, the C/O ratio in the ice is zero. Before 200 - 250 kyr, when the disc cools down enough for CO to freeze out, the C/O ratio in the ice in the rest of the disc is close to 0.2. It is determined by CO 2 and H 2 O ices, with a small contribution from the low-abundance CH 4 . After that, when CO freezes out and CO snowline appears, the C/O ratio in the outer disc region becomes close to the initial value both in the ice and in total. This region beyond the CO snowline is frequently referred to in relation to giant exoplanets with stellar C/O ratios (e.g., P. Mollière et al. 2020; Öberg and Wordsworth 2019; Ohno and Ueda 2021). This region would be a perfect location for planets to accrete pebbles covered with icy mantles with the primordial elemental ratio, which would directly become part of the planetary atmosphere. However, pebbles are not initially present in the disc, and their existence in these outer regions is not guaranteed. Pebbles are dust grains large enough to move relative to gas e.g., Lambrechts and Johansen 2012; Lenz, Klahr, and Birnstiel 2019, and a fraction of the grown dust in our modelling can be classified as pebbles. The properties of pebbles and composition of their ice mantles in the same setting of the FEOSAD model were studied by Topchieva et al. (2024). They show that pebbles</text>
<text><location><page_15><loc_52><loc_83><loc_93><loc_92></location>appear in the disc as early as 50 kyr after its formation, and exist in a wide region of the disc. This partially includes the region beyond the CO snowline, but only the area of CO enhancement, where CO dominates in the ice composition and thus the C/O ratio in the ice is close to unity (see lower panels in their Figure 3).</text>
<text><location><page_15><loc_52><loc_66><loc_93><loc_83></location>As shown by Topchieva et al. (2024), ices on pebbles are dominated by H 2 O and CO 2 . In this case, relatively high values of the C/O in the ice ( ≈ 0.5) correspond to the regions where there is more CO 2 , i.e. around the CO 2 snowlines. This is also the region where a prominent dust ring forms under the influence of the primary water snowline. It is characterised by accumulation of CO 2 and to lesser degree H 2 Oice, as well as vapours, and relatively high amount of grown dust in the ring. The dust ring is situated between the two regions with C/O ≈ 0.34 in the ice. It presents another favourable location for accreting the ice content with close-to-initial C/O ratio.</text>
<text><location><page_15><loc_52><loc_34><loc_94><loc_66></location>The total C/O ratio in the disc also changed significantly from the initial value due to dust drift that redistributes the ices. The matter becomes more carbon-rich as the grains bring CO and CH 4 from the outer disc parts. This effect was previously investigated by Stammler et al. (2017) and Krijt et al. (2018) in their modelling of CO dynamics and dust evolution. However, as was shown by Krijt et al. (2018) and Krijt et al. (2020), vertical settling of grown grains towards the midplane is responsible for depleting the upper layers in the outer disc of gas-phase oxygen, which cannot be captured within our thin-disc modelling. The panels in the second row of Figure 5 demonstrate strong enhancement of total C/O ratio in the intermediate disc regions. In model M2, the total C/O ratio between 10 - 100 au becomes ≈ 0.7 after ∼ 300 kyr, which is two times higher than the initial value. At the snowlines of CH 4 and CO 2 it approaches unity, mostly due to the accumulation of these species in the gas. In model M1, this process begins ≈ 200kyr earlier and consequently leads to even higher total C/O ratio. At 400 - 500 kyr, most of the disc between 5 and 100 au has total C/O ≳ 1 in model M2, which demonstrates the powerful impact of dust drift.</text>
<text><location><page_15><loc_52><loc_16><loc_94><loc_34></location>A distinctive feature of the produced C/O distributions is the peaks of total and ice-phase C/O ratios around the CO and CH 4 snowlines. In model M2, the increase of the C/O ratios becomes noticeable only after ≈ 450kyr, while in model M1 it starts to form around ≈ 300 kyr. Initial abundances of CH 4 and CO are lower than that of CO 2 and H 2 O. As these species accumulate at their snowlines, the abundances become comparable, leading to C/O in the ice ≈ 0.6 - 0.8 in model M2 and up to 1 in model M1. Similar accumulation is seen around CO 2 snowline, but unlike CO and CH 4 snowlines, it is also connected to the interaction with the dust ring structure and is considered in more detail in Section 3.5.</text>
<section_header_level_1><location><page_15><loc_52><loc_12><loc_74><loc_13></location>3.5 Two-dimentional structure</section_header_level_1>
<text><location><page_15><loc_52><loc_6><loc_94><loc_12></location>The disc is not axisymmetric even at later stages of its evolution (see Section 3.1 and Figure 1), which is also reflected in the distributions of volatiles and C/O ratios. Examples of 2D distributions of C/O in model M1 at two time instances are</text>
<figure>
<location><page_16><loc_8><loc_20><loc_93><loc_80></location>
<caption>Figure 6. Distributions of C/O ratios and gas/dust surface densities. Model M1 160 kyr (upper le/f_t) and 300 kyr (upper right); model M2 250 kyr (lower le/f_t) and 490 kyr (lower right). Dotted lines mark the positions of the snowlines for H 2 O(dark purple), CO 2 (magenta), CH 4 (green), and CO (yellow).</caption>
</figure>
<figure>
<location><page_17><loc_7><loc_68><loc_48><loc_90></location>
<caption>Figure 7. Averaged radial profiles of dust-to-gas ratio, the total C/O ratio, and CO 2 and H 2 Oin the gas and in the ice in model M2 at 490 kyr. The horizontal lines in the upper panel show the reference values for the C/O ratio (0.34) and dust-to-gas ratio (0.01).</caption>
</figure>
<figure>
<location><page_17><loc_8><loc_13><loc_47><loc_56></location>
<caption>Figure 8. Dependence between total C/O ratio and dust-to-gas mass ratio in models M1 (upper panel) and M2 (lower panel). Three time instances are shown. The dashed line shows the fitted log-linear dependence for 490 kyr.</caption>
</figure>
<text><location><page_17><loc_52><loc_59><loc_94><loc_92></location>shown in upper panels of Figure 6. The left-hand side group of panels shows the structure that is characteristic of the earlier phases, the same time instance as the upper row in Figure 1. The prominent spiral structure and a clump both have their reflection in the C/O ratios. The snowlines of CO 2 and CH 4 have clearly non-regular shape affected by the spiral pattern of the gas, with blobs of frozen methane inside the main snowline. Inside the clump, the temperature is higher (see Figure 1), and both CO 2 and water are in the gas. The separation between gas-phase and ice-phase C/O ratios is clear, but the total C/O ratio in the clump is similar to the surroundings and is only slightly above the primordial value. The gas-phase C/O ratio inside the clump is around 0.7, lower than in the surrounding gas at the same radial distances. Similar decrease of C/O ratio indicated by lowered CS/CO ratio was observed in the disc around DR Tau (Huang et al. 2024). Lifetime of the clump (several orbital periods) is too short for significant differentiation between gas- and ice-phase composition to develop, mainly because of insufficient numerical resolution. Focused studies with higher resolution are needed to explore the C/O ratio in the clumps as precursor of giant planets formed via disc fragmentation.</text>
<text><location><page_17><loc_52><loc_24><loc_94><loc_59></location>The upper right-hand side group of panels in Figure 6 features a later stage of the disc evolution in model M1. By 300kyr, the accretion rate and the average positions of the snowlines are stabilised (see Figure 5). The spiral structure in the gas is much weaker but still present at r > 10 au. The CO snowline is established at ≈ 80au. The 2D shape of the snowlines is more circular at 300 kyr, particularly for the less volatile CO 2 and H 2 O. The effect of spirals is still evident in the contours of the CO and CH 4 snowlines. At the outer side of these snowlines, at approximately 40 and 80 au, respectively, the C/O in the gas has local maxima, which are also seen in Figure 4. The one at 40 au also has a clearly spiral shape, repeating the pattern of the gas distribution. The radial span of these peaks is of the same scale as the dispersion of the respective snowline distance from the star. For example, at the CH 4 snowline the C/O peak is ≈ 10au wide, and the snowline is at 28 to 37 au distances. Similar accumulation powered by diffusion of water vapour was shown, e.g., by Drążkowska and Alibert (2017). In addition to diffusion, in our modelling, the species are delivered outward from the snowline due to the dynamic shape of the snowline and two-dimensional movement of gas and dust (see Molyarova et al. 2021, and Molyarova et al., in prep).</text>
<text><location><page_17><loc_52><loc_6><loc_94><loc_24></location>In more massive model M2, the shape of the snowlines remains complex for longer times. Examples of C/O distributions in M2 are shown in the lower panels of Figure 6. The radial scale is chosen so that CO 2 and H 2 O snowlines are seen in more detail. At 250 kyr, even the grown dust distribution still has spiral pattern, and CO 2 snowline is following its complex multi-armed shape. The accumulation of CO 2 at the snowline is also very efficient, but it does not strongly affect the C/O ratios, as it drives them to the value 0.5 of the CO 2 molecule, which is close to the intrinsic value. The complex-shaped region between these snowlines has the ice-phase C/O ≈ 0.7. This pattern moves and changes its shape, thus affecting all</text>
<text><location><page_18><loc_7><loc_87><loc_48><loc_92></location>radial distances between 10 and 20 au. The total C/O ratio is slightly above the ambient value, approaching 0.5, because CO 2 in both phases begins to accumulate in this region.</text>
<text><location><page_18><loc_7><loc_60><loc_48><loc_87></location>At later times, the C/O distribution becomes more complex due to the presence of disc substructures, particularly the dust rings. The lower right-hand side group of panels in Figure 6 show this in more details. First, the temperature and density variations across the dust rings lead to the formation of multiple CO 2 and H 2 Osnowlines. An additional annulus of icy CO 2 appears in a relatively cold region between the dense dust rings at 6 - 7 and 8 - 13 au. The dust rings are warmer due to higher optical depth and active heating by disk internal sources, and the inner edge of the 6 au ring is warm enough to sustain gas-phase CO 2 . Second, the accumulation of icy dust grains in the rings alters the total C/O ratio. The total C/O ratio anticorrelates with the distribution of grown dust grains: inside the dense rings it is close to the initial value of 0.34, while between the rings, it is higher and reaches 0.8 - 0.9. At the same time, neither ice-phase, nor gas-phase C/O ratio displays any noticeable variations at 12-25 au, but they do have variations at r < 12 au, following the dust ring pattern.</text>
<text><location><page_18><loc_7><loc_28><loc_48><loc_60></location>The total C/O ratio is defined by the combination of the ice- and the gas-phase component. Their relative contribution is proportional to the dust-to-gas mass ratio. This is illustrated in Figure 7. Within the rings, the total C/O is dominated by ices of oxygen-rich species CO 2 and H 2 O, particularly on grown dust accumulated in the pressure maxima. For example, there is a lot of water ice at 6 - 7 au, and its contribution to the total C/O is weighted with high dust-to-gas ratio of almost 0.1, so the resulting total C/O is lower than the initial value. At the same time, at ≈ 5au, dust-to-gas ratio is around 10 -4 . The dominant species there are in the gas, ice surface densities are 1 - 2 orders of magnitude lower, so the total C/O ratio goes up. In the wide dust ring at 9-13 au, both H 2 Oand CO 2 contribute, although at the warmer inner edge of the dust ring CO 2 is sublimated. The value of total C/O is approaches 0.34, also elevated by the presence of gas-phase CO and CH 4 . Beyond ≈ 10au, both H 2 O and CO 2 are frozen, and the variations of the total C/O ratio are clearly anticorrelated with the dust-to-gas ratio and the position of the rings. Between the dust rings, where contribution of ices is low, the C/O is determined by CO and CH 4 gases, and reaches values of ≈ 1.</text>
<text><location><page_18><loc_7><loc_6><loc_48><loc_28></location>The anticorrelation between the total C/O ratio and dustto-gas mass ratio is an interesting finding. It is illustrated in Figure 8 for both models. Only the points within the disc are shown, with Σ gas > 0.1 g cm -2 .The pattern obviously changes with time, but the anticorrelation persists. Model M1 demonstrates wider variety of C/O and dust-to-gas ratios. The disc points are grouped in tangled curved "branches", some of them steeper than others. The 'width' (or spread) of these branches is determined by the azimuthal substructures. In the axially symmetric parts of the disc the values of dust-to-gas ratio and the total C/O are similar at a given radial distance. Different branches, which can be closer to vertical or horizontal orientation, are the result of the radial variations in the ice-phase C/O ratio and in ice fraction relative to the dust silicate cores. Horizontal branches correspond to weak or absent anticorrelation.</text>
<text><location><page_18><loc_51><loc_74><loc_93><loc_92></location>For example, near the snowlines of carbon-rich species, C/O in the ice is high and close to that of the gas, which decreases the effect of dust mass fraction on the total C/O. In areas with no ices, the anticorrelation is also irrelevant, because the total C/O is determined entirely by the gas phase. Vertical branches, on the contrary, correspond to the strongest anticorrelation effect, which is expected in the regions between the snowlines, where ice-phase C/O is the lowest. The identified anticorrelation in Figure 8 is similar to the results of chemical population synthesis modelling (Cridland et al. 2019, 2020) showing that the more solids a planet accreted in the disc, the lower the C/O ratio is in its atmosphere.</text>
<text><location><page_18><loc_51><loc_60><loc_93><loc_74></location>We fit the data for t = 490kyr with a linear law (taking the logarithm of dust-to-gas ratio) and obtain the following fits: C/O = -0.056 - 0.25 log 10 ( ξ ) for model M1, and C/O = -0.18 - 0.27 log 10 ( ξ ) for model M2. Here, ξ is the dust-to-gas mass ratio. The correlation coefficients are -0.54 for M1 and -0.57 for M2. At the shown earlier times (160 and 350 kyr), the correlation coefficient changes between approximately -0.9 and -0.5, which indicates noticeable anticorrelation throughout the disc evolution.</text>
<text><location><page_18><loc_51><loc_34><loc_93><loc_60></location>We note that these C/O ratios only include the volatile component, without the contribution from the refractory material. Although the refractory material is typically considered as silicates, which are rich oxygen, it contains a significant amount of carbon, with the resulting C/O ≈ 0.5 (see Table 2 in Hensley and Draine 2021). This solid carbon can be subject to carbon grain destruction (Lee, Bergin, and Nomura 2010; Gail and Trieloff 2017; Wei et al. 2019), but this process should be treated separately, as it also affects the gas-phase carbon abundance. Taking refractory cores into account should affect the dependence between total C/O ratio and dust-to-gas ratio, as adding more rock would make the C/O ratio closer to 0.5. Thus the degree of the anticorrelation must be affected by the composition of rocky cores, but the anticorrelation itself should remain even when refractories are included, because the C/O ≈ 0.5 is still lower than typical C/O of the gas in most of the disc ( ⪆ 1).</text>
<text><location><page_18><loc_51><loc_6><loc_93><loc_34></location>Dust rings are detected in the majority of the observed protoplanetary discs (Long et al. 2018; Huang, Andrews, Dullemond, et al. 2018). They are considered as a plausible sites of planet formation (Carrera, Johansen, and Davies 2015; Yang, Johansen, and Carrera 2017; Li and Youdin 2021; Lee, Fuentes, and Hopkins 2022; Jiang and Ormel 2023). The anticorrelation between the total C/O ratio of the volatiles and dust-to-gas mass ratio that we point out is a logical consequence of ices having typically lower C/O ratios and being attached to dust grains. If planets are formed in the dust rings with high dustto-gas ratios (> 10 -2 ), either exclusively from solids, or with the inclusion of the dust component, this would imply that their material initially has lower C/O ratio of ≈ 0.5 and below. To reach higher C/O ratios up to unity and above, which are observed in many exoplanets, these planets would need to migrate and accrete carbon-rich gas from regions other than their immediate formation sites inside the dust rings. In case if planet formation occurs independently of the dust rings, e.g. in the GI, their material is not determined by this anticorrelation.</text>
<section_header_level_1><location><page_19><loc_7><loc_90><loc_19><loc_91></location>4. DISCUSSION</section_header_level_1>
<text><location><page_19><loc_7><loc_33><loc_49><loc_89></location>Our simulations present a wide range of C/O ratios in the disc in different phases evolving with time. For the atmospheres of giant exoplanets, a variety of C/O ratios were retrieved, too. Here we can compare them to identify the disc regions and times where the chemical and physical conditions for planet formation are consistent. Most of the exoplanets for which the atmospheric composition was retrieved have super-stellar C/O ratios (Hoch et al. 2023; Weiner Mansfield et al. 2024), which draws more attention to carbon-enriched areas. They are suggested to form by core accretion, and accreting mostly the gas, which is typically more carbon rich (beyond water snowline). A lot of planets are observed to have stellar or slightly super-stellar C/O (e.g., P. Mollière et al. 2020; Zhang et al. 2021; Smith et al. 2024; Sing et al. 2024; Nortmann et al. 2024, and many others). One way to form such planets is gravitational instability, which includes solids and gas together, thus undifferentiated matter is suitable for producing planets with stellar C/O. Disc fragmentation to clumps due to GI requires particular conditions (Meru and Bate 2010; Vorobyov 2013), and the direct collapse of gravitationally unstable clumps tends to produce rather massive objects (e.g. ≈ 5 M J planets and ≈ 60 - 70 M J brown dwarfs, see Figure 4 in Vorobyov, Zakhozhay, and Dunham 2013) at larger radial distances (> 10 - 100 au, see Vorobyov, Zakhozhay, and Dunham 2013; Kratter and Lodato 2016). GI can also assist the assemblage of planetary cores (Nayakshin 2010a, 2010b; Nayakshin, Helled, and Boley 2014; Vorobyov and Elbakyan 2019). A planet formed through core accretion can also accrete planetesimals, which can be covered with ice, and enrich the atmosphere with oxygen, making the C/O ratio close to the initial stellar value. There are particular exoplanets, where lower than stellar C/O ratio is observed in the atmosphere, such as β Pic b (GRAVITY Collaboration et al. 2020; Worthen et al. 2024), HD 209458 b (Xue et al. 2024), or HD 189733b (Fu et al. 2024). Such planets need even more enrichment in ices with low C/O, which makes the regions with low C/O in the ice also more attractive sites for planet formation.</text>
<text><location><page_19><loc_7><loc_6><loc_49><loc_33></location>Gravitational instability implies that the planet forms from a mix of gas and dust (Bodenheimer 1974), this is why it is suitable to explain the formation of planets with solar, or unaltered C/O ratios. In our modelling, GI would be associated with the total C/O ratio, which we find to be significantly variable, too. For GI to form a planet with a primordial C/O ratio, it has to occur during the first 100 kyr after the disc formation. At later times, the total C/O ratio changes, and the only region with the primordial C/O ratio is the very outer disc parts, at > 100 au, which is the part of a protoplanetary disc, where conditions for GI are the most consistent with the observed properties of these objects (Rafikov 2005). Planet formation through GI is indeed more likely at earlier evolutionary stages, when gas surface density is higher (Armitage 2010). We can highlight the areas where planet formation via GI is possible in our modelling as the regions where Q Toomre ≤ 1. They are shown in the upper panel of Figure 9 for model M1. These regions appear between ≈ 10-100au before ≈ 300 kyr.</text>
<text><location><page_19><loc_52><loc_80><loc_93><loc_92></location>However, at later times, clumps could appear in the disc as a result of an external perturbation, such as a stellar flyby (Thies et al. 2010). In this case, the planet would be formed from the material with altered C/O ratio, most probably with elevated amount of carbon, as the regions outside 5 - 10 au are typically more gravitationally unstable. This means that GI can produce planets with super-solar C/O ratios, if it is induced by external influence at later stages of disc evolution.</text>
<text><location><page_19><loc_52><loc_30><loc_94><loc_80></location>Core accretion is another most widely discussed scenario of giant planet formation. Accretion of gas should produce atmospheres with the C/O ratio close to the one in the gas phase of protoplanetary disc. However, dust grains are also accreted, so pebble and planetesimal accretion can enrich the atmosphere in volatile components (Mordasini et al. 2016; Danti, Bitsch, and Mah 2023). This makes atmospheric C/O ratio closer to the ice-phase C/O, but in case of gas giants, the amount of the solids needed to compensate the prevalence of carbon in the gas should be quite high, up to hundreds of Earth masses (GRAVITY Collaboration et al. 2020). The C/O ratios in exoplanetary atmospheres are often interpreted in terms of pebble accretion, so planets with stellar C/O ratios are assumed to form in the environments where solid phase C/O is unprocessed and thus close to the initial value. One of such locations is beyond CO snowline, where most of the carbon- and oxygen- bearing material is in the ice, e.g. for Jupiter (e.g. Öberg and Wordsworth 2019; Ohno and Ueda 2021). In our models, this is rather the vicinity of the CO 2 snowlines, and the pebbles beyond the CO snowline are mostly covered with carbon-rich CO ice (Topchieva et al. 2024). Additionally, the CO snowline is typically very far from the star (> 40 au), so such scenarios must rely on planet migration to obtain their current location. Interpretations relying on chemical modelling extend this region to include the area beyond CO 2 snowline due to additional chemical processing of CO in this region, e.g. HR 8799e (P. Mollière et al. 2020). This puts milder constraints on the original distance from the star where pebbles should be accreted and requires less migration. Modelling of planet formation and migration including pebble and gas accretion puts the formation location of planets with super-solar C/O ratios beyond water and CO 2 snowlines (Bitsch, Schneider, and Kreidberg 2022).</text>
<text><location><page_19><loc_52><loc_12><loc_94><loc_30></location>In our modelling, the C/O in the ice is close to initial value in the regions beyond CO snowline, excluding the area of CO accumulation. Between CO and CO 2 snowlines, it is lower, as our model does not include chemical processes apart from adsorption and desorption. However, there is another region with C/O in the ice close to initial. The vicinity of primary water snowline and the ring induced outside of it has values of C/O in the ice only slightly above the initial value of 0.34. It is surrounded by the snowlines of CO 2 . This region could be another favourable location for forming planets with the stellar C/O. As it is situated closer to the star, it would imply less migration.</text>
<text><location><page_19><loc_52><loc_6><loc_93><loc_12></location>Rare planets with lower than stellar C/O ratio, such as β Pic b (GRAVITY Collaboration et al. 2020; Reggiani et al. 2024), HD 209458 b (Xue et al. 2024), HD 189733b (Fu et al. 2024), or KELT-1 b, Kepler-13A b and WASP-79 b</text>
<text><location><page_20><loc_7><loc_71><loc_48><loc_92></location>(less precisely determined, see Hoch et al. 2023), need to have accreted a lot of oxygen-rich ice. Therefore, they are more likely to accrete solid material in the regions with the lowest ice-phase C/O ratios. The most suitable region would be at the distances between H 2 O and CO 2 snowlines, where ice mantles are made of pure water. However, in our modelling results, this region is very small, typically only a few au wide, as the snowlines are close to each other. This is because of steep temperature profile in this region, which is a result of the significant contribution of non-irradiation heating mechanisms, particularly viscous heating. Beyond CO 2 snowline, there are also regions with relatively low (0.2 - 0.3) C/O in the ice, but much more solids need to be accreted in such areas to compensate for the excess of carbon from the gas.</text>
<text><location><page_20><loc_7><loc_34><loc_48><loc_71></location>Let us summarise the above constraints on planet formation locations and mechanisms implied by our simulated C/O ratios. Core accretion is suitable for forming planets with high C/O ( ≈ 1) in the atmosphere around the snowlines of CO, CH 4 and CO 2 , or anywhere beyond CO 2 snowline if they did not accrete much solids. Planets with stellar or slightly superstellar C/O ratio need to accrete (a lot of ) oxygen-rich solids to compensate their initially high C/O inherited from the gas. The locations where this is possible is between CO 2 and CH 4 and between CO and CH 4 snowlines. Planets with low C/O ratio could accrete ices between H 2 O and CO 2 snowlines. Snowlines are favourable planetesimal formation sites, so the planets that accreted planetesimals/pebbles there can have altered C/O ratios. The values will be lower than the initial if they form at the water snowline, and higher if they form at the snowlines of carbon-rich species. At the same time, to obtain planets with stellar C/O formed at the snowlines, these planets would need to migrate and accrete matter in different regions of the disc to make their C/O ratio close to the initial stellar value. Alternatively, planets with stellar C/O ratio can form via disc fragmentation through GI at earlier stages. Dedicated modelling of planet formation accounting for evolution of dust and volatiles is necessary to put more particular constraints on planet formation scenarios.</text>
<text><location><page_20><loc_7><loc_6><loc_48><loc_34></location>Planetesimals play an important role in delivering the icephase elements to planetary atmospheres. To form planetesimals, additional physical process is needed, such as streaming instability (SI, see Youdin and Goodman 2005), which is not explicitly included in our modelling because of insufficient numerical resolution and simplified vertical disc structure. However, we can post-process the simulation results to check if the conditions for SI are fulfilled in some regions of the disc where dust-to-gas ratio and dust size are enhanced, following Vorobyov et al. (2024). Dense dust rings forming at later stages (see Section 3.1) seem to be an ideal location for triggering the SI, which would ultimately lead to formation of planetesimals and then planets in the disc. Triggering the SI requires specific relations between local dust-to-gas ratio and Stokes number (Yang, Johansen, and Carrera 2017). The criteria vary depending on the model, we adopt them from Li and Youdin (2021). Another criteria would be the requirement of volume density of dust to exceed that of gas in the midplane (Youdin and Goodman 2005). We do not apply it, as in our modelling,</text>
<figure>
<location><page_20><loc_52><loc_60><loc_92><loc_89></location>
<caption>Figure 9. The disc regions where the conditions for GI and SI are fulfilled in model M1. The regions and times where there is no instability are shaded in white. In the upper panel, the colour indicates the minimum value of Q Toomre at a given radius, if Q Toomre ≤ 1 . In the lower panel, the colour indicates the fraction of mass at a given radius where SI can be triggered according to Li and Youdin (2021) criterion. Positions of the snowlines are shown for reference in dashed lines.</caption>
</figure>
<figure>
<location><page_20><loc_52><loc_15><loc_91><loc_44></location>
<caption>Figure 10. Distribution of C/O ratios in the regions where gravitational and streaming instabilities are triggered. For GI, total C/O ratio is shown, for SI, the C/O ratios in the ice and in the gas. Black and grey points show the observed C/O ratios in two populations of exoplanets, the data is adopted from Hoch et al. (2023).</caption>
</figure>
<text><location><page_21><loc_7><loc_80><loc_49><loc_92></location>the residual value of α is 10 -3 which makes this condition unreachable outside of the dead zone. The regions in model M1 where the conditions of Li and Youdin (2021) are satisfied are shown in the lower panel of Figure 9. Most of the suitable regions are in the inner disc ( r < 20 au) inside the dust rings, and appear after 200 kyr. However, there are some suitable regions between 10 - 100 au at earlier times, where SI could be triggered in the spirals.</text>
<text><location><page_21><loc_7><loc_48><loc_49><loc_80></location>The regions where GI and SI are possible shown in Figure 9 are separated in space and time, and they have different characteristic C/O ratios. We can sum up all the volatiles in these regions (throughout the disc lifetime) to assess typical C/O ratios of the planet-forming material. For GI, we exclude the pre-disc phase ( t < 53 kyr) and consider the total C/O ratio, assuming both gas and solids are included in the forming planet. For SI, we separate the gas- and ice-phase C/O ratios. The formed planetesimals would only include the ices, however, if they form the planetary cores, these cores would also accrete gas. We note that the composition of the rocks, which are typically carbon-rich, is not included in our assessment. The resulting distributions of the C/O ratios in planet-forming regions are shown in Figure 10. For GI regions, the C/O distribution has a distinct and relatively narrow peak around 0.5. It is slightly higher than the initial value of 0.34. For SI regions, the ice-phase C/O is below 0.5, with major peaks at 0 and ≈ 0.2, and the gas-phase C/O has a broad distribution with multiple peaks between ≈ 0.2 - 1.4. Distributions of C/O ratios in the regions where GI and SI can be triggered are noticeably different.</text>
<text><location><page_21><loc_7><loc_7><loc_49><loc_48></location>It was shown by Hoch et al. (2023) that there are two different populations of C/O ratios observed in giant exoplanets. They find that directly imaged exoplanets have C/O ≈ 0.5-0.8, while transiting hot Jupiters have a wider variety of C/O ratios ( ≈ 0.3-1.7, see Figures 12 and 13 in Hoch et al. 2023), and suggest that these two populations could have different formation pathways. We add the C/O data of exoplanetary atmospheres compiled in Table 3 of Hoch et al. (2023) to Figure 10 (with arbitrary position at the y -axis). The narrow distribution of C/O ratios in the regions with GI is in step with the distribution of directly imaged exoplanets, albeit with a slightly shifted value due to our assumed initial conditions, while the wide range of C/O values in the regions of SI matches the variety of C/O ratios in transiting exoplanets. This could suggest that directly imaged exoplanets could form as a result of gravitational instability, which is also in line with their typically higher masses and orbital separations. At the same time, the transiting hot Jupiters could have experienced a lot of migration (Lin, Bodenheimer, and Richardson 1996; Dawson and Johnson 2018), during which they accrete material with a variety of C/O ratios both from the gas and solid phase. It suggests that they could also form in the core accretion scenario. The origin of wide separation planets was also investigated by Bergin et al. (2024), based on the comparison with the observed C/O > 1 in protoplanetary discs (including the full composition of solids). They conclude that both core accretion and gravitational instability can work as the formation mechanism of these planets.</text>
<text><location><page_21><loc_10><loc_6><loc_49><loc_7></location>Apart from (exo)planets, the C/O ratios can be measured</text>
<text><location><page_21><loc_52><loc_39><loc_94><loc_92></location>for the comets, which present the best preserved sample of the primordial composition of the ices in the Solar System. Spectroscopic measurements of molecular composition in the comae suggest that the C/O ratio of cometary ice is quite low, typically below 0.1 due to the dominance of water ice (A'Hearn et al. 2012; Seligman et al. 2022; Harrington Pinto et al. 2022). Although most comets are carbon-depleted, there are individual measurements of C/O in comets above 0.5, for example in C/2006 W3 Christensen and 29P/SchwassmannWachmann (Ootsubo et al. 2012; Seligman et al. 2022), or even close to 1 in C/2016 R2 (PanSTARRS) (Wierzchos and Womack 2018; McKay et al. 2019). Additionally, high value of C/O ≈ 1 was observed in the interstellar object 2I/Borisov (Bodewits et al. 2020). Our modelling results show the icephase C/O = 0 in the vicinity of the water snowline, as well as low values between the CO 2 and CH 4 snowlines ( ≈ 0.2) and between the CH 4 and CO snowlines ( ≈ 0.3). These are the locations where comets could originate from. However, in the vicinity of the CO 2 , CH 4 and CO ice lines themselves, the C/O ratio in the ice phase is much higher. The fact that carbon-rich cometary ices are extremely rare in the Solar System may indicate that planetesimals formed on ice lines from carbon-rich volatiles do not persist throughout the evolution of a planetary system. This means that they are likely to be included in larger bodies, which favours the snowline-aided planet formation scenarios (Drążkowska and Alibert 2017; Hyodo et al. 2021). This is also consistent with the abundance of exoplanets with high C/O (Weiner Mansfield et al. 2024), which could be formed around the snowlines of carbon-rich species. In the giant planets of the Solar System, the C/O ratios are not well constrained (Mousis, Cavalié, et al. 2024). However, the existing data suggest rather super-solar values for all giant planets except for Neptune (Cavalié et al. 2024); for Jupiter, the C/O ratio is assessed as ≈ 0.9 (Wong et al. 2004; Li et al. 2024).</text>
<text><location><page_21><loc_52><loc_10><loc_94><loc_39></location>Our model only considers four most abundant chemical species. However, there can be other more complex molecules in protoplanetary discs, which could affect the balance of carbon and oxygen. The most obvious candidate is methanol CH 3 OH, which has the abundance similar to methane in the protostellar cores (Karin I. Öberg et al. 2011). It was also observed in a protoplanetary disc around an erupting star V883 Ori (Lee et al. 2019). We do not consider it in the model as its binding energy is close to that of water, thus the snowlines would have similar positions, but the abundance is an order of magnitude lower. However, it could somewhat increase the local C/O ratio in the inner regions where there are no other carbon-bearing species, such as the ice in the ring at 1 au. Including methanol would alter the distribution of the C/O ratio. Interactions between the ices considered in the model could also affect the results. As was recently shown by Ligterink, Kipfer, and Gavino (2024), trapping of volatile species inside the mantles of less volatile ices could have a significant impact on the distribution of C/O ratios.</text>
<text><location><page_21><loc_52><loc_6><loc_94><loc_10></location>Another important process missing in our modelling is gas-phase and surface chemical reactions. They could significantly affect the distribution of C/O ratio in the gas and in</text>
<text><location><page_22><loc_7><loc_68><loc_48><loc_92></location>the ice, particularly with high level of cosmic ray ionisation (Eistrup, Walsh, and van Dishoeck 2016) or if carbon grain destruction is considered (Cridland, Eistrup, and van Dishoeck 2019). One particular mechanism is the transformation of CO to CO 2 on the surface of dust grains, which can lead to the depletion of CO from both gas and ice phases between CO and CO 2 snowlines (Molyarova et al. 2017; Bosman, Tielens, and van Dishoeck 2018). Considering this mechanism can change the conclusions about planet formation location (P. Mollière et al. 2020). Nevertheless, radial variations of the C/O ratio are necessary to explain molecular emission of discs with gaps (Leemker et al. 2024), and they can only be result of dust dynamics. In order to more consistently describe the distribution of molecules and elements in the disc, the models combining dust evolution and dynamics with more complex chemistry treatment are necessary.</text>
<text><location><page_22><loc_7><loc_36><loc_48><loc_68></location>Our simulations adopt the thin-disc approximation and focus on the midplane of the protoplanetary discs, in order to capture the essential physics of self-gravity, thermal balance, and dust evolution in a global modelling within reasonable computational times. This means that some relevant processes connected with the vertical structure are inevitably excluded. For example, vertical mixing and dust settling affect the C/O ratio in the upper layers of the disc (Krijt et al. 2018; Krijt et al. 2020). Dust settling is implicitly included in our modelling through separate scale heights of drown dust and gas (as well as small dust), affecting dust number density in the midplane. However, this approach does not allow to reproduce vertical stratification in dust properties and chemical composition, which is particularly relevant for the interpretation of molecular observations. Vertical structure is also relevant for the accretion of matter on forming giant planets, which should proceed in 3D manner through meridional flows (Morbidelli et al. 2014). Cridland, Bosman, and van Dishoeck (2020) showed that the C/O ratio in the atmospheres of giant planets is rather affected by the composition of the molecular layer than that of the midplane.</text>
<section_header_level_1><location><page_22><loc_7><loc_33><loc_18><loc_34></location>5. Conclusions</section_header_level_1>
<text><location><page_22><loc_7><loc_20><loc_48><loc_32></location>In this work, we studied the distribution of volatiles in a viscous self-gravitating protoplanetary disc with dust evolution using a thin-disc hydrodynamic code FEOSAD (Vorobyov et al. 2018; Molyarova et al. 2021). We calculated the C/O elemental ratio in the gas, in the ice, and in total, identified the key properties of the distribution of elements over 500 kyr of disc evolution and considered their implications for planet formation theory. Our main findings can be summarised as follows.</text>
<unordered_list>
<list_item><location><page_22><loc_8><loc_6><loc_48><loc_19></location>· The simulated C/O ratios in the regions where GI and SI conditions are fulfilled are consistent with the C/O ratios in two populations of exoplanets possibly formed in different mechanisms pointed out by Hoch et al. (2023). We show that narrow C/O distribution of directly imaged planets is consistent with their formation via gravitational instability, while a variety of C/O in transiting hot Jupiters is in line with their migration through varying C/O conditions after the formation via either core accretion or GI.</list_item>
<list_item><location><page_22><loc_52><loc_80><loc_93><loc_92></location>· The lower C/O ratio in the ice between the CO 2 , CH 4 and COsnowlines is consistent with the typical composition of Solar System comets, while the higher value of C/O ≈ 0.51 at these snowlines corresponds to the composition of rare carbon-rich comets. This may indicate that matter from the snowlines is hardly preserved during the evolution of the disc and planetary system, possibly due to the inclusion in to planets.</list_item>
<list_item><location><page_22><loc_52><loc_66><loc_93><loc_80></location>· The distribution of volatiles is affected by the disc substructures, such as rings and spirals, as well as by dust radial drift. Variations of physical conditions create multiple snowlines of CO 2 and H 2 O inside 10 au. Dust drift of icy grains brings the volatiles from the outer to the inner disc, enriching the inner disc with both C and O. It also creates a radial gradient of the total C/O ratio: its value is around 0.2 where water is not frozen, and 0.6 - 0.9 where it is icy, compared to the initial value of 0.34.</list_item>
<list_item><location><page_22><loc_52><loc_51><loc_93><loc_66></location>· Volatiles accumulate at their snowlines in both ice and gas phases due to the combined effect of dust drift and azimuthal variations of gas and dust radial velocities in a self-gravitating, non-axisymmetric disc. The species with low initial abundances, such as CH 4 (or methanol not considered here), can significantly affect C/O ratio, as their accumulation at the snowline creates a bump in C/O in all phases above 1.0. The total mass of the model affects the timescales and the magnitude of the accumulation by a factor of two.</list_item>
<list_item><location><page_22><loc_52><loc_39><loc_93><loc_51></location>· Forming planets can accrete gas with C/O > 1 beyond CO 2 snowline, ices with C/O ≈ 0.5-1 at the CO, CH 4 and CO 2 snowlines, ices with C/O ≈ 0.2 - 0.3 between these snowlines and ices with C/O = 0 between H 2 Oand CO 2 snowlines. Planets with stellar C/O would need to migrate through these regions to acquire necessary composition or form via GI at earlier stages from the mixture of gas and dust with unaltered C/O ratio.</list_item>
<list_item><location><page_22><loc_52><loc_31><loc_93><loc_39></location>· Dust-to-gas mass ratio and the total C/O ratio are systematically anticorrelated, because in dust-rich regions the volatile composition is close to that of the ice (which is lower), and in dust-poor regions, gas determines the C/O ratio.</list_item>
</unordered_list>
<text><location><page_22><loc_51><loc_17><loc_93><loc_30></location>The connection between protoplanetary disc components and exoplanets based on their composition should be more thoroughly investigated in the models focused on the planet formation process. We emphasise that these models should also take into account the effect of dust evolution and dynamics on the distribution of the elements in the planet-forming material. Inclusion of chemical processes and more accurate consideration of the bulk composition of dust grains could also affect the C/O ratios of the planet-forming environment.</text>
<section_header_level_1><location><page_22><loc_51><loc_13><loc_65><loc_15></location>Acknowledgement</section_header_level_1>
<text><location><page_22><loc_51><loc_6><loc_93><loc_13></location>Weare thankful to the anonymous referee for useful comments that helped to improve the manuscript. The computational results presented have been achieved using the Vienna Scientific Cluster (VSC) and the local computing facility of the Southern Federal University.</text>
<text><location><page_23><loc_7><loc_87><loc_49><loc_92></location>Funding Statement The work is supported by Russian Science Foundation grant 22-72-10029, https://rscf.ru/project/2272-10029/</text>
<text><location><page_23><loc_7><loc_84><loc_27><loc_85></location>Competing Interests None</text>
<text><location><page_23><loc_7><loc_77><loc_49><loc_82></location>Data Availability Statement The data underlying this article will be shared on reasonable request to the corresponding author.</text>
<section_header_level_1><location><page_23><loc_7><loc_74><loc_15><loc_75></location>References</section_header_level_1>
<text><location><page_23><loc_7><loc_69><loc_49><loc_74></location>A'Hearn, Michael F., Lori M. Feaga, H. Uwe Keller, Hideyo Kawakita, Donald L. Hampton, Jochen Kissel, Kenneth P. Klaasen, et al. 2012. Cometary Volatiles and the Origin of Comets. ApJ 758, no. 1 (October): 29. https: //doi.org/10.1088/0004-637X/758/1/29.</text>
<text><location><page_23><loc_7><loc_64><loc_49><loc_69></location>Aikawa, Yuri, Shoken M. Miyama, Takenori Nakano, and Toyoharu Umebayashi. 1996. Evolution of Molecular Abundance in Gaseous Disks around Young Stars: Depletion of CO Molecules. ApJ 467 (August): 684. https://doi.org/10.1086/177644.</text>
<text><location><page_23><loc_7><loc_57><loc_49><loc_63></location>Akimkin, Vitaly, Eduard Vorobyov, Yaroslav Pavlyuchenkov, and Olga Stoyanovskaya. 2020. Gravitoviscous protoplanetary discs with a dust component - IV. Disc outer edges, spectral indices, and opacity gaps. MNRAS 499, no. 4 (December): 5578-5597. https://doi.org/10.1093/mnras/ staa3134. arXiv: 2010.06566 [astro-ph.EP] .</text>
<text><location><page_23><loc_7><loc_55><loc_38><loc_56></location>Armitage, Philip J. 2010. Astrophysics of Planet Formation.</text>
<text><location><page_23><loc_7><loc_50><loc_49><loc_54></location>Armitage, Philip J., Mario Livio, and J. E. Pringle. 2001. Episodic accretion in magnetically layered protoplanetary discs. MNRAS 324, no. 3 (June): 705-711. https://doi.org/10.1046/j.1365-8711.2001.04356.x. arXiv: astro-ph/0101253 [astro-ph] .</text>
<text><location><page_23><loc_7><loc_41><loc_49><loc_49></location>Audard, M., P. Ábrahám, M. M. Dunham, J. D. Green, N. Grosso, K. Hamaguchi, J. H. Kastner, et al. 2014. Episodic Accretion in Young Stars. In Protostars and planets vi, edited by Henrik Beuther, Ralf S. Klessen, Cornelis P. Dullemond, and Thomas Henning, 387-410. January. https: / / doi . org / 10 . 2458 / azu _ uapress _ 9780816531240 - ch017. arXiv: 1401.3368 [astro-ph.SR] .</text>
<text><location><page_23><loc_7><loc_35><loc_49><loc_41></location>Bai, Xue-Ning, and James M. Stone. 2013. Wind-driven Accretion in Protoplanetary Disks. I. Suppression of the Magnetorotational Instability and Launching of the Magnetocentrifugal Wind. ApJ 769, no. 1 (May): 76. https://doi.org/10.1088/0004-637X/769/1/76. arXiv: 1301.0318 [astro-ph.EP] .</text>
<text><location><page_23><loc_7><loc_27><loc_49><loc_34></location>Banzatti, Andrea, Ilaria Pascucci, Arthur D. Bosman, Paola Pinilla, Colette Salyk, Gregory J. Herczeg, Klaus M. Pontoppidan, et al. 2020. Hints for Icy Pebble Migration Feeding an Oxygen-rich Chemistry in the Inner Planet-forming Region of Disks. ApJ 903, no. 2 (November): 124. https://doi.org/10.3847/1538-4357/abbc1a. arXiv: 2009.13525 [astro-ph.EP] .</text>
<text><location><page_23><loc_7><loc_19><loc_49><loc_26></location>Benneke, Björn, Heather A. Knutson, Joshua Lothringer, Ian J. M. Crossfield, Julianne I. Moses, Caroline Morley, Laura Kreidberg, et al. 2019. A sub-Neptune exoplanet with a low-metallicity methane-depleted atmosphere and Mie-scattering clouds. Nature Astronomy 3 (July): 813-821. https://doi.org/10.1038/s41550- 019- 0800- 5. arXiv: 1907.00449 [astro-ph.EP] .</text>
<text><location><page_23><loc_7><loc_13><loc_49><loc_18></location>Bergin, Edwin A., Richard A. Booth, Maria Jose Colmenares, and John D. Ilee. 2024. C/O Ratios and the formation of wide separation exoplanets. arXiv e-prints (June): arXiv:2406.12037. https://doi.org/10.48550/arXiv. 2406.12037. arXiv: 2406.12037 [astro-ph.EP] .</text>
<text><location><page_23><loc_7><loc_7><loc_49><loc_13></location>Bergin, Edwin A., Fujun Du, L. Ilsedore Cleeves, G. A. Blake, K. Schwarz, R. Visser, and K. Zhang. 2016. Hydrocarbon Emission Rings in Protoplanetary Disks Induced by Dust Evolution. ApJ 831, no. 1 (November): 101. https://doi.org/10.3847/0004-637X/831/1/101. arXiv: 1609.06337 [astro-ph.EP] .</text>
<text><location><page_23><loc_52><loc_90><loc_84><loc_91></location>Binney, James, and Scott Tremaine. 1987. Galactic dynamics.</text>
<text><location><page_23><loc_52><loc_86><loc_94><loc_90></location>Birnstiel, Tilman. 2023. Dust growth and evolution in protoplanetary disks. arXiv e-prints (December): arXiv:2312.13287. https://doi.org/10.48550/ arXiv.2312.13287. arXiv: 2312.13287 [astro-ph.EP] .</text>
<text><location><page_23><loc_52><loc_80><loc_94><loc_85></location>Bisschop, S. E., H. J. Fraser, K. I. Öberg, E. F. van Dishoeck, and S. Schlemmer. 2006. Desorption rates and sticking coefficients for CO and N2 interstellar ices. A&A 449, no. 3 (April): 1297-1309. https://doi.org/10. 1051/0004-6361:20054051. arXiv: astro-ph/0601082 [astro-ph] .</text>
<text><location><page_23><loc_52><loc_74><loc_93><loc_80></location>Bitsch, Bertram, Aaron David Schneider, and Laura Kreidberg. 2022. How drifting and evaporating pebbles shape giant planets. III. The formation of WASP-77A b and τ Boötis b. A&A 665 (September): A138. http s : / / doi . org / 10 . 1051 / 0004 - 6361 / 202243345. arXiv: 2207 . 06077 [astro-ph.EP] .</text>
<text><location><page_23><loc_52><loc_69><loc_94><loc_73></location>Bodenheimer, P. 1974. Calculations of the Early Evolution of Jupiter. Icarus 23, no. 3 (November): 319-325. https : / / doi . org / 10 . 1016 / 0019 1035(74)90050-5.</text>
<text><location><page_23><loc_52><loc_63><loc_94><loc_69></location>Bodewits, D., J. W. Noonan, P. D. Feldman, M. T. Bannister, D. Farnocchia, W. M. Harris, J. -Y. Li, K. E. Mandt, J. Wm. Parker, and Z. -X. Xing. 2020. The carbon monoxide-rich interstellar comet 2I/Borisov. Nature Astronomy 4 (April): 867-871. https://doi.org/10.1038/s41550-0201095-2. arXiv: 2004.08972 [astro-ph.EP] .</text>
<text><location><page_23><loc_52><loc_57><loc_94><loc_62></location>Booth, R. A., and J. D. Ilee. 2019. Planet-forming material in a protoplanetary disc: the interplay between chemical evolution and pebble drift. MNRAS 487, no. 3 (August): 3998-4011. https://doi.org/10.1093/mnras/stz1488. arXiv: 1905.12639 [astro-ph.EP] .</text>
<text><location><page_23><loc_52><loc_52><loc_94><loc_56></location>Booth, Richard A., Cathie J. Clarke, Nikku Madhusudhan, and John D. Ilee. 2017. Chemical enrichment of giant planets and discs due to pebble drift. MNRAS 469, no. 4 (August): 3994-4011. https://doi.org/10.1093/ mnras/stx1103. arXiv: 1705.03305 [astro-ph.EP] .</text>
<text><location><page_23><loc_52><loc_47><loc_93><loc_51></location>Bosman, A. D., A. J. Cridland, and Y. Miguel. 2019. Jupiter formed as a pebble pile around the N2 ice line. A&A 632 (December): L11. https://doi.org/ 10.1051/0004-6361/201936827. arXiv: 1911.11154 [astro-ph.EP] .</text>
<text><location><page_23><loc_52><loc_41><loc_94><loc_47></location>Bosman, Arthur D., Alexander G. G. M. Tielens, and Ewine F. van Dishoeck. 2018. Efficiency of radial transport of ices in protoplanetary disks probed with infrared observations: the case of CO2. A&A 611 (April): A80. https://doi.org/10.1051/0004-6361/201732056. arXiv: 1712.03989 [astro-ph.EP] .</text>
<text><location><page_23><loc_52><loc_35><loc_93><loc_40></location>Brown, Paul D., and S. B. Charnley. 1990. Chemical models of interstellar gas-grain processes. I. Modelling and the effect of accretion on gas abundances and mantle composition in dense clouds. MNRAS 244 (June): 432.</text>
<text><location><page_23><loc_52><loc_28><loc_94><loc_34></location>Brown-Sevilla, S. B., M. Keppler, M. Barraza-Alfaro, J. D. Melon Fuksman, N. Kurtovic, P. Pinilla, M. Feldt, et al. 2021. A multiwavelength analysis of the spiral arms in the protoplanetary disk around WaOph 6. A&A 654 (October): A35. https://doi.org/10.1051/0004-6361/202140783. arXiv: 2107.13560 [astro-ph.EP] .</text>
<text><location><page_23><loc_52><loc_23><loc_94><loc_28></location>Carrera, Daniel, Anders Johansen, and Melvyn B. Davies. 2015. How to form planetesimals from mm-sized chondrules and chondrule aggregates. A&A 579 (July): A43. https://doi.org/10.1051/0004-6361/201425120. arXiv: 1501.05314 [astro-ph.EP] .</text>
<text><location><page_23><loc_52><loc_16><loc_94><loc_22></location>Cavalié, Thibault, Jonathan Lunine, Olivier Mousis, and Ricardo Hueso. 2024. The Deep Oxygen Abundance in Solar System Giant Planets, with a New Derivation for Saturn. Space Sci. Rev. 220, no. 1 (January): 8. https://doi.org/10.1007/s11214-024-01045-6. arXiv: 2407.07515 [astro-ph.EP] .</text>
<text><location><page_23><loc_52><loc_9><loc_94><loc_15></location>Changeat, Q., B. Edwards, A. F. Al-Refaie, A. Tsiaras, J. W. Skinner, J. Y. K. Cho, K. H. Yip, et al. 2022. Five Key Exoplanet Questions Answered via the Analysis of 25 Hot-Jupiter Atmospheres in Eclipse. ApJS 260, no. 1 (May): 3. https://doi.org/10.3847/1538-4365/ac5cc2. arXiv: 2204.11729 [astro-ph.EP] .</text>
<text><location><page_24><loc_7><loc_85><loc_48><loc_91></location>Cleeves, L. Ilsedore, Karin I. Öberg, David J. Wilner, Jane Huang, Ryan A. Loomis, Sean M. Andrews, and V. V. Guzman. 2018. Constraining Gasphase Carbon, Oxygen, and Nitrogen in the IM Lup Protoplanetary Disk. ApJ 865, no. 2 (October): 155. https://doi.org/10.3847/15384357/aade96. arXiv: 1808.10682 [astro-ph.SR] .</text>
<text><location><page_24><loc_7><loc_81><loc_48><loc_85></location>Connelley, Michael S., and Bo Reipurth. 2018. A Near-infrared Spectroscopic Survey of FU Orionis Objects. ApJ 861, no. 2 (July): 145. https://doi. org/10.3847/1538-4357/aaba7b. arXiv: 1806.08880 [astro-ph.SR] .</text>
<text><location><page_24><loc_7><loc_74><loc_48><loc_80></location>Cridland, Alex J., Ewine F. van Dishoeck, Matthew Alessi, and Ralph E. Pudritz. 2019. Connecting planet formation and astrochemistry. A main sequence for C/O in hot exoplanetary atmospheres. A&A 632 (December): A63. https://doi.org/10.1051/0004- 6361/201936105. arXiv: 1910.13171 [astro-ph.EP] .</text>
<text><location><page_24><loc_10><loc_69><loc_48><loc_74></location>. 2020. Connecting planet formation and astrochemistry. C/Os and N/Os of warm giant planets and Jupiter analogues. A&A 642 (October): A229. https://doi.org/10.1051/0004-6361/202038767. arXiv: 2009. 02907 [astro-ph.EP] .</text>
<text><location><page_24><loc_7><loc_63><loc_48><loc_68></location>Cridland, Alexander J., Arthur D. Bosman, and Ewine F. van Dishoeck. 2020. Impact of vertical gas accretion on the carbon-to-oxygen ratio of gas giant atmospheres. A&A 635 (March): A68. https://doi.org/10.1051/ 0004-6361/201936858. arXiv: 2001.05808 [astro-ph.EP] .</text>
<text><location><page_24><loc_7><loc_57><loc_48><loc_63></location>Cridland, Alexander J., Christian Eistrup, and Ewine F. van Dishoeck. 2019. Connecting planet formation and astrochemistry. Refractory carbon depletion leading to super-stellar C/O in giant planetary atmospheres. A&A 627 (July): A127. https://doi.org/10.1051/0004-6361/201834378. arXiv: 1901.08896 [astro-ph.EP] .</text>
<text><location><page_24><loc_7><loc_51><loc_48><loc_56></location>Cuppen, H. M., C. Walsh, T. Lamberts, D. Semenov, R. T. Garrod, E. M. Penteado, and S. Ioppolo. 2017. Grain Surface Models and Data for Astrochemistry. Space Sci. Rev. 212, nos. 1-2 (October): 1-58. https: //doi.org/10.1007/s11214-016-0319-3.</text>
<text><location><page_24><loc_7><loc_46><loc_48><loc_50></location>Cuzzi, Jeffrey N., and Kevin J. Zahnle. 2004. Material Enhancement in Protoplanetary Nebulae by Particle Drift through Evaporation Fronts. ApJ 614, no. 1 (October): 490-496. https://doi.org/10.1086/423611. arXiv: astro-ph/0409276 [astro-ph] .</text>
<text><location><page_24><loc_7><loc_41><loc_48><loc_45></location>Danti, C., B. Bitsch, and J. Mah. 2023. Composition of giant planets: The roles of pebbles and planetesimals. A&A 679 (November): L7. https://doi.org/ 10.1051/0004-6361/202347501. arXiv: 2310.02886 [astro-ph.EP] .</text>
<text><location><page_24><loc_7><loc_37><loc_48><loc_41></location>Dawson, Rebekah I., and John Asher Johnson. 2018. Origins of Hot Jupiters. ARA&A 56 (September): 175-221. https://doi.org/10.1146/annurevastro-081817-051853. arXiv: 1801.06117 [astro-ph.EP] .</text>
<text><location><page_24><loc_7><loc_30><loc_48><loc_36></location>Dong, Ruobing, Eduard Vorobyov, Yaroslav Pavlyuchenkov, Eugene Chiang, and Hauyu Baobab Liu. 2016. Signatures of Gravitational Instability in Resolved Images of Protostellar Disks. ApJ 823, no. 2 (June): 141. https://doi.org/10.3847/0004-637X/823/2/141. arXiv: 1603.01618 [astro-ph.SR] .</text>
<text><location><page_24><loc_7><loc_27><loc_48><loc_30></location>Draine, B. T. 1978. Photoelectric heating of interstellar gas. ApJS 36 (April): 595-619. https://doi.org/10.1086/190513.</text>
<text><location><page_24><loc_7><loc_23><loc_48><loc_26></location>Drążkowska, J., and Y. Alibert. 2017. Planetesimal formation starts at the snow line. A&A 608 (December): A92. https://doi.org/10.1051/00046361/201731491. arXiv: 1710.00009 [astro-ph.EP] .</text>
<text><location><page_24><loc_7><loc_16><loc_48><loc_22></location>Dutrey, A., V. Wakelam, Y. Boehler, S. Guilloteau, F. Hersant, D. Semenov, E. Chapillon, et al. 2011. Chemistry in disks. V. Sulfur-bearing molecules in the protoplanetary disks surrounding LkCa15, MWC480, DM Tauri, and GO Tauri. A&A 535 (November): A104. https://doi.org/10.1051/ 0004-6361/201116931. arXiv: 1109.5870 [astro-ph.SR] .</text>
<text><location><page_24><loc_7><loc_9><loc_48><loc_15></location>Eistrup, Christian, Catherine Walsh, and Ewine F. van Dishoeck. 2016. Setting the volatile composition of (exo)planet-building material. Does chemical evolution in disk midplanes matter? A&A 595 (November): A83. htt ps://doi.org/10.1051/0004- 6361/201628509. arXiv: 1607.06710 [astro-ph.EP] .</text>
<text><location><page_24><loc_55><loc_88><loc_93><loc_91></location>. 2018. Molecular abundances and C/O ratios in chemically evolving planet-forming disk midplanes. A&A 613 (May): A14. https://doi.org/ 10.1051/0004-6361/201731302. arXiv: 1709.07863 [astro-ph.EP] .</text>
<text><location><page_24><loc_51><loc_82><loc_93><loc_87></location>Facchini, Stefano, Richard Teague, Jaehan Bae, Myriam Benisty, Miriam Keppler, and Andrea Isella. 2021. The Chemical Inventory of the Planethosting Disk PDS 70. AJ 162, no. 3 (September): 99. https://doi.org/10. 3847/1538-3881/abf0a4. arXiv: 2101.08369 [astro-ph.EP] .</text>
<text><location><page_24><loc_51><loc_77><loc_93><loc_82></location>Fedele, D., and C. Favre. 2020. Measuring elemental abundance ratios in protoplanetary disks at millimeter wavelengths. A&A 638 (June): A110. https://doi.org/10.1051/0004-6361/202037927. arXiv: 2005.03891 [astro-ph.SR] .</text>
<text><location><page_24><loc_51><loc_70><loc_93><loc_76></location>Fraser, Helen J., Mark P. Collings, Martin R. S. McCoustra, and David A. Williams. 2001. Thermal desorption of water ice in the interstellar medium. MNRAS 327, no. 4 (November): 1165-1172. https://doi. org/10.1046/j.1365-8711.2001.04835.x. arXiv: astro-ph/0107487 [astro-ph] .</text>
<text><location><page_24><loc_51><loc_63><loc_93><loc_69></location>Fu, Guangwei, Luis Welbanks, Drake Deming, Julie Inglis, Michael Zhang, Joshua Lothringer, Jegug Ih, et al. 2024. Hydrogen sulfide and metalenriched atmosphere for a Jupiter-mass exoplanet. arXiv e-prints (July): arXiv:2407.06163. https://doi.org/10.48550/arXiv.2407.06163. arXiv: 2407.06163 [astro-ph.EP] .</text>
<text><location><page_24><loc_51><loc_58><loc_93><loc_63></location>Gail, Hans-Peter, and Mario Trieloff. 2017. Spatial distribution of carbon dust in the early solar nebula and the carbon content of planetesimals. A&A 606 (September): A16. https://doi.org/10.1051/0004-6361/201730480. arXiv: 1707.07611 [astro-ph.EP] .</text>
<text><location><page_24><loc_51><loc_55><loc_93><loc_57></location>Gammie, Charles F. 1996. Layered Accretion in T Tauri Disks. ApJ 457 (January): 355. https://doi.org/10.1086/176735.</text>
<text><location><page_24><loc_51><loc_49><loc_93><loc_54></location>Gárate, Matias, Til Birnstiel, Joanna Drążkowska, and Sebastian Markus Stammler. 2020. Gas accretion damped by dust back-reaction at the snow line. A&A 635 (March): A149. https://doi.org/10.1051/00046361/201936067. arXiv: 1906.07708 [astro-ph.EP] .</text>
<text><location><page_24><loc_51><loc_42><loc_93><loc_49></location>GRAVITY Collaboration, M. Nowak, S. Lacour, P. Mollière, J. Wang, B. Charnay, E. F. van Dishoeck, et al. 2020. Peering into the formation history of β Pictoris b with VLTI/GRAVITY long-baseline interferometry. A&A 633 (January): A110. https://doi.org/10.1051/00046361/201936898. arXiv: 1912.04651 [astro-ph.EP] .</text>
<text><location><page_24><loc_51><loc_37><loc_93><loc_42></location>Gressel, Oliver, Neal J. Turner, Richard P. Nelson, and Colin P. McNally. 2015. Global Simulations of Protoplanetary Disks With Ohmic Resistivity and Ambipolar Diffusion. ApJ 801, no. 2 (March): 84. https://doi.org/ 10.1088/0004-637X/801/2/84. arXiv: 1501.05431 [astro-ph.EP] .</text>
<text><location><page_24><loc_51><loc_33><loc_93><loc_36></location>Gundlach, B., and J. Blum. 2015. The Stickiness of Micrometer-sized Waterice Particles. ApJ 798, no. 1 (January): 34. https://doi.org/10.1088/0004637X/798/1/34. arXiv: 1410.7199 [astro-ph.EP] .</text>
<text><location><page_24><loc_51><loc_27><loc_93><loc_32></location>Harrington Pinto, Olga, Maria Womack, Yanga Fernandez, and James Bauer. 2022. A Survey of CO, CO2, and H2O in Comets and Centaurs. PSJ 3, no. 11 (November): 247. https://doi.org/10.3847/PSJ/ac960d. arXiv: 2209.09985 [astro-ph.EP] .</text>
<text><location><page_24><loc_51><loc_24><loc_93><loc_26></location>Hartmann, L., and S. J. Kenyon. 1985. On the nature of FU Orionis objects. ApJ 299 (December): 462-478. https://doi.org/10.1086/163713.</text>
<text><location><page_24><loc_51><loc_20><loc_93><loc_23></location>Hasegawa, T. I., and E. Herbst. 1993. Three-Phase Chemical Models of Dense Interstellar Clouds - Gas Dust Particle Mantles and Dust Particle Surfaces. MNRAS 263 (August): 589. https://doi.org/10.1093/mnras/263.3.589.</text>
<text><location><page_24><loc_51><loc_14><loc_93><loc_19></location>Hensley, Brandon S., and B. T. Draine. 2021. Observational Constraints on the Physical Properties of Interstellar Dust in the Post-Planck Era. ApJ 906, no. 2 (January): 73. https://doi.org/10.3847/1538-4357/abc8f1. arXiv: 2009.00018 [astro-ph.GA] .</text>
<text><location><page_24><loc_51><loc_6><loc_93><loc_13></location>Hoch, Kielan K. W., Quinn M. Konopacky, Christopher A. Theissen, JeanBaptiste Ruffio, Travis S. Barman, Emily L. Rickman, Marshall D. Perrin, Bruce Macintosh, and Christian Marois. 2023. Assessing the C/O Ratio Formation Diagnostic: A Potential Trend with Companion Mass. AJ 166, no. 3 (September): 85. https://doi.org/10.3847/1538-3881/ace442. arXiv: 2212.04557 [astro-ph.EP] .</text>
<text><location><page_25><loc_7><loc_84><loc_49><loc_91></location>Huang, Jane, Sean M. Andrews, Cornelis P. Dullemond, Andrea Isella, Laura M. Pérez, Viviana V. Guzmán, Karin I. Öberg, et al. 2018. The Disk Substructures at High Angular Resolution Project (DSHARP). II. Characteristics of Annular Substructures. ApJ 869, no. 2 (December): L42. https : / / doi . org / 10 . 3847 / 2041 - 8213 / aaf 740. arXiv: 1812 . 04041 [astro-ph.EP] .</text>
<text><location><page_25><loc_7><loc_76><loc_49><loc_84></location>Huang, Jane, Sean M. Andrews, Laura M. Pérez, Zhaohuan Zhu, Cornelis P. Dullemond, Andrea Isella, Myriam Benisty, et al. 2018. The Disk Substructures at High Angular Resolution Project (DSHARP). III. Spiral Structures in the Millimeter Continuum of the Elias 27, IM Lup, and WaOph 6 Disks. ApJ 869, no. 2 (December): L43. https://doi.org/10. 3847/2041-8213/aaf7a0. arXiv: 1812.04193 [astro-ph.SR] .</text>
<text><location><page_25><loc_7><loc_70><loc_49><loc_76></location>Huang, Jane, Edwin A. Bergin, Romane Le Gal, Sean M. Andrews, Jaehan Bae, Luke Keyte, and J. A. Sturm. 2024. Constraints on the gas-phase C/O ratio of DR Tau's outer disk from CS, SO, and C2H observations. arXiv e-prints (July): arXiv:2407.01679. https://doi.org/10.48550/arXiv. 2407.01679. arXiv: 2407.01679 [astro-ph.EP] .</text>
<text><location><page_25><loc_7><loc_64><loc_49><loc_69></location>Hyodo, Ryuki, Tristan Guillot, Shigeru Ida, Satoshi Okuzumi, and Andrew N. Youdin. 2021. Planetesimal formation around the snow line. II. Dust or pebbles? A&A 646 (February): A14. https://doi.org/10.1051/00046361/202039894. arXiv: 2012.06700 [astro-ph.EP] .</text>
<text><location><page_25><loc_7><loc_59><loc_49><loc_63></location>Ilee, J. D., A. C. Boley, P. Caselli, R. H. Durisen, T. W. Hartquist, and J. M. C. Rawlings. 2011. Chemistry in a gravitationally unstable protoplanetary disc. MNRAS 417, no. 4 (November): 2950-2961. https://doi.org/10. 1111/j.1365-2966.2011.19455.x. arXiv: 1107.3041 [astro-ph.GA] .</text>
<text><location><page_25><loc_7><loc_53><loc_49><loc_58></location>Jiang, Haochang, and Chris W. Ormel. 2023. Efficient planet formation by pebble accretion in ALMA rings. MNRAS 518, no. 3 (January): 38773900. https://doi.org/10.1093/mnras/stac3275. arXiv: 2207.13002 [astro-ph.EP] .</text>
<text><location><page_25><loc_7><loc_48><loc_49><loc_52></location>Kadam, Kundan, Eduard Vorobyov, and Shantanu Basu. 2022. Primordial dusty rings and episodic outbursts in protoplanetary discs. MNRAS 516, no. 3 (November): 4448-4468. https://doi.org/10.1093/mnras/stac2455. arXiv: 2208.12105 [astro-ph.EP] .</text>
<text><location><page_25><loc_7><loc_41><loc_49><loc_47></location>Kadam, Kundan, Eduard Vorobyov, Zsolt Regály, Ágnes Kóspál, and Péter Ábrahám. 2019. Dynamical Gaseous Rings in Global Simulations of Protoplanetary Disk Formation. ApJ 882, no. 2 (September): 96. h ttps : / / doi . org / 10 . 3847 / 1538 - 4357 / ab378a. arXiv: 1908 . 02515 [astro-ph.SR] .</text>
<text><location><page_25><loc_10><loc_36><loc_49><loc_40></location>. 2020. Outbursts in Global Protoplanetary Disk Simulations. ApJ 895, no. 1 (May): 41. https://doi.org/10.3847/1538-4357/ab8bd8. arXiv: 2005.03578 [astro-ph.SR] .</text>
<text><location><page_25><loc_7><loc_30><loc_49><loc_36></location>Kama, M., S. Bruderer, E. F. van Dishoeck, M. Hogerheijde, C. P. Folsom, A. Miotello, D. Fedele, A. Belloche, R. Güsten, and F. Wyrowski. 2016. Volatile-carbon locking and release in protoplanetary disks. A study of TWHya and HD 100546. A&A 592 (August): A83. https://doi.org/10. 1051/0004-6361/201526991. arXiv: 1605.05093 [astro-ph.EP] .</text>
<text><location><page_25><loc_7><loc_24><loc_49><loc_29></location>Khorshid, N., M. Min, and J. M. Désert. 2023. Retrieving planet formation parameters of WASP-77Ab using SimAb. A&A 675 (July): A95. htt ps://doi.org/10.1051/0004- 6361/202245469. arXiv: 2311.15702 [astro-ph.EP] .</text>
<text><location><page_25><loc_7><loc_19><loc_49><loc_23></location>Kratter, Kaitlin, and Giuseppe Lodato. 2016. Gravitational Instabilities in Circumstellar Disks. ARA&A 54 (September): 271-311. https://doi. org/10.1146/annurev- astro- 081915- 023307. arXiv: 1603.01280 [astro-ph.SR] .</text>
<text><location><page_25><loc_7><loc_12><loc_49><loc_18></location>Krijt, Sebastiaan, Arthur D. Bosman, Ke Zhang, Kamber R. Schwarz, Fred J. Ciesla, and Edwin A. Bergin. 2020. CO Depletion in Protoplanetary Disks: A Unified Picture Combining Physical Sequestration and Chemical Processing. ApJ 899, no. 2 (August): 134. https://doi.org/10.3847/ 1538-4357/aba75d. arXiv: 2007.09517 [astro-ph.SR] .</text>
<text><location><page_25><loc_52><loc_85><loc_94><loc_91></location>Krijt, Sebastiaan, Kamber R. Schwarz, Edwin A. Bergin, and Fred J. Ciesla. 2018. Transport of CO in Protoplanetary Disks: Consequences of Pebble Formation, Settling, and Radial Drift. ApJ 864, no. 1 (September): 78. https : / / doi . org / 10 . 3847 / 1538 - 4357 / aad69b. arXiv: 1808 . 01840 [astro-ph.EP] .</text>
<text><location><page_25><loc_52><loc_81><loc_94><loc_85></location>Lambrechts, M., and A. Johansen. 2012. Rapid growth of gas-giant cores by pebble accretion. A&A 544 (August): A32. https://doi.org/10.1051/00046361/201219127. arXiv: 1205.3030 [astro-ph.EP] .</text>
<text><location><page_25><loc_52><loc_76><loc_93><loc_80></location>Lee, Eve J., J. R. Fuentes, and Philip F. Hopkins. 2022. Establishing Dust Rings and Forming Planets within Them. ApJ 937, no. 2 (October): 95. https://doi.org/10.3847/1538-4357/ac8cfe. arXiv: 2206.01219 [astro-ph.EP] .</text>
<text><location><page_25><loc_52><loc_70><loc_93><loc_75></location>Lee, Jeong-Eun, Edwin A. Bergin, and Hideko Nomura. 2010. The Solar Nebula on Fire: A Solution to the Carbon Deficit in the Inner Solar System. ApJ 710, no. 1 (February): L21-L25. https://doi.org/10.1088/ 2041-8205/710/1/L21. arXiv: 1001.0818 [astro-ph.GA] .</text>
<text><location><page_25><loc_52><loc_63><loc_94><loc_69></location>Lee, Jeong-Eun, Seokho Lee, Giseon Baek, Yuri Aikawa, Lucas Cieza, SungYong Yoon, Gregory Herczeg, Doug Johnstone, and Simon Casassus. 2019. The ice composition in the disk around V883 Ori revealed by its stellar outburst. Nature Astronomy 3 (February): 314-319. https://doi. org/10.1038/s41550-018-0680-0. arXiv: 1809.00353 [astro-ph.SR] .</text>
<text><location><page_25><loc_52><loc_57><loc_94><loc_63></location>Leemker, M., A. S. Booth, E. F. van Dishoeck, L. Wölfer, and B. Dent. 2024. Chemistry across dust and gas gaps in protoplanetary disks: modelling the co-spatial molecular rings in the HD 100546 disk. arXiv e-prints (May): arXiv:2405.10361. https://doi.org/10.48550/arXiv.2405.10361. arXiv: 2405.10361 [astro-ph.EP] .</text>
<text><location><page_25><loc_52><loc_51><loc_94><loc_56></location>Lenz, Christian T., Hubert Klahr, and Tilman Birnstiel. 2019. Planetesimal Population Synthesis: Pebble Flux-regulated Planetesimal Formation. ApJ 874, no. 1 (March): 36. https://doi.org/10.3847/1538-4357/ab05d9. arXiv: 1902.07089 [astro-ph.EP] .</text>
<text><location><page_25><loc_52><loc_44><loc_94><loc_50></location>Li, Cheng, Michael Allison, Sushil Atreya, Shawn Brueshaber, Leigh N. Fletcher, Tristan Guillot, Liming Li, et al. 2024. Super-adiabatic temperature gradient at Jupiter's equatorial zone and implications for the water abundance. Icarus 414 (May): 116028. https://doi.org/10.1016/j. icarus.2024.116028. arXiv: 2403.05363 [astro-ph.EP] .</text>
<text><location><page_25><loc_52><loc_40><loc_94><loc_44></location>Li, Rixin, and Andrew N. Youdin. 2021. Thresholds for Particle Clumping by the Streaming Instability. ApJ 919, no. 2 (October): 107. https://doi. org/10.3847/1538-4357/ac0e9f. arXiv: 2105.06042 [astro-ph.EP] .</text>
<text><location><page_25><loc_52><loc_33><loc_94><loc_39></location>Ligterink, N. F. W., K. A. Kipfer, and S. Gavino. 2024. Mind the Trap: Non-negligible effect of volatile trapping in ice on C/O ratios in protoplanetary disks and exoplanetary atmospheres. arXiv e-prints (June): arXiv:2406.16029. https://doi.org/10.48550/arXiv.2406.16029. arXiv: 2406.16029 [astro-ph.EP] .</text>
<text><location><page_25><loc_52><loc_29><loc_93><loc_33></location>Lin, D. N. C., P. Bodenheimer, and D. C. Richardson. 1996. Orbital migration of the planetary companion of 51 Pegasi to its present location. Nature 380, no. 6575 (April): 606-607. https://doi.org/10.1038/380606a0.</text>
<text><location><page_25><loc_52><loc_22><loc_94><loc_28></location>Line, Michael R., Matteo Brogi, Jacob L. Bean, Siddharth Gandhi, Joseph Zalesky, Vivien Parmentier, Peter Smith, et al. 2021. A solar C/O and sub-solar metallicity in a hot Jupiter atmosphere. Nature 598, no. 7882 (October): 580-584. https://doi.org/10.1038/s41586-021-03912-6. arXiv: 2110.14821 [astro-ph.EP] .</text>
<text><location><page_25><loc_52><loc_19><loc_93><loc_22></location>Lodders, Katharina. 2004. Jupiter Formed with More Tar than Ice. ApJ 611, no. 1 (August): 587-597. https://doi.org/10.1086/421970.</text>
<text><location><page_25><loc_52><loc_12><loc_94><loc_19></location>Long, Feng, Paola Pinilla, Gregory J. Herczeg, Daniel Harsono, Giovanni Dipierro, Ilaria Pascucci, Nathan Hendler, et al. 2018. Gaps and Rings in an ALMA Survey of Disks in the Taurus Star-forming Region. ApJ 869, no. 1 (December): 17. https://doi.org/10.3847/1538-4357/aae8e1. arXiv: 1810.06044 [astro-ph.SR] .</text>
<text><location><page_25><loc_52><loc_8><loc_94><loc_12></location>Madhusudhan, Nikku. 2012. C/O Ratio as a Dimension for Characterizing Exoplanetary Atmospheres. ApJ 758, no. 1 (October): 36. https://doi. org/10.1088/0004-637X/758/1/36. arXiv: 1209.2412 [astro-ph.EP] .</text>
<text><location><page_26><loc_7><loc_85><loc_49><loc_91></location>Madhusudhan, Nikku, Joseph Harrington, Kevin B. Stevenson, Sarah Nymeyer, Christopher J. Campo, Peter J. Wheatley, Drake Deming, et al. 2011. A high C/O ratio and weak thermal inversion in the atmosphere of exoplanet WASP-12b. Nature 469, no. 7328 (January): 64-67. https: //doi.org/10.1038/nature09602. arXiv: 1012.1603 [astro-ph.EP] .</text>
<text><location><page_26><loc_7><loc_80><loc_48><loc_85></location>Matter, A., F. C. Pignatale, and B. Lopez. 2020. Spatially resolving the chemical composition of the planet building blocks. MNRAS 497, no. 3 (September): 2540-2552. https://doi.org/10.1093/mnras/staa2137. arXiv: 2007.09385 [astro-ph.EP] .</text>
<text><location><page_26><loc_7><loc_73><loc_48><loc_79></location>McKay, Adam J., Michael A. DiSanti, Michael S. P. Kelley, Matthew M. Knight, Maria Womack, Kacper Wierzchos, Olga Harrington Pinto, et al. 2019. The Peculiar Volatile Composition of CO-dominated Comet C/2016 R2 (PanSTARRS). AJ 158, no. 3 (September): 128. https://doi. org/10.3847/1538-3881/ab32e4. arXiv: 1907.07208 [astro-ph.EP] .</text>
<text><location><page_26><loc_7><loc_67><loc_48><loc_73></location>Meru, Farzana, and Matthew R. Bate. 2010. Exploring the conditions required to form giant planets via gravitational instability in massive protoplanetary discs. MNRAS 406, no. 4 (August): 2279-2288. https: //doi.org/10.1111/j.1365- 2966.2010.16867.x. arXiv: 1004.3766 [astro-ph.EP] .</text>
<text><location><page_26><loc_7><loc_60><loc_48><loc_66></location>Meru, Farzana, Attila Juhász, John D. Ilee, Cathie J. Clarke, Giovanni P. Rosotti, and Richard A. Booth. 2017. On the Origin of the Spiral Morphology in the Elias 2-27 Circumstellar Disk. ApJ 839, no. 2 (April): L24. https://doi.org/10.3847/2041-8213/aa6837. arXiv: 1703.05338 [astro-ph.EP] .</text>
<text><location><page_26><loc_7><loc_52><loc_48><loc_59></location>Minissale, Marco, Yuri Aikawa, Edwin Bergin, Mathieu Bertin, Wendy A. Brown, Stephanie Cazaux, Steven B. Charnley, et al. 2022. Thermal Desorption of Interstellar Ices: A Review on the Controlling Parameters and Their Implications from Snowlines to Chemical Complexity. ACS Earth and Space Chemistry 6, no. 3 (March): 597-630. https://doi.org/10. 1021/acsearthspacechem.1c00357. arXiv: 2201.07512 [astro-ph.GA] .</text>
<text><location><page_26><loc_7><loc_45><loc_48><loc_51></location>Miotello, A., S. Facchini, E. F. van Dishoeck, P. Cazzoletti, L. Testi, J. P. Williams, M. Ansdell, S. van Terwisga, and N. van der Marel. 2019. Bright C2H emission in protoplanetary discs in Lupus: high volatile C/O > 1 ratios. A&A 631 (November): A69. https://doi.org/10.1051/00046361/201935441. arXiv: 1909.04477 [astro-ph.SR] .</text>
<text><location><page_26><loc_7><loc_38><loc_48><loc_44></location>Mollière, P., T. Stolker, S. Lacour, G. P. P. L. Otten, J. Shangguan, B. Charnay, T. Molyarova, et al. 2020. Retrieving scattering clouds and disequilibrium chemistry in the atmosphere of HR 8799e. A&A 640 (August): A131. https://doi.org/10.1051/0004-6361/202038325. arXiv: 2006.09394 [astro-ph.EP] .</text>
<text><location><page_26><loc_7><loc_32><loc_48><loc_38></location>Mollière, Paul, Tamara Molyarova, Bertram Bitsch, Thomas Henning, Aaron Schneider, Laura Kreidberg, Christian Eistrup, et al. 2022. Interpreting the Atmospheric Composition of Exoplanets: Sensitivity to Planet Formation Assumptions. ApJ 934, no. 1 (July): 74. https://doi.org/10.3847/ 1538-4357/ac6a56. arXiv: 2204.13714 [astro-ph.EP] .</text>
<text><location><page_26><loc_7><loc_25><loc_48><loc_31></location>Molyarova, Tamara, Vitaly Akimkin, Dmitry Semenov, Thomas Henning, Anton Vasyunin, and Dmitri Wiebe. 2017. Gas Mass Tracers in Protoplanetary Disks: CO is Still the Best. ApJ 849, no. 2 (November): 130. https://doi.org/10.3847/1538-4357/aa9227. arXiv: 1710.02993 [astro-ph.EP] .</text>
<text><location><page_26><loc_7><loc_18><loc_49><loc_24></location>Molyarova, Tamara, Eduard I. Vorobyov, Vitaly Akimkin, Aleksandr Skliarevskii, Dmitri Wiebe, and Manuel Güdel. 2021. Gravitoviscous Protoplanetary Disks with a Dust Component. V. The Dynamic Model for Freeze-out and Sublimation of Volatiles. ApJ 910, no. 2 (April): 153. https://doi.org/ 10.3847/1538-4357/abe2b0. arXiv: 2103.06045 [astro-ph.EP] .</text>
<text><location><page_26><loc_7><loc_12><loc_48><loc_18></location>Morbidelli, A., J. Szulágyi, A. Crida, E. Lega, B. Bitsch, T. Tanigawa, and K. Kanagawa. 2014. Meridional circulation of gas into gaps opened by giant planets in three-dimensional low-viscosity disks. Icarus 232 (April): 266-270. https://doi.org/10.1016/j.icarus.2014.01.010. arXiv: 1401.2925 [astro-ph.EP] .</text>
<text><location><page_26><loc_51><loc_85><loc_93><loc_91></location>Mordasini, C., R. van Boekel, P. Mollière, Th. Henning, and Björn Benneke. 2016. The Imprint of Exoplanet Formation History on Observable Present-day Spectra of Hot Jupiters. ApJ 832, no. 1 (November): 41. https://doi.org/10.3847/0004-637X/832/1/41. arXiv: 1609.03019 [astro-ph.EP] .</text>
<text><location><page_26><loc_51><loc_79><loc_93><loc_85></location>Moses, J. I., N. Madhusudhan, C. Visscher, and R. S. Freedman. 2013. Chemical Consequences of the C/O Ratio on Hot Jupiters: Examples from WASP-12b, CoRoT-2b, XO-1b, and HD 189733b. ApJ 763, no. 1 (January): 25. https://doi.org/10.1088/0004-637X/763/1/25. arXiv: 1211.2996 [astro-ph.EP] .</text>
<text><location><page_26><loc_51><loc_73><loc_93><loc_78></location>Mousis, Olivier, Sarah E. Anderson, Adrienn Luspay-Kuti, Kathleen E. Mandt, and Pierre Vernazza. 2024. Triton and Pluto: same origin but separated at birth. arXiv e-prints (June): arXiv:2406.03815. https://doi.org/10. 48550/arXiv.2406.03815. arXiv: 2406.03815 [astro-ph.EP] .</text>
<text><location><page_26><loc_51><loc_67><loc_93><loc_73></location>Mousis, Olivier, Thibault Cavalié, Jonathan I. Lunine, Kathleen E. Mandt, Ricardo Hueso, Artyom Aguichine, Antoine Schneeberger, et al. 2024. Recipes for Forming a Carbon-Rich Giant Planet. Space Sci. Rev. 220, no. 4 (June): 44. https://doi.org/10.1007/s11214-024-01071-4. arXiv: 2405.19748 [astro-ph.EP] .</text>
<text><location><page_26><loc_51><loc_60><loc_93><loc_66></location>Nasedkin, E., P. Mollière, S. Lacour, M. Nowak, L. Kreidberg, T. Stolker, J. J. Wang, et al. 2024. Four-of-a-kind? Comprehensive atmospheric characterisation of the HR 8799 planets with VLTI/GRAVITY. arXiv e-prints (April): arXiv:2404.03776. https://doi.org/10.48550/arXiv.2404. 03776. arXiv: 2404.03776 [astro-ph.EP] .</text>
<text><location><page_26><loc_51><loc_55><loc_93><loc_59></location>Nayakshin, Sergei. 2010a. Formation of planets by tidal downsizing of giant planet embryos. MNRAS 408, no. 1 (October): L36-L40. https://doi.org/ 10.1111/j.1745-3933.2010.00923.x. arXiv: 1007.4159 [astro-ph.EP] .</text>
<text><location><page_26><loc_55><loc_51><loc_93><loc_55></location>. 2010b. Grain sedimentation inside giant planet embryos. MNRAS 408, no. 4 (November): 2381-2396. https://doi.org/10.1111/j.13652966.2010.17289.x. arXiv: 1007.4162 [astro-ph.EP] .</text>
<text><location><page_26><loc_51><loc_46><loc_93><loc_50></location>Nayakshin, Sergei, Ravit Helled, and Aaron C. Boley. 2014. Core-assisted gas capture instability: a new mode of giant planet formation by gravitationally unstable discs. MNRAS 440, no. 4 (June): 3797-3808. https: //doi.org/10.1093/mnras/stu473. arXiv: 1403.1813 [astro-ph.EP] .</text>
<text><location><page_26><loc_51><loc_39><loc_93><loc_45></location>Nortmann, L., F. Lesjak, F. Yan, D. Cont, S. Czesla, A. Lavail, A. D. Rains, et al. 2024. CRIRES + transmission spectroscopy of WASP-127b. Detection of the resolved signatures of a supersonic equatorial jet and cool poles in a hot planet. arXiv e-prints (April): arXiv:2404.12363. https://doi.org/ 10.48550/arXiv.2404.12363. arXiv: 2404.12363 [astro-ph.EP] .</text>
<text><location><page_26><loc_51><loc_33><loc_93><loc_38></location>Öberg, K. I., F. van Broekhuizen, H. J. Fraser, S. E. Bisschop, E. F. van Dishoeck, and S. Schlemmer. 2005. Competition between CO and N2 Desorption from Interstellar Ices. ApJ 621, no. 1 (March): L33-L36. https://doi.org/10.1086/428901.</text>
<text><location><page_26><loc_51><loc_27><loc_93><loc_33></location>Öberg, Karin I., A. C. Adwin Boogert, Klaus M. Pontoppidan, Saskia van den Broek, Ewine F. van Dishoeck, Sandrine Bottinelli, Geoffrey A. Blake, and II Evans Neal J. 2011. The Spitzer Ice Legacy: Ice Evolution from Cores to Protostars. ApJ 740, no. 2 (October): 109. https://doi.org/10. 1088/0004-637X/740/2/109. arXiv: 1107.5825 [astro-ph.GA] .</text>
<text><location><page_26><loc_51><loc_21><loc_93><loc_26></location>Öberg, Karin I., Ruth Murray-Clay, and Edwin A. Bergin. 2011. The Effects of Snowlines on C/O in Planetary Atmospheres. ApJ 743, no. 1 (December): L16. https://doi.org/10.1088/2041-8205/743/1/L16. arXiv: 1110.5567 [astro-ph.GA] .</text>
<text><location><page_26><loc_51><loc_16><loc_93><loc_20></location>Öberg, Karin I., and Robin Wordsworth. 2019. Jupiter's Composition Suggests its Core Assembled Exterior to the N2 Snowline. AJ 158, no. 5 (November): 194. https://doi.org/10.3847/1538-3881/ab46a8. arXiv: 1909.11246 [astro-ph.EP] .</text>
<text><location><page_26><loc_51><loc_10><loc_93><loc_15></location>Ohno, Kazumasa, and Takahiro Ueda. 2021. Jupiter's 'cold' formation in the protosolar disk shadow. An explanation for the planet's uniformly enriched atmosphere. A&A 651 (July): L2. https://doi.org/10.1051/00046361/202141169. arXiv: 2106.09084 [astro-ph.EP] .</text>
<text><location><page_27><loc_7><loc_87><loc_49><loc_91></location>Okuzumi, Satoshi, and Shigenobu Hirose. 2011. Modeling Magnetorotational Turbulence in Protoplanetary Disks with Dead Zones. ApJ 742, no. 2 (December): 65. https://doi.org/10.1088/0004-637X/742/2/65. arXiv: 1108.4892 [astro-ph.EP] .</text>
<text><location><page_27><loc_7><loc_81><loc_49><loc_86></location>Okuzumi, Satoshi, and Ryo Tazaki. 2019. Nonsticky Ice at the Origin of the Uniformly Polarized Submillimeter Emission from the HL Tau Disk. ApJ 878, no. 2 (June): 132. https://doi.org/10.3847/1538-4357/ab204d. arXiv: 1904.03869 [astro-ph.EP] .</text>
<text><location><page_27><loc_7><loc_76><loc_49><loc_80></location>Ootsubo, Takafumi, Hideyo Kawakita, Saki Hamada, Hitomi Kobayashi, Mitsuru Yamaguchi, Fumihiko Usui, Takao Nakagawa, et al. 2012. AKARI Near-infrared Spectroscopic Survey for CO2 in 18 Comets. ApJ 752, no. 1 (June): 15. https://doi.org/10.1088/0004-637X/752/1/15.</text>
<text><location><page_27><loc_7><loc_70><loc_49><loc_75></location>Padoan, Paolo, Liubin Pan, Veli-Matti Pelkonen, Troels Haugboelle, and AAke Nordlund. 2024. Protoplanetary Disks from Pre-Main Sequence BondiHoyle Accretion. arXiv e-prints (May): arXiv:2405.07334. https://doi. org/10.48550/arXiv.2405.07334. arXiv: 2405.07334 [astro-ph.GA] .</text>
<text><location><page_27><loc_7><loc_66><loc_49><loc_69></location>Padovani, M., D. Galli, and A. E. Glassgold. 2009. Cosmic-ray ionization of molecular clouds. A&A 501, no. 2 (July): 619-631. https://doi.org/10. 1051/0004-6361/200911794. arXiv: 0904.4149 [astro-ph.SR] .</text>
<text><location><page_27><loc_7><loc_59><loc_50><loc_65></location>Pavlyuchenkov, Yaroslav, Vitaly Akimkin, Dmitri Wiebe, and Eduard Vorobyov. 2019. Revealing dust segregation in protoplanetary discs with the help of multifrequency spectral index maps. MNRAS 486, no. 3 (July): 39073914. https://doi.org/10.1093/mnras/stz1046. arXiv: 1904.05251 [astro-ph.IM] .</text>
<text><location><page_27><loc_7><loc_54><loc_49><loc_58></location>Pelkonen, Veli-Matti, Paolo Padoan, Mika Juvela, Troels Haugbølle, and Åke Nordlund. 2024. Origin and Evolution of Angular Momentum of Class II Disks. arXiv e-prints (May): arXiv:2405.06520. https://doi.org/10. 48550/arXiv.2405.06520. arXiv: 2405.06520 [astro-ph.SR] .</text>
<text><location><page_27><loc_7><loc_47><loc_49><loc_53></location>Pérez, Laura M., John M. Carpenter, Sean M. Andrews, Luca Ricci, Andrea Isella, Hendrik Linz, Anneila I. Sargent, et al. 2016. Spiral density waves in a young protoplanetary disk. Science 353, no. 6307 (September): 15191521. https://doi.org/10.1126/science.aaf8296. arXiv: 1610.05139 [astro-ph.GA] .</text>
<text><location><page_27><loc_7><loc_40><loc_49><loc_46></location>Piso, Ana-Maria A., Karin I. Öberg, Tilman Birnstiel, and Ruth A. MurrayClay. 2015. C/O and Snowline Locations in Protoplanetary Disks: The Effect of Radial Drift and Viscous Gas Accretion. ApJ 815, no. 2 (December): 109. https://doi.org/10.1088/0004-637X/815/2/109. arXiv: 1511.05563 [astro-ph.EP] .</text>
<text><location><page_27><loc_7><loc_32><loc_49><loc_39></location>Pontoppidan, K. M., C. Salyk, E. A. Bergin, S. Brittain, B. Marty, O. Mousis, and K. I. Öberg. 2014. Volatiles in Protoplanetary Disks. In Protostars and planets vi, edited by Henrik Beuther, Ralf S. Klessen, Cornelis P. Dullemond, and Thomas Henning, 363-385. January. https : / / doi . org/10.2458/azu_uapress_9780816531240-ch016. arXiv: 1401.2423 [astro-ph.EP] .</text>
<text><location><page_27><loc_7><loc_27><loc_49><loc_31></location>Przybilla, Norbert, Maria-Fernanda Nieva, and Keith Butler. 2008. A Cosmic Abundance Standard: Chemical Homogeneity of the Solar Neighborhood and the ISM Dust-Phase Composition. ApJ 688, no. 2 (December): L103. https://doi.org/10.1086/595618. arXiv: 0809.2403 [astro-ph] .</text>
<text><location><page_27><loc_7><loc_22><loc_49><loc_26></location>Rafikov, Roman R. 2005. Can Giant Planets Form by Direct Gravitational Instability? ApJ 621, no. 1 (March): L69-L72. https://doi.org/10.1086/ 428899. arXiv: astro-ph/0406469 [astro-ph] .</text>
<text><location><page_27><loc_7><loc_14><loc_49><loc_22></location>Rampinelli, L., S. Facchini, M. Leemker, J. Bae, M. Benisty, R. Teague, C. J. Law, K. I. Öberg, B. Portilla-Revelo, and A. J. Cridland. 2024. ALMA high-resolution observations unveil planet formation shaping molecular emission in the PDS 70 disk. arXiv e-prints (July): arXiv:2407.06272. https://doi.org/10.48550/arXiv.2407.06272. arXiv: 2407.06272 [astro-ph.EP] .</text>
<text><location><page_27><loc_52><loc_83><loc_94><loc_91></location>Reggiani, Henrique, Jhon Yana Galarza, Kevin C. Schlaufman, David K. Sing, Brian F. Healy, Andrew McWilliam, Joshua D. Lothringer, and Laurent Pueyo. 2024. Insight into the Formation of β Pic b through the Composition of Its Parent Protoplanetary Disk as Revealed by the β Pic Moving Group Member HD 181327. AJ 167, no. 1 (January): 45. https://doi.org/10.3847/1538-3881/ad0f93. arXiv: 2311.12210 [astro-ph.SR] .</text>
<text><location><page_27><loc_52><loc_78><loc_94><loc_82></location>Schneider, Aaron David, and Bertram Bitsch. 2021. How drifting and evaporating pebbles shape giant planets. I. Heavy element content and atmospheric C/O. A&A 654 (October): A71. https://doi.org/10.1051/00046361/202039640. arXiv: 2105.13267 [astro-ph.EP] .</text>
<text><location><page_27><loc_52><loc_72><loc_94><loc_77></location>Seager, S., L. J. Richardson, B. M. S. Hansen, K. Menou, J. Y. -K. Cho, and D. Deming. 2005. On the Dayside Thermal Emission of Hot Jupiters. ApJ 632, no. 2 (October): 1122-1131. https://doi.org/10.1086/444411. arXiv: astro-ph/0504212 [astro-ph] .</text>
<text><location><page_27><loc_52><loc_65><loc_94><loc_71></location>Seligman, Darryl Z., Leslie A. Rogers, Samuel H. C. Cabot, John W. Noonan, Theodore Kareta, Kathleen E. Mandt, Fred Ciesla, et al. 2022. The Volatile Carbon-to-oxygen Ratio as a Tracer for the Formation Locations of Interstellar Comets. PSJ 3, no. 7 (July): 150. https://doi.org/10. 3847/PSJ/ac75b5. arXiv: 2204.13211 [astro-ph.EP] .</text>
<text><location><page_27><loc_52><loc_57><loc_94><loc_65></location>Semenov, D., C. Favre, D. Fedele, S. Guilloteau, R. Teague, Th. Henning, A. Dutrey, E. Chapillon, F. Hersant, and V. Piétu. 2018. Chemistry in disks. XI. Sulfur-bearing species as tracers of protoplanetary disk physics and chemistry: the DM Tau case. A&A 617 (September): A28. https://doi.org/10.1051/0004-6361/201832980. arXiv: 1806.07707 [astro-ph.GA] .</text>
<text><location><page_27><loc_52><loc_52><loc_94><loc_57></location>Semenov, D., Th. Henning, Ch. Helling, M. Ilgner, and E. Sedlmayr. 2003. Rosseland and Planck mean opacities for protoplanetary discs. A&A 410 (November): 611-621. https://doi.org/10.1051/0004-6361:20031279. arXiv: astro-ph/0308344 [astro-ph] .</text>
<text><location><page_27><loc_52><loc_46><loc_94><loc_51></location>Semenov, D., and D. Wiebe. 2011. Chemical Evolution of Turbulent Protoplanetary Disks and the Solar Nebula. ApJS 196, no. 2 (October): 25. https://doi.org/10.1088/0067- 0049/196/2/25. arXiv: 1104.4358 [astro-ph.GA] .</text>
<text><location><page_27><loc_52><loc_43><loc_94><loc_46></location>Shakura, N. I., and R. A. Sunyaev. 1973. Black holes in binary systems. Observational appearance. A&A 24 (January): 337-355.</text>
<text><location><page_27><loc_52><loc_37><loc_94><loc_43></location>Sing, David K., Zafar Rustamkulov, Daniel P. Thorngren, Joanna K. Barstow, Pascal Tremblin, Catarina Alves de Oliveira, Tracy L. Beck, et al. 2024. A warm Neptune's methane reveals core mass and vigorous atmospheric mixing. arXiv e-prints (May): arXiv:2405.11027. https://doi.org/10. 48550/arXiv.2405.11027. arXiv: 2405.11027 [astro-ph.EP] .</text>
<text><location><page_27><loc_52><loc_29><loc_94><loc_36></location>Smith, Peter C. B., Michael R. Line, Jacob L. Bean, Matteo Brogi, Prune August, Luis Welbanks, Jean-Michel Desert, et al. 2024. A Combined Ground-based and JWST Atmospheric Retrieval Analysis: Both IGRINS and NIRSpec Agree that the Atmosphere of WASP-77A b Is Metal-poor. AJ 167, no. 3 (March): 110. https://doi.org/10.3847/1538-3881/ad17bf. arXiv: 2312.13069 [astro-ph.EP] .</text>
<text><location><page_27><loc_52><loc_22><loc_93><loc_28></location>Stammler, Sebastian Markus, Tilman Birnstiel, Olja Panić, Cornelis Petrus Dullemond, and Carsten Dominik. 2017. Redistribution of CO at the location of the CO ice line in evolving gas and dust disks. A&A 600 (April): A140. https://doi.org/10.1051/0004-6361/201629041. arXiv: 1701.02385 [astro-ph.EP] .</text>
<text><location><page_27><loc_52><loc_18><loc_93><loc_21></location>Stevenson, David J., and Jonathan I. Lunine. 1988. Rapid formation of Jupiter by diffusive redistribution of water vapor in the solar nebula. Icarus 75, no. 1 (July): 146-155. https://doi.org/10.1016/0019-1035(88)90133-9.</text>
<text><location><page_27><loc_52><loc_11><loc_94><loc_17></location>Swain, M. R., G. Tinetti, G. Vasisht, P. Deroo, C. Griffith, J. Bouwman, Pin Chen, et al. 2009. Water, Methane, and Carbon Dioxide Present in the Dayside Spectrum of the Exoplanet HD 209458b. ApJ 704, no. 2 (October): 1616-1621. https://doi.org/10.1088/0004-637X/704/2/1616. arXiv: 0908.4010 [astro-ph.EP] .</text>
<text><location><page_28><loc_7><loc_85><loc_48><loc_91></location>Testi, L., T. Birnstiel, L. Ricci, S. Andrews, J. Blum, J. Carpenter, C. Dominik, et al. 2014. Dust Evolution in Protoplanetary Disks. In Protostars and planets vi, edited by Henrik Beuther, Ralf S. Klessen, Cornelis P. Dullemond, and Thomas Henning, 339-361. January. https://doi.org/10.2458/azu_ uapress_9780816531240-ch015. arXiv: 1402.1354 [astro-ph.SR] .</text>
<text><location><page_28><loc_7><loc_80><loc_48><loc_85></location>Thiabaud, A., U. Marboeuf, Y. Alibert, I. Leya, and K. Mezger. 2015. Gas composition of the main volatile elements in protoplanetary discs and its implication for planet formation. A&A 574 (February): A138. https: //doi.org/10.1051/0004-6361/201424868.</text>
<text><location><page_28><loc_7><loc_73><loc_48><loc_79></location>Thies, Ingo, Pavel Kroupa, Simon P. Goodwin, Dimitrios Stamatellos, and Anthony P. Whitworth. 2010. Tidally Induced Brown Dwarf and Planet Formation in Circumstellar Disks. ApJ 717, no. 1 (July): 577-585. h ttps://doi.org/10.1088/0004- 637X/717/1/577. arXiv: 1005.3017 [astro-ph.SR] .</text>
<text><location><page_28><loc_7><loc_71><loc_48><loc_73></location>Tielens, A. G. G. M. 2005. The Physics and Chemistry of the Interstellar Medium.</text>
<text><location><page_28><loc_7><loc_66><loc_48><loc_71></location>Tong, Simin, Richard Alexander, and Giovanni Rosotti. 2024. A question of personalities: evolution of viscous and wind-driven protoplanetary discs in the presence of dead zones. MNRAS (July). https://doi.org/10.1093/ mnras/stae1748. arXiv: 2407.12209 [astro-ph.EP] .</text>
<text><location><page_28><loc_7><loc_63><loc_48><loc_65></location>Toomre, A. 1964. On the gravitational stability of a disk of stars. ApJ 139 (May): 1217-1238. https://doi.org/10.1086/147861.</text>
<text><location><page_28><loc_7><loc_57><loc_48><loc_62></location>Topchieva, A., T. Molyarova, V. Akimkin, L. Maksimova, and E. Vorobyov. 2024. Ices on pebbles in protoplanetary discs. MNRAS 530, no. 3 (May): 2731-2748. https://doi.org/10.1093/mnras/stae597. arXiv: 2403.02895 [astro-ph.EP] .</text>
<text><location><page_28><loc_7><loc_50><loc_48><loc_56></location>Turrini, D., E. Schisano, S. Fonte, S. Molinari, R. Politi, D. Fedele, O. Panić, M. Kama, Q. Changeat, and G. Tinetti. 2021. Tracing the Formation History of Giant Planets in Protoplanetary Disks with Carbon, Oxygen, Nitrogen, and Sulfur. ApJ 909, no. 1 (March): 40. https://doi.org/10. 3847/1538-4357/abd6e5. arXiv: 2012.14315 [astro-ph.EP] .</text>
<text><location><page_28><loc_7><loc_46><loc_48><loc_50></location>Umebayashi, T., and T. Nakano. 1981. Fluxes of Energetic Particles and the Ionization Rate in Very Dense Interstellar Clouds. PASJ 33 (January): 617.</text>
<text><location><page_28><loc_7><loc_41><loc_48><loc_45></location>Vallet, David, Anna C. Childs, Rebecca G. Martin, Mario Livio, and Stephen Lepp. 2023. Formation of super-Earths in icy dead zones around lowmass stars. MNRAS 519, no. 1 (February): L10-L14. https://doi.org/10. 1093/mnrasl/slac144. arXiv: 2211.07759 [astro-ph.EP] .</text>
<text><location><page_28><loc_7><loc_35><loc_48><loc_40></location>Visser, R., E. F. van Dishoeck, S. D. Doty, and C. P. Dullemond. 2009. The chemical history of molecules in circumstellar disks. I. Ices. A&A 495, no. 3 (March): 881-897. https://doi.org/10.1051/0004-6361/200810846. arXiv: 0901.1313 [astro-ph.SR] .</text>
<text><location><page_28><loc_7><loc_31><loc_48><loc_34></location>Vorobyov, E. I. 2013. Formation of giant planets and brown dwarfs on wide orbits. A&A 552 (April): A129. https://doi.org/10.1051/0004-6361/ 201220601. arXiv: 1302.1892 [astro-ph.EP] .</text>
<text><location><page_28><loc_7><loc_24><loc_51><loc_30></location>Vorobyov, Eduard I., Vitaly Akimkin, Olga Stoyanovskaya, Yaroslav Pavlyuchenkov, and Hauyu Baobab Liu. 2018. Early evolution of viscous and selfgravitating circumstellar disks with a dust component. A&A 614 (June): A98. https://doi.org/10.1051/0004-6361/201731690. arXiv: 1801.06898 [astro-ph.EP] .</text>
<text><location><page_28><loc_7><loc_18><loc_48><loc_23></location>Vorobyov, Eduard I., and Shantanu Basu. 2010. The Burst Mode of Accretion and Disk Fragmentation in the Early Embedded Stages of Star Formation. ApJ 719, no. 2 (August): 1896-1911. https://doi.org/10.1088/0004637X/719/2/1896. arXiv: 1007.2993 [astro-ph.SR] .</text>
<text><location><page_28><loc_7><loc_12><loc_48><loc_18></location>Vorobyov, Eduard I., and Vardan G. Elbakyan. 2019. Gravitoviscous protoplanetary disks with a dust component. II. Spatial distribution and growth of dust in a clumpy disk. A&A 631 (November): A1. https: / / doi . org / 10 . 1051 / 0004 - 6361 / 201936132. arXiv: 1908 . 10589 [astro-ph.SR] .</text>
<text><location><page_28><loc_7><loc_6><loc_48><loc_11></location>Vorobyov, Eduard I., Vardan G. Elbakyan, Michihiro Takami, and Hauyu B. Liu. 2020. Effect of luminosity outbursts on protoplanetary disk dynamics. A&A 643 (November): A13. https://doi.org/10.1051/00046361/202038122. arXiv: 2009.01888 [astro-ph.SR] .</text>
<text><location><page_28><loc_51><loc_87><loc_93><loc_91></location>Vorobyov, Eduard I., Sergey Khaibrakhmanov, Shantanu Basu, and Marc Audard. 2020. Accretion bursts in magnetized gas-dust protoplanetary disks. A&A 644 (December): A74. https://doi.org/10.1051/00046361/202039081. arXiv: 2011.00951 [astro-ph.SR] .</text>
<text><location><page_28><loc_51><loc_81><loc_93><loc_86></location>Vorobyov, Eduard I., D. N. C. Lin, and Manuel Guedel. 2015. The effect of external environment on the evolution of protostellar disks. A&A 573 (January): A5. https://doi.org/10.1051/0004-6361/201424583. arXiv: 1410.1743 [astro-ph.SR] .</text>
<text><location><page_28><loc_51><loc_76><loc_93><loc_80></location>Vorobyov, Eduard I., Zsolt Regaly, Manuel Guedel, and Doug N. C. Lin. 2016. An alternative model for the origin of gaps in circumstellar disks. A&A 587 (March): A146. https://doi.org/10.1051/0004-6361/201527701. arXiv: 1601.08089 [astro-ph.SR] .</text>
<text><location><page_28><loc_51><loc_69><loc_93><loc_75></location>Vorobyov, Eduard I., Aleksandr M. Skliarevskii, Manuel Guedel, and Tamara Molyarova. 2024. Primordial dust rings, hidden dust mass, and the first generation of planetesimals in gravitationally unstable protoplanetary disks. arXiv e-prints (April): arXiv:2404.16151. https://doi.org/10.48550/ arXiv.2404.16151. arXiv: 2404.16151 [astro-ph.EP] .</text>
<text><location><page_28><loc_51><loc_61><loc_93><loc_68></location>Vorobyov, Eduard I., Aleksandr M. Skliarevskii, Tamara Molyarova, Vitaly Akimkin, Yaroslav Pavlyuchenkov, Ágnes Kóspál, Hauyu Baobab Liu, Michihiro Takami, and Anastasiia Topchieva. 2022. Evolution of dust in protoplanetary disks of eruptive stars. A&A 658 (February): A191. https://doi.org/10.1051/0004-6361/202141932. arXiv: 2112.06004 [astro-ph.EP] .</text>
<text><location><page_28><loc_51><loc_55><loc_93><loc_60></location>Vorobyov, Eduard I., Olga V. Zakhozhay, and Michael M. Dunham. 2013. Fragmenting protostellar discs: properties and observational signatures. MNRAS 433, no. 4 (August): 3256-3273. https://doi.org/10.1093/ mnras/stt970. arXiv: 1306.4074 [astro-ph.SR] .</text>
<text><location><page_28><loc_51><loc_50><loc_93><loc_55></location>Wada, Koji, Hidekazu Tanaka, Toru Suyama, Hiroshi Kimura, and Tetsuo Yamamoto. 2009. Collisional Growth Conditions for Dust Aggregates. ApJ 702, no. 2 (September): 1490-1501. https://doi.org/10.1088/0004637X/702/2/1490.</text>
<text><location><page_28><loc_51><loc_43><loc_93><loc_49></location>Walsh, Catherine, Hideko Nomura, and Ewine van Dishoeck. 2015. The molecular composition of the planet-forming regions of protoplanetary disks across the luminosity regime. A&A 582 (October): A88. http s : / / doi . org / 10 . 1051 / 0004 - 6361 / 201526751. arXiv: 1507 . 08544 [astro-ph.EP] .</text>
<text><location><page_28><loc_51><loc_36><loc_93><loc_42></location>Weber, Philipp, Sebastián Pérez, Alice Zurlo, James Miley, Antonio Hales, Lucas Cieza, David Principe, et al. 2023. Spirals and Clumps in V960 Mon: Signs of Planet Formation via Gravitational Instability around an FU Ori Star? ApJ 952, no. 1 (July): L17. https://doi.org/10.3847/20418213/ace186. arXiv: 2307.13433 [astro-ph.EP] .</text>
<text><location><page_28><loc_51><loc_30><loc_93><loc_36></location>Wei, Chen-En, Hideko Nomura, Jeong-Eun Lee, Wing-Huen Ip, Catherine Walsh, and T. J. Millar. 2019. The Effect of Carbon Grain Destruction on the Chemical Structure of Protoplanetary Disks. ApJ 870, no. 2 (January): 129. https://doi.org/10.3847/1538- 4357/aaf390. arXiv: 1811.10194 [astro-ph.EP] .</text>
<text><location><page_28><loc_51><loc_26><loc_93><loc_29></location>Weidenschilling, S. J. 1977. Aerodynamics of solid bodies in the solar nebula. MNRAS 180 (July): 57-70. https://doi.org/10.1093/mnras/180.2.57.</text>
<text><location><page_28><loc_51><loc_20><loc_93><loc_26></location>Weiner Mansfield, Megan, Michael R. Line, Joost P. Wardenier, Matteo Brogi, Jacob L. Bean, Hayley Beltz, Peter Smith, et al. 2024. The metallicity and carbon-to-oxygen ratio of the ultra-hot Jupiter WASP-76b from Gemini-S/IGRINS. arXiv e-prints (May): arXiv:2405.09769. https://doi. org/10.48550/arXiv.2405.09769. arXiv: 2405.09769 [astro-ph.EP] .</text>
<text><location><page_28><loc_51><loc_14><loc_93><loc_19></location>Westley, M. S., R. A. Baragiola, R. E. Johnson, and G. A. Baratta. 1995. Photodesorption from low-temperature water ice in interstellar and circumsolar grains. Nature 373, no. 6513 (February): 405-407. https: //doi.org/10.1038/373405a0.</text>
<text><location><page_28><loc_51><loc_10><loc_93><loc_14></location>Wierzchos, K., and M. Womack. 2018. C/2016 R2 (PANSTARRS): A Comet Rich in CO and Depleted in HCN. AJ 156, no. 1 (July): 34. https://doi. org/10.3847/1538-3881/aac6bc. arXiv: 1805.06918 [astro-ph.EP] .</text>
<text><location><page_29><loc_7><loc_88><loc_49><loc_91></location>Winter, Andrew J., Myriam Benisty, and Sean M. Andrews. 2024. Planet formation regulated by galactic-scale interstellar turbulence. arXiv eprints (May): arXiv:2405.08451. arXiv: 2405.08451 [astro-ph.EP] .</text>
<text><location><page_29><loc_7><loc_81><loc_49><loc_87></location>Wong, Michael H., Paul R. Mahaffy, Sushil K. Atreya, Hasso B. Niemann, and Tobias C. Owen. 2004. Updated Galileo probe mass spectrometer measurements of carbon, oxygen, nitrogen, and sulfur on Jupiter. Icarus 171, no. 1 (September): 153-170. https://doi.org/10.1016/j.icarus.2004. 04.010.</text>
<text><location><page_29><loc_7><loc_76><loc_49><loc_80></location>Worthen, Kadin, Christine H. Chen, David R. Law, Cicero X. Lu, Kielan Hoch, Yiwei Chai, G. C. Sloan, et al. 2024. MIRI MRS Observations of β Pictoris. I. The Inner Dust, the Planet, and the Gas. ApJ 964, no. 2 (April): 168. https://doi.org/10.3847/1538-4357/ad2354.</text>
<text><location><page_29><loc_7><loc_69><loc_49><loc_75></location>Xue, Qiao, Jacob L. Bean, Michael Zhang, Luis Welbanks, Jonathan Lunine, and Prune August. 2024. JWST Transmission Spectroscopy of HD 209458b: A Supersolar Metallicity, a Very Low C/O, and No Evidence of CH4, HCN, or C2H2. ApJ 963, no. 1 (March): L5. https://doi.org/ 10.3847/2041-8213/ad2682. arXiv: 2310.03245 [astro-ph.EP] .</text>
<text><location><page_29><loc_7><loc_63><loc_49><loc_68></location>Yang, Chao-Chin, Anders Johansen, and Daniel Carrera. 2017. Concentrating small particles in protoplanetary disks through the streaming instability. A&A 606 (October): A80. https : / / doi . org / 10 . 1051 / 0004 - 6361 / 201630106. arXiv: 1611.07014 [astro-ph.EP] .</text>
<text><location><page_29><loc_7><loc_59><loc_49><loc_63></location>Youdin, Andrew N., and Jeremy Goodman. 2005. Streaming Instabilities in Protoplanetary Disks. ApJ 620, no. 1 (February): 459-469. https: //doi.org/10.1086/426895. arXiv: astro-ph/0409263 [astro-ph] .</text>
<text><location><page_29><loc_7><loc_52><loc_49><loc_58></location>Zhang, Yapeng, Ignas A. G. Snellen, Alexander J. Bohn, Paul Mollière, Christian Ginski, H. Jens Hoeijmakers, Matthew A. Kenworthy, et al. 2021. The 13 CO-rich atmosphere of a young accreting super-Jupiter. Nature 595, no. 7867 (July): 370-372. https://doi.org/10.1038/s41586-02103616-x. arXiv: 2107.06297 [astro-ph.EP] .</text>
<text><location><page_29><loc_7><loc_46><loc_49><loc_52></location>Zhu, Zhaohuan, Lee Hartmann, Charles F. Gammie, Laura G. Book, Jacob B. Simon, and Eric Engelhard. 2010. Long-term Evolution of Protostellar and Protoplanetary Disks. I. Outbursts. ApJ 713, no. 2 (April): 11341142. https : / / doi . org / 10 . 1088 / 0004 - 637X / 713 / 2 / 1134. arXiv: 1003.1759 [astro-ph.SR] .</text>
<text><location><page_29><loc_7><loc_39><loc_49><loc_45></location>Zhu, Zhaohuan, Yan-Fei Jiang, and James M. Stone. 2020. Global 3D radiation magnetohydrodynamic simulations for FU Ori's accretion disc and observational signatures of magnetic fields. MNRAS 495, no. 3 (January): 3494-3514. https://doi.org/10.1093/mnras/staa952. arXiv: 1912.01632 [astro-ph.EP] .</text>
</document>
[
{
"title": "C/O ratios in self-gravitating protoplanetary discs with dust evolution",
"content": "Tamara Molyarova, 1,2 Eduard Vorobyov, 1,2 and Vitaly Akimkin 1 1 Institute of Astronomy, Russian Academy of Sciences, 48 Pyatnitskaya St., Moscow, 119017, Russia 2 Research Institute of Physics, Southern Federal University, Stachki Ave. 194, Rostov-on-Don 344090, Russia Author for correspondence: Tamara Molyarova, Email: moliarova@sfedu.ru.",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "Elemental abundances, particularly the C/O ratio, are seen as a way to connect the composition of planetary atmospheres with planet formation scenario and the disc chemical environment. We model the chemical composition of gas and ices in a self-gravitating disc on timescales of 0.5 Myr since its formation to study the evolution of C/O ratio due to dust dynamics and growth, and phase transitions of the volatile species. We use the thin-disc hydrodynamic code FEOSAD, which includes disc self-gravity, thermal balance, dust evolution and turbulent diffusion, and treats dust as a dynamically different and evolving component interacting with the gas. It also describes freeze-out, sublimation and advection of four most abundant volatile species: H2O, CO2, CH4 and CO. We demonstrate the effect of gas and dust substructures such as spirals and rings on the distribution of volatiles and C/O ratios, including the formation of multiple snowlines of one species, and point out the anticorrelation between dust-to-gas ratio and total C/O ratio emerging due to the contribution of oxygen-rich ice mantles. We identify time and spatial locations where two distinct trigger mechanisms for planet formation are operating and differentiate them by C/O ratio range: wide range of the C/O ratios of 0 - 1.4 for streaming instability, and a much narrower range 0.3 - 0.6 for gravitational instability (with the initial value of 0.34). This conclusion is corroborated by observations, showing that transiting exoplanets, which possibly experienced migration through a variety of disc conditions, have significantly larger spread of C/O in comparison with directly imaged exoplanets likely formed in gravitationally unstable outer disk regions. We show that the ice-phase C/O ≈ 0.2 - 0.3 between the CO, CO2 and CH4 snowlines corresponds to the composition of the Solar system comets, that represent primordial planetesimals. Keywords: protoplanetary disc, volatiles, dust evolution",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "The protoplanetary disc matter can be roughly divided into three component: gaseous chemical species, solid dust particles, and icy mantles covering the surface of dust grains. Gas and solid particles become dynamically decoupled, as evolving dust grows and acquires relative velocities leading to the redistribution of elements in the disc and between the phases, and creating the premises for different chemical environments. When planets start to form, their properties, including chemical composition of the atmosphere, are inevitably affected by the location and the mechanism of their formation. This suggests that the origin of (exo)planets might be investigated using their observed chemical composition, and makes understanding the disc chemical evolution vital for creating a consistent planet formation theory. One of the key parameters that govern the chemical setup of a planetary atmosphere is the relation between the abundances of carbon and oxygen, often referred to as carbon-tooxygen ratio (hereafter C/O ratio). The variations of C/O ratio in the ice and gas phases at the snowlines of main disc volatiles (CO, CO 2 , and H 2 O) and the prospects of connecting them to planet formation were discussed in Öberg, Murray-Clay, and Bergin (2011) within a qualitative freeze-out model. Since then, C/O ratio received a lot of attention in this context. It was thoroughly investigated in modelling (see, e.g., Booth et al. 2017; Eistrup, Walsh, and van Dishoeck 2018; Cridland, Eistrup, and van Dishoeck 2019; Cridland et al. 2019; Crid- and, Bosman, and van Dishoeck 2020; Cridland et al. 2020; Krijt et al. 2020; Turrini et al. 2021; Schneider and Bitsch 2021). The connection of disc chemical composition with C/O in exoplanetary atmospheres was modelled using core accretion model (Thiabaud et al. 2015) and 'chain' planet population synthesis model (Mordasini et al. 2016). Paul Mollière et al. (2022) considered a simple formation retrieval pipeline and found that this task requires careful consideration of the model assumptions. The measurements of molecular abundances in the atmospheres of giant exoplanets obtained by a variety of modern facilities, such as HST, Spitzer, VLTI, JWST, Gemini, indicate a diversity of C/O ratios: from low C/O ratio values ( ≈ 0.4, below the solar value of 0.54; Benneke et al. 2019; GRAVITY Collaboration et al. 2020; Worthen et al. 2024; Xue et al. 2024) to stellar ( ≈ 0.5, close to solar; P. Mollière et al. 2020; Zhang et al. 2021; Smith et al. 2024) and close to or above unity (Swain et al. 2009; Madhusudhan et al. 2011). A variety of solar and super-solar C/O ratios is observed in four planets within the HR8799 system (Nasedkin et al. 2024). Chemical composition of the atmospheres of many hot Jupiters indicates high C/O>1 of the forming material (Moses et al. 2013). There is an observational evidence of young planets in PDS 70 disc accreting the material with C/O>1 (Facchini et al. 2021). The number of exoplanets with constrained atmospheric C/O ratios grows with large studies of multiple planets such as Changeat et al. (2022), which allows us to make some statistical conclusions. The population study of C/O ratios in exoplanetary atmospheres reveals that there are two populations with different elemental ratio, which are likely formed in different mechanisms (Hoch et al. 2023). Khorshid, Min, and Désert (2023) were able to restrict the formation scenario for WASP-77b based on the measured C/O ratio of the planet (Line et al. 2021) and the modelling of planet formation and migration. Elemental abundances in protoplanetary discs can be constrained from observations (Fedele and Favre 2020), and C/O ratio in the gas can be estimated. Spatially resolved observations can help distinguish between different C/O ratios spectroscopically (Matter, Pignatale, and Lopez 2020). Cleeves et al. (2018) report C/O ≈ 0.8 in the molecular layer of IM Lup disc. ALMA observations of hydrocarbons and sulphur-bearing species indicate C/O > 1 in the upper disc layers and in the outer disc in TW Hya and DM Tau (Dutrey et al. 2011; Bergin et al. 2016; Semenov et al. 2018) and for a population of discs in Lupus (Miotello et al. 2019). For the nearby discs the solar elemental composition with C/O ≈ 0.54 is usually expected, thus the observed higher values confirm redistribution of carbon and oxygen in discs. In addition to high C/O in disc atmospheres, there is evidence of both carbon and oxygen depletion from gas (Kama et al. 2016; Miotello et al. 2019). However, some of heavy oxygen carriers might not be observable, leading to overestimated C/O in disc observations (Walsh, Nomura, and van Dishoeck 2015). The volatile composition is also used to constrain the origin of bodies in the Solar System. Fraction of CO and CO 2 ices relative to water in cometary comae indicate their formation between the CO and CO 2 snowlines or exterior to the CO snowline (A'Hearn et al. 2012; Seligman et al. 2022). Abundances of CO and N 2 ices were used to analyse the original location of Pluto and Triton (Mousis, Anderson, et al. 2024). Observed elemental abundances were used to constrain the Jupiter formation scenario (Lodders 2004), relying also on abundances of nitrogen (Öberg and Wordsworth 2019; Bosman, Cridland, and Miguel 2019) and chemically inactive species like Ar. However, the model assumptions can lead to different interpretation of the observations: while Öberg and Wordsworth (2019) and Bosman, Cridland, and Miguel (2019) suggest that Jupiter formed outside N 2 snowline (at > 30 au), Ohno and Ueda (2021) consider the concept disc shadow, which allows Jupiter to form near its current location. Chemical processes other than freeze-out and sublimation at the snowlines can alter the composition of ice and gas as well. Due to gas-phase and surface reactions, snowlines can become important for the redistribution of elements. More detailed chemical modelling shows that the C/O ratio in the gas and in the ice depends also on the initial chemical setup and ionisation by cosmic rays and radioactive nuclei (Eistrup, Walsh, and van Dishoeck 2016, 2018). It directly affects the interpretation of observations. Another essential chemical process is CO depletion from the gas, resulting in its transformation to CO 2 ice (Bosman, Tielens, and van Dishoeck 2018). For example, stellar C/O ratio in the atmosphere of HR 8799e indicates that the planet accreted its material beyond CO snowline ( ≈ 45 au), but chemical modelling suggests that due to CO depletion, the C/O in the ice already approaches the stellar ratio beyond CO 2 snowline ( ≈ 20 au), which is closer to the star (P. Mollière et al. 2020). Another key process affecting the elemental ratio is dust drift, which leads to spatial segregation between the chemical constituents of the gas and the grains covered with ice. The distribution of CO in the gas and ice phases was studied within dynamical models of dust evolution (Stammler et al. 2017; Krijt et al. 2018). Even without chemical processes, dust evolution and dynamics can substantially alter C/O ratio in the atmospheres of forming planets (Booth et al. 2017). Some models combine chemical reactions treatment with dust evolution and transport, usually within 1D viscous models. Dust transport can have a strong effect on the abundances of volatiles in the inner disc regions (Bosman, Tielens, and van Dishoeck 2018). However, for discs with low turbulence and high cosmic ray ionisation rate, C/O ratio is rather defined by chemical evolution (Booth and Ilee 2019). In our previous work (Molyarova et al. 2021) we showed that the volatile species tend to concentrate around their snowlines both in the gas and more notably on the dust surface. This accumulation was found to be caused by effective transport of volatiles through the snowlines by azimuthal variations in the gas and dust radial and angular velocity, an effect that cannot be captured in 1D viscous disc models. Such accumulation should immediately affect the local C/O ratio, which suggests the connection between the snowlines of various volatiles and the formation of planets with altered C/O in their atmospheres. In this work, we follow the distribution of the main volatile species in the disc to investigate the distribution of C/O ratio in gas and ice in a 2D thin-disc hydrodynamic model. We study the effect of dust growth and dynamics on the elemental ratios and consider the role of the initial mass of the collapsing core on the distribution of volatiles. The paper is organised as follows. The main features of the used FEOSAD model are described in Section 2, with the details of the treatment of the volatiles given in Section 2.3. In Section 3 we describe the results of the simulations, focusing on distribution of the volatiles in Section 3.2, the C/O ratios in Section 3.3, and their evolution in Section 3.4. In Section 4, we discuss the implications of our results in the context of planet formation via different mechanisms. The main conclusions are listed in Section 5.",
"pages": [
1,
2
]
},
{
"title": "2. Model",
"content": "We use the global model of protoplanetary disc formation FEOSAD (Vorobyov et al. 2018), which includes disc selfgravity, dust evolution and interaction with gas (including backreaction of dust on gas), turbulent viscosity, adiabatic and radiative cooling and heating. It describes the formation of a protostar and a protoplanetary disc from a collapsing cloud in a 2D thin-disc approach. The model also includes freeze-out of main volatile species as in Molyarova et al. (2021), with the feedback from ice mantles on dust evolution via fragmentation velocity. Here we summarise the key characteristics of the model, more details can be found in the previous works (Vorobyov et al. 2018; Molyarova et al. 2021; Kadam, Vorobyov, and Basu 2022). The main difference from our previous study in Molyarova et al. (2021) is that here we consider the formation of dead zones via variable α -parameter of Shakura and Sunyaev and also include turbulent diffusion.",
"pages": [
2,
3
]
},
{
"title": "2.1 Gas evolution",
"content": "For the gas component, the hydrodynamic equations for mass, momentum, and internal energy conservation are the following where subscripts p and p ' denote the planar components ( r , ϕ ) in polar coordinates, Σ g is the gas mass surface density, e is the internal energy per surface area, P is the vertically integrated gas pressure calculated via the ideal equation of state as P = ( γ - 1) e with γ = 7/5, f is the friction force between gas and dust, v = vr ˆ r + v ϕ ˆ ϕ is the gas velocity in the disc plane, and ∇ = ˆ r ∂ / ∂ r + ˆ ϕ r -1 ∂ / ∂ϕ is the gradient along the planar coordinates of the disc. The gravitational acceleration in the disc plane, g = gr ˆ r + g ϕ ˆ ϕ , includes the gravity of the central protostar when formed and takes into account disc self-gravity of both gas and dust found by solving the Poisson integral (Binney and Tremaine 1987). Theconsideration of time-dependent energy balance (Eq. (3)) allows us to accurately calculate the midplane temperature T mp and is particularly important to describe the phase state of the volatiles and the level of turbulent viscosity. The terms Λ and Γ describe the rates of dust cooling and heating, respectively, by stellar and background irradiation. They are calculated based on the analytical solution of the radiation transfer equations in the vertical direction (Dong et al. 2016; Vorobyov et al. 2018) Here, σ is the Stefan-Boltzmann constant, τ P and τ R are the Planck and Rosseland mean optical depths to the disc midplane, calculated as τ = κΣ dust from Planck and Rosseland mean opacities κ P and κ R (Semenov et al. 2003) and total dust surface density Σ dust . Gas and dust temperatures are assumed to be equal, and the midplane temperature is linked with gas pressure as T mp = P µ / R Σ g, where µ = 2.3 is the mean molecular weight of the gas, and R is the universal gas constant. The irradiation temperature at the disc surface T irr is determined by both stellar and background irradiation. Stellar irradiation includes the luminosity from the photosphere of the protostar and accretion luminosity. The background radiation is assumed as a black body with the temperature of 15 K. For more details on the irradiation we refer to Vorobyov et al. (2018). Turbulent viscosity is described using the common α -parameter approach of Shakura and Sunyaev (1973). It is taken into account via the viscous stress tensor Π (see Vorobyov and Basu 2010, for explicit expressions for the components of the terms with Π ). The magnitude of kinematic viscosity is ν = α c s H , where c s is the sound speed and H is the vertical scale height of the gas disc calculated using an assumption of local hydrostatic equilibrium of a self-gravitating disc (see Vorobyov and Basu 2010, Appendix A). Here, we use the adaptive α approach implying accretion through a layered disc (Gammie 1996; Armitage, Livio, and Pringle 2001; Kadam, Vorobyov, and Basu 2022). Turbulence is assumed to be generated by magneto-rotational instability (MRI) which only develops in layers of the disc where the ionisation level is high enough. The MRI-active layer is characterised by its surface density Σ MRI and relatively high value of turbulent viscosity α MRI = 10 -3 . As thermal and photo-ionisation are not efficient enough for the relatively cold and dense matter in the disc at >0.5 au, the main process determining the thickness of the MRI-active layer is ionisation by cosmic rays. It is assumed to be constant Σ MRI = 100g cm -2 , which is the typical depth of Galactic cosmic rays penetration in the ISM (Umebayashi and Nakano 1981) and in protoplanetary discs (Padovani, Galli, and Glassgold 2009). The dead zone is characterised by surface density from the midplane Σ dz = Σ g/2Σ MRI with residual turbulence α dz . The turbulence in this layer is only hydrodynamic turbulence driven by the Maxwell stress in the active layer, and small value α dz = 10 -5 is adopted (Okuzumi and Hirose 2011). However, if local temperature exceeds the critical value of 1300 K, thermal ionisation becomes possible, the MRI develops and the dead zone is no longer dead, in which case α dz = 10 -1 (Zhu et al. 2010; Kadam et al. 2019). This value is higher than α MRI in the outer disc due to the different ionisation processes and local conditions. In the outer disc, the MRI can be suppressed by non-ideal magneto-hydrodynamic (MHD) effects such as ambipolar diffusion and Ohmic resistivity (Bai and Stone 2013; Gressel et al. 2015). In the dead zone in the inner disc, α can reach higher values when the MRI is triggered by thermal ionisation, as shown by 3D MHD simulations (Zhu, Jiang, and Stone 2020). The adopted parameterization makes use of an effective parameter α eff , which at any given location is calculated as",
"pages": [
3
]
},
{
"title": "2.2 Dust evolution",
"content": "Dust is described as consisting of two components: small grains that are dynamically coupled to the gas, with the mass surface density Σ d,sm , and grown grains with the mass surface density Σ d,gr that can move relative to the gas and change in size. The total dust surface density necessary for the calculation of the optical depths for Eq. (4) is Σ dust = ( Σ d,gr + Σ d,sm )/2. The factor 1/2 appears as the optical depths is calculated to the midplane. Each dust population has a power-law size distribution f ( a ) = dN / da = Ca -p with a normalisation constant C and a fixed exponent p = 3.5. Small dust has sizes between a min = 5 × 10 -7 cm and a ∗ = 10 -4 cm, grown dust has sizes between a ∗ and a max, which can vary due to dust coagulation and fragmentation. Dynamics of these dust components follows the continuity and momentum equations where u is the grown dust velocity. The term S ( a max) is responsible for the exchange of matter between the dust populations, as dust is converted from the small to grown component due to coagulation and back due to fragmentation. The details of the dust evolution model are presented in Vorobyov et al. (2018). The last term in Eqs. (6) and (7) is responsible for dust turbulent diffusion, similar to Vorobyov, Elbakyan, et al. (2020). The coefficient of turbulent diffusion D is related to the kinematic viscosity ν as D = ν /(1 + St 2 ) (Birnstiel 2023). Diffusion affects dust grains along with their ice mantles, as well as the gas-phase species (see Section 2.3). The innermost regions of the disc are challenging to simulate explicitly due to the Courant criterion: in the highly dynamic inner regions (fraction of au) the timescales are so short that the code demands very small time step in order to preserve stability. Therefore, the inner regions are represented by a sink cell, with a carefully chosen inflow-outflow boundary condition at the sink cell and a parametric description of the accretion onto the star (see Vorobyov et al. 2018, for details). In the simulations presented below, the radius of the sink cell is 0.62 au. We consider two disc models with different initial mass of the collapsing cloud, 0.66 and 1 M ⊙ . We note that about 10% of the gas mass that crosses the sink cell is assumed to be evacuated by jets and outflows, and the other 90% lands on to the star. A small amount of mass remains in the envelope by the end of simulations. In both models, the initial gas temperature T init = 15 K and the ratio of rotational to gravitational energy β = 0.28%. The simulations start from the collapse of a molecular cloud, with only small dust grains. The simulation continues until the age of the system becomes equal to 0.5 Myr. Masses of the central protostar and the disc by the end of simulation are M ⋆ = 0.4 M ⊙ and M disc = 0.22 M ⊙ in model M1 and M ⋆ = 0.58 M ⊙ and M disc = 0.35 M ⊙ in model M2. The disc masses are around 0.5 stellar masses, which makes them essentially self-gravitating. A number of recent studies develop the idea that the mass infall from the ambient ISM continues during the lifetime of the disc, including the Class II stage (Padoan et al. 2024; Pelkonen et al. 2024; Winter, Benisty, and Andrews 2024). Such models describe a Bondi-Hoyle accretion regime and are in good agreement with the observed properties of the disc population, such as accretion rates, masses and sizes. This input of matter can play an important role in disc evolution and planet formation process (Vorobyov, Lin, and Guedel 2015; Vorobyov et al. 2016). In the FEOSAD model, the mass infall to the disc can be accounted for (Vorobyov, Lin, and Guedel 2015; Vorobyov et al. 2016), but in the present simulation this effect is not included. The mass infall from the envelope continues until the cloud is depleted of matter. Because of the thin-disc geometry, the gravitational contraction of the cloud proceeds in the plane of the disc. The matter is landing on the disc outer edge and is transported towards the star by the combined action of gravitational and viscous torques. The infall on the disc inner regions is therefore neglected. This is a reasonable approximation, considering that most of the matter and angular momentum in a three-dimensional cloud is located at relatively large polar angles and a flared outer edge of the disc intercepts most of them (see, e.g., Visser et al. 2009, Figure 1). In our modelling, we do not consider the continuous Bondi-Hoyle accretion and concentrate on the internal disc processes.",
"pages": [
3,
4
]
},
{
"title": "2.3 Evolution of volatiles",
"content": "We follow the evolution of four main volatile species: H 2 O, CO 2 , CO and CH 4 . These are the most abundant carbon- and oxygen-bearing ices observed in protostellar cores (Karin I. Öberg et al. 2011). In the model, each of these species can be present in three states: in the gas, in the ice on the surface of small dust, and in the ice on the surface of grown dust. Each species s is described by its surface density in the gas Σ gas s , on small dust Σ sm s , and on grown dust Σ gr s . Their distributions in the disc can change through three main processes: advection together with the corresponding component (gas or small/grown dust); exchange of mantles between dust populations due to grain collisions; phase transitions, including adsorption from gas to dust, and thermal and photo-desorption. Initially, all ices are on small grains and no volatiles present in the gas. The treatment of volatiles is adopted from Molyarova et al. (2021), who describe the models in more details. Here we recap main features of the chemical model. Our chemical model only describes phase transitions, i.e. adsorption and desorption, which includes thermal desorption and photo-desorption by interstellar UV radiation. These reactions were shown to be he most important for gas-phase abundances of most species (Ilee et al. 2011). Due to high computational costs, no other chemical processes, either gasphase or surface reactions are included, although they also may have significant effect on the composition of both ice and gas (Semenov and Wiebe 2011). The chemical evolution of the surface densities of volatile species is calculated from the system of equations where the mass rate coefficients per disc unit area of adsorption λ s and desorption η s for the species s are calculated for local conditions at every hydrodynamic step, separately for small and grown dust populations. Eqs. (9)-(11) are solved in two steps: first, the right-hand side is considered without the advection term. The case of pure adsorption/desorption represented by the right-hand side of the equation can be solved analytically (see Appendix A in Molyarova et al. 2021). This is done at every hydrodynamic step, before the dust growth step, when ices on small and grown grains are redistributed proportionally to mass exchange between the dust populations. This is followed by a transport step, when the surface densities of the volatiles are changed according to the fluxes of their respective gas or dust components between the cells. Restricting chemical processes to only adsorption and desorption allows the chemical step to be calculated fast, which is very important for computationally demanding hydrodynamic simulations. For each dust population, the desorption rate is a sum of thermal desorption and photo-desorption η = η td + η pd (the indices 'sm' and 'gr' are omitted for convenience). We operate under the assumption of zeroth-order desorption, i.e. the desorption rate does not depend on the present amount of ice ( Σ sm s or Σ gr s ). It implies that only the upper layers of the ice mantle are able to sublimate, which is a more appropriate approach for thick mantles. This assumption better describes desorption of CO and H 2 Oin temperature programmed desorption (TPD) experiments (Fraser et al. 2001; K. I. Öberg et al. 2005; Bisschop et al. 2006) than first-order desorption, which is more suitable for thin mantles of several monolayers. The rate of thermal desorption is calculated as where N ss = 10 15 cm -2 is the surface density of binding sites (Cuppen et al. 2017), E b (K) is the binding energy of the species to the surface, µ s is the species molecular mass, m p is atomic mass unit, k B is the Boltzmann constant. We follow Hasegawa and Herbst (1993) and use binding energy to calculate the pre-exponential factor in Eq. (12), the same way it was done in Molyarova et al. (2021). However, this approach was recently demonstrated by Minissale et al. (2022) to underestimate the pre-exponential factor by a few orders of magnitude, as it does not account for the rotational part of the partition functions of desorbing molecules. Total surface area of dust grains (small or grown) per disc unit area ˜ σ tot (cm 2 cm -2 ) is calculated for the adopted power-law size distribution with p = 3.5 as Here, ρ s = 3 g cm -3 is the material density of silicate cores of the dust grains. The model includes photodesorption of volatiles by interstellar irradiation, which is mostly relevant in the outer disc regions with lower optical depth. We do not consider the UV radiation field produced by the star and the accretion region as a source of photodesorption, assuming that they do not reach disc midplane due to high optical depth. The photo-desorption rate is calculated as where F UV = F UV 0 G UV (photons cm -2 s -1 ) is the UV photon flux expressed in the units of standard UV field and Y = 3.5 × 10 -3 + 0.13 exp ( -336K/ T mp ) (mol photon -1 ) is the photodesorption yield adopted from Westley et al. (1995) for water ice. The intensity of the interstellar UV radiation field with G 0 = 1 is F UV 0 = 4.63 × 10 7 photon cm -2 s -1 (Draine 1978). We assume that the disc situated in a star-forming region is illuminated by a slightly elevated unattenuated UV field with G env = 5.5 G 0 . For the disc midplane, which is illuminated from above and below, this field scales with the UV optical depth τ UV towards the disc midplane as The optical depth can be calculated as τ UV = 0.5( κ sm Σ d,sm + κ gr Σ d,gr ), where κ sm = 10 4 cm 2 g -1 , κ gr = 2 × 10 2 cm 2 g -1 are typical values of absorption coefficients in the UV for small and grown grains (Pavlyuchenkov et al. 2019, Fig. 1). We calculate the adsorption rate λ following Brown and Charnley (1990). It is proportional to the total cross-section of dust grains per unit volume, which can be obtained from the total surface area of dust grains per unit disc surface ˜ σ tot. To change the normalisation to the 2D case, ˜ σ tot needs to be multiplied by 1/2 H . Another factor of 1/4 follows from the difference between cross-section and surface area of a sphere. As a result, the adsorption rate is calculated as A more detailed derivation of rate coefficients for adsorption and desorption is presented in Section 2.3 of Molyarova et al. (2021). Table 1 summarises the molecular parameters used in both models in this work. Binding energies for H 2 O, CO 2 , and CO are based on the experimental data from Cuppen et al. (2017) for desorption from crystalline water ice. The binding energy for methane is taken from Aikawa et al. (1996). The values of the initial abundances relative to water f s are based on the observations of ices in protostellar cores around the low-mass protostars (Karin I. Öberg et al. 2011). They are transformed to the mass fraction relative to gas assuming the water abundance of 5 × 10 -5 relative to gas number density. Total initial mass of ices comprises ≈ 8.5% of the total mass of refractory grain cores. This is relatively low compared to the estimates suggesting comparable masses of ices and refractories in the discs (Pontoppidan et al. 2014). However, the lower fraction of ices is more suitable in our approach, that suggests that ice mantles do not change mass and radius of dust grains (Molyarova et al. 2021). As we are mostly interested in the elemental ratios and relative abundances of the considered ices, lower ice fraction is an appropriate simplification. However, we note that dust dynamics can lead to accumulation of ices and ice mantles exceeding masses of silicate cores in some disc regions, as was shown previously in Molyarova et al. (2021). Ice mantles also provide feedback on dust evolution. The model includes the effect of ices on fragmentation velocity v frag , which is the the maximum collision velocity leading to sticking instead of fragmentation. According to laboratory experiments, icy grains have higher v frag than bare silicate grains by an order of magnitude (Wada et al. 2009; Gundlach and Blum 2015). In Molyarova et al. (2021) we used the values of fragmentation velocity v frag = 1.5 and 15 m s -1 for bare and icy grains, correspondingly. Here we follow Okuzumi and Tazaki (2019) and adopt lower values of v frag = 0.5 and 5 m s -1 , which are more relevant for grains consisting of µ m-sized monomers. These lower values of v frag will lead to higher importance of fragmentation compared to Molyarova et al. (2021). To determine if a dust grain should be considered icy or bare, we compare the local total surface density of all ices on grown dust divided by Σ d,gr with the threshold value K , which is calculated as i.e., an icy grain must have at least one monolayer of ice. Here, a ml is the thickness of the ice monolayer estimated as the size of a water molecule 3 × 10 -8 cm. The material density of ice ρ ice = 1 g cm -3 and the mean radius of a grown grain is calculated as √ a ∗ a max for the power-law distribution with p = 3.5.",
"pages": [
4,
5,
6
]
},
{
"title": "3. Results",
"content": "To understand the distribution of the species, we first need to consider the global evolution of the disc and its structure. The distribution of volatiles and the C/O ratio is very sensitive to gas and dust substructures appearing in the disc. Variations in temperature and pressure lead to the complex shape and temporal evolution of the snowlines. The dependence of dust fragmentation velocity on the presence of ice mantles implies the feedback from the volatiles on dust and (through backreaction) on gas. Our modelling starts with the gravitational collapse of a flattened, slowly rotating molecular cloud. The protoplanetary disc is formed after the formation of the protostar, when the in-spiraling layers of the contracting cloud hit the centrifugal barrier near the inner edge of the sink cell, at a time instance depending on the initial core mass. The disc and the protostar are formed ≈ 53 kyr after the beginning of the cloud collapse in model M1 and at ≈ 78kyr in model M2. If not stated otherwise, times are specified counting from the beginning of the simulation, e.g. the 100 kyr time instance for model M1 refers to a ≈ 50kyr old disc, as the first stage includes cloud collapse as well. An important characteristic of young stellar objects is their variable accretion rate. Our modelling produces accretion bursts with the magnitude of ∼ 100 L ⊙ occurring every ∼ 10 4 years during the first hundred thousands years of disc evolution. These burst parameters are in line with the episodic accretion scenario (Hartmann and Kenyon 1985) and resemble the observed phenomenon of FU Ori type stars (see Audard et al. 2014, for a review). The luminosity outbursts heat up the disc and typically shift the snowlines further away from the star. Although this effect is temporary, it can be reflected in the distributions of the volatiles, and leave long-term imprints in the observed dust properties (Vorobyov et al. 2022). The detailed analysis of the effect of such outbursts on the distribution of the elements is worthy of a separate study and lies beyond the scope of this paper.",
"pages": [
6
]
},
{
"title": "3.1 Dust and gas structures",
"content": "During the first hundred thousands years of evolution, protoplanetary discs change from highly asymmetric and dynamic objects to nearly axisymmetric and slowly evolving structures. Figure 1 shows the examples of different gas and dust substructures that are present in the disc at different stages. The snapshots are shown for model M1, they include a young disc (160 kyr), an intermediate state (350 kyr), and a more evolved and axisymmetric disc (490 kyr). Each of the selected time instances represent some characteristic morphological features addressed below. In model M2, similar structures appear, although sometimes at different evolution times. In this subsection, we consider model M1 as an example and discuss these features, highlighting the properties of dust and gas substructures that are most relevant for the distribution of the volatiles. The earliest phases of disc evolution are characterised by a large-scale spiral structure in both dust and gas, as well as episodic appearance of clumps. They are the result of gravitational instability (GI) in a massive disc, as in our modelling, the disc mass comprises more than 0.1 of the stellar mass, which is roughly a threshold of the disc stability against GI (Vorobyov 2013; Kratter and Lodato 2016). This is illustrated by the Toomre Q -parameter (Toomre 1964) in the third column of Figure 1: inside the spirals and the clump Q < 1, which indicates the dominance of self-gravity over Keplerian sheer and gas density. The spiral structures become less prominent with time as the disc looses mass and the Q -parameter increases. However, spirals persist in the model throughout the disc evolution up to 0.5 Myr. For example, at 350 kyr, a very tight spiral is present in the gas, starting at the gas and dust ring at ≈ 10au. At 490 kyr, the spiral pattern is weak and exists at > 10 au distances, which are not displayed in Figure 1. One- or two-armed grand design spirals are common 100 - 200 kyr after the disc formation in the models. The analogues of such spirals in the observed young protoplanetary discs around low-mass stars are found, for example, in Elias 227 or WaOph 6 (Pérez et al. 2016; Huang, Andrews, Pérez, et al. 2018). It is not yet clear if the observed spirals are the result of GI or caused by a perturbation from a companion planet or a (sub)stellar object (Meru et al. 2017; Brown-Sevilla et al. 2021). A spiral with multiple clumps produced by GI was recently observed in the disc around a FUor V960 Mon (Weber et al. 2023). Another important feature of gas and dust spatial distributions is ring-like structures at various scales. The system of rings starts to form as early as 100 kyr, and develops to the high-contrast multiple ring structure (1 - 3 orders of magnitude difference between surface densities in rings and gaps), which is evident in the middle and bottom rows of Figure 1. While gas rings are also common, the annual structures are more prominent in the dust surface density, as well as in dust size. Some of the dust rings correspond spatially to the gas rings, while others are barely reflected in the gas distribution. Overall, the difference between the gas and dust structures develops with time, as the result of dust growth and drift (Testi et al. 2014). Only some particular substructures, such as the ring between 1 - 2 au, are present in the distributions of both gas and dust components. Another location where prominent rings form in both dust and gas is the water snowline. Here, the water snowline is defined as the location in the disc midplane where the amount of water in the gas equals to its amount in the ice (on both dust populations). There are multiple location in the disc where this happens. Water is frozen in most of the disc beyond 5 - 10 au, and we will refer to the furthest snowline dividing these outer frozen region from the inner one with the gas-phase water as a primary snowline. Generally, the primary snowline is roughly circular, but it can have asymmetries due to the spiral structure and an additional snowline may appear, e.g., around a gravitationally bound clump (upper rows in Figure 1). Besides, at ≈ 260kyr, another region rich in water ice appears in the inner disc, creating a pattern of double or triple water snowline at later times (middle and lower rows in Figure 1). We will refer to these inner additional snowlines as the secondary snowlines. They circumcise a cold and dense gas-dust ring that forms at 1 - 2 au. As the disc cools down with time, the primary snowline position moves from ≈ 10au distance at 160 kyr, to ≈ 6au at 350 kyr and ≈ 4au at 490 kyr. Snowlines are known to be associated with the enhancement of dust and volatiles (Stevenson and Lunine 1988; Cuzzi and Zahnle 2004; Drążkowska and Alibert 2017; Molyarova et al. 2021). Dust enhancement was also shown to affect the distribution of gas and its accretion rate through the disc (Gárate et al. 2020) by means of dust back reaction, which is also accounted for in our modelling. In our models, an about 2 au wide dust ring is formed at the inner edge of the water snowline (at ≈ 10au) as soon as 50 kyr after the disc formation. Grown dust grains drifting towards the star through the snowline lose their mantles, their fragmentation velocity drops, rendering them more vulnerable to fragmentation. Consequently, the grain maximum size a max decreases, their drift towards the star slows down, which leads to the accumulation of grown dust, as well as small dust as a product of fragmentation. Increase of total dust density also leads to less efficient cooling and results in higher temperature inside the snowline (see the right panels in Figure 1). Later, at times > 200 kyr, several additional rings form outside the water snowline at distances up to 100 au, the most notable one being 1 - 2 au exterior to the primary snowline. Dust is trapped inside the gas pressure maxima in these rings, which is a self-supporting phenomenon as the temperature also increases inside the dust ring due to high optical depth. These rings are worthy of attention in the context of possible planet formation. Dust surface density and size are higher in the rings, with a max reaching centimetres, dust-to-gas ratio up to 0.1 - 0.2, and Stokes number up to 0.05 - 0.1. This could ease the development of the streaming instability, which typically requires pebble-size grains with St ≳ 0.01 and dustto-gas ratio ≳ 0.02 (Carrera, Johansen, and Davies 2015). Multiple ring-like structures are commonly observed in protoplanetary discs at a range of ages and display a variety of examples, with different widths, contrasts and numbers of rings (Huang, Andrews, Dullemond, et al. 2018, and many others). However, to directly compare the ring structures in the simulated dust surface density with the observed dust emission, require radiative transfer modelling is needed. Some of the observed ring structures could be produced by radial variation in dust size even in the absence of gaps in dust surface density (Akimkin et al. 2020). The most prominent dust rings are located in the vicinity of the water snowline: the ring outside the primary snowline at 5 - 8 au (depending on the time) and the ring at 1 - 2 au, inside the primary snowline, which at later times also contains water ice and additional snowlines. The main effect of the snowlines is the change in fragmentation velocity between mantled and non-mantled grains: dust size sharply decreases by ≈ 2 orders of magnitude for the latter. Immediately outside the water snowline, the midplane temperature is lower, due to lower dust opacity and thus more efficient cooling. Turbulent α is on the contrary, higher, providing more efficient radial transport of matter. It leads to lowering the gas surface density in this gap, which in turn increases α (see Eq. (5)), creating the positive feedback and further deepening the gap. One of the possible mechanisms to create the initial decrease in the gas surface density is dust back-reaction, which can affect the inward flow of gas at the snowline. This effect was investigated by Gárate et al. (2020) for different initial dust-to-gas ratios. The radial variation of α itself lead to the appearance of gas substructures (Tong, Alexander, and Rosotti 2024). The combined effect of lower temperature and density creates a pressure minimum, which dust tends to avoid. The dead zone, where α -parameter values are lower than 10 -3 , includes the ice-free inner disc and has an approximately two times larger radial span than the iceless region. The distribution of α -parameter is shown in the right column of Figure 1. Inside the primary water snowline, the values of α are the lowest due to high surface density of gas. The dead zone is not axisymmetric and reflects the spiral structures of the gas distribution, as it is sensitive to Σ g (see Eq. (5)). The spiral arms of the dead zone span to 15 - 25 au at 160 kyr, to 10 - 18 au at 350 kyr and to 10 - 14 au at 490 kyr. The comparison with the temperature distribution in Figure 1 indicates that α and T mp are anticorrelated, as higher viscosity provides faster accretion, hence lower surface densities and more efficient cooling. Similarly, a lot of dust accumulates in the dead zone, increasing opacity, which hampers cooling. The icy dust ring at 1 - 2 au is especially interesting in the context of planet formation. In this ring, dust-to-gas ratio exceeds 0.1, and surface densities of both gas and dust are increased by more than an order of magnitude compared to the adjacent regions. Due to the ice mantles that protect dust from fragmentation, the dust size reaches several cm, close to the values behind the water snowline. When the ring is established, it is self-supporting, in a sense that without external perturbation (e.g. sharp change in accretion flow from the outer disc), it can be stable for a long time, over tens of kyr. The presence of water ice inside the dead zone was investigated by Vallet et al. (2023). They showed that in the discs around lower-mass stars, the turbulent heating in the inner disc can be low enough to allow the existence of ices. In our modelling the cooling of the inner region is assisted by the inner dust ring. The freeze-out has a positive feedback on dust growth due to higher v frag , which helps dust grow larger and further accumulate toward the pressure maximum in the ring. So in our modelling ices coexist with a lot of grown dust in the dead zone. Such an icy inner region appears to be a promising place for the formation of the volatile rich planets.",
"pages": [
6,
7,
8
]
},
{
"title": "3.2 Distribution of volatiles",
"content": "Even in the absence of chemical reactions, over the years of disc evolution the distribution of volatiles significantly changes compared to the initial one. This is the result of both phase transitions and dust growth and advection. Dust drift brings ices from the outer disc, enriching the inner disc with volatiles. Snowlines provide the conditions favouring accumulation of ices and gas-phase species. The formation of the established disc structures, such as dust and gas rings and spirals (see Section 3.1) leads to a complex pattern of intermittent snowlines. In this Section, we describe the main features of the molecular distributions and their implications for the composition of dust and gas at early stages of protoplanetary disc evolution. Figures 2 and 3 show the azimuthally averaged radial profiles of the volatile species in models M1 and M2, respectively. As mentioned earlier, we define the snowline position as the location in the disc midplane where the amount of species in the gas equals to its amount in the ice (on both dust populations). This definition allows the snowline to have complex shape, characterised by different positions for each azimuthal direction. In the azimuthally averaged distributions shown in Figures 2 and 3, the position would only be approximate. The slope of the species distribution in the vicinity of the snowline reflects the degree of the axial asymmetry in the disc. Flatter distributions appear as the contributions sum up from the snowlines, the radial position of which vary at different azimuthal angles ϕ . In our model setup, the volatiles can either have no snowline in the disc, or have two or more snowlines depending on the local conditions. Snowlines are absent for more volatile species (CO and CH 4 ) at earlier times or during bright luminosity outbursts, when the disc is too hot for them to freeze out. When there are two snowlines, the inner snowline in the disc is the one determined by thermal desorption. We refer to it as the primary snowline. The outer snowline is determined by photo-desorption, it lies in the embedding envelope outside of the body of the disc where optical depth is low. It must be noted that the gas-phase species outside this photo-snowline are vulnerable to photo-dissociation by the UV radiation. This process is not explicitly included in our chemical model, but can be assessed as in Molyarova et al. (2021). More than two snowlines appear when the disc physical structure becomes more complex, mainly due to the presence of the ring-like structures. Species can freeze inside a cold dense dust ring, creating additional secondary snowlines, also governed by thermal desorption. The concepts of primary and secondary snowline is necessary for H 2 O and CO 2 , which have multiple snowlines at the later stages of disc evolution. For water, the inner icy region appears at ≈ 1au at 490 kyr (see right column of Figure 3) inside a dense dust ring. For CO 2 , the ring that appears at 7 au, outside the primary water snowline at 4 au, creates the inverted thermal structure in the region with T mp close to the typical CO2 sublimation temperature of 70 - 90 K. Increase in T mp in these ring is caused by higher optical depth and consequently lower cooling on the viscously heated midplane. Apart from the snowlines, the distributions of volatiles in Figures 2 and 3 present local radial variations in all of the species components, including total abundance of the species. The initial total abundance is kept only in the envelope. As the matter is being redistributed, abundances of all volatiles in the inner disc grow. Particularly, the process responsible for this is dust drift. It brings the grown ice-covered grains to the inner disc regions, where their ice mantles are sublimated and no longer move with the drifting silicate dust cores. The effect is stronger for less volatile species H 2 Oand CO 2 : their abundance in the gas grows by a factor of a few inside their primary snowlines. For CO and CH 4 the effect is weaker, because there is less grown dust outside their snowlines, and those snowlines are not very much established at earlier times. Thus their abundances inside the snowline only grow by a factor of unity, except for the immediate vicinity of the snowline. Moreover, both CO and CH 4 have a bump in the gas-phase distribution just outside the dust ring at 1 au, that is absent in H 2 Oand CO 2 . At the inner disc edge the abundances of CO and CH 4 in the gas are lower than the initial value. The abundance enhancements appear most notably at the snowlines, with local bumps in all three phases at later times (after ≈ 350kyr). They are produced by the combination of the dust radial drift and the azimuthal oscillations of dust and gas radial velocity, described in more detail in Molyarova et al. (2021) and Molyarova et al., in prep. By 500 kyr, the surface density of gas-phase H 2 Oexceeds the initial value by a factor of 5, of CO 2 - by a factor of 7, of CH 4 - by 3.5 and of CO by 2.5. Similar accumulation powered by turbulent diffusion was previously studied for CO molecule in axially symmetric model setup (Stammler et al. 2017; Krijt et al. 2018). Their results indicated similar enhancement in the gas phase by a factor of a few. In our non-axisymmetric approach, diffusion is not a necessary requirement, and the necessary transport is rather provided by azimuthal variations of radial velocity induced by disc self-gravity (Molyarova et al., in prep). Distributions of ices on small and grown dust are different as they are affected by dust growth and drift. In general, there is more grown dust than small by mass, and in most of the disc there is more ices on grown dust, particularly at later times. However, in the outer disc and at the earlier times, ices on small grains dominate. As initially all ices are on small dust, it seems inevitable that they will gradually move to the grown grains as small dust coagulates and turns into grown grains. However, near all the snowlines, amount of ices on small dust increases due to the effect of the spiral pattern. Ice abundances are determined by episodic sublimation and freeze-out in the warm spiral arms of the complex density and temperature pattern (see Section 3.3.2. in Molyarova et al. 2021). As adsorption preferably happens to the smaller grains due to their larger total surface area, there is much more ices on small dust in the wake of the spiral arms. This concerns the photo-desorption snowlines as well.",
"pages": [
10
]
},
{
"title": "3.3 C/O ratios",
"content": "Here we consider the relative amount of carbon and oxygen in the gas, on the surface of small and grown grains, and in total for all for all phases and disc locations. C/O ratio is seen as a perspective tracer of planet formation mechanisms (see Öberg, Murray-Clay, and Bergin 2011; Thiabaud et al. 2015). Therefore, we are especially interested in the regions of the disc and the volatile phase component where the C/O ratio declines from the total initial value (see below). Particularly, weare interested in C/O ratio noticeably above the initial value, as it was observed in atmospheres of exoplanets and suggests the formation of these planets in similarly carbon-enriched environments (Swain et al. 2009; Madhusudhan et al. 2011; Moses et al. 2013; Facchini et al. 2021). The C/O ratio is calculated as the total number of carbon atoms contained in the molecules, divided by the total number of oxygen atoms in the molecules. This calculation can be done separately for the species contained in the gas phase, species on dust surface, or for the species in all phases. Thus, we calculate the C/O ratio in the gas, in the ice, and total C/O, respectively. For the ice species, we include both ices on grown and small dust grains.In some disc regions, the amount of carbon and oxygen in in a particular phase is very low, e.g. in the ice phase inside the water snowline, where all the molecules are in the gas. For such cases, C/O ratio would not be representative of the chemical composition, so we exclude the corresponding computational cells from the consideration. We only display the C/O ratio if the mass of the volatiles in the phase is higher than 10 -3 of the total mass of volatiles at a given location. For the adopted molecular abundances based on Karin I. Öberg et al. (2011) and also used by Eistrup, Walsh, and van Dishoeck (2018), the baseline value is C/O = 0.34, which is in line with the gas-phase abundances in the ISM ( ≈ 0.36, Przybilla, Nieva, and Butler 2008). This value is different from the typical solar value of ≈ 0.5 (Przybilla, Nieva, and Butler 2008), which is commonly used as a reference (e.g., Öberg, Murray-Clay, and Bergin 2011; Semenov et al. 2018). First, local galactic abundances changed since the Solar system formation 4.6 billion years ago (see, e.g., Appendix A in Bergin et al. 2024, for the comparison). Second, the difference also stems from inclusion or non-inclusion of the dust component. The stellar atmospheric abundances comprise all the elements present in the medium where the star formed, while the elements available for the volatiles are only a fraction of that. The values of the elemental abundances can be determined separately for the volatile (gas and ices) and the refractory (rocky dust grain cores) components of the ISM (Hensley and Draine 2021). Here, we do not take into account carbon and oxygen from the rocky cores of dust grains. Due to the model assumptions, the initial ice-to-rock mass ratio in our model is quite low (0.08 compared to 2 - 4 in Pontoppidan et al. 2014). Therefore, adding the elements from solid dust grains to those contained in ices would be misleading, as the former would dominate in the resulting C/O ratio. In this work, we concentrate on considering the C/O ratios of the volatile component (ice and gas), and compare them with the initial value of 0.34. By the age of 490 yr, the C/O ratio in both models is significantly different from the initial value. This concerns the total C/O ratio, as well as the C/O ratio in the gas and in the ice. Let us analyse the distributions of C/O ratios that we can see from Figure 4, moving inwards from the envelope to the centre. The main features of the C/O distributions are the following: As expected, the key changes in the C/O ratio distribution are associated with the positions of the snowlines. There are two main processes. First, the freeze-out and desorption at the snowlines transfer the elements between phases, altering the C/O in the gas and in the ice. Second, the snowlines favour the accumulation of the respective volatiles both in the gas and in the ice, as was discussed in Section 3.2, pumping up the amount of both components and altering their proportion. Besides the snowlines, radial drift of grown grains (Weidenschilling 1977) transports the volatiles inwards. We discuss below which processes are responsible for the formation of the listed features. C/O in the outer disc. In the surrounding envelope, outside the disc, all volatiles are in the gas due to photo-desorption, and C/O in the gas is close to the initial value of 0.34. Around the COsnowline and beyond, the C/O in the gas is close to unity, and in the gas C/O is higher that the initial, which requires explanation. The region beyond the CO snowline is usually described as the place where all species are frozen, thus having a stellar C/O ratio in the solid phase and practically no carbon or oxygen in the gas (e.g., Öberg, Murray-Clay, and Bergin 2011; Öberg and Wordsworth 2019; P. Mollière et al. 2020). In our simulations, only in model M2 there is a region where less than 10 -3 of C and O is in the gas, and in model M1 such region is absent. Due to the asymmetric spiral structure that persists even at 490 kyr, even though most of CO is frozen beyond the snowline, there is a significant (> 10 -3 ) fraction of it in the gas. Additionally, the position of CO snowline itself is significantly affected by the dust drift, as it declines from the equilibrium between adsorption and desorption.The indistinctness of the CO snowline also helps CO to persist in the outer regions: all other species are ultimately frozen, so they are efficiently carried away to the inner disc via radial drift, while this works worse for only partially frozen CO. It creates relative overabundance of CO in the outer disc, elevating the total C/O ratio and later the ice-phase C/O ratio, when the preserved CO freezes out. Between the CO ice line and the envelope, there is a gradient of C/O in the gas due to photodesorption of the ices. The last molecule to be photo-desorbed is water, which returns oxygen to the gas phase at the farthest radial distance. C / O ≈ 1 around the CO snowline. Beyond the CH 4 snowline, CO dominates the composition of volatiles in the gas phase, leading to the gas-phase C/O close to unity, the value characteristic of the CO molecule. At the CO snowline, the CO dominates the ice-phase composition as well. While dust drift substantially lowered the abundances of CO 2 and H 2 Oby 490 kyr in these regions, CO ice accumulated at the snowline. This leads to the ice-phase C/O closer to 1, too, making the vicinity of the CO snowline a region where total amounts of C and O are similar. High C/O in the gas, low C/O in the ice. In the disc regions beyond the primary CO 2 snowline, only CO and CH 4 are in the gas, meaning the dominance of carbon and C/O > 1. Consequently, the C/O in the ice is generally lower, around 0.2, as the ices are mainly H 2 O and CO 2 rich in oxygen. This is consistent with the classical step-like picture of Öberg, Murray-Clay, and Bergin (2011), with the addition of carbonrich methane allowing the C/O > 1. The midplane C/O ratios we simulate are difficult to directly compare with observations, which mostly trace the molecular layer above the disc midplane. High C/O ratios in the gas are indeed observed in many protoplanetary discs (e.g. Miotello et al. 2019), but they are considered to be a natural consequence of dust settling (Krijt et al. 2018; Krijt et al. 2020). Dust inward drift could also enhance this effect in the outer disc regions, creating the radial gradient of total C/O ratio in the disc. Observations of CS and SO emission coming from close to the midplane layers potentially indicate the presence of such radial gradient of the gas-phase C/O in the PDS 70 disc (Rampinelli et al. 2024). Variations of total C/O. Throughout the disc, there are sharp changes in total C/O ratio not connected with any snowline. Most noticeable are the variations between CO 2 and CH 4 snowlines, where gas-phase and ice-phase C/O ratios are stable. These variations are associated with the disc substructures, particularly with the dense dust rings described in Section 3.1. The total C/O changes due to radial variations in dust-to-gas ratio: when it is higher, the total C/O is closer to the ice-phase C/O, and vice versa. Inside the dust-rich rings, H 2 Oand CO 2 ices are abundant, due to high surface density of dust relative to gas. At the same time, CO and CH 4 in the gas phase have similar surface densities inside dust rings and between them. Thus, in the dust rings, CO 2 and water are overabundant, leading to lower total C/O ratio. We consider this effect in more detail in Section 3.5. Variations of the total C/O ratio due to dust substructures are also present in the cold ring at ≈ 1au. C/O peaks at the snowlines. There are peaks of the C/O ratio in the ice right outside the snowlines of CO 2 , CH 4 and CO, produced by the accumulation of the respective ices (see Figures 2 and 3). In M2 model, the amount of CH 4 and CO ices at their respective snowlines becomes comparable with or even larger than those of CO 2 and H 2 O(compare the right panels of Figure 3), leading to C/O in the ice ≈ 0.6 - 0.9. In model M1, the accumulation is more prominent, so the C/O ratio in the ice approaches unity at CO snowline and > 1 at the methane snowline. These peaks distort the pattern of generally low C/O ratio in the ice and preset additional regions where carbon-rich planetesimals could be formed, and carbon-rich pebbles could be accreted onto forming protoplanets. At the snowlines of H 2 Oand CO 2 , the C/O in the ice approaches the C/O ratios of these molecules, 0 and 0.5, respectively. Lower C/O inside the water snowline. Inside the primary water snowline, the C/O ratio is generally lower than outside of it, close to the initial 0.34. Contrary to the outer regions depleted of ices due to the radial drift, the disc parts inside and around the water snowline are enriched in volatiles, and particularly of oxygen-rich water and CO 2 . Total and gas-phase C/O ratios vary from ≈ 0.2 to 0.6, as the secondary water snowlines add more substructure to the C/O distribution. The regions where the only ice is water and the ice-phase C/O = 0 are the inner cold dust ring at 1 au and the narrow annuli between water and CO 2 snowlines. The enrichment of the inner disc regions with oxygen as a result of dust radial drift is suggested by the resolved observations of molecules in protoplanetary discs (Banzatti et al. 2020). These characteristic features of the C/O distributions are similar in the two presented models. The main difference is the radial distances where the borders between the zones are located; they are closer to the star in the less massive and thus colder model M1. This is mainly due to slightly different masses of the central star accumulated throughout 490 kyr of non-identical protostellar accretion history, which lead to different luminosity and thermal structure (see upper panels of Figure 5). Particularly, the stellar masses and luminosities at this time instance are: 1.07 L ⊙ and 0.34 M ⊙ (for M1), 1.89 L ⊙ and 0.58 M ⊙ (for M2). The less massive model M1 demonstrates overall higher C/O ratio in both phases.",
"pages": [
11,
12,
13
]
},
{
"title": "3.4 Evolution of the snowlines and the C/O ratios",
"content": "The positions of the snowlines are crucial for the values of C/O ratios, both because of the direct change through the phase transitions and the associated accumulation of volatiles. They depend on the local gas and dust properties, particularly on temperature. They can also be shifted inward due to dust drift (Piso et al. 2015). Snowlines evolve as the disc structure changes with time. In this Section, we consider the co-evolution of the snowlines and the C/O ratios and discuss the mechanisms of species redistribution over the disc. The temporal evolution of the azimuthally averaged C/O ratios and the equilibrium positions of the snowlines is shown in Figure 5. It is evident from Figure 5 that the C/O ratio indeed follows the snowlines, particularly inside ≈ 10 au. The C/O structure changes throughout the disc evolution, and some key features only appear at later times. One of the key factors affecting the disc thermal structure is the luminosity of the central source. In the upper panels of Figure 5, we show the evolution of total luminosity, which directly affects the positions of the snowlines. The luminosity is the sum of stellar and accretion components, coming from the protorstar itself and gravitational potential energy of the accreted matter. The stellar luminosity gradually decreases as the protostar becomes more compact. Accretion luminosity depends on the accretion rate, which is highly variable as a result of magnetorotational and gravitational instabilities in the disc (Kadam et al. 2019, 2020; Vorobyov, Khaibrakhmanov, et al. 2020). The simulated episodic luminosity outbursts are similar to those occurring in the observed YSOs (Connelley and Reipurth 2018), with their amplitudes of tens and hundreds L ⊙ . In the more massive model M2, the outbursts are more frequent, brighter and occur until later times. The reasons behind this difference is the massive disc being more prone to both MRI and GI, which needs to be investigated in more detail in a separate study. Snowlines of the least volatile of the considered species, H 2 Oand CO 2 , exist in the model since the earliest phases of the disc formation. Even during bright luminosity outbursts ( ∼ 100 L ⊙ ) they do not disappear, but move farther away from the star. During the first ≈ 50kyr the disc is spreading out, it is highly asymmetric and dynamic, so the snowline positions oscillate. Later on, the disc generally cools down, and the snowlines gradually move toward the star (except during the outbursts). In model M2, between 130 and 500 kyr, the primary water snowline moves from 12 to 4.5 au; the primary CO 2 snowline moves from 23 to 7 au. In model M1, the water snowline moves from 9 to 4 au, and the CO 2 snowline moves from 17 to 4.5 au. Despite a factor of two different binding energies of H 2 O and CO 2 (5770 and 2360 K, respectively), the locations of their primary snowlines do not differ much due to the steep radial temperature gradient around these distances (see upper panels of Figures 3 and 2). Inside 10 - 20 au, T mp is determined by heating mechanisms other than external irradiation: viscous heating, gas work (PdV heating), heating by shocks and energy transport with advection. This also means that the water snowline is less sensitive to the level of irradiative heating, thus only slightly affected by the luminosity outbursts. The temperature change is particularly sharp at the water snowline(s), with the absolute value of the approximated power-law slope of 1.5 - 6. High surface density of dust in water-ice-free regions makes cooling less efficient and leads to locking up the produced heat and consequently higher temperatures. Besides, both these species have multiple snowlines due to the formation of ring-like substructures with the conditions close to the borderline between their frozen and gaseous state. These additional snowlines also affect the C/O distributions. Methane and CO are the more volatile species in our model. They either have zero or two snowlines in the disc. The snowlines are absent during the outbursts brighter than ≈ 100 L ⊙ for methane and ≈ 200 L ⊙ for CO. Until approximately 200 - 250 kyr, there is no established CO snowline inside the disc. For example, at 160 kyr (see lower left panel in Figures 3 and 2) there is CO ice on both small and grown dust, but their amount is an order of magnitude lower than that of the gas. Additionally, most of this ice is located at the outer disc edge, where gas and dust surface densities sharply drop. Similar distribution appears for CO and CH 4 during bright outbursts: their ices are present in some disc regions, but due to non-axisymmetric disc structure, they do not dominate in the averaged profiles, and there is no common snowline for the whole disc. There are no secondary snowlines of CO or CH 4 , because there is no prominent gas and dust structures in the outer disc where these species are frozen. Snowlines divide the disc into several zones with different characteristic C/O ratios. However, the chemical composition and C/O ratios in these zones change with time. One of the distinct zones is the region where water is not frozen, shaded with purple in Figure 4 and circumscribed by the dark purple dotted line in Figure 5. In this zone, all the species are in the gas phase, so the C/O ratio is initially close to 0.34. However, as the disc evolves, dust brings more volatiles from the outer disc. Particularly, the abundance of water grows most significantly, making C/O in the gas decrease with time. This happens because the main mechanism of the transport of the volatiles is dust radial drift, which works best in the regions where grown grains can sustain their mantles. Banzatti et al. (2020) suggest that dust growth and drift can be responsible for the observed anticorrelation between disc radius and H 2 O emission, implying that the inner regions of small discs with are enriched in water brought by efficiently drifting grains. Water is the least volatile species in our model, it is frozen in the largest part of the disc, thus its distribution is most strongly affected by the dust drift. At later times, this zone is divided into two, when a cold dense dust ring forms at 1 - 2 au, which happens at ≈ 460 kyr in model M2 and at ≈ 270 kyr in model M1. Interior to the ring, water abundance in the gas is around an order of magnitude lower than outside of it in both models. This happens because the inward flow of gasphase H 2 Ois 'blocked' by the cold ring where it freezes and accumulates with the grown grains in the pressure maximum. So the C/O ratio in the gas at r ⪅ 2au is determined by CO 2 and thus close to 0.5. The C/O ratio in the ice is not defined in the envelope, where all ices are photo-desorbed, and in the warmest inner disc, where even water is in the gas. In disc regions where only water is frozen, the C/O ratio in the ice is zero. Before 200 - 250 kyr, when the disc cools down enough for CO to freeze out, the C/O ratio in the ice in the rest of the disc is close to 0.2. It is determined by CO 2 and H 2 O ices, with a small contribution from the low-abundance CH 4 . After that, when CO freezes out and CO snowline appears, the C/O ratio in the outer disc region becomes close to the initial value both in the ice and in total. This region beyond the CO snowline is frequently referred to in relation to giant exoplanets with stellar C/O ratios (e.g., P. Mollière et al. 2020; Öberg and Wordsworth 2019; Ohno and Ueda 2021). This region would be a perfect location for planets to accrete pebbles covered with icy mantles with the primordial elemental ratio, which would directly become part of the planetary atmosphere. However, pebbles are not initially present in the disc, and their existence in these outer regions is not guaranteed. Pebbles are dust grains large enough to move relative to gas e.g., Lambrechts and Johansen 2012; Lenz, Klahr, and Birnstiel 2019, and a fraction of the grown dust in our modelling can be classified as pebbles. The properties of pebbles and composition of their ice mantles in the same setting of the FEOSAD model were studied by Topchieva et al. (2024). They show that pebbles appear in the disc as early as 50 kyr after its formation, and exist in a wide region of the disc. This partially includes the region beyond the CO snowline, but only the area of CO enhancement, where CO dominates in the ice composition and thus the C/O ratio in the ice is close to unity (see lower panels in their Figure 3). As shown by Topchieva et al. (2024), ices on pebbles are dominated by H 2 O and CO 2 . In this case, relatively high values of the C/O in the ice ( ≈ 0.5) correspond to the regions where there is more CO 2 , i.e. around the CO 2 snowlines. This is also the region where a prominent dust ring forms under the influence of the primary water snowline. It is characterised by accumulation of CO 2 and to lesser degree H 2 Oice, as well as vapours, and relatively high amount of grown dust in the ring. The dust ring is situated between the two regions with C/O ≈ 0.34 in the ice. It presents another favourable location for accreting the ice content with close-to-initial C/O ratio. The total C/O ratio in the disc also changed significantly from the initial value due to dust drift that redistributes the ices. The matter becomes more carbon-rich as the grains bring CO and CH 4 from the outer disc parts. This effect was previously investigated by Stammler et al. (2017) and Krijt et al. (2018) in their modelling of CO dynamics and dust evolution. However, as was shown by Krijt et al. (2018) and Krijt et al. (2020), vertical settling of grown grains towards the midplane is responsible for depleting the upper layers in the outer disc of gas-phase oxygen, which cannot be captured within our thin-disc modelling. The panels in the second row of Figure 5 demonstrate strong enhancement of total C/O ratio in the intermediate disc regions. In model M2, the total C/O ratio between 10 - 100 au becomes ≈ 0.7 after ∼ 300 kyr, which is two times higher than the initial value. At the snowlines of CH 4 and CO 2 it approaches unity, mostly due to the accumulation of these species in the gas. In model M1, this process begins ≈ 200kyr earlier and consequently leads to even higher total C/O ratio. At 400 - 500 kyr, most of the disc between 5 and 100 au has total C/O ≳ 1 in model M2, which demonstrates the powerful impact of dust drift. A distinctive feature of the produced C/O distributions is the peaks of total and ice-phase C/O ratios around the CO and CH 4 snowlines. In model M2, the increase of the C/O ratios becomes noticeable only after ≈ 450kyr, while in model M1 it starts to form around ≈ 300 kyr. Initial abundances of CH 4 and CO are lower than that of CO 2 and H 2 O. As these species accumulate at their snowlines, the abundances become comparable, leading to C/O in the ice ≈ 0.6 - 0.8 in model M2 and up to 1 in model M1. Similar accumulation is seen around CO 2 snowline, but unlike CO and CH 4 snowlines, it is also connected to the interaction with the dust ring structure and is considered in more detail in Section 3.5.",
"pages": [
13,
15
]
},
{
"title": "3.5 Two-dimentional structure",
"content": "The disc is not axisymmetric even at later stages of its evolution (see Section 3.1 and Figure 1), which is also reflected in the distributions of volatiles and C/O ratios. Examples of 2D distributions of C/O in model M1 at two time instances are shown in upper panels of Figure 6. The left-hand side group of panels shows the structure that is characteristic of the earlier phases, the same time instance as the upper row in Figure 1. The prominent spiral structure and a clump both have their reflection in the C/O ratios. The snowlines of CO 2 and CH 4 have clearly non-regular shape affected by the spiral pattern of the gas, with blobs of frozen methane inside the main snowline. Inside the clump, the temperature is higher (see Figure 1), and both CO 2 and water are in the gas. The separation between gas-phase and ice-phase C/O ratios is clear, but the total C/O ratio in the clump is similar to the surroundings and is only slightly above the primordial value. The gas-phase C/O ratio inside the clump is around 0.7, lower than in the surrounding gas at the same radial distances. Similar decrease of C/O ratio indicated by lowered CS/CO ratio was observed in the disc around DR Tau (Huang et al. 2024). Lifetime of the clump (several orbital periods) is too short for significant differentiation between gas- and ice-phase composition to develop, mainly because of insufficient numerical resolution. Focused studies with higher resolution are needed to explore the C/O ratio in the clumps as precursor of giant planets formed via disc fragmentation. The upper right-hand side group of panels in Figure 6 features a later stage of the disc evolution in model M1. By 300kyr, the accretion rate and the average positions of the snowlines are stabilised (see Figure 5). The spiral structure in the gas is much weaker but still present at r > 10 au. The CO snowline is established at ≈ 80au. The 2D shape of the snowlines is more circular at 300 kyr, particularly for the less volatile CO 2 and H 2 O. The effect of spirals is still evident in the contours of the CO and CH 4 snowlines. At the outer side of these snowlines, at approximately 40 and 80 au, respectively, the C/O in the gas has local maxima, which are also seen in Figure 4. The one at 40 au also has a clearly spiral shape, repeating the pattern of the gas distribution. The radial span of these peaks is of the same scale as the dispersion of the respective snowline distance from the star. For example, at the CH 4 snowline the C/O peak is ≈ 10au wide, and the snowline is at 28 to 37 au distances. Similar accumulation powered by diffusion of water vapour was shown, e.g., by Drążkowska and Alibert (2017). In addition to diffusion, in our modelling, the species are delivered outward from the snowline due to the dynamic shape of the snowline and two-dimensional movement of gas and dust (see Molyarova et al. 2021, and Molyarova et al., in prep). In more massive model M2, the shape of the snowlines remains complex for longer times. Examples of C/O distributions in M2 are shown in the lower panels of Figure 6. The radial scale is chosen so that CO 2 and H 2 O snowlines are seen in more detail. At 250 kyr, even the grown dust distribution still has spiral pattern, and CO 2 snowline is following its complex multi-armed shape. The accumulation of CO 2 at the snowline is also very efficient, but it does not strongly affect the C/O ratios, as it drives them to the value 0.5 of the CO 2 molecule, which is close to the intrinsic value. The complex-shaped region between these snowlines has the ice-phase C/O ≈ 0.7. This pattern moves and changes its shape, thus affecting all radial distances between 10 and 20 au. The total C/O ratio is slightly above the ambient value, approaching 0.5, because CO 2 in both phases begins to accumulate in this region. At later times, the C/O distribution becomes more complex due to the presence of disc substructures, particularly the dust rings. The lower right-hand side group of panels in Figure 6 show this in more details. First, the temperature and density variations across the dust rings lead to the formation of multiple CO 2 and H 2 Osnowlines. An additional annulus of icy CO 2 appears in a relatively cold region between the dense dust rings at 6 - 7 and 8 - 13 au. The dust rings are warmer due to higher optical depth and active heating by disk internal sources, and the inner edge of the 6 au ring is warm enough to sustain gas-phase CO 2 . Second, the accumulation of icy dust grains in the rings alters the total C/O ratio. The total C/O ratio anticorrelates with the distribution of grown dust grains: inside the dense rings it is close to the initial value of 0.34, while between the rings, it is higher and reaches 0.8 - 0.9. At the same time, neither ice-phase, nor gas-phase C/O ratio displays any noticeable variations at 12-25 au, but they do have variations at r < 12 au, following the dust ring pattern. The total C/O ratio is defined by the combination of the ice- and the gas-phase component. Their relative contribution is proportional to the dust-to-gas mass ratio. This is illustrated in Figure 7. Within the rings, the total C/O is dominated by ices of oxygen-rich species CO 2 and H 2 O, particularly on grown dust accumulated in the pressure maxima. For example, there is a lot of water ice at 6 - 7 au, and its contribution to the total C/O is weighted with high dust-to-gas ratio of almost 0.1, so the resulting total C/O is lower than the initial value. At the same time, at ≈ 5au, dust-to-gas ratio is around 10 -4 . The dominant species there are in the gas, ice surface densities are 1 - 2 orders of magnitude lower, so the total C/O ratio goes up. In the wide dust ring at 9-13 au, both H 2 Oand CO 2 contribute, although at the warmer inner edge of the dust ring CO 2 is sublimated. The value of total C/O is approaches 0.34, also elevated by the presence of gas-phase CO and CH 4 . Beyond ≈ 10au, both H 2 O and CO 2 are frozen, and the variations of the total C/O ratio are clearly anticorrelated with the dust-to-gas ratio and the position of the rings. Between the dust rings, where contribution of ices is low, the C/O is determined by CO and CH 4 gases, and reaches values of ≈ 1. The anticorrelation between the total C/O ratio and dustto-gas mass ratio is an interesting finding. It is illustrated in Figure 8 for both models. Only the points within the disc are shown, with Σ gas > 0.1 g cm -2 .The pattern obviously changes with time, but the anticorrelation persists. Model M1 demonstrates wider variety of C/O and dust-to-gas ratios. The disc points are grouped in tangled curved \"branches\", some of them steeper than others. The 'width' (or spread) of these branches is determined by the azimuthal substructures. In the axially symmetric parts of the disc the values of dust-to-gas ratio and the total C/O are similar at a given radial distance. Different branches, which can be closer to vertical or horizontal orientation, are the result of the radial variations in the ice-phase C/O ratio and in ice fraction relative to the dust silicate cores. Horizontal branches correspond to weak or absent anticorrelation. For example, near the snowlines of carbon-rich species, C/O in the ice is high and close to that of the gas, which decreases the effect of dust mass fraction on the total C/O. In areas with no ices, the anticorrelation is also irrelevant, because the total C/O is determined entirely by the gas phase. Vertical branches, on the contrary, correspond to the strongest anticorrelation effect, which is expected in the regions between the snowlines, where ice-phase C/O is the lowest. The identified anticorrelation in Figure 8 is similar to the results of chemical population synthesis modelling (Cridland et al. 2019, 2020) showing that the more solids a planet accreted in the disc, the lower the C/O ratio is in its atmosphere. We fit the data for t = 490kyr with a linear law (taking the logarithm of dust-to-gas ratio) and obtain the following fits: C/O = -0.056 - 0.25 log 10 ( ξ ) for model M1, and C/O = -0.18 - 0.27 log 10 ( ξ ) for model M2. Here, ξ is the dust-to-gas mass ratio. The correlation coefficients are -0.54 for M1 and -0.57 for M2. At the shown earlier times (160 and 350 kyr), the correlation coefficient changes between approximately -0.9 and -0.5, which indicates noticeable anticorrelation throughout the disc evolution. We note that these C/O ratios only include the volatile component, without the contribution from the refractory material. Although the refractory material is typically considered as silicates, which are rich oxygen, it contains a significant amount of carbon, with the resulting C/O ≈ 0.5 (see Table 2 in Hensley and Draine 2021). This solid carbon can be subject to carbon grain destruction (Lee, Bergin, and Nomura 2010; Gail and Trieloff 2017; Wei et al. 2019), but this process should be treated separately, as it also affects the gas-phase carbon abundance. Taking refractory cores into account should affect the dependence between total C/O ratio and dust-to-gas ratio, as adding more rock would make the C/O ratio closer to 0.5. Thus the degree of the anticorrelation must be affected by the composition of rocky cores, but the anticorrelation itself should remain even when refractories are included, because the C/O ≈ 0.5 is still lower than typical C/O of the gas in most of the disc ( ⪆ 1). Dust rings are detected in the majority of the observed protoplanetary discs (Long et al. 2018; Huang, Andrews, Dullemond, et al. 2018). They are considered as a plausible sites of planet formation (Carrera, Johansen, and Davies 2015; Yang, Johansen, and Carrera 2017; Li and Youdin 2021; Lee, Fuentes, and Hopkins 2022; Jiang and Ormel 2023). The anticorrelation between the total C/O ratio of the volatiles and dust-to-gas mass ratio that we point out is a logical consequence of ices having typically lower C/O ratios and being attached to dust grains. If planets are formed in the dust rings with high dustto-gas ratios (> 10 -2 ), either exclusively from solids, or with the inclusion of the dust component, this would imply that their material initially has lower C/O ratio of ≈ 0.5 and below. To reach higher C/O ratios up to unity and above, which are observed in many exoplanets, these planets would need to migrate and accrete carbon-rich gas from regions other than their immediate formation sites inside the dust rings. In case if planet formation occurs independently of the dust rings, e.g. in the GI, their material is not determined by this anticorrelation.",
"pages": [
15,
17,
18
]
},
{
"title": "4. DISCUSSION",
"content": "Our simulations present a wide range of C/O ratios in the disc in different phases evolving with time. For the atmospheres of giant exoplanets, a variety of C/O ratios were retrieved, too. Here we can compare them to identify the disc regions and times where the chemical and physical conditions for planet formation are consistent. Most of the exoplanets for which the atmospheric composition was retrieved have super-stellar C/O ratios (Hoch et al. 2023; Weiner Mansfield et al. 2024), which draws more attention to carbon-enriched areas. They are suggested to form by core accretion, and accreting mostly the gas, which is typically more carbon rich (beyond water snowline). A lot of planets are observed to have stellar or slightly super-stellar C/O (e.g., P. Mollière et al. 2020; Zhang et al. 2021; Smith et al. 2024; Sing et al. 2024; Nortmann et al. 2024, and many others). One way to form such planets is gravitational instability, which includes solids and gas together, thus undifferentiated matter is suitable for producing planets with stellar C/O. Disc fragmentation to clumps due to GI requires particular conditions (Meru and Bate 2010; Vorobyov 2013), and the direct collapse of gravitationally unstable clumps tends to produce rather massive objects (e.g. ≈ 5 M J planets and ≈ 60 - 70 M J brown dwarfs, see Figure 4 in Vorobyov, Zakhozhay, and Dunham 2013) at larger radial distances (> 10 - 100 au, see Vorobyov, Zakhozhay, and Dunham 2013; Kratter and Lodato 2016). GI can also assist the assemblage of planetary cores (Nayakshin 2010a, 2010b; Nayakshin, Helled, and Boley 2014; Vorobyov and Elbakyan 2019). A planet formed through core accretion can also accrete planetesimals, which can be covered with ice, and enrich the atmosphere with oxygen, making the C/O ratio close to the initial stellar value. There are particular exoplanets, where lower than stellar C/O ratio is observed in the atmosphere, such as β Pic b (GRAVITY Collaboration et al. 2020; Worthen et al. 2024), HD 209458 b (Xue et al. 2024), or HD 189733b (Fu et al. 2024). Such planets need even more enrichment in ices with low C/O, which makes the regions with low C/O in the ice also more attractive sites for planet formation. Gravitational instability implies that the planet forms from a mix of gas and dust (Bodenheimer 1974), this is why it is suitable to explain the formation of planets with solar, or unaltered C/O ratios. In our modelling, GI would be associated with the total C/O ratio, which we find to be significantly variable, too. For GI to form a planet with a primordial C/O ratio, it has to occur during the first 100 kyr after the disc formation. At later times, the total C/O ratio changes, and the only region with the primordial C/O ratio is the very outer disc parts, at > 100 au, which is the part of a protoplanetary disc, where conditions for GI are the most consistent with the observed properties of these objects (Rafikov 2005). Planet formation through GI is indeed more likely at earlier evolutionary stages, when gas surface density is higher (Armitage 2010). We can highlight the areas where planet formation via GI is possible in our modelling as the regions where Q Toomre ≤ 1. They are shown in the upper panel of Figure 9 for model M1. These regions appear between ≈ 10-100au before ≈ 300 kyr. However, at later times, clumps could appear in the disc as a result of an external perturbation, such as a stellar flyby (Thies et al. 2010). In this case, the planet would be formed from the material with altered C/O ratio, most probably with elevated amount of carbon, as the regions outside 5 - 10 au are typically more gravitationally unstable. This means that GI can produce planets with super-solar C/O ratios, if it is induced by external influence at later stages of disc evolution. Core accretion is another most widely discussed scenario of giant planet formation. Accretion of gas should produce atmospheres with the C/O ratio close to the one in the gas phase of protoplanetary disc. However, dust grains are also accreted, so pebble and planetesimal accretion can enrich the atmosphere in volatile components (Mordasini et al. 2016; Danti, Bitsch, and Mah 2023). This makes atmospheric C/O ratio closer to the ice-phase C/O, but in case of gas giants, the amount of the solids needed to compensate the prevalence of carbon in the gas should be quite high, up to hundreds of Earth masses (GRAVITY Collaboration et al. 2020). The C/O ratios in exoplanetary atmospheres are often interpreted in terms of pebble accretion, so planets with stellar C/O ratios are assumed to form in the environments where solid phase C/O is unprocessed and thus close to the initial value. One of such locations is beyond CO snowline, where most of the carbon- and oxygen- bearing material is in the ice, e.g. for Jupiter (e.g. Öberg and Wordsworth 2019; Ohno and Ueda 2021). In our models, this is rather the vicinity of the CO 2 snowlines, and the pebbles beyond the CO snowline are mostly covered with carbon-rich CO ice (Topchieva et al. 2024). Additionally, the CO snowline is typically very far from the star (> 40 au), so such scenarios must rely on planet migration to obtain their current location. Interpretations relying on chemical modelling extend this region to include the area beyond CO 2 snowline due to additional chemical processing of CO in this region, e.g. HR 8799e (P. Mollière et al. 2020). This puts milder constraints on the original distance from the star where pebbles should be accreted and requires less migration. Modelling of planet formation and migration including pebble and gas accretion puts the formation location of planets with super-solar C/O ratios beyond water and CO 2 snowlines (Bitsch, Schneider, and Kreidberg 2022). In our modelling, the C/O in the ice is close to initial value in the regions beyond CO snowline, excluding the area of CO accumulation. Between CO and CO 2 snowlines, it is lower, as our model does not include chemical processes apart from adsorption and desorption. However, there is another region with C/O in the ice close to initial. The vicinity of primary water snowline and the ring induced outside of it has values of C/O in the ice only slightly above the initial value of 0.34. It is surrounded by the snowlines of CO 2 . This region could be another favourable location for forming planets with the stellar C/O. As it is situated closer to the star, it would imply less migration. Rare planets with lower than stellar C/O ratio, such as β Pic b (GRAVITY Collaboration et al. 2020; Reggiani et al. 2024), HD 209458 b (Xue et al. 2024), HD 189733b (Fu et al. 2024), or KELT-1 b, Kepler-13A b and WASP-79 b (less precisely determined, see Hoch et al. 2023), need to have accreted a lot of oxygen-rich ice. Therefore, they are more likely to accrete solid material in the regions with the lowest ice-phase C/O ratios. The most suitable region would be at the distances between H 2 O and CO 2 snowlines, where ice mantles are made of pure water. However, in our modelling results, this region is very small, typically only a few au wide, as the snowlines are close to each other. This is because of steep temperature profile in this region, which is a result of the significant contribution of non-irradiation heating mechanisms, particularly viscous heating. Beyond CO 2 snowline, there are also regions with relatively low (0.2 - 0.3) C/O in the ice, but much more solids need to be accreted in such areas to compensate for the excess of carbon from the gas. Let us summarise the above constraints on planet formation locations and mechanisms implied by our simulated C/O ratios. Core accretion is suitable for forming planets with high C/O ( ≈ 1) in the atmosphere around the snowlines of CO, CH 4 and CO 2 , or anywhere beyond CO 2 snowline if they did not accrete much solids. Planets with stellar or slightly superstellar C/O ratio need to accrete (a lot of ) oxygen-rich solids to compensate their initially high C/O inherited from the gas. The locations where this is possible is between CO 2 and CH 4 and between CO and CH 4 snowlines. Planets with low C/O ratio could accrete ices between H 2 O and CO 2 snowlines. Snowlines are favourable planetesimal formation sites, so the planets that accreted planetesimals/pebbles there can have altered C/O ratios. The values will be lower than the initial if they form at the water snowline, and higher if they form at the snowlines of carbon-rich species. At the same time, to obtain planets with stellar C/O formed at the snowlines, these planets would need to migrate and accrete matter in different regions of the disc to make their C/O ratio close to the initial stellar value. Alternatively, planets with stellar C/O ratio can form via disc fragmentation through GI at earlier stages. Dedicated modelling of planet formation accounting for evolution of dust and volatiles is necessary to put more particular constraints on planet formation scenarios. Planetesimals play an important role in delivering the icephase elements to planetary atmospheres. To form planetesimals, additional physical process is needed, such as streaming instability (SI, see Youdin and Goodman 2005), which is not explicitly included in our modelling because of insufficient numerical resolution and simplified vertical disc structure. However, we can post-process the simulation results to check if the conditions for SI are fulfilled in some regions of the disc where dust-to-gas ratio and dust size are enhanced, following Vorobyov et al. (2024). Dense dust rings forming at later stages (see Section 3.1) seem to be an ideal location for triggering the SI, which would ultimately lead to formation of planetesimals and then planets in the disc. Triggering the SI requires specific relations between local dust-to-gas ratio and Stokes number (Yang, Johansen, and Carrera 2017). The criteria vary depending on the model, we adopt them from Li and Youdin (2021). Another criteria would be the requirement of volume density of dust to exceed that of gas in the midplane (Youdin and Goodman 2005). We do not apply it, as in our modelling, the residual value of α is 10 -3 which makes this condition unreachable outside of the dead zone. The regions in model M1 where the conditions of Li and Youdin (2021) are satisfied are shown in the lower panel of Figure 9. Most of the suitable regions are in the inner disc ( r < 20 au) inside the dust rings, and appear after 200 kyr. However, there are some suitable regions between 10 - 100 au at earlier times, where SI could be triggered in the spirals. The regions where GI and SI are possible shown in Figure 9 are separated in space and time, and they have different characteristic C/O ratios. We can sum up all the volatiles in these regions (throughout the disc lifetime) to assess typical C/O ratios of the planet-forming material. For GI, we exclude the pre-disc phase ( t < 53 kyr) and consider the total C/O ratio, assuming both gas and solids are included in the forming planet. For SI, we separate the gas- and ice-phase C/O ratios. The formed planetesimals would only include the ices, however, if they form the planetary cores, these cores would also accrete gas. We note that the composition of the rocks, which are typically carbon-rich, is not included in our assessment. The resulting distributions of the C/O ratios in planet-forming regions are shown in Figure 10. For GI regions, the C/O distribution has a distinct and relatively narrow peak around 0.5. It is slightly higher than the initial value of 0.34. For SI regions, the ice-phase C/O is below 0.5, with major peaks at 0 and ≈ 0.2, and the gas-phase C/O has a broad distribution with multiple peaks between ≈ 0.2 - 1.4. Distributions of C/O ratios in the regions where GI and SI can be triggered are noticeably different. It was shown by Hoch et al. (2023) that there are two different populations of C/O ratios observed in giant exoplanets. They find that directly imaged exoplanets have C/O ≈ 0.5-0.8, while transiting hot Jupiters have a wider variety of C/O ratios ( ≈ 0.3-1.7, see Figures 12 and 13 in Hoch et al. 2023), and suggest that these two populations could have different formation pathways. We add the C/O data of exoplanetary atmospheres compiled in Table 3 of Hoch et al. (2023) to Figure 10 (with arbitrary position at the y -axis). The narrow distribution of C/O ratios in the regions with GI is in step with the distribution of directly imaged exoplanets, albeit with a slightly shifted value due to our assumed initial conditions, while the wide range of C/O values in the regions of SI matches the variety of C/O ratios in transiting exoplanets. This could suggest that directly imaged exoplanets could form as a result of gravitational instability, which is also in line with their typically higher masses and orbital separations. At the same time, the transiting hot Jupiters could have experienced a lot of migration (Lin, Bodenheimer, and Richardson 1996; Dawson and Johnson 2018), during which they accrete material with a variety of C/O ratios both from the gas and solid phase. It suggests that they could also form in the core accretion scenario. The origin of wide separation planets was also investigated by Bergin et al. (2024), based on the comparison with the observed C/O > 1 in protoplanetary discs (including the full composition of solids). They conclude that both core accretion and gravitational instability can work as the formation mechanism of these planets. Apart from (exo)planets, the C/O ratios can be measured for the comets, which present the best preserved sample of the primordial composition of the ices in the Solar System. Spectroscopic measurements of molecular composition in the comae suggest that the C/O ratio of cometary ice is quite low, typically below 0.1 due to the dominance of water ice (A'Hearn et al. 2012; Seligman et al. 2022; Harrington Pinto et al. 2022). Although most comets are carbon-depleted, there are individual measurements of C/O in comets above 0.5, for example in C/2006 W3 Christensen and 29P/SchwassmannWachmann (Ootsubo et al. 2012; Seligman et al. 2022), or even close to 1 in C/2016 R2 (PanSTARRS) (Wierzchos and Womack 2018; McKay et al. 2019). Additionally, high value of C/O ≈ 1 was observed in the interstellar object 2I/Borisov (Bodewits et al. 2020). Our modelling results show the icephase C/O = 0 in the vicinity of the water snowline, as well as low values between the CO 2 and CH 4 snowlines ( ≈ 0.2) and between the CH 4 and CO snowlines ( ≈ 0.3). These are the locations where comets could originate from. However, in the vicinity of the CO 2 , CH 4 and CO ice lines themselves, the C/O ratio in the ice phase is much higher. The fact that carbon-rich cometary ices are extremely rare in the Solar System may indicate that planetesimals formed on ice lines from carbon-rich volatiles do not persist throughout the evolution of a planetary system. This means that they are likely to be included in larger bodies, which favours the snowline-aided planet formation scenarios (Drążkowska and Alibert 2017; Hyodo et al. 2021). This is also consistent with the abundance of exoplanets with high C/O (Weiner Mansfield et al. 2024), which could be formed around the snowlines of carbon-rich species. In the giant planets of the Solar System, the C/O ratios are not well constrained (Mousis, Cavalié, et al. 2024). However, the existing data suggest rather super-solar values for all giant planets except for Neptune (Cavalié et al. 2024); for Jupiter, the C/O ratio is assessed as ≈ 0.9 (Wong et al. 2004; Li et al. 2024). Our model only considers four most abundant chemical species. However, there can be other more complex molecules in protoplanetary discs, which could affect the balance of carbon and oxygen. The most obvious candidate is methanol CH 3 OH, which has the abundance similar to methane in the protostellar cores (Karin I. Öberg et al. 2011). It was also observed in a protoplanetary disc around an erupting star V883 Ori (Lee et al. 2019). We do not consider it in the model as its binding energy is close to that of water, thus the snowlines would have similar positions, but the abundance is an order of magnitude lower. However, it could somewhat increase the local C/O ratio in the inner regions where there are no other carbon-bearing species, such as the ice in the ring at 1 au. Including methanol would alter the distribution of the C/O ratio. Interactions between the ices considered in the model could also affect the results. As was recently shown by Ligterink, Kipfer, and Gavino (2024), trapping of volatile species inside the mantles of less volatile ices could have a significant impact on the distribution of C/O ratios. Another important process missing in our modelling is gas-phase and surface chemical reactions. They could significantly affect the distribution of C/O ratio in the gas and in the ice, particularly with high level of cosmic ray ionisation (Eistrup, Walsh, and van Dishoeck 2016) or if carbon grain destruction is considered (Cridland, Eistrup, and van Dishoeck 2019). One particular mechanism is the transformation of CO to CO 2 on the surface of dust grains, which can lead to the depletion of CO from both gas and ice phases between CO and CO 2 snowlines (Molyarova et al. 2017; Bosman, Tielens, and van Dishoeck 2018). Considering this mechanism can change the conclusions about planet formation location (P. Mollière et al. 2020). Nevertheless, radial variations of the C/O ratio are necessary to explain molecular emission of discs with gaps (Leemker et al. 2024), and they can only be result of dust dynamics. In order to more consistently describe the distribution of molecules and elements in the disc, the models combining dust evolution and dynamics with more complex chemistry treatment are necessary. Our simulations adopt the thin-disc approximation and focus on the midplane of the protoplanetary discs, in order to capture the essential physics of self-gravity, thermal balance, and dust evolution in a global modelling within reasonable computational times. This means that some relevant processes connected with the vertical structure are inevitably excluded. For example, vertical mixing and dust settling affect the C/O ratio in the upper layers of the disc (Krijt et al. 2018; Krijt et al. 2020). Dust settling is implicitly included in our modelling through separate scale heights of drown dust and gas (as well as small dust), affecting dust number density in the midplane. However, this approach does not allow to reproduce vertical stratification in dust properties and chemical composition, which is particularly relevant for the interpretation of molecular observations. Vertical structure is also relevant for the accretion of matter on forming giant planets, which should proceed in 3D manner through meridional flows (Morbidelli et al. 2014). Cridland, Bosman, and van Dishoeck (2020) showed that the C/O ratio in the atmospheres of giant planets is rather affected by the composition of the molecular layer than that of the midplane.",
"pages": [
19,
20,
21,
22
]
},
{
"title": "5. Conclusions",
"content": "In this work, we studied the distribution of volatiles in a viscous self-gravitating protoplanetary disc with dust evolution using a thin-disc hydrodynamic code FEOSAD (Vorobyov et al. 2018; Molyarova et al. 2021). We calculated the C/O elemental ratio in the gas, in the ice, and in total, identified the key properties of the distribution of elements over 500 kyr of disc evolution and considered their implications for planet formation theory. Our main findings can be summarised as follows. The connection between protoplanetary disc components and exoplanets based on their composition should be more thoroughly investigated in the models focused on the planet formation process. We emphasise that these models should also take into account the effect of dust evolution and dynamics on the distribution of the elements in the planet-forming material. Inclusion of chemical processes and more accurate consideration of the bulk composition of dust grains could also affect the C/O ratios of the planet-forming environment.",
"pages": [
22
]
},
{
"title": "Acknowledgement",
"content": "Weare thankful to the anonymous referee for useful comments that helped to improve the manuscript. The computational results presented have been achieved using the Vienna Scientific Cluster (VSC) and the local computing facility of the Southern Federal University. Funding Statement The work is supported by Russian Science Foundation grant 22-72-10029, https://rscf.ru/project/2272-10029/ Competing Interests None Data Availability Statement The data underlying this article will be shared on reasonable request to the corresponding author.",
"pages": [
22,
23
]
},
{
"title": "References",
"content": "A'Hearn, Michael F., Lori M. Feaga, H. Uwe Keller, Hideyo Kawakita, Donald L. Hampton, Jochen Kissel, Kenneth P. Klaasen, et al. 2012. Cometary Volatiles and the Origin of Comets. ApJ 758, no. 1 (October): 29. https: //doi.org/10.1088/0004-637X/758/1/29. Aikawa, Yuri, Shoken M. Miyama, Takenori Nakano, and Toyoharu Umebayashi. 1996. Evolution of Molecular Abundance in Gaseous Disks around Young Stars: Depletion of CO Molecules. ApJ 467 (August): 684. https://doi.org/10.1086/177644. Akimkin, Vitaly, Eduard Vorobyov, Yaroslav Pavlyuchenkov, and Olga Stoyanovskaya. 2020. Gravitoviscous protoplanetary discs with a dust component - IV. Disc outer edges, spectral indices, and opacity gaps. MNRAS 499, no. 4 (December): 5578-5597. https://doi.org/10.1093/mnras/ staa3134. arXiv: 2010.06566 [astro-ph.EP] . Armitage, Philip J. 2010. Astrophysics of Planet Formation. Armitage, Philip J., Mario Livio, and J. E. Pringle. 2001. Episodic accretion in magnetically layered protoplanetary discs. MNRAS 324, no. 3 (June): 705-711. https://doi.org/10.1046/j.1365-8711.2001.04356.x. arXiv: astro-ph/0101253 [astro-ph] . Audard, M., P. Ábrahám, M. M. Dunham, J. D. Green, N. Grosso, K. Hamaguchi, J. H. Kastner, et al. 2014. Episodic Accretion in Young Stars. In Protostars and planets vi, edited by Henrik Beuther, Ralf S. Klessen, Cornelis P. Dullemond, and Thomas Henning, 387-410. January. https: / / doi . org / 10 . 2458 / azu _ uapress _ 9780816531240 - ch017. arXiv: 1401.3368 [astro-ph.SR] . Bai, Xue-Ning, and James M. Stone. 2013. Wind-driven Accretion in Protoplanetary Disks. I. Suppression of the Magnetorotational Instability and Launching of the Magnetocentrifugal Wind. ApJ 769, no. 1 (May): 76. https://doi.org/10.1088/0004-637X/769/1/76. arXiv: 1301.0318 [astro-ph.EP] . Banzatti, Andrea, Ilaria Pascucci, Arthur D. Bosman, Paola Pinilla, Colette Salyk, Gregory J. Herczeg, Klaus M. Pontoppidan, et al. 2020. Hints for Icy Pebble Migration Feeding an Oxygen-rich Chemistry in the Inner Planet-forming Region of Disks. ApJ 903, no. 2 (November): 124. https://doi.org/10.3847/1538-4357/abbc1a. arXiv: 2009.13525 [astro-ph.EP] . Benneke, Björn, Heather A. Knutson, Joshua Lothringer, Ian J. M. Crossfield, Julianne I. Moses, Caroline Morley, Laura Kreidberg, et al. 2019. A sub-Neptune exoplanet with a low-metallicity methane-depleted atmosphere and Mie-scattering clouds. Nature Astronomy 3 (July): 813-821. https://doi.org/10.1038/s41550- 019- 0800- 5. arXiv: 1907.00449 [astro-ph.EP] . Bergin, Edwin A., Richard A. Booth, Maria Jose Colmenares, and John D. Ilee. 2024. C/O Ratios and the formation of wide separation exoplanets. arXiv e-prints (June): arXiv:2406.12037. https://doi.org/10.48550/arXiv. 2406.12037. arXiv: 2406.12037 [astro-ph.EP] . Bergin, Edwin A., Fujun Du, L. Ilsedore Cleeves, G. A. Blake, K. Schwarz, R. Visser, and K. Zhang. 2016. Hydrocarbon Emission Rings in Protoplanetary Disks Induced by Dust Evolution. ApJ 831, no. 1 (November): 101. https://doi.org/10.3847/0004-637X/831/1/101. arXiv: 1609.06337 [astro-ph.EP] . Binney, James, and Scott Tremaine. 1987. Galactic dynamics. Birnstiel, Tilman. 2023. Dust growth and evolution in protoplanetary disks. arXiv e-prints (December): arXiv:2312.13287. https://doi.org/10.48550/ arXiv.2312.13287. arXiv: 2312.13287 [astro-ph.EP] . Bisschop, S. E., H. J. Fraser, K. I. Öberg, E. F. van Dishoeck, and S. Schlemmer. 2006. Desorption rates and sticking coefficients for CO and N2 interstellar ices. A&A 449, no. 3 (April): 1297-1309. https://doi.org/10. 1051/0004-6361:20054051. arXiv: astro-ph/0601082 [astro-ph] . Bitsch, Bertram, Aaron David Schneider, and Laura Kreidberg. 2022. How drifting and evaporating pebbles shape giant planets. III. The formation of WASP-77A b and τ Boötis b. A&A 665 (September): A138. http s : / / doi . org / 10 . 1051 / 0004 - 6361 / 202243345. arXiv: 2207 . 06077 [astro-ph.EP] . Bodenheimer, P. 1974. Calculations of the Early Evolution of Jupiter. Icarus 23, no. 3 (November): 319-325. https : / / doi . org / 10 . 1016 / 0019 1035(74)90050-5. Bodewits, D., J. W. Noonan, P. D. Feldman, M. T. Bannister, D. Farnocchia, W. M. Harris, J. -Y. Li, K. E. Mandt, J. Wm. Parker, and Z. -X. Xing. 2020. The carbon monoxide-rich interstellar comet 2I/Borisov. Nature Astronomy 4 (April): 867-871. https://doi.org/10.1038/s41550-0201095-2. arXiv: 2004.08972 [astro-ph.EP] . Booth, R. A., and J. D. Ilee. 2019. Planet-forming material in a protoplanetary disc: the interplay between chemical evolution and pebble drift. MNRAS 487, no. 3 (August): 3998-4011. https://doi.org/10.1093/mnras/stz1488. arXiv: 1905.12639 [astro-ph.EP] . Booth, Richard A., Cathie J. Clarke, Nikku Madhusudhan, and John D. Ilee. 2017. Chemical enrichment of giant planets and discs due to pebble drift. MNRAS 469, no. 4 (August): 3994-4011. https://doi.org/10.1093/ mnras/stx1103. arXiv: 1705.03305 [astro-ph.EP] . Bosman, A. D., A. J. Cridland, and Y. Miguel. 2019. Jupiter formed as a pebble pile around the N2 ice line. A&A 632 (December): L11. https://doi.org/ 10.1051/0004-6361/201936827. arXiv: 1911.11154 [astro-ph.EP] . Bosman, Arthur D., Alexander G. G. M. Tielens, and Ewine F. van Dishoeck. 2018. Efficiency of radial transport of ices in protoplanetary disks probed with infrared observations: the case of CO2. A&A 611 (April): A80. https://doi.org/10.1051/0004-6361/201732056. arXiv: 1712.03989 [astro-ph.EP] . Brown, Paul D., and S. B. Charnley. 1990. Chemical models of interstellar gas-grain processes. I. Modelling and the effect of accretion on gas abundances and mantle composition in dense clouds. MNRAS 244 (June): 432. Brown-Sevilla, S. B., M. Keppler, M. Barraza-Alfaro, J. D. Melon Fuksman, N. Kurtovic, P. Pinilla, M. Feldt, et al. 2021. A multiwavelength analysis of the spiral arms in the protoplanetary disk around WaOph 6. A&A 654 (October): A35. https://doi.org/10.1051/0004-6361/202140783. arXiv: 2107.13560 [astro-ph.EP] . Carrera, Daniel, Anders Johansen, and Melvyn B. Davies. 2015. How to form planetesimals from mm-sized chondrules and chondrule aggregates. A&A 579 (July): A43. https://doi.org/10.1051/0004-6361/201425120. arXiv: 1501.05314 [astro-ph.EP] . Cavalié, Thibault, Jonathan Lunine, Olivier Mousis, and Ricardo Hueso. 2024. The Deep Oxygen Abundance in Solar System Giant Planets, with a New Derivation for Saturn. Space Sci. Rev. 220, no. 1 (January): 8. https://doi.org/10.1007/s11214-024-01045-6. arXiv: 2407.07515 [astro-ph.EP] . Changeat, Q., B. Edwards, A. F. Al-Refaie, A. Tsiaras, J. W. Skinner, J. Y. K. Cho, K. H. Yip, et al. 2022. Five Key Exoplanet Questions Answered via the Analysis of 25 Hot-Jupiter Atmospheres in Eclipse. ApJS 260, no. 1 (May): 3. https://doi.org/10.3847/1538-4365/ac5cc2. arXiv: 2204.11729 [astro-ph.EP] . Cleeves, L. Ilsedore, Karin I. Öberg, David J. Wilner, Jane Huang, Ryan A. Loomis, Sean M. Andrews, and V. V. Guzman. 2018. Constraining Gasphase Carbon, Oxygen, and Nitrogen in the IM Lup Protoplanetary Disk. ApJ 865, no. 2 (October): 155. https://doi.org/10.3847/15384357/aade96. arXiv: 1808.10682 [astro-ph.SR] . Connelley, Michael S., and Bo Reipurth. 2018. A Near-infrared Spectroscopic Survey of FU Orionis Objects. ApJ 861, no. 2 (July): 145. https://doi. org/10.3847/1538-4357/aaba7b. arXiv: 1806.08880 [astro-ph.SR] . Cridland, Alex J., Ewine F. van Dishoeck, Matthew Alessi, and Ralph E. Pudritz. 2019. Connecting planet formation and astrochemistry. A main sequence for C/O in hot exoplanetary atmospheres. A&A 632 (December): A63. https://doi.org/10.1051/0004- 6361/201936105. arXiv: 1910.13171 [astro-ph.EP] . . 2020. Connecting planet formation and astrochemistry. C/Os and N/Os of warm giant planets and Jupiter analogues. A&A 642 (October): A229. https://doi.org/10.1051/0004-6361/202038767. arXiv: 2009. 02907 [astro-ph.EP] . Cridland, Alexander J., Arthur D. Bosman, and Ewine F. van Dishoeck. 2020. Impact of vertical gas accretion on the carbon-to-oxygen ratio of gas giant atmospheres. A&A 635 (March): A68. https://doi.org/10.1051/ 0004-6361/201936858. arXiv: 2001.05808 [astro-ph.EP] . Cridland, Alexander J., Christian Eistrup, and Ewine F. van Dishoeck. 2019. Connecting planet formation and astrochemistry. Refractory carbon depletion leading to super-stellar C/O in giant planetary atmospheres. A&A 627 (July): A127. https://doi.org/10.1051/0004-6361/201834378. arXiv: 1901.08896 [astro-ph.EP] . Cuppen, H. M., C. Walsh, T. Lamberts, D. Semenov, R. T. Garrod, E. M. Penteado, and S. Ioppolo. 2017. Grain Surface Models and Data for Astrochemistry. Space Sci. Rev. 212, nos. 1-2 (October): 1-58. https: //doi.org/10.1007/s11214-016-0319-3. Cuzzi, Jeffrey N., and Kevin J. Zahnle. 2004. Material Enhancement in Protoplanetary Nebulae by Particle Drift through Evaporation Fronts. ApJ 614, no. 1 (October): 490-496. https://doi.org/10.1086/423611. arXiv: astro-ph/0409276 [astro-ph] . Danti, C., B. Bitsch, and J. Mah. 2023. Composition of giant planets: The roles of pebbles and planetesimals. A&A 679 (November): L7. https://doi.org/ 10.1051/0004-6361/202347501. arXiv: 2310.02886 [astro-ph.EP] . Dawson, Rebekah I., and John Asher Johnson. 2018. Origins of Hot Jupiters. ARA&A 56 (September): 175-221. https://doi.org/10.1146/annurevastro-081817-051853. arXiv: 1801.06117 [astro-ph.EP] . Dong, Ruobing, Eduard Vorobyov, Yaroslav Pavlyuchenkov, Eugene Chiang, and Hauyu Baobab Liu. 2016. Signatures of Gravitational Instability in Resolved Images of Protostellar Disks. ApJ 823, no. 2 (June): 141. https://doi.org/10.3847/0004-637X/823/2/141. arXiv: 1603.01618 [astro-ph.SR] . Draine, B. T. 1978. Photoelectric heating of interstellar gas. ApJS 36 (April): 595-619. https://doi.org/10.1086/190513. Drążkowska, J., and Y. Alibert. 2017. Planetesimal formation starts at the snow line. A&A 608 (December): A92. https://doi.org/10.1051/00046361/201731491. arXiv: 1710.00009 [astro-ph.EP] . Dutrey, A., V. Wakelam, Y. Boehler, S. Guilloteau, F. Hersant, D. Semenov, E. Chapillon, et al. 2011. Chemistry in disks. V. Sulfur-bearing molecules in the protoplanetary disks surrounding LkCa15, MWC480, DM Tauri, and GO Tauri. A&A 535 (November): A104. https://doi.org/10.1051/ 0004-6361/201116931. arXiv: 1109.5870 [astro-ph.SR] . Eistrup, Christian, Catherine Walsh, and Ewine F. van Dishoeck. 2016. Setting the volatile composition of (exo)planet-building material. Does chemical evolution in disk midplanes matter? A&A 595 (November): A83. htt ps://doi.org/10.1051/0004- 6361/201628509. arXiv: 1607.06710 [astro-ph.EP] . . 2018. Molecular abundances and C/O ratios in chemically evolving planet-forming disk midplanes. A&A 613 (May): A14. https://doi.org/ 10.1051/0004-6361/201731302. arXiv: 1709.07863 [astro-ph.EP] . Facchini, Stefano, Richard Teague, Jaehan Bae, Myriam Benisty, Miriam Keppler, and Andrea Isella. 2021. The Chemical Inventory of the Planethosting Disk PDS 70. AJ 162, no. 3 (September): 99. https://doi.org/10. 3847/1538-3881/abf0a4. arXiv: 2101.08369 [astro-ph.EP] . Fedele, D., and C. Favre. 2020. Measuring elemental abundance ratios in protoplanetary disks at millimeter wavelengths. A&A 638 (June): A110. https://doi.org/10.1051/0004-6361/202037927. arXiv: 2005.03891 [astro-ph.SR] . Fraser, Helen J., Mark P. Collings, Martin R. S. McCoustra, and David A. Williams. 2001. Thermal desorption of water ice in the interstellar medium. MNRAS 327, no. 4 (November): 1165-1172. https://doi. org/10.1046/j.1365-8711.2001.04835.x. arXiv: astro-ph/0107487 [astro-ph] . Fu, Guangwei, Luis Welbanks, Drake Deming, Julie Inglis, Michael Zhang, Joshua Lothringer, Jegug Ih, et al. 2024. Hydrogen sulfide and metalenriched atmosphere for a Jupiter-mass exoplanet. arXiv e-prints (July): arXiv:2407.06163. https://doi.org/10.48550/arXiv.2407.06163. arXiv: 2407.06163 [astro-ph.EP] . Gail, Hans-Peter, and Mario Trieloff. 2017. Spatial distribution of carbon dust in the early solar nebula and the carbon content of planetesimals. A&A 606 (September): A16. https://doi.org/10.1051/0004-6361/201730480. arXiv: 1707.07611 [astro-ph.EP] . Gammie, Charles F. 1996. Layered Accretion in T Tauri Disks. ApJ 457 (January): 355. https://doi.org/10.1086/176735. Gárate, Matias, Til Birnstiel, Joanna Drążkowska, and Sebastian Markus Stammler. 2020. Gas accretion damped by dust back-reaction at the snow line. A&A 635 (March): A149. https://doi.org/10.1051/00046361/201936067. arXiv: 1906.07708 [astro-ph.EP] . GRAVITY Collaboration, M. Nowak, S. Lacour, P. Mollière, J. Wang, B. Charnay, E. F. van Dishoeck, et al. 2020. Peering into the formation history of β Pictoris b with VLTI/GRAVITY long-baseline interferometry. A&A 633 (January): A110. https://doi.org/10.1051/00046361/201936898. arXiv: 1912.04651 [astro-ph.EP] . Gressel, Oliver, Neal J. Turner, Richard P. Nelson, and Colin P. McNally. 2015. Global Simulations of Protoplanetary Disks With Ohmic Resistivity and Ambipolar Diffusion. ApJ 801, no. 2 (March): 84. https://doi.org/ 10.1088/0004-637X/801/2/84. arXiv: 1501.05431 [astro-ph.EP] . Gundlach, B., and J. Blum. 2015. The Stickiness of Micrometer-sized Waterice Particles. ApJ 798, no. 1 (January): 34. https://doi.org/10.1088/0004637X/798/1/34. arXiv: 1410.7199 [astro-ph.EP] . Harrington Pinto, Olga, Maria Womack, Yanga Fernandez, and James Bauer. 2022. A Survey of CO, CO2, and H2O in Comets and Centaurs. PSJ 3, no. 11 (November): 247. https://doi.org/10.3847/PSJ/ac960d. arXiv: 2209.09985 [astro-ph.EP] . Hartmann, L., and S. J. Kenyon. 1985. On the nature of FU Orionis objects. ApJ 299 (December): 462-478. https://doi.org/10.1086/163713. Hasegawa, T. I., and E. Herbst. 1993. Three-Phase Chemical Models of Dense Interstellar Clouds - Gas Dust Particle Mantles and Dust Particle Surfaces. MNRAS 263 (August): 589. https://doi.org/10.1093/mnras/263.3.589. Hensley, Brandon S., and B. T. Draine. 2021. Observational Constraints on the Physical Properties of Interstellar Dust in the Post-Planck Era. ApJ 906, no. 2 (January): 73. https://doi.org/10.3847/1538-4357/abc8f1. arXiv: 2009.00018 [astro-ph.GA] . Hoch, Kielan K. W., Quinn M. Konopacky, Christopher A. Theissen, JeanBaptiste Ruffio, Travis S. Barman, Emily L. Rickman, Marshall D. Perrin, Bruce Macintosh, and Christian Marois. 2023. Assessing the C/O Ratio Formation Diagnostic: A Potential Trend with Companion Mass. AJ 166, no. 3 (September): 85. https://doi.org/10.3847/1538-3881/ace442. arXiv: 2212.04557 [astro-ph.EP] . Huang, Jane, Sean M. Andrews, Cornelis P. Dullemond, Andrea Isella, Laura M. Pérez, Viviana V. Guzmán, Karin I. Öberg, et al. 2018. The Disk Substructures at High Angular Resolution Project (DSHARP). II. Characteristics of Annular Substructures. ApJ 869, no. 2 (December): L42. https : / / doi . org / 10 . 3847 / 2041 - 8213 / aaf 740. arXiv: 1812 . 04041 [astro-ph.EP] . Huang, Jane, Sean M. Andrews, Laura M. Pérez, Zhaohuan Zhu, Cornelis P. Dullemond, Andrea Isella, Myriam Benisty, et al. 2018. The Disk Substructures at High Angular Resolution Project (DSHARP). III. Spiral Structures in the Millimeter Continuum of the Elias 27, IM Lup, and WaOph 6 Disks. ApJ 869, no. 2 (December): L43. https://doi.org/10. 3847/2041-8213/aaf7a0. arXiv: 1812.04193 [astro-ph.SR] . Huang, Jane, Edwin A. Bergin, Romane Le Gal, Sean M. Andrews, Jaehan Bae, Luke Keyte, and J. A. Sturm. 2024. Constraints on the gas-phase C/O ratio of DR Tau's outer disk from CS, SO, and C2H observations. arXiv e-prints (July): arXiv:2407.01679. https://doi.org/10.48550/arXiv. 2407.01679. arXiv: 2407.01679 [astro-ph.EP] . Hyodo, Ryuki, Tristan Guillot, Shigeru Ida, Satoshi Okuzumi, and Andrew N. Youdin. 2021. Planetesimal formation around the snow line. II. Dust or pebbles? A&A 646 (February): A14. https://doi.org/10.1051/00046361/202039894. arXiv: 2012.06700 [astro-ph.EP] . Ilee, J. D., A. C. Boley, P. Caselli, R. H. Durisen, T. W. Hartquist, and J. M. C. Rawlings. 2011. Chemistry in a gravitationally unstable protoplanetary disc. MNRAS 417, no. 4 (November): 2950-2961. https://doi.org/10. 1111/j.1365-2966.2011.19455.x. arXiv: 1107.3041 [astro-ph.GA] . Jiang, Haochang, and Chris W. Ormel. 2023. Efficient planet formation by pebble accretion in ALMA rings. MNRAS 518, no. 3 (January): 38773900. https://doi.org/10.1093/mnras/stac3275. arXiv: 2207.13002 [astro-ph.EP] . Kadam, Kundan, Eduard Vorobyov, and Shantanu Basu. 2022. Primordial dusty rings and episodic outbursts in protoplanetary discs. MNRAS 516, no. 3 (November): 4448-4468. https://doi.org/10.1093/mnras/stac2455. arXiv: 2208.12105 [astro-ph.EP] . Kadam, Kundan, Eduard Vorobyov, Zsolt Regály, Ágnes Kóspál, and Péter Ábrahám. 2019. Dynamical Gaseous Rings in Global Simulations of Protoplanetary Disk Formation. ApJ 882, no. 2 (September): 96. h ttps : / / doi . org / 10 . 3847 / 1538 - 4357 / ab378a. arXiv: 1908 . 02515 [astro-ph.SR] . . 2020. Outbursts in Global Protoplanetary Disk Simulations. ApJ 895, no. 1 (May): 41. https://doi.org/10.3847/1538-4357/ab8bd8. arXiv: 2005.03578 [astro-ph.SR] . Kama, M., S. Bruderer, E. F. van Dishoeck, M. Hogerheijde, C. P. Folsom, A. Miotello, D. Fedele, A. Belloche, R. Güsten, and F. Wyrowski. 2016. Volatile-carbon locking and release in protoplanetary disks. A study of TWHya and HD 100546. A&A 592 (August): A83. https://doi.org/10. 1051/0004-6361/201526991. arXiv: 1605.05093 [astro-ph.EP] . Khorshid, N., M. Min, and J. M. Désert. 2023. Retrieving planet formation parameters of WASP-77Ab using SimAb. A&A 675 (July): A95. htt ps://doi.org/10.1051/0004- 6361/202245469. arXiv: 2311.15702 [astro-ph.EP] . Kratter, Kaitlin, and Giuseppe Lodato. 2016. Gravitational Instabilities in Circumstellar Disks. ARA&A 54 (September): 271-311. https://doi. org/10.1146/annurev- astro- 081915- 023307. arXiv: 1603.01280 [astro-ph.SR] . Krijt, Sebastiaan, Arthur D. Bosman, Ke Zhang, Kamber R. Schwarz, Fred J. Ciesla, and Edwin A. Bergin. 2020. CO Depletion in Protoplanetary Disks: A Unified Picture Combining Physical Sequestration and Chemical Processing. ApJ 899, no. 2 (August): 134. https://doi.org/10.3847/ 1538-4357/aba75d. arXiv: 2007.09517 [astro-ph.SR] . Krijt, Sebastiaan, Kamber R. Schwarz, Edwin A. Bergin, and Fred J. Ciesla. 2018. Transport of CO in Protoplanetary Disks: Consequences of Pebble Formation, Settling, and Radial Drift. ApJ 864, no. 1 (September): 78. https : / / doi . org / 10 . 3847 / 1538 - 4357 / aad69b. arXiv: 1808 . 01840 [astro-ph.EP] . Lambrechts, M., and A. Johansen. 2012. Rapid growth of gas-giant cores by pebble accretion. A&A 544 (August): A32. https://doi.org/10.1051/00046361/201219127. arXiv: 1205.3030 [astro-ph.EP] . Lee, Eve J., J. R. Fuentes, and Philip F. Hopkins. 2022. Establishing Dust Rings and Forming Planets within Them. ApJ 937, no. 2 (October): 95. https://doi.org/10.3847/1538-4357/ac8cfe. arXiv: 2206.01219 [astro-ph.EP] . Lee, Jeong-Eun, Edwin A. Bergin, and Hideko Nomura. 2010. The Solar Nebula on Fire: A Solution to the Carbon Deficit in the Inner Solar System. ApJ 710, no. 1 (February): L21-L25. https://doi.org/10.1088/ 2041-8205/710/1/L21. arXiv: 1001.0818 [astro-ph.GA] . Lee, Jeong-Eun, Seokho Lee, Giseon Baek, Yuri Aikawa, Lucas Cieza, SungYong Yoon, Gregory Herczeg, Doug Johnstone, and Simon Casassus. 2019. The ice composition in the disk around V883 Ori revealed by its stellar outburst. Nature Astronomy 3 (February): 314-319. https://doi. org/10.1038/s41550-018-0680-0. arXiv: 1809.00353 [astro-ph.SR] . Leemker, M., A. S. Booth, E. F. van Dishoeck, L. Wölfer, and B. Dent. 2024. Chemistry across dust and gas gaps in protoplanetary disks: modelling the co-spatial molecular rings in the HD 100546 disk. arXiv e-prints (May): arXiv:2405.10361. https://doi.org/10.48550/arXiv.2405.10361. arXiv: 2405.10361 [astro-ph.EP] . Lenz, Christian T., Hubert Klahr, and Tilman Birnstiel. 2019. Planetesimal Population Synthesis: Pebble Flux-regulated Planetesimal Formation. ApJ 874, no. 1 (March): 36. https://doi.org/10.3847/1538-4357/ab05d9. arXiv: 1902.07089 [astro-ph.EP] . Li, Cheng, Michael Allison, Sushil Atreya, Shawn Brueshaber, Leigh N. Fletcher, Tristan Guillot, Liming Li, et al. 2024. Super-adiabatic temperature gradient at Jupiter's equatorial zone and implications for the water abundance. Icarus 414 (May): 116028. https://doi.org/10.1016/j. icarus.2024.116028. arXiv: 2403.05363 [astro-ph.EP] . Li, Rixin, and Andrew N. Youdin. 2021. Thresholds for Particle Clumping by the Streaming Instability. ApJ 919, no. 2 (October): 107. https://doi. org/10.3847/1538-4357/ac0e9f. arXiv: 2105.06042 [astro-ph.EP] . Ligterink, N. F. W., K. A. Kipfer, and S. Gavino. 2024. Mind the Trap: Non-negligible effect of volatile trapping in ice on C/O ratios in protoplanetary disks and exoplanetary atmospheres. arXiv e-prints (June): arXiv:2406.16029. https://doi.org/10.48550/arXiv.2406.16029. arXiv: 2406.16029 [astro-ph.EP] . Lin, D. N. C., P. Bodenheimer, and D. C. Richardson. 1996. Orbital migration of the planetary companion of 51 Pegasi to its present location. Nature 380, no. 6575 (April): 606-607. https://doi.org/10.1038/380606a0. Line, Michael R., Matteo Brogi, Jacob L. Bean, Siddharth Gandhi, Joseph Zalesky, Vivien Parmentier, Peter Smith, et al. 2021. A solar C/O and sub-solar metallicity in a hot Jupiter atmosphere. Nature 598, no. 7882 (October): 580-584. https://doi.org/10.1038/s41586-021-03912-6. arXiv: 2110.14821 [astro-ph.EP] . Lodders, Katharina. 2004. Jupiter Formed with More Tar than Ice. ApJ 611, no. 1 (August): 587-597. https://doi.org/10.1086/421970. Long, Feng, Paola Pinilla, Gregory J. Herczeg, Daniel Harsono, Giovanni Dipierro, Ilaria Pascucci, Nathan Hendler, et al. 2018. Gaps and Rings in an ALMA Survey of Disks in the Taurus Star-forming Region. ApJ 869, no. 1 (December): 17. https://doi.org/10.3847/1538-4357/aae8e1. arXiv: 1810.06044 [astro-ph.SR] . Madhusudhan, Nikku. 2012. C/O Ratio as a Dimension for Characterizing Exoplanetary Atmospheres. ApJ 758, no. 1 (October): 36. https://doi. org/10.1088/0004-637X/758/1/36. arXiv: 1209.2412 [astro-ph.EP] . Madhusudhan, Nikku, Joseph Harrington, Kevin B. Stevenson, Sarah Nymeyer, Christopher J. Campo, Peter J. Wheatley, Drake Deming, et al. 2011. A high C/O ratio and weak thermal inversion in the atmosphere of exoplanet WASP-12b. Nature 469, no. 7328 (January): 64-67. https: //doi.org/10.1038/nature09602. arXiv: 1012.1603 [astro-ph.EP] . Matter, A., F. C. Pignatale, and B. Lopez. 2020. Spatially resolving the chemical composition of the planet building blocks. MNRAS 497, no. 3 (September): 2540-2552. https://doi.org/10.1093/mnras/staa2137. arXiv: 2007.09385 [astro-ph.EP] . McKay, Adam J., Michael A. DiSanti, Michael S. P. Kelley, Matthew M. Knight, Maria Womack, Kacper Wierzchos, Olga Harrington Pinto, et al. 2019. The Peculiar Volatile Composition of CO-dominated Comet C/2016 R2 (PanSTARRS). AJ 158, no. 3 (September): 128. https://doi. org/10.3847/1538-3881/ab32e4. arXiv: 1907.07208 [astro-ph.EP] . Meru, Farzana, and Matthew R. Bate. 2010. Exploring the conditions required to form giant planets via gravitational instability in massive protoplanetary discs. MNRAS 406, no. 4 (August): 2279-2288. https: //doi.org/10.1111/j.1365- 2966.2010.16867.x. arXiv: 1004.3766 [astro-ph.EP] . Meru, Farzana, Attila Juhász, John D. Ilee, Cathie J. Clarke, Giovanni P. Rosotti, and Richard A. Booth. 2017. On the Origin of the Spiral Morphology in the Elias 2-27 Circumstellar Disk. ApJ 839, no. 2 (April): L24. https://doi.org/10.3847/2041-8213/aa6837. arXiv: 1703.05338 [astro-ph.EP] . Minissale, Marco, Yuri Aikawa, Edwin Bergin, Mathieu Bertin, Wendy A. Brown, Stephanie Cazaux, Steven B. Charnley, et al. 2022. Thermal Desorption of Interstellar Ices: A Review on the Controlling Parameters and Their Implications from Snowlines to Chemical Complexity. ACS Earth and Space Chemistry 6, no. 3 (March): 597-630. https://doi.org/10. 1021/acsearthspacechem.1c00357. arXiv: 2201.07512 [astro-ph.GA] . Miotello, A., S. Facchini, E. F. van Dishoeck, P. Cazzoletti, L. Testi, J. P. Williams, M. Ansdell, S. van Terwisga, and N. van der Marel. 2019. Bright C2H emission in protoplanetary discs in Lupus: high volatile C/O > 1 ratios. A&A 631 (November): A69. https://doi.org/10.1051/00046361/201935441. arXiv: 1909.04477 [astro-ph.SR] . Mollière, P., T. Stolker, S. Lacour, G. P. P. L. Otten, J. Shangguan, B. Charnay, T. Molyarova, et al. 2020. Retrieving scattering clouds and disequilibrium chemistry in the atmosphere of HR 8799e. A&A 640 (August): A131. https://doi.org/10.1051/0004-6361/202038325. arXiv: 2006.09394 [astro-ph.EP] . Mollière, Paul, Tamara Molyarova, Bertram Bitsch, Thomas Henning, Aaron Schneider, Laura Kreidberg, Christian Eistrup, et al. 2022. Interpreting the Atmospheric Composition of Exoplanets: Sensitivity to Planet Formation Assumptions. ApJ 934, no. 1 (July): 74. https://doi.org/10.3847/ 1538-4357/ac6a56. arXiv: 2204.13714 [astro-ph.EP] . Molyarova, Tamara, Vitaly Akimkin, Dmitry Semenov, Thomas Henning, Anton Vasyunin, and Dmitri Wiebe. 2017. Gas Mass Tracers in Protoplanetary Disks: CO is Still the Best. ApJ 849, no. 2 (November): 130. https://doi.org/10.3847/1538-4357/aa9227. arXiv: 1710.02993 [astro-ph.EP] . Molyarova, Tamara, Eduard I. Vorobyov, Vitaly Akimkin, Aleksandr Skliarevskii, Dmitri Wiebe, and Manuel Güdel. 2021. Gravitoviscous Protoplanetary Disks with a Dust Component. V. The Dynamic Model for Freeze-out and Sublimation of Volatiles. ApJ 910, no. 2 (April): 153. https://doi.org/ 10.3847/1538-4357/abe2b0. arXiv: 2103.06045 [astro-ph.EP] . Morbidelli, A., J. Szulágyi, A. Crida, E. Lega, B. Bitsch, T. Tanigawa, and K. Kanagawa. 2014. Meridional circulation of gas into gaps opened by giant planets in three-dimensional low-viscosity disks. Icarus 232 (April): 266-270. https://doi.org/10.1016/j.icarus.2014.01.010. arXiv: 1401.2925 [astro-ph.EP] . Mordasini, C., R. van Boekel, P. Mollière, Th. Henning, and Björn Benneke. 2016. The Imprint of Exoplanet Formation History on Observable Present-day Spectra of Hot Jupiters. ApJ 832, no. 1 (November): 41. https://doi.org/10.3847/0004-637X/832/1/41. arXiv: 1609.03019 [astro-ph.EP] . Moses, J. I., N. Madhusudhan, C. Visscher, and R. S. Freedman. 2013. Chemical Consequences of the C/O Ratio on Hot Jupiters: Examples from WASP-12b, CoRoT-2b, XO-1b, and HD 189733b. ApJ 763, no. 1 (January): 25. https://doi.org/10.1088/0004-637X/763/1/25. arXiv: 1211.2996 [astro-ph.EP] . Mousis, Olivier, Sarah E. Anderson, Adrienn Luspay-Kuti, Kathleen E. Mandt, and Pierre Vernazza. 2024. Triton and Pluto: same origin but separated at birth. arXiv e-prints (June): arXiv:2406.03815. https://doi.org/10. 48550/arXiv.2406.03815. arXiv: 2406.03815 [astro-ph.EP] . Mousis, Olivier, Thibault Cavalié, Jonathan I. Lunine, Kathleen E. Mandt, Ricardo Hueso, Artyom Aguichine, Antoine Schneeberger, et al. 2024. Recipes for Forming a Carbon-Rich Giant Planet. Space Sci. Rev. 220, no. 4 (June): 44. https://doi.org/10.1007/s11214-024-01071-4. arXiv: 2405.19748 [astro-ph.EP] . Nasedkin, E., P. Mollière, S. Lacour, M. Nowak, L. Kreidberg, T. Stolker, J. J. Wang, et al. 2024. Four-of-a-kind? Comprehensive atmospheric characterisation of the HR 8799 planets with VLTI/GRAVITY. arXiv e-prints (April): arXiv:2404.03776. https://doi.org/10.48550/arXiv.2404. 03776. arXiv: 2404.03776 [astro-ph.EP] . Nayakshin, Sergei. 2010a. Formation of planets by tidal downsizing of giant planet embryos. MNRAS 408, no. 1 (October): L36-L40. https://doi.org/ 10.1111/j.1745-3933.2010.00923.x. arXiv: 1007.4159 [astro-ph.EP] . . 2010b. Grain sedimentation inside giant planet embryos. MNRAS 408, no. 4 (November): 2381-2396. https://doi.org/10.1111/j.13652966.2010.17289.x. arXiv: 1007.4162 [astro-ph.EP] . Nayakshin, Sergei, Ravit Helled, and Aaron C. Boley. 2014. Core-assisted gas capture instability: a new mode of giant planet formation by gravitationally unstable discs. MNRAS 440, no. 4 (June): 3797-3808. https: //doi.org/10.1093/mnras/stu473. arXiv: 1403.1813 [astro-ph.EP] . Nortmann, L., F. Lesjak, F. Yan, D. Cont, S. Czesla, A. Lavail, A. D. Rains, et al. 2024. CRIRES + transmission spectroscopy of WASP-127b. Detection of the resolved signatures of a supersonic equatorial jet and cool poles in a hot planet. arXiv e-prints (April): arXiv:2404.12363. https://doi.org/ 10.48550/arXiv.2404.12363. arXiv: 2404.12363 [astro-ph.EP] . Öberg, K. I., F. van Broekhuizen, H. J. Fraser, S. E. Bisschop, E. F. van Dishoeck, and S. Schlemmer. 2005. Competition between CO and N2 Desorption from Interstellar Ices. ApJ 621, no. 1 (March): L33-L36. https://doi.org/10.1086/428901. Öberg, Karin I., A. C. Adwin Boogert, Klaus M. Pontoppidan, Saskia van den Broek, Ewine F. van Dishoeck, Sandrine Bottinelli, Geoffrey A. Blake, and II Evans Neal J. 2011. The Spitzer Ice Legacy: Ice Evolution from Cores to Protostars. ApJ 740, no. 2 (October): 109. https://doi.org/10. 1088/0004-637X/740/2/109. arXiv: 1107.5825 [astro-ph.GA] . Öberg, Karin I., Ruth Murray-Clay, and Edwin A. Bergin. 2011. The Effects of Snowlines on C/O in Planetary Atmospheres. ApJ 743, no. 1 (December): L16. https://doi.org/10.1088/2041-8205/743/1/L16. arXiv: 1110.5567 [astro-ph.GA] . Öberg, Karin I., and Robin Wordsworth. 2019. Jupiter's Composition Suggests its Core Assembled Exterior to the N2 Snowline. AJ 158, no. 5 (November): 194. https://doi.org/10.3847/1538-3881/ab46a8. arXiv: 1909.11246 [astro-ph.EP] . Ohno, Kazumasa, and Takahiro Ueda. 2021. Jupiter's 'cold' formation in the protosolar disk shadow. An explanation for the planet's uniformly enriched atmosphere. A&A 651 (July): L2. https://doi.org/10.1051/00046361/202141169. arXiv: 2106.09084 [astro-ph.EP] . Okuzumi, Satoshi, and Shigenobu Hirose. 2011. Modeling Magnetorotational Turbulence in Protoplanetary Disks with Dead Zones. ApJ 742, no. 2 (December): 65. https://doi.org/10.1088/0004-637X/742/2/65. arXiv: 1108.4892 [astro-ph.EP] . Okuzumi, Satoshi, and Ryo Tazaki. 2019. Nonsticky Ice at the Origin of the Uniformly Polarized Submillimeter Emission from the HL Tau Disk. ApJ 878, no. 2 (June): 132. https://doi.org/10.3847/1538-4357/ab204d. arXiv: 1904.03869 [astro-ph.EP] . Ootsubo, Takafumi, Hideyo Kawakita, Saki Hamada, Hitomi Kobayashi, Mitsuru Yamaguchi, Fumihiko Usui, Takao Nakagawa, et al. 2012. AKARI Near-infrared Spectroscopic Survey for CO2 in 18 Comets. ApJ 752, no. 1 (June): 15. https://doi.org/10.1088/0004-637X/752/1/15. Padoan, Paolo, Liubin Pan, Veli-Matti Pelkonen, Troels Haugboelle, and AAke Nordlund. 2024. Protoplanetary Disks from Pre-Main Sequence BondiHoyle Accretion. arXiv e-prints (May): arXiv:2405.07334. https://doi. org/10.48550/arXiv.2405.07334. arXiv: 2405.07334 [astro-ph.GA] . Padovani, M., D. Galli, and A. E. Glassgold. 2009. Cosmic-ray ionization of molecular clouds. A&A 501, no. 2 (July): 619-631. https://doi.org/10. 1051/0004-6361/200911794. arXiv: 0904.4149 [astro-ph.SR] . Pavlyuchenkov, Yaroslav, Vitaly Akimkin, Dmitri Wiebe, and Eduard Vorobyov. 2019. Revealing dust segregation in protoplanetary discs with the help of multifrequency spectral index maps. MNRAS 486, no. 3 (July): 39073914. https://doi.org/10.1093/mnras/stz1046. arXiv: 1904.05251 [astro-ph.IM] . Pelkonen, Veli-Matti, Paolo Padoan, Mika Juvela, Troels Haugbølle, and Åke Nordlund. 2024. Origin and Evolution of Angular Momentum of Class II Disks. arXiv e-prints (May): arXiv:2405.06520. https://doi.org/10. 48550/arXiv.2405.06520. arXiv: 2405.06520 [astro-ph.SR] . Pérez, Laura M., John M. Carpenter, Sean M. Andrews, Luca Ricci, Andrea Isella, Hendrik Linz, Anneila I. Sargent, et al. 2016. Spiral density waves in a young protoplanetary disk. Science 353, no. 6307 (September): 15191521. https://doi.org/10.1126/science.aaf8296. arXiv: 1610.05139 [astro-ph.GA] . Piso, Ana-Maria A., Karin I. Öberg, Tilman Birnstiel, and Ruth A. MurrayClay. 2015. C/O and Snowline Locations in Protoplanetary Disks: The Effect of Radial Drift and Viscous Gas Accretion. ApJ 815, no. 2 (December): 109. https://doi.org/10.1088/0004-637X/815/2/109. arXiv: 1511.05563 [astro-ph.EP] . Pontoppidan, K. M., C. Salyk, E. A. Bergin, S. Brittain, B. Marty, O. Mousis, and K. I. Öberg. 2014. Volatiles in Protoplanetary Disks. In Protostars and planets vi, edited by Henrik Beuther, Ralf S. Klessen, Cornelis P. Dullemond, and Thomas Henning, 363-385. January. https : / / doi . org/10.2458/azu_uapress_9780816531240-ch016. arXiv: 1401.2423 [astro-ph.EP] . Przybilla, Norbert, Maria-Fernanda Nieva, and Keith Butler. 2008. A Cosmic Abundance Standard: Chemical Homogeneity of the Solar Neighborhood and the ISM Dust-Phase Composition. ApJ 688, no. 2 (December): L103. https://doi.org/10.1086/595618. arXiv: 0809.2403 [astro-ph] . Rafikov, Roman R. 2005. Can Giant Planets Form by Direct Gravitational Instability? ApJ 621, no. 1 (March): L69-L72. https://doi.org/10.1086/ 428899. arXiv: astro-ph/0406469 [astro-ph] . Rampinelli, L., S. Facchini, M. Leemker, J. Bae, M. Benisty, R. Teague, C. J. Law, K. I. Öberg, B. Portilla-Revelo, and A. J. Cridland. 2024. ALMA high-resolution observations unveil planet formation shaping molecular emission in the PDS 70 disk. arXiv e-prints (July): arXiv:2407.06272. https://doi.org/10.48550/arXiv.2407.06272. arXiv: 2407.06272 [astro-ph.EP] . Reggiani, Henrique, Jhon Yana Galarza, Kevin C. Schlaufman, David K. Sing, Brian F. Healy, Andrew McWilliam, Joshua D. Lothringer, and Laurent Pueyo. 2024. Insight into the Formation of β Pic b through the Composition of Its Parent Protoplanetary Disk as Revealed by the β Pic Moving Group Member HD 181327. AJ 167, no. 1 (January): 45. https://doi.org/10.3847/1538-3881/ad0f93. arXiv: 2311.12210 [astro-ph.SR] . Schneider, Aaron David, and Bertram Bitsch. 2021. How drifting and evaporating pebbles shape giant planets. I. Heavy element content and atmospheric C/O. A&A 654 (October): A71. https://doi.org/10.1051/00046361/202039640. arXiv: 2105.13267 [astro-ph.EP] . Seager, S., L. J. Richardson, B. M. S. Hansen, K. Menou, J. Y. -K. Cho, and D. Deming. 2005. On the Dayside Thermal Emission of Hot Jupiters. ApJ 632, no. 2 (October): 1122-1131. https://doi.org/10.1086/444411. arXiv: astro-ph/0504212 [astro-ph] . Seligman, Darryl Z., Leslie A. Rogers, Samuel H. C. Cabot, John W. Noonan, Theodore Kareta, Kathleen E. Mandt, Fred Ciesla, et al. 2022. The Volatile Carbon-to-oxygen Ratio as a Tracer for the Formation Locations of Interstellar Comets. PSJ 3, no. 7 (July): 150. https://doi.org/10. 3847/PSJ/ac75b5. arXiv: 2204.13211 [astro-ph.EP] . Semenov, D., C. Favre, D. Fedele, S. Guilloteau, R. Teague, Th. Henning, A. Dutrey, E. Chapillon, F. Hersant, and V. Piétu. 2018. Chemistry in disks. XI. Sulfur-bearing species as tracers of protoplanetary disk physics and chemistry: the DM Tau case. A&A 617 (September): A28. https://doi.org/10.1051/0004-6361/201832980. arXiv: 1806.07707 [astro-ph.GA] . Semenov, D., Th. Henning, Ch. Helling, M. Ilgner, and E. Sedlmayr. 2003. Rosseland and Planck mean opacities for protoplanetary discs. A&A 410 (November): 611-621. https://doi.org/10.1051/0004-6361:20031279. arXiv: astro-ph/0308344 [astro-ph] . Semenov, D., and D. Wiebe. 2011. Chemical Evolution of Turbulent Protoplanetary Disks and the Solar Nebula. ApJS 196, no. 2 (October): 25. https://doi.org/10.1088/0067- 0049/196/2/25. arXiv: 1104.4358 [astro-ph.GA] . Shakura, N. I., and R. A. Sunyaev. 1973. Black holes in binary systems. Observational appearance. A&A 24 (January): 337-355. Sing, David K., Zafar Rustamkulov, Daniel P. Thorngren, Joanna K. Barstow, Pascal Tremblin, Catarina Alves de Oliveira, Tracy L. Beck, et al. 2024. A warm Neptune's methane reveals core mass and vigorous atmospheric mixing. arXiv e-prints (May): arXiv:2405.11027. https://doi.org/10. 48550/arXiv.2405.11027. arXiv: 2405.11027 [astro-ph.EP] . Smith, Peter C. B., Michael R. Line, Jacob L. Bean, Matteo Brogi, Prune August, Luis Welbanks, Jean-Michel Desert, et al. 2024. A Combined Ground-based and JWST Atmospheric Retrieval Analysis: Both IGRINS and NIRSpec Agree that the Atmosphere of WASP-77A b Is Metal-poor. AJ 167, no. 3 (March): 110. https://doi.org/10.3847/1538-3881/ad17bf. arXiv: 2312.13069 [astro-ph.EP] . Stammler, Sebastian Markus, Tilman Birnstiel, Olja Panić, Cornelis Petrus Dullemond, and Carsten Dominik. 2017. Redistribution of CO at the location of the CO ice line in evolving gas and dust disks. A&A 600 (April): A140. https://doi.org/10.1051/0004-6361/201629041. arXiv: 1701.02385 [astro-ph.EP] . Stevenson, David J., and Jonathan I. Lunine. 1988. Rapid formation of Jupiter by diffusive redistribution of water vapor in the solar nebula. Icarus 75, no. 1 (July): 146-155. https://doi.org/10.1016/0019-1035(88)90133-9. Swain, M. R., G. Tinetti, G. Vasisht, P. Deroo, C. Griffith, J. Bouwman, Pin Chen, et al. 2009. Water, Methane, and Carbon Dioxide Present in the Dayside Spectrum of the Exoplanet HD 209458b. ApJ 704, no. 2 (October): 1616-1621. https://doi.org/10.1088/0004-637X/704/2/1616. arXiv: 0908.4010 [astro-ph.EP] . Testi, L., T. Birnstiel, L. Ricci, S. Andrews, J. Blum, J. Carpenter, C. Dominik, et al. 2014. Dust Evolution in Protoplanetary Disks. In Protostars and planets vi, edited by Henrik Beuther, Ralf S. Klessen, Cornelis P. Dullemond, and Thomas Henning, 339-361. January. https://doi.org/10.2458/azu_ uapress_9780816531240-ch015. arXiv: 1402.1354 [astro-ph.SR] . Thiabaud, A., U. Marboeuf, Y. Alibert, I. Leya, and K. Mezger. 2015. Gas composition of the main volatile elements in protoplanetary discs and its implication for planet formation. A&A 574 (February): A138. https: //doi.org/10.1051/0004-6361/201424868. Thies, Ingo, Pavel Kroupa, Simon P. Goodwin, Dimitrios Stamatellos, and Anthony P. Whitworth. 2010. Tidally Induced Brown Dwarf and Planet Formation in Circumstellar Disks. ApJ 717, no. 1 (July): 577-585. h ttps://doi.org/10.1088/0004- 637X/717/1/577. arXiv: 1005.3017 [astro-ph.SR] . Tielens, A. G. G. M. 2005. The Physics and Chemistry of the Interstellar Medium. Tong, Simin, Richard Alexander, and Giovanni Rosotti. 2024. A question of personalities: evolution of viscous and wind-driven protoplanetary discs in the presence of dead zones. MNRAS (July). https://doi.org/10.1093/ mnras/stae1748. arXiv: 2407.12209 [astro-ph.EP] . Toomre, A. 1964. On the gravitational stability of a disk of stars. ApJ 139 (May): 1217-1238. https://doi.org/10.1086/147861. Topchieva, A., T. Molyarova, V. Akimkin, L. Maksimova, and E. Vorobyov. 2024. Ices on pebbles in protoplanetary discs. MNRAS 530, no. 3 (May): 2731-2748. https://doi.org/10.1093/mnras/stae597. arXiv: 2403.02895 [astro-ph.EP] . Turrini, D., E. Schisano, S. Fonte, S. Molinari, R. Politi, D. Fedele, O. Panić, M. Kama, Q. Changeat, and G. Tinetti. 2021. Tracing the Formation History of Giant Planets in Protoplanetary Disks with Carbon, Oxygen, Nitrogen, and Sulfur. ApJ 909, no. 1 (March): 40. https://doi.org/10. 3847/1538-4357/abd6e5. arXiv: 2012.14315 [astro-ph.EP] . Umebayashi, T., and T. Nakano. 1981. Fluxes of Energetic Particles and the Ionization Rate in Very Dense Interstellar Clouds. PASJ 33 (January): 617. Vallet, David, Anna C. Childs, Rebecca G. Martin, Mario Livio, and Stephen Lepp. 2023. Formation of super-Earths in icy dead zones around lowmass stars. MNRAS 519, no. 1 (February): L10-L14. https://doi.org/10. 1093/mnrasl/slac144. arXiv: 2211.07759 [astro-ph.EP] . Visser, R., E. F. van Dishoeck, S. D. Doty, and C. P. Dullemond. 2009. The chemical history of molecules in circumstellar disks. I. Ices. A&A 495, no. 3 (March): 881-897. https://doi.org/10.1051/0004-6361/200810846. arXiv: 0901.1313 [astro-ph.SR] . Vorobyov, E. I. 2013. Formation of giant planets and brown dwarfs on wide orbits. A&A 552 (April): A129. https://doi.org/10.1051/0004-6361/ 201220601. arXiv: 1302.1892 [astro-ph.EP] . Vorobyov, Eduard I., Vitaly Akimkin, Olga Stoyanovskaya, Yaroslav Pavlyuchenkov, and Hauyu Baobab Liu. 2018. Early evolution of viscous and selfgravitating circumstellar disks with a dust component. A&A 614 (June): A98. https://doi.org/10.1051/0004-6361/201731690. arXiv: 1801.06898 [astro-ph.EP] . Vorobyov, Eduard I., and Shantanu Basu. 2010. The Burst Mode of Accretion and Disk Fragmentation in the Early Embedded Stages of Star Formation. ApJ 719, no. 2 (August): 1896-1911. https://doi.org/10.1088/0004637X/719/2/1896. arXiv: 1007.2993 [astro-ph.SR] . Vorobyov, Eduard I., and Vardan G. Elbakyan. 2019. Gravitoviscous protoplanetary disks with a dust component. II. Spatial distribution and growth of dust in a clumpy disk. A&A 631 (November): A1. https: / / doi . org / 10 . 1051 / 0004 - 6361 / 201936132. arXiv: 1908 . 10589 [astro-ph.SR] . Vorobyov, Eduard I., Vardan G. Elbakyan, Michihiro Takami, and Hauyu B. Liu. 2020. Effect of luminosity outbursts on protoplanetary disk dynamics. A&A 643 (November): A13. https://doi.org/10.1051/00046361/202038122. arXiv: 2009.01888 [astro-ph.SR] . Vorobyov, Eduard I., Sergey Khaibrakhmanov, Shantanu Basu, and Marc Audard. 2020. Accretion bursts in magnetized gas-dust protoplanetary disks. A&A 644 (December): A74. https://doi.org/10.1051/00046361/202039081. arXiv: 2011.00951 [astro-ph.SR] . Vorobyov, Eduard I., D. N. C. Lin, and Manuel Guedel. 2015. The effect of external environment on the evolution of protostellar disks. A&A 573 (January): A5. https://doi.org/10.1051/0004-6361/201424583. arXiv: 1410.1743 [astro-ph.SR] . Vorobyov, Eduard I., Zsolt Regaly, Manuel Guedel, and Doug N. C. Lin. 2016. An alternative model for the origin of gaps in circumstellar disks. A&A 587 (March): A146. https://doi.org/10.1051/0004-6361/201527701. arXiv: 1601.08089 [astro-ph.SR] . Vorobyov, Eduard I., Aleksandr M. Skliarevskii, Manuel Guedel, and Tamara Molyarova. 2024. Primordial dust rings, hidden dust mass, and the first generation of planetesimals in gravitationally unstable protoplanetary disks. arXiv e-prints (April): arXiv:2404.16151. https://doi.org/10.48550/ arXiv.2404.16151. arXiv: 2404.16151 [astro-ph.EP] . Vorobyov, Eduard I., Aleksandr M. Skliarevskii, Tamara Molyarova, Vitaly Akimkin, Yaroslav Pavlyuchenkov, Ágnes Kóspál, Hauyu Baobab Liu, Michihiro Takami, and Anastasiia Topchieva. 2022. Evolution of dust in protoplanetary disks of eruptive stars. A&A 658 (February): A191. https://doi.org/10.1051/0004-6361/202141932. arXiv: 2112.06004 [astro-ph.EP] . Vorobyov, Eduard I., Olga V. Zakhozhay, and Michael M. Dunham. 2013. Fragmenting protostellar discs: properties and observational signatures. MNRAS 433, no. 4 (August): 3256-3273. https://doi.org/10.1093/ mnras/stt970. arXiv: 1306.4074 [astro-ph.SR] . Wada, Koji, Hidekazu Tanaka, Toru Suyama, Hiroshi Kimura, and Tetsuo Yamamoto. 2009. Collisional Growth Conditions for Dust Aggregates. ApJ 702, no. 2 (September): 1490-1501. https://doi.org/10.1088/0004637X/702/2/1490. Walsh, Catherine, Hideko Nomura, and Ewine van Dishoeck. 2015. The molecular composition of the planet-forming regions of protoplanetary disks across the luminosity regime. A&A 582 (October): A88. http s : / / doi . org / 10 . 1051 / 0004 - 6361 / 201526751. arXiv: 1507 . 08544 [astro-ph.EP] . Weber, Philipp, Sebastián Pérez, Alice Zurlo, James Miley, Antonio Hales, Lucas Cieza, David Principe, et al. 2023. Spirals and Clumps in V960 Mon: Signs of Planet Formation via Gravitational Instability around an FU Ori Star? ApJ 952, no. 1 (July): L17. https://doi.org/10.3847/20418213/ace186. arXiv: 2307.13433 [astro-ph.EP] . Wei, Chen-En, Hideko Nomura, Jeong-Eun Lee, Wing-Huen Ip, Catherine Walsh, and T. J. Millar. 2019. The Effect of Carbon Grain Destruction on the Chemical Structure of Protoplanetary Disks. ApJ 870, no. 2 (January): 129. https://doi.org/10.3847/1538- 4357/aaf390. arXiv: 1811.10194 [astro-ph.EP] . Weidenschilling, S. J. 1977. Aerodynamics of solid bodies in the solar nebula. MNRAS 180 (July): 57-70. https://doi.org/10.1093/mnras/180.2.57. Weiner Mansfield, Megan, Michael R. Line, Joost P. Wardenier, Matteo Brogi, Jacob L. Bean, Hayley Beltz, Peter Smith, et al. 2024. The metallicity and carbon-to-oxygen ratio of the ultra-hot Jupiter WASP-76b from Gemini-S/IGRINS. arXiv e-prints (May): arXiv:2405.09769. https://doi. org/10.48550/arXiv.2405.09769. arXiv: 2405.09769 [astro-ph.EP] . Westley, M. S., R. A. Baragiola, R. E. Johnson, and G. A. Baratta. 1995. Photodesorption from low-temperature water ice in interstellar and circumsolar grains. Nature 373, no. 6513 (February): 405-407. https: //doi.org/10.1038/373405a0. Wierzchos, K., and M. Womack. 2018. C/2016 R2 (PANSTARRS): A Comet Rich in CO and Depleted in HCN. AJ 156, no. 1 (July): 34. https://doi. org/10.3847/1538-3881/aac6bc. arXiv: 1805.06918 [astro-ph.EP] . Winter, Andrew J., Myriam Benisty, and Sean M. Andrews. 2024. Planet formation regulated by galactic-scale interstellar turbulence. arXiv eprints (May): arXiv:2405.08451. arXiv: 2405.08451 [astro-ph.EP] . Wong, Michael H., Paul R. Mahaffy, Sushil K. Atreya, Hasso B. Niemann, and Tobias C. Owen. 2004. Updated Galileo probe mass spectrometer measurements of carbon, oxygen, nitrogen, and sulfur on Jupiter. Icarus 171, no. 1 (September): 153-170. https://doi.org/10.1016/j.icarus.2004. 04.010. Worthen, Kadin, Christine H. Chen, David R. Law, Cicero X. Lu, Kielan Hoch, Yiwei Chai, G. C. Sloan, et al. 2024. MIRI MRS Observations of β Pictoris. I. The Inner Dust, the Planet, and the Gas. ApJ 964, no. 2 (April): 168. https://doi.org/10.3847/1538-4357/ad2354. Xue, Qiao, Jacob L. Bean, Michael Zhang, Luis Welbanks, Jonathan Lunine, and Prune August. 2024. JWST Transmission Spectroscopy of HD 209458b: A Supersolar Metallicity, a Very Low C/O, and No Evidence of CH4, HCN, or C2H2. ApJ 963, no. 1 (March): L5. https://doi.org/ 10.3847/2041-8213/ad2682. arXiv: 2310.03245 [astro-ph.EP] . Yang, Chao-Chin, Anders Johansen, and Daniel Carrera. 2017. Concentrating small particles in protoplanetary disks through the streaming instability. A&A 606 (October): A80. https : / / doi . org / 10 . 1051 / 0004 - 6361 / 201630106. arXiv: 1611.07014 [astro-ph.EP] . Youdin, Andrew N., and Jeremy Goodman. 2005. Streaming Instabilities in Protoplanetary Disks. ApJ 620, no. 1 (February): 459-469. https: //doi.org/10.1086/426895. arXiv: astro-ph/0409263 [astro-ph] . Zhang, Yapeng, Ignas A. G. Snellen, Alexander J. Bohn, Paul Mollière, Christian Ginski, H. Jens Hoeijmakers, Matthew A. Kenworthy, et al. 2021. The 13 CO-rich atmosphere of a young accreting super-Jupiter. Nature 595, no. 7867 (July): 370-372. https://doi.org/10.1038/s41586-02103616-x. arXiv: 2107.06297 [astro-ph.EP] . Zhu, Zhaohuan, Lee Hartmann, Charles F. Gammie, Laura G. Book, Jacob B. Simon, and Eric Engelhard. 2010. Long-term Evolution of Protostellar and Protoplanetary Disks. I. Outbursts. ApJ 713, no. 2 (April): 11341142. https : / / doi . org / 10 . 1088 / 0004 - 637X / 713 / 2 / 1134. arXiv: 1003.1759 [astro-ph.SR] . Zhu, Zhaohuan, Yan-Fei Jiang, and James M. Stone. 2020. Global 3D radiation magnetohydrodynamic simulations for FU Ori's accretion disc and observational signatures of magnetic fields. MNRAS 495, no. 3 (January): 3494-3514. https://doi.org/10.1093/mnras/staa952. arXiv: 1912.01632 [astro-ph.EP] .",
"pages": [
23,
24,
25,
26,
27,
28,
29
]
}
]
2024arXiv241205772M
https://arxiv.org/pdf/2412.05772.pdf
<document>
<section_header_level_1><location><page_1><loc_15><loc_81><loc_86><loc_85></location>WOLF-RAYET STARS - WHAT WE KNOW AND WHAT WE DON'T</section_header_level_1>
<text><location><page_1><loc_41><loc_75><loc_60><loc_77></location>O. V. Maryeva a *</text>
<text><location><page_1><loc_14><loc_71><loc_87><loc_74></location>a Astronomical Institute of the Czech Academy of Sciences, Fričova 298, 25165 Ondřejov, Czech Republic</text>
<text><location><page_1><loc_18><loc_57><loc_83><loc_68></location>Today, we have a sufficiently complete picture of what the Wolf-Rayet (WR) stars are. Predictions of stellar evolution theory are in a good agreement with their parameters, estimated from observational data using stellar atmospheres codes; predictions of population synthesis also agree well with number of known WR stars. This article provides an overview of the main historical milestones in the studies of WR stars, showing how we came to this understanding, and what questions are still unanswered.</text>
<text><location><page_1><loc_18><loc_51><loc_83><loc_54></location>Keywords: Stars: Wolf-Rayet - Stars: evolution - Stars: atmospheres - General: history and philosophy of astronomy</text>
<section_header_level_1><location><page_1><loc_40><loc_42><loc_61><loc_44></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_12><loc_16><loc_89><loc_40></location>Wolf-Rayet (WR) stars are a class of objects identified on the basis of their spectral features. WR spectra show strong emission lines of helium, nitrogen, carbon and oxygen in different stages of ionization. The width of these lines reaches tens of angstroms, the central intensities are sometimes 10-20 times the intensity of the continuum spectrum. A prominent feature of WRs are two bumps at 4650 1) and 5808 Å, which are groups of closely located lines of N II , N III , C III and C IV , which are visible even in low-resolution spectra. The lines are formed in an extended atmosphere - in the stellar wind, expelling at velocities of 10 2 -10 3 kms -1 . WR stars are characterized by a high mass-loss rate of several 10 -5 M ⊙ yr -1 and by high temperatures ( T eff ≳ 30000 K). WR are the final stage in the evolution of massive stars (stars with an initial mass ≳ 25 M ⊙ ) before the core-collapse supernova explosion.</text>
<text><location><page_2><loc_12><loc_56><loc_89><loc_90></location>With accumulation of observational data it became clear that emission spectra and WR bumps are intrinsic to objects of different natures. Therefore nowadays there is a clear distinction between WR stars and WR phenomenon , which occurs when fast-moving, hot plasma is expanding around a hot star (Gräfener et al., 2011; Vink, 2015). WR phenomenon may occur in the low mass stars after ejecting their outer layers in the planetary nebula phase and exposing their hot cores prior to the white-dwarf phase (Marcolino et al., 2007; Todt, 2009; van der Hucht et al., 1981). Central stars of planetary nebulae showing WR phenomenon in their spectra are denoted as [WR] (van der Hucht et al., 1981). Rare and still unique case of low mass object with WR phenomenon and the absence of a clearly detected circumstellar nebula is LAMOST J040901.83+323955.6 (Maryeva et al., 2024). It was selected as a WR star by Škoda et al. (2020), but Maryeva et al. (2024) showed that this object is 0 . 9 M ⊙ star caught in a rare transitional phase from post-AGB to CSPN. Another unique object with WR phenomenon is IRAS 00500+6713, formed as a result of a merger of two white dwarfs (Gvaramadze et al., 2019). The WR phenomenon is also observed in young supernovae (GalYam et al., 2014).</text>
<text><location><page_2><loc_12><loc_36><loc_89><loc_56></location>As we know now, massive stars showing WR spectra are also not a homogeneous class. They are split into two groups: very massive stars and classical WR stars . Very massive stars ( M ini ≥ 100M ⊙ ) experience strong stellar winds and show WR spectra already during core H-burning phase, when they are located on main sequence. All of them show signs of hydrogen on their surface and are classified as WNh objects (Crowther et al., 2010; Martins et al., 2023; Smith et al., 1996). Classical WR stars (usually just named WR stars ) stars are hydrogen-depleted objects that have evolved off the main sequence and suffered intense mass loss (Crowther, 2007; Shenar, 2024). Classical WR stars will be the subject of this review article.</text>
<text><location><page_2><loc_12><loc_20><loc_89><loc_36></location>The field of contemporary studies of WR stars is enormous and cannot be fully covered in a single small review like this one, so I'd like to recommend two other detailed reviews - Crowther (2007) and Shenar (2024) - for better understanding what WR stars are. On the other hand, here I will concentrate more on the historical aspects of WR studies, on how we have arrived to our present day understanding of them. Therefore Section 2 will cover the history of studies of WR stars, while Section 3 will highlight several topical problems and open questions about them.</text>
<section_header_level_1><location><page_2><loc_25><loc_16><loc_76><loc_17></location>2. FROM DISCOVERY TO UNDERSTANDING</section_header_level_1>
<text><location><page_2><loc_12><loc_10><loc_89><loc_14></location>In 1867 year astronomers from Paris observatory Charles Wolf and Georges Rayet discovered three stars in Cygnus constellation those spectra significantly</text>
<text><location><page_3><loc_12><loc_71><loc_89><loc_90></location>differ from other stars. In contrast with other stars, where usually absorption lines are dominant in spectra, discovered objects showed strong emission lines (Wolf & Rayet, 1867). The class of objects received its name in honor of the discoverers and gradually began to be replenished with new members. One hundred years after the discovery of WRs, by the end of 1960-s the number of known WRs in the Galaxy had reached over a hundred, totalling 127 (Roberts, 1962; Smith, 1968), while now is ∼ 700 (Rosslowe & Crowther, 2015). At the same time, in the late 1950-s middle 1960-s, work began to search for extragalactic WRs - WRs in the Large Magellanic Cloud (Westerlund & Rodgers, 1959; Westerlund & Smith, 1964).</text>
<text><location><page_3><loc_12><loc_46><loc_89><loc_70></location>Significant contribution to the understanding of the physics of WR stars was given by Carlyle S. Beals (1929). Beals found that some lines in the spectra of WR stars have a P Cygni profile and based on comparison with P Cygni profile lines in spectra of novae he suggested that WRs are surrounded by expanding envelopes (Beals, 1929). Beals also found a difference between WR stars and novae: the P Cygni profiles in the WR stars do not change over time. This allowed to propose that an outflow from WR stars occurs continuously (Beals, 1929). This was confirmed by Chandrasekhar (1934), who developed a solid footing for interpreting P Cygni profiles as arising in expanding atmospheres. Kosirev (1934) used the diagnostics developed by Chandrasekhar to estimate the mass loss and maximum outflow velocity of a WR star and found, respectively, ∼ 10 -5 M ⊙ yr -1 and ∼ 1000 kms -1 .</text>
<text><location><page_3><loc_12><loc_22><loc_89><loc_46></location>At the beginning of the 20th century, stellar spectroscopy developed in close cooperation with atomic physics. Investigating spectra of the star ζ Puppis Edward Pickering paid his attention to previously unknown lines 5411, 4541, 4200, 4100, 4026, 3924, 3858, 3813, 3782 Å (Pickering, 1897). He interpreted them as another series of hydrogen (besides Balmer) (Pickering, 1901). Line 4686 Å, first discovered during a solar eclipse and often found in WR spectra, was also considered to be a hydrogen line (Fowler, 1912). Although Fowler (1912) was able to obtain these spectral lines in the laboratory in 1912, during an experiment with a helium-filled tube, he considered their appearance to be a contribution from hydrogen impurity. It was not until 1913 that Niels Bohr's work (Bohr, 1913) explained the nature of these lines as ionized helium He II , confirming the atomic model (see Robotti (1983) for historical review).</text>
<text><location><page_3><loc_12><loc_10><loc_89><loc_22></location>In 1890 Pickering drew attention to the similarity of the spectra of WR stars and planetary nebula (Pickering, 1891). Due to this, in papers devoted to emission spectra of nebulae it is possible to find descriptions of spectra of WR stars. The lines at 4647, 4650, 5696, 5801 and 5812 Å were identified with the carbon by Wright (1918). Ira S. Bowen, who found an interpretation of the 'nebulium' lines as [O III ], gave a detailed list of emission lines observed in nebulae, which</text>
<text><location><page_4><loc_12><loc_85><loc_89><loc_90></location>includes an identification of almost all lines visible in the WRs (Bowen, 1928). In a 1934 paper Bowen (1934) suggests a significant role of the fluorescence for formation of the N III 4634, 46340 lines belonging to the WR bump.</text>
<text><location><page_4><loc_12><loc_54><loc_89><loc_84></location>Already the spectra of the first discovered WR stars showed a difference: the broad emission bands are located in different places (Huggins & Huggins, 1890; Wolf & Rayet, 1867). After the identification of all the main spectral lines, Beals (1933) proposed the splitting of WR into two groups. This classification was approved in 1938, the International Astronomical Union (IAU) divided the spectra of WR stars into types WN and WC, depending on whether the spectrum was dominated by lines of nitrogen or carbon-oxygen respectively (Beals, 1933; Swings, 1942). WN stars are believed to show the hydrogen burning products via the CNO cycle, while WC stars reveal the helium burning products via the tripleα cycle. Based on the strength of the emission lines and line ratios, WN stars can be further classified into the spectral subtypes WN2 to WN11, and WC stars into the spectral subtypes WC4 to WC9 (Crowther et al., 1998; Smith, 1968; Smith et al., 1990, 1994, 1996). Also, there are transition types from WN to WC, which are called WN/C, whose spectra show strong emission lines of carbon and nitrogen simultaneously (Conti & Massey, 1989).</text>
<text><location><page_4><loc_12><loc_28><loc_89><loc_54></location>In 1933 Victor A. Ambartsumian (1933) estimated He/H ratio for WR stars using intensities of H β and He II 4686 lines and found He/H ≳ 1 . 8 . Subsequent studies (Rublev (1972a) and references therein) confirmed excess of helium. Gamow (1943) suggested that the anomalous composition of WR stars was the result of nuclear processed material being visible on their surfaces. Despite these arguments in favor of far evolved status of WR stars, debates about it continued until the mid-1970s. There was alternative hypothesis, that WR are young and more massive than T Tau objects, before main sequence (Sahade, 1958; Underhill, 1968). It is important to mention the studies of Sergej V. Rublev, in which he developed methods for determining the temperatures and luminosities of WR stars and estimated the hydrogen abundances (Rublev, 1965, 1970, 1972b, 1975). These works played a significant role in the formation of the modern understanding of WR stars as far evolved massive objects.</text>
<section_header_level_1><location><page_4><loc_29><loc_23><loc_72><loc_25></location>2.1. Development of numerical methods</section_header_level_1>
<text><location><page_4><loc_12><loc_14><loc_89><loc_22></location>Contemporary progress in our understanding of hot stars and especially WR stars has been significantly defined by the progress of numerical modelling of stellar atmospheres, in turn enabled by the development of both physical approximations and numerical methods.</text>
<text><location><page_4><loc_12><loc_10><loc_89><loc_14></location>Victor V. Sobolev developed the theory of radiative transfer in moving media and applied it to the determination of physical conditions in the envelopes</text>
<text><location><page_5><loc_12><loc_79><loc_89><loc_90></location>of WR, Be, novae, etc. type stars (Sobolev, 1947, 1960). He showed that due to the presence of a velocity gradient, photons of spectral lines are able to leave the deep layers of the atmosphere directly, avoiding the usual diffusion process; this significantly simplifies the study of the radiative equilibrium of moving atmospheres. This theory became the basis for the transition to modern quantitative calculations of the stellar wind.</text>
<text><location><page_5><loc_12><loc_60><loc_89><loc_78></location>WRstars are essentially hot stellar cores surrounded by extended, low-density atmospheres. Due to their high luminosity the radiation in the stellar wind dominates over collisional processes, and the models of WR's atmospheres should be calculated taking into account the deviations from local thermodynamic equilibrium (non-LTE). Straightforward iterative procedure for the solution of radiative transfer equations Λ iteration - fails to converge in typical non-LTE conditions, as in scattering dominated media of large optical thickness, its convergence is extremely slow (see overview by Atanackovic-Vukmanovic (2004) and references therein).</text>
<text><location><page_5><loc_12><loc_46><loc_89><loc_60></location>Several approaches have been proposed in early 1970-s as a way to numerically solve radiative transfer (RT) problems in non-LTE case. The first fully self-consistent solution of RT problems for non-LTE case was achieved by Auer & Mihalas (1969) who developed the complete linearization method based on the Newton-Raphson scheme to solve the set of discretized structural equations in a robust manner that opened a way for significant progress in the modelling of stellar atmospheres.</text>
<text><location><page_5><loc_12><loc_32><loc_89><loc_46></location>A different approach is the 'core saturation method', introduced by Rybicki (1972) who implemented a modification of the Λ iteration method that made it practically usable for solving the RT and statistical equilibrium equations. In order to eliminate the cause of the slow convergence of simple Λ iteration (poor conditioning of the equations arising from a large number of scatterings), Rybicki proposed the elimination of scattering events in the line core as they do not contribute much to the transfer process.</text>
<text><location><page_5><loc_12><loc_18><loc_89><loc_32></location>Another approach is 'perturbation technique' proposed by Cannon (1973a,b) who was the first to use the idea of 'operator splitting' in radiative transfer computations. He replaced the full (exact) Λ operator by a simplified one (so called approximate Lambda operator, or ALO) and computed a small error term made by this approximation by using a perturbation technique. Cannon's operator perturbation technique was an efficient way to combine computational advantages of approximate solutions with the accuracy of exact methods.</text>
<text><location><page_5><loc_12><loc_10><loc_89><loc_18></location>In 1980-s computational approximations to accelerate the Λ iteration started to be used together with the operator perturbation technique introduced by Cannon (1973a,b) to simplify the direct solution, making the basis for the class of methods known as Accelerated Lambda Iteration. Wolf-Rainer Hamann (1985,</text>
<text><location><page_6><loc_12><loc_73><loc_89><loc_90></location>1986) adopted perturbation technique to approximate lambda operator, based on idea of 'core saturation' and presented test calculations for a typical WR star atmosphere. This effort became the foundation for the PoWR code (Hamann & Gräfener, 2003) that is still being actively developed and successfully applied for numerical modelling of various types of astrophysical objects. Independently, D. John Hillier (1990) presented a method based on tridiagonal Newton-Raphson operator, complete linearization scheme and perturbation technique for calculations of WN and WC atmosphere models. Over time, this method has grown into CMFGEN code (Hillier & Miller, 1998).</text>
<text><location><page_6><loc_12><loc_48><loc_89><loc_72></location>Since the mid-1990s, CMFGEN and PoWR have become a reliable instruments for studying hot stars with high mass-loss rates. Systematic study of Galactic WR stars started with a series of works by Paul Crowther titled 'Fundamental parameters of WR stars' and devoted to the analysis of properties of nitrogen-rich WR stars (WN) based on optical and ultraviolet (UV) data and numerical modelling using CMFGEN code (Crowther et al., 1995a,b,c,d,e). Advances in infrared (IR) spectroscopy also enabled studies of WR stars near the Galactic center (Liermann et al., 2010; Martins et al., 2007). Modeling of large sample of Galctic WN (Hamann et al., 2006) and WC (Sander et al., 2012) stars ( ≈ 126 objects in total) allowed to obtain a homogeneous set of stellar and atmospheric parameters for them, to determine the correct location of WR stars in the Hertzsprung-Russell (HR) diagram.</text>
<text><location><page_6><loc_12><loc_34><loc_89><loc_48></location>PoWR code was widely used for studying extragalactic objects. Among WR stars in nearby galaxies, the ones in Magellanic Clouds are most studied (Bestenlehner et al., 2014; Hainich et al., 2014, 2015). Sample of 74 WR stars belonged to M33 galaxy was analysed by Pritzkuleit (2020) (Pritzkuleit, 2020). Sander et al. (2014) investigated late-type WN stars in M31 galaxy (Sander et al., 2014). These studies extended our understanding of properties of WR stars in different galactic environments and metallicity-dependence of massive star winds.</text>
<section_header_level_1><location><page_6><loc_43><loc_29><loc_58><loc_31></location>2.2. Evolution</section_header_level_1>
<text><location><page_6><loc_12><loc_14><loc_89><loc_28></location>Modern view on WRs as products of evolution of massive stars was finally formed only in 1980s. Although by the 1980s WR stars were already recognised as descendants of massive OB stars, their exact evolutionary status with respect to the main sequence and other evolved massive stars was still an open question. WR stars were being proposed as possible progenitors of supernovae, and particularly the newly-discovered type Ib supernovae that lack hydrogen but are apparently associated with young massive stars.</text>
<text><location><page_6><loc_12><loc_10><loc_89><loc_14></location>The start of systematic X-ray observations with space-borne X-ray telescopes in 1960s significantly advanced the theory of close binary stars and their interac-</text>
<text><location><page_7><loc_12><loc_77><loc_89><loc_90></location>tions, and it led Bohdan Paczyński (1967) to the suggestion that WR stars are the products of binary evolution - namely, mass exchange in a binary system. In his scenario, a more massive star in binary system evolves faster and, upon exhaustion of hydrogen in the core, fills the Roche lobe and a rapidly loses its hydrogen envelope that outflows towards the companion. As a result, when the envelope is fully transferred, its hot helium core remains as a WR star with little hydrogen in the outermost layers (Paczyński, 1973).</text>
<text><location><page_7><loc_12><loc_54><loc_89><loc_76></location>The hypothesis of Paczyński was well accepted and it naturally explained the anomalous chemical composition of WR stars as naked stellar cores enriched with products of nuclear reactions. On the other hand, based on the similarity of Of and WN stars (Figure 1) and on the first estimates of the mass loss rate for hot massive stars based on UV observations Peter Conti (1975) suggested that the stellar wind of O stars is strong enough to remove the outer layers and uncover the stellar core, thus proposing a single star mass loss channel to form WR stars. In addition, brightest WR star, γ Vel, was inconsistent with Paczyński's hypothesis as, while being a binary, its orbit was clearly elliptical and its second component lacks any signs of previous mass exchange, both suggesting that there was no interaction between the components (Conti, 2015).</text>
<text><location><page_7><loc_12><loc_26><loc_89><loc_54></location>In 1984 Peter Conti (1984) introduced a new class of objects. In his talk at the IAU symposium, Peter Conti grouped S Doradus variables, Hubble-Sandage variables, η Carinae, P Cygni, and other similar stars together under the term 'luminous blue variables' (LBV). LBV are characterized by strong mass loss (around 10 -5 M ⊙ yr -1 ) and occasional giant eruptive events (Humphreys & Davidson, 1994). Conti suggested that massive O stars transit to WR stars through this short-duration LBV phase. This idea is consolidated as the 'Conti scenario' and becomes the main channel for WR star formation. Modern stellar evolution calculations are consistent with the 'Conti scenario' (Groh et al., 2014; Sander et al., 2019). According to numerical calculations the stars with initial mass of 60 M ⊙ spend 2 × 10 5 yr in the LBV phase - only 5% of their lifetime (Groh et al., 2014). Recently observed transition from LBV to WN8 spectral type in Romano' star is also a direct confirmation of 'Conti scenario' (Maryeva et al., 2019; Polcaro et al., 2016).</text>
<text><location><page_7><loc_12><loc_16><loc_89><loc_26></location>Similar scenario for WR formation was also proposed by Bisnovatyi-Kogan & Nadyozhin (1972), with significant mass loss on the red supergiant phase. They show that large inverse density gradient appears in the outer layers of a 30 M ⊙ star in this phase due to supercritical luminosity, resulting in powerful and rapid mass loss and formation of a stripped helium core.</text>
<text><location><page_7><loc_12><loc_10><loc_89><loc_16></location>In the last decade, Paczyński's hypothesis has again been discussed as a mechanism for the formation of WR stars. Analysing the spatial positions of LBV and WR stars, Nathan Smith (Smith, 2016, 2019; Smith & Tombleson, 2015) con-</text>
<figure>
<location><page_8><loc_23><loc_52><loc_78><loc_90></location>
<caption>Fig. 1: Comparison of spectra of J013340.19+303134.5 (WC4), J013352.43+304351.7 (WN8) and HD14947 (O4.5If) as illustration of significant differences in intensities of spectral lines for O stars and different sub-types of WR stars.</caption>
</figure>
<text><location><page_8><loc_12><loc_18><loc_89><loc_40></location>luded that both LBVs and WRs could be the products of binary evolution, LBVs being mass-gainers, and WRs - mass-donors, or stripped components. This idea was critically discussed by Humphreys et al. (2016) and Davidson et al. (2016). Now we may confidently conclude only that we do not have enough data for statistical analysis. Modern evolutionary codes allow detailed calculations of binary system evolution (see for example Götberg et al. (2018); Laplace et al. (2021)), thus making it possible to compare theoretical predictions for stripped stars with properties of real WR stars (Götberg et al., 2018). Our current understanding is that binary interaction scenario cannot properly describe all observed WR stars, but most probably takes place in a part of them (see Shenar (2024) and references therein).</text>
<text><location><page_8><loc_12><loc_10><loc_89><loc_18></location>Improving the statistics of binarity among WR stars is a crucial task. Among the first researchers who started to work on it was Virpi S. Niemelä, who determined the stellar masses in binary systems and discovered and analyzed the spectroscopic orbits of many binary systems with WR components (see for ex-</text>
<text><location><page_9><loc_12><loc_85><loc_89><loc_90></location>mple her works Niemela (1995, 2001)). Nowadays this work is being continued, both through spectroscopy (Chené et al., 2022) and using high-resolution imaging (Deshmukh et al., 2024).</text>
<section_header_level_1><location><page_9><loc_36><loc_79><loc_65><loc_80></location>3. CURRENT PROBLEMS</section_header_level_1>
<section_header_level_1><location><page_9><loc_34><loc_75><loc_67><loc_77></location>3.1. Search for new WR stars</section_header_level_1>
<text><location><page_9><loc_12><loc_44><loc_89><loc_74></location>In our Galaxy there are 679 WR stars currently discovered 2) (Rosslowe & Crowther, 2015), while the theory based on current Milky Way star formation rate and duration of WR phase predicts the numbers of WR stars ∼ 1200 (Rosslowe & Crowther, 2015). It means that significant part of WR stars are still not discovered. WR stars in the Galaxy align with spiral arms and the regions of star formation. Thus, the discovery of new WR stars through optical observations has been significantly limited due to the presence of dust extinction. Although the probability of finding a WR star during optical spectroscopic surveys remains (Maíz Apellániz et al., 2016; Zhang et al., 2020), the majority of discoveries of new WR stars now happens through the observations in IR range. For example, Mauerhan et al. (2011) identified 60 Galactic WR stars: candidates were selected using the photometry from Spitzer and Two Micron All Sky Survey (2MASS) surveys and confirmed by near-IR spectroscopy. Shara et al. (2012) expanded the list by adding 71 more stars, also preliminary selected from J and K band 2MASS photometry.</text>
<text><location><page_9><loc_12><loc_24><loc_89><loc_43></location>Moreover, the search for new WR stars is complicated by the effect of spectral mimicry - low mass [WR] objects masquerading as classical WR stars. Thanks to the results of Gaia mission (Gaia Collaboration et al., 2016) we now have reliable estimations of distances, and it helps to quickly understand the nature of individual objects and separate low-mass evolved stars from massive stars, and, moreover, to find the objects of unusual origin among low mass stars. For example, based on Gaia distance estimation Gvaramadze et al. (2019) demonstrated that IRAS00500+6713 is a product of merging of two white dwarfs, while the star has WO-type spectrum. Counterexample is PMR 5 star classified as [WN6] by Morgan et al. (2003) that was recently reclassified as a classical WN.</text>
<text><location><page_9><loc_12><loc_14><loc_89><loc_23></location>Systematic search for WR stars in nearby galaxies is ongoing since 1970-s. In 1972 first catalog of WR candidate stars located in M33 galaxy was published (Wray & Corso, 1972). Twenty-five objects were selected based on photometry in three narrow-band filters (one filter is centered on He II 4686 - the strongest emission line in WN-type WRs, one is centered on C III -IV ≈ 4650 - the strongest</text>
<figure>
<location><page_10><loc_12><loc_64><loc_90><loc_90></location>
<caption>Fig. 2: HR diagram and evolutionary tracks for the low metallicity massive stars ( Z = 0 . 002 ) from Georgy et al. (2013) (left panel) and Grasha et al. (2021) (right panel) evolutionary models. Geneva tracks do not produce massive WR stars at such low metallicities, while Stromlo tracks show rather different evolutionary path and produce WR stars with M ∗ > 70M ⊙ .</caption>
</figure>
<text><location><page_10><loc_12><loc_36><loc_89><loc_49></location>line in WC-type WRs, and one is centered on continuum). This technique of selection of WR candidates for following spectroscopy has proven itself successfully and is still widely used (Massey & Conti, 1983; Massey & Johnson, 1998; Neugent & Massey, 2011; Neugent et al., 2012). As of today, 211 WR are known in M33 galaxy (Massey et al., 2016) and 173 are in M31 (Neugent & Massey, 2023). Search for new WR and their following investigations are important for understanding the impact of metallicity on mass-loss rate.</text>
<section_header_level_1><location><page_10><loc_22><loc_31><loc_79><loc_33></location>3.2. Wolf-Rayet stars in low metallicity environment</section_header_level_1>
<text><location><page_10><loc_12><loc_16><loc_89><loc_30></location>As the opacity of stellar wind depends on the metallicity, mass-loss rates of massive stars decrease with Z . It means that in low metallicity environment single massive stars are unable to lose enough material through the stellar wind to become WR star. Thus, the mass loss on pre-WR stages is of fundamental importance for the final fate of massive stars, and may influence e.g. the formation of long-duration Gamma Ray Bursts (GRBs) or the yields from early stellar generations.</text>
<text><location><page_10><loc_12><loc_10><loc_89><loc_16></location>Metallicity impacts overall evolutionary behaviour of massive stars in a complicated way. Georgy et al. (2013) calculated evolutionary tracks for stars in a wide range of initial masses at low metallicity. According to their calculations,</text>
<text><location><page_11><loc_12><loc_71><loc_89><loc_90></location>metal-poor stars become colder after the end of hydrogen burning in the core and move along the track to the right side of the diagram. Unlike solar metallicity stars, metal-poor ones no longer return to the left side of the diagram (see Figure 2), even when the effects of stellar rotation are taken into account in the models. On the other hand, evolutionary tracks behave differently if the models implement Galactic Concordance abundances instead of solar, scaled-solar, or alpha-element enhanced abundances (Grasha et al., 2021) - the stars with the initial mass of M ∗ > 70M ⊙ with rotation V/V crit = 0 . 2 after the main sequence move to the right and then return to the left to the WR stars region, in contrast to predictions of Georgy et al. (2013).</text>
<text><location><page_11><loc_12><loc_50><loc_89><loc_70></location>Yarovova et al. (2023) used evolutionary tracks from Georgy et al. (2013) and Grasha et al. (2021) for the study of emission-line stellar-like object in the nearby low-metallicity ( Z ∼ 0 . 1 Z ⊙ ) dwarf galaxy NGC 4068. They found the best agreement between the modelled and observed spectra for the model assuming ionization by low-metallicity WR star of M ∗ ≈ 80M ⊙ mass, ionizing the nebula through the strong wind and enriching the interstellar medium with nitrogen. Thus, parameters of the object favor Grasha et al. (2021) predictions. Therefore, understanding of mass loss rate requires complex approach that includes both studies of massive stars in different metallicites, and studies of present-day chemical abundances.</text>
<section_header_level_1><location><page_11><loc_27><loc_44><loc_74><loc_45></location>3.3. Wolf-Rayet stars and red supergiants</section_header_level_1>
<text><location><page_11><loc_12><loc_10><loc_89><loc_42></location>Isolated massive stars with initial masses in 8 M ⊙ ≲ M ∗ ≲ 20 M ⊙ interval undergo the transition to Red Supergiants (RSG) after hydrogen in the core is exhausted. For them RSG is the final stage before Type-II supernova (SNII) explosion. More massive stars ( 40 M ⊙ ≲ M ∗ < 90 M ⊙ ) evolve through LBV phase before reaching WR phase, and finally explode as SN Type Ib/c. The evolution of stars with 25 M ⊙ ≲ M ∗ ≲ 35 M ⊙ initial masses is more complicated. Observed population of supernovae Type II progentiors in the local Universe shows that they are produced by the stars with initial masses M ∗ ≲ 18M ⊙ (Smartt, 2015), and it suggests that more massive stars should either directly collapse to black holes, or evolve leftwards in HR diagram, thus becoming Wolf-Rayet stars. It is supported by the luminosity distributions of cool supergiants in Large and Small Magellanic Clouds (Davies et al., 2018) that overlaps with those of apparently-single WR stars in both galaxies, thus suggesting a changing evolutionary sequence of massive stars with increasing initial mass. At the same time, between the RSG and WR regions in HR diagram lies the region of low luminosity LBVs and yellow hypergiants (YHGs) (Figure 3). Therefore, the evolution of 25 M ⊙ ≲ M ∗ ≲ 35 M ⊙</text>
<figure>
<location><page_12><loc_16><loc_56><loc_85><loc_90></location>
<caption>Fig. 3: Hertzsprung-Russell (HR) diagram (luminosities versus effective temperatures). Color horizontal lines show evolutionary tracks, with solid parts showing the phase when hydrogen burns in the stellar core. The tracks are taken from Ekström et al. (2012) (Ekström et al., 2012). ZAMS is zero age Main Sequence.</caption>
</figure>
<text><location><page_12><loc_12><loc_42><loc_42><loc_43></location>initial mass may look like that:</text>
<formula><location><page_12><loc_17><loc_38><loc_84><loc_40></location>O → RSG → Y HG (? ) → low LBV → WR ( for M ∗ ≈ 25 -35 M ⊙ )</formula>
<text><location><page_12><loc_12><loc_32><loc_89><loc_36></location>It means that the star with initial mass about 30 M ⊙ should pass through all four stages - do such objects really exist?</text>
<text><location><page_12><loc_12><loc_10><loc_89><loc_32></location>Groh et al. (2013) demonstrated that the stars with 20 -25 M ⊙ initial masses are unstable on low luminosity LBV stage, and explode as core-collapse supernovae (SN Type IIL/b). Prior to the explosion they show the spectrum similar to S Doradus during its minimum of brightness (WN11h spectral type). Maryeva et al. (2020) have recently found a Galactic object, Wray 15-906, with characteristics typical of a low luminosity LBV and the spectrum looking like WN11h. It was identified via the detection of its infrared circumstellar shell (of ≈ 2 pc in diameter) with the Wide-field Infrared Survey Explorer (WISE) and the Herschel Space Observatory. Maryeva et al. (2020) found that Wray 15-906 is a relatively low-luminosity, log( L ∗ / L ⊙ ) ≈ 5 . 4 , star with a temperature of 25 ± 2 kK, with position in HR diagram corresponding to post-red supergiant with initial mass</text>
<text><location><page_13><loc_12><loc_83><loc_89><loc_90></location>of ≈ 25M ⊙ . Its spectrum and properties are consistent with theoretical predictions of Groh et al. (2013). Thus, the monitoring of this object, and the search for similar ones, may improve our understanding of the link between WRs and RSGs.</text>
<section_header_level_1><location><page_13><loc_41><loc_77><loc_60><loc_79></location>4. CONCLUSION</section_header_level_1>
<text><location><page_13><loc_12><loc_58><loc_89><loc_75></location>Wolf-Rayet stars were discovered more than 150 years ago, and since then they were targets of numerous studies that combined astrophysical insights with advancements in atomic physics, development of numerical methods, and instrumental progress. Nowadays we understand quite well what Wolf-Rayet stars are, and they are no longer a hot topic in astrophysics. However they still are keystones for many directions of research, from theory of stellar atmospheres up to stellar populations in other galaxies and gravitational waves. Through the existence of the Wolf-Rayet phenomenon WR stars are linked, besides massive stars, with low mass evolved stars.</text>
<text><location><page_13><loc_12><loc_38><loc_89><loc_57></location>We expect new progress in understanding the Wolf-Rayet stars and massloss of massive stars in general from the upcoming advances in the instrumentation. For example, SITELLE imaging Fourier transform spectrometer at CanadaFrance-Hawaii telescope (Drissen et al., 2019) may provide spatially resolved line ratios and kinematics of WR nebulae thus hinting us on ionizing flux from the stars, and their mass loss history. James Webb Space Telescope will allow studying dust shells around WR stars (Lau et al., 2022), and to assess their role in enrichment of interstellar medium with organic compounds and carbonaceous dust. Spectropolarimetric surveys of WR and monitoring of variability of linear polarization are also still topical.</text>
<text><location><page_13><loc_12><loc_24><loc_89><loc_35></location>Acknowledgements I would like to thank the organizers of the conference for their kind invitation to present this overview. This research received funding from the European Union's Framework Programme for Research and Innovation Horizon 2020 (2014-2020) under the Marie Skłodowska-Curie Grant Agreement No. 823734 (POEMS project). The Astronomical Institute in Ondřejov is supported by the project RVO:67985815.</text>
<section_header_level_1><location><page_13><loc_43><loc_19><loc_58><loc_20></location>REFERENCES</section_header_level_1>
<text><location><page_13><loc_12><loc_14><loc_89><loc_17></location>Ambartsumian V. A., 1933, Izvestiya Glavnoj Astronomicheskoj Observatorii v Pulkove, 13</text>
<text><location><page_13><loc_12><loc_10><loc_82><loc_12></location>Atanackovic-Vukmanovic O., 2004, Serbian Astronomical Journal, 169, 1</text>
<text><location><page_14><loc_12><loc_10><loc_89><loc_90></location>Auer L. H., Mihalas D., 1969, ApJ, 158, 641 Beals C. S., 1929, MNRAS, 90, 202 Beals C. S., 1933, The Observatory, 56, 196 Bestenlehner J. M., et al., 2014, A&A, 570, A38 Bisnovatyi-Kogan G. S., Nadyozhin D. K., 1972, Ap&SS, 15, 353 Bohr N., 1913, Nature, 92, 231 Bowen I. S., 1928, ApJ, 67, 1 Bowen I. S., 1934, PASP, 46, 146 Cannon C. J., 1973a, J. Quant. Spec. Radiat. Transf., 13, 627 Cannon C. J., 1973b, ApJ, 185, 621 Chandrasekhar S., 1934, MNRAS, 94, 522 Chené A.-N., Mahy L., Gosset E., St-Louis N., Dsilva K., Manick R., 2022, MNRAS, 516, 1022 Conti P. S., 1975, Memoires of the Societe Royale des Sciences de Liege, 9, 193 Conti P. S., 1984, in Maeder A., Renzini A., eds, IAU Symposium Vol. 105, Observational Tests of the Stellar Evolution Theory. p. 233 Conti P. S., 2015, in Hamann W.-R., Sander A., Todt H., eds, Wolf-Rayet Stars. pp 347-350 Conti P. S., Massey P., 1989, ApJ, 337, 251 Crowther P. A., 2007, ARA&A, 45, 177 Crowther P. A., Hillier D. J., Smith L. J., 1995a, A&A, 293, 172 Crowther P. A., Hillier D. J., Smith L. J., 1995b, A&A, 293, 403 Crowther P. A., Smith L. J., Hillier D. J., Schmutz W., 1995c, A&A, 293, 427 Crowther P. A., Smith L. J., Hillier D. J., 1995d, A&A, 302, 457 Crowther P. A., Smith L. J., Willis A. J., 1995e, A&A, 304, 269 Crowther P. A., De Marco O., Barlow M. J., 1998, MNRAS, 296, 367</text>
<table>
<location><page_15><loc_12><loc_10><loc_90><loc_90></location>
</table>
<text><location><page_16><loc_12><loc_89><loc_61><loc_90></location>Hamann W. R., Gräfener G., 2003, A&A, 410, 993</text>
<text><location><page_16><loc_12><loc_14><loc_89><loc_87></location>Hamann W. R., Gräfener G., Liermann A., 2006, A&A, 457, 1015 Hillier D. J., 1990, A&A, 231, 116 Hillier D. J., Miller D. L., 1998, ApJ, 496, 407 Huggins W., Huggins M., 1890, Proceedings of the Royal Society of London Series I, 49, 33 Humphreys R. M., Davidson K., 1994, PASP, 106, 1025 Humphreys R. M., Weis K., Davidson K., Gordon M. S., 2016, ApJ, 825, 64 Kosirev N. A., 1934, MNRAS, 94, 430 Laplace E., Justham S., Renzo M., Götberg Y., Farmer R., Vartanyan D., de Mink S. E., 2021, A&A, 656, A58 Lau R. M., et al., 2022, Nature Astronomy, 6, 1308 Liermann A., Hamann W. R., Oskinova L. M., Todt H., Butler K., 2010, A&A, 524, A82 Maíz Apellániz J., et al., 2016, ApJS, 224, 4 Marcolino W. L. F., Hillier D. J., de Araujo F. X., Pereira C. B., 2007, ApJ, 654, 1068 Martins F., Genzel R., Hillier D. J., Eisenhauer F., Paumard T., Gillessen S., Ott T., Trippe S., 2007, A&A, 468, 233 Martins F., Schaerer D., Marques-Chaves R., Upadhyaya A., 2023, A&A, 678, A159 Maryeva O., Viotti R. F., Koenigsberger G., Calabresi M., Rossi C., Gualandi R., 2019, Galaxies, 7, 79 Maryeva O. V., Gvaramadze V. V., Kniazev A. Y., Berdnikov L. N., 2020, MNRAS, 498, 5093 Maryeva O., Abdulkarimova A., Karpov S., Moiseev A., Oparin D., 2024, MNRAS, 527, 11925</text>
<text><location><page_16><loc_12><loc_10><loc_54><loc_12></location>Massey P., Conti P. S., 1983, ApJ, 273, 576</text>
<text><location><page_17><loc_12><loc_17><loc_89><loc_90></location>Massey P., Johnson O., 1998, ApJ, 505, 793 Massey P., Neugent K. F., Smart B. M., 2016, AJ, 152, 62 Mauerhan J. C., Van Dyk S. D., Morris P. W., 2011, AJ, 142, 40 Morgan D. H., Parker Q. A., Cohen M., 2003, MNRAS, 346, 719 Neugent K. F., Massey P., 2011, ApJ, 733, 123 Neugent K. F., Massey P., 2023, AJ, 166, 68 Neugent K. F., Massey P., Georgy C., 2012, ApJ, 759, 11 Niemela V. S., 1995, in van der Hucht K. A., Williams P. M., eds, IAU Symposium Vol. 163, Wolf-Rayet Stars: Binaries; Colliding Winds; Evolution. p. 223 Niemela V., 2001, in Revista Mexicana de Astronomia y Astrofisica Conference Series. pp 23-26 Paczyński B., 1967, Acta Astron., 17, 355 Paczyński B., 1973, in Bappu M. K. V., Sahade J., eds, IAU Symposium Vol. 49, Wolf-Rayet and High-Temperature Stars. p. 143 Pickering E. C., 1891, Astronomische Nachrichten, 127, 1 Pickering E. C., 1897, ApJ, 5, 92 Pickering E. C., 1901, Harvard College Observatory Circular, 55, 1 Polcaro V. F., et al., 2016, AJ, 151, 149 Pritzkuleit M., 2020, Master's thesis, University of Potsdam, Germany Roberts M. S., 1962, AJ, 67, 79 Robotti N., 1983, Historical Studies in the Physical Sciences, 14, 123 Rosslowe C. K., Crowther P. A., 2015, MNRAS, 447, 2322 Rublev S. V., 1965, Soviet Ast., 9, 274 Rublev S. V., 1970, Astrofizicheskie Issledovaniia Izvestiya Spetsial'noj As-</text>
<text><location><page_17><loc_14><loc_15><loc_45><loc_17></location>trofizicheskoj Observatorii, 1, 25</text>
<text><location><page_17><loc_12><loc_10><loc_89><loc_14></location>Rublev S. V., 1972a, Astrofizicheskie Issledovaniia Izvestiya Spetsial'noj Astrofizicheskoj Observatorii, 4, 3</text>
<table>
<location><page_18><loc_12><loc_10><loc_90><loc_90></location>
</table>
<text><location><page_19><loc_12><loc_89><loc_48><loc_90></location>Underhill A. B., 1968, ARA&A, 6, 39</text>
<text><location><page_19><loc_12><loc_50><loc_89><loc_87></location>Vink J. S., 2015, in Hamann W.-R., Sander A., Todt H., eds, Wolf-Rayet Stars. pp 133-138 ( arXiv:1510.00227 ), doi:10.48550/arXiv.1510.00227 Westerlund B. E., Rodgers A. W., 1959, The Observatory, 79, 132 Westerlund B. E., Smith L. F., 1964, MNRAS, 128, 311 Wolf C. J. E., Rayet G., 1867, Academie des Sciences Paris Comptes Rendus, 65, 292 Wray J. D., Corso G. J., 1972, ApJ, 172, 577 Wright W. H., 1918, Publications of Lick Observatory, 13, 191 Yarovova A. D., Egorov O. V., Moiseev A. V., Maryeva O. V., 2023, MNRAS, 518, 2256 Zhang W., et al., 2020, ApJ, 902, 62 Škoda P., Podsztavek O., Tvrdík P., 2020, A&A, 643, A122 1981,</text>
<text><location><page_19><loc_12><loc_48><loc_82><loc_51></location>van der Hucht K. A., Conti P. S., Lundstrom I., Stenholm B., Space Sci. Rev., 28, 227</text>
</document>
[
{
"title": "WOLF-RAYET STARS - WHAT WE KNOW AND WHAT WE DON'T",
"content": "O. V. Maryeva a * a Astronomical Institute of the Czech Academy of Sciences, Fričova 298, 25165 Ondřejov, Czech Republic Today, we have a sufficiently complete picture of what the Wolf-Rayet (WR) stars are. Predictions of stellar evolution theory are in a good agreement with their parameters, estimated from observational data using stellar atmospheres codes; predictions of population synthesis also agree well with number of known WR stars. This article provides an overview of the main historical milestones in the studies of WR stars, showing how we came to this understanding, and what questions are still unanswered. Keywords: Stars: Wolf-Rayet - Stars: evolution - Stars: atmospheres - General: history and philosophy of astronomy",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "Wolf-Rayet (WR) stars are a class of objects identified on the basis of their spectral features. WR spectra show strong emission lines of helium, nitrogen, carbon and oxygen in different stages of ionization. The width of these lines reaches tens of angstroms, the central intensities are sometimes 10-20 times the intensity of the continuum spectrum. A prominent feature of WRs are two bumps at 4650 1) and 5808 Å, which are groups of closely located lines of N II , N III , C III and C IV , which are visible even in low-resolution spectra. The lines are formed in an extended atmosphere - in the stellar wind, expelling at velocities of 10 2 -10 3 kms -1 . WR stars are characterized by a high mass-loss rate of several 10 -5 M ⊙ yr -1 and by high temperatures ( T eff ≳ 30000 K). WR are the final stage in the evolution of massive stars (stars with an initial mass ≳ 25 M ⊙ ) before the core-collapse supernova explosion. With accumulation of observational data it became clear that emission spectra and WR bumps are intrinsic to objects of different natures. Therefore nowadays there is a clear distinction between WR stars and WR phenomenon , which occurs when fast-moving, hot plasma is expanding around a hot star (Gräfener et al., 2011; Vink, 2015). WR phenomenon may occur in the low mass stars after ejecting their outer layers in the planetary nebula phase and exposing their hot cores prior to the white-dwarf phase (Marcolino et al., 2007; Todt, 2009; van der Hucht et al., 1981). Central stars of planetary nebulae showing WR phenomenon in their spectra are denoted as [WR] (van der Hucht et al., 1981). Rare and still unique case of low mass object with WR phenomenon and the absence of a clearly detected circumstellar nebula is LAMOST J040901.83+323955.6 (Maryeva et al., 2024). It was selected as a WR star by Škoda et al. (2020), but Maryeva et al. (2024) showed that this object is 0 . 9 M ⊙ star caught in a rare transitional phase from post-AGB to CSPN. Another unique object with WR phenomenon is IRAS 00500+6713, formed as a result of a merger of two white dwarfs (Gvaramadze et al., 2019). The WR phenomenon is also observed in young supernovae (GalYam et al., 2014). As we know now, massive stars showing WR spectra are also not a homogeneous class. They are split into two groups: very massive stars and classical WR stars . Very massive stars ( M ini ≥ 100M ⊙ ) experience strong stellar winds and show WR spectra already during core H-burning phase, when they are located on main sequence. All of them show signs of hydrogen on their surface and are classified as WNh objects (Crowther et al., 2010; Martins et al., 2023; Smith et al., 1996). Classical WR stars (usually just named WR stars ) stars are hydrogen-depleted objects that have evolved off the main sequence and suffered intense mass loss (Crowther, 2007; Shenar, 2024). Classical WR stars will be the subject of this review article. The field of contemporary studies of WR stars is enormous and cannot be fully covered in a single small review like this one, so I'd like to recommend two other detailed reviews - Crowther (2007) and Shenar (2024) - for better understanding what WR stars are. On the other hand, here I will concentrate more on the historical aspects of WR studies, on how we have arrived to our present day understanding of them. Therefore Section 2 will cover the history of studies of WR stars, while Section 3 will highlight several topical problems and open questions about them.",
"pages": [
1,
2
]
},
{
"title": "2. FROM DISCOVERY TO UNDERSTANDING",
"content": "In 1867 year astronomers from Paris observatory Charles Wolf and Georges Rayet discovered three stars in Cygnus constellation those spectra significantly differ from other stars. In contrast with other stars, where usually absorption lines are dominant in spectra, discovered objects showed strong emission lines (Wolf & Rayet, 1867). The class of objects received its name in honor of the discoverers and gradually began to be replenished with new members. One hundred years after the discovery of WRs, by the end of 1960-s the number of known WRs in the Galaxy had reached over a hundred, totalling 127 (Roberts, 1962; Smith, 1968), while now is ∼ 700 (Rosslowe & Crowther, 2015). At the same time, in the late 1950-s middle 1960-s, work began to search for extragalactic WRs - WRs in the Large Magellanic Cloud (Westerlund & Rodgers, 1959; Westerlund & Smith, 1964). Significant contribution to the understanding of the physics of WR stars was given by Carlyle S. Beals (1929). Beals found that some lines in the spectra of WR stars have a P Cygni profile and based on comparison with P Cygni profile lines in spectra of novae he suggested that WRs are surrounded by expanding envelopes (Beals, 1929). Beals also found a difference between WR stars and novae: the P Cygni profiles in the WR stars do not change over time. This allowed to propose that an outflow from WR stars occurs continuously (Beals, 1929). This was confirmed by Chandrasekhar (1934), who developed a solid footing for interpreting P Cygni profiles as arising in expanding atmospheres. Kosirev (1934) used the diagnostics developed by Chandrasekhar to estimate the mass loss and maximum outflow velocity of a WR star and found, respectively, ∼ 10 -5 M ⊙ yr -1 and ∼ 1000 kms -1 . At the beginning of the 20th century, stellar spectroscopy developed in close cooperation with atomic physics. Investigating spectra of the star ζ Puppis Edward Pickering paid his attention to previously unknown lines 5411, 4541, 4200, 4100, 4026, 3924, 3858, 3813, 3782 Å (Pickering, 1897). He interpreted them as another series of hydrogen (besides Balmer) (Pickering, 1901). Line 4686 Å, first discovered during a solar eclipse and often found in WR spectra, was also considered to be a hydrogen line (Fowler, 1912). Although Fowler (1912) was able to obtain these spectral lines in the laboratory in 1912, during an experiment with a helium-filled tube, he considered their appearance to be a contribution from hydrogen impurity. It was not until 1913 that Niels Bohr's work (Bohr, 1913) explained the nature of these lines as ionized helium He II , confirming the atomic model (see Robotti (1983) for historical review). In 1890 Pickering drew attention to the similarity of the spectra of WR stars and planetary nebula (Pickering, 1891). Due to this, in papers devoted to emission spectra of nebulae it is possible to find descriptions of spectra of WR stars. The lines at 4647, 4650, 5696, 5801 and 5812 Å were identified with the carbon by Wright (1918). Ira S. Bowen, who found an interpretation of the 'nebulium' lines as [O III ], gave a detailed list of emission lines observed in nebulae, which includes an identification of almost all lines visible in the WRs (Bowen, 1928). In a 1934 paper Bowen (1934) suggests a significant role of the fluorescence for formation of the N III 4634, 46340 lines belonging to the WR bump. Already the spectra of the first discovered WR stars showed a difference: the broad emission bands are located in different places (Huggins & Huggins, 1890; Wolf & Rayet, 1867). After the identification of all the main spectral lines, Beals (1933) proposed the splitting of WR into two groups. This classification was approved in 1938, the International Astronomical Union (IAU) divided the spectra of WR stars into types WN and WC, depending on whether the spectrum was dominated by lines of nitrogen or carbon-oxygen respectively (Beals, 1933; Swings, 1942). WN stars are believed to show the hydrogen burning products via the CNO cycle, while WC stars reveal the helium burning products via the tripleα cycle. Based on the strength of the emission lines and line ratios, WN stars can be further classified into the spectral subtypes WN2 to WN11, and WC stars into the spectral subtypes WC4 to WC9 (Crowther et al., 1998; Smith, 1968; Smith et al., 1990, 1994, 1996). Also, there are transition types from WN to WC, which are called WN/C, whose spectra show strong emission lines of carbon and nitrogen simultaneously (Conti & Massey, 1989). In 1933 Victor A. Ambartsumian (1933) estimated He/H ratio for WR stars using intensities of H β and He II 4686 lines and found He/H ≳ 1 . 8 . Subsequent studies (Rublev (1972a) and references therein) confirmed excess of helium. Gamow (1943) suggested that the anomalous composition of WR stars was the result of nuclear processed material being visible on their surfaces. Despite these arguments in favor of far evolved status of WR stars, debates about it continued until the mid-1970s. There was alternative hypothesis, that WR are young and more massive than T Tau objects, before main sequence (Sahade, 1958; Underhill, 1968). It is important to mention the studies of Sergej V. Rublev, in which he developed methods for determining the temperatures and luminosities of WR stars and estimated the hydrogen abundances (Rublev, 1965, 1970, 1972b, 1975). These works played a significant role in the formation of the modern understanding of WR stars as far evolved massive objects.",
"pages": [
2,
3,
4
]
},
{
"title": "2.1. Development of numerical methods",
"content": "Contemporary progress in our understanding of hot stars and especially WR stars has been significantly defined by the progress of numerical modelling of stellar atmospheres, in turn enabled by the development of both physical approximations and numerical methods. Victor V. Sobolev developed the theory of radiative transfer in moving media and applied it to the determination of physical conditions in the envelopes of WR, Be, novae, etc. type stars (Sobolev, 1947, 1960). He showed that due to the presence of a velocity gradient, photons of spectral lines are able to leave the deep layers of the atmosphere directly, avoiding the usual diffusion process; this significantly simplifies the study of the radiative equilibrium of moving atmospheres. This theory became the basis for the transition to modern quantitative calculations of the stellar wind. WRstars are essentially hot stellar cores surrounded by extended, low-density atmospheres. Due to their high luminosity the radiation in the stellar wind dominates over collisional processes, and the models of WR's atmospheres should be calculated taking into account the deviations from local thermodynamic equilibrium (non-LTE). Straightforward iterative procedure for the solution of radiative transfer equations Λ iteration - fails to converge in typical non-LTE conditions, as in scattering dominated media of large optical thickness, its convergence is extremely slow (see overview by Atanackovic-Vukmanovic (2004) and references therein). Several approaches have been proposed in early 1970-s as a way to numerically solve radiative transfer (RT) problems in non-LTE case. The first fully self-consistent solution of RT problems for non-LTE case was achieved by Auer & Mihalas (1969) who developed the complete linearization method based on the Newton-Raphson scheme to solve the set of discretized structural equations in a robust manner that opened a way for significant progress in the modelling of stellar atmospheres. A different approach is the 'core saturation method', introduced by Rybicki (1972) who implemented a modification of the Λ iteration method that made it practically usable for solving the RT and statistical equilibrium equations. In order to eliminate the cause of the slow convergence of simple Λ iteration (poor conditioning of the equations arising from a large number of scatterings), Rybicki proposed the elimination of scattering events in the line core as they do not contribute much to the transfer process. Another approach is 'perturbation technique' proposed by Cannon (1973a,b) who was the first to use the idea of 'operator splitting' in radiative transfer computations. He replaced the full (exact) Λ operator by a simplified one (so called approximate Lambda operator, or ALO) and computed a small error term made by this approximation by using a perturbation technique. Cannon's operator perturbation technique was an efficient way to combine computational advantages of approximate solutions with the accuracy of exact methods. In 1980-s computational approximations to accelerate the Λ iteration started to be used together with the operator perturbation technique introduced by Cannon (1973a,b) to simplify the direct solution, making the basis for the class of methods known as Accelerated Lambda Iteration. Wolf-Rainer Hamann (1985, 1986) adopted perturbation technique to approximate lambda operator, based on idea of 'core saturation' and presented test calculations for a typical WR star atmosphere. This effort became the foundation for the PoWR code (Hamann & Gräfener, 2003) that is still being actively developed and successfully applied for numerical modelling of various types of astrophysical objects. Independently, D. John Hillier (1990) presented a method based on tridiagonal Newton-Raphson operator, complete linearization scheme and perturbation technique for calculations of WN and WC atmosphere models. Over time, this method has grown into CMFGEN code (Hillier & Miller, 1998). Since the mid-1990s, CMFGEN and PoWR have become a reliable instruments for studying hot stars with high mass-loss rates. Systematic study of Galactic WR stars started with a series of works by Paul Crowther titled 'Fundamental parameters of WR stars' and devoted to the analysis of properties of nitrogen-rich WR stars (WN) based on optical and ultraviolet (UV) data and numerical modelling using CMFGEN code (Crowther et al., 1995a,b,c,d,e). Advances in infrared (IR) spectroscopy also enabled studies of WR stars near the Galactic center (Liermann et al., 2010; Martins et al., 2007). Modeling of large sample of Galctic WN (Hamann et al., 2006) and WC (Sander et al., 2012) stars ( ≈ 126 objects in total) allowed to obtain a homogeneous set of stellar and atmospheric parameters for them, to determine the correct location of WR stars in the Hertzsprung-Russell (HR) diagram. PoWR code was widely used for studying extragalactic objects. Among WR stars in nearby galaxies, the ones in Magellanic Clouds are most studied (Bestenlehner et al., 2014; Hainich et al., 2014, 2015). Sample of 74 WR stars belonged to M33 galaxy was analysed by Pritzkuleit (2020) (Pritzkuleit, 2020). Sander et al. (2014) investigated late-type WN stars in M31 galaxy (Sander et al., 2014). These studies extended our understanding of properties of WR stars in different galactic environments and metallicity-dependence of massive star winds.",
"pages": [
4,
5,
6
]
},
{
"title": "2.2. Evolution",
"content": "Modern view on WRs as products of evolution of massive stars was finally formed only in 1980s. Although by the 1980s WR stars were already recognised as descendants of massive OB stars, their exact evolutionary status with respect to the main sequence and other evolved massive stars was still an open question. WR stars were being proposed as possible progenitors of supernovae, and particularly the newly-discovered type Ib supernovae that lack hydrogen but are apparently associated with young massive stars. The start of systematic X-ray observations with space-borne X-ray telescopes in 1960s significantly advanced the theory of close binary stars and their interac- tions, and it led Bohdan Paczyński (1967) to the suggestion that WR stars are the products of binary evolution - namely, mass exchange in a binary system. In his scenario, a more massive star in binary system evolves faster and, upon exhaustion of hydrogen in the core, fills the Roche lobe and a rapidly loses its hydrogen envelope that outflows towards the companion. As a result, when the envelope is fully transferred, its hot helium core remains as a WR star with little hydrogen in the outermost layers (Paczyński, 1973). The hypothesis of Paczyński was well accepted and it naturally explained the anomalous chemical composition of WR stars as naked stellar cores enriched with products of nuclear reactions. On the other hand, based on the similarity of Of and WN stars (Figure 1) and on the first estimates of the mass loss rate for hot massive stars based on UV observations Peter Conti (1975) suggested that the stellar wind of O stars is strong enough to remove the outer layers and uncover the stellar core, thus proposing a single star mass loss channel to form WR stars. In addition, brightest WR star, γ Vel, was inconsistent with Paczyński's hypothesis as, while being a binary, its orbit was clearly elliptical and its second component lacks any signs of previous mass exchange, both suggesting that there was no interaction between the components (Conti, 2015). In 1984 Peter Conti (1984) introduced a new class of objects. In his talk at the IAU symposium, Peter Conti grouped S Doradus variables, Hubble-Sandage variables, η Carinae, P Cygni, and other similar stars together under the term 'luminous blue variables' (LBV). LBV are characterized by strong mass loss (around 10 -5 M ⊙ yr -1 ) and occasional giant eruptive events (Humphreys & Davidson, 1994). Conti suggested that massive O stars transit to WR stars through this short-duration LBV phase. This idea is consolidated as the 'Conti scenario' and becomes the main channel for WR star formation. Modern stellar evolution calculations are consistent with the 'Conti scenario' (Groh et al., 2014; Sander et al., 2019). According to numerical calculations the stars with initial mass of 60 M ⊙ spend 2 × 10 5 yr in the LBV phase - only 5% of their lifetime (Groh et al., 2014). Recently observed transition from LBV to WN8 spectral type in Romano' star is also a direct confirmation of 'Conti scenario' (Maryeva et al., 2019; Polcaro et al., 2016). Similar scenario for WR formation was also proposed by Bisnovatyi-Kogan & Nadyozhin (1972), with significant mass loss on the red supergiant phase. They show that large inverse density gradient appears in the outer layers of a 30 M ⊙ star in this phase due to supercritical luminosity, resulting in powerful and rapid mass loss and formation of a stripped helium core. In the last decade, Paczyński's hypothesis has again been discussed as a mechanism for the formation of WR stars. Analysing the spatial positions of LBV and WR stars, Nathan Smith (Smith, 2016, 2019; Smith & Tombleson, 2015) con- luded that both LBVs and WRs could be the products of binary evolution, LBVs being mass-gainers, and WRs - mass-donors, or stripped components. This idea was critically discussed by Humphreys et al. (2016) and Davidson et al. (2016). Now we may confidently conclude only that we do not have enough data for statistical analysis. Modern evolutionary codes allow detailed calculations of binary system evolution (see for example Götberg et al. (2018); Laplace et al. (2021)), thus making it possible to compare theoretical predictions for stripped stars with properties of real WR stars (Götberg et al., 2018). Our current understanding is that binary interaction scenario cannot properly describe all observed WR stars, but most probably takes place in a part of them (see Shenar (2024) and references therein). Improving the statistics of binarity among WR stars is a crucial task. Among the first researchers who started to work on it was Virpi S. Niemelä, who determined the stellar masses in binary systems and discovered and analyzed the spectroscopic orbits of many binary systems with WR components (see for ex- mple her works Niemela (1995, 2001)). Nowadays this work is being continued, both through spectroscopy (Chené et al., 2022) and using high-resolution imaging (Deshmukh et al., 2024).",
"pages": [
6,
7,
8,
9
]
},
{
"title": "3.1. Search for new WR stars",
"content": "In our Galaxy there are 679 WR stars currently discovered 2) (Rosslowe & Crowther, 2015), while the theory based on current Milky Way star formation rate and duration of WR phase predicts the numbers of WR stars ∼ 1200 (Rosslowe & Crowther, 2015). It means that significant part of WR stars are still not discovered. WR stars in the Galaxy align with spiral arms and the regions of star formation. Thus, the discovery of new WR stars through optical observations has been significantly limited due to the presence of dust extinction. Although the probability of finding a WR star during optical spectroscopic surveys remains (Maíz Apellániz et al., 2016; Zhang et al., 2020), the majority of discoveries of new WR stars now happens through the observations in IR range. For example, Mauerhan et al. (2011) identified 60 Galactic WR stars: candidates were selected using the photometry from Spitzer and Two Micron All Sky Survey (2MASS) surveys and confirmed by near-IR spectroscopy. Shara et al. (2012) expanded the list by adding 71 more stars, also preliminary selected from J and K band 2MASS photometry. Moreover, the search for new WR stars is complicated by the effect of spectral mimicry - low mass [WR] objects masquerading as classical WR stars. Thanks to the results of Gaia mission (Gaia Collaboration et al., 2016) we now have reliable estimations of distances, and it helps to quickly understand the nature of individual objects and separate low-mass evolved stars from massive stars, and, moreover, to find the objects of unusual origin among low mass stars. For example, based on Gaia distance estimation Gvaramadze et al. (2019) demonstrated that IRAS00500+6713 is a product of merging of two white dwarfs, while the star has WO-type spectrum. Counterexample is PMR 5 star classified as [WN6] by Morgan et al. (2003) that was recently reclassified as a classical WN. Systematic search for WR stars in nearby galaxies is ongoing since 1970-s. In 1972 first catalog of WR candidate stars located in M33 galaxy was published (Wray & Corso, 1972). Twenty-five objects were selected based on photometry in three narrow-band filters (one filter is centered on He II 4686 - the strongest emission line in WN-type WRs, one is centered on C III -IV ≈ 4650 - the strongest line in WC-type WRs, and one is centered on continuum). This technique of selection of WR candidates for following spectroscopy has proven itself successfully and is still widely used (Massey & Conti, 1983; Massey & Johnson, 1998; Neugent & Massey, 2011; Neugent et al., 2012). As of today, 211 WR are known in M33 galaxy (Massey et al., 2016) and 173 are in M31 (Neugent & Massey, 2023). Search for new WR and their following investigations are important for understanding the impact of metallicity on mass-loss rate.",
"pages": [
9,
10
]
},
{
"title": "3.2. Wolf-Rayet stars in low metallicity environment",
"content": "As the opacity of stellar wind depends on the metallicity, mass-loss rates of massive stars decrease with Z . It means that in low metallicity environment single massive stars are unable to lose enough material through the stellar wind to become WR star. Thus, the mass loss on pre-WR stages is of fundamental importance for the final fate of massive stars, and may influence e.g. the formation of long-duration Gamma Ray Bursts (GRBs) or the yields from early stellar generations. Metallicity impacts overall evolutionary behaviour of massive stars in a complicated way. Georgy et al. (2013) calculated evolutionary tracks for stars in a wide range of initial masses at low metallicity. According to their calculations, metal-poor stars become colder after the end of hydrogen burning in the core and move along the track to the right side of the diagram. Unlike solar metallicity stars, metal-poor ones no longer return to the left side of the diagram (see Figure 2), even when the effects of stellar rotation are taken into account in the models. On the other hand, evolutionary tracks behave differently if the models implement Galactic Concordance abundances instead of solar, scaled-solar, or alpha-element enhanced abundances (Grasha et al., 2021) - the stars with the initial mass of M ∗ > 70M ⊙ with rotation V/V crit = 0 . 2 after the main sequence move to the right and then return to the left to the WR stars region, in contrast to predictions of Georgy et al. (2013). Yarovova et al. (2023) used evolutionary tracks from Georgy et al. (2013) and Grasha et al. (2021) for the study of emission-line stellar-like object in the nearby low-metallicity ( Z ∼ 0 . 1 Z ⊙ ) dwarf galaxy NGC 4068. They found the best agreement between the modelled and observed spectra for the model assuming ionization by low-metallicity WR star of M ∗ ≈ 80M ⊙ mass, ionizing the nebula through the strong wind and enriching the interstellar medium with nitrogen. Thus, parameters of the object favor Grasha et al. (2021) predictions. Therefore, understanding of mass loss rate requires complex approach that includes both studies of massive stars in different metallicites, and studies of present-day chemical abundances.",
"pages": [
10,
11
]
},
{
"title": "3.3. Wolf-Rayet stars and red supergiants",
"content": "Isolated massive stars with initial masses in 8 M ⊙ ≲ M ∗ ≲ 20 M ⊙ interval undergo the transition to Red Supergiants (RSG) after hydrogen in the core is exhausted. For them RSG is the final stage before Type-II supernova (SNII) explosion. More massive stars ( 40 M ⊙ ≲ M ∗ < 90 M ⊙ ) evolve through LBV phase before reaching WR phase, and finally explode as SN Type Ib/c. The evolution of stars with 25 M ⊙ ≲ M ∗ ≲ 35 M ⊙ initial masses is more complicated. Observed population of supernovae Type II progentiors in the local Universe shows that they are produced by the stars with initial masses M ∗ ≲ 18M ⊙ (Smartt, 2015), and it suggests that more massive stars should either directly collapse to black holes, or evolve leftwards in HR diagram, thus becoming Wolf-Rayet stars. It is supported by the luminosity distributions of cool supergiants in Large and Small Magellanic Clouds (Davies et al., 2018) that overlaps with those of apparently-single WR stars in both galaxies, thus suggesting a changing evolutionary sequence of massive stars with increasing initial mass. At the same time, between the RSG and WR regions in HR diagram lies the region of low luminosity LBVs and yellow hypergiants (YHGs) (Figure 3). Therefore, the evolution of 25 M ⊙ ≲ M ∗ ≲ 35 M ⊙ initial mass may look like that: It means that the star with initial mass about 30 M ⊙ should pass through all four stages - do such objects really exist? Groh et al. (2013) demonstrated that the stars with 20 -25 M ⊙ initial masses are unstable on low luminosity LBV stage, and explode as core-collapse supernovae (SN Type IIL/b). Prior to the explosion they show the spectrum similar to S Doradus during its minimum of brightness (WN11h spectral type). Maryeva et al. (2020) have recently found a Galactic object, Wray 15-906, with characteristics typical of a low luminosity LBV and the spectrum looking like WN11h. It was identified via the detection of its infrared circumstellar shell (of ≈ 2 pc in diameter) with the Wide-field Infrared Survey Explorer (WISE) and the Herschel Space Observatory. Maryeva et al. (2020) found that Wray 15-906 is a relatively low-luminosity, log( L ∗ / L ⊙ ) ≈ 5 . 4 , star with a temperature of 25 ± 2 kK, with position in HR diagram corresponding to post-red supergiant with initial mass of ≈ 25M ⊙ . Its spectrum and properties are consistent with theoretical predictions of Groh et al. (2013). Thus, the monitoring of this object, and the search for similar ones, may improve our understanding of the link between WRs and RSGs.",
"pages": [
11,
12,
13
]
},
{
"title": "4. CONCLUSION",
"content": "Wolf-Rayet stars were discovered more than 150 years ago, and since then they were targets of numerous studies that combined astrophysical insights with advancements in atomic physics, development of numerical methods, and instrumental progress. Nowadays we understand quite well what Wolf-Rayet stars are, and they are no longer a hot topic in astrophysics. However they still are keystones for many directions of research, from theory of stellar atmospheres up to stellar populations in other galaxies and gravitational waves. Through the existence of the Wolf-Rayet phenomenon WR stars are linked, besides massive stars, with low mass evolved stars. We expect new progress in understanding the Wolf-Rayet stars and massloss of massive stars in general from the upcoming advances in the instrumentation. For example, SITELLE imaging Fourier transform spectrometer at CanadaFrance-Hawaii telescope (Drissen et al., 2019) may provide spatially resolved line ratios and kinematics of WR nebulae thus hinting us on ionizing flux from the stars, and their mass loss history. James Webb Space Telescope will allow studying dust shells around WR stars (Lau et al., 2022), and to assess their role in enrichment of interstellar medium with organic compounds and carbonaceous dust. Spectropolarimetric surveys of WR and monitoring of variability of linear polarization are also still topical. Acknowledgements I would like to thank the organizers of the conference for their kind invitation to present this overview. This research received funding from the European Union's Framework Programme for Research and Innovation Horizon 2020 (2014-2020) under the Marie Skłodowska-Curie Grant Agreement No. 823734 (POEMS project). The Astronomical Institute in Ondřejov is supported by the project RVO:67985815.",
"pages": [
13
]
},
{
"title": "REFERENCES",
"content": "Ambartsumian V. A., 1933, Izvestiya Glavnoj Astronomicheskoj Observatorii v Pulkove, 13 Atanackovic-Vukmanovic O., 2004, Serbian Astronomical Journal, 169, 1 Auer L. H., Mihalas D., 1969, ApJ, 158, 641 Beals C. S., 1929, MNRAS, 90, 202 Beals C. S., 1933, The Observatory, 56, 196 Bestenlehner J. M., et al., 2014, A&A, 570, A38 Bisnovatyi-Kogan G. S., Nadyozhin D. K., 1972, Ap&SS, 15, 353 Bohr N., 1913, Nature, 92, 231 Bowen I. S., 1928, ApJ, 67, 1 Bowen I. S., 1934, PASP, 46, 146 Cannon C. J., 1973a, J. Quant. Spec. Radiat. Transf., 13, 627 Cannon C. J., 1973b, ApJ, 185, 621 Chandrasekhar S., 1934, MNRAS, 94, 522 Chené A.-N., Mahy L., Gosset E., St-Louis N., Dsilva K., Manick R., 2022, MNRAS, 516, 1022 Conti P. S., 1975, Memoires of the Societe Royale des Sciences de Liege, 9, 193 Conti P. S., 1984, in Maeder A., Renzini A., eds, IAU Symposium Vol. 105, Observational Tests of the Stellar Evolution Theory. p. 233 Conti P. S., 2015, in Hamann W.-R., Sander A., Todt H., eds, Wolf-Rayet Stars. pp 347-350 Conti P. S., Massey P., 1989, ApJ, 337, 251 Crowther P. A., 2007, ARA&A, 45, 177 Crowther P. A., Hillier D. J., Smith L. J., 1995a, A&A, 293, 172 Crowther P. A., Hillier D. J., Smith L. J., 1995b, A&A, 293, 403 Crowther P. A., Smith L. J., Hillier D. J., Schmutz W., 1995c, A&A, 293, 427 Crowther P. A., Smith L. J., Hillier D. J., 1995d, A&A, 302, 457 Crowther P. A., Smith L. J., Willis A. J., 1995e, A&A, 304, 269 Crowther P. A., De Marco O., Barlow M. J., 1998, MNRAS, 296, 367 Hamann W. R., Gräfener G., 2003, A&A, 410, 993 Hamann W. R., Gräfener G., Liermann A., 2006, A&A, 457, 1015 Hillier D. J., 1990, A&A, 231, 116 Hillier D. J., Miller D. L., 1998, ApJ, 496, 407 Huggins W., Huggins M., 1890, Proceedings of the Royal Society of London Series I, 49, 33 Humphreys R. M., Davidson K., 1994, PASP, 106, 1025 Humphreys R. M., Weis K., Davidson K., Gordon M. S., 2016, ApJ, 825, 64 Kosirev N. A., 1934, MNRAS, 94, 430 Laplace E., Justham S., Renzo M., Götberg Y., Farmer R., Vartanyan D., de Mink S. E., 2021, A&A, 656, A58 Lau R. M., et al., 2022, Nature Astronomy, 6, 1308 Liermann A., Hamann W. R., Oskinova L. M., Todt H., Butler K., 2010, A&A, 524, A82 Maíz Apellániz J., et al., 2016, ApJS, 224, 4 Marcolino W. L. F., Hillier D. J., de Araujo F. X., Pereira C. B., 2007, ApJ, 654, 1068 Martins F., Genzel R., Hillier D. J., Eisenhauer F., Paumard T., Gillessen S., Ott T., Trippe S., 2007, A&A, 468, 233 Martins F., Schaerer D., Marques-Chaves R., Upadhyaya A., 2023, A&A, 678, A159 Maryeva O., Viotti R. F., Koenigsberger G., Calabresi M., Rossi C., Gualandi R., 2019, Galaxies, 7, 79 Maryeva O. V., Gvaramadze V. V., Kniazev A. Y., Berdnikov L. N., 2020, MNRAS, 498, 5093 Maryeva O., Abdulkarimova A., Karpov S., Moiseev A., Oparin D., 2024, MNRAS, 527, 11925 Massey P., Conti P. S., 1983, ApJ, 273, 576 Massey P., Johnson O., 1998, ApJ, 505, 793 Massey P., Neugent K. F., Smart B. M., 2016, AJ, 152, 62 Mauerhan J. C., Van Dyk S. D., Morris P. W., 2011, AJ, 142, 40 Morgan D. H., Parker Q. A., Cohen M., 2003, MNRAS, 346, 719 Neugent K. F., Massey P., 2011, ApJ, 733, 123 Neugent K. F., Massey P., 2023, AJ, 166, 68 Neugent K. F., Massey P., Georgy C., 2012, ApJ, 759, 11 Niemela V. S., 1995, in van der Hucht K. A., Williams P. M., eds, IAU Symposium Vol. 163, Wolf-Rayet Stars: Binaries; Colliding Winds; Evolution. p. 223 Niemela V., 2001, in Revista Mexicana de Astronomia y Astrofisica Conference Series. pp 23-26 Paczyński B., 1967, Acta Astron., 17, 355 Paczyński B., 1973, in Bappu M. K. V., Sahade J., eds, IAU Symposium Vol. 49, Wolf-Rayet and High-Temperature Stars. p. 143 Pickering E. C., 1891, Astronomische Nachrichten, 127, 1 Pickering E. C., 1897, ApJ, 5, 92 Pickering E. C., 1901, Harvard College Observatory Circular, 55, 1 Polcaro V. F., et al., 2016, AJ, 151, 149 Pritzkuleit M., 2020, Master's thesis, University of Potsdam, Germany Roberts M. S., 1962, AJ, 67, 79 Robotti N., 1983, Historical Studies in the Physical Sciences, 14, 123 Rosslowe C. K., Crowther P. A., 2015, MNRAS, 447, 2322 Rublev S. V., 1965, Soviet Ast., 9, 274 Rublev S. V., 1970, Astrofizicheskie Issledovaniia Izvestiya Spetsial'noj As- trofizicheskoj Observatorii, 1, 25 Rublev S. V., 1972a, Astrofizicheskie Issledovaniia Izvestiya Spetsial'noj Astrofizicheskoj Observatorii, 4, 3 Underhill A. B., 1968, ARA&A, 6, 39 Vink J. S., 2015, in Hamann W.-R., Sander A., Todt H., eds, Wolf-Rayet Stars. pp 133-138 ( arXiv:1510.00227 ), doi:10.48550/arXiv.1510.00227 Westerlund B. E., Rodgers A. W., 1959, The Observatory, 79, 132 Westerlund B. E., Smith L. F., 1964, MNRAS, 128, 311 Wolf C. J. E., Rayet G., 1867, Academie des Sciences Paris Comptes Rendus, 65, 292 Wray J. D., Corso G. J., 1972, ApJ, 172, 577 Wright W. H., 1918, Publications of Lick Observatory, 13, 191 Yarovova A. D., Egorov O. V., Moiseev A. V., Maryeva O. V., 2023, MNRAS, 518, 2256 Zhang W., et al., 2020, ApJ, 902, 62 Škoda P., Podsztavek O., Tvrdík P., 2020, A&A, 643, A122 1981, van der Hucht K. A., Conti P. S., Lundstrom I., Stenholm B., Space Sci. Rev., 28, 227",
"pages": [
13,
14,
16,
17,
19
]
}
]
